wpmath0000001_5

Continuum_hypothesis.html

  1. { banana , apple , pear } \{\,\text{banana},\,\text{apple},\,\text{pear}\}
  2. { yellow , red , green } \{\,\text{yellow},\,\text{red},\,\text{green}\}
  3. 0 \aleph_{0}
  4. 2 0 2^{\aleph_{0}}
  5. S S
  6. 0 < | S | < 2 0 . \aleph_{0}<|S|<2^{\aleph_{0}}.\,
  7. 1 \aleph_{1}
  8. 0 \aleph_{0}
  9. 2 0 = 1 . 2^{\aleph_{0}}=\aleph_{1}.\,
  10. α \alpha\,
  11. 2 α = α + 1 . 2^{\aleph_{\alpha}}=\aleph_{\alpha+1}.
  12. ϕ \phi
  13. ( ϕ ¬ ϕ ) (\phi\neg\phi)
  14. λ \lambda\,
  15. κ \kappa\,
  16. λ < κ < 2 λ . \lambda<\kappa<2^{\lambda}.\,
  17. α + 1 = 2 α \aleph_{\alpha+1}=2^{\aleph_{\alpha}}
  18. α . \alpha.\,
  19. α = α \aleph_{\alpha}=\beth_{\alpha}
  20. α . \alpha.\,
  21. 2 0 + n 2^{\aleph_{0}+n}\,
  22. 2 0 + n = 2 2 0 + n 2^{\aleph_{0}+n}\,=\,2\cdot\,2^{\aleph_{0}+n}
  23. α \aleph_{\alpha}
  24. 2 α = α + 1 . 2^{\aleph_{\alpha}}=\aleph_{\alpha+1}.
  25. 2 κ > κ + 2^{\kappa}>\kappa^{+}\,
  26. κ . \kappa.\,
  27. 2 κ = κ + + 2^{\kappa}=\kappa^{++}\,
  28. κ \kappa\,
  29. A < B 2 A 2 B . A<B\to 2^{A}\leq 2^{B}.
  30. A < B 2 A < 2 B A<B\to 2^{A}<2^{B}\!
  31. α β \aleph_{\alpha}^{\aleph_{\beta}}
  32. β + 1 \aleph_{\beta+1}
  33. α \aleph_{\alpha}
  34. α + 1 \aleph_{\alpha+1}

Continuum_mechanics.html

  1. x x
  2. t t
  3. ρ ( x , t ) \rho(x,t)
  4. u ( x , t ) u(x,t)
  5. x x
  6. x = a ( t ) x=a(t)
  7. x = b ( t ) x=b(t)
  8. N = a ( t ) b ( t ) ρ ( x , t ) d x N=\int_{a(t)}^{b(t)}\rho(x,t)\,dx
  9. d N / d t = 0 dN/dt=0
  10. d N d t = d d t a ( t ) b ( t ) ρ ( x , t ) d x = a b ρ t d x + ρ ( b , t ) d b d t - ρ ( a , t ) d a d t = a b ρ t d x + ρ ( b , t ) u ( b , t ) - ρ ( a , t ) u ( a , t ) = a b ρ t + x ( ρ u ) d x \begin{array}[]{rcl}\frac{dN}{dt}&=&\frac{d}{dt}\int_{a(t)}^{b(t)}\rho(x,t)\,% dx\\ &=&\int_{a}^{b}\frac{\partial\rho}{\partial t}\,dx+\rho(b,t)\frac{db}{dt}-\rho% (a,t)\frac{da}{dt}\\ &=&\int_{a}^{b}\frac{\partial\rho}{\partial t}\,dx+\rho(b,t)u(b,t)-\rho(a,t)u(% a,t)\\ &=&\int_{a}^{b}\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(% \rho u)\,dx\end{array}
  11. [ a , b ] [a,b]
  12. x x
  13. ρ t + x ( ρ u ) = 0 \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}(\rho u)=0
  14. u = V ( ρ ) u=V(\rho)
  15. V V
  16. u = V ( ρ ) = 27.5 ln ( 142 / ρ ) u=V(\rho)=27.5\ln(142/\rho)
  17. ρ t + x [ ρ V ( ρ ) ] = 0 \frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}[\rho V(\rho)]=0
  18. ρ ( x , t ) \rho(x,t)
  19. \mathcal{B}
  20. t \ t
  21. κ t ( ) \ \kappa_{t}(\mathcal{B})
  22. 𝐱 = i = 1 3 x i 𝐞 i , \ \mathbf{x}=\sum_{i=1}^{3}x_{i}\mathbf{e}_{i},
  23. 𝐞 i \mathbf{e}_{i}
  24. 𝐗 \mathbf{X}
  25. 𝐱 = κ t ( 𝐗 ) . \mathbf{x}=\kappa_{t}(\mathbf{X}).
  26. κ t ( ) \kappa_{t}(\cdot)
  27. κ t ( ) \ \kappa_{t}(\cdot)
  28. 𝐅 C \mathbf{F}_{C}
  29. 𝐅 B \mathbf{F}_{B}
  30. \mathcal{F}
  31. = 𝐅 B + 𝐅 C \mathcal{F}=\mathbf{F}_{B}+\mathbf{F}_{C}
  32. 𝐓 ( 𝐧 , 𝐱 , t ) \mathbf{T}(\mathbf{n},\mathbf{x},t)
  33. t t\,\!
  34. 𝐱 \mathbf{x}
  35. 𝐧 \mathbf{n}
  36. d S dS\,\!
  37. 𝐧 \mathbf{n}
  38. S S\,\!
  39. d 𝐅 C d\mathbf{F}_{C}\,\!
  40. S S\,\!
  41. d 𝐅 C = 𝐓 ( 𝐧 ) d S d\mathbf{F}_{C}=\mathbf{T}^{(\mathbf{n})}\,dS
  42. 𝐓 ( 𝐧 ) \mathbf{T}^{(\mathbf{n})}
  43. S S\,\!
  44. d S dS\,\!
  45. 𝐅 C = S 𝐓 ( 𝐧 ) d S \mathbf{F}_{C}=\int_{S}\mathbf{T}^{(\mathbf{n})}\,dS
  46. 𝐛 ( 𝐱 , t ) \mathbf{b}(\mathbf{x},t)
  47. ρ ( 𝐱 , t ) \mathbf{\rho}(\mathbf{x},t)\,\!
  48. b i b_{i}\,\!
  49. p i p_{i}\,\!
  50. ρ b i = p i \rho b_{i}=p_{i}\,\!
  51. 𝐅 B = V 𝐛 d m = V ρ 𝐛 d V \mathbf{F}_{B}=\int_{V}\mathbf{b}\,dm=\int_{V}\rho\mathbf{b}\,dV
  52. \mathcal{M}
  53. = 𝐌 B + 𝐌 C \mathcal{M}=\mathbf{M}_{B}+\mathbf{M}_{C}
  54. = V 𝐚 d m = S 𝐓 d S + V ρ 𝐛 d V \mathcal{F}=\int_{V}\mathbf{a}\,dm=\int_{S}\mathbf{T}\,dS+\int_{V}\rho\mathbf{% b}\,dV
  55. = S 𝐫 × 𝐓 d S + V 𝐫 × ρ 𝐛 d V \mathcal{M}=\int_{S}\mathbf{r}\times\mathbf{T}\,dS+\int_{V}\mathbf{r}\times% \rho\mathbf{b}\,dV
  56. κ 0 ( ) \ \kappa_{0}(\mathcal{B})
  57. κ t ( ) \ \kappa_{t}(\mathcal{B})
  58. t = 0 \ t=0
  59. κ 0 ( ) \ \kappa_{0}(\mathcal{B})
  60. X i \ X_{i}
  61. 𝐗 \ \mathbf{X}
  62. t = 0 \ t=0
  63. κ 0 ( ) \kappa_{0}(\mathcal{B})
  64. χ ( ) \ \chi(\cdot)
  65. 𝐱 = χ ( 𝐗 , t ) \ \mathbf{x}=\chi(\mathbf{X},t)
  66. κ 0 ( ) \kappa_{0}(\mathcal{B})
  67. κ t ( ) \kappa_{t}(\mathcal{B})
  68. 𝐱 = x i 𝐞 i \ \mathbf{x}=x_{i}\mathbf{e}_{i}
  69. X \ X
  70. 𝐗 \ \mathbf{X}
  71. κ 0 ( ) \kappa_{0}(\mathcal{B})
  72. κ t ( ) \kappa_{t}(\mathcal{B})
  73. t \ t
  74. x i \ x_{i}
  75. P i j \ P_{ij\ldots}
  76. P i j = P i j ( 𝐗 , t ) \ P_{ij\ldots}=P_{ij\ldots}(\mathbf{X},t)
  77. P i j \ P_{ij\ldots}
  78. P i j \ P_{ij\ldots}
  79. 𝐗 \ \mathbf{X}
  80. d d t [ P i j ( 𝐗 , t ) ] = t [ P i j ( 𝐗 , t ) ] \ \frac{d}{dt}[P_{ij\ldots}(\mathbf{X},t)]=\frac{\partial}{\partial t}[P_{ij% \ldots}(\mathbf{X},t)]
  81. 𝐱 \ \mathbf{x}
  82. 𝐯 \ \mathbf{v}
  83. 𝐯 = 𝐱 ˙ = d 𝐱 d t = χ ( 𝐗 , t ) t \ \mathbf{v}=\dot{\mathbf{x}}=\frac{d\mathbf{x}}{dt}=\frac{\partial\chi(% \mathbf{X},t)}{\partial t}
  84. 𝐚 = 𝐯 ˙ = 𝐱 ¨ = d 2 𝐱 d t 2 = 2 χ ( 𝐗 , t ) t 2 \ \mathbf{a}=\dot{\mathbf{v}}=\ddot{\mathbf{x}}=\frac{d^{2}\mathbf{x}}{dt^{2}}% =\frac{\partial^{2}\chi(\mathbf{X},t)}{\partial t^{2}}
  85. χ ( ) \chi(\cdot)
  86. P i j ( ) \ P_{ij\ldots}(\cdot)
  87. χ ( ) \chi(\cdot)
  88. 𝐱 \mathbf{x}
  89. κ 0 ( ) \kappa_{0}(\mathcal{B})
  90. κ t ( ) \kappa_{t}(\mathcal{B})
  91. 𝐗 = χ - 1 ( 𝐱 , t ) \mathbf{X}=\chi^{-1}(\mathbf{x},t)
  92. 𝐱 \mathbf{x}
  93. κ t ( ) \kappa_{t}(\mathcal{B})
  94. 𝐗 \mathbf{X}
  95. κ 0 ( ) \kappa_{0}(\mathcal{B})
  96. J = | χ i X J | = | x i X J | 0 \ J=\left|\frac{\partial\chi_{i}}{\partial X_{J}}\right|=\left|\frac{\partial x% _{i}}{\partial X_{J}}\right|\neq 0
  97. P i j \ P_{ij\ldots}
  98. P i j = P i j ( 𝐗 , t ) = P i j [ χ - 1 ( 𝐱 , t ) , t ] = p i j ( 𝐱 , t ) \ P_{ij\ldots}=P_{ij\ldots}(\mathbf{X},t)=P_{ij\ldots}[\chi^{-1}(\mathbf{x},t)% ,t]=p_{ij\ldots}(\mathbf{x},t)
  99. P i j \ P_{ij\ldots}
  100. p i j \ p_{ij\ldots}
  101. p i j ( 𝐱 , t ) \ p_{ij\ldots}(\mathbf{x},t)
  102. d d t [ p i j ( 𝐱 , t ) ] = t [ p i j ( 𝐱 , t ) ] + x k [ p i j ( 𝐱 , t ) ] d x k d t \ \frac{d}{dt}[p_{ij\ldots}(\mathbf{x},t)]=\frac{\partial}{\partial t}[p_{ij% \ldots}(\mathbf{x},t)]+\frac{\partial}{\partial x_{k}}[p_{ij\ldots}(\mathbf{x}% ,t)]\frac{dx_{k}}{dt}
  103. p i j ( 𝐱 , t ) \ p_{ij\ldots}(\mathbf{x},t)
  104. 𝐱 \ \mathbf{x}
  105. 𝐱 \ \mathbf{x}
  106. P \ P
  107. 𝐮 ( 𝐗 , t ) = u i 𝐞 i \ \mathbf{u}(\mathbf{X},t)=u_{i}\mathbf{e}_{i}
  108. 𝐔 ( 𝐱 , t ) = U J 𝐄 J \ \mathbf{U}(\mathbf{x},t)=U_{J}\mathbf{E}_{J}
  109. 𝐮 ( 𝐗 , t ) = 𝐛 + 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}(\mathbf{X},t)-\mathbf{X}% \qquad\,\text{or}\qquad u_{i}=\alpha_{iJ}b_{J}+x_{i}-\alpha_{iJ}X_{J}
  110. 𝐔 ( 𝐱 , t ) = 𝐛 + 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}-\mathbf{X}(\mathbf{x},t)% \qquad\,\text{or}\qquad U_{J}=b_{J}+\alpha_{Ji}x_{i}-X_{J}\,
  111. α J i \ \alpha_{Ji}
  112. 𝐄 J \ \mathbf{E}_{J}
  113. 𝐞 i \mathbf{e}_{i}
  114. 𝐄 J 𝐞 i = α J i = α i J \ \mathbf{E}_{J}\cdot\mathbf{e}_{i}=\alpha_{Ji}=\alpha_{iJ}
  115. u i \ u_{i}
  116. U J \ U_{J}
  117. 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}
  118. 𝐞 i = α i J 𝐄 J \ \mathbf{e}_{i}=\alpha_{iJ}\mathbf{E}_{J}
  119. 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)
  120. 𝐛 = 0 \ \mathbf{b}=0
  121. 𝐄 J 𝐞 i = δ J i = δ i J \ \mathbf{E}_{J}\cdot\mathbf{e}_{i}=\delta_{Ji}=\delta_{iJ}
  122. 𝐮 ( 𝐗 , t ) = 𝐱 ( 𝐗 , t ) - 𝐗 or u i = x i - δ i J X J \ \mathbf{u}(\mathbf{X},t)=\mathbf{x}(\mathbf{X},t)-\mathbf{X}\qquad\,\text{or% }\qquad u_{i}=x_{i}-\delta_{iJ}X_{J}
  123. 𝐔 ( 𝐱 , t ) = 𝐱 - 𝐗 ( 𝐱 , t ) or U J = δ J i x i - 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}
  124. Ω \Omega
  125. Ω \partial\Omega
  126. Ω \Omega
  127. 𝐱 = s y m b o l χ ( 𝐗 ) = 𝐱 ( 𝐗 ) \mathbf{x}=symbol{\chi}(\mathbf{X})=\mathbf{x}(\mathbf{X})
  128. 𝐗 \mathbf{X}
  129. 𝐱 \mathbf{x}
  130. s y m b o l F = 𝐱 𝐗 = s y m b o l 𝐱 . symbol{F}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}}=\nabla symbol{\mathbf{% x}}~{}.
  131. f ( 𝐱 , t ) f(\mathbf{x},t)
  132. g ( 𝐱 , t ) g(\mathbf{x},t)
  133. h ( 𝐱 , t ) h(\mathbf{x},t)
  134. 𝐧 ( 𝐱 , t ) \mathbf{n}(\mathbf{x},t)
  135. Ω \partial\Omega
  136. 𝐯 ( 𝐱 , t ) \mathbf{v}(\mathbf{x},t)
  137. Ω \partial\Omega
  138. u n u_{n}
  139. 𝐧 \mathbf{n}
  140. d d t [ Ω f ( 𝐱 , t ) dV ] = Ω f ( 𝐱 , t ) [ u n ( 𝐱 , t ) - 𝐯 ( 𝐱 , t ) 𝐧 ( 𝐱 , t ) ] dA + Ω g ( 𝐱 , t ) dA + Ω h ( 𝐱 , t ) dV . \cfrac{d}{dt}\left[\int_{\Omega}f(\mathbf{x},t)~{}\,\text{dV}\right]=\int_{% \partial\Omega}f(\mathbf{x},t)[u_{n}(\mathbf{x},t)-\mathbf{v}(\mathbf{x},t)% \cdot\mathbf{n}(\mathbf{x},t)]~{}\,\text{dA}+\int_{\partial\Omega}g(\mathbf{x}% ,t)~{}\,\text{dA}+\int_{\Omega}h(\mathbf{x},t)~{}\,\text{dV}~{}.
  141. f ( 𝐱 , t ) f(\mathbf{x},t)
  142. g ( 𝐱 , t ) g(\mathbf{x},t)
  143. h ( 𝐱 , t ) h(\mathbf{x},t)
  144. ρ ˙ + ρ s y m b o l 𝐯 = 0 Balance of Mass ρ 𝐯 ˙ - s y m b o l s y m b o l σ - ρ 𝐛 = 0 Balance of Linear Momentum (Cauchy’s first law of motion) s y m b o l σ = s y m b o l σ T Balance of Angular Momentum (Cauchy’s second law of motion) ρ e ˙ - s y m b o l σ : ( s y m b o l 𝐯 ) + s y m b o l 𝐪 - ρ s = 0 Balance of Energy. {\begin{aligned}\displaystyle\dot{\rho}+\rho~{}symbol{\nabla}\cdot\mathbf{v}&% \displaystyle=0&&\displaystyle\qquad\,\text{Balance of Mass}\\ \displaystyle\rho~{}\dot{\mathbf{v}}-symbol{\nabla}\cdot symbol{\sigma}-\rho~{% }\mathbf{b}&\displaystyle=0&&\displaystyle\qquad\,\text{Balance of Linear % Momentum (Cauchy's first law of motion)}\\ \displaystyle symbol{\sigma}&\displaystyle=symbol{\sigma}^{T}&&\displaystyle% \qquad\,\text{Balance of Angular Momentum (Cauchy's second law of motion)}\\ \displaystyle\rho~{}\dot{e}-symbol{\sigma}:(symbol{\nabla}\mathbf{v})+symbol{% \nabla}\cdot\mathbf{q}-\rho~{}s&\displaystyle=0&&\displaystyle\qquad\,\text{% Balance of Energy.}\end{aligned}}
  145. ρ ( 𝐱 , t ) \rho(\mathbf{x},t)
  146. ρ ˙ \dot{\rho}
  147. ρ \rho
  148. 𝐯 ( 𝐱 , t ) \mathbf{v}(\mathbf{x},t)
  149. 𝐯 ˙ \dot{\mathbf{v}}
  150. 𝐯 \mathbf{v}
  151. s y m b o l σ ( 𝐱 , t ) symbol{\sigma}(\mathbf{x},t)
  152. 𝐛 ( 𝐱 , t ) \mathbf{b}(\mathbf{x},t)
  153. e ( 𝐱 , t ) e(\mathbf{x},t)
  154. e ˙ \dot{e}
  155. e e
  156. 𝐪 ( 𝐱 , t ) \mathbf{q}(\mathbf{x},t)
  157. s ( 𝐱 , t ) s(\mathbf{x},t)
  158. ρ det ( s y m b o l F ) - ρ 0 = 0 Balance of Mass ρ 0 𝐱 ¨ - s y m b o l \cdotsymbol P T - ρ 0 𝐛 = 0 Balance of Linear Momentum s y m b o l F \cdotsymbol P T = s y m b o l P \cdotsymbol F T Balance of Angular Momentum ρ 0 e ˙ - s y m b o l P T : s y m b o l F ˙ + s y m b o l 𝐪 - ρ 0 s = 0 Balance of Energy. {\begin{aligned}\displaystyle\rho~{}\det(symbol{F})-\rho_{0}&\displaystyle=0&&% \displaystyle\qquad\,\text{Balance of Mass}\\ \displaystyle\rho_{0}~{}\ddot{\mathbf{x}}-symbol{\nabla}_{\circ}\cdotsymbol{P}% ^{T}-\rho_{0}~{}\mathbf{b}&\displaystyle=0&&\displaystyle\qquad\,\text{Balance% of Linear Momentum}\\ \displaystyle symbol{F}\cdotsymbol{P}^{T}&\displaystyle=symbol{P}\cdotsymbol{F% }^{T}&&\displaystyle\qquad\,\text{Balance of Angular Momentum}\\ \displaystyle\rho_{0}~{}\dot{e}-symbol{P}^{T}:\dot{symbol{F}}+symbol{\nabla}_{% \circ}\cdot\mathbf{q}-\rho_{0}~{}s&\displaystyle=0&&\displaystyle\qquad\,\text% {Balance of Energy.}\end{aligned}}
  159. s y m b o l P symbol{P}
  160. ρ 0 \rho_{0}
  161. s y m b o l P = J s y m b o l σ \cdotsymbol F - T where J = det ( s y m b o l F ) symbol{P}=J~{}symbol{\sigma}\cdotsymbol{F}^{-T}~{}\,\text{where}~{}J=\det(% symbol{F})
  162. s y m b o l N symbol{N}
  163. s y m b o l N = s y m b o l P T = J s y m b o l F - 1 \cdotsymbol σ . symbol{N}=symbol{P}^{T}=J~{}symbol{F}^{-1}\cdotsymbol{\sigma}~{}.
  164. ρ det ( s y m b o l F ) - ρ 0 = 0 Balance of Mass ρ 0 𝐱 ¨ - s y m b o l \cdotsymbol N - ρ 0 𝐛 = 0 Balance of Linear Momentum s y m b o l F \cdotsymbol N = s y m b o l N T \cdotsymbol F T Balance of Angular Momentum ρ 0 e ˙ - s y m b o l N : s y m b o l F ˙ + s y m b o l 𝐪 - ρ 0 s = 0 Balance of Energy. {\begin{aligned}\displaystyle\rho~{}\det(symbol{F})-\rho_{0}&\displaystyle=0&&% \displaystyle\qquad\,\text{Balance of Mass}\\ \displaystyle\rho_{0}~{}\ddot{\mathbf{x}}-symbol{\nabla}_{\circ}\cdotsymbol{N}% -\rho_{0}~{}\mathbf{b}&\displaystyle=0&&\displaystyle\qquad\,\text{Balance of % Linear Momentum}\\ \displaystyle symbol{F}\cdotsymbol{N}&\displaystyle=symbol{N}^{T}\cdotsymbol{F% }^{T}&&\displaystyle\qquad\,\text{Balance of Angular Momentum}\\ \displaystyle\rho_{0}~{}\dot{e}-symbol{N}:\dot{symbol{F}}+symbol{\nabla}_{% \circ}\cdot\mathbf{q}-\rho_{0}~{}s&\displaystyle=0&&\displaystyle\qquad\,\text% {Balance of Energy.}\end{aligned}}
  165. s y m b o l 𝐯 = i , j = 1 3 v i x j 𝐞 i 𝐞 j = v i , j 𝐞 i 𝐞 j ; s y m b o l 𝐯 = i = 1 3 v i x i = v i , i ; s y m b o l s y m b o l S = i , j = 1 3 S i j x j 𝐞 i = σ i j , j 𝐞 i . symbol{\nabla}\mathbf{v}=\sum_{i,j=1}^{3}\frac{\partial v_{i}}{\partial x_{j}}% \mathbf{e}_{i}\otimes\mathbf{e}_{j}=v_{i,j}\mathbf{e}_{i}\otimes\mathbf{e}_{j}% ~{};~{}~{}symbol{\nabla}\cdot\mathbf{v}=\sum_{i=1}^{3}\frac{\partial v_{i}}{% \partial x_{i}}=v_{i,i}~{};~{}~{}symbol{\nabla}\cdot symbol{S}=\sum_{i,j=1}^{3% }\frac{\partial S_{ij}}{\partial x_{j}}~{}\mathbf{e}_{i}=\sigma_{ij,j}~{}% \mathbf{e}_{i}~{}.
  166. 𝐯 \mathbf{v}
  167. s y m b o l S symbol{S}
  168. 𝐞 i \mathbf{e}_{i}
  169. s y m b o l 𝐯 = i , j = 1 3 v i X j 𝐄 i 𝐄 j = v i , j 𝐄 i 𝐄 j ; s y m b o l 𝐯 = i = 1 3 v i X i = v i , i ; s y m b o l \cdotsymbol S = i , j = 1 3 S i j X j 𝐄 i = S i j , j 𝐄 i symbol{\nabla}_{\circ}\mathbf{v}=\sum_{i,j=1}^{3}\frac{\partial v_{i}}{% \partial X_{j}}\mathbf{E}_{i}\otimes\mathbf{E}_{j}=v_{i,j}\mathbf{E}_{i}% \otimes\mathbf{E}_{j}~{};~{}~{}symbol{\nabla}_{\circ}\cdot\mathbf{v}=\sum_{i=1% }^{3}\frac{\partial v_{i}}{\partial X_{i}}=v_{i,i}~{};~{}~{}symbol{\nabla}_{% \circ}\cdotsymbol{S}=\sum_{i,j=1}^{3}\frac{\partial S_{ij}}{\partial X_{j}}~{}% \mathbf{E}_{i}=S_{ij,j}~{}\mathbf{E}_{i}
  170. 𝐯 \mathbf{v}
  171. s y m b o l S symbol{S}
  172. 𝐄 i \mathbf{E}_{i}
  173. s y m b o l A : s y m b o l B = i , j = 1 3 A i j B i j = trace ( s y m b o l A s y m b o l B T ) . symbol{A}:symbol{B}=\sum_{i,j=1}^{3}A_{ij}~{}B_{ij}=\operatorname{trace}(% symbol{A}symbol{B}^{T})~{}.
  174. η \eta
  175. Ω \Omega
  176. Ω \partial\Omega
  177. η \eta
  178. Ω \Omega
  179. Ω \partial\Omega
  180. u n u_{n}
  181. Ω \Omega
  182. 𝐯 \mathbf{v}
  183. 𝐧 \mathbf{n}
  184. Ω \partial\Omega
  185. ρ \rho
  186. q ¯ \bar{q}
  187. r r
  188. d d t ( Ω ρ η dV ) Ω ρ η ( u n - 𝐯 𝐧 ) dA + Ω q ¯ dA + Ω ρ r dV . \cfrac{d}{dt}\left(\int_{\Omega}\rho~{}\eta~{}\,\text{dV}\right)\geq\int_{% \partial\Omega}\rho~{}\eta~{}(u_{n}-\mathbf{v}\cdot\mathbf{n})~{}\,\text{dA}+% \int_{\partial\Omega}\bar{q}~{}\,\text{dA}+\int_{\Omega}\rho~{}r~{}\,\text{dV}.
  189. q ¯ = - s y m b o l ψ ( 𝐱 ) 𝐧 \bar{q}=-symbol{\psi}(\mathbf{x})\cdot\mathbf{n}
  190. s y m b o l ψ ( 𝐱 ) = 𝐪 ( 𝐱 ) T ; r = s T symbol{\psi}(\mathbf{x})=\cfrac{\mathbf{q}(\mathbf{x})}{T}~{};~{}~{}r=\cfrac{s% }{T}
  191. 𝐪 \mathbf{q}
  192. s s
  193. T T
  194. 𝐱 \mathbf{x}
  195. t t
  196. d d t ( Ω ρ η dV ) Ω ρ η ( u n - 𝐯 𝐧 ) dA - Ω 𝐪 𝐧 T dA + Ω ρ s T dV . {\cfrac{d}{dt}\left(\int_{\Omega}\rho~{}\eta~{}\,\text{dV}\right)\geq\int_{% \partial\Omega}\rho~{}\eta~{}(u_{n}-\mathbf{v}\cdot\mathbf{n})~{}\,\text{dA}-% \int_{\partial\Omega}\cfrac{\mathbf{q}\cdot\mathbf{n}}{T}~{}\,\text{dA}+\int_{% \Omega}\cfrac{\rho~{}s}{T}~{}\,\text{dV}.}
  197. ρ η ˙ - s y m b o l ( 𝐪 T ) + ρ s T . {\rho~{}\dot{\eta}\geq-symbol{\nabla}\cdot\left(\cfrac{\mathbf{q}}{T}\right)+% \cfrac{\rho~{}s}{T}.}
  198. ρ ( e ˙ - T η ˙ ) - s y m b o l σ : s y m b o l 𝐯 - 𝐪 \cdotsymbol T T . {\rho~{}(\dot{e}-T~{}\dot{\eta})-symbol{\sigma}:symbol{\nabla}\mathbf{v}\leq-% \cfrac{\mathbf{q}\cdotsymbol{\nabla}T}{T}.}

Contraction_mapping.html

  1. 0 k < 1 0\leq k<1
  2. d ( f ( x ) , f ( y ) ) k d ( x , y ) . d(f(x),f(y))\leq k\,d(x,y).
  3. f : M N f:M\rightarrow N
  4. k < 1 k<1
  5. d ( f ( x ) , f ( y ) ) k d ( x , y ) d^{\prime}(f(x),f(y))\leq k\,d(x,y)
  6. k = 1 k=1
  7. f ( x ) - f ( y ) 2 x - y , f ( x ) - f ( y ) . \|f(x)-f(y)\|^{2}\leq\,\langle x-y,f(x)-f(y)\rangle.
  8. d ( x , y ) = x - y d(x,y)=\|x-y\|
  9. α \alpha
  10. α = 1 / 2 \alpha=1/2
  11. d ( f ( x ) , f ( y ) ) d ( x , y ) ; d(f(x),f(y))\leq d(x,y)\ ;
  12. d ( f ( f ( x ) ) , f ( x ) ) < d ( f ( x ) , x ) unless x = f ( x ) . d(f(f(x)),f(x))<d(f(x),x)\quad\,\text{unless}\quad x=f(x)\ .

