wpmath0000001_14

Mach_number.html

  1. M = u c , \mathrm{M}=\frac{u}{c},
  2. u u
  3. c c
  4. M = u c \mathrm{M}=\frac{u}{c}
  5. q = γ 2 p M 2 q=\frac{\gamma}{2}p\,\mathrm{M}^{2}
  6. M = 2 γ - 1 [ ( q c p + 1 ) γ - 1 γ - 1 ] \mathrm{M}=\sqrt{\frac{2}{\gamma-1}\left[\left(\frac{q_{c}}{p}+1\right)^{\frac% {\gamma-1}{\gamma}}-1\right]}\,
  7. γ \ \gamma\,
  8. p t p = [ γ + 1 2 M 2 ] ( γ γ - 1 ) [ γ + 1 ( 1 - γ + 2 γ M 2 ) ] ( 1 γ - 1 ) \frac{p_{t}}{p}=\left[\frac{\gamma+1}{2}\mathrm{M}^{2}\right]^{\left(}\frac{% \gamma}{\gamma-1}\right)\cdot\left[\frac{\gamma+1}{\left(1-\gamma+2\gamma\,% \mathrm{M}^{2}\right)}\right]^{\left(}\frac{1}{\gamma-1}\right)
  9. M = 5 [ ( q c p + 1 ) 2 7 - 1 ] \mathrm{M}=\sqrt{5\left[\left(\frac{q_{c}}{p}+1\right)^{\frac{2}{7}}-1\right]}\,
  10. M = 0.88128485 ( q c p + 1 ) ( 1 - 1 7 M 2 ) 2.5 \mathrm{M}=0.88128485\sqrt{\left(\frac{q_{c}}{p}+1\right)\left(1-\frac{1}{7\,% \mathrm{M}^{2}}\right)^{2.5}}

Macroeconomics.html

  1. M V = P Q M\cdot V=P\cdot Q

Magic_square.html

  1. n n
  2. n n
  3. n 2 {n^{2}}
  4. n 2 + 1 2 \frac{n^{2}+1}{2}
  5. ( n 2 + 1 2 ) n \left(\frac{n^{2}+1}{2}\right)n
  6. n ( ( I + J - 1 + n 2 ) mod n ) + ( ( I + 2 J - 2 ) mod n ) + 1 n((I+J-1+\left\lfloor\frac{n}{2}\right\rfloor)\,\bmod\,n)+((I+2J-2)\,\bmod\,n)+1
  7. \mathbb{C}
  8. \mathbb{R}

Magnet.html

  1. F = B 2 A 2 μ 0 F={{B^{2}A}\over{2\mu_{0}}}
  2. m = B 2 A 2 μ 0 g n m={{B^{2}A}\over{2\mu_{0}g_{n}}}
  3. F = μ q m 1 q m 2 4 π r 2 F={{\mu q_{m1}q_{m2}}\over{4\pi r^{2}}}
  4. F = μ 0 H 2 A 2 = B 2 A 2 μ 0 F=\frac{\mu_{0}H^{2}A}{2}=\frac{B^{2}A}{2\mu_{0}}
  5. F = [ B 0 2 A 2 ( L 2 + R 2 ) π μ 0 L 2 ] [ 1 x 2 + 1 ( x + 2 L ) 2 - 2 ( x + L ) 2 ] F=\left[\frac{B_{0}^{2}A^{2}\left(L^{2}+R^{2}\right)}{\pi\mu_{0}L^{2}}\right]% \left[{\frac{1}{x^{2}}}+{\frac{1}{(x+2L)^{2}}}-{\frac{2}{(x+L)^{2}}}\right]
  6. B 0 = μ 0 2 M B_{0}\,=\,\frac{\mu_{0}}{2}M
  7. R R
  8. t t
  9. t t
  10. F ( x ) = π μ 0 4 M 2 R 4 [ 1 x 2 + 1 ( x + 2 t ) 2 - 2 ( x + t ) 2 ] F(x)=\frac{\pi\mu_{0}}{4}M^{2}R^{4}\left[\frac{1}{x^{2}}+\frac{1}{(x+2t)^{2}}-% \frac{2}{(x+t)^{2}}\right]
  11. M M
  12. x x
  13. B 0 B_{0}
  14. M M
  15. B 0 = μ 0 M B_{0}=\mu_{0}M
  16. m = M V m=MV
  17. V V
  18. V = π R 2 t V=\pi R^{2}t
  19. t x t<<x
  20. F ( x ) = 3 π μ 0 2 M 2 R 4 t 2 1 x 4 = 3 μ 0 2 π M 2 V 2 1 x 4 = 3 μ 0 2 π m 1 m 2 1 x 4 F(x)=\frac{3\pi\mu_{0}}{2}M^{2}R^{4}t^{2}\frac{1}{x^{4}}=\frac{3\mu_{0}}{2\pi}% M^{2}V^{2}\frac{1}{x^{4}}=\frac{3\mu_{0}}{2\pi}m_{1}m_{2}\frac{1}{x^{4}}

Magnetic_field.html

  1. 𝐁 \mathbf{B}
  2. 𝐇 \mathbf{H}
  3. 𝐇 \mathbf{H}
  4. 𝐁 \mathbf{B}
  5. 𝐁 \mathbf{B}
  6. 𝐇 \mathbf{H}
  7. 𝐇 \mathbf{H}
  8. 𝐁 \mathbf{B}
  9. 𝐇 \mathbf{H}
  10. 𝐁 \mathbf{B}
  11. 𝐇 \mathbf{H}
  12. 𝐁 \mathbf{B}
  13. 𝐌 \mathbf{M}
  14. 𝐇 \mathbf{H}
  15. 𝐁 \mathbf{B}
  16. q q
  17. 𝐄 \mathbf{E}
  18. 𝐅 = q 𝐄 \mathbf{F}=q\mathbf{E}
  19. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) . \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}).
  20. 𝐯 \mathbf{v}
  21. 𝐁 \mathbf{B}
  22. 𝐁 \mathbf{B}
  23. 𝐁 \mathbf{B}
  24. 𝐇 \mathbf{H}
  25. 𝐁 \mathbf{B}
  26. 𝐇 \mathbf{H}
  27. 𝐇 \mathbf{H}
  28. 𝐁 \mathbf{B}
  29. 𝐁 \mathbf{B}
  30. 𝐇 \mathbf{H}
  31. 𝐁 \mathbf{B}
  32. 𝐁 \mathbf{B}
  33. 𝐇 \mathbf{H}
  34. 𝐦 \mathbf{m}
  35. 𝐦 \mathbf{m}
  36. 𝐦 \mathbf{m}
  37. 𝐦 \mathbf{m}
  38. 𝐇 \mathbf{H}
  39. 𝐁 \mathbf{B}
  40. 𝐇 \mathbf{H}
  41. 𝐌 \mathbf{M}
  42. 𝐇 \mathbf{H}
  43. 𝐄 \mathbf{E}
  44. 𝐇 \mathbf{H}
  45. 𝐇 \mathbf{H}
  46. 𝐇 \mathbf{H}
  47. 𝐇 \mathbf{H}
  48. 𝐦 \mathbf{m}
  49. q m q_{m}
  50. 𝐝 \mathbf{d}
  51. 𝐦 = q m 𝐝 \mathbf{m}=q_{m}\mathbf{d}
  52. 𝐇 \mathbf{H}
  53. 𝐇 \mathbf{H}
  54. 𝐌 \mathbf{M}
  55. 𝐁 \mathbf{B}
  56. m = I A m=IA
  57. A A
  58. 𝐁 \mathbf{B}
  59. 𝐁 \mathbf{B}
  60. 𝐁 \mathbf{B}
  61. 𝐁 \mathbf{B}
  62. 𝐁 \mathbf{B}
  63. 𝐁 \mathbf{B}
  64. S 𝐁 d 𝐀 = 0 , \oint_{S}\mathbf{B}\cdot\mathrm{d}\mathbf{A}=0,
  65. S S
  66. d 𝐀 d\mathbf{A}
  67. 𝐁 \mathbf{B}
  68. 𝐁 \mathbf{B}
  69. 𝐇 \mathbf{H}
  70. 𝐇 \mathbf{H}
  71. 𝐇 \mathbf{H}
  72. 𝐦 \mathbf{m}
  73. 𝐁 \mathbf{B}
  74. 𝐅 = ( 𝐦 𝐁 ) , \mathbf{F}=\mathbf{\nabla}\left(\mathbf{m}\cdot\mathbf{B}\right),
  75. \mathbf{∇}
  76. 𝐦 · 𝐁 \mathbf{m}·\mathbf{B}
  77. 𝐦 · 𝐁 \mathbf{m}·\mathbf{B}
  78. 𝐦 · 𝐁 = m B c o s ( θ ) \mathbf{m}·\mathbf{B}=mBcos(θ)
  79. m m
  80. B B
  81. 𝐦 \mathbf{m}
  82. 𝐁 \mathbf{B}
  83. θ θ
  84. 𝐦 \mathbf{m}
  85. 𝐁 \mathbf{B}
  86. 𝐁 \mathbf{B}
  87. 𝐦 · 𝐁 \mathbf{m}·\mathbf{B}
  88. 𝐦 \mathbf{m}
  89. τ \mathbf{τ}
  90. 𝐇 \mathbf{H}
  91. + q +q
  92. q −q
  93. 𝐇 \mathbf{H}
  94. 𝐦 \mathbf{m}
  95. θ θ
  96. 𝐇 \mathbf{H}
  97. 𝐦 \mathbf{m}
  98. 𝐁 \mathbf{B}
  99. τ \mathbf{τ}
  100. 𝐦 \mathbf{m}
  101. s y m b o l τ = 𝐦 × 𝐁 = μ 0 𝐦 × 𝐇 , symbol{\tau}=\mathbf{m}\times\mathbf{B}=\mu_{0}\mathbf{m}\times\mathbf{H},\,
  102. 𝐦 \mathbf{m}
  103. I I
  104. 𝐁 = μ 0 I 4 π wire d s y m b o l × 𝐫 ^ r 2 , \mathbf{B}=\frac{\mu_{0}I}{4\pi}\int_{\mathrm{wire}}\frac{dsymbol{\ell}\times% \mathbf{\hat{r}}}{r^{2}},
  105. d d\mathbf{ℓ}
  106. I I
  107. r r
  108. d d\mathbf{ℓ}
  109. \mathbf{r̂}
  110. 𝐫 \mathbf{r}
  111. I {I}
  112. 𝐁 \mathbf{B}
  113. 𝐁 d s y m b o l = μ 0 I enc , \oint\mathbf{B}\cdot dsymbol{\ell}=\mu_{0}I_{\mathrm{enc}},
  114. I {I}
  115. 𝐁 \mathbf{B}
  116. 𝐄 \mathbf{E}
  117. 𝐅 \mathbf{F}
  118. g r a d 𝐇 grad\mathbf{H}
  119. 𝐁 \mathbf{B}
  120. 𝐅 = q 𝐯 × 𝐁 , \mathbf{F}=q\mathbf{v}\times\mathbf{B},
  121. 𝐅 \mathbf{F}
  122. q q
  123. 𝐯 \mathbf{v}
  124. 𝐁 \mathbf{B}
  125. A A
  126. q q
  127. i i
  128. B B
  129. θ θ
  130. q q
  131. F = q v B sin θ , F=qvB\sin\theta,
  132. N N
  133. N = n A N=n\ell A
  134. f = F N = q v B n A sin θ = B i sin θ f=FN=qvBn\ell A\sin\theta=Bi\ell\sin\theta
  135. i = n q v A i=nqvA
  136. 𝐁 \mathbf{B}
  137. 𝐇 \mathbf{H}
  138. 𝐌 \mathbf{M}
  139. 𝐦 \mathbf{m}
  140. 𝐌 \mathbf{M}
  141. 𝐇 \mathbf{H}
  142. 𝐌 \mathbf{M}
  143. 𝐁 \mathbf{B}
  144. 𝐌 d s y m b o l = I b , \oint\mathbf{M}\cdot dsymbol{\ell}=I_{\mathrm{b}},
  145. S μ 0 𝐌 d 𝐀 = - q M \oint_{S}\mu_{0}\mathbf{M}\cdot\mathrm{d}\mathbf{A}=-q_{M}
  146. S S
  147. S S
  148. 𝐇 \mathbf{H}
  149. 𝐇 𝐁 μ 0 - 𝐌 , \mathbf{H}\ \equiv\ \frac{\mathbf{B}}{\mu_{0}}-\mathbf{M},
  150. 𝐇 \mathbf{H}
  151. 𝐇 d s y m b o l = ( 𝐁 μ 0 - 𝐌 ) d s y m b o l = I tot - I b = I f , \oint\mathbf{H}\cdot dsymbol{\ell}=\oint(\frac{\mathbf{B}}{\mu_{0}}-\mathbf{M}% )\cdot dsymbol{\ell}=I_{\mathrm{tot}}-I_{\mathrm{b}}=I_{\mathrm{f}},
  152. 𝐇 \mathbf{H}
  153. ( 𝐇 𝟏 - 𝐇 𝟐 ) = 𝐊 f × 𝐧 ^ , (\mathbf{H_{1}^{\parallel}}-\mathbf{H_{2}^{\parallel}})=\mathbf{K}\text{f}% \times\hat{\mathbf{n}},
  154. 𝐧 ^ \hat{\mathbf{n}}
  155. 𝐇 \mathbf{H}
  156. S μ 0 𝐇 d 𝐀 = S ( 𝐁 - μ 0 𝐌 ) d 𝐀 = ( 0 - ( - q M ) ) = q M , \oint_{S}\mu_{0}\mathbf{H}\cdot\mathrm{d}\mathbf{A}=\oint_{S}(\mathbf{B}-\mu_{% 0}\mathbf{M})\cdot\mathrm{d}\mathbf{A}=(0-(-q_{M}))=q_{M},
  157. 𝐇 \mathbf{H}
  158. 𝐇 = 𝐇 0 + 𝐇 d , \mathbf{H}=\mathbf{H}_{0}+\mathbf{H}_{d},\,
  159. 𝐇 \mathbf{H}
  160. 𝐇 \mathbf{H}
  161. 𝐁 \mathbf{B}
  162. 𝐁 \mathbf{B}
  163. 𝐌 \mathbf{M}
  164. 𝐁 \mathbf{B}
  165. 𝐌 \mathbf{M}
  166. 𝐁 \mathbf{B}
  167. 𝐌 \mathbf{M}
  168. 𝐁 = μ 𝐇 , \mathbf{B}=\mu\mathbf{H},
  169. μ μ
  170. 𝐇 \mathbf{H}
  171. 𝐁 \mathbf{B}
  172. 𝐁 \mathbf{B}
  173. 𝐇 \mathbf{H}
  174. 𝐁 \mathbf{B}
  175. 𝐇 \mathbf{H}
  176. 𝐁 = μ 𝐇 \mathbf{B}=μ\mathbf{H}
  177. μ μ
  178. u = 𝐁 𝐇 2 = 𝐁 𝐁 2 μ = μ 𝐇 𝐇 2 . u=\frac{\mathbf{B}\cdot\mathbf{H}}{2}=\frac{\mathbf{B}\cdot\mathbf{B}}{2\mu}=% \frac{\mu\mathbf{H}\cdot\mathbf{H}}{2}.
  179. μ μ
  180. δ W δW
  181. δ 𝐁 δ\mathbf{B}
  182. δ W = 𝐇 δ 𝐁 . \delta W=\mathbf{H}\cdot\delta\mathbf{B}.
  183. 𝐇 \mathbf{H}
  184. 𝐁 \mathbf{B}
  185. = - d Φ m d t , \mathcal{E}=-\frac{d\Phi_{\mathrm{m}}}{dt},
  186. \scriptstyle\mathcal{E}
  187. 𝐁 \mathbf{B}
  188. 𝐄 \mathbf{E}
  189. = - d Φ m d t \textstyle\mathcal{E}=-\frac{d\Phi_{m}}{dt}
  190. = Σ ( t ) ( 𝐄 ( 𝐫 , t ) + 𝐯 × 𝐁 ( 𝐫 , t ) ) d s y m b o l \textstyle=\oint_{\partial\Sigma(t)}\left(\mathbf{E}(\mathbf{r},\ t)+\mathbf{v% \times B}(\mathbf{r},\ t)\right)\cdot dsymbol{\ell}
  191. = - d d t Σ ( t ) d s y m b o l A 𝐁 ( 𝐫 , t ) , \textstyle=-\frac{d}{dt}\iint_{\Sigma(t)}dsymbol{A}\cdot\mathbf{B}(\mathbf{r},% \ t),
  192. 𝚺 ( t ) \mathbf{∂Σ}(t)
  193. 𝚺 ( t ) \mathbf{Σ}(t)
  194. d 𝐀 d\mathbf{A}
  195. 𝚺 ( t ) \mathbf{Σ}(t)
  196. d d\mathbf{ℓ}
  197. 𝐁 \mathbf{B}
  198. 𝐀 \mathbf{A}
  199. · 𝐀 \mathbf{∇}·\mathbf{A}
  200. 𝐀 \mathbf{A}
  201. 𝐁 \mathbf{B}
  202. 𝐁 \mathbf{B}
  203. × 𝐀 \mathbf{∇}×\mathbf{A}
  204. 𝐀 \mathbf{A}
  205. 𝐁 \mathbf{B}
  206. 𝐄 \mathbf{E}
  207. 𝐁 = 0 , \nabla\cdot\mathbf{B}=0,
  208. 𝐄 = ρ ϵ 0 , \nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}},
  209. × 𝐁 = μ 0 𝐉 + μ 0 ε 0 𝐄 t , \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t},
  210. × 𝐄 = - 𝐁 t , \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},
  211. 𝐉 \mathbf{J}
  212. ρ ρ
  213. 𝐁 \mathbf{B}
  214. 𝐄 \mathbf{E}
  215. 𝐁 \mathbf{B}
  216. 𝐄 \mathbf{E}
  217. 𝐁 \mathbf{B}
  218. 𝐇 \mathbf{H}
  219. 𝐃 \mathbf{D}
  220. 𝐁 = 0 , \nabla\cdot\mathbf{B}=0,
  221. 𝐃 = ρ f , \nabla\cdot\mathbf{D}=\rho_{\mathrm{f}},
  222. × 𝐇 = 𝐉 f + 𝐃 t , \nabla\times\mathbf{H}=\mathbf{J}_{\mathrm{f}}+\frac{\partial\mathbf{D}}{% \partial t},
  223. × 𝐄 = - 𝐁 t . \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}.
  224. 𝐁 \mathbf{B}
  225. 𝐇 \mathbf{H}
  226. 𝐄 \mathbf{E}
  227. 𝐃 \mathbf{D}
  228. 𝐀 \mathbf{A}
  229. φ φ
  230. 𝐁 = × 𝐀 , \mathbf{B}=\nabla\times\mathbf{A},
  231. 𝐄 = - φ - 𝐀 t . \mathbf{E}=-\nabla\varphi-\frac{\partial\mathbf{A}}{\partial t}.
  232. 𝐀 \mathbf{A}
  233. φ φ
  234. 𝐀 \mathbf{A}
  235. φ φ
  236. 𝐇 \mathbf{H}
  237. 𝐁 = μ 𝐇 \mathbf{B}=μ\mathbf{H}
  238. μ μ
  239. 𝐉 = σ 𝐄 \mathbf{J}=σ\mathbf{E}
  240. 𝐉 \mathbf{J}
  241. σ σ
  242. 𝐄 \mathbf{E}
  243. I = V R I=V⁄R
  244. Φ = F R m , \Phi=\frac{F}{R}_{\mathrm{m}},
  245. Φ = 𝐁 d 𝐀 \Phi=\int\mathbf{B}\cdot d\mathbf{A}
  246. F = 𝐇 d s y m b o l F=\int\mathbf{H}\cdot dsymbol{\ell}
  247. I I
  248. a a
  249. m = I a , m=Ia,\,
  250. 𝐦 \mathbf{m}
  251. a a
  252. I I
  253. m = I a m=Ia
  254. d d
  255. 𝐇 \mathbf{H}
  256. 𝐁 \mathbf{B}
  257. 𝐁 \mathbf{B}
  258. 𝐇 \mathbf{H}
  259. 𝐁 \mathbf{B}
  260. 𝐇 \mathbf{H}
  261. 𝐇 \mathbf{H}
  262. 𝐇 \mathbf{H}
  263. 𝐁 \mathbf{B}
  264. 𝐇 \mathbf{H}
  265. 𝐇 \mathbf{H}
  266. F = q v B s i n θ F=qvBsinθ
  267. 𝐦 \mathbf{m}
  268. 𝐁 \mathbf{B}
  269. 𝐇 \mathbf{H}
  270. 𝐁 \mathbf{B}

Magnetic_flux.html

  1. Φ Φ
  2. Φ B = 𝐁 𝐒 = B S cos θ , \Phi_{B}=\mathbf{B}\cdot\mathbf{S}=BS\cos\theta,
  3. d Φ B = 𝐁 d 𝐒 . d\Phi_{B}=\mathbf{B}\cdot d\mathbf{S}.
  4. Φ B = S 𝐁 d 𝐒 . \Phi_{B}=\iint\limits_{S}\mathbf{B}\cdot d\mathbf{S}.
  5. Φ B = S 𝐀 d s y m b o l , \Phi_{B}=\oint\limits_{\partial S}\mathbf{A}\cdot dsymbol{\ell},
  6. = Σ ( 𝐄 + 𝐯 × 𝐁 ) d s y m b o l = - d Φ B d t , \mathcal{E}=\oint_{\partial\Sigma}\left(\mathbf{E}+\mathbf{v\times B}\right)% \cdot dsymbol{\ell}=-{d\Phi_{B}\over dt},
  7. \mathcal{E}

Magnetic_mirror.html

  1. μ = m v 2 2 B \mu=\frac{mv_{\perp}^{2}}{2B}
  2. v v_{\perp}
  3. \mathcal{E}
  4. = q ϕ + 1 2 m v 2 + 1 2 m v 2 \mathcal{E}=q\phi+\frac{1}{2}mv_{\parallel}^{2}+\frac{1}{2}mv_{\perp}^{2}
  5. r mirror = B max B min r\text{mirror}=\frac{B\text{max}}{B\text{min}}
  6. v v > 1 r mirror \frac{v_{\perp}}{v}>\frac{1}{\sqrt{r\text{mirror}}}
  7. v v_{\perp}
  8. v v

Magnetism.html

  1. 𝐁 = μ 0 𝐇 , \mathbf{B}\ =\ \mu_{0}\mathbf{H},
  2. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) . \mathbf{B}\ =\ \mu_{0}(\mathbf{H}+\mathbf{M}).
  3. 𝐇 \mathbf{H}
  4. 𝐌 \mathbf{M}
  5. 𝐌 = χ 𝐇 , \mathbf{M}=\chi\mathbf{H},
  6. μ 0 ( 𝐇 + 𝐌 ) = μ 0 ( 1 + χ ) 𝐇 = μ r μ 0 𝐇 = μ 𝐇 . \mu_{0}(\mathbf{H}+\mathbf{M})\ =\ \mu_{0}(1+\chi)\mathbf{H}\ =\ \mu_{r}\mu_{0% }\mathbf{H}\ =\ \mu\mathbf{H}.
  7. 𝐌 \mathbf{M}
  8. 𝐇 \mathbf{H}
  9. 𝐅 = q ( 𝐯 × 𝐁 ) \mathbf{F}=q(\mathbf{v}\times\mathbf{B})
  10. q q
  11. F = q v B sin θ F=qvB\sin\theta\,
  12. θ \theta
  13. u A u_{A}
  14. u B u_{B}
  15. σ \sigma
  16. ψ ( 𝐫 1 , 𝐫 2 ) = 1 2 ( u A ( 𝐫 1 ) u B ( 𝐫 2 ) + u B ( 𝐫 1 ) u A ( 𝐫 2 ) ) \psi(\mathbf{r}_{1},\,\,\mathbf{r}_{2})=\frac{1}{\sqrt{2}}\,\,\left(u_{A}(% \mathbf{r}_{1})u_{B}(\mathbf{r}_{2})+u_{B}(\mathbf{r}_{1})u_{A}(\mathbf{r}_{2}% )\right)
  17. χ ( s 1 , s 2 ) \chi(s_{1},s_{2})
  18. χ ( s 1 , s 2 ) = 1 2 ( α ( s 1 ) β ( s 2 ) - β ( s 1 ) α ( s 2 ) ) \chi(s_{1},\,\,s_{2})=\frac{1}{\sqrt{2}}\,\,\left(\alpha(s_{1})\beta(s_{2})-% \beta(s_{1})\alpha(s_{2})\right)
  19. u A u_{A}
  20. u B u_{B}
  21. α ( + 1 / 2 ) = β ( - 1 / 2 ) = 1 \alpha(+1/2)=\beta(-1/2)=1
  22. α ( - 1 / 2 ) = β ( + 1 / 2 ) = 0 \alpha(-1/2)=\beta(+1/2)=0
  23. A = C s - 1 \mathrm{A=C\ s^{-1}}
  24. C = A s \mathrm{C=A\ s}
  25. U , Δ V , Δ ϕ , \Epsilon U,\ \Delta V,\ \Delta\phi,\ \Epsilon
  26. V = J C - 1 = kg A - 1 m 2 s - 3 \mathrm{V=J\ C^{-1}=kg\ A^{-1}m^{2}s^{-3}}
  27. R ; \Zeta ; \Chi R;\ \Zeta;\ \Chi
  28. Ω = V A - 1 = kg m 2 A - 2 s - 3 \mathrm{\Omega=V\ A^{-1}=kg\ m^{2}\ A^{-2}s^{-3}}
  29. ρ \ \rho
  30. Ω m = kg A - 2 m 3 s - 3 \mathrm{\Omega\ m=kg\ A^{-2}m^{3}s^{-3}}
  31. \Rho \ \Rho
  32. W = V A = kg m 2 s - 3 \mathrm{W=V\ A=kg\ m^{2}s^{-3}}
  33. C \ C
  34. F = C V - 1 = A 2 kg - 1 m - 2 s 4 \mathrm{F=C\ V^{-1}=A^{2}kg^{-1}m^{-2}s^{4}}
  35. \Epsilon \mathbf{\Epsilon}
  36. V m - 1 = C - 1 N = kg A - 1 m s - 3 \mathrm{V\ m^{-1}=C^{-1}N=kg\ A^{-1}m\ s^{-3}}
  37. 𝐃 \mathbf{D}
  38. C m - 2 = A m - 2 s \mathrm{C\ m^{-2}=A\ m^{-2}s}
  39. ε \varepsilon
  40. F m - 1 = A 2 kg - 1 m - 3 s 4 \mathrm{F\ m^{-1}=A^{2}kg^{-1}m^{-3}s^{4}}
  41. χ e \!\chi_{e}
  42. B ; G ; Υ B;\ G;\ \Upsilon
  43. S = Ω - 1 = kg - 1 A 2 m - 2 s 3 \ \mathrm{S=\Omega^{-1}=kg^{-1}A^{2}m^{-2}s^{3}}
  44. γ , κ , σ \gamma,\ \kappa,\ \sigma
  45. S m - 1 = A 2 kg - 1 m - 3 s 3 \mathrm{S\ m^{-1}=A^{2}kg^{-1}m^{-3}s^{3}}
  46. 𝐁 \ \mathbf{B}
  47. T = Wb m - 2 = kg A - 1 s - 2 \mathrm{T=Wb\ m^{-2}=kg\ A^{-1}s^{-2}}
  48. Φ \ \Phi
  49. Wb = V s = kg A - 1 m 2 s - 2 \mathrm{Wb=V\ s=kg\ A^{-1}m^{2}s^{-2}}
  50. 𝐇 \mathbf{H}
  51. A m - 1 \mathrm{A\ m^{-1}}
  52. L , \Mu L,\ \Mu
  53. H = Wb A - 1 = V A - 1 s = kg A - 2 m 2 s - 2 \mathrm{H=Wb\ A^{-1}=V\ A^{-1}s=kg\ A^{-2}m^{2}s^{-2}}
  54. μ \ \mu
  55. Hm - 1 = kg A - 2 m s - 2 \mathrm{Hm^{-1}=kg\ A^{-2}m\ s^{-2}}
  56. χ \ \chi

