wpmath0000013_3

Differential_stress.html

  1. σ 1 \sigma_{1}
  2. σ 3 \sigma_{3}
  3. σ D = σ 1 - σ 3 \!\sigma_{D}=\sigma_{1}-\sigma_{3}
  4. σ 3 \sigma_{3}
  5. σ 1 \sigma_{1}
  6. σ D = σ 3 - σ 1 \!\sigma_{D}=\sigma_{3}-\sigma_{1}
  7. σ 1 \sigma_{1}
  8. σ 3 \sigma_{3}

Diffusing-wave_spectroscopy.html

  1. g 2 ( τ ) = I ( t ) I ( t + τ ) t I ( t ) t 2 g_{2}(\tau)=\frac{\langle I(t)I(t+\tau)\rangle_{t}}{\langle I(t)\rangle_{t}^{2}}
  2. g 2 ( τ ) - 1 = [ d s P ( s ) exp ( - ( s / l * ) k 0 2 Δ r 2 ( τ ) ) ] 2 g_{2}(\tau)-1=[\int{dsP(s)\exp(-(s/l*)k_{0}^{2}\langle\Delta r^{2}(\tau)% \rangle)}]^{2}
  3. k 0 = 2 π n λ k_{0}=\frac{2\pi n}{\lambda}
  4. l * l*
  5. g 2 ( τ ) - 1 = exp ( - 2 γ Δ r 2 ( τ ) k 0 2 ) g_{2}(\tau)-1=\exp\left(-2\gamma\sqrt{\langle\Delta r^{2}(\tau)\rangle k_{0}^{% 2}}\right)
  6. g 2 ( τ ) = I ( t ) I ( t + τ ) p I ( t ) p 2 g_{2}(\tau)=\frac{\langle I(t)I(t+\tau)\rangle_{p}}{\langle I(t)\rangle_{p}^{2}}

Diffusion_current.html

  1. J = q n μ E + q D d n d x J=qn\mu E+qD\frac{dn}{dx}
  2. J e - Φ / V t = q D ( - n V t * d Φ d x + d n d x ) e - Φ / V t = q D d d x ( n e - Φ / V t ) Je^{-\Phi/V_{t}}=qD\left(\frac{-n}{V_{t}}*\frac{d\Phi}{dx}+\frac{dn}{dx}\right% )e^{-\Phi/V_{t}}=qD\frac{d}{dx}(ne^{-\Phi/V_{t}})
  3. J = q D n e - Φ / V t | 0 x d 0 x d e - Φ / V t d x J=\frac{qDne^{-\Phi/V_{t}}\big|_{0}^{x_{d}}}{\int_{0}^{x_{d}}e^{-\Phi/V_{t}}dx}
  4. J = q D N c e - Φ B / V t [ e V a / V t - 1 ] 0 x d e - Φ * / V t d x J=\frac{qDN_{c}e^{-\Phi_{B}/V_{t}}\left[e^{V_{a}/V_{t}}-1\right]}{\int_{0}^{x_% {d}}e^{-\Phi^{*}/V_{t}}dx}
  5. Φ * = Φ B + Φ i - V a \Phi^{*}=\Phi_{B}+\Phi_{i}-V_{a}
  6. Φ = - q N d 2 E s ( x - x d ) 2 \Phi=-\frac{qN_{d}}{2E_{s}(x-x_{d})^{2}}
  7. Φ * = q N d x E s ( x d - x 2 ) = ( Φ i - V a ) x x d \Phi^{*}=\frac{qN_{d}x}{E_{s}}\left(x_{d}-\frac{x}{2}\right)=(\Phi_{i}-V_{a})% \frac{x}{x_{d}}
  8. 0 x d e - Φ * / V t d x = x d Φ i - V a V t \int_{0}^{x_{d}}e^{-\Phi^{*}/V_{t}}dx=x_{d}\frac{\Phi_{i}-V_{a}}{V_{t}}
  9. J = q 2 D N c V t [ 2 q E s ( Φ i - V a ) N d ] 1 / 2 e - Φ B / V t ( e V a / V t - 1 ) J=\frac{q_{2}DN_{c}}{V_{t}}\left[\frac{2q}{E_{s}}(\Phi_{i}-V_{a})N_{d}\right]^% {1/2}e^{-\Phi_{B}/V_{t}}(e^{V_{a}/V_{t}}-1)
  10. E max = [ 2 q E s ( Φ i - V a ) N d ] 1 / 2 E_{\mathrm{max}}=\left[\frac{2q}{E_{s}}(\Phi_{i}-V_{a})N_{d}\right]^{1/2}
  11. J = q μ E max N c e - Φ B / V t ( e V a / V t - 1 ) J=q\mu E_{\mathrm{max}}N_{c}e^{-\Phi_{B}/V_{t}}(e^{V_{a}/V_{t}}-1)

Dikinase.html

  1. \rightleftharpoons

Dimethylphenylphosphine.html

  1. θ = 2 3 i = 1 3 θ i 2 \theta=\frac{2}{3}\sum_{i=1}^{3}\frac{\theta_{i}}{2}

Dini's_surface.html

  1. x = a cos ( u ) sin ( v ) x=a\cos\left(u\right)\sin\left(v\right)
  2. y = a sin ( u ) sin ( v ) y=a\sin\left(u\right)\sin\left(v\right)
  3. z = a ( cos ( v ) + ln ( tan ( v 2 ) ) ) + b u z=a\left(\cos\left(v\right)+\ln\left(\tan\left(\frac{v}{2}\right)\right)\right% )+bu

Dini_continuity.html

  1. X X
  2. n \mathbb{R}^{n}
  3. f : X X f:X\rightarrow X
  4. X X
  5. f f
  6. ω f ( t ) = sup d ( x , y ) t d ( f ( x ) , f ( y ) ) . \omega_{f}(t)=\sup_{d(x,y)\leq t}d(f(x),f(y)).\,
  7. f f
  8. 0 1 ω f ( t ) t d t < . \int_{0}^{1}\frac{\omega_{f}(t)}{t}\,dt<\infty.
  9. θ ( 0 , 1 ) \theta\in(0,1)
  10. i = 1 ω f ( θ i a ) < \sum_{i=1}^{\infty}\omega_{f}(\theta^{i}a)<\infty
  11. a a
  12. X X

Dinic's_algorithm.html

  1. O ( V 2 E ) O(V^{2}E)
  2. O ( V E 2 ) O(VE^{2})
  3. G = ( ( V , E ) , c , s , t ) G=((V,E),c,s,t)
  4. c ( u , v ) c(u,v)
  5. f ( u , v ) f(u,v)
  6. ( u , v ) (u,v)
  7. c f : V × V R + c_{f}\colon V\times V\to R^{+}
  8. ( u , v ) E (u,v)\in E
  9. c f ( u , v ) = c ( u , v ) - f ( u , v ) c_{f}(u,v)=c(u,v)-f(u,v)
  10. c f ( v , u ) = f ( u , v ) c_{f}(v,u)=f(u,v)
  11. c f ( u , v ) = 0 c_{f}(u,v)=0
  12. G f = ( ( V , E f ) , c f | E f , s , t ) G_{f}=((V,E_{f}),c_{f}|_{E_{f}},s,t)
  13. E f = { ( u , v ) V × V : c f ( u , v ) > 0 } E_{f}=\{(u,v)\in V\times V:c_{f}(u,v)>0\}
  14. s - t s-t
  15. G f G_{f}
  16. dist ( v ) \operatorname{dist}(v)
  17. s s
  18. v v
  19. G f G_{f}
  20. G f G_{f}
  21. G L = ( V , E L , c f | E L , s , t ) G_{L}=(V,E_{L},c_{f}|_{E_{L}},s,t)
  22. E L = { ( u , v ) E f : dist ( v ) = dist ( u ) + 1 } E_{L}=\{(u,v)\in E_{f}:\operatorname{dist}(v)=\operatorname{dist}(u)+1\}
  23. s - t s-t
  24. f f
  25. G = ( V , E L , s , t ) G^{\prime}=(V,E_{L}^{\prime},s,t)
  26. E L = { ( u , v ) : f ( u , v ) < c f | E L ( u , v ) } E_{L}^{\prime}=\{(u,v):f(u,v)<c_{f}|_{E_{L}}(u,v)\}
  27. s - t s-t
  28. G = ( ( V , E ) , c , s , t ) G=((V,E),c,s,t)
  29. s - t s-t
  30. f f
  31. f ( e ) = 0 f(e)=0
  32. e E e\in E
  33. G L G_{L}
  34. G f G_{f}
  35. G G
  36. dist ( t ) = \operatorname{dist}(t)=\infty
  37. f f
  38. f f\;^{\prime}
  39. G L G_{L}
  40. f \ f
  41. f f\;^{\prime}
  42. n - 1 n-1
  43. n n
  44. G L G_{L}
  45. O ( E ) O(E)
  46. O ( V E ) O(VE)
  47. O ( V 2 E ) O(V^{2}E)
  48. O ( E log V ) O(E\log V)
  49. O ( V E log V ) O(VE\log V)
  50. O ( E ) O(E)
  51. O ( E ) O(\sqrt{E})
  52. O ( V 2 / 3 ) O(V^{2/3})
  53. O ( min { V 2 / 3 , E 1 / 2 } E ) O(\min\{V^{2/3},E^{1/2}\}E)
  54. O ( V ) O(\sqrt{V})
  55. O ( V E ) O(\sqrt{V}E)
  56. G L G_{L}
  57. dist ( v ) \operatorname{dist}(v)
  58. G G
  59. G f G_{f}
  60. G L G_{L}
  61. { s , 1 , 3 , t } \{s,1,3,t\}
  62. { s , 1 , 4 , t } \{s,1,4,t\}
  63. { s , 2 , 4 , t } \{s,2,4,t\}
  64. | f | |f|
  65. { s , 2 , 4 , 3 , t } \{s,2,4,3,t\}
  66. | f | |f|
  67. t t
  68. G f G_{f}

Dipole_field_strength_in_free_space.html

  1. p = N 4 π d 2 \mbox{p}~{}=\frac{N}{4\cdot\pi\cdot d^{2}}
  2. p = E 2 R \mbox{p}~{}=\frac{E^{2}}{R}
  3. N 4 π d 2 = E 2 R \frac{N}{4\cdot\pi\cdot d^{2}}=\frac{E^{2}}{R}
  4. E = N R 2 π d \mbox{E}~{}=\frac{\sqrt{N}\cdot\sqrt{R}}{2\cdot\sqrt{\pi}\cdot d}
  5. 120 π 120\cdot\pi
  6. 2.15 dBi = 1.64 \mbox{2.15 dBi}~{}=1.64
  7. E = 1.64 N 120 π 2 π d 7 N d \mbox{E}~{}=\frac{\sqrt{1.64\cdot N}\cdot\sqrt{120\cdot\pi}}{2\cdot\sqrt{\pi}% \cdot d}\approx 7\cdot\frac{\sqrt{N}}{d}
  8. 1000 \sqrt{1000}
  9. E 222 N d \mbox{E}~{}\approx 222\cdot\frac{\sqrt{N}}{d}

Dipole_model_of_the_Earth's_magnetic_field.html

  1. B 0 B_{0}
  2. B 0 = 3.12 × 10 - 5 T B_{0}=3.12\times 10^{-5}\ \textrm{T}
  3. B r = - 2 B 0 ( R E r ) 3 cos θ B_{r}=-2B_{0}\left(\frac{R_{E}}{r}\right)^{3}\cos\theta
  4. B θ = - B 0 ( R E r ) 3 sin θ B_{\theta}=-B_{0}\left(\frac{R_{E}}{r}\right)^{3}\sin\theta
  5. | B | = B 0 ( R E r ) 3 1 + 3 cos 2 θ |B|=B_{0}\left(\frac{R_{E}}{r}\right)^{3}\sqrt{1+3\cos^{2}\theta}
  6. R E R_{E}
  7. r r
  8. R E R_{E}
  9. θ \theta
  10. λ \lambda
  11. θ \theta
  12. λ = π / 2 - θ \lambda=\pi/2-\theta
  13. θ \theta
  14. B r = - 2 B 0 R 3 sin λ B_{r}=-\frac{2B_{0}}{R^{3}}\sin\lambda
  15. B θ = B 0 R 3 cos λ B_{\theta}=\frac{B_{0}}{R^{3}}\cos\lambda
  16. | B | = B 0 R 3 1 + 3 sin 2 λ |B|=\frac{B_{0}}{R^{3}}\sqrt{1+3\sin^{2}\lambda}
  17. R R
  18. R = r / R E R=r/R_{E}
  19. Λ = arccos ( 1 / L ) \Lambda=\arccos\left(\sqrt{1/L}\right)
  20. L = 1 / cos 2 ( Λ ) L=1/\cos^{2}\left(\Lambda\right)
  21. Λ \Lambda
  22. L L
  23. Λ \Lambda
  24. λ \lambda

Direct_and_indirect_band_gaps.html

  1. α \alpha
  2. α A * h ν - E g \alpha\approx A^{*}\sqrt{h\nu-E_{\,\text{g}}}
  3. A * = q 2 x v c 2 ( 2 m r ) 3 / 2 λ 0 ϵ 0 3 n A^{*}=\frac{q^{2}x_{vc}^{2}(2m_{\,\text{r}})^{3/2}}{\lambda_{0}\epsilon_{0}% \hbar^{3}n}
  4. α \alpha
  5. ν \nu
  6. h h
  7. h ν h\nu
  8. ν \nu
  9. \hbar
  10. = h / 2 π \hbar=h/2\pi
  11. E g E_{\,\text{g}}
  12. A * A^{*}
  13. m r = m h * m e * m h * + m e * m_{\,\text{r}}=\frac{m_{\,\text{h}}^{*}m_{\,\text{e}}^{*}}{m_{\,\text{h}}^{*}+% m_{\,\text{e}}^{*}}
  14. m e * m_{\,\text{e}}^{*}
  15. m h * m_{\,\text{h}}^{*}
  16. m r m_{\,\text{r}}
  17. q q
  18. n n
  19. ϵ 0 \epsilon_{0}
  20. x v c x_{vc}
  21. α ( h ν - E g + E p ) 2 exp ( E p k T ) - 1 + ( h ν - E g - E p ) 2 1 - exp ( - E p k T ) \alpha\propto\frac{(h\nu-E_{\,\text{g}}+E_{\,\text{p}})^{2}}{\exp(\frac{E_{\,% \text{p}}}{kT})-1}+\frac{(h\nu-E_{\,\text{g}}-E_{\,\text{p}})^{2}}{1-\exp(-% \frac{E_{\,\text{p}}}{kT})}
  22. E p E_{\,\text{p}}
  23. k k
  24. T T
  25. h ν h\nu
  26. α 2 \alpha^{2}
  27. α = 0 \alpha=0
  28. h ν h\nu
  29. α 1 / 2 \alpha^{1/2}
  30. α = 0 \alpha=0
  31. E p 0 E_{\,\text{p}}\approx 0
  32. A * A^{*}

Direct_method_in_the_calculus_of_variations.html

  1. J : V ¯ J:V\to\bar{\mathbb{R}}
  2. V V
  3. ¯ = { } \bar{\mathbb{R}}=\mathbb{R}\cup\{\infty\}
  4. v V v\in V
  5. J ( v ) J ( u ) u V . J(v)\leq J(u)\forall u\in V.
  6. J J
  7. inf { J ( u ) | u V } > - . \inf\{J(u)|u\in V\}>-\infty.\,
  8. ( u n ) (u_{n})
  9. V V
  10. J ( u n ) inf { J ( u ) | u V } . J(u_{n})\to\inf\{J(u)|u\in V\}.
  11. ( u n ) (u_{n})
  12. J J
  13. ( u n ) (u_{n})
  14. ( u n k ) (u_{n_{k}})
  15. u 0 V u_{0}\in V
  16. τ \tau
  17. V V
  18. J J
  19. τ \tau
  20. J J
  21. lim inf n J ( u n ) J ( u 0 ) \liminf_{n\to\infty}J(u_{n})\geq J(u_{0})
  22. u n u 0 u_{n}\to u_{0}
  23. V V
  24. inf { J ( u ) | u V } = lim n J ( u n ) = lim k J ( u n k ) J ( u 0 ) inf { J ( u ) | u V } \inf\{J(u)|u\in V\}=\lim_{n\to\infty}J(u_{n})=\lim_{k\to\infty}J(u_{n_{k}})% \geq J(u_{0})\geq\inf\{J(u)|u\in V\}
  25. J ( u 0 ) = inf { J ( u ) | u V } J(u_{0})=\inf\{J(u)|u\in V\}
  26. V V
  27. W W
  28. ( u n ) (u_{n})
  29. V V
  30. u 0 u_{0}
  31. W W
  32. V V
  33. W W
  34. u 0 u_{0}
  35. V V
  36. J : V ¯ J:V\to\bar{\mathbb{R}}
  37. J J
  38. J J
  39. J J
  40. u n u 0 u_{n}\to u_{0}
  41. lim inf n J ( u n ) J ( u 0 ) \liminf_{n\to\infty}J(u_{n})\geq J(u_{0})
  42. J J
  43. J ( x ) α x q - β J(x)\geq\alpha\lVert x\rVert^{q}-\beta
  44. α > 0 \alpha>0
  45. q 1 q\geq 1
  46. β 0 \beta\geq 0
  47. J ( u ) = Ω F ( x , u ( x ) , u ( x ) ) d x J(u)=\int_{\Omega}F(x,u(x),\nabla u(x))dx
  48. Ω \Omega
  49. n \mathbb{R}^{n}
  50. F F
  51. Ω × m × m n \Omega\times\mathbb{R}^{m}\times\mathbb{R}^{mn}
  52. J J
  53. u : Ω m u:\Omega\to\mathbb{R}^{m}
  54. u ( x ) \nabla u(x)
  55. m n mn
  56. Ω \Omega
  57. C 2 C^{2}
  58. J J
  59. C 2 ( Ω , m ) C^{2}(\Omega,\mathbb{R}^{m})
  60. W 1 , p ( Ω , m ) W^{1,p}(\Omega,\mathbb{R}^{m})
  61. p > 1 p>1
  62. u u
  63. J J
  64. J ( u ) = Ω F ( x , u ( x ) , u ( x ) ) d x J(u)=\int_{\Omega}F(x,u(x),\nabla u(x))dx
  65. Ω n \Omega\subseteq\mathbb{R}^{n}
  66. F F
  67. J J
  68. W 1 , p ( Ω , m ) W^{1,p}(\Omega,\mathbb{R}^{m})
  69. F F
  70. ( y , p ) F ( x , y , p ) (y,p)\mapsto F(x,y,p)
  71. x Ω x\in\Omega
  72. x F ( x , y , p ) x\mapsto F(x,y,p)
  73. ( y , p ) m × m n (y,p)\in\mathbb{R}^{m}\times\mathbb{R}^{mn}
  74. F ( x , y , p ) a ( x ) p + b ( x ) F(x,y,p)\geq a(x)\cdot p+b(x)
  75. a L q ( Ω , m ) a\in L^{q}(\Omega,\mathbb{R}^{m})
  76. 1 / q + 1 / p = 1 1/q+1/p=1
  77. b L 1 ( Ω ) b\in L^{1}(\Omega)
  78. x Ω x\in\Omega
  79. ( y , p ) m × m n (y,p)\in\mathbb{R}^{m}\times\mathbb{R}^{mn}
  80. a ( x ) p a(x)\cdot p
  81. a ( x ) a(x)
  82. p p
  83. m n \mathbb{R}^{mn}
  84. p F ( x , y , p ) p\mapsto F(x,y,p)
  85. x Ω x\in\Omega
  86. y m y\in\mathbb{R}^{m}
  87. J J
  88. n = 1 n=1
  89. m = 1 m=1
  90. F F
  91. | F ( x , y , p ) | a ( x , | y | , | p | ) |F(x,y,p)|\leq a(x,|y|,|p|)
  92. ( x , y , p ) (x,y,p)
  93. a ( x , y , p ) a(x,y,p)
  94. y y
  95. p p
  96. x x
  97. J J
  98. ( x , y ) Ω × m (x,y)\in\Omega\times\mathbb{R}^{m}
  99. p F ( x , y , p ) p\mapsto F(x,y,p)
  100. m = 1 m=1
  101. n = 1 n=1
  102. J J
  103. F F
  104. p F ( x , y , p ) p\mapsto F(x,y,p)
  105. n n
  106. m m

Directed_algebraic_topology.html

  1. \sim
  2. X X
  3. D X DX
  4. X X
  5. X X
  6. π 1 ( X ) := D X / \pi_{1}(X)\ :=\ DX/\sim
  7. π 1 \pi_{1}
  8. C C
  9. π 0 ( C ) \pi_{0}(C)
  10. C C
  11. Σ \Sigma
  12. C C
  13. π 0 ( C ) \pi_{0}(C)
  14. C / Σ C/\Sigma
  15. C [ Σ - 1 ] C[\Sigma^{-1}]
  16. π 0 ( π 1 ( X ) ) \pi_{0}(\pi_{1}(X))
  17. X X
  18. [ 1 ] {}^{[1]}
  19. { * } [ 0 , 1 ] \{*\}\hookrightarrow[0,1]
  20. { 0 } [ 0 , 1 ] \{0\}\hookrightarrow[0,1]
  21. { 0 } × [ 0 , 1 ] [ 0 , 1 ] × { 0 } \{0\}\times[0,1]\cup[0,1]\times\{0\}
  22. C a t Cat

Director_circle.html

  1. a 2 + b 2 \sqrt{a^{2}+b^{2}}
  2. a a
  3. b b
  4. w i d 2 ( X , P i ) = C . \sum w_{i}\,d^{2}(X,P_{i})=C.

Dirichlet_eigenvalue.html

  1. { Δ u + λ u = 0 in Ω u | Ω = 0. \begin{cases}\Delta u+\lambda u=0&\rm{in\ }\Omega\\ u|_{\partial\Omega}=0.&\end{cases}
  2. Δ u = 2 u x 2 + 2 u y 2 . \Delta u=\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^% {2}}.
  3. 0 < λ 1 λ 2 , λ n , 0<\lambda_{1}\leq\lambda_{2}\leq\cdots,\quad\lambda_{n}\to\infty,
  4. H 0 2 ( Ω ) H^{2}_{0}(\Omega)
  5. L 2 ( Ω ) L^{2}(\Omega)
  6. λ 1 = inf u 0 Ω | u | 2 Ω | u | 2 , \lambda_{1}=\inf_{u\not=0}\frac{\int_{\Omega}|\nabla u|^{2}}{\int_{\Omega}|u|^% {2}},
  7. u H 0 1 ( Ω ) u\in H_{0}^{1}(\Omega)
  8. H 0 1 ( Ω ) H_{0}^{1}(\Omega)
  9. λ k = sup inf Ω | u | 2 Ω | u | 2 \lambda_{k}=\sup\inf\frac{\int_{\Omega}|\nabla u|^{2}}{\int_{\Omega}|u|^{2}}
  10. ϕ 1 , , ϕ k - 1 H 0 1 ( Ω ) \phi_{1},\dots,\phi_{k-1}\in H^{1}_{0}(\Omega)

Dirichlet_form.html

  1. ( u ) = n | u | 2 d x \mathcal{E}(u)=\int_{\mathbb{R}^{n}}|\nabla u|^{2}\;dx
  2. ( X , μ ) (X,\mu)
  3. : D × D \mathcal{E}:D\times D\to\mathbb{R}
  4. D D
  5. L 2 ( X , μ ) L^{2}(X,\mu)
  6. \mathcal{E}
  7. ( u , v ) = ( v , u ) \mathcal{E}(u,v)=\mathcal{E}(v,u)
  8. u , v D u,v\in D
  9. ( u , u ) 0 \mathcal{E}(u,u)\geq 0
  10. u D u\in D
  11. D D
  12. ( u , v ) := ( u , v ) L 2 ( X , μ ) + ( u , v ) (u,v)_{\mathcal{E}}:=(u,v)_{L^{2}(X,\mu)}+\mathcal{E}(u,v)
  13. u D u\in D
  14. u * = min ( max ( u , 0 ) , 1 ) D u_{*}=\min(\max(u,0),1)\in D
  15. ( u * , u * ) ( u , u ) \mathcal{E}(u_{*},u_{*})\leq\mathcal{E}(u,u)
  16. L 2 ( X , μ ) L^{2}(X,\mu)
  17. u ( u , u ) u\to\mathcal{E}(u,u)
  18. \mathcal{E}
  19. ( u ) := ( u , u ) \mathcal{E}(u):=\mathcal{E}(u,u)
  20. n \mathbb{R}^{n}
  21. ( u ) = n | u | 2 d x \mathcal{E}(u)=\int_{\mathbb{R}^{n}}|\nabla u|^{2}\;dx
  22. H 1 ( n ) H^{1}(\mathbb{R}^{n})
  23. ( u ) = n × n ( u ( y ) - u ( x ) ) 2 k ( x , y ) d x d y \mathcal{E}(u)=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}(u(y)-u(x))^{2}k(x,y)% \,\mathrm{d}x\mathrm{d}y
  24. k : n × n k:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}
  25. k k
  26. k ( x , y ) Λ | x - y | - n - s k(x,y)\leq\Lambda|x-y|^{-n-s}
  27. H ˙ s / 2 \dot{H}^{s/2}
  28. λ | x - y | - n - s k ( x , y ) \lambda|x-y|^{-n-s}\leq k(x,y)
  29. H ˙ s / 2 \dot{H}^{s/2}
  30. D L 2 ( n ) D\subset L^{2}(\mathbb{R}^{n})
  31. H s / 2 ( n ) H^{s/2}(\mathbb{R}^{n})
  32. ( u ) = ( A u , u ) d x , \mathcal{E}(u)=\int(A\nabla u,\nabla u)\;\mathrm{d}x,
  33. A ( x ) A(x)

Dirichlet_kernel.html

  1. D n ( x ) = k = - n n e i k x = 1 + 2 k = 1 n cos ( k x ) = sin ( ( n + 1 / 2 ) x ) sin ( x / 2 ) . D_{n}(x)=\sum_{k=-n}^{n}e^{ikx}=1+2\sum_{k=1}^{n}\cos(kx)=\frac{\sin\left(% \left(n+1/2\right)x\right)}{\sin(x/2)}.
  2. ( D n * f ) ( x ) = 1 2 π - π π f ( y ) D n ( x - y ) d y = k = - n n f ^ ( k ) e i k x , (D_{n}*f)(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)D_{n}(x-y)\,dy=\sum_{k=-n}^{n}% \hat{f}(k)e^{ikx},
  3. f ^ ( k ) = 1 2 π - π π f ( x ) e - i k x d x \hat{f}(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}\,dx
  4. D n L 1 = O ( log n ) \|D_{n}\|_{L^{1}}=O(\log n)
  5. D n D_{n}
  6. D n L 1 4 Si ( π ) + 8 π log n . \|D_{n}\|_{L^{1}}\geq 4\operatorname{Si}(\pi)+\frac{8}{\pi}\log n.
  7. f * ( 2 π δ ) = f f*(2\pi\delta)=f\,
  8. 2 π δ ( x ) k = - e i k x = ( 1 + 2 k = 1 cos ( k x ) ) . 2\pi\delta(x)\sim\sum_{k=-\infty}^{\infty}e^{ikx}=\left(1+2\sum_{k=1}^{\infty}% \cos(kx)\right).
  9. k = - n n e i k x = sin ( ( n + 1 / 2 ) x ) sin ( x / 2 ) \sum_{k=-n}^{n}e^{ikx}=\frac{\sin((n+1/2)x)}{\sin(x/2)}
  10. k = 0 n a r k = a 1 - r n + 1 1 - r . \sum_{k=0}^{n}ar^{k}=a\frac{1-r^{n+1}}{1-r}.
  11. k = - n n r k = r - n 1 - r 2 n + 1 1 - r . \sum_{k=-n}^{n}r^{k}=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}.
  12. r - n - 1 / 2 r - 1 / 2 1 - r 2 n + 1 1 - r = r - n - 1 / 2 - r n + 1 / 2 r - 1 / 2 - r 1 / 2 . \frac{r^{-n-1/2}}{r^{-1/2}}\cdot\frac{1-r^{2n+1}}{1-r}=\frac{r^{-n-1/2}-r^{n+1% /2}}{r^{-1/2}-r^{1/2}}.
  13. k = - n n e i k x = e - ( n + 1 / 2 ) i x - e ( n + 1 / 2 ) i x e - i x / 2 - e i x / 2 = - 2 i sin ( ( n + 1 / 2 ) x ) - 2 i sin ( x / 2 ) = sin ( ( n + 1 / 2 ) x ) sin ( x / 2 ) \sum_{k=-n}^{n}e^{ikx}=\frac{e^{-(n+1/2)ix}-e^{(n+1/2)ix}}{e^{-ix/2}-e^{ix/2}}% =\frac{-2i\sin((n+1/2)x)}{-2i\sin(x/2)}=\frac{\sin((n+1/2)x)}{\sin(x/2)}
  14. f ( x ) = 1 / 2 + k = 1 n cos ( k x ) . f(x)=1/2+\sum_{k=1}^{n}\cos(kx).
  15. 2 sin ( x / 2 ) 2\sin(x/2)\!
  16. cos ( a ) sin ( b ) = ( sin ( a + b ) - sin ( a - b ) ) / 2 \cos(a)\sin(b)=(\sin(a+b)-\sin(a-b))/2\!
  17. sin ( ( n + 1 / 2 ) x ) . \sin((n+1/2)x).\!
  18. k = 0 n e i k x = e i x n / 2 sin ( ( n / 2 + 1 / 2 ) x ) sin ( x / 2 ) \sum_{k=0}^{n}e^{ikx}=e^{ixn/2}\frac{\sin((n/2+1/2)x)}{\sin(x/2)}

Discharge_coefficient.html

  1. C d A = m ˙ 2 ρ Δ P C_{d}A=\dfrac{\dot{m}}{\sqrt{{2}{\rho}{\Delta}{P}}}
  2. C d C_{d}
  3. A A
  4. m ˙ \dot{m}
  5. ρ \rho
  6. Δ P \Delta P
  7. k k
  8. k = 1 C d 2 k=\dfrac{1}{C_{d}^{2}}
  9. Δ P \Delta P
  10. k k
  11. q q

Discontinuous-constituent_phrase_structure_grammar.html

  1. X α X\to\alpha
  2. α \alpha
  3. X α X\to\alpha
  4. X X
  5. α \alpha
  6. X α _ β X\to\alpha\_\beta
  7. X X
  8. α \alpha
  9. β \beta
  10. β \beta
  11. X X
  12. S V P N P s u b j S\to VP\ NP_{subj}
  13. V P I T V | T V _ N P o b j VP\to ITV~{}|~{}TV\ \_\ NP_{obj}
  14. N P J o h n | S u s a n | NP\to John~{}|~{}Susan~{}|~{}...
  15. I T V r a n | d a n c e d | ITV\to ran~{}|~{}danced~{}|~{}...
  16. T V s a w | m e t | TV\to saw~{}|~{}met~{}|~{}...
  17. S V P N P s u b j T V N P s u b j N P o b j s a w N P s u b j N P o b j s a w J o h n N P o b j s a w J o h n M a r y S\to VP\ NP_{subj}\to TV\ NP_{subj}NP_{obj}\to saw\ NP_{subj}\ NP_{obj}\to saw% \ John\ NP_{obj}\to saw\ John\ Mary

Discrete_Chebyshev_polynomials.html

  1. g d := ( g , g ) d 1 / 2 \left\|g\right\|_{d}:=(g,g)^{1/2}_{d}
  2. ( ϕ k , ϕ i ) d = 0 \left(\phi_{k},\phi_{i}\right)_{d}=0
  3. ϕ k d = 1. \left\|\phi_{k}\right\|_{d}=1.