Control_theory.html

  1. Y ( s ) = P ( s ) U ( s ) Y(s)=P(s)U(s)\,\!
  2. U ( s ) = C ( s ) E ( s ) U(s)=C(s)E(s)\,\!
  3. E ( s ) = R ( s ) - F ( s ) Y ( s ) . E(s)=R(s)-F(s)Y(s).\,\!
  4. Y ( s ) = ( P ( s ) C ( s ) 1 + F ( s ) P ( s ) C ( s ) ) R ( s ) = H ( s ) R ( s ) . Y(s)=\left(\frac{P(s)C(s)}{1+F(s)P(s)C(s)}\right)R(s)=H(s)R(s).
  5. H ( s ) = P ( s ) C ( s ) 1 + F ( s ) P ( s ) C ( s ) H(s)=\frac{P(s)C(s)}{1+F(s)P(s)C(s)}
  6. | P ( s ) C ( s ) | 1 |P(s)C(s)|\gg 1
  7. | F ( s ) | 1 |F(s)|\approx 1
  8. e ( t ) = r ( t ) - y ( t ) e(t)=r(t)-y(t)
  9. u ( t ) = K P e ( t ) + K I e ( t ) d t + K D d d t e ( t ) . u(t)=K_{P}e(t)+K_{I}\int e(t)\,\text{d}t+K_{D}\frac{\,\text{d}}{\,\text{d}t}e(% t).
  10. K P K_{P}
  11. K I K_{I}
  12. K D K_{D}
  13. u ( s ) = K P e ( s ) + K I 1 s e ( s ) + K D s e ( s ) u(s)=K_{P}e(s)+K_{I}\frac{1}{s}e(s)+K_{D}se(s)
  14. u ( s ) = ( K P + K I 1 s + K D s ) e ( s ) u(s)=\left(K_{P}+K_{I}\frac{1}{s}+K_{D}s\right)e(s)
  15. C ( s ) = ( K P + K I 1 s + K D s ) . C(s)=\left(K_{P}+K_{I}\frac{1}{s}+K_{D}s\right).
  16. C ( s ) = K ( 1 + 1 s T i ) ( 1 + s T d ) C(s)=K\left(1+\frac{1}{sT_{i}}\right)(1+sT_{d})
  17. F ( s ) = 1 1 + s T f F(s)=\frac{1}{1+sT_{f}}
  18. P ( s ) = A 1 + s T p P(s)=\frac{A}{1+sT_{p}}
  19. K = 1 A , T i = T f , T d = T p K=\frac{1}{A},T_{i}=T_{f},T_{d}=T_{p}
  20. x x
  21. ρ \rho
  22. x [ n ] = 0.5 n u [ n ] \ x[n]=0.5^{n}u[n]
  23. X ( z ) = 1 1 - 0.5 z - 1 \ X(z)=\frac{1}{1-0.5z^{-1}}
  24. z = 0.5 z=0.5
  25. x [ n ] = 1.5 n u [ n ] \ x[n]=1.5^{n}u[n]
  26. X ( z ) = 1 1 - 1.5 z - 1 \ X(z)=\frac{1}{1-1.5z^{-1}}
  27. z = 1.5 z=1.5
  28. R e [ λ ] < - λ ¯ Re[\lambda]<-\overline{\lambda}
  29. λ ¯ \overline{\lambda}
  30. R e [ λ ] < 0 Re[\lambda]<0
  31. m < ¨ m t p l > x ( t ) = - K x ( t ) - B x ˙ ( t ) m\ddot{<}mtpl>{{x}}(t)=-Kx(t)-B\dot{x}(t)

Convection.html

  1. G r / R e 2 1 Gr/Re^{2}\gg 1
  2. G r / R e 2 1 Gr/Re^{2}\ll 1

Convergence_of_random_variables.html

  1. X n = 1 n i = 1 n Y i , X_{n}=\frac{1}{n}\sum_{i=1}^{n}Y_{i}\,,
  2. ( Ω , , P ) \scriptstyle(\Omega,\mathcal{F},P)
  3. n n
  4. Z n = n σ ( X n - μ ) \scriptstyle Z_{n}=\frac{\sqrt{n}}{\sigma}(X_{n}-\mu)
  5. U ( 1 , 1 ) U(−1,1)
  6. Z n = 1 n i = 1 n X i \scriptstyle Z_{n}={\scriptscriptstyle\frac{1}{\sqrt{n}}}\sum_{i=1}^{n}X_{i}
  7. N ( 0 , 1 3 ) N(0,\frac{1}{3})
  8. n n
  9. X X
  10. lim n F n ( x ) = F ( x ) , \lim_{n\to\infty}F_{n}(x)=F(x),
  11. x 𝐑 x∈\mathbf{R}
  12. F F
  13. F F
  14. X X
  15. F F
  16. ( 0 , 1 n ) (0,\frac{1}{n})
  17. X = 0 X=0
  18. x 0 x≤0
  19. x 1 n x≥\frac{1}{n}
  20. n > 0 n>0
  21. F ( 0 ) = 1 F(0)=1
  22. n n
  23. x = 0 x=0
  24. F F
  25. X n 𝑑 X , X n 𝒟 X , X n X , X n 𝑑 X , \displaystyle X_{n}\ \xrightarrow{d}\ X,\ \ X_{n}\ \xrightarrow{\mathcal{D}}\ % X,\ \ X_{n}\ \xrightarrow{\mathcal{L}}\ X,\ \ X_{n}\ \xrightarrow{d}\ \mathcal% {L}_{X},
  26. X \scriptstyle\mathcal{L}_{X}
  27. X X
  28. X X
  29. X n 𝑑 𝒩 ( 0 , 1 ) X_{n}\,\xrightarrow{d}\,\mathcal{N}(0,\,1)
  30. k k
  31. X X
  32. lim n Pr ( X n A ) = Pr ( X A ) \lim_{n\to\infty}\operatorname{Pr}(X_{n}\in A)=\operatorname{Pr}(X\in A)
  33. X X
  34. X X
  35. E * h ( X n ) E h ( X ) \operatorname{E}^{*}h(X_{n})\to\operatorname{E}\,h(X)
  36. h h
  37. g g
  38. h ( X < s u b > n ) h(X<sub>n)
  39. g g
  40. X X
  41. g ( X ) g(X)
  42. X X
  43. Y Y
  44. X + Y X+Y
  45. X Y XY
  46. X X
  47. φ φ
  48. X X
  49. lim n Pr ( | X n - X | ε ) = 0. \lim_{n\to\infty}\Pr\big(|X_{n}-X|\geq\varepsilon\big)=0.
  50. ε > 0 ε>0
  51. δ > 0 δ>0
  52. n N n≥N
  53. ( S , d ) (S,d)
  54. ε > 0 , Pr ( d ( X n , X ) ε ) 0. \forall\varepsilon>0,\Pr\big(d(X_{n},X)\geq\varepsilon\big)\to 0.
  55. X n 𝑝 X \scriptstyle X_{n}\xrightarrow{p}X
  56. g ( X n ) 𝑝 g ( X ) \scriptstyle g(X_{n})\xrightarrow{p}g(X)
  57. d ( X , Y ) = inf { ε Align g t ; 0 : Pr ( | X - Y | Align g t ; ε ) ε } d(X,Y)=\inf\!\big\{\varepsilon&gt;0:\ \Pr\big(|X-Y|&gt;\varepsilon\big)\leq% \varepsilon\big\}
  58. d ( X , Y ) = 𝔼 [ min ( | X - Y | , 1 ) ] d(X,Y)=\mathbb{E}\left[\min(|X-Y|,1)\right]
  59. Pr ( lim n X n = X ) = 1. \operatorname{Pr}\!\left(\lim_{n\to\infty}\!X_{n}=X\right)=1.
  60. ( Ω , , Pr ) \scriptstyle(\Omega,\mathcal{F},\operatorname{Pr})
  61. Pr ( ω Ω : lim n X n ( ω ) = X ( ω ) ) = 1. \operatorname{Pr}\Big(\omega\in\Omega:\lim_{n\to\infty}X_{n}(\omega)=X(\omega)% \Big)=1.
  62. Pr ( lim inf n { ω Ω : | X n ( ω ) - X ( ω ) | < ε } ) = 1 for all ε > 0. \operatorname{Pr}\Big(\liminf_{n\to\infty}\big\{\omega\in\Omega:|X_{n}(\omega)% -X(\omega)|<\varepsilon\big\}\Big)=1\quad\,\text{for all}\quad\varepsilon>0.
  63. X n a . s . X . X_{n}\,\xrightarrow{\mathrm{a.s.}}\,X.
  64. Pr ( ω Ω : d ( X n ( ω ) , X ( ω ) ) n 0 ) = 1 \operatorname{Pr}\Big(\omega\in\Omega:\,d\big(X_{n}(\omega),X(\omega)\big)\,% \underset{n\to\infty}{\longrightarrow}\,0\Big)=1
  65. lim n X n ( ω ) = X ( ω ) , ω Ω . \lim_{n\to\infty}X_{n}(\omega)=X(\omega),\,\,\forall\omega\in\Omega.
  66. { ω Ω | lim n X n ( ω ) = X ( ω ) } = Ω . \big\{\omega\in\Omega\,|\,\lim_{n\to\infty}X_{n}(\omega)=X(\omega)\big\}=\Omega.
  67. r 1 r≥1
  68. r r
  69. lim n E ( | X n - X | r ) = 0 , \lim_{n\to\infty}\operatorname{E}\left(|X_{n}-X|^{r}\right)=0,
  70. r r
  71. r r
  72. X n L r X . X_{n}\,\xrightarrow{L^{r}}\,X.
  73. X n L r X X_{n}\xrightarrow{L^{r}}X
  74. lim n E [ | X n | r ] = E [ | X | r ] \lim_{n\to\infty}E[|X_{n}|^{r}]=E[|X|^{r}]
  75. X n 𝑝 X X_{n}\ \xrightarrow{p}\ X
  76. X n 𝑝 Y X_{n}\ \xrightarrow{p}\ Y
  77. X = Y X=Y
  78. X n a . s . X X_{n}\ \xrightarrow{a.s.}\ X
  79. X n a . s . Y X_{n}\ \xrightarrow{a.s.}\ Y
  80. X = Y X=Y
  81. X n L r X X_{n}\ \xrightarrow{L^{r}}\ X
  82. X n L r Y X_{n}\ \xrightarrow{L^{r}}\ Y
  83. X = Y X=Y
  84. X n 𝑝 X X_{n}\ \xrightarrow{p}\ X
  85. Y n 𝑝 Y Y_{n}\ \xrightarrow{p}\ Y
  86. a X n + b Y n 𝑝 a X + b Y aX_{n}+bY_{n}\ \xrightarrow{p}\ aX+bY
  87. a a
  88. b b
  89. X n Y n 𝑝 X Y X_{n}Y_{n}\xrightarrow{p}\ XY
  90. X n a . s . X X_{n}\ \xrightarrow{a.s.}\ X
  91. Y n a . s . Y Y_{n}\ \xrightarrow{a.s.}\ Y
  92. a X n + b Y n a . s . a X + b Y aX_{n}+bY_{n}\ \xrightarrow{a.s.}\ aX+bY
  93. a a
  94. b b
  95. X n Y n a . s . X Y X_{n}Y_{n}\xrightarrow{a.s.}\ XY
  96. X n L r X X_{n}\ \xrightarrow{L^{r}}\ X
  97. Y n L r Y Y_{n}\ \xrightarrow{L^{r}}\ Y
  98. a X n + b Y n L r a X + b Y aX_{n}+bY_{n}\ \xrightarrow{L^{r}}\ aX+bY
  99. a a
  100. b b
  101. L s s > r 1 L r a . s . 𝑝 𝑑 \begin{matrix}\xrightarrow{L^{s}}&\underset{s>r\geq 1}{\Rightarrow}&% \xrightarrow{L^{r}}&&\\ &&\Downarrow&&\\ \xrightarrow{a.s.}&\Rightarrow&\xrightarrow{\ p\ }&\Rightarrow&\xrightarrow{\ % d\ }\end{matrix}
  102. X n a . s . X X n 𝑝 X X_{n}\ \xrightarrow{a.s.}\ X\quad\Rightarrow\quad X_{n}\ \xrightarrow{p}\ X
  103. ( k n ) (k_{n})
  104. X n 𝑝 X X k n a . s . X X_{n}\ \xrightarrow{p}\ X\quad\Rightarrow\quad X_{k_{n}}\ \xrightarrow{a.s.}\ X
  105. X n 𝑝 X X n 𝑑 X X_{n}\ \xrightarrow{p}\ X\quad\Rightarrow\quad X_{n}\ \xrightarrow{d}\ X
  106. X n L r X X n 𝑝 X X_{n}\ \xrightarrow{L^{r}}\ X\quad\Rightarrow\quad X_{n}\ \xrightarrow{p}\ X
  107. X n L r X X n L s X , X_{n}\ \xrightarrow{L^{r}}\ X\quad\Rightarrow\quad X_{n}\ \xrightarrow{L^{s}}% \ X,
  108. X n 𝑑 c X n 𝑝 c , X_{n}\ \xrightarrow{d}\ c\quad\Rightarrow\quad X_{n}\ \xrightarrow{p}\ c,
  109. X n 𝑑 X , | X n - Y n | 𝑝 0 Y n 𝑑 X X_{n}\ \xrightarrow{d}\ X,\ \ |X_{n}-Y_{n}|\ \xrightarrow{p}\ 0\ \quad% \Rightarrow\quad Y_{n}\ \xrightarrow{d}\ X
  110. X n 𝑑 X , Y n 𝑑 c ( X n , Y n ) 𝑑 ( X , c ) X_{n}\ \xrightarrow{d}\ X,\ \ Y_{n}\ \xrightarrow{d}\ c\ \quad\Rightarrow\quad% (X_{n},Y_{n})\ \xrightarrow{d}\ (X,c)
  111. X n 𝑝 X , Y n 𝑝 Y ( X n , Y n ) 𝑝 ( X , Y ) X_{n}\ \xrightarrow{p}\ X,\ \ Y_{n}\ \xrightarrow{p}\ Y\ \quad\Rightarrow\quad% (X_{n},Y_{n})\ \xrightarrow{p}\ (X,Y)
  112. r 1 r≥1
  113. n 0 n≥0
  114. n ( | X n - X | > ε ) < , \sum_{n}\mathbb{P}\left(|X_{n}-X|>\varepsilon\right)<\infty,
  115. ε > 0 ε>0
  116. S n = X 1 + + X n S_{n}=X_{1}+\cdots+X_{n}\,
  117. X n a . s . X | X n | < Y E ( Y ) < } X n L 1 X \left.\begin{matrix}X_{n}\xrightarrow{a.s.}X\\ |X_{n}|<Y\\ \mathrm{E}(Y)<\infty\end{matrix}\right\}\quad\Rightarrow\quad X_{n}% \xrightarrow{L^{1}}X
  118. X n 𝑃 X X_{n}\xrightarrow{P}X

Conversion_of_units.html

  1. d i g i t s ¯ \overline{digits}
  2. 1 / 3 {1}/{3}
  3. 6 ¯ \overline{6}
  4. × 10 - 3 \times 10^{-}3
  5. 1 / 10 {1}/{10}
  6. 7 / 8 {7}/{8}
  7. 41 / 2 4{1}/{2}
  8. 1 / 10 \sqrt{{1}/{10}}
  9. 1 / 3 {1}/{3}
  10. 3 ¯ \overline{3}
  11. 1 / 3 {1}/{3}
  12. [ u v a l , u 1200 ] / [ u v a l , u 3937 ] {[u^{\prime}val^{\prime},u^{\prime}1200^{\prime}]}/{[u^{\prime}val^{\prime},u^% {\prime}3937^{\prime}]}
  13. 1 / 3 {1}/{3}
  14. 3 ¯ \overline{3}
  15. × 10 - 3 \times 10^{-}3
  16. 1 / 36 {1}/{36}
  17. 1 / 12 {1}/{12}
  18. 1 / 12 {1}/{12}
  19. 1 / 100 {1}/{100}
  20. 1 / [ u v a l , u 299792458 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}299792458^{\prime}]}
  21. 1 / [ u v a l , u 10000000 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}10000000^{\prime}]}
  22. 1 / 200 {1}/{200}
  23. [ u v a l , u 1200 ] / [ u v a l , u 3937 ] {[u^{\prime}val^{\prime},u^{\prime}1200^{\prime}]}/{[u^{\prime}val^{\prime},u^% {\prime}3937^{\prime}]}
  24. 21 / 4 2{1}/{4}
  25. 1 / 72.272 {1}/{72.272}
  26. 1 / 12 {1}/{12}
  27. 1 / 72 {1}/{72}
  28. 5 / 133 {5}/{133}
  29. 1 / 72 {1}/{72}
  30. 1 / 72.27 {1}/{72.27}
  31. [ u g a p s , u 0 , u 351 , u 4598 ] ¯ \overline{[u^{\prime}gaps^{\prime},u^{\prime}0^{\prime},u^{\prime}351^{\prime}% ,u^{\prime}4598^{\prime}]}
  32. 1 / 4 {1}/{4}
  33. 161 / 2 16{1}/{2}
  34. 1 / 1440 {1}/{1440}
  35. 8 ¯ \overline{8}
  36. × 10 5 \times 10^{−}5
  37. π / 4 {π}/{4}
  38. π / 4 {π}/{4}
  39. 1 / 4 {1}/{4}
  40. 1 / 10 {1}/{10}
  41. 1 / 10 {1}/{10}
  42. 311 / 2 31{1}/{2}
  43. 11 / 4 1{1}/{4}
  44. 1 / 16 {1}/{16}
  45. 1 / 384 {1}/{384}
  46. 1 / 2 {1}/{2}
  47. × 10 9 \times 10^{−}9
  48. 1 / 96 {1}/{96}
  49. 1 / 2 {1}/{2}
  50. 1 / 12 {1}/{12}
  51. × 10 6 \times 10^{−}6
  52. 1 / 288 {1}/{288}
  53. × 10 9 \times 10^{−}9
  54. 1 / [ u v a l , u 1824 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}1824^{\prime}]}
  55. 0.9964 / 12 {0.9964}/{12}
  56. 3 ¯ \overline{3}
  57. × 10 9 \times 10^{−}9
  58. 1 / 12 {1}/{12}
  59. 3 ¯ \overline{3}
  60. × 10 9 \times 10^{−}9
  61. 1 / 20 {1}/{20}
  62. 1 / 360 {1}/{360}
  63. × 10 9 \times 10^{−}9
  64. 1 / 456 {1}/{456}
  65. 1 / 576 {1}/{576}
  66. 1 / 5 {1}/{5}
  67. 1 / 8 {1}/{8}
  68. 1 / 8 {1}/{8}
  69. 1 / 24 {1}/{24}
  70. × 10 6 \times 10^{−}6
  71. 1 / 8 {1}/{8}
  72. 11 / 2 1{1}/{2}
  73. 1 / 480 {1}/{480}
  74. × 10 9 \times 10^{−}9
  75. 1 / 480 {1}/{480}
  76. 1 / 60 {1}/{60}
  77. 1 / 160 {1}/{160}
  78. 1 / 128 {1}/{128}
  79. 1 / 4 {1}/{4}
  80. 1 / 192 {1}/{192}
  81. × 10 9 \times 10^{−}9
  82. 1 / 48 {1}/{48}
  83. 1 / 8 {1}/{8}
  84. 1 / 8 {1}/{8}
  85. 1 / 64 {1}/{64}
  86. 1 / 8 {1}/{8}
  87. 1 / 8 {1}/{8}
  88. 3 / 4 {3}/{4}
  89. 1 / 2 {1}/{2}
  90. 1 / 4 {1}/{4}
  91. 1 / 32 {1}/{32}
  92. 1 / 4 {1}/{4}
  93. 1 / 4 {1}/{4}
  94. 1 / 2 {1}/{2}
  95. 5 / 8 {5}/{8}
  96. 1 / 2 {1}/{2}
  97. 1 / 6 {1}/{6}
  98. × 10 6 \times 10^{−}6
  99. 1 / 24 {1}/{24}
  100. × 10 6 \times 10^{−}6
  101. 1 / 6 {1}/{6}
  102. 2 π / [ u v a l , u 6400 ] {2π}/{[u^{\prime}val^{\prime},u^{\prime}6400^{\prime}]}
  103. 1 ° / 60 {1°}/{60}
  104. 1 ° / [ u v a l , u 3600 ] {1°}/{[u^{\prime}val^{\prime},u^{\prime}3600^{\prime}]}
  105. 1 / 100 {1}/{100}
  106. 1 / [ u v a l , u 10000 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}10000^{\prime}]}
  107. 1 / 360 {1}/{360}
  108. π / 180 {π}/{180}
  109. 1 / 400 {1}/{400}
  110. π / 200 {π}/{200}
  111. 221 / 2 22{1}/{2}
  112. 31 / 6 3{1}/{6}
  113. 3 ¯ \overline{3}
  114. 2711 / 32 27{11}/{32}
  115. 1 / [ u v a l , u 7000 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}7000^{\prime}]}
  116. 1 / 20 {1}/{20}
  117. 1 / 20 {1}/{20}
  118. 1 / 12 {1}/{12}
  119. 1 / 16 {1}/{16}
  120. 1 / 20 {1}/{20}
  121. 1 / 100 {1}/{100}
  122. 1 / 4 {1}/{4}
  123. 1 / 4 {1}/{4}
  124. 1 / 4 {1}/{4}
  125. 1 / 700 {1}/{700}
  126. ɡ < s u b > 0 ɡ<sub>0
  127. 1 / [ u v a l , u 1080 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}1080^{\prime}]}
  128. 3 ¯ \overline{3}
  129. 1 / 60 {1}/{60}
  130. 6 ¯ \overline{6}
  131. 1 / 100 {1}/{100}
  132. 1 / 4 {1}/{4}
  133. 1 / 96 {1}/{96}
  134. 1 / 100 {1}/{100}
  135. 1 / [ u v a l , u 1000 ] {1}/{[u^{\prime}val^{\prime},u^{\prime}1000^{\prime}]}
  136. G [ u U n i c o d e , u 10 f ] / c G{[u^{\prime}Unicode^{\prime},u^{\prime}\u{2}10f^{\prime}]}/{c}
  137. 6 ¯ \overline{6}
  138. × 10 5 \times 10^{−}5
  139. 5 ¯ \overline{5}
  140. × 10 6 \times 10^{−}6
  141. 3 ¯ \overline{3}
  142. × 10 4 \times 10^{−}4
  143. 7 ¯ \overline{7}
  144. × 10 1 \times 10^{−}1
  145. 4 ¯ \overline{4}
  146. 3 ¯ \overline{3}
  147. 3 ¯ \overline{3}
  148. × 10 - 7 \times 10^{-}7
  149. 8 ¯ \overline{8}
  150. × 10 - 8 \times 10^{-}8
  151. 3 ¯ \overline{3}
  152. × 10 - 6 \times 10^{-}6
  153. 6 ¯ \overline{6}
  154. × 10 - 5 \times 10^{-}5
  155. 6 ¯ \overline{6}
  156. × 10 5 \times 10^{−}5
  157. 3 ¯ \overline{3}
  158. × 10 4 \times 10^{−}4
  159. 4 ¯ \overline{4}
  160. × 10 1 \times 10^{−}1
  161. ɡ < s u b > 0 ɡ<sub>0
  162. m < s u b > e · α / · 2 m<sub>e·{α}/{{}^{2}·}
  163. μ \mu
  164. μ \mu
  165. ɡ < s u b > 0 ɡ<sub>0
  166. ɡ < s u b > 0 ɡ<sub>0
  167. 1 / 100 {1}/{100}
  168. ɡ < s u b > 0 ɡ<sub>0
  169. 2 / π ℎ{2}/{π}
  170. × 10 - 7 \times 10^{-}7
  171. 0.1 A · m / s / c {0.1A·m/s}/{c}
  172. 0.1 A · m / c {0.1A·m}/{c}
  173. 2 / 3 {2}/{3}
  174. 9 / 5 {9}/{5}
  175. 5 / 9 {5}/{9}
  176. 100 / 33 {100}/{33}
  177. 9 / 5 {9}/{5}
  178. 5 / 4 {5}/{4}
  179. 40 / 21 {40}/{21}
  180. 125 / 9 {125}/{9}
  181. 1 / 273.16 {1}/{273.16}
  182. × 10 1 2 \times 10^{1}2

Convex_hull.html

  1. S S
  2. x i x_{i}
  3. S S
  4. α i \alpha_{i}
  5. { i = 1 | S | α i x i | ( i : α i 0 ) i = 1 | S | α i = 1 } . \left\{\sum_{i=1}^{|S|}\alpha_{i}x_{i}\mathrel{\Bigg|}(\forall i:\alpha_{i}% \geq 0)\wedge\sum_{i=1}^{|S|}\alpha_{i}=1\right\}.
  6. S n S\in\mathbb{R}^{n}
  7. n \mathbb{R}^{n}
  8. x i x_{i}
  9. S S
  10. x i Conv ( S { x i } ) x_{i}\notin\operatorname{Conv}(S\setminus\{x_{i}\})
  11. Conv ( S ) \operatorname{Conv}(S)
  12. n \mathbb{R}^{n}
  13. S S
  14. S S
  15. S S
  16. S S
  17. O ( n d / 2 ) O(n^{\lfloor d/2\rfloor})
  18. { ( x , y ) y 1 1 + x 2 } \left\{(x,y)\mid y\geq\frac{1}{1+x^{2}}\right\}

Convex_set.html

  1. S S
  2. C C
  3. S S
  4. x x
  5. y y
  6. C C
  7. t t
  8. 0 , 11 0,11
  9. ( 1 t ) x + t y (1−t)x+ty
  10. C C
  11. x x
  12. y y
  13. C C
  14. C C
  15. x x
  16. y y
  17. C C
  18. C C
  19. 𝐑 \mathbf{R}
  20. 𝐑 \mathbf{R}
  21. S S
  22. n n
  23. r r
  24. r > 1 r>1
  25. n n
  26. S S
  27. k = 1 r λ k u k S . \sum_{k=1}^{r}\lambda_{k}u_{k}\in S.
  28. C C
  29. P P
  30. H H
  31. C C
  32. P P
  33. 0.5 A r e a ( R ) A r e a ( C ) 2 A r e a ( r ) 0.5Area(R)\leq Area(C)\leq 2Area(r)
  34. A A
  35. A A
  36. A A
  37. S C o n v ( S ) S⊆ Conv(S)
  38. S T S⊆T
  39. C o n v ( S ) C o n v ( T ) Conv(S) ⊆ Conv(T)
  40. C o n v ( C o n v ( S ) ) = C o n v ( S ) Conv(Conv(S))=Conv(S)
  41. C o n v ( S ) C o n v ( T ) = C o n v ( S T ) = C o n v ( C o n v ( S ) C o n v ( T ) ) Conv(S)∨Conv(T)=Conv(S∪T)=Conv(Conv(S) ∪ Conv(T))
  42. S < s u b > 1 S<sub>1
  43. n S n = { n x n : x n S n } . \sum_{n}S_{n}=\left\{\sum_{n}x_{n}:x_{n}\in S_{n}\right\}.
  44. 0
  45. Conv ( n S n ) = n Conv ( S n ) . \,\text{Conv}\left(\sum_{n}S_{n}\right)=\sum_{n}\,\text{Conv}\left(S_{n}\right).
  46. C C
  47. C C
  48. x < s u b > 0 x<sub>0
  49. S S
  50. S + = S+∅=∅

Convex_uniform_honeycomb.html

  1. C ~ 3 {\tilde{C}}_{3}
  2. B ~ 3 {\tilde{B}}_{3}
  3. A ~ 3 {\tilde{A}}_{3}
  4. C ~ 3 {\tilde{C}}_{3}
  5. B ~ 3 {\tilde{B}}_{3}
  6. B ~ 3 {\tilde{B}}_{3}
  7. A ~ 3 {\tilde{A}}_{3}
  8. C ~ 2 {\tilde{C}}_{2}
  9. I ~ 1 {\tilde{I}}_{1}
  10. H ~ 2 {\tilde{H}}_{2}
  11. I ~ 1 {\tilde{I}}_{1}
  12. A ~ 2 {\tilde{A}}_{2}
  13. I ~ 1 {\tilde{I}}_{1}
  14. I ~ 1 {\tilde{I}}_{1}
  15. I ~ 1 {\tilde{I}}_{1}
  16. I ~ 1 {\tilde{I}}_{1}
  17. 3 ¯ \overline{3}
  18. 3 ¯ \overline{3}
  19. B ~ 4 {\tilde{B}}_{4}
  20. 3 ¯ \overline{3}
  21. 3 ¯ \overline{3}
  22. A ~ 3 {\tilde{A}}_{3}
  23. 3 ¯ \overline{3}
  24. 3 ¯ \overline{3}
  25. 3 ¯ \overline{3}
  26. 3 ¯ \overline{3}
  27. C ~ 2 {\tilde{C}}_{2}
  28. A 1 A_{1}
  29. G ~ 2 {\tilde{G}}_{2}
  30. A 1 A_{1}
  31. A ~ 2 {\tilde{A}}_{2}
  32. A 1 A_{1}
  33. I ~ 1 {\tilde{I}}_{1}
  34. A 1 A_{1}
  35. A 1 A_{1}
  36. I 2 ( p ) I_{2}(p)
  37. I ~ 1 {\tilde{I}}_{1}
  38. C ~ 2 {\tilde{C}}_{2}
  39. C ~ 2 {\tilde{C}}_{2}
  40. A 1 A_{1}

Convolution.html

  1. ( f * g ) ( t ) (f*g)(t)\ \ \,
  2. = def - f ( τ ) g ( t - τ ) d τ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}f(\tau)\,g(t-\tau)\,d\tau
  3. = - f ( t - τ ) g ( τ ) d τ . =\int_{-\infty}^{\infty}f(t-\tau)\,g(\tau)\,d\tau.
  4. [ 0 , ) [0,\infty)
  5. ( f * g ) ( t ) = 0 t f ( τ ) g ( t - τ ) d τ for f , g : [ 0 , ) (f*g)(t)=\int_{0}^{t}f(\tau)\,g(t-\tau)\,d\tau\ \ \ \mathrm{for}\ \ f,g:[0,% \infty)\to\mathbb{R}
  6. τ . \tau.
  7. g ( τ ) g(\tau)
  8. g ( - τ ) . g(-\tau).
  9. g ( t - τ ) g(t-\tau)
  10. τ \tau
  11. f ( τ ) f(\tau)
  12. g ( - τ ) . g(-\tau).
  13. - δ ( τ ) g ( t - τ ) d τ = g ( t ) \int_{-\infty}^{\infty}\delta(\tau)\,g(t-\tau)\,d\tau=g(t)
  14. g ( τ ) , g(\tau),
  15. ( g ( - τ ) = g ( τ ) ) , (\ g(-\tau)=g(\tau)\ ),
  16. g ( t - τ ) g(t-\tau)
  17. f ( τ ) f(\tau)
  18. t , t,
  19. τ = 0 \tau=0
  20. f ( τ ) g ( t - τ ) , f(\tau)\cdot g(t-\tau),
  21. t t
  22. t , t,
  23. τ , \tau,
  24. f ( τ ) f(\tau)
  25. τ = 0. \tau=0.
  26. g ( τ ) = δ ( τ ) , g(\tau)=\delta(\tau),
  27. f ( t ) . f(t).
  28. g ( τ ) g(\tau)
  29. f ( t ) . f(t).
  30. t = - 0.5 , t=-0.5,
  31. t t
  32. τ = 0 \tau=0
  33. f ( u ) g ( x - u ) d u \int f(u)\cdot g(x-u)du
  34. 0 t φ ( s ) ψ ( t - s ) d s , 0 t < , \int_{0}^{t}\varphi(s)\psi(t-s)\,ds,\qquad 0\leq t<\infty,
  35. ( f * g T ) ( t ) t 0 t 0 + T [ k = - f ( τ + k T ) ] g T ( t - τ ) d τ , (f*g_{T})(t)\equiv\int_{t_{0}}^{t_{0}+T}\left[\sum_{k=-\infty}^{\infty}f(\tau+% kT)\right]g_{T}(t-\tau)\,d\tau,
  36. ( f * g ) [ n ] = def m = - f [ m ] g [ n - m ] (f*g)[n]\ \stackrel{\mathrm{def}}{=}\ \sum_{m=-\infty}^{\infty}f[m]\,g[n-m]
  37. = m = - f [ n - m ] g [ m ] . =\sum_{m=-\infty}^{\infty}f[n-m]\,g[m].
  38. { - M , - M + 1 , , M - 1 , M } \{-M,-M+1,\dots,M-1,M\}
  39. ( f * g ) [ n ] = m = - M M f [ n - m ] g [ m ] . (f*g)[n]=\sum_{m=-M}^{M}f[n-m]g[m].
  40. ( f * g N ) [ n ] m = 0 N - 1 ( k = - f [ m + k N ] ) g N [ n - m ] . (f*g_{N})[n]\equiv\sum_{m=0}^{N-1}\left(\sum_{k=-\infty}^{\infty}{f}[m+kN]% \right)g_{N}[n-m].\,
  41. ( f * g N ) [ n ] \displaystyle(f*g_{N})[n]
  42. ( f * g ) ( x ) = 𝐑 d f ( y ) g ( x - y ) d y = 𝐑 d f ( x - y ) g ( y ) d y , (f*g)(x)=\int_{\mathbf{R}^{d}}f(y)g(x-y)\,dy=\int_{\mathbf{R}^{d}}f(x-y)g(y)\,dy,
  43. 1 \ell^{1}
  44. f * g p f 1 g p . \|{f}*g\|_{p}\leq\|f\|_{1}\|g\|_{p}.\,
  45. 1 p + 1 q = 1 r + 1 , \frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1,
  46. f * g r f p g q , f p , g q , \left\|f*g\right\|_{r}\leq\left\|f\right\|_{p}\left\|g\right\|_{q},\quad f\in% \mathcal{L}^{p},\ g\in\mathcal{L}^{q},
  47. f * g r B p , q f p g q , f p , g q . \left\|f*g\right\|_{r}\leq B_{p,q}\left\|f\right\|_{p}\left\|g\right\|_{q},% \quad f\in\mathcal{L}^{p},\ g\in\mathcal{L}^{q}.
  48. f * g r C p , q f p g q , w \|f*g\|_{r}\leq C_{p,q}\|f\|_{p}\|g\|_{q,w}
  49. g q , w \|g\|_{q,w}
  50. L p , w × L q . w L r , w L^{p,w}\times L^{q.w}\to L^{r,w}
  51. 1 < p , q , r < 1<p,q,r<\infty
  52. f * g r , w C p , q f p , w g r , w . \|f*g\|_{r,w}\leq C_{p,q}\|f\|_{p,w}\|g\|_{r,w}.
  53. 𝐑 d f ( y ) g ( x - y ) d y . \int_{\mathbf{R}^{d}}{f}(y)g(x-y)\,dy.
  54. f * ( g * φ ) = ( f * g ) * φ f*(g*\varphi)=(f*g)*\varphi\,
  55. 𝐑 d f ( x ) d λ ( x ) = 𝐑 d 𝐑 d f ( x + y ) d μ ( x ) d ν ( y ) . \int_{\mathbf{R}^{d}}f(x)d\lambda(x)=\int_{\mathbf{R}^{d}}\int_{\mathbf{R}^{d}% }f(x+y)\,d\mu(x)\,d\nu(y).
  56. μ * ν μ ν \|\mu*\nu\|\leq\|\mu\|\|\nu\|\,
  57. f * g = g * f f*g=g*f\,
  58. f * ( g * h ) = ( f * g ) * h f*(g*h)=(f*g)*h\,
  59. f * ( g + h ) = ( f * g ) + ( f * h ) f*(g+h)=(f*g)+(f*h)\,
  60. a ( f * g ) = ( a f ) * g a(f*g)=(af)*g\,
  61. a {a}\,
  62. f * δ = f f*\delta=f\,
  63. S ( - 1 ) * S = δ . S^{(-1)}*S=\delta.\,
  64. f * g ¯ = f ¯ * g ¯ \overline{f*g}=\overline{f}*\overline{g}\!
  65. 𝐑 d ( f * g ) ( x ) d x = ( 𝐑 d f ( x ) d x ) ( 𝐑 d g ( x ) d x ) . \int_{\mathbf{R}^{d}}(f*g)(x)\,dx=\left(\int_{\mathbf{R}^{d}}f(x)\,dx\right)% \left(\int_{\mathbf{R}^{d}}g(x)\,dx\right).
  66. d d x ( f * g ) = d f d x * g = f * d g d x \frac{d}{dx}(f*g)=\frac{df}{dx}*g=f*\frac{dg}{dx}\,
  67. x i ( f * g ) = f x i * g = f * g x i . \frac{\partial}{\partial x_{i}}(f*g)=\frac{\partial f}{\partial x_{i}}*g=f*% \frac{\partial g}{\partial x_{i}}.
  68. d d x ( f * g ) = d f d x * g . \frac{d}{dx}({f}*g)=\frac{df}{dx}*g.
  69. D ( f * g ) = ( D f ) * g = f * ( D g ) . D(f*g)=(Df)*g=f*(Dg).\,
  70. { f * g } = k { f } { g } \mathcal{F}\{f*g\}=k\cdot\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}
  71. { f } \mathcal{F}\{f\}\,
  72. f f
  73. k k
  74. τ x ( f * g ) = ( τ x f ) * g = f * ( τ x g ) \tau_{x}({f}*g)=(\tau_{x}f)*g={f}*(\tau_{x}g)\,
  75. ( τ x f ) ( y ) = f ( y - x ) . (\tau_{x}f)(y)=f(y-x).\,
  76. f ( x y - 1 ) g ( y ) d λ ( y ) \textstyle{\int f(xy^{-1})g(y)\,d\lambda(y)}
  77. L h ( f * g ) = ( L h f ) * g . L_{h}(f*g)=(L_{h}f)*g.
  78. T f ( x ) = 1 2 π 𝐓 f ( y ) g ( x - y ) d y . T{f}(x)=\frac{1}{2\pi}\int_{\mathbf{T}}{f}(y)g(x-y)\,dy.
  79. g ¯ ( - y ) . \bar{g}(-y).\,
  80. h k ( x ) = e i k x , k , h_{k}(x)=e^{ikx},\quad k\in\mathbb{Z},\;
  81. ( μ * ν ) ( E ) = 1 E ( x y ) d μ ( x ) d ν ( y ) (\mu*\nu)(E)=\int\!\!\!\int 1_{E}(xy)\,d\mu(x)\,d\nu(y)
  82. μ * ν μ ν . \|\mu*\nu\|\leq\|\mu\|\|\nu\|.\,
  83. X Δ X X ϕ ψ X X X . X\xrightarrow{\Delta}X\otimes X\xrightarrow{\phi\otimes\psi}X\otimes X% \xrightarrow{\nabla}X.\,
  84. S * id X = id X * S = η ε . S*\operatorname{id}_{X}=\operatorname{id}_{X}*S=\eta\circ\varepsilon.

