wpmath0000014_13

Thurstonian_model.html

  1. z i j = μ j + ϵ i j z_{ij}=\mu_{j}+\epsilon_{ij}
  2. z i j 𝒩 ( β j , σ i 2 ) . z_{ij}\ \sim\ \mathcal{N}(\beta_{j},\,\sigma_{i}^{2}).
  3. β 𝒩 ( β * , Σ * ) . \beta\ \sim\ \mathcal{N}(\beta^{*},\Sigma^{*}).

Thymidylate_kinase.html

  1. \rightleftharpoons

Thyroid's_secretory_capacity.html

  1. G ^ T = β T ( D T + [ T S H ] ) ( 1 + K 41 [ T B G ] + K 42 [ T B P A ] ) [ F T 4 ] α T [ T S H ] \hat{G}_{T}={{\beta_{T}(D_{T}+[TSH])(1+K_{41}[TBG]+K_{42}[TBPA])[FT_{4}]}\over% {\alpha_{T}[TSH]}}
  2. G ^ T = β T ( D T + [ T S H ] ) [ T T 4 ] α T [ T S H ] \hat{G}_{T}={{\beta_{T}(D_{T}+[TSH])[TT_{4}]}\over{\alpha_{T}[TSH]}}
  3. G ^ T \hat{G}_{T}
  4. α T \alpha_{T}
  5. β T \beta_{T}

Tidal_barrage.html

  1. E = 1 2 A ρ g h 2 E\,=\,\tfrac{1}{2}\,A\,\rho\,g\,h^{2}

Tidal_stream_generator.html

  1. C P C_{P}
  2. P = ρ A V 3 2 C P P=\frac{\rho AV^{3}}{2}C_{P}
  3. C P C_{P}
  4. ρ \rho
  5. P = 0.22 ρ g Δ H max Q max P=0.22\,\rho\,g\,\Delta H\text{max}\,Q\text{max}
  6. ρ \rho
  7. Δ H max \Delta H\text{max}
  8. Q max Q\text{max}

Tikhonov's_theorem_(dynamical_systems).html

  1. d 𝐱 d t \displaystyle\frac{d\mathbf{x}}{dt}
  2. μ 0 \mu\to 0
  3. d 𝐱 d t \displaystyle\frac{d\mathbf{x}}{dt}
  4. 𝐠 ( 𝐱 , 𝐳 , t ) = 0. \mathbf{g}(\mathbf{x},\mathbf{z},t)=0.\,
  5. μ 0 , \mu\to 0,\,
  6. 𝐳 = φ ( 𝐱 , t ) \mathbf{z}=\varphi(\mathbf{x},t)
  7. d 𝐳 d t = 𝐠 ( 𝐱 , 𝐳 , t ) . \frac{d\mathbf{z}}{dt}=\mathbf{g}(\mathbf{x},\mathbf{z},t).

Tilting_theory.html

  1. = ker ( F ) \mathcal{F}=\ker(F)
  2. 𝒯 = ker ( F ) \mathcal{T}=\ker(F^{\prime})
  3. 𝒳 = ker ( G ) \mathcal{X}=\ker(G)
  4. 𝒴 = ker ( G ) \mathcal{Y}=\ker(G^{\prime})
  5. ( 𝒯 , ) (\mathcal{T},\mathcal{F})
  6. 𝒯 \mathcal{T}
  7. \mathcal{F}
  8. Hom ( 𝒯 , ) = 0 \operatorname{Hom}(\mathcal{T},\mathcal{F})=0
  9. 0 U M V 0 0\to U\to M\to V\to 0
  10. 𝒯 \mathcal{T}
  11. \mathcal{F}
  12. ( 𝒳 , 𝒴 ) (\mathcal{X},\mathcal{Y})
  13. 𝒯 \mathcal{T}
  14. 𝒴 \mathcal{Y}
  15. \mathcal{F}
  16. 𝒳 \mathcal{X}
  17. ( 𝒯 , ) (\mathcal{T},\mathcal{F})
  18. ( 𝒳 , 𝒴 ) (\mathcal{X},\mathcal{Y})
  19. 𝒯 = mod - A \mathcal{T}=\operatorname{mod}-A
  20. 𝒴 = mod - B \mathcal{Y}=\operatorname{mod}-B
  21. ( 𝒳 , 𝒴 ) (\mathcal{X},\mathcal{Y})
  22. 𝒳 \mathcal{X}
  23. 𝒴 \mathcal{Y}
  24. 0 A T 1 T n 0 0\to A\to T_{1}\to\dots\to T_{n}\to 0

Time-weighted_return.html

  1. 1 + r = M 1 - C 1 M 0 × M 2 - C 2 M 1 × M 3 - C 3 M 2 × × M n - 1 - C n - 1 M n - 2 × M n - C n M n - 1 1+r=\frac{M_{1}-C_{1}}{M_{0}}\times\frac{M_{2}-C_{2}}{M_{1}}\times\frac{M_{3}-% C_{3}}{M_{2}}\times\cdots\times\frac{M_{n-1}-C_{n-1}}{M_{n-2}}\times\frac{M_{n% }-C_{n}}{M_{n-1}}
  2. r r
  3. M 0 M_{0}
  4. M t M_{t}
  5. t t
  6. C t C_{t}
  7. M n M_{n}
  8. C t C_{t}
  9. t t
  10. n n
  11. n n
  12. C n C_{n}
  13. n n
  14. 1 + r = M 1 M 0 + C 0 × M 2 M 1 + C 1 × M 3 M 2 + C 2 × × M n - 1 M n - 2 + C n - 2 × M n M n - 1 + C n - 1 1+r=\frac{M_{1}}{M_{0}+C_{0}}\times\frac{M_{2}}{M_{1}+C_{1}}\times\frac{M_{3}}% {M_{2}+C_{2}}\times...\times\frac{M_{n-1}}{M_{n-2}+C_{n-2}}\times\frac{M_{n}}{% M_{n-1}+C_{n-1}}
  15. r r
  16. M 0 M_{0}
  17. M t M_{t}
  18. t t
  19. C t C_{t}
  20. M n M_{n}
  21. C t C_{t}
  22. t + 1 {t+1}
  23. n n
  24. r = M 2 - M 1 M 1 r=\frac{M_{2}-M_{1}}{M_{1}}
  25. M 2 M_{2}
  26. M 1 M_{1}
  27. r r
  28. 1 + r = M 2 M 1 1+r=\frac{M_{2}}{M_{1}}
  29. C 1 C_{1}
  30. M 1 M_{1}
  31. C 1 C_{1}
  32. r = M 2 - ( M 1 + C 1 ) M 1 + C 1 r=\frac{M_{2}-(M_{1}+C_{1})}{M_{1}+C_{1}}
  33. 1 + r = M 2 M 1 + C 1 1+r=\frac{M_{2}}{M_{1}+C_{1}}
  34. C 2 C_{2}
  35. M 2 M_{2}
  36. M 2 M_{2}
  37. C 2 C_{2}
  38. r = ( M 2 - C 2 ) - M 1 M 1 r=\frac{(M_{2}-C_{2})-M_{1}}{M_{1}}
  39. 1 + r = M 2 - C 2 M 1 1+r=\frac{M_{2}-C_{2}}{M_{1}}
  40. ( 1 + 1.0 ) ( 1 - 0.25 ) - 1 = ( 2.0 ) ( 0.75 ) - 1 = 1.5 - 1 = 0.5 = 50 % (1+1.0)(1-0.25)-1=(2.0)(0.75)-1=1.5-1=0.5=50\%

Time_evolution_of_integrals.html

  1. d d t a ( t ) b ( t ) f ( t , x ) d x = a ( t ) b ( t ) f ( t , x ) t d x + f ( t , b ( t ) ) b ( t ) - f ( t , a ( t ) ) a ( t ) \frac{d}{dt}\int_{a\left(t\right)}^{b\left(t\right)}f\left(t,x\right)dx=\int_{% a\left(t\right)}^{b\left(t\right)}\frac{\partial f\left(t,x\right)}{\partial t% }dx+f\left(t,b\left(t\right)\right)b^{\prime}\left(t\right)-f\left(t,a\left(t% \right)\right)a^{\prime}\left(t\right)
  2. Ω F d Ω \int_{\Omega}F\,d\Omega
  3. d d t Ω F d Ω = Ω F t d Ω + S C F d S \frac{d}{dt}\int_{\Omega}F\,d\Omega=\int_{\Omega}\frac{\partial F}{\partial t}% \,d\Omega+\int_{S}CF\,dS
  4. S F d S \int_{S}F\,dS
  5. d d t S F d S = S δ F δ t d S - S C B α α d S \frac{d}{dt}\int_{S}F\,dS=\int_{S}\frac{\delta F}{\delta t}\,dS-\int_{S}CB^{% \alpha}_{\alpha}\,dS
  6. δ / δ t {\delta}/{\delta}t
  7. B α α B^{\alpha}_{\alpha}
  8. B α α B^{\alpha}_{\alpha}
  9. F 1 F\equiv 1
  10. S d S \int_{S}\,dS
  11. d d t S S d S = - S C B α α d S \frac{d}{dt}\int_{S}S\,dS=-\int_{S}CB^{\alpha}_{\alpha}\,dS
  12. B α α B^{\alpha}_{\alpha}
  13. C B α α C\equiv B^{\alpha}_{\alpha}
  14. B α α = - 2 / R B^{\alpha}_{\alpha}=-2/R
  15. B α α = - 1 / R B^{\alpha}_{\alpha}=-1/R
  16. C B α α d t CB^{\alpha}_{\alpha}dt
  17. c d t cdt
  18. S F d S \int_{S}F\,dS
  19. d d t S F d S = S δ F δ t d S - S C B α α F d S + γ c d γ \frac{d}{dt}\int_{S}F\,dS=\int_{S}\frac{\delta F}{\delta t}\,dS-\int_{S}CB_{% \alpha}^{\alpha}F\,dS+\int_{\gamma}c\,d\gamma

Timeline_of_mathematical_innovation_in_South_and_West_Asia.html

  1. ( a , b , c ) (a,b,c)
  2. c 2 = a 2 + b 2 c^{2}=a^{2}+b^{2}
  3. a , b , c a,b,c
  4. 13500 2 + 12709 2 = 18541 2 13500^{2}+12709^{2}=18541^{2}

Timeline_of_the_far_future.html

  1. 10 10 26 10^{10^{26}}
  2. 10 10 50 10^{10^{50}}
  3. 10 10 56 10^{10^{56}}
  4. 10 10 76 10^{10^{76}}
  5. 10 10 120 10^{10^{120}}

Time–frequency_analysis_for_music_signals.html

  1. 𝐒𝐓𝐅𝐓 { x ( t ) } X ( t , f ) = - x ( τ ) w ( t - τ ) e - j 2 π f τ d τ \mathbf{STFT}\left\{x(t)\right\}\equiv X(t,f)=\int_{-\infty}^{\infty}x(\tau)w(% t-\tau)e^{-j2\pi f\tau}\,d\tau
  2. X ( n Δ t , m Δ f ) = p = n - Q n + Q x ( p Δ t ) e - j 2 π p m Δ t Δ f Δ t X(n\,\Delta t,m\,\Delta f)=\sum_{p=n-Q}^{n+Q}x(p\,\Delta t)e^{-j2\pi pm\,% \Delta t\,\Delta f}\,\Delta t
  3. t = n Δ t t=n\,\Delta t
  4. f = m Δ f f=m\,\Delta f
  5. τ = p Δ t \tau=p\,\Delta t
  6. B = Q Δ t B=Q\,\Delta t
  7. Δ t Δ f = 1 N , \Delta t\,\Delta f=\frac{1}{N},
  8. N 2 Q + 1 N\geq 2Q+1
  9. Δ < 1 2 f max \Delta<\frac{1}{2f_{\max}}
  10. f max f_{\max}
  11. 𝐬𝐩𝐞𝐜𝐭𝐫𝐨𝐠𝐫𝐚𝐦 ( t , f ) = | 𝐒𝐓𝐅𝐓 ( t , f ) | 2 \mathbf{spectrogram}(t,f)=\left|\mathbf{STFT}(t,f)\right|^{2}
  12. W x ( t , f ) W_{x}(t,f)
  13. 𝐖 x ( t , f ) = - x ( t + τ / 2 ) x * ( t - τ / 2 ) e - j 2 π τ f d τ , \mathbf{W}_{x}(t,f)=\int_{-\infty}^{\infty}x(t+\tau/2)x^{*}(t-\tau/2)e^{-j2\pi% \tau\,f}\,d\tau,

TLA+.html

  1. $\or$
  2. and \and
  3. ¬ \neg
  4. \Rightarrow
  5. \Leftrightarrow
  6. \equiv
  7. \forall
  8. \exists
  9. ϵ \epsilon
  10. P \Box P
  11. P \Diamond P
  12. P \Box\Diamond P
  13. P \Diamond\Box P

Toida's_conjecture.html

  1. n * \mathbb{Z}^{*}_{n}
  2. X = X ( n ; S ) \vec{X}=\vec{X}(\mathbb{Z}_{n};S)
  3. X \vec{X}

Toilet_paper_orientation.html

  1. p p
  2. q q
  3. | p - 1 / 2 | |p-1/2|
  4. 1 / n 1/\sqrt{n}
  5. M n ( p ) = { p / ( p - q ) + O ( r n ) , q < p ( ( q - p ) / q ) n + p / ( q - p ) + O ( r n ) , q > p M_{n}(p)=\begin{cases}p/(p-q)+O(r^{n}),&q<p\\ ((q-p)/q)n+p/(q-p)+O(r^{n}),&q>p\end{cases}
  6. r r

ToneScript.html

  1. n k n_{k}
  2. 1 < k < 6 1<k<6
  3. Z i Z_{i}
  4. D i D_{i}
  5. Z Z i , j ZZ_{i,j}
  6. f i , j f_{i,j}
  7. f i , j := n 1 [ + n 2 [ + n 3 [ + n 4 [ + n 5 [ + n 6 ] ] ] ] ] f_{i,j}:=n_{1}[+n_{2}[+n_{3}[+n_{4}[+n_{5}[+n_{6}]]]]]
  8. Z Z i , j := o n i , j / o f f i , j / f i , j ZZ_{i,j}:=on_{i,j}/off_{i,j}/f_{i,j}
  9. Z i := D i ( [ Z Z i , 1 [ , Z Z i , 2 [ , Z Z i , 3 [ , Z Z i , 4 [ , Z Z i , 5 [ , Z Z i , 6 ] ] ] ] ] ) Z_{i}:=D_{i}([ZZ_{i,1}[,ZZ_{i,2}[,ZZ_{i,3}[,ZZ_{i,4}[,ZZ_{i,5}[,ZZ_{i,6}]]]]])
  10. F i F_{i}
  11. L i L_{i}
  12. F r e q S c r i p t := F 1 FreqScript:=F_{1}
  13. L 1 [ , F 2 L_{1}[,F_{2}
  14. L 2 ] L_{2}]
  15. T o n e S c r i p t := F r e q S c r i p t ; Z 1 [ ; Z 2 ] ToneScript:=FreqScript;Z_{1}[;Z_{2}]

Top_tree.html

  1. \Re
  2. 𝒯 \mathcal{T}
  3. T \partial{T}
  4. \Re
  5. 𝒞 \mathcal{C}
  6. C . \partial{C}.
  7. 𝒞 \mathcal{C}
  8. I ( 𝒞 ) , I(\mathcal{C}),
  9. π ( 𝒞 ) \pi(\mathcal{C})
  10. 𝒞 \mathcal{C}
  11. π ( 𝒞 ) \pi(\mathcal{C})
  12. 𝒞 \mathcal{C}
  13. 𝒞 \mathcal{C}
  14. 𝒞 \mathcal{C}
  15. C \partial{C}
  16. 𝒞 . \mathcal{C}.
  17. 𝒞 \mathcal{C}
  18. 𝒞 \mathcal{C}
  19. π ( 𝒞 ) . \pi(\mathcal{C}).
  20. 𝒜 \mathcal{A}
  21. \mathcal{B}
  22. 𝒜 \mathcal{A}\cap\mathcal{B}
  23. 𝒜 \mathcal{A}\cup\mathcal{B}
  24. \Re
  25. 𝒯 \mathcal{T}
  26. 𝒪 ( log n ) \mathcal{O}(\log n)
  27. 𝒯 \mathcal{T}
  28. 𝒯 \mathcal{T}
  29. 𝒯 \mathcal{T}
  30. \Re
  31. \Re
  32. 𝒪 ( n ) \mathcal{O}(n)
  33. \Re
  34. 𝒯 , \mathcal{T},
  35. T \partial{T}
  36. \Re
  37. 𝒯 , \mathcal{T},
  38. T \partial{T}
  39. \Re
  40. 𝒯 ; \mathcal{T};
  41. \Re
  42. 𝒯 \mathcal{T}
  43. v v
  44. w w
  45. 𝒯 \mathcal{T}
  46. 𝒯 \mathcal{T}
  47. \Re
  48. \cup
  49. \Re
  50. ( v , w ) \cup{(v,w)}
  51. ( v , w ) {(v,w)}
  52. 𝒯 \mathcal{T}
  53. , \Re,
  54. 𝒯 \mathcal{T}
  55. 𝒯 \mathcal{T}
  56. \Re
  57. \Re
  58. S S
  59. S S
  60. S S
  61. 𝒞 \mathcal{C}
  62. C = S . \partial{C}=S.
  63. 𝒪 ( log n ) \mathcal{O}(\log n)
  64. 𝒪 ( log n ) \mathcal{O}(\log n)
  65. I ( 𝒞 ) I(\mathcal{C})
  66. ( 𝒜 , ) : (\mathcal{A},\mathcal{B}){:}
  67. 𝒜 \mathcal{A}
  68. \mathcal{B}
  69. 𝒞 \mathcal{C}
  70. 𝒜 \mathcal{A}
  71. \mathcal{B}
  72. 𝒜 . \mathcal{A}\cup\mathcal{B}.
  73. I ( 𝒞 ) I(\mathcal{C})
  74. I ( 𝒜 ) I(\mathcal{A})
  75. I ( ) . I(\mathcal{B}).
  76. ( 𝒞 ) : (\mathcal{C}){:}
  77. 𝒞 \mathcal{C}
  78. 𝒜 . \mathcal{A}\cup\mathcal{B}.
  79. I ( 𝒜 ) I(\mathcal{A})
  80. I ( ) I(\mathcal{B})
  81. I ( 𝒞 ) I(\mathcal{C})
  82. 𝒞 \mathcal{C}
  83. \Re
  84. ( 𝒞 ) (\mathcal{C})
  85. I ( 𝒜 ) I(\mathcal{A})
  86. I ( ) I(\mathcal{B})
  87. I ( 𝒞 ) I(\mathcal{C})
  88. I ( 𝒞 ) I(\mathcal{C})
  89. 𝒞 \mathcal{C}
  90. ( v , w ) : (v,w){:}
  91. 𝒞 \mathcal{C}
  92. ( v , w ) . (v,w).
  93. C = \partial{C}=\partial
  94. ( v , w ) . (v,w).
  95. I ( 𝒞 ) I(\mathcal{C})
  96. ( 𝒞 ) : (\mathcal{C}){:}
  97. 𝒞 \mathcal{C}
  98. ( v , w ) . (v,w).
  99. I ( 𝒞 ) I(\mathcal{C})
  100. 𝒞 \mathcal{C}
  101. ( 𝒞 ) : (\mathcal{C}){:}
  102. ( 𝒞 ) : (\mathcal{C}){:}
  103. 𝒞 \mathcal{C}
  104. π ( 𝒞 ) \pi(\mathcal{C})
  105. v v
  106. w w
  107. 𝒞 \mathcal{C}
  108. 𝒞 \mathcal{C}
  109. v v
  110. i . i.
  111. 𝒜 \mathcal{A}
  112. v A v\in\partial{A}
  113. π ( 𝒜 ) \pi(\mathcal{A})
  114. i i
  115. I I
  116. I I
  117. I I
  118. v v
  119. 𝒞 \mathcal{C}
  120. 𝒞 \mathcal{C}
  121. 𝒜 , \mathcal{A},
  122. \mathcal{B}
  123. a A B a\in\partial{A}\cap\partial{B}
  124. ( a ) (a)
  125. I I
  126. 𝒪 ( log n ) \mathcal{O}(\log n)
  127. O ( log n ) O(\log n)
  128. 𝒞 \mathcal{C}
  129. - . -\infty.
  130. v v
  131. w w
  132. 𝒞 = \mathcal{C}=
  133. ( v , w ) , (v,w),
  134. ( 𝒞 ) . (\mathcal{C}).
  135. x x
  136. v v
  137. w w
  138. 𝒪 ( log n ) \mathcal{O}(\log n)
  139. 𝒞 \mathcal{C}
  140. π ( 𝒞 ) . \pi(\mathcal{C}).
  141. 𝒞 \mathcal{C}
  142. 𝒜 \mathcal{A}
  143. 𝒞 , \mathcal{C},
  144. 𝒜 \mathcal{A}
  145. 𝒞 \mathcal{C}
  146. 𝒜 \mathcal{A}
  147. 𝒜 \mathcal{A}
  148. 𝒞 \mathcal{C}
  149. 𝒞 \mathcal{C}
  150. 𝒜 , \mathcal{A},
  151. \mathcal{B}
  152. 𝒞 \mathcal{C}
  153. 𝒜 \mathcal{A}
  154. \mathcal{B}
  155. 𝒞 \mathcal{C}
  156. v v
  157. w , w,
  158. 𝒞 \mathcal{C}
  159. ( v , w ) (v,w)
  160. 𝒞 \mathcal{C}
  161. v v
  162. 𝒪 ( log n ) \mathcal{O}(\log n)
  163. v v
  164. w w
  165. 𝒪 ( log n ) \mathcal{O}(\log n)
  166. ( v , w ) (v,w)
  167. 𝒞 \mathcal{C}
  168. 𝒞 \mathcal{C}
  169. 𝒞 \mathcal{C}
  170. 𝒪 ( log n ) \mathcal{O}(\log n)
  171. 𝒪 ( log n ) \mathcal{O}(\log n)
  172. O ( log 4 n ) O(\log^{4}n)
  173. I ( 𝒞 ) I(\mathcal{C})
  174. Θ ( log 2 n ) \Theta(\log^{2}n)
  175. O ( log 5 n ) O(\log^{5}n)
  176. I ( 𝒞 ) I(\mathcal{C})
  177. Θ ( log 2 n ) \Theta(\log^{2}n)
  178. 𝒯 \mathcal{T}
  179. ( 𝒯 ) , (\mathcal{T}),
  180. 𝒯 \mathcal{T}
  181. 𝒯 \mathcal{T}
  182. 𝒯 . \mathcal{T}.
  183. \Re
  184. . \Re.
  185. 𝒯 \mathcal{T}
  186. 𝒪 ( log n ) \mathcal{O}(\log n)
  187. 𝒪 ( log n ) \mathcal{O}(\log n)

Topological_category.html

  1. C C
  2. T : C 𝐒𝐞𝐭 T:C\to\mathbf{Set}
  3. C C
  4. T T
  5. 𝐒𝐞𝐭 \mathbf{Set}
  6. C C
  7. T - 1 x , x 𝐒𝐞𝐭 T^{-1}x,x\in\mathbf{Set}

Toric_code.html

  1. v v
  2. p p
  3. A v = i v σ i x , B p = i p σ i z . A_{v}=\prod_{i\in v}\sigma^{x}_{i},\,\,B_{p}=\prod_{i\in p}\sigma^{z}_{i}.
  4. i v i\in v
  5. v v
  6. i p i\in p
  7. p p
  8. A v | ψ = | ψ , v , B p | ψ = | ψ , p , A_{v}|\psi\rangle=|\psi\rangle,\,\,\forall v,\,\,B_{p}|\psi\rangle=|\psi% \rangle,\,\,\forall p,
  9. | ψ |\psi\rangle
  10. | ϕ |\phi\rangle
  11. A v | ϕ = - | ϕ A_{v}|\phi\rangle=-|\phi\rangle
  12. e e
  13. v v
  14. B p B_{p}
  15. m m
  16. H T C = - J v A v - J p B p , J > 0. H_{TC}=-J\sum_{v}A_{v}-J\sum_{p}B_{p},\,\,\,J>0.
  17. J J
  18. e e
  19. m m
  20. e e
  21. m m
  22. e e
  23. m m
  24. - 1 -1
  25. e e
  26. m m
  27. - 1 -1
  28. Z Z
  29. X X
  30. m m
  31. e e
  32. e e
  33. m m
  34. - 1 -1
  35. e e
  36. m m
  37. e e
  38. m m
  39. e × e = 1 , m × m = 1. e\times e=1,\,\,\,m\times m=1.
  40. 1 1
  41. e e
  42. m m
  43. ψ \psi
  44. e × m = ψ . e\times m=\psi.
  45. ψ \psi
  46. e e
  47. m m
  48. - 1 -1
  49. ψ \psi

Toroidal_moment.html

  1. T i = 1 10 c [ r i ( 𝐫 𝐉 ) - 2 r 2 J i ] d 3 x . T_{i}=\frac{1}{10c}\int[r_{i}(\mathbf{r}\cdot\mathbf{J})-2r^{2}J_{i}]\mathrm{d% }^{3}x.
  2. H - d ( σ 𝐄 ) - μ ( σ 𝐁 ) - a ( σ × 𝐁 ) , H\propto-d(\mathbf{\sigma}\cdot\mathbf{E})-\mu(\mathbf{\sigma}\cdot\mathbf{B})% -a(\mathbf{\sigma}\cdot\nabla\times\mathbf{B}),

Torpedo_Data_Computer.html

  1. v T a r g e t sin ( θ D e f l e c t i o n ) = v T o r p e d o sin ( θ B o w ) \frac{\left\|v_{Target}\right\|}{\sin(\theta_{Deflection})}=\frac{\left\|v_{% Torpedo}\right\|}{\sin(\theta_{Bow})}
  2. v T a r g e t sin ( θ D e f l e c t i o n ) = v T o r p e d o sin ( θ T r a c k - θ D e f l e c t i o n ) \frac{\left\|v_{Target}\right\|}{\sin(\theta_{Deflection})}=\frac{\left\|v_{% Torpedo}\right\|}{\sin(\theta_{Track}-\theta_{Deflection})}

Torsionless_module.html

  1. f M = Hom R ( M , R ) , f ( m ) 0. f\in M^{\ast}=\operatorname{Hom}_{R}(M,R),\quad f(m)\neq 0.
  2. M M = Hom R ( M , R ) , m ( f f ( m ) ) , m M , f M , M\to M^{\ast\ast}=\operatorname{Hom}_{R}(M^{\ast},R),\quad m\mapsto(f\mapsto f% (m)),m\in M,f\in M^{\ast},
  3. M R S M\otimes_{R}S

Total_active_reflection_coefficient.html

  1. Γ a t = i = 1 N | b i | 2 i = 1 N | a i | 2 . \Gamma^{t}_{a}=\frac{\sqrt{\sum_{i=1}^{N}|b_{i}|^{2}}}{\sqrt{\sum_{i=1}^{N}|a_% {i}|^{2}}}.
  2. [ b ] = [ S ] [ a ] . [b]=[S][a].
  3. [ S ] [S]
  4. [ a ] [a]
  5. [ b ] [b]

Totative.html

  1. n n
  2. k k
  3. n n
  4. 0 < a 1 < a 2 < a ϕ ( n ) < n , 0<a_{1}<a_{2}\cdots<a_{\phi(n)}<n,
  5. i = 1 ϕ ( n ) - 1 ( a i + 1 - a i ) 2 < C n 2 / ϕ ( n ) \sum_{i=1}^{\phi(n)-1}(a_{i+1}-a_{i})^{2}<Cn^{2}/\phi(n)

Tower_(mathematics).html

  1. \mathcal{I}
  2. 2 1 0 \cdots\rightarrow 2\rightarrow 1\rightarrow 0
  3. 𝒜 \mathcal{A}
  4. \mathcal{I}
  5. 𝒜 \mathcal{A}
  6. 𝒜 \mathcal{A}
  7. { A i } i 0 \{A_{i}\}_{i\geq 0}
  8. 𝒜 \mathcal{A}
  9. A i A j A_{i}\rightarrow A_{j}
  10. i > j i>j
  11. A i A j A k A_{i}\rightarrow A_{j}\rightarrow A_{k}
  12. A i A k A_{i}\rightarrow A_{k}
  13. M i = M M_{i}=M
  14. R R
  15. M M
  16. M i M j M_{i}\rightarrow M_{j}
  17. i > j i>j
  18. { M i } \{M_{i}\}

Tower_of_fields.html

  1. | F 2 | F 1 | F 0 . \begin{array}[]{c}\vdots\\ |\\ F_{2}\\ |\\ F_{1}\\ |\\ F_{0}.\end{array}
  2. F n + 1 = F n ( 2 1 / 2 n ) F_{n+1}=F_{n}\left(2^{1/2^{n}}\right)

Trace_distance.html

  1. T ( ρ , σ ) := 1 2 || ρ - σ || 1 = 1 2 Tr [ ( ρ - σ ) ( ρ - σ ) ] . T(\rho,\sigma):=\frac{1}{2}||\rho-\sigma||_{1}=\frac{1}{2}\mathrm{Tr}\left[% \sqrt{(\rho-\sigma)^{\dagger}(\rho-\sigma)}\right].
  2. T ( ρ , σ ) = 1 2 Tr [ ( ρ - σ ) 2 ] = 1 2 i | λ i | , T(\rho,\sigma)=\frac{1}{2}\mathrm{Tr}\left[\sqrt{(\rho-\sigma)^{2}}\right]=% \frac{1}{2}\sum_{i}|\lambda_{i}|,
  3. λ i \lambda_{i}
  4. ( ρ - σ ) (\rho-\sigma)
  5. T ( ρ , σ ) = max P Tr [ P ( ρ - σ ) ] , T(\rho,\sigma)=\max_{P}\mathrm{Tr}[P(\rho-\sigma)],
  6. P P
  7. P I P\leq I
  8. I I
  9. Tr [ P ( ρ - σ ) ] \mathrm{Tr}[P(\rho-\sigma)]
  10. P P
  11. ρ \rho
  12. σ \sigma
  13. ρ \rho
  14. σ \sigma
  15. 1 2 \frac{1}{2}
  16. p max = 1 2 ( 1 + T ( ρ , σ ) ) p_{\,\text{max}}=\frac{1}{2}(1+T(\rho,\sigma))
  17. T ( ρ , σ ) = 0 ρ = σ T(\rho,\sigma)=0\Leftrightarrow\rho=\sigma
  18. 0 T ( ρ , σ ) 1 0\leq T(\rho,\sigma)\leq 1
  19. T ( ρ , σ ) = 1 T(\rho,\sigma)=1
  20. ρ \rho
  21. σ \sigma
  22. T ( U ρ U , U σ U ) = T ( ρ , σ ) T(U\rho U^{\dagger},U\sigma U^{\dagger})=T(\rho,\sigma)
  23. Φ \Phi
  24. T ( Φ ( ρ ) , Φ ( σ ) ) T ( ρ , σ ) T(\Phi(\rho),\Phi(\sigma))\leq T(\rho,\sigma)
  25. T ( i p i ρ i , σ ) i p i T ( ρ i , σ ) T(\sum_{i}p_{i}\rho_{i},\sigma)\leq\sum_{i}p_{i}T(\rho_{i},\sigma)
  26. F ( ρ , σ ) F(\rho,\sigma)
  27. T ( ρ , σ ) T(\rho,\sigma)
  28. 1 - F ( ρ , σ ) T ( ρ , σ ) 1 - F ( ρ , σ ) 2 . 1-F(\rho,\sigma)\leq T(\rho,\sigma)\leq\sqrt{1-F(\rho,\sigma)^{2}}\,.
  29. ρ \rho
  30. σ \sigma

Traffic_congestion_reconstruction_with_Kerner's_three-phase_theory.html

  1. v g v_{g}
  2. τ d e l , j a m ( a ) \tau_{del,jam}^{(a)}
  3. 1 ρ m a x \frac{1}{\rho_{max}}
  4. ρ m a x \rho_{max}
  5. v g v_{g}
  6. v g = - 1 ρ m a x τ d e l , j a m ( a ) ( 1 ) v_{g}=-\frac{1}{\rho_{max}\tau_{del,jam}^{(a)}}\qquad\qquad(1)
  7. τ d e l , j a m ( a ) \tau_{del,jam}^{(a)}
  8. ρ m a x \rho_{max}
  9. v g v_{g}
  10. x d o w n ( s y n ) ( t ) x_{down}^{(syn)}(t)
  11. x u p ( s y n ) ( t ) x_{up}^{(syn)}(t)
  12. t t
  13. x d o w n ( j a m ) ( t ) x_{down}^{(jam)}(t)
  14. x u p ( j a m ) ( t ) x_{up}^{(jam)}(t)
  15. x u p ( s y n ) ( t ) = μ 1 n t 0 t ( q S ( t ) - q F ( t ) ) d t , t t 0 ( 2 ) x_{up}^{(syn)}(t)=\mu\frac{1}{n}\int_{t_{0}}^{t}(q_{S}(t)-q_{F}(t))dt,\ \ t% \geq t_{0}\qquad\qquad(2)
  16. q F q_{F}
  17. q S q_{S}
  18. μ \mu
  19. n n
  20. v ( j a m ) v^{(jam)}
  21. v ( j a m ) ( t ) = q 2 ( t ) - q 1 ( t ) ρ 2 ( t ) - ρ 1 ( t ) ( 3 ) v^{(jam)}(t)=\frac{q_{2}(t)-q_{1}(t)}{\rho_{2}(t)-\rho_{1}(t)}\qquad\qquad(3)
  22. q 1 q_{1}
  23. ρ 1 \rho_{1}
  24. q 2 q_{2}
  25. ρ 2 \rho_{2}
  26. q 1 q_{1}
  27. q 2 q_{2}
  28. ρ 1 \rho_{1}
  29. ρ 2 \rho_{2}
  30. v d o w n ( j a m ) = v g ( 4 ) v_{down}^{(jam)}=v_{g}\qquad\qquad(4)
  31. v g v_{g}
  32. t = t 1 t=t_{1}
  33. x ( t ) - x ( t 1 ) = v g ( t - t 1 ) , t > t 1 ( 5 ) x(t)-x(t_{1})=v_{g}(t-t_{1}),\ \ t>t_{1}\qquad\qquad(5)
  34. v g - 15 km/h v_{g}\approx-15\mbox{km/h}~{}
  35. v g v_{g}
  36. v g v_{g}
  37. v g v_{g}
  38. - 12 > v g > - 20 km/h -12>v_{g}>-20\mbox{km/h}~{}

Traffic_equations.html

  1. λ i \lambda_{i}
  2. γ i \gamma_{i}
  3. λ i = γ i + j = 1 m p j i λ j . \lambda_{i}=\gamma_{i}+\sum_{j=1}^{m}p_{ji}\lambda_{j}.
  4. λ ( I - P ) = γ , \lambda(I-P)=\gamma\,,
  5. λ i \lambda_{i}
  6. γ i \gamma_{i}
  7. λ i = j = 1 m p j i λ j . \lambda_{i}=\sum_{j=1}^{m}p_{ji}\lambda_{j}.