Disjoining_pressure.html

  1. Π D = - 1 A ( G x ) T , V , A \Pi_{D}=-{1\over A}\left(\frac{\partial G}{\partial x}\right)_{T,V,A}
  2. P = P 0 + Π D P=P_{0}+\Pi_{D}
  3. Π D = - A H 6 π δ 0 3 \Pi_{D}=-{{A_{H}}\over{6\pi\delta_{0}^{3}}}
  4. Π D ( x , ζ L ( x ) ) = d 2 ρ ζ L ( x ) - ζ S ( x + ρ ) + d z ω ( ρ , z ) {\Pi_{D}(x,{{\zeta}_{\,\text{L}}(x)})}=\int d^{2}\rho{\int_{\zeta\text{L}(x)-% \zeta\text{S}(x+\rho)}^{+\infty}}dz\omega(\rho,z)
  5. δ W total = W total ζ L δ ζ L + W total ζ L δ ζ L = 0 {\delta W_{\,\text{total}}}={{\partial W_{\,\text{total}}}\over{\partial{\zeta% \text{L}}}}\delta\zeta\text{L}+{{\partial W_{\,\text{total}}}\over{\partial% \zeta\text{L}^{^{\prime}}}}\delta\zeta\text{L}^{^{\prime}}=0

Disjunct_matrix.html

  1. t × n t\times n
  2. t × n t\times n
  3. M M
  4. d d
  5. S 1 S 2 [ n ] \forall S_{1}\neq S_{2}\subseteq[n]
  6. | S 1 | , | S 2 | d |S_{1}|,|S_{2}|\leq d
  7. j S 1 M j i S 2 M i \bigcup_{j\in S_{1}}M_{j}\neq\bigcup_{i\in S_{2}}M_{i}
  8. M M
  9. 𝐫 \mathbf{r}
  10. 𝐫 = M 𝐱 \mathbf{r}=M\mathbf{x}
  11. 𝐱 \mathbf{x}
  12. M j M_{j}
  13. j t h j^{th}
  14. M M
  15. S M j [ t ] S_{M_{j}}\subseteq[t]
  16. M j ( i ) = 1 M_{j}(i)=1
  17. i S M j i\in S_{M_{j}}
  18. S 𝐫 = j [ n ] , 𝐱 j = 1 S M j S_{\mathbf{r}}=\bigcup_{j\in[n],\mathbf{x}_{j}=1}S_{M_{j}}
  19. 𝐱 \mathbf{x}
  20. M M
  21. 𝐫 \mathbf{r}
  22. d d
  23. d d
  24. d d
  25. t × n t\times n
  26. M M
  27. M M
  28. d d
  29. T [ n ] T\subseteq[n]
  30. | T | d |T|\leq d
  31. S 𝐫 = j T S M j S_{\mathbf{r}}=\bigcup_{j\in T}S_{M_{j}}
  32. n 𝒪 ( d ) n^{\mathcal{O}(d)}
  33. t t
  34. n n
  35. M M
  36. S [ n ] \forall S\subseteq[n]
  37. | S | d |S|\leq d
  38. j S \forall j\notin S
  39. i \exists i
  40. M i , j = 1 M_{i,j}=1
  41. k S , M i , k = 0 \forall k\in S,M_{i,k}=0
  42. M a M_{a}
  43. a t h a^{th}
  44. M M
  45. S M a [ t ] S_{M_{a}}\subseteq[t]
  46. M a ( b ) = 1 M_{a}(b)=1
  47. b S M a b\in S_{M_{a}}
  48. M M
  49. d d
  50. S M j k S S M k S_{M_{j}}\not\subset\cup_{k\in S}S_{M_{k}}
  51. M M
  52. d d
  53. M M
  54. d d
  55. M M
  56. t t
  57. n n
  58. d d
  59. M M
  60. d d
  61. T 1 , T 2 [ n ] T_{1},T_{2}\in[n]
  62. T 1 T 2 T_{1}\neq T_{2}
  63. | T 1 | , | T 2 | d |T_{1}|,|T_{2}|\leq d
  64. i T 1 M i = i T 2 S M i \bigcup_{i\in T_{1}}M_{i}=\cup_{i\in T_{2}}S_{M_{i}}
  65. j T 2 T 1 \exists j\in T_{2}\setminus T_{1}
  66. S M j k T 1 T M k S_{M_{j}}\subseteq\bigcup_{k\in T_{1}}T_{M_{k}}
  67. M M
  68. d d
  69. M M
  70. d d
  71. \Box
  72. d d
  73. n n
  74. d d
  75. d d
  76. d d
  77. t t
  78. 𝒪 ( n t ) \mathcal{O}(nt)
  79. d d
  80. t t
  81. n n
  82. M M
  83. M 𝐱 = 𝐫 M\mathbf{x}=\mathbf{r}
  84. 𝐫 i = 1 \mathbf{r}_{i}=1
  85. j \exists j
  86. M i , j = 1 M_{i,j}=1
  87. 𝐱 j = 1 \mathbf{x}_{j}=1
  88. 1 i t 1\leq i\leq t
  89. 1 j n 1\leq j\leq n
  90. 𝐫 i = 0 \mathbf{r}_{i}=0
  91. j \forall j
  92. M i , j = 1 M_{i,j}=1
  93. 𝐱 j = 0 \mathbf{x}_{j}=0
  94. 𝐫 \mathbf{r}
  95. 𝐱 j \mathbf{x}_{j}
  96. M i , j = 1 M_{i,j}=1
  97. d d
  98. T = { j | 𝐱 j = 1 } T=\{j|\mathbf{x}_{j}=1\}
  99. | T | d |T|\leq d
  100. j T j\notin T
  101. 1 j n 1\leq j\leq n
  102. i i
  103. 1 i t 1\leq i\leq t
  104. M i , j = 1 M_{i,j}=1
  105. M i , l = 0 l T M_{i,l}=0\,\text{ }\forall l\in T
  106. 𝐫 i = 0 \mathbf{r}_{i}=0
  107. 𝐱 j = 0 \mathbf{x}_{j}=0
  108. 𝐫 { 0 , 1 } t , M \mathbf{r}\in\{0,1\}^{t},M
  109. j [ n ] j\in[n]
  110. 𝐱 j = 1 \mathbf{x}_{j}=1
  111. i = 1 t i=1\ldots t
  112. 𝐫 i = 0 \mathbf{r}_{i}=0
  113. j [ n ] j\in[n]
  114. M i , j = 1 M_{i,j}=1
  115. 𝐱 j = 0 \mathbf{x}_{j}=0
  116. 𝐫 i = 0 \mathbf{r}_{i}=0
  117. 𝐱 j \mathbf{x}_{j}
  118. i i
  119. 𝐱 j \mathbf{x}_{j}
  120. M i , j = 1 M_{i,j}=1
  121. 𝐱 j \mathbf{x}_{j}
  122. 𝐫 i = 1 \mathbf{r}_{i}=1
  123. 𝐱 j \mathbf{x}_{j}
  124. 𝐱 \mathbf{x}
  125. 𝒪 ( n t ) \mathcal{O}(nt)
  126. \Box
  127. M M
  128. d e d^{e}
  129. d + 1 d+1
  130. M 0 M_{0}^{\prime}
  131. M 1 M_{1}^{\prime}
  132. \cdots
  133. M d M_{d}^{\prime}
  134. M M
  135. e + 1 e+1
  136. 1 1
  137. M 0 - i = 1 d M i M_{0}^{\prime}-\cup_{i=1}^{d}M_{i}^{\prime}
  138. M M
  139. t × n t\times n
  140. d e d^{e}
  141. o ( S ) o(S)
  142. S S
  143. M M
  144. S S
  145. S S
  146. { 1 , 2 , , n } \{1,2,\cdots,n\}
  147. d d
  148. M M
  149. d e d^{e}
  150. S , T { 1 , 2 , , n } S,T\subseteq\{1,2,\cdots,n\}
  151. d d
  152. o ( S ) o(S)
  153. o ( T ) o(T)
  154. e + 1 e+1
  155. j \exists j
  156. j S j\in S
  157. j T j\notin T
  158. j j
  159. M M
  160. M M
  161. T T
  162. e + 1 e+1
  163. \ell
  164. M j = 1 M_{\ell j}=1
  165. M k = 0 M_{\ell k}=0
  166. k T k\in T
  167. d e d^{e}
  168. e e
  169. e 2 \lfloor\dfrac{e}{2}\rfloor
  170. d e d^{e}
  171. d d
  172. 1 d n 1\leq d\leq n
  173. M M
  174. t × n t\times n
  175. j [ n ] , | S M j | w min \forall j\in[n]\,\text{, }|S_{M_{j}}|\geq w_{\min}
  176. i j [ n ] , | S M i S M j | a max \forall i\neq j\in[n],|S_{M_{i}}\cap S_{M_{j}}|\leq a_{\max}
  177. a max w min t a_{\max}\leq w_{\min}\leq t
  178. M M
  179. d w min - 1 a max \geq d^{\prime}\left\lfloor\frac{w_{\min}-1}{a_{\max}}\right\rfloor
  180. d d
  181. i i
  182. w min w_{\min}
  183. a max a_{\max}
  184. S [ n ] , | S | d , j S S\subseteq[n],|S|\leq d,j\notin S
  185. M M
  186. i S and j S i\in S\,\text{ and }j\notin S
  187. i i
  188. j j
  189. a max d a max ( w min - 1 a max ) = w min - 1 < w min \leq a_{\max}\cdot d\leq a_{\max}\cdot(\frac{w_{\min}-1}{a_{\max}})=w_{\min}-1% <\,\text{ }w_{\min}
  190. j j
  191. S S
  192. j j
  193. \Box
  194. n × m n\times m
  195. 𝒪 ( poly ( n + m ) ) \mathcal{O}(\,\text{poly}(n+m))
  196. poly ( m ) \,\text{poly}(m)
  197. t ( d , n ) 𝒪 ( d 2 log n ) t(d,n)\leq\mathcal{O}(d^{2}\log n)
  198. d d
  199. 𝒪 ( d 2 log n ) \mathcal{O}(d^{2}\log n)
  200. t × n t\times n
  201. M M
  202. t = c d 2 log n t=cd^{2}\log n
  203. c c
  204. M M
  205. Ω ( d ) \Omega(d)
  206. M i , j { 0 , 1 } M_{i,j}\in\{0,1\}
  207. M i , j = 1 M_{i,j}=1
  208. 1 d \frac{1}{d}
  209. i [ t ] i\in[t]
  210. j [ n ] j\in[n]
  211. j [ n ] j\in[n]
  212. j t h j^{th}
  213. M M
  214. T j [ t ] T_{j}\subseteq[t]
  215. 𝔼 [ | T j | ] = t d \mathbb{E}[|T_{j}|]=\frac{t}{d}
  216. μ = 1 2 \mu=\frac{1}{2}
  217. Pr [ | T j | < t 2 d ] e - t 12 d = e - c d log n 12 n - 2 d [ \mathrm{Pr}[|T_{j}|<\frac{t}{2d}]\leq e^{\frac{-t}{12d}}=e^{\frac{-cd\log n}{1% 2}}\leq n^{-2d}[
  218. c 24 ] c\geq 24]
  219. Pr [ j \mathrm{Pr}[\exists j
  220. | T j | < t 2 d ] n n - 2 d n - d |T_{j}|<\frac{t}{2d}]\leq n\cdot n^{-2d}\leq n^{-d}
  221. Pr [ j \mathrm{Pr}[\forall j
  222. | T j | t 2 d ] 1 - n - d |T_{j}|\geq\frac{t}{2d}]\geq 1-n^{-d}
  223. w min t 2 d w_{\min}\geq\frac{t}{2d}
  224. 1 - n - d \geq 1-n^{-d}
  225. j k [ n ] j\neq k\in[n]
  226. i [ t ] i\in[t]
  227. Pr [ M i , j = M i , k = 1 ] = 1 d 2 \mathrm{Pr}[M_{i,j}=M_{i,k}=1]=\frac{1}{d^{2}}
  228. 𝔼 [ | T j T k | ] = t d 2 \mathbb{E}[|T_{j}\cap T_{k}|]=\frac{t}{d^{2}}
  229. Pr [ | T j T k | > 2 t d 2 ] e - t 3 d 2 = e - 2 log n n - 4 [ \mathrm{Pr}[|T_{j}\cap T_{k}|>\frac{2t}{d^{2}}]\leq e^{\frac{-t}{3d^{2}}}=e^{-% 2\log n}\leq n^{-4}[
  230. c 12 ] c\geq 12]
  231. ( j , k ) (j,k)
  232. Pr [ ( j , k ) \mathrm{Pr}[\exists(j,k)
  233. | T j T k | > 2 t d 2 ] n 2 n - 4 = n - 2 |T_{j}\cap T_{k}|>\frac{2t}{d^{2}}]\leq n^{2}\cdot n^{-4}=n^{-2}
  234. a max 2 t d 2 a_{\max}\leq\frac{2t}{d^{2}}
  235. w min t 2 d w_{\min}\geq\frac{t}{2d}
  236. 1 - n - d - n - 2 1 - 1 n \geq 1-n^{-d}-n^{-2}\geq 1-\frac{1}{n}
  237. c c
  238. 1 - 1 n 1-\frac{1}{n}
  239. 1 - 1 p o l y ( n ) 1-\frac{1}{poly(n)}
  240. d = t 2 d - 1 2 t d 2 d 4 d^{\prime}=\lfloor\frac{\frac{t}{2d}-1}{\frac{2t}{d^{2}}}\rfloor\approx\frac{d% }{4}
  241. d d
  242. 4 d 4d
  243. M M
  244. d d
  245. t = d 2 log n t=d^{2}\log n
  246. t ( d , n ) 𝒪 ( d 2 log n ) t(d,n)\leq\mathcal{O}(d^{2}\log n)
  247. \Box
  248. t ( d , n ) 𝒪 ( d 2 log 2 n ) t(d,n)\leq\mathcal{O}(d^{2}\log^{2}{n})
  249. log n \log n
  250. d d
  251. 𝒪 ( d 2 log 2 n ) \mathcal{O}(d^{2}\log^{2}{n})
  252. C { 0 , 1 } t , | C | = n C\subseteq\{0,1\}^{t},|C|=n
  253. C = { 𝐜 1 , , 𝐜 n } C=\{\mathbf{c}_{1},\ldots,\mathbf{c}_{n}\}
  254. M C M_{C}
  255. i t h i^{th}
  256. 𝐜 i \mathbf{c}_{i}
  257. C * C^{*}
  258. i C * , | 𝐜 i | w min \forall i\in C^{*}\,\text{, }|\mathbf{c}_{i}|\geq w_{\min}
  259. 𝐜 1 𝐜 2 C * , | { i | 𝐜 i 1 = 𝐜 i 2 = 1 } | a max \forall\mathbf{c}^{1}\neq\mathbf{c}^{2}\in C^{*}\,\text{, }|\{i|\mathbf{c}^{1}% _{i}=\mathbf{c}^{2}_{i}=1\}|\leq a_{\max}
  260. M C * M_{C^{*}}
  261. w min - 1 a max \lfloor\frac{w_{\min}-1}{a_{\max}}\rfloor
  262. C * = C o u t C i n C^{*}=C_{out}\circ C_{in}
  263. C o u t C_{out}
  264. [ q , k ] q [q,k]_{q}
  265. C i n = [ q ] { 0 , 1 } q C_{in}=[q]\rightarrow\{0,1\}^{q}
  266. i [ q ] i\in[q]
  267. c i n ( i ) = ( 0 , , 0 , 1 , 0 , , 0 ) c_{in}(i)=(0,\ldots,0,1,0,\ldots,0)
  268. i t h i^{th}
  269. n = q k n=q^{k}
  270. t = q 2 t=q^{2}
  271. w min = q w_{\min}=q
  272. k = 1 , q = 3 , C o u t = { ( 0 , 0 , 0 ) , ( 1 , 1 , 1 ) , ( 2 , 2 , 2 ) } k=1,q=3,C_{out}=\{(0,0,0),(1,1,1),(2,2,2)\}
  273. M C M_{C}
  274. C o u t C_{out}
  275. M C * M_{C^{*}}
  276. C * = C o u t C i n C^{*}=C_{out}\circ C_{in}
  277. M C = [ 0 1 2 0 1 2 0 1 2 ] M C * = [ 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 ] M_{C}=\begin{bmatrix}0&1&2\\ 0&1&2\\ 0&1&2\end{bmatrix}\quad\Rightarrow\quad M_{C^{*}}=\begin{bmatrix}0&0&1\\ 0&1&0\\ 1&0&0\\ 0&0&1\\ 0&1&0\\ 1&0&0\\ 0&0&1\\ 0&1&0\\ 1&0&0\end{bmatrix}
  278. M C * M_{C^{*}}
  279. q q
  280. ( i , j ) [ q ] × [ q ] (i,j)\in[q]\times[q]
  281. i i
  282. j j
  283. M ( i , j ) , k 1 = M ( i , j ) , k 2 = 1 M_{(i,j),k_{1}}=M_{(i,j),k_{2}}=1
  284. 𝐜 k 1 ( i ) = 𝐜 k 2 ( i ) = j \mathbf{c}_{k_{1}}(i)=\mathbf{c}_{k_{2}}(i)=j
  285. 𝐜 k 1 , 𝐜 k 2 C o u t \mathbf{c}_{k_{1}},\mathbf{c}_{k_{2}}\in C_{out}
  286. | M k 1 M k 2 | = q - Δ ( 𝐜 k 1 , 𝐜 k 2 ) |M_{k_{1}}\cap M_{k_{2}}|=q-\Delta(\mathbf{c}_{k_{1}},\mathbf{c}_{k_{2}})
  287. Δ ( 𝐜 k 1 , 𝐜 k 2 ) q - k + 1 \Delta(\mathbf{c}_{k_{1}},\mathbf{c}_{k_{2}})\geq q-k+1
  288. | M k 1 M k 2 | k - 1 |M_{k_{1}}\cap M_{k_{2}}|\leq k-1
  289. a max = k - 1 a_{\max}=k-1
  290. t = q 2 t=q^{2}
  291. M C * M_{C^{*}}
  292. q q
  293. q q
  294. C i n C_{in}
  295. q q
  296. w min = q w_{\min}=q
  297. d = d e f w min - 1 a max d=_{def}\lfloor\frac{w_{\min}-1}{a_{\max}}\rfloor
  298. M C * M_{C^{*}}
  299. d d
  300. q q
  301. k k
  302. q - 1 k - 1 = d \lfloor\frac{q-1}{k-1}\rfloor=d
  303. q k d \lfloor\frac{q}{k}\rfloor\approx d
  304. q k = n q^{k}=n
  305. k = log n log q log n k=\frac{\log n}{\log q}\leq\log n
  306. q k d q\approx kd
  307. t = q 2 t=q^{2}
  308. t = q 2 ( k d ) 2 ( d log n ) 2 t=q^{2}\approx(kd)^{2}\leq(d\log n)^{2}
  309. \Box
  310. t ( d , n ) ( d log n ) 2 t(d,n)\leq(d\log n)^{2}
  311. Ω ( d log n ) t ( d , n ) \Omega(d\log n)\leq t(d,n)
  312. t ( d , n ) 𝒪 ( d 2 log 2 n ) t(d,n)\leq\mathcal{O}(d^{2}\log^{2}{n})
  313. t ( d , n ) 𝒪 ( d 2 log n ) t(d,n)\leq\mathcal{O}(d^{2}\log n)
  314. t ( d , n ) 𝒪 ( d 2 log n ) t(d,n)\leq\mathcal{O}(d^{2}\log n)
  315. t ( d , n ) Ω ( d 2 log d log n ) t(d,n)\geq\Omega(\frac{d^{2}}{\log d}\log n)
  316. M 9 × 12 M_{9\times 12}
  317. M 9 × 12 = [ 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 ] M_{9\times 12}=\left[\begin{array}[]{cccccccccccc}0&0&0&0&0&0&1&1&1&1&0&0\\ 0&0&0&1&1&1&0&0&0&1&0&0\\ 1&1&1&0&0&0&0&0&0&1&0&0\\ 0&0&1&0&0&1&0&0&1&0&1&0\\ 0&1&0&0&1&0&0&1&0&0&1&0\\ 1&0&0&1&0&0&1&0&0&0&1&0\\ 0&1&0&1&0&0&0&0&1&0&0&1\\ 0&0&1&0&1&0&1&0&0&0&0&1\\ 1&0&0&0&0&1&0&1&0&0&0&1\end{array}\right]

Dissociative_substitution.html

  1. R a t e = k 1 [ L n M - L ] Rate={k_{1}[L_{n}M-L]}
  2. k 2 [ L ] [ I n t ] k - 1 [ L ] [ I n t ] + k 2 [ L ] [ I n t ] \frac{k_{2}[L^{\prime}][Int]}{k_{-}1[L][Int]+k_{2}[L^{\prime}][Int]}
  3. R a t e o v e r a l l = ( k 2 [ L ] [ I n t ] k - 1 [ L ] [ I n t ] + k 2 [ L ] [ I n t ] ) ( k 1 [ L n M - L ] ) = k 1 k 2 [ L ] [ L n M - L ] k - 1 [ L ] + k 2 [ L ] Rate_{overall}=(\frac{k_{2}[L^{\prime}][Int]}{k_{-1}[L][Int]+k_{2}[L^{\prime}]% [Int]})({k_{1}[L_{n}M-L]})=\frac{k_{1}k_{2}[L^{\prime}][L_{n}M-L]}{k_{-1}[L]+k% _{2}[L^{\prime}]}

Distance_from_a_point_to_a_line.html

  1. distance ( a x + b y + c = 0 , ( x 0 , y 0 ) ) = | a x 0 + b y 0 + c | a 2 + b 2 . \operatorname{distance}(ax+by+c=0,(x_{0},y_{0}))=\frac{|ax_{0}+by_{0}+c|}{% \sqrt{a^{2}+b^{2}}}.
  2. x = b ( b x 0 - a y 0 ) - a c a 2 + b 2 and y = a ( - b x 0 + a y 0 ) - b c a 2 + b 2 . x=\frac{b(bx_{0}-ay_{0})-ac}{a^{2}+b^{2}}\,\text{ and }y=\frac{a(-bx_{0}+ay_{0% })-bc}{a^{2}+b^{2}}.
  3. distance ( P 1 , P 2 , ( x 0 , y 0 ) ) = | ( y 2 - y 1 ) x 0 - ( x 2 - x 1 ) y 0 + x 2 y 1 - y 2 x 1 | ( y 2 - y 1 ) 2 + ( x 2 - x 1 ) 2 . \operatorname{distance}(P_{1},P_{2},(x_{0},y_{0}))=\frac{|(y_{2}-y_{1})x_{0}-(% x_{2}-x_{1})y_{0}+x_{2}y_{1}-y_{2}x_{1}|}{\sqrt{(y_{2}-y_{1})^{2}+(x_{2}-x_{1}% )^{2}}}.
  4. h = 2 A b h=\frac{2A}{b}
  5. A = 1 2 b h A=\frac{1}{2}bh
  6. y 0 - n x 0 - m = b a . \frac{y_{0}-n}{x_{0}-m}=\frac{b}{a}.
  7. a ( y 0 - n ) - b ( x 0 - m ) = 0 , a(y_{0}-n)-b(x_{0}-m)=0,
  8. a 2 ( y 0 - n ) 2 + b 2 ( x 0 - m ) 2 = 2 a b ( y 0 - n ) ( x 0 - m ) . a^{2}(y_{0}-n)^{2}+b^{2}(x_{0}-m)^{2}=2ab(y_{0}-n)(x_{0}-m).
  9. ( a ( x 0 - m ) + b ( y 0 - n ) ) 2 = a 2 ( x 0 - m ) 2 + 2 a b ( y 0 - n ) ( x 0 - m ) + b 2 ( y 0 - n ) 2 = ( a 2 + b 2 ) ( ( x 0 - m ) 2 + ( y 0 - n ) 2 ) (a(x_{0}-m)+b(y_{0}-n))^{2}=a^{2}(x_{0}-m)^{2}+2ab(y_{0}-n)(x_{0}-m)+b^{2}(y_{% 0}-n)^{2}=(a^{2}+b^{2})((x_{0}-m)^{2}+(y_{0}-n)^{2})
  10. ( a ( x 0 - m ) + b ( y 0 - n ) ) 2 = ( a x 0 + b y 0 - a m - b n ) 2 = ( a x 0 + b y 0 + c ) 2 (a(x_{0}-m)+b(y_{0}-n))^{2}=(ax_{0}+by_{0}-am-bn)^{2}=(ax_{0}+by_{0}+c)^{2}
  11. ( a 2 + b 2 ) ( ( x 0 - m ) 2 + ( y 0 - n ) 2 ) = ( a x 0 + b y 0 + c ) 2 (a^{2}+b^{2})((x_{0}-m)^{2}+(y_{0}-n)^{2})=(ax_{0}+by_{0}+c)^{2}
  12. d = ( x 0 - m ) 2 + ( y 0 - n ) 2 = | a x 0 + b y 0 + c | a 2 + b 2 . d=\sqrt{(x_{0}-m)^{2}+(y_{0}-n)^{2}}=\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^{2}+b^{2% }}}.
  13. | P R ¯ | | P S ¯ | = | T V ¯ | | T U ¯ | . \frac{|\overline{PR}|}{|\overline{PS}|}=\frac{|\overline{TV}|}{|\overline{TU}|}.
  14. | P R ¯ | = | y 0 - m | | B | A 2 + B 2 . |\overline{PR}|=\frac{|y_{0}-m||B|}{\sqrt{A^{2}+B^{2}}}.
  15. m = - A x 0 - C B , m=\frac{-Ax_{0}-C}{B},
  16. | P R ¯ | = | A x 0 + B y 0 + C | A 2 + B 2 . |\overline{PR}|=\frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}.
  17. D | T U ¯ | = | V U ¯ | | V T ¯ | D|\overline{TU}|=|\overline{VU}||\overline{VT}|
  18. | T U ¯ | |\overline{TU}|
  19. | V U ¯ | |\overline{VU}|
  20. | V T ¯ | |\overline{VT}|
  21. Q P \overrightarrow{QP}
  22. d = | Q P 𝐧 | 𝐧 . d=\frac{|\overrightarrow{QP}\cdot\mathbf{n}|}{\|\mathbf{n}\|}.
  23. Q P = ( x 0 - x 1 , y 0 - y 1 ) , \overrightarrow{QP}=(x_{0}-x_{1},y_{0}-y_{1}),
  24. Q P 𝐧 = a ( x 0 - x 1 ) + b ( y 0 - y 1 ) \overrightarrow{QP}\cdot\mathbf{n}=a(x_{0}-x_{1})+b(y_{0}-y_{1})
  25. 𝐧 = a 2 + b 2 , \|\mathbf{n}\|=\sqrt{a^{2}+b^{2}},
  26. d = | a ( x 0 - x 1 ) + b ( y 0 - y 1 ) | a 2 + b 2 . d=\frac{|a(x_{0}-x_{1})+b(y_{0}-y_{1})|}{\sqrt{a^{2}+b^{2}}}.
  27. c = - a x 1 - b y 1 c=-ax_{1}-by_{1}
  28. d = | a x 0 + b y 0 + c | a 2 + b 2 . d=\frac{|ax_{0}+by_{0}+c|}{\sqrt{a^{2}+b^{2}}}.
  29. x 0 , y 0 x_{0},y_{0}
  30. y = m x + k y=mx+k
  31. y = x 0 - x m + y 0 y=\frac{x_{0}-x}{m}+y_{0}
  32. m x + k = x 0 - x m + y 0 . mx+k=\frac{x_{0}-x}{m}+y_{0}.
  33. x = x 0 + m y 0 - m k m 2 + 1 . x=\frac{x_{0}+my_{0}-mk}{m^{2}+1}.
  34. y = m ( x 0 + m y 0 - m k ) m 2 + 1 + k . y=m\frac{(x_{0}+my_{0}-mk)}{m^{2}+1}+k.
  35. d = ( X 2 - X 1 ) 2 + ( Y 2 - Y 1 ) 2 d=\sqrt{(X_{2}-X_{1})^{2}+(Y_{2}-Y_{1})^{2}}
  36. d = ( x 0 + m y 0 - m k m 2 + 1 - x 0 ) 2 + ( m x 0 + m y 0 - m k m 2 + 1 + k - y 0 ) 2 . d=\sqrt{\left({\frac{x_{0}+my_{0}-mk}{m^{2}+1}-x_{0}}\right)^{2}+\left({m\frac% {x_{0}+my_{0}-mk}{m^{2}+1}+k-y_{0}}\right)^{2}}.
  37. 𝐱 = 𝐚 + t 𝐧 \mathbf{x}=\mathbf{a}+t\mathbf{n}
  38. 𝐱 \mathbf{x}
  39. 𝐧 \mathbf{n}
  40. 𝐚 \mathbf{a}
  41. 𝐱 \mathbf{x}
  42. 𝐚 \mathbf{a}
  43. t t
  44. 𝐩 \mathbf{p}
  45. distance ( 𝐱 = 𝐚 + t 𝐧 , 𝐩 ) = ( 𝐚 - 𝐩 ) - ( ( 𝐚 - 𝐩 ) 𝐧 ) 𝐧 . \operatorname{distance}(\mathbf{x}=\mathbf{a}+t\mathbf{n},\mathbf{p})=\|(% \mathbf{a}-\mathbf{p})-((\mathbf{a}-\mathbf{p})\cdot\mathbf{n})\mathbf{n}\|.
  46. 𝐚 - 𝐩 \mathbf{a}-\mathbf{p}
  47. 𝐩 \mathbf{p}
  48. 𝐚 \mathbf{a}
  49. ( 𝐚 - 𝐩 ) 𝐧 (\mathbf{a}-\mathbf{p})\cdot\mathbf{n}
  50. ( ( 𝐚 - 𝐩 ) 𝐧 ) 𝐧 ((\mathbf{a}-\mathbf{p})\cdot\mathbf{n})\mathbf{n}
  51. 𝐚 - 𝐩 \mathbf{a}-\mathbf{p}
  52. ( 𝐚 - 𝐩 ) - ( ( 𝐚 - 𝐩 ) 𝐧 ) 𝐧 (\mathbf{a}-\mathbf{p})-((\mathbf{a}-\mathbf{p})\cdot\mathbf{n})\mathbf{n}
  53. 𝐚 - 𝐩 \mathbf{a}-\mathbf{p}

Distance_of_closest_approach_of_ellipses_and_ellipsoids.html

  1. E 1 E_{1}
  2. E 2 E_{2}
  3. d d
  4. C 1 C_{1}^{\prime}
  5. E 2 E_{2}^{\prime}
  6. d d^{\prime}
  7. C 1 C_{1}^{\prime}
  8. E 2 E_{2}^{\prime}
  9. d d^{\prime}
  10. C 1 C_{1}^{\prime}
  11. E 2 E_{2}^{\prime}
  12. n n^{\prime}
  13. d d
  14. E 1 E_{1}
  15. E 2 E_{2}
  16. d d^{\prime}
  17. n n^{\prime}
  18. a 1 , b 1 , a 2 , b 2 a_{1},b_{1},a_{2},b_{2}
  19. k 1 k_{1}
  20. k 2 k_{2}
  21. d d
  22. d d
  23. E 1 E_{1}
  24. E 2 E_{2}
  25. k 1 k_{1}
  26. k 2 k_{2}

Distortion_(mathematics).html

  1. H ( z , f ) = lim sup r 0 max | h | = r | f ( z + h ) - f ( z ) | min | h | = r | f ( z + h ) - f ( z ) | H(z,f)=\limsup_{r\to 0}\frac{\max_{|h|=r}|f(z+h)-f(z)|}{\min_{|h|=r}|f(z+h)-f(% z)|}
  2. | D f ( x ) | 2 K ( x ) | J ( x , f ) | |Df(x)|^{2}\leq K(x)|J(x,f)|\,
  3. G ( x , f ) = { | J ( x , f ) - 2 / n D T f ( x ) D f ( x ) if J ( x , f ) 0 I if J ( x , f ) = 0. G(x,f)=\begin{cases}|J(x,f)^{-2/n}D^{T}f(x)Df(x)&\,\text{if }J(x,f)\not=0\\ I&\,\text{if }J(x,f)=0.\end{cases}
  4. K O ( x ) = sup ξ 0 G ( x ) ξ , ξ n / 2 | ξ | n , K O ( x ) = sup ξ 0 G - 1 ( x ) ξ , ξ n / 2 | ξ | n . K_{O}(x)=\sup_{\xi\not=0}\frac{\langle G(x)\xi,\xi\rangle^{n/2}}{|\xi|^{n}},% \quad K_{O}(x)=\sup_{\xi\not=0}\frac{\langle G^{-1}(x)\xi,\xi\rangle^{n/2}}{|% \xi|^{n}}.
  5. | D f ( x ) | n K O ( x ) | J ( x , f ) | |Df(x)|^{n}\leq K_{O}(x)|J(x,f)|\,

