wpmath0000004_6

Friedrichs_extension.html

  1. ξ T ξ 0 ξ dom T \langle\xi\mid T\xi\rangle\geq 0\quad\xi\in\operatorname{dom}\ T
  2. [ T ϕ ] ( x ) = - i , j x i { a i j ( x ) x j ϕ ( x ) } x U , ϕ C 0 ( U ) , [T\phi](x)=-\sum_{i,j}\partial_{x_{i}}\{a_{ij}(x)\partial_{x_{j}}\phi(x)\}% \quad x\in U,\phi\in\operatorname{C}_{0}^{\infty}(U),
  3. C 0 ( U ) L 2 ( U ) . \operatorname{C}_{0}^{\infty}(U)\subseteq L^{2}(U).
  4. [ a 11 ( x ) a 12 ( x ) a 1 n ( x ) a 21 ( x ) a 22 ( x ) a 2 n ( x ) a n 1 ( x ) a n 2 ( x ) a n n ( x ) ] \begin{bmatrix}a_{11}(x)&a_{12}(x)&\cdots&a_{1n}(x)\\ a_{21}(x)&a_{22}(x)&\cdots&a_{2n}(x)\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}(x)&a_{n2}(x)&\cdots&a_{nn}(x)\end{bmatrix}
  5. i , j a i j ( x ) c i c j ¯ 0 \sum_{i,j}a_{ij}(x)c_{i}\overline{c_{j}}\geq 0
  6. Q ( ξ , η ) = ξ T η + ξ η \operatorname{Q}(\xi,\eta)=\langle\xi\mid T\eta\rangle+\langle\xi\mid\eta\rangle
  7. Q ( ξ , ξ ) = ξ T ξ + ξ ξ ξ 2 . \operatorname{Q}(\xi,\xi)=\langle\xi\mid T\xi\rangle+\langle\xi\mid\xi\rangle% \geq\|\xi\|^{2}.
  8. dom T H \operatorname{dom}\ T\rightarrow H
  9. dom A = { ξ H 1 : ϕ ξ : η Q ( ξ , η ) is bounded linear. } \operatorname{dom}\ A=\{\xi\in H_{1}:\phi_{\xi}:\eta\mapsto\operatorname{Q}(% \xi,\eta)\mbox{ is bounded linear.}~{}\}
  10. Q ( ξ , η ) = A ξ η η H 1 \operatorname{Q}(\xi,\eta)=\langle A\xi\mid\eta\rangle\quad\eta\in H_{1}
  11. T S T\leq S
  12. dom ( S 1 / 2 ) dom ( T 1 / 2 ) \operatorname{dom}(S^{1/2})\subseteq\operatorname{dom}(T^{1/2})
  13. T 1 / 2 ξ T 1 / 2 ξ S 1 / 2 ξ S 1 / 2 ξ ξ dom ( S 1 / 2 ) \langle T^{1/2}\xi\mid T^{1/2}\xi\rangle\leq\langle S^{1/2}\xi\mid S^{1/2}\xi% \rangle\quad\forall\xi\in\operatorname{dom}(S^{1/2})
  14. T min T max , T_{\mathrm{min}}\leq T_{\mathrm{max}},
  15. T min S T max . T_{\mathrm{min}}\leq S\leq T_{\mathrm{max}}.

Frobenius_endomorphism.html

  1. p p
  2. p p
  3. R R
  4. p p
  5. F ( r ) = r p F(r)=r^{p}
  6. F ( r s ) = ( r s ) p = r p s p = F ( r ) F ( s ) , F(rs)=(rs)^{p}=r^{p}s^{p}=F(r)F(s)\ ,
  7. F ( 1 ) F(1)
  8. R R
  9. p p
  10. p ! p!
  11. q ! q!
  12. p ! k ! ( p - k ) ! , \frac{p!}{k!(p-k)!},
  13. 1 k p 1 1≤k≤p−1
  14. p p
  15. F ( r + s ) = ( r + s ) p = r p + s p = F ( r ) + F ( s ) . F(r+s)=(r+s)^{p}=r^{p}+s^{p}=F(r)+F(s)\ .
  16. φ : R S φ:R→S
  17. p p
  18. ϕ ( x p ) = ϕ ( x ) p . \phi(x^{p})=\phi(x)^{p}.
  19. R R
  20. S S
  21. ϕ F R = F S ϕ . \phi\circ F_{R}=F_{S}\circ\phi.
  22. p p
  23. R R
  24. F ( r ) = 0 F(r)=0
  25. r r
  26. p p
  27. r r
  28. p p
  29. R R
  30. R R
  31. p p
  32. K K
  33. F F
  34. t t
  35. q ( t ) / r ( t ) q(t)/r(t)
  36. p p
  37. t t
  38. p p
  39. p d e g ( q ) p d e g ( r ) pdeg(q)−pdeg(r)
  40. p p
  41. t t
  42. t t
  43. F F
  44. K K
  45. x x
  46. p p
  47. p p
  48. p p
  49. K K
  50. K K
  51. R R
  52. p > 0 p>0
  53. R R
  54. R R
  55. p p
  56. 𝐅 p e \mathbf{F}_{p^{e}}
  57. 𝐅 p e \mathbf{F}_{p^{e}}
  58. X p e - X X^{p^{e}}-X
  59. K K
  60. 𝐅 p e \mathbf{F}_{p^{e}}
  61. F F
  62. K K
  63. 𝐅 p e \mathbf{F}_{p^{e}}
  64. 𝐅 p e \mathbf{F}_{p^{e}}
  65. 𝐅 p e \mathbf{F}_{p^{e}}
  66. R R
  67. x , x p , x p 2 , x p 3 , . x,x^{p},x^{p^{2}},x^{p^{3}},\ldots.
  68. q q
  69. F F
  70. F F
  71. F F
  72. e e
  73. x x
  74. F F
  75. i i
  76. e e
  77. Gal ( 𝐅 q ¯ / 𝐅 q ) , \operatorname{Gal}\left(\overline{\mathbf{F}_{q}}/\mathbf{F}_{q}\right),
  78. 𝐙 ^ = lim n 𝐙 / n 𝐙 , \widehat{\mathbf{Z}}=\textstyle\underleftarrow{\lim}_{n}\mathbf{Z}/n\mathbf{Z},
  79. X X
  80. p > 0 p>0
  81. U = S p e c A U=SpecA
  82. X X
  83. A A
  84. V V
  85. U U
  86. U U
  87. V V
  88. V V
  89. X X
  90. X X
  91. X X
  92. X X
  93. S S
  94. S S
  95. S S
  96. A = 𝐅 p 2 A=\mathbf{F}_{p^{2}}
  97. X X
  98. S S
  99. S p e c A SpecA
  100. X S X→S
  101. A A
  102. a a
  103. 𝐅 p 2 \mathbf{F}_{p^{2}}
  104. b b
  105. 𝐅 p 2 \mathbf{F}_{p^{2}}
  106. b a = b a F ( b ) a = b p a . b\cdot a=ba\neq F(b)\cdot a=b^{p}a.
  107. b b
  108. 𝐅 p 2 \mathbf{F}_{p^{2}}
  109. A A
  110. 𝐅 p 2 \mathbf{F}_{p^{2}}
  111. S p e c A SpecA
  112. 𝐅 p 2 \mathbf{F}_{p^{2}}
  113. p p
  114. X X
  115. Y Y
  116. φ : X S φ:X→S
  117. S S
  118. X X
  119. S S
  120. φ φ
  121. S S
  122. S S
  123. X Y X→Y
  124. S S
  125. p > 0 p>0
  126. R = A [ X 1 , , X n ] / ( f 1 , , f m ) . R=A[X_{1},\ldots,X_{n}]/(f_{1},\ldots,f_{m}).
  127. c a α X α = c a α X α , c\cdot\sum a_{\alpha}X^{\alpha}=\sum ca_{\alpha}X^{\alpha},
  128. X = S p e c R X=SpecR
  129. S p e c R SpecR
  130. S p e c R S p e c A SpecR→SpecA
  131. c a α X α = F ( c ) a α X α = c p a α X α . c\cdot\sum a_{\alpha}X^{\alpha}=\sum F(c)a_{\alpha}X^{\alpha}=\sum c^{p}a_{% \alpha}X^{\alpha}.
  132. X X
  133. X X
  134. X ( p ) = X × S S F . X^{(p)}=X\times_{S}S_{F}.
  135. S S
  136. S S
  137. S S
  138. S S
  139. X Y X→Y
  140. S S
  141. X = S p e c R X=SpecR
  142. X ( p ) = Spec R A A F . X^{(p)}=\operatorname{Spec}R\otimes_{A}A_{F}.
  143. i ( α a i α X α ) b i = i α X α a i α p b i , \sum_{i}\left(\sum_{\alpha}a_{i\alpha}X^{\alpha}\right)\otimes b_{i}=\sum_{i}% \sum_{\alpha}X^{\alpha}\otimes a_{i\alpha}^{p}b_{i},
  144. c i ( α a i α X α ) b i = i ( α a i α X α ) b i c . c\cdot\sum_{i}\left(\sum_{\alpha}a_{i\alpha}X^{\alpha}\right)\otimes b_{i}=% \sum_{i}\left(\sum_{\alpha}a_{i\alpha}X^{\alpha}\right)\otimes b_{i}c.
  145. Spec A [ X 1 , , X n ] / ( f 1 ( p ) , , f m ( p ) ) , \operatorname{Spec}A[X_{1},\ldots,X_{n}]/\left(f_{1}^{(p)},\ldots,f_{m}^{(p)}% \right),
  146. f j = β f j β X β , f_{j}=\sum_{\beta}f_{j\beta}X^{\beta},
  147. f j ( p ) = β f j β p X β . f_{j}^{(p)}=\sum_{\beta}f_{j\beta}^{p}X^{\beta}.
  148. X X
  149. S S S′→S
  150. X ( p / S ) × S S ( X × S S ) ( p / S ) . X^{(p/S)}\times_{S}S^{\prime}\cong(X\times_{S}S^{\prime})^{(p/S^{\prime})}.
  151. S S
  152. F X / S : X X ( p ) F_{X/S}:X\to X^{(p)}
  153. F X / S = ( F X , 1 S ) . F_{X/S}=(F_{X},1_{S}).
  154. S S
  155. R = A [ X 1 , , X n ] / ( f 1 , , f m ) . R=A[X_{1},\ldots,X_{n}]/(f_{1},\ldots,f_{m}).
  156. R ( p ) = A [ X 1 , , X n ] / ( f 1 ( p ) , , f m ( p ) ) . R^{(p)}=A[X_{1},\ldots,X_{n}]/(f_{1}^{(p)},\ldots,f_{m}^{(p)}).
  157. i α X α a i α i α a i α X p α . \sum_{i}\sum_{\alpha}X^{\alpha}\otimes a_{i\alpha}\mapsto\sum_{i}\sum_{\alpha}% a_{i\alpha}X^{p\alpha}.
  158. F X / S × 1 S = F X × S S / S . F_{X/S}\times 1_{S^{\prime}}=F_{X\times_{S}S^{\prime}/S^{\prime}}.
  159. X S X→S
  160. X S X→S
  161. S S
  162. S S
  163. S S
  164. X X
  165. F X / S a : X ( p ) X × S S X F^{a}_{X/S}:X^{(p)}\to X\times_{S}S\cong X
  166. F X / S a = 1 X × F S . F^{a}_{X/S}=1_{X}\times F_{S}.
  167. R = A [ X 1 , , X n ] / ( f 1 , , f m ) , R=A[X_{1},\ldots,X_{n}]/(f_{1},\ldots,f_{m}),
  168. R ( p ) = A [ X 1 , , X n ] / ( f 1 , , f m ) A A F , R^{(p)}=A[X_{1},\ldots,X_{n}]/(f_{1},\ldots,f_{m})\otimes_{A}A_{F},
  169. i ( α a i α X α ) b i i α a i α b i p X α . \sum_{i}\left(\sum_{\alpha}a_{i\alpha}X^{\alpha}\right)\otimes b_{i}\mapsto% \sum_{i}\sum_{\alpha}a_{i\alpha}b_{i}^{p}X^{\alpha}.
  170. R ( p ) = A [ X 1 , , X n ] / ( f 1 ( p ) , , f m ( p ) ) , R^{(p)}=A[X_{1},\ldots,X_{n}]/\left(f_{1}^{(p)},\ldots,f_{m}^{(p)}\right),
  171. a α X α a α p X α . \sum a_{\alpha}X^{\alpha}\mapsto\sum a_{\alpha}^{p}X^{\alpha}.
  172. S S
  173. F S - 1 F_{S}^{-1}
  174. S F - 1 S_{F^{-1}}
  175. S S
  176. F S - 1 : S S F_{S}^{-1}:S\to S
  177. X X
  178. F S - 1 F_{S}^{-1}
  179. X ( 1 / p ) = X × S S F - 1 . X^{(1/p)}=X\times_{S}S_{F^{-1}}.
  180. R = A [ X 1 , , X n ] / ( f 1 , , f m ) , R=A[X_{1},\ldots,X_{n}]/(f_{1},\ldots,f_{m}),
  181. F S - 1 F_{S}^{-1}
  182. R ( 1 / p ) = A [ X 1 , , X n ] / ( f 1 , , f m ) A A F - 1 . R^{(1/p)}=A[X_{1},\ldots,X_{n}]/(f_{1},\ldots,f_{m})\otimes_{A}A_{F^{-1}}.
  183. f j = β f j β X β , f_{j}=\sum_{\beta}f_{j\beta}X^{\beta},
  184. f j ( 1 / p ) = β f j β 1 / p X β , f_{j}^{(1/p)}=\sum_{\beta}f_{j\beta}^{1/p}X^{\beta},
  185. R ( 1 / p ) A [ X 1 , , X n ] / ( f 1 ( 1 / p ) , , f m ( 1 / p ) ) . R^{(1/p)}\cong A[X_{1},\ldots,X_{n}]/(f_{1}^{(1/p)},\ldots,f_{m}^{(1/p)}).
  186. S S
  187. X X
  188. F X / S g : X ( 1 / p ) X × S S X F^{g}_{X/S}:X^{(1/p)}\to X\times_{S}S\cong X
  189. F X / S g = 1 X × F S - 1 . F^{g}_{X/S}=1_{X}\times F_{S}^{-1}.
  190. F S - 1 F_{S}^{-1}
  191. i ( α a i α X α ) b i i α a i α b i 1 / p X α . \sum_{i}\left(\sum_{\alpha}a_{i\alpha}X^{\alpha}\right)\otimes b_{i}\mapsto% \sum_{i}\sum_{\alpha}a_{i\alpha}b_{i}^{1/p}X^{\alpha}.
  192. { f j ( 1 / p ) } \{f_{j}^{(1/p)}\}
  193. a α X α a α 1 / p X α . \sum a_{\alpha}X^{\alpha}\mapsto\sum a_{\alpha}^{1/p}X^{\alpha}.
  194. S S
  195. S S
  196. S = S p e c k S=Speck
  197. X X
  198. X X
  199. X ( K ) X(K)
  200. 𝒪 X k ( x ) 𝐹 k ( x ) \mathcal{O}_{X}\to k(x)\xrightarrow{F}k(x)
  201. 𝒪 X F X / S a 𝒪 X k ( x ) \mathcal{O}_{X}\xrightarrow{F^{a}_{X/S}}\mathcal{O}_{X}\to k(x)
  202. L / K L/K
  203. L / K L/K
  204. K K
  205. K K
  206. φ φ
  207. q q
  208. Φ Φ
  209. L L
  210. φ φ
  211. L / K L/K
  212. L L
  213. Φ Φ
  214. L L
  215. K K
  216. f f
  217. L / K L/K
  218. L L
  219. L L
  220. s Φ ( x ) x q mod Φ . s_{\Phi}(x)\equiv x^{q}\mod\Phi.
  221. L / K L/K
  222. Φ Φ
  223. L L
  224. L / K L/K
  225. Φ Φ
  226. L L
  227. s Φ ( x ) x q mod Φ , s_{\Phi}(x)\equiv x^{q}\mod\Phi,
  228. q q
  229. Φ Φ
  230. 19 × 151 19×151
  231. ρ ρ
  232. 3 3
  233. ρ ρ
  234. ρ ρ
  235. 3 3
  236. ρ 3 + 3 ( 460 + 183 ρ - 354 ρ 2 - 979 ρ 3 - 575 ρ 4 ) \rho^{3}+3(460+183\rho-354\rho^{2}-979\rho^{3}-575\rho^{4})
  237. 𝐐 \mathbf{Q}
  238. 𝐐 \mathbf{Q}
  239. p p
  240. L / K L/K
  241. φ φ
  242. K K
  243. 𝐐 ( β ) \mathbf{Q}(β)
  244. 𝐐 \mathbf{Q}
  245. β β
  246. β 5 + β 4 - 4 β 3 - 3 β 2 + 3 β + 1 = 0 \beta^{5}+\beta^{4}-4\beta^{3}-3\beta^{2}+3\beta+1=0
  247. 𝐐 \mathbf{Q}
  248. 2 cos 2 π n 11 2\cos\tfrac{2\pi n}{11}
  249. n n
  250. β β
  251. β < s u p > 2 2 , β 3 3 β , β 5 5 β 3 + 5 β β<sup>2−2,β^{3}−3β,β^{5}−5β^{3}+5β

Frobenius_method.html

  1. z 2 u ′′ + p ( z ) z u + q ( z ) u = 0 z^{2}u^{\prime\prime}+p(z)zu^{\prime}+q(z)u=0
  2. u d u d z u^{\prime}\equiv{{du}\over{dz}}
  3. u ′′ d 2 u d z 2 u^{\prime\prime}\equiv{{d^{2}u}\over{dz^{2}}}
  4. z = 0 z=0
  5. z 2 z^{2}
  6. u ′′ + p ( z ) z u + q ( z ) z 2 u = 0 u^{\prime\prime}+{p(z)\over z}u^{\prime}+{q(z)\over z^{2}}u=0
  7. u ( z ) = k = 0 A k z k + r , ( A 0 0 ) u(z)=\sum_{k=0}^{\infty}A_{k}z^{k+r},\qquad(A_{0}\neq 0)
  8. u ( z ) = k = 0 ( k + r ) A k z k + r - 1 u^{\prime}(z)=\sum_{k=0}^{\infty}(k+r)A_{k}z^{k+r-1}
  9. u ′′ ( z ) = k = 0 ( k + r - 1 ) ( k + r ) A k z k + r - 2 u^{\prime\prime}(z)=\sum_{k=0}^{\infty}(k+r-1)(k+r)A_{k}z^{k+r-2}
  10. z 2 k = 0 ( k + r - 1 ) ( k + r ) A k z k + r - 2 + z p ( z ) k = 0 ( k + r ) A k z k + r - 1 + q ( z ) k = 0 A k z k + r z^{2}\sum_{k=0}^{\infty}(k+r-1)(k+r)A_{k}z^{k+r-2}+zp(z)\sum_{k=0}^{\infty}(k+% r)A_{k}z^{k+r-1}+q(z)\sum_{k=0}^{\infty}A_{k}z^{k+r}
  11. = k = 0 ( k + r - 1 ) ( k + r ) A k z k + r + p ( z ) k = 0 ( k + r ) A k z k + r + q ( z ) k = 0 A k z k + r =\sum_{k=0}^{\infty}(k+r-1)(k+r)A_{k}z^{k+r}+p(z)\sum_{k=0}^{\infty}(k+r)A_{k}% z^{k+r}+q(z)\sum_{k=0}^{\infty}A_{k}z^{k+r}
  12. = k = 0 [ ( k + r - 1 ) ( k + r ) A k z k + r + p ( z ) ( k + r ) A k z k + r + q ( z ) A k z k + r ] =\sum_{k=0}^{\infty}[(k+r-1)(k+r)A_{k}z^{k+r}+p(z)(k+r)A_{k}z^{k+r}+q(z)A_{k}z% ^{k+r}]
  13. = k = 0 [ ( k + r - 1 ) ( k + r ) + p ( z ) ( k + r ) + q ( z ) ] A k z k + r =\sum_{k=0}^{\infty}\left[(k+r-1)(k+r)+p(z)(k+r)+q(z)\right]A_{k}z^{k+r}
  14. = [ r ( r - 1 ) + p ( z ) r + q ( z ) ] A 0 z r + k = 1 [ ( k + r - 1 ) ( k + r ) + p ( z ) ( k + r ) + q ( z ) ] A k z k + r =\left[r(r-1)+p(z)r+q(z)\right]A_{0}z^{r}+\sum_{k=1}^{\infty}\left[(k+r-1)(k+r% )+p(z)(k+r)+q(z)\right]A_{k}z^{k+r}
  15. r ( r - 1 ) + p ( 0 ) r + q ( 0 ) = I ( r ) r\left(r-1\right)+p\left(0\right)r+q\left(0\right)=I(r)
  16. I ( k + r ) A k + j = 0 k - 1 ( j + r ) p ( k - j ) ( 0 ) + q ( k - j ) ( 0 ) ( k - j ) ! A j I(k+r)A_{k}+\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0)\over(k-j)!}A_{j}
  17. I ( k + r ) A k + j = 0 k - 1 ( j + r ) p ( k - j ) ( 0 ) + q ( k - j ) ( 0 ) ( k - j ) ! A j = 0 I(k+r)A_{k}+\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0)\over(k-j)!}A_{j}=0
  18. j = 0 k - 1 ( j + r ) p ( k - j ) ( 0 ) + q ( k - j ) ( 0 ) ( k - j ) ! A j = - I ( k + r ) A k \sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0)\over(k-j)!}A_{j}=-I(k+r)A_{k}
  19. 1 - I ( k + r ) j = 0 k - 1 ( j + r ) p ( k - j ) ( 0 ) + q ( k - j ) ( 0 ) ( k - j ) ! A j = A k {1\over-I(k+r)}\sum_{j=0}^{k-1}{(j+r)p^{(k-j)}(0)+q^{(k-j)}(0)\over(k-j)!}A_{j% }=A_{k}
  20. U r ( z ) = k = 0 A k z k + r U_{r}(z)=\sum_{k=0}^{\infty}A_{k}z^{k+r}
  21. z 2 U r ( z ) ′′ + p ( z ) z U r ( z ) + q ( z ) U r ( z ) = I ( r ) z r z^{2}U_{r}(z)^{\prime\prime}+p(z)zU_{r}(z)^{\prime}+q(z)U_{r}(z)=I(r)z^{r}
  22. z 2 f ′′ - z f + ( 1 - z ) f = 0 z^{2}f^{\prime\prime}-zf^{\prime}+(1-z)f=0\,
  23. f ′′ - 1 z f + 1 - z z 2 f = f ′′ - 1 z f + ( 1 z 2 - 1 z ) f = 0 f^{\prime\prime}-{1\over z}f^{\prime}+{1-z\over z^{2}}f=f^{\prime\prime}-{1% \over z}f^{\prime}+\left({1\over z^{2}}-{1\over z}\right)f=0
  24. f \displaystyle f
  25. k = 0 \displaystyle\sum_{k=0}^{\infty}
  26. ( k + 1 - 1 ) 2 A k - A k - 1 = k 2 A k - A k - 1 = 0 (k+1-1)^{2}A_{k}-A_{k-1}=k^{2}A_{k}-A_{k-1}=0
  27. A k = A k - 1 k 2 A_{k}=\frac{A_{k-1}}{k^{2}}
  28. A k / A k - 1 A_{k}/A_{k-1}
  29. y 2 = C y 1 ln x + k = 0 B k x k + r 2 y_{2}=Cy_{1}\ln x+\sum_{k=0}^{\infty}B_{k}x^{k+r_{2}}
  30. y 1 ( x ) y_{1}(x)
  31. r 2 r_{2}
  32. C C
  33. B k B_{k}
  34. C C
  35. 1 1

Frobenius_normal_form.html

  1. A = ( - 1 3 - 1 0 - 2 0 0 - 2 - 1 - 1 1 1 - 2 - 1 0 - 1 - 2 - 6 4 3 - 8 - 4 - 2 1 - 1 8 - 3 - 1 5 2 3 - 3 0 0 0 0 0 0 0 1 0 0 0 0 - 1 0 0 0 1 0 0 0 2 0 0 0 0 0 0 0 4 0 1 0 ) . A=\begin{pmatrix}-1&3&-1&0&-2&0&0&-2\\ -1&-1&1&1&-2&-1&0&-1\\ -2&-6&4&3&-8&-4&-2&1\\ -1&8&-3&-1&5&2&3&-3\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&-1&0&0&0\\ 1&0&0&0&2&0&0&0\\ 0&0&0&0&4&0&1&0\end{pmatrix}.
  2. μ = X 6 - 4 X 4 - 2 X 3 + 4 X 2 + 4 X + 1 \mu=X^{6}-4X^{4}-2X^{3}+4X^{2}+4X+1
  3. χ = X 8 - X 7 - 5 X 6 + 2 X 5 + 10 X 4 + 2 X 3 - 7 X 2 - 5 X - 1 \chi=X^{8}-X^{7}-5X^{6}+2X^{5}+10X^{4}+2X^{3}-7X^{2}-5X-1
  4. X 2 - X - 1 X^{2}-X-1
  5. e 1 e_{1}
  6. A k ( e 1 ) A^{k}(e_{1})
  7. k = 0 , 1 , , 5 k=0,1,\ldots,5
  8. μ \mu
  9. v = ( 3 , 4 , 8 , 0 , - 1 , 0 , 2 , - 1 ) v=(3,4,8,0,-1,0,2,-1)^{\top}
  10. w = ( 5 , 4 , 5 , 9 , - 1 , 1 , 1 , - 2 ) w=(5,4,5,9,-1,1,1,-2)^{\top}
  11. A v = w A\cdot v=w
  12. v v
  13. X 2 - X - 1 X^{2}-X-1
  14. μ \mu
  15. X 2 - X - 1 X^{2}-X-1
  16. μ \mu
  17. X 2 - X - 1 X^{2}-X-1
  18. μ = X 6 - 4 X 4 - 2 X 3 + 4 X 2 + 4 X + 1 \mu=X^{6}-4X^{4}-2X^{3}+4X^{2}+4X+1
  19. C = ( 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 - 4 0 0 0 1 0 0 0 - 4 0 0 0 0 1 0 0 2 0 0 0 0 0 1 0 4 0 0 0 0 0 0 1 0 ) . C=\begin{pmatrix}0&1&0&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&-1\\ 0&0&1&0&0&0&0&-4\\ 0&0&0&1&0&0&0&-4\\ 0&0&0&0&1&0&0&2\\ 0&0&0&0&0&1&0&4\\ 0&0&0&0&0&0&1&0\end{pmatrix}.
  20. v , w v,w
  21. A k ( e 1 ) A^{k}(e_{1})
  22. k = 0 , 1 , , 5 k=0,1,\ldots,5
  23. P = ( 3 5 1 - 1 0 0 - 4 0 4 4 0 - 1 - 1 - 2 - 3 - 5 8 5 0 - 2 - 5 - 2 - 11 - 6 0 9 0 - 1 3 - 2 0 0 - 1 - 1 0 0 0 1 - 1 4 0 1 0 0 0 0 - 1 1 2 1 0 1 - 1 0 2 - 6 - 1 - 2 0 0 1 - 1 4 - 2 ) one has A = P C P - 1 . P=\begin{pmatrix}3&5&1&-1&0&0&-4&0\\ 4&4&0&-1&-1&-2&-3&-5\\ 8&5&0&-2&-5&-2&-11&-6\\ 0&9&0&-1&3&-2&0&0\\ -1&-1&0&0&0&1&-1&4\\ 0&1&0&0&0&0&-1&1\\ 2&1&0&1&-1&0&2&-6\\ -1&-2&0&0&1&-1&4&-2\end{pmatrix}\quad\,\text{one has}\quad A=PCP^{-1}.
  24. ( C 0 0 U C 0 0 U C ) \begin{pmatrix}C&0&\cdots&0\\ U&C&\cdots&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&\cdots&U&C\end{pmatrix}
  25. P P
  26. U U
  27. P = x λ P=x−λ
  28. C = λ C=λ
  29. U = 1 U=1
  30. v v
  31. v , A ( v ) , A 2 ( v ) , , A d - 1 ( v ) , P ( A ) ( v ) , A ( P ( A ) ( v ) ) , , A d - 1 ( P ( A ) ( v ) ) , P 2 ( A ) ( v ) , , P k - 1 ( A ) ( v ) , , A d - 1 ( P k - 1 ( A ) ( v ) ) v,A(v),A^{2}(v),\ldots,A^{d-1}(v),~{}P(A)(v),A(P(A)(v)),\ldots,A^{d-1}(P(A)(v)% ),~{}P^{2}(A)(v),\ldots,~{}P^{k-1}(A)(v),\ldots,A^{d-1}(P^{k-1}(A)(v))
  32. d = d e g ( P ) d=deg(P)

Frobenius–Schur_indicator.html

  1. g G χ ( g 2 ) d μ \int_{g\in G}\chi(g^{2})\,d\mu
  2. 1 | G | g G χ ( g 2 ) . {1\over|G|}\sum_{g\in G}\chi(g^{2}).
  3. V V V\otimes V
  4. V S V V\otimes_{S}V
  5. V A V . V\otimes_{A}V.
  6. χ V S V ( g ) = 1 2 [ χ V ( g ) 2 + χ V ( g 2 ) ] \chi_{V\otimes_{S}V}(g)=\frac{1}{2}[\chi_{V}(g)^{2}+\chi_{V}(g^{2})]
  7. χ V A V ( g ) = 1 2 [ χ V ( g ) 2 - χ V ( g 2 ) ] \chi_{V\otimes_{A}V}(g)=\frac{1}{2}[\chi_{V}(g)^{2}-\chi_{V}(g^{2})]
  8. 1 | G | g G χ V S V ( g ) \frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_{S}V}(g)
  9. 1 | G | g G χ V A V ( g ) \frac{1}{|G|}\sum_{g\in G}\chi_{V\otimes_{A}V}(g)
  10. V S V V\otimes_{S}V
  11. V A V , V\otimes_{A}V,
  12. V V V\otimes V
  13. 1 | G | g G ρ ( g ) \frac{1}{|G|}\sum_{g\in G}\rho(g)
  14. 1 | G | g G ρ ( g n ) \frac{1}{|G|}\sum_{g\in G}\rho(g^{n})
  15. g G g n \sum_{g\in G}g^{n}

