wpmath0000005_3

Delta_rule.html

  1. j j\,
  2. g ( x ) g(x)\,
  3. j j\,
  4. i i\,
  5. w j i w_{ji}\,
  6. Δ w j i = α ( t j - y j ) g ( h j ) x i \Delta w_{ji}=\alpha(t_{j}-y_{j})g^{\prime}(h_{j})x_{i}\,
  7. α \alpha\,
  8. g ( x ) g(x)\,
  9. t j t_{j}\,
  10. h j h_{j}\,
  11. y j y_{j}\,
  12. x i x_{i}\,
  13. i i\,
  14. h j = x i w j i h_{j}=\sum x_{i}w_{ji}\,
  15. y j = g ( h j ) y_{j}=g(h_{j})\,
  16. Δ w j i = α ( t j - y j ) x i \Delta w_{ji}=\alpha(t_{j}-y_{j})x_{i}\,
  17. g ( h ) g(h)
  18. g ( h ) g^{\prime}(h)
  19. j j\,
  20. E = j 1 2 ( t j - y j ) 2 E=\sum_{j}\frac{1}{2}(t_{j}-y_{j})^{2}\,
  21. i i\,
  22. E w j i \frac{\partial E}{\partial w_{ji}}\,
  23. j j\,
  24. E w j i = ( 1 2 ( t j - y j ) 2 ) w j i \frac{\partial E}{\partial w_{ji}}=\frac{\partial\left(\frac{1}{2}\left(t_{j}-% y_{j}\right)^{2}\right)}{\partial w_{ji}}\,
  25. = ( 1 2 ( t j - y j ) 2 ) y j y j w j i =\frac{\partial\left(\frac{1}{2}\left(t_{j}-y_{j}\right)^{2}\right)}{\partial y% _{j}}\frac{\partial y_{j}}{\partial w_{ji}}\,
  26. = - ( t j - y j ) y j w j i =-\left(t_{j}-y_{j}\right)\frac{\partial y_{j}}{\partial w_{ji}}\,
  27. j j\,
  28. h j h_{j}\,
  29. = - ( t j - y j ) y j h j h j w j i =-\left(t_{j}-y_{j}\right)\frac{\partial y_{j}}{\partial h_{j}}\frac{\partial h% _{j}}{\partial w_{ji}}\,
  30. j j
  31. y j y_{j}\,
  32. g g\,
  33. h j h_{j}\,
  34. y j y_{j}\,
  35. h j h_{j}\,
  36. g g\,
  37. = - ( t j - y j ) g ( h j ) h j w j i =-\left(t_{j}-y_{j}\right)g^{\prime}(h_{j})\frac{\partial h_{j}}{\partial w_{% ji}}\,
  38. h j h_{j}\,
  39. k k\,
  40. w j k w_{jk}\,
  41. x k x_{k}\,
  42. = - ( t j - y j ) g ( h j ) ( k x k w j k ) w j i =-\left(t_{j}-y_{j}\right)g^{\prime}(h_{j})\frac{\partial\left(\sum_{k}x_{k}w_% {jk}\right)}{\partial w_{ji}}\,
  43. i i\,
  44. x i w j i x_{i}w_{ji}\,
  45. x i w j i w j i = x i \frac{\partial x_{i}w_{ji}}{\partial w_{ji}}=x_{i}\,
  46. E w j i = - ( t j - y j ) g ( h j ) x i \frac{\partial E}{\partial w_{ji}}=-\left(t_{j}-y_{j}\right)g^{\prime}(h_{j})x% _{i}\,
  47. α \alpha\,
  48. Δ w j i = α ( t j - y j ) g ( h j ) x i \Delta w_{ji}=\alpha(t_{j}-y_{j})g^{\prime}(h_{j})x_{i}\,

Deontic_logic.html

  1. O ( A B ) ( O A O B ) O(A\rightarrow B)\rightarrow(OA\rightarrow OB)
  2. P A ¬ O ¬ A PA\to\lnot O\lnot A
  3. O ¬ A O\lnot A
  4. ¬ P A \lnot PA
  5. \Box
  6. O A A . OA\to\Diamond A.
  7. ¬ ¬ \Diamond\equiv\lnot\Box\lnot
  8. \Box
  9. A B O A O B . \vdash A\to B\Rightarrow\ \vdash OA\to OB.
  10. O A O ( A / ) OA\equiv O(A/\top)
  11. \top
  12. \Box
  13. O A ( ¬ A s ) OA\equiv\Box(\lnot A\to s)
  14. O ( smoke ashtray ) O(\mathrm{smoke}\rightarrow\mathrm{ashtray})
  15. smoke O ( ashtray ) \mathrm{smoke}\rightarrow O(\mathrm{ashtray})
  16. O ( A B ) O(A\mid B)
  17. P ( A B ) P(A\mid B)

Dependence_analysis.html

  1. S 1 δ c S 2 S1\ \delta^{c}\ S2
  2. S 1 δ f S 2 S1\ \delta^{f}\ S2
  3. S 1 δ a S 2 S1\ \delta^{a}\ S2
  4. S 1 δ o S 2 S1\ \delta^{o}\ S2
  5. S 1 δ i S 2 S1\ \delta^{i}\ S2

Dependent_type.html

  1. A : 𝒰 A:\mathcal{U}
  2. 𝒰 \mathcal{U}
  3. B : A 𝒰 B:A\to\mathcal{U}
  4. a : A a:A
  5. B ( a ) : 𝒰 B(a):\mathcal{U}
  6. Π ( x : A ) B ( x ) \Pi_{(x:A)}B(x)
  7. Π ( x : A ) , B ( x ) \Pi(x:A),B(x)
  8. A B A\to B
  9. Π ( x : A ) B \Pi_{(x:A)}B
  10. A B A\to B
  11. Vec ( , n ) \mbox{Vec}~{}({\mathbb{R}},n)
  12. n n
  13. Π ( n : ) Vec ( , n ) \Pi_{(n:{\mathbb{N}})}\mbox{Vec}~{}({\mathbb{R}},n)
  14. Π ( n : ) \Pi_{(n:{\mathbb{N}})}\;{\mathbb{R}}
  15. {\mathbb{N}}\to{\mathbb{R}}
  16. Π ( A : 𝒰 ) A C \Pi_{(A:\mathcal{U})}A\to C
  17. Σ ( x : A ) B ( x ) \Sigma_{(x:A)}B(x)
  18. ( a , b ) : Σ ( x : A ) B ( x ) (a,b):\Sigma_{(x:A)}B(x)
  19. a : A a:A
  20. b : B ( a ) b:B(a)
  21. A × B A\times B
  22. Σ A B a : 𝒰 \Sigma AB_{a}:\mathcal{U}
  23. A : 𝒰 A:\mathcal{U}
  24. B : A 𝒰 B:A\rightarrow\mathcal{U}
  25. A A
  26. B a B_{a}
  27. Σ A B a \Sigma AB_{a}
  28. a b Σ ( λ n a + n = b ) a\leq b\iff\Sigma\mathbb{N}(\lambda n\rightarrow a+n=b)
  29. λ Π \lambda\Pi
  30. λ Π 2 \lambda\Pi 2
  31. λ Π \lambda\Pi
  32. \to
  33. \forall
  34. λ Π ω \lambda\Pi\omega
  35. λ Π 2 \lambda\Pi 2

Depletion_region.html

  1. n + N A = p + N D n+N_{A}=p+N_{D}\,
  2. N D N_{D}
  3. N A N_{A}
  4. n , p N D , N A n,p<<N_{D},N_{A}
  5. q N A w P q N D w N qN_{A}w_{P}\approx qN_{D}w_{N}\,
  6. w = w N + w P w=w_{N}+w_{P}
  7. W [ 2 ϵ r ϵ 0 q ( N A + N D N A N D ) ( V b i - V ) ] 1 2 W\approx\left[\frac{2\epsilon_{r}\epsilon_{0}}{q}\left(\frac{N_{A}+N_{D}}{N_{A% }N_{D}}\right)\left(V_{bi}-V\right)\right]^{\frac{1}{2}}
  8. ϵ r \epsilon_{r}
  9. V b i V_{bi}
  10. V V
  11. N A N_{A}
  12. Q = q N A w Q=qN_{A}w\,
  13. E m E_{m}
  14. E m = Q / A ϵ 0 = q N A w / A ϵ 0 , E_{m}=Q/A\epsilon_{0}=qN_{A}w/A\epsilon_{0},\,
  15. ϵ 0 \epsilon_{0}

Derivation_of_the_Cartesian_form_for_an_ellipse.html

  1. d 1 d_{1}
  2. d 2 d_{2}
  3. a a
  4. d 1 + d 2 = 2 a d_{1}+d_{2}=2a\,
  5. ( x + c ) 2 + y 2 + ( x - c ) 2 + y 2 = 2 a \sqrt{(x+c)^{2}+y^{2}}+\sqrt{(x-c)^{2}+y^{2}}=2a
  6. ( x + c ) 2 + y 2 = 2 a - ( x - c ) 2 + y 2 \sqrt{(x+c)^{2}+y^{2}}=2a-\sqrt{(x-c)^{2}+y^{2}}
  7. ( x + c ) 2 + y 2 = ( 2 a - ( x - c ) 2 + y 2 ) 2 (x+c)^{2}+y^{2}=\left(2a-\sqrt{(x-c)^{2}+y^{2}}\right)^{2}
  8. ( x + c ) 2 + y 2 = 4 a 2 - 4 a ( x - c ) 2 + y 2 + ( x - c ) 2 + y 2 (x+c)^{2}+y^{2}=4a^{2}-4a\sqrt{(x-c)^{2}+y^{2}}+(x-c)^{2}+y^{2}
  9. ( x + c ) 2 + y 2 - ( x - c ) 2 - y 2 - 4 a 2 = - 4 a ( x - c ) 2 + y 2 (x+c)^{2}+y^{2}-(x-c)^{2}-y^{2}-4a^{2}=-4a\sqrt{(x-c)^{2}+y^{2}}
  10. - 1 4 a ( ( x + c ) 2 + y 2 - 4 a 2 - ( x - c ) 2 - y 2 ) = ( x - c ) 2 + y 2 -{1\over 4a}((x+c)^{2}+y^{2}-4a^{2}-(x-c)^{2}-y^{2})=\sqrt{(x-c)^{2}+y^{2}}
  11. ( x - c ) 2 + y 2 = - 1 4 a ( ( x + c ) 2 + y 2 - 4 a 2 - ( x - c ) 2 - y 2 ) \sqrt{(x-c)^{2}+y^{2}}=-{1\over 4a}((x+c)^{2}+y^{2}-4a^{2}-(x-c)^{2}-y^{2})
  12. ( x - c ) 2 + y 2 = - 1 4 a ( x 2 + 2 x c + c 2 - 4 a 2 - x 2 + 2 x c - c 2 ) \sqrt{(x-c)^{2}+y^{2}}=-{1\over 4a}(x^{2}+2xc+c^{2}-4a^{2}-x^{2}+2xc-c^{2})
  13. ( x - c ) 2 + y 2 = - 1 4 a ( 4 x c - 4 a 2 ) \sqrt{(x-c)^{2}+y^{2}}=-{1\over 4a}(4xc-4a^{2})
  14. ( x - c ) 2 + y 2 = a - c a x \sqrt{(x-c)^{2}+y^{2}}=a-{c\over a}x
  15. ( x - c ) 2 + y 2 = a 2 - 2 x c + c 2 a 2 x 2 (x-c)^{2}+y^{2}=a^{2}-2xc+{c^{2}\over a^{2}}x^{2}
  16. x 2 - 2 x c + c 2 + y 2 = a 2 - 2 x c + c 2 a 2 x 2 x^{2}-2xc+c^{2}+y^{2}=a^{2}-2xc+{c^{2}\over a^{2}}x^{2}
  17. x 2 + c 2 + y 2 = a 2 + c 2 a 2 x 2 x^{2}+c^{2}+y^{2}=a^{2}+{c^{2}\over a^{2}}x^{2}
  18. a 2 - c 2 a^{2}-c^{2}\,
  19. x 2 - c 2 a 2 x 2 + y 2 = a 2 - c 2 x^{2}-{c^{2}\over a^{2}}x^{2}+y^{2}=a^{2}-c^{2}
  20. x 2 ( 1 - c 2 a 2 ) + y 2 = a 2 - c 2 x^{2}\left(1-{c^{2}\over a^{2}}\right)+y^{2}=a^{2}-c^{2}
  21. 1 = a 2 a 2 1={a^{2}\over a^{2}}
  22. x 2 ( a 2 - c 2 a 2 ) + y 2 = a 2 - c 2 x^{2}\left({a^{2}-c^{2}\over a^{2}}\right)+y^{2}=a^{2}-c^{2}
  23. x 2 a 2 + y 2 a 2 - c 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over a^{2}-c^{2}}=1
  24. d 1 = d 2 = a = c 2 + b 2 d_{1}=d_{2}=a=\sqrt{c^{2}+b^{2}}
  25. b 2 = a 2 - c 2 b^{2}=a^{2}-c^{2}\,
  26. x 2 a 2 + y 2 b 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1

Derived_set_(mathematics).html

  1. S S^{\prime}
  2. S S S^{\prime}\subseteq S
  3. S S
  4. S = S S=S^{\prime}
  5. = * {}^{*}=
  6. S * * S * S^{**}\subseteq S^{*}
  7. a S * a ( S { a } ) * a\in S^{*}\implies a\in(S\setminus\{a\})^{*}
  8. ( S T ) * S * T * (S\cup T)^{*}\subseteq S^{*}\cup T^{*}
  9. S T S * T * S\subseteq T\implies S^{*}\subseteq T^{*}
  10. = * {}^{*}=
  11. S * * S * S^{**}\subseteq S^{*}
  12. S * = ( S { a } ) * S^{*}=(S\setminus\{a\})^{*}
  13. ( S T ) * = S * T * \,(S\cup T)^{*}=S^{*}\cup T^{*}
  14. S * S S^{*}\subseteq S
  15. S * = S S^{*}=S^{\prime}\,\!
  16. X 0 = X \displaystyle X^{0}=X
  17. X α + 1 = ( X α ) \displaystyle X^{\alpha+1}=(X^{\alpha})^{\prime}
  18. X λ = α < λ X α \displaystyle X^{\lambda}=\bigcap_{\alpha<\lambda}X^{\alpha}

Deriving_the_Schwarzschild_solution.html

  1. ( r , θ , ϕ , t ) \left(r,\theta,\phi,t\right)
  2. t t
  3. t g μ ν = 0 \tfrac{\partial}{\partial t}g_{\mu\nu}=0
  4. t - t t\rightarrow-t
  5. T a b = 0 T_{ab}=0
  6. R a b = 0 R_{ab}=0
  7. R a b - R 2 g a b = 0 R_{ab}-\tfrac{R}{2}g_{ab}=0
  8. R = 0 R=0
  9. ( r , θ , ϕ , t ) ( r , θ , ϕ , - t ) (r,\theta,\phi,t)\rightarrow(r,\theta,\phi,-t)
  10. g μ 4 g_{\mu 4}
  11. μ 4 \mu\neq 4
  12. g μ 4 = x α x μ x β x 4 g α β = - g μ 4 g_{\mu 4}^{\prime}=\frac{\partial x^{\alpha}}{\partial x^{{}^{\prime}\mu}}% \frac{\partial x^{\beta}}{\partial x^{{}^{\prime}4}}g_{\alpha\beta}=-g_{\mu 4}
  13. μ 4 \mu\neq 4
  14. g μ 4 = g μ 4 g^{\prime}_{\mu 4}=g_{\mu 4}
  15. g μ 4 = 0 g_{\mu 4}=\,0
  16. μ 4 \mu\neq 4
  17. ( r , θ , ϕ , t ) ( r , θ , - ϕ , t ) (r,\theta,\phi,t)\rightarrow(r,\theta,-\phi,t)
  18. ( r , θ , ϕ , t ) ( r , - θ , ϕ , t ) (r,\theta,\phi,t)\rightarrow(r,-\theta,\phi,t)
  19. g μ 3 = 0 g_{\mu 3}=\,0
  20. μ 3 \mu\neq 3
  21. g μ 2 = 0 g_{\mu 2}=\,0
  22. μ 2 \mu\neq 2
  23. g μ ν = 0 g_{\mu\nu}=\,0
  24. μ ν \mu\neq\nu
  25. d s 2 = g 11 d r 2 + g 22 d θ 2 + g 33 d ϕ 2 + g 44 d t 2 ds^{2}=\,g_{11}\,dr^{2}+g_{22}\,d\theta^{2}+g_{33}\,d\phi^{2}+g_{44}\,dt^{2}
  26. t t
  27. t t
  28. θ \theta
  29. ϕ \phi
  30. g 11 g_{11}
  31. r r
  32. g 11 g_{11}
  33. g 11 = A ( r ) g_{11}=A\left(r\right)
  34. g 44 g_{44}
  35. g 44 = B ( r ) g_{44}=B\left(r\right)
  36. t t
  37. r r
  38. d l 2 = r 0 2 ( d θ 2 + sin 2 θ d ϕ 2 ) dl^{2}=r_{0}^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})
  39. r 0 r_{0}
  40. g ~ 22 \tilde{g}_{22}
  41. g ~ 33 \tilde{g}_{33}
  42. θ \theta
  43. ϕ \phi
  44. g ~ 22 ( d θ 2 + g ~ 33 g ~ 22 d ϕ 2 ) = r 0 2 ( d θ 2 + sin 2 θ d ϕ 2 ) \tilde{g}_{22}\left(d\theta^{2}+\frac{\tilde{g}_{33}}{\tilde{g}_{22}}\,d\phi^{% 2}\right)=r_{0}^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})
  45. g ~ 22 = r 0 2 \tilde{g}_{22}=r_{0}^{2}
  46. g ~ 33 = r 0 2 sin 2 θ \tilde{g}_{33}=r_{0}^{2}\sin^{2}\theta
  47. g 22 = r 2 g_{22}=\,r^{2}
  48. g 33 = r 2 sin 2 θ g_{33}=\,r^{2}\sin^{2}\theta
  49. d s 2 = A ( r ) d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 + B ( r ) d t 2 ds^{2}=A\left(r\right)dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}% +B\left(r\right)dt^{2}
  50. A A
  51. B B
  52. r r
  53. A A
  54. B B
  55. ( 1 , 2 , 3 , 4 ) = ( r , θ , ϕ , t ) (1,2,3,4)=(r,\theta,\phi,t)
  56. {}^{\prime}
  57. Γ i k 1 = [ A / ( 2 A ) 0 0 0 0 - r / A 0 0 0 0 - r sin 2 θ / A 0 0 0 0 - B / ( 2 A ) ] \Gamma^{1}_{ik}=\begin{bmatrix}A^{\prime}/\left(2A\right)&0&0&0\\ 0&-r/A&0&0\\ 0&0&-r\sin^{2}\theta/A&0\\ 0&0&0&-B^{\prime}/\left(2A\right)\end{bmatrix}
  58. Γ i k 2 = [ 0 1 / r 0 0 1 / r 0 0 0 0 0 - sin θ cos θ 0 0 0 0 0 ] \Gamma^{2}_{ik}=\begin{bmatrix}0&1/r&0&0\\ 1/r&0&0&0\\ 0&0&-\sin\theta\cos\theta&0\\ 0&0&0&0\end{bmatrix}
  59. Γ i k 3 = [ 0 0 1 / r 0 0 0 cot θ 0 1 / r cot θ 0 0 0 0 0 0 ] \Gamma^{3}_{ik}=\begin{bmatrix}0&0&1/r&0\\ 0&0&\cot\theta&0\\ 1/r&\cot\theta&0&0\\ 0&0&0&0\end{bmatrix}
  60. Γ i k 4 = [ 0 0 0 B / ( 2 B ) 0 0 0 0 0 0 0 0 B / ( 2 B ) 0 0 0 ] \Gamma^{4}_{ik}=\begin{bmatrix}0&0&0&B^{\prime}/\left(2B\right)\\ 0&0&0&0\\ 0&0&0&0\\ B^{\prime}/\left(2B\right)&0&0&0\end{bmatrix}
  61. A A
  62. B B
  63. R α β = 0 R_{\alpha\beta}=\,0
  64. Γ β α , ρ ρ - Γ ρ α , β ρ + Γ ρ λ ρ Γ β α λ - Γ β λ ρ Γ ρ α λ = 0 {\Gamma^{\rho}_{\beta\alpha,\rho}}-\Gamma^{\rho}_{\rho\alpha,\beta}+\Gamma^{% \rho}_{\rho\lambda}\Gamma^{\lambda}_{\beta\alpha}-\Gamma^{\rho}_{\beta\lambda}% \Gamma^{\lambda}_{\rho\alpha}=0
  65. 4 A B 2 - 2 r B ′′ A B + r A B B + r B A 2 = 0 4A^{\prime}B^{2}-2rB^{\prime\prime}AB+rA^{\prime}B^{\prime}B+rB^{\prime}{}^{2}% A=0
  66. r A B + 2 A 2 B - 2 A B - r B A = 0 rA^{\prime}B+2A^{2}B-2AB-rB^{\prime}A=0
  67. - 2 r B ′′ A B + r A B B + r B A 2 - 4 B A B = 0 -2rB^{\prime\prime}AB+rA^{\prime}B^{\prime}B+rB^{\prime}{}^{2}A-4B^{\prime}AB=0
  68. sin 2 θ \sin^{2}\theta
  69. A B + A B = 0 A ( r ) B ( r ) = K A^{\prime}B+AB^{\prime}=0\Rightarrow A(r)B(r)=K
  70. K K
  71. A ( r ) B ( r ) = K A(r)B(r)\,=K
  72. r A = A ( 1 - A ) rA^{\prime}=A(1-A)
  73. A ( r ) = ( 1 + 1 S r ) - 1 A(r)=\left(1+\frac{1}{Sr}\right)^{-1}
  74. S S
  75. d s 2 = ( 1 + 1 S r ) - 1 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) + K ( 1 + 1 S r ) d t 2 ds^{2}=\left(1+\frac{1}{Sr}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d% \phi^{2})+K\left(1+\frac{1}{Sr}\right)dt^{2}
  76. r r\rightarrow\infty
  77. K K
  78. S S
  79. d s ds
  80. 0 = δ d s d t d t = δ ( K E + P E g ) d t 0=\delta\int\frac{ds}{dt}dt=\delta\int(KE+PE_{g})dt
  81. K E KE
  82. P E g PE_{g}
  83. K K
  84. S S
  85. g 44 = K ( 1 + 1 S r ) - c 2 + 2 G m r = - c 2 ( 1 - 2 G m c 2 r ) g_{44}=K\left(1+\frac{1}{Sr}\right)\approx-c^{2}+\frac{2Gm}{r}=-c^{2}\left(1-% \frac{2Gm}{c^{2}r}\right)
  86. G G
  87. m m
  88. c c
  89. K = - c 2 K=\,-c^{2}
  90. 1 S = - 2 G m c 2 \frac{1}{S}=-\frac{2Gm}{c^{2}}
  91. A ( r ) = ( 1 - 2 G m c 2 r ) - 1 A(r)=\left(1-\frac{2Gm}{c^{2}r}\right)^{-1}
  92. B ( r ) = - c 2 ( 1 - 2 G m c 2 r ) B(r)=-c^{2}\left(1-\frac{2Gm}{c^{2}r}\right)
  93. d s 2 = ( 1 - 2 G m c 2 r ) - 1 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) - c 2 ( 1 - 2 G m c 2 r ) d t 2 ds^{2}=\left(1-\frac{2Gm}{c^{2}r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}% \theta d\phi^{2})-c^{2}\left(1-\frac{2Gm}{c^{2}r}\right)dt^{2}
  94. 2 G m c 2 = r s \frac{2Gm}{c^{2}}=r_{s}
  95. m m
  96. d s 2 = ( 1 - r s r ) - 1 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) - c 2 ( 1 - r s r ) d t 2 ds^{2}=\left(1-\frac{r_{s}}{r}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}% \theta d\phi^{2})-c^{2}\left(1-\frac{r_{s}}{r}\right)dt^{2}
  97. r r s r\rightarrow r_{s}
  98. r = 0 r=0
  99. r 1 r_{1}
  100. θ \theta
  101. ϕ \phi
  102. θ \theta
  103. ϕ \phi
  104. r 2 G m c 2 r\geq\frac{2Gm}{c^{2}}
  105. r = r 1 ( 1 + G m 2 c 2 r 1 ) 2 r=r_{1}\left(1+\frac{Gm}{2c^{2}r_{1}}\right)^{2}
  106. d r = d r 1 ( 1 - ( G m ) 2 4 c 4 r 1 2 ) dr=dr_{1}\left(1-\frac{(Gm)^{2}}{4c^{4}r_{1}^{2}}\right)
  107. ( 1 - 2 G m c 2 r ) = ( 1 - G m 2 c 2 r 1 ) 2 / ( 1 + G m 2 c 2 r 1 ) 2 \left(1-\frac{2Gm}{c^{2}r}\right)=\left(1-\frac{Gm}{2c^{2}r_{1}}\right)^{2}/% \left(1+\frac{Gm}{2c^{2}r_{1}}\right)^{2}
  108. x x
  109. y y
  110. z z
  111. x = r 1 sin ( θ ) cos ( ϕ ) , x=r_{1}\,\sin(\theta)\,\cos(\phi)\dots,
  112. y = r 1 sin ( θ ) sin ( ϕ ) , y=r_{1}\,\sin(\theta)\,\sin(\phi)\dots,
  113. z = r 1 cos ( θ ) z=r_{1}\,\cos(\theta)\dots
  114. d s 2 = ( 1 + G m 2 c 2 r 1 ) 4 ( d x 2 + d y 2 + d z 2 ) - c 2 d t 2 ( 1 - G m 2 c 2 r 1 ) 2 / ( 1 + G m 2 c 2 r 1 ) 2 ds^{2}=\left(1+\frac{Gm}{2c^{2}r_{1}}\right)^{4}(dx^{2}+dy^{2}+dz^{2})-c^{2}dt% ^{2}\left(1-\frac{Gm}{2c^{2}r_{1}}\right)^{2}/\left(1+\frac{Gm}{2c^{2}r_{1}}% \right)^{2}

Desargues_graph.html

  1. ( x - 3 ) ( x - 2 ) 4 ( x - 1 ) 5 ( x + 1 ) 5 ( x + 2 ) 4 ( x + 3 ) . (x-3)(x-2)^{4}(x-1)^{5}(x+1)^{5}(x+2)^{4}(x+3).\,