Distortion_free_energy_density.html

  1. T = 0 + d \mathcal{F}_{T}=\mathcal{F}_{0}+\mathcal{F}_{d}
  2. T \mathcal{F}_{T}
  3. 0 \mathcal{F}_{0}
  4. d \mathcal{F}_{d}
  5. d \mathcal{F}_{d}
  6. d = 1 2 K 1 ( 𝐧 ^ ) 2 + 1 2 K 2 ( 𝐧 ^ × 𝐧 ^ ) 2 + 1 2 K 3 ( 𝐧 ^ × × 𝐧 ^ ) 2 \mathcal{F}_{d}=\frac{1}{2}K_{1}(\nabla\cdot\mathbf{\hat{n}})^{2}+\frac{1}{2}K% _{2}(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}})^{2}+\frac{1}{2}K_{3}(% \mathbf{\hat{n}}\times\nabla\times\mathbf{\hat{n}})^{2}
  7. 𝐧 ^ \mathbf{\hat{n}}
  8. ( | 𝐧 ^ | = 1 ) (|\mathbf{\hat{n}}|=1)
  9. K i K_{i}
  10. 10 - 6 10^{-6}
  11. K 1 = K 2 = K 3 = K K_{1}=K_{2}=K_{3}=K
  12. d = 1 2 K ( ( 𝐧 ^ ) 2 + ( × 𝐧 ^ ) 2 ) = 1 2 K α n β α n β \mathcal{F}_{d}=\frac{1}{2}K((\nabla\cdot\mathbf{\hat{n}})^{2}+(\nabla\times% \mathbf{\hat{n}})^{2})=\frac{1}{2}K\partial_{\alpha}n_{\beta}\partial_{\alpha}% n_{\beta}
  13. 1 2 K 24 ( ( 𝐧 ^ ) 𝐧 ^ - 𝐧 ^ ( 𝐧 ^ ) ) \frac{1}{2}K_{24}\nabla\cdot((\mathbf{\hat{n}}\cdot\nabla)\mathbf{\hat{n}}-% \mathbf{\mathbf{\hat{n}}}(\nabla\cdot\mathbf{\hat{n}}))
  14. s = - 1 2 W ( 𝐧 ^ ν ^ ) 2 d S \mathcal{F}_{s}=-\oint\frac{1}{2}W(\mathbf{\hat{n}}\cdot\mathbf{\hat{\nu}})^{2% }\mathrm{d}S
  15. W W
  16. ν ^ \mathbf{\hat{\nu}}
  17. C h = k 2 ( 𝐧 ^ × 𝐧 ^ ) \mathcal{F}_{Ch}=k_{2}(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}})
  18. k 2 k_{2}
  19. T = 0 + 1 2 K 1 ( 𝐧 ^ ) 2 + 1 2 K 2 ( 𝐧 ^ × 𝐧 ^ + q 0 ) 2 + 1 2 K 3 ( 𝐧 ^ × × 𝐧 ^ ) 2 \mathcal{F}_{T}=\mathcal{F}_{0}+\frac{1}{2}K_{1}(\nabla\cdot\mathbf{\hat{n}})^% {2}+\frac{1}{2}K_{2}(\mathbf{\hat{n}}\cdot\nabla\times\mathbf{\hat{n}}+q_{0})^% {2}+\frac{1}{2}K_{3}(\mathbf{\hat{n}}\times\nabla\times\mathbf{\hat{n}})^{2}
  20. q 0 = 2 π / P 0 q_{0}=2\pi/P_{0}
  21. P 0 P_{0}
  22. 𝐧 ^ \mathbf{\hat{n}}
  23. χ \chi_{\perp}
  24. χ \chi_{\parallel}
  25. 𝐧 ^ \mathbf{\hat{n}}
  26. Δ χ χ - χ = N < P 2 ( cos θ ) > \Delta\chi\equiv\chi_{\parallel}-\chi_{\perp}=N<P_{2}(\cos{\theta})>
  27. W m a g n e t i c = 0 H ( - M sin θ - M cos θ ) d H = - H 2 2 ( χ + Δ χ cos θ 2 ) W_{magnetic}=\int_{0}^{H}(-M_{\perp}\sin{\theta}-M_{\parallel}\cos{\theta})\,% dH=-\frac{H^{2}}{2}(\chi_{\perp}+\Delta\chi\cos{\theta}^{2})
  28. M = H χ cos θ M_{\parallel}=H\chi_{\parallel}\cos{\theta}
  29. M = H χ sin θ M_{\perp}=H\chi_{\perp}\sin{\theta}
  30. - H 2 χ 2 -\frac{H^{2}\chi_{\perp}}{2}
  31. - Δ χ 2 [ 𝐇 𝐧 ^ ] 2 -\frac{\Delta\chi}{2}[\mathbf{H}\cdot\mathbf{\hat{n}}]^{2}
  32. - Δ ϵ 8 π [ 𝐄 𝐧 ^ ] -\frac{\Delta\epsilon}{8\pi}[\mathbf{E}\cdot\mathbf{\hat{n}}]
  33. Δ ϵ ϵ - ϵ \Delta\epsilon\equiv\epsilon_{\parallel}-\epsilon_{\perp}
  34. 𝐧 ^ \mathbf{\hat{n}}

Distributed_element_filter.html

  1. Δ ω = ω 2 - ω 1 \Delta\omega=\omega_{2}-\omega_{1}\,
  2. ω 0 = ω 1 ω 2 \omega_{0}=\sqrt{\omega_{1}\omega_{2}}
  3. Q = ω 0 Δ ω Q=\frac{\omega_{0}}{\Delta\omega}

Distributed_source_coding.html

  1. R X H ( X | Y ) , R_{X}\geq H(X|Y),\,
  2. R Y H ( Y | X ) , R_{Y}\geq H(Y|X),\,
  3. R X + R Y H ( X , Y ) . R_{X}+R_{Y}\geq H(X,Y).\,
  4. H ( X ) H(X)
  5. H ( Y ) H(Y)
  6. X X
  7. Y Y
  8. H ( X ) H(X)
  9. H ( Y ) H(Y)
  10. X X
  11. Y Y
  12. X X
  13. Y Y
  14. H ( X , Y ) H(X,Y)
  15. Y Y
  16. R Y = H ( Y ) R_{Y}=H(Y)
  17. Y Y
  18. H ( X | Y ) H(X|Y)
  19. X X
  20. X X
  21. Y Y
  22. Y Y
  23. X X
  24. D D
  25. X X
  26. X X
  27. Y Y
  28. 𝐱 \mathbf{x}
  29. 𝐲 \mathbf{y}
  30. 𝐱 \mathbf{x}
  31. 𝐲 \mathbf{y}
  32. R X + R Y 10 R_{X}+R_{Y}\geq 10
  33. R X 5 R_{X}\geq 5
  34. R Y 5 R_{Y}\geq 5
  35. R X + R Y = 10 R_{X}+R_{Y}=10
  36. R X + R Y = 10 R_{X}+R_{Y}=10
  37. X X
  38. Y Y
  39. R X = 3 R_{X}=3
  40. R Y = 7 R_{Y}=7
  41. R Y = 3 R_{Y}=3
  42. R X = 7 R_{X}=7
  43. X X
  44. Y Y
  45. X , Y { 0 , 1 } n X,Y\in\left\{0,1\right\}^{n}
  46. 𝐝 𝐇 ( X , Y ) t \mathbf{d_{H}}(X,Y)\leq t
  47. ( R 1 , R 2 ) (R_{1},R_{2})
  48. R 1 , R 2 n - k , R 1 + R 2 2 n - k R_{1},R_{2}\geq n-k,R_{1}+R_{2}\geq 2n-k
  49. R 1 R_{1}
  50. R 2 R_{2}
  51. k n - log ( i = 0 t ( n i ) ) k\leq n-\log(\sum_{i=0}^{t}{n\choose i})
  52. ( n , k , 2 t + 1 ) (n,k,2t+1)
  53. ( n , k , 2 t + 1 ) (n,k,2t+1)
  54. k n - log ( i = 0 t ( n i ) ) k\leq n-\log(\sum_{i=0}^{t}{n\choose i})
  55. 𝐂 \mathbf{C}
  56. k × n k\times n
  57. 𝐆 \mathbf{G}
  58. R 1 + R 2 = 2 n - k R_{1}+R_{2}=2n-k
  59. 𝐆 𝟏 \mathbf{G_{1}}
  60. ( n - R 1 ) (n-R_{1})
  61. 𝐆 \mathbf{G}
  62. 𝐆 𝟐 \mathbf{G_{2}}
  63. ( n - R 2 ) (n-R_{2})
  64. 𝐆 \mathbf{G}
  65. 𝐂 𝟏 \mathbf{C_{1}}
  66. 𝐂 𝟐 \mathbf{C_{2}}
  67. 𝐆 𝟏 \mathbf{G_{1}}
  68. 𝐆 𝟐 \mathbf{G_{2}}
  69. 𝐇 𝟏 \mathbf{H_{1}}
  70. 𝐇 𝟐 \mathbf{H_{2}}
  71. ( 𝐱 , 𝐲 ) \mathbf{(x,y)}
  72. 𝐬 𝟏 = 𝐇 𝟏 𝐱 \mathbf{s_{1}}=\mathbf{H_{1}}\mathbf{x}
  73. 𝐬 𝟐 = 𝐇 𝟐 𝐲 \mathbf{s_{2}}=\mathbf{H_{2}}\mathbf{y}
  74. 𝐱 \mathbf{x}
  75. 𝐲 \mathbf{y}
  76. 𝐱 = 𝐮 𝟏 𝐆 𝟏 + 𝐜 𝐬𝟏 \mathbf{x=u_{1}G_{1}+c_{s1}}
  77. 𝐲 = 𝐮 𝟐 𝐆 𝟐 + 𝐜 𝐬𝟐 \mathbf{y=u_{2}G_{2}+c_{s2}}
  78. 𝐜 𝐬𝟏 , 𝐜 𝐬𝟐 \mathbf{c_{s1},c_{s2}}
  79. 𝐬𝟏 , 𝐬𝟐 \mathbf{s1,s2}
  80. 𝐂 𝟏 , 𝐂 𝟐 \mathbf{C_{1},C_{2}}
  81. 𝐲 = 𝐱 + 𝐞 \mathbf{y=x+e}
  82. w ( 𝐞 ) t w(\mathbf{e})\leq t
  83. 𝐱 + 𝐲 = 𝐮𝐆 + 𝐜 𝐬 = 𝐞 \mathbf{x+y=uG+c_{s}=e}
  84. 𝐮 = [ 𝐮 𝟏 , 𝐮 𝟐 ] \mathbf{u=\left[u_{1},u_{2}\right]}
  85. 𝐜 𝐬 = 𝐜 𝐬𝟏 + 𝐜 𝐬𝟐 \mathbf{c_{s}=c_{s1}+c_{s2}}
  86. 𝐮 𝟏 , 𝐮 𝟐 { 0 , 1 } k \mathbf{u^{1},u^{2}}\in\left\{0,1\right\}^{k}
  87. 𝐮 𝟏 𝐆 + 𝐜 𝐬 = 𝐞 \mathbf{u^{1}G+c_{s}=e}
  88. 𝐮 𝟐 𝐆 + 𝐜 𝐬 = 𝐞 \mathbf{u^{2}G+c_{s}=e}
  89. ( 𝐮 𝟏 - 𝐮 𝟐 ) 𝐆 = 𝟎 \mathbf{(u^{1}-u^{2})G=0}
  90. 𝐂 \mathbf{C}
  91. 2 t + 1 2t+1
  92. 𝐮 𝟏 𝐆 \mathbf{u_{1}G}
  93. 𝐮 𝟐 𝐆 \mathbf{u_{2}G}
  94. 2 t + 1 \geq 2t+1
  95. w ( 𝐞 ) t w(\mathbf{e})\leq t
  96. 𝐮 𝟏 𝐆 + 𝐜 𝐬 = 𝐞 \mathbf{u^{1}G+c_{s}=e}
  97. 𝐮 𝟐 𝐆 + 𝐜 𝐬 = 𝐞 \mathbf{u^{2}G+c_{s}=e}
  98. d H ( 𝐮 𝟏 𝐆 , 𝐜 𝐬 ) t d_{H}(\mathbf{u^{1}G,c_{s}})\leq t
  99. d H ( 𝐮 𝟐 𝐆 , 𝐜 𝐬 ) t d_{H}(\mathbf{u^{2}G,c_{s}})\leq t
  100. d H ( 𝐮 𝟏 𝐆 , 𝐮 𝟐 𝐆 ) 2 t + 1 d_{H}(\mathbf{u^{1}G,u^{2}G})\geq 2t+1
  101. ( n , k , 2 t + 1 ) (n,k,2t+1)
  102. ( R 1 , R 2 ) (R_{1},R_{2})
  103. R 1 , R 2 n - k , R 1 + R 2 2 n - k R_{1},R_{2}\geq n-k,R_{1}+R_{2}\geq 2n-k
  104. R 1 R_{1}
  105. R 2 R_{2}
  106. k n - log ( i = 0 t ( n i ) ) k\leq n-\log(\sum_{i=0}^{t}{n\choose i})
  107. R X = 3 R_{X}=3
  108. R Y = 7 R_{Y}=7
  109. 𝐲 \mathbf{y}
  110. Y Y
  111. 𝐱 \mathbf{x}
  112. 𝐲 \mathbf{y}
  113. 𝐱 \mathbf{x}
  114. X X
  115. 𝐲 \mathbf{y}
  116. 𝐱 \mathbf{x}
  117. 𝐲 \mathbf{y}
  118. 𝐱 \mathbf{x}
  119. 𝐲 \mathbf{y}
  120. 𝐲 \mathbf{y}
  121. 𝐱 \mathbf{x}
  122. 𝐱 \mathbf{x}
  123. 𝐲 \mathbf{y}
  124. X X
  125. 𝐲 \mathbf{y}
  126. 𝐱 \mathbf{x}
  127. ( 7 , 4 , 3 ) (7,4,3)
  128. 𝐂 \mathbf{C}
  129. 𝐇 \mathbf{H}
  130. 𝐱 \mathbf{x}
  131. X X
  132. 𝐬 = 𝐇𝐱 \mathbf{s}=\mathbf{H}\mathbf{x}
  133. 𝐲 \mathbf{y}
  134. 𝐬 \mathbf{s}
  135. 𝐱 𝟏 \mathbf{x_{1}}
  136. 𝐱 𝟐 \mathbf{x_{2}}
  137. 𝐬 \mathbf{s}
  138. 𝐇𝐱 𝟏 = 𝐇𝐱 𝟐 \mathbf{H}\mathbf{x_{1}}=\mathbf{H}\mathbf{x_{2}}
  139. 𝐇 ( 𝐱 𝟏 - 𝐱 𝟐 ) = 0 \mathbf{H}(\mathbf{x_{1}}-\mathbf{x_{2}})=0
  140. ( 7 , 4 , 3 ) (7,4,3)
  141. d H ( 𝐱 𝟏 , 𝐱 𝟐 ) 3 d_{H}(\mathbf{x_{1}},\mathbf{x_{2}})\geq 3
  142. 𝐱 \mathbf{x}
  143. d H ( 𝐱 , 𝐲 ) 1 d_{H}(\mathbf{x},\mathbf{y})\leq 1
  144. R X = 7 R_{X}=7
  145. R Y = 3 R_{Y}=3
  146. X X
  147. Y Y
  148. 𝐂 𝟏 \mathbf{C_{1}}
  149. 𝐂 𝟐 \mathbf{C_{2}}
  150. 𝐬 𝟏 = 𝐇 𝟏 𝐱 \mathbf{s_{1}}=\mathbf{H_{1}}\mathbf{x}
  151. 𝐬 𝟐 = 𝐇 𝟐 𝐲 \mathbf{s_{2}}=\mathbf{H_{2}}\mathbf{y}
  152. 𝐱 𝟏 , 𝐲 𝟏 \mathbf{x_{1}},\mathbf{y_{1}}
  153. 𝐱 𝟐 , 𝐲 𝟐 \mathbf{x_{2}},\mathbf{y_{2}}
  154. 𝐇 𝟏 𝐱 𝟏 = 𝐇 𝟏 𝐱 𝟐 \mathbf{H_{1}}\mathbf{x_{1}}=\mathbf{H_{1}}\mathbf{x_{2}}
  155. 𝐇 𝟏 𝐲 𝟏 = 𝐇 𝟏 𝐲 𝟐 \mathbf{H_{1}}\mathbf{y_{1}}=\mathbf{H_{1}}\mathbf{y_{2}}
  156. w ( ) w()
  157. 𝐲 𝟏 = 𝐱 𝟏 + 𝐞 𝟏 \mathbf{y_{1}}=\mathbf{x_{1}}+\mathbf{e_{1}}
  158. w ( 𝐞 𝟏 ) 1 w(\mathbf{e_{1}})\leq 1
  159. 𝐲 𝟐 = 𝐱 𝟐 + 𝐞 𝟐 \mathbf{y_{2}}=\mathbf{x_{2}}+\mathbf{e_{2}}
  160. w ( 𝐞 𝟐 ) 1 w(\mathbf{e_{2}})\leq 1
  161. 𝐱 𝟏 + 𝐱 𝟐 𝐂 𝟏 \mathbf{x_{1}}+\mathbf{x_{2}}\in\mathbf{C_{1}}
  162. 𝐲 𝟏 + 𝐲 𝟐 = 𝐱 𝟏 + 𝐱 𝟐 + 𝐞 𝟑 𝐂 𝟐 \mathbf{y_{1}}+\mathbf{y_{2}}=\mathbf{x_{1}}+\mathbf{x_{2}}+\mathbf{e_{3}}\in% \mathbf{C_{2}}
  163. 𝐞 𝟑 = 𝐞 𝟐 + 𝐞 𝟏 \mathbf{e_{3}}=\mathbf{e_{2}}+\mathbf{e_{1}}
  164. w ( 𝐞 𝟑 ) 2 w(\mathbf{e_{3}})\leq 2
  165. 3 3
  166. 𝐂 𝟏 \mathbf{C_{1}}
  167. 𝐂 𝟐 \mathbf{C_{2}}
  168. ( 7 , 4 , 3 ) (7,4,3)
  169. 3 3
  170. 𝐆 \mathbf{G}
  171. 𝐆 𝟏 \mathbf{G_{1}}
  172. 𝐂 𝟏 \mathbf{C_{1}}
  173. 𝐆 \mathbf{G}
  174. 𝐆 𝟐 \mathbf{G_{2}}
  175. 𝐆 \mathbf{G}
  176. ( 5 × 7 ) (5\times 7)
  177. 2 N R 2^{NR}
  178. \mathcal{B}
  179. = 2 { 1 , , N R } \mathcal{B}=2^{\{1,...,NR\}}
  180. 𝒮 : { 1 , , N } \mathcal{S}:\{1,...,N\}\rightarrow\mathcal{B}
  181. C = n = 1 N 2 | 𝒮 ( n ) | C=\sum_{n=1}^{N}2^{|\mathcal{S}(n)|}
  182. J = D + λ C J=D+\lambda C
  183. a a
  184. n n
  185. 𝐱 1 , 𝐱 2 , , 𝐱 a { 0 , 1 } n \mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{a}\in\{0,1\}^{n}
  186. 𝐇 1 , 𝐇 2 , , 𝐇 s \mathbf{H}_{1},\mathbf{H}_{2},\cdots,\mathbf{H}_{s}
  187. m 1 × n , m 2 × n , , m a × n m_{1}\times n,m_{2}\times n,\cdots,m_{a}\times n
  188. 𝐬 1 = 𝐇 1 𝐱 1 , 𝐬 2 = 𝐇 2 𝐱 2 , , 𝐬 a = 𝐇 a 𝐱 a \mathbf{s}_{1}=\mathbf{H}_{1}\mathbf{x}_{1},\mathbf{s}_{2}=\mathbf{H}_{2}% \mathbf{x}_{2},\cdots,\mathbf{s}_{a}=\mathbf{H}_{a}\mathbf{x}_{a}
  189. m = m 1 + m 2 + m a m=m_{1}+m_{2}+\cdots m_{a}
  190. 2 n ( a n + 1 ) 2^{n}(an+1)
  191. 2 m 2 n ( a n + 1 ) 2^{m}\geq 2^{n}(an+1)
  192. a , n , m a,n,m
  193. a = 3 , n = 5 , m = 9 a=3,n=5,m=9
  194. n = 21 n=21
  195. m = 27 m=27
  196. 𝐐 1 = ( 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 ) , \mathbf{Q}_{1}=\begin{pmatrix}1\;0\;0\;0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;1\;0\;1\;% 1\;0\;0\;0\;0\\ 0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;0\;0\;1\;0\;0\;0\;0\;0\;1\;1\;1\\ 0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;0\;0\;1\;1\;1\;0\;1\;0\;1\;1\\ 0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;0\;1\;0\;0\;1\;1\;1\;1\;0\\ 0\;0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;1\;0\;1\;1\;0\;1\;1\;1\;1\\ 0\;0\;0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;1\;0\;0\;1\;1\;0\;1\end{pmatrix},
  197. 𝐐 2 = ( 0 0 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 1 0 ) , \mathbf{Q}_{2}=\begin{pmatrix}0\;0\;0\;1\;0\;1\;1\;0\;1\;1\;1\;1\;0\;1\;0\;0\;% 0\;1\;1\;1\;1\\ 1\;0\;0\;0\;1\;0\;1\;1\;0\;1\;1\;1\;1\;0\;1\;1\;1\;1\;0\;0\;0\\ 0\;1\;0\;0\;0\;1\;1\;1\;1\;0\;1\;1\;1\;0\;0\;0\;0\;0\;1\;0\;1\\ 1\;0\;1\;0\;0\;0\;1\;1\;1\;1\;0\;1\;0\;1\;1\;1\;0\;0\;1\;1\;1\\ 0\;1\;0\;1\;0\;0\;1\;1\;1\;1\;1\;0\;0\;0\;1\;0\;1\;1\;0\;1\;1\\ 0\;0\;1\;0\;1\;0\;0\;1\;1\;1\;1\;1\;1\;1\;0\;1\;0\;1\;1\;1\;0\end{pmatrix},
  198. 𝐐 3 = ( 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 ) , \mathbf{Q}_{3}=\begin{pmatrix}1\;0\;0\;1\;0\;1\;0\;0\;1\;1\;1\;0\;1\;0\;0\;1\;% 1\;1\;1\;1\;1\\ 1\;1\;0\;0\;1\;0\;0\;0\;0\;1\;1\;1\;0\;0\;1\;1\;1\;1\;1\;1\;1\\ 0\;1\;1\;0\;0\;1\;1\;0\;0\;0\;1\;1\;1\;1\;1\;1\;0\;1\;1\;1\;0\\ 1\;0\;1\;1\;0\;0\;1\;1\;0\;0\;0\;1\;0\;0\;1\;1\;1\;1\;0\;0\;1\\ 0\;1\;0\;1\;1\;0\;1\;1\;1\;0\;0\;0\;1\;0\;0\;1\;1\;0\;1\;0\;0\\ 0\;0\;1\;0\;1\;1\;0\;1\;1\;1\;0\;0\;1\;1\;1\;1\;0\;0\;0\;1\;1\end{pmatrix},
  199. 𝐆 = [ 𝟎 | 𝐈 9 ] \mathbf{G}=[\mathbf{0}|\mathbf{I}_{9}]
  200. 𝐆 = ( 𝐆 1 𝐆 2 𝐆 3 ) \mathbf{G}=\begin{pmatrix}\mathbf{G}_{1}\\ \mathbf{G}_{2}\\ \mathbf{G}_{3}\end{pmatrix}
  201. 𝐆 1 , 𝐆 2 , 𝐆 3 \mathbf{G}_{1},\mathbf{G}_{2},\mathbf{G}_{3}
  202. 𝐆 \mathbf{G}
  203. 𝐇 1 = ( 𝐆 1 𝐐 1 ) , 𝐇 2 = ( 𝐆 2 𝐐 2 ) , 𝐇 3 = ( 𝐆 3 𝐐 3 ) \mathbf{H}_{1}=\begin{pmatrix}\mathbf{G}_{1}\\ \mathbf{Q}_{1}\end{pmatrix},\mathbf{H}_{2}=\begin{pmatrix}\mathbf{G}_{2}\\ \mathbf{Q}_{2}\end{pmatrix},\mathbf{H}_{3}=\begin{pmatrix}\mathbf{G}_{3}\\ \mathbf{Q}_{3}\end{pmatrix}
  204. 𝐇 1 = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 1 ) , \mathbf{H}_{1}=\begin{pmatrix}0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;% 0\;0\;1\;0\;0\\ 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;0\\ 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\\ 1\;0\;0\;0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;1\;0\;1\;1\;0\;0\;0\;0\\ 0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;0\;0\;1\;0\;0\;0\;0\;0\;1\;1\;1\\ 0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;0\;0\;1\;1\;1\;0\;1\;0\;1\;1\\ 0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;0\;1\;0\;0\;1\;1\;1\;1\;0\\ 0\;0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;1\;0\;1\;1\;0\;1\;1\;1\;1\\ 0\;0\;0\;0\;0\;1\;0\;0\;0\;0\;1\;1\;0\;0\;1\;0\;0\;1\;1\;0\;1\end{pmatrix},
  205. 𝐇 2 = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 1 1 0 ) , \mathbf{H}_{2}=\begin{pmatrix}0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;% 0\;0\;0\;0\;0\\ 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;0\;0\;0\;0\\ 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;0\;0\;0\\ 0\;0\;0\;1\;0\;1\;1\;0\;1\;1\;1\;1\;0\;1\;0\;0\;0\;1\;1\;1\;1\\ 1\;0\;0\;0\;1\;0\;1\;1\;0\;1\;1\;1\;1\;0\;1\;1\;1\;1\;0\;0\;0\\ 0\;1\;0\;0\;0\;1\;1\;1\;1\;0\;1\;1\;1\;0\;0\;0\;0\;0\;1\;0\;1\\ 1\;0\;1\;0\;0\;0\;1\;1\;1\;1\;0\;1\;0\;1\;1\;1\;0\;0\;1\;1\;1\\ 0\;1\;0\;1\;0\;0\;1\;1\;1\;1\;1\;0\;0\;0\;1\;0\;1\;1\;0\;1\;1\\ 0\;0\;1\;0\;1\;0\;0\;1\;1\;1\;1\;1\;1\;1\;0\;1\;0\;1\;1\;1\;0\end{pmatrix},
  206. 𝐇 3 = ( 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 ) . \mathbf{H}_{3}=\begin{pmatrix}0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;0\;0\;0\;% 0\;0\;0\;0\;0\\ 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;0\;0\;0\;0\;0\;0\;0\\ 0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;0\;1\;0\;0\;0\;0\;0\;0\\ 1\;0\;0\;1\;0\;1\;0\;0\;1\;1\;1\;0\;1\;0\;0\;1\;1\;1\;1\;1\;1\\ 1\;1\;0\;0\;1\;0\;0\;0\;0\;1\;1\;1\;0\;0\;1\;1\;1\;1\;1\;1\;1\\ 0\;1\;1\;0\;0\;1\;1\;0\;0\;0\;1\;1\;1\;1\;1\;1\;0\;1\;1\;1\;0\\ 1\;0\;1\;1\;0\;0\;1\;1\;0\;0\;0\;1\;0\;0\;1\;1\;1\;1\;0\;0\;1\\ 0\;1\;0\;1\;1\;0\;1\;1\;1\;0\;0\;0\;1\;0\;0\;1\;1\;0\;1\;0\;0\\ 0\;0\;1\;0\;1\;1\;0\;1\;1\;1\;0\;0\;1\;1\;1\;1\;0\;0\;0\;1\;1\end{pmatrix}.

Divergence_(statistics).html

  1. D * ( p q ) = D ( q p ) . D^{*}(p\parallel q)=D(q\parallel p).
  2. D ( ( i ) p q ) \displaystyle D((\partial_{i})_{p}\parallel q)
  3. D [ i ] \displaystyle D[\partial_{i}\parallel\cdot]
  4. D [ i ] = D [ i ] = 0 , \displaystyle D[\partial_{i}\parallel\cdot]=D[\cdot\parallel\partial_{i}]=0,
  5. Γ i j , k ( D ) = - D [ i j k ] , \Gamma_{ij,k}^{(D)}=-D[\partial_{i}\partial_{j}\parallel\partial_{k}],
  6. D f ( p q ) = p ( x ) f ( q ( x ) p ( x ) ) d x D_{f}(p\parallel q)=\int p(x)f\bigg(\frac{q(x)}{p(x)}\bigg)dx
  7. D KL ( p q ) = p ( x ) ln ( p ( x ) q ( x ) ) d x D_{\mathrm{KL}}(p\parallel q)=\int p(x)\ln\left(\frac{p(x)}{q(x)}\right)dx
  8. H 2 ( p , q ) = 2 ( p ( x ) - q ( x ) ) 2 d x H^{2}(p,\,q)=2\int\Big(\sqrt{p(x)}-\sqrt{q(x)}\,\Big)^{2}dx
  9. D J ( p q ) = ( p ( x ) - q ( x ) ) ( ln p ( x ) - ln q ( x ) ) d x D_{J}(p\parallel q)=\int(p(x)-q(x))\big(\ln p(x)-\ln q(x)\big)dx
  10. D ( α ) ( p q ) = 4 1 - α 2 ( 1 - p ( x ) 1 - α 2 q ( x ) 1 + α 2 d x ) D^{(\alpha)}(p\parallel q)=\frac{4}{1-\alpha^{2}}\bigg(1-\int p(x)^{\frac{1-% \alpha}{2}}q(x)^{\frac{1+\alpha}{2}}dx\bigg)
  11. D e ( p q ) = p ( x ) ( ln p ( x ) - ln q ( x ) ) 2 d x D_{e}(p\parallel q)=\int p(x)\big(\ln p(x)-\ln q(x)\big)^{2}dx
  12. D χ 2 ( p q ) = 1 2 ( p ( x ) - q ( x ) ) 2 p ( x ) d x D_{\chi^{2}}(p\parallel q)=\frac{1}{2}\int\frac{(p(x)-q(x))^{2}}{p(x)}dx
  13. D α , β ( p q ) = 2 ( 1 - α ) ( 1 - β ) ( 1 - ( q ( x ) p ( x ) ) 1 - α 2 ) ( 1 - ( q ( x ) p ( x ) ) 1 - β 2 ) p ( x ) d x D_{\alpha,\beta}(p\parallel q)=\frac{2}{(1-\alpha)(1-\beta)}\int\Big(1-\Big(% \tfrac{q(x)}{p(x)}\Big)^{\!\!\frac{1-\alpha}{2}}\Big)\Big(1-\Big(\tfrac{q(x)}{% p(x)}\Big)^{\!\!\frac{1-\beta}{2}}\Big)p(x)dx

Division_algebra.html

  1. a * b = a b ¯ . a*b=\overline{ab}.
  2. q q ¯ = sum of squares q\overline{q}=\textrm{sum\ of\ squares}

DNSS_point.html

  1. - -
  2. - -
  3. max u ( t ) Ω 0 e - ρ t φ ( x ( t ) , u ( t ) ) d t \max_{u(t)\in\Omega}\int_{0}^{\infty}e^{-\rho t}\varphi\left(x(t),u(t)\right)dt
  4. x ˙ ( t ) = f ( x ( t ) , u ( t ) ) , x ( 0 ) = x 0 , \dot{x}(t)=f\left(x(t),u(t)\right),x(0)=x_{0},
  5. ρ > 0 \rho>0
  6. x ( t ) x(t)
  7. u ( t ) u(t)
  8. t t
  9. φ \varphi
  10. f f
  11. t t
  12. Ω \Omega
  13. t t
  14. ( x ( . ) , u ( . ) ) \left(x(.),u(.)\right)
  15. x x
  16. x 0 x_{0}
  17. x 0 x_{0}
  18. x > x 0 x>x_{0}
  19. x < x 0 x<x_{0}
  20. x 0 x_{0}
  21. φ ( x , u ) = x u , \varphi\left(x,u\right)=xu,
  22. f ( x , u ) = - x + u , f\left(x,u\right)=-x+u,
  23. Ω = [ - 1 , 1 ] \Omega=\left[-1,1\right]
  24. x 0 = 0 x_{0}=0
  25. x ( t ) x(t)
  26. ( 1 - e - t ) \left(1-e^{-t}\right)
  27. ( - 1 + e - t ) \left(-1+e^{-t}\right)
  28. x 0 < 0 x_{0}<0
  29. x ( t ) = - 1 + e - t ( x 0 + 1 ) x(t)=-1+e^{-t\left(x_{0}+1\right)}
  30. x 0 > 0 x_{0}>0
  31. x ( t ) = 1 + e - t ( x 0 - 1 ) x(t)=1+e^{-t\left(x_{0}-1\right)}

