wpmath0000003_11

Monoclinic_crystal_system.html

  1. 2 2
  2. 2 ¯ = m \bar{2}=m
  3. 2 / m 2/m\,\!

Monoidal_category.html

  1. 𝐂 \mathbf{C}
  2. : 𝐂 × 𝐂 𝐂 \otimes\colon\mathbf{C}\times\mathbf{C}\to\mathbf{C}
  3. I I
  4. α \alpha
  5. α A , B , C : ( A B ) C A ( B C ) \alpha_{A,B,C}\colon(A\otimes B)\otimes C\cong A\otimes(B\otimes C)
  6. I I
  7. λ \lambda
  8. ρ \rho
  9. λ A : I A A \lambda_{A}\colon I\otimes A\cong A
  10. ρ A : A I A \rho_{A}\colon A\otimes I\cong A
  11. A A
  12. B B
  13. C C
  14. D D
  15. 𝐂 \mathbf{C}
  16. A A
  17. B B
  18. 𝐂 \mathbf{C}
  19. α \alpha
  20. λ \lambda
  21. ρ \rho

Monomial.html

  1. x x
  2. x x
  3. x x
  4. x x
  5. n n
  6. x , y , z , x,y,z,
  7. x a y b z c x^{a}y^{b}z^{c}
  8. a , b , c a,b,c
  9. - 7 x 5 -7x^{5}
  10. ( 3 - 4 i ) x 4 y z 13 (3-4i)x^{4}yz^{13}
  11. x , y , z , x,y,z,
  12. ( ( n d ) ) \textstyle{\left(\!\!{n\choose d}\!\!\right)}
  13. ( ( n d ) ) = ( n + d - 1 d ) = ( d + ( n - 1 ) n - 1 ) = ( d + 1 ) × ( d + 2 ) × × ( d + n - 1 ) 1 × 2 × × ( n - 1 ) = 1 ( n - 1 ) ! ( d + 1 ) n - 1 ¯ . \left(\!\!{n\choose d}\!\!\right)={\left({{n+d-1}\atop{d}}\right)}={\left({{d+% (n-1)}\atop{n-1}}\right)}=\frac{(d+1)\times(d+2)\times\cdots\times(d+n-1)}{1% \times 2\times\cdots\times(n-1)}=\frac{1}{(n-1)!}(d+1)^{\overline{n-1}}.
  14. n - 1 n-1
  15. 1 ( n - 1 ) ! \tfrac{1}{(n-1)!}
  16. n = 3 n=3
  17. 1 2 ( d + 1 ) 2 ¯ = 1 2 ( d + 1 ) ( d + 2 ) \textstyle{\frac{1}{2}}(d+1)^{\overline{2}}=\textstyle{\frac{1}{2}}(d+1)(d+2)
  18. d d
  19. n n
  20. d d
  21. 1 ( 1 - t ) n . \frac{1}{(1-t)^{n}}.
  22. d d
  23. n n
  24. ( n + d n ) = ( n + d d ) . {\left({{n+d}\atop{n}}\right)}={\left({{n+d}\atop{d}}\right)}.
  25. d d
  26. n + 1 n+1
  27. d d
  28. n n
  29. x 1 x_{1}
  30. x 2 x_{2}
  31. x 3 x_{3}
  32. α = ( a , b , c ) \alpha=(a,b,c)
  33. x α = x 1 a x 2 b x 3 c x^{\alpha}=x_{1}^{a}\,x_{2}^{b}\,x_{3}^{c}
  34. a + b + c a+b+c
  35. x y z 2 xyz^{2}
  36. x α = 0 x^{\alpha}=0

Moore_machine.html

  1. ( S , S 0 , Σ , Λ , T , G ) (S,S_{0},\Sigma,\Lambda,T,G)
  2. S S
  3. S 0 S_{0}
  4. S S
  5. Σ \Sigma
  6. Λ \Lambda
  7. T : S × Σ S T:S\times\Sigma\rightarrow S
  8. G : S Λ G:S\rightarrow\Lambda
  9. S × Σ S\times\Sigma
  10. G G
  11. S S
  12. G G
  13. M M
  14. G ( s , σ ) G M ( s ) G(s,\sigma)\rightarrow G_{M}(s)
  15. ( s , σ ) (s,\sigma)
  16. G M ( s ) G_{M}(s)
  17. G M G_{M}
  18. M M
  19. s i s j s_{i}\rightarrow s_{j}
  20. s i s_{i}
  21. s j s_{j}
  22. s i s j s_{i}\rightarrow s_{j}
  23. ( s i , s j ) (s_{i},s_{j})
  24. s i s_{i}
  25. s j s_{j}
  26. s i s j s_{i}\rightarrow s_{j}
  27. ( n ; m ; p ) (n;m;p)
  28. S S
  29. n n
  30. m m
  31. p p
  32. S S
  33. S S
  34. S S
  35. ( n ; m ; p ) (n;m;p)
  36. S S
  37. n ( n - 1 ) 2 \tfrac{n(n-1)}{2}
  38. S S
  39. S S
  40. ( n ; m ; p ) (n;m;p)
  41. ( n - 1 ) ( n - 2 ) 2 + 1 \tfrac{(n-1)(n-2)}{2}+1
  42. S S
  43. ( n ; m ; p ) (n;m;p)
  44. ( n - 1 ) ( n - 2 ) 2 + 1 \tfrac{(n-1)(n-2)}{2}+1

Moore–Penrose_pseudoinverse.html

  1. A A
  2. K K
  3. , \mathbb{R},\,\mathbb{C}
  4. m × n m\times n
  5. K K
  6. M ( m , n ; K ) \mathrm{M}(m,n;K)
  7. A M ( m , n ; K ) A\in\mathrm{M}(m,n;K)
  8. A T A^{\mathrm{T}}
  9. A * A^{*}
  10. K = K=\mathbb{R}
  11. A * = A T A^{*}=A^{\mathrm{T}}
  12. A M ( m , n ; K ) A\in\mathrm{M}(m,n;K)
  13. im ( A ) \operatorname{im}(A)
  14. A A
  15. A A
  16. ker ( A ) \operatorname{ker}(A)
  17. A A
  18. n n
  19. I n M ( n , n ; K ) I_{n}\in\mathrm{M}(n,n;K)
  20. n × n n\times n
  21. A M ( m , n ; K ) A\in\mathrm{M}(m,n;K)
  22. A A
  23. A + M ( n , m ; K ) A^{+}\in\mathrm{M}(n,m;K)
  24. A A + A = A AA^{+}A=A\,\!
  25. A A
  26. A + A A + = A + A^{+}AA^{+}=A^{+}\,\!
  27. ( A A + ) * = A A + (AA^{+})^{*}=AA^{+}\,\!
  28. ( A + A ) * = A + A (A^{+}A)^{*}=A^{+}A\,\!
  29. A + A^{+}
  30. A A
  31. A + A^{+}
  32. A A
  33. A * A A^{*}A
  34. A + A^{+}
  35. A + = ( A * A ) - 1 A * . A^{+}=(A^{*}A)^{-1}A^{*}\,.
  36. A + A = I A^{+}A=I
  37. A A
  38. A A * AA^{*}
  39. A + A^{+}
  40. A + = A * ( A A * ) - 1 . A^{+}=A^{*}(AA^{*})^{-1}\,.
  41. A A + = I AA^{+}=I
  42. A A\,\!
  43. A + A^{+}\,\!
  44. A A\,\!
  45. A + A^{+}\,\!
  46. A A\,\!
  47. A + = A - 1 A^{+}=A^{-1}\,\!
  48. ( A + ) + = A (A^{+})^{+}=A\,\!
  49. ( A T ) + = ( A + ) T , ( A ¯ ) + = A + ¯ , ( A * ) + = ( A + ) * . (A^{\mathrm{T}})^{+}=(A^{+})^{\mathrm{T}},~{}~{}(\,\overline{A}\,)^{+}=% \overline{A^{+}},~{}~{}(A^{*})^{+}=(A^{+})^{*}.\,\!
  50. A A
  51. ( α A ) + = α - 1 A + (\alpha A)^{+}=\alpha^{-1}A^{+}\,\!
  52. α 0. \alpha\neq 0.
  53. A + = A + A + * A * A + = A * A + * A + A = A + * A * A A = A A * A + * A * = A * A A + A * = A + A A * \begin{array}[]{lclll}A^{+}&=&A^{+}&A^{+*}&A^{*}\\ A^{+}&=&A^{*}&A^{+*}&A^{+}\\ A&=&A^{+*}&A^{*}&A\\ A&=&A&A^{*}&A^{+*}\\ A^{*}&=&A^{*}&A&A^{+}\\ A^{*}&=&A^{+}&A&A^{*}\\ \end{array}
  54. A + = ( A * A ) + A * A^{+}=(A^{*}A)^{+}A^{*}\,\!
  55. A + = A * ( A A * ) + A^{+}=A^{*}(AA^{*})^{+}\,\!
  56. A * A A^{*}A
  57. A A * AA^{*}
  58. A M ( m , n ; K ) , B M ( n , p ; K ) A\in\mathrm{M}(m,n;K),~{}B\in\mathrm{M}(n,p;K)\,
  59. A A\,\!
  60. A * A = I n A^{*}A=I_{n}\,
  61. B B\,\!
  62. B B * = I n BB^{*}=I_{n}\,
  63. A A\,\!
  64. B B\,\!
  65. B = A * B=A^{*}\,\!
  66. B B
  67. A A
  68. ( A B ) + B + A + (AB)^{+}\equiv B^{+}A^{+}\,\!
  69. ( A A * ) + \displaystyle(AA^{*})^{+}
  70. P = A A + P=AA^{+}\,\!
  71. Q = A + A Q=A^{+}A\,\!
  72. P = P * P=P^{*}\,\!
  73. Q = Q * Q=Q^{*}\,\!
  74. P 2 = P P^{2}=P\,\!
  75. Q 2 = Q Q^{2}=Q\,\!
  76. P A = A = A Q PA=A=AQ\,\!
  77. A + P = A + = Q A + A^{+}P=A^{+}=QA^{+}\,\!
  78. P P\,\!
  79. A A\,\!
  80. A * A^{*}\,\!
  81. Q Q\,\!
  82. A * A^{*}\,\!
  83. A A\,\!
  84. ( I - P ) (I-P)\,\!
  85. A * A^{*}\,\!
  86. ( I - Q ) (I-Q)\,\!
  87. A A\,\!
  88. A : K n K m A:K^{n}\to K^{m}
  89. K K
  90. A + : K m K n A^{+}:K^{m}\to K^{n}
  91. \oplus
  92. \perp
  93. ker \operatorname{ker}
  94. ran \operatorname{ran}
  95. K n = ( ker A ) ker A K^{n}=(\operatorname{ker}A)^{\perp}\oplus\operatorname{ker}A
  96. K m = ran A ( ran A ) K^{m}=\operatorname{ran}A\oplus(\operatorname{ran}A)^{\perp}
  97. A : ( ker A ) ran A A:(\operatorname{ker}A)^{\perp}\to\operatorname{ran}A
  98. A + A^{+}
  99. ran A \operatorname{ran}A
  100. ( ran A ) (\operatorname{ran}A)^{\perp}
  101. A + b A^{+}b
  102. b b
  103. b b
  104. A A
  105. p ( b ) p(b)
  106. A A
  107. p ( b ) p(b)
  108. A A
  109. A + b A^{+}b
  110. A A
  111. ker ( A + ) = ker ( A * ) im ( A + ) = im ( A * ) \begin{aligned}\displaystyle\operatorname{ker}(A^{+})&\displaystyle=% \operatorname{ker}(A^{*})\\ \displaystyle\operatorname{im}(A^{+})&\displaystyle=\operatorname{im}(A^{*})% \end{aligned}
  112. A + = lim δ 0 ( A * A + δ I ) - 1 A * = lim δ 0 A * ( A A * + δ I ) - 1 A^{+}=\lim_{\delta\searrow 0}(A^{*}A+\delta I)^{-1}A^{*}=\lim_{\delta\searrow 0% }A^{*}(AA^{*}+\delta I)^{-1}
  113. ( A A * ) - 1 (AA^{*})^{-1}\,\!
  114. ( A * A ) - 1 (A^{*}A)^{-1}\,\!
  115. ( A n ) (A_{n})
  116. A A
  117. x x
  118. d d x A + ( x ) = - A + ( d d x A ) A + + A + A + T ( d d x A T ) ( 1 - A A + ) + ( 1 - A + A ) ( d d x A T ) A + T A + \frac{\mathrm{d}}{\mathrm{d}x}A^{+}(x)=-A^{+}\left(\frac{\mathrm{d}}{\mathrm{d% }x}A\right)A^{+}~{}+~{}A^{+}A^{+\,\text{T}}\left(\frac{\mathrm{d}}{\mathrm{d}x% }A\text{T}\right)\left(1-AA^{+}\right)~{}+~{}\left(1-A^{+}A\right)\left(\frac{% \,\text{d}}{\,\text{d}x}A\text{T}\right)A^{+\,\text{T}}A^{+}
  119. x x
  120. x x
  121. x x
  122. x + = { 0 , if x = 0 ; x - 1 , otherwise . x^{+}=\left\{\begin{matrix}0,&\mbox{if }~{}x=0;\\ x^{-1},&\mbox{otherwise}~{}.\end{matrix}\right.
  123. x + = { 0 T , if x = 0 ; x * x * x , otherwise . x^{+}=\left\{\begin{matrix}0^{\mathrm{T}},&\mbox{if }~{}x=0;\\ {x^{*}\over x^{*}x},&\mbox{otherwise}~{}.\end{matrix}\right.
  124. A A\,\!
  125. m n m\geq n
  126. A * A A^{*}A\,\!
  127. A + = ( A * A ) - 1 A * A^{+}=(A^{*}A)^{-1}A^{*}\,\!
  128. A + A^{+}\,\!
  129. A A\,\!
  130. A + A = I n A^{+}A=I_{n}\,\!
  131. A A\,\!
  132. m n m\leq n
  133. A A * AA^{*}
  134. A + = A * ( A A * ) - 1 A^{+}=A^{*}(AA^{*})^{-1}\,\!
  135. A + A^{+}\,\!
  136. A A\,\!
  137. A A + = I m AA^{+}=I_{m}\,\!
  138. A A\,\!
  139. A * A = I n A^{*}A=I_{n}\,\!
  140. A A * = I m AA^{*}=I_{m}\,\!
  141. A + = A * A^{+}=A^{*}\,\!
  142. C C\,\!
  143. \mathcal{F}
  144. C = Σ * C + = Σ + * \begin{aligned}\displaystyle C&\displaystyle=\mathcal{F}\cdot\Sigma\cdot% \mathcal{F}^{*}\\ \displaystyle C^{+}&\displaystyle=\mathcal{F}\cdot\Sigma^{+}\cdot\mathcal{F}^{% *}\end{aligned}
  145. r min ( m , n ) r\leq\min(m,n)
  146. A M ( m , n ; K ) A\in\mathrm{M}(m,n;K)\,\!
  147. A A\,\!
  148. A = B C A=BC\,\!
  149. B M ( m , r ; K ) B\in\mathrm{M}(m,r;K)\,\!
  150. C M ( r , n ; K ) C\in\mathrm{M}(r,n;K)\,\!
  151. r r
  152. A + = C + B + = C * ( C C * ) - 1 ( B * B ) - 1 B * A^{+}=C^{+}B^{+}=C^{*}(CC^{*})^{-1}(B^{*}B)^{-1}B^{*}\,\!
  153. K = K=\mathbb{R}\,\!
  154. K = K=\mathbb{C}\,\!
  155. A A * AA^{*}
  156. A * A A^{*}A
  157. A A\,\!
  158. A A\,\!
  159. A + = ( A * A ) - 1 A * A^{+}=(A^{*}A)^{-1}A^{*}\,\!
  160. A * A = R * R A^{*}A=R^{*}R\,\!
  161. R R\,\!
  162. A + = ( A * A ) - 1 A * ( A * A ) A + = A * R * R A + = A * A^{+}=(A^{*}A)^{-1}A^{*}\quad\Leftrightarrow\quad(A^{*}A)A^{+}=A^{*}\quad% \Leftrightarrow\quad R^{*}RA^{+}=A^{*}
  163. A * A A^{*}A\,\!
  164. A = Q R A=QR\,\!
  165. Q Q\,\,\!
  166. Q * Q = I Q^{*}Q=I
  167. R R\,\!
  168. A * A = ( Q R ) * ( Q R ) = R * Q * Q R = R * R A^{*}A\,=\,(QR)^{*}(QR)\,=\,R^{*}Q^{*}QR\,=\,R^{*}R
  169. R R
  170. A * A A^{*}A
  171. A + = A * ( A A * ) - 1 A^{+}=A^{*}(AA^{*})^{-1}\,\!
  172. A A
  173. A * A^{*}
  174. A = U Σ V * A=U\Sigma V^{*}
  175. A A
  176. A + = V Σ + U * A^{+}=V\Sigma^{+}U^{*}
  177. Σ \Sigma
  178. t = ε m a x ( m , n ) m a x ( Σ ) t=ε⋅max(m,n)⋅max(Σ)
  179. A A
  180. Σ \Sigma
  181. A A
  182. A i + 1 = 2 A i - A i A A i , A_{i+1}=2A_{i}-A_{i}AA_{i},\,
  183. A A
  184. A 0 A_{0}
  185. A 0 A = ( A 0 A ) * A_{0}A=(A_{0}A)^{*}
  186. A 0 = α A * A_{0}=\alpha A^{*}
  187. 0 < α < 2 / σ 1 2 ( A ) 0<\alpha<2/\sigma^{2}_{1}(A)
  188. σ 1 ( A ) \sigma_{1}(A)
  189. A A
  190. A i A_{i}
  191. A 0 A_{0}
  192. A 0 A = ( A 0 A ) * A_{0}A=(A_{0}A)^{*}
  193. A 0 := ( A * A + δ I ) - 1 A * A_{0}:=(A^{*}A+\delta I)^{-1}A^{*}
  194. A A
  195. A A * AA^{*}
  196. A A
  197. A * A A^{*}A
  198. A A
  199. A M ( m , n ; K ) A\in\mathrm{M}(m,n;K)\,\!
  200. A x = b , Ax=b,\,
  201. x x
  202. x K n \forall x\in K^{n}\,\!
  203. A x - b 2 A z - b 2 \|Ax-b\|_{2}\geq\|Az-b\|_{2}
  204. z = A + b z=A^{+}b
  205. 2 \|\cdot\|_{2}
  206. x = A + b + ( I - A + A ) w x=A^{+}b+(I-A^{+}A)w
  207. ( I - A + A ) (I-A^{+}A)
  208. z . z.
  209. B M ( m , p ; K ) B\in\mathrm{M}(m,p;K)
  210. X M ( n , p ; K ) \forall X\in\mathrm{M}(n,p;K)\,\!
  211. A X - B F A Z - B F \|AX-B\|_{\mathrm{F}}\geq\|AZ-B\|_{\mathrm{F}}
  212. Z = A + B Z=A^{+}B
  213. F \|\cdot\|_{\mathrm{F}}
  214. A x = b Ax=b\,
  215. x = A + b + [ I - A + A ] w x=A^{+}b+[I-A^{+}A]w
  216. A A + b = b AA^{+}b=b
  217. [ I - A + A ] [I-A^{+}A]
  218. A x = b , Ax=b,\,
  219. x 2 \|x\|_{2}
  220. A x = b Ax=b\,
  221. z = A + b z=A^{+}b
  222. z 2 x 2 \|z\|_{2}\leq\|x\|_{2}
  223. B M ( m , p ; K ) B\in\mathrm{M}(m,p;K)\,\!
  224. A X = B AX=B\,
  225. Z = A + B Z=A^{+}B
  226. Z F X F \|Z\|_{\mathrm{F}}\leq\|X\|_{\mathrm{F}}
  227. cond ( A ) = A A + . \mbox{cond}~{}(A)=\|A\|\|A^{+}\|.
  228. A A
  229. A : H < s u b > 1 H 2 A:H<sub>1→H_{2}

Morera's_theorem.html

  1. γ f ( z ) d z = 0 \oint_{\gamma}f(z)\,dz=0
  2. γ \gamma
  3. z D z\in D
  4. γ : [ 0 , 1 ] D \gamma:[0,1]\to D
  5. γ ( 0 ) = z 0 \gamma(0)=z_{0}
  6. γ ( 1 ) = z \gamma(1)=z
  7. F ( z ) = γ f ( ζ ) d ζ . F(z)=\int_{\gamma}f(\zeta)\,d\zeta.\,
  8. τ : [ 0 , 1 ] D \tau:[0,1]\to D
  9. τ ( 0 ) = z 0 \tau(0)=z_{0}
  10. τ ( 1 ) = z \tau(1)=z
  11. γ τ - 1 \gamma\tau^{-1}
  12. γ \gamma
  13. τ \tau
  14. γ f ( ζ ) d ζ + τ - 1 f ( ζ ) d ζ = γ τ - 1 f ( ζ ) d ζ = 0 \int_{\gamma}f(\zeta)\,d\zeta\,+\int_{\tau^{-1}}f(\zeta)\,d\zeta\,=\oint_{% \gamma\tau^{-1}}f(\zeta)\,d\zeta\,=0
  15. γ f ( ζ ) d ζ = τ f ( ζ ) d ζ . \int_{\gamma}f(\zeta)\,d\zeta\,=\int_{\tau}f(\zeta)\,d\zeta.\,
  16. C f n ( z ) d z = 0 \oint_{C}f_{n}(z)\,dz=0
  17. C f ( z ) d z = C lim n f n ( z ) d z = lim n C f n ( z ) d z = 0 \oint_{C}f(z)\,dz=\oint_{C}\lim_{n\to\infty}f_{n}(z)\,dz=\lim_{n\to\infty}% \oint_{C}f_{n}(z)\,dz=0
  18. ζ ( s ) = n = 1 1 n s \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}
  19. Γ ( α ) = 0 x α - 1 e - x d x . \Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}\,dx.
  20. C Γ ( α ) d α = 0 \oint_{C}\Gamma(\alpha)\,d\alpha=0
  21. C Γ ( α ) d α = C 0 x α - 1 e - x d x d α \oint_{C}\Gamma(\alpha)\,d\alpha=\oint_{C}\int_{0}^{\infty}x^{\alpha-1}e^{-x}% \,dx\,d\alpha
  22. 0 C x α - 1 e - x d α d x = 0 e - x C x α - 1 d α d x . \int_{0}^{\infty}\oint_{C}x^{\alpha-1}e^{-x}\,d\alpha\,dx=\int_{0}^{\infty}e^{% -x}\oint_{C}x^{\alpha-1}\,d\alpha\,dx.
  23. C x α - 1 d α = 0 , \oint_{C}x^{\alpha-1}\,d\alpha=0,
  24. T f ( z ) d z \oint_{\partial T}f(z)\,dz

Morse_theory.html

  1. f ( x ) = a + b x + c x 2 + d x 3 + f(x)=a+bx+cx^{2}+dx^{3}+\cdots
  2. x i ( b ) = 0 x_{i}(b)=0
  3. f ( x ) = f ( b ) - x 1 2 - - x α 2 + x α + 1 2 + + x n 2 f(x)=f(b)-x_{1}^{2}-\cdots-x_{\alpha}^{2}+x_{\alpha+1}^{2}+\cdots+x_{n}^{2}
  4. χ ( M ) \chi(M)
  5. ( - 1 ) γ C γ = χ ( M ) \sum(-1)^{\gamma}C^{\gamma}\,=\chi(M)
  6. b γ ( M ) b_{\gamma}(M)
  7. C γ - C γ - 1 ± + ( - 1 ) γ C 0 b γ ( M ) - b γ - 1 ( M ) ± + ( - 1 ) γ b 0 ( M ) . C^{\gamma}-C^{\gamma-1}\pm\cdots+(-1)^{\gamma}C^{0}\geq b_{\gamma}(M)-b_{% \gamma-1}(M)\pm\cdots+(-1)^{\gamma}b_{0}(M).
  8. γ { 0 , , n = dim M } , \gamma\in\{0,\dots,n=\dim M\},
  9. C γ b γ ( M ) . C^{\gamma}\geq b_{\gamma}(M).
  10. S n S^{n}
  11. ( i - , i + ) , (i_{-},i_{+}),

Motive_(algebraic_geometry).html

  1. X = i X i \scriptstyle X=\coprod_{i}X_{i}
  2. C o r r r ( k ) ( X , Y ) := i A d i + r ( X i × Y ) Corr^{r}(k)(X,Y):=\bigoplus_{i}A^{d_{i}+r}(X_{i}\times Y)
  3. α β := π X Z * ( π X Y * ( α ) π Y Z * ( β ) ) C o r r r + s ( X , Z ) \alpha\circ\beta:=\pi_{XZ*}(\pi^{*}_{XY}(\alpha)\cdot\pi^{*}_{YZ}(\beta))\in Corr% ^{r+s}(X,Z)
  4. F : S m P r o j ( k ) C o r r ( k ) X X f Γ f F:\begin{array}[]{rcl}SmProj(k)&\longrightarrow&Corr(k)\\ X&\longmapsto&X\\ f&\longmapsto&\Gamma_{f}\end{array}
  5. X Y := X Y \scriptstyle X\oplus Y:=X\coprod Y
  6. α + β := ( α , β ) A * ( X × X ) A * ( Y × Y ) A * ( ( X Y ) × ( X Y ) ) \alpha+\beta:=(\alpha,\beta)\in A^{*}(X\times X)\oplus A^{*}(Y\times Y)% \hookrightarrow A^{*}((X\coprod Y)\times(X\coprod Y))
  7. C h o w e f f ( k ) := S p l i t ( C o r r ( k ) ) Chow^{eff}(k):=Split(Corr(k))
  8. O b ( C h o w e f f ( k ) ) := { ( X , α ) | ( α : X X ) C o r r ( k ) such that α α = α } Ob(Chow^{eff}(k)):=\{(X,\alpha)\mbox{ }~{}|\mbox{ }~{}(\alpha:X\vdash X)\in Corr% (k)\mbox{ such that }~{}\alpha\circ\alpha=\alpha\}
  9. M o r ( ( X , α ) , ( Y , β ) ) := { f : X Y | f α = f = β f } Mor((X,\alpha),(Y,\beta)):=\{f:X\vdash Y\mbox{ }~{}|\mbox{ }~{}f\circ\alpha=f=% \beta\circ f\}
  10. h : S m P r o j ( k ) C o r r ( k ) X [ X ] := ( X , Δ ) X f [ f ] := Γ f X × Y h:\begin{array}[]{rcl}SmProj(k)&\longrightarrow&Corr(k)\\ X&\longmapsto&[X]:=(X,\Delta)_{X}\\ f&\longmapsto&[f]:=\Gamma_{f}\subset X\times Y\end{array}
  11. ( [ X ] , α ) ( [ Y ] , β ) := ( [ X Y ] , α + β ) ([X],\alpha)\oplus([Y],\beta):=([X\coprod Y],\alpha+\beta)
  12. ( [ X ] , α ) ( [ Y ] , β ) := ( X × Y , π X * α π Y * β ) , π X : ( X × Y ) × ( X × Y ) X × X , and π Y : ( X × Y ) × ( X × Y ) Y × Y ([X],\alpha)\otimes([Y],\beta):=(X\times Y,\pi_{X}^{*}\alpha\cdot\pi_{Y}^{*}% \beta),\qquad\pi_{X}:(X\times Y)\times(X\times Y)\to X\times X,\mbox{ and }~{}% \pi_{Y}:(X\times Y)\times(X\times Y)\to Y\times Y
  13. f 1 f 2 : ( X 1 , α 1 ) ( X 2 , α 2 ) ( Y 1 , β 1 ) ( Y 2 , β 2 ) , f 1 f 2 := π 1 * γ 1 π 2 * γ 2 f_{1}\otimes f_{2}:(X_{1},\alpha_{1})\otimes(X_{2},\alpha_{2})\vdash(Y_{1},% \beta_{1})\otimes(Y_{2},\beta_{2}),\qquad f_{1}\otimes f_{2}:=\pi^{*}_{1}% \gamma_{1}\cdot\pi^{*}_{2}\gamma_{2}
  14. L := ( 𝐏 1 , λ ) , λ := p t × 𝐏 1 A 1 ( 𝐏 1 × 𝐏 1 ) L:=(\mathbf{P}^{1},\lambda),\qquad\lambda:=pt\times\mathbf{P}^{1}\in A^{1}(% \mathbf{P}^{1}\times\mathbf{P}^{1})
  15. [ 𝐏 1 ] = 𝟏 L [\mathbf{P}^{1}]=\mathbf{1}\oplus L
  16. C h o w ( k ) := C h o w e f f ( k ) [ T ] Chow(k):=Chow^{eff}(k)[T]
  17. f : ( X , p , m ) ( Y , q , n ) , f C o r r n - m ( X , Y ) such that f p = f = q f f:(X,p,m)\to(Y,q,n),\quad f\in Corr^{n-m}(X,Y)\mbox{ such that }~{}f\circ p=f=% q\circ f

Mouse_keys.html

  1. action _ delta × mk _ max _ speed × ( i mk _ time _ to _ max ) 1000 + mk _ curve 1000 \mathrm{action\_delta}\times\mathrm{mk\_max\_speed}\times\left(\frac{i}{% \mathrm{mk\_time\_to\_max}}\right)^{\frac{1000+\mathrm{mk\_curve}}{1000}}

Moving_frame.html

  1. ( A , f ) = ( f f ) ( A , f ) . (A,f^{\prime})=(f\to f^{\prime})\circ(A,f).