Descent_direction.html

  1. 𝐩 n \mathbf{p}\in\mathbb{R}^{n}
  2. 𝐱 * \mathbf{x}^{*}
  3. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  4. 𝐱 * \mathbf{x}^{*}
  5. 𝐩 k n \mathbf{p}_{k}\in\mathbb{R}^{n}
  6. k k
  7. 𝐩 k \mathbf{p}_{k}
  8. 𝐩 k , f ( 𝐱 k ) < 0 \langle\mathbf{p}_{k},\nabla f(\mathbf{x}_{k})\rangle<0
  9. , \langle,\rangle
  10. 𝐩 k \mathbf{p}_{k}
  11. f \displaystyle f
  12. - f ( 𝐱 k ) , f ( 𝐱 k ) = - f ( 𝐱 k ) , f ( 𝐱 k ) < 0 \langle-\nabla f(\mathbf{x}_{k}),\nabla f(\mathbf{x}_{k})\rangle=-\langle% \nabla f(\mathbf{x}_{k}),\nabla f(\mathbf{x}_{k})\rangle<0
  13. P P
  14. d = - P f ( x ) d=-P\nabla f(x)
  15. x x

Descriptive_complexity_theory.html

  1. \exist \exist
  2. i {}^{i}
  3. i i
  4. i - 1 i-1
  5. exp 2 i - 2 ( n O ( 1 ) ) \exp_{2}^{i-2}(n^{O(1)})
  6. j i {}^{i}_{j}
  7. exp 2 i - 2 ( n O ( 1 ) ) ) Σ j P \exp_{2}^{i-2}(n^{O(1)}))^{\Sigma_{j}^{\rm P}}

Desorption.html

  1. R = r N x R=rN^{x}
  2. r r
  3. N N
  4. x x
  5. r r
  6. r = A e - E a / k T r=Ae^{{-E_{a}}/{kT}}
  7. A A
  8. ν \nu
  9. E a E_{a}
  10. k k
  11. T T

Dessin_d'enfant.html

  1. A 5 P S L ( 2 , 5 ) A_{5}\cong PSL(2,5)
  2. f ( x ) = - ( x - 1 ) 3 ( x - 9 ) 64 x = 1 - ( x 2 - 6 x - 3 ) 2 64 x . f(x)=-\frac{(x-1)^{3}(x-9)}{64x}=1-\frac{(x^{2}-6x-3)^{2}}{64x}.
  3. q ( x ) = p ( x ) - y 1 y 2 - y 1 , q(x)=\frac{p(x)-y_{1}}{y_{2}-y_{1}},
  4. p ( x ) = x 3 ( x 2 - 2 x + a ) 2 \displaystyle p(x)=x^{3}(x^{2}-2x+a)^{2}
  5. a = 1 7 ( 34 ± 6 21 ) . a=\frac{1}{7}(34\pm 6\sqrt{21}).
  6. 𝐐 ( 21 ) \mathbf{Q}(\sqrt{21})
  7. 𝐐 ( 21 ) \mathbf{Q}(\sqrt{21})

Destructive_dilemma.html

  1. P Q , R S , ¬ Q ¬ S ¬ P ¬ R \frac{P\to Q,R\to S,\neg Q\neg S}{\therefore\neg P\neg R}
  2. P Q P\to Q
  3. R S R\to S
  4. ¬ Q ¬ S \neg Q\neg S
  5. ¬ P ¬ R \neg P\neg R
  6. ( P Q ) , ( R S ) , ( ¬ Q ¬ S ) ( ¬ P ¬ R ) (P\to Q),(R\to S),(\neg Q\neg S)\vdash(\neg P\neg R)
  7. \vdash
  8. ¬ P ¬ R \neg P\neg R
  9. P Q P\to Q
  10. R S R\to S
  11. ¬ Q ¬ S \neg Q\neg S
  12. ( ( ( P Q ) and ( R S ) ) and ( ¬ Q ¬ S ) ) ( ¬ P ¬ R ) (((P\to Q)\and(R\to S))\and(\neg Q\neg S))\to(\neg P\neg R)
  13. P P
  14. Q Q
  15. R R
  16. S S
  17. ( A B ) and ( C D ) (A\rightarrow B)\and(C\rightarrow D)
  18. ¬ B ¬ D \neg B\neg D
  19. B ¬ D B\rightarrow\neg D
  20. ¬ D ¬ C \neg D\rightarrow\neg C
  21. B ¬ C B\rightarrow\neg C
  22. A B A\rightarrow B
  23. A ¬ C A\rightarrow\neg C
  24. ¬ A ¬ C \neg A\neg C
  25. ( ( P Q ) & ( R S ) ) & ( ¬ Q ¬ S ) ((P\rightarrow Q)\And(R\rightarrow S))\And(\neg Q\vee\neg S)
  26. ( P Q ) & ( R S ) (P\rightarrow Q)\And(R\rightarrow S)
  27. ( P Q ) (P\rightarrow Q)
  28. ( R S ) (R\rightarrow S)
  29. ( ¬ Q ¬ S ) (\neg Q\vee\neg S)
  30. ¬ ( ¬ P ¬ R ) \neg(\neg P\vee\neg R)
  31. ¬ ¬ P & ¬ ¬ R \neg\neg P\And\neg\neg R
  32. ¬ ¬ P \neg\neg P
  33. ¬ ¬ R \neg\neg R
  34. P P
  35. R R
  36. Q Q
  37. S S
  38. ¬ ¬ Q \neg\neg Q
  39. ¬ S \neg S
  40. S & ¬ S S\And\neg S
  41. ¬ P ¬ R \neg P\vee\neg R
  42. ( ( ( P Q ) & ( R S ) ) & ( ¬ Q ¬ S ) ) ) ¬ P ¬ R (((P\rightarrow Q)\And(R\rightarrow S))\And(\neg Q\vee\neg S)))\rightarrow\neg P% \vee\neg R

Dewetting.html

  1. S S
  2. S = γ gw - γ go - γ ow S\ =\gamma\text{gw}-\gamma\text{go}-\gamma\text{ow}
  3. γ gw \gamma\text{gw}
  4. γ go \gamma\text{go}
  5. γ ow \gamma\text{ow}
  6. S > 0 S>0
  7. S < 0 S<0
  8. S < 0 S<0

DeWitt_notation.html

  1. A , i [ ϕ ] = def δ δ ϕ α ( x ) A [ ϕ ] A_{,i}[\phi]\ \stackrel{\mathrm{def}}{=}\ \frac{\delta}{\delta\phi^{\alpha}(x)% }A[\phi]
  2. A i B i = def M α A α ( x ) B α ( x ) d d x A^{i}B_{i}\ \stackrel{\mathrm{def}}{=}\ \int_{M}\sum_{\alpha}A^{\alpha}(x)B_{% \alpha}(x)d^{d}x

Diagonal_intersection.html

  1. δ \displaystyle\delta
  2. X α α < δ \displaystyle\langle X_{\alpha}\mid\alpha<\delta\rangle
  3. δ \displaystyle\delta
  4. Δ α < δ X α , \displaystyle\Delta_{\alpha<\delta}X_{\alpha},
  5. { β < δ β α < β X α } . \displaystyle\{\beta<\delta\mid\beta\in\bigcap_{\alpha<\beta}X_{\alpha}\}.
  6. β \displaystyle\beta
  7. Δ α < δ X α \displaystyle\Delta_{\alpha<\delta}X_{\alpha}
  8. β \displaystyle\beta
  9. α < δ ( [ 0 , α ] X α ) , \displaystyle\bigcap_{\alpha<\delta}([0,\alpha]\cup X_{\alpha}),
  10. α \displaystyle\alpha

Diagonal_subgroup.html

  1. { ( g , , g ) G n : g G } . \{(g,\dots,g)\in G^{n}:g\in G\}.

Diamond_cubic.html

  1. 3 ¯ \overline{3}
  2. 4 ¯ \overline{4}
  3. π 3 16 0.34 \frac{\pi\sqrt{3}}{16}\approx 0.34
  4. 3 4 \frac{\sqrt{3}}{4}
  5. 2 2 \frac{\sqrt{2}}{2}
  6. 1 1
  7. 19 4 \frac{\sqrt{19}}{4}

Diceware.html

  1. 6 5 = 7 776 6^{5}=7\,776
  2. log 2 ( 6 5 ) \log_{2}(6^{5})

Dielectric_heating.html

  1. Q = ω ε r ′′ ε 0 E 2 , Q=\omega\cdot\varepsilon_{r}^{\prime\prime}\cdot\varepsilon_{0}\cdot E^{2},
  2. ω \omega
  3. ε r ′′ \varepsilon_{r}^{\prime\prime}
  4. ε 0 \varepsilon_{0}
  5. E E
  6. σ \sigma
  7. σ ω ε \sigma\ll\omega\varepsilon
  8. ε = ε r ′′ ε 0 \varepsilon=\varepsilon_{r}^{\prime\prime}\cdot\varepsilon_{0}

Dielectric_spectroscopy.html

  1. \leftrightarrow
  2. j t = j 0 ( exp ( α o f η ) - exp ( - α r f η ) ) j_{\,\text{t}}=j_{0}\left(\exp(\alpha_{\,\text{o}}\,f\,\eta)-\exp(-\alpha_{\,% \text{r}}\,f\,\eta)\right)
  3. η = E - E eq , f = F / ( R T ) , α o + α r = 1 \eta=E-E_{\,\text{eq}},\;f=F/(R\,T),\;\alpha_{\,\text{o}}+\alpha_{\,\text{r}}=1
  4. j 0 j_{0}
  5. α o \alpha_{\,\text{o}}
  6. α r \alpha_{\,\text{r}}
  7. j t v s . E j_{\,\text{t}}\;vs.\;E
  8. j t ( t ) = j t ( η ( t ) ) = j 0 ( exp ( α o f η ( t ) ) - exp ( - α r f η ( t ) ) ) j_{\,\text{t}}(t)=j_{\,\text{t}}(\eta(t))=j_{0}\,\left(\exp(\alpha_{\,\text{o}% }\,f\,\eta(t))-\exp(-\alpha_{\,\text{r}}\,f\,\eta(t))\right)
  9. R ct = 1 j t / η = 1 f j 0 ( α o exp ( α o f η ) + α r exp ( - α r f η ) ) R_{\,\text{ct}}=\frac{1}{\partial j_{\,\text{t}}/\partial\eta}=\frac{1}{f\,j_{% 0}\,\left(\alpha_{\,\text{o}}\,\exp(\alpha_{\,\text{o}}\,f\,\eta)+\alpha_{\,% \text{r}}\,\exp(-\alpha_{\,\text{r}}\,f\,\eta)\right)}
  10. R ct = 1 f j 0 R_{\,\text{ct}}=\frac{1}{f\,j_{0}}
  11. η = 0 \eta=0
  12. | |
  13. C dl C_{\,\text{dl}}
  14. Z dl ( ω ) = 1 i ω C dl Z_{\,\text{dl}}(\omega)=\frac{1}{\,\text{i}\,\omega\,C_{\,\text{dl}}}
  15. ω \omega
  16. i = - 1 \scriptstyle{\,\text{i}=\sqrt{-1}}
  17. Z ( ω ) = R t 1 + R t C dl i ω Z(\omega)=\frac{R_{\,\text{t}}}{1+R_{\,\text{t}}\,C_{\,\text{dl}}\,\,\text{i}% \,\omega}
  18. R t \scriptstyle{R_{\,\text{t}}}
  19. 1 / ( R t C dc ) \scriptstyle{1/(R_{\,\text{t}}\,C_{\,\text{dc}})}
  20. R Ω R_{\Omega}
  21. j 0 j_{0}
  22. η = 0 \eta=0

Dielectrophoresis.html

  1. r r
  2. ε p * \varepsilon_{p}^{*}
  3. ε m * \varepsilon_{m}^{*}
  4. F DEP = 2 π r 3 ε m Re { ε p * - ε m * ε p * + 2 ε m * } | E r m s | 2 \langle F_{\mathrm{DEP}}\rangle=2\pi r^{3}\varepsilon_{m}\textrm{Re}\left\{% \frac{\varepsilon^{*}_{p}-\varepsilon^{*}_{m}}{\varepsilon^{*}_{p}+2% \varepsilon^{*}_{m}}\right\}\nabla\left|\vec{E}_{rms}\right|^{2}
  5. r r
  6. l l
  7. ε p * \varepsilon_{p}^{*}
  8. ε m * \varepsilon_{m}^{*}
  9. F DEP = π r 2 l 3 ε m Re { ε p * - ε m * ε m * } | E | 2 F_{\mathrm{DEP}}=\frac{\pi r^{2}l}{3}\varepsilon_{m}\textrm{Re}\left\{\frac{% \varepsilon^{*}_{p}-\varepsilon^{*}_{m}}{\varepsilon^{*}_{m}}\right\}\nabla% \left|\vec{E}\right|^{2}
  10. ε * = ε + i σ ω \varepsilon^{*}=\varepsilon+\frac{i\sigma}{\omega}
  11. ε \varepsilon
  12. σ \sigma
  13. ω \omega
  14. i i
  15. ε 1 e f f * ( ω ) = ε 2 * ( r 2 r 1 ) 3 + 2 ε 1 * - ε 2 * ε 1 * + 2 ε 2 * ( r 2 r 1 ) 3 - ε 1 * - ε 2 * ε 1 * + 2 ε 2 * \varepsilon_{1eff}^{*}(\omega)=\varepsilon_{2}^{*}\frac{(\frac{r_{2}}{r_{1}})^% {3}+2\frac{\varepsilon_{1}^{*}-\varepsilon_{2}^{*}}{\varepsilon_{1}^{*}+2% \varepsilon_{2}^{*}}}{(\frac{r_{2}}{r_{1}})^{3}-\frac{\varepsilon_{1}^{*}-% \varepsilon_{2}^{*}}{\varepsilon_{1}^{*}+2\varepsilon_{2}^{*}}}

Differential_algebra.html

  1. : R R \partial:R\to R\,
  2. ( r 1 r 2 ) = ( r 1 ) r 2 + r 1 ( r 2 ) , \partial(r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2}),\,
  3. r 1 , r 2 R r_{1},r_{2}\in R
  4. M : R × R R M:R\times R\to R
  5. M = M ( × id ) + M ( id × ) . \partial\circ M=M\circ(\partial\times\operatorname{id})+M\circ(\operatorname{% id}\times\partial).
  6. f × g f\times g
  7. ( x , y ) (x,y)
  8. ( f ( x ) , g ( y ) ) (f(x),g(y))
  9. ( u v ) = ( u ) v - u ( v ) v 2 \partial\left(\frac{u}{v}\right)=\frac{\partial(u)\,v-u\,\partial(v)}{v^{2}}
  10. ( u v × v ) = ( u ) \partial\left(\frac{u}{v}\times v\right)=\partial(u)
  11. ( u v ) v + u v ( v ) = ( u ) \partial\left(\frac{u}{v}\right)\,v+\frac{u}{v}\,\partial(v)=\partial(u)
  12. ( u / v ) \partial(u/v)
  13. k = { u K : ( u ) = 0 } . k=\{u\in K:\partial(u)=0\}.
  14. k K k\in K
  15. x A x\in A
  16. ( k x ) = k x \partial(kx)=k\partial x
  17. η : K A \eta\colon K\to A
  18. M ( η × Id ) = M ( η × ) \partial\circ M\circ(\eta\times\operatorname{Id})=M\circ(\eta\times\partial)
  19. a , b K a,b\in K
  20. x , y A x,y\in A
  21. ( x y ) = ( x ) y + x ( y ) \partial(xy)=(\partial x)y+x(\partial y)
  22. ( a x + b y ) = a x + b y . \partial(ax+by)=a\,\partial x+b\,\partial y.
  23. 𝔤 \mathfrak{g}
  24. D : 𝔤 𝔤 D\colon\mathfrak{g}\to\mathfrak{g}
  25. D ( [ a , b ] ) = [ a , D ( b ) ] + [ D ( a ) , b ] D([a,b])=[a,D(b)]+[D(a),b]
  26. a 𝔤 a\in\mathfrak{g}
  27. 𝔤 \mathfrak{g}
  28. A A
  29. K K
  30. K K
  31. ( u ) = u \partial(u)=u
  32. R ( ( ξ - 1 ) ) = { n < r n ξ n | r n R } . R((\xi^{-1}))=\left\{\sum_{n<\infty}r_{n}\xi^{n}|r_{n}\in R\right\}.
  33. ( r ξ m ) ( s ξ n ) = k = 0 m r ( k s ) ( m k ) ξ m + n - k . (r\xi^{m})(s\xi^{n})=\sum_{k=0}^{m}r(\partial^{k}s){m\choose k}\xi^{m+n-k}.
  34. ( m k ) {m\choose k}
  35. ξ - 1 r = n = 0 ( - 1 ) n ( n r ) ξ - 1 - n \xi^{-1}r=\sum_{n=0}^{\infty}(-1)^{n}(\partial^{n}r)\xi^{-1-n}
  36. ( - 1 n ) = ( - 1 ) n {-1\choose n}=(-1)^{n}
  37. r ξ - 1 = n = 0 ξ - 1 - n ( n r ) . r\xi^{-1}=\sum_{n=0}^{\infty}\xi^{-1-n}(\partial^{n}r).

Differential_algebraic_equation.html

  1. F ( x ˙ ( t ) , x ( t ) , t ) = 0 F(\dot{x}(t),\,x(t),\,t)=0
  2. x : [ a , b ] \R n x:[a,b]\to\R^{n}
  3. x ( t ) = ( x 1 ( t ) , , x n ( t ) ) x(t)=(x_{1}(t),\dots,x_{n}(t))
  4. F = ( F 1 , , F n ) : \R 2 n + 1 \R n F=(F_{1},\dots,F_{n}):\R^{2n+1}\to\R^{n}
  5. F ( u , v , t ) u \frac{\partial F(u,v,t)}{\partial u}
  6. ( x , y ) (x,y)
  7. x ˙ ( t ) \displaystyle\dot{x}(t)
  8. x ( t ) \R n x(t)\in\R^{n}
  9. y ( t ) \R m y(t)\in\R^{m}
  10. f : \R n + m + 1 \R n f:\R^{n+m+1}\to\R^{n}
  11. g : \R n + m + 1 \R m . g:\R^{n+m+1}\to\R^{m}.
  12. ( x , y ) (x,y)
  13. F ( x ˙ , x , y , t ) = 0 F\left(\dot{x},x,y,t\right)=0
  14. x x
  15. \R n \R^{n}
  16. y y
  17. \R m \R^{m}
  18. t t
  19. F F
  20. n + m n+m
  21. n + m n+m
  22. n n
  23. F : \R ( 2 n + m + 1 ) \R ( n + m ) . F:\R^{(2n+m+1)}\to\R^{(n+m)}.
  24. F ( x ˙ ( t 0 ) , x ( t 0 ) , y ( t 0 ) , t 0 ) = 0. F\left(\dot{x}(t_{0}),\,x(t_{0}),y(t_{0}),t_{0}\right)=0.
  25. x ˙ \displaystyle\dot{x}
  26. λ \lambda
  27. x ˙ x + y ˙ y \displaystyle\dot{x}\,x+\dot{y}\,y
  28. u ˙ x + v ˙ y + u x ˙ + v y ˙ \displaystyle\dot{u}\,x+\dot{v}\,y+u\,\dot{x}+v\,\dot{y}
  29. L 2 λ ˙ - 3 g v = 0 L^{2}\dot{\lambda}-3gv=0
  30. E = 3 2 g y - 1 2 L 2 λ = 1 2 ( u 2 + v 2 ) + g y E=\tfrac{3}{2}gy-\tfrac{1}{2}L^{2}\lambda=\frac{1}{2}(u^{2}+v^{2})+gy
  31. ( x 0 , u 0 ) (x_{0},u_{0})
  32. y = ± L 2 - x 2 y=\pm\sqrt{L^{2}-x^{2}}
  33. y 0 y\neq 0
  34. v = - u x / y v=-ux/y
  35. λ = ( g y - u 2 - v 2 ) / L 2 \lambda=(gy-u^{2}-v^{2})/L^{2}
  36. x ˙ \displaystyle\dot{x}
  37. ( y 0 , v 0 ) (y_{0},v_{0})
  38. x ˙ \displaystyle\dot{x}
  39. x ˙ \displaystyle\dot{x}
  40. ( x ˙ , y ˙ ) (\dot{x},\,\dot{y})
  41. det ( y g ( x , y , t ) ) 0. \det\left(\partial_{y}g(x,y,t)\right)\neq 0.
  42. d x d t \displaystyle\frac{dx}{dt}
  43. 0 \displaystyle 0
  44. Σ \Sigma
  45. Σ = ( σ i , j ) \Sigma=(\sigma_{i,j})
  46. f i f_{i}
  47. x j x_{j}
  48. ( i , j ) (i,j)
  49. σ i , j \sigma_{i,j}
  50. x j x_{j}
  51. f i f_{i}
  52. - -\infty
  53. x j x_{j}
  54. f i f_{i}
  55. ( x 1 , x 2 , x 3 , x 4 , x 5 ) = ( x , y , u , v , λ ) (x_{1},x_{2},x_{3},x_{4},x_{5})=(x,y,u,v,\lambda)
  56. Σ = [ 1 - 0 - - - 1 - 0 - 0 - 1 - 0 - 0 - 1 0 0 0 - - - ] \Sigma=\begin{bmatrix}1&-&0^{\bullet}&-&-\\ -&1^{\bullet}&-&0&-\\ 0&-&1&-&0^{\bullet}\\ -&0&-&1^{\bullet}&0\\ 0^{\bullet}&0&-&-&-\end{bmatrix}

Differential_equation.html

  1. d y d x = f ( x ) \frac{dy}{dx}=f(x)
  2. d y d x = f ( x , y ) \frac{dy}{dx}=f(x,y)
  3. x 1 y x 1 + x 2 y x 2 = y x_{1}\frac{\partial y}{\partial x_{1}}+x_{2}\frac{\partial y}{\partial x_{2}}=y
  4. y + P ( x ) y = Q ( x ) y n y^{\prime}+P(x)y=Q(x)y^{n}\,
  5. d u d x = c u + x 2 . \frac{du}{dx}=cu+x^{2}.
  6. d 2 u d x 2 - x d u d x + u = 0. \frac{d^{2}u}{dx^{2}}-x\frac{du}{dx}+u=0.
  7. d 2 u d x 2 + ω 2 u = 0. \frac{d^{2}u}{dx^{2}}+\omega^{2}u=0.
  8. d u d x = u 2 + 4. \frac{du}{dx}=u^{2}+4.
  9. L d 2 u d x 2 + g sin u = 0. L\frac{d^{2}u}{dx^{2}}+g\sin u=0.
  10. u t + t u x = 0. \frac{\partial u}{\partial t}+t\frac{\partial u}{\partial x}=0.
  11. 2 u x 2 + 2 u y 2 = 0. \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0.
  12. u t = 6 u u x - 3 u x 3 . \frac{\partial u}{\partial t}=6u\frac{\partial u}{\partial x}-\frac{\partial^{% 3}u}{\partial x^{3}}.
  13. ( a , b ) (a,b)
  14. Z Z
  15. Z = [ l , m ] × [ n , p ] Z=[l,m]\times[n,p]
  16. ( a , b ) (a,b)
  17. Z Z
  18. d x d t = g ( x , t ) \frac{\mathrm{d}x}{\mathrm{d}t}=g(x,t)
  19. x ( t 0 ) = x 0 x(t_{0})=x_{0}
  20. g ( x , t ) g(x,t)
  21. g x \frac{\partial g}{\partial x}
  22. Z Z
  23. a a
  24. f n ( x ) d n y d x n + + f 1 ( x ) d y d x + f 0 ( x ) y = h ( x ) f_{n}(x)\frac{\mathrm{d}^{n}y}{\mathrm{d}x^{n}}+\cdots+f_{1}(x)\frac{\mathrm{d% }y}{\mathrm{d}x}+f_{0}(x)y=h(x)
  25. y ( x 0 ) = y 0 , y ( x 0 ) = y 0 , y ′′ ( x 0 ) = y 0 ′′ , y(x_{0})=y_{0},y^{\prime}(x_{0})=y^{\prime}_{0},y^{\prime\prime}(x_{0})=y^{% \prime\prime}_{0},\cdots
  26. f n ( x ) f_{n}(x)
  27. { f 0 , f 1 , } \{f_{0},f_{1},\cdots\}
  28. g g
  29. x 0 x_{0}
  30. y y
  31. k ( t ) t = s [ k ( t ) ] α - δ k ( t ) \frac{\partial k(t)}{\partial t}=s[k(t)]^{\alpha}-\delta k(t)

Differential_signaling.html

  1. V S V_{S}
  2. V S V_{S}\,
  3. V S - 0 V = V S V_{S}-0\,\mathrm{V}=V_{S}
  4. V S V_{S}\,
  5. V S - 0 V = V S V_{S}-0\,\mathrm{V}=V_{S}
  6. 0 V - V S = - V S 0\,\mathrm{V}-V_{S}=-V_{S}
  7. V S - ( - V S ) = 2 V S V_{S}-(-V_{S})=2V_{S}\,

Diffusion_capacitance.html

  1. V V
  2. I = I ( V ) I=I(V)
  3. τ F {\tau}_{F}
  4. Q Q
  5. Q = I ( V ) τ F Q=I(V){\tau}_{F}
  6. C d i f f C_{diff}
  7. C d i f f = d Q d V = d I ( V ) d V τ F C_{diff}=\begin{matrix}\frac{dQ}{dV}\end{matrix}=\begin{matrix}\frac{dI(V)}{dV% }\end{matrix}{\tau}_{F}
  8. τ F {\tau}_{F}

Digit_sum.html

  1. n = 0 log b x 1 b n ( x mod b n + 1 - x mod b n ) \sum_{n=0}^{\lfloor\log_{b}x\rfloor}\frac{1}{b^{n}}(x\bmod b^{n+1}-x\bmod b^{n})

Dilation_(metric_space).html

  1. f f
  2. d ( f ( x ) , f ( y ) ) = r d ( x , y ) d(f(x),f(y))=rd(x,y)
  3. ( x , y ) (x,y)
  4. d ( x , y ) d(x,y)
  5. x x
  6. y y
  7. r r

Dilemma.html

  1. A B , A C , B C , C A\vee B,A\Rightarrow C,B\Rightarrow C,\vdash C

Dimensional_deconstruction.html

  1. G × G × G G\times G\times G...

Dimensional_regularization.html

  1. 2 π d / 2 Γ ( d 2 ) \frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}
  2. d d p ( 2 π ) d 1 ( p 2 + m 2 ) 2 , \int\frac{d^{d}p}{(2\pi)^{d}}\frac{1}{\left(p^{2}+m^{2}\right)^{2}},
  3. 0 d p ( 2 π ) 4 - ε 2 π ( 4 - ε ) / 2 Γ ( 4 - ε 2 ) p 3 - ε ( p 2 + m 2 ) 2 = 2 ε - 4 π ε 2 - 1 sin ( π ε 2 ) Γ ( 1 - ε 2 ) m - ε = 1 8 π 2 ε - 1 16 π 2 ( ln m 2 4 π + γ ) + 𝒪 ( ε ) . \int_{0}^{\infty}\frac{dp}{(2\pi)^{4-\varepsilon}}\frac{2\pi^{(4-\varepsilon)/% 2}}{\Gamma\left(\frac{4-\varepsilon}{2}\right)}\frac{p^{3-\varepsilon}}{\left(% p^{2}+m^{2}\right)^{2}}=\frac{2^{\varepsilon-4}\pi^{\frac{\varepsilon}{2}-1}}{% \sin(\frac{\pi\varepsilon}{2})\Gamma(1-\frac{\varepsilon}{2})}m^{-\varepsilon}% =\frac{1}{8\pi^{2}\varepsilon}-\frac{1}{16\pi^{2}}\left(\ln\frac{m^{2}}{4\pi}+% \gamma\right)+\mathcal{O}(\varepsilon).