Trajectory_(fluid_mechanics).html

  1. v ( x , t ) \vec{v}(\vec{x},~{}t)
  2. d x d t = v ( x , t ) \frac{d\vec{x}}{dt}=\vec{v}(\vec{x},~{}t)
  3. v \vec{v}
  4. x \vec{x}
  5. d θ d t = u r cos ϕ \frac{d\theta}{dt}=\frac{u}{r\cos\phi}
  6. d ϕ d t = v r \frac{d\phi}{dt}=\frac{v}{r}
  7. θ \theta
  8. ϕ \phi

Transient_kinetic_isotope_fractionation.html

  1. S = S=
  2. P = P=
  3. E = E=
  4. C = C=
  5. B = B=
  6. C 12 {}^{12}\,\text{C}
  7. C 13 {}^{13}\,\text{C}
  8. S a b {}_{a}^{b}S
  9. a {}_{a}
  10. b {}^{b}
  11. 0 b a 0\leq b\leq a
  12. 2 {}_{2}
  13. N 14 15 N {}^{15}\,\text{N}^{14}\,\text{N}
  14. P 2 1 {}_{2}^{1}\,\text{P}
  15. x b x_{b}
  16. y d y_{d}
  17. S a b {}_{a}^{b}\,\text{S}
  18. P c d {}_{c}^{d}\,\text{P}
  19. b = 0 a x b S a b d = 0 c y d P c d . \sum_{b=0}^{a}x_{b}{{}_{a}^{b}}\ \,\text{S}\rightarrow\sum_{d=0}^{c}y_{d}{{}_{% c}^{d}}\ \,\text{P}.
  20. NO 3 - 14 + {}^{14}\,\text{NO}_{3}^{-}+
  21. NO 3 - 15 14 N 15 NO {}^{15}\,\text{NO}_{3}^{-}\rightarrow^{14}\,\text{N}^{15}\,\text{NO}
  22. S 1 0 + S 1 1 P 2 1 {{}_{1}^{0}}\,\text{S}+{{}_{1}^{1}}\,\text{S}\rightarrow{{}_{2}^{1}}\,\text{P}
  23. x 0 = x 1 = 1 x_{0}=x_{1}=1
  24. b = 0 b=0
  25. b = 1 b=1
  26. y 1 = 1 y_{1}=1
  27. P 2 1 {}_{2}^{1}\,\text{P}
  28. y 0 = y 2 = 0 y_{0}=y_{2}=0
  29. P 2 0 = 14 N 2 O {}_{2}^{0}\,\text{P}=^{14}\,\text{N}_{2}\,\text{O}
  30. P 2 2 = 15 N 2 O {}_{2}^{2}\,\text{P}=^{15}\,\text{N}_{2}\,\text{O}
  31. S β a b {}_{a}^{b}\,\text{S}^{\beta}
  32. S γ a b {}_{a}^{b}\,\text{S}^{\gamma}
  33. β \beta
  34. γ \gamma
  35. S a b {}_{a}^{b}\,\text{S}
  36. 1 b < a 1\leq b<a
  37. a 2 a\geq 2
  38. a a
  39. b b
  40. b = 1 b=1
  41. H 2 {}^{2}\,\text{H}
  42. 3 {}_{3}
  43. b = 2 b=2
  44. 3 {}_{3}
  45. 4 {}_{4}
  46. H 2 2 {}_{2}\,\text{H}_{2}
  47. CD 2 H 2 + 2 O 2 H 2 O + D 2 O + CO 2 , \,\text{CD}_{2}\,\text{H}_{2}+2\,\text{O}_{2}\rightarrow\,\text{H}_{2}\,\text{% O}+\,\text{D}_{2}\,\text{O}+\,\text{CO}_{2},
  48. S β 4 2 P 2 0 + P 2 2 {{}_{4}^{2}}\,\text{S}^{\beta}\rightarrow{{}_{2}^{0}}\,\text{P}+{{}_{2}^{2}}\,% \text{P}
  49. β \beta
  50. S β 4 2 {{}_{4}^{2}}\,\text{S}^{\beta}
  51. 1 b < a 1\leq b<a
  52. a 2 a\geq 2
  53. N 2 O \,\text{N}_{2}\,\text{O}
  54. S β 2 1 = N 14 15 NO {}_{2}^{1}\,\text{S}^{\beta}={{}^{15}}\,\text{N}^{14}\,\text{NO}
  55. S γ 2 1 = 14 N 15 NO {}_{2}^{1}\,\text{S}^{\gamma}=^{14}\,\text{N}^{15}\,\text{NO}
  56. u u
  57. b = 0 a β x b S β a b d = 0 c γ u γ y d P γ c d , \sum_{b=0}^{a}\sum_{\beta}x_{b}\ {{}_{a}^{b}}\,\text{S}^{\beta}\rightarrow\sum% _{d=0}^{c}\sum_{\gamma}u_{\gamma}y_{d}\ {{}_{c}^{d}}\,\text{P}^{\gamma},
  58. u γ = 1 u_{\gamma}=1
  59. NO 3 - 14 + 15 NO 3 - N 15 14 NO , {}^{14}\,\text{NO}_{3}^{-}+^{15}\,\text{NO}_{3}^{-}\rightarrow{{}^{14}}\,\text% {N}^{15}\,\text{NO},
  60. NO 3 - 14 + 15 NO 3 - N 14 15 NO , {}^{14}\,\text{NO}_{3}^{-}+^{15}\,\text{NO}_{3}^{-}\rightarrow{{}^{15}}\,\text% {N}^{14}\,\text{NO},
  61. S 1 0 + S 1 1 u β P β 2 1 + u γ P γ 2 1 , {{}_{1}^{0}}\,\text{S}+{{}_{1}^{1}}\,\text{S}\rightarrow u_{\beta}{{}_{2}^{1}}% \,\text{P}^{\beta}+u_{\gamma}{{}_{2}^{1}}\,\text{P}^{\gamma},
  62. u γ = 1 - u β u_{\gamma}=1-u_{\beta}
  63. n S n_{S}
  64. n P n_{P}
  65. j = 1 n S b j = 0 a j β j x b j S j β j a j b j h = 1 n P d h = 0 c h γ h u γ h y d h P h γ h c h d h . ( 1 ) \sum_{j=1}^{n_{S}}\sum_{b_{j}=0}^{a_{j}}\sum_{\beta_{j}}x_{b_{j}}\ {{}_{a_{j}}% ^{b_{j}}}\,\text{S}_{j}^{\beta_{j}}\rightarrow\sum_{h=1}^{n_{P}}\sum_{d_{h}=0}% ^{c_{h}}\sum_{\gamma_{h}}u_{\gamma_{h}}\ y_{d_{h}}\ {{}_{c_{h}}^{d_{h}}}\,% \text{P}_{h}^{\gamma_{h}}.\qquad\qquad(1)
  66. O 16 {}^{16}\,\text{O}
  67. O 18 {}^{18}\,\text{O}
  68. CH 2 18 O + 16 O 2 H 2 16 O + C 18 O 16 O \,\text{CH}_{2}^{18}\,\text{O}+^{16}\,\text{O}_{2}\rightarrow\,\text{H}_{2}^{1% 6}\,\text{O}+\,\text{C}^{18}\,\text{O}^{16}\,\text{O}
  69. S 1 1 1 + 2 0 S 2 P 1 1 0 + P 2 2 1 {}_{1}^{1}\,\text{S}_{1}+_{2}^{0}\,\text{S}_{2}\rightarrow{{}_{1}^{0}}\,\text{% P}_{1}+{{}_{2}^{1}}\,\text{P}_{2}
  70. S j \,\text{S}_{j}
  71. P h \,\text{P}_{h}
  72. j = 1 n S b j i = 0 a j i β j i x b j i S j β j i a j b j i + E k 2 ( i ) k 1 ( i ) C i k 3 ( i ) h = 1 n P d h i = 0 c h i γ h i u γ h i y d h i P h γ h i c h d h i + E , ( 2 ) \sum_{j=1}^{n_{S}}\sum_{b_{ji}=0}^{a_{ji}}\sum_{\beta_{ji}}x_{b_{ji}}\ {{}_{a_% {j}}^{b_{ji}}}\,\text{S}_{j}^{\beta_{ji}}+\,\text{E}\overset{\xrightarrow{\,% \text{k}_{1(i)}}}{\xleftarrow[\,\text{k}_{2(i)}]{}}\,\text{C}_{i}\xrightarrow{% \,\text{k}_{3(i)}}\sum_{h=1}^{n_{P}}\sum_{d_{hi}=0}^{c_{hi}}\sum_{\gamma_{hi}}% u_{\gamma_{hi}}y_{d_{hi}}\ {{}_{c_{h}}^{d_{hi}}}\,\text{P}_{h}^{\gamma_{hi}}+% \,\text{E},\qquad\qquad(2)
  73. i = 1 , , m i=1,...,m
  74. m m
  75. k 1 ( i ) k_{1(i)}
  76. k 2 ( i ) k_{2(i)}
  77. k 3 ( i ) k_{3(i)}
  78. 2 14 NO 3 - N 2 14 O , 2^{14}\,\text{NO}_{3}^{-}\rightarrow{{}^{14}}\,\text{N}_{2}\,\text{O},
  79. NO 3 - 14 + NO 3 - 15 N 15 14 N O , {}^{14}\,\text{NO}_{3}^{-}+{{}^{15}}\,\text{NO}_{3}^{-}\rightarrow{{}^{14}}\,% \text{N}^{15}\,\text{N}\,\text{O},
  80. NO 3 - 14 + NO 3 - 15 N 14 15 N O , {}^{14}\,\text{NO}_{3}^{-}+{{}^{15}}\,\text{NO}_{3}^{-}\rightarrow{{}^{15}}\,% \text{N}^{14}\,\text{N}\,\text{O},
  81. 2 NO 3 - 15 N 2 15 O , 2{{}^{15}}\,\text{NO}_{3}^{-}\rightarrow{{}^{15}}\,\text{N}_{2}\,\text{O},
  82. 2 1 0 S + E k 2 ( 1 ) k 1 ( 1 ) C 1 k 3 ( 1 ) P 2 0 + E , 2_{1}^{0}\,\text{S}+\,\text{E}\overset{\xrightarrow{\,\text{k}_{1(1)}}}{% \xleftarrow[\,\text{k}_{2(1)}]{}}\,\text{C}_{1}\xrightarrow{\,\text{k}_{3(1)}}% {{}_{2}^{0}}\,\text{P}+\,\text{E},
  83. S 1 0 + S 1 1 + E k 2 ( 2 ) k 1 ( 2 ) C 2 k 3 ( 2 ) u β P β 2 1 + u γ P γ 2 1 + E , {{}_{1}^{0}}\,\text{S}+{{}_{1}^{1}}\,\text{S}+\,\text{E}\overset{\xrightarrow{% \,\text{k}_{1(2)}}}{\xleftarrow[\,\text{k}_{2(2)}]{}}\,\text{C}_{2}% \xrightarrow{\,\text{k}_{3(2)}}u_{\beta}{{}_{2}^{1}}\,\text{P}^{\beta}+u_{% \gamma}{{}_{2}^{1}}\,\text{P}^{\gamma}+\,\text{E},
  84. 2 S 1 1 + E k 2 ( 3 ) k 1 ( 3 ) C 3 k 3 ( 3 ) P 2 2 + E , 2{{}_{1}^{1}}\,\text{S}+\,\text{E}\overset{\xrightarrow{\,\text{k}_{1(3)}}}{% \xleftarrow[\,\text{k}_{2(3)}]{}}\,\text{C}_{3}\xrightarrow{\,\text{k}_{3(3)}}% {{}_{2}^{2}}\,\text{P}+\,\text{E},
  85. j = 1 n S b j = 0 a j β j x b j a j = h = 1 n P d h = 0 c h γ h u γ h y d h c h , \sum_{j=1}^{n_{S}}\sum_{b_{j}=0}^{a_{j}}\sum_{\beta_{j}}x_{b_{j}}\ a_{j}=\sum_% {h=1}^{n_{P}}\sum_{d_{h}=0}^{c_{h}}\sum_{\gamma_{h}}u_{\gamma_{h}}\ y_{d_{h}}% \ c_{h},
  86. j = 1 n S b j = 0 a j β j x b j b j = h = 1 n P d h = 0 c h γ h u γ h y d h d h . \sum_{j=1}^{n_{S}}\sum_{b_{j}=0}^{a_{j}}\sum_{\beta_{j}}x_{b_{j}}\ b_{j}=\sum_% {h=1}^{n_{P}}\sum_{d_{h}=0}^{c_{h}}\sum_{\gamma_{h}}u_{\gamma_{h}}\ y_{d_{h}}% \ d_{h}.
  87. d [ S j β j a j b j ] d t = i x b j i [ k 2 ( i ) C i - k 1 ( i ) E S ¯ i ] ( 3 a ) \frac{\,\text{d}[{{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}]}{\,\text{d}t}=\sum_{i}% x_{b_{ji}}[\,\text{k}_{2(i)}C_{i}-\,\text{k}_{1(i)}E\overline{S}_{i}]\qquad% \qquad(3a)
  88. d C i d t = k 1 ( i ) E S ¯ i - [ k 2 ( i ) + k 3 ( i ) ] C i ( 3 b ) \frac{\,\text{d}C_{i}}{\,\text{d}t}=\,\text{k}_{1(i)}E\overline{S}_{i}-[\,% \text{k}_{2(i)}+\,\text{k}_{3(i)}]C_{i}\qquad\qquad(3b)
  89. d [ P h γ h c h d h ] d t = i u γ h i y d h i k 3 ( i ) C i ( 3 c ) \frac{\,\text{d}[{{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}]}{\,\text{d}t}=\sum_{i% }u_{\gamma_{hi}}y_{d_{hi}}\,\text{k}_{3(i)}C_{i}\qquad\qquad(3c)
  90. d E d t = z d B d t - i d C i d t ( 3 d ) \frac{\,\text{d}E}{\,\text{d}t}=z\frac{\,\text{d}B}{\,\text{d}t}-\sum_{i}\frac% {\,\text{d}C_{i}}{\,\text{d}t}\qquad\qquad(3d)
  91. d B d t = Y h d h γ h d [ P h γ h c h d h ] d t - μ B ( 3 e ) \frac{\,\text{d}B}{\,\text{d}t}=Y\sum_{h}\sum_{d_{h}}\sum_{\gamma_{h}}\frac{\,% \text{d}[{{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}]}{\,\text{d}t}-\mu B\qquad% \qquad(3e)
  92. i = 1 , , m i=1,...,m
  93. S ¯ i \overline{S}_{i}
  94. μ \mu
  95. R S b j , β j * * ( t ) = S j β j a j b j ( t ) S j a j 0 ( t ) R^{**}_{S_{b_{j},\beta_{j}}}(t)=\frac{{{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}(t)% }{{}^{0}_{a_{j}}S_{j}(t)}
  96. R P d h , γ h * * ( t ) = P h γ h c h d h ( t ) P h c h 0 ( t ) R^{**}_{P_{d_{h},\gamma_{h}}}(t)=\frac{{{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}(% t)}{{}^{0}_{c_{h}}P_{h}(t)}
  97. R S j * ( t ) = b j 0 β j b j q M S j b j S j β j a j b j ( t ) b j a j β j ( a j - b j ) p M S j b j S j β j a j b j ( t ) R^{*}_{S_{j}}(t)=\frac{\displaystyle\sum_{b_{j}\neq 0}\sum_{\beta_{j}}\frac{b_% {j}q}{{}^{b_{j}}M_{S_{j}}}\ {{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}(t)}{% \displaystyle\sum_{b_{j}\neq a_{j}}\sum_{\beta_{j}}\frac{(a_{j}-b_{j})p}{{}^{b% _{j}}M_{S_{j}}}\ {{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}(t)}
  98. R P h * ( t ) = d h 0 γ h d h q M P h d h P h γ h c h d h ( t ) d h c h γ h ( c h - d h ) p M P h d h P h γ h c h d h ( t ) R^{*}_{P_{h}}(t)=\frac{\displaystyle\sum_{d_{h}\neq 0}\sum_{\gamma_{h}}\frac{d% _{h}q}{{}^{d_{h}}M_{P_{h}}}\ {{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}(t)}{% \displaystyle\sum_{d_{h}\neq c_{h}}\sum_{\gamma_{h}}\frac{(c_{h}-d_{h})p}{{}^{% d_{h}}M_{P_{h}}}\ {{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}(t)}
  99. M S j b j {}^{b_{j}}M_{S_{j}}
  100. M P h d h {}^{d_{h}}M_{P_{h}}
  101. R S ( t ) = j b j 0 β j b j q M S j b j S j β j a j b j ( t ) j b j a j β j ( a j - b j ) p M S j b j S j β j a j b j ( t ) , R_{S}(t)=\frac{\displaystyle\sum_{j}\sum_{b_{j}\neq 0}\sum_{\beta_{j}}\frac{b_% {j}q}{{}^{b_{j}}M_{S_{j}}}\ {{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}(t)}{% \displaystyle\sum_{j}\sum_{b_{j}\neq a_{j}}\sum_{\beta_{j}}\frac{(a_{j}-b_{j})% p}{{}^{b_{j}}M_{S_{j}}}\ {{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}(t)},
  102. R P ( t ) = h d h 0 γ h d h q M P h d h P h γ h c h d h ( t ) h d h c h γ h ( c h - d h ) p M P h d h P h γ h c h d h ( t ) . R_{P}(t)=\frac{\displaystyle\sum_{h}\sum_{d_{h}\neq 0}\sum_{\gamma_{h}}\frac{d% _{h}q}{{}^{d_{h}}M_{P_{h}}}\ {{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}(t)}{% \displaystyle\sum_{h}\sum_{d_{h}\neq c_{h}}\sum_{\gamma_{h}}\frac{(c_{h}-d_{h}% )p}{{}^{d_{h}}M_{P_{h}}}\ {{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}(t)}.
  103. δ S ( t ) = ( R S ( t ) R s t d - 1 ) 1000 , ( 4 a ) \delta_{S}(t)=\left(\frac{R_{S}(t)}{R_{std}}-1\right)1000,\qquad\qquad(4a)
  104. δ P ( t ) = ( R P ( t ) R s t d - 1 ) 1000. ( 4 b ) \delta_{P}(t)=\left(\frac{R_{P}(t)}{R_{std}}-1\right)1000.\qquad\qquad(4b)
  105. R s t d R_{std}
  106. I R P ( t ) = h d h 0 γ h d h q M P h d h d [ P h γ h c h d h ( t ) ] d t h d h c h γ h ( c h - d h ) p M P h d h d [ P h γ h c h d h ( t ) ] d t ( 5 ) IR_{P}(t)=\cfrac{\displaystyle\sum_{h}\sum_{d_{h}\neq 0}\sum_{\gamma_{h}}% \cfrac{d_{h}q}{{}^{d_{h}}M_{P_{h}}}\ \cfrac{\,\text{d}[{{}^{d_{h}}_{c_{h}}}P^{% \gamma_{h}}_{h}(t)]}{\,\text{d}t}}{\displaystyle\sum_{h}\sum_{d_{h}\neq c_{h}}% \sum_{\gamma_{h}}\cfrac{(c_{h}-d_{h})p}{{}^{d_{h}}M_{P_{h}}}\ \cfrac{\,\text{d% }[{{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}(t)]}{\,\text{d}t}}\qquad\qquad(5)
  107. α ( t ) = I R P ( t ) R S ( t ) ( 6 ) \alpha(t)=\frac{IR_{P}(t)}{R_{S}(t)}\qquad\qquad(6)
  108. ϵ ( t ) = 1 - α ( t ) ( 7 ) \epsilon(t)=1-\alpha(t)\qquad\qquad(7)
  109. d [ S j β j a j b j ] d t = i x b j i [ k 2 ( i ) C i - k 1 ( i ) E S ¯ i ] ( 8 a ) \frac{\,\text{d}[{{}^{b_{j}}_{a_{j}}}S^{\beta_{j}}_{j}]}{\,\text{d}t}=\sum_{i}% x_{b_{ji}}[\,\text{k}_{2(i)}C_{i}-\,\text{k}_{1(i)}E\overline{S}_{i}]\qquad% \qquad(8a)
  110. d C i d t = k 1 ( i ) E S ¯ i - [ k 2 ( i ) + k 3 ( i ) ] C i ( 8 b ) \frac{\,\text{d}C_{i}}{\,\text{d}t}=\,\text{k}_{1(i)}E\overline{S}_{i}-[\,% \text{k}_{2(i)}+\,\text{k}_{3(i)}]C_{i}\qquad\qquad(8b)
  111. d [ P h γ h c h d h ] d t = i u γ h i y d h i k 3 ( i ) C i ( 8 c ) \frac{\,\text{d}[{{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}]}{\,\text{d}t}=\sum_{i% }u_{\gamma_{hi}}y_{d_{hi}}\,\text{k}_{3(i)}C_{i}\qquad\qquad(8c)
  112. d E d t = - i d C i d t ( 8 d ) \frac{\,\text{d}E}{\,\text{d}t}=-\sum_{i}\frac{\,\text{d}C_{i}}{\,\text{d}t}% \qquad\qquad(8d)
  113. d [ S j β j a j b j ] d t - i = 1 m x b j i k 3 ( i ) E 0 S ¯ i S ¯ i + K i ( 1 + p i S ¯ p K p ) ( 9 a ) \frac{\,\text{d}[{{}_{a_{j}}^{b_{j}}}S_{j}^{\beta_{j}}]}{\,\text{d}t}\simeq-% \sum_{i=1}^{m}\frac{x_{b_{ji}}\,\text{k}_{3(i)}E_{0}\overline{S}_{i}}{% \overline{S}_{i}+K_{i}\left(1+\displaystyle\sum_{p\neq i}\dfrac{\overline{S}_{% p}}{K_{p}}\right)}\qquad\qquad(9a)
  114. d [ P h γ h c h d h ] d t i = 1 m u γ h i y d h i k 3 ( i ) E 0 S ¯ i S ¯ i + K i ( 1 + p i S ¯ p K p ) ( 9 b ) \frac{\,\text{d}[{{}^{d_{h}}_{c_{h}}}P^{\gamma_{h}}_{h}]}{\,\text{d}t}\simeq% \sum_{i=1}^{m}\frac{u_{\gamma_{hi}}y_{d_{hi}}\,\text{k}_{3(i)}E_{0}\overline{S% }_{i}}{\overline{S}_{i}+K_{i}\left(1+\displaystyle\sum_{p\neq i}\dfrac{% \overline{S}_{p}}{K_{p}}\right)}\qquad\qquad(9b)
  115. 2 {}_{2}
  116. 2 {}_{2}
  117. N 2 14 O N 2 14 , {}^{14}\,\text{N}_{2}\,\text{O}\rightarrow{{}^{14}}\,\text{N}_{2},
  118. N 15 14 N O N 14 N 15 , {{}^{14}}\,\text{N}^{15}\,\text{N}\,\text{O}\rightarrow{{}^{14}}\,\text{N}{{}^% {15}}\,\text{N},
  119. N 14 15 N O N 14 N 15 , {{}^{15}}\,\text{N}^{14}\,\text{N}\,\text{O}\rightarrow{{}^{14}}\,\text{N}{{}^% {15}}\,\text{N},
  120. S 2 0 + E k 1 ( 1 ) k 2 ( 1 ) C 1 k 3 ( 1 ) P 2 0 + E , {}_{2}^{0}\,\text{S}+\,\text{E}\underset{\,\text{k}_{2(1)}}{\overset{\,\text{k% }_{1(1)}}{\rightleftarrows}}\,\text{C}_{1}\overset{\,\text{k}_{3(1)}}{% \rightarrow}{{}_{2}^{0}}\,\text{P}+\,\text{E},
  121. S β 2 1 + E k 1 ( 2 ) k 2 ( 2 ) C 2 k 3 ( 2 ) P 2 1 + E , {{}_{2}^{1}}\,\text{S}^{\beta}+\,\text{E}\underset{\,\text{k}_{2(2)}}{\overset% {\,\text{k}_{1(2)}}{\rightleftarrows}}\,\text{C}_{2}\overset{\,\text{k}_{3(2)}% }{\rightarrow}{{}_{2}^{1}}\,\text{P}+\,\text{E},
  122. S γ 2 1 + E k 1 ( 3 ) k 2 ( 3 ) C 3 k 3 ( 3 ) P 2 1 + E , {{}_{2}^{1}}\,\text{S}^{\gamma}+\,\text{E}\underset{\,\text{k}_{2(3)}}{% \overset{\,\text{k}_{1(3)}}{\rightleftarrows}}\,\text{C}_{3}\overset{\,\text{k% }_{3(3)}}{\rightarrow}{{}_{2}^{1}}\,\text{P}+\,\text{E},
  123. S 2 2 = 15 N 2 O {}_{2}^{2}\,\text{S}=^{15}\,\text{N}_{2}\,\text{O}
  124. 2 {}_{2}
  125. 2 {}_{2}
  126. ( k 1 ( 3 ) k 1 ( 2 ) (k_{1(3)}\equiv k_{1(2)}
  127. k 2 ( 3 ) k 2 ( 2 ) k_{2(3)}\equiv k_{2(2)}
  128. k 3 ( 3 ) k 3 ( 2 ) ) k_{3(3)}\equiv k_{3(2)})
  129. d [ S 2 0 ] d t = k 2 ( 1 ) C 1 - k 1 ( 1 ) S 2 0 E \frac{\,\text{d}[{{}^{0}_{2}}S]}{\,\text{d}t}=\,\text{k}_{2(1)}C_{1}-\,\text{k% }_{1(1)}{{}^{0}_{2}}SE
  130. d [ S β 2 1 ] d t = k 2 ( 2 ) C 2 - k 1 ( 2 ) S β 2 1 E \frac{\,\text{d}[{{}^{1}_{2}}S^{\beta}]}{\,\text{d}t}=\,\text{k}_{2(2)}C_{2}-% \,\text{k}_{1(2)}{{}^{1}_{2}}S^{\beta}E
  131. d [ S γ 2 1 ] d t = k 2 ( 2 ) C 3 - k 1 ( 2 ) S γ 2 1 E \frac{\,\text{d}[{{}^{1}_{2}}S^{\gamma}]}{\,\text{d}t}=\,\text{k}_{2(2)}C_{3}-% \,\text{k}_{1(2)}{{}^{1}_{2}}S^{\gamma}E
  132. d C 1 d t = k 1 ( 1 ) S 2 0 E - ( k 2 ( 1 ) + k 3 ( 1 ) ) C 1 \frac{\,\text{d}C_{1}}{\,\text{d}t}=\,\text{k}_{1(1)}{{}^{0}_{2}}SE-(\,\text{k% }_{2(1)}+\,\text{k}_{3(1)})C_{1}
  133. d C 2 d t = k 1 ( 2 ) S β 2 1 E - ( k 2 ( 2 ) + k 3 ( 2 ) ) C 2 \frac{\,\text{d}C_{2}}{\,\text{d}t}=\,\text{k}_{1(2)}{{}^{1}_{2}}S^{\beta}E-(% \,\text{k}_{2(2)}+\,\text{k}_{3(2)})C_{2}
  134. d C 3 d t = k 1 ( 2 ) S γ 2 1 E - ( k 2 ( 2 ) + k 3 ( 2 ) ) C 3 \frac{\,\text{d}C_{3}}{\,\text{d}t}=\,\text{k}_{1(2)}{{}^{1}_{2}}S^{\gamma}E-(% \,\text{k}_{2(2)}+\,\text{k}_{3(2)})C_{3}
  135. d [ P 2 0 ] d t = k 3 ( 1 ) C 1 \frac{\,\text{d}[{{}_{2}^{0}}P]}{\,\text{d}t}=\,\text{k}_{3(1)}C_{1}
  136. d [ P 2 1 ] d t = k 3 ( 2 ) ( C 2 + C 3 ) \frac{\,\text{d}[{{}_{2}^{1}}P]}{\,\text{d}t}=\,\text{k}_{3(2)}(C_{2}+C_{3})
  137. d E d t = z d B d t - d C 1 d t - d C 2 d t - d C 3 d t \frac{\,\text{d}E}{\,\text{d}t}=z\frac{\,\text{d}B}{\,\text{d}t}-\frac{\,\text% {d}C_{1}}{\,\text{d}t}-\frac{\,\text{d}C_{2}}{\,\text{d}t}-\frac{\,\text{d}C_{% 3}}{\,\text{d}t}
  138. d B d t = Y ( d [ P 2 0 ] d t + d [ P 2 1 ] d t ) - μ B \frac{\,\text{d}B}{\,\text{d}t}=Y\left(\frac{\,\text{d}[{{}_{2}^{0}}P]}{\,% \text{d}t}+\frac{\,\text{d}[{{}_{2}^{1}}P]}{\,\text{d}t}\right)-\mu B
  139. R P ( t ) = 15 P 2 1 ( t ) 14 P 2 1 ( t ) + 29 P 2 0 ( t ) R_{P}(t)=\frac{15\ {{}_{2}^{1}}P(t)}{14\ {{}_{2}^{1}}P(t)+29\ {{}_{2}^{0}}P(t)}
  140. I R P ( t ) = 15 ( C 2 + C 3 ) k 3 ( 2 ) 29 C 1 k 3 ( 1 ) + 14 ( C 2 + C 3 ) k 3 ( 2 ) , IR_{P}(t)=\dfrac{15\ (C_{2}+C_{3})\,\text{k}_{3(2)}}{29\ C_{1}\,\text{k}_{3(1)% }+14\ (C_{2}+C_{3})\,\text{k}_{3(2)}},
  141. R S ( t ) = 165 S 2 1 154 S 2 1 + 315 S 2 0 R_{S}(t)=\frac{165\ {{}_{2}^{1}}S}{154\ {{}_{2}^{1}}S+315\ {{}_{2}^{0}}S}
  142. α ( t ) = 7 ( C 2 + C 3 ) k 3 ( 2 ) [ 45 S 2 0 + 22 S 2 1 ] 11 [ 29 C 1 k 3 ( 1 ) + 14 ( C 2 + C 3 ) k 3 ( 2 ) ] S 2 1 \alpha(t)=\frac{7\ (C_{2}+C_{3})\,\text{k}_{3(2)}[45\ {{}_{2}^{0}}S+22\ {{}_{2% }^{1}}S]}{11\ [29\ C_{1}\,\text{k}_{3(1)}+14\ (C_{2}+C_{3})\,\text{k}_{3(2)}]% \ {{}_{2}^{1}}S}
  143. d [ S 2 0 ] d t - k 3 ( 1 ) E 0 S 2 0 S 2 0 + K 1 ( 1 + S β 2 1 K 2 + S γ 2 1 K 2 ) \frac{\,\text{d}[{{}^{0}_{2}}S]}{\,\text{d}t}\simeq-\frac{\,\text{k}_{3(1)}E_{% 0}{{}^{0}_{2}}S}{{}^{0}_{2}S+K_{1}\left(1+\dfrac{{{}^{1}_{2}}S^{\beta}}{K_{2}}% +\dfrac{{{}^{1}_{2}}S^{\gamma}}{K_{2}}\right)}
  144. d [ S β 2 1 ] d t - k 3 ( 2 ) E 0 S β 2 1 S β 2 1 + K 2 ( 1 + S 2 0 K 1 + S γ 2 1 K 2 ) \frac{\,\text{d}[{{}^{1}_{2}}S^{\beta}]}{\,\text{d}t}\simeq-\frac{\,\text{k}_{% 3(2)}E_{0}{{}^{1}_{2}}S^{\beta}}{{}^{1}_{2}S^{\beta}+K_{2}\left(1+\dfrac{{{}^{% 0}_{2}}S}{K_{1}}+\dfrac{{{}^{1}_{2}}S^{\gamma}}{K_{2}}\right)}
  145. d [ S γ 2 1 ] d t - k 3 ( 2 ) E 0 S γ 2 1 S γ 2 1 + K 2 ( 1 + S 2 0 K 1 + S β 2 1 K 2 ) \frac{\,\text{d}[{{}^{1}_{2}}S^{\gamma}]}{\,\text{d}t}\simeq-\frac{\,\text{k}_% {3(2)}E_{0}{{}^{1}_{2}}S^{\gamma}}{{}^{1}_{2}S^{\gamma}+K_{2}\left(1+\dfrac{{{% }^{0}_{2}}S}{K_{1}}+\dfrac{{{}^{1}_{2}}S^{\beta}}{K_{2}}\right)}
  146. d P 2 0 d t = - d [ S 2 0 ] d t \frac{\,\text{d }{{}_{2}^{0}}P}{\,\text{d}t}=-\frac{\,\text{d}[{{}^{0}_{2}}S]}% {\,\text{d}t}
  147. d P 2 1 d t = - d [ S β 2 1 ] d t - d [ S γ 2 1 ] d t \frac{\,\text{d }{{}_{2}^{1}}P}{\,\text{d}t}=-\frac{\,\text{d}[{{}^{1}_{2}}S^{% \beta}]}{\,\text{d}t}-\frac{\,\text{d}[{{}^{1}_{2}}S^{\gamma}]}{\,\text{d}t}
  148. R P ( t ) = 15 P 2 1 14 P 2 1 + 29 P 2 0 R_{P}(t)=\frac{15{{}_{2}^{1}}P}{14{{}_{2}^{1}}P+29\ {{}_{2}^{0}}P}
  149. I R P ( t ) = 15 K 1 k 3 ( 2 ) S 2 1 29 K 2 k 3 ( 1 ) S 2 0 + 14 K 1 k 3 ( 2 ) S 2 1 IR_{P}(t)=\dfrac{15K_{1}\,\text{k}_{3(2)}{{}_{2}^{1}}S}{29K_{2}\,\text{k}_{3(1% )}{{}_{2}^{0}}S+14K_{1}\,\text{k}_{3(2)}{{}_{2}^{1}}S}
  150. R S ( t ) = 465 S 2 1 14 [ 63 S 2 0 + 31 S 2 1 ] R_{S}(t)=\frac{465\ {{}_{2}^{1}}S}{14[63\ {{}_{2}^{0}}S+31\ {{}_{2}^{1}}S]}
  151. α ( t ) = 14 K 1 k 3 ( 2 ) [ 63 S 2 0 + 31 S 2 1 ] 31 [ 29 K 2 k 3 ( 1 ) S 2 0 + 14 K 1 k 3 ( 2 ) S 2 0 ] \alpha(t)=\frac{14K_{1}\,\text{k}_{3(2)}[63\ {{}_{2}^{0}}S+31\ {{}_{2}^{1}}S]}% {31[29K_{2}\,\text{k}_{3(1)}\ {{}_{2}^{0}}S+14K_{1}\,\text{k}_{3(2)}\ {{}_{2}^% {0}}S]}
  152. 3 {}_{3}
  153. 2 {}_{2}