Multi-index_notation.html

  1. α = ( α 1 , α 2 , , α n ) \alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})
  2. 0 n \mathbb{N}^{n}_{0}
  3. α , β 0 n \alpha,\beta\in\mathbb{N}^{n}_{0}
  4. x = ( x 1 , x 2 , , x n ) n x=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}
  5. α ± β = ( α 1 ± β 1 , α 2 ± β 2 , , α n ± β n ) \alpha\pm\beta=(\alpha_{1}\pm\beta_{1},\,\alpha_{2}\pm\beta_{2},\ldots,\,% \alpha_{n}\pm\beta_{n})
  6. α β α i β i i { 1 , , n } \alpha\leq\beta\quad\Leftrightarrow\quad\alpha_{i}\leq\beta_{i}\quad\forall\,i% \in\{1,\ldots,n\}
  7. | α | = α 1 + α 2 + + α n |\alpha|=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}
  8. α ! = α 1 ! α 2 ! α n ! \alpha!=\alpha_{1}!\cdot\alpha_{2}!\cdots\alpha_{n}!
  9. ( α β ) = ( α 1 β 1 ) ( α 2 β 2 ) ( α n β n ) = α ! β ! ( α - β ) ! {\left({{\alpha}\atop{\beta}}\right)}={\left({{\alpha_{1}}\atop{\beta_{1}}}% \right)}{\left({{\alpha_{2}}\atop{\beta_{2}}}\right)}\cdots{\left({{\alpha_{n}% }\atop{\beta_{n}}}\right)}=\frac{\alpha!}{\beta!(\alpha-\beta)!}
  10. ( k α ) = k ! α 1 ! α 2 ! α n ! = k ! α ! {\left({{k}\atop{\alpha}}\right)}=\frac{k!}{\alpha_{1}!\alpha_{2}!\cdots\alpha% _{n}!}=\frac{k!}{\alpha!}
  11. k := | α | 0 k:=|\alpha|\in\mathbb{N}_{0}\,\!
  12. x α = x 1 α 1 x 2 α 2 x n α n x^{\alpha}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\ldots x_{n}^{\alpha_{n}}
  13. α = 1 α 1 2 α 2 n α n \partial^{\alpha}=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}\ldots% \partial_{n}^{\alpha_{n}}
  14. i α i := α i / x i α i \partial_{i}^{\alpha_{i}}:=\partial^{\alpha_{i}}/\partial x_{i}^{\alpha_{i}}
  15. x , y , h n x,y,h\in\mathbb{C}^{n}
  16. n \mathbb{R}^{n}
  17. α , ν 0 n \alpha,\nu\in\mathbb{N}_{0}^{n}
  18. f , g , a α : n f,g,a_{\alpha}\colon\mathbb{C}^{n}\to\mathbb{C}
  19. n \mathbb{R}^{n}\to\mathbb{R}
  20. ( i = 1 n x i ) k = | α | = k ( k α ) x α \biggl(\sum_{i=1}^{n}x_{i}\biggr)^{k}=\sum_{|\alpha|=k}{\left({{k}\atop{\alpha% }}\right)}\,x^{\alpha}
  21. ( x + y ) α = ν α ( α ν ) x ν y α - ν . (x+y)^{\alpha}=\sum_{\nu\leq\alpha}{\left({{\alpha}\atop{\nu}}\right)}\,x^{\nu% }y^{\alpha-\nu}.
  22. x + y x+y
  23. α α
  24. α ( f g ) = ν α ( α ν ) ν f α - ν g . \partial^{\alpha}(fg)=\sum_{\nu\leq\alpha}{\left({{\alpha}\atop{\nu}}\right)}% \,\partial^{\nu}f\,\partial^{\alpha-\nu}g.
  25. f ( x + h ) = α 0 n α f ( x ) α ! h α . f(x+h)=\sum_{\alpha\in\mathbb{N}^{n}_{0}}{\frac{\partial^{\alpha}f(x)}{\alpha!% }h^{\alpha}}.
  26. f ( x + h ) = | α | n α f ( x ) α ! h α + R n ( x , h ) , f(x+h)=\sum_{|\alpha|\leq n}{\frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha}}+% R_{n}(x,h),
  27. R n ( x , h ) = ( n + 1 ) | α | = n + 1 h α α ! 0 1 ( 1 - t ) n α f ( x + t h ) d t . R_{n}(x,h)=(n+1)\sum_{|\alpha|=n+1}\frac{h^{\alpha}}{\alpha!}\int_{0}^{1}(1-t)% ^{n}\partial^{\alpha}f(x+th)\,dt.
  28. P ( ) = | α | N a α ( x ) α . P(\partial)=\sum_{|\alpha|\leq N}{}{a_{\alpha}(x)\partial^{\alpha}}.
  29. Ω n \Omega\subset\mathbb{R}^{n}
  30. Ω u ( α v ) d x = ( - 1 ) | α | Ω ( α u ) v d x . \int_{\Omega}{}{u(\partial^{\alpha}v)}\,dx=(-1)^{|\alpha|}\int_{\Omega}{(% \partial^{\alpha}u)v\,dx}.
  31. α , β 0 n \alpha,\beta\in\mathbb{N}^{n}_{0}
  32. x = ( x 1 , , x n ) x=(x_{1},\ldots,x_{n})
  33. α x β = { β ! ( β - α ) ! x β - α if α β , 0 otherwise. \partial^{\alpha}x^{\beta}=\begin{cases}\frac{\beta!}{(\beta-\alpha)!}x^{\beta% -\alpha}&\hbox{if}\,\,\alpha\leq\beta,\\ 0&\hbox{otherwise.}\end{cases}
  34. d α d x α x β = { β ! ( β - α ) ! x β - α if α β , 0 otherwise. ( 1 ) \frac{d^{\alpha}}{dx^{\alpha}}x^{\beta}=\begin{cases}\frac{\beta!}{(\beta-% \alpha)!}x^{\beta-\alpha}&\hbox{if}\,\,\alpha\leq\beta,\\ 0&\hbox{otherwise.}\end{cases}\qquad(1)
  35. α = ( α 1 , , α n ) \alpha=(\alpha_{1},\ldots,\alpha_{n})
  36. β = ( β 1 , , β n ) \beta=(\beta_{1},\ldots,\beta_{n})
  37. x = ( x 1 , , x n ) x=(x_{1},\ldots,x_{n})
  38. α x β = | α | x 1 α 1 x n α n x 1 β 1 x n β n = α 1 x 1 α 1 x 1 β 1 α n x n α n x n β n . \begin{aligned}\displaystyle\partial^{\alpha}x^{\beta}&\displaystyle=\frac{% \partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdots\partial x_{n}^{\alpha_{% n}}}x_{1}^{\beta_{1}}\cdots x_{n}^{\beta_{n}}\\ &\displaystyle=\frac{\partial^{\alpha_{1}}}{\partial x_{1}^{\alpha_{1}}}x_{1}^% {\beta_{1}}\cdots\frac{\partial^{\alpha_{n}}}{\partial x_{n}^{\alpha_{n}}}x_{n% }^{\beta_{n}}.\end{aligned}
  39. x i β i x_{i}^{\beta_{i}}
  40. x i x_{i}
  41. / x i \partial/\partial x_{i}
  42. d / d x i d/dx_{i}
  43. α x β \partial^{\alpha}x^{\beta}
  44. d α i d x i α i x i β i = β i ! ( β i - α i ) ! x i β i - α i \frac{d^{\alpha_{i}}}{dx_{i}^{\alpha_{i}}}x_{i}^{\beta_{i}}=\frac{\beta_{i}!}{% (\beta_{i}-\alpha_{i})!}x_{i}^{\beta_{i}-\alpha_{i}}
  45. i i
  46. \Box

Multidimensional_scaling.html

  1. I I
  2. δ i , j := \delta_{i,j}:=
  3. i i
  4. j j
  5. Δ := ( δ 1 , 1 δ 1 , 2 δ 1 , I δ 2 , 1 δ 2 , 2 δ 2 , I δ I , 1 δ I , 2 δ I , I ) . \Delta:=\begin{pmatrix}\delta_{1,1}&\delta_{1,2}&\cdots&\delta_{1,I}\\ \delta_{2,1}&\delta_{2,2}&\cdots&\delta_{2,I}\\ \vdots&\vdots&&\vdots\\ \delta_{I,1}&\delta_{I,2}&\cdots&\delta_{I,I}\end{pmatrix}.
  6. Δ \Delta
  7. I I
  8. x 1 , , x I N x_{1},\ldots,x_{I}\in\mathbb{R}^{N}
  9. x i - x j δ i , j \|x_{i}-x_{j}\|\approx\delta_{i,j}
  10. i , j 1 , , I i,j\in{1,\dots,I}
  11. \|\cdot\|
  12. I I
  13. N \mathbb{R}^{N}
  14. N N
  15. x i x_{i}
  16. I I
  17. x i x_{i}
  18. x i - x j \|x_{i}-x_{j}\|
  19. \mathbb{R}
  20. N \mathbb{R}^{N}
  21. N N
  22. \mathbb{R}
  23. N N
  24. x i x_{i}
  25. ( x 1 , , x I ) (x_{1},\ldots,x_{I})
  26. min x 1 , , x I i < j ( x i - x j - δ i , j ) 2 . \min_{x_{1},\ldots,x_{I}}\sum_{i<j}(\|x_{i}-x_{j}\|-\delta_{i,j})^{2}.\,
  27. Q = N ( N - 1 ) / 2 Q=N(N-1)/2

Multidisciplinary_design_optimization.html

  1. 𝐱 \mathbf{x}
  2. J ( 𝐱 ) J(\mathbf{x})
  3. 𝐠 ( 𝐱 ) 𝟎 \mathbf{g}(\mathbf{x})\leq\mathbf{0}
  4. 𝐡 ( 𝐱 ) = 𝟎 \mathbf{h}(\mathbf{x})=\mathbf{0}
  5. 𝐱 l b 𝐱 𝐱 u b \mathbf{x}_{lb}\leq\mathbf{x}\leq\mathbf{x}_{ub}
  6. J J
  7. 𝐱 \mathbf{x}
  8. 𝐠 \mathbf{g}
  9. 𝐡 \mathbf{h}
  10. 𝐱 l b \mathbf{x}_{lb}
  11. 𝐱 u b \mathbf{x}_{ub}

Multimagic_square.html

  1. a + i = 2 e a+i=2e
  2. a 2 + i 2 = 2 e 2 a^{2}+i^{2}=2e^{2}
  3. ( a - i ) 2 = 2 ( a 2 + i 2 ) - ( a + i ) 2 = 4 e 2 - 4 e 2 = 0 (a-i)^{2}=2(a^{2}+i^{2})-(a+i)^{2}=4e^{2}-4e^{2}=0
  4. a = e = i a=e=i

Multinomial_theorem.html

  1. ( x 1 + x 2 + + x m ) n = k 1 + k 2 + + k m = n ( n k 1 , k 2 , , k m ) 1 t m x t k t , (x_{1}+x_{2}+\cdots+x_{m})^{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}x_{t}^{k_{t}}\,,
  2. ( 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}!}
  3. ( a + b + c ) 3 = a 3 + b 3 + c 3 + 3 a 2 b + 3 a 2 c + 3 b 2 a + 3 b 2 c + 3 c 2 a + 3 c 2 b + 6 a b c . (a+b+c)^{3}=a^{3}+b^{3}+c^{3}+3a^{2}b+3a^{2}c+3b^{2}a+3b^{2}c+3c^{2}a+3c^{2}b+% 6abc.
  4. a 2 b 0 c 1 a^{2}b^{0}c^{1}
  5. ( 3 2 , 0 , 1 ) = 3 ! 2 ! 0 ! 1 ! = 6 2 1 1 = 3 {3\choose 2,0,1}=\frac{3!}{2!\cdot 0!\cdot 1!}=\frac{6}{2\cdot 1\cdot 1}=3
  6. a 1 b 1 c 1 a^{1}b^{1}c^{1}
  7. ( 3 1 , 1 , 1 ) = 3 ! 1 ! 1 ! 1 ! = 6 1 1 1 = 6 {3\choose 1,1,1}=\frac{3!}{1!\cdot 1!\cdot 1!}=\frac{6}{1\cdot 1\cdot 1}=6
  8. ( x 1 + + x m ) n = | α | = n ( n α ) x α (x_{1}+\cdots+x_{m})^{n}=\sum_{|\alpha|=n}{n\choose\alpha}x^{\alpha}
  9. ( x 1 + x 2 + + x m + x m + 1 ) n = ( x 1 + x 2 + + ( x m + x m + 1 ) ) n (x_{1}+x_{2}+\cdots+x_{m}+x_{m+1})^{n}=(x_{1}+x_{2}+\cdots+(x_{m}+x_{m+1}))^{n}
  10. = k 1 + k 2 + + k m - 1 + K = n ( n k 1 , k 2 , , k m - 1 , K ) x 1 k 1 x 2 k 2 x m - 1 k m - 1 ( x m + x m + 1 ) K =\sum_{k_{1}+k_{2}+\cdots+k_{m-1}+K=n}{n\choose k_{1},k_{2},\ldots,k_{m-1},K}x% _{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}(x_{m}+x_{m+1})^{K}
  11. = k 1 + k 2 + + k m - 1 + K = n ( n k 1 , k 2 , , k m - 1 , K ) x 1 k 1 x 2 k 2 x m - 1 k m - 1 k m + k m + 1 = K ( K k m , k m + 1 ) x m k m x m + 1 k m + 1 =\sum_{k_{1}+k_{2}+\cdots+k_{m-1}+K=n}{n\choose k_{1},k_{2},\ldots,k_{m-1},K}x% _{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}\sum_{k_{m}+k_{m+1}=K}{K% \choose k_{m},k_{m+1}}x_{m}^{k_{m}}x_{m+1}^{k_{m+1}}
  12. = k 1 + k 2 + + k m - 1 + k m + k m + 1 = n ( n k 1 , k 2 , , k m - 1 , k m , k m + 1 ) x 1 k 1 x 2 k 2 x m - 1 k m - 1 x m k m x m + 1 k m + 1 =\sum_{k_{1}+k_{2}+\cdots+k_{m-1}+k_{m}+k_{m+1}=n}{n\choose k_{1},k_{2},\ldots% ,k_{m-1},k_{m},k_{m+1}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}x_{m}% ^{k_{m}}x_{m+1}^{k_{m+1}}
  13. ( n k 1 , k 2 , , k m - 1 , K ) ( K k m , k m + 1 ) = ( n k 1 , k 2 , , k m - 1 , k m , k m + 1 ) , {n\choose k_{1},k_{2},\ldots,k_{m-1},K}{K\choose k_{m},k_{m+1}}={n\choose k_{1% },k_{2},\ldots,k_{m-1},k_{m},k_{m+1}},
  14. n ! k 1 ! k 2 ! k m - 1 ! K ! K ! k m ! k m + 1 ! = n ! k 1 ! k 2 ! k m + 1 ! . \frac{n!}{k_{1}!k_{2}!\cdots k_{m-1}!K!}\frac{K!}{k_{m}!k_{m+1}!}=\frac{n!}{k_% {1}!k_{2}!\cdots k_{m+1}!}.
  15. ( 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}!},
  16. = ( k 1 k 1 ) ( k 1 + k 2 k 2 ) ( k 1 + k 2 + + k m k m ) = i = 1 m ( j = 1 i k j k i ) ={k_{1}\choose k_{1}}{k_{1}+k_{2}\choose k_{2}}\cdots{k_{1}+k_{2}+\cdots+k_{m}% \choose k_{m}}=\prod_{i=1}^{m}{\sum_{j=1}^{i}k_{j}\choose k_{i}}
  17. k 1 + k 2 + + k m = n ( n k 1 , k 2 , , k m ) x 1 k 1 x 2 k 2 x m k m = ( x 1 + x 2 + + x m ) n , \sum_{k_{1}+k_{2}+\cdots+k_{m}=n}{n\choose k_{1},k_{2},\ldots,k_{m}}x_{1}^{k_{% 1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}=(x_{1}+x_{2}+\cdots+x_{m})^{n}\,,
  18. k 1 + k 2 + + k m = n ( n k 1 , k 2 , , k m ) = m n . \sum_{k_{1}+k_{2}+\cdots+k_{m}=n}{n\choose k_{1},k_{2},\ldots,k_{m}}=m^{n}\,.
  19. # n , m = ( n + m - 1 m - 1 ) . \#_{n,m}={n+m-1\choose m-1}\,.
  20. n m k i n m , i = 1 m k i = n , \left\lfloor\frac{n}{m}\right\rfloor\leq k_{i}\leq\left\lceil\frac{n}{m}\right% \rceil,\ \sum_{i=1}^{m}{k_{i}}=n,
  21. ( N n 1 ) N\choose n_{1}
  22. ( N - n 1 n 2 ) N-n_{1}\choose n_{2}
  23. ( N - n 1 - n 2 n 3 ) N-n_{1}-n_{2}\choose n_{3}
  24. ( N n 1 ) ( N - n 1 n 2 ) ( N - n 1 - n 2 n 3 ) = N ! ( N - n 1 ) ! n 1 ! ( N - n 1 ) ! ( N - n 1 - n 2 ) ! n 2 ! ( N - n 1 - n 2 ) ! ( N - n 1 - n 2 - n 3 ) ! n 3 ! . {N\choose n_{1}}{N-n_{1}\choose n_{2}}{N-n_{1}-n_{2}\choose n_{3}}...=\frac{N!% }{(N-n_{1})!n_{1}!}\frac{(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}\frac{(N-n_{1}-n_{% 2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}....
  25. ( 11 1 , 4 , 4 , 2 ) = 11 ! 1 ! 4 ! 4 ! 2 ! = 34650. {11\choose 1,4,4,2}=\frac{11!}{1!\,4!\,4!\,2!}=34650.

Multipactor_effect.html

  1. d d
  2. ω \omega
  3. V 0 V_{0}
  4. E 0 E_{0}
  5. V 0 V_{0}
  6. d d
  7. a ( t ) = F ( t ) m a(t)=\frac{F(t)}{m}
  8. x ¨ ( t ) = q E 0 m sin ( ω t ) \ddot{x}(t)=\frac{qE_{0}}{m}~{}\sin(\omega t)
  9. x ( t ) = - q E 0 m ω 2 sin ( ω t ) + q E 0 m ω t - d 2 x(t)=-\frac{qE_{0}}{m\omega^{2}}\sin(\omega t)+\frac{qE_{0}}{m\omega}t-\frac{d% }{2}
  10. t 1 2 = π ω t_{\frac{1}{2}}=\frac{\pi}{\omega}
  11. x ( t ) x(t)
  12. x ( t 1 2 ) = - q E 0 m ω 2 sin ( ω t 1 2 ) + q E 0 m ω t 1 2 - d 2 x(t_{\frac{1}{2}})=-\frac{qE_{0}}{m\omega^{2}}\sin(\omega t_{\frac{1}{2}})+% \frac{qE_{0}}{m\omega}t_{\frac{1}{2}}-\frac{d}{2}
  13. d 2 = - q E 0 m ω 2 sin ( ω π ω ) + q E 0 m ω π ω - d 2 \frac{d}{2}=-\frac{qE_{0}}{m\omega^{2}}\sin(\omega\frac{\pi}{\omega})+\frac{qE% _{0}}{m\omega}\frac{\pi}{\omega}-\frac{d}{2}
  14. f f
  15. f d = 1 2 π q V 0 m fd=\frac{1}{2\sqrt{\pi}}\sqrt{\frac{qV_{0}}{m}}
  16. f d fd

Multiple_choice.html

  1. 2 x + 3 = 4 2x+3=4

Multiplicative_order.html

  1. 4 0 = 1 = 0 × 7 + 1 1 ( mod 7 ) 4 1 = 4 = 0 × 7 + 4 4 ( mod 7 ) 4 2 = 16 = 2 × 7 + 2 2 ( mod 7 ) 4 3 = 64 = 9 × 7 + 1 1 ( mod 7 ) 4 4 = 256 = 36 × 7 + 4 4 ( mod 7 ) 4 5 = 1024 = 146 × 7 + 2 2 ( mod 7 ) \begin{array}[]{llll}4^{0}&=1&=0\times 7+1&\equiv 1\;\;(\mathop{{\rm mod}}7)\\ 4^{1}&=4&=0\times 7+4&\equiv 4\;\;(\mathop{{\rm mod}}7)\\ 4^{2}&=16&=2\times 7+2&\equiv 2\;\;(\mathop{{\rm mod}}7)\\ 4^{3}&=64&=9\times 7+1&\equiv 1\;\;(\mathop{{\rm mod}}7)\\ 4^{4}&=256&=36\times 7+4&\equiv 4\;\;(\mathop{{\rm mod}}7)\\ 4^{5}&=1024&=146\times 7+2&\equiv 2\;\;(\mathop{{\rm mod}}7)\\ \end{array}
  2. …etc… \,\text{...etc...}

Multiply_perfect_number.html

  1. o ( X ϵ ) o(X^{\epsilon})

Multiply–accumulate_operation.html

  1. a a + ( b × c ) \ a\leftarrow a+(b\times c)
  2. x < s u p > 2 y 2 x<sup>2−y^{2}

Multistage_rocket.html

  1. I sp I_{\mathrm{sp}}
  2. T / m e g 0 \ T/m_{\mathrm{e}}g_{\mathrm{0}}
  3. T = - I sp g 0 × d m d t T=-I_{\mathrm{sp}}g_{\mathrm{0}}\times\frac{dm}{dt}
  4. T W R = T m g 0 TWR=\frac{T}{mg_{\mathrm{0}}}
  5. I sp g 0 T × ( m 0 - m f ) \frac{I_{\mathrm{sp}}g_{\mathrm{0}}}{T}\times(m_{\mathrm{0}}-m_{\mathrm{f}})
  6. m 0 m_{\mathrm{0}}
  7. m f m_{\mathrm{f}}
  8. h bo = I sp g 0 m e × ( m f ln ( m f / m 0 ) + m 0 - m f ) h_{\mathrm{bo}}=\frac{I_{\mathrm{sp}}g_{\mathrm{0}}}{m_{\mathrm{e}}}\times(m_{% \mathrm{f}}~{}\mathrm{ln}(m_{\mathrm{f}}/m_{\mathrm{0}})+m_{\mathrm{0}}-m_{% \mathrm{f}})
  9. v bo = I sp g 0 m 0 m f - g 0 m e ( m 0 - m f ) v_{\mathrm{bo}}=\frac{I_{\mathrm{sp}}g_{\mathrm{0}}m_{\mathrm{0}}}{m_{\mathrm{% f}}}-\frac{g_{\mathrm{0}}}{m_{\mathrm{e}}}(m_{\mathrm{0}}-m_{\mathrm{f}})
  10. η = m E + m p + m PL m E + m PL \eta=\frac{m_{\mathrm{E}}+m_{\mathrm{p}}+m_{\mathrm{PL}}}{m_{\mathrm{E}}+m_{% \mathrm{PL}}}
  11. m E m_{\mathrm{E}}
  12. m p m_{\mathrm{p}}
  13. m PL m_{\mathrm{PL}}
  14. ϵ = m E m E + m P \epsilon=\frac{m_{\mathrm{E}}}{m_{\mathrm{E}}+m_{\mathrm{P}}}
  15. λ = m PL m E + m P \lambda=\frac{m_{\mathrm{PL}}}{m_{\mathrm{E}}+m_{\mathrm{P}}}
  16. η = 1 + λ ϵ + λ \eta=\frac{1+\lambda}{\epsilon+\lambda}
  17. m p = I tot / ( g * I sp ) m_{\mathrm{p}}=I_{\mathrm{tot}}/(g*I_{\mathrm{sp}})
  18. m ox / m fuel m_{\mathrm{ox}}/m_{\mathrm{fuel}}
  19. m ox m_{\mathrm{ox}}
  20. m fuel m_{\mathrm{fuel}}
  21. i = 1 n \prod_{i=1}^{n}
  22. Δ v = i = 1 n V e ln ( M initial M final ) \Delta v=\sum_{i=1}^{n}V_{\mathrm{e}}\cdot\ln\left(\frac{M_{\mathrm{initial}}}% {M_{\mathrm{final}}}\right)
  23. V e V_{\mathrm{e}}
  24. M initial M_{\mathrm{initial}}
  25. M final M_{\mathrm{final}}
  26. V e V_{\mathrm{e}}
  27. Δ v = n V e ln ( M ratio ) \Delta v=nV_{\mathrm{e}}\cdot\ln(M_{\mathrm{ratio}})

Multivariable_calculus.html

  1. f ( x , y ) = x 2 y x 4 + y 2 f(x,y)=\frac{x^{2}y}{x^{4}+y^{2}}
  2. y = x 2 y=x^{2}
  3. f ( x , y ) f(x,y)
  4. f f
  5. x x
  6. y y
  7. f f
  8. y y
  9. x x
  10. f f
  11. f ( x , y ) = { y x - y if 1 x > y 0 x y - x if 1 y > x 0 1 - x if x = y > 0 0 else . f(x,y)=\begin{cases}\frac{y}{x}-y&\,\text{if }1\geq x>y\geq 0\\ \frac{x}{y}-x&\,\text{if }1\geq y>x\geq 0\\ 1-x&\,\text{if }x=y>0\\ 0&\,\text{else}.\end{cases}
  12. f y ( x ) := f ( x , y ) f_{y}(x):=f(x,y)
  13. x x
  14. y y
  15. f x f_{x}
  16. f f
  17. x x
  18. y y
  19. f f
  20. f ( 1 n , 1 n ) f\left(\frac{1}{n},\frac{1}{n}\right)
  21. n n
  22. f ( 0 , 0 ) = 0 f(0,0)=0
  23. f f
  24. lim n f ( 1 n , 1 n ) = 1. \lim_{n\to\infty}f\left(\frac{1}{n},\frac{1}{n}\right)=1.
  25. \nabla
  26. f : n f:\mathbb{R}\to\mathbb{R}^{n}
  27. f : 2 n f:\mathbb{R}^{2}\to\mathbb{R}^{n}
  28. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  29. f : m n f:\mathbb{R}^{m}\to\mathbb{R}^{n}

Mutation_(genetic_algorithm).html

  1. 1 l \frac{1}{l}
  2. l l
  3. 1 1

Mutual_information.html

  1. I ( X ; Y ) = y Y x X p ( x , y ) log ( p ( x , y ) p ( x ) p ( y ) ) , I(X;Y)=\sum_{y\in Y}\sum_{x\in X}p(x,y)\log{\left(\frac{p(x,y)}{p(x)\,p(y)}% \right)},\,\!
  2. p ( x ) p(x)
  3. p ( y ) p(y)
  4. I ( X ; Y ) = Y X p ( x , y ) log ( p ( x , y ) p ( x ) p ( y ) ) d x d y , I(X;Y)=\int_{Y}\int_{X}p(x,y)\log{\left(\frac{p(x,y)}{p(x)\,p(y)}\right)}\;dx% \,dy,
  5. p ( x ) p(x)
  6. p ( y ) p(y)
  7. log ( p ( x , y ) p ( x ) p ( y ) ) = log 1 = 0. \log{\left(\frac{p(x,y)}{p(x)\,p(y)}\right)}=\log 1=0.\,\!
  8. I ( X ; Y ) = H ( X ) - H ( X | Y ) = H ( Y ) - H ( Y | X ) = H ( X ) + H ( Y ) - H ( X , Y ) = H ( X , Y ) - H ( X | Y ) - H ( Y | X ) \begin{aligned}\displaystyle I(X;Y)&\displaystyle{}=H(X)-H(X|Y)\\ &\displaystyle{}=H(Y)-H(Y|X)\\ &\displaystyle{}=H(X)+H(Y)-H(X,Y)\\ &\displaystyle{}=H(X,Y)-H(X|Y)-H(Y|X)\end{aligned}
  9. H ( X ) \ H(X)
  10. H ( Y ) \ H(Y)
  11. H ( X ) H ( X | Y ) \ H(X)\geq H(X|Y)
  12. I ( X ; Y ) \displaystyle I(X;Y)
  13. I ( X ; Y ) = D KL ( p ( x , y ) p ( x ) p ( y ) ) . I(X;Y)=D_{\mathrm{KL}}(p(x,y)\|p(x)p(y)).
  14. I ( X ; Y ) = y p ( y ) x p ( x | y ) log 2 p ( x | y ) p ( x ) = y p ( y ) D KL ( p ( x | y ) p ( x ) ) = 𝔼 Y { D KL ( p ( x | y ) p ( x ) ) } . \begin{aligned}\displaystyle I(X;Y)&\displaystyle{}=\sum_{y}p(y)\sum_{x}p(x|y)% \log_{2}\frac{p(x|y)}{p(x)}\\ &\displaystyle{}=\sum_{y}p(y)\;D_{\mathrm{KL}}(p(x|y)\|p(x))\\ &\displaystyle{}=\mathbb{E}_{Y}\{D_{\mathrm{KL}}(p(x|y)\|p(x))\}.\end{aligned}
  15. D KL ( p ( x | y ) p ( x ) ) D_{\mathrm{KL}}(p(x|y)\|p(x))
  16. d ( X , Y ) = H ( X , Y ) - I ( X ; Y ) = H ( X ) + H ( Y ) - 2 I ( X ; Y ) = H ( X | Y ) + H ( Y | X ) d(X,Y)=H(X,Y)-I(X;Y)=H(X)+H(Y)-2I(X;Y)=H(X|Y)+H(Y|X)
  17. X , Y X,Y
  18. 0 d ( X , Y ) H ( X , Y ) 0\leq d(X,Y)\leq H(X,Y)
  19. D ( X , Y ) = d ( X , Y ) / H ( X , Y ) 1. D(X,Y)=d(X,Y)/H(X,Y)\leq 1.
  20. D ( X , Y ) = 1 - I ( X ; Y ) / H ( X , Y ) . D(X,Y)=1-I(X;Y)/H(X,Y).
  21. D ( X , Y ) = 1 - I ( X ; Y ) max ( H ( X ) , H ( Y ) ) D^{\prime}(X,Y)=1-\frac{I(X;Y)}{\max(H(X),H(Y))}
  22. I ( X ; Y | Z ) = 𝔼 Z ( I ( X ; Y ) | Z ) = z Z y Y x X p Z ( z ) p X , Y | Z ( x , y | z ) log p X , Y | Z ( x , y | z ) p X | Z ( x | z ) p Y | Z ( y | z ) , I(X;Y|Z)=\mathbb{E}_{Z}\big(I(X;Y)|Z\big)=\sum_{z\in Z}\sum_{y\in Y}\sum_{x\in X% }p_{Z}(z)p_{X,Y|Z}(x,y|z)\log\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)},
  23. I ( X ; Y | Z ) = z Z y Y x X p X , Y , Z ( x , y , z ) log p Z ( z ) p X , Y , Z ( x , y , z ) p X , Z ( x , z ) p Y , Z ( y , z ) . I(X;Y|Z)=\sum_{z\in Z}\sum_{y\in Y}\sum_{x\in X}p_{X,Y,Z}(x,y,z)\log\frac{p_{Z% }(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}.
  24. I ( X ; Y | Z ) 0 I(X;Y|Z)\geq 0
  25. I ( X 1 ; X 1 ) = H ( X 1 ) I(X_{1};X_{1})=H(X_{1})
  26. n > 1 , n>1,
  27. I ( X 1 ; ; X n ) = I ( X 1 ; ; X n - 1 ) - I ( X 1 ; ; X n - 1 | X n ) , I(X_{1};\,...\,;X_{n})=I(X_{1};\,...\,;X_{n-1})-I(X_{1};\,...\,;X_{n-1}|X_{n}),
  28. I ( X 1 ; ; X n - 1 | X n ) = 𝔼 X n ( I ( X 1 ; ; X n - 1 ) | X n ) . I(X_{1};\,...\,;X_{n-1}|X_{n})=\mathbb{E}_{X_{n}}\big(I(X_{1};\,...\,;X_{n-1})% |X_{n}\big).
  29. n 3. n\geq 3.
  30. C X Y = I ( X ; Y ) H ( Y ) and C Y X = I ( X ; Y ) H ( X ) . C_{XY}=\frac{I(X;Y)}{H(Y)}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}C_{YX}=\frac{I(% X;Y)}{H(X)}.
  31. R = I ( X ; Y ) H ( X ) + H ( Y ) R=\frac{I(X;Y)}{H(X)+H(Y)}
  32. R max = min ( H ( X ) , H ( Y ) ) H ( X ) + H ( Y ) R_{\max}=\frac{\min(H(X),H(Y))}{H(X)+H(Y)}
  33. U ( X , Y ) = 2 R = 2 I ( X ; Y ) H ( X ) + H ( Y ) U(X,Y)=2R=2\frac{I(X;Y)}{H(X)+H(Y)}
  34. I ( X ; Y ) min [ H ( X ) , H ( Y ) ] \frac{I(X;Y)}{\min\left[H(X),H(Y)\right]}
  35. I ( X ; Y ) H ( X , Y ) . \frac{I(X;Y)}{H(X,Y)}\;.
  36. I ( X ; Y ) H ( X ) H ( Y ) . \frac{I(X;Y)}{\sqrt{H(X)H(Y)}}\;.
  37. I ( X ; Y ) = y Y x X p ( x , y ) log p ( x , y ) p ( x ) p ( y ) , I(X;Y)=\sum_{y\in Y}\sum_{x\in X}p(x,y)\log\frac{p(x,y)}{p(x)\,p(y)},
  38. ( x , y ) (x,y)
  39. p ( x , y ) p(x,y)
  40. { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) } \{(1,1),(2,2),(3,3)\}
  41. { ( 1 , 3 ) , ( 2 , 1 ) , ( 3 , 2 ) } \{(1,3),(2,1),(3,2)\}
  42. I ( X ; Y ) = y Y x X w ( x , y ) p ( x , y ) log p ( x , y ) p ( x ) p ( y ) , I(X;Y)=\sum_{y\in Y}\sum_{x\in X}w(x,y)p(x,y)\log\frac{p(x,y)}{p(x)\,p(y)},
  43. w ( x , y ) w(x,y)
  44. p ( x , y ) p(x,y)
  45. w ( 1 , 1 ) w(1,1)
  46. w ( 2 , 2 ) w(2,2)
  47. w ( 3 , 3 ) w(3,3)
  48. { ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3 ) } \{(1,1),(2,2),(3,3)\}
  49. { ( 1 , 3 ) , ( 2 , 1 ) , ( 3 , 2 ) } \{(1,3),(2,1),(3,2)\}
  50. I K ( X ; Y ) = K ( X ) - K ( X | Y ) . I_{K}(X;Y)=K(X)-K(X|Y).
  51. I K ( X ; Y ) I K ( Y ; X ) I_{K}(X;Y)\approx I_{K}(Y;X)
  52. i , j p i j log p i j p i p j \sum_{i,j}p_{ij}\log\frac{p_{ij}}{p_{i}p_{j}}
  53. p X = 1 , Y = 1 - p X = 1 p Y = 1 p_{X=1,Y=1}-p_{X=1}p_{Y=1}

Myriagon.html

  1. A = 2500 a 2 cot π 10000 A=2500a^{2}\cot\frac{\pi}{10000}

Nabla_symbol.html

  1. \nabla
  2. \nabla
  3. \nabla
  4. \nabla
  5. \nabla

Nagell–Lutz_theorem.html

  1. y 2 = x 3 + a x 2 + b x + c y^{2}=x^{3}+ax^{2}+bx+c
  2. D = - 4 a 3 c + a 2 b 2 + 18 a b c - 4 b 3 - 27 c 2 . D=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}.
  3. y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}

Naive_Bayes_spam_filtering.html

  1. Pr ( S | W ) = Pr ( W | S ) Pr ( S ) Pr ( W | S ) Pr ( S ) + Pr ( W | H ) Pr ( H ) \Pr(S|W)=\frac{\Pr(W|S)\cdot\Pr(S)}{\Pr(W|S)\cdot\Pr(S)+\Pr(W|H)\cdot\Pr(H)}
  2. Pr ( S | W ) \Pr(S|W)
  3. Pr ( S ) \Pr(S)
  4. Pr ( W | S ) \Pr(W|S)
  5. Pr ( H ) \Pr(H)
  6. Pr ( W | H ) \Pr(W|H)
  7. Pr ( S ) = 0.8 ; Pr ( H ) = 0.2 \Pr(S)=0.8;\Pr(H)=0.2
  8. Pr ( S ) = 0.5 ; Pr ( H ) = 0.5 \Pr(S)=0.5;\Pr(H)=0.5
  9. Pr ( S | W ) = Pr ( W | S ) Pr ( W | S ) + Pr ( W | H ) \Pr(S|W)=\frac{\Pr(W|S)}{\Pr(W|S)+\Pr(W|H)}
  10. Pr ( W | S ) \Pr(W|S)
  11. Pr ( W | H ) \Pr(W|H)
  12. p = p 1 p 2 p N p 1 p 2 p N + ( 1 - p 1 ) ( 1 - p 2 ) ( 1 - p N ) p=\frac{p_{1}p_{2}\cdots p_{N}}{p_{1}p_{2}\cdots p_{N}+(1-p_{1})(1-p_{2})% \cdots(1-p_{N})}
  13. p p
  14. p 1 p_{1}
  15. p ( S | W 1 ) p(S|W_{1})
  16. p 2 p_{2}
  17. p ( S | W 2 ) p(S|W_{2})
  18. p N p_{N}
  19. p ( S | W N ) p(S|W_{N})
  20. 1 p - 1 = ( 1 - p 1 ) ( 1 - p 2 ) ( 1 - p N ) p 1 p 2 p N \frac{1}{p}-1=\frac{(1-p_{1})(1-p_{2})\dots(1-p_{N})}{p_{1}p_{2}\dots p_{N}}
  21. ln ( 1 p - 1 ) = i = 1 N [ ln ( 1 - p i ) - ln p i ] \ln\left(\frac{1}{p}-1\right)=\sum_{i=1}^{N}\left[\ln(1-p_{i})-\ln p_{i}\right]
  22. η = i = 1 N [ ln ( 1 - p i ) - ln p i ] \eta=\sum_{i=1}^{N}\left[\ln(1-p_{i})-\ln p_{i}\right]
  23. 1 p - 1 = e η \frac{1}{p}-1=e^{\eta}
  24. p = 1 1 + e η p=\frac{1}{1+e^{\eta}}
  25. Pr ( S | W ) = s Pr ( S ) + n Pr ( S | W ) s + n \Pr^{\prime}(S|W)=\frac{s\cdot\Pr(S)+n\cdot\Pr(S|W)}{s+n}
  26. Pr ( S | W ) \Pr^{\prime}(S|W)
  27. s s
  28. Pr ( S ) \Pr(S)
  29. n n
  30. Pr ( S | W ) \Pr(S|W)
  31. Pr ( S ) \Pr(S)
  32. P r ( S ) Pr(S)
  33. p = C - 1 ( - 2 ln ( p 1 p 2 p N ) , 2 N ) p=C^{-1}(-2\ln(p_{1}p_{2}\cdots p_{N}),2N)\,

Nakhchivan_Autonomous_Republic.html

  1. ( 0.7599 0.8630 0.7011 ) 1 3 = 0.772. (0.7599\cdot 0.8630\cdot 0.7011)^{\frac{1}{3}}=0.772.

