wpmath0000001_1

Angular_momentum.html

  1. m m
  2. v v
  3. p = m v , p=mv,
  4. I I
  5. ω \omega
  6. L = I ω . L=I\omega.
  7. I = r 2 m I=r^{2}m
  8. ω = v / r \omega=v/r
  9. L = ( r 2 m ) ( v r ) , L=(r^{2}m)\left(\frac{v}{r}\right),
  10. L = r m v , L=rmv,
  11. r r
  12. p = m v p=mv
  13. v v
  14. = r ω =r\omega
  15. L = r m v , L=rmv_{\perp},
  16. v = v sin θ v_{\perp}=v\sin\theta
  17. L = r m v sin θ , L=rmv\sin{\theta},
  18. L = r sin θ m v , L=r\sin{\theta}mv,
  19. L = r m v , L=r_{\perp}mv,
  20. r = r sin θ r_{\perp}=r\sin{\theta}
  21. ( l e n g t h o f m o m e n t a r m ) × ( l i n e a r m o m e n t u m ) (lengthofmomentarm)×(linearmomentum)
  22. 𝐋 = I s y m b o l ω , \mathbf{L}=Isymbol{\omega},
  23. I = r 2 m I=r^{2}m
  24. s y m b o l ω = 𝐫 × 𝐯 r 2 symbol{\omega}=\frac{\mathbf{r}\times\mathbf{v}}{r^{2}}
  25. 𝐫 \mathbf{r}
  26. r = | 𝐫 | r=\left|\mathbf{r}\right|
  27. 𝐯 \mathbf{v}
  28. m m
  29. 𝐋 = ( r 2 m ) ( 𝐫 × 𝐯 r 2 ) , \mathbf{L}=(r^{2}m)\left(\frac{\mathbf{r}\times\mathbf{v}}{r^{2}}\right),
  30. 𝐋 = m ( 𝐫 × 𝐯 ) , \mathbf{L}=m(\mathbf{r}\times\mathbf{v}),
  31. 𝐋 = 𝐫 × m 𝐯 = 𝐫 × 𝐩 , \begin{aligned}\displaystyle\mathbf{L}&\displaystyle=\mathbf{r}\times m\mathbf% {v}\\ &\displaystyle=\mathbf{r}\times\mathbf{p},\end{aligned}
  32. 𝐫 \mathbf{r}
  33. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  34. 𝐋 \mathbf{L}
  35. 𝐫 \mathbf{r}
  36. 𝐩 \mathbf{p}
  37. 𝐋 \mathbf{L}
  38. 𝐫 \mathbf{r}
  39. 𝐩 \mathbf{p}
  40. 𝐮 ^ \mathbf{\hat{u}}
  41. ω \omega
  42. ω 𝐮 ^ = s y m b o l ω , \omega\mathbf{\hat{u}}=symbol{\omega},
  43. ω = v r , \omega=\frac{v_{\perp}}{r},
  44. v v_{\perp}
  45. 𝐋 \displaystyle\mathbf{L}
  46. 𝐋 = r m v 𝐮 ^ \mathbf{L}=rmv\mathbf{\hat{u}}
  47. r r
  48. ( amount of inertia ) × ( amount of displacement ) = amount of (inertia·displacement) mass × velocity = momentum m × v = p \begin{aligned}\displaystyle(\,\text{amount of inertia})\times(\,\text{amount % of displacement})&\displaystyle=\,\text{amount of (inertia·displacement)}\\ \displaystyle\,\text{mass}\times\,\text{velocity}&\displaystyle=\,\text{% momentum}\\ \displaystyle m\times v&\displaystyle=p\\ \end{aligned}
  49. ( moment arm ) × ( amount of inertia ) × ( amount of displacement ) = moment of (inertia·displacement) length × mass × velocity = moment of momentum r × m × v = L \begin{aligned}\displaystyle(\,\text{moment arm})\times(\,\text{amount of % inertia})\times(\,\text{amount of displacement})&\displaystyle=\,\text{moment % of (inertia·displacement)}\\ \displaystyle\,\text{length}\times\,\text{mass}\times\,\text{velocity}&% \displaystyle=\,\text{moment of momentum}\\ \displaystyle r\times m\times v&\displaystyle=L\\ \end{aligned}
  50. L = r m v L=rmv
  51. m m
  52. r r
  53. v v
  54. r r
  55. ω \omega
  56. v = r ω , v=r\omega,
  57. L = r m r ω . L=rmr\omega.
  58. L = r 2 m ω , L=r^{2}m\omega,
  59. r 2 m r^{2}m
  60. I = k 2 m , I=k^{2}m,
  61. k k
  62. m m
  63. m m
  64. I = r 2 m I=r^{2}m
  65. r r
  66. m i m_{i}
  67. i I i = i r i 2 m i \sum_{i}I_{i}=\sum_{i}r_{i}^{2}m_{i}
  68. 𝐅 = m 𝐚 , \mathbf{F}=m\mathbf{a},
  69. s y m b o l τ = I s y m b o l α , symbol{\tau}=Isymbol{\alpha},
  70. s y m b o l τ = I d s y m b o l ω d t . symbol{\tau}=I\frac{dsymbol{\omega}}{dt}.
  71. s y m b o l τ = d ( I s y m b o l ω ) d t symbol{\tau}=\frac{d(Isymbol{\omega})}{dt}
  72. s y m b o l τ d t = d ( I s y m b o l ω ) , symbol{\tau}dt=d(Isymbol{\omega}),
  73. s y m b o l τ d t = I s y m b o l ω + c o n s t a n t . \int symbol{\tau}dt=Isymbol{\omega}+constant.
  74. s y m b o l τ = 𝟎 , symbol{\tau}=\mathbf{0},
  75. 𝐋 = 𝟎 + c o n s t a n t . \mathbf{L}=\mathbf{0}+constant.
  76. 𝐋 = I s y m b o l ω \mathbf{L}=Isymbol{\omega}
  77. d 𝐋 d t = d I d t s y m b o l ω + I d s y m b o l ω d t . \frac{d\mathbf{L}}{dt}=\frac{dI}{dt}symbol{\omega}+I\frac{dsymbol{\omega}}{dt}.
  78. d I d t \frac{dI}{dt}
  79. d 𝐋 d t = 0 + I d s y m b o l ω d t , \frac{d\mathbf{L}}{dt}=0+I\frac{dsymbol{\omega}}{dt},
  80. d 𝐋 d t = I s y m b o l α . \frac{d\mathbf{L}}{dt}=Isymbol{\alpha}.
  81. d 𝐋 d t = 0 \frac{d\mathbf{L}}{dt}=0
  82. s y m b o l τ = 𝐫 × 𝐅 = 𝟎 , symbol{\tau}=\mathbf{r}\times\mathbf{F}=\mathbf{0},
  83. 𝐫 \mathbf{r}
  84. 𝐅 \mathbf{F}
  85. 𝐡 = 𝐫 × 𝐯 , \mathbf{h}=\mathbf{r}\times\mathbf{v},
  86. 𝐋 = m 𝐡 . \mathbf{L}=m\mathbf{h}.
  87. d 𝐋 = 𝐫 × d m 𝐯 = 𝐫 × ρ ( 𝐫 ) d V 𝐯 = d V 𝐫 × ρ ( 𝐫 ) 𝐯 d\mathbf{L}=\mathbf{r}\times dm\mathbf{v}=\mathbf{r}\times\rho(\mathbf{r})dV% \mathbf{v}=dV\mathbf{r}\times\rho(\mathbf{r})\mathbf{v}
  88. 𝐋 = V d V 𝐫 × ρ ( 𝐫 ) 𝐯 \mathbf{L}=\int_{V}dV\mathbf{r}\times\rho(\mathbf{r})\mathbf{v}
  89. m i m_{i}
  90. i i
  91. 𝐑 i \mathbf{R}_{i}
  92. i i
  93. 𝐕 i \mathbf{V}_{i}
  94. i i
  95. 𝐑 \mathbf{R}
  96. 𝐕 \mathbf{V}
  97. 𝐫 i \mathbf{r}_{i}
  98. i i
  99. 𝐯 i \mathbf{v}_{i}
  100. i i
  101. M = i m i . M=\sum_{i}m_{i}.
  102. M 𝐑 = i m i 𝐑 i . M\mathbf{R}=\sum_{i}m_{i}\mathbf{R}_{i}.
  103. 𝐑 i = 𝐑 + 𝐫 i \mathbf{R}_{i}=\mathbf{R}+\mathbf{r}_{i}
  104. 𝐕 i = 𝐕 + 𝐯 i . \mathbf{V}_{i}=\mathbf{V}+\mathbf{v}_{i}.
  105. 𝐑 i \mathbf{R}_{i}
  106. 𝐋 \displaystyle\mathbf{L}
  107. 𝐕 i \mathbf{V}_{i}
  108. 𝐋 \displaystyle\mathbf{L}
  109. i m i 𝐫 i = 𝟎 \sum_{i}m_{i}\mathbf{r}_{i}=\mathbf{0}
  110. 𝐫 i = 𝐑 i - 𝐑 \mathbf{r}_{i}=\mathbf{R}_{i}-\mathbf{R}
  111. m i 𝐫 i = m i ( 𝐑 i - 𝐑 ) m_{i}\mathbf{r}_{i}=m_{i}(\mathbf{R}_{i}-\mathbf{R})
  112. i m i 𝐫 i = i m i ( 𝐑 i - 𝐑 ) = i ( m i 𝐑 i - m i 𝐑 ) = i m i 𝐑 i - i m i 𝐑 = i m i 𝐑 i - ( i m i ) 𝐑 = i m i 𝐑 i - M 𝐑 \begin{aligned}\displaystyle\sum_{i}m_{i}\mathbf{r}_{i}&\displaystyle=\sum_{i}% m_{i}(\mathbf{R}_{i}-\mathbf{R})\\ &\displaystyle=\sum_{i}(m_{i}\mathbf{R}_{i}-m_{i}\mathbf{R})\\ &\displaystyle=\sum_{i}m_{i}\mathbf{R}_{i}-\sum_{i}m_{i}\mathbf{R}\\ &\displaystyle=\sum_{i}m_{i}\mathbf{R}_{i}-\left(\sum_{i}m_{i}\right)\mathbf{R% }\\ &\displaystyle=\sum_{i}m_{i}\mathbf{R}_{i}-M\mathbf{R}\end{aligned}
  113. = 𝟎 , =\mathbf{0},
  114. i m i 𝐯 i . \sum_{i}m_{i}\mathbf{v}_{i}.
  115. i m i 𝐫 i = 𝟎 \sum_{i}m_{i}\mathbf{r}_{i}=\mathbf{0}
  116. i m i 𝐯 i = 𝟎 , \sum_{i}m_{i}\mathbf{v}_{i}=\mathbf{0},
  117. 𝐋 = i 𝐑 × m i 𝐕 + i 𝐫 i × m i 𝐯 i . \mathbf{L}=\sum_{i}\mathbf{R}\times m_{i}\mathbf{V}+\sum_{i}\mathbf{r}_{i}% \times m_{i}\mathbf{v}_{i}.
  118. i 𝐑 × m i 𝐕 = 𝐑 × i m i 𝐕 = 𝐑 × M 𝐕 , \sum_{i}\mathbf{R}\times m_{i}\mathbf{V}=\mathbf{R}\times\sum_{i}m_{i}\mathbf{% V}=\mathbf{R}\times M\mathbf{V},
  119. 𝐋 = M ( 𝐑 × 𝐕 ) + i [ m i ( 𝐫 i × 𝐯 i ) ] , = R 2 R 2 M ( 𝐑 × 𝐕 ) + i [ r i 2 r i 2 m i ( 𝐫 i × 𝐯 i ) ] , = R 2 M ( 𝐑 × 𝐕 R 2 ) + i [ r i 2 m i ( 𝐫 i × 𝐯 i r i 2 ) ] , \begin{aligned}\displaystyle\mathbf{L}&\displaystyle=M(\mathbf{R}\times\mathbf% {V})+\sum_{i}[m_{i}(\mathbf{r}_{i}\times\mathbf{v}_{i})],\\ &\displaystyle=\frac{R^{2}}{R^{2}}M(\mathbf{R}\times\mathbf{V})+\sum_{i}\left[% \frac{r_{i}^{2}}{r_{i}^{2}}m_{i}(\mathbf{r}_{i}\times\mathbf{v}_{i})\right],\\ &\displaystyle=R^{2}M\left(\frac{\mathbf{R}\times\mathbf{V}}{R^{2}}\right)+% \sum_{i}\left[r_{i}^{2}m_{i}\left(\frac{\mathbf{r}_{i}\times\mathbf{v}_{i}}{r_% {i}^{2}}\right)\right],\\ \end{aligned}
  120. I I
  121. s y m b o l ω symbol{\omega}
  122. 𝐫 i = 𝐯 i = 𝟎 , \mathbf{r}_{i}=\mathbf{v}_{i}=\mathbf{0},
  123. 𝐫 = 𝐑 , \mathbf{r}=\mathbf{R},
  124. 𝐯 = 𝐕 , \mathbf{v}=\mathbf{V},
  125. m = M , m=M,
  126. i 𝐫 i × m i 𝐯 i = 𝟎 , \sum_{i}\mathbf{r}_{i}\times m_{i}\mathbf{v}_{i}=\mathbf{0},
  127. i I i s y m b o l ω i = 𝟎 , \sum_{i}I_{i}symbol{\omega}_{i}=\mathbf{0},
  128. 𝐋 = 𝐑 × m 𝐕 = I R s y m b o l ω R . \mathbf{L}=\mathbf{R}\times m\mathbf{V}=I_{R}symbol{\omega}_{R}.
  129. 𝐕 = 𝟎 , \mathbf{V}=\mathbf{0},
  130. 𝐑 × M 𝐕 = 𝟎 , \mathbf{R}\times M\mathbf{V}=\mathbf{0},
  131. I R s y m b o l ω R = 𝟎 , I_{R}symbol{\omega}_{R}=\mathbf{0},
  132. 𝐋 = i 𝐫 i × m i 𝐯 i = i I i s y m b o l ω i . \mathbf{L}=\sum_{i}\mathbf{r}_{i}\times m_{i}\mathbf{v}_{i}=\sum_{i}I_{i}% symbol{\omega}_{i}.
  133. 𝐋 = 𝐫 𝐩 , \mathbf{L}=\mathbf{r}\wedge\mathbf{p}\,,
  134. 𝐋 = ( x p y - y p x ) 𝐞 x 𝐞 y + ( y p z - z p y ) 𝐞 y 𝐞 z + ( z p x - x p z ) 𝐞 z 𝐞 x = L x y 𝐞 x 𝐞 y + L y z 𝐞 y 𝐞 z + L z x 𝐞 z 𝐞 x , \begin{array}[]{rl}\mathbf{L}&=\left(xp_{y}-yp_{x}\right)\mathbf{e}_{x}\wedge% \mathbf{e}_{y}+\left(yp_{z}-zp_{y}\right)\mathbf{e}_{y}\wedge\mathbf{e}_{z}+% \left(zp_{x}-xp_{z}\right)\mathbf{e}_{z}\wedge\mathbf{e}_{x}\\ &=L_{xy}\mathbf{e}_{x}\wedge\mathbf{e}_{y}+L_{yz}\mathbf{e}_{y}\wedge\mathbf{e% }_{z}+L_{zx}\mathbf{e}_{z}\wedge\mathbf{e}_{x}\,,\end{array}
  135. L i j = x i p j - x j p i . L_{ij}=x_{i}p_{j}-x_{j}p_{i}\,.
  136. L i j = I i j k ω k . L_{ij}=I_{ijk\ell}\omega_{k\ell}\,.
  137. M α β = X α P β - X β P α M_{\alpha\beta}=X_{\alpha}\ P_{\beta}-X_{\beta}P_{\alpha}
  138. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}
  139. \hbar
  140. n ^ \hat{n}
  141. L n ^ L_{\hat{n}}
  142. , - 2 , - , 0 , , 2 , \ldots,-2\hbar,-\hbar,0,\hbar,2\hbar,\ldots
  143. S n ^ S_{\hat{n}}
  144. J n ^ J_{\hat{n}}
  145. , - 3 2 , - , - 1 2 , 0 , 1 2 , , 3 2 , \ldots,-\frac{3}{2}\hbar,-\hbar,-\frac{1}{2}\hbar,0,\frac{1}{2}\hbar,\hbar,% \frac{3}{2}\hbar,\ldots
  146. L 2 L^{2}
  147. = L x 2 + L y 2 + L z 2 =L_{x}^{2}+L_{y}^{2}+L_{z}^{2}
  148. ( 2 n ( n + 1 ) ) (\hbar^{2}n(n+1))
  149. n = 0 , 1 , 2 , n=0,1,2,\ldots
  150. S 2 S^{2}
  151. J 2 J^{2}
  152. ( 2 n ( n + 1 ) ) (\hbar^{2}n(n+1))
  153. n = 0 , 1 2 , 1 , 3 2 , n=0,\frac{1}{2},1,\frac{3}{2},\ldots
  154. \hbar
  155. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}
  156. r x r_{x}
  157. r y r_{y}
  158. r z r_{z}
  159. p x p_{x}
  160. p y p_{y}
  161. p z p_{z}
  162. L x L y L y L x L_{x}L_{y}\neq L_{y}L_{x}
  163. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}
  164. R ( n ^ , ϕ ) exp ( - i ϕ 𝐉 𝐧 ^ ) R(\hat{n},\phi)\equiv\exp\left(-\frac{i}{\hbar}\phi\,\mathbf{J}\cdot\hat{% \mathbf{n}}\right)
  165. ϕ \phi
  166. 𝐧 ^ \hat{\mathbf{n}}
  167. 𝐩 = m 𝐯 = 𝐏 - e 𝐀 \mathbf{p}=m\mathbf{v}=\mathbf{P}-e\mathbf{A}
  168. 𝐊 = 𝐫 × ( 𝐏 - e 𝐀 ) \mathbf{K}=\mathbf{r}\times(\mathbf{P}-e\mathbf{A})
  169. < v a r > r <var>r

Angular_velocity.html

  1. ω = d ϕ d t \omega=\frac{d\phi}{dt}
  2. v = r d ϕ d t \mathrm{v}_{\perp}=r\,\frac{d\phi}{dt}
  3. v = | 𝐯 | sin ( θ ) \mathrm{v}_{\perp}=|\mathrm{\mathbf{v}}|\,\sin(\theta)
  4. ω = | 𝐯 | sin ( θ ) | 𝐫 | \omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}{|\mathrm{\mathbf{r}}|}
  5. u u
  6. ω \vec{\omega}
  7. s y m b o l ω = d ϕ d t u symbol\omega=\frac{d\phi}{dt}u
  8. u u
  9. s y m b o l ω = | v | sin ( θ ) | r | u symbol\omega=\frac{|v|\sin(\theta)}{|r|}u
  10. s y m b o l ω = r × v | r | 2 symbol\omega=\frac{r\times v}{|r|^{2}}
  11. ω 2 \omega_{2}
  12. F 2 F_{2}
  13. ω 1 \omega_{1}
  14. F 1 F_{1}
  15. ω 1 + ω 2 \omega_{1}+\omega_{2}
  16. F 1 F_{1}
  17. R = e W t R=e^{Wt}
  18. R = I + W d t + 1 2 ( W d t ) 2 + R=I+W\cdot dt+{1\over 2}(W\cdot dt)^{2}+...
  19. ( I + W 1 d t ) ( I + W 2 d t ) = ( I + W 2 d t ) ( I + W 1 d t ) (I+W_{1}\cdot dt)(I+W_{2}\cdot dt)=(I+W_{2}\cdot dt)(I+W_{1}\cdot dt)
  20. ω 1 + ω 2 = ω 2 + ω 1 \omega_{1}+\omega_{2}=\omega_{2}+\omega_{1}
  21. ω 1 + ω 2 = ω 2 + ω 1 \omega_{1}+\omega_{2}=\omega_{2}+\omega_{1}
  22. s y m b o l ω = 𝐫 × 𝐯 | 𝐫 | 2 symbol\omega=\frac{\mathbf{r}\times\mathbf{v}}{|\mathrm{\mathbf{r}}|^{2}}
  23. s y m b o l ω = s y m b o l e × s y m b o l e ˙ | 𝐞 | 2 symbol\omega=\frac{symbole\times\dot{symbole}}{|{\mathbf{e}}|^{2}}
  24. s y m b o l ω = 𝐞 1 × 𝐞 ˙ 1 = 𝐞 2 × 𝐞 ˙ 2 = 𝐞 3 × 𝐞 ˙ 3 . symbol\omega=\mathbf{e}_{1}\times\dot{\mathbf{e}}_{1}=\mathbf{e}_{2}\times\dot% {\mathbf{e}}_{2}=\mathbf{e}_{3}\times\dot{\mathbf{e}}_{3}.
  25. ω \omega
  26. s y m b o l ω = α ˙ u 1 + β ˙ u 2 + γ ˙ u 3 symbol\omega=\dot{\alpha}u_{1}+\dot{\beta}u_{2}+\dot{\gamma}u_{3}
  27. s y m b o l ω = ( α ˙ sin β sin γ + β ˙ cos γ ) I + ( α ˙ sin β cos γ - β ˙ sin γ ) J + ( α ˙ cos β + γ ˙ ) K symbol\omega=(\dot{\alpha}\sin\beta\sin\gamma+\dot{\beta}\cos\gamma)I+(\dot{% \alpha}\sin\beta\cos\gamma-\dot{\beta}\sin\gamma)J+(\dot{\alpha}\cos\beta+\dot% {\gamma})K
  28. I , J , K I,J,K
  29. r \vec{r}
  30. ω \vec{\omega}
  31. d r ( t ) d t = ω × r \frac{d\vec{r}(t)}{dt}=\vec{\omega}\times\vec{r}
  32. ω \omega
  33. W ( t ) = ( 0 - ω z ( t ) ω y ( t ) ω z ( t ) 0 - ω x ( t ) - ω y ( t ) ω x ( t ) 0 ) W(t)=\begin{pmatrix}0&-\omega_{z}(t)&\omega_{y}(t)\\ \omega_{z}(t)&0&-\omega_{x}(t)\\ -\omega_{y}(t)&\omega_{x}(t)&0\\ \end{pmatrix}
  34. ( ω × ) (\vec{\omega}\times)
  35. ω ( t ) × r ( t ) = W ( t ) r ( t ) \vec{\omega}(t)\times\vec{r}(t)=W(t)\vec{r}(t)
  36. d r ( t ) d t = W r \frac{d\vec{r}(t)}{dt}=W\cdot\vec{r}
  37. d A ( t ) d t = W A ( t ) \frac{dA(t)}{dt}=W\cdot A(t)
  38. W = d A ( t ) d t A - 1 ( t ) W=\frac{dA(t)}{dt}\cdot A^{-1}(t)
  39. d A ( t ) d t = W A ( t ) \frac{dA(t)}{dt}=W\cdot A(t)
  40. d A ( t ) A = W d t \frac{dA(t)}{A}=W\cdot{dt}
  41. W W
  42. A ( t ) = e W t A ( 0 ) A(t)=e^{W\cdot t}A(0)
  43. W = d R ( t ) d t R T W=\frac{dR(t)}{dt}\cdot{R^{T}}
  44. W T = - W W^{T}=-W
  45. T \mathcal{R}\mathcal{R}^{T}
  46. = T \mathcal{I}=\mathcal{R}\mathcal{R}^{T}
  47. 0 = d d t T + d T d t 0=\frac{d\mathcal{R}}{dt}\mathcal{R}^{T}+\mathcal{R}\frac{d\mathcal{R}^{T}}{dt}
  48. 0 = d d t T + ( d d t T ) T = W + W T 0=\frac{d\mathcal{R}}{dt}\mathcal{R}^{T}+\left(\frac{d\mathcal{R}}{dt}\mathcal% {R}^{T}\right)^{T}=W+W^{T}
  49. W ( t ) = ( 0 - ω z ( t ) ω y ( t ) ω z ( t ) 0 - ω x ( t ) - ω y ( t ) ω x ( t ) 0 ) W(t)=\begin{pmatrix}0&-\omega_{z}(t)&\omega_{y}(t)\\ \omega_{z}(t)&0&-\omega_{x}(t)\\ -\omega_{y}(t)&\omega_{x}(t)&0\\ \end{pmatrix}
  50. ω \vec{\omega}
  51. s y m b o l ω = [ ω x , ω y , ω z ] symbol\omega=[\omega_{x},\omega_{y},\omega_{z}]
  52. t t
  53. 𝐫 ( t ) \mathbf{r}(t)
  54. 𝐯 ( t ) \mathbf{v}(t)
  55. 𝐯 = W 𝐫 \mathbf{v}=W\mathbf{r}
  56. t t
  57. 𝐯 \mathbf{v}
  58. 𝐫 \mathbf{r}
  59. V V
  60. ω \omega
  61. B ( 𝐫 , 𝐬 ) = ( W 𝐫 ) 𝐬 B(\mathbf{r},\mathbf{s})=(W\mathbf{r})\cdot\mathbf{s}
  62. \cdot
  63. L L
  64. Λ 2 V \Lambda^{2}V
  65. L ( 𝐫 𝐬 ) = B ( 𝐫 , 𝐬 ) L(\mathbf{r}\wedge\mathbf{s})=B(\mathbf{r},\mathbf{s})
  66. 𝐫 𝐬 Λ 2 V \mathbf{r}\wedge\mathbf{s}\in\Lambda^{2}V
  67. 𝐫 \mathbf{r}
  68. 𝐬 \mathbf{s}
  69. ( W 𝐫 ) 𝐬 = L * ( 𝐫 𝐬 ) (W\mathbf{r})\cdot\mathbf{s}=L^{*}\cdot(\mathbf{r}\wedge\mathbf{s})
  70. ω := * L * \omega:=*L^{*}
  71. ( W 𝐫 ) 𝐬 = * ( * L * 𝐫 𝐬 ) = * ( ω 𝐫 𝐬 ) = * ( ω 𝐫 ) 𝐬 = ( ω × 𝐫 ) 𝐬 (W\mathbf{r})\cdot\mathbf{s}=*(*L^{*}\wedge\mathbf{r}\wedge\mathbf{s})=*(% \omega\wedge\mathbf{r}\wedge\mathbf{s})=*(\omega\wedge\mathbf{r})\cdot\mathbf{% s}=(\omega\times\mathbf{r})\cdot\mathbf{s}
  72. ω × 𝐫 := * ( ω 𝐫 ) \omega\times\mathbf{r}:=*(\omega\wedge\mathbf{r})
  73. 𝐬 \mathbf{s}
  74. W 𝐫 = ω × 𝐫 W\mathbf{r}=\omega\times\mathbf{r}
  75. 𝐑 i = 𝐑 + 𝐫 i \mathbf{R}_{i}=\mathbf{R}+\mathbf{r}_{i}
  76. 𝐫 i \mathbf{r}_{i}
  77. 𝐫 i \mathbf{r}_{i}
  78. 𝐫 i o \mathcal{R}\mathbf{r}_{io}
  79. \mathcal{R}
  80. 𝐫 i o \mathbf{r}_{io}
  81. \mathcal{R}
  82. 𝐫 i o \mathbf{r}_{io}
  83. 𝐫 i \mathbf{r}_{i}
  84. 𝐑 i = 𝐑 + 𝐫 i o \mathbf{R}_{i}=\mathbf{R}+\mathcal{R}\mathbf{r}_{io}
  85. 𝐕 i = 𝐕 + d d t 𝐫 i o \mathbf{V}_{i}=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathbf{r}_{io}
  86. \mathcal{R}
  87. = T \mathcal{I}=\mathcal{R}^{T}\mathcal{R}
  88. 𝐕 i = 𝐕 + d d t 𝐫 i o \mathbf{V}_{i}=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{I}\mathbf{r}_{io}
  89. 𝐕 i = 𝐕 + d d t T 𝐫 i o \mathbf{V}_{i}=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}^{T}\mathcal{R}% \mathbf{r}_{io}
  90. 𝐕 i = 𝐕 + d d t T 𝐫 i \mathbf{V}_{i}=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}^{T}\mathbf{r}_{i}
  91. 𝐕 i = 𝐕 + W 𝐫 i \mathbf{V}_{i}=\mathbf{V}+W\mathbf{r}_{i}
  92. W = d d t T W=\frac{d\mathcal{R}}{dt}\mathcal{R}^{T}
  93. ω \vec{\omega}
  94. s y m b o l ω = [ ω x , ω y , ω z ] symbol\omega=[\omega_{x},\omega_{y},\omega_{z}]
  95. 𝐕 i = 𝐕 + s y m b o l ω × 𝐫 i \mathbf{V}_{i}=\mathbf{V}+symbol\omega\times\mathbf{r}_{i}
  96. 𝐯 1 \mathbf{v}_{1}
  97. 𝐯 2 \mathbf{v}_{2}
  98. s y m b o l ω 1 symbol{\omega}_{1}
  99. s y m b o l ω 2 symbol{\omega}_{2}
  100. 𝐯 1 + s y m b o l ω 1 × 𝐫 1 = 𝐯 2 + s y m b o l ω 2 × 𝐫 2 \mathbf{v}_{1}+symbol{\omega}_{1}\times\mathbf{r}_{1}=\mathbf{v}_{2}+symbol{% \omega}_{2}\times\mathbf{r}_{2}
  101. 𝐯 2 = 𝐯 1 + s y m b o l ω 1 × 𝐫 = 𝐯 1 + s y m b o l ω 1 × ( 𝐫 1 - 𝐫 2 ) \mathbf{v}_{2}=\mathbf{v}_{1}+symbol{\omega}_{1}\times\mathbf{r}=\mathbf{v}_{1% }+symbol{\omega}_{1}\times(\mathbf{r}_{1}-\mathbf{r}_{2})
  102. ( s y m b o l ω 1 - s y m b o l ω 2 ) × 𝐫 2 = 0 (symbol{\omega}_{1}-symbol{\omega}_{2})\times\mathbf{r}_{2}=0
  103. 𝐫 2 \mathbf{r}_{2}
  104. s y m b o l ω 1 = s y m b o l ω 2 symbol{\omega}_{1}=symbol{\omega}_{2}