Convolution_theorem.html

  1. f \ f
  2. g \ g
  3. f * g \ f*g
  4. \otimes
  5. \ \mathcal{F}
  6. { f } \ \mathcal{F}\{f\}
  7. { g } \ \mathcal{F}\{g\}
  8. f \ f
  9. g \ g
  10. { f * g } = { f } { g } \mathcal{F}\{f*g\}=\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}
  11. \cdot
  12. { f g } = { f } * { g } \mathcal{F}\{f\cdot g\}=\mathcal{F}\{f\}*\mathcal{F}\{g\}
  13. - 1 \mathcal{F}^{-1}
  14. f * g = - 1 { { f } { g } } f*g=\mathcal{F}^{-1}\big\{\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}\big\}
  15. f g = - 1 { { f } * { g } } f\cdot g=\mathcal{F}^{-1}\big\{\mathcal{F}\{f\}*\mathcal{F}\{g\}\big\}
  16. 2 π \ 2\pi
  17. 2 π \ \sqrt{2\pi}
  18. F F
  19. f f
  20. G G
  21. g g
  22. F ( ν ) = { f } = n f ( x ) e - 2 π i x ν d x F(\nu)=\mathcal{F}\{f\}=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\nu}\,\mathrm% {d}x
  23. G ( ν ) = { g } = n g ( x ) e - 2 π i x ν d x , G(\nu)=\mathcal{F}\{g\}=\int_{\mathbb{R}^{n}}g(x)e^{-2\pi ix\cdot\nu}\,\mathrm% {d}x,
  24. h h
  25. f f
  26. g g
  27. h ( z ) = n f ( x ) g ( z - x ) d x . h(z)=\int\limits_{\mathbb{R}^{n}}f(x)g(z-x)\,\mathrm{d}x.
  28. | f ( x ) g ( z - x ) | d z d x = | f ( x ) | | g ( z - x ) | d z d x = | f ( x ) | g 1 d x = f 1 g 1 . \int\!\!\int|f(x)g(z-x)|\,dz\,dx=\int|f(x)|\int|g(z-x)|\,dz\,dx=\int|f(x)|\,\|% g\|_{1}\,dx=\|f\|_{1}\|g\|_{1}.
  29. h L 1 ( n ) h\in L^{1}(\mathbb{R}^{n})
  30. H H
  31. H ( ν ) = { h } = n h ( z ) e - 2 π i z ν d z = n n f ( x ) g ( z - x ) d x e - 2 π i z ν d z . \begin{aligned}\displaystyle H(\nu)=\mathcal{F}\{h\}&\displaystyle=\int_{% \mathbb{R}^{n}}h(z)e^{-2\pi iz\cdot\nu}\,dz\\ &\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}f(x)g(z-x)\,dx\,e^{-2% \pi iz\cdot\nu}\,dz.\end{aligned}
  32. | f ( x ) g ( z - x ) e - 2 π i z ν | = | f ( x ) g ( z - x ) | |f(x)g(z-x)e^{-2\pi iz\cdot\nu}|=|f(x)g(z-x)|
  33. H ( ν ) = n f ( x ) ( n g ( z - x ) e - 2 π i z ν d z ) d x . H(\nu)=\int_{\mathbb{R}^{n}}f(x)\left(\int_{\mathbb{R}^{n}}g(z-x)e^{-2\pi iz% \cdot\nu}\,dz\right)\,dx.
  34. y = z - x y=z-x
  35. d y = d z dy=dz
  36. H ( ν ) = n f ( x ) ( n g ( y ) e - 2 π i ( y + x ) ν d y ) d x H(\nu)=\int_{\mathbb{R}^{n}}f(x)\left(\int_{\mathbb{R}^{n}}g(y)e^{-2\pi i(y+x)% \cdot\nu}\,dy\right)\,dx
  37. = n f ( x ) e - 2 π i x ν ( n g ( y ) e - 2 π i y ν d y ) d x =\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\nu}\left(\int_{\mathbb{R}^{n}}g(y)e% ^{-2\pi iy\cdot\nu}\,dy\right)\,dx
  38. = n f ( x ) e - 2 π i x ν d x n g ( y ) e - 2 π i y ν d y . =\int_{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot\nu}\,dx\int_{\mathbb{R}^{n}}g(y)e^{% -2\pi iy\cdot\nu}\,dy.
  39. F ( ν ) F(\nu)
  40. G ( ν ) G(\nu)
  41. H ( ν ) = F ( ν ) G ( ν ) , H(\nu)=F(\nu)\cdot G(\nu),
  42. - 1 { f * g } = - 1 { f } - 1 { g } \mathcal{F}^{-1}\{f*g\}=\mathcal{F}^{-1}\{f\}\cdot\mathcal{F}^{-1}\{g\}
  43. - 1 { f g } = - 1 { f } * - 1 { g } \mathcal{F}^{-1}\{f\cdot g\}=\mathcal{F}^{-1}\{f\}*\mathcal{F}^{-1}\{g\}
  44. f * g = { - 1 { f } - 1 { g } } f*g=\mathcal{F}\big\{\mathcal{F}^{-1}\{f\}\cdot\mathcal{F}^{-1}\{g\}\big\}
  45. f g = { - 1 { f } * - 1 { g } } f\cdot g=\mathcal{F}\big\{\mathcal{F}^{-1}\{f\}*\mathcal{F}^{-1}\{g\}\big\}
  46. x x
  47. y y
  48. x * y = D T F T - 1 [ D T F T { x } D T F T { y } ] , x*y=\scriptstyle{DTFT}^{-1}\displaystyle\big[\scriptstyle{DTFT}\displaystyle\{% x\}\cdot\ \scriptstyle{DTFT}\displaystyle\{y\}\big],
  49. x x
  50. y y
  51. x N * y , x_{N}*y,
  52. x N x_{N}
  53. x N [ n ] = def m = - x [ n - m N ] . x_{N}[n]\ \stackrel{\,\text{def}}{=}\sum_{m=-\infty}^{\infty}x[n-mN].
  54. x N * y \displaystyle x_{N}*y
  55. D T F T { x N } \scriptstyle{DTFT}\displaystyle\{x_{N}\}
  56. D T F T { x N } ( f ) = 1 N k = - ( D F T { x N } [ k ] ) δ ( f - k / N ) \scriptstyle{DTFT}\displaystyle\{x_{N}\}(f)=\frac{1}{N}\sum_{k=-\infty}^{% \infty}\left(\scriptstyle{DFT}\displaystyle\{x_{N}\}[k]\right)\cdot\delta\left% (f-k/N\right)
  57. D T F T { y } ( f ) \scriptstyle{DTFT}\displaystyle\{y\}(f)
  58. D T F T { x N } D T F T { y } = 1 N k = - D F T { x N } [ k ] D T F T { y } ( k / N ) D F T { y N } [ k ] δ ( f - k / N ) \scriptstyle{DTFT}\displaystyle\{x_{N}\}\cdot\scriptstyle{DTFT}\displaystyle\{% y\}=\frac{1}{N}\sum_{k=-\infty}^{\infty}\scriptstyle{DFT}\displaystyle\{x_{N}% \}[k]\cdot\underbrace{\scriptstyle{DTFT}\displaystyle\{y\}(k/N)}_{\scriptstyle% {DFT}\displaystyle\{y_{N}\}[k]}\cdot\delta\left(f-k/N\right)
  59. ( x N * y ) [ n ] = 0 1 1 N k = - D F T { x N } [ k ] D F T { y N } [ k ] δ ( f - k / N ) e i 2 π f n d f = 1 N k = - D F T { x N } [ k ] D F T { y N } [ k ] 0 1 δ ( f - k / N ) e i 2 π f n d f = 1 N k = 0 N - 1 D F T { x N } [ k ] D F T { y N } [ k ] e i 2 π n N k = D F T - 1 [ D F T { x N } D F T { y N } ] , \begin{aligned}\displaystyle(x_{N}*y)[n]&\displaystyle=\int_{0}^{1}\frac{1}{N}% \sum_{k=-\infty}^{\infty}\scriptstyle{DFT}\displaystyle\{x_{N}\}[k]\cdot% \scriptstyle{DFT}\displaystyle\{y_{N}\}[k]\cdot\delta\left(f-k/N\right)\cdot e% ^{i2\pi fn}df\\ &\displaystyle=\frac{1}{N}\sum_{k=-\infty}^{\infty}\scriptstyle{DFT}% \displaystyle\{x_{N}\}[k]\cdot\scriptstyle{DFT}\displaystyle\{y_{N}\}[k]\cdot% \int_{0}^{1}\delta\left(f-k/N\right)\cdot e^{i2\pi fn}df\\ &\displaystyle=\frac{1}{N}\sum_{k=0}^{N-1}\scriptstyle{DFT}\displaystyle\{x_{N% }\}[k]\cdot\scriptstyle{DFT}\displaystyle\{y_{N}\}[k]\cdot e^{i2\pi\frac{n}{N}% k}\\ &\displaystyle=\scriptstyle{DFT}^{-1}\displaystyle\big[\scriptstyle{DFT}% \displaystyle\{x_{N}\}\cdot\scriptstyle{DFT}\displaystyle\{y_{N}\}\big],\end{aligned}

Convolutional_code.html

  1. y i j = k = 0 h k j x i - k , y_{i}^{j}=\sum_{k=0}^{\infty}h^{j}_{k}x_{i-k},
  2. x x\,
  3. y j y^{j}\,
  4. j j\,
  5. h j h^{j}\,
  6. j j\,
  7. H 1 ( z ) = 1 + z - 1 + z - 2 , H_{1}(z)=1+z^{-1}+z^{-2},\,
  8. H 2 ( z ) = z - 1 + z - 2 , H_{2}(z)=z^{-1}+z^{-2},\,
  9. H 3 ( z ) = 1 + z - 2 . H_{3}(z)=1+z^{-2}.\,
  10. H 1 ( z ) = 1 + z - 1 + z - 3 1 - z - 2 - z - 3 , H_{1}(z)=\frac{1+z^{-1}+z^{-3}}{1-z^{-2}-z^{-3}},\,
  11. H 2 ( z ) = 1. H_{2}(z)=1.\,
  12. m m\,
  13. m = max i p o l y d e g ( H i ( 1 / z ) ) m=\max_{i}polydeg(H_{i}(1/z))\,
  14. f ( z ) = P ( z ) / Q ( z ) f(z)=P(z)/Q(z)\,
  15. p o l y d e g ( f ) = max ( d e g ( P ) , d e g ( Q ) ) polydeg(f)=\max(deg(P),deg(Q))\,
  16. m m\,
  17. H i ( 1 / z ) H_{i}(1/z)\,
  18. K = m + 1 K=m+1\,
  19. t = d - 1 2 . t=\left\lfloor\frac{d-1}{2}\right\rfloor.

Conway's_Game_of_Life.html

  1. c c

Copenhagen_interpretation.html

  1. Ψ \Psi
  2. ( | dead + | alive ) / 2 (|\,\text{dead}\rangle+|\,\text{alive}\rangle)/\sqrt{2}
  3. ( | dead + | alive ) / 2 (|\,\text{dead}\rangle+|\,\text{alive}\rangle)/\sqrt{2}
  4. | alive |\,\text{alive}\rangle

Coprime_integers.html

  1. gcd ( a , b ) = 1 \gcd(a,b)=1\;
  2. ( a , b ) = 1 , (a,b)=1,\;
  3. a b a\perp b
  4. gcd ( n a - 1 , n b - 1 ) = n gcd ( a , b ) - 1. \gcd(n^{a}-1,n^{b}-1)=n^{\gcd(a,b)}-1.
  5. p p
  6. 1 / p 1/p
  7. 1 / p 2 1/p^{2}
  8. 1 - 1 / p 2 1-1/p^{2}
  9. prime p ( 1 - 1 p 2 ) = ( prime p 1 1 - p - 2 ) - 1 = 1 ζ ( 2 ) = 6 π 2 0.607927102 61 % . \prod_{\,\text{prime }p}\left(1-\frac{1}{p^{2}}\right)=\left(\prod_{\,\text{% prime }p}\frac{1}{1-p^{-2}}\right)^{-1}=\frac{1}{\zeta(2)}=\frac{6}{\pi^{2}}% \approx 0.607927102\approx 61\%.
  10. { 1 , 2 , , N } \{1,2,\ldots,N\}
  11. 6 / π 2 6/\pi^{2}
  12. N N\to\infty
  13. P N P_{N}
  14. 6 / π 2 6/\pi^{2}
  15. ( m , n ) (m,n)
  16. m > n m>n
  17. ( 2 , 1 ) (2,1)
  18. ( 3 , 1 ) (3,1)
  19. ( m , n ) (m,n)
  20. ( 2 m - n , m ) (2m-n,m)
  21. ( 2 m + n , m ) (2m+n,m)
  22. ( m + 2 n , n ) (m+2n,n)

Coriolis_effect.html

  1. s y m b o l a C = - 2 s y m b o l Ω × v symbol{a}_{C}=-2\,symbol{\Omega\times v}
  2. s y m b o l a C symbol{a}_{C}
  3. s y m b o l v symbol{v}\,
  4. s y m b o l F C = - 2 m s y m b o l Ω × v symbol{F}_{C}=-2\,m\,symbol{\Omega\times v}
  5. s y m b o l Ω × v = | s y m b o l i s y m b o l j s y m b o l k Ω x Ω y Ω z v x v y v z | = ( Ω y v z - Ω z v y Ω z v x - Ω x v z Ω x v y - Ω y v x ) , symbol{\Omega\times v}=\begin{vmatrix}symbol{i}&symbol{j}&symbol{k}\\ \Omega_{x}&\Omega_{y}&\Omega_{z}\\ v_{x}&v_{y}&v_{z}\end{vmatrix}\ =\begin{pmatrix}\Omega_{y}v_{z}-\Omega_{z}v_{y% }\\ \Omega_{z}v_{x}-\Omega_{x}v_{z}\\ \Omega_{x}v_{y}-\Omega_{y}v_{x}\end{pmatrix}\ ,
  6. - s y m b o l Ω \timessymbol v -symbol\Omega\timessymbol v
  7. - s y m b o l Ω \timessymbol v -symbol\Omega\timessymbol v
  8. f = 2 ω sin φ f=2\omega\sin\varphi\,
  9. R o = U f L . Ro=\frac{U}{fL}.
  10. s y m b o l Ω = ω ( 0 cos φ sin φ ) , symbol{\Omega}=\omega\begin{pmatrix}0\\ \cos\varphi\\ \sin\varphi\end{pmatrix}\ ,
  11. s y m b o l v = ( v e v n v u ) , symbol{v}=\begin{pmatrix}v_{e}\\ v_{n}\\ v_{u}\end{pmatrix}\ ,
  12. s y m b o l a C = - 2 s y m b o l Ω × v = 2 ω ( v n sin φ - v u cos φ - v e sin φ v e cos φ ) . symbol{a}_{C}=-2symbol{\Omega\times v}=2\,\omega\,\begin{pmatrix}v_{n}\sin% \varphi-v_{u}\cos\varphi\\ -v_{e}\sin\varphi\\ v_{e}\cos\varphi\end{pmatrix}\ .
  13. s y m b o l v = ( v e v n ) , symbol{v}=\begin{pmatrix}v_{e}\\ v_{n}\end{pmatrix}\ ,
  14. s y m b o l a c = ( v n - v e ) f , symbol{a}_{c}=\begin{pmatrix}v_{n}\\ -v_{e}\end{pmatrix}\ f\ ,
  15. f = 2 ω sin φ f=2\omega\sin\varphi\,
  16. s y m b o l Ω = ω ( 0 1 0 ) , symbol{\Omega}=\omega\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}\ ,
  17. s y m b o l v = ( v e v n v u ) , symbol{v}=\begin{pmatrix}v_{e}\\ v_{n}\\ v_{u}\end{pmatrix}\ ,
  18. s y m b o l a C = - 2 s y m b o l Ω × v = 2 ω ( - v u 0 v e ) . symbol{a}_{C}=-2symbol{\Omega\times v}=2\,\omega\,\begin{pmatrix}-v_{u}\\ 0\\ v_{e}\end{pmatrix}\ .
  19. s y m b o l F f \displaystyle symbol{F_{f}}
  20. v v\,
  21. R R
  22. R = v f R=\frac{v}{f}\,
  23. f f
  24. 2 Ω sin φ 2\Omega\sin\varphi
  25. φ \varphi
  26. 2 π / f 2\pi/f
  27. f f
  28. f f
  29. f f
  30. 𝐫 A ( t ) = v t ( cos ( θ ) , sin ( θ ) ) . \mathbf{r}_{A}(t)=vt\ \left(\cos(\theta),\ \sin(\theta)\right)\ .
  31. 𝐫 B ( t ) = v t ( cos ( θ - ω t ) , sin ( θ - ω t ) ) , \mathbf{r}_{B}(t)=vt\ \left(\cos(\theta-\omega t),\ \sin(\theta-\omega t)% \right)\ ,
  32. 𝐚 B = 𝐚 A \mathbf{a}_{B}=\mathbf{a}_{A}
  33. - 2 s y m b o l Ω × 𝐯 B {}-2symbol\Omega\times\mathbf{v}_{B}
  34. - s y m b o l Ω × ( s y m b o l Ω × 𝐫 B ) {}-symbol\Omega\times(symbol\Omega\times\mathbf{r}_{B})
  35. - d s y m b o l Ω d t × 𝐫 B , {}-\frac{dsymbol\Omega}{dt}\times\mathbf{r}_{B}\ ,
  36. s y m b o l Ω × 𝐫 𝐁 symbol{\Omega}\mathbf{\times r_{B}}
  37. = | s y m b o l i s y m b o l j s y m b o l k 0 0 ω v t cos α v t sin α 0 | =\begin{vmatrix}symbol{i}&symbol{j}&symbol{k}\\ 0&0&\omega\\ vt\cos\alpha&vt\sin\alpha&0\end{vmatrix}
  38. = ω t v ( - sin α , cos α ) , =\omega tv\left(-\sin\alpha,\cos\alpha\right)\ ,
  39. s y m b o l Ω × ( s y m b o l Ω × 𝐫 𝐁 ) symbol{\Omega\ \times}\left(symbol{\Omega}\mathbf{\times r_{B}}\right)
  40. = | s y m b o l i s y m b o l j s y m b o l k 0 0 ω - ω t v sin α ω t v cos α 0 | , =\begin{vmatrix}symbol{i}&symbol{j}&symbol{k}\\ 0&0&\omega\\ -\omega tv\sin\alpha&\omega tv\cos\alpha&0\end{vmatrix}\ \ ,
  41. 𝐚 Cfgl \mathbf{a_{\mathrm{Cfgl}}}
  42. = ω 2 v t ( cos α , sin α ) = ω 2 𝐫 𝐁 ( t ) . =\omega^{2}vt\left(\cos\alpha,\sin\alpha\right)=\omega^{2}\mathbf{r_{B}}(t)\ .
  43. 𝐯 𝐁 = d 𝐫 𝐁 ( t ) d t = ( v cos α + ω t v sin α , \mathbf{v_{B}}=\frac{d\mathbf{r_{B}}(t)}{dt}=(v\cos\alpha+\omega t\ v\sin\alpha,
  44. v sin α - ω t v cos α , 0 ) , \ v\sin\alpha-\omega t\ v\cos\alpha,\ 0)\ \ ,
  45. s y m b o l Ω × 𝐯 𝐁 symbol{\Omega}\mathbf{\times v_{B}}
  46. = | s y m b o l i s y m b o l j s y m b o l k 0 0 ω v cos α v sin α + ω t v sin α - ω t v cos α 0 | , =\begin{vmatrix}\!symbol{i}&\!symbol{j}&\!symbol{k}\\ 0&0&\omega\\ v\cos\alpha&v\sin\alpha&\\ \;+\omega t\ v\sin\alpha&\;-\omega t\ v\cos\alpha&0\end{vmatrix}\ \ ,
  47. 𝐚 Cor \mathbf{a_{\mathrm{Cor}}}
  48. = - 2 [ - ω v ( sin α - ω t cos α ) , =-2\left[-\omega v\left(\sin\alpha-\omega t\cos\alpha\right),\right.
  49. \color w h i t e ω v ( cos α + ω t sin α ) ] \left.{\color{white}...}\ \omega v\left(\cos\alpha+\omega t\sin\alpha\right)\right]
  50. = 2 ω v ( sin α , - cos α ) =2\omega v\left(\sin\alpha,\ -\cos\alpha\right)
  51. - 2 ω 2 𝐫 𝐁 ( t ) . -2\omega^{2}\mathbf{r_{B}}(t)\ .
  52. 𝐚 Cptl = - ω 2 𝐫 𝐁 ( t ) , \mathbf{a_{\mathrm{Cptl}}}=-\omega^{2}\mathbf{r_{B}}(t)\ ,
  53. 𝐚 𝐂 \mathbf{a_{C\perp}}
  54. = 2 ω v ( sin α , - cos α ) . =2\omega v\left(\sin\alpha,\ -\cos\alpha\right)\ .

Corona.html

  1. p p
  2. p = n K B T p=nK_{B}T
  3. n n
  4. K B K_{B}
  5. T T
  6. k = 20 ( 2 π ) 3 / 2 ( k B T ) 5 / 2 k B m e 1 / 2 e 4 ln Λ 1.8 10 - 10 T 5 / 2 ln Λ W m - 1 K - 1 k=20\left(\frac{2}{\pi}\right)^{3/2}\frac{\left(k_{B}T\right)^{5/2}k_{B}}{m_{e% }^{1/2}e^{4}\ln\Lambda}\approx 1.8~{}10^{-10}~{}\frac{T^{5/2}}{\ln\Lambda}~{}% Wm^{-1}K^{-1}
  7. k B k_{B}
  8. T T
  9. m e m_{e}
  10. e e
  11. ln Λ = ln ( 12 π n λ D 3 ) \ln\Lambda=\ln\left(12\pi n\lambda_{D}^{3}\right)
  12. λ D = k B T 4 π n e 2 \lambda_{D}=\sqrt{\frac{k_{B}T}{4\pi ne^{2}}}
  13. n n
  14. ln Λ \ln\Lambda
  15. q q
  16. x x
  17. q t = 0.9 10 - 11 2 T 7 / 2 x 2 \frac{\partial q}{\partial t}=0.9~{}10^{-11}~{}\frac{\partial^{2}T^{7/2}}{% \partial x^{2}}

Correspondence_principle.html

  1. Δ E n = h T ( E n ) . \Delta E_{n}={h\over T(E_{n})}.
  2. E n E_{n}
  3. E n + 1 E_{n+1}
  4. Δ E 1 r 3 2 E 3 2 . \Delta E\propto{1\over r^{3\over 2}}\propto E^{3\over 2}.
  5. r ¯ \overline{r}
  6. E 1 r 1 L 2 E\propto{1\over r}\propto{1\over L^{2}}
  7. Δ E 1 ( L + ) 2 - 1 L 2 - 2 L 3 - E 3 2 . \Delta E\propto{1\over(L+\hbar)^{2}}-{1\over L^{2}}\approx-{2\hbar\over L^{3}}% \propto-E^{3\over 2}.
  8. L = n h 2 π = n . L={nh\over 2\pi}=n\hbar~{}.
  9. d J d E = T {dJ\over dE}=T
  10. Δ E = E n + 1 - E n = d E d J ( J n + 1 - J n ) = 1 T Δ J \Delta E=E_{n+1}-E_{n}={dE\over dJ}(J_{n+1}-J_{n})={1\over T}\,\Delta J
  11. J = 0 T p d x d t d t J=\int_{0}^{T}p{dx\over dt}\,dt
  12. J k = h n k . J_{k}=hn_{k}.\,
  13. r , θ r,\theta
  14. θ \theta
  15. 0 2 π L d θ = 2 π L = n h . \int_{0}^{2\pi}Ld\theta=2\pi L=nh.
  16. E = ( n + 1 / 2 ) ω , n = 0 , 1 , 2 , 3 , , E=(n+1/2)\hbar\omega,\ n=0,1,2,3,\dots~{},
  17. E = m ω 2 A 2 2 . E=\frac{m\omega^{2}A^{2}}{2}.
  18. n = E ω - 1 2 = m ω A 2 2 - 1 2 n=\frac{E}{\hbar\cdot\omega}-\frac{1}{2}=\frac{m\omega A^{2}}{2\hbar}-\frac{1}% {2}
  19. E = m 0 1 - v 2 / c 2 c 2 , E=\frac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}c^{2}~{},
  20. v v
  21. m 0 m_{0}
  22. c c
  23. v v
  24. E 0 = m 0 c 2 . E_{0}=m_{0}c^{2}.
  25. T = E - E 0 = m 0 c 2 1 - v 2 / c 2 - m 0 c 2 T=E-E_{0}=\frac{m_{0}c^{2}}{\sqrt{1-v^{2}/c^{2}}}\ -\ m_{0}c^{2}
  26. ( 1 + x ) n 1 + n x (1+x)^{n}\approx 1+nx
  27. | x | 1 |x|\ll 1
  28. v c v\ll c
  29. T T
  30. = m 0 c 2 ( 1 1 - v 2 / c 2 - 1 ) =m_{0}c^{2}\left(\frac{1}{\sqrt{1-v^{2}/c^{2}}}-1\right)
  31. = m 0 c 2 ( ( 1 - v 2 / c 2 ) - 1 2 - 1 ) =m_{0}c^{2}\left(\left(1-v^{2}/c^{2}\right)^{-\frac{1}{2}}-1\right)
  32. m 0 c 2 ( ( 1 - ( - 1 2 ) v 2 / c 2 ) - 1 ) \approx m_{0}c^{2}\left((1-(-\begin{matrix}\frac{1}{2}\end{matrix})v^{2}/c^{2}% )-1\right)
  33. = m 0 c 2 ( 1 2 v 2 / c 2 ) =m_{0}c^{2}\left(\begin{matrix}\frac{1}{2}\end{matrix}v^{2}/c^{2}\right)
  34. = 1 2 m 0 v 2 =\begin{matrix}\frac{1}{2}\end{matrix}m_{0}v^{2}

Cosmic_censorship_hypothesis.html

  1. M M
  2. J J
  3. V eff ( r , e , l ) = - M r + l 2 - a 2 ( e 2 - 1 ) 2 r 2 - M ( l - a e ) 2 r 3 , a J M V_{\rm{eff}}(r,e,l)=-\frac{M}{r}+\frac{l^{2}-a^{2}(e^{2}-1)}{2r^{2}}-\frac{M(l% -ae)^{2}}{r^{3}},~{}~{}~{}a\equiv\frac{J}{M}
  4. r r
  5. e e
  6. l l
  7. a < 1 a<1
  8. a < 1 a<1
  9. l = 2 M e l=2Me
  10. ( e 2 - 1 ) / 2 (e^{2}-1)/2
  11. ( e 2 - 1 ) / 2 (e^{2}-1)/2
  12. M < | Q | M<|Q|
  13. r = 0 r=0
  14. R a b = 2 ϕ a ϕ b R_{ab}=2\phi_{a}\phi_{b}
  15. d s 2 = - ( 1 + 2 σ ) d v 2 + 2 d v d r + r ( r - 2 σ v ) ( d θ 2 + sin 2 θ d ϕ 2 ) , ϕ = 1 2 ln ( 1 - 2 σ v r ) , ds^{2}=-(1+2\sigma)dv^{2}+2dv\,dr+r(r-2\sigma v)\left(d\theta^{2}+\sin^{2}% \theta\,d\phi^{2}\right),\quad\phi=\frac{1}{2}\ln\left(1-\frac{2\sigma v}{r}% \right),
  16. σ \sigma