Transimpedance_amplifier.html

  1. - I p = V out R f -I\text{p}=\frac{V\text{out}}{R\text{f}}
  2. V out I p = - R f \frac{V\text{out}}{I\text{p}}=-R\text{f}
  3. β = X Ci R f + X Ci = 1 1 + R f C i s \beta=\frac{X\text{Ci}}{R\text{f}+X\text{Ci}}={1\over 1+R\text{f}C\text{i}s}
  4. X Ci X\text{Ci}
  5. V out = - I p R f 1 + 1 A OL β V\text{out}=-\frac{I\text{p}R\text{f}}{1+{1\over A\text{OL}\beta}}
  6. A OL A\text{OL}
  7. V out = - I p R f V\text{out}=-{I\text{p}R\text{f}}
  8. A OL A\text{OL}
  9. β \beta
  10. β = 1 + R f C f s 1 + R f ( C i + C f ) s \beta=\frac{1+R\text{f}C\text{f}s}{1+R\text{f}{(C\text{i}+C\text{f})}s}
  11. f Cf = 1 2 π R f C f f\text{Cf}={1\over 2\pi R\text{f}C\text{f}}
  12. f zf = 1 2 π R f ( C i + C f ) f\text{zf}={1\over 2\pi R\text{f}{(C\text{i}+C\text{f})}}

Transition_of_state.html

  1. V = 1 2 k x 2 V=\dfrac{1}{2}kx^{2}
  2. Ψ ( x , t ) = ψ ( r ) exp ( - i E t ) \Psi(x,t)=\psi(r)\exp\left(-i\dfrac{E}{\hbar}t\right)
  3. i Ψ t = E Ψ \textstyle i\hbar\frac{\partial\Psi}{\partial t}=E\Psi
  4. Ψ ( x , t ) = c 0 ( t ) Ψ 0 ( x , t ) + c 1 ( t ) Ψ 1 ( x , t ) \Psi(x,t)=c_{0}(t)\Psi_{0}(x,t)+c_{1}(t)\Psi_{1}(x,t)
  5. Ψ ( x , t ) | Ψ ( x , t ) = 1 | c 0 ( t ) | 2 + | c 1 ( t ) | 2 = 1 \langle\Psi(x,t)|\Psi(x,t)\rangle=1\Leftrightarrow\sqrt{|c_{0}(t)|^{2}+|c_{1}(% t)|^{2}}=1
  6. V ( x ) = 1 2 k x 2 + e ϵ ( t ) x V(x)=\dfrac{1}{2}kx^{2}+e\epsilon(t)x
  7. ( - 2 2 m 2 x 2 + V ( x ) ) Ψ ( x , t ) = i Ψ ( x , t ) t \left(-\dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}}{\partial x^{2}}+V(x)\right)% \Psi(x,t)=i\hbar\dfrac{\partial\Psi(x,t)}{\partial t}
  8. i Ψ ( x , t ) t = i ( ψ 0 exp ( - i E 0 t ) ( c 0 ( t ) - i E 0 c 0 ( t ) ) + ψ 1 exp ( - i E 1 t ) ( c 1 ( t ) - i E 1 c 1 ( t ) ) ) i\hbar\dfrac{\partial\Psi(x,t)}{\partial t}=i\hbar\left(\psi_{0}\exp\left(-i% \dfrac{E_{0}t}{\hbar}\right)\left({c_{0}}^{\prime}(t)-i\dfrac{E_{0}}{\hbar}c_{% 0}(t)\right)+\psi_{1}\exp\left(-i\dfrac{E_{1}t}{\hbar}\right)\left({c_{1}}^{% \prime}(t)-i\dfrac{E_{1}}{\hbar}c_{1}(t)\right)\right)
  9. i Ψ ( x , t ) t = i ( Ψ 0 ( c 0 ( t ) - i E 0 c 0 ( t ) ) + Ψ 1 ( c 1 ( t ) - i E 1 c 1 ( t ) ) ) i\hbar\dfrac{\partial\Psi(x,t)}{\partial t}=i\hbar\left(\Psi_{0}\left({c_{0}}^% {\prime}(t)-i\dfrac{E_{0}}{\hbar}c_{0}(t)\right)+\Psi_{1}\left({c_{1}}^{\prime% }(t)-i\dfrac{E_{1}}{\hbar}c_{1}(t)\right)\right)
  10. H ^ Ψ ( x , t ) = ( E 0 c 0 ( t ) Ψ 0 ( x , t ) + E 1 c 1 ( t ) Ψ 1 ( x , t ) + e ϵ ( t ) x Ψ ( x , t ) ) \hat{H}\Psi(x,t)=\left(E_{0}c_{0}(t)\Psi_{0}(x,t)+E_{1}c_{1}(t)\Psi_{1}(x,t)+e% \epsilon(t)x\Psi(x,t)\right)
  11. e ϵ ( t ) x Ψ ( x , t ) = i ( c 1 ( t ) Ψ 1 ( x , t ) + c 0 ( t ) Ψ 0 ( x , t ) ) e\epsilon(t)x\Psi(x,t)=i\hbar(c_{1}^{\prime}(t)\Psi_{1}(x,t)+c_{0}^{\prime}(t)% \Psi_{0}(x,t))
  12. e ϵ ( t ) ( c 1 ( t ) x | Ψ 1 ( x , t ) + c 0 ( t ) x | Ψ 0 ( x , t ) = i ( c 1 ( t ) | Ψ 1 ( x , t ) + c 0 ( t ) x | Ψ 0 ( x , t ) ) e\epsilon(t)(c_{1}(t)x|\Psi_{1}(x,t)\rangle+c_{0}(t)x|\Psi_{0}(x,t)\rangle=i% \hbar(c_{1}^{\prime}(t)|\Psi_{1}(x,t)\rangle+c_{0}^{\prime}(t)x|\Psi_{0}(x,t)\rangle)
  13. Ψ 1 | \langle\Psi_{1}|
  14. e ϵ ( t ) ( c 1 ( t ) Ψ 1 | x | Ψ 1 + c 0 ( t ) Ψ 1 | x | Ψ 0 ) = i ( c 1 ( t ) Ψ 1 | Ψ 1 + c 0 ( t ) Ψ 1 | Ψ 0 ) e\epsilon(t)(c_{1}(t)\langle\Psi_{1}|x|\Psi_{1}\rangle+c_{0}(t)\langle\Psi_{1}% |x|\Psi_{0}\rangle)=i\hbar\left(c_{1}^{\prime}(t)\langle\Psi_{1}|\Psi_{1}% \rangle+c_{0}^{\prime}(t)\langle\Psi_{1}|\Psi_{0}\rangle\right)
  15. Ψ 1 | Ψ 0 = 0 \langle\Psi_{1}|\Psi_{0}\rangle=0
  16. Ψ 1 | Ψ 1 = 1 \langle\Psi_{1}|\Psi_{1}\rangle=1
  17. e ϵ ( t ) ( c 1 ( t ) Ψ 1 | x | Ψ 1 + c 0 ( t ) Ψ 1 | x | Ψ 0 ) = i c 1 ( t ) e\epsilon(t)\left(c_{1}(t)\langle\Psi_{1}|x|\Psi_{1}\rangle+c_{0}(t)\langle% \Psi_{1}|x|\Psi_{0}\rangle\right)=i\hbar c_{1}^{\prime}(t)
  18. Ψ 1 | x | Ψ 0 \langle\Psi_{1}|x|\Psi_{0}\rangle
  19. e ϵ ( t ) Ψ 1 | x | Ψ 0 = i c 1 ( t ) e\epsilon(t)\langle\Psi_{1}|x|\Psi_{0}\rangle=i\hbar c_{1}^{\prime}(t)
  20. e ϵ ( t ) exp ( - i E 0 - E 1 t ) ψ 1 | x | ψ 0 = i c 1 ( t ) e\epsilon(t)\exp\left(-i\dfrac{E_{0}-E_{1}}{\hbar}t\right)\langle\psi_{1}|x|% \psi_{0}\rangle=i\hbar c_{1}^{\prime}(t)
  21. e ψ 1 | x | ψ 0 e\langle\psi_{1}|x|\psi_{0}\rangle
  22. μ 01 \mu_{01}
  23. c 1 ( t ) = μ 01 i ϵ ( t ) exp ( - i E 0 - E 1 t ) c 1 ( t ) = μ 01 i 0 t ϵ ( t ) exp ( - i E 0 - E 1 t ) d t c_{1}^{\prime}(t)=\dfrac{\mu_{01}}{i\hbar}\epsilon(t)\exp\left(-i\dfrac{E_{0}-% E_{1}}{\hbar}t\right)\Rightarrow c_{1}(t^{\prime})=\dfrac{\mu_{01}}{i\hbar}% \int_{0}^{t^{\prime}}\epsilon(t)\exp\left(-i\dfrac{E_{0}-E_{1}}{\hbar}t\right)% \mathrm{d}t
  24. ϵ ( t ) = ϵ 0 exp ( i ω t ) \epsilon(t)=\epsilon_{0}\exp(i\omega t)
  25. c 1 ( t ) = μ 01 ϵ 0 i 0 t d t exp ( - i ( E 0 - E 1 - ω ) t ) c_{1}(t^{\prime})=\dfrac{\mu_{01}\epsilon_{0}}{i\hbar}\int_{0}^{t^{\prime}}% \mathrm{d}t\exp\left(-i\left(\dfrac{E_{0}-E_{1}}{\hbar}-\omega\right)t\right)
  26. c 1 ( t ) = μ 01 ϵ 0 i 0 t d t exp ( - i E 0 - E 1 t ) exp ( i ω t ) = - + d t exp ( - i E 0 - E 1 t ) H ( t t - 1 2 ) exp ( i ω t ) c_{1}(t^{\prime})=\dfrac{\mu_{01}\epsilon_{0}}{i\hbar}\int_{0}^{t^{\prime}}% \mathrm{d}t\exp\left(-i\frac{E_{0}-E_{1}}{\hbar}t\right)\exp({i\omega t})=\int% _{-\infty}^{+\infty}\mathrm{d}t\exp\left(-i\frac{E_{0}-E_{1}}{\hbar}t\right)H% \left(\frac{t}{t^{\prime}}-\frac{1}{2}\right)\exp\left(i\omega t\right)
  27. t - t t\rightarrow-t
  28. c 1 ( t ) = - + d t exp ( i E 0 - E 1 t ) H ( - t t - 1 2 ) exp ( - i 2 π ν t ) c_{1}(t^{\prime})=\int_{-\infty}^{+\infty}\mathrm{d}t\exp\left(i\frac{E_{0}-E_% {1}}{\hbar}t\right)H\left(-\frac{t}{t^{\prime}}-\frac{1}{2}\right)\exp\left(-i% 2\pi\nu t\right)
  29. H H
  30. c 1 ( t ) = TF [ exp ( i E 0 - E 1 t ) H ( - t t - 1 2 ) ] = TF [ exp ( i E 0 - E 1 t ) ] TF [ H ( - t t - 1 2 ) ] c_{1}(t^{\prime})=\mathrm{TF}\left[\exp{\left(i\frac{E_{0}-E_{1}}{\hbar}t% \right)}H\left(-\frac{t}{t^{\prime}}-\dfrac{1}{2}\right)\right]=\mathrm{TF}% \left[\exp{\left(i\frac{E_{0}-E_{1}}{\hbar}t\right)}\right]\otimes\mathrm{TF}% \left[H\left(-\frac{t}{t^{\prime}}-\dfrac{1}{2}\right)\right]
  31. c 1 ( t ) = δ ( f - E 0 - E 1 h ) TF [ H ( t t - 1 2 ) ] c_{1}(t^{\prime})=\delta\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\otimes\mathrm{TF% }\left[H\left(\frac{t}{t^{\prime}}-\dfrac{1}{2}\right)\right]
  32. c 1 ( t ) = δ ( f - E 0 - E 1 h ) ( exp ( i π f t ) sinc ( t f ) ) c_{1}(t^{\prime})=\delta\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\otimes(\exp({i% \pi ft^{\prime}})\mathrm{sinc}(t^{\prime}f))
  33. \otimes
  34. c 1 ( t ) = exp ( i π ( f - E 0 - E 1 h ) t ) sinc ( t ( f - E 0 - E 1 h ) ) c_{1}(t^{\prime})=\exp\left({i\pi\left(f-\dfrac{E_{0}-E_{1}}{h}\right)t^{% \prime}}\right)\mathrm{sinc}\left(t^{\prime}\left(f-\dfrac{E_{0}-E_{1}}{h}% \right)\right)
  35. P k = n k c n * ( t ) c n ( t ) P_{k}=\sum_{n\neq k}c_{n}^{*}(t)c_{n}(t)
  36. | c 1 ( t ) | 2 |c_{1}(t)|^{2}
  37. | c 1 ( t ) | 2 |c_{1}(t)|^{2}
  38. | c 1 ( t ) | 2 = sinc 2 ( t ( f - E 0 - E 1 h ) ) |c_{1}(t)|^{2}=\mathrm{sinc}^{2}\left(t\left(f-\dfrac{E_{0}-E_{1}}{h}\right)\right)

Transport_coefficient.html

  1. γ \gamma
  2. γ = 0 A ˙ ( t ) A ˙ ( 0 ) d t \gamma=\int_{0}^{\infty}\langle\dot{A}(t)\dot{A}(0)\rangle dt
  3. A A
  4. \langle\cdot\rangle
  5. t t
  6. 2 t γ = | A ( t ) - A ( 0 ) | 2 2t\gamma=\langle|A(t)-A(0)|^{2}\rangle
  7. η = 1 k B T V 0 d t σ x y ( 0 ) σ x y ( t ) \eta=\frac{1}{k_{B}TV}\int_{0}^{\infty}dt\langle\sigma_{xy}(0)\sigma_{xy}(t)\rangle

Transvectant.html

  1. t r Ω r ( Q 1 Q n ) tr\Omega^{r}(Q_{1}\otimes\cdots\otimes Q_{n})

Transverse_Mercator:_Bowring_series.html

  1. a a
  2. b b
  3. k 0 k_{0}
  4. ϕ \scriptstyle\phi
  5. ω \scriptstyle\omega
  6. m m
  7. ϕ \scriptstyle\phi
  8. ε = 2 r - 1 ( r - 1 ) 2 = a 2 - b 2 b 2 \varepsilon\;=\;\frac{2r-1}{(r-1)^{2}}=\;\frac{a^{2}-b^{2}}{b^{2}}\;
  9. c = cos ϕ s = sin ϕ c=\cos\phi\qquad s=\sin\phi
  10. ν = a 1 + ε 1 + ε c 2 \nu\;=\;a\sqrt{\frac{1+\varepsilon}{1+\varepsilon c^{2}}}
  11. z = ε ω 3 c 5 6 z=\frac{\varepsilon\omega^{3}c^{5}}{6}
  12. tan θ 2 = 2 s c sin 2 ( ω / 2 ) s 2 + c 2 cos ω \tan\theta_{2}\;=\;\frac{2sc\sin^{2}(\omega/2)}{s^{2}+c^{2}\cos\omega}
  13. E = k 0 ν [ tanh - 1 ( c sin ω ) + z ( 1 + ω 2 10 ( 36 c 2 - 29 ) ) ] \,\text{E}\;=\;k_{0}\nu\left[{\tanh}^{-1}(c\sin\omega)+z\left(1+\frac{\omega^{% 2}}{10}(36c^{2}-29)\right)\right]
  14. N = k 0 [ m + ν θ 2 + z ν ω s 4 ( 9 + 4 ε c 2 - 11 ω 2 + 20 ω 2 c 2 ) ] \,\text{N}\;=\;k_{0}\left[m+\nu\theta_{2}+\frac{z\nu\omega s}{4}(9+4% \varepsilon c^{2}-11\omega^{2}+20\omega^{2}c^{2})\right]\,
  15. θ 2 \theta_{2}
  16. ϕ \scriptstyle\phi^{\prime}
  17. k 0 k_{0}
  18. c 1 = cos ϕ s 1 = sin ϕ c_{1}=\cos\phi^{\prime}\qquad s_{1}=\sin\phi^{\prime}
  19. ν 1 = a 1 + ε 1 + ε c 1 2 \nu_{1}\;=\;a\sqrt{\frac{1+\varepsilon}{1+\varepsilon{c_{1}}^{2}}}
  20. x = E k 0 ν 1 x\;=\;\frac{\,\text{E}}{k_{0}\nu_{1}}
  21. tan θ 4 = sinh x c 1 \tan\theta_{4}\;=\;\frac{\sinh x}{c_{1}}
  22. tan θ 5 = tan ϕ cos θ 4 \tan\theta_{5}\;=\;\tan\phi^{\prime}\cos\theta_{4}
  23. ϕ = ( 1 + ε c 1 2 ) [ θ 5 - ε 24 x 4 tan ϕ ( 9 - 10 c 1 2 ) ] - ε c 1 2 ϕ \phi\;=\;\left(1+\varepsilon{c_{1}}^{2}\right)\left[\theta_{5}-\frac{% \varepsilon}{24}x^{4}\tan\phi^{\prime}(9-10{c_{1}}^{2})\right]-\varepsilon{c_{% 1}}^{2}\phi^{\prime}
  24. ω = θ 4 - ε 60 x 3 c 1 ( 10 - 4 x 2 c 1 2 + x 2 c 1 2 ) \omega\;=\;\theta_{4}-\frac{\varepsilon}{60}x^{3}c_{1}\left(10-\frac{4x^{2}}{{% c_{1}}^{2}}+x^{2}{c_{1}}^{2}\right)
  25. θ 4 \theta_{4}
  26. θ 5 \theta_{5}
  27. ϕ \scriptstyle\phi^{\prime}
  28. ϕ \scriptstyle\phi
  29. ω \omega
  30. n = a - b a + b = 1 2 r - 1 n\;=\;\frac{a-b}{a+b}\;=\;\frac{1}{2r-1}
  31. tan ψ = ( r - 1 r ) tan ϕ = ( 1 - n 1 + n ) tan ϕ \tan\psi\quad=\quad\left(\frac{r-1}{r}\right)\tan\phi\quad=\quad\left(\frac{1-% n}{1+n}\right)\tan\phi
  32. p = 1 - 3 4 n cos 2 ψ q = 3 4 n sin 2 ψ p=1-\frac{3}{4}n\cos 2\psi\qquad q=\frac{3}{4}n\sin 2\psi
  33. Z = ( 1 - 3 8 n 2 ) ( p + q i ) 2 / 3 where i = - 1 Z=\left(1-\frac{3}{8}n^{2}\right)(p+qi)^{2/3}\qquad\,\text{ where }\;i=\sqrt{-1}
  34. ψ \psi
  35. θ \theta
  36. meridian distance = a θ 1 + n ( 1 + n 2 8 ) 2 \,\text{meridian distance}\;=\quad\frac{a\theta}{1+n}\left(1+\frac{n^{2}}{8}% \right)^{2}
  37. θ = π 2 \theta=\frac{\pi}{2}
  38. θ \theta
  39. p = 1 - 33 20 n cos 2 θ q = 33 20 n sin 2 θ p^{\prime}=1-\frac{33}{20}n\cos 2\theta\qquad q^{\prime}=\frac{33}{20}n\sin 2\theta
  40. Z = 5 4 ( 1 - 9 16 n 2 ) ( p + q i ) 8 / 33 Z^{\prime}=\frac{5}{4}\left(1-\frac{9}{16}n^{2}\right)(p^{\prime}+q^{\prime}i)% ^{8/33}
  41. θ \theta
  42. ψ \psi
  43. ϕ \scriptstyle\phi
  44. ϵ \epsilon
  45. k 0 k_{0}
  46. ν \scriptstyle\nu
  47. 10 - 8 10^{-8}
  48. θ 2 \theta_{2}
  49. ψ \psi
  50. θ \theta

Transverse_Mercator:_Redfearn_series.html

  1. a a
  2. 1 / f 1/f
  3. b b
  4. e e
  5. f f
  6. n n
  7. e e^{\prime}
  8. f = a - b a , e 2 = 2 f - f 2 , e 2 = e 2 1 - e 2 b = a ( 1 - f ) = a ( 1 - e 2 ) 1 / 2 , n = a - b a + b . \begin{aligned}\displaystyle f&\displaystyle=\frac{a-b}{a},\qquad e^{2}=2f-f^{% 2},\qquad e^{\prime 2}=\frac{e^{2}}{1-e^{2}}\\ \displaystyle b&\displaystyle=a(1-f)=a(1-e^{2})^{1/2},\qquad n=\frac{a-b}{a+b}% .\end{aligned}
  9. ρ ( ϕ ) \rho(\phi)
  10. ϕ \phi
  11. ν ( ϕ ) \nu(\phi)
  12. ν ( ϕ ) = a 1 - e 2 sin 2 ϕ , ρ ( ϕ ) = ν 3 ( 1 - e 2 ) a 2 . \nu(\phi)=\frac{a}{\sqrt{1-e^{2}\sin^{2}\phi}},\qquad\rho(\phi)=\frac{\nu^{3}(% 1-e^{2})}{a^{2}}.
  13. β ( ϕ ) \beta(\phi)
  14. η ( ϕ ) \eta(\phi)
  15. β ( ϕ ) = ν ( ϕ ) ρ ( ϕ ) , η 2 = β - 1 = e 2 cos 2 ϕ . \beta(\phi)=\frac{\nu(\phi)}{\rho(\phi)},\qquad\eta^{2}=\beta-1={e^{\prime 2}% \cos^{2}\!\phi}.
  16. s = sin ϕ , c = cos ϕ , t = tan ϕ . s=\sin\phi,\qquad c=\cos\phi,\qquad t=\tan\phi.
  17. m ( ϕ ) m(\phi)
  18. ϕ \phi
  19. m ( ϕ ) \displaystyle m(\phi)
  20. n 3 n^{3}
  21. e 6 e^{6}
  22. B 0 \displaystyle B_{0}
  23. m p = m ( π / 2 ) = π B 0 / 2 . m_{p}=m(\pi/2)=\pi B_{0}/2\,.
  24. M M
  25. ϕ 0 = M / B 0 \phi_{0}=M/B_{0}
  26. ϕ n = ϕ n - 1 + M - m ( ϕ n - 1 ) B 0 , n = 1 , 2 , 3 , \displaystyle\phi_{n}=\phi_{n-1}+\frac{M-m(\phi_{n-1})}{B_{0}},\qquad n=1,2,3,\ldots
  27. | M - m ( ϕ n - 1 ) | < 0.01 |M-m(\phi_{n-1})|<0.01
  28. M M
  29. μ = π M 2 m p . \mu=\frac{\pi M}{2m_{p}}.
  30. M M
  31. ϕ \displaystyle\phi
  32. O ( n 4 ) O(n^{4})
  33. D 2 \displaystyle D_{2}
  34. R R
  35. x = R λ , y = R ψ , x=R\lambda,\qquad\qquad y=R\psi,
  36. ψ \psi
  37. ψ \displaystyle\psi
  38. ψ \displaystyle\psi
  39. ϕ \phi
  40. λ \lambda
  41. ψ \psi
  42. λ \lambda
  43. ψ \psi
  44. λ \lambda
  45. ζ = ψ + i λ \zeta=\psi+i\lambda
  46. f ( ζ ) f(\zeta)
  47. f ( ζ ) f(\zeta)
  48. λ = 0 \lambda=0
  49. y + i x \displaystyle y+ix
  50. f ( ψ + i .0 ) f(\psi+i.0)
  51. m ( ϕ ) m(\phi)
  52. A n A_{n}
  53. f ( ζ ) f(\zeta)
  54. m ( ϕ ) m(\phi)
  55. ψ \psi
  56. ϕ \phi
  57. ψ \psi
  58. ϕ \phi
  59. x x
  60. y y
  61. x x
  62. y y
  63. λ \lambda
  64. ω \omega
  65. x / a x/a
  66. x x
  67. y y
  68. k k
  69. γ \gamma
  70. ϕ \phi
  71. λ \lambda
  72. k k
  73. γ \gamma
  74. x x
  75. y y
  76. λ \lambda
  77. λ \lambda
  78. ϕ \phi
  79. y y
  80. λ = 0 \lambda=0
  81. x ( λ , ϕ ) \displaystyle x(\lambda,\phi)
  82. ( x , y ) (x,y)
  83. ( 0 , y ) (0,y)
  84. k 0 k_{0}
  85. m = y / k 0 m=y/k_{0}
  86. ϕ 1 \phi_{1}
  87. μ \displaystyle\mu
  88. ϕ 1 \phi_{1}
  89. λ ( x , y ) \displaystyle\lambda(x,y)
  90. k k
  91. k k
  92. k 0 k_{0}
  93. λ = 0 \lambda=0
  94. x = 0 x=0
  95. γ \gamma
  96. k ( λ , ϕ ) \displaystyle k(\lambda,\phi)
  97. W 3 \displaystyle W_{3}
  98. x = 0 x=0
  99. y = 0 y=0
  100. λ \lambda
  101. ϕ \phi
  102. x x
  103. λ \lambda
  104. E 0 E0
  105. N 0 N0
  106. ϕ 0 \phi_{0}
  107. k 0 k_{0}
  108. E \displaystyle E
  109. x x
  110. E - E 0 E-E0
  111. m ( ϕ 1 ) = y k 0 = N - N 0 k 0 + m ( ϕ 0 ) m(\phi_{1})=\frac{y}{k_{0}}=\frac{N-N0}{k_{0}}+m(\phi_{0})
  112. x = 0 x=0
  113. y = 0 , y=0,
  114. λ \lambda
  115. x / a x/a
  116. λ \lambda
  117. x / a x/a
  118. β \beta
  119. η \eta
  120. n n
  121. e e^{\prime}
  122. n n

Trapezoid_graph.html

  1. O ( n log n ) {O}(n\log n)
  2. P = ( X , < ) P=(X,<)
  3. P = ( X , < ) P=(X,<)
  4. G = ( X , E ) G=(X,E)
  5. O ( n 2 ) O(n^{2})
  6. O ( n 2 log n ) {O}(n^{2}\log n)
  7. ( x 1 , , x k ) (x_{1},\ldots,x_{k})
  8. O ( n log k - 1 n ) {O}(n\log^{k-1}n)
  9. O ( n 2 log n ) {O}(n^{2}\log n)
  10. O ( n k ) {O}(nk)
  11. O ( n log n ) {O}(n\log n)
  12. O ( n ) {O}(n)
  13. O ( log n ) {O}(\log n)
  14. O ( n log n ) {O}(n\log n)
  15. G {G}
  16. F {F}
  17. G {G}
  18. G {G}
  19. G {G^{\prime}}
  20. F {F}
  21. G {G^{\prime}}
  22. G {G}
  23. F {F}
  24. F {F}
  25. O ( n 2 ) O(n^{2})
  26. O ( n ( n + m ) ) O(n(n+m))
  27. m m
  28. G {G}
  29. G {G}

Trapezoidal_rule_(differential_equations).html

  1. y = f ( t , y ) . y^{\prime}=f(t,y).
  2. y n + 1 = y n + 1 2 h ( f ( t n , y n ) + f ( t n + 1 , y n + 1 ) ) , y_{n+1}=y_{n}+\tfrac{1}{2}h\Big(f(t_{n},y_{n})+f(t_{n+1},y_{n+1})\Big),
  3. h = t n + 1 - t n h=t_{n+1}-t_{n}
  4. y n + 1 y_{n+1}
  5. t n t_{n}
  6. t n + 1 t_{n+1}
  7. y ( t n + 1 ) - y ( t n ) = t n t n + 1 f ( t , y ( t ) ) d t . y(t_{n+1})-y(t_{n})=\int_{t_{n}}^{t_{n+1}}f(t,y(t))\,\mathrm{d}t.
  8. t n t n + 1 f ( t , y ( t ) ) d t 1 2 h ( f ( t n , y ( t n ) ) + f ( t n + 1 , y ( t n + 1 ) ) ) . \int_{t_{n}}^{t_{n+1}}f(t,y(t))\,\mathrm{d}t\approx\tfrac{1}{2}h\Big(f(t_{n},y% (t_{n}))+f(t_{n+1},y(t_{n+1}))\Big).
  9. y n y ( t n ) y_{n}\approx y(t_{n})
  10. y n + 1 y ( t n + 1 ) y_{n+1}\approx y(t_{n+1})
  11. τ n \tau_{n}
  12. | τ n | 1 12 h 3 max t | y ′′′ ( t ) | . |\tau_{n}|\leq\tfrac{1}{12}h^{3}\max_{t}|y^{\prime\prime\prime}(t)|.
  13. O ( h 2 ) O(h^{2})
  14. h h
  15. { z Re ( z ) < 0 } . \{z\in\mathbb{C}\mid\operatorname{Re}(z)<0\}.

Trapping_region.html

  1. ϕ t \phi_{t}
  2. D D
  3. N N
  4. ϕ t ( N ) int ( N ) \phi_{t}(N)\subset\mathrm{int}(N)
  5. t > 0 t>0

Tree_of_primitive_Pythagorean_triples.html

  1. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  2. A = [ 1 - 2 2 2 - 1 2 2 - 2 3 ] B = [ 1 2 2 2 1 2 2 2 3 ] C = [ - 1 2 2 - 2 1 2 - 2 2 3 ] \begin{array}[]{lcr}A=\begin{bmatrix}1&-2&2\\ 2&-1&2\\ 2&-2&3\end{bmatrix}&B=\begin{bmatrix}1&2&2\\ 2&1&2\\ 2&2&3\end{bmatrix}&C=\begin{bmatrix}-1&2&2\\ -2&1&2\\ -2&2&3\end{bmatrix}\end{array}
  3. x n + 3 - 3 x n + 2 + 3 x n + 1 - x n = 0 x_{n+3}-3x_{n+2}+3x_{n+1}-x_{n}=0\,
  4. λ 3 - 3 λ 2 + 3 λ - 1 = 0. \lambda^{3}-3\lambda^{2}+3\lambda-1=0.\,
  5. x n + 3 - 5 x n + 2 - 5 x n + 1 + x n = 0 , x_{n+3}-5x_{n+2}-5x_{n+1}+x_{n}=0,\,
  6. a = m 2 - n 2 , a=m^{2}-n^{2},\,
  7. b = 2 m n , b=2mn,\,
  8. c = m 2 + n 2 , c=m^{2}+n^{2},\,
  9. [ 2 - 1 1 0 ] , [ 2 1 1 0 ] , [ 1 2 0 1 ] , \begin{array}[]{lcr}\begin{bmatrix}2&-1\\ 1&0\end{bmatrix},&\begin{bmatrix}2&1\\ 1&0\end{bmatrix},&\begin{bmatrix}1&2\\ 0&1\end{bmatrix},\end{array}
  10. a = u v , a=uv,\,
  11. b = u 2 - v 2 2 , b=\frac{u^{2}-v^{2}}{2},
  12. c = u 2 + v 2 2 , c=\frac{u^{2}+v^{2}}{2},
  13. A = [ 2 1 - 1 - 2 2 2 - 2 1 3 ] B = [ 2 1 1 2 - 2 2 2 - 1 3 ] C = [ 2 - 1 1 2 2 2 2 1 3 ] \begin{array}[]{lcr}A^{\prime}=\begin{bmatrix}2&1&-1\\ -2&2&2\\ -2&1&3\end{bmatrix}&B^{\prime}=\begin{bmatrix}2&1&1\\ 2&-2&2\\ 2&-1&3\end{bmatrix}&C^{\prime}=\begin{bmatrix}2&-1&1\\ 2&2&2\\ 2&1&3\end{bmatrix}\end{array}

Triangular_decomposition.html

  1. S S
  2. S S
  3. S S
  4. S S
  5. S S
  6. 𝐤 \mathbf{k}
  7. F R F⊂R
  8. 𝐤 \mathbf{k}
  9. V ( F ) V(F)
  10. ( F ) = i = 1 e sat ( T i ) . \sqrt{(F)}=\bigcap_{i=1}^{e}\sqrt{\mathrm{sat}(T_{i})}.
  11. V ( F ) V(F)
  12. V ( F ) = i = 1 e W ( T i ) . V(F)=\bigcup_{i=1}^{e}W(T_{i}).
  13. R R
  14. sat ( T i ) \sqrt{\mathrm{sat}(T_{i})}
  15. 𝐤 \mathbf{k}
  16. S S
  17. R R
  18. Z 𝐤 ( S ) = Z 𝐤 ( S 1 ) Z 𝐤 ( S e ) Z_{\mathbf{k}}(S)=Z_{\mathbf{k}}(S_{1})\cup\cdots\cup Z_{\mathbf{k}}(S_{e})
  19. S S
  20. S S
  21. 𝐊 \mathbf{K}
  22. 𝐤 \mathbf{k}
  23. 𝐊 \mathbf{K}
  24. 𝐤 \mathbf{k}
  25. F F
  26. V V
  27. V V
  28. V = V 1 V e . V=V_{1}\cup\cdots\cup V_{e}.
  29. V < s u b > i V j V<sub>i⊈V_{j}
  30. V V
  31. C C
  32. F \langle F\rangle
  33. F F
  34. g g
  35. F \langle F\rangle
  36. C C
  37. C C
  38. F \langle F\rangle
  39. C F C\subset\langle F\rangle
  40. g g
  41. F F
  42. C C
  43. V ( F ) V(F)

Tricomi–Carlitz_polynomials.html

  1. l n ( x ) = ( - 1 ) n L n ( x - n ) ( x ) . \displaystyle l_{n}(x)=(-1)^{n}L_{n}^{(x-n)}(x).

Trirectangular_tetrahedron.html

  1. 1 h 2 = 1 a 2 + 1 b 2 + 1 c 2 . \frac{1}{h^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}.
  2. T 0 T_{0}
  3. T 0 = a b c 2 h . T_{0}=\frac{abc}{2h}.
  4. T 0 T_{0}
  5. T 1 T_{1}
  6. T 2 T_{2}
  7. T 3 T_{3}
  8. T 0 2 = T 1 2 + T 2 2 + T 3 2 . T_{0}^{2}=T_{1}^{2}+T_{2}^{2}+T_{3}^{2}.