Nazca_Plate.html

  1. M w M_{w}

Necklace_polynomial.html

  1. M ( α , n ) M(α,n)
  2. α n = d | n d M ( α , d ) . \alpha^{n}=\sum_{d\,|\,n}d\,M(\alpha,d).
  3. M ( α , n ) = 1 n d | n μ ( n d ) α d M(\alpha,n)={1\over n}\sum_{d\,|\,n}\mu\left({n\over d}\right)\alpha^{d}
  4. μ μ
  5. n n
  6. α α
  7. n n
  8. α α
  9. n n
  10. α α
  11. α α
  12. M ( 1 , n ) \displaystyle M(1,n)

Needham–Schroeder_protocol.html

  1. A S : A , B , N A A\rightarrow S:\left.A,B,N_{A}\right.
  2. S A : { N A , K A B , B , { K A B , A } K B S } K A S S\rightarrow A:\{N_{A},K_{AB},B,\{K_{AB},A\}_{K_{BS}}\}_{K_{AS}}
  3. K A B {K_{AB}}
  4. K B S {K_{BS}}
  5. A B : { K A B , A } K B S A\rightarrow B:\{K_{AB},A\}_{K_{BS}}
  6. B A : { N B } K A B B\rightarrow A:\{N_{B}\}_{K_{AB}}
  7. K A B {K_{AB}}
  8. A B : { N B - 1 } K A B A\rightarrow B:\{N_{B}-1\}_{K_{AB}}
  9. { K A B , A } K B S \{K_{AB},A\}_{K_{BS}}
  10. A B : A A\rightarrow B:A
  11. B A : { A , 𝐍 𝐁 } K B S B\rightarrow A:\{A,\mathbf{N_{B}^{\prime}}\}_{K_{BS}}
  12. A S : A , B , N A , { A , 𝐍 𝐁 } K B S A\rightarrow S:\left.A,B,N_{A},\{A,\mathbf{N_{B}^{\prime}}\}_{K_{BS}}\right.
  13. S A : { N A , K A B , B , { K A B , A , 𝐍 𝐁 } K B S } K A S S\rightarrow A:\{N_{A},K_{AB},B,\{K_{AB},A,\mathbf{N_{B}^{\prime}}\}_{K_{BS}}% \}_{K_{AS}}
  14. N B N_{B}^{\prime}
  15. N B N_{B}
  16. { K A B , A } K B S \{K_{AB},A\}_{K_{BS}}
  17. { K A B , A , 𝐍 𝐁 } K B S \{K_{AB},A,\mathbf{N_{B}^{\prime}}\}_{K_{BS}}
  18. K B S K_{BS}
  19. A S : A , B A\rightarrow S:\left.A,B\right.
  20. S A : { K P B , B } K S S S\rightarrow A:\{K_{PB},B\}_{K_{SS}}
  21. B S : B , A B\rightarrow S:\left.B,A\right.
  22. S B : { K P A , A } K S S S\rightarrow B:\{K_{PA},A\}_{K_{SS}}
  23. A B : { N A , A } K P B A\rightarrow B:\{N_{A},A\}_{K_{PB}}
  24. B A : { N A , N B } K P A B\rightarrow A:\{N_{A},N_{B}\}_{K_{PA}}
  25. A B : { N B } K P B A\rightarrow B:\{N_{B}\}_{K_{PB}}
  26. A I : { N A , A } K P I A\rightarrow I:\{N_{A},A\}_{K_{PI}}
  27. I B : { N A , A } K P B I\rightarrow B:\{N_{A},A\}_{K_{PB}}
  28. B I : { N A , N B } K P A B\rightarrow I:\{N_{A},N_{B}\}_{K_{PA}}
  29. I A : { N A , N B } K P A I\rightarrow A:\{N_{A},N_{B}\}_{K_{PA}}
  30. A I : { N B } K P I A\rightarrow I:\{N_{B}\}_{K_{PI}}
  31. I B : { N B } K P B I\rightarrow B:\{N_{B}\}_{K_{PB}}
  32. B A : { N A , N B } K P A B\rightarrow A:\{N_{A},N_{B}\}_{K_{PA}}
  33. B A : { N A , N B , B } K P A B\rightarrow A:\{N_{A},N_{B},B\}_{K_{PA}}

Negation_normal_form.html

  1. ¬ \lnot
  2. \land
  3. \lor
  4. a ( b ¬ c ) a\land(b\lor\lnot c)
  5. ( a b ) ( a ¬ c ) (a\land b)\lor(a\land\lnot c)
  6. ¬ ( x . G ) x . ¬ G \lnot(\forall x.G)\to\exists x.\lnot G
  7. ¬ ( x . G ) x . ¬ G \lnot(\exists x.G)\to\forall x.\lnot G
  8. ¬ ¬ G G \lnot\lnot G\to G
  9. ¬ ( G 1 G 2 ) ( ¬ G 1 ) ( ¬ G 2 ) \lnot(G_{1}\land G_{2})\to(\lnot G_{1})\lor(\lnot G_{2})
  10. ¬ ( G 1 G 2 ) ( ¬ G 1 ) ( ¬ G 2 ) \lnot(G_{1}\lor G_{2})\to(\lnot G_{1})\land(\lnot G_{2})
  11. ( A B ) C (A\vee B)\wedge C
  12. ( A ( ¬ B C ) ¬ C ) D (A\wedge(\lnot B\vee C)\wedge\lnot C)\vee D
  13. A ¬ B A\vee\lnot B
  14. A ¬ B A\wedge\lnot B
  15. A B A\Rightarrow B
  16. ¬ ( A B ) \lnot(A\vee B)
  17. ¬ ( A B ) \lnot(A\wedge B)
  18. ¬ ( A ¬ C ) \lnot(A\vee\lnot C)
  19. ¬ A B \lnot A\vee B
  20. ¬ A ¬ B \lnot A\wedge\lnot B
  21. ¬ A ¬ B \lnot A\vee\lnot B
  22. ¬ A C \lnot A\wedge C

Negative_temperature.html

  1. T = d q rev d S T=\frac{dq_{\mathrm{rev}}}{dS}
  2. E = ϵ i = 1 N σ i = ϵ j E=\epsilon\sum_{i=1}^{N}\sigma_{i}=\epsilon\cdot j
  3. Ω E = ( N ( N + j ) / 2 ) = N ! ( N + j 2 ) ! ( N - j 2 ) ! . \Omega_{E}={\left({{N}\atop{(N+j)/2}}\right)}=\frac{N!}{\left(\frac{N+j}{2}% \right)!\left(\frac{N-j}{2}\right)!}.
  4. S = k B ln Ω E S=k_{B}\ln\Omega_{E}
  5. β = 1 k B δ 2 ϵ [ S ] 2 ϵ \beta=\frac{1}{k_{B}}\frac{\delta_{2\epsilon}[S]}{2\epsilon}
  6. β = 1 2 ϵ ( ln Ω E + ϵ - ln Ω E - ϵ ) \beta=\frac{1}{2\epsilon}(\ln\Omega_{E+\epsilon}-\ln\Omega_{E-\epsilon})
  7. β = 1 2 ϵ ln ( ( N + j - 1 2 ) ! ( N - j + 1 2 ) ! ( N + j + 1 2 ) ! ( N - j - 1 2 ) ! ) \beta=\frac{1}{2\epsilon}\ln\left(\frac{\left(\frac{N+j-1}{2}\right)!\left(% \frac{N-j+1}{2}\right)!}{\left(\frac{N+j+1}{2}\right)!\left(\frac{N-j-1}{2}% \right)!}\right)
  8. β = 1 2 ϵ ln ( N - j + 1 N + j + 1 ) . \beta=\frac{1}{2\epsilon}\ln\left(\frac{N-j+1}{N+j+1}\right).
  9. T ( E ) = 2 ϵ k B [ ln ( ( N + 1 ) ϵ - E ( N + 1 ) ϵ + E ) ] - 1 . T(E)=\frac{2\epsilon}{k_{B}}\left[\ln\left(\frac{(N+1)\epsilon-E}{(N+1)% \epsilon+E}\right)\right]^{-1}.
  10. H = ( h ν - μ ) a a . H=(h\nu-\mu)a^{\dagger}a.\,
  11. ρ = exp ( - β H ) 𝐓𝐫 ( exp ( - β H ) ) . \rho=\frac{\exp(-\beta H)}{\mathbf{Tr}(\exp(-\beta H))}.
  12. H ^ - H ^ \hat{H}\rightarrow-\hat{H}

Negentropy.html

  1. J ( p x ) = S ( ϕ x ) - S ( p x ) J(p_{x})=S(\phi_{x})-S(p_{x})\,
  2. S ( ϕ x ) S(\phi_{x})
  3. p x p_{x}
  4. S ( p x ) S(p_{x})
  5. p x p_{x}
  6. S ( p x ) = - p x ( u ) log p x ( u ) d u S(p_{x})=-\int p_{x}(u)\log p_{x}(u)du
  7. J = S max - S = - Φ = - k ln Z J=S_{\max}-S=-\Phi=-k\ln Z\,
  8. J J
  9. Φ \Phi
  10. Z Z
  11. k k

Neighbor_joining.html

  1. Q Q
  2. i j i\neq j
  3. Q ( i , j ) Q(i,j)
  4. n n
  5. Q Q
  6. Q ( i , j ) = ( n - 2 ) d ( i , j ) - k = 1 n d ( i , k ) - k = 1 n d ( j , k ) Q(i,j)=(n-2)d(i,j)-\sum_{k=1}^{n}d(i,k)-\sum_{k=1}^{n}d(j,k)
  7. d ( i , j ) d(i,j)
  8. i i
  9. j j
  10. δ ( g , u ) = d ( f , g ) - δ ( f , u ) \delta(g,u)=d(f,g)-\delta(f,u)\quad
  11. f f
  12. g g
  13. u u
  14. f f
  15. u u
  16. g g
  17. u u
  18. δ ( f , u ) \delta(f,u)
  19. δ ( g , u ) \delta(g,u)
  20. u u
  21. k k
  22. f f
  23. g g
  24. n n
  25. n - 3 n-3
  26. Q Q
  27. Q Q
  28. n × n n\times n
  29. ( n - 1 ) × ( n - 1 ) (n-1)\times(n-1)
  30. O ( n 3 ) O(n^{3})
  31. ( a , b , c , d , e ) (a,b,c,d,e)
  32. Q Q
  33. Q ( a , b ) = - 50 Q(a,b)=-50
  34. Q Q
  35. a a
  36. b b
  37. u u
  38. a a
  39. b b
  40. u u
  41. δ ( a , u ) = 2 \delta(a,u)=2
  42. δ ( b , u ) = 3 \delta(b,u)=3
  43. u u
  44. a a
  45. b b
  46. d ( u , c ) = 7 d(u,c)=7
  47. d ( u , d ) = 7 d(u,d)=7
  48. d ( u , e ) = 6 d(u,e)=6
  49. u u
  50. c c
  51. d d
  52. e e
  53. Q Q
  54. - 28 -28
  55. u u
  56. c c
  57. v v
  58. δ ( u , v ) = 3 \delta(u,v)=3
  59. δ ( c , v ) = 4 \delta(c,v)=4
  60. v v
  61. d d
  62. e e
  63. Q Q
  64. d d
  65. b b
  66. 2 + 2 + 3 + 3 = 10 2+2+3+3=10

Nerve_of_a_covering.html

  1. j J U j . \bigcap_{j\in J}U_{j}\neq\varnothing.

Nest_algebra.html

  1. n n
  2. n \mathbb{C}^{n}
  3. e 1 , e 2 , , e n e_{1},e_{2},\dots,e_{n}
  4. j = 0 , 1 , 2 , , n j=0,1,2,\dots,n
  5. S j S_{j}
  6. j j
  7. n \mathbb{C}^{n}
  8. j j
  9. e 1 , , e j e_{1},\dots,e_{j}
  10. N = { ( 0 ) = S 0 , S 1 , S 2 , , S n - 1 , S n = n } ; N=\{(0)=S_{0},S_{1},S_{2},\dots,S_{n-1},S_{n}=\mathbb{C}^{n}\};
  11. M S S MS\subseteq S

Net_force.html

  1. F \scriptstyle\vec{F}
  2. F 1 \scriptstyle\vec{F}_{1}
  3. F 2 \scriptstyle\vec{F}_{2}
  4. F \scriptstyle\vec{F}
  5. 𝐅 = 𝐁 - 𝐀 = ( B x - A x , B y - A y , B z - A z ) . \mathbf{F}=\mathbf{B}-\mathbf{A}=(B_{x}-A_{x},B_{y}-A_{y},B_{z}-A_{z}).
  6. | 𝐅 | = ( B x - A x ) 2 + ( B y - A y ) 2 + ( B z - A z ) 2 . |\mathbf{F}|=\sqrt{(B_{x}-A_{x})^{2}+(B_{y}-A_{y})^{2}+(B_{z}-A_{z})^{2}}.
  7. 𝐅 = 𝐅 1 + 𝐅 2 = 𝐁 - 𝐀 + 𝐃 - 𝐀 , \mathbf{F}=\mathbf{F}_{1}+\mathbf{F}_{2}=\mathbf{B}-\mathbf{A}+\mathbf{D}-% \mathbf{A},
  8. 𝐅 = 𝐅 1 + 𝐅 2 = 2 ( 𝐁 + 𝐃 2 - 𝐀 ) = 2 ( 𝐄 - 𝐀 ) , \mathbf{F}=\mathbf{F}_{1}+\mathbf{F}_{2}=2(\frac{\mathbf{B}+\mathbf{D}}{2}-% \mathbf{A})=2(\mathbf{E}-\mathbf{A}),
  9. F \scriptstyle\vec{F}
  10. m \scriptstyle m
  11. a = F m \vec{a}={\vec{F}\over m}
  12. α = τ I \vec{\alpha}={\vec{\tau}\over I}
  13. τ \scriptstyle\vec{\tau}
  14. I \scriptstyle I
  15. F \scriptstyle\vec{F}
  16. τ = r × F \vec{\tau}=\vec{r}\times\vec{F}
  17. τ = F k \ \tau=Fk
  18. r \scriptstyle\vec{r}
  19. k \scriptstyle k
  20. F \scriptstyle\vec{F}
  21. r \scriptstyle\vec{r}
  22. I \scriptstyle I
  23. I = m r 2 / 2 \scriptstyle I=mr^{2}/2
  24. τ τ
  25. r × F R = i = 1 N ( r i × F i ) \vec{r}\times\vec{F}_{R}=\sum_{i=1}^{N}(\vec{r}_{i}\times\vec{F}_{i})
  26. F R \scriptstyle\vec{F}_{R}
  27. r \scriptstyle\vec{r}
  28. F i \scriptstyle\vec{F}_{i}
  29. r i \scriptstyle\vec{r}_{i}
  30. F 1 \scriptstyle\vec{F}_{1}
  31. F 2 \scriptstyle\vec{F}_{2}
  32. F 1 \scriptstyle\vec{F}_{1}
  33. F R \scriptstyle\vec{F}_{R}
  34. F 2 \scriptstyle\vec{F}_{2}
  35. F R \scriptstyle\vec{F}_{R}
  36. τ = F d \scriptstyle\tau=Fd
  37. d \scriptstyle\ d

Network_congestion.html

  1. x i x_{i}
  2. i i
  3. C l C_{l}
  4. l l
  5. r l i r_{li}
  6. i i
  7. l l
  8. x x
  9. c c
  10. R R
  11. U ( x ) U(x)
  12. x x
  13. max x i U ( x i ) \max\limits_{x}\sum_{i}U(x_{i})
  14. R x c Rx\leq c
  15. p l p_{l}
  16. y i = l p l r l i , y_{i}=\sum_{l}p_{l}r_{li},
  17. p l p_{l}
  18. l l

Neutralino.html

  1. χ ~ 1 0 , , χ ~ 4 0 \tilde{\chi}_{1}^{0},\ldots,\tilde{\chi}_{4}^{0}
  2. χ ~ i ± \tilde{\chi}_{i}^{\pm}

Nevanlinna_theory.html

  1. N ( r , f ) = 0 r ( n ( t , f ) - n ( 0 , f ) ) d t t + n ( 0 , f ) log r . N(r,f)=\int\limits_{0}^{r}\left(n(t,f)-n(0,f)\right)\dfrac{dt}{t}+n(0,f)\log r.\,
  2. m ( r , f ) = 1 2 π 0 2 π log + | f ( r e i θ ) | d θ . m(r,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\log^{+}\left|f(re^{i\theta})\right|d% \theta.\,
  3. T ( r , f ) = m ( r , f ) + N ( r , f ) . T(r,f)=m(r,f)+N(r,f).\,
  4. 0 r d t t ( 1 π | z | t | f | 2 ( 1 + | f | 2 ) 2 d m ) = T ( r , f ) + O ( 1 ) , \int_{0}^{r}\frac{dt}{t}\left(\frac{1}{\pi}\int_{|z|\leq t}\frac{|f^{\prime}|^% {2}}{(1+|f|^{2})^{2}}dm\right)=T(r,f)+O(1),\,
  5. log M ( r , f ) = log max | z | r | f ( z ) | \log M(r,f)=\log\max_{|z|\leq r}|f(z)|\,
  6. T ( r , f ) log + M ( r , f ) T(r,f)\leq\log^{+}M(r,f)\,
  7. log M ( r , f ) ( R + r R - r ) T ( R , f ) , \log M(r,f)\leq\left(\dfrac{R+r}{R-r}\right)T(R,f),\,
  8. ρ ( f ) = lim sup r log + T ( r , f ) log r . \rho(f)=\limsup_{r\rightarrow\infty}\dfrac{\log^{+}T(r,f)}{\log r}.
  9. N ( r , a , f ) = N ( r , 1 f - a ) , m ( r , a , f ) = m ( r , 1 f - a ) . \quad N(r,a,f)=N\left(r,\dfrac{1}{f-a}\right),\quad m(r,a,f)=m\left(r,\dfrac{1% }{f-a}\right).\,
  10. T ( r , f ) = N ( r , a , f ) + m ( r , a , f ) + O ( 1 ) , T(r,f)=N(r,a,f)+m(r,a,f)+O(1),\,
  11. T ( r , f g ) T ( r , f ) + T ( r , g ) + O ( 1 ) , T ( r , f + g ) T ( r , f ) + T ( r , g ) + O ( 1 ) , T ( r , 1 / f ) = T ( r , f ) + O ( 1 ) , T ( r , f m ) = m T ( r , f ) + O ( 1 ) , \begin{array}[]{lcl}T(r,fg)&\leq&T(r,f)+T(r,g)+O(1),\\ T(r,f+g)&\leq&T(r,f)+T(r,g)+O(1),\\ T(r,1/f)&=&T(r,f)+O(1),\\ T(r,f^{m})&=&mT(r,f)+O(1),\end{array}
  12. N ¯ \overline{N}
  13. N 1 ( r , f ) = 2 N ( r , f ) - N ( r , f ) + N ( r , 1 f ) = N ( r , f ) + N ¯ ( r , f ) + N ( r , 1 f ) . N_{1}(r,f)=2N(r,f)-N(r,f^{\prime})+N\left(r,\dfrac{1}{f^{\prime}}\right)=N(r,f% )+\overline{N}(r,f)+N\left(r,\dfrac{1}{f^{\prime}}\right).\,
  14. j = 1 k m ( r , a j , f ) 2 T ( r , f ) - N 1 ( r , f ) + S ( r , f ) . \sum_{j=1}^{k}m(r,a_{j},f)\leq 2T(r,f)-N_{1}(r,f)+S(r,f).\,
  15. ( k - 2 ) T ( r , f ) j = 1 k N ¯ ( r , a j , f ) + S ( r , f ) , (k-2)T(r,f)\leq\sum_{j=1}^{k}\overline{N}(r,a_{j},f)+S(r,f),\,
  16. δ ( a , f ) = lim inf r m ( r , a , f ) T ( r , f ) = 1 - lim sup r N ( r , a , f ) T ( r , f ) . \delta(a,f)=\liminf_{r\rightarrow\infty}\frac{m(r,a,f)}{T(r,f)}=1-\limsup_{r% \rightarrow\infty}\dfrac{N(r,a,f)}{T(r,f)}.\,
  17. a δ ( a , f ) 2 , \sum_{a}\delta(a,f)\leq 2,\,
  18. T ( r , f ) 2 T ( r , f ) + S ( r , f ) , T(r,f^{\prime})\leq 2T(r,f)+S(r,f),\,

New_riddle_of_induction.html

  1. x quus y = { x + y for x , y < 57 5 for x 57 or y 57 x\,\text{ quus }y=\begin{cases}x+y&\,\text{for }x,y<57\\ 5&\,\text{for }x\geq 57\,\text{ or }y\geq 57\end{cases}

Newman–Shanks–Williams_prime.html

  1. S 2 m + 1 = ( 1 + 2 ) 2 m + 1 + ( 1 - 2 ) 2 m + 1 2 . S_{2m+1}=\frac{\left(1+\sqrt{2}\right)^{2m+1}+\left(1-\sqrt{2}\right)^{2m+1}}{% 2}.
  2. S 0 = 1 S_{0}=1\,
  3. S 1 = 1 S_{1}=1\,
  4. S n = 2 S n - 1 + S n - 2 for all n 2. S_{n}=2S_{n-1}+S_{n-2}\qquad\,\text{for all }n\geq 2.

Newtonian_fluid.html

  1. τ \tau
  2. v v
  3. v \nabla v
  4. τ \tau
  5. v \nabla v
  6. τ = μ ( v ) \mathbf{\tau}=\mathbf{\mu}(\nabla v)
  7. μ \mu
  8. τ = μ d u d y \tau=\mu\frac{du}{dy}
  9. τ \tau
  10. μ \mu
  11. d u d y \frac{du}{dy}
  12. τ i j = μ ( v i x j + v j x i ) \tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}% {\partial x_{i}}\right)
  13. x j x_{j}
  14. j j
  15. v i v_{i}
  16. i i
  17. τ i j \tau_{ij}
  18. j j
  19. i i
  20. σ \mathbf{\sigma}
  21. p p
  22. σ i j = - p δ i j + μ ( v i x j + v j x i ) \mathbf{\sigma}_{ij}=-p\delta_{ij}+\mu\left(\frac{\partial v_{i}}{\partial x_{% j}}+\frac{\partial v_{j}}{\partial x_{i}}\right)
  23. μ \mu
  24. μ i j \mu_{ij}
  25. d 𝐅 = μ i j 𝐝𝐒 × rot 𝐮 {d}\mathbf{F}{=}\mu_{ij}\,\mathbf{dS}\times\mathrm{rot}\,\mathbf{u}
  26. μ i j \mu_{ij}

NICAM.html

  1. x 9 + x 4 + 1. x^{9}+x^{4}+1.
  2. 111111111. 111111111.

Nicholas_Metropolis.html

  1. e - E / k T e^{-E/kT}
  2. E E
  3. T T
  4. k k

Nick_Bostrom.html

  1. f sim = f p N H ( f p N H ) + H f_{\textrm{sim}}=\frac{f_{\textrm{p}}NH}{(f_{\textrm{p}}NH)+H}
  2. f p f_{\textrm{p}}
  3. N N
  4. f p f_{\textrm{p}}
  5. H H
  6. f sim f_{\textrm{sim}}
  7. f 1 f_{\textrm{1}}
  8. N 1 N_{\textrm{1}}
  9. N = f 1 N=f_{\textrm{1}}
  10. N 1 N_{\textrm{1}}
  11. f sim = f p f 1 N 1 ( f p f 1 N 1 ) + 1 f_{\textrm{sim}}=\frac{f_{\textrm{p}}f_{\textrm{1}}N_{\textrm{1}}}{(f_{\textrm% {p}}f_{\textrm{1}}N_{\textrm{1}})+1}
  12. N 1 N_{\textrm{1}}
  13. f p f_{\textrm{p}}
  14. f 1 f_{\textrm{1}}
  15. f sim f_{\textrm{sim}}

Nicole_Oresme.html

  1. ( 1 / 2 ) log 2 n (1/2)\log_{2}n

Noise_reduction.html

  1. δ i \delta_{i}
  2. i i
  3. [ 0 , 1 ] [0,1]
  4. i i
  5. ( x ( i ) = c | x ( j ) j δ i ) e - β 2 λ j δ i ( c - x ( j ) ) 2 \mathbb{P}(x(i)=c|x(j)\forall j\in\delta i)\propto e^{-\frac{\beta}{2\lambda}% \sum_{j\in\delta i}(c-x(j))^{2}}
  6. β 0 \beta\geq 0
  7. λ \lambda

Nominal_interest_rate.html

  1. r r
  2. R R
  3. i i
  4. ( 1 + r ) = ( 1 + R ) / ( 1 + i ) (1+r)=(1+R)/(1+i)\,
  5. i i
  6. r R - i r\approx R-i\,
  7. r = ( 1 + i / n ) n - 1 r\ =\ (1+i/n)^{n}-1

Nominal_rigidity.html

  1. P f P = θ \frac{P_{f}}{P}=\theta
  2. P = P f a P m 1 - a P=P_{f}^{a}P_{m}^{1-a}
  3. θ {\theta}
  4. P f P f a P m 1 - a = θ \frac{P_{f}}{P_{f}^{a}P_{m}^{1-a}}=\theta
  5. P f 1 - a = P m 1 - a θ P_{f}^{1-a}=P_{m}^{1-a}\theta
  6. P f = P m θ 1 1 - a P_{f}=P_{m}\theta^{\frac{1}{1-a}}
  7. P f = P m θ ( Y ) 1 1 - a P_{f}=P_{m}\theta(Y)^{\frac{1}{1-a}}
  8. θ {\theta}

Non-abelian_gauge_transformation.html

  1. a * b = b * a a*b=b*a\,
  2. t a t^{a}
  3. [ t a , t b ] = t a t b - t b t a = C a b c t c . \left[t^{a},t^{b}\right]=t^{a}t^{b}-t^{b}t^{a}=C^{abc}t^{c}.
  4. C a b c C^{abc}
  5. T r ( t a t b ) = 1 2 δ a b . Tr(t^{a}t^{b})=\frac{1}{2}\delta^{ab}.
  6. ω \omega
  7. ω = e x p ( θ a t a ) \omega=exp(\theta^{a}t^{a})
  8. θ a \theta^{a}
  9. φ ( x ) \varphi(x)
  10. T ( ω ) T(\omega)
  11. φ ( x ) φ ( x ) = T ( ω ) φ ( x ) . \varphi(x)\to\varphi^{\prime}(x)=T(\omega)\varphi(x).
  12. T ( ω ) T(\omega)
  13. \mathcal{L}
  14. φ ( x ) \varphi(x)
  15. μ φ ( x ) \partial_{\mu}\varphi(x)
  16. = ( φ ( x ) , μ φ ( x ) ) . \mathcal{L}=\mathcal{L}\big(\varphi(x),\partial_{\mu}\varphi(x)\big).
  17. ω \omega
  18. μ T ( ω ) φ ( x ) = T ( ω ) μ φ ( x ) . \partial_{\mu}T(\omega)\varphi(x)=T(\omega)\partial_{\mu}\varphi(x).
  19. φ \varphi
  20. ( φ , φ ) (\varphi,\varphi)
  21. ( μ φ , μ φ ) (\partial_{\mu}\varphi,\partial_{\mu}\varphi)
  22. ( φ , μ φ ) (\varphi,\partial_{\mu}\varphi)
  23. ( φ ( x ) , μ φ ( x ) ) = ( T ( ω ) φ , T ( ω ) μ φ ) . \mathcal{L}\big(\varphi(x),\partial_{\mu}\varphi(x)\big)=\mathcal{L}\big(T(% \omega)\varphi,T(\omega)\partial_{\mu}\varphi\big).

Non-analytic_smooth_function.html

  1. f ( x ) = { exp ( - 1 / x ) if x > 0 , 0 if x 0 , f(x)=\begin{cases}\exp(-1/x)&\,\text{if }x>0,\\ 0&\,\text{if }x\leq 0,\end{cases}
  2. f ( n ) ( x ) = { p n ( x ) x 2 n f ( x ) if x > 0 , 0 if x 0 , f^{(n)}(x)=\begin{cases}\displaystyle\frac{p_{n}(x)}{x^{2n}}\,f(x)&\,\text{if % }x>0,\\ 0&\,\text{if }x\leq 0,\end{cases}
  3. p n + 1 ( x ) = x 2 p n ( x ) - ( 2 n x - 1 ) p n ( x ) , n . p_{n+1}(x)=x^{2}p_{n}^{\prime}(x)-(2nx-1)p_{n}(x),\qquad n\in\mathbb{N}.
  4. lim x 0 e - 1 / x x m = 0 , \lim_{x\searrow 0}\frac{e^{-1/x}}{x^{m}}=0,
  5. lim x 0 f ( n ) ( x ) - f ( n ) ( 0 ) x - 0 = lim x 0 p n ( x ) x 2 n + 1 e - 1 / x = 0. \lim_{x\searrow 0}\frac{f^{(n)}(x)-f^{(n)}(0)}{x-0}=\lim_{x\searrow 0}\frac{p_% {n}(x)}{x^{2n+1}}\,e^{-1/x}=0.
  6. 1 x m = x ( 1 x ) m + 1 ( m + 1 ) ! x n = 0 1 n ! ( 1 x ) n = ( m + 1 ) ! x exp ( 1 x ) , x > 0 , \frac{1}{x^{m}}=x\Bigl(\frac{1}{x}\Bigr)^{m+1}\leq(m+1)!\,x\sum_{n=0}^{\infty}% \frac{1}{n!}\Bigl(\frac{1}{x}\Bigr)^{n}=(m+1)!\,x\exp\Bigl(\frac{1}{x}\Bigr),% \qquad x>0,
  7. lim x 0 e - 1 / x x m ( m + 1 ) ! lim x 0 x = 0. \lim_{x\searrow 0}\frac{e^{-1/x}}{x^{m}}\leq(m+1)!\lim_{x\searrow 0}x=0.
  8. f ( n + 1 ) ( x ) = ( p n ( x ) x 2 n - 2 n p n ( x ) x 2 n + 1 + p n ( x ) x 2 n + 2 ) f ( x ) = x 2 p n ( x ) - ( 2 n x - 1 ) p n ( x ) x 2 n + 2 f ( x ) = p n + 1 ( x ) x 2 ( n + 1 ) f ( x ) , \begin{aligned}\displaystyle f^{(n+1)}(x)&\displaystyle=\biggl(\frac{p^{\prime% }_{n}(x)}{x^{2n}}-2n\frac{p_{n}(x)}{x^{2n+1}}+\frac{p_{n}(x)}{x^{2n+2}}\biggr)% f(x)\\ &\displaystyle=\frac{x^{2}p^{\prime}_{n}(x)-(2nx-1)p_{n}(x)}{x^{2n+2}}f(x)\\ &\displaystyle=\frac{p_{n+1}(x)}{x^{2(n+1)}}f(x),\end{aligned}
  9. lim x 0 f ( n ) ( x ) - f ( n ) ( 0 ) x - 0 = lim x 0 p n ( x ) x 2 n + 1 e - 1 / x = 0. \lim_{x\searrow 0}\frac{f^{(n)}(x)-f^{(n)}(0)}{x-0}=\lim_{x\searrow 0}\frac{p_% {n}(x)}{x^{2n+1}}\,e^{-1/x}=0.
  10. n = 0 f ( n ) ( 0 ) n ! x n = n = 0 0 n ! x n = 0 , x , \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}=\sum_{n=0}^{\infty}\frac{0}{n!}x% ^{n}=0,\qquad x\in\mathbb{R},
  11. { 0 } z exp ( - 1 / z ) , \mathbb{C}\setminus\{0\}\ni z\mapsto\exp(-1/z)\in\mathbb{C},
  12. F ( x ) := k A e - k cos ( k x ) . F(x):=\sum_{k\in A}e^{-\sqrt{k}}\cos(kx)\ .
  13. k A e - k k n \sum_{k\in A}e^{-\sqrt{k}}k^{n}
  14. F ( n ) ( x ) := k A e - k k n cos ( k x ) = k A k > q e - k k n + k A k q e - k k n cos ( k x ) e - n n n + O ( q n ) ( as n ) F^{(n)}(x):=\sum_{k\in A}e^{-\sqrt{k}}k^{n}\cos(kx)=\sum_{k\in A\atop k>q}e^{-% \sqrt{k}}k^{n}+\sum_{k\in A\atop k\leq q}e^{-\sqrt{k}}k^{n}\cos(kx)\geq e^{-% \sqrt{n}}n^{n}+O(q^{n})\quad(\mathrm{as}\;n\to\infty)
  15. lim sup n ( | F ( n ) ( x ) | n ! ) 1 / n = + , \limsup_{n\to\infty}\left(\frac{|F^{(n)}(x)|}{n!}\right)^{1/n}=+\infty\,,
  16. g ( x ) = f ( x ) f ( x ) + f ( 1 - x ) , x , g(x)=\frac{f(x)}{f(x)+f(1-x)},\qquad x\in\mathbb{R},
  17. f n ( k ) ( 0 ) = { α n if k = n , 0 otherwise, k , n 0 . f_{n}^{(k)}(0)=\begin{cases}\alpha_{n}&\,\text{if }k=n,\\ 0&\,\text{otherwise,}\end{cases}\qquad k,n\in\mathbb{N}_{0}.
  18. F ( x ) = n = 0 f n ( x ) , x , F(x)=\sum_{n=0}^{\infty}f_{n}(x),\qquad x\in\mathbb{R},
  19. n = 0 f n ( k ) n = 0 k + 1 | α n | n ! λ n n - k ψ n ( k ) + n = k + 2 1 n ! 1 λ n n - k - 2 1 | α n | λ n 1 ψ n ( k ) λ n 1 < , \sum_{n=0}^{\infty}\|f_{n}^{(k)}\|_{\infty}\leq\sum_{n=0}^{k+1}\frac{|\alpha_{% n}|}{n!\,\lambda_{n}^{n-k}}\|\psi_{n}^{(k)}\|_{\infty}+\sum_{n=k+2}^{\infty}% \frac{1}{n!}\underbrace{\frac{1}{\lambda_{n}^{n-k-2}}}_{\leq\,1}\underbrace{% \frac{|\alpha_{n}|}{\lambda_{n}}}_{\leq\,1}\underbrace{\frac{\|\psi_{n}^{(k)}% \|_{\infty}}{\lambda_{n}}}_{\leq\,1}<\infty,
  20. n x Ψ r ( x ) = f ( r 2 - x 2 ) \mathbb{R}^{n}\ni x\mapsto\Psi_{r}(x)=f(r^{2}-\|x\|^{2})
  21. Ψ r ( 0 ) > 0 \Psi_{r}(0)>0

Non-equilibrium_thermodynamics.html

  1. E i E_{i}
  2. I i I_{i}
  3. I i = S / E i . I_{i}=\partial{S}/\partial{E_{i}}.
  4. k b M = S - i ( I i E i ) , \ k_{b}M=S-\sum_{i}(I_{i}E_{i}),
  5. k b \ k_{b}
  6. k b d M = i ( E i d I i ) . \ k_{b}\,dM=\sum_{i}(E_{i}\,dI_{i}).
  7. d S = 1 T d U + p T d V - i = 1 s μ i T d N i dS=\frac{1}{T}dU+\frac{p}{T}dV-\sum_{i=1}^{s}\frac{\mu_{i}}{T}dN_{i}
  8. d S dS
  9. T T
  10. p p
  11. i t h i^{th}
  12. μ i \mu_{i}
  13. U U
  14. V V
  15. i t h i^{th}
  16. N i N_{i}
  17. U U
  18. V V
  19. N i N_{i}
  20. T T
  21. p p
  22. μ i \mu_{i}
  23. J i J_{i}
  24. J i J_{i}
  25. F i F_{i}
  26. ( σ ) (\sigma)
  27. σ = i J i F i x i \sigma=\sum_{i}J_{i}\frac{\partial F_{i}}{\partial x_{i}}
  28. L L
  29. J i = j L i j F j x j J_{i}=\sum_{j}L_{ij}\frac{\partial F_{j}}{\partial x_{j}}
  30. σ = i , j L i j F i x i F j x j \sigma=\sum_{i,j}L_{ij}\frac{\partial F_{i}}{\partial x_{i}}\frac{\partial F_{% j}}{\partial x_{j}}
  31. L L
  32. L L

Non-heart-beating_donation.html

  1. } \bigg\}
  2. } \bigg\}

Non-measurable_set.html

  1. / \mathbb{Q}/\mathbb{Z}
  2. X S X\subset S

Nonagon.html

  1. A = 9 4 a 2 cot π 9 6.18182 a 2 . A=\frac{9}{4}a^{2}\cot\frac{\pi}{9}\simeq 6.18182\,a^{2}.
  2. \scriptstyle\angle{}
  3. \scriptstyle\angle{}

Nonfirstorderizability.html

  1. X ( x , y ( X x X y A x y ) x ¬ X x x y ( X x A x y X y ) ) \exists X(\exists x,y(Xx\land Xy\land Axy)\land\exists x\neg Xx\land\forall x% \,\forall y(Xx\land Axy\rightarrow Xy))
  2. X ( x , y ( X x X y ( y = x + 1 x = y + 1 ) ) x ¬ X x x y ( X x ( y = x + 1 x = y + 1 ) X y ) ) \exists X(\exists x,y(Xx\land Xy\land(y=x+1\lor x=y+1))\land\exists x\neg Xx% \land\forall x\,\forall y(Xx\land(y=x+1\lor x=y+1)\rightarrow Xy))

Nonholonomic_system.html

  1. i = 1 n a s , i d q i + a s , t d t = 0 ( s = 1 , 2 , , k ) \sum_{i=1}^{n}a_{s,i}dq_{i}+a_{s,t}dt=0~{}~{}~{}~{}(s=1,2,...,k)
  2. n n
  3. k k
  4. q i q_{i}
  5. a s , i a_{s,i}
  6. i = 1 n a s , i δ q i = 0 ( s = 1 , 2 , , k ) . \sum_{i=1}^{n}a_{s,i}\delta q_{i}=0~{}~{}~{}~{}(s=1,2,...,k).