Fugacity.html

  1. f f
  2. P = 100 atm P=100\,\rm atm
  3. f = 97.03 atm f=97.03\,\rm atm
  4. ϕ \phi\,
  5. ϕ = f / P \phi=f/P\,
  6. ϕ \phi\,
  7. ϕ \phi
  8. d μ = d G = - S d T + V d P {d\mu}=dG=-SdT+VdP
  9. P P
  10. P P
  11. μ μ d μ = P P V ¯ d P \int_{\mu^{\circ}}^{\mu}{d\mu}=\int_{P^{\circ}}^{P}{\bar{V}dP}
  12. V ¯ = < m t p l > R T P \bar{V}=\frac{<}{m}tpl>{{RT}}{P}
  13. μ - μ = P P < m t p l > R T P d P = R T ln P P \mu-\mu^{\circ}=\int_{P^{\circ}}^{P}{\frac{<}{m}tpl>{{RT}}{P}dP}=RT\ln\frac{P}% {{P^{\circ}}}
  14. μ = μ + R T ln P < m t p l > P \mu=\mu^{\circ}+RT\ln\frac{P}{<}mtpl>{{P^{\circ}}}
  15. P P
  16. P P V ¯ d P \int_{P^{\circ}}^{P}{\bar{V}dP}
  17. μ = μ + R T ln f < m t p l > f \mu=\mu^{\circ}+RT\ln\frac{f}{<}mtpl>{{f^{\circ}}}
  18. f = < m t p l > f exp ( μ - μ R T ) f=<mtpl>{{f^{\circ}}}\exp\left(\frac{\mu-\mu^{\circ}}{RT}\right)
  19. T T\,
  20. f f\,
  21. d ln f f 0 = d G R T = V ¯ d P R T d\ln{f\over f_{0}}={dG\over RT}={{\bar{V}dP}\over RT}\,
  22. G G\,
  23. R R\,
  24. V ¯ \bar{V}\,
  25. f 0 f_{0}\,
  26. f = P f=P
  27. f 2 f 1 = exp ( 1 R T G 1 G 2 d G ) = exp ( 1 R T P 1 P 2 V ¯ d P ) {f_{2}\over f_{1}}=\exp\left({1\over RT}\int_{G_{1}}^{G_{2}}dG\right)=\exp% \left({1\over RT}\int_{P_{1}}^{P_{2}}\bar{V}\,dP\right)\,
  28. f 2 f 1 = p 2 p 1 {f_{2}\over f_{1}}={p_{2}\over p_{1}}
  29. f = P f=P
  30. P 0 P\to 0
  31. lim P 0 f P = 1 \mathop{\lim}_{P\to 0}\frac{f}{P}=1
  32. f f
  33. Φ \Phi
  34. Φ = < m t p l > P V ¯ - R T P \Phi=\frac{<}{m}tpl>{{P\bar{V}-RT}}{P}
  35. Φ \Phi
  36. V V
  37. T T
  38. P P
  39. V ¯ = < m t p l > R T P + Φ \bar{V}=\frac{<}{m}tpl>{{RT}}{P}+\Phi
  40. Φ \Phi
  41. μ μ d μ = P P V ¯ d P = P P < m t p l > R T P d P + P P Φ d P \int_{\mu^{\circ}}^{\mu}{d\mu}=\int_{P^{\circ}}^{P}{\bar{V}dP}=\int_{P^{\circ}% }^{P}{\frac{<}{m}tpl>{{RT}}{P}dP}+\int_{P^{\circ}}^{P}{\Phi dP}
  42. μ = μ + R T ln P < m t p l > P + P P Φ d P \mu=\mu^{\circ}+RT\ln\frac{P}{<}mtpl>{{P^{\circ}}}+\int_{P^{\circ}}^{P}{\Phi dP}
  43. μ = μ + R T ln f < m t p l > f \mu=\mu^{\circ}+RT\ln\frac{f}{<}mtpl>{{f^{\circ}}}
  44. μ + R T ln f < m t p l > f = μ + R T ln P P + P P Φ d P \mu^{\circ}+RT\ln\frac{f}{<}mtpl>{{f^{\circ}}}=\mu^{\circ}+RT\ln\frac{P}{{P^{% \circ}}}+\int_{P^{\circ}}^{P}{\Phi dP}
  45. R T ln f < m t p l > f - R T ln P P = P P Φ d P \Rightarrow RT\ln\frac{f}{<}mtpl>{{f^{\circ}}}-RT\ln\frac{P}{{P^{\circ}}}=\int% _{P^{\circ}}^{P}{\Phi dP}
  46. R T ln < m t p l > f P P f = P P Φ d P RT\ln\frac{<}{m}tpl>{{fP^{\circ}}}{{Pf^{\circ}}}=\int_{P^{\circ}}^{P}{\Phi dP}
  47. P 0 P^{\circ}\to 0
  48. lim P 0 f = P \mathop{\lim}_{P^{\circ}\to 0}f^{\circ}=P^{\circ}
  49. R T ln f P = 0 P Φ d P RT\ln\frac{f}{P}=\int_{0}^{P}{\Phi dP}
  50. ϕ \phi
  51. ϕ \phi
  52. ln ϕ = 1 < m t p l > R T 0 P Φ d P \ln\phi=\frac{1}{<}mtpl>{{RT}}\int_{0}^{P}{\Phi dP}
  53. Φ \Phi
  54. P P
  55. P P
  56. f = ϕ P f=\phi P\,
  57. P P
  58. T T
  59. μ = μ + R T ln f < m t p l > P \mu=\mu^{\circ}+RT\ln\frac{f}{<}mtpl>{{P^{\circ}}}
  60. f liq ( T , P ) = ϕ sat P sat exp [ P sat P v liq R T d P ] f^{\rm liq}(T,P)=\phi^{\rm sat}P^{\rm sat}{\rm exp}\left[\int_{P^{\rm sat}}^{P% }{v_{\rm liq}\over RT}dP\right]
  61. v liq v_{\rm liq}
  62. ϕ sat \phi^{\rm sat}
  63. P sat P^{\rm sat}

Fuglede's_theorem.html

  1. T N * = ( N T ) * = ( T N ) * = N * T . TN^{*}=(NT)^{*}=(TN)^{*}=N^{*}T.\,
  2. N = i λ i P i N=\sum_{i}\lambda_{i}P_{i}\,
  3. N * = i λ ¯ i P i . N^{*}=\sum_{i}{\bar{\lambda}_{i}}P_{i}.
  4. N = σ ( N ) λ d P ( λ ) . N=\int_{\sigma(N)}\lambda dP(\lambda).\,
  5. ρ = i λ ¯ P Ω i . \rho=\sum_{i}{\bar{\lambda}}P_{\Omega_{i}}.
  6. P Ω i P_{\Omega_{i}}
  7. \mathbb{C}
  8. e λ ¯ M T = T e λ ¯ N . e^{\bar{\lambda}M}T=Te^{\bar{\lambda}N}.
  9. F ( λ ) = e λ M * T e - λ N * . F(\lambda)=e^{\lambda M^{*}}Te^{-\lambda N^{*}}.
  10. e λ M * [ e - λ ¯ M T e λ ¯ N ] e - λ N * = U ( λ ) T V ( λ ) - 1 e^{\lambda M^{*}}\left[e^{-\bar{\lambda}M}Te^{\bar{\lambda}N}\right]e^{-% \lambda N^{*}}=U(\lambda)TV(\lambda)^{-1}
  11. U ( λ ) = e λ M * - λ ¯ M U(\lambda)=e^{\lambda M^{*}-\bar{\lambda}M}
  12. V ( λ ) = e λ N * - λ ¯ N V(\lambda)=e^{\lambda N^{*}-\bar{\lambda}N}
  13. U ( λ ) * = e λ ¯ M - λ M * = U ( λ ) - 1 U(\lambda)^{*}=e^{\bar{\lambda}M-\lambda M^{*}}=U(\lambda)^{-1}
  14. F ( λ ) T λ . \|F(\lambda)\|\leq\|T\|\ \forall\lambda.
  15. T = [ 0 0 T 0 ] and N = [ N 0 0 M ] . T^{\prime}=\begin{bmatrix}0&0\\ T&0\end{bmatrix}\quad\mbox{and}~{}\quad N^{\prime}=\begin{bmatrix}N&0\\ 0&M\end{bmatrix}.
  16. T ( N ) * = ( N ) * T . T^{\prime}(N^{\prime})^{*}=(N^{\prime})^{*}T^{\prime}.\,
  17. S - 1 M * S = N * . S^{-1}M^{*}S=N^{*}.\,
  18. S * M ( S - 1 ) * = N . S^{*}M(S^{-1})^{*}=N.\,
  19. S * M ( S - 1 ) * = S - 1 M S S S * M ( S S * ) - 1 = M . S^{*}M(S^{-1})^{*}=S^{-1}MS\quad\Rightarrow\quad SS^{*}M(SS^{*})^{-1}=M.
  20. N = S * M ( S * ) - 1 = V R M R - 1 V * = V M V * . N=S^{*}M(S^{*})^{-1}=VRMR^{-1}V^{*}=VMV^{*}.
  21. ( M N ) ( M N ) * = M N ( N M ) * = M N M * N * . (MN)(MN)^{*}=MN(NM)^{*}=MNM^{*}N^{*}.\,
  22. = M M * N N * = M * M N * N . =MM^{*}NN^{*}=M^{*}MN^{*}N.\,
  23. = M * N * M N = ( M N ) * M N . =M^{*}N^{*}MN=(MN)^{*}MN.\,

Full_and_faithful_functors.html

  1. F X , Y : Hom 𝒞 ( X , Y ) Hom 𝒟 ( F ( X ) , F ( Y ) ) F_{X,Y}\colon\mathrm{Hom}_{\mathcal{C}}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal{% D}}(F(X),F(Y))
  2. F ( X ) F ( Y ) F(X)\cong F(Y)
  3. X Y X\cong Y

Function_problem.html

  1. P P
  2. R ( x , y ) R(x,y)
  3. Σ \Sigma
  4. R Σ * × Σ * R\subset\Sigma^{*}\times\Sigma^{*}
  5. P P
  6. x x
  7. y y
  8. ( x , y ) R (x,y)\in R
  9. y y
  10. φ \varphi
  11. x 1 , , x n x_{1},\ldots,x_{n}
  12. x i { TRUE , FALSE } x_{i}\rightarrow\{\,\text{TRUE},\,\text{FALSE}\}
  13. φ \varphi
  14. TRUE \,\text{TRUE}
  15. R ( x , y ) R(x,y)
  16. x x
  17. y y
  18. ( x , y ) (x,y)
  19. 𝐅 ( 𝐍𝐏 ) \mathbf{F}(\mathbf{NP})
  20. φ \varphi
  21. x 1 x_{1}
  22. x 1 x_{1}
  23. x 2 x_{2}
  24. x 1 x_{1}
  25. Π R \Pi_{R}
  26. Π S \Pi_{S}
  27. Π R \Pi_{R}
  28. Π S \Pi_{S}
  29. f f
  30. g g
  31. x x
  32. R R
  33. y y
  34. S S
  35. x x
  36. R R
  37. f ( x ) f(x)
  38. S S
  39. ( f ( x ) , y ) S ( x , g ( x , y ) ) R . (f(x),y)\in S\implies(x,g(x,y))\in R.
  40. Π R \Pi_{R}
  41. Π R \Pi_{R}
  42. 𝐅 ( 𝐍𝐏 ) \mathbf{F}(\mathbf{NP})
  43. 𝐏 = 𝐍𝐏 \mathbf{P}=\mathbf{NP}
  44. 𝐅𝐏 = 𝐅𝐍𝐏 \mathbf{FP}=\mathbf{FNP}
  45. R ( x , y ) R(x,y)
  46. x x
  47. y y
  48. ( x , y ) R (x,y)\in R
  49. 𝐓𝐅𝐍𝐏 = 𝐅 ( 𝐍𝐏 co-NP ) \mathbf{TFNP}=\mathbf{F}(\mathbf{NP}\cap\,\textbf{co-NP})
  50. 𝐍𝐏 = co-NP \mathbf{NP}=\,\textbf{co-NP}

Fundamental_class.html

  1. H r ( M ; 𝐙 ) 𝐙 H_{r}(M;\mathbf{Z})\cong\mathbf{Z}
  2. H n ( M , 𝐙 ) 𝐙 H_{n}(M,\mathbf{Z})\cong\mathbf{Z}
  3. 𝐙 H n ( M , 𝐙 ) \mathbf{Z}\to H_{n}(M,\mathbf{Z})
  4. ω , [ M ] = M ω , \langle\omega,[M]\rangle=\int_{M}\omega\ ,
  5. H n ( M , 𝐙 ) 𝐙 H_{n}(M,\mathbf{Z})\ncong\mathbf{Z}
  6. 𝐙 2 \mathbf{Z}_{2}
  7. H n ( M ; 𝐙 2 ) = 𝐙 2 H_{n}(M;\mathbf{Z}_{2})=\mathbf{Z}_{2}
  8. 𝐙 2 \mathbf{Z}_{2}
  9. 𝐙 2 \mathbf{Z}_{2}
  10. 𝐙 2 \mathbf{Z}_{2}
  11. H n ( M , M ) 𝐙 H_{n}(M,\partial M)\cong\mathbf{Z}
  12. G G
  13. q 0 q\geq 0
  14. [ M ] : H q ( M ; G ) H n - q ( M ; G ) [M]\cap:H^{q}(M;G)\rightarrow H_{n-q}(M;G)
  15. q q
  16. H * ( M ; G ) H n - * ( M ; G ) H^{*}(M;G)\cong H_{n-*}(M;G)

Fundamental_representation.html

  1. Alt k n \operatorname{Alt}^{k}\ {\mathbb{C}}^{n}

Fundamental_theorem_of_curves.html

  1. κ \kappa
  2. τ \tau

Fundamental_theorem_of_linear_algebra.html

  1. A = U Σ V T A=U\Sigma V^{\mathrm{T}}
  2. A 𝐑 m × n A\in\mathbf{R}^{m\times n}
  3. A A
  4. m m
  5. n n
  6. im ( A ) \mathrm{im}(A)
  7. range ( A ) \mathrm{range}(A)
  8. 𝐑 m \mathbf{R}^{m}
  9. r r
  10. r r
  11. U U
  12. ker ( A ) \mathrm{ker}(A)
  13. null ( A ) \mathrm{null}(A)
  14. 𝐑 n \mathbf{R}^{n}
  15. n - r n-r
  16. ( n - r ) (n-r)
  17. V V
  18. im ( A T ) \mathrm{im}(A^{\mathrm{T}})
  19. range ( A T ) \mathrm{range}(A^{\mathrm{T}})
  20. 𝐑 n \mathbf{R}^{n}
  21. r r
  22. r r
  23. V V
  24. ker ( A T ) \mathrm{ker}(A^{\mathrm{T}})
  25. null ( A T ) \mathrm{null}(A^{\mathrm{T}})
  26. 𝐑 m \mathbf{R}^{m}
  27. m - r m-r
  28. ( m - r ) (m-r)
  29. U U
  30. 𝐑 n \mathbf{R}^{n}
  31. ker ( A ) = ( im ( A T ) ) \mathrm{ker}(A)=(\mathrm{im}(A^{\mathrm{T}}))^{\perp}
  32. 𝐑 m \mathbf{R}^{m}
  33. ker ( A T ) = ( im ( A ) ) \mathrm{ker}(A^{\mathrm{T}})=(\mathrm{im}(A))^{\perp}
  34. A : V W A\colon V\to W
  35. A * : W * V * A^{*}\colon W^{*}\to V^{*}
  36. A * A^{*}
  37. A A

Fusion_tree.html

  1. x b r x b r - 1 x b 1 x_{b_{r}}x_{b_{r-1}}\cdots x_{b_{1}}
  2. i = 1 r \textstyle\sum_{i=1}^{r}

G-test.html

  1. G = 2 i O i ln ( O i E i ) , G=2\sum_{i}{O_{i}\cdot\ln\left(\frac{O_{i}}{E_{i}}\right)},
  2. χ 2 = i ( O i - E i ) 2 E i . \chi^{2}=\sum_{i}{\frac{\left(O_{i}-E_{i}\right)^{2}}{E_{i}}}.
  3. O i > 2 E i O_{i}>2\cdot E_{i}
  4. N = i j O i j N=\sum_{ij}{O_{ij}}\;
  5. π i j = O i j N \;\pi_{ij}=\frac{O_{ij}}{N}\;
  6. π i . = j O i j N \;\pi_{i.}=\frac{\sum_{j}O_{ij}}{N}\;
  7. π . j = i O i j N \;\pi_{.j}=\frac{\sum_{i}O_{ij}}{N}\;
  8. G = 2 N i j π i j ( ln ( π i j ) - ln ( π i . ) - ln ( π . j ) ) , G=2\cdot N\cdot\sum_{ij}{\pi_{ij}\left(\ln(\pi_{ij})-\ln(\pi_{i.})-\ln(\pi_{.j% })\right)},
  9. G = 2 N [ H ( r ) + H ( c ) - H ( r , c ) ] , G=2\cdot N\cdot\left[H(r)+H(c)-H(r,c)\right],
  10. G = 2 N M I ( r , c ) , G=2\cdot N\cdot MI(r,c)\,,
  11. X X\,
  12. H ( X ) = - x Supp ( X ) p ( x ) log p ( x ) , H(X)=-{\sum_{x\in\,\text{Supp}(X)}p(x)\log p(x)}\,,
  13. M I ( r , c ) = H ( r ) + H ( c ) - H ( r , c ) MI(r,c)=H(r)+H(c)-H(r,c)\,

G2_manifold.html

  1. G 2 G_{2}
  2. ϕ \phi
  3. ψ = * ϕ \psi=*\phi
  4. G 2 G_{2}
  5. G 2 G_{2}
  6. G 2 G_{2}
  7. G 2 G_{2}
  8. G 2 G_{2}
  9. G 2 G_{2}
  10. G 2 G_{2}

Galaxy_morphological_classification.html

  1. n n
  2. T T

Gas_centrifuge.html

  1. W SWU W_{\mathrm{SWU}}
  2. F F
  3. x f x_{f}
  4. P P
  5. x p x_{p}
  6. T T
  7. x t x_{t}
  8. W SWU = P V ( x p ) + T V ( x t ) - F V ( x f ) W_{\mathrm{SWU}}=P\cdot V\left(x_{p}\right)+T\cdot V(x_{t})-F\cdot V(x_{f})
  9. V ( x ) V\left(x\right)
  10. V ( x ) = ( 1 - 2 x ) ln ( 1 - x x ) V(x)=(1-2x)\cdot\ln\left(\frac{1-x}{x}\right)

Gaseous_diffusion.html

  1. Rate 1 Rate 2 = M 2 M 1 = 352.041206 349.034348 = 1.004298... {\mbox{Rate}~{}_{1}\over\mbox{Rate}~{}_{2}}=\sqrt{M_{2}\over M_{1}}=\sqrt{352.% 041206\over 349.034348}=1.004298...

Gaussian_units.html

  1. F = Q 1 Q 2 r 2 F=\frac{Q_{1}Q_{2}}{r^{2}}
  2. F = 1 4 π ϵ 0 Q 1 Q 2 r 2 = k e Q 1 Q 2 r 2 F=\frac{1}{4\pi\epsilon_{0}}\frac{Q_{1}Q_{2}}{r^{2}}=k\text{e}\frac{Q_{1}Q_{2}% }{r^{2}}
  3. μ 0 ϵ 0 = 1 / c 2 \mu_{0}\epsilon_{0}=1/c^{2}
  4. 𝐃 = 4 π ρ f \nabla\cdot\mathbf{D}=4\pi\rho\text{f}
  5. 𝐃 = ρ f \nabla\cdot\mathbf{D}=\rho\text{f}
  6. 𝐄 = 4 π ρ \nabla\cdot\mathbf{E}=4\pi\rho
  7. 𝐄 = ρ / ϵ 0 \nabla\cdot\mathbf{E}=\rho/\epsilon_{0}
  8. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  9. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  10. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}
  11. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  12. × 𝐇 = 4 π c 𝐉 f + 1 c 𝐃 t \nabla\times\mathbf{H}=\frac{4\pi}{c}\mathbf{J}\text{f}+\frac{1}{c}\frac{% \partial\mathbf{D}}{\partial t}
  13. × 𝐇 = 𝐉 f + 𝐃 t \nabla\times\mathbf{H}=\mathbf{J}\text{f}+\frac{\partial\mathbf{D}}{\partial t}
  14. × 𝐁 = 4 π c 𝐉 + 1 c 𝐄 t \nabla\times\mathbf{B}=\frac{4\pi}{c}\mathbf{J}+\frac{1}{c}\frac{\partial% \mathbf{E}}{\partial t}
  15. × 𝐁 = μ 0 𝐉 + 1 c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}
  16. 𝐅 = q ( 𝐄 + 1 c 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\frac{1}{c}\mathbf{v}\times\mathbf{B}\right)
  17. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)
  18. 𝐅 = q 1 q 2 r 2 𝐫 ^ \mathbf{F}=\frac{q_{1}q_{2}}{r^{2}}\mathbf{\hat{r}}
  19. 𝐅 = 1 4 π ϵ 0 q 1 q 2 r 2 𝐫 ^ \mathbf{F}=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}\mathbf{\hat{r}}
  20. 𝐄 = q r 2 𝐫 ^ \mathbf{E}=\frac{q}{r^{2}}\mathbf{\hat{r}}
  21. 𝐄 = 1 4 π ϵ 0 q r 2 𝐫 ^ \mathbf{E}=\frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\mathbf{\hat{r}}
  22. 𝐁 = 1 c I d 𝐥 × 𝐫 ^ r 2 \mathbf{B}=\frac{1}{c}\oint\frac{Id\mathbf{l}\times\mathbf{\hat{r}}}{r^{2}}
  23. 𝐁 = μ 0 4 π I d 𝐥 × 𝐫 ^ r 2 \mathbf{B}=\frac{\mu_{0}}{4\pi}\oint\frac{Id\mathbf{l}\times\mathbf{\hat{r}}}{% r^{2}}
  24. 𝐃 = 𝐄 + 4 π 𝐏 \mathbf{D}=\mathbf{E}+4\pi\mathbf{P}
  25. 𝐃 = ϵ 0 𝐄 + 𝐏 \mathbf{D}=\epsilon_{0}\mathbf{E}+\mathbf{P}
  26. 𝐏 = χ e 𝐄 \mathbf{P}=\chi\text{e}\mathbf{E}
  27. 𝐏 = χ e ϵ 0 𝐄 \mathbf{P}=\chi\text{e}\epsilon_{0}\mathbf{E}
  28. 𝐃 = ϵ 𝐄 \mathbf{D}=\epsilon\mathbf{E}
  29. 𝐃 = ϵ 𝐄 \mathbf{D}=\epsilon\mathbf{E}
  30. ϵ = 1 + 4 π χ e \epsilon=1+4\pi\chi\text{e}
  31. ϵ / ϵ 0 = 1 + χ e \epsilon/\epsilon_{0}=1+\chi\text{e}
  32. ϵ \epsilon
  33. ϵ 0 \epsilon_{0}
  34. χ e \chi\text{e}
  35. ϵ \epsilon
  36. ϵ / ϵ 0 \epsilon/\epsilon_{0}
  37. χ e \chi_{e}
  38. χ eSI = 4 π χ eG \chi\text{e}\text{SI}=4\pi\chi\text{e}\text{G}
  39. 𝐁 = 𝐇 + 4 π 𝐌 \mathbf{B}=\mathbf{H}+4\pi\mathbf{M}
  40. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) \mathbf{B}=\mu_{0}(\mathbf{H}+\mathbf{M})
  41. 𝐌 = χ m 𝐇 \mathbf{M}=\chi\text{m}\mathbf{H}
  42. 𝐌 = χ m 𝐇 \mathbf{M}=\chi\text{m}\mathbf{H}
  43. 𝐁 = μ 𝐇 \mathbf{B}=\mu\mathbf{H}
  44. 𝐁 = μ 𝐇 \mathbf{B}=\mu\mathbf{H}
  45. μ = 1 + 4 π χ m \mu=1+4\pi\chi\text{m}
  46. μ / μ 0 = 1 + χ m \mu/\mu_{0}=1+\chi\text{m}
  47. μ \mu
  48. μ 0 \mu_{0}
  49. χ m \chi\text{m}
  50. μ \mu
  51. μ / μ 0 \mu/\mu_{0}
  52. χ m \chi\text{m}
  53. χ mSI = 4 π χ mG \chi\text{m}\text{SI}=4\pi\chi\text{m}\text{G}
  54. 𝐄 = - ϕ \mathbf{E}=-\nabla\phi
  55. 𝐄 = - ϕ \mathbf{E}=-\nabla\phi
  56. 𝐄 = - ϕ - 1 c 𝐀 t \mathbf{E}=-\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}
  57. 𝐄 = - ϕ - 𝐀 t \mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}
  58. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  59. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  60. c c
  61. 1 ϵ 0 μ 0 \frac{1}{\sqrt{\epsilon_{0}\mu_{0}}}
  62. ( 𝐄 , φ ) \left(\mathbf{E},\varphi\right)
  63. 4 π ϵ 0 ( 𝐄 , φ ) \sqrt{4\pi\epsilon_{0}}\left(\mathbf{E},\varphi\right)
  64. 𝐃 \mathbf{D}
  65. 4 π ϵ 0 𝐃 \sqrt{\frac{4\pi}{\epsilon_{0}}}\mathbf{D}
  66. ( q , ρ , I , 𝐉 , 𝐏 , 𝐩 ) \left(q,\rho,I,\mathbf{J},\mathbf{P},\mathbf{p}\right)
  67. 1 4 π ϵ 0 ( q , ρ , I , 𝐉 , 𝐏 , 𝐩 ) \frac{1}{\sqrt{4\pi\epsilon_{0}}}\left(q,\rho,I,\mathbf{J},\mathbf{P},\mathbf{% p}\right)
  68. ( 𝐁 , Φ m , 𝐀 ) \left(\mathbf{B},\Phi\text{m},\mathbf{A}\right)
  69. 4 π μ 0 ( 𝐁 , Φ m , 𝐀 ) \sqrt{\frac{4\pi}{\mu_{0}}}\left(\mathbf{B},\Phi\text{m},\mathbf{A}\right)
  70. 𝐇 \mathbf{H}
  71. 4 π μ 0 𝐇 \sqrt{4\pi\mu_{0}}\mathbf{H}
  72. ( 𝐦 , 𝐌 ) \left(\mathbf{m},\mathbf{M}\right)
  73. μ 0 4 π ( 𝐦 , 𝐌 ) \sqrt{\frac{\mu_{0}}{4\pi}}\left(\mathbf{m},\mathbf{M}\right)
  74. ( ϵ , μ ) \left(\epsilon,\mu\right)
  75. ( ϵ ϵ 0 , μ μ 0 ) \left(\frac{\epsilon}{\epsilon_{0}},\frac{\mu}{\mu_{0}}\right)
  76. ( χ e , χ m ) \left(\chi\text{e},\chi\text{m}\right)
  77. 1 4 π ( χ e , χ m ) \frac{1}{4\pi}\left(\chi\text{e},\chi\text{m}\right)
  78. ( σ , S , C ) \left(\sigma,S,C\right)
  79. 1 4 π ϵ 0 ( σ , S , C ) \frac{1}{4\pi\epsilon_{0}}\left(\sigma,S,C\right)
  80. ( ρ , R , L ) \left(\rho,R,L\right)
  81. 4 π ϵ 0 ( ρ , R , L ) 4\pi\epsilon_{0}\left(\rho,R,L\right)

Gauss–Newton_algorithm.html

  1. S ( s y m b o l β ) = i = 1 m r i ( s y m b o l β ) 2 . S(symbol\beta)=\sum_{i=1}^{m}r_{i}(symbol\beta)^{2}.
  2. s y m b o l β ( 0 ) symbol\beta^{(0)}
  3. s y m b o l β ( s + 1 ) = s y m b o l β ( s ) - ( 𝐉 𝐫 𝖳 𝐉 𝐫 ) - 1 𝐉 𝐫 𝖳 𝐫 ( s y m b o l β ( s ) ) symbol\beta^{(s+1)}=symbol\beta^{(s)}-\left(\mathbf{J_{r}}^{\mathsf{T}}\mathbf% {J_{r}}\right)^{-1}\mathbf{J_{r}}^{\mathsf{T}}\mathbf{r}(symbol\beta^{(s)})
  4. ( 𝐉 𝐫 ) i j = r i ( s y m b o l β ( s ) ) β j , (\mathbf{J_{r}})_{ij}=\frac{\partial r_{i}(symbol\beta^{(s)})}{\partial\beta_{% j}},
  5. 𝖳 {}^{\mathsf{T}}
  6. s y m b o l β ( s + 1 ) = s y m b o l β ( s ) - ( 𝐉 𝐫 ) - 1 𝐫 ( s y m b o l β ( s ) ) symbol\beta^{(s+1)}=symbol\beta^{(s)}-\left(\mathbf{J_{r}}\right)^{-1}\mathbf{% r}(symbol\beta^{(s)})
  7. r i ( s y m b o l β ) = y i - f ( x i , s y m b o l β ) . r_{i}(symbol\beta)=y_{i}-f(x_{i},symbol\beta).
  8. s y m b o l β ( s + 1 ) = s y m b o l β ( s ) + ( 𝐉 𝐟 𝖳 𝐉 𝐟 ) - 1 𝐉 𝐟 𝖳 𝐫 ( s y m b o l β ( s ) ) . symbol\beta^{(s+1)}=symbol\beta^{(s)}+\left(\mathbf{J_{f}}^{\mathsf{T}}\mathbf% {J_{f}}\right)^{-1}\mathbf{J_{f}}^{\mathsf{T}}\mathbf{r}(symbol\beta^{(s)}).
  9. 𝐫 ( s y m b o l β ) 𝐫 ( s y m b o l β s ) + 𝐉 𝐫 ( s y m b o l β s ) Δ \mathbf{r}(symbol\beta)\approx\mathbf{r}(symbol\beta^{s})+\mathbf{J_{r}}(% symbol\beta^{s})\Delta
  10. Δ = s y m b o l β - s y m b o l β s . \Delta=symbol\beta-symbol\beta^{s}.
  11. 𝐦𝐢𝐧 𝐫 ( s y m b o l β s ) + 𝐉 𝐫 ( s y m b o l β s ) Δ 2 2 \mathbf{min}\|\mathbf{r}(symbol\beta^{s})+\mathbf{J_{r}}(symbol\beta^{s})% \Delta\|_{2}^{2}
  12. rate = V max [ S ] K M + [ S ] \,\text{rate}=\frac{V\text{max}[S]}{K_{M}+[S]}
  13. V max V\text{max}
  14. K M K_{M}
  15. x i x_{i}
  16. y i y_{i}
  17. [ S ] [S]
  18. i = 1 , , 7. i=1,\dots,7.
  19. β 1 = V max \beta_{1}=V\text{max}
  20. β 2 = K M . \beta_{2}=K_{M}.
  21. β 1 \beta_{1}
  22. β 2 \beta_{2}
  23. r i = y i - β 1 x i β 2 + x i r_{i}=y_{i}-\frac{\beta_{1}x_{i}}{\beta_{2}+x_{i}}
  24. i = 1 , , 7 i=1,\dots,7
  25. 𝐉 𝐫 \mathbf{J_{r}}
  26. r i r_{i}
  27. β j \beta_{j}
  28. 7 × 2 7\times 2
  29. i i
  30. r i β 1 = - x i β 2 + x i , r i β 2 = β 1 x i ( β 2 + x i ) 2 . \frac{\partial r_{i}}{\partial\beta_{1}}=-\frac{x_{i}}{\beta_{2}+x_{i}},\ % \frac{\partial r_{i}}{\partial\beta_{2}}=\frac{\beta_{1}x_{i}}{\left(\beta_{2}% +x_{i}\right)^{2}}.
  31. β 1 \beta_{1}
  32. β 2 \beta_{2}
  33. β ^ 1 = 0.362 \hat{\beta}_{1}=0.362
  34. β ^ 2 = 0.556 \hat{\beta}_{2}=0.556
  35. 𝐉 𝐫 𝖳 𝐉 𝐫 \mathbf{J_{r}^{\mathsf{T}}J_{r}}
  36. m = 2 m=2
  37. n = 1 n=1
  38. r 1 ( β ) = β + 1 r 2 ( β ) = λ β 2 + β - 1. \begin{aligned}\displaystyle r_{1}(\beta)&\displaystyle=\beta+1\\ \displaystyle r_{2}(\beta)&\displaystyle=\lambda\beta^{2}+\beta-1.\end{aligned}
  39. β = 0 \beta=0
  40. β = - 1 \beta=-1
  41. λ = 2 \lambda=2
  42. S ( 0 ) = 1 2 + ( - 1 ) 2 = 2 S(0)=1^{2}+(-1)^{2}=2
  43. S ( - 1 ) = 0 S(-1)=0
  44. λ = 0 \lambda=0
  45. s y m b o l β symbol\beta
  46. s y m b o l β ( s + 1 ) = s y m b o l β ( s ) - 𝐇 - 1 𝐠 symbol\beta^{(s+1)}=symbol\beta^{(s)}-\mathbf{H}^{-1}\mathbf{g}\,
  47. S = i = 1 m r i 2 S=\sum_{i=1}^{m}r_{i}^{2}
  48. g j = 2 i = 1 m r i r i β j . g_{j}=2\sum_{i=1}^{m}r_{i}\frac{\partial r_{i}}{\partial\beta_{j}}.
  49. g j g_{j}
  50. β k \beta_{k}
  51. H j k = 2 i = 1 m ( r i β j r i β k + r i 2 r i β j β k ) . H_{jk}=2\sum_{i=1}^{m}\left(\frac{\partial r_{i}}{\partial\beta_{j}}\frac{% \partial r_{i}}{\partial\beta_{k}}+r_{i}\frac{\partial^{2}r_{i}}{\partial\beta% _{j}\partial\beta_{k}}\right).
  52. H j k 2 i = 1 m J i j J i k H_{jk}\approx 2\sum_{i=1}^{m}J_{ij}J_{ik}
  53. J i j = r i β j J_{ij}=\frac{\partial r_{i}}{\partial\beta_{j}}
  54. 𝐠 = 2 𝐉 𝐫 𝖳 𝐫 , 𝐇 2 𝐉 𝐫 𝖳 𝐉 𝐫 . \mathbf{g}=2\mathbf{J}_{\mathbf{r}}^{\mathsf{T}}\mathbf{r},\quad\mathbf{H}% \approx 2\mathbf{J}_{\mathbf{r}}^{\mathsf{T}}\mathbf{J_{r}}.\,
  55. s y m b o l β ( s + 1 ) = s y m b o l β ( s ) + Δ ; Δ = - ( 𝐉 𝐫 𝖳 𝐉 𝐫 ) - 1 𝐉 𝐫 𝖳 𝐫 . symbol{\beta}^{(s+1)}=symbol\beta^{(s)}+\Delta;\quad\Delta=-\left(\mathbf{J_{r% }}^{\mathsf{T}}\mathbf{J_{r}}\right)^{-1}\mathbf{J_{r}}^{\mathsf{T}}\mathbf{r}.
  56. | r i 2 r i β j β k | | r i β j r i β k | \left|r_{i}\frac{\partial^{2}r_{i}}{\partial\beta_{j}\partial\beta_{k}}\right|% \ll\left|\frac{\partial r_{i}}{\partial\beta_{j}}\frac{\partial r_{i}}{% \partial\beta_{k}}\right|
  57. r i r_{i}
  58. 2 r i β j β k \frac{\partial^{2}r_{i}}{\partial\beta_{j}\partial\beta_{k}}
  59. S ( s y m b o l β s ) S(symbol\beta^{s})
  60. S ( s y m b o l β s + α Δ ) < S ( s y m b o l β s ) S(symbol\beta^{s}+\alpha\Delta)<S(symbol\beta^{s})
  61. α > 0 \alpha>0
  62. α \alpha
  63. s y m b o l β s + 1 = s y m b o l β s + α Δ symbol\beta^{s+1}=symbol\beta^{s}+\alpha\ \Delta
  64. α \alpha
  65. α \alpha
  66. 0 < α < 1 0<\alpha<1
  67. α \alpha
  68. ( 𝐉 𝐓 𝐉 + λ 𝐃 ) Δ = - 𝐉 T 𝐫 \left(\mathbf{J^{T}J+\lambda D}\right)\Delta=-\mathbf{J}^{T}\mathbf{r}
  69. λ + \lambda\to+\infty
  70. Δ / λ - 𝐉 T 𝐫 \Delta/\lambda\to-\mathbf{J}^{T}\mathbf{r}
  71. - 𝐉 T 𝐫 -\mathbf{J}^{T}\mathbf{r}
  72. λ \lambda
  73. λ \lambda
  74. 2 S β j β k \frac{\partial^{2}S}{\partial\beta_{j}\partial\beta_{k}}
  75. r i β j \frac{\partial r_{i}}{\partial\beta_{j}}