Anisotropy.html

  1. γ ( Ω i , Ω v ) \gamma(\Omega_{i},\Omega_{v})
  2. P ( Ω i ) = Ω v γ ( Ω i , Ω v ) n ^ d Ω v ^ P(\Omega_{i})=\int_{\Omega_{v}}\gamma(\Omega_{i},\Omega_{v})\hat{n}\cdot d\hat% {\Omega_{v}}
  3. A ( Ω i , Ω v ) = γ ( Ω i , Ω v ) P ( Ω i ) A(\Omega_{i},\Omega_{v})=\frac{\gamma(\Omega_{i},\Omega_{v})}{P(\Omega_{i})}
  4. Ω v \Omega_{v}
  5. Ω i \Omega_{i}

Antenna_gain.html

  1. G = E a n t e n n a D . G=E_{antenna}\cdot D.
  2. P i n P_{in}
  3. E a n t e n n a E_{antenna}
  4. P o = E a n t e n n a P i n P_{o}=E_{antenna}\cdot P_{in}
  5. ( θ , ϕ ) (\theta,\phi)
  6. θ \theta
  7. ϕ \phi
  8. ( θ , ϕ ) (\theta,\phi)
  9. U ( θ , ϕ ) U(\theta,\phi)
  10. P o = - π π - π / 2 π / 2 U ( θ , ϕ ) d θ d ϕ . P_{o}=\int_{-\pi}^{\pi}\int_{-\pi/2}^{\pi/2}U(\theta,\phi)d\theta d\phi.
  11. U ¯ \overline{U}
  12. U ¯ = P o 4 π \overline{U}=\frac{P_{o}}{4\pi}~{}~{}
  13. = E a n t e n n a P i n 4 π =\frac{E_{antenna}\cdot P_{in}}{4\pi}
  14. P o P_{o}
  15. D ( θ , ϕ ) D(\theta,\phi)
  16. U ( θ , ϕ ) U(\theta,\phi)
  17. U ¯ \overline{U}
  18. D ( θ , ϕ ) = U ( θ , ϕ ) U ¯ . D(\theta,\phi)=\frac{U(\theta,\phi)}{\overline{U}}.
  19. D D
  20. D = max θ , ϕ D ( θ , ϕ ) . D=\max_{\theta,\phi}D(\theta,\phi).
  21. G ( θ , ϕ ) G(\theta,\phi)
  22. U ( θ , ϕ ) U(\theta,\phi)
  23. P i n / 4 π P_{in}/4\pi
  24. G ( θ , ϕ ) = U ( θ , ϕ ) P i n / 4 π G(\theta,\phi)=\frac{U(\theta,\phi)}{P_{in}/4\pi}
  25. = E a n t e n n a U ( θ , ϕ ) U ¯ =E_{antenna}\cdot\frac{U(\theta,\phi)}{\overline{U}}
  26. U ¯ \overline{U}
  27. = E a n t e n n a D ( θ , ϕ ) =E_{antenna}\cdot D(\theta,\phi)
  28. D ( θ , ϕ ) . D(\theta,\phi).
  29. G G
  30. E a n t e n n a E_{antenna}
  31. θ \theta
  32. ϕ \phi
  33. G = E a n t e n n a D . G=E_{antenna}\cdot D.
  34. P i n = V 2 R e { 1 Z i n } P_{in}=V^{2}\cdot Re\left\{\frac{1}{Z_{in}}\right\}
  35. G d B i = 10 log 10 ( G ) G_{dBi}=10\cdot\log_{10}\left(G\right)
  36. G d B d = 10 log 10 ( G 1.64 ) G_{dBd}=10\cdot\log_{10}\left(\frac{G}{1.64}\right)
  37. G d B d = G d B i - 2.15 d B G_{dBd}=G_{dBi}-2.15dB
  38. U U
  39. G θ = 4 π ( U θ P in ) G_{\theta}=4\pi\left(\frac{U_{\theta}}{P_{\mathrm{in}}}\right)
  40. G ϕ = 4 π ( U ϕ P in ) G_{\phi}=4\pi\left(\frac{U_{\phi}}{P_{\mathrm{in}}}\right)
  41. U θ U_{\theta}
  42. U ϕ U_{\phi}
  43. G = G θ + G ϕ G=G_{\theta}+G_{\phi}
  44. U = B 0 sin 3 ( θ ) U=B_{0}\,\sin^{3}(\theta)
  45. U max = B 0 U_{\mathrm{max}}=B_{0}
  46. P rad = 0 2 π 0 π U ( θ , ϕ ) sin ( θ ) d θ d ϕ = 2 π B 0 0 π sin 4 ( θ ) d θ = B 0 ( 3 π 2 4 ) P_{\mathrm{rad}}=\int_{0}^{2\pi}\int_{0}^{\pi}U(\theta,\phi)\sin(\theta)\,d% \theta\,d\phi=2\pi B_{0}\int_{0}^{\pi}\sin^{4}(\theta)\,d\theta=B_{0}\left(% \frac{3\pi^{2}}{4}\right)
  47. D = 4 π ( U max P rad ) = 4 π [ B 0 B 0 ( 3 π 2 4 ) ] = 16 3 π = 1.698 D=4\pi\left(\frac{U_{\mathrm{max}}}{P_{\mathrm{rad}}}\right)=4\pi\left[\frac{B% _{0}}{B_{0}\left(\frac{3\pi^{2}}{4}\right)}\right]=\frac{16}{3\pi}=1.698
  48. G = E a n t e n n a D = ( 1 ) ( 1.698 ) = 1.698 G=E_{antenna}\,D=(1)(1.698)=1.698
  49. G d B i = 10 log 10 ( 1.698 ) = 2.30 dBi G_{dBi}=10\,\log_{10}(1.698)=2.30\,\mathrm{dBi}
  50. G d B d = 10 log 10 ( 1.698 / 1.64 ) = 0.15 dBd G_{dBd}=10\,\log_{10}(1.698/1.64)=0.15\,\mathrm{dBd}

Antenna_noise_temperature.html

  1. T A = 3.468 F λ 2 10 G / 10 T_{A}=3.468F{{\lambda}^{2}}10^{G/10}
  2. F F
  3. λ \lambda
  4. G G

Anthropic_principle.html

  1. 5 + 2 k 5+2k

Anti-realism.html

  1. x . ϕ ( x ) \exists x.\phi(x)
  2. t t
  3. ϕ \phi

Antiderivative.html

  1. f f
  2. F F
  3. f f
  4. F F
  5. = =
  6. f f
  7. a b f ( x ) d x = F ( b ) - F ( a ) . \int_{a}^{b}f(x)\,\mathrm{d}x=F(b)-F(a).
  8. f ( x ) d x . \int f(x)\,\mathrm{d}x.
  9. F ( x ) = { - 1 x + C 1 x < 0 - 1 x + C 2 x > 0 F(x)=\begin{cases}-\frac{1}{x}+C_{1}\quad x<0\\ -\frac{1}{x}+C_{2}\quad x>0\end{cases}
  10. f ( x ) = 1 / x 2 f(x)=1/x^{2}
  11. ( - , 0 ) ( 0 , ) . (-\infty,0)\cup(0,\infty).
  12. F ( x ) = 0 x f ( t ) d t . F(x)=\int_{0}^{x}f(t)\,\mathrm{d}t.
  13. e - x 2 d x , sin x 2 d x , sin x x d x , 1 ln x d x , x x d x . \int e^{-x^{2}}\,\mathrm{d}x,\qquad\int\sin x^{2}\,\mathrm{d}x,\qquad\int\frac% {\sin x}{x}\,\mathrm{d}x,\qquad\int\frac{1}{\ln x}\,\mathrm{d}x,\qquad\int x^{% x}\,\mathrm{d}x.
  14. f - 1 f^{-1}
  15. f f
  16. f f
  17. f - 1 f^{-1}
  18. x 0 x x 0 x 1 x 0 x n - 1 f ( x n ) d x n d x 2 d x 1 = x 0 x f ( t ) ( x - t ) n - 1 ( n - 1 ) ! d t . \int_{x_{0}}^{x}\int_{x_{0}}^{x_{1}}\dots\int_{x_{0}}^{x_{n-1}}f(x_{n})\,% \mathrm{d}x_{n}\dots\,\mathrm{d}x_{2}\,\mathrm{d}x_{1}=\int_{x_{0}}^{x}f(t)% \frac{(x-t)^{n-1}}{(n-1)!}\,\mathrm{d}t.
  19. g ( x ) = F ( x ) - C x g(x)=F(x)-Cx
  20. 0 = g ( c ) = f ( c ) - C . 0=g^{\prime}(c)=f(c)-C.
  21. a = x 0 < x 1 < x 2 < < x n = b a=x_{0}<x_{1}<x_{2}<\dots<x_{n}=b
  22. x i * [ x i - 1 , x i ] x_{i}^{*}\in[x_{i-1},x_{i}]
  23. i = 1 n f ( x i * ) ( x i - x i - 1 ) \displaystyle\sum_{i=1}^{n}f(x_{i}^{*})(x_{i}-x_{i-1})
  24. x i * x_{i}^{*}

Antimatter_rocket.html

  1. I sp I_{\,\text{sp}}
  2. η e \eta_{e}
  3. I sp I_{\,\text{sp}}
  4. η e \eta_{e}
  5. I sp I_{\,\text{sp}}
  6. η e \eta_{e}
  7. M 0 M_{0}
  8. M 1 M_{1}
  9. M 0 M 1 \frac{M_{0}}{M_{1}}
  10. Δ v \Delta v
  11. I sp I_{\,\text{sp}}
  12. M 0 M 1 \frac{M_{0}}{M_{1}}
  13. Δ v \Delta v
  14. I sp I_{\,\text{sp}}
  15. M 0 M 1 = ( 1 + Δ v c 1 - Δ v c ) c 2 I sp \frac{M_{0}}{M_{1}}=\left(\frac{1+\frac{\Delta v}{c}}{1-\frac{\Delta v}{c}}% \right)^{\frac{c}{2I_{\,\text{sp}}}}
  16. c c
  17. I sp I_{\,\text{sp}}
  18. I sp I_{\,\text{sp}}
  19. c c
  20. d M ship M ship = - d v ( 1 - I sp v c 2 ) ( 1 - v 2 c 2 ) ( - I sp c 2 v 2 + ( 1 + a ) v + a I sp ) \frac{dM_{\,\text{ship}}}{M_{\,\text{ship}}}=\frac{-dv(1-I_{\,\text{sp}}\frac{% v}{c^{2}})}{(1-\frac{v^{2}}{c^{2}})(-\frac{I_{\,\text{sp}}}{c^{2}v2}+(1+a)v+aI% _{\,\text{sp}})}
  21. M ship M_{\,\text{ship}}
  22. a a
  23. a a
  24. v I sp v\sim I_{\,\text{sp}}
  25. ( 1 - I sp v c 2 ) ( 1 - v 2 c 2 ) (1-\frac{I_{\,\text{sp}}v}{c^{2}})\sim(1-\frac{v^{2}}{c^{2}})
  26. d M ship M ship = - d v ( - I sp c 2 v 2 + ( 1 - a ) v + a I sp ) \frac{dM_{\,\text{ship}}}{M_{\,\text{ship}}}=\frac{-dv}{(-\frac{I_{\,\text{sp}% }}{c^{2}v^{2}}+(1-a)v+aI_{\,\text{sp}})}
  27. M 0 M_{0}
  28. M 1 M_{1}
  29. v i = 0 v_{i}=0
  30. v f = Δ v v_{f}=\Delta v
  31. M 0 M 1 = ( ( - 2 I sp Δ v / c 2 + 1 - a - ( 1 - a ) 2 + 4 a I sp 2 / c 2 ) ( 1 - a + ( 1 - a ) 2 + 4 a I sp 2 / c 2 ) ( - 2 I sp Δ v / c 2 + 1 - a + ( 1 - a ) 2 + 4 a I sp 2 / c 2 ) ( 1 - a - ( 1 - a ) 2 + 4 a I sp 2 / c 2 ) ) 1 ( 1 - a ) 2 + 4 a I sp 2 / c 2 \frac{M_{0}}{M_{1}}=\left(\frac{(-2I_{\,\text{sp}}\Delta v/c^{2}+1-a-\sqrt{(1-% a)^{2}+4aI_{\,\text{sp}}^{2}/c^{2}})(1-a+\sqrt{(1-a)^{2}+4aI_{\,\text{sp}}^{2}% /c^{2}})}{(-2I_{\,\text{sp}}\Delta v/c^{2}+1-a+\sqrt{(1-a)^{2}+4aI_{\,\text{sp% }}^{2}/c^{2}})(1-a-\sqrt{(1-a)^{2}+4aI_{\,\text{sp}}^{2}/c^{2}})}\right)^{% \frac{1}{\sqrt{(1-a)^{2}+4aI_{\,\text{sp}}^{2}/c^{2}}}}
  32. c c

Antiparticle.html

  1. | p , σ , n |p,\sigma,n\rangle
  2. C P T | p , σ , n = ( - 1 ) J - σ | p , - σ , n c , CPT\ |p,\sigma,n\rangle\ =\ (-1)^{J-\sigma}\ |p,-\sigma,n^{c}\rangle,
  3. T | p , σ , n | - p , - σ , n , T\ |p,\sigma,n\rangle\ \propto\ |-p,-\sigma,n\rangle,
  4. C P | p , σ , n | - p , σ , n c , CP\ |p,\sigma,n\rangle\ \propto\ |-p,\sigma,n^{c}\rangle,
  5. C | p , σ , n | p , σ , n c , C\ |p,\sigma,n\rangle\ \propto\ |p,\sigma,n^{c}\rangle,
  6. ψ ( x ) = k u k ( x ) a k e - i E ( k ) t , \psi(x)=\sum_{k}u_{k}(x)a_{k}e^{-iE(k)t},\,
  7. H = k E ( k ) a k a k , H=\sum_{k}E(k)a^{\dagger}_{k}a_{k},\,
  8. b k = a k and b k = a k , b_{k\prime}=a^{\dagger}_{k}\ \mathrm{and}\ b^{\dagger}_{k\prime}=a_{k},\,
  9. ψ ( x ) = k + u k ( x ) a k e - i E ( k ) t + k - u k ( x ) b k e - i E ( k ) t , \psi(x)=\sum_{k_{+}}u_{k}(x)a_{k}e^{-iE(k)t}+\sum_{k_{-}}u_{k}(x)b^{\dagger}_{% k}e^{-iE(k)t},\,
  10. H = k + E k a k a k + k - | E ( k ) | b k b k + E 0 , H=\sum_{k_{+}}E_{k}a^{\dagger}_{k}a_{k}+\sum_{k_{-}}|E(k)|b^{\dagger}_{k}b_{k}% +E_{0},\,
  11. a k | 0 = 0 a_{k}|0\rangle=0
  12. b k | 0 = 0 b_{k}|0\rangle=0

Antiprism.html

  1. ( cos k π n , sin k π n , ( - 1 ) k h ) \left(\cos\frac{k\pi}{n},\sin\frac{k\pi}{n},(-1)^{k}h\right)
  2. 2 h 2 = cos π n - cos 2 π n . 2h^{2}=\cos\frac{\pi}{n}-\cos\frac{2\pi}{n}.
  3. V = n 4 cos 2 π 2 n - 1 sin 3 π 2 n 12 sin 2 π n a 3 V=\frac{n\sqrt{4\cos^{2}\frac{\pi}{2n}-1}\sin\frac{3\pi}{2n}}{12\sin^{2}\frac{% \pi}{n}}\;a^{3}
  4. A = n 2 ( cot π n + 3 ) a 2 . A=\frac{n}{2}(\cot{\frac{\pi}{n}}+\sqrt{3})a^{2}.

Antisymmetric_relation.html

  1. a , b X , R ( a , b ) and R ( b , a ) a = b \forall a,b\in X,\ R(a,b)\and R(b,a)\;\Rightarrow\;a=b
  2. a , b X , R ( a , b ) and a b ¬ R ( b , a ) . \forall a,b\in X,\ R(a,b)\and a\neq b\Rightarrow\lnot R(b,a).
  3. A B and B A A = B A\subseteq B\and B\subseteq A\Rightarrow A=B

Aperture.html

  1. Area = π ( D 2 ) 2 = π ( f 2 N ) 2 \mathrm{Area}=\pi\left({D\over 2}\right)^{2}=\pi\left({f\over 2N}\right)^{2}

APL_(programming_language).html

  1. i = 4 7 i \displaystyle\sum\limits_{i=4}^{7}i
  2. O ( R 2 ) O(R^{2})\,\!

Apollos.html

  1. 𝔓 \mathfrak{P}

Apparent_magnitude.html

  1. m x - m x , 0 = - 2.5 log 10 ( F x F x , 0 ) m_{x}-m_{x,0}=-2.5\log_{10}\left(\frac{F_{x}}{F_{x,0}}\right)\,
  2. F x F_{x}\!\,
  3. m x , 0 m_{x,0}
  4. F x , 0 F_{x,0}
  5. 2.512 \approx 2.512
  6. m 1 - m 2 = Δ m m_{1}-m_{2}=\Delta m
  7. F 2 / F 1 2.512 Δ m F_{2}/F_{1}\approx 2.512^{\Delta m}
  8. x = m 1 - m 2 = ( - 12.74 ) - ( - 26.74 ) = 14.00 x=m_{1}-m_{2}=(-12.74)-(-26.74)=14.00
  9. v b = 2.512 x = 2.512 14.00 400 , 000 v_{b}=2.512^{x}=2.512^{14.00}\approx 400,000
  10. 2.512 - m f = 2.512 - m 1 + 2.512 - m 2 2.512^{-m_{f}}=2.512^{-m_{1}}+2.512^{-m_{2}}\!
  11. m f m_{f}
  12. m f = - log 2.512 ( 2.512 - m 1 + 2.512 - m 2 ) m_{f}=-\log_{2.512}\left(2.512^{-m_{1}}+2.512^{-m_{2}}\right)\!
  13. m f m_{f}
  14. m 1 m_{1}
  15. m 2 m_{2}
  16. λ \lambda
  17. μ m \mu m
  18. Δ λ / λ \Delta\lambda/\lambda
  19. F x , 0 F_{x,0}
  20. F x , 0 F_{x,0}
  21. ( 10 - 20 erg/s/cm 2 /Hz ) (10^{-20}\,\text{ erg/s/cm}^{2}\,\text{/Hz})

Aqua_regia.html

  1. \rightleftharpoons

Arago_spot.html

  1. F = d 2 λ 1 F=\frac{d^{2}}{\ell\lambda}\gtrsim 1
  2. U ( P 1 ) = A e 𝐢 k r 0 r 0 S e 𝐢 k r 1 r 1 K ( χ ) d S , U(P_{1})=\frac{Ae^{\mathbf{i}kr_{0}}}{r_{0}}\int_{S}\frac{e^{\mathbf{i}kr_{1}}% }{r_{1}}K(\chi)\,dS,
  3. K ( χ ) K(\chi)
  4. K ( χ ) = 𝐢 2 λ ( 1 + cos ( χ ) ) K(\chi)=\frac{\mathbf{i}}{2\lambda}(1+\cos(\chi))
  5. k = 2 π λ k=\frac{2\pi}{\lambda}
  6. U ( P 1 ) = - 𝐢 λ A e 𝐢 k ( g + b ) g b 2 π a e 𝐢 k 1 2 ( 1 g + 1 b ) r 2 r d r . U(P_{1})=-\frac{\mathbf{i}}{\lambda}\frac{Ae^{\mathbf{i}k(g+b)}}{gb}2\pi\int_{% a}^{\infty}e^{\mathbf{i}k\frac{1}{2}\left(\frac{1}{g}+\frac{1}{b}\right)r^{2}}% r\,dr.
  7. χ \chi
  8. U ( P 1 ) = A e 𝐢 k g g b b 2 + a 2 e 𝐢 k b 2 + a 2 . U(P_{1})=\frac{Ae^{\mathbf{i}kg}}{g}\frac{b}{\sqrt{b^{2}+a^{2}}}e^{\mathbf{i}k% \sqrt{b^{2}+a^{2}}}.
  9. I 0 = | A e 𝐢 k g g | 2 I_{0}=\left|\frac{Ae^{\mathbf{i}kg}}{g}\right|^{2}
  10. I = | U ( P 1 ) | 2 I=\left|U(P_{1})\right|^{2}
  11. I = b 2 b 2 + a 2 I 0 . I=\frac{b^{2}}{b^{2}+a^{2}}I_{0}.
  12. g ( r , θ ) g(r,\theta)
  13. U ( P 1 ) 0 2 π 0 g ( r , θ ) e 𝐢 π ρ 2 λ ( 1 g + 1 b ) ρ d ρ d θ . U(P_{1})\propto\int_{0}^{2\pi}\int_{0}^{\infty}g(r,\theta)e^{\frac{\mathbf{i}% \pi\rho^{2}}{\lambda}\left(\frac{1}{g}+\frac{1}{b}\right)}\rho\,d\rho\,d\theta.
  14. θ 1 \theta_{1}
  15. r = s r=s
  16. r = t r=t
  17. R ( θ 1 ) e π 𝐢 s 2 / 2 - e π 𝐢 t 2 / 2 . R(\theta_{1})\propto e^{\pi\mathbf{i}s^{2}/2}-e^{\pi\mathbf{i}t^{2}/2}.
  18. I ( θ 1 ) I(\theta_{1})
  19. 2 π 2\pi
  20. U ( P 1 , r ) J 0 2 ( π r d λ b ) U(P_{1},r)\propto J_{0}^{2}\left(\frac{\pi rd}{\lambda b}\right)
  21. Δ r r 2 + λ g b g + b - r . \Delta r\approx\sqrt{r^{2}+\lambda\frac{gb}{g+b}}-r.

Arbitrage.html

  1. P ( V t 0 ) = 1 and P ( V t 0 ) > 0 P(V_{t}\geq 0)=1\,\text{ and }P(V_{t}\neq 0)>0\,
  2. V 0 = 0 V_{0}=0
  3. V t V_{t}

Archimedean_spiral.html

  1. r = a + b θ \,r=a+b\theta
  2. r = a + b θ 1 / c . r=a+b\theta^{1/c}.

Archimedes.html

  1. 1 / 7 {1}/{7}
  2. 10 / 71 {10}/{71}
  3. 265 / 153 {265}/{153}
  4. 1351 / 780 {1351}/{780}
  5. 4 / 3 {4}/{3}
  6. 1 / 4 {1}/{4}
  7. n = 0 4 - n = 1 + 4 - 1 + 4 - 2 + 4 - 3 + = 4 3 . \sum_{n=0}^{\infty}4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots={4\over 3}.\;
  8. 1 / 3 {1}/{3}
  9. × 10 6 3 \times 10^{6}3
  10. π \pi
  11. 223 / 71 {223}/{71}
  12. 22 / 7 {22}/{7}
  13. r r
  14. θ θ
  15. r = a + b θ \,r=a+b\theta
  16. a a
  17. b b
  18. 4 / 3 {4}/{3}
  19. π \pi
  20. r r
  21. π \pi
  22. r r
  23. π \pi
  24. r r
  25. π \pi
  26. r r
  27. r r
  28. 1 / 4 {1}/{4}
  29. × 10 2 06544 \times 10^{2}06544
  30. × 10 6 3 \times 10^{6}3

