wpmath0000012_2

Commutation_theorem.html

  1. J a Ω = a * Ω Ja\Omega=a^{*}\Omega
  2. J M J M . JMJ\subseteq M^{\prime}.
  3. J M J = M JMJ=M^{\prime}
  4. H = K i K , H=K\oplus iK,
  5. J M J M JM^{\prime}J\subseteq M
  6. 2 ( Γ ) \ell^{2}(\Gamma)
  7. ( λ ( g ) f ) ( x ) = f ( g - 1 x ) , ( ρ ( g ) f ) ( x ) = f ( x g ) (\lambda(g)f)(x)=f(g^{-1}x),\,\,(\rho(g)f)(x)=f(xg)
  8. 2 ( Γ ) \ell^{2}(\Gamma)
  9. λ ( Γ ) ′′ = ρ ( Γ ) , ρ ( Γ ) ′′ = λ ( Γ ) . \lambda(\Gamma)^{\prime\prime}=\rho(\Gamma)^{\prime},\,\,\rho(\Gamma)^{\prime% \prime}=\lambda(\Gamma)^{\prime}.
  10. J f ( g ) = f ( g - 1 ) ¯ . Jf(g)=\overline{f(g^{-1})}.
  11. A = A , A^{\prime}=A,
  12. U g f ( x ) = f ( g - 1 x ) , U_{g}f(x)=f(g^{-1}x),
  13. H 1 = H 2 ( Γ ) , H_{1}=H\otimes\ell^{2}(\Gamma),
  14. M = A Γ M=A\rtimes\Gamma
  15. A I A\otimes I
  16. U g λ ( g ) U_{g}\otimes\lambda(g)
  17. Ω = 1 δ 1 \Omega=1\otimes\delta_{1}
  18. τ ( λ a + μ b ) = λ τ ( a ) + μ τ ( b ) \tau(\lambda a+\mu b)=\lambda\tau(a)+\mu\tau(b)
  19. τ ( u a u * ) = τ ( a ) \tau(uau^{*})=\tau(a)
  20. M 0 = { a M | τ ( a * a ) < } M_{0}=\{a\in M|\tau(a^{*}a)<\infty\}
  21. ( a , b ) = τ ( b * a ) . (a,b)=\tau(b^{*}a).
  22. J a = a * Ja=a^{*}
  23. J M J = M JMJ=M^{\prime}
  24. \supseteq
  25. 𝔄 \mathfrak{A}
  26. 𝔄 \mathfrak{A}
  27. 𝔄 \mathfrak{A}
  28. 𝔄 \mathfrak{A}
  29. \cap
  30. \cap
  31. \cap
  32. 𝔄 \mathfrak{A}
  33. 𝔄 \mathfrak{A}
  34. λ ( a ) x = a x , ρ ( a ) x = x a . \lambda(a)x=ax,\,\,\rho(a)x=xa.
  35. λ ( 𝔄 ) ′′ = ρ ( 𝔄 ) \lambda(\mathfrak{A})^{\prime\prime}=\rho(\mathfrak{A})^{\prime}
  36. M = λ ( 𝔄 ) ′′ , M=\lambda(\mathfrak{A})^{\prime\prime},
  37. J M J = M JMJ=M^{\prime}
  38. 𝔄 \mathfrak{A}
  39. 𝔄 \mathfrak{A}
  40. 𝔅 \mathfrak{B}
  41. 𝔅 \mathfrak{B}
  42. 𝔄 \mathfrak{A}
  43. 𝔅 \mathfrak{B}
  44. 𝔅 \mathfrak{B}
  45. τ ( x ) = ( a , a ) \tau(x)=(a,a)
  46. M 0 = 𝔅 . M_{0}=\mathfrak{B}.

Compact_stencil.html

  1. f ( x 0 ) = f ( x 0 + h ) - f ( x 0 - h ) 2 h + O ( h 2 ) f^{\prime}(x_{0})=\frac{f\left(x_{0}+h\right)-f\left(x_{0}-h\right)}{2h}+O% \left(h^{2}\right)
  2. f ( x 0 ) = f ( x 0 + h ) - f ( x 0 ) h - f ( 2 ) ( x 0 ) 2 ! h - f ( 3 ) ( x 0 ) 3 ! h 2 - f ( 4 ) ( x 0 ) 4 ! h 3 + \begin{array}[]{l}f^{\prime}(x_{0})=\frac{f\left(x_{0}+h\right)-f(x_{0})}{h}-% \frac{f^{(2)}(x_{0})}{2!}h-\frac{f^{(3)}(x_{0})}{3!}h^{2}-\frac{f^{(4)}(x_{0})% }{4!}h^{3}+\cdots\end{array}
  3. h h
  4. - h -h
  5. f ( x 0 ) = - f ( x 0 - h ) - f ( x 0 ) h + f ( 2 ) ( x 0 ) 2 ! h - f ( 3 ) ( x 0 ) 3 ! h 2 + f ( 4 ) ( x 0 ) 4 ! h 3 + \begin{array}[]{l}f^{\prime}(x_{0})=-\frac{f\left(x_{0}-h\right)-f(x_{0})}{h}+% \frac{f^{(2)}(x_{0})}{2!}h-\frac{f^{(3)}(x_{0})}{3!}h^{2}+\frac{f^{(4)}(x_{0})% }{4!}h^{3}+\cdots\end{array}
  6. h h
  7. 2 f ( x 0 ) = f ( x 0 + h ) - f ( x 0 ) h - f ( x 0 - h ) - f ( x 0 ) h - 2 f ( 3 ) ( x 0 ) 3 ! h 2 + \begin{array}[]{l}2f^{\prime}(x_{0})=\frac{f\left(x_{0}+h\right)-f(x_{0})}{h}-% \frac{f\left(x_{0}-h\right)-f(x_{0})}{h}-2\frac{f^{(3)}(x_{0})}{3!}h^{2}+% \cdots\end{array}
  8. f ( x 0 ) = f ( x 0 + h ) - f ( x 0 - h ) 2 h - f ( 3 ) ( x 0 ) 3 ! h 2 + \begin{array}[]{l}f^{\prime}(x_{0})=\frac{f\left(x_{0}+h\right)-f\left(x_{0}-h% \right)}{2h}-\frac{f^{(3)}(x_{0})}{3!}h^{2}+\cdots\end{array}
  9. f ( x 0 ) = f ( x 0 + h ) - f ( x 0 - h ) 2 h + O ( h 2 ) \begin{array}[]{l}f^{\prime}(x_{0})=\frac{f\left(x_{0}+h\right)-f\left(x_{0}-h% \right)}{2h}+O\left(h^{2}\right)\end{array}
  10. f ( 2 ) ( x 0 ) = f ( x 0 + h ) + f ( x 0 - h ) - 2 f ( x 0 ) h 2 + O ( h 2 ) \begin{array}[]{l}f^{(2)}(x_{0})=\frac{f\left(x_{0}+h\right)+f\left(x_{0}-h% \right)-2f(x_{0})}{h^{2}}+O\left(h^{2}\right)\end{array}
  11. f ( x 0 ) = f ( x 0 + h ) - f ( x 0 ) h - f ( 2 ) ( x 0 ) 2 ! h - f ( 3 ) ( x 0 ) 3 ! h 2 - f ( 4 ) ( x 0 ) 4 ! h 3 + \begin{array}[]{l}f^{\prime}(x_{0})=\frac{f\left(x_{0}+h\right)-f(x_{0})}{h}-% \frac{f^{(2)}(x_{0})}{2!}h-\frac{f^{(3)}(x_{0})}{3!}h^{2}-\frac{f^{(4)}(x_{0})% }{4!}h^{3}+\cdots\end{array}
  12. h h
  13. - h -h
  14. f ( x 0 ) = - f ( x 0 - h ) - f ( x 0 ) h + f ( 2 ) ( x 0 ) 2 ! h - f ( 3 ) ( x 0 ) 3 ! h 2 + f ( 4 ) ( x 0 ) 4 ! h 3 + \begin{array}[]{l}f^{\prime}(x_{0})=-\frac{f\left(x_{0}-h\right)-f(x_{0})}{h}+% \frac{f^{(2)}(x_{0})}{2!}h-\frac{f^{(3)}(x_{0})}{3!}h^{2}+\frac{f^{(4)}(x_{0})% }{4!}h^{3}+\cdots\end{array}
  15. h h
  16. 0 = f ( x 0 + h ) - f ( x 0 ) h + f ( x 0 - h ) - f ( x 0 ) h - 2 f ( 2 ) ( x 0 ) 2 ! h - 2 f ( 4 ) ( x 0 ) 4 ! h 3 + \begin{array}[]{l}0=\frac{f\left(x_{0}+h\right)-f(x_{0})}{h}+\frac{f\left(x_{0% }-h\right)-f(x_{0})}{h}-2\frac{f^{(2)}(x_{0})}{2!}h-2\frac{f^{(4)}(x_{0})}{4!}% h^{3}+\cdots\end{array}
  17. f ( 2 ) ( x 0 ) = f ( x 0 + h ) + f ( x 0 - h ) - 2 f ( x 0 ) h 2 - 2 f ( 4 ) ( x 0 ) 4 ! h 2 + \begin{array}[]{l}f^{(2)}(x_{0})=\frac{f\left(x_{0}+h\right)+f\left(x_{0}-h% \right)-2f(x_{0})}{h^{2}}-2\frac{f^{(4)}(x_{0})}{4!}h^{2}+\cdots\end{array}
  18. f ( 2 ) ( x 0 ) = f ( x 0 + h ) + f ( x 0 - h ) - 2 f ( x 0 ) h 2 + O ( h 2 ) \begin{array}[]{l}f^{(2)}(x_{0})=\frac{f\left(x_{0}+h\right)+f\left(x_{0}-h% \right)-2f(x_{0})}{h^{2}}+O\left(h^{2}\right)\end{array}

Compacton.html

  1. u t + ( u m ) x + ( u n ) x x x = 0 u_{t}+(u^{m})_{x}+(u^{n})_{xxx}=0\,
  2. u t + ( u 2 ) x + ( u 2 ) x x x = 0 u_{t}+(u^{2})_{x}+(u^{2})_{xxx}=0\,
  3. u ( x , t ) = { 4 λ 3 cos 2 ( ( x - λ t ) / 4 ) if | x - λ t | 2 π , 0 if | x - λ t | 2 π . u(x,t)=\begin{cases}\dfrac{4\lambda}{3}\cos^{2}((x-\lambda t)/4)&\,\text{if }|% x-\lambda t|\leq 2\pi,\\ \\ 0&\,\text{if }|x-\lambda t|\geq 2\pi.\end{cases}

Compaq_Evo_N1020v.html

  1. P = C V f 2 \ P={C}{V}{{}^{2}}{f}
  2. \therefore\!\,

Complementarity_theory.html

  1. i i

Completely_randomized_design.html

  1. Y i , j = μ + T i + random error Y_{i,j}=\mu+T_{i}+\mathrm{random\ error}
  2. Y ¯ \bar{Y}
  3. Y ¯ i - Y ¯ \bar{Y}_{i}-\bar{Y}
  4. Y ¯ i \bar{Y}_{i}

Complex_analytic_space.html

  1. \mathbb{C}
  2. ¯ \underline{\mathbb{C}}
  3. \mathbb{C}
  4. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  5. ¯ \underline{\mathbb{C}}
  6. U U
  7. n \mathbb{C}^{n}
  8. f 1 , , f k f_{1},\dots,f_{k}
  9. U U
  10. X = V ( f 1 , , f k ) X=V(f_{1},\dots,f_{k})
  11. X = { x f 1 ( x ) = = f k ( x ) = 0 } X=\{x\mid f_{1}(x)=\cdots=f_{k}(x)=0\}
  12. X X
  13. 𝒪 X \mathcal{O}_{X}
  14. X X
  15. 𝒪 U / ( f 1 , , f k ) \mathcal{O}_{U}/(f_{1},\ldots,f_{k})
  16. 𝒪 U \mathcal{O}_{U}
  17. U U
  18. \mathbb{C}
  19. ( X , 𝒪 X ) (X,\mathcal{O}_{X})
  20. \mathbb{C}
  21. ( X , 𝒪 X ) (X,\mathcal{O}_{X})

Complexity_index.html

  1. 𝖢 \mathsf{C}
  2. 𝖢 \mathsf{C}
  3. 𝒳 \mathcal{X}
  4. 𝖢 \mathsf{C}
  5. s y m b o l S symbolS
  6. 𝖢 \mathsf{C}
  7. c c^{\prime}
  8. c c^{\prime}
  9. c c^{\prime}
  10. c c^{\prime}
  11. c + c^{+}
  12. ( c ) + \left(c^{\prime}\right)^{+}
  13. 𝖢 \mathsf{C}
  14. 𝔛 \mathfrak{X}
  15. s y m b o l S : 𝖢 { , 𝔛 } 2 𝔛 symbolS:\mathsf{C}\cup\{\emptyset,\mathfrak{X}\}\mapsto 2^{\mathfrak{X}}
  16. c s y m b o l S ( c ) = c\cap{symbolS}(c)=\emptyset
  17. c 𝖢 c\in\mathsf{C}
  18. c + = c \cupsymbol S ( c ) c^{+}=c\cupsymbol S(c)
  19. c 𝖢 c^{\prime}\in\mathsf{C}
  20. c c c^{\prime}\not\subseteq c
  21. c + ( c ) + c^{+}\subseteq\left(c^{\prime}\right)^{+}
  22. up ( c ) \mathrm{up}(c)
  23. c 2 up ( c 1 ) c_{2}\in\mathrm{up}(c_{1})
  24. c 2 s y m b o l S ( c 1 ) c_{2}\cap{symbolS}(c_{1})\neq\emptyset
  25. s y m b o l S ( c ) {symbolS}(c)
  26. s y m b o l S s y m b o l S {symbolS}^{\prime}\neq{symbolS}
  27. s y m b o l S ( c ) s y m b o l S ( c ) symbolS^{\prime}(c)\subseteq symbolS(c)
  28. c 𝖢 c\in\mathsf{C}
  29. c ( c ) + c\subseteq\left(c^{\prime}\right)^{+}
  30. s y m b o l S ( c ) c = {symbolS}(c)\cap c^{\prime}=\emptyset
  31. c up ( c ) c^{\prime}\not\in\mathrm{up}(c)
  32. s y m b o l S ( c ) {symbolS}(c)
  33. up ( c ) \mathrm{up}(c)
  34. c + c^{+}
  35. ( c ) + (c^{\prime})^{+}
  36. c , s y m b o l S ( c ) c^{\prime},{symbolS}(c)
  37. c 1 c_{1}
  38. c 2 c_{2}
  39. c 1 c 2 s y m b o l S ( c 2 ) c_{1}\subset c_{2}\cup{symbolS}(c_{2})
  40. c 2 s y m b o l S ( c 1 ) = c_{2}\cap{symbolS}(c_{1})=\emptyset
  41. s y m b o l S {symbolS}
  42. { c 1 } up ( c 1 ) - { c 2 } \{c_{1}\}\cup\mathrm{up}(c_{1})-\{c_{2}\}
  43. s y m b o l S ( c ) {symbolS}(c)
  44. s y m b o l S symbolS
  45. { x 1 , x 2 , x 3 } \{x_{1},x_{2},x_{3}\}
  46. c 0 c_{0}
  47. c 1 , c 2 , c 3 , c 4 c_{1},c_{2},c_{3},c_{4}
  48. c i c_{i}
  49. c 0 c_{0}
  50. c 0 { x 1 , x 2 , x 3 } c_{0}\cup\{x_{1},x_{2},x_{3}\}
  51. c 3 c_{3}
  52. c 1 c_{1}
  53. x 1 x_{1}
  54. x 3 x_{3}
  55. c 2 c_{2}
  56. x 2 x_{2}
  57. x 3 x_{3}
  58. c 4 c_{4}
  59. x 1 x_{1}
  60. x 2 x_{2}
  61. x 4 x_{4}
  62. c 0 c_{0}
  63. c 0 c_{0}
  64. D 𝖢 = sup s y m b o l S , c # s y m b o l S ( c ) \mathrm{D}_{\mathsf{C}}=\sup_{{symbolS},c}\#{symbolS}(c)
  65. 𝖢 \mathsf{C}
  66. s y m b o l S symbolS
  67. 𝔛 \mathfrak{X}
  68. 𝔛 \mathfrak{X}
  69. 𝔛 \mathfrak{X}
  70. D 𝖢 \mathrm{D}_{\mathsf{C}}
  71. D 𝖵 C \mathrm{D}_{\mathsf{V}C}
  72. D 𝖢 D 𝖵 C + 1 \mathrm{D}_{\mathsf{C}}\leq\mathrm{D}_{\mathsf{V}C}+1
  73. 2 \mathbb{R}^{2}
  74. D 𝖢 = 2 \mathrm{D}_{\mathsf{C}}=2
  75. \mathbb{R}
  76. x 1 , x 2 x_{1},x_{2}
  77. \mathbb{R}
  78. 𝖢 = { c 1 , c 2 , c 3 , c 4 } \mathsf{C}=\{c_{1},c_{2},c_{3},c_{4}\}
  79. 𝔛 = { x 1 , x 2 , x 3 } \mathfrak{X}=\{x_{1},x_{2},x_{3}\}
  80. + +
  81. x j x_{j}
  82. c i c_{i}
  83. - -
  84. c i c_{i}
  85. \bigcirc
  86. x 1 x_{1}
  87. x 2 x_{2}
  88. x 3 x_{3}
  89. c 1 = c_{1}=
  90. - \bigcirc\!\!\!\!\!-
  91. - \bigcirc\!\!\!\!\!-
  92. - -
  93. c 2 = c_{2}=
  94. - \bigcirc\!\!\!\!\!-
  95. + +
  96. + +
  97. c 3 = c_{3}=
  98. + +
  99. - \bigcirc\!\!\!\!\!-
  100. + +
  101. c 4 = c_{4}=
  102. + +
  103. + +
  104. + +
  105. D 𝖢 = 2 \mathrm{D}_{\mathsf{C}}=2
  106. 𝐒 \mathbf{S}
  107. 𝐒 ( c 1 ) = { x 1 , x 2 } , 𝐒 ( c 2 ) = { x 1 } , 𝐒 ( c 3 ) = { x 2 } , 𝐒 ( c 4 ) = \mathbf{S}(c_{1})=\{x_{1},x_{2}\},\mathbf{S}(c_{2})=\{x_{1}\},\mathbf{S}(c_{3}% )=\{x_{2}\},\mathbf{S}(c_{4})=\emptyset
  108. 𝐒 ( c 1 ) = { x 3 } , 𝐒 ( c 2 ) = { x 1 } , 𝐒 ( c 3 ) = { x 2 } , 𝐒 ( c 4 ) = \mathbf{S}(c_{1})=\{x_{3}\},\mathbf{S}(c_{2})=\{x_{1}\},\mathbf{S}(c_{3})=\{x_% {2}\},\mathbf{S}(c_{4})=\emptyset
  109. x 1 x_{1}
  110. x 2 x_{2}
  111. x 3 x_{3}
  112. c 1 = c_{1}=
  113. - -
  114. - -
  115. - \bigcirc\!\!\!\!\!-
  116. c 2 = c_{2}=
  117. - \bigcirc\!\!\!\!\!-
  118. + +
  119. + +
  120. c 3 = c_{3}=
  121. + +
  122. - \bigcirc\!\!\!\!\!-
  123. + +
  124. c 4 = c_{4}=
  125. + +
  126. + +
  127. + +

Composite_image_filter.html

  1. Z i \scriptstyle Z_{\mathrm{i}}
  2. Z iT \scriptstyle Z_{\mathrm{iT}}
  3. Z i Π \scriptstyle Z_{\mathrm{i\Pi}}
  4. Z iT m \scriptstyle Z_{\mathrm{iT}m}
  5. Z i Π \scriptstyle Z_{\mathrm{i\Pi}}
  6. Z iT m \scriptstyle Z_{\mathrm{iT}m}
  7. Z iT \scriptstyle Z_{\mathrm{iT}}
  8. Z i Π m \scriptstyle Z_{\mathrm{i\Pi}m}
  9. Z i m \scriptstyle Z_{\mathrm{i}m}
  10. Z i m m \scriptstyle Z_{\mathrm{i}mm^{\prime}}

Composite_laminates.html

  1. ε 0 = [ ε x 0 ε y 0 τ x y 0 ] T \varepsilon^{0}=\begin{bmatrix}\varepsilon^{0}_{x}&\varepsilon^{0}_{y}&\tau^{0% }_{xy}\end{bmatrix}^{T}
  2. κ = [ κ x κ y κ x y ] T \kappa=\begin{bmatrix}\kappa_{x}&\kappa_{y}&\kappa_{xy}\end{bmatrix}^{T}
  3. x x
  4. y y
  5. [ θ 1 , θ 2 , θ N ] \begin{bmatrix}\theta_{1},&\theta_{2},&\dots&\theta_{N}\end{bmatrix}
  6. [ σ ] = 𝐐 [ ε ] [\sigma]=\mathbf{Q}[\varepsilon]
  7. Q 11 * = Q 11 cos 4 θ + 2 ( Q 12 + 2 Q 66 ) sin 2 θ cos 2 θ + Q 22 sin 4 θ Q^{*}_{11}=Q_{11}\cos^{4}\theta+2(Q_{12}+2Q_{66})\sin^{2}\theta\cos^{2}\theta+% Q_{22}\sin^{4}\theta
  8. Q 22 * = Q 11 sin 4 θ + 2 ( Q 12 + 2 Q 66 ) sin 2 θ cos 2 θ + Q 22 cos 4 θ Q^{*}_{22}=Q_{11}\sin^{4}\theta+2(Q_{12}+2Q_{66})\sin^{2}\theta\cos^{2}\theta+% Q_{22}\cos^{4}\theta
  9. Q 12 * = ( Q 11 + Q 22 - 4 Q 66 ) sin 2 θ cos 2 θ + Q 12 ( sin 4 θ + cos 4 θ ) Q^{*}_{12}=(Q_{11}+Q_{22}-4Q_{66})\sin^{2}\theta\cos^{2}\theta+Q_{12}(\sin^{4}% \theta+\cos^{4}\theta)
  10. Q 66 * = ( Q 11 + Q 22 - 2 Q 12 - 2 Q 66 ) sin 2 θ cos 2 θ + Q 66 ( sin 4 θ + cos 4 θ ) Q^{*}_{66}=(Q_{11}+Q_{22}-2Q_{12}-2Q_{66})\sin^{2}\theta\cos^{2}\theta+Q_{66}(% \sin^{4}\theta+\cos^{4}\theta)
  11. Q 16 * = ( Q 11 - Q 12 - 2 Q 66 ) cos 3 θ sin θ - ( Q 22 - Q 12 - 2 Q 66 ) cos θ sin 3 θ Q^{*}_{16}=(Q_{11}-Q_{12}-2Q_{66})\cos^{3}\theta\sin\theta-(Q_{22}-Q_{12}-2Q_{% 66})\cos\theta\sin^{3}\theta
  12. Q 26 * = ( Q 11 - Q 12 - 2 Q 66 ) cos θ sin 3 θ - ( Q 22 - Q 12 - 2 Q 66 ) cos 3 θ sin θ Q^{*}_{26}=(Q_{11}-Q_{12}-2Q_{66})\cos\theta\sin^{3}\theta-(Q_{22}-Q_{12}-2Q_{% 66})\cos^{3}\theta\sin\theta
  13. [ σ ] * = 𝐐 * [ ε ] * [\sigma]^{*}=\mathbf{Q}^{*}[\varepsilon]^{*}
  14. ε = ε 0 + κ z \varepsilon=\varepsilon^{0}+\kappa\cdot z
  15. [ 𝐍 𝐌 ] = [ 𝐀 𝐁 𝐁 𝐃 ] [ ε 0 κ ] \begin{bmatrix}\mathbf{N}\\ \mathbf{M}\end{bmatrix}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{B}&\mathbf{D}\end{bmatrix}\begin{bmatrix}\varepsilon^{0}\\ \kappa\end{bmatrix}
  16. 𝐀 = j = 1 N 𝐐 * ( z j - z j - 1 ) \mathbf{A}=\sum^{N}_{j=1}\mathbf{Q}^{*}\left(z_{j}-z_{j-1}\right)
  17. 𝐁 = 1 2 j = 1 N 𝐐 * ( z j 2 - z j - 1 2 ) \mathbf{B}=\frac{1}{2}\sum^{N}_{j=1}\mathbf{Q}^{*}\left(z^{2}_{j}-z^{2}_{j-1}\right)
  18. 𝐃 = 1 3 j = 1 N 𝐐 * ( z j 3 - z j - 1 3 ) \mathbf{D}=\frac{1}{3}\sum^{N}_{j=1}\mathbf{Q}^{*}\left(z^{3}_{j}-z^{3}_{j-1}\right)

Computable_real_function.html

  1. f : f\colon\mathbb{R}\to\mathbb{R}
  2. { x i } i = 1 \{x_{i}\}_{i=1}^{\infty}
  3. { f ( x i ) } i = 1 \{f(x_{i})\}_{i=1}^{\infty}
  4. f : f\colon\mathbb{R}\to\mathbb{R}
  5. d : d\colon\mathbb{N}\to\mathbb{N}
  6. | x - y | < 1 d ( n ) |x-y|<{1\over d(n)}
  7. | f ( x ) - f ( y ) | < 1 n |f(x)-f(y)|<{1\over n}
  8. n . \mathbb{R}^{n}.
  9. D D
  10. n . \mathbb{R}^{n}.
  11. f : D f\colon D\to\mathbb{R}
  12. n n
  13. ( { x i 1 } i = 1 , { x i n } i = 1 ) \left(\{x_{i\,1}\}_{i=1}^{\infty},\ldots\{x_{i\,n}\}_{i=1}^{\infty}\right)
  14. ( i ) ( x i 1 , x i n ) D , (\forall i)\quad(x_{i\,1},\ldots x_{i\,n})\in D\qquad,
  15. { f ( x i ) } i = 1 \{f(x_{i})\}_{i=1}^{\infty}

Computational_lithography.html

  1. C D = k 1 λ N A CD=k_{1}\cdot\frac{\lambda}{NA}
  2. C D \,CD
  3. λ \,\lambda
  4. N A \,NA
  5. k 1 \,k_{1}

Computing_the_permanent.html

  1. perm ( A ) = σ S n i = 1 n a i , σ ( i ) . \operatorname{perm}(A)=\sum_{\sigma\in S_{n}}\prod_{i=1}^{n}a_{i,\sigma(i)}.
  2. A k A_{k}
  3. P ( A k ) P(A_{k})
  4. A k A_{k}
  5. Σ k \Sigma_{k}
  6. P ( A k ) P(A_{k})
  7. A k A_{k}
  8. perm ( A ) = k = 0 n - 1 ( - 1 ) k Σ k . \operatorname{perm}(A)=\sum_{k=0}^{n-1}(-1)^{k}\Sigma_{k}.
  9. perm ( A ) = ( - 1 ) n S { 1 , , n } ( - 1 ) | S | i = 1 n j S a i j . \operatorname{perm}(A)=(-1)^{n}\sum_{S\subseteq\{1,\dots,n\}}(-1)^{|S|}\prod_{% i=1}^{n}\sum_{j\in S}a_{ij}.
  10. O ( 2 n n 2 ) O(2^{n}n^{2})
  11. O ( 2 n n ) O(2^{n}n)
  12. S S
  13. perm ( A ) = [ δ ( k = 1 m δ k ) j = 1 m i = 1 m δ i a i j ] / 2 m - 1 , \operatorname{perm}(A)=\left[\sum_{\delta}\left(\prod_{k=1}^{m}\delta_{k}% \right)\prod_{j=1}^{m}\sum_{i=1}^{m}\delta_{i}a_{ij}\right]/2^{m-1},
  14. 2 m - 1 2^{m-1}
  15. δ = ( δ 1 = 1 , δ 2 , , δ m ) { ± 1 } m \delta=(\delta_{1}=1,\delta_{2},\dots,\delta_{m})\in\{\pm 1\}^{m}
  16. T T
  17. per T ( A ) = det A \operatorname{per}\,T(A)=\det A
  18. n × n n\times n
  19. A A
  20. C C
  21. G C G\setminus C
  22. K 3 , 3 K_{3,3}
  23. ( - 1 ) 1 ( mod 2 ) . (-1)\equiv 1\;\;(\mathop{{\rm mod}}2).
  24. 2 k 2^{k}
  25. O ( n 4 k - 3 ) O(n^{4k-3})
  26. k 2 k\geq 2
  27. p p
  28. p = 3 p=3
  29. perm ( A ) = ( - 1 ) m Σ U { 1 , , m } det ( A U ) . det ( A U ¯ ) , \operatorname{perm}(A)=(-1)^{m}\Sigma_{U\subseteq\{1,\dots,m\}}\det(A_{U}).% \det(A_{\bar{U}}),
  30. A U A_{U}
  31. A A
  32. A A
  33. U U
  34. U ¯ \bar{U}
  35. U U
  36. { 1 , , m } . \{1,\dots,m\}.
  37. A A
  38. perm ( A - 1 ) det ( A ) 2 = perm ( A ) \operatorname{perm}(A^{-1})\det(A)^{2}=\operatorname{perm}(A)
  39. U U
  40. U U
  41. U T U = I U^{T}U=I\,
  42. perm ( U ) 2 = det ( U + V ) det ( U ) \operatorname{perm}(U)^{2}=\det(U+V)\det(U)
  43. V V
  44. U U
  45. M ( G ) M(G)
  46. G G
  47. e e
  48. G G
  49. G G
  50. G e G\setminus e
  51. ρ = M ( G ) M ( G e ) \rho=\frac{M(G)}{M(G\setminus e)}
  52. M ( G ) M(G)
  53. ρ M ( G e ) \rho M(G\setminus e)
  54. M ( G e ) M(G\setminus e)

Concentrated_solar_power.html

  1. η \eta
  2. η R e c e i v e r \eta_{Receiver}
  3. η C a r n o t \eta_{Carnot}
  4. η = η Receiver η Carnot \eta=\eta_{\mathrm{Receiver}}\cdot\eta_{\mathrm{Carnot}}
  5. η Carnot = 1 - T 0 T H \eta_{\mathrm{Carnot}}=1-\frac{T^{0}}{T_{H}}
  6. η Receiver = Q absorbed - Q lost Q solar \eta_{\mathrm{Receiver}}=\frac{Q_{\mathrm{absorbed}}-Q_{\mathrm{lost}}}{Q_{% \mathrm{solar}}}
  7. Q solar Q_{\mathrm{solar}}
  8. Q absorbed Q_{\mathrm{absorbed}}
  9. Q lost Q_{\mathrm{lost}}
  10. η O p t i c s \eta_{Optics}
  11. α \alpha
  12. Q solar = η Optics I C A Q_{\mathrm{solar}}=\eta_{\mathrm{Optics}}ICA
  13. Q absorbed = α Q solar Q_{\mathrm{absorbed}}=\alpha Q_{\mathrm{solar}}
  14. ϵ \epsilon
  15. Q lost = A ϵ σ T H 4 Q_{\mathrm{lost}}=A\epsilon\sigma T_{H}^{4}
  16. η Optics \eta_{\mathrm{Optics}}
  17. α \alpha
  18. ϵ \epsilon
  19. η = ( 1 - σ T H 4 I C ) ( 1 - T 0 T H ) \eta=\left(1-\frac{\sigma T_{H}^{4}}{IC}\right)\cdot\left(1-\frac{T^{0}}{T_{H}% }\right)
  20. T max = ( I C σ ) 0.25 T_{\mathrm{max}}=\left({\frac{IC}{\sigma}}\right)^{0.25}
  21. d η d T H ( T opt ) = 0 \frac{d\eta}{dT_{H}}(T_{\mathrm{opt}})=0
  22. T o p t 5 - ( 0.75 T 0 ) T opt 4 - T 0 I C 4 σ = 0 T_{opt}^{5}-(0.75T^{0})T_{\mathrm{opt}}^{4}-\frac{T^{0}IC}{4\sigma}=0

Concurrence_(quantum_computing).html

  1. 𝒞 ( ρ ) max ( 0 , λ 1 - λ 2 - λ 3 - λ 4 ) \mathcal{C}(\rho)\equiv\max(0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4})
  2. λ 1 , , λ 4 \lambda_{1},...,\lambda_{4}
  3. R = ρ ρ ~ ρ R=\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}
  4. ρ ~ = ( σ y σ y ) ρ * ( σ y σ y ) \tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})
  5. ρ \rho
  6. σ y \sigma_{y}
  7. λ i \lambda_{i}
  8. ρ ρ ~ \rho\tilde{\rho}
  9. λ i \lambda_{i}
  10. S L ( 2 , ) 2 SL(2,\mathbb{C})^{\otimes 2}

Concurrent_estimation.html

  1. B 0 B_{0}
  2. B 1 , , B n B_{1},\ldots,B_{n}
  3. n n
  4. n n

Condensation_lemma.html

  1. ( X , ) ( L α , ) (X,\in)\prec(L_{\alpha},\in)
  2. β α \beta\leq\alpha
  3. X = L β X=L_{\beta}
  4. L β L_{\beta}
  5. Σ 1 \Sigma_{1}
  6. α = ω 1 \alpha=\omega_{1}