GC-content.html

  1. G + C A + T + G + C × 100 \cfrac{G+C}{A+T+G+C}\times\ 100
  2. A + T G + C \cfrac{A+T}{G+C}

Gear_ratio.html

  1. ω A \omega_{A}\!
  2. ω B \omega_{B}\!
  3. v = r A ω A = r B ω B , v=r_{A}\omega_{A}=r_{B}\omega_{B},\!
  4. 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}}.
  5. R = ω A ω B = N B N A . R=\frac{\omega_{A}}{\omega_{B}}=\frac{N_{B}}{N_{A}}.
  6. p = 2 t . p=2t.\!
  7. p = 2 π r A N A . p=\frac{2\pi r_{A}}{N_{A}}.
  8. p = 2 π r A N A = 2 π r B N B . p=\frac{2\pi r_{A}}{N_{A}}=\frac{2\pi r_{B}}{N_{B}}.
  9. r B r A = N B N A . \frac{r_{B}}{r_{A}}=\frac{N_{B}}{N_{A}}.
  10. R = ω A ω B = r B r A , R=\frac{\omega_{A}}{\omega_{B}}=\frac{r_{B}}{r_{A}},
  11. R = ω A ω B = N B N A . R=\frac{\omega_{A}}{\omega_{B}}=\frac{N_{B}}{N_{A}}.
  12. R = T B T A , R=\frac{T_{B}}{T_{A}},
  13. 𝑀𝐴 = T B T A . \mathit{MA}=\frac{T_{B}}{T_{A}}.
  14. ω A ω I = N I N A , ω I ω B = N B N I . \frac{\omega_{A}}{\omega_{I}}=\frac{N_{I}}{N_{A}},\quad\frac{\omega_{I}}{% \omega_{B}}=\frac{N_{B}}{N_{I}}.
  15. R = ω A ω B = N B N A . R=\frac{\omega_{A}}{\omega_{B}}=\frac{N_{B}}{N_{A}}.
  16. d = c t g r t × g r d d=\frac{c_{t}}{gr_{t}\times gr_{d}}
  17. v c = c t × v e g r t × g r d v_{c}=\frac{c_{t}\times v_{e}}{gr_{t}\times gr_{d}}

Gelfond–Schneider_constant.html

  1. 2 2 = 2.6651441426902251886502972498731 , 2^{\sqrt{2}}=2.6651441426902251886502972498731\ldots,
  2. 2 2 = 2 2 = 1.6325269 . \sqrt{2^{\sqrt{2}}}=\sqrt{2}^{\sqrt{2}}=1.6325269\ldots.
  3. 2 2 \sqrt{2}^{\sqrt{2}}
  4. ( 2 2 ) 2 = ( 2 ) ( 2 2 ) = ( 2 ) 2 = 2 \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=\left(\sqrt{2}\right)^{\left(}% \sqrt{2}\sqrt{2}\right)=\left(\sqrt{2}\right)^{2}=2

Gene_expression_programming.html

  1. ( a - b ) ( c + d ) \sqrt{(a-b)(c+d)}\,
  2. t = h ( n max - 1 ) + 1 t=h(n_{\max}-1)+1
  3. d w = h n max d_{w}=hn_{\max}
  4. t = h ( n max - 1 ) + 1 t=h(n_{\max}-1)+1\,

General_Leibniz_rule.html

  1. ( f g ) ( n ) = k = 0 n ( n k ) f ( k ) g ( n - k ) (fg)^{(n)}=\sum_{k=0}^{n}{n\choose k}f^{(k)}g^{(n-k)}
  2. ( n k ) = n ! k ! ( n - k ) ! {n\choose k}={n!\over k!(n-k)!}
  3. ( f 1 f 2 f m ) ( n ) = k 1 + k 2 + + k m = n ( n k 1 , k 2 , , k m ) 1 t m f t ( k t ) , \left(f_{1}f_{2}\cdots f_{m}\right)^{(n)}=\sum_{k_{1}+k_{2}+\cdots+k_{m}=n}{n% \choose k_{1},k_{2},\ldots,k_{m}}\prod_{1\leq t\leq m}f_{t}^{(k_{t})}\,,
  4. t = 1 m k t = n \sum_{t=1}^{m}k_{t}=n
  5. ( n k 1 , k 2 , , k m ) = n ! k 1 ! k 2 ! k m ! {n\choose k_{1},k_{2},\ldots,k_{m}}=\frac{n!}{k_{1}!\,k_{2}!\cdots k_{m}!}
  6. α ( f g ) = { β : β α } ( α β ) ( β f ) ( α - β g ) . \partial^{\alpha}(fg)=\sum_{\{\beta\,:\,\beta\leq\alpha\}}{\alpha\choose\beta}% (\partial^{\beta}f)(\partial^{\alpha-\beta}g).
  7. R = P Q R=P\circ Q
  8. R ( x , ξ ) = e - x , ξ R ( e x , ξ ) . R(x,\xi)=e^{-{\langle x,\xi\rangle}}R(e^{\langle x,\xi\rangle}).
  9. R ( x , ξ ) = α 1 α ! ( ξ ) α P ( x , ξ ) ( x ) α Q ( x , ξ ) . R(x,\xi)=\sum_{\alpha}{1\over\alpha!}\left({\partial\over\partial\xi}\right)^{% \alpha}P(x,\xi)\left({\partial\over\partial x}\right)^{\alpha}Q(x,\xi).

General_linear_model.html

  1. 𝐘 = 𝐗𝐁 + 𝐔 , \mathbf{Y}=\mathbf{X}\mathbf{B}+\mathbf{U},
  2. Y i = β 0 + β 1 X i 1 + β 2 X i 2 + + β p X i p + ϵ i . Y_{i}=\beta_{0}+\beta_{1}X_{i1}+\beta_{2}X_{i2}+\ldots+\beta_{p}X_{ip}+% \epsilon_{i}.

Generalized_continued_fraction.html

  1. x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + x=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cfrac{a_{% 4}}{b_{4}+\ddots\,}}}}
  2. x 0 = A 0 B 0 = b 0 , x 1 = A 1 B 1 = b 1 b 0 + a 1 b 1 , x 2 = A 2 B 2 = b 2 ( b 1 b 0 + a 1 ) + a 2 b 0 b 2 b 1 + a 2 , x_{0}=\frac{A_{0}}{B_{0}}=b_{0},\qquad x_{1}=\frac{A_{1}}{B_{1}}=\frac{b_{1}b_% {0}+a_{1}}{b_{1}},\qquad x_{2}=\frac{A_{2}}{B_{2}}=\frac{b_{2}(b_{1}b_{0}+a_{1% })+a_{2}b_{0}}{b_{2}b_{1}+a_{2}},\qquad\cdots\,
  3. A n = b n A n - 1 + a n A n - 2 , B n = b n B n - 1 + a n B n - 2 , A_{n}=b_{n}A_{n-1}+a_{n}A_{n-2},\qquad B_{n}=b_{n}B_{n-1}+a_{n}B_{n-2},\,
  4. a 0 . a_{0}.\,
  5. n 1 d 1 . n_{1}\over d_{1}.
  6. n 2 d 2 . n_{2}\over d_{2}.
  7. n 3 d 3 , {n_{3}\over d_{3}},
  8. π \pi
  9. t a n x tanx
  10. tan ( x ) = x 1 + - x 2 3 + - x 2 5 + - x 2 7 + \tan(x)=\cfrac{x}{1+\cfrac{-x^{2}}{3+\cfrac{-x^{2}}{5+\cfrac{-x^{2}}{7+{}% \ddots}}}}
  11. x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + x=b_{0}+\frac{a_{1}}{b_{1}+}\,\frac{a_{2}}{b_{2}+}\,\frac{a_{3}}{b_{3}+}\cdots
  12. x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + x=b_{0}+\frac{a_{1}\mid}{\mid b_{1}}+\frac{a_{2}\mid}{\mid b_{2}}+\frac{a_{3}% \mid}{\mid b_{3}}+\cdots\,
  13. x = b 0 + K i = 1 a i b i . x=b_{0}+\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{b_{i}}.\,
  14. b 0 + K i = 1 a i b i b_{0}+\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{b_{i}}\,
  15. x n = b 0 + K i = 1 𝑛 a i b i x_{n}=b_{0}+\underset{i=1}{\overset{n}{\mathrm{K}}}\frac{a_{i}}{b_{i}}\,
  16. A n - 1 B n - A n B n - 1 = ( - 1 ) n a 1 a 2 a n = Π i = 1 n ( - a i ) A_{n-1}B_{n}-A_{n}B_{n-1}=(-1)^{n}a_{1}a_{2}\cdots a_{n}=\Pi_{i=1}^{n}(-a_{i})
  17. n - 1 n-1
  18. n n
  19. A n A_{n}
  20. B n B_{n}
  21. = b n A n - 1 B n - 1 + a n A n - 1 B n - 2 - b n A n - 1 B n - 1 - a n A n - 2 B n - 1 \displaystyle=b_{n}A_{n-1}B_{n-1}+a_{n}A_{n-1}B_{n-2}-b_{n}A_{n-1}B_{n-1}-a_{n% }A_{n-2}B_{n-1}
  22. A n - 1 B n - A n B n - 1 = ( - 1 ) n a 1 a 2 a n = Π i = 1 n ( - a i ) A_{n-1}B_{n}-A_{n}B_{n-1}=(-1)^{n}a_{1}a_{2}\cdots a_{n}=\Pi_{i=1}^{n}(-a_{i})\,
  23. x n - 1 - x n = A n - 1 B n - 1 - A n B n = ( - 1 ) n a 1 a 2 a n B n B n - 1 = Π i = 1 n ( - a i ) B n B n - 1 . x_{n-1}-x_{n}=\frac{A_{n-1}}{B_{n-1}}-\frac{A_{n}}{B_{n}}=(-1)^{n}\frac{a_{1}a% _{2}\cdots a_{n}}{B_{n}B_{n-1}}=\frac{\Pi_{i=1}^{n}(-a_{i})}{B_{n}B_{n-1}}.\,
  24. b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + = b 0 + c 1 a 1 c 1 b 1 + c 1 c 2 a 2 c 2 b 2 + c 2 c 3 a 3 c 3 b 3 + c 3 c 4 a 4 c 4 b 4 + b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cfrac{a_{4}% }{b_{4}+\ddots\,}}}}=b_{0}+\cfrac{c_{1}a_{1}}{c_{1}b_{1}+\cfrac{c_{1}c_{2}a_{2% }}{c_{2}b_{2}+\cfrac{c_{2}c_{3}a_{3}}{c_{3}b_{3}+\cfrac{c_{3}c_{4}a_{4}}{c_{4}% b_{4}+\ddots\,}}}}
  25. b 0 + K i = 1 a i b i = b 0 + K i = 1 1 c i b i b_{0}+\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{b_{i}}=b_{0}+% \underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{1}{c_{i}b_{i}}\,
  26. b 0 + K i = 1 a i b i = b 0 + K i = 1 d i a i 1 b_{0}+\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{b_{i}}=b_{0}+% \underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{d_{i}a_{i}}{1}\,
  27. x = b 0 + K i = 1 a i b i x=b_{0}+\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{b_{i}}\,
  28. x = K i = 1 1 b i x=\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{1}{b_{i}}\,
  29. f ( z ) = K i = 1 1 z . f(z)=\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{1}{z}.\,
  30. f ( z ) - f n ( z ) = K i = 1 a i ( z ) b i ( z ) - K i = 1 𝑛 a i ( z ) b i ( z ) f(z)-f_{n}(z)=\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}(z)}{b_{i% }(z)}-\underset{i=1}{\overset{n}{\mathrm{K}}}\frac{a_{i}(z)}{b_{i}(z)}\,
  31. x = K i = 1 a i 1 x=\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{1}\,
  32. x even = a 1 1 + a 2 - a 2 a 3 1 + a 3 + a 4 - a 4 a 5 1 + a 5 + a 6 - a 6 a 7 1 + a 7 + a 8 - x_{\mathrm{even}}=\cfrac{a_{1}}{1+a_{2}-\cfrac{a_{2}a_{3}}{1+a_{3}+a_{4}-% \cfrac{a_{4}a_{5}}{1+a_{5}+a_{6}-\cfrac{a_{6}a_{7}}{1+a_{7}+a_{8}-\ddots}}}}\,
  33. x odd = a 1 - a 1 a 2 1 + a 2 + a 3 - a 3 a 4 1 + a 4 + a 5 - a 5 a 6 1 + a 6 + a 7 - a 7 a 8 1 + a 8 + a 9 - x_{\mathrm{odd}}=a_{1}-\cfrac{a_{1}a_{2}}{1+a_{2}+a_{3}-\cfrac{a_{3}a_{4}}{1+a% _{4}+a_{5}-\cfrac{a_{5}a_{6}}{1+a_{6}+a_{7}-\cfrac{a_{7}a_{8}}{1+a_{8}+a_{9}-% \ddots}}}}\,
  34. a 1 , a 2 , . . . a_{1},a_{2},\,\text{ . . .}
  35. b 1 , b 2 , . . . b_{1},b_{2},\,\text{ . . .}
  36. a k a_{k}
  37. b k b_{k}
  38. k k
  39. x = b 0 + K i = 1 a i b i x=b_{0}+\underset{i=1}{\overset{\infty}{\mathrm{K}}}\frac{a_{i}}{b_{i}}\,
  40. A - 1 \displaystyle A_{-1}
  41. x n = A n B n . x_{n}=\frac{A_{n}}{B_{n}}.\,
  42. x = 1 + 1 1 + 1 1 + 1 1 + 1 1 + x=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots\,}}}}
  43. w = f ( z ) = a + b z c + d z , w=f(z)=\frac{a+bz}{c+dz},\,
  44. f ( z ) = z d z 2 + c z = a + b z f(z)=z\Rightarrow dz^{2}+cz=a+bz\,
  45. z = g ( w ) = - a + c w b - d w z=g(w)=\frac{-a+cw}{b-dw}\,
  46. w = f ( z ) = a c + d z , w=f(z)=\frac{a}{c+dz},\,
  47. τ 0 ( z ) = b 0 + z , τ 1 ( z ) = a 1 b 1 + z , τ 2 ( z ) = a 2 b 2 + z , τ 3 ( z ) = a 3 b 3 + z , \tau_{0}(z)=b_{0}+z,\quad\tau_{1}(z)=\frac{a_{1}}{b_{1}+z},\quad\tau_{2}(z)=% \frac{a_{2}}{b_{2}+z},\quad\tau_{3}(z)=\frac{a_{3}}{b_{3}+z},\quad\cdots\,
  48. s y m b o l \Tau s y m b o l 1 ( z ) = τ 0 τ 1 ( z ) = τ 0 ( τ 1 ( z ) ) , s y m b o l \Tau s y m b o l 2 ( z ) = τ 0 τ 1 τ 2 ( z ) = τ 0 ( τ 1 ( τ 2 ( z ) ) ) , symbol{\Tau}_{symbol{1}}(z)=\tau_{0}\circ\tau_{1}(z)=\tau_{0}(\tau_{1}(z)),% \quad symbol{\Tau}_{symbol{2}}(z)=\tau_{0}\circ\tau_{1}\circ\tau_{2}(z)=\tau_{% 0}(\tau_{1}(\tau_{2}(z))),\,
  49. s y m b o l \Tau s y m b o l 1 ( z ) = τ 0 τ 1 ( z ) = b 0 + a 1 b 1 + z s y m b o l \Tau s y m b o l 2 ( z ) = τ 0 τ 1 τ 2 ( z ) = b 0 + a 1 b 1 + a 2 b 2 + z \begin{aligned}\displaystyle symbol{\Tau}_{symbol{1}}(z)&\displaystyle=\tau_{0% }\circ\tau_{1}(z)&\displaystyle=&\displaystyle\quad b_{0}+\cfrac{a_{1}}{b_{1}+% z}\\ \displaystyle symbol{\Tau}_{symbol{2}}(z)&\displaystyle=\tau_{0}\circ\tau_{1}% \circ\tau_{2}(z)&\displaystyle=&\displaystyle\quad b_{0}+\cfrac{a_{1}}{b_{1}+% \cfrac{a_{2}}{b_{2}+z}}\end{aligned}
  50. s y m b o l \Tau s y m b o l n ( z ) = τ 0 τ 1 τ 2 τ n ( z ) = b 0 + K i = 1 𝑛 a i b i symbol{\Tau}_{symbol{n}}(z)=\tau_{0}\circ\tau_{1}\circ\tau_{2}\circ\cdots\circ% \tau_{n}(z)=b_{0}+\underset{i=1}{\overset{n}{\mathrm{K}}}\frac{a_{i}}{b_{i}}\,
  51. s y m b o l \Tau s y m b o l n ( 0 ) = s y m b o l \Tau s y m b o l n + 1 ( ) = b 0 + K i = 1 𝑛 a i b i . symbol{\Tau}_{symbol{n}}(0)=symbol{\Tau}_{symbol{n+1}}(\infty)=b_{0}+\underset% {i=1}{\overset{n}{\mathrm{K}}}\frac{a_{i}}{b_{i}}.\,
  52. x n = b 0 + K i = 1 𝑛 a i b i = A n B n = s y m b o l \Tau s y m b o l n ( 0 ) = s y m b o l \Tau s y m b o l n + 1 ( ) x_{n}=b_{0}+\underset{i=1}{\overset{n}{\mathrm{K}}}\frac{a_{i}}{b_{i}}=\frac{A% _{n}}{B_{n}}=symbol{\Tau}_{symbol{n}}(0)=symbol{\Tau}_{symbol{n+1}}(\infty)\,
  53. s y m b o l \Tau s y m b o l n ( z ) \displaystyle symbol{\Tau}_{symbol{n}}(z)
  54. A n - 1 B n - 1 A n B n A n - 1 A n B n - 1 B n = k \frac{A_{n-1}}{B_{n-1}}\approx\frac{A_{n}}{B_{n}}\quad\Rightarrow\quad\frac{A_% {n-1}}{A_{n}}\approx\frac{B_{n-1}}{B_{n}}=k\,
  55. s y m b o l \Tau s y m b o l n ( z ) = z A n - 1 + A n z B n - 1 + B n = A n B n ( z A n - 1 A n + 1 z B n - 1 B n + 1 ) A n B n ( z k + 1 z k + 1 ) = A n B n symbol{\Tau}_{symbol{n}}(z)=\frac{zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}=\frac{A_{n}}% {B_{n}}\left(\frac{z\frac{A_{n-1}}{A_{n}}+1}{z\frac{B_{n-1}}{B_{n}}+1}\right)% \approx\frac{A_{n}}{B_{n}}\left(\frac{zk+1}{zk+1}\right)=\frac{A_{n}}{B_{n}}\,
  56. x = 1 + z 1 + z 1 + z 1 + z 1 + x=1+\cfrac{z}{1+\cfrac{z}{1+\cfrac{z}{1+\cfrac{z}{1+\ddots}}}}\,
  57. a 0 + a 0 a 1 + a 0 a 1 a 2 + + a 0 a 1 a 2 a n = a 0 1 - a 1 1 + a 1 - a 2 1 + a 2 - a n 1 + a n . a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots+a_{0}a_{1}a_{2}\cdots a_{n}=\frac{a_{0% }}{1-}\frac{a_{1}}{1+a_{1}-}\frac{a_{2}}{1+a_{2}-}\cdots\frac{a_{n}}{1+a_{n}}.\,
  58. 1 u 1 + 1 u 2 + 1 u 3 + + 1 u n = 1 u 1 - u 1 2 u 1 + u 2 - u 2 2 u 2 + u 3 - u n - 1 2 u n - 1 + u n , \frac{1}{u_{1}}+\frac{1}{u_{2}}+\frac{1}{u_{3}}+\cdots+\frac{1}{u_{n}}=\frac{1% }{u_{1}-}\frac{u_{1}^{2}}{u_{1}+u_{2}-}\frac{u_{2}^{2}}{u_{2}+u_{3}-}\cdots% \frac{u_{n-1}^{2}}{u_{n-1}+u_{n}},\,
  59. 1 a 0 + x a 0 a 1 + x 2 a 0 a 1 a 2 + + x n a 0 a 1 a 2 a n = 1 a 0 - a 0 x a 1 + x - a 1 x a 2 + x - a n - 1 x a n + x . \frac{1}{a_{0}}+\frac{x}{a_{0}a_{1}}+\frac{x^{2}}{a_{0}a_{1}a_{2}}+\cdots+% \frac{x^{n}}{a_{0}a_{1}a_{2}\ldots a_{n}}=\frac{1}{a_{0}-}\frac{a_{0}x}{a_{1}+% x-}\frac{a_{1}x}{a_{2}+x-}\cdots\frac{a_{n-1}x}{a_{n}+x}.\,
  60. e x = x 0 0 ! + x 1 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + = 1 + x 1 - 1 x 2 + x - 2 x 3 + x - 3 x 4 + x - e^{x}=\frac{x^{0}}{0!}+\frac{x^{1}}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+% \frac{x^{4}}{4!}+\cdots=1+\cfrac{x}{1-\cfrac{1x}{2+x-\cfrac{2x}{3+x-\cfrac{3x}% {4+x-\ddots}}}}
  61. log ( 1 + x ) = x 1 1 - x 2 2 + x 3 3 - x 4 4 + = x 1 - 0 x + 1 2 x 2 - 1 x + 2 2 x 3 - 2 x + 3 2 x 4 - 3 x + \log(1+x)=\frac{x^{1}}{1}-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+% \cdots=\cfrac{x}{1-0x+\cfrac{1^{2}x}{2-1x+\cfrac{2^{2}x}{3-2x+\cfrac{3^{2}x}{4% -3x+\ddots}}}}
  62. tan - 1 x y = x y 1 y 2 + ( 1 x y ) 2 3 y 2 - 1 x 2 + ( 3 x y ) 2 5 y 2 - 3 x 2 + ( 5 x y ) 2 7 y 2 - 5 x 2 + = x 1 y + ( 1 x ) 2 3 y + ( 2 x ) 2 5 y + ( 3 x ) 2 7 y + \tan^{-1}\cfrac{x}{y}=\cfrac{xy}{1y^{2}+\cfrac{(1xy)^{2}}{3y^{2}-1x^{2}+\cfrac% {(3xy)^{2}}{5y^{2}-3x^{2}+\cfrac{(5xy)^{2}}{7y^{2}-5x^{2}+\ddots}}}}=\cfrac{x}% {1y+\cfrac{(1x)^{2}}{3y+\cfrac{(2x)^{2}}{5y+\cfrac{(3x)^{2}}{7y+\ddots}}}}
  63. e x / y = 1 + 2 x 2 y - x + x 2 6 y + x 2 10 y + x 2 14 y + x 2 18 y + x 2 22 y + ; e 2 = 7 + 2 5 + 1 7 + 1 9 + 1 11 + e^{x/y}=1+\cfrac{2x}{2y-x+\cfrac{x^{2}}{6y+\cfrac{x^{2}}{10y+\cfrac{x^{2}}{14y% +\cfrac{x^{2}}{18y+\cfrac{x^{2}}{22y+\ddots}}}}}};e^{2}=7+\cfrac{2}{5+\cfrac{1% }{7+\cfrac{1}{9+\cfrac{1}{11+\ddots}}}}
  64. log ( 1 + x y ) = x y + 1 x 2 + 1 x 3 y + 2 x 2 + 2 x 5 y + 3 x 2 + = 2 x 2 y + x - ( 1 x ) 2 3 ( 2 y + x ) - ( 2 x ) 2 5 ( 2 y + x ) - ( 3 x ) 2 7 ( 2 y + x ) - \log\left(1+\frac{x}{y}\right)=\cfrac{x}{y+\cfrac{1x}{2+\cfrac{1x}{3y+\cfrac{2% x}{2+\cfrac{2x}{5y+\cfrac{3x}{2+\ddots}}}}}}=\cfrac{2x}{2y+x-\cfrac{(1x)^{2}}{% 3(2y+x)-\cfrac{(2x)^{2}}{5(2y+x)-\cfrac{(3x)^{2}}{7(2y+x)-\ddots}}}}
  65. log 2 = log ( 1 + 1 ) = 1 1 + 1 2 + 1 3 + 2 2 + 2 5 + 3 2 + = 2 3 - 1 2 9 - 2 2 15 - 3 2 21 - \log 2=\log(1+1)=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+% \cfrac{3}{2+\ddots}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2}}{15-\cfrac{3% ^{2}}{21-\ddots}}}}
  66. π \pi
  67. π \pi
  68. π \pi
  69. π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + = n = 0 4 ( - 1 ) n 2 n + 1 = 4 1 - 4 3 + 4 5 - 4 7 + - \pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\ddots}}}}=% \sum_{n=0}^{\infty}\frac{4(-1)^{n}}{2n+1}=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-% \frac{4}{7}+-\cdots
  70. π = 3 + 1 2 6 + 3 2 6 + 5 2 6 + = 3 - n = 1 ( - 1 ) n n ( n + 1 ) ( 2 n + 1 ) = 3 + 1 1 2 3 - 1 2 3 5 + 1 3 4 7 - + \pi=3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+\cfrac{5^{2}}{6+\ddots}}}=3-\sum_{n=1}^{% \infty}\frac{(-1)^{n}}{n(n+1)(2n+1)}=3+\frac{1}{1\cdot 2\cdot 3}-\frac{1}{2% \cdot 3\cdot 5}+\frac{1}{3\cdot 4\cdot 7}-+\cdots
  71. π \pi
  72. π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + = 4 - 1 + 1 6 - 1 34 + 16 3145 - 4 4551 + 1 6601 - 1 38341 + - \pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}=4-1% +\frac{1}{6}-\frac{1}{34}+\frac{16}{3145}-\frac{4}{4551}+\frac{1}{6601}-\frac{% 1}{38341}+-\cdots
  73. π \pi
  74. π \pi
  75. π = 6 sin - 1 ( 1 2 ) = n = 0 3 ( 2 n n ) 16 n ( 2 n + 1 ) = 3 16 0 1 + 6 16 1 3 + 18 16 2 5 + 60 16 3 7 + \pi=6\sin^{-1}\left(\frac{1}{2}\right)=\sum_{n=0}^{\infty}\frac{3\cdot{\left({% {2n}\atop{n}}\right)}}{16^{n}(2n+1)}=\frac{3}{16^{0}\cdot 1}+\frac{6}{16^{1}% \cdot 3}+\frac{18}{16^{2}\cdot 5}+\frac{60}{16^{3}\cdot 7}+\cdots\!
  76. π = 16 tan - 1 1 5 - 4 tan - 1 1 239 = 16 5 + 1 2 15 + 2 2 25 + 3 2 35 + - 4 239 + 1 2 717 + 2 2 1195 + 3 2 1673 + . \pi=16\tan^{-1}\cfrac{1}{5}-4\tan^{-1}\cfrac{1}{239}=\cfrac{16}{5+\cfrac{1^{2}% }{15+\cfrac{2^{2}}{25+\cfrac{3^{2}}{35+\ddots}}}}-\cfrac{4}{239+\cfrac{1^{2}}{% 717+\cfrac{2^{2}}{1195+\cfrac{3^{2}}{1673+\ddots}}}}.
  77. z m n = ( x n + y ) m n = x m + m y n x n - m + ( n - m ) y 2 x m + ( n + m ) y 3 n x n - m + ( 2 n - m ) y 2 x m + ( 2 n + m ) y 5 n x n - m + ( 3 n - m ) y 2 x m + \sqrt[n]{z^{m}}=\sqrt[n]{(x^{n}+y)^{m}}=x^{m}+\cfrac{my}{nx^{n-m}+\cfrac{(n-m)% y}{2x^{m}+\cfrac{(n+m)y}{3nx^{n-m}+\cfrac{(2n-m)y}{2x^{m}+\cfrac{(2n+m)y}{5nx^% {n-m}+\cfrac{(3n-m)y}{2x^{m}+\ddots}}}}}}
  78. z m n = x m + 2 x m m y n ( 2 z - y ) - m y - ( 1 2 n 2 - m 2 ) y 2 3 n ( 2 z - y ) - ( 2 2 n 2 - m 2 ) y 2 5 n ( 2 z - y ) - ( 3 2 n 2 - m 2 ) y 2 7 n ( 2 z - y ) - ( 4 2 n 2 - m 2 ) y 2 9 n ( 2 z - y ) - . \sqrt[n]{z^{m}}=x^{m}+\cfrac{2x^{m}\cdot my}{n(2z-y)-my-\cfrac{(1^{2}n^{2}-m^{% 2})y^{2}}{3n(2z-y)-\cfrac{(2^{2}n^{2}-m^{2})y^{2}}{5n(2z-y)-\cfrac{(3^{2}n^{2}% -m^{2})y^{2}}{7n(2z-y)-\cfrac{(4^{2}n^{2}-m^{2})y^{2}}{9n(2z-y)-\ddots}}}}}.
  79. z = x 2 + y = x + y 2 x + y 2 x + 3 y 6 x + 3 y 2 x + = x + 2 x y 2 ( 2 z - y ) - y - 1 3 y 2 6 ( 2 z - y ) - 3 5 y 2 10 ( 2 z - y ) - \sqrt{z}=\sqrt{x^{2}+y}=x+\cfrac{y}{2x+\cfrac{y}{2x+\cfrac{3y}{6x+\cfrac{3y}{2% x+\ddots}}}}=x+\cfrac{2x\cdot y}{2(2z-y)-y-\cfrac{1\cdot 3y^{2}}{6(2z-y)-% \cfrac{3\cdot 5y^{2}}{10(2z-y)-\ddots}}}
  80. z = x 2 + y = x + y 2 x + y 2 x + y 2 x + y 2 x + = x + 2 x y 2 ( 2 z - y ) - y - y 2 2 ( 2 z - y ) - y 2 2 ( 2 z - y ) - . \sqrt{z}=\sqrt{x^{2}+y}=x+\cfrac{y}{2x+\cfrac{y}{2x+\cfrac{y}{2x+\cfrac{y}{2x+% \ddots}}}}=x+\cfrac{2x\cdot y}{2(2z-y)-y-\cfrac{y^{2}}{2(2z-y)-\cfrac{y^{2}}{2% (2z-y)-\ddots}}}.
  81. 2 3 = 1 + 1 3 + 2 2 + 4 9 + 5 2 + 7 15 + 8 2 + 10 21 + 11 2 + = 1 + 2 1 9 - 1 - 2 4 27 - 5 7 45 - 8 10 63 - 11 13 81 - . \sqrt[3]{2}=1+\cfrac{1}{3+\cfrac{2}{2+\cfrac{4}{9+\cfrac{5}{2+\cfrac{7}{15+% \cfrac{8}{2+\cfrac{10}{21+\cfrac{11}{2+\ddots}}}}}}}}=1+\cfrac{2\cdot 1}{9-1-% \cfrac{2\cdot 4}{27-\cfrac{5\cdot 7}{45-\cfrac{8\cdot 10}{63-\cfrac{11\cdot 13% }{81-\ddots}}}}}.
  82. 2 3 = 5 4 + 0.5 50 + 2 5 + 4 150 + 5 5 + 7 250 + 8 5 + 10 350 + 11 5 + = 5 4 + 2.5 1 253 - 1 - 2 4 759 - 5 7 1265 - 8 10 1771 - . \sqrt[3]{2}=\cfrac{5}{4}+\cfrac{0.5}{50+\cfrac{2}{5+\cfrac{4}{150+\cfrac{5}{5+% \cfrac{7}{250+\cfrac{8}{5+\cfrac{10}{350+\cfrac{11}{5+\ddots}}}}}}}}=\cfrac{5}% {4}+\cfrac{2.5\cdot 1}{253-1-\cfrac{2\cdot 4}{759-\cfrac{5\cdot 7}{1265-\cfrac% {8\cdot 10}{1771-\ddots}}}}.
  83. 100 5 = 5 2 + 3 250 + 12 5 + 18 750 + 27 5 + 33 1250 + 42 5 + = 5 2 + 5 3 1265 - 3 - 12 18 3795 - 27 33 6325 - 42 48 8855 - . \sqrt[5]{100}=\cfrac{5}{2}+\cfrac{3}{250+\cfrac{12}{5+\cfrac{18}{750+\cfrac{27% }{5+\cfrac{33}{1250+\cfrac{42}{5+\ddots}}}}}}=\cfrac{5}{2}+\cfrac{5\cdot 3}{12% 65-3-\cfrac{12\cdot 18}{3795-\cfrac{27\cdot 33}{6325-\cfrac{42\cdot 48}{8855-% \ddots}}}}.
  84. 2 12 = 1 + 1 12 + 11 2 + 13 36 + 23 2 + 25 60 + 35 2 + 37 84 + 47 2 + = 1 + 2 1 36 - 1 - 11 13 108 - 23 25 180 - 35 37 252 - 47 49 324 - . \sqrt[12]{2}=1+\cfrac{1}{12+\cfrac{11}{2+\cfrac{13}{36+\cfrac{23}{2+\cfrac{25}% {60+\cfrac{35}{2+\cfrac{37}{84+\cfrac{47}{2+\ddots}}}}}}}}=1+\cfrac{2\cdot 1}{% 36-1-\cfrac{11\cdot 13}{108-\cfrac{23\cdot 25}{180-\cfrac{35\cdot 37}{252-% \cfrac{47\cdot 49}{324-\ddots}}}}}.
  85. 2 7 12 = 1 + 7 12 + 5 2 + 19 36 + 17 2 + 31 60 + 29 2 + 43 84 + 41 2 + = 1 + 2 7 36 - 7 - 5 19 108 - 17 31 180 - 29 43 252 - 41 55 324 - . \sqrt[12]{2^{7}}=1+\cfrac{7}{12+\cfrac{5}{2+\cfrac{19}{36+\cfrac{17}{2+\cfrac{% 31}{60+\cfrac{29}{2+\cfrac{43}{84+\cfrac{41}{2+\ddots}}}}}}}}=1+\cfrac{2\cdot 7% }{36-7-\cfrac{5\cdot 19}{108-\cfrac{17\cdot 31}{180-\cfrac{29\cdot 43}{252-% \cfrac{41\cdot 55}{324-\ddots}}}}}.
  86. 2 7 12 = 1 2 3 12 - 7153 12 = 3 2 - 0.5 7153 4 3 12 - 11 7153 6 - 13 7153 12 3 12 - 23 7153 6 - 25 7153 20 3 12 - 35 7153 6 - 37 7153 28 3 12 - 47 7153 6 - \sqrt[12]{2^{7}}=\cfrac{1}{2}\sqrt[12]{3^{12}-7153}=\cfrac{3}{2}-\cfrac{0.5% \cdot 7153}{4\cdot 3^{12}-\cfrac{11\cdot 7153}{6-\cfrac{13\cdot 7153}{12\cdot 3% ^{12}-\cfrac{23\cdot 7153}{6-\cfrac{25\cdot 7153}{20\cdot 3^{12}-\cfrac{35% \cdot 7153}{6-\cfrac{37\cdot 7153}{28\cdot 3^{12}-\cfrac{47\cdot 7153}{6-% \ddots}}}}}}}}
  87. 2 7 12 = 3 2 - 3 7153 12 ( 2 19 + 3 12 ) + 7153 - 11 13 7153 2 36 ( 2 19 + 3 12 ) - 23 25 7153 2 60 ( 2 19 + 3 12 ) - 35 37 7153 2 84 ( 2 19 + 3 12 ) - . \sqrt[12]{2^{7}}=\cfrac{3}{2}-\cfrac{3\cdot 7153}{12(2^{19}+3^{12})+7153-% \cfrac{11\cdot 13\cdot 7153^{2}}{36(2^{19}+3^{12})-\cfrac{23\cdot 25\cdot 7153% ^{2}}{60(2^{19}+3^{12})-\cfrac{35\cdot 37\cdot 7153^{2}}{84(2^{19}+3^{12})-% \ddots}}}}.