Area.html

  1. π \pi
  2. π \pi
  3. ( x i , y i ) (x_{i},y_{i})
  4. A = 1 2 | i = 0 n - 1 ( x i y i + 1 - x i + 1 y i ) | A=\frac{1}{2}|\sum_{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})|
  5. l l
  6. w w
  7. A = l w A=lw
  8. l = w l=w
  9. s s
  10. A = b h A=bh
  11. A = 1 2 b h A=\frac{1}{2}bh
  12. r r
  13. r r
  14. π r πr
  15. r × π r r×πr
  16. A = 2 - r r r 2 - x 2 d x = π r 2 A\;=\;2\int_{-r}^{r}\sqrt{r^{2}-x^{2}}\,dx\;=\;\pi r^{2}
  17. x x
  18. y y
  19. A = π x y A=\pi xy\,\!
  20. r r
  21. 1 2 B h \tfrac{1}{2}Bh
  22. s ( s - a ) ( s - b ) ( s - c ) \sqrt{s(s-a)(s-b)(s-c)}
  23. s = 1 2 ( a + b + c ) s=\tfrac{1}{2}(a+b+c)
  24. 1 2 a b sin ( C ) \tfrac{1}{2}ab\sin(C)
  25. C C
  26. a a
  27. b b
  28. 1 2 ( x 1 y 2 + x 2 y 3 + x 3 y 1 - x 2 y 1 - x 3 y 2 - x 1 y 3 ) \tfrac{1}{2}(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})
  29. i + b 2 - 1 i+\frac{b}{2}-1
  30. A = a b f ( x ) d x A=\int_{a}^{b}f(x)\,dx
  31. A = a b ( f ( x ) - g ( x ) ) d x A=\int_{a}^{b}(f(x)-g(x))\,dx
  32. f ( x ) f(x)
  33. A = 1 2 r 2 d θ A={1\over 2}\int r^{2}\,d\theta
  34. u ( t ) = ( x ( t ) , y ( t ) ) \vec{u}(t)=(x(t),y(t))
  35. u ( t 0 ) = u ( t 1 ) \vec{u}(t_{0})=\vec{u}(t_{1})
  36. t 0 t 1 x y ˙ d t = - t 0 t 1 y x ˙ d t = 1 2 t 0 t 1 ( x y ˙ - y x ˙ ) d t \oint_{t_{0}}^{t_{1}}x\dot{y}\,dt=-\oint_{t_{0}}^{t_{1}}y\dot{x}\,dt={1\over 2% }\oint_{t_{0}}^{t_{1}}(x\dot{y}-y\dot{x})\,dt
  37. 1 2 t 0 t 1 u × u ˙ d t . {1\over 2}\oint_{t_{0}}^{t_{1}}\vec{u}\times\dot{\vec{u}}\,dt.
  38. f ( x ) - g ( x ) = a x 2 + b x + c f(x)-g(x)=ax^{2}+bx+c
  39. Δ = b 2 - 4 a c . \Delta=b^{2}-4ac.
  40. A = Δ Δ 6 a 2 , a 0. A=\frac{\Delta\sqrt{\Delta}}{6a^{2}},\qquad a\neq 0.
  41. π r ( r + r 2 + h 2 ) \pi r\left(r+\sqrt{r^{2}+h^{2}}\right)
  42. π r 2 + π r l \pi r^{2}+\pi rl
  43. π r ( r + l ) \pi r(r+l)\,\!
  44. π r 2 \pi r^{2}
  45. π r l \pi rl
  46. 6 s 2 6s^{2}
  47. 2 π r ( r + h ) 2\pi r(r+h)
  48. π \pi
  49. π \pi
  50. B + P L 2 B+\frac{PL}{2}
  51. 2 ( w + h + w h ) 2(\ell w+\ell h+wh)
  52. \ell
  53. z = f ( x , y ) , z=f(x,y),
  54. ( x , y ) D 2 (x,y)\in D\subset\mathbb{R}^{2}
  55. D D
  56. A = D ( f x ) 2 + ( f y ) 2 + 1 d x d y . A=\iint_{D}\sqrt{\left(\frac{\partial f}{\partial x}\right)^{2}+\left(\frac{% \partial f}{\partial y}\right)^{2}+1}\,dx\,dy.
  57. 𝐫 = 𝐫 ( u , v ) , \mathbf{r}=\mathbf{r}(u,v),
  58. 𝐫 \mathbf{r}
  59. ( u , v ) D 2 (u,v)\in D\subset\mathbb{R}^{2}
  60. A = D | 𝐫 u × 𝐫 v | d u d v . A=\iint_{D}\left|\frac{\partial\mathbf{r}}{\partial u}\times\frac{\partial% \mathbf{r}}{\partial v}\right|\,du\,dv.
  61. 3 4 s 2 \frac{\sqrt{3}}{4}s^{2}\,\!
  62. s s
  63. s ( s - a ) ( s - b ) ( s - c ) \sqrt{s(s-a)(s-b)(s-c)}\,\!
  64. s s
  65. a a
  66. b b
  67. c c
  68. 1 2 a b sin ( C ) \tfrac{1}{2}ab\sin(C)\,\!
  69. a a
  70. b b
  71. C C
  72. 1 2 b h \tfrac{1}{2}bh\,\!
  73. b b
  74. h h
  75. 1 2 b a 2 - b 2 4 = b 4 4 a 2 - b 2 \frac{1}{2}b\sqrt{a^{2}-\frac{b^{2}}{4}}=\frac{b}{4}\sqrt{4a^{2}-b^{2}}
  76. a a
  77. b b
  78. 1 2 a b \tfrac{1}{2}ab
  79. a a
  80. b b
  81. b h bh\,\!
  82. b b
  83. h h
  84. ( a + b ) h 2 \frac{(a+b)h}{2}\,\!
  85. a a
  86. b b
  87. h h
  88. 3 2 3 s 2 \frac{3}{2}\sqrt{3}s^{2}\,\!
  89. s s
  90. 2 ( 1 + 2 ) s 2 2(1+\sqrt{2})s^{2}\,\!
  91. s s
  92. 1 4 n l 2 cot ( π / n ) \frac{1}{4}nl^{2}\cdot\cot(\pi/n)\,\!
  93. l l
  94. n n
  95. 1 4 n p 2 cot ( π / n ) \frac{1}{4n}p^{2}\cdot\cot(\pi/n)\,\!
  96. p p
  97. n n
  98. 1 2 n R 2 sin ( 2 π / n ) = n r 2 tan ( π / n ) \frac{1}{2}nR^{2}\cdot\sin(2\pi/n)=nr^{2}\tan(\pi/n)\,\!
  99. R R
  100. r r
  101. n n
  102. 1 2 a p = 1 2 n s a \tfrac{1}{2}ap=\tfrac{1}{2}nsa\,\!
  103. n n
  104. s s
  105. a a
  106. p p
  107. π r 2 or π d 2 4 \pi r^{2}\ \,\text{or}\ \frac{\pi d^{2}}{4}\,\!
  108. r r
  109. d d
  110. θ 2 r 2 or L r 2 \frac{\theta}{2}r^{2}\ \,\text{or}\ \frac{L\cdot r}{2}\,\!
  111. r r
  112. θ \theta
  113. L L
  114. π a b \pi ab\,\!
  115. a a
  116. b b
  117. 2 π r ( r + h ) 2\pi r(r+h)\,\!
  118. r r
  119. h h
  120. 2 π r h 2\pi rh\,\!
  121. r r
  122. h h
  123. 4 π r 2 or π d 2 4\pi r^{2}\ \,\text{or}\ \pi d^{2}\,\!
  124. r r
  125. d d
  126. B + P L 2 B+\frac{PL}{2}\,\!
  127. B B
  128. P P
  129. L L
  130. B + P L 2 B+\frac{PL}{2}\,\!
  131. B B
  132. P P
  133. L L
  134. 4 π A \frac{4}{\pi}A\,\!
  135. A A
  136. π 4 C \frac{\pi}{4}C\,\!
  137. C C
  138. 4 π A L 2 , 4\pi A\leq L^{2},
  139. π 3 3 \frac{\pi}{3\sqrt{3}}
  140. 1 12 3 , \frac{1}{12\sqrt{3}},

Arithmetic_coding.html

  1. - log 2 ( p i ) = - log 2 ( 0.6 ) - log 2 ( 0.1 ) - log 2 ( 0.1 ) = 7.381 bits \sum-\log_{2}(p_{i})=-\log_{2}(0.6)-\log_{2}(0.1)-\log_{2}(0.1)=7.381\,\text{ bits}
  2. D A B D D B DABDDB
  3. 6 5 × 3 + 6 4 × 0 + 6 3 × 1 + 6 2 × 3 + 6 1 × 3 + 6 0 × 1 = 23671 6^{5}\times 3+6^{4}\times 0+6^{3}\times 1+6^{2}\times 3+6^{1}\times 3+6^{0}% \times 1=23671
  4. L = \displaystyle\mathrm{L}=
  5. L = i = 1 n n n - i C i k = 1 i - 1 f k \mathrm{L}=\sum_{i=1}^{n}n^{n-i}C_{i}{\prod_{k=1}^{i-1}f_{k}}
  6. C i \scriptstyle C_{i}
  7. f k \scriptstyle f_{k}
  8. f k \scriptstyle f_{k}
  9. U = L + k = 1 n f k \mathrm{U}=\mathrm{L}+\prod_{k=1}^{n}f_{k}
  10. log 2 ( n n ) = n log 2 ( n ) \scriptstyle\log_{2}(n^{n})\;=\;n\log_{2}(n)
  11. log 2 ( k = 1 n f k ) \scriptstyle\log_{2}\left(\prod_{k=1}^{n}f_{k}\right)
  12. k = 1 n f k = k = 1 A f k f k \prod_{k=1}^{n}f_{k}=\prod_{k=1}^{A}f_{k}^{f_{k}}
  13. n log 2 ( n ) - i = 1 A f i log 2 ( f i ) n\log_{2}(n)-\sum_{i=1}^{A}f_{i}\log_{2}(f_{i})
  14. 1 - [ - 0.625 log 2 ( 0.625 ) + - 0.375 log 2 ( 0.375 ) ] 4.6 % . 1-[-0.625\log_{2}(0.625)+-0.375\log_{2}(0.375)]\approx 4.6\%.
  15. 1 - [ - 0.95 log 2 ( 0.95 ) + - 0.05 log 2 ( 0.05 ) ] 71.4 % . 1-[-0.95\log_{2}(0.95)+-0.05\log_{2}(0.05)]\approx 71.4\%.

Arithmetic_function.html

  1. χ ( n ) = ( - 4 n ) = { 0 if n is even , 1 if n 1 mod 4 , - 1 if n 3 mod 4 , \chi(n)=\left(\frac{-4}{n}\right)=\begin{cases}\;\;\,0&\,\text{if }n\,\text{ % is even},\\ \;\;\,1&\,\text{if }n\equiv 1\mod 4,\\ -1&\,\text{if }n\equiv 3\mod 4,\end{cases}
  2. ( - 4 n ) (\tfrac{-4}{n})
  3. p f ( p ) \sum_{p}f(p)\;
  4. p f ( p ) \prod_{p}f(p)\;
  5. p f ( p ) = f ( 2 ) + f ( 3 ) + f ( 5 ) + \sum_{p}f(p)=f(2)+f(3)+f(5)+\cdots
  6. p f ( p ) = f ( 2 ) f ( 3 ) f ( 5 ) . \prod_{p}f(p)=f(2)f(3)f(5)\cdots.
  7. p k f ( p k ) \sum_{p^{k}}f(p^{k})\;
  8. p k f ( p k ) \prod_{p^{k}}f(p^{k})\;
  9. p k f ( p k ) = f ( 2 ) + f ( 3 ) + f ( 4 ) + f ( 5 ) + f ( 7 ) + f ( 8 ) + f ( 9 ) + \sum_{p^{k}}f(p^{k})=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots
  10. d n f ( d ) \sum_{d\mid n}f(d)\;
  11. d n f ( d ) \prod_{d\mid n}f(d)\;
  12. d 12 f ( d ) = f ( 1 ) f ( 2 ) f ( 3 ) f ( 4 ) f ( 6 ) f ( 12 ) . \prod_{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).
  13. p n f ( p ) \sum_{p\mid n}f(p)\;
  14. p n f ( p ) \prod_{p\mid n}f(p)\;
  15. p 18 f ( p ) = f ( 2 ) + f ( 3 ) , \sum_{p\mid 18}f(p)=f(2)+f(3),
  16. p k n f ( p k ) \sum_{p^{k}\mid n}f(p^{k})\;
  17. p k n f ( p k ) \prod_{p^{k}\mid n}f(p^{k})\;
  18. p k 24 f ( p k ) = f ( 2 ) f ( 3 ) f ( 4 ) f ( 8 ) . \prod_{p^{k}\mid 24}f(p^{k})=f(2)f(3)f(4)f(8).
  19. n = p 1 a 1 p k a k n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}
  20. n = p p ν p ( n ) . n=\prod_{p}p^{\nu_{p}(n)}.
  21. i i
  22. i i
  23. i i
  24. i i
  25. i i
  26. σ k ( n ) = i = 1 ω ( n ) p i ( a i + 1 ) k - 1 p i k - 1 = i = 1 ω ( n ) ( 1 + p i k + p i 2 k + + p i a i k ) . \sigma_{k}(n)=\prod_{i=1}^{\omega(n)}\frac{p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}=% \prod_{i=1}^{\omega(n)}\left(1+p_{i}^{k}+p_{i}^{2k}+\cdots+p_{i}^{a_{i}k}% \right).
  27. τ ( n ) = d ( n ) = ( 1 + a 1 ) ( 1 + a 2 ) ( 1 + a ω ( n ) ) . \tau(n)=d(n)=(1+a_{1})(1+a_{2})\cdots(1+a_{\omega(n)}).
  28. φ ( n ) = n p n ( 1 - 1 p ) = n ( p 1 - 1 p 1 ) ( p 2 - 1 p 2 ) ( p ω ( n ) - 1 p ω ( n ) ) . \varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)=n\left(\frac{p_{1}-1}{p_% {1}}\right)\left(\frac{p_{2}-1}{p_{2}}\right)\cdots\left(\frac{p_{\omega(n)}-1% }{p_{\omega(n)}}\right).
  29. J k ( n ) = n k p n ( 1 - 1 p k ) = n k ( p 1 k - 1 p 1 k ) ( p 2 k - 1 p 2 k ) ( p ω ( n ) k - 1 p ω ( n ) k ) . J_{k}(n)=n^{k}\prod_{p\mid n}\left(1-\frac{1}{p^{k}}\right)=n^{k}\left(\frac{p% ^{k}_{1}-1}{p^{k}_{1}}\right)\left(\frac{p^{k}_{2}-1}{p^{k}_{2}}\right)\cdots% \left(\frac{p^{k}_{\omega(n)}-1}{p^{k}_{\omega(n)}}\right).
  30. μ ( n ) = { ( - 1 ) ω ( n ) = ( - 1 ) Ω ( n ) if ω ( n ) = Ω ( n ) 0 if ω ( n ) Ω ( n ) . \mu(n)=\begin{cases}(-1)^{\omega(n)}=(-1)^{\Omega(n)}&\,\text{if }\;\omega(n)=% \Omega(n)\\ 0&\,\text{if }\;\omega(n)\neq\Omega(n).\end{cases}
  31. n 1 τ ( n ) q n = q n 1 ( 1 - q n ) 24 . \sum_{n\geq 1}\tau(n)q^{n}=q\prod_{n\geq 1}(1-q^{n})^{24}.
  32. c q ( n ) = gcd ( a , q ) = 1 1 a q e 2 π i a q n . c_{q}(n)=\sum_{\stackrel{1\leq a\leq q}{\gcd(a,q)=1}}e^{2\pi i\tfrac{a}{q}n}.
  33. c q ( n ) c r ( n ) = c q r ( n ) . c_{q}(n)c_{r}(n)=c_{qr}(n).\;
  34. λ ( n ) = ( - 1 ) Ω ( n ) . \lambda(n)=(-1)^{\Omega(n)}.\;
  35. χ 0 ( a ) = { 1 if gcd ( a , n ) = 1 , 0 if gcd ( a , n ) 1. \chi_{0}(a)=\begin{cases}1&\,\text{if }\gcd(a,n)=1,\\ 0&\,\text{if }\gcd(a,n)\neq 1.\end{cases}
  36. ( a n ) = ( a p 1 ) a 1 ( a p 2 ) a 2 ( a p ω ( n ) ) a ω ( n ) . \Bigg(\frac{a}{n}\Bigg)=\left(\frac{a}{p_{1}}\right)^{a_{1}}\left(\frac{a}{p_{% 2}}\right)^{a_{2}}\cdots\left(\frac{a}{p_{\omega(n)}}\right)^{a_{\omega(n)}}.
  37. ( a p ) (\tfrac{a}{p})
  38. ( a p ) = { 0 if a 0 ( mod p ) + 1 if a 0 ( mod p ) and for some integer x , a x 2 ( mod p ) - 1 if there is no such x . \left(\frac{a}{p}\right)=\begin{cases}\;\;\,0\,\text{ if }a\equiv 0\;\;(% \mathop{{\rm mod}}p)\\ +1\,\text{ if }a\not\equiv 0\;\;(\mathop{{\rm mod}}p)\,\text{ and for some % integer }x,\;a\equiv x^{2}\;\;(\mathop{{\rm mod}}p)\\ -1\,\text{ if there is no such }x.\end{cases}
  39. ( a 1 ) = 1. \left(\frac{a}{1}\right)=1.
  40. π \pi
  41. π ( x ) = p x 1 \pi(x)=\sum_{p\leq x}1
  42. Π ( x ) = p k x 1 k . \Pi(x)=\sum_{p^{k}\leq x}\frac{1}{k}.
  43. ϑ ( x ) = p x log p , \vartheta(x)=\sum_{p\leq x}\log p,
  44. ψ ( x ) = p k x log p . \psi(x)=\sum_{p^{k}\leq x}\log p.
  45. Λ ( n ) = { log p if n = 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , = p k is a prime power 0 if n = 1 , 6 , 10 , 12 , 14 , 15 , 18 , 20 , 21 , is not a prime power . \Lambda(n)=\begin{cases}\log p&\,\text{if }n=2,3,4,5,7,8,9,11,13,16,\ldots=p^{% k}\,\text{ is a prime power}\\ 0&\,\text{if }n=1,6,10,12,14,15,18,20,21,\dots\;\;\;\;\,\text{ is not a prime % power}.\end{cases}
  46. p ( n ) = | { ( a 1 , a 2 , a k ) : 0 < a 1 a 2 a k and n = a 1 + a 2 + + a k } | . p(n)=|\left\{(a_{1},a_{2},\dots a_{k}):0<a_{1}\leq a_{2}\leq\cdots\leq a_{k}\;% \and\;n=a_{1}+a_{2}+\cdots+a_{k}\right\}|.
  47. a λ ( n ) 1 ( mod n ) a^{\lambda(n)}\equiv 1\;\;(\mathop{{\rm mod}}n)
  48. λ ( n ) = { ϕ ( n ) if n = 2 , 3 , 4 , 5 , 7 , 9 , 11 , 13 , 17 , 19 , 23 , 25 , 27 , 1 2 ϕ ( n ) if n = 8 , 16 , 32 , 64 , \lambda(n)=\begin{cases}\;\;\phi(n)&\,\text{if }n=2,3,4,5,7,9,11,13,17,19,23,2% 5,27,\dots\\ \tfrac{1}{2}\phi(n)&\,\text{if }n=8,16,32,64,\dots\end{cases}
  49. λ ( p 1 a 1 p 2 a 2 p ω ( n ) a ω ( n ) ) = lcm [ λ ( p 1 a 1 ) , λ ( p 2 a 2 ) , , λ ( p ω ( n ) a ω ( n ) ) ] . \lambda(p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega(n)}^{a_{\omega(n)}})=% \operatorname{lcm}[\lambda(p_{1}^{a_{1}}),\;\lambda(p_{2}^{a_{2}}),\dots,% \lambda(p_{\omega(n)}^{a_{\omega(n)}})].
  50. r k ( n ) = | { ( a 1 , a 2 , , a k ) : n = a 1 2 + a 2 2 + + a k 2 } | r_{k}(n)=|\{(a_{1},a_{2},\dots,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\cdots+a_{k}^{2}\}|
  51. A ( x ) := n x a ( n ) . A(x):=\sum_{n\leq x}a(n).
  52. lim inf n d ( n ) = 2 \liminf_{n\to\infty}d(n)=2
  53. lim sup n log d ( n ) log log n log n = log 2 \limsup_{n\to\infty}\frac{\log d(n)\log\log n}{\log n}=\log 2
  54. lim n d ( 1 ) + d ( 2 ) + + d ( n ) log ( 1 ) + log ( 2 ) + + log ( n ) = 1. \lim_{n\to\infty}\frac{d(1)+d(2)+\cdots+d(n)}{\log(1)+\log(2)+\cdots+\log(n)}=1.
  55. n x f ( n ) n x g ( n ) \sum_{n\leq x}f(n)\sim\sum_{n\leq x}g(n)
  56. F a ( s ) := n = 1 a ( n ) n s . F_{a}(s):=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}.
  57. ζ ( s ) n = 1 μ ( n ) n s = 1 , s > 0. \zeta(s)\,\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=1,\;\;\mathfrak{R}\,s>0.
  58. F a ( s ) F b ( s ) = ( m = 1 a ( m ) m s ) ( n = 1 b ( n ) n s ) . F_{a}(s)F_{b}(s)=\left(\sum_{m=1}^{\infty}\frac{a(m)}{m^{s}}\right)\left(\sum_% {n=1}^{\infty}\frac{b(n)}{n^{s}}\right).
  59. c ( n ) := i j = n a ( i ) b ( j ) = i n a ( i ) b ( n i ) , c(n):=\sum_{ij=n}a(i)b(j)=\sum_{i\mid n}a(i)b\left(\frac{n}{i}\right),
  60. F c ( s ) = F a ( s ) F b ( s ) . F_{c}(s)=F_{a}(s)F_{b}(s).\;
  61. a * b a*b
  62. g ( n ) = d n f ( d ) . g(n)=\sum_{d\mid n}f(d).\;
  63. f ( n ) = d n μ ( n d ) g ( d ) . f(n)=\sum_{d\mid n}\mu\left(\frac{n}{d}\right)g(d).
  64. δ n μ ( δ ) = δ n λ ( n δ ) | μ ( δ ) | = { 1 if n = 1 0 if n 1. \sum_{\delta\mid n}\mu(\delta)=\sum_{\delta\mid n}\lambda\left(\frac{n}{\delta% }\right)|\mu(\delta)|=\begin{cases}&1\,\text{ if }n=1\\ &0\,\text{ if }n\neq 1.\end{cases}
  65. δ n φ ( δ ) = n . \sum_{\delta\mid n}\varphi(\delta)=n.
  66. φ ( n ) = δ n μ ( n δ ) δ = n δ n μ ( δ ) δ . \varphi(n)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\delta=n\sum_{% \delta\mid n}\frac{\mu(\delta)}{\delta}.
  67. d n J k ( d ) = n k . \sum_{d\mid n}J_{k}(d)=n^{k}.\,
  68. J k ( n ) = δ n μ ( n δ ) δ k = n k δ n μ ( δ ) δ k . J_{k}(n)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\delta^{k}=n^{k}% \sum_{\delta\mid n}\frac{\mu(\delta)}{\delta^{k}}.
  69. δ n δ s J r ( δ ) J s ( n δ ) = J r + s ( n ) \sum_{\delta\mid n}\delta^{s}J_{r}(\delta)J_{s}\left(\frac{n}{\delta}\right)=J% _{r+s}(n)
  70. δ n φ ( δ ) d ( n δ ) = σ ( n ) . \sum_{\delta\mid n}\varphi(\delta)d\left(\frac{n}{\delta}\right)=\sigma(n).
  71. δ n | μ ( δ ) | = 2 ω ( n ) . \sum_{\delta\mid n}|\mu(\delta)|=2^{\omega(n)}.
  72. | μ ( n ) | = δ n μ ( n δ ) 2 ω ( δ ) . |\mu(n)|=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)2^{\omega(\delta)}.
  73. δ n 2 ω ( δ ) = d ( n 2 ) . \sum_{\delta\mid n}2^{\omega(\delta)}=d(n^{2}).
  74. 2 ω ( n ) = δ n μ ( n δ ) d ( δ 2 ) . 2^{\omega(n)}=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)d(\delta^{2}).
  75. δ n d ( δ 2 ) = d 2 ( n ) . \sum_{\delta\mid n}d(\delta^{2})=d^{2}(n).
  76. d ( n 2 ) = δ n μ ( n δ ) d 2 ( δ ) . d(n^{2})=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)d^{2}(\delta).
  77. δ n d ( n δ ) 2 ω ( δ ) = d 2 ( n ) . \sum_{\delta\mid n}d\left(\frac{n}{\delta}\right)2^{\omega(\delta)}=d^{2}(n).
  78. δ n λ ( δ ) = { 1 if n is a square 0 if n is not square. \sum_{\delta\mid n}\lambda(\delta)=\begin{cases}&1\,\text{ if }n\,\text{ is a % square }\\ &0\,\text{ if }n\,\text{ is not square.}\end{cases}
  79. δ n Λ ( δ ) = log n . \sum_{\delta\mid n}\Lambda(\delta)=\log n.
  80. Λ ( n ) = δ n μ ( n δ ) log ( δ ) . \Lambda(n)=\sum_{\delta\mid n}\mu\left(\frac{n}{\delta}\right)\log(\delta).
  81. If k 4 , r k ( n ) > 0. \,\text{If }k\geq 4,\;\;\;r_{k}(n)>0.
  82. r 2 ( n ) = 4 d n χ ( d ) , r_{2}(n)=4\sum_{d\mid n}\chi(d),\;
  83. r 4 ( n ) = 8 4 d d n d = 8 ( 2 + ( - 1 ) n ) 2 d d n d = { 8 σ ( n ) if n is odd 24 σ ( n 2 ν ) if n is even , r_{4}(n)=8\sum_{\stackrel{d\mid n}{4\,\nmid\,d}}d=8(2+(-1)^{n})\sum_{\stackrel% {d\mid n}{2\,\nmid\,d}}d=\begin{cases}8\sigma(n)&\,\text{if }n\,\text{ is odd % }\\ 24\sigma\left(\frac{n}{2^{\nu}}\right)&\,\text{if }n\,\text{ is even }\end{% cases},
  84. r 6 ( n ) = 16 d n χ ( n d ) d 2 - 4 d n χ ( d ) d 2 . r_{6}(n)=16\sum_{d\mid n}\chi\left(\frac{n}{d}\right)d^{2}-4\sum_{d\mid n}\chi% (d)d^{2}.
  85. σ k * ( n ) = ( - 1 ) n d n ( - 1 ) d d k = { d n d k = σ k ( n ) if n is odd 2 d d n d k - 2 d d n d k if n is even . \sigma_{k}^{*}(n)=(-1)^{n}\sum_{d\mid n}(-1)^{d}d^{k}=\begin{cases}\sum_{d\mid n% }d^{k}=\sigma_{k}(n)&\,\text{if }n\,\text{ is odd }\\ \sum_{\stackrel{d\mid n}{2\,\mid\,d}}d^{k}-\sum_{\stackrel{d\mid n}{2\,\nmid\,% d}}d^{k}&\,\text{if }n\,\text{ is even}.\end{cases}
  86. r 8 ( n ) = 16 σ 3 * ( n ) . r_{8}(n)=16\sigma_{3}^{*}(n).\;
  87. r 24 ( n ) = 16 691 σ 11 * ( n ) + 128 691 { ( - 1 ) n - 1 259 τ ( n ) - 512 τ ( n 2 ) } r_{24}(n)=\frac{16}{691}\sigma_{11}^{*}(n)+\frac{128}{691}\left\{(-1)^{n-1}259% \tau(n)-512\tau\left(\frac{n}{2}\right)\right\}
  88. ( n = 0 a n x n ) ( n = 0 b n x n ) = i = 0 j = 0 a i b j x i + j = n = 0 ( i = 0 n a i b n - i ) x n = n = 0 c n x n . \left(\sum_{n=0}^{\infty}a_{n}x^{n}\right)\left(\sum_{n=0}^{\infty}b_{n}x^{n}% \right)=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{i}b_{j}x^{i+j}=\sum_{n=0}^{% \infty}\left(\sum_{i=0}^{n}a_{i}b_{n-i}\right)x^{n}=\sum_{n=0}^{\infty}c_{n}x^% {n}.
  89. c n = i = 0 n a i b n - i c_{n}=\sum_{i=0}^{n}a_{i}b_{n-i}\;
  90. σ 3 ( n ) = 1 5 { 6 n σ 1 ( n ) - σ 1 ( n ) + 12 0 < k < n σ 1 ( k ) σ 1 ( n - k ) } . \sigma_{3}(n)=\frac{1}{5}\left\{6n\sigma_{1}(n)-\sigma_{1}(n)+12\sum_{0<k<n}% \sigma_{1}(k)\sigma_{1}(n-k)\right\}.\;
  91. σ 5 ( n ) = 1 21 { 10 ( 3 n - 1 ) σ 3 ( n ) + σ 1 ( n ) + 240 0 < k < n σ 1 ( k ) σ 3 ( n - k ) } . \sigma_{5}(n)=\frac{1}{21}\left\{10(3n-1)\sigma_{3}(n)+\sigma_{1}(n)+240\sum_{% 0<k<n}\sigma_{1}(k)\sigma_{3}(n-k)\right\}.\;
  92. σ 7 ( n ) = 1 20 { 21 ( 2 n - 1 ) σ 5 ( n ) - σ 1 ( n ) + 504 0 < k < n σ 1 ( k ) σ 5 ( n - k ) } = σ 3 ( n ) + 120 0 < k < n σ 3 ( k ) σ 3 ( n - k ) . \begin{aligned}\displaystyle\sigma_{7}(n)&\displaystyle=\frac{1}{20}\left\{21(% 2n-1)\sigma_{5}(n)-\sigma_{1}(n)+504\sum_{0<k<n}\sigma_{1}(k)\sigma_{5}(n-k)% \right\}\\ &\displaystyle=\sigma_{3}(n)+120\sum_{0<k<n}\sigma_{3}(k)\sigma_{3}(n-k).\end{aligned}
  93. σ 9 ( n ) = 1 11 { 10 ( 3 n - 2 ) σ 7 ( n ) + σ 1 ( n ) + 480 0 < k < n σ 1 ( k ) σ 7 ( n - k ) } = 1 11 { 21 σ 5 ( n ) - 10 σ 3 ( n ) + 5040 0 < k < n σ 3 ( k ) σ 5 ( n - k ) } . \begin{aligned}\displaystyle\sigma_{9}(n)&\displaystyle=\frac{1}{11}\left\{10(% 3n-2)\sigma_{7}(n)+\sigma_{1}(n)+480\sum_{0<k<n}\sigma_{1}(k)\sigma_{7}(n-k)% \right\}\\ &\displaystyle=\frac{1}{11}\left\{21\sigma_{5}(n)-10\sigma_{3}(n)+5040\sum_{0<% k<n}\sigma_{3}(k)\sigma_{5}(n-k)\right\}.\end{aligned}
  94. τ ( n ) = 65 756 σ 11 ( n ) + 691 756 σ 5 ( n ) - 691 3 0 < k < n σ 5 ( k ) σ 5 ( n - k ) , \tau(n)=\frac{65}{756}\sigma_{11}(n)+\frac{691}{756}\sigma_{5}(n)-\frac{691}{3% }\sum_{0<k<n}\sigma_{5}(k)\sigma_{5}(n-k),\;
  95. p ( n ) = 1 n 1 k n σ ( k ) p ( n - k ) . p(n)=\frac{1}{n}\sum_{1\leq k\leq n}\sigma(k)p(n-k).
  96. ( a 2 ) = { 0 if a is even ( - 1 ) a 2 - 1 8 if a is odd. \left(\frac{a}{2}\right)=\begin{cases}\;\;\,0&\,\text{ if }a\,\text{ is even}% \\ (-1)^{\frac{a^{2}-1}{8}}&\,\text{ if }a\,\text{ is odd. }\end{cases}
  97. h ( D ) = 1 D r = 1 | D | r ( D r ) = 1 2 - ( D 2 ) r = 1 | D | / 2 ( D r ) . \begin{aligned}\displaystyle h(D)&\displaystyle=\frac{1}{D}\sum_{r=1}^{|D|}r% \left(\frac{D}{r}\right)\\ &\displaystyle=\frac{1}{2-\left(\tfrac{D}{2}\right)}\sum_{r=1}^{|D|/2}\left(% \frac{D}{r}\right).\end{aligned}
  98. r 3 ( | D | ) = 12 ( 1 - ( D 2 ) ) h ( D ) . r_{3}(|D|)=12\left(1-\left(\frac{D}{2}\right)\right)h(D).
  99. H n = 1 + 1 2 + 1 3 + + 1 n H_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}
  100. σ ( n ) H n + e H n log H n \sigma(n)\leq H_{n}+e^{H_{n}}\log H_{n}
  101. σ ( n ) < e γ n log log n \sigma(n)<e^{\gamma}n\log\log n\,
  102. p ν p ( n ) = Ω ( n ) . \sum_{p}\nu_{p}(n)=\Omega(n).\;
  103. ψ ( x ) = n x Λ ( n ) . \psi(x)=\sum_{n\leq x}\Lambda(n).\;
  104. Π ( x ) = n x Λ ( n ) log n . \Pi(x)=\sum_{n\leq x}\frac{\Lambda(n)}{\log n}.\;
  105. e θ ( x ) = p x p . e^{\theta(x)}=\prod_{p\leq x}p.\;
  106. e ψ ( x ) = lcm [ 1 , 2 , , x ] . e^{\psi(x)}=\operatorname{lcm}[1,2,\dots,\lfloor x\rfloor].\;
  107. gcd ( k , n ) = 1 1 k n gcd ( k - 1 , n ) = φ ( n ) d ( n ) . \sum_{\stackrel{1\leq k\leq n}{\gcd(k,n)=1}}\gcd(k-1,n)=\varphi(n)d(n).
  108. gcd ( k 1 , n ) = 1 1 k 1 , k 2 , , k s n gcd ( k 1 - 1 , k 2 , , k s , n ) = φ ( n ) σ s - 1 ( n ) . \sum_{\stackrel{1\leq k_{1},k_{2},\dots,k_{s}\leq n}{\gcd(k_{1},n)=1}}\gcd(k_{% 1}-1,k_{2},\dots,k_{s},n)=\varphi(n)\sigma_{s-1}(n).
  109. gcd ( k 1 , k 2 , , k s , n ) = 1 1 k 1 , k 2 , , k s n gcd ( k 1 - a 1 , k 2 - a 2 , , k s - a s , n ) s = J s ( n ) d ( n ) , \sum_{\stackrel{1\leq k_{1},k_{2},\dots,k_{s}\leq n}{\gcd(k_{1},k_{2},\dots,k_% {s},n)=1}}\gcd(k_{1}-a_{1},k_{2}-a_{2},\dots,k_{s}-a_{s},n)^{s}=J_{s}(n)d(n),
  110. gcd ( k , m ) = 1 1 k m gcd ( k 2 - 1 , m 1 ) gcd ( k 2 - 1 , m 2 ) = φ ( n ) d 2 m 2 d 1 m 1 φ ( gcd ( d 1 , d 2 ) ) 2 ω ( lcm ( d 1 , d 2 ) ) , \sum_{\stackrel{1\leq k\leq m}{\gcd(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_% {2})=\varphi(n)\sum_{\stackrel{d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi(\gcd(% d_{1},d_{2}))2^{\omega(\operatorname{lcm}(d_{1},d_{2}))},
  111. gcd ( k , n ) = 1 1 k n f ( gcd ( k - 1 , n ) ) = φ ( n ) d n ( μ * f ) ( d ) φ ( d ) , \sum_{\stackrel{1\leq k\leq n}{\gcd(k,n)=1}}f(\gcd(k-1,n))=\varphi(n)\sum_{d% \mid n}\frac{(\mu*f)(d)}{\varphi(d)},
  112. ( m n ) ( n m ) = ( - 1 ) ( m - 1 ) ( n - 1 ) / 4 . \left(\frac{m}{n}\right)\left(\frac{n}{m}\right)=(-1)^{(m-1)(n-1)/4}.
  113. | λ ( n ) | μ ( n ) = λ ( n ) | μ ( n ) | = μ ( n ) , |\lambda(n)|\mu(n)=\lambda(n)|\mu(n)|=\mu(n),
  114. λ ( n ) μ ( n ) = | μ ( n ) | = μ 2 ( n ) . \lambda(n)\mu(n)=|\mu(n)|=\mu^{2}(n).
  115. λ ( n ) ϕ ( n ) . \lambda(n)\mid\phi(n).
  116. λ ( n ) = ϕ ( n ) if and only if n = { 1 , 2 , 4 ; 3 , 5 , 7 , 9 , 11 , i.e. p k where p is an odd prime ; 6 , 10 , 14 , 18 , i.e. 2 p k where p is an odd prime . \lambda(n)=\phi(n)\,\text{ if and only if }n=\begin{cases}1,2,4;\\ 3,5,7,9,11,\ldots\,\text{ i.e. }p^{k}\,\text{ where }p\,\text{ is an odd prime% };\\ 6,10,14,18,\ldots\,\text{ i.e. }2p^{k}\,\text{ where }p\,\text{ is an odd % prime}.\end{cases}
  117. 2 ω ( n ) d ( n ) 2 Ω ( n ) . 2^{\omega(n)}\leq d(n)\leq 2^{\Omega(n)}.\;
  118. 6 π 2 < ϕ ( n ) σ ( n ) n 2 < 1. \frac{6}{\pi^{2}}<\frac{\phi(n)\sigma(n)}{n^{2}}<1.\;
  119. c q ( n ) = μ ( q gcd ( q , n ) ) ϕ ( q gcd ( q , n ) ) ϕ ( q ) = δ gcd ( q , n ) μ ( q δ ) δ . \begin{aligned}\displaystyle c_{q}(n)&\displaystyle=\frac{\mu\left(\frac{q}{% \gcd(q,n)}\right)}{\phi\left(\frac{q}{\gcd(q,n)}\right)}\phi(q)\\ &\displaystyle=\sum_{\delta\mid\gcd(q,n)}\mu\left(\frac{q}{\delta}\right)% \delta.\end{aligned}
  120. ϕ ( q ) = δ q μ ( q δ ) δ . \phi(q)=\sum_{\delta\mid q}\mu\left(\frac{q}{\delta}\right)\delta.
  121. c q ( 1 ) = μ ( q ) . c_{q}(1)=\mu(q).\;
  122. c q ( q ) = ϕ ( q ) . c_{q}(q)=\phi(q).\;
  123. δ n d 3 ( δ ) = ( δ n d ( δ ) ) 2 . \sum_{\delta\mid n}d^{\;3}(\delta)=\left(\sum_{\delta\mid n}d(\delta)\right)^{% 2}.\;
  124. d ( u v ) = δ gcd ( u , v ) μ ( δ ) d ( u δ ) d ( v δ ) . d(uv)=\sum_{\delta\mid\gcd(u,v)}\mu(\delta)d\left(\frac{u}{\delta}\right)d% \left(\frac{v}{\delta}\right).\;
  125. σ k ( u ) σ k ( v ) = δ gcd ( u , v ) δ k σ k ( u v δ 2 ) . \sigma_{k}(u)\sigma_{k}(v)=\sum_{\delta\mid\gcd(u,v)}\delta^{k}\sigma_{k}\left% (\frac{uv}{\delta^{2}}\right).\;
  126. τ ( u ) τ ( v ) = δ gcd ( u , v ) δ 11 τ ( u v δ 2 ) , \tau(u)\tau(v)=\sum_{\delta\mid\gcd(u,v)}\delta^{11}\tau\left(\frac{uv}{\delta% ^{2}}\right),\;