Cosmological_constant.html

  1. R μ ν - 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν , R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}={8\pi G\over c^{4}}T_{% \mu\nu},
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi
  6. Ω Λ 0.7 \Omega_{\Lambda}\simeq 0.7
  7. M pl 4 M_{\rm pl}^{4}

Cost_accounting.html

  1. throughput = sales revenue - totally variable costs \,\text{throughput}=\,\text{sales revenue}-\,\text{totally variable costs}
  2. throughput accounting ratio = return factory hours \,\text{throughput accounting ratio}=\frac{\,\text{return}}{\text{factory % hours}}
  3. Contribution Margin Ratio = ( Sales - Variable Costs ) / Sales \,\text{Contribution Margin Ratio}=(\,\text{Sales - Variable Costs})/\,\text{Sales}

Costas_loop.html

  1. m 2 ( t ) = 1 m^{2}(t)=1
  2. m 2 ( t ) = 1 m^{2}(t)=1
  3. f 1 , 2 ( θ ( t ) ) f^{1,2}(\theta(t))
  4. θ ˙ 1 , 2 ( t ) \dot{\theta}^{1,2}(t)
  5. - 90 o -90^{o}
  6. - π 2 -\frac{\pi}{2}
  7. \bigotimes
  8. x ˙ = A x + b ξ ( t ) , σ = c * x , \begin{array}[]{ll}\dot{x}=Ax+b\xi(t),&\sigma=c^{*}x,\end{array}
  9. A A
  10. x ( t ) x(t)
  11. b b
  12. c c
  13. θ ˙ 2 ( t ) = ω f r e e 2 + L G ( t ) , t [ 0 , T ] , \begin{array}[]{ll}\dot{\theta}^{2}(t)=\omega^{2}_{free}+LG(t),&t\in[0,T],\end% {array}
  14. ω f r e e 2 \omega^{2}_{free}
  15. L L
  16. θ ˙ 1 ( t ) ω 1 . \dot{\theta}^{1}(t)\equiv\omega^{1}.
  17. x ˙ = A x + b f 1 ( θ 1 ( t ) ) f 2 ( θ 2 ( t ) ) , θ ˙ 2 = ω f r e e 2 + L c * x . \begin{array}[]{ll}\dot{x}=Ax+bf^{1}(\theta^{1}(t))f^{2}(\theta^{2}(t)),&\dot{% \theta}^{2}=\omega^{2}_{free}+Lc^{*}x.\end{array}
  18. f 1 ( θ 1 ( t ) ) = cos ( ω 1 t ) , f 2 ( θ 2 ( t ) ) = sin ( ω 2 t ) f 1 ( θ 1 ( t ) ) 2 f 2 ( θ 2 ( t ) ) f 2 ( θ 2 ( t ) - π 2 ) = - 1 8 ( 2 sin ( 2 ω 2 t ) + sin ( 2 ω 2 t - 2 ω 1 t ) + sin ( 2 ω 2 t + 2 ω 1 t ) ) \begin{array}[]{l}f^{1}\big(\theta^{1}(t)\big)=\cos\big(\omega^{1}t\big),f^{2}% \big(\theta^{2}(t)\big)=\sin\big(\omega^{2}t\big)\\ f^{1}\big(\theta^{1}(t)\big)^{2}f^{2}\big(\theta^{2}(t)\big)f^{2}\big(\theta^{% 2}(t)-\frac{\pi}{2}\big)=-\frac{1}{8}\Big(2\sin(2\omega^{2}t)+\sin(2\omega^{2}% t-2\omega^{1}t)+\sin(2\omega^{2}t+2\omega^{1}t)\Big)\end{array}
  19. φ ( θ 1 ( t ) - θ 2 ( t ) ) = 1 8 sin ( 2 ω 1 - 2 ω 2 ) \varphi(\theta^{1}(t)-\theta^{2}(t))=\frac{1}{8}\sin(2\omega^{1}-2\omega^{2})
  20. φ ( θ ) \varphi(\theta)
  21. f 1 ( θ ) f^{1}(\theta)
  22. f 2 ( θ ) f^{2}(\theta)
  23. g ( t ) g(t)
  24. G ( t ) G(t)
  25. x ˙ = A x + b φ ( Δ θ ) , Δ θ ˙ = ω f r e e 2 - ω 1 + L c * x , Δ θ = θ 2 - θ 1 . \begin{array}[]{ll}\dot{x}=Ax+b\varphi(\Delta\theta),&\Delta\dot{\theta}=% \omega^{2}_{free}-\omega^{1}+Lc^{*}x,\\ \Delta\theta=\theta^{2}-\theta^{1}.&\end{array}

Cotangent_space.html

  1. i f i g i \sum_{i}f_{i}g_{i}\,
  2. X ( f ) = X f X(f)=\mathcal{L}_{X}f
  3. f * : T x M T f ( x ) N f_{*}\colon T_{x}M\to T_{f(x)}N
  4. f * : T f ( x ) * N T x * M f^{*}\colon T_{f(x)}^{*}N\to T_{x}^{*}M
  5. ( f * θ ) ( X x ) = θ ( f * X x ) (f^{*}\theta)(X_{x})=\theta(f_{*}X_{x})
  6. f * d g = d ( g f ) . f^{*}\mathrm{d}g=\mathrm{d}(g\circ f).

Coulomb.html

  1. 1 C = 1 A 1 s 1\,\text{ C}=1\,\text{ A}\cdot 1\,\text{ s}
  2. 1 C = 1 F 1 V 1\,\text{ C}=1\,\text{ F}\cdot 1\,\text{ V}
  3. × 10 2 3 \times 10^{2}3
  4. × 10 5 \times 10^{−}5
  5. × 10 - 10 \times 10^{-}10

Countable_set.html

  1. G : 𝐍 × 𝐍 n 𝐍 A n G:\mathbf{N}\times\mathbf{N}\to\bigcup_{n\in\mathbf{N}}A_{n}

Cox's_theorem.html

  1. f ( f ( x ) ) = x , f(f(x))=x,\,
  2. P ( A and B ) = g ( P ( A ) , P ( B | A ) ) P(A\ \mbox{and}~{}\ B)=g(P(A),P(B|A))
  3. y f ( f ( z ) y ) = z f ( f ( y ) z ) y\,f\left({f(z)\over y}\right)=z\,f\left({f(y)\over z}\right)

Cross_section_(physics).html

  1. σ σ
  2. σ σ
  3. σ = μ n \sigma=\frac{\mu}{n}
  4. σ σ
  5. μ μ
  6. n n
  7. σ = 1 n Φ ( - d Φ d z ) \sigma=\frac{1}{n\Phi}\left(-\frac{\mathrm{d}\Phi}{\mathrm{d}z}\right)
  8. d Φ −dΦ
  9. d z dz
  10. Φ Φ
  11. σ σ
  12. σ = 1 n I A d W d z \sigma=\frac{1}{nIA}\frac{\mathrm{d}W}{\mathrm{d}z}
  13. d W dW
  14. I I
  15. A A
  16. σ σ
  17. σ σ
  18. d σ / d Ω dσ/dΩ
  19. d σ / d Ω dσ/dΩ
  20. σ σ
  21. Ω Ω
  22. z z
  23. θ θ
  24. φ φ
  25. φ φ
  26. d σ / d Ω dσ/dΩ
  27. ( θ , φ ) (θ,φ)
  28. d w dw
  29. d σ d Ω = 1 n A I Ω d w d z \frac{\mathrm{d}\sigma}{d\Omega}=\frac{1}{nAI\Omega}\frac{\mathrm{d}w}{\mathrm% {d}z}
  30. Ω Ω
  31. σ σ
  32. d σ / d Ω dσ/dΩ
  33. 4 π
  34. σ = 4 π d σ d Ω d Ω = 0 2 π 0 π d σ d Ω sin θ d θ d φ \sigma=\oint_{4\pi}\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\,\mathrm{d}\Omega% =\int_{0}^{2\pi}\int_{0}^{\pi}\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\sin% \theta\,\mathrm{d}\theta\,\mathrm{d}\varphi
  35. σ σ
  36. 2 50 µ m 2–50µm
  37. b b
  38. m b mb
  39. μ b μb
  40. d σ = d 2 b d\sigma=d^{2}b
  41. b b
  42. | d σ d Ω | \left|\frac{d\sigma}{d\Omega}\right|
  43. φ \varphi
  44. b sin ϑ | d b d ϑ | \frac{b}{\sin\vartheta}\left|\frac{db}{d\vartheta}\right|
  45. ϑ \vartheta
  46. k k
  47. ϕ - ( 𝐫 ) r e i k z \phi_{-}(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}e^{ikz}
  48. z z
  49. 𝐫 \mathbf{r}
  50. ϕ + ( 𝐫 ) r f ( θ , ϕ ) e i k r r \phi_{+}(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}f(\theta,\phi)\frac{% e^{ikr}}{r}
  51. f f
  52. ϕ ( 𝐫 ) r ϕ - ( 𝐫 ) + ϕ + ( 𝐫 ) \phi(\mathbf{r})\stackrel{r\to\infty}{\longrightarrow}\phi_{-}(\mathbf{r})+% \phi_{+}(\mathbf{r})
  53. d σ d Ω ( θ , ϕ ) = | f ( θ , ϕ ) | 2 \frac{d\sigma}{d\Omega}(\theta,\phi)=|f(\theta,\phi)|^{2}
  54. I r I\text{r}
  55. I i I\text{i}
  56. N σ 1 N\sigma\ll 1
  57. I r = I i N σ I\text{r}=I\text{i}N\sigma\,
  58. σ = I r I i 1 N = Probability of interaction × 1 N \sigma={{I\text{r}}\over{I\text{i}}}{{1}\over{N}}={\hbox{Probability of % interaction}}\times{{1}\over{N}}
  59. d σ d Ω = ( 2 π ) 4 m i m f p f p i | T f i | 2 , {d\sigma\over d\Omega}=(2\pi)^{4}m_{i}m_{f}{p_{f}\over p_{i}}|T_{fi}|^{2},
  60. S f i = δ f i - 2 π i δ ( E f - E i ) δ ( 𝐩 i - 𝐩 f ) T f i S_{fi}=\delta_{fi}-2\pi i\delta(E_{f}-E_{i})\delta(\mathbf{p}_{i}-\mathbf{p}_{% f})T_{fi}
  61. δ \delta
  62. d z dz
  63. Φ Φ
  64. d Φ d z = - n σ Φ \frac{\mathrm{d}\Phi}{\mathrm{d}z}=-n\sigma\Phi
  65. σ σ
  66. Φ = Φ 0 e - n σ z \Phi=\Phi_{0}\mathrm{e}^{-n\sigma z}
  67. σ = σ a + σ s + σ l . \sigma=\sigma_{\mathrm{a}}+\sigma_{\mathrm{s}}+\sigma_{\mathrm{l}}.
  68. A λ = C σ , A_{\lambda}=C\ell\sigma,
  69. \ell
  70. A λ = - log 𝒯 . A_{\lambda}=-\log\mathcal{T}.
  71. σ = σ A + σ S + σ L . \sigma=\sigma\text{A}+\sigma\text{S}+\sigma\text{L}.
  72. A λ = C l σ A_{\lambda}=Cl\sigma
  73. A λ = - log 𝒯 . A_{\lambda}=-\log\mathcal{T}.
  74. L V = 3.9 C σ scat . L\text{V}=\frac{3.9}{C\sigma\text{scat}}.
  75. R R
  76. r r
  77. b b
  78. ϑ \vartheta
  79. | d σ d Ω | = 1 4 ( r + R ) 2 \left|\frac{d\sigma}{d\Omega}\right|=\frac{1}{4}(r+R)^{2}
  80. σ tot = π ( r + R ) 2 . \sigma\text{tot}=\pi\;(r+R)^{2}\ .
  81. r + R r+R
  82. r r
  83. α \alpha
  84. d x = r cos α d α dx=r\cos\alpha d\alpha
  85. 2 α 2\alpha
  86. θ = π - 2 α \theta=\pi-2\alpha
  87. I I
  88. d x dx
  89. I d σ = I d x ( x ) = I r cos α d α = I r 2 sin ( θ / 2 ) d θ = I d σ d θ d θ Id\sigma=Idx(x)=Ir\cos\alpha d\alpha=I\frac{r}{2}\sin(\theta/2)d\theta=I\frac{% d\sigma}{d\theta}d\theta
  90. ( d Ω = d θ ) (d\Omega=d\theta)
  91. d σ d θ = r 2 sin ( θ / 2 ) \frac{d\sigma}{d\theta}=\frac{r}{2}\sin(\theta/2)
  92. θ = π \theta=\pi
  93. θ = 0 \theta=0
  94. σ = 0 2 π d σ d θ d θ = 0 2 π r 2 ( sin θ / 2 ) d θ = - r cos ( θ / 2 ) | 0 2 π = 2 r \sigma=\int_{0}^{2\pi}\frac{d\sigma}{d\theta}d\theta=\int_{0}^{2\pi}\frac{r}{2% }(\sin\theta/2)d\theta=-r\cos(\theta/2)\bigg|_{0}^{2\pi}=2r
  95. r r
  96. σ = π r 2 \sigma=\pi r^{2}
  97. a a
  98. r r
  99. ϕ \phi
  100. r = a sin α r=a\sin\alpha
  101. d r = a cos α d α dr=a\cos\alpha d\alpha
  102. d σ = d r ( r ) × r d φ = a 2 2 sin ( θ / 2 ) cos ( θ / 2 ) d θ d φ d\sigma=dr(r)\times rd\varphi=\frac{a^{2}}{2}\sin(\theta/2)\cos(\theta/2)d% \theta d\varphi
  103. d Ω = sin ( θ ) d θ d φ d\Omega=\sin(\theta)d\theta d\varphi
  104. sin ( θ ) = 2 sin ( θ / 2 ) cos ( θ / 2 ) \sin(\theta)=2\sin(\theta/2)\cos(\theta/2)
  105. d σ d Ω = a 2 4 \frac{d\sigma}{d\Omega}=\frac{a^{2}}{4}
  106. σ = 4 π d σ d Ω d Ω = π a 2 \sigma=\oint_{4\pi}\frac{d\sigma}{d\Omega}d\Omega=\pi a^{2}

Crystal_oscillator.html

  1. Z ( s ) = ( 1 s C 1 + s L 1 + R 1 ) | | ( 1 s C 0 ) , Z(s)=\left({\frac{1}{s\cdot C_{1}}+s\cdot L_{1}+R_{1}}\right)||\left({\frac{1}% {s\cdot C_{0}}}\right),
  2. Z ( s ) = s 2 + s R 1 L 1 + ω s 2 ( s C 0 ) [ s 2 + s R 1 L 1 + ω p 2 ] Z(s)=\frac{s^{2}+s\frac{R_{1}}{L_{1}}+{\omega_{s}}^{2}}{(s\cdot C_{0})[s^{2}+s% \frac{R_{1}}{L_{1}}+{\omega_{p}}^{2}]}
  3. ω s = 1 L 1 C 1 , ω p = C 1 + C 0 L 1 C 1 C 0 = ω s 1 + C 1 C 0 ω s ( 1 + C 1 2 C 0 ) ( C 0 C 1 ) \Rightarrow\omega_{s}=\frac{1}{\sqrt{L_{1}\cdot C_{1}}},\quad\omega_{p}=\sqrt{% \frac{C_{1}+C_{0}}{L_{1}\cdot C_{1}\cdot C_{0}}}=\omega_{s}\sqrt{1+\frac{C_{1}% }{C_{0}}}\approx\omega_{s}\left(1+\frac{C_{1}}{2C_{0}}\right)\quad(C_{0}\gg C_% {1})
  4. s = j ω s=j\omega
  5. ω s \omega_{s}
  6. ω p \omega_{p}

Crystal_structure.html

  1. 2 {}_{2}
  2. 2 {}_{2}
  3. d = 2 π / | 𝐠 m n | d=2\pi/|\mathbf{g}_{\ell mn}|
  4. d m n = a 2 + m 2 + n 2 d_{\ell mn}=\frac{a}{\sqrt{\ell^{2}+m^{2}+n^{2}}}
  5. 1 d 2 = h 2 + k 2 + l 2 a 2 \frac{1}{d^{2}}=\frac{h^{2}+k^{2}+l^{2}}{a^{2}}
  6. 1 d 2 = h 2 + k 2 a 2 + l 2 c 2 \frac{1}{d^{2}}=\frac{h^{2}+k^{2}}{a^{2}}+\frac{l^{2}}{c^{2}}
  7. 1 d 2 = 4 3 ( h 2 + h k + k 2 a 2 ) + l 2 c 2 \frac{1}{d^{2}}=\frac{4}{3}\bigg(\frac{h^{2}+hk+k^{2}}{a^{2}}\bigg)+\frac{l^{2% }}{c^{2}}
  8. 1 d 2 = ( h 2 + k 2 + l 2 ) sin 2 α + 2 ( h k + k l + h l ) ( cos 2 α - cos α ) a 2 ( 1 - 3 cos 2 α + 2 cos 3 α ) \frac{1}{d^{2}}=\frac{(h^{2}+k^{2}+l^{2})\sin^{2}\alpha+2(hk+kl+hl)(\cos^{2}% \alpha-\cos\alpha)}{a^{2}(1-3\cos^{2}\alpha+2\cos^{3}\alpha)}
  9. 1 d 2 = h 2 a 2 + k 2 b 2 + l 2 c 2 \frac{1}{d^{2}}=\frac{h^{2}}{a^{2}}+\frac{k^{2}}{b^{2}}+\frac{l^{2}}{c^{2}}
  10. 1 d 2 = 1 sin 2 β ( h 2 a 2 + k 2 sin 2 β b 2 + l 2 c 2 - 2 h l cos β a c ) \frac{1}{d^{2}}=\frac{1}{\sin^{2}\beta}\bigg(\frac{h^{2}}{a^{2}}+\frac{k^{2}% \sin^{2}\beta}{b^{2}}+\frac{l^{2}}{c^{2}}-\frac{2hl\cos\beta}{ac}\bigg)
  11. 4 × 4 3 π r 3 16 2 r 3 = π 3 2 = 0.7405 \frac{4\times\frac{4}{3}\pi r^{3}}{16\sqrt{2}r^{3}}=\frac{\pi}{3\sqrt{2}}=0.7405

Cube.html

  1. max { | x - x 0 | , | y - y 0 | , | z - z 0 | } = a . \max\{|x-x_{0}|,|y-y_{0}|,|z-z_{0}|\}=a.
  2. a a
  3. 6 a 2 6a^{2}\,
  4. a 3 a^{3}\,
  5. 2 a \sqrt{2}a
  6. 3 a \sqrt{3}a
  7. 3 2 a \frac{\sqrt{3}}{2}a
  8. a 2 \frac{a}{\sqrt{2}}
  9. a 2 \frac{a}{2}
  10. π 2 \frac{\pi}{2}
  11. a × a × a a\times a\times a
  12. 2 / 2 \scriptstyle\sqrt{2}/2
  13. 1 / 3 {1}/{3}
  14. 1 / 6 {1}/{6}

Cuboctahedron.html

  1. A = ( 6 + 2 3 ) a 2 9.4641016 a 2 A=\left(6+2\sqrt{3}\right)a^{2}\approx 9.4641016a^{2}
  2. V = 5 3 2 a 3 2.3570226 a 3 . V=\frac{5}{3}\sqrt{2}a^{3}\approx 2.3570226a^{3}.

Cumulative_distribution_function.html

  1. F X ( x ) = P ( X x ) , F_{X}(x)=\operatorname{P}(X\leq x),
  2. F X ( x ) = - x f X ( t ) d t . F_{X}(x)=\int_{-\infty}^{x}f_{X}(t)\,dt.
  3. P ( X = b ) = F X ( b ) - lim x b - F X ( x ) . \operatorname{P}(X=b)=F_{X}(b)-\lim_{x\to b^{-}}F_{X}(x).
  4. lim x - F ( x ) = 0 , lim x + F ( x ) = 1. \lim_{x\to-\infty}F(x)=0,\quad\lim_{x\to+\infty}F(x)=1.
  5. F ( x ) = P ( X x ) = x i x P ( X = x i ) = x i x p ( x i ) . F(x)=\operatorname{P}(X\leq x)=\sum_{x_{i}\leq x}\operatorname{P}(X=x_{i})=% \sum_{x_{i}\leq x}p(x_{i}).
  6. F ( b ) - F ( a ) = P ( a < X b ) = a b f ( x ) d x F(b)-F(a)=\operatorname{P}(a<X\leq b)=\int_{a}^{b}f(x)\,dx
  7. X X
  8. X X
  9. F ( x ) = { 0 : x < 0 x : 0 x < 1 1 : x 1. F(x)=\begin{cases}0&:\ x<0\\ x&:\ 0\leq x<1\\ 1&:\ x\geq 1.\end{cases}
  10. X X
  11. X X
  12. F ( x ) = { 0 : x < 0 1 / 2 : 0 x < 1 1 : x 1. F(x)=\begin{cases}0&:\ x<0\\ 1/2&:\ 0\leq x<1\\ 1&:\ x\geq 1.\end{cases}
  13. F ¯ ( x ) = P ( X > x ) = 1 - F ( x ) . \bar{F}(x)=\operatorname{P}(X>x)=1-F(x).
  14. p = P ( T t ) = P ( T > t ) = 1 - F T ( t ) . p=\operatorname{P}(T\geq t)=\operatorname{P}(T>t)=1-F_{T}(t).
  15. F ¯ ( x ) \bar{F}(x)
  16. S ( x ) S(x)
  17. F ¯ ( x ) 𝔼 ( X ) x . \bar{F}(x)\leq\frac{\mathbb{E}(X)}{x}.
  18. x , F ¯ ( x ) 0 x\to\infty,\bar{F}(x)\to 0
  19. F ¯ ( x ) = o ( 1 / x ) \bar{F}(x)=o(1/x)
  20. 𝔼 ( X ) \mathbb{E}(X)
  21. c > 0 c>0
  22. 𝔼 ( X ) = 0 x f ( x ) d x 0 c x f ( x ) d x + c c f ( x ) d x \mathbb{E}(X)=\int_{0}^{\infty}xf(x)dx\geq\int_{0}^{c}xf(x)dx+c\int_{c}^{% \infty}f(x)dx
  23. F ¯ ( c ) = c f ( x ) d x \bar{F}(c)=\int_{c}^{\infty}f(x)dx
  24. 0 c F ¯ ( c ) 𝔼 ( X ) - 0 c x f ( x ) d x 0 as c 0\leq c\bar{F}(c)\leq\mathbb{E}(X)-\int_{0}^{c}xf(x)dx\to 0\,\text{ as }c\to\infty
  25. F - 1 ( y ) , y [ 0 , 1 ] , F^{-1}(y),y\in[0,1],
  26. x x
  27. F ( x ) = y F(x)=y
  28. y [ 0 , 1 ] y\in[0,1]
  29. F - 1 ( y ) = inf { x : F ( x ) y } . F^{-1}(y)=\inf\{x\in\mathbb{R}:F(x)\geq y\}.
  30. F - 1 ( 0.5 ) F^{-1}(0.5)
  31. τ = F - 1 ( 0.95 ) \tau=F^{-1}(0.95)
  32. τ \tau
  33. F - 1 F^{-1}
  34. F - 1 ( F ( x ) ) x F^{-1}(F(x))\leq x
  35. F ( F - 1 ( y ) ) y F(F^{-1}(y))\geq y
  36. F - 1 ( y ) x F^{-1}(y)\leq x
  37. y F ( x ) y\leq F(x)
  38. Y Y
  39. U [ 0 , 1 ] U[0,1]
  40. F - 1 ( Y ) F^{-1}(Y)
  41. F F
  42. { X α } \{X_{\alpha}\}
  43. F F
  44. Y α Y_{\alpha}
  45. Y α Y_{\alpha}
  46. U [ 0 , 1 ] U[0,1]
  47. F - 1 ( Y α ) = X α F^{-1}(Y_{\alpha})=X_{\alpha}
  48. α \alpha
  49. F F
  50. F ( x , y ) = P ( X x , Y y ) , F(x,y)=\operatorname{P}(X\leq x,Y\leq y),
  51. 0 F ( x 1 , , x n ) 1 0\leq F(x_{1},...,x_{n})\leq 1
  52. lim x 1 , , x n + F ( x 1 , , x n ) = 1 \lim_{x_{1},...,x_{n}\rightarrow+\infty}F(x_{1},...,x_{n})=1
  53. lim x i - F ( x 1 , , x n ) = 0 , for all i \lim_{x_{i}\rightarrow-\infty}F(x_{1},...,x_{n})=0,\quad\mbox{for all }~{}i

Cumulus_cloud.html

  1. t = 18 η E g w r 0 t={18\eta\over Egwr_{0}}
  2. t t
  3. η \eta
  4. E E
  5. w w
  6. r 0 r_{0}

Curium.html

  1. Pu 94 239 + 2 4 He 96 242 Cm + 0 1 n \mathrm{{}^{239\!\,}_{\ 94}Pu\ +\ ^{4}_{2}He\ \longrightarrow\ ^{242}_{\ 96}Cm% \ +\ ^{1}_{0}n}
  2. Cm 96 242 94 238 Pu + 2 4 He \mathrm{{}^{242}_{\ 96}Cm\ \longrightarrow\ ^{238}_{\ 94}Pu\ +\ ^{4}_{2}He}
  3. Pu 94 239 + 2 4 He 96 240 Cm + 3 0 1 n \mathrm{{}^{239}_{\ 94}Pu\ +\ ^{4}_{2}He\ \longrightarrow\ ^{240}_{\ 96}Cm\ +% \ 3\ ^{1}_{0}n}
  4. 3 ¯ \overline{3}
  5. × 10 - 3 \times 10^{-}3
  6. U 92 238 ( n , γ ) 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}
  7. Pu 94 239 2 ( n , γ ) 94 241 Pu β - 14.35 yr 95 241 Am ( n , γ ) 95 242 Am β - 16.02 h 96 242 Cm \mathrm{{}^{239}_{\ 94}Pu\ \xrightarrow{2(n,\gamma)}\ ^{241}_{\ 94}Pu\ % \xrightarrow[14.35\ yr]{\beta^{-}}\ ^{241}_{\ 95}Am\ \xrightarrow{(n,\gamma)}% \ ^{242}_{\ 95}Am\ \xrightarrow[16.02\ h]{\beta^{-}}\ ^{242}_{\ 96}Cm}
  8. Pu 94 239 4 ( n , γ ) 94 243 Pu β - 4 , 956 h 95 243 Am ( n , γ ) 95 244 Am β - 10.1 h 96 244 Cm \mathrm{{}^{239}_{\ 94}Pu\ \xrightarrow{4(n,\gamma)}\ ^{243}_{\ 94}Pu\ % \xrightarrow[4,956\ h]{\beta^{-}}\ ^{243}_{\ 95}Am\ \xrightarrow{(n,\gamma)}\ % ^{244}_{\ 95}Am\ \xrightarrow[10.1\ h]{\beta^{-}}\ ^{244}_{\ 96}Cm}
  9. Cm 96 244 18.11 yr 𝛼 94 240 Pu \mathrm{{}^{244}_{\ 96}Cm\ \xrightarrow[18.11\ yr]{\alpha}\ ^{240}_{\ 94}Pu}
  10. Cm 96 A + 0 1 n 96 A + 1 Cm + γ \mathrm{{}^{A}_{96}Cm\ +\ ^{1}_{0}n\ \longrightarrow\ ^{A+1}_{\ \ 96}Cm\ +\ \gamma}
  11. Cf 98 252 2.645 yr 𝛼 96 248 Cm \mathrm{{}^{252}_{\ 98}Cf\ \xrightarrow[2.645\ yr]{\alpha}\ ^{248}_{\ 96}Cm}
  12. Bk 97 249 β - 330 d 98 249 Cf 351 yr 𝛼 96 245 Cm \mathrm{{}^{249}_{\ 97}Bk\ \xrightarrow[330\ d]{\beta^{-}}\ ^{249}_{\ 98}Cf\ % \xrightarrow[351\ yr]{\alpha}\ ^{245}_{\ 96}Cm}
  13. CmF 3 + 3 Li Cm + 3 LiF \mathrm{CmF_{3}\ +\ 3\ Li\ \longrightarrow\ Cm\ +\ 3\ LiF}
  14. 4 CmO 2 Δ T 2 Cm 2 O 3 + O 2 \mathrm{4\ CmO_{2}\ \xrightarrow{\Delta T}\ 2\ Cm_{2}O_{3}\ +\ O_{2}}
  15. 2 CmO 2 + H 2 Cm 2 O 3 + H 2 O \mathrm{2\ CmO_{2}\ +\ H_{2}\ \longrightarrow\ Cm_{2}O_{3}\ +\ H_{2}O}
  16. 2 CmF 3 + F 2 2 CmF 4 \mathrm{2\ CmF_{3}\ +\ F_{2}\ \longrightarrow\ 2\ CmF_{4}}
  17. CmCl 3 + 3 NH 4 I CmI 3 + 3 NH 4 Cl \mathrm{CmCl_{3}\ +\ 3\ NH_{4}I\ \longrightarrow\ CmI_{3}\ +\ 3\ NH_{4}Cl}
  18. CmCl 3 + H 2 O CmOCl + 2 HCl \mathrm{CmCl_{3}\ +\ \ H_{2}O\ \longrightarrow\ CmOCl\ +\ 2\ HCl}