Generalized_eigenvector.html

  1. A A
  2. V V
  3. ϕ \phi
  4. L ( V ) L(V)
  5. V V
  6. A A
  7. ϕ \phi
  8. A A
  9. V V
  10. A A
  11. λ i \lambda_{i}
  12. ( A - λ i I ) (A-\lambda_{i}I)
  13. λ i \lambda_{i}
  14. A A
  15. x i x_{i}
  16. λ i \lambda_{i}
  17. ( A - λ i I ) (A-\lambda_{i}I)
  18. V V
  19. A A
  20. V V
  21. J J
  22. A A
  23. A A
  24. J J
  25. x = A x , x^{\prime}=Ax,
  26. A A
  27. u u
  28. λ \lambda
  29. n n
  30. n n
  31. A A
  32. ( A - λ I ) u = 0 (A-\lambda I)u=0
  33. I I
  34. n n
  35. n n
  36. 0
  37. n n
  38. u u
  39. ( A - λ I ) (A-\lambda I)
  40. A A
  41. n n
  42. A A
  43. D D
  44. M M
  45. A A
  46. D = M - 1 A M D=M^{-1}AM
  47. D D
  48. A A
  49. M M
  50. A A
  51. A A
  52. n n
  53. A A
  54. x m x_{m}
  55. A A
  56. λ \lambda
  57. ( A - λ I ) m x m = 0 (A-\lambda I)^{m}x_{m}=0
  58. ( A - λ I ) m - 1 x m 0. (A-\lambda I)^{m-1}x_{m}\neq 0.
  59. n n
  60. n n
  61. A A
  62. n n
  63. J J
  64. M M
  65. J = M - 1 A M J=M^{-1}AM
  66. M M
  67. A A
  68. λ \lambda
  69. μ \mu
  70. A A
  71. μ \mu
  72. λ \lambda
  73. A A
  74. λ \lambda
  75. λ \lambda
  76. A = ( 1 1 0 1 ) . A=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}.
  77. λ = 1 \lambda=1
  78. V V
  79. A - λ I A-\lambda I
  80. v 1 = ( 1 0 ) v_{1}=\begin{pmatrix}1\\ 0\end{pmatrix}
  81. v 2 v_{2}
  82. ( A - λ I ) v 2 = v 1 . (A-\lambda I)v_{2}=v_{1}.
  83. ( ( 1 1 0 1 ) - 1 ( 1 0 0 1 ) ) ( v 21 v 22 ) = ( 0 1 0 0 ) ( v 21 v 22 ) = ( 1 0 ) . \left(\begin{pmatrix}1&1\\ 0&1\end{pmatrix}-1\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\right)\begin{pmatrix}v_{21}\\ v_{22}\end{pmatrix}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}v_{21}\\ v_{22}\end{pmatrix}=\begin{pmatrix}1\\ 0\end{pmatrix}.
  84. v 22 = 1. v_{22}=1.
  85. v 21 v_{21}
  86. v 2 = ( a 1 ) v_{2}=\begin{pmatrix}a\\ 1\end{pmatrix}
  87. ( A - λ I ) v 2 = ( 0 1 0 0 ) ( a 1 ) = ( 1 0 ) = v 1 , (A-\lambda I)v_{2}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}a\\ 1\end{pmatrix}=\begin{pmatrix}1\\ 0\end{pmatrix}=v_{1},
  88. v 2 v_{2}
  89. ( A - λ I ) v 1 = ( 0 1 0 0 ) ( 1 0 ) = ( 0 0 ) = 0 , (A-\lambda I)v_{1}=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\begin{pmatrix}1\\ 0\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}=0,
  90. v 1 v_{1}
  91. v 1 v_{1}
  92. v 2 v_{2}
  93. V V
  94. A = ( 1 0 0 0 0 3 1 0 0 0 6 3 2 0 0 10 6 3 2 0 15 10 6 3 2 ) A=\begin{pmatrix}1&0&0&0&0\\ 3&1&0&0&0\\ 6&3&2&0&0\\ 10&6&3&2&0\\ 15&10&6&3&2\end{pmatrix}
  95. λ 1 = 1 \lambda_{1}=1
  96. λ 2 = 2 \lambda_{2}=2
  97. μ 1 = 2 \mu_{1}=2
  98. μ 2 = 3 \mu_{2}=3
  99. γ 1 = 1 \gamma_{1}=1
  100. γ 2 = 1 \gamma_{2}=1
  101. A A
  102. x 1 x_{1}
  103. λ 1 \lambda_{1}
  104. x 2 x_{2}
  105. λ 1 \lambda_{1}
  106. y 1 y_{1}
  107. λ 2 \lambda_{2}
  108. y 2 y_{2}
  109. y 3 y_{3}
  110. λ 2 \lambda_{2}
  111. ( A - 1 I ) x 1 = ( 0 0 0 0 0 3 0 0 0 0 6 3 1 0 0 10 6 3 1 0 15 10 6 3 1 ) ( 0 3 - 9 9 - 3 ) = ( 0 0 0 0 0 ) = 0 , (A-1I)x_{1}=\begin{pmatrix}0&0&0&0&0\\ 3&0&0&0&0\\ 6&3&1&0&0\\ 10&6&3&1&0\\ 15&10&6&3&1\end{pmatrix}\begin{pmatrix}0\\ 3\\ -9\\ 9\\ -3\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ 0\\ 0\end{pmatrix}=0,
  112. ( A - 1 I ) x 2 = ( 0 0 0 0 0 3 0 0 0 0 6 3 1 0 0 10 6 3 1 0 15 10 6 3 1 ) ( 1 - 15 30 - 1 - 45 ) = ( 0 3 - 9 9 - 3 ) = x 1 , (A-1I)x_{2}=\begin{pmatrix}0&0&0&0&0\\ 3&0&0&0&0\\ 6&3&1&0&0\\ 10&6&3&1&0\\ 15&10&6&3&1\end{pmatrix}\begin{pmatrix}1\\ -15\\ 30\\ -1\\ -45\end{pmatrix}=\begin{pmatrix}0\\ 3\\ -9\\ 9\\ -3\end{pmatrix}=x_{1},
  113. ( A - 2 I ) y 1 = ( - 1 0 0 0 0 3 - 1 0 0 0 6 3 0 0 0 10 6 3 0 0 15 10 6 3 0 ) ( 0 0 0 0 9 ) = ( 0 0 0 0 0 ) = 0 , (A-2I)y_{1}=\begin{pmatrix}-1&0&0&0&0\\ 3&-1&0&0&0\\ 6&3&0&0&0\\ 10&6&3&0&0\\ 15&10&6&3&0\end{pmatrix}\begin{pmatrix}0\\ 0\\ 0\\ 0\\ 9\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ 0\\ 0\end{pmatrix}=0,
  114. ( A - 2 I ) y 2 = ( - 1 0 0 0 0 3 - 1 0 0 0 6 3 0 0 0 10 6 3 0 0 15 10 6 3 0 ) ( 0 0 0 3 0 ) = ( 0 0 0 0 9 ) = y 1 , (A-2I)y_{2}=\begin{pmatrix}-1&0&0&0&0\\ 3&-1&0&0&0\\ 6&3&0&0&0\\ 10&6&3&0&0\\ 15&10&6&3&0\end{pmatrix}\begin{pmatrix}0\\ 0\\ 0\\ 3\\ 0\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ 0\\ 9\end{pmatrix}=y_{1},
  115. ( A - 2 I ) y 3 = ( - 1 0 0 0 0 3 - 1 0 0 0 6 3 0 0 0 10 6 3 0 0 15 10 6 3 0 ) ( 0 0 1 - 2 0 ) = ( 0 0 0 3 0 ) = y 2 . (A-2I)y_{3}=\begin{pmatrix}-1&0&0&0&0\\ 3&-1&0&0&0\\ 6&3&0&0&0\\ 10&6&3&0&0\\ 15&10&6&3&0\end{pmatrix}\begin{pmatrix}0\\ 0\\ 1\\ -2\\ 0\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ 3\\ 0\end{pmatrix}=y_{2}.
  116. A A
  117. { x 1 , x 2 } = { ( 0 3 - 9 9 - 3 ) ( 1 - 15 30 - 1 - 45 ) } , { y 1 , y 2 , y 3 } = { ( 0 0 0 0 9 ) ( 0 0 0 3 0 ) ( 0 0 1 - 2 0 ) } . \left\{x_{1},x_{2}\right\}=\left\{\begin{pmatrix}0\\ 3\\ -9\\ 9\\ -3\end{pmatrix}\begin{pmatrix}1\\ -15\\ 30\\ -1\\ -45\end{pmatrix}\right\},\left\{y_{1},y_{2},y_{3}\right\}=\left\{\begin{% pmatrix}0\\ 0\\ 0\\ 0\\ 9\end{pmatrix}\begin{pmatrix}0\\ 0\\ 0\\ 3\\ 0\end{pmatrix}\begin{pmatrix}0\\ 0\\ 1\\ -2\\ 0\end{pmatrix}\right\}.
  118. J J
  119. A A
  120. M = ( x 1 x 2 y 1 y 2 y 3 ) = ( 0 1 0 0 0 3 - 15 0 0 0 - 9 30 0 0 1 9 - 1 0 3 - 2 - 3 - 45 9 0 0 ) , M=\begin{pmatrix}x_{1}&x_{2}&y_{1}&y_{2}&y_{3}\end{pmatrix}=\begin{pmatrix}0&1% &0&0&0\\ 3&-15&0&0&0\\ -9&30&0&0&1\\ 9&-1&0&3&-2\\ -3&-45&9&0&0\end{pmatrix},
  121. J = ( 1 1 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 2 1 0 0 0 0 2 ) , J=\begin{pmatrix}1&1&0&0&0\\ 0&1&0&0&0\\ 0&0&2&1&0\\ 0&0&0&2&1\\ 0&0&0&0&2\end{pmatrix},
  122. M M
  123. A A
  124. M M
  125. A A
  126. A M = M J AM=MJ
  127. x m x_{m}
  128. A A
  129. λ \lambda
  130. x m x_{m}
  131. { x m , x m - 1 , , x 1 } \left\{x_{m},x_{m-1},\dots,x_{1}\right\}
  132. x j = ( A - λ I ) m - j x m = ( A - λ I ) x j + 1 ( j = 1 , 2 , , m - 1 ) . x_{j}=(A-\lambda I)^{m-j}x_{m}=(A-\lambda I)x_{j+1}\qquad(j=1,2,\dots,m-1).
  133. x j x_{j}
  134. λ \lambda
  135. x m - 1 , x m - 2 , , x 1 x_{m-1},x_{m-2},\ldots,x_{1}
  136. x m x_{m}
  137. λ i \lambda_{i}
  138. A A
  139. μ i \mu_{i}
  140. ( A - λ i I ) , ( A - λ i I ) 2 , , ( A - λ i I ) m i (A-\lambda_{i}I),(A-\lambda_{i}I)^{2},\ldots,(A-\lambda_{i}I)^{m_{i}}
  141. m i m_{i}
  142. ( A - λ i I ) m i (A-\lambda_{i}I)^{m_{i}}
  143. n - μ i n-\mu_{i}
  144. A A
  145. A A
  146. ρ k = r a n k ( A - λ i I ) k - 1 - r a n k ( A - λ i I ) k ( k = 1 , 2 , , m i ) . \rho_{k}=rank(A-\lambda_{i}I)^{k-1}-rank(A-\lambda_{i}I)^{k}\qquad(k=1,2,% \ldots,m_{i}).
  147. ρ k \rho_{k}
  148. λ i \lambda_{i}
  149. A A
  150. r a n k ( A - λ i I ) 0 = r a n k ( I ) = n rank(A-\lambda_{i}I)^{0}=rank(I)=n
  151. V V
  152. A A
  153. A A
  154. λ i \lambda_{i}
  155. μ i \mu_{i}
  156. λ i : \lambda_{i}:
  157. n - μ i n-\mu_{i}
  158. m i m_{i}
  159. ρ k \rho_{k}
  160. ( k = 1 , , m i ) (k=1,\ldots,m_{i})
  161. λ i \lambda_{i}
  162. A = ( 5 1 - 2 4 0 5 2 2 0 0 5 3 0 0 0 4 ) A=\begin{pmatrix}5&1&-2&4\\ 0&5&2&2\\ 0&0&5&3\\ 0&0&0&4\end{pmatrix}
  163. λ 1 = 5 \lambda_{1}=5
  164. μ 1 = 3 \mu_{1}=3
  165. λ 2 = 4 \lambda_{2}=4
  166. μ 2 = 1 \mu_{2}=1
  167. λ 1 \lambda_{1}
  168. n - μ 1 = 4 - 3 = 1 n-\mu_{1}=4-3=1
  169. ( A - 5 I ) = ( 0 1 - 2 4 0 0 2 2 0 0 0 3 0 0 0 - 1 ) , r a n k ( A - 5 I ) = 3. (A-5I)=\begin{pmatrix}0&1&-2&4\\ 0&0&2&2\\ 0&0&0&3\\ 0&0&0&-1\end{pmatrix},\qquad rank(A-5I)=3.
  170. ( A - 5 I ) 2 = ( 0 0 2 - 8 0 0 0 4 0 0 0 - 3 0 0 0 1 ) , r a n k ( A - 5 I ) 2 = 2. (A-5I)^{2}=\begin{pmatrix}0&0&2&-8\\ 0&0&0&4\\ 0&0&0&-3\\ 0&0&0&1\end{pmatrix},\qquad rank(A-5I)^{2}=2.
  171. ( A - 5 I ) 3 = ( 0 0 0 14 0 0 0 - 4 0 0 0 3 0 0 0 - 1 ) , r a n k ( A - 5 I ) 3 = 1. (A-5I)^{3}=\begin{pmatrix}0&0&0&14\\ 0&0&0&-4\\ 0&0&0&3\\ 0&0&0&-1\end{pmatrix},\qquad rank(A-5I)^{3}=1.
  172. m 1 m_{1}
  173. ( A - 5 I ) m 1 (A-5I)^{m_{1}}
  174. n - μ 1 = 1 n-\mu_{1}=1
  175. m 1 = 3 m_{1}=3
  176. ρ 3 = r a n k ( A - 5 I ) 2 - r a n k ( A - 5 I ) 3 = 2 - 1 = 1 , \rho_{3}=rank(A-5I)^{2}-rank(A-5I)^{3}=2-1=1,
  177. ρ 2 = r a n k ( A - 5 I ) 1 - r a n k ( A - 5 I ) 2 = 3 - 2 = 1 , \rho_{2}=rank(A-5I)^{1}-rank(A-5I)^{2}=3-2=1,
  178. ρ 1 = r a n k ( A - 5 I ) 0 - r a n k ( A - 5 I ) 1 = 4 - 3 = 1. \rho_{1}=rank(A-5I)^{0}-rank(A-5I)^{1}=4-3=1.
  179. λ 1 \lambda_{1}
  180. x 3 x_{3}
  181. λ 1 \lambda_{1}
  182. x 3 x_{3}
  183. x 3 = ( x 31 x 32 x 33 x 34 ) . x_{3}=\begin{pmatrix}x_{31}\\ x_{32}\\ x_{33}\\ x_{34}\end{pmatrix}.
  184. ( A - 5 I ) 3 x 3 = ( 0 0 0 14 0 0 0 - 4 0 0 0 3 0 0 0 - 1 ) ( x 31 x 32 x 33 x 34 ) = ( 14 x 34 - 4 x 34 3 x 34 - x 34 ) = ( 0 0 0 0 ) (A-5I)^{3}x_{3}=\begin{pmatrix}0&0&0&14\\ 0&0&0&-4\\ 0&0&0&3\\ 0&0&0&-1\end{pmatrix}\begin{pmatrix}x_{31}\\ x_{32}\\ x_{33}\\ x_{34}\end{pmatrix}=\begin{pmatrix}14x_{34}\\ -4x_{34}\\ 3x_{34}\\ -x_{34}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ 0\end{pmatrix}
  185. ( A - 5 I ) 2 x 3 = ( 0 0 2 - 8 0 0 0 4 0 0 0 - 3 0 0 0 1 ) ( x 31 x 32 x 33 x 34 ) = ( 2 x 33 - 8 x 34 4 x 34 - 3 x 34 x 34 ) ( 0 0 0 0 ) . (A-5I)^{2}x_{3}=\begin{pmatrix}0&0&2&-8\\ 0&0&0&4\\ 0&0&0&-3\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}x_{31}\\ x_{32}\\ x_{33}\\ x_{34}\end{pmatrix}=\begin{pmatrix}2x_{33}-8x_{34}\\ 4x_{34}\\ -3x_{34}\\ x_{34}\end{pmatrix}\neq\begin{pmatrix}0\\ 0\\ 0\\ 0\end{pmatrix}.
  186. x 34 = 0 x_{34}=0
  187. x 33 0 x_{33}\neq 0
  188. x 31 x_{31}
  189. x 32 x_{32}
  190. x 31 = x 32 = x 34 = 0 , x 33 = 1 x_{31}=x_{32}=x_{34}=0,x_{33}=1
  191. x 3 = ( 0 0 1 0 ) x_{3}=\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix}
  192. λ 1 = 5 \lambda_{1}=5
  193. x 31 x_{31}
  194. x 32 x_{32}
  195. x 33 x_{33}
  196. x 33 0 x_{33}\neq 0
  197. x 2 x_{2}
  198. x 1 x_{1}
  199. x 2 = ( A - 5 I ) x 3 = ( - 2 2 0 0 ) , x_{2}=(A-5I)x_{3}=\begin{pmatrix}-2\\ 2\\ 0\\ 0\end{pmatrix},
  200. x 1 = ( A - 5 I ) x 2 = ( 2 0 0 0 ) . x_{1}=(A-5I)x_{2}=\begin{pmatrix}2\\ 0\\ 0\\ 0\end{pmatrix}.
  201. λ 2 = 4 \lambda_{2}=4
  202. y 1 = ( - 14 4 - 3 1 ) . y_{1}=\begin{pmatrix}-14\\ 4\\ -3\\ 1\end{pmatrix}.
  203. A A
  204. { x 3 , x 2 , x 1 , y 1 } = { ( 0 0 1 0 ) ( - 2 2 0 0 ) ( 2 0 0 0 ) ( - 14 4 - 3 1 ) } . \left\{x_{3},x_{2},x_{1},y_{1}\right\}=\left\{\begin{pmatrix}0\\ 0\\ 1\\ 0\end{pmatrix}\begin{pmatrix}-2\\ 2\\ 0\\ 0\end{pmatrix}\begin{pmatrix}2\\ 0\\ 0\\ 0\end{pmatrix}\begin{pmatrix}-14\\ 4\\ -3\\ 1\end{pmatrix}\right\}.
  205. x 1 , x 2 x_{1},x_{2}
  206. x 3 x_{3}
  207. λ 1 \lambda_{1}
  208. y 1 y_{1}
  209. λ 2 \lambda_{2}
  210. ρ k \rho_{k}
  211. A A
  212. M M
  213. A A
  214. A A
  215. M M
  216. M M
  217. M M
  218. M M
  219. V V
  220. ϕ \phi
  221. L ( V ) L(V)
  222. V V
  223. A A
  224. ϕ \phi
  225. f ( λ ) f(\lambda)
  226. A A
  227. f ( λ ) f(\lambda)
  228. f ( λ ) = ± ( λ - λ 1 ) μ 1 ( λ - λ 2 ) μ 2 ( λ - λ r ) μ r , f(\lambda)=\pm(\lambda-\lambda_{1})^{\mu_{1}}(\lambda-\lambda_{2})^{\mu_{2}}% \cdots(\lambda-\lambda_{r})^{\mu_{r}},
  229. λ 1 , λ 2 , , λ r \lambda_{1},\lambda_{2},\ldots,\lambda_{r}
  230. A A
  231. μ i \mu_{i}
  232. λ i \lambda_{i}
  233. A A
  234. J J
  235. λ i \lambda_{i}
  236. μ i \mu_{i}
  237. λ i \lambda_{i}
  238. λ i \lambda_{i}
  239. J J
  240. A A
  241. A A
  242. A A
  243. J J
  244. J = M - 1 A M J=M^{-1}AM
  245. M M
  246. A A
  247. A = ( 0 4 2 - 3 8 3 4 - 8 - 2 ) . A=\begin{pmatrix}0&4&2\\ -3&8&3\\ 4&-8&-2\end{pmatrix}.
  248. A A
  249. ( λ - 2 ) 3 = 0 (\lambda-2)^{3}=0
  250. λ = 2 \lambda=2
  251. r a n k ( A - 2 I ) = 1 rank(A-2I)=1
  252. r a n k ( A - 2 I ) 2 = 0 = n - μ . rank(A-2I)^{2}=0=n-\mu.
  253. ρ 2 = 1 \rho_{2}=1
  254. ρ 1 = 2 \rho_{1}=2
  255. A A
  256. { x 2 , x 1 } \left\{x_{2},x_{1}\right\}
  257. { y 1 } \left\{y_{1}\right\}
  258. M = ( y 1 x 1 x 2 ) M=\begin{pmatrix}y_{1}&x_{1}&x_{2}\end{pmatrix}
  259. M = ( 2 2 0 1 3 0 0 - 4 1 ) , M=\begin{pmatrix}2&2&0\\ 1&3&0\\ 0&-4&1\end{pmatrix},
  260. J = ( 2 0 0 0 2 1 0 0 2 ) , J=\begin{pmatrix}2&0&0\\ 0&2&1\\ 0&0&2\end{pmatrix},
  261. M M
  262. A A
  263. M M
  264. A A
  265. A M = M J AM=MJ
  266. M M
  267. J J
  268. M M
  269. J J
  270. A A
  271. A A
  272. M = ( y 1 x 1 x 2 x 3 ) = ( - 14 2 - 2 0 4 0 2 0 - 3 0 0 1 1 0 0 0 ) . M=\begin{pmatrix}y_{1}&x_{1}&x_{2}&x_{3}\end{pmatrix}=\begin{pmatrix}-14&2&-2&% 0\\ 4&0&2&0\\ -3&0&0&1\\ 1&0&0&0\end{pmatrix}.
  273. A A
  274. J = ( 4 0 0 0 0 5 1 0 0 0 5 1 0 0 0 5 ) , J=\begin{pmatrix}4&0&0&0\\ 0&5&1&0\\ 0&0&5&1\\ 0&0&0&5\end{pmatrix},
  275. A M = M J AM=MJ
  276. A A
  277. A A
  278. D = M - 1 A M , D=M^{-1}AM,
  279. D = ( λ 1 0 0 0 λ 2 0 0 0 λ n ) , D=\begin{pmatrix}\lambda_{1}&0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{n}\end{pmatrix},
  280. D k = ( λ 1 k 0 0 0 λ 2 k 0 0 0 λ n k ) D^{k}=\begin{pmatrix}\lambda_{1}^{k}&0&\cdots&0\\ 0&\lambda_{2}^{k}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\lambda_{n}^{k}\end{pmatrix}
  281. A A
  282. A A
  283. D k D^{k}
  284. D k D^{k}
  285. M M
  286. M - 1 M^{-1}
  287. A A
  288. x = ( x 1 ( t ) x 2 ( t ) x n ( t ) ) , x = ( x 1 ( t ) x 2 ( t ) x n ( t ) ) , x=\begin{pmatrix}x_{1}(t)\\ x_{2}(t)\\ \vdots\\ x_{n}(t)\end{pmatrix},\quad x^{\prime}=\begin{pmatrix}x_{1}^{\prime}(t)\\ x_{2}^{\prime}(t)\\ \vdots\\ x_{n}^{\prime}(t)\end{pmatrix},
  289. A = ( a i j ) . A=(a_{ij}).
  290. A A
  291. a i j = 0 a_{ij}=0
  292. i j i\neq j
  293. x 1 = k 1 e a 11 t x_{1}=k_{1}e^{a_{11}t}
  294. x 2 = k 2 e a 22 t x_{2}=k_{2}e^{a_{22}t}
  295. \vdots
  296. x n = k n e a n n t . x_{n}=k_{n}e^{a_{nn}t}.
  297. A A
  298. A A
  299. D = M - 1 A M D=M^{-1}AM
  300. M M
  301. A A
  302. A = M D M - 1 A=MDM^{-1}
  303. M - 1 x = D ( M - 1 x ) M^{-1}x^{\prime}=D(M^{-1}x)
  304. y 1 = k 1 e λ 1 t y_{1}=k_{1}e^{\lambda_{1}t}
  305. y 2 = k 2 e λ 2 t y_{2}=k_{2}e^{\lambda_{2}t}
  306. \vdots
  307. y n = k n e λ n t . y_{n}=k_{n}e^{\lambda_{n}t}.
  308. x x
  309. A A
  310. M M
  311. A A
  312. J = M - 1 A M J=M^{-1}AM
  313. A A
  314. y = J y y^{\prime}=Jy
  315. λ i \lambda_{i}
  316. J J
  317. ϵ i \epsilon_{i}
  318. J J
  319. y n y_{n}
  320. y n = k n e λ n t y_{n}=k_{n}e^{\lambda_{n}t}
  321. y n y_{n}
  322. y n - 1 y_{n-1}
  323. y y
  324. x x