Arithmetic_mean.html

  1. a 1 , , a n . a_{1},\ldots,a_{n}.
  2. A A
  3. A = 1 n i = 1 n a i A=\frac{1}{n}\sum_{i=1}^{n}a_{i}
  4. 2500 + 2700 + 2400 + 2300 + 2550 + 2650 + 2750 + 2450 + 2600 + 2400 10 = 2530. \frac{2500+2700+2400+2300+2550+2650+2750+2450+2600+2400}{10}=2530.
  5. x ¯ \bar{x}
  6. x x
  7. n n
  8. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  9. x 1 , , x n x_{1},\ldots,x_{n}
  10. x ¯ \bar{x}
  11. ( x 1 - x ¯ ) + + ( x n - x ¯ ) = 0 (x_{1}-\bar{x})+\cdots+(x_{n}-\bar{x})=0
  12. x i - x ¯ x_{i}-\bar{x}
  13. x 1 , , x n x_{1},\ldots,x_{n}
  14. ( x i - x ¯ ) 2 (x_{i}-\bar{x})^{2}
  15. 1 , 2 , 3 , 4 {1,2,3,4}
  16. 2.5 2.5
  17. 1 , 2 , 4 , 8 , 16 {1,2,4,8,16}
  18. 3 3
  19. 5 5
  20. ( 3 + 5 ) 2 = 4 \frac{(3+5)}{2}=4
  21. ( 1 2 3 ) + ( 1 2 5 ) = 4 \left(\frac{1}{2}\cdot 3\right)+\left(\frac{1}{2}\cdot 5\right)=4
  22. ( 2 3 3 ) + ( 1 3 5 ) = 11 3 \left(\frac{2}{3}\cdot 3\right)+\left(\frac{1}{3}\cdot 5\right)=\frac{11}{3}
  23. ( 2 / 3 ) (2/3)
  24. ( 1 / 3 ) (1/3)
  25. 1 2 \frac{1}{2}
  26. 1 n \frac{1}{n}
  27. n n

Arithmetic–geometric_mean.html

  1. x x
  2. y y
  3. x x
  4. y y
  5. x x
  6. y y
  7. x y xy
  8. a 1 \displaystyle a_{1}
  9. x x
  10. y y
  11. a n + 1 \displaystyle a_{n+1}
  12. x x
  13. y y
  14. M ( x , y ) M(x,y)
  15. a g m ( x , y ) agm(x,y)
  16. a 1 \displaystyle a_{1}
  17. a 2 \displaystyle a_{2}
  18. n n
  19. a < s u b > n a<sub>n
  20. x y x≠y
  21. M ( x , y ) M(x,y)
  22. x x
  23. y y
  24. x x
  25. y y
  26. r 0 r≥0
  27. M ( r x , r y ) = r M ( x , y ) M(rx,ry)=rM(x,y)
  28. M ( x , y ) M(x,y)
  29. M ( x , y ) \displaystyle M(x,y)
  30. K ( k ) K(k)
  31. K ( k ) = 0 π 2 d θ 1 - k 2 sin 2 ( θ ) K(k)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}(\theta)}}
  32. 1 M ( 1 , 2 ) = G = 0.8346268 \frac{1}{M(1,\sqrt{2})}=G=0.8346268\dots
  33. g n a n g_{n}\leqslant a_{n}
  34. g n + 1 = g n a n g n g n = g n g_{n+1}=\sqrt{g_{n}\cdot a_{n}}\geqslant\sqrt{g_{n}\cdot g_{n}}=g_{n}
  35. x x
  36. y y
  37. g g
  38. lim n g n = g \lim_{n\to\infty}g_{n}=g
  39. a n = g n + 1 2 g n a_{n}=\frac{g_{n+1}^{2}}{g_{n}}
  40. lim n a n = lim n g n + 1 2 g n = g 2 g = g \lim_{n\to\infty}a_{n}=\lim_{n\to\infty}\frac{g_{n+1}^{2}}{g_{n}}=\frac{g^{2}}% {g}=g
  41. I ( x , y ) = 0 π / 2 d θ x 2 cos 2 θ + y 2 sin 2 θ , I(x,y)=\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{x^{2}\cos^{2}\theta+y^{2}\sin^{2}% \theta}},
  42. θ \theta^{\prime}
  43. sin θ = 2 x sin θ ( x + y ) + ( x - y ) sin 2 θ , \sin\theta=\frac{2x\sin\theta^{\prime}}{(x+y)+(x-y)\sin^{2}\theta^{\prime}},
  44. I ( x , y ) = 0 π / 2 d θ ( 1 2 ( x + y ) ) 2 cos 2 θ + ( x y ) 2 sin 2 θ = I ( 1 2 ( x + y ) , x y ) . \begin{aligned}\displaystyle I(x,y)&\displaystyle=\int_{0}^{\pi/2}\frac{d% \theta^{\prime}}{\sqrt{\bigl(\frac{1}{2}(x+y)\bigr)^{2}\cos^{2}\theta^{\prime}% +\bigl(\sqrt{xy}\bigr)^{2}\sin^{2}\theta^{\prime}}}\\ &\displaystyle=I\bigl(\tfrac{1}{2}(x+y),\sqrt{xy}\bigr).\end{aligned}
  45. I ( x , y ) = I ( a 1 , g 1 ) = I ( a 2 , g 2 ) = = I ( M ( x , y ) , M ( x , y ) ) = π / ( 2 M ( x , y ) ) . \begin{aligned}\displaystyle I(x,y)&\displaystyle=I(a_{1},g_{1})=I(a_{2},g_{2}% )=\cdots\\ &\displaystyle=I\bigl(M(x,y),M(x,y)\bigr)=\pi/\bigr(2M(x,y)\bigl).\end{aligned}
  46. I ( z , z ) = π / ( 2 z ) I(z,z)=\pi/(2z)
  47. M ( x , y ) = π / ( 2 I ( x , y ) ) . M(x,y)=\pi/\bigl(2I(x,y)\bigr).

Arity.html

  1. x y mod z x^{y}\mod z
  2. a = f ( b + c ) a=f(b+c)

Array_data_structure.html

  1. 𝐀 = [ 1 2 3 4 5 6 7 8 9 ] . \mathbf{A}=\begin{bmatrix}1&2&3\\ 4&5&6\\ 7&8&9\end{bmatrix}.

Arrhenius_equation.html

  1. k k
  2. T T
  3. A A
  4. E a E_{a}
  5. R R
  6. k = A e - E a / ( R T ) k=Ae^{-E_{a}/(RT)}
  7. k = A e - E a / ( k B T ) k=Ae^{-E_{a}/(k_{B}T)}
  8. E a E_{a}
  9. R R
  10. k B k_{B}
  11. T T
  12. A A
  13. k k
  14. A A
  15. e - E a / ( R T ) e^{{-E_{a}}/{(RT)}}\ \
  16. exp ( - E a / R T ) \exp(-E_{a}/RT)\
  17. ln ( k ) = - E a R 1 T + ln ( A ) \ln(k)=\frac{-E_{a}}{R}\frac{1}{T}+\ln(A)
  18. y = m x + c y=mx+c
  19. E a - R [ ln k ( 1 / T ) ] P \ E_{a}\equiv-R\left[\frac{\partial\ln k}{\partial~{}(1/T)}\right]_{P}
  20. k = A ( T / T 0 ) n e - E a / ( R T ) k=A(T/T_{0})^{n}e^{{-E_{a}}/{(RT)}}
  21. k = A exp [ - ( E a R T ) β ] k=A\exp\left[-\left(\frac{E_{a}}{RT}\right)^{\beta}\right]
  22. E a E_{a}
  23. e - E a R T \ e^{\frac{-E_{a}}{RT}}
  24. k = k B T h e - Δ G R T \ k=\frac{k_{B}T}{h}e^{-\frac{\Delta G^{\ddagger}}{RT}}
  25. Δ G \Delta G^{\ddagger}
  26. k B k_{B}
  27. h h

Artificial_neural_network.html

  1. f : X Y \textstyle f:X\rightarrow Y
  2. X \textstyle X
  3. X \textstyle X
  4. Y \textstyle Y
  5. f ( x ) \textstyle f(x)
  6. g i ( x ) \textstyle g_{i}(x)
  7. f ( x ) = K ( i w i g i ( x ) ) \textstyle f(x)=K\left(\sum_{i}w_{i}g_{i}(x)\right)
  8. K \textstyle K
  9. g i \textstyle g_{i}
  10. g = ( g 1 , g 2 , , g n ) \textstyle g=(g_{1},g_{2},\ldots,g_{n})
  11. f \textstyle f
  12. x \textstyle x
  13. h \textstyle h
  14. g \textstyle g
  15. f \textstyle f
  16. F = f ( G ) \textstyle F=f(G)
  17. G = g ( H ) \textstyle G=g(H)
  18. H = h ( X ) \textstyle H=h(X)
  19. X \textstyle X
  20. g \textstyle g
  21. h \textstyle h
  22. f \textstyle f
  23. F \textstyle F
  24. f * F \textstyle f^{*}\in F
  25. C : F \textstyle C:F\rightarrow\mathbb{R}
  26. f * \textstyle f^{*}
  27. C ( f * ) C ( f ) \textstyle C(f^{*})\leq C(f)
  28. f F \textstyle\forall f\in F
  29. C \textstyle C
  30. f \textstyle f
  31. C = E [ ( f ( x ) - y ) 2 ] \textstyle C=E\left[(f(x)-y)^{2}\right]
  32. ( x , y ) \textstyle(x,y)
  33. 𝒟 \textstyle\mathcal{D}
  34. N \textstyle N
  35. 𝒟 \textstyle\mathcal{D}
  36. C ^ = 1 N i = 1 N ( f ( x i ) - y i ) 2 \textstyle\hat{C}=\frac{1}{N}\sum_{i=1}^{N}(f(x_{i})-y_{i})^{2}
  37. N \textstyle N\rightarrow\infty
  38. 𝒟 \textstyle\mathcal{D}
  39. ( x , y ) , x X , y Y \textstyle(x,y),x\in X,y\in Y
  40. f : X Y \textstyle f:X\rightarrow Y
  41. f ( x ) \textstyle f(x)
  42. y \textstyle y
  43. x \textstyle x
  44. x \textstyle x
  45. f \textstyle f
  46. f ( x ) = a \textstyle f(x)=a
  47. a \textstyle a
  48. C = E [ ( x - f ( x ) ) 2 ] \textstyle C=E[(x-f(x))^{2}]
  49. a \textstyle a
  50. x \textstyle x
  51. f ( x ) \textstyle f(x)
  52. x \textstyle x
  53. t \textstyle t
  54. y t \textstyle y_{t}
  55. x t \textstyle x_{t}
  56. c t \textstyle c_{t}
  57. s 1 , , s n S \textstyle{s_{1},...,s_{n}}\in S
  58. a 1 , , a m A \textstyle{a_{1},...,a_{m}}\in A
  59. P ( c t | s t ) \textstyle P(c_{t}|s_{t})
  60. P ( x t | s t ) \textstyle P(x_{t}|s_{t})
  61. P ( s t + 1 | s t , a t ) \textstyle P(s_{t+1}|s_{t},a_{t})
  62. y i = e x i j = 1 c e x j y_{i}=\frac{e^{x_{i}}}{\sum_{j=1}^{c}e^{x_{j}}}
  63. x 2 \scriptstyle x_{2}
  64. y q \scriptstyle y_{q}
  65. y q = ( x i * w i q ) \scriptstyle y_{q}=\sum(x_{i}*w_{iq})

Ascending_chain_condition.html

  1. a 1 a 2 a 3 , a_{1}\,\leq\,a_{2}\,\leq\,a_{3}\,\leq\,\cdots,
  2. a n = a n + 1 = a n + 2 = . a_{n}=a_{n+1}=a_{n+2}=\cdots.
  3. a 3 a 2 a 1 \cdots\,\leq\,a_{3}\,\leq\,a_{2}\,\leq\,a_{1}

Ascorbic_acid.html

  1. [ α ] D 20 = + 23 \textstyle[\alpha]_{D}^{20}=+23

Associative_algebra.html

  1. r ( x y ) = ( r x ) y = x ( r y ) r\cdot(xy)=(r\cdot x)y=x(r\cdot y)
  2. 1 x = x = x 1 1x=x=x1
  3. x ( y z ) = ( x y ) z x(yz)=(xy)z\,
  4. η : R A \eta\colon R\to A
  5. r x = η ( r ) x r\cdot x=\eta(r)x
  6. η : R A \eta\colon R\to A
  7. ϕ : A 1 A 2 \phi:A_{1}\to A_{2}
  8. ϕ ( r x ) = r ϕ ( x ) \phi(r\cdot x)=r\cdot\phi(x)
  9. ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) \phi(x+y)=\phi(x)+\phi(y)\,
  10. ϕ ( x y ) = ϕ ( x ) ϕ ( y ) \phi(xy)=\phi(x)\phi(y)\,
  11. ϕ ( 1 ) = 1 \phi(1)=1\,
  12. M : A × A A M:A\times A\rightarrow A
  13. M ( Id × M ) = M ( M × Id ) M\circ(\mbox{Id}\times M)=M\circ(M\times\mbox{Id})
  14. \circ
  15. ( M ( Id × M ) ) ( x , y , z ) = M ( x , M ( y , z ) ) (M\circ(\mbox{Id}\times M))(x,y,z)=M(x,M(y,z))
  16. η : K A \eta:K\rightarrow A
  17. M ( Id × η ) = s ; M ( η × Id ) = t M\circ(\mbox{Id}\times\eta)=s;\ M\circ(\eta\times\mbox{Id})=t
  18. t : K × A A , ( k , a ) k a t:K\times A\rightarrow A,\ \left(k,a\right)\mapsto ka
  19. s : A × K A , ( a , k ) k a s:A\times K\rightarrow A,\ \left(a,k\right)\mapsto ka
  20. σ : A End ( V ) \sigma:A\rightarrow\mathrm{End}(V)
  21. τ : A End ( W ) \tau:A\rightarrow\mathrm{End}(W)
  22. ρ : x σ ( x ) τ ( x ) \rho:x\mapsto\sigma(x)\otimes\tau(x)
  23. ρ ( x ) ( v w ) = ( σ ( x ) ( v ) ) ( τ ( x ) ( w ) ) . \rho(x)(v\otimes w)=(\sigma(x)(v))\otimes(\tau(x)(w)).
  24. ρ ( k x ) = σ ( k x ) τ ( k x ) = k σ ( x ) k τ ( x ) = k 2 ( σ ( x ) τ ( x ) ) = k 2 ρ ( x ) \rho(kx)=\sigma(kx)\otimes\tau(kx)=k\sigma(x)\otimes k\tau(x)=k^{2}(\sigma(x)% \otimes\tau(x))=k^{2}\rho(x)
  25. ρ = ( σ τ ) Δ . \rho=(\sigma\otimes\tau)\circ\Delta.
  26. x ρ ( x ) = σ ( x ) Id + W Id V τ ( x ) x\mapsto\rho(x)=\sigma(x)\otimes\mbox{Id}~{}_{W}+\mbox{Id}~{}_{V}\otimes\tau(x)
  27. ρ ( x ) ( v w ) = ( σ ( x ) v ) w + v ( τ ( x ) w ) \rho(x)(v\otimes w)=(\sigma(x)v)\otimes w+v\otimes(\tau(x)w)
  28. ρ ( x y ) = σ ( x ) σ ( y ) Id + W Id V τ ( x ) τ ( y ) \rho(xy)=\sigma(x)\sigma(y)\otimes\mbox{Id}~{}_{W}+\mbox{Id}~{}_{V}\otimes\tau% (x)\tau(y)
  29. ρ ( x ) ρ ( y ) = σ ( x ) σ ( y ) Id + W σ ( x ) τ ( y ) + σ ( y ) τ ( x ) + Id V τ ( x ) τ ( y ) \rho(x)\rho(y)=\sigma(x)\sigma(y)\otimes\mbox{Id}~{}_{W}+\sigma(x)\otimes\tau(% y)+\sigma(y)\otimes\tau(x)+\mbox{Id}~{}_{V}\otimes\tau(x)\tau(y)

Associative_property.html

  1. ( 2 + 3 ) + 4 = 2 + ( 3 + 4 ) = 9 (2+3)+4=2+(3+4)=9\,
  2. 2 × ( 3 × 4 ) = ( 2 × 3 ) × 4 = 24. 2\times(3\times 4)=(2\times 3)\times 4=24.
  3. ( x + y ) + z = x + ( y + z ) = x + y + z ( x y ) z = x ( y z ) = x y z } for all x , y , z . \left.\begin{matrix}(x+y)+z=x+(y+z)=x+y+z\\ (x\,y)z=x(y\,z)=x\,y\,z\end{matrix}\right\}\mbox{for all }~{}x,y,z\in\mathbb{R}.
  4. gcd ( gcd ( x , y ) , z ) = gcd ( x , gcd ( y , z ) ) = gcd ( x , y , z ) lcm ( lcm ( x , y ) , z ) = lcm ( x , lcm ( y , z ) ) = lcm ( x , y , z ) } for all x , y , z . \left.\begin{matrix}\operatorname{gcd}(\operatorname{gcd}(x,y),z)=% \operatorname{gcd}(x,\operatorname{gcd}(y,z))=\operatorname{gcd}(x,y,z)\\ \operatorname{lcm}(\operatorname{lcm}(x,y),z)=\operatorname{lcm}(x,% \operatorname{lcm}(y,z))=\operatorname{lcm}(x,y,z)\end{matrix}\right\}\mbox{ % for all }~{}x,y,z\in\mathbb{Z}.
  5. ( A B ) C = A ( B C ) = A B C ( A B ) C = A ( B C ) = A B C } for all sets A , B , C . \left.\begin{matrix}(A\cap B)\cap C=A\cap(B\cap C)=A\cap B\cap C\\ (A\cup B)\cup C=A\cup(B\cup C)=A\cup B\cup C\end{matrix}\right\}\mbox{for all % sets }~{}A,B,C.
  6. ( f g ) h = f ( g h ) = f g h for all f , g , h S . (f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h\qquad\mbox{for all }~{}f,g,% h\in S.
  7. ( f g ) h = f ( g h ) = f g h (f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h
  8. ( P ( Q R ) ) ( ( P Q ) R ) (P(QR))\Leftrightarrow((PQ)R)
  9. ( P and ( Q and R ) ) ( ( P and Q ) and R ) , (P\and(Q\and R))\Leftrightarrow((P\and Q)\and R),
  10. \Leftrightarrow
  11. ( P ( Q R ) ) ( ( P Q ) R ) (P(QR))\leftrightarrow((PQ)R)
  12. ( ( P Q ) R ) ( P ( Q R ) ) ((PQ)R)\leftrightarrow(P(QR))
  13. ( ( P and Q ) and R ) ( P and ( Q and R ) ) ((P\and Q)\and R)\leftrightarrow(P\and(Q\and R))
  14. ( P and ( Q and R ) ) ( ( P and Q ) and R ) (P\and(Q\and R))\leftrightarrow((P\and Q)\and R)
  15. ( ( P Q ) R ) ( P ( Q R ) ) ((P\leftrightarrow Q)\leftrightarrow R)\leftrightarrow(P\leftrightarrow(Q% \leftrightarrow R))
  16. ( P ( Q R ) ) ( ( P Q ) R ) (P\leftrightarrow(Q\leftrightarrow R))\leftrightarrow((P\leftrightarrow Q)% \leftrightarrow R)
  17. * *
  18. ( x * y ) * z x * ( y * z ) for some x , y , z S . (x*y)*z\neq x*(y*z)\qquad\mbox{for some }~{}x,y,z\in S.
  19. ( 5 - 3 ) - 2 5 - ( 3 - 2 ) (5-3)-2\,\neq\,5-(3-2)
  20. ( 4 / 2 ) / 2 4 / ( 2 / 2 ) (4/2)/2\,\neq\,4/(2/2)
  21. 2 ( 1 2 ) ( 2 1 ) 2 2^{(1^{2})}\,\neq\,(2^{1})^{2}
  22. ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) + ( 1 - 1 ) + = 0 (1-1)+(1-1)+(1-1)+(1-1)+(1-1)+(1-1)+\dots\,=\,0
  23. 1 + ( - 1 + 1 ) + ( - 1 + 1 ) + ( - 1 + 1 ) + ( - 1 + 1 ) + ( - 1 + 1 ) + ( - 1 + = 1 1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+\dots\,=\,1
  24. x * y * z = ( x * y ) * z w * x * y * z = ( ( w * x ) * y ) * z etc. } for all w , x , y , z S \left.\begin{matrix}x*y*z=(x*y)*z\\ w*x*y*z=((w*x)*y)*z\\ \mbox{etc.}\end{matrix}\right\}\mbox{for all }~{}w,x,y,z\in S
  25. x * y * z = x * ( y * z ) w * x * y * z = w * ( x * ( y * z ) ) etc. } for all w , x , y , z S \left.\begin{matrix}x*y*z=x*(y*z)\\ w*x*y*z=w*(x*(y*z))\\ \mbox{etc.}\end{matrix}\right\}\mbox{for all }~{}w,x,y,z\in S
  26. x - y - z = ( x - y ) - z for all x , y , z ; x-y-z=(x-y)-z\qquad\mbox{for all }~{}x,y,z\in\mathbb{R};
  27. x / y / z = ( x / y ) / z for all x , y , z with y 0 , z 0. x/y/z=(x/y)/z\qquad\qquad\quad\mbox{for all }~{}x,y,z\in\mathbb{R}\mbox{ with % }~{}y\neq 0,z\neq 0.
  28. ( f x y ) = ( ( f x ) y ) (f\,x\,y)=((f\,x)\,y)
  29. x y z = x ( y z ) . x^{y^{z}}=x^{(y^{z})}.\,
  30. ( x y ) z = x ( y z ) . (x^{y})^{z}=x^{(yz)}.\,
  31. = ( ) \mathbb{Z}\rightarrow\mathbb{Z}\rightarrow\mathbb{Z}=\mathbb{Z}\rightarrow(% \mathbb{Z}\rightarrow\mathbb{Z})
  32. x y x - y = x ( y x - y ) x\mapsto y\mapsto x-y=x\mapsto(y\mapsto x-y)
  33. a × ( b × c ) ( a × b ) × c for some a , b , c 3 \vec{a}\times(\vec{b}\times\vec{c})\neq(\vec{a}\times\vec{b})\times\vec{c}% \qquad\mbox{ for some }~{}\vec{a},\vec{b},\vec{c}\in\mathbb{R}^{3}
  34. ( x + y ) / 2 + z 2 x + ( y + z ) / 2 2 for all x , y , z with x z . {(x+y)/2+z\over 2}\neq{x+(y+z)/2\over 2}\qquad\mbox{for all }~{}x,y,z\in% \mathbb{R}\mbox{ with }~{}x\neq z.
  35. ( A \ B ) \ C (A\backslash B)\backslash C
  36. A \ ( B \ C ) A\backslash(B\backslash C)

Asterisk.html

  1. * : A k A n - k *:A^{k}\rightarrow A^{n-k}
  2. z ¯ \bar{z}
  3. * = - { 0 } . \mathbb{C}^{*}=\mathbb{C}-\{0\}.