Magneto-optic_effect.html

  1. ε = ( ε x x ε x y + i g z ε x z - i g y ε x y - i g z ε y y ε y z + i g x ε x z + i g y ε y z - i g x ε z z ) \varepsilon=\begin{pmatrix}\varepsilon_{xx}^{\prime}&\varepsilon_{xy}^{\prime}% +ig_{z}&\varepsilon_{xz}^{\prime}-ig_{y}\\ \varepsilon_{xy}^{\prime}-ig_{z}&\varepsilon_{yy}^{\prime}&\varepsilon_{yz}^{% \prime}+ig_{x}\\ \varepsilon_{xz}^{\prime}+ig_{y}&\varepsilon_{yz}^{\prime}-ig_{x}&\varepsilon_% {zz}^{\prime}\\ \end{pmatrix}
  2. 𝐃 = ε 𝐄 = ε 𝐄 + i 𝐄 × 𝐠 \mathbf{D}=\varepsilon\mathbf{E}=\varepsilon^{\prime}\mathbf{E}+i\mathbf{E}% \times\mathbf{g}
  3. ε \varepsilon^{\prime}
  4. 𝐠 = ( g x , g y , g z ) \mathbf{g}=(g_{x},g_{y},g_{z})
  5. ε \varepsilon^{\prime}
  6. 𝐠 = ε 0 χ ( m ) 𝐇 \mathbf{g}=\varepsilon_{0}\chi^{(m)}\mathbf{H}
  7. χ ( m ) \chi^{(m)}\!
  8. ε \varepsilon^{\prime}
  9. ε \varepsilon^{\prime}
  10. ε = ( ε 1 + i g z 0 - i g z ε 1 0 0 0 ε 2 ) \varepsilon=\begin{pmatrix}\varepsilon_{1}&+ig_{z}&0\\ -ig_{z}&\varepsilon_{1}&0\\ 0&0&\varepsilon_{2}\\ \end{pmatrix}
  11. 1 / μ ( ε 1 ± g z ) 1/\sqrt{\mu(\varepsilon_{1}\pm g_{z})}
  12. v p = 1 ϵ μ v_{p}=\frac{1}{\sqrt{\epsilon\mu}}
  13. v p v_{p}
  14. ϵ \epsilon
  15. μ \mu
  16. 𝐇 \mathbf{H}

Magnetohydrodynamics.html

  1. 𝐁 \mathbf{B}
  2. 𝐄 \mathbf{E}
  3. 𝐯 \mathbf{v}
  4. 𝐉 \mathbf{J}
  5. ρ \rho
  6. p p
  7. t t
  8. ρ t + ( ρ 𝐯 ) = 0. \frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{v}\right)=0.
  9. ρ ( t + 𝐯 ) 𝐯 = 𝐉 × 𝐁 - p . \rho\left(\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla\right)\mathbf{v}=% \mathbf{J}\times\mathbf{B}-\nabla p.
  10. 𝐉 × 𝐁 \mathbf{J}\times\mathbf{B}
  11. 1 2 ( 𝐁 𝐁 ) = ( 𝐁 ) 𝐁 + 𝐁 × ( × 𝐁 ) \frac{1}{2}\nabla(\mathbf{B}\cdot\mathbf{B})=(\mathbf{B}\cdot\nabla)\mathbf{B}% +\mathbf{B}\times(\nabla\times\mathbf{B})
  12. 𝐉 × 𝐁 = ( 𝐁 ) 𝐁 μ 0 - ( B 2 2 μ 0 ) , \mathbf{J}\times\mathbf{B}=\frac{\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}}% {\mu_{0}}-\nabla\left(\frac{B^{2}}{2\mu_{0}}\right),
  13. 𝐄 + 𝐯 × 𝐁 = 0. \mathbf{E}+\mathbf{v}\times\mathbf{B}=0.
  14. 𝐁 t = - × 𝐄 . \frac{\partial\mathbf{B}}{\partial t}=-\nabla\times\mathbf{E}.
  15. μ 0 𝐉 = × 𝐁 . \mu_{0}\mathbf{J}=\nabla\times\mathbf{B}.
  16. 𝐁 = 0. \nabla\cdot\mathbf{B}=0.
  17. d d t ( p ρ γ ) = 0 , \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{p}{\rho^{\gamma}}\right)=0,
  18. γ = 5 / 3 \gamma=5/3
  19. k B \vec{k}\|\vec{B}
  20. k B \vec{k}\|\vec{B}
  21. B B
  22. k B \vec{k}\perp\vec{B}
  23. B B
  24. E E
  25. η 0 \eta\neq 0
  26. h = - 1 h=-1
  27. h = 0 h=0
  28. h = - 1 h=-1
  29. h 0 h\neq 0
  30. 1 + 3 1+3
  31. V I 0 VI_{0}
  32. V I 0 VI_{0}

Magnetopause.html

  1. ( ρ v 2 ) s w ( 4 B ( r ) 2 2 μ 0 ) m (\rho v^{2})_{sw}\approx\left(\frac{4B(r)^{2}}{2\mu_{0}}\right)_{m}
  2. ρ \rho
  3. v v
  4. 1 / r 3 1/r^{3}
  5. B ( r ) = B 0 / r 3 B(r)=B_{0}/r^{3}
  6. B 0 B_{0}
  7. [ T m 3 ] [T\cdot m^{3}]
  8. ρ v 2 2 B 0 2 r 6 μ 0 \rho v^{2}\approx\frac{2B_{0}^{2}}{r^{6}\mu_{0}}
  9. r 2 B 0 2 μ 0 ρ v 2 6 r\approx\sqrt[6]{\frac{2B_{0}^{2}}{\mu_{0}\rho v^{2}}}
  10. {}_{\oplus}

Magnetoplasmadynamic_thruster.html

  1. a 2 a^{2}

Magnetoresistance.html

  1. 𝐯 = μ ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{v}=\mu\left(\mathbf{E}+\mathbf{v\times B}\right),
  2. 𝐯 = μ 1 + ( μ B ) 2 ( 𝐄 + μ 𝐄 × 𝐁 + μ 2 ( 𝐁 𝐄 ) 𝐁 ) = μ 1 + ( μ B ) 2 ( 𝐄 + μ 𝐄 × 𝐁 ) + μ 𝐄 , \mathbf{v}=\frac{\mu}{1+(\mu B)^{2}}\left(\mathbf{E}+\mu\mathbf{E\times B}+\mu% ^{2}(\mathbf{B\cdot E})\mathbf{B}\right)=\frac{\mu}{1+(\mu B)^{2}}\left(% \mathbf{E}_{\perp}+\mu\mathbf{E\times B}\right)+\mu\mathbf{E}_{\parallel},
  3. φ = ψ - θ \varphi=\psi-\theta
  4. ρ ( φ ) = ρ + ( ρ - ρ ) cos 2 φ \rho(\varphi)=\rho_{\perp}+(\rho_{\parallel}-\rho_{\perp})\cos^{2}\varphi
  5. ρ \rho
  6. ρ , \rho_{\parallel,\perp}
  7. φ = 0 \varphi=0
  8. 90 90^{\circ}
  9. ρ \rho
  10. ψ , θ \psi,\theta

Magnetosphere.html

  1. R P R_{P}
  2. B s u r f B_{surf}
  3. V S W V_{SW}
  4. R C F = R P ( B s u r f 2 μ 0 ρ V S W 2 ) 1 6 R_{CF}=R_{P}\left(\frac{B_{surf}^{2}}{\mu_{0}\rho V_{SW}^{2}}\right)^{\frac{1}% {6}}
  5. R C F R P R_{CF}\gg R_{P}
  6. R C F R P R_{CF}\ll R_{P}
  7. R C F R P R_{CF}\approx R_{P}

Main_sequence.html

  1. T eff T_{\rm eff}
  2. ϵ = L / M \epsilon=L/M
  3. L M 3.5 \begin{smallmatrix}L\ \propto\ M^{3.5}\end{smallmatrix}
  4. τ MS 10 10 years [ M M ] [ L L ] = 10 10 years [ M M ] - 2.5 \begin{smallmatrix}\tau_{\rm MS}\ \approx\ 10^{10}\,\text{years}\cdot\left[% \frac{M}{M_{\bigodot}}\right]\cdot\left[\frac{L_{\bigodot}}{L}\right]\ =\ 10^{% 10}\,\text{years}\cdot\left[\frac{M}{M_{\bigodot}}\right]^{-2.5}\end{smallmatrix}
  5. M \begin{smallmatrix}M_{\bigodot}\end{smallmatrix}
  6. L \begin{smallmatrix}L_{\bigodot}\end{smallmatrix}
  7. τ MS \tau_{\rm MS}

Majority_function.html

  1. Majority ( p 1 , , p n ) = 1 2 + ( i = 1 n p i ) - 1 / 2 n . \operatorname{Majority}\left(p_{1},\dots,p_{n}\right)=\left\lfloor\frac{1}{2}+% \frac{\left(\sum_{i=1}^{n}p_{i}\right)-1/2}{n}\right\rfloor.

Mandelbrot_set.html

  1. z n + 1 = z n 2 + c z_{n+1}=z_{n}^{2}+c
  2. M M
  3. P c : P_{c}:\mathbb{C}\to\mathbb{C}
  4. P c : z z 2 + c , P_{c}:z\mapsto z^{2}+c,
  5. c c
  6. c c
  7. ( 0 , P c ( 0 ) , P c ( P c ( 0 ) ) , P c ( P c ( P c ( 0 ) ) ) , ) (0,P_{c}(0),P_{c}(P_{c}(0)),P_{c}(P_{c}(P_{c}(0))),\ldots)
  8. P c ( z ) P_{c}(z)
  9. z = 0 z=0
  10. c c
  11. P c n ( z ) P_{c}^{n}(z)
  12. P c ( z ) P_{c}(z)
  13. P c ( z ) P_{c}(z)
  14. M = { c : s , n , | P c n ( 0 ) | s } . M=\left\{c\in\mathbb{C}:\exists s\in\mathbb{R},\forall n\in\mathbb{N},|P_{c}^{% n}(0)|\leq s\right\}.
  15. s = 2 s=2
  16. c c
  17. | P c n ( 0 ) | |P_{c}^{n}(0)|
  18. P c ( z ) P_{c}(z)
  19. c c
  20. P c P_{c}
  21. c c
  22. | P c n ( 0 ) | 2 |P_{c}^{n}(0)|\leq 2
  23. n 0 n\geq 0
  24. P c n ( 0 ) P_{c}^{n}(0)
  25. M M
  26. z λ z ( 1 - z ) , λ [ 1 , 4 ] . z\mapsto\lambda z(1-z),\quad\lambda\in[1,4].\,
  27. c = λ 2 ( 1 - λ 2 ) . c=\frac{\lambda}{2}\left(1-\frac{\lambda}{2}\right).
  28. M M
  29. M M
  30. M M
  31. c c
  32. c . c.
  33. c c
  34. P c P_{c}
  35. c = μ 2 ( 1 - μ 2 ) c=\frac{\mu}{2}\left(1-\frac{\mu}{2}\right)
  36. μ \mu
  37. c = - 3 / 4 c=-3/4
  38. c c
  39. P c P_{c}
  40. p q \textstyle\frac{p}{q}
  41. c p q = e 2 π i p q 2 ( 1 - e 2 π i p q 2 ) . c_{\frac{p}{q}}=\frac{e^{2\pi i\frac{p}{q}}}{2}\left(1-\frac{e^{2\pi i\frac{p}% {q}}}{2}\right).
  42. p q \textstyle\frac{p}{q}
  43. q q
  44. p q \textstyle\frac{p}{q}
  45. q q
  46. α \alpha
  47. U 0 , , U q - 1 U_{0},\dots,U_{q-1}
  48. P c P_{c}
  49. U j U_{j}
  50. U j + p ( mod q ) U_{j+p\,(\operatorname{mod}q)}
  51. c p q c_{\frac{p}{q}}
  52. p q \textstyle\frac{p}{q}
  53. α \alpha
  54. P c P_{c}
  55. M M
  56. P c ( z ) P_{c}(z)
  57. P c P_{c}
  58. ( 0 ) = 0 (0)=0
  59. Q n ( c ) Q^{n}(c)
  60. Q n + 1 ( c ) = Q n ( c ) 2 + c Q^{n+1}(c)=Q^{n}(c)^{2}+c
  61. Q n ( c ) Q^{n}(c)
  62. 2 n - 1 2^{n-1}
  63. Q n ( c ) = 0 , n = 1 , 2 , 3 , Q^{n}(c)=0,n=1,2,3,...
  64. M M
  65. π \pi
  66. π \pi
  67. π \pi
  68. p q \textstyle\frac{p}{q}
  69. 1 q 2 \textstyle\frac{1}{q^{2}}
  70. 1 q \textstyle\frac{1}{q}
  71. - 3 4 + i ϵ \textstyle-\frac{3}{4}+i\epsilon
  72. - 3 4 \textstyle-\frac{3}{4}
  73. - 3 4 \textstyle-\frac{3}{4}
  74. π \pi
  75. z z d + c . z\mapsto z^{d}+c.
  76. z z 3 + 3 k z + c z\mapsto z^{3}+3kz+c
  77. z z ¯ 2 + c . z\mapsto\bar{z}^{2}+c\,.
  78. z ( | ( z ) | + i | ( z ) | ) 2 + c . z\mapsto(|\Re\left(z\right)|+i|\Im\left(z\right)|)^{2}+c\,.
  79. c c
  80. P c P_{c}
  81. c c
  82. c c
  83. z z
  84. P c P_{c}
  85. z = x + i y z=x+iy
  86. z 2 = x 2 + i 2 x y - y 2 z^{2}=x^{2}+i2xy-y^{2}
  87. c = x 0 + i y 0 c=x_{0}+iy_{0}
  88. x = Re ( z 2 + c ) = x 2 - y 2 + x 0 x=\mathop{\mathrm{Re}}(z^{2}+c)=x^{2}-y^{2}+x_{0}
  89. y = Im ( z 2 + c ) = 2 x y + y 0 . y=\mathop{\mathrm{Im}}(z^{2}+c)=2xy+y_{0}.
  90. ν \nu
  91. ϕ ( z ) = lim n ( log | z n | / P n ) , \phi(z)=\lim_{n\to\infty}(\log|z_{n}|/P^{n}),\,
  92. log | z n | / P n = log ( N ) / P ν ( z ) , \log|z_{n}|/P^{n}=\log(N)/P^{\nu(z)},\,
  93. ν ( z ) \nu(z)
  94. ν ( z ) = n - log P ( log | z n | / log ( N ) ) , \nu(z)=n-\log_{P}(\log|z_{n}|/\log(N)),\,
  95. ν ( z ) \nu(z)
  96. M M
  97. b = lim n 2 P c n ( c ) ln P c n ( c ) c P c n ( c ) b=\lim_{n\to\infty}2\cdot\frac{\mid{P_{c}^{n}(c)\mid\cdot\ln\mid{P_{c}^{n}(c)}% }\mid}{\mid\frac{\partial}{\partial{c}}P_{c}^{n}(c)\mid}
  98. P c ( z ) P_{c}(z)\,
  99. P c n ( c ) P_{c}^{n}(c)
  100. P c ( z ) z P_{c}(z)\to z
  101. z 2 + c z z^{2}+c\to z
  102. z = c z=c
  103. P c 0 ( c ) = c P_{c}^{0}(c)=c
  104. P c n + 1 ( c ) = P c n ( c ) 2 + c P_{c}^{n+1}(c)=P_{c}^{n}(c)^{2}+c
  105. c P c n ( c ) \frac{\partial}{\partial{c}}P_{c}^{n}(c)
  106. P c n ( c ) P_{c}^{n}(c)
  107. c P c 0 ( c ) = 1 \frac{\partial}{\partial{c}}P_{c}^{0}(c)=1
  108. c P c n + 1 ( c ) = 2 P c n ( c ) c P c n ( c ) + 1 \frac{\partial}{\partial{c}}P_{c}^{n+1}(c)=2\cdot{}P_{c}^{n}(c)\cdot\frac{% \partial}{\partial{c}}P_{c}^{n}(c)+1
  109. ϕ ( z ) \phi(z)
  110. | ϕ ( z ) | |\phi^{\prime}(z)|
  111. ϕ ( z ) / | ϕ ( z ) | \phi(z)/|\phi^{\prime}(z)|
  112. b = 1 - | z P c p ( z 0 ) | 2 | c z P c p ( z 0 ) + z z P c p ( z 0 ) c P c p ( z 0 ) 1 - z P c p ( z 0 ) | b=\frac{1-\left|{\frac{\partial}{\partial{z}}P_{c}^{p}(z_{0})}\right|^{2}}{% \left|{\frac{\partial}{\partial{c}}\frac{\partial}{\partial{z}}P_{c}^{p}(z_{0}% )+\frac{\partial}{\partial{z}}\frac{\partial}{\partial{z}}P_{c}^{p}(z_{0})% \frac{\frac{\partial}{\partial{c}}P_{c}^{p}(z_{0})}{1-\frac{\partial}{\partial% {z}}P_{c}^{p}(z_{0})}}\right|}
  113. p p
  114. c c
  115. P c ( z ) P_{c}(z)
  116. P c ( z ) = z 2 + c P_{c}(z)=z^{2}+c
  117. P c p ( z 0 ) P_{c}^{p}(z_{0})
  118. p p
  119. P c ( z ) z P_{c}(z)\to z
  120. P c 0 ( z ) = z 0 P_{c}^{0}(z)=z_{0}
  121. z 0 z_{0}
  122. p p
  123. P c ( z ) z P_{c}(z)\to z
  124. P c 0 ( z ) = c P_{c}^{0}(z)=c
  125. z 0 z_{0}
  126. z 0 = P c p ( z 0 ) z_{0}=P_{c}^{p}(z_{0})
  127. c z P c p ( z 0 ) \frac{\partial}{\partial{c}}\frac{\partial}{\partial{z}}P_{c}^{p}(z_{0})
  128. z z P c p ( z 0 ) \frac{\partial}{\partial{z}}\frac{\partial}{\partial{z}}P_{c}^{p}(z_{0})
  129. c P c p ( z 0 ) \frac{\partial}{\partial{c}}P_{c}^{p}(z_{0})
  130. z P c p ( z 0 ) \frac{\partial}{\partial{z}}P_{c}^{p}(z_{0})
  131. P c p ( z ) P_{c}^{p}(z)
  132. z 0 z_{0}
  133. z 0 z_{0}
  134. p p
  135. z 0 z_{0}
  136. z 0 z_{0}
  137. P c ( z ) P_{c}(z)
  138. p p
  139. p = ( x - 1 4 ) 2 + y 2 p=\sqrt{\left(x-\frac{1}{4}\right)^{2}+y^{2}}
  140. x < p - 2 p 2 + 1 4 x<p-2p^{2}+\frac{1}{4}
  141. ( x + 1 ) 2 + y 2 < 1 16 (x+1)^{2}+y^{2}<\frac{1}{16}
  142. q = ( x - 1 4 ) 2 + y 2 q=\left(x-\frac{1}{4}\right)^{2}+y^{2}
  143. q ( q + ( x - 1 4 ) ) < 1 4 y 2 . q\left(q+\left(x-\frac{1}{4}\right)\right)<\frac{1}{4}y^{2}.
  144. z n + 1 = z n 2 + c z_{n+1}=z_{n}^{2}+c
  145. ( z n + ϵ ) 2 + c = z n 2 + 2 z n ϵ + ϵ 2 + c (z_{n}+\epsilon)^{2}+c=z_{n}^{2}+2z_{n}\epsilon+\epsilon^{2}+c
  146. z n + 1 + 2 z n ϵ + ϵ 2 z_{n+1}+2z_{n}\epsilon+\epsilon^{2}
  147. ϵ n + 1 = 2 z n ϵ n + ϵ n 2 \epsilon_{n+1}=2z_{n}\epsilon_{n}+\epsilon_{n}^{2}
  148. z λ z ( 1 - z ) z\mapsto\lambda z(1-z)
  149. λ , z \lambda,z

Many-valued_logic.html

  1. K 3 S K_{3}^{S}
  2. B 3 I B_{3}^{I}
  3. G k G_{k}
  4. 0 , 1 k - 1 , 2 k - 1 , k - 2 k - 1 , 1 0,\tfrac{1}{k-1},\tfrac{2}{k-1},\ldots\tfrac{k-2}{k-1},1
  5. G 3 G_{3}
  6. 0 , 1 2 , 1 0,\tfrac{1}{2},1
  7. G 4 G_{4}
  8. 0 , 1 3 , 2 3 , 1 0,\tfrac{1}{3},\tfrac{2}{3},1
  9. G G_{\infty}
  10. [ 0 , 1 ] [0,1]
  11. \wedge
  12. \vee
  13. u v := min { u , v } u\wedge v:=\min\{u,v\}
  14. u v := max { u , v } u\vee v:=\max\{u,v\}
  15. ¬ G \neg_{G}
  16. G \rightarrow_{G}
  17. ¬ G u = { 1 , if u = 0 0 , if u > 0 \neg_{G}u=\begin{cases}1,&\,\text{if }u=0\\ 0,&\,\text{if }u>0\end{cases}
  18. u G v = { 1 , if u v 0 , if u > v u\rightarrow_{G}v=\begin{cases}1,&\,\text{if }u\leq v\\ 0,&\,\text{if }u>v\end{cases}
  19. L \rightarrow_{L}
  20. ¬ L \neg_{L}
  21. ¬ L u := 1 - u \neg_{L}u:=1-u
  22. u L v := m i n { 1 , 1 - u + v } u\rightarrow_{L}v:=min\{1,1-u+v\}
  23. L 3 L_{3}
  24. 0 , 1 2 , 1 0,\tfrac{1}{2},1
  25. L L_{\infty}
  26. [ 0 , 1 ] [0,1]
  27. 0 , 1 v - 1 , 2 v - 1 , , v - 2 v - 1 , 1 0,\tfrac{1}{v-1},\tfrac{2}{v-1},\ldots,\tfrac{v-2}{v-1},1
  28. L v L_{v}
  29. L L_{\infty}
  30. L 0 L_{\aleph_{0}}
  31. [ 0 , 1 ] [0,1]
  32. L L_{\infty}
  33. L 0 L_{\aleph_{0}}
  34. [ 0 , 1 ] [0,1]
  35. \odot
  36. Π \rightarrow_{\Pi}
  37. u v := u v u\odot v:=uv
  38. u Π v := { 1 , if u v v u , if u > v u\rightarrow_{\Pi}v:=\begin{cases}1,&\,\text{if }u\leq v\\ \frac{v}{u},&\,\text{if }u>v\end{cases}
  39. 0 ¯ \overline{0}
  40. ¬ Π \neg_{\Pi}
  41. Π \wedge_{\Pi}
  42. ¬ Π u := u Π 0 ¯ \neg_{\Pi}u:=u\rightarrow_{\Pi}\overline{0}
  43. u Π v := u ( u Π v ) u\wedge_{\Pi}v:=u\odot(u\rightarrow_{\Pi}v)
  44. P m P_{m}
  45. L v L_{v}
  46. G k G_{k}
  47. 0 , 1 m - 1 , 2 m - 1 , , m - 2 m - 1 , 1 0,\tfrac{1}{m-1},\tfrac{2}{m-1},\ldots,\tfrac{m-2}{m-1},1
  48. ¬ P \neg_{P}
  49. P \vee_{P}
  50. ¬ P u := { 1 , if u = 0 u - 1 m - 1 , if u 0 \neg_{P}u:=\begin{cases}1,&\,\text{if }u=0\\ u-\frac{1}{m-1},&\,\text{if }u\not=0\end{cases}
  51. u P v := m a x { u , v } u\vee_{P}v:=max\{u,v\}

Many-worlds_interpretation.html

  1. O i | O j = δ i j \langle O_{i}|O_{j}\rangle=\delta_{ij}
  2. O i O_{i}
  3. Ψ A \Psi_{A}
  4. Ψ B \Psi_{B}
  5. Ψ = ( a Ψ A + b Ψ B ) \Psi=(a\Psi_{A}+b\Psi_{B})
  6. b a b\ll a