Condensation_point.html

  1. 1 \aleph_{1}

Conditional_mutual_information.html

  1. I ( x ; z | y ) I(x;z|y)
  2. I ( y ; z | x ) I(y;z|x)
  3. I ( x ; y | z ) I(x;y|z)
  4. X , X,
  5. Y , Y,
  6. Z , Z,
  7. I ( X ; Y | Z ) = 𝔼 Z ( I ( X ; Y ) | Z ) = z Z p Z ( z ) y Y x X p X , Y | Z ( x , y | z ) log p X , Y | Z ( x , y | z ) p X | Z ( x | z ) p Y | Z ( y | z ) , I(X;Y|Z)=\mathbb{E}_{Z}\big(I(X;Y)|Z\big)=\sum_{z\in Z}p_{Z}(z)\sum_{y\in Y}% \sum_{x\in X}p_{X,Y|Z}(x,y|z)\log\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y% |z)},
  8. p p
  9. I ( X ; Y | Z ) = z Z y Y x X p X , Y , Z ( x , y , z ) log p Z ( z ) p X , Y , Z ( x , y , z ) p X , Z ( x , z ) p Y , Z ( y , z ) . I(X;Y|Z)=\sum_{z\in Z}\sum_{y\in Y}\sum_{x\in X}p_{X,Y,Z}(x,y,z)\log\frac{p_{Z% }(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}.
  10. I ( X ; Y | Z ) = H ( X , Z ) + H ( Y , Z ) - H ( X , Y , Z ) - H ( Z ) = H ( X | Z ) - H ( X | Y , Z ) . I(X;Y|Z)=H(X,Z)+H(Y,Z)-H(X,Y,Z)-H(Z)=H(X|Z)-H(X|Y,Z).
  11. I ( X ; Y | Z ) = I ( X ; Y , Z ) - I ( X ; Z ) I(X;Y|Z)=I(X;Y,Z)-I(X;Z)
  12. I ( X ; Y , Z ) = I ( X ; Z ) + I ( X ; Y | Z ) I(X;Y,Z)=I(X;Z)+I(X;Y|Z)
  13. I ( X ; Y | Z ) = H ( Z | X ) + H ( X ) + H ( Z | Y ) + H ( Y ) - H ( Z | X , Y ) - H ( X , Y ) - H ( Z ) = I ( X ; Y ) + H ( Z | X ) + H ( Z | Y ) - H ( Z | X , Y ) - H ( Z ) I(X;Y|Z)=H(Z|X)+H(X)+H(Z|Y)+H(Y)-H(Z|X,Y)-H(X,Y)-H(Z)=I(X;Y)+H(Z|X)+H(Z|Y)-H(Z% |X,Y)-H(Z)
  14. I ( X ; Y | Z ) - I ( X ; Y ) I(X;Y|Z)-I(X;Y)
  15. I ( X ; Y | Z ) 0 I(X;Y|Z)\geq 0
  16. I ( X ; Y | Z ) = D KL [ p ( X , Y , Z ) p ( X | Z ) p ( Y | Z ) p ( Z ) ] . I(X;Y|Z)=D_{\mathrm{KL}}[p(X,Y,Z)\|p(X|Z)p(Y|Z)p(Z)].
  17. I ( X ; Y | Z ) = z Z p ( Z = z ) D KL [ p ( X , Y | z ) p ( X | z ) p ( Y | z ) ] , I(X;Y|Z)=\sum_{z\in Z}p(Z=z)D_{\mathrm{KL}}[p(X,Y|z)\|p(X|z)p(Y|z)],
  18. I ( X ; Y | Z ) = y Y p ( Y = y ) D KL [ p ( X , Z | y ) p ( X | Z ) p ( Z | y ) ] . I(X;Y|Z)=\sum_{y\in Y}p(Y=y)D_{\mathrm{KL}}[p(X,Z|y)\|p(X|Z)p(Z|y)].
  19. ( Ω , , 𝔓 ) (\Omega,\mathcal{F},\mathfrak{P})
  20. Ω \Omega
  21. 𝔓 \mathfrak{P}
  22. \mathcal{F}
  23. X * 𝔓 = 𝔓 ( X - 1 ( ) ) . X_{*}\mathfrak{P}=\mathfrak{P}\big(X^{-1}(\cdot)\big).
  24. supp X = supp X * 𝔓 . \mathrm{supp}\,X=\mathrm{supp}\,X_{*}\mathfrak{P}.
  25. M M
  26. Ω , \Omega,
  27. M , M\in\mathcal{F},
  28. x supp X . x\in\mathrm{supp}\,X.
  29. 𝔓 ( M | X = x ) = lim U x 𝔓 ( M { X U } ) 𝔓 ( { X U } ) and 𝔓 ( M | X ) = M d 𝔓 ( ω | X = X ( ω ) ) , \mathfrak{P}(M|X=x)=\lim_{U\ni x}\frac{\mathfrak{P}(M\cap\{X\in U\})}{% \mathfrak{P}(\{X\in U\})}\qquad\textrm{and}\qquad\mathfrak{P}(M|X)=\int_{M}d% \mathfrak{P}\big(\omega|X=X(\omega)\big),
  30. U U
  31. x x
  32. I ( X ; Y | Z ) = Ω log d 𝔓 ( ω | X , Z ) d 𝔓 ( ω | Y , Z ) d 𝔓 ( ω | Z ) d 𝔓 ( ω | X , Y , Z ) d 𝔓 ( ω ) , I(X;Y|Z)=\int_{\Omega}\log\frac{d\mathfrak{P}(\omega|X,Z)\,d\mathfrak{P}(% \omega|Y,Z)}{d\mathfrak{P}(\omega|Z)\,d\mathfrak{P}(\omega|X,Y,Z)}d\mathfrak{P% }(\omega),
  33. I ( A ; B | C ) , I(A;B|C),
  34. A , A,
  35. B , B,
  36. C C
  37. I ( A 0 , A 1 ; B 1 , B 2 , B 3 | C 0 , C 1 ) . I(A_{0},A_{1};B_{1},B_{2},B_{3}|C_{0},C_{1}).
  38. \wedge
  39. I ( X 1 ; ; X n + 1 ) = I ( X 1 ; ; X n ) - I ( X 1 ; ; X n | X n + 1 ) , I(X_{1};\ldots;X_{n+1})=I(X_{1};\ldots;X_{n})-I(X_{1};\ldots;X_{n}|X_{n+1}),
  40. I ( X 1 ; ; X n | X n + 1 ) = 𝔼 X n + 1 ( I ( X 1 ; ; X n ) | X n + 1 ) . I(X_{1};\ldots;X_{n}|X_{n+1})=\mathbb{E}_{X_{n+1}}\big(I(X_{1};\ldots;X_{n})|X% _{n+1}\big).
  41. 2 n - 1 2^{n}-1

Conditionality_principle.html

  1. ( E h , x h ) (E_{h},x_{h})

Conditioning_(probability).html

  1. ( Y = 0 | X = x ) = ( 7 x ) ( 10 x ) = 7 ! ( 10 - x ) ! ( 7 - x ) ! 10 ! \mathbb{P}(Y=0|X=x)=\frac{{\left({{7}\atop{x}}\right)}}{{\left({{10}\atop{x}}% \right)}}=\frac{7!(10-x)!}{(7-x)!10!}
  2. ( Y = 0 | X ) = { ( 7 X ) / ( 10 X ) for X 7 , 0 for X > 7. \mathbb{P}(Y=0|X)=\begin{cases}{\left({{7}\atop{X}}\right)}/{\left({{10}\atop{% X}}\right)}&\,\text{for }X\leq 7,\\ 0&\,\text{for }X>7.\end{cases}
  3. 𝔼 ( ( Y = 0 | X ) ) = x ( Y = 0 | X = x ) ( X = x ) = ( Y = 0 ) , \mathbb{E}(\mathbb{P}(Y=0|X))=\sum_{x}\mathbb{P}(Y=0|X=x)\mathbb{P}(X=x)=% \mathbb{P}(Y=0),
  4. x = 0 7 ( 7 x ) ( 10 x ) 1 2 10 ( 10 x ) = 1 8 , \sum_{x=0}^{7}\frac{{\left({{7}\atop{x}}\right)}}{{\left({{10}\atop{x}}\right)% }}\cdot\frac{1}{2^{10}}{\left({{10}\atop{x}}\right)}=\frac{1}{8},
  5. 𝔼 ( Y | X = x ) = 3 10 x \mathbb{E}(Y|X=x)=\frac{3}{10}x
  6. 𝔼 ( Y | X ) = 3 10 X . \mathbb{E}(Y|X)=\frac{3}{10}X.
  7. 𝔼 ( 𝔼 ( Y | X ) ) = x 𝔼 ( Y | X = x ) ( X = x ) = 𝔼 ( Y ) , \mathbb{E}(\mathbb{E}(Y|X))=\sum_{x}\mathbb{E}(Y|X=x)\mathbb{P}(X=x)=\mathbb{E% }(Y),
  8. x = 0 10 3 10 x 1 2 10 ( 10 x ) = 3 2 , \sum_{x=0}^{10}\tfrac{3}{10}x\cdot\tfrac{1}{2^{10}}{\left({{10}\atop{x}}\right% )}=\tfrac{3}{2},
  9. 𝔼 ( 3 10 X ) = 3 10 𝔼 ( X ) = 3 10 5 = 3 2 , \mathbb{E}\left(\tfrac{3}{10}X\right)=\tfrac{3}{10}\mathbb{E}(X)=\tfrac{3}{10}% \cdot 5=\tfrac{3}{2},
  10. Ω = { X = x 1 } { X = x 2 } \Omega=\{X=x_{1}\}\uplus\{X=x_{2}\}\uplus\dots
  11. x 1 , x 2 , x_{1},x_{2},\dots
  12. ( Y = y | X = x ) = ( 3 y ) ( 7 x - y ) ( 10 x ) = ( x y ) ( 10 - x 3 - y ) ( 10 3 ) \mathbb{P}(Y=y|X=x)=\frac{{\left({{3}\atop{y}}\right)}{\left({{7}\atop{x-y}}% \right)}}{{\left({{10}\atop{x}}\right)}}=\frac{{\left({{x}\atop{y}}\right)}{% \left({{10-x}\atop{3-y}}\right)}}{{\left({{10}\atop{3}}\right)}}
  13. n R R + W n\frac{R}{R+W}
  14. x = 0 10 ( Y = y | X = x ) ( X = x ) = ( Y = y ) = 1 2 3 ( 3 y ) \sum_{x=0}^{10}\mathbb{P}(Y=y|X=x)\mathbb{P}(X=x)=\mathbb{P}(Y=y)=\frac{1}{2^{% 3}}{\left({{3}\atop{y}}\right)}
  15. f X , Y ( x , y ) = { 1 2 π 1 - x 2 - y 2 if x 2 + y 2 < 1 , 0 otherwise . f_{X,Y}(x,y)=\begin{cases}\frac{1}{2\pi\sqrt{1-x^{2}-y^{2}}}&\,\text{if }x^{2}% +y^{2}<1,\\ 0&\,\text{otherwise}.\end{cases}
  16. f X ( x ) = - + f X , Y ( x , y ) d y = - 1 - x 2 + 1 - x 2 d y 2 π 1 - x 2 - y 2 ; f_{X}(x)=\int_{-\infty}^{+\infty}f_{X,Y}(x,y)\,\mathrm{d}y=\int_{-\sqrt{1-x^{2% }}}^{+\sqrt{1-x^{2}}}\frac{\mathrm{d}y}{2\pi\sqrt{1-x^{2}-y^{2}}}\,;
  17. f X ( x ) = { 0.5 for - 1 < x < 1 , 0 otherwise , f_{X}(x)=\begin{cases}0.5&\,\text{for }-1<x<1,\\ 0&\,\text{otherwise},\end{cases}
  18. f Y | X = 0.5 ( y ) = f X , Y ( 0.5 , y ) f X ( 0.5 ) = { 1 π 0.75 - y 2 for - 0.75 < y < 0.75 , 0 otherwise . f_{Y|X=0.5}(y)=\frac{f_{X,Y}(0.5,y)}{f_{X}(0.5)}=\begin{cases}\frac{1}{\pi% \sqrt{0.75-y^{2}}}&\,\text{for }-\sqrt{0.75}<y<\sqrt{0.75},\\ 0&\,\text{otherwise}.\end{cases}
  19. ( Y 0.75 | X = 0.5 ) = - 0.75 f Y | X = 0.5 ( y ) d y = - 0.75 0.75 d y π 0.75 - y 2 = 1 2 + 1 π arcsin 0.75 = 5 6 . \mathbb{P}(Y\leq 0.75|X=0.5)=\int_{-\infty}^{0.75}f_{Y|X=0.5}(y)\,\mathrm{d}y=% \int_{-\sqrt{0.75}}^{0.75}\frac{\mathrm{d}y}{\pi\sqrt{0.75-y^{2}}}=\tfrac{1}{2% }+\tfrac{1}{\pi}\arcsin\sqrt{0.75}=\tfrac{5}{6}.
  20. ( Y y | X = x ) = 1 2 + 1 π arcsin y 1 - x 2 \mathbb{P}(Y\leq y|X=x)=\tfrac{1}{2}+\tfrac{1}{\pi}\arcsin\frac{y}{\sqrt{1-x^{% 2}}}
  21. - 1 - x 2 < y < 1 - x 2 \textstyle-\sqrt{1-x^{2}}<y<\sqrt{1-x^{2}}
  22. ( Y y | X ) = { 0 for X 2 1 - y 2 and y < 0 , 1 2 + 1 π arcsin y 1 - X 2 for X 2 < 1 - y 2 , 1 for X 2 1 - y 2 and y > 0. \mathbb{P}(Y\leq y|X)=\begin{cases}0&\,\text{for }X^{2}\geq 1-y^{2}\,\text{ % and }y<0,\\ \frac{1}{2}+\frac{1}{\pi}\arcsin\frac{y}{\sqrt{1-X^{2}}}&\,\text{for }X^{2}<1-% y^{2},\\ 1&\,\text{for }X^{2}\geq 1-y^{2}\,\text{ and }y>0.\end{cases}
  23. 𝔼 ( ( Y y | X ) ) = - + ( Y y | X = x ) f X ( x ) d x = ( Y y ) , \mathbb{E}(\mathbb{P}(Y\leq y|X))=\int_{-\infty}^{+\infty}\mathbb{P}(Y\leq y|X% =x)f_{X}(x)\,\mathrm{d}x=\mathbb{P}(Y\leq y),
  24. ( Y 0.75 | X = 0.5 ) = lim ε 0 + ( Y 0.75 | 0.5 - ε < X < 0.5 + ε ) = lim ε 0 + ( Y 0.75 , 0.5 - ε < X < 0.5 + ε ) ( 0.5 - ε < X < 0.5 + ε ) = lim ε 0 + 0.5 - ε 0.5 + ε d x - 0.75 d y f X , Y ( x , y ) 0.5 - ε 0.5 + ε d x f X ( x ) . \begin{aligned}\displaystyle\mathbb{P}(Y\leq 0.75|X=0.5)&\displaystyle=\lim_{% \varepsilon\to 0+}\mathbb{P}(Y\leq 0.75|0.5-\varepsilon<X<0.5+\varepsilon)\\ &\displaystyle=\lim_{\varepsilon\to 0+}\frac{\mathbb{P}(Y\leq 0.75,0.5-% \varepsilon<X<0.5+\varepsilon)}{\mathbb{P}(0.5-\varepsilon<X<0.5+\varepsilon)}% \\ &\displaystyle=\lim_{\varepsilon\to 0+}\frac{\int_{0.5-\varepsilon}^{0.5+% \varepsilon}\mathrm{d}x\int_{-\infty}^{0.75}\mathrm{d}y\,f_{X,Y}(x,y)}{\int_{0% .5-\varepsilon}^{0.5+\varepsilon}\mathrm{d}x\,f_{X}(x)}.\end{aligned}
  25. | Z | \displaystyle|Z|
  26. 𝔼 ( | Z | | X = x ) = 2 π 1 - x 2 \mathbb{E}(|Z||X=x)=\frac{2}{\pi}\sqrt{1-x^{2}}
  27. 𝔼 ( 𝔼 ( | Z | | X ) ) = - + 𝔼 ( | Z | | X = x ) f X ( x ) d x = 𝔼 ( | Z | ) , \mathbb{E}(\mathbb{E}(|Z||X))=\int_{-\infty}^{+\infty}\mathbb{E}(|Z||X=x)f_{X}% (x)\,\mathrm{d}x=\mathbb{E}(|Z|),
  28. - 1 + 1 2 π 1 - x 2 d x 2 = 1 2 , \int_{-1}^{+1}\frac{2}{\pi}\sqrt{1-x^{2}}\cdot\frac{\mathrm{d}x}{2}=\tfrac{1}{% 2},
  29. F Y | X = x ( y ) = ( Y y | X = x ) = 1 2 + 1 π arcsin y 1 - x 2 F_{Y|X=x}(y)=\mathbb{P}(Y\leq y|X=x)=\frac{1}{2}+\frac{1}{\pi}\arcsin\frac{y}{% \sqrt{1-x^{2}}}
  30. - + f Y | X = x ( y ) f X ( x ) d x = f Y ( y ) , \displaystyle\int_{-\infty}^{+\infty}f_{Y|X=x}(y)f_{X}(x)\,\mathrm{d}x=f_{Y}(y),
  31. 0.75 \sqrt{0.75}
  32. ( A | B ) = ( A B ) / ( B ) \textstyle\mathbb{P}(A|B)=\mathbb{P}(A\cap B)/\mathbb{P}(B)
  33. ( A | B ) = lim n ( A B n ) / ( B n ) \textstyle\mathbb{P}(A|B)=\lim_{n\to\infty}\mathbb{P}(A\cap B_{n})/\mathbb{P}(% B_{n})
  34. B 1 B 2 \textstyle B_{1}\supset B_{2}\supset\dots
  35. B 1 B 2 = B \textstyle B_{1}\cap B_{2}\cap\dots=B
  36. f 1 ( y ) = { 3 y for 0 y 1 / 3 , 1.5 ( 1 - y ) for 1 / 3 y 2 / 3 , 0.5 for 2 / 3 y 1 , f_{1}(y)=\begin{cases}3y&\,\text{for }0\leq y\leq 1/3,\\ 1.5(1-y)&\,\text{for }1/3\leq y\leq 2/3,\\ 0.5&\,\text{for }2/3\leq y\leq 1,\end{cases}
  37. I = { 1 if Y 1 / 3 , 0 otherwise , I=\begin{cases}1&\,\text{if }Y\leq 1/3,\\ 0&\,\text{otherwise},\end{cases}
  38. g 1 ( x ) = { 1 for 0 < x < 0.5 , 0 for x = 0.5 , 1 / 3 for 0.5 < x < 1. g_{1}(x)=\begin{cases}1&\,\text{for }0<x<0.5,\\ 0&\,\text{for }x=0.5,\\ 1/3&\,\text{for }0.5<x<1.\end{cases}
  39. g 1 ( x ) = lim ε 0 + ( Y 1 / 3 | x - ε X x + ε ) , g_{1}(x)=\lim_{\varepsilon\to 0+}\mathbb{P}(Y\leq 1/3|x-\varepsilon\leq X\leq x% +\varepsilon)\,,
  40. 1 ( X < 0.5 ) + 0 ( X = 0.5 ) + 1 3 ( X > 0.5 ) = 1 1 6 + 0 1 3 + 1 3 ( 1 6 + 1 3 ) = 1 3 , 1\cdot\mathbb{P}(X<0.5)+0\cdot\mathbb{P}(X=0.5)+\frac{1}{3}\cdot\mathbb{P}(X>0% .5)=1\cdot\frac{1}{6}+0\cdot\frac{1}{3}+\frac{1}{3}\cdot\left(\frac{1}{6}+% \frac{1}{3}\right)=\frac{1}{3},
  41. 0 1 g 2 ( f 2 ( y ) ) d y = 1 3 . \int_{0}^{1}g_{2}(f_{2}(y))\,\mathrm{d}y=\tfrac{1}{3}.
  42. g = d ν d μ , g=\frac{\mathrm{d}\nu}{\mathrm{d}\mu},
  43. μ ( B ) \displaystyle\mu(B)
  44. B . B\subset\mathbb{R}.
  45. ν ( B ) = ( X B | Y 1 3 ) ( Y 1 3 ) = 1 3 ( X B | Y 1 3 ) . \nu(B)=\mathbb{P}(X\in B|Y\leq\tfrac{1}{3})\mathbb{P}(Y\leq\tfrac{1}{3})=% \tfrac{1}{3}\mathbb{P}(X\in B|Y\leq\tfrac{1}{3}).
  46. 𝔼 ( Y - h 1 ( X ) ) 2 \displaystyle\mathbb{E}(Y-h_{1}(X))^{2}
  47. ( a - x 3 ) 2 + 2 ( a - 1 + 2 x 3 ) 2 \left(a-\frac{x}{3}\right)^{2}+2\left(a-1+\frac{2x}{3}\right)^{2}
  48. a = 2 - x 3 , a=\frac{2-x}{3},
  49. 1 3 a 2 - 5 9 a \frac{1}{3}a^{2}-\frac{5}{9}a
  50. a = 5 6 . a=\tfrac{5}{6}.
  51. h 1 ( x ) = { x / 3 for 0 < x < 0.5 , 5 / 6 for x = 0.5 , ( 2 - x ) / 3 for 0.5 < x < 1 , h_{1}(x)=\begin{cases}x/3&\,\text{for }0<x<0.5,\\ 5/6&\,\text{for }x=0.5,\\ (2-x)/3&\,\text{for }0.5<x<1,\end{cases}
  52. h 1 ( x ) = lim ε 0 + 𝔼 ( Y | x - ε X x + ε ) , h_{1}(x)=\lim_{\varepsilon\to 0+}\mathbb{E}(Y|x-\varepsilon\leq X\leq x+% \varepsilon),
  53. 0 1 h 1 ( f 1 ( y ) ) d y = 0 1 / 6 3 y 3 d y + \displaystyle\int_{0}^{1}h_{1}(f_{1}(y))\,\mathrm{d}y=\int_{0}^{1/6}\frac{3y}{% 3}\,\mathrm{d}y+
  54. h = d ν d μ , h=\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\,,
  55. μ ( B ) \displaystyle\mu(B)
  56. B . B\subset\mathbb{R}.
  57. F Y | X = 3 4 ( y ) = ( Y y | X = 3 4 ) = lim ε 0 + ( Y y | 3 4 - ε X 3 4 + ε ) = { 0 for - < y < 1 4 , 1 6 for y = 1 4 , 1 3 for 1 4 < y < 1 2 , 2 3 for y = 1 2 , 1 for 1 2 < y < , F_{Y|X=\frac{3}{4}}(y)=\mathbb{P}\left(Y\leq y|X=\tfrac{3}{4}\right)=\lim_{% \varepsilon\to 0^{+}}\mathbb{P}\left(Y\leq y|\tfrac{3}{4}-\varepsilon\leq X% \leq\tfrac{3}{4}+\varepsilon\right)=\begin{cases}0&\,\text{for }-\infty<y<% \tfrac{1}{4},\\ \tfrac{1}{6}&\,\text{for }y=\tfrac{1}{4},\\ \tfrac{1}{3}&\,\text{for }\tfrac{1}{4}<y<\tfrac{1}{2},\\ \tfrac{2}{3}&\,\text{for }y=\tfrac{1}{2},\\ 1&\,\text{for }\tfrac{1}{2}<y<\infty,\end{cases}
  58. F Y | X = 3 4 ( y ) = ( Y y | X = 3 4 ) = { 0 for - < y < 1 4 , 1 3 for 1 4 y < 1 2 , 1 for 1 2 y < F_{Y|X=\frac{3}{4}}(y)=\mathbb{P}\left(Y\leq y|X=\tfrac{3}{4}\right)=\begin{% cases}0&\,\text{for }-\infty<y<\tfrac{1}{4},\\ \tfrac{1}{3}&\,\text{for }\tfrac{1}{4}\leq y<\tfrac{1}{2},\\ 1&\,\text{for }\tfrac{1}{2}\leq y<\infty\end{cases}
  59. 𝔼 ( I - g ( X ) ) 2 \displaystyle\mathbb{E}(I-g(X))^{2}

Conductor-discriminant_formula.html

  1. L / K L/K
  2. Irr ( G ) \mathrm{Irr}(G)
  3. G = G ( L / K ) G=G(L/K)
  4. L / K L/K
  5. G G
  6. 𝔡 L / K = χ Irr ( G ) 𝔣 ( χ ) χ ( 1 ) , \mathfrak{d}_{L/K}=\prod_{\chi\in\mathrm{Irr}(G)}\mathfrak{f}(\chi)^{\chi(1)},
  7. 𝔣 ( χ ) \mathfrak{f}(\chi)
  8. χ \chi
  9. L = 𝐐 ( ζ p n ) / 𝐐 L=\mathbf{Q}(\zeta_{p^{n}})/\mathbf{Q}
  10. G G
  11. ( 𝐙 / p n ) × (\mathbf{Z}/p^{n})^{\times}
  12. ( p ) (p)
  13. 𝔣 ( χ ) \mathfrak{f}(\chi)
  14. 𝔣 ( p ) ( χ ) \mathfrak{f}_{(p)}(\chi)
  15. G G
  16. χ \chi
  17. 1 = χ ( 1 ) 1=\chi(1)
  18. χ \chi
  19. 𝔭 \mathfrak{p}
  20. L χ = L ker ( χ ) / 𝐐 L^{\chi}=L^{\mathrm{ker}(\chi)}/\mathbf{Q}
  21. ( p ) n p (p)^{n_{p}}
  22. n p n_{p}
  23. U 𝐐 p ( n p ) N L 𝔭 χ / 𝐐 p ( U L 𝔭 χ ) U_{\mathbf{Q}_{p}}^{(n_{p})}\subseteq N_{L^{\chi}_{\mathfrak{p}}/\mathbf{Q}_{p% }}(U_{L^{\chi}_{\mathfrak{p}}})
  24. p > 2 p>2
  25. G ( L 𝔭 / 𝐐 p ) = G ( L / 𝐐 p ) = ( 𝐙 / p n ) × G(L_{\mathfrak{p}}/\mathbf{Q}_{p})=G(L/\mathbf{Q}_{p})=(\mathbf{Z}/p^{n})^{\times}
  26. φ ( p n ) \varphi(p^{n})
  27. U 𝐐 p / U 𝐐 p ( k ) = ( 𝐙 / p k ) × U_{\mathbf{Q}_{p}}/U^{(k)}_{\mathbf{Q}_{p}}=(\mathbf{Z}/p^{k})^{\times}
  28. 𝔣 ( p ) ( χ ) = ( p φ ( p n ) ( n - 1 / ( p - 1 ) ) ) \mathfrak{f}_{(p)}(\chi)=(p^{\varphi(p^{n})(n-1/(p-1))})
  29. i = 0 n - 1 ( φ ( p n ) - φ ( p i ) ) = n φ ( p n ) - 1 - ( p - 1 ) i = 0 n - 2 p i = n φ ( p n ) - p n - 1 . \sum_{i=0}^{n-1}(\varphi(p^{n})-\varphi(p^{i}))=n\varphi(p^{n})-1-(p-1)\sum_{i% =0}^{n-2}p^{i}=n\varphi(p^{n})-p^{n-1}.

Conductor_of_an_abelian_variety.html

  1. f P = 2 u P + t P + δ P , f_{P}=2u_{P}+t_{P}+\delta_{P},\,
  2. δ P \delta_{P}\in\mathbb{N}
  3. f = P P f P . f=\prod_{P}P^{f_{P}}.
  4. u P = t P = 0 u_{P}=t_{P}=0
  5. f P = δ P = 0 f_{P}=\delta_{P}=0
  6. u P = 0 u_{P}=0
  7. δ P = 0 \delta_{P}=0

Confidence_and_prediction_bands.html

  1. f ^ ( x ) ± w ( x ) \hat{f}(x)\pm w(x)
  2. Pr ( f ^ ( x ) - w ( x ) f ( x ) f ^ ( x ) + w ( x ) ) = 1 - α , {\rm Pr}\Big(\hat{f}(x)-w(x)\leq f(x)\leq\hat{f}(x)+w(x)\Big)=1-\alpha,
  3. f ^ ( x ) \hat{f}(x)
  4. f ^ ( x ) ± w ( x ) \hat{f}(x)\pm w(x)
  5. Pr ( f ^ ( x ) - w ( x ) f ( x ) f ^ ( x ) + w ( x ) for all x ) = 1 - α . {\rm Pr}\Big(\hat{f}(x)-w(x)\leq f(x)\leq\hat{f}(x)+w(x)\;\;\;\;\,\text{ for % all }x\Big)=1-\alpha.
  6. f ^ ( x ) ± w ( x ) \hat{f}(x)\pm w(x)
  7. Pr ( f ^ ( x ) - w ( x ) y * f ^ ( x ) + w ( x ) ) = 1 - α , {\rm Pr}\Big(\hat{f}(x)-w(x)\leq y^{*}\leq\hat{f}(x)+w(x)\Big)=1-\alpha,
  8. f ^ ( x ) \hat{f}(x)

Confusion_of_the_inverse.html

  1. P ( malignant | positive ) \displaystyle{}\qquad P(\,\text{malignant}|\,\text{positive})
  2. P ( ill ) = 1 % = 0.01 and P ( well ) = 99 % = 0.99. P(\,\text{ill})=1\%=0.01\,\text{ and }P(\,\text{well})=99\%=0.99.
  3. P ( positive | well ) = 1 % , and P ( negative | well ) = 99 % . P(\,\text{positive}|\,\text{well})=1\%,\,\text{ and }P(\,\text{negative}|\,% \text{well})=99\%.
  4. P ( negative | ill ) = 1 % and P ( positive | ill ) = 99 % . P(\,\text{negative}|\,\text{ill})=1\%\,\text{ and }P(\,\text{positive}|\,\text% {ill})=99\%.
  5. P ( well negative ) = P ( well ) × P ( negative | well ) = 99 % × 99 % = 98.01 % . P(\,\text{well}\cap\,\text{negative})=P(\,\text{well})\times P(\,\text{% negative}|\,\text{well})=99\%\times 99\%=98.01\%.
  6. P ( ill positive ) = P ( ill ) × P ( positive | ill ) = 1 % × 99 % = 0.99 % . P(\,\text{ill}\cap\,\text{positive})=P(\,\text{ill})\times P(\,\text{positive}% |\,\text{ill})=1\%\times 99\%=0.99\%.
  7. P ( well positive ) = P ( well ) × P ( positive | well ) = 99 % × 1 % = 0.99 % . P(\,\text{well}\cap\,\text{positive})=P(\,\text{well})\times P(\,\text{% positive}|\,\text{well})=99\%\times 1\%=0.99\%.
  8. P ( ill negative ) = P ( ill ) × P ( negative | ill ) = 1 % × 1 % = 0.01 % . P(\,\text{ill}\cap\,\text{negative})=P(\,\text{ill})\times P(\,\text{negative}% |\,\text{ill})=1\%\times 1\%=0.01\%.
  9. P ( positive ) \displaystyle P(\,\text{positive})
  10. P ( ill | positive ) = P ( ill positive ) P ( positive ) = 0.99 % 1.98 % = 50 % . P(\,\text{ill}|\,\text{positive})=\frac{P(\,\text{ill}\cap\,\text{positive})}{% P(\,\text{positive})}=\frac{0.99\%}{1.98\%}=50\%.