Doob_decomposition_theorem.html

  1. A n = k = 1 n ( 𝔼 [ X k | k - 1 ] - X k - 1 ) A_{n}=\sum_{k=1}^{n}\bigl(\mathbb{E}[X_{k}\,|\,\mathcal{F}_{k-1}]-X_{k-1}\bigr)
  2. M n = X 0 + k = 1 n ( X k - 𝔼 [ X k | k - 1 ] ) , M_{n}=X_{0}+\sum_{k=1}^{n}\bigl(X_{k}-\mathbb{E}[X_{k}\,|\,\mathcal{F}_{k-1}]% \bigr),
  3. 𝔼 [ M n - M n - 1 | n - 1 ] = 0 \mathbb{E}[M_{n}-M_{n-1}\,|\,\mathcal{F}_{n-1}]=0
  4. 𝔼 [ Y n | n - 1 ] = Y n - 1 \mathbb{E}[Y_{n}\,|\,\mathcal{F}_{n-1}]=Y_{n-1}
  5. 𝔼 [ Y n | n - 1 ] = Y n \mathbb{E}[Y_{n}\,|\,\mathcal{F}_{n-1}]=Y_{n}
  6. 𝔼 [ X k | k - 1 ] X k - 1 \mathbb{E}[X_{k}\,|\,\mathcal{F}_{k-1}]\geq X_{k-1}
  7. A n = k = 1 n ( 𝔼 [ X k ] - X k - 1 ) , n 0 , A_{n}=\sum_{k=1}^{n}\bigl(\mathbb{E}[X_{k}]-X_{k-1}\bigr),\quad n\in\mathbb{N}% _{0},
  8. M n = X 0 + k = 1 n ( X k - 𝔼 [ X k ] ) , n 0 . M_{n}=X_{0}+\sum_{k=1}^{n}\bigl(X_{k}-\mathbb{E}[X_{k}]\bigr),\quad n\in% \mathbb{N}_{0}.
  9. A n = - k = 0 n - 1 X k A_{n}=-\sum_{k=0}^{n-1}X_{k}
  10. M n = k = 0 n X k , n 0 , M_{n}=\sum_{k=0}^{n}X_{k},\quad n\in\mathbb{N}_{0},
  11. τ max := { N if A N = 0 , min { n { 0 , , N - 1 } A n + 1 < 0 } if A N < 0. \tau_{\,\text{max}}:=\begin{cases}N&\,\text{if }A_{N}=0,\\ \min\{n\in\{0,\dots,N-1\}\mid A_{n+1}<0\}&\,\text{if }A_{N}<0.\end{cases}

Doob–Dynkin_lemma.html

  1. σ \sigma
  2. σ \sigma
  3. σ \sigma
  4. Ω \Omega
  5. f : Ω R n f:\Omega\rightarrow R^{n}
  6. σ \sigma
  7. f f
  8. f - 1 ( S ) f^{-1}(S)
  9. S S
  10. X , Y : Ω R n X,Y:\Omega\rightarrow R^{n}
  11. σ ( X ) \sigma(X)
  12. σ \sigma
  13. X X
  14. Y Y
  15. σ ( X ) \sigma(X)
  16. Y = g ( X ) Y=g(X)
  17. g : R n R n g:R^{n}\rightarrow R^{n}
  18. Y Y
  19. σ ( X ) \sigma(X)
  20. Y - 1 ( S ) σ ( X ) Y^{-1}(S)\in\sigma(X)
  21. S S
  22. σ ( Y ) σ ( X ) \sigma(Y)\subset\sigma(X)
  23. X , Y : Ω R n X,Y:\Omega\rightarrow R^{n}
  24. σ ( X ) \sigma(X)
  25. σ ( Y ) \sigma(Y)
  26. σ \sigma
  27. X X
  28. Y Y
  29. Y = g ( X ) Y=g(X)
  30. g : R n R n g:R^{n}\rightarrow R^{n}
  31. σ ( Y ) σ ( X ) \sigma(Y)\subset\sigma(X)

Doomsday_conjecture.html

  1. Ext A * s , * ( Z / p Z , Z / p Z ) . \,\text{Ext}_{A_{*}}^{s,*}(Z/pZ,Z/pZ).\,

Double_diffusive_convection.html

  1. . U = 0 {\nabla}.U=0
  2. U t + U . U = ν 2 U - g ( β Δ S - α Δ T ) k . \frac{\partial U}{\partial t}+U.\nabla U=\nu{\nabla}^{2}U-g(\beta\Delta S-% \alpha\Delta T)k.
  3. T t + U . Δ T = k T 2 T . \frac{\partial T}{\partial t}+U.\Delta T=k_{T}{\nabla}^{2}T.
  4. S t + U . Δ S = k s 2 s . \frac{\partial S}{\partial t}+U.\Delta S=k_{s}{\nabla}^{2}s.
  5. x = X H , z = Z H , u = U k T / H , w = W k T / H , S * = S - S B S T - S B , T * = T - T B T T - T B , t * = t H 2 / k T . x=\frac{X}{H},z=\frac{Z}{H},u=\frac{U}{k_{T}/H},w=\frac{W}{k_{T}/H},S^{*}=% \frac{S-S_{B}}{S_{T}-S_{B}},T^{*}=\frac{T-T_{B}}{T_{T}-T_{B}},t^{*}=\frac{t}{H% ^{2}/k_{T}}.
  6. . u = 0 {\nabla}.u=0
  7. u t * + u . u = P r 2 u - [ P r R a T ( S * R ρ - T * ) ] k . \frac{\partial u}{\partial t^{*}}+u.\nabla u=Pr{\nabla}^{2}u-[PrRa_{T}(\frac{S% ^{*}}{R_{\rho}}-T^{*})]k.
  8. T * t * + u . Δ T * = 2 T * . \frac{\partial T^{*}}{\partial t^{*}}+u.\Delta T^{*}={\nabla}^{2}T^{*}.
  9. S * t * + u . Δ S * = P r S c 2 S * . \frac{\partial S^{*}}{\partial t^{*}}+u.\Delta S^{*}=\frac{Pr}{Sc}{\nabla}^{2}% S^{*}.
  10. R ρ = α Δ T β Δ S , R a T = g β Δ S H 3 ν k T , P r = ν α , S c = ν β . R_{\rho}=\frac{\alpha\Delta T}{\beta\Delta S},Ra_{T}=\frac{g\beta\Delta SH^{3}% }{\nu k_{T}},Pr=\frac{\nu}{\alpha},Sc=\frac{\nu}{\beta}.
  11. R f = α T β S . R_{f}=\frac{\alpha T^{\prime}}{\beta S^{\prime}}.

Double_groupoid.html

  1. ( H , V ) \Box(H,V)
  2. ( h v v h ) \begin{pmatrix}&h&\\ v&&v^{\prime}\\ &h^{\prime}&\end{pmatrix}
  3. ( h v v h ) 1 ( h w w h ′′ ) = ( h v w v w h ′′ ) \begin{pmatrix}&h&\\ v&&v^{\prime}\\ &h^{\prime}&\end{pmatrix}\circ_{1}\begin{pmatrix}&h^{\prime}&\\ w&&w^{\prime}\\ &h^{\prime\prime}&\end{pmatrix}=\begin{pmatrix}&h&\\ vw&&v^{\prime}w^{\prime}\\ &h^{\prime\prime}&\end{pmatrix}
  4. ( h v v h ) 2 ( k v v ′′ k ) = ( h k v v ′′ h k ) \begin{pmatrix}&h&\\ v&&v^{\prime}\\ &h^{\prime}&\end{pmatrix}\circ_{2}\begin{pmatrix}&k&\\ v^{\prime}&&v^{\prime\prime}\\ &k^{\prime}&\end{pmatrix}=\begin{pmatrix}&hk&\\ v&&v^{\prime\prime}\\ &h^{\prime}k^{\prime}&\end{pmatrix}
  5. ( X , A , C ) (X,A,C)
  6. C A X C\subseteq A\subseteq X
  7. ρ ( X , A , C ) \rho(X,A,C)

Double_layer_potential.html

  1. u ( 𝐱 ) = - 1 4 π S ρ ( 𝐲 ) ν 1 | 𝐱 - 𝐲 | d σ ( 𝐲 ) u(\mathbf{x})=\frac{-1}{4\pi}\int_{S}\rho(\mathbf{y})\frac{\partial}{\partial% \nu}\frac{1}{|\mathbf{x}-\mathbf{y}|}\,d\sigma(\mathbf{y})
  2. u ( 𝐱 ) = S ρ ( 𝐲 ) ν P ( 𝐱 - 𝐲 ) d σ ( 𝐲 ) u(\mathbf{x})=\int_{S}\rho(\mathbf{y})\frac{\partial}{\partial\nu}P(\mathbf{x}% -\mathbf{y})\,d\sigma(\mathbf{y})

Double_turnstile.html

  1. \vDash
  2. \models
  3. \vdash
  4. \vDash
  5. \models
  6. Γ φ \Gamma\vDash\varphi
  7. 𝒜 Γ \mathcal{A}\models\Gamma
  8. φ \vDash\varphi
  9. φ \varphi

Doubling-oriented_Doche–Icart–Kohel_curve.html

  1. y 2 = x 3 - x 2 - 16 x y^{2}=x^{3}-x^{2}-16x
  2. K K
  3. a K a\in K
  4. y 2 = x 3 + a x 2 + 16 a x y^{2}=x^{3}+ax^{2}+16ax
  5. Z Y 2 = X 3 + a Z X 2 + 16 a X Z 2 , ZY^{2}=X^{3}+aZX^{2}+16aXZ^{2},
  6. x = X Z x=\frac{X}{Z}
  7. y = Y Z y=\frac{Y}{Z}
  8. θ = ( 0 : 1 : 0 ) \theta=(0:1:0)
  9. θ = - θ \theta=-\theta
  10. P = ( x , y ) P=(x,y)
  11. P ! = O P!=O
  12. K = K=\mathbb{Q}
  13. K = K=\mathbb{Q}
  14. x , y x,y
  15. X , Y , Z , Z Z X,Y,Z,ZZ
  16. x = X Z x=\frac{X}{Z}
  17. y = Y Z Z y=\frac{Y}{ZZ}
  18. Z Z = Z 2 ZZ=Z^{2}
  19. Y 2 = Z X 3 + a Z 2 X 2 + 16 a Z 3 X Y^{2}=ZX^{3}+aZ^{2}X^{2}+16aZ^{3}X
  20. P = ( X : Y : Z : Z Z ) P=(X:Y:Z:ZZ)
  21. - P = ( X : - Y : Z : Z Z ) -P=(X:-Y:Z:ZZ)
  22. ( X : Y : Z : Z 2 ) = ( λ X : λ 2 Y : λ Z : λ 2 Z 2 ) (X:Y:Z:Z^{2})=(\lambda X:\lambda^{2}Y:\lambda Z:\lambda^{2}Z^{2})
  23. λ \lambda

Doubly_logarithmic_tree.html

  1. h > 1 h>1
  2. 2 2 h - 2 2^{2^{h-2}}
  3. n \sqrt{n}

Douglas'_lemma.html

  1. im ( A ) im ( B ) . \,\text{im}(A)\subseteq\,\text{im}(B).\,
  2. A A * λ 2 B B * AA^{*}\leq\lambda^{2}BB^{*}
  3. λ 0. \lambda\geq 0.\,
  4. C 2 = inf { μ : A A * μ B B * } . \|C\|^{2}=\inf\{\mu:\,AA^{*}\leq\mu BB^{*}\}.
  5. im ( C ) im ( B * ) ¯ \,\text{im}(C)\subseteq\overline{\,\text{im}(B^{*})}

DR-DP-Matrix.html

  1. D R T / D V / S DR_{T/D}^{V/S}
  2. D R T V DR_{T}^{V}
  3. D R D S DR_{D}^{S}
  4. D R T S DR_{T}^{S}
  5. D P T / D V / S DP_{T/D}^{V/S}
  6. D P T V DP_{T}^{V}
  7. D P D S DP_{D}^{S}
  8. D P T S DP_{T}^{S}

Draft:Heisenberg_Hamiltonian.html

  1. H = - J e x 2 n l s n s n + l H=-\frac{J_{ex}}{2}\sum_{\vec{n}}\sum_{\vec{l}}\vec{s}_{\vec{n}}\cdot\vec{s}_{% \vec{n}+\vec{l}}
  2. J e x J_{ex}
  3. S n S_{n}
  4. S l S_{l}

Draper_point.html

  1. ν p e a k \nu_{peak}
  2. ν p e a k = 2.821 h k T \nu_{peak}={2.821\over h}kT

Drummond_geometry.html

  1. [ A v g ( H ( 1 ) , L ( 1 ) , C ( 1 ) ) + A v g ( H ( 2 ) , L ( 2 ) , C ( 2 ) ) + A v g ( H ( 3 ) , L ( 3 ) , C ( 3 ) ) ] / 3 [Avg(H(1),L(1),C(1))+Avg(H(2),L(2),C(2))+Avg(H(3),L(3),C(3))]/3

Dry_Lake_Wind_Power_Project.html

  1. 132 , 450 MWh ( 8760 h/yr ) × ( 63 MW ) 24 % \frac{132,450\mbox{ MWh}~{}}{(8760\mbox{ h/yr}~{})\times(63\mbox{ MW}~{})}% \approx{24\%}
  2. 2 , 000 , 000 tons × 63 MW 1000 MW = 126 , 000 tons \frac{2,000,000\mbox{ tons}~{}\times 63\mbox{ MW}~{}}{1000\mbox{ MW}~{}}=126,0% 00\mbox{ tons}~{}
  3. 818 , 000 , 000 gallons × 63 MW 1000 MW = 51 , 534 , 000 gallons \frac{818,000,000\mbox{ gallons}~{}\times 63\mbox{ MW}~{}}{1000\mbox{ MW}~{}}=% 51,534,000\mbox{ gallons}~{}

Du_Bois_singularity.html

  1. X X
  2. Y Y
  3. π : Z X \pi:Z\to X
  4. X X
  5. Y Y
  6. X X
  7. E E
  8. X X
  9. Z Z
  10. X X
  11. 𝒪 X R π * 𝒪 E \mathcal{O}_{X}\to R\pi_{*}\mathcal{O}_{E}

Du_Noüy_ring_method.html

  1. F F
  2. γ \gamma
  3. F = 2 π ( r i + r a ) γ F=2\pi\cdot(r_{i}+r_{a})\cdot\gamma
  4. r i r_{i}
  5. r a r_{a}

Du_Val_singularity.html

  1. w 2 + x 2 + y n + 1 = 0 w^{2}+x^{2}+y^{n+1}=0
  2. w 2 + y ( x 2 + y n - 2 ) = 0 ( n 4 ) w^{2}+y(x^{2}+y^{n-2})=0\qquad(n\geq 4)
  3. w 2 + x 3 + y 4 = 0 w^{2}+x^{3}+y^{4}=0
  4. w 2 + x ( x 2 + y 3 ) = 0 w^{2}+x(x^{2}+y^{3})=0
  5. w 2 + x 3 + y 5 = 0. w^{2}+x^{3}+y^{5}=0.

Dual_matroid.html

  1. M M
  2. M M^{\ast}
  3. M M
  4. M M
  5. M M
  6. ( M ) = M (M^{\ast})^{\ast}=M
  7. M M
  8. M M
  9. M M^{\ast}
  10. r r
  11. M M
  12. E E
  13. r ( S ) = r ( E S ) + | S | - r ( E ) r^{\ast}(S)=r(E\setminus S)+|S|-r(E)
  14. M M
  15. M x M\setminus x
  16. x x
  17. M M
  18. M / x M/x
  19. x x
  20. M M
  21. M X = ( M / x ) M\setminus X=(M^{\ast}/x)^{\ast}
  22. M / X = ( M x ) M/X=(M^{\ast}\setminus x)^{\ast}
  23. U n r U{}^{r}_{n}
  24. U n n - r U{}^{n-r}_{n}

Dual_of_BCH_is_an_independent_source.html

  1. \ell
  2. \ell
  3. 1 - 2 - 1-2^{-\ell}
  4. C F 2 n C\subseteq F_{2}^{n}
  5. C C^{\perp}
  6. + 1 \ell+1
  7. C C
  8. \ell
  9. k × l k\times l
  10. x F 2 k x\in F_{2}^{k}
  11. x M xM
  12. F 2 l F_{2}^{l}
  13. M 1 M_{1}
  14. M 2 M_{2}
  15. M 1 M_{1}
  16. x M xM
  17. x 1 M 1 + x 2 M 2 x_{1}M_{1}+x_{2}M_{2}
  18. x 1 x_{1}
  19. x 2 x_{2}
  20. x 1 x_{1}
  21. M 2 M_{2}
  22. x 2 x_{2}
  23. M 1 M_{1}
  24. y = x M y=xM
  25. x 1 M 1 + x 2 M 2 x_{1}M_{1}+x_{2}M_{2}
  26. x 1 , x 2 x_{1},x_{2}
  27. x 1 M 1 = y - x 2 M 2 x_{1}M_{1}=y-x_{2}M_{2}
  28. k - l k-l
  29. x 2 F 2 k x_{2}\in F_{2}^{k}
  30. 2 k - l 2^{k-l}
  31. x 1 M 1 = b x_{1}M_{1}=b
  32. M 1 M_{1}
  33. x 1 x_{1}
  34. 2 k - l 2^{k-l}
  35. [ n = 2 m , n - 1 - d - 2 / 2 m , d ] 2 [n=2^{m},n-1-\lceil{d-2}/2\rceil m,d]_{2}
  36. C C^{\perp}
  37. C C
  38. \ell
  39. O ( n / 2 ) O(n^{\lfloor\ell/2\rfloor})
  40. ( + 1 - 2 ) / 2 log n + 1 \lceil{(\ell+1-2)/{2}}\rceil\log n+1
  41. d = ( - 1 ) / 2 log n + 1 = / 2 log n + 1 d=\lceil{(\ell-1)}/2\rceil\log n+1=\lfloor\ell/2\rfloor\log n+1
  42. C C
  43. 2 d = O ( n / 2 ) 2^{d}=O(n^{\lfloor\ell/2\rfloor})

Duane–Hunt_law.html

  1. ν max = e V h \nu_{\rm max}=\frac{eV}{h}\,
  2. λ min = h c e V \lambda_{\rm min}=\frac{hc}{eV}
  3. λ min 1239.8 pm V in kV \lambda_{\rm min}\approx\frac{1239.8\,\text{ pm}}{V\,\text{ in kV}}\,

Dudley's_theorem.html

  1. d X ( s , t ) = 𝐄 [ | X s - X t | 2 ] . d_{X}(s,t)=\sqrt{\mathbf{E}\big[|X_{s}-X_{t}|^{2}]}.\,
  2. 𝐄 [ sup t T X t ] 24 0 + log N ( T , d X ; ε ) d ε . \mathbf{E}\left[\sup_{t\in T}X_{t}\right]\leq 24\int_{0}^{+\infty}\sqrt{\log N% (T,d_{X};\varepsilon)}\,\mathrm{d}\varepsilon.

Duhem–Margules_equation.html

  1. ( d ln P A d ln x A ) T , P = ( d ln P B d ln x B ) T , P \left(\frac{d\,\ln\,P_{A}}{d\,\ln\,x_{A}}\right)_{T,P}=\left(\frac{d\,\ln\,P_{% B}}{d\,\ln\,x_{B}}\right)_{T,P}

Dulong–Petit_law.html

  1. c M = K cM=K
  2. m / M = N m/M=N
  3. ( C / m ) M = K (C/m)M=K
  4. C ( M / m ) = C / N = K = 3 R C(M/m)=C/N=K=3R
  5. C / N = 3 R C/N=3R
  6. 1 / 2 {1}/{2}
  7. 1 / 2 {1}/{2}
  8. F = N ε 0 + k B T α log ( 1 - e - ω α / k B T ) F=N\varepsilon_{0}+k_{B}T\sum_{\alpha}\log\left(1-e^{-\hbar\omega_{\alpha}/k_{% B}T}\right)
  9. k B T ω α . k_{B}T\gg\hbar\omega_{\alpha}.\,
  10. 1 - e - ω α / k B T ω α / k B T 1-e^{-\hbar\omega_{\alpha}/k_{B}T}\approx\hbar\omega_{\alpha}/k_{B}T\,
  11. F = N ε 0 + k B T α log ( ω α k B T ) . F=N\varepsilon_{0}+k_{B}T\sum_{\alpha}\log\left(\frac{\hbar\omega_{\alpha}}{k_% {B}T}\right).
  12. log ω ¯ = 1 M α log ω α , \log\bar{\omega}=\frac{1}{M}\sum_{\alpha}\log\omega_{\alpha},
  13. F = N ε 0 - M k B T log k B T + M k B T log ω ¯ . F=N\varepsilon_{0}-Mk_{B}T\log k_{B}T+Mk_{B}T\log\hbar\bar{\omega}.\,
  14. E = F - T ( F T ) V , E=F-T\left(\frac{\partial F}{\partial T}\right)_{V},
  15. E = N ε 0 + M k B T . E=N\varepsilon_{0}+Mk_{B}T.\,
  16. C V = ( E T ) V = M k B , C_{V}=\left(\frac{\partial E}{\partial T}\right)_{V}=Mk_{B},

Dumas_method_of_molecular_weight_determination.html

  1. P V = n R T PV=nRT\,

Dyadic_distribution.html

  1. f ( u ) = 2 - n u , u U f(u)=2^{-n_{u}},\quad u\in U

Dyck_graph.html

  1. ( x - 3 ) ( x - 1 ) 9 ( x + 1 ) 9 ( x + 3 ) ( x 2 - 5 ) 6 (x-3)(x-1)^{9}(x+1)^{9}(x+3)(x^{2}-5)^{6}

Dynamic_errors_of_numerical_methods_of_ODE_discretization.html

  1. 𝐃 = ln ρ ( h λ ) \,\textbf{D}=\ln\rho(h\lambda)
  2. 𝐃 \,\textbf{D}
  3. 𝐃 R \,\textbf{D}_{R}
  4. 𝐃 I \,\textbf{D}_{I}

Dynamic_perfect_hashing.html

  1. 0 j s ( M ) s j 32 M 2 s ( M ) + 4 M . \sum_{0\leq j\leq s(M)}s_{j}\leq\frac{32M^{2}}{s(M)}+4M.

Dynamic_speckle.html

  1. M I = M O C ( i , j ) * ( i - j ) 2 MI=\sum{MOC(i,j)*(i-j)^{2}}\,\!
  2. F u j i i ( x , y ) = k = 1 N I k ( x , y ) - I k + 1 ( x , y ) I k ( x , y ) + I k + 1 ( x , y ) Fujii(x,y)=\sum_{k=1}^{N}\frac{I_{k}(x,y)-I_{k+1}(x,y)}{I_{k}(x,y)+I_{k+1}(x,y% )}\,\!
  3. D G ( x , y ) = k = 1 N l = 1 N I k ( x , y ) - I k + l ( x , y ) DG(x,y)=\sum_{k=1}^{N}\sum_{l=1}^{N}{I_{k}(x,y)-I_{k+l}(x,y)}\,\!
  4. D ( k ) = m = 1 M n = 1 N E ( m , n , k + 1 ) - E ( m , n , k ) D(k)=\sum_{m=1}^{M}\sum_{n=1}^{N}{E(m,n,k+1)-E(m,n,k)}\,\!

Dynamic_structure_factor.html

  1. S ( k , ω ) S(\vec{k},\omega)
  2. k \vec{k}
  3. q \vec{q}
  4. ω \omega
  5. ω \hbar\omega
  6. S ( k , ω ) 1 2 π - F ( k , t ) exp ( i ω t ) d t S(\vec{k},\omega)\equiv\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\vec{k},t)\mbox{% exp}~{}(i\omega t)dt
  7. F ( k , t ) F(\vec{k},t)
  8. G ( r , t ) G(\vec{r},t)
  9. F ( k , t ) G ( r , t ) exp ( - i k r ) d r F(\vec{k},t)\equiv\int G(\vec{r},t)\exp(-i\vec{k}\cdot\vec{r})d\vec{r}
  10. ρ \rho
  11. F ( k , t ) = 1 N ρ k ( t ) ρ - k F(\vec{k},t)=\frac{1}{N}\langle\rho_{\vec{k}}(t)\rho_{-\vec{k}}\rangle
  12. d 2 σ d Ω d ω = a 2 ( E f E i ) 1 / 2 S ( k , ω ) \frac{d^{2}\sigma}{d\Omega d\omega}=a^{2}\left(\frac{E_{f}}{E_{i}}\right)^{1/2% }S(\vec{k},\omega)
  13. a a
  14. N N
  15. G ( r , t ) = 1 N i = 1 N j = 1 N δ [ r + r - r j ( t ) ] δ [ r - r i ( 0 ) ] d r G(\vec{r},t)=\langle\frac{1}{N}\int\sum_{i=1}^{N}\sum_{j=1}^{N}\delta[\vec{r}^% {\prime}+\vec{r}-\vec{r}_{j}(t)]\delta[\vec{r}^{\prime}-\vec{r}_{i}(0)]d\vec{r% }^{\prime}\rangle
  16. G ( r , t ) = 1 N ρ ( r + r , t ) ρ ( r , 0 ) d r G(\vec{r},t)=\langle\frac{1}{N}\int\rho(\vec{r}^{\prime}+\vec{r},t)\rho(\vec{r% }^{\prime},0)d\vec{r}^{\prime}\rangle

Dyson's_transform.html

  1. A = { a 1 < a 2 < } A=\{a_{1}<a_{2}<\cdots\}
  2. B = { 0 = b 1 < b 2 < } B=\{0=b_{1}<b_{2}<\cdots\}
  3. A = A { B + { e } } A^{\prime}=A\cup\{B+\{e\}\}
  4. B = B { A - { e } } \,B^{\prime}=B\cap\{A-\{e\}\}
  5. A + B A + B A^{\prime}+B^{\prime}\subset A+B
  6. { e } + B A \{e\}+B^{\prime}\subset A^{\prime}
  7. 0 B 0\in B^{\prime}
  8. A ( m ) + B ( m - e ) = A ( m ) + B ( m - e ) A^{\prime}(m)+B^{\prime}(m-e)=A(m)+B(m-e)

Earth_bulge.html

  1. d 2 = ( R + h ) 2 - R 2 = 2 R h + h 2 d^{2}=(R+h)^{2}-R^{2}=2\cdot R\cdot h+h^{2}
  2. d 2 R h d\approx\sqrt{2\cdot R\cdot h}
  3. d 2 6378 h 112.9 h d\approx\sqrt{2\cdot 6378\cdot h}\approx 112.9\cdot\sqrt{\cdot h}
  4. d 3.57 h d\approx 3.57\cdot\sqrt{h}
  5. d 1.23 h d\approx 1.23\cdot\sqrt{h}
  6. d 2 k R h d\approx\sqrt{2\cdot k\cdot R\cdot h}
  7. d 4.12 h d\approx 4.12\cdot\sqrt{h}
  8. d 1.41 h d\approx 1.41\cdot\sqrt{h}
  9. d 4.12 1500 = 160 km. d\approx 4.12\cdot\sqrt{1500}=160\mbox{ km.}

Earthquake_duration_magnitude.html

  1. M d = 2.49 l o g 10 ( T ) - 2.31 + S t a t i o n C o r r e c t i o n F a c t o r Md=2.49log10(T)-2.31+StationCorrectionFactor
  2. M d = 0.80 l o g 10 ( T ) 2 + 1.7 l o g 10 ( T ) - 0.87 Md=0.80log10(T)^{2}+1.7log10(T)-0.87
  3. M L = 0.936 M d - 0.16 + / - 0.22 ML=0.936Md-0.16+/-0.22

Eckmann–Hilton_duality.html

  1. X X
  2. X × I Y X\times I\to Y
  3. X Y I X\to Y^{I}
  4. Y I Y^{I}
  5. I I
  6. Y Y
  7. I I
  8. X × I X\times I
  9. Y I Y^{I}
  10. Σ X \Sigma X
  11. X × I X\times I
  12. Ω Y \Omega Y
  13. Y I Y^{I}
  14. Σ X , Y = X , Ω Y \langle\Sigma X,Y\rangle=\langle X,\Omega Y\rangle
  15. p : E B p\colon E\to B
  16. i : A X i\colon A\to X
  17. F E B F\to E\to B
  18. Ω 2 B Ω F Ω E Ω B F E B \cdots\to\Omega^{2}B\to\Omega F\to\Omega E\to\Omega B\to F\to E\to B\,
  19. A X X / A A\to X\to X/A
  20. A X X / A Σ A Σ X Σ ( X / A ) Σ 2 A . A\to X\to X/A\to\Sigma A\to\Sigma X\to\Sigma\left(X/A\right)\to\Sigma^{2}A\to% \cdots.\,
  21. π n ( X , p ) S n , X \pi_{n}(X,p)\cong\langle S^{n},X\rangle
  22. K ( G , n ) K(G,n)
  23. H n ( X ; G ) X , K ( G , n ) H^{n}(X;G)\cong\langle X,K(G,n)\rangle

Ecological_Debt_Day.html

  1. ( World Biocapacity / World Ecological Footprint ) × 365 = Ecological Debt Day (\,\text{World Biocapacity}/\,\text{World Ecological Footprint})\times 365=\,% \text{Ecological Debt Day}

Ecological_efficiency.html

  1. I / P n I/P_{n}
  2. A / I A/I
  3. P n + 1 / A P_{n+1}/A
  4. P n + 1 / I P_{n+1}/I
  5. P n + 1 / P n P_{n+1}/P_{n}
  6. * *

Edge_crush_test.html

  1. R = 0.01 × F ¯ max R=0.01\times\overline{F}_{\mathrm{max}}
  2. F ¯ max \overline{F}_{\mathrm{max}}
  3. \color B l u e B C T = 5.876 × \color R e d E C T × U × d {\color{Blue}BCT}=5.876\times{\color{Red}ECT}\times\sqrt{U\times d}

Edwards_curve.html

  1. x 2 + y 2 = 1 + d x 2 y 2 x^{2}+y^{2}=1+dx^{2}y^{2}\,
  2. d K { 0 , 1 } d\in K\setminus\{0,1\}
  3. x 2 + y 2 = c 2 ( 1 + d x 2 y 2 ) x^{2}+y^{2}=c^{2}(1+dx^{2}y^{2})\,
  4. ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 y 2 + x 2 y 1 1 + d x 1 x 2 y 1 y 2 , y 1 y 2 - x 1 x 2 1 - d x 1 x 2 y 1 y 2 ) (x_{1},y_{1})+(x_{2},y_{2})=\left(\frac{x_{1}y_{2}+x_{2}y_{1}}{1+dx_{1}x_{2}y_% {1}y_{2}},\frac{y_{1}y_{2}-x_{1}x_{2}}{1-dx_{1}x_{2}y_{1}y_{2}}\right)\,
  5. x 2 + y 2 = 1 + 2 x 2 y 2 {\displaystyle}x^{2}+y^{2}=1+2x^{2}y^{2}
  6. P 1 = ( 1 , 0 ) P_{1}=(1,0)
  7. x 3 = x 1 y 2 + y 1 x 2 1 + d x 1 x 2 y 1 y 2 = 1 x_{3}=\frac{x_{1}y_{2}+y_{1}x_{2}}{1+dx_{1}x_{2}y_{1}y_{2}}=1
  8. y 3 = y 1 y 2 - x 1 x 2 1 - d x 1 x 2 y 1 y 2 = 0 y_{3}=\frac{y_{1}y_{2}-x_{1}x_{2}}{1-dx_{1}x_{2}y_{1}y_{2}}=0
  9. x 2 + y 2 = 1 {\displaystyle}x^{2}+y^{2}=1
  10. P 1 = ( x 1 , y 1 ) = ( sin α 1 , cos α 1 ) {\displaystyle}P_{1}=(x_{1},y_{1})=(\sin{\alpha_{1}},\cos{\alpha_{1}})
  11. P 2 = ( x 2 , y 2 ) = ( sin α 2 , cos α 2 ) {\displaystyle}P_{2}=(x_{2},y_{2})=(\sin{\alpha_{2}},\cos{\alpha_{2}})
  12. x 3 = sin ( α 1 + α 2 ) = sin α 1 cos α 2 + sin α 2 cos α 1 = x 1 y 2 + x 2 y 1 {\displaystyle}x_{3}=\sin({\alpha}_{1}+{\alpha}_{2})=\sin{\alpha}_{1}\cos{% \alpha}_{2}+\sin{\alpha}_{2}\cos{\alpha}_{1}=x_{1}y_{2}+x_{2}y_{1}
  13. y 3 = cos ( α 1 + α 2 ) = cos α 1 cos α 2 - sin α 1 sin α 2 = y 1 y 2 - x 1 x 2 . {\displaystyle}y_{3}=\cos({\alpha}_{1}+{\alpha}_{2})=\cos{\alpha}_{1}\cos{% \alpha}_{2}-\sin{\alpha}_{1}\sin{\alpha}_{2}=y_{1}y_{2}-x_{1}x_{2}.
  14. ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 y 2 + x 2 y 1 , y 1 y 2 - x 1 x 2 ) {\displaystyle}(x_{1},y_{1})+(x_{2},y_{2})=(x_{1}y_{2}+x_{2}y_{1},y_{1}y_{2}-x% _{1}x_{2})
  15. ( X 2 + Y 2 ) Z 2 = Z 4 + d X 2 Y 2 (X^{2}+Y^{2})Z^{2}=Z^{4}+dX^{2}Y^{2}
  16. 2 ( x 1 , y 1 ) \displaystyle 2(x_{1},y_{1})
  17. x 2 + y 2 = 1 + 2 x 2 y 2 {\displaystyle}x^{2}+y^{2}=1+2x^{2}y^{2}
  18. x 3 = 2 x 1 y 1 1 + d x 1 2 y 1 2 = 0 x_{3}=\frac{2x_{1}y_{1}}{1+dx_{1}^{2}y_{1}^{2}}=0
  19. y 3 = y 1 2 - x 1 2 1 - d x 1 2 y 1 2 = - 1 y_{3}=\frac{y_{1}^{2}-x_{1}^{2}}{1-dx_{1}^{2}y_{1}^{2}}=-1
  20. 3 ( x 1 , y 1 ) = ( ( x 1 2 + y 1 2 ) - ( 2 y 1 ) 2 4 ( x 1 2 - 1 ) x 1 2 - ( x 1 2 - y 1 2 ) 2 x 1 , ( x 1 2 + y 1 2 ) - 2 ( x 1 ) 2 - 4 ( y 1 2 - 1 ) y 1 2 + ( x 1 2 - y 1 2 ) 2 y 1 ) . 3(x_{1},y_{1})=\left(\frac{(x_{1}^{2}+y_{1}^{2})-(2y_{1})^{2}}{4(x_{1}^{2}-1)x% _{1}^{2}-(x_{1}^{2}-y_{1}^{2})^{2}}x_{1},\frac{(x_{1}^{2}+y_{1}^{2})-2(x_{1})^% {2}}{-4(y_{1}^{2}-1)y_{1}^{2}+(x_{1}^{2}-y_{1}^{2})^{2}}y_{1}\right).\,
  21. x 3 = ( x 1 2 + y 1 2 ) - ( 2 y 1 ) 2 4 ( x 1 2 - 1 ) x 1 2 - ( x 1 2 - y 1 2 ) 2 x 1 = - 1 x_{3}=\frac{(x_{1}^{2}+y_{1}^{2})-(2y_{1})^{2}}{4(x_{1}^{2}-1)x_{1}^{2}-(x_{1}% ^{2}-y_{1}^{2})^{2}}x_{1}=-1
  22. y 3 = ( x 1 2 + y 1 2 - 2 ( x 1 ) 2 - 4 ( y 1 2 - 1 ) y 1 2 + ( x 1 2 - y 1 2 ) 2 y 1 = 0 y_{3}=\frac{(x_{1}^{2}+y_{1}^{2}-2(x_{1})^{2}}{-4(y_{1}^{2}-1)y_{1}^{2}+(x_{1}% ^{2}-y_{1}^{2})^{2}}y_{1}=0