Truck_lane_restriction.html

  1. I = 1 r H C I=\frac{1}{rHC}
  2. f H ( h ) = { λ 1 e - h λ 1 , h τ e τ ( λ 0 - λ 1 ) λ 0 e - h λ 0 , h > τ f_{H}(h)=\begin{cases}\lambda_{1}e^{-h\lambda_{1}},&h\leq\tau\\ e^{\tau(\lambda_{0}-\lambda_{1})}\lambda_{0}e^{-h\lambda_{0}},&h>\tau\end{cases}
  3. U = D + ( w v k j w + v ) U=D+(\frac{wvkj}{w+v})

True_strength_index.html

  1. T S I ( c 0 , r , s ) = 100 E M A ( E M A ( m , r ) , s ) E M A ( E M A ( | m | , r ) , s ) TSI(c_{0},r,s)=100\frac{EMA(EMA(m,r),s)}{EMA(EMA(|m|,r),s)}
  2. E M A ( m 0 , n ) = 2 n + 1 [ m 0 - E M A ( m 1 , n ) ] + E M A ( m 1 , n ) EMA(m_{0},n)=\frac{2}{n+1}\left[m_{0}-EMA(m_{1},n)\right]+EMA(m_{1},n)

Truncated_24-cell_honeycomb.html

  1. F ~ 4 {\tilde{F}}_{4}
  2. B ~ 4 {\tilde{B}}_{4}
  3. C ~ 4 {\tilde{C}}_{4}
  4. D ~ 4 {\tilde{D}}_{4}
  5. D ~ 4 {\tilde{D}}_{4}
  6. F ~ 4 {\tilde{F}}_{4}
  7. F ~ 4 {\tilde{F}}_{4}
  8. C ~ 4 {\tilde{C}}_{4}
  9. B ~ 4 {\tilde{B}}_{4}
  10. D ~ 4 {\tilde{D}}_{4}

Truncus_(mathematics).html

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

Tschuprow's_T.html

  1. T = ϕ 2 ( r - 1 ) ( c - 1 ) T=\sqrt{\frac{\phi^{2}}{\sqrt{(r-1)(c-1)}}}
  2. π i j \pi_{ij}
  3. ( i , j ) (i,j)
  4. π i + = j = 1 c π i j \pi_{i+}=\sum_{j=1}^{c}\pi_{ij}
  5. π + j = i = 1 r π i j . \pi_{+j}=\sum_{i=1}^{r}\pi_{ij}.
  6. ϕ 2 = i = 1 r j = 1 c ( π i j - π i + π + j ) 2 π i + π + j , \phi^{2}=\sum_{i=1}^{r}\sum_{j=1}^{c}\frac{(\pi_{ij}-\pi_{i+}\pi_{+j})^{2}}{% \pi_{i+}\pi_{+j}},
  7. T = ϕ 2 ( r - 1 ) ( c - 1 ) . T=\sqrt{\frac{\phi^{2}}{\sqrt{(r-1)(c-1)}}}.
  8. π i j = π i + π + j \pi_{ij}=\pi_{i+}\pi_{+j}
  9. π i j > 0 \pi_{ij}>0
  10. T ^ = i = 1 r j = 1 c ( p i j - p i + p + j ) 2 p i + p + j ( r - 1 ) ( c - 1 ) , \hat{T}=\sqrt{\frac{\sum_{i=1}^{r}\sum_{j=1}^{c}\frac{(p_{ij}-p_{i+}p_{+j})^{2% }}{p_{i+}p_{+j}}}{\sqrt{(r-1)(c-1)}}},
  11. p i j = n i j / n p_{ij}=n_{ij}/n
  12. ( i , j ) (i,j)
  13. χ 2 \chi^{2}
  14. T ^ = χ 2 / n ( r - 1 ) ( c - 1 ) . \hat{T}=\sqrt{\frac{\chi^{2}/n}{\sqrt{(r-1)(c-1)}}}.

Tseitin_transformation.html

  1. C = A B C=A\cdot B
  2. ( A ¯ B ¯ C ) ( A C ¯ ) ( B C ¯ ) (\overline{A}\vee\overline{B}\vee C)\wedge(A\vee\overline{C})\wedge(B\vee% \overline{C})
  3. C = A B ¯ C=\overline{A\cdot B}
  4. ( A ¯ B ¯ C ¯ ) ( A C ) ( B C ) (\overline{A}\vee\overline{B}\vee\overline{C})\wedge(A\vee C)\wedge(B\vee C)
  5. C = A + B C=A+B
  6. ( A B C ¯ ) ( A ¯ C ) ( B ¯ C ) (A\vee B\vee\overline{C})\wedge(\overline{A}\vee C)\wedge(\overline{B}\vee C)
  7. C = A + B ¯ C=\overline{A+B}
  8. ( A B C ) ( A ¯ C ¯ ) ( B ¯ C ¯ ) (A\vee B\vee C)\wedge(\overline{A}\vee\overline{C})\wedge(\overline{B}\vee% \overline{C})
  9. C = A ¯ C=\overline{A}
  10. ( A ¯ C ¯ ) ( A C ) (\overline{A}\vee\overline{C})\wedge(A\vee C)
  11. C = A B C=A\oplus B
  12. ( A ¯ B ¯ C ¯ ) ( A B C ¯ ) ( A B ¯ C ) ( A ¯ B C ) (\overline{A}\vee\overline{B}\vee\overline{C})\wedge(A\vee B\vee\overline{C})% \wedge(A\vee\overline{B}\vee C)\wedge(\overline{A}\vee B\vee C)
  13. y = x 1 ¯ x 2 + x 1 x 2 ¯ + x 2 ¯ x 3 y=\overline{x1}\cdot x2+x1\cdot\overline{x2}+\overline{x2}\cdot x3
  14. x 2 x_{2}
  15. g a t e 1 gate1
  16. ( g a t e 1 x 1 ) ( g a t e 1 ¯ x 1 ¯ ) (gate1\vee x1)\wedge(\overline{gate1}\vee\overline{x1})
  17. g a t e 2 gate2
  18. ( g a t e 2 ¯ g a t e 1 ) ( g a t e 2 ¯ x 2 ) ( x 2 ¯ g a t e 2 g a t e 1 ¯ ) (\overline{gate2}\vee gate1)\wedge(\overline{gate2}\vee x2)\wedge(\overline{x2% }\vee gate2\vee\overline{gate1})
  19. g a t e 3 gate3
  20. ( g a t e 3 x 2 ) ( g a t e 3 ¯ x 2 ¯ ) (gate3\vee x2)\wedge(\overline{gate3}\vee\overline{x2})
  21. g a t e 4 gate4
  22. ( g a t e 4 ¯ x 1 ) ( g a t e 4 ¯ g a t e 3 ) ( g a t e 3 ¯ g a t e 4 x 1 ¯ ) (\overline{gate4}\vee x1)\wedge(\overline{gate4}\vee gate3)\wedge(\overline{% gate3}\vee gate4\vee\overline{x1})
  23. g a t e 5 gate5
  24. ( g a t e 5 x 2 ) ( g a t e 5 ¯ x 2 ¯ ) (gate5\vee x2)\wedge(\overline{gate5}\vee\overline{x2})
  25. g a t e 6 gate6
  26. ( g a t e 6 ¯ g a t e 5 ) ( g a t e 6 ¯ x 3 ) ( x 3 ¯ g a t e 6 g a t e 5 ¯ ) (\overline{gate6}\vee gate5)\wedge(\overline{gate6}\vee x3)\wedge(\overline{x3% }\vee gate6\vee\overline{gate5})
  27. g a t e 7 gate7
  28. ( g a t e 7 g a t e 2 ¯ ) ( g a t e 7 g a t e 4 ¯ ) ( g a t e 2 g a t e 7 ¯ g a t e 4 ) (gate7\vee\overline{gate2})\wedge(gate7\vee\overline{gate4})\wedge(gate2\vee% \overline{gate7}\vee gate4)
  29. g a t e 8 gate8
  30. ( g a t e 8 g a t e 6 ¯ ) ( g a t e 8 g a t e 7 ¯ ) ( g a t e 6 g a t e 8 ¯ g a t e 7 ) (gate8\vee\overline{gate6})\wedge(gate8\vee\overline{gate7})\wedge(gate6\vee% \overline{gate8}\vee gate7)
  31. g a t e 8 gate8
  32. ( g a t e 8 ) (gate8)
  33. ( g a t e 1 x 1 ) ( g a t e 1 ¯ x 1 ¯ ) ( g a t e 2 ¯ g a t e 1 ) ( g a t e 2 ¯ x 2 ) (gate1\vee x1)\wedge(\overline{gate1}\vee\overline{x1})\wedge(\overline{gate2}% \vee gate1)\wedge(\overline{gate2}\vee x2)\wedge
  34. ( x 2 ¯ g a t e 2 g a t e 1 ¯ ) ( g a t e 3 x 2 ) ( g a t e 3 ¯ x 2 ¯ ) ( g a t e 4 ¯ x 1 ) (\overline{x2}\vee gate2\vee\overline{gate1})\wedge(gate3\vee x2)\wedge(% \overline{gate3}\vee\overline{x2})\wedge(\overline{gate4}\vee x1)\wedge
  35. ( g a t e 4 ¯ g a t e 3 ) ( g a t e 3 ¯ g a t e 4 x 1 ¯ ) ( g a t e 5 x 2 ) (\overline{gate4}\vee gate3)\wedge(\overline{gate3}\vee gate4\vee\overline{x1}% )\wedge(gate5\vee x2)\wedge
  36. ( g a t e 5 ¯ x 2 ¯ ) ( g a t e 6 ¯ g a t e 5 ) ( g a t e 6 ¯ x 3 ) (\overline{gate5}\vee\overline{x2})\wedge(\overline{gate6}\vee gate5)\wedge(% \overline{gate6}\vee x3)\wedge
  37. ( x 3 ¯ g a t e 6 g a t e 5 ¯ ) ( g a t e 7 g a t e 2 ¯ ) ( g a t e 7 g a t e 4 ¯ ) (\overline{x3}\vee gate6\vee\overline{gate5})\wedge(gate7\vee\overline{gate2})% \wedge(gate7\vee\overline{gate4})\wedge
  38. ( g a t e 2 g a t e 7 ¯ g a t e 4 ) ( g a t e 8 g a t e 6 ¯ ) ( g a t e 8 g a t e 7 ¯ ) (gate2\vee\overline{gate7}\vee gate4)\wedge(gate8\vee\overline{gate6})\wedge(% gate8\vee\overline{gate7})\wedge
  39. ( g a t e 6 g a t e 8 ¯ g a t e 7 ) ( g a t e 8 ) = 1 (gate6\vee\overline{gate8}\vee gate7)\wedge(gate8)=1
  40. ( x 1 , x 2 , x 3 ) = ( 0 , 0 , 1 ) (x1,x2,x3)=(0,0,1)
  41. ( x 1 x 2 x 3 ¯ ) (x1\vee x2\vee\overline{x3})
  42. ( C ( A B ) ) ( C ¯ ( A ¯ B ¯ ) ) (C\rightarrow(A\vee B))\wedge(\overline{C}\rightarrow(\overline{A}\wedge% \overline{B}))
  43. ( C ¯ ( A B ) ) ( C ( A ¯ B ¯ ) ) (\overline{C}\vee(A\vee B))\wedge(C\vee(\overline{A}\wedge\overline{B}))
  44. ( C ¯ A B ) ( ( C A ¯ ) ( C B ¯ ) ) (\overline{C}\vee A\vee B)\wedge((C\vee\overline{A})\wedge(C\vee\overline{B}))
  45. ( C ¯ A B ) ( C A ¯ ) ( C B ¯ ) (\overline{C}\vee A\vee B)\wedge(C\vee\overline{A})\wedge(C\vee\overline{B})
  46. ( C A ¯ ) ( C ¯ A ) (C\rightarrow\overline{A})\wedge(\overline{C}\rightarrow A)
  47. ( C ¯ A ¯ ) ( C A ) (\overline{C}\vee\overline{A})\wedge(C\vee A)
  48. ( C ( A B ) ¯ ) ( C ¯ ( A B ) ) (C\rightarrow\overline{(A\vee B)})\wedge(\overline{C}\rightarrow(A\vee B))
  49. ( C ¯ ( A ¯ B ¯ ) ) ¯ ¯ ( C A B ) ) \overline{\overline{(\overline{C}\vee(\overline{A}\wedge\overline{B}))}}\wedge% (C\vee A\vee B))
  50. ( C ( A B ) ) ¯ ( C A B ) ) \overline{(C\wedge(A\vee B))}\wedge(C\vee A\vee B))
  51. ( A C ) ( A B ) ¯ ( C A B ) ) \overline{(A\wedge C)\vee(A\vee B)}\wedge(C\vee A\vee B))
  52. ( A ¯ C ¯ ) ( A ¯ B ¯ ) ( C A B ) ) (\overline{A}\vee\overline{C})\wedge(\overline{A}\wedge\overline{B})\wedge(C% \vee A\vee B))

Tube_domain.html

  1. a = ( z 1 , , z n ) = ( x 1 + i y 1 , , x n + i y n ) = ( x 1 , , x n ) + i ( y 1 , , y n ) = x + i y . a=(z_{1},\dots,z_{n})=(x_{1}+iy_{1},\dots,x_{n}+iy_{n})=(x_{1},\dots,x_{n})+i(% y_{1},\dots,y_{n})=x+iy.
  2. T A = { z = x + i y n x A } . T_{A}=\{z=x+iy\in\mathbb{C}^{n}\mid x\in A\}.
  3. ch T A = T ch A . \operatorname{ch}\,T_{A}=T_{\operatorname{ch}\,A}.
  4. n | F ( x + i y ) | p d y < \int_{\mathbb{R}^{n}}|F(x+iy)|^{p}\,dy<\infty
  5. sup x A n | f ( t ) | 2 e - 4 π x t d t < . \sup_{x\in A}\int_{\mathbb{R}^{n}}|f(t)|^{2}e^{-4\pi x\cdot t}\,dt<\infty.
  6. F ( x + i y ) = n e 2 π z t f ( t ) d t . F(x+iy)=\int_{\mathbb{R}^{n}}e^{2\pi z\cdot t}f(t)\,dt.
  7. x A t x A for all t > 0. x\in A\implies tx\in A\ \ \ \,\text{for all}\ t>0.
  8. lim y 0 F ( x + i y ) \lim_{y\to 0}F(x+iy)

Tug_of_war_(astronomy).html

  1. F = G m 1 m 2 d 2 F=G\cdot\frac{m_{1}\cdot m_{2}}{d^{2}}
  2. F p = G m M p d p 2 F_{p}=\frac{G\cdot m\cdot M_{p}}{d_{p}^{2}}
  3. F s = G m M s d s 2 F_{s}=\frac{G\cdot m\cdot M_{s}}{d_{s}^{2}}
  4. F p F s = M p d s 2 M s d p 2 \frac{F_{p}}{F_{s}}=\frac{M_{p}\cdot d_{s}^{2}}{M_{s}\cdot d_{p}^{2}}
  5. F p F s = 1.9 10 27 ( 778.3 ) 2 1.989 10 30 ( 1.883 ) 2 163 \frac{F_{p}}{F_{s}}=\frac{1.9\cdot 10^{27}\cdot(778.3)^{2}}{1.989\cdot 10^{30}% \cdot(1.883)^{2}}\approx 163

Turingery.html

  1. ψ \psi
  2. ψ \psi
  3. ψ \psi
  4. ψ \psi
  5. ψ \psi
  6. μ \mu
  7. μ \mu
  8. χ \chi
  9. χ \chi
  10. χ \chi
  11. χ \chi
  12. χ \chi
  13. χ \chi
  14. ψ \psi
  15. μ \mu
  16. χ \chi
  17. ψ \psi
  18. χ \chi
  19. ψ \psi
  20. χ \chi
  21. ψ \psi
  22. χ \chi
  23. ψ \psi
  24. χ \chi
  25. ψ \psi
  26. ψ \psi
  27. χ \chi
  28. χ \chi
  29. ψ \psi
  30. χ \chi
  31. χ \chi
  32. χ \chi
  33. χ \chi
  34. χ \chi
  35. χ \chi
  36. ψ \psi
  37. χ \chi

Tversky_index.html

  1. S ( X , Y ) = | X Y | | X Y | + α | X - Y | + β | Y - X | S(X,Y)=\frac{|X\cap Y|}{|X\cap Y|+\alpha|X-Y|+\beta|Y-X|}
  2. X - Y X-Y
  3. α , β 0 \alpha,\beta\geq 0
  4. α = β = 1 \alpha=\beta=1
  5. α = β = 0.5 \alpha=\beta=0.5
  6. α \alpha
  7. β \beta
  8. α + β = 1 \alpha+\beta=1
  9. S ( X , Y ) = | X Y | | X Y | + β ( α a + ( 1 - α ) b ) S(X,Y)=\frac{|X\cap Y|}{|X\cap Y|+\beta\left(\alpha a+(1-\alpha)b\right)}
  10. a = min ( | X - Y | , | Y - X | ) a=\min\left(|X-Y|,|Y-X|\right)
  11. b = max ( | X - Y | , | Y - X | ) b=\max\left(|X-Y|,|Y-X|\right)
  12. α \alpha
  13. β \beta
  14. α \alpha
  15. | X - Y | |X-Y|
  16. | Y - X | |Y-X|
  17. β \beta
  18. | X Y | |X\,\triangle\,Y\,|
  19. | X Y | |X\cap Y|

Twisted_Poincaré_duality.html

  1. o ( M ) o(M)
  2. H * ( M ; w ) H^{*}(M;\mathbb{R}^{w})
  3. H * ( M ; o ( M ) ) H^{*}(M;o(M))
  4. θ : H d ( M ; o ( M ) ) \theta:H^{d}(M;o(M))\to\mathbb{R}
  5. : H * ( M ; ) H d - * ( M , o ( M ) ) H d ( M , o ( M ) ) \cup:H^{*}(M;\mathbb{R})\otimes H^{d-*}(M,o(M))\to H^{d}(M,o(M))\simeq\mathbb{R}

Two-balloon_experiment.html

  1. f i = 1 L i [ k K T ( L i L i 0 ) 2 - p V ] . f_{i}={1\over L_{i}}\left[kKT\left({L_{i}\over L_{i}^{0}}\right)^{2}-pV\right].
  2. f i = ( C 1 / L i ) ( λ i 2 - C 2 p ) f_{i}=\left(C_{1}/L_{i}\right)\left(\lambda_{i}^{2}-C_{2}p\right)
  3. λ r 2 = ( t / t 0 ) 2 = C 2 p \lambda_{r}^{2}=(t/t_{0})^{2}=C_{2}p
  4. r r
  5. t 1 r 2 t\propto\frac{1}{r^{2}}
  6. t t 0 = ( r 0 r ) 2 \frac{t}{t_{0}}=\left(\frac{r_{0}}{r}\right)^{2}
  7. p = 1 C 2 ( r 0 r ) 4 p=\frac{1}{C_{2}}\left(\frac{r_{0}}{r}\right)^{4}
  8. \propto
  9. f t ( r / r 0 2 ) [ 1 - ( r 0 / r ) 6 ] . f_{t}\propto(r/r_{0}^{2})\left[1-(r_{0}/r)^{6}\right].
  10. P in - P out P = f t π r 2 = C r 0 2 r [ 1 - ( r 0 r ) 6 ] P_{\mathrm{in}}-P_{\mathrm{out}}\equiv P=\frac{f_{t}}{\pi r^{2}}=\frac{C}{r_{0% }^{2}r}\left[1-\left(\frac{r_{0}}{r}\right)^{6}\right]
  11. r = r p = 7 1 / 6 r 0 1.38 r 0 r=r_{p}=7^{1/6}r_{0}\approx 1.38r_{0}

Two-moment_decision_model.html

  1. w - μ w σ w , \tfrac{w-\mu_{w}}{\sigma_{w}},
  2. μ w + σ w x \mu_{w}+\sigma_{w}x
  3. E u ( w ) = - u ( μ w + σ w x ) f ( x ) d x v ( μ w , σ w ) , \operatorname{E}u(w)=\int_{-\infty}^{\infty}\!u(\mu_{w}+\sigma_{w}x)f(x)\,dx% \equiv v(\mu_{w},\sigma_{w}),

Two-state_trajectory.html

  1. + / - +/-
  2. X ( t ) X(t)
  3. t , t,
  4. X ( t ) = c off X(t)=c_{\mathrm{off}}
  5. X ( t ) = c on X(t)=c_{\mathrm{on}}

Two-way_analysis_of_variance.html

  1. I I
  2. J J
  3. ( i , j ) (i,j)
  4. I × J I\times J
  5. ( i , j ) (i,j)
  6. n i j n_{ij}
  7. k k
  8. n i + = j = 1 J n i j n_{i+}=\sum_{j=1}^{J}n_{ij}
  9. n + j = i = 1 I n i j n_{+j}=\sum_{i=1}^{I}n_{ij}
  10. n = i , j n i j = i n i + = j n + j n=\sum_{i,j}n_{ij}=\sum_{i}n_{i+}=\sum_{j}n_{+j}
  11. K K
  12. i , j n i j = K \forall i,j\;n_{ij}=K
  13. i , j n i j = n i + × n + j n \forall i,j\;n_{ij}=\frac{n_{i+}\times n_{+j}}{n}
  14. n n
  15. Y i j k Y_{ijk}
  16. y i j k y_{ijk}
  17. k k
  18. ( i , j ) (i,j)
  19. μ i j \mu_{ij}
  20. σ 2 \sigma^{2}
  21. Y i j k | μ i j , σ 2 i . i . d . 𝒩 ( μ i j , σ 2 ) Y_{ijk}\,|\,\mu_{ij},\sigma^{2}\;\overset{i.i.d.}{\sim}\;\mathcal{N}(\mu_{ij},% \sigma^{2})
  22. μ i j = μ + α i + β j + γ i j \mu_{ij}=\mu+\alpha_{i}+\beta_{j}+\gamma_{ij}
  23. μ \mu
  24. α i \alpha_{i}
  25. i i
  26. β j \beta_{j}
  27. j j
  28. γ i j \gamma_{ij}
  29. ( i , j ) (i,j)
  30. ϵ i j k \epsilon_{ijk}
  31. n n
  32. Y i j k = μ i j + ϵ i j k with ϵ i j k i . i . d . 𝒩 ( 0 , σ 2 ) Y_{ijk}=\mu_{ij}+\epsilon_{ijk}\,\text{ with }\epsilon_{ijk}\overset{i.i.d.}{% \sim}\mathcal{N}(0,\sigma^{2})
  33. i α i = j β j = i j γ i j = 0 \sum_{i}\alpha_{i}=\sum_{j}\beta_{j}=\sum_{i}\sum_{j}\gamma_{ij}=0

Type-1_OWA_operators.html

  1. α \alpha
  2. F ( X ) F(X)
  3. X X
  4. { W i } i = 1 n \left\{{W^{i}}\right\}_{i=1}^{n}
  5. U = [ 0 , 1 ] U=[0,1]
  6. Φ \Phi
  7. Φ : F ( X ) × × F ( X ) F ( X ) \Phi\colon F(X)\times\cdots\times F(X)\longrightarrow F(X)
  8. ( A 1 , , A n ) Y (A^{1},\cdots,A^{n})\mapsto Y
  9. μ Y ( y ) = sup k = 1 n w ¯ i a σ ( i ) = y ( μ W 1 ( w 1 ) μ W n ( w n ) μ A 1 ( a 1 ) μ A n ( a n ) ) \mu_{Y}(y)=\displaystyle\sup_{\displaystyle\sum_{k=1}^{n}\bar{w}_{i}a_{\sigma(% i)}=y}\left({\begin{array}[]{*{1}l}\mu_{W^{1}}(w_{1})\wedge\cdots\wedge\mu_{W^% {n}}(w_{n})\wedge\mu_{A^{1}}(a_{1})\wedge\cdots\wedge\mu_{A^{n}}(a_{n})\end{% array}}\right)
  10. w ¯ i = w i i = 1 n w i \bar{w}_{i}=\frac{w_{i}}{\sum_{i=1}^{n}{w_{i}}}
  11. σ : { 1 , , n } { 1 , , n } \sigma\colon\{1,\cdots,n\}\longrightarrow\{1,\cdots,n\}
  12. a σ ( i ) a σ ( i + 1 ) , i = 1 , , n - 1 a_{\sigma(i)}\geq a_{\sigma(i+1)},\ \forall i=1,\cdots,n-1
  13. a σ ( i ) a_{\sigma(i)}
  14. i i
  15. { a 1 , , a n } \left\{{a_{1},\cdots,a_{n}}\right\}
  16. { W i } i = 1 n \left\{{W^{i}}\right\}_{i=1}^{n}
  17. U = [ 0 , 1 ] U=[0,\;\;1]
  18. α [ 0 , 1 ] \alpha\in[0,\;1]
  19. α \alpha
  20. α \alpha
  21. { W α i } i = 1 n \left\{{W_{\alpha}^{i}}\right\}_{i=1}^{n}
  22. α \alpha
  23. { A i } i = 1 n \left\{{A^{i}}\right\}_{i=1}^{n}
  24. Φ α ( A α 1 , , A α n ) = { i = 1 n w i a σ ( i ) i = 1 n w i | w i W α i , a i A α i , i = 1 , , n } \Phi_{\alpha}\left({A_{\alpha}^{1},\ldots,A_{\alpha}^{n}}\right)=\left\{{\frac% {\sum\limits_{i=1}^{n}{w_{i}a_{\sigma(i)}}}{\sum\limits_{i=1}^{n}{w_{i}}}\left% |{w_{i}\in W_{\alpha}^{i},\;a_{i}}\right.\in A_{\alpha}^{i},\;i=1,\ldots,n}\right\}
  25. W α i = { w | μ W i ( w ) α } , A α i = { x | μ A i ( x ) α } W_{\alpha}^{i}=\{w|\mu_{W_{i}}(w)\geq\alpha\},A_{\alpha}^{i}=\{x|\mu_{A_{i}}(x% )\geq\alpha\}
  26. σ : { 1 , , n } { 1 , , n } \sigma:\{\;1,\cdots,n\;\}\to\{\;1,\cdots,n\;\}
  27. a σ ( i ) a σ ( i + 1 ) , i = 1 , , n - 1 a_{\sigma(i)}\geq a_{\sigma(i+1)},\;\forall\;i=1,\cdots,n-1
  28. a σ ( i ) a_{\sigma(i)}
  29. i i
  30. { a 1 , , a n } \left\{{a_{1},\cdots,a_{n}}\right\}
  31. { W i } i = 1 n \left\{{W^{i}}\right\}_{i=1}^{n}
  32. U = [ 0 , 1 ] U=[0,\;\;1]
  33. A 1 , , A n A^{1},\cdots,A^{n}
  34. Y = G Y=G
  35. Y Y
  36. G G
  37. α \alpha
  38. α \alpha
  39. Φ α ( A α 1 , , A α n ) \Phi_{\alpha}\left({A_{\alpha}^{1},\cdots,A_{\alpha}^{n}}\right)
  40. μ G ( x ) = α : x Φ α ( A α 1 , , A α n ) α α \mu_{G}(x)=\mathop{\vee}\limits_{\alpha:x\in\Phi_{\alpha}\left({A_{\alpha}^{1}% ,\cdots,A_{\alpha}^{n}}\right)_{\alpha}}\alpha
  41. Φ α ( A α 1 , , A α n ) - = min W α - i w i W α + i A α - i a i A α + i i = 1 n w i a σ ( i ) / i = 1 n w i \Phi_{\alpha}\left({A_{\alpha}^{1},\cdots,A_{\alpha}^{n}}\right)_{-}=\mathop{% \min}\limits_{\begin{array}[]{l}W_{\alpha-}^{i}\leq w_{i}\leq W_{\alpha+}^{i}A% _{\alpha-}^{i}\leq a_{i}\leq A_{\alpha+}^{i}\end{array}}\sum\limits_{i=1}^{n}{% w_{i}a_{\sigma(i)}/\sum\limits_{i=1}^{n}{w_{i}}}
  42. Φ α ( A α 1 , , A α n ) + = max W α - i w i W α + i A α - i a i A α + i i = 1 n w i a σ ( i ) / i = 1 n w i \Phi_{\alpha}\left({A_{\alpha}^{1},\cdots,A_{\alpha}^{n}}\right)_{+}=\mathop{% \max}\limits_{\begin{array}[]{l}W_{\alpha-}^{i}\leq w_{i}\leq W_{\alpha+}^{i}A% _{\alpha-}^{i}\leq a_{i}\leq A_{\alpha+}^{i}\end{array}}\sum\limits_{i=1}^{n}{% w_{i}a_{\sigma(i)}/\sum\limits_{i=1}^{n}{w_{i}}}
  43. α \alpha
  44. α [ 0 , 1 ] \alpha\in[0,1]
  45. ρ α + i 0 \rho_{\alpha+}^{i_{0}^{\ast}}
  46. i 0 = 1 i_{0}=1
  47. ρ α + i 0 A α + σ ( i 0 ) \rho_{\alpha+}^{i_{0}}\geq A_{\alpha+}^{\sigma(i_{0})}
  48. ρ α + i 0 \rho_{\alpha+}^{i_{0}}
  49. i 0 i 0 + 1 i_{0}\leftarrow i_{0}+1
  50. ρ α - i 0 \rho_{\alpha-}^{i_{0}^{\ast}}
  51. i 0 = 1 i_{0}=1
  52. ρ α - i 0 A α - σ ( i 0 ) \rho_{\alpha-}^{i_{0}}\geq A_{\alpha-}^{\sigma(i_{0})}
  53. ρ α - i 0 \rho_{\alpha-}^{i_{0}}
  54. i 0 i 0 + 1 i_{0}\leftarrow i_{0}+1
  55. G G
  56. [ ρ α - i 0 , ρ α + i 0 ] \left[{\rho_{\alpha-}^{i_{0}^{\ast}},\;\rho_{\alpha+}^{i_{0}^{\ast}}}\right]
  57. μ G ( x ) = α : x [ ρ α - i 0 , ρ α + i 0 ] α \mu_{G}(x)=\mathop{\vee}\limits_{\alpha:x\in\left[{\rho_{\alpha-}^{i_{0}^{\ast% }},\;\rho_{\alpha+}^{i_{0}^{\ast}}}\right]}\alpha

Typing_environment.html

  1. Γ \Gamma
  2. x , τ \langle x,\tau\rangle
  3. x : τ x:\tau
  4. x x
  5. τ \tau

Ultrasensitivity.html

  1. A + B A B \ A+\ B\rightleftharpoons\ AB
  2. B T B_{T}
  3. B T B_{T}
  4. K d K_{d}
  5. B T B_{T}
  6. A T A_{T}
  7. A T A_{T}
  8. B T B_{T}
  9. A T A_{T}
  10. B T B_{T}
  11. B T B_{T}
  12. A T A_{T}
  13. A T A_{T}
  14. A T A_{T}
  15. B T B_{T}
  16. K d K_{d}
  17. A T A_{T}
  18. W + E 1 a 1 d 1 W E 1 k 1 W + E 1 W+E_{1}\overset{a_{1}}{\underset{}{}}{d_{1}}\rightleftharpoons WE_{1}\overset{% k_{1}}{\rightarrow}W^{\prime}+E_{1}
  19. W + E 2 a 2 d 2 W E 2 k 2 W + E 2 W^{\prime}+E_{2}\overset{a_{2}}{\underset{}{}}{d_{2}}\rightleftharpoons W^{% \prime}E_{2}\overset{k_{2}}{\rightarrow}W+E_{2}
  20. d [ W ] d t = - a 1 [ W ] [ E 1 ] + d 1 [ W E 1 ] + k 2 [ W E 2 ] \frac{d[W]}{dt}=-a_{1}[W][E_{1}]+d_{1}[WE_{1}]+k_{2}[W^{\prime}E_{2}]
  21. d [ E 1 ] d t = a 1 [ W ] [ E 1 ] - ( d 1 + k 1 ) [ W E 1 ] \frac{d[E_{1}]}{dt}=a_{1}[W][E_{1}]-(d_{1}+k_{1})[WE_{1}]
  22. d [ W ] d t = - a 2 [ W ] [ E 2 ] + d 2 [ W E 2 ] + k 1 [ W E 1 ] \frac{d[W^{\prime}]}{dt}=-a_{2}[W^{\prime}][E_{2}]+d_{2}[W^{\prime}E_{2}]+k1[% WE_{1}]
  23. d [ E 2 ] d t = a 2 [ W ] [ E 2 ] - ( d 2 + k 2 ) [ W E 2 ] \frac{d[E_{2}]}{dt}=a_{2}[W^{\prime}][E_{2}]-(d_{2}+k_{2})[W^{\prime}E_{2}]
  24. [ W T ] = [ W ] + [ W ] + [ W E 1 ] + [ W E 2 ] [W_{T}]=[W]+[W^{\prime}]+[WE_{1}]+[W^{\prime}E_{2}]
  25. [ E 1 T ] = [ E 1 ] + [ W E 1 ] [E_{1T}]=[E_{1}]+[WE_{1}]
  26. [ E 2 T ] = [ E 2 ] + [ W E 2 ] [E_{2T}]=[E_{2}]+[W^{\prime}E_{2}]
  27. [ W T ] [ E 1 ] [W_{T}]\gg[E_{1}]
  28. [ E 2 ] [E_{2}]
  29. [ W E 1 ] [WE_{1}]
  30. [ W E 2 ] [W^{\prime}E_{2}]
  31. V 1 / V 2 V_{1}/V_{2}
  32. W = [ W ] / [ W T ] W=[W]/[W_{T}]
  33. W = 1 - W W=1-W^{\prime}
  34. V 1 V 2 = W ( 1 - W + K 1 ) ( 1 - W ) ( W + K 2 ) , \frac{V_{1}}{V_{2}}=\frac{W^{\prime}\left(1-W^{\prime}+K_{1}\right)}{\left(1-W% ^{\prime}\right)\left(W^{\prime}+K_{2}\right)},
  35. k 1 [ W E 1 ] = k 2 [ W E 2 ] k_{1}[WE_{1}]=k_{2}[W^{\prime}E_{2}]
  36. V 1 = k 1 [ E 1 T ] , V_{1}=k_{1}[E_{1T}],
  37. V 2 = k 2 [ E 2 T ] , V_{2}=k_{2}[E_{2T}],
  38. K 1 = d 1 + k 1 a 1 [ W T ] = K M 1 [ W T ] , K_{1}=\frac{d_{1}+k_{1}}{a_{1}[W_{T}]}=\frac{K_{M1}}{[W_{T}]},
  39. K 2 = d 2 + k 2 a 2 [ W T ] = K M 2 [ W T ] . K_{2}=\frac{d_{2}+k_{2}}{a_{2}[W_{T}]}=\frac{K_{M2}}{[W_{T}]}.
  40. V 1 / V 2 V_{1}/V_{2}
  41. W W^{\prime}
  42. W W
  43. V 1 / V 2 V_{1}/V_{2}
  44. [ L ] [L]
  45. θ \theta
  46. θ = [ L ] n K d + [ L ] n = [ L ] n ( K A ) n + [ L ] n , \theta=\frac{[L]^{n}}{K_{d}+[L]^{n}}=\frac{[L]^{n}}{(K_{A})^{n}+[L]^{n}},
  47. n n

Uncertainty_exponent.html

  1. ϵ \epsilon
  2. f ( ϵ ) f(\epsilon)
  3. ε \varepsilon
  4. f ( ε ) ε γ f(\varepsilon)\sim\varepsilon^{\gamma}\,
  5. γ \gamma
  6. γ = lim ε 0 ln f ( ε ) ln ε \gamma=\lim_{\varepsilon\to 0}\frac{\ln f(\varepsilon)}{\ln\varepsilon}
  7. D 0 = N - γ D_{0}=N-\gamma\,

Unemployment_in_the_United_Kingdom.html

  1. Official UK Unemployment Rate = All those aged 16 and over classed as unemployed All those aged 16 and over classed as employed or unemployed {\mathrm{Official\ UK\ Unemployment\ Rate=\dfrac{All\ those\ aged\ 16\ and\ % over\ classed\ as\ unemployed}{All\ those\ aged\ 16\ and\ over\ classed\ as\ % employed\ or\ unemployed}}}

Uniform_limit_theorem.html

  1. sin n ( x ) \scriptstyle\scriptstyle\sin^{n}(x)
  2. d Y ( f 0 ( x ) , f 0 ( y ) ) < ϵ , y U d_{Y}(f_{0}(x),f_{0}(y))<\epsilon,\qquad\forall y\in U
  3. d Y ( f N ( t ) , f 0 ( t ) ) < ϵ 3 , t X d_{Y}(f_{N}(t),f_{0}(t))<\frac{\epsilon}{3},\qquad\forall t\in X
  4. d Y ( f N ( x ) , f N ( y ) ) < ϵ 3 , y U d_{Y}(f_{N}(x),f_{N}(y))<\frac{\epsilon}{3},\qquad\forall y\in U
  5. d Y ( f 0 ( x ) , f 0 ( y ) ) \displaystyle d_{Y}(f_{0}(x),f_{0}(y))

Uniform_matroid.html

  1. U n r U{}^{r}_{n}
  2. n n
  3. r r
  4. r r
  5. r + 1 r+1
  6. S S
  7. min ( | S | , r ) \min(|S|,r)
  8. r r
  9. r r
  10. r + 1 r+1
  11. U n 2 U{}^{2}_{n}
  12. n n
  13. U n r U{}^{r}_{n}
  14. U n n - r U{}^{n-r}_{n}
  15. r = n / 2 r=n/2
  16. U n r U{}^{r}_{n}
  17. r < n r<n
  18. U n - 1 r U{}^{r}_{n-1}
  19. r > 0 r>0
  20. U n - 1 r - 1 U{}^{r-1}_{n-1}
  21. U n r U{}^{r}_{n}
  22. n n
  23. r r
  24. n n
  25. ( r + 1 ) (r+1)
  26. n n
  27. U n 2 U{}^{2}_{n}
  28. n - 1 n-1
  29. n n
  30. U 4 2 U{}^{2}_{4}
  31. U 5 2 U{}^{2}_{5}
  32. r { 0 , n } r\in\{0,n\}
  33. U n r U{}^{r}_{n}
  34. U 4 2 U{}^{2}_{4}
  35. U n 1 U{}^{1}_{n}
  36. n n
  37. U n n - 1 U{}^{n-1}_{n}
  38. n n
  39. U n 0 U{}^{0}_{n}
  40. n n
  41. U n n U{}^{n}_{n}
  42. n n
  43. U n r U{}^{r}_{n}
  44. 1 < r < n - 1 1<r<n-1
  45. U 4 2 U{}^{2}_{4}
  46. n n

Uniform_module.html

  1. M / ( N 1 N 2 ) M/(N_{1}\cap N_{2})
  2. N 1 / ( N 1 N 2 ) N 2 / ( N 1 N 2 ) = { 0 } . N_{1}/(N_{1}\cap N_{2})\cap N_{2}/(N_{1}\cap N_{2})=\{0\}.
  3. i = 1 n U i \oplus_{i=1}^{n}U_{i}
  4. i = 1 m V i \oplus_{i=1}^{m}V_{i}
  5. i = 1 n U i \oplus_{i=1}^{n}U_{i}
  6. N 1 + N 2 = M N_{1}+N_{2}=M

Uniformization_(probability_theory).html

  1. P := ( p i j ) i , j P:=(p_{ij})_{i,j}
  2. p i j = { q i j / γ if i j 1 - j i q i j / γ if i = j p_{ij}=\begin{cases}q_{ij}/\gamma&\,\text{ if }i\neq j\\ 1-\sum_{j\neq i}q_{ij}/\gamma&\,\text{ if }i=j\end{cases}
  3. γ max i | q i i | . \gamma\geq\max_{i}|q_{ii}|.
  4. P = I + 1 γ Q . P=I+\frac{1}{\gamma}Q.
  5. π ( t ) = n = 0 π ( 0 ) P n ( γ t ) n n ! e - γ t . \pi(t)=\sum_{n=0}^{\infty}\pi(0)P^{n}\frac{(\gamma t)^{n}}{n!}e^{-\gamma t}.