Curl_(mathematics).html

  1. 𝐧 ^ \scriptstyle\mathbf{\hat{n}}
  2. 𝐧 ^ \scriptstyle\mathbf{\hat{n}}
  3. 𝐧 ^ \scriptstyle\mathbf{\hat{n}}
  4. ( × 𝐅 ) 𝐧 ^ = def lim A 0 ( 1 | A | C 𝐅 d 𝐫 ) (\nabla\times\mathbf{F})\cdot\mathbf{\hat{n}}\ \overset{\underset{\mathrm{def}% }{}}{=}\lim_{A\to 0}\left(\frac{1}{|A|}\oint_{C}\mathbf{F}\cdot d\mathbf{r}\right)
  5. C 𝐅 d 𝐫 \scriptstyle\oint_{C}\mathbf{F}\cdot d\mathbf{r}
  6. ν ^ \scriptstyle\mathbf{\hat{\nu}}
  7. 𝐧 ^ \scriptstyle\mathbf{\hat{n}}
  8. ω ^ \scriptstyle\mathbf{\hat{\omega}}
  9. { 𝐧 ^ , ν ^ , ω ^ } \scriptstyle\{\mathbf{\hat{n}},\mathbf{\hat{\nu}},\mathbf{\hat{\omega}}\}
  10. ( curl 𝐅 ) 1 = 1 h 2 h 3 ( ( h 3 F 3 ) u 2 - ( h 2 F 2 ) u 3 ) ({\rm curl}\,\mathbf{F})\,_{1}=\frac{1}{h_{2}h_{3}}\left(\frac{\partial(h_{3}F% _{3})}{\partial u_{2}}-\frac{\partial(h_{2}F_{2})}{\partial u_{3}}\right)\,
  11. ( curl 𝐅 ) 2 = 1 h 3 h 1 ( ( h 1 F 1 ) u 3 - ( h 3 F 3 ) u 1 ) ({\rm curl}\,\mathbf{F})\,_{2}=\frac{1}{h_{3}h_{1}}\left(\frac{\partial(h_{1}F% _{1})}{\partial u_{3}}-\frac{\partial(h_{3}F_{3})}{\partial u_{1}}\right)\,
  12. ( curl 𝐅 ) 3 = 1 h 1 h 2 ( ( h 2 F 2 ) u 1 - ( h 1 F 1 ) u 2 ) ({\rm curl}\,\mathbf{F})\,_{3}=\frac{1}{h_{1}h_{2}}\left(\frac{\partial(h_{2}F% _{2})}{\partial u_{1}}-\frac{\partial(h_{1}F_{1})}{\partial u_{2}}\right)\,
  13. ( curl 𝐅 ) k ({\rm curl}\,\mathbf{F})\,_{k}
  14. h i = j = 1 3 ( x j u i ) 2 h_{i}=\sqrt{\sum\limits_{j=1}^{3}\left(\frac{\partial x_{j}}{\partial u_{i}}% \right)^{2}}
  15. | 𝐢 𝐣 𝐤 x y z F x F y F z | \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \\ {\frac{\partial}{\partial x}}&{\frac{\partial}{\partial y}}&{\frac{\partial}{% \partial z}}\\ \\ F_{x}&F_{y}&F_{z}\end{vmatrix}
  16. ( F z y - F y z ) 𝐢 + ( F x z - F z x ) 𝐣 + ( F y x - F x y ) 𝐤 \left(\frac{\partial F_{z}}{\partial y}-\frac{\partial F_{y}}{\partial z}% \right)\mathbf{i}+\left(\frac{\partial F_{x}}{\partial z}-\frac{\partial F_{z}% }{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_{y}}{\partial x}-\frac{% \partial F_{x}}{\partial y}\right)\mathbf{k}
  17. ( × 𝐅 ) k = ϵ k m F m (\nabla\times\mathbf{F})^{k}=\epsilon^{k\ell m}\partial_{\ell}F_{m}
  18. ( × 𝐅 ) = s y m b o l e k ϵ k m F m (\nabla\times\mathbf{F})=symbol{e}_{k}\epsilon^{k\ell m}\partial_{\ell}F_{m}
  19. × 𝐅 = [ ( 𝐝 F ) ] \nabla\times\mathbf{F}=\left[\star\left({\mathbf{d}}F^{\flat}\right)\right]^{\sharp}
  20. \scriptstyle\flat
  21. \scriptstyle\sharp
  22. \scriptstyle\star
  23. 𝐅 ( x , y , z ) = y s y m b o l x ^ - x s y m b o l y ^ . \mathbf{F}(x,y,z)=ysymbol{\hat{x}}-xsymbol{\hat{y}}.
  24. × 𝐅 = 0 s y m b o l x ^ + 0 s y m b o l y ^ + [ x ( - x ) - y y ] s y m b o l z ^ = - 2 s y m b o l z ^ \nabla\times\mathbf{F}=0symbol{\hat{x}}+0symbol{\hat{y}}+\left[{\frac{\partial% }{\partial x}}(-x)-{\frac{\partial}{\partial y}}y\right]symbol{\hat{z}}=-2% symbol{\hat{z}}
  25. 𝐅 ( x , y , z ) = - x 2 s y m b o l y ^ . \mathbf{F}(x,y,z)=-x^{2}symbol{\hat{y}}.
  26. × 𝐅 = 0 s y m b o l x ^ + 0 s y m b o l y ^ + x ( - x 2 ) s y m b o l z ^ = - 2 x s y m b o l z ^ . {\nabla}\times\mathbf{F}=0symbol{\hat{x}}+0symbol{\hat{y}}+{\frac{\partial}{% \partial x}}(-x^{2})symbol{\hat{z}}=-2xsymbol{\hat{z}}.
  27. × ( 𝐯 × 𝐅 ) = [ ( 𝐅 ) + 𝐅 ] 𝐯 - [ ( 𝐯 ) + 𝐯 ] 𝐅 . \nabla\times\left(\mathbf{v\times F}\right)=\left[\left(\mathbf{\nabla\cdot F}% \right)+\mathbf{F\cdot\nabla}\right]\mathbf{v}-\left[\left(\mathbf{\nabla\cdot v% }\right)+\mathbf{v\cdot\nabla}\right]\mathbf{F}\ .
  28. 𝐯 × ( × 𝐅 ) = F ( 𝐯 𝐅 ) - ( 𝐯 ) 𝐅 , \mathbf{v\ \times}\left(\mathbf{\nabla\times F}\right)=\nabla_{F}\left(\mathbf% {v\cdot F}\right)-\left(\mathbf{v\cdot\nabla}\right)\mathbf{F}\ ,
  29. × ( × 𝐅 ) = ( 𝐅 ) - 2 𝐅 , \nabla\times\left(\mathbf{\nabla\times F}\right)=\mathbf{\nabla}(\mathbf{% \nabla\cdot F})-\nabla^{2}\mathbf{F}\ ,
  30. × ( ϕ ) = s y m b o l 0 \nabla\times(\nabla\phi)=symbol{0}
  31. × ( φ 𝐅 ) = φ × 𝐅 + φ × 𝐅 \nabla\times(\varphi\mathbf{F})=\nabla\varphi\times\mathbf{F}+\varphi\nabla% \times\mathbf{F}
  32. a 1 d x + a 2 d y + a 3 d z ; a_{1}\,dx+a_{2}\,dy+a_{3}\,dz;
  33. a 12 d x d y + a 13 d x d z + a 23 d y d z ; a_{12}\,dx\wedge dy+a_{13}\,dx\wedge dz+a_{23}\,dy\wedge dz;
  34. a 123 d x d y d z . a_{123}\,dx\wedge dy\wedge dz.
  35. d x d y , dx\wedge dy,
  36. d x d y = - d y d x dx\wedge dy=-dy\wedge dx
  37. ω ( k ) = i 1 < i 2 < < i k ; i ν 1 , , n a i 1 , , i k d x i 1 d x i k , \omega^{(k)}=\sum_{i_{1}<i_{2}<...<i_{k};\,\,\forall i_{\nu}\in 1,...,n}\,a_{i% _{1},...,i_{k}}\,dx_{i_{1}}\wedge...\wedge dx_{i_{k}},
  38. d ω ( k ) = j = 1 ; i 1 < < i k n a i 1 , , i k x j d x j d x i 1 d x i k . d\,\omega^{(k)}=\sum_{j=1;\,i_{1}<...<i_{k}}^{n}\frac{\partial a_{i_{1},...,i_% {k}}}{\partial x_{j}}\,dx_{j}\,\wedge dx_{i_{1}}\wedge...\wedge dx_{i_{k}}.
  39. 2 x y = 2 y x , \frac{\partial^{2}}{\partial x\partial y}=\frac{\partial^{2}}{\partial y% \partial x},
  40. Ω k ( 𝐑 3 ) \Omega^{k}(\mathbf{R}^{3})
  41. 0 𝑑 Ω 0 ( 𝐑 3 ) 𝑑 Ω 1 ( 𝐑 3 ) 𝑑 Ω 2 ( 𝐑 3 ) 𝑑 Ω 3 ( 𝐑 3 ) 𝑑 0. 0\overset{d}{\to}\Omega^{0}(\mathbf{R}^{3})\overset{d}{\to}\Omega^{1}(\mathbf{% R}^{3})\overset{d}{\to}\Omega^{2}(\mathbf{R}^{3})\overset{d}{\to}\Omega^{3}(% \mathbf{R}^{3})\overset{d}{\to}0.
  42. Ω k ( 𝐑 n ) \Omega^{k}(\mathbf{R}^{n})
  43. Λ k ( 𝐑 n ) \Lambda^{k}(\mathbf{R}^{n})
  44. ( n k ) ; \textstyle{{\left({{n}\atop{k}}\right)}};
  45. Ω k ( 𝐑 3 ) = 0 \Omega^{k}(\mathbf{R}^{3})=0
  46. a x d x + a y d y + a z d z a_{x}\,dx+a_{y}\,dy+a_{z}\,dz
  47. ( a x , a y , a z ) . (a_{x},a_{y},a_{z}).
  48. d y d z dy\wedge dz
  49. d z d x = - d x d z , dz\wedge dx=-dx\wedge dz,
  50. d x d y . dx\wedge dy.
  51. a x d x + a y d y + a z d z a_{x}\,dx+a_{y}\,dy+a_{z}\,dz
  52. a z d x d y + a y d z d x + a x d y d z . a_{z}\,dx\wedge dy+a_{y}\,dz\wedge dx+a_{x}\,dy\wedge dz.
  53. div curl v = 0 \mathrm{div\,\,curl\,\,}\vec{v}\,=0
  54. v . \vec{v}.
  55. ω ( 2 ) = i < k = 1 , , 4 a i , k d x i d x k , \omega^{(2)}=\sum_{i<k=1,...,4}a_{i,k}dx_{i}\wedge dx_{k},
  56. ( n 2 ) = 1 2 n ( n - 1 ) \textstyle{{\left({{n}\atop{2}}\right)}=\frac{1}{2}n(n-1)}
  57. n = 1 2 n ( n - 1 ) , \textstyle{n=\frac{1}{2}n(n-1)},

Currying.html

  1. f ( x , y ) = y / x f(x,y)=y/x
  2. f ( 2 , 3 ) f(2,3)
  3. x x
  4. 2 2
  5. y y
  6. g ( y ) g(y)
  7. g ( y ) = f ( 2 , y ) = y / 2 g(y)=f(2,y)=y/2
  8. y y
  9. 3 3
  10. g ( 3 ) = f ( 2 , 3 ) = 3 / 2 g(3)=f(2,3)=3/2
  11. f ( x , y ) = y x f(x,y)=\frac{y}{x}
  12. h ( x ) = y f ( x , y ) h(x)=y\mapsto f(x,y)
  13. f f
  14. y z \scriptstyle y\mapsto z
  15. g ( y ) = h ( 2 ) = y f ( 2 , y ) g(y)=h(2)=y\mapsto f(2,y)
  16. f \scriptstyle f
  17. f : ( X × Y ) Z \scriptstyle f\colon(X\times Y)\to Z
  18. curry ( f ) : X ( Y Z ) \scriptstyle\,\text{curry}(f)\colon X\to(Y\to Z)
  19. curry ( f ) \scriptstyle\,\text{curry}(f)
  20. X \scriptstyle X
  21. Y Z \scriptstyle Y\to Z
  22. X ( Y Z ) \scriptstyle X\to(Y\to Z)
  23. X Y Z \scriptstyle X\to Y\to Z
  24. f x , y \scriptstyle f\;\langle x,y\rangle
  25. curry ( f ) x y \scriptstyle\,\text{curry}(f)\;x\;y
  26. A B × C \scriptstyle A^{B\times C}
  27. B × C \scriptstyle B\times C
  28. A A
  29. ( A C ) B \scriptstyle\left(A^{C}\right)^{B}
  30. B \scriptstyle B
  31. C \scriptstyle C
  32. A \scriptstyle A
  33. f : ( X × Y ) Z \scriptstyle f\colon(X\times Y)\to Z
  34. g : X Z Y \scriptstyle g\colon X\to Z^{Y}
  35. hom ( A × B , C ) hom ( A , C B ) . \hom(A\times B,C)\cong\hom(A,C^{B}).
  36. ( A and B ) C A ( B C ) \scriptstyle(A\and B)\to C\Leftrightarrow A\to(B\to C)
  37. f : ( X × Y × Z ) N \scriptstyle f\colon(X\times Y\times Z)\to N
  38. curry ( f ) : X ( Y ( Z N ) ) \scriptstyle\,\text{curry}(f)\colon X\to(Y\to(Z\to N))
  39. f ( 1 , 2 , 3 ) \scriptstyle f(1,2,3)
  40. f curried ( 1 ) ( 2 ) ( 3 ) \scriptstyle f\text{curried}(1)(2)(3)
  41. f curried ( 1 ) \scriptstyle f\text{curried}(1)
  42. f \scriptstyle f
  43. partial ( f ) : ( Y × Z ) N \scriptstyle\,\text{partial}(f)\colon(Y\times Z)\to N
  44. f partial ( 2 , 3 ) \scriptstyle f\text{partial}(2,3)

Curvature.html

  1. κ = 1 R . \kappa=\frac{1}{R}.
  2. κ = d 𝐓 d s . \kappa=\left\|\frac{d\mathbf{T}}{ds}\right\|.
  3. d 𝐓 / d s d\mathbf{T}/ds
  4. κ \kappa
  5. γ 2 = x ( t ) 2 + y ( t ) 2 0 \|\gamma^{\prime}\|^{2}=x^{\prime}(t)^{2}+y^{\prime}(t)^{2}\not=0
  6. γ 2 = x ( s ) 2 + y ( s ) 2 = 1. \|\gamma^{\prime}\|^{2}=x^{\prime}(s)^{2}+y^{\prime}(s)^{2}=1.
  7. 𝐓 ( s ) = γ ( s ) , 𝐓 ( s ) = k ( s ) 𝐍 ( s ) , κ ( s ) = 𝐓 ( s ) = γ ′′ ( s ) = | k ( s ) | , R ( s ) = 1 κ ( s ) . \mathbf{T}(s)=\gamma^{\prime}(s),\quad\mathbf{T}^{\prime}(s)=k(s)\mathbf{N}(s)% ,\quad\kappa(s)=\|\mathbf{T}^{\prime}(s)\|=\|\gamma^{\prime\prime}(s)\|=\left|% k(s)\right|,\quad R(s)=\frac{1}{\kappa(s)}.
  8. κ = | x y ′′ - y x ′′ | ( x 2 + y 2 ) 3 / 2 , \kappa=\frac{|x^{\prime}y^{\prime\prime}-y^{\prime}x^{\prime\prime}|}{(x^{% \prime 2}+y^{\prime 2})^{3/2}},
  9. k = x y ′′ - y x ′′ ( x 2 + y 2 ) 3 / 2 . k=\frac{x^{\prime}y^{\prime\prime}-y^{\prime}x^{\prime\prime}}{(x^{\prime 2}+y% ^{\prime 2})^{3/2}}.
  10. k = m x 2 + y 2 , m = x y ′′ - y x ′′ ( x 2 + y 2 ) 1 / 2 = ( x ′′ , y ′′ ) ( - y , x ) ( - y , x ) = 𝐭 𝐭 ^ . k=\frac{m}{x^{\prime 2}+y^{\prime 2}},\ \ \ m=\frac{x^{\prime}y^{\prime\prime}% -y^{\prime}x^{\prime\prime}}{(x^{\prime 2}+y^{\prime 2})^{1/2}}=(x^{\prime% \prime},y^{\prime\prime})\cdot\frac{(-y^{\prime},x^{\prime})}{\|(-y^{\prime},x% ^{\prime})\|}=\mathbf{t}^{\prime}\cdot\mathbf{\hat{t}}_{\bot}.
  11. k = det ( γ , γ ′′ ) γ 3 , κ = | det ( γ , γ ′′ ) | γ 3 . k=\frac{\det(\gamma^{\prime},\gamma^{\prime\prime})}{\|\gamma^{\prime}\|^{3}},% \ \ \ \kappa=\frac{|\det(\gamma^{\prime},\gamma^{\prime\prime})|}{\|\gamma^{% \prime}\|^{3}}.
  12. y = f ( x ) y=f(x)
  13. κ = | y ′′ | ( 1 + y 2 ) 3 / 2 \kappa=\frac{|y^{\prime\prime}|}{(1+y^{\prime 2})^{3/2}}
  14. k = y ′′ ( 1 + y 2 ) 3 / 2 k=\frac{y^{\prime\prime}}{(1+y^{\prime 2})^{3/2}}
  15. κ | d 2 y d x 2 | \kappa\approx\left|\frac{d^{2}y}{dx^{2}}\right|
  16. r ( θ ) r(\theta)
  17. κ ( θ ) = | r 2 + 2 r 2 - r r ′′ | ( r 2 + r 2 ) 3 / 2 \kappa(\theta)=\frac{|r^{2}+2r^{\prime 2}-rr^{\prime\prime}|}{\left(r^{2}+r^{% \prime 2}\right)^{3/2}}
  18. θ \theta
  19. x = 1 , x ′′ = 0 , y = 2 t , y ′′ = 2. x^{\prime}=1,\quad x^{\prime\prime}=0,\quad y^{\prime}=2t,\quad y^{\prime% \prime}=2.
  20. κ ( t ) = | x y ′′ - y x ′′ ( x 2 + y 2 ) 3 / 2 | = 1 2 - ( 2 t ) ( 0 ) ( 1 + ( 2 t ) 2 ) 3 / 2 = 2 ( 1 + 4 t 2 ) 3 / 2 . \kappa(t)=\left|\frac{x^{\prime}y^{\prime\prime}-y^{\prime}x^{\prime\prime}}{(% {x^{\prime 2}+y^{\prime 2}})^{3/2}}\right|={1\cdot 2-(2t)(0)\over(1+(2t)^{2})^% {3/2}}={2\over(1+4t^{2})^{3/2}}.
  21. γ ( t ) = ( cos ( 3 t ) sin ( 2 t ) ) . \gamma(t)\,=\,\begin{pmatrix}\cos(3t)\\ \sin(2t)\end{pmatrix}\,.
  22. k ( t ) = 6 cos ( t ) ( 8 cos ( t ) 4 - 10 cos ( t ) 2 + 5 ) ( 232 cos ( t ) 4 - 97 cos ( t ) 2 + 13 - 144 cos ( t ) 6 ) 3 / 2 . k(t)=\frac{6\cos(t)(8\cos(t)^{4}-10\cos(t)^{2}+5)}{(232\cos(t)^{4}-97\cos(t)^{% 2}+13-144\cos(t)^{6})^{3/2}}\,.
  23. 𝐓 ( s ) = γ ( s ) \mathbf{T}(s)=\gamma^{\prime}(s)
  24. κ ( s ) = 𝐓 ( s ) = γ ′′ ( s ) . \kappa(s)=\|\mathbf{T}^{\prime}(s)\|=\|\gamma^{\prime\prime}(s)\|.
  25. 𝐍 ( s ) = 𝐓 ( s ) 𝐓 ( s ) . \mathbf{N}(s)=\frac{\mathbf{T}^{\prime}(s)}{\|\mathbf{T}^{\prime}(s)\|}.
  26. κ ( s ) = 1 R ( s ) . \kappa(s)=\frac{1}{R(s)}.
  27. κ = ( z ′′ y - y ′′ z ) 2 + ( x ′′ z - z ′′ x ) 2 + ( y ′′ x - x ′′ y ) 2 ( x 2 + y 2 + z 2 ) 3 / 2 , \kappa=\frac{\sqrt{(z^{\prime\prime}y^{\prime}-y^{\prime\prime}z^{\prime})^{2}% +(x^{\prime\prime}z^{\prime}-z^{\prime\prime}x^{\prime})^{2}+(y^{\prime\prime}% x^{\prime}-x^{\prime\prime}y^{\prime})^{2}}}{(x^{\prime 2}+y^{\prime 2}+z^{% \prime 2})^{3/2}},
  28. κ = | γ × γ ′′ | | γ | 3 \kappa=\frac{|\gamma^{\prime}\times\gamma^{\prime\prime}|}{|\gamma^{\prime}|^{% 3}}
  29. × \times
  30. κ = det ( ( γ , γ ′′ ) t ( γ , γ ′′ ) ) γ 3 = γ 2 γ ′′ 2 - ( γ γ ′′ ) 2 γ 3 . \kappa=\frac{\sqrt{\det\left((\gamma^{\prime},\gamma^{\prime\prime})^{t}(% \gamma^{\prime},\gamma^{\prime\prime})\right)}}{\|\gamma^{\prime}\|^{3}}=\frac% {\sqrt{\|\gamma^{\prime}\|^{2}\|\gamma^{\prime\prime}\|^{2}-(\gamma^{\prime}% \cdot\gamma^{\prime\prime})^{2}}}{\|\gamma^{\prime}\|^{3}}.
  31. κ ( P ) = lim Q P 24 ( s ( P , Q ) - d ( P , Q ) ) s ( P , Q ) 3 \kappa(P)=\lim_{Q\to P}\sqrt{\frac{24\left(s(P,Q)-d(P,Q)\right)}{s(P,Q)^{3}}}
  32. ( 𝐓 𝐭 𝐮 ) = ( 0 κ g κ n - κ g 0 τ r - κ n - τ r 0 ) ( 𝐓 𝐭 𝐮 ) \begin{pmatrix}\mathbf{T^{\prime}}\\ \mathbf{t^{\prime}}\\ \mathbf{u^{\prime}}\end{pmatrix}=\begin{pmatrix}0&\kappa_{g}&\kappa_{n}\\ -\kappa_{g}&0&\tau_{r}\\ -\kappa_{n}&-\tau_{r}&0\end{pmatrix}\begin{pmatrix}\mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{pmatrix}
  33. K = lim r 0 + 3 2 π r - C ( r ) π r 3 . K=\lim_{r\to 0^{+}}3\frac{2\pi r-C(r)}{\pi r^{3}}.
  34. I I ( X , X ) = N ( X X ) I\!I(X,X)=N\cdot(\nabla_{X}X)
  35. I I ( X , X ) = k 1 ( X u 1 ) 2 + k 2 ( X u 2 ) 2 . I\!I(X,X)=k_{1}(X\cdot u_{1})^{2}+k_{2}(X\cdot u_{2})^{2}.

Cutback_technique.html

  1. P ( x ) P(x)
  2. P ( y ) P(y)
  3. P P
  4. x x
  5. y y
  6. 2 m 2m

Cutoff_frequency.html

  1. 1 / 2 0.707 \scriptstyle\sqrt{1/2}\ \approx\ 0.707
  2. H ( s ) = 1 1 + α s H(s)=\frac{1}{1+\alpha s}
  3. s = - 1 / α s=-1/α
  4. j ω
  5. | H ( j ω ) | = | 1 1 + α j ω | = 1 1 + α 2 ω 2 \left|H(j\omega)\right|=\left|\frac{1}{1+\alpha j\omega}\right|=\sqrt{\frac{1}% {1+\alpha^{2}\omega^{2}}}
  6. | H ( j ω c ) | = 1 2 = 1 1 + α 2 ω c 2 \left|H(j\omega_{\mathrm{c}})\right|=\frac{1}{\sqrt{2}}=\sqrt{\frac{1}{1+% \alpha^{2}\omega_{\mathrm{c}}^{2}}}
  7. ω c = 1 α \omega_{\mathrm{c}}=\frac{1}{\alpha}
  8. s s
  9. ω ω
  10. j j
  11. ω c = c ( n π a ) 2 + ( m π b ) 2 , \omega_{c}=c\sqrt{\left(\frac{n\pi}{a}\right)^{2}+\left(\frac{m\pi}{b}\right)^% {2}},
  12. n , m 0 n,m\geq 0
  13. n , m 0 n,m\geq 0
  14. n = m = 0 n=m=0
  15. n , m 1 n,m\geq 1
  16. ω c = c χ 01 r = c 2.4048 r , \omega_{c}=c\frac{\chi_{01}}{r}=c\frac{2.4048}{r},
  17. r r
  18. χ 01 \chi_{01}
  19. J 0 ( r ) J_{0}(r)
  20. ω c = c χ 11 r = c 1.8412 r \omega_{c}=c\frac{\chi_{11}}{r}=c\frac{1.8412}{r}
  21. ( 2 - 1 c 2 2 t 2 ) ψ ( 𝐫 , t ) = 0 , \left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial{t}^{2}}\right)% \psi(\mathbf{r},t)=0,
  22. ψ ( x , y , z , t ) = ψ ( x , y , z ) e i ω t . \psi(x,y,z,t)=\psi(x,y,z)e^{i\omega t}.
  23. ( 2 + ω 2 c 2 ) ψ ( x , y , z ) = 0. (\nabla^{2}+\frac{\omega^{2}}{c^{2}})\psi(x,y,z)=0.
  24. ψ \psi
  25. ψ ( x , y , z , t ) = ψ ( x , y ) e i ( ω t - k z z ) , \psi(x,y,z,t)=\psi(x,y)e^{i\left(\omega t-k_{z}z\right)},
  26. k z k_{z}
  27. ( T 2 - k z 2 + ω 2 c 2 ) ψ ( x , y , z ) = 0 , (\nabla_{T}^{2}-k_{z}^{2}+\frac{\omega^{2}}{c^{2}})\psi(x,y,z)=0,
  28. ψ ( x , y , z , t ) = ψ 0 e i ( ω t - k z z - k x x - k y y ) . \psi(x,y,z,t)=\psi_{0}e^{i\left(\omega t-k_{z}z-k_{x}x-k_{y}y\right)}.
  29. ω 2 c 2 = k x 2 + k y 2 + k z 2 \frac{\omega^{2}}{c^{2}}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}
  30. k x = n π a , k_{x}=\frac{n\pi}{a},
  31. k y = m π b , k_{y}=\frac{m\pi}{b},
  32. ω 2 c 2 = ( n π a ) 2 + ( m π b ) 2 + k z 2 , \frac{\omega^{2}}{c^{2}}=\left(\frac{n\pi}{a}\right)^{2}+\left(\frac{m\pi}{b}% \right)^{2}+k_{z}^{2},
  33. ω c \omega_{c}
  34. k z k_{z}
  35. ω c = c ( n π a ) 2 + ( m π b ) 2 \omega_{c}=c\sqrt{\left(\frac{n\pi}{a}\right)^{2}+\left(\frac{m\pi}{b}\right)^% {2}}

Cyclic_group.html

  1. / m / n \mathbb{Z}/m{\mathbb{Z}}\otimes\mathbb{Z}/n{\mathbb{Z}}
  2. H o m ( / m , / n ) Hom(\mathbb{Z}/m{\mathbb{Z}},\mathbb{Z}/n{\mathbb{Z}})
  3. / g c d ( m , n ) \mathbb{Z}/gcd(m,n){\mathbb{Z}}
  4. R / I R R / J R / ( I + J ) R/I\otimes_{R}R/J\cong R/(I+J)
  5. / n \mathbb{Z}/n{\mathbb{Z}}

Cyclic_redundancy_check.html

  1. crc ( x y z ) = crc ( x ) crc ( y ) crc ( z ) \operatorname{crc}(x\oplus y\oplus z)=\operatorname{crc}(x)\oplus\operatorname% {crc}(y)\oplus\operatorname{crc}(z)
  2. 2 r - 1 2^{r}-1
  3. g ( x ) = p ( x ) ( 1 + x ) g(x)=p(x)(1+x)
  4. p ( x ) p(x)
  5. r - 1 r-1
  6. 2 r - 1 - 1 2^{r-1}-1
  7. g ( x ) g(x)
  8. x 0 x^{0}
  9. x 4 + x + 1 x^{4}+x+1
  10. x 4 + x^{4}+
  11. 0 x 3 + 0 x 2 + 1 x 1 + 1 x 0 0x^{3}+0x^{2}+1x^{1}+1x^{0}
  12. 1 x 0 + 1 x 1 + 0 x 2 + 0 x 3 1x^{0}+1x^{1}+0x^{2}+0x^{3}
  13. + x 4 +x^{4}
  14. 1 x 4 + 0 x 3 + 0 x 2 + 1 x 1 1x^{4}+0x^{3}+0x^{2}+1x^{1}
  15. + x 0 +x^{0}

Cycloid.html

  1. x = r ( t - sin t ) y = r ( 1 - cos t ) \begin{aligned}\displaystyle x&\displaystyle=r(t-\sin t)\\ \displaystyle y&\displaystyle=r(1-\cos t)\end{aligned}
  2. x = r cos - 1 ( 1 - y r ) - y ( 2 r - y ) . x=r\cos^{-1}\left(1-\frac{y}{r}\right)-\sqrt{y(2r-y)}.
  3. 0 t 2 π . 0\leq t\leq 2\pi.
  4. \infty
  5. - -\infty
  6. ( d y d x ) 2 = 2 r y - 1. \left(\frac{dy}{dx}\right)^{2}=\frac{2r}{y}-1.
  7. P 1 P_{1}
  8. P 2 P_{2}
  9. P 1 P_{1}
  10. P 2 P_{2}
  11. P 1 P_{1}
  12. P 2 P_{2}
  13. P 1 O 1 Q ^ = P 2 O 2 Q ^ \widehat{P1O1Q}=\widehat{P2O2Q}
  14. O 1 Q P 1 ^ = O 2 Q P 2 ^ \widehat{O1QP1}=\widehat{O2QP2}
  15. O 1 Q P 2 ^ + O 2 Q P 2 ^ = π \widehat{O1QP2}+\widehat{O2QP2}=\pi
  16. P 1 Q P 2 ^ = π \widehat{P1QP2}=\pi
  17. Q P 2 A ^ = 1 2 P 2 O 2 Q ^ \widehat{QP2A}=\frac{1}{2}\widehat{P2O2Q}
  18. Q P 1 A ^ = 1 2 Q O 1 R ^ = \widehat{QP1A}=\frac{1}{2}\widehat{QO1R}=
  19. = 1 2 Q O 1 P 1 ^ =\frac{1}{2}\widehat{QO1P1}
  20. P 1 O 1 Q ^ \widehat{P1O1Q}
  21. Q O 2 P 2 ^ \widehat{QO2P2}
  22. Q P 1 A ^ = Q P 2 A ^ \widehat{QP1A}=\widehat{QP2A}
  23. x = r ( t - sin t ) y = r ( 1 - cos t ) \begin{aligned}\displaystyle x&\displaystyle=r(t-\sin t)\\ \displaystyle y&\displaystyle=r(1-\cos t)\end{aligned}
  24. d x d t = r ( 1 - cos t ) \frac{dx}{dt}=r(1-\cos t)
  25. 0 t 2 π . 0\leq t\leq 2\pi.
  26. d x d t = r ( 1 - cos t ) \frac{dx}{dt}=r(1-\cos t)
  27. A = t = 0 t = 2 π y d x = t = 0 t = 2 π r 2 ( 1 - cos t ) 2 d t = r 2 ( 3 2 t - 2 sin t + 1 2 cos t sin t ) | t = 0 t = 2 π = 3 π r 2 . \begin{aligned}\displaystyle A&\displaystyle=\int_{t=0}^{t=2\pi}y\,dx=\int_{t=% 0}^{t=2\pi}r^{2}(1-\cos t)^{2}dt\\ &\displaystyle=\left.r^{2}\left(\frac{3}{2}t-2\sin t+\frac{1}{2}\cos t\sin t% \right)\right|_{t=0}^{t=2\pi}\\ &\displaystyle=3\pi r^{2}.\end{aligned}
  28. S = 0 2 π [ ( d y d t ) 2 + ( d x d t ) 2 ] 1 2 d t = 0 2 π r 2 - 2 cos ( t ) d t = 0 2 π 2 r sin t 2 d t = 8 r . \begin{aligned}\displaystyle S&\displaystyle=\int_{0}^{2\pi}\left[\left(\frac{% \operatorname{d}\!y}{\operatorname{d}\!t}\right)^{2}+\left(\frac{\operatorname% {d}\!x}{\operatorname{d}\!t}\right)^{2}\right]^{\frac{1}{2}}\operatorname{d}\!% t\\ &\displaystyle=\int_{0}^{2\pi}r\sqrt{2-2\cos(t)}\,\operatorname{d}\!t\\ &\displaystyle=\int_{0}^{2\pi}2\,r\sin\frac{t}{2}\,\operatorname{d}\!t\\ &\displaystyle=8\,r.\end{aligned}
  29. x \displaystyle x