Conic_section.html

  1. x 2 + y 2 = a 2 x^{2}+y^{2}=a^{2}\,
  2. 0 0\,
  3. 0 0\,
  4. a a\,
  5. \infty
  6. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  7. 1 - b 2 a 2 \sqrt{1-\frac{b^{2}}{a^{2}}}
  8. a 2 - b 2 \sqrt{a^{2}-b^{2}}
  9. b 2 a \frac{b^{2}}{a}
  10. b 2 a 2 - b 2 \frac{b^{2}}{\sqrt{a^{2}-b^{2}}}
  11. y 2 = 4 a x y^{2}=4ax\,
  12. 1 1\,
  13. - -\,
  14. 2 a 2a\,
  15. 2 a 2a\,
  16. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  17. 1 + b 2 a 2 \sqrt{1+\frac{b^{2}}{a^{2}}}
  18. a 2 + b 2 \sqrt{a^{2}+b^{2}}
  19. b 2 a \frac{b^{2}}{a}
  20. b 2 a 2 + b 2 \frac{b^{2}}{\sqrt{a^{2}+b^{2}}}
  21. p e = pe=\ell\,
  22. a e = c . ae=c.\,
  23. F F
  24. L L
  25. F F
  26. e e
  27. F F
  28. e e
  29. L L
  30. 0 < e < 1 0<e<1
  31. e = 1 e=1
  32. e > 1 e>1
  33. a / e a/e
  34. a a
  35. a e a\,e
  36. e = 0 e=0
  37. a a
  38. e e
  39. x 2 + y 2 = 1 x^{2}+y^{2}=1
  40. y = i w , y=iw,
  41. x 2 - w 2 = 1 x^{2}-w^{2}=1
  42. x 1 2 + x 2 2 + + x k 2 - x k + 1 2 - - x k + l 2 , x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2}-x_{k+1}^{2}-\cdots-x_{k+l}^{2},
  43. k + l + m = n . k+l+m=n.
  44. x 2 + y 2 x^{2}+y^{2}
  45. x 2 x^{2}
  46. x 2 - y 2 x^{2}-y^{2}
  47. x 2 + y 2 , x^{2}+y^{2},
  48. x 2 x^{2}
  49. x 2 - y 2 x^{2}-y^{2}
  50. 0 | tr | / 2 < 1 , 0\leq|\operatorname{tr}|/2<1,
  51. | tr | / 2 = 1 , |\operatorname{tr}|/2=1,
  52. | tr | / 2 > 1 , |\operatorname{tr}|/2>1,
  53. A x 2 + B x y + C y 2 + D x + E y + F = 0 with A , B , C not all zero. Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,\text{ with }A,B,C\,\text{ not all zero.}\,
  54. 𝐏 5 . \mathbf{P}^{5}.
  55. B 2 - 4 A C . B^{2}-4AC.\,
  56. B 2 - 4 A C < 0 B^{2}-4AC<0
  57. A = C A=C
  58. B = 0 B=0
  59. B 2 - 4 A C = 0 B^{2}-4AC=0
  60. B 2 - 4 A C > 0 B^{2}-4AC>0
  61. A + C = 0 A+C=0
  62. B 2 - 4 A C < 0 B^{2}-4AC<0
  63. x 2 + y 2 + 10 = 0 x^{2}+y^{2}+10=0
  64. [ x y ] . [ A B / 2 B / 2 C ] . [ x y ] + D x + E y + F = 0. \begin{bmatrix}x&y\end{bmatrix}.\begin{bmatrix}A&B/2\\ B/2&C\end{bmatrix}.\begin{bmatrix}x\\ y\end{bmatrix}+Dx+Ey+F=0.
  65. ( x - a y - c ) T ( A B 2 B 2 C ) ( x - a y - c ) = G \left(\begin{array}[]{c}x-a\\ y-c\end{array}\right)^{T}\left(\begin{array}[]{cc}A&\frac{B}{2}\\ \frac{B}{2}&C\end{array}\right)\left(\begin{array}[]{c}x-a\\ y-c\end{array}\right)=G
  66. D + 2 a A + B c = 0 , E + 2 C c + B a = 0 , D+2aA+Bc=0,\,E+2Cc+Ba=0,\,
  67. G = A a 2 + C c 2 + B a c - F G=Aa^{2}+Cc^{2}+Bac-F
  68. [ x y 1 ] . [ A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ] . [ x y 1 ] = 0. \begin{bmatrix}x&y&1\end{bmatrix}.\begin{bmatrix}A&B/2&D/2\\ B/2&C&E/2\\ D/2&E/2&F\end{bmatrix}.\begin{bmatrix}x\\ y\\ 1\end{bmatrix}=0.
  69. A x 2 + B x y + C y 2 + D x + E y + F = 0 Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,
  70. A x 2 + B x y + C y 2 = - ( D x + E y + F ) . Ax^{2}+Bxy+Cy^{2}=-(Dx+Ey+F).\,
  71. z = A x 2 + B x y + C y 2 z=Ax^{2}+Bxy+Cy^{2}
  72. z = - ( D x + E y + F ) . z=-(Dx+Ey+F).
  73. D = E = 0 D=E=0
  74. z = - 1 z=-1
  75. A x 2 + B x y + C y 2 + D x + E y + F = 0 , Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,\,
  76. e = 2 ( A - C ) 2 + B 2 η ( A + C ) + ( A - C ) 2 + B 2 , e=\sqrt{\frac{2\sqrt{(A-C)^{2}+B^{2}}}{\eta(A+C)+\sqrt{(A-C)^{2}+B^{2}}}},
  77. x x
  78. y y
  79. y = x y=x
  80. y = x y=−x
  81. ( a c o s θ , a s i n θ ) (acosθ,asinθ)
  82. ( a c o s θ , b s i n θ ) (acosθ,bsinθ)
  83. ( a s e c θ , b t a n θ ) (asecθ,btanθ)
  84. ( ± a c o s h u , b s i n h u ) (±acoshu,bsinhu)
  85. [ A B / 2 B / 2 C ] \begin{bmatrix}A&B/2\\ B/2&C\end{bmatrix}
  86. r = l 1 - e cos θ r=\frac{l}{1-e\cos\theta}
  87. x 2 + y 2 = ( l + e x ) ( x - l e 1 - e 2 l 1 - e 2 ) 2 + ( 1 - e 2 ) y 2 l 2 = 1 \begin{aligned}\displaystyle\sqrt{x^{2}+y^{2}}=\left(l+ex\right)\\ \displaystyle\Rightarrow\left(\frac{x-\frac{le}{1-e^{2}}}{\frac{l}{1-e^{2}}}% \right)^{2}+\frac{\left(1-e^{2}\right)y^{2}}{l^{2}}=1\end{aligned}
  88. c = ( l e 1 - e 2 ) c=\left(\frac{le}{1-e^{2}}\right)
  89. x 2 + y 2 = r 2 x^{2}+y^{2}=r^{2}\,
  90. ( x - a 2 - b 2 ) 2 a 2 + y 2 b 2 = 1 \frac{\left(x-\sqrt{a^{2}-b^{2}}\right)^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  91. y 2 = 4 a ( x + a ) y^{2}=4a\left(x+a\right)
  92. ( x + a 2 + b 2 ) 2 a 2 - y 2 b 2 = 1 \frac{\left(x+\sqrt{a^{2}+b^{2}}\right)^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  93. A 1 x 2 + A 2 y 2 + A 3 z 2 + 2 B 1 x y + 2 B 2 x z + 2 B 3 y z = 0. A_{1}x^{2}+A_{2}y^{2}+A_{3}z^{2}+2B_{1}xy+2B_{2}xz+2B_{3}yz=0.
  94. [ x y z ] . [ A 1 B 1 B 2 B 1 A 2 B 3 B 2 B 3 A 3 ] . [ x y z ] = 0. \begin{bmatrix}x&y&z\end{bmatrix}.\begin{bmatrix}A_{1}&B_{1}&B_{2}\\ B_{1}&A_{2}&B_{3}\\ B_{2}&B_{3}&A_{3}\end{bmatrix}.\begin{bmatrix}x\\ y\\ z\end{bmatrix}=0.
  95. M = [ A 1 B 1 B 2 B 1 A 2 B 3 B 2 B 3 A 3 ] M=\begin{bmatrix}A_{1}&B_{1}&B_{2}\\ B_{1}&A_{2}&B_{3}\\ B_{2}&B_{3}&A_{3}\end{bmatrix}
  96. Δ = det ( M ) = det ( [ A 1 B 1 B 2 B 1 A 2 B 3 B 2 B 3 A 3 ] ) \Delta=\det(M)=\det\left(\begin{bmatrix}A_{1}&B_{1}&B_{2}\\ B_{1}&A_{2}&B_{3}\\ B_{2}&B_{3}&A_{3}\end{bmatrix}\right)
  97. [ x y z ] . [ 1 0 0 0 - 1 0 0 0 0 ] . [ x y z ] = 0 \begin{bmatrix}x&y&z\end{bmatrix}.\begin{bmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\end{bmatrix}.\begin{bmatrix}x\\ y\\ z\end{bmatrix}=0
  98. { x 2 - y 2 = 0 } = { ( x + y ) ( x - y ) = 0 } = { x + y = 0 } { x - y = 0 } . \{x^{2}-y^{2}=0\}=\{(x+y)(x-y)=0\}=\{x+y=0\}\cup\{x-y=0\}.\,
  99. { x 2 + 2 x y + y 2 = 0 } = { ( x + y ) 2 = 0 } = { x + y = 0 } { x + y = 0 } = { x + y = 0 } . \{x^{2}+2xy+y^{2}=0\}=\{(x+y)^{2}=0\}=\{x+y=0\}\cup\{x+y=0\}=\{x+y=0\}.\,
  100. δ = det ( [ A 1 B 1 B 1 A 2 ] ) \delta=\det\left(\begin{bmatrix}A_{1}&B_{1}\\ B_{1}&A_{2}\end{bmatrix}\right)
  101. r = l < m t p l > 1 + e cos θ r=\frac{l}{<}mtpl>{{1+e\cos\theta}}
  102. λ C 1 + μ C 2 \lambda C_{1}+\mu C_{2}
  103. C 1 C_{1}
  104. C 2 C_{2}
  105. λ C 1 + μ C 2 . \lambda C_{1}+\mu C_{2}.
  106. ( λ , μ ) (\lambda,\mu)
  107. det ( λ C 1 + μ C 2 ) = 0 \det(\lambda C_{1}+\mu C_{2})=0
  108. λ \lambda
  109. μ \mu
  110. C 0 C_{0}
  111. C 0 C_{0}

Conical_combination.html

  1. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}\,
  2. α 1 x 1 + α 2 x 2 + + α n x n \alpha_{1}x_{1}+\alpha_{2}x_{2}+\cdots+\alpha_{n}x_{n}
  3. α i \alpha_{i}\,
  4. α i 0. \alpha_{i}\geq 0.
  5. coni ( S ) = { i = 1 k α i x i | x i S , α i , α i 0 , i , k = 1 , 2 , } . \operatorname{coni}(S)=\Bigl\{\sum_{i=1}^{k}\alpha_{i}x_{i}\;\Big|\;x_{i}\in S% ,\,\alpha_{i}\in\mathbb{R},\,\alpha_{i}\geq 0,i,k=1,2,\dots\Bigr\}.

Conjugate_Fourier_series.html

  1. f ( θ ) = 1 2 a 0 + n = 1 ( a n cos n θ + b n sin n θ ) f(\theta)=\tfrac{1}{2}a_{0}+\sum_{n=1}^{\infty}\left(a_{n}\cos n\theta+b_{n}% \sin n\theta\right)
  2. F ( z ) = 1 2 a 0 + n = 1 ( a n - i b n ) z n F(z)=\tfrac{1}{2}a_{0}+\sum_{n=1}^{\infty}(a_{n}-ib_{n})z^{n}
  3. z = e i θ z=e^{i\theta}
  4. f ~ ( θ ) = n = 1 ( a n sin n θ - b n cos n θ ) . \tilde{f}(\theta)=\sum_{n=1}^{\infty}\left(a_{n}\sin n\theta-b_{n}\cos n\theta% \right).

Constant-energy_surface.html

  1. k \vec{k}

Constant_k_filter.html

  1. k 2 = Z Y k^{2}=\frac{Z}{Y}
  2. k = i ω L i ω C = L C k=\sqrt{\frac{i\omega L}{i\omega C}}=\sqrt{\frac{L}{C}}
  3. lim Z , Y 0 Z i = k \lim_{Z,Y\to 0}Z_{\mathrm{i}}=k
  4. Z iT 2 = Z 2 + k 2 {Z_{\mathrm{iT}}}^{2}=Z^{2}+k^{2}
  5. 1 Z i Π 2 = Y i Π 2 = Y 2 + 1 k 2 \frac{1}{{Z_{\mathrm{i\Pi}}}^{2}}={Y_{\mathrm{i\Pi}}}^{2}=Y^{2}+\frac{1}{k^{2}}
  6. Z iT 2 = - ( ω L ) 2 + L C {Z_{\mathrm{iT}}}^{2}=-(\omega L)^{2}+\frac{L}{C}
  7. ω c = 1 L C \omega_{c}=\frac{1}{\sqrt{LC}}
  8. Z iT = L ω c 2 - ω 2 Z_{\mathrm{iT}}=L\sqrt{\omega_{c}^{2}-\omega^{2}}
  9. Z iT = i L ω 2 - ω c 2 Z_{\mathrm{iT}}=iL\sqrt{\omega^{2}-\omega_{c}^{2}}
  10. γ = sinh - 1 Z k \gamma=\sinh^{-1}\frac{Z}{k}
  11. γ n = n γ \gamma_{n}=n\gamma\,\!
  12. γ = α + i β = 0 + i sin - 1 ω ω c \gamma=\alpha+i\beta=0+i\sin^{-1}\frac{\omega}{\omega_{c}}
  13. γ = α + i β = cosh - 1 ω ω c + i π 2 \gamma=\alpha+i\beta=\cosh^{-1}\frac{\omega}{\omega_{c}}+i\frac{\pi}{2}

Constant_problem.html

  1. | a 1 x 1 + + a n x n | = 0. |a_{1}x_{1}+{}\cdots{}+a_{n}x_{n}|=0.\,

Constrained_generalized_inverse.html

  1. x x
  2. A x = b ( with given A \R m × n and b \R m ) Ax=b\qquad(\,\text{with given }A\in\R^{m\times n}\,\text{ and }b\in\R^{m})
  3. L L
  4. \R m \R^{m}
  5. L L
  6. P L P_{L}
  7. A x = b x L Ax=b\qquad x\in L
  8. ( A P L ) x = b x \R m (AP_{L})x=b\qquad x\in\R^{m}
  9. L L
  10. \R m \R^{m}
  11. ( A P L ) (AP_{L})
  12. A A
  13. m = n m=n
  14. ( A P L ) (AP_{L})
  15. L L
  16. A A
  17. A A
  18. L L
  19. A L ( - 1 ) := P L ( A P L + P L ) - 1 , A_{L}^{(-1)}:=P_{L}(AP_{L}+P_{L^{\perp}})^{-1},

Contact_type.html

  1. Σ \Sigma
  2. ( M , ω ) (M,\omega)
  3. α \alpha
  4. j * ( ω ) = d α j^{*}(\omega)=d\alpha
  5. ( Σ , α ) (\Sigma,\alpha)
  6. j : Σ M j:\Sigma\to M

Container_(type_theory).html

  1. λ ( s , f ) . ( s , g f ) \lambda\left(s,f\right).\left(s,g\circ f\right)

Continuant_(mathematics).html

  1. K ( 0 ) = 1 ; K(0)=1;\,
  2. K ( 1 ) = a 1 ; K(1)=a_{1};\,
  3. K ( n ) = a n K ( n - 1 ) + K ( n - 2 ) . K(n)=a_{n}K(n-1)+K(n-2).\,
  4. K ( 0 ) = 1 ; K(0)=1;\,
  5. K ( 1 ) = a 1 ; K(1)=a_{1};\,
  6. K ( n ) = a n K ( n - 1 ) - b n - 1 c n - 1 K ( n - 2 ) . K(n)=a_{n}K(n-1)-b_{n-1}c_{n-1}K(n-2).\,
  7. [ a 0 ; a 1 , a 2 , ] [a_{0};a_{1},a_{2},\ldots]
  8. K ( n + 1 , ( a 0 , , a n ) ) K ( n , ( a 1 , , a n ) ) . \frac{K(n+1,(a_{0},\ldots,a_{n}))}{K(n,(a_{1},\ldots,a_{n}))}.
  9. ( a 1 b 1 0 0 0 c 1 a 2 b 2 0 0 0 c 2 a 3 0 0 0 0 0 a n - 1 b n - 1 0 0 0 c n - 1 a n ) . \begin{pmatrix}a_{1}&b_{1}&0&\ldots&0&0\\ c_{1}&a_{2}&b_{2}&\ldots&0&0\\ 0&c_{2}&a_{3}&\ldots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\ldots&a_{n-1}&b_{n-1}\\ 0&0&0&\ldots&c_{n-1}&a_{n}\end{pmatrix}.

Continuous_functions_on_a_compact_Hausdorff_space.html

  1. f = sup x X | f ( x ) | , \|f\|=\sup_{x\in X}|f(x)|,

Continuous_game.html

  1. G = ( P , 𝐂 , 𝐔 ) G=(P,\mathbf{C},\mathbf{U})
  2. P = 1 , 2 , 3 , , n P={1,2,3,\ldots,n}
  3. n n\,
  4. 𝐂 = ( C 1 , C 2 , , C n ) \mathbf{C}=(C_{1},C_{2},\ldots,C_{n})
  5. C i C_{i}\,
  6. i i\,
  7. 𝐔 = ( u 1 , u 2 , , u n ) \mathbf{U}=(u_{1},u_{2},\ldots,u_{n})
  8. u i : 𝐂 \R u_{i}:\mathbf{C}\to\R
  9. i i\,
  10. Δ i \Delta_{i}\,
  11. C i C_{i}\,
  12. s y m b o l σ = ( σ 1 , σ 2 , , σ n ) symbol{\sigma}=(\sigma_{1},\sigma_{2},\ldots,\sigma_{n})
  13. σ i Δ i \sigma_{i}\in\Delta_{i}\,
  14. s y m b o l σ - i symbol{\sigma}_{-i}
  15. i i
  16. i i\,
  17. b i b_{i}
  18. b i b_{i}\,
  19. i i
  20. b i ( σ - i ) b_{i}(\sigma_{-i})\,
  21. σ - i \sigma_{-i}
  22. 𝐛 ( s y m b o l σ ) = b 1 ( σ - 1 ) × b 2 ( σ - 2 ) × × b n ( σ - n ) \mathbf{b}(symbol{\sigma})=b_{1}(\sigma_{-1})\times b_{2}(\sigma_{-2})\times% \cdots\times b_{n}(\sigma_{-n})
  23. s y m b o l σ * symbol{\sigma}*
  24. s y m b o l σ * 𝐛 ( s y m b o l σ * ) symbol{\sigma}*\in\mathbf{b}(symbol{\sigma}*)
  25. C i C_{i}\,
  26. u i : 𝐂 \R u_{i}:\mathbf{C}\to\R
  27. u i ( 𝐬 ) = k 1 = 1 m 1 k n = 1 m n a i , k 1 k n f 1 ( s 1 ) f n ( s n ) u_{i}(\mathbf{s})=\sum_{k_{1}=1}^{m_{1}}\ldots\sum_{k_{n}=1}^{m_{n}}a_{i\,,\,k% _{1}\ldots k_{n}}f_{1}(s_{1})\ldots f_{n}(s_{n})
  28. 𝐬 𝐂 \mathbf{s}\in\mathbf{C}
  29. s i C i s_{i}\in C_{i}
  30. a i , k 1 k n \R a_{i\,,\,k_{1}\ldots k_{n}}\in\R
  31. f i , k : C i \R f_{i\,,\,k}:C_{i}\to\R
  32. C i C_{i}\,
  33. \R \R\,
  34. m i + 1 m_{i}+1\,
  35. C X = C Y = [ 0 , 1 ] C_{X}=C_{Y}=\left[0,1\right]
  36. C X C_{X}\,
  37. C Y C_{Y}\,
  38. x x\,
  39. y y\,
  40. H ( x , y ) = u x ( x , y ) = - u y ( x , y ) H(x,y)=u_{x}(x,y)=-u_{y}(x,y)\,
  41. H ( x , y ) = ( x - y ) 2 H(x,y)=(x-y)^{2}\,
  42. b X ( y ) = { 1 , if y [ 0 , 1 / 2 ) 0 or 1 , if y = 1 / 2 0 , if y ( 1 / 2 , 1 ] b_{X}(y)=\begin{cases}1,&\mbox{if }~{}y\in\left[0,1/2\right)\\ 0\,\text{ or }1,&\mbox{if }~{}y=1/2\\ 0,&\mbox{if }~{}y\in\left(1/2,1\right]\end{cases}
  43. b Y ( x ) = x b_{Y}(x)=x\,
  44. b X ( y ) b_{X}(y)\,
  45. b Y ( x ) b_{Y}(x)\,
  46. v = 𝔼 [ H ( x , y ) ] v=\mathbb{E}[H(x,y)]
  47. v = μ X 2 - 2 μ X 1 μ Y 1 + μ Y 2 v=\mu_{X2}-2\mu_{X1}\mu_{Y1}+\mu_{Y2}\,
  48. μ X N = 𝔼 [ x N ] \mu_{XN}=\mathbb{E}[x^{N}]
  49. μ X 1 \mu_{X1}\,
  50. μ X 2 \mu_{X2}
  51. μ X 1 μ X 2 μ X 1 2 μ X 2 μ Y 1 μ Y 2 μ Y 1 2 μ Y 2 \begin{aligned}\displaystyle\mu_{X1}\geq\mu_{X2}\\ \displaystyle\mu_{X1}^{2}\leq\mu_{X2}\end{aligned}\qquad\begin{aligned}% \displaystyle\mu_{Y1}\geq\mu_{Y2}\\ \displaystyle\mu_{Y1}^{2}\leq\mu_{Y2}\end{aligned}
  52. v v\,
  53. μ i 1 = μ i 2 or μ i 1 2 = μ i 2 \mu_{i1}=\mu_{i2}\,\text{ or }\mu_{i1}^{2}=\mu_{i2}
  54. μ i 1 = μ i 2 \mu_{i1}=\mu_{i2}\,
  55. b Y ( μ X 1 , μ X 2 ) b_{Y}(\mu_{X1},\mu_{X2})\,
  56. y = μ X 1 y=\mu_{X1}\,
  57. b Y b_{Y}\,
  58. b x b_{x}\,
  59. ( μ X 1 * , μ X 2 * , μ Y 1 * , μ Y 2 * ) = ( 1 / 2 , 1 / 2 , 1 / 2 , 1 / 4 ) (\mu_{X1}*,\mu_{X2}*,\mu_{Y1}*,\mu_{Y2}*)=(1/2,1/2,1/2,1/4)\,
  60. C X = C Y = [ 0 , 1 ] C_{X}=C_{Y}=\left[0,1\right]
  61. C X C_{X}\,
  62. C Y C_{Y}\,
  63. x x\,
  64. y y\,
  65. H ( x , y ) = u x ( x , y ) = - u y ( x , y ) H(x,y)=u_{x}(x,y)=-u_{y}(x,y)\,
  66. H ( x , y ) = ( 1 + x ) ( 1 + y ) ( 1 - x y ) ( 1 + x y ) 2 . H(x,y)=\frac{(1+x)(1+y)(1-xy)}{(1+xy)^{2}}.
  67. f * ( x ) = 2 π x ( 1 + x ) g * ( y ) = 2 π y ( 1 + y ) . f^{*}(x)=\frac{2}{\pi\sqrt{x}(1+x)}\qquad g^{*}(y)=\frac{2}{\pi\sqrt{y}(1+y)}.
  68. 4 / π 4/\pi
  69. C X = C Y = [ 0 , 1 ] C_{X}=C_{Y}=\left[0,1\right]
  70. C X C_{X}\,
  71. C Y C_{Y}\,
  72. x x\,
  73. y y\,
  74. H ( x , y ) = u x ( x , y ) = - u y ( x , y ) H(x,y)=u_{x}(x,y)=-u_{y}(x,y)\,
  75. H ( x , y ) = n = 0 1 2 n ( 2 x n - ( ( 1 - x 3 ) n - ( x 3 ) n ) ) ( 2 y n - ( ( 1 - y 3 ) n - ( y 3 ) n ) ) H(x,y)=\sum_{n=0}^{\infty}\frac{1}{2^{n}}\left(2x^{n}-\left(\left(1-\frac{x}{3% }\right)^{n}-\left(\frac{x}{3}\right)^{n}\right)\right)\left(2y^{n}-\left(% \left(1-\frac{y}{3}\right)^{n}-\left(\frac{y}{3}\right)^{n}\right)\right)

Continuous_mapping_theorem.html

  1. X n 𝑑 X g ( X n ) 𝑑 g ( X ) ; X_{n}\ \xrightarrow{d}\ X\quad\Rightarrow\quad g(X_{n})\ \xrightarrow{d}\ g(X);
  2. X n 𝑝 X g ( X n ) 𝑝 g ( X ) ; X_{n}\ \xrightarrow{p}\ X\quad\Rightarrow\quad g(X_{n})\ \xrightarrow{p}\ g(X);
  3. X n a s X g ( X n ) a s g ( X ) . X_{n}\ \xrightarrow{\!\!as\!\!}\ X\quad\Rightarrow\quad g(X_{n})\ \xrightarrow% {\!\!as\!\!}\ g(X).
  4. X n 𝑑 X X_{n}\xrightarrow{d}X
  5. lim sup n Pr ( X n F ) Pr ( X F ) for every closed set F . \limsup_{n\to\infty}\operatorname{Pr}(X_{n}\in F)\leq\operatorname{Pr}(X\in F)% \,\text{ for every closed set }F.
  6. g - 1 ( F ) ¯ g - 1 ( F ) D g . \overline{g^{-1}(F)}\ \subset\ g^{-1}(F)\cup D_{g}\ .
  7. Pr ( g ( X n ) F ) = Pr ( X n g - 1 ( F ) ) Pr ( X n g - 1 ( F ) ¯ ) , \operatorname{Pr}\big(g(X_{n})\in F\big)=\operatorname{Pr}\big(X_{n}\in g^{-1}% (F)\big)\leq\operatorname{Pr}\big(X_{n}\in\overline{g^{-1}(F)}\big),
  8. Pr ( X g - 1 ( F ) ¯ ) Pr ( X g - 1 ( F ) D g ) \displaystyle\operatorname{Pr}\big(X\in\overline{g^{-1}(F)}\big)\leq% \operatorname{Pr}\big(X\in g^{-1}(F)\cup D_{g}\big)\leq
  9. lim sup n Pr ( g ( X n ) F ) Pr ( g ( X ) F ) , \limsup_{n\to\infty}\operatorname{Pr}\big(g(X_{n})\in F\big)\leq\operatorname{% Pr}\big(g(X)\in F\big),
  10. B δ = { x S | x D g : y S : | x - y | < δ , | g ( x ) - g ( y ) | > ε } . B_{\delta}=\big\{x\in S\ \big|\ x\notin D_{g}:\ \exists y\in S:\ |x-y|<\delta,% \,|g(x)-g(y)|>\varepsilon\big\}.
  11. Pr ( | g ( X n ) - g ( X ) | > ε ) Pr ( | X n - X | δ ) + Pr ( X B δ ) + Pr ( X D g ) . \operatorname{Pr}\big(\big|g(X_{n})-g(X)\big|>\varepsilon\big)\leq% \operatorname{Pr}\big(|X_{n}-X|\geq\delta\big)+\operatorname{Pr}(X\in B_{% \delta})+\operatorname{Pr}(X\in D_{g}).
  12. lim n Pr ( | g ( X n ) - g ( X ) | > ε ) = 0 , \lim_{n\to\infty}\operatorname{Pr}\big(\big|g(X_{n})-g(X)\big|>\varepsilon\big% )=0,
  13. lim n X n ( ω ) = X ( ω ) lim n g ( X n ( ω ) ) = g ( X ( ω ) ) \lim_{n\to\infty}X_{n}(\omega)=X(\omega)\quad\Rightarrow\quad\lim_{n\to\infty}% g(X_{n}(\omega))=g(X(\omega))
  14. Pr ( lim n g ( X n ) = g ( X ) ) \displaystyle\operatorname{Pr}\Big(\lim_{n\to\infty}g(X_{n})=g(X)\Big)

Continuous_reactor.html

  1. q x = U A ( T p - T j ) q_{x}=UA(T_{p}-T_{j})

Control_table.html

  1. O ( 1 ) O(1)\,

Controlled_invariant_subspace.html

  1. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) . \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t).

Convection–diffusion_equation.html

  1. c t = ( D c ) - ( v c ) + R \frac{\partial c}{\partial t}=\nabla\cdot(D\nabla c)-\nabla\cdot(\vec{v}c)+R
  2. v \vec{v}
  3. v \vec{v}
  4. v \vec{v}
  5. v \vec{v}
  6. \nabla
  7. \nabla\cdot
  8. ( D c ) \nabla\cdot(D\nabla c)
  9. - ( v c ) -\nabla\cdot(\vec{v}c)
  10. c t = D 2 c - v c . \frac{\partial c}{\partial t}=D\nabla^{2}c-\vec{v}\cdot\nabla c.
  11. c / t = 0 \partial c/\partial t=0
  12. 0 = ( D c ) - ( v c ) + R . 0=\nabla\cdot(D\nabla c)-\nabla\cdot(\vec{v}c)+R.
  13. c t + j = R , \frac{\partial c}{\partial t}+\nabla\cdot\vec{j}=R,
  14. j \vec{j}
  15. j diffusion = - D c \vec{j}_{\,\text{diffusion}}=-D\,\nabla c
  16. j advective = v c \vec{j}_{\,\text{advective}}=\vec{v}\,c
  17. j = j diffusion + j advective = - D c + v c . \vec{j}=\vec{j}_{\,\text{diffusion}}+\vec{j}_{\,\text{advective}}=-D\,\nabla c% +\vec{v}\,c.
  18. c t + ( - D c + v c ) = R . \frac{\partial c}{\partial t}+\nabla\cdot\left(-D\,\nabla c+\vec{v}\,c\right)=R.
  19. v \vec{v}
  20. v \vec{v}
  21. v \vec{v}
  22. c t = ( D c ) - ( ζ - 1 F c ) + R \frac{\partial c}{\partial t}=\nabla\cdot(D\nabla c)-\nabla\cdot(\zeta^{-1}% \vec{F}c)+R
  23. F \vec{F}
  24. ζ \zeta
  25. ζ - 1 \zeta^{-1}
  26. F = - U \vec{F}=-\nabla U
  27. c exp ( - D - 1 ζ - 1 U ) c\propto\exp(-D^{-1}\zeta^{-1}U)
  28. ζ \zeta
  29. D ζ = k B T D\zeta=k_{B}T
  30. v \vec{v}
  31. 𝐌 t = μ ρ 2 𝐌 - 𝐯 𝐌 + ( 𝐟 - P ) \frac{\partial\mathbf{M}}{\partial t}=\frac{\mu}{\rho}\nabla^{2}\mathbf{M}-% \mathbf{v}\cdot\nabla\mathbf{M}+(\mathbf{f}-\nabla\,\text{P})
  32. ρ \rho
  33. μ \mu
  34. 𝐉 n - q = - D n n - n μ n 𝐄 \frac{\mathbf{J}_{n}}{-q}=-D_{n}\nabla n-n\mu_{n}\mathbf{E}
  35. 𝐉 p q = - D p p + p μ p 𝐄 \frac{\mathbf{J}_{p}}{q}=-D_{p}\nabla p+p\mu_{p}\mathbf{E}
  36. n t = - 𝐉 n - q + R \frac{\partial n}{\partial t}=-\nabla\cdot\frac{\mathbf{J}_{n}}{-q}+R
  37. p t = - 𝐉 p q + R \frac{\partial p}{\partial t}=-\nabla\cdot\frac{\mathbf{J}_{p}}{q}+R
  38. μ n \mu_{n}
  39. μ p \mu_{p}
  40. D n = μ n k B T / q , D p = μ p k B T / q , D_{n}=\mu_{n}k_{B}T/q,\quad D_{p}=\mu_{p}k_{B}T/q,
  41. 𝐉 n , drift / ( - q ) = - n μ n 𝐄 , 𝐉 p , drift / q = p μ p 𝐄 \mathbf{J}_{n,\,\text{drift}}/(-q)=-n\mu_{n}\mathbf{E},\qquad\mathbf{J}_{p,\,% \text{drift}}/q=p\mu_{p}\mathbf{E}
  42. 𝐉 n , diffusion / ( - q ) = - D n n , 𝐉 p , diffusion / q = - D p p . \mathbf{J}_{n,\,\text{diffusion}}/(-q)=-D_{n}\nabla n,\qquad\mathbf{J}_{p,\,% \text{diffusion}}/q=-D_{p}\nabla p.