Dirac_comb.html

  1. III T ( t ) = def k = - δ ( t - k T ) = 1 T III ( t T ) \operatorname{III}_{T}(t)\ \stackrel{\mathrm{def}}{=}\ \sum_{k=-\infty}^{% \infty}\delta(t-kT)=\frac{1}{T}\operatorname{III}\left(\frac{t}{T}\right)
  2. III ( t ) \operatorname{III}(t)
  3. III T ( t ) = 1 T k = - e 2 π i k t T . \operatorname{III}_{T}(t)=\frac{1}{T}\sum_{k=-\infty}^{\infty}e^{2\pi ik\frac{% t}{T}}.
  4. 1 1
  5. δ \delta
  6. comb T { 1 } = III T = rep T { δ } \operatorname{comb}_{T}\left\{1\right\}\,=\,\operatorname{III}_{T}\,=\,% \operatorname{rep}_{T}\left\{\delta\right\}
  7. comb T { f ( t ) } = def k = - f ( k T ) δ ( t - k T ) \operatorname{comb}_{T}\left\{f(t)\right\}\,\stackrel{\mathrm{def}}{=}\,\sum_{% k=-\infty}^{\infty}\,f(kT)\,\delta(t-kT)
  8. rep T { g ( t ) } = def k = - g ( t - k T ) \operatorname{rep}_{T}\left\{g(t)\right\}\,\stackrel{\mathrm{def}}{=}\,\sum_{k% =-\infty}^{\infty}\,g(t-kT)
  9. f ( t ) f(t)
  10. III T \operatorname{III}_{T}
  11. f ( t ) f(t)
  12. III T \operatorname{III}_{T}
  13. δ ( t ) = 1 a δ ( t a ) \delta(t)=\frac{1}{a}\,\,\delta\left(\frac{t}{a}\right)
  14. a a
  15. III a T ( t ) = 1 a III T ( t a ) \operatorname{III}_{aT}\left(t\right)\,=\,\frac{1}{a}\,\,\operatorname{III}_{T% }\left(\frac{t}{a}\right)
  16. a a
  17. III T \operatorname{III}_{T}
  18. III T ( t ) \ \operatorname{III}_{T}(t)
  19. T T
  20. III T ( t + T ) = III T ( t ) \operatorname{III}_{T}(t+T)=\operatorname{III}_{T}(t)\,
  21. III T ( t ) = n = - + c n e 2 π i n t T \operatorname{III}_{T}(t)=\sum_{n=-\infty}^{+\infty}c_{n}e^{2\pi in\frac{t}{T}}
  22. c n = 1 T t 0 t 0 + T III T ( t ) e - 2 π i n t T d t ( - < t 0 < + ) = 1 T - T 2 T 2 III T ( t ) e - 2 π i n t T d t = 1 T - T 2 T 2 δ ( t ) e - 2 π i n t T d t = 1 T e - 2 π i n 0 T = 1 T . \begin{aligned}\displaystyle c_{n}&\displaystyle=\frac{1}{T}\int_{t_{0}}^{t_{0% }+T}\operatorname{III}_{T}(t)e^{-2\pi in\frac{t}{T}}\,dt\quad(-\infty<t_{0}<+% \infty)\\ &\displaystyle=\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}\operatorname{III}_% {T}(t)e^{-2\pi in\frac{t}{T}}\,dt\\ &\displaystyle=\frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}\delta(t)e^{-2\pi in% \frac{t}{T}}\,dt\\ &\displaystyle=\frac{1}{T}e^{-2\pi in\frac{0}{T}}\\ &\displaystyle=\frac{1}{T}.\end{aligned}
  23. III T ( t ) = 1 T n = - e 2 π i n t T . \operatorname{III}_{T}(t)=\frac{1}{T}\sum_{n=-\infty}^{\infty}e^{2\pi in\frac{% t}{T}}.
  24. f f
  25. 1 T \frac{1}{T}
  26. III T ( t ) 1 T III 1 T ( f ) = n = - e - i 2 π f n T . \operatorname{III}_{T}(t)\quad\stackrel{\mathcal{F}}{\longleftrightarrow}\quad% \frac{1}{T}\operatorname{III}_{\frac{1}{T}}(f)\quad=\sum_{n=-\infty}^{\infty}e% ^{-i2\pi fnT}.
  27. III ( t ) III ( f ) \operatorname{III}(t)\quad\stackrel{\mathcal{F}}{\longleftrightarrow}\quad% \operatorname{III}(f)
  28. III T ( t ) 2 π T III 2 π T ( ω ) = 1 2 π n = - e - i ω n T . \operatorname{III}_{T}(t)\quad\stackrel{\mathcal{F}}{\longleftrightarrow}\quad% \frac{\sqrt{2\pi}}{T}\operatorname{III}_{\frac{2\pi}{T}}(\omega)\quad=\frac{1}% {\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-i\omega nT}.\,
  29. ( III T x ) ( t ) = k = - x ( t ) δ ( t - k T ) = k = - x ( k T ) δ ( t - k T ) . (\operatorname{III}_{T}x)(t)=\sum_{k=-\infty}^{\infty}x(t)\delta(t-kT)=\sum_{k% =-\infty}^{\infty}x(kT)\delta(t-kT).
  30. III T x 1 T III 1 T * X \operatorname{III}_{T}x\quad\stackrel{\mathcal{F}}{\longleftrightarrow}\quad% \frac{1}{T}\operatorname{III}_{\frac{1}{T}}*X
  31. δ ( t - k T ) \delta(t-kT)
  32. k T kT
  33. ( III 1 T * X ) ( f ) = k = - X ( f - k T ) (\operatorname{III}_{\frac{1}{T}}*X)(f)=\sum_{k=-\infty}^{\infty}X\left(f-% \frac{k}{T}\right)
  34. x x
  35. ( - B , B ) (-B,B)
  36. 1 / 2 B 1/2B
  37. III 1 2 B x 2 B III 2 B * X \operatorname{III}_{\frac{1}{2B}}x\quad\stackrel{\mathcal{F}}{% \longleftrightarrow}\quad 2B\,\operatorname{III}_{2B}*X
  38. 1 2 B Π ( t 2 B ) ( 2 B III 2 B * X ) = X \frac{1}{2B}\Pi\left(\frac{t}{2B}\right)(2B\,\operatorname{III}_{2B}*X)=X
  39. π \pi
  40. - -\infty
  41. + +\infty
  42. π \pi
  43. π \pi
  44. π \pi

Dirac_large_numbers_hypothesis.html

  1. G 1 / t G\propto 1/t\,
  2. M t 2 M\propto t^{2}
  3. R U r e r H r e 10 42 , \frac{R_{U}}{r_{e}}\approx\frac{r_{H}}{r_{e}}\approx 10^{42},
  4. r e = e 2 4 π ϵ 0 m e c 2 , r_{e}=\frac{e^{2}}{4\pi\epsilon_{0}m_{e}c^{2}},
  5. r H = e 2 4 π ϵ 0 m H c 2 , r_{H}=\frac{e^{2}}{4\pi\epsilon_{0}m_{H}c^{2}},
  6. m H c 2 = G m e 2 r e m_{H}c^{2}=\frac{Gm_{e}^{2}}{r_{e}}
  7. e 2 4 π ϵ 0 G m e 2 N 10 42 . \frac{e^{2}}{4\pi\epsilon_{0}Gm_{e}^{2}}\approx\sqrt{N}\approx 10^{42}.
  8. G = ( c 3 M U ) t , G=\left(\frac{c^{3}}{M_{U}}\right)t,
  9. c t r e 10 40 , \frac{ct}{r_{e}}\approx 10^{40},
  10. c c
  11. 4 π ϵ 0 G m p m e e 2 10 - 40 . \frac{4\pi\epsilon_{0}Gm_{p}m_{e}}{e^{2}}\approx 10^{-40}.
  12. e e
  13. m p m_{p}
  14. m e m_{e}
  15. 4 π ϵ 0 4\pi\epsilon_{0}
  16. G G
  17. G 1 / t G\approx 1/t\,

Direct_comparison_test.html

  1. b n \sum b_{n}
  2. 0 a n b n 0\leq a_{n}\leq b_{n}
  3. n > N n>N
  4. a n \sum a_{n}
  5. b n \sum b_{n}
  6. 0 b n a n 0\leq b_{n}\leq a_{n}
  7. a n \sum a_{n}
  8. b n \sum b_{n}
  9. | a n | | b n | |a_{n}|\leq|b_{n}|
  10. a n \sum a_{n}
  11. b n \sum b_{n}
  12. | b n | | a n | |b_{n}|\leq|a_{n}|
  13. a n \sum a_{n}
  14. a n \sum a_{n}
  15. c n \sum c_{n}
  16. | c n | \sum|c_{n}|
  17. a n \sum a_{n}
  18. b n \sum b_{n}
  19. b n \sum b_{n}
  20. | b n | \sum|b_{n}|
  21. | a n | | b n | |a_{n}|\leq|b_{n}|
  22. S n = | a 1 | + | a 2 | + + | a n | , T n = | b 1 | + | b 2 | + + | b n | . S_{n}=|a_{1}|+|a_{2}|+\ldots+|a_{n}|,\ T_{n}=|b_{1}|+|b_{2}|+\ldots+|b_{n}|.
  23. b n \sum b_{n}
  24. lim n T n = T \lim_{n\to\infty}T_{n}=T
  25. T n T_{n}
  26. T n T T_{n}\leq T
  27. 0 S n = | a 1 | + | a 2 | + + | a n | | b 1 | + | b 2 | + + | b n | T . 0\leq S_{n}=|a_{1}|+|a_{2}|+\ldots+|a_{n}|\leq|b_{1}|+|b_{2}|+\ldots+|b_{n}|% \leq T.
  28. S n S_{n}
  29. a n \sum a_{n}
  30. [ a , b ) [a,b)
  31. + +\infty
  32. a b g ( x ) d x \int_{a}^{b}g(x)\,dx
  33. 0 f ( x ) g ( x ) 0\leq f(x)\leq g(x)
  34. a x < b a\leq x<b
  35. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  36. a b f ( x ) d x a b g ( x ) d x . \int_{a}^{b}f(x)\,dx\leq\int_{a}^{b}g(x)\,dx.
  37. a b g ( x ) d x \int_{a}^{b}g(x)\,dx
  38. 0 g ( x ) f ( x ) 0\leq g(x)\leq f(x)
  39. a x < b a\leq x<b
  40. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  41. b n \sum b_{n}
  42. a n > 0 a_{n}>0
  43. b n > 0 b_{n}>0
  44. a n + 1 a n b n + 1 b n \frac{a_{n+1}}{a_{n}}\leq\frac{b_{n+1}}{b_{n}}
  45. a n \sum a_{n}

Direct_integral.html

  1. L μ 2 ( X , H ) . L^{2}_{\mu}(X,H).
  2. { X n } 1 n ω \{X_{n}\}_{1\leq n\leq\omega}
  3. H x = 𝐇 n x X n H_{x}=\mathbf{H}_{n}\quad x\in X_{n}
  4. 𝐇 n = { n if n < ω 2 if n = ω \mathbf{H}_{n}=\left\{\begin{matrix}\mathbb{C}^{n}&\mbox{ if }~{}n<\omega\\ \ell^{2}&\mbox{ if }~{}n=\omega\end{matrix}\right.
  5. X H x d μ ( x ) \int^{\oplus}_{X}H_{x}\ d\mu(x)
  6. s t = X s ( x ) t ( x ) d μ ( x ) \langle s\mid t\rangle=\int_{X}\langle s(x)\mid t(x)\rangle d\mu(x)
  7. s ( d μ d ν ) 1 / 2 s s\mapsto\bigg(\frac{d\mu}{d\nu}\bigg)^{1/2}s
  8. X H x d ν ( x ) X H x d μ ( x ) . \int^{\oplus}_{X}H_{x}\ d\nu(x)\rightarrow\int^{\oplus}_{X}H_{x}\ d\mu(x).
  9. X H x d μ ( x ) k H k \int^{\oplus}_{X}H_{x}\ d\mu(x)\cong\bigoplus_{k\in\mathbb{N}}H_{k}
  10. H = k H k H=\bigoplus_{k\in\mathbb{N}}H_{k}
  11. [ T 11 T 12 T 1 n T 21 T 22 T 2 n T n 1 T n 2 T n n ] . \begin{bmatrix}T_{11}&T_{12}&\cdots&T_{1n}&\cdots\\ T_{21}&T_{22}&\cdots&T_{2n}&\cdots\\ \vdots&\vdots&\ddots&\vdots&\cdots\\ T_{n1}&T_{n2}&\cdots&T_{nn}&\cdots\\ \vdots&\vdots&\cdots&\vdots&\ddots\end{bmatrix}.
  12. [ λ 1 0 0 0 λ 2 0 0 0 λ n ] . \begin{bmatrix}\lambda_{1}&0&\cdots&0&\cdots\\ 0&\lambda_{2}&\cdots&0&\cdots\\ \vdots&\vdots&\ddots&\vdots&\cdots\\ 0&0&\cdots&\lambda_{n}&\cdots\\ \vdots&\vdots&\cdots&\vdots&\ddots\end{bmatrix}.
  13. ess - sup x X T x < \operatorname{ess-sup}_{x\in X}\|T_{x}\|<\infty
  14. X T x d μ ( x ) L ( X H x d μ ( x ) ) \int^{\oplus}_{X}\ T_{x}d\mu(x)\in\operatorname{L}\bigg(\int^{\oplus}_{X}H_{x}% \ d\mu(x)\bigg)
  15. [ X T x d μ ( x ) ] ( X s x d μ ( x ) ) = X T x ( s x ) d μ ( x ) . \bigg[\int^{\oplus}_{X}\ T_{x}d\mu(x)\bigg]\bigg(\int^{\oplus}_{X}\ s_{x}d\mu(% x)\bigg)=\int^{\oplus}_{X}\ T_{x}(s_{x})d\mu(x).
  16. ϕ : L μ ( X ) L ( X H x d μ ( x ) ) \phi:L^{\infty}_{\mu}(X)\rightarrow\operatorname{L}\bigg(\int^{\oplus}_{X}H_{x% }\ d\mu(x)\bigg)
  17. λ X λ x d μ ( x ) \lambda\mapsto\int^{\oplus}_{X}\ \lambda_{x}d\mu(x)
  18. X H x d μ ( x ) . \int_{X}^{\oplus}H_{x}d\mu(x).\quad
  19. U : H X H x d μ ( x ) U:H\rightarrow\int_{X}^{\oplus}H_{x}d\mu(x)
  20. X H x d μ ( x ) , Y K y d ν ( y ) \int_{X}^{\oplus}H_{x}d\mu(x),\quad\int_{Y}^{\oplus}K_{y}d\nu(y)
  21. φ : X - E Y - F \varphi:X-E\rightarrow Y-F
  22. K ϕ ( x ) = H x almost everywhere K_{\phi(x)}=H_{x}\quad\mbox{almost everywhere}~{}
  23. A x L ( H x ) A_{x}\subseteq\operatorname{L}(H_{x})
  24. W * ( { S x : S D } ) = A x \operatorname{W^{*}}(\{S_{x}:S\in D\})=A_{x}
  25. X A x d μ ( x ) \int_{X}^{\oplus}A_{x}d\mu(x)
  26. X T x d μ ( x ) \int_{X}^{\oplus}T_{x}d\mu(x)
  27. [ X A x d μ ( x ) ] = X A x d μ ( x ) . \bigg[\int_{X}^{\oplus}A_{x}d\mu(x)\bigg]^{\prime}=\int_{X}^{\oplus}A^{\prime}% _{x}d\mu(x).
  28. 𝐙 ( A ) = A A \mathbf{Z}(A)=A\cap A^{\prime}
  29. 1 = i E i 1=\sum_{i\in\mathbb{N}}E_{i}
  30. A = i A E i A=\bigoplus_{i\in\mathbb{N}}AE_{i}
  31. H = X H x d μ ( x ) H=\int_{X}^{\oplus}H_{x}d\mu(x)
  32. 𝐀 = X A x d μ ( x ) \mathbf{A}=\int^{\oplus}_{X}A_{x}d\mu(x)
  33. { π x } x X \{\pi_{x}\}_{x\in X}
  34. π ( a ) = X π x ( a ) d μ ( x ) , a A . \pi(a)=\int_{X}^{\oplus}\pi_{x}(a)d\mu(x),\quad\forall a\in A.

Directional_statistics.html

  1. p ( x ) p(x)
  2. θ = x w = x mod 2 π ( - π , π ] \theta=x_{w}=x\mod 2\pi\ \ \in(-\pi,\pi]
  3. p w ( θ ) = k = - p ( θ + 2 π k ) . p_{w}(\theta)=\sum_{k=-\infty}^{\infty}{p(\theta+2\pi k)}.
  4. F F
  5. p w ( θ ) = k 1 = - k F = - p ( θ + 2 π k 1 𝐞 1 + + 2 π k F 𝐞 F ) p_{w}(\vec{\theta})=\sum_{k_{1}=-\infty}^{\infty}\cdots\sum_{k_{F}=-\infty}^{% \infty}{p(\vec{\theta}+2\pi k_{1}\mathbf{e}_{1}+\dots+2\pi k_{F}\mathbf{e}_{F})}
  6. 𝐞 k = ( 0 , , 0 , 1 , 0 , , 0 ) 𝖳 \mathbf{e}_{k}=(0,\dots,0,1,0,\dots,0)^{\mathsf{T}}
  7. k k
  8. f ( θ ; μ , κ ) = e κ cos ( θ - μ ) 2 π I 0 ( κ ) f(\theta;\mu,\kappa)=\frac{e^{\kappa\cos(\theta-\mu)}}{2\pi I_{0}(\kappa)}
  9. I 0 I_{0}
  10. U ( θ ) = 1 / 2 π . U(\theta)=1/2\pi.\,
  11. W N ( θ ; μ , σ ) = 1 σ 2 π k = - exp [ - ( θ - μ - 2 π k ) 2 2 σ 2 ] = 1 2 π ϑ ( θ - μ 2 π , i σ 2 2 π ) WN(\theta;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\sum^{\infty}_{k=-\infty}\exp% \left[\frac{-(\theta-\mu-2\pi k)^{2}}{2\sigma^{2}}\right]=\frac{1}{2\pi}% \vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^{2}}{2\pi}\right)
  12. ϑ ( θ , τ ) \vartheta(\theta,\tau)
  13. ϑ ( θ , τ ) = n = - ( w 2 ) n q n 2 \vartheta(\theta,\tau)=\sum_{n=-\infty}^{\infty}(w^{2})^{n}q^{n^{2}}
  14. w e i π θ w\equiv e^{i\pi\theta}
  15. q e i π τ . q\equiv e^{i\pi\tau}.
  16. W C ( θ ; θ 0 , γ ) = n = - γ π ( γ 2 + ( θ + 2 π n - θ 0 ) 2 ) = 1 2 π sinh γ cosh γ - cos ( θ - θ 0 ) WC(\theta;\theta_{0},\gamma)=\sum_{n=-\infty}^{\infty}\frac{\gamma}{\pi(\gamma% ^{2}+(\theta+2\pi n-\theta_{0})^{2})}=\frac{1}{2\pi}\,\,\frac{\sinh\gamma}{% \cosh\gamma-\cos(\theta-\theta_{0})}
  17. γ \gamma
  18. θ 0 \theta_{0}
  19. f W L ( θ ; μ , c ) = n = - c 2 π e - c / 2 ( θ + 2 π n - μ ) ( θ + 2 π n - μ ) 3 / 2 f_{WL}(\theta;\mu,c)=\sum_{n=-\infty}^{\infty}\sqrt{\frac{c}{2\pi}}\,\frac{e^{% -c/2(\theta+2\pi n-\mu)}}{(\theta+2\pi n-\mu)^{3/2}}
  20. θ + 2 π n - μ 0 \theta+2\pi n-\mu\leq 0
  21. c c
  22. μ \mu
  23. θ ¯ \scriptstyle\bar{\theta}
  24. s ¯ \scriptstyle\bar{s}
  25. c ¯ 0 \scriptstyle\bar{c}\not=0
  26. s ¯ = 1 3 ( sin ( 355 ) + sin ( 5 ) + sin ( 15 ) ) = 1 3 ( - 0.087 + 0.087 + 0.259 ) 0.086 \bar{s}=\frac{1}{3}\left(\sin(355^{\circ})+\sin(5^{\circ})+\sin(15^{\circ})% \right)=\frac{1}{3}\left(-0.087+0.087+0.259\right)\approx 0.086
  27. c ¯ = 1 3 ( cos ( 355 ) + cos ( 5 ) + cos ( 15 ) ) = 1 3 ( 0.996 + 0.996 + 0.966 ) 0.986 \bar{c}=\frac{1}{3}\left(\cos(355^{\circ})+\cos(5^{\circ})+\cos(15^{\circ})% \right)=\frac{1}{3}\left(0.996+0.996+0.966\right)\approx 0.986
  28. θ ¯ = { arctan ( s ¯ c ¯ ) s ¯ > 0 , c ¯ > 0 arctan ( s ¯ c ¯ ) + 180 c ¯ < 0 arctan ( s ¯ c ¯ ) + 360 s ¯ < 0 , c ¯ > 0 } = arctan ( 0.086 0.986 ) = arctan ( 0.087 ) = 5 . \bar{\theta}=\left.\begin{cases}\arctan\left(\frac{\bar{s}}{\bar{c}}\right)&% \bar{s}>0,\ \bar{c}>0\\ \arctan\left(\frac{\bar{s}}{\bar{c}}\right)+180^{\circ}&\bar{c}<0\\ \arctan\left(\frac{\bar{s}}{\bar{c}}\right)+360^{\circ}&\bar{s}<0,\ \bar{c}>0% \end{cases}\right\}=\arctan\left(\frac{0.086}{0.986}\right)=\arctan(0.087)=5^{% \circ}.
  29. z = cos ( θ ) + i sin ( θ ) = e i θ z=\cos(\theta)+i\,\sin(\theta)=e^{i\theta}
  30. θ \theta
  31. ρ ¯ = 1 N n = 1 N z n . \overline{\mathbf{\rho}}=\frac{1}{N}\sum_{n=1}^{N}z_{n}.
  32. θ ¯ = Arg ( ρ ¯ ) . \overline{\theta}=\mathrm{Arg}(\overline{\mathbf{\rho}}).
  33. R ¯ = | ρ ¯ | \overline{R}=|\overline{\mathbf{\rho}}|
  34. ρ ¯ = R ¯ e i θ ¯ . \overline{\mathbf{\rho}}=\overline{R}\,e^{i\overline{\theta}}.
  35. m n = E ( z n ) = Γ P ( θ ) z n d θ m_{n}=E(z^{n})=\int_{\Gamma}P(\theta)z^{n}d\theta\,
  36. Γ \Gamma
  37. 2 π 2\pi
  38. P ( θ ) P(\theta)
  39. P ( θ ) P(\theta)
  40. m ¯ n = 1 N i = 1 N z i n . \overline{m}_{n}=\frac{1}{N}\sum_{i=1}^{N}z_{i}^{n}.
  41. ρ = m 1 \rho=m_{1}\,
  42. R = | m 1 | R=|m_{1}|\,
  43. θ μ = Arg ( m 1 ) . \theta_{\mu}=\mathrm{Arg}(m_{1}).\,
  44. R n = | m n | R_{n}=|m_{n}|\,
  45. ( n θ μ ) mod 2 π (n\theta_{\mu})\mod 2\pi
  46. Var ( z ) ¯ = 1 - R ¯ \overline{\mathrm{Var}(z)}=1-\overline{R}\,
  47. Var ( z ) = 1 - R \mathrm{Var}(z)=1-R\,
  48. S ( z ) = ln ( 1 / R 2 ) = - 2 ln ( R ) S(z)=\sqrt{\ln(1/R^{2})}=\sqrt{-2\ln(R)}\,
  49. S ¯ ( z ) = ln ( 1 / R ¯ 2 ) = - 2 ln ( R ¯ ) \overline{S}(z)=\sqrt{\ln(1/{\overline{R}}^{2})}=\sqrt{-2\ln({\overline{R}})}\,
  50. S ( z ) S(z)
  51. S ( z ) 2 = 2 V a r ( z ) S(z)^{2}=2\mathrm{Var}(z)
  52. δ = 1 - R 2 2 R 2 \delta=\frac{1-R_{2}}{2R^{2}}
  53. δ ¯ = 1 - R ¯ 2 2 R ¯ 2 \overline{\delta}=\frac{1-{\overline{R}_{2}}}{2{\overline{R}}^{2}}
  54. z n = e i θ n z_{n}=e^{i\theta_{n}}
  55. z ¯ = 1 N n = 1 N z n \overline{z}=\frac{1}{N}\sum_{n=1}^{N}z_{n}
  56. z ¯ = C ¯ + i S ¯ \overline{z}=\overline{C}+i\overline{S}
  57. C ¯ = 1 N n = 1 N cos ( θ n ) and S ¯ = 1 N n = 1 N sin ( θ n ) \overline{C}=\frac{1}{N}\sum_{n=1}^{N}\cos(\theta_{n})\,\text{ and }\overline{% S}=\frac{1}{N}\sum_{n=1}^{N}\sin(\theta_{n})
  58. z ¯ = R ¯ e i θ ¯ \overline{z}=\overline{R}e^{i\overline{\theta}}
  59. R ¯ = C ¯ 2 + S ¯ 2 and θ ¯ = ArcTan ( S ¯ , C ¯ ) . \overline{R}=\sqrt{{\overline{C}}^{2}+{\overline{S}}^{2}}\,\,\,\mathrm{and}\,% \,\,\,\overline{\theta}=\mathrm{ArcTan}(\overline{S},\overline{C}).
  60. θ ¯ \overline{\theta}
  61. P ( C ¯ , S ¯ ) d C ¯ d S ¯ = P ( R ¯ , θ ¯ ) d R ¯ d θ ¯ = Γ Γ n = 1 N [ P ( θ n ) d θ n ] P(\overline{C},\overline{S})\,d\overline{C}\,d\overline{S}=P(\overline{R},% \overline{\theta})\,d\overline{R}\,d\overline{\theta}=\int_{\Gamma}...\int_{% \Gamma}\prod_{n=1}^{N}\left[P(\theta_{n})\,d\theta_{n}\right]
  62. Γ \Gamma
  63. 2 π 2\pi
  64. S ¯ \overline{S}
  65. C ¯ \overline{C}
  66. R ¯ \overline{R}
  67. θ ¯ \overline{\theta}
  68. [ C ¯ , S ¯ ] [\overline{C},\overline{S}]

