wpmath0000002_5

Drag_equation.html

  1. F D = 1 2 ρ u 2 C D A F_{D}\,=\,\tfrac{1}{2}\,\rho\,u^{2}\,C_{D}\,A
  2. F D F_{D}
  3. ρ \rho
  4. u u
  5. A A
  6. C D C_{D}
  7. u 2 u^{2}
  8. f a ( F D , u , A , ρ , ν ) = 0. f_{a}(F_{D},\,u,\,A,\,\rho,\,\nu)\,=\,0.\,
  9. Re = u A ν \mathrm{Re}\,=\,\frac{u\,\sqrt{A}}{\nu}
  10. C D = F D 1 2 ρ A u 2 . C_{D}\,=\,\frac{F_{D}}{\frac{1}{2}\,\rho\,A\,u^{2}}.
  11. f b ( F D 1 2 ρ A u 2 , u A ν ) = 0. f_{b}\left(\frac{F_{D}}{\frac{1}{2}\,\rho\,A\,u^{2}},\,\frac{u\,\sqrt{A}}{\nu}% \right)\,=\,0.
  12. F D 1 2 ρ A u 2 = f c ( u A ν ) \frac{F_{D}}{\frac{1}{2}\,\rho\,A\,u^{2}}\,=\,f_{c}\left(\frac{u\,\sqrt{A}}{% \nu}\right)
  13. F D = 1 2 ρ A u 2 f c ( R e ) , F_{D}\,=\,\tfrac{1}{2}\,\rho\,A\,u^{2}\,f_{c}(R_{e}),\,
  14. C D = f c ( R e ) . C_{D}\,=\,f_{c}(R_{e}).

Drift_velocity.html

  1. u = μ E , u=\mu E,
  2. u u
  3. μ μ
  4. E E
  5. u = j n q , u={j\over nq},
  6. u u
  7. j j
  8. n n
  9. q q
  10. u = m σ Δ V ρ e f , u={m\;\sigma\Delta V\over\rho ef\ell},
  11. u u
  12. m m
  13. Δ V ΔV
  14. ρ ρ
  15. e e
  16. f f
  17. σ σ
  18. n n
  19. u \displaystyle u
  20. u = A electron m 3 m 2 C electron = C s 1 m C = m s u=\dfrac{\,\text{A}}{\dfrac{\,\text{electron}}{\,\text{m}^{3}}{\cdot}\,\text{m% }^{2}\cdot\dfrac{\,\text{C}}{\,\text{electron}}}=\dfrac{\dfrac{\,\text{C}}{s}}% {\dfrac{1}{\,\text{m}}{\cdot}\,\text{C}}=\dfrac{\,\text{m}}{\,\text{s}}
  21. F F
  22. A = 1 2 F 2 2 π | u | = 2.1 × 10 - 6 m A=\frac{1}{2}F\frac{2\sqrt{2}}{\pi}|u|=2.1\times 10^{-6}\,\text{ m}

Droop_quota.html

  1. ( Total Valid Poll ( Seats + 1 ) ) + 1 \left(\frac{\mbox{Total Valid Poll}~{}}{\left(\mbox{Seats}~{}+1\right)}\right)+1
  2. 100 2 + 1 + 1 = 34 1 3 \frac{100}{2+1}+1=34\frac{1}{3}

Dual-modulus_prescaler.html

  1. f o N = f r f = N f r \frac{f_{o}}{N}=f_{r}\Rightarrow f=Nf_{r}
  2. f o = 40 N f r f_{o}=40Nf_{r}
  3. f o = f r [ M ( N - A ) + ( M + 1 ) A ] f o = f r ( M N + A ) \begin{aligned}&\displaystyle f_{o}=f_{r}\left[{M(N-A)+(M+1)A}\right]\\ \displaystyle\Rightarrow&\displaystyle f_{o}=f_{r}\left(MN+A\right)\end{aligned}
  4. N \displaystyle N

Dual_(category_theory).html

  1. g f g\circ f
  2. f g f\circ g
  3. f : A B f\colon A\to B
  4. f g = f h f\circ g=f\circ h
  5. g = h g=h
  6. g f = h f g\circ f=h\circ f
  7. g = h . g=h.
  8. f : B A f\colon B\to A

Ductility.html

  1. ε f \varepsilon_{f}
  2. q q

Dummy_variable_(statistics).html

  1. P i = 1 1 + e - z i = e z i 1 + e z i P_{i}=\frac{1}{1+e^{-z_{i}}}\ =\frac{e^{z_{i}}}{1+e^{z_{i}}}
  2. L i = ln ( P i 1 - P i ) = z i = α 1 + α 2 X i . L_{i}=\ln\left(\frac{P_{i}}{1-P_{i}}\right)=z_{i}=\alpha_{1}+\alpha_{2}X_{i}.

DVB-T.html

  1. f s = 8 7 B f_{s}=\frac{8}{7}B
  2. B B

Dynamic_equilibrium.html

  1. c = k p c=kp\,
  2. K c = [ CH 3 CO 2 - ] [ H + ] [ CH 3 CO 2 H ] K_{c}=\mathrm{\frac{[CH_{3}CO_{2}^{-}][H^{+}]}{[CH_{3}CO_{2}H]}}
  3. K P = P ( N 2 O 4 ) P ( NO 2 ) 2 K_{P}=\mathrm{\frac{P(N_{2}O_{4})}{P(NO_{2})^{2}}}
  4. A B A\rightleftharpoons B
  5. d [ A ] d t = - k f [ A ] t + k b [ B ] t \frac{d[A]}{dt}=-k_{f}[A]_{t}+k_{b}[B]_{t}
  6. d [ A ] d t = - k f [ A ] t + k b ( [ A ] 0 - [ A ] t ) \frac{d[A]}{dt}=-k_{f}[A]_{t}+k_{b}\left([A]_{0}-[A]_{t}\right)
  7. [ A ] t = k b + k f e - ( k f + k b ) t k f + k b [ A ] 0 [A]_{t}=\frac{k_{b}+k_{f}e^{-\left(k_{f}+k_{b}\right)t}}{k_{f}+k_{b}}[A]_{0}
  8. [ A ] = k b k f + k b [ A ] 0 ; [ B ] = k f k f + k b [ A ] 0 [A]_{\infty}=\frac{k_{b}}{k_{f}+k_{b}}[A]_{0};[B]_{\infty}=\frac{k_{f}}{k_{f}+% k_{b}}[A]_{0}
  9. t 10 k f + k b t\gtrapprox\frac{10}{k_{f}+k_{b}}
  10. K = [ B ] e q [ A ] e q K=\frac{[B]_{eq}}{[A]_{eq}}
  11. K = k f k f + k b [ A ] 0 k b k f + k b [ A ] 0 = k f k b K=\frac{\frac{k_{f}}{k_{f}+k_{b}}[A]_{0}}{\frac{k_{b}}{k_{f}+k_{b}}[A]_{0}}=% \frac{k_{f}}{k_{b}}
  12. K = ( k f k b ) 1 × ( k f k b ) 2 K=\left(\frac{k_{f}}{k_{b}}\right)_{1}\times\left(\frac{k_{f}}{k_{b}}\right)_{% 2}\dots

Dynamic_mechanical_analysis.html

  1. σ = σ 0 sin ( t ω + δ ) \sigma=\sigma_{0}\sin(t\omega+\delta)\,
  2. ε = ε 0 sin ( t ω ) \varepsilon=\varepsilon_{0}\sin(t\omega)
  3. ω \omega
  4. t t
  5. δ \delta
  6. σ ( t ) = E ϵ ( t ) σ 0 sin ( ω t + δ ) = E ϵ 0 sin ω t δ = 0 \sigma(t)=E\epsilon(t)\implies\sigma_{0}\sin{(\omega t+\delta)}=E\epsilon_{0}% \sin{\omega t}\implies\delta=0
  7. σ ( t ) = K d ϵ d t σ 0 sin ( ω t + δ ) = E ϵ 0 ω cos ω t δ = π 2 \sigma(t)=K\frac{d\epsilon}{dt}\implies\sigma_{0}\sin{(\omega t+\delta)}=E% \epsilon_{0}\omega\cos{\omega t}\implies\delta=\frac{\pi}{2}
  8. E = σ 0 ε 0 cos δ E^{\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\cos\delta
  9. E ′′ = σ 0 ε 0 sin δ E^{\prime\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\sin\delta
  10. δ = arctan E ′′ E \delta=\arctan\frac{E^{\prime\prime}}{E^{\prime}}
  11. G G^{\prime}
  12. G ′′ G^{\prime\prime}
  13. E * E^{*}
  14. G * G^{*}
  15. E * = E + i E ′′ E^{*}=E^{\prime}+iE^{\prime\prime}\,
  16. G * = G + i G ′′ G^{*}=G^{\prime}+iG^{\prime\prime}\,
  17. i 2 = - 1 {i}^{2}=-1\,
  18. tan ( δ ) \tan(\delta)
  19. tan ( δ ) \tan(\delta)
  20. E ′′ = E τ 0 ω τ 0 2 ω 2 + 1 , E^{\prime\prime}=\frac{E\tau_{0}\omega}{\tau_{0}^{2}\omega^{2}+1},
  21. τ 0 = η / E \tau_{0}=\eta/E
  22. 1 / τ 0 1/\tau_{0}

Dynamic_programming.html

  1. t = 0 , 1 , 2 , , T t=0,1,2,\ldots,T
  2. c t c_{t}
  3. t t
  4. u ( c t ) = ln ( c t ) u(c_{t})=\ln(c_{t})
  5. b b
  6. 0 < b < 1 0<b<1
  7. k t k_{t}
  8. t t
  9. k 0 > 0 k_{0}>0
  10. k t + 1 = A k t a - c t k_{t+1}=Ak^{a}_{t}-c_{t}
  11. A A
  12. 0 < a < 1 0<a<1
  13. max t = 0 T b t ln ( c t ) \max\sum_{t=0}^{T}b^{t}\ln(c_{t})
  14. k t + 1 = A k t a - c t 0 k_{t+1}=Ak^{a}_{t}-c_{t}\geq 0
  15. t = 0 , 1 , 2 , , T t=0,1,2,\ldots,T
  16. c 0 , c 1 , c 2 , , c T c_{0},c_{1},c_{2},\ldots,c_{T}
  17. k 0 k_{0}
  18. V t ( k ) V_{t}(k)
  19. t = 0 , 1 , 2 , , T , T + 1 t=0,1,2,\ldots,T,T+1
  20. k k
  21. t t
  22. V T + 1 ( k ) = 0 V_{T+1}(k)=0
  23. t = 0 , 1 , 2 , , T t=0,1,2,\ldots,T
  24. V t ( k t ) = max ( ln ( c t ) + b V t + 1 ( k t + 1 ) ) subject to k t + 1 = A k t a - c t 0 V_{t}(k_{t})\,=\,\max\left(\ln(c_{t})+bV_{t+1}(k_{t+1})\right)\,\text{ subject% to }k_{t+1}=Ak^{a}_{t}-c_{t}\geq 0
  25. c t c_{t}
  26. k t + 1 k_{t+1}
  27. t t
  28. k t k_{t}
  29. c t c_{t}
  30. k t + 1 k_{t+1}
  31. k k
  32. V T + 1 ( k ) V_{T+1}(k)
  33. V T ( k ) V_{T}(k)
  34. V 0 ( k ) V_{0}(k)
  35. V T - j + 1 ( k ) V_{T-j+1}(k)
  36. V T - j ( k ) V_{T-j}(k)
  37. ln ( c T - j ) + b V T - j + 1 ( A k a - c T - j ) \ln(c_{T-j})+bV_{T-j+1}(Ak^{a}-c_{T-j})
  38. c T - j c_{T-j}
  39. A k a - c T - j 0 Ak^{a}-c_{T-j}\geq 0
  40. t = T - j t=T-j
  41. V T - j ( k ) = a i = 0 j a i b i ln k + v T - j V_{T-j}(k)\,=\,a\sum_{i=0}^{j}a^{i}b^{i}\ln k+v_{T-j}
  42. v T - j v_{T-j}
  43. t = T - j t=T-j
  44. c T - j ( k ) = 1 i = 0 j a i b i A k a c_{T-j}(k)\,=\,\frac{1}{\sum_{i=0}^{j}a^{i}b^{i}}Ak^{a}
  45. c T ( k ) = A k a c_{T}(k)\,=\,Ak^{a}
  46. c T - 1 ( k ) = A k a 1 + a b c_{T-1}(k)\,=\,\frac{Ak^{a}}{1+ab}
  47. c T - 2 ( k ) = A k a 1 + a b + a 2 b 2 c_{T-2}(k)\,=\,\frac{Ak^{a}}{1+ab+a^{2}b^{2}}
  48. c 2 ( k ) = A k a 1 + a b + a 2 b 2 + + a T - 2 b T - 2 c_{2}(k)\,=\,\frac{Ak^{a}}{1+ab+a^{2}b^{2}+\ldots+a^{T-2}b^{T-2}}
  49. c 1 ( k ) = A k a 1 + a b + a 2 b 2 + + a T - 2 b T - 2 + a T - 1 b T - 1 c_{1}(k)\,=\,\frac{Ak^{a}}{1+ab+a^{2}b^{2}+\ldots+a^{T-2}b^{T-2}+a^{T-1}b^{T-1}}
  50. c 0 ( k ) = A k a 1 + a b + a 2 b 2 + + a T - 2 b T - 2 + a T - 1 b T - 1 + a T b T c_{0}(k)\,=\,\frac{Ak^{a}}{1+ab+a^{2}b^{2}+\ldots+a^{T-2}b^{T-2}+a^{T-1}b^{T-1% }+a^{T}b^{T}}
  51. T T
  52. Ω ( n 2 ) \Omega(n^{2})
  53. Ω ( n ) \Omega(n)
  54. Ω ( n ) \Omega(n)
  55. Ω ( n ) \Omega(n)
  56. n n
  57. O ( n ( log n ) 2 ) O(n(\log n)^{2})
  58. O ( n log n ) O(n\log n)
  59. n n
  60. [ 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 ] and [ 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 ] and [ 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 ] and [ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 ] . \begin{bmatrix}0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\end{bmatrix}\,\text{ and }\begin{bmatrix}0&0&1&1\\ 0&0&1&1\\ 1&1&0&0\\ 1&1&0&0\end{bmatrix}\,\text{ and }\begin{bmatrix}1&1&0&0\\ 0&0&1&1\\ 1&1&0&0\\ 0&0&1&1\end{bmatrix}\,\text{ and }\begin{bmatrix}1&0&0&1\\ 0&1&1&0\\ 0&1&1&0\\ 1&0&0&1\end{bmatrix}.
  61. ( n n / 2 ) n {\textstyle\left({{n}\atop{n/2}}\right)}^{n}
  62. n = 6 n=6
  63. k k
  64. n / 2 n/2
  65. n / 2 n/2
  66. f ( ( n / 2 , n / 2 ) , ( n / 2 , n / 2 ) , ( n / 2 , n / 2 ) ) f((n/2,n/2),(n/2,n/2),\ldots(n/2,n/2))
  67. n n
  68. n n
  69. ( n n / 2 ) {\textstyle\left({{n}\atop{n/2}}\right)}
  70. ( 0 , 1 ) (0,1)
  71. ( 1 , 0 ) (1,0)
  72. 1 , 2 , 90 , 297200 , 116963796250 , 6736218287430460752 , 1,\,2,\,90,\,297200,\,116963796250,\,6736218287430460752,\ldots
  73. q ( A ) = min ( q ( B ) , q ( C ) , q ( D ) ) + c ( A ) q(A)=\min(q(B),q(C),q(D))+c(A)\,
  74. q ( i , j ) = { j < 1 or j > n c ( i , j ) i = 1 min ( q ( i - 1 , j - 1 ) , q ( i - 1 , j ) , q ( i - 1 , j + 1 ) ) + c ( i , j ) otherwise. q(i,j)=\begin{cases}\infty&j<1\,\text{ or }j>n\\ c(i,j)&i=1\\ \min(q(i-1,j-1),q(i-1,j),q(i-1,j+1))+c(i,j)&\,\text{otherwise.}\end{cases}
  75. O ( n k 2 ) O(nk^{2})
  76. O ( n k log k ) O(nk\log k)
  77. x x
  78. W ( n - 1 , x - 1 ) W(n-1,x-1)
  79. x x
  80. W ( n , k - x ) W(n,k-x)
  81. x x
  82. max ( W ( n - 1 , x - 1 ) , W ( n , k - x ) ) \max(W(n-1,x-1),W(n,k-x))
  83. x x
  84. x x
  85. O ( n k ) O(nk)
  86. k k
  87. k k
  88. k = 37 k=37
  89. m m
  90. f ( t , n ) f(t,n)
  91. m m
  92. t t
  93. n n
  94. f ( t , 0 ) = f ( 0 , n ) = 1 f(t,0)=f(0,n)=1
  95. t , n 0 t,n\geq 0
  96. a a
  97. m m
  98. 1 1
  99. a a
  100. t - 1 t-1
  101. n - 1 n-1
  102. m m
  103. a + 1 a+1
  104. k k
  105. t - 1 t-1
  106. n n
  107. f ( t , n ) = f ( t - 1 , n - 1 ) + f ( t - 1 , n ) f(t,n)=f(t-1,n-1)+f(t-1,n)
  108. x x
  109. f ( x , n ) k f(x,n)\geq k
  110. { f ( t , i ) : 0 i n } \{f(t,i):0\leq i\leq n\}
  111. t t
  112. O ( n x ) O(nx)
  113. n = 1 n=1
  114. O ( n k ) O(n\sqrt{k})
  115. f ( t , n ) = i = 0 n ( t i ) f(t,n)=\sum_{i=0}^{n}{{\left({{t}\atop{i}}\right)}}
  116. O ( n ) O(n)
  117. ( t i + 1 ) = ( t i ) t - i i + 1 {\left({{t}\atop{i+1}}\right)}={\left({{t}\atop{i}}\right)}\frac{t-i}{i+1}
  118. i 0 i\geq 0
  119. f ( t , n ) f ( t + 1 , n ) f(t,n)\leq f(t+1,n)
  120. t 0 t\geq 0
  121. t t
  122. x x
  123. O ( n log k ) O(n\log k)
  124. s i i , j sii,j
  125. m 1 , 33 = 1000 m1,33=1000
  126. s 1 , 33 = 22 s1,33=22

Dynamo_theory.html

  1. 𝐁 t = η 2 𝐁 + × ( 𝐮 × 𝐁 ) \frac{\partial\mathbf{B}}{\partial t}=\eta\nabla^{2}\mathbf{B}+\nabla\times(% \mathbf{u}\times\mathbf{B})
  2. η = 1 / σ μ \eta=1/\sigma\mu
  3. σ \sigma
  4. μ \mu
  5. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  6. 𝐮 = 0 \nabla\cdot\mathbf{u}=0
  7. D 𝐮 D t = - p + ν 2 𝐮 + ρ 𝐠 + 2 𝛀 × 𝐮 + 𝛀 × 𝛀 × 𝐑 + 𝐉 × 𝐁 \frac{D\mathbf{u}}{Dt}=-\nabla p+\nu\nabla^{2}\mathbf{u}+\rho^{\prime}\mathbf{% g}+2\mathbf{\Omega}\times\mathbf{u}+\mathbf{\Omega}\times\mathbf{\Omega}\times% \mathbf{R}+\mathbf{J}\times\mathbf{B}
  8. ν \nu
  9. ρ \rho^{\prime}
  10. ρ = α Δ T \rho^{\prime}=\alpha\Delta T
  11. Ω \Omega
  12. 𝐉 \mathbf{J}
  13. T t = κ 2 T + ϵ \frac{\partial T}{\partial t}=\kappa\nabla^{2}T+\epsilon
  14. κ = k / ρ c p \kappa=k/\rho c_{p}
  15. c p c_{p}
  16. ρ \rho
  17. ϵ \epsilon
  18. R a = g α T D 3 ν κ , E = ν Ω D 2 , P r = ν κ , P m = ν η Ra=\frac{g\alpha TD^{3}}{\nu\kappa},E=\frac{\nu}{\Omega D^{2}},Pr=\frac{\nu}{% \kappa},Pm=\frac{\nu}{\eta}
  19. B = ( ρ Ω / σ ) 1 / 2 B=(\rho\Omega/\sigma)^{1/2}

Dynkin_diagram.html

  1. B n B_{n}
  2. C n C_{n}
  3. B C n . BC_{n}.
  4. B 4 B_{4}
  5. 𝔰 𝔬 2 4 + 1 = 𝔰 𝔬 9 , \mathfrak{so}_{2\cdot 4+1}=\mathfrak{so}_{9},
  6. A 2 A_{2}
  7. 2 π / 3 2\pi/3
  8. 𝔰 𝔩 2 + 1 = 𝔰 𝔩 3 \mathfrak{sl}_{2+1}=\mathfrak{sl}_{3}
  9. S 3 S_{3}
  10. r 1 , r 2 ( r 1 ) 2 = ( r 2 ) 2 = ( r i r j ) 3 = 1 . \left\langle r_{1},r_{2}\mid(r_{1})^{2}=(r_{2})^{2}=(r_{i}r_{j})^{3}=1\right\rangle.
  11. n 1 n\geq 1
  12. A n , A_{n},
  13. n 2 n\geq 2
  14. B n , B_{n},
  15. n 3 n\geq 3
  16. C n , C_{n},
  17. n 4 n\geq 4
  18. D n , D_{n},
  19. E n E_{n}
  20. n = 6. n=6.
  21. n = 0 n=0
  22. n = 1 , n=1,
  23. A 1 B 1 C 1 A_{1}\cong B_{1}\cong C_{1}
  24. B 2 C 2 B_{2}\cong C_{2}
  25. D 2 A 1 × A 1 D_{2}\cong A_{1}\times A_{1}
  26. D 3 A 3 D_{3}\cong A_{3}
  27. E 3 A 1 × A 2 E_{3}\cong A_{1}\times A_{2}
  28. E 4 A 4 E_{4}\cong A_{4}
  29. E 5 D 5 E_{5}\cong D_{5}
  30. n > 1 n>1
  31. n > 1 n>1
  32. i C n \bigwedge^{i}C^{n}
  33. i = 1 , , n i=1,\dots,n
  34. i C n n - i C n . \bigwedge^{i}C^{n}\mapsto\bigwedge^{n-i}C^{n}.
  35. 𝔰 𝔩 n + 1 , \mathfrak{sl}_{n+1},
  36. T - T T T\mapsto-T^{\mathrm{T}}
  37. 𝔰 𝔬 2 n , \mathfrak{so}_{2n},
  38. A 3 D 3 , \mathrm{A}_{3}\cong\mathrm{D}_{3},
  39. D 2 A 1 × A 1 , \mathrm{D}_{2}\cong\mathrm{A}_{1}\times\mathrm{A}_{1},
  40. B 2 C 2 \mathrm{B}_{2}\cong\mathrm{C}_{2}
  41. A 2 n - 1 C n A_{2n-1}\to C_{n}
  42. D n + 1 B n D_{n+1}\to B_{n}
  43. D 4 G 2 D_{4}\to G_{2}
  44. D 4 B 3 D_{4}\to B_{3}
  45. E 6 F 4 E_{6}\to F_{4}
  46. A ~ 2 n - 1 C ~ n \tilde{A}_{2n-1}\to\tilde{C}_{n}
  47. D ~ n + 1 B ~ n \tilde{D}_{n+1}\to\tilde{B}_{n}
  48. D ~ 4 G ~ 2 \tilde{D}_{4}\to\tilde{G}_{2}
  49. E ~ 6 F ~ 4 \tilde{E}_{6}\to\tilde{F}_{4}
  50. D 5 E 5 \mathrm{D}_{5}\cong\mathrm{E}_{5}
  51. A n , D n , E n A_{n},D_{n},E_{n}
  52. A = [ 2 a 12 a 21 2 ] A=\left[\begin{matrix}2&a_{12}\\ a_{21}&2\end{matrix}\right]
  53. A = ( a i j ) A=(a_{ij})
  54. a i i = 2 a_{ii}=2
  55. a i j 0 a_{ij}\leq 0
  56. a i j = 0 a_{ij}=0
  57. a j i = 0 a_{ji}=0
  58. [ 2 a 12 a 21 2 ] \left[\begin{matrix}2&a_{12}\\ a_{21}&2\end{matrix}\right]
  59. [ 2 0 0 2 ] \left[\begin{smallmatrix}2&0\\ 0&2\end{smallmatrix}\right]
  60. [ 2 - 1 - 1 2 ] \left[\begin{smallmatrix}2&-1\\ -1&2\end{smallmatrix}\right]
  61. [ 2 - 2 - 1 2 ] \left[\begin{smallmatrix}2&-2\\ -1&2\end{smallmatrix}\right]
  62. A 3 {A}_{3}
  63. [ 2 - 1 - 2 2 ] \left[\begin{smallmatrix}2&-1\\ -2&2\end{smallmatrix}\right]
  64. A 3 {A}_{3}
  65. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-\sqrt{2}\\ -\sqrt{2}&2\end{smallmatrix}\right]
  66. [ 2 - 1 - 3 2 ] \left[\begin{smallmatrix}2&-1\\ -3&2\end{smallmatrix}\right]
  67. D 4 {D}_{4}
  68. [ 2 - 3 - 3 2 ] \left[\begin{smallmatrix}2&-\sqrt{3}\\ -\sqrt{3}&2\end{smallmatrix}\right]
  69. [ 2 - 2 - 2 2 ] \left[\begin{smallmatrix}2&-2\\ -2&2\end{smallmatrix}\right]
  70. A ~ 3 {\tilde{A}}_{3}
  71. [ 2 - 1 - 4 2 ] \left[\begin{smallmatrix}2&-1\\ -4&2\end{smallmatrix}\right]
  72. D ~ 4 {\tilde{D}}_{4}
  73. [ 2 - 1 - 5 2 ] \left[\begin{smallmatrix}2&-1\\ -5&2\end{smallmatrix}\right]
  74. [ 2 - 2 - 3 2 ] \left[\begin{smallmatrix}2&-2\\ -3&2\end{smallmatrix}\right]
  75. [ 2 - 1 - 6 2 ] \left[\begin{smallmatrix}2&-1\\ -6&2\end{smallmatrix}\right]
  76. [ 2 - 1 - 7 2 ] \left[\begin{smallmatrix}2&-1\\ -7&2\end{smallmatrix}\right]
  77. [ 2 - 2 - 4 2 ] \left[\begin{smallmatrix}2&-2\\ -4&2\end{smallmatrix}\right]
  78. [ 2 - 1 - 8 2 ] \left[\begin{smallmatrix}2&-1\\ -8&2\end{smallmatrix}\right]
  79. [ 2 - 3 - 3 2 ] \left[\begin{smallmatrix}2&-3\\ -3&2\end{smallmatrix}\right]
  80. [ 2 - b - a 2 ] \left[\begin{smallmatrix}2&-b\\ -a&2\end{smallmatrix}\right]
  81. A 1 + {A}_{1+}
  82. B 2 + {B}_{2+}
  83. C 2 + {C}_{2+}
  84. D 2 + {D}_{2+}
  85. E 3 - 8 {E}_{3-8}
  86. G 2 {G}_{2}
  87. F 4 {F}_{4}
  88. X l ( 1 ) , X l ( 2 ) , X_{l}^{(1)},X_{l}^{(2)},
  89. X l ( 3 ) , X_{l}^{(3)},
  90. X l ( 1 ) , X_{l}^{(1)},
  91. A ~ 5 = A 5 ( 1 ) = A 5 + \tilde{A}_{5}=A_{5}^{(1)}=A_{5}^{+}
  92. n 3 n\geq 3
  93. B n B_{n}
  94. n 4 n\geq 4
  95. D n D_{n}
  96. A ~ 1 + {\tilde{A}}_{1+}
  97. B ~ 3 + {\tilde{B}}_{3+}
  98. C ~ 2 + {\tilde{C}}_{2+}
  99. D ~ 4 + {\tilde{D}}_{4+}
  100. A ~ 1 {\tilde{A}}_{1}
  101. A 1 ( 1 ) {A}_{1}^{(1)}
  102. A 2 ( 2 ) {A}_{2}^{(2)}
  103. A ~ 2 {\tilde{A}}_{2}
  104. A 2 ( 1 ) {A}_{2}^{(1)}
  105. C ~ 2 {\tilde{C}}_{2}
  106. C 2 ( 1 ) {C}_{2}^{(1)}
  107. D 5 ( 2 ) {D}_{5}^{(2)}
  108. A 4 ( 2 ) {A}_{4}^{(2)}
  109. G ~ 2 {\tilde{G}}_{2}
  110. G 2 ( 1 ) {G}_{2}^{(1)}
  111. D 4 ( 3 ) {D}_{4}^{(3)}
  112. A ~ 3 {\tilde{A}}_{3}
  113. A 3 ( 1 ) {A}_{3}^{(1)}
  114. B ~ 3 {\tilde{B}}_{3}
  115. B 3 ( 1 ) {B}_{3}^{(1)}
  116. A 5 ( 2 ) {A}_{5}^{(2)}
  117. C ~ 3 {\tilde{C}}_{3}
  118. C 3 ( 1 ) {C}_{3}^{(1)}
  119. D 6 ( 2 ) {D}_{6}^{(2)}
  120. A 6 ( 2 ) {A}_{6}^{(2)}
  121. A ~ 4 {\tilde{A}}_{4}
  122. A 4 ( 1 ) {A}_{4}^{(1)}
  123. B ~ 4 {\tilde{B}}_{4}
  124. B 4 ( 1 ) {B}_{4}^{(1)}
  125. A 7 ( 2 ) {A}_{7}^{(2)}
  126. C ~ 4 {\tilde{C}}_{4}
  127. C 4 ( 1 ) {C}_{4}^{(1)}
  128. D 7 ( 2 ) {D}_{7}^{(2)}
  129. A 8 ( 2 ) {A}_{8}^{(2)}
  130. D ~ 4 {\tilde{D}}_{4}
  131. D 4 ( 1 ) {D}_{4}^{(1)}
  132. F ~ 4 {\tilde{F}}_{4}
  133. F 4 ( 1 ) {F}_{4}^{(1)}
  134. E 6 ( 2 ) {E}_{6}^{(2)}
  135. A ~ 5 {\tilde{A}}_{5}
  136. A 5 ( 1 ) {A}_{5}^{(1)}
  137. B ~ 5 {\tilde{B}}_{5}
  138. B 5 ( 1 ) {B}_{5}^{(1)}
  139. A 9 ( 2 ) {A}_{9}^{(2)}
  140. C ~ 5 {\tilde{C}}_{5}
  141. C 5 ( 1 ) {C}_{5}^{(1)}
  142. D 8 ( 2 ) {D}_{8}^{(2)}
  143. A 10 ( 2 ) {A}_{10}^{(2)}
  144. D ~ 5 {\tilde{D}}_{5}
  145. D 5 ( 1 ) {D}_{5}^{(1)}
  146. A ~ 6 {\tilde{A}}_{6}
  147. A 6 ( 1 ) {A}_{6}^{(1)}
  148. B ~ 6 {\tilde{B}}_{6}
  149. B 6 ( 1 ) {B}_{6}^{(1)}
  150. A 11 ( 2 ) {A}_{11}^{(2)}
  151. C ~ 6 {\tilde{C}}_{6}
  152. C 6 ( 1 ) {C}_{6}^{(1)}
  153. D 9 ( 2 ) {D}_{9}^{(2)}
  154. A 12 ( 2 ) {A}_{12}^{(2)}
  155. D ~ 6 {\tilde{D}}_{6}
  156. D 6 ( 1 ) {D}_{6}^{(1)}
  157. E ~ 6 {\tilde{E}}_{6}
  158. E 6 ( 1 ) {E}_{6}^{(1)}
  159. A ~ 7 {\tilde{A}}_{7}
  160. A 7 ( 1 ) {A}_{7}^{(1)}
  161. B ~ 7 {\tilde{B}}_{7}
  162. B 7 ( 1 ) {B}_{7}^{(1)}
  163. A 13 ( 2 ) {A}_{13}^{(2)}
  164. C ~ 7 {\tilde{C}}_{7}
  165. C 7 ( 1 ) {C}_{7}^{(1)}
  166. D 10 ( 2 ) {D}_{10}^{(2)}
  167. A 14 ( 2 ) {A}_{14}^{(2)}
  168. D ~ 7 {\tilde{D}}_{7}
  169. D 7 ( 1 ) {D}_{7}^{(1)}
  170. E ~ 7 {\tilde{E}}_{7}
  171. E 7 ( 1 ) {E}_{7}^{(1)}
  172. A ~ 8 {\tilde{A}}_{8}
  173. A 8 ( 1 ) {A}_{8}^{(1)}
  174. B ~ 8 {\tilde{B}}_{8}
  175. B 8 ( 1 ) {B}_{8}^{(1)}
  176. A 15 ( 2 ) {A}_{15}^{(2)}
  177. C ~ 8 {\tilde{C}}_{8}
  178. C 8 ( 1 ) {C}_{8}^{(1)}
  179. D 11 ( 2 ) {D}_{11}^{(2)}
  180. A 16 ( 2 ) {A}_{16}^{(2)}
  181. D ~ 8 {\tilde{D}}_{8}
  182. D 8 ( 1 ) {D}_{8}^{(1)}
  183. E ~ 8 {\tilde{E}}_{8}
  184. E 8 ( 1 ) {E}_{8}^{(1)}
  185. A ~ 9 {\tilde{A}}_{9}
  186. A 9 ( 1 ) {A}_{9}^{(1)}
  187. B ~ 9 {\tilde{B}}_{9}
  188. B 9 ( 1 ) {B}_{9}^{(1)}
  189. A 17 ( 2 ) {A}_{17}^{(2)}
  190. C ~ 9 {\tilde{C}}_{9}
  191. C 9 ( 1 ) {C}_{9}^{(1)}
  192. D 12 ( 2 ) {D}_{12}^{(2)}
  193. A 18 ( 2 ) {A}_{18}^{(2)}
  194. D ~ 9 {\tilde{D}}_{9}
  195. D 9 ( 1 ) {D}_{9}^{(1)}
  196. A E n {AE}_{n}
  197. B E n {BE}_{n}
  198. C E n {CE}_{n}
  199. D E n {DE}_{n}
  200. A E 3 {AE}_{3}
  201. A E 4 {AE}_{4}
  202. A E 5 {AE}_{5}
  203. B E 5 {BE}_{5}
  204. C E 5 {CE}_{5}
  205. A E 6 {AE}_{6}
  206. B E 6 {BE}_{6}
  207. C E 6 {CE}_{6}
  208. D E 6 {DE}_{6}
  209. A E 7 {AE}_{7}
  210. B E 7 {BE}_{7}
  211. C E 7 {CE}_{7}
  212. D E 7 {DE}_{7}
  213. A E 8 {AE}_{8}
  214. B E 8 {BE}_{8}
  215. C E 8 {CE}_{8}
  216. D E 8 {DE}_{8}
  217. A E 9 {AE}_{9}
  218. B E 9 {BE}_{9}
  219. C E 9 {CE}_{9}
  220. D E 9 {DE}_{9}
  221. B E 10 {BE}_{10}
  222. C E 10 {CE}_{10}
  223. D E 10 {DE}_{10}
  224. E 10 {E}_{10}
  225. A 2 A_{2}
  226. C 2 C_{2}
  227. G 2 G_{2}
  228. A ~ 2 {\tilde{A}}_{2}
  229. C ~ 2 {\tilde{C}}_{2}
  230. G ~ 2 {\tilde{G}}_{2}
  231. A 3 A_{3}
  232. B 3 B_{3}
  233. C 3 C_{3}
  234. A 4 A_{4}
  235. B 4 B_{4}
  236. C 4 C_{4}
  237. D 4 D_{4}
  238. F 4 F_{4}
  239. A ~ 3 {\tilde{A}}_{3}
  240. B ~ 3 {\tilde{B}}_{3}
  241. C ~ 3 {\tilde{C}}_{3}
  242. A ~ 4 {\tilde{A}}_{4}
  243. B ~ 4 {\tilde{B}}_{4}
  244. C ~ 4 {\tilde{C}}_{4}
  245. D ~ 4 {\tilde{D}}_{4}
  246. F ~ 4 {\tilde{F}}_{4}
  247. A 5 A_{5}
  248. B 5 B_{5}
  249. D 5 D_{5}
  250. A 6 A_{6}
  251. B 6 B_{6}
  252. D 6 D_{6}
  253. E 6 E_{6}
  254. A ~ 5 {\tilde{A}}_{5}
  255. B ~ 5 {\tilde{B}}_{5}
  256. D ~ 5 {\tilde{D}}_{5}
  257. A ~ 6 {\tilde{A}}_{6}
  258. B ~ 6 {\tilde{B}}_{6}
  259. D ~ 6 {\tilde{D}}_{6}
  260. E ~ 6 {\tilde{E}}_{6}
  261. A ~ 7 {\tilde{A}}_{7}
  262. B ~ 7 {\tilde{B}}_{7}
  263. D ~ 7 {\tilde{D}}_{7}
  264. E ~ 7 {\tilde{E}}_{7}
  265. E ~ 8 {\tilde{E}}_{8}