Normal_function.html

  1. f ( α ) = α f(\alpha)=\aleph_{\alpha}
  2. f ( α ) = α f(\alpha)=\beth_{\alpha}

Normal_mapping.html

  1. 0.3 2 + 0.4 2 + ( - 0.866 ) 2 = 1 0.3^{2}+0.4^{2}+(-0.866)^{2}=1
  2. t = t × M 3 x 3 × V 3 x 3 t^{\prime}=t\times M_{3x3}\times V_{3x3}
  3. b = b × M 3 x 3 × V 3 x 3 b^{\prime}=b\times M_{3x3}\times V_{3x3}
  4. n = n × ( M 3 x 3 × V 3 x 3 ) - 1 T = n × M 3 x 3 - 1 T × V 3 x 3 - 1 T n^{\prime}=n\times(M_{3x3}\times V_{3x3})^{-1T}=n\times M_{3x3}^{-1T}\times V_% {3x3}^{-1T}

Normal_mode.html

  1. x ¨ \scriptstyle\ddot{x}
  2. m x ¨ 1 = - k x 1 + k ( x 2 - x 1 ) = - 2 k x 1 + k x 2 m\ddot{x}_{1}=-kx_{1}+k(x_{2}-x_{1})=-2kx_{1}+kx_{2}\,\!
  3. m x ¨ 2 = - k x 2 + k ( x 1 - x 2 ) = - 2 k x 2 + k x 1 m\ddot{x}_{2}=-kx_{2}+k(x_{1}-x_{2})=-2kx_{2}+kx_{1}\,\!
  4. x 1 ( t ) = A 1 e i ω t x_{1}(t)=A_{1}e^{i\omega t}\,\!
  5. x 2 ( t ) = A 2 e i ω t x_{2}(t)=A_{2}e^{i\omega t}\,\!
  6. - ω 2 m A 1 e i ω t = - 2 k A 1 e i ω t + k A 2 e i ω t -\omega^{2}mA_{1}e^{i\omega t}=-2kA_{1}e^{i\omega t}+kA_{2}e^{i\omega t}\,\!
  7. - ω 2 m A 2 e i ω t = k A 1 e i ω t - 2 k A 2 e i ω t -\omega^{2}mA_{2}e^{i\omega t}=kA_{1}e^{i\omega t}-2kA_{2}e^{i\omega t}\,\!
  8. ( ω 2 m - 2 k ) A 1 + k A 2 = 0 (\omega^{2}m-2k)A_{1}+kA_{2}=0\,\!
  9. k A 1 + ( ω 2 m - 2 k ) A 2 = 0 kA_{1}+(\omega^{2}m-2k)A_{2}=0\,\!
  10. [ ω 2 m - 2 k k k ω 2 m - 2 k ] ( A 1 A 2 ) = 0 \begin{bmatrix}\omega^{2}m-2k&k\\ k&\omega^{2}m-2k\end{bmatrix}\begin{pmatrix}A_{1}\\ A_{2}\end{pmatrix}=0
  11. ( ω 2 m - 2 k ) 2 - k 2 = 0 (\omega^{2}m-2k)^{2}-k^{2}=0\,\!
  12. ω \omega
  13. ω 1 = k m , \omega_{1}=\sqrt{\frac{k}{m}},
  14. ω 2 = 3 k m . \omega_{2}=\sqrt{\frac{3k}{m}}.
  15. η 1 = ( x 1 1 ( t ) x 2 1 ( t ) ) = c 1 ( 1 1 ) cos ( ω 1 t + φ 1 ) \vec{\eta}_{1}=\begin{pmatrix}x^{1}_{1}(t)\\ x^{1}_{2}(t)\end{pmatrix}=c_{1}\begin{pmatrix}1\\ 1\end{pmatrix}\cos{(\omega_{1}t+\varphi_{1})}
  16. η 2 = ( x 1 2 ( t ) x 2 2 ( t ) ) = c 2 ( 1 - 1 ) cos ( ω 2 t + φ 2 ) \vec{\eta}_{2}=\begin{pmatrix}x^{2}_{1}(t)\\ x^{2}_{2}(t)\end{pmatrix}=c_{2}\begin{pmatrix}1\\ -1\end{pmatrix}\cos{(\omega_{2}t+\varphi_{2})}
  17. Ψ ( t ) = f ( x , y , z ) ( A cos ( ω t ) + B sin ( ω t ) ) \Psi(t)=f(x,y,z)(A\cos(\omega t)+B\sin(\omega t))
  18. E ( v ) = 1 2 h v + h v e h v / k T - 1 E(v)=\frac{1}{2}hv+\frac{hv}{e^{hv/kT}-1}
  19. E ( v ) = k T [ 1 + 1 12 ( h v k T ) 2 + O ( h v k T ) 4 + ] E(v)=kT\left[1+\frac{1}{12}\left(\frac{hv}{kT}\right)^{2}+O\left(\frac{hv}{kT}% \right)^{4}+\cdots\right]
  20. ( S E ) N , V = 1 T \left(\frac{\partial S}{\partial E}\right)_{N,V}=\frac{1}{T}
  21. S ( v ) \displaystyle S\left(v\right)
  22. F ( v ) = E - T S = k T log ( 1 - e - h v k T ) F(v)=E-TS=kT\log\left(1-e^{-\frac{hv}{kT}}\right)
  23. F ( v ) = k T log ( h v k T ) F(v)=kT\log\left(\frac{hv}{kT}\right)
  24. f ( v ) d v = 3 N \int f(v)\,dv=3N
  25. U = f ( v ) E ( v ) d v U=\int f(v)E(v)\,dv
  26. | ψ \ |\psi\rangle
  27. ψ ( x , t ) \ \psi(x,t)
  28. ψ \ \psi
  29. P ( x , t ) = | ψ ( x , t ) | 2 \ P(x,t)=|\psi(x,t)|^{2}
  30. ω = E n / \omega=E_{n}/\hbar
  31. | ψ ( t ) = n | n n | ψ ( t = 0 ) e - i E n t / |\psi(t)\rangle=\sum_{n}|n\rangle\left\langle n|\psi(t=0)\right\rangle e^{-iE_% {n}t/\hbar}

Normal_probability_plot.html

  1. z i = Φ - 1 ( i - a n + 1 - 2 a ) , z_{i}=\Phi^{-1}\left(\frac{i-a}{n+1-2a}\right),
  2. i = 1 , 2 , , n i=1,2,...,n
  3. a = 3 / 8 a=3/8
  4. n 10 n≤ 10
  5. Φ 1 Φ^{−1}
  6. Φ 1 Φ^{−1}

Normalizing_constant.html

  1. p ( x ) = e - x 2 / 2 , x ( - , ) p(x)=e^{-x^{2}/2},x\in(-\infty,\infty)
  2. - p ( x ) d x = - e - x 2 / 2 d x = 2 π , \int_{-\infty}^{\infty}p(x)\,dx=\int_{-\infty}^{\infty}e^{-x^{2}/2}\,dx=\sqrt{% 2\pi\,},
  3. φ ( x ) \varphi(x)
  4. φ ( x ) = 1 2 π p ( x ) = 1 2 π e - x 2 / 2 \varphi(x)=\frac{1}{\sqrt{2\pi\,}}p(x)=\frac{1}{\sqrt{2\pi\,}}e^{-x^{2}/2}
  5. - φ ( x ) d x = - 1 2 π e - x 2 / 2 d x = 1 \int_{-\infty}^{\infty}\varphi(x)\,dx=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2% \pi\,}}e^{-x^{2}/2}\,dx=1
  6. φ ( x ) \varphi(x)
  7. 1 2 π \frac{1}{\sqrt{2\pi\,}}
  8. p ( x ) p(x)
  9. n = 0 λ n n ! = e λ , \sum_{n=0}^{\infty}\frac{\lambda^{n}}{n!}=e^{\lambda},
  10. f ( n ) = λ n e - λ n ! f(n)=\frac{\lambda^{n}e^{-\lambda}}{n!}
  11. P ( H 0 | D ) = P ( D | H 0 ) P ( H 0 ) P ( D ) P(H_{0}|D)=\frac{P(D|H_{0})P(H_{0})}{P(D)}
  12. P ( H 0 | D ) P ( D | H 0 ) P ( H 0 ) . P(H_{0}|D)\propto P(D|H_{0})P(H_{0}).
  13. P ( H 0 | D ) = P ( D | H 0 ) P ( H 0 ) i P ( D | H i ) P ( H i ) . P(H_{0}|D)=\frac{P(D|H_{0})P(H_{0})}{\displaystyle\sum_{i}P(D|H_{i})P(H_{i})}.
  14. P ( D ) = i P ( D | H i ) P ( H i ) P(D)=\sum_{i}P(D|H_{i})P(H_{i})\;
  15. f i , f j = δ i , j \langle f_{i},\,f_{j}\rangle=\,\delta_{i,j}

Nowhere_continuous_function.html

  1. f ( x ) = lim k ( lim j ( cos ( k ! π x ) ) 2 j ) f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)\right)^{2j}\right)

NTRUEncrypt.html

  1. R = [ X ] / ( X N - 1 ) \ R=\mathbb{Z}[X]/(X^{N}-1)
  2. 𝐚 = a 0 + a 1 X + a 2 X 2 + + a N - 2 X N - 2 + a N - 1 X N - 1 \,\textbf{a}=a_{0}+a_{1}X+a_{2}X^{2}+\cdots+a_{N-2}X^{N-2}+a_{N-1}X^{N-1}
  3. N - 1 \ N-1
  4. f , g , m \ \mathcal{L}_{f},\mathcal{L}_{g},\mathcal{L}_{m}
  5. r \ \mathcal{L}_{r}
  6. N - 1 \ N-1
  7. N - 1 \ N-1
  8. X N - 1 \ X^{N}-1
  9. 𝐟 L f \,\textbf{f}\in L_{f}
  10. 𝐟 𝐟 p = 1 ( mod p ) \ \,\textbf{f}\cdot\,\textbf{f}_{p}=1\;\;(\mathop{{\rm mod}}p)
  11. 𝐟 𝐟 q = 1 ( mod q ) \ \,\textbf{f}\cdot\,\textbf{f}_{q}=1\;\;(\mathop{{\rm mod}}q)
  12. 𝐟 p \ \mathbf{f}_{p}
  13. 𝐡 = p 𝐟 q 𝐠 ( mod q ) . \,\textbf{h}=p\,\textbf{f}_{q}\cdot\,\textbf{g}\;\;(\mathop{{\rm mod}}q).
  14. 𝐟 = - 1 + X + X 2 - X 4 + X 6 + X 9 - X 10 \,\textbf{f}=-1+X+X^{2}-X^{4}+X^{6}+X^{9}-X^{10}
  15. 𝐠 = - 1 + X 2 + X 3 + X 5 - X 8 - X 10 \,\textbf{g}=-1+X^{2}+X^{3}+X^{5}-X^{8}-X^{10}
  16. 𝐟 p = 1 + 2 X + 2 X 3 + 2 X 4 + X 5 + 2 X 7 + X 8 + 2 X 9 ( mod 3 ) \,\textbf{f}_{p}=1+2X+2X^{3}+2X^{4}+X^{5}+2X^{7}+X^{8}+2X^{9}\;\;(\mathop{{\rm mod% }}3)
  17. 𝐟 q = 5 + 9 X + 6 X 2 + 16 X 3 + 4 X 4 + 15 X 5 + 16 X 6 + 22 X 7 + 20 X 8 + 18 X 9 + 30 X 10 ( mod 32 ) \,\textbf{f}_{q}=5+9X+6X^{2}+16X^{3}+4X^{4}+15X^{5}+16X^{6}+22X^{7}+20X^{8}+18% X^{9}+30X^{10}\;\;(\mathop{{\rm mod}}32)
  18. 𝐡 = p 𝐟 q 𝐠 ( mod 32 ) = 8 - 7 X - 10 X 2 - 12 X 3 + 12 X 4 - 8 X 5 + 15 X 6 - 13 X 7 + 12 X 8 - 13 X 9 + 16 X 10 ( mod 32 ) \,\textbf{h}=p\,\textbf{f}_{q}\cdot\,\textbf{g}\;\;(\mathop{{\rm mod}}32)=8-7X% -10X^{2}-12X^{3}+12X^{4}-8X^{5}+15X^{6}-13X^{7}+12X^{8}-13X^{9}+16X^{10}\;\;(% \mathop{{\rm mod}}32)
  19. 𝐞 = 𝐫 𝐡 + 𝐦 ( mod q ) \,\textbf{e}=\,\textbf{r}\cdot\,\textbf{h}+\,\textbf{m}\;\;(\mathop{{\rm mod}}q)
  20. 𝐦 = - 1 + X 3 - X 4 - X 8 + X 9 + X 10 \,\textbf{m}=-1+X^{3}-X^{4}-X^{8}+X^{9}+X^{10}
  21. 𝐫 = - 1 + X 2 + X 3 + X 4 - X 5 - X 7 \,\textbf{r}=-1+X^{2}+X^{3}+X^{4}-X^{5}-X^{7}
  22. 𝐞 = 𝐫 𝐡 + 𝐦 ( mod 32 ) = 14 + 11 X + 26 X 2 + 24 X 3 + 14 X 4 + 16 X 5 + 30 X 6 + 7 X 7 + 25 X 8 + 6 X 9 + 19 X 10 ( mod 32 ) \,\textbf{e}=\,\textbf{r}\cdot\,\textbf{h}+\,\textbf{m}\;\;(\mathop{{\rm mod}}% 32)=14+11X+26X^{2}+24X^{3}+14X^{4}+16X^{5}+30X^{6}+7X^{7}+25X^{8}+6X^{9}+19X^{% 10}\;\;(\mathop{{\rm mod}}32)
  23. 𝐚 = 𝐟 𝐞 ( mod q ) \,\textbf{a}=\,\textbf{f}\cdot\,\textbf{e}\;\;(\mathop{{\rm mod}}q)
  24. 𝐚 = 𝐟 𝐞 ( mod q ) \,\textbf{a}=\,\textbf{f}\cdot\,\textbf{e}\;\;(\mathop{{\rm mod}}q)
  25. 𝐚 = 𝐟 ( 𝐫 𝐡 + 𝐦 ) ( mod q ) \,\textbf{a}=\,\textbf{f}\cdot(\,\textbf{r}\cdot\,\textbf{h}+\,\textbf{m})\;\;% (\mathop{{\rm mod}}q)
  26. 𝐚 = 𝐟 ( 𝐫 p 𝐟 q 𝐠 + 𝐦 ) ( mod q ) \,\textbf{a}=\,\textbf{f}\cdot(\,\textbf{r}\cdot p\,\textbf{f}_{q}\cdot\,% \textbf{g}+\,\textbf{m})\;\;(\mathop{{\rm mod}}q)
  27. 𝐚 = p 𝐫 𝐠 + 𝐟 𝐦 ( mod q ) \,\textbf{a}=p\,\textbf{r}\cdot\,\textbf{g}+\,\textbf{f}\cdot\,\textbf{m}\;\;(% \mathop{{\rm mod}}q)
  28. p 𝐫 𝐠 + 𝐟 𝐦 \ p\,\textbf{r}\cdot\,\textbf{g}+\,\textbf{f}\cdot\,\textbf{m}
  29. 𝐛 = 𝐚 ( mod p ) = 𝐟 𝐦 ( mod p ) \,\textbf{b}=\,\textbf{a}\;\;(\mathop{{\rm mod}}p)=\,\textbf{f}\cdot\,\textbf{% m}\;\;(\mathop{{\rm mod}}p)
  30. p 𝐫 𝐠 ( mod p ) = 0 \ p\,\textbf{r}\cdot\,\textbf{g}\;\;(\mathop{{\rm mod}}p)=0
  31. ( 𝐟 p ) \ \left(\,\textbf{f}_{p}\right)
  32. 𝐟 p \ \,\textbf{f}_{p}
  33. 𝐜 = 𝐟 p 𝐛 = 𝐟 p 𝐟 𝐦 ( mod p ) \,\textbf{c}=\,\textbf{f}_{p}\cdot\,\textbf{b}=\,\textbf{f}_{p}\cdot\,\textbf{% f}\cdot\,\textbf{m}\;\;(\mathop{{\rm mod}}p)
  34. 𝐜 = 𝐦 ( mod p ) \,\textbf{c}=\,\textbf{m}\;\;(\mathop{{\rm mod}}p)
  35. 𝐟 𝐟 p = 1 ( mod p ) \ \,\textbf{f}\cdot\,\textbf{f}_{p}=1\;\;(\mathop{{\rm mod}}p)
  36. 𝐟 p \ \,\textbf{f}_{p}
  37. 𝐚 = 𝐟 𝐞 ( mod 32 ) = 3 - 7 X - 10 X 2 - 11 X 3 + 10 X 4 + 7 X 5 + 6 X 6 + 7 X 7 + 5 X 8 - 3 X 9 - 7 X 10 ( mod 32 ) , \,\textbf{a}=\,\textbf{f}\cdot\,\textbf{e}\;\;(\mathop{{\rm mod}}32)=3-7X-10X^% {2}-11X^{3}+10X^{4}+7X^{5}+6X^{6}+7X^{7}+5X^{8}-3X^{9}-7X^{10}\;\;(\mathop{{% \rm mod}}32),
  38. 𝐛 = 𝐚 ( mod 3 ) = - X - X 2 + X 3 + X 4 + X 5 + X 7 - X 8 - X 10 ( mod 3 ) \,\textbf{b}=\,\textbf{a}\;\;(\mathop{{\rm mod}}3)=-X-X^{2}+X^{3}+X^{4}+X^{5}+% X^{7}-X^{8}-X^{10}\;\;(\mathop{{\rm mod}}3)
  39. 𝐛 = 𝐟 𝐦 ( mod 3 ) \ \,\textbf{b}=\,\textbf{f}\cdot\,\textbf{m}\;\;(\mathop{{\rm mod}}3)
  40. 𝐟 p \ \,\textbf{f}_{p}
  41. 𝐜 = 𝐟 p 𝐛 = 𝐟 p 𝐟 𝐦 ( mod 3 ) = 𝐦 ( mod 3 ) \,\textbf{c}=\,\textbf{f}_{p}\cdot\,\textbf{b}=\,\textbf{f}_{p}\cdot\,\textbf{% f}\cdot\,\textbf{m}\;\;(\mathop{{\rm mod}}3)=\,\textbf{m}\;\;(\mathop{{\rm mod% }}3)
  42. 𝐜 = - 1 + X 3 - X 4 - X 8 + X 9 + X 10 \,\textbf{c}=-1+X^{3}-X^{4}-X^{8}+X^{9}+X^{10}
  43. 𝐟 𝐡 ( mod q ) \ \,\textbf{f}^{{}^{\prime}}\cdot\,\textbf{h}\;\;(\mathop{{\rm mod}}q)
  44. 𝐠 𝐡 - 1 ( mod q ) \ \,\textbf{g}^{{}^{\prime}}\cdot\,\textbf{h}^{-1}\;\;(\mathop{{\rm mod}}q)
  45. 𝐟 𝐡 = 𝐠 ( mod q ) \ \,\textbf{f}\cdot\,\textbf{h}=\,\textbf{g}\;\;(\mathop{{\rm mod}}q)
  46. 𝐟 1 \ \,\textbf{f}_{1}
  47. 𝐟 2 \ \,\textbf{f}_{2}
  48. 𝐟 = 𝐟 1 + 𝐟 2 \ \,\textbf{f}=\,\textbf{f}_{1}+\,\textbf{f}_{2}
  49. ( 𝐟 1 + 𝐟 2 ) 𝐡 = 𝐠 ( mod q ) \left(\,\textbf{f}_{1}+\,\textbf{f}_{2}\right)\cdot\,\textbf{h}=\,\textbf{g}\;% \;(\mathop{{\rm mod}}q)
  50. 𝐟 1 𝐡 = 𝐠 - 𝐟 2 𝐡 ( mod q ) \,\textbf{f}_{1}\cdot\,\textbf{h}=\,\textbf{g}-\,\textbf{f}_{2}\cdot\,\textbf{% h}\;\;(\mathop{{\rm mod}}q)
  51. 𝐟 1 \ \,\textbf{f}_{1}
  52. 𝐟 2 \ \,\textbf{f}_{2}
  53. 1 2 N \ \frac{1}{2}N
  54. 𝐟 1 \ \,\textbf{f}_{1}
  55. 1 2 N \ \frac{1}{2}N
  56. 𝐟 2 \ \,\textbf{f}_{2}
  57. 𝐟 1 𝐡 ( mod q ) \,\textbf{f}_{1}\cdot\,\textbf{h}\;\;(\mathop{{\rm mod}}q)
  58. 𝐟 1 \ \,\textbf{f}_{1}
  59. - 𝐟 2 𝐡 ( mod q ) \ -\,\textbf{f}_{2}\cdot\,\textbf{h}\;\;(\mathop{{\rm mod}}q)
  60. 𝐟 1 \ \,\textbf{f}_{1}
  61. 𝐟 2 \ \,\textbf{f}_{2}
  62. 𝐟 1 𝐡 = 𝐠 - 𝐟 2 𝐡 ( mod q ) \ \,\textbf{f}_{1}\cdot\,\textbf{h}=\,\textbf{g}-\,\textbf{f}_{2}\cdot\,% \textbf{h}\;\;(\mathop{{\rm mod}}q)
  63. 𝐞 = c 𝐡 + c \ \,\textbf{e}=c\,\textbf{h}+c
  64. c = 0 ( mod p ) , c < q 2 \ c=0\;\;(\mathop{{\rm mod}}p),c<\frac{q}{2}
  65. 2 c > q 2 \ 2c>\frac{q}{2}
  66. 𝐚 = 𝐟 𝐞 ( mod q ) \ \,\textbf{a}=\,\textbf{f}\cdot\,\textbf{e}\;\;(\mathop{{\rm mod}}q)
  67. 𝐚 = 𝐟 ( c 𝐡 + c ) ( mod q ) \,\textbf{a}=\,\textbf{f}\left(c\,\textbf{h}+c\right)\;\;(\mathop{{\rm mod}}q)
  68. 𝐚 = c 𝐠 + c 𝐟 ( mod q ) \,\textbf{a}=c\,\textbf{g}+c\,\textbf{f}\;\;(\mathop{{\rm mod}}q)
  69. 𝐚 = c 𝐠 + c 𝐟 - q K \,\textbf{a}=c\,\textbf{g}+c\,\textbf{f}-qK
  70. K = k i x i \ K=\sum k_{i}x^{i}
  71. k i = { 1 if the i t h coefficient of 𝐟 and 𝐠 is 1 - 1 if the i t h coefficient of 𝐟 and 𝐠 is - 1 0 Otherwise k_{i}=\begin{cases}1\ \ \qquad\,\text{if the}\ i^{th}\ \,\text{coefficient of}% \ \,\textbf{f}\ \,\text{and}\ \,\textbf{g}\ \,\text{is}\ 1\\ -1\qquad\,\text{if the}\ i^{th}\ \,\text{coefficient of}\ \,\textbf{f}\ \,% \text{and}\ \,\textbf{g}\ \,\text{is}\ -1\\ 0\ \ \qquad\,\text{Otherwise}\end{cases}
  72. 𝐟 = - 1 + X + X 2 - X 4 + X 6 + X 9 - X 10 \,\textbf{f}=-1+X+X^{2}-X^{4}+X^{6}+X^{9}-X^{10}
  73. 𝐠 = - 1 + X 2 + X 3 + X 5 - X 8 - X 10 \,\textbf{g}=-1+X^{2}+X^{3}+X^{5}-X^{8}-X^{10}
  74. K = - 1 + X 2 - X 10 \ K=-1+X^{2}-X^{10}
  75. c 𝐠 + c 𝐟 - q K ( mod p ) \ c\,\textbf{g}+c\,\textbf{f}-qK\;\;(\mathop{{\rm mod}}p)
  76. 𝐟 p \ \,\textbf{f}_{p}
  77. 𝐦 = c 𝐟 p 𝐠 + c 𝐟 p 𝐟 - q 𝐟 p K ( mod p ) \,\textbf{m}=c\,\textbf{f}_{p}\cdot\,\textbf{g}+c\,\textbf{f}_{p}\cdot\,% \textbf{f}-q\,\textbf{f}_{p}\cdot K\;\;(\mathop{{\rm mod}}p)
  78. 𝐦 = c 𝐡 + c - q 𝐟 p K ( mod p ) \,\textbf{m}=c\,\textbf{h}+c-q\,\textbf{f}_{p}\cdot K\;\;(\mathop{{\rm mod}}p)
  79. 𝐦 = - q 𝐟 p K ( mod p ) \,\textbf{m}=-q\,\textbf{f}_{p}\cdot K\;\;(\mathop{{\rm mod}}p)
  80. 𝐟 = - q K 𝐦 - 1 ( mod p ) \ \,\textbf{f}=-qK\cdot\,\textbf{m}^{-1}\;\;(\mathop{{\rm mod}}p)
  81. 𝐟 = 1 + p 𝐅 \ \,\textbf{f}=1+p\,\textbf{F}
  82. 𝐟 - 1 = 1 ( mod p ) \ \,\textbf{f}^{-1}=1\;\;(\mathop{{\rm mod}}p)
  83. 𝐟 p \ \,\textbf{f}_{p}

Object_language.html

  1. S \ S
  2. G \ G
  3. S \ S
  4. G \ G

Oblique_projection.html

  1. ( x , y , z ) (x,y,z)
  2. x y xy
  3. ( x + a z , y + b z , 0 ) (x+az,y+bz,0)
  4. a a
  5. b b
  6. a = b = 0 a=b=0
  7. a a
  8. b b
  9. x x
  10. y y
  11. z z
  12. z z
  13. x y xy
  14. z z
  15. P P
  16. P ( x y z ) = ( x + 0.5 z cos α y + 0.5 z sin α 0 ) P\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}x+0.5\cdot z\cdot\cos\alpha\\ y+0.5\cdot z\cdot\sin\alpha\\ 0\end{pmatrix}
  17. α \alpha
  18. P = [ 1 0 0.5 cos α 0 1 0.5 sin α 0 0 0 ] P=\begin{bmatrix}1&0&0.5\cdot\cos\alpha\\ 0&1&0.5\cdot\sin\alpha\\ 0&0&0\end{bmatrix}

Oblivious_transfer.html

  1. m 0 , m 1 m_{0},m_{1}
  2. d d
  3. N , e N,e
  4. \Rightarrow
  5. N , e N,e
  6. x 0 , x 1 x_{0},x_{1}
  7. \Rightarrow
  8. x 0 , x 1 x_{0},x_{1}
  9. k , b , x b k,b,x_{b}
  10. b { 0 , 1 } b\in\{0,1\}
  11. x b { x 0 , x 1 } x_{b}\in\{x_{0},x_{1}\}
  12. k k
  13. v v
  14. \Leftarrow
  15. v = ( x b + k e ) mod N v=(x_{b}+k^{e})\mod N
  16. k k
  17. x b x_{b}
  18. k 0 = ( v - x 0 ) d mod N k 1 = ( v - x 1 ) d mod N \begin{aligned}\displaystyle k_{0}&\displaystyle=(v-x_{0})^{d}\mod N\\ \displaystyle k_{1}&\displaystyle=(v-x_{1})^{d}\mod N\end{aligned}
  19. k k
  20. m 0 = m 0 + k 0 m 1 = m 1 + k 1 \begin{aligned}\displaystyle m^{\prime}_{0}=m_{0}+k_{0}\\ \displaystyle m^{\prime}_{1}=m_{1}+k_{1}\end{aligned}
  21. \Rightarrow
  22. m 0 , m 1 m^{\prime}_{0},m^{\prime}_{1}
  23. m b = m b - k m_{b}=m^{\prime}_{b}-k
  24. m b m^{\prime}_{b}
  25. x b x_{b}
  26. m 0 , m 1 m_{0},m_{1}
  27. N N
  28. e e
  29. d d
  30. x 0 , x 1 x_{0},x_{1}
  31. b b
  32. x b x_{b}
  33. k k
  34. x b x_{b}
  35. v = ( x b + k e ) mod N v=(x_{b}+k^{e})\mod N
  36. x 0 x_{0}
  37. x 1 x_{1}
  38. k k
  39. k 0 = ( v - x 0 ) d mod N k_{0}=(v-x_{0})^{d}\mod N
  40. k 1 = ( v - x 1 ) d mod N k_{1}=(v-x_{1})^{d}\mod N
  41. k k
  42. k k
  43. m 0 = m 0 + k 0 m^{\prime}_{0}=m_{0}+k_{0}
  44. m 1 = m 1 + k 1 m^{\prime}_{1}=m_{1}+k_{1}
  45. k k
  46. m b = m b - k m_{b}=m^{\prime}_{b}-k

Octagonal_number.html

  1. x n = n 2 + 4 k = 1 n - 1 k = 3 n 2 - 2 n . x_{n}=n^{2}+4\sum_{k=1}^{n-1}k=3n^{2}-2n.
  2. x n , x_{n},
  3. n = 3 x n + 1 + 1 3 . n=\frac{\sqrt{3x_{n}+1}+1}{3}.