Mariner_1.html

  1. R ˙ n ¯ \bar{\dot{R}_{n}}

Market_capitalization.html

  1. M C = N × P MC=N\times P

Markov_chain.html

  1. Pr ( X n + 1 = x X 1 = x 1 , X 2 = x 2 , , X n = x n ) = Pr ( X n + 1 = x X n = x n ) \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n})=\Pr(X_{n+1}=x% \mid X_{n}=x_{n})
  2. Pr ( X 1 = x 1 , , X n = x n ) > 0 \Pr(X_{1}=x_{1},...,X_{n}=x_{n})>0
  3. Pr ( X n + 1 = x X n = x n ) \Pr(X_{n+1}=x\mid X_{n}=x_{n})
  4. Pr ( X 1 = x 1 ) \Pr(X_{1}=x_{1})
  5. Pr ( X n = x | X 1 = x 1 ) \Pr(X_{n}=x|X_{1}=x_{1})
  6. x 1 x_{1}
  7. Pr ( X 1 = y ) = [ x 1 = y ] \Pr(X_{1}=y)=[x_{1}=y]
  8. [ P ] [P]
  9. Pr ( X n + 1 = b | X n = a ) \Pr(X_{n+1}=b|X_{n}=a)
  10. Pr ( X n + 1 = b | X 1 = x , , X n = a ) \Pr(X_{n+1}=b|X_{1}=x,...,X_{n}=a)
  11. Pr ( X n + 1 = x X n = y ) = Pr ( X n = x X n - 1 = y ) \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)\,
  12. Pr ( X n = x n X n - 1 = x n - 1 , X n - 2 = x n - 2 , , X 1 = x 1 ) \displaystyle\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots,X_{1}=x% _{1})
  13. P = [ 0.9 0.075 0.025 0.15 0.8 0.05 0.25 0.25 0.5 ] . P=\begin{bmatrix}0.9&0.075&0.025\\ 0.15&0.8&0.05\\ 0.25&0.25&0.5\end{bmatrix}.
  14. x ( n + 3 ) = x ( n + 2 ) P = ( x ( n + 1 ) P ) P = x ( n + 1 ) P 2 = ( x ( n ) P 2 ) P = x ( n ) P 3 = [ 0 1 0 ] [ 0.9 0.075 0.025 0.15 0.8 0.05 0.25 0.25 0.5 ] 3 = [ 0 1 0 ] [ 0.7745 0.17875 0.04675 0.3575 0.56825 0.07425 0.4675 0.37125 0.16125 ] = [ 0.3575 0.56825 0.07425 ] . \begin{aligned}\displaystyle x^{(n+3)}&\displaystyle=x^{(n+2)}P=\left(x^{(n+1)% }P\right)P\\ \\ &\displaystyle=x^{(n+1)}P^{2}=\left(x^{(n)}P^{2}\right)P\\ &\displaystyle=x^{(n)}P^{3}\\ &\displaystyle=\begin{bmatrix}0&1&0\end{bmatrix}\begin{bmatrix}0.9&0.075&0.025% \\ 0.15&0.8&0.05\\ 0.25&0.25&0.5\end{bmatrix}^{3}\\ &\displaystyle=\begin{bmatrix}0&1&0\end{bmatrix}\begin{bmatrix}0.7745&0.17875&% 0.04675\\ 0.3575&0.56825&0.07425\\ 0.4675&0.37125&0.16125\\ \end{bmatrix}\\ &\displaystyle=\begin{bmatrix}0.3575&0.56825&0.07425\end{bmatrix}.\end{aligned}
  15. lim N P N = [ 0.625 0.3125 0.0625 0.625 0.3125 0.0625 0.625 0.3125 0.0625 ] \lim_{N\to\infty}\,P^{N}=\begin{bmatrix}0.625&0.3125&0.0625\\ 0.625&0.3125&0.0625\\ 0.625&0.3125&0.0625\\ \end{bmatrix}
  16. p i j ( n ) = Pr ( X n = j X 0 = i ) p_{ij}^{(n)}=\Pr(X_{n}=j\mid X_{0}=i)\,
  17. p i j = Pr ( X 1 = j X 0 = i ) . p_{ij}=\Pr(X_{1}=j\mid X_{0}=i).\,
  18. p i j ( n ) = Pr ( X k + n = j X k = i ) p_{ij}^{(n)}=\Pr(X_{k+n}=j\mid X_{k}=i)\,
  19. p i j = Pr ( X k + 1 = j X k = i ) . p_{ij}=\Pr(X_{k+1}=j\mid X_{k}=i).\,
  20. Pr ( X n = j ) = r S p r j Pr ( X n - 1 = r ) = r S p r j ( n ) Pr ( X 0 = r ) . \Pr(X_{n}=j)=\sum_{r\in S}p_{rj}\Pr(X_{n-1}=r)=\sum_{r\in S}p_{rj}^{(n)}\Pr(X_% {0}=r).
  21. Pr ( X n i j = j X 0 = i ) = p i j ( n i j ) > 0. \Pr(X_{n_{ij}}=j\mid X_{0}=i)=p_{ij}^{(n_{ij})}>0.\,
  22. k = gcd { n > 0 : Pr ( X n = i X 0 = i ) > 0 } k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}
  23. Pr ( X n = i X 0 = i ) > 0. \Pr(X_{n^{\prime}}=i\mid X_{0}=i)>0.
  24. T i = inf { n 1 : X n = i X 0 = i } . T_{i}=\inf\{n\geq 1:X_{n}=i\mid X_{0}=i\}.
  25. f i i ( n ) = Pr ( T i = n ) f_{ii}^{(n)}=\Pr(T_{i}=n)
  26. Pr ( T i < ) = n = 1 f i i ( n ) < 1. \Pr(T_{i}<{\infty})=\sum_{n=1}^{\infty}f_{ii}^{(n)}<1.
  27. M i = E [ T i ] = n = 1 n f i i ( n ) . M_{i}=E[T_{i}]=\sum_{n=1}^{\infty}n\cdot f_{ii}^{(n)}.\,
  28. n = 0 p i i ( n ) = . \sum_{n=0}^{\infty}p_{ii}^{(n)}=\infty.
  29. p i i = 1 and p i j = 0 for i j . p_{ii}=1\,\text{ and }p_{ij}=0\,\text{ for }i\not=j.
  30. p i j p_{ij}
  31. s y m b o l π symbol{\pi}
  32. j S \forall j\in S
  33. 0 π j 1. 0\leq\pi_{j}\leq 1.
  34. j S π j = 1. \sum_{j\in S}\pi_{j}=1.
  35. π j = i S π i p i j . \pi_{j}=\sum_{i\in S}\pi_{i}p_{ij}.
  36. π j = C M j , \pi_{j}=\frac{C}{M_{j}}\,,
  37. C C
  38. lim n p i j ( n ) = C M j . \lim_{n\rightarrow\infty}p_{ij}^{(n)}=\frac{C}{M_{j}}.
  39. π \pi
  40. C i C_{i}
  41. π i \pi_{i}
  42. π i \pi_{i}
  43. lim n p j j ( n ) = C M j \lim_{n\rightarrow\infty}p_{jj}^{(n)}=\frac{C}{M_{j}}
  44. lim n p i j ( n ) = C f i j M j . \lim_{n\rightarrow\infty}p_{ij}^{(n)}=C\frac{f_{ij}}{M_{j}}.
  45. lim n p i i ( n ) \lim_{n\rightarrow\infty}p_{ii}^{(n)}
  46. lim n p i i ( k n + r ) \lim_{n\rightarrow\infty}p_{ii}^{(kn+r)}
  47. s y m b o l π symbol{\pi}
  48. π j = i S π i Pr ( X n + 1 = j X n = i ) \pi_{j}=\sum_{i\in S}\pi_{i}\,\Pr(X_{n+1}=j\mid X_{n}=i)
  49. s y m b o l π symbol{\pi}
  50. p i j = Pr ( X n + 1 = j X n = i ) . p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).\,
  51. π 𝐏 = π . \pi\mathbf{P}=\pi.\,
  52. π = e i e i \pi=\frac{e}{\sum_{i}{e_{i}}}
  53. i π i = 1 \textstyle\sum_{i}\pi_{i}=1
  54. π i \textstyle\pi_{i}
  55. i 1 π i = 1 \textstyle\sum_{i}1\cdot\pi_{i}=1
  56. lim k 𝐏 k = 𝟏 π \lim_{k\rightarrow\infty}\mathbf{P}^{k}=\mathbf{1}\pi
  57. lim k 𝐏 k \scriptstyle\lim_{k\to\infty}\mathbf{P}^{k}
  58. lim k 𝐏 k \scriptstyle\lim\limits_{k\to\infty}\mathbf{P}^{k}
  59. 𝐏 = ( 0 1 1 0 ) 𝐏 2 k = I 𝐏 2 k + 1 = 𝐏 \mathbf{P}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\qquad\mathbf{P}^{2k}=I\qquad\mathbf{P}^{2k+1}=\mathbf{P}
  60. ( 1 2 1 2 ) ( 0 1 1 0 ) = ( 1 2 1 2 ) \begin{pmatrix}\frac{1}{2}&\frac{1}{2}\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}\end{pmatrix}
  61. 𝐐 = lim k 𝐏 k . \scriptstyle\mathbf{Q}=\lim_{k\to\infty}\mathbf{P}^{k}.
  62. 𝐐𝐏 = 𝐐 . \mathbf{QP}=\mathbf{Q}.
  63. 𝐐 ( 𝐏 - 𝐈 n ) = 𝟎 n , n , \mathbf{Q}(\mathbf{P}-\mathbf{I}_{n})=\mathbf{0}_{n,n},
  64. 𝐐 = f ( 𝟎 n , n ) [ f ( 𝐏 - 𝐈 n ) ] - 1 . \mathbf{Q}=f(\mathbf{0}_{n,n})[f(\mathbf{P}-\mathbf{I}_{n})]^{-1}.
  65. π = π 𝐏 \mathbf{\pi}=\mathbf{\pi P}
  66. 𝐏 = 𝐔 𝚺 𝐔 - 1 . \mathbf{P}=\mathbf{U\Sigma U}^{-1}.
  67. 𝐱 T = i = 1 n a i 𝐮 i \mathbf{x}^{T}=\sum_{i=1}^{n}a_{i}\mathbf{u}_{i}
  68. π ( k ) = 𝐱 ( 𝐔 𝚺 𝐔 - 1 ) ( 𝐔 𝚺 𝐔 - 1 ) ( 𝐔 𝚺 𝐔 - 1 ) \mathbf{\pi}^{(k)}=\mathbf{x}(\mathbf{U\Sigma U}^{-1})(\mathbf{U\Sigma U}^{-1}% )\cdots(\mathbf{U\Sigma U}^{-1})
  69. = 𝐱𝐔 𝚺 k 𝐔 - 1 =\mathbf{xU\Sigma}^{k}\mathbf{U}^{-1}
  70. = ( a 1 𝐮 1 T + a 2 𝐮 2 T + + a n 𝐮 n T ) 𝐔 𝚺 k 𝐔 - 1 , =(a_{1}\mathbf{u}_{1}^{T}+a_{2}\mathbf{u}_{2}^{T}+\cdots+a_{n}\mathbf{u}_{n}^{% T})\mathbf{U\Sigma}^{k}\mathbf{U}^{-1},
  71. = a 1 λ 1 k 𝐮 1 + a 2 λ 2 k 𝐮 2 + + a n λ n k 𝐮 n , =a_{1}\lambda_{1}^{k}\mathbf{u}_{1}+a_{2}\lambda_{2}^{k}\mathbf{u}_{2}+\cdots+% a_{n}\lambda_{n}^{k}\mathbf{u}_{n},
  72. = λ 1 k { a 1 𝐮 1 + a 2 ( λ 2 λ 1 ) k 𝐮 2 + a 3 ( λ 3 λ 1 ) k 𝐮 3 + + a n ( λ n λ 1 ) k 𝐮 n } . =\lambda_{1}^{k}\left\{a_{1}\mathbf{u}_{1}+a_{2}\left(\frac{\lambda_{2}}{% \lambda_{1}}\right)^{k}\mathbf{u}_{2}+a_{3}\left(\frac{\lambda_{3}}{\lambda_{1% }}\right)^{k}\mathbf{u}_{3}+\cdots+a_{n}\left(\frac{\lambda_{n}}{\lambda_{1}}% \right)^{k}\mathbf{u}_{n}\right\}.
  73. π i Pr ( X n + 1 = j X n = i ) = π j Pr ( X n + 1 = i X n = j ) \pi_{i}\Pr(X_{n+1}=j\mid X_{n}=i)=\pi_{j}\Pr(X_{n+1}=i\mid X_{n}=j)
  74. p i j p_{ij}
  75. π i p i j = π j p j i . \pi_{i}p_{ij}=\pi_{j}p_{ji}\,.
  76. i π i Pr ( X n + 1 = j X n = i ) = i π j Pr ( X n + 1 = i X n = j ) = π j i Pr ( X n + 1 = i X n = j ) = π j , \begin{aligned}\displaystyle\sum_{i}\pi_{i}\Pr(X_{n+1}=j\mid X_{n}=i)&% \displaystyle=\sum_{i}\pi_{j}\Pr(X_{n+1}=i\mid X_{n}=j)\\ &\displaystyle=\pi_{j}\sum_{i}\Pr(X_{n+1}=i\mid X_{n}=j)=\pi_{j}\,,\end{aligned}
  77. Pr ( X n = i , X n + 1 = j ) = Pr ( X n + 1 = i , X n = j ) . \Pr(X_{n}=i,X_{n+1}=j)=\Pr(X_{n+1}=i,X_{n}=j)\,.
  78. If τ A = inf { n 0 : X n A } , then P z ( τ A < ) > 0 for all z . \,\text{If }\tau_{A}=\inf\{n\geq 0:X_{n}\in A\},\,\text{ then }P_{z}(\tau_{A}<% \infty)>0\,\text{ for all }z.
  79. If x A and C B , then p ( x , C ) ε ρ ( C ) . \,\text{If }x\in A\,\text{ and }C\subset B,\,\text{ then }p(x,C)\geq% \varepsilon\rho(C).
  80. i i
  81. N N
  82. i i
  83. k i k_{i}
  84. α k i + 1 - α N \frac{\alpha}{k_{i}}+\frac{1-\alpha}{N}
  85. 1 - α N \frac{1-\alpha}{N}
  86. α \alpha

Masoretic_Text.html

  1. 𝔐 \mathfrak{M}

Mass.html

  1. a = M m g . a=\frac{M}{m}g.
  2. W n n , W_{n}\propto n,
  3. W n n = W m m \frac{W_{n}}{n}=\frac{W_{m}}{m}
  4. W n W m = n m . \frac{W_{n}}{W_{m}}=\frac{n}{m}.
  5. ounce pound = W 144 W 1728 = 144 1728 = 1 12 . \frac{\mathrm{ounce}}{\mathrm{pound}}=\frac{W_{144}}{W_{1728}}=\frac{144}{1728% }=\frac{1}{12}.
  6. μ = 4 π 2 distance 3 time 2 gravitational mass \mu=4\pi^{2}\frac{\,\text{distance}^{3}}{\,\text{time}^{2}}\propto\,\text{% gravitational mass}
  7. Distance Time 2 {\,\text{Distance}}\propto{\,\text{Time}^{2}}
  8. 1.2 π 2 10 - 5 AU 3 y 2 = 3.986 10 14 m 3 s 2 1.2\pi^{2}\cdot 10^{-5}\frac{\,\text{AU}^{3}}{\,\text{y}^{2}}=3.986\cdot 10^{1% 4}\frac{\,\text{m}^{3}}{\,\text{s}^{2}}
  9. 𝐠 = - μ 𝐑 ^ | 𝐑 | 2 \mathbf{g}=-\mu\frac{\hat{\mathbf{R}}}{|\mathbf{R}|^{2}}
  10. 𝐅 AB = - G M A M B 𝐑 ^ AB | 𝐑 AB | 2 \mathbf{F}_{\,\text{AB}}=-GM_{\,\text{A}}M_{\,\text{B}}\frac{\hat{\mathbf{R}}_% {\,\text{AB}}}{|\mathbf{R}_{\,\text{AB}}|^{2}}
  11. F = M g F=Mg
  12. 𝐅 = m 𝐚 , \mathbf{F}=m\mathbf{a},
  13. 𝐅 𝟏𝟐 \displaystyle\mathbf{F_{12}}
  14. 𝐅 12 = - 𝐅 21 ; \mathbf{F}_{12}=-\mathbf{F}_{21};
  15. m 1 = m 2 | 𝐚 2 | | 𝐚 1 | . m_{1}=m_{2}\frac{|\mathbf{a}_{2}|}{|\mathbf{a}_{1}|}\!.
  16. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  17. K = 1 2 m | 𝐯 | 2 K=\dfrac{1}{2}m|\mathbf{v}|^{2}
  18. m relative = γ ( m rest ) m_{\mathrm{relative}}=\gamma(m_{\mathrm{rest}})\!
  19. γ \gamma
  20. γ = 1 1 - v 2 / c 2 \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}
  21. ( m rest ) c 2 = E total 2 - ( | 𝐩 | c ) 2 . (m_{\mathrm{rest}})c^{2}=\sqrt{E_{\mathrm{total}}^{2}-(|\mathbf{p}|c)^{2}}.\!
  22. E rest = ( m rest ) c 2 E_{\mathrm{rest}}=(m_{\mathrm{rest}})c^{2}\!
  23. E total = ( m relative ) c 2 E_{\mathrm{total}}=(m_{\mathrm{relative}})c^{2}\!
  24. d d t ( L x ˙ i ) = m x ¨ i \frac{\mathrm{d}}{\mathrm{d}t}\ \left(\,\frac{\partial L}{\partial\dot{x}_{i}}% \,\right)\ =\ m\,\ddot{x}_{i}
  25. i t Ψ ( 𝐫 , t ) = ( - 2 2 m 2 + V ( 𝐫 ) ) Ψ ( 𝐫 , t ) i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=\left(-\frac{\hbar^{2}}{% 2m}\nabla^{2}+V(\mathbf{r})\right)\Psi(\mathbf{r},\,t)
  26. ( - i γ μ μ + m ) ψ = 0 (-i\gamma^{\mu}\partial_{\mu}+m)\psi=0\,
  27. G ψ ψ ¯ ϕ ψ G_{\psi}\overline{\psi}\phi\psi
  28. E 2 = p 2 c 2 + m 2 c 4 E^{2}=p^{2}c^{2}+m^{2}c^{4}\;
  29. E = m c 2 1 - v 2 c 2 . E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.

Mass_noun.html

  1. X U p [ C U M p ( X ) x , y [ X ( x ) X ( y ) ¬ ( x = y ) ] x , y [ X ( x ) X ( y ) X ( x y ) ] ] \forall X\subseteq U_{p}[CUM_{p}(X)\Leftrightarrow\exists x,y[X(x)\,\wedge\,X(% y)\,\wedge\,\neg(x=y)]\;\wedge\;\forall x,y[X(x)\,\wedge\,X(y)\Rightarrow X(x% \,\oplus\,y)]]

Mathematical_analysis.html

  1. ( M , d ) (M,d)
  2. M M
  3. d d
  4. M M
  5. d : M × M d\colon M\times M\rightarrow\mathbb{R}
  6. x , y , z M x,y,z\in M
  7. d ( x , y ) = 0 d(x,y)=0\,
  8. x = y x=y\,
  9. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)\,
  10. d ( x , z ) d ( x , y ) + d ( y , z ) d(x,z)\leq d(x,y)+d(y,z)
  11. z = x z=x
  12. d ( x , y ) 0 d(x,y)\geq 0
  13. lim n a n = x . \lim_{n\to\infty}a_{n}=x.
  14. n n
  15. n \mathbb{R}^{n}
  16. [ 0 , 1 ] \left[0,1\right]
  17. X X
  18. σ \sigma

Mathematical_constant.html

  1. e e
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi
  6. π \pi
  7. F = 1 4 π ε 0 | q 1 q 2 | r 2 . F=\frac{1}{4\pi\varepsilon_{0}}\frac{\left|q_{1}q_{2}\right|}{r^{2}}.
  8. ε 0 {\varepsilon_{0}}
  9. 4 π r 2 {4\pi r^{2}}
  10. π \pi
  11. π \pi
  12. e e
  13. e e
  14. e = lim n ( 1 + 1 n ) n e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
  15. e e
  16. R R
  17. e e
  18. n n
  19. n n
  20. n n
  21. 1 / e 1/e
  22. n n
  23. e e
  24. n n
  25. p n = 1 - 1 1 ! + 1 2 ! - 1 3 ! + + ( - 1 ) n 1 n ! p_{n}=1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^{n}\frac{1}{n!}
  26. n n
  27. 1 / e 1/e
  28. e e
  29. 2 \sqrt{2}
  30. 2 \sqrt{2}
  31. i i
  32. i i
  33. P ( x ) P(x)
  34. i i
  35. i −i
  36. i i
  37. j j
  38. ι ι
  39. j j
  40. i i
  41. i i
  42. ζ ( 3 ) = 1 + 1 2 3 + 1 3 3 + 1 4 3 + \zeta(3)=1+\frac{1}{2^{3}}+\frac{1}{3^{3}}+\frac{1}{4^{3}}+\cdots
  43. F ( n ) = φ n - ( 1 - φ ) n 5 F\left(n\right)=\frac{\varphi^{n}-(1-\varphi)^{n}}{\sqrt{5}}
  44. 1 + 5 2 . \scriptstyle\frac{1+\sqrt{5}}{2}.
  45. γ \gamma
  46. γ \gamma
  47. 1 11 21 1211 111221 312211 \begin{matrix}1\\ 11\\ 21\\ 1211\\ 111221\\ 312211\\ \vdots\end{matrix}
  48. r = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + , r=a_{0}+\dfrac{1}{a_{1}+\dfrac{1}{a_{2}+\dfrac{1}{a_{3}+\cdots}}},
  49. A = lim n k = 1 n k k n n 2 / 2 + n / 2 + 1 / 12 e - n 2 / 4 A=\lim_{n\rightarrow\infty}\frac{\prod_{k=1}^{n}k^{k}}{n^{n^{2}/2+n/2+1/12}e^{% -n^{2}/4}}
  50. c = j = 1 10 - j ! = 0. 110001 3 ! digits 000000000000000001 4 ! digits 000 c=\sum_{j=1}^{\infty}10^{-j!}=0.\underbrace{\overbrace{110001}^{3!\,\text{ % digits}}000000000000000001}_{4!\,\text{ digits}}000\dots\,
  51. C 10 = 0. \color b l u e 12 \color b l u e 34 \color b l u e 56 \color b l u e 78 \color b l u e 910 \color b l u e 1112 \color b l u e 1314 \color b l u e 1516 C_{10}=0.{\color{blue}{1}}2{\color{blue}{3}}4{\color{blue}{5}}6{\color{blue}{7% }}8{\color{blue}{9}}10{\color{blue}{11}}12{\color{blue}{13}}14{\color{blue}{15% }}16\dots
  52. y ( x ) = - 2 y + e - x y^{\prime}(x)=-2y+e^{-x}\,
  53. cos x d x = sin x + C \int\cos x\ dx=\sin x+C
  54. f ( x , y ) x = 0 \frac{\partial f(x,y)}{\partial x}=0
  55. G = 3 3 3 3 } 64 layers G=\left.\begin{matrix}3\underbrace{\uparrow\ldots\uparrow}3\\ \underbrace{\vdots}\\ 3\uparrow\uparrow\uparrow\uparrow 3\end{matrix}\right\}\,\text{64 layers}
  56. a , b , c , a,b,c,\dots\,
  57. α , β , γ , \alpha,\beta,\,\gamma,\dots\,
  58. E B E_{B}\,
  59. β * \beta*\,
  60. B 2 B_{2}\,
  61. C b C_{b}
  62. 0 \aleph_{0}
  63. googol = 10 100 , googolplex = 10 googol = 10 10 100 \mathrm{googol}=10^{100}\,\ ,\ \mathrm{googolplex}=10^{\mathrm{googol}}=10^{10% ^{100}}\,
  64. 1 \sqrt{–1}
  65. π \pi
  66. 2 \sqrt{2}
  67. 3 \sqrt{3}
  68. 5 \sqrt{5}
  69. γ \gamma
  70. ϕ \phi
  71. Λ \Lambda
  72. β \beta
  73. λ \lambda
  74. σ \sigma
  75. λ \lambda
  76. μ \mu
  77. β \beta
  78. Λ \Lambda
  79. ζ ( 3 ) \zeta(3)
  80. θ \theta
  81. ρ \rho
  82. μ \mu
  83. α \alpha
  84. ψ \psi
  85. δ \delta

Mathematical_formulation_of_quantum_mechanics.html

  1. h h
  2. H H
  3. [ u b r a k e t , u b r a k e t , u 3 d 5 , u 3 c 8 ] [u^{\prime}braket^{\prime},u^{\prime}braket^{\prime},u^{\prime}\u{0}3d5^{% \prime},u^{\prime}\u{0}3c8^{\prime}]
  4. H H
  5. H H
  6. H H
  7. A A
  8. [ u b r a k e t , u k e t , u 3 c 8 ] H [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{0}3c8^{\prime}]∈H
  9. ψ A ψ \langle\psi\mid A\mid\psi\rangle
  10. A A
  11. ψ ψ
  12. A A
  13. A A
  14. A A
  15. A A
  16. ρ ρ
  17. A A
  18. ρ ρ
  19. tr ( A ρ ) \operatorname{tr}(A\rho)
  20. H H
  21. [ u b r a k e t , u k e t , u 3 c 8 ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{0}3c8^{\prime}]
  22. tr ( A ρ ψ ) = ψ A ψ \operatorname{tr}(A\rho_{\psi})=\left\langle\psi\mid A\mid\psi\right\rangle
  23. 𝐑 \mathbf{R}
  24. [ u b r a k e t , u k e t , u 3 c 8 , u ( , u t , u ) ] [u^{\prime}braket^{\prime},u^{\prime}ket^{\prime},u^{\prime}\u{0}3c8^{\prime},% u^{\prime}(^{\prime},u^{\prime}t^{\prime},u^{\prime})^{\prime}]
  25. t t
  26. H H
  27. i i
  28. ħ ħ
  29. H H
  30. U ( t ) U(t)
  31. H H H→H
  32. | ψ ( t + s ) = U ( t ) | ψ ( s ) \left|\psi(t+s)\right\rangle=U(t)\left|\psi(s)\right\rangle
  33. s , t s,t
  34. H H
  35. U ( t ) = e - ( i / ) t H U(t)=e^{-(i/\hbar)tH}
  36. H H
  37. U ( t ) = 𝒯 [ exp ( - i t 0 t d t H ( t ) ) ] , U(t)=\mathcal{T}\left[\exp\left(-\frac{i}{\hbar}\int_{t_{0}}^{t}\,{\rm d}t^{% \prime}\,H(t^{\prime})\right)\right]\,,
  38. 𝒯 {\mathcal{T}}
  39. B 1 ( t 1 ) B 2 ( t 2 ) B n ( t n ) B_{1}(t_{1})\cdot B_{2}(t_{2})\cdot\dots\cdot B_{n}(t_{n})
  40. B i 1 ( t i 1 ) B i 2 ( t i 2 ) B i n ( t i n ) B_{i_{1}}(t_{i_{1}})\cdot B_{i_{2}}(t_{i_{2}})\cdot\dots\cdot B_{i_{n}}(t_{i_{% n}})
  41. t i 1 t i 2 t i n . t_{i_{1}}\geq t_{i_{2}}\geq\dots\geq t_{i_{n}}\,.
  42. | ψ = | ψ ( 0 ) \left|\psi\right\rangle=\left|\psi(0)\right\rangle
  43. A ( t ) = U ( - t ) A U ( t ) . A(t)=U(-t)AU(t).\quad
  44. ψ A ( t ) ψ = ψ ( t ) A ψ ( t ) \langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle
  45. A = A ( t ) A=A(t)
  46. V V
  47. H int ( t ) e ( i / ) t H 0 V e ( - i / ) t H 0 H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_{0}}\,V\,e^{{(-i/\hbar})tH_{0}}
  48. s s
  49. s s
  50. H E H−E
  51. H H
  52. H E H−E
  53. 𝐫 \mathbf{r}
  54. t t
  55. ψ = ψ ( 𝐫 , t ) ψ=ψ(\mathbf{r},t)
  56. ψ = ψ ( 𝐫 , t , σ ) ψ=ψ(\mathbf{r},t,σ)
  57. σ σ
  58. σ = - S , - ( S - 1 ) , , 0 , , + ( S - 1 ) , + S . \sigma=-S\hbar,-(S-1)\hbar,\dots,0,\dots,+(S-1)\hbar,+S\hbar\,.
  59. S S
  60. ( 2 S + 1 ) (2S+1)
  61. S = 0 , 1 , 2... S=0, 1, 2...
  62. S = 1 / 2 , 3 / 2 , 5 / 2 , S={1}/{2},{3}/{2},{5}/{2}, ...
  63. N N
  64. N N
  65. N N
  66. + 1 +1
  67. 1 −1
  68. S = 1 / 2 S=1/2
  69. S = 1 S=1
  70. d = 2 d=2
  71. A A
  72. ψ ψ
  73. A A
  74. A = λ d E A ( λ ) , A=\int\lambda\,d\operatorname{E}_{A}(\lambda),
  75. A A
  76. B B
  77. 𝐑 \mathbf{R}
  78. B B
  79. ψ E A ψ . \langle\psi\mid\operatorname{E}_{A}\psi\rangle.
  80. B B
  81. B B
  82. B B
  83. n n
  84. A A
  85. A A
  86. E A ( B ) = | ψ i ψ i | , \operatorname{E}_{A}(B)=|\psi_{i}\rangle\langle\psi_{i}|,
  87. B B
  88. | ψ |\psi\rangle\,
  89. ψ E A ψ \langle\psi\mid\operatorname{E}_{A}\psi\rangle
  90. ψ | ψ i ψ i ψ = | ψ ψ i | 2 . \langle\psi|\psi_{i}\rangle\langle\psi_{i}\mid\psi\rangle=|\langle\psi\mid\psi% _{i}\rangle|^{2}.
  91. | ψ i ψ i | |\psi_{i}\rangle\langle\psi_{i}|\,
  92. F i F i * F_{i}F_{i}^{*}\,
  93. | ψ i ψ i | ψ |\psi_{i}\rangle\langle\psi_{i}|\psi\rangle\,
  94. F i | ψ . F_{i}|\psi\rangle.\,
  95. F < s u b > i F i * F<sub>iF_{i}*

Mathematical_induction.html

  1. 0 + 1 + 2 + + n = n ( n + 1 ) 2 . 0+1+2+\cdots+n=\frac{n(n+1)}{2}\,.
  2. 0 = 0 ( 0 + 1 ) 2 . 0=\frac{0\cdot(0+1)}{2}\,.
  3. ( 0 + 1 + 2 + + k ) + ( k + 1 ) = ( k + 1 ) ( ( k + 1 ) + 1 ) 2 . (0+1+2+\cdots+k)+(k+1)=\frac{(k+1)((k+1)+1)}{2}.
  4. k ( k + 1 ) 2 + ( k + 1 ) . \frac{k(k+1)}{2}+(k+1)\,.
  5. k ( k + 1 ) 2 + ( k + 1 ) \displaystyle\frac{k(k+1)}{2}+(k+1)
  6. P . [ [ P ( 0 ) ( k ) . [ P ( k ) P ( k + 1 ) ] ] ( n ) . P ( n ) ] \forall P.\,[[P(0)\land\forall(k\in\mathbb{N}).\,[P(k)\Rightarrow P(k+1)]]% \Rightarrow\forall(n\in\mathbb{N}).\,P(n)]
  7. \mathbb{N}
  8. \mathbb{N}
  9. n n 3 n < n ! < n n 2 n for n 6. {n^{n}\over 3^{n}}<n!<{n^{n}\over 2^{n}}\mbox{ for }~{}n\geq 6.
  10. ( f g ) = f g + g f . (fg)^{\prime}=f^{\prime}g+g^{\prime}f.\!
  11. ( f g ) / ( f g ) = f / f + g / g . (fg)^{\prime}/(fg)=f^{\prime}/f+g^{\prime}/g.\!
  12. ( f 1 f 2 f 3 f n ) (f_{1}f_{2}f_{3}\cdots f_{n})^{\prime}\!
  13. = ( f 1 f 2 f 3 f n ) + ( f 1 f 2 f 3 f n ) + ( f 1 f 2 f 3 f n ) + + ( f 1 f 2 f n - 1 f n ) . =(f_{1}^{\prime}f_{2}f_{3}\cdots f_{n})+(f_{1}f_{2}^{\prime}f_{3}\cdots f_{n})% +(f_{1}f_{2}f_{3}^{\prime}\cdots f_{n})+\cdots+(f_{1}f_{2}\cdots f_{n-1}f_{n}^% {\prime}).
  14. ( f 1 f 2 f 3 f n ) / ( f 1 f 2 f 3 f n ) (f_{1}f_{2}f_{3}\cdots f_{n})^{\prime}/(f_{1}f_{2}f_{3}\cdots f_{n})\!
  15. = ( f 1 / f 1 ) + ( f 2 / f 2 ) + ( f 3 / f 3 ) + + ( f n / f n ) . =(f_{1}^{\prime}/f_{1})+(f_{2}^{\prime}/f_{2})+(f_{3}^{\prime}/f_{3})+\cdots+(% f_{n}^{\prime}/f_{n}).
  16. ( 1 ) = 0 (1)^{\prime}=0\!
  17. f 1 = f 1 . f_{1}^{\prime}=f_{1}^{\prime}\!.
  18. k ( P ( k ) P ( k + 1 ) ) \forall k(P(k)\to P(k+1))
  19. k ( P ( k - 1 ) P ( k ) ) \forall k(P(k-1)\to P(k))
  20. k ( P ( k ) P ( 2 k ) P ( 2 k + 1 ) ) \forall k(P(k)\to P(2k)\land P(2k+1))
  21. k ( P ( k 2 ) P ( k ) ) \forall k\left(P\left(\left\lfloor\frac{k}{2}\right\rfloor\right)\to P(k)\right)
  22. k ( P ( k ) P ( k ) ) \forall k\left(P\left(\left\lfloor\sqrt{k}\right\rfloor\right)\to P(k)\right)

Mathematical_logic.html

  1. L ω 1 , ω L_{\omega_{1},\omega}
  2. L ω 1 , ω L_{\omega_{1},\omega}
  3. ( x = 0 ) ( x = 1 ) ( x = 2 ) . (x=0)\lor(x=1)\lor(x=2)\lor\cdots.