Conventional_electrical_unit.html

  1. 90 {}_{90}
  2. × 10 9 \times 10^{9}
  3. J 90 {}_{J–90}
  4. J {}_{J}
  5. K 90 {}_{K–90}
  6. K {}_{K}
  7. 90 {}_{90}
  8. J 90 {}_{J–90}
  9. 90 {}_{90}
  10. K 90 {}_{K–90}
  11. 90 {}_{90}
  12. J 90 {}_{J–90}
  13. J {}_{J}
  14. × 10 8 \times 10^{−}8
  15. 90 {}_{90}
  16. K {}_{K}
  17. K 90 {}_{K–90}
  18. × 10 8 \times 10^{−}8
  19. 90 {}_{90}
  20. 90 {}_{90}
  21. 90 {}_{90}
  22. 90 {}_{90}
  23. × 10 8 \times 10^{−}8
  24. 90 {}_{90}
  25. 90 {}_{90}
  26. 90 {}_{90}
  27. 90 {}_{90}
  28. 90 {}_{90}
  29. × 10 8 \times 10^{−}8
  30. 90 {}_{90}
  31. 90 {}_{90}
  32. 90 {}_{90}
  33. 90 {}_{90}
  34. 90 {}_{90}
  35. 90 {}_{90}
  36. × 10 8 \times 10^{−}8
  37. 90 {}_{90}
  38. 90 {}_{90}
  39. 90 {}_{90}
  40. 90 {}_{90}
  41. 90 {}_{90}
  42. × 10 8 \times 10^{−}8
  43. 90 {}_{90}
  44. 90 {}_{90}
  45. 90 {}_{90}
  46. × 10 8 \times 10^{−}8
  47. c = e = = 1 c=e=\hbar=1
  48. 1 4 π ϵ 0 = μ 0 4 π = α \frac{1}{4\pi\epsilon_{0}}=\frac{\mu_{0}}{4\pi}=\alpha
  49. c c\,
  50. 1 1\,
  51. 1 1\,
  52. 1 α \frac{1}{\alpha}
  53. 1 α \frac{1}{\alpha}
  54. 1 1\,
  55. 299792458 299792458
  56. h h\,
  57. 2 π 2\pi\,
  58. 2 π α \frac{2\pi}{\alpha}
  59. 2 π 2\pi\,
  60. 2 π 2\pi\,
  61. 2 π α \frac{2\pi}{\alpha}
  62. 4 × 10 - 18 ( 25812.807 ) ( 483597.9 ) 2 \frac{4\times 10^{-18}}{(25812.807)(483597.9)^{2}}
  63. = h 2 π \hbar=\frac{h}{2\pi}
  64. 1 1\,
  65. 1 α \frac{1}{\alpha}
  66. 1 1\,
  67. 1 1\,
  68. 1 α \frac{1}{\alpha}
  69. 2 × 10 - 18 π ( 25812.807 ) ( 483597.9 ) 2 \frac{2\times 10^{-18}}{\pi(25812.807)(483597.9)^{2}}
  70. e e\,
  71. α \sqrt{\alpha}\,
  72. 1 1\,
  73. 1 1\,
  74. 1 1\,
  75. 1 1\,
  76. 2 × 10 - 9 ( 25812.807 ) ( 483597.9 ) \frac{2\times 10^{-9}}{(25812.807)(483597.9)}
  77. K J = 2 e h K_{J}=\frac{2e}{h}\,
  78. α π \frac{\sqrt{\alpha}}{\pi}\,
  79. α π \frac{\alpha}{\pi}\,
  80. 1 π \frac{1}{\pi}\,
  81. 1 π \frac{1}{\pi}\,
  82. α π \frac{\alpha}{\pi}\,
  83. 483597.9 × 10 9 483597.9\times 10^{9}\,
  84. R K = h e 2 R_{K}=\frac{h}{e^{2}}\,
  85. 2 π α \frac{2\pi}{\alpha}\,
  86. 2 π α \frac{2\pi}{\alpha}\,
  87. 2 π 2\pi\,
  88. 2 π 2\pi\,
  89. 2 π α \frac{2\pi}{\alpha}\,
  90. 25812.807 25812.807\,
  91. Z 0 = 2 α R K Z_{0}=2\alpha R_{K}\,
  92. 4 π 4\pi\,
  93. 4 π 4\pi\,
  94. 4 π α 4\pi\alpha\,
  95. 4 π α 4\pi\alpha\,
  96. 4 π 4\pi\,
  97. 2 α ( 25812.807 ) 2\alpha(25812.807)\,
  98. ε 0 = 1 Z 0 c \varepsilon_{0}=\frac{1}{Z_{0}c}\,
  99. 1 4 π \frac{1}{4\pi}\,
  100. 1 4 π \frac{1}{4\pi}\,
  101. 1 4 π \frac{1}{4\pi}\,
  102. 1 4 π \frac{1}{4\pi}\,
  103. 1 4 π \frac{1}{4\pi}\,
  104. 1 2 α ( 25812.807 ) ( 299792458 ) \frac{1}{2\alpha(25812.807)(299792458)}
  105. μ 0 = Z 0 c \mu_{0}=\frac{Z_{0}}{c}\,
  106. 4 π 4\pi\,
  107. 4 π 4\pi\,
  108. 4 π α 2 4\pi\alpha^{2}\,
  109. 4 π α 2 4\pi\alpha^{2}\,
  110. 4 π 4\pi\,
  111. 2 α ( 25812.807 ) 299792458 \frac{2\alpha(25812.807)}{299792458}
  112. G G\,
  113. 1 1\,
  114. 1 1\,
  115. 1 1\,
  116. - -\,
  117. - -\,
  118. - -\,
  119. m e m_{e}\,
  120. - -\,
  121. - -\,
  122. - -\,
  123. 1 1\,
  124. 1 1\,
  125. - -\,
  126. E h = α 2 m e c 2 E_{h}=\alpha^{2}m_{e}c^{2}\,
  127. - -\,
  128. - -\,
  129. - -\,
  130. 1 1\,
  131. α 2 \alpha^{2}\,
  132. - -\,
  133. R = E h 2 h c R_{\infty}=\frac{E_{h}}{2hc}\,
  134. - -\,
  135. - -\,
  136. - -\,
  137. α 4 π \frac{\alpha}{4\pi}\,
  138. α 3 4 π \frac{\alpha^{3}}{4\pi}\,
  139. - -\,
  140. - -\,
  141. - -\,
  142. - -\,
  143. - -\,
  144. - -\,
  145. 9 192 631 770 9\ 192\ 631\ 770\,

Conway_triangle_notation.html

  1. S = b c sin A = a c sin B = a b sin C S=bc\sin A=ac\sin B=ab\sin C\,
  2. S φ = S cot φ . S_{\varphi}=S\cot\varphi.\,
  3. S A = S cot A = b c cos A = b 2 + c 2 - a 2 2 S_{A}=S\cot A=bc\cos A=\frac{b^{2}+c^{2}-a^{2}}{2}\,
  4. S B = S cot B = a c cos B = a 2 + c 2 - b 2 2 S_{B}=S\cot B=ac\cos B=\frac{a^{2}+c^{2}-b^{2}}{2}\,
  5. S C = S cot C = a b cos C = a 2 + b 2 - c 2 2 S_{C}=S\cot C=ab\cos C=\frac{a^{2}+b^{2}-c^{2}}{2}\,
  6. S ω = S cot ω = a 2 + b 2 + c 2 2 S_{\omega}=S\cot\omega=\frac{a^{2}+b^{2}+c^{2}}{2}\,
  7. ω \omega\,
  8. S π 3 = S cot π 3 = S 3 3 S_{\frac{\pi}{3}}=S\cot{\frac{\pi}{3}}=S\frac{\sqrt{3}}{3}\,
  9. S 2 φ = S φ 2 - S 2 2 S φ S φ 2 = S φ + S φ 2 + S 2 S_{2\varphi}=\frac{S_{\varphi}^{2}-S^{2}}{2S_{\varphi}}\quad\quad S_{\frac{% \varphi}{2}}=S_{\varphi}+\sqrt{S_{\varphi}^{2}+S^{2}}\,
  10. φ \varphi
  11. 0 < φ < π 0<\varphi<\pi\,
  12. S ϑ + φ = S ϑ S φ - S 2 S ϑ + S φ S ϑ - φ = S ϑ S φ + S 2 S φ - S ϑ S_{\vartheta+\varphi}=\frac{S_{\vartheta}S_{\varphi}-S^{2}}{S_{\vartheta}+S_{% \varphi}}\quad\quad S_{\vartheta-\varphi}=\frac{S_{\vartheta}S_{\varphi}+S^{2}% }{S_{\varphi}-S_{\vartheta}}\,
  13. sin A = S b c = S S A 2 + S 2 cos A = S A b c = S A S A 2 + S 2 tan A = S S A \sin A=\frac{S}{bc}=\frac{S}{\sqrt{S_{A}^{2}+S^{2}}}\quad\quad\cos A=\frac{S_{% A}}{bc}=\frac{S_{A}}{\sqrt{S_{A}^{2}+S^{2}}}\quad\quad\tan A=\frac{S}{S_{A}}\,
  14. cyclic S A = S A + S B + S C = S ω \sum\text{cyclic}S_{A}=S_{A}+S_{B}+S_{C}=S_{\omega}\,
  15. S 2 = b 2 c 2 - S A 2 = a 2 c 2 - S B 2 = a 2 b 2 - S C 2 S^{2}=b^{2}c^{2}-S_{A}^{2}=a^{2}c^{2}-S_{B}^{2}=a^{2}b^{2}-S_{C}^{2}\,
  16. S B S C = S 2 - a 2 S A S A S C = S 2 - b 2 S B S A S B = S 2 - c 2 S C S_{B}S_{C}=S^{2}-a^{2}S_{A}\quad\quad S_{A}S_{C}=S^{2}-b^{2}S_{B}\quad\quad S_% {A}S_{B}=S^{2}-c^{2}S_{C}\,
  17. S A S B S C = S 2 ( S ω - 4 R 2 ) S ω = s 2 - r 2 - 4 r R S_{A}S_{B}S_{C}=S^{2}(S_{\omega}-4R^{2})\quad\quad S_{\omega}=s^{2}-r^{2}-4rR\,
  18. s = a + b + c 2 s=\frac{a+b+c}{2}\,
  19. a + b + c = S r a+b+c=\frac{S}{r}\,
  20. sin A sin B sin C = S 4 R 2 cos A cos B cos C = S ω - 4 R 2 4 R 2 \sin A\sin B\sin C=\frac{S}{4R^{2}}\quad\quad\cos A\cos B\cos C=\frac{S_{% \omega}-4R^{2}}{4R^{2}}
  21. cyclic sin A = S 2 R r = s R cyclic cos A = r + R R cyclic tan A = S S ω - 4 R 2 = tan A tan B tan C \sum\text{cyclic}\sin A=\frac{S}{2Rr}=\frac{s}{R}\quad\quad\sum\text{cyclic}% \cos A=\frac{r+R}{R}\quad\quad\sum\text{cyclic}\tan A=\frac{S}{S_{\omega}-4R^{% 2}}=\tan A\tan B\tan C\,
  22. cyclic a 2 S A = a 2 S A + b 2 S B + c 2 S C = 2 S 2 cyclic a 4 = 2 ( S ω 2 - S 2 ) \sum\text{cyclic}a^{2}S_{A}=a^{2}S_{A}+b^{2}S_{B}+c^{2}S_{C}=2S^{2}\quad\quad% \sum\text{cyclic}a^{4}=2(S_{\omega}^{2}-S^{2})\,
  23. cyclic S A 2 = S ω 2 - 2 S 2 cyclic S B S C = S 2 cyclic b 2 c 2 = S ω 2 + S 2 \sum\text{cyclic}S_{A}^{2}=S_{\omega}^{2}-2S^{2}\quad\quad\sum\text{cyclic}S_{% B}S_{C}=S^{2}\quad\quad\sum\text{cyclic}b^{2}c^{2}=S_{\omega}^{2}+S^{2}\,
  24. D 2 = cyclic a 2 S A ( p a K p - q a K q ) 2 D^{2}=\sum\text{cyclic}a^{2}S_{A}\left(\frac{p_{a}}{K_{p}}-\frac{q_{a}}{K_{q}}% \right)^{2}\,
  25. K p = cyclic a 2 S A = 2 S 2 K q = cyclic S B S C = S 2 K_{p}=\sum\text{cyclic}a^{2}S_{A}=2S^{2}\quad\quad K_{q}=\sum\text{cyclic}S_{B% }S_{C}=S^{2}\,
  26. D 2 \displaystyle D^{2}
  27. O H = 9 R 2 - 2 S ω . OH=\sqrt{9R^{2}-2S_{\omega}\,}.

Conway–Maxwell–Poisson_distribution.html

  1. Pr ( X = x ) = f ( x ; λ , ν ) = λ x ( x ! ) ν 1 Z ( λ , ν ) , \Pr(X=x)=f(x;\lambda,\nu)=\frac{\lambda^{x}}{(x!)^{\nu}}\frac{1}{Z(\lambda,\nu% )},
  2. λ > 0 \lambda>0
  3. ν \nu
  4. Z ( λ , ν ) = j = 0 λ j ( j ! ) ν . Z(\lambda,\nu)=\sum_{j=0}^{\infty}\frac{\lambda^{j}}{(j!)^{\nu}}.
  5. Z ( λ , ν ) Z(\lambda,\nu)
  6. Z ( λ , ν ) Z(\lambda,\nu)
  7. ν \nu
  8. Pr ( X = x - 1 ) Pr ( X = x ) = x ν λ . \frac{\Pr(X=x-1)}{\Pr(X=x)}=\frac{x^{\nu}}{\lambda}.
  9. ν = 1 \nu=1
  10. ν \nu\to\infty
  11. λ / ( 1 + λ ) \lambda/(1+\lambda)
  12. ν = 0 \nu=0
  13. 1 - λ 1-\lambda
  14. λ < 1 \lambda<1
  15. E [ X r + 1 ] = { λ E [ X + 1 ] 1 - ν if r = 0 λ d d λ E [ X r ] + E [ X ] E [ X r ] if r > 0. \operatorname{E}[X^{r+1}]=\begin{cases}\lambda\,\operatorname{E}[X+1]^{1-\nu}&% \,\text{ if }r=0\\ \lambda\,\frac{d}{d\lambda}\operatorname{E}[X^{r}]+\operatorname{E}[X]% \operatorname{E}[X^{r}]&\,\text{ if }r>0.\\ \end{cases}
  16. log p x - 1 p x = - log λ + ν log x \log\frac{p_{x-1}}{p_{x}}=-\log\lambda+\nu\log x
  17. p x p_{x}
  18. ( X = x ) \mathbb{P}(X=x)
  19. x x
  20. x - 1 x-1
  21. log x \log x
  22. log ( p ^ x - 1 / p ^ x ) \log(\hat{p}_{x-1}/\hat{p}_{x})
  23. log x \log x
  24. 𝕍 [ log p ^ x - 1 p ^ x ] 1 n p x + 1 n p x - 1 \mathbb{V}\left[\log\frac{\hat{p}_{x-1}}{\hat{p}_{x}}\right]\approx\frac{1}{np% _{x}}+\frac{1}{np_{x-1}}
  25. cov ( log p ^ x - 1 p ^ x , log p ^ x p ^ x + 1 ) - 1 n p x \,\text{cov}\left(\log\frac{\hat{p}_{x-1}}{\hat{p}_{x}},\log\frac{\hat{p}_{x}}% {\hat{p}_{x+1}}\right)\approx-\frac{1}{np_{x}}
  26. ( λ , ν x 1 , , x n ) = λ S 1 exp ( - ν S 2 ) Z - n ( λ , ν ) \mathcal{L}(\lambda,\nu\mid x_{1},\dots,x_{n})=\lambda^{S_{1}}\exp(-\nu S_{2})% Z^{-n}(\lambda,\nu)
  27. S 1 = i = 1 n x i S_{1}=\sum_{i=1}^{n}x_{i}
  28. S 2 = i = 1 n log x i ! S_{2}=\sum_{i=1}^{n}\log x_{i}!
  29. 𝔼 [ X ] = X ¯ \mathbb{E}[X]=\bar{X}
  30. 𝔼 [ log X ! ] = log X ! ¯ \mathbb{E}[\log X!]=\overline{\log X!}
  31. X X
  32. log X ! \log X!
  33. λ \lambda
  34. ν \nu
  35. 𝔼 [ f ( x ) ] = j = 0 f ( j ) λ j ( j ! ) ν Z ( λ , ν ) . \mathbb{E}[f(x)]=\sum_{j=0}^{\infty}f(j)\frac{\lambda^{j}}{(j!)^{\nu}Z(\lambda% ,\nu)}.
  36. λ ^ \hat{\lambda}
  37. ν ^ \hat{\nu}
  38. λ \lambda
  39. μ = λ 1 / ν \mu=\lambda^{1/\nu}
  40. μ \mu

Coordinate_conditions.html

  1. 0 = Γ β γ α g β γ . 0=\Gamma^{\alpha}_{\beta\gamma}g^{\beta\gamma}\!.
  2. g μ ν g_{\mu\nu}\!
  3. g 01 = g 02 = g 03 = 0 g_{01}=g_{02}=g_{03}=0\!
  4. g 00 = - 1 g_{00}=-1\!
  5. g μ ν g_{\mu\nu}\!
  6. g α β , γ η β γ = k g μ ν , α η μ ν . g_{\alpha\beta,\gamma}\eta^{\beta\gamma}=k\,g_{\mu\nu,\alpha}\eta^{\mu\nu}\,.
  7. g α β = , β k g μ ν η μ ν , γ η α γ . g^{\alpha\beta}{}_{,\beta}=k\,g^{\mu\nu}{}_{,\gamma}\eta_{\mu\nu}\eta^{\alpha% \gamma}\,.

Corner_transfer_matrix.html

  1. E = a l l f a c e s ϵ ( σ i , σ j , σ k , σ l ) , E=\sum_{all\atop faces}\epsilon\left(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{% l}\right),
  2. Z N = a l l s p i n s a l l f a c e s w ( σ i , σ j , σ k , σ l ) , Z_{N}=\sum_{all\atop spins}\prod_{all\atop faces}w\left(\sigma_{i},\sigma_{j},% \sigma_{k},\sigma_{l}\right),
  3. w ( σ i , σ j , σ k , σ l ) = exp ( - ϵ ( σ i , σ j , σ k , σ l ) / k B T ) . w\left(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l}\right)=\exp\left(-\epsilon% \left(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l}\right)/k_{B}T\right).
  4. A σ | σ = δ ( σ 1 , σ 1 ) i n t e r i o r s p i n s a l l f a c e s w ( σ i , σ j , σ k , σ l ) . A_{\sigma|\sigma^{\prime}}=\delta\left(\sigma_{1},\sigma_{1}^{\prime}\right)% \sum_{interior\atop spins}\prod_{all\atop faces}w\left(\sigma_{i},\sigma_{j},% \sigma_{k},\sigma_{l}\right).
  5. σ 1 = + 1 σ 1 = - 1 A = [ | A + | 0 | - - - | - - - | 0 | A - | ] σ 1 = + 1 σ 1 = - 1 \begin{array}[]{cccc}&&\begin{array}[]{ccccc}\sigma_{1}^{\prime}=+1&&&&\sigma_% {1}^{\prime}=-1\end{array}\\ A&=&\left[\begin{array}[]{ccccccc}&&&|\\ &A_{+}&&|&&0\\ &&&|\\ -&-&-&|&-&-&-\\ &&&|\\ &0&&|&&A_{-}\\ &&&|\end{array}\right]&\begin{array}[]{c}\sigma_{1}=+1\\ \\ \\ \\ \sigma_{1}=-1\end{array}\end{array}
  6. Z N = σ , σ , σ ′′ , σ ′′′ A σ | σ A σ | σ ′′ A σ ′′ | σ ′′′ A σ ′′′ | σ = tr A 4 . Z_{N}=\sum_{\sigma,\sigma^{\prime},\sigma^{\prime\prime},\sigma^{\prime\prime% \prime}}A_{\sigma|\sigma^{\prime}}A_{\sigma^{\prime}|\sigma^{\prime\prime}}A_{% \sigma^{\prime\prime}|\sigma^{\prime\prime\prime}}A_{\sigma^{\prime\prime% \prime}|\sigma}=\textrm{tr}A^{4}.
  7. ( U i ) σ | σ = δ ( σ 1 , σ 1 ) δ ( σ i - 1 , σ i - 1 ) w ( σ i , σ i + 1 , σ i , σ i - 1 ) δ ( σ i + 1 , σ i + 1 ) δ ( σ m , σ m ) , \left(U_{i}\right)_{\sigma|\sigma^{\prime}}=\delta\left(\sigma_{1},\sigma_{1}^% {\prime}\right)\cdots\delta\left(\sigma_{i-1},\sigma_{i-1}^{\prime}\right)w% \left(\sigma_{i},\sigma_{i+1},\sigma_{i}^{\prime},\sigma_{i-1}\right)\delta% \left(\sigma_{i+1},\sigma_{i+1}^{\prime}\right)\cdots\delta\left(\sigma_{m},% \sigma_{m}^{\prime}\right),
  8. ( U m ) σ | σ = δ ( σ 1 , σ 1 ) δ ( σ m - 1 , σ m - 1 ) w ( σ m , + 1 , σ m , σ m - 1 ) , \left(U_{m}\right)_{\sigma|\sigma^{\prime}}=\delta\left(\sigma_{1},\sigma_{1}^% {\prime}\right)\cdots\delta\left(\sigma_{m-1},\sigma_{m-1}^{\prime}\right)w% \left(\sigma_{m},+1,\sigma_{m}^{\prime},\sigma_{m-1}\right),
  9. ( U m + 1 ) σ | σ = δ ( σ 1 , σ 1 ) δ ( σ m , σ m ) w ( + 1 , + 1 , + 1 , σ m ) . \left(U_{m+1}\right)_{\sigma|\sigma^{\prime}}=\delta\left(\sigma_{1},\sigma_{1% }^{\prime}\right)\cdots\delta\left(\sigma_{m},\sigma_{m}^{\prime}\right)w\left% (+1,+1,+1,\sigma_{m}\right).
  10. A = F 2 F m + 1 , A=F_{2}\cdots F_{m+1},
  11. F j = U m + 1 U j . F_{j}=U_{m+1}\cdots U_{j}.
  12. ( A * ) σ | σ = δ ( σ 1 , σ 1 ) A σ 2 , , σ m | σ 2 , , σ m , \left(A^{*}\right)_{\sigma|\sigma^{\prime}}=\delta\left(\sigma_{1},\sigma_{1}^% {\prime}\right)A_{\sigma_{2},\ldots,\sigma_{m}|\sigma_{2}^{\prime},\ldots,% \sigma_{m}^{\prime}},
  13. ( A * * ) σ | σ = δ ( σ 1 , σ 1 ) δ ( σ 2 , σ 2 ) A σ 3 , , σ m | σ 3 , , σ m , \left(A^{**}\right)_{\sigma|\sigma^{\prime}}=\delta\left(\sigma_{1},\sigma_{1}% ^{\prime}\right)\delta\left(\sigma_{2},\sigma_{2}^{\prime}\right)A_{\sigma_{3}% ,\ldots,\sigma_{m}|\sigma_{3}^{\prime},\ldots,\sigma_{m}^{\prime}},
  14. A * = I 2 A 2 m - 1 = [ A 0 0 A ] , A^{*}=I_{2}\otimes A_{2^{m-1}}=\left[\begin{array}[]{cc}A&0\\ 0&A\end{array}\right],
  15. A * * = I 2 I 2 A 2 m - 2 = [ A 0 0 0 0 A 0 0 0 0 A 0 0 0 0 A ] . A^{**}=I_{2}\otimes I_{2}\otimes A_{2^{m-2}}=\left[\begin{array}[]{cccc}A&0&0&% 0\\ 0&A&0&0\\ 0&0&A&0\\ 0&0&0&A\end{array}\right].
  16. A * = F 3 F m + 1 , A * * = F 4 F m + 1 A^{*}=F_{3}\cdots F_{m+1},A^{**}=F_{4}\cdots F_{m+1}
  17. A = F 2 A * , A * = F 3 A * * \Rightarrow A=F_{2}A^{*},A^{*}=F_{3}A^{**}
  18. A = A * ( A * * ) - 1 U 2 A * , \Rightarrow A=A^{*}\left(A^{**}\right)^{-1}U_{2}A^{*},
  19. w ( σ i , σ j , σ k , σ l ) = w ( σ k , σ j , σ i , σ l ) = w ( σ i , σ l , σ k , σ j ) , w\left(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l}\right)=w\left(\sigma_{k},% \sigma_{j},\sigma_{i},\sigma_{l}\right)=w\left(\sigma_{i},\sigma_{l},\sigma_{k% },\sigma_{j}\right),
  20. A = α m P A d P T , A=\alpha_{m}PA_{d}P^{T},
  21. A * = α m - 1 P * A d * ( P * ) T , A^{*}=\alpha_{m-1}P^{*}A_{d}^{*}\left(P^{*}\right)^{T},
  22. A * * = α m - 2 P * * A d * * ( P * * ) T , A^{**}=\alpha_{m-2}P^{**}A_{d}^{**}\left(P^{**}\right)^{T},
  23. A t = κ R A d R T , A_{t}=\kappa RA_{d}R^{T},
  24. κ = α m α m - 2 α m - 1 2 , \kappa=\frac{\alpha_{m}\alpha_{m-2}}{\alpha_{m-1}^{2}},
  25. R = ( P * ) T P , R=\left(P^{*}\right)^{T}P,
  26. A t = A d * ( R * ) T ( A d * * ) - 1 U 2 R * A d * . A_{t}=A_{d}^{*}\left(R^{*}\right)^{T}\left(A_{d}^{**}\right)^{-1}U_{2}R^{*}A_{% d}^{*}.
  27. σ 1 = 1 Z N a l l s p i n s σ 1 a l l f a c e s w ( σ i , σ j , σ k , σ l ) . \left\langle\sigma_{1}\right\rangle=\frac{1}{Z_{N}}\sum_{all\atop spins}\sigma% _{1}\prod_{all\atop faces}w\left(\sigma_{i},\sigma_{j},\sigma_{k},\sigma_{l}% \right).
  28. S = [ I 0 0 - I ] , S=\left[\begin{array}[]{cc}I&0\\ 0&-I\end{array}\right],
  29. σ 1 = tr S A 4 tr A 4 = tr α m 4 P S A 4 P T tr α m 4 P A 4 P T = tr S A d 4 tr A d 4 . \left\langle\sigma_{1}\right\rangle=\frac{\textrm{tr}SA^{4}}{\textrm{tr}A^{4}}% =\frac{\textrm{tr}\alpha_{m}^{4}PSA^{4}P^{T}}{\textrm{tr}\alpha_{m}^{4}PA^{4}P% ^{T}}=\frac{\textrm{tr}SA_{d}^{4}}{\textrm{tr}A_{d}^{4}}.
  30. κ = lim N ( Z N ) 1 / N . \kappa=\lim_{N\rightarrow\infty}\left(Z_{N}\right)^{1/N}.
  31. κ = lim N ( α m 4 tr A d 4 ) 1 / N α m 4 / N , \kappa=\lim_{N\rightarrow\infty}\left(\alpha_{m}^{4}\textrm{tr}A_{d}^{4}\right% )^{1/N}\sim\alpha_{m}^{4/N},
  32. α m κ m ( m + 1 ) / 2 , \alpha_{m}\sim\kappa^{m\left(m+1\right)/2},

Coronal_seismology.html

  1. ω K = 2 k z B 2 μ ( ρ i + ρ e ) \omega_{K}=\sqrt{\frac{2k_{z}B^{2}}{\mu(\rho_{i}+\rho_{e})}}
  2. m m
  3. ω S = k z 2 B 2 μ ρ e \omega_{S}=\sqrt{\frac{k_{z}^{2}B^{2}}{\mu\rho_{e}}}
  4. m m
  5. ω L = k z 2 ( C s 2 C A 2 C s 2 + C A 2 ) \omega_{L}=\sqrt{k^{2}_{z}\left(\frac{C_{s}^{2}C_{A}^{2}}{C_{s}^{2}+C_{A}^{2}}% \right)}
  6. C s C_{s}
  7. C A C_{A}
  8. ω A = k z 2 B 2 μ ρ i \omega_{A}=\sqrt{\frac{k_{z}^{2}B^{2}}{\mu\rho_{i}}}

Correlation_attack.html

  1. x 1 x_{1}
  2. x 2 x_{2}
  3. x 3 x_{3}
  4. F ( x 1 , x 2 , x 3 ) = ( x 1 x 2 ) ( ¬ x 1 x 3 ) F(x_{1},x_{2},x_{3})=(x_{1}\wedge x_{2})\oplus(\neg x_{1}\wedge x_{3})
  5. x 1 x_{1}
  6. x 2 x_{2}
  7. x 1 x_{1}
  8. x 3 x_{3}
  9. x 1 x_{1}
  10. x 2 x_{2}
  11. x 3 x_{3}
  12. F ( x 1 , x 2 , x 3 ) F(x_{1},x_{2},x_{3})
  13. x 3 x_{3}
  14. x 3 x_{3}
  15. F ( x 1 , x 2 , x 3 ) F(x_{1},x_{2},x_{3})
  16. x 3 = F ( x 1 , x 2 , x 3 ) x_{3}=F(x_{1},x_{2},x_{3})
  17. c 1 , c 2 , c 3 , , c n c_{1},c_{2},c_{3},\ldots,c_{n}
  18. p 1 , p 2 , p 3 , p_{1},p_{2},p_{3},\ldots
  19. c i = p i F ( x 1 i , x 2 i , x 3 i ) c_{i}=p_{i}\oplus F(x_{1i},x_{2i},x_{3i})
  20. i = 1 , 2 , 3 , , n i=1,2,3,\ldots,n
  21. x 1 i x_{1i}
  22. i i
  23. p 1 , p 2 , p 3 , , p 32 p_{1},p_{2},p_{3},\ldots,p_{32}
  24. c 1 , c 2 , c 3 , , c 32 c_{1},c_{2},c_{3},\ldots,c_{32}
  25. p 1 , p 2 , p 3 , , p 32 p_{1},p_{2},p_{3},\ldots,p_{32}
  26. F ( x 1 i , x 2 i , x 3 i ) F(x_{1i},x_{2i},x_{3i})
  27. i = 1 , 2 , 3 , , 32 i=1,2,3,\ldots,32
  28. x 2 x_{2}
  29. x 2 x_{2}
  30. x 1 x_{1}
  31. F ( x 1 , x 2 , x 3 ) F(x_{1},x_{2},x_{3})
  32. x 1 x 2 x_{1}\oplus x_{2}
  33. 2 8 × 8 = 18446744073709551616 2^{8\times 8}=18446744073709551616
  34. 2 8 + 2 7 × 8 = 72057594037928192 2^{8}+2^{7\times 8}=72057594037928192
  35. 2 2 × 8 + 2 6 × 8 = 281474976776192 2^{2\times 8}+2^{6\times 8}=281474976776192
  36. 2 3 × 8 + 2 5 × 8 = 1099528404992 2^{3\times 8}+2^{5\times 8}=1099528404992
  37. 2 4 × 8 + 2 4 × 8 = 8589934592 2^{4\times 8}+2^{4\times 8}=8589934592
  38. 2 5 × 8 + 2 3 × 8 = 1099528404992 2^{5\times 8}+2^{3\times 8}=1099528404992
  39. 2 6 × 8 + 2 2 × 8 = 281474976776192 2^{6\times 8}+2^{2\times 8}=281474976776192
  40. 2 7 × 8 + 2 8 = 72057594037928192 2^{7\times 8}+2^{8}=72057594037928192
  41. F ( x 1 , , x n ) F(x_{1},\ldots,x_{n})

Correlation_clustering.html

  1. G = ( V , E ) G=(V,E)
  2. k k

Correlation_swap.html

  1. N corr N_{\,\text{corr}}
  2. ρ strike \rho_{\,\text{strike}}
  3. ρ realized \rho_{\,\text{realized }}
  4. N corr ( ρ realized - ρ strike ) N_{\,\text{corr}}(\rho_{\,\text{realized}}-\rho_{\,\text{strike}})
  5. w i w_{i}
  6. n n
  7. ρ i , j \rho_{i,j}
  8. ρ realized := i j w i w j ρ i , j i j w i w j \rho_{\,\text{realized }}:=\frac{\sum_{i\neq j}{w_{i}w_{j}\rho_{i,j}}}{\sum_{i% \neq j}{w_{i}w_{j}}}
  9. ρ i , j \rho_{i,j}
  10. ρ realized = 2 n ( n - 1 ) i < j ρ i , j \rho_{\,\text{realized }}=\frac{2}{n(n-1)}\sum_{i<j}{\rho_{i,j}}

Cotangent_complex.html

  1. f * Ω Y / Z Ω X / Z Ω X / Y 0. f^{*}\Omega_{Y/Z}\to\Omega_{X/Z}\to\Omega_{X/Y}\to 0.
  2. I / I 2 f * Ω Y / Z Ω X / Z 0. I/I^{2}\to f^{*}\Omega_{Y/Z}\to\Omega_{X/Z}\to 0.
  3. L 0 X / Y = i * Ω V / Y , L^{X/Y}_{0}=i^{*}\Omega_{V/Y},
  4. L 1 X / Y = J / J 2 = i * J L^{X/Y}_{1}=J/J^{2}=i^{*}J
  5. L i X / Y = 0 L^{X/Y}_{i}=0
  6. L 1 X / Y L 0 X / Y L^{X/Y}_{1}\to L^{X/Y}_{0}
  7. 𝒪 V \mathcal{O}_{V}
  8. d : 𝒪 V Ω V / Y d:\mathcal{O}_{V}\to\Omega_{V/Y}
  9. 𝐋 f * L Y / Z L X / Z L X / Y 𝐋 f * L Y / Z [ 1 ] . \mathbf{L}f^{*}L^{Y/Z}_{\bullet}\to L^{X/Z}_{\bullet}\to L^{X/Y}_{\bullet}\to% \mathbf{L}f^{*}L^{Y/Z}_{\bullet}[1].
  10. L B / A B C L C / A L C / B . L^{B/A}\otimes_{B}C\to L^{C/A}\to L^{C/B}.
  11. L C / B ( L B / A B C ) [ 1 ] L^{C/B}\to(L^{B/A}\otimes_{B}C)[1]
  12. f : A B f\colon A\rightarrow B
  13. L f L^{f}
  14. L B / A L^{B/A}
  15. M B / / B M_{B//B}
  16. A 𝑓 B 𝑔 C A\xrightarrow{f}B\xrightarrow{g}C
  17. L B / A B C L C / A L C / B ( L B / A B C ) [ 1 ] L^{B/A}\otimes_{B}C\rightarrow L^{C/A}\rightarrow L^{C/B}\rightarrow(L^{B/A}% \otimes_{B}C)[1]
  18. L B A C / C B A L C / A , L^{B\otimes_{A}C/C}\cong B\otimes_{A}L^{C/A},
  19. L B A C / A ( L B / A A C ) ( B A L C / A ) . L^{B\otimes_{A}C/A}\cong(L^{B/A}\otimes_{A}C)\oplus(B\otimes_{A}L^{C/A}).
  20. I / I 2 I/I^{2}
  21. I / I 2 [ 1 ] I/I^{2}[1]

Cotes's_spiral.html

  1. 1 r = A cos ( k θ + ε ) \frac{1}{r}=A\cos\left(k\theta+\varepsilon\right)
  2. 1 r = A cosh ( k θ + ε ) \frac{1}{r}=A\cosh\left(k\theta+\varepsilon\right)
  3. 1 r = A θ + ε \frac{1}{r}=A\theta+\varepsilon
  4. F ( r ) = μ r 3 F(r)=\frac{\mu}{r^{3}}
  5. k 2 = 1 - μ h 2 k^{2}=1-\frac{\mu}{h^{2}}
  6. k 2 = μ h 2 - 1 k^{2}=\frac{\mu}{h^{2}}-1
  7. 1 r = A θ + ε . \frac{1}{r}=A\theta+\varepsilon.