Eells–Kuiper_manifold.html

  1. R n R^{n}
  2. n 2 \frac{n}{2}
  3. R P ( 2 ) RP(2)
  4. n 4 n\geq 4
  5. C P 2 CP^{2}
  6. n = 4 n=4
  7. H P 2 HP^{2}
  8. n = 8 n=8
  9. M M
  10. n n
  11. M M
  12. f : M R f:M\to R
  13. C 3 C^{3}
  14. M M
  15. M n M^{n}
  16. F F
  17. M M
  18. c c
  19. F F
  20. s s
  21. c = s + 2 c=s+2
  22. M n M^{n}
  23. S n S^{n}
  24. c = s + 1 c=s+1
  25. M n M^{n}
  26. n = 2 , 4 , 8 n=2,4,8
  27. 16 16

Effective_diffusion_coefficient.html

  1. D eff = f {D^{\mathrm{eff}}}=f
  2. + ( 1 - f ) +(1-f)
  3. D eff = {D^{\mathrm{eff}}}=
  4. f = q d f=\tfrac{q}{d}
  5. q = q=
  6. d = d=
  7. = =

Effective_mass_(spring–mass_system).html

  1. m m
  2. u u
  3. M M
  4. m u 2 / 2 mu^{2}/2
  5. m m
  6. M M
  7. M M
  8. T = m 1 2 u 2 d m T=\int_{m}\frac{1}{2}u^{2}\,dm
  9. d m = ( d y L ) m dm=\left(\frac{dy}{L}\right)m
  10. L L
  11. T = 0 L 1 2 u 2 ( d y L ) m T=\int_{0}^{L}\frac{1}{2}u^{2}\left(\frac{dy}{L}\right)m\!
  12. = 1 2 m L 0 L u 2 d y =\frac{1}{2}\frac{m}{L}\int_{0}^{L}u^{2}\,dy
  13. u = v y L u=\frac{vy}{L}
  14. T = 1 2 m L 0 L ( v y L ) 2 d y T=\frac{1}{2}\frac{m}{L}\int_{0}^{L}\left(\frac{vy}{L}\right)^{2}\,dy
  15. = 1 2 m L 3 v 2 0 L y 2 d y =\frac{1}{2}\frac{m}{L^{3}}v^{2}\int_{0}^{L}y^{2}\,dy
  16. = 1 2 m L 3 v 2 [ y 3 3 ] 0 L =\frac{1}{2}\frac{m}{L^{3}}v^{2}\left[\frac{y^{3}}{3}\right]_{0}^{L}
  17. = 1 2 m 3 v 2 =\frac{1}{2}\frac{m}{3}v^{2}
  18. 1 2 m v 2 , \frac{1}{2}mv^{2},
  19. x x
  20. L = T - V L=T-V
  21. = 1 2 m 3 x ˙ 2 + 1 2 M x ˙ 2 - 1 2 k x 2 - m g x 2 - M g x =\frac{1}{2}\frac{m}{3}\dot{x}^{2}+\frac{1}{2}M\dot{x}^{2}-\frac{1}{2}kx^{2}-% \frac{mgx}{2}-Mgx
  22. g g
  23. ( m 3 + M ) x ¨ = - k x - m g 2 - M g \left(\frac{m}{3}+M\right)\ddot{x}=-kx-\frac{mg}{2}-Mg
  24. x eq x_{\mathrm{eq}}
  25. x eq = - 1 k ( m g 2 + M g ) x_{\mathrm{eq}}=-\frac{1}{k}\left(\frac{mg}{2}+Mg\right)
  26. x ¯ = x - x eq \bar{x}=x-x_{\mathrm{eq}}
  27. ( m 3 + M ) x ¯ ¨ = - k x ¯ \left(\frac{m}{3}+M\right)\ddot{\bar{x}}=-k\bar{x}
  28. τ = 2 π ( M + m / 3 k ) 1 / 2 \tau=2\pi\left(\frac{M+m/3}{k}\right)^{1/2}
  29. 2 π ( m k ) 1 / 2 2\pi\left(\frac{m}{k}\right)^{1/2}
  30. ρ ( x ) \rho(x)
  31. T = m 1 2 u 2 d m T=\int_{m}\frac{1}{2}u^{2}\,dm
  32. = 0 L 1 2 u 2 ρ ( x ) d x =\int_{0}^{L}\frac{1}{2}u^{2}\rho(x)\,dx
  33. = 0 L 1 2 ( v x L ) 2 ρ ( x ) d x =\int_{0}^{L}\frac{1}{2}\left(\frac{vx}{L}\right)^{2}\rho(x)\,dx
  34. = 1 2 [ 0 L x 2 L 2 ρ ( x ) d x ] v 2 =\frac{1}{2}\left[\int_{0}^{L}\frac{x^{2}}{L^{2}}\rho(x)\,dx\right]v^{2}
  35. m eff = 0 L x 2 L 2 ρ ( x ) d x m_{\mathrm{eff}}=\int_{0}^{L}\frac{x^{2}}{L^{2}}\rho(x)\,dx
  36. m eff m m_{\mathrm{eff}}\leq m
  37. m eff = m m_{\mathrm{eff}}=m
  38. M / m M/m
  39. M / m M/m
  40. m / 3 m/3

Effective_Polish_space.html

  1. 4 \mathbb{N}^{4}
  2. P ( i , j , k , m ) d ( c i , c j ) m k + 1 P(i,j,k,m)\equiv d(c_{i},c_{j})\leq\frac{m}{k+1}
  3. Q ( i , j , k , m ) d ( c i , c j ) < m k + 1 Q(i,j,k,m)\equiv d(c_{i},c_{j})<\frac{m}{k+1}

Efficiency_(statistics).html

  1. e ( T ) = 1 / ( θ ) var ( T ) e(T)=\frac{1/\mathcal{I}(\theta)}{\mathrm{var}(T)}
  2. ( θ ) \mathcal{I}(\theta)
  3. e ( T ) = 1 e(T)=1
  4. N N
  5. μ \mu
  6. X n 𝒩 ( μ , 1 ) . X_{n}\sim\mathcal{N}(\mu,1).
  7. X ¯ \overline{X}
  8. X 1 , X 2 , , X N X_{1},X_{2},\ldots,X_{N}
  9. X ¯ = 1 N n = 1 N X n 𝒩 ( μ , 1 N ) . \overline{X}=\frac{1}{N}\sum_{n=1}^{N}X_{n}\sim\mathcal{N}\left(\mu,\frac{1}{N% }\right).
  10. X ~ \widetilde{X}
  11. μ \mu
  12. N N
  13. μ \mu
  14. π / 2 N , {\pi}/{2N},
  15. X ~ 𝒩 ( μ , π 2 N ) . \widetilde{X}\sim\mathcal{N}\left(\mu,\frac{\pi}{2N}\right).
  16. N N
  17. e ( X ~ ) = ( 1 N ) ( π 2 N ) - 1 = 2 / π 64 % . e\left(\widetilde{X}\right)=\left(\frac{1}{N}\right)\left(\frac{\pi}{2N}\right% )^{-1}=2/\pi\approx 64\%.
  18. N N
  19. N , N,
  20. T 1 T_{1}
  21. T 2 T_{2}
  22. θ \theta
  23. T 1 T_{1}
  24. T 2 T_{2}
  25. θ \theta
  26. T 2 T_{2}
  27. T 1 T_{1}
  28. T 2 T_{2}
  29. E [ ( T 1 - θ ) 2 ] E [ ( T 2 - θ ) 2 ] \mathrm{E}\left[(T_{1}-\theta)^{2}\right]\leq\mathrm{E}\left[(T_{2}-\theta)^{2% }\right]
  30. θ \theta
  31. e ( T 1 , T 2 ) = E [ ( T 2 - θ ) 2 ] E [ ( T 1 - θ ) 2 ] e(T_{1},T_{2})=\frac{\mathrm{E}\left[(T_{2}-\theta)^{2}\right]}{\mathrm{E}% \left[(T_{1}-\theta)^{2}\right]}
  32. e e
  33. θ \theta
  34. e e
  35. T 1 T_{1}
  36. θ \theta
  37. E ( σ μ ) 2 \mathrm{E}\equiv\left(\frac{\sigma}{\mu}\right)^{2}
  38. E 1 E 2 = s 1 2 s 2 2 \frac{\mathrm{E}_{1}}{\mathrm{E}_{2}}=\frac{s_{1}^{2}}{s_{2}^{2}}
  39. s 1 2 = n 1 σ 2 , s 2 2 = n 2 σ 2 s_{1}^{2}=n_{1}\sigma^{2},\,s_{2}^{2}=n_{2}\sigma^{2}
  40. E 1 E 2 = n 1 n 2 \frac{\mathrm{E}_{1}}{\mathrm{E}_{2}}=\frac{n_{1}}{n_{2}}

Ehrenfest_equations.html

  1. s s
  2. v v
  3. s s
  4. d s = ( s T ) P d T + ( s P ) T d P ds=\left({{{\partial s}\over{\partial T}}}\right)_{P}dT+\left({{{\partial s}% \over{\partial P}}}\right)_{T}dP
  5. ( s T ) P = c P T , ( s P ) T = - ( v T ) P \left({{{\partial s}\over{\partial T}}}\right)_{P}={{c_{P}}\over T},\left({{{% \partial s}\over{\partial P}}}\right)_{T}=-\left({{{\partial v}\over{\partial T% }}}\right)_{P}
  6. d s i = c i P T d T - ( v i T ) P d P d{s_{i}}={{c_{iP}}\over T}dT-\left({{{\partial v_{i}}\over{\partial T}}}\right% )_{P}dP
  7. i = 1 i=1
  8. i = 2 i=2
  9. d s 1 = d s 2 {ds_{1}}={ds_{2}}
  10. ( c 2 P - c 1 P ) d T T = [ ( v 2 T ) P - ( v 1 T ) P ] d P \left({c_{2P}-c_{1P}}\right){{dT}\over T}=\left[{\left({{{\partial v_{2}}\over% {\partial T}}}\right)_{P}-\left({{{\partial v_{1}}\over{\partial T}}}\right)_{% P}}\right]dP
  11. Δ c P = T Δ ( ( v T ) P ) d P d T {\Delta c_{P}=T\cdot\Delta\left({\left({{{\partial v}\over{\partial T}}}\right% )_{P}}\right)\cdot{{dP}\over{dT}}}
  12. Δ c V = - T Δ ( ( P T ) v ) d v d T {\Delta c_{V}=-T\cdot\Delta\left({\left({{{\partial P}\over{\partial T}}}% \right)_{v}}\right)\cdot{{dv}\over{dT}}}
  13. v v
  14. P P
  15. Δ ( v T ) P = Δ ( ( P T ) v ) d v d P {\Delta\left({{{\partial v}\over{\partial T}}}\right)_{P}=\Delta\left({\left({% {{\partial P}\over{\partial T}}}\right)_{v}}\right)\cdot{{dv}\over{dP}}}
  16. T T
  17. P P
  18. Δ ( v T ) P = - Δ ( ( v P ) T ) d P d T {\Delta\left({{{\partial v}\over{\partial T}}}\right)_{P}=-\Delta\left({\left(% {{{\partial v}\over{\partial P}}}\right)_{T}}\right)\cdot{{dP}\over{dT}}}

Ehrenfeucht–Mycielski_sequence.html

  1. lim i f ( i ) i = 1 2 . \lim_{i\rightarrow\infty}\frac{f(i)}{i}=\frac{1}{2}.
  2. f ( i ) i = 1 2 + O ( log log i i ) \scriptstyle\frac{f(i)}{i}=\frac{1}{2}+O(\sqrt{\frac{\log\log i}{i}})

Eigenmode_expansion.html

  1. E ( x , y , z ) = E ( x , y ) e ( i β z ) \textstyle E(x,y,z)=E(x,y)e^{(i\beta z)}
  2. e x p ( i ω t ) \scriptstyle exp(i\omega t)
  3. E ( x , y ) \scriptstyle E(x,y)
  4. β \scriptstyle\beta
  5. E ( x , y , z ) = k = 1 M ( a k e ( i β k z ) + b k e ( - i β k z ) ) E k ( x , y ) E(x,y,z)=\sum_{k=1}^{M}{(a_{k}e^{(i\beta_{k}z)}+b_{k}e^{(-i\beta_{k}z)})E_{k}(% x,y)}
  6. H ( x , y , z ) = k = 1 M ( a k e ( i β k z ) + b k e ( - i β k z ) ) H k ( x , y ) H(x,y,z)=\sum_{k=1}^{M}{(a_{k}e^{(i\beta_{k}z)}+b_{k}e^{(-i\beta_{k}z)})H_{k}(% x,y)}

Eigenvalues_and_eigenvectors_of_the_second_derivative.html

  1. \infty
  2. x [ 0 , L ] x\in[0,L]
  3. λ j = - j 2 π 2 L 2 \lambda_{j}=-\frac{j^{2}\pi^{2}}{L^{2}}
  4. v j ( x ) = 2 L sin ( j π x L ) v_{j}(x)=\sqrt{\frac{2}{L}}\sin(\frac{j\pi x}{L})
  5. λ j = - ( j - 1 ) 2 π 2 L 2 \lambda_{j}=-\frac{(j-1)^{2}\pi^{2}}{L^{2}}
  6. v j ( x ) = { L - 1 2 j = 1 2 L cos ( ( j - 1 ) π x L ) o t h e r w i s e v_{j}(x)=\left\{\begin{array}[]{lr}L^{-\frac{1}{2}}&j=1\\ \sqrt{\frac{2}{L}}\cos(\frac{(j-1)\pi x}{L})&otherwise\end{array}\right.
  7. λ j = { - j 2 π 2 L 2 j is even. - ( j + 1 ) 2 π 2 L 2 j is odd. \lambda_{j}=\left\{\begin{array}[]{lr}-\frac{j^{2}\pi^{2}}{L^{2}}&\mbox{j is % even.}\\ -\frac{(j+1)^{2}\pi^{2}}{L^{2}}&\mbox{j is odd.}\end{array}\right.
  8. 0
  9. j 2 π 2 L 2 \frac{j^{2}\pi^{2}}{L^{2}}
  10. j = 1 , 2 , j=1,2,\ldots
  11. v j ( x ) = { L - 1 2 if j = 1. 2 L sin ( j π x L ) if j is even. 2 L cos ( ( j + 1 ) π x L ) otherwise if j is odd. v_{j}(x)=\begin{cases}L^{-\frac{1}{2}}&\mbox{if }~{}j=1.\\ \sqrt{\frac{2}{L}}\sin(\frac{j\pi x}{L})&\mbox{ if j is even.}\\ \sqrt{\frac{2}{L}}\cos(\frac{(j+1)\pi x}{L})&\mbox{ otherwise if j is odd.}% \end{cases}
  12. λ j = - ( 2 j - 1 ) 2 π 2 4 L 2 \lambda_{j}=-\frac{(2j-1)^{2}\pi^{2}}{4L^{2}}
  13. v j ( x ) = 2 L sin ( ( 2 j - 1 ) π x 2 L ) v_{j}(x)=\sqrt{\frac{2}{L}}\sin(\frac{(2j-1)\pi x}{2L})
  14. λ j = - ( 2 j - 1 ) 2 π 2 4 L 2 \lambda_{j}=-\frac{(2j-1)^{2}\pi^{2}}{4L^{2}}
  15. v j ( x ) = 2 L cos ( ( 2 j - 1 ) π x 2 L ) v_{j}(x)=\sqrt{\frac{2}{L}}\cos(\frac{(2j-1)\pi x}{2L})
  16. λ j = - 4 h 2 sin ( π j 2 ( n + 1 ) ) 2 \lambda_{j}=-\frac{4}{h^{2}}\sin(\frac{\pi j}{2(n+1)})^{2}
  17. v i , j = 2 n + 1 sin ( i j π n + 1 ) v_{i,j}=\sqrt{\frac{2}{n+1}}\sin(\frac{ij\pi}{n+1})
  18. λ j = - 4 h 2 sin ( π ( j - 1 ) 2 n ) 2 \lambda_{j}=-\frac{4}{h^{2}}\sin(\frac{\pi(j-1)}{2n})^{2}
  19. v i , j = { n - 1 2 j = 1 2 n cos ( π ( j - 1 ) ( i - 1 2 ) n ) o t h e r w i s e v_{i,j}=\begin{cases}n^{-\frac{1}{2}}&j=1\\ \sqrt{\frac{2}{n}}\cos(\frac{\pi(j-1)(i-\frac{1}{2})}{n})&otherwise\end{cases}
  20. λ j = { - 4 h 2 sin ( π ( j - 1 ) ) 2 n ) 2 if j is odd. - 4 h 2 sin ( π j 2 n ) 2 if j is even. \lambda_{j}=\begin{cases}-\frac{4}{h^{2}}\sin(\frac{\pi(j-1))}{2n})^{2}&\mbox{% if j is odd.}\\ -\frac{4}{h^{2}}\sin(\frac{\pi j}{2n})^{2}&\mbox{ if j is even.}\end{cases}
  21. v i , j = { n - 1 2 if j = 1. n - 1 2 ( - 1 ) i if j = n and n is even. 2 n sin ( π ( i - 0.5 ) j n ) otherwise if j is even. 2 n cos ( π ( i - 0.5 ) ( j - 1 ) n ) otherwise if j is odd. v_{i,j}=\begin{cases}n^{-\frac{1}{2}}&\mbox{if }~{}j=1.\\ n^{-\frac{1}{2}}(-1)^{i}&\mbox{if }~{}j=n\mbox{ and n is even.}\\ \sqrt{\frac{2}{n}}\sin(\frac{\pi(i-0.5)j}{n})&\mbox{ otherwise if j is even.}% \\ \sqrt{\frac{2}{n}}\cos(\frac{\pi(i-0.5)(j-1)}{n})&\mbox{ otherwise if j is odd% .}\end{cases}
  22. λ j = - 4 h 2 sin ( π ( j - 1 2 ) 2 n + 1 ) 2 \lambda_{j}=-\frac{4}{h^{2}}\sin(\frac{\pi(j-\frac{1}{2})}{2n+1})^{2}
  23. v i , j = 2 n + 0.5 sin ( π i ( 2 j - 1 ) 2 n + 1 ) v_{i,j}=\sqrt{\frac{2}{n+0.5}}\sin(\frac{\pi i(2j-1)}{2n+1})
  24. λ j = - 4 h 2 sin ( π ( j - 1 2 ) 2 n + 1 ) 2 \lambda_{j}=-\frac{4}{h^{2}}\sin(\frac{\pi(j-\frac{1}{2})}{2n+1})^{2}
  25. v i , j = 2 n + 0.5 cos ( π ( i - 0.5 ) ( 2 j - 1 ) 2 n + 1 ) v_{i,j}=\sqrt{\frac{2}{n+0.5}}\cos(\frac{\pi(i-0.5)(2j-1)}{2n+1})
  26. v k + 1 - 2 v k + v k - 1 h 2 = λ v k , k = 1 , , n , v 0 = v n + 1 = 0. \frac{v_{k+1}-2v_{k}+v_{k-1}}{h^{2}}=\lambda v_{k},\ k=1,...,n,\ v_{0}=v_{n+1}% =0.
  27. v k + 1 = ( 2 + h 2 λ ) v k - v k - 1 . v_{k+1}=(2+h^{2}\lambda)v_{k}-v_{k-1}.\!
  28. 2 α = ( 2 + h 2 λ ) 2\alpha=(2+h^{2}\lambda)
  29. v 1 0 v_{1}\neq 0
  30. v v
  31. v 1 = 1 v_{1}=1
  32. v 0 = 0 v_{0}=0\,\!
  33. v 1 = 1. v_{1}=1.\,\!
  34. v k + 1 = 2 α v k - v k - 1 v_{k+1}=2\alpha v_{k}-v_{k-1}\,\!
  35. α \alpha
  36. v k + 1 = U k ( α ) v_{k+1}=U_{k}(\alpha)\,\!
  37. U k U_{k}
  38. v n + 1 = 0 v_{n+1}=0
  39. U k ( α ) = 0 U_{k}(\alpha)=0\,\!
  40. 2 α = ( 2 + h 2 λ ) 2\alpha=(2+h^{2}\lambda)
  41. α k = cos ( k π n + 1 ) . \alpha_{k}=\cos(\frac{k\pi}{n+1}).\,\!
  42. λ \lambda
  43. 2 cos ( k π n + 1 ) = h 2 λ k + 2 2\cos(\frac{k\pi}{n+1})=h^{2}\lambda_{k}+2\,\!
  44. λ k = - 2 h 2 ( 1 - cos ( k π n + 1 ) ) . \lambda_{k}=-\frac{2}{h^{2}}(1-\cos(\frac{k\pi}{n+1})).\,\!
  45. λ k = - 4 h 2 ( sin 2 ( k π 2 ( n + 1 ) ) ) . \lambda_{k}=-\frac{4}{h^{2}}(\sin^{2}(\frac{k\pi}{2(n+1)})).\,\!
  46. v k + 1 - 2 v k + v k - 1 h 2 = λ v k , k = 1 , , n , v 0.5 = v n + 0.5 = 0. \frac{v_{k+1}-2v_{k}+v_{k-1}}{h^{2}}=\lambda v_{k},\ k=1,...,n,\ v^{\prime}_{0% .5}=v^{\prime}_{n+0.5}=0.\,\!
  47. v 0 v_{0}\,\!
  48. v n + 1 v_{n+1}\,\!
  49. v 0.5 := v 1 - v 0 h , v n + 0.5 := v n + 1 - v n h v^{\prime}_{0.5}:=\frac{v_{1}-v_{0}}{h},\ v^{\prime}_{n+0.5}:=\frac{v_{n+1}-v_% {n}}{h}\,\!
  50. v 1 - v 0 = 0 , v n + 1 - v n = 0. v_{1}-v_{0}=0,\ v_{n+1}-v_{n}=0.
  51. w k = v k + 1 - v k , k = 0 , , n w_{k}=v_{k+1}-v_{k},\ k=0,...,n\,\!
  52. v k + 1 - 2 v k + v k - 1 h 2 \displaystyle\frac{v_{k+1}-2v_{k}+v_{k-1}}{h^{2}}
  53. w n = w 0 = 0 w_{n}=w_{0}=0
  54. n - 1 n-1
  55. h h
  56. w 1 0 w_{1}\neq 0
  57. λ k = - 4 h 2 ( sin 2 ( k π n ) ) , k = 1 , , n - 1. \lambda_{k}=-\frac{4}{h^{2}}(\sin^{2}(\frac{k\pi}{n})),\ k=1,...,n-1.
  58. n - 1 n-1
  59. n n
  60. w 1 0 w_{1}\neq 0
  61. v k = c o n s t a n t k = 0 , , n + 1 , v_{k}=constant\ \forall\ k=0,...,n+1,
  62. 0
  63. λ k = - 4 h 2 ( sin 2 ( ( k - 1 ) π n ) ) , k = 1 , , n . \lambda_{k}=-\frac{4}{h^{2}}(\sin^{2}(\frac{(k-1)\pi}{n})),\ k=1,...,n.
  64. v k + 1 - 2 v k + v k - 1 h 2 = λ v k , k = 1 , , n , v 0 = v n + 0.5 = 0. \frac{v_{k+1}-2v_{k}+v_{k-1}}{h^{2}}=\lambda v_{k},\ k=1,...,n,\ v_{0}=v^{% \prime}_{n+0.5}=0.
  65. v n + 0.5 := v n + 1 - v n h . v^{\prime}_{n+0.5}:=\frac{v_{n+1}-v_{n}}{h}.
  66. v j + 0.5 , j = 0 , , n . v_{j+0.5},\ j=0,...,n.
  67. v k + 0.5 = 2 β v k - v k - 0.5 , for some β v_{k+0.5}=2\beta v_{k}-v_{k-0.5},\,\text{ for some }\beta\,\!
  68. v 0 = 0 v_{0}=0
  69. v 0.5 0 v_{0.5}\neq 0
  70. v 0.5 v_{0.5}
  71. v 0.5 = 1. v_{0.5}=1.
  72. v k = 2 β v k - 0.5 - v k - 1 v_{k}=2\beta v_{k-0.5}-v_{k-1}\,\!
  73. v k + 1 = 2 β v k + 0.5 - v k . v_{k+1}=2\beta v_{k+0.5}-v_{k}.\,\!
  74. v k + 1 = ( 4 β 2 - 2 ) v k - v k - 1 . v_{k+1}=(4\beta^{2}-2)v_{k}-v_{k-1}.\,\!
  75. h 2 λ + 2 = ( 4 β 2 - 2 ) . h^{2}\lambda+2=(4\beta^{2}-2).\,\!
  76. λ \lambda
  77. λ = 4 ( β 2 - 1 ) h 2 . \lambda=\frac{4(\beta^{2}-1)}{h^{2}}.
  78. v n + 1 = U 2 n + 1 ( β ) , v n = U 2 n - 1 ( β ) , v_{n+1}=U_{2n+1}(\beta),\ v_{n}=U_{2n-1}(\beta),\,\!
  79. U k ( β ) U_{k}(\beta)
  80. U 2 n + 1 ( β ) - U 2 n - 1 ( β ) = 0. U_{2n+1}(\beta)-U_{2n-1}(\beta)=0.\,\!
  81. T k ( β ) T_{k}(\beta)
  82. U k ( β ) - U k - 2 ( β ) = T k ( β ) . U_{k}(\beta)-U_{k-2}(\beta)=T_{k}(\beta).\,\!
  83. T 2 n + 1 ( β ) = 0 , λ = 4 ( β 2 - 1 ) h 2 . T_{2n+1}(\beta)=0,\ \lambda=\frac{4(\beta^{2}-1)}{h^{2}}.\,\!
  84. β k = cos ( π ( k - 0.5 ) 2 n + 1 ) , k = 1 , , 2 n + 1 \beta_{k}=\cos(\frac{\pi(k-0.5)}{2n+1}),\ k=1,...,2n+1\,\!
  85. λ k = 4 h 2 ( cos ( π ( k - 0.5 ) 2 n + 1 ) 2 - 1 ) = - 4 h 2 sin ( π ( k - 0.5 ) 2 n + 1 ) 2 . \begin{aligned}\displaystyle\lambda_{k}&\displaystyle=\frac{4}{h^{2}}(\cos(% \frac{\pi(k-0.5)}{2n+1})^{2}-1)\\ &\displaystyle=-\frac{4}{h^{2}}\sin(\frac{\pi(k-0.5)}{2n+1})^{2}.\end{aligned}
  86. λ k = - 4 h 2 sin ( π ( k - 0.5 ) 2 n + 1 ) 2 , k = 1 , , n . \lambda_{k}=-\frac{4}{h^{2}}\sin(\frac{\pi(k-0.5)}{2n+1})^{2},\ k=1,...,n.