E6_(mathematics).html

  1. 𝔢 6 \mathfrak{e}_{6}
  2. ( 3 ; 3 ; 3 ) ({3};{3};{3})
  3. 3 {3}
  4. ( 2 3 , - 1 3 , - 1 3 ) , ( - 1 3 , 2 3 , - 1 3 ) , ( - 1 3 , - 1 3 , 2 3 ) \left(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right),\ \left(-\frac{1}{3},\frac{% 2}{3},-\frac{1}{3}\right),\ \left(-\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right)
  5. ( 3 ¯ ; 3 ¯ ; 3 ¯ ) (\bar{{3}};\bar{{3}};\bar{{3}})
  6. 3 ¯ \bar{{3}}
  7. ( - 2 3 , 1 3 , 1 3 ) , ( 1 3 , - 2 3 , 1 3 ) , ( 1 3 , 1 3 , - 2 3 ) \left(-\frac{2}{3},\frac{1}{3},\frac{1}{3}\right),\ \left(\frac{1}{3},-\frac{2% }{3},\frac{1}{3}\right),\ \left(\frac{1}{3},\frac{1}{3},-\frac{2}{3}\right)
  8. ( 1 3 , - 2 3 , 1 3 ; - 2 3 , 1 3 , 1 3 ; - 2 3 , 1 3 , 1 3 ) \left(\frac{1}{3},-\frac{2}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3% };-\frac{2}{3},\frac{1}{3},\frac{1}{3}\right)
  9. 4 × ( 5 2 ) 4\times\begin{pmatrix}5\\ 2\end{pmatrix}
  10. ( ± 1 , ± 1 , 0 , 0 , 0 , 0 ) (\pm 1,\pm 1,0,0,0,0)
  11. ( ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 1 2 , ± 3 2 ) . \left(\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},% \pm{\sqrt{3}\over 2}\right).
  12. 4 × ( 5 2 ) 4\times\begin{pmatrix}5\\ 2\end{pmatrix}
  13. spin ( 10 ) \operatorname{spin}(10)
  14. [ 1 - 1 0 0 0 0 0 1 - 1 0 0 0 0 0 1 - 1 0 0 0 0 0 1 1 0 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 3 2 0 0 0 1 - 1 0 ] \left[\begin{smallmatrix}1&-1&0&0&0&0\\ 0&1&-1&0&0&0\\ 0&0&1&-1&0&0\\ 0&0&0&1&1&0\\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{\sqrt{3% }}{2}\\ 0&0&0&1&-1&0\\ \end{smallmatrix}\right]
  15. [ 2 - 1 0 0 0 0 - 1 2 - 1 0 0 0 0 - 1 2 - 1 0 - 1 0 0 - 1 2 - 1 0 0 0 0 - 1 2 0 0 0 - 1 0 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&-1\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&0\\ 0&0&-1&0&0&2\end{smallmatrix}\right]
  16. E ~ 6 ( q ) \tilde{E}_{6}(q)
  17. | E 6 ( q ) | = 1 gcd ( 3 , q - 1 ) q 36 ( q 12 - 1 ) ( q 9 - 1 ) ( q 8 - 1 ) ( q 6 - 1 ) ( q 5 - 1 ) ( q 2 - 1 ) |E_{6}(q)|=\frac{1}{\mathrm{gcd}(3,q-1)}q^{36}(q^{12}-1)(q^{9}-1)(q^{8}-1)(q^{% 6}-1)(q^{5}-1)(q^{2}-1)
  18. | E 6 2 ( q ) | = 1 gcd ( 3 , q + 1 ) q 36 ( q 12 - 1 ) ( q 9 + 1 ) ( q 8 - 1 ) ( q 6 - 1 ) ( q 5 + 1 ) ( q 2 - 1 ) |{}^{2}\!E_{6}(q)|=\frac{1}{\mathrm{gcd}(3,q+1)}q^{36}(q^{12}-1)(q^{9}+1)(q^{8% }-1)(q^{6}-1)(q^{5}+1)(q^{2}-1)
  19. 16 ¯ \bar{16}
  20. 78 45 0 16 - 3 16 ¯ 3 + 1 0 . 78\rightarrow 45_{0}\oplus 16_{-3}\oplus\bar{16}_{3}+1_{0}.

Early_stopping.html

  1. X X
  2. Y Y
  3. ρ \rho
  4. Z = X × Y Z=X\times Y
  5. f ρ f_{\rho}
  6. f ρ ( x ) = Y y d ρ ( y | x ) , x X f_{\rho}(x)=\int_{Y}yd\rho(y|x),x\in X
  7. ρ ( y | x ) \rho(y|x)
  8. x x
  9. ρ \rho
  10. X n X\subseteq\mathbb{R}^{n}
  11. Y = Y=\mathbb{R}
  12. 𝐳 = { ( x i , y i ) X × Y : i = 1 , , m } Z m \mathbf{z}=\left\{(x_{i},y_{i})\in X\times Y:i=1,\dots,m\right\}\in Z^{m}
  13. ρ \rho
  14. ( f ) = X × Y ( f ( x ) - y ) 2 d ρ \mathcal{E}(f)=\int_{X\times Y}\left(f(x)-y\right)^{2}d\rho
  15. f f
  16. \mathcal{H}
  17. \mathcal{E}
  18. ρ \rho
  19. 𝐳 ( f ) = 1 m i = 1 m ( f ( x i ) - y i ) 2 . \mathcal{E}_{\mathbf{z}}(f)=\frac{1}{m}\sum_{i=1}^{m}\left(f(x_{i})-y_{i}% \right)^{2}.
  20. f t f_{t}
  21. f t 𝐳 f_{t}^{\mathbf{z}}
  22. γ t \gamma_{t}
  23. f t f_{t}
  24. f ρ f_{\rho}
  25. f t 𝐳 f_{t}^{\mathbf{z}}
  26. ( f t 𝐳 ) - ( f ρ ) \mathcal{E}(f_{t}^{\mathbf{z}})-\mathcal{E}(f_{\rho})
  27. ( f t 𝐳 ) - ( f ρ ) = [ ( f t 𝐳 ) - ( f t ) ] + [ ( f t ) - ( f ρ ) ] \mathcal{E}(f_{t}^{\mathbf{z}})-\mathcal{E}(f_{\rho})=\left[\mathcal{E}(f_{t}^% {\mathbf{z}})-\mathcal{E}(f_{t})\right]+\left[\mathcal{E}(f_{t})-\mathcal{E}(f% _{\rho})\right]
  28. 2 {}_{2}
  29. L 2 L_{2}
  30. 2 {}_{2}

Earth's_magnetic_field.html

  1. 𝐁 t = η 2 𝐁 + × ( 𝐮 × 𝐁 ) \frac{\partial\mathbf{B}}{\partial t}=\eta\nabla^{2}\mathbf{B}+\nabla\times(% \mathbf{u}\times\mathbf{B})
  2. 𝐮 \mathbf{u}
  3. 𝐁 \mathbf{B}
  4. η = 1 / σ μ η=1/σμ
  5. σ σ
  6. μ μ
  7. 𝐁 / t ∂\mathbf{B}/∂t
  8. < s u p > 2 ∇<sup>2

Earth_radius.html

  1. R R_{\oplus}
  2. a a
  3. b b
  4. a q aq
  5. q q
  6. q = a 3 ω 2 G M q=\frac{a^{3}\omega^{2}}{GM}\,\!
  7. ω \omega
  8. G G
  9. M M
  10. a a
  11. b b
  12. φ \varphi\,\!
  13. R = R ( φ ) = ( a 2 cos φ ) 2 + ( b 2 sin φ ) 2 ( a cos φ ) 2 + ( b sin φ ) 2 R=R(\varphi)=\sqrt{\frac{(a^{2}\cos\varphi)^{2}+(b^{2}\sin\varphi)^{2}}{(a\cos% \varphi)^{2}+(b\sin\varphi)^{2}}}\,\!
  14. a a
  15. b b
  16. φ \varphi\,\!
  17. M = M ( φ ) = ( a b ) 2 ( ( a cos φ ) 2 + ( b sin φ ) 2 ) 3 / 2 M=M(\varphi)=\frac{(ab)^{2}}{((a\cos\varphi)^{2}+(b\sin\varphi)^{2})^{3/2}}\,\!
  18. φ \varphi\,\!
  19. N = N ( φ ) = a 2 ( a cos φ ) 2 + ( b sin φ ) 2 N=N(\varphi)=\frac{a^{2}}{\sqrt{(a\cos\varphi)^{2}+(b\sin\varphi)^{2}}}\,\!
  20. b 2 a \frac{b^{2}}{a}\,\!
  21. a 2 b \frac{a^{2}}{b}\,\!
  22. φ \varphi\,\!
  23. R a = M N = a 2 b ( a cos φ ) 2 + ( b sin φ ) 2 R_{a}=\sqrt{MN}=\frac{a^{2}b}{(a\cos\varphi)^{2}+(b\sin\varphi)^{2}}\,\!
  24. α \alpha\,\!
  25. φ \varphi\,\!
  26. R c = 1 cos 2 α M + sin 2 α N R_{c}=\frac{{}_{1}}{\frac{\cos^{2}\alpha}{M}+\frac{\sin^{2}\alpha}{N}}\,\!
  27. φ \varphi\,\!
  28. R m = 2 1 M + 1 N R_{m}=\frac{{}_{2}}{\frac{1}{M}+\frac{1}{N}}\,\!
  29. a = \textstyle a=
  30. b = \textstyle b=
  31. R 1 R_{1}
  32. R 1 = 2 a + b 3 R_{1}=\frac{2a+b}{3}\,\!
  33. R 2 R_{2}
  34. R 2 = a 2 + a b 2 a 2 - b 2 ln ( a + a 2 - b 2 b ) 2 = a 2 2 + b 2 2 tanh - 1 e e = A 4 π R_{2}=\sqrt{\frac{a^{2}+\frac{ab^{2}}{\sqrt{a^{2}-b^{2}}}\ln{\left(\frac{a+% \sqrt{a^{2}-b^{2}}}{b}\right)}}{2}}=\sqrt{\frac{a^{2}}{2}+\frac{b^{2}}{2}\frac% {\tanh^{-1}e}{e}}=\sqrt{\frac{A}{4\pi}}\,\!
  35. e 2 = ( a 2 - b 2 ) / a 2 e^{2}=(a^{2}-b^{2})/a^{2}
  36. A A
  37. R 3 R_{3}
  38. R 3 = a 2 b 3 R_{3}=\sqrt[3]{a^{2}b}\,\!
  39. M r = 2 π 0 π 2 a 2 cos 2 φ + b 2 sin 2 φ d φ M_{r}=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\sqrt{{a^{2}}\cos^{2}\varphi+{b^{2}% }\sin^{2}\varphi}\,d\varphi
  40. M r = 2 π 0 π 2 M ( φ ) d φ M_{r}=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\!M(\varphi)\,d\varphi\!
  41. M r [ a 3 / 2 + b 3 / 2 2 ] 2 / 3 M_{r}\approx\left[\frac{a^{3/2}+b^{3/2}}{2}\right]^{2/3}\,
  42. M r a 2 + b 2 2 M_{r}\approx\sqrt{\frac{a^{2}+b^{2}}{2}}\,\!
  43. M r a + b 2 M_{r}\approx\frac{a+b}{2}\,\!

Economic_equilibrium.html

  1. Q s \displaystyle Q_{s}

Economic_Value_Added.html

  1. 𝐸𝑉𝐴 = ( r - c ) K = 𝑁𝑂𝑃𝐴𝑇 - c K \mathit{EVA}\ =\ (r-c)\cdot K\ =\ \mathit{NOPAT}-c\cdot K
  2. r = 𝑁𝑂𝑃𝐴𝑇 K r={\mathit{NOPAT}\over K}
  3. c c\,
  4. K K\,
  5. M V A = V - K 0 = t = 1 E V A t ( 1 + c ) t MVA=V-K_{0}=\sum_{t=1}^{\infty}{EVA_{t}\over(1+c)^{t}}

Eddington_luminosity.html

  1. d u d t = - p ρ - Φ = 0 \frac{du}{dt}=-\frac{\nabla p}{\rho}-\nabla\Phi=0
  2. u u
  3. p p
  4. ρ \rho
  5. Φ \Phi
  6. F rad F_{\rm rad}
  7. - p ρ = κ c F rad . -\frac{\nabla p}{\rho}=\frac{\kappa}{c}F_{\rm rad}\,.
  8. κ \kappa
  9. κ = σ T / m p \kappa=\sigma_{\rm T}/m_{\rm p}
  10. σ T \sigma_{\rm T}
  11. m p m_{\rm p}
  12. S S
  13. L = S F rad d S = S c κ Φ d S . L=\int_{S}F_{\rm rad}\cdot dS=\int_{S}\frac{c}{\kappa}\nabla\Phi\cdot dS\,.
  14. L = c κ S Φ d S = c κ V 2 Φ d V = 4 π G c κ V ρ d V = 4 π G M c κ L=\frac{c}{\kappa}\int_{S}\nabla\Phi\cdot dS=\frac{c}{\kappa}\int_{V}\nabla^{2% }\Phi\,dV=\frac{4\pi Gc}{\kappa}\int_{V}\rho\,dV=\frac{4\pi GMc}{\kappa}
  15. M M
  16. L Edd = 4 π G M m p c σ T 1.26 × 10 31 ( M M ) W = 3.2 × 10 4 ( M M ) L \begin{aligned}\displaystyle L_{\rm Edd}&\displaystyle=\frac{4\pi GMm_{\rm p}c% }{\sigma_{\rm T}}\\ &\displaystyle\cong 1.26\times 10^{31}\left(\frac{M}{M_{\bigodot}}\right){\rm W% }=3.2\times 10^{4}\left(\frac{M}{M_{\bigodot}}\right)L_{\bigodot}\end{aligned}
  17. M M
  18. L L

Eddington–Finkelstein_coordinates.html

  1. ( t , r , θ , ϕ ) (t,r,\theta,\phi)
  2. d s 2 = - ( 1 - 2 G M r ) d t 2 + ( 1 - 2 G M r ) - 1 d r 2 + r 2 d Ω 2 ds^{2}=-\left(1-\frac{2GM}{r}\right)dt^{2}+\left(1-\frac{2GM}{r}\right)^{-1}dr% ^{2}+r^{2}d\Omega^{2}
  3. d Ω 2 d θ 2 + sin 2 θ d ϕ 2 . d\Omega^{2}\equiv d\theta^{2}+\sin^{2}\theta\,d\phi^{2}.
  4. G = 6.67 × 10 - 11 m 3 kg - 1 s - 2 = 2.48 × 10 - 36 kg - 1 s G=6.67\times 10^{-11}\ \rm{m}^{3}\ \rm{kg}^{-1}\ \rm{s}^{-2}=2.48\times 10^{-3% 6}\ \rm{kg}^{-1}\ \rm{s}
  5. r * r^{*}
  6. r * = r + 2 G M ln | r 2 G M - 1 | . r^{*}=r+2GM\ln\left|\frac{r}{2GM}-1\right|.
  7. d r * d r = ( 1 - 2 G M r ) - 1 . \frac{dr^{*}}{dr}=\left(1-\frac{2GM}{r}\right)^{-1}.
  8. v = t + r * v=t+r^{*}
  9. d s 2 = - ( 1 - 2 G M r ) d v 2 + 2 d v d r + r 2 d Ω 2 . ds^{2}=-\left(1-\frac{2GM}{r}\right)dv^{2}+2dvdr+r^{2}d\Omega^{2}.
  10. d Ω 2 = d θ 2 + sin ( θ ) 2 d ϕ 2 d\Omega^{2}=d\theta^{2}+\sin(\theta)^{2}d\phi^{2}
  11. u = t - r * u=t-r^{*}
  12. d s 2 = - ( 1 - 2 G M r ) d u 2 - 2 d u d r + r 2 d Ω 2 . ds^{2}=-\left(1-\frac{2GM}{r}\right)du^{2}-2dudr+r^{2}d\Omega^{2}.
  13. v - 2 r * v-2r^{*}
  14. u - 2 r * u-2r^{*}
  15. t = t ± ( r * - r ) t^{\prime}=t\pm(r^{*}-r)\,
  16. t t^{\prime}
  17. d s 2 = - ( 1 - 2 G M r ) d t 2 ± 4 G M r d t d r + ( 1 + 2 G M r ) d r 2 + r 2 d Ω 2 = ( - d t 2 + d r 2 + r 2 d Ω 2 ) + 2 G M r ( d t ± d r ) 2 ds^{2}=-\left(1-\frac{2GM}{r}\right)dt^{\prime 2}\pm\frac{4GM}{r}dt^{\prime}dr% +\left(1+\frac{2GM}{r}\right)dr^{2}+r^{2}d\Omega^{2}=(-dt^{\prime 2}+dr^{2}+r^% {2}d\Omega^{2})+\frac{2GM}{r}(dt^{\prime}\pm dr)^{2}
  18. r ( τ ) = 2 G M τ r(\tau)=\sqrt{2GM\tau}
  19. v ( τ ) = r ( τ ) r ( τ ) - 2 G M d τ v(\tau)=\int\frac{r(\tau)}{r(\tau)-2GM}d\tau
  20. = C + τ + 2 2 G M τ + 4 G M ln ( τ 2 G M - 1 ) ~{}~{}~{}=C+\tau+2\sqrt{2GM\tau}+4GM\ln\left(\sqrt{\frac{\tau}{2GM}}-1\right)

Edmonds–Karp_algorithm.html

  1. f / c f/c
  2. f f
  3. c c
  4. u u
  5. v v
  6. c f ( u , v ) = c ( u , v ) - f ( u , v ) c_{f}(u,v)=c(u,v)-f(u,v)
  7. u u
  8. v v
  9. min ( c f ( A , D ) , c f ( D , E ) , c f ( E , G ) ) = \min(c_{f}(A,D),c_{f}(D,E),c_{f}(E,G))=
  10. min ( 3 - 0 , 2 - 0 , 1 - 0 ) = \min(3-0,2-0,1-0)=
  11. min ( 3 , 2 , 1 ) = 1 \min(3,2,1)=1
  12. A , D , E , G A,D,E,G
  13. min ( c f ( A , D ) , c f ( D , F ) , c f ( F , G ) ) = \min(c_{f}(A,D),c_{f}(D,F),c_{f}(F,G))=
  14. min ( 3 - 1 , 6 - 0 , 9 - 0 ) = \min(3-1,6-0,9-0)=
  15. min ( 2 , 6 , 9 ) = 2 \min(2,6,9)=2
  16. A , D , F , G A,D,F,G
  17. min ( c f ( A , B ) , c f ( B , C ) , c f ( C , D ) , c f ( D , F ) , c f ( F , G ) ) = \min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,D),c_{f}(D,F),c_{f}(F,G))=
  18. min ( 3 - 0 , 4 - 0 , 1 - 0 , 6 - 2 , 9 - 2 ) = \min(3-0,4-0,1-0,6-2,9-2)=
  19. min ( 3 , 4 , 1 , 4 , 7 ) = 1 \min(3,4,1,4,7)=1
  20. A , B , C , D , F , G A,B,C,D,F,G
  21. min ( c f ( A , B ) , c f ( B , C ) , c f ( C , E ) , c f ( E , D ) , c f ( D , F ) , c f ( F , G ) ) = \min(c_{f}(A,B),c_{f}(B,C),c_{f}(C,E),c_{f}(E,D),c_{f}(D,F),c_{f}(F,G))=
  22. min ( 3 - 1 , 4 - 1 , 2 - 0 , 0 - ( - 1 ) , 6 - 3 , 9 - 3 ) = \min(3-1,4-1,2-0,0-(-1),6-3,9-3)=
  23. min ( 2 , 3 , 2 , 1 , 3 , 6 ) = 1 \min(2,3,2,1,3,6)=1
  24. A , B , C , E , D , F , G A,B,C,E,D,F,G
  25. { A , B , C , E } \{A,B,C,E\}
  26. { D , F , G } \{D,F,G\}
  27. c ( A , D ) + c ( C , D ) + c ( E , G ) = 3 + 1 + 1 = 5. c(A,D)+c(C,D)+c(E,G)=3+1+1=5.

Edmund_Landau.html

  1. k = 1 μ ( k ) k = 0 \scriptstyle\sum\limits_{k=1}^{\infty}\frac{\mu(k)}{k}\,=\,0

Edwin_McMillan.html

  1. U 92 238 + 0 1 n 92 239 U β - 23 min 93 239 Np β - 2.355 days 94 239 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

Effective_mass_(solid-state_physics).html

  1. E ( 𝐤 ) E(\mathbf{k})
  2. E ( 𝐤 ) = E 0 + 2 𝐤 2 2 m * E(\mathbf{k})=E_{0}+\frac{\hbar^{2}\mathbf{k}^{2}}{2m^{*}}
  3. E ( 𝐤 ) E(\mathbf{k})
  4. 𝐤 \mathbf{k}
  5. e {}_{e}
  6. e {}_{e}
  7. E ( 𝐤 ) = E 0 + 2 2 m x * ( k x - k 0 , x ) 2 + 2 2 m y * ( k y - k 0 , y ) 2 + 2 2 m z * ( k z - k 0 , z ) 2 E(\mathbf{k})=E_{0}+\frac{\hbar^{2}}{2m_{x}^{*}}(k_{x}-k_{0,x})^{2}+\frac{% \hbar^{2}}{2m_{y}^{*}}(k_{y}-k_{0,y})^{2}+\frac{\hbar^{2}}{2m_{z}^{*}}(k_{z}-k% _{0,z})^{2}
  8. 𝐚 = d d t 𝐯 g = d d t ( k ω ( 𝐤 ) ) = k d ω ( 𝐤 ) d t = k ( d 𝐤 d t k ω ( 𝐤 ) ) , \mathbf{a}=\frac{\operatorname{d}}{\operatorname{d}t}\,\mathbf{v}\text{g}=% \frac{\operatorname{d}}{\operatorname{d}t}(\nabla_{k}\,\omega(\mathbf{k}))=% \nabla_{k}\frac{\operatorname{d}\omega(\mathbf{k})}{\operatorname{d}t}=\nabla_% {k}\left(\frac{\operatorname{d}\mathbf{k}}{\operatorname{d}t}\cdot\nabla_{k}\,% \omega(\mathbf{k})\right),
  9. 𝐅 = d 𝐩 crystal d t = d 𝐤 d t , \mathbf{F}=\frac{\operatorname{d}\mathbf{p}_{\,\text{crystal}}}{\operatorname{% d}t}=\hbar\frac{\operatorname{d}\mathbf{k}}{\operatorname{d}t},
  10. ħ ħ
  11. 1 / 2 π 1/2π
  12. 𝐚 = k ( 𝐅 k ω ( 𝐤 ) ) . \mathbf{a}=\nabla_{k}\left(\frac{\mathbf{F}}{\hbar}\cdot\nabla_{k}\,\omega(% \mathbf{k})\right).
  13. i i
  14. a i = ( 1 2 ω ( 𝐤 ) k i k j ) F j = ( 1 2 2 E ( 𝐤 ) k i k j ) F j , a_{i}=\left(\frac{1}{\hbar}\,\frac{\partial^{2}\omega(\mathbf{k})}{\partial k_% {i}\partial k_{j}}\right)\!F_{j}=\left(\frac{1}{\hbar^{2}}\,\frac{\partial^{2}% E(\mathbf{k})}{\partial k_{i}\partial k_{j}}\right)\!F_{j},
  15. a i a_{i}
  16. i i
  17. 𝐚 \mathbf{a}
  18. F j F_{j}
  19. j j
  20. 𝐅 \mathbf{F}
  21. k i k_{i}
  22. k j k_{j}
  23. i i
  24. j j
  25. 𝐤 \mathbf{k}
  26. E E
  27. j j
  28. j j
  29. [ M inert - 1 ] i j = - 2 2 E k i k j . \left[M_{\rm inert}^{-1}\right]_{ij}=\hbar^{-2}\frac{\partial^{2}E}{\partial k% _{i}\partial k_{j}}\,.
  30. 𝐤 \mathbf{k}
  31. T = | 2 π m e B | T=\left|\frac{2\pi m}{eB}\right|
  32. A A
  33. 𝐤 - s p a c e \mathbf{k}-space
  34. E E
  35. m * ( E , B ^ , k B ^ ) = 2 2 π E A ( E , B ^ , k B ^ ) m^{*}(E,\hat{B},k_{\hat{B}})=\frac{\hbar^{2}}{2\pi}\cdot\frac{\partial}{% \partial E}A\left(E,\hat{B},k_{\hat{B}}\right)
  36. g ( E ) = g v m * π 2 \scriptstyle g(E)\;=\;\frac{g_{v}m^{*}}{\pi\hbar^{2}}
  37. n e = N C exp ( - E C - E F k T ) n_{e}=N_{C}\exp\left(-\frac{E_{\rm C}-E_{\rm F}}{kT}\right)
  38. N C = 2 ( 2 π m e * k T h 2 ) 3 2 \quad N_{C}=2\left(\frac{2\pi m_{e}^{*}kT}{h^{2}}\right)^{\frac{3}{2}}
  39. n h = N V exp ( - E F - E V k T ) , N V = 2 ( 2 π m h * k T h 2 ) 3 2 n_{h}=N_{V}\exp\left(-\frac{E_{\rm F}-E_{\rm V}}{kT}\right),\quad N_{V}=2\left% (\frac{2\pi m_{h}^{*}kT}{h^{2}}\right)^{\frac{3}{2}}
  40. f c = e B 2 π m * \scriptstyle f_{c}\;=\;\frac{eB}{2\pi m^{*}}
  41. c v \scriptstyle c_{v}
  42. τ \tau
  43. v = μ E \scriptstyle\vec{v}\;=\;\left\|\mu\right\|\cdot\vec{E}
  44. μ = e τ m * \scriptstyle\left\|\mu\right\|\;=\;\frac{e\tau}{\left\|m^{*}\right\|}
  45. e \scriptstyle e

Effusion.html

  1. T T
  2. v rms v_{\rm rms}
  3. 3 2 k B T = 1 2 m v rms 2 \frac{3}{2}k_{\rm B}T=\frac{1}{2}mv_{\rm rms}^{2}
  4. k B k_{\rm B}
  5. Rate of effusion of gas 1 Rate of effusion of gas 2 = M 2 M 1 {\mbox{Rate of effusion of gas}~{}_{1}\over\mbox{Rate of effusion of gas}~{}_{% 2}}=\sqrt{M_{2}\over M_{1}}
  6. M 1 M_{1}
  7. M 2 M_{2}

EH.html

  1. E h E_{h}

Eigenfunction.html

  1. A A
  2. f f
  3. A f = λ f Af=\lambda f
  4. λ λ
  5. f f
  6. A A
  7. A = d 2 d x 2 A=\frac{d^{2}}{dx^{2}}
  8. k k
  9. s i n ( k x ) sin(kx)
  10. c o s ( k x ) cos(kx)
  11. f ( 0 ) = 0 f(0)=0
  12. f ( 0 ) = 3 f(0)=3
  13. f ( t ) f(t)
  14. y ( t ) = λ f ( t ) y(t)=λ f(t)
  15. λ λ
  16. t t
  17. d d t ( a f + b g ) = a d f d t + b d g d t , f , g C , a , b 𝐑 . \frac{d}{dt}(af+bg)=a\frac{df}{dt}+b\frac{dg}{dt},\qquad f,g\in C^{\infty},% \quad a,b\in\mathbf{R}.
  18. D D
  19. D f = λ f Df=\lambda f
  20. d d t \frac{d}{dt}
  21. d d t f ( t ) = λ f ( t ) \frac{d}{dt}f(t)=\lambda f(t)
  22. t t
  23. λ λ
  24. f ( t ) = A e λ t . f(t)=Ae^{\lambda t}.
  25. λ λ
  26. d d t \frac{d}{dt}
  27. h ( x , t ) h(x,t)
  28. x x
  29. t t
  30. h h
  31. 2 h t 2 = c 2 2 h x 2 , \frac{\partial^{2}h}{\partial t^{2}}=c^{2}\frac{\partial^{2}h}{\partial x^{2}},
  32. c c
  33. h ( x , t ) h(x,t)
  34. X ( x ) T ( t ) X(x)T(t)
  35. d 2 d x 2 X = - ω 2 c 2 X d 2 d t 2 T = - ω 2 T . \frac{d^{2}}{dx^{2}}X=-\frac{\omega^{2}}{c^{2}}X\qquad\frac{d^{2}}{dt^{2}}T=-% \omega^{2}T.
  36. - ω 2 c 2 -\tfrac{\omega^{2}}{c^{2}}
  37. ω ω
  38. c c
  39. X ( x ) = sin ( ω x c + φ ) , X(x)=\sin\left(\frac{\omega x}{c}+\varphi\right),
  40. T ( t ) = sin ( ω t + ψ ) , T(t)=\sin(\omega t+\psi),
  41. φ φ
  42. ψ ψ
  43. X ( x ) = 0 X(x)=0
  44. x = 0 x=0
  45. x = L x=L
  46. s i n ( φ ) = 0 sin(φ)=0
  47. φ = 0 φ=0
  48. sin ( ω L c ) = 0. \sin\left(\frac{\omega L}{c}\right)=0.
  49. ω ω
  50. n n
  51. h ( x , t ) = sin ( n π x L ) sin ( ω n t ) . h(x,t)=\sin\left(\frac{n\pi x}{L}\right)\sin(\omega_{n}t).
  52. n n
  53. ( n 1 ) (n−1)
  54. H ψ = E ψ , H\psi=E\psi,
  55. H = - 2 2 m 2 + V ( 𝐫 , t ) H=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r},t)
  56. ψ ( t ) = k e - i E k t φ k , \psi(t)=\sum_{k}e^{-\frac{iE_{k}t}{\hbar}}\varphi_{k},
  57. H H
  58. H H
  59. A A
  60. 0 = f i , f j = d 𝐫 f i ¯ f j 0=\langle f_{i},f_{j}\rangle=\int d\mathbf{r}\overline{f_{i}}f_{j}
  61. f < s u b > i ¯ \overline{f<sub>i}

Einstein_notation.html

  1. y = i = 1 3 c i x i = c 1 x 1 + c 2 x 2 + c 3 x 3 y=\sum_{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}
  2. y = c i x i . y=c_{i}x^{i}\,.
  3. ( x , y , z ) (x,y,z)
  4. μ , ν , μ,ν,...
  5. i , j , i,j,...
  6. v = v i e i = [ e 1 e 2 e n ] [ v 1 v 2 v n ] \,v=v^{i}e_{i}=\begin{bmatrix}e_{1}&e_{2}&\cdots&e_{n}\end{bmatrix}\begin{% bmatrix}v^{1}\\ v^{2}\\ \vdots\\ v^{n}\end{bmatrix}
  7. w = w i e i = [ w 1 w 2 w n ] [ e 1 e 2 e n ] \,w=w_{i}e^{i}=\begin{bmatrix}w_{1}&w_{2}&\cdots&w_{n}\end{bmatrix}\begin{% bmatrix}e^{1}\\ e^{2}\\ \vdots\\ e^{n}\end{bmatrix}
  8. V V * V\to V^{*}
  9. [ v 1 v k ] . \begin{bmatrix}v_{1}&\cdots&v_{k}\end{bmatrix}.
  10. [ w 1 w k ] \begin{bmatrix}w^{1}\\ \vdots\\ w^{k}\end{bmatrix}
  11. V V V\otimes V
  12. 𝐞 i j = 𝐞 i 𝐞 j \mathbf{e}_{ij}=\mathbf{e}_{i}\otimes\mathbf{e}_{j}
  13. 𝐓 \mathbf{T}
  14. V V V\otimes V
  15. 𝐓 = T i j 𝐞 i j \mathbf{T}=T^{ij}\mathbf{e}_{ij}
  16. V * V^{*}
  17. V V
  18. 𝐞 i ( 𝐞 j ) = δ j i . \mathbf{e}^{i}(\mathbf{e}_{j})=\delta^{i}_{j}.
  19. δ \delta
  20. Hom ( V , W ) = V * W \mathrm{Hom}(V,W)=V^{*}\otimes W
  21. A m n A_{mn}
  22. A m n A^{m}{}_{n}
  23. 𝐮 𝐯 = u j v j \mathbf{u}\cdot\mathbf{v}=u_{j}v^{j}
  24. 𝐮 × 𝐯 = u j v k ϵ i 𝐞 i j k \mathbf{u}\times\mathbf{v}=u^{j}v^{k}\epsilon^{i}{}_{jk}\mathbf{e}_{i}
  25. ϵ i = j k δ i l ϵ l j k \epsilon^{i}{}_{jk}=\delta^{il}\epsilon_{ljk}
  26. ϵ i j k \epsilon_{ijk}
  27. ϵ \epsilon
  28. ϵ i j k \epsilon^{i}{}_{jk}
  29. ϵ i j k \epsilon_{ijk}
  30. A i j A_{ij}
  31. B j k B_{jk}
  32. 𝐂 i k = ( 𝐀 𝐁 ) i k = j = 1 N A i j B j k \mathbf{C}_{ik}=(\mathbf{A}\,\mathbf{B})_{ik}=\sum_{j=1}^{N}A_{ij}B_{jk}
  33. C i = k A i B j j k C^{i}{}_{k}=A^{i}{}_{j}\,B^{j}{}_{k}
  34. A i j A^{i}{}_{j}
  35. A i i A^{i}{}_{i}
  36. u i u^{i}
  37. v j v_{j}
  38. A i = j u i v j = ( u v ) i j A^{i}{}_{j}=u^{i}\,v_{j}=(uv)^{i}{}_{j}
  39. g μ ν g_{\mu\nu}
  40. T β α T^{\alpha}_{\beta}
  41. T μ α = g μ σ T σ α T^{\mu\alpha}=g^{\mu\sigma}T^{\;\alpha}_{\sigma}
  42. T μ β = g μ σ T β σ T_{\mu\beta}=g_{\mu\sigma}T^{\sigma}_{\;\beta}

Elasticity_(physics).html

  1. F F
  2. x x
  3. F = - k x , F=-kx,
  4. k k
  5. σ σ
  6. ε \varepsilon
  7. σ = E ε , \sigma=E\varepsilon,
  8. E E
  9. s y m b o l σ = 𝒢 ( s y m b o l F ) \ symbol{\sigma}=\mathcal{G}(symbol{F})
  10. s y m b o l σ ˙ = 𝖣 : s y m b o l F ˙ . \dot{symbol{\sigma}}=\mathsf{D}:\dot{symbol{F}}\,.
  11. s y m b o l σ = 1 J W s y m b o l F \cdotsymbol F T where J := \detsymbol F . symbol{\sigma}=\cfrac{1}{J}~{}\cfrac{\partial W}{\partial symbol{F}}% \cdotsymbol{F}^{T}\quad\,\text{where}\quad J:=\detsymbol{F}\,.