Coulomb_wave_function.html

  1. ( - 2 2 + Z r ) ψ k ( r ) = k 2 2 ψ k ( r ) , \left(-\frac{\nabla^{2}}{2}+\frac{Z}{r}\right)\psi_{\vec{k}}(\vec{r})=\frac{k^% {2}}{2}\psi_{\vec{k}}(\vec{r})\,,
  2. Z = Z 1 Z 2 Z=Z_{1}Z_{2}
  3. Z = - 1 Z=-1
  4. k 2 k^{2}
  5. ξ = r + r k ^ , ζ = r - r k ^ ( k ^ = k / k ) . \xi=r+\vec{r}\cdot\hat{k},\quad\zeta=r-\vec{r}\cdot\hat{k}\qquad(\hat{k}=\vec{% k}/k)\,.
  6. ψ k ( ± ) ( r ) = 1 ( 2 π ) 3 / 2 Γ ( 1 ± i η ) e - π η / 2 e i k r M ( i η , 1 , ± i k r - i k r ) , \psi_{\vec{k}}^{(\pm)}(\vec{r})=\frac{1}{(2\pi)^{3/2}}\Gamma(1\pm i\eta)e^{-% \pi\eta/2}e^{i\vec{k}\cdot\vec{r}}M(\mp i\eta,1,\pm ikr-i\vec{k}\cdot\vec{r})\,,
  7. M ( a , b , z ) F 1 1 ( a ; b ; z ) M(a,b,z)\equiv{}_{1}\!F_{1}(a;b;z)
  8. η = Z / k \eta=Z/k
  9. Γ ( z ) \Gamma(z)
  10. ψ k ( ± ) ( r ) 1 ( 2 π ) 3 / 2 e i k r ( r k ) , \psi_{\vec{k}}^{(\pm)}(\vec{r})\rightarrow\frac{1}{(2\pi)^{3/2}}e^{i\vec{k}% \cdot\vec{r}}\qquad(\vec{r}\cdot\vec{k}\rightarrow\mp\infty)\,,
  11. k \vec{k}
  12. ψ k ( ± ) \psi_{\vec{k}}^{(\pm)}
  13. ψ k ( + ) = ψ - k ( - ) * . \psi_{\vec{k}}^{(+)}=\psi_{-\vec{k}}^{(-)*}\,.
  14. ψ k ( r ) \psi_{\vec{k}}(\vec{r})
  15. w ( η , ρ ) w_{\ell}(\eta,\rho)
  16. ρ = k r \rho=kr
  17. ψ k ( r ) = 1 ( 2 π ) 3 / 2 1 r = 0 m = - 4 π ( - i ) w ( η , ρ ) Y m ( r ^ ) Y m ( k ^ ) . \psi_{\vec{k}}(\vec{r})=\frac{1}{(2\pi)^{3/2}}\frac{1}{r}\sum_{\ell=0}^{\infty% }\sum_{m=-\ell}^{\ell}4\pi(-i)^{\ell}w_{\ell}(\eta,\rho)Y_{\ell}^{m}(\hat{r})Y% _{\ell}^{m\ast}(\hat{k})\,.
  18. ψ k m ( r ) = ψ k ( r ) Y m ( k ) d k ^ = R k ( r ) Y m ( r ^ ) , R k ( r ) = 2 π ( - i ) 1 r w ( η , ρ ) . \psi_{k\ell m}(\vec{r})=\int\psi_{\vec{k}}(\vec{r})Y_{\ell}^{m}(\vec{k})d\hat{% k}=R_{k\ell}(r)Y_{\ell}^{m}(\hat{r}),\qquad R_{k\ell}(r)=\sqrt{\frac{2}{\pi}}(% -i)^{\ell}\frac{1}{r}w_{\ell}(\eta,\rho).
  19. w ( η , ρ ) w_{\ell}(\eta,\rho)
  20. Y m ( r ^ ) Y_{\ell}^{m}(\hat{r})
  21. d 2 w d ρ 2 + ( 1 - 2 η ρ - ( + 1 ) ρ 2 ) w = 0 . \frac{d^{2}w_{\ell}}{d\rho^{2}}+\left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}% {\rho^{2}}\right)w_{\ell}=0\,.
  22. x = 2 i ρ x=2i\rho
  23. F ( η , ρ ) F_{\ell}(\eta,\rho)
  24. G ( η , ρ ) G_{\ell}(\eta,\rho)
  25. F ( η , ρ ) = 2 e - π η / 2 | Γ ( + 1 + i η ) | ( 2 + 1 ) ! ρ + 1 e i ρ M ( + 1 i η , 2 + 2 , ± 2 i ρ ) . F_{\ell}(\eta,\rho)=\frac{2^{\ell}e^{-\pi\eta/2}|\Gamma(\ell+1+i\eta)|}{(2\ell% +1)!}\rho^{\ell+1}e^{\mp i\rho}M(\ell+1\mp i\eta,2\ell+2,\pm 2i\rho)\,.
  26. 0 R k ( r ) R k ( r ) r 2 d r = δ ( k - k ) \int_{0}^{\infty}R_{k\ell}^{\ast}(r)R_{k^{\prime}\ell}(r)r^{2}dr=\delta(k-k^{% \prime})
  27. Z = - 1 Z=-1
  28. 0 R k ( r ) R n ( r ) r 2 d r = 0 \int_{0}^{\infty}R_{k\ell}^{\ast}(r)R_{n\ell}(r)r^{2}dr=0

Counter_automaton.html

  1. Γ \Gamma\,
  2. { a n b n : n } \{\ a^{n}b^{n}:n\in\mathbb{N}\}

Counting_points_on_elliptic_curves.html

  1. E ( 𝔽 q ) E(\mathbb{F}_{q})
  2. 𝔽 q \mathbb{F}_{q}
  3. E ( 𝔽 q ) E(\mathbb{F}_{q})
  4. 𝔽 q \mathbb{F}_{q}
  5. E ( 𝔽 q ) E(\mathbb{F}_{q})
  6. | | E ( 𝔽 q ) | - ( q + 1 ) | 2 q . ||E(\mathbb{F}_{q})|-(q+1)|\leq 2\sqrt{q}.\,
  7. 𝔽 q \mathbb{F}_{q}
  8. y 2 = x 3 + A x + B . y^{2}=x^{3}+Ax+B.\,
  9. 𝔽 5 \mathbb{F}_{5}
  10. x x
  11. x 3 + x + 1 x^{3}+x+1
  12. y y
  13. 0 \quad 0
  14. 1 1
  15. ± 1 \pm 1
  16. ( 0 , 1 ) , ( 0 , 4 ) (0,1),(0,4)
  17. 1 \quad 1
  18. 3 3
  19. - -
  20. - -
  21. 2 \quad 2
  22. 1 1
  23. ± 1 \pm 1
  24. ( 2 , 1 ) , ( 2 , 4 ) (2,1),(2,4)
  25. 3 \quad 3
  26. 1 1
  27. ± 1 \pm 1
  28. ( 3 , 1 ) , ( 3 , 4 ) (3,1),(3,4)
  29. 4 \quad 4
  30. 4 4
  31. ± 2 \pm 2
  32. ( 4 , 2 ) , ( 4 , 3 ) (4,2),(4,3)
  33. x = 4 x=4
  34. y 2 = x 3 + x + 1 y^{2}=x^{3}+x+1
  35. 4 4
  36. y = ± 2 y=\pm 2
  37. ( 4 , 2 ) , ( 4 , 3 ) (4,2),(4,3)
  38. 4 4
  39. x x
  40. ( 4 , 2 ) (4,2)
  41. y y
  42. ( 4 , 3 ) (4,3)
  43. y y
  44. - 2 -2
  45. 3 3
  46. 𝔽 5 \mathbb{F}_{5}
  47. E ( 𝔽 5 ) E(\mathbb{F}_{5})
  48. x 𝔽 q x\in\mathbb{F}_{q}
  49. P = ( x , y ) E ( 𝔽 q ) P=(x,y)\in E(\mathbb{F}_{q})
  50. x x
  51. x 3 + A x + B x^{3}+Ax+B
  52. 𝔽 q \mathbb{F}_{q}
  53. y y
  54. | E ( 𝔽 q ) | |E(\mathbb{F}_{q})|
  55. ( q + 1 - 2 q , q + 1 + 2 q ) (q+1-2\sqrt{q},q+1+2\sqrt{q})
  56. M M
  57. M P = O MP=O
  58. E ( 𝔽 q ) E(\mathbb{F}_{q})
  59. M M
  60. M M^{\prime}
  61. M P = M P = O MP=M^{\prime}P=O
  62. E ( 𝔽 q ) E(\mathbb{F}_{q})
  63. M M
  64. M P = O MP=O
  65. 4 q 4\sqrt{q}
  66. E ( 𝔽 q ) E(\mathbb{F}_{q})
  67. 4 q 4 4\sqrt[4]{q}
  68. m m
  69. m > q 4 m>\sqrt[4]{q}
  70. j = 0 j=0
  71. m m
  72. P j j P P_{j}\leftarrow jP
  73. L 1 L\leftarrow 1
  74. Q ( q + 1 ) P Q\leftarrow(q+1)P
  75. Q + k ( 2 m P ) Q+k(2mP)
  76. j \exists j
  77. Q + k ( 2 m P ) = ± P j Q+k(2mP)=\pm P_{j}
  78. x x
  79. M q + 1 + 2 m k j M\leftarrow q+1+2mk\mp j
  80. M P = O MP=O
  81. M M
  82. p 1 , , p r p_{1},\ldots,p_{r}
  83. M M
  84. i r i\leq r
  85. M p i P = O \frac{M}{p_{i}}P=O
  86. M M p i M\leftarrow\frac{M}{p_{i}}
  87. i i + 1 i\leftarrow i+1
  88. L lcm ( L , M ) L\leftarrow\operatorname{lcm}(L,M)
  89. M M
  90. P P
  91. L L
  92. N N
  93. ( q + 1 - 2 q , q + 1 + 2 q ) (q+1-2\sqrt{q},q+1+2\sqrt{q})
  94. P P
  95. N N
  96. E ( 𝔽 q ) E(\mathbb{F}_{q})
  97. a a
  98. | a | 2 m 2 |a|\leq 2m^{2}
  99. a 0 a_{0}
  100. a 1 a_{1}
  101. - m < a 0 m and - m a 1 m s.t. a = a 0 + 2 m a 1 . -m<a_{0}\leq m\mbox{ and }~{}-m\leq a_{1}\leq m\mbox{ s.t. }~{}a=a_{0}+2ma_{1}.
  102. ( j + 1 ) P (j+1)P
  103. j P jP
  104. P P
  105. j P jP
  106. m m
  107. 2 m P 2mP
  108. m P mP
  109. Q Q
  110. log ( q + 1 ) \log(q+1)
  111. w w
  112. w w
  113. q + 1 q+1
  114. j P jP
  115. 2 m P 2mP
  116. Q + k ( 2 m P ) Q+k(2mP)
  117. Q + ( k + 1 ) ( 2 m P ) Q+(k+1)(2mP)
  118. 2 m P 2mP
  119. M M
  120. p i p_{i}
  121. M p i O \frac{M}{p_{i}}\neq O
  122. M M
  123. P P
  124. M P = O MP=O
  125. P P
  126. M M
  127. M ¯ \bar{M}
  128. M M
  129. M ¯ P = O \bar{M}P=O
  130. M M
  131. P P
  132. x x
  133. j P jP
  134. j j
  135. - j -j
  136. + j +j
  137. O ( q 4 ) O(\sqrt[4]{q})
  138. O ( log 2 q ) O(\log^{2}{q})
  139. E ( 𝔽 q ) E(\mathbb{F}_{q})
  140. | E ( 𝔽 q ) | |E(\mathbb{F}_{q})|
  141. | E ( 𝔽 q ) | |E(\mathbb{F}_{q})|
  142. N > 4 q N>4\sqrt{q}
  143. | E ( 𝔽 q ) | |E(\mathbb{F}_{q})|
  144. 1 , , s \ell_{1},\ldots,\ell_{s}
  145. 4 q 4\sqrt{q}
  146. ψ \psi_{\ell}
  147. | E ( 𝔽 q ) | |E(\mathbb{F}_{q})|
  148. \ell
  149. n = log q n=\log{q}
  150. O ( n 2 M ( n 3 ) / log n ) = O ( n 5 + o ( 1 ) ) O(n^{2}M(n^{3})/\log{n})=O(n^{5+o(1)})
  151. M ( n ) M(n)
  152. O ( n 3 ) O(n^{3})
  153. 1 , , s \ell_{1},\ldots,\ell_{s}
  154. \ell
  155. ϕ 2 - t ϕ + q = 0 \phi^{2}-t\phi+q=0
  156. 𝔽 \mathbb{F}_{\ell}
  157. \ell
  158. \ell
  159. E / 𝔽 q E/\mathbb{F}_{q}
  160. Ψ ( X , Y ) \Psi_{\ell}(X,Y)
  161. Ψ ( X , j ( E ) ) \Psi_{\ell}(X,j(E))
  162. 𝔽 q \mathbb{F}_{q}
  163. j ( E ) j(E)
  164. E E
  165. \ell
  166. ψ \psi_{\ell}
  167. O ( ) O(\ell)
  168. ψ \psi_{\ell}
  169. O ( 2 ) O(\ell^{2})
  170. Ψ ( X , Y ) \Psi_{\ell}(X,Y)
  171. E E
  172. O ( n 2 M ( n 2 ) / log n ) = O ( n 4 + o ( 1 ) ) O(n^{2}M(n^{2})/\log{n})=O(n^{4+o(1)})
  173. n = log q n=\log{q}
  174. O ( n 3 log n ) O(n^{3}\log{n})
  175. O ( n 4 ) O(n^{4})
  176. E ( 𝔽 q ) E(\mathbb{F}_{q})

Counting_process.html

  1. N ( t ) = M ( t ) + A ( t ) N(t)=M(t)+A(t)

Coupling_from_the_past.html

  1. M M
  2. S S
  3. π \pi
  4. π \pi
  5. μ \mu
  6. f : S S f:S\to S
  7. s S s\in S
  8. f ( s ) f(s)
  9. M M
  10. s s
  11. f ( s ) f(s)
  12. f ( s ) f(s^{\prime})
  13. s s s\neq s^{\prime}
  14. f j f_{j}
  15. j j\in\mathbb{Z}
  16. μ \mu
  17. x x
  18. π \pi
  19. f j f_{j}
  20. x x
  21. f - 1 ( x ) f_{-1}(x)
  22. π \pi
  23. π \pi
  24. M M
  25. f f
  26. F j := f - 1 f - 2 f - j . F_{j}:=f_{-1}\circ f_{-2}\circ\cdots\circ f_{-j}.
  27. F j ( x ) F_{j}(x)
  28. π \pi
  29. j j\in\mathbb{N}
  30. n n\in\mathbb{N}
  31. F n F_{n}
  32. S S
  33. F n ( x ) = F n ( y ) F_{n}(x)=F_{n}(y)
  34. y S y\in S
  35. x x
  36. F n ( x ) F_{n}(x)
  37. n n\in\mathbb{N}
  38. F n ( S ) F_{n}(S)
  39. μ \mu
  40. n n
  41. F n F_{n}
  42. μ \mu
  43. | F n ( S ) | = 1 |F_{n}(S)|=1
  44. | | |\cdot|
  45. S S
  46. \leq
  47. s 0 s_{0}
  48. s 1 s_{1}
  49. s S s\in S
  50. s 0 s s 1 s_{0}\leq s\leq s_{1}
  51. μ \mu
  52. f : S S f:S\to S
  53. | F n ( S ) | = 1 |F_{n}(S)|=1
  54. F n ( s 0 ) = F n ( s 1 ) F_{n}(s_{0})=F_{n}(s_{1})
  55. F n F_{n}
  56. n := n 0 n:=n_{0}
  57. n 0 n_{0}
  58. f - 1 , , f - n f_{-1},\dots,f_{-n}
  59. F n ( s 0 ) F_{n}(s_{0})
  60. F n ( s 0 ) = F n ( s 1 ) F_{n}(s_{0})=F_{n}(s_{1})
  61. F n ( s 0 ) F n ( s 1 ) F_{n}(s_{0})\neq F_{n}(s_{1})
  62. n n
  63. f - j f_{-j}

Courant_minimax_principle.html

  1. λ k = min C max ( x = 1 C x = 0 ) A x , x , \lambda_{k}=\min\limits_{C}\max\limits_{{\left({{\|x\|=1}\atop{Cx=0}}\right)}}% \langle Ax,x\rangle,

Covariance_group.html

  1. O ( 1 , 3 ) O(1,3)
  2. μ F μ ν = 4 π j ν \partial_{\mu}F^{\mu\nu}=4\pi j^{\nu}
  3. O ( 1 , 3 ) O(1,3)
  4. ( i γ μ μ - m ) ψ = 0 (i\gamma^{\mu}\partial_{\mu}-m)\psi=0
  5. O ( 1 , 3 ) O(1,3)

Covering_code.html

  1. q 2 q\geq 2
  2. n 1 n\geq 1
  3. R 0 R\geq 0
  4. C Q n C\subseteq Q^{n}
  5. y Q n y\in Q^{n}
  6. x C x\in C
  7. d H ( x , y ) R d_{H}(x,y)\leq R
  8. Q n Q^{n}
  9. K q ( n , R ) K_{q}(n,R)
  10. K q ( n , 1 ) q n - 1 / ( n - 1 ) K_{q}(n,1)\geq q^{n-1}/(n-1)
  11. K q ( n , n - 2 ) q 2 / ( n - 1 ) K_{q}(n,n-2)\geq q^{2}/(n-1)
  12. Q n Q^{n}
  13. n = 1 2 ( 3 k - 1 ) n=\tfrac{1}{2}(3^{k}-1)
  14. K q ( n , R ) K_{q}(n,R)

Covering_number.html

  1. K K
  2. ( X , d ) (X,d)
  3. ε > 0 \varepsilon>0
  4. ε \varepsilon
  5. x X x\in X
  6. B ε ( x ) B_{\varepsilon}(x)
  7. N ε pack ( K ) N^{\,\text{pack}}_{\varepsilon}(K)
  8. x i K x_{i}\in K
  9. B ε ( x i ) B_{\varepsilon}(x_{i})
  10. N ε int ( K ) N^{\,\text{int}}_{\varepsilon}(K)
  11. x i K x_{i}\in K
  12. B ε ( x i ) B_{\varepsilon}(x_{i})
  13. K K
  14. N ε ext ( K ) N^{\,\text{ext}}_{\varepsilon}(K)
  15. x i X x_{i}\in X
  16. B ε ( x i ) B_{\varepsilon}(x_{i})
  17. K K
  18. N ε met ( K ) N^{\,\text{met}}_{\varepsilon}(K)
  19. x i K x_{i}\in K
  20. ε \varepsilon
  21. d ( x i , x j ) ε d(x_{i},x_{j})\geq\varepsilon
  22. i j i\neq j
  23. ε > 0 \varepsilon>0
  24. N 2 ε met ( K ) N ε pack ( K ) N ε ext ( K ) N ε int ( K ) N ε met ( K ) \displaystyle N^{\,\text{met}}_{2\varepsilon}(K)\leq N^{\,\text{pack}}_{% \varepsilon}(K)\leq N^{\,\text{ext}}_{\varepsilon}(K)\leq N^{\,\text{int}}_{% \varepsilon}(K)\leq N^{\,\text{met}}_{\varepsilon}(K)
  25. N ε * ( K ) N^{*}_{\varepsilon}(K)
  26. ε \varepsilon
  27. K K
  28. * = pack , ext , met *=\,\text{pack},\,\text{ext},\,\text{met}
  29. K K
  30. * = int *=\,\text{int}

Covering_problem_of_Rado.html

  1. F ( X ) = inf S sup I | I | | S | , F(X)=\inf_{S}\sup_{I}\frac{|I|}{|S|},
  2. 1 8.4797 F ( square ) 1 4 - 1 384 , \frac{1}{8.4797}\leq F(\textrm{square})\leq\frac{1}{4}-\frac{1}{384},
  3. f ( X ) 1 6 f(X)\geq\frac{1}{6}

Coxeter's_loxodromic_sequence_of_tangent_circles.html

  1. k = φ + φ 2.89005 , k=\varphi+\sqrt{\varphi}\approx 2.89005\ ,
  2. ( 1 + x + x 2 + x 3 ) 2 = 2 ( 1 + x 2 + x 4 + x 6 ) , (1+x+x^{2}+x^{3})^{2}=2(1+x^{2}+x^{4}+x^{6})\ ,
  3. cos - 1 ( - 1 φ ) 128.173 . \cos^{-1}\left(\frac{-1}{\varphi}\right)\approx 128.173^{\circ}\ .

CP_violation.html

  1. a a
  2. b b
  3. a ¯ \bar{a}
  4. b ¯ \bar{b}
  5. a b a\rightarrow b
  6. a ¯ b ¯ \bar{a}\rightarrow\bar{b}
  7. M M
  8. M ¯ \bar{M}
  9. M = | M | e i θ M=|M|e^{i\theta}
  10. e i ϕ e^{i\phi}
  11. M ¯ \bar{M}
  12. M M
  13. e - i ϕ e^{-i\phi}
  14. M = | M | e i θ e i ϕ M=|M|e^{i\theta}e^{i\phi}
  15. M ¯ = | M | e i θ e - i ϕ \bar{M}=|M|e^{i\theta}e^{-i\phi}
  16. | M | 2 |M|^{2}
  17. a b a\rightarrow b
  18. M = | M 1 | e i θ 1 e i ϕ 1 + | M 2 | e i θ 2 e i ϕ 2 M=|M_{1}|e^{i\theta_{1}}e^{i\phi_{1}}+|M_{2}|e^{i\theta_{2}}e^{i\phi_{2}}
  19. M ¯ = | M 1 | e i θ 1 e - i ϕ 1 + | M 2 | e i θ 2 e - i ϕ 2 \bar{M}=|M_{1}|e^{i\theta_{1}}e^{-i\phi_{1}}+|M_{2}|e^{i\theta_{2}}e^{-i\phi_{% 2}}
  20. | M | 2 - | M ¯ | 2 = 4 | M 1 | | M 2 | sin ( θ 1 - θ 2 ) sin ( ϕ 1 - ϕ 2 ) |M|^{2}-|\bar{M}|^{2}=4|M_{1}||M_{2}|\sin(\theta_{1}-\theta_{2})\sin(\phi_{1}-% \phi_{2})
  21. = - 1 4 F μ ν F μ ν - n f g 2 θ 32 π 2 F μ ν F ~ μ ν + ψ ¯ ( i γ μ D μ - m e i θ γ 5 ) ψ {\mathcal{L}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{n_{f}g^{2}\theta}{32\pi^{% 2}}F_{\mu\nu}\tilde{F}^{\mu\nu}+\bar{\psi}(i\gamma^{\mu}D_{\mu}-me^{i\theta^{% \prime}\gamma_{5}})\psi
  22. θ ~ \scriptstyle{\tilde{\theta}}

Critical_field.html

  1. H c H_{c}
  2. T c T_{c}
  3. C s u p e r - C n o r m a l = T 4 π ( d H c d T ) T = T c 2 C_{super}-C_{normal}={T\over 4\pi}\left(\frac{dH_{c}}{dT}\right)^{2}_{T=T_{c}}
  4. H c 1 H_{c1}
  5. H c 2 H_{c2}

Cross-polarized_wave_generation.html

  1. ω ( ) = ω ( ) + ω ( ) - ω ( ) \omega^{(\perp)}~{}=~{}\omega^{(\|)}~{}+~{}\omega^{(\|)}~{}-~{}\omega^{(\|)}
  2. d B d z = - i γ | A | 2 A \frac{dB}{dz}=-i\gamma_{\perp}|A|^{2}A
  3. γ \gamma_{\|}
  4. γ \gamma_{\perp}
  5. χ x x x x ( 3 ) \chi^{(3)}_{xxxx}
  6. χ ( 3 ) \chi^{(3)}
  7. A = A 0 e - i γ | | | A | 2 L A=A_{0}e^{-i\gamma_{||}|A|^{2}L}
  8. B = A 0 γ γ | | ( e - i γ | | | A | 2 L - 1 ) B=A_{0}\frac{\gamma_{\perp}}{\gamma_{||}}(e^{-i\gamma_{||}|A|^{2}L}-1)
  9. η \eta
  10. η = | B ( L ) | 2 | A 0 | 2 = I o u t I i n = 4 ( γ γ | | ) 2 sin 2 ( γ | | | A | 2 L / 2 ) \eta=\frac{|B(L)|^{2}}{|A_{0}|^{2}}=\frac{I_{out}}{I_{in}}=4(\frac{\gamma_{% \perp}}{\gamma_{||}})^{2}\sin^{2}(\gamma_{||}|A|^{2}L/2)
  11. γ | A | 2 L 1 \gamma_{\|}|A|^{2}L\leq 1
  12. η = ( γ ) 2 | A | 4 L 2 γ 2 I i n 2 L 2 \eta=({\gamma_{\perp}})^{2}|A|^{4}L^{2}\propto\gamma_{\perp}^{2}I_{in}^{2}L^{2}
  13. γ | A | 2 L \gamma_{\|}|A|^{2}L

Crossing_number_(graph_theory).html

  1. c r ( G ) cr(G)
  2. G G
  3. G G
  4. n n
  5. cr ( K m , n ) n 2 n - 1 2 m 2 m - 1 2 \textrm{cr}(K_{m,n})\leq\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{% n-1}{2}\right\rfloor\left\lfloor\frac{m}{2}\right\rfloor\left\lfloor\frac{m-1}% {2}\right\rfloor
  6. cr ( K p ) 1 4 p 2 p - 1 2 p - 2 2 p - 3 2 , \textrm{cr}(K_{p})\leq\tfrac{1}{4}\left\lfloor\tfrac{p}{2}\right\rfloor\left% \lfloor\tfrac{p-1}{2}\right\rfloor\left\lfloor\tfrac{p-2}{2}\right\rfloor\left% \lfloor\tfrac{p-3}{2}\right\rfloor,
  7. 1 , 3 , 9 , 18 , 36 , 60 , 100 , 150 1,3,9,18,36,60,100,150
  8. p = 5 , , 12 p=5,...,12
  9. p 10 p≤10
  10. p = 11 , 12. p=11,12.
  11. 1 , 3 , 9 , 19 , 36 , 62 , 102 , 153 1,3,9,19,36,62,102,153
  12. lim p cr ( K p ) 64 p 4 = 1. \lim_{p\to\infty}\textrm{cr}(K_{p})\tfrac{64}{p^{4}}=1.
  13. n n
  14. n n
  15. n 16 n≤16
  16. k k
  17. c r ( G ) cr(G)
  18. G G
  19. n n
  20. e e
  21. e > 7 n e>7n
  22. cr ( G ) e 3 29 n 2 . \operatorname{cr}(G)\geq\frac{e^{3}}{29n^{2}}.
  23. 29 29
  24. 7 7
  25. 4 4
  26. 29 29
  27. 64 64
  28. 2 r 2r
  29. e 4 n e≥4n
  30. cr ( G ) c r e r + 2 n r + 1 . \operatorname{cr}(G)\geq c_{r}\frac{e^{r+2}}{n^{r+1}}.
  31. G G
  32. n n
  33. e e
  34. cr ( G ) e - 3 n . \operatorname{cr}(G)\geq e-3n.
  35. G G
  36. c r ( G ) cr(G)
  37. G G
  38. e c r ( G ) e−cr(G)
  39. n n
  40. e c r ( G ) 3 n e−cr(G)≤3n
  41. e c r ( G ) 3 n 6 e−cr(G)≤3n−6
  42. n 3 n≥3
  43. G G
  44. G G
  45. H H
  46. p p
  47. G G
  48. H H
  49. H H
  50. H H
  51. H H
  52. G G
  53. H H
  54. cr H e H - 3 n H . \operatorname{cr}_{H}\geq e_{H}-3n_{H}.
  55. 𝐄 [ cr H ] 𝐄 [ e H ] - 3 𝐄 [ n H ] . \mathbf{E}[\operatorname{cr}_{H}]\geq\mathbf{E}[e_{H}]-3\mathbf{E}[n_{H}].
  56. n n
  57. G G
  58. p p
  59. H H
  60. G G
  61. H H
  62. H H
  63. G G
  64. H H
  65. G G
  66. c r ( G ) cr(G)
  67. G G
  68. p 4 cr ( G ) p 2 e - 3 p n . p^{4}\operatorname{cr}(G)\geq p^{2}e-3pn.
  69. e > 4 n e>4n
  70. cr ( G ) e 3 64 n 2 . \operatorname{cr}(G)\geq\frac{e^{3}}{64n^{2}}.
  71. 64 64
  72. 33.75 33.75
  73. e > 7.5 n e>7.5n

Crossing_number_(knot_theory).html

  1. 1 N ( cr ( K 1 ) + cr ( K 2 ) ) cr ( K 1 + K 2 ) \frac{1}{N}(\mathrm{cr}(K_{1})+\mathrm{cr}(K_{2}))\leq\mathrm{cr}(K_{1}+K_{2})

Crown_graph.html

  1. σ ( n ) = min { k n ( k k / 2 ) } , \sigma(n)=\min\left\{\,k\mid n\leq{\left({{k}\atop{\lfloor k/2\rfloor}}\right)% }\,\right\},
  2. \square

Cure.html

  1. S ( t ) = p + [ ( 1 - p ) × S * ( t ) ] S(t)=p+[(1-p)\times S^{*}(t)]
  2. S ( t ) S(t)
  3. p p
  4. S * ( t ) S^{*}(t)

Curvelet.html

  1. 2 - j 2^{-j}
  2. 2 - j / 2 2^{-j/2}
  3. n n
  4. n n
  5. O ( 1 / n 1 / 2 ) O(1/n^{1/2})
  6. O ( 1 / n ) O(1/n)
  7. O ( ( l o g ( n ) ) 3 / n 2 ) O({(log(n))}^{3}/{n^{2}})
  8. O ( n 2 l o g ( n ) ) O(n^{2}log(n))
  9. n × n n\times n
  10. ϕ \phi
  11. N j = 4 2 j 2 N_{j}=4\cdot 2^{\left\lceil\frac{j}{2}\right\rceil}
  12. 2 - j 2^{-j}
  13. s y m b o l ξ = ( ξ 1 , ξ 2 ) T symbol{\xi}=(\xi_{1},\xi_{2})^{T}
  14. r = ξ 1 2 + ξ 2 2 , ω = arctan ξ 1 ξ 2 r=\sqrt{\xi_{1}^{2}+\xi_{2}^{2}},\omega=\arctan\frac{\xi_{1}}{\xi_{2}}
  15. ϕ ^ j , 0 , 0 := 2 - 3 j 4 W ( 2 - j r ) V ~ N j ( ω ) , r 0 , ω [ 0 , 2 π ) , j N 0 \hat{\phi}_{j,0,0}:=2^{\frac{-3j}{4}}W(2^{-j}r)\tilde{V}_{N_{j}}(\omega),r\geq 0% ,\omega\in[0,2\pi),j\in N_{0}
  16. W W
  17. V ~ N j \tilde{V}_{N_{j}}
  18. W ( r ) W(r)
  19. ( 0 , ) (0,\infty)
  20. V ~ N j \tilde{V}_{N_{j}}
  21. V ~ N j \tilde{V}_{N_{j}}
  22. j = - | W ( 2 - j r ) | 2 = 1 , r ( 0 , ) . \sum_{j=-\infty}^{\infty}\left|W(2^{-j}r)\right|^{2}=1,r\in(0,\infty).
  23. N N
  24. N N
  25. 2 π 2\pi
  26. V ~ N \tilde{V}_{N}
  27. [ - 2 π N , 2 π N ] \left[\frac{-2\pi}{N},\frac{2\pi}{N}\right]
  28. l = 0 N - 1 V ~ N 2 ( ω - 2 π l N ) = 1 \sum_{l=0}^{N-1}\tilde{V}^{2}_{N}(\omega-\frac{2\pi l}{N})=1
  29. ω [ 0 , 2 π ) \omega\in\left[0,2\pi\right)
  30. V ~ N \tilde{V}_{N}
  31. 2 π 2\pi
  32. V ( N ω 2 π ) V(\frac{N\omega}{2\pi})
  33. l = 0 N j - 1 | 2 3 j 4 ϕ ^ j , 0 , 0 ( r , ω - 2 π l N j ) | 2 = | W ( 2 - j r ) | 2 l = 0 N j - 1 V ~ N j 2 ( ω - 2 π l N ) = | W ( 2 - j r ) | 2 \sum_{l=0}^{N_{j}-1}\left|2^{\frac{3j}{4}}\hat{\phi}_{j,0,0}(r,\omega-\frac{2% \pi l}{N_{j}})\right|^{2}=\left|W(2^{-j}r)\right|^{2}\sum_{l=0}^{N_{j}-1}% \tilde{V}^{2}_{N_{j}}(\omega-\frac{2\pi l}{N})=\left|W(2^{-j}r)\right|^{2}
  34. ϕ ^ - 1 := W 0 ( | ξ | ) w i t h \hat{\phi}_{-1}:=W_{0}(\left|\xi\right|)with
  35. W 0 2 ( r ) 2 := 1 - j = 0 W ( 2 - j r ) 2 W_{0}^{2}(r)^{2}:=1-\sum_{j=0}^{\infty}W(2^{-j}r)^{2}

Cut-off_factor.html

  1. C u t H o s e L e n g t h = H o s e A s s e m b l y O v e r a l l L e n g t h - C 1 - C 2 Cut\ Hose\ Length\ =Hose\ Assembly\ Overall\ Length-C1-C2

Cyano_radical.html

  1. N + S = J N+S=J
  2. J + I = F J+I=F

Cyclotomic_character.html

  1. χ p : G 𝐙 p × \chi_{p}:G\rightarrow\mathbf{Z}_{p}^{\times}
  2. θ = ζ n a g , n \theta=\zeta_{n}^{a_{g,n}}
  3. χ p \chi_{p}
  4. χ : G 𝐐 GL 1 ( 𝐙 ) \chi_{\ell}:G_{\mathbf{Q}}\rightarrow\operatorname{GL}_{1}(\mathbf{Z}_{\ell})

Czesław_Lejewski.html

  1. ( F a F b ) F d (FaFb)\leftrightarrow Fd
  2. ( x = c ) and ( x exists ) (x=c)\and(x\,\text{ exists})
  3. x ( x exists ) \forall x(x\,\text{ exists})
  4. x ( x does not exist ) \exists x\,(x\,\text{ does not exist})
  5. x F x ( x F x ) \forall x\,Fx\rightarrow(\exists x\,Fx)
  6. x ( x exists F x ) x ( x exists and F x ) \forall x(x\,\text{ exists}\rightarrow Fx)\rightarrow\exists x\,(x\,\text{ % exists and }Fx)

Čech-to-derived_functor_spectral_sequence.html

  1. \mathcal{F}
  2. 𝔘 \mathfrak{U}
  3. 𝔘 \mathfrak{U}
  4. q ( X , ) \mathcal{H}^{q}(X,\mathcal{F})
  5. \mathcal{F}
  6. H q ( U , ) H^{q}(U,\mathcal{F})
  7. 𝒢 \mathcal{G}
  8. H ˇ p ( 𝔘 , 𝒢 ) \check{H}^{p}(\mathfrak{U},\mathcal{G})
  9. 𝒢 \mathcal{G}
  10. 𝔘 \mathfrak{U}
  11. E 2 p , q = H ˇ p ( 𝔘 , q ( X , ) ) H p + q ( X , ) . E^{p,q}_{2}=\check{H}^{p}(\mathfrak{U},\mathcal{H}^{q}(X,\mathcal{F}))% \Rightarrow H^{p+q}(X,\mathcal{F}).
  12. 𝔘 \mathfrak{U}
  13. \mathcal{F}
  14. 𝔘 \mathfrak{U}

DAD–SAS_model.html

  1. π = μ - b Y + b Y - 1 + h ( Δ i W + Δ ϵ e ) \pi=\mu-bY+bY_{-1}+h(\Delta i^{W}+\Delta\epsilon^{e})
  2. π = ϵ + π W - b Y + b Y - 1 + γ Δ Y W + δ Δ G - f ( Δ i W + Δ ϵ e ) \pi=\epsilon+\pi^{W}-bY+bY_{-1}+\gamma\Delta Y^{W}+\delta\Delta G-f(\Delta i^{% W}+\Delta\epsilon^{e})
  3. π = π e + λ ( Y - Y * ) \pi=\pi^{e}+\lambda(Y-Y*)

Dark-energy-dominated_era.html

  1. a ( t ) exp ( H t ) a(t)\propto\exp(Ht)
  2. H = 8 π G ρ full / 3 = Λ / 3 . H=\sqrt{8\pi G\rho_{\mathrm{full}}/3}=\sqrt{\Lambda/3}.