Generalized_linear_model.html

  1. E ( 𝐘 ) = s y m b o l μ = g - 1 ( 𝐗 s y m b o l β ) \operatorname{E}(\mathbf{Y})=symbol{\mu}=g^{-1}(\mathbf{X}symbol{\beta})
  2. Var ( 𝐘 ) = V ( s y m b o l μ ) = V ( g - 1 ( 𝐗 s y m b o l β ) ) . \operatorname{Var}(\mathbf{Y})=\operatorname{V}(symbol{\mu})=\operatorname{V}(% g^{-1}(\mathbf{X}symbol{\beta})).
  3. s y m b o l θ symbol\theta
  4. τ \tau
  5. f Y ( 𝐲 | s y m b o l θ , τ ) = h ( 𝐲 , τ ) exp ( 𝐛 ( s y m b o l θ ) T 𝐓 ( y ) - A ( s y m b o l θ ) d ( τ ) ) . f_{Y}(\mathbf{y}|symbol\theta,\tau)=h(\mathbf{y},\tau)\exp{\left(\frac{\mathbf% {b}(symbol\theta)^{\rm T}\mathbf{T}(y)-A(symbol\theta)}{d(\tau)}\right)}.\,\!
  6. τ \tau
  7. h ( 𝐲 , τ ) h(\mathbf{y},\tau)
  8. 𝐛 ( s y m b o l θ ) \mathbf{b}(symbol\theta)
  9. 𝐓 ( y ) \mathbf{T}(y)
  10. A ( s y m b o l θ ) A(symbol\theta)
  11. d ( τ ) d(\tau)
  12. Y Y
  13. θ \theta
  14. f Y ( y | θ , τ ) = h ( y , τ ) exp ( b ( θ ) T ( y ) - A ( θ ) d ( τ ) ) . f_{Y}(y|\theta,\tau)=h(y,\tau)\exp{\left(\frac{b(\theta)T(y)-A(\theta)}{d(\tau% )}\right)}.\,\!
  15. s y m b o l θ symbol\theta
  16. 𝐛 ( s y m b o l θ ) \mathbf{b}(symbol\theta)
  17. s y m b o l θ symbol\theta
  18. s y m b o l θ symbol\theta^{\prime}
  19. s y m b o l θ = 𝐛 ( s y m b o l θ ) symbol\theta=\mathbf{b}(symbol\theta^{\prime})
  20. A ( s y m b o l θ ) A(symbol\theta)
  21. 𝐛 ( s y m b o l θ ) \mathbf{b}(symbol\theta^{\prime})
  22. 𝐓 ( y ) \mathbf{T}(y)
  23. τ \tau
  24. s y m b o l θ symbol\theta
  25. s y m b o l μ = E ( 𝐘 ) = A ( s y m b o l θ ) . symbol\mu=\operatorname{E}(\mathbf{Y})=\nabla A(symbol\theta).\,\!
  26. Y Y
  27. θ \theta
  28. μ = E ( Y ) = A ( θ ) . \mu=\operatorname{E}(Y)=A^{\prime}(\theta).\,\!
  29. Var ( 𝐘 ) = T A ( s y m b o l θ ) d ( τ ) . \operatorname{Var}(\mathbf{Y})=\nabla\nabla^{\rm T}A(symbol\theta)d(\tau).\,\!
  30. Y Y
  31. θ \theta
  32. Var ( Y ) = A ′′ ( θ ) d ( τ ) . \operatorname{Var}(Y)=A^{\prime\prime}(\theta)d(\tau).\,\!
  33. η = 𝐗 s y m b o l β . \eta=\mathbf{X}symbol{\beta}.\,
  34. θ \theta
  35. θ \theta
  36. μ \mu
  37. θ = b ( μ ) \theta=b(\mu)
  38. μ \mu
  39. b ( μ ) b(\mu)
  40. b ( μ ) = θ = 𝐗 s y m b o l β b(\mu)=\theta=\mathbf{X}symbol{\beta}
  41. 𝐗 T 𝐘 \mathbf{X}^{\rm T}\mathbf{Y}
  42. s y m b o l β symbol{\beta}
  43. ( - , + ) (-\infty,+\infty)
  44. 𝐗 s y m b o l β = μ \mathbf{X}symbol{\beta}=\mu\,\!
  45. μ = 𝐗 s y m b o l β \mu=\mathbf{X}symbol{\beta}\,\!
  46. ( 0 , + ) (0,+\infty)
  47. 𝐗 s y m b o l β = - μ - 1 \mathbf{X}symbol{\beta}=-\mu^{-1}\,\!
  48. μ = - ( 𝐗 s y m b o l β ) - 1 \mu=-(\mathbf{X}symbol{\beta})^{-1}\,\!
  49. ( 0 , + ) (0,+\infty)
  50. 𝐗 s y m b o l β = - μ - 2 \mathbf{X}symbol{\beta}=-\mu^{-2}\,\!
  51. μ = ( - 𝐗 s y m b o l β ) - 1 / 2 \mu=(-\mathbf{X}symbol{\beta})^{-1/2}\,\!
  52. 0 , 1 , 2 , 0,1,2,\ldots
  53. 𝐗 s y m b o l β = ln ( μ ) \mathbf{X}symbol{\beta}=\ln{(\mu)}\,\!
  54. μ = exp ( 𝐗 s y m b o l β ) \mu=\exp{(\mathbf{X}symbol{\beta})}\,\!
  55. { 0 , 1 } \{0,1\}
  56. 𝐗 s y m b o l β = ln ( μ 1 - μ ) \mathbf{X}symbol{\beta}=\ln{\left(\frac{\mu}{1-\mu}\right)}\,\!
  57. μ = exp ( 𝐗 s y m b o l β ) 1 + exp ( 𝐗 s y m b o l β ) = 1 1 + exp ( - 𝐗 s y m b o l β ) \mu=\frac{\exp{(\mathbf{X}symbol{\beta})}}{1+\exp{(\mathbf{X}symbol{\beta})}}=% \frac{1}{1+\exp{(-\mathbf{X}symbol{\beta})}}\,\!
  58. 0 , 1 , , N 0,1,\ldots,N
  59. [ 0 , K ) [0,K)
  60. [ 0 , 1 ] [0,1]
  61. [ 0 , N ] [0,N]
  62. [ 0 , 1 ] [0,1]
  63. s y m b o l β ( t + 1 ) = s y m b o l β ( t ) + 𝒥 - 1 ( s y m b o l β ( t ) ) u ( s y m b o l β ( t ) ) , symbol\beta^{(t+1)}=symbol\beta^{(t)}+\mathcal{J}^{-1}(symbol\beta^{(t)})u(% symbol\beta^{(t)}),
  64. 𝒥 ( s y m b o l β ( t ) ) \mathcal{J}(symbol\beta^{(t)})
  65. u ( s y m b o l β ( t ) ) u(symbol\beta^{(t)})
  66. s y m b o l β ( t + 1 ) = s y m b o l β ( t ) + - 1 ( s y m b o l β ( t ) ) u ( s y m b o l β ( t ) ) , symbol\beta^{(t+1)}=symbol\beta^{(t)}+\mathcal{I}^{-1}(symbol\beta^{(t)})u(% symbol\beta^{(t)}),
  67. ( s y m b o l β ( t ) ) \mathcal{I}(symbol\beta^{(t)})
  68. g ( p ) = ln ( p 1 - p ) . g(p)=\ln\left({p\over 1-p}\right).
  69. [ 0 , 1 ] [0,1]
  70. Φ \Phi
  71. g ( p ) = Φ - 1 ( p ) . g(p)=\Phi^{-1}(p).\,\!
  72. l o g ( l o g ( 1 p ) ) log(−log(1−p))
  73. [ 0 , 1 ] [0,1]
  74. p = 0.5 p=0.5
  75. Var ( Y i ) = τ μ i ( 1 - μ i ) \operatorname{Var}(Y_{i})=\tau\mu_{i}(1-\mu_{i})\,\!
  76. g ( μ m ) = η m = β 0 + X 1 β 1 + + X p β p + γ 2 + + γ m = η 1 + γ 2 + + γ m g(\mu_{m})=\eta_{m}=\beta_{0}+X_{1}\beta_{1}+\ldots+X_{p}\beta_{p}+\gamma_{2}+% \ldots+\gamma_{m}=\eta_{1}+\gamma_{2}+\ldots+\gamma_{m}\,
  77. μ m = P ( Y m ) \mu_{m}=\mathrm{P}(Y\leq m)\,
  78. g ( μ m ) = η m = β m , 0 + X 1 β m , 1 + + X p β m , p g(\mu_{m})=\eta_{m}=\beta_{m,0}+X_{1}\beta_{m,1}+\ldots+X_{p}\beta_{m,p}\,
  79. μ m = P ( Y = m Y { 1 , m } ) \mu_{m}=\mathrm{P}(Y=m\mid Y\in\{1,m\})\,
  80. var ( Y i ) = τ μ i , \operatorname{var}(Y_{i})=\tau\mu_{i},\,
  81. η = β 0 + f 1 ( x 1 ) + f 2 ( x 2 ) + \eta=\beta_{0}+f_{1}(x_{1})+f_{2}(x_{2})+\ldots\,\!

Generalized_signal_averaging.html

  1. M = 1 9 [ 1 1 1 1 1 1 1 1 1 ] M=\frac{1}{9}\begin{bmatrix}1&1&1\\ 1&1&1\\ 1&1&1\\ \end{bmatrix}
  2. M = 1 10 [ 1 1 1 1 2 1 1 1 1 ] M=\frac{1}{10}\begin{bmatrix}1&1&1\\ 1&2&1\\ 1&1&1\\ \end{bmatrix}
  3. M = 1 16 [ 1 2 1 2 4 2 1 2 1 ] M=\frac{1}{16}\begin{bmatrix}1&2&1\\ 2&4&2\\ 1&2&1\\ \end{bmatrix}

Generalized_singular_value_decomposition.html

  1. 𝔽 = \mathbb{F}=\mathbb{R}
  2. 𝔽 = \mathbb{F}=\mathbb{C}
  3. A 𝔽 m × n A\in\mathbb{F}^{m\times n}
  4. B 𝔽 p × n B\in\mathbb{F}^{p\times n}
  5. A = U Σ 1 [ X , 0 ] Q * A=U\Sigma_{1}[X,0]Q^{*}
  6. B = V Σ 2 [ X , 0 ] Q * B=V\Sigma_{2}[X,0]Q^{*}
  7. U 𝔽 m × m , V 𝔽 p × p U\in\mathbb{F}^{m\times m},V\in\mathbb{F}^{p\times p}
  8. Q 𝔽 n × n Q\in\mathbb{F}^{n\times n}
  9. X 𝔽 r × r X\in\mathbb{F}^{r\times r}
  10. r = r a n k ( [ A * , B * ] ) r=rank([A^{*},B^{*}])
  11. Σ 1 𝔽 m × r \Sigma_{1}\in\mathbb{F}^{m\times r}
  12. Σ 2 𝔽 p × r \Sigma_{2}\in\mathbb{F}^{p\times r}
  13. Σ 2 \Sigma_{2}
  14. Σ 1 T Σ 1 = α 1 2 , , α r 2 \Sigma_{1}^{T}\Sigma_{1}=\lceil\alpha_{1}^{2},\dots,\alpha_{r}^{2}\rfloor
  15. Σ 2 T Σ 2 = β 1 2 , , β r 2 \Sigma_{2}^{T}\Sigma_{2}=\lceil\beta_{1}^{2},\dots,\beta_{r}^{2}\rfloor
  16. Σ 1 T Σ 1 + Σ 2 T Σ 2 = I r \Sigma_{1}^{T}\Sigma_{1}+\Sigma_{2}^{T}\Sigma_{2}=I_{r}
  17. 0 α i , β i 1 0\leq\alpha_{i},\beta_{i}\leq 1
  18. σ i = α i / β i \sigma_{i}=\alpha_{i}/\beta_{i}
  19. A A
  20. B B
  21. B B
  22. U U
  23. V V
  24. A B - 1 AB^{-1}
  25. B = I B=I
  26. M = U Σ V * M=U\Sigma V^{*}\,
  27. U * W u U = V * W v V = I . U^{*}W_{u}U=V^{*}W_{v}V=I.
  28. U U
  29. V V
  30. W u W_{u}
  31. W v W_{v}
  32. W u W_{u}
  33. W v W_{v}

Generalized_taxicab_number.html

  1. Taxicab ( 1 , 2 , 2 ) = 4 = 1 + 3 = 2 + 2. \mathrm{Taxicab}(1,2,2)=4=1+3=2+2.
  2. Taxicab ( 2 , 2 , 2 ) = 50 = 1 2 + 7 2 = 5 2 + 5 2 . \mathrm{Taxicab}(2,2,2)=50=1^{2}+7^{2}=5^{2}+5^{2}.
  3. Taxicab ( 3 , 2 , 2 ) = 1729 = 1 3 + 12 3 = 9 3 + 10 3 \mathrm{Taxicab}(3,2,2)=1729=1^{3}+12^{3}=9^{3}+10^{3}
  4. Taxicab ( 4 , 2 , 2 ) = 635318657 = 59 4 + 158 4 = 133 4 + 134 4 . \mathrm{Taxicab}(4,2,2)=635318657=59^{4}+158^{4}=133^{4}+134^{4}.

Geodetic_datum.html

  1. ϕ \ \phi
  2. λ \ \lambda
  3. h h
  4. ϕ \ \phi
  5. ϕ \ \phi^{\prime}
  6. a a
  7. f f
  8. ( λ , ϕ ) \ (\lambda,\phi)
  9. ( ϕ , λ ) \ (\phi,\lambda)

Geometric_group_theory.html

  1. \mathbb{R}
  2. S L ( n , ) SL(n,\mathbb{R})

Geometric_quantization.html

  1. ( M , Ω ) (M,\Omega)
  2. f C ( M ) f\in C^{\infty}(M)
  3. M M
  4. f ^ \widehat{f}
  5. L M L\to M
  6. L M L\to M
  7. R R
  8. R = i Ω R=i\Omega
  9. T T
  10. M M
  11. Ω ( v , v ) = 0 \Omega(v,v^{\prime})=0
  12. v , v T v,v^{\prime}\in T
  13. [ v , v ] Γ ( T ) [v,v^{\prime}]\in\Gamma(T)
  14. v , v Γ ( T ) v,v^{\prime}\in\Gamma(T)
  15. A M A_{M}
  16. M M
  17. f ^ \widehat{f}
  18. f f
  19. X f X_{f}
  20. [ X f , T ] T [X_{f},T]\subset T
  21. A M A_{M}
  22. M M
  23. f f + i 1 / 2 X f f\mapsto f\cdot+i\hbar^{1/2}\mathcal{L}_{X_{f}}
  24. X \mathcal{L}_{X}

Geometrical_optics.html

  1. n 1 n_{1}
  2. n 2 n_{2}
  3. n 1 sin θ 1 = n 2 sin θ 2 n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}
  4. θ 1 \theta_{1}
  5. θ 2 \theta_{2}
  6. v 1 sin θ 2 = v 2 sin θ 1 v_{1}\sin\theta_{2}\ =v_{2}\sin\theta_{1}
  7. v 1 v_{1}
  8. v 2 v_{2}
  9. f f
  10. S 1 S_{1}
  11. 1 S 1 + 1 S 2 = 1 f \frac{1}{S_{1}}+\frac{1}{S_{2}}=\frac{1}{f}
  12. S 2 S_{2}
  13. M = - S 2 S 1 = f f - S 1 M=-\frac{S_{2}}{S_{1}}=\frac{f}{f-S_{1}}
  14. u ( t , x ) a ( t , x ) e i ( k x - ω t ) u(t,x)\approx a(t,x)e^{i(k\cdot x-\omega t)}
  15. k , ω k,\omega
  16. a ( t , x ) a(t,x)
  17. a 0 ( t , x ) e i φ ( t , x ) / ε . a_{0}(t,x)e^{i\varphi(t,x)/\varepsilon}.
  18. φ ( t , x ) / ε \varphi(t,x)/\varepsilon
  19. k := x φ k:=\nabla_{x}\varphi
  20. ω := - t φ \omega:=-\partial_{t}\varphi
  21. a 0 a_{0}
  22. ε \varepsilon\,
  23. ( t , x ) × n (t,x)\in\mathbb{R}\times\mathbb{R}^{n}
  24. L ( t , x ) u := ( 2 t 2 - c ( x ) 2 Δ ) u ( t , x ) = 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = 0 L(\partial_{t},\nabla_{x})u:=\left(\frac{\partial^{2}}{\partial t^{2}}-c(x)^{2% }\Delta\right)u(t,x)=0,\;\;u(0,x)=u_{0}(x),\;\;u_{t}(0,x)=0
  25. u ( t , x ) a ε ( t , x ) e i φ ( t , x ) / ε = j = 0 i j ε j a j ( t , x ) e i φ ( t , x ) / ε . u(t,x)\sim a_{\varepsilon}(t,x)e^{i\varphi(t,x)/\varepsilon}=\sum_{j=0}^{% \infty}i^{j}\varepsilon^{j}a_{j}(t,x)e^{i\varphi(t,x)/\varepsilon}.
  26. L ( t , x ) ( e i φ ( t , x ) / ε ) a ε ( t , x ) = e i φ ( t , x ) / ε ( ( i ε ) 2 L ( φ t , x φ ) a ε + 2 i ε V ( t , x ) a ε + i ε ( a ε L ( t , x ) φ ) + L ( t , x ) a ε ) L(\partial_{t},\nabla_{x})(e^{i\varphi(t,x)/\varepsilon})a_{\varepsilon}(t,x)=% e^{i\varphi(t,x)/\varepsilon}\left(\left(\frac{i}{\varepsilon}\right)^{2}L(% \varphi_{t},\nabla_{x}\varphi)a_{\varepsilon}+\frac{2i}{\varepsilon}V(\partial% _{t},\nabla_{x})a_{\varepsilon}+\frac{i}{\varepsilon}(a_{\varepsilon}L(% \partial_{t},\nabla_{x})\varphi)+L(\partial_{t},\nabla_{x})a_{\varepsilon}\right)
  27. V ( t , x ) := φ t t - c 2 ( x ) j φ x j x j V(\partial_{t},\nabla_{x}):=\frac{\partial\varphi}{\partial t}\frac{\partial}{% \partial t}-c^{2}(x)\sum_{j}\frac{\partial\varphi}{\partial x_{j}}\frac{% \partial}{\partial x_{j}}
  28. ε \varepsilon
  29. O ( ε - 2 ) O(\varepsilon^{-2})
  30. 0 = L ( φ t , x φ ) = ( φ t ) 2 - c ( x ) 2 ( x φ ) 2 . 0=L(\varphi_{t},\nabla_{x}\varphi)=(\varphi_{t})^{2}-c(x)^{2}(\nabla_{x}% \varphi)^{2}.
  31. ε - 1 \varepsilon^{-1}
  32. 2 V a 0 + ( L φ ) a 0 = 0 2Va_{0}+(L\varphi)a_{0}=0
  33. k := x φ k:=\nabla_{x}\varphi
  34. ω := - φ t \omega:=-\varphi_{t}
  35. e i ( k x - ω t ) e^{i(k\cdot x-\omega t)}
  36. c c
  37. a 0 a_{0}
  38. φ \varphi

Geometric–harmonic_mean.html

  1. g n + 1 = g n h n g_{n+1}=\sqrt{g_{n}h_{n}}
  2. h n + 1 = 2 1 g n + 1 h n h_{n+1}=\frac{2}{\frac{1}{g_{n}}+\frac{1}{h_{n}}}
  3. M ( x , y ) = 1 A G ( 1 x , 1 y ) M(x,y)=\frac{1}{AG(\frac{1}{x},\frac{1}{y})}
  4. min ( x , y ) H ( x , y ) H G ( x , y ) G ( x , y ) G A ( x , y ) A ( x , y ) max ( x , y ) \min(x,y)\leq H(x,y)\leq HG(x,y)\leq G(x,y)\leq GA(x,y)\leq A(x,y)\leq\max(x,y)

George_Johnstone_Stoney.html

  1. m S = e 2 4 π ε 0 G = α m P m_{S}=\sqrt{\frac{e^{2}}{4\pi\varepsilon_{0}G}}=\sqrt{\alpha}\,m_{P}
  2. P = m c G m c 2 \ell_{P}=\sqrt{\frac{\hbar}{mc}\cdot\frac{Gm}{c^{2}}}
  3. S = e 2 4 π ε 0 m c 2 G m c 2 \ell_{S}=\sqrt{\frac{e^{2}}{4\pi\varepsilon_{0}mc^{2}}\cdot\frac{Gm}{c^{2}}}
  4. \hbar

George_Kingsley_Zipf.html

  1. P n 1 / n a P_{n}\sim 1/n^{a}

Gerbe.html

  1. H 3 ( X , ) H^{3}(X,\mathbb{Z})
  2. 𝒪 X * \mathcal{O}_{X}^{*}

Gerstenhaber_algebra.html

  1. ( - 1 ) ( | a | - 1 ) ( | c | - 1 ) [ a , [ b , c ] ] + ( - 1 ) ( | b | - 1 ) ( | a | - 1 ) [ b , [ c , a ] ] + ( - 1 ) ( | c | - 1 ) ( | b | - 1 ) [ c , [ a , b ] ] = 0. (-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b% |-1)}[c,[a,b]]=0.\,

Ghost_condensate.html

  1. S = d 4 x [ a X 2 - b X ] S=\int d^{4}x\left[aX^{2}-bX\right]
  2. X = def 1 2 η μ ν μ ϕ ν ϕ X\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi% \partial_{\nu}\phi

Giant_magnetoresistance.html

  1. δ H = R ( 0 ) - R ( H ) R ( H ) \delta_{H}=\frac{R(0)-R(H)}{R(H)}
  2. w = - J ( 𝐌 1 𝐌 2 ) . w=-J(\mathbf{M}_{1}\cdot\mathbf{M}_{2}).
  3. R = R 0 + Δ R sin 2 θ 2 , R=R_{0}+\Delta R\sin^{2}\frac{\theta}{2},
  4. ρ F ± = 2 ρ F 1 ± β , \rho_{F\pm}=\frac{2\rho_{F}}{1\pm\beta},
  5. δ H = Δ R R = R - R R = ( ρ F + - ρ F - ) 2 ( 2 ρ F + + χ ρ N ) ( 2 ρ F - + χ ρ N ) . \delta_{H}=\frac{\Delta R}{R}=\frac{R_{\uparrow\downarrow}-R_{\uparrow\uparrow% }}{R_{\uparrow\uparrow}}=\frac{(\rho_{F+}-\rho_{F-})^{2}}{(2\rho_{F+}+\chi\rho% _{N})(2\rho_{F-}+\chi\rho_{N})}.
  6. χ ρ N ρ F ± \chi\rho_{N}\ll\rho_{F\pm}
  7. δ H = β 2 1 - β 2 . \delta_{H}=\frac{\beta^{2}}{1-\beta^{2}}.
  8. Δ μ = β 1 - β 2 e E 0 l s e z / l s , \Delta\mu=\frac{\beta}{1-\beta^{2}}eE_{0}l_{s}e^{z/l_{s}},
  9. Δ E = β 2 1 - β 2 e E 0 l s e z / l s , \Delta E=\frac{\beta^{2}}{1-\beta^{2}}eE_{0}l_{s}e^{z/l_{s}},
  10. R i = β ( μ - μ ) 2 e j = β 2 l s N ρ N 1 + ( 1 - β 2 ) l s N ρ N / ( l s F ρ F ) , R_{i}=\frac{\beta(\mu_{\uparrow\downarrow}-\mu_{\uparrow\uparrow})}{2ej}=\frac% {\beta^{2}l_{sN}\rho_{N}}{1+(1-\beta^{2})l_{sN}\rho_{N}/(l_{sF}\rho_{F})},
  11. Δ G = Δ G S V + Δ G f ( 1 - e β t / λ ) , \Delta G=\Delta G_{SV}+\Delta G_{f}(1-e^{\beta t/\lambda}),
  12. δ H ( d N ) = δ H 0 exp ( - d N / λ N ) 1 + d N / d 0 , \delta_{H}(d_{N})=\delta_{H0}\frac{\exp\left(-d_{N}/\lambda_{N}\right)}{1+d_{N% }/d_{0}},
  13. δ H ( d F ) = δ H 1 1 - exp ( - d F / λ F ) 1 + d F / d 0 . \delta_{H}(d_{F})=\delta_{H1}\frac{1-\exp\left(-d_{F}/\lambda_{F}\right)}{1+d_% {F}/d_{0}}.
  14. ρ , = 2 ρ F 1 ± β \rho_{\uparrow,\downarrow}=\frac{2\rho_{F}}{1\pm\beta}

Gilbreath's_conjecture.html

  1. { p n } \{p_{n}\}
  2. p n p_{n}
  3. { d n } \{d_{n}\}
  4. d n = p n + 1 - p n d_{n}=p_{n+1}-p_{n}
  5. n n
  6. k k
  7. { d n k } \{d_{n}^{k}\}
  8. d n k = | d n + 1 k - 1 - d n k - 1 | d_{n}^{k}=|d_{n+1}^{k-1}-d_{n}^{k-1}|
  9. a k = d 1 k a_{k}={d_{1}^{k}}
  10. k k
  11. d 1 k d_{1}^{k}
  12. k n = 3.4 × 10 11 k\leq n=3.4\times 10^{11}