Eight-dimensional_space.html

  1. S 7 = { x 8 : x = r } . S^{7}=\left\{x\in\mathbb{R}^{8}:\|x\|=r\right\}.
  2. V 8 = π 4 24 R 8 V_{8}\,=\frac{\pi^{4}}{24}\,R^{8}
  3. x y x y \|xy\|\leq\|x\|\|y\|
  4. \mathbb{C}\otimes\mathbb{H}
  5. C 2 ( ) C\ell_{2}(\mathbb{C})

Einstein_function.html

  1. x 2 e x ( e x - 1 ) 2 \frac{x^{2}e^{x}}{(e^{x}-1)^{2}}
  2. x e x - 1 \frac{x}{e^{x}-1}
  3. log ( 1 - e - x ) \log(1-e^{-x})
  4. x e x - 1 - log ( 1 - e - x ) \frac{x}{e^{x}-1}-\log(1-e^{-x})

Einstein–Infeld–Hoffmann_equations.html

  1. a A \displaystyle\vec{a}_{A}
  2. x A \vec{x}_{A}
  3. v A = d x A / d t \vec{v}_{A}=d\vec{x}_{A}/dt
  4. a A = d 2 x A / d t 2 \vec{a}_{A}=d^{2}\vec{x}_{A}/dt^{2}
  5. r A B = | x A - x B | r_{AB}=|\vec{x}_{A}-\vec{x}_{B}|
  6. n A B = ( x A - x B ) / r A B \vec{n}_{AB}=(\vec{x}_{A}-\vec{x}_{B})/r_{AB}
  7. m A m_{A}
  8. c c
  9. G G

Electric_dipole_transition.html

  1. H 0 H_{0}
  2. 𝐄 ( 𝐫 , t ) = E 0 𝐳 ^ cos ( k y - ω t ) , 𝐁 ( 𝐫 , t ) = B 0 𝐱 ^ cos ( k y - ω t ) . {\mathbf{E}}({\mathbf{r}},t)=E_{0}{\hat{\mathbf{z}}}\cos(ky-\omega t),\ \ \ {% \mathbf{B}}({\mathbf{r}},t)=B_{0}{\hat{\mathbf{x}}}\cos(ky-\omega t).
  3. H ( t ) = H 0 + W ( t ) . H(t)\ =\ H_{0}+W(t).
  4. W ( t ) W(t)
  5. W D E ( t ) = q E 0 m ω p z sin ω t . W_{DE}(t)=\frac{qE_{0}}{m\omega}p_{z}\sin\omega t.\,
  6. H 0 + W D E ( t ) H_{0}+W_{DE}(t)
  7. W ( t ) W(t)
  8. W D M ( t ) = q 2 m ( L x + 2 S x ) B 0 cos ω t W_{DM}(t)=\frac{q}{2m}(L_{x}+2S_{x})B_{0}\cos\omega t\,

Electrical_cardiometry.html

  1. Z ( t ) = Z 0 + Δ Z R + Δ Z C Z(t)=Z_{0}+\Delta Z_{R}+\Delta Z_{C}
  2. d Z ( t ) d t \frac{dZ(t)}{dt}
  3. d Z ( t ) d t \frac{dZ(t)}{dt}
  4. d Z ( t ) d t \frac{dZ(t)}{dt}
  5. v ¯ \bar{v}
  6. S V T E B = C P v ¯ F T F T SV_{TEB}=C_{P}\cdot\bar{v}_{FT}\cdot FT
  7. v ¯ \bar{v}

Electro_Thermal_Dynamic_Stripping_Process.html

  1. ρ c ¯ ( T t ) = λ ¯ ( 2 T t 2 + 1 r T r ) + 1 σ ¯ ( I 2 π L r ) 2 \bar{\rho\,\!c}\left(\frac{\partial T}{\partial t}\right)\ =\bar{\lambda}\left% (\frac{\partial^{2}T}{\partial t^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}% \right)+\frac{1}{\bar{\sigma}}\left(\frac{I}{2{\pi}Lr}\right)^{2}
  2. ρ c ¯ \bar{\rho\,\!c}
  3. λ ¯ \bar{\lambda\,\!}
  4. σ ¯ \bar{\sigma\,\!}
  5. ρ c ¯ ( T t ) = λ ¯ ( 2 T t 2 + 1 r T r ) + 1 σ ¯ ( I 2 π L r ) 2 - ρ w c w Q 2 π L r 1 r T r C o n v e c t i o n \bar{\rho\,\!c}\left(\frac{\partial T}{\partial t}\right)\ =\bar{\lambda}\left% (\frac{\partial^{2}T}{\partial t^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}% \right)+\frac{1}{\bar{\sigma}}\left(\frac{I}{2{\pi}Lr}\right)^{2}-\underbrace{% \rho_{w}\,\!c_{w}\frac{Q}{2{\pi}Lr}\frac{1}{r}\frac{\partial T}{\partial r}}_{Convection}
  6. ρ w \rho_{w}
  7. c w \!c_{w}

Electrocatalyst.html

  1. \rightleftarrows
  2. \rightleftarrows

Electrochemical_regeneration.html

  1. a d s o r p t i v e c a p a c i t y b e f o r e a d s o r p t i o n a d s o r p t i v e c a p a c i t y a f t e r a d s o r p t i o n a n d e l e c t r o c h e m i c a l r e g e n e r a t i o n × 100 \frac{adsorptive\;capacity\;before\;adsorption}{adsorptive\;capacity\;after\;% adsorption\;and\;electrochemical\;regeneration}\times 100

Electrofiltration.html

  1. F W = 6 π η r H ν F_{W}=6\cdot\pi\cdot\eta\cdot\,\text{r}_{H}\cdot\nu
  2. F E = 4 π ε 0 ε r r H ζ E F_{E}=4\cdot\pi\cdot\varepsilon_{0}\cdot\varepsilon_{r}\cdot\,\text{r}_{H}% \cdot\zeta\cdot\,\text{E}
  3. ν \nu
  4. η \eta
  5. ε 0 \varepsilon_{0}
  6. ε r \varepsilon_{r}
  7. ζ \zeta
  8. t V L = η α c c ( E crit - E ) E crit 2 ( Δ P H + P e ) A 2 V L \frac{t}{V_{L}}=\frac{\eta\cdot\alpha\text{c}\cdot c\cdot\frac{\left(E\text{% crit}-E\right)}{E\text{crit}}}{2\cdot\left(\Delta P_{H}+P_{e}\right)\cdot A^{2% }}\cdot V_{L}
  9. α c \alpha\text{c}
  10. Δ P H \Delta P_{H}

Electromagnetic_lock.html

  1. B B
  2. l l
  3. I I
  4. N N
  5. B = μ 0 μ r I N l B=\frac{\mu_{0}\mu_{r}IN}{l}
  6. F F
  7. S S
  8. F = B 2 S 2 μ 0 F=\frac{B^{2}S}{2\mu_{0}}
  9. μ 0 \mu_{0}
  10. μ r \mu_{r}

Electron-longitudinal_acoustic_phonon_interaction.html

  1. M d 2 d t 2 u n = - k 0 ( u n - 1 + u n + 1 - 2 u n ) M\frac{d^{2}}{dt^{2}}u_{n}=-k_{0}(u_{n-1}+u_{n+1}-2u_{n})
  2. u n u_{n}
  3. u l u_{l}
  4. u l = x l - l a u_{l}=x_{l}-la
  5. x l x_{l}
  6. u n = A e i q l a - ω t u_{n}=Ae^{iqla-\omega t}
  7. Q q = 1 N l u l e - i q a l Q_{q}=\frac{1}{\sqrt{N}}\sum_{l}u_{l}e^{-iqal}
  8. u l = 1 N q Q q e i q a l u_{l}=\frac{1}{\sqrt{N}}\sum_{q}Q_{q}e^{iqal}
  9. u l u_{l}
  10. u l = 1 2 N q ( Q q e i q a l + Q q e - i q a l ) u_{l}=\frac{1}{2\sqrt{N}}\sum_{q}(Q_{q}e^{iqal}+Q^{\dagger}_{q}e^{-iqal})
  11. a q = q 2 M ω q ( M ω q Q - q - i P q ) , a q = q 2 M ω q ( M ω q Q - q + i P q ) a^{\dagger}_{q}=\frac{q}{\sqrt{2M\hbar\omega_{q}}}(M\omega_{q}Q_{-q}-iP_{q}),% \;a_{q}=\frac{q}{\sqrt{2M\hbar\omega_{q}}}(M\omega_{q}Q_{-q}+iP_{q})
  12. Q q Q_{q}
  13. Q q = 2 M ω q ( a - q + a q ) Q_{q}=\sqrt{\frac{\hbar}{2M\omega_{q}}}(a^{\dagger}_{-q}+a_{q})
  14. u l u_{l}
  15. u l = q 2 M N ω q ( a q e i q a l + a q e - i q a l ) u_{l}=\sum_{q}\sqrt{\frac{\hbar}{2MN\omega_{q}}}(a_{q}e^{iqal}+a^{\dagger}_{q}% e^{-iqal})
  16. u ( r ) = q 2 M N ω q e q [ a q e i q r + a q e - i q r ] u(r)=\sum_{q}\sqrt{\frac{\hbar}{2MN\omega_{q}}}e_{q}[a_{q}e^{iq\cdot r}+a^{% \dagger}_{q}e^{-iq\cdot r}]
  17. e q e_{q}
  18. H e l H_{el}
  19. H e l = D a c δ V V = D a c d i v u ( r ) H_{el}=D_{ac}\frac{\delta V}{V}=D_{ac}\,div\,u(r)
  20. D a c D_{ac}
  21. H e l = D a c q 2 M N ω q ( i e q q ) [ a q e i q r - a q e - i q r ] H_{el}=D_{ac}\sum_{q}\sqrt{\frac{\hbar}{2MN\omega_{q}}}(ie_{q}\cdot q)[a_{q}e^% {iq\cdot r}-a^{\dagger}_{q}e^{-iq\cdot r}]
  22. | k |k\rangle
  23. | k |k^{\prime}\rangle
  24. P ( k , k ) = 2 π k , q | H e l | k , q 2 δ [ ε ( k ) - ε ( k ) ω q ] P(k,k^{\prime})=\frac{2\pi}{\hbar}\mid\langle k^{\prime},q^{\prime}|H_{el}|\ k% ,q\rangle\mid^{2}\delta[\varepsilon(k^{\prime})-\varepsilon(k)\mp\hbar\omega_{% q}]
  25. = 2 π | D a c q 2 M N ω q ( i e q q ) n q + 1 2 1 2 1 L 3 d 3 r u k ( r ) u k ( r ) e i ( k - k ± q ) r | 2 δ [ ε ( k ) - ε ( k ) ω q ] =\frac{2\pi}{\hbar}\left|D_{ac}\sum_{q}\sqrt{\frac{\hbar}{2MN\omega_{q}}}(ie_{% q}\cdot q)\sqrt{n_{q}+\frac{1}{2}\mp\frac{1}{2}}\,\frac{1}{L^{3}}\int d^{3}r\,% u^{\ast}_{k^{\prime}}(r)u_{k}(r)e^{i(k-k^{\prime}\pm q)\cdot r}\right|^{2}% \delta[\varepsilon(k^{\prime})-\varepsilon(k)\mp\hbar\omega_{q}]
  26. P ( k , k ) = 2 π ( D a c q 2 M N ω q | q | n q + 1 2 1 2 I ( k , k ) δ k , k ± q ) 2 δ [ ε ( k ) - ε ( k ) ω q ] , P(k,k^{\prime})=\frac{2\pi}{\hbar}\left(D_{ac}\sum_{q}\sqrt{\frac{\hbar}{2MN% \omega_{q}}}|q|\sqrt{n_{q}+\frac{1}{2}\mp\frac{1}{2}}\,I(k,k^{\prime})\delta_{% k^{\prime},k\pm q}\right)^{2}\delta[\varepsilon(k^{\prime})-\varepsilon(k)\mp% \hbar\omega_{q}],
  27. I ( k , k ) = Ω Ω d 3 r u k ( r ) u k ( r ) I(k,k^{\prime})=\Omega\int_{\Omega}d^{3}r\,u^{\ast}_{k^{\prime}}(r)u_{k}(r)
  28. Ω \Omega
  29. P ( k , k ) = { 2 π D a c 2 2 M N ω q | q | 2 n q ( k = k + q ; absorption ) , 2 π D a c 2 2 M N ω q | q | 2 ( n q + 1 ) ( k = k - q ; emission ) . P(k,k^{\prime})=\begin{cases}\frac{2\pi}{\hbar}D_{ac}^{2}\frac{\hbar}{2MN% \omega_{q}}|q|^{2}n_{q}&(k^{\prime}=k+q;\,\text{absorption}),\\ \frac{2\pi}{\hbar}D_{ac}^{2}\frac{\hbar}{2MN\omega_{q}}|q|^{2}(n_{q}+1)&(k^{% \prime}=k-q;\,\text{emission}).\end{cases}

Electron_nuclear_double_resonance.html

  1. 0 = EZ + NZ + HFS + Q \mathcal{H}_{\mathrm{0}}=\mathcal{H}_{\mathrm{EZ}}+\mathcal{H}_{\mathrm{NZ}}+% \mathcal{H}_{\mathrm{HFS}}+\mathcal{H}_{\mathrm{Q}}
  2. B 1 \mathrm{B}_{\mathrm{1}}
  3. B 2 \mathrm{B}_{\mathrm{2}}
  4. ν 1 \nu_{\mathrm{1}}
  5. ν 2 \nu_{\mathrm{2}}
  6. Δ M I = ± 1 \Delta M_{I}=\pm 1
  7. Δ M S = 0 \Delta M_{S}=0
  8. ν n \nu_{\mathrm{n}}
  9. ν 1 = | ν n - a / 2 | \nu_{\mathrm{1}}=|\nu_{\mathrm{n}}-a/2|
  10. ν 2 = | ν n + a / 2 | \nu_{\mathrm{2}}=|\nu_{\mathrm{n}}+a/2|
  11. γ e 2 B 1 2 T 1 e T 2 e 1 \gamma_{e}^{2}B_{1}^{2}T_{1e}T_{2e}\geq{1}
  12. γ n 2 B 2 2 T 1 n T 2 n 1 \gamma_{n}^{2}B_{2}^{2}T_{1n}T_{2n}\geq{1}
  13. γ e \gamma_{\mathrm{e}}
  14. γ n \gamma_{\mathrm{n}}
  15. B 1 B_{1}
  16. B 2 B_{2}
  17. T 1 e T_{\mathrm{1e}}
  18. T 1 n T_{\mathrm{1n}}
  19. T 2 e T_{\mathrm{2e}}
  20. T 2 n T_{\mathrm{2n}}

Elementary_cellular_automaton.html

  1. 1 + 2 x ( 1 + x ) ( 1 - 2 x ) \frac{1+2x}{(1+x)(1-2x)}
  2. a ( t ) = ( 4 2 t - ( - 1 ) t ) / 3 a(t)=(4\cdot 2^{t}-(-1)^{t})/3
  3. 1 ( 1 - x ) ( 1 - 4 x ) \frac{1}{(1-x)(1-4x)}
  4. a ( t ) = ( 4 4 t - 1 ) / 3 a(t)=(4\cdot 4^{t}-1)/3
  5. 1 + 7 x ( 1 - x 2 ) ( 1 - 16 x 2 ) \frac{1+7x}{(1-x^{2})(1-16x^{2})}
  6. a ( t ) = ( 22 4 t - 6 ( - 4 ) t - 4 + 3 ( - 1 ) t ) / 15 a(t)=(22\cdot 4^{t}-6(-4)^{t}-4+3(-1)^{t})/15
  7. a ( t ) = { 1 , if t = 0 7 , if t = 1 ( 1 + 5 4 n ) / 3 , if t is even > 0 ( 10 + 11 4 n ) / 6 , if t is odd > 1 a(t)=\begin{cases}1,&\mbox{if }~{}t=0\\ 7,&\mbox{if }~{}t=1\\ (1+5\cdot 4^{n})/3,&\mbox{if }~{}t\mbox{ is even }~{}>0\\ (10+11\cdot 4^{n})/6,&\mbox{if }~{}t\mbox{ is odd }~{}>1\end{cases}
  8. ( 1 + 2 x ) ( 1 + 5 x - 16 x 4 ) ( 1 - x 2 ) ( 1 - 16 x 2 ) \frac{(1+2x)(1+5x-16x^{4})}{(1-x^{2})(1-16x^{2})}
  9. 1 + 7 x + 12 x 2 - 4 x 3 ( 1 - x 2 ) ( 1 - 16 x 2 ) \frac{1+7x+12x^{2}-4x^{3}}{(1-x^{2})(1-16x^{2})}
  10. 1 + 3 x + 4 x 2 + 12 x 3 + 8 x 4 - 8 x 5 ( 1 - x 2 ) ( 1 - 16 x 4 ) \frac{1+3x+4x^{2}+12x^{3}+8x^{4}-8x^{5}}{(1-x^{2})(1-16x^{4})}
  11. 1 + 3 x ( 1 - x 2 ) ( 1 - 4 x ) \frac{1+3x}{(1-x^{2})(1-4x)}
  12. 1 ( 1 - x ) ( 1 - 2 x ) \frac{1}{(1-x)(1-2x)}
  13. a ( t ) = 2 2 t - 1 a(t)=2\cdot 2^{t}-1
  14. 1 + 2 x ( 1 - x ) ( 1 - 4 x ) \frac{1+2x}{(1-x)(1-4x)}
  15. a ( t ) = 2 4 t - 1 a(t)=2\cdot 4^{t}-1

Elementary_effects_method.html

  1. k k
  2. Y Y
  3. Y = f ( X 1 , X 2 , X k ) . Y=f(X_{1},X_{2},...X_{k}).
  4. μ \mu
  5. σ \sigma
  6. p p
  7. Ω \Omega
  8. k k
  9. p p
  10. ( k + 1 ) (k+1)
  11. Δ \Delta
  12. { 0 , 1 / ( p - 1 ) , 2 / ( p - 1 ) , , 1 } \{0,1/(p-1),2/(p-1),...,1\}
  13. d i ( X ) = Y ( X 1 , , X i - 1 , X i + Δ , X i + 1 , , X k ) - Y ( 𝐗 ) Δ d_{i}(X)=\frac{Y(X_{1},\ldots,X_{i-1},X_{i}+\Delta,X_{i+1},\ldots,X_{k})-Y(% \mathbf{X})}{\Delta}
  14. 𝐗 = ( X 1 , X 2 , X k ) \mathbf{X}=(X_{1},X_{2},...X_{k})
  15. Ω \Omega
  16. Ω \Omega
  17. i = 1 , , k . i=1,\ldots,k.
  18. r r
  19. d i ( X ( 1 ) ) , d i ( X ( 2 ) ) , , d i ( X ( r ) ) d_{i}\left(X^{(1)}\right),d_{i}\left(X^{(2)}\right),\ldots,d_{i}\left(X^{(r)}\right)
  20. r r
  21. X ( 1 ) , X ( 2 ) , , X ( r ) X^{(1)},X^{(2)},\ldots,X^{(r)}
  22. r r
  23. p p
  24. p p
  25. Δ \Delta
  26. p p
  27. Δ \Delta
  28. p / [ 2 ( p - 1 ) ] p/[2(p-1)]
  29. Δ \Delta
  30. μ \mu
  31. σ \sigma
  32. μ i = 1 r j = 1 r d i ( X ( j ) ) \mu_{i}=\frac{1}{r}\sum_{j=1}^{r}d_{i}\left(X^{(j)}\right)
  33. σ i = 1 ( r - 1 ) j = 1 r ( d i ( X ( j ) ) - μ i ) 2 \sigma_{i}=\sqrt{\frac{1}{(r-1)}\sum_{j=1}^{r}\left(d_{i}\left(X^{(j)}\right)-% \mu_{i}\right)^{2}}
  34. μ \mu
  35. σ \sigma
  36. μ * \mu^{*}
  37. μ i * = 1 r j = 1 r | d i ( X ( j ) ) | \mu_{i}^{*}=\frac{1}{r}\sum_{j=1}^{r}\left|d_{i}\left(X^{(j)}\right)\right|
  38. μ * \mu^{*}
  39. μ \mu

Elias_Bassalygo_bound.html

  1. C C
  2. q q
  3. n n
  4. [ q ] n [q]^{n}
  5. q q
  6. n n
  7. 𝒜 q n , \mathcal{A}_{q}^{n}\,\text{,}
  8. 𝒜 q \mathcal{A}_{q}
  9. q q
  10. R R
  11. C C
  12. δ \delta
  13. B q ( s y m b o l y , ρ n ) = { s y m b o l x [ q ] n | Δ ( s y m b o l x , s y m b o l y ) ρ n } B_{q}(symbol{y},\rho n)=\{symbol{x}\in[q]^{n}|\Delta(symbol{x},symbol{y})\leq% \rho n\}
  14. ρ n \rho n
  15. s y m b o l y symbol{y}
  16. V o l q ( s y m b o l y , ρ n ) = | B q ( s y m b o l y , ρ n ) | Vol_{q}(symbol{y},\rho n)=|B_{q}(symbol{y},\rho n)|
  17. ρ n \rho n
  18. s y m b o l y symbol{y}
  19. | B q ( s y m b o l y , ρ n ) | = | B q ( s y m b o l 0 , ρ n ) | |B_{q}(symbol{y},\rho n)|=|B_{q}(symbol{0},\rho n)|
  20. n n
  21. R R
  22. δ \delta
  23. R 1 - H q ( J q ( δ ) ) + o ( 1 ) , R\leq 1-H_{q}(J_{q}(\delta))+o(1)\,\text{,}
  24. H q ( x ) def - x log q x q - 1 - ( 1 - x ) log q ( 1 - x ) H_{q}(x)\equiv\text{def}-x\cdot\log_{q}{x\over{q-1}}-(1-x)\cdot\log_{q}{(1-x)}
  25. J q ( δ ) def ( 1 - 1 q ) ( 1 - 1 - q δ q - 1 ) J_{q}(\delta)\equiv\text{def}\left(1-{1\over q}\right)\left(1-\sqrt{1-{q\delta% \over{q-1}}}\right)
  26. C [ q ] n C\subseteq[q]^{n}
  27. 0 e n 0\leq e\leq n
  28. e e
  29. | C | V o l q ( 0 , e ) q n {|C|Vol_{q}(0,e)}\over{q^{n}}
  30. y [ q ] n y\in[q]^{n}
  31. C C
  32. y y
  33. e e
  34. | B q ( y , e ) C | |B_{q}(y,e)\cap C|
  35. V o l q ( y , e ) | C | q n Vol_{q}(y,e){{|C|}\over{q^{n}}}
  36. y y
  37. y y
  38. | B q ( y , e ) C | V o l q ( y , e ) | C | q n = | C | V o l q ( 0 , e ) q n |B_{q}(y,e)\cap C|\geq Vol_{q}(y,e){{|C|}\over{q^{n}}}={{|C|Vol_{q}(0,e)}\over% {q^{n}}}
  39. e = n J q ( δ ) - 1 e=nJ_{q}(\delta)-1
  40. B B
  41. B | C | V o l ( 0 , e ) q n B\geq{{|C|Vol(0,e)}\over{q^{n}}}
  42. B q d n B\leq qdn
  43. C q n d q n V o l q ( 0 , e ) q n ( 1 - H q ( J q ( δ ) ) + o ( 1 ) ) \mid C\mid\leq qnd\cdot{{q^{n}}\over{Vol_{q}(0,e)}}\leq q^{n(1-H_{q}(J_{q}(% \delta))+o(1))}
  44. V o l q ( 0 , d - 1 2 ) q H q ( δ 2 ) n - o ( n ) Vol_{q}(0,\lfloor{{d-1}\over 2}\rfloor)\leq q^{H_{q}({\delta\over 2})n-o(n)}
  45. d = 2 e + 1 d=2e+1
  46. δ = d n \delta={d\over n}
  47. R = log q | C | n 1 - H q ( J q ( δ ) ) + o ( 1 ) R={\log_{q}{|C|}\over n}\leq 1-H_{q}(J_{q}(\delta))+o(1)

Elliptic_curve_only_hash.html

  1. n n
  2. M M
  3. n n
  4. M 0 , , M n - 1 M_{0},\ldots,M_{n-1}
  5. P P
  6. P P
  7. P i P_{i}
  8. X 1 X_{1}
  9. X 2 X_{2}
  10. H H
  11. P i : = P ( M i , i ) X 1 : = P ( n ) X 2 : = P * ( M i , n ) Q : = i = 0 n - 1 P i + X 1 + X 2 R : = f ( Q ) \begin{aligned}\displaystyle P_{i}&\displaystyle{}:=P(M_{i},i)\\ \displaystyle X_{1}&\displaystyle{}:=P^{\prime}(n)\\ \displaystyle X_{2}&\displaystyle{}:=P^{*}(M_{i},n)\\ \displaystyle Q&\displaystyle{}:=\sum_{i=0}^{n-1}P_{i}+X_{1}+X_{2}\\ \displaystyle R&\displaystyle{}:=f(Q)\end{aligned}
  12. P P^{\prime}
  13. P * P^{*}
  14. Q Q
  15. R R
  16. X 283 + X 12 + X 7 + X 5 + 1 X^{283}+X^{12}+X^{7}+X^{5}+1
  17. X 409 + X 87 + 1 X^{409}+X^{87}+1
  18. X 571 + X 10 + X 5 + X 2 + 1 X^{571}+X^{10}+X^{5}+X^{2}+1
  19. 2 128 2^{128}
  20. P i s P_{i}^{\prime}s
  21. f n f_{n}
  22. n n
  23. E E
  24. 𝐅 \mathbf{F}
  25. E E
  26. 𝐅 \mathbf{F}
  27. O O
  28. f n ( X 1 , , X N ) f_{n}(X_{1},\ldots,X_{N})
  29. ( x 1 , y 1 ) + + ( x n , y n ) = O (x_{1},y_{1})+\ldots+(x_{n},y_{n})=O
  30. 2 n - 2 2^{n-2}
  31. M = ( M 1 , M 2 , M 3 , M 4 , M 5 , M 6 ) M^{\prime}=(M_{1},M_{2},M_{3},M_{4},M_{5},M_{6})
  32. ( M 0 , M 1 ) (M_{0},M_{1})
  33. M 2 M_{2}
  34. M 2 := M 0 + M 1 M_{2}:=M_{0}+M_{1}
  35. P ( M 0 , 0 ) + P ( M 1 , 1 ) + P ( M 2 , 2 ) P(M_{0},0)+P(M_{1},1)+P(M_{2},2)
  36. ( M 3 , M 4 ) (M_{3},M_{4})
  37. M 5 := M 3 + M 4 M_{5}:=M_{3}+M_{4}
  38. Q - X 1 - X 2 - P ( M 3 , 3 ) - P ( M 4 , 4 ) - P ( M 5 , 5 ) Q-X_{1}-X_{2}-P(M_{3},3)-P(M_{4},4)-P(M_{5},5)
  39. X 1 X_{1}
  40. X 2 X_{2}
  41. X 2 X_{2}
  42. ( M 1 , M 2 , M 3 , M 4 , M 5 , M 6 ) (M_{1},M_{2},M_{3},M_{4},M_{5},M_{6})
  43. Q = i = 0 5 P ( M i , i ) + X 1 + X 2 Q=\sum_{i=0}^{5}P(M_{i},i)+X_{1}+X_{2}
  44. 2 283 2^{283}
  45. K = 2 142 K=2^{142}
  46. 2 143 2^{143}
  47. 2 409 2^{409}
  48. K = 2 205 K=2^{205}
  49. 2 206 . 2^{206}.
  50. 2 571 2^{571}
  51. K = 2 286 K=2^{286}
  52. 2 287 . 2^{287}.

Elliptic_curve_primality.html

  1. ( / n ) * (\mathbb{Z}/n\mathbb{Z})^{*}
  2. E ( / n ) E(\mathbb{Z}/n\mathbb{Z})
  3. O ( ( log n ) 5 + ε ) O((\log n)^{5+\varepsilon})\,
  4. ε > 0 \varepsilon>0
  5. 4 + ε 4+\varepsilon
  6. p p
  7. p - 1 p-1
  8. L u c a s ( 140057 ) . Lucas(140057).
  9. y 2 = x 3 + a x + b ( mod N ) y^{2}=x^{3}+ax+b\;\;(\mathop{{\rm mod}}N)
  10. / N \mathbb{Z}/N\mathbb{Z}
  11. ( N 1 / 4 + 1 ) 2 (N^{1/4}+1)^{2}
  12. p N p\leq\sqrt{N}
  13. E p E_{p}
  14. m p m_{p}
  15. E p E_{p}
  16. m p p + 1 + 2 p = ( p + 1 ) 2 ( N 1 / 4 + 1 ) 2 < q m_{p}\leq p+1+2\sqrt{p}=(\sqrt{p}+1)^{2}\leq(N^{1/4}+1)^{2}<q
  17. gcd ( q , m p ) = 1 \gcd{(q,m_{p})}=1
  18. u q 1 ( mod m p ) uq\equiv 1\;\;(\mathop{{\rm mod}}m_{p})
  19. P p P_{p}
  20. E p E_{p}
  21. ( m / q ) P p = u q ( m / q ) P p = u m P p = 0 (m/q)P_{p}=uq(m/q)P_{p}=umP_{p}=0\,
  22. m P p mP_{p}
  23. p N p\mid N
  24. ( m / q ) P p 0 (m/q)P_{p}\neq 0
  25. b y 2 - x 3 - a x ( mod N ) b\equiv y^{2}-x^{3}-ax\;\;(\mathop{{\rm mod}}N)
  26. y 2 = x 3 + a x + b y^{2}=x^{3}+ax+b
  27. m = k q m=kq
  28. k 2 k\geq 2
  29. m P 0 mP\neq 0
  30. m P = 0 mP=0
  31. k P 0 kP\neq 0
  32. [ p + 1 - 2 p , p + 1 + 2 p ] [p+1-2\sqrt{p},p+1+2\sqrt{p}]
  33. D = π π ¯ D=\pi\bar{\pi}
  34. a 2 + | D | b 2 = 4 N a^{2}+|D|b^{2}=4N\,
  35. / N \mathbb{Z}/N\mathbb{Z}
  36. | E ( / N ) | = N + 1 - π - π ¯ = N + 1 - a . |E(\mathbb{Z}/N\mathbb{Z})|=N+1-\pi-\bar{\pi}=N+1-a.\,
  37. ( D N ) = 1 \left(\frac{D}{N}\right)=1
  38. ( D N ) \left(\frac{D}{N}\right)
  39. ( D ) \mathbb{Q}(\sqrt{D})
  40. ( D N ) = 1 \left(\frac{D}{N}\right)=1
  41. a 2 + | D | b 2 = 4 N a^{2}+|D|b^{2}=4N\,
  42. m = N + 1 - a m=N+1-a
  43. q > ( N 1 / 4 + 1 ) 2 q>(N^{1/4}+1)^{2}
  44. D - 3 D\neq-3
  45. D - 4 D\neq-4
  46. H D ( X ) H_{D}(X)
  47. H D ( X ) H_{D}(X)
  48. H D ( X ) H_{D}(X)
  49. H D ( X ) H_{D}(X)
  50. y 2 = x 3 - 3 k c 2 r x + 2 k c 3 r , where k = j j - 1728 , y^{2}=x^{3}-3kc^{2r}x+2kc^{3r},\,\text{ where }k=\frac{j}{j-1728},
  51. | E ( / N ) | = N + 1 - a |E(\mathbb{Z}/N\mathbb{Z})|=N+1-a
  52. | E ( / N ) | = N + 1 + a |E(\mathbb{Z}/N\mathbb{Z})|=N+1+a
  53. N = 167 N=167
  54. ( D / N ) = 1 (D/N)=1
  55. 4 N = a 2 + | D | b 2 4N=a^{2}+|D|b^{2}
  56. ( D / N ) = ( - 43 / 167 ) = 1 (D/N)=(-43/167)=1
  57. 4 ( 167 ) = 25 2 + ( 43 ) ( 1 2 ) 4\cdot(167)=25^{2}+(43)(1^{2})
  58. m = N + 1 - a m=N+1-a
  59. m = 167 + 1 - 25 = 143 m=167+1-25=143
  60. q > ( N 1 / 4 + 1 ) 2 q>(N^{1/4}+1)^{2}
  61. s > ( N 1 / 4 + 1 ) 2 s>(N^{1/4}+1)^{2}
  62. p i p_{i}
  63. m / p i P P m/p_{i}P\neq P_{\infty}
  64. p i p_{i}
  65. 143 > ( 167 1 / 4 + 1 ) 2 143>(167^{1/4}+1)^{2}
  66. ( 143 / 11 ) P = 13 P and ( 143 / 13 ) P = 11 P . (143/11)P=13P\,\text{ and }(143/13)P=11P.
  67. j - 960 3 ( mod 167 ) 107 ( mod 167 ) . j\equiv-960^{3}\;\;(\mathop{{\rm mod}}167)\equiv 107\;\;(\mathop{{\rm mod}}167% ).\,
  68. k = j 1728 - j ( mod 167 ) 158 ( mod 167 ) k=\frac{j}{1728-j}\;\;(\mathop{{\rm mod}}167)\equiv 158\;\;(\mathop{{\rm mod}}% 167)
  69. y 2 = x 3 + 3 k c 2 x + 2 k c 3 y^{2}=x^{3}+3kc^{2}x+2kc^{3}
  70. 𝔽 167 \mathbb{F}_{167}
  71. r = 0 , 3 k 140 ( mod 167 ) r=0,3k\equiv 140\;\;(\mathop{{\rm mod}}167)
  72. 2 k 149 ( mod 167 ) 2k\equiv 149\;\;(\mathop{{\rm mod}}167)
  73. y 2 = x 3 + 140 x + 149 ( mod 167 ) y^{2}=x^{3}+140x+149\;\;(\mathop{{\rm mod}}167)
  74. P P_{\infty}
  75. 1 - O ( 2 - N 1 log log n ) 1-O\left(2^{-N\frac{1}{\log\log n}}\right)
  76. π ( x ) \pi(x)
  77. c 1 , c 2 > 0 : π ( x + x ) - π ( x ) c 2 x log c 1 x \exists c_{1},c_{2}>0:\pi(x+\sqrt{x})-\pi(x)\geq\frac{c_{2}\sqrt{x}}{\log^{c_{% 1}}x}
  78. O ( k 2 ) O(k^{2})
  79. O ( k 4 ) O(k^{4})
  80. c 1 c_{1}
  81. c 2 c_{2}
  82. [ x , x + 2 x ] , x 2 [x,x+\sqrt{2x}],x\geq 2
  83. c 1 x ( log x ) - c 2 c_{1}\sqrt{x}(\log x)^{-c_{2}}
  84. O ( log 10 + c 2 n ) O(\log^{10+c_{2}}n)
  85. O ( ( log N ) 6 + ϵ ) O((\log N)^{6+\epsilon})
  86. ϵ > 0 \epsilon>0
  87. N = 2 k n - 1 N=2^{k}n-1
  88. k , n , k 2 k,n\in\mathbb{Z},k\geq 2
  89. 2 k n - 1 2^{k}n-1
  90. y 2 = x 3 - m x y^{2}=x^{3}-mx
  91. m m\in\mathbb{Z}
  92. m 0 ( mod p ) m\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  93. p 3 ( mod 4 ) p\equiv 3\;\;(\mathop{{\rm mod}}4)
  94. p + 1 = 2 k n , k , k 2 , n p+1=2^{k}n,k\in\mathbb{Z},k\geq 2,n
  95. ( 𝔽 p ) = p + 1 (\mathbb{F}_{p})=p+1
  96. ( 𝔽 p ) 2 k n (\mathbb{F}_{p})\cong\mathbb{Z}_{2^{k}n}
  97. ( 𝔽 p ) 2 2 k - 1 n (\mathbb{F}_{p})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2^{k-1}n}
  98. p 3 ( mod 4 ) p\equiv 3\;\;(\mathop{{\rm mod}}4)
  99. 2 k 2^{k}
  100. E ( 𝔽 p ) 2 k n E(\mathbb{F}_{p})\cong\mathbb{Z}_{2^{k}n}
  101. 2 k 2^{k}
  102. λ > 1 \lambda>1
  103. n p / λ n\leq\sqrt{p}/\lambda
  104. λ p > ( p 1 / 4 + 1 ) 2 \lambda\sqrt{p}>(p^{1/4}+1)^{2}\,
  105. gcd ( S i , p ) = 1 \gcd{(S_{i},p)}=1
  106. S k 0 ( mod p ) S_{k}\equiv 0\;\;(\mathop{{\rm mod}}p)
  107. S i S_{i}
  108. S 0 = x S_{0}=x
  109. x x\in\mathbb{Z}
  110. ( x N ) = - 1 (\frac{x}{N})=-1
  111. ( x 3 - y 2 N ) = 1 (\frac{x^{3}-y^{2}}{N})=1
  112. m = x 3 - y 2 x ( mod N ) m=\frac{x^{3}-y^{2}}{x}\;\;(\mathop{{\rm mod}}N)
  113. y 2 = x 3 - m x y^{2}=x^{3}-mx
  114. m 0 ( mod N ) m\not\equiv 0\;\;(\mathop{{\rm mod}}N)
  115. Q = P Q=P_{\infty}
  116. S i S_{i}
  117. S i S_{i}
  118. gcd ( S i , N ) > 1 \gcd({S_{i},N})>1
  119. 1 i k - 1 1\leq i\leq k-1
  120. S k 0 ( mod N ) S_{k}\equiv 0\;\;(\mathop{{\rm mod}}N)
  121. ( m N ) = ( x 3 - y 2 x N ) = ( x N ) ( x 3 - y 2 N ) = - 1 1 = - 1. \left(\frac{m}{N}\right)=\left(\frac{\frac{x^{3}-y^{2}}{x}}{N}\right)=\left(% \frac{x}{N}\right)\left(\frac{x^{3}-y^{2}}{N}\right)=-1\cdot 1=-1.
  122. 2 k 2^{k}
  123. 2 k d 2^{k}d
  124. 2 k 2^{k}
  125. 2 k 2^{k}
  126. S 1 S_{1}
  127. 2 k n - 1 2^{k}n-1

Elliptical_distribution.html

  1. e i t μ Ψ ( t Σ t ) e^{it^{\prime}\mu}\Psi(t^{\prime}\Sigma t)\,
  2. μ \mu
  3. Σ \Sigma
  4. Ψ \Psi
  5. Ψ \Psi
  6. f ( x ) = k g ( ( x - μ ) Σ - 1 ( x - μ ) ) f(x)=k\cdot g((x-\mu)^{\prime}\Sigma^{-1}(x-\mu))
  7. k k
  8. x x
  9. n n
  10. μ \mu
  11. Σ \Sigma
  12. g g
  13. f ( x ) f(x)
  14. g ( z ) = e - z / 2 g(z)=e^{-z/2}
  15. x x
  16. e - z / 2 > 0 e^{-z/2}>0
  17. z z
  18. g ( z ) = 0 g(z)=0
  19. z z
  20. μ . \mu.