Unital_map.html

  1. ϕ \phi
  2. ϕ ( I ) = I . \phi(I)=I.\,
  3. ϕ \phi
  4. ϕ ( ρ ) = i E i ρ E i . \phi(\rho)=\sum_{i}E_{i}\rho E_{i}^{\dagger}.\,
  5. E i E_{i}
  6. ϕ \phi
  7. i E i E i = I . \sum_{i}E_{i}E_{i}^{\dagger}=I.\,

United_States_of_America_Mathematical_Talent_Search.html

  1. 4 × 5 × 5 = 100 4\times 5\times 5=100
  2. 3 × 5 × 5 = 75 3\times 5\times 5=75

Unrooted_binary_tree.html

  1. ( 2 n - 5 ) ! ! = ( 2 n - 4 ) ! ( n - 2 ) ! 2 n - 2 . (2n-5)!!=\frac{(2n-4)!}{(n-2)!2^{n-2}}.

Unscented_transform.html

  1. [ x , y ] [x,y]
  2. x x
  3. y y
  4. T T
  5. m m
  6. M M
  7. T m Tm
  8. T M T T TMT^{T}
  9. ( m , M ) (m,M)
  10. M M
  11. m m
  12. M M
  13. f ( x , y ) f(x,y)
  14. f ( x , y ) f(x,y)
  15. n + 1 n+1
  16. n n
  17. s 1 = [ 0 , 2 ] T , s 2 = [ - 3 2 , - 1 2 ] T , s 3 = [ 3 2 , - 1 2 ] T s_{1}=\left[0,\sqrt{2}\right]^{T},\quad s_{2}=\left[-\sqrt{3\over 2},-\sqrt{1% \over 2}\right]^{T},\quad s_{3}=\left[\sqrt{3\over 2},-\sqrt{1\over 2}\right]^% {T}
  18. s = 0 s=0
  19. S = I S=I
  20. ( x , X ) (x,X)
  21. X X
  22. x x
  23. n n
  24. 2 n 2n
  25. ± n X \pm\sqrt{nX}
  26. X X
  27. ( m , M ) (m,M)
  28. m = [ 12.3 , 7.6 ] T , M = [ 1.44 0 0 2.89 ] m=[12.3,7.6]^{T},\quad M=\begin{bmatrix}1.44&0\\ 0&2.89\end{bmatrix}
  29. f ( x , y ) [ r , θ ] f(x,y)\rightarrow[r,\theta]
  30. r = ( x 2 + y 2 ) , θ = arctan ( y / x ) r=\sqrt{(x^{2}+y^{2})},\quad\theta=\arctan(y/x)
  31. M 1 / 2 = [ 1.2 0 0 1.7 ] M^{1/2}=\begin{bmatrix}1.2&0\\ 0&1.7\end{bmatrix}
  32. m m
  33. m 1 = [ 0 , 2.40 ] + [ 12.3 , 7.6 ] = [ 12.3 , 10.0 ] m_{1}=[0,2.40]+[12.3,7.6]=[12.3,10.0]
  34. m 2 = [ - 1.47 , - 1.20 ] + [ 12.3 , 7.6 ] = [ 10.8 , 6.40 ] m_{2}=[-1.47,-1.20]+[12.3,7.6]=[10.8,6.40]
  35. m 3 = [ 1.47 , - 1.20 ] + [ 12.3 , 7.6 ] = [ 13.8 , 6.40 ] m_{3}=[1.47,-1.20]+[12.3,7.6]=[13.8,6.40]
  36. f ( ) f()
  37. m + 1 = f ( 12.3 , 10.0 ) = [ 15.85 , 0.68 ] {m^{+}}_{1}=f(12.3,10.0)=[15.85,0.68]
  38. m + 2 = f ( 10.8 , 6.40 ) = [ 12.58 , 0.53 ] {m^{+}}_{2}=f(10.8,6.40)=[12.58,0.53]
  39. m + 3 = f ( 13.8 , 6.40 ) = [ 15.18 , 0.44 ] {m^{+}}_{3}=f(13.8,6.40)=[15.18,0.44]
  40. m U T = 1 3 Σ i = 1 3 m + i m_{UT}=\frac{1}{3}\Sigma^{3}_{i=1}{m^{+}}_{i}
  41. m U T = [ 14.539 , 0.551 ] m_{UT}=[14.539,0.551]
  42. M U T = 1 3 Σ i = 1 3 ( m + i - m U T ) 2 M_{UT}=\frac{1}{3}\Sigma^{3}_{i=1}({m^{+}}_{i}-m_{UT})^{2}
  43. M U T = [ 2.00 0.0443 0.0443 0.0104 ] M_{UT}=\begin{bmatrix}2.00&0.0443\\ 0.0443&0.0104\end{bmatrix}
  44. m linear = f ( 12.3 , 7.6 ) = [ 14.46 , 0.554 ] T m_{\mbox{linear}}~{}=f(12.3,7.6)=[14.46,0.554]^{T}
  45. M linear = f M f T = [ 1.927 0.047 0.047 0.011 ] M_{\mbox{linear}}~{}=\nabla_{f}M\nabla_{f}^{T}=\begin{bmatrix}1.927&0.047\\ 0.047&0.011\end{bmatrix}
  46. n + 1 n+1
  47. 2 n 2n
  48. ( m , M ) (m,M)
  49. M = [ 1.44 0 0 2.89 ] M=\begin{bmatrix}1.44&0\\ 0&2.89\end{bmatrix}
  50. m = [ 12.3 , 7.6 ] m=[12.3,7.6]
  51. ( 2 M ) 1 / 2 = 2 * [ 1.2 0 0 1.7 ] = [ 1.697 0 0 2.404 ] (2M)^{1/2}=\sqrt{2}*\begin{bmatrix}1.2&0\\ 0&1.7\end{bmatrix}=\begin{bmatrix}1.697&0\\ 0&2.404\end{bmatrix}
  52. m 1 = [ 12.3 , 7.6 ] + [ 1.697 , 0 ] = [ 13.997 , 7.6 ] m_{1}=[12.3,7.6]+[1.697,0]=[13.997,7.6]
  53. m 2 = [ 12.3 , 7.6 ] - [ 1.697 , 0 ] = [ 10.603 , 7.6 ] m_{2}=[12.3,7.6]-[1.697,0]=[10.603,7.6]
  54. m 3 = [ 12.3 , 7.6 ] + [ 0 , 2.404 ] = [ 12.3 , 10.004 ] m_{3}=[12.3,7.6]+[0,2.404]=[12.3,10.004]
  55. m 4 = [ 12.3 , 7.6 ] - [ 0 , 2.404 ] = [ 12.3 , 5.196 ] m_{4}=[12.3,7.6]-[0,2.404]=[12.3,5.196]
  56. ( m , M ) (m,M)
  57. f ( ) f()
  58. m + 1 = [ 15.927 , 0.497 ] {m^{+}}_{1}=[15.927,0.497]
  59. m + 2 = [ 13.045 , 0.622 ] {m^{+}}_{2}=[13.045,0.622]
  60. m + 3 = [ 15.854 , 0.683 ] {m^{+}}_{3}=[15.854,0.683]
  61. m + 4 = [ 13.352 , 0.400 ] {m^{+}}_{4}=[13.352,0.400]
  62. m U T = 1 4 Σ i = 1 4 m i m_{UT}=\frac{1}{4}\Sigma^{4}_{i=1}{m^{\prime}}_{i}
  63. m U T = [ 14.545 , 0.550 ] m_{UT}=[14.545,0.550]
  64. M U T = 1 4 Σ i = 1 4 ( m + i - m U T ) 2 M_{UT}=\frac{1}{4}\Sigma^{4}_{i=1}({m^{+}}_{i}-m_{UT})^{2}
  65. M U T = [ 1.823 0.043 0.043 0.012 ] M_{UT}=\begin{bmatrix}1.823&0.043\\ 0.043&0.012\end{bmatrix}

Ushiki's_theorem.html

  1. F : n n F:\mathbb{C}^{n}\to\mathbb{C}^{n}

Usual_hypotheses.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. 𝔽 = { t } t 0 \mathbb{F}=\{\mathcal{F}_{t}\}_{t\geq 0}
  3. 0 \mathcal{F}_{0}
  4. \mathbb{P}
  5. 𝔽 \mathbb{F}

V-statistic.html

  1. T ( F n ) T(F_{n})
  2. ( F n ) (F_{n})
  3. T ( F ) = ( x - μ ) k d F ( x ) T(F)=\int(x-\mu)^{k}\,dF(x)
  4. μ = E [ X ] \mu=E[X]
  5. T n = m k = T ( F n ) = 1 n i = 1 n ( x i - x ¯ ) k . T_{n}=m_{k}=T(F_{n})=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{k}.
  6. T ( F ) = i = 1 k ( A i d F - p i ) 2 p i , T(F)=\sum_{i=1}^{k}\frac{(\int_{A_{i}}\,dF-p_{i})^{2}}{p_{i}},
  7. T ( F ) = ( F ( x ) - F 0 ( x ) ) 2 w ( x ; F 0 ) d F 0 ( x ) , T(F)=\int(F(x)-F_{0}(x))^{2}\,w(x;F_{0})\,dF_{0}(x),
  8. w ( x ; F 0 ) = [ F 0 ( x ) ( 1 - F 0 ( x ) ) ] - 1 w(x;F_{0})=[F_{0}(x)(1-F_{0}(x))]^{-1}
  9. V m n = 1 n m i 1 = 1 n i m = 1 n h ( x i 1 , x i 2 , , x i m ) , V_{mn}=\frac{1}{n^{m}}\sum_{i_{1}=1}^{n}\cdots\sum_{i_{m}=1}^{n}h(x_{i_{1}},x_% {i_{2}},\dots,x_{i_{m}}),
  10. V 2 , n = 1 n 2 i = 1 n j = 1 n h ( x i , x j ) . V_{2,n}=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}h(x_{i},x_{j}).
  11. V 2 , n = 1 n 2 i = 1 n j = 1 n 1 2 ( x i - x j ) 2 = 1 n i = 1 n ( x i - x ¯ ) 2 , V_{2,n}=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(x_{i}-x_{j})^{2% }=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2},
  12. s 2 = ( n 2 ) - 1 i < j 1 2 ( x i - x j ) 2 = 1 n - 1 i = 1 n ( x i - x ¯ ) 2 s^{2}={n\choose 2}^{-1}\sum_{i<j}\frac{1}{2}(x_{i}-x_{j})^{2}=\frac{1}{n-1}% \sum_{i=1}^{n}(x_{i}-\bar{x})^{2}
  13. σ 2 \sigma^{2}
  14. ( μ 4 - σ 4 ) / n (\mu_{4}-\sigma^{4})/n
  15. μ 4 = E ( X - E ( X ) ) 4 \mu_{4}=E(X-E(X))^{4}
  16. E [ h 2 ( X 1 , X 2 ) ] < , E | h ( X 1 , X 1 ) | < , E[h^{2}(X_{1},X_{2})]<\infty,\,E|h(X_{1},X_{1})|<\infty,
  17. E [ h ( x , X 1 ) ] 0 E[h(x,X_{1})]\equiv 0
  18. n V 2 , n d k = 1 λ k Z k 2 , nV_{2,n}{\stackrel{d}{\longrightarrow}}\sum_{k=1}^{\infty}\lambda_{k}Z^{2}_{k},
  19. Z k Z_{k}
  20. λ k \lambda_{k}

V2_ratio.html

  1. V R 2 = ( V n V 0 ) P n - 1 i = 0 n ( V i V i p - 1 ) 2 n + 1 V^{2}_{R}=\frac{\big(\frac{V_{n}}{V_{0}}\big)^{\frac{P}{n}}-1}{\sqrt{\frac{% \sum_{i=0}^{n}{\big(\frac{V_{i}}{V_{i}^{p}}-1\big)^{2}}}{n}}+1}
  2. V i V_{i}
  3. i i
  4. V 0 V_{0}
  5. V n V_{n}
  6. V i p V_{i}^{p}
  7. i i
  8. n n
  9. P P

Vaidya_metric.html

  1. ( 1 ) d s 2 = - ( 1 - 2 M r ) d t 2 + ( 1 - 2 M r ) - 1 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . (1)\quad ds^{2}=-\Big(1-\frac{2M}{r}\Big)dt^{2}+\Big(1-\frac{2M}{r}\Big)^{-1}% dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\;.
  2. r = 2 M r=2M
  3. u u
  4. ( 2 ) t = u + r + 2 M ln ( r 2 M - 1 ) d t = d u + ( 1 - 2 M r ) - 1 d r , (2)\quad t=u+r+2M\ln\Big(\frac{r}{2M}-1\Big)\qquad\Rightarrow\quad dt=du+\Big(% 1-\frac{2M}{r}\Big)^{-1}dr\;,
  5. ( 3 ) d s 2 = - ( 1 - 2 M r ) d u 2 - 2 d u d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ; (3)\quad ds^{2}=-\Big(1-\frac{2M}{r}\Big)du^{2}-2dudr+r^{2}(d\theta^{2}+\sin^{% 2}\theta\,d\phi^{2})\;;
  6. v v
  7. ( 4 ) t = v - r - 2 M ln ( r 2 M - 1 ) d t = d v - ( 1 - 2 M r ) - 1 d r , (4)\quad t=v-r-2M\ln\Big(\frac{r}{2M}-1\Big)\qquad\Rightarrow\quad dt=dv-\Big(% 1-\frac{2M}{r}\Big)^{-1}dr\;,
  8. ( 5 ) d s 2 = - ( 1 - 2 M r ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . (5)\quad ds^{2}=-\Big(1-\frac{2M}{r}\Big)dv^{2}+2dvdr+r^{2}(d\theta^{2}+\sin^{% 2}\theta\,d\phi^{2})\;.
  9. M M
  10. M ( u ) M(u)
  11. M ( v ) M(v)
  12. ( 6 ) d s 2 = - ( 1 - 2 M ( u ) r ) d u 2 - 2 d u d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , (6)\quad ds^{2}=-\Big(1-\frac{2M(u)}{r}\Big)du^{2}-2dudr+r^{2}(d\theta^{2}+% \sin^{2}\theta\,d\phi^{2})\;,
  13. ( 7 ) d s 2 = - ( 1 - 2 M ( v ) r ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . (7)\quad ds^{2}=-\Big(1-\frac{2M(v)}{r}\Big)dv^{2}+2dvdr+r^{2}(d\theta^{2}+% \sin^{2}\theta\,d\phi^{2})\;.
  14. ( 8 ) d s 2 = 2 M ( u ) r d u 2 + d s 2 ( flat ) = 2 M ( v ) r d v 2 + d s 2 ( flat ) , (8)\quad ds^{2}=\frac{2M(u)}{r}du^{2}+ds^{2}(\,\text{flat})=\frac{2M(v)}{r}dv^% {2}+ds^{2}(\,\text{flat})\,,
  15. d s 2 ( flat ) = - d u 2 - 2 d u d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) = - d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) = - d t 2 + d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}(\,\text{flat})=-du^{2}-2dudr+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2% })=-dv^{2}+2dvdr+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})=-dt^{2}+dr^{2}+r% ^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})
  16. ( 9 ) R u u = - 2 M ( u ) , u r 2 , (9)\quad R_{uu}=-2\frac{M(u)_{,\,u}}{r^{2}}\,,
  17. R = g a b R a b = 0 R=g^{ab}R_{ab}=0
  18. G a b = R a b = 8 π T a b G_{ab}=R_{ab}=8\pi T_{ab}
  19. T a b T_{ab}
  20. ( 10 ) T a b = - M ( u ) , u 4 π r 2 l a l b , l a d x a = - d u , (10)\quad T_{ab}=-\frac{M(u)_{,\,u}}{4\pi r^{2}}l_{a}l_{b}\;,\qquad l_{a}dx^{a% }=-du\;,
  21. l a = - a u l_{a}=-\partial_{a}u
  22. l a = g a b l b l^{a}=g^{ab}l_{b}
  23. T a b T_{ab}
  24. - M ( u ) , u 4 π r 2 -\frac{M(u)_{,\,u}}{4\pi r^{2}}
  25. ( 11 ) T a b k a k b 0 , (11)\quad T_{ab}k^{a}k^{b}\geq 0\;,
  26. M ( u ) , u < 0 M(u)_{,\,u}<0
  27. ( 12 ) Ψ 2 = - M ( u ) r 3 Φ 22 = - M ( u ) , u r 2 . (12)\quad\Psi_{2}=-\frac{M(u)}{r^{3}}\qquad\Phi_{22}=-\frac{M(u)_{\,,\,u}}{r^{% 2}}\;.
  28. ( 13 ) θ ( ) = - ( ρ + ρ ¯ ) = 2 r , θ ( n ) = μ + μ ¯ = - r + 2 M ( u ) r 2 . (13)\quad\theta_{(\ell)}=-(\rho+\bar{\rho})=\frac{2}{r}\,,\quad\theta_{(n)}=% \mu+\bar{\mu}=\frac{-r+2M(u)}{r^{2}}\;.
  29. F := 1 - 2 M ( u ) r F:=1-\frac{2M(u)}{r}
  30. ( L = 0 , θ ˙ = 0 , ϕ ˙ = 0 ) (L=0,\dot{\theta}=0,\dot{\phi}=0)
  31. L = 0 = - F u ˙ 2 + 2 u ˙ r ˙ , L=0=-F\dot{u}^{2}+2\dot{u}\dot{r}\,,
  32. λ \lambda
  33. u ˙ = 0 and r ˙ = F 2 u ˙ . \dot{u}=0\quad\,\text{and}\quad\dot{r}=\frac{F}{2}\dot{u}\;.
  34. u u
  35. t t
  36. r r
  37. u ˙ = 0 \dot{u}=0
  38. r r
  39. r ˙ = F 2 u ˙ \dot{r}=\frac{F}{2}\dot{u}
  40. u ˙ = 0 \dot{u}=0
  41. r ˙ = F 2 u ˙ \dot{r}=\frac{F}{2}\dot{u}
  42. l a = ( 0 , 1 , 0 , 0 ) , n a = ( 1 , - F 2 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc θ ) , l^{a}=(0,1,0,0)\,,\quad n^{a}=(1,-\frac{F}{2},0,0)\,,\quad m^{a}=\frac{1}{% \sqrt{2}\,r}(0,0,1,i\,\csc\theta)\,,
  43. l a = ( - 1 , 0 , 0 , 0 ) , n a = ( - F 2 , - 1 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin θ ) . l_{a}=(-1,0,0,0)\,,\quad n_{a}=(-\frac{F}{2},-1,0,0)\,,\quad m_{a}=\frac{r}{% \sqrt{2}}(0,0,1,\sin\theta)\,.
  44. κ = σ = τ = 0 , ν = λ = π = 0 , ε = 0 \kappa=\sigma=\tau=0\,,\quad\nu=\lambda=\pi=0\,,\quad\varepsilon=0
  45. ρ = - 1 r , μ = - r + 2 M ( u ) 2 r 2 , α = - β = - 2 cot θ 4 r , γ = M ( u ) 2 r 2 . \rho=-\frac{1}{r}\,,\quad\mu=\frac{-r+2M(u)}{2r^{2}}\,,\quad\alpha=-\beta=% \frac{-\sqrt{2}\cot\theta}{4r}\,,\quad\gamma=\frac{M(u)}{2r^{2}}\,.
  46. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = - M ( u ) r 3 , \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,,\quad\Psi_{2}=-\frac{M(u)}{r^{3}}\,,
  47. Φ 00 = Φ 10 = Φ 20 = Φ 11 = Φ 12 = Λ = 0 , Φ 22 = - M ( u ) , u r 2 , \Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{11}=\Phi_{12}=\Lambda=0\,,\quad\Phi_{22}=-% \frac{M(u)_{\,,\,u}}{r^{2}}\,,
  48. Ψ 2 \Psi_{2}
  49. Φ 22 0 \Phi_{22}\neq 0
  50. G := 1 - 2 M r G:=1-\frac{2M}{r}
  51. u ˙ = 0 \dot{u}=0
  52. r ˙ = - G 2 u ˙ \dot{r}=-\frac{G}{2}\dot{u}
  53. l a = ( 0 , 1 , 0 , 0 ) , n a = ( 1 , - G 2 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc θ ) , l^{a}=(0,1,0,0)\,,\quad n^{a}=(1,-\frac{G}{2},0,0)\,,\quad m^{a}=\frac{1}{% \sqrt{2}\,r}(0,0,1,i\,\csc\theta)\,,
  54. l a = ( - 1 , 0 , 0 , 0 ) , n a = ( - G 2 , - 1 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin θ ) . l_{a}=(-1,0,0,0)\,,\quad n_{a}=(-\frac{G}{2},-1,0,0)\,,\quad m_{a}=\frac{r}{% \sqrt{2}}(0,0,1,\sin\theta)\,.
  55. κ = σ = τ = 0 , ν = λ = π = 0 , ε = 0 \kappa=\sigma=\tau=0\,,\quad\nu=\lambda=\pi=0\,,\quad\varepsilon=0
  56. ρ = - 1 r , μ = - r + 2 M 2 r 2 , α = - β = - 2 cot θ 4 r , γ = M 2 r 2 , \rho=-\frac{1}{r}\,,\quad\mu=\frac{-r+2M}{2r^{2}}\,,\quad\alpha=-\beta=\frac{-% \sqrt{2}\cot\theta}{4r}\,,\quad\gamma=\frac{M}{2r^{2}}\,,
  57. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = - M r 3 , \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,,\quad\Psi_{2}=-\frac{M}{r^{3}}\,,
  58. Φ 00 = Φ 10 = Φ 20 = Φ 11 = Φ 12 = Φ 22 = Λ = 0 . \Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{11}=\Phi_{12}=\Phi_{22}=\Lambda=0\,.
  59. Ψ 2 \Psi_{2}
  60. ( 14 ) R v v = 2 M ( v ) , v r 2 , (14)\quad R_{vv}=2\frac{M(v)_{,\,v}}{r^{2}}\,,
  61. R = 0 R=0
  62. ( 15 ) T a b = M ( v ) , v 4 π r 2 n a n b , n a d x a = - d v . (15)\quad T_{ab}=\frac{M(v)_{,\,v}}{4\pi r^{2}}\,n_{a}n_{b}\;,\qquad n_{a}dx^{% a}=-dv\;.
  63. M ( v ) , v 4 π r 2 \frac{M(v)_{,\,v}}{4\pi r^{2}}
  64. M ( v ) , v > 0 M(v)_{,\,v}>0
  65. ( 16 ) Ψ 2 = - M ( v ) r 3 Φ 00 = M ( v ) , v r 2 . (16)\quad\Psi_{2}=-\frac{M(v)}{r^{3}}\qquad\Phi_{00}=\frac{M(v)_{\,,\,v}}{r^{2% }}\;.
  66. ( 17 ) θ ( ) = - ( ρ + ρ ¯ ) = r - 2 M ( v ) r 2 , θ ( n ) = μ + μ ¯ = - 2 r . (17)\quad\theta_{(\ell)}=-(\rho+\bar{\rho})=\frac{r-2M(v)}{r^{2}}\,,\quad% \theta_{(n)}=\mu+\bar{\mu}=-\frac{2}{r}\;.
  67. r = 2 M ( v ) r=2M(v)
  68. θ ( ) = 0 , θ ( n ) < 0 \theta_{(\ell)}=0\;,\theta_{(n)}<0
  69. F ~ := 1 - 2 M ( v ) r \tilde{F}:=1-\frac{2M(v)}{r}
  70. L = - F ~ v ˙ 2 + 2 v ˙ r ˙ , L=-\tilde{F}\dot{v}^{2}+2\dot{v}\dot{r}\,,
  71. v ˙ = 0 \dot{v}=0
  72. r ˙ = F ~ 2 v ˙ \dot{r}=\frac{\tilde{F}}{2}\dot{v}
  73. v v
  74. l a = ( 1 , F ~ 2 , 0 , 0 ) , n a = ( 0 , - 1 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc θ ) , l^{a}=(1,\frac{\tilde{F}}{2},0,0)\,,\quad n^{a}=(0,-1,0,0)\,,\quad m^{a}=\frac% {1}{\sqrt{2}\,r}(0,0,1,i\,\csc\theta)\,,
  75. l a = ( - F ~ 2 , 1 , 0 , 0 ) , n a = ( - 1 , 0 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin θ ) . l_{a}=(-\frac{\tilde{F}}{2},1,0,0)\,,\quad n_{a}=(-1,0,0,0)\,,\quad m_{a}=% \frac{r}{\sqrt{2}}(0,0,1,\sin\theta)\,.
  76. κ = σ = τ = 0 , ν = λ = π = 0 , γ = 0 \kappa=\sigma=\tau=0\,,\quad\nu=\lambda=\pi=0\,,\quad\gamma=0
  77. ρ = - r + 2 M ( v ) 2 r 2 , μ = - 1 r , α = - β = - 2 cot θ 4 r , ε = M ( v ) 2 r 2 . \rho=\frac{-r+2M(v)}{2r^{2}}\,,\quad\mu=-\frac{1}{r}\,,\quad\alpha=-\beta=% \frac{-\sqrt{2}\cot\theta}{4r}\,,\quad\varepsilon=\frac{M(v)}{2r^{2}}\,.
  78. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = - M ( v ) r 3 , \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,,\quad\Psi_{2}=-\frac{M(v)}{r^{3}}\,,
  79. Φ 10 = Φ 20 = Φ 11 = Φ 12 = Φ 22 = Λ = 0 , Φ 00 = M ( v ) , v r 2 . \Phi_{10}=\Phi_{20}=\Phi_{11}=\Phi_{12}=\Phi_{22}=\Lambda=0\,,\quad\Phi_{00}=% \frac{M(v)_{\,,\,v}}{r^{2}}\;.
  80. Ψ 2 \Psi_{2}
  81. Φ 00 \Phi_{00}
  82. G := 1 - 2 M r G:=1-\frac{2M}{r}
  83. v ˙ = 0 \dot{v}=0
  84. r ˙ = G 2 v ˙ \dot{r}=\frac{G}{2}\dot{v}
  85. l a = ( 1 , G 2 , 0 , 0 ) , n a = ( 0 , - 1 , 0 , 0 ) , m a = 1 2 r ( 0 , 0 , 1 , i csc θ ) , l^{a}=(1,\frac{G}{2},0,0)\,,\quad n^{a}=(0,-1,0,0)\,,\quad m^{a}=\frac{1}{% \sqrt{2}\,r}(0,0,1,i\,\csc\theta)\,,
  86. l a = ( - G 2 , 1 , 0 , 0 ) , n a = ( - 1 , 0 , 0 , 0 ) , m a = r 2 ( 0 , 0 , 1 , sin θ ) . l_{a}=(-\frac{G}{2},1,0,0)\,,\quad n_{a}=(-1,0,0,0)\,,\quad m_{a}=\frac{r}{% \sqrt{2}}(0,0,1,\sin\theta)\,.
  87. κ = σ = τ = 0 , ν = λ = π = 0 , γ = 0 \kappa=\sigma=\tau=0\,,\quad\nu=\lambda=\pi=0\,,\quad\gamma=0
  88. ρ = - r + 2 M 2 r 2 , μ = - 1 r , α = - β = - 2 cot θ 4 r , ε = M 2 r 2 , \rho=\frac{-r+2M}{2r^{2}}\,,\quad\mu=-\frac{1}{r}\,,\quad\alpha=-\beta=\frac{-% \sqrt{2}\cot\theta}{4r}\,,\quad\varepsilon=\frac{M}{2r^{2}}\,,
  89. Ψ 0 = Ψ 1 = Ψ 3 = Ψ 4 = 0 , Ψ 2 = - M r 3 , \Psi_{0}=\Psi_{1}=\Psi_{3}=\Psi_{4}=0\,,\quad\Psi_{2}=-\frac{M}{r^{3}}\,,
  90. Φ 00 = Φ 10 = Φ 20 = Φ 11 = Φ 12 = Φ 22 = Λ = 0 . \Phi_{00}=\Phi_{10}=\Phi_{20}=\Phi_{11}=\Phi_{12}=\Phi_{22}=\Lambda=0\,.
  91. Ψ 2 \Psi_{2}
  92. Ψ 2 \Psi_{2}
  93. M M
  94. M ( u ) M(u)
  95. R a b = 0 R_{ab}=0
  96. R a b = 8 π T a b R_{ab}=8\pi T_{ab}
  97. Φ 00 = M ( u ) , u r 2 \Phi_{00}=\frac{M(u)_{\,,\,u}}{r^{2}}
  98. x μ x^{\mu}
  99. ( 18 ) g μ ν = η μ ν - 2 m ( u ( x ) ) r ( x ) - 3 σ μ ( x ) σ ν ( x ) (18)\quad g_{\mu\nu}=\eta_{\mu\nu}-2m(u(x))\,\,r(x)^{-3}\,\,\sigma_{\mu}(x)% \sigma_{\nu}(x)\!
  100. ( 19 ) r ( x ) = σ μ ( x ) λ μ ( u ( x ) ) (19)\quad r(x)=\sigma_{\mu}(x)\,\,\lambda^{\mu}(u(x))\!
  101. ( 20 ) σ μ ( x ) = X μ ( u ( x ) ) - x μ , η μ ν σ μ ( x ) σ ν ( x ) = 0 (20)\quad\sigma^{\mu}(x)=X^{\mu}(u(x))-x^{\mu},\quad\eta_{\mu\nu}\sigma^{\mu}(% x)\sigma^{\nu}(x)=0\!
  102. η μ ν \eta_{\mu\nu}
  103. m ( u ) m(u)
  104. u u
  105. d u 2 = η μ ν d X μ d X μ , du^{2}=\eta_{\mu\nu}\,dX^{\mu}dX^{\mu},
  106. X μ ( u ) X^{\mu}(u)
  107. λ μ ( u ) = d X μ ( u ) / d u \lambda^{\mu}(u)=dX^{\mu}(u)/du
  108. σ μ ( x ) \sigma_{\mu}(x)
  109. u ( x ) u(x)
  110. X μ ( u ) , X^{\mu}(u),
  111. λ μ ( u ( x ) ) μ u ( x ) = 1. \lambda^{\mu}(u(x))\,\partial_{\mu}u(x)=1.
  112. g μ ν g_{\mu\nu}
  113. g μ ν g_{\mu\nu}
  114. P μ = m ( u ) λ μ ( u ) P^{\mu}=m(u)\,\lambda^{\mu}(u)
  115. - d P μ / d u ; -dP^{\mu}/du;
  116. A ( u ) A(u)
  117. B ( u ) B(u)
  118. m ( u ) , λ μ ( u ) , σ μ ( u ) , m(u),\lambda^{\mu}(u),\sigma_{\mu}(u),
  119. θ ( u ) \theta(u)
  120. λ μ \lambda^{\mu}
  121. ( 18 ) d s 2 = - ( 1 - 2 M ( u ) r + Q ( u ) r 2 ) d u 2 - 2 d u d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) , (18)\quad ds^{2}=-\Big(1-\frac{2M(u)}{r}+\frac{Q(u)}{r^{2}}\Big)du^{2}-2dudr+r% ^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\;,
  122. ( 19 ) d s 2 = - ( 1 - 2 M ( v ) r + Q ( v ) r 2 ) d v 2 + 2 d v d r + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . (19)\quad ds^{2}=-\Big(1-\frac{2M(v)}{r}+\frac{Q(v)}{r^{2}}\Big)dv^{2}+2dvdr+r% ^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\;.