Mathematical_model.html

  1. V : 3 V\!:\mathbb{R}^{3}\!\rightarrow\mathbb{R}
  2. 𝐫 : 3 \mathbf{r}\!:\mathbb{R}\rightarrow\mathbb{R}^{3}
  3. - d 2 𝐫 ( t ) d t 2 m = V [ 𝐫 ( t ) ] x 𝐱 ^ + V [ 𝐫 ( t ) ] y 𝐲 ^ + V [ 𝐫 ( t ) ] z 𝐳 ^ , -\frac{\mathrm{d}^{2}\mathbf{r}(t)}{\mathrm{d}t^{2}}m=\frac{\partial V[\mathbf% {r}(t)]}{\partial x}\mathbf{\hat{x}}+\frac{\partial V[\mathbf{r}(t)]}{\partial y% }\mathbf{\hat{y}}+\frac{\partial V[\mathbf{r}(t)]}{\partial z}\mathbf{\hat{z}},
  4. m d 2 𝐫 ( t ) d t 2 = - V [ 𝐫 ( t ) ] . m\frac{\mathrm{d}^{2}\mathbf{r}(t)}{\mathrm{d}t^{2}}=-\nabla V[\mathbf{r}(t)].
  5. max U ( x 1 , x 2 , , x n ) \max U(x_{1},x_{2},\ldots,x_{n})
  6. i = 1 n p i x i M . \sum_{i=1}^{n}p_{i}x_{i}\leq M.
  7. x i 0 i { 1 , 2 , , n } x_{i}\geq 0\;\;\;\forall i\in\{1,2,\ldots,n\}

Mathematical_optimization.html

  1. \to
  2. 𝐱 - 𝐱 * δ ; \|\mathbf{x}-\mathbf{x}^{*}\|\leq\delta;\,
  3. f ( 𝐱 * ) f ( 𝐱 ) f(\mathbf{x}^{*})\leq f(\mathbf{x})
  4. min x ( x 2 + 1 ) \min_{x\in\mathbb{R}}\;(x^{2}+1)
  5. x 2 + 1 x^{2}+1
  6. \mathbb{R}
  7. 1 1
  8. x = 0 x=0
  9. max x 2 x \max_{x\in\mathbb{R}}\;2x
  10. arg min x ( - , - 1 ] x 2 + 1 , \underset{x\in(-\infty,-1]}{\operatorname{arg\,min}}\;x^{2}+1,
  11. arg min 𝑥 x 2 + 1 , subject to: x ( - , - 1 ] . \underset{x}{\operatorname{arg\,min}}\;x^{2}+1,\;\,\text{subject to:}\;x\in(-% \infty,-1].
  12. ( - , - 1 ] (-\infty,-1]
  13. arg max x [ - 5 , 5 ] , y x cos ( y ) , \underset{x\in[-5,5],\;y\in\mathbb{R}}{\operatorname{arg\,max}}\;x\cos(y),
  14. arg max x , y x cos ( y ) , subject to: x [ - 5 , 5 ] , y , \underset{x,\;y}{\operatorname{arg\,max}}\;x\cos(y),\;\,\text{subject to:}\;x% \in[-5,5],\;y\in\mathbb{R},
  15. ( x , y ) (x,y)
  16. x cos ( y ) x\cos(y)
  17. [ - 5 , 5 ] [-5,5]

Mathematics.html

  1. p q p\Rightarrow q\,
  2. 1 , 2 , 3 , 1,2,3,\ldots\!
  3. , - 2 , - 1 , 0 , 1 , 2 \ldots,-2,-1,0,1,2\,\ldots\!
  4. - 2 , 2 3 , 1.21 -2,\frac{2}{3},1.21\,\!
  5. - e , 2 , 3 , π -e,\sqrt{2},3,\pi\,\!
  6. 2 , i , - 2 + 3 i , 2 e i 4 π 3 2,i,-2+3i,2e^{i\frac{4\pi}{3}}\,\!
  7. ( 1 , 2 , 3 ) ( 1 , 3 , 2 ) ( 2 , 1 , 3 ) ( 2 , 3 , 1 ) ( 3 , 1 , 2 ) ( 3 , 2 , 1 ) \begin{matrix}(1,2,3)&(1,3,2)\\ (2,1,3)&(2,3,1)\\ (3,1,2)&(3,2,1)\end{matrix}

MathML.html

  1. a x 2 + b x + c ax^{2}+bx+c
  2. sin ( x ) \sin(x)
  3. x + 5 x+5
  4. a x 2 + b x + c ax^{2}+bx+c
  5. x = - b ± b 2 - 4 a c 2 a x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}

Max_Born.html

  1. p q - q p = h 2 π i I pq-qp={h\over 2\pi i}I

Max_Planck.html

  1. E = h ν E=h\nu
  2. h h
  3. ν \nu
  4. h ν h\nu
  5. h h
  6. ν \nu
  7. h ν h\nu

Maximal-ratio_combining.html

  1. y y
  2. n n
  3. n - C N ( 0 , I N × N ) n-CN(0,I_{N\times N})
  4. s ~ = a r g m i n s Q P S K | s ^ - s | 2 , \tilde{s}=argmin_{s\in QPSK}|\hat{s}-s|^{2},
  5. s ^ \hat{s}
  6. s ^ = ( h * h ) - 1 h * y . \hat{s}=(h^{*}h)^{-1}h^{*}y.
  7. s ^ = h 0 * y 0 + h 1 * y 1 + + h N - 1 * y N - 1 | h 0 | 2 + | h 1 | 2 + + | h N - 1 | 2 , \hat{s}=\frac{h_{0}^{*}y_{0}+h_{1}^{*}y_{1}+...+h_{N-1}^{*}y_{N-1}}{|h_{0}|^{2% }+|h_{1}|^{2}+...+|h_{N-1}|^{2}},

Maximal_ideal.html

  1. 4 4\mathbb{Z}
  2. 2 2\mathbb{Z}
  3. 2 / 4 2\mathbb{Z}/4\mathbb{Z}

Maximum_usable_frequency.html

  1. MUF = critical frequency cos θ \,\text{MUF}=\frac{\,\text{critical frequency}}{\cos\theta}

Maxwell's_equations.html

  1. 𝐄 \mathbf{E}
  2. 𝐁 \mathbf{B}
  3. ρ ρ
  4. 𝐉 \mathbf{J}
  5. c c
  6. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  7. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  8. Σ 𝐄 d s y m b o l = - d d t Σ 𝐁 d 𝐒 \oint_{\partial\Sigma}\mathbf{E}\cdot\mathrm{d}symbol{\ell}=-\frac{d}{dt}\iint% _{\Sigma}\mathbf{B}\cdot\mathrm{d}\mathbf{S}
  9. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  10. Σ 𝐁 d s y m b o l = μ 0 Σ 𝐉 d 𝐒 + μ 0 ε 0 d d t Σ 𝐄 d 𝐒 \oint_{\partial\Sigma}\mathbf{B}\cdot\mathrm{d}symbol{\ell}=\mu_{0}\iint_{% \Sigma}\mathbf{J}\cdot\mathrm{d}\mathbf{S}+\mu_{0}\varepsilon_{0}\frac{d}{dt}% \iint_{\Sigma}\mathbf{E}\cdot\mathrm{d}\mathbf{S}
  11. × 𝐁 = μ 0 ( 𝐉 + ε 0 𝐄 t ) \nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}\right)
  12. · ∇·
  13. × ∇×
  14. ρ ρ
  15. 𝐉 \mathbf{J}
  16. Ω Ω
  17. Ω ∂Ω
  18. Σ Σ
  19. Σ ∂Σ
  20. Ω ∂Ω
  21. Ω \iiint_{\Omega}
  22. Ω Ω
  23. Σ \iint_{\Sigma}
  24. Σ Σ
  25. Σ \oint_{\partial\Sigma}
  26. Σ ∂Σ
  27. ρ ρ
  28. Ω Ω
  29. Ω Ω
  30. Q = Ω ρ d V , Q=\iiint_{\Omega}\rho\,\mathrm{d}V\,,
  31. d V dV
  32. 𝐉 \mathbf{J}
  33. Σ Σ
  34. I = Σ 𝐉 d 𝐒 , I=\iint_{\Sigma}\mathbf{J}\cdot\mathrm{d}\mathbf{S}\,,
  35. d 𝐒 d\mathbf{S}
  36. S S
  37. Σ Σ
  38. 𝐀 \mathbf{A}
  39. 𝐒 \mathbf{S}
  40. Ω Ω
  41. Ω ∂Ω
  42. ( + ) (+)
  43. ( ) (−)
  44. 𝐅 \mathbf{F}
  45. 𝐅 \mathbf{F}
  46. 𝐄 \mathbf{E}
  47. 𝐁 \mathbf{B}
  48. Ω ∂Ω
  49. · 𝐄 ∇·\mathbf{E}
  50. · 𝐁 ∇·\mathbf{B}
  51. Σ Σ
  52. Σ ∂Σ
  53. 𝐅 \mathbf{F}
  54. 𝐄 \mathbf{E}
  55. 𝐁 \mathbf{B}
  56. 𝐧 \mathbf{n}
  57. Σ 𝐄 d s y m b o l , Σ 𝐁 d s y m b o l , \oint_{\partial\Sigma}\mathbf{E}\cdot\mathrm{d}symbol{\ell},\quad\oint_{% \partial\Sigma}\mathbf{B}\cdot\mathrm{d}symbol{\ell}\,,
  58. d [ 𝐮 𝐞𝐥𝐥 ] d\mathbf{[u^{\prime}ell^{\prime}]}
  59. × 𝐄 , × 𝐁 . \nabla\times\mathbf{E},\quad\nabla\times\mathbf{B}\,.
  60. 𝐄 t , 𝐁 t . \frac{\partial\mathbf{E}}{\partial t},\quad\frac{\partial\mathbf{B}}{\partial t}.
  61. d d t Σ 𝐁 d 𝐒 = Σ 𝐁 t d 𝐒 , \frac{d}{dt}\iint_{\Sigma}\mathbf{B}\cdot\mathrm{d}\mathbf{S}=\iint_{\Sigma}% \frac{\partial\mathbf{B}}{\partial t}\cdot\mathrm{d}\mathbf{S}\,,
  62. ρ = 0 ρ=0
  63. 𝐉 = 𝟎 \mathbf{J}=\mathbf{0}
  64. 𝐄 \displaystyle\nabla\cdot\mathbf{E}
  65. ( × ) (∇×)
  66. 1 c 2 2 𝐄 t 2 - 2 𝐄 = 0 1 c 2 2 𝐁 t 2 - 2 𝐁 = 0 \frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}-\nabla^{2}\mathbf% {E}=0\,\quad\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{B}}{\partial t^{2}}-% \nabla^{2}\mathbf{B}=0\,
  67. c = 1 μ 0 ε 0 = 2.99792458 × 10 8 m s - 1 c=\frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}=2.99792458\times 10^{8}\,\mathrm{m~{% }s}^{-1}
  68. v p = 1 μ 0 μ r ε 0 ε r v\text{p}=\frac{1}{\sqrt{\mu_{0}\mu\text{r}\varepsilon_{0}\varepsilon\text{r}}}
  69. c c
  70. 𝐄 \mathbf{E}
  71. 𝐁 \mathbf{B}
  72. c c
  73. 𝐄 \mathbf{E}
  74. 𝐁 \mathbf{B}
  75. 𝐃 = ρ f \nabla\cdot\mathbf{D}=\rho\text{f}
  76. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  77. Σ 𝐄 d s y m b o l = - d d t Σ 𝐁 d 𝐒 \oint_{\partial\Sigma}\mathbf{E}\cdot\mathrm{d}symbol{\ell}=-\frac{d}{dt}\iint% _{\Sigma}\mathbf{B}\cdot\mathrm{d}\mathbf{S}
  78. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  79. Σ 𝐇 d s y m b o l = Σ 𝐉 f d 𝐒 + d d t Σ 𝐃 d 𝐒 \oint_{\partial\Sigma}\mathbf{H}\cdot\mathrm{d}symbol{\ell}=\iint_{\Sigma}% \mathbf{J}\text{f}\cdot\mathrm{d}\mathbf{S}+\frac{d}{dt}\iint_{\Sigma}\mathbf{% D}\cdot\mathrm{d}\mathbf{S}
  80. × 𝐇 = 𝐉 f + 𝐃 t \nabla\times\mathbf{H}=\mathbf{J}\text{f}+\frac{\partial\mathbf{D}}{\partial t}
  81. Q = Q f + Q b = Ω ( ρ f + ρ b ) d V = Ω ρ d V Q=Q\text{f}+Q\text{b}=\iiint_{\Omega}\left(\rho\text{f}+\rho\text{b}\right)\,% \mathrm{d}V=\iiint_{\Omega}\rho\,\mathrm{d}V
  82. I = I f + I b = Σ ( 𝐉 f + 𝐉 b ) d 𝐒 = Σ 𝐉 d 𝐒 I=I\text{f}+I\text{b}=\iint_{\Sigma}\left(\mathbf{J}\text{f}+\mathbf{J}\text{b% }\right)\cdot\mathrm{d}\mathbf{S}=\iint_{\Sigma}\mathbf{J}\cdot\mathrm{d}% \mathbf{S}
  83. 𝐉 \mathbf{J}
  84. 𝐉 f \mathbf{J}_{{}_{f}}
  85. 𝐉 b \mathbf{J}_{{}_{b}}
  86. ρ ρ
  87. ρ f ρ_{{}_{f}}
  88. ρ b ρ_{{}_{b}}
  89. 𝐃 \mathbf{D}
  90. 𝐇 \mathbf{H}
  91. 𝐄 \mathbf{E}
  92. 𝐁 \mathbf{B}
  93. 𝐏 \mathbf{P}
  94. 𝐏 \mathbf{P}
  95. 𝐏 \mathbf{P}
  96. 𝐏 \mathbf{P}
  97. 𝐌 \mathbf{M}
  98. 𝐏 \mathbf{P}
  99. 𝐌 \mathbf{M}
  100. 𝐃 ( 𝐫 , t ) = ε 0 𝐄 ( 𝐫 , t ) + 𝐏 ( 𝐫 , t ) \mathbf{D}(\mathbf{r},t)=\varepsilon_{0}\mathbf{E}(\mathbf{r},t)+\mathbf{P}(% \mathbf{r},t)
  101. 𝐇 ( 𝐫 , t ) = 1 μ 0 𝐁 ( 𝐫 , t ) - 𝐌 ( 𝐫 , t ) , \mathbf{H}(\mathbf{r},t)=\frac{1}{\mu_{0}}\mathbf{B}(\mathbf{r},t)-\mathbf{M}(% \mathbf{r},t),
  102. 𝐏 \mathbf{P}
  103. 𝐌 \mathbf{M}
  104. 𝐏 \mathbf{P}
  105. 𝐌 \mathbf{M}
  106. ρ b = - 𝐏 , \rho\text{b}=-\nabla\cdot\mathbf{P},
  107. 𝐉 b = × 𝐌 + 𝐏 t . \mathbf{J}\text{b}=\nabla\times\mathbf{M}+\frac{\partial\mathbf{P}}{\partial t}.
  108. ρ = ρ b + ρ f , \rho=\rho\text{b}+\rho\text{f},
  109. 𝐉 = 𝐉 b + 𝐉 f , \mathbf{J}=\mathbf{J}\text{b}+\mathbf{J}\text{f},
  110. 𝐃 \mathbf{D}
  111. 𝐇 \mathbf{H}
  112. 𝐃 \mathbf{D}
  113. 𝐄 \mathbf{E}
  114. 𝐇 \mathbf{H}
  115. 𝐁 \mathbf{B}
  116. 𝐏 \mathbf{P}
  117. 𝐌 \mathbf{M}
  118. 𝐃 = ε 0 𝐄 , 𝐇 = 1 μ 0 𝐁 \mathbf{D}=\varepsilon_{0}\mathbf{E},\quad\mathbf{H}=\frac{1}{\mu_{0}}\mathbf{B}
  119. 𝐃 = ε 𝐄 , 𝐇 = 1 μ 𝐁 \mathbf{D}=\varepsilon\mathbf{E}\,,\quad\mathbf{H}=\frac{1}{\mu}\mathbf{B}
  120. ε ε
  121. μ μ
  122. ε ε
  123. μ μ
  124. ε ε
  125. μ μ
  126. ε ε
  127. μ μ
  128. 𝐃 \mathbf{D}
  129. 𝐏 \mathbf{P}
  130. 𝐄 \mathbf{E}
  131. 𝐁 \mathbf{B}
  132. 𝐇 \mathbf{H}
  133. 𝐌 \mathbf{M}
  134. 𝐃 \mathbf{D}
  135. 𝐇 \mathbf{H}
  136. 𝐄 \mathbf{E}
  137. 𝐁 \mathbf{B}
  138. 𝐄 \mathbf{E}
  139. 𝐁 \mathbf{B}
  140. 𝐉 f = σ 𝐄 . \mathbf{J}\text{f}=\sigma\mathbf{E}\,.
  141. 𝐄 = 4 π ρ \nabla\cdot\mathbf{E}=4\pi\rho
  142. 𝐃 = 4 π ρ f \nabla\cdot\mathbf{D}=4\pi\rho\text{f}
  143. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  144. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}
  145. × 𝐁 = 1 c ( 4 π 𝐉 + 𝐄 t ) \nabla\times\mathbf{B}=\frac{1}{c}\left(4\pi\mathbf{J}+\frac{\partial\mathbf{E% }}{\partial t}\right)
  146. × 𝐇 = 1 c ( 4 π 𝐉 f + 𝐃 t ) \nabla\times\mathbf{H}=\frac{1}{c}\left(4\pi\mathbf{J}\text{f}+\frac{\partial% \mathbf{D}}{\partial t}\right)
  147. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  148. × 𝐄 + 𝐁 t = 0 \nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}=0
  149. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  150. × 𝐁 - 1 c 2 𝐄 t = μ 0 𝐉 \nabla\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=% \mu_{0}\mathbf{J}
  151. 𝐁 = × 𝐀 \mathbf{B}=\mathbf{\nabla}\times\mathbf{A}
  152. 𝐄 = - φ - 𝐀 t \mathbf{E}=-\mathbf{\nabla}\varphi-\frac{\partial\mathbf{A}}{\partial t}
  153. 2 φ + t ( 𝐀 ) = - ρ ε 0 \nabla^{2}\varphi+\frac{\partial}{\partial t}\left(\mathbf{\nabla}\cdot\mathbf% {A}\right)=-\frac{\rho}{\varepsilon_{0}}
  154. 𝐀 + ( 𝐀 + 1 c 2 φ t ) = μ 0 𝐉 \Box\mathbf{A}+\mathbf{\nabla}\left(\mathbf{\nabla}\cdot\mathbf{A}+\frac{1}{c^% {2}}\frac{\partial\varphi}{\partial t}\right)=\mu_{0}\mathbf{J}
  155. 𝐁 = × 𝐀 \mathbf{B}=\mathbf{\nabla}\times\mathbf{A}
  156. 𝐄 = - φ - 𝐀 t \mathbf{E}=-\mathbf{\nabla}\varphi-\frac{\partial\mathbf{A}}{\partial t}
  157. 𝐀 + 1 c 2 φ t = 0 \mathbf{\nabla}\cdot\mathbf{A}+\frac{1}{c^{2}}\frac{\partial\varphi}{\partial t% }=0
  158. φ = ρ ε 0 \Box\varphi=\frac{\rho}{\varepsilon_{0}}
  159. 𝐀 = μ 0 𝐉 \Box\mathbf{A}=\mu_{0}\mathbf{J}
  160. [ α F β γ ] = 0 \partial_{[\alpha}F_{\beta\gamma]}=0
  161. α F β α = μ 0 J β \partial_{\alpha}F^{\beta\alpha}=\mu_{0}J^{\beta}
  162. F α β = [ α A β ] F_{\alpha\beta}=\partial_{[\alpha}A_{\beta]}
  163. α [ β A α ] = μ 0 J β \partial_{\alpha}\partial^{[\beta}A^{\alpha]}=\mu_{0}J^{\beta}
  164. F α β = [ α A β ] F_{\alpha\beta}=\partial_{[\alpha}A_{\beta]}
  165. α A α = 0 \partial_{\alpha}A^{\alpha}=0
  166. A α = μ 0 J α \Box A^{\alpha}=\mu_{0}J^{\alpha}
  167. [ α F β γ ] = [ α F β γ ] = 0 \partial_{[\alpha}F_{\beta\gamma]}=\nabla_{[\alpha}F_{\beta\gamma]}=0
  168. α ( - g F β α ) = μ 0 J β \nabla_{\alpha}(\sqrt{-g}F^{\beta\alpha})=\mu_{0}J^{\beta}
  169. F α β = [ α A β ] = [ α A β ] F_{\alpha\beta}=\partial_{[\alpha}A_{\beta]}=\nabla_{[\alpha}A_{\beta]}
  170. α ( - g [ β A α ] ) = μ 0 J β \nabla_{\alpha}(\sqrt{-g}\nabla^{[\beta}A^{\alpha]})=\mu_{0}J^{\beta}
  171. F α β = [ α A β ] = [ α A β ] , F_{\alpha\beta}=\partial_{[\alpha}A_{\beta]}=\nabla_{[\alpha}A_{\beta]},
  172. α A α = 0 \nabla_{\alpha}A^{\alpha}=0
  173. A α - R α A β β = - μ 0 J α \Box A^{\alpha}-R^{\alpha}{}_{\beta}A^{\beta}=-\mu_{0}J^{\alpha}
  174. d F = 0 \mathrm{d}F=0
  175. d * F = μ 0 J \mathrm{d}{*}F=\mu_{0}J
  176. F = d A F=\mathrm{d}A
  177. d * d A = μ 0 J \mathrm{d}{*}\mathrm{d}A=\mu_{0}J
  178. F = d A F=\mathrm{d}A
  179. d A = 0 \mathrm{d}{\star}A=0
  180. A = μ 0 J {\star}\Box A=\mu_{0}J
  181. φ \varphi
  182. 𝐀 \mathbf{A}
  183. = 1 c 2 2 t 2 - 2 \Box=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}
  184. F α β F_{\alpha\beta}
  185. A α A_{\alpha}
  186. J α J^{\alpha}
  187. α \partial_{\alpha}
  188. x α x^{\alpha}
  189. ( x α ) = ( c t , x , y , z ) (x^{\alpha})=(ct,x,y,z)
  190. η α β = diag ( 1 , - 1 , - 1 , - 1 ) \eta_{\alpha\beta}=\mathrm{diag}(1,-1,-1,-1)
  191. = α α \Box=\partial_{\alpha}\partial^{\alpha}
  192. x α x^{\alpha}
  193. α \nabla_{\alpha}
  194. R α β R_{\alpha\beta}
  195. g α β g_{\alpha\beta}
  196. = α α \Box=\nabla_{\alpha}\nabla^{\alpha}
  197. F = F α β d x α d x β F=F_{\alpha\beta}dx^{\alpha}\wedge dx^{\beta}
  198. A = A α d x α A=A_{\alpha}dx^{\alpha}
  199. J J
  200. d \mathrm{d}
  201. * , {*},{\star}
  202. * {*}
  203. = ( - d * d - d d ) \Box=(-{\star}\mathrm{d}{*}\mathrm{d}-\mathrm{d}{\star}\mathrm{d}{\star})
  204. 𝐄 = 𝐁 = 𝟎 \mathbf{E}=\mathbf{B}=\mathbf{0}
  205. 𝐄 = c o n s t a n t \mathbf{E}=constant
  206. 𝐁 = c o n s t a n t \mathbf{B}=constant
  207. 𝐄 \mathbf{E}
  208. 𝐁 \mathbf{B}
  209. 1 / μ 0 ε 0 \scriptstyle{1/\sqrt{\mu_{0}\varepsilon_{0}}}
  210. 𝐏 \mathbf{P}
  211. 𝐌 \mathbf{M}
  212. 𝐁 0 ∇⋅\mathbf{B}≠0
  213. 𝐁 = 0 ∇⋅\mathbf{B}=0
  214. 𝐇 0 ∇⋅\mathbf{H}≠0