Octahedral_number.html

  1. O n O_{n}
  2. O n = n ( 2 n 2 + 1 ) 3 . O_{n}={n(2n^{2}+1)\over 3}.
  3. z ( z + 1 ) 2 ( z - 1 ) 4 = n = 1 O n z n = z + 6 z 2 + 19 z 3 + . \frac{z(z+1)^{2}}{(z-1)^{4}}=\sum_{n=1}^{\infty}O_{n}z^{n}=z+6z^{2}+19z^{3}+\cdots.
  4. O n O_{n}
  5. O n = P n - 1 + P n . O_{n}=P_{n-1}+P_{n}.
  6. O n O_{n}
  7. T n T_{n}
  8. O n + 4 T n - 1 = T 2 n - 1 . O_{n}+4T_{n-1}=T_{2n-1}.
  9. O n = T n + 2 T n - 1 + T n - 2 . O_{n}=T_{n}+2T_{n-1}+T_{n-2}.
  10. O n + 2 T n - 1 = n 3 . O_{n}+2T_{n-1}=n^{3}.
  11. O n - O n - 1 = C 4 , n = n 2 + ( n - 1 ) 2 . O_{n}-O_{n-1}=C_{4,n}=n^{2}+(n-1)^{2}.
  12. O n + O n - 1 = ( 2 n - 1 ) ( 2 n 2 - 2 n + 3 ) 3 O_{n}+O_{n-1}=\frac{(2n-1)(2n^{2}-2n+3)}{3}

Odds_ratio.html

  1. N E , N_{E},
  2. D E D_{E}
  3. H E H_{E}
  4. N N E , N_{NE},
  5. D N E D_{NE}
  6. H N E H_{NE}
  7. N E = D E + H E N_{E}=D_{E}+H_{E}
  8. D E D_{E}
  9. H E H_{E}
  10. D N E D_{NE}
  11. H N E H_{NE}
  12. D E / N E D_{E}/N_{E}
  13. N E = D E + H E N_{E}=D_{E}+H_{E}
  14. D N E / N N E . D_{NE}/N_{NE}.
  15. R R = D E / N E D N E / N N E , RR=\frac{D_{E}/N_{E}}{D_{NE}/N_{NE}}\,,
  16. R R = D E N N E D N E N E = D E / D N E N E / N N E . RR=\frac{D_{E}N_{NE}}{D_{NE}N_{E}}=\frac{D_{E}/D_{NE}}{N_{E}/N_{NE}}.
  17. D E / H E , D_{E}/H_{E}\,,
  18. D N E / H N E . D_{NE}/H_{NE}\,.
  19. O R = D E / H E D N E / H N E , OR=\frac{D_{E}/H_{E}}{D_{NE}/H_{NE}}\,,
  20. O R = D E H N E D N E H E = D E / D N E H E / H N E . OR=\frac{D_{E}H_{NE}}{D_{NE}H_{E}}=\frac{D_{E}/D_{NE}}{H_{E}/H_{NE}}.
  21. D E H E , D_{E}\ll H_{E},
  22. D E + H E H E ; D_{E}+H_{E}\approx H_{E};
  23. D E / ( D E + H E ) D E / H E , D_{E}/(D_{E}+H_{E})\approx D_{E}/H_{E},
  24. N E H E N_{E}\approx H_{E}
  25. N N E H N E , N_{NE}\approx H_{NE},
  26. N E / N N E H E / H N E , N_{E}/N_{NE}\approx H_{E}/H_{NE},
  27. D E / D N E , D_{E}/D_{NE},
  28. H E / H N E , H_{E}/H_{NE},
  29. N E H E N_{E}\approx H_{E}
  30. N N E H N E , N_{NE}\approx H_{NE},
  31. N E / N N E H E / H N E . N_{E}/N_{NE}\approx H_{E}/H_{NE}.
  32. H E / H N E , H_{E}/H_{NE},
  33. N E / N N E , N_{E}/N_{NE},
  34. D E / D N E , D_{E}/D_{NE},
  35. p 1 / ( 1 - p 1 ) p 2 / ( 1 - p 2 ) = p 1 / q 1 p 2 / q 2 = p 1 q 2 p 2 q 1 , {p_{1}/(1-p_{1})\over p_{2}/(1-p_{2})}={p_{1}/q_{1}\over p_{2}/q_{2}}=\frac{\;% p_{1}q_{2}\;}{\;p_{2}q_{1}\;},
  36. p 11 p_{11}
  37. p 10 p_{10}
  38. p 01 p_{01}
  39. p 00 p_{00}
  40. p 11 / ( p 11 + p 10 ) p_{11}/(p_{11}+p_{10})
  41. p 10 / ( p 11 + p 10 ) p_{10}/(p_{11}+p_{10})
  42. p 01 / ( p 01 + p 00 ) p_{01}/(p_{01}+p_{00})
  43. p 00 / ( p 01 + p 00 ) p_{00}/(p_{01}+p_{00})
  44. p 11 / ( p 11 + p 10 ) p 10 / ( p 11 + p 10 ) / p 01 / ( p 01 + p 00 ) p 00 / ( p 01 + p 00 ) = p 11 p 00 p 10 p 01 . {\dfrac{p_{11}/(p_{11}+p_{10})}{p_{10}/(p_{11}+p_{10})}\bigg/\dfrac{p_{01}/(p_% {01}+p_{00})}{p_{00}/(p_{01}+p_{00})}}=\dfrac{p_{11}p_{00}}{p_{10}p_{01}}.
  45. p 11 / ( p 11 + p 01 ) p_{11}/(p_{11}+p_{01})
  46. p 10 / ( p 10 + p 00 ) p_{10}/(p_{10}+p_{00})
  47. p 01 / ( p 11 + p 01 ) p_{01}/(p_{11}+p_{01})
  48. p 00 / ( p 10 + p 00 ) p_{00}/(p_{10}+p_{00})
  49. p 11 / ( p 11 + p 01 ) p 01 / ( p 11 + p 01 ) / p 10 / ( p 10 + p 00 ) p 00 / ( p 10 + p 00 ) = p 11 p 00 p 10 p 01 . {\dfrac{p_{11}/(p_{11}+p_{01})}{p_{01}/(p_{11}+p_{01})}\bigg/\dfrac{p_{10}/(p_% {10}+p_{00})}{p_{00}/(p_{10}+p_{00})}}=\dfrac{p_{11}p_{00}}{p_{10}p_{01}}.
  50. p x p y p_{x}p_{y}
  51. p x ( 1 - p y ) p_{x}(1-p_{y})
  52. ( 1 - p x ) p y (1-p_{x})p_{y}
  53. ( 1 - p x ) ( 1 - p y ) (1-p_{x})(1-p_{y})
  54. p 11 = 1 + ( p 1 + p 1 ) ( R - 1 ) - S 2 ( R - 1 ) p_{11}=\frac{1+(p_{1\cdot}+p_{\cdot 1})(R-1)-S}{2(R-1)}
  55. S = ( 1 + ( p 1 + p 1 ) ( R - 1 ) ) 2 + 4 R ( 1 - R ) p 1 p 1 . S=\sqrt{(1+(p_{1\cdot}+p_{\cdot 1})(R-1))^{2}+4R(1-R)p_{1\cdot}p_{\cdot 1}}.
  56. 0.9 / 0.1 0.2 / 0.8 = 0.9 × 0.8 0.1 × 0.2 = 0.72 0.02 = 36. {0.9/0.1\over 0.2/0.8}=\frac{\;0.9\times 0.8\;}{\;0.1\times 0.2\;}={0.72\over 0% .02}=36.
  57. log ( p 11 p 00 p 01 p 10 ) = log ( p 11 ) + log ( p 00 ) - log ( p 10 ) - log ( p 01 ) . {\log\left(\frac{p_{11}p_{00}}{p_{01}p_{10}}\right)=\log(p_{11})+\log(p_{00}% \big)-\log(p_{10})-\log(p_{01})}.\,
  58. n 11 n_{11}
  59. n 10 n_{10}
  60. n 01 n_{01}
  61. n 00 n_{00}
  62. p ^ 11 \hat{p}_{11}
  63. p ^ 10 \hat{p}_{10}
  64. p ^ 01 \hat{p}_{01}
  65. p ^ 00 \hat{p}_{00}
  66. L = log ( p ^ 11 p ^ 00 p ^ 10 p ^ 01 ) = log ( n 11 n 00 n 10 n 01 ) {L=\log\left(\dfrac{\hat{p}_{11}\hat{p}_{00}}{\hat{p}_{10}\hat{p}_{01}}\right)% =\log\left(\dfrac{n_{11}n_{00}}{n_{10}n_{01}}\right)}
  67. X 𝒩 ( log ( O R ) , σ 2 ) . X\ \sim\ \mathcal{N}(\log(OR),\,\sigma^{2}).\,
  68. SE = 1 n 11 + 1 n 10 + 1 n 01 + 1 n 00 {{\rm SE}=\sqrt{\dfrac{1}{n_{11}}+\dfrac{1}{n_{10}}+\dfrac{1}{n_{01}}+\dfrac{1% }{n_{00}}}}
  69. β ^ x \hat{\beta}_{x}
  70. exp ( β x ) = P ( Y = 1 X = 1 , Z 1 , , Z p ) / P ( Y = 0 X = 1 , Z 1 , , Z p ) P ( Y = 1 X = 0 , Z 1 , , Z p ) / P ( Y = 0 X = 0 , Z 1 , , Z p ) , \exp(\beta_{x})=\frac{P(Y=1\mid X=1,Z_{1},\ldots,Z_{p})/P(Y=0\mid X=1,Z_{1},% \ldots,Z_{p})}{P(Y=1\mid X=0,Z_{1},\ldots,Z_{p})/P(Y=0\mid X=0,Z_{1},\ldots,Z_% {p})},
  71. exp ( β ^ x ) \exp(\hat{\beta}_{x})
  72. exp ( β ^ x ) \exp(\hat{\beta}_{x})
  73. f p 11 / ( p 11 + p 10 ) fp_{11}/(p_{11}+p_{10})
  74. f p 10 ( p 11 + p 10 ) fp_{10}(p_{11}+p_{10})
  75. ( 1 - f ) p 01 / ( p 01 + p 00 ) (1-f)p_{01}/(p_{01}+p_{00})
  76. ( 1 - f ) p 00 / ( p 01 + p 00 ) (1-f)p_{00}/(p_{01}+p_{00})
  77. R R O R 1 - R C + ( R C × O R ) RR\approx\frac{OR}{1-R_{C}+(R_{C}\times OR)}

Okun's_law.html

  1. ( Y - Y ¯ ) / Y ¯ = c ( u - u ¯ ) (Y-\overline{Y})/\overline{Y}=c(u-\overline{u})
  2. Y Y
  3. Y ¯ \overline{Y}
  4. u u
  5. u ¯ \overline{u}
  6. c c
  7. Y ¯ \overline{Y}
  8. u ¯ \overline{u}
  9. Δ Y / Y = k - c Δ u \Delta Y/Y=k-c\Delta u\,
  10. Y Y
  11. c c
  12. Δ Y \Delta Y
  13. Δ u \Delta u
  14. k k
  15. Δ Y / Y = 0.03 - 2 Δ u . \Delta Y/Y=0.03-2\Delta u.\,
  16. ( Y ¯ - Y ) / Y ¯ = 1 - Y / Y ¯ = c ( u - u ¯ ) (\overline{Y}-Y)/\overline{Y}=1-Y/\overline{Y}=c(u-\overline{u})
  17. - 1 + Y / Y ¯ = c ( u ¯ - u ) . -1+Y/\overline{Y}=c(\overline{u}-u).
  18. Δ ( Y / Y ¯ ) = ( Y + Δ Y ) / ( Y ¯ + Δ Y ¯ ) - Y / Y ¯ = c ( Δ u ¯ - Δ u ) . \Delta(Y/\overline{Y})=(Y+\Delta Y)/(\overline{Y}+\Delta\overline{Y})-Y/% \overline{Y}=c(\Delta\overline{u}-\Delta u).
  19. ( Y ¯ Δ Y - Y Δ Y ¯ ) / ( Y ¯ ( Y ¯ + Δ Y ¯ ) ) = c ( Δ u ¯ - Δ u ) . (\overline{Y}\Delta Y-Y\Delta\overline{Y})/(\overline{Y}(\overline{Y}+\Delta% \overline{Y}))=c(\Delta\overline{u}-\Delta u).
  20. ( Y ¯ + Δ Y ¯ ) / Y (\overline{Y}+\Delta\overline{Y})/Y
  21. ( Y ¯ Δ Y - Y Δ Y ¯ ) / ( Y ¯ Y ) = Δ Y / Y - Δ Y ¯ / Y ¯ c ( Δ u ¯ - Δ u ) (\overline{Y}\Delta Y-Y\Delta\overline{Y})/(\overline{Y}Y)=\Delta Y/Y-\Delta% \overline{Y}/\overline{Y}\approx c(\Delta\overline{u}-\Delta u)
  22. Δ Y / Y Δ Y ¯ / Y ¯ + c ( Δ u ¯ - Δ u ) . \Delta Y/Y\approx\Delta\overline{Y}/\overline{Y}+c(\Delta\overline{u}-\Delta u).
  23. Δ u ¯ \Delta\overline{u}
  24. Δ Y ¯ / Y ¯ \Delta\overline{Y}/\overline{Y}
  25. k k
  26. Δ Y / Y k - c Δ u . \Delta Y/Y\approx k-c\Delta u.

Omega_constant.html

  1. Ω e Ω = 1. \Omega\,e^{\Omega}=1.\,
  2. e - Ω = Ω , e^{-\Omega}=\Omega,\,
  3. ln Ω = - Ω . \ln\Omega=-\Omega.\,
  4. Ω n + 1 = e - Ω n . \Omega_{n+1}=e^{-\Omega_{n}}.\,
  5. Ω n + 1 = 1 + Ω n 1 + e Ω n , \Omega_{n+1}=\frac{1+\Omega_{n}}{1+e^{\Omega_{n}}},
  6. f ( x ) = 1 + x 1 + e x , f(x)=\frac{1+x}{1+e^{x}},
  7. Ω = 1 - + d t ( e t - t ) 2 + π 2 - 1. \Omega=\frac{1}{\displaystyle\int_{-\infty}^{+\infty}\frac{\,dt}{(e^{t}-t)^{2}% +\pi^{2}}}-1.
  8. p q = Ω \frac{p}{q}=\Omega
  9. 1 = p e ( p q ) q 1=\frac{pe^{\left(\frac{p}{q}\right)}}{q}
  10. e = ( q p ) ( q p ) = q q p q p e=\left(\frac{q}{p}\right)^{\left(\frac{q}{p}\right)}=\sqrt[p]{\frac{q^{q}}{p^% {q}}}

Omnidirectional_antenna.html

  1. G = e D G=eD
  2. sin b θ / b θ \sin{b\theta}/{b\theta}
  3. D = 10 log 10 ( 101.5 H P B W - 0.00272 ( H P B W ) 2 ) d B . D=10\log_{10}{\left({101.5\over{HPBW-0.00272(HPBW)^{2}}}\right)}\;\;dB.

On-Line_Encyclopedia_of_Integer_Sequences.html

  1. 1 5 , 1 4 , 1 3 , 2 5 , 1 2 , 3 5 , 2 3 , 3 4 , 4 5 \textstyle{1\over 5},{1\over 4},{1\over 3},{2\over 5},{1\over 2},{3\over 5},{2% \over 3},{3\over 4},{4\over 5}
  2. ζ ( n + 2 ) ζ ( n ) \textstyle{{\zeta(n+2)}\over{\zeta(n)}}

One-sided_limit.html

  1. lim x a + f ( x ) \lim_{x\to a^{+}}f(x)
  2. lim x a f ( x ) \lim_{x\downarrow a}\,f(x)
  3. lim x a f ( x ) \lim_{x\searrow a}\,f(x)
  4. lim x > a f ( x ) \lim_{x\underset{>}{\to}a}f(x)
  5. lim x a - f ( x ) \lim_{x\to a^{-}}f(x)
  6. lim x a f ( x ) \lim_{x\uparrow a}\,f(x)
  7. lim x a f ( x ) \lim_{x\nearrow a}\,f(x)
  8. lim x < a f ( x ) \lim_{x\underset{<}{\to}a}f(x)
  9. lim x a f ( x ) \lim_{x\to a}f(x)\,
  10. ε > 0 δ > 0 x I ( 0 < x - a < δ | f ( x ) - L | < ε ) \forall\varepsilon>0\;\exists\delta>0\;\forall x\in I\;(0<x-a<\delta% \Rightarrow|f(x)-L|<\varepsilon)
  11. ε > 0 δ > 0 x I ( 0 < a - x < δ | f ( x ) - L | < ε ) \forall\varepsilon>0\;\exists\delta>0\;\forall x\in I\;(0<a-x<\delta% \Rightarrow|f(x)-L|<\varepsilon)
  12. I I
  13. f f
  14. lim x 0 + 1 1 + 2 - 1 / x = 1 , \lim_{x\to 0^{+}}{1\over 1+2^{-1/x}}=1,
  15. lim x 0 - 1 1 + 2 - 1 / x = 0. \lim_{x\to 0^{-}}{1\over 1+2^{-1/x}}=0.

One-way_function.html

  1. P r [ f ( A ( f ( x ) ) ) = f ( x ) ] < 1 p ( n ) Pr[f(A(f(x)))=f(x)]<\frac{1}{p(n)}
  2. O ( 2 ( log N ) 1 / 3 ( log log N ) 2 / 3 ) O(2^{{(\log N)^{1/3}(\log\log N})^{2/3}})
  3. log N \log N
  4. N = p q N=pq
  5. p p
  6. q q
  7. Rabin N ( x ) x 2 mod N \,\text{Rabin}_{N}(x)\triangleq x^{2}\mod N
  8. N N
  9. N N
  10. p p
  11. q q

Onsager_reciprocal_relations.html

  1. d U = T d S - P d V + μ d M dU=T\,dS-P\,dV+\mu\,dM
  2. μ \mu
  3. ρ \rho
  4. d u = T d s + μ d ρ du=T\,ds+\mu\,d\rho
  5. d s = ( 1 / T ) d u + ( - μ / T ) d ρ ds=(1/T)\,du+(-\mu/T)\,d\rho
  6. u u
  7. ρ \rho
  8. 1 / T 1/T
  9. - μ / T -\mu/T
  10. U U
  11. M M
  12. ρ t + 𝐉 ρ = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{J}_{\rho}=0
  13. u t + 𝐉 u = 0 \frac{\partial u}{\partial t}+\nabla\cdot\mathbf{J}_{u}=0
  14. 𝐉 ρ \mathbf{J}_{\rho}
  15. 𝐉 u \mathbf{J}_{u}
  16. s t + 𝐉 s = s c t \frac{\partial s}{\partial t}+\nabla\cdot\mathbf{J}_{s}=\frac{\partial s_{c}}{% \partial t}
  17. s c t \frac{\partial s_{c}}{\partial t}
  18. 𝐉 u = - k T \mathbf{J}_{u}=-k\,\nabla T
  19. T T \nabla T\ll T
  20. 𝐉 u = k T 2 ( 1 / T ) \mathbf{J}_{u}=kT^{2}\nabla(1/T)
  21. 𝐉 ρ = - D ρ \mathbf{J}_{\rho}=-D\,\nabla\rho
  22. 𝐉 ρ = D ( - μ / T ) \mathbf{J}_{\rho}=D^{\prime}\,\nabla(-\mu/T)\!
  23. 𝐉 u = L u u ( 1 / T ) + L u ρ ( - μ / T ) \mathbf{J}_{u}=L_{uu}\,\nabla(1/T)+L_{u\rho}\,\nabla(-\mu/T)
  24. 𝐉 ρ = L ρ u ( 1 / T ) + L ρ ρ ( - μ / T ) \mathbf{J}_{\rho}=L_{\rho u}\,\nabla(1/T)+L_{\rho\rho}\,\nabla(-\mu/T)
  25. 𝐉 α = β L α β f β \mathbf{J}_{\alpha}=\sum_{\beta}L_{\alpha\beta}\,\nabla f_{\beta}
  26. u u
  27. μ \mu
  28. f u = ( 1 / T ) f_{u}=(1/T)
  29. f ρ = ( - μ / T ) f_{\rho}=(-\mu/T)
  30. L α β L_{\alpha\beta}
  31. s t = ( 1 / T ) u t + ( - μ / T ) ρ t \frac{\partial s}{\partial t}=(1/T)\frac{\partial u}{\partial t}+(-\mu/T)\frac% {\partial\rho}{\partial t}
  32. 𝐉 s = ( 1 / T ) 𝐉 u + ( - μ / T ) 𝐉 ρ = β 𝐉 α f α \mathbf{J}_{s}=(1/T)\mathbf{J}_{u}+(-\mu/T)\mathbf{J}_{\rho}=\sum_{\beta}% \mathbf{J}_{\alpha}f_{\alpha}
  33. s c t = 𝐉 u ( 1 / T ) + 𝐉 ρ ( - μ / T ) = α 𝐉 α f α \frac{\partial s_{c}}{\partial t}=\mathbf{J}_{u}\cdot\nabla(1/T)+\mathbf{J}_{% \rho}\cdot\nabla(-\mu/T)=\sum_{\alpha}\mathbf{J}_{\alpha}\cdot\nabla f_{\alpha}
  34. s c t = α β L α β ( f α ) ( f β ) \frac{\partial s_{c}}{\partial t}=\sum_{\alpha}\sum_{\beta}L_{\alpha\beta}(% \nabla f_{\alpha})(\nabla f_{\beta})
  35. L α β L_{\alpha\beta}
  36. L α β L_{\alpha\beta}
  37. L u ρ \ L_{u\rho}
  38. L ρ u \ L_{\rho u}
  39. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  40. S ( x 1 , x 2 , , x n ) S(x_{1},x_{2},\ldots,x_{n})
  41. w = A exp ( S / k ) w=A\exp(S/k)
  42. x 1 , x 2 , , x n {x_{1},x_{2},\ldots,x_{n}}
  43. w = A e - 1 2 β i k x i x k ; β i k = - 1 k 2 S x i x k , w=Ae^{-\frac{1}{2}\beta_{ik}x_{i}x_{k}}\,;\;\;\;\;\ \beta_{ik}=-\frac{1}{k}% \frac{\partial^{2}S}{\partial x_{i}\partial x_{k}}\,,
  44. β i k \beta_{ik}
  45. x ˙ i = - λ i k x k \dot{x}_{i}=-\lambda_{ik}x_{k}
  46. X i = - 1 k S x i X_{i}=-\frac{1}{k}\frac{\partial S}{\partial x_{i}}
  47. X i = β i k x k X_{i}=\beta_{ik}x_{k}
  48. x ˙ i = - γ i k X k \dot{x}_{i}=-\gamma_{ik}X_{k}
  49. γ i k = λ i l β l k - 1 \gamma_{ik}=\lambda_{il}\beta^{-1}_{lk}
  50. γ \gamma
  51. γ i k = γ k i \gamma_{ik}=\gamma_{ki}
  52. ξ i ( t ) \xi_{i}(t)
  53. Ξ i ( t ) \Xi_{i}(t)
  54. x i x_{i}
  55. X i X_{i}
  56. x 1 , x 2 , x_{1},x_{2},\ldots
  57. t = 0 t=0
  58. ξ ˙ i ( t ) = - γ i k Ξ k \dot{\xi}_{i}(t)=-\gamma_{ik}\Xi_{k}
  59. x i ( t ) x k ( 0 ) = x i ( - t ) x k ( 0 ) = x i ( 0 ) x k ( t ) \langle x_{i}(t)x_{k}(0)\rangle=\langle x_{i}(-t)x_{k}(0)\rangle=\langle x_{i}% (0)x_{k}(t)\rangle
  60. ξ i ( t ) \xi_{i}(t)
  61. ξ i ( t ) x k = x i ξ k ( t ) \langle\xi_{i}(t)x_{k}\rangle=\langle x_{i}\xi_{k}(t)\rangle
  62. t t
  63. γ i l Ξ l ( t ) x k = γ k l x i Ξ l ( t ) \gamma_{il}\langle\Xi_{l}(t)x_{k}\rangle=\gamma_{kl}\langle x_{i}\Xi_{l}(t)\rangle
  64. t = 0 t=0
  65. γ i l X l x k = γ k l X l x i \gamma_{il}\langle X_{l}x_{k}\rangle=\gamma_{kl}\langle X_{l}x_{i}\rangle
  66. X i x k = δ i k \langle X_{i}x_{k}\rangle=\delta_{ik}

Operator_topologies.html

  1. T n - T 0 \|T_{n}-T\|\to 0
  2. T n x - T x H \|T_{n}x-Tx\|_{H}
  3. T n T T_{n}\to T
  4. T n x T x T_{n}x\to Tx
  5. T n T T_{n}\to T
  6. T n x T x T_{n}x\to Tx
  7. F ( T n x ) F ( T x ) F(T_{n}x)\to F(Tx)
  8. T n T T_{n}\to T

Opposite_category.html

  1. ( C o p ) o p = C (C^{op})^{op}=C
  2. ( C × D ) o p C o p × D o p (C\times D)^{op}\cong C^{op}\times D^{op}
  3. ( Funct ( C , D ) ) o p Funct ( C o p , D o p ) (\mathrm{Funct}(C,D))^{op}\cong\mathrm{Funct}(C^{op},D^{op})
  4. ( F G ) o p ( G o p F o p ) (F\downarrow G)^{op}\cong(G^{op}\downarrow F^{op})

Optical_cavity.html

  1. 0 ( 1 - L R 1 ) ( 1 - L R 2 ) 1. 0\leqslant\left(1-\frac{L}{R_{1}}\right)\left(1-\frac{L}{R_{2}}\right)% \leqslant 1.
  2. g 1 = 1 - L R 1 , g 2 = 1 - L R 2 g_{1}=1-\frac{L}{R_{1}},\qquad g_{2}=1-\frac{L}{R_{2}}

Optical_coating.html

  1. n 1 = n 0 n S n_{1}=\sqrt{n_{0}n_{S}}
  2. n 1 n_{1}
  3. n 0 n_{0}
  4. n S n_{S}

Optical_coherence_tomography.html

  1. I S I_{S}
  2. I = k 1 I S + k 2 I S + 2 ( k 1 I S ) ( k 2 I S ) R e [ γ ( τ ) ] ( 1 ) I=k_{1}I_{S}+k_{2}I_{S}+2\sqrt{\left(k_{1}I_{S}\right)\cdot\left(k_{2}I_{S}% \right)}\cdot Re\left[\gamma\left(\tau\right)\right]\qquad(1)
  3. k 1 + k 2 < 1 k_{1}+k_{2}<1
  4. γ ( τ ) \gamma(\tau)
  5. τ \tau
  6. γ ( τ ) = exp [ - ( π Δ ν τ 2 ln 2 ) 2 ] exp ( - j 2 π ν 0 τ ) ( 2 ) \gamma\left(\tau\right)=\exp\left[-\left(\frac{\pi\Delta\nu\tau}{2\sqrt{\ln 2}% }\right)^{2}\right]\cdot\exp\left(-j2\pi\nu_{0}\tau\right)\qquad\quad(2)
  7. Δ ν \Delta\nu
  8. ν 0 \nu_{0}
  9. f D o p p = 2 ν 0 v s c ( 3 ) f_{Dopp}=\frac{2\cdot\nu_{0}\cdot v_{s}}{c}\qquad\qquad\qquad\qquad\qquad% \qquad\qquad\quad(3)
  10. ν 0 \nu_{0}
  11. v s v_{s}
  12. c c
  13. l c \,{l_{c}}
  14. = 2 ln 2 π λ 0 2 Δ λ =\frac{2\ln 2}{\pi}\cdot\frac{\lambda_{0}^{2}}{\Delta\lambda}
  15. 0.44 λ 0 2 Δ λ ( 4 ) \approx 0.44\cdot\frac{\lambda_{0}^{2}}{\Delta\lambda}\qquad\qquad\qquad\qquad% \qquad\qquad\qquad\qquad(4)

Optimal_control.html

  1. J = Φ [ 𝐱 ( t 0 ) , t 0 , 𝐱 ( t f ) , t f ] + t 0 t f [ 𝐱 ( t ) , 𝐮 ( t ) , t ] d t J=\Phi\,[\,\,\textbf{x}(t_{0}),t_{0},\,\textbf{x}(t_{f}),t_{f}\,]+\int_{t_{0}}% ^{t_{f}}\mathcal{L}\,[\,\,\textbf{x}(t),\,\textbf{u}(t),t\,]\,\operatorname{d}t
  2. 𝐱 ˙ ( t ) = 𝐚 [ 𝐱 ( t ) , 𝐮 ( t ) , t ] , \dot{\,\textbf{x}}(t)=\,\textbf{a}\,[\,\,\textbf{x}(t),\,\textbf{u}(t),t\,],
  3. 𝐛 [ 𝐱 ( t ) , 𝐮 ( t ) , t ] 𝟎 , \,\textbf{b}\,[\,\,\textbf{x}(t),\,\textbf{u}(t),t\,]\leq\,\textbf{0},
  4. s y m b o l ϕ [ 𝐱 ( t 0 ) , t 0 , 𝐱 ( t f ) , t f ] = 0 symbol{\phi}\,[\,\,\textbf{x}(t_{0}),t_{0},\,\textbf{x}(t_{f}),t_{f}\,]=0
  5. 𝐱 ( t ) \,\textbf{x}(t)
  6. 𝐮 ( t ) \,\textbf{u}(t)
  7. t t
  8. t 0 t_{0}
  9. t f t_{f}
  10. Φ \Phi
  11. \mathcal{L}
  12. [ 𝐱 * ( t * ) , 𝐮 * ( t * ) , t * ] [\,\textbf{x}^{*}(t^{*}),\,\textbf{u}^{*}(t^{*}),t^{*}]
  13. J = 1 2 𝐱 T ( t f ) 𝐒 f 𝐱 ( t f ) + 1 2 t 0 t f [ 𝐱 T ( t ) 𝐐 ( t ) 𝐱 ( t ) + 𝐮 T ( t ) 𝐑 ( t ) 𝐮 ( t ) ] d t J=\tfrac{1}{2}\mathbf{x}^{\,\text{T}}(t_{f})\mathbf{S}_{f}\mathbf{x}(t_{f})+% \tfrac{1}{2}\int_{t_{0}}^{t_{f}}[\,\mathbf{x}^{\,\text{T}}(t)\mathbf{Q}(t)% \mathbf{x}(t)+\mathbf{u}^{\,\text{T}}(t)\mathbf{R}(t)\mathbf{u}(t)\,]\,% \operatorname{d}t
  14. 𝐱 ˙ ( t ) = 𝐀 ( t ) 𝐱 ( t ) + 𝐁 ( t ) 𝐮 ( t ) , \dot{\mathbf{x}}(t)=\mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t),
  15. 𝐱 ( t 0 ) = 𝐱 0 \mathbf{x}(t_{0})=\mathbf{x}_{0}
  16. 𝐀 \mathbf{A}
  17. 𝐁 \mathbf{B}
  18. , 𝐐 ,\mathbf{Q}
  19. 𝐑 \mathbf{R}
  20. t f t_{f}\rightarrow\infty
  21. J = 1 2 0 [ 𝐱 T ( t ) 𝐐𝐱 ( t ) + 𝐮 T ( t ) 𝐑𝐮 ( t ) ] d t J=\tfrac{1}{2}\int_{0}^{\infty}[\,\mathbf{x}^{\,\text{T}}(t)\mathbf{Q}\mathbf{% x}(t)+\mathbf{u}^{\,\text{T}}(t)\mathbf{R}\mathbf{u}(t)\,]\,\operatorname{d}t
  22. 𝐱 ˙ ( t ) = 𝐀𝐱 ( t ) + 𝐁𝐮 ( t ) , \dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t),
  23. 𝐱 ( t 0 ) = 𝐱 0 \mathbf{x}(t_{0})=\mathbf{x}_{0}
  24. 𝐐 \mathbf{Q}
  25. 𝐑 \mathbf{R}
  26. 𝐐 \mathbf{Q}
  27. 𝐑 \mathbf{R}
  28. 𝐐 \mathbf{Q}
  29. 𝐑 \mathbf{R}
  30. ( 𝐀 , 𝐁 ) (\mathbf{A},\mathbf{B})
  31. 𝐮 ( t ) = - 𝐊 ( t ) 𝐱 ( t ) \mathbf{u}(t)=-\mathbf{K}(t)\mathbf{x}(t)
  32. 𝐊 ( t ) \mathbf{K}(t)
  33. 𝐊 ( t ) = 𝐑 - 1 𝐁 T 𝐒 ( t ) , \mathbf{K}(t)=\mathbf{R}^{-1}\mathbf{B}^{\,\text{T}}\mathbf{S}(t),
  34. 𝐒 ( t ) \mathbf{S}(t)
  35. 𝐒 ˙ ( t ) = - 𝐒 ( t ) 𝐀 - 𝐀 T 𝐒 ( t ) + 𝐒 ( t ) 𝐁𝐑 - 1 𝐁 T 𝐒 ( t ) - 𝐐 \dot{\mathbf{S}}(t)=-\mathbf{S}(t)\mathbf{A}-\mathbf{A}^{\,\text{T}}\mathbf{S}% (t)+\mathbf{S}(t)\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{\,\text{T}}\mathbf{S}(t)% -\mathbf{Q}
  36. 𝐒 ( t f ) = 𝐒 f \mathbf{S}(t_{f})=\mathbf{S}_{f}
  37. 𝟎 = - 𝐒𝐀 - 𝐀 T 𝐒 + 𝐒𝐁𝐑 - 1 𝐁 T 𝐒 - 𝐐 \mathbf{0}=-\mathbf{S}\mathbf{A}-\mathbf{A}^{\,\text{T}}\mathbf{S}+\mathbf{S}% \mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{\,\text{T}}\mathbf{S}-\mathbf{Q}
  38. 𝐀 \mathbf{A}
  39. 𝐁 \mathbf{B}
  40. 𝐐 \mathbf{Q}
  41. 𝐑 \mathbf{R}
  42. 𝐱 ˙ = H / \partialsymbol λ s y m b o l λ ˙ = - H / 𝐱 \begin{array}[]{lcl}\dot{\,\textbf{x}}&=&\partial H/\partialsymbol{\lambda}\\ \dot{symbol{\lambda}}&=&-\partial H/\partial\,\textbf{x}\end{array}
  43. H = + s y m b o l λ T 𝐚 - s y m b o l μ T 𝐛 H=\mathcal{L}+symbol{\lambda}^{\,\text{T}}\,\textbf{a}-symbol{\mu}^{\,\text{T}% }\,\textbf{b}
  44. s y m b o l λ symbol{\lambda}
  45. F ( 𝐳 ) F(\,\textbf{z})\,
  46. 𝐠 ( 𝐳 ) = 𝟎 𝐡 ( 𝐳 ) 𝟎 \begin{array}[]{lcl}\,\textbf{g}(\,\textbf{z})&=&\,\textbf{0}\\ \,\textbf{h}(\,\textbf{z})&\leq&\,\textbf{0}\end{array}
  47. λ ( t ) \lambda(t)
  48. λ ( t ) \lambda(t)
  49. λ ( t ) \lambda(t)
  50. λ ( t ) \lambda(t)
  51. 0
  52. T T
  53. 0
  54. x 0 x_{0}
  55. x ( t ) x(t)
  56. u ( t ) 2 / x ( t ) u(t)^{2}/x(t)
  57. p p
  58. T T
  59. Π \Pi
  60. Π = t = 0 T - 1 [ p u t - u t 2 x t ] \Pi=\sum_{t=0}^{T-1}\left[pu_{t}-\frac{u_{t}^{2}}{x_{t}}\right]
  61. x t x_{t}
  62. x t + 1 - x t = - u t x_{t+1}-x_{t}=-u_{t}\!
  63. H = p u t - u t 2 x t - λ t + 1 u t H=pu_{t}-\frac{u_{t}^{2}}{x_{t}}-\lambda_{t+1}u_{t}
  64. H u t = p - λ t + 1 - 2 u t x t = 0 \frac{\partial H}{\partial u_{t}}=p-\lambda_{t+1}-2\frac{u_{t}}{x_{t}}=0
  65. λ t + 1 - λ t = - H x t = - ( u t x t ) 2 \lambda_{t+1}-\lambda_{t}=-\frac{\partial H}{\partial x_{t}}=-\left(\frac{u_{t% }}{x_{t}}\right)^{2}
  66. T T
  67. λ T = 0 \lambda_{T}=0\!
  68. x t x_{t}
  69. λ t \lambda_{t}
  70. λ t = λ t + 1 + ( p - λ t + 1 ) 2 4 \lambda_{t}=\lambda_{t+1}+\frac{(p-\lambda_{t+1})^{2}}{4}
  71. x t + 1 = x t 2 - p + λ t + 1 2 x_{t+1}=x_{t}\frac{2-p+\lambda_{t+1}}{2}
  72. x t x_{t}
  73. u t u_{t}
  74. Π \Pi
  75. Π = 0 T [ p u ( t ) - u ( t ) 2 x ( t ) ] d t \Pi=\int_{0}^{T}\left[pu(t)-\frac{u(t)^{2}}{x(t)}\right]dt
  76. x ( t ) x(t)
  77. x ˙ ( t ) = - u ( t ) \dot{x}(t)=-u(t)
  78. H = p u ( t ) - u ( t ) 2 x ( t ) - λ ( t ) u ( t ) H=pu(t)-\frac{u(t)^{2}}{x(t)}-\lambda(t)u(t)
  79. H u = p - λ ( t ) - 2 u ( t ) x ( t ) = 0 \frac{\partial H}{\partial u}=p-\lambda(t)-2\frac{u(t)}{x(t)}=0
  80. λ ˙ ( t ) = - H x = - ( u ( t ) x ( t ) ) 2 \dot{\lambda}(t)=-\frac{\partial H}{\partial x}=-\left(\frac{u(t)}{x(t)}\right% )^{2}
  81. T T
  82. λ ( T ) = 0 \lambda(T)=0
  83. u ( t ) u(t)
  84. λ ( t ) \lambda(t)
  85. λ ˙ ( t ) = - ( p - λ ( t ) ) 2 4 \dot{\lambda}(t)=-\frac{(p-\lambda(t))^{2}}{4}
  86. u ( t ) = x ( t ) p - λ ( t ) 2 u(t)=x(t)\frac{p-\lambda(t)}{2}