Vakhitov–Kolokolov_stability_criterion.html

  1. u ( x , t ) = ϕ ω ( x ) e - i ω t u(x,t)=\phi_{\omega}(x)e^{-i\omega t}\,
  2. ω \omega\,
  3. d d ω Q ( ω ) < 0 , \frac{d}{d\omega}Q(\omega)<0,
  4. Q ( ω ) Q(\omega)\,
  5. ϕ ω ( x ) e - i ω t \phi_{\omega}(x)e^{-i\omega t}\,
  6. i t u ( x , t ) = - 2 x 2 u ( x , t ) + g ( | u ( x , t ) | 2 ) u ( x , t ) , i\frac{\partial}{\partial t}u(x,t)=-\frac{\partial^{2}}{\partial x^{2}}u(x,t)+% g(|u(x,t)|^{2})u(x,t),
  7. x \R x\in\R\,
  8. t \R t\in\R
  9. g C ( \R ) g\in C^{\infty}(\R)
  10. u ( x , t ) u(x,t)\,
  11. Q ( u ) = 1 2 \R | u ( x , t ) | 2 d x Q(u)=\frac{1}{2}\int_{\R}|u(x,t)|^{2}\,dx
  12. g g\,
  13. u ( x , t ) = ϕ ω ( x ) e - i ω t u(x,t)=\phi_{\omega}(x)e^{-i\omega t}\,
  14. ω \R \omega\in\R
  15. ϕ ω ( x ) \phi_{\omega}(x)\,
  16. x x\,
  17. ϕ ω ( x ) \phi_{\omega}(x)\,
  18. H 1 ( \R n ) H^{1}(\R^{n})
  19. ω \omega\,
  20. d d ω Q ( ϕ ω ) < 0 , \frac{d}{d\omega}Q(\phi_{\omega})<0,
  21. ω \omega\,
  22. ω \omega\,
  23. t u + x 3 u + x f ( u ) = 0 \partial_{t}u+\partial_{x}^{3}u+\partial_{x}f(u)=0\,

Van_Cittert–Zernike_theorem.html

  1. Γ 12 ( u , v , 0 ) \Gamma_{12}(u,v,0)
  2. Γ 12 ( u , v , 0 ) = I ( l , m ) e - 2 π i ( u l + v m ) d l d m \Gamma_{12}(u,v,0)=\iint I(l,m)e^{-2\pi i(ul+vm)}\,dl\,dm
  3. l l
  4. m m
  5. I I
  6. E ( t ) E(t)
  7. Γ 12 ( τ ) = lim T 1 2 T - T T E 1 ( t ) E 2 * ( t - τ ) d t \Gamma_{12}(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}E_{1}(t)E_{2}^{*}(% t-\tau)dt
  8. τ \tau
  9. E ( t ) E(t)
  10. τ = 0 \tau=0
  11. τ \tau
  12. γ 12 ( τ ) = Γ 12 ( τ ) I 1 I 2 \gamma_{12}(\tau)=\frac{\Gamma_{12}(\tau)}{\sqrt{I_{1}}\sqrt{I_{2}}}
  13. P 1 P_{1}
  14. P 2 P_{2}
  15. ( l , m ) (l,m)
  16. P 1 P_{1}
  17. P 2 P_{2}
  18. P 1 P_{1}
  19. P 1 P_{1}
  20. P 2 P_{2}
  21. l l
  22. m m
  23. E 1 ( l , m , t ) = A ( l , m , t - R 1 c ) e - i ω ( t - R 1 c ) R 1 E_{1}(l,m,t)=A\left(l,m,t-\frac{R_{1}}{c}\right)\frac{e^{-i\omega\left(t-\frac% {R_{1}}{c}\right)}}{R_{1}}
  24. R 1 R_{1}
  25. P 1 P_{1}
  26. ω \omega
  27. A A
  28. P 2 P_{2}
  29. E 2 ( l , m , t ) = A ( l , m , t - R 2 c ) e - i ω ( t - R 2 c ) R 2 E_{2}(l,m,t)=A\left(l,m,t-\frac{R_{2}}{c}\right)\frac{e^{-i\omega\left(t-\frac% {R_{2}}{c}\right)}}{R_{2}}
  30. P 1 P_{1}
  31. P 2 P_{2}
  32. E 1 ( l , m , t ) E 2 * ( l , m , t ) = A ( l , m , t - R 1 c ) A * ( l , m , t - R 2 c ) × e - i ω ( t - R 1 c ) R 1 × e i ω ( t - R 2 c ) R 2 \big\langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\big\rangle=\Bigg\langle A\left(l,m,t-% \frac{R_{1}}{c}\right)A^{*}\left(l,m,t-\frac{R_{2}}{c}\right)\Bigg\rangle% \times\frac{e^{-i\omega\left(t-\frac{R_{1}}{c}\right)}}{R_{1}}\times\frac{e^{i% \omega\left(t-\frac{R_{2}}{c}\right)}}{R_{2}}
  33. R 1 c \frac{R_{1}}{c}
  34. E 1 ( l , m , t ) E 2 * ( l , m , t ) = A ( l , m , t ) A * ( l , m , t - R 2 - R 1 c ) × e i ω ( R 1 - R 2 c ) R 1 R 2 \big\langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\big\rangle=\Bigg\langle A(l,m,t)A^{*}% \left(l,m,t-\frac{R_{2}-R_{1}}{c}\right)\Bigg\rangle\times\frac{e^{i\omega% \left(\frac{R_{1}-R_{2}}{c}\right)}}{R_{1}R_{2}}
  35. R 1 R_{1}
  36. R 2 R_{2}
  37. t t
  38. t t
  39. P 1 P_{1}
  40. P 2 P_{2}
  41. E 1 ( l , m , t ) E 2 * ( l , m , t ) = A ( l , m , t ) A * ( l , m , t ) × e i ω ( R 1 - R 2 c ) R 1 R 2 \langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\rangle=\langle A(l,m,t)A^{*}(l,m,t)% \rangle\times\frac{e^{i\omega\left(\frac{R_{1}-R_{2}}{c}\right)}}{R_{1}R_{2}}
  42. A ( l , m , t ) A * ( l , m , t ) \langle A(l,m,t)A^{*}(l,m,t)\rangle
  43. I ( l , m ) I(l,m)
  44. E 1 ( l , m , t ) E 2 * ( l , m , t ) = I ( l , m ) e i ω ( R 1 - R 2 c ) R 1 R 2 \langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\rangle=I(l,m)\frac{e^{i\omega\left(\frac{% R_{1}-R_{2}}{c}\right)}}{R_{1}R_{2}}
  45. Γ 12 ( u , v , 0 ) = source I ( l , m ) e i ω ( R 1 - R 2 c ) R 1 R 2 d S \Gamma_{12}(u,v,0)=\iint_{\textrm{source}}I(l,m)\frac{e^{i\omega\left(\frac{R_% {1}-R_{2}}{c}\right)}}{R_{1}R_{2}}\,dS
  46. A 1 ( l , m , t ) A 2 * ( l , m , t ) \langle A_{1}(l,m,t)A_{2}^{*}(l,m,t)\rangle
  47. R 2 - R 1 R_{2}-R_{1}
  48. u , v , l u,v,l
  49. m m
  50. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  51. P 2 = ( x 2 , y 2 ) P_{2}=(x_{2},y_{2})
  52. R 1 = R 2 + x 1 2 + y 1 2 R_{1}=\sqrt{R^{2}+x_{1}^{2}+y_{1}^{2}}\,
  53. R 2 = R 2 + x 2 2 + y 2 2 R_{2}=\sqrt{R^{2}+x_{2}^{2}+y_{2}^{2}}\,
  54. R R
  55. R 1 R_{1}
  56. R 2 R_{2}
  57. R 2 - R 1 = R 1 + x 2 2 R 2 + y 2 2 R 2 - R 1 + x 1 2 R 2 + y 1 2 R 2 R_{2}-R_{1}=R\sqrt{1+\frac{x_{2}^{2}}{R^{2}}+\frac{y_{2}^{2}}{R^{2}}}-R\sqrt{1% +\frac{x_{1}^{2}}{R^{2}}+\frac{y_{1}^{2}}{R^{2}}}
  58. x 1 , x 2 , y 1 x_{1},x_{2},y_{1}
  59. y 2 y_{2}
  60. R R
  61. R 2 - R 1 = R ( 1 + 1 2 ( x 2 2 + y 2 2 R 2 ) ) - R ( 1 + 1 2 ( x 1 2 + y 1 2 R 2 ) ) R_{2}-R_{1}=R\left(1+\frac{1}{2}\left(\frac{x_{2}^{2}+y_{2}^{2}}{R^{2}}\right)% \right)-R\left(1+\frac{1}{2}\left(\frac{x_{1}^{2}+y_{1}^{2}}{R^{2}}\right)\right)
  62. R 2 - R 1 = 1 2 R ( ( x 2 - x 1 ) ( x 2 + x 1 ) + ( y 2 - y 1 ) ( y 2 + y 1 ) ) R_{2}-R_{1}=\frac{1}{2R}\left((x_{2}-x_{1})(x_{2}+x_{1})+(y_{2}-y_{1})(y_{2}+y% _{1})\right)
  63. 1 2 ( x 2 + x 1 ) \frac{1}{2}(x_{2}+x_{1})
  64. x x
  65. P 1 P_{1}
  66. P 2 P_{2}
  67. 1 2 R ( x 2 + x 1 ) \frac{1}{2R}(x_{2}+x_{1})
  68. l l
  69. m = 1 2 R ( y 2 + y 1 ) m=\frac{1}{2R}(y_{2}+y_{1})
  70. u u
  71. x x
  72. P 1 P_{1}
  73. P 2 P_{2}
  74. u = ω 2 π c ( x 1 - x 2 ) u=\frac{\omega}{2\pi c}(x_{1}-x_{2})
  75. v v
  76. P 1 P_{1}
  77. P 2 P_{2}
  78. y y
  79. v = ω 2 π c ( y 1 - y 2 ) v=\frac{\omega}{2\pi c}(y_{1}-y_{2})
  80. R 2 - R 1 = 2 π c ω ( u l + v m ) R_{2}-R_{1}=\frac{2\pi c}{\omega}(ul+vm)
  81. x 1 , x 2 , y 1 , x_{1},x_{2},y_{1},
  82. y 2 y_{2}
  83. R R
  84. R 1 R 2 R R_{1}\simeq R_{2}\simeq R
  85. d S dS
  86. R 2 d l d m R^{2}\,dl\,dm
  87. Γ 12 ( u , v , 0 ) = source I ( l , m ) e - i ω c 2 π c ω ( u l + v m ) d l d m \Gamma_{12}(u,v,0)=\iint_{\textrm{source}}I(l,m)e^{-\frac{i\omega}{c}\frac{2% \pi c}{\omega}(ul+vm)}\,dl\,dm
  88. Γ 12 ( u , v , 0 ) = source I ( l , m ) e - 2 π i ( u l + v m ) d l d m \Gamma_{12}(u,v,0)=\iint_{\textrm{source}}I(l,m)e^{-2\pi i(ul+vm)}\,dl\,dm
  89. Γ 12 ( u , v , 0 ) = I ( l , m ) e - 2 π i ( u l + v m ) d l d m \Gamma_{12}(u,v,0)=\iint I(l,m)e^{-2\pi i(ul+vm)}\,dl\,dm
  90. a a
  91. b b
  92. P 1 P_{1}
  93. P 2 P_{2}
  94. P 1 P_{1}
  95. E 1 = E a 1 + E b 1 E_{1}=E_{a1}+E_{b1}
  96. P 2 P_{2}
  97. E 2 = E a 2 + E b 2 E_{2}=E_{a2}+E_{b2}
  98. E 1 ( t ) E 2 * ( t - τ ) = ( E a 1 ( t ) + E b 1 ( t ) ) ( E a 2 * ( t - τ ) + E b 2 * ( t - τ ) ) \langle E_{1}(t)E_{2}^{*}(t-\tau)\rangle=\langle(E_{a1}(t)+E_{b1}(t))(E_{a2}^{% *}(t-\tau)+E_{b2}^{*}(t-\tau))\rangle
  99. E 1 ( t ) E 2 * ( t - τ ) = E a 1 ( t ) E a 2 * ( t - τ ) + E a 1 ( t ) E b 2 * ( t - τ ) + E b 1 ( t ) E a 2 * ( t - τ ) + E b 1 ( t ) E b 2 * ( t - τ ) \langle E_{1}(t)E_{2}^{*}(t-\tau)\rangle=\langle E_{a1}(t)E_{a2}^{*}(t-\tau)% \rangle+\langle E_{a1}(t)E_{b2}^{*}(t-\tau)\rangle+\langle E_{b1}(t)E_{a2}^{*}% (t-\tau)\rangle+\langle E_{b1}(t)E_{b2}^{*}(t-\tau)\rangle
  100. a a
  101. b b
  102. R x 1 - x 2 R\gg x_{1}-x_{2}
  103. R y 1 - y 2 R\gg y_{1}-y_{2}
  104. D D
  105. R D 2 λ R\gg\frac{D^{2}}{\lambda}
  106. 4 × 10 10 4\times 10^{10}
  107. l l
  108. m m
  109. 1 2 ( x 1 + x 2 ) / R \frac{1}{2}(x_{1}+x_{2})/R
  110. 1 2 ( y 1 + y 2 ) / R \frac{1}{2}(y_{1}+y_{2})/R
  111. R 1 2 ( x 1 + x 2 ) R\gg\frac{1}{2}(x_{1}+x_{2})
  112. R 1 2 ( y 1 + y 2 ) R\gg\frac{1}{2}(y_{1}+y_{2})
  113. Δ ν \Delta\nu
  114. ν \nu
  115. Δ ν ν 1 \frac{\Delta\nu}{\nu}\lesssim 1
  116. Δ ν ν 1 l u \frac{\Delta\nu}{\nu}\ll\frac{1}{lu}
  117. l l
  118. u u
  119. ( R 2 - R 1 ) / c (R_{2}-R_{1})/c
  120. t t
  121. K ( l , m , P , ν ) K(l,m,P,\nu)
  122. Γ 12 ( l , m , 0 ) = λ 2 I ( l , m ) K ( l , m , P 1 , ν ) K * ( l , m , P 2 , ν ) d S \Gamma_{12}(l,m,0)=\lambda^{2}\iint I(l,m)K(l,m,P_{1},\nu)K^{*}(l,m,P_{2},\nu)% \,dS
  123. U ( l , m , P 1 ) i λ K ( l , m , P 1 , ν ) I ( l , m ) U(l,m,P_{1})\equiv i\lambda K(l,m,P_{1},\nu)\sqrt{I(l,m)}
  124. Γ 12 ( l , m , 0 ) = U ( l , m , P 1 ) U * ( l , m , P 2 ) d S \Gamma_{12}(l,m,0)=\iint U(l,m,P_{1})U^{*}(l,m,P_{2})\,dS
  125. K ( l , m , P , ν ) = - i e i k R λ R K(l,m,P,\nu)=-\frac{ie^{ikR}}{\lambda R}
  126. s s

Van_Laar_equation.html

  1. H e x = b 1 X 1 b 2 X 2 b 1 X 1 + b 2 X 2 ( a 1 b 1 - a 2 b 2 ) 2 H^{ex}=\frac{b_{1}X_{1}b_{2}X_{2}}{b_{1}X_{1}+b_{2}X_{2}}\left(\frac{\sqrt{a_{% 1}}}{b_{1}}-\frac{\sqrt{a_{2}}}{b_{2}}\right)^{2}
  2. G e x R T = A 12 X 1 A 21 X 2 A 12 X 1 + A 21 X 2 \frac{G^{ex}}{RT}=\frac{A_{12}X_{1}A_{21}X_{2}}{A_{12}X_{1}+A_{21}X_{2}}
  3. { ln γ 1 = A 12 ( A 21 X 2 A 12 X 1 + A 21 X 2 ) 2 ln γ 2 = A 21 ( A 12 X 1 A 12 X 1 + A 21 X 2 ) 2 \left\{\begin{matrix}\ln\ \gamma_{1}=A_{12}\left(\frac{A_{21}X_{2}}{A_{12}X_{1% }+A_{21}X_{2}}\right)^{2}\\ \ln\ \gamma_{2}=A_{21}\left(\frac{A_{12}X_{1}}{A_{12}X_{1}+A_{21}X_{2}}\right)% ^{2}\end{matrix}\right.
  4. ln ( γ 1 ) \ln\left(\gamma_{1}^{\infty}\right)
  5. ln ( γ 2 ) \ln\left(\gamma_{2}^{\infty}\right)
  6. { ln γ 1 = A x 2 2 ln γ 2 = A x 1 2 \left\{\begin{matrix}\ln\ \gamma_{1}=Ax^{2}_{2}\\ \ln\ \gamma_{2}=Ax^{2}_{1}\end{matrix}\right.

Variability_function.html

  1. X | { d } X|\{d\}
  2. 𝔇 \mathfrak{D}
  3. X | { d } X|\{d\}
  4. 𝔛 \mathfrak{X}
  5. 𝔇 \mathfrak{D}
  6. X | { d } X|\{d\}
  7. 𝔇 0 ϵ 𝔇 \mathfrak{D}_{0}\epsilon\mathfrak{D}
  8. 𝔛 ( 𝔇 0 ) = d ϵ 𝔇 0 𝔛 ( { d } ) \mathfrak{X}(\mathfrak{D}_{0})=\cup_{d\epsilon\mathfrak{D}_{0}}\mathfrak{X}(\{% d\})

Variable-mass_system.html

  1. 𝐅 ext + 𝐯 rel d m d t = m d 𝐯 d t \mathbf{F}_{\mathrm{ext}}+\mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{% d}t}=m{\mathrm{d}\mathbf{v}\over\mathrm{d}t}
  2. 𝐩 1 = m 𝐯 + 𝐮 d m \mathbf{p}_{\mathrm{1}}=m\mathbf{v}+\mathbf{u}\mathrm{d}m
  3. 𝐩 2 = ( m + d m ) ( 𝐯 + d 𝐯 ) = m 𝐯 + m d 𝐯 + 𝐯 d m + d m d 𝐯 \mathbf{p}_{\mathrm{2}}=(m+\mathrm{d}m)(\mathbf{v}+\mathrm{d}\mathbf{v})=m% \mathbf{v}+m\mathrm{d}\mathbf{v}+\mathbf{v}\mathrm{d}m+\mathrm{d}m\mathrm{d}% \mathbf{v}
  4. d 𝐩 = 𝐩 2 - 𝐩 1 = ( m 𝐯 + m d 𝐯 + 𝐯 d m ) - ( m 𝐯 + 𝐮 d m ) = m d 𝐯 - ( 𝐮 - 𝐯 ) d m \mathrm{d}\mathbf{p}=\mathbf{p}_{\mathrm{2}}-\mathbf{p}_{\mathrm{1}}=(m\mathbf% {v}+m\mathrm{d}\mathbf{v}+\mathbf{v}\mathrm{d}m)-(m\mathbf{v}+\mathbf{u}% \mathrm{d}m)=m\mathrm{d}\mathbf{v}-(\mathbf{u}-\mathbf{v})\mathrm{d}m
  5. 𝐅 net = d 𝐩 d t = m d 𝐯 - ( 𝐮 - 𝐯 ) d m d t = m d 𝐯 d t - ( 𝐮 - 𝐯 ) d m d t \mathbf{F}_{\mathrm{net}}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=\frac{m% \mathrm{d}\mathbf{v}-(\mathbf{u}-\mathbf{v})\mathrm{d}m}{\mathrm{d}t}=m\frac{% \mathrm{d}\mathbf{v}}{\mathrm{d}t}-(\mathbf{u}-\mathbf{v})\frac{\mathrm{d}m}{% \mathrm{d}t}
  6. 𝐅 ext + 𝐯 rel d m d t = m d 𝐯 d t \mathbf{F}_{\mathrm{ext}}+\mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{% d}t}=m{\mathrm{d}\mathbf{v}\over\mathrm{d}t}
  7. 𝐩 1 = m 𝐯 \mathbf{p}_{\mathrm{1}}=m\mathbf{v}
  8. 𝐩 2 = ( m + d m ) ( 𝐯 + d 𝐯 ) + 𝐮 ( - d m ) = m 𝐯 + m d 𝐯 + 𝐯 d m + d m d 𝐯 - 𝐮 d m \mathbf{p}_{\mathrm{2}}=(m+\mathrm{d}m)(\mathbf{v}+\mathrm{d}\mathbf{v})+% \mathbf{u}(-\mathrm{d}m)=m\mathbf{v}+m\mathrm{d}\mathbf{v}+\mathbf{v}\mathrm{d% }m+\mathrm{d}m\mathrm{d}\mathbf{v}-\mathbf{u}\mathrm{d}m
  9. d 𝐩 = 𝐩 2 - 𝐩 1 = ( m 𝐯 + m d 𝐯 + 𝐯 d m - 𝐮 d m ) - ( m 𝐯 ) = m d 𝐯 - ( 𝐮 - 𝐯 ) d m \mathrm{d}\mathbf{p}=\mathbf{p}_{\mathrm{2}}-\mathbf{p}_{\mathrm{1}}=(m\mathbf% {v}+m\mathrm{d}\mathbf{v}+\mathbf{v}\mathrm{d}m-\mathbf{u}\mathrm{d}m)-(m% \mathbf{v})=m\mathrm{d}\mathbf{v}-(\mathbf{u}-\mathbf{v})\mathrm{d}m
  10. 𝐅 ext + 𝐯 rel d m d t = m 𝐚 \mathbf{F}_{\mathrm{ext}}+\mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{% d}t}=m\mathbf{a}
  11. 𝐅 ext + 𝐯 rel d m d t = m 𝐚 cm \mathbf{F}_{\mathrm{ext}}+\mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{% d}t}=m\mathbf{a}_{\mathrm{cm}}
  12. 𝐅 thrust = 𝐯 rel d m d t \mathbf{F}_{\mathrm{thrust}}=\mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{% \mathrm{d}t}
  13. 𝐅 ext + 𝐅 thrust = m 𝐚 cm \mathbf{F}_{\mathrm{ext}}+\mathbf{F}_{\mathrm{thrust}}=m\mathbf{a}_{\mathrm{cm}}
  14. 𝐅 net = m 𝐚 cm \mathbf{F}_{\mathrm{net}}=m\mathbf{a}_{\mathrm{cm}}
  15. 𝐯 rel d m d t = m d 𝐯 d t \mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t}=m\frac{\mathrm{d}% \mathbf{v}}{\mathrm{d}t}
  16. - v rel d m d t = m d v d t -v_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t}=m{\mathrm{d}v\over\mathrm{d}t}
  17. - v rel d m = m d v -v_{\mathrm{rel}}\mathrm{d}m=m\mathrm{d}v\,
  18. - v rel m 0 m 1 d m m = v 0 v 1 d v -v_{\mathrm{rel}}\int_{m_{0}}^{m_{1}}\frac{\mathrm{d}m}{m}=\int_{v_{0}}^{v_{1}% }\mathrm{d}v
  19. v rel ln m 0 m 1 = v 1 - v 0 v_{\mathrm{rel}}\ln{\frac{m_{0}}{m_{1}}}=v_{1}-v_{0}
  20. Δ v = v rel ln m 0 m 1 \Delta v=v_{\mathrm{rel}}\ln\frac{m_{0}}{m_{1}}

Variable_pathlength_cell.html

  1. A = ε c A=\varepsilon\ell c
  2. y = m x + b y=mx+b
  3. A = m + b A=m\ell+b
  4. m = A m={A\over\ell}
  5. m = ε c m=\varepsilon c

Vector_algebra_relations.html

  1. 𝐀 2 = A 1 2 + A 2 2 + A 3 2 \|\mathbf{A}\|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}
  2. 𝐀 2 = ( 𝐀 𝐀 ) \|\mathbf{A}\|^{2}=(\mathbf{A\cdot A})
  3. 𝐀 𝐁 𝐀 𝐁 1 \frac{\mathbf{A\cdot B}}{\|\mathbf{A}\|\|\mathbf{B}\|}\leq 1
  4. 𝐀 + 𝐁 𝐀 + 𝐁 \|\mathbf{A+B}\|\leq\|\mathbf{A}\|+\|\mathbf{B}\|
  5. 𝐀 - 𝐁 𝐀 - 𝐁 \|\mathbf{A-B}\|\geq\|\mathbf{A}\|-\|\mathbf{B}\|
  6. sin θ = 𝐀 × 𝐁 𝐀 𝐁 ( - π < θ π ) \sin\theta=\frac{\|\mathbf{A\times B}\|}{\|\mathbf{A}\|\|\mathbf{B}\|}\ \ (-% \pi<\theta\leq\pi)
  7. cos θ = 𝐀 𝐁 𝐀 𝐁 ( - π < θ π ) \cos\theta=\frac{\mathbf{A\cdot B}}{\|\mathbf{A}\|\|\mathbf{B}\|}\ \ (-\pi<% \theta\leq\pi)
  8. 𝐀 × 𝐁 2 + ( 𝐀 𝐁 ) 2 = 𝐀 2 𝐁 2 \|\mathbf{A\times B}\|^{2}+(\mathbf{A\cdot B})^{2}=\|\mathbf{A}\|^{2}\|\mathbf% {B}\|^{2}
  9. cos α = A x A x 2 + A y 2 + A z 2 = A x 𝐀 , \cos\alpha=\frac{A_{x}}{\sqrt{A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}}=\frac{A_{x}}{\|% \mathbf{A}\|}\ ,
  10. 𝐀 = 𝐀 ( cos α 𝐢 ^ + cos β 𝐣 ^ + cos γ 𝐤 ^ ) , \mathbf{A}=\|\mathbf{A}\|\left(\cos\alpha\ \hat{\mathbf{i}}+\cos\beta\ \hat{% \mathbf{j}}+\cos\gamma\ \hat{\mathbf{k}}\right)\ ,
  11. 𝐢 ^ , 𝐣 ^ , 𝐤 ^ \hat{\mathbf{i}},\ \hat{\mathbf{j}},\ \hat{\mathbf{k}}
  12. Σ = A B sin θ , \Sigma=AB\ \sin\theta\ ,
  13. Σ = 𝐀 × 𝐁 = 𝐀 2 𝐁 2 - ( 𝐀 𝐁 ) 2 . \Sigma=\|\mathbf{A\times B}\|=\sqrt{\|\mathbf{A}\|^{2}\|\mathbf{B}\|^{2}-(% \mathbf{A\cdot B})^{2}}\ .
  14. Σ 2 = ( 𝐀 𝐀 ) ( 𝐁 𝐁 ) - ( 𝐀 𝐁 ) ( 𝐁 𝐀 ) = Γ ( 𝐀 , 𝐁 ) , \Sigma^{2}=(\mathbf{A\cdot A})(\mathbf{B\cdot B})-(\mathbf{A\cdot B})(\mathbf{% B\cdot A})=\Gamma(\mathbf{A},\ \mathbf{B})\ ,
  15. Γ ( 𝐀 , 𝐁 ) = | 𝐀 𝐀 𝐀 𝐁 𝐁 𝐀 𝐁 𝐁 | . \Gamma(\mathbf{A},\ \mathbf{B})=\begin{vmatrix}\mathbf{A\cdot A}&\mathbf{A% \cdot B}\\ \mathbf{B\cdot A}&\mathbf{B\cdot B}\end{vmatrix}\ .
  16. V 2 = Γ ( 𝐀 , 𝐁 , 𝐂 ) = | 𝐀 𝐀 𝐀 𝐁 𝐀 𝐂 𝐁 𝐀 𝐁 𝐁 𝐁 𝐂 𝐂 𝐀 𝐂 𝐁 𝐂 𝐂 | . V^{2}=\Gamma(\mathbf{A},\ \mathbf{B},\ \mathbf{C})=\begin{vmatrix}\mathbf{A% \cdot A}&\mathbf{A\cdot B}&\mathbf{A\cdot C}\\ \mathbf{B\cdot A}&\mathbf{B\cdot B}&\mathbf{B\cdot C}\\ \mathbf{C\cdot A}&\mathbf{C\cdot B}&\mathbf{C\cdot C}\end{vmatrix}\ .
  17. c ( 𝐀 + 𝐁 ) = c 𝐀 + c 𝐁 c(\mathbf{A}+\mathbf{B})=c\mathbf{A}+c\mathbf{B}
  18. 𝐀 + 𝐁 = 𝐁 + 𝐀 \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}
  19. 𝐀 + ( 𝐁 + 𝐂 ) = ( 𝐀 + 𝐁 ) + 𝐂 \mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}
  20. 𝐀 𝐁 = 𝐁 𝐀 \mathbf{A}\cdot\mathbf{B}=\mathbf{B}\cdot\mathbf{A}
  21. 𝐀 × 𝐁 = - 𝐁 × 𝐀 \mathbf{A}\times\mathbf{B}=\mathbf{-B}\times\mathbf{A}
  22. ( 𝐀 + 𝐁 ) 𝐂 = 𝐀 𝐂 + 𝐁 𝐂 \left(\mathbf{A}+\mathbf{B}\right)\cdot\mathbf{C}=\mathbf{A}\cdot\mathbf{C}+% \mathbf{B}\cdot\mathbf{C}
  23. ( 𝐀 + 𝐁 ) × 𝐂 = 𝐀 × 𝐂 + 𝐁 × 𝐂 \left(\mathbf{A}+\mathbf{B}\right)\times\mathbf{C}=\mathbf{A}\times\mathbf{C}+% \mathbf{B}\times\mathbf{C}
  24. 𝐀 ( 𝐁 × 𝐂 ) = 𝐁 ( 𝐂 × 𝐀 ) = 𝐂 ( 𝐀 × 𝐁 ) \mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\mathbf{B}\cdot\left(% \mathbf{C}\times\mathbf{A}\right)=\mathbf{C}\cdot\left(\mathbf{A}\times\mathbf% {B}\right)
  25. = | A x B x C x A y B y C y A z B z C z | = [ 𝐀 , 𝐁 , 𝐂 ] =\left|\begin{array}[]{ccc}A_{x}&B_{x}&C_{x}\\ A_{y}&B_{y}&C_{y}\\ A_{z}&B_{z}&C_{z}\end{array}\right|=[\mathbf{A,\ B,\ C}]
  26. 𝐀 × ( 𝐁 × 𝐂 ) = ( 𝐀 𝐂 ) 𝐁 - ( 𝐀 𝐁 ) 𝐂 \mathbf{A\times}\left(\mathbf{B}\times\mathbf{C}\right)=\left(\mathbf{A}\cdot% \mathbf{C}\right)\mathbf{B}-\left(\mathbf{A}\cdot\mathbf{B}\right)\mathbf{C}
  27. ( 𝐀 × 𝐁 ) ( 𝐂 × 𝐃 ) = ( 𝐀 𝐂 ) ( 𝐁 𝐃 ) - ( 𝐁 𝐂 ) ( 𝐀 𝐃 ) \mathbf{\left(A\times B\right)\cdot}\left(\mathbf{C}\times\mathbf{D}\right)=% \left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\cdot\mathbf{D}\right)-% \left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right)
  28. ( 𝐀 × 𝐁 ) ( 𝐀 × 𝐁 ) = | 𝐀 × 𝐁 | 𝟐 = ( 𝐀 𝐀 ) ( 𝐁 𝐁 ) - ( 𝐀 𝐁 ) 𝟐 \mathbf{(A\times B)\cdot(A\times B)=|A\times B|^{2}=(A\cdot A)(B\cdot B)-(A% \cdot B)^{2}}
  29. [ 𝐀 , 𝐁 , 𝐂 ] 𝐃 = ( 𝐀 𝐃 ) ( 𝐁 × 𝐂 ) + ( 𝐁 𝐃 ) ( 𝐂 × 𝐀 ) + ( 𝐂 𝐃 ) ( 𝐀 × 𝐁 ) [\mathbf{A},\mathbf{B},\mathbf{C}]\mathbf{D}=\left(\mathbf{A}\cdot\mathbf{D}% \right)\left(\mathbf{B}\times\mathbf{C}\right)+\left(\mathbf{B}\cdot\mathbf{D}% \right)\left(\mathbf{C}\times\mathbf{A}\right)+\left(\mathbf{C}\cdot\mathbf{D}% \right)\left(\mathbf{A}\times\mathbf{B}\right)
  30. ( 𝐀 × 𝐁 ) × ( 𝐂 × 𝐃 ) = [ 𝐀 , 𝐁 , 𝐃 ] 𝐂 - [ 𝐀 , 𝐁 , 𝐂 ] 𝐃 = [ 𝐀 , 𝐂 , 𝐃 ] 𝐁 - [ 𝐁 , 𝐂 , 𝐃 ] 𝐀 (\mathbf{A}\times\mathbf{B})\times(\mathbf{C}\times\mathbf{D})=[\mathbf{A},% \mathbf{B},\mathbf{D}]\mathbf{C}-[\mathbf{A},\mathbf{B},\mathbf{C}]\mathbf{D}=% [\mathbf{A},\mathbf{C},\mathbf{D}]\mathbf{B}-[\mathbf{B},\mathbf{C},\mathbf{D}% ]\mathbf{A}
  31. 𝐃 = 𝐃 ( 𝐁 × 𝐂 ) [ 𝐀 , 𝐁 , 𝐂 ] 𝐀 + 𝐃 ( 𝐂 × 𝐀 ) [ 𝐀 , 𝐁 , 𝐂 ] 𝐁 + 𝐃 ( 𝐀 × 𝐁 ) [ 𝐀 , 𝐁 , 𝐂 ] 𝐂 . \mathbf{D}=\frac{\mathbf{D\cdot(B\times C)}}{[\mathbf{A,\ B,\ C}]}\ \mathbf{A}% +\frac{\mathbf{D\cdot(C\times A)}}{[\mathbf{A,\ B,\ C}]}\ \mathbf{B}+\frac{% \mathbf{D\cdot(A\times B)}}{[\mathbf{A,\ B,\ C}]}\ \mathbf{C}\ .

Vector_model_of_the_atom.html

  1. J = L + S . {J}={L}+{S}.
  2. [ L ^ a , L ^ b ] = i ε a b c L ^ c \displaystyle[\hat{L}_{a},\hat{L}_{b}]=i\hbar\varepsilon_{abc}\hat{L}_{c}
  3. L ^ a \hat{L}_{a}
  4. L ^ 2 \hat{L}^{2}
  5. S ^ a \hat{S}_{a}
  6. S ^ 2 \hat{S}^{2}
  7. J ^ a \hat{J}_{a}
  8. J ^ 2 \hat{J}^{2}
  9. [ L ^ a , L ^ 2 ] = 0 \displaystyle[\hat{L}_{a},\hat{L}^{2}]=0
  10. L ^ 2 = L ^ x 2 + L ^ y 2 + L ^ z 2 L L = L 2 = L x 2 + L y 2 + L z 2 , \displaystyle\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}\,\,% \rightleftharpoons\,\,{L}\cdot{L}=L^{2}=L_{x}^{2}+L_{y}^{2}+L_{z}^{2},
  11. | L | = ( + 1 ) , L z = m , \displaystyle|{L}|=\hbar\sqrt{\ell(\ell+1)},\quad L_{z}=m_{\ell}\hbar,
  12. \scriptstyle\ell
  13. m { - , - ( - 1 ) - 1 , } , { 0 , 1 n - 1 } \displaystyle m_{\ell}\in\{-\ell,-(\ell-1)\cdots\ell-1,\ell\},\quad\ell\in\{0,% 1\cdots n-1\}
  14. \scriptstyle\ell
  15. \scriptstyle\ell
  16. \scriptstyle\ell
  17. 2 + 1 \scriptstyle 2\ell+1
  18. L = m L=m\hbar
  19. J = J 1 + J 2 {J}={J}_{1}+{J}_{2}\,\!
  20. J z = J 1 z + J 2 z J_{z}=J_{1z}+J_{2z}\,\!
  21. J z = m j J 1 z = m j 1 J 2 z = m j 2 \begin{aligned}&\displaystyle{J}_{z}=m_{j}\hbar\\ &\displaystyle{J}_{1z}=m_{j_{1}}\hbar\\ &\displaystyle{J}_{2z}=m_{j_{2}}\hbar\\ \end{aligned}\,\!
  22. | J | = j ( j + 1 ) \displaystyle|{J}|=\hbar\sqrt{j(j+1)}
  23. j { | j 1 - j 2 | , | j 1 - j 2 | - 1 j 1 + j 2 - 1 , j 1 + j 2 } j\in\{|j_{1}-j_{2}|,|j_{1}-j_{2}|-1\cdots j_{1}+j_{2}-1,j_{1}+j_{2}\}\,\!