Maxwell–Boltzmann_distribution.html

  1. erf ( x 2 a ) - 2 π x e - x 2 / ( 2 a 2 ) a \textrm{erf}\left(\frac{x}{\sqrt{2}a}\right)-\sqrt{\frac{2}{\pi}}\frac{xe^{-x^% {2}/(2a^{2})}}{a}
  2. μ = 2 a 2 π \mu=2a\sqrt{\frac{2}{\pi}}
  3. 2 a \sqrt{2}a
  4. σ 2 = a 2 ( 3 π - 8 ) π \sigma^{2}=\frac{a^{2}(3\pi-8)}{\pi}
  5. γ 1 = 2 2 ( 16 - 5 π ) ( 3 π - 8 ) 3 / 2 \gamma_{1}=\frac{2\sqrt{2}(16-5\pi)}{(3\pi-8)^{3/2}}
  6. γ 2 = 4 ( - 96 + 40 π - 3 π 2 ) ( 3 π - 8 ) 2 \gamma_{2}=4\frac{(-96+40\pi-3\pi^{2})}{(3\pi-8)^{2}}
  7. ln ( a 2 π ) + γ - 1 2 \ln(a\sqrt{2\pi})+\gamma-\frac{1}{2}
  8. f ( v ) = ( m 2 π k T ) 3 4 π v 2 e - m v 2 2 k T , f(v)=\sqrt{\left(\frac{m}{2\pi kT}\right)^{3}}\,4\pi v^{2}e^{-\frac{mv^{2}}{2% kT}},
  9. m m
  10. k T kT
  11. v v
  12. a = k T / m a=\sqrt{kT/m}
  13. a = k T / m a=\sqrt{kT/m}
  14. k T v f ( v ) + f ( v ) ( m v 2 - 2 k T ) = 0 , kTvf^{\prime}(v)+f(v)\left(mv^{2}-2kT\right)=0,
  15. f ( 1 ) = 2 π e - m 2 k T ( m k T ) 3 / 2 f(1)=\sqrt{\frac{2}{\pi}}e^{-\frac{m}{2kT}}\left(\frac{m}{kT}\right)^{3/2}
  16. a 2 x f ( x ) + ( x 2 - 2 a 2 ) f ( x ) = 0 , a^{2}xf^{\prime}(x)+\left(x^{2}-2a^{2}\right)f(x)=0,
  17. f ( 1 ) = 2 π e - 1 2 a 2 a 3 . f(1)=\frac{\sqrt{\frac{2}{\pi}}e^{-\frac{1}{2a^{2}}}}{a^{3}}.
  18. v 2 = ( 0 v 2 f ( v ) d v ) 1 / 2 = 3 k T m = 3 R T M = 3 2 v p \sqrt{\langle v^{2}\rangle}=\left(\int_{0}^{\infty}v^{2}\,f(v)\,dv\right)^{1/2% }=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3}{2}}v_{p}
  19. 0.886 v = v p < v < v 2 = 1.085 v . 0.886\langle v\rangle=v_{p}<\langle v\rangle<\sqrt{\langle v^{2}\rangle}=1.085% \langle v\rangle.
  20. E = p 2 2 m E=\frac{p^{2}}{2m}
  21. f 𝐩 ( p x , p y , p z ) = c Z exp [ - p x 2 + p y 2 + p z 2 2 m k T ] f_{\mathbf{p}}(p_{x},p_{y},p_{z})=\frac{c}{Z}\exp\left[-\frac{p_{x}^{2}+p_{y}^% {2}+p_{z}^{2}}{2mkT}\right]
  22. c = Z ( 2 π m k T ) 3 / 2 c=\frac{Z}{(2\pi mkT)^{3/2}}
  23. p x p_{x}
  24. p y p_{y}
  25. p z p_{z}
  26. m k T mkT
  27. a = m k T a=\sqrt{mkT}
  28. d 3 𝐩 d^{3}\textbf{p}
  29. d E dE
  30. E = | 𝐩 | 2 / 2 m E=|\textbf{p}|^{2}/2m
  31. d E dE
  32. d 3 𝐩 = 4 π | 𝐩 | 2 d | 𝐩 | = 4 π m 2 m E d E . d^{3}\textbf{p}=4\pi|\textbf{p}|^{2}d|\textbf{p}|=4\pi m\sqrt{2mE}dE.
  33. E E
  34. f E ( E ) d E = 1 ( 2 π m k T ) 3 / 2 e - E / k T 4 π m 2 m E d E = 2 E π ( 1 k T ) 3 / 2 exp ( - E k T ) d E f_{E}(E)dE=\frac{1}{(2\pi mkT)^{3/2}}e^{-E/kT}4\pi m\sqrt{2mE}dE=2\sqrt{\frac{% E}{\pi}}\left(\frac{1}{kT}\right)^{3/2}\exp\left(\frac{-E}{kT}\right)dE
  35. f ϵ ( ϵ ) d ϵ = ϵ π k T exp [ - ϵ k T ] d ϵ f_{\epsilon}(\epsilon)\,d\epsilon=\sqrt{\frac{\epsilon}{\pi kT}}~{}\exp\left[% \frac{-\epsilon}{kT}\right]\,d\epsilon
  36. ϵ \epsilon
  37. f 𝐯 d 3 v = f 𝐩 ( d p d v ) 3 d 3 v f_{\mathbf{v}}d^{3}v=f_{\mathbf{p}}\left(\frac{dp}{dv}\right)^{3}d^{3}v
  38. f 𝐯 ( v x , v y , v z ) d v x d v y d v z . f_{\mathbf{v}}\left(v_{x},v_{y},v_{z}\right)\,dv_{x}\,dv_{y}\,dv_{z}.
  39. v x v_{x}
  40. v y v_{y}
  41. v z v_{z}
  42. k T m \frac{kT}{m}
  43. f v ( v x , v y , v z ) = f v ( v x ) f v ( v y ) f v ( v z ) f_{v}\left(v_{x},v_{y},v_{z}\right)=f_{v}(v_{x})f_{v}(v_{y})f_{v}(v_{z})
  44. f v ( v i ) = m 2 π k T exp [ - m v i 2 2 k T ] . f_{v}(v_{i})=\sqrt{\frac{m}{2\pi kT}}\exp\left[\frac{-mv_{i}^{2}}{2kT}\right].
  45. μ v x = μ v y = μ v z = 0 \mu_{v_{x}}=\mu_{v_{y}}=\mu_{v_{z}}=0
  46. σ v x = σ v y = σ v z = k T m \sigma_{v_{x}}=\sigma_{v_{y}}=\sigma_{v_{z}}=\sqrt{\frac{kT}{m}}
  47. μ 𝐯 = 𝟎 \mu_{\mathbf{v}}={\mathbf{0}}
  48. σ 𝐯 = 3 k T m \sigma_{\mathbf{v}}=\sqrt{\frac{3kT}{m}}
  49. v = v x 2 + v y 2 + v z 2 v=\sqrt{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}
  50. d v x d v y d v z = v 2 sin θ d v d θ d ϕ dv_{x}\,dv_{y}\,dv_{z}=v^{2}\sin\theta\,dv\,d\theta\,d\phi
  51. ϕ \phi
  52. θ \theta
  53. 2 π 2\pi
  54. π \pi

MD5.html

  1. F ( B , C , D ) = ( B C ) ( ¬ B D ) F(B,C,D)=(B\wedge{C})\vee(\neg{B}\wedge{D})
  2. G ( B , C , D ) = ( B D ) ( C ¬ D ) G(B,C,D)=(B\wedge{D})\vee(C\wedge\neg{D})
  3. H ( B , C , D ) = B C D H(B,C,D)=B\oplus C\oplus D
  4. I ( B , C , D ) = C ( B ¬ D ) I(B,C,D)=C\oplus(B\vee\neg{D})
  5. , , , ¬ \oplus,\wedge,\vee,\neg

Mean.html

  1. μ = x P ( x ) \mu=\sum xP(x)
  2. 2 n 2^{n}
  3. 1 2 n \tfrac{1}{2^{n}}
  4. x ¯ \bar{x}
  5. x ¯ \bar{x}
  6. μ \mu
  7. μ x \mu_{x}
  8. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  9. x ¯ \bar{x}
  10. x ¯ = x 1 + x 2 + + x n n \bar{x}=\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}
  11. 4 + 36 + 45 + 50 + 75 5 = 210 5 = 42. \frac{4+36+45+50+75}{5}=\frac{210}{5}=42.
  12. x ¯ = ( i = 1 n x i ) 1 n \bar{x}=\left(\prod_{i=1}^{n}{x_{i}}\right)^{\tfrac{1}{n}}
  13. ( 4 × 36 × 45 × 50 × 75 ) / 5 1 = 24 300 000 5 = 30. (4\times 36\times 45\times 50\times 75)^{{}^{1}/_{5}}=\sqrt[5]{24\;300\;000}=30.
  14. x ¯ = n ( i = 1 n 1 x i ) - 1 \bar{x}=n\cdot\left(\sum_{i=1}^{n}\frac{1}{x_{i}}\right)^{-1}
  15. 5 1 4 + 1 36 + 1 45 + 1 50 + 1 75 = 5 1 3 = 15. \frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45}+\tfrac{1}{50}+\tfrac{1}{75}}% =\frac{5}{\;\tfrac{1}{3}\;}=15.
  16. A M G M H M AM\geq GM\geq HM\,
  17. x ¯ ( m ) = ( 1 n i = 1 n x i m ) 1 m \bar{x}(m)=\left(\frac{1}{n}\cdot\sum_{i=1}^{n}{x_{i}^{m}}\right)^{\tfrac{1}{m}}
  18. m m\rightarrow\infty
  19. x i x_{i}
  20. m = 2 m=2
  21. m = 1 m=1
  22. m 0 m\rightarrow 0
  23. m = - 1 m=-1
  24. m - m\rightarrow-\infty
  25. x i x_{i}
  26. x ¯ = f - 1 ( 1 n i = 1 n f ( x i ) ) \bar{x}=f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^{n}{f(x_{i})}}\right)
  27. f ( x ) = x f(x)=x
  28. f ( x ) = 1 x f(x)=\frac{1}{x}
  29. f ( x ) = x m f(x)=x^{m}
  30. f ( x ) = ln x f(x)=\ln x
  31. x ¯ = i = 1 n w i x i i = 1 n w i . \bar{x}=\frac{\sum_{i=1}^{n}{w_{i}\cdot x_{i}}}{\sum_{i=1}^{n}{w_{i}}}.
  32. w i w_{i}
  33. x ¯ = 2 n i = ( n / 4 ) + 1 3 n / 4 x i \bar{x}={2\over n}\sum_{i=(n/4)+1}^{3n/4}{x_{i}}
  34. y ave y_{\,\text{ave}}
  35. f ( x ) f(x)
  36. y ave ( a , b ) = a b f ( x ) d x b - a y_{\,\text{ave}}(a,b)=\frac{\int\limits_{a}^{b}\!f(x)\,dx\,}{b-a}
  37. x ¯ N { μ , σ 2 n } . \bar{x}\thicksim N\left\{\mu,\frac{\sigma^{2}}{n}\right\}.

Mean_time_between_failures.html

  1. Mean time between failures = MTBF = ( start of downtime - start of uptime ) number of failures . \,\text{Mean time between failures}=\,\text{MTBF}=\frac{\sum{(\,\text{start of% downtime}-\,\text{start of uptime})}}{\,\text{number of failures}}.\!
  2. MTBF = θ . \,\text{MTBF}=\theta.\!
  3. MTBF = 0 t f ( t ) d t \,\text{MTBF}=\int_{0}^{\infty}tf(t)\,dt\!
  4. 0 f ( t ) d t = 1. \int_{0}^{\infty}f(t)\,dt=1.\!
  5. MTTF B 10 0.1 n o p , \,\text{MTTF}\approx\frac{B_{10}}{0.1n_{op}},
  6. MTTFd B 10 d 0.1 n o p , \,\text{MTTFd}\approx\frac{B_{10d}}{0.1n_{op}},

Mean_time_between_outages.html

  1. M T B O = M T B F 1 - F F A S MTBO=\frac{MTBF}{1-FFAS}

Mean_value_theorem.html

  1. f ( c ) = f ( b ) - f ( a ) b - a . f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}.
  2. lim h 0 f ( x + h ) - f ( x ) h \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
  3. g ( a ) = g ( b ) f ( a ) - r a = f ( b ) - r b r ( b - a ) = f ( b ) - f ( a ) r = f ( b ) - f ( a ) b - a \begin{aligned}\displaystyle g(a)=g(b)&\displaystyle\iff f(a)-ra=f(b)-rb\\ &\displaystyle\iff r(b-a)=f(b)-f(a)\\ &\displaystyle\iff r=\frac{f(b)-f(a)}{b-a}\cdot\end{aligned}
  4. f ( c ) = g ( c ) + r = 0 + r = f ( b ) - f ( a ) b - a f^{\prime}(c)=g^{\prime}(c)+r=0+r=\frac{f(b)-f(a)}{b-a}
  5. 0 = f ( c ) = f ( b ) - f ( a ) b - a . 0=f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}.
  6. ( f ( b ) - f ( a ) ) g ( c ) = ( g ( b ) - g ( a ) ) f ( c ) . (f(b)-f(a))g\,^{\prime}(c)=(g(b)-g(a))f\,^{\prime}(c).\,
  7. f ( c ) g ( c ) = f ( b ) - f ( a ) g ( b ) - g ( a ) \frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\cdot
  8. [ a , b ] 2 t ( f ( t ) , g ( t ) ) , \begin{array}[]{rcl}[a,b]&\longrightarrow&\mathbb{R}^{2}\\ t&\mapsto&\bigl(f(t),g(t)\bigr),\end{array}
  9. t ( t 3 , 1 - t 2 ) , t\mapsto(t^{3},1-t^{2}),
  10. h ( a ) = h ( b ) f ( a ) - r g ( a ) = f ( b ) - r g ( b ) r ( g ( b ) - g ( a ) ) = f ( b ) - f ( a ) r = f ( b ) - f ( a ) g ( b ) - g ( a ) . \begin{aligned}\displaystyle h(a)=h(b)&\displaystyle\iff f(a)-r\,g(a)=f(b)-r\,% g(b)\\ &\displaystyle\iff r\,(g(b)-g(a))=f(b)-f(a)\\ &\displaystyle\iff r=\frac{f(b)-f(a)}{g(b)-g(a)}.\end{aligned}
  11. 0 = h ( c ) = f ( c ) - r g ( c ) ( g ( b ) - g ( a ) ) f ( c ) = ( g ( b ) - g ( a ) ) r g ( c ) = ( f ( b ) - f ( a ) ) g ( c ) 0=h^{\prime}(c)=f^{\prime}(c)-r\,g^{\prime}(c)\Rightarrow(g(b)-g(a))\,f^{% \prime}(c)=(g(b)-g(a))\,r\,g^{\prime}(c)=(f(b)-f(a))\,g^{\prime}(c)
  12. f f
  13. g g
  14. h h
  15. ( a , b ) (a,b)
  16. [ a , b ] [a,b]
  17. D ( x ) = | f ( x ) g ( x ) h ( x ) f ( a ) g ( a ) h ( a ) f ( b ) g ( b ) h ( b ) | D(x)=\left|\begin{array}[]{ccc}f(x)&g(x)&h(x)\\ f(a)&g(a)&h(a)\\ f(b)&g(b)&h(b)\end{array}\right|
  18. c ( a , b ) c\in(a,b)
  19. D ( c ) = 0 D^{\prime}(c)=0
  20. D ( x ) = | f ( x ) g ( x ) h ( x ) f ( a ) g ( a ) h ( a ) f ( b ) g ( b ) h ( b ) | D^{\prime}(x)=\left|\begin{array}[]{ccc}f^{\prime}(x)&g^{\prime}(x)&h^{\prime}% (x)\\ f(a)&g(a)&h(a)\\ f(b)&g(b)&h(b)\end{array}\right|
  21. h ( x ) = 1 h(x)=1
  22. h ( x ) = 1 h(x)=1
  23. g ( x ) = x g(x)=x
  24. D ( a ) D(a)
  25. D ( b ) D(b)
  26. D ( a ) = D ( b ) = 0 D(a)=D(b)=0
  27. c ( a , b ) c\in(a,b)
  28. D ( c ) = 0 D^{\prime}(c)=0
  29. g ( 1 ) - g ( 0 ) = g ( c ) g(1)-g(0)=g^{\prime}(c)\!
  30. f ( y ) - f ( x ) = f ( ( 1 - c ) x + c y ) ( y - x ) f(y)-f(x)=\nabla f((1-c)x+cy)\cdot(y-x)
  31. | f ( y ) - f ( x ) | | f ( ( 1 - c ) x + c y ) | | y - x | . |f(y)-f(x)|\leq|\nabla f((1-c)x+cy)|\,|y-x|.
  32. | g ( y ) | = | g ( y ) - g ( x ) | ( 0 ) | y - x | = 0 |g(y)|=|g(y)-g(x)|\leq(0)|y-x|=0
  33. f i ( x + h ) - f i ( x ) = f i ( x + t i h ) h . f_{i}(x+h)-f_{i}(x)=\nabla f_{i}(x+t_{i}h)\cdot h.\,
  34. f i ( x + h ) - f i ( x ) = f i ( x + t * h ) h . f_{i}(x+h)-f_{i}(x)=\nabla f_{i}(x+t^{*}h)\cdot h.\,
  35. f 1 ( x ) = - sin ( x ) \,f_{1}^{\prime}(x)=-\sin(x)
  36. f 2 ( x ) = cos ( x ) \,f_{2}^{\prime}(x)=\cos(x)
  37. f ( x + h ) - f ( x ) = f ( x + t * h ) h . f(x+h)-f(x)=f^{\prime}(x+t^{*}h)\cdot h.\,
  38. f ( x + h ) - f ( x ) = x x + h f ( u ) d u = ( 0 1 f ( x + t h ) d t ) h . f(x+h)-f(x)=\int_{x}^{x+h}f^{\prime}(u)du=\left(\int_{0}^{1}f^{\prime}(x+th)\,% dt\right)\cdot h.
  39. 0 1 f ( x + t h ) d t . \int_{0}^{1}f^{\prime}(x+th)\,dt.
  40. (*) f ( x + h ) - f ( x ) = ( 0 1 D f ( x + t h ) d t ) h , \,\text{(*)}\qquad f(x+h)-f(x)=\left(\int_{0}^{1}Df(x+th)\,dt\right)\cdot h,
  41. (**) f ( x + h ) - f ( x ) M h . \,\text{(**)}\qquad\|f(x+h)-f(x)\|\leq M\|h\|.
  42. f i ( x + h ) - f i ( x ) = g i ( 1 ) - g i ( 0 ) = 0 1 g i ( t ) d t = 0 1 ( j = 1 n f i x j ( x + t h ) h j ) d t = j = 1 n ( 0 1 f i x j ( x + t h ) d t ) h j . f_{i}(x+h)-f_{i}(x)\,=\,g_{i}(1)-g_{i}(0)=\int_{0}^{1}g_{i}^{\prime}(t)dt=\int% _{0}^{1}\left(\sum_{j=1}^{n}\frac{\partial f_{i}}{\partial x_{j}}(x+th)h_{j}% \right)\,dt=\sum_{j=1}^{n}\left(\int_{0}^{1}\frac{\partial f_{i}}{\partial x_{% j}}(x+th)\,dt\right)h_{j}.
  43. f i x j \frac{\partial f_{i}}{\partial x_{j}}
  44. f ( x + h ) - f ( x ) = 0 1 ( D f ( x + t h ) h ) d t 0 1 D f ( x + t h ) h d t M h . \|f(x+h)-f(x)\|=\left\|\int_{0}^{1}(Df(x+th)\cdot h)\,dt\right\|\leq\int_{0}^{% 1}\|Df(x+th)\|\cdot\|h\|\,dt\leq M\|h\|.
  45. (***) a b v ( t ) d t a b v ( t ) d t . \,\text{(***)}\qquad\left\|\int_{a}^{b}v(t)\,dt\right\|\leq\int_{a}^{b}\|v(t)% \|\,dt.
  46. u := a b v ( t ) d t . u:=\int_{a}^{b}v(t)\,dt.
  47. u 2 = u , u = a b v ( t ) d t , u = a b v ( t ) , u d t a b v ( t ) u d t = u a b v ( t ) d t , \|u\|^{2}=\langle u,u\rangle=\left\langle\int_{a}^{b}v(t)dt,u\right\rangle=% \int_{a}^{b}\langle v(t),u\rangle\,dt\leq\int_{a}^{b}\|v(t)\|\cdot\|u\|\,dt=\|% u\|\int_{a}^{b}\|v(t)\|\,dt,
  48. u a b v ( t ) d t \|u\|\leq\int_{a}^{b}\|v(t)\|\,dt
  49. φ \varphi
  50. a b G ( t ) φ ( t ) d t = G ( x ) a b φ ( t ) d t . \int_{a}^{b}G(t)\varphi(t)\,dt=G(x)\int_{a}^{b}\varphi(t)\,dt.
  51. a b G ( t ) d t = G ( x ) ( b - a ) . \int_{a}^{b}G(t)\,dt=\ G(x)(b-a).\,
  52. 1 b - a a b G ( t ) d t = G ( x ) , \frac{1}{b-a}\int_{a}^{b}G(t)\,dt=\ G(x),
  53. φ ( t ) 0 \varphi(t)\geq 0
  54. φ ( t ) \varphi(t)
  55. m I = a b m φ ( t ) d t a b G ( t ) φ ( t ) d t a b M φ ( t ) d t = M I , mI=\int_{a}^{b}m\varphi(t)\,dt\leq\int^{b}_{a}G(t)\varphi(t)\,dt\leq\int_{a}^{% b}M\varphi(t)\,dt=MI,
  56. I := a b φ ( t ) d t I:=\int^{b}_{a}\varphi(t)\,dt
  57. φ ( t ) \varphi(t)
  58. m 1 I a b G ( t ) φ ( t ) d t M . m\leq\frac{1}{I}\int^{b}_{a}G(t)\varphi(t)\,dt\leq M.
  59. G ( x ) = 1 I a b G ( t ) φ ( t ) d t . G(x)=\frac{1}{I}\int^{b}_{a}G(t)\varphi(t)\,dt.
  60. a b G ( t ) φ ( t ) d t = G ( a + ) a x φ ( t ) d t . \int_{a}^{b}G(t)\varphi(t)\,dt=G(a^{+})\int_{a}^{x}\varphi(t)\,dt.
  61. G ( a + ) G(a^{+})
  62. lim x a + G ( x ) {\lim_{x\to a^{+}}G(x)}
  63. a b G ( t ) φ ( t ) d t = G ( a + ) a x φ ( t ) d t + G ( b - ) x b φ ( t ) d t . \int_{a}^{b}G(t)\varphi(t)\,dt=G(a^{+})\int_{a}^{x}\varphi(t)\,dt+G(b^{-})\int% _{x}^{b}\varphi(t)\,dt.
  64. G G
  65. G G
  66. n n
  67. G : [ 0 , 2 π ] n 2 G:[0,2\pi]^{n}\to\mathbb{R}^{2}
  68. G ( x 1 , , x n ) = ( s i n ( x 1 + + x n ) , c o s ( x 1 + + x n ) ) G(x_{1},\cdots,x_{n})=\left(sin(x_{1}+\cdots+x_{n}),cos(x_{1}+\cdots+x_{n})\right)
  69. G G
  70. [ 0 , 2 π ] n G ( x 1 , , x n ) d x 1 d x n = ( 0 , 0 ) \int_{[0,2\pi]^{n}}G(x_{1},\cdots,x_{n})dx_{1}\cdots dx_{n}=(0,0)
  71. G = ( 0 , 0 ) G=(0,0)
  72. | G | = 1 |G|=1
  73. f Z ( x ) = Pr ( Y > x ) - Pr ( X > x ) E [ Y ] - E [ X ] , x 0. f_{Z}(x)={\Pr(Y>x)-\Pr(X>x)\over{\rm E}[Y]-{\rm E}[X]}\,,\qquad x\geq 0.
  74. E [ g ( Y ) ] - E [ g ( X ) ] = E [ g ( Z ) ] [ E ( Y ) - E ( X ) ] . {\rm E}[g(Y)]-{\rm E}[g(X)]={\rm E}[g^{\prime}(Z)]\,[{\rm E}(Y)-{\rm E}(X)].
  75. Re ( f ( u ) ) = Re ( f ( b ) - f ( a ) b - a ) , \mathrm{Re}(f^{\prime}(u))=\mathrm{Re}\left(\frac{f(b)-f(a)}{b-a}\right),
  76. Im ( f ( v ) ) = Im ( f ( b ) - f ( a ) b - a ) . \mathrm{Im}(f^{\prime}(v))=\mathrm{Im}\left(\frac{f(b)-f(a)}{b-a}\right).

Measurable_function.html

  1. [ a , ] [a,\infty]
  2. f - 1 ( E ) := { x X | f ( x ) E } Σ , E T . f^{-1}(E):=\{x\in X|\;f(x)\in E\}\in\Sigma,\;\;\forall E\in T.
  3. f : ( X , Σ ) ( Y , T ) f\colon(X,\Sigma)\rightarrow(Y,T)
  4. f : ( 𝐑 , ) ( 𝐑 , ) f:(\mathbf{R},\mathcal{L})\to(\mathbf{R},\mathcal{B})
  5. \mathcal{L}
  6. \mathcal{B}
  7. Y π X Y\stackrel{\pi}{\to}X
  8. f : ( 𝐑 , ) ( 𝐂 , 𝐂 ) f:(\mathbf{R},\mathcal{L})\to(\mathbf{C},\mathcal{B}_{\mathbf{C}})
  9. \mathcal{L}
  10. 𝐂 \mathcal{B}_{\mathbf{C}}
  11. f : X 𝐑 f:X\to\mathbf{R}
  12. f f
  13. { f > α } = { x X : f ( x ) > α } \{f>\alpha\}=\{x\in X:f(x)>\alpha\}
  14. α \alpha
  15. { f α } , { f < α } , { f α } \{f\geq\alpha\},\{f<\alpha\},\{f\leq\alpha\}
  16. α \alpha
  17. f : X f:X\to\mathbb{C}
  18. 𝟏 A ( x ) = { 1 if x A 0 otherwise \mathbf{1}_{A}(x)=\begin{cases}1&\,\text{ if }x\in A\\ 0&\,\text{ otherwise}\end{cases}

Measure_(mathematics).html

  1. n n
  2. 0 , 11 0,11
  3. X X
  4. σ σ
  5. μ μ
  6. X X
  7. Σ Σ
  8. σ σ
  9. X X
  10. μ μ
  11. Σ Σ
  12. E E
  13. Σ Σ
  14. μ ( E ) 0 μ(E)≥0
  15. μ ( ) = 0 μ(∅)=0
  16. σ σ
  17. { E i } i \{E_{i}\}_{i\in\mathbb{N}}
  18. Σ Σ
  19. μ ( k = 1 E k ) = k = 1 μ ( E k ) \mu(\bigcup_{k=1}^{\infty}E_{k})=\sum_{k=1}^{\infty}\mu(E_{k})
  20. E E
  21. μ ( E ) = μ ( E ) = μ ( E ) + μ ( ) \mu(E)=\mu(E\cup\varnothing)=\mu(E)+\mu(\varnothing)
  22. μ ( ) = μ ( E ) - μ ( E ) = 0 \mu(\varnothing)=\mu(E)-\mu(E)=0
  23. μ μ
  24. ± ±∞
  25. μ μ
  26. ( X , Σ ) (X,Σ)
  27. Σ Σ
  28. ( X , Σ X ) \left(X,\Sigma_{X}\right)
  29. ( Y , Σ Y ) \left(Y,\Sigma_{Y}\right)
  30. f : X Y f:X\to Y
  31. Y Y
  32. B Σ Y B\in\Sigma_{Y}
  33. X X
  34. f ( - 1 ) ( B ) Σ X f^{(-1)}(B)\in\Sigma_{X}
  35. ( X , Σ , μ ) (X,Σ,μ)
  36. μ ( X ) = 1 μ(X)=1
  37. μ ( S ) μ(S)
  38. S S
  39. 𝐑 \mathbf{R}
  40. 𝐑 \mathbf{R}
  41. μ ( 0 , 11 ) = 1 μ(0,11)=1
  42. S S
  43. a a
  44. μ μ
  45. μ ( E 1 ) μ ( E 2 ) . \mu(E_{1})\leq\mu(E_{2}).
  46. μ μ
  47. Σ Σ
  48. μ ( i = 1 E i ) i = 1 μ ( E i ) . \mu\left(\bigcup_{i=1}^{\infty}E_{i}\right)\leq\sum_{i=1}^{\infty}\mu(E_{i}).
  49. n n
  50. μ ( i = 1 E i ) = lim i μ ( E i ) . \mu\left(\bigcup_{i=1}^{\infty}E_{i}\right)=\lim_{i\to\infty}\mu(E_{i}).
  51. μ μ
  52. μ ( i = 1 E i ) = lim i μ ( E i ) . \mu\left(\bigcap_{i=1}^{\infty}E_{i}\right)=\lim_{i\to\infty}\mu(E_{i}).
  53. n 𝐍 n∈\mathbf{N}
  54. ( X , Σ , μ ) (X,Σ,μ)
  55. μ ( X ) μ(X)
  56. μ μ
  57. 1 μ ( X ) μ \frac{1}{\mu(X)}\mu
  58. μ μ
  59. X X
  60. k k , k + 1 kk,k+1
  61. k k
  62. X X
  63. μ ( X ) = 0 μ(X)=0
  64. Y Y
  65. X X
  66. X X
  67. Y Y
  68. μ ( Y ) μ(Y)
  69. μ ( X ) μ(X)
  70. I I
  71. i I i\in I
  72. i I r i = sup { i J r i : | J | < 0 , J I } . \sum_{i\in I}r_{i}=\sup\left\{\sum_{i\in J}r_{i}:|J|<\aleph_{0},J\subseteq I% \right\}.
  73. μ μ
  74. Σ Σ
  75. κ κ
  76. X α X_{\alpha}
  77. α λ X α Σ \bigcup_{\alpha\in\lambda}X_{\alpha}\in\Sigma
  78. μ ( α λ X α ) = α λ μ ( X α ) . \mu\left(\bigcup_{\alpha\in\lambda}X_{\alpha}\right)=\sum_{\alpha\in\lambda}% \mu\left(X_{\alpha}\right).
  79. κ κ