Optimal_solutions_for_Rubik's_Cube.html

  1. G 0 = L , R , F , B , U , D G_{0}=\langle L,R,F,B,U,D\rangle
  2. G 1 = L , R , F , B , U 2 , D 2 G_{1}=\langle L,R,F,B,U^{2},D^{2}\rangle
  3. G 2 = L , R , F 2 , B 2 , U 2 , D 2 G_{2}=\langle L,R,F^{2},B^{2},U^{2},D^{2}\rangle
  4. G 3 = L 2 , R 2 , F 2 , B 2 , U 2 , D 2 G_{3}=\langle L^{2},R^{2},F^{2},B^{2},U^{2},D^{2}\rangle
  5. G 4 = { 1 } G_{4}=\{1\}
  6. G i + 1 G i G_{i+1}\setminus G_{i}
  7. G 0 G_{0}
  8. G 1 G 0 G_{1}\setminus G_{0}
  9. G 1 G_{1}
  10. G 2 G_{2}
  11. G 3 G_{3}
  12. G 4 G_{4}
  13. G 0 G_{0}
  14. G 1 G 0 , G 2 G 1 , G 3 G 2 G_{1}\setminus G_{0},G_{2}\setminus G_{1},G_{3}\setminus G_{2}
  15. G 3 G_{3}
  16. G 2 G 1 G_{2}\setminus G_{1}
  17. G 0 = U , D , L , R , F , B G_{0}=\langle U,D,L,R,F,B\rangle
  18. G 1 = U , D , L 2 , R 2 , F 2 , B 2 G_{1}=\langle U,D,L^{2},R^{2},F^{2},B^{2}\rangle
  19. G 2 = { 1 } G_{2}=\{1\}
  20. G 1 G 0 G_{1}\setminus G_{0}
  21. G 1 G_{1}
  22. G 1 G_{1}
  23. G 1 G 0 G_{1}\setminus G_{0}
  24. G 1 G_{1}
  25. G 1 G 0 G_{1}\setminus G_{0}
  26. G 1 G_{1}
  27. G 1 G_{1}
  28. G 1 G_{1}

Option_time_value.html

  1. max [ ( S - K ) , 0 ] \max[(S-K),0]
  2. ( S - K ) + (S-K)^{+}
  3. max [ ( K - S ) , 0 ] \max[(K-S),0]
  4. ( K - S ) + (K-S)^{+}

Order_(group_theory).html

  1. | G | |G|
  2. | a | |a|
  3. a b = ( a b ) - 1 = b - 1 a - 1 = b a ab=(ab)^{-1}=b^{-1}a^{-1}=ba
  4. 2 + 2 + 2 = 6 0 ( mod 6 ) 2+2+2=6\equiv 0\;\;(\mathop{{\rm mod}}6)
  5. a = { a k : k } \langle a\rangle=\{a^{k}:k\in\mathbb{Z}\}
  6. ord ( a ) = ord ( a ) . \operatorname{ord}(a)=\operatorname{ord}(\langle a\rangle).
  7. ord ( a b ) = ord ( b a ) \operatorname{ord}(ab)=\operatorname{ord}(ba)
  8. S y m ( ) Sym(\mathbb{Z})
  9. | G | = | Z ( G ) | + i d i |G|=|Z(G)|+\sum_{i}d_{i}\;

Order_theory.html

  1. a a * a = 0 a\leq a^{*}\Rightarrow a=0

Ordered_ring.html

  1. | a | := { a , if 0 a , - a , otherwise , |a|:=\begin{cases}a,&\mbox{if }~{}0\leq a,\\ -a,&\mbox{otherwise}~{},\end{cases}

Orders_of_approximation.html

  1. x n + 1 x^{n+1}
  2. O ( x n + 1 ) . O(x^{n+1}).
  3. x = [ 0 , 1 , 2 ] x=[0,1,2]\,
  4. y = [ 3 , 3 , 5 ] y=[3,3,5]\,
  5. y f ( x ) = 3.67 y\sim f(x)=3.67\,
  6. x = [ 0 , 1 , 2 ] x=[0,1,2]\,
  7. y = [ 3 , 3 , 5 ] y=[3,3,5]\,
  8. y f ( x ) = x + 2.67 y\sim f(x)=x+2.67\,
  9. x = [ 0 , 1 , 2 ] x=[0,1,2]\,
  10. y = [ 3 , 3 , 5 ] y=[3,3,5]\,
  11. y f ( x ) = x 2 - x + 3 y\sim f(x)=x^{2}-x+3\,

Orthogonal_functions.html

  1. f f
  2. g g
  3. f , g \langle f,g\rangle
  4. f , g = f ( x ) * g ( x ) d x \langle f,g\rangle=\int f(x)^{*}g(x)\,dx
  5. f \vec{f}
  6. g \vec{g}
  7. f \vec{f}
  8. g \vec{g}

OSCAR.html

  1. f d f_{d}
  2. f u f_{u}
  3. f f
  4. v v
  5. c c
  6. 3 × 10 8 3\times 10^{8}
  7. Δ f = f × v c \Delta f=f\times\frac{v}{c}
  8. f d = f ( 1 + v c ) f_{d}=f(1+\frac{v}{c})
  9. f u = f ( 1 - v c ) f_{u}=f(1-\frac{v}{c})

Otway–Rees_protocol.html

  1. A B : M , A , B , { N A , M , A , B } K A S A\rightarrow B:M,A,B,\{N_{A},M,A,B\}_{K_{AS}}
  2. B S : M , A , B , { N A , M , A , B } K A S , { N B , M , A , B } K B S B\rightarrow S:M,A,B,\{N_{A},M,A,B\}_{K_{AS}},\{N_{B},M,A,B\}_{K_{BS}}
  3. S B : M , { N A , K A B } K A S , { N B , K A B } K B S S\rightarrow B:M,\{N_{A},K_{AB}\}_{K_{AS}},\{N_{B},K_{AB}\}_{K_{BS}}
  4. B A : M , { N A , K A B } K A S B\rightarrow A:M,\{N_{A},K_{AB}\}_{K_{AS}}
  5. K A B K_{AB}
  6. K A B K^{\prime}_{AB}
  7. K A B K^{\prime}_{AB}
  8. K A B K_{AB}

Outer_automorphism_group.html

  1. | Out ( G ) | |\mbox{Out}~{}(G)|
  2. n p | n ( 1 - 1 p ) n\prod_{p|n}\left(1-\frac{1}{p}\right)
  3. n 6 : Out ( S n ) = 1 n 3 , n 6 : Out ( A n ) = C 2 Out ( S 6 ) = C 2 Out ( A 6 ) = C 2 × C 2 \begin{aligned}\displaystyle n\neq 6:\mathrm{Out}(S_{n})&\displaystyle=1\\ \displaystyle n\geq 3,\ n\neq 6:\mathrm{Out}(A_{n})&\displaystyle=C_{2}\\ \displaystyle\mathrm{Out}(S_{6})&\displaystyle=C_{2}\\ \displaystyle\mathrm{Out}(A_{6})&\displaystyle=C_{2}\times C_{2}\end{aligned}
  4. 𝔤 \mathfrak{g}
  5. Aut ( 𝔤 ) \operatorname{Aut}(\mathfrak{g})
  6. Inn ( 𝔤 ) \operatorname{Inn}(\mathfrak{g})
  7. Out ( 𝔤 ) \operatorname{Out}(\mathfrak{g})
  8. 1 Inn ( 𝔤 ) Aut ( 𝔤 ) Out ( 𝔤 ) 1 1\;\xrightarrow{}\;\operatorname{Inn}(\mathfrak{g})\;\xrightarrow{}\;% \operatorname{Aut}(\mathfrak{g})\;\xrightarrow{}\;\operatorname{Out}(\mathfrak% {g})\;\xrightarrow{}\;1
  9. σ : G Aut ( G ) . \sigma\colon G\to\operatorname{Aut}(G).
  10. Z ( G ) G 𝜎 Aut ( G ) Out ( G ) . Z(G)\hookrightarrow G\overset{\sigma}{\to}\operatorname{Aut}(G)% \twoheadrightarrow\operatorname{Out}(G).
  11. Out ( F n ) \scriptstyle\operatorname{Out}(F_{n})

Outer_measure.html

  1. A + x = { a + x : a A } A+x=\{a+x:a\in A\}
  2. φ ( i = 1 A i ) = i = 1 φ ( A i ) . \varphi\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}\varphi(A_{% i}).
  3. X X
  4. φ : 2 X [ 0 , ] , \varphi:2^{X}\rightarrow[0,\infty],
  5. X X
  6. φ ( ) = 0 \varphi(\varnothing)=0
  7. A A
  8. B B
  9. X X
  10. A B implies φ ( A ) φ ( B ) . A\subseteq B\quad\,\text{implies}\quad\varphi(A)\leq\varphi(B).
  11. X X
  12. φ ( j = 1 A j ) j = 1 φ ( A j ) . \varphi\left(\bigcup_{j=1}^{\infty}A_{j}\right)\leq\sum_{j=1}^{\infty}\varphi(% A_{j}).
  13. E E
  14. X X
  15. A A
  16. X X
  17. φ ( A ) = φ ( A E ) + φ ( A E c ) . \varphi(A)=\varphi(A\cap E)+\varphi(A\cap E^{c}).
  18. ( X , d ) (X,d)
  19. φ φ
  20. X X
  21. φ φ
  22. φ ( E F ) = φ ( E ) + φ ( F ) \varphi(E\cup F)=\varphi(E)+\varphi(F)
  23. d ( E , F ) = inf { d ( x , y ) : x E , y F } > 0 , d(E,F)=\inf\{d(x,y):x\in E,y\in F\}>0,
  24. φ φ
  25. φ φ
  26. X X
  27. X X
  28. φ φ
  29. X X
  30. σ σ
  31. X X
  32. C C
  33. X X
  34. p p
  35. C C
  36. C C
  37. p p
  38. φ ( E ) = inf { i = 0 p ( A i ) | E i = 0 A i , i , A i C } . \varphi(E)=\inf\biggl\{\sum_{i=0}^{\infty}p(A_{i})\,\bigg|\,E\subseteq\bigcup_% {i=0}^{\infty}A_{i},\forall i\in\mathbb{N},A_{i}\in C\biggr\}.
  39. C C
  40. E E
  41. φ φ
  42. X X
  43. ( X , d ) (X,d)
  44. C C
  45. X X
  46. p p
  47. C C
  48. δ > 0 δ>0
  49. C δ = { A C : diam ( A ) δ } C_{\delta}=\{A\in C:\operatorname{diam}(A)\leq\delta\}
  50. φ δ ( E ) = inf { i = 0 p ( A i ) | E i = 0 A i , i , A i C δ } . \varphi_{\delta}(E)=\inf\biggl\{\sum_{i=0}^{\infty}p(A_{i})\,\bigg|\,E% \subseteq\bigcup_{i=0}^{\infty}A_{i},\forall i\in\mathbb{N},A_{i}\in C_{\delta% }\biggr\}.
  51. δ δ δ≤δ
  52. δ δ
  53. lim δ 0 φ δ ( E ) = φ 0 ( E ) [ 0 , ] \lim_{\delta\rightarrow 0}\varphi_{\delta}(E)=\varphi_{0}(E)\in[0,\infty]
  54. φ < s u b > 0 φ<sub>0

Owen_Willans_Richardson.html

  1. s = A T 1 / 2 e - b / T s=A\,T^{1/2}\,e^{-b/T}

Oxyanion.html

  1. K 1 = [ HCrO 4 - ] [ CrO 4 2 + ] [ H + ] K_{1}=\frac{[\mathrm{HCrO_{4}^{-}}]}{[\mathrm{CrO_{4}^{2+}}][\mathrm{H^{+}}]}
  2. K 2 = [ Cr 2 O 7 2 - ] [ HCrO 4 - ] 2 K_{2}=\frac{[\mathrm{Cr_{2}O_{7}^{2-}}]}{[\mathrm{HCrO_{4}^{-}}]^{2}}

Oxygen_cycle.html

  1. 6 CO 2 + 6 H 2 O + energy C 6 H 12 O 6 + 6 O 2 \mathrm{6\ CO_{2}+6H_{2}O+energy\longrightarrow C_{6}H_{12}O_{6}+6\ O_{2}}
  2. 2 H 2 O + energy 4 H + O 2 \mathrm{2\ H_{2}O+energy\longrightarrow 4\ H+O_{2}}
  3. 2 N 2 O + energy 4 N + O 2 \mathrm{2\ N_{2}O+energy\longrightarrow 4\ N+O_{2}}
  4. 4 FeO + O 2 2 Fe 2 O 3 \mathrm{4\ FeO+O_{2}\longrightarrow 2\ Fe_{2}O_{3}}
  5. O 2 + uv light 2 O ( λ 200 nm ) \mathrm{O_{2}+uv~{}light\longrightarrow 2~{}O}\qquad(\lambda\lesssim 200~{}\,% \text{nm})
  6. O + O 2 O 3 \mathrm{O+O_{2}\longrightarrow O_{3}}
  7. O 3 + uv light O 2 + O ( λ 300 nm ) \mathrm{O_{3}+uv~{}light\longrightarrow O_{2}+O}\qquad(\lambda\lesssim 300~{}% \,\text{nm})

P-value.html

  1. X X
  2. H H
  3. P r ( X | H ) , Pr(X|H),
  4. X X
  5. x x
  6. P r ( X = x | H ) = 0. Pr(X=x|H)=0.
  7. P r ( H | X ) , Pr(H|X),
  8. P r ( H ) , Pr(H),
  9. P r ( X ) , Pr(X),
  10. H H
  11. { X x } \{X\geq x\}
  12. { X x } \{X\leq x\}
  13. { X x } \{X\leq x\}
  14. { X x } \{X\geq x\}
  15. P r ( X x | H ) Pr(X\geq x|H)
  16. P r ( X x | H ) Pr(X\leq x|H)
  17. 2 min { P r ( X x | H ) , P r ( X x | H ) } 2\min\{Pr(X\leq x|H),Pr(X\geq x|H)\}
  18. H H
  19. α \alpha
  20. α \alpha
  21. α \alpha
  22. x x
  23. x x
  24. [ 0 , 1 ] [0,1]
  25. x x
  26. α \alpha
  27. P r ( Reject H | H ) = P r ( p α ) = α Pr(\mathrm{Reject}\;H|H)=Pr(p\leq\alpha)=\alpha
  28. X X
  29. 1 / 2 8 = 1 / 256 0.0039 1/2^{8}=1/256\approx 0.0039
  30. 0.0199 0.02. 0.0199\approx 0.02.
  31. Prob ( 14 heads ) + Prob ( 15 heads ) + + Prob ( 20 heads ) = 1 2 20 [ ( 20 14 ) + ( 20 15 ) + + ( 20 20 ) ] = 60 , 460 1 , 048 , 576 0.058 \begin{aligned}&\displaystyle\operatorname{Prob}(14\,\text{ heads})+% \operatorname{Prob}(15\,\text{ heads})+\cdots+\operatorname{Prob}(20\,\text{ % heads})\\ &\displaystyle=\frac{1}{2^{20}}\left[{\left({{20}\atop{14}}\right)}+{\left({{2% 0}\atop{15}}\right)}+\cdots+{\left({{20}\atop{20}}\right)}\right]=\frac{60,\!4% 60}{1,\!048,\!576}\approx 0.058\end{aligned}
  32. 1 / ( 8 4 ) = 1 / 70 0.014 , 1/{\left({{8}\atop{4}}\right)}=1/70\approx 0.014,
  33. 1 / ( 6 3 ) = 1 / 20 = 0.05 , 1/{\left({{6}\atop{3}}\right)}=1/20=0.05,

Paillier_cryptosystem.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. m 1 + m 2 m_{1}+m_{2}
  4. gcd ( p q , ( p - 1 ) ( q - 1 ) ) = 1 \gcd(pq,(p-1)(q-1))=1
  5. n = p q n=pq
  6. λ = lcm ( p - 1 , q - 1 ) \lambda=\operatorname{lcm}(p-1,q-1)
  7. g g
  8. g n 2 * g\in\mathbb{Z}^{*}_{n^{2}}
  9. n n
  10. g g
  11. μ = ( L ( g λ mod n 2 ) ) - 1 mod n \mu=(L(g^{\lambda}\bmod n^{2}))^{-1}\bmod n
  12. L L
  13. L ( u ) = u - 1 n L(u)=\frac{u-1}{n}
  14. a b \frac{a}{b}
  15. a a
  16. b b
  17. a a
  18. b b
  19. v 0 v\geq 0
  20. a v b a\geq vb
  21. ( n , g ) (n,g)
  22. ( λ , μ ) . (\lambda,\mu).
  23. g = n + 1 , λ = φ ( n ) , g=n+1,\lambda=\varphi(n),
  24. μ = φ ( n ) - 1 mod n \mu=\varphi(n)^{-1}\bmod n
  25. φ ( n ) = ( p - 1 ) ( q - 1 ) \varphi(n)=(p-1)(q-1)
  26. m m
  27. m n m\in\mathbb{Z}_{n}
  28. r r
  29. r n * r\in\mathbb{Z}^{*}_{n}
  30. c = g m r n mod n 2 c=g^{m}\cdot r^{n}\bmod n^{2}
  31. c c
  32. c n 2 * c\in\mathbb{Z}^{*}_{n^{2}}
  33. m = L ( c λ mod n 2 ) μ mod n m=L(c^{\lambda}\bmod n^{2})\cdot\mu\bmod n
  34. n 2 n^{2}
  35. D ( E ( m 1 , r 1 ) E ( m 2 , r 2 ) mod n 2 ) = m 1 + m 2 mod n . D(E(m_{1},r_{1})\cdot E(m_{2},r_{2})\bmod n^{2})=m_{1}+m_{2}\bmod n.\,
  36. D ( E ( m 1 , r 1 ) g m 2 mod n 2 ) = m 1 + m 2 mod n . D(E(m_{1},r_{1})\cdot g^{m_{2}}\bmod n^{2})=m_{1}+m_{2}\bmod n.\,
  37. D ( E ( m 1 , r 1 ) m 2 mod n 2 ) = m 1 m 2 mod n , D(E(m_{1},r_{1})^{m_{2}}\bmod n^{2})=m_{1}m_{2}\bmod n,\,
  38. D ( E ( m 2 , r 2 ) m 1 mod n 2 ) = m 1 m 2 mod n . D(E(m_{2},r_{2})^{m_{1}}\bmod n^{2})=m_{1}m_{2}\bmod n.\,
  39. D ( E ( m 1 , r 1 ) k mod n 2 ) = k m 1 mod n . D(E(m_{1},r_{1})^{k}\bmod n^{2})=km_{1}\bmod n.\,
  40. ( 1 + n ) x = k = 0 x ( x k ) n k = 1 + n x + ( x 2 ) n 2 + higher powers of n (1+n)^{x}=\sum_{k=0}^{x}{x\choose k}n^{k}=1+nx+{x\choose 2}n^{2}+\,\text{% higher powers of }n
  41. ( 1 + n ) x 1 + n x ( mod n 2 ) (1+n)^{x}\equiv 1+nx\;\;(\mathop{{\rm mod}}n^{2})
  42. y = ( 1 + n ) x mod n 2 y=(1+n)^{x}\bmod n^{2}
  43. x y - 1 n ( mod n ) x\equiv\frac{y-1}{n}\;\;(\mathop{{\rm mod}}n)
  44. L ( ( 1 + n ) x mod n 2 ) x ( mod n ) L((1+n)^{x}\bmod n^{2})\equiv x\;\;(\mathop{{\rm mod}}n)
  45. L L
  46. L ( u ) = u - 1 n L(u)=\frac{u-1}{n}
  47. x n x\in\mathbb{Z}_{n}

Palatini_variation.html

  1. g μ ν \scriptstyle{g_{\mu\nu}}
  2. g μ ν \scriptstyle{g_{\mu\nu}}
  3. Γ β μ α \scriptstyle{\Gamma^{\alpha}_{\,\beta\mu}}
  4. Γ β μ α \scriptstyle{\Gamma^{\alpha}_{\,\beta\mu}}
  5. g μ ν \scriptstyle{g_{\mu\nu}}

Paley–Wiener_theorem.html

  1. f ( ζ ) = - F ( x ) e i x ζ d x f(\zeta)=\int_{-\infty}^{\infty}F(x)e^{ix\zeta}\,dx
  2. f ( ζ ) = 0 F ( x ) e i x ζ d x f(\zeta)=\int_{0}^{\infty}F(x)e^{ix\zeta}\,dx
  3. - | f ( ξ + i η ) | 2 d ξ 0 | F ( x ) | 2 d x \int_{-\infty}^{\infty}\left|f(\xi+i\eta)\right|^{2}\,d\xi\leq\int_{0}^{\infty% }|F(x)|^{2}\,dx
  4. lim η 0 + - | f ( ξ + i η ) - f ( ξ ) | 2 d ξ = 0. \lim_{\eta\to 0^{+}}\int_{-\infty}^{\infty}\left|f(\xi+i\eta)-f(\xi)\right|^{2% }\,d\xi=0.
  5. sup η > 0 - | f ( ξ + i η ) | 2 d ξ = C < \sup_{\eta>0}\int_{-\infty}^{\infty}\left|f(\xi+i\eta)\right|^{2}\,d\xi=C<\infty
  6. H 2 ( 𝐑 ) = L 2 ( 𝐑 + ) . \mathcal{F}H^{2}(\mathbf{R})=L^{2}(\mathbf{R_{+}}).
  7. f ( ζ ) = - A A F ( x ) e i x ζ d x f(\zeta)=\int_{-A}^{A}F(x)e^{ix\zeta}\,dx
  8. | f ( ζ ) | C e A | ζ | , |f(\zeta)|\leq Ce^{A|\zeta|},
  9. - | f ( ξ + i η ) | 2 d ξ < . \int_{-\infty}^{\infty}|f(\xi+i\eta)|^{2}\,d\xi<\infty.
  10. v ( f ) = v x ( f ( x ) ) v(f)=v_{x}(f(x))
  11. v ^ ( s ) = ( 2 π ) - n 2 v x ( e - i x , s ) \hat{v}(s)=(2\pi)^{-\frac{n}{2}}v_{x}\left(e^{-i\langle x,s\rangle}\right)
  12. | F ( z ) | C ( 1 + | z | ) N e B | Im ( z ) | |F(z)|\leq C(1+|z|)^{N}e^{B|\,\text{Im}(z)|}
  13. | F ( z ) | C N ( 1 + | z | ) - N e B | Im ( z ) | |F(z)|\leq C_{N}(1+|z|)^{-N}e^{B|\,\text{Im}(z)|}
  14. H ( x ) = sup y K x , y . H(x)=\sup_{y\in K}\langle x,y\rangle.
  15. | v ^ ( ζ ) | C m ( 1 + | ζ | ) N e H ( Im ( ζ ) ) |\hat{v}(\zeta)|\leq C_{m}(1+|\zeta|)^{N}e^{H(\,\text{Im}(\zeta))}

Pappus's_centroid_theorem.html

  1. A = s d . A=sd.\,
  2. A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r . A=(2\pi r)(2\pi R)=4\pi^{2}Rr.\,
  3. V = A d . V=Ad.\,
  4. V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 . V=(\pi r^{2})(2\pi R)=2\pi^{2}Rr^{2}.\,

Parabolic_coordinates.html

  1. ( σ , τ ) (\sigma,\tau)
  2. x = σ τ x=\sigma\tau\,
  3. y = 1 2 ( τ 2 - σ 2 ) y=\frac{1}{2}\left(\tau^{2}-\sigma^{2}\right)
  4. σ \sigma
  5. 2 y = x 2 σ 2 - σ 2 2y=\frac{x^{2}}{\sigma^{2}}-\sigma^{2}
  6. + y +y
  7. τ \tau
  8. 2 y = - x 2 τ 2 + τ 2 2y=-\frac{x^{2}}{\tau^{2}}+\tau^{2}
  9. - y -y
  10. ( σ , τ ) (\sigma,\tau)
  11. h σ = h τ = σ 2 + τ 2 h_{\sigma}=h_{\tau}=\sqrt{\sigma^{2}+\tau^{2}}
  12. d A = ( σ 2 + τ 2 ) d σ d τ dA=\left(\sigma^{2}+\tau^{2}\right)d\sigma d\tau
  13. 2 Φ = 1 σ 2 + τ 2 ( 2 Φ σ 2 + 2 Φ τ 2 ) \nabla^{2}\Phi=\frac{1}{\sigma^{2}+\tau^{2}}\left(\frac{\partial^{2}\Phi}{% \partial\sigma^{2}}+\frac{\partial^{2}\Phi}{\partial\tau^{2}}\right)
  14. 𝐅 \nabla\cdot\mathbf{F}
  15. × 𝐅 \nabla\times\mathbf{F}
  16. ( σ , τ ) (\sigma,\tau)
  17. z z
  18. x = σ τ cos φ x=\sigma\tau\cos\varphi
  19. y = σ τ sin φ y=\sigma\tau\sin\varphi
  20. z = 1 2 ( τ 2 - σ 2 ) z=\frac{1}{2}\left(\tau^{2}-\sigma^{2}\right)
  21. z z
  22. ϕ \phi
  23. tan φ = y x \tan\varphi=\frac{y}{x}
  24. σ \sigma
  25. 2 z = x 2 + y 2 σ 2 - σ 2 2z=\frac{x^{2}+y^{2}}{\sigma^{2}}-\sigma^{2}
  26. + z +z
  27. τ \tau
  28. 2 z = - x 2 + y 2 τ 2 + τ 2 2z=-\frac{x^{2}+y^{2}}{\tau^{2}}+\tau^{2}
  29. - z -z
  30. g i j = [ σ 2 + τ 2 0 0 0 σ 2 + τ 2 0 0 0 σ 2 τ 2 ] g_{ij}=\begin{bmatrix}\sigma^{2}+\tau^{2}&0&0\\ 0&\sigma^{2}+\tau^{2}&0\\ 0&0&\sigma^{2}\tau^{2}\end{bmatrix}
  31. h σ = σ 2 + τ 2 h_{\sigma}=\sqrt{\sigma^{2}+\tau^{2}}
  32. h τ = σ 2 + τ 2 h_{\tau}=\sqrt{\sigma^{2}+\tau^{2}}
  33. h φ = σ τ h_{\varphi}=\sigma\tau\,
  34. h σ h_{\sigma}
  35. h τ h_{\tau}
  36. d V = h σ h τ h φ d σ d τ d φ = σ τ ( σ 2 + τ 2 ) d σ d τ d φ dV=h_{\sigma}h_{\tau}h_{\varphi}\,d\sigma\,d\tau\,d\varphi=\sigma\tau\left(% \sigma^{2}+\tau^{2}\right)\,d\sigma\,d\tau\,d\varphi
  37. 2 Φ = 1 σ 2 + τ 2 [ 1 σ σ ( σ Φ σ ) + 1 τ τ ( τ Φ τ ) ] + 1 σ 2 τ 2 2 Φ φ 2 \nabla^{2}\Phi=\frac{1}{\sigma^{2}+\tau^{2}}\left[\frac{1}{\sigma}\frac{% \partial}{\partial\sigma}\left(\sigma\frac{\partial\Phi}{\partial\sigma}\right% )+\frac{1}{\tau}\frac{\partial}{\partial\tau}\left(\tau\frac{\partial\Phi}{% \partial\tau}\right)\right]+\frac{1}{\sigma^{2}\tau^{2}}\frac{\partial^{2}\Phi% }{\partial\varphi^{2}}
  38. 𝐅 \nabla\cdot\mathbf{F}
  39. × 𝐅 \nabla\times\mathbf{F}
  40. ( σ , τ , ϕ ) (\sigma,\tau,\phi)

Paraconsistent_logic.html

  1. P ¬ P P\land\neg P
  2. P P\,
  3. P A P\lor A
  4. ¬ P \neg P\,
  5. A A\,
  6. A A B A\vdash A\lor B
  7. A B , ¬ A B A\lor B,\neg A\vdash B
  8. Γ A ; A B Γ B \Gamma\vdash A;A\vdash B\Rightarrow\Gamma\vdash B
  9. A B A\vdash B
  10. A B \vdash A\Rightarrow B
  11. A ( B ¬ B ) ¬ A A\to(B\wedge\neg B)\vdash\neg A
  12. A B A A\vdash B\to A
  13. ¬ ¬ A A \neg\neg A\vdash A
  14. V V\,
  15. V ( A , 1 ) V(A,1)\,
  16. A A\,
  17. V ( A , 0 ) V(A,0)\,
  18. A A\,
  19. V ( ¬ A , 1 ) V ( A , 0 ) V(\neg A,1)\Leftrightarrow V(A,0)
  20. V ( ¬ A , 0 ) V ( A , 1 ) V(\neg A,0)\Leftrightarrow V(A,1)
  21. V ( A B , 1 ) V ( A , 1 ) o r V ( B , 1 ) V(A\lor B,1)\Leftrightarrow V(A,1)\ or\ V(B,1)
  22. V ( A B , 0 ) V ( A , 0 ) a n d V ( B , 0 ) V(A\lor B,0)\Leftrightarrow V(A,0)\ and\ V(B,0)
  23. Γ A \Gamma\vDash A
  24. A A\,
  25. Γ \Gamma\,
  26. V V\,
  27. V ( A , 1 ) V(A,1)\,
  28. V ( A , 0 ) V(A,0)\,
  29. V ( B , 1 ) V(B,1)\,
  30. A ¬ A \vdash A\lor\neg A
  31. A ¬ A A\land\neg A\vdash
  32. ¬ ¬ A A \neg\neg A\vdash A
  33. A ¬ ¬ A A\vdash\neg\neg A

Parallelogram_law.html

  1. 2 ( A B ) 2 + 2 ( B C ) 2 = ( A C ) 2 + ( B D ) 2 2(AB)^{2}+2(BC)^{2}=(AC)^{2}+(BD)^{2}\,
  2. 2 ( A B ) 2 + 2 ( B C ) 2 = 2 ( A C ) 2 2(AB)^{2}+2(BC)^{2}=2(AC)^{2}\,
  3. ( A B ) 2 + ( B C ) 2 + ( C D ) 2 + ( D A ) 2 = ( A C ) 2 + ( B D ) 2 + 4 x 2 . (AB)^{2}+(BC)^{2}+(CD)^{2}+(DA)^{2}=(AC)^{2}+(BD)^{2}+4x^{2}.\,
  4. 2 x 2 + 2 y 2 = x + y 2 + x - y 2 . 2\|x\|^{2}+2\|y\|^{2}=\|x+y\|^{2}+\|x-y\|^{2}.\,
  5. x 2 = x , x . \|x\|^{2}=\langle x,x\rangle.\,
  6. x + y 2 = x + y , x + y = x , x + x , y + y , x + y , y , \|x+y\|^{2}=\langle x+y,x+y\rangle=\langle x,x\rangle+\langle x,y\rangle+% \langle y,x\rangle+\langle y,y\rangle,\,
  7. x - y 2 = x - y , x - y = x , x - x , y - y , x + y , y . \|x-y\|^{2}=\langle x-y,x-y\rangle=\langle x,x\rangle-\langle x,y\rangle-% \langle y,x\rangle+\langle y,y\rangle.\,
  8. x + y 2 + x - y 2 = 2 x , x + 2 y , y = 2 x 2 + 2 y 2 , \|x+y\|^{2}+\|x-y\|^{2}=2\langle x,x\rangle+2\langle y,y\rangle=2\|x\|^{2}+2\|% y\|^{2},\,
  9. x , y = 0 \langle x,\ y\rangle=0
  10. x + y 2 = x , x + x , y + y , x + y , y = x 2 + y 2 , \|x+y\|^{2}=\langle x,x\rangle+\langle x,y\rangle+\langle y,x\rangle+\langle y% ,y\rangle=\|x\|^{2}+\|y\|^{2},
  11. x p = ( i = 1 n | x i | p ) 1 / p , \|x\|_{p}=\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{1/p},
  12. x i x_{i}
  13. x x
  14. x , y = x + y 2 - x - y 2 4 , \langle x,y\rangle={\|x+y\|^{2}-\|x-y\|^{2}\over 4},\,
  15. x + y 2 - x 2 - y 2 2 or x 2 + y 2 - x - y 2 2 . {\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\over 2}\,\text{ or }{\|x\|^{2}+\|y\|^{2}-\|x-% y\|^{2}\over 2}.\,
  16. x , y = x + y 2 - x - y 2 4 + i i x - y 2 - i x + y 2 4 . \langle x,y\rangle={\|x+y\|^{2}-\|x-y\|^{2}\over 4}+i{\|ix-y\|^{2}-\|ix+y\|^{2% }\over 4}.
  17. x , y x,\ y\,
  18. x , y \displaystyle\langle x,y\rangle