Davenport–Schinzel_sequence.html

  1. 2 n α ( n ) - O ( n ) λ 3 ( n ) 2 n α ( n ) + O ( n α ( n ) ) 2n\alpha(n)-O(n)\leq\lambda_{3}(n)\leq 2n\alpha(n)+O(n\sqrt{\alpha(n)})
  2. λ s ( n ) = n 2 1 t ! α ( n ) t ( 1 + o ( 1 ) ) \lambda_{s}(n)=n\cdot 2^{\frac{1}{t!}\alpha(n)^{t}(1+o(1))}
  3. λ s ( n ) < n 2 1 t ! α ( n ) t log α ( n ) ( 1 + o ( 1 ) ) \lambda_{s}(n)<n\cdot 2^{\frac{1}{t!}\alpha(n)^{t}\log\alpha(n)(1+o(1))}
  4. λ s ( 1 ) = 1 \lambda_{s}(1)=1\,
  5. λ s ( 2 ) = s + 1 \lambda_{s}(2)=s+1\,
  6. λ s ( 3 ) = 3 s - 2 + ( s mod 2 ) \lambda_{s}(3)=3s-2+(s\,\bmod\,2)
  7. λ s ( 4 ) = 6 s - 2 + ( s mod 2 ) . \lambda_{s}(4)=6s-2+(s\,\bmod\,2).

Davenport–Schmidt_theorem.html

  1. x α 2 + y α + z = 0. x\alpha^{2}+y\alpha+z=0.\,
  2. H ( α ) = max { | x | , | y | , | z | } . H(\alpha)=\max\{|x|,|y|,|z|\}.\,
  3. | ξ - α | < C H ( α ) - 3 , |\xi-\alpha|<CH(\alpha)^{-3},\,
  4. C = { C 0 if | ξ | < 1 C 0 ξ 2 if | ξ | > 1. C=\left\{\begin{array}[]{ c l }C_{0}&\textrm{if}\ |\xi|<1\\ C_{0}\xi^{2}&\textrm{if}\ |\xi|>1.\end{array}\right.

Davey–Stewartson_equation.html

  1. u u\,
  2. ϕ \phi\,
  3. i u t + c 0 u x x + u y y = c 1 | u | 2 u + c 2 u ϕ x , iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi_{x},\,
  4. ϕ x x + c 3 ϕ y y = ( | u | 2 ) x . \phi_{xx}+c_{3}\phi_{yy}=(|u|^{2})_{x}.\,
  5. i u t + u x x + 2 k | u | 2 u = 0. iu_{t}+u_{xx}+2k|u|^{2}u=0.\,

De_Moivre–Laplace_theorem.html

  1. [ u r a d i c a l , u n p ( 1 - p ) ] [u^{\prime}radical^{\prime},u^{\prime}np(1-p)^{\prime}]
  2. ( n k ) p k q n - k 1 2 π n p q e - ( k - n p ) 2 2 n p q , p + q = 1 , p , q > 0 {n\choose k}\,p^{k}q^{n-k}\simeq\frac{1}{\sqrt{2\pi npq}}\,e^{-\frac{(k-np)^{2% }}{2npq}},\qquad p+q=1,\ p,q>0
  3. k = n p + x n p q k=np+x\sqrt{npq}
  4. k n p \tfrac{k}{n}\to p
  5. 1 2 π n p q e - ( k + 1 - n p ) 2 2 n p q 1 2 π n p q e - ( k - n p ) 2 2 n p q = e 2 n p - 2 k - 1 2 n p q e - x 2 n p q e 0 = 1 as n . \frac{\frac{1}{\sqrt{2\pi npq}}e^{\frac{-(k+1-np)^{2}}{2npq}}}{\frac{1}{\sqrt{% 2\pi npq}}e^{\frac{-(k-np)^{2}}{2npq}}}=e^{\frac{2np-2k-1}{2npq}}\simeq e^{% \frac{-x}{2\sqrt{npq}}}\simeq e^{0}=1\qquad\,\text{as }n\to\infty.
  6. n ! n n e - n 2 π n as n . n!\simeq n^{n}e^{-n}\sqrt{2\pi n}\qquad\,\text{as }n\to\infty.
  7. ( n k ) p k q n - k \displaystyle{n\choose k}p^{k}q^{n-k}
  8. k n p \tfrac{k}{n}\to p
  9. ( n k ) p k q n - k \displaystyle{n\choose k}p^{k}q^{n-k}
  10. ln ( 1 + x ) x - x 2 2 + x 3 3 - \ln\left(1+x\right)\simeq x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-...
  11. ( n k ) p k q n - k \displaystyle{n\choose k}p^{k}q^{n-k}

De_Sitter_invariant_special_relativity.html

  1. μ m \mu m
  2. Λ Λ
  3. Λ \Lambda

Debt_deflation.html

  1. 50 % × 10 % = 5 % , 50\%\times 10\%=5\%,
  2. 300 % × 10 % = 30 % , 300\%\times 10\%=30\%,

Debt_ratio.html

  1. Debt ratio = Total Debt Total Assets \mbox{Debt ratio}~{}=\frac{\mbox{Total Debt}~{}}{\mbox{Total Assets}~{}}
  2. Debt ratio = Total Liability Total Assets \mbox{Debt ratio}~{}=\frac{\mbox{Total Liability}~{}}{\mbox{Total Assets}~{}}

Decision_list.html

  1. r r
  2. f 1 f_{1}
  3. b 1 b_{1}
  4. f 2 f_{2}
  5. b 2 b_{2}
  6. f r f_{r}
  7. b r b_{r}
  8. f i f_{i}
  9. i i
  10. b i b_{i}
  11. i i
  12. i { 1... r } i\in\{1...r\}
  13. f r f_{r}
  14. k k
  15. k k

Decoherence-free_subspaces.html

  1. H ^ = H ^ S I ^ B + I ^ S H ^ B + H ^ I \hat{H}=\hat{H}_{S}\otimes\hat{I}_{B}+\hat{I}_{S}\otimes\hat{H}_{B}+\hat{H}_{I}
  2. H ^ I \hat{H}_{I}
  3. H ^ I = i S ^ i B ^ i , \hat{H}_{I}=\sum_{i}\hat{S}_{i}\otimes\hat{B}_{i},
  4. S ^ i ( B ^ i ) \hat{S}_{i}\big(\hat{B}_{i}\big)
  5. H ^ S ( H ^ B ) \hat{H}_{S}\big(\hat{H}_{B}\big)
  6. I ^ S ( I ^ B ) \hat{I}_{S}\big(\hat{I}_{B}\big)
  7. ~ S S \tilde{\mathcal{H}}_{S}\subset\mathcal{H}_{S}
  8. S \mathcal{H}_{S}
  9. | ϕ \forall|\phi\rangle
  10. S ^ i | ϕ = s i | ϕ , s i \hat{S}_{i}|\phi\rangle=s_{i}|\phi\rangle,s_{i}\in\mathbb{C}
  11. | ϕ \forall|\phi\rangle
  12. ~ S \mathcal{\tilde{H}}_{S}
  13. S ^ i 𝒪 S B ( S B ) \forall\hat{S}_{i}\in\mathcal{O}_{SB}(\mathcal{H}_{SB})
  14. S B \mathcal{H}_{SB}
  15. ~ S \mathcal{\tilde{H}}_{S}
  16. H ^ S \hat{H}_{S}
  17. | ϕ |\phi\rangle
  18. ~ S \mathcal{\tilde{H}}_{S}
  19. ~ S \mathcal{\tilde{H}}_{S}
  20. H ^ S \hat{H}_{S}
  21. ~ S = s p a n [ { | ϕ k } k = 1 N ] \mathcal{\tilde{H}}_{S}=span\big[\big\{|\phi_{k}\rangle\big\}_{k=1}^{N}\big]
  22. ~ S \mathcal{\tilde{H}}_{S}
  23. S ^ i 𝒪 S B ( S B ) \hat{S}_{i}\in\mathcal{O}_{SB}(\mathcal{H}_{SB})
  24. ~ S S \mathcal{\tilde{H}}_{S}\subset\mathcal{H}_{S}
  25. S \mathcal{H}_{S}
  26. S \mathcal{H}_{S}
  27. 𝐀 l = ( g l 𝐔 ~ 𝟎 𝟎 𝐀 ¯ l ) , g l = a j k | 𝐔 C | j \mathbf{A}_{l}=\begin{pmatrix}g_{l}\mathbf{\tilde{U}}&\mathbf{0}\\ \mathbf{0}&\mathbf{\bar{A}}_{l}\end{pmatrix},\quad g_{l}=\sqrt{a_{j}}\langle k% |\mathbf{U}_{C}|j\rangle
  28. 𝐔 C = 𝑒𝑥𝑝 ( - i 𝐇 C t ) \mathbf{U}_{C}=\mathit{exp}\big(\frac{-i\mathbf{H}_{C}t}{\hbar}\big)
  29. 𝐇 C \mathbf{H}_{C}
  30. 𝐔 ~ \mathbf{\tilde{U}}
  31. ~ S S \mathcal{\tilde{H}}_{S}\subset\mathcal{H}_{S}
  32. 𝐀 ¯ l \mathbf{\bar{A}}_{l}
  33. ~ S \mathcal{\tilde{H}^{\bot}}_{S}
  34. ~ S \mathcal{\tilde{H}}_{S}
  35. 𝐀 ¯ l \mathbf{\bar{A}}_{l}
  36. ~ S \mathcal{\tilde{H}^{\bot}}_{S}
  37. ~ S \mathcal{\tilde{H}}_{S}
  38. ~ S \mathcal{\tilde{H}^{\bot}}_{S}
  39. { | j } j = 1 N \big\{|j\rangle\big\}_{j=1}^{N}
  40. ~ S \mathcal{\tilde{H}}_{S}
  41. 𝐀 ¯ l | j = g l 𝐔 ~ | j , l . \mathbf{\bar{A}}_{l}|j\rangle=g_{l}\mathbf{\tilde{U}}|j\rangle,\quad\forall{l}.
  42. 𝐔 ~ \mathbf{\tilde{U}}
  43. l \mathbf{\mathit{l}}
  44. g l \mathbf{\mathit{g}}_{l}
  45. { | j } j = 1 N \big\{|j\rangle\big\}_{j=1}^{N}
  46. ~ S \mathcal{\tilde{H}}_{S}
  47. | ψ ~ S |\psi\rangle\in\mathcal{\tilde{H}}_{S}
  48. | ψ = j = 1 N b j | j , b j . |\psi\rangle=\sum_{j=1}^{N}b_{j}|j\rangle,\quad b_{j}\in\mathbb{C}.
  49. 𝐀 ¯ l \mathbf{\bar{A}}_{l}
  50. | ψ |\psi\rangle
  51. 𝐀 ¯ l | ψ \displaystyle\mathbf{\bar{A}}_{l}|\psi\rangle
  52. | ψ |\psi\rangle
  53. ρ i n i t i a l = | ψ ψ | \rho_{initial}=|\psi\rangle\langle\psi|
  54. ρ f i n a l \displaystyle\rho_{final}
  55. ρ f i n a l \mathbf{\mathit{\rho}}_{final}
  56. 𝐔 ~ \mathbf{\tilde{U}}
  57. ~ S \mathcal{\tilde{H}}_{S}
  58. ~ S \mathcal{\tilde{H}}_{S}
  59. L D [ ρ ] = 0 \mathbf{\mathit{L}}_{D}[\rho]=0
  60. ρ \mathbf{\mathit{\rho}}
  61. { | j } j = 1 N \big\{|j\rangle\big\}_{j=1}^{N}
  62. ~ S S \mathcal{\tilde{H}}_{S}\subset\mathcal{H}_{S}
  63. S \mathcal{H}_{S}
  64. ~ S \mathcal{\tilde{H}}_{S}
  65. | j \forall{|j\rangle}
  66. 𝐅 α | j = λ α | j , α . \mathbf{F}_{\alpha}|j\rangle=\lambda_{\alpha}|j\rangle,\quad\forall\alpha.
  67. | j |j\rangle
  68. { 𝐅 α } α = 1 M = N × N . \big\{\mathbf{F}_{\alpha}\big\}_{\alpha=1}^{M=N\times{N}}.
  69. ~ S \mathcal{\tilde{H}}_{S}
  70. ρ \mathbf{\rho}
  71. ( 𝑇𝑟 [ ρ ] = 1 ) \big(\mathbf{\mathit{Tr}}[\rho]=1\big)
  72. d × d d\times d
  73. \mathcal{H}
  74. ( ( ) ) \big(\mathcal{B(\mathcal{H})}\big)
  75. S = { ρ i } i = 1 n ~ S S=\big\{\rho_{i}\big\}_{i=1}^{n}\in\mathcal{\tilde{H}}_{S}
  76. S \mathcal{H}_{S}
  77. n < d \mathbf{\mathit{n}}<\mathbf{\mathit{d}}
  78. S \mathbf{\mathit{S}}
  79. S \mathbf{\mathit{S}}
  80. ζ \mathbf{\zeta}
  81. ρ i , ρ j S ( i j ) \mathbf{\rho}_{i},\mathbf{\rho}_{j}\in\mathit{S}\big(i\neq j\big)
  82. ζ \mathbf{\zeta}
  83. ρ i , ρ j \mathbf{\rho}_{i},\mathbf{\rho}_{j}
  84. S \mathbf{\mathit{S}}
  85. ζ \mathbf{\zeta}
  86. ρ i , ρ j S \mathbf{\rho}_{i},\mathbf{\rho}_{j}\in\mathit{S}
  87. ζ \mathbf{\zeta}
  88. S \mathbf{\mathit{S}}
  89. ζ \mathbf{\zeta}
  90. ρ , ρ S \forall\mathbf{\rho,\rho^{\prime}}\in\mathit{S}
  91. x + \mathit{x}\in\mathbb{R}^{+}
  92. ζ ( ρ - x ρ ) 1 = ρ - x ρ 1 . \big\|\mathbf{\zeta}\big(\mathbf{\rho}-\mathit{x}\mathbf{\rho^{\prime}}\big)% \big\|_{1}=\big\|\mathbf{\rho}-\mathit{x}\mathbf{\rho^{\prime}}\big\|_{1}.
  93. ζ \mathbf{\zeta}
  94. S \mathbf{\mathit{S}}
  95. ζ \mathbf{\zeta}
  96. C = span [ { | j k } ] C=\operatorname{span}\big[\big\{|j_{k}\rangle\big\}\big]
  97. { | j k } \big\{|j_{k}\rangle\big\}
  98. C \mathbf{\mathit{C}}
  99. 𝐑 \mathbf{R}
  100. C \mathbf{\mathit{C}}
  101. { 𝐑 r } . \big\{\mathbf{R}_{r}\big\}.
  102. C \mathbf{\mathit{C}}
  103. { 𝐀 l } \big\{\mathbf{A}_{l}\big\}
  104. { 𝐑 r } . \big\{\mathbf{R}_{r}\big\}.
  105. C \mathbf{\mathit{C}}
  106. C \mathbf{\mathit{C}}
  107. 𝐑 r 𝐔 ~ S , r \mathbf{R}_{r}\propto\mathbf{\tilde{U}}_{S}^{\dagger},\forall{r}
  108. 𝐔 ~ S \mathbf{\tilde{U}}_{S}^{\dagger}
  109. { | 0 1 | 0 2 , | 0 1 | 1 2 , | 1 1 | 0 2 , | 1 1 | 1 2 } \big\{|0\rangle_{1}\otimes|0\rangle_{2},|0\rangle_{1}\otimes|1\rangle_{2},|1% \rangle_{1}\otimes|0\rangle_{2},|1\rangle_{1}\otimes|1\rangle_{2}\big\}
  110. ϕ \mathbf{\mathit{\phi}}
  111. | 0 1 | 0 2 \displaystyle|0\rangle_{1}\otimes|0\rangle_{2}
  112. | 0 1 | 1 2 , | 1 1 | 0 2 |0\rangle_{1}\otimes|1\rangle_{2},|1\rangle_{1}\otimes|0\rangle_{2}
  113. e i ϕ \mathbf{\mathit{e}}^{i\phi}
  114. | ψ |\psi\rangle
  115. | 0 E \displaystyle|0_{E}\rangle
  116. | ψ E = l | 0 E + m | 1 E , l , m . |\psi_{E}\rangle=l|0_{E}\rangle+m|1_{E}\rangle,\quad l,m\in\mathbb{C}.
  117. | ψ E l | 0 1 e i ϕ | 1 2 + e i ϕ m | 1 1 | 0 2 = e i ϕ | ψ E . |\psi_{E}\rangle\longrightarrow l|0\rangle_{1}\otimes e^{i\phi}|1\rangle_{2}+e% ^{i\phi}m|1\rangle_{1}\otimes|0\rangle_{2}=e^{i\phi}|\psi_{E}\rangle.
  118. | ψ E |\psi_{E}\rangle
  119. { | 0 1 | 1 2 , | 1 1 | 0 2 } \big\{|0\rangle_{1}\otimes|1\rangle_{2},|1\rangle_{1}\otimes|0\rangle_{2}\big\}
  120. { | 0 1 | 0 2 } , { | 1 1 | 1 2 } \big\{|0\rangle_{1}\otimes|0\rangle_{2}\big\},\big\{|1\rangle_{1}\otimes|1% \rangle_{2}\big\}
  121. C \mathcal{H}_{C}
  122. C = j = 1 N ( i = 1 l N j i ) . \mathcal{H}_{C}=\oplus_{j=1}^{N}(\otimes_{i=1}^{l_{N}}\mathcal{H}_{ji}).
  123. j i \mathcal{H}_{ji}
  124. j i \mathcal{H}_{ji}
  125. α | 0 + β | 1 \alpha|0\rangle+\beta|1\rangle
  126. α | 0 1 + β | 1 2 α | 0 1 | 1 2 + β | 1 1 | 0 2 . \alpha|0\rangle_{1}+\beta|1\rangle_{2}\longrightarrow\alpha|0\rangle_{1}% \otimes|1\rangle_{2}+\beta|1\rangle_{1}\otimes|0\rangle_{2}.
  127. α | 0 1 + β | 1 2 ( α | 0 1 + β | 1 2 ) | ψ . \alpha|0\rangle_{1}+\beta|1\rangle_{2}\longrightarrow\big(\alpha|0\rangle_{1}+% \beta|1\rangle_{2}\big)\otimes|\psi\rangle.
  128. | ψ |\psi\rangle

Definite_quadratic_form.html

  1. V V
  2. V V
  3. Q Q
  4. B B
  5. Q ( x ) = B ( x , x ) \,Q(x)=B(x,x)
  6. B ( x , y ) = B ( y , x ) = 1 2 ( Q ( x + y ) - Q ( x ) - Q ( y ) ) \,B(x,y)=B(y,x)=\tfrac{1}{2}(Q(x+y)-Q(x)-Q(y))
  7. V = 2 V=\mathbb{R}^{2}
  8. Q ( x ) = c 1 x 1 2 + c 2 x 2 2 Q(x)=c_{1}{x_{1}}^{2}+c_{2}{x_{2}}^{2}\,
  9. x = ( x < s u b > 1 , x 2 ) x=(x<sub>1,x_{2})

Deflection_(engineering).html

  1. w w
  2. d 2 w ( x ) d x 2 = M ( x ) E ( x ) I ( x ) \cfrac{\mathrm{d}^{2}w(x)}{\mathrm{d}x^{2}}=\frac{M(x)}{E(x)I(x)}
  3. x x
  4. E E
  5. I I
  6. M M
  7. q q
  8. E I d 4 w ( x ) d x 4 = q ( x ) EI~{}\cfrac{\mathrm{d}^{4}w(x)}{\mathrm{d}x^{4}}=q(x)
  9. δ \delta
  10. ϕ \phi
  11. δ B = F L 3 3 E I \delta_{B}=\frac{FL^{3}}{3EI}
  12. ϕ B = F L 2 2 E I \phi_{B}=\frac{FL^{2}}{2EI}
  13. F F
  14. L L
  15. E E
  16. I I
  17. x x
  18. δ x = F x 2 6 E I ( 3 L - x ) \delta_{x}=\frac{Fx^{2}}{6EI}(3L-x)
  19. ϕ x = F x 2 E I ( 2 L - x ) \phi_{x}=\frac{Fx}{2EI}(2L-x)
  20. x = L x=L
  21. δ x \delta_{x}
  22. ϕ x \phi_{x}
  23. δ B \delta_{B}
  24. ϕ B \phi_{B}
  25. δ B = q L 4 8 E I \delta_{B}=\frac{qL^{4}}{8EI}
  26. ϕ B = q L 3 6 E I \phi_{B}=\frac{qL^{3}}{6EI}
  27. q q
  28. L L
  29. E E
  30. I I
  31. x x
  32. δ x = q x 2 24 E I ( 6 L 2 - 4 L x + x 2 ) \delta_{x}=\frac{qx^{2}}{24EI}(6L^{2}-4Lx+x^{2})
  33. ϕ x = q x 6 E I ( 3 L 2 - 3 L x + x 2 ) \phi_{x}=\frac{qx}{6EI}(3L^{2}-3Lx+x^{2})
  34. δ C = F L 3 48 E I \delta_{C}=\frac{FL^{3}}{48EI}
  35. F F
  36. L L
  37. E E
  38. I I
  39. x x
  40. δ x = F x 48 E I ( 3 L 2 - 4 x 2 ) \delta_{x}=\frac{Fx}{48EI}(3L^{2}-4x^{2})
  41. 0 x L 2 0\leq x\leq\frac{L}{2}
  42. a a
  43. δ m a x = F a ( L 2 - a 2 ) 3 / 2 9 3 L E I \delta_{max}=\frac{Fa(L^{2}-a^{2})^{3/2}}{9\sqrt{3}LEI}
  44. F F
  45. L L
  46. E E
  47. I I
  48. a a
  49. a L / 2 a\leq L/2
  50. x 1 x_{1}
  51. x 1 = L 2 - a 2 3 x_{1}=\sqrt{\frac{L^{2}-a^{2}}{3}}
  52. δ C = 5 q L 4 384 E I \delta_{C}=\frac{5qL^{4}}{384EI}
  53. q q
  54. L L
  55. E E
  56. I I
  57. x x
  58. δ x = q x 24 E I ( L 3 - 2 L x 2 + x 3 ) \delta_{x}=\frac{qx}{24EI}(L^{3}-2Lx^{2}+x^{3})
  59. N N
  60. m m
  61. N m 2 \frac{N}{m^{2}}
  62. m 4 m^{4}
  63. l b f lb_{f}
  64. i n in
  65. l b f i n 2 \frac{lb_{f}}{in^{2}}
  66. i n 4 in^{4}
  67. k g f kg_{f}
  68. k g f m 2 \frac{kg_{f}}{m^{2}}