Electric-field_screening.html

  1. 𝐅 = q 1 q 2 4 π ϵ 0 ϵ R | 𝐫 | 2 𝐫 ^ \mathbf{F}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}\epsilon_{R}\left|\mathbf{r}% \right|^{2}}\hat{\mathbf{r}}
  2. - 2 [ Δ ϕ ( r ) ] = 1 ϵ 0 [ Q δ ( r ) - e Δ ρ ( r ) ] -\nabla^{2}[\Delta\phi(r)]=\frac{1}{\epsilon_{0}}[Q\delta(r)-e\Delta\rho(r)]
  3. ρ j ( r ) = ρ j ( 0 ) ( r ) exp [ e ϕ ( r ) k B T ] \rho_{j}(r)=\rho_{j}^{(0)}(r)\;\exp\!\left[\frac{e\phi(r)}{k_{B}T}\right]
  4. e Δ ρ ϵ 0 k 0 2 Δ ϕ e\Delta\rho\simeq\epsilon_{0}k_{0}^{2}\Delta\phi
  5. k 0 = def ρ e 2 ϵ 0 k B T k_{0}\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{\rho e^{2}}{\epsilon_{0}k_{B}T}}
  6. Δ μ = Δ T - e Δ ϕ = 0 \Delta\mu=\Delta T-e\Delta\phi=0
  7. ρ = 2 1 ( 2 π ) 3 4 3 π k F 3 , E F = 2 k F 2 2 m , ρ E F 3 / 2 . \rho=2\frac{1}{(2\pi)^{3}}\frac{4}{3}\pi k_{F}^{3}\quad,\quad E_{F}=\frac{% \hbar^{2}k_{F}^{2}}{2m}\quad,\quad\rho\propto E_{F}^{3/2}.
  8. Δ ρ 3 ρ 2 E F Δ E F \Delta\rho\simeq\frac{3\rho}{2E_{F}}\Delta E_{F}
  9. e Δ ρ ϵ 0 k 0 2 Δ ϕ e\Delta\rho\simeq\epsilon_{0}k_{0}^{2}\Delta\phi
  10. k 0 = def 3 e 2 ρ 2 ϵ 0 E F = m e 2 k f ϵ 0 π 2 2 k_{0}\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{3e^{2}\rho}{2\epsilon_{0}E_{F}}% }=\sqrt{\frac{me^{2}k_{f}}{\epsilon_{0}\pi^{2}\hbar^{2}}}
  11. [ 2 - k 0 2 ] ϕ ( r ) = - Q ϵ 0 δ ( r ) \left[\nabla^{2}-k_{0}^{2}\right]\phi(r)=-\frac{Q}{\epsilon_{0}}\delta(r)
  12. ϕ ( r ) = Q 4 π ϵ 0 r e - k 0 r \phi(r)=\frac{Q}{4\pi\epsilon_{0}r}e^{-k_{0}r}
  13. ϵ ( r ) = e - k 0 r \epsilon(r)=e^{-k_{0}r}

Electric_generator.html

  1. V G V_{G}
  2. R G R_{G}

Electric_motor.html

  1. 𝐅 = I s y m b o l × 𝐁 \mathbf{F}=Isymbol{\ell}\times\mathbf{B}\,\!
  2. 𝐅 = 𝐉 × 𝐁 \mathbf{F}=\mathbf{J}\times\mathbf{B}
  3. P e m = r p m × T 5252 P_{em}=\frac{rpm\times T}{5252}
  4. P e m = a n g u l a r s p e e d × T P_{em}={angularspeed\times T}
  5. P e m = F × v P_{em}=F\times{v}
  6. P a i r g a p = R r s * I r 2 P_{airgap}=\frac{R_{r}}{s}*I_{r}^{2}
  7. η = P m P e \eta=\frac{P_{m}}{P_{e}}
  8. η \eta
  9. P e P_{e}
  10. P m P_{m}
  11. P e = I V P_{e}=IV
  12. P m = T ω P_{m}=T\omega
  13. V V
  14. I I
  15. T T
  16. ω \omega
  17. G = ω r e s i s t a n c e × r e l u c t a n c e = ω μ σ A m A e l m l e G=\frac{\omega}{resistance\times reluctance}=\frac{\omega\mu\sigma A_{m}A_{e}}% {l_{m}l_{e}}
  18. G G
  19. A m , A e A_{m},A_{e}
  20. l m , l e l_{m},l_{e}
  21. μ \mu
  22. ω \omega

Electrical_conductor.html

  1. R R
  2. G G
  3. R = ρ A , R=\rho\frac{\ell}{A},
  4. G = σ A . G=\sigma\frac{A}{\ell}.
  5. \ell
  6. ρ = 1 / σ \rho=1/\sigma

Electrical_element.html

  1. I I
  2. V V
  3. Q Q
  4. Φ \Phi
  5. d Q = - I d t dQ=-I\,dt
  6. d Φ = V d t d\Phi=V\,dt
  7. Φ \Phi
  8. Q Q
  9. Φ \Phi
  10. R R
  11. d V = R d I dV=R\,dI
  12. C C
  13. d Q = C d V dQ=C\,dV
  14. L L
  15. d Φ = L d I d\Phi=L\,dI
  16. f ( V , I ) = 0 f(V,I)=0
  17. f ( V , Q ) = 0 f(V,Q)=0
  18. f ( Φ , I ) = 0 f(\Phi,I)=0
  19. f ( Φ , Q ) = 0 f(\Phi,Q)=0
  20. f ( x , y ) f(x,y)
  21. V = f ( I ) V=f(I)
  22. V = I = 0 V=I=0
  23. [ V 1 I 2 ] = [ 0 n - n 0 ] [ I 1 V 2 ] \begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}0&n\\ -n&0\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}
  24. [ V 1 V 2 ] = [ 0 - r r 0 ] [ I 1 I 2 ] \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}0&-r\\ r&0\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}

Electrical_reactance.html

  1. X \scriptstyle{X}
  2. X \scriptstyle{X}
  3. R \scriptstyle{R}
  4. Z \scriptstyle{Z}
  5. Z = R + j X Z=R+jX
  6. Z Z
  7. R R
  8. R = ( Z ) {R=\Re{(Z)}}
  9. X X
  10. X = ( Z ) {X=\Im{(Z)}}
  11. j j
  12. i i
  13. j j
  14. i i
  15. X C \scriptstyle{X_{C}}
  16. X L \scriptstyle{X_{L}}
  17. X \scriptstyle{X}
  18. X = X L - X C = ω L - 1 ω C {X=X_{L}-X_{C}=\omega L-\frac{1}{\omega C}}
  19. X C \scriptstyle{X_{C}}
  20. X L \scriptstyle{X_{L}}
  21. ω \omega
  22. 2 π 2\pi
  23. X L \scriptstyle{X_{L}}
  24. X C \scriptstyle{X_{C}}
  25. X C \scriptstyle{X_{C}}
  26. X > 0 \scriptstyle X>0
  27. X = 0 \scriptstyle X=0
  28. X < 0 \scriptstyle X<0
  29. X C \scriptstyle{X_{C}}
  30. f \scriptstyle{f}
  31. C \scriptstyle{C}
  32. X C = 1 ω C = 1 2 π f C X_{C}=\frac{1}{\omega C}=\frac{1}{2\pi fC}
  33. X L \scriptstyle{X_{L}}
  34. f \scriptstyle{f}
  35. L \scriptstyle{L}
  36. X L = ω L = 2 π f L X_{L}=\omega L=2\pi fL
  37. L \scriptstyle{L}
  38. A \scriptstyle{A}
  39. f \scriptstyle{f}
  40. I L = A ω L = A 2 π f L . I_{L}={A\over\omega L}={A\over 2\pi fL}.
  41. L \scriptstyle{L}
  42. A \scriptstyle{A}
  43. f \scriptstyle{f}
  44. I L = A π 2 8 ω L = A π 16 f L I_{L}={A\pi^{2}\over 8\omega L}={A\pi\over 16fL}
  45. X L = 16 π f L X_{L}={16\over\pi}fL
  46. \scriptstyle{\mathcal{E}}
  47. B \scriptstyle{B}
  48. = - d Φ B d t \mathcal{E}=-{{d\Phi_{B}}\over dt}
  49. N \scriptstyle N
  50. = - N d Φ B d t \mathcal{E}=-N{d\Phi_{B}\over dt}
  51. π / 2 \scriptstyle{\pi/2}
  52. π / 2 \scriptstyle{\pi/2}
  53. e ± j π 2 e^{\pm j{\pi\over 2}}
  54. Z ~ C = 1 ω C e j ( - π 2 ) = j ( - 1 ω C ) = - j X C Z ~ L = ω L e j π 2 = j ω L = j X L \begin{aligned}\displaystyle\tilde{Z}_{C}&\displaystyle={1\over\omega C}e^{j(-% {\pi\over 2})}=j\left({-\frac{1}{\omega C}}\right)=-jX_{C}\\ \displaystyle\tilde{Z}_{L}&\displaystyle=\omega Le^{j{\pi\over 2}}=j\omega L=% jX_{L}\end{aligned}
  55. π / 2 \scriptstyle{\pi/2}

Electrocardiography.html

  1. V W = 1 3 ( R A + L A + L L ) V_{W}=\frac{1}{3}(RA+LA+LL)
  2. I = L A - R A I=LA-RA
  3. I I = L L - R A II=LL-RA
  4. I I I = L L - L A III=LL-LA
  5. a V R = R A - 1 2 ( L A + L L ) = 3 2 ( R A - V W ) aVR=RA-\frac{1}{2}(LA+LL)=\frac{3}{2}(RA-V_{W})
  6. a V L = L A - 1 2 ( R A + L L ) = 3 2 ( L A - V W ) aVL=LA-\frac{1}{2}(RA+LL)=\frac{3}{2}(LA-V_{W})
  7. a V F = L L - 1 2 ( R A + L A ) = 3 2 ( L L - V W ) aVF=LL-\frac{1}{2}(RA+LA)=\frac{3}{2}(LL-V_{W})

Electrochemical_potential.html

  1. μ ¯ \bar{\mu}
  2. μ ¯ i = μ i + z i F Φ \bar{\mu}_{i}=\mu_{i}+z_{i}F\Phi
  3. μ ¯ i \bar{\mu}_{i}
  4. μ i \mu_{i}
  5. z i z_{i}
  6. F F
  7. Φ \Phi
  8. z i z_{i}
  9. μ ¯ i = μ i \bar{\mu}_{i}=\mu_{i}

Electromagnet.html

  1. 𝐉 d 𝐀 = 𝐇 d 𝐥 \int\mathbf{J}\cdot d\mathbf{A}=\oint\mathbf{H}\cdot d\mathbf{l}
  2. N I = H core L core + H gap L gap NI=H_{\mathrm{core}}L_{\mathrm{core}}+H_{\mathrm{gap}}L_{\mathrm{gap}}\,
  3. N I = B ( L core μ + L gap μ 0 ) ( 1 ) NI=B\left(\frac{L_{\mathrm{core}}}{\mu}+\frac{L_{\mathrm{gap}}}{\mu_{0}}\right% )\qquad\qquad\qquad\qquad(1)\,
  4. μ = B / H \mu=B/H\,
  5. μ 0 = 4 π ( 10 - 7 ) N A - 2 \mu_{0}=4\pi(10^{-7})\ \mathrm{N}\cdot\mathrm{A}^{-2}
  6. A \mathrm{A}
  7. μ r = μ / μ 0 2000 - 6000 \mu_{r}=\mu/\mu_{0}\approx 2000-6000\,
  8. F = B 2 A 2 μ 0 ( 2 ) F=\frac{B^{2}A}{2\mu_{0}}\qquad\qquad\qquad\qquad\qquad\qquad(2)\,
  9. F A = B s a t 2 2 μ 0 1000 kPa = 10 6 N / m 2 = 145 lbf in - 2 \frac{F}{A}=\frac{B_{sat}^{2}}{2\mu_{0}}\approx 1000\ \mathrm{kPa}=10^{6}% \mathrm{N/m^{2}}=145\ \mathrm{lbf}\cdot\mathrm{in}^{-2}\,
  10. B = N I μ L ( 3 ) B=\frac{NI\mu}{L}\qquad\qquad\qquad\qquad\qquad\qquad(3)\,
  11. F = μ 2 N 2 I 2 A 2 μ 0 L 2 ( 4 ) F=\frac{\mu^{2}N^{2}I^{2}A}{2\mu_{0}L^{2}}\qquad\qquad\qquad\qquad\qquad(4)\,
  12. m = N I A L m=\frac{NIA}{L}
  13. F = μ 0 m 1 m 2 4 π r 2 F=\frac{\mu_{0}m_{1}m_{2}}{4\pi r^{2}}
  14. q 𝐯 × 𝐁 q\mathbf{v}\times\mathbf{B}\,
  15. A A\,
  16. B B\,
  17. F F\,
  18. H H\,
  19. I I\,
  20. L L\,
  21. L core + L gap L_{\mathrm{core}}+L_{\mathrm{gap}}\,
  22. L core L_{\mathrm{core}}\,
  23. L gap L_{\mathrm{gap}}\,
  24. m 1 , m 2 m_{1},m_{2}\,
  25. μ \mu\,
  26. μ 0 \mu_{0}\,
  27. μ r \mu_{r}\,
  28. N N\,
  29. r r\,

Electron_affinity.html

  1. E EA E vac - E C E_{\rm EA}\equiv E_{\rm vac}-E_{\rm C}
  2. W = E vac - E F W=E_{\rm vac}-E_{\rm F}

Electron_degeneracy_pressure.html

  1. P = 2 3 E t o t V = 2 3 2 k F 5 10 π 2 m e = ( 3 π 2 ) 2 / 3 2 5 m e ρ N 5 / 3 , P=\frac{2}{3}\frac{E_{tot}}{V}=\frac{2}{3}\frac{\hbar^{2}k_{\rm{F}}^{5}}{10\pi% ^{2}m_{\rm{e}}}=\frac{(3\pi^{2})^{2/3}\hbar^{2}}{5m_{\rm{e}}}\rho_{N}^{5/3},
  2. \hbar
  3. m e m_{\rm e}
  4. ρ N \rho_{N}
  5. k = 2 π λ k=\frac{2\pi}{\lambda}
  6. E = p 2 2 m = 2 k 2 2 m E=\frac{p^{2}}{2m}=\frac{\hbar^{2}k^{2}}{2m}
  7. P = N k T / V P=NkT/V
  8. n / V n/V
  9. Δ x Δ p 2 \Delta x\Delta p\geq\frac{\hbar}{2}

Electron_density.html

  1. ρ ( 𝐫 ) \rho(\mathbf{r})
  2. ρ ( 𝐫 ) = μ ν P μ ν ϕ μ ( 𝐫 ) ϕ ν ( 𝐫 ) \rho(\mathbf{r})=\sum_{\mu}\sum_{\nu}P_{\mu\nu}\phi_{\mu}(\mathbf{r})\phi_{\nu% }(\mathbf{r})

Electron_diffraction.html

  1. I 𝐠 = | ψ 𝐠 | 2 | F 𝐠 | 2 . I_{\mathbf{g}}=\left|\psi_{\mathbf{g}}\right|^{2}\propto\left|F_{\mathbf{g}}% \right|^{2}.
  2. ψ 𝐠 \psi_{\mathbf{g}}
  3. F 𝐠 F_{\mathbf{g}}
  4. F 𝐠 = i f i e - 2 π i 𝐠 𝐫 i F_{\mathbf{g}}=\sum_{i}f_{i}e^{-2\pi i\mathbf{g}\cdot\mathbf{r}_{i}}
  5. 𝐠 \mathbf{g}
  6. 𝐫 i \mathbf{r}_{i}
  7. i i
  8. f i f_{i}
  9. f i f_{i}
  10. λ = h p . \lambda=\frac{h}{p}.
  11. h h
  12. p p
  13. λ \lambda
  14. U U
  15. v = 2 e U m 0 v=\sqrt{\frac{2eU}{m_{0}}}
  16. m 0 m_{0}
  17. e e
  18. λ = h p = h m 0 v = h 2 m 0 e U . \lambda=\frac{h}{p}=\frac{h}{m_{0}v}=\frac{h}{\sqrt{2m_{0}eU}}.
  19. p = 2 m 0 Δ E + Δ E 2 c 2 = 2 m 0 Δ E 1 + Δ E 2 m 0 c 2 p=\sqrt{2m_{0}\Delta E+\frac{\Delta E^{2}}{c^{2}}}=\sqrt{2m_{0}\Delta E}\sqrt{% 1+\frac{\Delta E}{2m_{0}c^{2}}}
  20. λ = h 2 m 0 e U 1 1 + e U 2 m 0 c 2 \lambda=\frac{h}{\sqrt{2m_{0}eU}}\frac{1}{\sqrt{1+\frac{eU}{2m_{0}c^{2}}}}
  21. c c

Elementary_charge.html

  1. 𝐞 \mathbf{e}
  2. q q
  3. 1 / 2 {1}/{2}
  4. 1 / 3 {1}/{3}
  5. 1 / 3 {1}/{3}
  6. 1 / 3 {1}/{3}
  7. 1 / 3 {1}/{3}
  8. e = F N A e=\frac{F}{N_{\mathrm{A}}}
  9. K J = 2 e h K_{\mathrm{J}}=\frac{2e}{h}
  10. R K = h e 2 . R_{\mathrm{K}}=\frac{h}{e^{2}}.
  11. e = 2 R K K J . e=\frac{2}{R_{\mathrm{K}}K_{\mathrm{J}}}.
  12. e 2 = 2 h α μ 0 c = 2 h α ϵ 0 c e^{2}=\frac{2h\alpha}{\mu_{0}c}=2h\alpha\epsilon_{0}c

Ellipsoid.html

  1. x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 , {x^{2}\over a^{2}}+{y^{2}\over b^{2}}+{z^{2}\over c^{2}}=1,
  2. a > b > c a>b>c
  3. a = b > c a=b>c
  4. a = b < c a=b<c
  5. a = b = c a=b=c
  6. ( 𝐱 - 𝐯 ) T A ( 𝐱 - 𝐯 ) = 1 , (\mathbf{x-v})^{\mathrm{T}}\!A\,(\mathbf{x-v})=1,
  7. a - 2 a^{-2}
  8. b - 2 b^{-2}
  9. c - 2 c^{-2}
  10. x = a cos u cos v , y = b cos u sin v , z = c sin u ; \begin{aligned}\displaystyle x&\displaystyle=a\,\cos u\cos v,\\ \displaystyle y&\displaystyle=b\,\cos u\sin v,\\ \displaystyle z&\displaystyle=c\,\sin u;\end{aligned}\,\!
  11. - π / 2 u + π / 2 , - π v + π . -{\pi}/{2}\leq u\leq+{\pi}/{2},\qquad-\pi\leq v\leq+\pi.\!\,\!
  12. V = 4 3 π a b c , V=\frac{4}{3}\pi abc,\!
  13. V max = 8 3 3 a b c , V min = 8 a b c . V_{\max}=\frac{8}{3\sqrt{3}}abc,\qquad V_{\min}=8abc.
  14. S = 2 π c 2 + 2 π a b sin ϕ ( E ( ϕ , k ) sin 2 ϕ + F ( ϕ , k ) cos 2 ϕ ) , S=2\pi c^{2}+\frac{2\pi ab}{\sin\phi}\left(E(\phi,k)\,\sin^{2}\phi+F(\phi,k)\,% \cos^{2}\phi\right),
  15. cos ϕ = c a , k 2 = a 2 ( b 2 - c 2 ) b 2 ( a 2 - c 2 ) , a b c , \cos\phi=\frac{c}{a},\qquad k^{2}=\frac{a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})% },\qquad a\geq b\geq c,
  16. S oblate = 2 π a 2 ( 1 + 1 - e 2 e tanh - 1 e ) where e 2 = 1 - c 2 a 2 ( c < a ) , S_{\rm oblate}=2\pi a^{2}\left(1+\frac{1-e^{2}}{e}\tanh^{-1}e\right)\quad\mbox% {where}~{}\quad e^{2}=1-\frac{c^{2}}{a^{2}}\quad(c<a),
  17. S prolate = 2 π a 2 ( 1 + c a e sin - 1 e ) where e 2 = 1 - a 2 c 2 ( c > a ) , S_{\rm prolate}=2\pi a^{2}\left(1+\frac{c}{ae}\sin^{-1}e\right)\quad\qquad% \mbox{where}~{}\;\quad e^{2}=1-\frac{a^{2}}{c^{2}}\quad(c>a),
  18. S oblate S_{\rm oblate}
  19. S 4 π ( a p b p + a p c p + b p c p 3 ) 1 / p . S\approx 4\pi\!\left(\frac{a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3}\right)^{1/p}.\,\!
  20. m = ρ V = ρ 4 3 π a b c m=\rho V=\rho\frac{4}{3}\pi abc\,\!
  21. I xx = 1 5 m ( b 2 + c 2 ) , I yy = 1 5 m ( c 2 + a 2 ) , I zz = 1 5 m ( a 2 + b 2 ) , I_{\mathrm{xx}}=\frac{1}{5}m(b^{2}+c^{2}),\qquad I_{\mathrm{yy}}=\frac{1}{5}m(% c^{2}+a^{2}),\qquad I_{\mathrm{zz}}=\frac{1}{5}m(a^{2}+b^{2}),
  22. I xy = I yz = I zx = 0. I_{\mathrm{xy}}=I_{\mathrm{yz}}=I_{\mathrm{zx}}=0.\,\!
  23. x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 , {x^{2}\over a^{2}}+{y^{2}\over b^{2}}+{z^{2}\over c^{2}}=1,
  24. r 2 cos 2 θ sin 2 ϕ a 2 + r 2 sin 2 θ sin 2 ϕ b 2 + r 2 cos 2 ϕ c 2 = 1 , {r^{2}\cos^{2}\!\theta\,\sin^{2}\!\phi\over a^{2}}+{r^{2}\sin^{2}\!\theta\,% \sin^{2}\!\phi\over b^{2}}+{r^{2}\cos^{2}\!\phi\over c^{2}}=1,
  25. r 2 cos 2 θ a 2 + r 2 sin 2 θ b 2 + z 2 c 2 = 1 , {r^{2}\cos^{2}\!\theta\over a^{2}}+{r^{2}\sin^{2}\!\theta\over b^{2}}+{z^{2}% \over c^{2}}=1,

Elliptic_geometry.html

  1. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  2. e a r = cos a + r sin a , r 2 = - 1. e^{ar}=\cos a+r\sin a,\quad r^{2}=-1.
  3. a = 0 a=0
  4. r r
  5. e a r , 0 a < π e^{ar},\quad 0\leq a<\pi
  6. e a r e^{ar}
  7. a a
  8. u u
  9. q u q v , q\mapsto uqv,
  10. u u
  11. v v
  12. u u
  13. v v
  14. u = 1 u=1
  15. v = 1 v=1
  16. u u
  17. { u e a r : 0 a < π } \{ue^{ar}:0\leq a<\pi\}
  18. { e a r u : 0 a < π } \{e^{ar}u:0\leq a<\pi\}
  19. r r
  20. u u
  21. ( x , x ) (x, −x)
  22. u u
  23. λ u λu
  24. λ λ
  25. d ( u , v ) = arccos ( | u v | u v ) ; d(u,v)=\arccos\left(\frac{|u\cdot v|}{\|u\|\ \|v\|}\right);
  26. d ( λ u , μ v ) = d ( u , v ) d(λu, μv)=d(u,v)
  27. λ λ
  28. μ μ
  29. n n
  30. δ ( u , v ) = 2 u - v ( 1 + u 2 ) ( 1 + v 2 ) \delta(u,v)=\frac{2\|u-v\|}{\sqrt{(1+\|u\|^{2})(1+\|v\|^{2})}}
  31. u u
  32. v v
  33. \|\cdot\|
  34. δ ( u , ) = δ ( , u ) = 2 1 + u 2 . \delta(u,\infty)=\delta(\infty,u)=\frac{2}{\sqrt{1+\|u\|^{2}}}.
  35. d ( u , v ) = 2 arcsin ( δ ( u , v ) 2 ) . d(u,v)=2\arcsin\left(\frac{\delta(u,v)}{2}\right).
  36. u u
  37. u −u
  38. v v
  39. v v

Elliptical_galaxy.html

  1. 10 × ( 1 - b a ) 10\times\left(1-\frac{b}{a}\right)

Elo_rating_system.html

  1. p p
  2. d p d_{p}
  3. ( w + 400 + x + 400 + y - 400 + z - 400 ) 4 \textstyle\displaystyle{\frac{\left(w+400+x+400+y-400+z-400\right)}{4}}
  4. [ w + x + y + z + 400 ( 2 ) - 400 ( 2 ) ] 4 \textstyle\displaystyle{\frac{\left[w+x+y+z+400\left(2\right)-400\left(2\right% )\right]}{4}}
  5. Performance rating = Total of opponents’ ratings + 400 × ( Wins - Losses ) Games \textstyle\,\text{Performance rating}=\frac{\,\text{Total of opponents' % ratings }+400\times(\,\text{Wins}-\,\text{Losses})}{\,\text{Games}}
  6. Performance rating = 1000 + 400 × ( 1 ) 1 = 1400 \textstyle\displaystyle\,\text{Performance rating}=\frac{1000+400\times(1)}{1}% =1400
  7. Performance rating = 2000 + 400 × ( 2 ) 2 = 1400 \textstyle\displaystyle\,\text{Performance rating}=\frac{2000+400\times(2)}{2}% =1400
  8. Performance rating = 1000 + 400 × ( 0 ) 1 = 1000 \textstyle\displaystyle\,\text{Performance rating}=\frac{1000+400\times(0)}{1}% =1000
  9. d p d_{p}
  10. p p
  11. p p
  12. d p d_{p}
  13. K = 800 / ( N e + m ) K=800/(N_{e}+m)\,
  14. A F = min { 100 + 4 N W + 2 N D + N R , 150 } AF=\operatorname{min}\{100+4N_{W}+2N_{D}+N_{R},150\}
  15. N W N_{W}
  16. N D N_{D}
  17. N R N_{R}
  18. R A R_{A}
  19. R B R_{B}
  20. E A = 1 1 + 10 ( R B - R A ) / 400 . E_{A}=\frac{1}{1+10^{(R_{B}-R_{A})/400}}.
  21. E B = 1 1 + 10 ( R A - R B ) / 400 . E_{B}=\frac{1}{1+10^{(R_{A}-R_{B})/400}}.
  22. E A = Q A Q A + Q B E_{A}=\frac{Q_{A}}{Q_{A}+Q_{B}}
  23. E B = Q B Q A + Q B , E_{B}=\frac{Q_{B}}{Q_{A}+Q_{B}},
  24. Q A = 10 R A / 400 Q_{A}=10^{R_{A}/400}
  25. Q B = 10 R B / 400 Q_{B}=10^{R_{B}/400}
  26. Q A / Q B Q_{A}/Q_{B}
  27. E A + E B = 1 E_{A}+E_{B}=1
  28. E A E_{A}
  29. S A S_{A}
  30. R A = R A + K ( S A - E A ) . R_{A}^{\prime}=R_{A}+K(S_{A}-E_{A}).
  31. E a = 1 1 + 10 R b - R a 400 E_{a}=\frac{1}{1+10^{\tfrac{R_{b}-R_{a}}{400}}}
  32. E b = 1 1 + 10 R a - R b 400 E_{b}=\frac{1}{1+10^{\tfrac{R_{a}-R_{b}}{400}}}

Elongation_(astronomy).html

  1. T = 2 π ω = 2 π ω p - ω e = 2 π 2 π T p - 2 π T e = T e T e T p - 1 T={2\pi\over\omega}={2\pi\over\omega_{p}-\omega_{e}}={2\pi\over{2\pi\over T_{p% }}-{2\pi\over T_{e}}}={T_{e}\over{T_{e}\over T_{p}}-1}

Emission_spectrum.html

  1. E photon = h ν E_{\,\text{photon}}=h\nu
  2. E photon E_{\,\text{photon}}
  3. ν \nu
  4. h h

Emmy_Noether.html

  1. A 1 A 2 A 3 . A_{1}\subset A_{2}\subset A_{3}\subset\cdots.
  2. A 1 A 2 A 3 . A_{1}\supset A_{2}\supset A_{3}\supset\cdots.
  3. A n = A m A_{n}=A_{m}

Empty_function.html

  1. f A : A . f_{A}:\varnothing\rightarrow A.

Empty_product.html

  1. P m = i = 1 m a i = a 1 a m P_{m}=\prod_{i=1}^{m}a_{i}=a_{1}\cdots a_{m}
  2. P m = a m P m - 1 P_{m}=a_{m}\cdot P_{m-1}
  3. P 1 = a 1 P_{1}=a_{1}
  4. P 0 = 1 P_{0}=1
  5. P 1 P_{1}
  6. P 0 P_{0}
  7. ln ( 1 ) = 0 \ln(1)=0
  8. e 0 = 1 e^{0}=1
  9. i x i = e i ln x i \prod_{i}x_{i}=e^{\sum_{i}\ln x_{i}}
  10. i I X i = { g : I i I X i | i g ( i ) X i } . \prod_{i\in I}X_{i}=\{g:I\to\bigcup_{i\in I}X_{i}\ |\ \forall i\ g(i)\in X_{i}\}.
  11. f f_{\varnothing}
  12. × \varnothing\times\varnothing
  13. \varnothing\to\varnothing
  14. \varnothing
  15. × = \varnothing\times\varnothing=\varnothing
  16. = { f : } = { } . \prod_{\varnothing}{}=\{f_{\varnothing}:\varnothing\to\varnothing\}=\{% \varnothing\}.
  17. = { ( ) } , \prod_{\varnothing}{}=\{()\},

Empty_sum.html

  1. s m = i = 1 m a i = a 1 + + a m s_{m}=\sum_{i=1}^{m}a_{i}=a_{1}+...+a_{m}
  2. s m = a m + s m - 1 s_{m}=a_{m}+s_{m-1}
  3. s 1 = a 1 s_{1}=a_{1}
  4. s 0 = 0 s_{0}=0
  5. s 1 s_{1}
  6. s 0 s_{0}

Enceladus.html

  1. Q t i d = 63 ρ n 5 r 4 e 2 38 u q Q_{tid}=\frac{63\rho n^{5}r^{4}e^{2}}{38uq}

Engine_displacement.html

  1. displacement = π 4 × bore × 2 stroke × number of cylinders \mbox{displacement}~{}={\pi\over 4}\times\mbox{bore}~{}^{2}\times\mbox{stroke}% ~{}\times\mbox{number of cylinders}~{}
  2. displacement = bore × 2 0.7854 × stroke × number of cylinders \mbox{displacement}~{}=\mbox{bore}~{}^{2}\times 0.7854\times\mbox{stroke}~{}% \times\mbox{number of cylinders}~{}

English_numerals.html

  1. 0 \aleph_{0}

Enharmonic.html

  1. twelve fifths seven octaves = ( 3 2 ) 12 / 2 7 = 3 12 2 19 = 531441 524288 = 1.0136432647705078125 \frac{\hbox{twelve fifths}}{\hbox{seven octaves}}=\left(\tfrac{3}{2}\right)^{1% 2}\!\!\bigg/\,2^{7}=\frac{3^{12}}{2^{19}}=\frac{531441}{524288}=1.013643264770% 5078125\!
  2. x x
  3. 2 x 2x
  4. 8 x 5 = 1.6 x . \frac{8x}{5}=1.6x.\!
  5. ( 5 4 ) 2 x = ( 25 16 ) x = 1.5625 x \left(\frac{5}{4}\right)^{2}x=\left(\frac{25}{16}\right)x=1.5625x
  6. 128 125 \frac{128}{125}
  7. 2 8 12 x = 2 2 3 x 1.5874 x 2^{\frac{8}{12}}x=2^{\frac{2}{3}}x\approx 1.5874x

Enriched_category.html

  1. α \alpha
  2. λ \lambda
  3. ρ \rho
  4. ( , I , α , λ , ρ ) (⊗,I,α,λ,ρ)
  5. I I
  6. T a a id a = id T ( a ) , T_{aa}\circ\operatorname{id}_{a}=\operatorname{id}_{T(a)},

Enthalpy_of_neutralization.html

  1. Q = m c p Δ T Q=mc_{p}\Delta T
  2. Δ H = - Q n \Delta H=-\frac{Q}{n}
  3. H C N + N a O H N a C N + H 2 O ; Δ H = - 12 k J / m o l HCN+NaOH\rightarrow NaCN+H_{2}O;\Delta H=-12kJ/mol
  4. N a O H ( a q ) + H C l ( a q ) N a C l ( a q ) + H 2 O ( l ) NaOH_{(aq)}+HCl_{(aq)}\rightarrow NaCl_{(aq)}+H_{2}O_{(l)}
  5. H ( a q ) + + O H ( a q ) - H 2 O ( l ) H^{+}_{(aq)}+OH^{-}_{(aq)}\rightarrow H_{2}O_{(l)}

Enumeration.html

  1. \mathbb{N}
  2. f : f:\mathbb{N}\to\mathbb{N}
  3. \mathbb{Z}
  4. f ( x ) := { - ( x + 1 ) / 2 , if x is odd x / 2 , if x is even . f(x):=\begin{cases}-(x+1)/2,&\mbox{if }~{}x\mbox{ is odd}\\ x/2,&\mbox{if }~{}x\mbox{ is even}~{}.\end{cases}
  5. f : f:\mathbb{N}\to\mathbb{Z}
  6. f : { 1 , 2 , , n } S f:\{1,2,\dots,n\}\to S
  7. \mathbb{N}
  8. ω 1 \omega_{1}
  9. ω 1 \omega_{1}
  10. \mathbb{N}

Epsilon_Indi.html

  1. × 10 2 7 \times 10^{2}7
  2. × 10 6 \times 10^{6}
  3. 4 π 2 T 2 = G ( M + m ) R 3 \begin{smallmatrix}\frac{4\pi^{2}}{T^{2}}=\frac{G(M+m)}{R^{3}}\end{smallmatrix}
  4. R = G ( M + m ) T 2 4 π 2 3 \begin{smallmatrix}R=\sqrt[3]{\frac{G(M+m)T^{2}}{4\pi^{2}}}\end{smallmatrix}
  5. 77 2 + 38 2 + 4 2 = 86 \begin{smallmatrix}\sqrt{77^{2}\ +\ 38^{2}\ +\ 4^{2}}\ =\ 86\end{smallmatrix}
  6. m = M v + 5 ( ( log 10 3.63 ) - 1 ) = 2.6 \begin{smallmatrix}m\ =\ M_{v}\ +\ 5\cdot((\log_{10}\ 3.63)\ -\ 1)\ =\ 2.6\end% {smallmatrix}

Equaliser_(mathematics).html

  1. Eq ( f , g ) := { x X f ( x ) = g ( x ) } . \mathrm{Eq}(f,g):=\{x\in X\mid f(x)=g(x)\}\mbox{.}~{}\!
  2. Eq ( ) := { x X f , g , f ( x ) = g ( x ) } . \mathrm{Eq}(\mathcal{F}):=\{x\in X\mid\forall{f,g\,}{\in}\,\mathcal{F},\;f(x)=% g(x)\}\mbox{.}~{}\!
  3. \mathcal{F}
  4. f e q = g e q f\circ eq=g\circ eq
  5. f m = g m f\circ m=g\circ m
  6. e q u = m eq\circ u=m
  7. m : O X m:O\rightarrow X
  8. f f
  9. g g
  10. f m = g m f\circ m=g\circ m

Equality_(mathematics).html

  1. 1 / 2 1/2
  2. 2 / 4 2/4
  3. { A , B , C } \{\,\text{A},\,\text{B},\,\text{C}\}\,
  4. { 1 , 2 , 3 } \{1,2,3\}\,
  5. A 1 , B 2 , C 3. \,\text{A}\mapsto 1,\,\text{B}\mapsto 2,\,\text{C}\mapsto 3.
  6. A 3 , B 2 , C 1 , \,\text{A}\mapsto 3,\,\text{B}\mapsto 2,\,\text{C}\mapsto 1,

Equatorial_bulge.html

  1. F c = M v 2 / R F_{c}=Mv^{2}/R
  2. f f
  3. f = a e - a p a = 5 4 ω 2 a 3 G M = 15 π 4 1 G T 2 ρ f=\frac{a_{e}-a_{p}}{a}={5\over 4}{\omega^{2}a^{3}\over GM}={15\pi\over 4}{1% \over GT^{2}\rho}
  4. a e = a ( 1 + f 3 ) a_{e}=a(1+{f\over 3})
  5. a p = a ( 1 - 2 f 3 ) a_{p}=a(1-{2f\over 3})
  6. a a
  7. ω = 2 π T \omega={2\pi\over T}
  8. T T
  9. G G
  10. M 4 3 π ρ a 3 M\simeq{4\over 3}\pi\rho a^{3}
  11. ρ \rho

Equilateral_polygon.html

  1. d 1 a 2 \frac{d_{1}}{a}\leq 2
  2. d 2 a > 3 . \frac{d_{2}}{a}>\sqrt{3}.