Givens_rotation.html

  1. G ( i , j , θ ) = [ 1 0 0 0 0 c - s 0 0 s c 0 0 0 0 1 ] G(i,j,\theta)=\begin{bmatrix}1&\cdots&0&\cdots&0&\cdots&0\\ \vdots&\ddots&\vdots&&\vdots&&\vdots\\ 0&\cdots&c&\cdots&-s&\cdots&0\\ \vdots&&\vdots&\ddots&\vdots&&\vdots\\ 0&\cdots&s&\cdots&c&\cdots&0\\ \vdots&&\vdots&&\vdots&\ddots&\vdots\\ 0&\cdots&0&\cdots&0&\cdots&1\end{bmatrix}
  2. c = c o s θ c=cosθ
  3. s = s i n θ s=sinθ
  4. i i
  5. j j
  6. g k k \displaystyle g_{k\,k}
  7. j > i j>i
  8. G ( i , j , θ ) 𝐱 G(i,j,θ)\mathbf{x}
  9. 𝐱 \mathbf{x}
  10. ( i , j ) (i,j)
  11. θ θ
  12. G ( i , j , θ ) G(i,j,θ)
  13. A A
  14. G A G A
  15. i i
  16. j j
  17. A A
  18. a a
  19. b b
  20. c = c o s θ c=cosθ
  21. s = s i n θ s=sinθ
  22. [ c - s s c ] [ a b ] = [ r 0 ] . \begin{bmatrix}c&-s\\ s&c\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix}=\begin{bmatrix}r\\ 0\end{bmatrix}.
  23. θ θ
  24. c c
  25. s s
  26. r r
  27. r a 2 + b 2 c a / r s - b / r . \begin{aligned}\displaystyle r&\displaystyle{}\leftarrow\sqrt{a^{2}+b^{2}}\\ \displaystyle c&\displaystyle{}\leftarrow a/r\\ \displaystyle s&\displaystyle{}\leftarrow-b/r.\end{aligned}
  28. r r
  29. r r
  30. | x | s g n ( y ) |x|⋅sgn(y)
  31. A = [ 6 5 0 5 1 4 0 4 3 ] A=\begin{bmatrix}6&5&0\\ 5&1&4\\ 0&4&3\\ \end{bmatrix}
  32. G 1 = [ c - s 0 s c 0 0 0 1 ] G_{1}=\begin{bmatrix}c&-s&0\\ s&c&0\\ 0&0&1\\ \end{bmatrix}
  33. [ c - s 0 s c 0 0 0 1 ] [ 6 5 0 5 1 4 0 4 3 ] \begin{bmatrix}c&-s&0\\ s&c&0\\ 0&0&1\\ \end{bmatrix}\begin{bmatrix}6&5&0\\ 5&1&4\\ 0&4&3\\ \end{bmatrix}
  34. r \displaystyle r
  35. c c
  36. s s
  37. A A
  38. A = [ 7.8102 4.4813 2.5607 0 - 2.4327 3.0729 0 4 3 ] A=\begin{bmatrix}7.8102&4.4813&2.5607\\ 0&-2.4327&3.0729\\ 0&4&3\\ \end{bmatrix}
  39. G 2 = [ 1 0 0 0 c - s 0 s c ] G_{2}=\begin{bmatrix}1&0&0\\ 0&c&-s\\ 0&s&c\\ \end{bmatrix}
  40. [ 1 0 0 0 c - s 0 s c ] [ 7.8102 4.4813 2.5607 0 - 2.4327 3.0729 0 4 3 ] \begin{bmatrix}1&0&0\\ 0&c&-s\\ 0&s&c\\ \end{bmatrix}\begin{bmatrix}7.8102&4.4813&2.5607\\ 0&-2.4327&3.0729\\ 0&4&3\\ \end{bmatrix}
  41. r \displaystyle r
  42. c c
  43. s s
  44. R = [ 7.8102 4.4813 2.5607 0 4.6817 0.9664 0 0 - 4.1843 ] R=\begin{bmatrix}7.8102&4.4813&2.5607\\ 0&4.6817&0.9664\\ 0&0&-4.1843\\ \end{bmatrix}
  45. Q Q
  46. Q = G 1 T G 2 T Q=G_{1}^{T}\,G_{2}^{T}
  47. Q = [ 0.7682 0.3327 0.5470 0.6402 - 0.3992 - 0.6564 0 0.8544 - 0.5196 ] Q=\begin{bmatrix}0.7682&0.3327&0.5470\\ 0.6402&-0.3992&-0.6564\\ 0&0.8544&-0.5196\\ \end{bmatrix}
  48. e i , e j e_{i},e_{j}
  49. B i j = e i e j B_{ij}=e_{i}\wedge e_{j}
  50. v = e - ( θ / 2 ) ( e i e j ) u e ( θ / 2 ) ( e i e j ) v=e^{-(\theta/2)(e_{i}\wedge e_{j})}ue^{(\theta/2)(e_{i}\wedge e_{j})}
  51. e ( θ / 2 ) ( e i e j ) = cos ( θ / 2 ) + sin ( θ / 2 ) e i e j e^{(\theta/2)(e_{i}\wedge e_{j})}=\cos(\theta/2)+\sin(\theta/2)e_{i}\wedge e_{j}
  52. R X ( θ ) = [ 1 0 0 0 cos θ - sin θ 0 sin θ cos θ ] \displaystyle R_{X}(\theta)=\begin{bmatrix}1&0&0\\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta\end{bmatrix}
  53. R Y ( θ ) R_{Y}(\theta)
  54. R Y ( θ ) R_{Y}(\theta)
  55. R Y ( θ ) = [ cos θ 0 sin θ 0 1 0 - sin θ 0 cos θ ] \begin{aligned}\\ \displaystyle R_{Y}(\theta)=\begin{bmatrix}\cos\theta&0&\sin\theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\end{bmatrix}\end{aligned}
  56. R Y ( θ ) R_{Y}(\theta)
  57. R Y ( θ ) = [ cos θ 0 - sin θ 0 1 0 sin θ 0 cos θ ] \displaystyle R_{Y}(\theta)=\begin{bmatrix}\cos\theta&0&-\sin\theta\\ 0&1&0\\ \sin\theta&0&\cos\theta\end{bmatrix}
  58. R Z ( θ ) = [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] \begin{aligned}\\ \displaystyle R_{Z}(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}\end{aligned}
  59. g f f g g∘f≠f∘g
  60. R = R Y ( θ 3 ) . R X ( θ 2 ) . R Z ( θ 1 ) R=R_{Y}(\theta_{3}).R_{X}(\theta_{2}).R_{Z}(\theta_{1})
  61. Y P R = ( θ 3 , θ 2 , θ 1 ) YPR=(\theta_{3},\theta_{2},\theta_{1})
  62. g f g∘f
  63. f f
  64. g g
  65. f f
  66. g g
  67. [ c 2 - c 1 s 2 s 1 s 2 c 3 s 2 c 3 c 2 c 1 - s 3 s 1 - c 2 c 3 s 1 - c 1 s 3 s 2 s 3 c 3 s 1 + c 1 c 2 s 3 c 3 c 1 - c 2 s 3 s 1 ] \begin{bmatrix}c_{2}&-c_{1}s_{2}&s_{1}s_{2}\\ c_{3}s_{2}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{2}c_{3}s_{1}-c_{1}s_{3}\\ s_{2}s_{3}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}\end{bmatrix}
  68. [ c 2 c 3 - c 3 s 2 c 1 + s 3 s 1 c 3 s 2 s 1 + s 3 c 1 s 2 c 1 c 2 - c 2 s 1 - s 3 c 2 s 3 s 2 c 1 + c 3 s 1 - s 3 s 2 s 1 + c 3 c 1 ] \begin{bmatrix}c_{2}c_{3}&-c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{% 1}\\ s_{2}&c_{1}c_{2}&-c_{2}s_{1}\\ -s_{3}c_{2}&s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}\end{bmatrix}
  69. [ c 2 s 1 s 2 c 1 s 2 s 2 s 3 c 3 c 1 - c 2 s 3 s 1 - c 3 s 1 - c 1 c 2 s 3 - c 3 s 2 c 3 c 2 s 1 + c 1 s 3 c 3 c 2 c 1 - s 3 s 1 ] \begin{bmatrix}c_{2}&s_{1}s_{2}&c_{1}s_{2}\\ s_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\ -c_{3}s_{2}&c_{3}c_{2}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}\end{bmatrix}
  70. [ c 3 c 2 - s 3 c 1 + c 3 s 2 s 1 s 3 s 1 + c 3 s 2 c 1 s 3 c 2 c 3 c 1 + s 3 s 2 s 1 - c 3 s 1 + s 3 s 2 c 1 - s 2 c 2 s 1 c 2 c 1 ] \begin{bmatrix}c_{3}c_{2}&-s_{3}c_{1}+c_{3}s_{2}s_{1}&s_{3}s_{1}+c_{3}s_{2}c_{% 1}\\ s_{3}c_{2}&c_{3}c_{1}+s_{3}s_{2}s_{1}&-c_{3}s_{1}+s_{3}s_{2}c_{1}\\ -s_{2}&c_{2}s_{1}&c_{2}c_{1}\end{bmatrix}
  71. [ c 3 c 1 - c 2 s 3 s 1 s 2 s 3 c 3 s 1 + s 3 c 2 c 1 s 1 s 2 c 2 - c 1 s 2 - c 2 c 3 s 1 - c 1 s 3 c 3 s 2 c 3 c 2 c 1 - s 3 s 1 ] \begin{bmatrix}c_{3}c_{1}-c_{2}s_{3}s_{1}&s_{2}s_{3}&c_{3}s_{1}+s_{3}c_{2}c_{1% }\\ s_{1}s_{2}&c_{2}&-c_{1}s_{2}\\ -c_{2}c_{3}s_{1}-c_{1}s_{3}&c_{3}s_{2}&c_{3}c_{2}c_{1}-s_{3}s_{1}\end{bmatrix}
  72. [ c 3 c 1 - s 3 s 2 s 1 - s 3 c 2 c 3 s 1 + s 3 s 2 c 1 s 3 c 1 + c 3 s 2 s 1 c 3 c 2 s 3 s 1 - c 3 s 2 c 1 - c 2 s 1 s 2 c 2 c 1 ] \begin{bmatrix}c_{3}c_{1}-s_{3}s_{2}s_{1}&-s_{3}c_{2}&c_{3}s_{1}+s_{3}s_{2}c_{% 1}\\ s_{3}c_{1}+c_{3}s_{2}s_{1}&c_{3}c_{2}&s_{3}s_{1}-c_{3}s_{2}c_{1}\\ -c_{2}s_{1}&s_{2}&c_{2}c_{1}\end{bmatrix}
  73. [ c 3 c 2 c 1 - s 3 s 1 - c 3 s 2 c 2 c 3 s 1 + c 1 s 3 c 1 s 2 c 2 s 1 s 2 - c 3 s 1 - c 1 c 2 s 3 s 2 s 3 c 3 c 1 - c 2 s 3 s 1 ] \begin{bmatrix}c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}&c_{2}c_{3}s_{1}+c_{1}s_{% 3}\\ c_{1}s_{2}&c_{2}&s_{1}s_{2}\\ -c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}\end{bmatrix}
  74. [ c 2 c 1 - s 2 c 2 s 1 c 3 s 2 c 1 + s 3 s 1 c 3 c 2 c 3 s 2 s 1 - s 3 c 1 s 3 s 2 c 1 - c 3 s 1 s 3 c 2 s 3 s 2 s 1 + c 3 c 1 ] \begin{bmatrix}c_{2}c_{1}&-s_{2}&c_{2}s_{1}\\ c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}c_{2}&c_{3}s_{2}s_{1}-s_{3}c_{1}\\ s_{3}s_{2}c_{1}-c_{3}s_{1}&s_{3}c_{2}&s_{3}s_{2}s_{1}+c_{3}c_{1}\end{bmatrix}
  75. [ c 3 c 2 c 1 - s 3 s 1 - c 2 s 1 c 3 - c 1 s 3 c 3 s 2 c 3 s 1 + c 1 c 2 s 3 c 3 c 1 - c 2 s 3 s 1 s 2 s 3 - c 1 s 2 s 1 s 2 c 2 ] \begin{bmatrix}c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{2}s_{1}c_{3}-c_{1}s_{3}&c_{3}s_{% 2}\\ c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}&s_{2}s_{3}\\ -c_{1}s_{2}&s_{1}s_{2}&c_{2}\end{bmatrix}
  76. [ c 2 c 1 - c 2 s 1 s 2 s 3 s 2 c 1 + c 3 s 1 - s 3 s 2 s 1 + c 3 c 1 - s 3 c 2 - c 3 s 2 c 1 + s 3 s 1 c 3 s 2 s 1 + s 3 c 1 c 3 c 2 ] \begin{bmatrix}c_{2}c_{1}&-c_{2}s_{1}&s_{2}\\ s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}&-s_{3}c_{2}\\ -c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}&c_{3}c_{2}\end{bmatrix}
  77. [ c 3 c 1 - c 2 s 1 s 3 - c 3 s 1 - c 1 c 2 s 3 s 2 s 3 c 2 c 3 s 1 + c 1 s 3 c 3 c 2 c 1 - s 3 s 1 - c 3 s 2 s 1 s 2 c 1 s 2 c 2 ] \begin{bmatrix}c_{3}c_{1}-c_{2}s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{% 3}\\ c_{2}c_{3}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}\\ s_{1}s_{2}&c_{1}s_{2}&c_{2}\end{bmatrix}
  78. [ c 3 c 1 + s 3 s 2 s 1 - c 3 s 1 + s 3 s 2 c 1 s 3 c 2 c 2 s 1 c 2 c 1 - s 2 - s 3 c 1 + c 3 s 2 s 1 s 3 s 1 + c 3 s 2 c 1 c 3 c 2 ] \begin{bmatrix}c_{3}c_{1}+s_{3}s_{2}s_{1}&-c_{3}s_{1}+s_{3}s_{2}c_{1}&s_{3}c_{% 2}\\ c_{2}s_{1}&c_{2}c_{1}&-s_{2}\\ -s_{3}c_{1}+c_{3}s_{2}s_{1}&s_{3}s_{1}+c_{3}s_{2}c_{1}&c_{3}c_{2}\end{bmatrix}

Gliese_876.html

  1. M V = 4.83 \scriptstyle M_{V_{\odot}}=4.83
  2. L V L V = 10 0.4 ( M V - M V ) \scriptstyle\frac{L_{V_{\ast}}}{L_{V_{\odot}}}=10^{0.4\left(M_{V_{\odot}}-M_{V% _{\ast}}\right)}
  3. ± \pm

Global_section_functor.html

  1. Sh ( X , 𝒞 ) \mathrm{Sh}(X,\mathcal{C})
  2. 𝒞 \mathcal{C}
  3. \mathcal{F}
  4. Γ ( X , ) \Gamma(X,\mathcal{F})
  5. 𝒞 \mathcal{C}
  6. 𝒞 \mathcal{C}
  7. ¯ X {\underline{\mathbb{Z}}}_{X}
  8. Γ ( X , ¯ X ) = π 0 ( X ) \Gamma(X,{\underline{\mathbb{Z}}}_{X})=\mathbb{Z}^{\pi_{0}(X)}
  9. 𝒪 X \mathcal{O}_{X}
  10. Γ ( X , 𝒪 X ) = \Gamma(X,\mathcal{O}_{X})=\mathbb{C}
  11. 𝒪 ( i ) , i \mathcal{O}(i),i\in\mathbb{Z}
  12. k n \mathbb{P}_{k}^{n}
  13. Γ ( X , 𝒪 ( d ) ) = k d [ X 0 , , X n ] \Gamma(X,\mathcal{O}(d))=k_{d}[X_{0},\ldots,X_{n}]
  14. d 0 d\geq 0
  15. d < 0 d<0

Glomerulus_(kidney).html

  1. G F R = K f [ ( P gc - P bc ) - ( π gc - π bc ) ] \ GFR=K_{\mathrm{f}}[(P_{\mathrm{gc}}-P_{\mathrm{bc}})-(\pi_{\mathrm{gc}}-\pi_% {\mathrm{bc}})]

Glutamate_dehydrogenase.html

  1. K m K_{m}
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

Go_ranks_and_ratings.html

  1. S E ( A ) = 1 e D / a + 1 S_{E}(A)=\frac{1}{e^{D/a}+1}
  2. R B - R A R_{B}-R_{A}\,
  3. S E ( B ) = 1 - S E ( A ) S_{E}(B)=1-S_{E}(A)\,
  4. R n = R o + K ( S - S E ) R_{n}=R_{o}+K(S-S_{E})\,

Goddard–Thorn_theorem.html

  1. V I I 1 , 1 V_{II_{1,1}}
  2. I ^ I 1 , 1 \hat{I}I_{1,1}
  3. I I 1 , 1 II_{1,1}
  4. V I I 1 , 1 V_{II_{1,1}}
  5. I I 1 , 1 II_{1,1}
  6. V V I I 1 , 1 V\otimes V_{II_{1,1}}
  7. P r 1 P^{1}_{r}
  8. I I 1 , 1 II_{1,1}
  9. V I I 1 , 1 V_{II_{1,1}}
  10. P r 1 P^{1}_{r}
  11. V 1 - ( r , r ) / 2 V^{1-(r,r)/2}
  12. V 1 2 V^{1}\oplus\mathbb{R}^{2}

Gordon_Jenkins.html

  1. 2 4 \textstyle\frac{2}{4}

Graded_category.html

  1. 𝒜 \mathcal{A}
  2. 𝒜 \mathcal{A}
  3. 𝒞 \mathcal{C}
  4. F : 𝒞 𝒜 F:\mathcal{C}\rightarrow\mathcal{A}
  5. 𝒞 \mathcal{C}
  6. 𝔾 \mathbb{G}
  7. 𝒮 = { S g : g G } \mathcal{S}=\{S_{g}:g\in G\}
  8. 𝒞 \mathcal{C}
  9. S 1 S_{1}
  10. 𝒜 \mathcal{A}
  11. S g S h = S g h S_{g}S_{h}=S_{gh}
  12. g , h 𝔾 g,h\in\mathbb{G}
  13. S g S_{g}
  14. g 𝔾 g\in\mathbb{G}
  15. ( 𝒞 , 𝒮 ) (\mathcal{C},\mathcal{S})
  16. 𝔾 \mathbb{G}

Graded_vector_space.html

  1. V = n V n V=\bigoplus_{n\in\mathbb{N}}V_{n}
  2. V n V_{n}
  3. V n V_{n}
  4. V = i I V i . V=\bigoplus_{i\in I}V_{i}.
  5. \mathbb{N}
  6. \mathbb{N}
  7. / 2 \mathbb{Z}/2\mathbb{Z}
  8. ( / 2 ) (\mathbb{Z}/2\mathbb{Z})
  9. f ( V i ) W i f(V_{i})\subseteq W_{i}
  10. f ( V j ) W i + j f(V_{j})\subseteq W_{i+j}
  11. f ( V i + j ) W j f(V_{i+j})\subseteq W_{j}
  12. f ( V j ) = 0 f(V_{j})=0\,
  13. V W V\otimes W
  14. ( V W ) i = { j , k | j + k = i } V j W k . (V\otimes W)_{i}=\bigoplus_{\{j,k|j+k=i\}}V_{j}\otimes W_{k}.

Gradient-index_optics.html

  1. n = f ( x , y , z ) n=f(x,y,z)
  2. L = S o S n d s L=\int_{S_{o}}^{S}n\,ds
  3. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  4. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Graham_number.html

  1. 22.5 × ( earnings per share ) × ( book value per share ) \sqrt{22.5\times(\,\text{earnings per share})\times(\,\text{book value per % share})}
  2. 15 × 1.5 × ( net income shares outstanding ) × ( shareholders equity shares outstanding ) \sqrt{15\times 1.5\times\left(\frac{\,\text{net income}}{\,\text{shares % outstanding}}\right)\times\left(\frac{\mathrm{shareholders^{\prime}\ equity}}{% \,\text{shares outstanding}}\right)}

Gramian_matrix.html

  1. v 1 , , v n v_{1},\dots,v_{n}
  2. G i j = v i , v j G_{ij}=\langle v_{i},v_{j}\rangle
  3. G = V T V G=V^{\mathrm{T}}V
  4. G = V V G=V^{\dagger}V
  5. v k v_{k}
  6. { i ( ) , i = 1 , , n } \{\ell_{i}(\cdot),\,i=1,\dots,n\}
  7. [ t 0 , t f ] [t_{0},t_{f}]
  8. G = [ G i j ] G=[G_{ij}]
  9. G i j = t 0 t f i ( τ ) j ¯ ( τ ) d τ . G_{ij}=\int_{t_{0}}^{t_{f}}\ell_{i}(\tau)\bar{\ell_{j}}(\tau)\,d\tau.
  10. v 1 , , v n v_{1},\dots,v_{n}
  11. G i j = B ( v i , v j ) G_{ij}=B(v_{i},v_{j})\,
  12. P = U D U * = ( U D ) ( U D ) * P=UDU^{*}=(U\sqrt{D})(U\sqrt{D})^{*}
  13. U D U\sqrt{D}
  14. G ( x 1 , , x n ) = | x 1 , x 1 x 1 , x 2 x 1 , x n x 2 , x 1 x 2 , x 2 x 2 , x n x n , x 1 x n , x 2 x n , x n | . G(x_{1},\dots,x_{n})=\begin{vmatrix}\langle x_{1},x_{1}\rangle&\langle x_{1},x% _{2}\rangle&\dots&\langle x_{1},x_{n}\rangle\\ \langle x_{2},x_{1}\rangle&\langle x_{2},x_{2}\rangle&\dots&\langle x_{2},x_{n% }\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \langle x_{n},x_{1}\rangle&\langle x_{n},x_{2}\rangle&\dots&\langle x_{n},x_{n% }\rangle\end{vmatrix}.
  15. G ( x 1 , , x n ) = x 1 x n 2 . G(x_{1},\dots,x_{n})=\|x_{1}\wedge\cdots\wedge x_{n}\|^{2}.

Grand_canonical_ensemble.html

  1. µ µ
  2. T T
  3. V V
  4. µ V T µVT
  5. P P
  6. P = e Ω + μ N - E k T , P=e^{\frac{\Omega+\mu N-E}{kT}},
  7. N N
  8. E E
  9. k k
  10. Ω Ω
  11. Ω Ω
  12. µ , V , T µ,V,T
  13. Ω Ω
  14. Ω ( µ , V , T ) Ω(µ,V,T)
  15. P = e Ω + μ 1 N 1 + μ 2 N 2 + + μ s N s - E k T , P=e^{\frac{\Omega+\mu_{1}N_{1}+\mu_{2}N_{2}+\ldots+\mu_{s}N_{s}-E}{kT}},
  16. s s
  17. P = 1 𝒵 e ( μ N - E ) / ( k T ) \textstyle P=\frac{1}{\mathcal{Z}}e^{(\mu N-E)/(kT)}
  18. 𝒵 = e - Ω / ( k T ) \textstyle\mathcal{Z}=e^{-\Omega/(kT)}
  19. Ω Ω
  20. d Ω = - S d T - N 1 d μ 1 - N s d μ s - p d V . d\Omega=-SdT-\langle N_{1}\rangle d\mu_{1}\ldots-\langle N_{s}\rangle d\mu_{s}% -\langle p\rangle dV.
  21. E ⟨E⟩
  22. Ω Ω
  23. d E = T d S + μ 1 d N 1 + μ s d N s - p d V . d\langle E\rangle=TdS+\mu_{1}d\langle N_{1}\rangle\ldots+\mu_{s}d\langle N_{s}% \rangle-\langle p\rangle dV.
  24. E 2 - E 2 = k T 2 E T + k T μ 1 E μ 1 + k T μ 2 E μ 2 + , \langle E^{2}\rangle-\langle E\rangle^{2}=kT^{2}\frac{\partial\langle E\rangle% }{\partial T}+kT\mu_{1}\frac{\partial\langle E\rangle}{\partial\mu_{1}}+kT\mu_% {2}\frac{\partial\langle E\rangle}{\partial\mu_{2}}+\ldots,
  25. N 1 2 - N 1 2 = k T N 1 μ 1 . \langle N_{1}^{2}\rangle-\langle N_{1}\rangle^{2}=kT\frac{\partial\langle N_{1% }\rangle}{\partial\mu_{1}}.
  26. N 1 N 2 - N 1 N 2 = k T N 2 μ 1 = k T N 1 μ 2 . \langle N_{1}N_{2}\rangle-\langle N_{1}\rangle\langle N_{2}\rangle=kT\frac{% \partial\langle N_{2}\rangle}{\partial\mu_{1}}=kT\frac{\partial\langle N_{1}% \rangle}{\partial\mu_{2}}.
  27. N 1 E - N 1 E = k T E μ 1 , \langle N_{1}E\rangle-\langle N_{1}\rangle\langle E\rangle=kT\frac{\partial% \langle E\rangle}{\partial\mu_{1}},
  28. Ω = - k T ln ( microstates e μ N - E k T ) \Omega=-kT\ln\Big(\sum_{\rm microstates}e^{\frac{\mu N-E}{kT}}\Big)
  29. N N
  30. N ϵ
  31. ϵ ϵ
  32. N = 1 N=1
  33. Ω = - k T ln ( N = 0 1 e N μ - N ϵ k T ) = - k T ln ( 1 + e μ - ϵ k T ) \begin{aligned}\displaystyle\Omega&\displaystyle=-kT\ln\Big(\sum_{N=0}^{1}e^{% \frac{N\mu-N\epsilon}{kT}}\Big)\\ &\displaystyle=-kT\ln\Big(1+e^{\frac{\mu-\epsilon}{kT}}\Big)\end{aligned}
  34. N N
  35. N N
  36. Ω \displaystyle\Omega
  37. N = - Ω μ \scriptstyle\langle N\rangle=-\tfrac{\partial\Omega}{\partial\mu}
  38. Ω = - k T ln ( N = 0 1 N ! e N μ - N ϵ k T ) = - k T ln ( e e μ - ϵ k T ) = - k T e μ - ϵ k T , \begin{aligned}\displaystyle\Omega&\displaystyle=-kT\ln\Big(\sum_{N=0}^{\infty% }\frac{1}{N!}e^{\frac{N\mu-N\epsilon}{kT}}\Big)\\ &\displaystyle=-kT\ln\Big(e^{e^{\frac{\mu-\epsilon}{kT}}}\Big)\\ &\displaystyle=-kTe^{\frac{\mu-\epsilon}{kT}},\end{aligned}
  39. N = - Ω μ \scriptstyle\langle N\rangle=-\tfrac{\partial\Omega}{\partial\mu}
  40. ϕ ϕ
  41. q −q
  42. Ω = - k T ln ( e μ N 0 - E 0 k T + e μ N 0 - μ - E 0 - Δ E I - q ϕ k T + e μ N 0 + μ - E 0 + Δ E A + q ϕ k T ) . = E 0 - μ N 0 - k T ln ( 1 + e - μ - Δ E I - q ϕ k T + e μ + Δ E A + q ϕ k T ) . \begin{aligned}\displaystyle\Omega&\displaystyle=-kT\ln\Big(e^{\frac{\mu N_{0}% -E_{0}}{kT}}+e^{\frac{\mu N_{0}-\mu-E_{0}-\Delta E_{\rm I}-q\phi}{kT}}+e^{% \frac{\mu N_{0}+\mu-E_{0}+\Delta E_{\rm A}+q\phi}{kT}}\Big).\\ &\displaystyle=E_{0}-\mu N_{0}-kT\ln\Big(1+e^{\frac{-\mu-\Delta E_{\rm I}-q% \phi}{kT}}+e^{\frac{\mu+\Delta E_{\rm A}+q\phi}{kT}}\Big).\\ \end{aligned}
  43. q ϕ µ −qϕ−µ
  44. q ϕ µ = W −qϕ−µ=W
  45. q ϕ −qϕ
  46. µ µ
  47. 600 m e V 600meV
  48. µ N µN
  49. µ = 0 µ=0
  50. µ µ
  51. N N
  52. N = ( p a r t i c l e n u m b e r - a n t i p a r t i c l e n u m b e r ) N=(particlenumber-antiparticlenumber)
  53. ρ̂ ρ̂
  54. ρ ^ = exp ( 1 k T ( Ω + μ 1 N ^ 1 + + μ s N ^ s - H ^ ) ) , \hat{\rho}=\exp\big(\tfrac{1}{kT}(\Omega+\mu_{1}\hat{N}_{1}+\ldots+\mu_{s}\hat% {N}_{s}-\hat{H})\big),
  55. Ĥ Ĥ
  56. e x p exp
  57. Ω Ω
  58. T r ρ̂ = 1 Trρ̂=1
  59. e - Ω k T = Tr exp ( 1 k T ( μ 1 N ^ 1 + + μ s N ^ s - H ^ ) ) . e^{-\frac{\Omega}{kT}}=\operatorname{Tr}\exp\big(\tfrac{1}{kT}(\mu_{1}\hat{N}_% {1}+\ldots+\mu_{s}\hat{N}_{s}-\hat{H})\big).
  60. Ĥ Ĥ
  61. i i
  62. ρ ^ = i e Ω + μ 1 N 1 , i + + μ s N s , i - E i k T | ψ i ψ i | \hat{\rho}=\sum_{i}e^{\frac{\Omega+\mu_{1}N_{1,i}+\ldots+\mu_{s}N_{s,i}-E_{i}}% {kT}}|\psi_{i}\rangle\langle\psi_{i}|
  63. e - Ω k T = i e μ 1 N 1 , i + + μ s N s , i - E i k T . e^{-\frac{\Omega}{kT}}=\sum_{i}e^{\frac{\mu_{1}N_{1,i}+\ldots+\mu_{s}N_{s,i}-E% _{i}}{kT}}.
  64. i i
  65. n n
  66. n n
  67. n = 3 N n=3N
  68. ρ = 1 h n C e Ω + μ 1 N 1 + + μ s N s - E k T , \rho=\frac{1}{h^{n}C}e^{\frac{\Omega+\mu_{1}N_{1}+\ldots+\mu_{s}N_{s}-E}{kT}},
  69. E E
  70. h h
  71. e n e r g y × t i m e energy×time
  72. ρ ρ
  73. C C
  74. Ω Ω
  75. ρ ρ
  76. e - Ω k T = N 1 = 0 N s = 0 1 h n C e μ 1 N 1 + + μ s N s - E k T d p 1 d q n e^{-\frac{\Omega}{kT}}=\sum_{N_{1}=0}^{\infty}\ldots\sum_{N_{s}=0}^{\infty}% \int\ldots\int\frac{1}{h^{n}C}e^{\frac{\mu_{1}N_{1}+\ldots+\mu_{s}N_{s}-E}{kT}% }\,dp_{1}\ldots dq_{n}
  77. C C
  78. C = 1 C=1
  79. C = N 1 ! N 2 ! N s ! , C=N_{1}!N_{2}!\ldots N_{s}!,
  80. C C
  81. i i
  82. N i N_{i}
  83. μ i \mu_{i}
  84. µ V A T µVAT
  85. h = 1 e n e r g y u n i t t × × t i m e u n i t t h=1energyunitt××timeunitt
  86. h h
  87. N N
  88. N N

Granular_computing.html

  1. t e m p temp
  2. c l u b club
  3. p ( Y = y j | X = x i ) α p(Y=y_{j}|X=x_{i})\geq\alpha
  4. α = 1 \alpha=1
  5. X = x i Y = y j X=x_{i}\rightarrow Y=y_{j}
  6. X = x i X=x_{i}
  7. Y = y j Y=y_{j}
  8. X X
  9. { x 1 , x 2 , x 3 , x 4 } \{x_{1},x_{2},x_{3},x_{4}\}
  10. { X 1 , X 2 } \{X_{1},X_{2}\}
  11. Y Y
  12. { y 1 , y 2 , y 3 , y 4 } \{y_{1},y_{2},y_{3},y_{4}\}
  13. { Y 1 , Y 2 } \{Y_{1},Y_{2}\}
  14. X = x i Y = y j X=x_{i}\rightarrow Y=y_{j}
  15. x i x_{i}
  16. y j y_{j}
  17. x i x_{i}
  18. p ( Y = y j | X = x i ) < 1 p(Y=y_{j}|X=x_{i})<1
  19. X = X 1 Y = Y 1 X=X_{1}\leftrightarrow Y=Y_{1}
  20. X = X 2 Y = Y 2 X=X_{2}\leftrightarrow Y=Y_{2}
  21. X 1 X_{1}
  22. Y 1 Y_{1}
  23. X 2 X_{2}
  24. Y 2 Y_{2}
  25. P 1 P_{1}
  26. P 2 P_{2}
  27. P 3 P_{3}
  28. P 4 P_{4}
  29. P 5 P_{5}
  30. O 1 O_{1}
  31. O 2 O_{2}
  32. O 3 O_{3}
  33. O 4 O_{4}
  34. O 5 O_{5}
  35. O 6 O_{6}
  36. O 7 O_{7}
  37. O 8 O_{8}
  38. O 9 O_{9}
  39. O 10 O_{10}
  40. P = { P 1 , P 2 , P 3 , P 4 , P 5 } P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}
  41. { { O 1 , O 2 } { O 3 , O 7 , O 10 } { O 4 } { O 5 } { O 6 } { O 8 } { O 9 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{7},O_{10}\}\\ \{O_{4}\}\\ \{O_{5}\}\\ \{O_{6}\}\\ \{O_{8}\}\\ \{O_{9}\}\end{cases}
  42. { O 1 , O 2 } \{O_{1},O_{2}\}
  43. { O 3 , O 7 , O 10 } \{O_{3},O_{7},O_{10}\}
  44. P 1 P_{1}
  45. { { O 1 , O 2 } { O 3 , O 5 , O 7 , O 9 , O 10 } { O 4 , O 6 , O 8 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{5},O_{7},O_{9},O_{10}\}\\ \{O_{4},O_{6},O_{8}\}\end{cases}
  46. [ x ] Q = { Q 1 , Q 2 , Q 3 , , Q N } [x]_{Q}=\{Q_{1},Q_{2},Q_{3},\dots,Q_{N}\}
  47. Q i Q_{i}
  48. Q Q
  49. Q Q
  50. P 1 P_{1}
  51. [ x ] Q [x]_{Q}
  52. Q 1 = { O 1 , O 2 } Q_{1}=\{O_{1},O_{2}\}
  53. Q 2 = { O 3 , O 5 , O 7 , O 9 , O 10 } Q_{2}=\{O_{3},O_{5},O_{7},O_{9},O_{10}\}
  54. Q 3 = { O 4 , O 6 , O 8 } Q_{3}=\{O_{4},O_{6},O_{8}\}
  55. Q Q
  56. P P
  57. γ P ( Q ) \gamma_{P}(Q)
  58. γ P ( Q ) = | i = 1 N P ¯ Q i | | 𝕌 | 1 \gamma_{P}(Q)=\frac{\left|\sum_{i=1}^{N}{\underline{P}}Q_{i}\right|}{\left|% \mathbb{U}\right|}\leq 1
  59. Q i Q_{i}
  60. [ x ] Q [x]_{Q}
  61. P P
  62. P ¯ Q i {\underline{P}}Q_{i}
  63. P P
  64. Q i Q_{i}
  65. [ x ] Q [x]_{Q}
  66. P P
  67. Q Q
  68. [ x ] Q [x]_{Q}
  69. [ x ] P [x]_{P}
  70. γ P ( Q ) \gamma_{P}(Q)
  71. P P
  72. Q Q
  73. P 1 P_{1}
  74. P 2 P_{2}
  75. P 3 P_{3}
  76. P 4 P_{4}
  77. P 5 P_{5}
  78. O 1 O_{1}
  79. O 2 O_{2}
  80. O 3 O_{3}
  81. O 4 O_{4}
  82. O 5 O_{5}
  83. O 6 O_{6}
  84. O 7 O_{7}
  85. O 8 O_{8}
  86. O 9 O_{9}
  87. O 10 O_{10}
  88. Q = { P 4 , P 5 } Q=\{P_{4},P_{5}\}
  89. P = { P 2 , P 3 } P=\{P_{2},P_{3}\}
  90. [ x ] Q [x]_{Q}
  91. [ x ] P [x]_{P}
  92. [ x ] Q [x]_{Q}
  93. [ x ] P [x]_{P}
  94. [ x ] Q [x]_{Q}
  95. [ x ] P [x]_{P}
  96. { { O 1 , O 2 } { O 3 , O 7 , O 10 } { O 4 , O 5 , O 8 } { O 6 , O 9 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{7},O_{10}\}\\ \{O_{4},O_{5},O_{8}\}\\ \{O_{6},O_{9}\}\end{cases}
  97. { { O 1 , O 2 } { O 3 , O 7 , O 10 } { O 4 , O 6 } { O 5 , O 9 } { O 8 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{7},O_{10}\}\\ \{O_{4},O_{6}\}\\ \{O_{5},O_{9}\}\\ \{O_{8}\}\end{cases}
  98. [ x ] Q [x]_{Q}
  99. [ x ] P [x]_{P}
  100. { O 1 , O 2 , O 3 , O 7 , O 8 , O 10 } \{O_{1},O_{2},O_{3},O_{7},O_{8},O_{10}\}
  101. Q Q
  102. P P
  103. γ P ( Q ) = 6 / 10 \gamma_{P}(Q)=6/10
  104. Q = { P 4 } Q=\{P_{4}\}
  105. P = { P 2 , P 3 } P=\{P_{2},P_{3}\}
  106. Q = { P 4 , P 5 } Q=\{P_{4},P_{5}\}
  107. Q = { P 4 } Q=\{P_{4}\}
  108. [ x ] Q [x]_{Q}
  109. [ x ] Q [x]_{Q}
  110. [ x ] P [x]_{P}
  111. [ x ] Q [x]_{Q}
  112. [ x ] P [x]_{P}
  113. [ x ] Q [x]_{Q}
  114. [ x ] P [x]_{P}
  115. { { O 1 , O 2 , O 3 , O 7 , O 10 } { O 4 , O 5 , O 6 , O 8 , O 9 } \begin{cases}\{O_{1},O_{2},O_{3},O_{7},O_{10}\}\\ \{O_{4},O_{5},O_{6},O_{8},O_{9}\}\end{cases}
  116. { { O 1 , O 2 } { O 3 , O 7 , O 10 } { O 4 , O 6 } { O 5 , O 9 } { O 8 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{7},O_{10}\}\\ \{O_{4},O_{6}\}\\ \{O_{5},O_{9}\}\\ \{O_{8}\}\end{cases}
  117. [ x ] Q [x]_{Q}
  118. [ x ] Q [x]_{Q}
  119. [ x ] P [x]_{P}
  120. { O 1 , O 2 , , O 10 } \{O_{1},O_{2},\ldots,O_{10}\}
  121. Q Q
  122. P P
  123. γ P ( Q ) = 1 \gamma_{P}(Q)=1
  124. [ x ] P [x]_{P}
  125. [ x ] Q [x]_{Q}
  126. P Q P\rightarrow Q
  127. [ x ] Q [x]_{Q}
  128. [ x ] Q [x]_{Q}