Dirichlet's_principle.html

  1. u ( x ) u(x)
  2. Δ u + f = 0 \Delta u+f=0\,
  3. Ω \Omega
  4. n \mathbb{R}^{n}
  5. u = g on Ω , u=g\,\text{ on }\partial\Omega,\,
  6. E [ v ( x ) ] = Ω ( 1 2 | v | 2 - v f ) d x E[v(x)]=\int_{\Omega}\left(\frac{1}{2}|\nabla v|^{2}-vf\right)\,\mathrm{d}x
  7. v v
  8. v = g v=g
  9. Ω \partial\Omega

Dirichlet_conditions.html

  1. a n = 1 2 π - π π f ( x ) e - i n x d x . a_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\,dx.
  2. n = - a n e i n x = 1 2 ( f ( x + ) + f ( x - ) ) \sum_{n=-\infty}^{\infty}a_{n}e^{inx}=\frac{1}{2}(f(x+)+f(x-))
  3. f ( x + ) = lim y x + f ( y ) f(x+)=\lim_{y\to x^{+}}f(y)
  4. f ( x - ) = lim y x - f ( y ) f(x-)=\lim_{y\to x^{-}}f(y)
  5. 1 2 ( f ( x + ) + f ( x - ) ) = f ( x ) \frac{1}{2}(f(x+)+f(x-))=f(x)

Dirichlet_integral.html

  1. 0 sin ω ω d ω = π 2 \int_{0}^{\infty}\frac{\sin\omega}{\omega}\,d\omega=\frac{\pi}{2}
  2. 0 sin t t d t = 0 { sin t } ( s ) d s = 0 1 s 2 + 1 d s = arctan s | 0 = π 2 , \int_{0}^{\infty}\frac{\sin t}{t}\,dt=\int_{0}^{\infty}\mathcal{L}\{\sin t\}(s% )\;ds=\int_{0}^{\infty}\frac{1}{s^{2}+1}\,ds=\arctan s\bigg|_{0}^{\infty}=% \frac{\pi}{2},
  3. { sin t } ( s ) \mathcal{L}\{\sin t\}(s)
  4. sin t \sin t
  5. 1 s 2 + 1 \tfrac{1}{s^{2}+1}
  6. ( I 1 = 0 0 e - s t sin t d t d s ) = ( I 2 = 0 0 e - s t sin t d s d t ) , \left(I_{1}=\int_{0}^{\infty}{\int_{0}^{\infty}e^{-st}\sin t\,dt}\,ds\right)=% \left(I_{2}=\int_{0}^{\infty}{\int_{0}^{\infty}e^{-st}\sin t\,ds}\,dt\right),
  7. ( I 1 = 0 1 s 2 + 1 d s = π 2 ) = ( I 2 = 0 sin t 1 t d t ) , provided s > 0. \left(I_{1}=\int_{0}^{\infty}{\frac{1}{s^{2}+1}}\,ds=\frac{\pi}{2}\right)=% \left(I_{2}=\int_{0}^{\infty}\sin t\,\frac{1}{t}\,dt\right)\,\text{, provided % }s>0.
  8. a \!a
  9. f ( a ) = 0 e - a ω sin ω ω d ω ; f(a)=\int_{0}^{\infty}e^{-a\omega}\frac{\sin\omega}{\omega}d\omega;
  10. f ( 0 ) \!f(0)
  11. a \!a
  12. d f d a = d d a 0 e - a ω sin ω ω d ω = 0 a e - a ω sin ω ω d ω = - 0 e - a ω sin ω d ω = - { sin ω } ( a ) . \frac{df}{da}=\frac{d}{da}\int_{0}^{\infty}e^{-a\omega}\frac{\sin\omega}{% \omega}d\omega=\int_{0}^{\infty}\frac{\partial}{\partial a}e^{-a\omega}\frac{% \sin\omega}{\omega}d\omega=-\int_{0}^{\infty}e^{-a\omega}\sin\omega\,d\omega=-% \mathcal{L}\{\sin\omega\}(a).
  13. e i ω = cos ω + i sin ω , \!e^{i\omega}=\cos\omega+i\sin\omega,
  14. e i ω = sin ω , \Im e^{i\omega}=\sin\omega,
  15. \Im
  16. d f d a = - 0 e - a ω e i ω d ω = 1 - a + i = - a - i a 2 + 1 = - 1 a 2 + 1 , given that a > 0. \therefore\frac{df}{da}=-\Im\int_{0}^{\infty}e^{-a\omega}e^{i\omega}d\omega=% \Im\frac{1}{-a+i}=\Im\frac{-a-i}{a^{2}+1}=\frac{-1}{a^{2}+1}\,\text{, given % that }a>0.
  17. a \!a
  18. f ( a ) = - d a a 2 + 1 = A - arctan a , f(a)=\int\frac{-da}{a^{2}+1}=A-\arctan a,
  19. A \!A
  20. f ( + ) = 0 A = arctan ( + ) = π 2 + m π , f(+\infty)=0\therefore A=\arctan(+\infty)=\frac{\pi}{2}+m\pi,
  21. f ( 0 ) = lim a 0 + f ( a ) = π 2 + m π - arctan 0 = π 2 + n π , \therefore f(0)=\lim_{a\to 0^{+}}f(a)=\frac{\pi}{2}+m\pi-\arctan 0=\frac{\pi}{% 2}+n\pi,
  22. n \!n
  23. 0 < 0 sin x x d x < 0 π sin x x d x < π 0<\int_{0}^{\infty}\frac{\sin x}{x}dx<\int_{0}^{\pi}\frac{\sin x}{x}dx<\pi
  24. [ 0 , ] [0,\infty]
  25. 0 sin x x d x = n = 0 n = 2 π n 2 π ( n + 1 ) sin x x d x = n = 0 n = 0 2 π sin x 2 π n + x d x > n = 0 n = 0 2 π sin x 2 π ( n + 1 ) d x = 0 \int_{0}^{\infty}\frac{\sin x}{x}dx=\sum_{n=0}^{n=\infty}\int_{2\pi n}^{2\pi(n% +1)}\frac{\sin x}{x}dx=\sum_{n=0}^{n=\infty}\int_{0}^{2\pi}\frac{\sin x}{2\pi n% +x}dx>\sum_{n=0}^{n=\infty}\int_{0}^{2\pi}\frac{\sin x}{2\pi(n+1)}dx=0
  26. 0 sin x x d x = 0 π sin x x d x + π sin x x d x \int_{0}^{\infty}\frac{\sin x}{x}dx=\int_{0}^{\pi}\frac{\sin x}{x}dx+\int_{\pi% }^{\infty}\frac{\sin x}{x}dx
  27. n = 1 π ( 2 n - 1 ) π ( 2 n + 1 ) sin x x d x < n = 1 1 2 π n ( π ( 2 n - 1 ) 2 n π sin x d x + 2 n π π ( 2 n + 1 ) sin x d x ) = n = 1 1 2 π n π ( 2 n - 1 ) π ( 2 n + 1 ) sin x d x \sum_{n=1}^{\infty}\int_{\pi(2n-1)}^{\pi(2n+1)}\frac{\sin x}{x}\ dx<\sum_{n=1}% ^{\infty}\frac{1}{2\pi n}(\int_{\pi(2n-1)}^{2n\pi}\sin x\ dx+\int_{2n\pi}^{\pi% (2n+1)}\sin x\ dx)=\sum_{n=1}^{\infty}\frac{1}{2\pi n}\int_{\pi(2n-1)}^{\pi(2n% +1)}\sin x\ dx
  28. π ( 2 n - 1 ) π ( 2 n + 1 ) sin x d x = 0 \int_{\pi(2n-1)}^{\pi(2n+1)}\sin x\ dx=0
  29. π sin x x d x < 0 \int_{\pi}^{\infty}\frac{\sin x}{x}dx<0
  30. sin x / x \!{\sin x}/{x}
  31. 0 sin x x d x = - 0 sin x x d x = - 0 - sin x x d x , \int_{0}^{\infty}\frac{\sin x}{x}\,dx=\int_{-\infty}^{0}\frac{\sin x}{x}\,dx=-% \int_{0}^{-\infty}\frac{\sin x}{x}\,dx,
  32. 0 sin b ω ω d ω = 0 b sin b ω b ω d ( b ω ) = 0 sgn b × sin x x d x = sgn b 0 sin x x d x = π 2 sgn b \int_{0}^{\infty}\frac{\sin b\,\omega}{\omega}\,d\omega=\int_{0}^{b\,\infty}% \frac{\sin b\,\omega}{b\,\omega}\,d(b\,\omega)=\int_{0}^{\operatorname{sgn}b% \times\infty}\frac{\sin x}{x}\,dx=\operatorname{sgn}b\int_{0}^{\infty}\frac{% \sin x}{x}\,dx=\frac{\pi}{2}\,\operatorname{sgn}b
  33. f ( z ) = e i z z f(z)=\frac{e^{iz}}{z}
  34. g ( z ) = e i z z + i ϵ g(z)=\frac{e^{iz}}{z+i\epsilon}
  35. ϵ 0 \epsilon\rightarrow 0
  36. 0 = γ g ( z ) d z = - R R e i x x + i ϵ d x + 0 π e i ( R e i θ + θ ) R e i θ + i ϵ i R d θ 0=\int_{\gamma}g(z)dz=\int_{-R}^{R}\frac{e^{ix}}{x+i\epsilon}dx+\int_{0}^{\pi}% \frac{e^{i(Re^{i\theta}+\theta)}}{Re^{i\theta}+i\epsilon}iRd\theta
  37. ϵ \epsilon
  38. 0 = P . V . e i x x d x - π i - δ ( x ) e i x d x 0=\mathrm{P.V.}\int\frac{e^{ix}}{x}dx-\pi i\int_{-\infty}^{\infty}\delta(x)e^{% ix}dx
  39. sinc ( x ) \mathrm{sinc}(x)
  40. sinc ( 0 ) = 1 \mathrm{sinc}(0)=1
  41. lim ϵ 0 ϵ sin ( x ) x d x = 0 sin ( x ) x d x = π 2 \lim_{\epsilon\rightarrow 0}\int_{\epsilon}^{\infty}\frac{\sin(x)}{x}dx=\int_{% 0}^{\infty}\frac{\sin(x)}{x}dx=\frac{\pi}{2}
  42. D n ( x ) = k = - n n e 2 π i k x = sin ( ( 2 n + 1 ) π x ) sin ( π x ) D_{n}(x)=\sum_{k=-n}^{n}e^{2\pi ikx}=\frac{\sin\left(\left(2n+1\right)\pi x% \right)}{\sin(\pi x)}
  43. D n ( - x ) = D n ( x ) D_{n}(-x)=D_{n}(x)
  44. x x
  45. - 1 / 2 1 / 2 D n ( x ) d x = | k | < n - 1 / 2 1 / 2 e 2 π i k x d x = 1 + 0 < | k | < n 1 2 π i k ( e π i k - e - π i k ) = 1 + 0 < | k | < n sin ( π k ) π k = 1 \int_{-1/2}^{1/2}D_{n}(x)dx=\sum_{|k|<n}\int_{-1/2}^{1/2}e^{2\pi ikx}dx=1+\sum% _{0<|k|<n}\frac{1}{2\pi ik}\left(e^{\pi ik}-e^{-\pi ik}\right)=1+\sum_{0<|k|<n% }\frac{\sin(\pi k)}{\pi k}=1
  46. sin ( π k ) = 0 \sin(\pi k)=0
  47. k k\in\mathbb{Z}
  48. f ( x ) = 1 π x - 1 sin ( π x ) f(x)=\frac{1}{\pi x}-\frac{1}{\sin(\pi x)}
  49. [ 0 , 1 2 ] \left[0,\frac{1}{2}\right]
  50. | f ( x ) | A |f(x)|\leq A
  51. x x
  52. A 0 A\in\mathbb{R}_{\geq 0}
  53. | 0 1 2 f ( x ) sin ( ( 2 n + 1 ) π x ) d x | A | 0 1 2 sin ( ( 2 n + 1 ) π x ) d x | = A ( 2 n + 1 ) π 0 \left|\int_{0}^{\frac{1}{2}}f(x)\sin((2n+1)\pi x)dx\right|\leq A\left|\int_{0}% ^{\frac{1}{2}}\sin((2n+1)\pi x)dx\right|=\frac{A}{(2n+1)\pi}\rightarrow 0
  54. n n\rightarrow\infty
  55. So 0 sin ( x ) x d x = lim n 0 ( 2 n + 1 ) π 2 sin ( x ) x d x = lim n 0 1 2 sin ( ( 2 n + 1 ) π x ) x d x by substituting x ( 2 n + 1 ) π x . = π lim n 0 1 2 sin ( ( 2 n + 1 ) π x ) ( f ( x ) + 1 sin ( π x ) ) d x = π lim n ( 0 1 2 f ( x ) sin ( 2 n + 1 ) π x ) d x + 1 2 - 1 2 1 2 D n ( x ) d x ) = π 2 by the above. \begin{aligned}\displaystyle\,\text{So }\int_{0}^{\infty}\frac{\sin(x)}{x}dx&% \displaystyle=\lim_{n\rightarrow\infty}\int_{0}^{(2n+1)\frac{\pi}{2}}\frac{% \sin(x)}{x}dx\\ &\displaystyle=\lim_{n\rightarrow\infty}\int_{0}^{\frac{1}{2}}\frac{\sin((2n+1% )\pi x)}{x}dx\,\text{ by substituting }x\mapsto(2n+1)\pi x.\\ &\displaystyle=\pi\lim_{n\rightarrow\infty}\int_{0}^{\frac{1}{2}}\sin((2n+1)% \pi x)\left(f(x)+\frac{1}{\sin(\pi x)}\right)dx\\ &\displaystyle=\pi\lim_{n\rightarrow\infty}\left(\int_{0}^{\frac{1}{2}}f(x)% \sin(2n+1)\pi x)dx+\frac{1}{2}\int_{-\frac{1}{2}}^{\frac{1}{2}}D_{n}(x)dx% \right)\\ &\displaystyle=\frac{\pi}{2}\,\text{ by the above.}\end{aligned}

Discount_function.html

  1. f ( t ) f(t)
  2. c ( t ) c(t)
  3. U ( { c t } t = 0 ) = t = 0 f ( t ) u ( c t ) U(\{c_{t}\}_{t=0}^{\infty})=\sum_{t=0}^{\infty}{f(t)u(c_{t})}
  4. U ( { c ( t ) } t = 0 ) = 0 f ( t ) u ( c ( t ) ) d t U(\{c(t)\}_{t=0}^{\infty})=\int_{0}^{\infty}{f(t)u(c(t))dt}

Discounted_utility.html

  1. β , \beta,
  2. β t u ( x t ) . \beta^{t}u(x_{t}).
  3. 0 < β < 1. 0<\beta<1.
  4. t = 0 T β t u ( x t ) , \sum_{t=0}^{T}\beta^{t}u(x_{t}),
  5. u ( x t ) u(x_{t})
  6. x x
  7. t t
  8. β \beta
  9. β \beta
  10. β ( t ) \beta(t)
  11. β \beta
  12. β < 1 \beta<1

Discrete-time_Fourier_transform.html

  1. X 2 π ( ω ) = n = - x [ n ] e - i ω n . X_{2\pi}(\omega)=\sum_{n=-\infty}^{\infty}x[n]\,e^{-i\omega n}.
  2. T x ( n T ) = x [ n ] T\cdot x(nT)=x[n]
  3. f \scriptstyle f
  4. X 1 / T ( f ) = X 2 π ( 2 π f T ) = def n = - T x ( n T ) x [ n ] e - i 2 π f T n = Poisson f . k = - X ( f - k / T ) . X_{1/T}(f)=X_{2\pi}(2\pi fT)\ \stackrel{\mathrm{def}}{=}\sum_{n=-\infty}^{% \infty}\underbrace{T\cdot x(nT)}_{x[n]}\ e^{-i2\pi fTn}\;\stackrel{\mathrm{% Poisson\;f.}}{=}\;\sum_{k=-\infty}^{\infty}X\left(f-k/T\right).
  5. e - i 2 π f T n e^{-i2\pi fTn}
  6. δ ( t - n T ) . \scriptstyle\delta(t-nT).
  7. X 1 / T ( f ) = { n = - x [ n ] δ ( t - n T ) } X_{1/T}(f)=\mathcal{F}\left\{\sum_{n=-\infty}^{\infty}x[n]\cdot\delta(t-nT)\right\}
  8. X 1 / T ( f ) \scriptstyle X_{1/T}(f)
  9. 1 N T , \scriptstyle\frac{1}{NT},
  10. 1 T / 1 N T = N . \scriptstyle\frac{1}{T}/\frac{1}{NT}=N.
  11. x [ n ] e - i 2 π f T n \scriptstyle x[n]\cdot e^{-i2\pi fTn}
  12. f = k N T . \scriptstyle f=\frac{k}{NT}.
  13. X 1 / T ( k N T ) \scriptstyle X_{1/T}(\frac{k}{NT})
  14. n = - x [ n ] δ ( t - n T ) = k = - X [ k ] e i 2 π k N T t Fourier series k = - X [ k ] δ ( f - k N T ) DTFT of a periodic sequence , \sum_{n=-\infty}^{\infty}x[n]\cdot\delta(t-nT)=\underbrace{\sum_{k=-\infty}^{% \infty}X[k]\cdot e^{i2\pi\frac{k}{NT}t}}_{\,\text{Fourier series}}\quad% \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad\underbrace{\sum_{k=-\infty}^{% \infty}X[k]\ \cdot\ \delta\left(f-\frac{k}{NT}\right)}_{\,\text{DTFT of a % periodic sequence}},
  15. X [ k ] \displaystyle X[k]
  16. n = - x [ n ] δ ( t - n T ) = - 1 { X 1 / T ( f ) } = def - X 1 / T ( f ) e i 2 π f t d f . \sum_{n=-\infty}^{\infty}x[n]\cdot\delta(t-nT)=\mathcal{F}^{-1}\left\{X_{1/T}(% f)\right\}\ \stackrel{\mathrm{def}}{=}\int_{-\infty}^{\infty}X_{1/T}(f)\cdot e% ^{i2\pi ft}df.
  17. x [ n ] \displaystyle x[n]
  18. X 1 / T ( k N T ) X k \displaystyle\underbrace{X_{1/T}\left(\frac{k}{NT}\right)}_{X_{k}}
  19. x N [ n ] = def m = - x [ n - m N ] . x_{N}[n]\ \stackrel{\,\text{def}}{=}\ \sum_{m=-\infty}^{\infty}x[n-mN].
  20. X k = n = 0 N - 1 x [ n ] e - i 2 π k n N . X_{k}=\sum_{n=0}^{N-1}x[n]\cdot e^{-i2\pi\frac{kn}{N}}.
  21. L = 64. L=64.
  22. x * y = DTFT - 1 [ DTFT { x } DTFT { y } ] . x*y\ =\ \scriptstyle\,\text{DTFT}^{-1}\displaystyle\left[\scriptstyle\,\text{% DTFT}\displaystyle\{x\}\cdot\scriptstyle\,\text{DTFT}\displaystyle\{y\}\right].
  23. x N * y = DTFT - 1 [ DTFT { x N } DTFT { y } ] = DFT - 1 [ DFT { x N } DFT { y N } ] . x_{N}*y\ =\ \scriptstyle\,\text{DTFT}^{-1}\displaystyle\left[\scriptstyle\,% \text{DTFT}\displaystyle\{x_{N}\}\cdot\scriptstyle\,\text{DTFT}\displaystyle\{% y\}\right]\ =\ \scriptstyle\,\text{DFT}^{-1}\displaystyle\left[\scriptstyle\,% \text{DFT}\displaystyle\{x_{N}\}\cdot\scriptstyle\,\text{DFT}\displaystyle\{y_% {N}\}\right].
  24. x N * y = DFT - 1 [ DFT { x } DFT { y } ] . x_{N}*y\ =\ \scriptstyle\,\text{DFT}^{-1}\displaystyle\left[\scriptstyle\,% \text{DFT}\displaystyle\{x\}\cdot\scriptstyle\,\text{DFT}\displaystyle\{y\}% \right].
  25. X ( z ) = n = - x [ n ] z - n , X(z)=\sum_{n=-\infty}^{\infty}x[n]\,z^{-n},
  26. e i ω . e^{i\omega}.
  27. X ( e i ω ) , 0 ω 2 π X(e^{i\omega}),0\leq\omega\leq 2\pi
  28. X ( e i ω ) X(e^{i\omega})
  29. X 2 π ( ω ) = def k = - X ( ω - 2 π k ) . X_{2\pi}(\omega)\ \stackrel{\mathrm{def}}{=}\sum_{k=-\infty}^{\infty}X(\omega-% 2\pi k).
  30. δ [ n ] \delta[n]
  31. X 2 π ( ω ) = 1 X_{2\pi}(\omega)=1
  32. δ [ n - M ] \delta[n-M]
  33. X 2 π ( ω ) = e - i ω M X_{2\pi}(\omega)=e^{-i\omega M}
  34. m = - δ [ n - M m ] \sum_{m=-\infty}^{\infty}\delta[n-Mm]\!
  35. X 2 π ( ω ) = m = - e - i ω M m = 2 π M k = - δ ( ω - 2 π k M ) X_{2\pi}(\omega)=\sum_{m=-\infty}^{\infty}e^{-i\omega Mm}=\frac{2\pi}{M}\sum_{% k=-\infty}^{\infty}\delta\left(\omega-\frac{2\pi k}{M}\right)\,
  36. X ( ω ) = 2 π M k = - ( M - 1 ) / 2 ( M - 1 ) / 2 δ ( ω - 2 π k M ) X(\omega)=\frac{2\pi}{M}\sum_{k=-(M-1)/2}^{(M-1)/2}\delta\left(\omega-\frac{2% \pi k}{M}\right)\,
  37. X ( ω ) = 2 π M k = - M / 2 + 1 M / 2 δ ( ω - 2 π k M ) X(\omega)=\frac{2\pi}{M}\sum_{k=-M/2+1}^{M/2}\delta\left(\omega-\frac{2\pi k}{% M}\right)\,
  38. u [ n ] u[n]
  39. X 2 π ( ω ) = 1 1 - e - i ω + π k = - δ ( ω - 2 π k ) X_{2\pi}(\omega)=\frac{1}{1-e^{-i\omega}}+\pi\sum_{k=-\infty}^{\infty}\delta(% \omega-2\pi k)\!
  40. X ( ω ) = 1 1 - e - i ω + π δ ( ω ) X(\omega)=\frac{1}{1-e^{-i\omega}}+\pi\cdot\delta(\omega)\!
  41. 1 / ( 1 - e - i ω ) 1/(1-e^{-i\omega})
  42. a n u [ n ] a^{n}u[n]
  43. X 2 π ( ω ) = 1 1 - a e - i ω X_{2\pi}(\omega)=\frac{1}{1-ae^{-i\omega}}\!
  44. 0 < | a | < 1 0<|a|<1
  45. e - i a n e^{-ian}
  46. X ( ω ) = 2 π δ ( ω + a ) , X(\omega)=2\pi\cdot\delta(\omega+a),
  47. X 2 π ( ω ) = 2 π k = - δ ( ω + a - 2 π k ) X_{2\pi}(\omega)=2\pi\sum_{k=-\infty}^{\infty}\delta(\omega+a-2\pi k)
  48. cos ( a n ) \cos(an)
  49. X ( ω ) = π [ δ ( ω + a ) + δ ( ω - a ) ] , X(\omega)=\pi[\delta(\omega+a)+\delta(\omega-a)],
  50. X 2 π ( ω ) = π k = - [ δ ( ω - a - 2 π k ) + δ ( ω + a + 2 π k ) ] X_{2\pi}(\omega)=\pi\sum_{k=-\infty}^{\infty}\left[\delta(\omega-a-2\pi k)+% \delta(\omega+a+2\pi k)\right]
  51. sin ( a n ) \sin(an)
  52. X 2 π ( ω ) = π i k = - [ δ ( ω - a - 2 π k ) - δ ( ω + a + 2 π k ) ] X_{2\pi}(\omega)=\frac{\pi}{i}\sum_{k=-\infty}^{\infty}\left[\delta(\omega-a-2% \pi k)-\delta(\omega+a+2\pi k)\right]
  53. rect [ n - M / 2 M ] \operatorname{rect}\left[{n-M/2\over M}\right]
  54. X ( ω ) = sin [ ω ( M + 1 ) / 2 ] sin ( ω / 2 ) e - i ω M 2 X(\omega)={\sin[\omega(M+1)/2]\over\sin(\omega/2)}\,e^{-\frac{i\omega M}{2}}\!
  55. sinc ( a + n ) \operatorname{sinc}(a+n)
  56. X ( ω ) = e i a ω X(\omega)=e^{ia\omega}
  57. W sinc 2 ( W n ) W\cdot\operatorname{sinc}^{2}(Wn)\,
  58. X ( ω ) = tri ( ω 2 π W ) X(\omega)=\operatorname{tri}\left({\omega\over 2\pi W}\right)
  59. W sinc ( W n ) W\cdot\operatorname{sinc}(Wn)
  60. X ( ω ) = rect ( ω 2 π W ) X(\omega)=\operatorname{rect}\left({\omega\over 2\pi W}\right)
  61. { 0 n = 0 ( - 1 ) n n elsewhere \begin{cases}0&n=0\\ \frac{(-1)^{n}}{n}&\mbox{elsewhere}\end{cases}
  62. X ( ω ) = j ω X(\omega)=j\omega
  63. W ( n + a ) { cos [ π W ( n + a ) ] - sinc [ W ( n + a ) ] } \frac{W}{(n+a)}\left\{\cos[\pi W(n+a)]-\operatorname{sinc}[W(n+a)]\right\}
  64. X ( ω ) = j ω rect ( ω π W ) e j a ω X(\omega)=j\omega\cdot\operatorname{rect}\left({\omega\over\pi W}\right)e^{ja\omega}
  65. { π 2 n = 0 ( - 1 ) n - 1 π n 2 otherwise \begin{cases}\frac{\pi}{2}&n=0\\ \frac{(-1)^{n}-1}{\pi n^{2}}&\mbox{ otherwise}\end{cases}
  66. X ( ω ) = | ω | X(\omega)=|\omega|
  67. { 0 ; n even 2 π n ; n odd \begin{cases}0;&n\,\text{ even}\\ \frac{2}{\pi n};&n\,\text{ odd}\end{cases}
  68. X ( ω ) = { j ω < 0 0 ω = 0 - j ω > 0 X(\omega)=\begin{cases}j&\omega<0\\ 0&\omega=0\\ -j&\omega>0\end{cases}
  69. C ( A + B ) 2 π sinc [ A - B 2 π n ] sinc [ A + B 2 π n ] \frac{C(A+B)}{2\pi}\cdot\operatorname{sinc}\left[\frac{A-B}{2\pi}n\right]\cdot% \operatorname{sinc}\left[\frac{A+B}{2\pi}n\right]
  70. X ( ω ) = X(\omega)=
  71. * *\!
  72. x [ n ] e i a n x[n]e^{ian}\!
  73. X ( e i ( ω - a ) ) X(e^{i(\omega-a)})\!
  74. x [ n / k ] x[n/k]\!
  75. X ( e i ( k ω ) ) X(e^{i(k\omega)})\!
  76. n i x [ n ] \frac{n}{i}x[n]\!
  77. d X ( e i ω ) d ω \frac{dX(e^{i\omega})}{d\omega}\!
  78. i n x [ n ] \frac{i}{n}x[n]\!
  79. - π ω X ( e i ϑ ) d ϑ \int_{-\pi}^{\omega}X(e^{i\vartheta})d\vartheta\!
  80. x [ n ] * y [ n ] x[n]*y[n]\!
  81. X ( e i ω ) Y ( e i ω ) X(e^{i\omega})\cdot Y(e^{i\omega})\!
  82. x [ n ] y [ n ] x[n]\cdot y[n]\!
  83. 1 2 π - π π X ( e i ϑ ) Y ( e i ( ω - ϑ ) ) d ϑ \frac{1}{2\pi}\int_{-\pi}^{\pi}{X(e^{i\vartheta})\cdot Y(e^{i(\omega-\vartheta% )})d\vartheta}\!
  84. ρ x y [ n ] = x [ - n ] * * y [ n ] \rho_{xy}[n]=x[-n]^{*}*y[n]\!
  85. R x y ( ω ) = X ( e i ω ) * Y ( e i ω ) R_{xy}(\omega)=X(e^{i\omega})^{*}\cdot Y(e^{i\omega})\!
  86. E = n = - x [ n ] y * [ n ] E=\sum_{n=-\infty}^{\infty}{x[n]\cdot y^{*}[n]}\!
  87. E = 1 2 π - π π X ( e i ω ) Y * ( e i ω ) d ω E=\frac{1}{2\pi}\int_{-\pi}^{\pi}{X(e^{i\omega})\cdot Y^{*}(e^{i\omega})d% \omega}\!
  88. { n = - T x ( n T ) δ ( t - n T ) } \displaystyle\mathcal{F}\left\{\sum_{n=-\infty}^{\infty}T\cdot x(nT)\cdot% \delta(t-nT)\right\}