Cyclotron.html

  1. f = q B 2 π m f=\frac{qB}{2\pi m}
  2. F C \scriptstyle F_{C}
  3. F C = m v 2 r F_{C}={mv^{2}\over r}\;
  4. m \scriptstyle m
  5. v \scriptstyle v
  6. r \scriptstyle r
  7. F B \scriptstyle F_{B}
  8. B \scriptstyle B
  9. F B = q v B F_{B}=qvB\;
  10. q \scriptstyle q
  11. r = R \scriptstyle r\;=\;R
  12. m v 2 R = q v B {mv^{2}\over R}=qvB\;
  13. E = 1 2 m v 2 = q 2 B 2 R 2 2 m E={1\over 2}mv^{2}=\frac{q^{2}B^{2}R^{2}}{2m}\;
  14. B \scriptstyle B
  15. R \scriptstyle R
  16. m = m 0 1 - ( v c ) 2 = m 0 1 - β 2 = γ m 0 m=\frac{m_{0}}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}=\frac{m_{0}}{\sqrt{1-% \beta^{2}}}=\gamma{m_{0}}
  17. m 0 m_{0}
  18. β = v c \beta=\frac{v}{c}
  19. γ = 1 1 - β 2 = 1 1 - ( v c ) 2 \gamma=\frac{1}{\sqrt{1-\beta^{2}}}=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^% {2}}}
  20. f = q B 2 π γ m 0 = f 0 γ = f 0 1 - β 2 = f 0 1 - ( v c ) 2 f=\frac{qB}{2\pi\gamma m_{0}}=\frac{f_{0}}{\gamma}={f_{0}}{\sqrt{1-\beta^{2}}}% ={f_{0}}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
  21. ω = 2 π f = q B γ m 0 = ω 0 γ = ω 0 1 - β 2 = ω 0 1 - ( v c ) 2 \omega={2\pi f}=\frac{qB}{\gamma m_{0}}=\frac{\omega_{0}}{\gamma}={\omega_{0}}% {\sqrt{1-\beta^{2}}}={\omega_{0}}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
  22. f 0 f_{0}
  23. ω 0 \omega_{0}
  24. r = v ω = β c ω = γ β m 0 c q B r=\frac{v}{\omega}=\frac{\beta c}{\omega}=\frac{\gamma\beta m_{0}c}{qB}
  25. ω r = v = β c \omega r=v=\beta c
  26. f = f 0 γ = f 0 1 - β 2 f=\frac{f_{0}}{\gamma}={f_{0}}{\sqrt{1-\beta^{2}}}
  27. f 0 f_{0}
  28. β = v c \beta=\frac{v}{c}
  29. r = γ m 0 v q B r=\frac{\gamma m_{0}v}{qB}
  30. f = f 0 γ f=\frac{f_{0}}{\gamma}
  31. B B
  32. B = γ B 0 B=\gamma B_{0}
  33. r = m 0 v q B 0 r=\frac{m_{0}v}{qB_{0}}
  34. v v

CYK_algorithm.html

  1. Θ ( n 3 | G | ) \Theta(n^{3}\cdot|G|)
  2. A α A\rightarrow\alpha
  3. A B C A\rightarrow BC
  4. P [ i , j , k ] P[i,j,k]
  5. i i
  6. j j
  7. R k R_{k}
  8. P Q R P\to Q\;R
  9. Q Q
  10. R R
  11. P P
  12. S 𝑁𝑃 𝑉𝑃 𝑉𝑃 𝑉𝑃 𝑃𝑃 𝑉𝑃 V 𝑁𝑃 𝑉𝑃 𝑒𝑎𝑡𝑠 𝑃𝑃 P 𝑁𝑃 𝑁𝑃 𝐷𝑒𝑡 N 𝑁𝑃 𝑠ℎ𝑒 V 𝑒𝑎𝑡𝑠 P 𝑤𝑖𝑡ℎ N 𝑓𝑖𝑠ℎ N 𝑓𝑜𝑟𝑘 𝐷𝑒𝑡 𝑎 \begin{array}[]{lcl}\mathit{S}&\to&\mathit{NP}\;\mathit{VP}\\ \mathit{VP}&\to&\mathit{VP}\;\mathit{PP}\\ \mathit{VP}&\to&\mathit{V}\;\mathit{NP}\\ \mathit{VP}&\to&\,\textit{eats}\\ \mathit{PP}&\to&\mathit{P}\;\mathit{NP}\\ \mathit{NP}&\to&\mathit{Det}\;\mathit{N}\\ \mathit{NP}&\to&\,\textit{she}\\ \mathit{V}&\to&\,\textit{eats}\\ \mathit{P}&\to&\,\textit{with}\\ \mathit{N}&\to&\,\textit{fish}\\ \mathit{N}&\to&\,\textit{fork}\\ \mathit{Det}&\to&\,\textit{a}\end{array}
  13. P [ i , j , k ] P[i,j,k]
  14. i i
  15. j j
  16. g g
  17. g 2 g^{2}
  18. 2 2 g 2^{2g}
  19. Θ ( n 3 | G | ) \Theta(n^{3}\cdot|G|)
  20. O ( n 2.38 | G | ) O(n^{2.38}\cdot|G|)
  21. O ( n 3 - ε | G | ) O(n^{3-\varepsilon}\cdot|G|)
  22. ( n × n ) (n\times n)
  23. O ( n 3 - ε / 3 ) O(n^{3-\varepsilon/3})

D.html

  1. \partial

Dalton's_law.html

  1. P total = i = 1 n p i P_{\,\text{total}}=\sum_{i=1}^{n}{p_{i}}
  2. P total = p 1 + + p n P_{\,\text{total}}=p_{1}+\cdots+p_{n}
  3. p 1 , p 2 , , p n p_{1},\ p_{2},\dots,\ p_{n}
  4. p i = P total y i \ p_{i}=P_{\,\text{total}}y_{i}
  5. y i y_{i}
  6. P i = P total C i 1 , 000 , 000 P_{i}=\frac{P_{\,\text{total}}C_{i}}{1,000,000}
  7. C i C_{i}

Daniel_Bernoulli.html

  1. 1 2 ρ u 2 + P = constant \tfrac{1}{2}\rho u^{2}+P=\,\text{constant}

Dark_matter.html

  1. w w

Data_compression_ratio.html

  1. Compression Ratio = Uncompressed Size Compressed Size {\rm Compression\;Ratio}=\frac{\rm Uncompressed\;Size}{\rm Compressed\;Size}
  2. Space Savings = 1 - Compressed Size Uncompressed Size {\rm Space\;Savings}=1-\frac{\rm Compressed\;Size}{\rm Uncompressed\;Size}
  3. Compression Ratio = Uncompressed Data Rate Compressed Data Rate {\rm Compression\;Ratio}=\frac{\rm Uncompressed\;Data\;Rate}{\rm Compressed\;% Data\;Rate}
  4. Data Rate Savings = 1 - Compressed Data Rate Uncompressed Data Rate {\rm Data\;Rate\;Savings}=1-\frac{\rm Compressed\;Data\;Rate}{\rm Uncompressed% \;Data\;Rate}

Data_Encryption_Standard.html

  1. E K ( P ) = C E K ¯ ( P ¯ ) = C ¯ E_{K}(P)=C\Leftrightarrow E_{\overline{K}}(\overline{P})=\overline{C}
  2. x ¯ \overline{x}
  3. x . x.
  4. E K E_{K}
  5. K . K.
  6. P P
  7. C C
  8. E K ( E K ( P ) ) = P E_{K}(E_{K}(P))=P
  9. E K = D K . E_{K}=D_{K}.
  10. K 1 K_{1}
  11. K 2 K_{2}
  12. E K 1 ( E K 2 ( P ) ) = P E_{K_{1}}(E_{K_{2}}(P))=P
  13. E K 2 = D K 1 . E_{K_{2}}=D_{K_{1}}.
  14. { E K } \{E_{K}\}
  15. K K

Data_signaling_rate.html

  1. i = 1 m log 2 n i T i \sum_{i=1}^{m}\frac{\log_{2}{n_{i}}}{T_{i}}

Data_stream.html

  1. ( s , Δ ) (s,\Delta)
  2. s s
  3. Δ \Delta

Data_warehouse.html

  1. t c h _ r e q _ s u c c e s s _ c i t y = t c h _ r e q _ s u c c e s s _ b t s 1 + t c h _ r e q _ s u c c e s s _ b t s 2 + t c h _ r e q _ s u c c e s s _ b t s 3 tch\_req\_success\_city=tch\_req\_success\_bts1+tch\_req\_success\_bts2+tch\_% req\_success\_bts3
  2. a v g _ t c h _ r e q _ s u c c e s s _ c i t y = ( t c h _ r e q _ s u c c e s s _ b t s 1 + t c h _ r e q _ s u c c e s s _ b t s 2 + t c h _ r e q _ s u c c e s s _ b t s 3 ) / 3 avg\_tch\_req\_success\_city=(tch\_req\_success\_bts1+tch\_req\_success\_bts2+% tch\_req\_success\_bts3)/3

DBm.html

  1. x \displaystyle x
  2. x = 30 + 10 log 10 P 1 W \begin{aligned}\displaystyle x&\displaystyle=30+10\log_{10}\frac{P}{1\mathrm{W% }}\end{aligned}
  3. P = 1 mW 10 x 10 P = 1 W 10 x - 30 10 \begin{aligned}\displaystyle P&\displaystyle=1\,\text{mW}\cdot 10^{\frac{x}{10% }}\\ \displaystyle P&\displaystyle=1\,\text{W}\cdot 10^{\frac{x-30}{10}}\end{aligned}

De_Broglie–Bohm_theory.html

  1. q q
  2. q k q^{k}
  3. Q Q
  4. 𝐐 k \mathbf{Q}_{k}
  5. N N
  6. ϕ ( x ) \phi(x)
  7. m k d q k d t ( t ) = k Im ln ψ ( q , t ) = Im ( k ψ ψ ) ( q , t ) = m k j k ψ * ψ = Re ( P ^ k Ψ Ψ ) m_{k}\frac{dq^{k}}{dt}(t)=\hbar\nabla_{k}\operatorname{Im}\ln\psi(q,t)=\hbar% \operatorname{Im}\left(\frac{\nabla_{k}\psi}{\psi}\right)(q,t)=\frac{m_{k}{j}_% {k}}{\psi^{*}\psi}=\mathrm{Re}\left(\frac{{\hat{P}}_{k}\Psi}{\Psi}\right)
  8. j {j}
  9. P ^ {\hat{P}}
  10. ψ ( q , t ) \psi(q,t)
  11. i t ψ ( q , t ) = - i = 1 N 2 2 m i i 2 ψ ( q , t ) + V ( q ) ψ ( q , t ) i\hbar\frac{\partial}{\partial t}\psi(q,t)=-\sum_{i=1}^{N}\frac{\hbar^{2}}{2m_% {i}}\nabla_{i}^{2}\psi(q,t)+V(q)\psi(q,t)
  12. H = 1 2 m i p ^ i 2 + V ( q ^ ) H=\sum\frac{1}{2m_{i}}\hat{p}_{i}^{2}+V(\hat{q})
  13. | ψ ( q , t ) | 2 |\psi(q,t)|^{2}
  14. t t
  15. q ( t ) Q q(t)\in Q
  16. ψ ( q , t ) \psi(q,t)\in\mathbb{C}
  17. Q Q
  18. q ( t ) Q q(t)\in Q
  19. ψ ( q , t ) \psi(q,t)\in\mathbb{C}
  20. q ( t ) Q q(t)\in Q
  21. 3 \mathbb{R}^{3}
  22. N N
  23. 3 \mathbb{R}^{3}
  24. 3 N \mathbb{R}^{3N}
  25. 𝐐 \mathbf{Q}
  26. ψ \psi
  27. 3 \mathbb{R}^{3}
  28. d 𝐐 d t ( t ) = m Im ( ψ ψ ) ( 𝐐 , t ) \frac{d\mathbf{Q}}{dt}(t)=\frac{\hbar}{m}\operatorname{Im}\left(\frac{\nabla% \psi}{\psi}\right)(\mathbf{Q},t)
  29. 𝐐 k \mathbf{Q}_{k}
  30. k k
  31. d 𝐐 k d t ( t ) = m k Im ( k ψ ψ ) ( 𝐐 1 , 𝐐 2 , , 𝐐 N , t ) \frac{d\mathbf{Q}_{k}}{dt}(t)=\frac{\hbar}{m_{k}}\operatorname{Im}\left(\frac{% \nabla_{k}\psi}{\psi}\right)(\mathbf{Q}_{1},\mathbf{Q}_{2},\ldots,\mathbf{Q}_{% N},t)
  32. N N
  33. 3 \mathbb{R}^{3}
  34. V V
  35. 3 \mathbb{R}^{3}
  36. i t ψ = - 2 2 m 2 ψ + V ψ i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi
  37. ψ \psi
  38. V V
  39. 3 N \mathbb{R}^{3N}
  40. i t ψ = - k = 1 N 2 2 m k k 2 ψ + V ψ i\hbar\frac{\partial}{\partial t}\psi=-\sum_{k=1}^{N}\frac{\hbar^{2}}{2m_{k}}% \nabla_{k}^{2}\psi+V\psi
  41. | ψ | 2 |\psi|^{2}
  42. | ψ | 2 |\psi|^{2}
  43. | ψ | 2 |\psi|^{2}
  44. | ψ | 2 |\psi|^{2}
  45. | ψ | 2 |\psi|^{2}
  46. ψ ( t , q I , q II ) \psi(t,q^{\mathrm{I}},q^{\mathrm{II}})
  47. q I q^{\mathrm{I}}
  48. q II q^{\mathrm{II}}
  49. Q I ( t ) Q^{\mathrm{I}}(t)
  50. Q II ( t ) Q^{\mathrm{II}}(t)
  51. ψ I ( t , q I ) = ψ ( t , q I , Q II ( t ) ) . \psi^{\mathrm{I}}(t,q^{\mathrm{I}})=\psi(t,q^{\mathrm{I}},Q^{\mathrm{II}}(t)).\,
  52. Q ( t ) = ( Q I ( t ) , Q II ( t ) ) Q(t)=(Q^{\mathrm{I}}(t),Q^{\mathrm{II}}(t))
  53. Q I ( t ) Q^{\mathrm{I}}(t)
  54. ψ \psi
  55. ψ I \psi^{\mathrm{I}}
  56. Q ( t ) Q(t)
  57. ψ ( t , ) \psi(t,\cdot)
  58. Q I ( t ) Q^{\mathrm{I}}(t)
  59. Q II ( t ) Q^{\mathrm{II}}(t)
  60. ψ I ( t , ) \psi^{\mathrm{I}}(t,\cdot)
  61. ψ ( t , q I , q II ) = ψ I ( t , q I ) ψ II ( t , q II ) \psi(t,q^{\mathrm{I}},q^{\mathrm{II}})=\psi^{\mathrm{I}}(t,q^{\mathrm{I}})\psi% ^{\mathrm{II}}(t,q^{\mathrm{II}})\,
  62. ψ I \psi^{\mathrm{I}}
  63. ψ I \psi^{\mathrm{I}}
  64. ψ \psi
  65. ψ ( t , q I , q II ) = ψ I ( t , q I ) ψ II ( t , q II ) + ϕ ( t , q I , q II ) , \psi(t,q^{\mathrm{I}},q^{\mathrm{II}})=\psi^{\mathrm{I}}(t,q^{\mathrm{I}})\psi% ^{\mathrm{II}}(t,q^{\mathrm{II}})+\phi(t,q^{\mathrm{I}},q^{\mathrm{II}}),\,
  66. ϕ \phi
  67. ϕ ( t , q I , Q II ( t ) ) = 0 \phi(t,q^{\mathrm{I}},Q^{\mathrm{II}}(t))=0
  68. t t
  69. q I q^{\mathrm{I}}
  70. ψ I \psi^{\mathrm{I}}
  71. ψ I \psi^{\mathrm{I}}
  72. | ψ | 2 |\psi|^{2}
  73. 2 \mathbb{C}^{2}
  74. d 𝐐 k d t ( t ) = m k I m ( ( ψ , D k ψ ) ( ψ , ψ ) ) ( 𝐐 1 , 𝐐 2 , , 𝐐 N , t ) i t ψ = ( - k = 1 N 2 2 m k D k 2 + V - k = 1 N μ k 𝐒 k / S k 𝐁 ( 𝐪 k ) ) ψ \begin{aligned}\displaystyle\frac{d\mathbf{Q}_{k}}{dt}(t)&\displaystyle=\frac{% \hbar}{m_{k}}Im\left(\frac{(\psi,D_{k}\psi)}{(\psi,\psi)}\right)(\mathbf{Q}_{1% },\mathbf{Q}_{2},\ldots,\mathbf{Q}_{N},t)\\ \displaystyle i\hbar\frac{\partial}{\partial t}\psi&\displaystyle=\left(-\sum_% {k=1}^{N}\frac{\hbar^{2}}{2m_{k}}D_{k}^{2}+V-\sum_{k=1}^{N}\mu_{k}\mathbf{S}_{% k}/{S}_{k}\cdot\mathbf{B}(\mathbf{q}_{k})\right)\psi\end{aligned}
  75. μ k \mu_{k}
  76. k k
  77. 𝐒 k \mathbf{S}_{k}
  78. k k
  79. S k {S}_{k}
  80. S k = 1 / 2 {S}_{k}=1/2
  81. D k = k - i e k c 𝐀 ( 𝐪 k ) D_{k}=\nabla_{k}-\frac{ie_{k}}{c\hbar}\mathbf{A}(\mathbf{q}_{k})
  82. 𝐁 \mathbf{B}
  83. 𝐀 \mathbf{A}
  84. 3 \mathbb{R}^{3}
  85. e k e_{k}
  86. k k
  87. ( , ) (\cdot,\cdot)
  88. d \mathbb{C}^{d}
  89. ( ϕ , ψ ) = s = 1 d ϕ s * ψ s . (\phi,\psi)=\sum_{s=1}^{d}\phi_{s}^{*}\psi_{s}.
  90. ψ : 9 × 2 2 3 \psi:\mathbb{R}^{9}\times\mathbb{R}\to\mathbb{C}^{2}\otimes\mathbb{C}^{2}% \otimes\mathbb{C}^{3}
  91. | ψ | 2 |\psi|^{2}
  92. | ψ | 2 |\psi|^{2}
  93. | ψ | 2 |\psi|^{2}
  94. Δ x \Delta x
  95. Δ p \Delta p
  96. Δ x Δ p h . \Delta x\Delta p\gtrsim h.
  97. R - 1 2 R . R^{-1}\nabla^{2}R\rightarrow\infty.
  98. E = ω E=\hbar\omega\;
  99. 𝐩 = 𝐤 \mathbf{p}=\hbar\mathbf{k}\;
  100. ψ ( 𝐱 , t ) = A e i ( 𝐤 𝐱 - ω t ) \psi(\mathbf{x},t)=Ae^{i(\mathbf{k}\cdot\mathbf{x}-\omega t)}
  101. i 𝐤 = ψ / ψ i\mathbf{k}=\nabla\psi/\psi
  102. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  103. 𝐯 = m I m ( ψ ψ ) \mathbf{v}=\frac{\hbar}{m}Im\left(\frac{\nabla\psi}{\psi}\right)
  104. - ρ t = ( ρ v ψ ) -\frac{\partial\rho}{\partial t}=\nabla\cdot(\rho v^{\psi})
  105. ρ = | ψ | 2 \rho=|\psi|^{2}
  106. ψ ( 𝐱 , t ) = R ( 𝐱 , t ) e i S ( 𝐱 , t ) / . \psi(\mathbf{x},t)=R(\mathbf{x},t)e^{iS(\mathbf{x},t)/\hbar}.
  107. R 2 ( 𝐱 , t ) R^{2}(\mathbf{x},t)
  108. ρ ( 𝐱 , t ) = | ψ ( 𝐱 , t ) | 2 \rho(\mathbf{x},t)=|\psi(\mathbf{x},t)|^{2}
  109. - ρ ( 𝐱 , t ) t = ( ρ ( 𝐱 , t ) S ( 𝐱 , t ) m ) -\frac{\partial\rho(\mathbf{x},t)}{\partial t}=\nabla\cdot\left(\rho(\mathbf{x% },t)\frac{\nabla S(\mathbf{x},t)}{m}\right)
  110. S ( 𝐱 , t ) t = - [ V + 1 2 m ( S ( 𝐱 , t ) ) 2 - 2 2 m 2 R ( 𝐱 , t ) R ( 𝐱 , t ) ] . \frac{\partial S(\mathbf{x},t)}{\partial t}=-\left[V+\frac{1}{2m}(\nabla S(% \mathbf{x},t))^{2}-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R(\mathbf{x},t)}{R(% \mathbf{x},t)}\right].
  111. V - 2 2 m 2 R R V-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R}{R}
  112. S m . \frac{\nabla S}{m}.
  113. V V
  114. R R
  115. S m \frac{\nabla S}{m}
  116. H H
  117. f f
  118. f ^ \hat{f}
  119. ( v ( f ) ) ( q ) = Re ( ψ , i [ H , f ^ ] ψ ) ( ψ , ψ ) ( q ) (v(f))(q)=\mathrm{Re}\frac{(\psi,\frac{i}{\hbar}[H,\hat{f}]\psi)}{(\psi,\psi)}% (q)
  120. ( v , w ) (v,w)
  121. ψ \psi
  122. 𝐱 ˙ ( t ) = [ S ( 𝐱 ( t ) , t ) ) ] / m \mathbf{\dot{x}}(t)=[\nabla S(\mathbf{x}(t),t))]/m
  123. 𝐱 ( t = 0 ) \mathbf{x}(t=0)
  124. S S
  125. ψ \psi
  126. ρ ( 𝐱 ( t ) ) \rho(\mathbf{x}(t))
  127. d 3 x d^{3}x
  128. | ψ ( 𝐱 ( t ) ) | 2 |\psi(\mathbf{x}(t))|^{2}

De_Bruijn–Newman_constant.html

  1. × 10 6 \times 10^{−}6
  2. × 10 9 \times 10^{−}9
  3. × 10 9 \times 10^{−}9
  4. × 10 12 \times 10^{−}12
  5. H ( λ , z ) H(\lambda,z)
  6. F ( e λ x Φ ) F(e^{\lambda x}\Phi)
  7. ξ ( 1 / 2 + i z ) = A π ( λ ) - 1 - e - 1 4 λ ( x - z ) 2 H ( λ , x ) d x \xi(1/2+iz)=A\sqrt{\pi}(\lambda)^{-1}\int_{-\infty}^{\infty}e^{\frac{-1}{4% \lambda}(x-z)^{2}}H(\lambda,x)\,dx
  8. H ( 0 , x ) = ξ ( 1 / 2 + i x ) H(0,x)=\xi(1/2+ix)
  9. H ( z , λ ) = B π ( λ ) - 1 - e - 1 4 λ ( x - z ) 2 ξ ( 1 / 2 + i x ) d x H(z,\lambda)=B\sqrt{\pi}(\lambda)^{-1}\int_{-\infty}^{\infty}e^{\frac{-1}{4% \lambda}(x-z)^{2}}\xi(1/2+ix)\,dx

De_Moivre's_formula.html

  1. ( cos x + i sin x ) n = cos ( n x ) + i sin ( n x ) , (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx),
  2. e i x = cos x + i sin x e^{ix}=\cos x+i\sin x\,
  3. ( e i x ) n = e i n x . \left(e^{ix}\right)^{n}=e^{inx}.
  4. e i ( n x ) = cos ( n x ) + i sin ( n x ) . e^{i(nx)}=\cos(nx)+i\sin(nx).\,
  5. ( cos x + i sin x ) 2 = cos 2 x + 2 i sin x cos x - sin 2 x = ( cos 2 x - sin 2 x ) + i ( 2 sin x cos x ) = cos ( 2 x ) + i sin ( 2 x ) \begin{aligned}\displaystyle(\cos x+i\sin x)^{2}&\displaystyle=\cos^{2}x+2i% \sin x\cos x-\sin^{2}x=(\cos^{2}x-\sin^{2}x)+i(2\sin x\cos x)\\ &\displaystyle=\cos(2x)+i\sin(2x)\end{aligned}
  6. cos 2 x - sin 2 x = cos ( 2 x ) \cos^{2}x-\sin^{2}x=\cos(2x)
  7. 2 sin x cos x = sin ( 2 x ) . 2\sin x\cos x=\sin(2x).
  8. ( cos x + i sin x ) n = cos ( n x ) + i sin ( n x ) . (\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).
  9. ( cos x + i sin x ) k = cos ( k x ) + i sin ( k x ) . \left(\cos x+i\sin x\right)^{k}=\cos\left(kx\right)+i\sin\left(kx\right).\,
  10. ( cos x + i sin x ) k + 1 \displaystyle\left(\cos x+i\sin x\right)^{k+1}
  11. ( cos x + i sin x ) - n = [ ( cos x + i sin x ) n ] - 1 = [ cos ( n x ) + i sin ( n x ) ] - 1 = cos ( - n x ) + i sin ( - n x ) . ( * ) \begin{aligned}\displaystyle\left(\cos x+i\sin x\right)^{-n}&\displaystyle=% \left[\left(\cos x+i\sin x\right)^{n}\right]^{-1}\\ &\displaystyle=\left[\cos(nx)+i\sin(nx)\right]^{-1}\\ &\displaystyle=\cos(-nx)+i\sin(-nx).\qquad(*)\\ \end{aligned}
  12. z - 1 = z ¯ | z | 2 z^{-1}=\frac{\bar{z}}{|z|^{2}}
  13. sin n x = k = 0 n ( n k ) cos k x sin n - k x sin ( 1 2 ( n - k ) π ) \sin nx=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\cos^{k}x\,\sin^{n-k}x\,\sin% \left(\frac{1}{2}(n-k)\pi\right)
  14. cos n x = k = 0 n ( n k ) cos k x sin n - k x cos ( 1 2 ( n - k ) π ) . \cos nx=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\cos^{k}x\,\sin^{n-k}x\,\cos% \left(\frac{1}{2}(n-k)\pi\right).
  15. cos ( 2 x ) \displaystyle\cos(2x)
  16. π \pi
  17. ( cos z + i sin z ) w \left(\cos z+i\sin z\right)^{w}
  18. cos ( w z ) + i sin ( w z ) \cos(wz)+i\sin(wz)\,
  19. cos ( w z ) + i sin ( w z ) \cos(wz)+i\sin(wz)
  20. ( cos z + i sin z ) w . \left(\cos z+i\sin z\right)^{w}.\,
  21. z = r ( cos x + i sin x ) , z=r\left(\cos x+i\sin x\right),\,
  22. r 1 / n [ cos ( x + k 2 π n ) + i sin ( x + k 2 π n ) ] r^{1/n}\left[\cos\left(\frac{x+k\cdot 2\pi}{n}\right)+i\sin\left(\frac{x+k% \cdot 2\pi}{n}\right)\right]
  23. cosh x + sinh x = e x \cosh x+\sinh x=e^{x}
  24. n n\in\mathbb{Z}
  25. ( cosh x + sinh x ) n = cosh n x + sinh n x (\cosh x+\sinh x)^{n}=\cosh nx+\sinh nx
  26. n n\in\mathbb{Q}
  27. ( cosh x + sinh x ) n (\cosh x+\sinh x)^{n}
  28. cosh n x + sinh n x \cosh nx+\sinh nx
  29. d + a 𝐢 ^ + b 𝐣 ^ + c 𝐤 ^ d+a\mathbf{\hat{i}}+b\mathbf{\hat{j}}+c\mathbf{\hat{k}}
  30. q = k ( cos θ + ϵ sin θ ) q=k(\cos\theta+\epsilon\sin\theta)
  31. 0 θ < 2 π 0\leq\theta<2\pi
  32. k = d 2 + a 2 + b 2 + c 2 k=\sqrt{d^{2}+a^{2}+b^{2}+c^{2}}
  33. cos θ = d k \cos\theta=\frac{d}{k}
  34. sin θ = ± a 2 + b 2 + c 2 k \sin\theta=\pm\frac{\sqrt{a^{2}+b^{2}+c^{2}}}{k}
  35. a 2 + b 2 + c 2 0 a^{2}+b^{2}+c^{2}\neq 0
  36. ϵ = ± a 𝐢 ^ + b 𝐣 ^ + c 𝐤 ^ a 2 + b 2 + c 2 \epsilon=\pm\frac{a\mathbf{\hat{i}}+b\mathbf{\hat{j}}+c\mathbf{\hat{k}}}{\sqrt% {a^{2}+b^{2}+c^{2}}}
  37. q n = k n ( cos n θ + ϵ sin n θ ) q^{n}=k^{n}(\cos n\theta+\epsilon\sin n\theta)
  38. Q = 1 + 𝐢 ^ + 𝐣 ^ + 𝐤 ^ Q=1+\mathbf{\hat{i}}+\mathbf{\hat{j}}+\mathbf{\hat{k}}
  39. Q = 2 ( cos 60 + ϵ sin 60 ) where ϵ = 𝐢 ^ + 𝐣 ^ + 𝐤 ^ 3 Q=2(\cos 60^{\circ}+\epsilon\sin 60^{\circ})\qquad\mathrm{where}\;\epsilon=% \frac{\mathbf{\hat{i}}+\mathbf{\hat{j}}+\mathbf{\hat{k}}}{\sqrt{3}}
  40. q q
  41. q = 2 3 ( cos θ + ϵ sin θ ) θ = 20 , 140 , 260 q=\sqrt[3]{2}(\cos\theta+\epsilon\sin\theta)\qquad\theta=20^{\circ},140^{\circ% },260^{\circ}

De_Morgan's_laws.html

  1. ¬ ( P Q ) ( ¬ P ) ( ¬ Q ) \neg(P\land Q)\iff(\neg P)\lor(\neg Q)
  2. ¬ ( P Q ) ( ¬ P ) ( ¬ Q ) \neg(P\lor Q)\iff(\neg P)\land(\neg Q)
  3. \land
  4. \lor
  5. ¬ ( P and Q ) ( ¬ P ¬ Q ) \neg(P\and Q)\vdash(\neg P\neg Q)
  6. ¬ ( P Q ) ( ¬ P and ¬ Q ) \neg(PQ)\vdash(\neg P\and\neg Q)
  7. ¬ ( P and Q ) ¬ P ¬ Q \frac{\neg(P\and Q)}{\therefore\neg P\neg Q}
  8. ¬ ( P Q ) ¬ P and ¬ Q \frac{\neg(PQ)}{\therefore\neg P\and\neg Q}
  9. ¬ ( P and Q ) \displaystyle\neg(P\and Q)
  10. P P
  11. Q Q
  12. ( P and Q ) \displaystyle(P\and Q)
  13. A B ¯ \displaystyle\overline{A\cup B}
  14. A ¯ \overline{A}
  15. i I A i ¯ \displaystyle\overline{\bigcap_{i\in I}A_{i}}
  16. A B ¯ A ¯ + B ¯ \overline{A\cdot B}\equiv\overline{A}+\overline{B}
  17. A + B ¯ A ¯ B ¯ , \overline{A+B}\equiv\overline{A}\cdot\overline{B},
  18. \cdot
  19. + +
  20. o v e r b a r ¯ \overline{overbar}
  21. ¬ ( A B ) \neg(A\lor B)
  22. ( ¬ A ) ( ¬ B ) (\neg A)\wedge(\neg B)
  23. ¬ ( A B ) \neg(A\land B)
  24. ( ¬ A ) ( ¬ B ) (\neg A)\lor(\neg B)
  25. ( A B ) c = A c B c (A\cap B)^{c}=A^{c}\cup B^{c}
  26. ( A B ) c A c B c (A\cap B)^{c}\subseteq A^{c}\cup B^{c}
  27. A c B c ( A B ) c A^{c}\cup B^{c}\subseteq(A\cap B)^{c}
  28. x ( A B ) c x\in(A\cap B)^{c}
  29. x A B x\not\in A\cap B
  30. A B = { y | y A and y B } A\cap B=\{y|y\in A\,\text{ and }y\in B\}
  31. x A x\not\in A
  32. x B x\not\in B
  33. x A x\not\in A
  34. x A c x\in A^{c}
  35. x A c B c x\in A^{c}\cup B^{c}
  36. x B x\not\in B
  37. x B c x\in B^{c}
  38. x A c B c x\in A^{c}\cup B^{c}
  39. x ( \forall x(
  40. x ( A B ) c x\in(A\cap B)^{c}
  41. x A c B c ) x\in A^{c}\cup B^{c})
  42. ( A B ) c A c B c (A\cap B)^{c}\subseteq A^{c}\cup B^{c}
  43. x A c B c x\in A^{c}\cup B^{c}
  44. x ( A B ) c x\not\in(A\cap B)^{c}
  45. x A B x\in A\cap B
  46. x A x\in A
  47. x B x\in B
  48. x A c x\not\in A^{c}
  49. x B c x\not\in B^{c}
  50. x A c B c x\not\in A^{c}\cup B^{c}
  51. x A c B c x\in A^{c}\cup B^{c}
  52. x ( A B ) c x\not\in(A\cap B)^{c}
  53. x ( A B ) c x\in(A\cap B)^{c}
  54. x ( \forall x(
  55. x A c B c x\in A^{c}\cup B^{c}
  56. x ( A B ) c ) x\in(A\cap B)^{c})
  57. A c B c ( A B ) c A^{c}\cup B^{c}\subseteq(A\cap B)^{c}
  58. A c B c ( A B ) c A^{c}\cup B^{c}\subseteq(A\cap B)^{c}
  59. ( A B ) c A c B c (A\cap B)^{c}\subseteq A^{c}\cup B^{c}
  60. ( A B ) c = A c B c (A\cap B)^{c}=A^{c}\cup B^{c}
  61. ( A B ) c = A c B c (A\cup B)^{c}=A^{c}\cap B^{c}
  62. P d \mbox{P}~{}^{d}
  63. P ( p , q , ) d = ¬ P ( ¬ p , ¬ q , ) . \mbox{P}~{}^{d}(p,q,...)=\neg P(\neg p,\neg q,\dots).
  64. x P ( x ) ¬ x ¬ P ( x ) , \forall x\,P(x)\equiv\neg\exists x\,\neg P(x),
  65. x P ( x ) ¬ x ¬ P ( x ) . \exists x\,P(x)\equiv\neg\forall x\,\neg P(x).
  66. x P ( x ) P ( a ) P ( b ) P ( c ) \forall x\,P(x)\equiv P(a)\land P(b)\land P(c)
  67. x P ( x ) P ( a ) P ( b ) P ( c ) . \exists x\,P(x)\equiv P(a)\lor P(b)\lor P(c).\,
  68. P ( a ) P ( b ) P ( c ) ¬ ( ¬ P ( a ) ¬ P ( b ) ¬ P ( c ) ) P(a)\land P(b)\land P(c)\equiv\neg(\neg P(a)\lor\neg P(b)\lor\neg P(c))
  69. P ( a ) P ( b ) P ( c ) ¬ ( ¬ P ( a ) ¬ P ( b ) ¬ P ( c ) ) , P(a)\lor P(b)\lor P(c)\equiv\neg(\neg P(a)\land\neg P(b)\land\neg P(c)),
  70. p ¬ ¬ p , \Box p\equiv\neg\Diamond\neg p,
  71. p ¬ ¬ p . \Diamond p\equiv\neg\Box\neg p.\,