Astrobiology.html

  1. N = R * × f p × n e × f l × f i × f c × L N=R^{*}~{}\times~{}f_{p}~{}\times~{}n_{e}~{}\times~{}f_{l}~{}\times~{}f_{i}~{}% \times~{}f_{c}~{}\times~{}L

Astronomical_unit.html

  1. 91 / 2 9{1}/{2}
  2. 641 / 6 64{1}/{6}
  3. A = cot α . A={\cot\alpha}.
  4. 91 / 2 9{1}/{2}
  5. A 3 = G M D 2 k 2 A^{3}=\frac{GM_{\odot}D^{2}}{k^{2}}

Asymptote.html

  1. f ( x ) = 1 x f(x)=\frac{1}{x}
  2. ( x , 1 x ) \left(x,\frac{1}{x}\right)
  3. x x
  4. y y
  5. x x
  6. 1 x \frac{1}{x}
  7. x x
  8. y y
  9. lim x a - f ( x ) = ± \lim_{x\to a^{-}}f(x)=\pm\infty
  10. lim x a + f ( x ) = ± . \lim_{x\to a^{+}}f(x)=\pm\infty.
  11. f ( x ) = { 1 x if x > 0 , 5 if x 0. f(x)=\begin{cases}\frac{1}{x}&\mbox{if }~{}x>0,\\ 5&\mbox{if }~{}x\leq 0.\end{cases}
  12. y = arctan ( x ) . y=\arctan(x).
  13. lim x - f ( x ) = c \lim_{x\rightarrow-\infty}f(x)=c
  14. lim x + f ( x ) = c \lim_{x\rightarrow+\infty}f(x)=c
  15. lim x - arctan ( x ) = - π 2 \lim_{x\rightarrow-\infty}\arctan(x)=-\frac{\pi}{2}
  16. lim x + arctan ( x ) = π 2 . \lim_{x\rightarrow+\infty}\arctan(x)=\frac{\pi}{2}.
  17. lim x - 1 x 2 + 1 = lim x + 1 x 2 + 1 = 0. \lim_{x\to-\infty}\frac{1}{x^{2}+1}=\lim_{x\to+\infty}\frac{1}{x^{2}+1}=0.
  18. lim x + [ f ( x ) - ( m x + n ) ] = 0 or lim x - [ f ( x ) - ( m x + n ) ] = 0. \lim_{x\to+\infty}\left[f(x)-(mx+n)\right]=0\,\mbox{ or }~{}\lim_{x\to-\infty}% \left[f(x)-(mx+n)\right]=0.
  19. lim x ± [ f ( x ) - x ] \lim_{x\to\pm\infty}\left[f(x)-x\right]
  20. = lim x ± [ ( x + 1 x ) - x ] =\lim_{x\to\pm\infty}\left[\left(x+\frac{1}{x}\right)-x\right]
  21. = lim x ± 1 x = 0. =\lim_{x\to\pm\infty}\frac{1}{x}=0.
  22. m = def lim x a f ( x ) / x m\stackrel{\,\text{def}}{=}\lim_{x\rightarrow a}f(x)/x
  23. - -\infty
  24. + +\infty
  25. n = def lim x a ( f ( x ) - m x ) n\stackrel{\,\text{def}}{=}\lim_{x\rightarrow a}(f(x)-mx)
  26. m = lim x + f ( x ) / x = lim x + 2 x 2 + 3 x + 1 x 2 = 2 m=\lim_{x\rightarrow+\infty}f(x)/x=\lim_{x\rightarrow+\infty}\frac{2x^{2}+3x+1% }{x^{2}}=2
  27. n = lim x + ( f ( x ) - m x ) = lim x + ( 2 x 2 + 3 x + 1 x - 2 x ) = 3 n=\lim_{x\rightarrow+\infty}(f(x)-mx)=\lim_{x\rightarrow+\infty}\left(\frac{2x% ^{2}+3x+1}{x}-2x\right)=3
  28. m = lim x + f ( x ) / x = lim x + ln x x = 0 m=\lim_{x\rightarrow+\infty}f(x)/x=\lim_{x\rightarrow+\infty}\frac{\ln x}{x}=0
  29. n = lim x + ( f ( x ) - m x ) = lim x + ln x n=\lim_{x\rightarrow+\infty}(f(x)-mx)=\lim_{x\rightarrow+\infty}\ln x
  30. y = 0 y=0
  31. f ( x ) = 1 x 2 + 1 f(x)=\frac{1}{x^{2}+1}
  32. y = 0 y=0
  33. f ( x ) = 2 x 2 + 7 3 x 2 + x + 12 f(x)=\frac{2x^{2}+7}{3x^{2}+x+12}
  34. y = 2 3 y=\frac{2}{3}
  35. f ( x ) = x 2 + x + 1 x f(x)=\frac{x^{2}+x+1}{x}
  36. y = x + 1 y=x+1
  37. f ( x ) = 2 x 4 3 x 2 + 1 f(x)=\frac{2x^{4}}{3x^{2}+1}
  38. none \mbox{none}~{}
  39. f ( x ) = x 2 - 5 x + 6 x 3 - 3 x 2 + 2 x = ( x - 2 ) ( x - 3 ) x ( x - 1 ) ( x - 2 ) f(x)=\frac{x^{2}-5x+6}{x^{3}-3x^{2}+2x}=\frac{(x-2)(x-3)}{x(x-1)(x-2)}
  40. f ( x ) = ( x 2 + x + 1 ) / ( x + 1 ) f(x)=(x^{2}+x+1)/(x+1)
  41. y = x y=x
  42. x = 1 , 2 , 3 , 4 , 5 , 6 x=1,2,3,4,5,6
  43. f ( x ) = x 2 + x + 1 x + 1 = x + 1 x + 1 f(x)=\frac{x^{2}+x+1}{x+1}=x+\frac{1}{x+1}
  44. lim t b ( x 2 ( t ) + y 2 ( t ) ) = . \lim_{t\rightarrow b}(x^{2}(t)+y^{2}(t))=\infty.
  45. a x + b y + c = 0 ax+by+c=0
  46. | a x ( t ) + b y ( t ) + c | a 2 + b 2 \frac{|ax(t)+by(t)+c|}{\sqrt{a^{2}+b^{2}}}
  47. | a x ( γ ( t ) ) + b y ( γ ( t ) ) + c | a 2 + b 2 \frac{|ax(\gamma(t))+by(\gamma(t))+c|}{\sqrt{a^{2}+b^{2}}}
  48. t ( t , f ( t ) ) . t\mapsto(t,f(t)).
  49. n n
  50. y = x 3 + 2 x 2 + 3 x + 4 x y=\frac{x^{3}+2x^{2}+3x+4}{x}
  51. P ( x , y ) = P n ( x , y ) + P n - 1 ( x , y ) + + P 1 ( x , y ) + P 0 P(x,y)=P_{n}(x,y)+P_{n-1}(x,y)+\cdots+P_{1}(x,y)+P_{0}
  52. Q x ( b , a ) x + Q y ( b , a ) y + P n - 1 ( b , a ) = 0 Q^{\prime}_{x}(b,a)x+Q^{\prime}_{y}(b,a)y+P_{n-1}(b,a)=0
  53. Q x ( b , a ) Q^{\prime}_{x}(b,a)
  54. Q y ( b , a ) Q^{\prime}_{y}(b,a)
  55. Q x ( b , a ) = Q y ( b , a ) = 0 Q^{\prime}_{x}(b,a)=Q^{\prime}_{y}(b,a)=0
  56. P n - 1 ( b , a ) 0 P_{n-1}(b,a)\neq 0
  57. Q x ( b , a ) = Q y ( b , a ) = P n - 1 ( b , a ) = 0 , Q^{\prime}_{x}(b,a)=Q^{\prime}_{y}(b,a)=P_{n-1}(b,a)=0,
  58. | x | 1 , | y | 1 |x|\leq 1,|y|\leq 1
  59. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  60. y = ± b a x . y=\pm\frac{b}{a}x.
  61. x 2 a 2 - y 2 b 2 = 0. \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=0.
  62. x 2 a 2 - y 2 b 2 - z 2 c 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1
  63. x 2 a 2 - y 2 b 2 - z 2 c 2 = 0. \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=0.
  64. P d ( x , y , z ) + P d - 2 ( x , y , z ) + P 0 = 0 , P_{d}(x,y,z)+P_{d-2}(x,y,z)+\cdots P_{0}=0,
  65. P i P_{i}
  66. i i
  67. P d - 1 = 0 P_{d-1}=0
  68. P d ( x , y , z ) = 0 P_{d}(x,y,z)=0

Atbash.html

  1. 0 , 1 , . . , m - 1 0,1,..,m-1
  2. a = b = ( m - 1 ) a=b=(m-1)
  3. E ( x ) = D ( x ) = ( ( m - 1 ) x + ( m - 1 ) ) mod m \mbox{E}~{}(x)=\mbox{D}~{}(x)=((m-1)x+(m-1))\mod{m}
  4. E ( x ) \displaystyle\mbox{E}~{}(x)
  5. 1 , 2 , . . , m 1,2,..,m
  6. E ( x ) = ( - x mod m ) + 1 \mbox{E}~{}(x)=(-x\mod{m})+1

Atlas_(topology).html

  1. φ \varphi
  2. ( U , φ ) (U,\varphi)
  3. { ( U α , φ α ) } \{(U_{\alpha},\varphi_{\alpha})\}
  4. U α = M \bigcup U_{\alpha}=M
  5. { ( U α , φ α ) } \{(U_{\alpha},\varphi_{\alpha})\}
  6. M M
  7. ( U β , φ β ) (U_{\beta},\varphi_{\beta})
  8. { ( U α , φ α ) } ( U β , φ β ) M \{(U_{\alpha},\varphi_{\alpha})\}\cup(U_{\beta},\varphi_{\beta})\subset M
  9. ( U α , φ α ) (U_{\alpha},\varphi_{\alpha})
  10. ( U β , φ β ) (U_{\beta},\varphi_{\beta})
  11. U α U β U_{\alpha}\cap U_{\beta}
  12. τ α , β : φ α ( U α U β ) φ β ( U α U β ) \tau_{\alpha,\beta}:\varphi_{\alpha}(U_{\alpha}\cap U_{\beta})\to\varphi_{% \beta}(U_{\alpha}\cap U_{\beta})
  13. τ α , β = φ β φ α - 1 . \tau_{\alpha,\beta}=\varphi_{\beta}\circ\varphi_{\alpha}^{-1}.
  14. φ α \varphi_{\alpha}
  15. φ β \varphi_{\beta}
  16. τ α , β \tau_{\alpha,\beta}
  17. C k C^{k}
  18. 𝒢 {\mathcal{G}}
  19. 𝒢 {\mathcal{G}}

Atle_Selberg.html

  1. ( s ) = 1 2 \Re(s)=\tfrac{1}{2}
  2. ϑ ( x ) log ( x ) + p x log ( p ) ϑ ( x p ) = 2 x log ( x ) + O ( x ) \vartheta\left(x\right)\log\left(x\right)+\sum\limits_{p\leq x}{\log\left(p% \right)}\vartheta\left({\frac{x}{p}}\right)=2x\log\left(x\right)+O\left(x\right)
  3. ϑ ( x ) = p x log ( p ) \vartheta\left(x\right)=\sum\limits_{p\leq x}{\log\left(p\right)}
  4. p p

Atmospheric_pressure.html

  1. p = p 0 ( 1 - L h T 0 ) g M R L p 0 ( 1 - g h c p T 0 ) c p M R p 0 exp ( - g M h R T 0 ) , p=p_{0}\cdot\left(1-\frac{L\cdot h}{T_{0}}\right)^{\frac{g\cdot M}{R\cdot L}}% \approx p_{0}\cdot\left(1-\frac{g\cdot h}{c_{p}\cdot T_{0}}\right)^{\frac{c_{p% }\cdot M}{R}}\approx p_{0}\cdot\exp\left(-\frac{g\cdot M\cdot h}{R\cdot T_{0}}% \right),

Atom_probe.html

  1. F = n e ϕ F=ne\nabla\phi
  2. m a = q ϕ ma=q\nabla\phi
  3. a = q m ϕ a=\frac{q}{m}\nabla\phi
  4. E = 1 2 m U ion 2 = - n e V 1 E=\frac{1}{2}mU_{\mathrm{ion}}^{2}=-neV_{1}
  5. U = 2 n e V 1 m U=\sqrt{\frac{2neV_{1}}{m}}
  6. m n = - 2 e V 1 U 2 \frac{m}{n}=-\frac{2eV_{1}}{U^{2}}
  7. U = f t U=\frac{f}{t}
  8. m n = - 2 e V 1 ( t f ) 2 \frac{m}{n}=-2eV_{1}\left(\frac{t}{f}\right)^{2}
  9. M = r s c r e e n r t i p . M=\frac{r_{screen}}{r_{tip}}.

Atomic,_molecular,_and_optical_physics.html

  1. k = n π L k=\frac{n\pi}{L}
  2. n 1 \scriptstyle n\in\mathbb{N}_{1}
  3. E = E 0 sin ( n π L x ) E=E_{0}\sin\left(\frac{n\pi}{L}x\right)\,\!
  4. ν \nu
  5. h ν h\nu

Atomic_mass_unit.html

  1. 1 12 \frac{1}{12}
  2. 1 u = m u = 1 12 m ( C 12 ) 1\mathrm{u}=m_{\mathrm{u}}=\frac{1}{12}m\left({}^{12}\mathrm{C}\right)

Atomic_orbital.html

  1. ψ ( x , y , z ) ψ(x,y,z)
  2. n n
  3. m m
  4. = 0 , 1 , 2 ℓ=0,1,2
  5. 3 3
  6. n n
  7. ( r , θ , φ ) (r, θ, φ)
  8. ( x , y , z ) (x, y, z)
  9. ψ ( r , θ , φ ) = R ( r ) Θ ( θ ) Φ ( φ ) ψ(r, θ, φ)=R(r) Θ(θ) Φ(φ)
  10. m m
  11. R ( r ) R(r)
  12. X type y X\,\mathrm{type}^{y}
  13. n n
  14. y y
  15. n = 1 n=1
  16. = 0 ℓ=0
  17. n = 1 , 2 , 3 , 4 , 5 , n=1, 2, 3, 4, 5, …
  18. n n
  19. n n
  20. n n
  21. n n
  22. n n
  23. 0 n 0 - 1 0\leq\ell\leq n_{0}-1
  24. n = 1 n=1
  25. = 0 \ell=0
  26. n = 2 n=2
  27. = 0 \ell=0
  28. = 1 \ell=1
  29. m m_{\ell}
  30. \ell
  31. 0 \ell_{0}
  32. m m_{\ell}
  33. - 0 m 0 -\ell_{0}\leq m_{\ell}\leq\ell_{0}
  34. m m_{\ell}
  35. = 0 ℓ=0
  36. = 1 ℓ=1
  37. = 2 ℓ=2
  38. = 3 ℓ=3
  39. = 4 ℓ=4
  40. n = 1 n=1
  41. m = 0 m_{\ell}=0
  42. n = 2 n=2
  43. n = 3 n=3
  44. n = 4 n=4
  45. n = 5 n=5
  46. n n
  47. \ell
  48. n n
  49. \ell
  50. n = 2 n=2
  51. = 0 \ell=0
  52. 1 / 2 {1}/{2}
  53. 1 / 2 {1}/{2}
  54. m = + 1 m=+1
  55. m = 1 m=−1
  56. n n
  57. m m
  58. p z = p 0 p_{z}=p_{0}
  59. p x = 1 2 ( p 1 + p - 1 ) p_{x}=\frac{1}{\sqrt{2}}\left(p_{1}+p_{-1}\right)
  60. p y = 1 i 2 ( p 1 - p - 1 ) p_{y}=\frac{1}{i\sqrt{2}}\left(p_{1}-p_{-1}\right)
  61. = 1 ℓ=1
  62. ( n = 6 , = 0 , m = 0 ) (n=6,ℓ=0,m=0)
  63. n > 1 n>1
  64. ψ ( r , θ , φ ) ψ(r, θ, φ)
  65. ψ ψ
  66. ψ ( r , θ , φ ) ψ(r, θ, φ)
  67. ψ ( r , θ , φ ) ψ(r, θ, φ)
  68. ψ ( r , θ , φ ) ψ(r, θ, φ)
  69. m m
  70. m −m
  71. m m
  72. m m
  73. m + m ⟨m⟩+⟨−m⟩
  74. m m ⟨m⟩−⟨−m⟩
  75. m = 0 m=0
  76. = 0 ℓ=0
  77. n n
  78. m m
  79. n n
  80. n n
  81. Z Z
  82. Z Z
  83. n n
  84. = 0 \ell=0
  85. n = 1 n=1
  86. n = 2 n=2
  87. n n
  88. n = 2 n=2
  89. n = 3 n=3
  90. n n
  91. n n
  92. n n
  93. m = ± 1 m=±1
  94. = 0 ℓ=0
  95. = 1 ℓ=1
  96. = 2 ℓ=2
  97. = 3 ℓ=3
  98. m = 0 m=0
  99. m = 0 m=0
  100. m = ± 1 m=±1
  101. m = 0 m=0
  102. m = ± 1 m=±1
  103. m = ± 2 m=±2
  104. m = 0 m=0
  105. m = ± 1 m=±1
  106. m = ± 2 m=±2
  107. m = ± 3 m=±3
  108. n = 1 n=1
  109. n = 2 n=2
  110. n = 3 n=3
  111. n = 4 n=4
  112. n = 5 n=5
  113. n = 6 n=6
  114. n = 7 n=7
  115. ψ ( r , θ ) ψ(r, θ)
  116. ψ ( r , θ , φ ) ψ(r, θ, φ)
  117. u 01 u_{01}
  118. u 02 u_{02}
  119. u 03 u_{03}
  120. u 11 u_{11}
  121. u 12 u_{12}
  122. u 13 u_{13}
  123. u 21 u_{21}
  124. u 22 u_{22}
  125. u 23 u_{23}
  126. n n
  127. n = 1 n=1
  128. n n
  129. n n
  130. n n
  131. \ell
  132. n n
  133. n n
  134. \ell
  135. \ell
  136. = 2 \ell=2
  137. = 3 \ell=3
  138. \ell
  139. n n
  140. n n
  141. \ell
  142. r r
  143. s s
  144. s s
  145. n n
  146. Z Z
  147. Z Z
  148. n = 1 n=1
  149. v = Z α c v=Z\alpha c
  150. Z Z
  151. α \alpha
  152. c c
  153. 𝐙 > 𝟏𝟑𝟕 \mathbf{Z>137}
  154. Z Z
  155. Z Z
  156. Z Z
  157. Z Z
  158. n = 1 n=1
  159. = 0 ℓ=0
  160. m < s u b > = 0 m<sub>ℓ=0

Attenuation.html

  1. α \alpha
  2. Attenuation = α [ dB / ( MHz cm ) ] [ cm ] f [ MHz ] \,\text{Attenuation}=\alpha[\,\text{dB}/(\,\text{MHz}\cdot\,\text{cm})]\cdot% \ell[\,\text{cm}]\cdot\,\text{f}[\,\text{MHz}]
  3. α ( dB / ( MHz cm ) ) \alpha(\,\text{dB}/(\,\text{MHz}\cdot\,\text{cm}))
  4. Attenuation (dB) = 10 × log 10 ( Input intensity (W) Output intensity (W) ) \,\text{Attenuation (dB)}=10\times\log_{10}\left(\frac{\,\text{Input intensity% (W)}}{\,\text{Output intensity (W)}}\right)

Auger_electron_spectroscopy.html

  1. E k i n = E Core State - E B - E C E_{kin}=E_{\,\text{Core State}}-E_{B}-E_{C}^{\prime}
  2. E Core State E_{\,\text{Core State}}
  3. E B E_{B}
  4. E C E_{C}^{\prime}
  5. K L 1 L 2 , 3 KL_{1}L_{2,3}
  6. K L 1 L 2 , 3 KL_{1}L_{2,3}
  7. L 1 L_{1}
  8. L 2 , 3 L_{2,3}
  9. E A B C = E A ( Z ) - 0.5 [ E B ( Z ) + E B ( Z + 1 ) ] - 0.5 [ E C ( Z ) + E C ( Z + 1 ) ] E_{ABC}=E_{A}(Z)-0.5[E_{B}(Z)+E_{B}(Z+1)]-0.5[E_{C}(Z)+E_{C}(Z+1)]
  10. E i ( Z ) E_{i}(Z)
  11. i i
  12. E i ( Z + 1 ) E_{i}(Z+1)
  13. E A B C = E A - E B - E C - F ( B C : x ) + R x i n + R x e x E_{ABC}=E_{A}-E_{B}-E_{C}-F(BC:x)+R_{xin}+R_{xex}
  14. F ( B C : x ) F(BC:x)
  15. E i E_{i}
  16. Δ V = k sin ( ω t ) \Delta V=k\sin(\omega t)
  17. I ( V + k sin ( ω t ) ) I(V+k\sin(\omega t))
  18. I ( V + k sin ( ω t ) ) I 0 + I ( V + k sin ( ω t ) ) + O ( I ′′ ) I(V+k\sin(\omega t))\approx I_{0}+I^{\prime}(V+k\sin(\omega t))+O(I^{\prime% \prime})
  19. I I^{\prime}
  20. d N d E \frac{dN}{dE}
  21. ω A \omega_{A}
  22. ω X \omega_{X}
  23. ω A = 1 - ω X = 1 - W X W X + W A \omega_{A}=1-\omega_{X}=1-\frac{W_{X}}{W_{X}+W_{A}}
  24. W X W_{X}
  25. W A W_{A}
  26. σ a x ( E ) = 1.3 × 10 13 b C E p \sigma_{ax}(E)=1.3\times 10^{13}b\frac{C}{E_{p}}
  27. E p E_{p}
  28. σ a x \sigma_{ax}
  29. σ ( E ) = σ a x [ 1 + r m ( E p , α ) ] \sigma(E)=\sigma_{ax}[1+r_{m}(E_{p},\alpha)]
  30. Y ( t ) = N x × δ t × σ ( E , t ) [ 1 - ω X ] exp ( - t cos θ λ ) × I ( t ) × T × d ( Ω ) 4 π Y(t)=N_{x}\times\delta t\times\sigma(E,t)[1-\omega_{X}]\exp(-t\cos\frac{\theta% }{\lambda})\times I(t)\times T\times\frac{d(\Omega)}{4\pi}

Augustin-Louis_Cauchy.html

  1. C f ( z ) d z = 0 , \oint_{C}f(z)dz=0,
  2. f ( z ) = ϕ ( z ) + B 1 z - a + B 2 ( z - a ) 2 + + B n ( z - a ) n , B i , z , a , f(z)=\phi(z)+\frac{B_{1}}{z-a}+\frac{B_{2}}{(z-a)^{2}}+\cdots+\frac{B_{n}}{(z-% a)^{n}},\quad B_{i},z,a\in\mathbb{C},
  3. Res z = a f ( z ) = lim z a ( z - a ) f ( z ) , \underset{z=a}{\mathrm{Res}}f(z)=\lim_{z\rightarrow a}(z-a)f(z),
  4. f ( a ) = 1 2 π i C f ( z ) z - a d z , f(a)=\frac{1}{2\pi i}\oint_{C}\frac{f(z)}{z-a}dz,
  5. 1 2 π i C f ( z ) d z = k = 1 n Res z = a k f ( z ) , \frac{1}{2\pi i}\oint_{C}f(z)dz=\sum_{k=1}^{n}\underset{z=a_{k}}{\mathrm{Res}}% f(z),
  6. δ - ϵ \delta-\epsilon

Australian_Broadcasting_Corporation.html

  1. \uparrow
  2. \uparrow
  3. \uparrow
  4. \uparrow
  5. \uparrow
  6. \uparrow
  7. \downarrow
  8. \downarrow
  9. \downarrow
  10. \downarrow
  11. \downarrow
  12. \downarrow
  13. \downarrow