Deformation_(mechanics).html

  1. 𝐱 = s y m b o l F ( 𝐗 ) \mathbf{x}=symbol{F}(\mathbf{X})
  2. 𝐗 \mathbf{X}
  3. s y m b o l ε 𝐗 ( 𝐱 - 𝐗 ) = s y m b o l F - s y m b o l I , symbol{\varepsilon}\doteq\cfrac{\partial}{\partial\mathbf{X}}\left(\mathbf{x}-% \mathbf{X}\right)=symbol{F}-symbol{I},
  4. s y m b o l I symbol{I}
  5. e = Δ L L = - L L \ e=\frac{\Delta L}{L}=\frac{\ell-L}{L}
  6. e \ e
  7. L L
  8. \ \ell
  9. λ = L \ \lambda=\frac{\ell}{L}
  10. e = - L L = λ - 1 \ e=\frac{\ell-L}{L}=\lambda-1
  11. δ ε = δ \ \delta\varepsilon=\frac{\delta\ell}{\ell}
  12. δ ε = L δ ε = ln ( L ) = ln ( λ ) = ln ( 1 + e ) = e - e 2 / 2 + e 3 / 3 - \ \begin{aligned}\displaystyle\int\delta\varepsilon&\displaystyle=\int_{L}^{% \ell}\frac{\delta\ell}{\ell}\\ \displaystyle\varepsilon&\displaystyle=\ln\left(\frac{\ell}{L}\right)=\ln(% \lambda)\\ &\displaystyle=\ln(1+e)\\ &\displaystyle=e-e^{2}/2+e^{3}/3-\cdots\\ \end{aligned}
  13. ε G = 1 2 ( 2 - L 2 L 2 ) = 1 2 ( λ 2 - 1 ) \ \varepsilon_{G}=\frac{1}{2}\left(\frac{\ell^{2}-L^{2}}{L^{2}}\right)=\frac{1% }{2}(\lambda^{2}-1)
  14. ε E = 1 2 ( 2 - L 2 2 ) = 1 2 ( 1 - 1 λ 2 ) \ \varepsilon_{E}=\frac{1}{2}\left(\frac{\ell^{2}-L^{2}}{\ell^{2}}\right)=% \frac{1}{2}\left(1-\frac{1}{\lambda^{2}}\right)
  15. d x × d y dx\times dy
  16. length ( A B ) = d x \mathrm{length}(AB)=dx\,
  17. length ( a b ) = ( d x + u x x d x ) 2 + ( u y x d x ) 2 = d x 1 + 2 u x x + ( u x x ) 2 + ( u y x ) 2 \begin{aligned}\displaystyle\mathrm{length}(ab)&\displaystyle=\sqrt{\left(dx+% \frac{\partial u_{x}}{\partial x}dx\right)^{2}+\left(\frac{\partial u_{y}}{% \partial x}dx\right)^{2}}\\ &\displaystyle=dx~{}\sqrt{1+2\frac{\partial u_{x}}{\partial x}+\left(\frac{% \partial u_{x}}{\partial x}\right)^{2}+\left(\frac{\partial u_{y}}{\partial x}% \right)^{2}}\\ \end{aligned}\,\!
  18. length ( a b ) d x + u x x d x \mathrm{length}(ab)\approx dx+\frac{\partial u_{x}}{\partial x}dx
  19. x x\,\!
  20. ε x = extension original length = length ( a b ) - length ( A B ) length ( A B ) = u x x \varepsilon_{x}=\frac{\,\text{extension}}{\,\text{original length}}=\frac{% \mathrm{length}(ab)-\mathrm{length}(AB)}{\mathrm{length}(AB)}=\frac{\partial u% _{x}}{\partial x}
  21. y y\,\!
  22. z z\,\!
  23. ε y = u y y , ε z = u z z \varepsilon_{y}=\frac{\partial u_{y}}{\partial y}\quad,\qquad\varepsilon_{z}=% \frac{\partial u_{z}}{\partial z}\,\!
  24. γ x y \gamma_{xy}
  25. A C ¯ \overline{AC}\,\!
  26. A B ¯ \overline{AB}\,\!
  27. γ x y = α + β \gamma_{xy}=\alpha+\beta\,\!
  28. tan α = u y x d x d x + u x x d x = u y x 1 + u x x tan β = u x y d y d y + u y y d y = u x y 1 + u y y \begin{aligned}\displaystyle\tan\alpha&\displaystyle=\frac{\tfrac{\partial u_{% y}}{\partial x}dx}{dx+\tfrac{\partial u_{x}}{\partial x}dx}=\frac{\tfrac{% \partial u_{y}}{\partial x}}{1+\tfrac{\partial u_{x}}{\partial x}}\\ \displaystyle\tan\beta&\displaystyle=\frac{\tfrac{\partial u_{x}}{\partial y}% dy}{dy+\tfrac{\partial u_{y}}{\partial y}dy}=\frac{\tfrac{\partial u_{x}}{% \partial y}}{1+\tfrac{\partial u_{y}}{\partial y}}\end{aligned}
  29. u x x 1 ; u y y 1 \cfrac{\partial u_{x}}{\partial x}\ll 1~{};~{}~{}\cfrac{\partial u_{y}}{% \partial y}\ll 1
  30. α \alpha\,\!
  31. β \beta\,\!
  32. 1 \ll 1\,\!
  33. tan α α , tan β β \tan\alpha\approx\alpha,~{}\tan\beta\approx\beta\,\!
  34. α u y x ; β u x y \alpha\approx\cfrac{\partial u_{y}}{\partial x}~{};~{}~{}\beta\approx\cfrac{% \partial u_{x}}{\partial y}
  35. γ x y = α + β = u y x + u x y \gamma_{xy}=\alpha+\beta=\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x% }}{\partial y}\,\!
  36. x x\,\!
  37. y y\,\!
  38. u x u_{x}\,\!
  39. u y u_{y}\,\!
  40. γ x y = γ y x \gamma_{xy}=\gamma_{yx}\,\!
  41. y y\,\!
  42. z z\,\!
  43. x x\,\!
  44. z z\,\!
  45. γ y z = γ z y = u y z + u z y , γ z x = γ x z = u z x + u x z \gamma_{yz}=\gamma_{zy}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}% }{\partial y}\quad,\qquad\gamma_{zx}=\gamma_{xz}=\frac{\partial u_{z}}{% \partial x}+\frac{\partial u_{x}}{\partial z}\,\!
  46. γ \gamma\,\!
  47. s y m b o l ε ¯ ¯ = [ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ ε x x γ x y / 2 γ x z / 2 γ y x / 2 ε y y γ y z / 2 γ z x / 2 γ z y / 2 ε z z ] \underline{\underline{symbol{\varepsilon}}}=\left[\begin{matrix}\varepsilon_{% xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz}\\ \end{matrix}\right]=\left[\begin{matrix}\varepsilon_{xx}&\gamma_{xy}/2&\gamma_% {xz}/2\\ \gamma_{yx}/2&\varepsilon_{yy}&\gamma_{yz}/2\\ \gamma_{zx}/2&\gamma_{zy}/2&\varepsilon_{zz}\\ \end{matrix}\right]\,\!
  48. 𝐱 ( 𝐗 , t ) = s y m b o l F ( t ) 𝐗 + 𝐜 ( t ) \mathbf{x}(\mathbf{X},t)=symbol{F}(t)\cdot\mathbf{X}+\mathbf{c}(t)
  49. 𝐱 \mathbf{x}
  50. 𝐗 \mathbf{X}
  51. t t
  52. s y m b o l F symbol{F}
  53. 𝐜 \mathbf{c}
  54. [ x 1 ( X 1 , X 2 , X 3 , t ) x 2 ( X 1 , X 2 , X 3 , t ) x 3 ( X 1 , X 2 , X 3 , t ) ] = [ F 11 ( t ) F 12 ( t ) F 13 ( t ) F 21 ( t ) F 22 ( t ) F 23 ( t ) F 31 ( t ) F 32 ( t ) F 33 ( t ) ] [ X 1 X 2 X 3 ] + [ c 1 ( t ) c 2 ( t ) c 3 ( t ) ] \begin{bmatrix}x_{1}(X_{1},X_{2},X_{3},t)\\ x_{2}(X_{1},X_{2},X_{3},t)\\ x_{3}(X_{1},X_{2},X_{3},t)\end{bmatrix}=\begin{bmatrix}F_{11}(t)&F_{12}(t)&F_{% 13}(t)\\ F_{21}(t)&F_{22}(t)&F_{23}(t)\\ F_{31}(t)&F_{32}(t)&F_{33}(t)\end{bmatrix}\begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\end{bmatrix}+\begin{bmatrix}c_{1}(t)\\ c_{2}(t)\\ c_{3}(t)\end{bmatrix}
  55. s y m b o l F = s y m b o l F ( 𝐗 , t ) symbol{F}=symbol{F}(\mathbf{X},t)
  56. 𝐜 = 𝐜 ( 𝐗 , t ) \mathbf{c}=\mathbf{c}(\mathbf{X},t)
  57. s y m b o l F symbol{F}
  58. 𝐱 ( 𝐗 , t ) = s y m b o l Q ( t ) 𝐗 + 𝐜 ( t ) \mathbf{x}(\mathbf{X},t)=symbol{Q}(t)\cdot\mathbf{X}+\mathbf{c}(t)
  59. s y m b o l Q \cdotsymbol Q T = s y m b o l Q T s y m b o l Q = s y m b o l 1 symbol{Q}\cdotsymbol{Q}^{T}=symbol{Q}^{T}\cdot symbol{Q}=symbol{\mathit{1}}
  60. [ x 1 ( X 1 , X 2 , X 3 , t ) x 2 ( X 1 , X 2 , X 3 , t ) x 3 ( X 1 , X 2 , X 3 , t ) ] = [ Q 11 ( t ) Q 12 ( t ) Q 13 ( t ) Q 21 ( t ) Q 22 ( t ) Q 23 ( t ) Q 31 ( t ) Q 32 ( t ) Q 33 ( t ) ] [ X 1 X 2 X 3 ] + [ c 1 ( t ) c 2 ( t ) c 3 ( t ) ] \begin{bmatrix}x_{1}(X_{1},X_{2},X_{3},t)\\ x_{2}(X_{1},X_{2},X_{3},t)\\ x_{3}(X_{1},X_{2},X_{3},t)\end{bmatrix}=\begin{bmatrix}Q_{11}(t)&Q_{12}(t)&Q_{% 13}(t)\\ Q_{21}(t)&Q_{22}(t)&Q_{23}(t)\\ Q_{31}(t)&Q_{32}(t)&Q_{33}(t)\end{bmatrix}\begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\end{bmatrix}+\begin{bmatrix}c_{1}(t)\\ c_{2}(t)\\ c_{3}(t)\end{bmatrix}
  61. κ 0 ( ) \ \kappa_{0}(\mathcal{B})
  62. κ t ( ) \ \kappa_{t}(\mathcal{B})
  63. 𝐮 ( 𝐗 , t ) = 𝐛 ( 𝐗 , t ) + 𝐱 ( 𝐗 , t ) - 𝐗 or u i = α i J b J + x i - α i J X J \ \mathbf{u}(\mathbf{X},t)=\mathbf{b}(\mathbf{X},t)+\mathbf{x}(\mathbf{X},t)-% \mathbf{X}\qquad\,\text{or}\qquad u_{i}=\alpha_{iJ}b_{J}+x_{i}-\alpha_{iJ}X_{J}
  64. 𝐔 ( 𝐱 , t ) = 𝐛 ( 𝐱 , t ) + 𝐱 - 𝐗 ( 𝐱 , t ) or U J = b J + α J i x i - X J \ \mathbf{U}(\mathbf{x},t)=\mathbf{b}(\mathbf{x},t)+\mathbf{x}-\mathbf{X}(% \mathbf{x},t)\qquad\,\text{or}\qquad U_{J}=b_{J}+\alpha_{Ji}x_{i}-X_{J}\,
  65. 𝐄 J 𝐞 i = α J i = α i J \ \mathbf{E}_{J}\cdot\mathbf{e}_{i}=\alpha_{Ji}=\alpha_{iJ}
  66. u i = α i J U J or U J = α J i u i \ u_{i}=\alpha_{iJ}U_{J}\qquad\,\text{or}\qquad U_{J}=\alpha_{Ji}u_{i}
  67. 𝐞 i = α i J 𝐄 J \ \mathbf{e}_{i}=\alpha_{iJ}\mathbf{E}_{J}
  68. 𝐮 ( 𝐗 , t ) = u i 𝐞 i = u i ( α i J 𝐄 J ) = U J 𝐄 J = 𝐔 ( 𝐱 , t ) \mathbf{u}(\mathbf{X},t)=u_{i}\mathbf{e}_{i}=u_{i}(\alpha_{iJ}\mathbf{E}_{J})=% U_{J}\mathbf{E}_{J}=\mathbf{U}(\mathbf{x},t)
  69. 𝐄 J 𝐞 i = δ J i = δ i J \ \mathbf{E}_{J}\cdot\mathbf{e}_{i}=\delta_{Ji}=\delta_{iJ}
  70. 𝐮 ( 𝐗 , t ) = 𝐱 ( 𝐗 , t ) - 𝐗 or u i = x i - δ i J X J = x i - X i \ \mathbf{u}(\mathbf{X},t)=\mathbf{x}(\mathbf{X},t)-\mathbf{X}\qquad\,\text{or% }\qquad u_{i}=x_{i}-\delta_{iJ}X_{J}=x_{i}-X_{i}
  71. 𝐔 ( 𝐱 , t ) = 𝐱 - 𝐗 ( 𝐱 , t ) or U J = δ J i x i - X J = x J - X J \ \mathbf{U}(\mathbf{x},t)=\mathbf{x}-\mathbf{X}(\mathbf{x},t)\qquad\,\text{or% }\qquad U_{J}=\delta_{Ji}x_{i}-X_{J}=x_{J}-X_{J}
  72. 𝐗 𝐮 \ \nabla_{\mathbf{X}}\mathbf{u}
  73. 𝐮 ( 𝐗 , t ) = 𝐱 ( 𝐗 , t ) - 𝐗 𝐗 𝐮 = 𝐗 𝐱 - 𝐈 𝐗 𝐮 = 𝐅 - 𝐈 \begin{aligned}\displaystyle\mathbf{u}(\mathbf{X},t)&\displaystyle=\mathbf{x}(% \mathbf{X},t)-\mathbf{X}\\ \displaystyle\nabla_{\mathbf{X}}\mathbf{u}&\displaystyle=\nabla_{\mathbf{X}}% \mathbf{x}-\mathbf{I}\\ \displaystyle\nabla_{\mathbf{X}}\mathbf{u}&\displaystyle=\mathbf{F}-\mathbf{I}% \\ \end{aligned}
  74. u i = x i - δ i J X J = x i - X i u i X K = x i X K - δ i K \begin{aligned}\displaystyle u_{i}&\displaystyle=x_{i}-\delta_{iJ}X_{J}=x_{i}-% X_{i}\\ \displaystyle\frac{\partial u_{i}}{\partial X_{K}}&\displaystyle=\frac{% \partial x_{i}}{\partial X_{K}}-\delta_{iK}\\ \end{aligned}
  75. 𝐅 \mathbf{F}
  76. 𝐱 𝐔 \ \nabla_{\mathbf{x}}\mathbf{U}
  77. 𝐔 ( 𝐱 , t ) = 𝐱 - 𝐗 ( 𝐱 , t ) 𝐱 𝐔 = 𝐈 - 𝐱 𝐗 𝐱 𝐔 = 𝐈 - 𝐅 - 1 \begin{aligned}\displaystyle\mathbf{U}(\mathbf{x},t)&\displaystyle=\mathbf{x}-% \mathbf{X}(\mathbf{x},t)\\ \displaystyle\nabla_{\mathbf{x}}\mathbf{U}&\displaystyle=\mathbf{I}-\nabla_{% \mathbf{x}}\mathbf{X}\\ \displaystyle\nabla_{\mathbf{x}}\mathbf{U}&\displaystyle=\mathbf{I}-\mathbf{F}% ^{-1}\\ \end{aligned}
  78. U J = δ J i x i - X J = x J - X J U J x k = δ J k - X J x k \begin{aligned}\displaystyle U_{J}&\displaystyle=\delta_{Ji}x_{i}-X_{J}=x_{J}-% X_{J}\\ \displaystyle\frac{\partial U_{J}}{\partial x_{k}}&\displaystyle=\delta_{Jk}-% \frac{\partial X_{J}}{\partial x_{k}}\\ \end{aligned}
  79. 𝐞 1 , 𝐞 2 \mathbf{e}_{1},\mathbf{e}_{2}
  80. s y m b o l F = F 11 𝐞 1 𝐞 1 + F 12 𝐞 1 𝐞 2 + F 21 𝐞 2 𝐞 1 + F 22 𝐞 2 𝐞 2 + 𝐞 3 𝐞 3 symbol{F}=F_{11}\mathbf{e}_{1}\otimes\mathbf{e}_{1}+F_{12}\mathbf{e}_{1}% \otimes\mathbf{e}_{2}+F_{21}\mathbf{e}_{2}\otimes\mathbf{e}_{1}+F_{22}\mathbf{% e}_{2}\otimes\mathbf{e}_{2}+\mathbf{e}_{3}\otimes\mathbf{e}_{3}
  81. s y m b o l F = [ F 11 F 12 0 F 21 F 22 0 0 0 1 ] symbol{F}=\begin{bmatrix}F_{11}&F_{12}&0\\ F_{21}&F_{22}&0\\ 0&0&1\end{bmatrix}
  82. s y m b o l F = s y m b o l R \cdotsymbol U = [ cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ] [ λ 1 0 0 0 λ 2 0 0 0 1 ] symbol{F}=symbol{R}\cdotsymbol{U}=\begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}\lambda_{1}&0&0\\ 0&\lambda_{2}&0\\ 0&0&1\end{bmatrix}
  83. θ \theta
  84. λ 1 \lambda_{1}
  85. λ 2 \lambda_{2}
  86. det ( s y m b o l F ) = 1 \det(symbol{F})=1
  87. F 11 F 22 - F 12 F 21 = 1 F_{11}F_{22}-F_{12}F_{21}=1
  88. λ 1 λ 2 = 1 \lambda_{1}\lambda_{2}=1
  89. 𝐞 1 \mathbf{e}_{1}
  90. λ 1 = 1 \lambda_{1}=1
  91. s y m b o l F 𝐞 1 = 𝐞 1 symbol{F}\cdot\mathbf{e}_{1}=\mathbf{e}_{1}
  92. F 11 𝐞 1 + F 21 𝐞 2 = 𝐞 1 F 11 = 1 ; F 21 = 0 F_{11}\mathbf{e}_{1}+F_{21}\mathbf{e}_{2}=\mathbf{e}_{1}\quad\implies\quad F_{% 11}=1~{};~{}~{}F_{21}=0
  93. F 11 F 22 - F 12 F 21 = 1 F 22 = 1 F_{11}F_{22}-F_{12}F_{21}=1\quad\implies\quad F_{22}=1
  94. γ := F 12 \gamma:=F_{12}\,
  95. s y m b o l F = [ 1 γ 0 0 1 0 0 0 1 ] symbol{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}
  96. s y m b o l F 𝐞 2 = F 12 𝐞 1 + F 22 𝐞 2 = γ 𝐞 1 + 𝐞 2 s y m b o l F ( 𝐞 2 𝐞 2 ) = γ 𝐞 1 𝐞 2 + 𝐞 2 𝐞 2 symbol{F}\cdot\mathbf{e}_{2}=F_{12}\mathbf{e}_{1}+F_{22}\mathbf{e}_{2}=\gamma% \mathbf{e}_{1}+\mathbf{e}_{2}\quad\implies\quad symbol{F}\cdot(\mathbf{e}_{2}% \otimes\mathbf{e}_{2})=\gamma\mathbf{e}_{1}\otimes\mathbf{e}_{2}+\mathbf{e}_{2% }\otimes\mathbf{e}_{2}
  97. 𝐞 i 𝐞 i = s y m b o l 1 \mathbf{e}_{i}\otimes\mathbf{e}_{i}=symbol{\mathit{1}}
  98. s y m b o l F = s y m b o l 1 + γ 𝐞 1 𝐞 2 symbol{F}=symbol{\mathit{1}}+\gamma\mathbf{e}_{1}\otimes\mathbf{e}_{2}

Degree_diameter_problem.html

  1. n d , k M d , k n_{d,k}\leq M_{d,k}
  2. M d , k M_{d,k}
  3. M d , k = { 1 + d ( d - 1 ) k - 1 d - 2 if d > 2 2 k + 1 if d = 2 M_{d,k}=\begin{cases}1+d\frac{(d-1)^{k}-1}{d-2}&\,\text{ if }d>2\\ 2k+1&\,\text{ if }d=2\end{cases}
  4. M d , k = d k + O ( d k - 1 ) M_{d,k}=d^{k}+O(d^{k-1})
  5. μ k = lim inf d n d , k d k \mu_{k}=\liminf_{d\to\infty}\frac{n_{d,k}}{d^{k}}
  6. μ k = 1 \mu_{k}=1
  7. μ 1 = μ 2 = μ 3 = μ 5 = 1 \mu_{1}=\mu_{2}=\mu_{3}=\mu_{5}=1
  8. μ 4 1 / 4 \mu_{4}\geq 1/4
  9. μ k 1.6 k \mu_{k}\geq 1.6^{k}
  10. μ k = 1 \mu_{k}=1

Dehn_function.html

  1. G = X | R ( * ) G=\langle X|R\rangle\qquad(*)
  2. w = u 1 r 1 u 1 - 1 u m r m u m - 1 in F ( X ) , w=u_{1}r_{1}u_{1}^{-1}\cdots u_{m}r_{m}u_{m}^{-1}\,\text{ in }F(X),
  3. f : [ 0 , ) f:\mathbb{N}\to[0,\infty)
  4. Dehn ( n ) = max { Area ( w ) : w = 1 in G , | w | n , w freely reduced . } {\rm Dehn}(n)=\max\{{\rm Area}(w):w=1\,\text{ in }G,|w|\leq n,w\,\text{ freely% reduced}.\}
  5. f , g : [ 0 , ) f,g:\mathbb{N}\to[0,\infty)
  6. f ( n ) C g ( C n + C ) + C n + C f(n)\leq Cg(Cn+C)+Cn+C
  7. G = a 1 , a 2 , b 1 , b n | [ a 1 , b 1 ] [ a 2 , b 2 ] = 1 G=\langle a_{1},a_{2},b_{1},b_{n}|[a_{1},b_{1}][a_{2},b_{2}]=1\rangle
  8. k \mathbb{Z}^{k}
  9. B ( 1 , 2 ) = a , b | b - 1 a b = a 2 B(1,2)=\langle a,b|b^{-1}ab=a^{2}\rangle
  10. H 3 = a , b , t | [ a , t ] = [ b , t ] = 1 , [ a , b ] = t 2 H_{3}=\langle a,b,t|[a,t]=[b,t]=1,[a,b]=t^{2}\rangle
  11. H 2 k + 1 = a 1 , b 1 , , a k , b k , t | [ a i , b i ] = t , [ a i , t ] = [ b i , t ] = 1 , i = 1 , , k , [ a i , b j ] = 1 , i j H_{2k+1}=\langle a_{1},b_{1},\dots,a_{k},b_{k},t|[a_{i},b_{i}]=t,[a_{i},t]=[b_% {i},t]=1,i=1,\dots,k,[a_{i},b_{j}]=1,i\neq j\rangle
  12. G = a , t | ( t - 1 a - 1 t ) a ( t - 1 a t ) = a 2 G=\langle a,t|(t^{-1}a^{-1}t)a(t^{-1}at)=a^{2}\rangle
  13. [ 2 , ) [2,\infty)
  14. F k ϕ F_{k}\rtimes_{\phi}\mathbb{Z}
  15. O ( f ( n ) 4 ) O(\sqrt[4]{f(n)})

Delta-functor.html

  1. 0 M M M ′′ 0 0\rightarrow M^{\prime}\rightarrow M\rightarrow M^{\prime\prime}\rightarrow 0
  2. δ n : T n ( M ′′ ) T n + 1 ( M ) \delta^{n}:T^{n}(M^{\prime\prime})\rightarrow T^{n+1}(M^{\prime})
  3. 0 M M M ′′ 0 0\rightarrow M^{\prime}\rightarrow M\rightarrow M^{\prime\prime}\rightarrow 0
  4. F 0 : S 0 T 0 F_{0}:S^{0}\rightarrow T^{0}

Delta_set.html

  1. { S n } n = 0 \{S_{n}\}_{n=0}^{\infty}
  2. d i : S n + 1 S n d_{i}:S_{n+1}\rightarrow S_{n}
  3. d i d j = d j - 1 d i d_{i}\circ d_{j}=d_{j-1}\circ d_{i}
  4. Δ \Delta
  5. Δ \Delta
  6. { f n : S n T n } n = 0 \{f_{n}:S_{n}\rightarrow T_{n}\}_{n=0}^{\infty}
  7. f n d i = d i f n + 1 f_{n}\circ d_{i}=d_{i}\circ f_{n+1}
  8. Δ \Delta
  9. Δ \Delta
  10. Δ \Delta
  11. | S | = ( n = 0 S n × Δ n ) / |S|=\left(\coprod_{n=0}^{\infty}S_{n}\times\Delta^{n}\right)/_{\sim}
  12. ( σ , d i t ) ( d i σ , t ) for all σ S n , t Δ n - 1 . (\sigma,d^{i}t)\sim(d_{i}\sigma,t)\quad\,\text{ for all}\quad\sigma\in S_{n},t% \in\Delta^{n-1}.
  13. Δ n \Delta^{n}
  14. d i : Δ n - 1 Δ n d^{i}:\Delta^{n-1}\rightarrow\Delta^{n}
  15. Δ \Delta
  16. | S | 0 | S | 1 | S | , |S|_{0}\subset|S|_{1}\subset\cdots\subset|S|,
  17. | S | N = ( n = 0 N S n × Δ n ) / |S|_{N}=\left(\coprod_{n=0}^{N}S_{n}\times\Delta^{n}\right)/_{\sim}
  18. ( S , ) (\mathbb{Z}S,\partial)
  19. ( S ) n = S n , (\mathbb{Z}S)_{n}=\mathbb{Z}\langle S_{n}\rangle,
  20. S n S_{n}
  21. n = d 0 - d 1 + d 2 - + ( - 1 ) n d n . \partial_{n}=d_{0}-d_{1}+d_{2}-\cdots+(-1)^{n}d_{n}.
  22. sing ( X ) \mathrm{sing}(X)
  23. σ : Δ n X . \sigma:\Delta^{n}\rightarrow X.
  24. sing n ( X ) \mathrm{sing}_{n}(X)
  25. d i : sing i + 1 ( X ) sing i ( X ) d_{i}:\mathrm{sing}_{i+1}(X)\rightarrow\mathrm{sing}_{i}(X)
  26. d i ( σ ) = σ d i , d_{i}(\sigma)=\sigma\circ d^{i},
  27. Δ \Delta
  28. S 1 S^{1}
  29. S 1 S^{1}
  30. S 0 = { v } , S 1 = { e } , S_{0}=\{v\},\quad S_{1}=\{e\},
  31. S n = S_{n}=\varnothing
  32. d 0 , d 1 : S 1 S 0 , d_{0},d_{1}:S_{1}\rightarrow S_{0},
  33. d 0 ( e ) = d 1 ( e ) = v . d_{0}(e)=d_{1}(e)=v.\quad
  34. Δ \Delta
  35. | S | S 1 |S|\cong S^{1}
  36. ( S , ) (\mathbb{Z}S,\partial)
  37. 0 e 1 v 0 , 0\longrightarrow\mathbb{Z}\langle e\rangle\stackrel{\partial_{1}}{% \longrightarrow}\mathbb{Z}\langle v\rangle\longrightarrow 0,
  38. 1 ( e ) = d 0 ( e ) - d 1 ( e ) = v - v = 0. \partial_{1}(e)=d_{0}(e)-d_{1}(e)=v-v=0.
  39. n = 0 \partial_{n}=0
  40. H 0 ( S ) = ker 0 im 1 = v \Z , H_{0}(\mathbb{Z}S)=\frac{\ker\partial_{0}}{\mathrm{im}\partial_{1}}=\mathbb{Z}% \langle v\rangle\cong\Z,
  41. H 1 ( S ) = ker 1 im 2 = e \Z . H_{1}(\mathbb{Z}S)=\frac{\ker\partial_{1}}{\mathrm{im}\partial_{2}}=\mathbb{Z}% \langle e\rangle\cong\Z.
  42. Δ \Delta
  43. Δ \Delta
  44. Δ \Delta

Demon_algorithm.html

  1. Δ E \Delta E
  2. Δ E \Delta E
  3. | Δ E | |\Delta E|
  4. Δ E < 0 \Delta E<0
  5. Δ E \Delta E
  6. E d > Δ E E_{d}>\Delta E

Depolarizer_(optics).html

  1. δ ( y ) = 2 π λ [ n 2 - n 1 ] ( 2 y - a ) \delta(y)=\frac{2\pi}{\lambda}[n_{2}-n_{1}]\left(2y-a\right)
  2. y y

Depth–slope_product.html

  1. R h R_{h}
  2. h h
  3. R h = A P R_{h}=\frac{A}{P}
  4. A A
  5. P P
  6. A = b h A=bh
  7. b b
  8. P = b + 2 h P=b+2h
  9. P b P\rightarrow b
  10. R h b h b = h R_{h}\rightarrow\frac{bh}{b}=h
  11. ρ \rho
  12. g g
  13. h h
  14. ρ g \rho g
  15. h h
  16. P b P_{b}
  17. P b = ρ g h P_{b}=\rho gh\,
  18. α \alpha
  19. / ( τ b ) /\left(\tau_{b}\right)
  20. α \alpha
  21. τ b = ρ g h sin ( α ) \tau_{b}=\rho gh\sin{\left(\alpha\right)}\,
  22. α \alpha
  23. sin ( α ) tan ( α ) \sin{\left(\alpha\right)}\approx\tan{\left(\alpha\right)}
  24. α \alpha
  25. S S
  26. tan ( α ) S \tan{\left(\alpha\right)}\equiv S
  27. τ b = ρ g h S \tau_{b}=\rho ghS\,
  28. τ b h S \tau_{b}\propto hS
  29. τ b = 10000 [ kg m 2 s 2 ] h S \tau_{b}=10000\left[\frac{\mathrm{kg}}{\mathrm{m}^{2}\ \mathrm{s}^{2}}\right]hS