Discrete_Laplace_operator.html

  1. G = ( V , E ) G=(V,E)
  2. V \scriptstyle V
  3. E \scriptstyle E
  4. ϕ : V R \phi\colon V\to R
  5. Δ \Delta
  6. ϕ \phi
  7. ( Δ ϕ ) ( v ) = w : d ( w , v ) = 1 [ ϕ ( v ) - ϕ ( w ) ] (\Delta\phi)(v)=\sum_{w:\,d(w,v)=1}\left[\phi(v)-\phi(w)\right]\,
  8. d ( w , v ) d(w,v)
  9. ϕ \phi
  10. Δ ϕ \Delta\phi
  11. ( Δ ϕ ) ( v ) (\Delta\phi)(v)
  12. γ : E R \gamma\colon E\to R
  13. ( Δ γ ϕ ) ( v ) = w : d ( w , v ) = 1 γ w v [ ϕ ( v ) - ϕ ( w ) ] (\Delta_{\gamma}\phi)(v)=\sum_{w:\,d(w,v)=1}\gamma_{wv}\left[\phi(v)-\phi(w)\right]
  14. γ w v \gamma_{wv}
  15. w v E wv\in E
  16. ( M ϕ ) ( v ) = 1 deg v w : d ( w , v ) = 1 ϕ ( w ) . (M\phi)(v)=\frac{1}{\deg v}\sum_{w:\,d(w,v)=1}\phi(w).
  17. Δ f ( x , y ) f ( x - h , y ) + f ( x + h , y ) + f ( x , y - h ) + f ( x , y + h ) - 4 f ( x , y ) h 2 , \Delta f(x,y)\approx\frac{f(x-h,y)+f(x+h,y)+f(x,y-h)+f(x,y+h)-4f(x,y)}{h^{2}},\,
  18. { ( x - h , y ) , ( x , y ) , ( x + h , y ) , ( x , y - h ) , ( x , y + h ) } . \{(x-h,y),(x,y),(x+h,y),(x,y-h),(x,y+h)\}.\,
  19. D x 2 = [ 1 - 2 1 ] \vec{D}^{2}_{x}=\begin{bmatrix}1&-2&1\end{bmatrix}
  20. 𝐃 x y 2 = [ 0 1 0 1 - 4 1 0 1 0 ] \mathbf{D}^{2}_{xy}=\begin{bmatrix}0&1&0\\ 1&-4&1\\ 0&1&0\end{bmatrix}
  21. 𝐃 x y 2 = [ 0.5 1 0.5 1 - 6 1 0.5 1 0.5 ] \mathbf{D}^{2}_{xy}=\begin{bmatrix}0.5&1&0.5\\ 1&-6&1\\ 0.5&1&0.5\end{bmatrix}
  22. 𝐃 x y z 2 \mathbf{D}^{2}_{xyz}
  23. [ 0 0 0 0 1 0 0 0 0 ] \begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix}
  24. [ 0 1 0 1 - 6 1 0 1 0 ] \begin{bmatrix}0&1&0\\ 1&-6&1\\ 0&1&0\end{bmatrix}
  25. [ 0 0 0 0 1 0 0 0 0 ] \begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix}
  26. n n
  27. a x 1 , x 2 , , x n a_{x_{1},x_{2},\dots,x_{n}}
  28. 𝐃 x 1 , x 2 , , x n 2 , \mathbf{D}^{2}_{x_{1},x_{2},\dots,x_{n}},
  29. a x 1 , x 2 , , x n = { - 2 n , if s = n 1 , if s = n - 1 0 , otherwise a_{x_{1},x_{2},\dots,x_{n}}=\left\{\begin{array}[]{ll}-2n,&\,\text{if }s=n\\ 1,&\,\text{if }s=n-1\\ 0,&\,\text{otherwise}\end{array}\right.
  30. x i x_{i}
  31. - 1 -1
  32. 0
  33. 1 1
  34. i : t h i:th
  35. s s
  36. i i
  37. x i = 0 x_{i}=0
  38. 𝐃 x y 2 = [ 1 1 1 1 - 8 1 1 1 1 ] . \mathbf{D}^{2}_{xy}=\begin{bmatrix}1&1&1\\ 1&-8&1\\ 1&1&1\end{bmatrix}.
  39. γ 2 = ( 1 - γ ) 5 2 + γ × 2 = ( 1 - γ ) [ 0 1 0 1 - 4 1 0 1 0 ] + γ [ 1 / 2 0 1 / 2 0 - 2 0 1 / 2 0 1 / 2 ] \nabla^{2}_{\gamma}=(1-\gamma)\nabla^{2}_{5}+\gamma\nabla^{2}_{\times}=(1-% \gamma)\begin{bmatrix}0&1&0\\ 1&-4&1\\ 0&1&0\end{bmatrix}+\gamma\begin{bmatrix}1/2&0&1/2\\ 0&-2&0\\ 1/2&0&1/2\end{bmatrix}
  40. γ 1 , γ 2 2 = ( 1 - γ 1 - γ 2 ) 7 2 + γ 1 + 3 2 + γ 2 × 3 2 ) \nabla^{2}_{\gamma_{1},\gamma_{2}}=(1-\gamma_{1}-\gamma_{2})\,\nabla_{7}^{2}+% \gamma_{1}\,\nabla_{+^{3}}^{2}+\gamma_{2}\,\nabla_{\times^{3}}^{2})
  41. ( 7 2 f ) 0 , 0 , 0 = f - 1 , 0 , 0 + f + 1 , 0 , 0 + f 0 , - 1 , 0 + f 0 , + 1 , 0 + f 0 , 0 , - 1 + f 0 , 0 , + 1 - 6 f 0 , 0 , 0 (\nabla_{7}^{2}f)_{0,0,0}=f_{-1,0,0}+f_{+1,0,0}+f_{0,-1,0}+f_{0,+1,0}+f_{0,0,-% 1}+f_{0,0,+1}-6f_{0,0,0}
  42. ( + 3 2 f ) 0 , 0 , 0 = 1 4 ( f - 1 , - 1 , 0 + f - 1 , + 1 , 0 + f + 1 , - 1 , 0 + f + 1 , + 1 , 0 + f - 1 , 0 , - 1 + f - 1 , 0 , + 1 + f + 1 , 0 , - 1 + f + 1 , 0 , + 1 + f 0 , - 1 , - 1 + f 0 , - 1 , + 1 + f 0 , + 1 , - 1 + f 0 , + 1 , + 1 - 12 f 0 , 0 , 0 ) , (\nabla_{+^{3}}^{2}f)_{0,0,0}=\frac{1}{4}(f_{-1,-1,0}+f_{-1,+1,0}+f_{+1,-1,0}+% f_{+1,+1,0}+f_{-1,0,-1}+f_{-1,0,+1}+f_{+1,0,-1}+f_{+1,0,+1}+f_{0,-1,-1}+f_{0,-% 1,+1}+f_{0,+1,-1}+f_{0,+1,+1}-12f_{0,0,0}),
  43. ( × 3 2 f ) 0 , 0 , 0 = 1 4 ( f - 1 , - 1 , - 1 + f - 1 , - 1 , + 1 + f - 1 , + 1 , - 1 + f - 1 , + 1 , + 1 + f + 1 , - 1 , - 1 + f + 1 , - 1 , + 1 + f + 1 , + 1 , - 1 + f + 1 , + 1 , + 1 - 8 f 0 , 0 , 0 ) . (\nabla_{\times^{3}}^{2}f)_{0,0,0}=\frac{1}{4}(f_{-1,-1,-1}+f_{-1,-1,+1}+f_{-1% ,+1,-1}+f_{-1,+1,+1}+f_{+1,-1,-1}+f_{+1,-1,+1}+f_{+1,+1,-1}+f_{+1,+1,+1}-8f_{0% ,0,0}).
  44. Δ = I - M \Delta=I-M
  45. [ 0 , 2 ] [0,2]
  46. [ - 1 , 1 ] [-1,1]
  47. λ 1 \lambda_{1}
  48. 2 F x 2 = lim ϵ 0 [ F ( x + ϵ ) - F ( x ) ] + [ F ( x - ϵ ) - F ( x ) ] ϵ 2 . \frac{\partial^{2}F}{\partial x^{2}}=\lim_{\epsilon\rightarrow 0}\frac{[F(x+% \epsilon)-F(x)]+[F(x-\epsilon)-F(x)]}{\epsilon^{2}}.
  49. P : V R P:V\rightarrow R
  50. ϕ \phi
  51. ( P ϕ ) ( v ) = P ( v ) ϕ ( v ) . (P\phi)(v)=P(v)\phi(v).\,
  52. H = Δ + P H=\Delta+P
  53. G ( v , w ; λ ) = δ v | 1 H - λ | δ w G(v,w;\lambda)=\left\langle\delta_{v}\left|\frac{1}{H-\lambda}\right|\delta_{w% }\right\rangle
  54. δ w \delta_{w}
  55. δ w ( v ) = δ w v \delta_{w}(v)=\delta_{wv}
  56. w V w\in V
  57. λ \lambda
  58. ( H - λ ) G ( v , w ; λ ) = δ w ( v ) . (H-\lambda)G(v,w;\lambda)=\delta_{w}(v).\,
  59. Δ ϕ = ϕ , \Delta\phi=\phi,
  60. Δ ϕ = ϕ - 2. \Delta\phi=\phi-2.

Disjoint_union_(topology).html

  1. X = i X i X=\coprod_{i}X_{i}
  2. φ i : X i X \varphi_{i}:X_{i}\to X\,
  3. φ i ( x ) = ( x , i ) \varphi_{i}(x)=(x,i)
  4. φ i - 1 ( U ) \varphi_{i}^{-1}(U)

Dispersion_(water_waves).html

  1. η ( x , t ) = a sin ( θ ( x , t ) ) , \eta(x,t)=a\sin\left(\theta(x,t)\right),\,
  2. θ = 2 π ( x λ - t T ) = k x - ω t , \theta=2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)=kx-\omega t,
  3. k = 2 π λ k=\frac{2\pi}{\lambda}
  4. ω = 2 π T , \omega=\frac{2\pi}{T},
  5. ω 2 = Ω 2 ( k ) . \omega^{2}=\Omega^{2}(k).\,
  6. x = λ T t + λ 2 π θ 0 = ω k t + θ 0 k . x=\frac{\lambda}{T}\,t+\frac{\lambda}{2\pi}\,\theta_{0}=\frac{\omega}{k}\,t+% \frac{\theta_{0}}{k}.
  7. c p = λ T = ω k = Ω ( k ) k , c_{p}=\frac{\lambda}{T}=\frac{\omega}{k}=\frac{\Omega(k)}{k},
  8. c p = g h (shallow water), c_{p}=\sqrt{gh}\qquad\scriptstyle\,\text{(shallow water),}\,
  9. c p = g λ 2 π = g 2 π T (deep water), c_{p}=\sqrt{\frac{g\lambda}{2\pi}}=\frac{g}{2\pi}T\qquad\scriptstyle\,\text{(% deep water),}
  10. λ = g 2 π T 2 (deep water). \lambda=\frac{g}{2\pi}T^{2}\qquad\scriptstyle\,\text{(deep water).}
  11. c g = Λ g τ g . c_{g}=\frac{\Lambda_{g}}{\tau_{g}}.
  12. 3 / 4 {3}/{4}
  13. 1 / 2 {1}/{2}
  14. No. of waves in space No. of waves in time = Λ g / λ τ g / T = Λ g τ g T λ = c g c p . \tfrac{\,\text{No. of waves in space}}{\,\text{No. of waves in time}}=\frac{% \Lambda_{g}/\lambda}{\tau_{g}/T}=\frac{\Lambda_{g}}{\tau_{g}}\cdot\frac{T}{% \lambda}=\frac{c_{g}}{c_{p}}.
  15. η = a sin ( k 1 x - ω 1 t ) + a sin ( k 2 x - ω 2 t ) , \eta=a\,\sin\left(k_{1}x-\omega_{1}t\right)+a\,\sin\left(k_{2}x-\omega_{2}t% \right),
  16. ω 1 2 = Ω 2 ( k 1 ) \omega_{1}^{2}=\Omega^{2}(k_{1})\,
  17. ω 2 2 = Ω 2 ( k 2 ) . \omega_{2}^{2}=\Omega^{2}(k_{2}).\,
  18. η = [ 2 a cos ( k 1 - k 2 2 x - ω 1 - ω 2 2 t ) ] sin ( k 1 + k 2 2 x - ω 1 + ω 2 2 t ) . \eta=\left[2\,a\,\cos\left(\frac{k_{1}-k_{2}}{2}x-\frac{\omega_{1}-\omega_{2}}% {2}t\right)\right]\;\cdot\;\sin\left(\frac{k_{1}+k_{2}}{2}x-\frac{\omega_{1}+% \omega_{2}}{2}t\right).
  19. c g = lim k 1 k 2 ω 1 - ω 2 k 1 - k 2 = lim k 1 k 2 Ω ( k 1 ) - Ω ( k 2 ) k 1 - k 2 = d Ω ( k ) d k . c_{g}=\lim_{k_{1}\,\to\,k_{2}}\frac{\omega_{1}-\omega_{2}}{k_{1}-k_{2}}=\lim_{% k_{1}\,\to\,k_{2}}\frac{\Omega(k_{1})-\Omega(k_{2})}{k_{1}-k_{2}}=\frac{\,% \text{d}\Omega(k)}{\,\text{d}k}.
  20. Ω ( k ) \displaystyle\Omega(k)
  21. g k = 2 π g λ \sqrt{gk}=\sqrt{\frac{2\pi\,g}{\lambda}}
  22. k g h = 2 π λ g h k\sqrt{gh}=\frac{2\pi}{\lambda}\sqrt{gh}
  23. g k tanh ( k h ) = 2 π g λ tanh ( 2 π h λ ) \begin{aligned}&\displaystyle\sqrt{gk\,\tanh\left(kh\right)}\\ &\displaystyle=\sqrt{\frac{2\pi g}{\lambda}\tanh\left(\frac{2\pi h}{\lambda}% \right)}\end{aligned}
  24. c p = λ T = ω k \displaystyle c_{p}=\frac{\lambda}{T}=\frac{\omega}{k}
  25. g k = g ω = g 2 π T \sqrt{\frac{g}{k}}=\frac{g}{\omega}=\frac{g}{2\pi}T
  26. g h \sqrt{gh}
  27. g k tanh ( k h ) \sqrt{\frac{g}{k}\tanh\left(kh\right)}
  28. c g = Ω k \displaystyle c_{g}=\frac{\partial\Omega}{\partial k}
  29. 1 2 g k = 1 2 g ω = g 4 π T \frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}\frac{g}{\omega}=\frac{g}{4\pi}T
  30. g h \sqrt{gh}
  31. 1 2 c p ( 1 + 2 k h sinh ( 2 k h ) ) \frac{1}{2}c_{p}\left(1+\frac{2kh}{\sinh\left(2kh\right)}\right)
  32. c g c p \displaystyle\frac{c_{g}}{c_{p}}
  33. 1 2 \displaystyle\frac{1}{2}
  34. 1 \displaystyle 1
  35. 1 2 ( 1 + 2 k h sinh ( 2 k h ) ) \frac{1}{2}\left(1+\frac{2kh}{\sinh\left(2kh\right)}\right)
  36. λ \displaystyle\lambda
  37. g 2 π T 2 \frac{g}{2\pi}T^{2}
  38. T g h T\sqrt{gh}
  39. ( 2 π T ) 2 = 2 π g λ tanh ( 2 π h λ ) \displaystyle\left(\frac{2\pi}{T}\right)^{2}=\frac{2\pi g}{\lambda}\tanh\left(% \frac{2\pi h}{\lambda}\right)
  40. ω 2 = ( g k + σ ρ k 3 ) tanh ( k h ) , \omega^{2}=\left(gk+\frac{\sigma}{\rho}k^{3}\right)\tanh(kh),
  41. Ω 2 ( k ) = g k ( ρ - ρ ) ρ coth ( k h ) + ρ coth ( k h ) , \Omega^{2}(k)=\frac{g\,k(\rho-\rho^{\prime})}{\rho\,\coth(kh)+\rho^{\prime}\,% \coth(kh^{\prime})},
  42. Ω 2 ( k ) = ρ - ρ ρ + ρ g k . \Omega^{2}(k)=\frac{\rho-\rho^{\prime}}{\rho+\rho^{\prime}}\,g\,k.
  43. c p = c g = g ( h + H ) . c_{p}=c_{g}=\sqrt{g(h+H)}.
  44. ω 2 = g k [ 1 + ( k A ) 2 ] . \omega^{2}=gk\left[1+(kA)^{2}\right].
  45. ω 2 = Ω 2 ( k ) , \omega^{2}=\Omega^{2}(k),\,
  46. ( ω - k V ) 2 = Ω 2 ( k ) , \left(\omega-k\cdot V\right)^{2}=\Omega^{2}(k),

Dispersive_mass_transfer.html

  1. J = - E d c d x , J=-E\frac{dc}{dx},
  2. E = α U . E=\alpha U.

Dissociation_(chemistry).html

  1. K a = [ A ] [ B ] [ AB ] K_{a}=\mathrm{\frac{[A][B]}{[AB]}}
  2. i i
  3. n n
  4. i = 1 + α ( n - 1 ) i=1+\alpha(n-1)
  5. n = 2 n=2
  6. i = 1 + α i=1+\alpha
  7. K p = p ( NO 2 ) 2 p ( N 2 O 4 ) K_{p}=\mathrm{\frac{p(NO_{2})^{2}}{p(N_{2}O_{4})}}
  8. K p = ( p T ) 2 ( x ( NO 2 ) ) 2 ( p T ) ( x ( N 2 O 4 ) ) = ( p T ) ( x ( NO 2 ) ) 2 ( x ( N 2 O 4 ) ) K_{p}=\mathrm{\frac{(p_{T})^{2}(x(NO_{2}))^{2}}{(p_{T})(x(N_{2}O_{4}))}=\frac{% (p_{T})(x(NO_{2}))^{2}}{(x(N_{2}O_{4}))}}
  9. K p = ( p T ) ( 4 α 2 ) ( 1 + α ) ( 1 - α ) = ( p T ) ( 4 α 2 ) ( 1 - α 2 ) K_{p}=\frac{(\mathrm{p_{T}})(4\alpha^{2})}{(1+\alpha)(1-\alpha)}=\frac{(% \mathrm{p_{T}})(4\alpha^{2})}{(1-\alpha^{2})}
  10. HA H + + A - \mathrm{HA\rightleftharpoons H^{+}+A^{-}}
  11. K a = [ H + ] [ A - ] [ HA ] K_{\mathrm{a}}=\mathrm{\frac{[H^{+}][A^{-}]}{[HA]}}
  12. HA + H 2 O H 3 O + + A - \mathrm{HA+H_{2}O\rightleftharpoons H_{3}O^{+}+A^{-}}
  13. K a = [ H 3 O + ] [ A - ] [ HA ] K_{\mathrm{a}}=\mathrm{\frac{[H_{3}O^{+}][A^{-}]}{[HA]}}
  14. [ H 2 O ] [H_{2}O]

Distance_modulus.html

  1. μ = m - M \mu=m-M
  2. m m
  3. M M
  4. d d
  5. log 10 ( d ) = 1 + μ 5 \log_{10}(d)=1+\frac{\mu}{5}
  6. μ = 5 log 10 ( d ) - 5 \mu=5\log_{10}(d)-5
  7. M M
  8. d d
  9. L ( d ) = L ( 10 ) ( d 10 ) 2 L(d)=\frac{L(10)}{(\frac{d}{10})^{2}}
  10. m = - 2.5 log 10 L ( d ) m=-2.5\log_{10}L(d)
  11. M = - 2.5 log 10 L ( 10 ) M=-2.5\log_{10}L(10)
  12. m - M = 5 log 10 ( d ) - 5 = μ m-M=5\log_{10}(d)-5=\mu
  13. d d
  14. 5 log 10 ( d ) - 5 = μ 5\log_{10}(d)-5=\mu
  15. d = 10 μ 5 + 1 d=10^{\frac{\mu}{5}+1}
  16. δ d = 0.2 ln ( 10 ) 10 0.2 μ + 1 δ μ = 0.461 d δ μ \delta d=0.2\ln(10)10^{0.2\mu+1}\delta\mu=0.461d\ \delta\mu
  17. ( m - M ) v {(m-M)}_{v}
  18. ( m - M ) 0 {(m-M)}_{0}

Distortion_(optics).html

  1. x d = x u ( 1 + K 1 r 2 + K 2 r 4 + ) + ( P 2 ( r 2 + 2 x u 2 ) + 2 P 1 x u y u ) ( 1 + P 3 r 2 + P 4 r 4 ) x_{\mathrm{d}}=x_{\mathrm{u}}(1+K_{1}r^{2}+K_{2}r^{4}+\cdots)+(P_{2}(r^{2}+2x_% {\mathrm{u}}^{2})+2P_{1}x_{\mathrm{u}}y_{\mathrm{u}})(1+P_{3}r^{2}+P_{4}r^{4}\cdots)
  2. y d = y u ( 1 + K 1 r 2 + K 2 r 4 + ) + ( P 1 ( r 2 + 2 y u 2 ) + 2 P 2 x u y u ) ( 1 + P 3 r 2 + P 4 r 4 ) y_{\mathrm{d}}=y_{\mathrm{u}}(1+K_{1}r^{2}+K_{2}r^{4}+\cdots)+(P_{1}(r^{2}+2y_% {\mathrm{u}}^{2})+2P_{2}x_{\mathrm{u}}y_{\mathrm{u}})(1+P_{3}r^{2}+P_{4}r^{4}\cdots)
  3. ( x d , y d ) (x_{\mathrm{d}},\ y_{\mathrm{d}})
  4. ( x u , y u ) (x_{\mathrm{u}},\ y_{\mathrm{u}})
  5. ( x c , y c ) (x_{\mathrm{c}},\ y_{\mathrm{c}})
  6. K n K_{n}
  7. n th n^{\mathrm{th}}
  8. P n P_{n}
  9. n th n^{\mathrm{th}}
  10. r r
  11. ( x u - x c ) 2 + ( y u - y c ) 2 \sqrt{(x_{\mathrm{u}}-x_{\mathrm{c}})^{2}+(y_{\mathrm{u}}-y_{\mathrm{c}})^{2}}
  12. ...
  13. K 1 K_{1}
  14. r r