Vector_optimization.html

  1. C - min x S f ( x ) C\operatorname{-}\min_{x\in S}f(x)
  2. f : X Z f:X\to Z
  3. Z Z
  4. C Z C\subseteq Z
  5. X X
  6. S X S\subseteq X
  7. x ¯ S \bar{x}\in S
  8. x S x\in S
  9. f ( x ) - f ( x ¯ ) - int C f(x)-f(\bar{x})\not\in-\operatorname{int}C
  10. x ¯ S \bar{x}\in S
  11. x S x\in S
  12. f ( x ) - f ( x ¯ ) - C \ { 0 } f(x)-f(\bar{x})\not\in-C\backslash\{0\}
  13. x ¯ S \bar{x}\in S
  14. x ¯ \bar{x}
  15. C ~ \tilde{C}
  16. C \ { 0 } int C ~ C\backslash\{0\}\subseteq\operatorname{int}\tilde{C}
  17. + d - min x M f ( x ) \mathbb{R}^{d}_{+}\operatorname{-}\min_{x\in M}f(x)
  18. f : X d f:X\to\mathbb{R}^{d}
  19. + d \mathbb{R}^{d}_{+}
  20. d \mathbb{R}^{d}

Vehicle-specific_power.html

  1. V S P = v × ( a + g × sin ϕ + ψ ) + ζ × v 3 VSP=v\times(a+g\times\sin{\phi}+\psi)+\zeta\times v^{3}
  2. V S P = p o w e r m a s s = d d t ( E k i n e t i c + E p o t e n t i a l ) + F r o l l i n g v + F a e r o d y n a m i c v + F i n t e r n a l v m VSP=\frac{power}{mass}=\frac{{\operatorname{d}\over\operatorname{d}t}(E_{% kinetic}+E_{potential})+F_{rolling}\cdot v+F_{aerodynamic}\cdot v+F_{internal}% \cdot v}{m}

Velocity_Moments.html

  1. m p q = x = 1 M y = 1 N x p y q P x y m_{pq}=\sum_{x=1}^{M}\sum_{y=1}^{N}x^{p}y^{q}P_{xy}
  2. M M
  3. N N
  4. P x y P_{xy}
  5. ( x , y ) (x,y)
  6. x p y q x^{p}y^{q}
  7. v m p q μ γ vm_{pq\mu\gamma}
  8. v m p q μ γ = i = 2 i m a g e s x = 1 M y = 1 N U ( i , μ , γ ) C ( i , p , g ) P i x y vm_{pq\mu\gamma}=\sum_{i=2}^{images}\sum_{x=1}^{M}\sum_{y=1}^{N}U(i,\mu,\gamma% )C(i,p,g)P_{i_{xy}}
  9. M M
  10. N N
  11. i m a g e s images
  12. P i x y P_{i_{xy}}
  13. ( x , y ) (x,y)
  14. i i
  15. C ( i , p , q ) C(i,p,q)
  16. C ( i , p , q ) = ( x - x i ¯ ) p ( y - y i ¯ ) q C(i,p,q)=(x-\overline{x_{i}})^{p}(y-\overline{y_{i}})^{q}
  17. x i ¯ \overline{x_{i}}
  18. x x
  19. i i
  20. y y
  21. U ( i , μ , γ ) U(i,\mu,\gamma)
  22. U ( i , μ , γ ) = ( x i ¯ - x i - 1 ¯ ) μ ( y i ¯ - y i - 1 ¯ ) γ U(i,\mu,\gamma)=(\overline{x_{i}}-\overline{x_{i-1}})^{\mu}(\overline{y_{i}}-% \overline{y_{i-1}})^{\gamma}
  23. x i - 1 ¯ \overline{x_{i-1}}
  24. x x
  25. i - 1 i-1
  26. y y
  27. v m p q μ γ ¯ = v m p q μ γ A * I \overline{vm_{pq\mu\gamma}}=\frac{vm_{pq\mu\gamma}}{A*I}
  28. A A
  29. I I
  30. A m n = m + 1 π x y [ V m n ( r , θ ) ] * P x y A_{mn}=\frac{m+1}{\pi}\sum_{x}\sum_{y}[V_{mn}(r,\theta)]^{*}P_{xy}
  31. * {}^{*}
  32. m m
  33. 0
  34. \infty
  35. n n
  36. m - | n | m-|n|
  37. | n | < m |n|<m
  38. P x y P_{xy}
  39. ( x , y ) (x,y)
  40. x 2 + y 2 1 x^{2}+y^{2}\leq 1
  41. x x
  42. y y
  43. r r
  44. θ \theta
  45. ( x , y ) (x,y)
  46. V m n ( r , θ ) V_{mn}(r,\theta)
  47. V m n ( r , θ ) = R m n ( r ) e j n θ V_{mn}(r,\theta)=R_{mn}(r)e^{jn\theta}
  48. R m n ( r ) = s = 0 m - | n | 2 ( - 1 ) s F ( m , n , s , r ) R_{mn}(r)=\sum_{s=0}^{\frac{m-|n|}{2}}(-1)^{s}F(m,n,s,r)
  49. F ( m , n , s , r ) = ( m - s ) ! s ! ( m + | n | 2 - s ) ! ( m - | n | 2 - s ) ! r m - 2 s F(m,n,s,r)=\frac{(m-s)!}{s!(\frac{m+|n|}{2}-s)!(\frac{m-|n|}{2}-s)!}r^{m-2s}
  50. A m n μ γ A_{mn\mu\gamma}
  51. A m n μ γ = m + 1 π i = 2 i m a g e s x = 1 y = 1 U ( i , μ , γ ) [ V m n ( r , θ ) ] * P i x y A_{mn\mu\gamma}=\frac{m+1}{\pi}\sum_{i=2}^{images}\sum_{x=1}\sum_{y=1}U(i,\mu,% \gamma)[V_{mn}(r,\theta)]^{*}P_{i_{xy}}
  52. i m a g e s images
  53. P i x y P_{i_{xy}}
  54. ( x , y ) (x,y)
  55. i i
  56. U ( i , μ , γ ) U(i,\mu,\gamma)
  57. [ V m n ( r , θ ) ] * [V_{mn}(r,\theta)]^{*}
  58. A m n μ γ ¯ = A m n μ γ A * I \overline{A_{mn\mu\gamma}}=\frac{A_{mn\mu\gamma}}{A*I}
  59. A A
  60. I I

Vertical_pressure_variation.html

  1. Δ P = - ρ g Δ h \Delta P=-\rho g\Delta h
  2. W = w A α . W=\frac{w\ A}{\alpha}.
  3. ρ = ( m P ) / ( k T ) \rho=(mP)/(kT)
  4. P h = P 0 e ( - m g h ) / ( k T ) P_{h}=P_{0}e^{(-mgh)/(kT)}
  5. z = ( - R T / g ) ln ( P / P 0 ) z=(-RT/g)\ln(P/P_{0})
  6. z = ( T 0 / L ) ( ( P / P 0 ) - L R / g - 1 ) z=(T_{0}/L)((P/P_{0})^{-LR/g}-1)

Vibration_of_plates.html

  1. N α β , β = J 1 u ¨ α M α β , α β - q ( x , t ) = J 1 w ¨ - J 3 w ¨ , α α \begin{aligned}\displaystyle N_{\alpha\beta,\beta}&\displaystyle=J_{1}~{}\ddot% {u}_{\alpha}\\ \displaystyle M_{\alpha\beta,\alpha\beta}-q(x,t)&\displaystyle=J_{1}~{}\ddot{w% }-J_{3}~{}\ddot{w}_{,\alpha\alpha}\end{aligned}
  2. u α u_{\alpha}
  3. w w
  4. q q
  5. N α β := - h h σ α β d x 3 and M α β := - h h x 3 σ α β d x 3 . N_{\alpha\beta}:=\int_{-h}^{h}\sigma_{\alpha\beta}~{}dx_{3}\quad\,\text{and}% \quad M_{\alpha\beta}:=\int_{-h}^{h}x_{3}~{}\sigma_{\alpha\beta}~{}dx_{3}\,.
  6. 2 h 2h
  7. σ α β \sigma_{\alpha\beta}
  8. u ˙ i := u i t ; u ¨ i := 2 u i t 2 ; u i , α := u i x α ; u i , α β := 2 u i x α x β \dot{u}_{i}:=\frac{\partial u_{i}}{\partial t}~{};~{}~{}\ddot{u}_{i}:=\frac{% \partial^{2}u_{i}}{\partial t^{2}}~{};~{}~{}u_{i,\alpha}:=\frac{\partial u_{i}% }{\partial x_{\alpha}}~{};~{}~{}u_{i,\alpha\beta}:=\frac{\partial^{2}u_{i}}{% \partial x_{\alpha}\partial x_{\beta}}
  9. x 3 x_{3}
  10. x 1 x_{1}
  11. x 2 x_{2}
  12. 2 h 2h
  13. ρ \rho
  14. J 1 := - h h ρ d x 3 = 2 ρ h and J 3 := - h h x 3 2 ρ d x 3 = 2 3 ρ h 3 . J_{1}:=\int_{-h}^{h}\rho~{}dx_{3}=2\rho h\quad\,\text{and}\quad J_{3}:=\int_{-% h}^{h}x_{3}^{2}~{}\rho~{}dx_{3}=\frac{2}{3}\rho h^{3}\,.
  15. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] . \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}\,.
  16. ε α β \varepsilon_{\alpha\beta}
  17. ε α β = 1 2 ( u α , β + u β , α ) - x 3 w , α β . \varepsilon_{\alpha\beta}=\frac{1}{2}(u_{\alpha,\beta}+u_{\beta,\alpha})-x_{3}% \,w_{,\alpha\beta}\,.
  18. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 w , 22 w , 12 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w_{,11}\\ w_{,22}\\ w_{,12}\end{bmatrix}
  19. u α β u_{\alpha\beta}
  20. D 2 2 w = - q ( x , t ) - 2 ρ h w ¨ . D\nabla^{2}\nabla^{2}w=-q(x,t)-2\rho h\ddot{w}\,.
  21. μ Δ Δ w + q ^ + ρ w t t = 0 . \mu\Delta\Delta w+\hat{q}+\rho w_{tt}=0\,.
  22. D 2 2 w = - 2 ρ h w ¨ D\nabla^{2}\nabla^{2}w=-2\rho h\ddot{w}
  23. μ Δ Δ w + ρ w t t = 0 . \mu\Delta\Delta w+\rho w_{tt}=0\,.
  24. U = Ω [ ( Δ w ) 2 + ( 1 - μ ) ( w x x w y y - w x y 2 ) ] d x d y U=\int_{\Omega}[(\Delta w)^{2}+(1-\mu)(w_{xx}w_{yy}-w_{xy}^{2})]\,dx\,dy
  25. T = ρ 2 Ω w t 2 d x d y . T=\frac{\rho}{2}\int_{\Omega}w_{t}^{2}\,dx\,dy.
  26. ρ w t t + μ Δ Δ w = 0. \rho w_{tt}+\mu\Delta\Delta w=0.\,
  27. w = w ( r , t ) w=w(r,t)
  28. 2 w 1 r r ( r w r ) . \nabla^{2}w\equiv\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w% }{\partial r}\right)\,.
  29. 2 h 2h
  30. 1 r r [ r r { 1 r r ( r w r ) } ] = - 2 ρ h D 2 w t 2 . \frac{1}{r}\frac{\partial}{\partial r}\left[r\frac{\partial}{\partial r}\left% \{\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w}{\partial r}% \right)\right\}\right]=-\frac{2\rho h}{D}\frac{\partial^{2}w}{\partial t^{2}}\,.
  31. 4 w r 4 + 2 r 3 w r 3 - 1 r 2 2 w r 2 + 1 r 3 w r = - 2 ρ h D 2 w t 2 . \frac{\partial^{4}w}{\partial r^{4}}+\frac{2}{r}\frac{\partial^{3}w}{\partial r% ^{3}}-\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial r^{2}}+\frac{1}{r^{3}}\frac% {\partial w}{\partial r}=-\frac{2\rho h}{D}\frac{\partial^{2}w}{\partial t^{2}% }\,.
  32. w ( r , t ) = W ( r ) F ( t ) . w(r,t)=W(r)F(t)\,.
  33. 1 β W [ d 4 W d r 4 + 2 r d 3 W d r 3 - 1 r 2 d 2 W d r 2 + 1 r 3 d W d r ] = - 1 F d 2 F d t 2 = ω 2 \frac{1}{\beta W}\left[\frac{d^{4}W}{dr^{4}}+\frac{2}{r}\frac{d^{3}W}{dr^{3}}-% \frac{1}{r^{2}}\frac{d^{2}W}{dr^{2}}+\frac{1}{r^{3}}\frac{dW}{dr}\right]=-% \frac{1}{F}\cfrac{d^{2}F}{dt^{2}}=\omega^{2}
  34. ω 2 \omega^{2}
  35. β := 2 ρ h / D \beta:=2\rho h/D
  36. F ( t ) = Re [ A e i ω t + B e - i ω t ] . F(t)=\,\text{Re}[Ae^{i\omega t}+Be^{-i\omega t}]\,.
  37. d 4 W d r 4 + 2 r d 3 W d r 3 - 1 r 2 d 2 W d r 2 + 1 r 3 d W d r = λ 4 W \frac{d^{4}W}{dr^{4}}+\frac{2}{r}\frac{d^{3}W}{dr^{3}}-\frac{1}{r^{2}}\frac{d^% {2}W}{dr^{2}}+\frac{1}{r^{3}}\cfrac{dW}{dr}=\lambda^{4}W
  38. λ 4 := β ω 2 \lambda^{4}:=\beta\omega^{2}
  39. W ( r ) = C 1 J 0 ( λ r ) + C 2 I 0 ( λ r ) W(r)=C_{1}J_{0}(\lambda r)+C_{2}I_{0}(\lambda r)
  40. J 0 J_{0}
  41. I 0 I_{0}
  42. C 1 C_{1}
  43. C 2 C_{2}
  44. a a
  45. W ( r ) = 0 and d W d r = 0 at r = a . W(r)=0\quad\,\text{and}\quad\cfrac{dW}{dr}=0\quad\,\text{at}\quad r=a\,.
  46. J 0 ( λ a ) I 1 ( λ a ) + I 0 ( λ a ) J 1 ( λ a ) = 0 . J_{0}(\lambda a)I_{1}(\lambda a)+I_{0}(\lambda a)J_{1}(\lambda a)=0\,.
  47. λ n \lambda_{n}
  48. ω n = λ n 2 / β \omega_{n}=\lambda_{n}^{2}/\beta
  49. w ( r , t ) = n = 1 C n [ J 0 ( λ n r ) - J 0 ( λ n a ) I 0 ( λ n a ) I 0 ( λ n r ) ] [ A n e i ω n t + B n e - i ω n t ] . w(r,t)=\sum_{n=1}^{\infty}C_{n}\left[J_{0}(\lambda_{n}r)-\frac{J_{0}(\lambda_{% n}a)}{I_{0}(\lambda_{n}a)}I_{0}(\lambda_{n}r)\right][A_{n}e^{i\omega_{n}t}+B_{% n}e^{-i\omega_{n}t}]\,.
  50. ω n \omega_{n}
  51. C 1 C_{1}
  52. r = 0 r=0
  53. A n A_{n}
  54. B n B_{n}
  55. a × b a\times b
  56. ( x 1 , x 2 ) (x_{1},x_{2})
  57. 2 h 2h
  58. x 3 x_{3}
  59. w ( x 1 , x 2 , t ) = W ( x 1 , x 2 ) F ( t ) . w(x_{1},x_{2},t)=W(x_{1},x_{2})F(t)\,.
  60. 2 2 w = w , 1111 + 2 w , 1212 + w , 2222 = [ 4 W x 1 4 + 2 4 W x 1 2 x 2 2 + 4 W x 2 4 ] F ( t ) \nabla^{2}\nabla^{2}w=w_{,1111}+2w_{,1212}+w_{,2222}=\left[\frac{\partial^{4}W% }{\partial x_{1}^{4}}+2\frac{\partial^{4}W}{\partial x_{1}^{2}\partial x_{2}^{% 2}}+\frac{\partial^{4}W}{\partial x_{2}^{4}}\right]F(t)
  61. w ¨ = W ( x 1 , x 2 ) d 2 F d t 2 . \ddot{w}=W(x_{1},x_{2})\frac{d^{2}F}{dt^{2}}\,.
  62. D 2 ρ h W [ 4 W x 1 4 + 2 4 W x 1 2 x 2 2 + 4 W x 2 4 ] = - 1 F d 2 F d t 2 = ω 2 \frac{D}{2\rho hW}\left[\frac{\partial^{4}W}{\partial x_{1}^{4}}+2\frac{% \partial^{4}W}{\partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}W}{% \partial x_{2}^{4}}\right]=-\frac{1}{F}\frac{d^{2}F}{dt^{2}}=\omega^{2}
  63. ω 2 \omega^{2}
  64. t t
  65. x 1 , x 2 x_{1},x_{2}
  66. F ( t ) = A e i ω t + B e - i ω t . F(t)=Ae^{i\omega t}+Be^{-i\omega t}\,.
  67. 4 W x 1 4 + 2 4 W x 1 2 x 2 2 + 4 W x 2 4 = 2 ρ h ω 2 D W = : λ 4 W \frac{\partial^{4}W}{\partial x_{1}^{4}}+2\frac{\partial^{4}W}{\partial x_{1}^% {2}\partial x_{2}^{2}}+\frac{\partial^{4}W}{\partial x_{2}^{4}}=\frac{2\rho h% \omega^{2}}{D}W=:\lambda^{4}W
  68. λ 2 = ω 2 ρ h D . \lambda^{2}=\omega\sqrt{\frac{2\rho h}{D}}\,.
  69. W m n ( x 1 , x 2 ) = sin m π x 1 a sin n π x 2 b . W_{mn}(x_{1},x_{2})=\sin\frac{m\pi x_{1}}{a}\sin\frac{n\pi x_{2}}{b}\,.
  70. w ( x 1 , x 2 , t ) = 0 \displaystyle w(x_{1},x_{2},t)=0
  71. λ 2 = π 2 ( m 2 a 2 + n 2 b 2 ) . \lambda^{2}=\pi^{2}\left(\frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}}\right)\,.
  72. λ 2 \lambda^{2}
  73. ω m n = D π 4 2 ρ h ( m 2 a 2 + n 2 b 2 ) . \omega_{mn}=\sqrt{\frac{D\pi^{4}}{2\rho h}}\left(\frac{m^{2}}{a^{2}}+\frac{n^{% 2}}{b^{2}}\right)\,.
  74. w ( x 1 , x 2 , t ) = m = 1 n = 1 sin m π x 1 a sin n π x 2 b ( A m n e i ω m n t + B m n e - i ω m n t ) . w(x_{1},x_{2},t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sin\frac{m\pi x_{1}}{a% }\sin\frac{n\pi x_{2}}{b}\left(A_{mn}e^{i\omega_{mn}t}+B_{mn}e^{-i\omega_{mn}t% }\right)\,.
  75. A m n A_{mn}
  76. B m n B_{mn}
  77. w ( x 1 , x 2 , 0 ) = φ ( x 1 , x 2 ) on x 1 [ 0 , a ] and w t ( x 1 , x 2 , 0 ) = ψ ( x 1 , x 2 ) on x 2 [ 0 , b ] w(x_{1},x_{2},0)=\varphi(x_{1},x_{2})\quad\,\text{on}\quad x_{1}\in[0,a]\quad% \,\text{and}\quad\frac{\partial w}{\partial t}(x_{1},x_{2},0)=\psi(x_{1},x_{2}% )\quad\,\text{on}\quad x_{2}\in[0,b]
  78. A m n = 4 a b 0 a 0 b φ ( x 1 , x 2 ) sin m π x 1 a sin n π x 2 b d x 1 d x 2 B m n = 4 a b ω m n 0 a 0 b ψ ( x 1 , x 2 ) sin m π x 1 a sin n π x 2 b d x 1 d x 2 . \begin{aligned}\displaystyle A_{mn}&\displaystyle=\frac{4}{ab}\int_{0}^{a}\int% _{0}^{b}\varphi(x_{1},x_{2})\sin\frac{m\pi x_{1}}{a}\sin\frac{n\pi x_{2}}{b}dx% _{1}dx_{2}\\ \displaystyle B_{mn}&\displaystyle=\frac{4}{ab\omega_{mn}}\int_{0}^{a}\int_{0}% ^{b}\psi(x_{1},x_{2})\sin\frac{m\pi x_{1}}{a}\sin\frac{n\pi x_{2}}{b}dx_{1}dx_% {2}\,.\end{aligned}

Vibrational_temperature.html

  1. θ v i b = h ν k B \theta_{vib}=\frac{h\nu}{k_{B}}
  2. k B k_{B}
  3. v ~ \tilde{v}
  4. θ v i b \theta_{vib}

Video_copy_detection.html

  1. a ( t ) = i = 1 N K ( i ) ( I ( i , t - 1 ) ) 2 a(t)=\sum_{i=1}^{N}K(i)(I(i,t-1))^{2}
  2. S ( t ) = ( r 1 , r 2 , , r N ) S(t)=(r_{1},r_{2},\cdots,r_{N})
  3. D ( t ) = 1 T 1 = t - T 2 t + T 2 | R ( i ) - C ( i ) | D(t)=\frac{1}{T}\sum_{1=t-\frac{T}{2}}^{t+\frac{T}{2}}\begin{vmatrix}R(i)-C(i)% \end{vmatrix}

Viennot's_geometric_construction.html

  1. σ S n \sigma\in S_{n}
  2. σ = ( 1 2 n σ 1 σ 2 σ n ) , \sigma=\begin{pmatrix}1&2&\cdots&n\\ \sigma_{1}&\sigma_{2}&\cdots&\sigma_{n}\end{pmatrix},
  3. ( i , σ i ) (i,\sigma_{i})
  4. σ = ( 1 2 3 4 5 6 7 8 3 8 1 2 4 7 5 6 ) . \sigma=\begin{pmatrix}1&2&3&4&5&6&7&8\\ 3&8&1&2&4&7&5&6\end{pmatrix}.
  5. σ \sigma
  6. σ - 1 \sigma^{-1}
  7. σ \sigma
  8. σ - 1 \sigma^{-1}
  9. y = x y=x

Virbhadra–Ellis_lens_equation.html

  1. ( β ) \left(\beta\right)
  2. ( θ ) \left(\theta\right)
  3. ( α ^ ) (\hat{\alpha})
  4. ( D d s ) \left(D_{ds}\right)
  5. ( D s ) \left(D_{s}\right)
  6. tan β = tan θ - D d s D s [ tan θ + tan ( α ^ - θ ) ] \tan\beta=\tan\theta-\frac{D_{ds}}{D_{s}}\left[\tan\theta+\tan\left(\hat{% \alpha}-\theta\right)\right]

Vojta's_conjecture.html

  1. F F
  2. X / F X/F
  3. D D
  4. X X
  5. H H
  6. X X
  7. K X K_{X}
  8. X X
  9. h H h_{H}
  10. h K X h_{K_{X}}
  11. v v
  12. F F
  13. λ D , v \lambda_{D,v}
  14. S S
  15. F F
  16. ϵ > 0 \epsilon>0
  17. C C
  18. U X U\subseteq X
  19. v S λ D , v ( P ) + h K X ( P ) ϵ h H ( P ) + C for all P U ( F ) . \sum_{v\in S}\lambda_{D,v}(P)+h_{K_{X}}(P)\leq\epsilon h_{H}(P)+C\quad\hbox{% for all }P\in U(F).
  20. X = N X=\mathbb{P}^{N}
  21. K X - ( N + 1 ) H K_{X}\sim-(N+1)H
  22. v S λ D , v ( P ) ( N + 1 + ϵ ) h H ( P ) + C \sum_{v\in S}\lambda_{D,v}(P)\leq(N+1+\epsilon)h_{H}(P)+C
  23. P U ( F ) P\in U(F)
  24. X X
  25. D D
  26. S S
  27. X D X\setminus D
  28. X X
  29. K X K_{X}
  30. X X
  31. S = S=\emptyset
  32. X ( F ) X(F)
  33. X X
  34. P P
  35. X ( F ¯ ) X(\overline{F})
  36. F ( P ) / F F(P)/F
  37. λ D , v \lambda_{D,v}

Voorhoeve_index.html

  1. V I ( f ) V_{I}(f)
  2. I I
  3. V I ( f ) = 1 2 π a b | d d t Arg f ( t ) | d t = 1 2 π a b | Im ( f f ) | d t . V_{I}(f)=\frac{1}{2\pi}\int_{a}^{b}\!\left|\frac{d}{dt}{\rm Arg}\,f(t)\right|% \,\,dt\,=\frac{1}{2\pi}\int_{a}^{b}\!\left|{\rm Im}\left(\frac{f^{\prime}}{f}% \right)\right|\,dt.
  4. N I ( f ) N_{I}(f)
  5. I I
  6. N I ( f ) N_{I}(f)
  7. N I N_{I}
  8. V I ( f ) V I ( f ) + 1 2 . V_{I}(f)\leq V_{I}(f^{\prime})+\frac{1}{2}.
  9. C C

Voronoi_pole.html

  1. V p V_{p}
  2. p P p\in P
  3. V p V_{p}
  4. V p V_{p}
  5. p p
  6. u ¯ \bar{u}
  7. p p
  8. u ¯ \bar{u}
  9. v v
  10. V p V_{p}
  11. p p
  12. u ¯ \bar{u}
  13. p p
  14. v v
  15. π 2 \frac{\pi}{2}
  16. x x
  17. V p V_{p}
  18. y y
  19. q q
  20. z z

Vortex_core_line.html

  1. s y m b o l v ( s y m b o l x , t ) \nabla symbol{v}(symbol{x},t)

Wagner's_gene_network_model.html

  1. n n
  2. N N
  3. N N
  4. ( R ) (R)
  5. t t
  6. S ( t ) S(t)
  7. S ( t ) := ( S 1 ( t ) , , S N ( t ) ) S(t)\ :=\ (S_{1}(t),\ ...,\ S_{N}(t))
  8. s i ( t ) s_{i}(t)
  9. S ( t ) S(t)
  10. { - 1 , 1 } \{-1,1\}
  11. S ( 0 ) S(0)
  12. t = 0 t=0
  13. S l ( t + S_{l}(t+
  14. ) = )=
  15. [ j = 1 N w i j S j ( t ) ] [\sum_{j=1}^{N}w_{ij}S_{j}(t)]
  16. [ h i ] ( t ) [h_{i}](t)
  17. S l ( t + S_{l}(t+
  18. G l G_{l}
  19. ( x ) (x)
  20. h i ( t ) h_{i}(t)
  21. w i j w_{ij}
  22. G i G_{i}
  23. ( x ) = - 1 (x)=-1
  24. x < 0 ; 1 x<0;\ 1
  25. x > 0 ; 0 x>0;\ 0
  26. x = 0 x=0
  27. ( x ) = 2 / ( 1 + e - a x ) - 1 (x)=2/(1+e^{-ax})-1
  28. R R

Wahba's_problem.html

  1. J ( 𝐑 ) = 1 2 k = 1 N a k || 𝐰 k - 𝐑𝐯 k || 2 J(\mathbf{R})=\frac{1}{2}\sum_{k=1}^{N}a_{k}||\mathbf{w}_{k}-\mathbf{R}\mathbf% {v}_{k}||^{2}
  2. 𝐰 k \mathbf{w}_{k}
  3. 𝐯 k \mathbf{v}_{k}
  4. 𝐑 \mathbf{R}
  5. a k a_{k}
  6. 𝐁 \mathbf{B}
  7. 𝐁 = i = 1 n a i 𝐰 i 𝐯 i T \mathbf{B}=\sum_{i=1}^{n}a_{i}\mathbf{w}_{i}{\mathbf{v}_{i}}^{T}
  8. 𝐁 \mathbf{B}
  9. 𝐁 = 𝐔𝐒𝐕 T \mathbf{B}=\mathbf{U}\mathbf{S}\mathbf{V}^{T}
  10. 𝐑 = 𝐔𝐌𝐕 T \mathbf{R}=\mathbf{U}\mathbf{M}\mathbf{V}^{T}
  11. 𝐌 = diag ( [ 1 1 det ( 𝐔 ) det ( 𝐕 ) ] ) \mathbf{M}=\operatorname{diag}(\begin{bmatrix}1&1&\det(\mathbf{U})\det(\mathbf% {V})\end{bmatrix})

Wald's_maximin_model.html

  1. v * := max d D min s S ( d ) f ( d , s ) v^{*}:=\max_{d\in D}\min_{s\in S(d)}f(d,s)
  2. D D
  3. S ( d ) S(d)
  4. d d
  5. f ( d , s ) f(d,s)
  6. d d
  7. s s
  8. max \max
  9. S ( d ) S(d)
  10. S ( d ) S(d)
  11. f ( d , s ) f(d,s)
  12. s s
  13. S ( d ) S(d)
  14. v * := max d D , z { z : z f ( d , s ) , s S ( d ) } v^{*}:=\max_{d\in D,\,z\in\mathbb{R}}\{z:z\leq f(d,s),\forall s\in S(d)\}
  15. \mathbb{R}
  16. d d
  17. v ( d ) := min s S ( d ) f ( d , s ) , d D v(d):=\min_{s\in S(d)}f(d,s)\ ,\ d\in D
  18. d d
  19. max \max
  20. min \min
  21. v := min d D max s S ( d ) f ( d , s ) . v^{\circ}:=\min_{d\in D}\max_{s\in S(d)}f(d,s).
  22. v := min d D , z { z : z f ( d , s ) , s S ( d ) } v^{\circ}:=\min_{d\in D,\,z\in\mathbb{R}}\{z:z\geq f(d,s),\forall s\in S(d)\}
  23. max \max
  24. min \min
  25. min x max y { x 2 - y 2 } \min_{x\in\mathbb{R}}\max_{y\in\mathbb{R}}\ \{x^{2}-y^{2}\}
  26. \mathbb{R}
  27. D = S ( d ) = D=S(d)=\mathbb{R}
  28. f ( d , s ) = d 2 - s 2 f(d,s)=d^{2}-s^{2}
  29. ( x , y ) = ( 0 , 0 ) (x,y)=(0,0)
  30. min d D max s S r ( d , s ) \min_{d\in D}\max_{s\in S}r(d,s)
  31. r ( d , s ) := max d D f ( d , s ) - f ( d , s ) r(d,s):=\max_{d\,^{\prime}\in D}f(d\,^{\prime},s)-f(d,s)
  32. f ( d , s ) f(d,s)
  33. ( d , s ) (d,s)
  34. S ( d ) , d D , S(d),d\in D,
  35. D D
  36. S S
  37. max d D min s S d i s t ( d , s ) \max_{d\in D}\min_{s\in S}dist(d,s)
  38. d i s t ( d , s ) dist(d,s)
  39. s s
  40. d d
  41. S ( d ) S(d)
  42. d d
  43. min d D max s S d i s t ( d , s ) \min_{d\in D}\max_{s\in S}dist(d,s)
  44. X X
  45. g g
  46. X X
  47. X X
  48. g ( x ) 0 g(x)\leq 0
  49. x x
  50. max Y X { | Y | : g ( x ) 0 , x Y } . \max_{Y\subseteq X}\ \{|Y|:g(x)\leq 0,\forall x\in Y\}.
  51. v * := max d D min s S ( d ) { f ( d , s ) : g ( d , s ) 0 , s S ( d ) } . v^{*}:=\max_{d\in D}\min_{s\in S(d)}\ \{f(d,s):g(d,s)\leq 0,\forall s\in S(d)\}.
  52. v * := max d D , z { z : z f ( d , s ) , g ( d , s ) 0 , s S ( d ) } . v^{*}:=\max_{d\in D,\,z\in\mathbb{R}}\{z:z\leq f(d,s),g(d,s)\leq 0,\forall s% \in S(d)\}.
  53. max d D min s S ( d ) f ( d , s ) = max d , d ′′ min a s b f ( d , s ) = max { min a s b f ( d , s ) , min a s b f ( d ′′ , s ) } \max_{d\in D}\min_{s\in S(d)}f(d,s)=\max_{d\,^{\prime},d\,^{\prime\prime}}\ % \min_{a\leq s\leq b}f(d,s)=\max\ \{\min_{a\leq s\leq b}f(d\,^{\prime},s),\min_% {a\leq s\leq b}f(d\,^{\prime\prime},s)\}
  54. S ( d ) = { 1 , 2 , , k } S(d)=\{1,2,\dots,k\}
  55. d D d\in D
  56. max d D min s S ( d ) f ( d , s ) \displaystyle\max_{d\in D}\min_{s\in S(d)}f(d,s)
  57. f s ( d ) f ( d , s ) f_{s}(d)\equiv f(d,s)
  58. d n d\in\mathbb{R}^{n}
  59. f s , s = 1 , 2 , , k , f_{s},s=1,2,\dots,k,
  60. d D d\in D
  61. d d