Parameterized_complexity.html

  1. k k
  2. k k
  3. k k
  4. k k
  5. F P T FPT
  6. x x
  7. k k
  8. x x
  9. k k
  10. k k
  11. L Σ * × 𝒩 L\subseteq\Sigma^{*}\times\mathcal{N}
  12. Σ \Sigma
  13. L L
  14. ( x , k ) L (x,k)\in L
  15. f ( k ) | x | O ( 1 ) f(k)\cdot|x|^{O(1)}
  16. f f
  17. k k
  18. O ( k n + 1.274 k ) O(kn+1.274^{k})
  19. n n
  20. k k
  21. f ( k ) | x | O ( 1 ) f(k)\cdot{|x|}^{O(1)}
  22. f f
  23. 2 O ( k ) 2^{O(k)}
  24. f ( n , k ) f(n,k)
  25. n k n^{k}
  26. f ( k ) | x | f(k)\cdot|x|
  27. f f
  28. m m
  29. k k
  30. O ( 2 k m ) O(2^{k}m)
  31. k k
  32. n n
  33. O ( 2 k n ) O(2^{k}n)
  34. k k
  35. f ( k ) n O ( 1 ) f(k)n^{O(1)}
  36. k = 3 k=3
  37. f ( k ) + | x | O ( 1 ) f(k)+|x|^{O(1)}
  38. ( x , k ) (x,k)
  39. ( x , k ) L (x,k)\in L
  40. \subseteq
  41. i j i\leq j
  42. W [ t ] W[t]
  43. t t
  44. d d
  45. d t d\geq t
  46. W [ t , d ] W[t,d]
  47. W [ t ] = d t W [ t , d ] W[t]=\bigcup_{d\geq t}W[t,d]
  48. t t
  49. d d
  50. d d
  51. t t
  52. k k
  53. k k
  54. t t
  55. W [ t ] W[t]
  56. t t
  57. t t
  58. k k
  59. k k
  60. O ( f ( k ) log n ) O(f(k)\cdot\log n)
  61. ( x , k ) (x,k)
  62. exp ( o ( n ) ) m O ( 1 ) \exp(o(n))m^{O(1)}
  63. n f ( k ) n^{f(k)}
  64. f f

Parametric_equation.html

  1. x = cos t y = sin t \begin{aligned}\displaystyle x&\displaystyle=\cos t\\ \displaystyle y&\displaystyle=\sin t\end{aligned}
  2. y = x 2 y=x^{2}\,
  3. x = t , y = t 2 for - < t < . x=t,y=t^{2}\quad\mathrm{for}-\infty<t<\infty.\,
  4. x 2 + y 2 = 1. x^{2}+y^{2}=1.\,
  5. ( cos ( t ) , sin ( t ) ) for 0 t < 2 π . (\cos(t),\;\sin(t))\quad\mathrm{for}\ 0\leq t<2\pi.\,
  6. x = 1 - t 2 1 + t 2 y = 2 t 1 + t 2 . \begin{aligned}\displaystyle x&\displaystyle=\frac{1-t^{2}}{1+t^{2}}\\ \displaystyle y&\displaystyle=\frac{2t}{1+t^{2}}\end{aligned}.
  7. ( - 1 , 0 ) (-1,0)
  8. t t
  9. x x
  10. y y
  11. t t
  12. x = a cos t y = b sin t . \begin{aligned}\displaystyle x&\displaystyle=a\,\cos t\\ \displaystyle y&\displaystyle=b\,\sin t.\end{aligned}
  13. x = X c + a cos t cos φ - b sin t sin φ y = Y c + a cos t sin φ + b sin t cos φ \begin{aligned}\displaystyle x&\displaystyle=X_{c}+a\,\cos t\,\cos\varphi-b\,% \sin t\,\sin\varphi\\ \displaystyle y&\displaystyle=Y_{c}+a\,\cos t\,\sin\varphi+b\,\sin t\,\cos% \varphi\end{aligned}
  14. ( X c , Y c ) (X_{c},Y_{c})
  15. φ \varphi
  16. X X
  17. tan t 2 = u . \tan\frac{t}{2}=u.
  18. x = a sec t + h y = b tan t + k \begin{aligned}\displaystyle x&\displaystyle=a\sec t+h\\ \displaystyle y&\displaystyle=b\tan t+k\end{aligned}
  19. x = a 1 + t 2 1 - t 2 + h y = b 2 t 1 - t 2 + k \begin{aligned}\displaystyle x&\displaystyle=a\frac{1+t^{2}}{1-t^{2}}+h\\ \displaystyle y&\displaystyle=b\frac{2t}{1-t^{2}}+k\end{aligned}
  20. x = b tan t + h y = a sec t + k or , rationally , x = b 2 t 1 - t 2 + h y = a 1 + t 2 1 - t 2 + k \begin{matrix}x=b\tan t+h\\ y=a\sec t+k\\ \end{matrix}\qquad\mathrm{or,rationally,}\qquad\begin{matrix}x=b\frac{2t}{1-t^% {2}}+h\\ y=a\frac{1+t^{2}}{1-t^{2}}+k\\ \end{matrix}
  21. x ( θ ) \displaystyle x(\theta)
  22. x \displaystyle x
  23. x = cos ( a t ) - cos ( b t ) j y = sin ( c t ) - sin ( d t ) k \begin{aligned}\displaystyle x&\displaystyle=\cos(at)-\cos(bt)^{j}\\ \displaystyle y&\displaystyle=\sin(ct)-\sin(dt)^{k}\end{aligned}
  24. x = i cos ( a t ) - cos ( b t ) sin ( c t ) y = j sin ( d t ) - sin ( e t ) \begin{aligned}\displaystyle x&\displaystyle=i\cos(at)-\cos(bt)\sin(ct)\\ \displaystyle y&\displaystyle=j\sin(dt)-\sin(et)\end{aligned}
  25. x \displaystyle x
  26. 𝐫 ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) = ( a cos ( t ) , a sin ( t ) , b t ) , \mathbf{r}(t)=(x(t),y(t),z(t))=(a\cos(t),a\sin(t),bt),
  27. x = cos [ t ] [ R + r cos ( u ) ] , y = sin [ t ] [ R + r cos ( u ) ] , z = r sin [ u ] . \begin{aligned}\displaystyle x&\displaystyle=\cos[t]\left[R+r\cos(u)\right],\\ \displaystyle y&\displaystyle=\sin[t]\left[R+r\cos(u)\right],\\ \displaystyle z&\displaystyle=r\sin[u].\end{aligned}
  28. v ( t ) = r ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) = ( - a sin ( t ) , a cos ( t ) , b ) v(t)=r^{\prime}(t)=(x^{\prime}(t),y^{\prime}(t),z^{\prime}(t))=(-a\sin(t),a% \cos(t),b)\,
  29. a ( t ) = r ′′ ( t ) = ( x ′′ ( t ) , y ′′ ( t ) , z ′′ ( t ) ) = ( - a cos ( t ) , - a sin ( t ) , 0 ) a(t)=r^{\prime\prime}(t)=(x^{\prime\prime}(t),y^{\prime\prime}(t),z^{\prime% \prime}(t))=(-a\cos(t),-a\sin(t),0)\,
  30. y = f ( x ) y=f(x)\,\!
  31. y = m x + b y=mx+b\,\!
  32. f ( x , y ) = 0 f(x,y)=0\,\!
  33. ( x - a ) 2 + ( y - b ) 2 = r 2 \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}
  34. x = x ( t ) w ( t ) x=\frac{x(t)}{w(t)}
  35. y = y ( t ) w ( t ) y=\frac{y(t)}{w(t)}
  36. x = a 0 + a 1 t ; x=a_{0}+a_{1}t;\,\!
  37. y = b 0 + b 1 t y=b_{0}+b_{1}t\,\!
  38. x = a + r cos t ; x=a+r\,\cos t;\,\!
  39. y = b + r sin t y=b+r\,\sin t\,\!
  40. t t
  41. x = x ( t ) , y = y ( t ) . x=x(t),\ y=y(t).
  42. x = p ( t ) r ( t ) , y = q ( t ) r ( t ) , x=\frac{p(t)}{r(t)},\qquad y=\frac{q(t)}{r(t)},
  43. p , q , r p,q,r
  44. t t
  45. x r ( t ) p ( t ) xr(t)–p(t)
  46. y r ( t ) q ( t ) yr(t)–q(t)
  47. x = a cos ( t ) y = a sin ( t ) \begin{aligned}\displaystyle x&\displaystyle=a\cos(t)\\ \displaystyle y&\displaystyle=a\sin(t)\end{aligned}
  48. x a \displaystyle\frac{x}{a}
  49. a = 2 m n , b = m 2 - n 2 , c = m 2 + n 2 , a=2mn,\ \ b=m^{2}-n^{2},\ \ c=m^{2}+n^{2},

Parasitic_drag.html

  1. F d r a g = 1 2 ρ V 2 A s C D F_{drag}=\frac{1}{2}\rho V^{2}A_{s}C_{D}
  2. C D = C D , o + C D , i C_{D}=C_{D,o}+C_{D,i}
  3. C D , i = K C L 2 C_{D,i}=KC_{L}^{2}
  4. C f C_{f}
  5. C f τ w 1 2 ρ U 2 , C_{f}\equiv\frac{\tau_{w}}{\frac{1}{2}\,\rho\,U_{\infty}^{2}},
  6. τ w \tau_{w}
  7. ρ \rho
  8. U U_{\infty}
  9. C f = 2 d θ d x . C_{f}=2\frac{d\theta}{dx}.
  10. C f = 0.074 R e 0.2 , C_{f}=\frac{0.074}{Re^{0.2}},
  11. R e Re

Parseval's_identity.html

  1. n = - | c n | 2 = 1 2 π - π π | f ( x ) | 2 d x , \sum_{n=-\infty}^{\infty}|c_{n}|^{2}=\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{2}% \,dx,
  2. c n = 1 2 π - π π f ( x ) e - i n x d x . c_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\mathrm{e}^{-inx}\,dx.
  3. - | f ^ ( ξ ) | 2 d ξ = - | f ( x ) | 2 d x . \int_{-\infty}^{\infty}|\hat{f}(\xi)|^{2}\,d\xi=\int_{-\infty}^{\infty}|f(x)|^% {2}\,dx.
  4. e m , e n = { 1 if m = n 0 if m n . \langle e_{m},e_{n}\rangle=\begin{cases}1&\mbox{if}~{}\ m=n\\ 0&\mbox{if}~{}\ m\not=n.\end{cases}
  5. n | x , e n | 2 = x 2 . \sum_{n}|\langle x,e_{n}\rangle|^{2}=\|x\|^{2}.
  6. x 2 = x , x = v B | x , v | 2 . \|x\|^{2}=\langle x,x\rangle=\sum_{v\in B}\left|\langle x,v\rangle\right|^{2}.

Partial_discharge.html

  1. q = C b Δ ( V c ) q=C_{b}\Delta(V_{c})

Particle-in-cell.html

  1. 𝐱 k + 1 - 𝐱 k Δ t = 𝐯 k + 1 / 2 , \frac{\mathbf{x}_{k+1}-\mathbf{x}_{k}}{\Delta t}=\mathbf{v}_{k+1/2},
  2. 𝐯 k + 1 / 2 - 𝐯 k - 1 / 2 Δ t = q m ( 𝐄 k + 𝐯 k + 1 / 2 + 𝐯 k - 1 / 2 2 × 𝐁 k ) , \frac{\mathbf{v}_{k+1/2}-\mathbf{v}_{k-1/2}}{\Delta t}=\frac{q}{m}\left(% \mathbf{E}_{k}+\frac{\mathbf{v}_{k+1/2}+\mathbf{v}_{k-1/2}}{2}\times\mathbf{B}% _{k}\right),
  3. k k
  4. k + 1 k+1
  5. t k + 1 = t k + Δ t t_{k+1}=t_{k}+\Delta t
  6. t k t_{k}
  7. 𝐱 k + 1 = 𝐱 k + Δ t 𝐯 k + 1 / 2 , \mathbf{x}_{k+1}=\mathbf{x}_{k}+{\Delta t}\mathbf{v}_{k+1/2},
  8. 𝐯 k + 1 / 2 = 𝐮 + q 𝐄 k , \mathbf{v}_{k+1/2}=\mathbf{u}^{\prime}+q^{\prime}\mathbf{E}_{k},
  9. 𝐮 = 𝐮 + ( 𝐮 + ( 𝐮 × 𝐡 ) ) × 𝐬 , \mathbf{u}^{\prime}=\mathbf{u}+(\mathbf{u}+(\mathbf{u}\times\mathbf{h}))\times% \mathbf{s},
  10. 𝐮 = 𝐯 k - 1 / 2 + q 𝐄 k , \mathbf{u}=\mathbf{v}_{k-1/2}+q^{\prime}\mathbf{E}_{k},
  11. 𝐡 = q 𝐁 k , \mathbf{h}=q^{\prime}\mathbf{B}_{k},
  12. 𝐬 = 2 𝐡 / ( 1 + h 2 ) \mathbf{s}=2\mathbf{h}/(1+h^{2})
  13. q = Δ t × ( q / 2 m ) q^{\prime}=\Delta t\times(q/2m)
  14. S ( 𝐱 - 𝐗 ) , S(\mathbf{x}-\mathbf{X}),
  15. 𝐱 \mathbf{x}
  16. 𝐗 \mathbf{X}
  17. 𝐄 ( 𝐱 ) = i 𝐄 i S ( 𝐱 i - 𝐱 ) , \mathbf{E}(\mathbf{x})=\sum_{i}\mathbf{E}_{i}S(\mathbf{x}_{i}-\mathbf{x}),
  18. i i
  19. Δ x \Delta x
  20. Δ t \Delta t
  21. Δ x < 3.4 λ D , \Delta x<3.4\lambda_{D},
  22. Δ t 2 ω p e - 1 , \Delta t\leq 2\omega_{pe}^{-1},
  23. Δ t 0.1 ω p e - 1 , \Delta t\leq 0.1\omega_{pe}^{-1},
  24. ω p e - 1 \omega_{pe}^{-1}
  25. λ D \lambda_{D}
  26. Δ t < Δ x / c , \Delta t<\Delta x/c,
  27. Δ x λ D \Delta x\sim\lambda_{D}
  28. c c

Particle_displacement.html

  1. δ = t 𝐯 d t \mathbf{\delta}=\int_{t}\mathbf{v}\,\mathrm{d}t
  2. δ ( 𝐫 , t ) = δ cos ( 𝐤 𝐫 - ω t + φ δ , 0 ) , \delta(\mathbf{r},\,t)=\delta\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi_{% \delta,0}),
  3. φ δ , 0 \varphi_{\delta,0}
  4. v ( 𝐫 , t ) = δ t ( 𝐫 , t ) = ω δ cos ( 𝐤 𝐫 - ω t + φ δ , 0 + π 2 ) = v cos ( 𝐤 𝐫 - ω t + φ v , 0 ) , v(\mathbf{r},\,t)=\frac{\partial\delta}{\partial t}(\mathbf{r},\,t)=\omega% \delta\cos\!\left(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi_{\delta,0}+\frac{% \pi}{2}\right)=v\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi_{v,0}),
  5. p ( 𝐫 , t ) = - ρ c 2 δ x ( 𝐫 , t ) = ρ c 2 k x δ cos ( 𝐤 𝐫 - ω t + φ δ , 0 + π 2 ) = p cos ( 𝐤 𝐫 - ω t + φ p , 0 ) , p(\mathbf{r},\,t)=-\rho c^{2}\frac{\partial\delta}{\partial x}(\mathbf{r},\,t)% =\rho c^{2}k_{x}\delta\cos\!\left(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi_{% \delta,0}+\frac{\pi}{2}\right)=p\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+% \varphi_{p,0}),
  6. φ v , 0 \varphi_{v,0}
  7. φ p , 0 \varphi_{p,0}
  8. v ^ ( 𝐫 , s ) = v s cos φ v , 0 - ω sin φ v , 0 s 2 + ω 2 , \hat{v}(\mathbf{r},\,s)=v\frac{s\cos\varphi_{v,0}-\omega\sin\varphi_{v,0}}{s^{% 2}+\omega^{2}},
  9. p ^ ( 𝐫 , s ) = p s cos φ p , 0 - ω sin φ p , 0 s 2 + ω 2 . \hat{p}(\mathbf{r},\,s)=p\frac{s\cos\varphi_{p,0}-\omega\sin\varphi_{p,0}}{s^{% 2}+\omega^{2}}.
  10. φ v , 0 = φ p , 0 \varphi_{v,0}=\varphi_{p,0}
  11. z ( 𝐫 , s ) = | z ( 𝐫 , s ) | = | p ^ ( 𝐫 , s ) v ^ ( 𝐫 , s ) | = p v = ρ c 2 k x ω . z(\mathbf{r},\,s)=|z(\mathbf{r},\,s)|=\left|\frac{\hat{p}(\mathbf{r},\,s)}{% \hat{v}(\mathbf{r},\,s)}\right|=\frac{p}{v}=\frac{\rho c^{2}k_{x}}{\omega}.
  12. δ = v ω , \delta=\frac{v}{\omega},
  13. δ = p ω z ( 𝐫 , s ) . \delta=\frac{p}{\omega z(\mathbf{r},\,s)}.

Particle_horizon.html

  1. η \eta
  2. c c
  3. t t
  4. η = 0 t d t a ( t ) \eta=\int_{0}^{t}\frac{dt^{\prime}}{a(t^{\prime})}
  5. a ( t ) a(t)
  6. t = 0 t=0
  7. η ( t 0 ) = η 0 = 1.48 × 10 18 s \eta(t_{0})=\eta_{0}=1.48\times 10^{18}\ {\rm s}
  8. η 0 \eta_{0}
  9. a ( t 0 ) = 1 a(t_{0})=1
  10. t t
  11. a ( t ) H p ( t ) = a ( t ) 0 t c d t a ( t ) a(t)H_{p}(t)=a(t)\int_{0}^{t}\frac{cdt^{\prime}}{a(t^{\prime})}
  12. t = t 0 t=t_{0}
  13. H p ( t 0 ) = c η 0 = 14.4 Gpc = 46.9 billion light years H_{p}(t_{0})=c\eta_{0}=14.4\ {\rm Gpc}=46.9\ {\rm billion\ light\ years}
  14. ρ i \rho_{i}
  15. p i p_{i}
  16. p i = ω i ρ i p_{i}=\omega_{i}\rho_{i}
  17. ρ \rho
  18. p p
  19. H = a ˙ a H=\frac{\dot{a}}{a}
  20. ρ c = 3 8 π H 2 \rho_{c}=\frac{3}{8\pi}H^{2}
  21. Ω i = ρ i ρ c \Omega_{i}=\frac{\rho_{i}}{\rho_{c}}
  22. Ω = ρ ρ c = Ω i \Omega=\frac{\rho}{\rho_{c}}=\sum\Omega_{i}
  23. z z
  24. 1 + z = a 0 a ( t ) 1+z=\frac{a_{0}}{a(t)}
  25. t 0 t_{0}
  26. z = 0 z=0
  27. 1 1
  28. H ( z ) = H 0 Ω i 0 ( 1 + z ) n i H(z)=H_{0}\sqrt{\sum\Omega_{i0}(1+z)^{n_{i}}}
  29. n i = 3 ( 1 + ω i ) n_{i}=3(1+\omega_{i})
  30. The particle horizon H p exists if and only if N > 2 \,\text{The particle horizon }H_{p}\,\text{ exists if and only if }N>2
  31. N N
  32. n i n_{i}
  33. a ˙ > 0 \dot{a}>0
  34. d H p d t = H p ( z ) H ( z ) + c \frac{dH_{p}}{dt}=H_{p}(z)H(z)+c
  35. c c
  36. 1 1
  37. t t
  38. z z
  39. H p ( t CMB ) = c η CMB = 284 Mpc = 8.9 × 10 - 3 H p ( t 0 ) H_{p}(t_{\rm CMB})=c\eta_{\rm CMB}=284\ {\rm Mpc}=8.9\times 10^{-3}H_{p}(t_{0})
  40. a CMB H p ( t CMB ) = 261 kpc a_{\rm CMB}H_{p}(t_{\rm CMB})=261\ {\rm kpc}
  41. 284 Mpc 14.4 Gpc 284\,\text{ Mpc}\ll 14.4\,\text{ Gpc}
  42. f = H p ( t CMB ) / H p ( t 0 ) f=H_{p}(t_{\rm CMB})/H_{p}(t_{0})
  43. θ 7.1 \theta\sim 7.1^{\circ}

Particle_in_a_one-dimensional_lattice.html

  1. a a
  2. a a
  3. ψ ( x ) = e i k x u ( x ) . \psi(x)=e^{ikx}u(x).
  4. u ( x ) u(x)
  5. u ( x + a ) = u ( x ) u(x+a)=u(x)
  6. L L
  7. L a L≫a
  8. ψ ( 0 ) = ψ ( L ) . \psi(0)=\psi(L).
  9. N N
  10. a N = L aN=L
  11. k k
  12. ψ ( 0 ) = e i k 0 u ( 0 ) = e i k L u ( L ) = ψ ( L ) \psi(0)=e^{ik\cdot 0}u(0)=e^{ikL}u(L)=\psi(L)
  13. u ( 0 ) = e i k L u ( L ) = e i k L u ( N a ) e i k L = 1 u(0)=e^{ikL}u(L)=e^{ikL}u(Na)\to e^{ikL}=1
  14. k L = 2 π n k = 2 π L n ( n = 0 , ± 1 , , ± N 2 ) . \Rightarrow kL=2\pi n\to k={2\pi\over L}n\qquad\left(n=0,\pm 1,\cdots,\pm{N% \over 2}\right).
  15. u ( x ) u(x)
  16. For 0 < x < ( a - b ) \mathrm{For}\quad 0<x<(a-b)
  17. - 2 2 m ψ x x = E ψ {-\hbar^{2}\over 2m}\psi_{xx}=E\psi
  18. ψ = A e i α x + A e - i α x ( α 2 = 2 m E 2 ) \Rightarrow\psi=Ae^{i\alpha x}+A^{\prime}e^{-i\alpha x}\quad\left(\alpha^{2}={% 2mE\over\hbar^{2}}\right)
  19. For - b < x < 0 \mathrm{For}\quad-b<x<0
  20. - 2 2 m ψ x x = ( E + V 0 ) ψ {-\hbar^{2}\over 2m}\psi_{xx}=(E+V_{0})\psi
  21. ψ = B e i β x + B e - i β x ( β 2 = 2 m ( E + V 0 ) 2 ) . \Rightarrow\psi=Be^{i\beta x}+B^{\prime}e^{-i\beta x}\quad\left(\beta^{2}={2m(% E+V_{0})\over\hbar^{2}}\right).
  22. ψ ( 0 < x < a - b ) = A e i α x + A e - i α x = e i k x ( A e i ( α - k ) x + A e - i ( α + k ) x ) \psi(0<x<a-b)=Ae^{i\alpha x}+A^{\prime}e^{-i\alpha x}=e^{ikx}\cdot\left(Ae^{i(% \alpha-k)x}+A^{\prime}e^{-i(\alpha+k)x}\right)\,\!
  23. u ( 0 < x < a - b ) = A e i ( α - k ) x + A e - i ( α + k ) x . \Rightarrow u(0<x<a-b)=Ae^{i(\alpha-k)x}+A^{\prime}e^{-i(\alpha+k)x}.\,\!
  24. u ( - b < x < 0 ) = B e i ( β - k ) x + B e - i ( β + k ) x . u(-b<x<0)=Be^{i(\beta-k)x}+B^{\prime}e^{-i(\beta+k)x}.
  25. ψ ( 0 - ) = ψ ( 0 + ) ψ ( 0 - ) = ψ ( 0 + ) . \psi(0^{-})=\psi(0^{+})\qquad\psi^{\prime}(0^{-})=\psi^{\prime}(0^{+}).
  26. u ( x ) u(x)
  27. u ( x ) u′(x)
  28. u ( - b ) = u ( a - b ) u ( - b ) = u ( a - b ) . u(-b)=u(a-b)\qquad u^{\prime}(-b)=u^{\prime}(a-b).
  29. ( 1 1 - 1 - 1 α - α - β β e i ( α - k ) ( a - b ) e - i ( α + k ) ( a - b ) - e - i ( β - k ) b - e i ( β + k ) b ( α - k ) e i ( α - k ) ( a - b ) - ( α + k ) e - i ( α + k ) ( a - b ) - ( β - k ) e - i ( β - k ) b ( β + k ) e i ( β + k ) b ) ( A A B B ) = ( 0 0 0 0 ) . \begin{pmatrix}1&1&-1&-1\\ \alpha&-\alpha&-\beta&\beta\\ e^{i(\alpha-k)(a-b)}&e^{-i(\alpha+k)(a-b)}&-e^{-i(\beta-k)b}&-e^{i(\beta+k)b}% \\ (\alpha-k)e^{i(\alpha-k)(a-b)}&-(\alpha+k)e^{-i(\alpha+k)(a-b)}&-(\beta-k)e^{-% i(\beta-k)b}&(\beta+k)e^{i(\beta+k)b}\end{pmatrix}\begin{pmatrix}A\\ A^{\prime}\\ B\\ B^{\prime}\end{pmatrix}=\begin{pmatrix}0\\ 0\\ 0\\ 0\end{pmatrix}.
  30. cos ( k a ) = cos ( β b ) cos [ α ( a - b ) ] - α 2 + β 2 2 α β sin ( β b ) sin [ α ( a - b ) ] . \cos(ka)=\cos(\beta b)\cos[\alpha(a-b)]-{\alpha^{2}+\beta^{2}\over 2\alpha% \beta}\sin(\beta b)\sin[\alpha(a-b)].
  31. b 0 ; V 0 ; V 0 b = constant b\to 0;\quad V_{0}\to\infty;\quad V_{0}b=\mathrm{constant}
  32. β 2 b = constant ; α 2 b 0 \Rightarrow\beta^{2}b=\mathrm{constant};\quad\alpha^{2}b\to 0
  33. β b 0 ; sin ( β b ) β b ; cos ( β b ) 1. \Rightarrow\beta b\to 0;\quad\sin(\beta b)\to\beta b;\quad\cos(\beta b)\to 1.
  34. cos ( k a ) = cos ( α a ) - P sin ( α a ) α a , P = m V 0 b a 2 . \cos(ka)=\cos(\alpha a)-P\frac{\sin(\alpha a)}{\alpha a},\qquad P=\frac{mV_{0}% ba}{\hbar^{2}}.
  35. V ( x ) = A n = - δ ( x - n a ) . V(x)=A\cdot\sum_{n=-\infty}^{\infty}\delta(x-n\cdot a).
  36. A A
  37. a a
  38. V ( x ) = K V ~ ( K ) e i K x , V(x)=\sum_{K}\tilde{V}(K)\cdot e^{i\cdot K\cdot x},
  39. V ~ ( K ) = 1 a - a / 2 a / 2 d x V ( x ) e - i K x = 1 a - a / 2 a / 2 d x n = - A δ ( x - n a ) e - i K x = A a \tilde{V}(K)=\frac{1}{a}\int_{-a/2}^{a/2}dx\,V(x)\,e^{-i\cdot K\cdot x}=\frac{% 1}{a}\int_{-a/2}^{a/2}dx\sum_{n=-\infty}^{\infty}A\cdot\delta(x-na)\,e^{-i\,K% \,x}=\frac{A}{a}
  40. ψ k ( x ) = e i k x u k ( x ) \psi_{k}(x)=e^{ikx}u_{k}(x)
  41. u k ( x ) u_{k}(x)
  42. u k ( x ) = K u ~ k ( K ) e i K x . u_{k}(x)=\sum_{K}\tilde{u}_{k}(K)e^{iKx}.
  43. ψ k ( x ) = K u ~ k ( K ) e i ( k + K ) x . \psi_{k}(x)=\sum_{K}\tilde{u}_{k}(K)\,e^{i(k+K)x}.
  44. [ 2 ( k + K ) 2 2 m - E k ] u ~ k ( K ) + K V ~ ( K - K ) u ~ k ( K ) = 0 \left[\frac{\hbar^{2}(k+K)^{2}}{2m}-E_{k}\right]\cdot\tilde{u}_{k}(K)+\sum_{K^% {\prime}}\tilde{V}(K-K^{\prime})\,\tilde{u}_{k}(K^{\prime})=0
  45. [ 2 ( k + K ) 2 2 m - E k ] u ~ k ( K ) + A a K u ~ k ( K ) = 0 \left[\frac{\hbar^{2}(k+K)^{2}}{2m}-E_{k}\right]\cdot\tilde{u}_{k}(K)+\frac{A}% {a}\sum_{K^{\prime}}\tilde{u}_{k}(K^{\prime})=0
  46. f ( k ) := K u ~ k ( K ) f(k):=\sum_{K^{\prime}}\tilde{u}_{k}(K^{\prime})
  47. [ 2 ( k + K ) 2 2 m - E k ] u ~ k ( K ) + A a f ( k ) = 0 \left[\frac{\hbar^{2}(k+K)^{2}}{2m}-E_{k}\right]\cdot\tilde{u}_{k}(K)+\frac{A}% {a}f(k)=0
  48. u ~ k ( K ) \tilde{u}_{k}(K)
  49. u ~ k ( K ) = 2 m 2 A a f ( k ) 2 m E k 2 - ( k + K ) 2 = 2 m 2 A a 2 m E k 2 - ( k + K ) 2 f ( k ) \tilde{u}_{k}(K)=\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}f(k)}{\frac{2mE_{k}}{% \hbar^{2}}-(k+K)^{2}}=\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}}{\frac{2mE_{k}}{% \hbar^{2}}-(k+K)^{2}}\,f(k)
  50. K K
  51. K u ~ k ( K ) = K 2 m 2 A a 2 m E k 2 - ( k + K ) 2 f ( k ) \sum_{K}\tilde{u}_{k}(K)=\sum_{K}\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}}{\frac{% 2mE_{k}}{\hbar^{2}}-(k+K)^{2}}\,f(k)
  52. f ( k ) = K 2 m 2 A a 2 m E k 2 - ( k + K ) 2 f ( k ) f(k)=\sum_{K}\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}}{\frac{2mE_{k}}{\hbar^{2}}-% (k+K)^{2}}\,f(k)
  53. f ( k ) f(k)
  54. 1 = K 2 m 2 A a 2 m E k 2 - ( k + K ) 2 1=\sum_{K}\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}}{\frac{2mE_{k}}{\hbar^{2}}-(k+% K)^{2}}
  55. 2 2 m a A = K 1 2 m E k 2 - ( k + K ) 2 \frac{\hbar^{2}}{2m}\frac{a}{A}=\sum_{K}\frac{1}{\frac{2mE_{k}}{\hbar^{2}}-(k+% K)^{2}}
  56. α 2 := 2 m E k 2 \alpha^{2}:=\frac{2mE_{k}}{\hbar^{2}}
  57. 2 2 m a A = K 1 α 2 - ( k + K ) 2 \frac{\hbar^{2}}{2m}\frac{a}{A}=\sum_{K}\frac{1}{\alpha^{2}-(k+K)^{2}}
  58. K K
  59. K K
  60. 2 π a \frac{2\pi}{a}
  61. 2 2 m a A = n = - 1 α 2 - ( k + 2 π n a ) 2 \frac{\hbar^{2}}{2m}\frac{a}{A}=\sum_{n=-\infty}^{\infty}\frac{1}{\alpha^{2}-(% k+\frac{2\pi n}{a})^{2}}
  62. 2 2 m a A \displaystyle\frac{\hbar^{2}}{2m}\frac{a}{A}
  63. cot ( x ) = n = - 1 n π + x \cot(x)=\sum_{n=-\infty}^{\infty}\frac{1}{n\pi+x}
  64. 2 2 m a A = - a 4 α [ cot ( k a 2 - α a 2 ) - cot ( k a 2 + α a 2 ) ] \frac{\hbar^{2}}{2m}\frac{a}{A}=-\frac{a}{4\alpha}\left[\cot\left(\tfrac{ka}{2% }-\tfrac{\alpha a}{2}\right)-\cot\left(\tfrac{ka}{2}+\tfrac{\alpha a}{2}\right% )\right]
  65. c o t cot
  66. s i n sin
  67. c o t cot
  68. cos ( k a ) = cos ( α a ) + m A 2 α sin ( α a ) \cos(ka)=\cos(\alpha a)+\frac{mA}{\hbar^{2}\alpha}\sin(\alpha a)
  69. α α
  70. k k
  71. 1 −1
  72. 1 1
  73. α α