Measures_of_national_income_and_output.html

  1. GDP = C + G + I + ( X - M ) \mathrm{GDP}=C+G+I+\left(\mathrm{X}-M\right)

Mechanical_advantage.html

  1. M A = F B F A = a b . MA=\frac{F_{B}}{F_{A}}=\frac{a}{b}.
  2. P = T A ω A = T B ω B , P=T_{A}\omega_{A}=T_{B}\omega_{B},\!
  3. M A = T B T A = ω A ω B . MA=\frac{T_{B}}{T_{A}}=\frac{\omega_{A}}{\omega_{B}}.
  4. v = r A ω A = r B ω B , v=r_{A}\omega_{A}=r_{B}\omega_{B},\!
  5. ω A ω B = r B r A = N B N A . \frac{\omega_{A}}{\omega_{B}}=\frac{r_{B}}{r_{A}}=\frac{N_{B}}{N_{A}}.
  6. M A = r B r A = N B N A . MA=\frac{r_{B}}{r_{A}}=\frac{N_{B}}{N_{A}}.
  7. v = r A ω A = r B ω B , v=r_{A}\omega_{A}=r_{B}\omega_{B},\!
  8. ω A ω B = r B r A = N B N A . \frac{\omega_{A}}{\omega_{B}}=\frac{r_{B}}{r_{A}}=\frac{N_{B}}{N_{A}}.
  9. M A = T B T A = N B N A . MA=\frac{T_{B}}{T_{A}}=\frac{N_{B}}{N_{A}}.
  10. M A = T B T A = r B r A . MA=\frac{T_{B}}{T_{A}}=\frac{r_{B}}{r_{A}}.
  11. M A = F B F A = 7 13 = 0.54. MA=\frac{F_{B}}{F_{A}}=\frac{7}{13}=0.54.
  12. L = 2 R + S + K , L=2R+S+K,\!
  13. L ˙ = 2 R ˙ + S ˙ = 0 , \dot{L}=2\dot{R}+\dot{S}=0,
  14. S ˙ = - 2 R ˙ . \dot{S}=-2\dot{R}.
  15. V A V B = S ˙ - R ˙ = 2 , \frac{V_{A}}{V_{B}}=\frac{\dot{S}}{-\dot{R}}=2,
  16. F A V A = F B V B . F_{A}V_{A}=F_{B}V_{B}.\!
  17. M A = F B F A = V A V B = 2. MA=\frac{F_{B}}{F_{A}}=\frac{V_{A}}{V_{B}}=2.\!
  18. M A = F B F A = V A V B = n . MA=\frac{F_{B}}{F_{A}}=\frac{V_{A}}{V_{B}}=n.\!
  19. P = F i n v i n = F o u t v o u t . P=F_{in}v_{in}=F_{out}v_{out}.
  20. I M A = F o u t F i n . IMA=\frac{F_{out}}{F_{in}}.
  21. I M A = F o u t F i n = v i n v o u t . IMA=\frac{F_{out}}{F_{in}}=\frac{v_{in}}{v_{out}}.
  22. A M A = F o u t F i n , AMA=\frac{F_{out}}{F_{in}},
  23. η = A M A I M A . \eta=\frac{AMA}{IMA}.

Mechanics.html

  1. F = m a F=ma
  2. F = γ m a F=\gamma ma
  3. γ \gamma

Median.html

  1. μ 1 / 2 , \mu_{1/2},
  2. P ( X m ) 1 2 and P ( X m ) 1 2 \operatorname{P}(X\leq m)\geq\frac{1}{2}\,\text{ and }\operatorname{P}(X\geq m% )\geq\frac{1}{2}\,\!
  3. ( - , m ] d F ( x ) 1 2 and [ m , ) d F ( x ) 1 2 \int_{(-\infty,m]}dF(x)\geq\frac{1}{2}\,\text{ and }\int_{[m,\infty)}dF(x)\geq% \frac{1}{2}\,\!
  4. P ( X m ) = P ( X m ) = - m f ( x ) d x = 1 2 . \operatorname{P}(X\leq m)=\operatorname{P}(X\geq m)=\int_{-\infty}^{m}f(x)\,dx% =\frac{1}{2}.\,\!
  5. E ( | X - c | ) E(\left|X-c\right|)\,
  6. X ~ \tilde{X}
  7. X ¯ \bar{X}
  8. | X ~ - X ¯ | σ ( 3 / 5 ) 1 / 2 \frac{\left|\tilde{X}-\bar{X}\right|}{\sigma}\leq(3/5)^{1/2}
  9. | X ~ - mode | σ 3 1 / 2 . \frac{\left|\tilde{X}-\mathrm{mode}\right|}{\sigma}\leq 3^{1/2}.
  10. | μ - m | = | E ( X - m ) | \displaystyle\left|\mu-m\right|=\left|\mathrm{E}(X-m)\right|
  11. a E ( | X - a | ) . a\mapsto\mathrm{E}(\left|X-a\right|).\,
  12. μ - m = E ( X - m ) \displaystyle\left\|\mu-m\right\|=\left\|\mathrm{E}(X-m)\right\|
  13. a E ( X - a ) . a\mapsto\mathrm{E}(\left\|X-a\right\|).\,
  14. f ( E ( x ) ) E ( f ( x ) ) f(E(x))\leq E(f(x))
  15. f ( m ) Median ( f ( x ) ) f(m)\leq\operatorname{Median}(f(x))
  16. f - 1 ( ( - , t ] ) = { x R | f ( x ) t } f^{-1}((-\infty,t])=\{x\in R|f(x)\leq t\}
  17. f ( x ) f(x)
  18. m m
  19. 1 4 n f ( m ) 2 \frac{1}{4nf(m)^{2}}
  20. m m
  21. n n
  22. p p
  23. p p
  24. p p
  25. p ( 1 - p ) n f ( x p ) 2 \frac{p(1-p)}{nf(x_{p})^{2}}
  26. f ( x p ) f(x_{p})
  27. p p
  28. ( 2 f ( x ) ) - 2 (2f(x))^{-2}
  29. n - 1 2 ( ν - m ) n^{-\frac{1}{2}}(\nu-m)
  30. ν \nu
  31. k k
  32. n - 1 4 n^{-\frac{1}{4}}
  33. N = 2 n + 1 N=2n+1
  34. 4 n π ( 2 n + 1 ) \frac{4n}{\pi(2n+1)}
  35. n n
  36. 2 π . \frac{2}{\pi}.
  37. C D = 1 n | m - x | m CD=\frac{1}{n}\frac{\sum|m-x|}{m}
  38. exp [ log ( t a t b ) - z α ( v a r [ log ( t a t b ) ] ) 0.5 ] \exp\left[\log\left(\frac{t_{a}}{t_{b}}\right)-z_{\alpha}\left(var\left[\log% \left(\frac{t_{a}}{t_{b}}\right)\right]\right)^{0.5}\right]
  39. v a r [ log ( t a ) ] = ( s a 2 t a 2 + ( x a - x ¯ t a ) 2 - 1 ) n var[\log(t_{a})]=\frac{\left(\frac{s_{a}^{2}}{t_{a}^{2}}+\left(\frac{x_{a}-% \bar{x}}{t_{a}}\right)^{2}-1\right)}{n}
  40. v a r [ log ( t a / t b ) ] = v a r [ log ( t a ) ] + v a r [ log ( t b ) ] - 2 r ( v a r [ log ( t a ) ] v a r [ log ( t b ) ] ) 0.5 var[\log(t_{a}/t_{b})]=var[\log(t_{a})]+var[\log(t_{b})]-2r(var[\log(t_{a})]% var[\log(t_{b})])^{0.5}
  41. d i a = | x i a - x ¯ a | d_{ia}=|x_{ia}-\bar{x}_{a}|
  42. d i b = | x i b - x ¯ b | d_{ib}=|x_{ib}-\bar{x}_{b}|
  43. a E ( X - a ) , a\mapsto\mathrm{E}(\left\|X-a\right\|),\,
  44. x x
  45. y y
  46. x x

Meissner_effect.html

  1. χ v \chi_{v}
  2. λ M := h / ( M c ) \lambda_{M}:=h/(Mc)
  3. λ M \lambda_{M}

Mel_scale.html

  1. f f
  2. m m
  3. m = 2595 log 10 ( 1 + f 700 ) m=2595\log_{10}\left(1+\frac{f}{700}\right)
  4. m = 2595 log 10 ( 1 + f 700 ) = 1127 log e ( 1 + f 700 ) m=2595\log_{10}\left(1+\frac{f}{700}\right)=1127\log_{e}\left(1+\frac{f}{700}\right)
  5. f = 700 ( 10 m / 2595 - 1 ) = 700 ( e m / 1127 - 1 ) f=700(10^{m/2595}-1)=700(e^{m/1127}-1)
  6. m = 1000 log ( 2 ) log ( 1 + f 1000 ) m=\frac{1000}{\log(2)}\log\left(1+\frac{f}{1000}\right)
  7. m = 2410 log 10 ( 1.6 × 10 - 3 f + 1 ) m=2410\log_{10}(1.6\times 10^{-3}f+1)

Melting.html

  1. T g = H d S d + R ln ( 1 - f c f c ) , T_{g}=\frac{H_{d}}{S_{d}+R\ln(\frac{1-f_{c}}{f_{c}})},

Melting_point.html

  1. Δ S = Δ H T \Delta S=\frac{\Delta H}{T}
  2. E = 4 π 2 m ν 2 u 2 = k B T E=4\pi^{2}m\nu^{2}~{}u^{2}=k_{B}T
  3. T m = 4 π 2 m ν 2 c 2 a 2 k B . T_{m}=\cfrac{4\pi^{2}m\nu^{2}c^{2}a^{2}}{k_{B}}.
  4. T m = m ν 2 c 2 a 2 k B . T_{m}=\cfrac{m\nu^{2}c^{2}a^{2}}{k_{B}}.
  5. T m = 2 π m c 2 a 2 θ D 2 k B h 2 T_{m}=\cfrac{2\pi mc^{2}a^{2}\theta_{D}^{2}k_{B}}{h^{2}}

Mercalli_intensity_scale.html

  1. M w M_{w}

Mercator_projection.html

  1. tan α R cos ϕ δ λ R δ ϕ , tan β = δ x δ y , \tan\alpha\approx\frac{R\cos\phi\,\delta\lambda}{R\,\delta\phi},\qquad\qquad% \tan\beta=\frac{\delta x}{\delta y},
  2. k ( ϕ ) = P M P M = δ x R cos ϕ δ λ , \quad k(\phi)\;=\;\frac{P^{\prime}M^{\prime}}{PM}\;=\;\frac{\delta x}{R\cos% \phi\,\delta\lambda},
  3. h ( ϕ ) = P K P K = δ y R δ ϕ . \quad h(\phi)\;=\;\frac{P^{\prime}K^{\prime}}{PK}\;=\;\frac{\delta y}{R\delta% \phi\,}.
  4. tan β = R sec ϕ y ( ϕ ) tan α , k = sec ϕ , h = y ( ϕ ) R . \tan\beta=\frac{R\sec\phi}{y^{\prime}(\phi)}\tan\alpha\,,\qquad k=\sec\phi\,,% \qquad h=\frac{y^{\prime}(\phi)}{R}.
  5. y ( ϕ ) = R sec ϕ , y^{\prime}(\phi)=R\sec\phi,
  6. x = R ( λ - λ 0 ) , y = R ln [ tan ( π 4 + ϕ 2 ) ] . \begin{aligned}\displaystyle x&\displaystyle=R(\lambda-\lambda_{0}),\qquad y&% \displaystyle=R\ln\left[\tan\left(\frac{\pi}{4}+\frac{\phi}{2}\right)\right].% \end{aligned}
  7. λ = λ 0 + x R , ϕ = 2 tan - 1 [ exp ( y R ) ] - π 2 . \begin{aligned}\displaystyle\lambda&\displaystyle=\lambda_{0}+\frac{x}{R},% \qquad\phi&\displaystyle=2\tan^{-1}\left[\exp\left(\frac{y}{R}\right)\right]-% \frac{\pi}{2}\,.\end{aligned}
  8. y = R 2 ln [ 1 + sin ϕ 1 - sin ϕ ] = R ln [ 1 + sin ϕ cos ϕ ] = R ln ( sec ϕ + tan ϕ ) = R tanh - 1 ( sin ϕ ) = R sinh - 1 ( tan ϕ ) = R cosh - 1 ( sec ϕ ) = R gd ( ϕ ) - 1 . \begin{aligned}\displaystyle y&\displaystyle=&\displaystyle\frac{R}{2}\ln\left% [\frac{1+\sin\phi}{1-\sin\phi}\right]&\displaystyle=&\displaystyle{R}\ln\left[% \frac{1+\sin\phi}{\cos\phi}\right]&\displaystyle=R\ln\left(\sec\phi+\tan\phi% \right)\\ &\displaystyle=&\displaystyle R\tanh^{-1}\!\left(\sin\phi\right)&\displaystyle% =&\displaystyle R\sinh^{-1}\!\left(\tan\phi\right)&\displaystyle=R\cosh^{-1}\!% \left(\sec\phi\right)=R\;\mbox{gd}~{}^{-1}(\phi).\end{aligned}
  9. ϕ \displaystyle\phi
  10. x = π R ( λ - λ 0 ) 180 , y = R ln [ tan ( 45 + ϕ 2 ) ] . \displaystyle x=\frac{\pi R(\lambda^{\circ}-\lambda^{\circ}_{0})}{180},\qquad% \quad y=R\ln\left[\tan\left(45+\frac{\phi^{\circ}}{2}\right)\right].
  11. x = W 2 π ( λ - λ 0 ) , y = W 2 π ln [ tan ( π 4 + ϕ 2 ) ] . \begin{aligned}\displaystyle x&\displaystyle=\frac{W}{2\pi}\left(\lambda-% \lambda_{0}\right),\qquad\quad y=\frac{W}{2\pi}\ln\left[\tan\left(\frac{\pi}{4% }+\frac{\phi}{2}\right)\right].\end{aligned}
  12. ϕ = tan - 1 [ sinh ( y R ) ] = tan - 1 [ sinh π ] = tan - 1 [ 11.5487 ] = 85.05113 . \phi=\tan^{-1}\left[\sinh\left(\frac{y}{R}\right)\right]=\tan^{-1}\left[\sinh% \pi\right]=\tan^{-1}\left[11.5487\right]=85.05113^{\circ}.
  13. δ s δ s = P Q P Q = P M P M = k = P K P K = h = sec ϕ . \frac{\delta s^{\prime}}{\delta s}=\frac{P^{\prime}Q^{\prime}}{PQ}=\frac{P^{% \prime}M^{\prime}}{PM}=k=\frac{P^{\prime}K^{\prime}}{PK}=h=\sec\phi.
  14. k = sec ϕ = cosh ( y R ) = cosh ( 2 π y W ) . k=\sec\phi=\cosh\left(\frac{y}{R}\right)=\cosh\left(\frac{2\pi y}{W}\right).
  15. x = 0.99 R λ y = 0.99 R ln tan ( π 4 + ϕ 2 ) k = 0.99 sec ϕ . x=0.99R\lambda\qquad y=0.99R\ln\tan\!\left(\frac{\pi}{4}+\frac{\phi}{2}\right)% \qquad k\;=0.99\sec\phi.
  16. x = R ( λ - λ 0 ) , y = R ln [ tan ( π 4 + ϕ 2 ) ( 1 - e sin ϕ 1 + e sin ϕ ) e / 2 ] , k = sec ϕ 1 - e 2 sin 2 ϕ . \begin{aligned}\displaystyle x&\displaystyle=R\left(\lambda-\lambda_{0}\right)% ,\\ \displaystyle y&\displaystyle=R\ln\left[\tan\left(\frac{\pi}{4}+\frac{\phi}{2}% \right)\left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)^{e/2}\right],\\ \displaystyle k&\displaystyle=\sec\phi\sqrt{1-e^{2}\sin^{2}\phi}.\end{aligned}
  17. m 12 = a | ϕ 1 - ϕ 2 | . m_{12}=a|\phi_{1}-\phi_{2}|.
  18. m 12 = a | tan - 1 [ sinh ( y 1 R ) ] - tan - 1 [ sinh ( y 2 R ) ] | , m_{12}=a\left|\tan^{-1}\left[\sinh\left(\frac{y_{1}}{R}\right)\right]-\tan^{-1% }\left[\sinh\left(\frac{y_{2}}{R}\right)\right]\right|,
  19. r 12 = a sec α | ϕ 1 - ϕ 2 | = a sec α Δ ϕ . r_{12}=a\sec\alpha\,|\phi_{1}-\phi_{2}|=a\,\sec\alpha\;\Delta\phi.
  20. r 12 = a sec α | tan - 1 sinh ( y 1 R ) - tan - 1 sinh ( y 2 R ) | . r_{12}=a\sec\alpha\left|\tan^{-1}\sinh\left(\frac{y_{1}}{R}\right)-\tan^{-1}% \sinh\left(\frac{y_{2}}{R}\right)\right|.

Merge_sort.html

  1. α = - 1 + k = 0 1 2 k + 1 0.2645. \alpha=-1+\sum_{k=0}^{\infty}\frac{1}{2^{k}+1}\approx 0.2645.
  2. O ( n ) O(n)
  3. O ( 1 ) O(1)
  4. O ( n l o g n ) O(nlogn)
  5. O ( n l o g n ) O(nlogn)
  6. n n
  7. m m
  8. 5 n + 12 n + o ( m ) 5n+12n+o(m)
  9. O ( ( n + m ) l o g ( n + m ) ) O((n+m)log(n+m))
  10. O ( n l o g < s u p > 2 n ) O(nlog<sup>2n)
  11. Θ ( n l o g n ) Θ(nlogn)
  12. Θ ( l o g n ) Θ(logn)
  13. Θ ( n / l o g n ) Θ(n/logn)

Merkle–Hellman_knapsack_cryptosystem.html

  1. A A
  2. A A
  3. A A
  4. B B
  5. A A
  6. A m A_{m}
  7. A m A_{m}
  8. q > i = 1 n w i q>\sum_{i=1}^{n}w_{i}
  9. α i \alpha_{i}
  10. α i s y m b o l \alpha_{i}symbol{\in}
  11. c = i = 1 n α i β i . c=\sum_{i=1}^{n}\alpha_{i}\beta_{i}.
  12. c = i = 1 n α i β i . c=\sum_{i=1}^{n}\alpha_{i}\beta_{i}.
  13. c c s ( mod q ) . c^{\prime}\equiv cs\;\;(\mathop{{\rm mod}}q).
  14. c c s i = 1 n α i β i s ( mod q ) . c^{\prime}\equiv cs\equiv\sum_{i=1}^{n}\alpha_{i}\beta_{i}s\;\;(\mathop{{\rm mod% }}q).
  15. β i s w i r s w i ( mod q ) . \beta_{i}s\equiv w_{i}rs\equiv w_{i}\;\;(\mathop{{\rm mod}}q).
  16. c i = 1 n α i w i ( mod q ) . c^{\prime}\equiv\sum_{i=1}^{n}\alpha_{i}w_{i}\;\;(\mathop{{\rm mod}}q).
  17. i = 1 n α i w i \sum_{i=1}^{n}\alpha_{i}w_{i}
  18. c = i = 1 n α i w i . c^{\prime}=\sum_{i=1}^{n}\alpha_{i}w_{i}.
  19. w = 706 \sum w=706
  20. [ 1 , q ) [1,q)
  21. q q
  22. 0

Mersenne_prime.html

  1. 2 a b - 1 = ( 2 a - 1 ) ( 1 + 2 a + 2 2 a + 2 3 a + + 2 ( b - 1 ) a ) = ( 2 b - 1 ) ( 1 + 2 b + 2 2 b + 2 3 b + + 2 ( a - 1 ) b ) . \begin{aligned}\displaystyle 2^{ab}-1&\displaystyle=(2^{a}-1)\cdot\left(1+2^{a% }+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right)\\ &\displaystyle=(2^{b}-1)\cdot\left(1+2^{b}+2^{2b}+2^{3b}+\cdots+2^{(a-1)b}% \right).\end{aligned}
  2. 1 , 193 {}^{1,193}
  3. 2 n - 1 2^{n}-1
  4. b n - 1 b^{n}-1
  5. b 2 b\neq 2
  6. n > 1 n>1
  7. b n - 1 b^{n}-1
  8. b - 1 b-1
  9. b - 1 b-1
  10. b - 1 b-1
  11. b b
  12. 2 n - 1 2^{n}-1
  13. 0 n - 1 0^{n}-1
  14. b = 1 + i b=1+i
  15. b = 1 - i b=1-i
  16. n n
  17. ( 1 + i ) n - 1 (1+i)^{n}-1
  18. ( 1 + i ) n - 1 (1+i)^{n}-1
  19. n n
  20. b n - 1 b^{n}-1
  21. b - 1 b-1
  22. b n - 1 b - 1 \frac{b^{n}-1}{b-1}
  23. b = 10 b=10
  24. n n
  25. b = - 12 b=-12
  26. n n
  27. n n
  28. b b
  29. b n - 1 b - 1 \frac{b^{n}-1}{b-1}
  30. b p r i m e ( n ) - 1 b - 1 \frac{b^{prime(n)}-1}{b-1}
  31. a n - b n a - b \frac{a^{n}-b^{n}}{a-b}
  32. ( b + 1 ) n - b n (b+1)^{n}-b^{n}
  33. ( b + 1 ) n - b n (b+1)^{n}-b^{n}
  34. ( b + 1 ) p r i m e ( n ) - b p r i m e ( n ) (b+1)^{prime(n)}-b^{prime(n)}

Mersenne_Twister.html

  1. ( trunc v ( x i ) , trunc v ( x i + 1 ) , , trunc v ( x i + k - 1 ) ) ( 0 i < P ) (\,\text{trunc}_{v}(x_{i}),\,\,\text{trunc}_{v}(x_{i+1}),\,...,\,\,\text{trunc% }_{v}(x_{i+k-1}))\quad(0\leq i<P)
  2. x i x_{i}
  3. x i T x_{i}T
  4. T T
  5. x k + n := x k + m ( x k u x k + 1 l ) A k = 0 , 1 , x_{k+n}:=x_{k+m}\oplus({x_{k}}^{u}\mid{x_{k+1}}^{l})A\qquad\qquad k=0,1,\ldots
  6. \mid
  7. \oplus
  8. x k u {x_{k}}^{u}
  9. w - r w-r
  10. x k x_{k}
  11. x k + 1 l x_{k+1}^{l}
  12. r r
  13. x k + 1 x_{k+1}
  14. A = ( 0 I w - 1 a w - 1 ( a w - 2 , , a 0 ) ) A=\begin{pmatrix}0&I_{w-1}\\ a_{w-1}&(a_{w-2},\ldots,a_{0})\end{pmatrix}
  15. s y m b o l x A = { s y m b o l x 1 x 0 = 0 ( s y m b o l x 1 ) s y m b o l a x 0 = 1 symbol{x}A=\begin{cases}symbol{x}\gg 1&x_{0}=0\\ (symbol{x}\gg 1)\oplus symbol{a}&x_{0}=1\end{cases}
  16. s + t w / 2 - 1 s+t\geq\lfloor w/2\rfloor-1
  17. B = ( 0 I w 0 0 I w 0 0 I w 0 0 0 0 I w - r S 0 0 0 ) m -th row B=\begin{pmatrix}0&I_{w}&\cdots&0&0\\ \vdots&&&&\\ I_{w}&\vdots&\ddots&\vdots&\vdots\\ \vdots&&&&\\ 0&0&\cdots&I_{w}&0\\ 0&0&\cdots&0&I_{w-r}\\ S&0&\cdots&0&0\end{pmatrix}\begin{matrix}\\ \\ \leftarrow m\hbox{-th row}\\ \\ \\ \\ \end{matrix}
  18. S = ( 0 I r I w - r 0 ) A S=\begin{pmatrix}0&I_{r}\\ I_{w-r}&0\end{pmatrix}A

Meson.html

  1. 2 / 3 {2}/{3}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 1 / 2 {1}/{2}
  7. 1 / 2 {1}/{2}
  8. 1 / 2 {1}/{2}
  9. P = ( - 1 ) L + 1 P=\left(-1\right)^{L+1}
  10. | q q ¯ = | q ¯ q |q\bar{q}\rangle=|\bar{q}q\rangle
  11. | q q ¯ = - | q ¯ q |q\bar{q}\rangle=-|\bar{q}q\rangle
  12. | q 1 q 2 ¯ = | q 1 ¯ q 2 |q_{1}\bar{q_{2}}\rangle=|\bar{q_{1}}q_{2}\rangle
  13. | q 1 q 2 ¯ = - | q 1 ¯ q 2 |q_{1}\bar{q_{2}}\rangle=-|\bar{q_{1}}q_{2}\rangle
  14. 2 / 3 {2}/{3}
  15. 1 / 3 {1}/{3}
  16. I 3 = 1 2 [ ( n u - n u ¯ ) - ( n d - n d ¯ ) ] , I_{3}=\frac{1}{2}[(n_{u}-n_{\bar{u}})-(n_{d}-n_{\bar{d}})],
  17. 1 / 2 {1}/{2}
  18. 1 / 2 {1}/{2}
  19. Q = I 3 + 1 2 ( B + S + C + B + T ) , Q=I_{3}+\frac{1}{2}(B+S+C+B^{\prime}+T),
  20. S = - ( n s - n s ¯ ) S=-(n_{s}-n_{\bar{s}})
  21. C = + ( n c - n c ¯ ) C=+(n_{c}-n_{\bar{c}})
  22. B = - ( n b - n b ¯ ) B^{\prime}=-(n_{b}-n_{\bar{b}})
  23. T = + ( n t - n t ¯ ) , T=+(n_{t}-n_{\bar{t}}),
  24. 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_{u}-n_{\bar{u}})+(n_{c}-n_{\bar{c}})+(n_{t}-n_{\bar{t}})]-% \frac{1}{3}[(n_{d}-n_{\bar{d}})+(n_{s}-n_{\bar{s}})+(n_{b}-n_{\bar{b}})].
  25. u u ¯ - d d ¯ 2 \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}

Meta-analysis.html

  1. δ = μ t - μ c σ , \delta=\frac{\mu_{t}-\mu_{c}}{\sigma},
  2. μ t \mu_{t}
  3. μ c \mu_{c}
  4. σ 2 \sigma^{2}

Meteoroid.html

  1. N ( > D ) = 37 D - 2.7 N(>D)=37D^{-2.7}

Metre.html

  1. 33 / 8 3{3}/{8}
  2. 393 / 8 39{3}/{8}
  3. λ = c n f \lambda=\frac{c}{nf}
  4. 10 {}^{10}
  5. 5 {}^{5}
  6. 1 299792458 \frac{1}{299792458}
  7. 1 10000000 \frac{1}{10000000}
  8. 4 {}^{−4}
  9. 1 10000000 \frac{1}{10000000}
  10. 4 {}^{−4}
  11. 5 {}^{−5}
  12. 7 {}^{−7}
  13. 9 {}^{−9}
  14. 1 299792458 \frac{1}{299792458}
  15. 10 {}^{−10}
  16. 10 {}^{10}
  17. 10 {}^{−10}
  18. 10000 254 \frac{10000}{254}
  19. 10000 9144 \frac{10000}{9144}
  20. 33 / 8 3{3}/{8}
  21. 1 / 2 {1}/{2}