Wallis'_integrals.html

  1. ( W n ) n (W_{n})_{\,n\,\in\,\mathbb{N}\,}
  2. W n = 0 π 2 sin n ( x ) d x , W_{n}=\int_{0}^{\frac{\pi}{2}}\sin^{n}(x)\,dx,
  3. x = π 2 - t x=\frac{\pi}{2}-t
  4. W n = 0 π 2 cos n ( x ) d x W_{n}=\int_{0}^{\frac{\pi}{2}}\cos^{n}(x)\,dx
  5. W 0 W_{0}
  6. W 1 W_{1}
  7. W 2 W_{2}
  8. W 3 W_{3}
  9. W 4 W_{4}
  10. W 5 W_{5}
  11. W 6 W_{6}
  12. W 7 W_{7}
  13. W 8 W_{8}
  14. π 2 \frac{\pi}{2}
  15. 1 1
  16. π 4 \frac{\pi}{4}
  17. 2 3 \frac{2}{3}
  18. 3 π 16 \frac{3\pi}{16}
  19. 8 15 \frac{8}{15}
  20. 5 π 32 \frac{5\pi}{32}
  21. 16 35 \frac{16}{35}
  22. 35 π 256 \frac{35\pi}{256}
  23. ( W n ) \ (W_{n})
  24. n n\in\,\mathbb{N}
  25. W n > 0 \ W_{n}>0
  26. W n - W n + 1 = 0 π 2 sin n ( x ) d x - 0 π 2 sin n + 1 ( x ) d x = 0 π 2 sin n ( x ) [ 1 - sin ( x ) ] d x 0 W_{n}-W_{n+1}=\int_{0}^{\frac{\pi}{2}}\sin^{n}(x)\,dx-\int_{0}^{\frac{\pi}{2}}% \sin^{n+1}(x)\,dx=\int_{0}^{\frac{\pi}{2}}\sin^{n}(x)\,[1-\sin(x)]\,dx\geqslant 0
  27. ( W n ) \ (W_{n})
  28. x x
  29. sin 2 ( x ) = 1 - cos 2 ( x ) \quad\sin^{2}(x)=1-\cos^{2}(x)
  30. n 2 n\geqslant 2
  31. 0 π 2 sin n ( x ) d x = 0 π 2 sin n - 2 ( x ) [ 1 - cos 2 ( x ) ] d x \int_{0}^{\frac{\pi}{2}}\sin^{n}(x)\,dx=\int_{0}^{\frac{\pi}{2}}\sin^{n-2}(x)% \left[1-\cos^{2}(x)\right]\,dx
  32. 0 π 2 sin n ( x ) d x = 0 π 2 sin n - 2 ( x ) d x - 0 π 2 sin n - 2 ( x ) cos 2 ( x ) d x \int_{0}^{\frac{\pi}{2}}\sin^{n}(x)\,dx=\int_{0}^{\frac{\pi}{2}}\sin^{n-2}(x)% \,dx-\int_{0}^{\frac{\pi}{2}}\sin^{n-2}(x)\cos^{2}(x)\,dx
  33. ( 𝟏 ) \mathbf{(1)}
  34. u ( x ) = cos ( x ) sin n - 2 ( x ) u^{\prime}(x)=\cos(x)\sin^{n-2}(x)
  35. u ( x ) = 1 n - 1 sin n - 1 ( x ) u(x)=\frac{1}{n-1}\sin^{n-1}(x)
  36. v ( x ) = cos ( x ) v(x)=\cos(x)
  37. v ( x ) = - sin ( x ) v^{\prime}(x)=-\sin(x)
  38. 0 π 2 sin n - 2 ( x ) cos 2 ( x ) d x = [ 1 n - 1 sin n - 1 ( x ) cos ( x ) ] 0 π 2 + 0 π 2 1 n - 1 sin n - 1 ( x ) sin ( x ) d x = 0 + 1 n - 1 W n \int_{0}^{\frac{\pi}{2}}\sin^{n-2}(x)\cos^{2}(x)\,dx=\left[\frac{1}{n-1}\sin^{% n-1}(x)\cos(x)\right]_{0}^{\frac{\pi}{2}}+\int_{0}^{\frac{\pi}{2}}\ \frac{1}{n% -1}\sin^{n-1}(x)\sin(x)\,dx=0+{1\over{n-1}}\,W_{n}
  39. ( 𝟏 ) \mathbf{(1)}
  40. W n = W n - 2 - 1 n - 1 W n W_{n}=W_{n-2}-{1\over{n-1}}\,W_{n}
  41. ( 1 + 1 n - 1 ) W n = W n - 2 \qquad\left(1+\frac{1}{n-1}\right)W_{n}=W_{n-2}
  42. ( 𝟐 ) \mathbf{(2)}
  43. n W n = ( n - 1 ) W n - 2 n\,W_{n}=(n-1)\,W_{n-2}\qquad\,
  44. n 2 n\geqslant 2\qquad\,
  45. W n W_{n}
  46. W n - 2 W_{n-2}
  47. W 0 W_{0}
  48. W 1 W_{1}
  49. ( W n ) \ (W_{n})
  50. n n
  51. n = 2 p \quad n=2\,p
  52. W 2 p = 2 p - 1 2 p × 2 p - 3 2 p - 2 × × 1 2 W 0 = 2 p 2 p × 2 p - 1 2 p × 2 p - 2 2 p - 2 × 2 p - 3 2 p - 2 × × 2 2 × 1 2 W 0 = ( 2 p ) ! 2 2 p ( p ! ) 2 π 2 \quad W_{2\,p}=\frac{2\,p-1}{2\,p}\times\frac{2\,p-3}{2\,p-2}\times\cdots% \times\frac{1}{2}\,W_{0}=\frac{2\,p}{2\,p}\times\frac{2\,p-1}{2\,p}\times\frac% {2\,p-2}{2\,p-2}\times\frac{2\,p-3}{2\,p-2}\times\cdots\times\frac{2}{2}\times% \frac{1}{2}\,W_{0}=\frac{(2\,p)!}{2^{2\,p}\,(p!)^{2}}\frac{\pi}{2}
  53. n = 2 p + 1 \quad n=2\,p+1
  54. W 2 p + 1 = 2 p 2 p + 1 2 p - 2 2 p - 1 2 3 W 1 = 2 2 p ( p ! ) 2 ( 2 p + 1 ) ! \quad W_{2\,p+1}=\frac{2\,p}{2\,p+1}\,\frac{2\,p-2}{2\,p-1}\cdots\frac{2}{3}\,% W_{1}=\frac{2^{2\,p}\,(p!)^{2}}{(2\,p+1)!}~{}
  55. B ( x , y ) = 0 1 t x - 1 ( 1 - t ) y - 1 d t = Γ ( x ) Γ ( y ) Γ ( x + y ) B(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}\,dt=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
  56. Γ ( z ) = 0 t z - 1 e - t d t \Gamma(z)=\int_{0}^{\infty}t^{z-1}\,e^{-t}\,dt
  57. { t = sin 2 ( u ) 1 - t = cos 2 ( u ) d t = 2 sin ( u ) cos ( u ) d u \quad\left\{\begin{matrix}t=\sin^{2}(u)\\ 1-t=\cos^{2}(u)\\ dt=2\sin(u)\cos(u)\,du\end{matrix}\right.
  58. B ( a , b ) = 2 0 π 2 sin 2 a - 1 ( u ) cos 2 b - 1 ( u ) d u B(a,b)=2\int_{0}^{\frac{\pi}{2}}\sin^{2a-1}(u)\cos^{2b-1}(u)\,du
  59. Γ ( 1 2 ) = π \Gamma(\tfrac{1}{2})=\sqrt{\pi}
  60. W n = 1 2 B ( n + 1 2 , 1 2 ) = π 2 Γ ( n + 1 2 ) Γ ( n 2 + 1 ) . W_{n}={\frac{1}{2}}B(\frac{n+1}{2},\frac{1}{2})=\frac{\sqrt{\pi}}{2}\frac{% \Gamma(\tfrac{n+1}{2})}{\Gamma(\tfrac{n}{2}+1)}.
  61. ( 𝟐 ) \mathbf{(2)}
  62. W n + 1 W n \ W_{n+1}\sim W_{n}
  63. n n\in\,\mathbb{N}
  64. W n + 2 W n + 1 W n \ W_{n+2}\leqslant W_{n+1}\leqslant W_{n}
  65. W n + 2 W n W n + 1 W n 1 \frac{W_{n+2}}{W_{n}}\leqslant\frac{W_{n+1}}{W_{n}}\leqslant 1
  66. W n > 0 \ W_{n}>0
  67. n + 1 n + 2 W n + 1 W n 1 \frac{n+1}{n+2}\leqslant\frac{W_{n+1}}{W_{n}}\leqslant 1
  68. ( 𝟐 ) \mathbf{(2)}
  69. W n + 1 W n 1 \frac{W_{n+1}}{W_{n}}\to 1
  70. W n + 1 W n \ W_{n+1}\sim W_{n}
  71. W n W n + 1 W_{n}W_{n+1}
  72. W n π 2 n W_{n}\sim\sqrt{\frac{\pi}{2\,n}}\quad
  73. lim n n W n = π / 2 \quad\lim_{n\rightarrow\infty}\sqrt{n}\,W_{n}=\sqrt{\pi/2}\quad
  74. n ! C n ( n e ) n \ n\,!\sim C\,\sqrt{n}\left(\frac{n}{\mathrm{e}}\right)^{n}
  75. C \R * \ C\in\R^{*}
  76. C \ C
  77. W 2 p W_{2\,p}
  78. W 2 p π 4 p = π 2 1 p W_{2\,p}\sim\sqrt{\frac{\pi}{4\,p}}=\frac{\sqrt{\pi}}{2}\,\frac{1}{\sqrt{p}}
  79. ( 𝟑 ) \mathbf{(3)}
  80. W 2 p W_{2\,p}
  81. W 2 p = ( 2 p ) ! 2 2 p ( p ! ) 2 π 2 C ( 2 p e ) 2 p 2 p 2 2 p C 2 ( p e ) 2 p ( p ) 2 π 2 W_{2\,p}=\frac{(2\,p)!}{2^{2\,p}\,(p\,!)^{2}}\,\frac{\pi}{2}\sim\frac{C\,\left% (\frac{2\,p}{\mathrm{e}}\right)^{2p}\,\sqrt{2\,p}}{2^{2p}\,C^{2}\,\left(\frac{% p}{\mathrm{e}}\right)^{2p}\,\left(\sqrt{p}\right)^{2}}\,\frac{\pi}{2}
  82. W 2 p π C 2 1 p W_{2\,p}\sim\frac{\pi}{C\,\sqrt{2}}\,\frac{1}{\sqrt{p}}
  83. ( 𝟒 ) \mathbf{(4)}
  84. ( 𝟑 ) \mathbf{(3)}
  85. ( 𝟒 ) \mathbf{(4)}
  86. π C 2 1 p π 2 1 p \frac{\pi}{C\,\sqrt{2}}\,\frac{1}{\sqrt{p}}\sim\frac{\sqrt{\pi}}{2}\,\frac{1}{% \sqrt{p}}
  87. π C 2 = π 2 \frac{\pi}{C\,\sqrt{2}}=\frac{\sqrt{\pi}}{2}
  88. C = 2 π C=\sqrt{2\,\pi}
  89. n ! 2 π n ( n e ) n \ n\,!\sim\sqrt{2\,\pi\,n}\,\left(\frac{n}{\mathrm{e}}\right)^{n}
  90. n * u + u n ( 1 - u / n ) n e - u \forall n\in\mathbb{N}^{*}\quad\forall u\in\mathbb{R}_{+}\quad u\leqslant n% \quad\Rightarrow\quad(1-u/n)^{n}\leqslant e^{-u}
  91. n * u + e - u ( 1 + u / n ) - n \forall n\in\mathbb{N}^{*}\quad\forall u\in\mathbb{R}_{+}\qquad e^{-u}% \leqslant(1+u/n)^{-n}
  92. u / n = t \quad u/n=t
  93. t [ 0 , 1 ] t\in[0,1]
  94. 1 - t e - t 1-t\leqslant e^{-t}
  95. e - t ( 1 + t ) - 1 e^{-t}\leqslant(1+t)^{-1}
  96. e t 1 + t e^{t}\geqslant 1+t
  97. t e t - 1 - t t\mapsto e^{t}-1-t
  98. u = x 2 u=x^{2}
  99. 0 n ( 1 - x 2 / n ) n d x 0 n e - x 2 d x 0 + e - x 2 d x 0 + ( 1 + x 2 / n ) - n d x \int_{0}^{\sqrt{n}}(1-x^{2}/n)^{n}dx\leqslant\int_{0}^{\sqrt{n}}e^{-x^{2}}dx% \leqslant\int_{0}^{+\infty}e^{-x^{2}}dx\leqslant\int_{0}^{+\infty}(1+x^{2}/n)^% {-n}dx
  100. n n\to\infty
  101. x = n sin t x=\sqrt{n}\,\sin\,t
  102. π / 2 \pi/2
  103. n W 2 n + 1 \sqrt{n}\,W_{2n+1}
  104. x = n tan t x=\sqrt{n}\,\tan\,t
  105. 0
  106. π / 2 \pi/2
  107. n W 2 n - 2 \sqrt{n}\,W_{2n-2}
  108. lim n + n W n = π / 2 \lim_{n\rightarrow+\infty}\sqrt{n}\;W_{n}=\sqrt{\pi/2}
  109. 0 + e - x 2 d x = π / 2 \int_{0}^{+\infty}e^{-x^{2}}dx=\sqrt{\pi}/2
  110. B ( x , y ) = 2 0 π / 2 ( sin θ ) 2 x - 1 ( cos θ ) 2 y - 1 d θ , Re ( x ) > 0 , Re ( y ) > 0 B(x,y)=2\int_{0}^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta,\qquad% \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0\!
  111. x = n + 1 2 x=\frac{n+1}{2}
  112. y = 1 2 y=\frac{1}{2}
  113. B ( n + 1 2 , 1 2 ) = 2 0 π / 2 ( sin θ ) n ( cos θ ) 0 d θ = 2 0 π / 2 ( sin θ ) n d θ = 2 W n B\left(\frac{n+1}{2},\frac{1}{2}\right)=2\int_{0}^{\pi/2}(\sin\theta)^{n}(\cos% \theta)^{0}\,d\theta=2\int_{0}^{\pi/2}(\sin\theta)^{n}\,d\theta=2W_{n}
  114. W n = 1 2 B ( n + 1 2 , 1 2 ) W_{n}=\frac{1}{2}B\left(\frac{n+1}{2},\frac{1}{2}\right)
  115. B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}
  116. W n = 1 2 Γ ( n + 1 2 ) Γ ( 1 2 ) Γ ( n + 1 2 + 1 2 ) = Γ ( n + 1 2 ) Γ ( 1 2 ) 2 Γ ( n + 2 2 ) W_{n}=\frac{1}{2}\frac{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{1}{2}% \right)}{\Gamma\left(\frac{n+1}{2}+\frac{1}{2}\right)}=\frac{\Gamma\left(\frac% {n+1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{2\,\Gamma\left(\frac{n+2}{2}% \right)}
  117. n n
  118. n = 2 p + 1 n=2p+1
  119. W 2 p + 1 = Γ ( p + 1 ) Γ ( 1 2 ) 2 Γ ( p + 1 + 1 2 ) = p ! Γ ( 1 2 ) ( 2 p + 1 ) Γ ( p + 1 2 ) = 2 p p ! ( 2 p + 1 ) < t h > = 4 p ( p ! ) 2 ( 2 p + 1 ) ! w h e r e a s f o r e v e n < m a t h > n W_{2p+1}=\frac{\Gamma\left(p+1\right)\Gamma\left(\frac{1}{2}\right)}{2\,\Gamma% \left(p+1+\frac{1}{2}\right)}=\frac{p!\Gamma\left(\frac{1}{2}\right)}{(2p+1)\,% \Gamma\left(p+\frac{1}{2}\right)}=\frac{2^{p}\;p!}{(2p+1)<th>}=\frac{4^{p}\;(p% !)^{2}}{(2p+1)!}whereasforeven<math>n
  120. n = 2 p n=2p
  121. W 2 p = Γ ( p + 1 2 ) Γ ( 1 2 ) 2 Γ ( p + 1 ) = ( 2 p - 1 ) < t h > π 2 p + 1 p ! = ( 2 p ) ! 4 p ( p ! ) 2 π 2 = = N o t e = = T h e s a m e p r o p e r t i e s l e a d t o [ [ W a l l i s p r o d u c t | W a l l i s p r o d u c t ] ] , w h i c h e x p r e s s e s < m a t h > π 2 W_{2p}=\frac{\Gamma\left(p+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{2% \,\Gamma\left(p+1\right)}=\frac{(2p-1)<th>\;\pi}{2^{p+1}\;p!}=\frac{(2p)!}{4^{% p}\;(p!)^{2}}\cdot\frac{\pi}{2}\par ==Note==\par Thesamepropertiesleadto[[% Wallis_{p}roduct|Wallisproduct]],whichexpresses<math>\frac{\pi}{2}\,
  122. π \pi

Ward's_method.html

  1. d i j = d ( { X i } , { X j } ) = X i - X j 2 . d_{ij}=d(\{X_{i}\},\{X_{j}\})={\|X_{i}-X_{j}\|^{2}}.
  2. C i C_{i}
  3. C j C_{j}
  4. C i C_{i}
  5. C j C_{j}
  6. d i j d_{ij}
  7. d i k d_{ik}
  8. d j k d_{jk}
  9. C i C_{i}
  10. C j C_{j}
  11. C k C_{k}
  12. d ( i j ) k d_{(ij)k}
  13. C i C j C_{i}\cup C_{j}
  14. C k C_{k}
  15. d ( i j ) k d_{(ij)k}
  16. d ( i j ) k = α i d i k + α j d j k + β d i j + γ | d i k - d j k | , d_{(ij)k}=\alpha_{i}d_{ik}+\alpha_{j}d_{jk}+\beta d_{ij}+\gamma|d_{ik}-d_{jk}|,
  17. α i , α j , β , \alpha_{i},\alpha_{j},\beta,
  18. γ \gamma
  19. d i j d_{ij}
  20. C i , C j , C_{i},C_{j},
  21. C k C_{k}
  22. n i , n j , n_{i},n_{j},
  23. n k n_{k}
  24. d ( C i C j , C k ) = n i + n k n i + n j + n k d ( C i , C k ) + n j + n k n i + n j + n k d ( C j , C k ) - n k n i + n j + n k d ( C i , C j ) . d(C_{i}\cup C_{j},C_{k})=\frac{n_{i}+n_{k}}{n_{i}+n_{j}+n_{k}}\;d(C_{i},C_{k})% +\frac{n_{j}+n_{k}}{n_{i}+n_{j}+n_{k}}\;d(C_{j},C_{k})-\frac{n_{k}}{n_{i}+n_{j% }+n_{k}}\;d(C_{i},C_{j}).
  25. α l = n i + n k n i + n j + n k , β = - n k n i + n j + n k , γ = 0. \alpha_{l}=\frac{n_{i}+n_{k}}{n_{i}+n_{j}+n_{k}},\qquad\beta=\frac{-n_{k}}{n_{% i}+n_{j}+n_{k}},\qquad\gamma=0.

Ward–Takahashi_identity.html

  1. ( k ; p 1 p n ; q 1 q n ) = ϵ μ ( k ) μ ( k ; p 1 p n ; q 1 q n ) \mathcal{M}(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})=\epsilon_{\mu}(k)\mathcal{M% }^{\mu}(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})
  2. ϵ μ ( k ) \!\epsilon_{\mu}(k)
  3. μ \mu
  4. p 1 p n p_{1}\cdots p_{n}
  5. q 1 q n q_{1}\cdots q_{n}
  6. 0 \mathcal{M}_{0}
  7. k μ μ ( k ; p 1 p n ; q 1 q n ) = e i [ 0 ( p 1 p n ; q 1 ( q i - k ) q n ) k_{\mu}\mathcal{M}^{\mu}(k;p_{1}\cdots p_{n};q_{1}\cdots q_{n})=e\sum_{i}\left% [\mathcal{M}_{0}(p_{1}\cdots p_{n};q_{1}\cdots(q_{i}-k)\cdots q_{n})\right.
  8. - 0 ( p 1 ( p i + k ) p n ; q 1 q n ) ] \left.-\mathcal{M}_{0}(p_{1}\cdots(p_{i}+k)\cdots p_{n};q_{1}\cdots q_{n})\right]
  9. \mathcal{M}
  10. ( k ) = ϵ μ ( k ) μ ( k ) \mathcal{M}(k)=\epsilon_{\mu}(k)\mathcal{M}^{\mu}(k)
  11. k \!k
  12. ϵ μ ( k ) \!\epsilon_{\mu}(k)
  13. k μ μ ( k ) = 0 k_{\mu}\mathcal{M}^{\mu}(k)=0
  14. δ ϵ \delta_{\epsilon}
  15. δ ϵ ( e i S ) 𝒟 ϕ = 0 \int\delta_{\epsilon}\left(\mathcal{F}e^{iS}\right)\mathcal{D}\phi=0
  16. \mathcal{F}
  17. δ ϵ S = ( μ ϵ ) J μ d d x = - ϵ μ J μ d d x \delta_{\epsilon}S=\int\left(\partial_{\mu}\epsilon\right)J^{\mu}\mathrm{d}^{d% }x=-\int\epsilon\partial_{\mu}J^{\mu}\mathrm{d}^{d}x
  18. δ ϵ - i ϵ μ J μ d d x = 0 \langle\delta_{\epsilon}\mathcal{F}\rangle-i\int\epsilon\langle\mathcal{F}% \partial_{\mu}J^{\mu}\rangle\mathrm{d}^{d}x=0
  19. μ J μ = 0 \partial_{\mu}J^{\mu}=0
  20. δ ϵ ( e i ( S + S g f ) ) 𝒟 ϕ = 0 \int\delta_{\epsilon}\left(\mathcal{F}e^{i\left(S+S_{gf}\right)}\right)% \mathcal{D}\phi=0
  21. δ ϵ ( e i S ) 𝒟 ϕ = ϵ λ e i S d d x \int\delta_{\epsilon}\left(\mathcal{F}e^{iS}\right)\mathcal{D}\phi=\int% \epsilon\lambda\mathcal{F}e^{iS}\mathrm{d}^{d}x

Wavelet_transform_modulus_maxima_method.html

  1. f ( t ) = a 0 + a 1 ( t - t i ) + a 2 ( t - t i ) 2 + + a h ( t - t i ) h i f(t)=a_{0}+a_{1}(t-t_{i})+a_{2}(t-t_{i})^{2}+\cdots+a_{h}(t-t_{i})^{h_{i}}\,
  2. t t
  3. t i t_{i}
  4. h i h_{i}
  5. G ( t , a , b ) = a ( 2 π ) - 1 / 2 ( t - b ) e ( - ( t - b ) 2 2 a 2 ) G^{\prime}(t,a,b)=\frac{a}{(2\pi)^{-1/2}}(t-b)e^{\left(\frac{-(t-b)^{2}}{2a^{2% }}\right)}\,
  6. a > 0 a>0
  7. b b\in\mathbb{R}
  8. X w ( a , b ) = 1 a - x ( t ) ψ ( t - b a ) d t X_{w}(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}x(t)\psi^{\ast}\left(\frac% {t-b}{a}\right)\,dt
  9. ψ ( t ) \psi(t)
  10. {}^{\ast}
  11. X w ( a , b ) X_{w}(a,b)
  12. h i h_{i}

Weak_duality.html

  1. ( x 1 , x 2 , . , x n ) (x_{1},x_{2},....,x_{n})
  2. ( y 1 , y 2 , . , y m ) (y_{1},y_{2},....,y_{m})
  3. i = 1 m b i y i j = 1 n c j x j \sum_{i=1}^{m}b_{i}y_{i}\leq\sum_{j=1}^{n}c_{j}x_{j}
  4. c j c_{j}
  5. b i b_{i}
  6. x x
  7. y y
  8. g ( y ) f ( x ) g(y)\leq f(x)
  9. f f
  10. g g

Weak_equivalence_(homotopy_theory).html

  1. p : A B p:A\rightarrow B\,
  2. p n : H n ( A ) H n ( B ) . p_{n}:H_{n}(A)\rightarrow H_{n}(B).\,
  3. π n ( X , x ) π n ( Y , y ) \pi_{n}(X,x)\cong\pi_{n}(Y,y)\,

Weak_Hopf_algebra.html

  1. ( H , μ , η , Δ , ε ) (H,\mu,\eta,\Delta,\varepsilon)
  2. k k
  3. H H
  4. ( H , μ , η ) (H,\mu,\eta)
  5. μ : H H H \mu:H\otimes H\rightarrow H
  6. η : k H \eta:k\rightarrow H
  7. ( H , Δ , ε ) (H,\Delta,\varepsilon)
  8. Δ : H H H \Delta:H\rightarrow H\otimes H
  9. ε : H k \varepsilon:H\rightarrow k
  10. Δ μ = ( μ μ ) ( id H σ H , H id H ) ( Δ Δ ) \Delta\circ\mu=(\mu\otimes\mu)\circ(\mathrm{id}_{H}\otimes\sigma_{H,H}\otimes% \mathrm{id}_{H})\circ(\Delta\otimes\Delta)
  11. ε μ ( μ id H ) = ( ε ε ) ( μ μ ) ( id H Δ id H ) = ( ε ε ) ( μ μ ) ( id H Δ o p id H ) \varepsilon\circ\mu\circ(\mu\otimes\mathrm{id}_{H})=(\varepsilon\otimes% \varepsilon)\circ(\mu\otimes\mu)\circ(\mathrm{id}_{H}\otimes\Delta\otimes% \mathrm{id}_{H})=(\varepsilon\otimes\varepsilon)\circ(\mu\otimes\mu)\circ(% \mathrm{id}_{H}\otimes\Delta^{op}\otimes\mathrm{id}_{H})
  12. ( Δ id H ) Δ η = ( id H μ id H ) ( Δ Δ ) ( η η ) = ( id H μ o p id H ) ( Δ Δ ) ( η η ) (\Delta\otimes\mathrm{id}_{H})\circ\Delta\circ\eta=(\mathrm{id}_{H}\otimes\mu% \otimes\mathrm{id}_{H})\circ(\Delta\otimes\Delta)\circ(\eta\otimes\eta)=(% \mathrm{id}_{H}\otimes\mu^{op}\otimes\mathrm{id}_{H})\circ(\Delta\otimes\Delta% )\circ(\eta\otimes\eta)
  13. σ V , W : V W W V : v w w v \sigma_{V,W}:V\otimes W\rightarrow W\otimes V:v\otimes w\mapsto w\otimes v
  14. μ o p = μ σ H , H \mu^{op}=\mu\circ\sigma_{H,H}
  15. Δ o p = σ H , H Δ \Delta^{op}=\sigma_{H,H}\circ\Delta
  16. ( U V ) W U ( V W ) (U\otimes V)\otimes W\cong U\otimes(V\otimes W)
  17. V k V k V V\otimes k\cong V\cong k\otimes V
  18. ( H , μ , η , Δ , ε , S ) (H,\mu,\eta,\Delta,\varepsilon,S)
  19. ( H , μ , η , Δ , ε ) (H,\mu,\eta,\Delta,\varepsilon)
  20. S : H H S:H\to H
  21. μ ( id H S ) Δ = ( ε id H ) ( μ id H ) ( id H σ H , H ) ( Δ id H ) ( η id H ) \mu\circ(\mathrm{id}_{H}\otimes S)\circ\Delta=(\varepsilon\otimes\mathrm{id}_{% H})\circ(\mu\otimes\mathrm{id}_{H})\circ(\mathrm{id}_{H}\otimes\sigma_{H,H})% \circ(\Delta\otimes\mathrm{id}_{H})\circ(\eta\otimes\mathrm{id}_{H})
  22. μ ( S id H ) Δ = ( id H ε ) ( id H μ ) ( σ H , H id H ) ( id H Δ ) ( id H η ) \mu\circ(S\otimes\mathrm{id}_{H})\circ\Delta=(\mathrm{id}_{H}\otimes% \varepsilon)\circ(\mathrm{id}_{H}\otimes\mu)\circ(\sigma_{H,H}\otimes\mathrm{% id}_{H})\circ(\mathrm{id}_{H}\otimes\Delta)\circ(\mathrm{id}_{H}\otimes\eta)
  23. S = μ ( μ id H ) ( S id H S ) ( Δ id H ) Δ S=\mu\circ(\mu\otimes\mathrm{id}_{H})\circ(S\otimes\mathrm{id}_{H}\otimes S)% \circ(\Delta\otimes\mathrm{id}_{H})\circ\Delta
  24. G = ( G 0 , G 1 ) G=(G_{0},G_{1})
  25. K [ G ] K[G]
  26. g G 1 g\in G_{1}
  27. μ : K [ G ] K [ G ] K [ G ] by μ ( g h ) = { g h if target(h) = source(g) 0 otherwise \mu:K[G]\otimes K[G]\to K[G]~{}\,\text{by}~{}\mu(g\otimes h)=\left\{\begin{% array}[]{cl}g\circ h&\,\text{if target(h) = source(g)}\\ 0&\,\text{otherwise}\end{array}\right.
  28. η : k K [ G ] by η ( 1 ) = X G 0 id X \eta:k\to K[G]~{}\,\text{by}~{}\eta(1)=\sum_{X\in G_{0}}\mathrm{id}_{X}
  29. Δ : K [ G ] K [ G ] K [ G ] by Δ ( g ) = g g for all g G 1 \Delta:K[G]\to K[G]\otimes K[G]~{}\,\text{by}~{}\Delta(g)=g\otimes g~{}\,\text% {for all}~{}g\in G_{1}
  30. ε : K [ G ] k by ε ( g ) = 1 for all g G 1 \varepsilon:K[G]\to k~{}\,\text{by}~{}\varepsilon(g)=1~{}\,\text{for all}~{}g% \in G_{1}
  31. S : K [ G ] K [ G ] by S ( g ) = g - 1 for all g G 1 S:K[G]\to K[G]~{}\,\text{by}~{}S(g)=g^{-1}~{}\,\text{for all}~{}g\in G_{1}

Weapon_target_assignment_problem.html

  1. i = 1 , , m i=1,\ldots,m
  2. W i W_{i}
  3. i i
  4. j = 1 , , n j=1,\ldots,n
  5. V j V_{j}
  6. p i j p_{ij}
  7. min j = 1 n ( V j i = 1 m q i j x i j ) \min\sum_{j=1}^{n}\left(V_{j}\prod_{i=1}^{m}q_{ij}^{x_{ij}}\right)
  8. j = 1 n x i j W i for i = 1 , , m , \sum_{j=1}^{n}x_{ij}\leq W_{i}\,\text{ for }i=1,\ldots,m,\,
  9. x i j 0 and integer for i = 1 , , m and j = 1 , , n . x_{ij}\geq 0\,\text{ and integer for }i=1,\ldots,m\,\text{ and }j=1,\ldots,n.
  10. x i j x_{ij}
  11. i i
  12. j j
  13. q i j q_{ij}
  14. 1 - p i j 1-p_{ij}
  15. V 1 = 5 V_{1}=5
  16. V 2 = 10 V_{2}=10
  17. V 3 = 20 V_{3}=20
  18. 20 ( 0.6 ) ( 0.5 ) = 6 20(0.6)(0.5)=6
  19. 10 ( 0.4 ) ( 0.8 ) 2 = 2.56 10(0.4)(0.8)^{2}=2.56
  20. 5 ( 0.7 ) 3 = 1.715 5(0.7)^{3}=1.715
  21. 6 + 2.56 + 1.715 = 10.275 6+2.56+1.715=10.275

Web_(differential_geometry).html

  1. 𝒮 = ( 𝒮 1 , , 𝒮 n ) \mathcal{S}=(\mathcal{S}^{1},\dots,\mathcal{S}^{n})
  2. 𝒞 = ( 𝒞 1 , , 𝒞 n ) \mathcal{C}=(\mathcal{C}^{1},\dots,\mathcal{C}^{n})
  3. M = X n r M=X^{nr}
  4. D X n r D\subset X^{nr}

Weber_modular_function.html

  1. q = e 2 π i τ q=e^{2\pi i\tau}
  2. 𝔣 ( τ ) = q - 1 48 n > 0 ( 1 + q n - 1 2 ) = e - π i 24 η ( τ + 1 2 ) η ( τ ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) 𝔣 1 ( τ ) = q - 1 48 n > 0 ( 1 - q n - 1 2 ) = η ( τ 2 ) η ( τ ) 𝔣 2 ( τ ) = 2 q - 1 24 n > 0 ( 1 + q n ) = 2 η ( 2 τ ) η ( τ ) \begin{aligned}\displaystyle\mathfrak{f}(\tau)&\displaystyle=q^{-\frac{1}{48}}% \prod_{n>0}(1+q^{n-\frac{1}{2}})=e^{-\frac{\pi\rm{i}}{24}}\frac{\eta\big(\frac% {\tau+1}{2}\big)}{\eta(\tau)}=\frac{\eta^{2}(\tau)}{\eta\big(\tfrac{\tau}{2}% \big)\eta(2\tau)}\\ \displaystyle\mathfrak{f}_{1}(\tau)&\displaystyle=q^{-\frac{1}{48}}\prod_{n>0}% (1-q^{n-\frac{1}{2}})=\frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)}\\ \displaystyle\mathfrak{f}_{2}(\tau)&\displaystyle=\sqrt{2}\,q^{-\frac{1}{24}}% \prod_{n>0}(1+q^{n})=\frac{\sqrt{2}\,\eta(2\tau)}{\eta(\tau)}\end{aligned}
  3. η ( τ ) \eta(\tau)
  4. 𝔣 ( τ ) 𝔣 1 ( τ ) 𝔣 2 ( τ ) = 2 \mathfrak{f}(\tau)\mathfrak{f}_{1}(\tau)\mathfrak{f}_{2}(\tau)=\sqrt{2}
  5. q = e π i τ q=e^{\pi i\tau}
  6. 𝔣 ( τ ) = θ 3 ( 0 , q ) η ( τ ) 𝔣 1 ( τ ) = θ 4 ( 0 , q ) η ( τ ) 𝔣 2 ( τ ) = θ 2 ( 0 , q ) η ( τ ) \begin{aligned}\displaystyle\mathfrak{f}(\tau)&\displaystyle=\sqrt{\frac{% \theta_{3}(0,q)}{\eta(\tau)}}\\ \displaystyle\mathfrak{f}_{1}(\tau)&\displaystyle=\sqrt{\frac{\theta_{4}(0,q)}% {\eta(\tau)}}\\ \displaystyle\mathfrak{f}_{2}(\tau)&\displaystyle=\sqrt{\frac{\theta_{2}(0,q)}% {\eta(\tau)}}\\ \end{aligned}
  7. 𝔣 1 ( τ ) 8 + 𝔣 2 ( τ ) 8 = 𝔣 ( τ ) 8 \mathfrak{f}_{1}(\tau)^{8}+\mathfrak{f}_{2}(\tau)^{8}=\mathfrak{f}(\tau)^{8}
  8. θ 2 ( 0 , q ) 4 + θ 4 ( 0 , q ) 4 = θ 3 ( 0 , q ) 4 \theta_{2}(0,q)^{4}+\theta_{4}(0,q)^{4}=\theta_{3}(0,q)^{4}
  9. j ( τ ) = ( x + 16 ) 3 x j(\tau)=\frac{(x+16)^{3}}{x}
  10. x i = 𝔣 ( τ ) 24 , 𝔣 1 ( τ ) 24 , 𝔣 2 ( τ ) 24 x_{i}=\mathfrak{f}(\tau)^{24},\mathfrak{f}_{1}(\tau)^{24},\mathfrak{f}_{2}(% \tau)^{24}
  11. j ( τ ) = 32 ( θ 2 ( 0 , q ) 8 + θ 3 ( 0 , q ) 8 + θ 4 ( 0 , q ) 8 ) 3 ( θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) ) 8 j(\tau)=32\frac{\Big(\theta_{2}(0,q)^{8}+\theta_{3}(0,q)^{8}+\theta_{4}(0,q)^{% 8}\Big)^{3}}{\Big(\theta_{2}(0,q)\theta_{3}(0,q)\theta_{4}(0,q)\Big)^{8}}
  12. j ( τ ) = ( 𝔣 ( τ ) 16 + 𝔣 1 ( τ ) 16 + 𝔣 2 ( τ ) 16 2 ) 3 j(\tau)=\left(\frac{\mathfrak{f}(\tau)^{16}+\mathfrak{f}_{1}(\tau)^{16}+% \mathfrak{f}_{2}(\tau)^{16}}{2}\right)^{3}