Graph_homomorphism.html

  1. f f
  2. G = ( V , E ) G=(V,E)
  3. G = ( V , E ) G^{\prime}=(V^{\prime},E^{\prime})
  4. f : G G f:G\rightarrow G^{\prime}
  5. f : V V f:V\rightarrow V^{\prime}
  6. G G
  7. G G^{\prime}
  8. { u , v } E \{u,v\}\in E
  9. { f ( u ) , f ( v ) } E \{f(u),f(v)\}\in E^{\prime}
  10. f : G G f:G\rightarrow G^{\prime}
  11. ( f ( u ) , f ( v ) ) (f(u),f(v))
  12. G G^{\prime}
  13. ( u , v ) (u,v)
  14. G G
  15. f : G H f:G\rightarrow H
  16. G H G\rightarrow H
  17. G ↛ H G\not\rightarrow H
  18. G H G\rightarrow H
  19. G G
  20. H H
  21. H H
  22. f : G G f:G\rightarrow G^{\prime}
  23. f f
  24. G G
  25. G G^{\prime}
  26. G G G\rightarrow G^{\prime}
  27. G G G^{\prime}\rightarrow G
  28. G G
  29. H H
  30. G G
  31. r : G H r:G\rightarrow H
  32. r ( x ) = x r(x)=x
  33. x x
  34. H H
  35. f : G K k f:G\rightarrow K_{k}
  36. H G K k H\rightarrow G\rightarrow K_{k}
  37. H K k H\rightarrow K_{k}
  38. H G H\rightarrow G
  39. H G H\rightarrow G
  40. C g G C_{g}\rightarrow G
  41. G H G\rightarrow H

Graphene.html

  1. 1 / 2 {1}/{2}
  2. E = ± γ 0 2 ( 1 + 4 cos 2 k y a 2 + 4 cos k y a 2 cos k x 3 a 2 ) E=\pm\sqrt{\gamma_{0}^{2}\left(1+4\cos^{2}{\frac{k_{y}a}{2}}+4\cos{\frac{k_{y}% a}{2}}\cdot\cos{\frac{k_{x}\sqrt{3}a}{2}}\right)}
  3. v F σ ψ ( 𝐫 ) = E ψ ( 𝐫 ) . v_{F}\,\vec{\sigma}\cdot\nabla\psi(\mathbf{r})\,=\,E\psi(\mathbf{r}).
  4. σ \vec{\sigma}
  5. ψ ( 𝐫 ) \psi(\mathbf{r})
  6. E = v F k x 2 + k y 2 E=\hbar v_{F}\sqrt{k_{x}^{2}+k_{y}^{2}}
  7. 4 e 2 / h 4e^{2}/h
  8. 4 e 2 / ( π h ) 4e^{2}/{(\pi}h)
  9. 4 e 2 / h 4e^{2}/h
  10. σ x y \sigma_{xy}
  11. e 2 / h e^{2}/h
  12. σ x y = ± 4 ( N + 1 / 2 ) e 2 / h \sigma_{xy}=\pm{4\cdot\left(N+1/2\right)e^{2}}/h
  13. σ x y = ± 4 N e 2 / h \sigma_{xy}=\pm{4\cdot N\cdot e^{2}}/h
  14. σ x y = ν e 2 / h \sigma_{xy}=\nu e^{2}/h
  15. ν = 0 , ± 1 , ± 4 \nu=0,\pm{1},\pm{4}
  16. ν = 3 \nu=3
  17. ν = 1 / 3 \nu=1/3
  18. ν = 0 , ± 1 , ± 3 , ± 4 \nu=0,\pm 1,\pm 3,\pm 4

Graphic_matroid.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. x A B x\in A\setminus B
  6. B { x } B\cup\{x\}
  7. G G
  8. F F
  9. G G
  10. F F
  11. A A
  12. B B
  13. A A
  14. B B
  15. C C
  16. A A
  17. B B
  18. C C
  19. B B
  20. B B
  21. F F
  22. G G
  23. M ( G ) M(G)
  24. M ( G ) M(G)
  25. G G
  26. M ( G ) M(G)
  27. G G
  28. M ( G ) M(G)
  29. X X
  30. G G
  31. r ( X ) = n - c r(X)=n-c
  32. n n
  33. G G
  34. c c
  35. X X
  36. cl ( S ) \operatorname{cl}(S)
  37. S S
  38. M ( G ) M(G)
  39. S S
  40. S S
  41. G G
  42. S S
  43. S S
  44. cl ( S ) \operatorname{cl}(S)
  45. x y x\leq y
  46. x x
  47. y y
  48. K n K_{n}
  49. K n K_{n}
  50. n n
  51. G G
  52. G G
  53. e e
  54. + 1 +1
  55. - 1 -1
  56. - 1 -1
  57. M ( G ) M(G)
  58. U 4 2 U{}^{2}_{4}
  59. M ( K 5 ) M(K_{5})
  60. M ( K 3 , 3 ) M(K_{3,3})
  61. K 5 K_{5}
  62. K 3 , 3 K_{3,3}
  63. M ( K 5 ) M(K_{5})
  64. M ( K 3 , 3 ) M(K_{3,3})
  65. M ( K 5 ) M(K_{5})
  66. M ( K 3 , 3 ) M(K_{3,3})
  67. K 5 K_{5}
  68. K 3 , 3 K_{3,3}
  69. M ( G ) M(G)
  70. G G
  71. G G
  72. M ( G ) M(G)
  73. G G
  74. G G

Graviphoton.html

  1. A μ A_{\mu}
  2. g μ 5 g_{\mu 5}

Graviscalar.html

  1. ϕ \phi
  2. g 55 g_{55}

Gravitational_time_dilation.html

  1. g ( h ) g(h)
  2. h = 0 h=0
  3. T d ( h ) = exp [ 1 c 2 0 h g ( h ) d h ] T_{d}(h)=\exp\left[\frac{1}{c^{2}}\int_{0}^{h}g(h^{\prime})dh^{\prime}\right]
  4. T d ( h ) T_{d}(h)
  5. h h
  6. g ( h ) g(h)
  7. h h
  8. c c
  9. exp \exp
  10. g ( h ) = c 2 / ( H + h ) g(h)=c^{2}/(H+h)
  11. H H
  12. T d ( h ) = e ln ( H + h ) - ln H = H + h H T_{d}(h)=e^{\ln(H+h)-\ln H}=\tfrac{H+h}{H}
  13. g g
  14. g h gh
  15. c 2 c^{2}
  16. T d = 1 + g h / c 2 T_{d}=1+gh/c^{2}
  17. t 0 = t f 1 - 2 G M r c 2 = t f 1 - r 0 r t_{0}=t_{f}\sqrt{1-\frac{2GM}{rc^{2}}}=t_{f}\sqrt{1-\frac{r_{0}}{r}}
  18. t 0 t_{0}
  19. t f t_{f}
  20. G G
  21. M M
  22. r r
  23. c c
  24. r 0 = 2 G M / c 2 r_{0}=2GM/c^{2}
  25. M M
  26. 3 2 r 0 \tfrac{3}{2}r_{0}
  27. t 0 = t f 1 - 3 2 r 0 r . t_{0}=t_{f}\sqrt{1-\frac{3}{2}\!\cdot\!\frac{r_{0}}{r}}\,.

Greek_alphabet.html

  1. A {}^{A}
  2. ϵ \epsilon\,\!
  3. ε \varepsilon\,\!
  4. ϕ \textstyle\phi\,\!
  5. φ \textstyle\varphi\,\!

Green's_identities.html

  1. 𝐅 = ψ φ \mathbf{F}=ψ∇φ
  2. φ φ
  3. ψ ψ
  4. φ φ
  5. ψ ψ
  6. U ( ψ Δ φ + φ ψ ) d V = U ψ ( φ n ) d S = U ψ φ d 𝐒 \int_{U}\left(\psi\Delta\varphi+\nabla\varphi\cdot\nabla\psi\right)\,dV=\oint_% {\partial U}\psi\left(\nabla\varphi\cdot{n}\right)\,dS=\oint_{\partial U}\psi% \nabla\varphi\cdot d\mathbf{S}
  7. Δ \Delta
  8. U ∂U
  9. U U
  10. 𝐧 \mathbf{n}
  11. d S dS
  12. d d
  13. 𝐒 \mathbf{S}
  14. ψ ψ
  15. φ φ
  16. u u
  17. v v
  18. 𝐅 = ψ 𝚪 \mathbf{F}=ψ\mathbf{Γ}
  19. U ( ψ 𝚪 + 𝚪 ψ ) d V = U ψ ( 𝚪 n ) d S = U ψ 𝚪 d 𝐒 . \int_{U}\left(\psi\nabla\cdot\mathbf{\Gamma}+\mathbf{\Gamma}\cdot\nabla\psi% \right)\,dV=\oint_{\partial U}\psi\left(\mathbf{\Gamma}\cdot{n}\right)\,dS=% \oint_{\partial U}\psi\mathbf{\Gamma}\cdot d\mathbf{S}.
  20. φ φ
  21. ψ ψ
  22. ε ε
  23. 𝐅 = ψ ε φ φ ε ψ \mathbf{F}=ψε∇φ−φε∇ψ
  24. U [ ψ ( ϵ φ ) - φ ( ϵ ψ ) ] d V = U ϵ ( ψ φ 𝐧 - φ ψ 𝐧 ) d S . \int_{U}\left[\psi\nabla\cdot\left(\epsilon\nabla\varphi\right)-\varphi\nabla% \cdot\left(\epsilon\nabla\psi\right)\right]\,dV=\oint_{\partial U}\epsilon% \left(\psi{\partial\varphi\over\partial\mathbf{n}}-\varphi{\partial\psi\over% \partial\mathbf{n}}\right)\,dS.
  25. ε = 1 ε=1
  26. U ( ψ Δ φ - φ Δ ψ ) d V = U ( ψ φ 𝐧 - φ ψ 𝐧 ) d S . \int_{U}\left(\psi\Delta\varphi-\varphi\Delta\psi\right)\,dV=\oint_{\partial U% }\left(\psi{\partial\varphi\over\partial\mathbf{n}}-\varphi{\partial\psi\over% \partial\mathbf{n}}\right)\,dS.
  27. φ / 𝐧 ∂φ/∂\mathbf{n}
  28. φ φ
  29. 𝐧 \mathbf{n}
  30. d S dS
  31. φ 𝐧 = φ 𝐧 = 𝐧 φ . {\partial\varphi\over\partial\mathbf{n}}=\nabla\varphi\cdot\mathbf{n}=\nabla_{% \mathbf{n}}\varphi.
  32. φ = G φ=G
  33. G G
  34. Δ \Delta
  35. Δ G ( 𝐱 , η ) = δ ( 𝐱 - η ) . \Delta G(\mathbf{x},\mathbf{\eta})=\delta(\mathbf{x}-\mathbf{\eta}).
  36. G ( 𝐱 , η ) = - 1 4 π 𝐱 - η . G(\mathbf{x},\mathbf{\eta})=\frac{-1}{4\pi\|\mathbf{x}-\mathbf{\eta}\|}.
  37. ψ ψ
  38. U U
  39. U [ G ( 𝐲 , η ) Δ ψ ( 𝐲 ) ] d V 𝐲 - ψ ( η ) = U [ G ( 𝐲 , η ) ψ 𝐧 ( 𝐲 ) - ψ ( 𝐲 ) G ( 𝐲 , η ) 𝐧 ] d S 𝐲 . \int_{U}\left[G(\mathbf{y},\mathbf{\eta})\Delta\psi(\mathbf{y})\right]\,dV_{% \mathbf{y}}-\psi(\mathbf{\eta})=\oint_{\partial U}\left[G(\mathbf{y},\mathbf{% \eta}){\partial\psi\over\partial\mathbf{n}}(\mathbf{y})-\psi(\mathbf{y}){% \partial G(\mathbf{y},\mathbf{\eta})\over\partial\mathbf{n}}\right]\,dS_{% \mathbf{y}}.
  40. ψ ψ
  41. ψ ( η ) = U [ ψ ( 𝐲 ) G ( 𝐲 , η ) 𝐧 - G ( 𝐲 , η ) ψ 𝐧 ( 𝐲 ) ] d S 𝐲 . \psi(\mathbf{\eta})=\oint_{\partial U}\left[\psi(\mathbf{y})\frac{\partial G(% \mathbf{y},\mathbf{\eta})}{\partial\mathbf{n}}-G(\mathbf{y},\mathbf{\eta})% \frac{\partial\psi}{\partial\mathbf{n}}(\mathbf{y})\right]\,dS_{\mathbf{y}}.
  42. G G
  43. U U
  44. ψ ( η ) = U ψ ( 𝐲 ) G ( 𝐲 , η ) 𝐧 d S 𝐲 . \psi(\mathbf{\eta})=\oint_{\partial U}\psi(\mathbf{y})\frac{\partial G(\mathbf% {y},\mathbf{\eta})}{\partial\mathbf{n}}\,dS_{\mathbf{y}}.
  45. ψ ψ
  46. G G
  47. M u Δ v d V + M grad u , grad v d V = M u N v d V ~ M ( u Δ v - v Δ u ) d V = M ( u N v - v N u ) d V ~ \begin{aligned}\displaystyle\int_{M}u\Delta v\,dV+\int_{M}\langle\operatorname% {grad}\ u,\operatorname{grad}\ v\rangle\,dV&\displaystyle=\int_{\partial M}% uNvd\widetilde{V}\\ \displaystyle\int_{M}\left(u\Delta v-v\Delta u\right)\,dV&\displaystyle=\int_{% \partial M}(uNv-vNu)d\widetilde{V}\end{aligned}
  48. u u
  49. v v
  50. M M
  51. d V dV
  52. d V ~ d\widetilde{V}
  53. M M
  54. N N
  55. p m Δ q m - q m Δ p m = ( p m q m - q m p m ) , p_{m}\Delta q_{m}-q_{m}\Delta p_{m}=\nabla\cdot\left(p_{m}\nabla q_{m}-q_{m}% \nabla p_{m}\right),
  56. 𝐏 ( × × 𝐐 ) - 𝐐 ( × × 𝐏 ) = ( 𝐐 × × 𝐏 - 𝐏 × × 𝐐 ) . \mathbf{P}\cdot\left(\nabla\times\nabla\times\mathbf{Q}\right)-\mathbf{Q}\cdot% \left(\nabla\times\nabla\times\mathbf{P}\right)=\nabla\cdot\left(\mathbf{Q}% \times\nabla\times\mathbf{P}-\mathbf{P}\times\nabla\times\mathbf{Q}\right).
  57. 𝐏 Δ 𝐐 - 𝐐 Δ 𝐏 + 𝐐 [ ( 𝐏 ) ] - 𝐏 [ ( 𝐐 ) ] = ( 𝐏 × × 𝐐 - 𝐐 × × 𝐏 ) . \mathbf{P}\cdot\Delta\mathbf{Q}-\mathbf{Q}\cdot\Delta\mathbf{P}+\mathbf{Q}% \cdot\left[\nabla\left(\nabla\cdot\mathbf{P}\right)\right]-\mathbf{P}\cdot% \left[\nabla\left(\nabla\cdot\mathbf{Q}\right)\right]=\nabla\cdot\left(\mathbf% {P}\times\nabla\times\mathbf{Q}-\mathbf{Q}\times\nabla\times\mathbf{P}\right).
  58. 𝐐 [ ( 𝐏 ) ] - 𝐏 [ ( 𝐐 ) ] , \mathbf{Q}\cdot\left[\nabla\left(\nabla\cdot\mathbf{P}\right)\right]-\mathbf{P% }\cdot\left[\nabla\left(\nabla\cdot\mathbf{Q}\right)\right],
  59. 𝐏 = m p m 𝐞 ^ m , 𝐐 = m q m 𝐞 ^ m . \mathbf{P}=\sum_{m}p_{m}\hat{\mathbf{e}}_{m},\qquad\mathbf{Q}=\sum_{m}q_{m}% \hat{\mathbf{e}}_{m}.
  60. m [ p m Δ q m - q m Δ p m ] = m ( p m q m - q m p m ) . \sum_{m}\left[p_{m}\Delta q_{m}-q_{m}\Delta p_{m}\right]=\sum_{m}\nabla\cdot% \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right).
  61. m [ p m Δ q m - q m Δ p m ] = 𝐏 Δ 𝐐 - 𝐐 Δ 𝐏 . \sum_{m}\left[p_{m}\Delta q_{m}-q_{m}\Delta p_{m}\right]=\mathbf{P}\cdot\Delta% \mathbf{Q}-\mathbf{Q}\cdot\Delta\mathbf{P}.
  62. m ( p m q m - q m p m ) = ( m p m q m - m q m p m ) . \sum_{m}\nabla\cdot\left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)=\nabla% \cdot\left(\sum_{m}p_{m}\nabla q_{m}-\sum_{m}q_{m}\nabla p_{m}\right).
  63. ( 𝐏 𝐐 ) = ( 𝐏 ) 𝐐 + ( 𝐐 ) 𝐏 + 𝐏 × × 𝐐 + 𝐐 × × 𝐏 , \nabla\left(\mathbf{P}\cdot\mathbf{Q}\right)=\left(\mathbf{P}\cdot\nabla\right% )\mathbf{Q}+\left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}+\mathbf{P}\times% \nabla\times\mathbf{Q}+\mathbf{Q}\times\nabla\times\mathbf{P},
  64. ( 𝐏 𝐐 ) = m p m q m = m p m q m + m q m p m . \nabla\left(\mathbf{P}\cdot\mathbf{Q}\right)=\nabla\sum_{m}p_{m}q_{m}=\sum_{m}% p_{m}\nabla q_{m}+\sum_{m}q_{m}\nabla p_{m}.
  65. p m p_{m}
  66. q m q_{m}
  67. m p m q m = ( 𝐏 ) 𝐐 + 𝐏 × × 𝐐 , \sum_{m}p_{m}\nabla q_{m}=\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf% {P}\times\nabla\times\mathbf{Q},
  68. m q m p m = ( 𝐐 ) 𝐏 + 𝐐 × × 𝐏 . \sum_{m}q_{m}\nabla p_{m}=\left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}+\mathbf% {Q}\times\nabla\times\mathbf{P}.
  69. m p m q m - m q m p m = ( 𝐏 ) 𝐐 + 𝐏 × × 𝐐 - ( 𝐐 ) 𝐏 - 𝐐 × × 𝐏 . \sum_{m}p_{m}\nabla q_{m}-\sum_{m}q_{m}\nabla p_{m}=\left(\mathbf{P}\cdot% \nabla\right)\mathbf{Q}+\mathbf{P}\times\nabla\times\mathbf{Q}-\left(\mathbf{Q% }\cdot\nabla\right)\mathbf{P}-\mathbf{Q}\times\nabla\times\mathbf{P}.
  70. \color O l i v e G r e e n 𝐏 Δ 𝐐 - 𝐐 Δ 𝐏 = [ ( 𝐏 ) 𝐐 + 𝐏 × × 𝐐 - ( 𝐐 ) 𝐏 - 𝐐 × × 𝐏 ] . \color{OliveGreen}\mathbf{P}\cdot\Delta\mathbf{Q}-\mathbf{Q}\cdot\Delta\mathbf% {P}=\nabla\cdot\left[\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P}% \times\nabla\times\mathbf{Q}-\left(\mathbf{Q}\cdot\nabla\right)\mathbf{P}-% \mathbf{Q}\times\nabla\times\mathbf{P}\right].
  71. × ( 𝐏 × 𝐐 ) = ( 𝐐 ) 𝐏 - ( 𝐏 ) 𝐐 + 𝐏 ( 𝐐 ) - 𝐐 ( 𝐏 ) ; \nabla\times\left(\mathbf{P}\times\mathbf{Q}\right)=\left(\mathbf{Q}\cdot% \nabla\right)\mathbf{P}-\left(\mathbf{P}\cdot\nabla\right)\mathbf{Q}+\mathbf{P% }\left(\nabla\cdot\mathbf{Q}\right)-\mathbf{Q}\left(\nabla\cdot\mathbf{P}% \right);
  72. 𝐏 Δ 𝐐 - 𝐐 Δ 𝐏 = [ 𝐏 ( 𝐐 ) - 𝐐 ( 𝐏 ) - × ( 𝐏 × 𝐐 ) + 𝐏 × × 𝐐 - 𝐐 × × 𝐏 ] . \mathbf{P}\cdot\Delta\mathbf{Q}-\mathbf{Q}\cdot\Delta\mathbf{P}=\nabla\cdot% \left[\mathbf{P}\left(\nabla\cdot\mathbf{Q}\right)-\mathbf{Q}\left(\nabla\cdot% \mathbf{P}\right)-\nabla\times\left(\mathbf{P}\times\mathbf{Q}\right)+\mathbf{% P}\times\nabla\times\mathbf{Q}-\mathbf{Q}\times\nabla\times\mathbf{P}\right].
  73. \color O l i v e G r e e n 𝐏 Δ 𝐐 - 𝐐 Δ 𝐏 = [ 𝐏 ( 𝐐 ) - 𝐐 ( 𝐏 ) + 𝐏 × × 𝐐 - 𝐐 × × 𝐏 ] . \color{OliveGreen}\mathbf{P}\cdot\Delta\mathbf{Q}-\mathbf{Q}\cdot\Delta\mathbf% {P}=\nabla\cdot\left[\mathbf{P}\left(\nabla\cdot\mathbf{Q}\right)-\mathbf{Q}% \left(\nabla\cdot\mathbf{P}\right)+\mathbf{P}\times\nabla\times\mathbf{Q}-% \mathbf{Q}\times\nabla\times\mathbf{P}\right].
  74. Δ ( 𝐏 𝐐 ) = 𝐏 Δ 𝐐 - 𝐐 Δ 𝐏 + 2 [ ( 𝐐 ) 𝐏 + 𝐐 × × 𝐏 ] . \Delta\left(\mathbf{P}\cdot\mathbf{Q}\right)=\mathbf{P}\cdot\Delta\mathbf{Q}-% \mathbf{Q}\cdot\Delta\mathbf{P}+2\nabla\cdot\left[\left(\mathbf{Q}\cdot\nabla% \right)\mathbf{P}+\mathbf{Q}\times\nabla\times\mathbf{P}\right].
  75. 𝐏 [ ( 𝐐 ) ] - 𝐐 [ ( 𝐏 ) ] = [ 𝐏 ( 𝐐 ) - 𝐐 ( 𝐏 ) ] . \mathbf{P}\cdot\left[\nabla\left(\nabla\cdot\mathbf{Q}\right)\right]-\mathbf{Q% }\cdot\left[\nabla\left(\nabla\cdot\mathbf{P}\right)\right]=\nabla\cdot\left[% \mathbf{P}\left(\nabla\cdot\mathbf{Q}\right)-\mathbf{Q}\left(\nabla\cdot% \mathbf{P}\right)\right].

Green's_relations.html

  1. S 1 a = { s a s S 1 } S^{1}a=\{sa\mid s\in S^{1}\}
  2. { s a s S } { a } \{sa\mid s\in S\}\cup\{a\}
  3. S a { a } Sa\cup\{a\}
  4. a S 1 = { a s s S 1 } aS^{1}=\{as\mid s\in S^{1}\}
  5. a S { a } aS\cup\{a\}
  6. S 1 a S 1 S^{1}aS^{1}
  7. S a S a S S a { a } SaS\cup aS\cup Sa\cup\{a\}
  8. 𝔩 \mathfrak{l}
  9. 𝔯 \mathfrak{r}
  10. 𝔣 \mathfrak{f}
  11. a b ( 𝔩 ) a\equiv b(\mathfrak{l})
  12. \mathcal{R}
  13. H 2 H = H^{2}\cap H=\emptyset
  14. H 2 = H H^{2}=H

Green–Schwarz_mechanism.html

  1. F 6 , F 4 R 2 , F 2 R 4 , R 6 F^{6},\ F^{4}R^{2},\ F^{2}R^{4},\ R^{6}
  2. S G S = B 2 X 8 S_{GS}=\int B_{2}\wedge X_{8}
  3. B 2 B_{2}
  4. X 8 X_{8}
  5. F 4 , F 2 R 2 , R 4 F^{4},\ F^{2}R^{2},\ R^{4}
  6. B 2 B_{2}
  7. F ( 2 ) F_{(2)}
  8. S G S S_{GS}

Grigory_Margulis.html

  1. {}^{\prime}
  2. {}^{\prime}

Gromov's_theorem_on_groups_of_polynomial_growth.html

  1. G = G 1 G 2 . G=G_{1}\supseteq G_{2}\supseteq\ldots.
  2. d ( G ) = k 1 k rank ( G k / G k + 1 ) d(G)=\sum_{k\geq 1}k\ \operatorname{rank}(G_{k}/G_{k+1})

Groombridge_1618.html

  1. S e f f = S e f f + a T + b T 2 + c T 3 + d T 4 \scriptstyle S_{eff_{\ast}}=S_{eff_{\odot}}+aT_{\ast}+bT_{\ast}^{2}+cT_{\ast}^% {3}+dT_{\ast}^{4}
  2. S e f f \scriptstyle S_{eff_{\ast}}
  3. d = ( L / L S e f f ) 0.5 A U \scriptstyle{d={\left(\frac{{L_{\ast}}/{L_{\odot}}}{S_{eff_{\ast}}}\right)^{0.% 5}AU}}
  4. 0.4 A U \scriptstyle{0.4AU}

Grothendieck_universe.html

  1. { x α } α I \{x_{\alpha}\}_{\alpha\in I}
  2. α I x α \bigcup_{\alpha\in I}x_{\alpha}
  3. x U x\in U
  4. y x y\subseteq x
  5. y U y\in U
  6. y P ( x ) y\in P(x)
  7. y x y\subseteq x
  8. P ( x ) U P(x)\in U
  9. x U x\in U
  10. y U y\in U
  11. V ω V_{\omega}
  12. 𝐜 ( U ) = sup x U | x | \mathbf{c}(U)=\sup_{x\in U}|x|
  13. 𝐜 ( U ) = 0 \mathbf{c}(U)=\aleph_{0}
  14. \bigcup
  15. n \bigcup_{n}
  16. 0 \aleph_{0}
  17. V ω V_{\omega}

Ground_loop_(electricity).html

  1. R G \scriptstyle R_{G}
  2. R G = 0 \scriptstyle R_{G}\;=\;0
  3. V G \scriptstyle V_{G}
  4. V o u t = V 2 \scriptstyle V_{out}\;=\;V_{2}
  5. R G 0 \scriptstyle R_{G}\neq 0
  6. R 1 \scriptstyle R_{1}
  7. I 1 \scriptstyle I_{1}
  8. R G \scriptstyle R_{G}
  9. V G = I 1 R G \scriptstyle V_{G}\;=\;I_{1}R_{G}
  10. R G \scriptstyle R_{G}
  11. V o u t = V 2 - V G = V 2 - R G R G + R 1 V 1 . V_{out}=V_{2}-V_{G}=V_{2}-\frac{R_{G}}{R_{G}+R_{1}}V_{1}.\,
  12. V G \scriptstyle V_{G}
  13. V 2 = V S2 - V G2 V_{2}=V\text{S2}-V\text{G2}\,
  14. V G2 = V G1 - I R V\text{G2}=V\text{G1}-IR\,
  15. V S2 = V S1 V\text{S2}=V\text{S1}\,
  16. V 2 = V S1 - ( V G1 - I R ) V_{2}=V\text{S1}-(V\text{G1}-IR)\,
  17. V 2 = V 1 + I R V_{2}=V_{1}+IR\,

Growing_degree-day.html

  1. G D D = T max + T min 2 - T base . GDD=\frac{T_{\mathrm{max}}+T_{\mathrm{min}}}{2}-T_{\mathrm{base}}.
  2. 23 + 12 2 - 10 = 7.5 \frac{23+12}{2}-10=7.5
  3. 13 + 10 2 - 10 = 1.5 \frac{13+10}{2}-10=1.5

GRS_80.html

  1. a a
  2. b b
  3. ( b / a ) (b/a)
  4. f f
  5. a a
  6. G M GM
  7. J 2 J_{2}
  8. ω \omega
  9. f f
  10. a = 6 378 137 m a=6\,378\,137\,\mathrm{m}
  11. G M = 3986005 × 10 8 m 3 / s 2 GM=3986005\times 10^{8}\,\mathrm{m^{3}/s^{2}}
  12. J 2 = 108 263 × 10 - 8 J_{2}=108\,263\times 10^{-8}
  13. ω = 7 292 115 × 10 - 11 s - 1 \omega=7\,292\,115\times 10^{-11}\,\mathrm{s^{-1}}
  14. f f
  15. 1 / f 1/f
  16. b b
  17. b / a b/a
  18. R 1 = ( 2 a + b ) / 3 R_{1}=(2a+b)/3
  19. ( a 2 b ) 1 / 3 (a^{2}b)^{1/3}
  20. a 2 - b 2 \sqrt{a^{2}-b^{2}}
  21. a 2 - b 2 / a \sqrt{a^{2}-b^{2}}/a
  22. a 2 / b a^{2}/b
  23. b 2 / a b^{2}/a
  24. e 2 = a 2 - b 2 a 2 = 3 J 2 + 4 15 ω 2 a 3 G M e 3 2 q 0 , e^{2}=\frac{a^{2}-b^{2}}{a^{2}}=3J_{2}+\frac{4}{15}\frac{\omega^{2}a^{3}}{GM}% \frac{e^{3}}{2q_{0}},
  25. 2 q 0 = ( 1 + 3 e 2 ) arctan e - 3 e 2q_{0}=\left(1+\frac{3}{e^{\prime 2}}\right)\arctan e^{\prime}-\frac{3}{e^{% \prime}}
  26. e = e / 1 - e 2 e^{\prime}=e/\sqrt{1-e^{2}}
  27. e 2 = 0.00669 43800 22903 41574 95749 48586 28930 62124 43890 e^{2}=0.00669\,43800\,22903\,41574\,95749\,48586\,28930\,62124\,43890\,\ldots
  28. f = 1 / 298.25722 21008 82711 24316 28366 . f=1/298.25722\,21008\,82711\,24316\,28366\,\ldots.

Guard_(computer_science).html

  1. f ( x ) = { 1 if x > 0 0 otherwise f(x)=\left\{\begin{matrix}1&\mbox{if }~{}x>0\\ 0&\mbox{otherwise}\end{matrix}\right.