Autocorrelation.html

  1. R ( s , t ) = E [ ( X t - μ t ) ( X s - μ s ) ] σ t σ s , R(s,t)=\frac{\operatorname{E}[(X_{t}-\mu_{t})(X_{s}-\mu_{s})]}{\sigma_{t}% \sigma_{s}}\,,
  2. R ( τ ) = E [ ( X t - μ ) ( X t + τ - μ ) ] σ 2 , R(\tau)=\frac{\operatorname{E}[(X_{t}-\mu)(X_{t+\tau}-\mu)]}{\sigma^{2}},\,
  3. R ( τ ) = R ( - τ ) . R(\tau)=R(-\tau).\,
  4. f ( t ) f(t)
  5. R f f ( τ ) R_{ff}(\tau)
  6. f ( t ) f(t)
  7. τ \tau
  8. R f f ( τ ) = ( f * g - 1 ( f ¯ ) ) ( τ ) = - f ( u + τ ) f ¯ ( u ) d u = - f ( u ) f ¯ ( u - τ ) d u R_{ff}(\tau)=(f*g_{-1}(\overline{f}))(\tau)=\int_{-\infty}^{\infty}f(u+\tau)% \overline{f}(u)\,{\rm d}u=\int_{-\infty}^{\infty}f(u)\overline{f}(u-\tau)\,{% \rm d}u
  9. f ¯ \overline{f}
  10. g - 1 g_{-1}
  11. f f
  12. g - 1 ( f ) ( u ) = f ( - u ) g_{-1}(f)(u)=f(-u)
  13. * *
  14. f ¯ = f \overline{f}=f
  15. u u
  16. R R
  17. l l
  18. y ( n ) y(n)
  19. R y y ( l ) = n Z y ( n ) y ¯ ( n - l ) . R_{yy}(l)=\sum_{n\in Z}y(n)\,\overline{y}(n-l).
  20. R f f ( τ ) = E [ f ( t ) f ¯ ( t - τ ) ] R_{ff}(\tau)=\operatorname{E}\left[f(t)\overline{f}(t-\tau)\right]
  21. R y y ( l ) = E [ y ( n ) y ¯ ( n - l ) ] . R_{yy}(l)=\operatorname{E}\left[y(n)\,\overline{y}(n-l)\right].
  22. t t
  23. n n
  24. R f f ( τ ) = lim T 1 T 0 T f ( t + τ ) f ¯ ( t ) d t R_{ff}(\tau)=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}f(t+\tau)% \overline{f}(t)\,{\rm d}t
  25. R y y ( l ) = lim N 1 N n = 0 N - 1 y ( n ) y ¯ ( n - l ) . R_{yy}(l)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}y(n)\,\overline{% y}(n-l).
  26. R ( j , k , ) = n , q , r x n , q , r x n - j , q - k , r - . R(j,k,\ell)=\sum_{n,q,r}x_{n,q,r}\,x_{n-j,q-k,r-\ell}.
  27. R ( i ) = R ( - i ) R(i)=R(-i)
  28. R f ( - τ ) = R f ( τ ) R_{f}(-\tau)=R_{f}(\tau)\,
  29. f f
  30. R f ( - τ ) = R f * ( τ ) R_{f}(-\tau)=R_{f}^{*}(\tau)\,
  31. f f
  32. τ \tau
  33. | R f ( τ ) | R f ( 0 ) |R_{f}(\tau)|\leq R_{f}(0)
  34. τ \tau
  35. τ = 0 \tau=0
  36. τ \tau
  37. R ( τ ) = - S ( f ) e j 2 π f τ d f R(\tau)=\int_{-\infty}^{\infty}S(f)e^{j2\pi f\tau}\,{\rm d}f
  38. S ( f ) = - R ( τ ) e - j 2 π f τ d τ . S(f)=\int_{-\infty}^{\infty}R(\tau)e^{-j2\pi f\tau}\,{\rm d}\tau.
  39. R ( τ ) = - S ( f ) cos ( 2 π f τ ) d f R(\tau)=\int_{-\infty}^{\infty}S(f)\cos(2\pi f\tau)\,{\rm d}f
  40. S ( f ) = - R ( τ ) cos ( 2 π f τ ) d τ . S(f)=\int_{-\infty}^{\infty}R(\tau)\cos(2\pi f\tau)\,{\rm d}\tau.
  41. R x x ( j ) = n x n x ¯ n - j R_{xx}(j)=\sum_{n}x_{n}\,\overline{x}_{n-j}
  42. x = ( 2 , 3 , 1 ) x=(2,3,1)
  43. x ( 0 ) = 2 , x ( 1 ) = 3 , x ( 2 ) = 1 x(0)=2,x(1)=3,x(2)=1
  44. x ( i ) = 0 x(i)=0
  45. i i
  46. R x x = ( 2 , 9 , 14 , 9 , 2 ) R_{xx}=(2,9,14,9,2)
  47. R x x ( 0 ) = 14 , R_{xx}(0)=14,
  48. R x x ( - 1 ) = R x x ( 1 ) = 9 , R_{xx}(-1)=R_{xx}(1)=9,
  49. R x x ( - 2 ) = R x x ( 2 ) = 2 , R_{xx}(-2)=R_{xx}(2)=2,
  50. x = ( , 2 , 3 , 1 , 2 , 3 , 1 , ) , x=(...,2,3,1,2,3,1,...),
  51. R x x = ( , 14 , 11 , 11 , 14 , 11 , 11 , ) R_{xx}=(...,14,11,11,14,11,11,...)
  52. x . x.
  53. n l o g ( n ) nlog(n)
  54. X ( t ) X(t)
  55. R ( τ ) = I F F T S S ( f ) R(τ)=IFFTSS(f)
  56. τ τ
  57. τ τ
  58. X ( t ) X(t)
  59. n l o g ( n ) nlog(n)
  60. n n
  61. { X 1 , X 2 , , X n } \{X_{1},\,X_{2},\,\ldots,\,X_{n}\}
  62. R ^ ( k ) = 1 ( n - k ) σ 2 t = 1 n - k ( X t - μ ) ( X t + k - μ ) \hat{R}(k)=\frac{1}{(n-k)\sigma^{2}}\sum_{t=1}^{n-k}(X_{t}-\mu)(X_{t+k}-\mu)
  63. k < n k<n
  64. μ \mu
  65. σ 2 \sigma^{2}
  66. μ \mu
  67. σ 2 \sigma^{2}
  68. n - k n-k
  69. n n
  70. { X 1 , X 2 , , X n - k } \{X_{1},\,X_{2},\,\ldots,\,X_{n-k}\}
  71. { X k + 1 , X k + 2 , , X n } \{X_{k+1},\,X_{k+2},\,\ldots,\,X_{n}\}
  72. k k
  73. X X
  74. χ 2 \chi^{2}

Automorphism.html

  1. μ \mu
  2. λ \lambda

Availability.html

  1. A = E [ Uptime ] E [ Uptime ] + E [ Downtime ] A=\frac{E[\mathrm{Uptime}]}{E[\mathrm{Uptime}]+E[\mathrm{Downtime}]}
  2. X ( t ) X(t)
  3. X ( t ) = { 1 , sys functions at time t 0 , otherwise X(t)=\begin{cases}1,&\mbox{sys functions at time }~{}t\\ 0,&\mbox{otherwise}\end{cases}
  4. A ( t ) = Pr [ X ( t ) = 1 ] = E [ X ( t ) ] . A(t)=\Pr[X(t)=1]=E[X(t)].
  5. c > 0 c>0
  6. A c = 1 c 0 c A ( t ) d t . A_{c}=\frac{1}{c}\int_{0}^{c}A(t)\,dt.
  7. A = lim c A c . A=\lim_{c\rightarrow\infty}A_{c}.
  8. [ 0 , c ] [0,c]
  9. A = lim c A c = lim c 1 c 0 c A ( t ) d t , c > 0. A_{\infty}=\lim_{c\rightarrow\infty}A_{c}=\lim_{c\rightarrow\infty}\frac{1}{c}% \int_{0}^{c}A(t)\,dt,\quad c>0.

Average.html

  1. A M = 1 n i = 1 n a i = 1 n ( a 1 + a 2 + + a n ) AM=\frac{1}{n}\sum_{i=1}^{n}a_{i}=\frac{1}{n}\left(a_{1}+a_{2}+\cdots+a_{n}\right)
  2. G M = i = 1 n a i n = a 1 a 2 a n n GM=\sqrt[n]{\prod_{i=1}^{n}a_{i}}=\sqrt[n]{a_{1}a_{2}\cdots a_{n}}
  3. G M = 2 8 = 4 GM=\sqrt{2\cdot 8}=4
  4. H M = 1 1 n i = 1 n 1 a i = n 1 a 1 + 1 a 2 + + 1 a n HM=\frac{1}{\frac{1}{n}\sum_{i=1}^{n}\frac{1}{a_{i}}}=\frac{n}{\frac{1}{a_{1}}% +\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}}
  5. 2 1 60 + 1 40 = 48 \frac{2}{\frac{1}{60}+\frac{1}{40}}=48
  6. A M G M H M AM\geq GM\geq HM
  7. x ¯ = 1 n i = 1 n x i = 1 n ( x 1 + + x n ) \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{1}{n}(x_{1}+\cdots+x_{n})
  8. ( i = 1 n x i ) 1 n = x 1 x 2 x n n \bigg(\prod_{i=1}^{n}x_{i}\bigg)^{\frac{1}{n}}=\sqrt[n]{x_{1}\cdot x_{2}\cdots x% _{n}}
  9. n 1 x 1 + 1 x 2 + + 1 x n \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}}
  10. 1 n i = 1 n x i 2 = 1 n ( x 1 2 + x 2 2 + + x n 2 ) \sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}}=\sqrt{\frac{1}{n}\left(x_{1}^{2}+x_{% 2}^{2}+\cdots+x_{n}^{2}\right)}
  11. 1 n i = 1 n x i 3 3 = 1 n ( x 1 3 + x 2 3 + + x n 3 ) 3 \sqrt[3]{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{3}}=\sqrt[3]{\frac{1}{n}\left(x_{1}^{% 3}+x_{2}^{3}+\cdots+x_{n}^{3}\right)}
  12. 1 n i = 1 n x i p p \sqrt[p]{\frac{1}{n}\cdot\sum_{i=1}^{n}x_{i}^{p}}
  13. i = 1 n w i x i i = 1 n w i = w 1 x 1 + w 2 x 2 + + w n x n w 1 + w 2 + + w n \frac{\sum_{i=1}^{n}w_{i}x_{i}}{\sum_{i=1}^{n}w_{i}}=\frac{w_{1}x_{1}+w_{2}x_{% 2}+\cdots+w_{n}x_{n}}{w_{1}+w_{2}+\cdots+w_{n}}
  14. 1 2 ( max x + min x ) \frac{1}{2}\left(\max x+\min x\right)
  15. y = f - 1 ( 1 n [ f ( x 1 ) + f ( x 2 ) + + f ( x n ) ] ) y=f^{-1}\left(\frac{1}{n}\left[f(x_{1})+f(x_{2})+\cdots+f(x_{n})\right]\right)

AVL_tree.html

  1. μ 1 2 \scriptstyle\mu\leq\tfrac{1}{2}
  2. μ \mu
  3. 0 μ 1 2 0\leq\mu\leq\tfrac{1}{2}
  4. N N
  5. 1 2 - μ | N l | | N | + 1 1 2 + μ \tfrac{1}{2}-\mu\leq\tfrac{|N_{l}|}{|N|+1}\leq\tfrac{1}{2}+\mu
  6. μ \mu
  7. | N | |N|
  8. N N
  9. N l N_{l}
  10. N N
  11. O ( l o g n ) O(logn)
  12. O ( l o g n ) O(logn)
  13. O ( l o g n ) O(logn)
  14. O ( l o g n ) O(logn)
  15. O ( l o g n ) O(logn)
  16. O ( l o g n ) O(logn)
  17. n n
  18. log φ ( 5 ( n + 2 ) ) - 2 = log 2 ( 5 ( n + 2 ) ) log 2 ( φ ) - 2 = log φ ( 2 ) log 2 ( 5 ( n + 2 ) ) - 2 1.44 log 2 ( n + 2 ) - 0.328 \log_{\varphi}(\sqrt{5}(n+2))-2={\log_{2}(\sqrt{5}(n+2))\over\log_{2}(\varphi)% }-2=\log_{\varphi}(2)\cdot\log_{2}(\sqrt{5}(n+2))-2\approx 1.44\log_{2}(n+2)-0% .328
  19. φ \varphi
  20. 2 log 2 ( n + 1 ) 2\log_{2}(n+1)

Avogadro_constant.html

  1. L L
  2. n 0 n_{0}
  3. n 0 = p 0 N A R T 0 n_{0}=\frac{p_{0}N_{\rm A}}{RT_{0}}
  4. A {}_{A}
  5. R = k B N A = 8.314 4621 ( 75 ) J mol - 1 K - 1 R=k_{\rm B}N_{\rm A}=8.314\,4621(75)\ {\rm J\,mol^{-1}\,K^{-1}}\,
  6. F = N A e = 96 485.3365 ( 21 ) C mol - 1 . F=N_{\rm A}e=96\,485.3365(21)\ {\rm C\,mol^{-1}}.\,
  7. 1 u = M u N A = 1.660 538 921 ( 73 ) × 10 - 27 kg 1\ {\rm u}=\frac{M_{\rm u}}{N_{\rm A}}=1.660\,538\,921(73)\times 10^{-27}\ {% \rm kg}
  8. N A = F e N_{\rm A}=\frac{F}{e}
  9. r {}_{r}
  10. F = A r M u I t m . F=\frac{A_{\rm r}M_{\rm u}It}{m}.
  11. 90 {}_{90}
  12. r {}_{r}
  13. u {}_{u}
  14. e {}_{e}
  15. N A = A r ( e ) M u m e . N_{\rm A}=\frac{A_{\rm r}({\rm e})M_{\rm u}}{m_{\rm e}}.
  16. r {}_{r}
  17. u {}_{u}
  18. m e = 2 R h c α 2 . m_{\rm e}=\frac{2R_{\infty}h}{c\alpha^{2}}.
  19. A {}_{A}
  20. r {}_{r}
  21. × 10 4 \times 10^{–}4
  22. × 10 10 \times 10^{–}10
  23. u {}_{u}
  24. {}_{∞}
  25. × 10 12 \times 10^{–}12
  26. × 10 34 \times 10^{–}34
  27. × 10 8 \times 10^{–}8
  28. × 10 3 \times 10^{–}3
  29. × 10 10 \times 10^{–}10
  30. A {}_{A}
  31. × 10 2 3 \times 10^{2}3
  32. × 10 8 \times 10^{–}8
  33. m {}_{m}
  34. N A = V m V atom N_{\rm A}=\frac{V_{\rm m}}{V_{\rm atom}}
  35. V atom = V cell n V_{\rm atom}=\frac{V_{\rm cell}}{n}
  36. 220 {}_{220}
  37. 220 {}_{220}
  38. r {}_{r}
  39. m {}_{m}
  40. V m = A r M u ρ V_{\rm m}=\frac{A_{\rm r}M_{\rm u}}{\rho}
  41. u {}_{u}
  42. × 10 8 \times 10^{−}8
  43. h = c α 2 A r ( e ) M u 2 R N A . h=\frac{c\alpha^{2}A_{\rm r}({\rm e})M_{\rm u}}{2R_{\infty}N_{\rm A}}.
  44. × 10 7 \times 10^{−}7

Axiom.html

  1. ϕ \phi
  2. χ \chi
  3. ψ \psi
  4. ¬ \neg
  5. \to\,
  6. ϕ ( ψ ϕ ) \phi\to(\psi\to\phi)
  7. ( ϕ ( ψ χ ) ) ( ( ϕ ψ ) ( ϕ χ ) ) (\phi\to(\psi\to\chi))\to((\phi\to\psi)\to(\phi\to\chi))
  8. ( ¬ ϕ ¬ ψ ) ( ψ ϕ ) . (\lnot\phi\to\lnot\psi)\to(\psi\to\phi).
  9. A A
  10. B B
  11. C C
  12. A ( B A ) A\to(B\to A)
  13. ( A ¬ B ) ( C ( A ¬ B ) ) (A\to\lnot B)\to(C\to(A\to\lnot B))
  14. 𝔏 \mathfrak{L}\,
  15. x x\,
  16. x = x x=x\,
  17. x , x\,,
  18. x = x x=x\,
  19. x = x x=x\,
  20. = =\,
  21. ϕ \phi\,
  22. 𝔏 \mathfrak{L}\,
  23. x x\,
  24. t t\,\!
  25. x x\,
  26. ϕ \phi\,
  27. x ϕ ϕ t x \forall x\,\phi\to\phi^{x}_{t}
  28. ϕ t x \phi^{x}_{t}
  29. ϕ \phi\,
  30. t t\,\!
  31. x x\,
  32. P P\,
  33. x x\,
  34. t t\,\!
  35. P ( t ) P(t)\,
  36. x ϕ ϕ t x \forall x\phi\to\phi^{x}_{t}
  37. ϕ \phi\,
  38. 𝔏 \mathfrak{L}\,
  39. x x\,
  40. t t\,\!
  41. x x\,
  42. ϕ \phi\,
  43. ϕ t x x ϕ \phi^{x}_{t}\to\exists x\,\phi
  44. 𝔏 N T = { 0 , S } \mathfrak{L}_{NT}=\{0,S\}\,
  45. 0 0\,
  46. S S\,
  47. x . ¬ ( S x = 0 ) \forall x.\lnot(Sx=0)
  48. x . y . ( S x = S y x = y ) \forall x.\forall y.(Sx=Sy\to x=y)
  49. ( ( ϕ ( 0 ) x . ( ϕ ( x ) ϕ ( S x ) ) ) x . ϕ ( x ) ((\phi(0)\land\forall x.\,(\phi(x)\to\phi(Sx)))\to\forall x.\phi(x)
  50. 𝔏 N T \mathfrak{L}_{NT}\,
  51. ϕ \phi
  52. 𝔑 = 𝒩 , 0 , S \mathfrak{N}=\langle\mathcal{N},0,S\rangle\,
  53. 𝒩 \mathcal{N}\,
  54. S S\,
  55. 0 0\,
  56. Λ \Lambda\,
  57. Σ \Sigma\,
  58. { ( Γ , ϕ ) } \{(\Gamma,\phi)\}\,
  59. ϕ \phi
  60. if Σ ϕ then Σ ϕ \,\text{if }\Sigma\models\phi\,\text{ then }\Sigma\vdash\phi
  61. Σ \Sigma\,
  62. Σ \Sigma\,
  63. Σ \Sigma\,
  64. ϕ \phi\,
  65. ϕ \phi\,
  66. ¬ ϕ \lnot\phi\,

Axiom_of_choice.html

  1. ( S i ) i I (S_{i})_{i\in I}
  2. ( x i ) i I (x_{i})_{i\in I}
  3. x i S i x_{i}\in S_{i}
  4. i I i\in I
  5. X [ X f : X X A X ( f ( A ) A ) ] . \forall X\left[\emptyset\notin X\implies\exists f\colon X\rightarrow\bigcup X% \quad\forall A\in X\,(f(A)\in A)\right]\,.
  6. G S G_{S}
  7. S S

Axiom_of_empty_set.html

  1. x y ¬ ( y x ) \exists x\,\forall y\,\lnot(y\in x)

Axiom_of_extensionality.html

  1. A B ( X ( X A X B ) A = B ) \forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\Rightarrow A=B)
  2. A B ( A = B X ( X A X B ) ) \forall A\,\forall B\,(A=B\Rightarrow\forall X\,(X\in A\iff X\in B))
  3. A X ( X A P ( X ) ) \exists A\,\forall X\,(X\in A\iff P(X)\,)
  4. A A
  5. P P
  6. A A
  7. A B ( X ( X A X B ) Y ( A Y B Y ) ) \forall A\,\forall B\,(\forall X\,(X\in A\iff X\in B)\Rightarrow\forall Y\,(A% \in Y\iff B\in Y)\,)
  8. B A B\in A
  9. A A
  10. B A B\in A
  11. A A
  12. A B ( X ( X A ) [ Y ( Y A Y B ) A = B ] ) . \forall A\,\forall B\,(\exists X\,(X\in A)\Rightarrow[\forall Y\,(Y\in A\iff Y% \in B)\Rightarrow A=B]\,).
  13. A A
  14. A A
  15. A A

Axiom_of_pairing.html

  1. A B C D [ D C ( D = A D = B ) ] \forall A\,\forall B\,\exists C\,\forall D\,[D\in C\iff(D=AD=B)]
  2. a a
  3. b b
  4. ( a , b ) = { { a } , { a , b } } . (a,b)=\{\{a\},\{a,b\}\}.\,
  5. ( a , b ) = ( c , d ) a = c and b = d . (a,b)=(c,d)\iff a=c\and b=d.
  6. ( a 1 , , a n ) = ( ( a 1 , , a n - 1 ) , a n ) . (a_{1},\ldots,a_{n})=((a_{1},\ldots,a_{n-1}),a_{n}).\!
  7. A 1 A n C D [ D C ( D = A 1 D = A n ) ] \forall A_{1}\,\ldots\,\forall A_{n}\,\exists C\,\forall D\,[D\in C\iff(D=A_{1% }\cdots D=A_{n})]
  8. A B C D [ D C ( D A D = B ) ] \forall A\,\forall B\,\exists C\,\forall D\,[D\in C\iff(D\in AD=B)]

Axiom_of_power_set.html

  1. A P B [ B P C ( C B C A ) ] \forall A\,\exists P\,\forall B\,[B\in P\iff\forall C\,(C\in B\Rightarrow C\in A)]
  2. 𝒫 ( A ) \mathcal{P}(A)
  3. 𝒫 ( A ) \mathcal{P}(A)
  4. 𝒫 ( A ) \mathcal{P}(A)
  5. A A
  6. 𝒫 ( A ) \mathcal{P}(A)
  7. A A
  8. \subseteq
  9. \in
  10. 𝒫 ( A ) \mathcal{P}(A)
  11. X X
  12. Y Y
  13. X × Y = { ( x , y ) : x X y Y } . X\times Y=\{(x,y):x\in X\land y\in Y\}.
  14. x , y X Y x,y\in X\cup Y
  15. { x } , { x , y } 𝒫 ( X Y ) \{x\},\{x,y\}\in\mathcal{P}(X\cup Y)
  16. ( x , y ) = { { x } , { x , y } } 𝒫 ( 𝒫 ( X Y ) ) (x,y)=\{\{x\},\{x,y\}\}\in\mathcal{P}(\mathcal{P}(X\cup Y))
  17. X × Y 𝒫 ( 𝒫 ( X Y ) ) . X\times Y\subseteq\mathcal{P}(\mathcal{P}(X\cup Y)).
  18. X 1 × × X n = ( X 1 × × X n - 1 ) × X n . X_{1}\times\cdots\times X_{n}=(X_{1}\times\cdots\times X_{n-1})\times X_{n}.

Axiom_of_regularity.html

  1. x ( x y x ( y x = ) ) \forall x\,(x\neq\varnothing\rightarrow\exists y\in x\,(y\cap x=\varnothing))
  2. { ( n , α ) | n ω α is an ordinal } . \{(n,\alpha)|n\in\omega\land\alpha\,\text{ is an ordinal }\}\,.
  3. ( n - 1 ) n (n-1)\in n
  4. ( n - 2 ) ( n - 1 ) (n-2)\in(n-1)
  5. ( n - k - 1 ) ( n - k ) (n-k-1)\in(n-k)
  6. a R b : b S a aRb:\Leftrightarrow b\in S\cap a
  7. α V α \bigcup_{\alpha}V_{\alpha}\!
  8. α V α \bigcup_{\alpha}V_{\alpha}\!
  9. x ( x y x ( y x = ) ) \forall x\,(x\neq\emptyset\rightarrow\exists y\in x\,(y\cap x=\emptyset))

Axiom_of_union.html

  1. A B c ( c B D ( c D and D A ) ) \forall A\,\exists B\,\forall c\,(c\in B\iff\exists D\,(c\in D\and D\in A)\,)
  2. A A
  3. A \bigcup A
  4. A \bigcup A
  5. A B A\cup B
  6. { A , B } \bigcup\{A,B\}
  7. A \bigcap A

Axiom_schema_of_replacement.html

  1. w 1 , , w n A ( [ x A \displaystyle\forall w_{1},\ldots,w_{n}\,\forall A\,([\forall x\in A
  2. w 1 , , w n [ ( x y ϕ ( x , y , w 1 , , w n ) ) A B x A y B ϕ ( x , y , w 1 , , w n ) ] \forall w_{1},\ldots,w_{n}\,[(\forall x\,\exists\,y\phi(x,y,w_{1},\ldots,w_{n}% ))\Rightarrow\forall A\,\exists B\,\forall x\in A\,\exists y\in B\,\phi(x,y,w_% {1},\ldots,w_{n})]
  3. w 1 , , w n A B x A [ y ϕ ( x , y , w 1 , , w n ) y B ϕ ( x , y , w 1 , , w n ) ] \forall w_{1},\ldots,w_{n}\,\forall A\,\exists B\,\forall x\in A\,[\exists y% \phi(x,y,w_{1},\ldots,w_{n})\Rightarrow\exists y\in B\,\phi(x,y,w_{1},\ldots,w% _{n})]
  4. ω \aleph_{\omega}
  5. A B C ( C B [ C A and θ ( C ) ] ) \forall A\,\exists B\,\forall C\,(C\in B\Leftrightarrow[C\in A\and\theta(C)])
  6. V δ V_{\delta}

Axiom_schema_of_specification.html

  1. w 1 , , w n A B x ( x B [ x A and φ ( x , w 1 , , w n , A ) ] ) \forall w_{1},\ldots,w_{n}\,\forall A\,\exists B\,\forall x\,(x\in B% \Leftrightarrow[x\in A\and\varphi(x,w_{1},\ldots,w_{n},A)])
  2. A B C ( C B D [ D A and C = F ( D ) ] ) \forall A\,\exists B\,\forall C\,(C\in B\iff\exists D\,[D\in A\and C=F(D)])
  3. w 1 , , w n B x ( x B φ ( x , w 1 , , w n ) ) \forall w_{1},\ldots,w_{n}\,\exists B\,\forall x\,(x\in B\Leftrightarrow% \varphi(x,w_{1},\ldots,w_{n}))
  4. D C ( [ C D ] [ P ( C ) and E ( C E ) ] ) , \exists D\forall C\,([C\in D]\iff[P(C)\and\exists E\,(C\in E)])\,,
  5. D A ( E [ A E ] B [ E ( B E ) and C ( C B [ C A and C D ] ) ] ) , \forall D\forall A\,(\exists E\,[A\in E]\implies\exists B\,[\exists E\,(B\in E% )\and\forall C\,(C\in B\iff[C\in A\and C\in D])])\,,
  6. A B ( [ E ( A E ) and C ( C B C A ) ] E [ B E ] ) , \forall A\forall B\,([\exists E\,(A\in E)\and\forall C\,(C\in B\implies C\in A% )]\implies\exists E\,[B\in E])\,,

Azimuth.html

  1. α \alpha
  2. a z = 90.0 - 180.0 / π a t a n 2 ( X 2 - X 1 , Y 2 - Y 1 ) az=90.0-180.0/{\pi}\ {atan2\ (X2-X1,Y2-Y1)}
  3. ϕ 1 \phi_{1}
  4. ϕ 2 \phi_{2}
  5. α \alpha
  6. tan α = sin L ( cos ϕ 1 ) ( tan ϕ 2 ) - ( sin ϕ 1 ) ( cos L ) \tan\alpha=\frac{\sin L}{(\cos\phi_{1})(\tan\phi_{2})-(\sin\phi_{1})(\cos L)}
  7. f f
  8. e 2 = f ( 2 - f ) e^{2}=f(2-f)\,
  9. 1 - e 2 = ( 1 - f ) 2 1-e^{2}=(1-f)^{2}\,
  10. Λ = ( 1 - e 2 ) tan ϕ 2 tan ϕ 1 + e 2 1 + ( 1 - e 2 ) ( tan ϕ 2 ) 2 1 + ( 1 - e 2 ) ( tan ϕ 1 ) 2 \Lambda=(1-e^{2})\frac{\tan\phi_{2}}{\tan\phi_{1}}+e^{2}\sqrt{\cfrac{1+(1-e^{2% })(\tan\phi_{2})^{2}}{1+(1-e^{2})(\tan\phi_{1})^{2}}}
  11. tan α = sin L ( Λ - cos L ) sin ϕ 1 \tan\alpha=\frac{\sin L}{(\Lambda-\cos L)\sin\phi_{1}}
  12. ϕ 1 \phi_{1}
  13. tan α = sin L ( 1 - e 2 ) tan ϕ 2 \tan\alpha=\frac{\sin L}{(1-e^{2})\tan\phi_{2}}
  14. ϕ 2 \phi_{2}
  15. θ \theta
  16. ϕ \phi