Dittography.html

  1. 𝔓 \mathfrak{P}

Divisor_(algebraic_geometry).html

  1. ( f ) := z ν R ( f ) s ν z ν (f):=\sum_{z_{\nu}\in R(f)}s_{\nu}z_{\nu}
  2. s ν := { a if z ν is a zero of order a - a if z ν is a pole of order a . s_{\nu}:=\left\{\begin{array}[]{rl}a&\ \,\text{if }z_{\nu}\,\text{ is a zero % of order }a\\ -a&\ \,\text{if }z_{\nu}\,\text{ is a pole of order }a.\end{array}\right.
  3. Z n Z [ Z ] \sum_{Z}n_{Z}[Z]
  4. ord Z ( f ) = length 𝒪 η ( 𝒪 η / f 𝒪 η ) \operatorname{ord}_{Z}(f)=\operatorname{length}_{\mathcal{O}_{\eta}}({\mathcal% {O}_{\eta}}/f{\mathcal{O}_{\eta}})
  5. A / f A A/fA
  6. ord ( f ) = length A ( A / f A ) \operatorname{ord}(f)=\operatorname{length}_{A}(A/fA)
  7. f / g f/g
  8. ord : Q ( A ) - { 0 } \operatorname{ord}:Q(A)-\{0\}\to\mathbb{Z}
  9. ( f ) = Z ord Z ( f ) [ Z ] (f)=\sum_{Z}\operatorname{ord}_{Z}(f)[Z]
  10. [ Z ] = V i Z length 𝒪 Z ( 𝒪 Z , η i ) [ V i ] [Z]=\sum_{V_{i}\subset Z}\operatorname{length}_{\mathcal{O}_{Z}}({\mathcal{O}_% {Z,\eta_{i}}})[V_{i}]
  11. V i V_{i}
  12. η i \eta_{i}
  13. V i V_{i}
  14. f = g / h f=g/h
  15. ( f ) = ( g ) - ( h ) (f)=(g)-(h)
  16. ( f ) (f)
  17. Γ ( U , O ( D ) ) = { f k ( X ) | f = 0 or ( f ) + D 0 on U } \Gamma(U,O(D))=\{f\in k(X)|f=0\,\text{ or }(f)+D\geq 0\,\text{ on }U\}
  18. f f g f\mapsto fg
  19. z 2 = x y z^{2}=xy
  20. Cl ( X ) = Div ( X ) / { ( f ) | f k ( X ) * } \operatorname{Cl}(X)=\operatorname{Div}(X)/\{(f)|f\in k(X)^{*}\}
  21. 1 [ Z ] Cl ( X ) j * Cl ( X - Z ) 0 \mathbb{Z}\overset{1\mapsto[Z]}{\to}\operatorname{Cl}(X)\overset{j^{*}}{\to}% \operatorname{Cl}(X-Z)\to 0
  22. j * j^{*}
  23. j : X - Z X j:X-Z\hookrightarrow X
  24. Cl ( X ) j * Cl ( X - Z ) \operatorname{Cl}(X)\overset{j^{*}}{\underset{}{}}{\sim}\to\operatorname{Cl}(X% -Z)
  25. A k ( Z ) A k ( X ) A k ( X - Z ) 0 A_{k}(Z)\to A_{k}(X)\to A_{k}(X-Z)\to 0
  26. A k A_{k}
  27. Cl ( 𝐏 n ) j * Cl ( 𝐏 n - H ) 0 \mathbb{Z}\to\operatorname{Cl}(\mathbf{P}^{n})\overset{j^{*}}{\to}% \operatorname{Cl}(\mathbf{P}^{n}-H)\to 0
  28. Cl ( 𝐏 n - H ) = Cl ( 𝐀 n ) = 0 \operatorname{Cl}(\mathbf{P}^{n}-H)=\operatorname{Cl}(\mathbf{A}^{n})=0
  29. 𝐀 n \mathbf{A}^{n}
  30. Cl ( 𝐏 n ) \operatorname{Cl}(\mathbf{P}^{n})\simeq\mathbb{Z}
  31. ( s ) (s)
  32. ( s ) (s)
  33. ( s ) = V i X ord V i ( s ) [ V i ] , (s)=\sum_{V_{i}\subset X}\operatorname{ord}_{V_{i}}(s)[V_{i}],
  34. ord V i ( s ) \operatorname{ord}_{V_{i}}(s)
  35. 𝒪 X , η i \mathcal{O}_{X,\eta_{i}}
  36. Pic ( X ) Cl ( X ) \operatorname{Pic}(X)\to\operatorname{Cl}(X)
  37. D 𝒪 ( D ) D\mapsto\mathcal{O}(D)
  38. X reg X_{\,\text{reg}}
  39. X reg X X_{\,\text{reg}}\hookrightarrow X
  40. j * : Cl ( X ) Cl ( X reg ) = Pic ( X reg ) , j^{*}:\operatorname{Cl}(X)\overset{\sim}{\to}\operatorname{Cl}(X_{\,\text{reg}% })=\operatorname{Pic}(X_{\,\text{reg}}),
  41. X - X reg X-X_{\,\text{reg}}
  42. K X K_{X}
  43. 𝒪 ( j * K X ) = Ω X reg dim X \mathcal{O}(j^{*}K_{X})=\Omega^{\dim X}_{X_{\,\text{reg}}}
  44. U i {U_{i}}
  45. f i f_{i}
  46. U i U_{i}
  47. f i f_{i}
  48. f i f_{i}
  49. 1 𝒪 X * M X * M X * / 𝒪 X * 1 1\to\mathcal{O}^{*}_{X}\to M^{*}_{X}\to M^{*}_{X}/\mathcal{O}^{*}_{X}\to 1
  50. Γ ( X , ) \Gamma(X,\bullet)
  51. 1 Γ ( X , O X * ) Γ ( X , M X * ) Γ ( X , M X * / 𝒪 X * ) H 1 ( X , 𝒪 X * ) 1\to\Gamma(X,O^{*}_{X})\to\Gamma(X,M^{*}_{X})\to\Gamma(X,M^{*}_{X}/\mathcal{O}% ^{*}_{X})\to H^{1}(X,\mathcal{O}^{*}_{X})
  52. Γ ( X , M X * ) Γ ( X , M X * / 𝒪 X * ) \Gamma(X,M^{*}_{X})\to\Gamma(X,M^{*}_{X}/\mathcal{O}^{*}_{X})
  53. 𝒪 X ( D ) \mathcal{O}_{X}(D)
  54. ( D ) \mathcal{L}(D)
  55. ( D ) \mathcal{L}(D)
  56. x X x\in X
  57. D x Γ ( x , M X * / 𝒪 X * ) D_{x}\in\Gamma(x,M^{*}_{X}/\mathcal{O}^{*}_{X})
  58. 𝒪 X \mathcal{O}_{X}
  59. M X M_{X}
  60. 𝒪 X \mathcal{O}_{X}
  61. D ( D ) D\mapsto\mathcal{L}(D)
  62. D D
  63. X X
  64. ( D ) \mathcal{L}(D)
  65. ( D ) \mathcal{L}(D)
  66. ( D ) \mathcal{L}(D)
  67. Γ ( X , ( D ) ) \mathbb{P}\Gamma(X,\mathcal{L}(D))
  68. D D
  69. X X
  70. k k
  71. Γ ( X , ( D ) ) \Gamma(X,\mathcal{L}(D))
  72. k k
  73. k k
  74. D D
  75. \mathbb{Q}
  76. \mathbb{R}
  77. \mathbb{Q}
  78. \mathbb{Q}
  79. \mathbb{Q}
  80. \mathbb{Q}
  81. \mathbb{Q}
  82. D = a j Z j D=\sum a_{j}Z_{j}
  83. \mathbb{Q}
  84. [ a j ] Z j \sum[a_{j}]Z_{j}
  85. [ a j ] [a_{j}]
  86. a j a_{j}
  87. I ( D ) I(D)
  88. U i = Spec A i U_{i}=\operatorname{Spec}A_{i}
  89. f i A i f_{i}\in A_{i}
  90. D U i D\cap U_{i}
  91. f i = 0 f_{i}=0
  92. I ( D ) | U i = A / f i A I(D)|_{U_{i}}=A/f_{i}A
  93. D + D D+D^{\prime}
  94. f g = 0 fg=0
  95. R R R\to R^{\prime}
  96. D × R R D\times_{R}R^{\prime}
  97. X × R R X\times_{R}R^{\prime}
  98. f : X X f:X^{\prime}\to X
  99. D = D × X X D^{\prime}=D\times_{X}X^{\prime}
  100. I ( D ) = f * ( I ( D ) ) I(D^{\prime})=f^{*}(I(D))
  101. I ( D ) - 1 𝒪 X - I(D)^{-1}\otimes_{\mathcal{O}_{X}}-
  102. 0 I ( D ) 𝒪 X 𝒪 D 0 0\to I(D)\to\mathcal{O}_{X}\to\mathcal{O}_{D}\to 0
  103. 0 𝒪 X I ( D ) - 1 I ( D ) - 1 𝒪 D 0 0\to\mathcal{O}_{X}\to I(D)^{-1}\to I(D)^{-1}\otimes\mathcal{O}_{D}\to 0
  104. 𝒪 X \mathcal{O}_{X}
  105. I ( D ) - 1 I(D)^{-1}
  106. I ( D ) - 1 I(D)^{-1}
  107. L L
  108. 𝒪 X \mathcal{O}_{X}
  109. L / 𝒪 X L/\mathcal{O}_{X}
  110. s = 0 s=0
  111. Γ ( D , 𝒪 D ) \Gamma(D,\mathcal{O}_{D})
  112. deg D \operatorname{deg}D
  113. Spec R \operatorname{Spec}R
  114. D + D D+D^{\prime}
  115. deg ( D + D ) = deg ( D ) + deg ( D ) \operatorname{deg}(D+D^{\prime})=\operatorname{deg}(D)+\operatorname{deg}(D^{% \prime})
  116. f : X X f:X^{\prime}\to X
  117. deg ( f * D ) = deg ( f ) deg ( D ) \operatorname{deg}(f^{*}D)=\operatorname{deg}(f)\operatorname{deg}(D)
  118. deg ( D × R R ) = deg ( D ) \operatorname{deg}(D\times_{R}R^{\prime})=\operatorname{deg}(D)

DLVO_theory.html

  1. k B T k_{B}T
  2. a a
  3. Z Z
  4. r r
  5. ϵ r \epsilon_{r}
  6. n n
  7. β U ( r ) = Z 2 λ B ( exp ( κ a ) 1 + κ a ) 2 exp ( - κ r ) r , \beta U(r)=Z^{2}\lambda_{B}\,\left(\frac{\exp(\kappa a)}{1+\kappa a}\right)^{2% }\,\frac{\exp(-\kappa r)}{r},
  8. λ B \lambda_{B}
  9. κ - 1 \kappa^{-1}
  10. κ 2 = 4 π λ B n \kappa^{2}=4\pi\lambda_{B}n
  11. β - 1 = k B T \beta^{-1}=k_{B}T
  12. T T
  13. w ( r ) = - 2 π C ρ 1 z = D z = d z x = 0 x = x d x ( z 2 + x 2 ) 3 = 2 π C ρ 1 4 D d z z 4 = - π C ρ 1 / 6 D 3 w(r)=-2\pi\,C\rho_{1}\,\int_{z=D}^{z=\infty\,}dz\int_{x=0}^{x=\infty\,}\frac{% xdx}{(z^{2}+x^{2})^{3}}=\frac{2\pi C\rho_{1}}{4}\int_{D}^{\infty}\frac{dz}{z^{% 4}}=-\pi C\rho_{1}/6D^{3}
  14. ρ 1 \rho_{1}
  15. W ( D ) = - 2 π C ρ 1 ρ 2 12 z = 0 z = 2 R ( 2 R - z ) z d z ( D + z ) 3 - π 2 C ρ 1 ρ 2 R 6 D W(D)=-\frac{2\pi C\rho_{1}\rho_{2}}{12}\int_{z=0}^{z=2R}\frac{(2R-z)zdz}{(D+z)% ^{3}}\approx-\frac{\pi^{2}C\rho_{1}\rho_{2}R}{6D}
  16. ρ 2 \rho_{2}
  17. A = ( π 2 C ρ 1 ρ 2 ) A=\left(\pi^{2}C\rho_{1}\rho_{2}\right)
  18. W ( D ) = - A R 6 D W(D)=-\frac{AR}{6D}
  19. W ( D ) = - A 6 D R 1 R 2 ( R 1 + R 2 ) W(D)=-\frac{A}{6D}\frac{R_{1}R_{2}}{(R_{1}+R_{2})}
  20. W ( D ) = - A R 6 D W(D)=-\frac{AR}{6D}
  21. W ( D ) = - A 12 π D 2 W(D)=-\frac{A}{12\pi D^{2}}
  22. 1 / κ 1/\kappa
  23. κ = ( i ρ i e 2 z i 2 / ϵ r ϵ 0 k B T ) 1 / 2 \kappa=(\sum_{i}\rho_{\infty i}e^{2}z^{2}_{i}/\epsilon_{r}\epsilon_{0}k_{B}T)^% {1/2}
  24. ρ i \rho_{\infty i}
  25. ε 0 \varepsilon_{0}
  26. ϵ r \epsilon_{r}
  27. W = ( 64 k B T ρ γ 2 / κ ) e - κ D W=(64k_{B}T\rho_{\infty}\gamma^{2}/\kappa)e^{-\kappa D}
  28. γ \gamma
  29. γ = tanh ( z e ψ 0 4 k T ) \gamma=\tanh\left(\frac{ze\psi_{0}}{4kT}\right)
  30. ψ 0 \psi_{0}
  31. W = ( 64 π k B T R ρ γ 2 / κ 2 ) e - κ D W=(64\pi k_{B}TR\rho_{\infty}\gamma^{2}/\kappa^{2})e^{-\kappa D}
  32. W ( D ) = W ( D ) A + W ( D ) R W\left(D\right)=W(D)_{A}+W(D)_{R}\,

Dobinski's_formula.html

  1. 1 e k = 0 k n k ! . {1\over e}\sum_{k=0}^{\infty}{k^{n}\over k!}.
  2. x = 0 x=0
  3. 1 e k = x k n ( k - x ) ! = k = 0 n ( n k ) B k x n - k {1\over e}\sum_{k=x}^{\infty}{k^{n}\over(k-x)!}=\sum_{k=0}^{n}{n\choose k}B_{k% }x^{n-k}
  4. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) (x)_{n}=x(x-1)(x-2)\cdots(x-n+1)\,
  5. π ( x ) | π | , \sum_{\pi}(x)_{|\pi|},\,
  6. x n = π ( x ) | π | . x^{n}=\sum_{\pi}(x)_{|\pi|}.\,
  7. E ( X n ) = π E ( ( X ) | π | ) . E(X^{n})=\sum_{\pi}E((X)_{|\pi|}).\,
  8. E ( X n ) = π 1 , E(X^{n})=\sum_{\pi}1,\,
  9. n m n ! \textstyle\sum\frac{n^{m}}{n!}

Dodecagonal_number.html

  1. 5 n 2 - 4 n ; n > 0 5n^{2}-4n;n>0
  2. D n = n 2 + 4 ( n 2 - n ) D_{n}=n^{2}+4(n^{2}-n)

Dodgson_condensation.html

  1. i , j 1 , n i,j\neq 1,n
  2. b i , j = | a i , j a i , j + 1 a i + 1 , j a i + 1 , j + 1 | . b_{i,j}=\begin{vmatrix}a_{i,j}&a_{i,j+1}\\ a_{i+1,j}&a_{i+1,j+1}\end{vmatrix}.
  3. c i , j = | b i , j b i , j + 1 b i + 1 , j b i + 1 , j + 1 | / a i + 1 , j + 1 c_{i,j}=\begin{vmatrix}b_{i,j}&b_{i,j+1}\\ b_{i+1,j}&b_{i+1,j+1}\end{vmatrix}/a_{i+1,j+1}
  4. | - 2 - 1 - 1 - 4 - 1 - 2 - 1 - 6 - 1 - 1 2 4 2 1 - 3 - 8 | . \begin{vmatrix}-2&-1&-1&-4\\ -1&-2&-1&-6\\ -1&-1&2&4\\ 2&1&-3&-8\end{vmatrix}.
  5. [ | - 2 - 1 - 1 - 2 | | - 1 - 1 - 2 - 1 | | - 1 - 4 - 1 - 6 | | - 1 - 2 - 1 - 1 | | - 2 - 1 - 1 2 | | - 1 - 6 2 4 | | - 1 - 1 2 1 | | - 1 2 1 - 3 | | 2 4 - 3 - 8 | ] = [ 3 - 1 2 - 1 - 5 8 1 1 - 4 ] . \begin{bmatrix}\begin{vmatrix}-2&-1\\ -1&-2\end{vmatrix}&\begin{vmatrix}-1&-1\\ -2&-1\end{vmatrix}&\begin{vmatrix}-1&-4\\ -1&-6\end{vmatrix}\\ \\ \begin{vmatrix}-1&-2\\ -1&-1\end{vmatrix}&\begin{vmatrix}-2&-1\\ -1&2\end{vmatrix}&\begin{vmatrix}-1&-6\\ 2&4\end{vmatrix}\\ \\ \begin{vmatrix}-1&-1\\ 2&1\end{vmatrix}&\begin{vmatrix}-1&2\\ 1&-3\end{vmatrix}&\begin{vmatrix}2&4\\ -3&-8\end{vmatrix}\end{bmatrix}=\begin{bmatrix}3&-1&2\\ -1&-5&8\\ 1&1&-4\end{bmatrix}.
  6. [ | 3 - 1 - 1 - 5 | | - 1 2 - 5 8 | | - 1 - 5 1 1 | | - 5 8 1 - 4 | ] = [ - 16 2 4 12 ] . \begin{bmatrix}\begin{vmatrix}3&-1\\ -1&-5\end{vmatrix}&\begin{vmatrix}-1&2\\ -5&8\end{vmatrix}\\ \\ \begin{vmatrix}-1&-5\\ 1&1\end{vmatrix}&\begin{vmatrix}-5&8\\ 1&-4\end{vmatrix}\end{bmatrix}=\begin{bmatrix}-16&2\\ 4&12\end{bmatrix}.
  7. [ - 2 - 1 - 1 2 ] \begin{bmatrix}-2&-1\\ -1&2\end{bmatrix}
  8. [ 8 - 2 - 4 6 ] \begin{bmatrix}8&-2\\ -4&6\end{bmatrix}
  9. [ | 8 - 2 - 4 6 | ] = [ 40 ] . \begin{bmatrix}\begin{vmatrix}8&-2\\ -4&6\end{vmatrix}\end{bmatrix}=\begin{bmatrix}40\end{bmatrix}.
  10. [ - 8 ] \begin{bmatrix}-8\end{bmatrix}
  11. [ 2 - 1 2 1 - 3 1 2 1 - 1 2 1 - 1 - 2 - 1 - 1 2 1 - 1 - 2 - 1 1 - 2 - 1 - 1 2 ] [ 5 - 5 - 3 - 1 - 3 - 3 - 3 3 3 3 3 - 1 - 5 - 3 - 1 - 5 ] [ - 30 6 - 12 0 0 6 6 - 6 8 ] . \begin{bmatrix}2&-1&2&1&-3\\ 1&2&1&-1&2\\ 1&-1&-2&-1&-1\\ 2&1&-1&-2&-1\\ 1&-2&-1&-1&2\end{bmatrix}\to\begin{bmatrix}5&-5&-3&-1\\ -3&-3&-3&3\\ 3&3&3&-1\\ -5&-3&-1&-5\end{bmatrix}\to\begin{bmatrix}-30&6&-12\\ 0&0&6\\ 6&-6&8\end{bmatrix}.
  12. [ 1 2 1 - 1 2 1 - 1 - 2 - 1 - 1 2 1 - 1 - 2 - 1 1 - 2 - 1 - 1 2 2 - 1 2 1 - 3 ] [ - 3 - 3 - 3 3 3 3 3 - 1 - 5 - 3 - 1 - 5 3 - 5 1 1 ] [ 0 0 6 6 - 6 8 - 17 8 - 4 ] [ 0 12 18 40 ] [ 36 ] . \begin{bmatrix}1&2&1&-1&2\\ 1&-1&-2&-1&-1\\ 2&1&-1&-2&-1\\ 1&-2&-1&-1&2\\ 2&-1&2&1&-3\end{bmatrix}\to\begin{bmatrix}-3&-3&-3&3\\ 3&3&3&-1\\ -5&-3&-1&-5\\ 3&-5&1&1\end{bmatrix}\to\begin{bmatrix}0&0&6\\ 6&-6&8\\ -17&8&-4\end{bmatrix}\to\begin{bmatrix}0&12\\ 18&40\end{bmatrix}\to\begin{bmatrix}36\end{bmatrix}.
  13. M = ( m i , j ) i , j = 1 k M=(m_{i,j})_{i,j=1}^{k}
  14. 1 i , j k 1\leq i,j\leq k
  15. M i j M_{i}^{j}
  16. M M
  17. i i
  18. j j
  19. 1 i , j , p , q k 1\leq i,j,p,q\leq k
  20. M i , j p , q M_{i,j}^{p,q}
  21. M M
  22. i i
  23. j j
  24. p p
  25. q q
  26. det ( M ) det ( M 1 , k 1 , k ) = det ( M 1 1 ) det ( M k k ) - det ( M 1 k ) det ( M k 1 ) . \det(M)\det(M_{1,k}^{1,k})=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(% M_{k}^{1}).
  27. det ( M ) = det ( M 1 1 ) det ( M k k ) - det ( M 1 k ) det ( M k 1 ) det ( M 1 , k 1 , k ) . \det(M)=\frac{\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1})}{% \det(M_{1,k}^{1,k})}.
  28. A A
  29. n n
  30. k k
  31. k = 1 k=1
  32. A A
  33. k k
  34. A A
  35. k × k k\times k
  36. A A
  37. k = n k=n
  38. n n
  39. A A
  40. a i , j = ( - 1 ) i + j det ( M i j ) a_{i,j}=(-1)^{i+j}\det(M_{i}^{j})
  41. ( i , j ) (i,j)
  42. M M
  43. k × k k\times k
  44. M M^{\prime}
  45. M = ( a 1 , 1 0 0 0 0 a k , 1 a 1 , 2 1 0 0 0 a k , 2 a 1 , 3 0 1 0 0 a k , 3 a 1 , 4 0 0 1 0 a k , 4 a 1 , k - 1 0 0 0 1 a k , k - 1 a 1 , k 0 0 0 0 a k , k ) . M^{\prime}=\begin{pmatrix}a_{1,1}&0&0&0&\ldots&0&a_{k,1}\\ a_{1,2}&1&0&0&\ldots&0&a_{k,2}\\ a_{1,3}&0&1&0&\ldots&0&a_{k,3}\\ a_{1,4}&0&0&1&\ldots&0&a_{k,4}\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\ a_{1,k-1}&0&0&0&\ldots&1&a_{k,k-1}\\ a_{1,k}&0&0&0&\ldots&0&a_{k,k}\end{pmatrix}.
  46. M M^{\prime}
  47. A A
  48. det ( M M ) \det(MM^{\prime})
  49. M M MM^{\prime}
  50. M M = ( det ( M ) m 1 , 2 m 1 , 3 m 1 , k - 1 0 0 m 2 , 2 m 2 , 3 m 2 , k - 1 0 0 m 3 , 2 m 3 , 3 m 3 , k - 1 0 0 m k - 1 , 2 m k - 1 , 3 m k - 1 , k - 1 0 0 m k , 2 m k , 3 m k , k - 1 det ( M ) ) MM^{\prime}=\begin{pmatrix}\det(M)&m_{1,2}&m_{1,3}&\ldots&m_{1,k-1}&0\\ 0&m_{2,2}&m_{2,3}&\ldots&m_{2,k-1}&0\\ 0&m_{3,2}&m_{3,3}&\ldots&m_{3,k-1}&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots&\vdots\\ 0&m_{k-1,2}&m_{k-1,3}&\ldots&m_{k-1,k-1}&0\\ 0&m_{k,2}&m_{k,3}&\ldots&m_{k,k-1}&\det(M)\end{pmatrix}
  51. m i , j m_{i,j}
  52. ( i , j ) (i,j)
  53. M M
  54. det ( M ) 2 det ( M 1 , k 1 , k ) \det(M)^{2}\cdot\det(M_{1,k}^{1,k})
  55. det ( M ) det ( M ) \det(M)\cdot\det(M^{\prime})
  56. det ( M ) = a 1 , 1 a k , k - a k , 1 a 1 , k = det ( M 1 1 ) det ( M k k ) - det ( M 1 k ) det ( M k 1 ) , \det(M^{\prime})=a_{1,1}a_{k,k}-a_{k,1}a_{1,k}=\det(M_{1}^{1})\det(M_{k}^{k})-% \det(M_{1}^{k})\det(M_{k}^{1}),
  57. det ( M M ) \det(MM^{\prime})
  58. det ( M ) \det(M)
  59. k 2 k^{2}
  60. ( m i , j ) i , j = 1 k (m_{i,j})_{i,j=1}^{k}
  61. 1 , 2 , 7 , 42 , 429 , 7436 , 1,2,7,42,429,7436,\cdots

Dolby_Stereo.html

  1. 1 1
  2. 0
  3. 1 2 \frac{1}{\sqrt{2}}
  4. + j 1 2 +j\frac{1}{\sqrt{2}}
  5. 0
  6. 1 1
  7. 1 2 \frac{1}{\sqrt{2}}
  8. - j 1 2 -j\frac{1}{\sqrt{2}}

Dolby_Surround.html

  1. 1 1
  2. 0
  3. 2 2 \frac{\sqrt{2}}{2}
  4. j 2 2 j\frac{\sqrt{2}}{2}
  5. 0
  6. 1 1
  7. 2 2 \frac{\sqrt{2}}{2}
  8. - j 2 2 -j\frac{\sqrt{2}}{2}
  9. π / 2 {π}/{2}

Domatic_number.html

  1. G = ( V , E ) G=(V,E)
  2. V V
  3. V 1 V_{1}
  4. V 2 V_{2}
  5. V K V_{K}
  6. V 1 V_{1}
  7. V 2 V_{2}
  8. V 3 V_{3}
  9. δ \delta
  10. G G
  11. G G
  12. δ + 1 \delta+1
  13. v v
  14. δ \delta
  15. N N
  16. v v
  17. V i V_{i}
  18. N N
  19. N N
  20. V i V_{i}
  21. | N | = δ + 1 |N|=\delta+1
  22. δ = 2 \delta=2
  23. δ \delta
  24. V 1 = V V_{1}=V
  25. G G
  26. K K
  27. G G
  28. K K
  29. O ( log | V | ) O(\log|V|)
  30. ( 1 - ϵ ) ln | V | (1-\epsilon)\ln|V|
  31. ϵ > 0 \epsilon>0
  32. n O ( log log n ) n^{O(\log\log n)}

Dot_gain.html

  1. D G = a print - a form DG=a_{\,\text{print}}-a_{\,\text{form}}
  2. a print a_{\,\text{print}}
  3. a form a_{\,\text{form}}
  4. n n
  5. gain 𝑇𝐸 = a form ( 1 - a 𝑣𝑓 ) \mathrm{gain}_{\mathit{TE}}=a_{\mathrm{form}}\cdot(1-a_{\mathit{vf}})
  6. a 𝑣𝑓 a_{\mathit{vf}}
  7. gain 𝐺𝑅𝐿 = { a form - a 𝑤𝑓 , for a form a 𝑤𝑓 2 Δ 0 , 50 a form ( 1 - a form ) , for a 𝑤𝑓 < a form < a 𝑣𝑓 a form - a 𝑣𝑓 , for a form a 𝑣𝑓 \mathrm{gain}_{\mathit{GRL}}=\left\{\begin{array}[]{ll}a_{\mathrm{form}}-a_{% \mathit{wf}},&\mathrm{for}\ a_{\mathrm{form}}\leq a_{\mathit{wf}}\\ \\ 2\cdot\Delta_{0,50}\sqrt{a_{\mathrm{form}}(1-a_{\mathrm{form}})},&\mathrm{for}% \ a_{\mathit{wf}}<a_{\mathrm{form}}<a_{\mathit{vf}}\\ \\ a_{\mathrm{form}}-a_{\mathit{vf}},&\mathrm{for}\ a_{\mathrm{form}}\geq a_{% \mathit{vf}}\end{array}\right.
  8. Δ 0 , 50 \Delta_{0,50}
  9. a w f a_{wf}
  10. a 𝑤𝑓 = { 4 Δ 0 , 50 2 1 + 4 Δ 0 , 50 2 , for Δ 0 , 50 < 0 0 , for Δ 0 , 50 0 a_{\mathit{wf}}=\left\{\begin{array}[]{ll}\frac{4\Delta_{0,50}^{2}}{1+4\Delta_% {0,50}^{2}},&\mathrm{for}\ \Delta_{0,50}<0\\ \\ 0,&\mathrm{for}\ \Delta_{0,50}\geq 0\end{array}\right.
  11. a v f a_{vf}
  12. a 𝑣𝑓 = { 1 , for Δ 0 , 50 0 1 1 + 4 Δ 0 , 50 2 , for Δ 0 , 50 > 0 a_{\mathit{vf}}=\left\{\begin{array}[]{ll}1,&\mathrm{for}\ \Delta_{0,50}\leq 0% \\ \\ \frac{1}{1+4\Delta_{0,50}^{2}},&\mathrm{for}\ \Delta_{0,50}>0\end{array}\right.
  13. Δ 0 , 50 = 0 \Delta_{0,50}=0
  14. a 𝑤𝑓 a_{\mathit{wf}}
  15. a 𝑣𝑓 a_{\mathit{vf}}
  16. Δ 0 , 50 \Delta_{0,50}