Metric_space.html

  1. ( M , d ) (M,d)
  2. M M
  3. d d
  4. M M
  5. d : M × M d\colon M\times M\rightarrow\mathbb{R}
  6. x , y , z M x,y,z\in M
  7. d ( x , y ) 0 d(x,y)\geq 0
  8. d ( x , y ) = 0 d(x,y)=0\,
  9. \iff
  10. x = y x=y\,
  11. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)\,
  12. d ( x , z ) d ( x , y ) + d ( y , z ) d(x,z)\leq d(x,y)+d(y,z)
  13. x , y M x,y\in M
  14. d ( x , y ) + d ( y , x ) d ( x , x ) (by triangle inequality) \displaystyle d(x,y)+d(y,x)\geq d(x,x)\ \,\text{(by triangle inequality)}
  15. d d
  16. d d
  17. M M
  18. d ( x , y ) = | y - x | d(x,y)=|y-x|
  19. n n
  20. d ( x , y ) = | log ( y / x ) | d(x,y)=|\log(y/x)|
  21. d ( x , y ) = y - x d(x,y)=\lVert y-x\rVert
  22. x x
  23. y y
  24. d ( x , y ) = x + y d(x,y)=\lVert x\rVert+\lVert y\rVert
  25. x x
  26. y y
  27. d ( x , x ) = 0 d(x,x)=0
  28. . \lVert.\rVert
  29. f f
  30. S S
  31. 0
  32. S S
  33. d ( x , y ) = f ( x ) + f ( y ) d(x,y)=f(x)+f(y)
  34. x x
  35. y y
  36. d ( x , x ) = 0 d(x,x)=0
  37. ( M , d ) (M,d)
  38. X X
  39. M M
  40. ( X , d ) (X,d)
  41. d d
  42. X × X X\times X
  43. d ( x , y ) = 0 d(x,y)=0
  44. x = y x=y
  45. d ( x , y ) = 1 d(x,y)=1
  46. M M
  47. M M
  48. X X
  49. M M
  50. f : X M f\colon X\rightarrow M
  51. M M
  52. d ( f , g ) = sup x X d ( f ( x ) , g ( x ) ) d(f,g)=\sup_{x\in X}d(f(x),g(x))
  53. f f
  54. g g
  55. sup \sup
  56. M M
  57. X X
  58. M M
  59. G G
  60. V V
  61. G G
  62. d ( x , y ) d(x,y)
  63. x x
  64. y y
  65. u u
  66. v v
  67. u u
  68. v v
  69. ( X , d ) (X,d)
  70. f : [ 0 , ) [ 0 , ) f\colon[0,\infty)\rightarrow[0,\infty)
  71. f ( x ) = 0 f(x)=0
  72. x = 0 x=0
  73. f d f\circ d
  74. X X
  75. f f
  76. A A
  77. ( X , d ) (X,d)
  78. d ( f ( x ) , f ( y ) ) d(f(x),f(y))
  79. A A
  80. m m
  81. n n
  82. d ( X , Y ) = rank ( Y - X ) d(X,Y)=\mathrm{rank}(Y-X)
  83. x x
  84. M M
  85. r > 0 r>0
  86. r r
  87. x x
  88. B ( x ; r ) = { y M : d ( x , y ) < r } . B(x;r)=\{y\in M:d(x,y)<r\}.
  89. U U
  90. M M
  91. x x
  92. U U
  93. r > 0 r>0
  94. B ( x ; r ) B(x;r)
  95. U U
  96. x x
  97. M M
  98. x x
  99. x n x_{n}
  100. M M
  101. x M x\in M
  102. ϵ > 0 \epsilon>0
  103. d ( x n , x ) < ϵ d(x_{n},x)<\epsilon
  104. n > N n>N
  105. A A
  106. M M
  107. A A
  108. M M
  109. A A
  110. M M
  111. M M
  112. d ( x n , x m ) 0 d(x_{n},x_{m})\to 0
  113. n n
  114. m m
  115. y M y\in M
  116. d ( x n , y ) 0 d(x_{n},y)\to 0
  117. d ( x , y ) = | x - y | d(x,y)=|x-y|
  118. X X
  119. M M
  120. X X
  121. M M
  122. M M
  123. M M
  124. M M
  125. x , y M x,y\in M
  126. f : [ 0 , 1 ] M f\colon[0,1]\to M
  127. f ( 0 ) = x f(0)=x
  128. f ( 1 ) = y f(1)=y
  129. d 1 ( x , y ) < δ d 2 ( f ( x ) , f ( y ) ) < ε . d_{1}(x,y)<\delta\Rightarrow d_{2}(f(x),f(y))<\varepsilon.
  130. d 1 ( x , y ) < δ d 2 ( f ( x ) , f ( y ) ) < ε for all x , y M 1 . d_{1}(x,y)<\delta\Rightarrow d_{2}(f(x),f(y))<\varepsilon\quad\mbox{for all}~{% }\quad x,y\in M_{1}.
  131. d 2 ( f ( x ) , f ( y ) ) K d 1 ( x , y ) for all x , y M 1 . d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad\mbox{for all}~{}\quad x,y\in M_{1}.
  132. d ( f ( x ) , f ( y ) ) < d ( x , y ) for all x y M 1 d(f(x),f(y))<d(x,y)\quad\mbox{for all}~{}\quad x\neq y\in M_{1}
  133. d 2 ( f ( x ) , f ( y ) ) = d 1 ( x , y ) for all x , y M 1 d_{2}(f(x),f(y))=d_{1}(x,y)\quad\mbox{for all}~{}\quad x,y\in M_{1}
  134. 1 A d 2 ( f ( x ) , f ( y ) ) - B d 1 ( x , y ) A d 2 ( f ( x ) , f ( y ) ) + B for all x , y M 1 \frac{1}{A}d_{2}(f(x),f(y))-B\leq d_{1}(x,y)\leq Ad_{2}(f(x),f(y))+B\,\text{ % for all }x,y\in M_{1}
  135. d ( x , S ) = inf { d ( x , s ) : s S } d(x,S)=\inf\{d(x,s):s\in S\}
  136. inf \inf
  137. d ( x , S ) d ( x , y ) + d ( y , S ) , d(x,S)\leq d(x,y)+d(y,S),
  138. x d ( x , S ) x\mapsto d(x,S)
  139. d H ( S , T ) = max { sup { d ( s , T ) : s S } , sup { d ( t , S ) : t T } } d_{H}(S,T)=\max\{\sup\{d(s,T):s\in S\},\sup\{d(t,S):t\in T\}\}
  140. sup \sup
  141. ( M 1 , d 1 ) , , ( M n , d n ) (M_{1},d_{1}),\ldots,(M_{n},d_{n})
  142. ( M 1 × × M n , N ( d 1 , , d n ) ) \Big(M_{1}\times\ldots\times M_{n},N(d_{1},\ldots,d_{n})\Big)
  143. N ( d 1 , , d n ) ( ( x 1 , , x n ) , ( y 1 , , y n ) ) = N ( d 1 ( x 1 , y 1 ) , , d n ( x n , y n ) ) , N(d_{1},...,d_{n})\Big((x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n})\Big)=N\Big(d_% {1}(x_{1},y_{1}),\ldots,d_{n}(x_{n},y_{n})\Big),
  144. d ( x , y ) = i = 1 1 2 i d i ( x i , y i ) 1 + d i ( x i , y i ) . d(x,y)=\sum_{i=1}^{\infty}\frac{1}{2^{i}}\frac{d_{i}(x_{i},y_{i})}{1+d_{i}(x_{% i},y_{i})}.
  145. 𝐑 𝐑 \mathbf{R}^{\mathbf{R}}
  146. ( M , d ) (M,d)
  147. d : M × M R + d\colon M\times M\rightarrow R^{+}
  148. N ( d , d ) N(d,d)
  149. M × M M\times M
  150. d ( [ x ] , [ y ] ) = inf { d ( p 1 , q 1 ) + d ( p 2 , q 2 ) + + d ( p n , q n ) } d^{\prime}([x],[y])=\inf\{d(p_{1},q_{1})+d(p_{2},q_{2})+\cdots+d(p_{n},q_{n})\}
  151. ( p 1 , p 2 , , p n ) (p_{1},p_{2},\dots,p_{n})
  152. ( q 1 , q 2 , , q n ) (q_{1},q_{2},\dots,q_{n})
  153. [ p 1 ] = [ x ] [p_{1}]=[x]
  154. [ q n ] = [ y ] [q_{n}]=[y]
  155. [ q i ] = [ p i + 1 ] , i = 1 , 2 , , n - 1 [q_{i}]=[p_{i+1}],i=1,2,\dots,n-1
  156. d ( [ x ] , [ y ] ) = 0 d^{\prime}([x],[y])=0
  157. [ x ] = [ y ] [x]=[y]
  158. f : ( M , d ) ( X , δ ) f:(M,d)\longrightarrow(X,\delta)
  159. δ ( f ( x ) , f ( y ) ) d ( x , y ) \delta(f(x),f(y))\leq d(x,y)
  160. x y , x\sim y,
  161. f ¯ : M / X \overline{f}\colon M/\sim\longrightarrow X
  162. f ¯ ( [ x ] ) = f ( x ) \overline{f}([x])=f(x)
  163. f ¯ : ( M / , d ) ( X , δ ) . \overline{f}\colon(M/\sim,d^{\prime})\longrightarrow(X,\delta).
  164. \infty
  165. ( , ) (\mathbb{R},\geq)
  166. a b a\to b
  167. a b a\geq b
  168. + +
  169. 0
  170. R * R^{*}
  171. ( M , d ) (M,d)
  172. M * M^{*}
  173. R * R^{*}
  174. Ob ( M * ) := M \operatorname{Ob}(M^{*}):=M
  175. X , Y M X,Y\in M
  176. Hom ( X , Y ) := d ( X , Y ) Ob ( R * ) \operatorname{Hom}(X,Y):=d(X,Y)\in\operatorname{Ob}(R^{*})
  177. Hom ( Y , Z ) Hom ( X , Y ) Hom ( X , Z ) \operatorname{Hom}(Y,Z)\otimes\operatorname{Hom}(X,Y)\to\operatorname{Hom}(X,Z)
  178. R * R^{*}
  179. d ( y , z ) + d ( x , y ) d ( x , z ) d(y,z)+d(x,y)\geq d(x,z)
  180. 0 Hom ( X , X ) 0\to\operatorname{Hom}(X,X)
  181. 0 d ( X , X ) 0\geq d(X,X)
  182. R * R^{*}

Metrization_theorem.html

  1. ( X , τ ) (X,\tau)
  2. d : X × X [ 0 , ) d\colon X\times X\to[0,\infty)
  3. τ \tau
  4. [ 0 , 1 ] [0,1]^{\mathbb{N}}
  5. 𝕌 ( ) \mathbb{U}(\mathcal{H})
  6. \mathcal{H}

Metropolis–Hastings_algorithm.html

  1. Q ( x | y ) Q(x|y)
  2. Q ( x | y ) = Q ( y | x ) Q(x|y)=Q(y|x)
  3. Q ( x | y ) Q(x|y)
  4. Q ( x | x t ) Q(x^{\prime}|x_{t})
  5. α \alpha
  6. P ( x ) \displaystyle P(x)
  7. P ( x ) \displaystyle P(x)
  8. P ( x ) \displaystyle P(x)
  9. P ( x ) \displaystyle P(x)
  10. P ( x ) \displaystyle P(x)
  11. P ( x x ) P(x\rightarrow x^{\prime})
  12. π ( x ) P ( x x ) = π ( x ) P ( x x ) \pi(x)P(x\rightarrow x^{\prime})=\pi(x^{\prime})P(x^{\prime}\rightarrow x)
  13. P ( x ) P ( x x ) = P ( x ) P ( x x ) P(x)P(x\rightarrow x^{\prime})=P(x^{\prime})P(x^{\prime}\rightarrow x)
  14. P ( x x ) P ( x x ) = P ( x ) P ( x ) \frac{P(x\rightarrow x^{\prime})}{P(x^{\prime}\rightarrow x)}=\frac{P(x^{% \prime})}{P(x)}
  15. g ( x x ) \displaystyle g(x\rightarrow x^{\prime})
  16. A ( x x ) \displaystyle A(x\rightarrow x^{\prime})
  17. P ( x x ) = g ( x x ) A ( x x ) P(x\rightarrow x^{\prime})=g(x\rightarrow x^{\prime})A(x\rightarrow x^{\prime})
  18. A ( x x ) A ( x x ) = P ( x ) P ( x ) g ( x x ) g ( x x ) \frac{A(x\rightarrow x^{\prime})}{A(x^{\prime}\rightarrow x)}=\frac{P(x^{% \prime})}{P(x)}\frac{g(x^{\prime}\rightarrow x)}{g(x\rightarrow x^{\prime})}
  19. A ( x x ) = min ( 1 , P ( x ) P ( x ) g ( x x ) g ( x x ) ) A(x\rightarrow x^{\prime})=\min\left(1,\frac{P(x^{\prime})}{P(x)}\frac{g(x^{% \prime}\rightarrow x)}{g(x\rightarrow x^{\prime})}\right)
  20. g ( x x ) \displaystyle g(x\rightarrow x^{\prime})
  21. A ( x x ) \displaystyle A(x\rightarrow x^{\prime})
  22. P ( x ) P(x)
  23. g ( x x ) \displaystyle g(x\rightarrow x^{\prime})
  24. x t x_{t}\,
  25. x x^{\prime}\,
  26. Q ( x x t ) Q(x^{\prime}\mid x_{t})\,
  27. a = a 1 a 2 a=a_{1}a_{2}\,
  28. a 1 = P ( x ) P ( x t ) a_{1}=\frac{P(x^{\prime})}{P(x_{t})}\,\!
  29. x x^{\prime}\,
  30. x t x_{t}\,
  31. a 2 = Q ( x t x ) Q ( x x t ) a_{2}=\frac{Q(x_{t}\mid x^{\prime})}{Q(x^{\prime}\mid x_{t})}
  32. x t x_{t}\,
  33. x x^{\prime}\,
  34. x t + 1 \displaystyle x_{t+1}
  35. If a 1 : x t + 1 = x , \begin{matrix}\mbox{If }~{}a\geq 1:&\\ &x_{t+1}=x^{\prime},\end{matrix}
  36. else x t + 1 = { x with probability a x t with probability 1 - a . \begin{matrix}\mbox{else}&\\ &x_{t+1}=\left\{\begin{array}[]{lr}x^{\prime}&\mbox{ with probability }~{}a\\ x_{t}&\mbox{ with probability }~{}1-a.\end{array}\right.\end{matrix}
  37. x 0 \displaystyle x_{0}
  38. x x
  39. P ( x ) P(x)
  40. P ( x ) \displaystyle P(x)
  41. Q ( x x t ) P ( x ) Q(x^{\prime}\mid x_{t})\approx P(x^{\prime})\,\!
  42. Q \displaystyle Q
  43. σ 2 \displaystyle\sigma^{2}
  44. N \displaystyle N
  45. N \displaystyle N
  46. σ 2 \displaystyle\sigma^{2}
  47. P ( x ) \displaystyle P(x)
  48. σ 2 \displaystyle\sigma^{2}
  49. a 1 \displaystyle a_{1}

Michael_Atiyah.html

  1. K ( B G ) R ( G ) . K(BG)\cong R(G)^{\wedge}.
  2. X G X_{G}
  3. K G ( X ) K ( X G ) . K_{G}(X)^{\wedge}\cong K(X_{G}).

Microclimate.html

  1. v v
  2. Fr = v N h Fr c , \mathrm{Fr}=\frac{v}{Nh}\geq\mathrm{Fr}_{c},
  3. Fr \mathrm{Fr}
  4. N N
  5. h h
  6. Fr c \mathrm{Fr}_{c}

Microfluidics.html

  1. J i , j J_{i,j}
  2. λ i \lambda_{i}
  3. K i K_{i}

Minimax.html

  1. max ( a , b ) = - min ( - a , - b ) \max(a,b)=-\min(-a,-b)
  2. δ \delta
  3. θ Θ \theta\in\Theta
  4. R ( θ , δ ) R(\theta,\delta)
  5. δ ~ \tilde{\delta}
  6. sup θ R ( θ , δ ~ ) = inf δ sup θ R ( θ , δ ) . \sup_{\theta}R(\theta,\tilde{\delta})=\inf_{\delta}\sup_{\theta}R(\theta,% \delta).
  7. Π \Pi
  8. Θ R ( θ , δ ) d Π ( θ ) . \int_{\Theta}R(\theta,\delta)\,d\Pi(\theta).

Minimum_spanning_tree.html

  1. \cup
  2. n n^{\prime}
  3. n / 2 m / n n^{\prime}/2^{m/n^{\prime}}
  4. log * n \log*{n}
  5. 2 ( r 2 ) 2^{r\choose 2}
  6. r 2 r^{2}
  7. 2 r 2 2^{r^{2}}
  8. r 2 r^{2}
  9. r 4 r^{4}
  10. ( r 4 ) ( 2 r 2 ) = r 2 ( r 2 + 2 ) {(r^{4})}^{(2^{r^{2}})}=r^{2^{(r^{2}+2)}}
  11. ( r 2 ) ! (r^{2})!
  12. ( r 2 ) (r^{2})
  13. ( r 2 + 1 ) ! (r^{2}+1)!
  14. 2 ( r 2 ) r 2 ( r 2 + 2 ) ( r 2 + 1 ) ! 2^{r\choose 2}\cdot r^{2^{(r^{2}+2)}}\cdot(r^{2}+1)!
  15. 2 2 r 2 + o ( r ) 2^{2^{r^{2}+o(r)}}
  16. r = log log log n r=\log\log\log n
  17. O ( log n ) O(\log n)
  18. F F
  19. F ( 0 ) > 0 F^{\prime}(0)>0
  20. ζ ( 3 ) / F ( 0 ) \zeta(3)/F^{\prime}(0)
  21. ζ \zeta
  22. [ 0 , 1 ] [0,1]

Minkowski's_theorem.html

  1. f : S 2 , ( x , y ) ( x mod 2 , y mod 2 ) f:S\to\mathbb{R}^{2},(x,y)\mapsto(x\;\bmod\;2,y\;\bmod\;2)
  2. f ( S ) f(S)
  3. f ( p 1 ) = f ( p 2 ) f(p_{1})=f(p_{2})
  4. p 1 , p 2 p_{1},p_{2}
  5. p 2 = p 1 + ( 2 i , 2 j ) p_{2}=p_{1}+(2i,2j)
  6. - p 1 -p_{1}
  7. - p 1 -p_{1}
  8. p 2 p_{2}
  9. 1 2 ( - p 1 + p 2 ) = 1 2 ( - p 1 + p 1 + ( 2 i , 2 j ) ) = ( i , j ) \frac{1}{2}\left(-p_{1}+p_{2}\right)=\frac{1}{2}\left(-p_{1}+p_{1}+(2i,2j)% \right)=(i,j)

Minute_and_second_of_arc.html

  1. 1 / 60 {1}/{60}
  2. 1 / 360 {1}/{360}
  3. 1 / 21 , 600 {1}/{21,600}
  4. π / 10 , 800 {π}/{10,800}
  5. 4 π ( 10 , 800 π ) 2 = 466 , 560 , 000 π 4\pi\left(\frac{10,800}{\pi}\right)^{2}=\frac{466,560,000}{\pi}
  6. 1 / 60 {1}/{60}
  7. 1 / 3 , 600 {1}/{3,600}
  8. 1 / 1 , 296 , 000 {1}/{1,296,000}
  9. π / 648 , 000 {π}/{648,000}
  10. 1 / 206 , 265 {1}/{206,265}
  11. ^ \hat{{}^{\prime}}
  12. 1 / 360 {1}/{360}
  13. 1 / 60 {1}/{60}
  14. ^ \hat{{}^{\prime}}
  15. 1 / 60 {1}/{60}
  16. 1 / 2 {1}/{2}
  17. 1 / 4 {1}/{4}
  18. 1 / 8 {1}/{8}
  19. m / 60 {m}/{60}
  20. 1 / 60 {1}/{60}
  21. 1 / 1 , 000 {1}/{1,000}
  22. 1 / 4 {1}/{4}
  23. 1 / 3 , 600 {1}/{3,600}

MIPS_instruction_set.html

  1. 2 s h a m t 2^{shamt}
  2. $ d = $ t s h a m t + ( n = 1 shamt 2 32 - n ) ( $ t 31 ) \$d=\$t>>shamt+\left(\sum_{n=1}^{\,\text{shamt}}2^{32-n}\right)\cdot\left(\$t>% >31\right)
  3. 2 s h a m t 2^{shamt}
  4. 2 $ s 2^{\$s}
  5. 2 $ s 2^{\$s}
  6. $ d = $ t $ s + ( n = 1 $ s 2 32 - n ) ( $ t 31 ) \$d=\$t>>\$s+\left(\sum_{n=1}^{\$\,\text{s}}2^{32-n}\right)\cdot\left(\$t>>31\right)
  7. 2 $ s 2^{\$s}

Mirror.html

  1. θ \theta
  2. θ \theta

Mode_volume.html

  1. 4 V 2 π 2 4V^{2}\over\pi^{2}
  2. V 2 2 ( g g + 2 ) {V^{2}\over 2}\left({g\over g+2}\right)

Model_theory.html

  1. Π i I A i / U \Pi_{i\in I}A_{i}/U
  2. u v ( u E v v E u ) \forall u\forall v(uEv\rightarrow vEu)
  3. u ¬ ( u E u ) \forall u\neg(uEu)
  4. u 1 u 2 u n ( t = t ) \forall u_{1}u_{2}\dots u_{n}(t=t^{\prime})
  5. ¬ , , , \neg,\land,\lor,\rightarrow
  6. v \forall v
  7. v \exists v
  8. φ = u v ( w ( x × w = u × v ) ( w ( x × w = u ) w ( x × w = v ) ) ) x 0 x 1 , {\varphi\;=\;\forall u\forall v(\exists w(x\times w=u\times v)\rightarrow(% \exists w(x\times w=u)\lor\exists w(x\times w=v)))\land x\neq 0\land x\neq 1,}
  9. ψ = u v ( ( u × v = x ) ( u = x ) ( v = x ) ) x 0 x 1. \psi\;=\;\forall u\forall v((u\times v=x)\rightarrow(u=x)\lor(v=x))\land x\neq 0% \land x\neq 1.
  10. 𝒩 \mathcal{N}
  11. \models
  12. 𝒩 φ ( n ) n \mathcal{N}\models\varphi(n)\iff n
  13. 𝒩 ψ ( n ) n \mathcal{N}\models\psi(n)\iff n
  14. T \mathcal{M}\models T
  15. T \mathcal{M}\models T
  16. \mathcal{M}
  17. x 1 x n ( ϕ ( x 1 , , x n ) ψ ( x 1 , , x n ) ) \forall x_{1}\dots\forall x_{n}(\phi(x_{1},\dots,x_{n})\leftrightarrow\psi(x_{% 1},\dots,x_{n}))
  18. 𝒜 \mathcal{A}
  19. \mathcal{B}
  20. 𝒜 \mathcal{A}
  21. \mathcal{B}
  22. v 1 v m ψ ( x 1 , , x n , v 1 , , v m ) \exists v_{1}\dots\exists v_{m}\psi(x_{1},\dots,x_{n},v_{1},\dots,v_{m})
  23. 0 \aleph_{0}
  24. 0 \aleph_{0}
  25. 0 \aleph_{0}
  26. 0 \aleph_{0}
  27. 0 \aleph_{0}
  28. 0 \aleph_{0}
  29. 𝔄 \mathfrak{A}
  30. 𝔄 \mathfrak{A}
  31. L L
  32. M M
  33. n n
  34. M n M^{n}
  35. A A
  36. A A
  37. n n
  38. A A
  39. S n ( A ) S_{n}(A)
  40. m M n m\in M^{n}
  41. ϕ \phi
  42. A A
  43. x 1 , , x n x_{1},\ldots,x_{n}
  44. M ϕ ( m ) M\models\phi(m)
  45. m m
  46. A A
  47. n n
  48. p p
  49. N N
  50. M M
  51. a N n a\in N^{n}
  52. p p
  53. a a
  54. A A
  55. M M
  56. n n
  57. A A
  58. A [ x 1 , , x n ] A[x_{1},\ldots,x_{n}]
  59. A [ x 1 , , x n ] A[x_{1},\ldots,x_{n}]
  60. { p : f ( x ) = 0 p } \{p:f(x)=0\in p\}
  61. { p : f ( x ) 0 p } \{p:f(x)\neq 0\in p\}

Modular_arithmetic.html

  1. a b ( mod n ) , a\equiv b\;\;(\mathop{{\rm mod}}n),\,
  2. 38 14 ( mod 12 ) 38\equiv 14\;\;(\mathop{{\rm mod}}12)\,
  3. - 8 \displaystyle-8
  4. a b ( mod n ) a\equiv b\;\;(\mathop{{\rm mod}}n)\,
  5. a a
  6. b b
  7. n n
  8. 38 14 ( mod 12 ) 38\equiv 14\;\;(\mathop{{\rm mod}}12)\,
  9. 38 - 14 = 24 38-14=24
  10. a 1 b 1 ( mod n ) a_{1}\equiv b_{1}\;\;(\mathop{{\rm mod}}n)
  11. a 2 b 2 ( mod n ) , a_{2}\equiv b_{2}\;\;(\mathop{{\rm mod}}n),
  12. a 1 + a 2 b 1 + b 2 ( mod n ) a_{1}+a_{2}\equiv b_{1}+b_{2}\;\;(\mathop{{\rm mod}}n)\,
  13. a 1 - a 2 b 1 - b 2 ( mod n ) a_{1}-a_{2}\equiv b_{1}-b_{2}\;\;(\mathop{{\rm mod}}n)\,
  14. a 1 , a 2 , b 1 , b 2 , n a_{1},a_{2},b_{1},b_{2},n\,
  15. a 1 a 2 b 1 b 2 ( mod n ) . a_{1}a_{2}\equiv b_{1}b_{2}\;\;(\mathop{{\rm mod}}n).\,
  16. 14 m o d 12 14mod12
  17. 14 m o d 12 = 2 14mod12=2
  18. A B ( mod n ) A\equiv B\;\;(\mathop{{\rm mod}}n)
  19. A mod n = B mod n . A\!\!\!\!\mod n=B\!\!\!\!\mod n.
  20. ( a mod n ) ( b mod n ) a b ( mod n ) , (a\!\!\!\!\mod n)\,(b\!\!\!\!\mod n)\equiv ab\;\;(\mathop{{\rm mod}}n),
  21. ( ( a mod n ) ( b mod n ) ) mod n = ( a b ) mod n . ((a\!\!\!\!\mod n)\,(b\!\!\!\!\mod n))\!\!\!\!\mod n=(ab)\!\!\!\!\mod n.
  22. a ¯ n \overline{a}_{n}
  23. { , a - 2 n , a - n , a , a + n , a + 2 n , } \left\{\ldots,a-2n,a-n,a,a+n,a+2n,\ldots\right\}
  24. [ a ] \displaystyle[a]
  25. / n \mathbb{Z}/n\mathbb{Z}
  26. / n \mathbb{Z}/n
  27. n \mathbb{Z}_{n}
  28. n \mathbb{Z}_{n}
  29. / n = { a ¯ n | a } . \mathbb{Z}/n\mathbb{Z}=\left\{\overline{a}_{n}|a\in\mathbb{Z}\right\}.
  30. / n \mathbb{Z}/n\mathbb{Z}
  31. / n = { 0 ¯ n , 1 ¯ n , 2 ¯ n , , n - 1 ¯ n } . \mathbb{Z}/n\mathbb{Z}=\left\{\overline{0}_{n},\overline{1}_{n},\overline{2}_{% n},\ldots,\overline{n-1}_{n}\right\}.
  32. / n \mathbb{Z}/n\mathbb{Z}
  33. \mathbb{Z}
  34. a ¯ 0 = { a } \overline{a}_{0}=\left\{a\right\}
  35. / n \mathbb{Z}/n\mathbb{Z}
  36. a ¯ n + b ¯ n = ( a + b ) ¯ n \overline{a}_{n}+\overline{b}_{n}=\overline{(a+b)}_{n}
  37. a ¯ n - b ¯ n = ( a - b ) ¯ n \overline{a}_{n}-\overline{b}_{n}=\overline{(a-b)}_{n}
  38. a ¯ n b ¯ n = ( a b ) ¯ n . \overline{a}_{n}\overline{b}_{n}=\overline{(ab)}_{n}.
  39. / n \mathbb{Z}/n\mathbb{Z}
  40. / 24 \mathbb{Z}/24\mathbb{Z}
  41. 12 ¯ 24 + 21 ¯ 24 = 9 ¯ 24 \overline{12}_{24}+\overline{21}_{24}=\overline{9}_{24}
  42. / n \mathbb{Z}/n\mathbb{Z}
  43. \mathbb{Z}
  44. n n\mathbb{Z}
  45. 0 0\mathbb{Z}
  46. { 0 } \left\{0\right\}
  47. / n \mathbb{Z}/n\mathbb{Z}
  48. n n\mathbb{Z}
  49. n n
  50. a ¯ n \overline{a}_{n}
  51. / n \mathbb{Z}/n\mathbb{Z}
  52. / n \mathbb{Z}/n\mathbb{Z}
  53. / 0 \mathbb{Z}/0\mathbb{Z}
  54. \mathbb{Z}