Equilateral_triangle.html

  1. α = β = γ \alpha=\beta=\gamma
  2. A = 3 4 a 2 A=\frac{\sqrt{3}}{4}a^{2}
  3. p = 3 a p=3a\,\!
  4. R = 3 3 a R=\frac{\sqrt{3}}{3}a
  5. r = 3 6 a r=\frac{\sqrt{3}}{6}a
  6. r = R 2 r=\frac{R}{2}
  7. h = 3 2 a h=\frac{\sqrt{3}}{2}a
  8. A = h 2 3 A=\frac{h^{2}}{\sqrt{3}}
  9. h 3 \frac{h}{3}
  10. R = 2 h 3 R=\frac{2h}{3}
  11. r = h 3 r=\frac{h}{3}
  12. a = b = c \displaystyle a=b=c
  13. a 2 + b 2 + c 2 = a b + b c + c a \displaystyle a^{2}+b^{2}+c^{2}=ab+bc+ca
  14. a b c = ( a + b - c ) ( a - b + c ) ( - a + b + c ) (Lehmus) \displaystyle abc=(a+b-c)(a-b+c)(-a+b+c)\quad\,\text{(Lehmus)}
  15. 1 a + 1 b + 1 c = 25 R r - 2 r 2 4 R r \displaystyle\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{\sqrt{25Rr-2r^{2}}}{4Rr}
  16. s = 2 R + ( 3 3 - 4 ) r (Blundon) \displaystyle s=2R+(3\sqrt{3}-4)r\quad\,\text{(Blundon)}
  17. s 2 = 3 r 2 + 12 R r \displaystyle s^{2}=3r^{2}+12Rr
  18. s 2 = 3 3 T \displaystyle s^{2}=3\sqrt{3}T
  19. s = 3 3 r \displaystyle s=3\sqrt{3}r
  20. s = 3 3 2 R \displaystyle s=\frac{3\sqrt{3}}{2}R
  21. A = B = C = 60 \displaystyle A=B=C=60^{\circ}
  22. cos A + cos B + cos C = 3 2 \displaystyle\cos{A}+\cos{B}+\cos{C}=\frac{3}{2}
  23. sin A 2 sin B 2 sin C 2 = 1 8 \displaystyle\sin{\frac{A}{2}}\sin{\frac{B}{2}}\sin{\frac{C}{2}}=\frac{1}{8}
  24. A = a 2 + b 2 + c 2 4 3 (Weizenbock) \displaystyle A=\frac{a^{2}+b^{2}+c^{2}}{4\sqrt{3}}\quad\,\text{(Weizenbock)}
  25. A = 3 4 ( a b c ) 2 3 \displaystyle A=\frac{\sqrt{3}}{4}(abc)^{{}^{\frac{2}{3}}}
  26. R = 2 r (Chapple-Euler) \displaystyle R=2r\quad\,\text{(Chapple-Euler)}
  27. 9 R 2 = a 2 + b 2 + c 2 \displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}
  28. r = r a + r b + r c 9 \displaystyle r=\frac{r_{a}+r_{b}+r_{c}}{9}
  29. r a = r b = r c \displaystyle r_{a}=r_{b}=r_{c}
  30. 4 ( p 2 + q 2 + r 2 ) x 2 + y 2 + z 2 . 4(p^{2}+q^{2}+r^{2})\geq x^{2}+y^{2}+z^{2}.
  31. π 3 3 \frac{\pi}{3\sqrt{3}}
  32. 1 12 3 , \frac{1}{12\sqrt{3}},
  33. 7 9 A 1 A 2 9 7 . \frac{7}{9}\leq\frac{A_{1}}{A_{2}}\leq\frac{9}{7}.
  34. ω \omega
  35. z 1 + ω z 2 + ω 2 z 3 = 0. z_{1}+\omega z_{2}+\omega^{2}z_{3}=0.
  36. 3 ( p 4 + q 4 + t 4 + a 4 ) = ( p 2 + q 2 + t 2 + a 2 ) 2 . \displaystyle 3(p^{4}+q^{4}+t^{4}+a^{4})=(p^{2}+q^{2}+t^{2}+a^{2})^{2}.
  37. 4 ( p 2 + q 2 + t 2 ) = 5 a 2 \displaystyle 4(p^{2}+q^{2}+t^{2})=5a^{2}
  38. 16 ( p 4 + q 4 + t 4 ) = 11 a 4 . \displaystyle 16(p^{4}+q^{4}+t^{4})=11a^{4}.
  39. p = q + t \displaystyle p=q+t
  40. q 2 + q t + t 2 = a 2 ; \displaystyle q^{2}+qt+t^{2}=a^{2};
  41. z = t 2 + t q + q 2 t + q , z=\frac{t^{2}+tq+q^{2}}{t+q},
  42. t 3 - q 3 t 2 - q 2 \tfrac{t^{3}-q^{3}}{t^{2}-q^{2}}
  43. 1 q + 1 t = 1 y , \frac{1}{q}+\frac{1}{t}=\frac{1}{y},
  44. A = 3 4 a 2 A=\frac{\sqrt{3}}{4}a^{2}
  45. A = 1 2 a h . A=\frac{1}{2}ah.
  46. ( a 2 ) 2 + h 2 = a 2 \left(\frac{a}{2}\right)^{2}+h^{2}=a^{2}
  47. h = 3 2 a . h=\frac{\sqrt{3}}{2}a.
  48. A = 3 4 a 2 . A=\frac{\sqrt{3}}{4}a^{2}.
  49. A = 1 2 a b sin C . A=\frac{1}{2}ab\sin C.
  50. A = 1 2 a b sin 60 . A=\frac{1}{2}ab\sin 60^{\circ}.
  51. 3 2 \tfrac{\sqrt{3}}{2}
  52. A = 1 2 a b × 3 2 = 3 4 a b = 3 4 a 2 A=\frac{1}{2}ab\times\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}ab=\frac{\sqrt{3}}{4% }a^{2}

Equivalent_potential_temperature.html

  1. ( θ e ) \left(\theta_{e}\right)
  2. θ e \theta_{e}
  3. θ e \theta_{e}
  4. θ e \theta_{e}
  5. θ e = θ L exp [ ( 3036 T L - 1.78 ) r ( 1 + 0.448 r ) ] \theta_{e}=\theta_{L}\exp\left[\left(\frac{3036}{T_{L}}-1.78\right)r\left(1+0.% 448r\right)\right]
  6. θ L = T ( p 0 p - e ) κ d ( T T L ) 0.28 r \theta_{L}=T\left(\frac{p_{0}}{p-e}\right)^{\kappa_{d}}\left(\frac{T}{T_{L}}% \right)^{0.28r}
  7. T L = 1 1 T d - 56 + log e ( T / T d ) 800 + 56 T_{L}=\frac{1}{\frac{1}{T_{d}-56}+\frac{\log_{e}(T/T_{d})}{800}}+56
  8. θ L \theta_{L}
  9. T L T_{L}
  10. T T
  11. p p
  12. T d T_{d}
  13. p p
  14. p p
  15. p 0 p_{0}
  16. e e
  17. θ L \theta_{L}
  18. κ d = R d / c p d \kappa_{d}=R_{d}/c_{pd}
  19. r r
  20. θ e θ L exp [ r s ( T L ) L v ( T L ) c p d T L ] \theta_{e}\approx\theta_{L}\exp\left[\frac{r_{s}(T_{L})L_{v}(T_{L})}{c_{pd}T_{% L}}\right]
  21. r s ( T L ) r_{s}(T_{L})
  22. T L T_{L}
  23. L v ( T L ) L_{v}(T_{L})
  24. T L T_{L}
  25. c p d c_{pd}
  26. T L T_{L}
  27. θ e = T e ( p 0 p ) R d c p d ( T + L v c p d r ) ( p 0 p ) R d c p d \theta_{e}=T_{e}\left(\frac{p_{0}}{p}\right)^{\frac{R_{d}}{c_{pd}}}\approx% \left(T+\frac{L_{v}}{c_{pd}}r\right)\left(\frac{p_{0}}{p}\right)^{\frac{R_{d}}% {c_{pd}}}
  28. T e T_{e}
  29. R d R_{d}
  30. θ e z > 0 \frac{\partial\theta_{e}}{\partial z}>0
  31. θ e z < 0 \frac{\partial\theta_{e}}{\partial z}<0

Equivalent_rectangular_bandwidth.html

  1. { ERBS ( 0 ) = 0 d f d ERBS ( f ) = ERB ( f ) \begin{cases}\mathrm{ERBS}(0)=0\\ \frac{df}{d\mathrm{ERBS}(f)}=\mathrm{ERB}(f)\\ \end{cases}
  2. ERBS ( f ) = 11.17 ln ( f + 0.312 f + 14.675 ) + 43.0 \mathrm{ERBS}(f)=11.17\cdot\ln\left(\frac{f+0.312}{f+14.675}\right)+43.0
  3. ERBS ( f ) = 11.17268 ln ( 1 + 46.06538 f f + 14678.49 ) \mathrm{ERBS}(f)=11.17268\cdot\ln\left(1+\frac{46.06538\cdot f}{f+14678.49}\right)
  4. f = 676170.4 47.06538 - e 0.08950404 ERBS ( f ) - 14678.49 f=\frac{676170.4}{47.06538-e^{0.08950404\cdot\mathrm{ERBS}(f)}}-14678.49
  5. ERBS ( f ) = 21.4 l o g 10 ( 1 + 0.00437 f ) \mathrm{ERBS}(f)=21.4\cdot log_{10}(1+0.00437\cdot f)

ER.html

  1. ϵ r \epsilon_{r}

Erdős–Borwein_constant.html

  1. E = n = 1 1 2 n - 1 1.606695152415291763 E=\sum_{n=1}^{\infty}\frac{1}{2^{n}-1}\approx 1.606695152415291763\dots
  2. E = n = 1 1 2 n 2 2 n + 1 2 n - 1 E=\sum_{n=1}^{\infty}\frac{1}{2^{n^{2}}}\frac{2^{n}+1}{2^{n}-1}
  3. E = m = 1 n = 1 1 2 m n E=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{2^{mn}}
  4. E = 1 + n = 1 1 2 n ( 2 n - 1 ) E=1+\sum_{n=1}^{\infty}\frac{1}{2^{n}(2^{n}-1)}
  5. E = n = 1 σ 0 ( n ) 2 n E=\sum_{n=1}^{\infty}\frac{\sigma_{0}(n)}{2^{n}}

Erdős–Ko–Rado_theorem.html

  1. n 2 r n\geq 2r
  2. A A
  3. { 1 , 2 , , n } \{1,2,...,n\}
  4. r r
  5. A A
  6. ( n - 1 r - 1 ) . {\left({{n-1}\atop{r-1}}\right)}.
  7. A A
  8. A A
  9. r r
  10. r r
  11. r r
  12. | A | r ! ( n - r ) ! r ( n - 1 ) ! |A|r!(n-r)!\leq r(n-1)!
  13. | A | r ( n - 1 ) ! r ! ( n - r ) ! = ( n - 1 r - 1 ) . |A|\leq\frac{r(n-1)!}{r!(n-r)!}={n-1\choose r-1}.
  14. { 1 , 2 , , n } \{1,2,...,n\}
  15. { 1 , 2 , , n } \{1,2,...,n\}

Ergodic_theory.html

  1. T : X X \quad T:X\to X\quad
  2. ( X , Σ , μ ) \;(X,\Sigma,\mu)\;
  3. μ ( X ) = 1 \mu(X)=1
  4. f ^ ( x ) = lim n 1 n k = 0 n - 1 f ( T k x ) . \hat{f}(x)=\lim_{n\rightarrow\infty}\;\frac{1}{n}\sum_{k=0}^{n-1}f\left(T^{k}x% \right).
  5. f ¯ = 1 μ ( X ) f d μ . (For a probability space, μ ( X ) = 1. ) \bar{f}=\frac{1}{\mu(X)}\int f\,d\mu.\quad\,\text{ (For a probability space, }% \mu(X)=1.)
  6. f ^ L 1 ( μ ) . \hat{f}\in L^{1}(\mu).\,
  7. f ^ T = f ^ \hat{f}\circ T=\hat{f}\,
  8. f ^ d μ = f d μ . \int\hat{f}\,d\mu=\int f\,d\mu.
  9. f ¯ = f ^ \bar{f}=\hat{f}\,
  10. lim n 1 n k = 0 n - 1 f ( T k x ) = 1 μ ( X ) f d μ \lim_{n\rightarrow\infty}\;\frac{1}{n}\sum_{k=0}^{n-1}f\left(T^{k}x\right)=% \frac{1}{\mu(X)}\int f\,d\mu
  11. E ( f | 𝒞 ) E(f|\mathcal{C})
  12. 𝒞 \mathcal{C}
  13. 𝒞 \mathcal{C}
  14. lim n 1 n k = 0 n - 1 f ( T k x ) = E ( f ) . \lim_{n\rightarrow\infty}\;\frac{1}{n}\sum_{k=0}^{n-1}f\left(T^{k}x\right)=E(f).
  15. lim N 1 N n = 0 N - 1 U n x = P x , \lim_{N\to\infty}{1\over N}\sum_{n=0}^{N-1}U^{n}x=Px,
  16. 1 N n = 0 N - 1 U n \frac{1}{N}\sum_{n=0}^{N-1}U^{n}
  17. x H x\in H
  18. ker ( I - U ) \ker(I-U)
  19. Ran ( I - U ) ¯ \overline{\mathrm{Ran}(I-U)}
  20. N N
  21. lim N 1 N n = 0 N - 1 U n ( I - U ) = lim N 1 N ( I - U N ) = 0 \lim_{N\to\infty}{1\over N}\sum_{n=0}^{N-1}U^{n}(I-U)=\lim_{N\to\infty}{1\over N% }(I-U^{N})=0
  22. U f ( x ) = f ( T x ) Uf(x)=f(Tx)\,
  23. 1 T 0 T U t d t \frac{1}{T}\int_{0}^{T}U_{t}\,dt
  24. μ ( A ) μ ( X ) = 1 μ ( X ) χ A d μ = lim n 1 n k = 0 n - 1 χ A ( T k x ) \frac{\mu(A)}{\mu(X)}=\frac{1}{\mu(X)}\int\chi_{A}\,d\mu=\lim_{n\rightarrow% \infty}\;\frac{1}{n}\sum_{k=0}^{n-1}\chi_{A}\left(T^{k}x\right)
  25. R 1 + + R n n μ ( X ) μ ( A ) (almost surely) \frac{R_{1}+\cdots+R_{n}}{n}\rightarrow\frac{\mu(X)}{\mu(A)}\quad\mbox{(almost% surely)}~{}

Erlang_distribution.html

  1. γ ( k , λ x ) ( k - 1 ) ! = 1 - n = 0 k - 1 1 n ! e - λ x ( λ x ) n \scriptstyle\frac{\gamma(k,\,\lambda x)}{(k\,-\,1)!}\;=\;1\,-\,\sum_{n=0}^{k-1% }\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}
  2. k λ \scriptstyle\frac{k}{\lambda}\,
  3. 1 λ ( k - 1 ) \scriptstyle\frac{1}{\lambda}(k\,-\,1)\,
  4. k 1 \scriptstyle k\;\geq\;1\,
  5. k λ 2 \scriptstyle\frac{k}{\lambda^{2}}\,
  6. 2 k \scriptstyle\frac{2}{\sqrt{k}}
  7. 1 k \scriptstyle\frac{1}{k}
  8. 6 k \scriptstyle\frac{6}{k}
  9. ( 1 - k ) ψ ( k ) + ln [ Γ ( k ) λ ] + k \scriptstyle(1\,-\,k)\psi(k)\,+\,\ln\left[\frac{\Gamma(k)}{\lambda}\right]\,+\,k
  10. ( 1 - t λ ) - k \scriptstyle\left(1\,-\,\frac{t}{\lambda}\right)^{-k}\,
  11. t < λ \scriptstyle t\;<\;\lambda\,
  12. ( 1 - i t λ ) - k \scriptstyle\left(1\,-\,\frac{it}{\lambda}\right)^{-k}\,
  13. x ( 0 , ) x\;\in\;(0,\,\infty)
  14. k k
  15. λ \lambda
  16. μ \mu
  17. k k
  18. k k
  19. μ \mu
  20. f ( x ; k , λ ) = λ k x k - 1 e - λ x ( k - 1 ) ! for x , λ 0 , f(x;k,\lambda)={\lambda^{k}x^{k-1}e^{-\lambda x}\over(k-1)!}\quad\mbox{for }~{% }x,\lambda\geq 0,
  21. λ \lambda
  22. μ \mu
  23. μ = 1 / λ \mu=1/\lambda
  24. f ( x ; k , μ ) = x k - 1 e - x μ μ k ( k - 1 ) ! for x , μ 0. f(x;k,\mu)=\frac{x^{k-1}e^{-\frac{x}{\mu}}}{\mu^{k}(k-1)!}\quad\mbox{for }~{}x% ,\mu\geq 0.
  25. μ \mu
  26. F ( x ; k , λ ) = γ ( k , λ x ) ( k - 1 ) ! , F(x;k,\lambda)=\frac{\gamma(k,\lambda x)}{(k-1)!},
  27. γ ( ) \gamma()
  28. F ( x ; k , λ ) = 1 - n = 0 k - 1 1 n ! e - λ x ( λ x ) n . F(x;k,\lambda)=1-\sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}.
  29. x f ( x ) + ( λ x + 1 - k ) f ( x ) = 0 xf^{\prime}(x)+(\lambda x+1-k)f(x)=0
  30. f ( 1 ) = e - λ λ k Γ ( k ) f(1)=\frac{e^{-\lambda}\lambda^{k}}{\Gamma(k)}
  31. k λ ( 1 - 1 3 k + 0.2 ) \dfrac{k}{\lambda}\left(1-\dfrac{1}{3k+0.2}\right)
  32. k λ \tfrac{k}{\lambda}
  33. U ( 0 , 1 ] U\in(0,1]
  34. E ( k , λ ) - 1 λ ln i = 1 k U i E(k,\lambda)\approx-\frac{1}{\lambda}\ln\prod_{i=1}^{k}U_{i}
  35. X X
  36. λ / k \lambda/k
  37. k > 1 k>1
  38. x x
  39. x = 0 x=0
  40. λ \lambda
  41. x x
  42. X Erlang ( k , λ ) \scriptstyle X\;\sim\;\mathrm{Erlang}(k,\,\lambda)\,
  43. a X Erlang ( k , λ a ) \scriptstyle a\cdot X\;\sim\;\mathrm{Erlang}\left(k,\,\frac{\lambda}{a}\right)\,
  44. a \scriptstyle a\in\mathbb{R}
  45. lim k 1 σ k ( Erlang ( k , λ ) - μ k ) 𝑑 N ( 0 , 1 ) \scriptstyle\lim_{k\to\infty}\frac{1}{\sigma_{k}}\left(\mathrm{Erlang}(k,\,% \lambda)\,-\,\mu_{k}\right)\;\xrightarrow{d}\;N(0,\,1)\,
  46. X Erlang ( k 1 , λ ) \scriptstyle X\;\sim\;\mathrm{Erlang}(k_{1},\,\lambda)\,
  47. Y Erlang ( k 2 , λ ) \scriptstyle Y\;\sim\;\mathrm{Erlang}(k_{2},\,\lambda)\,
  48. X + Y Erlang ( k 1 + k 2 , λ ) \scriptstyle X\,+\,Y\;\sim\;\mathrm{Erlang}(k_{1}\,+\,k_{2},\,\lambda)\,
  49. X i \scriptstyle X_{i}\;\sim\;
  50. i = 1 k X i Erlang ( k , λ ) \scriptstyle\sum_{i=1}^{k}{X_{i}}\;\sim\;\mathrm{Erlang}(k,\,\lambda)\,
  51. X Γ ( k , 1 λ ) \scriptstyle X\;\sim\;\Gamma\left(k,\,\frac{1}{\lambda}\right)\,
  52. X Erlang ( k , λ ) \scriptstyle X\;\sim\;\mathrm{Erlang}(k,\,\lambda)\,
  53. U Exponential ( λ ) \scriptstyle U\;\sim\;\mathrm{Exponential}(\lambda)\,
  54. V Erlang ( n , λ ) \scriptstyle V\;\sim\;\mathrm{Erlang}(n,\,\lambda)\,
  55. U V Pareto ( 1 , n ) \scriptstyle\frac{U}{V}\;\sim\;\mathrm{Pareto}(1,\,n)

Erlangen_program.html

  1. O ( n , 1 ) / C 2 , \mathrm{O}(n,1)/\mathrm{C}_{2},

Error_function.html

  1. erf ( x ) = 2 π 0 x e - t 2 d t . \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\,\mathrm{d}t.
  2. erfc ( x ) = 1 - erf ( x ) = 2 π x e - t 2 d t = e - x 2 erfcx ( x ) , \begin{aligned}\displaystyle\operatorname{erfc}(x)&\displaystyle=1-% \operatorname{erf}(x)\\ &\displaystyle=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t\\ &\displaystyle=e^{-x^{2}}\operatorname{erfcx}(x),\end{aligned}
  3. erfc ( x ) \operatorname{erfc}(x)
  4. erfc ( x ) = 2 π 0 π 2 exp ( - x 2 sin 2 θ ) d θ . \begin{aligned}\displaystyle\operatorname{erfc}(x)&\displaystyle=\frac{2}{\pi}% \int_{0}^{\frac{\pi}{2}}\exp\left(-\frac{x^{2}}{\sin^{2}\theta}\right)d\theta.% \end{aligned}
  5. erfi ( x ) = - i erf ( i x ) = 2 π e x 2 D ( x ) . \operatorname{erfi}(x)=-i\operatorname{erf}(ix)=\frac{2}{\sqrt{\pi}}e^{x^{2}}D% (x).
  6. w ( z ) = e - z 2 erfc ( - i z ) = erfcx ( - i z ) . w(z)=e^{-z^{2}}\operatorname{erfc}(-iz)=\operatorname{erfcx}(-iz).
  7. Φ \Phi
  8. Φ ( x ) = 1 2 + 1 2 erf ( x / 2 ) = 1 2 erfc ( - x / 2 ) . \Phi(x)=\frac{1}{2}+\frac{1}{2}\operatorname{erf}\left(x/\sqrt{2}\right)=\frac% {1}{2}\operatorname{erfc}\left(-x/\sqrt{2}\right).
  9. x σ 2 \frac{x}{\sigma\sqrt{2}}
  10. σ \sigma
  11. erf ( - z ) = - erf ( z ) \operatorname{erf}(-z)=-\operatorname{erf}(z)
  12. e - t 2 e^{-t^{2}}
  13. erf ( z ¯ ) = erf ( z ) ¯ \operatorname{erf}(\overline{z})=\overline{\operatorname{erf}(z)}
  14. z ¯ \overline{z}
  15. erf ( z ) = 2 π n = 0 ( - 1 ) n z 2 n + 1 n ! ( 2 n + 1 ) = 2 π ( z - z 3 3 + z 5 10 - z 7 42 + z 9 216 - ) \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{% 2n+1}}{n!(2n+1)}=\frac{2}{\sqrt{\pi}}\left(z-\frac{z^{3}}{3}+\frac{z^{5}}{10}-% \frac{z^{7}}{42}+\frac{z^{9}}{216}-\ \cdots\right)
  16. erf ( z ) = 2 π n = 0 ( z k = 1 n - ( 2 k - 1 ) z 2 k ( 2 k + 1 ) ) = 2 π n = 0 z 2 n + 1 k = 1 n - z 2 k \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\left(z\prod_{k=1% }^{n}{\frac{-(2k-1)z^{2}}{k(2k+1)}}\right)=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{% \infty}\frac{z}{2n+1}\prod_{k=1}^{n}\frac{-z^{2}}{k}
  17. - ( 2 k - 1 ) z 2 k ( 2 k + 1 ) \frac{-(2k-1)z^{2}}{k(2k+1)}
  18. d dz erf ( z ) = 2 π e - z 2 . \frac{\rm d}{{\rm d}z}\,\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\,e^{-z^{2}}.
  19. z erf ( z ) + e - z 2 π . z\,\operatorname{erf}(z)+\frac{e^{-z^{2}}}{\sqrt{\pi}}.
  20. erf ( k ) ( z ) = ( - 1 ) k - 1 2 ( k + 1 ) / 2 π 𝐻𝑒 k - 1 ( 2 z ) exp - z 2 , k = 1 , 2 , {\operatorname{erf}}^{(k)}(z)={(-1)^{k-1}2^{(k+1)/2}\over\sqrt{\pi}}\mathit{He% }_{k-1}\Big(\sqrt{2}z\Big)\exp-z^{2},\qquad k=1,2,\dots
  21. 𝐻𝑒 \mathit{He}
  22. x x
  23. erf ( x ) = 2 π sgn ( x ) 1 - e - x 2 ( 1 - 1 12 ( 1 - e - x 2 ) - 7 480 ( 1 - e - x 2 ) 2 - 5 896 ( 1 - e - x 2 ) 3 - 787 276480 ( 1 - e - x 2 ) 4 - ) = 2 π sgn ( x ) 1 - e - x 2 ( π 2 + k = 1 c k e - k x 2 ) . \begin{aligned}\displaystyle\operatorname{erf}(x)&\displaystyle=\frac{2}{\sqrt% {\pi}}\operatorname{sgn}(x)\sqrt{1-e^{-x^{2}}}\left(1-\frac{1}{12}(1-e^{-x^{2}% })-\frac{7}{480}(1-e^{-x^{2}})^{2}-\frac{5}{896}(1-e^{-x^{2}})^{3}-\frac{787}{% 276480}(1-e^{-x^{2}})^{4}-\ \cdots\right)\\ &\displaystyle=\frac{2}{\sqrt{\pi}}\operatorname{sgn}(x)\sqrt{1-e^{-x^{2}}}% \left(\frac{\sqrt{\pi}}{2}+\sum_{k=1}^{\infty}c_{k}e^{-k\,x^{2}}\right).\end{aligned}
  24. c 1 = 31 200 c_{1}=\frac{31}{200}
  25. c 2 = - 341 8000 c_{2}=-\frac{341}{8000}
  26. x = ± 1 , 3796 \textstyle x=\pm 1{,}3796
  27. 3 , 6127 10 - 3 \textstyle 3{,}6127\cdot 10^{-3}
  28. erf ( x ) 2 π sgn ( x ) 1 - e - x 2 ( π 2 + 31 200 e - x 2 - 341 8000 e - 2 x 2 ) . \operatorname{erf}(x)\approx\frac{2}{\sqrt{\pi}}\operatorname{sgn}(x)\sqrt{1-e% ^{-x^{2}}}\left(\frac{\sqrt{\pi}}{2}+\frac{31}{200}\,e^{-x^{2}}-\frac{341}{800% 0}\,e^{-2\,x^{2}}\right).
  29. erf - 1 ( z ) = k = 0 c k 2 k + 1 ( π 2 z ) 2 k + 1 , \operatorname{erf}^{-1}(z)=\sum_{k=0}^{\infty}\frac{c_{k}}{2k+1}\left(\frac{% \sqrt{\pi}}{2}z\right)^{2k+1},\,\!
  30. c k = m = 0 k - 1 c m c k - 1 - m ( m + 1 ) ( 2 m + 1 ) = { 1 , 1 , 7 6 , 127 90 , 4369 2520 , 34807 16200 , } . c_{k}=\sum_{m=0}^{k-1}\frac{c_{m}c_{k-1-m}}{(m+1)(2m+1)}=\left\{1,1,\frac{7}{6% },\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}.
  31. erf - 1 ( z ) = 1 2 π ( z + π 12 z 3 + 7 π 2 480 z 5 + 127 π 3 40320 z 7 + 4369 π 4 5806080 z 9 + 34807 π 5 182476800 z 11 + ) . \operatorname{erf}^{-1}(z)=\tfrac{1}{2}\sqrt{\pi}\left(z+\frac{\pi}{12}z^{3}+% \frac{7\pi^{2}}{480}z^{5}+\frac{127\pi^{3}}{40320}z^{7}+\frac{4369\pi^{4}}{580% 6080}z^{9}+\frac{34807\pi^{5}}{182476800}z^{11}+\cdots\right).
  32. erfc - 1 ( 1 - z ) = erf - 1 ( z ) . \operatorname{erfc}^{-1}(1-z)=\operatorname{erf}^{-1}(z).
  33. erfc ( x ) = e - x 2 x π [ 1 + n = 1 ( - 1 ) n 1 3 5 ( 2 n - 1 ) ( 2 x 2 ) n ] = e - x 2 x π n = 0 ( - 1 ) n ( 2 n - 1 ) ! ! ( 2 x 2 ) n , \operatorname{erfc}(x)=\frac{e^{-x^{2}}}{x\sqrt{\pi}}\left[1+\sum_{n=1}^{% \infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{(2x^{2})^{n}}\right]=\frac{e% ^{-x^{2}}}{x\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\frac{(2n-1)!!}{(2x^{2})^{n}% },\,
  34. N 𝒩 N\in\mathcal{N}
  35. erfc ( x ) = e - x 2 x π n = 0 N - 1 ( - 1 ) n ( 2 n - 1 ) ! ! ( 2 x 2 ) n + R N ( x ) \operatorname{erfc}(x)=\frac{e^{-x^{2}}}{x\sqrt{\pi}}\sum_{n=0}^{N-1}(-1)^{n}% \frac{(2n-1)!!}{(2x^{2})^{n}}+R_{N}(x)\,
  36. R N ( x ) = O ( x - 2 N + 1 e - x 2 ) R_{N}(x)=O(x^{-2N+1}e^{-x^{2}})
  37. x . x\to\infty.
  38. R N ( x ) := ( - 1 ) N π 2 - 2 N + 1 ( 2 N ) ! N ! x t - 2 N e - t 2 d t , R_{N}(x):=\frac{(-1)^{N}}{\sqrt{\pi}}2^{-2N+1}\frac{(2N)!}{N!}\int_{x}^{\infty% }t^{-2N}e^{-t^{2}}\,\mathrm{d}t,
  39. e - t 2 = - ( 2 t ) - 1 ( e - t 2 ) e^{-t^{2}}=-(2t)^{-1}(e^{-t^{2}})^{\prime}
  40. erfc ( z ) = z π e - z 2 1 z 2 + a 1 1 + a 2 z 2 + a 3 1 + a m = m 2 . \operatorname{erfc}(z)=\frac{z}{\sqrt{\pi}}e^{-z^{2}}\cfrac{1}{z^{2}+\cfrac{a_% {1}}{1+\cfrac{a_{2}}{z^{2}+\cfrac{a_{3}}{1+\cdots}}}}\qquad a_{m}=\frac{m}{2}.
  41. erf [ b - a c 1 + 2 a 2 d 2 ] = - d x erf ( a x + b ) 2 π d 2 exp [ - ( x + c ) 2 2 d 2 ] , a , b , c , d \operatorname{erf}\left[\frac{b-ac}{\sqrt{1+2a^{2}d^{2}}}\right]=\int\limits_{% -\infty}^{\infty}{\rm d}x\frac{\operatorname{erf}\left(ax+b\right)}{\sqrt{2\pi d% ^{2}}}\exp{\left[-\frac{(x+c)^{2}}{2d^{2}}\right]},\ \ a,b,c,d\in\mathbb{R}
  42. erf ( x ) 1 - 1 ( 1 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 ) 4 \operatorname{erf}(x)\approx 1-\frac{1}{(1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x% ^{4})^{4}}
  43. erf ( x ) 1 - ( a 1 t + a 2 t 2 + a 3 t 3 ) e - x 2 , t = 1 1 + p x \operatorname{erf}(x)\approx 1-(a_{1}t+a_{2}t^{2}+a_{3}t^{3})e^{-x^{2}},\quad t% =\frac{1}{1+px}
  44. erf ( x ) 1 - 1 ( 1 + a 1 x + a 2 x 2 + + a 6 x 6 ) 16 \operatorname{erf}(x)\approx 1-\frac{1}{(1+a_{1}x+a_{2}x^{2}+\cdots+a_{6}x^{6}% )^{16}}
  45. erf ( x ) 1 - ( a 1 t + a 2 t 2 + + a 5 t 5 ) e - x 2 , t = 1 1 + p x \operatorname{erf}(x)\approx 1-(a_{1}t+a_{2}t^{2}+\cdots+a_{5}t^{5})e^{-x^{2}}% ,\quad t=\frac{1}{1+px}
  46. erf ( x ) sgn ( x ) 1 - exp ( - x 2 4 / π + a x 2 1 + a x 2 ) \operatorname{erf}(x)\approx\operatorname{sgn}(x)\sqrt{1-\exp\left(-x^{2}\frac% {4/\pi+ax^{2}}{1+ax^{2}}\right)}
  47. a = 8 ( π - 3 ) 3 π ( 4 - π ) 0.140012. a=\frac{8(\pi-3)}{3\pi(4-\pi)}\approx 0.140012.
  48. erf - 1 ( x ) sgn ( x ) ( 2 π a + ln ( 1 - x 2 ) 2 ) 2 - ln ( 1 - x 2 ) a - ( 2 π a + ln ( 1 - x 2 ) 2 ) . \operatorname{erf}^{-1}(x)\approx\operatorname{sgn}(x)\sqrt{\sqrt{\left(\frac{% 2}{\pi a}+\frac{\ln(1-x^{2})}{2}\right)^{2}-\frac{\ln(1-x^{2})}{a}}-\left(% \frac{2}{\pi a}+\frac{\ln(1-x^{2})}{2}\right)}.
  49. erfc ( x ) 1 2 e - 2 x 2 + 1 2 e - x 2 e - x 2 , x > 0 \operatorname{erfc}(x)\leq\frac{1}{2}e^{-2x^{2}}+\frac{1}{2}e^{-x^{2}}\leq e^{% -x^{2}},\qquad x>0\,
  50. erfc ( x ) 1 6 e - x 2 + 1 2 e - 4 3 x 2 , x > 0 . \operatorname{erfc}(x)\approx\frac{1}{6}e^{-x^{2}}+\frac{1}{2}e^{-\frac{4}{3}x% ^{2}},\qquad x>0\,.
  51. erfc ( x ) 2 e π β - 1 β e - β x 2 , x 0 , β > 1 , \operatorname{erfc}(x)\geq\sqrt{\frac{2e}{\pi}}\frac{\sqrt{\beta-1}}{\beta}e^{% -\beta x^{2}},\qquad x\geq 0,\,\beta>1,
  52. 1.2 × 10 - 7 1.2\times 10^{-7}
  53. erf ( x ) = { 1 - τ for x 0 τ - 1 for x < 0 \operatorname{erf}(x)=\begin{cases}1-\tau&\,\text{for }x\geq 0\\ \tau-1&\,\text{for }x<0\end{cases}
  54. τ = t exp ( - x 2 - 1.26551223 + 1.00002368 t + 0.37409196 t 2 + 0.09678418 t 3 - 0.18628806 t 4 + 0.27886807 t 5 - 1.13520398 t 6 + 1.48851587 t 7 - 0.82215223 t 8 + 0.17087277 t 9 ) \begin{aligned}\displaystyle\tau=&\displaystyle t\cdot\exp\left(-x^{2}-1.26551% 223+1.00002368t+0.37409196t^{2}+0.09678418t^{3}\right.\\ &\displaystyle\left.{}-0.18628806t^{4}+0.27886807t^{5}-1.13520398t^{6}+1.48851% 587\cdot t^{7}\right.\\ &\displaystyle\left.{}-0.82215223t^{8}+0.17087277t^{9}\right)\end{aligned}
  55. t = 1 1 + 0.5 | x | . t=\frac{1}{1+0.5|x|}.
  56. σ \textstyle\sigma
  57. erf ( a σ 2 ) \textstyle\operatorname{erf}\,\left(\,\frac{a}{\sigma\sqrt{2}}\,\right)
  58. X Norm [ μ , σ ] X\sim\operatorname{Norm}[\mu,\sigma]
  59. L < μ L<\mu
  60. Pr [ X L ] = 1 2 + 1 2 erf ( L - μ 2 σ ) A exp ( - B ( L - μ σ ) 2 ) \Pr[X\leq L]=\frac{1}{2}+\frac{1}{2}\operatorname{erf}\left(\frac{L-\mu}{\sqrt% {2}\sigma}\right)\approx A\exp\left(-B\left(\frac{L-\mu}{\sigma}\right)^{2}\right)
  61. μ - L σ ln k \mu-L\geq\sigma\sqrt{\ln{k}}
  62. Pr [ X L ] A exp ( - B ln k ) = A k B \Pr[X\leq L]\leq A\exp(-B\ln{k})=\frac{A}{k^{B}}
  63. k k\to\infty
  64. Φ ( x ) = 1 2 π - x e - t 2 2 d t = 1 2 [ 1 + erf ( x 2 ) ] = 1 2 erfc ( - x 2 ) \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{\tfrac{-t^{2}}{2}}\,\mathrm{% d}t=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)\right% ]=\frac{1}{2}\,\operatorname{erfc}\left(-\frac{x}{\sqrt{2}}\right)
  65. erf ( x ) \displaystyle\operatorname{erf}(x)
  66. Q ( x ) = 1 2 - 1 2 erf ( x 2 ) = 1 2 erfc ( x 2 ) . Q(x)=\frac{1}{2}-\frac{1}{2}\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)=% \frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right).
  67. Φ \textstyle\Phi\,
  68. probit ( p ) = Φ - 1 ( p ) = 2 erf - 1 ( 2 p - 1 ) = - 2 erfc - 1 ( 2 p ) . \operatorname{probit}(p)=\Phi^{-1}(p)=\sqrt{2}\,\operatorname{erf}^{-1}(2p-1)=% -\sqrt{2}\,\operatorname{erfc}^{-1}(2p).
  69. erf ( x ) = 2 x π 1 F 1 ( 1 2 , 3 2 , - x 2 ) . \operatorname{erf}(x)=\frac{2x}{\sqrt{\pi}}\,_{1}F_{1}\left(\tfrac{1}{2},% \tfrac{3}{2},-x^{2}\right).
  70. erf ( x ) = sgn ( x ) P ( 1 2 , x 2 ) = sgn ( x ) π γ ( 1 2 , x 2 ) . \operatorname{erf}(x)=\operatorname{sgn}(x)P\left(\tfrac{1}{2},x^{2}\right)={% \operatorname{sgn}(x)\over\sqrt{\pi}}\gamma\left(\tfrac{1}{2},x^{2}\right).
  71. sgn ( x ) \textstyle\operatorname{sgn}(x)
  72. E n ( x ) = n ! π 0 x e - t n d t = n ! π p = 0 ( - 1 ) p x n p + 1 ( n p + 1 ) p ! . E_{n}(x)=\frac{n!}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{n}}\,\mathrm{d}t=\frac{n!}{% \sqrt{\pi}}\sum_{p=0}^{\infty}(-1)^{p}\frac{x^{np+1}}{(np+1)p!}\,.
  73. E 0 ( x ) = x e π \textstyle E_{0}(x)=\frac{x}{e\sqrt{\pi}}
  74. E n ( x ) = Γ ( n ) ( Γ ( 1 n ) - Γ ( 1 n , x n ) ) π , x > 0. E_{n}(x)=\frac{\Gamma(n)\left(\Gamma\left(\frac{1}{n}\right)-\Gamma\left(\frac% {1}{n},x^{n}\right)\right)}{\sqrt{\pi}},\quad\quad x>0.
  75. erf ( x ) = 1 - Γ ( 1 2 , x 2 ) π . \operatorname{erf}(x)=1-\frac{\Gamma\left(\frac{1}{2},x^{2}\right)}{\sqrt{\pi}}.
  76. i n erfc ( z ) = z i n - 1 erfc ( ζ ) d ζ . \mathrm{i}^{n}\operatorname{erfc}\,(z)=\int_{z}^{\infty}\mathrm{i}^{n-1}% \operatorname{erfc}\,(\zeta)\;\mathrm{d}\zeta.\,
  77. i n erfc ( z ) = j = 0 ( - z ) j 2 n - j j ! Γ ( 1 + n - j 2 ) , \mathrm{i}^{n}\operatorname{erfc}\,(z)=\sum_{j=0}^{\infty}\frac{(-z)^{j}}{2^{n% -j}j!\Gamma\left(1+\frac{n-j}{2}\right)}\,,
  78. i 2 m erfc ( - z ) = - i 2 m erfc ( z ) + q = 0 m z 2 q 2 2 ( m - q ) - 1 ( 2 q ) ! ( m - q ) ! \mathrm{i}^{2m}\operatorname{erfc}(-z)=-\mathrm{i}^{2m}\operatorname{erfc}\,(z% )+\sum_{q=0}^{m}\frac{z^{2q}}{2^{2(m-q)-1}(2q)!(m-q)!}
  79. i 2 m + 1 erfc ( - z ) = i 2 m + 1 erfc ( z ) + q = 0 m z 2 q + 1 2 2 ( m - q ) - 1 ( 2 q + 1 ) ! ( m - q ) ! . \mathrm{i}^{2m+1}\operatorname{erfc}(-z)=\mathrm{i}^{2m+1}\operatorname{erfc}% \,(z)+\sum_{q=0}^{m}\frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)!(m-q)!}\,.
  80. i erf ( i x ) i\operatorname{erf}(ix)