Guarded_Command_Language.html

  1. \rightarrow
  2. \rightarrow
  3. \rightarrow
  4. \rightarrow
  5. \rightarrow
  6. \rightarrow
  7. \rightarrow
  8. \rightarrow
  9. \rightarrow
  10. \rightarrow
  11. \rightarrow
  12. \rightarrow
  13. \rightarrow
  14. \rightarrow
  15. \rightarrow

Gunning_fog_index.html

  1. 0.4 [ ( words sentences ) + 100 ( complex words words ) ] 0.4\left[\left(\frac{\mbox{words}~{}}{\mbox{sentences}~{}}\right)+100\left(% \frac{\mbox{complex words}~{}}{\mbox{words}~{}}\right)\right]

Gyrobifastigium.html

  1. V = ( 3 2 ) a 3 0.866025... a 3 V=(\frac{\sqrt{3}}{2})a^{3}\approx 0.866025...a^{3}
  2. A = ( 4 + 3 ) a 2 5.73205... a 2 A=(4+\sqrt{3})a^{2}\approx 5.73205...a^{2}

Gyroelongated_triangular_bicupola.html

  1. V = 2 ( 5 3 + 1 + 3 ) a 3 4.69456... a 3 V=\sqrt{2}(\frac{5}{3}+\sqrt{1+\sqrt{3}})a^{3}\approx 4.69456...a^{3}
  2. A = ( 6 + 5 3 ) a 2 14.6603... a 2 A=(6+5\sqrt{3})a^{2}\approx 14.6603...a^{2}

Gyroelongated_triangular_cupola.html

  1. V = ( 1 3 61 2 + 18 3 + 30 1 + 3 ) a 3 3.51605... a 3 V=(\frac{1}{3}\sqrt{\frac{61}{2}+18\sqrt{3}+30\sqrt{1+\sqrt{3}}})a^{3}\approx 3% .51605...a^{3}
  2. A = ( 3 + 11 3 2 ) a 2 12.5263... a 2 A=(3+\frac{11\sqrt{3}}{2})a^{2}\approx 12.5263...a^{2}

H-cobordism.html

  1. M W and N W M\hookrightarrow W\quad\mbox{and}~{}\quad N\hookrightarrow W
  2. f : W [ a , b ] f:W\to[a,b]
  3. f - 1 ( [ c , c ] ) f^{-1}([c,c^{\prime}])
  4. W c W_{c^{\prime}}
  5. W c W_{c}
  6. ( i - 1 ) + ( n - j ) dim W - 1 = n - 1 (i-1)+(n-j)\leq\dim\partial W-1=n-1
  7. i j i\leq j
  8. ( C * , * ) (C_{*},\partial_{*})
  9. C k C_{k}
  10. k : C k C k - 1 \partial_{k}:C_{k}\to C_{k-1}
  11. h α k h_{\alpha}^{k}
  12. β < h α k | h β k - 1 > h β k - 1 \sum_{\beta}<h_{\alpha}^{k}|h_{\beta}^{k-1}>h_{\beta}^{k-1}
  13. < h α k | h β k - 1 Align g t ; <h_{\alpha}^{k}|h_{\beta}^{k-1}&gt;
  14. h α k h_{\alpha}^{k}
  15. h β k + 1 h_{\beta}^{k+1}
  16. k + 1 h β k + 1 = ± h α k \partial_{k+1}h_{\beta}^{k+1}=\pm h_{\alpha}^{k}
  17. ( α , β ) (\alpha,\beta)
  18. k + 1 \partial_{k+1}
  19. ± 1 \pm 1
  20. π k ( W , M ) \pi_{k}(W,M)
  21. D k + 1 D^{k+1}
  22. dim W - 1 = n - 1 2 ( k + 1 ) \dim\partial W-1=n-1\geq 2(k+1)
  23. k \partial_{k}
  24. h α k h_{\alpha}^{k}
  25. h β k h_{\beta}^{k}
  26. h α k h_{\alpha}^{k}
  27. h α k ± h β k h_{\alpha}^{k}\pm h_{\beta}^{k}
  28. C k C_{k}
  29. H * ( W , M ; ) = 0 H_{*}(W,M;\mathbb{Z})=0
  30. C k coker k + 1 im k + 1 C_{k}\cong\mathrm{coker}\,\partial_{k+1}\oplus\mathrm{im}\,\partial_{k+1}
  31. C k C_{k}
  32. k \partial_{k}
  33. ± 1 \pm 1
  34. M W M\hookrightarrow W
  35. M W M\hookrightarrow W
  36. N W N\hookrightarrow W
  37. π π 1 ( M ) π 1 ( W ) π 1 ( N ) . \pi\cong\pi_{1}(M)\cong\pi_{1}(W)\cong\pi_{1}(N).
  38. m M , n N m\in M,n\in N

Haag's_theorem.html

  1. ( H 1 , { O 1 i } ) (H_{1},\{O^{i}_{1}\})
  2. ( H 2 , { O 2 i } ) (H_{2},\{O^{i}_{2}\})
  3. H n H_{n}
  4. { O n i } \{O^{i}_{n}\}
  5. U U
  6. H 1 H_{1}
  7. H 2 H_{2}
  8. O 1 j { O 1 i } O^{j}_{1}\in\{O^{i}_{1}\}
  9. O 2 j = U O 1 j U - 1 { O 2 i } O^{j}_{2}=UO^{j}_{1}U^{-1}\in\{O^{i}_{2}\}
  10. H R H_{R}
  11. H F H_{F}

Hadamard_transform.html

  1. 2 m 2^{m}
  2. 2 × 2 × × 2 × 2 2\times 2\times\cdots\times 2\times 2
  3. H m = 1 2 ( H m - 1 H m - 1 H m - 1 - H m - 1 ) = H 1 H m - 1 H_{m}=\frac{1}{\sqrt{2}}\begin{pmatrix}H_{m-1}&H_{m-1}\\ H_{m-1}&-H_{m-1}\end{pmatrix}=H_{1}\otimes H_{m-1}
  4. k = i = 0 m - 1 k i 2 i = k m - 1 2 m - 1 + k m - 2 2 m - 2 + + k 1 2 + k 0 k=\sum^{m-1}_{i=0}{k_{i}2^{i}}=k_{m-1}2^{m-1}+k_{m-2}2^{m-2}+\cdots+k_{1}2+k_{0}
  5. n = i = 0 m - 1 n i 2 i = n m - 1 2 m - 1 + n m - 2 2 m - 2 + + n 1 2 + n 0 n=\sum^{m-1}_{i=0}{n_{i}2^{i}}=n_{m-1}2^{m-1}+n_{m-2}2^{m-2}+\cdots+n_{1}2+n_{0}
  6. k = n = 0 k=n=0
  7. ( H m ) k , n = 1 2 m 2 ( - 1 ) j k j n j \left(H_{m}\right)_{k,n}=\frac{1}{2^{\frac{m}{2}}}(-1)^{\sum_{j}k_{j}n_{j}}
  8. 2 × 2 × × 2 × 2 \scriptstyle 2\,\times\,2\,\times\,\cdots\,\times\,2\,\times\,2
  9. H 0 = + 1 H 1 = 1 2 ( 1 1 1 - 1 ) \begin{aligned}\displaystyle H_{0}=&\displaystyle+1\\ \displaystyle H_{1}=\frac{1}{\sqrt{2}}&\displaystyle\begin{pmatrix}\begin{% array}[]{rr}1&1\\ 1&-1\end{array}\end{pmatrix}\end{aligned}
  10. H 2 = 1 2 ( 1 1 1 1 1 - 1 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 ) H 3 = 1 2 3 2 ( 1 1 1 1 1 1 1 1 1 - 1 1 - 1 1 - 1 1 - 1 1 1 - 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 1 - 1 - 1 1 1 1 1 1 - 1 - 1 - 1 - 1 1 - 1 1 - 1 - 1 1 - 1 1 1 1 - 1 - 1 - 1 - 1 1 1 1 - 1 - 1 1 - 1 1 1 - 1 ) ( H n ) i , j = 1 2 n 2 ( - 1 ) i j \begin{aligned}\displaystyle H_{2}=\frac{1}{2}&\displaystyle\begin{pmatrix}% \begin{array}[]{rrrr}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\end{array}\end{pmatrix}\\ \displaystyle H_{3}=\frac{1}{2^{\frac{3}{2}}}&\displaystyle\begin{pmatrix}% \begin{array}[]{rrrrrrrr}1&1&1&1&1&1&1&1\\ 1&-1&1&-1&1&-1&1&-1\\ 1&1&-1&-1&1&1&-1&-1\\ 1&-1&-1&1&1&-1&-1&1\\ 1&1&1&1&-1&-1&-1&-1\\ 1&-1&1&-1&-1&1&-1&1\\ 1&1&-1&-1&-1&-1&1&1\\ 1&-1&-1&1&-1&1&1&-1\end{array}\end{pmatrix}\\ \displaystyle(H_{n})_{i,j}=\frac{1}{2^{\frac{n}{2}}}&\displaystyle(-1)^{i\cdot j% }\end{aligned}
  11. i j i\cdot j
  12. n 2 \scriptstyle n\geq 2
  13. ( H n ) 3 , 2 = ( - 1 ) 3 2 = ( - 1 ) ( 1 , 1 ) ( 1 , 0 ) = ( - 1 ) 1 + 0 = ( - 1 ) 1 = - 1 \scriptstyle({H_{n}})_{3,2}\;=\;(-1)^{3\cdot 2}\;=\;(-1)^{(1,1)\cdot(1,0)}\;=% \;(-1)^{1+0}\;=\;(-1)^{1}\;=\;-1
  14. ( H n ) 0 , 0 \scriptstyle({H_{n}})_{0,0}
  15. | 0 |0\rangle
  16. | 1 |1\rangle
  17. | 0 |0\rangle
  18. | 1 |1\rangle
  19. H = | 0 + | 1 2 0 | + | 0 - | 1 2 1 | H=\frac{|0\rangle+|1\rangle}{\sqrt{2}}\langle 0|+\frac{|0\rangle-|1\rangle}{% \sqrt{2}}\langle 1|
  20. H 1 = 1 2 ( 1 1 1 - 1 ) H_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}
  21. | 0 , | 1 |0\rangle,|1\rangle
  22. | 0 |0\rangle
  23. | 0 , | 1 |0\rangle,|1\rangle
  24. H ( | 1 ) = 1 2 | 0 - 1 2 | 1 H(|1\rangle)=\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle
  25. H ( | 0 ) = 1 2 | 0 + 1 2 | 1 H(|0\rangle)=\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle
  26. H ( 1 2 | 0 + 1 2 | 1 ) = 1 2 ( | 0 + | 1 ) + 1 2 ( | 0 - | 1 ) = | 0 H\left(\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle\right)=\frac{1}% {2}(|0\rangle+|1\rangle)+\frac{1}{2}(|0\rangle-|1\rangle)=|0\rangle
  27. H ( 1 2 | 0 - 1 2 | 1 ) = 1 2 ( | 0 + | 1 ) - 1 2 ( | 0 - | 1 ) = | 1 H\left(\frac{1}{\sqrt{2}}|0\rangle-\frac{1}{\sqrt{2}}|1\rangle\right)=\frac{1}% {2}(|0\rangle+|1\rangle)-\frac{1}{2}(|0\rangle-|1\rangle)=|1\rangle
  28. | 0 = [ 1 0 ] |0\rangle=\begin{bmatrix}1\\ 0\\ \end{bmatrix}
  29. | 1 = [ 0 1 ] |1\rangle=\begin{bmatrix}0\\ 1\\ \end{bmatrix}

Haldane's_dilemma.html

  1. ( 1 / 2 ) 10 (1/2)^{10}
  2. I = ln ( s 0 / S ) I=\ln(s_{0}/S)
  3. s 0 s_{0}
  4. S S
  5. 1 - S 1-S
  6. 1 - s 0 1-s_{0}
  7. s 0 - S s_{0}-S
  8. s 0 S s_{0}\approx S
  9. I s 0 - S I\approx s_{0}-S
  10. i t h i^{th}
  11. d i d_{i}
  12. ( 1 - d i ) \prod\left(1-d_{i}\right)
  13. exp ( - d i ) \exp\left(-\sum d_{i}\right)
  14. d i d_{i}
  15. d i \sum d_{i}
  16. d i = s 0 i - S d_{i}=s_{0i}-S
  17. s 0 i s_{0i}
  18. i t h i^{th}
  19. S S
  20. i t h i^{th}
  21. D i D_{i}
  22. d i d_{i}
  23. D i D_{i}
  24. p 0 p_{0}
  25. p n p_{n}
  26. q n q_{n}
  27. n th n^{\mbox{th}}~{}
  28. p n 2 p_{n}^{2}
  29. 2 p n q n 2p_{n}q_{n}
  30. q n 2 q_{n}^{2}
  31. 1 - λ K 1-\lambda K
  32. 1 - K 1-K
  33. K K
  34. n th n^{\mbox{th}}~{}
  35. d n = 2 λ K p n q n + K q n 2 = K q n [ 2 λ + ( 1 - 2 λ ) q n ] d_{n}=2\lambda Kp_{n}q_{n}+Kq_{n}^{2}=Kq_{n}[2\lambda+(1-2\lambda)q_{n}]
  36. D = K 0 q n [ 2 λ + ( 1 - 2 λ ) q n ] . D=K\sum_{0}^{\infty}q_{n}\;[2\lambda+(1-2\lambda)q_{n}].
  37. d q n = - K p n q n [ λ + ( 1 - 2 λ ) q n ] dq_{n}=-Kp_{n}q_{n}[\lambda+(1-2\lambda)q_{n}]
  38. D = 0 q 0 [ 2 λ + ( 1 - 2 λ ) q ] ( 1 - q ) [ λ + ( 1 - 2 λ ) q ] d q = 1 1 - λ 0 q 0 [ 1 1 - q + λ ( 1 - 2 λ ) λ + ( 1 - 2 λ ) q ] d q . D=\int_{0}^{q_{{}_{0}}}\frac{[2\lambda+(1-2\lambda)q]}{(1-q)[\lambda+(1-2% \lambda)q]}dq=\frac{1}{1-\lambda}\int_{0}^{q_{{}_{0}}}\left[\frac{1}{1-q}+% \frac{\lambda(1-2\lambda)}{\lambda+(1-2\lambda)q}\right]dq.
  39. D = 1 1 - λ [ - ln p 0 + λ ln ( 1 - λ - ( 1 - 2 λ ) p 0 λ ) ] 1 1 - λ [ - ln p 0 + λ ln ( 1 - λ λ ) ] D=\frac{1}{1-\lambda}\left[-\mbox{ln }~{}p_{0}+\lambda\mbox{ ln }~{}\left(% \frac{1-\lambda-(1-2\lambda)p_{0}}{\lambda}\right)\right]\approx\frac{1}{1-% \lambda}\left[-\mbox{ln }~{}p_{0}+\lambda\mbox{ ln }~{}\left(\frac{1-\lambda}{% \lambda}\right)\right]
  40. p 0 p_{0}
  41. D = 0 q 0 2 - q ( 1 - q ) 2 = 0 q 0 [ 1 1 - q + 1 ( 1 - q ) 2 ] d q = p 0 - 1 - ln p 0 + O ( λ K ) . D=\int_{0}^{q_{{}_{0}}}\frac{2-q}{(1-q)^{2}}=\int_{0}^{q_{{}_{0}}}\left[\frac{% 1}{1-q}+\frac{1}{(1-q)^{2}}\right]dq=p_{0}^{-1}-\mbox{ ln }~{}p_{0}+O(\lambda K).
  42. I = 30 / 300 = 0.1 I=30/300=0.1

Half-period_ratio.html

  1. τ = ω 2 ω 1 \tau=\frac{\omega_{2}}{\omega_{1}}
  2. ( τ ) > 0 \Im(\tau)>0

Half-space_(geometry).html

  1. a 1 x 1 + a 2 x 2 + + a n x n > b a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}>b
  2. a 1 x 1 + a 2 x 2 + + a n x n b a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}\geq b

Hall–Janko_graph.html

  1. ( x - 36 ) ( x - 6 ) 36 ( x + 4 ) 63 (x-36)(x-6)^{36}(x+4)^{63}

Ham_sandwich_theorem.html

  1. n n
  2. n n
  3. ( n 1 ) (n−1)
  4. n = 3 n=3
  5. n = 3 n=3
  6. n n
  7. n = 3 n=3
  8. n = 3 n=3
  9. n n
  10. S S
  11. ( n 1 ) (n−1)
  12. n n
  13. n \mathbb{R}^{n}
  14. p p
  15. S S
  16. p p
  17. p p
  18. S S
  19. π ( p ) π(p)
  20. p p
  21. f f
  22. ( n 1 ) (n−1)
  23. S S
  24. ( n 1 ) (n−1)
  25. n - 1 \mathbb{R}^{n-1}
  26. f ( p ) = ( f(p)=(
  27. π ( p ) π(p)
  28. π ( p ) π(p)
  29. π ( p ) ) π(p))
  30. f f
  31. p p
  32. q q
  33. S S
  34. f ( p ) = f ( q ) f(p)=f(q)
  35. p p
  36. q q
  37. π ( p ) π(p)
  38. π ( q ) π(q)
  39. f ( p ) = f ( q ) f(p)=f(q)
  40. π ( p ) π(p)
  41. π ( q ) π(q)
  42. i = 1 , 2 , , n 1 i=1,2,...,n−1
  43. π ( p ) π(p)
  44. π ( q ) π(q)
  45. n n
  46. X X
  47. X X
  48. f : S n × X f\colon S^{n}\times X\to\mathbb{R}
  49. p p
  50. n n
  51. f ( s , x ) = 0 f(s,x)=0
  52. X X
  53. f ( s , x ) > 0 f(s,x)>0
  54. n + 1 n+1
  55. X X
  56. X X
  57. f ( x ) = 0 f(x)=0
  58. X X
  59. f ( x ) > 0 f(x)>0
  60. i > 0 i>0
  61. i i
  62. x x
  63. n n
  64. O ( n l o g n ) O(nlogn)
  65. O O
  66. O ( n ) O(n)
  67. d d
  68. d d
  69. ( d 1 ) (d−1)
  70. ( k + n n ) - 1 {\left({{k+n}\atop{n}}\right)}-1
  71. ( k + n n ) - 1 {\left({{k+n}\atop{n}}\right)}-1
  72. ( x , y ) ( x , y , x 2 , y 2 , x y ) (x,y)\to(x,y,x^{2},y^{2},xy)

Hamiltonian_constraint.html

  1. τ \tau
  2. t t
  3. x x
  4. τ \tau
  5. τ \tau
  6. τ \tau
  7. τ \tau
  8. x x
  9. t t
  10. τ \tau
  11. x ( τ ) , t ( τ ) x(\tau),\;\;\;\;t(\tau)
  12. τ \tau
  13. τ \tau
  14. τ = f ( τ ) \tau^{\prime}=f(\tau)
  15. x ( τ ) x(\tau)
  16. t ( τ ) t(\tau)
  17. τ \tau
  18. S = d τ [ d x d τ p + d t d τ p t - λ ( p t + p 2 2 m + 1 2 m ω 2 x 2 ) ] . S=\int d\tau\Big[{dx\over d\tau}p+{dt\over d\tau}p_{t}-\lambda\Big(p_{t}+{p^{2% }\over 2m}+{1\over 2}m\omega^{2}x^{2}\Big)\Big].
  19. x x
  20. t t
  21. p p
  22. p t p_{t}
  23. S = d τ [ d x d τ p + d t d τ p t - ( x , t ; p , p t ) ] S=\int d\tau\Big[{dx\over d\tau}p+{dt\over d\tau}p_{t}-\mathcal{H}(x,t;p,p_{t}% )\Big]
  24. \mathcal{H}
  25. ( x , t , λ ; p , p t ) = λ ( p t + p 2 2 m + 1 2 m ω 2 x 2 ) . \mathcal{H}(x,t,\lambda;p,p_{t})=\lambda\Big(p_{t}+{p^{2}\over 2m}+{1\over 2}m% \omega^{2}x^{2}\Big).
  26. λ \lambda
  27. λ = 0 {\partial\mathcal{H}\over\partial\lambda}=0
  28. C = p t + p 2 2 m + 1 2 m ω 2 x 2 = 0. C=p_{t}+{p^{2}\over 2m}+{1\over 2}m\omega^{2}x^{2}=0.
  29. C C
  30. λ \lambda
  31. τ \tau
  32. x x
  33. t t
  34. p p
  35. p t p_{t}
  36. λ C \lambda C
  37. λ \lambda
  38. τ \tau
  39. d x d τ = { x , λ C } , d p d τ = { p , λ C } d t d τ = { t , λ C } , d p t d τ = { p t , λ C } {dx\over d\tau}=\{x,\lambda C\},\;\;\;\;{dp\over d\tau}=\{p,\lambda C\}\;\;\;% \;\;\;{dt\over d\tau}=\{t,\lambda C\},\;\;\;\;{dp_{t}\over d\tau}=\{p_{t},% \lambda C\}
  40. d F ( x , p , t , p t ) d τ = { F ( x , p , t , p t ) , λ C } {dF(x,p,t,p_{t})\over d\tau}=\{F(x,p,t,p_{t}),\lambda C\}
  41. F F
  42. x ( τ ) x(\tau)
  43. t ( τ ) t(\tau)
  44. p ( τ ) p(\tau)
  45. x ( τ ) x(\tau)
  46. t ( τ ) t(\tau)
  47. τ \tau
  48. t ( τ ) t(\tau)
  49. x ( τ ) x(\tau)
  50. d x d τ = p , d p d τ = - x ; d t d τ = p t , d p t d τ = t . {dx\over d\tau}={\partial\mathcal{H}\over\partial p},\;\;\;\;{dp\over d\tau}=-% {\partial\mathcal{H}\over\partial x};\;\;\;\;\;\;{dt\over d\tau}={\partial% \mathcal{H}\over\partial p_{t}},\;\;\;\;{dp_{t}\over d\tau}={\partial\mathcal{% H}\over\partial t}.
  51. d x d τ = λ p m , d p d τ = - λ m ω 2 x ; d t d τ = λ , d p t d τ = 0 , {dx\over d\tau}=\lambda{p\over m},\;\;\;\;{dp\over d\tau}=-\lambda m\omega^{2}% x;\;\;\;\;\;\;{dt\over d\tau}=\lambda,\;\;\;\;{dp_{t}\over d\tau}=0,
  52. t t
  53. d x / d t dx/dt
  54. d p / d t dp/dt
  55. d x d t = d x d τ / d t d τ = λ p / m λ = p m {dx\over dt}={dx\over d\tau}\Big/{dt\over d\tau}={\lambda p/m\over\lambda}={p% \over m}
  56. d p d t = d p d τ / d t d τ = - λ m ω 2 x λ = - m ω 2 x . {dp\over dt}={dp\over d\tau}\Big/{dt\over d\tau}={-\lambda m\omega^{2}x\over% \lambda}=-m\omega^{2}x.
  57. d 2 x d t 2 = - ω 2 x . {d^{2}x\over dt^{2}}=-\omega^{2}x.
  58. d p t / d τ = d p t / d τ / d p t / d τ = 0 dp_{t}/d\tau=dp_{t}/d\tau\big/dp_{t}/d\tau=0
  59. p t = C o n s t . p_{t}=Const.
  60. τ \tau
  61. C = p t + C ( x , p ) C=p_{t}+C^{\prime}(x,p)
  62. S = d τ [ d x d τ p + d t d τ p t - λ ( p t + C ( x , p ) ) ] S=\int d\tau\Big[{dx\over d\tau}p+{dt\over d\tau}p_{t}-\lambda(p_{t}+C^{\prime% }(x,p))\Big]
  63. = d τ [ d x d τ p - d t d τ C ( x , p ) ] =\int d\tau\Big[{dx\over d\tau}p-{dt\over d\tau}C^{\prime}(x,p)\Big]
  64. = d t [ d x d t p - p 2 2 m + 1 2 m ω 2 x 2 ] =\int dt\Big[{dx\over dt}p-{p^{2}\over 2m}+{1\over 2}m\omega^{2}x^{2}\Big]
  65. q a b ( x ) q_{ab}(x)
  66. K a b ( x ) K^{ab}(x)
  67. q a b q_{ab}
  68. π a b = q ( K a b - q a b K c c ) \pi^{ab}=\sqrt{q}(K^{ab}-q^{ab}K_{c}^{c})
  69. H = d e t ( q ) ( K a b K a b - ( K a a ) 2 - R ) H=\sqrt{det(q)}(K_{ab}K^{ab}-(K_{a}^{a})^{2}-\;^{3}R)
  70. R 3 \;{}^{3}R
  71. q a b ( x ) q_{ab}(x)
  72. S U ( 2 ) SU(2)
  73. A a i A_{a}^{i}
  74. E ~ i a \tilde{E}_{i}^{a}
  75. E ~ i a = d e t ( q ) E i a \tilde{E}_{i}^{a}=\sqrt{det(q)}E_{i}^{a}
  76. d e t ( q ) q a b = E ~ i a E ~ j b δ i j det(q)q^{ab}=\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\delta^{ij}
  77. i i
  78. S U ( 2 ) SU(2)
  79. A a i = Γ a i - i K a i A_{a}^{i}=\Gamma_{a}^{i}-iK_{a}^{i}
  80. Γ a i \Gamma_{a}^{i}
  81. Γ a i j \Gamma_{a\;\;i}^{\;\;j}
  82. Γ a i = Γ a j k ϵ j k i \Gamma_{a}^{i}=\Gamma_{ajk}\epsilon^{jki}
  83. K a i = K a b E ~ a i / d e t ( q ) K_{a}^{i}=K_{ab}\tilde{E}^{ai}/\sqrt{det(q)}
  84. H = ϵ i j k F a b k E ~ i a E ~ j b d e t ( q ) H={\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\over\sqrt{det(q)}}
  85. F a b k F_{ab}^{k}
  86. A a i A_{a}^{i}
  87. 1 / d e t ( q ) 1/\sqrt{det(q)}
  88. H = 0 H=0
  89. H ~ = d e t ( q ) H = ϵ i j k F a b k E ~ i a E ~ j b = 0 \tilde{H}=\sqrt{det(q)}H=\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{j% }^{b}=0
  90. ( + , + , + , + ) (+,+,+,+)
  91. t t
  92. A a i = Γ a i + β K a i A_{a}^{i}=\Gamma_{a}^{i}+\beta K_{a}^{i}
  93. H = - ζ ϵ i j k F a b k E ~ i a E ~ j b d e t ( q ) + 2 ζ β 2 - 1 β 2 ( E ~ i a E ~ j b - E ~ j a E ~ i b ) d e t ( q ) ( A a i - Γ a i ) ( A b j - Γ b j ) = H E + H H=-\zeta{\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\over\sqrt{% det(q)}}+2{\zeta\beta^{2}-1\over\beta^{2}}{(\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}% -\tilde{E}_{j}^{a}\tilde{E}_{i}^{b})\over\sqrt{det(q)}}(A_{a}^{i}-\Gamma_{a}^{% i})(A_{b}^{j}-\Gamma_{b}^{j})=H_{E}+H^{\prime}
  94. β \beta
  95. ζ \zeta
  96. Γ a i \Gamma_{a}^{i}
  97. β = i \beta=i
  98. H E H_{E}
  99. β = ± 1 \beta=\pm 1
  100. 1 / d e t ( q ) 1/\sqrt{det(q)}
  101. β \beta
  102. 1 / d e t ( q ) 1/\sqrt{det(q)}
  103. { A c k , V } = ϵ a b c ϵ i j k E ~ i a E ~ j b d e t ( q ) \{A_{c}^{k},V\}={\epsilon_{abc}\epsilon^{ijk}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b% }\over\sqrt{det(q)}}
  104. V V
  105. V = d 3 x d e t ( q ) = 1 6 d 3 x | E ~ i a E ~ j b E ~ k c ϵ i j k ϵ a b c | V=\int d^{3}x\sqrt{det(q)}={1\over 6}\int d^{3}x\sqrt{|\tilde{E}_{i}^{a}\tilde% {E}_{j}^{b}\tilde{E}_{k}^{c}\epsilon^{ijk}\epsilon_{abc}|}
  106. H E = { A c k , V } F a b k ϵ ~ a b c H_{E}=\{A_{c}^{k},V\}F_{ab}^{k}\tilde{\epsilon}^{abc}
  107. Γ a i \Gamma_{a}^{i}
  108. E ~ i a \tilde{E}^{a}_{i}
  109. D a E ~ i a = 0 D_{a}\tilde{E}^{a}_{i}=0
  110. c g a b = 0 \nabla_{c}g_{ab}=0
  111. K = d 3 x K a i E ~ i a K=\int d^{3}xK_{a}^{i}\tilde{E}_{i}^{a}
  112. K a i = K a b E ~ a i / d e t ( q ) K_{a}^{i}=K_{ab}\tilde{E}^{ai}/\sqrt{det(q)}
  113. K a i = { A a i , K } K_{a}^{i}=\{A_{a}^{i},K\}
  114. A a i - Γ a i = β K a i = β { A a i , K } A_{a}^{i}-\Gamma_{a}^{i}=\beta K_{a}^{i}=\beta\{A_{a}^{i},K\}
  115. A a i A_{a}^{i}
  116. K K
  117. H = ϵ a b c ϵ i j k { A a i , K } { A b j , K } { A c k , V } H^{\prime}=\epsilon^{abc}\epsilon_{ijk}\{A_{a}^{i},K\}\{A_{b}^{j},K\}\{A_{c}^{% k},V\}
  118. K K
  119. K K
  120. K = - { V , d 3 x H E } K=-\{V,\int d^{3}xH_{E}\}

Hamiltonian_system.html

  1. H ( s y m b o l q , s y m b o l p , t ) H(symbol{q},symbol{p},t)
  2. s y m b o l r symbol{r}
  3. s y m b o l p symbol{p}
  4. s y m b o l q symbol{q}
  5. s y m b o l p symbol{p}
  6. s y m b o l q symbol{q}
  7. s y m b o l r = ( s y m b o l q , s y m b o l p ) symbol{r}=(symbol{q},symbol{p})
  8. d s y m b o l p d t = - H s y m b o l q \displaystyle\frac{dsymbol{p}}{dt}=-\frac{\partial H}{\partial symbol{q}}
  9. s y m b o l r ( t ) symbol{r}(t)
  10. s y m b o l r ( 0 ) = s y m b o l r 0 2 N symbol{r}(0)=symbol{r}_{0}\in\mathbb{R}^{2N}
  11. H ( s y m b o l q , s y m b o l p , t ) = H ( s y m b o l q , s y m b o l p ) H(symbol{q},symbol{p},t)=H(symbol{q},symbol{p})
  12. d H d t = H s y m b o l p ( - H s y m b o l q ) + H s y m b o l q H s y m b o l p + 0 = 0 \frac{dH}{dt}=\frac{\partial H}{\partial symbol{p}}\cdot\left(-\frac{\partial H% }{\partial symbol{q}}\right)+\frac{\partial H}{\partial symbol{q}}\cdot\frac{% \partial H}{\partial symbol{p}}+0=0
  13. H = E H=E
  14. s y m b o l p = p symbol{p}=p
  15. s y m b o l q = x symbol{q}=x
  16. H = p 2 2 m + 1 2 k x 2 H=\frac{p^{2}}{2m}+\frac{1}{2}kx^{2}
  17. s y m b o l r H ( s y m b o l r ) = [ s y m b o l q H ( s y m b o l q , s y m b o l p ) s y m b o l p H ( s y m b o l q , s y m b o l p ) ] \nabla_{symbol{r}}H(symbol{r})=\begin{bmatrix}\partial_{s}ymbol{q}H(symbol{q},% symbol{p})\\ \partial_{s}ymbol{p}H(symbol{q},symbol{p})\\ \end{bmatrix}
  18. d s y m b o l r d t = S N s y m b o l r H ( s y m b o l r ) \frac{dsymbol{r}}{dt}=S_{N}\cdot\nabla_{symbol{r}}H(symbol{r})
  19. S N = [ 0 I N - I N 0 ] S_{N}=\begin{bmatrix}0&I_{N}\\ -I_{N}&0\\ \end{bmatrix}