Double_beta_decay.html

  1. × 10 2 1 \times 10^{2}1
  2. ± \pm
  3. ± \pm
  4. ± \pm
  5. ± \pm
  6. ± \pm
  7. Γ = G | M | 2 | m β β | 2 , \Gamma=~{}~{}~{}~{}{G|M|^{2}|m_{\beta\beta}|^{2}},
  8. G G
  9. M M
  10. m β β = i = 1 3 m i U e i 2 . m_{\beta\beta}=\sum_{i=1}^{3}m_{i}U^{2}_{ei}.
  11. × 10 2 5 \times 10^{2}5
  12. × 10 2 5 \times 10^{2}5
  13. × 10 2 5 \times 10^{2}5
  14. × 10 2 5 \times 10^{2}5

Double_hashing.html

  1. h 1 h_{1}
  2. h 2 h_{2}
  3. T T
  4. h ( i , k ) = ( h 1 ( k ) + i h 2 ( k ) ) mod | T | . h(i,k)=(h_{1}(k)+i\cdot h_{2}(k))\mod|T|.
  5. h 1 h_{1}
  6. h 2 h_{2}
  7. T T
  8. n n
  9. T T
  10. T T
  11. α = n | T | \alpha=\frac{n}{|T|}
  12. h 1 h_{1}
  13. h 2 h_{2}
  14. T T
  15. T T
  16. h 1 h_{1}
  17. h 2 h_{2}
  18. k k
  19. ( i + 1 ) (i+1)
  20. h ( i , k ) = ( h 1 ( k ) + i h 2 ( k ) ) mod | T | . h(i,k)=(h_{1}(k)+i\cdot h_{2}(k))\mod|T|.
  21. T T
  22. α : 1 > α > 0 \alpha:1>\alpha>0
  23. T T
  24. 1 1 - α \frac{1}{1-\alpha}
  25. 1 1 - α \frac{1}{1-\alpha}
  26. α < 0.319 \alpha<0.319
  27. k k
  28. k = c log n k=c\log n
  29. c c
  30. h 2 ( k ) = 5 - ( k mod 7 ) h_{2}(k)=5-(k\mod 7)
  31. h 2 ( k ) = ( k mod 7 ) + 1 h_{2}(k)=(k\mod 7)+1

Double_or_Quits.html

  1. A * 2 = 2 A*2=2
  2. 2 * 2 = 4 2*2=4
  3. 4 * 2 = 8 4*2=8
  4. 8 * 2 = 16 ; 16 - 13 = 3 8*2=16;16-13=3
  5. 3 * 2 = 6 3*2=6
  6. 6 * 2 = 12 = Q 6*2=12=Q
  7. Q * 2 = 24 ; 24 - 13 = 11 = J Q*2=24;24-13=11=J
  8. J * 2 = 22 ; 22 - 13 = 9 J*2=22;22-13=9
  9. 9 * 2 = 18 ; 18 - 13 = 5 9*2=18;18-13=5
  10. 5 * 2 = 10 5*2=10
  11. 10 * 2 = 20 ; 20 - 13 = 7 10*2=20;20-13=7
  12. 7 * 2 = 14 ; 14 - 13 = 1 = A 7*2=14;14-13=1=A

Doublet–triplet_splitting_problem.html

  1. E 6 E_{6}
  2. S U ( 2 ) SU(2)
  3. μ \mu
  4. d 2 θ λ H 5 ¯ Σ H 5 + μ H 5 ¯ H 5 \int d^{2}\theta\;\lambda H_{\bar{5}}\Sigma H_{5}+\mu H_{\bar{5}}H_{5}
  5. Σ \Sigma
  6. Σ \Sigma
  7. Σ = diag ( 2 , 2 , 2 , - 3 , - 3 ) f \langle\Sigma\rangle=\rm{diag}(2,2,2,-3,-3)f
  8. d 2 θ ( 2 λ f + μ ) H 3 ¯ H 3 + ( - 3 λ f + μ ) H 2 ¯ H 2 \int d^{2}\theta\;(2\lambda f+\mu)H_{\bar{3}}H_{3}+(-3\lambda f+\mu)H_{\bar{2}% }H_{2}
  9. f f
  10. 10 16 10^{16}
  11. μ 3 λ f ± 100 GeV \mu\sim 3\lambda f\pm 100\mbox{GeV}~{}
  12. 10 14 10^{14}
  13. Σ \Sigma
  14. Σ = diag ( i σ 2 f 3 , i σ 2 f 3 , i σ 2 f 3 , i σ 2 f 2 , i σ 2 f 2 ) \langle\Sigma\rangle=\mbox{diag}~{}(i\sigma_{2}f_{3},i\sigma_{2}f_{3},i\sigma_% {2}f_{3},i\sigma_{2}f_{2},i\sigma_{2}f_{2})
  15. f 2 f_{2}
  16. f 3 f_{3}
  17. Σ \Sigma
  18. f 2 = 0 f_{2}=0
  19. 5 ( 1 , 2 ) 1 2 ( 3 , 1 ) - 1 3 5\rightarrow(1,2)_{1\over 2}\oplus(3,1)_{-{1\over 3}}
  20. 5 ¯ ( 1 , 2 ) - 1 2 ( 3 ¯ , 1 ) 1 3 \bar{5}\rightarrow(1,2)_{-{1\over 2}}\oplus(\bar{3},1)_{1\over 3}
  21. 10 ( 1 , 2 ) 1 2 ( 1 , 2 ) - 1 2 ( 3 , 1 ) - 1 3 ( 3 ¯ , 1 ) 1 3 10\rightarrow(1,2)_{1\over 2}\oplus(1,2)_{-{1\over 2}}\oplus(3,1)_{-{1\over 3}% }\oplus(\bar{3},1)_{1\over 3}
  22. T ( 3 , 1 ) - 1 3 T(3,1)_{-\frac{1}{3}}
  23. T ¯ ( 3 ¯ , 1 ) 1 3 \bar{T}(\bar{3},1)_{\frac{1}{3}}
  24. S U ( 5 ) SU(5)

Drawdown_(economics).html

  1. X = ( X ( t ) , t 0 ) X=(X(t),t\geq 0)
  2. D ( T ) D(T)
  3. D ( T ) = max { 0 , max t ( 0 , T ) X ( t ) - X ( T ) } D(T)=\max\left\{0,\max_{t\in(0,T)}X(t)-X(T)\right\}
  4. T T
  5. MDD ( T ) = max τ ( 0 , T ) [ max t ( 0 , τ ) X ( t ) - X ( τ ) ] \,\text{MDD}(T)=\max_{\tau\in(0,T)}\left[\max_{t\in(0,\tau)}X(t)-X(\tau)\right]
  6. X ( t ) = μ t + σ W ( t ) , X(t)=\mu t+\sigma W(t),
  7. W ( t ) W(t)
  8. μ > 0 \mu>0
  9. μ = 0 \mu=0
  10. μ < 0 \mu<0

Dual_code.html

  1. C 𝔽 q n C\subset\mathbb{F}_{q}^{n}
  2. C = { x 𝔽 q n x , c = 0 c C } C^{\perp}=\{x\in\mathbb{F}_{q}^{n}\mid\langle x,c\rangle=0\;\forall c\in C\}
  3. x , c = i = 1 n x i c i \langle x,c\rangle=\sum_{i=1}^{n}x_{i}c_{i}
  4. dim C + dim C = n . \dim C+\dim C^{\perp}=n.
  5. c > 1 c>1

Dual_pair.html

  1. ( X , Y , , ) (X,Y,\langle,\rangle)
  2. X X
  3. Y Y
  4. F F
  5. , : X × Y F \langle,\rangle:X\times Y\to F
  6. x X { 0 } y Y : x , y 0 \forall x\in X\setminus\{0\}\quad\exists y\in Y:\langle x,y\rangle\neq 0
  7. y Y { 0 } x X : x , y 0 \forall y\in Y\setminus\{0\}\quad\exists x\in X:\langle x,y\rangle\neq 0
  8. , \langle,\rangle
  9. X X
  10. Y Y
  11. X X
  12. X * X^{*}
  13. , : X * × X F : ( φ , x ) φ ( x ) \langle\cdot,\cdot\rangle:X^{*}\times X\rightarrow F:(\varphi,x)\mapsto\varphi% (x)
  14. x X x\in X
  15. y Y y\in Y
  16. x , y = 0. \langle x,y\rangle=0.
  17. M X M\subseteq X
  18. N Y N\subseteq Y
  19. M M
  20. N N
  21. V V
  22. V * V^{*}
  23. x , f := f ( x ) x V , f V * \langle x,f\rangle:=f(x)\qquad x\in V\mbox{ , }~{}f\in V^{*}
  24. E E
  25. E E^{\prime}
  26. x , f := f ( x ) x E , f E \langle x,f\rangle:=f(x)\qquad x\in E\mbox{ , }~{}f\in E^{\prime}
  27. ( X , Y , , ) (X,Y,\langle,\rangle)
  28. ( Y , X , , ) (Y,X,\langle,\rangle^{\prime})
  29. , : ( y , x ) x , y \langle,\rangle^{\prime}:(y,x)\to\langle x,y\rangle
  30. E E
  31. E β E^{\beta}
  32. x , y := i = 1 x i y i x E , y E β \langle x,y\rangle:=\sum_{i=1}^{\infty}x_{i}y_{i}\quad x\in E,y\in E^{\beta}
  33. ( X , Y , , ) (X,Y,\langle,\rangle)
  34. X X
  35. Y * Y^{*}
  36. x ( y x , y ) x\mapsto(y\mapsto\langle x,y\rangle)
  37. Y Y
  38. X * X^{*}
  39. X X
  40. Y Y

Dual_representation.html

  1. G G
  2. ρ ρ
  3. V V
  4. ρ * ρ*
  5. V * V*
  6. ρ * ( g ) ρ*(g)
  7. ρ * ( g ) ρ*(g)
  8. g G g∈G
  9. 𝐠 \mathbf{g}
  10. π π
  11. V V
  12. π * π*
  13. V * V*
  14. π * ( X ) π*(X)
  15. X 𝐠 X∈\mathbf{g}
  16. V V
  17. V * V*
  18. V V
  19. V * V*
  20. φ φ
  21. v v
  22. φ ( v ) φ(v)
  23. φ , v φ ( v ) = φ T v \langle\varphi,v\rangle\equiv\varphi(v)=\varphi^{T}v
  24. T T
  25. ρ * ( g ) φ , ρ ( g ) v = φ , v . \langle{\rho}^{*}(g)\varphi,\rho(g)v\rangle=\langle\varphi,v\rangle.
  26. ρ * ( g ) φ , ρ ( g ) v = ρ ( g - 1 ) T φ , ρ ( g ) v = ( ρ ( g - 1 ) T φ ) T ρ ( g ) v = φ T ρ ( g - 1 ) ρ ( g ) v = φ T v = φ , v \langle{\rho}^{*}(g)\varphi,\rho(g)v\rangle=\langle\rho(g^{-1})^{T}\varphi,% \rho(g)v\rangle=(\rho(g^{-1})^{T}\varphi)^{T}\rho(g)v=\varphi^{T}\rho(g^{-1})% \rho(g)v=\varphi^{T}v=\langle\varphi,v\rangle
  27. Π Π
  28. π π
  29. π ( X ) = d d t Π ( e t X ) | t = 0 . \pi(X)=\frac{d}{dt}\Pi(e^{tX})|_{t=0}.
  30. Π * Π*
  31. Π Π
  32. π * π*
  33. π * ( X ) = d d t Π * ( e t X ) | t = 0 = d d t Π ( e - t X ) T | t = 0 = - π ( X ) T . \pi^{*}(X)=\frac{d}{dt}\Pi^{*}(e^{tX})|_{t=0}=\frac{d}{dt}\Pi(e^{-tX})^{T}|_{t% =0}=-\pi(X)^{T}.

Dual_topology.html

  1. ( X , Y , , ) (X,Y,\langle,\rangle)
  2. X X
  3. τ \tau
  4. ( X , τ ) Y . (X,\tau)^{\prime}\simeq Y.
  5. ( X , τ ) (X,\tau)^{\prime}
  6. ( X , τ ) (X,\tau)
  7. ( X , τ ) Y (X,\tau)^{\prime}\simeq Y
  8. Ψ : Y ( X , τ ) , y ( x x , y ) . \Psi:Y\to(X,\tau)^{\prime},\quad y\mapsto(x\mapsto\langle x,y\rangle).
  9. τ \tau
  10. X X
  11. Ψ \Psi
  12. x x , y x\mapsto\langle x,y\rangle
  13. X X
  14. y y
  15. X X^{\prime}
  16. X X^{\prime}
  17. ( X , X ) (X,X^{\prime})
  18. X X
  19. X X^{\prime}
  20. τ \tau
  21. X X
  22. X X^{\prime}

DuPont_analysis.html

  1. ROA = Net income Sales × Sales Total assets = Net income Total assets \,\text{ROA}=\frac{\,\text{Net income}}{\,\text{Sales}}\times\frac{\,\text{% Sales}}{\,\text{Total assets}}=\frac{\,\text{Net income}}{\,\text{Total assets}}
  2. ROE = Net income Equity = Net income Pretax income × Pretax income EBIT × EBIT Sales × Sales Assets × Assets Equity \,\text{ROE}=\frac{\,\text{Net income}}{\,\text{Equity}}=\frac{\,\text{Net % income}}{\,\text{Pretax income}}\times\frac{\,\text{Pretax income}}{\,\text{% EBIT}}\times\frac{\,\text{EBIT}}{\,\text{Sales}}\times\frac{\,\text{Sales}}{\,% \text{Assets}}\times\frac{\,\text{Assets}}{\,\text{Equity}}
  3. ROE = Net income Sales × Sales Assets × Assets Equity \,\text{ROE}=\frac{\,\text{Net income}}{\,\text{Sales}}\times\frac{\,\text{% Sales}}{\,\text{Assets}}\times\frac{\,\text{Assets}}{\,\text{Equity}}

Duration_gap.html

  1. D u r a t i o n g a p = d u r a t i o n o f e a r n i n g a s s e t s - d u r a t i o n o f p a y i n g l i a b i l i t i e s × p a y i n g l i a b i l i t i e s e a r n i n g a s s e t s Duration\ gap=duration\ of\ earning\ assets\ -\ duration\ of\ paying\ % liabilities\ \times\ \frac{paying\ liabilities}{earning\ assets}
  2. 0 = 1 - 2 × 1 , 000 , 000 2 , 000 , 000 0=1-2\times\frac{1,000,000}{2,000,000}

Dyadic_transformation.html

  1. d : [ 0 , 1 ) [ 0 , 1 ) d:[0,1)\to[0,1)^{\infty}
  2. x ( x 0 , x 1 , x 2 , ) x\mapsto(x_{0},x_{1},x_{2},\ldots)
  3. x 0 = x x_{0}=x
  4. n 0 , x n + 1 = ( 2 x n ) mod 1 \forall n\geq 0,x_{n+1}=(2x_{n})\bmod 1
  5. f ( x ) = { 2 x 0 x < 0.5 2 x - 1 0.5 x < 1. f(x)=\begin{cases}2x&0\leq x<0.5\\ 2x-1&0.5\leq x<1.\end{cases}
  6. z n + 1 = 4 z n ( 1 - z n ) z_{n+1}=4z_{n}(1-z_{n})
  7. z n = sin 2 ( 2 π x n ) . z_{n}=\sin^{2}(2\pi x_{n}).
  8. 11 24 11 12 5 6 2 3 1 3 2 3 1 3 , \frac{11}{24}\mapsto\frac{11}{12}\mapsto\frac{5}{6}\mapsto\frac{2}{3}\mapsto% \frac{1}{3}\mapsto\frac{2}{3}\mapsto\frac{1}{3}\mapsto\cdots,
  9. 2 - n 2^{-n}

Dyck_language.html

  1. Σ = { [ , ] } \Sigma=\{[,]\}
  2. Σ * \Sigma^{*}
  3. u Σ * u\in\Sigma^{*}
  4. | u | |u|
  5. i n s e r t : Σ * × { 0 } Σ * insert:\Sigma^{*}\times\mathbb{N}\cup\{0\}\rightarrow\Sigma^{*}
  6. d e l e t e : Σ * × Σ * delete:\Sigma^{*}\times\mathbb{N}\rightarrow\Sigma^{*}
  7. i n s e r t ( u , j ) insert(u,j)
  8. u u
  9. [ ] []
  10. j j
  11. d e l e t e ( u , j ) delete(u,j)
  12. u u
  13. [ ] []
  14. j j
  15. i n s e r t ( u , j ) insert(u,j)
  16. j > | u | j>|u|
  17. d e l e t e ( u , j ) delete(u,j)
  18. j > | u | - 2 j>|u|-2
  19. R R
  20. Σ * \Sigma^{*}
  21. a , b Σ * a,b\in\Sigma^{*}
  22. ( a , b ) R (a,b)\in R
  23. i n s e r t insert
  24. d e l e t e delete
  25. a a
  26. b b
  27. R R
  28. i n s e r t insert
  29. d e l e t e delete
  30. Σ * \Sigma^{*}
  31. ϵ \epsilon
  32. Cl ( ϵ ) \operatorname{Cl}(\epsilon)
  33. i m b a l a n c e : Σ * ( { 0 } ) imbalance:\Sigma^{*}\rightarrow\left(\mathbb{N}\cup\{0\}\right)
  34. i m b a l a n c e ( u ) = | u | [ - | u | ] imbalance(u)=|u|_{[}-|u|_{]}
  35. u Σ * u\in\Sigma^{*}
  36. | u | [ |u|_{[}
  37. | u | ] |u|_{]}
  38. u u
  39. i m b a l a n c e imbalance
  40. i m b a l a n c e ( u ) imbalance(u)
  41. u u
  42. { u Σ * | i m b a l a n c e ( u ) = 0 and i m b a l a n c e ( v ) 0 for all prefixes v of u } \{u\in\Sigma^{*}|imbalance(u)=0\,\text{ and }imbalance(v)\geq 0\,\text{ for % all prefixes }v\,\text{ of }u\}
  43. Σ * \Sigma^{*}
  44. Σ * / R \Sigma^{*}/R
  45. Cl ( ϵ ) \operatorname{Cl}(\epsilon)
  46. 1 1
  47. u = Cl ( [ ) u=\operatorname{Cl}([)
  48. v = Cl ( ] ) v=\operatorname{Cl}(])
  49. u v = Cl ( [ ] ) = 1 Cl ( ] [ ) = v u uv=\operatorname{Cl}([])=1\neq\operatorname{Cl}(][)=vu
  50. u v = 1 uv=1
  51. u u
  52. v v
  53. Σ * / R \Sigma^{*}/R
  54. Cl ( [ ) \operatorname{Cl}([)
  55. Cl ( ] ) \operatorname{Cl}(])
  56. T C 0 TC^{0}

Dynamometer_car.html

  1. P = F d t P=\frac{F\cdot d}{t}
  2. P = F d t = F V P=F\cdot\frac{d}{t}=F\cdot V
  3. P = 50 , 000 l b 30 m i h r 5280 f t m i h r 3600 s = 2 , 200 , 000 f t l b s P=50,000lb\cdot\frac{30mi}{hr}\cdot\frac{5280ft}{mi}\cdot\frac{hr}{3600s}=2,20% 0,000\frac{ft\cdot lb}{s}
  4. P = 2 , 200 , 000 f t l b s 1 h p 550 f t l b / s = 4 , 000 h p P=2,200,000\frac{ft\cdot lb}{s}\cdot\frac{1hp}{550ft\cdot lb/s}=4,000hp

Dyson_series.html

  1. H H
  2. H H
  3. V V
  4. ħ ħ
  5. U U
  6. Ψ ( t ) = U ( t , t 0 ) Ψ ( t 0 ) \Psi(t)=U(t,t_{0})\Psi(t_{0})
  7. U ( t , t ) = I , U(t,t)=I,
  8. U ( t , t 0 ) = U ( t , t 1 ) U ( t 1 , t 0 ) , U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}),
  9. U - 1 ( t , t 0 ) = U ( t 0 , t ) , U^{-1}(t,t_{0})=U(t_{0},t),
  10. i d d t U ( t , t 0 ) Ψ ( t 0 ) = V ( t ) U ( t , t 0 ) Ψ ( t 0 ) . i\frac{d}{dt}U(t,t_{0})\Psi(t_{0})=V(t)U(t,t_{0})\Psi(t_{0}).
  11. U ( t , t 0 ) = 1 - i t 0 t d t 1 V ( t 1 ) U ( t 1 , t 0 ) . U(t,t_{0})=1-i\int_{t_{0}}^{t}{dt_{1}\ V(t_{1})U(t_{1},t_{0})}.
  12. U ( t , t 0 ) = 1 - i t 0 t d t 1 V ( t 1 ) + ( - i ) 2 t 0 t d t 1 t 0 t 1 d t 2 V ( t 1 ) V ( t 2 ) + + ( - i ) n t 0 t d t 1 t 0 t 1 d t 2 t 0 t n - 1 d t n V ( t 1 ) V ( t 2 ) V ( t n ) + . \begin{array}[]{lcl}U(t,t_{0})&=&1-i\int_{t_{0}}^{t}{dt_{1}V(t_{1})}+(-i)^{2}% \int_{t_{0}}^{t}{dt_{1}\int_{t_{0}}^{t_{1}}{dt_{2}V(t_{1})V(t_{2})}}+\cdots\\ &&{}+(-i)^{n}\int_{t_{0}}^{t}{dt_{1}\int_{t_{0}}^{t_{1}}{dt_{2}\cdots\int_{t_{% 0}}^{t_{n-1}}{dt_{n}V(t_{1})V(t_{2})\cdots V(t_{n})}}}+\cdots.\end{array}
  13. 𝒯 \mathcal{T}
  14. U n ( t , t 0 ) = ( - i ) n t 0 t d t 1 t 0 t 1 d t 2 t 0 t n - 1 d t n 𝒯 V ( t 1 ) V ( t 2 ) V ( t n ) . U_{n}(t,t_{0})=(-i)^{n}\int_{t_{0}}^{t}{dt_{1}\int_{t_{0}}^{t_{1}}{dt_{2}% \cdots\int_{t_{0}}^{t_{n-1}}{dt_{n}\mathcal{T}V(t_{1})V(t_{2})\cdots V(t_{n})}% }}.
  15. S n = t 0 t d t 1 t 0 t 1 d t 2 t 0 t n - 1 d t n K ( t 1 , t 2 , , t n ) . S_{n}=\int_{t_{0}}^{t}{dt_{1}\int_{t_{0}}^{t_{1}}{dt_{2}\cdots\int_{t_{0}}^{t_% {n-1}}{dt_{n}K(t_{1},t_{2},\dots,t_{n})}}}.
  16. I n = t 0 t d t 1 t 0 t d t 2 t 0 t d t n K ( t 1 , t 2 , , t n ) . I_{n}=\int_{t_{0}}^{t}{dt_{1}\int_{t_{0}}^{t}{dt_{2}\cdots\int_{t_{0}}^{t}{dt_% {n}K(t_{1},t_{2},\dots,t_{n})}}}.
  17. S n S_{n}
  18. S n = 1 n ! I n . S_{n}=\frac{1}{n!}I_{n}.
  19. U n = ( - i ) n n ! t 0 t d t 1 t 0 t d t 2 t 0 t d t n 𝒯 V ( t 1 ) V ( t 2 ) V ( t n ) . U_{n}=\frac{(-i)^{n}}{n!}\int_{t_{0}}^{t}{dt_{1}\int_{t_{0}}^{t}{dt_{2}\cdots% \int_{t_{0}}^{t}{dt_{n}\mathcal{T}V(t_{1})V(t_{2})\cdots V(t_{n})}}}.
  20. U ( t , t 0 ) = n = 0 U n ( t , t 0 ) = 𝒯 e - i t 0 t d τ V ( τ ) . U(t,t_{0})=\sum_{n=0}^{\infty}U_{n}(t,t_{0})=\mathcal{T}e^{-i\int_{t_{0}}^{t}{% d\tau V(\tau)}}.
  21. | Ψ ( t ) = n = 0 ( - i ) n n ! ( k = 1 n t 0 t d t k ) 𝒯 { k = 1 n e i H 0 t k V e - i H 0 t k } | Ψ ( t 0 ) . |\Psi(t)\rangle=\sum_{n=0}^{\infty}{(-i)^{n}\over n!}\left(\prod_{k=1}^{n}\int% _{t_{0}}^{t}dt_{k}\right)\mathcal{T}\left\{\prod_{k=1}^{n}e^{iH_{0}t_{k}}Ve^{-% iH_{0}t_{k}}\right\}|\Psi(t_{0})\rangle.
  22. ψ f ; t f ψ i ; t i = n = 0 ( - i ) n d t 1 d t n t f t 1 t n t i ψ f ; t f e - i H 0 ( t f - t 1 ) V e - i H 0 ( t 1 - t 2 ) V e - i H 0 ( t n - t i ) ψ i ; t i . \langle\psi_{f};t_{f}\mid\psi_{i};t_{i}\rangle=\sum_{n=0}^{\infty}(-i)^{n}% \underbrace{\int dt_{1}\cdots dt_{n}}_{t_{f}\,\geq\,t_{1}\,\geq\,\cdots\,\geq% \,t_{n}\,\geq\,t_{i}}\,\langle\psi_{f};t_{f}\mid e^{-iH_{0}(t_{f}-t_{1})}Ve^{-% iH_{0}(t_{1}-t_{2})}\cdots Ve^{-iH_{0}(t_{n}-t_{i})}\mid\psi_{i};t_{i}\rangle.