B-spline.html

  1. S n , t ( x ) = i α i B i , n ( x ) S_{n,t}(x)=\sum_{i}\alpha_{i}B_{i,n}(x)
  2. B i , 1 ( x ) := { 1 if t i x < t i + 1 0 otherwise B_{i,1}(x):=\left\{\begin{matrix}1&\mathrm{if}\quad t_{i}\leq x<t_{i+1}\\ 0&\mathrm{otherwise}\end{matrix}\right.
  3. B i , k ( x ) := x - t i t i + k - 1 - t i B i , k - 1 ( x ) + t i + k - x t i + k - t i + 1 B i + 1 , k - 1 ( x ) . B_{i,k}(x):=\frac{x-t_{i}}{t_{i+k-1}-t_{i}}B_{i,k-1}(x)+\frac{t_{i+k}-x}{t_{i+% k}-t_{i+1}}B_{i+1,k-1}(x).
  4. B j , 1 ( x ) B_{j,1}(x)
  5. x - t i t i + k - 1 - t i \frac{x-t_{i}}{t_{i+k-1}-t_{i}}
  6. t i t_{i}
  7. t i + k - 1 t_{i+k-1}
  8. t i + k - x t i + k - t i + 1 \frac{t_{i+k}-x}{t_{i+k}-t_{i+1}}
  9. t i + 1 t_{i+1}
  10. t i + k t_{i+k}
  11. B i , 2 ( x ) B_{i,2}(x)
  12. x = t i x=t_{i}
  13. x = t i + 1 x=t_{i+1}
  14. x = t i + 2 x=t_{i+2}
  15. 0 0 0 B i - 2 , 3 B i - 1 , 2 B i , 1 B i - 1 , 3 B i , 2 0 B i , 3 0 0 \begin{matrix}&&0\\ &0&\\ 0&&B_{i-2,3}\\ &B_{i-1,2}&\\ B_{i,1}&&B_{i-1,3}\\ &B_{i,2}&\\ 0&&B_{i,3}\\ &0&\\ &&0\\ \end{matrix}
  16. B 1 = x 2 / 2 0 x 1 B_{1}=x^{2}/2\qquad 0\leq x\leq 1
  17. B 2 = ( - 2 x 2 + 6 x - 3 ) / 2 1 x 2 B_{2}=(-2x^{2}+6x-3)/2\qquad 1\leq x\leq 2
  18. B 3 = ( 3 - x ) 2 / 2 2 x 3 B_{3}=(3-x)^{2}/2\qquad 2\leq x\leq 3
  19. At x=1 , B 1 = B 2 = 0.5 ; d B 1 d x = d B 2 d x = 1 \mbox{At x=1}~{},B_{1}=B_{2}=0.5;\frac{dB_{1}}{dx}=\frac{dB_{2}}{dx}=1
  20. At x=2 , B 2 = B 3 = 0.5 ; d B 2 d x = d B 3 d x = - 1 \mbox{At x=2}~{},B_{2}=B_{3}=0.5;\frac{dB_{2}}{dx}=\frac{dB_{3}}{dx}=-1
  21. d 2 B 1 d x 2 = 1 , d 2 B 2 d x 2 = - 2 , d 2 B 3 d x 2 = 1 , \frac{d^{2}B_{1}}{dx^{2}}=1,\frac{d^{2}B_{2}}{dx^{2}}=-2,\frac{d^{2}B_{3}}{dx^% {2}}=1,
  22. B i , n , t ( x ) = x - t i h n [ 0 , , n ] ( . - t i ) + n - 1 B_{i,n,t}(x)=\frac{x-t_{i}}{h}n[0,\dots,n](.-t_{i})^{n-1}_{+}
  23. ( t - x ) + n - 1 (t-x)^{n-1}_{+}
  24. ( t - x ) + n - 1 (t-x)^{n-1}_{+}
  25. 𝐛 \,\textbf{b}
  26. 𝐱 = [ 𝐛 1 , 0 , 0 , 𝐛 2 , 0 , 0 , 𝐛 3 , 0 , 0 , . , 𝐛 n , 0 , 0 ] \,\textbf{x}=[\,\textbf{b}_{1},0,0,\,\textbf{b}_{2},0,0,\,\textbf{b}_{3},0,0,.% ...,\,\textbf{b}_{n},0,0]
  27. 𝐱 \,\textbf{x}
  28. 𝐡 = [ 1 / 3 , 1 / 3 , 1 / 3 ] \,\textbf{h}=[1/3,1/3,1/3]
  29. 𝐲 = 𝐱 * 𝐡 * 𝐡 \,\textbf{y}=\,\textbf{x}*\,\textbf{h}*\,\textbf{h}
  30. d B i , k ( x ) d x = ( k - 1 ) ( - B i + 1 , k - 1 ( x ) t i + k - t i + 1 + B i . k - 1 ( x ) t i + k - 1 - t i ) \frac{dB_{i,k}(x)}{dx}=(k-1)\left(\frac{-B_{i+1,k-1}(x)}{t_{i+k}-t_{i+1}}+% \frac{B_{i.k-1}(x)}{t_{i+k-1}-t_{i}}\right)
  31. d d x i α i B i , k = i = r - k + 2 s - 1 ( k - 1 ) α i - α i - 1 t i + k - 1 - t i B i , k - 1 o n [ t r . t s ] \frac{d}{dx}\sum_{i}\alpha_{i}B_{i,k}=\sum_{i=r-k+2}^{s-1}(k-1)\frac{\alpha_{i% }-\alpha_{i-1}}{t_{i+k-1}-t_{i}}B_{i,k-1}\ on[t_{r}.t_{s}]
  32. U = a l l x { W ( x ) [ y ( x ) - i α i B i , k , t ( x ) ] } 2 U=\sum_{allx}\left\{W(x)\left[y(x)-\sum_{i}\alpha_{i}B_{i,k,t}(x)\right]\right% \}^{2}
  33. α i \alpha_{i}
  34. C ( u ) = i = 1 k N i , n w i P i i = 1 k N i , n w i C(u)=\frac{\sum_{i=1}^{k}{N_{i,n}w_{i}P}_{i}}{\sum_{i=1}^{k}{N_{i,n}w_{i}}}
  35. C ( u ) = i = 1 k R i , n ( u ) P i C(u)=\sum_{i=1}^{k}R_{i,n}(u)P_{i}
  36. R i , n ( u ) = N i , n ( u ) w i j = 1 k N j , n ( u ) w j R_{i,n}(u)={N_{i,n}(u)w_{i}\over\sum_{j=1}^{k}N_{j,n}(u)w_{j}}
  37. S ( u , v ) = i = 1 k j = 1 l R i , j ( u , v ) P i , j S(u,v)=\sum_{i=1}^{k}\sum_{j=1}^{l}R_{i,j}(u,v)P_{i,j}
  38. R i , j ( u , v ) = N i , n ( u ) N j , m ( v ) w i , j p = 1 k q = 1 l N p , n ( u ) N q , m ( v ) w p , q R_{i,j}(u,v)=\frac{N_{i,n}(u)N_{j,m}(v)w_{i,j}}{\sum_{p=1}^{k}\sum_{q=1}^{l}N_% {p,n}(u)N_{q,m}(v)w_{p,q}}

B-tree.html

  1. d d
  2. 2 d 2d
  3. d d
  4. d + 1 d+1
  5. 2 d 2d
  6. 2 d 2d
  7. d d
  8. d d
  9. d - 1 d-1
  10. d d
  11. 2 d 2d
  12. ( d + 1 ) (d+1)
  13. ( 2 d + 1 ) (2d+1)
  14. ( 2 d + 1 ) (2d+1)
  15. N N
  16. log 2 N \lceil\log_{2}N\rceil
  17. log 2 ( 1 , 000 , 000 ) = 20 \lceil\log_{2}(1,000,000)\rceil=20
  18. log 2 N \log_{2}N
  19. log b N \log_{b}N
  20. b b
  21. b = 100 b=100
  22. log b 1 , 000 , 000 = 3 \log_{b}1,000,000=3
  23. m / 2 {m}/{2}
  24. log m ( n + 1 ) . \lceil\log_{m}(n+1)\rceil.
  25. h log d ( n + 1 2 ) . h\leq\left\lfloor\log_{d}\left(\frac{n+1}{2}\right)\right\rfloor.
  26. i i
  27. i i

Baire_category_theorem.html

  1. U n U_{n}
  2. n U n \bigcap_{n}U_{n}
  3. \mathbb{R}
  4. d ( x , y ) = 1 n + 1 d(x,y)=\frac{1}{n+1}
  5. n n
  6. x x
  7. y y
  8. X \scriptstyle X
  9. U n \scriptstyle U_{n}
  10. U n \scriptstyle\bigcap U_{n}
  11. W \scriptstyle W
  12. X \scriptstyle X
  13. x \scriptstyle x
  14. U n \scriptstyle U_{n}
  15. U 1 \scriptstyle U_{1}
  16. W \scriptstyle W
  17. U 1 \scriptstyle U_{1}
  18. x 1 \scriptstyle x_{1}
  19. 0 < r 1 < 1 \scriptstyle 0\;<\;r_{1}\;<\;1
  20. B ¯ ( x 1 , r 1 ) W U 1 \overline{B}(x_{1},r_{1})\subset W\cap U_{1}
  21. B ( x , r ) \scriptstyle B(x,r)
  22. B ¯ ( x , r ) \scriptstyle\overline{B}(x,r)
  23. x \scriptstyle x
  24. r \scriptstyle r
  25. U n \scriptstyle U_{n}
  26. x n \scriptstyle x_{n}
  27. 0 < r n < 1 n \scriptstyle 0\;<\;r_{n}\;<\;\frac{1}{n}
  28. B ¯ ( x n , r n ) B ( x n - 1 , r n - 1 ) U n \overline{B}(x_{n},r_{n})\subset B(x_{n-1},r_{n-1})\cap U_{n}
  29. x n B ( x m , r m ) \scriptstyle x_{n}\;\in\;B(x_{m},r_{m})
  30. n > m \scriptstyle n\;>\;m
  31. x n \scriptstyle x_{n}
  32. x n \scriptstyle x_{n}
  33. x x
  34. n \scriptstyle n
  35. x B ¯ ( x n , r n ) . x\in\overline{B}(x_{n},r_{n}).
  36. x W \scriptstyle x\;\in\;W
  37. x U n \scriptstyle x\;\in\;U_{n}
  38. n \scriptstyle n

Balanced_line.html

  1. Z 0 Z_{0}
  2. Z 0 = 1 π μ ϵ ln ( l R + ( l R ) 2 - 1 ) , Z_{0}=\frac{1}{\pi}\sqrt{\frac{\mu}{\epsilon}}\ln\left(\frac{l}{R}+\sqrt{\left% (\frac{l}{R}\right)^{2}-1}~{}\right),
  3. l l
  4. R R
  5. μ \mu
  6. ϵ \epsilon
  7. Z 0 = 120 ϵ r ln ( 2 l R ) , Z_{0}=\frac{120}{\sqrt{\epsilon_{r}}}\ln\left(\frac{2l}{R}\right),
  8. ϵ r \epsilon_{r}

Baltimore_Ravens.html

  1. W i n s + 1 2 T i e s G a m e s \frac{Wins+\frac{1}{2}Ties}{Games}

Banach_algebra.html

  1. x , y A : x y x y \forall x,y\in A:\|x\,y\|\ \leq\|x\|\,\|y\|
  2. A A
  3. A e A_{e}
  4. A e A_{e}
  5. A e A_{e}
  6. C 0 ( X ) C_{0}(X)
  7. C 0 ( X ) C_{0}(X)
  8. C 0 ( X ) C_{0}(X)
  9. x y - y x 𝟏 xy-yx\neq\mathbf{1}
  10. σ ( x ) \sigma(x)
  11. σ ( x ) \sigma(x)
  12. sup { | λ | : λ σ ( x ) } = lim n x n 1 / n . \sup\{|\lambda|:\lambda\in\sigma(x)\}=\lim_{n\to\infty}\|x^{n}\|^{1/n}.
  13. σ ( x ) . \sigma(x).
  14. σ ( f ( x ) ) = f ( σ ( x ) ) . \sigma(f(x))=f(\sigma(x)).
  15. σ ( f ) = { f ( t ) : t X } . \sigma(f)=\{f(t):t\in X\}.
  16. 𝔪 \mathfrak{m}
  17. A / 𝔪 A/\mathfrak{m}
  18. σ ( x ) = σ ( x ^ ) \sigma(x)=\sigma(\hat{x})
  19. x ^ \hat{x}
  20. x ^ \hat{x}
  21. x ^ ( χ ) = χ ( x ) . \hat{x}(\chi)=\chi(x).
  22. x ^ , \hat{x},
  23. σ ( x ^ ) = { χ ( x ) : χ Δ ( A ) } \sigma(\hat{x})=\{\chi(x):\chi\in\Delta(A)\}

Banach_fixed-point_theorem.html

  1. d ( T ( x ) , T ( y ) ) q d ( x , y ) d(T(x),T(y))\leq qd(x,y)
  2. d ( x * , x n ) q n 1 - q d ( x 1 , x 0 ) , d ( x * , x n + 1 ) q 1 - q d ( x n + 1 , x n ) , d ( x * , x n + 1 ) q d ( x * , x n ) . \begin{array}[]{rcl}d(x^{*},x_{n})&\leq&\frac{q^{n}}{1-q}d(x_{1},x_{0}),\\ d(x^{*},x_{n+1})&\leq&\frac{q}{1-q}d(x_{n+1},x_{n}),\\ d(x^{*},x_{n+1})&\leq&qd(x^{*},x_{n}).\end{array}
  3. d ( x 1 + 1 , x 1 ) = d ( x 2 , x 1 ) = d ( T ( x 1 ) , T ( x 0 ) ) q d ( x 1 , x 0 ) . d(x_{1+1},x_{1})=d(x_{2},x_{1})=d(T(x_{1}),T(x_{0}))\leq qd(x_{1},x_{0}).
  4. d ( x ( k + 1 ) + 1 , x k + 1 ) = d ( x k + 2 , x k + 1 ) = d ( T ( x k + 1 ) , T ( x k ) ) q d ( x k + 1 , x k ) q q k d ( x 1 , x 0 ) Induction Hypothesis = q k + 1 d ( x 1 , x 0 ) . \begin{array}[]{rclll}d(x_{(k+1)+1},x_{k+1})&=&d(x_{k+2},x_{k+1})\\ &=&d(T(x_{k+1}),T(x_{k}))\\ &\leq&qd(x_{k+1},x_{k})\\ &\leq&qq^{k}d(x_{1},x_{0})&&\,\text{Induction Hypothesis}\\ &=&q^{k+1}d(x_{1},x_{0}).\end{array}
  5. d ( x m , x n ) d ( x m , x m - 1 ) + d ( x m - 1 , x m - 2 ) + + d ( x n + 1 , x n ) Triangle Inequality q m - 1 d ( x 1 , x 0 ) + q m - 2 d ( x 1 , x 0 ) + + q n d ( x 1 , x 0 ) Lemma 1 = q n d ( x 1 , x 0 ) k = 0 m - n - 1 q k q n d ( x 1 , x 0 ) k = 0 q k = q n d ( x 1 , x 0 ) ( 1 1 - q ) Geometric Series \begin{array}[]{rclll}d(x_{m},x_{n})&\leq&d(x_{m},x_{m-1})+d(x_{m-1},x_{m-2})+% \cdots+d(x_{n+1},x_{n})&&\,\text{Triangle Inequality}\\ &\leq&q^{m-1}d(x_{1},x_{0})+q^{m-2}d(x_{1},x_{0})+\cdots+q^{n}d(x_{1},x_{0})&&% \,\text{Lemma 1}\\ &=&q^{n}d(x_{1},x_{0})\sum_{k=0}^{m-n-1}q^{k}\\ &\leq&q^{n}d(x_{1},x_{0})\sum_{k=0}^{\infty}q^{k}\\ &=&q^{n}d(x_{1},x_{0})\left(\frac{1}{1-q}\right)&&\,\text{Geometric Series}% \end{array}
  6. q N < ε ( 1 - q ) d ( x 1 , x 0 ) . q^{N}<\frac{\varepsilon(1-q)}{d(x_{1},x_{0})}.
  7. d ( x m , x n ) q n d ( x 1 , x 0 ) ( 1 1 - q ) < ( ε ( 1 - q ) d ( x 1 , x 0 ) ) d ( x 1 , x 0 ) ( 1 1 - q ) = ε . d(x_{m},x_{n})\leq q^{n}d(x_{1},x_{0})\left(\frac{1}{1-q}\right)<\left(\frac{% \varepsilon(1-q)}{d(x_{1},x_{0})}\right)d(x_{1},x_{0})\left(\frac{1}{1-q}% \right)=\varepsilon.
  8. lim n x n = lim n T ( x n - 1 ) \lim_{n\to\infty}x_{n}=\lim_{n\to\infty}T(x_{n-1})
  9. lim n x n = T ( lim n x n - 1 ) . \lim_{n\to\infty}x_{n}=T\left(\lim_{n\to\infty}x_{n-1}\right).
  10. 0 d ( x * , y ) = d ( T ( x * ) , T ( y ) ) q d ( x * , y ) . 0\leq d(x^{*},y)=d(T(x^{*}),T(y))\leq qd(x^{*},y).
  11. d ( x , y ) d ( x , T ( x ) ) + d ( T ( x ) , T ( y ) ) + d ( T ( y ) , y ) d ( x , T ( x ) ) + q d ( x , y ) + d ( T ( y ) , y ) \begin{array}[]{rl}d(x,y)&\leq d(x,T(x))+d(T(x),T(y))+d(T(y),y)\\ &\leq d(x,T(x))+qd(x,y)+d(T(y),y)\end{array}
  12. d ( x , y ) d ( T ( x ) , x ) + d ( T ( y ) , y ) 1 - q , d(x,y)\leq\frac{d(T(x),x)+d(T(y),y)}{1-q},
  13. d ( T n ( x 0 ) , T m ( x 0 ) ) d ( T ( T n ( x 0 ) ) , T n ( x 0 ) ) + d ( T ( T m ( x 0 ) ) , T m ( x 0 ) ) 1 - q , = d ( T n ( T ( x 0 ) ) , T n ( x 0 ) ) + d ( T m ( T ( x 0 ) ) , T m ( x 0 ) ) 1 - q q n d ( T ( x 0 ) , x 0 ) + q m d ( T ( x 0 ) , x 0 ) 1 - q = q n + q m 1 - q d ( T ( x 0 ) , x 0 ) \begin{array}[]{rcl}d(T^{n}(x_{0}),T^{m}(x_{0}))&\leq&\frac{d(T(T^{n}(x_{0})),% T^{n}(x_{0}))+d(T(T^{m}(x_{0})),T^{m}(x_{0}))}{1-q},\\ &=&\frac{d(T^{n}(T(x_{0})),T^{n}(x_{0}))+d(T^{m}(T(x_{0})),T^{m}(x_{0}))}{1-q}% \\ &\leq&\frac{q^{n}d(T(x_{0}),x_{0})+q^{m}d(T(x_{0}),x_{0})}{1-q}\\ &=&\frac{q^{n}+q^{m}}{1-q}d(T(x_{0}),x_{0})\end{array}
  14. d ( T n ( x 0 ) , x * ) q n 1 - q d ( T ( x 0 ) , x 0 ) d(T^{n}(x_{0}),x^{*})\leq\frac{q^{n}}{1-q}d(T(x_{0}),x_{0})
  15. n d ( T n ( x ) , T n ( y ) ) < . \sum\nolimits_{n}d(T^{n}(x),T^{n}(y))<\infty.

Banach_space.html

  1. X X
  2. 𝐑 \mathbf{R}
  3. 𝐂 \mathbf{C}
  4. X X
  5. x x
  6. X X
  7. lim n x n = x , \lim_{n\to\infty}x_{n}=x,
  8. lim n x n - x X = 0. \lim_{n\to\infty}\left\|x_{n}-x\right\|_{X}=0.
  9. X X
  10. X X
  11. n = 1 v n X < implies that n = 1 v n converges in X . \sum_{n=1}^{\infty}\|v_{n}\|_{X}<\infty\quad\,\text{implies that}\quad\sum_{n=% 1}^{\infty}v_{n}\ \ \,\text{converges in}\ \ X.
  12. 𝐑 \mathbf{R}
  13. 𝐂 \mathbf{C}
  14. X X
  15. Y Y
  16. 𝐊 \mathbf{K}
  17. 𝐊 \mathbf{K}
  18. T : X Y T:X→Y
  19. B ( X , Y ) B(X,Y)
  20. X X
  21. X X
  22. B ( X , Y ) B(X,Y)
  23. T = sup { T x Y x X , x X 1 } . \|T\|=\sup\left\{\|Tx\|_{Y}\mid x\in X,\ \|x\|_{X}\leq 1\right\}.
  24. Y Y
  25. B ( X , Y ) B(X,Y)
  26. X X
  27. B ( X ) = B ( X , X ) B(X)=B(X,X)
  28. X X
  29. Y Y
  30. T : X Y T:X→Y
  31. T T
  32. X X
  33. Y Y
  34. X X
  35. Y Y
  36. T T
  37. [ u ! ! ] T ( x ) [ u ! ! ] = [ u ! ! ] x [ u ! ! ] [u^{\prime}!!^{\prime}]T(x)[u^{\prime}!!^{\prime}]=[u^{\prime}!!^{\prime}]x[u^% {\prime}!!^{\prime}]
  38. x x
  39. X X
  40. d ( X , Y ) d(X,Y)
  41. X X
  42. Y Y
  43. X X
  44. Y Y
  45. X X
  46. Y Y
  47. T : X Y T:X→Y
  48. T ( X ) T(X)
  49. Y Y
  50. Z Z
  51. X X
  52. Z Z
  53. Z Z
  54. Y Y
  55. Y Y
  56. X X
  57. Y Y
  58. X X
  59. X X
  60. Y Y
  61. X X
  62. X ^ \widehat{X}
  63. X × Y X×Y
  64. ( x , y ) 1 = x + y , ( x , y ) = max ( x , y ) \|(x,y)\|_{1}=\|x\|+\|y\|,\qquad\|(x,y)\|_{\infty}=\max(\|x\|,\|y\|)
  65. X × Y X×Y
  66. X Y X⊕Y
  67. M M
  68. X X
  69. X / M X/M
  70. x + M = inf m M x + m . \|x+M\|=\inf\limits_{m\in M}\|x+m\|.
  71. X / M X/M
  72. X X
  73. X X
  74. X / M X/M
  75. x x
  76. X X
  77. x + M x+M
  78. 1 1
  79. M = X M=X
  80. M M
  81. X X
  82. X X
  83. M M
  84. P P
  85. X X
  86. M M
  87. X X
  88. M M
  89. K e r ( P ) Ker(P)
  90. P P
  91. X X
  92. Y Y
  93. T B ( X , Y ) T∈B(X,Y)
  94. T T
  95. T = T 1 π , T : X 𝜋 X / Ker ( T ) T 1 Y T=T_{1}\circ\pi,\ \ \ T:X\ \overset{\pi}{\longrightarrow}\ X/\operatorname{Ker% }(T)\ \overset{T_{1}}{\longrightarrow}\ Y
  96. π π
  97. x + K e r ( T ) x+Ker(T)
  98. T ( x ) T(x)
  99. Y Y
  100. X / K e r ( T ) X/ Ker(T)
  101. T ( X ) T(X)
  102. 𝐍 \mathbf{N}
  103. C ( K ) C(K)
  104. K K
  105. f C ( K ) = max { | f ( x ) | : x K } , f C ( K ) . \|f\|_{C(K)}=\max\{|f(x)|:x\in K\},\quad f\in C(K).
  106. C ( K ) C(K)
  107. X X
  108. M M
  109. H H
  110. 𝐊 = 𝐑 , 𝐂 \mathbf{K}=\mathbf{R},\mathbf{C}
  111. x H = x , x , \|x\|_{H}=\sqrt{\langle x,x\rangle},
  112. , : H × H 𝐊 \langle\cdot,\cdot\rangle:H\times H\to\mathbf{K}
  113. x , y H : y , x \displaystyle\forall x,y\in H:\quad\langle y,x\rangle
  114. A A
  115. 𝐊 = 𝐑 \mathbf{K}=\mathbf{R}
  116. 𝐂 \mathbf{C}
  117. 𝐊 \mathbf{K}
  118. ( a , b ) A × A a b A (a,b)∈A×A→ab∈A
  119. A A
  120. [ u ! ! ] a b [ u ! ! ] [ u ! ! ] a [ u ! ! ] [ u ! ! ] b [ u ! ! ] [u^{\prime}!!^{\prime}]ab[u^{\prime}!!^{\prime}]≤[u^{\prime}!!^{\prime}]a[u^{% \prime}!!^{\prime}][u^{\prime}!!^{\prime}]b[u^{\prime}!!^{\prime}]
  121. a , b A a,b∈A
  122. C ( K ) C(K)
  123. A ( 𝐃 ) A(\mathbf{D})
  124. 𝐃 𝐂 \mathbf{D}⊂\mathbf{C}
  125. 𝐃 ¯ \overline{\mathbf{D}}
  126. 𝐃 ¯ \overline{\mathbf{D}}
  127. A ( 𝐃 ) A(\mathbf{D})
  128. C ( 𝐃 ¯ ) C(\overline{\mathbf{D}})
  129. A ( 𝐓 ) A(\mathbf{T})
  130. 𝐓 \mathbf{T}
  131. 𝐓 \mathbf{T}
  132. X X
  133. B ( X ) B(X)
  134. X X
  135. A A
  136. B ( H ) B(H)
  137. H H
  138. B ( H ) B(H)
  139. C ( K ) C(K)
  140. K K
  141. f f
  142. f ¯ \overline{f}
  143. X X
  144. 𝐊 \mathbf{K}
  145. X X
  146. 𝐊 \mathbf{K}
  147. X = B ( X , 𝐊 ) X′=B(X,\mathbf{K})
  148. 𝐊 \mathbf{K}
  149. X X′
  150. X X
  151. X X
  152. 𝐊 = 𝐑 , 𝐂 \mathbf{K}=\mathbf{R},\mathbf{C}
  153. Y X Y⊆X
  154. p : X 𝐑 p:X→\mathbf{R}
  155. f : Y 𝐊 f:Y→\mathbf{K}
  156. R e ( f ( y ) ) p ( y ) Re(f(y))≤p(y)
  157. y y
  158. Y Y
  159. F : X 𝐊 F:X→\mathbf{K}
  160. F | Y = f , and x X , Re ( F ( x ) ) p ( x ) . F|_{Y}=f,\quad\,\text{and}\quad\forall x\in X,\ \ \operatorname{Re}(F(x))\leq p% (x).
  161. x x
  162. X X
  163. f f
  164. X X
  165. f ( x ) = x X , f X 1. f(x)=\|x\|_{X},\quad\|f\|_{X^{\prime}}\leq 1.
  166. x x
  167. 𝟎 \mathbf{0}
  168. f f
  169. x x
  170. S S
  171. X X
  172. S S
  173. X X
  174. S S
  175. X X
  176. S S
  177. 𝟎 \mathbf{0}
  178. X X
  179. M M
  180. N N
  181. X X′
  182. X X
  183. M M
  184. N N
  185. M M
  186. X X
  187. M M
  188. M = { x X : x ( m ) = 0 , m M } . M^{\perp}=\left\{x^{\prime}\in X^{\prime}:x^{\prime}(m)=0,\ \forall m\in M% \right\}.
  189. M M
  190. X / M X/M
  191. X X
  192. X X′
  193. X X
  194. X X′
  195. X X
  196. X X
  197. X X
  198. x x′
  199. X X′
  200. X X
  201. Y Y
  202. X X
  203. Y Y
  204. X X
  205. X X
  206. 𝐊 \mathbf{K}
  207. X X′
  208. X X
  209. X X′
  210. X X′
  211. X X′
  212. x X x ( x ) , x X x′ ∈X′→x′(x),x∈X
  213. X X
  214. X X
  215. B B′
  216. f f
  217. f ( x ) = n 𝐍 x n y n , x = { x n } c 0 , and f ( c 0 ) = y 1 . f(x)=\sum_{n\in\mathbf{N}}x_{n}y_{n},\qquad x=\{x_{n}\}\in c_{0},\ \ \,\text{% and}\ \ \|f\|_{(c_{0})^{\prime}}=\|y\|_{\ell_{1}}.
  218. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  219. y y
  220. H H
  221. x H f y ( x ) = x , y x\in H\to f_{y}(x)=\langle x,y\rangle
  222. H H
  223. H H
  224. y y
  225. H H
  226. H H
  227. H H′
  228. K K
  229. M ( K ) M(K)
  230. C ( K ) C(K)
  231. P ( K ) P(K)
  232. M ( K ) M(K)
  233. M ( K ) M(K)
  234. P ( K ) P(K)
  235. K K
  236. K K
  237. K K
  238. K K
  239. L L
  240. C ( K ) C(K)
  241. C ( L ) C(L)
  242. K K
  243. L L
  244. C ( K ) C(K)
  245. C ( L ) C(L)
  246. = 2 =2
  247. C ( K ) C(K)
  248. K K
  249. I x = ker δ x = { f C ( K ) : f ( x ) = 0 } , x K . I_{x}=\ker\delta_{x}=\{f\in C(K):f(x)=0\},\quad x\in K.
  250. A A′
  251. K K
  252. Ξ Ξ
  253. C ( K ) C(K)
  254. K K
  255. C ( K ) C(K)
  256. K K
  257. C ( K ) C(K)
  258. A A
  259. C ( K ) C(K)
  260. K K
  261. A A
  262. X X
  263. X X′′
  264. X X′
  265. X X
  266. X X
  267. { F X : X X ′′ F X ( x ) ( f ) = f ( x ) x X , f X \begin{cases}F_{X}:X\to X^{\prime\prime}\\ F_{X}(x)(f)=f(x)&\forall x\in X,\forall f\in X^{\prime}\end{cases}
  268. X X′
  269. X X′′
  270. X X
  271. X X′′
  272. f f
  273. x x
  274. X X
  275. X X
  276. X X′′
  277. X X
  278. X X
  279. X X′′
  280. X X
  281. X X
  282. X X′′
  283. x x′′
  284. X X
  285. sup j x j x ′′ , x ′′ ( f ) = lim j f ( x j ) , f X . \sup_{j}\|x_{j}\|\leq\|x^{\prime\prime}\|,\ \ x^{\prime\prime}(f)=\lim_{j}f(x_% {j}),\quad f\in X^{\prime}.
  286. X X′
  287. X X
  288. Y Y
  289. F F
  290. X X
  291. Y Y
  292. x x
  293. X X
  294. X X
  295. U U
  296. 𝟎 \mathbf{0}
  297. X X
  298. T T
  299. F F
  300. U U
  301. sup T F sup x U T ( x ) Y < . \sup_{T\in F}\sup_{x\in U}\;\|T(x)\|_{Y}<\infty.
  302. X X
  303. Y Y
  304. T : X Y T:X→Y
  305. T T
  306. T T
  307. X X
  308. Y Y
  309. T B ( X , Y ) T∈B(X,Y)
  310. T T
  311. Y Y
  312. X / K e r ( T ) X/ Ker(T)
  313. T ( X ) T(X)
  314. X X
  315. X X
  316. X X
  317. T : X Y T:X→Y
  318. T T
  319. X × Y X×Y
  320. T T
  321. X X
  322. { F X : X X ′′ F X ( x ) ( f ) = f ( x ) x X , f X \begin{cases}F_{X}:X\to X^{\prime\prime}\\ F_{X}(x)(f)=f(x)&\forall x\in X,\forall f\in X^{\prime}\end{cases}
  323. X X
  324. X X
  325. X X
  326. X X
  327. Y Y
  328. Y Y
  329. X X
  330. X X
  331. X X′
  332. X X
  333. X X
  334. X X′
  335. Y Y′
  336. Y Y
  337. Y Y
  338. X X
  339. X X′
  340. X X′
  341. X X
  342. X X′′
  343. X X
  344. X X
  345. X X′′
  346. X / X X′′ /X
  347. J / J J′′ /J
  348. J J
  349. X X
  350. X X
  351. X X
  352. H H
  353. H H
  354. B B
  355. B B
  356. X X
  357. 𝐑 \mathbf{R}
  358. f f
  359. X X′
  360. [ u ! ! ] f [ u ! ! ] [u^{\prime}!!^{\prime}]f[u^{\prime}!!^{\prime}]
  361. X X
  362. X X
  363. f f
  364. X X′
  365. x x
  366. X X
  367. [ u ! ! ] x [ u ! ! ] 1 [u^{\prime}!!^{\prime}]x[u^{\prime}!!^{\prime}]≤1
  368. f ( x ) = [ u ! ! ] f [ u ! ! ] . f(x)=[u^{\prime}!!^{\prime}]f[u^{\prime}!!^{\prime}].
  369. X X
  370. X X′
  371. X X
  372. X X
  373. x X x∈X
  374. f ( x ) f(x)
  375. f f
  376. X X′
  377. L ( f ) L(f)
  378. f f
  379. X X′
  380. X X′
  381. f X f∈X′
  382. f ( x ) f(x)
  383. x x
  384. X X
  385. X X
  386. L L
  387. X X′
  388. L L
  389. X X
  390. L L
  391. X X
  392. X X
  393. μ μ
  394. 𝟎 \mathbf{0}
  395. c 1 c≥1
  396. c - 2 k = 1 n x k 2 Ave ± k = 1 n ± x k 2 c 2 k = 1 n x k 2 c^{-2}\sum_{k=1}^{n}\left\|x_{k}\right\|^{2}\leq\operatorname{Ave}_{\pm}\left% \|\sum_{k=1}^{n}\pm x_{k}\right\|^{2}\leq c^{2}\sum_{k=1}^{n}\left\|x_{k}% \right\|^{2}
  397. n n
  398. X X
  399. ± 1 ±1
  400. n n
  401. n n
  402. n n
  403. X X
  404. X X
  405. Y Y
  406. Z Z
  407. Z Z
  408. X X
  409. X X
  410. 2 2
  411. K K
  412. C ( K ) C(K)
  413. C ( 0 , 11 ) ) C(0,11))
  414. K K
  415. 1 , α = { γ : 1 γ α } \langle 1,\alpha\rangle=\{\gamma\ :\ 1\leq\gamma\leq\alpha\}
  416. α α
  417. C ( K ) C(K)
  418. α , β α,β
  419. α β α≤β
  420. C ( 1 , ω ) , C ( 1 , ω ω ) , C ( 1 , ω ω 2 ) , C ( 1 , ω ω 3 ) , , C ( 1 , ω ω ω ) , C(\langle 1,\omega\rangle),\ C(\langle 1,\omega^{\omega}\rangle),\ C(\langle 1% ,\omega^{\omega^{2}}\rangle),\ C(\langle 1,\omega^{\omega^{3}}\rangle),\cdots,% C(\langle 1,\omega^{\omega^{\omega}}\rangle),\cdots
  421. 𝐊 = 𝐑 , 𝐂 \mathbf{K}=\mathbf{R},\mathbf{C}
  422. X X
  423. I I
  424. a a , b aa,b
  425. p , q p,q
  426. Σ Σ
  427. Ξ Ξ
  428. μ μ
  429. | μ | |μ|
  430. x 2 = ( i = 1 n | x i | 2 ) 1 2 \|x\|_{2}=\left(\sum_{i=1}^{n}|x_{i}|^{2}\right)^{\frac{1}{2}}
  431. [ u s u , u p = , u n , u b = , u p ] ℓ[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}n^{\prime},u^{\prime}b% =^{\prime},u^{\prime}p^{\prime}]
  432. [ u s u , u p = , u n , u b = , u q ] ℓ[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}n^{\prime},u^{\prime}b% =^{\prime},u^{\prime}q^{\prime}]
  433. x p = ( i = 1 n | x i | p ) 1 p \|x\|_{p}=\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{\frac{1}{p}}
  434. [ u s u , u p = , u n , u b = 21 e ] ℓ[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}n^{\prime},u^{\prime}b% =\u{2}21e^{\prime}]
  435. [ u s u , u p = , u n , u b = 1 ] ℓ[u^{\prime}su^{\prime},u^{\prime}p=^{\prime},u^{\prime}n^{\prime},u^{\prime}b% =1^{\prime}]
  436. x = max 1 i n | x i | \|x\|_{\infty}=\max\nolimits_{1\leq i\leq n}|x_{i}|
  437. x p = ( i = 1 | x i | p ) 1 p \|x\|_{p}=\left(\sum_{i=1}^{\infty}|x_{i}|^{p}\right)^{\frac{1}{p}}
  438. x 1 = i = 1 | x i | \|x\|_{1}=\sum_{i=1}^{\infty}|x_{i}|
  439. b a ba
  440. x = sup i | x i | \|x\|_{\infty}=\sup\nolimits_{i}|x_{i}|
  441. c c
  442. x = sup i | x i | \|x\|_{\infty}=\sup\nolimits_{i}|x_{i}|
  443. x = sup i | x i | \|x\|_{\infty}=\sup\nolimits_{i}|x_{i}|
  444. c c
  445. b v bv
  446. x b v = | x 1 | + i = 1 | x i + 1 - x i | \|x\|_{bv}=|x_{1}|+\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|
  447. x b v 0 = i = 1 | x i + 1 - x i | \|x\|_{bv_{0}}=\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|
  448. b s bs
  449. b a ba
  450. x b s = sup n | i = 1 n x i | \|x\|_{bs}=\sup\nolimits_{n}\left|\sum_{i=1}^{n}x_{i}\right|
  451. c s cs
  452. x b s = sup n | i = 1 n x i | \|x\|_{bs}=\sup\nolimits_{n}\left|\sum_{i=1}^{n}x_{i}\right|
  453. c c
  454. B ( X , Ξ ) B(X,Ξ)
  455. b a ( Ξ ) ba(Ξ)
  456. f B = sup x X | f ( x ) | \|f\|_{B}=\sup\nolimits_{x\in X}|f(x)|
  457. C ( X ) C(X)
  458. r c a ( X ) rca(X)
  459. x C ( X ) = max x X | f ( x ) | \|x\|_{C(X)}=\max\nolimits_{x\in X}|f(x)|
  460. b a ( Ξ ) ba(Ξ)
  461. μ b a = sup A Σ | μ | ( A ) \|\mu\|_{ba}=\sup\nolimits_{A\in\Sigma}|\mu|(A)
  462. c a ( Σ ) ca(Σ)
  463. μ b a = sup A Σ | μ | ( A ) \|\mu\|_{ba}=\sup\nolimits_{A\in\Sigma}|\mu|(A)
  464. b a ( Σ ) ba(Σ)
  465. r c a ( Σ ) rca(Σ)
  466. μ b a = sup A Σ | μ | ( A ) \|\mu\|_{ba}=\sup\nolimits_{A\in\Sigma}|\mu|(A)
  467. c a ( Σ ) ca(Σ)
  468. f p = ( | f | p d μ ) 1 p \|f\|_{p}=\left(\int|f|^{p}\,d\mu\right)^{\frac{1}{p}}
  469. f 1 = | f | d μ \|f\|_{1}=\int|f|\,d\mu
  470. μ μ
  471. σ σ
  472. B V ( I ) BV(I)
  473. f B V = V f ( I ) + lim x a + f ( x ) \|f\|_{BV}=V_{f}(I)+\lim\nolimits_{x\to a^{+}}f(x)
  474. f f
  475. N B V ( I ) NBV(I)
  476. f B V = V f ( I ) \|f\|_{BV}=V_{f}(I)
  477. N B V ( I ) NBV(I)
  478. B V ( I ) BV(I)
  479. lim x a + f ( x ) = 0 \lim\nolimits_{x\to a^{+}}f(x)=0
  480. A C ( I ) AC(I)
  481. f B V = V f ( I ) + lim x a + f ( x ) \|f\|_{BV}=V_{f}(I)+\lim\nolimits_{x\to a^{+}}f(x)
  482. r c a ( a a , b ) ) rca(aa,b))
  483. f = i = 0 n sup x [ a , b ] | f ( i ) ( x ) | \|f\|=\sum_{i=0}^{n}\sup\nolimits_{x\in[a,b]}\left|f^{(i)}(x)\right|
  484. 𝐑 < s u p > n C ( a a , b ) ) \mathbf{R}<sup>n⊕C(aa,b))
  485. X < s u p > X<sup> ∗
  486. K K
  487. L L
  488. C ( X ) C(X)
  489. α = ω < s u p > β n α=ω<sup> βn