Erythrocyte_sedimentation_rate.html

  1. ESR ( m m / h ) Age ( 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠 ) + 10 ( 𝑖𝑓 𝑓𝑒𝑚𝑎𝑙𝑒 ) 2 {\rm ESR}\ (mm/h)\leq\frac{{\rm Age}\ ({\it in\ years})+10\ ({\it if\ female})% }{2}

Essential_singularity.html

  1. lim z a f ( z ) \lim_{z\to a}f(z)
  2. lim z a 1 f ( z ) \lim_{z\to a}\frac{1}{f(z)}
  3. lim z a f ( z ) \lim_{z\to a}f(z)
  4. lim z a 1 f ( z ) \lim_{z\to a}\frac{1}{f(z)}
  5. lim z a f ( z ) \lim_{z\to a}f(z)
  6. lim z a 1 f ( z ) \lim_{z\to a}\frac{1}{f(z)}
  7. lim z a f ( z ) \lim_{z\to a}f(z)
  8. lim z a 1 f ( z ) \lim_{z\to a}\frac{1}{f(z)}
  9. a a
  10. f ( z ) ( z - a ) n f(z)(z-a)^{n}
  11. n > 0 n>0
  12. a a
  13. f ( z ) f(z)

Eternity.html

  1. \infty

Ethylene_oxide.html

  1. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖱𝖬𝗀𝖡𝗋 𝖱 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖬𝗀𝖡𝗋 𝖧 𝟤 𝖮 𝖱 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖧 \mathsf{(CH_{2}CH_{2})O+RMgBr}\rightarrow\mathsf{R\!\!-\!\!CH_{2}CH_{2}\!\!-\!% \!OMgBr\ \xrightarrow{H_{2}O}\ R\!\!-\!\!CH_{2}CH_{2}\!\!-\!\!OH}
  2. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖱𝖫𝗂 𝖱 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖫𝗂 𝖧 𝟤 𝖮 𝖱 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖧 \mathsf{(CH_{2}CH_{2})O+RLi}\rightarrow\mathsf{R\!\!-\!\!CH_{2}CH_{2}\!\!-\!\!% OLi\ \xrightarrow{H_{2}O}\ R\!\!-\!\!CH_{2}CH_{2}\!\!-\!\!OH}
  3. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖧𝖭𝖮 𝟥 𝖧𝖮 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖭𝖮 𝟤 + 𝖧𝖭𝖮 𝟥 - 𝖧 𝟤 𝖮 𝖮 𝟤 𝖭𝖮 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖭𝖮 𝟤 \mathsf{(CH_{2}CH_{2})O+HNO_{3}}\rightarrow\mathsf{HO\!\!-\!\!CH_{2}CH_{2}\!\!% -\!\!ONO_{2}\ \xrightarrow[-H_{2}O]{+\ HNO_{3}}\ O_{2}NO\!\!-\!\!CH_{2}CH_{2}% \!\!-\!\!ONO_{2}}
  4. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 𝟤𝟢𝟢 𝗈 𝖢 , 𝖠𝗅 𝟤 𝖮 𝟥 𝖢𝖧 𝟥 𝖢𝖧𝖮 \mathsf{(CH_{2}CH_{2})O\ \xrightarrow{200\ ^{o}C,\ Al_{2}O_{3}}\ CH_{3}CHO}
  5. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖧 𝟤 𝟪𝟢 𝗈 𝖢 , 𝖭𝗂 𝖢 𝟤 𝖧 𝟧 𝖮𝖧 \mathsf{(CH_{2}CH_{2})O+H_{2}\ \xrightarrow{80\ ^{o}C,\ Ni}\ C_{2}H_{5}OH}
  6. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖧 𝟤 𝖹𝗇 + 𝖢𝖧 𝟥 𝖢𝖮𝖮𝖧 𝖢𝖧 𝟤 = 𝖢𝖧 𝟤 + 𝖧 𝟤 𝖮 \mathsf{(CH_{2}CH_{2})O+H_{2}\ \xrightarrow{Zn\ +\ CH_{3}COOH}\ CH_{2}\!\!=\!% \!CH_{2}+H_{2}O}
  7. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖮 𝟤 𝖠𝗀𝖭𝖮 𝟥 𝖧𝖮𝖢𝖧 𝟤 𝖢𝖮𝖮𝖧 \mathsf{(CH_{2}CH_{2})O+O_{2}\ \xrightarrow{AgNO_{3}}\ HOCH_{2}COOH}
  8. 𝗇 ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 𝖲𝗇𝖢𝗅 𝟦 ( - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮 - ) 𝗇 \mathsf{n(CH_{2}CH_{2})O\ \xrightarrow{SnCl_{4}}\ (-\!CH_{2}CH_{2}\!\!-\!\!O\!% -)_{n}}
  9. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖢𝖮 + 𝖧 𝟤 𝖢𝖧𝖮 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖧 + 𝖧 𝟤 𝖧𝖮 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮𝖧 \mathsf{(CH_{2}CH_{2})O+CO+H_{2}}\rightarrow\mathsf{CHO\!\!-\!\!CH_{2}CH_{2}\!% \!-\!\!OH\ \xrightarrow{+H_{2}}\ HO\!\!-\!\!CH_{2}CH_{2}CH_{2}\!\!-\!\!OH}
  10. 𝟤 𝖢 𝖧 𝟤 = 𝖢𝖧 𝟤 + 𝖮 𝟤 𝖠𝗀 2 ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 \mathsf{2CH_{2}\!\!=\!\!CH_{2}+O_{2}\ \xrightarrow{Ag}\ 2(CH_{2}CH_{2})O}
  11. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖢𝖮 𝟤 ( 𝖮 - 𝖢𝖧 𝟤 𝖢𝖧 𝟤 - 𝖮 ) 𝖢 = 𝖮 + 𝖧 𝟤 𝖮 - 𝖢𝖮 𝟤 𝖧𝖮𝖢𝖧 𝟤 𝖢𝖧 𝟤 𝖮𝖧 \mathsf{(CH_{2}CH_{2})O+CO_{2}}\rightarrow\mathsf{(O\!\!-\!\!CH_{2}CH_{2}\!\!-% \!\!O)C\!\!=\!\!O\ \xrightarrow[-CO_{2}]{+H_{2}O}\ HOCH_{2}CH_{2}OH}
  12. ( 𝖢𝖧 𝟤 𝖢𝖧 𝟤 ) 𝖮 + 𝖧𝖢𝖭 𝖧𝖮𝖢𝖧 𝟤 𝖢𝖧 𝟤 𝖢𝖭 - 𝖧 𝟤 𝖮 𝖢𝖧 𝟤 = 𝖢𝖧 - 𝖢𝖭 \mathsf{(CH_{2}CH_{2})O+HCN}\rightarrow\mathsf{HOCH_{2}CH_{2}CN\ \xrightarrow[% -H_{2}O]{\ }CH_{2}\!\!=\!\!CH\!\!-\!\!CN}

Ethylenediaminetetraacetic_acid.html

  1. \overrightarrow{\leftarrow}

Euclid's_Elements.html

  1. 10 3 ( 5 - 5 ) = 5 + 5 6 . \sqrt{\tfrac{10}{3(5-\sqrt{5})}}=\sqrt{\tfrac{5+\sqrt{5}}{6}}.

Euler's_criterion.html

  1. a p - 1 2 { 1 ( mod p ) if there is an integer x such that a x 2 ( mod p ) - 1 ( mod p ) if there is no such integer. a^{\tfrac{p-1}{2}}\equiv\begin{cases}\;\;\,1\;\;(\mathop{{\rm mod}}p)&\,\text{% if there is an integer }x\,\text{ such that }a\equiv x^{2}\;\;(\mathop{{\rm mod% }}p)\\ -1\;\;(\mathop{{\rm mod}}p)&\,\text{ if there is no such integer.}\end{cases}
  2. ( a p ) a ( p - 1 ) / 2 ( mod p ) . \left(\frac{a}{p}\right)\equiv a^{(p-1)/2}\;\;(\mathop{{\rm mod}}p).
  3. a p - 1 1 ( mod p ) a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p)
  4. ( a p - 1 2 - 1 ) ( a p - 1 2 + 1 ) 0 ( mod p ) . (a^{\tfrac{p-1}{2}}-1)(a^{\tfrac{p-1}{2}}+1)\equiv 0\;\;(\mathop{{\rm mod}}p).
  5. a p - 1 2 x 2 p - 1 2 x p - 1 1 ( mod p ) . a^{\tfrac{p-1}{2}}\equiv{x^{2}}^{\tfrac{p-1}{2}}\equiv x^{p-1}\equiv 1\;\;(% \mathop{{\rm mod}}p).

Euler's_four-square_identity.html

  1. ( a 1 2 + a 2 2 + a 3 2 + a 4 2 ) ( b 1 2 + b 2 2 + b 3 2 + b 4 2 ) = (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^% {2})=\,
  2. ( a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 ) 2 + (a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4})^{2}+\,
  3. ( a 1 b 2 - a 2 b 1 + a 3 b 4 - a 4 b 3 ) 2 + (a_{1}b_{2}-a_{2}b_{1}+a_{3}b_{4}-a_{4}b_{3})^{2}+\,
  4. ( a 1 b 3 - a 2 b 4 - a 3 b 1 + a 4 b 2 ) 2 + (a_{1}b_{3}-a_{2}b_{4}-a_{3}b_{1}+a_{4}b_{2})^{2}+\,
  5. ( a 1 b 4 + a 2 b 3 - a 3 b 2 - a 4 b 1 ) 2 . (a_{1}b_{4}+a_{2}b_{3}-a_{3}b_{2}-a_{4}b_{1})^{2}.\,
  6. a k a_{k}
  7. b k b_{k}
  8. a k a_{k}
  9. - a k -a_{k}
  10. b k b_{k}
  11. - b k -b_{k}
  12. ( a 1 2 + a 2 2 + a 3 2 + + a n 2 ) ( b 1 2 + b 2 2 + b 3 2 + + b n 2 ) = c 1 2 + c 2 2 + c 3 2 + + c n 2 (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+...+a_{n}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+..% .+b_{n}^{2})=c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+...+c_{n}^{2}\,
  13. c i c_{i}
  14. a i a_{i}
  15. b i b_{i}
  16. c i c_{i}
  17. n = 2 m n=2^{m}
  18. ( a 1 2 + a 2 2 + a 3 2 + a 4 2 ) ( b 1 2 + b 2 2 + b 3 2 + b 4 2 ) = (a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2})(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^% {2})=\,
  19. ( a 1 b 4 + a 2 b 3 + a 3 b 2 + a 4 b 1 ) 2 + (a_{1}b_{4}+a_{2}b_{3}+a_{3}b_{2}+a_{4}b_{1})^{2}+\,
  20. ( a 1 b 3 - a 2 b 4 + a 3 b 1 - a 4 b 2 ) 2 + (a_{1}b_{3}-a_{2}b_{4}+a_{3}b_{1}-a_{4}b_{2})^{2}+\,
  21. ( a 1 b 2 + a 2 b 1 + a 3 u 1 b 1 2 + b 2 2 - a 4 u 2 b 1 2 + b 2 2 ) 2 + \left(a_{1}b_{2}+a_{2}b_{1}+\frac{a_{3}u_{1}}{b_{1}^{2}+b_{2}^{2}}-\frac{a_{4}% u_{2}}{b_{1}^{2}+b_{2}^{2}}\right)^{2}+\,
  22. ( a 1 b 1 - a 2 b 2 - a 4 u 1 b 1 2 + b 2 2 - a 3 u 2 b 1 2 + b 2 2 ) 2 \left(a_{1}b_{1}-a_{2}b_{2}-\frac{a_{4}u_{1}}{b_{1}^{2}+b_{2}^{2}}-\frac{a_{3}% u_{2}}{b_{1}^{2}+b_{2}^{2}}\right)^{2}\,
  23. u 1 = b 1 2 b 4 - 2 b 1 b 2 b 3 - b 2 2 b 4 u_{1}=b_{1}^{2}b_{4}-2b_{1}b_{2}b_{3}-b_{2}^{2}b_{4}
  24. u 2 = b 1 2 b 3 + 2 b 1 b 2 b 4 - b 2 2 b 3 u_{2}=b_{1}^{2}b_{3}+2b_{1}b_{2}b_{4}-b_{2}^{2}b_{3}
  25. u 1 2 + u 2 2 = ( b 1 2 + b 2 2 ) 2 ( b 3 2 + b 4 2 ) u_{1}^{2}+u_{2}^{2}=(b_{1}^{2}+b_{2}^{2})^{2}(b_{3}^{2}+b_{4}^{2})

Euler_characteristic.html

  1. χ \chi
  2. χ \chi
  3. χ = V - E + F \chi=V-E+F\,\!
  4. V - E + F = 2. V-E+F=2.\,\!
  5. d f d_{f}
  6. d v V - E + d f F = 2 D . d_{v}V-E+d_{f}F=2D.
  7. V - E + F V-E+F
  8. V - E + F - C = 1 V-E+F-C=1
  9. χ = k 0 - k 1 + k 2 - k 3 + , \chi=k_{0}-k_{1}+k_{2}-k_{3}+\cdots,
  10. χ = k 0 - k 1 + k 2 - k 3 + , \chi=k_{0}-k_{1}+k_{2}-k_{3}+\cdots,
  11. χ = b 0 - b 1 + b 2 - b 3 + . \chi=b_{0}-b_{1}+b_{2}-b_{3}+\cdots.
  12. χ \chi
  13. n \mathbb{R}^{n}
  14. χ ( M N ) = χ ( M ) + χ ( N ) . \chi(M\sqcup N)=\chi(M)+\chi(N).
  15. χ ( M N ) = χ ( M ) + χ ( N ) - χ ( M N ) . \chi(M\cup N)=\chi(M)+\chi(N)-\chi(M\cap N).
  16. χ ( M × N ) = χ ( M ) χ ( N ) . \chi(M\times N)=\chi(M)\cdot\chi(N).
  17. M ~ M , \tilde{M}\to M,
  18. χ ( M ~ ) = k χ ( M ) . \chi(\tilde{M})=k\cdot\chi(M).
  19. p : E B p\colon E\to B
  20. χ ( E ) = χ ( F ) χ ( B ) . \chi(E)=\chi(F)\cdot\chi(B).
  21. τ : H * ( B ) H * ( E ) \tau\colon H_{*}(B)\to H_{*}(E)
  22. p * : H * ( E ) H * ( B ) p_{*}\colon H_{*}(E)\to H_{*}(B)
  23. p * τ = χ ( F ) 1. p_{*}\circ\tau=\chi(F)\cdot 1.
  24. χ = 2 - 2 g . \chi=2-2g.
  25. χ = 2 - k . \chi=2-k.
  26. \scriptstyle\mathcal{F}
  27. χ ( ) = Σ ( - 1 ) i h i ( X , ) , \chi(\mathcal{F})=\Sigma(-1)^{i}h^{i}(X,\mathcal{F}),
  28. h i ( X , ) \scriptstyle h^{i}(X,\mathcal{F})
  29. \scriptstyle\mathcal{F}

Euler_integral.html

  1. B ( x , y ) = 0 1 t x - 1 ( 1 - t ) y - 1 d t = Γ ( x ) Γ ( y ) Γ ( x + y ) \mathrm{B}(x,y)=\int_{0}^{1}t^{x-1}(1-t)^{y-1}\,dt=\frac{\Gamma(x)\Gamma(y)}{% \Gamma(x+y)}
  2. Γ ( z ) = 0 t z - 1 e - t d t \Gamma(z)=\int_{0}^{\infty}t^{z-1}\,e^{-t}\,dt
  3. B ( n , m ) = ( n - 1 ) ! ( m - 1 ) ! ( n + m - 1 ) ! = n + m n m ( n + m n ) \mathrm{B}(n,m)={(n-1)!(m-1)!\over(n+m-1)!}={n+m\over nm{n+m\choose n}}
  4. Γ ( n ) = ( n - 1 ) ! \Gamma(n)=(n-1)!\,

Euler_line.html

  1. sin ( 2 A ) sin ( B - C ) x + sin ( 2 B ) sin ( C - A ) y + sin ( 2 C ) sin ( A - B ) z = 0. \sin(2A)\sin(B-C)x+\sin(2B)\sin(C-A)y+\sin(2C)\sin(A-B)z=0.\,
  2. cos A : cos B : cos C \cos A:\cos B:\cos C
  3. sec A : sec B : sec C = cos B cos C : cos C cos A : cos A cos B ) \sec A:\sec B:\sec C=\cos B\cos C:\cos C\cos A:\cos A\cos B)
  4. cos A + t cos B cos C : cos B + t cos C cos A : cos C + t cos A cos B \cos A+t\cos B\cos C:\cos B+t\cos C\cos A:\cos C+t\cos A\cos B\,
  5. cos A : cos B : cos C , \cos A:\cos B:\cos C,
  6. t = 0. t=0.
  7. cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B , \cos A+\cos B\cos C:\cos B+\cos C\cos A:\cos C+\cos A\cos B,
  8. t = 1. t=1.
  9. cos A + 2 cos B cos C : cos B + 2 cos C cos A : cos C + 2 cos A cos B , \cos A+2\cos B\cos C:\cos B+2\cos C\cos A:\cos C+2\cos A\cos B,
  10. t = 2. t=2.
  11. cos A - cos B cos C : cos B - cos C cos A : cos C - cos A cos B , \cos A-\cos B\cos C:\cos B-\cos C\cos A:\cos C-\cos A\cos B,
  12. t = - 1. t=-1.
  13. m 1 , m_{1},
  14. m 2 , m_{2},
  15. m 3 , m_{3},
  16. m E m_{E}
  17. m 1 m 2 + m 1 m 3 + m 1 m E + m 2 m 3 + m 2 m E + m 3 m E m_{1}m_{2}+m_{1}m_{3}+m_{1}m_{E}+m_{2}m_{3}+m_{2}m_{E}+m_{3}m_{E}
  18. + 3 m 1 m 2 m 3 m E + 3 = 0. +3m_{1}m_{2}m_{3}m_{E}+3=0.
  19. m E = - m 1 m 2 + m 1 m 3 + m 2 m 3 + 3 m 1 + m 2 + m 3 + 3 m 1 m 2 m 3 . m_{E}=-\frac{m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}+3}{m_{1}+m_{2}+m_{3}+3m_{1}m_{2}% m_{3}}.
  20. tan B tan C = 3. \tan B\tan C=3.
  21. G H = 2 G O ; GH=2GO;
  22. O H = 3 G O . OH=3GO.
  23. O N = N H , O G = 2 G N , N H = 3 G N . ON=NH,\quad OG=2\cdot GN,\quad NH=3GN.
  24. G O 2 = R 2 - 1 9 ( a 2 + b 2 + c 2 ) . GO^{2}=R^{2}-\tfrac{1}{9}(a^{2}+b^{2}+c^{2}).
  25. O H 2 = 9 R 2 - ( a 2 + b 2 + c 2 ) ; OH^{2}=9R^{2}-(a^{2}+b^{2}+c^{2});
  26. G H 2 = 4 R 2 - 4 9 ( a 2 + b 2 + c 2 ) . GH^{2}=4R^{2}-\tfrac{4}{9}(a^{2}+b^{2}+c^{2}).
  27. P P
  28. E E
  29. P P
  30. P P
  31. L L
  32. E E
  33. L L
  34. L L
  35. P P
  36. C C
  37. E = C E=C
  38. P P
  39. E E

Euler_pseudoprime.html

  1. a ( n - 1 ) / 2 ± 1 ( mod n ) a^{(n-1)/2}\equiv\pm 1\;\;(\mathop{{\rm mod}}n)
  2. a ( n - 1 ) / 2 ( a n ) ( mod n ) a^{(n-1)/2}\equiv\left(\frac{a}{n}\right)\;\;(\mathop{{\rm mod}}n)

Euler–Jacobi_pseudoprime.html

  1. a ( n - 1 ) / 2 ( a n ) ( mod n ) a^{(n-1)/2}\equiv\left(\frac{a}{n}\right)\;\;(\mathop{{\rm mod}}n)
  2. ( a n ) \left(\frac{a}{n}\right)