ELSV_formula.html

  1. λ g \lambda_{g}
  2. h g ; k 1 , , k n h_{g;k_{1},\dots,k_{n}}
  3. h g ; k 1 , , k n = m ! i = 1 n k i k i k i ! ¯ g , n c ( E * ) ( 1 - k 1 ψ 1 ) ( 1 - k n ψ n ) . h_{g;k_{1},\dots,k_{n}}=m!\prod_{i=1}^{n}\frac{k_{i}^{k_{i}}}{k_{i}!}\int_{% \overline{\mathcal{M}}_{g,n}}\frac{c(E^{*})}{(1-k_{1}\psi_{1})\cdots(1-k_{n}% \psi_{n})}.
  4. m = k i + n + 2 g - 2 m=\sum k_{i}+n+2g-2
  5. ¯ g , n \overline{\mathcal{M}}_{g,n}
  6. h g ; k 1 , , k n h_{g;k_{1},\dots,k_{n}}
  7. h g ; k 1 , , k n h_{g;k_{1},\dots,k_{n}}
  8. ( τ 1 , , τ m , σ ) (\tau_{1},\dots,\tau_{m},\sigma)
  9. τ 1 τ m σ \tau_{1}\cdots\tau_{m}\sigma
  10. τ 1 , , τ n \tau_{1},\dots,\tau_{n}
  11. h g ; k 1 , , k n h_{g;k_{1},\dots,k_{n}}
  12. h g ; k h_{g;k}
  13. ( τ 1 , , τ k + 2 g - 1 ) (\tau_{1},\dots,\tau_{k+2g-1})
  14. h g ; k h_{g;k}
  15. ¯ g , n {\overline{\mathcal{M}}}_{g,n}
  16. ¯ g , n {\overline{\mathcal{M}}}_{g,n}
  17. λ j = c j ( E ) H 2 j ( ¯ g , n , 𝐐 ) . \lambda_{j}=c_{j}(E)\in H^{2j}({\overline{\mathcal{M}}}_{g,n},\mathbf{Q}).
  18. c ( E * ) = 1 - λ 1 + λ 2 - + ( - 1 ) g λ g . c(E^{*})=1-\lambda_{1}+\lambda_{2}-\cdots+(-1)^{g}\lambda_{g}.
  19. 1 \mathcal{L}_{1}
  20. n \mathcal{L}_{n}
  21. ¯ g , n {\overline{\mathcal{M}}}_{g,n}
  22. i \mathcal{L}_{i}
  23. i \mathcal{L}_{i}
  24. ψ i = c 1 ( i ) H 2 ( ¯ g , n , 𝐐 ) . \psi_{i}=c_{1}(\mathcal{L}_{i})\in H^{2}({\overline{\mathcal{M}}}_{g,n},% \mathbf{Q}).
  25. 1 / ( 1 - k i ψ i ) 1/(1-k_{i}\psi_{i})
  26. 1 + k i ψ i + k i 2 ψ i 2 + 1+k_{i}\psi_{i}+k_{i}^{2}\psi_{i}^{2}+\cdots
  27. ¯ g , n c ( E * ) ( 1 - k 1 ψ 1 ) ( 1 - k n ψ n ) \int_{\overline{\mathcal{M}}_{g,n}}\frac{c(E^{*})}{(1-k_{1}\psi_{1})\cdots(1-k% _{n}\psi_{n})}
  28. k 1 d 1 k n d n k_{1}^{d_{1}}\cdots k_{n}^{d_{n}}
  29. ¯ g , n ( - 1 ) j λ j ψ 1 d 1 ψ n d n , \int_{\overline{\mathcal{M}}_{g,n}}(-1)^{j}\lambda_{j}\psi_{1}^{d_{1}}\cdots% \psi_{n}^{d_{n}},
  30. j = 3 g - 3 - n - d i j=3g-3-n-\sum d_{i}
  31. h g ; k 1 , , k n m ! i = 1 n k i ! k i k i \frac{h_{g;k_{1},\dots,k_{n}}}{m!}\prod_{i=1}^{n}\frac{k_{i}!}{k_{i}^{k_{i}}}
  32. ¯ g , n c ( E * ) ( 1 - k 1 ψ 1 ) ( 1 - k n ψ n ) = ¯ 1 , 1 1 - λ 1 1 - k 1 ψ 1 = [ ¯ 1 , 1 ψ 1 ] k 1 - [ ¯ 1 , 1 λ 1 ] . \int_{\overline{\mathcal{M}}_{g,n}}\frac{c(E^{*})}{(1-k_{1}\psi_{1})\cdots(1-k% _{n}\psi_{n})}=\int_{\overline{\mathcal{M}}_{1,1}}\frac{1-\lambda_{1}}{1-k_{1}% \psi_{1}}=\left[\int_{\overline{\mathcal{M}}_{1,1}}\psi_{1}\right]k_{1}-\left[% \int_{\overline{\mathcal{M}}_{1,1}}\lambda_{1}\right].
  33. h 1 ; k = ( k + 1 ) ! k k k ! ¯ 1 , 1 1 - λ 1 1 - k ψ 1 = ( k + 1 ) k k { [ ¯ 1 , 1 ψ 1 ] k - [ ¯ 1 , 1 λ 1 ] } . h_{1;k}=(k+1)!\frac{k^{k}}{k!}\int_{\overline{\mathcal{M}}_{1,1}}\frac{1-% \lambda_{1}}{1-k\psi_{1}}=(k+1)k^{k}\left\{\left[\int_{\overline{\mathcal{M}}_% {1,1}}\psi_{1}\right]k-\left[\int_{\overline{\mathcal{M}}_{1,1}}\lambda_{1}% \right]\right\}.
  34. ¯ 1 , 1 ψ 1 = ¯ 1 , 1 λ 1 = 1 24 . \int_{\overline{\mathcal{M}}_{1,1}}\psi_{1}=\int_{\overline{\mathcal{M}}_{1,1}% }\lambda_{1}=\frac{1}{24}.
  35. h 1 ; k = ( k 2 - 1 ) k k 24 . h_{1;k}=\frac{(k^{2}-1)k^{k}}{24}.
  36. ¯ g , n {\overline{\mathcal{M}}}_{g,n}
  37. g ; k 1 , , k n \mathcal{M}_{g;k_{1},\dots,k_{n}}
  38. k 1 k n k_{1}\dots k_{n}
  39. f g ; k 1 , , k n f\in\mathcal{M}_{g;k_{1},\dots,k_{n}}
  40. h g ; k 1 , , k n h_{g;k_{1},\dots,k_{n}}
  41. g ; k 1 , , k n \mathcal{M}_{g;k_{1},\dots,k_{n}}
  42. ¯ g , n \overline{\mathcal{M}}_{g,n}
  43. z z k 1 , , z z k n z\mapsto z^{k_{1}},\dots,z\mapsto z^{k_{n}}
  44. g ; k 1 , , k n \mathcal{M}_{g;k_{1},\dots,k_{n}}

EMF_measurement.html

  1. | E | = E x 2 + E y 2 + E z 2 |E|=\sqrt{E_{x}^{2}+E_{y}^{2}+E_{z}^{2}}
  2. | H | = H x 2 + H y 2 + H z 2 |H|=\sqrt{H_{x}^{2}+H_{y}^{2}+H_{z}^{2}}
  3. | E | = E x 2 + E y 2 + E z 2 |E|=\sqrt{E_{x}^{2}+E_{y}^{2}+E_{z}^{2}}
  4. f ( θ , ϕ ) = A sin ( θ ) f(\theta,\phi)=A\cdot\sin(\theta)

Endoreversible_thermodynamics.html

  1. η = 1 - T L T H \eta=1-\sqrt{\frac{T_{L}}{T_{H}}}
  2. T c T_{c}
  3. T h T_{h}
  4. η \eta
  5. η \eta
  6. η \eta
  7. T H T L \sqrt{T_{H}T_{L}}
  8. T H T_{H}
  9. T H T L \sqrt{T_{H}T_{L}}

Energy_factor.html

  1. E F = i = 1 6 M i C p i ( 135 F - 58 F ) Q d m , EF=\sum\limits_{i=1}^{6}\frac{M_{i}C_{pi}\left(135^{\circ}F-58^{\circ}F\right)% }{Q_{dm}},
  2. Q d m Q_{dm}
  3. M i M_{i}
  4. C p i C_{pi}

Energy_operator.html

  1. E ^ = i t \hat{E}=i\hbar\frac{\partial}{\partial t}\,\!
  2. Ψ ( 𝐫 , t ) \Psi\left(\mathbf{r},t\right)\,\!
  3. E = H = T + V E=H=T+V\,\!
  4. E ^ = H ^ E ^ Ψ = H ^ Ψ \begin{aligned}&\displaystyle\hat{E}=\hat{H}\\ &\displaystyle\hat{E}\Psi=\hat{H}\Psi\\ \end{aligned}\,\!
  5. i t Ψ ( 𝐫 , t ) = H ^ Ψ ( 𝐫 , t ) i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=\hat{H}\Psi(\mathbf{r},t% )\,\!
  6. H ^ \hat{H}
  7. E 2 = ( p c ) 2 + ( m c 2 ) 2 E^{2}=(pc)^{2}+(mc^{2})^{2}\,\!
  8. E ^ 2 = c 2 p ^ 2 + ( m c 2 ) 2 E ^ 2 Ψ = c 2 p ^ 2 Ψ + ( m c 2 ) 2 Ψ \begin{aligned}&\displaystyle\hat{E}^{2}=c^{2}\hat{p}^{2}+(mc^{2})^{2}\\ &\displaystyle\hat{E}^{2}\Psi=c^{2}\hat{p}^{2}\Psi+(mc^{2})^{2}\Psi\\ \end{aligned}\,\!
  9. 2 Ψ t 2 = c 2 2 Ψ - ( m c 2 ) 2 Ψ \frac{\partial^{2}\Psi}{\partial t^{2}}=c^{2}\nabla^{2}\Psi-\left(\frac{mc^{2}% }{\hbar}\right)^{2}\Psi\,\!
  10. Ψ = e i ( k x - ω t ) \Psi=e^{i(kx-\omega t)}\,\!
  11. Ψ t = - i ω e i ( k x - ω t ) = - i ω Ψ \frac{\partial\Psi}{\partial t}=-i\omega e^{i(kx-\omega t)}=-i\omega\Psi\,\!
  12. E = ω E=\hbar\omega\,\!
  13. Ψ t = - i E Ψ \frac{\partial\Psi}{\partial t}=-i\frac{E}{\hbar}\Psi\,\!
  14. E Ψ = i Ψ t E\Psi=i\hbar\frac{\partial\Psi}{\partial t}\,\!
  15. E = i t E=i\hbar\frac{\partial}{\partial t}\,\!
  16. E ^ = i t \hat{E}=i\hbar\frac{\partial}{\partial t}\,\!
  17. E ^ \hat{E}\,\!
  18. E ^ Ψ = i t Ψ = E Ψ \hat{E}\Psi=i\hbar\frac{\partial}{\partial t}\Psi=E\Psi\,\!
  19. Ψ = e i ( k r - ω t ) \Psi=e^{i({k}\cdot{r}-\omega t)}\,\!

Ensemble_learning.html

  1. y = argmax c j C h i H P ( c j | h i ) P ( T | h i ) P ( h i ) y=\mathrm{argmax}_{c_{j}\in C}\sum_{h_{i}\in H}{P(c_{j}|h_{i})P(T|h_{i})P(h_{i% })}
  2. y y
  3. C C
  4. H H
  5. P P
  6. T T
  7. H H
  8. H H
  9. argmax \mathrm{argmax}
  10. P ( c j | h i ) P(c_{j}|h_{i})
  11. P ( T | h i ) P(T|h_{i})
  12. P ( h i ) P(h_{i})
  13. P ( T | H ) P(T|H)

Entanglement-assisted_stabilizer_formalism.html

  1. Π n \Pi^{n}
  2. n n
  3. 𝒮 Π n \mathcal{S}\subset\Pi^{n}
  4. n - k = 2 c + s n-k=2c+s
  5. { Z ¯ 1 , , Z ¯ s + c , X ¯ s + 1 , , X ¯ s + c } \left\{\bar{Z}_{1},\ldots,\bar{Z}_{s+c},\bar{X}_{s+1},\ldots,\bar{X}_{s+c}\right\}
  6. 𝒮 \mathcal{S}
  7. [ Z ¯ i , Z ¯ j ] = 0 i , j , \left[\bar{Z}_{i},\bar{Z}_{j}\right]=0\ \ \ \ \ \forall i,j,
  8. [ X ¯ i , X ¯ j ] = 0 i , j , \left[\bar{X}_{i},\bar{X}_{j}\right]=0\ \ \ \ \ \forall i,j,
  9. [ X ¯ i , Z ¯ j ] = 0 i j , \left[\bar{X}_{i},\bar{Z}_{j}\right]=0\ \ \ \ \ \forall i\neq j,
  10. { X ¯ i , Z ¯ i } = 0 i . \left\{\bar{X}_{i},\bar{Z}_{i}\right\}=0\ \ \ \ \ \forall i.
  11. 𝒮 \mathcal{S}
  12. s s
  13. c c
  14. + 1 +1
  15. { X X , Z Z } \left\{XX,ZZ\right\}
  16. { X , Z } \left\{X,Z\right\}
  17. { X X , Z Z } \left\{XX,ZZ\right\}
  18. 𝒮 \mathcal{S}
  19. 𝒮 I \mathcal{S}_{I}
  20. 𝒮 E \mathcal{S}_{E}
  21. 𝒮 I \mathcal{S}_{I}
  22. 𝒮 \mathcal{S}
  23. 𝒮 I = { Z ¯ 1 , , Z ¯ s } \mathcal{S}_{I}=\left\{\bar{Z}_{1},\ldots,\bar{Z}_{s}\right\}
  24. 𝒮 E \mathcal{S}_{E}
  25. 𝒮 E = { Z ¯ s + 1 , , Z ¯ s + c , X ¯ s + 1 , , X ¯ s + c } \mathcal{S}_{E}=\left\{\bar{Z}_{s+1},\ldots,\bar{Z}_{s+c},\bar{X}_{s+1},\ldots% ,\bar{X}_{s+c}\right\}
  26. 𝒮 I \mathcal{S}_{I}
  27. 𝒮 E \mathcal{S}_{E}
  28. Π n \mathcal{E}\subset\Pi^{n}
  29. E 1 , E 2 E_{1},E_{2}\in\mathcal{E}
  30. E 1 E 2 𝒮 I ( Π n - 𝒵 ( 𝒮 I , 𝒮 E ) ) . E_{1}^{\dagger}E_{2}\in\mathcal{S}_{I}\cup\left(\Pi^{n}-\mathcal{Z}\left(\left% \langle\mathcal{S}_{I},\mathcal{S}_{E}\right\rangle\right)\right).
  31. { Z 1 , , Z s , Z s + 1 | Z 1 , , Z s + c | Z c , X s + 1 | X 1 , , X s + c | X c } . \left\{Z_{1},\ldots,Z_{s},Z_{s+1}|Z_{1},\ldots,Z_{s+c}|Z_{c},X_{s+1}|X_{1},% \ldots,X_{s+c}|X_{c}\right\}.
  32. { Z ¯ 1 , , Z ¯ s , Z ¯ s + 1 | Z 1 , , Z ¯ s + c | Z c , X ¯ s + 1 | X 1 , , X ¯ s + c | X c } . \left\{\bar{Z}_{1},\ldots,\bar{Z}_{s},\bar{Z}_{s+1}|Z_{1},\ldots,\bar{Z}_{s+c}% |Z_{c},\bar{X}_{s+1}|X_{1},\ldots,\bar{X}_{s+c}|X_{c}\right\}.
  33. k k
  34. n n
  35. c c
  36. k / n k/n
  37. ( k / n , c / n ) \left(k/n,c/n\right)
  38. c c
  39. k k
  40. n n
  41. [ n , k ; c ] \left[n,k;c\right]
  42. ( k - c ) / n \left(k-c\right)/n
  43. n n
  44. k k
  45. c c
  46. Π 4 \Pi^{4}
  47. Z X Z I Z Z I Z X Y X I X X I X \begin{array}[c]{cccc}Z&X&Z&I\\ Z&Z&I&Z\\ X&Y&X&I\\ X&X&I&X\end{array}
  48. g 1 = Z X Z I g 2 = Z Z I Z g 3 = Y X X Z g 4 = Z Y Y X \begin{array}[c]{cccccc}g_{1}&=&Z&X&Z&I\\ g_{2}&=&Z&Z&I&Z\\ g_{3}&=&Y&X&X&Z\\ g_{4}&=&Z&Y&Y&X\end{array}
  49. { g 1 , g 2 } = [ g 1 , g 3 ] = [ g 1 , g 4 ] = [ g 2 , g 3 ] = [ g 2 , g 4 ] = [ g 3 , g 4 ] = 0. \left\{g_{1},g_{2}\right\}=\left[g_{1},g_{3}\right]=\left[g_{1},g_{4}\right]=% \left[g_{2},g_{3}\right]=\left[g_{2},g_{4}\right]=\left[g_{3},g_{4}\right]=0.
  50. X I I I Z I I I I Z I I I I Z I \begin{array}[c]{cccc}X&I&I&I\\ Z&I&I&I\\ I&Z&I&I\\ I&I&Z&I\end{array}
  51. X Z I I | X I I I Z I I I I Z I I I I Z I \begin{array}[c]{c}X\\ Z\\ I\\ I\end{array}\left|\begin{array}[c]{cccc}X&I&I&I\\ Z&I&I&I\\ I&Z&I&I\\ I&I&Z&I\end{array}\right.
  52. | Φ + B A | 00 A | ψ A . \left|\Phi^{+}\right\rangle^{BA}\left|00\right\rangle^{A}\left|\psi\right% \rangle^{A}.
  53. | ψ A \left|\psi\right\rangle^{A}
  54. X Z I I | Z X Z I Z Z I Z Y X X Z Z Y Y X \begin{array}[c]{c}X\\ Z\\ I\\ I\end{array}\left|\begin{array}[c]{cccc}Z&X&Z&I\\ Z&Z&I&Z\\ Y&X&X&Z\\ Z&Y&Y&X\end{array}\right.
  55. H = [ 1 0 1 0 1 1 0 1 0 1 0 0 0 0 0 0 | 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 ] . H=\left[\left.\begin{array}[c]{cccc}1&0&1&0\\ 1&1&0&1\\ 0&1&0&0\\ 0&0&0&0\end{array}\right|\begin{array}[c]{cccc}0&1&0&0\\ 0&0&0&0\\ 1&1&1&0\\ 1&1&0&1\end{array}\right].
  56. Z Z
  57. X X
  58. Π n \Pi^{n}
  59. i i
  60. j j
  61. i i
  62. j j
  63. X X
  64. j j
  65. i i
  66. Z Z
  67. i i
  68. i i
  69. Z Z
  70. i i
  71. X X
  72. i i
  73. i i
  74. X X
  75. i i
  76. Z Z
  77. i i
  78. j j
  79. i i
  80. j j
  81. X X
  82. Z Z
  83. X X
  84. [ 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 | 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 ] . \left[\left.\begin{array}[c]{cccc}0&1&1&0\\ 1&1&0&1\\ 1&0&0&0\\ 0&0&0&0\end{array}\right|\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 1&1&1&0\\ 1&1&0&1\end{array}\right].
  85. X X
  86. Z Z
  87. Z Z
  88. Z Z
  89. [ 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 | 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 ] . \left[\left.\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&1\\ 1&1&1&0\\ 0&1&0&0\end{array}\right|\begin{array}[c]{cccc}1&1&1&0\\ 0&1&0&0\\ 1&0&0&0\\ 1&0&0&1\end{array}\right].
  90. [ 0 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 | 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 1 ] . \left[\left.\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&1\\ 1&1&1&0\\ 1&1&0&0\end{array}\right|\begin{array}[c]{cccc}1&0&0&0\\ 0&1&0&0\\ 1&1&1&0\\ 1&1&1&1\end{array}\right].
  91. [ 1 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 | 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 1&1&1&0\\ 1&1&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&1&0&1\\ 1&1&1&0\\ 1&1&1&0\end{array}\right].
  92. [ 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 1 | 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 1 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&1&1&0\\ 1&1&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 1&0&1&1\\ 1&0&1&1\end{array}\right].
  93. Z Z
  94. X X
  95. [ 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 | 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&1&1&0\\ 0&1&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&0&1&1\\ 0&0&1&1\end{array}\right].
  96. [ 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 | 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&1&1&1\\ 0&1&1&1\end{array}\right].
  97. [ 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 | 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&1&1&0\\ 0&1&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&1&0&0\\ 0&1&0&0\end{array}\right].
  98. [ 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 | 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&1&0&0\\ 0&1&0&0\end{array}\right].
  99. [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&1&0&0\\ 0&1&1&0\end{array}\right].
  100. [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&1&0&0\\ 0&0&0&1\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&1&0\end{array}\right].
  101. [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ] . \left[\left.\begin{array}[c]{cccc}1&0&0&0\\ 0&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{array}\right|\begin{array}[c]{cccc}0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right].

Enthalpy_of_fusion.html

  1. ( x 2 ) (x_{2})
  2. ( T 𝑓𝑢𝑠 ) (T_{\mathit{fus}})
  3. ln x 2 = - Δ H 𝑓𝑢𝑠 R ( 1 T - 1 T 𝑓𝑢𝑠 ) \ln x_{2}=-\frac{\Delta H^{\circ}_{\mathit{fus}}}{R}\left(\frac{1}{T}-\frac{1}% {T_{\mathit{fus}}}\right)
  4. x 2 = exp ( - 28100 J mol - 1 8.314 J K mol - 1 - 1 ( 1 298 - 1 442 ) ) = 0.0248 x_{2}=\exp{\left(-\frac{28100\mbox{ J mol}~{}^{-1}}{8.314\mbox{ J K}~{}^{-1}% \mbox{ mol}~{}^{-1}}\left(\frac{1}{298}-\frac{1}{442}\right)\right)}=0.0248
  5. 0.0248 * 1000 g 18.053 mol - 1 1 - 0.0248 * 151.17 mol = - 1 213.4 \frac{0.0248*\frac{1000\mbox{ g}~{}}{18.053\mbox{ mol}~{}^{-1}}}{1-0.0248}*151% .17\mbox{ mol}~{}^{-1}=213.4
  6. μ s o l i d = μ s o l u t i o n \mu^{\circ}_{solid}=\mu^{\circ}_{solution}\,
  7. μ s o l i d = μ l i q u i d + R T ln X 2 \mu^{\circ}_{solid}=\mu^{\circ}_{liquid}+RT\ln X_{2}\,
  8. R R\,
  9. T T\,
  10. R T ln X 2 = - ( μ l i q u i d - μ s o l i d ) RT\ln X_{2}=-(\mu^{\circ}_{liquid}-\mu^{\circ}_{solid})\,
  11. Δ G 𝑓𝑢𝑠 = - ( μ l i q u i d - μ s o l i d ) \Delta G^{\circ}_{\mathit{fus}}=-(\mu^{\circ}_{liquid}-\mu^{\circ}_{solid})\,
  12. R T ln X 2 = - ( Δ G 𝑓𝑢𝑠 ) RT\ln X_{2}=-(\Delta G^{\circ}_{\mathit{fus}})\,
  13. ( ( Δ G 𝑓𝑢𝑠 T ) T ) p = - Δ H 𝑓𝑢𝑠 T 2 \left(\frac{\partial(\frac{\Delta G^{\circ}_{\mathit{fus}}}{T})}{\partial T}% \right)_{p\,}=-\frac{\Delta H^{\circ}_{\mathit{fus}}}{T^{2}}
  14. ( ( ln X 2 ) T ) = Δ H 𝑓𝑢𝑠 R T 2 \left(\frac{\partial(\ln X_{2})}{\partial T}\right)=\frac{\Delta H^{\circ}_{% \mathit{fus}}}{RT^{2}}
  15. ln X 2 = Δ H 𝑓𝑢𝑠 R T 2 * δ T \partial\ln X_{2}=\frac{\Delta H^{\circ}_{\mathit{fus}}}{RT^{2}}*\delta T
  16. x 2 = 1 x 2 = x 2 δ ln X 2 = ln x 2 = T 𝑓𝑢𝑠 T Δ H 𝑓𝑢𝑠 R T 2 * Δ T \int^{x_{2}=x_{2}}_{x_{2}=1}\delta\ln X_{2}=\ln x_{2}=\int_{T_{\mathit{fus}}}^% {T}\frac{\Delta H^{\circ}_{\mathit{fus}}}{RT^{2}}*\Delta T
  17. ln x 2 = - Δ H 𝑓𝑢𝑠 R ( 1 T - 1 T 𝑓𝑢𝑠 ) \ln x_{2}=-\frac{\Delta H^{\circ}_{\mathit{fus}}}{R}\left(\frac{1}{T}-\frac{1}% {T_{\mathit{fus}}}\right)

Entropy_exchange.html

  1. ϕ \phi\,
  2. ρ Q \rho_{Q}\,
  3. Q Q\,
  4. S ( ρ , ϕ ) S [ Q , R ] = S ( ρ Q R ) S(\rho,\phi)\equiv S[Q^{\prime},R^{\prime}]=S(\rho_{QR}^{\prime})
  5. S ( ρ Q R ) S(\rho_{QR}^{\prime})\,
  6. Q Q\,
  7. R R\,
  8. ϕ \phi\,
  9. ρ Q R = | Q R Q R | \rho_{QR}=|QR\rangle\langle QR|\quad
  10. Tr R [ ρ Q R ] = ρ Q \mathrm{Tr}_{R}[\rho_{QR}]=\rho_{Q}\quad
  11. ρ Q R = ϕ [ ρ Q R ] \rho_{QR}^{\prime}=\phi[\rho_{QR}]\quad

Enumerations_of_specific_permutation_classes.html

  1. ( n 2 ) + 1 {n\choose 2}+1
  2. 2 n - 1 2^{n-1}

Envelope_(waves).html

  1. F ( x , t ) = sin [ 2 π ( x λ - Δ λ - ( f + Δ f ) t ) ] F(x,\ t)=\sin\left[2\pi\left(\frac{x}{\lambda-\Delta\lambda}-(f+\Delta f)t% \right)\right]
  2. + sin [ 2 π ( x λ + Δ λ - ( f - Δ f ) t ) ] +\sin\left[2\pi\left(\frac{x}{\lambda+\Delta\lambda}-(f-\Delta f)t\right)\right]
  3. 2 cos [ 2 π ( x λ m o d - Δ f t ) ] sin [ 2 π ( x λ - f t ) ] , \approx 2\cos\left[2\pi\left(\frac{x}{\lambda_{mod}}-\Delta f\ t\right)\right]% \ \sin\left[2\pi\left(\frac{x}{\lambda}-f\ t\right)\right]\ ,
  4. λ m o d = λ 2 Δ λ . \lambda_{mod}=\frac{\lambda^{2}}{\Delta\lambda}\ .
  5. ξ C = ( x λ - f t ) , \xi_{C}=\left(\frac{x}{\lambda}-f\ t\right)\ ,
  6. ξ E = ( x λ m o d - Δ f t ) , \xi_{E}=\left(\frac{x}{\lambda_{mod}}-\Delta f\ t\right)\ ,
  7. ( x λ - f t ) = ( x + Δ x λ - f ( t + Δ t ) ) , \left(\frac{x}{\lambda}-f\ t\right)=\left(\frac{x+\Delta x}{\lambda}-f(t+% \Delta t)\right)\ ,
  8. v p = Δ x Δ t = λ f . v_{p}=\frac{\Delta x}{\Delta t}=\lambda f\ .
  9. v g = Δ x Δ t = λ m o d Δ f = λ 2 Δ f Δ λ . v_{g}=\frac{\Delta x}{\Delta t}=\lambda_{mod}\Delta f=\lambda^{2}\frac{\Delta f% }{\Delta\lambda}\ .
  10. k = 2 π λ . k=\frac{2\pi}{\lambda}\ .
  11. Δ k = | d k d λ | Δ λ = 2 π Δ λ λ 2 , \Delta k=\left|\frac{dk}{d\lambda}\right|\Delta\lambda=2\pi\frac{\Delta\lambda% }{\lambda^{2}}\ ,
  12. v g = 2 π Δ f Δ k = Δ ω Δ k , v_{g}=\frac{2\pi\Delta f}{\Delta k}=\frac{\Delta\omega}{\Delta k}\ ,
  13. v g = d ω ( k ) d k . v_{g}=\frac{d\omega(k)}{dk}\ .
  14. ω = c 0 k \omega=c_{0}k
  15. ψ n 𝐤 ( 𝐫 ) = e i 𝐤 𝐫 u n 𝐤 ( 𝐫 ) , \psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(% \mathbf{r})\ ,
  16. ψ ( 𝐫 ) = 𝐤 F ( 𝐤 ) e i 𝐤 𝐫 u 𝐤 ( 𝐫 ) , \psi(\mathbf{r})=\sum_{\mathbf{k}}F(\mathbf{k})e^{i\mathbf{k\cdot r}}u_{% \mathbf{k}}(\mathbf{r})\ ,
  17. ψ ( 𝐫 ) ( 𝐤 F ( 𝐤 ) e i 𝐤 𝐫 ) u 𝐤 = 𝐤 𝟎 ( 𝐫 ) = F ( 𝐫 ) u 𝐤 = 𝐤 𝟎 ( 𝐫 ) . \psi(\mathbf{r})\approx\left(\sum_{\mathbf{k}}F(\mathbf{k})e^{i\mathbf{k\cdot r% }}\right)u_{\mathbf{k=k_{0}}}(\mathbf{r})=F(\mathbf{r})u_{\mathbf{k=k_{0}}}(% \mathbf{r})\ .
  18. I 1 = I 0 sin 2 ( π d sin α λ ) / ( π d sin α λ ) 2 , I_{1}=I_{0}\sin^{2}\left(\frac{\pi d\sin\alpha}{\lambda}\right)/\left(\frac{% \pi d\sin\alpha}{\lambda}\right)^{2}\ ,
  19. I q = I 1 sin 2 ( q π g sin α λ ) / sin 2 ( π g sin α λ ) , I_{q}=I_{1}\sin^{2}\left(\frac{q\pi g\sin\alpha}{\lambda}\right)/\sin^{2}\left% (\frac{\pi g\sin\alpha}{\lambda}\right)\ ,

Epicyclic_frequency.html

  1. Ω \Omega
  2. κ \kappa
  3. κ 2 2 Ω R d d R ( R 2 Ω ) \kappa^{2}\equiv\frac{2\Omega}{R}\frac{d}{dR}(R^{2}\Omega)
  4. κ 2 \kappa^{2}
  5. 6 G M / c 2 6GM/c^{2}
  6. κ = Ω \kappa=\Omega