Decibel.html

  1. L P = 1 2 ln ( P P 0 ) Np = 10 log 10 ( P P 0 ) dB . L_{P}=\frac{1}{2}\ln\!\left(\frac{P}{P_{0}}\right)\!~{}\mathrm{Np}=10\log_{10}% \!\left(\frac{P}{P_{0}}\right)\!~{}\mathrm{dB}.
  2. P = 10 L P 10 dB P 0 . P=10^{\frac{L_{P}}{10\,\mathrm{dB}}}P_{0}.
  3. L F = ln ( F F 0 ) Np = 10 log 10 ( F 2 F 0 2 ) dB = 20 log 10 ( F F 0 ) dB . L_{F}=\ln\!\left(\frac{F}{F_{0}}\right)\!~{}\mathrm{Np}=10\log_{10}\!\left(% \frac{F^{2}}{F_{0}^{2}}\right)\!~{}\mathrm{dB}=20\log_{10}\left(\frac{F}{F_{0}% }\right)\!~{}\mathrm{dB}.
  4. F = 10 L F 20 dB F 0 . F=10^{\frac{L_{F}}{20\,\mathrm{dB}}}F_{0}.
  5. G dB = 20 log 10 ( V V 0 ) dB , G_{\mathrm{dB}}=20\log_{10}\!\left(\frac{V}{V_{0}}\right)\!~{}\mathrm{dB},
  6. G dB = 10 log 10 ( 1000 W 1 W ) = 30. G_{\mathrm{dB}}=10\log_{10}\bigg(\frac{1000~{}\mathrm{W}}{1~{}\mathrm{W}}\bigg% )=30.
  7. G dB = 20 log 10 ( 31.62 V 1 V ) = 30. G_{\mathrm{dB}}=20\log_{10}\bigg(\frac{31.62~{}\mathrm{V}}{1~{}\mathrm{V}}% \bigg)=30.
  8. G dB = 10 log 10 ( 0.001 W 10 W ) = - 40. G_{\mathrm{dB}}=10\log_{10}\bigg(\frac{0.001~{}\mathrm{W}}{10~{}\mathrm{W}}% \bigg)=-40.
  9. G = 10 3 10 × 1 = 1.99526... 2. G=10^{\frac{3}{10}}\times 1=1.99526...\approx 2.
  10. L p = 20 log 10 ( p rms p ref ) dB , L_{p}=20\log_{10}\!\left(\frac{p_{\mathrm{rms}}}{p_{\mathrm{ref}}}\right)\!~{}% \mathrm{dB},
  11. 0.6 V 0.7746 V - 2.218 dBV \sqrt{0.6}\,\mathrm{V}\,\approx 0.7746\,\mathrm{V}\,\approx-2.218\,\mathrm{dBV}
  12. V = 600 Ω 0.001 W V=\sqrt{600\,\Omega\cdot 0.001\,\mathrm{W}}
  13. 1 Np = 20 log 10 e dB 8.685889638 dB 1\ {\rm Np}=20\log_{10}e\ {\rm dB}\approx 8{.}685889638\ {\rm dB}\,

Decimal.html

  1. 0.0123123123 = 123 10000 k = 0 0.001 k = 123 10000 1 1 - 0.001 = 123 9990 = 41 3330 0.0123123123\cdots=\frac{123}{10000}\sum_{k=0}^{\infty}0.001^{k}=\frac{123}{10% 000}\ \frac{1}{1-0.001}=\frac{123}{9990}=\frac{41}{3330}
  2. x = sign i a i 10 i x=\mathop{\rm sign}\sum_{i\in\mathbb{Z}}a_{i}\,10^{i}
  3. { + , - } \in\{+,-\}

Dedekind_cut.html

  1. 2 \sqrt{2}
  2. A = { a : a 2 < 2 or a < 0 } , A=\{a\in\mathbb{Q}:a^{2}<2\,\text{ or }a<0\},
  3. B = { b : b 2 > 2 and b > 0 } . B=\{b\in\mathbb{Q}:b^{2}>2\,\text{ and }b>0\}.
  4. x x\,
  5. x 2 < 2 x^{2}<2\,
  6. y y\,
  7. x < y x<y\,
  8. y 2 < 2 . y^{2}<2\,.
  9. y = 2 x + 2 x + 2 y=\frac{2x+2}{x+2}\,
  10. r r\,
  11. x x\,
  12. A A
  13. r < x 2 r<x^{2}\,

Definable_real_number.html

  1. π \pi
  2. m n ( φ ( n , m ) n m < a ) . \forall m\,\forall n\,(\varphi(n,m)\iff\frac{n}{m}<a).

Delaunay_triangulation.html

  1. 4 π 3 3 2.418 \frac{4\pi}{3\sqrt{3}}\approx 2.418
  2. | A x A y A x 2 + A y 2 1 B x B y B x 2 + B y 2 1 C x C y C x 2 + C y 2 1 D x D y D x 2 + D y 2 1 | = | A x - D x A y - D y ( A x 2 - D x 2 ) + ( A y 2 - D y 2 ) B x - D x B y - D y ( B x 2 - D x 2 ) + ( B y 2 - D y 2 ) C x - D x C y - D y ( C x 2 - D x 2 ) + ( C y 2 - D y 2 ) | > 0 \begin{vmatrix}A_{x}&A_{y}&A_{x}^{2}+A_{y}^{2}&1\\ B_{x}&B_{y}&B_{x}^{2}+B_{y}^{2}&1\\ C_{x}&C_{y}&C_{x}^{2}+C_{y}^{2}&1\\ D_{x}&D_{y}&D_{x}^{2}+D_{y}^{2}&1\end{vmatrix}=\begin{vmatrix}A_{x}-D_{x}&A_{y% }-D_{y}&(A_{x}^{2}-D_{x}^{2})+(A_{y}^{2}-D_{y}^{2})\\ B_{x}-D_{x}&B_{y}-D_{y}&(B_{x}^{2}-D_{x}^{2})+(B_{y}^{2}-D_{y}^{2})\\ C_{x}-D_{x}&C_{y}-D_{y}&(C_{x}^{2}-D_{x}^{2})+(C_{y}^{2}-D_{y}^{2})\end{% vmatrix}>0

Delta_(letter).html

  1. y 2 - y 1 x 2 - x 1 = Δ y Δ x , {y_{2}-y_{1}\over x_{2}-x_{1}}={\Delta y\over\Delta x},
  2. Δ f = i = 1 n 2 f x i 2 \Delta f=\sum_{i=1}^{n}{\frac{\partial^{2}f}{\partial x_{i}^{2}}}
  3. Δ = b 2 - 4 a c \Delta=b^{2}-4ac\,\!
  4. Δ = 1 2 a b sin C \Delta=\tfrac{1}{2}ab\sin{C}

Delta_modulation.html

  1. m ( t ) = A cos ( ω t ) m(t)={A\cos(\omega t)}
  2. | m ˙ ( t ) | m a x = ω A |\dot{m}(t)|_{max}=\omega A
  3. | m ˙ ( t ) | m a x = ω A < σ f s |\dot{m}(t)|_{max}=\omega A<\sigma f_{s}
  4. A m a x = σ f s ω A_{max}={\sigma f_{s}\over\omega}

Demand_factor.html

  1. f D e m a n d ( t ) = Demand Maximum possible demand f_{Demand}(\it{t})=\frac{\,\text{Demand}}{\,\text{Maximum possible demand}}
  2. f D e m a n d = Maximum load in given time period Maximum possible load f_{Demand}=\frac{\,\text{Maximum load in given time period}}{\,\text{Maximum % possible load}}
  3. f L o a d = Average load Maximum load in given time period f_{Load}=\frac{\,\text{Average load}}{\,\text{Maximum load in given time % period}}

Demography.html

  1. P o p u l a t i o n t + 1 = P o p u l a t i o n t + N a t u r a l i n c r e a s e t + N e t m i g r a t i o n t Population_{t+1}=Population_{t}+Naturalincrease_{t}+Netmigration_{t}
  2. N a t u r a l i n c r e a s e t = B i r t h s t - D e a t h s t Naturalincrease_{t}=Births_{t}-Deaths_{t}
  3. N e t m i g r a t i o n t = I m m i g r a t i o n t - E m i g r a t i o n t Netmigration_{t}=Immigration_{t}-Emigration_{t}

Denotational_semantics.html

  1. i F i ( { } ) . \bigsqcup_{i\in\mathbb{N}}F^{i}(\{\}).

Density.html

  1. ρ = m V , \rho=\frac{m}{V},
  2. d V dV
  3. r r
  4. m = V ρ ( r ) d V . m=\int_{V}\rho(\vec{r})\,dV.
  5. ρ = M P R T , \rho=\frac{MP}{RT},\,
  6. M M
  7. P P
  8. R R
  9. T T
  10. ρ = ρ T 0 < m t p l > ( 1 + α Δ T ) \rho=\frac{{\rho_{T_{0}}}}{<}mtpl>{{(1+\alpha\cdot\Delta T)}}
  11. ρ T 0 \rho_{T_{0}}
  12. α \alpha
  13. T 0 T_{0}
  14. ρ = i ϱ i \rho=\sum_{i}\varrho_{i}\,
  15. ρ = i ρ i V i V = i ρ i φ i = i ρ i V i i V i + i V i E \rho=\sum_{i}\rho_{i}\frac{V_{i}}{V}\,=\sum_{i}\rho_{i}\varphi_{i}=\sum_{i}% \rho_{i}\frac{V_{i}}{\sum_{i}V_{i}+\sum_{i}V_{i}^{E}}
  16. V E ¯ i = R T ( l n ( γ i ) ) P \bar{V^{E}}_{i}=RT\frac{\partial(ln(\gamma_{i}))}{\partial P}

Density_matrix.html

  1. | ψ 1 |\psi_{1}\rangle
  2. | ψ 2 |\psi_{2}\rangle
  3. | ψ 3 |\psi_{3}\rangle
  4. ρ ^ = i p i | ψ i ψ i | , \hat{\rho}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|,
  5. { | ψ i } \{|\psi_{i}\rangle\}
  6. i p i = 1 \sum_{i}p_{i}=1
  7. { | u m } \{|u_{m}\rangle\}
  8. ρ m n = i p i u m | ψ i ψ i | u n = u m | ρ ^ | u n . \rho_{mn}=\sum_{i}p_{i}\langle u_{m}|\psi_{i}\rangle\langle\psi_{i}|u_{n}% \rangle=\langle u_{m}|\hat{\rho}|u_{n}\rangle.
  9. ρ ^ = m n | u m ρ m n u n | . \hat{\rho}=\sum_{mn}|u_{m}\rangle\rho_{mn}\langle u_{n}|.
  10. A ^ \hat{A}
  11. A A
  12. A \langle A\rangle
  13. A = i p i ψ i | A ^ | ψ i = m n u m | ρ ^ | u n u n | A ^ | u m = m n ρ m n A n m = tr ( ρ A ) . \langle A\rangle=\sum_{i}p_{i}\langle\psi_{i}|\hat{A}|\psi_{i}\rangle=\sum_{mn% }\langle u_{m}|\hat{\rho}|u_{n}\rangle\langle u_{n}|\hat{A}|u_{m}\rangle=\sum_% {mn}\rho_{mn}A_{nm}=\operatorname{tr}(\rho A).
  14. | ψ i |\psi_{i}\rangle
  15. A A
  16. | ψ |\psi\rangle
  17. | ψ |\psi\rangle
  18. | ψ 1 |\psi_{1}\rangle
  19. | ψ 2 |\psi_{2}\rangle
  20. | ψ = ( | ψ 1 + | ψ 2 ) / 2 |\psi\rangle=(|\psi_{1}\rangle+|\psi_{2}\rangle)/\sqrt{2}
  21. ρ \rho
  22. t r ( ρ 2 ) = 1 tr(\rho^{2})=1
  23. [ 0.5 0 0 0.5 ] \begin{bmatrix}0.5&0\\ 0&0.5\\ \end{bmatrix}
  24. [ 1 0 0 0 ] \begin{bmatrix}1&0\\ 0&0\\ \end{bmatrix}
  25. | R |R\rangle
  26. | L |L\rangle
  27. ( | R + | L ) / 2 (|R\rangle+|L\rangle)/\sqrt{2}
  28. ( | R - | L ) / 2 (|R\rangle-|L\rangle)/\sqrt{2}
  29. α | R + β | L \alpha|R\rangle+\beta|L\rangle
  30. ( | R + | L ) / 2 (|R\rangle+|L\rangle)/\sqrt{2}
  31. | R |R\rangle
  32. | L |L\rangle
  33. | R |R\rangle
  34. | L |L\rangle
  35. | R |R\rangle
  36. | L |L\rangle
  37. ( | R + | L ) / 2 (|R\rangle+|L\rangle)/\sqrt{2}
  38. α | R + β | L \alpha|R\rangle+\beta|L\rangle
  39. α | R + β | L \alpha|R\rangle+\beta|L\rangle
  40. | R |R\rangle
  41. | L |L\rangle
  42. ( | R , L + | L , R ) / 2 (|R,L\rangle+|L,R\rangle)/\sqrt{2}
  43. | ψ |\psi\rangle
  44. \mathcal{H}
  45. ψ | F ( A ) | ψ . \langle\psi|F(A)|\psi\rangle\,.
  46. | ψ |\psi\rangle
  47. | ϕ |\phi\rangle
  48. | ψ |\psi\rangle
  49. | ϕ |\phi\rangle
  50. | ξ |\xi\rangle
  51. ξ | F ( A ) | ξ . \langle\xi|F(A)|\xi\rangle\,.
  52. tr [ ρ F ( A ) ] , \operatorname{tr}[\rho F(A)]\,,
  53. ρ = p | ψ ψ | + ( 1 - p ) | ϕ ϕ | . \rho=p|\psi\rangle\langle\psi|+(1-p)|\phi\rangle\langle\phi|\,.
  54. ρ = 1 2 | R R | + 1 2 | L L | . \rho=\tfrac{1}{2}|R\rangle\langle R|+\tfrac{1}{2}|L\rangle\langle L|.
  55. ρ = j p j | ψ j ψ j | \rho=\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|
  56. | ψ j \textstyle|\psi_{j}\rangle
  57. | 1 , , | k \textstyle|1\rangle,...,|k\rangle
  58. | i \textstyle|i\rangle
  59. ρ = | i i | \textstyle\rho=|i\rangle\langle i|
  60. U * U = I U^{*}U=I
  61. | ψ i p i = j u i j | ψ j p j . |\psi_{i}^{\prime}\rangle\sqrt{p_{i}^{\prime}}=\sum_{j}u_{ij}|\psi_{j}\rangle% \sqrt{p_{j}}.
  62. ρ = ρ 2 \;\rho=\rho^{2}
  63. | ψ j \textstyle|\psi_{j}\rangle
  64. ρ = j p j | ψ j ψ j | . \rho=\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|.
  65. A = j p j ψ j | A | ψ j = j p j tr ( | ψ j ψ j | A ) = j tr ( p j | ψ j ψ j | A ) = tr ( j p j | ψ j ψ j | A ) = tr ( ρ A ) , \langle A\rangle=\sum_{j}p_{j}\langle\psi_{j}|A|\psi_{j}\rangle=\sum_{j}p_{j}% \operatorname{tr}\left(|\psi_{j}\rangle\langle\psi_{j}|A\right)=\sum_{j}% \operatorname{tr}\left(p_{j}|\psi_{j}\rangle\langle\psi_{j}|A\right)=% \operatorname{tr}\left(\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|A\right)=% \operatorname{tr}(\rho A),
  66. tr \operatorname{tr}
  67. A = i a i | a i a i | = i a i P i , A=\sum_{i}a_{i}|a_{i}\rangle\langle a_{i}|=\sum_{i}a_{i}P_{i},
  68. P i = | a i a i | P_{i}=|a_{i}\rangle\langle a_{i}|
  69. ρ = i P i ρ P i . \;\rho^{\prime}=\sum_{i}P_{i}\rho P_{i}.
  70. ρ i = P i ρ P i tr [ ρ P i ] . \rho_{i}^{\prime}=\frac{P_{i}\rho P_{i}}{\operatorname{tr}[\rho P_{i}]}.
  71. | a i \textstyle|a_{i}\rangle
  72. S S
  73. ρ \rho
  74. ρ \rho
  75. ρ \rho
  76. ρ = i λ i | φ i φ i | \rho=\sum_{i}\lambda_{i}|\varphi_{i}\rangle\langle\varphi_{i}|
  77. | φ i |\varphi_{i}\rangle
  78. λ i > 0 \lambda_{i}>0
  79. λ i = 1 \sum\lambda_{i}=1
  80. ρ \rho
  81. S = - i λ i ln λ i = - tr ( ρ ln ρ ) . S=-\sum_{i}\lambda_{i}\ln\,\lambda_{i}=-\operatorname{tr}(\rho\ln\rho)\quad.
  82. S ( ρ = i p i ρ i ) = H ( p i ) + i p i S ( ρ i ) S\left(\rho=\sum_{i}p_{i}\rho_{i}\right)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})
  83. ρ i \rho_{i}
  84. H ( p ) H(p)
  85. i ρ t = [ H , ρ ] , i\hbar\frac{\partial\rho}{\partial t}=[H,\rho]~{},
  86. i d A ( H ) d t = - [ H , A ( H ) ] , i\hbar\frac{dA^{(H)}}{dt}=-[H,A^{(H)}]~{},
  87. A ( H ) ( t ) A^{(H)}(t)
  88. A \langle A\rangle
  89. ρ ( t ) = e - i H t / ρ ( 0 ) e i H t / . \rho(t)=e^{-iHt/\hbar}\rho(0)e^{iHt/\hbar}.
  90. W ( x , p ) = def 1 π - ψ * ( x + y ) ψ ( x - y ) e 2 i p y / d y . W(x,p)\stackrel{\mathrm{def}}{=}\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\psi^% {*}(x+y)\psi(x-y)e^{2ipy/\hbar}\,dy~{}.
  91. W ( q , p , t ) t = - { { W ( q , p , t ) , H ( q , p ) } } , \frac{\partial W(q,p,t)}{\partial t}=-\{\{W(q,p,t),H(q,p)\}\}~{},
  92. ρ A B \rho_{AB}
  93. ρ A = tr B ρ A B \rho_{A}=\operatorname{tr}_{B}\rho_{AB}
  94. tr B \operatorname{tr}_{B}
  95. ρ A B = ρ A ρ B \rho_{AB}=\rho_{A}\otimes\rho_{B}

Depth_of_field.html

  1. N ( 1 + m ) N\left(1+m\right)
  2. f f
  3. N N
  4. c c
  5. H H
  6. H f 2 N c . H\approx\frac{f^{2}}{Nc}\,.
  7. s s
  8. s s
  9. D N D_{\mathrm{N}}
  10. D F D_{\mathrm{F}}
  11. D N H s H + s D_{\mathrm{N}}\approx\frac{Hs}{H+s}
  12. D F H s H - s for s < H . D_{\mathrm{F}}\approx\frac{Hs}{H-s}\,\text{ for }s<H\,.
  13. D F - D N D_{\mathrm{F}}-D_{\mathrm{N}}
  14. DOF 2 H s 2 H 2 - s 2 for s < H . \mathrm{DOF}\approx\frac{2Hs^{2}}{H^{2}-s^{2}}\,\text{ for }s<H\,.
  15. H H
  16. DOF 2 N c f 2 s 2 f 4 - N 2 c 2 s 2 . \mathrm{DOF}\approx\frac{2Ncf^{2}s^{2}}{f^{4}-N^{2}c^{2}s^{2}}\,.
  17. D F = D_{\mathrm{F}}=\infty
  18. D N = H 2 . D_{\mathrm{N}}=\frac{H}{2}\,.
  19. s H s\geq H
  20. s s
  21. m m
  22. DOF 2 N c m + 1 m 2 , \mathrm{DOF}\approx 2Nc\,\frac{m+1}{m^{2}}\,,
  23. s H s\ll H
  24. DOF 2 N c ( 1 + m / P ) m 2 , \mathrm{DOF}\approx\frac{2Nc\left(1+m/P\right)}{m^{2}}\,,
  25. P P
  26. P = 0.5 P=0.5
  27. P = 2 P=2
  28. m = 1.0 m=1.0
  29. DOF = 6 N c \mathrm{DOF}=6Nc
  30. DOF = 3 N c \mathrm{DOF}=3Nc
  31. DOF = 4 N c \mathrm{DOF}=4Nc
  32. D N D_{\mathrm{N}}
  33. D F D_{\mathrm{F}}
  34. s = 2 D N D F D N + D F , s=\frac{2D_{\mathrm{N}}D_{\mathrm{F}}}{D_{\mathrm{N}}+D_{\mathrm{F}}}\,,
  35. N f 2 c D F - D N 2 D N D F . N\approx\frac{f^{2}}{c}\frac{D_{\mathrm{F}}-D_{\mathrm{N}}}{2D_{\mathrm{N}}D_{% \mathrm{F}}}\,.
  36. s = 2 D N s=2D_{\mathrm{N}}
  37. N f 2 c 1 2 D N . N\approx\frac{f^{2}}{c}\frac{1}{2D_{\mathrm{N}}}\,.
  38. v N v_{\mathrm{N}}
  39. v F v_{\mathrm{F}}
  40. v v
  41. v v N + v F 2 = v F + v N - v F 2 . v\approx\frac{v_{\mathrm{N}}+v_{\mathrm{F}}}{2}=v_{\mathrm{F}}+\frac{v_{% \mathrm{N}}-v_{\mathrm{F}}}{2}\,.
  42. N v N - v F 2 c . N\approx\frac{v_{\mathrm{N}}-v_{\mathrm{F}}}{2c}\,.
  43. v N - v F v_{\mathrm{N}}\,-\,v_{\mathrm{F}}
  44. s s
  45. D D
  46. x d = | D - s | . x_{\mathrm{d}}=\left|D-s\right|\,.
  47. b b
  48. x d x_{\mathrm{d}}
  49. m s m_{\mathrm{s}}
  50. f f
  51. N N
  52. d d
  53. b = f m s N x d s ± x d = d m s x d D . b=\frac{fm_{\mathrm{s}}}{N}\frac{x_{\mathrm{d}}}{s\pm x_{\mathrm{d}}}=dm_{% \mathrm{s}}\frac{x_{\mathrm{d}}}{D}\,.
  54. b c b\leq c
  55. b f m s N , b\approx\frac{fm_{\mathrm{s}}}{N}\,,
  56. s s
  57. s s
  58. v v
  59. D F D_{\mathrm{F}}
  60. D N D_{\mathrm{N}}
  61. v F v_{\mathrm{F}}
  62. v N v_{\mathrm{N}}
  63. v v
  64. d d
  65. c c
  66. D N D_{\mathrm{N}}
  67. D F D_{\mathrm{F}}
  68. v N - v v N = c d \frac{v_{\mathrm{N}}-v}{v_{\mathrm{N}}}=\frac{c}{d}
  69. v - v F v F = c d . \frac{v-v_{\mathrm{F}}}{v_{\mathrm{F}}}=\frac{c}{d}\,.
  70. N N
  71. f f
  72. d d
  73. N = f d ; N=\frac{f}{d}\,;
  74. v N - v v N = v - v F v F = N c f . \frac{v_{\mathrm{N}}-v}{v_{\mathrm{N}}}=\frac{v-v_{\mathrm{F}}}{v_{\mathrm{F}}% }=\frac{Nc}{f}\,.
  75. v N v_{\mathrm{N}}
  76. v F v_{\mathrm{F}}
  77. v N = f v f - N c v_{\mathrm{N}}=\frac{fv}{f-Nc}
  78. v F = f v f + N c . v_{\mathrm{F}}=\frac{fv}{f+Nc}\,.
  79. v v
  80. s s
  81. 1 s + 1 v = 1 f ; \frac{1}{s}+\frac{1}{v}=\frac{1}{f}\,;
  82. v N v_{\mathrm{N}}
  83. v F v_{\mathrm{F}}
  84. 1 D N + 1 v N = 1 f \frac{1}{D_{\mathrm{N}}}+\frac{1}{v}_{\mathrm{N}}=\frac{1}{f}\,
  85. 1 D F + 1 v F = 1 f ; \frac{1}{D_{\mathrm{F}}}+\frac{1}{v}_{\mathrm{F}}=\frac{1}{f}\,;
  86. v v
  87. v N v_{\mathrm{N}}
  88. v F v_{\mathrm{F}}
  89. D N = s f 2 f 2 + N c ( s - f ) D_{\mathrm{N}}=\frac{sf^{2}}{f^{2}+Nc(s-f)}
  90. D F = s f 2 f 2 - N c ( s - f ) . D_{\mathrm{F}}=\frac{sf^{2}}{f^{2}-Nc(s-f)}\,.
  91. s s
  92. D F D_{\mathrm{F}}
  93. s = H = f 2 N c + f , s=H=\frac{f^{2}}{Nc}+f,
  94. H H
  95. D N = f 2 / ( N c ) + f 2 = H 2 . D_{\mathrm{N}}=\frac{f^{2}/(Nc)+f}{2}=\frac{H}{2}\,.
  96. H H
  97. H f 2 N c . H\approx\frac{f^{2}}{Nc}\,.
  98. D N = H s H + ( s - f ) D_{\mathrm{N}}=\frac{Hs}{H+(s-f)}
  99. D F = H s H - ( s - f ) . D_{\mathrm{F}}=\frac{Hs}{H-(s-f)}\,.
  100. D F - D N D_{\mathrm{F}}-D_{\mathrm{N}}
  101. DOF = 2 H s ( s - f ) H 2 - ( s - f ) 2 for s < H . \mathrm{DOF}=\frac{2Hs(s-f)}{H^{2}-(s-f)^{2}}\,\text{ for }s<H\,.
  102. m m
  103. m = f ( s - f ) ; m=\frac{f}{\left(s-f\right)}\,;
  104. m h m_{\mathrm{h}}
  105. m h = f ( H - f ) . m_{\mathrm{h}}=\frac{f}{\left(H-f\right)}\,.
  106. f 2 / N c + f f^{\,2}/Nc+f\,
  107. H H
  108. m h = N c f . m_{\mathrm{h}}=\frac{Nc}{f}\,.
  109. m m
  110. s = m + 1 m f s=\frac{m+1}{m}f
  111. s - f = f m s-f=\frac{f}{m}
  112. DOF = 2 f ( m + 1 ) / m ( f m ) / ( N c ) - ( N c ) / ( f m ) , \mathrm{DOF}=\frac{2f(m+1)/m}{(fm)/(Nc)-(Nc)/(fm)}\,,
  113. N c m f \frac{Ncm}{f}
  114. DOF = 2 N c ( m + 1 ) m 2 - ( N c f ) 2 . \mathrm{DOF}=\frac{2Nc\left(m+1\right)}{m^{2}-\left(\frac{Nc}{f}\right)^{2}}\,.
  115. f f
  116. m h m_{\mathrm{h}}
  117. DOF = 2 N c ( m + 1 ) m 2 - m h 2 . \mathrm{DOF}=\frac{2Nc\left(m+1\right)}{m^{2}-m^{2}_{\mathrm{h}}}\,.
  118. s H s\ll H
  119. m h 2 m 2 m^{2}_{\mathrm{h}}\ll m^{2}
  120. DOF 2 N c m + 1 m 2 , \mathrm{DOF}\approx 2Nc\,\frac{m+1}{m^{2}}\,,
  121. D N H s H + s D_{\mathrm{N}}\approx\frac{Hs}{H+s}
  122. D F H s H - s for s < H , D_{\mathrm{F}}\approx\frac{Hs}{H-s}\,\text{ for }s<H\,,
  123. DOF 2 H s 2 H 2 - s 2 for s < H . \mathrm{DOF}\approx\frac{2Hs^{2}}{H^{2}-s^{2}}\,\text{ for }s<H\,.
  124. s H s\geq H
  125. s s
  126. DOF 2 N c m + 1 m 2 , \mathrm{DOF}\approx 2Nc\,\frac{m+1}{m^{2}}\,,
  127. s - D N = N c s ( s - f ) f 2 + N c ( s - f ) , s-D_{\mathrm{N}}=\frac{Ncs(s-f)}{f^{2}+Nc(s-f)}\,,
  128. D F - s = N c s ( s - f ) f 2 - N c ( s - f ) . D_{\mathrm{F}}-s=\frac{Ncs(s-f)}{f^{2}-Nc(s-f)}\,.
  129. s - D N D F - s = f 2 - N c ( s - f ) f 2 + N c ( s - f ) . \frac{s-D_{\mathrm{N}}}{D_{\mathrm{F}}-s}=\frac{f^{2}-Nc(s-f)}{f^{2}+Nc(s-f)}\,.
  130. f s f\ll s
  131. s - D N D F - s f 2 - N c s f 2 + N c s = H - s H + s . \frac{s-D_{\mathrm{N}}}{D_{\mathrm{F}}-s}\approx\frac{f^{2}-Ncs}{f^{2}+Ncs}=% \frac{H-s}{H+s}\,.
  132. s H / 3 s\approx H/3
  133. m = f s - f ; m=\frac{f}{s-f}\,;
  134. s - D N D F - s = m - N c / f m + N c / f . \frac{s-D_{\mathrm{N}}}{D_{\mathrm{F}}-s}=\frac{m-Nc/f}{m+Nc/f}\,.
  135. DOF 2 N c m + 1 m 2 . \mathrm{DOF}\approx 2Nc\,\frac{m+1}{m^{2}}\,.
  136. m m
  137. DOF 2 N c m 2 . \mathrm{DOF}\approx\frac{2Nc}{m^{2}}\,.
  138. DOF 2 DOF 1 N 2 c 2 N 1 c 1 ( m 1 m 2 ) 2 . \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{N_{2}\,c_{2}}{N_{1}\,c_{% 1}}\left(\frac{m_{1}}{m_{2}}\right)^{2}\,.
  139. l l
  140. m 2 m 1 = c 2 c 1 = l 2 l 1 . \frac{m_{2}}{m_{1}}=\frac{c_{2}}{c_{1}}=\frac{l_{2}}{l_{1}}.
  141. DOF 2 DOF 1 c 2 c 1 ( m 1 m 2 ) 2 = l 2 l 1 ( l 1 l 2 ) 2 = l 1 l 2 , \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{c_{2}}{c_{1}}\left(\frac% {m_{1}}{m_{2}}\right)^{2}=\frac{l_{2}}{l_{1}}\left(\frac{l_{1}}{l_{2}}\right)^% {2}=\frac{l_{1}}{l_{2}}\,,
  142. m m
  143. N 2 N 1 c 1 c 2 ( m 2 m 1 ) 2 = l 2 l 1 . \frac{N_{2}}{N_{1}}\approx\frac{c_{1}}{c_{2}}\left(\frac{m_{2}}{m_{1}}\right)^% {2}=\frac{l_{2}}{l_{1}}\,.
  144. DOF 2 DOF 1 c 2 c 1 = l 2 l 1 , \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{c_{2}}{c_{1}}=\frac{l_{2% }}{l_{1}}\,,
  145. f / N f/N
  146. d d
  147. DOF = 2 s ( d m ) / c - c / ( d m ) , \mathrm{DOF}=\frac{2s}{(dm)/c-c/(dm)}\,,
  148. s s
  149. DOF 2 s c d m . \mathrm{DOF}\approx\frac{2sc}{dm}\,.
  150. s 2 = s 1 s_{2}=s_{1}
  151. DOF 2 DOF 1 c 2 c 1 d 1 d 2 m 1 m 2 = l 2 l 1 d 1 d 2 l 1 l 2 = d 1 d 2 , \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{c_{2}}{c_{1}}\frac{d_{1}% }{d_{2}}\frac{m_{1}}{m_{2}}=\frac{l_{2}}{l_{1}}\frac{d_{1}}{d_{2}}\frac{l_{1}}% {l_{2}}=\frac{d_{1}}{d_{2}}\,,
  152. N c Nc
  153. D N D_{\mathrm{N}}
  154. D F D_{\mathrm{F}}
  155. s = 2 D N D F D N + D F , s=\frac{2D_{\mathrm{N}}D_{\mathrm{F}}}{D_{\mathrm{N}}+D_{\mathrm{F}}}\,,
  156. s s
  157. N = f 2 c D F - D N D F ( D N - f ) + D N ( D F - f ) . N=\frac{f^{2}}{c}\frac{D_{\mathrm{F}}-D_{\mathrm{N}}}{D_{\mathrm{F}}(D_{% \mathrm{N}}-f)+D_{\mathrm{N}}(D_{\mathrm{F}}-f)}\,.
  158. N f 2 c D F - D N 2 D N D F . N\approx\frac{f^{2}}{c}\frac{D_{\mathrm{F}}-D_{\mathrm{N}}}{2D_{\mathrm{N}}D_{% \mathrm{F}}}\,.
  159. s s
  160. N N
  161. s s
  162. D F D_{\mathrm{F}}
  163. D F D_{\mathrm{F}}
  164. s = 2 D N . s=2D_{\mathrm{N}}\,.
  165. N N
  166. D F D_{\mathrm{F}}
  167. D F D_{\mathrm{F}}
  168. N = f 2 c 1 2 D N - f f 2 c 1 2 D N . N=\frac{f^{2}}{c}\frac{1}{2D_{\mathrm{N}}-f}\approx\frac{f^{2}}{c}\frac{1}{2D_% {\mathrm{N}}}\,.
  169. v N - v v N = N c f \frac{v_{\mathrm{N}}-v}{v_{\mathrm{N}}}=\frac{Nc}{f}
  170. v - v F v F = N c f \frac{v-v_{\mathrm{F}}}{v_{\mathrm{F}}}=\frac{Nc}{f}
  171. v v
  172. v = 2 v N v F v N + v F , v=\frac{2v_{\mathrm{N}}v_{\mathrm{F}}}{v_{\mathrm{N}}+v_{\mathrm{F}}}\,,
  173. N N
  174. N = f c v N - v F v N + v F . N=\frac{f}{c}\frac{v_{\mathrm{N}}-v_{\mathrm{F}}}{v_{\mathrm{N}}+v_{\mathrm{F}% }}\,.
  175. v v N + v F 2 = v F + v N - v F 2 . v\approx\frac{v_{\mathrm{N}}+v_{\mathrm{F}}}{2}=v_{\mathrm{F}}+\frac{v_{% \mathrm{N}}-v_{\mathrm{F}}}{2}\,.
  176. v N - v F v_{\mathrm{N}}\,-\,v_{\mathrm{F}}
  177. v N + v F = 2 v v_{\mathrm{N}}+v_{\mathrm{F}}=2v\,\!
  178. N N
  179. N f v v N - v F 2 c . N\approx\frac{f}{v}\frac{v_{\mathrm{N}}-v_{\mathrm{F}}}{2c}\,.
  180. v = ( m + 1 ) f v=\left(m+1\right)f
  181. m m
  182. N N
  183. N 1 1 + m v N - v F 2 c . N\approx\frac{1}{1+m}\frac{v_{\mathrm{N}}-v_{\mathrm{F}}}{2c}\,.
  184. m m
  185. N v N - v F 2 c . N\approx\frac{v_{\mathrm{N}}-v_{\mathrm{F}}}{2c}\,.
  186. v v
  187. N N
  188. v N - v F v_{\mathrm{N}}\,-\,v_{\mathrm{F}}
  189. v F = f v_{\mathrm{F}}=f\,\!
  190. c c
  191. D D
  192. b b
  193. b = f m s N D - s D . b=\frac{fm_{\mathrm{s}}}{N}\frac{D-s}{D}\,.
  194. c c
  195. D D
  196. b = f m s N | D - s | D . b=\frac{fm_{\mathrm{s}}}{N}\frac{\left|D-s\right|}{D}\,.
  197. x d = | D - s | ; x_{\mathrm{d}}=\left|D-s\right|;
  198. b = f m s N x d D . b=\frac{fm_{\mathrm{s}}}{N}\frac{x_{\mathrm{d}}}{D}\,.
  199. b = f m s N x d s ± x d , b=\frac{fm_{\mathrm{s}}}{N}\frac{x_{\mathrm{d}}}{s\pm x_{\mathrm{d}}},
  200. b f m s N . b\approx\frac{fm_{\mathrm{s}}}{N}\,.
  201. s = m s + 1 m s f , s=\frac{m_{\mathrm{s}}+1}{m_{\mathrm{s}}}f,
  202. b = f m s N x d m s + 1 m s f ± x d = f m s 2 N x d ( m s + 1 ) f ± m s x d . b=\frac{fm_{\mathrm{s}}}{N}\frac{x_{\mathrm{d}}}{\frac{m_{\mathrm{s}}+1}{m_{% \mathrm{s}}}f\pm x_{\mathrm{d}}}=\frac{fm_{\mathrm{s}}^{2}}{N}\frac{x_{\mathrm% {d}}}{\left(m_{\mathrm{s}}+1\right)f\pm m_{\mathrm{s}}x_{\mathrm{d}}}\,.
  203. b b
  204. f f
  205. d b d f = ± m s 3 x d 2 N [ ( m s + 1 ) f ± m s x d ] 2 . \frac{\mathrm{d}b}{\mathrm{d}f}=\frac{\pm m_{\mathrm{s}}^{3}x_{\mathrm{d}}^{2}% }{N\left[\left(m_{\mathrm{s}}+1\right)f\pm m_{\mathrm{s}}x_{\mathrm{d}}\right]% ^{2}}\,.
  206. m d = v s D = ( m s + 1 ) f D , m_{\mathrm{d}}=\frac{v_{\mathrm{s}}}{D}=\frac{\left(m_{\mathrm{s}}+1\right)f}{% D},
  207. v s v_{\mathrm{s}}
  208. y y
  209. m d y = ( m s + 1 ) f y D . m_{\mathrm{d}}y=\frac{\left(m_{\mathrm{s}}+1\right)fy}{D}\,.
  210. b m d y = m s m s + 1 x d N y , \frac{b}{m_{\mathrm{d}}y}=\frac{m_{\mathrm{s}}}{m_{\mathrm{s}}+1}\frac{x_{% \mathrm{d}}}{Ny},
  211. DOF N = N c ( 1 + m / P ) m 2 [ 1 + ( N c ) / ( f m ) ] \mathrm{DOF_{N}}=\frac{Nc(1+m/P)}{m^{2}[1+(Nc)/(fm)]}
  212. DOF F = N c ( 1 + m / P ) m 2 [ 1 - ( N c ) / ( f m ) ] , \mathrm{DOF_{F}}=\frac{Nc(1+m/P)}{m^{2}[1-(Nc)/(fm)]}\,,
  213. P P
  214. DOF = 2 f ( 1 / m + 1 / P ) ( f m ) / ( N c ) - ( N c ) / ( f m ) . \mathrm{DOF}=\frac{2f(1/m+1/P)}{(fm)/(Nc)-(Nc)/(fm)}\,.
  215. s H s\ll H
  216. DOF 2 N c ( 1 + m / P ) m 2 . \mathrm{DOF}\approx\frac{2Nc(1+m/P)}{m^{2}}\,.
  217. DOF 2 N c m ( 1 m + 1 P ) . \mathrm{DOF}\approx\frac{2Nc}{m}\left(\frac{1}{m}+\frac{1}{P}\right)\,.
  218. 1 / P 1/P
  219. 1 / m 1/m