Euler–Lagrange_equation.html

  1. S ( s y m b o l q ) = a b L ( t , s y m b o l q ( t ) , s y m b o l q ( t ) ) d t \displaystyle S(symbolq)=\int_{a}^{b}L(t,symbolq(t),symbolq^{\prime}(t))\,% \mathrm{d}t
  2. s y m b o l q symbolq
  3. s y m b o l q : [ a , b ] \displaystyle symbolq\colon[a,b]\subset\mathbb{R}
  4. s y m b o l q symbolq
  5. s y m b o l q ( a ) = s y m b o l x a symbolq(a)=symbolx_{a}
  6. s y m b o l q ( b ) = s y m b o l x b symbolq(b)=symbolx_{b}
  7. s y m b o l q symbolq^{\prime}
  8. s y m b o l q symbolq
  9. q : [ a , b ] T q ( t ) X t v = q ( t ) \begin{aligned}\displaystyle q^{\prime}\colon[a,b]&\displaystyle\to T_{q(t)}X% \\ \displaystyle t&\displaystyle\mapsto v=q^{\prime}(t)\end{aligned}
  10. T X = x X { x } × T x X TX=\bigcup_{x\in X}\{x\}\times T_{x}X
  11. L : [ a , b ] × T X ( t , x , v ) L ( t , x , v ) . \begin{aligned}\displaystyle L\colon[a,b]\times TX&\displaystyle\to\mathbb{R}% \\ \displaystyle(t,x,v)&\displaystyle\mapsto L(t,x,v).\end{aligned}
  12. L q i ( t , s y m b o l q ( t ) , s y m b o l q ( t ) ) - d d t L q ˙ i ( t , s y m b o l q ( t ) , s y m b o l q ( t ) ) = 0 for i = 1 , , n . \frac{\partial L}{\partial q_{i}}(t,symbolq(t),symbolq^{\prime}(t))-\frac{% \mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial\dot{q}_{i}}(t,symbolq(t),% symbolq^{\prime}(t))=0\quad\,\text{for }i=1,\dots,n.
  13. f f
  14. f ( a ) = A f(a)=A
  15. f ( b ) = B f(b)=B
  16. J = a b F ( x , f ( x ) , f ( x ) ) d x . J=\int_{a}^{b}F(x,f(x),f^{\prime}(x))\,\mathrm{d}x\ .\,\!
  17. F F
  18. f f
  19. f f
  20. J J
  21. f f
  22. J J
  23. f f
  24. g ε ( x ) = f ( x ) + ε η ( x ) g_{\varepsilon}(x)=f(x)+\varepsilon\eta(x)
  25. ε η ( x ) \varepsilon\eta(x)
  26. f f
  27. ε \varepsilon
  28. η ( x ) \eta(x)
  29. η ( a ) = η ( b ) = 0 \eta(a)=\eta(b)=0
  30. J ε = a b F ( x , g ε ( x ) , g ε ( x ) ) d x = a b F ε d x J_{\varepsilon}=\int_{a}^{b}F(x,g_{\varepsilon}(x),g_{\varepsilon}^{\prime}(x)% )\,\mathrm{d}x=\int_{a}^{b}F_{\varepsilon}\,\mathrm{d}x\,\!
  31. F ε = F ( x , g ε ( x ) , g ε ( x ) ) F_{\varepsilon}=F(x,\,g_{\varepsilon}(x),\,g_{\varepsilon}^{\prime}(x))
  32. J ε J_{\varepsilon}
  33. d J ε d ε = d d ε a b F ε d x = a b d F ε d ε d x \frac{\mathrm{d}J_{\varepsilon}}{\mathrm{d}\varepsilon}=\frac{\mathrm{d}}{% \mathrm{d}\varepsilon}\int_{a}^{b}F_{\varepsilon}\,\mathrm{d}x=\int_{a}^{b}% \frac{\mathrm{d}F_{\varepsilon}}{\mathrm{d}\varepsilon}\,\mathrm{d}x
  34. d F ε d ε = d x d ε F ε x + d g ε d ε F ε g ε + d g ε d ε F ε g ε = d g ε d ε F ε g ε + d g ε d ε F ε g ε = η ( x ) F ε g ε + η ( x ) F ε g ε . \begin{aligned}\displaystyle\frac{\mathrm{d}F_{\varepsilon}}{\mathrm{d}% \varepsilon}&\displaystyle=\frac{\mathrm{d}x}{\mathrm{d}\varepsilon}\frac{% \partial F_{\varepsilon}}{\partial x}+\frac{\mathrm{d}g_{\varepsilon}}{\mathrm% {d}\varepsilon}\frac{\partial F_{\varepsilon}}{\partial g_{\varepsilon}}+\frac% {\mathrm{d}g_{\varepsilon}^{\prime}}{\mathrm{d}\varepsilon}\frac{\partial F_{% \varepsilon}}{\partial g_{\varepsilon}^{\prime}}\\ &\displaystyle=\frac{\mathrm{d}g_{\varepsilon}}{\mathrm{d}\varepsilon}\frac{% \partial F_{\varepsilon}}{\partial g_{\varepsilon}}+\frac{\mathrm{d}g^{\prime}% _{\varepsilon}}{\mathrm{d}\varepsilon}\frac{\partial F_{\varepsilon}}{\partial g% ^{\prime}_{\varepsilon}}\\ &\displaystyle=\eta(x)\frac{\partial F_{\varepsilon}}{\partial g_{\varepsilon}% }+\eta^{\prime}(x)\frac{\partial F_{\varepsilon}}{\partial g_{\varepsilon}^{% \prime}}\ .\\ \end{aligned}
  35. d J ε d ε = a b [ η ( x ) F ε g ε + η ( x ) F ε g ε ] d x . \frac{\mathrm{d}J_{\varepsilon}}{\mathrm{d}\varepsilon}=\int_{a}^{b}\left[\eta% (x)\frac{\partial F_{\varepsilon}}{\partial g_{\varepsilon}}+\eta^{\prime}(x)% \frac{\partial F_{\varepsilon}}{\partial g_{\varepsilon}^{\prime}}\,\right]\,% \mathrm{d}x\ .
  36. d J ε d ε | ε = 0 = a b [ η ( x ) F f + η ( x ) F f ] d x = 0 . \frac{\mathrm{d}J_{\varepsilon}}{\mathrm{d}\varepsilon}\bigg|_{\varepsilon=0}=% \int_{a}^{b}\left[\eta(x)\frac{\partial F}{\partial f}+\eta^{\prime}(x)\frac{% \partial F}{\partial f^{\prime}}\,\right]\,\mathrm{d}x=0\ .
  37. a b [ F f - d d x F f ] η ( x ) d x + [ η ( x ) F f ] a b = 0 . \int_{a}^{b}\left[\frac{\partial F}{\partial f}-\frac{\mathrm{d}}{\mathrm{d}x}% \frac{\partial F}{\partial f^{\prime}}\right]\eta(x)\,\mathrm{d}x+\left[\eta(x% )\frac{\partial F}{\partial f^{\prime}}\right]_{a}^{b}=0\ .
  38. η ( a ) = η ( b ) = 0 \eta(a)=\eta(b)=0
  39. a b [ F f - d d x F f ] η ( x ) d x = 0 . \int_{a}^{b}\left[\frac{\partial F}{\partial f}-\frac{\mathrm{d}}{\mathrm{d}x}% \frac{\partial F}{\partial f^{\prime}}\right]\eta(x)\,\mathrm{d}x=0\ .\,\!
  40. F f - d d x F f = 0 . \frac{\partial F}{\partial f}-\frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial F}{% \partial f^{\prime}}=0\ .
  41. J = a b F ( t , y ( t ) , y ( t ) ) d t J=\int^{b}_{a}F(t,y(t),y^{\prime}(t))\,\mathrm{d}t
  42. C 1 ( [ a , b ] ) C^{1}([a,b])
  43. y ( a ) = A y(a)=A
  44. y ( b ) = B y(b)=B
  45. n n
  46. [ a , b ] [a,b]
  47. n n
  48. t 0 = a , t 1 , t 2 , , t n = b t_{0}=a,t_{1},t_{2},\ldots,t_{n}=b
  49. Δ t = t k - t k - 1 \Delta t=t_{k}-t_{k-1}
  50. y ( t ) y(t)
  51. ( t 0 , y 0 ) , , ( t n , y n ) (t_{0},y_{0}),\ldots,(t_{n},y_{n})
  52. y 0 = A y_{0}=A
  53. y n = B y_{n}=B
  54. n - 1 n-1
  55. J ( y 1 , , y n - 1 ) k = 0 n - 1 F ( t k , y k , y k + 1 - y k Δ t ) Δ t . J(y_{1},\ldots,y_{n-1})\approx\sum^{n-1}_{k=0}F\left(t_{k},y_{k},\frac{y_{k+1}% -y_{k}}{\Delta t}\right)\Delta t.
  56. t 0 , , t n t_{0},\ldots,t_{n}
  57. J ( y 1 , , y n ) y m = 0. \frac{\partial J(y_{1},\ldots,y_{n})}{\partial y_{m}}=0.
  58. J y m = F y ( t m , y m , y m + 1 - y m Δ t ) Δ t + F y ( t m - 1 , y m - 1 , y m - y m - 1 Δ t ) - F y ( t m , y m , y m + 1 - y m Δ t ) . \frac{\partial J}{\partial y_{m}}=F_{y}\left(t_{m},y_{m},\frac{y_{m+1}-y_{m}}{% \Delta t}\right)\Delta t+F_{y^{\prime}}\left(t_{m-1},y_{m-1},\frac{y_{m}-y_{m-% 1}}{\Delta t}\right)-F_{y^{\prime}}\left(t_{m},y_{m},\frac{y_{m+1}-y_{m}}{% \Delta t}\right).
  59. Δ t \Delta t
  60. J y m Δ t = F y ( t m , y m , y m + 1 - y m Δ t ) - 1 Δ t [ F y ( t m , y m , y m + 1 - y m Δ t ) - F y ( t m - 1 , y m - 1 , y m - y m - 1 Δ t ) ] , \frac{\partial J}{\partial y_{m}\Delta t}=F_{y}\left(t_{m},y_{m},\frac{y_{m+1}% -y_{m}}{\Delta t}\right)-\frac{1}{\Delta t}\left[F_{y^{\prime}}\left(t_{m},y_{% m},\frac{y_{m+1}-y_{m}}{\Delta t}\right)-F_{y^{\prime}}\left(t_{m-1},y_{m-1},% \frac{y_{m}-y_{m-1}}{\Delta t}\right)\right],
  61. Δ t 0 \Delta t\to 0
  62. F y - d d t F y = 0. F_{y}-\frac{\mathrm{d}}{\mathrm{d}t}F_{y^{\prime}}=0.
  63. δ J / δ y \delta J/\delta y
  64. J J
  65. ( f ) = a b 1 + ( f ( x ) ) 2 d x , \ell(f)=\int_{a}^{b}\sqrt{1+(f^{\prime}(x))^{2}}\,\mathrm{d}x,
  66. L ( x , y , y ) y = y 1 + y 2 and L ( x , y , y ) y = 0. \frac{\partial L(x,y,y^{\prime})}{\partial y^{\prime}}=\frac{y^{\prime}}{\sqrt% {1+y^{\prime 2}}}\quad\,\text{and}\quad\frac{\partial L(x,y,y^{\prime})}{% \partial y}=0.
  67. d d x f ( x ) 1 + ( f ( x ) ) 2 = 0 f ( x ) 1 + ( f ( x ) ) 2 = C = constant f ( x ) = C 1 - C 2 := A f ( x ) = A x + B \begin{aligned}\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\frac{f^{\prime}(x)}% {\sqrt{1+(f^{\prime}(x))^{2}}}&\displaystyle=0\\ \displaystyle\frac{f^{\prime}(x)}{\sqrt{1+(f^{\prime}(x))^{2}}}&\displaystyle=% C=\,\text{constant}\\ \displaystyle\Rightarrow f^{\prime}(x)&\displaystyle=\frac{C}{\sqrt{1-C^{2}}}:% =A\\ \displaystyle\Rightarrow f(x)&\displaystyle=Ax+B\end{aligned}
  68. T T
  69. V V
  70. L = T - V L=T-V
  71. L q \frac{\partial L}{\partial q}
  72. L q ˙ \frac{\partial L}{\partial\dot{q}}
  73. d d t L q ˙ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial\dot{q}}
  74. q ˙ \dot{q}
  75. L q = d d t L q ˙ \frac{\partial L}{\partial q}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{% \partial\dot{q}}
  76. q ˙ \dot{q}
  77. S = t 0 t 1 L ( t , 𝐱 ( t ) , 𝐱 ˙ ( t ) ) d t S=\int_{t_{0}}^{t_{1}}L(t,\mathbf{x}(t),\mathbf{\dot{x}}(t))\,\mathrm{d}t
  78. L ( t , 𝐱 , 𝐯 ) = 1 2 m i = 1 3 v i 2 - U ( 𝐱 ) , L(t,\mathbf{x},\mathbf{v})=\frac{1}{2}m\sum_{i=1}^{3}v_{i}^{2}-U(\mathbf{x}),
  79. L ( t , 𝐱 , 𝐯 ) x i = - U ( 𝐱 ) x i = F i ( 𝐱 ) and L ( t , 𝐱 , 𝐯 ) v i = m v i = p i , \frac{\partial L(t,\mathbf{x},\mathbf{v})}{\partial x_{i}}=-\frac{\partial U(% \mathbf{x})}{\partial x_{i}}=F_{i}(\mathbf{x})\quad\,\text{and}\quad\frac{% \partial L(t,\mathbf{x},\mathbf{v})}{\partial v_{i}}=mv_{i}=p_{i},
  80. F i ( 𝐱 ( t ) ) = d d t m x ˙ i ( t ) = m x ¨ i ( t ) , F_{i}(\mathbf{x}(t))=\frac{\mathrm{d}}{\mathrm{d}t}m\dot{x}_{i}(t)=m\ddot{x}_{% i}(t),
  81. 𝐅 ( 𝐱 ( t ) ) = m 𝐱 ¨ ( t ) \mathbf{F}(\mathbf{x}(t))=m\mathbf{\ddot{x}}(t)
  82. 𝐅 = d 𝐩 d t \mathbf{F}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}
  83. I [ f ] = x 0 x 1 ( x , f , f , f ′′ , , f ( n ) ) d x ; f := d f d x , f ′′ := d 2 f d x 2 , f ( n ) := d n f d x n I[f]=\int_{x_{0}}^{x_{1}}\mathcal{L}(x,f,f^{\prime},f^{\prime\prime},\dots,f^{% (n)})~{}\mathrm{d}x~{};~{}~{}f^{\prime}:=\cfrac{\mathrm{d}f}{\mathrm{d}x},~{}f% ^{\prime\prime}:=\cfrac{\mathrm{d}^{2}f}{\mathrm{d}x^{2}},~{}f^{(n)}:=\cfrac{% \mathrm{d}^{n}f}{\mathrm{d}x^{n}}
  84. f - d d x ( f ) + d 2 d x 2 ( f ′′ ) - + ( - 1 ) n d n d x n ( f ( n ) ) = 0 \cfrac{\partial\mathcal{L}}{\partial f}-\cfrac{\mathrm{d}}{\mathrm{d}x}\left(% \cfrac{\partial\mathcal{L}}{\partial f^{\prime}}\right)+\cfrac{\mathrm{d}^{2}}% {\mathrm{d}x^{2}}\left(\cfrac{\partial\mathcal{L}}{\partial f^{\prime\prime}}% \right)-\dots+(-1)^{n}\cfrac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\left(\cfrac{% \partial\mathcal{L}}{\partial f^{(n)}}\right)=0
  85. n - 1 n-1
  86. f ( i ) , i { 0 , , n - 1 } f^{(i)},i\in\{0,...,n-1\}
  87. f ( n ) f^{(n)}
  88. f 1 , f 2 , , f n f_{1},f_{2},\dots,f_{n}
  89. x x
  90. I [ f 1 , f 2 , , f n ] = x 0 x 1 ( x , f 1 , f 2 , , f n , f 1 , f 2 , , f n ) d x ; f i := d f i d x I[f_{1},f_{2},\dots,f_{n}]=\int_{x_{0}}^{x_{1}}\mathcal{L}(x,f_{1},f_{2},\dots% ,f_{n},f_{1}^{\prime},f_{2}^{\prime},\dots,f_{n}^{\prime})~{}\mathrm{d}x~{};~{% }~{}f_{i}^{\prime}:=\cfrac{\mathrm{d}f_{i}}{\mathrm{d}x}
  91. f i - d d x ( f i ) = 0 \begin{aligned}\displaystyle\cfrac{\partial\mathcal{L}}{\partial f_{i}}-\cfrac% {\mathrm{d}}{\mathrm{d}x}\left(\cfrac{\partial\mathcal{L}}{\partial f_{i}^{% \prime}}\right)=0\end{aligned}
  92. I [ f ] = Ω ( x 1 , , x n , f , f x 1 , , f x n ) d 𝐱 ; f x i := f x i I[f]=\int_{\Omega}\mathcal{L}(x_{1},\dots,x_{n},f,f_{x_{1}},\dots,f_{x_{n}})\,% \mathrm{d}\mathbf{x}\,\!~{};~{}~{}f_{x_{i}}:=\cfrac{\partial f}{\partial x_{i}}
  93. f - i = 1 n x i f x i = 0. \frac{\partial\mathcal{L}}{\partial f}-\sum_{i=1}^{n}\frac{\partial}{\partial x% _{i}}\frac{\partial\mathcal{L}}{\partial f_{x_{i}}}=0.\,\!
  94. \mathcal{L}
  95. I [ f 1 , f 2 , , f m ] = Ω ( x 1 , , x n , f 1 , , f m , f 1 , 1 , , f 1 , n , , f m , 1 , , f m , n ) d 𝐱 ; f j , i := f j x i I[f_{1},f_{2},\dots,f_{m}]=\int_{\Omega}\mathcal{L}(x_{1},\dots,x_{n},f_{1},% \dots,f_{m},f_{1,1},\dots,f_{1,n},\dots,f_{m,1},\dots,f_{m,n})\,\mathrm{d}% \mathbf{x}\,\!~{};~{}~{}f_{j,i}:=\cfrac{\partial f_{j}}{\partial x_{i}}
  96. f 1 - i = 1 n x i f 1 , i \displaystyle\frac{\partial\mathcal{L}}{\partial f_{1}}-\sum_{i=1}^{n}\frac{% \partial}{\partial x_{i}}\frac{\partial\mathcal{L}}{\partial f_{1,i}}
  97. I [ f ] \displaystyle I[f]
  98. f - x 1 ( f , 1 ) - x 2 ( f , 2 ) + 2 x 1 2 ( f , 11 ) + 2 x 1 x 2 ( f , 12 ) + 2 x 2 2 ( f , 22 ) - + ( - 1 ) n n x 2 n ( f , 22 2 ) = 0 \begin{aligned}\displaystyle\frac{\partial\mathcal{L}}{\partial f}&% \displaystyle-\frac{\partial}{\partial x_{1}}\left(\frac{\partial\mathcal{L}}{% \partial f_{,1}}\right)-\frac{\partial}{\partial x_{2}}\left(\frac{\partial% \mathcal{L}}{\partial f_{,2}}\right)+\frac{\partial^{2}}{\partial x_{1}^{2}}% \left(\frac{\partial\mathcal{L}}{\partial f_{,11}}\right)+\frac{\partial^{2}}{% \partial x_{1}\partial x_{2}}\left(\frac{\partial\mathcal{L}}{\partial f_{,12}% }\right)+\frac{\partial^{2}}{\partial x_{2}^{2}}\left(\frac{\partial\mathcal{L% }}{\partial f_{,22}}\right)\\ &\displaystyle-\dots+(-1)^{n}\frac{\partial^{n}}{\partial x_{2}^{n}}\left(% \frac{\partial\mathcal{L}}{\partial f_{,22\dots 2}}\right)=0\end{aligned}
  99. f + i = 1 n ( - 1 ) i i x μ 1 x μ i ( f , μ 1 μ i ) = 0 \frac{\partial\mathcal{L}}{\partial f}+\sum_{i=1}^{n}(-1)^{i}\frac{\partial^{i% }}{\partial x_{\mu_{1}}\dots\partial x_{\mu_{i}}}\left(\frac{\partial\mathcal{% L}}{\partial f_{,\mu_{1}\dots\mu_{i}}}\right)=0
  100. μ 1 μ i \mu_{1}\dots\mu_{i}
  101. μ 1 μ i \mu_{1}\dots\mu_{i}
  102. I [ f 1 , , f p ] \displaystyle I[f_{1},\ldots,f_{p}]
  103. μ 1 μ j \mu_{1}\dots\mu_{j}
  104. f i + j = 1 n ( - 1 ) j j x μ 1 x μ j ( f i , μ 1 μ j ) = 0 \frac{\partial\mathcal{L}}{\partial f_{i}}+\sum_{j=1}^{n}(-1)^{j}\frac{% \partial^{j}}{\partial x_{\mu_{1}}\dots\partial x_{\mu_{j}}}\left(\frac{% \partial\mathcal{L}}{\partial f_{i,\mu_{1}\dots\mu_{j}}}\right)=0
  105. μ 1 μ j \mu_{1}\dots\mu_{j}
  106. j = 0 n ( - 1 ) j μ 1 μ j j ( f i , μ 1 μ j ) = 0 \sum_{j=0}^{n}(-1)^{j}\partial_{\mu_{1}\ldots\mu_{j}}^{j}\left(\frac{\partial% \mathcal{L}}{\partial f_{i,\mu_{1}\dots\mu_{j}}}\right)=0
  107. M M
  108. C ( [ a , b ] ) C^{\infty}([a,b])
  109. f : [ a , b ] M f:[a,b]\to M
  110. S : C ( [ a , b ] ) S:C^{\infty}([a,b])\to\mathbb{R}
  111. S [ f ] = a b ( L f ˙ ) ( t ) d t S[f]=\int_{a}^{b}(L\circ\dot{f})(t)\,\mathrm{d}t
  112. L : T M L:TM\to\mathbb{R}
  113. d S f = 0 \mathrm{d}S_{f}=0
  114. t [ a , b ] t\in[a,b]
  115. ( x i , X i ) (x^{i},X^{i})
  116. f ˙ ( t ) \dot{f}(t)
  117. dim M \dim M
  118. i : d d t F X i | f ˙ ( t ) = F x i | f ˙ ( t ) \forall i:\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial F}{\partial X^{i}}\bigg% |_{\dot{f}(t)}=\frac{\partial F}{\partial x^{i}}\bigg|_{\dot{f}(t)}

Euler–Mascheroni_constant.html

  1. γ \gamma
  2. γ = lim n ( k = 1 n 1 k - ln ( n ) ) = 1 ( 1 x - 1 x ) d x . \gamma=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln(n)\right)=% \int_{1}^{\infty}\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx.
  3. x \lfloor x\rfloor
  4. γ \gamma
  5. γ \gamma
  6. γ \gamma
  7. γ \gamma
  8. γ \gamma
  9. γ \gamma
  10. γ \gamma
  11. γ \gamma
  12. - γ = Γ ( 1 ) = Ψ ( 1 ) . \ -\gamma=\Gamma^{\prime}(1)=\Psi(1).
  13. - γ = lim z 0 { Γ ( z ) - 1 z } = lim z 0 { Ψ ( z ) + 1 z } . -\gamma=\lim_{z\to 0}\left\{\Gamma(z)-\frac{1}{z}\right\}=\lim_{z\to 0}\left\{% \Psi(z)+\frac{1}{z}\right\}.
  14. lim z 0 1 z { 1 Γ ( 1 + z ) - 1 Γ ( 1 - z ) } = 2 γ \lim_{z\to 0}\frac{1}{z}\left\{\frac{1}{\Gamma(1+z)}-\frac{1}{\Gamma(1-z)}% \right\}=2\gamma
  15. lim z 0 1 z { 1 Ψ ( 1 - z ) - 1 Ψ ( 1 + z ) } = π 2 3 γ 2 . \lim_{z\to 0}\frac{1}{z}\left\{\frac{1}{\Psi(1-z)}-\frac{1}{\Psi(1+z)}\right\}% =\frac{\pi^{2}}{3\gamma^{2}}.
  16. γ = lim n { Γ ( 1 n ) Γ ( n + 1 ) n 1 + 1 / n Γ ( 2 + n + 1 n ) - n 2 n + 1 } \gamma=\lim_{n\to\infty}\left\{\frac{\Gamma(\frac{1}{n})\Gamma(n+1)\,n^{1+1/n}% }{\Gamma(2+n+\frac{1}{n})}-\frac{n^{2}}{n+1}\right\}
  17. γ = lim m k = 1 m ( m k ) ( - 1 ) k k ln ( Γ ( k + 1 ) ) . \gamma=\lim\limits_{m\to\infty}\sum_{k=1}^{m}{m\choose k}\frac{(-1)^{k}}{k}\ln% (\Gamma(k+1)).
  18. γ \gamma
  19. γ \displaystyle\gamma
  20. γ \displaystyle\gamma
  21. γ = lim s 1 + n = 1 ( 1 n s - 1 s n ) = lim s 1 ( ζ ( s ) - 1 s - 1 ) = lim s 0 ζ ( 1 + s ) + ζ ( 1 - s ) 2 \gamma=\lim_{s\to 1^{+}}\sum_{n=1}^{\infty}\left(\frac{1}{n^{s}}-\frac{1}{s^{n% }}\right)=\lim_{s\to 1}\left(\zeta(s)-\frac{1}{s-1}\right)=\lim_{s\to 0}\frac{% \zeta(1+s)+\zeta(1-s)}{2}
  22. γ = lim n 1 n k = 1 n ( n k - n k ) . \begin{aligned}\displaystyle\gamma=\lim_{n\to\infty}\frac{1}{n}\,\sum_{k=1}^{n% }\left(\left\lceil\frac{n}{k}\right\rceil-\frac{n}{k}\right).\end{aligned}
  23. γ = k = 1 n 1 k - ln n - m = 2 ζ ( m , n + 1 ) m \gamma=\sum_{k=1}^{n}\frac{1}{k}-\ln n-\sum_{m=2}^{\infty}\frac{\zeta(m,n+1)}{m}
  24. H n = ln n + γ + 1 2 n - 1 12 n 2 + 1 120 n 4 - ε H_{n}=\ln n+\gamma+\frac{1}{2n}-\frac{1}{12n^{2}}+\frac{1}{120n^{4}}-\varepsilon
  25. 0 < ε < 1 252 n 6 . 0<\varepsilon<\frac{1}{252n^{6}}.
  26. γ \gamma
  27. γ \displaystyle\gamma
  28. H x H_{x}
  29. γ \gamma
  30. 0 e - x 2 ln x d x = - 1 4 ( γ + 2 ln 2 ) π \int_{0}^{\infty}{e^{-x^{2}}\ln x}\,dx=-\tfrac{1}{4}(\gamma+2\ln 2)\sqrt{\pi}
  31. 0 e - x ln 2 x d x = γ 2 + π 2 6 . \int_{0}^{\infty}{e^{-x}\ln^{2}x}\,dx=\gamma^{2}+\frac{\pi^{2}}{6}.
  32. γ \gamma
  33. γ = 0 1 0 1 x - 1 ( 1 - x y ) ln ( x y ) d x d y = n = 1 ( 1 n - ln n + 1 n ) . \gamma=\int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1-x\,y)\ln(x\,y)}\,dx\,dy=\sum_{n=1% }^{\infty}\left(\frac{1}{n}-\ln\frac{n+1}{n}\right).
  34. ln ( 4 π ) = 0 1 0 1 x - 1 ( 1 + x y ) ln ( x y ) d x d y = n = 1 ( - 1 ) n - 1 ( 1 n - ln n + 1 n ) . \ln\left(\frac{4}{\pi}\right)=\int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1+x\,y)\ln(x% \,y)}\,dx\,dy=\sum_{n=1}^{\infty}(-1)^{n-1}\left(\frac{1}{n}-\ln\frac{n+1}{n}% \right).
  35. ln ( 4 π ) \ln\left(\frac{4}{\pi}\right)
  36. n = 1 N 1 ( n ) + N 0 ( n ) 2 n ( 2 n + 1 ) = γ \sum_{n=1}^{\infty}\frac{N_{1}(n)+N_{0}(n)}{2n(2n+1)}=\gamma
  37. n = 1 N 1 ( n ) - N 0 ( n ) 2 n ( 2 n + 1 ) = ln ( 4 π ) \sum_{n=1}^{\infty}\frac{N_{1}(n)-N_{0}(n)}{2n(2n+1)}=\ln\left(\frac{4}{\pi}\right)
  38. γ = 0 1 1 1 + x n = 1 x 2 n - 1 d x . \gamma=\int_{0}^{1}\frac{1}{1+x}\sum_{n=1}^{\infty}x^{2^{n}-1}\,dx.
  39. γ \gamma
  40. γ = k = 1 [ 1 k - ln ( 1 + 1 k ) ] . \gamma=\sum_{k=1}^{\infty}\left[\frac{1}{k}-\ln\left(1+\frac{1}{k}\right)% \right].
  41. γ \gamma
  42. γ = 1 - k = 2 ( - 1 ) k log 2 k k + 1 . \gamma=1-\sum_{k=2}^{\infty}(-1)^{k}\frac{\lfloor\log_{2}k\rfloor}{k+1}.
  43. γ = k = 2 ( - 1 ) k log 2 k k = 1 2 - 1 3 + 2 ( 1 4 - 1 5 + 1 6 - 1 7 ) + 3 ( 1 8 - 1 9 + 1 10 - 1 11 + - 1 15 ) + {\gamma=\sum_{k=2}^{\infty}(-1)^{k}\frac{\left\lfloor\log_{2}k\right\rfloor}{k% }=\frac{1}{2}-\frac{1}{3}+2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{% 7}\right)+3\left(\frac{1}{8}-\frac{1}{9}+\frac{1}{10}-\frac{1}{11}+\dots-\frac% {1}{15}\right)+\dots}
  44. log 2 \log_{2}
  45. \lfloor\,\rfloor
  46. γ + ζ ( 2 ) = k = 2 ( 1 k 2 - 1 k ) = k = 2 k - k 2 k k 2 = 1 2 + 2 3 + 1 2 2 k = 1 2 × 2 k k + 2 2 + 1 3 2 k = 1 3 × 2 k k + 3 2 + . {\gamma+\zeta(2)=\sum_{k=2}^{\infty}\left(\frac{1}{\lfloor\sqrt{k}\rfloor^{2}}% -\frac{1}{k}\right)=\sum_{k=2}^{\infty}\frac{k-\lfloor\sqrt{k}\rfloor^{2}}{k% \lfloor\sqrt{k}\rfloor^{2}}=\frac{1}{2}+\frac{2}{3}+\frac{1}{2^{2}}\sum_{k=1}^% {2\times 2}\frac{k}{k+2^{2}}+\frac{1}{3^{2}}\sum_{k=1}^{3\times 2}\frac{k}{k+3% ^{2}}+\dots}.
  47. γ = ln π - 4 ln Γ ( 3 4 ) + 4 π k = 1 ( - 1 ) k + 1 ln ( 2 k + 1 ) 2 k + 1 . \gamma=\ln\pi-4\ln\Gamma(\tfrac{3}{4})+\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k% +1}\frac{\ln(2k+1)}{2k+1}.
  48. γ = lim n ( ln n - p n ln p p - 1 ) . \begin{aligned}\displaystyle\gamma=\lim_{n\to\infty}\left(\ln n-\sum_{p\leq n}% \frac{\ln p}{p-1}\right)\end{aligned}.
  49. γ = lim n [ k = 1 n 1 k - ln k = 1 n k ] - ln 2 \gamma=\lim_{n\rightarrow\infty}\left[\sum_{k=1}^{n}\frac{1}{k}-\ln\sqrt{\sum_% {k=1}^{n}k}\right]-\ln\sqrt{2}
  50. γ \gamma
  51. H n H_{n}
  52. γ H n - ln ( n ) - 1 < m t p l > 2 n + 1 12 n 2 - 1 120 n 4 + \gamma\sim H_{n}-\ln\left(n\right)-\frac{1}{<}mtpl>{{2n}}+\frac{1}{{12n^{2}}}-% \frac{1}{{120n^{4}}}+...
  53. γ H n - ln ( n + 1 2 + 1 < m t p l > 24 n - 1 48 n 3 + ) \gamma\sim H_{n}-\ln\left({n+\frac{1}{2}+\frac{1}{<}mtpl>{{24n}}-\frac{1}{{48n% ^{3}}}+...}\right)
  54. γ H n - ln ( n ) + ln ( n + 1 ) 2 - 1 6 n ( n + 1 ) + 1 30 n 2 ( n + 1 ) 2 - \gamma\sim H_{n}-\frac{{\ln\left(n\right)+\ln\left({n+1}\right)}}{2}-\frac{1}{% {6n\left({n+1}\right)}}+\frac{1}{{30n^{2}\left({n+1}\right)^{2}}}-...
  55. z ln ( 1 - z ) = n = 0 C n z n , | z | < 1 , \frac{z}{\ln(1-z)}=\sum_{n=0}^{\infty}C_{n}z^{n},\quad|z|<1,
  56. C n C_{n}
  57. C 0 = - 1 , k = 0 n C k n + 1 - k = 0 , n = 1 , 2 , 3 , C_{0}=-1\;,\quad\sum_{k=0}^{n}\frac{C_{k}}{n+1-k}=0,\quad n=1,2,3,\dots
  58. n 1 n\geq 1
  59. 1 2 \tfrac{1}{2}
  60. 1 12 \tfrac{1}{12}
  61. 1 24 \tfrac{1}{24}
  62. 19 720 \tfrac{19}{720}
  63. 3 160 \tfrac{3}{160}
  64. 863 60480 \tfrac{863}{60480}
  65. 275 24192 \tfrac{275}{24192}
  66. 33953 3628800 \tfrac{33953}{3628800}
  67. 8183 1036800 \tfrac{8183}{1036800}
  68. 3250433 479001600 \tfrac{3250433}{479001600}
  69. C n = 1 n ln 2 n - 𝒪 ( 1 n ln 3 n ) , n , C_{n}=\frac{1}{n\ln^{2}n}-\mathcal{O}\left(\frac{1}{n\ln^{3}n}\right),\quad n% \to\infty,
  70. C n = 0 d x ( 1 + x ) n ( ln 2 x + π 2 ) , n = 0 , 1 , 2 , . C_{n}=\int_{0}^{\infty}\frac{dx}{(1+x)^{n}\left(\ln^{2}x+\pi^{2}\right)},\quad n% =0,1,2,\dots.
  71. γ = 0 ln ( 1 + x ) ln 2 x + π 2 d x x 2 = - ln ( 1 + e - x ) x 2 + π 2 e x d x . \gamma=\int_{0}^{\infty}\frac{\ln(1+x)}{\ln^{2}x+\pi^{2}}\cdot\frac{dx}{x^{2}}% =\int_{-\infty}^{\infty}\frac{\ln(1+e^{-x})}{x^{2}+\pi^{2}}\,e^{x}\,dx.
  72. H n \displaystyle H_{n}
  73. 1 \displaystyle 1
  74. γ \gamma^{\prime}
  75. e γ = lim n 1 ln p n i = 1 n p i p i - 1 . e^{\gamma}=\lim_{n\to\infty}\frac{1}{\ln p_{n}}\prod_{i=1}^{n}\frac{p_{i}}{p_{% i}-1}.
  76. e 1 + γ / 2 2 π = n = 1 e - 1 + 1 / ( 2 n ) ( 1 + 1 n ) n \frac{e^{1+\gamma/2}}{\sqrt{2\,\pi}}=\prod_{n=1}^{\infty}e^{-1+1/(2\,n)}\,% \left(1+\frac{1}{n}\right)^{n}
  77. e 3 + 2 γ 2 π = n = 1 e - 2 + 2 / n ( 1 + 2 n ) n . \frac{e^{3+2\gamma}}{2\,\pi}=\prod_{n=1}^{\infty}e^{-2+2/n}\,\left(1+\frac{2}{% n}\right)^{n}.
  78. e γ = ( 2 1 ) 1 / 2 ( 2 2 1 3 ) 1 / 3 ( 2 3 4 1 3 3 ) 1 / 4 ( 2 4 4 4 1 3 6 5 ) 1 / 5 e^{\gamma}=\left(\frac{2}{1}\right)^{1/2}\left(\frac{2^{2}}{1\cdot 3}\right)^{% 1/3}\left(\frac{2^{3}\cdot 4}{1\cdot 3^{3}}\right)^{1/4}\left(\frac{2^{4}\cdot 4% ^{4}}{1\cdot 3^{6}\cdot 5}\right)^{1/5}\cdots
  79. k = 0 n ( k + 1 ) ( - 1 ) k + 1 ( n k ) . \prod_{k=0}^{n}(k+1)^{(-1)^{k+1}{n\choose k}}.
  80. γ \gamma
  81. γ \gamma
  82. γ α = lim n [ k = 1 n 1 k α - 1 n 1 x α d x ] , \gamma_{\alpha}=\lim_{n\to\infty}\left[\sum_{k=1}^{n}\frac{1}{k^{\alpha}}-\int% _{1}^{n}\frac{1}{x^{\alpha}}\,dx\right],
  83. c f = lim n [ k = 1 n f ( k ) - 1 n f ( x ) d x ] c_{f}=\lim_{n\to\infty}\left[\sum_{k=1}^{n}f(k)-\int_{1}^{n}f(x)\,dx\right]
  84. f n ( x ) = ln n x x f_{n}(x)=\frac{\ln^{n}x}{x}
  85. f a ( x ) = x - a f_{a}(x)=x^{-a}
  86. γ f a = ( a - 1 ) ζ ( a ) - 1 a - 1 \gamma_{f_{a}}=\frac{(a-1)\zeta(a)-1}{a-1}
  87. γ = lim a 1 [ ζ ( a ) - 1 a - 1 ] \gamma=\lim_{a\to 1}\left[\zeta(a)-\frac{1}{a-1}\right]
  88. γ ( a , q ) = lim x ( 0 < n x n a ( mod q ) 1 n - log x q ) . \gamma(a,q)=\lim_{x\to\infty}\left(\sum_{0<n\leq x\atop n\equiv a\;\;(\mathop{% {\rm mod}}q)}\frac{1}{n}-\frac{\log x}{q}\right).
  89. γ ( 0 , q ) = γ - log q q , \gamma(0,q)=\frac{\gamma-\log q}{q},
  90. a = 0 q - 1 γ ( a , q ) = γ , \sum_{a=0}^{q-1}\gamma(a,q)=\gamma,
  91. q γ ( a , q ) = γ - j = 1 q - 1 e - 2 π a i j / q log ( 1 - e 2 π i j / q ) , q\gamma(a,q)=\gamma-\sum_{j=1}^{q-1}e^{-2\pi aij/q}\log(1-e^{2\pi ij/q}),
  92. g c d ( a , q ) = d gcd(a,q)=d
  93. q γ ( a , q ) = q d γ ( a / d , q / d ) - log d . q\gamma(a,q)=\frac{q}{d}\gamma(a/d,q/d)-\log d.
  94. γ \gamma

Eutectic_system.html

  1. Liquid eutectic temperature cooling α solid solution + β solid solution \,\text{Liquid}\xrightarrow[\,\text{cooling}]{\,\text{eutectic temperature}}% \alpha\,\,\,\text{solid solution}+\beta\,\,\,\text{solid solution}
  2. 63 {}_{63}
  3. 37 {}_{37}
  4. n n\!
  5. G = H - T S { H = G + T S ( G T ) P = - S H = G - T ( G T ) P . G=H-TS\Rightarrow{\left\{\begin{array}[]{l}H=G+TS\\ \\ {\left({\frac{\partial G}{\partial T}}\right)_{P}=-S}\end{array}\right.}% \Rightarrow H=G-T\left({\frac{\partial G}{\partial T}}\right)_{P}.
  6. ( G / T T ) P = 1 T ( G T ) P - 1 T 2 G = - 1 T 2 ( G - T ( G T ) P ) = - H T 2 \left({\frac{\partial G/T}{\partial T}}\right)_{P}=\frac{1}{T}\left({\frac{% \partial G}{\partial T}}\right)_{P}-\frac{1}{T^{2}}G=-\frac{1}{T^{2}}\left({G-% T\left({\frac{\partial G}{\partial T}}\right)_{P}}\right)=-\frac{H}{T^{2}}
  7. μ i \mu_{i}
  8. μ i = μ i + R T ln a i a μ i + R T ln x i \mu_{i}=\mu_{i}^{\circ}+RT\ln\frac{a_{i}}{a}\approx\mu_{i}^{\circ}+RT\ln x_{i}
  9. μ i = 0 \mu_{i}=0
  10. μ i \mu_{i}^{\circ}
  11. μ i = μ i + R T ln x i = 0 μ i = - R T ln x i . \mu_{i}=\mu_{i}^{\circ}+RT\ln x_{i}=0\Rightarrow\mu_{i}^{\circ}=-RT\ln x_{i}.
  12. ( μ i / T T ) P = T ( R ln x i ) R ln x i = - H i T + K \begin{array}[]{l}\left({\frac{\partial\mu_{i}/T}{\partial T}}\right)_{P}=% \frac{\partial}{\partial T}\left({R\ln x_{i}}\right)\Rightarrow R\ln x_{i}=-% \frac{H_{i}^{\circ}}{T}+K\\ \\ \end{array}
  13. T T^{\circ}
  14. H H^{\circ}
  15. x i = 1 T = T i K = H i T i x_{i}=1\Rightarrow T=T_{i}^{\circ}\Rightarrow K=\frac{H_{i}^{\circ}}{T_{i}^{% \circ}}
  16. R ln x i = - H i T + H i T i R\ln x_{i}=-\frac{H_{i}^{\circ}}{T}+\frac{H_{i}^{\circ}}{T_{i}^{\circ}}
  17. { ln x i + H i R T - H i R T i = 0 i = 1 n x i = 1 \begin{array}[]{l}\left\{{{\begin{array}[]{*{20}c}{\ln x_{i}+\frac{H_{i}^{% \circ}}{RT}-\frac{H_{i}^{\circ}}{RT_{i}^{\circ}}=0}\\ {\sum\limits_{i=1}^{n}{x_{i}=1}}\\ \end{array}}}\right.\\ \\ \end{array}
  18. { i < n ln x i + H i R T - H i R T i = 0 ln ( 1 - i = 1 n - 1 x i ) + H n R T - H n R T n = 0 \begin{array}[]{l}\left\{{{\begin{array}[]{*{20}c}{\forall i<n\Rightarrow\ln x% _{i}+\frac{H_{i}^{\circ}}{RT}-\frac{H_{i}^{\circ}}{RT_{i}^{\circ}}=0}\\ {\ln\left({1-\sum\limits_{i=1}^{n-1}{x_{i}}}\right)+\frac{H_{n}^{\circ}}{RT}-% \frac{H_{n}^{\circ}}{RT_{n}^{\circ}}=0}\\ \end{array}}}\right.\\ \\ \end{array}
  19. [ Δ x 1 Δ x 2 Δ x 3 Δ x n - 1 Δ T ] = [ 1 / x 1 0 0 0 0 - H 1 R T 2 0 1 / x 2 0 0 0 - H 2 R T 2 0 0 1 / x 3 0 0 - H 3 R T 2 0 0 0 0 - H 4 R T 2 0 0 0 0 1 / x n - 1 - H n - 1 R T 2 - 1 1 - 1 = 1 n - 1 x i - 1 1 - 1 = 1 n - 1 x i - 1 1 - 1 = 1 n - 1 x i - 1 1 - 1 = 1 n - 1 x i - 1 1 - 1 = 1 n - 1 x i - H n R T 2 ] - 1 . [ ln x 1 + H 1 R T - H 1 R T 1 ln x 2 + H 2 R T - H 2 R T 2 ln x 3 + H 3 R T - H 3 R T 3 ln x n - 1 + H n - 1 R T - H n - 1 R T n - 1 i ln ( 1 - i = 1 n - 1 x i ) + H n R T - H n R T n ] \begin{array}[]{c}\left[{{\begin{array}[]{*{20}c}{\Delta x_{1}}\\ {\Delta x_{2}}\\ {\Delta x_{3}}\\ \vdots\\ {\Delta x_{n-1}}\\ {\Delta T}\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{1/x_{1}}&0&0&0&0&{-\frac{% H_{1}^{\circ}}{RT^{2}}}\\ 0&{1/x_{2}}&0&0&0&{-\frac{H_{2}^{\circ}}{RT^{2}}}\\ 0&0&{1/x_{3}}&0&0&{-\frac{H_{3}^{\circ}}{RT^{2}}}\\ 0&0&0&\ddots&0&{-\frac{H_{4}^{\circ}}{RT^{2}}}\\ 0&0&0&0&{1/x_{n-1}}&{-\frac{H_{n-1}^{\circ}}{RT^{2}}}\\ {\frac{-1}{1-\sum\limits_{1=1}^{n-1}{x_{i}}}}&{\frac{-1}{1-\sum\limits_{1=1}^{% n-1}{x_{i}}}}&{\frac{-1}{1-\sum\limits_{1=1}^{n-1}{x_{i}}}}&{\frac{-1}{1-\sum% \limits_{1=1}^{n-1}{x_{i}}}}&{\frac{-1}{1-\sum\limits_{1=1}^{n-1}{x_{i}}}}&{-% \frac{H_{n}^{\circ}}{RT^{2}}}\\ \end{array}}}\right]^{-1}.\left[{{\begin{array}[]{*{20}c}{\ln x_{1}+\frac{H_{1% }^{\circ}}{RT}-\frac{H_{1}^{\circ}}{RT_{1}^{\circ}}}\\ {\ln x_{2}+\frac{H_{2}^{\circ}}{RT}-\frac{H_{2}^{\circ}}{RT_{2}^{\circ}}}\\ {\ln x_{3}+\frac{H_{3}^{\circ}}{RT}-\frac{H_{3}^{\circ}}{RT_{3}^{\circ}}}\\ \vdots\\ {\ln x_{n-1}+\frac{H_{n-1}^{\circ}}{RT}-\frac{H_{n-1}^{\circ}}{RT_{n-1i}^{% \circ}}}\\ {\ln\left({1-\sum\limits_{i=1}^{n-1}{x_{i}}}\right)+\frac{H_{n}^{\circ}}{RT}-% \frac{H_{n}^{\circ}}{RT_{n}^{\circ}}}\\ \end{array}}}\right]\end{array}

Evanescent_wave.html

  1. k 1 d ln 1 δ k\propto\frac{1}{d}\ln{\frac{1}{\delta}}
  2. k k
  3. d d
  4. δ \delta
  5. 𝐤 = k y 𝐲 ^ + k x 𝐱 ^ = i α 𝐲 ^ + β 𝐱 ^ , \mathbf{k}\ =\ k_{y}\hat{\mathbf{y}}+k_{x}\hat{\mathbf{x}}\ =\ i\alpha\hat{% \mathbf{y}}+\beta\hat{\mathbf{x}},
  6. 𝐄 ( 𝐫 , t ) = Re { E ( 𝐫 ) e i ω t } 𝐳 ^ \mathbf{E}(\mathbf{r},t)=\mathrm{Re}\left\{E(\mathbf{r})e^{i\omega t}\right\}% \mathbf{\hat{z}}
  7. 𝐳 ^ \scriptstyle\mathbf{\hat{z}}
  8. E ( 𝐫 ) = E o e - i ( i α y + β x ) = E o e α y - i β x E(\mathbf{r})=E_{o}e^{-i(i\alpha y+\beta x)}=E_{o}e^{\alpha y-i\beta x}

Evangelista_Torricelli.html

  1. d y d t = - k u ( y ) y \frac{dy}{dt}=-k\sqrt{u(y)y}
  2. a 0 , a 1 , a 2 a_{0},a_{1},a_{2}\cdots
  3. ( a 0 - a 1 ) + ( a 1 - a 2 ) + (a_{0}-a_{1})+(a_{1}-a_{2})+\cdots
  4. a 0 - L a_{0}-L

Evapotranspiration.html

  1. Δ S = P - E T - Q - D \Delta S=P-ET-Q-D\,\!
  2. E T = P - Δ S - Q - D ET=P-\Delta S-Q-D\,\!
  3. λ E = R n - G - H \lambda E=R_{n}-G-H\,\!