Particle_in_a_ring.html

  1. S 1 S^{1}
  2. - 2 2 m 2 ψ = E ψ -\frac{\hbar^{2}}{2m}\nabla^{2}\psi=E\psi
  3. 2 = 1 R 2 2 θ 2 \nabla^{2}=\frac{1}{R^{2}}\frac{\partial^{2}}{\partial\theta^{2}}
  4. θ \ \theta
  5. 2 π 2\pi
  6. 0 2 π | ψ ( θ ) | 2 R d θ = 1 \int_{0}^{2\pi}\left|\psi(\theta)\right|^{2}\,Rd\theta=1
  7. ψ ( θ ) = ψ ( θ + 2 π ) \ \psi(\theta)=\ \psi(\theta+2\pi)
  8. ψ ± ( θ ) = 1 2 π R e ± i R 2 m E θ \psi_{\pm}(\theta)=\frac{1}{\sqrt{2\pi R}}\,e^{\pm i\frac{R}{\hbar}\sqrt{2mE}\theta}
  9. E E
  10. e ± i r 2 m E θ = e ± i r 2 m E ( θ + 2 π ) e^{\pm i\frac{r}{\hbar}\sqrt{2mE}\theta}=e^{\pm i\frac{r}{\hbar}\sqrt{2mE}(% \theta+2\pi)}
  11. e ± i 2 π r 2 m E = 1 = e i 2 π n e^{\pm i2\pi\frac{r}{\hbar}\sqrt{2mE}}=1=e^{i2\pi n}
  12. ψ ( θ ) = 1 2 π e ± i n θ \psi(\theta)=\frac{1}{\sqrt{2\pi}}\,e^{\pm in\theta}
  13. E n = n 2 2 2 m r 2 E_{n}=\frac{n^{2}\hbar^{2}}{2mr^{2}}
  14. n = 0 , ± 1 , ± 2 , ± 3 , n=0,\pm 1,\pm 2,\pm 3,\ldots
  15. n > 0 n>0
  16. e ± i n θ \ e^{\pm in\theta}
  17. 2 × ( 2 n + 1 ) 2\times(2n+1)
  18. 4 n + 2 4n+2

Particle_in_a_spherically_symmetric_potential.html

  1. H ^ = p ^ 2 2 m 0 + V ( r ) \hat{H}=\frac{\hat{p}^{2}}{2m_{0}}+V(r)
  2. m 0 m_{0}
  3. p ^ \hat{p}
  4. V ( r ) V(r)
  5. r r
  6. r r
  7. θ \theta
  8. ϕ \phi
  9. r r
  10. V ( r ) V(r)
  11. ψ ( r , θ , ϕ ) = R ( r ) Θ ( θ ) Φ ( ϕ ) \psi(r,\theta,\phi)=R(r)\Theta(\theta)\Phi(\phi)\,\;
  12. ψ ( r , θ , ϕ ) = R ( r ) Y l m ( θ , ϕ ) . \psi(r,\theta,\phi)=R(r)Y_{lm}(\theta,\phi).\,
  13. R ( r ) R(r)\,\;
  14. p ^ 2 2 m 0 = - 2 2 m 0 2 = - 2 2 m 0 r 2 [ r ( r 2 r ) - l ^ 2 ] . \frac{\hat{p}^{2}}{2m_{0}}=-\frac{\hbar^{2}}{2m_{0}}\nabla^{2}=-\frac{\hbar^{2% }}{2m_{0}\,r^{2}}\left[\frac{\partial}{\partial r}\Big(r^{2}\frac{\partial}{% \partial r}\Big)-\hat{l}^{2}\right].
  15. l ^ 2 Y l m ( θ , ϕ ) { - 1 sin 2 θ [ sin θ θ ( sin θ θ ) + 2 ϕ 2 ] } Y l m ( θ , ϕ ) = l ( l + 1 ) Y l m ( θ , ϕ ) . \hat{l}^{2}Y_{lm}(\theta,\phi)\equiv\left\{-\frac{1}{\sin^{2}\theta}\left[\sin% \theta\frac{\partial}{\partial\theta}\Big(\sin\theta\frac{\partial}{\partial% \theta}\Big)+\frac{\partial^{2}}{\partial\phi^{2}}\right]\right\}Y_{lm}(\theta% ,\phi)=l(l+1)Y_{lm}(\theta,\phi).
  16. { - 2 2 m 0 r 2 d d r ( r 2 d d r ) + 2 l ( l + 1 ) 2 m 0 r 2 + V ( r ) } R ( r ) = E R ( r ) . \left\{-{\hbar^{2}\over 2m_{0}r^{2}}{d\over dr}\left(r^{2}{d\over dr}\right)+{% \hbar^{2}l(l+1)\over 2m_{0}r^{2}}+V(r)\right\}R(r)=ER(r).
  17. T r 1 r 2 d d r r 2 d d r = 1 r d 2 d r 2 r . T_{r}\equiv\frac{1}{r^{2}}\frac{d}{dr}r^{2}\frac{d}{dr}=\frac{1}{r}\frac{d^{2}% }{dr^{2}}r.
  18. u ( r ) = def r R ( r ) u(r)\ \stackrel{\mathrm{def}}{=}\ rR(r)
  19. - 2 2 m 0 r d 2 d r 2 ( r R ( r ) ) + 2 l ( l + 1 ) 2 m 0 r 2 R ( r ) + V ( r ) R ( r ) = E R ( r ) , -{\hbar^{2}\over 2m_{0}r}{d^{2}\over dr^{2}}\left(rR(r)\right)+{\hbar^{2}l(l+1% )\over 2m_{0}r^{2}}R(r)+V(r)R(r)=ER(r),
  20. - 2 2 m 0 d 2 u d r 2 + V eff ( r ) u ( r ) = E u ( r ) -{\hbar^{2}\over 2m_{0}}{d^{2}u\over dr^{2}}+V_{\mathrm{eff}}(r)u(r)=Eu(r)
  21. V eff ( r ) = V ( r ) + 2 l ( l + 1 ) 2 m 0 r 2 , V_{\mathrm{eff}}(r)=V(r)+{\hbar^{2}l(l+1)\over 2m_{0}r^{2}},
  22. \infty
  23. V ( r ) = V 0 V(r)=V_{0}
  24. r < r 0 r<r_{0}
  25. V 0 V_{0}
  26. E - V 0 E-V_{0}
  27. ρ = def k r , k = def 2 m 0 E 2 \rho\ \stackrel{\mathrm{def}}{=}\ kr,\qquad k\ \stackrel{\mathrm{def}}{=}\ % \sqrt{2m_{0}E\over\hbar^{2}}
  28. J ( ρ ) = def ρ R ( r ) J(\rho)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\rho}R(r)
  29. ρ 2 d 2 J d ρ 2 + ρ d J d ρ + [ ρ 2 - ( l + 1 2 ) 2 ] J = 0 \rho^{2}{d^{2}J\over d\rho^{2}}+\rho{dJ\over d\rho}+\left[\rho^{2}-\left(l+% \frac{1}{2}\right)^{2}\right]J=0
  30. J l + 1 / 2 ( ρ ) J_{l+1/2}(\rho)
  31. R ( r ) = j l ( k r ) = def π / ( 2 k r ) J l + 1 / 2 ( k r ) R(r)=j_{l}(kr)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\pi/(2kr)}J_{l+1/2}(kr)
  32. m 0 m_{0}
  33. [ 0 , ) [0,\infty)
  34. ψ ( 𝐫 ) = j l ( k r ) Y l m ( θ , ϕ ) \psi(\mathbf{r})=j_{l}(kr)Y_{lm}(\theta,\phi)
  35. k = def 2 m 0 E / k\ \stackrel{\mathrm{def}}{=}\ \sqrt{2m_{0}E}/\hbar
  36. j l j_{l}
  37. Y l m Y_{lm}
  38. exp ( i 𝐤 𝐫 ) \exp(i\mathbf{k}\cdot\mathbf{r})
  39. V ( r ) = V 0 V(r)=V_{0}
  40. r < r 0 r<r_{0}
  41. V ( r ) = 0 V(r)=0
  42. r 0 r_{0}
  43. R ( r ) = A j l ( 2 m 0 ( E - V 0 ) 2 r ) , r < r 0 R(r)=Aj_{l}\left(\sqrt{2m_{0}(E-V_{0})\over\hbar^{2}}r\right),\qquad r<r_{0}
  44. V 0 < E < 0 V_{0}<E<0
  45. R ( r ) = B h l ( 1 ) ( i - 2 m 0 E 2 r ) , r > r 0 R(r)=Bh^{(1)}_{l}\left(i\sqrt{-2m_{0}E\over\hbar^{2}}r\right),\qquad r>r_{0}
  46. r = r 0 r=r_{0}
  47. V 0 = 0 V_{0}=0
  48. \infty
  49. u l , k u_{l,k}
  50. j l j_{l}
  51. E k l = u l , k 2 2 2 m 0 r 0 2 E_{kl}={u_{l,k}^{2}\hbar^{2}\over 2m_{0}r_{0}^{2}}
  52. u l , k u_{l,k}
  53. l = 0 l=0
  54. j 0 ( x ) = sin x x j_{0}(x)=\frac{\sin x}{x}
  55. u 0 , k = k π u_{0,k}=k\pi
  56. E k 0 = ( k π ) 2 2 2 m 0 r 0 2 = k 2 h 2 8 m 0 r 0 2 E_{k0}={(k\pi)^{2}\hbar^{2}\over 2m_{0}r_{0}^{2}}={k^{2}h^{2}\over 8m_{0}r_{0}% ^{2}}
  57. V ( r ) = 1 2 m 0 ω 2 r 2 . V(r)=\frac{1}{2}m_{0}\omega^{2}r^{2}.
  58. E n = ω ( n + N 2 ) with n = 0 , 1 , , , E_{n}=\hbar\omega\Bigl(n+\frac{N}{2}\Bigr)\quad\,\text{with}\quad n=0,1,\ldots% ,\infty,
  59. [ - 2 2 m 0 d 2 d r 2 + 2 l ( l + 1 ) 2 m 0 r 2 + 1 2 m 0 ω 2 r 2 - ω ( n + 3 2 ) ] u ( r ) = 0. \left[-{\hbar^{2}\over 2m_{0}}{d^{2}\over dr^{2}}+{\hbar^{2}l(l+1)\over 2m_{0}% r^{2}}+\frac{1}{2}m_{0}\omega^{2}r^{2}-\hbar\omega\bigl(n+\tfrac{3}{2}\bigr)% \right]u(r)=0.
  60. γ m 0 ω \gamma\equiv\frac{m_{0}\omega}{\hbar}
  61. u ( r ) = r R ( r ) u(r)=rR(r)\,
  62. R n , l ( r ) = N n l r l e - 1 2 γ r 2 L 1 2 ( n - l ) ( l + 1 2 ) ( γ r 2 ) , R_{n,l}(r)=N_{nl}\,r^{l}\,e^{-\frac{1}{2}\gamma r^{2}}\;L^{(l+\frac{1}{2})}_{% \frac{1}{2}(n-l)}(\gamma r^{2}),
  63. L k ( α ) ( γ r 2 ) L^{(\alpha)}_{k}(\gamma r^{2})
  64. N n l = [ 2 n + l + 2 γ l + 3 2 π 1 2 ] 1 2 [ [ 1 2 ( n - l ) ] ! [ 1 2 ( n + l ) ] ! ( n + l + 1 ) ! ] 1 2 . N_{nl}=\left[\frac{2^{n+l+2}\,\gamma^{l+\frac{3}{2}}}{\pi^{\frac{1}{2}}}\right% ]^{\frac{1}{2}}\left[\frac{[\frac{1}{2}(n-l)]!\;[\frac{1}{2}(n+l)]!}{(n+l+1)!}% \right]^{\frac{1}{2}}.
  65. Y l m ( θ , ϕ ) Y_{lm}(\theta,\phi)\,
  66. l = n , n - 2 , , l min with l min = { 1 if n odd 0 if n even l=n,n-2,\ldots,l_{\min}\quad\hbox{with}\quad l_{\min}=\begin{cases}1&\mathrm{% if}\;n\;\mathrm{odd}\\ 0&\mathrm{if}\;n\;\mathrm{even}\end{cases}
  67. γ = 2 ν \gamma=2\nu\,
  68. y = γ r with γ m 0 ω , y=\sqrt{\gamma}r\quad\hbox{with}\quad\gamma\equiv\frac{m_{0}\omega}{\hbar},
  69. [ d 2 d y 2 - l ( l + 1 ) y 2 - y 2 + 2 n - 3 ] v ( y ) = 0 \left[{d^{2}\over dy^{2}}-{l(l+1)\over y^{2}}-y^{2}+2n-3\right]v(y)=0
  70. v ( y ) = u ( y / γ ) v(y)=u\left(y/\sqrt{\gamma}\right)
  71. v ( y ) = y l + 1 e - y 2 / 2 f ( y ) . v(y)=y^{l+1}e^{-y^{2}/2}f(y).
  72. [ d 2 d y 2 + 2 ( l + 1 y - y ) d d y + 2 n - 2 l ] f ( y ) = 0 , \left[{d^{2}\over dy^{2}}+2\left(\frac{l+1}{y}-y\right)\frac{d}{dy}+2n-2l% \right]f(y)=0,
  73. y l + 1 e - y 2 / 2 y^{l+1}e^{-y^{2}/2}
  74. x = y 2 x=y^{2}\,
  75. y = x y=\sqrt{x}
  76. d d y = d x d y d d x = 2 y d d x = 2 x d d x , and \frac{d}{dy}=\frac{dx}{dy}\frac{d}{dx}=2y\frac{d}{dx}=2\sqrt{x}\frac{d}{dx},\,% \text{ and }
  77. d 2 d y 2 = d d y ( 2 y d d x ) = 4 x d 2 d x 2 + 2 d d x . \frac{d^{2}}{dy^{2}}=\frac{d}{dy}\left(2y\frac{d}{dx}\right)=4x\frac{d^{2}}{dx% ^{2}}+2\frac{d}{dx}.
  78. x d 2 g d x 2 + ( ( l + 1 2 ) + 1 - x ) d g d x + 1 2 ( n - l ) g ( x ) = 0 x\frac{d^{2}g}{dx^{2}}+\Big((l+\frac{1}{2})+1-x\Big)\frac{dg}{dx}+\frac{1}{2}(% n-l)g(x)=0
  79. g ( x ) f ( x ) g(x)\equiv f(\sqrt{x})\,\;
  80. k ( n - l ) / 2 k\equiv(n-l)/2\,
  81. g ( x ) = L k ( l + 1 2 ) ( x ) . g(x)=L_{k}^{(l+\frac{1}{2})}(x).
  82. n l n\geq l\,
  83. u ( r ) = r R ( r ) u(r)=rR(r)\,
  84. R n , l ( r ) = N n l r l e - 1 2 γ r 2 L 1 2 ( n - l ) ( l + 1 2 ) ( γ r 2 ) . R_{n,l}(r)=N_{nl}\,r^{l}\,e^{-\frac{1}{2}\gamma r^{2}}\;L^{(l+\frac{1}{2})}_{% \frac{1}{2}(n-l)}(\gamma r^{2}).
  85. 0 r 2 | R ( r ) | 2 d r = 1. \int^{\infty}_{0}r^{2}|R(r)|^{2}\,dr=1.
  86. q = γ r 2 q=\gamma r^{2}\,\;
  87. d q = 2 γ r d r dq=2\gamma r\,dr\,\;
  88. N n l 2 2 γ l + 3 2 0 q l + 1 2 e - q [ L 1 2 ( n - l ) ( l + 1 2 ) ( q ) ] 2 d q = 1. \frac{N^{2}_{nl}}{2\gamma^{l+{3\over 2}}}\int^{\infty}_{0}q^{l+{1\over 2}}e^{-% q}\left[L^{(l+\frac{1}{2})}_{\frac{1}{2}(n-l)}(q)\right]^{2}\,dq=1.
  89. N n l 2 2 γ l + 3 2 Γ [ 1 2 ( n + l + 1 ) + 1 ] [ 1 2 ( n - l ) ] ! = 1. \frac{N^{2}_{nl}}{2\gamma^{l+{3\over 2}}}\cdot\frac{\Gamma[\frac{1}{2}(n+l+1)+% 1]}{[\frac{1}{2}(n-l)]!}=1.
  90. N n l = 2 γ l + 3 2 ( n - l 2 ) ! Γ ( n + l 2 + 3 2 ) N_{nl}=\sqrt{\frac{2\,\gamma^{l+{3\over 2}}\,(\frac{n-l}{2})!}{\Gamma(\frac{n+% l}{2}+\frac{3}{2})}}
  91. Γ [ 1 2 + ( n + l 2 + 1 ) ] = π ( n + l + 1 ) ! ! 2 n + l 2 + 1 = π ( n + l + 1 ) ! 2 n + l + 1 [ 1 2 ( n + l ) ] ! , \Gamma\left[{1\over 2}+\left(\frac{n+l}{2}+1\right)\right]=\frac{\sqrt{\pi}(n+% l+1)!!}{2^{\frac{n+l}{2}+1}}=\frac{\sqrt{\pi}(n+l+1)!}{2^{n+l+1}[\frac{1}{2}(n% +l)]!},
  92. N n l = [ 2 n + l + 2 γ l + 3 2 [ 1 2 ( n - l ) ] ! [ 1 2 ( n + l ) ] ! π 1 2 ( n + l + 1 ) ! ] 1 2 = 2 ( γ π ) 1 4 ( 2 γ ) 2 2 γ ( n - l ) ! ! ( n + l + 1 ) ! ! . N_{nl}=\left[\frac{2^{n+l+2}\,\gamma^{l+{3\over 2}}\,[{1\over 2}(n-l)]!\;[{1% \over 2}(n+l)]!}{\;\pi^{1\over 2}(n+l+1)!}\right]^{1\over 2}=\sqrt{2}\left(% \frac{\gamma}{\pi}\right)^{1\over 4}\,({2\gamma})^{\ell\over 2}\,\sqrt{\frac{2% \gamma(n-l)!!}{(n+l+1)!!}}.
  93. V ( r ) = - 1 4 π ϵ 0 Z e 2 r V(r)=-\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}
  94. E h = m e ( e 2 4 π ε 0 ) 2 and a 0 = 4 π ε 0 2 m e e 2 . E_{\textrm{h}}=m_{\textrm{e}}\left(\frac{e^{2}}{4\pi\varepsilon_{0}\hbar}% \right)^{2}\quad\hbox{and}\quad a_{0}={{4\pi\varepsilon_{0}\hbar^{2}}\over{m_{% \textrm{e}}e^{2}}}.
  95. y = Z r / a 0 y=Zr/a_{0}\,
  96. W = E / ( Z 2 E h ) W=E/(Z^{2}E_{\textrm{h}})\,
  97. [ - 1 2 d 2 d y 2 + 1 2 l ( l + 1 ) y 2 - 1 y ] u l = W u l . \left[-\frac{1}{2}\frac{d^{2}}{dy^{2}}+\frac{1}{2}\frac{l(l+1)}{y^{2}}-\frac{1% }{y}\right]u_{l}=Wu_{l}.
  98. α 2 - 2 W \alpha\equiv 2\sqrt{-2W}
  99. x α y x\equiv\alpha y
  100. [ d 2 d x 2 - l ( l + 1 ) x 2 + 2 α x - 1 4 ] u l = 0 , with x 0. \left[\frac{d^{2}}{dx^{2}}-\frac{l(l+1)}{x^{2}}+\frac{2}{\alpha x}-\frac{1}{4}% \right]u_{l}=0,\quad\,\text{with }x\geq 0.
  101. x x\rightarrow\infty
  102. exp [ - x / 2 ] \exp[-x/2]
  103. exp [ x / 2 ] \exp[x/2]
  104. x 0 x\rightarrow 0
  105. u l ( x ) = x l + 1 e - x / 2 f l ( x ) . u_{l}(x)=x^{l+1}e^{-x/2}f_{l}(x).\,
  106. [ x d 2 d x 2 + ( 2 l + 2 - x ) d d x + ( ν - l - 1 ) ] f l ( x ) = 0 with ν = ( - 2 W ) - 1 2 . \left[x\frac{d^{2}}{dx^{2}}+(2l+2-x)\frac{d}{dx}+(\nu-l-1)\right]f_{l}(x)=0% \quad\hbox{with}\quad\nu=(-2W)^{-\frac{1}{2}}.
  107. ν - l - 1 \nu-l-1
  108. L k ( 2 l + 1 ) ( x ) , k = 0 , 1 , , L^{(2l+1)}_{k}(x),\qquad k=0,1,\ldots,
  109. W = - 1 2 n 2 with n k + l + 1. W=-\frac{1}{2n^{2}}\quad\hbox{with}\quad n\equiv k+l+1.
  110. n l + 1 n\geq l+1
  111. l n - 1 l\leq n-1
  112. α = 2 / n \alpha=2/n
  113. R n l ( r ) = N n l ( 2 Z r n a 0 ) l e - Z r n a 0 L n - l - 1 ( 2 l + 1 ) ( 2 Z r n a 0 ) , R_{nl}(r)=N_{nl}\left(\frac{2Zr}{na_{0}}\right)^{l}\;e^{-{\textstyle\frac{Zr}{% na_{0}}}}\;L^{(2l+1)}_{n-l-1}\left(\frac{2Zr}{na_{0}}\right),
  114. N n l = [ ( 2 Z n a 0 ) 3 ( n - l - 1 ) ! 2 n [ ( n + l ) ! ] 3 ] 1 2 N_{nl}=\left[\left(\frac{2Z}{na_{0}}\right)^{3}\cdot\frac{(n-l-1)!}{2n[(n+l)!]% ^{3}}\right]^{1\over 2}
  115. E = - Z 2 2 n 2 E h , n = 1 , 2 , . E=-\frac{Z^{2}}{2n^{2}}E_{\textrm{h}},\qquad n=1,2,\ldots.
  116. 0 x 2 l + 2 e - x [ L n - l - 1 ( 2 l + 1 ) ( x ) ] 2 d x = 2 n ( n + l ) ! ( n - l - 1 ) ! . \int_{0}^{\infty}x^{2l+2}e^{-x}\left[L^{(2l+1)}_{n-l-1}(x)\right]^{2}dx=\frac{% 2n(n+l)!}{(n-l-1)!}.
  117. L ¯ n + k ( k ) = ( - 1 ) k ( n + k ) ! L n ( k ) \bar{L}^{(k)}_{n+k}=(-1)^{k}(n+k)!L^{(k)}_{n}

Particle_velocity.html

  1. 𝐯 = δ t \mathbf{v}=\frac{\partial\mathbf{\delta}}{\partial t}
  2. δ ( 𝐫 , t ) = δ m cos ( 𝐤 𝐫 - ω t + φ δ , 0 ) , \delta(\mathbf{r},\,t)=\delta_{\mathrm{m}}\cos(\mathbf{k}\cdot\mathbf{r}-% \omega t+\varphi_{\delta,0}),
  3. φ δ , 0 \varphi_{\delta,0}
  4. v ( 𝐫 , t ) = δ t ( 𝐫 , t ) = ω δ m cos ( 𝐤 𝐫 - ω t + φ δ , 0 + π 2 ) = v m cos ( 𝐤 𝐫 - ω t + φ v , 0 ) , v(\mathbf{r},\,t)=\frac{\partial\delta}{\partial t}(\mathbf{r},\,t)=\omega% \delta_{\mathrm{m}}\cos\!\left(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi_{% \delta,0}+\frac{\pi}{2}\right)=v_{\mathrm{m}}\cos(\mathbf{k}\cdot\mathbf{r}-% \omega t+\varphi_{v,0}),
  5. p ( 𝐫 , t ) = - ρ c 2 δ x ( 𝐫 , t ) = ρ c 2 k x δ m cos ( 𝐤 𝐫 - ω t + φ δ , 0 + π 2 ) = p m cos ( 𝐤 𝐫 - ω t + φ p , 0 ) , p(\mathbf{r},\,t)=-\rho c^{2}\frac{\partial\delta}{\partial x}(\mathbf{r},\,t)% =\rho c^{2}k_{x}\delta_{\mathrm{m}}\cos\!\left(\mathbf{k}\cdot\mathbf{r}-% \omega t+\varphi_{\delta,0}+\frac{\pi}{2}\right)=p_{\mathrm{m}}\cos(\mathbf{k}% \cdot\mathbf{r}-\omega t+\varphi_{p,0}),
  6. φ v , 0 \varphi_{v,0}
  7. φ p , 0 \varphi_{p,0}
  8. v ^ ( 𝐫 , s ) = v m s cos φ v , 0 - ω sin φ v , 0 s 2 + ω 2 , \hat{v}(\mathbf{r},\,s)=v_{\mathrm{m}}\frac{s\cos\varphi_{v,0}-\omega\sin% \varphi_{v,0}}{s^{2}+\omega^{2}},
  9. p ^ ( 𝐫 , s ) = p m s cos φ p , 0 - ω sin φ p , 0 s 2 + ω 2 . \hat{p}(\mathbf{r},\,s)=p_{\mathrm{m}}\frac{s\cos\varphi_{p,0}-\omega\sin% \varphi_{p,0}}{s^{2}+\omega^{2}}.
  10. φ v , 0 = φ p , 0 \varphi_{v,0}=\varphi_{p,0}
  11. z m ( 𝐫 , s ) = | z ( 𝐫 , s ) | = | p ^ ( 𝐫 , s ) v ^ ( 𝐫 , s ) | = p m v m = ρ c 2 k x ω . z_{\mathrm{m}}(\mathbf{r},\,s)=|z(\mathbf{r},\,s)|=\left|\frac{\hat{p}(\mathbf% {r},\,s)}{\hat{v}(\mathbf{r},\,s)}\right|=\frac{p_{\mathrm{m}}}{v_{\mathrm{m}}% }=\frac{\rho c^{2}k_{x}}{\omega}.
  12. v m = ω δ m , v_{\mathrm{m}}=\omega\delta_{\mathrm{m}},
  13. v m = p m z m ( 𝐫 , s ) . v_{\mathrm{m}}=\frac{p_{\mathrm{m}}}{z_{\mathrm{m}}(\mathbf{r},\,s)}.
  14. L v = ln ( v v 0 ) Np = 2 log 10 ( v v 0 ) B = 20 log 10 ( v v 0 ) dB , L_{v}=\ln\!\left(\frac{v}{v_{0}}\right)\!~{}\mathrm{Np}=2\log_{10}\!\left(% \frac{v}{v_{0}}\right)\!~{}\mathrm{B}=20\log_{10}\!\left(\frac{v}{v_{0}}\right% )\!~{}\mathrm{dB},
  15. v 0 = 5 × 10 - 8 m / s . v_{0}=5\times 10^{-8}~{}\mathrm{m/s}.

Partition_of_a_set.html

  1. P \emptyset\notin P
  2. A P A = X \bigcup_{A\in P}A=X
  3. A , B P A,B\in P
  4. A B A\neq B
  5. A B = A\cap B=\emptyset
  6. \emptyset
  7. n - 2 n-2
  8. B n + 1 = k = 0 n ( n k ) B k B_{n+1}=\sum_{k=0}^{n}{n\choose k}B_{k}
  9. n = 0 B n n ! z n = e e z - 1 . \sum_{n=0}^{\infty}\frac{B_{n}}{n!}z^{n}=e^{e^{z}-1}.
  10. C n = 1 n + 1 ( 2 n n ) . C_{n}={1\over n+1}{2n\choose n}.

Partition_of_an_interval.html

  1. x 0 , , x n x_{0},\ldots,x_{n}
  2. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  3. [ a , b ] [a,b]
  4. y 0 , , y m y_{0},\ldots,y_{m}
  5. s 0 , , s m - 1 s_{0},\ldots,s_{m-1}
  6. [ a , b ] [a,b]
  7. y 0 , , y m y_{0},\ldots,y_{m}
  8. s 0 , , s m - 1 s_{0},\ldots,s_{m-1}
  9. x 0 , , x n x_{0},\ldots,x_{n}
  10. t 0 , , t n - 1 t_{0},\ldots,t_{n-1}
  11. i i
  12. 0 i n 0\leq i\leq n
  13. r ( i ) r(i)
  14. x i = y r ( i ) x_{i}=y_{r(i)}
  15. t i = s j t_{i}=s_{j}
  16. j j
  17. r ( i ) j r ( i + 1 ) - 1 r(i)\leq j\leq r(i+1)-1

Paschen's_law.html

  1. V B = a p d ln ( p d ) + b V_{B}=\frac{apd}{\ln(pd)+b}
  2. V V
  3. p p
  4. d d
  5. a a
  6. b b
  7. a a
  8. b b
  9. p d pd
  10. p d = e 1 - b pd=e^{1-b}
  11. p d pd
  12. p d pd
  13. p d pd
  14. d d
  15. x = 0 x=0
  16. E e E_{e}
  17. E I E_{I}
  18. x x
  19. α \alpha
  20. α \alpha
  21. Γ e \Gamma_{e}
  22. Γ e ( x = d ) = Γ e ( x = 0 ) e α d ( 1 ) \Gamma_{e}(x=d)=\Gamma_{e}(x=0)\,\mathrm{e}^{\alpha d}\qquad\qquad(1)
  23. d d
  24. α \alpha
  25. Γ e ( d ) - Γ e ( 0 ) = Γ e ( 0 ) ( e α d - 1 ) ( 2 ) \Gamma_{e}(d)-\Gamma_{e}(0)=\Gamma_{e}(0)\left(\mathrm{e}^{\alpha d}-1\right)% \qquad\qquad(2)
  26. Γ i ( 0 ) - Γ i ( d ) = Γ e ( 0 ) ( e α d - 1 ) ( 3 ) \Gamma_{i}(0)-\Gamma_{i}(d)=\Gamma_{e}(0)\left(\mathrm{e}^{\alpha d}-1\right)% \qquad\qquad(3)
  27. Γ i \Gamma_{i}
  28. Γ e ( 0 ) = γ Γ i ( 0 ) ( 4 ) \Gamma_{e}(0)=\gamma\Gamma_{i}(0)\qquad\qquad(4)
  29. γ \gamma
  30. Γ i ( d ) = 0 \Gamma_{i}(d)=0
  31. α d = ln ( 1 + 1 γ ) ( 5 ) \alpha d=\ln\left(1+\frac{1}{\gamma}\right)\qquad\qquad(5)
  32. α \alpha
  33. P P
  34. σ \sigma
  35. A A
  36. P = N σ A = x λ ( 6 ) P=\frac{N\sigma}{A}=\frac{x}{\lambda}\qquad\qquad(6)
  37. x x
  38. λ \lambda
  39. σ \sigma
  40. r r
  41. N N
  42. p V = N k B T ( 7 ) pV=Nk_{B}T\qquad\qquad(7)
  43. p p
  44. V V
  45. k B k_{B}
  46. T T
  47. σ = π ( r a + r b ) 2 \sigma=\pi(r_{a}+r_{b})^{2}
  48. r I r_{I}
  49. σ = π r I 2 \sigma=\pi r_{I}^{2}
  50. λ \lambda
  51. λ = k B T p π r I 2 = 1 L p ( 8 ) \lambda=\frac{k_{B}T}{p\pi r_{I}^{2}}=\frac{1}{L\cdot p}\qquad\qquad(8)
  52. L L
  53. x x
  54. d Γ e ( x ) = - Γ e ( x ) d x λ e ( 9 ) \mathrm{d}\Gamma_{e}(x)=-\Gamma_{e}(x)\,\frac{\mathrm{d}x}{\lambda_{e}}\qquad% \qquad(9)
  55. Γ e ( x ) = Γ e ( 0 ) exp ( - x λ e ) ( 10 ) \Gamma_{e}(x)=\Gamma_{e}(0)\,\exp{\left(-\frac{x}{\lambda_{e}}\right)}\qquad% \qquad(10)
  56. λ > x \lambda>x
  57. x x
  58. P ( λ > x ) = Γ e ( x ) Γ e ( 0 ) = exp ( - x λ e ) ( 11 ) P(\lambda>x)=\frac{\Gamma_{e}(x)}{\Gamma_{e}(0)}=\exp{\left(-\frac{x}{\lambda_% {e}}\right)}\qquad\qquad(11)
  59. α \alpha
  60. α = P ( λ > λ I ) λ e = 1 λ e exp ( - λ I λ e ) = 1 λ e exp ( - E I E e ) ( 12 ) \alpha=\frac{P(\lambda>\lambda_{I})}{\lambda_{e}}=\frac{1}{\lambda_{e}}\exp% \left(\mbox{-}~{}\frac{\lambda_{I}}{\lambda_{e}}\right)=\frac{1}{\lambda_{e}}% \exp\left(\mbox{-}~{}\frac{E_{I}}{E_{e}}\right)\qquad\qquad(12)
  61. E E
  62. \mathcal{E}
  63. Q Q
  64. E = λ Q ( 13 ) E=\lambda Q\mathcal{E}\qquad\qquad(13)
  65. = U d \mathcal{E}=\frac{U}{d}
  66. U U
  67. Q Q
  68. e e
  69. α = L p exp ( - L p d E I e U ) ( 14 ) \alpha=L\cdot p\,\exp\left(\mbox{-}~{}\frac{L\cdot p\cdot d\cdot E_{I}}{eU}% \right)\qquad\qquad(14)
  70. U U
  71. U breakdown U_{\mathrm{breakdown}}
  72. U breakdown = L p d E I e ( ln ( L p d ) - ln ( ln ( 1 + γ - 1 ) ) ) ( 15 ) U_{\mathrm{breakdown}}=\frac{L\cdot p\cdot d\cdot E_{I}}{e\left(\ln(L\cdot p% \cdot d)-\ln\left(\ln\left(1+\gamma^{-1}\right)\right)\right)}\qquad\qquad(15)
  73. L = k B T π r I 2 \textstyle L=\frac{k_{B}T}{\pi r_{I}^{2}}
  74. d d
  75. α \alpha
  76. α d 1 ( 16 ) \alpha d\geq 1\qquad\qquad(16)
  77. α d = 1 \alpha d=1
  78. U breakdown Townsend = L p d E I ln ( L p d ) = d E I λ e ln ( d λ e ) ( 17 ) U_{\mathrm{breakdown\,Townsend}}=\frac{L\cdot p\cdot d\cdot E_{I}}{\ln(L\cdot p% \cdot d)}=\frac{d\cdot E_{I}}{\lambda_{e}\,\ln\left(\frac{d}{\lambda_{e}}% \right)}\qquad\qquad(17)
  79. Γ e ( x = 0 ) 0 \Gamma_{e}(x=0)\neq 0
  80. γ \gamma