Derivative.html

  1. f ( x ) f(x)
  2. x x
  3. f f
  4. x x
  5. x x
  6. y y
  7. f f
  8. x x
  9. y y
  10. x x
  11. y y
  12. x x
  13. y = f ( x ) = m x + b y=f(x)=mx+b
  14. m m
  15. b b
  16. m m
  17. m = change in y change in x = Δ y Δ x , m=\frac{\,\text{change in }y}{\,\text{change in }x}=\frac{\Delta y}{\Delta x},
  18. Δ Δ
  19. y + Δ y = f ( x + Δ x ) = m ( x + Δ x ) + b = m x + m Δ x + b = y + m Δ x . y+\Delta y=f\left(x+\Delta x\right)=m\left(x+\Delta x\right)+b=mx+m\Delta x+b=% y+m\Delta x.
  20. y + Δ y = y + m Δ x , y+\Delta y=y+m\Delta x,
  21. Δ y = m Δ x . \Delta y=m\Delta x.
  22. f f
  23. y y
  24. x x
  25. x x
  26. Δ y / Δ x Δy/Δx
  27. Δ x Δx
  28. x x
  29. d x dx
  30. y y
  31. x x
  32. d y d x \frac{dy}{dx}\,\!
  33. x x
  34. f ( x ) f(x)
  35. f [ u ( ] ( x ) f[u^{\prime}(^{\prime}](x)
  36. f f
  37. a a
  38. f f
  39. a a
  40. ( a , f ( a ) ) (a,f(a))
  41. f f
  42. y y
  43. x x
  44. a a
  45. f f
  46. ( a , f ( a ) ) (a,f(a))
  47. ( a , f ( a ) ) (a,f(a))
  48. ( a + h , f ( a + h ) ) (a+h,f(a+h))
  49. h h
  50. h h
  51. m m
  52. y y
  53. x x
  54. m = Δ f ( a ) Δ a = f ( a + h ) - f ( a ) ( a + h ) - ( a ) = f ( a + h ) - f ( a ) h . m=\frac{\Delta f(a)}{\Delta a}=\frac{f(a+h)-f(a)}{(a+h)-(a)}=\frac{f(a+h)-f(a)% }{h}.
  55. h h
  56. ( a , f ( a ) ) (a,f(a))
  57. f f
  58. a a
  59. f ( a ) = lim h 0 f ( a + h ) - f ( a ) h . f^{\prime}(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.
  60. f f
  61. a a
  62. f ( a ) f′(a)
  63. lim h 0 f ( a + h ) - f ( a ) - f ( a ) h h = 0 , \lim_{h\to 0}\frac{f(a+h)-f(a)-f^{\prime}(a)\cdot h}{h}=0,
  64. f f
  65. a a
  66. f ( a + h ) f ( a ) + f ( a ) h f(a+h)\approx f(a)+f^{\prime}(a)h
  67. f f
  68. a a
  69. h h
  70. h h
  71. Q ( h ) Q(h)
  72. h h
  73. Q ( h ) = f ( a + h ) - f ( a ) h . Q(h)=\frac{f(a+h)-f(a)}{h}.
  74. Q ( h ) Q(h)
  75. ( a , f ( a ) ) (a,f(a))
  76. ( a + h , f ( a + h ) ) (a+h,f(a+h))
  77. f f
  78. Q Q
  79. h = 0 h=0
  80. l i m h 0 Q ( h ) lim_{h→0}Q(h)
  81. Q ( 0 ) Q(0)
  82. Q Q
  83. f f
  84. a a
  85. a a
  86. Q ( 0 ) Q(0)
  87. Q ( h ) Q(h)
  88. h = 0 h=0
  89. h h
  90. Q Q
  91. h h
  92. Q Q
  93. h = 0 h=0
  94. y = f ( x ) y=f(x)
  95. x x
  96. y ∆\frac{y}{∆}
  97. x ∆x
  98. y = f ( x + x ) - f ( x ) ∆y=f(x+∆x)-f(x)
  99. f f
  100. f f
  101. x = 3 x=3
  102. h h
  103. f ( 3 ) f(3)
  104. f ( 3 ) = lim h 0 f ( 3 + h ) - f ( 3 ) h = lim h 0 ( 3 + h ) 2 - 3 2 h = lim h 0 9 + 6 h + h 2 - 9 h = lim h 0 6 h + h 2 h = lim h 0 ( 6 + h ) . f^{\prime}(3)=\lim_{h\to 0}\frac{f(3+h)-f(3)}{h}=\lim_{h\to 0}\frac{(3+h)^{2}-% 3^{2}}{h}=\lim_{h\to 0}\frac{9+6h+h^{2}-9}{h}=\lim_{h\to 0}\frac{6h+h^{2}}{h}=% \lim_{h\to 0}{(6+h)}.
  105. 6 + h 6+h
  106. h 0 h≠0
  107. h = 0 h=0
  108. h = 0 h=0
  109. h h
  110. 6 + h 6+h
  111. h h
  112. lim h 0 ( 6 + h ) = 6 + 0 = 6. \lim_{h\to 0}{(6+h)}=6+0=6.
  113. x = 3 x=3
  114. f ( 3 ) = 6 f′(3)=6
  115. x = a x=a
  116. f ( a ) = 2 a f′(a)=2a
  117. y = f ( x ) y=f(x)
  118. a a
  119. f f
  120. a a
  121. a a
  122. f f
  123. x x
  124. a a
  125. x x
  126. a a
  127. f f
  128. a a
  129. h h
  130. a + h a+h
  131. a a
  132. a + h a+h
  133. h h
  134. h h
  135. a + h a+h
  136. a a
  137. a + h a+h
  138. y = | x | y=|x|
  139. x = 0 x=0
  140. h h
  141. h h
  142. h h
  143. h h
  144. x = 0 x=0
  145. x = 0 x=0
  146. f f
  147. f f
  148. f f
  149. a a
  150. f f
  151. a a
  152. a a
  153. f f
  154. a a
  155. f ( x ) f′(x)
  156. f f
  157. f f
  158. f f
  159. f f
  160. f f
  161. a a
  162. f ( a ) f′(a)
  163. f ( a ) f′(a)
  164. f f
  165. f f
  166. D D
  167. D ( f ) D(f)
  168. f ( x ) f′(x)
  169. D ( f ) D(f)
  170. a a
  171. D ( f ) ( a ) = f ( a ) D(f)(a)=f′(a)
  172. f ( x ) = 2 x f(x)=2x
  173. f f
  174. 1 \displaystyle 1
  175. D D
  176. D ( x 1 ) \displaystyle D(x\mapsto 1)
  177. D D
  178. D D
  179. D D
  180. D D
  181. x 2 x x↦2x
  182. f ( x ) f(x)
  183. f ( 1 ) = 2 f(1)=2
  184. f ( 2 ) = 4 f(2)=4
  185. f f
  186. f ( x ) f′(x)
  187. f ( x ) f′(x)
  188. f ( x ) f′′(x)
  189. f f
  190. f ( x ) f′′′(x)
  191. f f
  192. n n
  193. ( n - 1 ) (n-1)
  194. n n
  195. n n
  196. x ( t ) x(t)
  197. t t
  198. x x
  199. x x
  200. x ( t ) x′(t)
  201. x x
  202. f f
  203. f f
  204. f ( x ) = { + x 2 , if x 0 - x 2 , if x 0. f(x)=\begin{cases}+x^{2},&\,\text{if }x\geq 0\\ -x^{2},&\,\text{if }x\leq 0.\end{cases}
  205. f f
  206. f ( x ) = { + 2 x , if x 0 - 2 x , if x 0. f^{\prime}(x)=\begin{cases}+2x,&\,\text{if }x\geq 0\\ -2x,&\,\text{if }x\leq 0.\end{cases}
  207. f ( x ) f′(x)
  208. k k
  209. k k
  210. ( k + 1 ) (k+1)
  211. k k
  212. k k
  213. k k
  214. k k
  215. n n
  216. n n
  217. f f
  218. x x
  219. x x
  220. f f
  221. f ( x + h ) f ( x ) + f ( x ) h + 1 2 f ′′ ( x ) h 2 f(x+h)\approx f(x)+f^{\prime}(x)h+\tfrac{1}{2}f^{\prime\prime}(x)h^{2}
  222. lim h 0 f ( x + h ) - f ( x ) - f ( x ) h - 1 2 f ′′ ( x ) h 2 h 2 = 0. \lim_{h\to 0}\frac{f(x+h)-f(x)-f^{\prime}(x)h-\frac{1}{2}f^{\prime\prime}(x)h^% {2}}{h^{2}}=0.
  223. f f
  224. f f
  225. x + h x+h
  226. x x
  227. x = 0 x=0
  228. x = 0 x=0
  229. d y d x , d f d x ( x ) , or d d x f ( x ) , \frac{dy}{dx},\quad\frac{df}{dx}(x),\;\;\mathrm{or}\;\;\frac{d}{dx}f(x),
  230. d n y d x n , d n f d x n ( x ) , or d n d x n f ( x ) \frac{d^{n}y}{dx^{n}},\quad\frac{d^{n}f}{dx^{n}}(x),\;\;\mathrm{or}\;\;\frac{d% ^{n}}{dx^{n}}f(x)
  231. d 2 y d x 2 = d d x ( d y d x ) . \frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right).
  232. d y d x | x = a = d y d x ( a ) . \left.\frac{dy}{dx}\right|_{x=a}=\frac{dy}{dx}(a).
  233. d y d x = d y d u d u d x . \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}.
  234. ( f ) = f ′′ (f^{\prime})^{\prime}=f^{\prime\prime}\,
  235. ( f ′′ ) = f ′′′ . (f^{\prime\prime})^{\prime}=f^{\prime\prime\prime}.
  236. f iv f^{\mathrm{iv}}\,\!
  237. f ( 4 ) . f^{(4)}.
  238. y ˙ \dot{y}
  239. y ¨ \ddot{y}
  240. x + x ˙ x+\dot{x}
  241. y + y ˙ y+\dot{y}
  242. y ˙ x ˙ . \frac{\dot{y}}{\dot{x}}\,.
  243. x 2 + y 2 = 1 \displaystyle x^{2}+y^{2}=1
  244. x 2 + y 2 = 1 x^{2}+y^{2}=1
  245. 2 x x ˙ + 2 y y ˙ = 0 2x\dot{x}+2y\dot{y}=0
  246. y ˙ x ˙ = - 2 x 2 y \frac{\dot{y}}{\dot{x}}=-\frac{2x}{2y}
  247. y ˙ x ˙ = - x y \frac{\dot{y}}{\dot{x}}=-\frac{x}{y}
  248. x ˙ \dot{x}
  249. x ˙ \dot{x}
  250. x ¨ \ddot{x}
  251. d x d t \frac{dx}{dt}
  252. d x d t \frac{dx}{dt}
  253. D x y D_{x}y\,
  254. D x f ( x ) D_{x}f(x)\,
  255. f ( x ) = x r , f(x)=x^{r},
  256. f ( x ) = r x r - 1 , f^{\prime}(x)=rx^{r-1},
  257. f ( x ) = x 1 / 4 f(x)=x^{1/4}
  258. f ( x ) = ( 1 / 4 ) x - 3 / 4 , f^{\prime}(x)=(1/4)x^{-3/4},
  259. d d x e x = e x . \frac{d}{dx}e^{x}=e^{x}.
  260. d d x a x = ln ( a ) a x . \frac{d}{dx}a^{x}=\ln(a)a^{x}.
  261. d d x ln ( x ) = 1 x , x > 0. \frac{d}{dx}\ln(x)=\frac{1}{x},\qquad x>0.
  262. d d x log a ( x ) = 1 x ln ( a ) . \frac{d}{dx}\log_{a}(x)=\frac{1}{x\ln(a)}.
  263. d d x sin ( x ) = cos ( x ) . \frac{d}{dx}\sin(x)=\cos(x).
  264. d d x cos ( x ) = - sin ( x ) . \frac{d}{dx}\cos(x)=-\sin(x).
  265. d d x tan ( x ) = sec 2 ( x ) = 1 cos 2 ( x ) = 1 + tan 2 ( x ) . \frac{d}{dx}\tan(x)=\sec^{2}(x)=\frac{1}{\cos^{2}(x)}=1+\tan^{2}(x).
  266. d d x arcsin ( x ) = 1 1 - x 2 , - 1 < x < 1. \frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^{2}}},-1<x<1.
  267. d d x arccos ( x ) = - 1 1 - x 2 , - 1 < x < 1. \frac{d}{dx}\arccos(x)=-\frac{1}{\sqrt{1-x^{2}}},-1<x<1.
  268. d d x arctan ( x ) = 1 < m t p l > 1 + x 2 \frac{d}{dx}\arctan(x)=\frac{1}{<}mtpl>{{1+x^{2}}}
  269. f = 0. f^{\prime}=0.\,
  270. ( α f + β g ) = α f + β g (\alpha f+\beta g)^{\prime}=\alpha f^{\prime}+\beta g^{\prime}\,
  271. α \alpha
  272. β \beta
  273. ( f g ) = f g + f g (fg)^{\prime}=f^{\prime}g+fg^{\prime}\,
  274. d d r π r 2 = 2 π r . \frac{d}{dr}\pi r^{2}=2\pi r.\,
  275. ( f g ) = f g - f g g 2 \left(\frac{f}{g}\right)^{\prime}=\frac{f^{\prime}g-fg^{\prime}}{g^{2}}
  276. f ( x ) = h ( g ( x ) ) f(x)=h(g(x))
  277. f ( x ) = h ( g ( x ) ) g ( x ) . f^{\prime}(x)=h^{\prime}(g(x))\cdot g^{\prime}(x).\,
  278. f ( x ) = x 4 + sin ( x 2 ) - ln ( x ) e x + 7 f(x)=x^{4}+\sin(x^{2})-\ln(x)e^{x}+7\,
  279. f ( x ) \displaystyle f^{\prime}(x)
  280. 𝐲 ( t ) = ( y 1 ( t ) , , y n ( t ) ) . \mathbf{y}^{\prime}(t)=(y^{\prime}_{1}(t),\ldots,y^{\prime}_{n}(t)).
  281. 𝐲 ( t ) = lim h 0 𝐲 ( t + h ) - 𝐲 ( t ) h , \mathbf{y}^{\prime}(t)=\lim_{h\to 0}\frac{\mathbf{y}(t+h)-\mathbf{y}(t)}{h},
  282. y 1 ( t ) 𝐞 1 + + y n ( t ) 𝐞 n y^{\prime}_{1}(t)\mathbf{e}_{1}+\cdots+y^{\prime}_{n}(t)\mathbf{e}_{n}
  283. f ( x , y ) = x 2 + x y + y 2 . f(x,y)=x^{2}+xy+y^{2}.\,
  284. f ( x , y ) = f x ( y ) = x 2 + x y + y 2 . f(x,y)=f_{x}(y)=x^{2}+xy+y^{2}.\,
  285. x f x , x\mapsto f_{x},\,
  286. f x ( y ) = x 2 + x y + y 2 . f_{x}(y)=x^{2}+xy+y^{2}.\,
  287. f a ( y ) = a 2 + a y + y 2 . f_{a}(y)=a^{2}+ay+y^{2}.\,
  288. f a ( y ) = a + 2 y . f_{a}^{\prime}(y)=a+2y.\,
  289. f y ( x , y ) = x + 2 y . \frac{\partial f}{\partial y}(x,y)=x+2y.
  290. f x i ( a 1 , , a n ) = lim h 0 f ( a 1 , , a i + h , , a n ) - f ( a 1 , , a i , , a n ) h . \frac{\partial f}{\partial x_{i}}(a_{1},\ldots,a_{n})=\lim_{h\to 0}\frac{f(a_{% 1},\ldots,a_{i}+h,\ldots,a_{n})-f(a_{1},\ldots,a_{i},\ldots,a_{n})}{h}.
  291. f a 1 , , a i - 1 , a i + 1 , , a n ( x i ) = f ( a 1 , , a i - 1 , x i , a i + 1 , , a n ) , f_{a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n}}(x_{i})=f(a_{1},\ldots,a_{i-1},x_% {i},a_{i+1},\ldots,a_{n}),
  292. d f a 1 , , a i - 1 , a i + 1 , , a n d x i ( a i ) = f x i ( a 1 , , a n ) . \frac{df_{a_{1},\ldots,a_{i-1},a_{i+1},\ldots,a_{n}}}{dx_{i}}(a_{i})=\frac{% \partial f}{\partial x_{i}}(a_{1},\ldots,a_{n}).
  293. f ( a ) = ( f x 1 ( a ) , , f x n ( a ) ) . \nabla f(a)=\left(\frac{\partial f}{\partial x_{1}}(a),\ldots,\frac{\partial f% }{\partial x_{n}}(a)\right).
  294. 𝐯 = ( v 1 , , v n ) . \mathbf{v}=(v_{1},\ldots,v_{n}).
  295. D 𝐯 f ( 𝐱 ) = lim h 0 f ( 𝐱 + h 𝐯 ) - f ( 𝐱 ) h . D_{\mathbf{v}}{f}(\mathbf{x})=\lim_{h\rightarrow 0}{\frac{f(\mathbf{x}+h% \mathbf{v})-f(\mathbf{x})}{h}}.
  296. f ( 𝐱 + ( k / λ ) ( λ 𝐮 ) ) - f ( 𝐱 ) k / λ = λ f ( 𝐱 + k 𝐮 ) - f ( 𝐱 ) k . \frac{f(\mathbf{x}+(k/\lambda)(\lambda\mathbf{u}))-f(\mathbf{x})}{k/\lambda}=% \lambda\cdot\frac{f(\mathbf{x}+k\mathbf{u})-f(\mathbf{x})}{k}.
  297. D 𝐯 f ( s y m b o l x ) = j = 1 n v j f x j . D_{\mathbf{v}}{f}(symbol{x})=\sum_{j=1}^{n}v_{j}\frac{\partial f}{\partial x_{% j}}.
  298. f ( 𝐚 + 𝐯 ) f ( 𝐚 ) + f ( 𝐚 ) 𝐯 . f(\mathbf{a}+\mathbf{v})\approx f(\mathbf{a})+f^{\prime}(\mathbf{a})\mathbf{v}.
  299. f ( 𝐚 + 𝐯 ) - f ( 𝐚 ) f ( 𝐚 ) 𝐯 . f(\mathbf{a}+\mathbf{v})-f(\mathbf{a})\approx f^{\prime}(\mathbf{a})\mathbf{v}.
  300. f ( 𝐚 + 𝐯 + 𝐰 ) - f ( 𝐚 + 𝐯 ) - f ( 𝐚 + 𝐰 ) + f ( 𝐚 ) f ( 𝐚 + 𝐯 ) 𝐰 - f ( 𝐚 ) 𝐰 . f(\mathbf{a}+\mathbf{v}+\mathbf{w})-f(\mathbf{a}+\mathbf{v})-f(\mathbf{a}+% \mathbf{w})+f(\mathbf{a})\approx f^{\prime}(\mathbf{a}+\mathbf{v})\mathbf{w}-f% ^{\prime}(\mathbf{a})\mathbf{w}.
  301. 0 \displaystyle 0
  302. lim h 0 f ( a + h ) - f ( a ) - f ( a ) h h = 0. \lim_{h\to 0}\frac{f(a+h)-f(a)-f^{\prime}(a)h}{h}=0.
  303. lim h 0 | f ( a + h ) - f ( a ) - f ( a ) h | | h | = 0 \lim_{h\to 0}\frac{|f(a+h)-f(a)-f^{\prime}(a)h|}{|h|}=0
  304. lim 𝐡 0 f ( 𝐚 + 𝐡 ) - f ( 𝐚 ) - f ( 𝐚 ) 𝐡 𝐡 = 0. \lim_{\mathbf{h}\to 0}\frac{\lVert f(\mathbf{a}+\mathbf{h})-f(\mathbf{a})-f^{% \prime}(\mathbf{a})\mathbf{h}\rVert}{\lVert\mathbf{h}\rVert}=0.
  305. f ( 𝐚 ) = Jac 𝐚 = ( f i x j ) i j . f^{\prime}(\mathbf{a})=\operatorname{Jac}_{\mathbf{a}}=\left(\frac{\partial f_% {i}}{\partial x_{j}}\right)_{ij}.
  306. f ( a + h ) f ( a ) + f ( a ) h . f(a+h)\approx f(a)+f^{\prime}(a)h.
  307. x f ( a ) + f ( a ) ( x - a ) x\mapsto f(a)+f^{\prime}(a)(x-a)
  308. D k f : n L k ( n × × n , m ) D^{k}f:\mathbb{R}^{n}\to L^{k}(\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n},% \mathbb{R}^{m})
  309. f ( 𝐱 ) f ( 𝐚 ) + ( D f ) ( 𝐱 ) + ( D 2 f ) ( Δ ( 𝐱 - 𝐚 ) ) + = f ( 𝐚 ) + ( D f ) ( 𝐱 - 𝐚 ) + ( D 2 f ) ( 𝐱 - 𝐚 , 𝐱 - 𝐚 ) + = f ( 𝐚 ) + i ( D f ) i ( 𝐱 - 𝐚 ) i + j , k ( D 2 f ) j k ( 𝐱 - 𝐚 ) j ( 𝐱 - 𝐚 ) k + \begin{aligned}\displaystyle f(\mathbf{x})&\displaystyle\approx f(\mathbf{a})+% (Df)(\mathbf{x})+(D^{2}f)(\Delta(\mathbf{x-a}))+\cdots\\ &\displaystyle=f(\mathbf{a})+(Df)(\mathbf{x-a})+(D^{2}f)(\mathbf{x-a},\mathbf{% x-a})+\cdots\\ &\displaystyle=f(\mathbf{a})+\sum_{i}(Df)_{i}(\mathbf{x-a})^{i}+\sum_{j,k}(D^{% 2}f)_{jk}(\mathbf{x-a})^{j}(\mathbf{x-a})^{k}+\cdots\end{aligned}
  310. a a