Equatorial_Rossby_wave.html

  1. f = β y f=\beta y
  2. β = f y \beta=\frac{\partial f}{\partial y}
  3. ϕ t + c 2 ( v y + u x ) = 0 \frac{\partial\phi}{\partial t}+c^{2}\left(\frac{\partial v}{\partial y}+\frac% {\partial u}{\partial x}\right)=0
  4. D u D t - v β y = - ϕ x \frac{Du}{Dt}-v\beta y=-\frac{\partial\phi}{\partial x}
  5. D v D t + u β y = - ϕ y \frac{Dv}{Dt}+u\beta y=-\frac{\partial\phi}{\partial y}
  6. { u , v , ϕ } = { ϕ ^ } e i ( k x + l y - ω t ) \begin{Bmatrix}u,v,\phi\end{Bmatrix}=\begin{Bmatrix}\hat{\phi}\end{Bmatrix}e^{% i(kx+ly-\omega t)}
  7. ω = - β k / ( k 2 + ( 2 n + 1 ) β / c ) \omega=-\beta k/(k^{2}+(2n+1)\beta/c)
  8. c 2 = g H c^{2}=gH
  9. ω / k = - c / ( 2 n + 1 ) \omega/k=-c/(2n+1)
  10. ω = - β / k , \omega=-\beta/k,\,
  11. c g = β / k 2 c_{g}=\beta/k^{2}
  12. D D t β y + ζ H = 0 \frac{D}{Dt}\frac{\beta y+\zeta}{H}=0
  13. e i ( k x + m z - ω t ) e^{i(kx+mz-\omega t)}

Equilibrium_chemistry.html

  1. \rightharpoonup
  2. \leftharpoondown
  3. δ G r = ( G ξ ) T , P ; δ G r ( E q ) = 0 \delta G_{r}=\left(\frac{\partial G}{\partial\xi}\right)_{T,P};\delta G_{r}(Eq% )=0
  4. μ i = ( G N i ) T , P \mu_{i}=\left(\frac{\partial G}{\partial N_{i}}\right)_{T,P}
  5. j n j R e a c t a n t j k m k P r o d u c t k \sum_{j}n_{j}Reactant_{j}\rightleftharpoons\sum_{k}m_{k}Product_{k}
  6. δ G r = k m k μ k - j n j μ j \delta G_{r}=\sum_{k}m_{k}\mu_{k}\,-\sum_{j}n_{j}\mu_{j}
  7. μ i = μ i + R T ln a i \mu_{i}=\mu_{i}^{\ominus}+RT\ln a_{i}
  8. j n j ( μ j + R T ln a j ) = k m k ( μ k + R T ln a k ) \sum_{j}n_{j}(\mu_{j}^{\ominus}+RT\ln a_{j})=\sum_{k}m_{k}(\mu_{k}^{\ominus}+% RT\ln a_{k})
  9. k m k μ k - j n j μ j = - R T ( k ln a k m k - j ln a j n j ) \sum_{k}m_{k}\mu_{k}^{\ominus}-\sum_{j}n_{j}\mu_{j}^{\ominus}=-RT\left(\sum_{k% }\ln{a_{k}}^{m_{k}}-\sum_{j}\ln{a_{j}}^{n_{j}}\right)
  10. Δ G = - R T l n K . \Delta G^{\ominus}=-RTlnK.
  11. Δ G = k m k μ k - j n j μ j \Delta G^{\ominus}=\sum_{k}m_{k}\mu_{k}^{\ominus}-\sum_{j}n_{j}\mu_{j}^{\ominus}
  12. ln K = k ln a k m k - j ln a j n j ; K = k a k m k j a j n j \ln K=\sum_{k}\ln{a_{k}}^{m_{k}}-\sum_{j}\ln{a_{j}}^{n_{j}};K=\frac{\prod_{k}{% a_{k}}^{m_{k}}}{\prod_{j}{a_{j}}^{n_{j}}}
  13. a i = [ A i ] γ i a_{i}=[A_{i}]\gamma_{i}
  14. K = k a k m k j a j n j = k ( [ A k ] γ k ) m k j ( [ A j ] γ j ) n j = k [ A k ] m k j [ A j ] n j × k γ k m k j γ j n j = k [ A k ] m k j [ A j ] n j × Γ K=\frac{\prod_{k}{a_{k}}^{m_{k}}}{\prod_{j}{a_{j}}^{n_{j}}}=\frac{\prod_{k}% \left([A_{k}]\gamma_{k}\right)^{m_{k}}}{\prod_{j}\left([A_{j}]\gamma_{j}\right% )^{n_{j}}}=\frac{\prod_{k}[A_{k}]^{m_{k}}}{\prod_{j}[A_{j}]^{n_{j}}}\times% \frac{\prod_{k}{\gamma_{k}}^{m_{k}}}{\prod_{j}{\gamma_{j}}^{n_{j}}}=\frac{% \prod_{k}[A_{k}]^{m_{k}}}{\prod_{j}[A_{j}]^{n_{j}}}\times\Gamma
  15. K = k [ A k ] m k j [ A j ] n j K=\frac{\prod_{k}[A_{k}]^{m_{k}}}{\prod_{j}[A_{j}]^{n_{j}}}
  16. α A + β B σ S + τ T \alpha A+\beta B...\rightleftharpoons\sigma S+\tau T...
  17. K = [ S ] σ [ T ] τ [ A ] α [ B ] β K=\frac{[S]^{\sigma}[T]^{\tau}...}{[A]^{\alpha}[B]^{\beta}...}
  18. d ln K d T = Δ H R T 2 or d ln K d ( 1 / T ) = - Δ H R \frac{d\ln K}{dT}\ =\frac{\Delta H^{\ominus}}{RT^{2}}\mbox{ or }~{}\frac{d\ln K% }{d(1/T)}\ =-\frac{\Delta H^{\ominus}}{R}
  19. ( H T ) p = C p \left(\frac{\partial H}{\partial T}\right)_{p}=C_{p}
  20. μ = μ + R T ln f p \mu=\mu^{\ominus}+RT\ln\frac{f}{p^{\ominus}}
  21. f = p Φ f=p\Phi
  22. N 2 + 3 H 2 2 N H 3 ; K = f N H 3 2 f N 2 f H 2 3 N_{2}+3H_{2}\leftrightharpoons 2NH_{3};K=\frac{{f_{NH_{3}}}^{2}}{f_{N_{2}}{f_{% H_{2}}}^{3}}
  23. A 2 - + H + H A - ; K 1 = [ H A - ] [ H + ] [ A 2 - ] A^{2-}+H^{+}\rightleftharpoons HA^{-};K_{1}=\frac{[HA^{-}]}{[H^{+}][A^{2-}]}
  24. H A - + H + H 2 A ; K 2 = [ H 2 A ] [ H + ] [ H A - ] HA^{-}+H^{+}\rightleftharpoons H_{2}A;K_{2}=\frac{[H_{2}A]}{[H^{+}][HA^{-}]}
  25. A 2 - + 2 H + H 2 A ; β 2 = [ H 2 A ] [ H + ] 2 [ A 2 - ] A^{2-}+2H^{+}\rightleftharpoons H_{2}A;\beta_{2}=\frac{[H_{2}A]}{[H^{+}]^{2}[A% ^{2-}]}
  26. α A + β B A α B β ; K α β = [ A α B β ] [ A ] α [ B ] β \alpha A+\beta B\ldots\rightleftharpoons A_{\alpha}B_{\beta}\ldots;K_{\alpha% \beta\ldots}=\frac{[A_{\alpha}B_{\beta}\ldots]}{[A]^{\alpha}[B]^{\beta}\ldots}
  27. T A = [ A ] + [ A α B β ] = [ A ] + ( α K α β [ A ] α [ B ] β ) T_{A}=[A]+\sum[A_{\alpha}B_{\beta}\ldots]=[A]+\sum\left(\alpha K_{\alpha\beta}% \ldots[A]^{\alpha}[B]^{\beta}\ldots\right)
  28. Δ \Delta
  29. T A ( i ) T_{A}(i)
  30. T B ( i ) T_{B}(i)
  31. U = i = n p i = 1 w i ( y i o b s e r v e d - y i c a l c u l a t e d ) 2 U=\sum^{i=1}_{i=np}w_{i}\left(y_{i}^{observed}-y_{i}^{calculated}\right)^{2}
  32. β p q r = [ M p L q H r ] [ M ] p [ L ] q [ H ] r \beta_{pqr}=\mathrm{\frac{[M_{p}L_{q}H_{r}]}{[M]^{p}[L]^{q}[H]^{r}}}
  33. F e 2 + + C e 4 + F e 3 + + C e 3 + ; K = [ F e 3 + ] [ C e 3 + ] [ F e 2 + ] [ C e 4 + ] Fe^{2+}+Ce^{4+}\rightleftharpoons Fe^{3+}+Ce^{3+};K=\frac{[Fe^{3+}][Ce^{3+}]}{% [Fe^{2+}][Ce^{4+}]}
  34. F e 3 + + e - F e 2 + Fe^{3+}+e^{-}\rightleftharpoons Fe^{2+}
  35. C e 4 + + e - C e 3 + Ce^{4+}+e^{-}\rightleftharpoons Ce^{3+}
  36. Δ G = - R T ln K \Delta G^{\ominus}=-RT\ln K\,
  37. Δ G = Δ G F e + Δ G C e \Delta G^{\ominus}=\Delta G^{\ominus}_{Fe}+\Delta G^{\ominus}_{Ce}
  38. Δ G F e = - n F E F e 0 ; Δ G C e = - n F E C e 0 \Delta G^{\ominus}_{Fe}=-nFE^{0}_{Fe};\Delta G^{\ominus}_{Ce}=-nFE^{0}_{Ce}
  39. Δ G = Δ G + R T ln Q \Delta G=\Delta G^{\ominus}+RT\ln Q
  40. E F e = E F e 0 + R T n F ln [ F e 3 + ] [ F e 2 + ] E_{Fe}=E_{Fe}^{0}+\frac{RT}{nF}\ln\frac{[Fe^{3+}]}{[Fe^{2+}]}
  41. E = E 0 - R T n F ln [ reduced species ] [ oxidized species ] E=E^{0}-\frac{RT}{nF}\ln\frac{[\,\text{reduced species}]}{[\,\text{oxidized % species}]}
  42. E = E 0 + R T n F ln [ oxidized species ] [ reduced species ] E=E^{0}+\frac{RT}{nF}\ln\frac{[\,\text{oxidized species}]}{[\,\text{reduced % species}]}
  43. α A + β B + n e - σ S + τ T \alpha A+\beta B...+ne^{-}\rightleftharpoons\sigma S+\tau T...
  44. E = E + R T n F ln { S } σ { T } τ { A } α { B } β E=E^{\ominus}+\frac{RT}{nF}\ln\frac{{\{S\}}^{\sigma}{\{T\}}^{\tau}...}{{\{A\}}% ^{\alpha}{\{B\}}^{\beta}...}
  45. l n K = - R T ln ( k a k m k ( s o l u t i o n ) a ( s o l i d ) ) lnK=-RT\ln\left(\frac{\sum_{k}{a_{k}}^{m_{k}}(solution)}{a(solid)}\right)
  46. l n K = - R T ln ( k a k m k ( s o l u t i o n ) ) lnK=-RT\ln\left(\sum_{k}{a_{k}}^{m_{k}}(solution)\right)
  47. K S P = k a k m k K_{SP}=\prod_{k}{{a_{k}}^{m_{k}}}
  48. K S P = [ N a + ] 2 [ S O 4 2 - ] K_{SP}=[Na^{+}]^{2}[SO_{4}^{2-}]
  49. log p = log [ s o l u t e ] organic phase [ s o l u t e ] aqueous phase \log p=\log\frac{[solute]_{\mbox{organic phase}}~{}}{[solute]_{\mbox{aqueous % phase}}~{}}
  50. K d = a s a m K_{d}=\frac{a_{s}}{a_{m}}
  51. ν ¯ \bar{\nu}
  52. ν ¯ 1 1 + f K d . \bar{\nu}\propto\frac{1}{1+fK_{d}}.

Equipollence_(geometry).html

  1. A B C D . AB\bumpeq CD.
  2. A B + B A 0. AB+BA\bumpeq 0.
  3. A B n . C D , AB\bumpeq n.CD,

Equitable_coloring.html

  1. 1 + n / 2 \scriptstyle 1+\lceil n/2\rceil
  2. 1 + Δ / 2 1+\lceil\Delta/2\rceil

Equivalent_oxide_thickness.html

  1. EOT = t high-k ( k SiO 2 k high-k ) \mathrm{EOT}=t\text{high-k}\left(\frac{k_{\,\text{SiO}_{2}}}{k\text{high-k}}\right)
  2. ϵ 0 ϵ SiO 2 A EOT = ϵ 0 ϵ high-k A t high-k = C \epsilon_{0}\,\epsilon_{\,\text{SiO}_{2}}\frac{A}{\mathrm{EOT}}=\epsilon_{0}\,% \epsilon\text{high-k}\,\frac{A}{t\text{high-k}}=C

Equivalent_width.html

  1. W λ = ( 1 - F λ / F 0 ) d λ W_{\lambda}=\int(1-F_{\lambda}/F_{0})d\lambda
  2. F 0 F_{0}
  3. F λ F_{\lambda}
  4. W λ W_{\lambda}
  5. W λ W_{\lambda}

Erdős_arcsine_law.html

  1. 2 π arcsin u . \frac{2}{\pi}\arcsin{\sqrt{u}}.

Erdős–Rado_theorem.html

  1. exp r ( κ ) + ( κ + ) κ r + 1 \exp_{r}(\kappa)^{+}\longrightarrow(\kappa^{+})^{r+1}_{\kappa}

Ericksen_number.html

  1. 𝐸𝑟 = μ v L K \mathit{Er}=\frac{\mu vL}{K}
  2. μ \mu
  3. v v
  4. L L
  5. K K

Ernst_angle.html

  1. θ E \theta_{E}
  2. cos ( θ E ) = e - ( d 1 + a t ) / T 1 \cos(\theta_{E})=e^{-(d_{1}+a_{t})/T_{1}}
  3. cos ( θ E ) = e - T R / T 1 \cos(\theta_{E})=e^{-T_{R}/T_{1}}
  4. θ E = arccos ( e - T R T 1 ) . \theta_{E}=\arccos\left(e^{-\frac{T_{R}}{T_{1}}}\right).

Errera_graph.html

  1. - ( x 2 - 2 x - 5 ) ( x 2 + x - 1 ) 2 ( x 3 - 4 x 2 - 9 x + 10 ) ( x 4 + 2 x 3 - 7 x 2 - 18 x - 9 ) 2 -(x^{2}-2x-5)(x^{2}+x-1)^{2}(x^{3}-4x^{2}-9x+10)(x^{4}+2x^{3}-7x^{2}-18x-9)^{2}

Error_correction_model.html

  1. Y Y
  2. X X
  3. C t C_{t}
  4. Y t Y_{t}
  5. C t = 0.9 Y t C_{t}=0.9Y_{t}
  6. C t = β Y t + ϵ t C_{t}=\beta Y_{t}+\epsilon_{t}
  7. Y t Y_{t}
  8. C t C_{t}
  9. Y t Y_{t}
  10. Δ Y t \Delta Y_{t}
  11. C t C_{t}
  12. Δ C t = 0.5 Δ Y t \Delta C_{t}=0.5\Delta Y_{t}
  13. Δ C t = C t - C t - 1 \Delta C_{t}=C_{t}-C_{t-1}
  14. Δ C t = 0.5 Δ Y t - 0.2 ( C t - 1 - 0.9 Y t - 1 ) + ϵ t \Delta C_{t}=0.5\Delta Y_{t}-0.2(C_{t-1}-0.9Y_{t-1})+\epsilon_{t}
  15. Y t Y_{t}
  16. C t C_{t}
  17. ϵ t \epsilon_{t}
  18. C t - 1 = 0.9 Y t - 1 C_{t-1}=0.9Y_{t-1}
  19. Y t Y_{t}
  20. C t C_{t}
  21. C t C_{t}
  22. Y t Y_{t}
  23. C t C_{t}
  24. C t - 1 C_{t-1}

Errors-in-variables_models.html

  1. y t = α + β x t * + ε t , t = 1 , , T , y_{t}=\alpha+\beta x_{t}^{*}+\varepsilon_{t}\,,\quad t=1,\ldots,T,
  2. x t = x t * + η t , x_{t}=x^{*}_{t}+\eta_{t}\,,
  3. β ^ = 1 T t = 1 T ( x t - x ¯ ) ( y t - y ¯ ) 1 T t = 1 T ( x t - x ¯ ) 2 , \hat{\beta}=\frac{\tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})(y_{t}-\bar{y})}{% \tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})^{2}}\,,
  4. β ^ 𝑝 Cov [ x t , y t ] Var [ x t ] = β σ x * 2 σ x * 2 + σ η 2 = β 1 + σ η 2 / σ x * 2 . \hat{\beta}\ \xrightarrow{p}\ \frac{\operatorname{Cov}[\,x_{t},y_{t}\,]}{% \operatorname{Var}[\,x_{t}\,]}=\frac{\beta\sigma^{2}_{x^{*}}}{\sigma_{x^{*}}^{% 2}+\sigma_{\eta}^{2}}=\frac{\beta}{1+\sigma_{\eta}^{2}/\sigma_{x^{*}}^{2}}\,.
  5. β x = Cov [ x t , y t ] Var [ x t ] . \beta_{x}=\frac{\operatorname{Cov}[\,x_{t},y_{t}\,]}{\operatorname{Var}[\,x_{t% }\,]}.
  6. { x = x * + η , y = y * + ε , y * = g ( x * , w | θ ) , \begin{cases}x=x^{*}+\eta,\\ y=y^{*}+\varepsilon,\\ y^{*}=g(x^{*}\!,w\,|\,\theta),\end{cases}
  7. η x * , \scriptstyle\eta\,\perp\,x^{*},
  8. E [ η | x * ] = 0 , \scriptstyle\operatorname{E}[\eta|x^{*}]\,=\,0,
  9. η x , \scriptstyle\eta\,\perp\,x,
  10. { y t = α + β x t * + ε t , x t = x t * + η t , \begin{cases}y_{t}=\alpha+\beta x_{t}^{*}+\varepsilon_{t},\\ x_{t}=x_{t}^{*}+\eta_{t},\end{cases}
  11. ( x t , y t ) t = 1 T \scriptstyle(x_{t},\,y_{t})_{t=1}^{T}
  12. { y t = α + β x t * + ε t , x t = x t * + η t . \begin{cases}y_{t}=\alpha+\beta^{\prime}x_{t}^{*}+\varepsilon_{t},\\ x_{t}=x_{t}^{*}+\eta_{t}.\end{cases}
  13. { y t = g ( x t * ) + ε t , x t = x t * + η t . \begin{cases}y_{t}=g(x^{*}_{t})+\varepsilon_{t},\\ x_{t}=x^{*}_{t}+\eta_{t}.\end{cases}
  14. g ( x * ) = a + b ln ( e c x * + d ) g(x^{*})=a+b\ln\big(e^{cx^{*}}+d\big)
  15. f x * ( x ) = { A e - B e C x + C D x ( e C x + E ) - F , if d > 0 A e - B x 2 + C x if d = 0 f_{x^{*}}(x)=\begin{cases}Ae^{-Be^{Cx}+CDx}(e^{Cx}+E)^{-F},&\,\text{if}\ d>0\\ Ae^{-Bx^{2}+Cx}&\,\text{if}\ d=0\end{cases}
  16. { x 1 t = x t * + η 1 t , x 2 t = x t * + η 2 t , \begin{cases}x_{1t}=x^{*}_{t}+\eta_{1t},\\ x_{2t}=x^{*}_{t}+\eta_{2t},\end{cases}
  17. φ ^ η j ( v ) = φ ^ x j ( v , 0 ) φ ^ x j * ( v ) , where φ ^ x j ( v 1 , v 2 ) = 1 T t = 1 T e i v 1 x 1 t j + i v 2 x 2 t j , φ ^ x j * ( v ) = exp 0 v φ ^ x j ( 0 , v 2 ) / v 1 φ ^ x j ( 0 , v 2 ) d v 2 , \displaystyle\hat{\varphi}_{\eta_{j}}(v)=\frac{\hat{\varphi}_{x_{j}}(v,0)}{% \hat{\varphi}_{x^{*}_{j}}(v)},\quad\,\text{where }\hat{\varphi}_{x_{j}}(v_{1},% v_{2})=\frac{1}{T}\sum_{t=1}^{T}e^{iv_{1}x_{1tj}+iv_{2}x_{2tj}},\ \ \hat{% \varphi}_{x^{*}_{j}}(v)=\exp\int_{0}^{v}\frac{\partial\hat{\varphi}_{x_{j}}(0,% v_{2})/\partial v_{1}}{\hat{\varphi}_{x_{j}}(0,v_{2})}dv_{2},
  18. f ^ x ( x ) = 1 ( 2 π ) k - C C - C C e - i u x φ ^ x ( u ) d u . \hat{f}_{x}(x)=\frac{1}{(2\pi)^{k}}\int_{-C}^{C}\cdots\int_{-C}^{C}e^{-iu^{% \prime}x}\hat{\varphi}_{x}(u)du.
  19. { y t = j = 1 k β j g j ( x t * ) + j = 1 β k + j w j t + ε t , x 1 t = x t * + η 1 t , x 2 t = x t * + η 2 t , \begin{cases}y_{t}=\textstyle\sum_{j=1}^{k}\beta_{j}g_{j}(x^{*}_{t})+\sum_{j=1% }^{\ell}\beta_{k+j}w_{jt}+\varepsilon_{t},\\ x_{1t}=x^{*}_{t}+\eta_{1t},\\ x_{2t}=x^{*}_{t}+\eta_{2t},\end{cases}
  20. β ^ = ( E ^ [ ξ t ξ t ] ) - 1 E ^ [ ξ t y t ] , \hat{\beta}=\big(\hat{\operatorname{E}}[\,\xi_{t}\xi_{t}^{\prime}\,]\big)^{-1}% \hat{\operatorname{E}}[\,\xi_{t}y_{t}\,],
  21. ξ t = ( g 1 ( x t * ) , , g k ( x t * ) , w 1 , t , , w l , t ) . \xi_{t}^{\prime}=(g_{1}(x^{*}_{t}),\cdots,g_{k}(x^{*}_{t}),w_{1,t},\cdots,w_{l% ,t}).
  22. E [ w t h ( x t * ) ] = 1 2 π - φ h ( - u ) ψ w ( u ) d u , \operatorname{E}[\,w_{t}h(x^{*}_{t})\,]=\frac{1}{2\pi}\int_{-\infty}^{\infty}% \varphi_{h}(-u)\psi_{w}(u)du,
  23. φ h ( u ) = e i u x h ( x ) d x \varphi_{h}(u)=\int e^{iux}h(x)dx
  24. ψ w ( u ) = E [ w t e i u x * ] = E [ w t e i u x 1 t ] E [ e i u x 1 t ] exp 0 u i E [ x 2 t e i v x 1 t ] E [ e i v x 1 t ] d v \psi_{w}(u)=\operatorname{E}[\,w_{t}e^{iux^{*}}\,]=\frac{\operatorname{E}[w_{t% }e^{iux_{1t}}]}{\operatorname{E}[e^{iux_{1t}}]}\exp\int_{0}^{u}i\frac{% \operatorname{E}[x_{2t}e^{ivx_{1t}}]}{\operatorname{E}[e^{ivx_{1t}}]}dv
  25. β ^ \scriptstyle\hat{\beta}
  26. g ^ ( x ) = E ^ [ y t K h ( x t * - x ) ] E ^ [ K h ( x t * - x ) ] , \hat{g}(x)=\frac{\hat{\operatorname{E}}[\,y_{t}K_{h}(x^{*}_{t}-x)\,]}{\hat{% \operatorname{E}}[\,K_{h}(x^{*}_{t}-x)\,]},

Estrada_index.html

  1. G = ( V , E ) G=(V,E)
  2. | V | = n |V|=n
  3. λ 1 λ 2 λ n \lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{n}
  4. A A
  5. E E ( G ) = j = 1 n e λ j EE(G)=\sum_{j=1}^{n}e^{\lambda_{j}}
  6. i i
  7. E E ( i ) = k = 0 ( A k ) i i k ! EE(i)=\sum_{k=0}^{\infty}\frac{(A^{k})_{ii}}{k!}
  8. E E ( i ) = ( e A ) i i = j = 1 n [ φ ( i ) ] 2 e λ j EE(i)=(e^{A})_{ii}=\sum_{j=1}^{n}[\varphi(i)]^{2}e^{\lambda_{j}}
  9. φ j ( i ) \varphi_{j}(i)
  10. i i
  11. j j
  12. λ j \lambda_{j}
  13. E E ( G ) = t r ( e A ) EE(G)=tr(e^{A})

Euler's_laws_of_motion.html

  1. 𝐩 = m 𝐯 cm \mathbf{p}=m\mathbf{v}_{\rm cm}
  2. 𝐅 = m 𝐚 cm \mathbf{F}=m\mathbf{a}_{\rm cm}
  3. 𝐌 = d 𝐋 d t \mathbf{M}={d\mathbf{L}\over dt}
  4. 𝐌 = 𝐫 cm × 𝐚 cm m + I s y m b o l α \mathbf{M}=\mathbf{r}_{\rm cm}\times\mathbf{a}_{\rm cm}m+Isymbol{\alpha}
  5. 𝐅 B = V 𝐛 d m = V 𝐛 ρ d V \mathbf{F}_{B}=\int_{V}\mathbf{b}\,dm=\int_{V}\mathbf{b}\rho\,dV
  6. 𝐌 = 𝐌 B + 𝐌 C \mathbf{M}=\mathbf{M}_{B}+\mathbf{M}_{C}
  7. 𝐅 = V 𝐚 d m = V 𝐚 ρ d V = S 𝐭 d S + V 𝐛 ρ d V \mathbf{F}=\int_{V}\mathbf{a}\,dm=\int_{V}\mathbf{a}\rho\,dV=\int_{S}\mathbf{t% }dS+\int_{V}\mathbf{b}\rho\,dV
  8. 𝐌 = S 𝐫 × 𝐭 d S + V 𝐫 × 𝐛 ρ d V . \mathbf{M}=\int_{S}\mathbf{r}\times\mathbf{t}dS+\int_{V}\mathbf{r}\times% \mathbf{b}\rho\,dV.
  9. d 𝐩 d t = 𝐅 d d t V ρ 𝐯 d V = S 𝐭 d S + V 𝐛 ρ d V . \begin{aligned}\displaystyle\frac{d\mathbf{p}}{dt}&\displaystyle=\mathbf{F}\\ \displaystyle\frac{d}{dt}\int_{V}\rho\mathbf{v}\,dV&\displaystyle=\int_{S}% \mathbf{t}dS+\int_{V}\mathbf{b}\rho\,dV.\\ \end{aligned}
  10. d 𝐋 d t = 𝐌 d d t V 𝐫 × ρ 𝐯 d V = S 𝐫 × 𝐭 d S + V 𝐫 × 𝐛 ρ d V . \begin{aligned}\displaystyle\frac{d\mathbf{L}}{dt}&\displaystyle=\mathbf{M}\\ \displaystyle\frac{d}{dt}\int_{V}\mathbf{r}\times\rho\mathbf{v}\,dV&% \displaystyle=\int_{S}\mathbf{r}\times\mathbf{t}dS+\int_{V}\mathbf{r}\times% \mathbf{b}\rho\,dV.\\ \end{aligned}

Eulerian_poset.html

  1. μ P ( x , y ) = ( - 1 ) | y | - | x | for all x y . \mu_{P}(x,y)=(-1)^{|y|-|x|}\,\text{ for all }x\leq y.
  2. h k = h d - k h_{k}=h_{d-k}\,

Euler–Lotka_equation.html

  1. 1 = a = 1 ω λ - a ( a ) b ( a ) 1=\sum_{a=1}^{\omega}\lambda^{-a}\ell(a)b(a)
  2. λ \lambda
  3. B ( t ) = 0 t B ( t - a ) ( a ) b ( a ) d a . B(t)=\int_{0}^{t}B(t-a)\ell(a)b(a)\,da.
  4. Q e r t = 0 t Q e r ( t - a ) ( a ) b ( a ) d a Qe^{rt}=\int_{0}^{t}Qe^{r(t-a)}\ell(a)b(a)\,da
  5. 1 = 0 t e - r a ( a ) b ( a ) d a . 1=\int_{0}^{t}e^{-ra}\ell(a)b(a)\,da.
  6. 1 = a = α β e - r a ( a ) b ( a ) 1=\sum_{a=\alpha}^{\beta}e^{-ra}\ell(a)b(a)
  7. α \alpha
  8. β \beta
  9. 1 = a = 1 ω λ - a ( a ) b ( a ) 1=\sum_{a=1}^{\omega}\lambda^{-a}\ell(a)b(a)
  10. ω \omega
  11. [ f 0 f 1 f 2 f 3 f ω - 1 s 0 0 0 0 0 0 s 1 0 0 0 0 0 s 2 0 0 0 0 0 0 0 0 0 s ω - 2 0 ] \begin{bmatrix}f_{0}&f_{1}&f_{2}&f_{3}&\ldots&f_{\omega-1}\\ s_{0}&0&0&0&\ldots&0\\ 0&s_{1}&0&0&\ldots&0\\ 0&0&s_{2}&0&\ldots&0\\ 0&0&0&\ddots&\ldots&0\\ 0&0&0&\ldots&s_{\omega-2}&0\end{bmatrix}
  12. s i s_{i}
  13. f i f_{i}
  14. s i = i + 1 / i s_{i}=\ell_{i+1}/\ell_{i}
  15. i i
  16. f i = s i b i + 1 f_{i}=s_{i}b_{i+1}
  17. i + 1 i+1
  18. i + 1 i+1
  19. 𝐧 𝐢 + 𝟏 = 𝐋𝐧 𝐢 = λ 𝐧 𝐢 \mathbf{n_{i+1}}=\mathbf{Ln_{i}}=\lambda\mathbf{n_{i}}
  20. n i n_{i}
  21. λ \lambda
  22. n i , t n_{i,t}
  23. i i
  24. t t
  25. n 1 , t + 1 = λ n 1 , t n_{1,t+1}=\lambda n_{1,t}
  26. n 1 , t + 1 = s 0 n 0 , t n_{1,t+1}=s_{0}n_{0,t}
  27. n 1 , t = s 0 λ n 0 , t . n_{1,t}=\frac{s_{0}}{\lambda}n_{0,t}.\,
  28. n 2 , t = s 1 λ n 1 , t = s 0 s 1 λ 2 n 0 , t . n_{2,t}=\frac{s_{1}}{\lambda}n_{1,t}=\frac{s_{0}s_{1}}{\lambda^{2}}n_{0,t}.
  29. n i , t = s 0 s i - 1 λ i n 0 , t . n_{i,t}=\frac{s_{0}\cdots s_{i-1}}{\lambda^{i}}n_{0,t}.
  30. n 0 , t + 1 = f 0 n 0 , t + + f ω - 1 n ω - 1 , t = λ n 0 , t . n_{0,t+1}=f_{0}n_{0,t}+\cdots+f_{\omega-1}n_{\omega-1,t}=\lambda n_{0,t}.
  31. n i , t n_{i,t}
  32. λ n 0 , t = ( f 0 + f 1 s 0 λ + + f ω - 1 s 0 s ω - 2 λ ω - 1 ) n ( 0 , t ) . \lambda n_{0,t}=\left(f_{0}+f_{1}\frac{s_{0}}{\lambda}+\cdots+f_{\omega-1}% \frac{s_{0}\cdots s_{\omega-2}}{\lambda^{\omega-1}}\right)n_{(0,t)}.
  33. 1 = s 0 b 1 λ + s 0 s 1 b 2 λ 2 + + s 0 s ω - 1 b ω λ ω . 1=\frac{s_{0}b_{1}}{\lambda}+\frac{s_{0}s_{1}b_{2}}{\lambda^{2}}+\cdots+\frac{% s_{0}\cdots s_{\omega-1}b_{\omega}}{\lambda^{\omega}}.
  34. s i = i + 1 / i s_{i}=\ell_{i+1}/\ell_{i}
  35. s 0 s i = 1 0 2 1 i + 1 i = i + 1 . s_{0}\ldots s_{i}=\frac{\ell_{1}}{\ell_{0}}\frac{\ell_{2}}{\ell_{1}}\cdots% \frac{\ell_{i+1}}{\ell_{i}}=\ell_{i+1}.
  36. i = 1 ω i b i λ i = 1 , \sum_{i=1}^{\omega}\frac{\ell_{i}b_{i}}{\lambda^{i}}=1,