Even.html

  1. f ( x ) = f ( - x ) f(x)=f(-x)

Exact_sequence.html

  1. G 0 f 1 G 1 f 2 G 2 f 3 f n G n G_{0}\;\xrightarrow{f_{1}}\;G_{1}\;\xrightarrow{f_{2}}\;G_{2}\;\xrightarrow{f_% {3}}\;\cdots\;\xrightarrow{f_{n}}\;G_{n}
  2. im ( f k ) = ker ( f k + 1 ) \mathrm{im}(f_{k})=\mathrm{ker}(f_{k+1})
  3. 0 A 𝑓 B 𝑔 C 0 0\rightarrow A~{}\overset{f}{\rightarrow}~{}B~{}\overset{g}{\rightarrow}~{}C\rightarrow 0
  4. A 𝑓 B 𝑔 C A\;\overset{f}{\hookrightarrow}\;B\;\overset{g}{\twoheadrightarrow}\;C
  5. C B / Im ( f ) C\cong B/\operatorname{Im}(f)
  6. 0 A 𝑓 B 𝑔 C 0 0\;\xrightarrow{}\;A\;\xrightarrow{f}\;B\;\xrightarrow{g}\;C\;\xrightarrow{}\;0
  7. 2 / 2 \mathbb{Z}\;\overset{2\cdot}{\hookrightarrow}\;\mathbb{Z}\twoheadrightarrow% \mathbb{Z}/2\mathbb{Z}
  8. \hookrightarrow
  9. \twoheadrightarrow
  10. 2 / 2 2\mathbb{Z}\;{\hookrightarrow}\;\mathbb{Z}\twoheadrightarrow\mathbb{Z}/2% \mathbb{Z}
  11. 2 n 2 n 2n\mapsto 2n
  12. 2 2\mathbb{Z}
  13. 2 2\mathbb{Z}
  14. \mathbb{Z}
  15. \mathbb{Z}
  16. n 2 n n\mapsto 2n
  17. 2 2\mathbb{Z}
  18. \mathbb{Z}
  19. 0 2 / 2 0 0\to\mathbb{Z}\;\xrightarrow{2\cdot}\;\mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to 0
  20. \mathbb{Z}
  21. 1 N G G / N 1 1\to N\to G\to G/N\to 1
  22. 1 C n D 2 n C 2 1 1\to C_{n}\to D_{2n}\to C_{2}\to 1
  23. C n C_{n}
  24. n n
  25. D 2 n D_{2n}
  26. 2 n 2n
  27. 1 grad curl curl div div 𝕃 2 \mathbb{H}_{1}\ \xrightarrow{\mbox{grad}~{}}\ \mathbb{H}_{\mbox{curl}}~{}\ % \xrightarrow{\mbox{curl}~{}}\ \mathbb{H}_{\mbox{div}}~{}\ \xrightarrow{\mbox{% div}~{}}\ \mathbb{L}_{2}
  28. curl ( grad f ) \displaystyle\mbox{curl}~{}\,(\mbox{grad}~{}\,f)
  29. curl \mathbb{H}_{\mbox{curl}}~{}
  30. div \mathbb{H}_{\mbox{div}}~{}
  31. A 1 A 2 A 3 A 4 A 5 A 6 A_{1}\to A_{2}\to A_{3}\to A_{4}\to A_{5}\to A_{6}
  32. C k ker ( A k A k + 1 ) im ( A k - 1 A k ) C_{k}\cong\ker(A_{k}\to A_{k+1})\cong\operatorname{im}(A_{k-1}\to A_{k})
  33. C k coker ( A k - 2 A k - 1 ) C_{k}\cong\operatorname{coker}(A_{k-2}\to A_{k-1})
  34. H / im f H H/{\left\langle\operatorname{im}f\right\rangle}^{H}
  35. A k + 1 A_{k+1}
  36. A k A k + 1 A_{k}\rightarrow A_{k+1}
  37. A k - 1 A k A k + 1 A_{k-1}\rightarrow A_{k}\rightarrow A_{k+1}
  38. 0 C k A k C k + 1 0 0\rightarrow C_{k}\rightarrow A_{k}\rightarrow C_{k+1}\rightarrow 0

Examples_of_Markov_chains.html

  1. P < m t p l > m o v e l e f t = 1 2 + 1 2 ( x c + | x | ) P_{<}mtpl>{{move~{}left}}=\tfrac{1}{2}+\tfrac{1}{2}\left(\tfrac{x}{c+|x|}\right)
  2. P < m t p l > m o v e r i g h t = 1 - P m o v e l e f t P_{<}mtpl>{{move~{}right}}=1-P_{{move~{}left}}
  3. 1 6 , 1 4 , 1 2 , 3 4 , 5 6 \tfrac{1}{6},\tfrac{1}{4},\tfrac{1}{2},\tfrac{3}{4},\tfrac{5}{6}
  4. P = [ 0.9 0.1 0.5 0.5 ] P=\begin{bmatrix}0.9&0.1\\ 0.5&0.5\end{bmatrix}
  5. 𝐱 ( 0 ) = [ 1 0 ] \mathbf{x}^{(0)}=\begin{bmatrix}1&0\end{bmatrix}
  6. 𝐱 ( 1 ) = 𝐱 ( 0 ) P = [ 1 0 ] [ 0.9 0.1 0.5 0.5 ] = [ 0.9 0.1 ] \mathbf{x}^{(1)}=\mathbf{x}^{(0)}P=\begin{bmatrix}1&0\end{bmatrix}\begin{% bmatrix}0.9&0.1\\ 0.5&0.5\end{bmatrix}=\begin{bmatrix}0.9&0.1\end{bmatrix}
  7. 𝐱 ( 2 ) = 𝐱 ( 1 ) P = 𝐱 ( 0 ) P 2 = [ 1 0 ] [ 0.9 0.1 0.5 0.5 ] 2 = [ 0.86 0.14 ] \mathbf{x}^{(2)}=\mathbf{x}^{(1)}P=\mathbf{x}^{(0)}P^{2}=\begin{bmatrix}1&0% \end{bmatrix}\begin{bmatrix}0.9&0.1\\ 0.5&0.5\end{bmatrix}^{2}=\begin{bmatrix}0.86&0.14\end{bmatrix}
  8. 𝐱 ( 2 ) = 𝐱 ( 1 ) P = [ 0.9 0.1 ] [ 0.9 0.1 0.5 0.5 ] = [ 0.86 0.14 ] \mathbf{x}^{(2)}=\mathbf{x}^{(1)}P=\begin{bmatrix}0.9&0.1\end{bmatrix}\begin{% bmatrix}0.9&0.1\\ 0.5&0.5\end{bmatrix}=\begin{bmatrix}0.86&0.14\end{bmatrix}
  9. 𝐱 ( n ) = 𝐱 ( n - 1 ) P \mathbf{x}^{(n)}=\mathbf{x}^{(n-1)}P
  10. 𝐱 ( n ) = 𝐱 ( 0 ) P n \mathbf{x}^{(n)}=\mathbf{x}^{(0)}P^{n}
  11. 𝐪 = lim n 𝐱 ( n ) \mathbf{q}=\lim_{n\to\infty}\mathbf{x}^{(n)}
  12. P = [ 0.9 0.1 0.5 0.5 ] 𝐪 P = 𝐪 ( 𝐪 is unchanged by P .) = 𝐪 I 𝐪 ( P - I ) = 𝟎 = 𝐪 ( [ 0.9 0.1 0.5 0.5 ] - [ 1 0 0 1 ] ) = 𝐪 [ - 0.1 0.1 0.5 - 0.5 ] \begin{matrix}P&=&\begin{bmatrix}0.9&0.1\\ 0.5&0.5\end{bmatrix}\\ \mathbf{q}P&=&\mathbf{q}&\mbox{(}~{}\mathbf{q}\mbox{ is unchanged by }~{}P% \mbox{.)}\\ &=&\mathbf{q}I\\ \mathbf{q}(P-I)&=&\mathbf{0}\\ &=&\mathbf{q}\left(\begin{bmatrix}0.9&0.1\\ 0.5&0.5\end{bmatrix}-\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\right)\\ &=&\mathbf{q}\begin{bmatrix}-0.1&0.1\\ 0.5&-0.5\end{bmatrix}\end{matrix}
  13. [ q 1 q 2 ] [ - 0.1 0.1 0.5 - 0.5 ] = [ 0 0 ] \begin{bmatrix}q_{1}&q_{2}\end{bmatrix}\begin{bmatrix}-0.1&0.1\\ 0.5&-0.5\end{bmatrix}=\begin{bmatrix}0&0\end{bmatrix}
  14. - 0.1 q 1 + 0.5 q 2 = 0 -0.1q_{1}+0.5q_{2}=0
  15. q 1 + q 2 = 1. q_{1}+q_{2}=1.
  16. [ q 1 q 2 ] = [ 0.833 0.167 ] \begin{bmatrix}q_{1}&q_{2}\end{bmatrix}=\begin{bmatrix}0.833&0.167\end{bmatrix}

Exclusive_or.html

  1. A B \scriptstyle A\oplus B
  2. A B C \scriptstyle A\oplus B\oplus C
  3. ~{}\oplus~{}
  4. ~{}\Leftrightarrow~{}
  5. p q p\oplus q
  6. \wedge
  7. \lor
  8. ¬ \lnot
  9. p q = ( p q ) ¬ ( p q ) \begin{matrix}p\oplus q&=&(p\lor q)\land\lnot(p\land q)\end{matrix}
  10. p q p\oplus q
  11. p q = ( p ¬ q ) ( ¬ p q ) \begin{matrix}p\oplus q&=&(p\land\lnot q)\lor(\lnot p\land q)\end{matrix}
  12. ¬ \lnot
  13. \wedge
  14. \lor
  15. p q = ( p ¬ q ) ( ¬ p q ) = ( ( p ¬ q ) ¬ p ) and ( ( p ¬ q ) q ) = ( ( p ¬ p ) ( ¬ q ¬ p ) ) ( ( p q ) ( ¬ q q ) ) = ( ¬ p ¬ q ) ( p q ) = ¬ ( p q ) ( p q ) \begin{matrix}p\oplus q&=&(p\land\lnot q)&\lor&(\lnot p\land q)\\ &=&((p\land\lnot q)\lor\lnot p)&\and&((p\land\lnot q)\lor q)\\ &=&((p\lor\lnot p)\land(\lnot q\lor\lnot p))&\land&((p\lor q)\land(\lnot q\lor q% ))\\ &=&(\lnot p\lor\lnot q)&\land&(p\lor q)\\ &=&\lnot(p\land q)&\land&(p\lor q)\end{matrix}
  16. p q p\oplus q
  17. p q = ¬ ( ( p q ) ( ¬ p ¬ q ) ) \begin{matrix}p\oplus q&=&\lnot((p\land q)\lor(\lnot p\land\lnot q))\end{matrix}
  18. p q = ( p ¬ q ) ( ¬ p q ) = p q ¯ + p ¯ q = ( p q ) ( ¬ p ¬ q ) = ( p + q ) ( p ¯ + q ¯ ) = ( p q ) ¬ ( p q ) = ( p + q ) ( p q ¯ ) \begin{matrix}p\oplus q&=&(p\land\lnot q)&\lor&(\lnot p\land q)&=&p\overline{q% }+\overline{p}q\\ \\ &=&(p\lor q)&\land&(\lnot p\lor\lnot q)&=&(p+q)(\overline{p}+\overline{q})\\ \\ &=&(p\lor q)&\land&\lnot(p\land q)&=&(p+q)(\overline{pq})\end{matrix}
  19. \wedge
  20. \lor
  21. ( { T , F } , ) (\{T,F\},\wedge)
  22. ( { T , F } , ) (\{T,F\},\lor)
  23. ( { T , F } , ) (\{T,F\},\oplus)
  24. \wedge
  25. \oplus
  26. { T , F } \{T,F\}
  27. F 2 F_{2}
  28. ( , ) (\land,\lor)
  29. F F
  30. T T
  31. F 2 F_{2}
  32. F 2 F_{2}
  33. r = p q r = p q ( mod 2 ) r = p q r = p + q ( mod 2 ) \begin{matrix}r=p\land q&\Leftrightarrow&r=p\cdot q\;\;(\mathop{{\rm mod}}2)\\ \\ r=p\oplus q&\Leftrightarrow&r=p+q\;\;(\mathop{{\rm mod}}2)\\ \end{matrix}
  34. p p
  35. q q
  36. p + q p+q
  37. \oplus
  38. \lor
  39. ¯ \underline{\lor}
  40. ˙ \dot{\vee}
  41. A B A\oplus B
  42. \Leftrightarrow
  43. B A B\oplus A
  44. \Leftrightarrow
  45. A ~{}A
  46. ~{}~{}~{}\oplus~{}~{}~{}
  47. ( B C ) (B\oplus C)
  48. \Leftrightarrow
  49. ( A B ) (A\oplus B)
  50. ~{}~{}~{}\oplus~{}~{}~{}
  51. C ~{}C
  52. ~{}~{}~{}\oplus~{}~{}~{}
  53. \Leftrightarrow
  54. \Leftrightarrow
  55. ~{}~{}~{}\oplus~{}~{}~{}
  56. A ~{}A~{}
  57. ~{}\oplus~{}
  58. A ~{}A~{}
  59. \Leftrightarrow
  60. 0 ~{}0~{}
  61. \nLeftrightarrow
  62. A ~{}A~{}
  63. ~{}\oplus~{}
  64. \Leftrightarrow
  65. \nLeftrightarrow
  66. A B A\rightarrow B
  67. \nRightarrow
  68. ( A C ) (A\oplus C)
  69. \rightarrow
  70. ( B C ) (B\oplus C)
  71. \nRightarrow
  72. \Leftrightarrow
  73. \rightarrow
  74. A and B A\and B
  75. \nRightarrow
  76. A B A\oplus B
  77. \nRightarrow
  78. A B A\oplus B
  79. \Rightarrow
  80. A B AB
  81. \Rightarrow
  82. ( \Z / 2 \Z ) n (\Z/2\Z)^{n}
  83. A B C D E A\oplus B\oplus C\oplus D\oplus E

Existential_quantification.html

  1. \mathbb{N}
  2. n P ( n , n , 25 ) \exists{n}{\in}\mathbb{N}\,P(n,n,25)
  3. n ( Q ( n ) P ( n , n , 25 ) ) \exists{n}{\in}\mathbb{N}\,\big(Q(n)\;\!\;\!{\wedge}\;\!\;\!P(n,n,25)\big)
  4. ¬ \lnot
  5. x 𝐗 P ( x ) \exists{x}{\in}\mathbf{X}\,P(x)
  6. ¬ x 𝐗 P ( x ) \lnot\ \exists{x}{\in}\mathbf{X}\,P(x)
  7. x 𝐗 P ( x ) \exists{x}{\in}\mathbf{X}\,P(x)
  8. x 𝐗 ¬ P ( x ) \forall{x}{\in}\mathbf{X}\,\lnot P(x)
  9. ¬ x 𝐗 P ( x ) x 𝐗 ¬ P ( x ) \lnot\ \exists{x}{\in}\mathbf{X}\,P(x)\equiv\ \forall{x}{\in}\mathbf{X}\,\lnot P% (x)
  10. ¬ x 𝐗 P ( x ) x 𝐗 ¬ P ( x ) ¬ x 𝐗 P ( x ) x 𝐗 ¬ P ( x ) \lnot\ \exists{x}{\in}\mathbf{X}\,P(x)\equiv\ \forall{x}{\in}\mathbf{X}\,\lnot P% (x)\not\equiv\ \lnot\ \forall{x}{\in}\mathbf{X}\,P(x)\equiv\ \exists{x}{\in}% \mathbf{X}\,\lnot P(x)
  11. x 𝐗 P ( x ) ¬ x 𝐗 P ( x ) \nexists{x}{\in}\mathbf{X}\,P(x)\equiv\lnot\ \exists{x}{\in}\mathbf{X}\,P(x)
  12. x 𝐗 P ( x ) Q ( x ) ( x 𝐗 P ( x ) x 𝐗 Q ( x ) ) \exists{x}{\in}\mathbf{X}\,P(x)Q(x)\to\ (\exists{x}{\in}\mathbf{X}\,P(x)% \exists{x}{\in}\mathbf{X}\,Q(x))
  13. P ( a ) x 𝐗 P ( x ) P(a)\to\ \exists{x}{\in}\mathbf{X}\,P(x)
  14. x 𝐗 P ( x ) ( ( P ( c ) Q ) Q ) \exists{x}{\in}\mathbf{X}\,P(x)\to\ ((P(c)\to\ Q)\to\ Q)
  15. P ( c ) Q P(c)\to\ Q
  16. x P ( x ) \exists{x}{\in}\emptyset\,P(x)
  17. \emptyset

Exosphere.html

  1. A A
  2. l l
  3. p p
  4. T T
  5. n = p A l R T n=\frac{pAl}{RT}
  6. R R
  7. p = m A n g A p=\frac{m_{A}ng}{A}
  8. m A m_{A}
  9. l = R T m A g l=\frac{RT}{m_{A}g}
  10. Kn ( h E B ) 1 \mathrm{Kn}(h_{EB})\simeq 1

Exotic_matter.html

  1. E = m c 2 1 - | 𝐯 | 2 c 2 E=\frac{m\cdot c^{2}}{\sqrt{1-\frac{\left|\mathbf{v}\right|^{2}}{c^{2}}}}
  2. m m

Experimental_mathematics.html

  1. 1 / c 10 1/c^{10}
  2. k = 1 1 k 2 ( 1 + 1 2 + 1 3 + + 1 k ) 2 = 17 π 4 360 . \displaystyle\sum_{k=1}^{\infty}\frac{1}{k^{2}}\left(1+\frac{1}{2}+\frac{1}{3}% +\cdots+\frac{1}{k}\right)^{2}=\frac{17\pi^{4}}{360}.
  3. 0 cos ( 2 x ) n = 1 cos ( x n ) d x π 8 . \int_{0}^{\infty}\cos(2x)\prod_{n=1}^{\infty}\cos\left(\frac{x}{n}\right)% \mathrm{d}x\approx\frac{\pi}{8}.

Exponential_growth.html

  1. x t = x 0 ( 1 + r ) t x_{t}=x_{0}(1+r)^{t}
  2. x ( t ) = a b t / τ x(t)=a\cdot b^{t/\tau}\,
  3. x ( 0 ) = a , x(0)=a\,,
  4. x ( t + τ ) = a b t + τ τ = a b t τ b τ τ = x ( t ) b . x(t+\tau)=a\cdot b^{\frac{t+\tau}{\tau}}=a\cdot b^{\frac{t}{\tau}}\cdot b^{% \frac{\tau}{\tau}}=x(t)\cdot b\,.
  5. x ( 1 hr ) = 1 2 6 = 64. x(1\,\text{ hr})=1\cdot 2^{6}=64.
  6. x ( t ) = x 0 e k t = x 0 e t / τ = x 0 2 t / T = x 0 ( 1 + r 100 ) t / p , x(t)=x_{0}\cdot e^{kt}=x_{0}\cdot e^{t/\tau}=x_{0}\cdot 2^{t/T}=x_{0}\cdot% \left(1+\frac{r}{100}\right)^{t/p},
  7. k = 1 τ = ln 2 T = ln ( 1 + r 100 ) p k=\frac{1}{\tau}=\frac{\ln 2}{T}=\frac{\ln\left(1+\frac{r}{100}\right)}{p}\,
  8. T 70 / r T\simeq 70/r
  9. x ( t ) = x 0 ( 1 + r ) t x(t)=x_{0}(1+r)^{t}
  10. log x ( t ) = log x 0 + t log ( 1 + r ) . \log x(t)=\log x_{0}+t\cdot\log(1+r).
  11. x ( t ) = x ( 0 ) e k t \scriptstyle x(t)=x(0)e^{kt}
  12. d x d t = k x \!\,\frac{dx}{dt}=kx
  13. x ( 0 ) . x(0).\,
  14. d x d t = k x \frac{dx}{dt}=kx
  15. d x x = k d t \Rightarrow\frac{dx}{x}=k\,dt
  16. x ( 0 ) x ( t ) d x x = k 0 t d t \Rightarrow\int_{x(0)}^{x(t)}\frac{dx}{x}=k\int_{0}^{t}\,dt
  17. ln x ( t ) x ( 0 ) = k t . \Rightarrow\ln\frac{x(t)}{x(0)}=kt.
  18. x ( t ) = x ( 0 ) e k t \Rightarrow x(t)=x(0)e^{kt}\,
  19. x t = a x t - 1 x_{t}=a\cdot x_{t-1}
  20. x t = x 0 a t , x_{t}=x_{0}\cdot a^{t},
  21. lim t t α a e t = 0. \lim_{t\rightarrow\infty}{t^{\alpha}\over ae^{t}}=0.

Exponential_map_(Riemannian_geometry).html

  1. M M
  2. p p
  3. M M
  4. M M
  5. p p
  6. p p
  7. p p
  8. u , v y = u v y 2 \langle u,v\rangle_{y}=\frac{uv}{y^{2}}
  9. | | y |\cdot|_{y}
  10. s ( t ) = 0 t | x | y ( τ ) d τ = 0 t | x | 1 + τ x d τ = | x | 0 t d τ 1 + τ x = | x | x ln | 1 + t x | s(t)=\int_{0}^{t}|x|_{y(\tau)}d\tau=\int_{0}^{t}\frac{|x|}{1+\tau x}d\tau=|x|% \int_{0}^{t}\frac{d\tau}{1+\tau x}=\frac{|x|}{x}\ln|1+tx|
  11. y ( s ) = e s x / | x | y(s)=e^{sx/|x|}
  12. exp 1 ( x ) = y ( | x | 1 ) = y ( | x | ) \exp_{1}(x)=y(|x|_{1})=y(|x|)
  13. dist ( a , b ) = | ln ( b / a ) | \operatorname{dist}(a,b)=|\ln(b/a)|

Exponentiation.html

  1. b n = b × × b n b^{n}=\underbrace{b\times\cdots\times b}_{n}
  2. b 1 = b b^{1}=b
  3. b n + 1 = b n b b^{n+1}=b^{n}\cdot b
  4. b m + n = b m b n b^{m+n}=b^{m}\cdot b^{n}
  5. b - n = 1 / b n b^{-n}=1/b^{n}
  6. b n = b n + 1 / b , n 1. b^{n}={b^{n+1}}/{b},\quad n\geq 1.
  7. b 0 = b 1 / b = 1 b - 1 = b 0 / b = 1 / b \begin{aligned}\displaystyle b^{0}&\displaystyle={b^{1}}/{b}=1\\ \displaystyle b^{-1}&\displaystyle={b^{0}}/{b}={1}/{b}\end{aligned}
  8. b - n = 1 / b n . b^{-n}={1}/{b^{n}}.
  9. b m + n \displaystyle b^{m+n}
  10. b p q = b ( p q ) ( b p ) q = b ( p q ) = b p q . b^{p^{q}}=b^{(p^{q})}\neq(b^{p})^{q}=b^{(p\cdot q)}=b^{p\cdot q}.
  11. b n \sqrt{bn}
  12. \sqrt{}
  13. x = b 1 / n x=b^{1/n}
  14. x n = b x^{n}=b
  15. x n = b 1 n × b 1 n × × b 1 n n = b ( 1 n + 1 n + + 1 n ) = b n n = b 1 = b . x^{n}=\underbrace{b^{\frac{1}{n}}\times b^{\frac{1}{n}}\times\cdots\times b^{% \frac{1}{n}}}_{n}=b^{\left(\frac{1}{n}+\frac{1}{n}+\cdots+\frac{1}{n}\right)}=% b^{\frac{n}{n}}=b^{1}=b.
  16. b 1 / n b^{1/n}
  17. b u v = ( b u ) 1 v = b u v b^{\frac{u}{v}}=\left(b^{u}\right)^{\frac{1}{v}}=\sqrt[v]{b^{u}}
  18. ( b r ) s = b r s (b^{r})^{s}=b^{r\cdot s}
  19. b x = lim r ( ) x b r ( b + , x ) b^{x}=\lim_{r(\in\mathbb{Q})\to x}b^{r}\quad(b\in\mathbb{R}^{+},\,x\in\mathbb{% R})
  20. x x
  21. [ b 3 , b 4 ] [b^{3},b^{4}]
  22. [ b 3.1 , b 3.2 ] [b^{3.1},b^{3.2}]
  23. [ b 3.14 , b 3.15 ] [b^{3.14},b^{3.15}]
  24. [ b 3.141 , b 3.142 ] [b^{3.141},b^{3.142}]
  25. [ b 3.1415 , b 3.1416 ] [b^{3.1415},b^{3.1416}]
  26. [ b 3.14159 , b 3.14160 ] [b^{3.14159},b^{3.14160}]
  27. b π b^{\pi}
  28. e e
  29. exp ( x ) = lim n ( 1 + x n ) n \exp(x)=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}
  30. exp ( x + y ) = exp ( x ) exp ( y ) \exp(x+y)=\exp(x)\cdot\exp(y)
  31. b = e ln b b=e^{\ln b}
  32. b x = ( e ln b ) x = e x ln b b^{x}=(e^{\ln b})^{x}=e^{x\cdot\ln b}
  33. z = x + i y z=x+iy
  34. i 2 = - 1 i^{2}=-1
  35. x = r c o s θ x=rcosθ
  36. y = r s i n θ y=rsinθ
  37. x + i y = r ( cos θ + i sin θ ) . x+iy=r(\cos\theta+i\sin\theta).
  38. i 2 = - 1 i^{2}=-1
  39. z 1 z 2 = ( x 1 + i y 1 ) ( x 2 + i y 2 ) = ( x 1 x 2 - y 1 y 2 ) + i ( x 1 y 2 + x 2 y 1 ) . z_{1}z_{2}=(x_{1}+iy_{1})(x_{2}+iy_{2})=(x_{1}x_{2}-y_{1}y_{2})+i(x_{1}y_{2}+x% _{2}y_{1}).
  40. { z : e z = 1 } = { 2 k π i : k } \{z:e^{z}=1\}=\{2k\pi i:k\in\mathbb{Z}\}
  41. { z : e z = w } = { v + 2 k π i : k } \{z:e^{z}=w\}=\{v+2k\pi i:k\in\mathbb{Z}\}
  42. cos ( z ) = e i z + e - i z 2 ; sin ( z ) = e i z - e - i z 2 i \cos(z)=\frac{e^{iz}+e^{-iz}}{2};\qquad\sin(z)=\frac{e^{iz}-e^{-iz}}{2i}
  43. e i ( x + y ) = e i x e i y e^{i(x+y)}=e^{ix}\cdot e^{iy}
  44. ( b z ) u = b z u (b^{z})^{u}=b^{zu}
  45. ( e 2 π i ) i = 1 i = 1 e - 2 π = e 2 π i i (e^{2\pi i})^{i}=1^{i}=1\neq e^{-2\pi}=e^{2\pi i\cdot i}
  46. z z
  47. u u
  48. w z = e z log w w^{z}=e^{z\log w}
  49. e 1 + 2 π i n = e 1 e 2 π i n = e 1 = e e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e
  50. ( e 1 + 2 π i n ) 1 + 2 π i n = e \left(e^{1+2\pi in}\right)^{1+2\pi in}=e
  51. e 1 + 4 π i n - 4 π 2 n 2 = e e^{1+4\pi in-4\pi^{2}n^{2}}=e
  52. e 1 e 4 π i n e - 4 π 2 n 2 = e e^{1}e^{4\pi in}e^{-4\pi^{2}n^{2}}=e
  53. e - 4 π 2 n 2 = 1 e^{-4\pi^{2}n^{2}}=1
  54. ( e z ) w e z w \scriptstyle(e^{z})^{w}\;\neq\;e^{zw}
  55. ( e z ) w = e ( z + 2 π i n ) w \scriptstyle(e^{z})^{w}\;=\;e^{(z\,+\,2\pi in)w}
  56. x 0 = 1 x^{0}=1
  57. x X x\in X
  58. x n + 1 = x n x x^{n+1}=x^{n}x
  59. x X x\in X
  60. A 0 A^{0}
  61. A - n = ( A - 1 ) n A^{-n}=(A^{-1})^{n}
  62. A 2 x A^{2}x
  63. A n x A^{n}x
  64. A n A^{n}
  65. d / d x d/dx
  66. f ( x ) f(x)
  67. ( d / d x ) f ( x ) = f ( x ) (d/dx)f(x)=f^{\prime}(x)
  68. ( d d x ) n f ( x ) = d n d x n f ( x ) = f ( n ) ( x ) . \left(\frac{d}{dx}\right)^{n}f(x)=\frac{d^{n}}{dx^{n}}f(x)=f^{(n)}(x).
  69. F 2 = { 0 , 1 } F_{2}=\{0,1\}
  70. 0 + 1 = 1 + 0 = 1 0+1=1+0=1
  71. 0 + 0 = 1 + 1 = 0 0+0=1+1=0
  72. 0 0 = 1 0 = 0 1 = 0 0\cdot 0=1\cdot 0=0\cdot 1=0
  73. 1 1 = 1 1\cdot 1=1
  74. p x = 0 px=0
  75. F 2 F_{2}
  76. p = 2 p=2
  77. f ( x ) = x p f(x)=x^{p}
  78. ( x + y ) p = x p + y p (x+y)^{p}=x^{p}+y^{p}
  79. x 1 = x x n = x n - 1 x for n > 1 \begin{aligned}\displaystyle x^{1}&\displaystyle=x\\ \displaystyle x^{n}&\displaystyle=x^{n-1}x\quad\hbox{for }n>1\end{aligned}
  80. ( x i x j ) x k = x i ( x j x k ) (power-associative property) x m + n = x m x n ( x m ) n = x m n \begin{aligned}\displaystyle(x^{i}x^{j})x^{k}&\displaystyle=x^{i}(x^{j}x^{k})% \quad\,\text{(power-associative property)}\\ \displaystyle x^{m+n}&\displaystyle=x^{m}x^{n}\\ \displaystyle(x^{m})^{n}&\displaystyle=x^{mn}\end{aligned}
  81. x 1 = 1 x = x (two-sided identity) x 0 = 1 \begin{aligned}\displaystyle x1&\displaystyle=1x=x\quad\,\text{(two-sided % identity)}\\ \displaystyle x^{0}&\displaystyle=1\end{aligned}
  82. x x - 1 = x - 1 x = 1 (two-sided inverse) ( x y ) z = x ( y z ) (associative) x - n = ( x - 1 ) n x m - n = x m x - n \begin{aligned}\displaystyle xx^{-1}&\displaystyle=x^{-1}x=1\quad\,\text{(two-% sided inverse)}\\ \displaystyle(xy)z&\displaystyle=x(yz)\quad\,\text{(associative)}\\ \displaystyle x^{-n}&\displaystyle=\left(x^{-1}\right)^{n}\\ \displaystyle x^{m-n}&\displaystyle=x^{m}x^{-n}\end{aligned}
  83. ( x y ) n = x n y n (xy)^{n}=x^{n}y^{n}
  84. i V i \bigoplus_{i\in\mathbb{N}}V_{i}
  85. S N { f : N S } S^{N}\equiv\{f\colon N\to S\}
  86. | X | |X|
  87. 0 0 0^{0}
  88. a 0 x 0 + + a n x n a_{0}x^{0}+\cdots+a_{n}x^{n}
  89. a n a_{n}
  90. [ x ] \mathbb{R}[x]
  91. x 0 x^{0}
  92. x 0 x^{0}
  93. p ( x ) p(x)
  94. p ( x ) p(x)
  95. x 0 x_{0}
  96. ev x 0 : [ x ] \operatorname{ev}_{x_{0}}:\mathbb{R}[x]\to\mathbb{R}
  97. ev x 0 ( x 1 ) = x 0 \operatorname{ev}_{x_{0}}(x^{1})=x_{0}
  98. ev x 0 ( x 0 ) = 1. \operatorname{ev}_{x_{0}}(x^{0})=1.
  99. x 0 = 1 x^{0}=1
  100. ( 1 + x ) n = k = 0 n ( n k ) x k (1+x)^{n}=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}x^{k}
  101. x 0 = 1 x^{0}=1
  102. 1 1 - x = n = 0 x n \frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}
  103. e x = n = 0 x n n ! e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}
  104. d d x x n = n x n - 1 \frac{d}{dx}x^{n}=nx^{n-1}
  105. lim t 0 + t t = 1 , lim t 0 + ( e - 1 t 2 ) t = 0 , lim t 0 + ( e - 1 t 2 ) - t = + , lim t 0 + ( e - 1 t ) a t = e - a \lim_{t\to 0^{+}}{t}^{t}=1,\quad\lim_{t\to 0^{+}}\left(e^{-\frac{1}{t^{2}}}% \right)^{t}=0,\quad\lim_{t\to 0^{+}}\left(e^{-\frac{1}{t^{2}}}\right)^{-t}=+% \infty,\quad\lim_{t\to 0^{+}}\left(e^{-\frac{1}{t}}\right)^{at}=e^{-a}
  106. 0 0 0^{0}
  107. 0 0 = 1 0^{0}=1
  108. 0 0 0^{0}
  109. 0 0 \frac{0}{0}
  110. 0 0 = 1 0^{0}=1
  111. lim t 0 + f ( t ) g ( t ) = 1 \scriptstyle\lim_{t\to 0^{+}}f(t)^{g(t)}\;=\;1
  112. lim t 0 + f ( t ) = lim t 0 + g ( t ) = 0 \scriptstyle\lim_{t\to 0^{+}}f(t)\;=\;\lim_{t\to 0^{+}}g(t)\;=\;0
  113. ( e - 1 / t ) t \scriptstyle(e^{-1/t})^{t}
  114. 0 0 0^{0}
  115. 0 0 0^{0}
  116. 0 0 0^{0}
  117. 0 0 = 1 0^{0}=1
  118. 0 0 0^{0}
  119. 0 0 0^{0}
  120. 0 0 0^{0}
  121. 0 0 0^{0}
  122. 0 0 0^{0}
  123. f ( x ) g ( x ) \scriptstyle f(x)^{g(x)}
  124. f ( x ) , g ( x ) 0 \scriptstyle f(x),g(x)\to 0
  125. A [ ( X i ) i I ] A[(X_{i})_{i\in I}]
  126. A A

Exposure_(photography).html

  1. H e = E e t , H_{\mathrm{e}}=E_{\mathrm{e}}t,
  2. H v = E v t , H_{\mathrm{v}}=E_{\mathrm{v}}t,

Exposure_value.html

  1. E v E_{v}
  2. EV = log 2 N 2 t , \mathrm{EV}=\log_{2}{\frac{N^{2}}{t}}\,,
  3. H v = E v t , H_{\mathrm{v}}=E_{\mathrm{v}}\cdot t\,,
  4. S S
  5. EV S = EV 100 + log 2 S 100 . \mathrm{EV}_{S}=\mathrm{EV}_{100}+\log_{2}\frac{S}{100}\,.
  6. EV 400 = EV 100 + log 2 400 100 = EV 100 + 2 . \mathrm{EV}_{400}=\mathrm{EV}_{100}+\log_{2}\frac{400}{100}=\mathrm{EV}_{100}+% 2\,.
  7. EV 50 = EV 100 + log 2 50 100 = EV 100 - 1 . \mathrm{EV}_{50}=\mathrm{EV}_{100}+\log_{2}\frac{50}{100}=\mathrm{EV}_{100}-1\,.
  8. EV 100 = EV S - log 2 S 100 . \mathrm{EV}_{100}=\mathrm{EV}_{S}-\log_{2}\frac{S}{100}\,.
  9. f f
  10. f f
  11. N 2 t = L S K , \frac{N^{2}}{t}=\frac{L\cdot S}{K}\,,
  12. EV = log 2 L S K . \mathrm{EV}=\log_{2}{\frac{L\cdot S}{K}}\,.
  13. N 2 t = E S C , \frac{N^{2}}{t}=\frac{E\cdot S}{C}\,,
  14. EV = log 2 E S C . \mathrm{EV}=\log_{2}{\frac{E\cdot S}{C}}\,.
  15. E v = A v + T v , E_{v}=A_{v}+T_{v}\,,
  16. A v = log 2 A 2 A_{v}=\log_{2}A^{2}
  17. T v = log 2 ( 1 / T ) , T_{v}=\log_{2}(1/T)\,,
  18. L = 2 EV - 3 . L=2^{\mathrm{EV}-3}\,.
  19. E = 2.5 × 2 EV . E=2.5\times 2^{\mathrm{EV}}\,.