Bandwidth_(signal_processing).html

  1. B = 2 W B=2W
  2. B B
  3. W W
  4. W W
  5. B B
  6. 1 2 \frac{1}{\sqrt{2}}
  7. % B = 100 × f H - f L f c = 200 × f H - f L f H + f L \%B=100\times\frac{f_{H}-f_{L}}{f_{c}}=200\times\frac{f_{H}-f_{L}}{f_{H}+f_{L}}
  8. f L = 0 f_{L}=0
  9. B = f H / f L , B=f_{H}/f_{L},
  10. B : 1 B:1
  11. % B = 200 × f H - f L f H + f L = p \%B=200\times\frac{f_{H}-f_{L}}{f_{H}+f_{L}}=p
  12. B = 200 + p 200 - p B=\frac{200+p}{200-p}

Barber_paradox.html

  1. ( x ) ( man ( x ) ( y ) ( man ( y ) ( shaves ( x , y ) ¬ shaves ( y , y ) ) ) ) (\exists x)(\,\text{man}(x)\wedge(\forall y)(\,\text{man}(y)\rightarrow(\,% \text{shaves}(x,y)\leftrightarrow\neg\,\text{shaves}(y,y))))
  2. ( ) (\forall)
  3. shaves ( x , x ) ¬ shaves ( x , x ) \,\text{shaves}(x,x)\leftrightarrow\neg\,\text{shaves}(x,x)
  4. a ¬ a a\leftrightarrow\neg a

Barnard's_Star.html

  1. m = 4.83 + 5 ( ( log 10 1.834 ) - 1 ) = 1.15 \begin{smallmatrix}m=4.83+5\cdot((\log_{10}1.834)-1)=1.15\end{smallmatrix}

Barrage_jamming.html

  1. 10 7 10^{7}

Baryon.html

  1. 1 / 3 {1}/{3}
  2. 1 / 3 {1}/{3}
  3. 1 / 3 {1}/{3}
  4. 1 / 3 {1}/{3}
  5. 1 / 3 {1}/{3}
  6. 1 / 3 {1}/{3}
  7. 1 / 3 {1}/{3}
  8. 1 / 3 {1}/{3}
  9. 1 / 3 {1}/{3}
  10. 1 / 3 {1}/{3}
  11. 2 / 3 {2}/{3}
  12. 1 / 3 {1}/{3}
  13. 3 / 2 {3}/{2}
  14. 3 / 2 {3}/{2}
  15. 1 / 2 {1}/{2}
  16. 1 / 2 {1}/{2}
  17. 3 / 2 {3}/{2}
  18. 1 / 2 {1}/{2}
  19. 1 / 2 {1}/{2}
  20. 1 / 2 {1}/{2}
  21. I 3 = 1 2 [ ( n u - n u ¯ ) - ( n d - n d ¯ ) ] , I_{\mathrm{3}}=\frac{1}{2}[(n_{\mathrm{u}}-n_{\mathrm{\bar{u}}})-(n_{\mathrm{d% }}-n_{\mathrm{\bar{d}}})],
  22. Q = I 3 + 1 2 ( B + S + C + B + T ) , Q=I_{\mathrm{3}}+\frac{1}{2}(B+S+C+B^{\prime}+T),
  23. S = - ( n s - n s ¯ ) , S=-(n_{\mathrm{s}}-n_{\mathrm{\bar{s}}}),
  24. C = + ( n c - n c ¯ ) , C=+(n_{\mathrm{c}}-n_{\mathrm{\bar{c}}}),
  25. B = - ( n b - n b ¯ ) , B^{\prime}=-(n_{\mathrm{b}}-n_{\mathrm{\bar{b}}}),
  26. T = + ( n t - n t ¯ ) , T=+(n_{\mathrm{t}}-n_{\mathrm{\bar{t}}}),
  27. Q = 2 3 [ ( n u - n u ¯ ) + ( n c - n c ¯ ) + ( n t - n t ¯ ) ] - 1 3 [ ( n d - n d ¯ ) + ( n s - n s ¯ ) + ( n b - n b ¯ ) ] . Q=\frac{2}{3}[(n_{\mathrm{u}}-n_{\mathrm{\bar{u}}})+(n_{\mathrm{c}}-n_{\mathrm% {\bar{c}}})+(n_{\mathrm{t}}-n_{\mathrm{\bar{t}}})]-\frac{1}{3}[(n_{\mathrm{d}}% -n_{\mathrm{\bar{d}}})+(n_{\mathrm{s}}-n_{\mathrm{\bar{s}}})+(n_{\mathrm{b}}-n% _{\mathrm{\bar{b}}})].
  28. 1 / 2 {1}/{2}
  29. 1 / 2 {1}/{2}
  30. 1 / 2 {1}/{2}
  31. 1 / 2 {1}/{2}
  32. 1 / 2 {1}/{2}
  33. 1 / 2 {1}/{2}
  34. 3 / 2 {3}/{2}
  35. 3 / 2 {3}/{2}
  36. 1 / 2 {1}/{2}
  37. 1 / 2 {1}/{2}
  38. 3 / 2 {3}/{2}
  39. 1 / 2 {1}/{2}
  40. 1 / 2 {1}/{2}
  41. 1 / 2 {1}/{2}
  42. 1 / 2 {1}/{2}
  43. 1 / 2 {1}/{2}
  44. 1 / 2 {1}/{2}
  45. 3 / 2 {3}/{2}
  46. 1 / 2 {1}/{2}
  47. 3 / 2 {3}/{2}
  48. 1 / 2 {1}/{2}
  49. 5 / 2 {5}/{2}
  50. 3 / 2 {3}/{2}
  51. 5 / 2 {5}/{2}
  52. 3 / 2 {3}/{2}
  53. 7 / 2 {7}/{2}
  54. 5 / 2 {5}/{2}
  55. 7 / 2 {7}/{2}
  56. 5 / 2 {5}/{2}
  57. 3 / 2 {3}/{2}
  58. 3 / 2 {3}/{2}
  59. 3 / 2 {3}/{2}
  60. 5 / 2 {5}/{2}
  61. 3 / 2 {3}/{2}
  62. 1 / 2 {1}/{2}
  63. 5 / 2 {5}/{2}
  64. 3 / 2 {3}/{2}
  65. 1 / 2 {1}/{2}
  66. 7 / 2 {7}/{2}
  67. 5 / 2 {5}/{2}
  68. 3 / 2 {3}/{2}
  69. 1 / 2 {1}/{2}
  70. 7 / 2 {7}/{2}
  71. 5 / 2 {5}/{2}
  72. 3 / 2 {3}/{2}
  73. 1 / 2 {1}/{2}
  74. 9 / 2 {9}/{2}
  75. 7 / 2 {7}/{2}
  76. 5 / 2 {5}/{2}
  77. 3 / 2 {3}/{2}
  78. 9 / 2 {9}/{2}
  79. 7 / 2 {7}/{2}
  80. 5 / 2 {5}/{2}
  81. 3 / 2 {3}/{2}
  82. 1 / 2 {1}/{2}
  83. 3 / 2 {3}/{2}
  84. 1 / 2 {1}/{2}
  85. 3 / 2 {3}/{2}
  86. 3 / 2 {3}/{2}
  87. 1 / 2 {1}/{2}
  88. 3 / 2 {3}/{2}
  89. P = ( - 1 ) L . P=(-1)^{L}.
  90. 1 / 2 {1}/{2}
  91. 3 / 2 {3}/{2}
  92. 1 / 2 {1}/{2}
  93. 3 / 2 {3}/{2}
  94. 1 / 2 {1}/{2}
  95. 1 / 2 {1}/{2}
  96. 2 / 3 {2}/{3}
  97. 2 / 3 {2}/{3}
  98. 1 / 3 {1}/{3}

Base_(topology).html

  1. χ ( x , X ) \chi(x,X)
  2. χ ( X ) sup { χ ( x , X ) : x X } . \chi(X)\triangleq\sup\{\chi(x,X):x\in X\}.
  3. 𝒩 \mathcal{N}
  4. 𝒩 \mathcal{N}
  5. B B B^{\prime}\subseteq B
  6. | B | w ( X ) |B^{\prime}|\leq w(X)
  7. N N N^{\prime}\subseteq N
  8. | N | χ ( x , X ) |N^{\prime}|\leq\chi(x,X)
  9. f ′′′ B { f ′′ U : U B } f^{\prime\prime\prime}B\triangleq\{f^{\prime\prime}U:U\in B\}
  10. ( X , τ ) (X,\tau)
  11. ( X , τ ) (X,\tau^{\prime})
  12. w ( X , τ ) n w ( X , τ ) w(X,\tau^{\prime})\leq nw(X,\tau)
  13. n w ( f ( X ) ) = w ( f ( X ) ) w ( X ) 0 nw(f(X))=w(f(X))\leq w(X)\leq\aleph_{0}
  14. { U ξ } ξ κ , \left\{U_{\xi}\right\}_{\xi\in\kappa},
  15. { V ξ } ξ κ + \left\{V_{\xi}\right\}_{\xi\in\kappa^{+}}
  16. α < κ + : V α ξ < α V ξ . \forall\alpha<\kappa^{+}:\qquad V_{\alpha}\setminus\bigcup_{\xi<\alpha}V_{\xi}% \neq\varnothing.
  17. x V α ξ < α V ξ , x\in V_{\alpha}\setminus\bigcup_{\xi<\alpha}V_{\xi},
  18. V α ξ < α V ξ . V_{\alpha}\setminus\bigcup_{\xi<\alpha}V_{\xi}.
  19. V β ξ < α V ξ V β V α , V_{\beta}\setminus\bigcup_{\xi<\alpha}V_{\xi}\subseteq V_{\beta}\setminus V_{% \alpha},

Baseband.html

  1. Z ( t ) = I ( t ) + j Q ( t ) Z(t)=I(t)+jQ(t)\,
  2. I ( t ) I(t)
  3. Q ( t ) Q(t)
  4. j j
  5. I ( t ) I(t)
  6. Q ( t ) Q(t)
  7. I ( t ) cos ( ω t ) - Q ( t ) sin ( ω t ) = Re { Z ( t ) e j ω t } I(t)\cos(\omega t)-Q(t)\sin(\omega t)=\mathrm{Re}\{Z(t)e^{j\omega t}\}\,
  8. ω \omega

Basis_(linear_algebra).html

  1. a v + b w = 0 av+bw=0
  2. a = 0 , b = 0. a=0,b=0.
  3. a ( 1 , 1 ) + b ( - 1 , 2 ) = ( 0 , 0 ) a(1,1)+b(-1,2)=(0,0)\,
  4. ( a - b , a + 2 b ) = ( 0 , 0 ) (a-b,a+2b)=(0,0)\,
  5. a - b = 0 a-b=0\;
  6. a + 2 b = 0. a+2b=0.\,
  7. 3 b = 0 3b=0\;
  8. b = 0. b=0.\,
  9. a = 0. a=0.\,
  10. r ( 1 , 1 ) + s ( - 1 , 2 ) = ( a , b ) . r(1,1)+s(-1,2)=(a,b).\,
  11. r - s = a r-s=a\,
  12. r + 2 s = b . r+2s=b.\,
  13. 3 s = b - a , 3s=b-a,\,
  14. s = ( b - a ) / 3 , s=(b-a)/3,\,
  15. r = s + a = ( ( b - a ) / 3 ) + a = ( b + 2 a ) / 3. r=s+a=((b-a)/3)+a=(b+2a)/3.\,
  16. det [ 1 - 1 1 2 ] = 3 0. \det\begin{bmatrix}1&-1\\ 1&2\end{bmatrix}=3\neq 0.
  17. c 00 c_{00}
  18. x = ( x n ) x=(x_{n})
  19. x = sup n | x n | . \|x\|=\sup_{n}|x_{n}|.
  20. 0 2 π | f ( x ) | 2 d x < . \int_{0}^{2\pi}\left|f(x)\right|^{2}\,dx<\infty.
  21. lim n 0 2 π | a 0 + k = 1 n ( a k cos ( k x ) + b k sin ( k x ) ) - f ( x ) | 2 d x = 0 \lim_{n\rightarrow\infty}\int_{0}^{2\pi}\biggl|a_{0}+\sum_{k=1}^{n}\bigl(a_{k}% \cos(kx)+b_{k}\sin(kx)\bigr)-f(x)\biggr|^{2}\,dx=0
  22. n + 1 n+1
  23. n + 1 n+1

Basis_function.html

  1. { 2 sin ( 2 π n x ) | n } { 2 cos ( 2 π n x ) | n } { 1 } \{\sqrt{2}\sin(2\pi nx)\;|\;n\in\mathbb{N}\}\cup\{\sqrt{2}\cos(2\pi nx)\;|\;n% \in\mathbb{N}\}\cup\{1\}

Baud.html

  1. T s = 1 f s , T_{s}={1\over f_{s}},
  2. f s = R N . f_{\mathrm{s}}={R\over N}.
  3. R = f s N R=f_{\mathrm{s}}N\quad
  4. N = log 2 ( M ) . \quad N=\log_{2}(M).

Bayes'_rule.html

  1. A 1 A_{1}
  2. A 2 A_{2}
  3. B B
  4. A 1 A_{1}
  5. A 2 A_{2}
  6. B B
  7. Λ \Lambda
  8. B B
  9. A 1 A_{1}
  10. A 2 A_{2}
  11. A A
  12. P ( A | B ) P ( A ) P ( B | A ) P(A|B)\propto P(A)P(B|A)
  13. A A
  14. B B
  15. A A
  16. A A
  17. A 1 A_{1}
  18. A 2 A_{2}
  19. B B
  20. A 1 A_{1}
  21. A 2 A_{2}
  22. B B
  23. A 1 A_{1}
  24. A 2 A_{2}
  25. B B
  26. A 1 : A 2 A_{1}:A_{2}
  27. B B
  28. A 1 : A 2 A_{1}:A_{2}
  29. Λ \Lambda
  30. O ( A 1 : A 2 | B ) = Λ ( A 1 : A 2 | B ) O ( A 1 : A 2 ) , O(A_{1}:A_{2}|B)=\Lambda(A_{1}:A_{2}|B)\cdot O(A_{1}:A_{2}),
  31. Λ ( A 1 : A 2 | B ) = P ( B | A 1 ) P ( B | A 2 ) . \Lambda(A_{1}:A_{2}|B)=\frac{P(B|A_{1})}{P(B|A_{2})}.
  32. O ( A 1 : A 2 ) = P ( A 1 ) P ( A 2 ) , O(A_{1}:A_{2})=\frac{P(A_{1})}{P(A_{2})},
  33. O ( A 1 : A 2 | B ) = P ( A 1 | B ) P ( A 2 | B ) . O(A_{1}:A_{2}|B)=\frac{P(A_{1}|B)}{P(A_{2}|B)}.
  34. A 1 = A A_{1}=A
  35. A 2 = ¬ A A_{2}=\neg A
  36. O ( A ) = O ( A : ¬ A ) O(A)=O(A:\neg A)
  37. A A
  38. A A
  39. O ( A | B ) = O ( A ) Λ ( A | B ) , O(A|B)=O(A)\cdot\Lambda(A|B),
  40. A A
  41. A A
  42. A A
  43. B B
  44. A 1 = A A_{1}=A
  45. A 2 = ¬ A A_{2}=\neg A
  46. B B
  47. A A
  48. A A
  49. ¬ A \neg A
  50. A A
  51. B B
  52. B B
  53. A A
  54. ¬ A \neg A
  55. A A
  56. B B
  57. A A
  58. B B
  59. P ( A | B ) = P ( A ) P ( B | A ) / P ( B ) P(A|B)=P(A)P(B|A)/P(B)
  60. P ( A | B ) P ( A ) P ( B | A ) P(A|B)\propto P(A)P(B|A)
  61. A A
  62. B B
  63. A A
  64. P ( B ) P(B)
  65. P ( A | B ) = P ( B | A ) P ( A ) P ( B ) P(A|B)=\frac{P(B|A)\,P(A)}{P(B)}\cdot\,
  66. A 1 A_{1}
  67. A 2 A_{2}
  68. B B
  69. C C
  70. O ( A 1 : A 2 | B C ) = Λ ( A 1 : A 2 | B C ) Λ ( A 1 : A 2 | B ) O ( A 1 : A 2 ) , O(A_{1}:A_{2}|B\cap C)=\Lambda(A_{1}:A_{2}|B\cap C)\cdot\Lambda(A_{1}:A_{2}|B)% \cdot O(A_{1}:A_{2}),
  71. Λ ( A 1 : A 2 | B ) = P ( B | A 1 ) P ( B | A 2 ) , \Lambda(A_{1}:A_{2}|B)=\frac{P(B|A_{1})}{P(B|A_{2})},
  72. Λ ( A 1 : A 2 | B C ) = P ( C | A 1 B ) P ( C | A 2 B ) . \Lambda(A_{1}:A_{2}|B\cap C)=\frac{P(C|A_{1}\cap B)}{P(C|A_{2}\cap B)}.
  73. A A
  74. ¬ A \neg A
  75. O ( A | B , C ) = Λ ( A | B C ) Λ ( A | B ) O ( A ) . O(A|B,C)=\Lambda(A|B\cap C)\cdot\Lambda(A|B)\cdot O(A).
  76. P ( A 1 | B ) = 1 P ( B ) P ( B | A 1 ) P ( A 1 ) , P(A_{1}|B)=\frac{1}{P(B)}\cdot P(B|A_{1})\cdot P(A_{1}),
  77. P ( A 2 | B ) = 1 P ( B ) P ( B | A 2 ) P ( A 2 ) . P(A_{2}|B)=\frac{1}{P(B)}\cdot P(B|A_{2})\cdot P(A_{2}).
  78. P ( A 1 | B ) P ( A 2 | B ) = P ( B | A 1 ) P ( B | A 2 ) P ( A 1 ) P ( A 2 ) . \frac{P(A_{1}|B)}{P(A_{2}|B)}=\frac{P(B|A_{1})}{P(B|A_{2})}\cdot\frac{P(A_{1})% }{P(A_{2})}.
  79. O ( A 1 : A 2 | B ) P ( A 1 | B ) P ( A 2 | B ) O(A_{1}:A_{2}|B)\triangleq\frac{P(A_{1}|B)}{P(A_{2}|B)}
  80. O ( A 1 : A 2 ) P ( A 1 ) P ( A 2 ) O(A_{1}:A_{2})\triangleq\frac{P(A_{1})}{P(A_{2})}
  81. Λ ( A 1 : A 2 | B ) P ( B | A 1 ) P ( B | A 2 ) , \Lambda(A_{1}:A_{2}|B)\triangleq\frac{P(B|A_{1})}{P(B|A_{2})},
  82. O ( A 1 : A 2 | B ) = Λ ( A 1 : A 2 | B ) O ( A 1 : A 2 ) . O(A_{1}:A_{2}|B)=\Lambda(A_{1}:A_{2}|B)\cdot O(A_{1}:A_{2}).
  83. 0.5 % = 1 200 \textstyle 0.5\%=\frac{1}{200}
  84. 99.5 % = 199 200 \textstyle 99.5\%=\frac{199}{200}
  85. 0.99 0.01 = 99 : 1 \textstyle\frac{0.99}{0.01}=99:1
  86. 1 × 99 : 199 × 1 = 99 : 199 \textstyle 1\times 99:199\times 1=99:199
  87. 100 : 200 = 1 : 2 \textstyle 100:200=1:2