Derivation_of_the_Routh_array.html

  1. f ( x ) \displaystyle f(x)
  2. f ( x ) = 0 f(x)=0\,
  3. N N\,
  4. f ( x ) = 0 f(x)=0\,
  5. P P\,
  6. f ( x ) = 0 f(x)=0\,
  7. N + P = n ( 3 ) N+P=n\quad(3)\,
  8. f ( x ) f(x)\,
  9. f ( x ) = ρ ( x ) e j θ ( x ) ( 4 ) f(x)=\rho(x)e^{j\theta(x)}\quad(4)\,
  10. ρ ( x ) = 𝔢 2 [ f ( x ) ] + 𝔪 2 [ f ( x ) ] ( 5 ) \rho(x)=\sqrt{\mathfrak{Re}^{2}[f(x)]+\mathfrak{Im}^{2}[f(x)]}\quad(5)
  11. θ ( x ) = tan - 1 ( 𝔪 [ f ( x ) ] / 𝔢 [ f ( x ) ] ) ( 6 ) \theta(x)=\tan^{-1}\big(\mathfrak{Im}[f(x)]/\mathfrak{Re}[f(x)]\big)\quad(6)
  12. θ ( x ) = θ r 1 ( x ) + θ r 2 ( x ) + + θ r n ( x ) ( 7 ) \theta(x)=\theta_{r_{1}}(x)+\theta_{r_{2}}(x)+\cdots+\theta_{r_{n}}(x)\quad(7)\,
  13. θ r i ( x ) = ( x - r i ) ( 8 ) \theta_{r_{i}}(x)=\angle(x-r_{i})\quad(8)\,
  14. f ( x ) = 0 f(x)=0\,
  15. θ r i ( x ) | x = j \displaystyle\theta_{r_{i}}(x)\big|_{x=j\infty}
  16. θ r i ( x ) | x = - j = ( - 𝔢 [ r i ] , - ) = lim ϕ tan - 1 ϕ = π 2 ( 10 ) \theta_{r_{i}}(x)\big|_{x=-j\infty}=\angle(-\mathfrak{Re}[r_{i}],-\infty)=\lim% _{\phi\to\infty}\tan^{-1}\phi=\frac{\pi}{2}\quad(10)\,
  17. f ( x ) = 0 f(x)=0\,
  18. θ r i ( x ) | x = j = ( - 𝔢 [ r i ] , ) = lim ϕ tan - 1 ϕ = π 2 ( 11 ) \theta_{r_{i}}(x)\big|_{x=j\infty}=\angle(-\mathfrak{Re}[r_{i}],\infty)=\lim_{% \phi\to\infty}\tan^{-1}\phi=\frac{\pi}{2}\,\quad(11)
  19. θ r i ( x ) | x = - j = ( - 𝔢 [ r i ] , - ) = lim ϕ - tan - 1 ϕ = - π 2 ( 12 ) \theta_{r_{i}}(x)\big|_{x=-j\infty}=\angle(-\mathfrak{Re}[r_{i}],-\infty)=\lim% _{\phi\to-\infty}\tan^{-1}\phi=-\frac{\pi}{2}\,\quad(12)
  20. θ r i ( x ) | x = - j x = j = - π \theta_{r_{i}}(x)\Big|_{x=-j\infty}^{x=j\infty}=-\pi\,
  21. f ( x ) f(x)\,
  22. θ r i ( x ) | x = - j x = j = π \theta_{r_{i}}(x)\Big|_{x=-j\infty}^{x=j\infty}=\pi\,
  23. f ( x ) f(x)\,
  24. θ ( x ) | x = j = ( x - r 1 ) | x = j + ( x - r 2 ) | x = j + + ( x - r n ) | x = j = π 2 N - π 2 P ( 13 ) \theta(x)\big|_{x=j\infty}=\angle(x-r_{1})\big|_{x=j\infty}+\angle(x-r_{2})% \big|_{x=j\infty}+\cdots+\angle(x-r_{n})\big|_{x=j\infty}=\frac{\pi}{2}N-\frac% {\pi}{2}P\quad(13)\,
  25. θ ( x ) | x = - j = ( x - r 1 ) | x = - j + ( x - r 2 ) | x = - j + + ( x - r n ) | x = - j = - π 2 N + π 2 P ( 14 ) \theta(x)\big|_{x=-j\infty}=\angle(x-r_{1})\big|_{x=-j\infty}+\angle(x-r_{2})% \big|_{x=-j\infty}+\cdots+\angle(x-r_{n})\big|_{x=-j\infty}=-\frac{\pi}{2}N+% \frac{\pi}{2}P\quad(14)\,
  26. Δ = 1 π θ ( x ) | - j j ( 15 ) \Delta=\frac{1}{\pi}\theta(x)\Big|_{-j\infty}^{j\infty}\quad(15)\,
  27. N - P = Δ ( 16 ) N-P=\Delta\quad(16)\,
  28. N = n + Δ 2 N=\frac{n+\Delta}{2}\,
  29. P = n - Δ 2 ( 17 ) P=\frac{n-\Delta}{2}\quad(17)\,
  30. f ( x ) f(x)\,
  31. n n\,
  32. Δ \Delta\,
  33. N N\,
  34. P P\,
  35. tan ( θ ) \tan(\theta)\,
  36. θ \theta\,
  37. x = ± x=\pm\infty\,
  38. θ = θ ( x ) \theta=\theta(x)\,
  39. π / 2 \pi/2\,
  40. tan ( θ ) \tan(\theta)\,
  41. θ \theta\,
  42. x x\,
  43. θ a = θ ( x ) | x = j a \theta_{a}=\theta(x)|_{x=ja}\,
  44. θ b = θ ( x ) | x = j b \theta_{b}=\theta(x)|_{x=jb}\,
  45. π \pi\,
  46. θ ( x ) \theta(x)\,
  47. π \pi\,
  48. θ \theta\,
  49. + +\infty\,
  50. - -\infty\,
  51. - -\infty\,
  52. + +\infty\,
  53. x x\,
  54. θ ( x ) \theta(x)\,
  55. π \pi\,
  56. θ \theta\,
  57. π \pi\,
  58. x = j a x=ja\,
  59. x = j b x=jb\,
  60. tan θ ( x ) = 𝔪 [ f ( x ) ] / 𝔢 [ f ( x ) ] \tan\theta(x)=\mathfrak{Im}[f(x)]/\mathfrak{Re}[f(x)]\,
  61. - -\infty\,
  62. + +\infty\,
  63. + +\infty\,
  64. - -\infty\,
  65. x x\,
  66. θ ( x ) | - j j \theta(x)\Big|_{-j\infty}^{j\infty}\,
  67. π \pi\,
  68. 𝔪 [ f ( x ) ] / 𝔢 [ f ( x ) ] \mathfrak{Im}[f(x)]/\mathfrak{Re}[f(x)]\,
  69. - -\infty\,
  70. + +\infty\,
  71. 𝔪 [ f ( x ) ] / 𝔢 [ f ( x ) ] \mathfrak{Im}[f(x)]/\mathfrak{Re}[f(x)]\,
  72. + +\infty\,
  73. - -\infty\,
  74. x x\,
  75. ( - j , + j ) (-j\infty,+j\infty\,)
  76. x = ± j x=\pm j\infty
  77. tan [ θ ( x ) ] \tan[\theta(x)]\,
  78. - cot ( θ ) -\cot(\theta)\,
  79. θ \theta\,
  80. θ a = π / 2 ± i π \theta_{a}=\pi/2\pm i\pi\,
  81. N N\,
  82. P P\,
  83. Δ \Delta\,
  84. π / 2 \pi/2\,
  85. π / 2 \pi/2\,
  86. θ \theta\,
  87. f ( x ) f(x)\,
  88. tan [ θ ] = 𝔪 [ f ( x ) ] / 𝔢 [ f ( x ) ] \tan[\theta]=\mathfrak{Im}[f(x)]/\mathfrak{Re}[f(x)]\,
  89. ( + j , - j ) (+j\infty,-j\infty)\,
  90. tan [ θ ( x ) ] = tan [ θ + π / 2 ] = - cot [ θ ( x ) ] = - 𝔢 [ f ( x ) ] / 𝔪 [ f ( x ) ] ( 18 ) \tan[\theta^{\prime}(x)]=\tan[\theta+\pi/2]=-\cot[\theta(x)]=-\mathfrak{Re}[f(% x)]/\mathfrak{Im}[f(x)]\quad(18)\,
  91. Δ \Delta\,
  92. x x\,
  93. - j -j\infty\,
  94. + j +j\infty\,
  95. θ ( x ) \theta(x)\,
  96. θ ( x ) \theta^{\prime}(x)\,
  97. θ a \theta_{a}\,
  98. π \pi\,
  99. f ( x ) f(x)\,
  100. f ( x ) = a 0 x n + b 0 x n - 1 + a 1 x n - 2 + b 1 x n - 3 + ( 19 ) f(x)=a_{0}x^{n}+b_{0}x^{n-1}+a_{1}x^{n-2}+b_{1}x^{n-3}+\cdots\quad(19)\,
  101. f ( j ω ) \displaystyle f(j\omega)
  102. n n\,
  103. f ( j ω ) = ( - 1 ) n / 2 [ a 0 ω n + a 1 ω n - 2 + a 2 ω n - 4 + ] ( 22 ) + j ( - 1 ) ( n / 2 ) - 1 [ b 0 ω n - 1 + b 1 ω n - 3 + b 2 ω n - 5 + ] \begin{aligned}\displaystyle f(j\omega)&\displaystyle=(-1)^{n/2}\big[a_{0}% \omega^{n}+a_{1}\omega^{n-2}+a_{2}\omega^{n-4}+\cdots\big]&\displaystyle{}% \quad(22)\\ &\displaystyle+j(-1)^{(n/2)-1}\big[b_{0}\omega^{n-1}+b_{1}\omega^{n-3}+b_{2}% \omega^{n-5}+\cdots\big]&\\ \end{aligned}
  104. n n\,
  105. f ( j ω ) \displaystyle f(j\omega)
  106. n n\,
  107. N + P N+P\,
  108. N + P N+P\,
  109. N - P N-P\,
  110. n n\,
  111. N - P N-P\,
  112. N - P N-P\,
  113. θ \theta\,
  114. π \pi\,
  115. tan ( θ ) \tan(\theta)\,
  116. n n\,
  117. tan ( θ ) = tan ( θ + π ) = - cot ( θ ) \tan(\theta^{\prime})=\tan(\theta+\pi)=-\cot(\theta)\,
  118. n n\,
  119. n n\,
  120. Δ = I - + - 𝔪 [ f ( x ) ] 𝔢 [ f ( x ) ] = I - + b 0 ω n - 1 - b 1 ω n - 3 + a 0 ω n - a 1 ω n - 2 + ( 24 ) \Delta=I_{-\infty}^{+\infty}\frac{-\mathfrak{Im}[f(x)]}{\mathfrak{Re}[f(x)]}=I% _{-\infty}^{+\infty}\frac{b_{0}\omega^{n-1}-b_{1}\omega^{n-3}+\cdots}{a_{0}% \omega^{n}-a_{1}\omega^{n-2}+\ldots}\quad(24)\,
  121. n n\,
  122. Δ = I - + 𝔢 [ f ( x ) ] 𝔪 [ f ( x ) ] = I - + b 0 ω n - 1 - b 1 ω n - 3 + a 0 ω n - a 1 ω n - 2 + ( 25 ) \Delta=I_{-\infty}^{+\infty}\frac{\mathfrak{Re}[f(x)]}{\mathfrak{Im}[f(x)]}=I_% {-\infty}^{+\infty}\frac{b_{0}\omega^{n-1}-b_{1}\omega^{n-3}+\ldots}{a_{0}% \omega^{n}-a_{1}\omega^{n-2}+\ldots}\quad(25)\,
  123. Δ = I - + b 0 ω n - 1 - b 1 ω n - 3 + a 0 ω n - a 1 ω n - 2 + ( 26 ) \Delta=I_{-\infty}^{+\infty}\frac{b_{0}\omega^{n-1}-b_{1}\omega^{n-3}+\ldots}{% a_{0}\omega^{n}-a_{1}\omega^{n-2}+\ldots}\quad(26)\,
  124. Δ = I - + f 2 ( x ) f 1 ( x ) \Delta=I_{-\infty}^{+\infty}\frac{f_{2}(x)}{f_{1}(x)}\,
  125. f 1 ( x ) , f 2 ( x ) , , f m ( x ) f_{1}(x),f_{2}(x),\dots,f_{m}(x)\,
  126. f k ( x ) = 0 f_{k}(x)=0\,
  127. f k - 1 ( x ) 0 f_{k-1}(x)\neq 0\,
  128. f k + 1 ( x ) 0 f_{k+1}(x)\neq 0\,
  129. sign [ f k - 1 ( x ) ] = - sign [ f k + 1 ( x ) ] \operatorname{sign}[f_{k-1}(x)]=-\operatorname{sign}[f_{k+1}(x)]\,
  130. f m ( x ) 0 f_{m}(x)\neq 0\,
  131. - < x < -\infty<x<\infty\,
  132. V ( x ) V(x)\,
  133. f 1 ( x ) , f 2 ( x ) , , f m ( x ) f_{1}(x),f_{2}(x),\dots,f_{m}(x)\,
  134. x x\,
  135. Δ = I - + f 2 ( x ) f 1 ( x ) = V ( - ) - V ( + ) ( 27 ) \Delta=I_{-\infty}^{+\infty}\frac{f_{2}(x)}{f_{1}(x)}=V(-\infty)-V(+\infty)% \quad(27)\,
  136. f 1 ( x ) f_{1}(x)\,
  137. f 2 ( x ) f_{2}(x)\,
  138. f 1 ( x ) / f 2 ( x ) f_{1}(x)/f_{2}(x)\,
  139. f 3 ( x ) f_{3}(x)\,
  140. f 2 ( x ) / f 3 ( x ) f_{2}(x)/f_{3}(x)\,
  141. f 4 ( x ) f_{4}(x)\,
  142. f 1 ( x ) = q 1 ( x ) f 2 ( x ) - f 3 ( x ) ( 28 ) \displaystyle f_{1}(x)=q_{1}(x)f_{2}(x)-f_{3}(x)\quad(28)
  143. f k - 1 ( x ) = q k - 1 ( x ) f k ( x ) - f k + 1 ( x ) f_{k-1}(x)=q_{k-1}(x)f_{k}(x)-f_{k+1}(x)\,
  144. f m ( x ) f_{m}(x)\,
  145. f 1 ( x ) , f 2 ( x ) , , f m - 1 ( x ) f_{1}(x),f_{2}(x),\dots,f_{m-1}(x)\,
  146. f 3 ( ω ) = a 0 b 0 f 2 ( ω ) - f 1 ( ω ) ( 29 ) f_{3}(\omega)=\frac{a_{0}}{b_{0}}f_{2}(\omega)-f_{1}(\omega)\quad(29)\,
  147. c 0 c_{0}\,
  148. - c 1 -c_{1}\,
  149. c 2 c_{2}\,
  150. - c 3 -c_{3}\,
  151. f 3 ( ω ) = c 0 ω n - 2 - c 1 ω n - 4 + c 2 ω n - 6 - ( 30 ) f_{3}(\omega)=c_{0}\omega^{n-2}-c_{1}\omega^{n-4}+c_{2}\omega^{n-6}-\cdots% \quad(30)\,
  152. c 0 = a 1 - a 0 b 0 b 1 = b 0 a 1 - a 1 b 0 b 0 ; c 1 = a 2 - a 0 b 0 b 2 = b 0 a 2 - a 0 b 2 b 0 ; ( 31 ) c_{0}=a_{1}-\frac{a_{0}}{b_{0}}b_{1}=\frac{b_{0}a_{1}-a_{1}b_{0}}{b_{0}};c_{1}% =a_{2}-\frac{a_{0}}{b_{0}}b_{2}=\frac{b_{0}a_{2}-a_{0}b_{2}}{b_{0}};\ldots% \quad(31)\,
  153. f 4 ( ω ) = b 0 c 0 f 3 ( ω ) - f 2 ( ω ) ( 32 ) f_{4}(\omega)=\frac{b_{0}}{c_{0}}f_{3}(\omega)-f_{2}(\omega)\quad(32)\,
  154. f 4 ( ω ) f_{4}(\omega)\,
  155. d 0 d_{0}\,
  156. - d 1 -d_{1}\,
  157. d 2 d_{2}\,
  158. - d 3 -d_{3}\,
  159. f 4 ( ω ) = d 0 ω n - 3 - d 1 ω n - 5 + d 2 ω n - 7 - ( 33 ) f_{4}(\omega)=d_{0}\omega^{n-3}-d_{1}\omega^{n-5}+d_{2}\omega^{n-7}-\cdots% \quad(33)\,
  160. d 0 = b 1 - b 0 c 0 c 1 = c 0 b 1 - b 1 c 0 c 0 ; d 1 = b 2 - b 0 c 0 c 2 = c 0 b 2 - b 0 c 2 c 0 ; ( 34 ) d_{0}=b_{1}-\frac{b_{0}}{c_{0}}c_{1}=\frac{c_{0}b_{1}-b_{1}c_{0}}{c_{0}};d_{1}% =b_{2}-\frac{b_{0}}{c_{0}}c_{2}=\frac{c_{0}b_{2}-b_{0}c_{2}}{c_{0}};\ldots% \quad(34)\,
  161. f 1 ( ω ) f_{1}(\omega)\,
  162. f 2 ( ω ) f_{2}(\omega)\,
  163. f n + 1 ( ω ) f_{n+1}(\omega)\,
  164. n n\,
  165. f 1 ( x ) , f 2 ( x ) , , f m ( x ) f_{1}(x),f_{2}(x),\dots,f_{m}(x)\,
  166. f 1 ( x ) , f 2 ( x ) , , f m ( x ) f_{1}(x),f_{2}(x),\dots,f_{m}(x)\,
  167. ω = ± \omega=\pm\infty\,
  168. ω \omega\,
  169. ω \omega\,
  170. f 1 ( x ) , f 2 ( x ) , f_{1}(x),f_{2}(x),\dots
  171. f m ( x ) f_{m}(x)\,
  172. a 0 a_{0}\,
  173. b 0 b_{0}\,
  174. c 0 c_{0}\,
  175. d 0 d_{0}\,
  176. f 1 ( x ) f_{1}(x)\,
  177. f 2 ( x ) f_{2}(x)\,
  178. f m ( x ) f_{m}(x)\,
  179. ω = ± \omega=\pm\infty\,
  180. V ( + ) = V ( a 0 , b 0 , c 0 , d 0 , ) V(+\infty)=V(a_{0},b_{0},c_{0},d_{0},\dots)\,
  181. V ( + ) V(+\infty)\,
  182. a 0 n a_{0}\infty^{n}\,
  183. b 0 n - 1 b_{0}\infty^{n-1}\,
  184. c 0 n - 2 c_{0}\infty^{n-2}\,
  185. a 0 a_{0}\,
  186. b 0 b_{0}\,
  187. c 0 c_{0}\,
  188. d 0 d_{0}\,
  189. V ( - ) = V ( a 0 , - b 0 , c 0 , - d 0 , ) V(-\infty)=V(a_{0},-b_{0},c_{0},-d_{0},...)\,
  190. V ( - ) V(-\infty)\,
  191. a 0 ( - ) n a_{0}(-\infty)^{n}\,
  192. b 0 ( - ) n - 1 b_{0}(-\infty)^{n-1}\,
  193. c 0 ( - ) n - 2 c_{0}(-\infty)^{n-2}\,
  194. a 0 a_{0}\,
  195. - b 0 -b_{0}\,
  196. c 0 c_{0}\,
  197. - d 0 -d_{0}\,
  198. a 0 a_{0}\,
  199. b 0 b_{0}\,
  200. c 0 c_{0}\,
  201. d 0 d_{0}\,
  202. n n\,
  203. V ( + ) + V ( - ) = n V(+\infty)+V(-\infty)=n\,
  204. V ( a 0 , b 0 , c 0 , d 0 , ) V(a_{0},b_{0},c_{0},d_{0},\dots)\,
  205. a 0 a_{0}\,
  206. b 0 b_{0}\,
  207. V ( a 0 , - b 0 , c 0 , - d 0 , ) V(a_{0},-b_{0},c_{0},-d_{0},\dots)\,
  208. a 0 a_{0}\,
  209. - b 0 -b_{0}\,
  210. n n\,
  211. n n\,
  212. Δ = V ( - ) - V ( + ) \Delta=V(-\infty)-V(+\infty)\,
  213. n = V ( + ) + V ( - ) n=V(+\infty)+V(-\infty)\,
  214. P = ( n - Δ / 2 ) P=(n-\Delta/2)\,
  215. P = V ( + ) = V ( a 0 , b 0 , c 0 , d 0 , ) P=V(+\infty)=V(a_{0},b_{0},c_{0},d_{0},\dots)\,
  216. f ( z ) f(z)\,
  217. 𝔢 ( r i ) > 0 \mathfrak{Re}(r_{i})>0\,
  218. P = 0 P=0\,
  219. V ( a 0 , b 0 , c 0 , d 0 , ) = 0 V(a_{0},b_{0},c_{0},d_{0},\dots)=0\,
  220. f ( z ) f(z)\,

Dexter_electron_transfer.html

  1. k E T k_{ET}
  2. k E T J exp [ - 2 r L ] k_{ET}\varpropto J\mathrm{exp}\left[\frac{-2r}{L}\right]
  3. r r
  4. L L
  5. J J
  6. J = f D ( λ ) ϵ A ( λ ) λ 4 d λ J=\int f_{\rm D}(\lambda)\,\epsilon_{\rm A}(\lambda)\,\lambda^{4}\,d\lambda

Dextrorotation_and_levorotation.html

  1. [ α ] = α c l , [\alpha]=\frac{\alpha}{c\cdot l},

Diaconescu's_theorem.html

  1. P P\,
  2. U = { x { 0 , 1 } : ( x = 0 ) P } U=\{x\in\{0,1\}:(x=0)\vee P\}
  3. V = { x { 0 , 1 } : ( x = 1 ) P } . V=\{x\in\{0,1\}:(x=1)\vee P\}.
  4. U = { { 0 , 1 } , if P { 0 } , if ¬ P U=\begin{cases}\{0,1\},&\mbox{if }~{}P\\ \{0\},&\mbox{if }~{}\neg P\end{cases}
  5. V V\,
  6. { U , V } \{U,V\}\,
  7. f f\,
  8. [ f ( U ) U ] [ f ( V ) V ] . [f(U)\in U]\wedge[f(V)\in V].\,
  9. [ ( f ( U ) = 0 ) P ] [ ( f ( V ) = 1 ) P ] [(f(U)=0)\vee P]\wedge[(f(V)=1)\vee P]\,
  10. f ( U ) f ( V ) P . f(U)\neq f(V)\vee P.
  11. P ( U = V ) P\to(U=V)
  12. P ( f ( U ) = f ( V ) ) P\to(f(U)=f(V))\,
  13. ( f ( U ) f ( V ) ) ¬ P . (f(U)\neq f(V))\to\neg P.
  14. ¬ P P . \neg P\vee P.

Dialectica_interpretation.html

  1. A A
  2. A D ( x ; y ) A_{D}(x;y)
  3. x x
  4. y y
  5. A A
  6. A A
  7. x y A D ( x ; y ) \exists x\forall yA_{D}(x;y)
  8. A A
  9. A A
  10. A A
  11. t t
  12. A D ( t ; y ) A_{D}(t;y)
  13. A D ( x ; y ) A_{D}(x;y)
  14. A A
  15. P P
  16. ( P ) D P ( A B ) D ( x , v ; y , w ) A D ( x ; y ) B D ( v ; w ) ( A B ) D ( x , v , z ; y , w ) ( z = 0 A D ( x ; y ) ) ( z 0 B D ( v ; w ) ) ( A B ) D ( f , g ; x , w ) A D ( x ; f x w ) B D ( g x ; w ) ( z A ) D ( x , z ; y ) A D ( x ; y ) ( z A ) D ( f ; y , z ) A D ( f z ; y ) \begin{array}[]{lcl}(P)_{D}&\equiv&P\\ (A\wedge B)_{D}(x,v;y,w)&\equiv&A_{D}(x;y)\wedge B_{D}(v;w)\\ (A\vee B)_{D}(x,v,z;y,w)&\equiv&(z=0\rightarrow A_{D}(x;y))\wedge(z\neq 0\to B% _{D}(v;w))\\ (A\rightarrow B)_{D}(f,g;x,w)&\equiv&A_{D}(x;fxw)\rightarrow B_{D}(gx;w)\\ (\exists zA)_{D}(x,z;y)&\equiv&A_{D}(x;y)\\ (\forall zA)_{D}(f;y,z)&\equiv&A_{D}(fz;y)\end{array}
  17. A A
  18. t t
  19. A D ( t ; y ) A_{D}(t;y)
  20. t t
  21. A D ( t ; y ) A_{D}(t;y)
  22. A A
  23. t t
  24. A A A A\rightarrow A\wedge A

Diamond_graph.html

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

Differential_game.html

  1. u ( t ) u(t)
  2. u ( t ) , v ( t ) u(t),v(t)

Differential_pulley.html

  1. M A = 2 R R - r = 2 1 - r R M\!A=\frac{2R}{R-r}=\frac{2}{1-\frac{r}{R}}
  2. F Z = F L M A , h = s M A . F_{\mathrm{Z}}=\frac{F_{\mathrm{L}}}{M\!A}\quad,\quad h=\frac{s}{M\!A}\quad.
  3. R R R\frac{R}{−}
  4. F < s u b > L 2 F\frac{<sub>L}{2}

Diffusion.html

  1. 𝐉 \mathbf{J}
  2. N N
  3. Δ S \Delta S
  4. ν \nu
  5. Δ t \Delta t
  6. Δ N = ( 𝐉 , ν ) Δ S Δ t + o ( Δ S Δ t ) , \Delta N=(\mathbf{J},\nu)\Delta S\Delta t+o(\Delta S\Delta t)\,,
  7. ( 𝐉 , ν ) (\mathbf{J},\nu)
  8. o ( ) o(...)
  9. Δ 𝐒 = ν Δ S \Delta\mathbf{S}=\nu\Delta S
  10. Δ N = ( 𝐉 , Δ 𝐒 ) Δ t + o ( Δ 𝐒 Δ t ) . \Delta N=(\mathbf{J},\Delta\mathbf{S})\Delta t+o(\Delta\mathbf{S}\Delta t)\,.
  11. N N
  12. n n
  13. n t = - 𝐉 + W , \frac{\partial n}{\partial t}=-\nabla\cdot\mathbf{J}+W\,,
  14. W W
  15. ( 𝐉 ( x ) , ν ( x ) ) = 0 (\mathbf{J}(x),\nu(x))=0
  16. ν \nu
  17. x x
  18. 𝐉 = - D n , J i = - D n x i . \mathbf{J}=-D\nabla n\ ,\;\;J_{i}=-D\frac{\partial n}{\partial x_{i}}\ .
  19. n ( x , t ) t = ( D n ( x , t ) ) = D Δ n ( x , t ) , \frac{\partial n(x,t)}{\partial t}=\nabla\cdot(D\nabla n(x,t))=D\Delta n(x,t)\ ,
  20. Δ \Delta
  21. Δ n ( x , t ) = i 2 n ( x , t ) x i 2 . \Delta n(x,t)=\sum_{i}\frac{\partial^{2}n(x,t)}{\partial x_{i}^{2}}\ .
  22. - n -\nabla n
  23. 𝐉 i = j L i j X j , \mathbf{J}_{i}=\sum_{j}L_{ij}X_{j}\,,
  24. 𝐉 i \mathbf{J}_{i}
  25. X j X_{j}
  26. X i = grad s ( n ) n i , X_{i}={\rm grad}\frac{\partial s(n)}{\partial n_{i}}\ ,
  27. n i n_{i}
  28. n 0 = u n_{0}=u
  29. n i n_{i}
  30. X 0 = grad 1 T , X i = - grad μ i T ( i > 0 ) , X_{0}={\rm grad}\frac{1}{T}\ ,\;\;\;X_{i}=-{\rm grad}\frac{\mu_{i}}{T}\;(i>0),
  31. d s = 1 T d u - i 1 μ i T d n i {\rm d}s=\frac{1}{T}{\rm d}u-\sum_{i\geq 1}\frac{\mu_{i}}{T}{\rm d}n_{i}
  32. μ i \mu_{i}
  33. X i = k 0 2 s ( n ) n i n k | n = n * grad n k , X_{i}=\sum_{k\geq 0}\left.\frac{\partial^{2}s(n)}{\partial n_{i}\partial n_{k}% }\right|_{n=n^{*}}{\rm grad}n_{k}\ ,
  34. L i j L_{ij}
  35. n i t = - div 𝐉 i = - j 0 L i j div X j = k 0 [ - j 0 L i j 2 s ( n ) n j n k | n = n * ] Δ n k . \frac{\partial n_{i}}{\partial t}=-{\rm div}\mathbf{J}_{i}=-\sum_{j\geq 0}L_{% ij}{\rm div}X_{j}=\sum_{k\geq 0}\left[-\sum_{j\geq 0}L_{ij}\left.\frac{% \partial^{2}s(n)}{\partial n_{j}\partial n_{k}}\right|_{n=n^{*}}\right]\Delta n% _{k}\ .
  36. D i k D_{ik}
  37. - ( 1 / T ) μ j -(1/T)\nabla\mu_{j}
  38. D i k = 1 T j 1 L i j μ j ( n , T ) n k | n = n * D_{ik}=\frac{1}{T}\sum_{j\geq 1}L_{ij}\left.\frac{\partial\mu_{j}(n,T)}{% \partial n_{k}}\right|_{n=n^{*}}
  39. j L i j X j \sum_{j}L_{ij}X_{j}
  40. c i t = j D i j Δ c j . \frac{\partial c_{i}}{\partial t}=\sum_{j}D_{ij}\Delta c_{j}\,.
  41. D 12 0 D_{12}\neq 0
  42. c 2 = = c n = 0 c_{2}=\ldots=c_{n}=0
  43. c 2 / t = D 12 Δ c 1 \partial c_{2}/\partial t=D_{12}\Delta c_{1}
  44. D 12 Δ c 1 ( x ) < 0 D_{12}\Delta c_{1}(x)<0
  45. c 2 ( x ) c_{2}(x)
  46. D = μ k B T D=\mu\,k_{B}T
  47. 𝔪 \mathfrak{m}
  48. - R T 1 n n = - R T ( ln ( n / n eq ) ) . -RT\frac{1}{n}\nabla n=-RT\nabla(\ln(n/n^{\rm eq})).
  49. q φ . q\nabla\varphi.
  50. 𝐉 = 𝔪 exp ( μ - μ 0 R T ) ( - μ + ( external force per gram particle ) ) , \mathbf{J}=\mathfrak{m}\exp\left(\frac{\mu-\mu_{0}}{RT}\right)(-\nabla\mu+(% \mbox{external force per gram particle}~{}))\,,
  51. a = exp ( μ - μ 0 R T ) a=\exp\left(\frac{\mu-\mu_{0}}{RT}\right)
  52. 𝐉 = 𝔪 a ( - μ + ( external force per gram particle ) ) . \mathbf{J}=\mathfrak{m}a(-\nabla\mu+(\mbox{external force per gram particle}~{% }))\,.
  53. a = n / n + o ( n / n ) a=n/n^{\ominus}+o(n/n^{\ominus})
  54. n n^{\ominus}
  55. n / n n/n^{\ominus}
  56. ( n / n ) t = [ 𝔪 a ( μ - ( external force per gram particle ) ) ] \frac{\partial(n/n^{\ominus})}{\partial t}=\nabla\cdot[\mathfrak{m}a(\nabla\mu% -(\mbox{external force per gram particle}~{}))]
  57. 𝐉 i = 𝔪 𝔦 a i j L i j X j , \mathbf{J}_{i}=\mathfrak{m_{i}}a_{i}\sum_{j}L_{ij}X_{j}\,,
  58. 𝔪 𝔦 \mathfrak{m_{i}}
  59. a i a_{i}
  60. L i j L_{ij}
  61. X j X_{j}
  62. X j = - μ j T X_{j}=-{\rm\nabla}\frac{\mu_{j}}{T}
  63. X j = - R n j n j X_{j}=-R\frac{{\rm\nabla}n_{j}}{n_{j}}
  64. n i t = j ( D i j n i n j n j ) . \frac{\partial n_{i}}{\partial t}=\sum_{j}\nabla\cdot\left(D_{ij}\frac{n_{i}}{% n_{j}}\nabla n_{j}\right)\,.
  65. D i j D_{ij}
  66. A 1 , A 2 , A m A_{1},A_{2},\ldots A_{m}
  67. c 1 , c 2 , c m c_{1},c_{2},\ldots c_{m}
  68. z = c 0 z=c_{0}
  69. c i c_{i}
  70. A i A_{i}
  71. 𝐉 i = - D i [ z c i - c i z ] . \mathbf{J}_{i}=-D_{i}[z\nabla c_{i}-c_{i}\nabla z]\,.
  72. c i t = - div 𝐉 i = D i [ z Δ c i - c i Δ z ] . \frac{\partial c_{i}}{\partial t}=-\mathrm{div}\mathbf{J}_{i}=D_{i}[z\Delta c_% {i}-c_{i}\Delta z]\,.
  73. z = b - i = 1 n c i , z=b-\sum_{i=1}^{n}c_{i}\,,
  74. ( b - c ) c - c ( b - c ) = b c (b-c)\nabla c-c\nabla(b-c)=b\nabla c
  75. 𝐉 i = - j D i j [ c j c i - c i c j ] \mathbf{J}_{i}=-\sum_{j}D_{ij}[c_{j}\nabla c_{i}-c_{i}\nabla c_{j}]
  76. c i t = j D i j [ c j Δ c i - c i Δ c j ] \frac{\partial c_{i}}{\partial t}=\sum_{j}D_{ij}[c_{j}\Delta c_{i}-c_{i}\Delta c% _{j}]
  77. D i j = D j i 0 D_{ij}=D_{ji}\geq 0
  78. c 0 c_{0}
  79. 𝐉 = - D n m \mathbf{J}=-D\nabla n^{m}
  80. n t = D Δ n m , \frac{\partial n}{\partial t}=D\Delta n^{m}\,,
  81. v = - k μ p v=-\frac{k}{\mu}\nabla p
  82. p n γ p\sim n^{\gamma}
  83. D D
  84. J = - D n / x J=-D{\partial n}/{\partial x}
  85. D = 1 3 v T = 2 3 k B 3 π 3 m T 3 / 2 P d 2 , D=\frac{1}{3}\ell v_{T}=\frac{2}{3}\sqrt{\frac{k_{\rm B}^{3}}{\pi^{3}m}}\frac{% T^{3/2}}{Pd^{2}}\,,
  86. \ell
  87. = k B T 2 π d 2 P , v T = 8 k B T π m . \ell=\frac{k_{\rm B}T}{\sqrt{2}\pi d^{2}P}\,,\;\;\;v_{T}=\sqrt{\frac{8k_{\rm B% }T}{\pi m}}\,.
  88. D AB = 2 3 k B 3 π 3 1 2 m A + 1 2 m B 4 T 3 / 2 P ( d A + d B ) 2 , D_{\rm AB}=\frac{2}{3}\sqrt{\frac{k_{\rm B}^{3}}{\pi^{3}}}\sqrt{\frac{1}{2m_{% \rm A}}+\frac{1}{2m_{\rm B}}}\frac{4T^{3/2}}{P(d_{\rm A}+d_{\rm B})^{2}}\,,
  89. f i ( x , c , t ) f_{i}(x,c,t)
  90. C i ( x , t ) = 1 n i c c f ( x , c , t ) d c C_{i}(x,t)=\frac{1}{n_{i}}\int_{c}cf(x,c,t)\,dc
  91. C i ( x , t ) C_{i}(x,t)
  92. n i ( x , t ) = c f i ( x , c , t ) d c n_{i}(x,t)=\int_{c}f_{i}(x,c,t)\,dc
  93. i m i n i C i ( x , t ) \sum_{i}m_{i}n_{i}C_{i}(x,t)
  94. i ( n i m i C i 2 ( x , t ) 2 + c m i ( c i - C i ( x , t ) ) 2 2 f i ( x , c , t ) d c ) \sum_{i}\left(n_{i}\frac{m_{i}C^{2}_{i}(x,t)}{2}+\int_{c}\frac{m_{i}(c_{i}-C_{% i}(x,t))^{2}}{2}f_{i}(x,c,t)\,dc\right)
  95. 3 2 k B T = 1 n c m i ( c i - C i ( x , t ) ) 2 2 f i ( x , c , t ) d c \frac{3}{2}k_{\rm B}T=\frac{1}{n}\int_{c}\frac{m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}% f_{i}(x,c,t)\,dc
  96. P = k B n T P=k_{\rm B}nT
  97. n = i n i n=\sum_{i}n_{i}
  98. C 1 - C 2 C_{1}-C_{2}
  99. C 1 - C 2 = - n 2 n 1 n 2 D 12 { ( n 1 n ) + n 1 n 2 ( m 2 - m 1 ) n ( m 1 n 1 + m 2 n 2 ) P - m 1 n 1 m 2 n 2 P ( m 1 n 1 + m 2 n 2 ) ( F 1 - F 2 ) + k T 1 T T } C_{1}-C_{2}=-\frac{n^{2}}{n_{1}n_{2}}D_{12}\left\{\nabla\left(\frac{n_{1}}{n}% \right)+\frac{n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}\nabla P-\frac% {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}(F_{1}-F_{2})+k_{T}\frac{1}{T}% \nabla T\right\}
  100. F i F_{i}
  101. k T k_{T}
  102. ( n 1 n ) \nabla\left(\frac{n_{1}}{n}\right)
  103. n 1 n 2 ( m 2 - m 1 ) n ( m 1 n 1 + m 2 n 2 ) P \frac{n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}\nabla P
  104. m 1 n 1 m 2 n 2 P ( m 1 n 1 + m 2 n 2 ) ( F 1 - F 2 ) \frac{m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}(F_{1}-F_{2})
  105. k T 1 T T k_{T}\frac{1}{T}\nabla T
  106. D 12 = 3 2 n ( d 1 + d 2 ) 2 [ k T ( m 1 + m 2 ) 2 π m 1 m 2 ] 1 / 2 D_{12}=\frac{3}{2n(d_{1}+d_{2})^{2}}\left[\frac{kT(m_{1}+m_{2})}{2\pi m_{1}m_{% 2}}\right]^{1/2}
  107. D 12 = 3 8 n A 1 ( ν ) Γ ( 3 - 2 ν - 1 ) [ k T ( m 1 + m 2 ) 2 π m 1 m 2 ] 1 / 2 ( 2 k T κ 12 ) 2 ν - 1 D_{12}=\frac{3}{8nA_{1}({\nu})\Gamma(3-\frac{2}{\nu-1})}\left[\frac{kT(m_{1}+m% _{2})}{2\pi m_{1}m_{2}}\right]^{1/2}\left(\frac{2kT}{\kappa_{12}}\right)^{% \frac{2}{\nu-1}}
  108. κ 12 r - ν \kappa_{12}r^{-\nu}
  109. A 1 ( ν ) A_{1}({\nu})
  110. V = i ρ i C i ρ . V=\frac{\sum_{i}\rho_{i}C_{i}}{\rho}\,.
  111. ρ i = m i n i \rho_{i}=m_{i}n_{i}
  112. ρ = i ρ i \rho=\sum_{i}\rho_{i}
  113. v i = C i - V v_{i}=C_{i}-V
  114. i ρ i v i = 0 \sum_{i}\rho_{i}v_{i}=0
  115. ρ i t + ( ρ i V ) + ( ρ i v i ) = W i , \frac{\partial\rho_{i}}{\partial t}+\nabla(\rho_{i}V)+\nabla(\rho_{i}v_{i})=W_% {i}\,,
  116. W i W_{i}
  117. i W i = 0 \sum_{i}W_{i}=0
  118. ( ρ i V ) \nabla(\rho_{i}V)
  119. ( ρ i v i ) \nabla(\rho_{i}v_{i})
  120. v i = - ( j = 1 N D i j 𝐝 j + D i ( T ) ( ln T ) ) ; v_{i}=-\left(\sum_{j=1}^{N}D_{ij}\mathbf{d}_{j}+D_{i}^{(T)}\nabla(\ln T)\right% )\,;
  121. 𝐝 j = X j + ( X j - Y j ) ( ln P ) + 𝐠 j ; \mathbf{d}_{j}=\nabla X_{j}+(X_{j}-Y_{j})\nabla(\ln P)+\mathbf{g}_{j}\,;
  122. 𝐠 j = ρ P ( Y j k = 1 N Y k ( f k - f j ) ) . \mathbf{g}_{j}=\frac{\rho}{P}\left(Y_{j}\sum_{k=1}^{N}Y_{k}(f_{k}-f_{j})\right% )\,.
  123. D i j D_{ij}
  124. D i ( T ) D_{i}^{(T)}
  125. f i f_{i}
  126. X i = P i / P X_{i}=P_{i}/P
  127. P i P_{i}
  128. Y i = ρ i / ρ Y_{i}=\rho_{i}/\rho
  129. i X i = i Y i = 1 \sum_{i}X_{i}=\sum_{i}Y_{i}=1