Extended_Euclidean_algorithm.html

  1. a x + b y = gcd ( a , b ) . ax+by=\gcd(a,b).
  2. q 1 , , q k q_{1},\ldots,q_{k}
  3. r 0 , , r k + 1 r_{0},\ldots,r_{k+1}
  4. r 0 = a r 1 = b r i + 1 = r i - 1 - q i r i and 0 r i + 1 < | r i | \begin{array}[]{l}r_{0}=a\\ r_{1}=b\\ \ldots\\ r_{i+1}=r_{i-1}-q_{i}r_{i}\quad\text{and}\quad 0\leq r_{i+1}<|r_{i}|\\ \ldots\end{array}
  5. r i + 1 r_{i+1}
  6. r i - 1 r_{i-1}
  7. r i . r_{i}.
  8. r k + 1 r_{k+1}
  9. r k . r_{k}.
  10. r 0 = a r 1 = b s 0 = 1 s 1 = 0 t 0 = 0 t 1 = 1 r i + 1 = r i - 1 - q i r i and 0 r i + 1 < | r i | (this defines q i ) s i + 1 = s i - 1 - q i s i t i + 1 = t i - 1 - q i t i \begin{array}[]{l}r_{0}=a\qquad r_{1}=b\\ s_{0}=1\qquad s_{1}=0\\ t_{0}=0\qquad t_{1}=1\\ \ldots\\ r_{i+1}=r_{i-1}-q_{i}r_{i}\quad\text{and}\quad 0\leq r_{i+1}<|r_{i}|\qquad\,% \text{(this defines }q_{i}\,\text{)}\\ s_{i+1}=s_{i-1}-q_{i}s_{i}\\ t_{i+1}=t_{i-1}-q_{i}t_{i}\\ \ldots\end{array}
  11. r k + 1 = 0 r_{k+1}=0
  12. r k r_{k}
  13. a = r 0 a=r_{0}
  14. b = r 1 . b=r_{1}.
  15. s k s_{k}
  16. t k , t_{k},
  17. gcd ( a , b ) = r k = a s k + b t k \gcd(a,b)=r_{k}=as_{k}+bt_{k}
  18. s k + 1 = ± b gcd ( a , b ) s_{k+1}=\pm\frac{b}{\gcd(a,b)}
  19. t k + 1 = ± a gcd ( a , b ) t_{k+1}=\pm\frac{a}{\gcd(a,b)}
  20. | s k | < b gcd ( a , b ) and | t k | < a gcd ( a , b ) . |s_{k}|<\frac{b}{\gcd(a,b)}\quad\,\text{and}\quad|t_{k}|<\frac{a}{\gcd(a,b)}.
  21. [ u g r e e n , u 240 ] [u^{\prime}green^{\prime},u^{\prime}240^{\prime}]
  22. [ u g r e e n , u 46 ] [u^{\prime}green^{\prime},u^{\prime}46^{\prime}]
  23. [ u r e d , u 2 ] [u^{\prime}red^{\prime},u^{\prime}2^{\prime}]
  24. [ u r e d , u 0 ] [u^{\prime}red^{\prime},u^{\prime}0^{\prime}]
  25. [ u m a g e n t a , u 2129 ] × [ u g r e e n , u 240 ] + [ u m a g e n t a , u 47 ] × [ u g r e e n , u 46 ] = [ u r e d , u 2 ] [u^{\prime}magenta^{\prime},u^{\prime}\u{2}2129^{\prime}]×[u^{\prime}green^{% \prime},u^{\prime}240^{\prime}]+[u^{\prime}magenta^{\prime},u^{\prime}47^{% \prime}]×[u^{\prime}green^{\prime},u^{\prime}46^{\prime}]=[u^{\prime}red^{% \prime},u^{\prime}2^{\prime}]
  26. [ u c y a n , u 23 ] [u^{\prime}cyan^{\prime},u^{\prime}23^{\prime}]
  27. [ u c y a n , u 212120 ] [u^{\prime}cyan^{\prime},u^{\prime}\u{2}212120^{\prime}]
  28. [ u g r e e n , u 46 ] [u^{\prime}green^{\prime},u^{\prime}46^{\prime}]
  29. [ u g r e e n , u 240 ] [u^{\prime}green^{\prime},u^{\prime}240^{\prime}]
  30. [ u r e d , u 2 ] [u^{\prime}red^{\prime},u^{\prime}2^{\prime}]
  31. 0 r i + 1 < | r i | , 0\leq r_{i+1}<|r_{i}|,
  32. r i r_{i}
  33. r k + 1 = 0. r_{k+1}=0.
  34. r i + 1 = r i - 1 - r i q i , r_{i+1}=r_{i-1}-r_{i}q_{i},
  35. ( r i - 1 , r i ) (r_{i-1},r_{i})
  36. ( r i , r i + 1 ) . (r_{i},r_{i+1}).
  37. a = r 0 , b = r 1 a=r_{0},b=r_{1}
  38. r k , r k + 1 = 0. r_{k},r_{k+1}=0.
  39. r k r_{k}
  40. a = r 0 a=r_{0}
  41. b = r 1 , b=r_{1},
  42. a s i + b t i = r i as_{i}+bt_{i}=r_{i}
  43. i > 1 i>1
  44. r i + 1 = r i - 1 - r i q i = ( a s i - 1 + b t i - 1 ) - ( a s i + b t i ) q i = ( a s i - 1 - a s i q i ) + ( b t i - 1 - b t i q i ) = a s i + 1 + b t i + 1 . r_{i+1}=r_{i-1}-r_{i}q_{i}=(as_{i-1}+bt_{i-1})-(as_{i}+bt_{i})q_{i}=(as_{i-1}-% as_{i}q_{i})+(bt_{i-1}-bt_{i}q_{i})=as_{i+1}+bt_{i+1}.
  45. s k s_{k}
  46. t k t_{k}
  47. A i = ( s i - 1 s i t i - 1 t i ) . A_{i}=\begin{pmatrix}s_{i-1}&s_{i}\\ t_{i-1}&t_{i}\end{pmatrix}\,.
  48. A i + 1 = A i . ( 0 1 1 - q i ) . A_{i+1}=A_{i}\,.\,\begin{pmatrix}0&1\\ 1&-q_{i}\end{pmatrix}\,.
  49. A 1 A_{1}
  50. A i A_{i}
  51. ( - 1 ) i - 1 . (-1)^{i-1}.
  52. s k t k + 1 - t k s k + 1 = ( - 1 ) k . s_{k}t_{k+1}-t_{k}s_{k+1}=(-1)^{k}.
  53. s k + 1 s_{k+1}
  54. t k + 1 t_{k+1}
  55. a s k + 1 + b t k + 1 = 0 as_{k+1}+bt_{k+1}=0
  56. s k + 1 s_{k+1}
  57. t k + 1 t_{k+1}
  58. 0 r i + 1 < | r i | 0\leq r_{i+1}<|r_{i}|
  59. deg r i + 1 < deg r i . \deg r_{i+1}<\deg r_{i}.
  60. a s + b t = gcd ( a , b ) as+bt=\gcd(a,b)
  61. deg s < deg b - deg ( gcd ( a , b ) ) , deg t < deg a - deg ( gcd ( a , b ) ) . \deg s<\deg b-\deg(\gcd(a,b)),\quad\deg t<\deg a-\deg(\gcd(a,b)).
  62. r k . r_{k}.
  63. r k , r_{k},
  64. r i r_{i}
  65. a s + b t = Res ( a , b ) , as+bt=\operatorname{Res}(a,b),
  66. Res ( a , b ) \operatorname{Res}(a,b)
  67. a b \frac{a}{b}
  68. a a
  69. b b
  70. b b
  71. - t -t
  72. - t 1 \frac{-t}{1}
  73. - t s \frac{-t}{s}
  74. t - s \frac{t}{-s}
  75. a b = - t s \frac{a}{b}=-\frac{t}{s}
  76. n n
  77. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  78. n n
  79. n n
  80. a a
  81. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  82. n n
  83. n n
  84. a a
  85. n n
  86. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  87. n n
  88. a a
  89. n n
  90. s s
  91. t t
  92. n s + a t = 1 ns+at=1
  93. n n
  94. a t = 1 mod n . at=1\mod n.
  95. t t
  96. t t
  97. n n
  98. a a
  99. n n
  100. n n
  101. t t
  102. t < 0 t<0
  103. q q
  104. p p
  105. d d
  106. L L
  107. K K
  108. p p
  109. d d
  110. K [ X ] / p , K[X]/\langle p\rangle,
  111. d d
  112. L L
  113. L L
  114. p p
  115. L L
  116. d d
  117. K K
  118. K K
  119. p p
  120. a a
  121. p p
  122. gcd ( a , b , c ) = gcd ( gcd ( a , b ) , c ) \gcd(a,b,c)=\gcd(\gcd(a,b),c)
  123. d = gcd ( a , b , c ) d=\gcd(a,b,c)
  124. d d
  125. a a
  126. b b
  127. gcd ( a , b ) = k d \gcd(a,b)=kd
  128. k k
  129. d d
  130. c c
  131. c = j d c=jd
  132. j j
  133. u = gcd ( k , j ) u=\gcd(k,j)
  134. u u
  135. u d | a , b , c ud|a,b,c
  136. d d
  137. u u
  138. u d = gcd ( g c d ( a , b ) , c ) ud=\gcd(gcd(a,b),c)
  139. n a + m b = gcd ( a , b ) na+mb=\gcd(a,b)
  140. x x
  141. y y
  142. x gcd ( a , b ) + y c = gcd ( a , b , c ) x\gcd(a,b)+yc=\gcd(a,b,c)
  143. x ( n a + m b ) + y c = ( x n ) a + ( x m ) b + y c = gcd ( a , b , c ) . x(na+mb)+yc=(xn)a+(xm)b+yc=\gcd(a,b,c).\,
  144. gcd ( a 1 , a 2 , , a n ) = gcd ( a 1 , gcd ( a 2 , gcd ( a 3 , , gcd ( a n - 1 , a n ) ) ) ) , \gcd(a_{1},a_{2},\dots,a_{n})=\gcd(a_{1},\,\gcd(a_{2},\,\gcd(a_{3},\dots,\gcd(% a_{n-1}\,,a_{n})))\dots),

Exterior_algebra.html

  1. 𝐞 1 = [ 1 0 ] , 𝐞 2 = [ 0 1 ] . {\mathbf{e}}_{1}=\begin{bmatrix}1\\ 0\end{bmatrix},\quad{\mathbf{e}}_{2}=\begin{bmatrix}0\\ 1\end{bmatrix}.
  2. 𝐯 = [ a b ] = a 𝐞 1 + b 𝐞 2 , 𝐰 = [ c d ] = c 𝐞 1 + d 𝐞 2 {\mathbf{v}}=\begin{bmatrix}a\\ b\end{bmatrix}=a{\mathbf{e}}_{1}+b{\mathbf{e}}_{2},\quad{\mathbf{w}}=\begin{% bmatrix}c\\ d\end{bmatrix}=c{\mathbf{e}}_{1}+d{\mathbf{e}}_{2}
  3. Area = | det [ 𝐯 𝐰 ] | = | det [ a c b d ] | = | a d - b c | . \,\text{Area}=\left|\det\begin{bmatrix}{\mathbf{v}}&{\mathbf{w}}\end{bmatrix}% \right|=\left|\det\begin{bmatrix}a&c\\ b&d\end{bmatrix}\right|=\left|ad-bc\right|.
  4. 𝐯 𝐰 \displaystyle{\mathbf{v}}\wedge{\mathbf{w}}
  5. 𝐮 = u 1 𝐞 1 + u 2 𝐞 2 + u 3 𝐞 3 \mathbf{u}=u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}+u_{3}\mathbf{e}_{3}
  6. 𝐯 = v 1 𝐞 1 + v 2 𝐞 2 + v 3 𝐞 3 \mathbf{v}=v_{1}\mathbf{e}_{1}+v_{2}\mathbf{e}_{2}+v_{3}\mathbf{e}_{3}
  7. 𝐮 𝐯 = ( u 1 v 2 - u 2 v 1 ) ( 𝐞 1 𝐞 2 ) + ( u 3 v 1 - u 1 v 3 ) ( 𝐞 3 𝐞 1 ) + ( u 2 v 3 - u 3 v 2 ) ( 𝐞 2 𝐞 3 ) \mathbf{u}\wedge\mathbf{v}=(u_{1}v_{2}-u_{2}v_{1})(\mathbf{e}_{1}\wedge\mathbf% {e}_{2})+(u_{3}v_{1}-u_{1}v_{3})(\mathbf{e}_{3}\wedge\mathbf{e}_{1})+(u_{2}v_{% 3}-u_{3}v_{2})(\mathbf{e}_{2}\wedge\mathbf{e}_{3})
  8. 𝐰 = w 1 𝐞 1 + w 2 𝐞 2 + w 3 𝐞 3 , \mathbf{w}=w_{1}\mathbf{e}_{1}+w_{2}\mathbf{e}_{2}+w_{3}\mathbf{e}_{3},
  9. 𝐮 𝐯 𝐰 = ( u 1 v 2 w 3 + u 2 v 3 w 1 + u 3 v 1 w 2 - u 1 v 3 w 2 - u 2 v 1 w 3 - u 3 v 2 w 1 ) ( 𝐞 1 𝐞 2 𝐞 3 ) \mathbf{u}\wedge\mathbf{v}\wedge\mathbf{w}=(u_{1}v_{2}w_{3}+u_{2}v_{3}w_{1}+u_% {3}v_{1}w_{2}-u_{1}v_{3}w_{2}-u_{2}v_{1}w_{3}-u_{3}v_{2}w_{1})(\mathbf{e}_{1}% \wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3})
  10. Λ ( V ) := T ( V ) / I . \Lambda(V):=T(V)/I.\,
  11. α β = α β ( mod I ) , \alpha\wedge\beta=\alpha\otimes\beta\;\;(\mathop{{\rm mod}}I),
  12. ( T 0 ( V ) T 1 ( V ) ) I = 0 , \left(T^{0}(V)\oplus T^{1}(V)\right)\cap I=0,
  13. K K
  14. V V
  15. T ( V ) T(V)
  16. K K
  17. V V
  18. Λ ( V ) Λ(V)
  19. 0 = ( x + y ) ( x + y ) = x x + x y + y x + y y = x y + y x 0=(x+y)\wedge(x+y)=x\wedge x+x\wedge y+y\wedge x+y\wedge y=x\wedge y+y\wedge x
  20. x y = - y x . x\wedge y=-y\wedge x.
  21. x σ ( 1 ) x σ ( 2 ) x σ ( k ) = sgn ( σ ) x 1 x 2 x k , x_{\sigma(1)}\wedge x_{\sigma(2)}\wedge\dots\wedge x_{\sigma(k)}=\operatorname% {sgn}(\sigma)x_{1}\wedge x_{2}\wedge\dots\wedge x_{k},
  22. x 1 x 2 x k , x i V , i = 1 , 2 , , k . x_{1}\wedge x_{2}\wedge\dots\wedge x_{k},\quad x_{i}\in V,i=1,2,\dots,k.
  23. α = e 1 e 2 + e 3 e 4 . \alpha=e_{1}\wedge e_{2}+e_{3}\wedge e_{4}.
  24. { e i 1 e i 2 e i k 1 i 1 < i 2 < < i k n } \{e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{k}}\mid 1\leq i_{1}<i_{2}<% \cdots<i_{k}\leq n\}
  25. v 1 v k v_{1}\wedge\cdots\wedge v_{k}
  26. dim ( Λ k ( V ) ) = ( n k ) \operatorname{dim}(\Lambda^{k}(V))={\left({{n}\atop{k}}\right)}
  27. Λ ( V ) = Λ 0 ( V ) Λ 1 ( V ) Λ 2 ( V ) Λ n ( V ) \Lambda(V)=\Lambda^{0}(V)\oplus\Lambda^{1}(V)\oplus\Lambda^{2}(V)\oplus\cdots% \oplus\Lambda^{n}(V)
  28. α = α ( 1 ) + α ( 2 ) + + α ( s ) \alpha=\alpha^{(1)}+\alpha^{(2)}+\cdots+\alpha^{(s)}
  29. α ( i ) = α 1 ( i ) α k ( i ) , i = 1 , 2 , , s . \alpha^{(i)}=\alpha^{(i)}_{1}\wedge\cdots\wedge\alpha^{(i)}_{k},\quad i=1,2,% \dots,s.
  30. α = i , j a i j e i e j \alpha=\sum_{i,j}a_{ij}e_{i}\wedge e_{j}
  31. α α 𝑝 0 \underset{p}{\underbrace{\alpha\wedge\cdots\wedge\alpha}}\not=0
  32. α α p + 1 = 0. \underset{p+1}{\underbrace{\alpha\wedge\cdots\wedge\alpha}}=0.
  33. Λ ( V ) = Λ 0 ( V ) Λ 1 ( V ) Λ 2 ( V ) Λ n ( V ) \Lambda(V)=\Lambda^{0}(V)\oplus\Lambda^{1}(V)\oplus\Lambda^{2}(V)\oplus\cdots% \oplus\Lambda^{n}(V)
  34. ( Λ k ( V ) ) ( Λ p ( V ) ) \sub Λ k + p ( V ) . \left(\Lambda^{k}(V)\right)\wedge\left(\Lambda^{p}(V)\right)\sub\Lambda^{k+p}(% V).
  35. K K
  36. Λ 0 ( V ) = K \Lambda^{0}(V)=K
  37. Λ 1 ( V ) = V . \Lambda^{1}(V)=V.
  38. α β = ( - 1 ) k p β α . \alpha\wedge\beta=(-1)^{kp}\beta\wedge\alpha.
  39. V V
  40. K K
  41. Λ ( V ) Λ(V)
  42. v v = 0 v∧v=0
  43. v V v∈V
  44. Λ ( V ) Λ(V)
  45. K K
  46. V V
  47. V V
  48. Λ ( V ) Λ(V)
  49. K K
  50. A A
  51. K K
  52. j : V A j:V→A
  53. j ( v ) j ( v ) = 0 j(v)j(v)=0
  54. v v
  55. V V
  56. f : Λ ( V ) A f:Λ(V)→A
  57. j ( v ) = f ( i ( v ) ) j(v)=f(i(v))
  58. v v
  59. V V
  60. i i
  61. V V
  62. Λ ( V ) Λ(V)
  63. V V
  64. V V
  65. V V
  66. T ( V ) T(V)
  67. I I
  68. T ( V ) T(V)
  69. v v v⊗v
  70. v v
  71. V V
  72. Λ ( V ) Λ(V)
  73. Λ ( V ) = T ( V ) / I \Lambda(V)=T(V)/I
  74. Λ ( V ) ) Λ(V))
  75. Λ ( V ) Λ(V)
  76. V V
  77. V V
  78. Λ ( V ) Λ(V)
  79. Λ ( V ) Λ(V)
  80. Λ ( V ) Λ(V)
  81. f : V k X f:V^{k}\to X
  82. f ( v 1 , , v k ) = 0 f(v_{1},\ldots,v_{k})=0
  83. w : V k Λ k ( V ) w:V^{k}\to\Lambda^{k}(V)
  84. f : V k K f:V^{k}\to K
  85. ω η = Alt ( ω η ) . \omega\wedge\eta=\operatorname{Alt}(\omega\otimes\eta).
  86. ω η = ( k + m ) ! k ! m ! Alt ( ω η ) \omega\wedge\eta=\frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega\otimes\eta)
  87. Alt ( ω ) ( x 1 , , x k ) = 1 k ! σ S k sgn ( σ ) ω ( x σ ( 1 ) , , x σ ( k ) ) . \operatorname{Alt}(\omega)(x_{1},\ldots,x_{k})=\frac{1}{k!}\sum_{\sigma\in S_{% k}}\operatorname{sgn}(\sigma)\,\omega(x_{\sigma(1)},\ldots,x_{\sigma(k)}).
  88. ω η ( x 1 , , x k + m ) = σ S h k , m sgn ( σ ) ω ( x σ ( 1 ) , , x σ ( k ) ) η ( x σ ( k + 1 ) , , x σ ( k + m ) ) , {\omega\wedge\eta(x_{1},\ldots,x_{k+m})}=\sum_{\sigma\in Sh_{k,m}}% \operatorname{sgn}(\sigma)\,\omega(x_{\sigma(1)},\ldots,x_{\sigma(k)})\eta(x_{% \sigma(k+1)},\ldots,x_{\sigma(k+m)}),
  89. Δ ( x 1 x k ) = p = 0 k σ S h p , k - p sgn ( σ ) ( x σ ( 1 ) x σ ( p ) ) ( x σ ( p + 1 ) x σ ( k ) ) . \Delta(x_{1}\wedge\dots\wedge x_{k})=\sum_{p=0}^{k}\sum_{\sigma\in Sh_{p,k-p}}% \operatorname{sgn}(\sigma)(x_{\sigma(1)}\wedge\dots\wedge x_{\sigma(p)})% \otimes(x_{\sigma(p+1)}\wedge\dots\wedge x_{\sigma(k)}).
  90. Δ ( x 1 ) = 1 x 1 + x 1 1 , \Delta(x_{1})=1\otimes x_{1}+x_{1}\otimes 1,
  91. Δ ( x 1 x 2 ) = 1 ( x 1 x 2 ) + x 1 x 2 - x 2 x 1 + ( x 1 x 2 ) 1. \Delta(x_{1}\wedge x_{2})=1\otimes(x_{1}\wedge x_{2})+x_{1}\otimes x_{2}-x_{2}% \otimes x_{1}+(x_{1}\wedge x_{2})\otimes 1.
  92. ( α β ) ( x 1 x k ) = ( α β ) ( Δ ( x 1 x k ) ) (\alpha\wedge\beta)(x_{1}\wedge\dots\wedge x_{k})=(\alpha\otimes\beta)\left(% \Delta(x_{1}\wedge\dots\wedge x_{k})\right)
  93. i α : Λ k V Λ k - 1 V . i_{\alpha}:\Lambda^{k}V\rightarrow\Lambda^{k-1}V.
  94. ( i α w ) ( u 1 , u 2 , u k - 1 ) = w ( α , u 1 , u 2 , , u k - 1 ) . (i_{\alpha}{w})(u_{1},u_{2}\dots,u_{k-1})={w}(\alpha,u_{1},u_{2},\dots,u_{k-1}).
  95. i α : Λ k V Λ k - 1 V . i_{\alpha}:\Lambda^{k}V\rightarrow\Lambda^{k-1}V.
  96. i α ( a b ) = ( i α a ) b + ( - 1 ) deg a a ( i α b ) . i_{\alpha}(a\wedge b)=(i_{\alpha}a)\wedge b+(-1)^{\deg a}a\wedge(i_{\alpha}b).
  97. i α i α = 0. i_{\alpha}\circ i_{\alpha}=0.
  98. i α i β = - i β i α . i_{\alpha}\circ i_{\beta}=-i_{\beta}\circ i_{\alpha}.
  99. Λ k ( V * ) Λ n ( V ) Λ n - k ( V ) \Lambda^{k}(V^{*})\otimes\Lambda^{n}(V)\to\Lambda^{n-k}(V)\,
  100. i α β = i β i α . i_{\alpha\wedge\beta}=i_{\beta}\circ i_{\alpha}.\,
  101. α Λ k ( V * ) i α σ Λ n - k ( V ) . \alpha\in\Lambda^{k}(V^{*})\mapsto i_{\alpha}\sigma\in\Lambda^{n-k}(V).\,
  102. * : Λ k ( V ) Λ n - k ( V ) . *:\Lambda^{k}(V)\rightarrow\Lambda^{n-k}(V).\,
  103. * * : Λ k ( V ) Λ k ( V ) = ( - 1 ) k ( n - k ) + q I *\circ*:\Lambda^{k}(V)\to\Lambda^{k}(V)=(-1)^{k(n-k)+q}I
  104. v 1 v k , w 1 w k = det ( v i , w j ) , \left\langle v_{1}\wedge\cdots\wedge v_{k},w_{1}\wedge\cdots\wedge w_{k}\right% \rangle=\det(\langle v_{i},w_{j}\rangle),
  105. e i 1 e i k , i 1 < < i k , e_{i_{1}}\wedge\cdots\wedge e_{i_{k}},\quad i_{1}<\cdots<i_{k},
  106. x 𝐯 , 𝐰 = 𝐯 , i x 𝐰 \langle x\wedge\mathbf{v},\mathbf{w}\rangle=\langle\mathbf{v},i_{x^{\flat}}% \mathbf{w}\rangle
  107. x ( y ) = x , y x^{\flat}(y)=\langle x,y\rangle
  108. 𝐱 𝐯 , 𝐰 = 𝐯 , i 𝐱 𝐰 \langle\mathbf{x}\wedge\mathbf{v},\mathbf{w}\rangle=\langle\mathbf{v},i_{% \mathbf{x}^{\flat}}\mathbf{w}\rangle
  109. \simeq
  110. 𝐱 ( 𝐲 ) = 𝐱 , 𝐲 \mathbf{x}^{\flat}(\mathbf{y})=\langle\mathbf{x},\mathbf{y}\rangle
  111. Λ ( f ) : Λ ( V ) Λ ( W ) \Lambda(f):\Lambda(V)\rightarrow\Lambda(W)
  112. Λ ( f ) | Λ 1 ( V ) = f : V = Λ 1 ( V ) W = Λ 1 ( W ) . \Lambda(f)|_{\Lambda^{1}(V)}=f:V=\Lambda^{1}(V)\rightarrow W=\Lambda^{1}(W).
  113. Λ ( f ) ( x 1 x k ) = f ( x 1 ) f ( x k ) . \Lambda(f)(x_{1}\wedge\dots\wedge x_{k})=f(x_{1})\wedge\dots\wedge f(x_{k}).
  114. Λ k ( f ) = Λ ( f ) Λ k ( V ) : Λ k ( V ) Λ k ( W ) . \Lambda^{k}(f)=\Lambda(f)_{\Lambda^{k}(V)}:\Lambda^{k}(V)\rightarrow\Lambda^{k% }(W).
  115. 0 U V W 0 0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0
  116. 0 Λ 1 ( U ) Λ ( V ) Λ ( V ) Λ ( W ) 0 0\to\Lambda^{1}(U)\wedge\Lambda(V)\to\Lambda(V)\rightarrow\Lambda(W)\rightarrow 0
  117. 0 Λ ( U ) Λ ( V ) . 0\to\Lambda(U)\to\Lambda(V).
  118. Λ ( V W ) Λ ( V ) Λ ( W ) . \Lambda(V\oplus W)\cong\Lambda(V)\otimes\Lambda(W).
  119. Λ k ( V W ) p + q = k Λ p ( V ) Λ q ( W ) . \Lambda^{k}(V\oplus W)\cong\bigoplus_{p+q=k}\Lambda^{p}(V)\otimes\Lambda^{q}(W).
  120. 0 U V W 0 0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0
  121. 0 = F 0 F 1 F k F k + 1 = Λ k ( V ) 0=F^{0}\subseteq F^{1}\subseteq\cdots\subseteq F^{k}\subseteq F^{k+1}=\Lambda^% {k}(V)
  122. F p + 1 / F p = Λ k - p ( U ) Λ p ( W ) F^{p+1}/F^{p}=\Lambda^{k-p}(U)\otimes\Lambda^{p}(W)
  123. 0 U Λ k - 1 ( W ) Λ k ( V ) Λ k ( W ) 0 0\rightarrow U\otimes\Lambda^{k-1}(W)\rightarrow\Lambda^{k}(V)\rightarrow% \Lambda^{k}(W)\rightarrow 0
  124. 0 Λ k ( U ) Λ k ( V ) Λ k - 1 ( U ) W 0 0\rightarrow\Lambda^{k}(U)\rightarrow\Lambda^{k}(V)\rightarrow\Lambda^{k-1}(U)% \otimes W\rightarrow 0
  125. v 1 v r , v i V . v_{1}\otimes\dots\otimes v_{r},\quad v_{i}\in V.
  126. Alt ( v 1 v r ) = 1 r ! σ 𝔖 r sgn ( σ ) v σ ( 1 ) v σ ( r ) \operatorname{Alt}(v_{1}\otimes\dots\otimes v_{r})=\frac{1}{r!}\sum_{\sigma\in% \mathfrak{S}_{r}}\operatorname{sgn}(\sigma)v_{\sigma(1)}\otimes\dots\otimes v_% {\sigma(r)}
  127. ^ \widehat{\otimes}
  128. t ^ s = Alt ( t s ) . t\widehat{\otimes}s=\operatorname{Alt}(t\otimes s).
  129. A ( V ) Λ ( V ) . A(V)\cong\Lambda(V).
  130. t = t i 1 i 2 i r 𝐞 i 1 𝐞 i 2 𝐞 i r t=t^{i_{1}i_{2}\dots i_{r}}\,{\mathbf{e}}_{i_{1}}\otimes{\mathbf{e}}_{i_{2}}% \otimes\dots\otimes{\mathbf{e}}_{i_{r}}
  131. t ^ s = 1 ( r + p ) ! σ 𝔖 r + p sgn ( σ ) t i σ ( 1 ) i σ ( r ) s i σ ( r + 1 ) i σ ( r + p ) 𝐞 i 1 𝐞 i 2 𝐞 i r + p . t\widehat{\otimes}s=\frac{1}{(r+p)!}\sum_{\sigma\in{\mathfrak{S}}_{r+p}}% \operatorname{sgn}(\sigma)t^{i_{\sigma(1)}\dots i_{\sigma(r)}}s^{i_{\sigma(r+1% )}\dots i_{\sigma(r+p)}}{\mathbf{e}}_{i_{1}}\otimes{\mathbf{e}}_{i_{2}}\otimes% \dots\otimes{\mathbf{e}}_{i_{r+p}}.
  132. ( t ^ s ) i 1 i r + p = t [ i 1 i r s i r + 1 i r + p ] . (t\widehat{\otimes}s)^{i_{1}\dots i_{r+p}}=t^{[i_{1}\dots i_{r}}s^{i_{r+1}% \dots i_{r+p}]}.
  133. t = t i 0 i 1 i r - 1 t=t^{i_{0}i_{1}\dots i_{r-1}}
  134. ( i α t ) i 1 i r - 1 = r j = 0 n α j t j i 1 i r - 1 . (i_{\alpha}t)^{i_{1}\dots i_{r-1}}=r\sum_{j=0}^{n}\alpha_{j}t^{ji_{1}\dots i_{% r-1}}.
  135. u v u∧v
  136. u v w u∧v∧w
  137. : Λ p + 1 L Λ p L \partial:\Lambda^{p+1}L\to\Lambda^{p}L
  138. ( x 1 x p + 1 ) = 1 p + 1 j < ( - 1 ) j + + 1 [ x j , x ] x 1 x ^ j x ^ x p + 1 . \partial(x_{1}\wedge\cdots\wedge x_{p+1})=\frac{1}{p+1}\sum_{j<\ell}(-1)^{j+% \ell+1}[x_{j},x_{\ell}]\wedge x_{1}\wedge\cdots\wedge\hat{x}_{j}\wedge\cdots% \wedge\hat{x}_{\ell}\wedge\cdots\wedge x_{p+1}.