Modulation.html

  1. M = 2 N M=2^{N}
  2. f S f_{S}
  3. N f S Nf_{S}

Modus_ponens.html

  1. P Q , P Q \frac{P\to Q,\;P}{\therefore Q}
  2. P Q , P Q P\to Q,\;P\;\;\vdash\;\;Q
  3. ( ( P Q ) P ) Q ((P\to Q)\land P)\to Q

Modus_tollens.html

  1. P P
  2. Q Q
  3. Q Q
  4. P P
  5. P Q , ¬ Q ¬ P \frac{P\to Q,\neg Q}{\therefore\neg P}
  6. P Q P\to Q
  7. ¬ Q ¬ P {\neg Q}{\to\neg P}
  8. ¬ Q \neg Q
  9. P Q P\to Q
  10. ¬ Q \neg Q
  11. ¬ P \neg P
  12. P Q , ¬ Q ¬ P P\to Q,\neg Q\vdash\neg P
  13. \vdash
  14. ¬ P \neg P
  15. P Q P\to Q
  16. ¬ Q \neg Q
  17. ( ( P Q ) and ¬ Q ) ¬ P ((P\to Q)\and\neg Q)\to\neg P
  18. P P
  19. Q Q
  20. Γ P Q Γ ¬ Q Γ ¬ P \frac{\Gamma\vdash P\to Q~{}~{}~{}\Gamma\vdash\neg Q}{\Gamma\vdash\neg P}
  21. P Q P\subseteq Q
  22. x Q x\notin Q
  23. x P \therefore x\notin P
  24. x : P ( x ) Q ( x ) \forall x:~{}P(x)\to Q(x)
  25. x : ¬ Q ( x ) \exists x:~{}\neg Q(x)
  26. x : ¬ P ( x ) \therefore\exists x:~{}\neg P(x)
  27. P Q P\rightarrow Q
  28. ¬ Q \neg Q
  29. ¬ P Q \neg PQ
  30. ¬ P \neg P
  31. P Q P\rightarrow Q
  32. ¬ Q \neg Q
  33. P P
  34. Q Q
  35. Q and ¬ Q Q\and\neg Q
  36. ¬ P \neg P

Moiré_pattern.html

  1. p p
  2. p + δ p p+\delta p
  3. 0 < δ < 1 0<\delta<1
  4. p 2 \frac{p}{2}
  5. n n
  6. n δ p n\cdot\delta p
  7. n n
  8. n δ p = p 2 n\cdot\delta p=\frac{p}{2}
  9. n = p 2 δ p . n=\frac{p}{2\delta p}.
  10. d = n p = p 2 2 δ p d=n\cdot p=\frac{p^{2}}{2\delta p}
  11. 2 d = p 2 δ p 2d=\frac{p^{2}}{\delta p}
  12. δ p \delta p
  13. f = 1 + sin ( k x ) 2 f=\frac{1+\sin(kx)}{2}
  14. k k
  15. 2 π 2\pi
  16. Δ x \Delta x
  17. k Δ x = 2 π k\Delta x=2\pi
  18. Δ x = 2 π k \Delta x=\frac{2\pi}{k}
  19. f 1 = 1 + sin ( k 1 x ) 2 f_{1}=\frac{1+\sin(k_{1}x)}{2}
  20. f 2 = 1 + sin ( k 2 x ) 2 f_{2}=\frac{1+\sin(k_{2}x)}{2}
  21. k 1 k 2 k_{1}\approx k_{2}
  22. f 3 = f 1 + f 2 2 f_{3}=\frac{f_{1}+f_{2}}{2}
  23. = 1 2 + sin ( k 1 x ) + sin ( k 2 x ) 4 =\frac{1}{2}+\frac{\sin(k_{1}x)+\sin(k_{2}x)}{4}
  24. = 1 + sin ( A x ) cos ( B x ) 2 =\frac{1+\sin(Ax)\cos(Bx)}{2}
  25. A = k 1 + k 2 2 A=\frac{k_{1}+k_{2}}{2}
  26. B = k 1 - k 2 2 . B=\frac{k_{1}-k_{2}}{2}.
  27. f 3 f_{3}
  28. A A
  29. k 1 k_{1}
  30. k 2 k_{2}
  31. cos ( B x ) \cos(Bx)
  32. k 1 k_{1}
  33. k 2 k_{2}
  34. p p
  35. α \alpha
  36. d = p sin α d=\frac{p}{\sin\alpha}
  37. d d
  38. α \alpha
  39. p p
  40. α 2 \frac{\alpha}{2}
  41. D D
  42. 2 D 2D
  43. d ( 1 + cos α ) d(1+\cos\alpha)
  44. p p
  45. ( 2 D ) 2 = d 2 ( 1 + cos α ) 2 + p 2 (2D)^{2}=d^{2}(1+\cos\alpha)^{2}+p^{2}
  46. ( 2 D ) 2 = p 2 sin 2 α ( 1 + cos α ) 2 + p 2 = p 2 ( ( 1 + cos α ) 2 sin 2 α + 1 ) (2D)^{2}=\frac{p^{2}}{\sin^{2}\alpha}(1+\cos\alpha)^{2}+p^{2}=p^{2}\cdot\left(% \frac{(1+\cos\alpha)^{2}}{\sin^{2}\alpha}+1\right)
  47. ( 2 D ) 2 = 2 p 2 1 + cos α sin 2 α (2D)^{2}=2p^{2}\cdot\frac{1+\cos\alpha}{\sin^{2}\alpha}
  48. D = p 2 / sin α 2 . D=\frac{p}{2}/\sin\frac{\alpha}{2}.
  49. α \alpha
  50. α < π 6 \alpha<\frac{\pi}{6}
  51. sin α α \sin\alpha\approx\alpha
  52. cos α 1 \cos\alpha\approx 1
  53. D p α . D\approx\frac{p}{\alpha}.
  54. α \alpha
  55. α = 0 \alpha=0
  56. α \alpha
  57. α p D \alpha\approx\frac{p}{D}

Mole_fraction.html

  1. x i x_{i}
  2. n i n_{i}
  3. n t o t n_{tot}
  4. x i = n i n t o t x_{i}=\frac{n_{i}}{n_{tot}}
  5. i = 1 N n i = n t o t ; i = 1 N x i = 1 \sum_{i=1}^{N}n_{i}=n_{tot};\;\sum_{i=1}^{N}x_{i}=1
  6. N i N_{i}
  7. N t o t N_{tot}
  8. χ \chi
  9. x x
  10. y y
  11. w i w_{i}
  12. w i = x i M i M w_{i}=x_{i}\cdot\frac{M_{i}}{M}
  13. M i M_{i}
  14. i i
  15. M M
  16. w i = x i M i i x i M i w_{i}=x_{i}\cdot\frac{M_{i}}{\sum_{i}x_{i}M_{i}}
  17. ρ i \rho_{i}
  18. x i = ρ i ρ M M i x_{i}=\frac{\rho_{i}}{\rho}\cdot\frac{M}{M_{i}}
  19. M M
  20. ρ i = x i ρ M i M \rho_{i}=x_{i}\rho\cdot\frac{M_{i}}{M}
  21. c i c_{i}
  22. c i = < m t p l > x i ρ M = x i c c_{i}=\frac{<}{m}tpl>{{x_{i}\cdot\rho}}{{M}}=x_{i}c
  23. c i = < m t p l > x i ρ i x i M i c_{i}=\frac{<}{m}tpl>{{x_{i}\cdot\rho}}{{\sum_{i}x_{i}M_{i}}}
  24. M M
  25. ρ \rho
  26. m i m_{i}
  27. M i M_{i}
  28. x i = < m t p l > m i M i i m i M i x_{i}=\frac{<}{m}tpl>{{\frac{{m_{i}}}{{M_{i}}}}}{{\sum_{i}\frac{{m_{i}}}{{M_{i% }}}}}

Molecular_diffusion.html

  1. N A = - D A B d C A d x N_{A}=-D_{AB}\frac{dC_{A}}{dx}
  2. N A = - N B N_{A}=-N_{B}
  3. d P A d x = - d P B d x \frac{dP_{A}}{dx}=-\frac{dP_{B}}{dx}
  4. P A V = n A R T P_{A}V=n_{A}RT
  5. P A = C A R T P_{A}=C_{A}RT
  6. N A = - D A B 1 R T d P A d x N_{A}=-D_{AB}\frac{1}{RT}\frac{dP_{A}}{dx}
  7. N B = - D B A 1 R T d P B d x = D A B 1 R T d P A d x N_{B}=-D_{BA}\frac{1}{RT}\frac{dP_{B}}{dx}=D_{AB}\frac{1}{RT}\frac{dP_{A}}{dx}
  8. N A = - D R T ( P A 2 - P A 1 ) x 2 - x 1 N_{A}=-\frac{D}{RT}\frac{(P_{A2}-P_{A1})}{x_{2}-x_{1}}

Molecular_orbital.html

  1. Ψ = c a ψ a + c b ψ b \Psi=c_{a}\psi_{a}+c_{b}\psi_{b}
  2. Ψ * = c a ψ a - c b ψ b \Psi^{*}=c_{a}\psi_{a}-c_{b}\psi_{b}
  3. Ψ \Psi
  4. Ψ * \Psi^{*}
  5. ψ a \psi_{a}
  6. ψ b \psi_{b}
  7. c a c_{a}
  8. c b c_{b}

Molybdenum.html

  1. N 2 + 8 H + + 8 e - + 16 ATP + 16 H 2 O 2 NH 3 + H 2 + 16 ADP + 16 P i \mathrm{N_{2}+8\ H^{+}+8\ e^{-}+16\ ATP+16\ H_{2}O\longrightarrow 2\ NH_{3}+H_% {2}+16\ ADP+16\ P_{i}}

Momentum.html

  1. 𝐩 = m 𝐯 . \mathbf{p}=m\mathbf{v}.
  2. p p
  3. m m
  4. v v
  5. p = m v . p=mv.
  6. p = p 1 + p 2 = m 1 v 1 + m 2 v 2 . \begin{aligned}\displaystyle p&\displaystyle=p_{1}+p_{2}\\ &\displaystyle=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}
  7. r cm = m 1 r 1 + m 2 r 2 + m 1 + m 2 + . r\text{cm}=\frac{m_{1}r_{1}+m_{2}r_{2}+\cdots}{m_{1}+m_{2}+\cdots}.
  8. p = m v cm . p=mv\text{cm}.
  9. F F
  10. Δ t Δt
  11. Δ p = F Δ t . \Delta p=F\Delta t\,.
  12. F F
  13. F = d p d t . F=\frac{dp}{dt}.
  14. Δ p = t 1 t 2 F ( t ) d t . \Delta p=\int_{t_{1}}^{t_{2}}F(t)\,dt\,.
  15. m m
  16. F = m d v d t = m a , F=m\frac{dv}{dt}=ma,
  17. d p 1 d t = - d p 2 d t , \frac{dp_{1}}{dt}=-\frac{dp_{2}}{dt},
  18. d d t ( p 1 + p 2 ) = 0. \frac{d}{dt}\left(p_{1}+p_{2}\right)=0.
  19. m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 . m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}.
  20. x x
  21. u u
  22. x = x - u t . x^{\prime}=x-ut\,.
  23. d x / d t = v dx/dt=v
  24. v = d x d t = v - u . v^{\prime}=\frac{dx^{\prime}}{dt}=v-u\,.
  25. u u
  26. a = d v d t = a . a^{\prime}=\frac{dv^{\prime}}{dt}=a\,.
  27. m 1 u 1 + m 2 u 2 \displaystyle m_{1}u_{1}+m_{2}u_{2}
  28. m m
  29. v v
  30. v / 2 v/2
  31. v / 2 v/2
  32. v v
  33. v 1 = u 2 v 2 = u 1 . \begin{aligned}\displaystyle v_{1}&\displaystyle=u_{2}\\ \displaystyle v_{2}&\displaystyle=u_{1}\,.\end{aligned}
  34. v 1 = ( m 1 - m 2 m 1 + m 2 ) u 1 + ( 2 m 2 m 1 + m 2 ) u 2 v_{1}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)u_{1}+\left(\frac{2m_{2}}{m_% {1}+m_{2}}\right)u_{2}\,
  35. v 2 = ( m 2 - m 1 m 1 + m 2 ) u 2 + ( 2 m 1 m 1 + m 2 ) u 1 . v_{2}=\left(\frac{m_{2}-m_{1}}{m_{1}+m_{2}}\right)u_{2}+\left(\frac{2m_{1}}{m_% {1}+m_{2}}\right)u_{1}\,.
  36. m 1 u 1 = ( m 1 + m 2 ) v , m_{1}u_{1}=\left(m_{1}+m_{2}\right)v\,,
  37. v = m 1 m 1 + m 2 u 1 . v=\frac{m_{1}}{m_{1}+m_{2}}u_{1}\,.
  38. v ) v)
  39. C R = bounce height drop height . C\text{R}=\sqrt{\frac{\,\text{bounce height}}{\,\text{drop height}}}\,.
  40. x , y , z x,y,z
  41. x x
  42. y y
  43. z z
  44. 𝐯 = ( v x , v y , v z ) . \mathbf{v}=\left(v_{x},v_{y},v_{z}\right).
  45. 𝐩 = ( p x , p y , p z ) . \mathbf{p}=\left(p_{x},p_{y},p_{z}\right).
  46. p p
  47. v v
  48. 𝐩 \mathbf{p}
  49. 𝐯 \mathbf{v}
  50. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  51. p x = m v x p y = m v y p z = m v z . \begin{aligned}\displaystyle p_{x}&\displaystyle=mv_{x}\\ \displaystyle p_{y}&\displaystyle=mv_{y}\\ \displaystyle p_{z}&\displaystyle=mv_{z}.\end{aligned}
  52. v 2 = v x 2 + v y 2 + v z 2 . v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\,.
  53. m ( t ) m(t)
  54. t t
  55. p ( t ) = m ( t ) v ( t ) p(t)=m(t)v(t)
  56. F F
  57. p ( t ) p(t)
  58. F = d p / d t F=dp/dt
  59. d ( m v ) / d t d(mv)/dt
  60. F = m ( t ) d v d t + v ( t ) d m d t . F=m(t)\frac{dv}{dt}+v(t)\frac{dm}{dt}.\qquad\mathrm{}
  61. F = m ( t ) d v d t - u d m d t , F=m(t)\frac{dv}{dt}-u\frac{dm}{dt},
  62. u u
  63. v v
  64. c c
  65. v v
  66. x x
  67. t = t x = x - v t \begin{aligned}\displaystyle t^{\prime}&\displaystyle=t\\ \displaystyle x^{\prime}&\displaystyle=x-vt\end{aligned}
  68. t = γ ( t - v x c 2 ) x = γ ( x - v t ) \begin{aligned}\displaystyle t^{\prime}&\displaystyle=\gamma\left(t-\frac{vx}{% c^{2}}\right)\\ \displaystyle x^{\prime}&\displaystyle=\gamma\left(x-vt\right)\end{aligned}
  69. γ γ
  70. γ = 1 1 - ( v / c ) 2 . \gamma=\frac{1}{\sqrt{1-(v/c)^{2}}}\,.
  71. m m
  72. m = γ m 0 ; m=\gamma m_{0}\,;
  73. 𝐩 = γ m 0 𝐯 , \mathbf{p}=\gamma m_{0}\mathbf{v}\,,
  74. 𝐅 = d 𝐩 d t . \mathbf{F}=\frac{d\mathbf{p}}{dt}\,.
  75. 𝐑 \mathbf{R}
  76. τ τ
  77. c 2 d τ 2 = c 2 d t 2 - d x 2 - d y 2 - d z 2 , c^{2}d\tau^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}\,,
  78. - 1 -1
  79. 𝐔 𝐕 = U 0 V 0 - U 1 V 1 - U 2 V 2 - U 3 V 3 . \mathbf{U}\cdot\mathbf{V}=U_{0}V_{0}-U_{1}V_{1}-U_{2}V_{2}-U_{3}V_{3}\,.
  80. 𝐔 d 𝐑 d τ = γ d 𝐑 d t , \mathbf{U}\equiv\frac{d\mathbf{R}}{d\tau}=\gamma\frac{d\mathbf{R}}{dt}\,,
  81. 𝐏 = m 0 𝐔 , \mathbf{P}=m_{0}\mathbf{U}\,,
  82. 𝐑 = ( c t , x , y , z ) \mathbf{R}=(ct,x,y,z)
  83. 𝐏 = γ m 0 ( c , 𝐯 ) = ( m c , 𝐩 ) . \mathbf{P}=\gamma m_{0}\left(c,\mathbf{v}\right)=(mc,\mathbf{p})\,.
  84. 𝐏 = ( E c , 𝐩 ) . \mathbf{P}=\left(\frac{E}{c},\mathbf{p}\right)\,.
  85. || 𝐏 || 2 = 𝐏 𝐏 = γ 2 m 0 2 ( c 2 - v 2 ) = ( m 0 c ) 2 , ||\mathbf{P}||^{2}=\mathbf{P}\cdot\mathbf{P}=\gamma^{2}m_{0}^{2}(c^{2}-v^{2})=% (m_{0}c)^{2}\,,
  86. E = p c . E=pc\,.
  87. 𝐩 = i = - i , \mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla\,,
  88. 𝐩 ψ ( p ) = p ψ ( p ) , \mathbf{p}\psi(p)=p\psi(p)\,,
  89. λ λ
  90. p = h / λ . p=h/\lambda.\,
  91. T T
  92. V V
  93. = T - V . \mathcal{L}=T-V\,.
  94. N N
  95. d d t ( q ˙ j ) - q j = 0 . \frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{q}_{j}}\right)-\frac% {\partial\mathcal{L}}{\partial q_{j}}=0\,.
  96. 𝚷 \mathbf{Π}
  97. p j = q ˙ j . p_{j}=\frac{\partial\mathcal{L}}{\partial\dot{q}_{j}}\,.
  98. p j = constant . p_{j}=\,\text{constant}\,.
  99. ( 𝐪 , 𝐩 , t ) = 𝐩 𝐪 ˙ - ( 𝐪 , 𝐪 ˙ , t ) , \mathcal{H}\left(\mathbf{q},\mathbf{p},t\right)=\mathbf{p}\cdot\dot{\mathbf{q}% }-\mathcal{L}\left(\mathbf{q},\dot{\mathbf{q}},t\right)\,,
  100. q ˙ i = p i - p ˙ i = q i - t = d d t . \begin{aligned}\displaystyle\dot{q}_{i}&\displaystyle=\frac{\partial\mathcal{H% }}{\partial p_{i}}\\ \displaystyle-\dot{p}_{i}&\displaystyle=\frac{\partial\mathcal{H}}{\partial q_% {i}}\\ \displaystyle-\frac{\partial\mathcal{L}}{\partial t}&\displaystyle=\frac{d% \mathcal{H}}{dt}\,.\end{aligned}
  101. q q
  102. 𝐄 \mathbf{E}
  103. 𝐁 \mathbf{B}
  104. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) . \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}).
  105. 𝐠 = 1 μ 0 c 2 𝐄 × 𝐁 , \mathbf{g}=\frac{1}{\mu_{0}c^{2}}\mathbf{E}\times\mathbf{B}\,,
  106. c c
  107. 𝐒 \mathbf{S}
  108. 𝐠 = 𝐒 c 2 . \mathbf{g}=\frac{\mathbf{S}}{c^{2}}\,.
  109. V V
  110. V V
  111. d 𝐏 mech d t = V ( ρ 𝐄 + 𝐉 × 𝐁 ) d V . \frac{d\mathbf{P}\text{mech}}{dt}=\int_{V}\left(\rho\mathbf{E}+\mathbf{J}% \times\mathbf{B}\right)dV\,.
  112. 𝐏 field = 1 μ 0 c 2 V 𝐄 × 𝐁 d V , \mathbf{P}\text{field}=\frac{1}{\mu_{0}c^{2}}\int_{V}\mathbf{E}\times\mathbf{B% }dV\,,
  113. i i
  114. d d t ( 𝐏 mech + 𝐏 field ) i = S j T i j n j d S . \frac{d}{dt}\left(\mathbf{P}\text{mech}+\mathbf{P}\text{field}\right)_{i}=% \oint_{S}\sum_{j}T_{ij}n_{j}dS\,.
  115. S S
  116. S S
  117. T i j ϵ 0 ( E i E j - 1 2 δ i j E 2 ) + 1 μ 0 ( B i B j - 1 2 δ i j B 2 ) . T_{ij}\equiv\epsilon_{0}\left(E_{i}E_{j}-\frac{1}{2}\delta_{ij}E^{2}\right)+% \frac{1}{\mu_{0}}\left(B_{i}B_{j}-\frac{1}{2}\delta_{ij}B^{2}\right)\,.
  118. 𝐠 = 1 c 2 𝐄 × 𝐇 = 𝐒 c 2 , \mathbf{g}=\frac{1}{c^{2}}\mathbf{E}\times\mathbf{H}=\frac{\mathbf{S}}{c^{2}}\,,
  119. 𝐇 \mathbf{H}
  120. 𝐌 \mathbf{M}
  121. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) . \mathbf{B}=\mu_{0}\left(\mathbf{H}+\mathbf{M}\right)\,.
  122. q q
  123. m 𝐯 m\mathbf{v}
  124. 𝐩 \mathbf{p}
  125. 𝐏 \mathbf{P}
  126. 𝐫 \mathbf{r}
  127. 𝐏 \mathbf{P}
  128. φ ( 𝐫 , t ) φ(\mathbf{r},t)
  129. 𝐀 ( 𝐫 , t ) \mathbf{A}(\mathbf{r},t)
  130. = m 2 𝐫 ˙ 𝐫 ˙ + q 𝐀 𝐫 ˙ - q φ \mathcal{L}=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot% \mathbf{\dot{r}}-q\varphi\,\!
  131. = - m c 2 γ - 1 + e 𝐀 𝐫 ˙ - e φ \mathcal{L}=-mc^{2}\gamma^{-1}+e\mathbf{A}\cdot\dot{\mathbf{r}}-e\varphi\,\!
  132. P = L 𝐫 ˙ {P}=\frac{\partial L}{\partial\dot{\mathbf{r}}}
  133. 𝐏 = m 𝐯 + q 𝐀 \mathbf{P}=m\mathbf{v}+q\mathbf{A}
  134. 𝐏 = γ m 𝐫 ˙ + e 𝐀 \mathbf{P}=\gamma m\dot{\mathbf{r}}+e\mathbf{A}
  135. p = m 𝐫 ˙ {p}=m\mathbf{\dot{r}}
  136. m 𝐯 = 𝐏 - q 𝐀 m\mathbf{v}=\mathbf{P}-q\mathbf{A}
  137. 𝐏 - e 𝐀 = γ m 𝐫 ˙ \mathbf{P}-e\mathbf{A}=\gamma m\dot{\mathbf{r}}
  138. = T + V = 𝐩 2 2 m + V = ( 𝐏 - q 𝐀 ) 2 2 m + q φ \begin{aligned}\displaystyle\mathcal{H}&\displaystyle=T+V\\ &\displaystyle=\frac{\mathbf{p}^{2}}{2m}+V\\ &\displaystyle=\frac{(\mathbf{P}-q\mathbf{A})^{2}}{2m}+q\varphi\end{aligned}
  139. = 𝐏 𝐫 ˙ - = γ m c 2 + e φ = c 2 ( 𝐏 - e 𝐀 ) 2 + ( m c 2 ) 2 + e φ \begin{aligned}\displaystyle\mathcal{H}&\displaystyle=\mathbf{P}\cdot\dot{% \mathbf{r}}-\mathcal{L}\\ &\displaystyle=\gamma mc^{2}+e\varphi\\ &\displaystyle=\sqrt{c^{2}(\mathbf{P}-e\mathbf{A})^{2}+(mc^{2})^{2}}+e\varphi% \end{aligned}
  140. = 𝐯 \mathbf{ṙ}=\mathbf{v}
  141. q q
  142. φ φ
  143. 𝐀 \mathbf{A}
  144. 𝐏 \mathbf{P}
  145. \mathcal{H}
  146. V V
  147. V = e φ V=eφ
  148. T T
  149. 𝐩 \mathbf{p}
  150. 𝐩 = 𝐏 q 𝐀 \mathbf{p}=\mathbf{P}−q\mathbf{A}
  151. [ p j , p k ] = i e c ϵ j k B \left[p_{j},p_{k}\right]=\frac{i\hbar e}{c}\epsilon_{jk\ell}B_{\ell}
  152. ρ ρ
  153. 𝐯 \mathbf{v}
  154. t t
  155. 𝐫 \mathbf{r}
  156. ρ 𝐯 ρ\mathbf{v}
  157. ρ 𝐠 ρ\mathbf{g}
  158. 𝐠 \mathbf{g}
  159. p p
  160. - p + ρ 𝐠 = 0 . -\nabla p+\rho\mathbf{g}=0\,.
  161. 𝐯 / t ∂\mathbf{v}/∂t
  162. D D t t + 𝐯 \cdotsymbol . \frac{D}{Dt}\equiv\frac{\partial}{\partial t}+\mathbf{v}\cdotsymbol{\nabla}\,.
  163. ρ D 𝐯 / D t ρD\mathbf{v}/Dt
  164. τ τ
  165. x x
  166. z z
  167. x x
  168. z z
  169. σ zx = - μ v x z , \sigma\text{zx}=-\mu\frac{\partial v\text{x}}{\partial z}\,,
  170. μ μ
  171. ρ D 𝐯 D t = - s y m b o l p + μ 2 𝐯 + ρ 𝐠 . \rho\frac{D\mathbf{v}}{Dt}=-symbol{\nabla}p+\mu\nabla^{2}\mathbf{v}+\rho% \mathbf{g}.\,
  172. i i
  173. j j
  174. σ \mathbf{σ}
  175. ρ D 𝐯 D t = s y m b o l s y m b o l σ + 𝐟 , \rho\frac{D\mathbf{v}}{Dt}=symbol{\nabla}\cdot symbol{\sigma}+\mathbf{f}\,,
  176. 𝐟 \mathbf{f}
  177. p p
  178. 2 p t 2 = c 2 2 p , \frac{\partial^{2}p}{\partial t^{2}}=c^{2}\nabla^{2}p\,,
  179. c c
  180. ρ v < s u b > j ρv<sub>j