E6B.html

  1. π \pi
  2. Δ a = sin - 1 ( V w sin ( w - d ) V a ) \Delta a=\sin^{-1}\left(\frac{V_{w}\sin(w-d)}{V_{a}}\right)
  3. V g = V a 2 + V w 2 - 2 V a V w cos ( d - w + Δ a ) V_{g}=\sqrt{V_{a}^{2}+V_{w}^{2}-2V_{a}V_{w}\cos(d-w+\Delta a)}
  4. Δ a = 180 deg π sin - 1 ( V w V a sin ( π ( w - d ) 180 deg ) ) \Delta a=\frac{180\deg}{\pi}\sin^{-1}\left(\frac{V_{w}}{V_{a}}\sin\left(\frac{% \pi(w-d)}{180\deg}\right)\right)
  5. V g = V a 2 + V w 2 - 2 V a V w cos ( π ( d - w + Δ a ) 180 deg ) V_{g}=\sqrt{V_{a}^{2}+V_{w}^{2}-2V_{a}V_{w}\cos\left(\frac{\pi(d-w+\Delta a)}{% 180\deg}\right)}

E8_lattice.html

  1. Γ 8 = { ( x i ) 8 ( + 1 2 ) 8 : i x i 0 ( mod 2 ) } . \Gamma_{8}=\left\{(x_{i})\in\mathbb{Z}^{8}\cup(\mathbb{Z}+\tfrac{1}{2})^{8}:{% \textstyle\sum_{i}}x_{i}\equiv 0\;(\mbox{mod }~{}2)\right\}.
  2. Γ 8 = { ( x i ) 8 ( + 1 2 ) 8 : i x i 2 x 1 2 x 2 2 x 3 2 x 4 2 x 5 2 x 6 2 x 7 2 x 8 ( mod 2 ) } . \Gamma_{8}^{\prime}=\left\{(x_{i})\in\mathbb{Z}^{8}\cup(\mathbb{Z}+\tfrac{1}{2% })^{8}:{{\textstyle\sum_{i}}x_{i}}\equiv 2x_{1}\equiv 2x_{2}\equiv 2x_{3}% \equiv 2x_{4}\equiv 2x_{5}\equiv 2x_{6}\equiv 2x_{7}\equiv 2x_{8}\;(\mbox{mod % }~{}2)\right\}.
  3. Γ 8 = { ( x i ) 8 : i x i 0 ( mod 2 ) } { ( x i ) ( + 1 2 ) 8 : i x i 1 ( mod 2 ) } . \Gamma_{8}^{\prime}=\left\{(x_{i})\in\mathbb{Z}^{8}:{{\textstyle\sum_{i}}x_{i}% }\equiv 0(\mbox{mod }~{}2)\right\}\cup\left\{(x_{i})\in(\mathbb{Z}+\tfrac{1}{2% })^{8}:{{\textstyle\sum_{i}}x_{i}}\equiv 1(\mbox{mod }~{}2)\right\}.
  4. [ 2 - 1 0 0 0 0 0 1 / 2 0 1 - 1 0 0 0 0 1 / 2 0 0 1 - 1 0 0 0 1 / 2 0 0 0 1 - 1 0 0 1 / 2 0 0 0 0 1 - 1 0 1 / 2 0 0 0 0 0 1 - 1 1 / 2 0 0 0 0 0 0 1 1 / 2 0 0 0 0 0 0 0 1 / 2 ] . \left[\begin{smallmatrix}2&-1&0&0&0&0&0&1/2\\ 0&1&-1&0&0&0&0&1/2\\ 0&0&1&-1&0&0&0&1/2\\ 0&0&0&1&-1&0&0&1/2\\ 0&0&0&0&1&-1&0&1/2\\ 0&0&0&0&0&1&-1&1/2\\ 0&0&0&0&0&0&1&1/2\\ 0&0&0&0&0&0&0&1/2\end{smallmatrix}\right].
  5. | W ( E 8 ) | = 696729600 = 4 ! 6 ! 8 ! . |W(\mathrm{E}_{8})|=696729600=4!\cdot 6!\cdot 8!.
  6. H 4 = 1 2 [ 1 1 1 1 1 - 1 1 - 1 1 1 - 1 - 1 1 - 1 - 1 1 ] . H_{4}=\tfrac{1}{2}\left[\begin{smallmatrix}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\\ \end{smallmatrix}\right].
  7. E ~ 8 {\tilde{E}}_{8}
  8. ( 5 6 , 1 6 , 1 6 , 1 6 , 1 6 , 1 6 , 1 6 , 1 6 ) (\tfrac{5}{6},\tfrac{1}{6},\tfrac{1}{6},\tfrac{1}{6},\tfrac{1}{6},\tfrac{1}{6}% ,\tfrac{1}{6},\tfrac{1}{6})
  9. 2 2 3 \tfrac{2\sqrt{2}}{3}
  10. π 4 384 0.25367. \frac{\pi^{4}}{384}\cong 0.25367.
  11. Θ Λ ( τ ) = x Λ e i π τ x 2 Im τ > 0. \Theta_{\Lambda}(\tau)=\sum_{x\in\Lambda}e^{i\pi\tau\|x\|^{2}}\qquad\mathrm{Im% }\,\tau>0.
  12. q = e i π τ q=e^{i\pi\tau}
  13. Θ Γ 8 ( τ ) = 1 + 240 n = 1 σ 3 ( n ) q 2 n \Theta_{\Gamma_{8}}(\tau)=1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{2n}
  14. Θ Γ 8 ( τ ) = 1 + 240 q 2 + 2160 q 4 + 6720 q 6 + 17520 q 8 + 30240 q 10 + 60480 q 12 + O ( q 14 ) . \Theta_{\Gamma_{8}}(\tau)=1+240\,q^{2}+2160\,q^{4}+6720\,q^{6}+17520\,q^{8}+30% 240\,q^{10}+60480\,q^{12}+O(q^{14}).
  15. Θ Γ 8 ( τ ) = 1 2 ( θ 2 ( q ) 8 + θ 3 ( q ) 8 + θ 4 ( q ) 8 ) \Theta_{\Gamma_{8}}(\tau)=\frac{1}{2}\left(\theta_{2}(q)^{8}+\theta_{3}(q)^{8}% +\theta_{4}(q)^{8}\right)
  16. θ 2 ( q ) = n = - q ( n + 1 2 ) 2 θ 3 ( q ) = n = - q n 2 θ 4 ( q ) = n = - ( - 1 ) n q n 2 . \theta_{2}(q)=\sum_{n=-\infty}^{\infty}q^{(n+\frac{1}{2})^{2}}\qquad\theta_{3}% (q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}\qquad\theta_{4}(q)=\sum_{n=-\infty}^{% \infty}(-1)^{n}q^{n^{2}}.
  17. Λ = 1 2 { x n : x mod 2 C } . \Lambda=\tfrac{1}{\sqrt{2}}\left\{x\in\mathbb{Z}^{n}:x\,\bmod\,2\in C\right\}.
  18. ± e i \pm e_{i}
  19. ± e 0 ± e a ± e b ± e c \pm e_{0}\pm e_{a}\pm e_{b}\pm e_{c}
  20. ± e p ± e q ± e r ± e s \pm e_{p}\pm e_{q}\pm e_{r}\pm e_{s}
  21. E 7 E_{7}
  22. E 8 E_{8}
  23. G 2 ( 2 ) G_{2}(2)
  24. E 7 E_{7}

Earnings_per_share.html

  1. Earnings Per Share = Profit- Preferred Dividends Weighted Average Common Shares \mbox{Earnings Per Share}~{}=\frac{\mbox{Profit- Preferred Dividends}~{}}{% \mbox{Weighted Average Common Shares}~{}}
  2. Earnings Per Share = Net Income - Preferred Dividends Average Common Shares \mbox{Earnings Per Share}~{}=\frac{\mbox{Net Income - Preferred Dividends}~{}}% {\mbox{Average Common Shares}~{}}
  3. Earnings Per Share = Income from Continuing Operations - Preferred dividends Weighted Average Common Shares \mbox{Earnings Per Share}~{}=\frac{\mbox{Income from Continuing Operations - % Preferred dividends}~{}}{\mbox{Weighted Average Common Shares}~{}}

Ease_(programming_language).html

  1. P ( ) Q ( ) ; \parallel P()\parallel Q();
  2. / / P ( ) ; \big/\!\!/P();
  3. i f o r n : P ( i ) ; \parallel{i}\;{for}\;{n}:P(i);

Eb::N0.html

  1. C / N = E b / N 0 f b B C/N=E_{b}/N_{0}\cdot\frac{f_{b}}{B}
  2. CNR dB = 10 log 10 ( E b / N 0 ) + 10 log 10 ( f b B ) \,\text{CNR}_{\,\text{dB}}=10\log_{10}(E_{b}/N_{0})+10\log_{10}\left(\frac{f_{% b}}{B}\right)
  3. N 0 / 2 N_{0}/2
  4. E b N 0 = E s ρ N 0 \frac{E_{b}}{N_{0}}=\frac{E_{s}}{\rho N_{0}}
  5. ρ \rho
  6. E s N 0 = E b N 0 log 2 M \frac{E_{s}}{N_{0}}=\frac{E_{b}}{N_{0}}\log_{2}M
  7. E s N 0 = C N B f s \frac{E_{s}}{N_{0}}=\frac{C}{N}\frac{B}{f_{s}}
  8. I < B log 2 ( 1 + S N ) I<B\log_{2}\left(1+\frac{S}{N}\right)
  9. R B = 2 R l < log 2 ( 1 + 2 R l E b N 0 ) {R\over B}=2R_{l}<\log_{2}\left(1+2R_{l}\frac{E_{b}}{N_{0}}\right)
  10. E b N 0 > 2 2 R l - 1 2 R l \frac{E_{b}}{N_{0}}>\frac{2^{2R_{l}}-1}{2R_{l}}
  11. E b N 0 > ln ( 2 ) \frac{E_{b}}{N_{0}}>\ln(2)

Eccentricity_(mathematics).html

  1. ε \varepsilon
  2. e = sin β sin α , 0 < α < 90 , 0 β 90 , e=\frac{\sin\beta}{\sin\alpha},\ \ 0<\alpha<90^{\circ},\ 0\leq\beta\leq 90^{% \circ}\ ,
  3. β = 0 \beta=0
  4. β = α \beta=\alpha
  5. e = c a e=\frac{c}{a}
  6. ε \varepsilon
  7. ϵ \epsilon
  8. x 2 + y 2 = r 2 x^{2}+y^{2}=r^{2}
  9. 0
  10. 0
  11. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  12. 1 - b 2 a 2 \sqrt{1-\frac{b^{2}}{a^{2}}}
  13. a 2 - b 2 \sqrt{a^{2}-b^{2}}
  14. x 2 = 4 a y x^{2}=4ay
  15. 1 1
  16. - -
  17. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  18. 1 + b 2 a 2 \sqrt{1+\frac{b^{2}}{a^{2}}}
  19. a 2 + b 2 \sqrt{a^{2}+b^{2}}
  20. A x 2 + B x y + C y 2 + D x + E y + F = 0 , Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,
  21. e = 2 ( A - C ) 2 + B 2 η ( A + C ) + ( A - C ) 2 + B 2 e=\sqrt{\frac{2\sqrt{(A-C)^{2}+B^{2}}}{\eta(A+C)+\sqrt{(A-C)^{2}+B^{2}}}}
  22. η = 1 \eta=1
  23. [ A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ] \begin{bmatrix}A&B/2&D/2\\ B/2&C&E/2\\ D/2&E/2&F\end{bmatrix}
  24. η = - 1 \eta=-1
  25. e e
  26. 1 - b 2 a 2 \sqrt{1-\frac{b^{2}}{a^{2}}}
  27. e e
  28. e e^{\prime}
  29. a 2 b 2 - 1 \sqrt{\frac{a^{2}}{b^{2}}-1}
  30. e 1 - e 2 \frac{e}{\sqrt{1-e^{2}}}
  31. e ′′ = m e^{\prime\prime}=\sqrt{m}
  32. a 2 - b 2 a 2 + b 2 \frac{\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}}
  33. e 2 - e 2 \frac{e}{\sqrt{2-e^{2}}}
  34. α \alpha
  35. cos - 1 ( b a ) \cos^{-1}\left(\frac{b}{a}\right)
  36. sin - 1 e \sin^{-1}e
  37. e = a d . e=\frac{a}{d}.
  38. e = g ( 2 - g ) . e=\sqrt{g(2-g)}.
  39. r max r\text{max}
  40. r min r\text{min}
  41. e = r max - r min r max + r min = r max - r min 2 a . e=\frac{r\text{max}-r\text{min}}{r\text{max}+r\text{min}}=\frac{r\text{max}-r% \text{min}}{2a}.
  42. 2 \sqrt{2}
  43. 1 / r 1/r

Ecohydrology.html

  1. θ \theta
  2. S S
  3. n n
  4. θ = n S \theta=nS
  5. n Z r d s ( t ) d t = R ( t ) - I ( t ) - Q [ s ( t ) , t ] - E [ s ( t ) ] - L [ s ( t ) ] nZ_{r}\frac{ds(t)}{dt}=R(t)-I(t)-Q[s(t),t]-E[s(t)]-L[s(t)]
  6. n n
  7. s s
  8. Z r Z_{r}
  9. d s ( t ) / d t ds(t)/dt
  10. R R
  11. I I
  12. Q Q
  13. E E
  14. L L

Econometric_model.html

  1. C t = a + b Y t - 1 + e t , C_{t}=a+bY_{t-1}+e_{t},

Economic_batch_quantity.html

  1. Economic batch quantity = 2 annual demand setup costs inventory carrying cost per unit \,\text{Economic batch quantity}=\sqrt{\frac{2\cdot\,\text{annual demand}\cdot% \,\text{setup costs}}{\,\text{inventory carrying cost per unit}}}

Edge-graceful_labeling.html

  1. V ( u ) = Σ E ( e ) mod | V ( G ) | V(u)=\Sigma E(e)\mod|V(G)|
  2. { 1 , 2 q } \{1,2\dots q\}
  3. { 0 , 1 p - 1 } \{0,1\dots p-1\}
  4. C m C_{m}
  5. C 5 C_{5}
  6. q ( q + 1 ) \;q(q+1)
  7. p ( p - 1 ) 2 \frac{p(p-1)}{2}
  8. q ( q + 1 ) p ( p - 1 ) 2 mod p . q(q+1)\equiv\frac{p(p-1)}{2}\mod p.
  9. S m S_{m}
  10. F m F_{m}
  11. T m , n T_{m,n}
  12. K n K_{n}
  13. n = 2 mod 4 n=2\mod 4

Edge-of-the-wedge_theorem.html

  1. | z | = 1 |z|=1
  2. r < | z | < 1 r<|z|<1
  3. 1 < | z | < R 1<|z|<R
  4. f ( z ) = - a n z n , g ( z ) = - b n z n f(z)=\sum_{-\infty}^{\infty}a_{n}z^{n},\,\,\,\,g(z)=\sum_{-\infty}^{\infty}b_{% n}z^{n}
  5. f ( θ ) = - a n e i n θ , g ( θ ) = - b n e i n θ . f(\theta)=\sum_{-\infty}^{\infty}a_{n}e^{in\theta},\,\,\,\,g(\theta)=\sum_{-% \infty}^{\infty}b_{n}e^{in\theta}.
  6. a n = b n a_{n}=b_{n}
  7. r < | z | < R r<|z|<R
  8. I = ( a , b ) I=(a,b)
  9. f + , f - f_{+},\,\,\ f_{-}
  10. ( a , b ) × ( 0 , R ) (a,b)\times(0,R)
  11. ( a , b ) × ( - R , 0 ) (a,b)\times(-R,0)
  12. | f ± ( x + i y ) | < C | y | - N |f_{\pm}(x+iy)|<C|y|^{-N}
  13. T ± T_{\pm}
  14. f ± f_{\pm}
  15. T ± , ϕ = lim ϵ 0 f ( x ± i ϵ ) ϕ ( x ) d x . \langle T_{\pm},\phi\rangle=\lim_{\epsilon\downarrow 0}\int f(x\pm i\epsilon)% \phi(x)\,dx.
  16. f ± ( z ) f_{\pm}(z)
  17. ( N + 1 ) (N+1)
  18. f ± f_{\pm}
  19. ( a , b ) × ( - R , R ) (a,b)\times(-R,R)
  20. F , ϕ = f ( x + i y ) ϕ ( x , y ) d x d y , \langle F,\phi\rangle=\int\int f(x+iy)\phi(x,y)\,dx\,dy,
  21. f ± f_{\pm}
  22. F z ¯ F_{\overline{z}}
  23. 1 2 ( T + - T - ) {1\over 2}(T_{+}-T_{-})
  24. T + = T - T_{+}=T_{-}
  25. F z ¯ = 0. F_{\overline{z}}=0.
  26. ( a , b ) × ( - R , R ) (a,b)\times(-R,R)
  27. ( π z ) - 1 (\pi z)^{-1}
  28. / z ¯ \partial/\partial\overline{z}
  29. f f
  30. g g
  31. D = z z , D=z{\partial\over\partial z},
  32. D k F = f , D k G = g D^{k}F=f,\,\,\,D^{k}G=g
  33. C C
  34. R n R^{n}
  35. E × i C E\times iC
  36. E × - i C E\times-iC
  37. W E W W\cup E\cup W^{\prime}

Edge_space.html

  1. G := ( V , E ) G:=(V,E)
  2. 𝒱 ( G ) \mathcal{V}(G)
  3. / 2 := { 0 , 1 } \mathbb{Z}/2\mathbb{Z}:=\{0,1\}
  4. V / 2 V\rightarrow\mathbb{Z}/2\mathbb{Z}
  5. 𝒱 ( G ) \mathcal{V}(G)
  6. 𝒱 ( G ) \mathcal{V}(G)
  7. ( G ) \mathcal{E}(G)
  8. / 2 \mathbb{Z}/2\mathbb{Z}
  9. E E
  10. ( G ) \mathcal{E}(G)
  11. P + Q := P Q P , Q ( G ) P+Q:=P\triangle Q\qquad P,Q\in\mathcal{E}(G)
  12. 0 P := P ( G ) 0\cdot P:=\emptyset\qquad P\in\mathcal{E}(G)
  13. 1 P := P P ( G ) 1\cdot P:=P\qquad P\in\mathcal{E}(G)
  14. ( G ) \mathcal{E}(G)
  15. 𝒱 ( G ) \mathcal{V}(G)
  16. ( G ) \mathcal{E}(G)
  17. H H
  18. G G
  19. H : ( G ) 𝒱 ( G ) H:\mathcal{E}(G)\to\mathcal{V}(G)
  20. G G
  21. v u vu
  22. v v
  23. u u
  24. H ( v u ) = v + u H(vu)=v+u

Effective_action.html

  1. ϕ \phi
  2. V ( ϕ ) V(\phi)
  3. V ( ϕ ) V(\phi)
  4. E [ J ] = i ln Z [ J ] E[J]=i\ln Z[J]
  5. E [ J ] = - ln Z [ J ] E[J]=-\ln Z[J]\,
  6. ϕ ( x 1 ) ϕ ( x n ) c o n = ( - i ) n + 1 δ n E δ J ( x 1 ) δ J ( x n ) | J = 0 \langle\phi(x_{1})\cdots\phi(x_{n})\rangle_{con}=(-i)^{n+1}\left.\frac{\delta^% {n}E}{\delta J(x_{1})\cdots\delta J(x_{n})}\right|_{J=0}
  7. ϕ i 1 ϕ i n con = ( - i ) n + 1 E , i 1 i n | J = 0 \langle\phi^{i_{1}}\cdots\phi^{i_{n}}\rangle\text{con}=(-i)^{n+1}E^{,i_{1}% \dots i_{n}}|_{J=0}
  8. ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x 3 ) = ϕ ( x 1 ) ϕ ( x 2 ) ϕ ( x 3 ) con + ϕ ( x 1 ) ϕ ( x 2 ) con ϕ ( x 3 ) con + ϕ ( x 1 ) ϕ ( x 3 ) con ϕ ( x 2 ) con + ϕ ( x 1 ) con ϕ ( x 2 ) ϕ ( x 3 ) con + ϕ ( x 1 ) c o n ϕ ( x 2 ) con ϕ ( x 3 ) con \begin{aligned}&\displaystyle{}\quad\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})% \rangle\\ &\displaystyle=\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\rangle\text{con}+% \langle\phi(x_{1})\phi(x_{2})\rangle\text{con}\langle\phi(x_{3})\rangle\text{% con}+\langle\phi(x_{1})\phi(x_{3})\rangle\text{con}\langle\phi(x_{2})\rangle% \text{con}\\ &\displaystyle+\langle\phi(x_{1})\rangle\text{con}\langle\phi(x_{2})\phi(x_{3}% )\rangle\text{con}+\langle\phi(x_{1})\rangle_{con}\langle\phi(x_{2})\rangle% \text{con}\langle\phi(x_{3})\rangle\text{con}\end{aligned}
  9. ϕ = - δ δ J E [ J ] \phi=-{\delta\over\delta J}E[J]
  10. ϕ i = - E , i \phi^{i}=-E^{,i}\,
  11. Γ [ ϕ ] = - J , ϕ - E [ J ] \Gamma[\phi]=-\langle J,\phi\rangle-E[J]\,
  12. Γ [ ϕ ] = - J i ϕ i - E [ J ] \Gamma[\phi]=-J_{i}\phi^{i}-E[J]\,
  13. ϕ = - δ δ J E [ J ] \phi=-{\delta\over\delta J}E[J]
  14. J = - δ δ ϕ Γ [ ϕ ] J=-{\delta\over\delta\phi}\Gamma[\phi]
  15. J i = - Γ , i . J_{i}=-\Gamma_{,i}.\,
  16. ϕ = 0 \langle\phi\rangle=0
  17. ϕ = 0 \langle\phi\rangle=0
  18. ϕ = 0 \langle\phi\rangle=0
  19. ϕ = 0 \langle\phi\rangle=0
  20. ϕ = ϕ - ϕ \phi^{\prime}=\phi-\langle\phi\rangle
  21. Γ [ ϕ ] = S [ ϕ ] + 1 2 T r [ ln S ( 2 ) [ ϕ ] ] + . \Gamma[\phi]=S[\phi]+\frac{1}{2}Tr\left[\ln{S^{(2)}[\phi]}\right]+\cdots.

Effective_half-life.html

  1. λ e = λ p + λ b {\lambda_{e}}\,={\lambda_{p}}\,+{\lambda_{b}}\,
  2. t 1 / 2 = ln ( 2 ) λ e t_{1/2}=\frac{\ln(2)}{\lambda_{e}}
  3. 1 t 1 / 2 e = 1 t 1 / 2 p + 1 t 1 / 2 b \frac{1}{t_{1/2e}}=\frac{1}{t_{1/2p}}+\frac{1}{t_{1/2b}}
  4. t 1 / 2 e = t 1 / 2 p × t 1 / 2 b t 1 / 2 p + t 1 / 2 b t_{1/2e}=\frac{t_{1/2p}\times t_{1/2b}}{t_{1/2p}+t_{1/2b}}

Effective_population_size.html

  1. p p^{\prime}
  2. p p
  3. var ( p p ) = p ( 1 - p ) 2 N . \operatorname{var}(p^{\prime}\mid p)={p(1-p)\over 2N}.
  4. var ^ ( p p ) \widehat{\operatorname{var}}(p^{\prime}\mid p)
  5. N e ( v ) N_{e}^{(v)}
  6. var ^ ( p p ) \widehat{\operatorname{var}}(p^{\prime}\mid p)
  7. var ( p p ) \operatorname{var}(p^{\prime}\mid p)
  8. N N
  9. N e ( v ) = p ( 1 - p ) 2 var ^ ( p ) . N_{e}^{(v)}={p(1-p)\over 2\widehat{\operatorname{var}}(p)}.
  10. 1 N e = 1 t i = 1 t 1 N i {1\over N_{e}}={1\over t}\sum_{i=1}^{t}{1\over N_{i}}
  11. 1 N e {1\over N_{e}}
  12. = 1 10 + 1 100 + 1 50 + 1 80 + 1 20 + 1 500 6 ={\begin{matrix}\frac{1}{10}\end{matrix}+\begin{matrix}\frac{1}{100}\end{% matrix}+\begin{matrix}\frac{1}{50}\end{matrix}+\begin{matrix}\frac{1}{80}\end{% matrix}+\begin{matrix}\frac{1}{20}\end{matrix}+\begin{matrix}\frac{1}{500}\end% {matrix}\over 6}
  13. = 0.1945 6 ={0.1945\over 6}
  14. = 0.032416667 =0.032416667
  15. N e N_{e}
  16. = 30.8 =30.8
  17. N e = N + 1 2 N_{e}=N+\begin{matrix}\frac{1}{2}\end{matrix}
  18. N e = N + D 2 N_{e}=N+\begin{matrix}\frac{D}{2}\end{matrix}
  19. N e = N + 1 2 N N_{e}=N+\begin{matrix}\frac{1}{2}\approx N\end{matrix}
  20. var ( k ) = k ¯ = 2. \operatorname{var}(k)=\bar{k}=2.
  21. var ( k ) > 2. \operatorname{var}(k)>2.
  22. N e ( v ) = 4 N - 2 D 2 + var ( k ) N_{e}^{(v)}={4N-2D\over 2+\operatorname{var}(k)}
  23. N e ( v ) = N e ( F ) = 4 N m N f N m + N f N_{e}^{(v)}=N_{e}^{(F)}={4N_{m}N_{f}\over N_{m}+N_{f}}
  24. N e N_{e}
  25. = 4 × 80 × 20 80 + 20 ={4\times 80\times 20\over 80+20}
  26. = 6400 100 ={6400\over 100}
  27. = 64 =64
  28. F t = 1 N ( 1 + F t - 2 2 ) + ( 1 - 1 N ) F t - 1 . F_{t}=\frac{1}{N}\left(\frac{1+F_{t-2}}{2}\right)+\left(1-\frac{1}{N}\right)F_% {t-1}.
  29. 1 - F t = P t = P 0 ( 1 - 1 2 N ) t . 1-F_{t}=P_{t}=P_{0}\left(1-\frac{1}{2N}\right)^{t}.
  30. P t + 1 P t = 1 - 1 2 N . \frac{P_{t+1}}{P_{t}}=1-\frac{1}{2N}.
  31. P t + 1 P t = 1 - 1 2 N e ( F ) . \frac{P_{t+1}}{P_{t}}=1-\frac{1}{2N_{e}^{(F)}}.
  32. N e ( F ) = 1 2 ( 1 - P t + 1 P t ) N_{e}^{(F)}=\frac{1}{2\left(1-\frac{P_{t+1}}{P_{t}}\right)}
  33. v i = v_{i}=
  34. i i
  35. i = \ell_{i}=
  36. i i
  37. N 0 = N_{0}=
  38. T = i = 0 i v i = T=\sum_{i=0}^{\infty}\ell_{i}v_{i}=
  39. N e ( F ) = N 0 T 1 + i i + 1 2 v i + 1 2 ( 1 i + 1 - 1 i ) . N_{e}^{(F)}=\frac{N_{0}T}{1+\sum_{i}\ell_{i+1}^{2}v_{i+1}^{2}(\frac{1}{\ell_{i% +1}}-\frac{1}{\ell_{i}})}.
  40. 1 N e ( F ) = 1 4 T { 1 N 0 f + 1 N 0 m + i ( i + 1 f ) 2 ( v i + 1 f ) 2 ( 1 i + 1 f - 1 i f ) + i ( i + 1 m ) 2 ( v i + 1 m ) 2 ( 1 i + 1 m - 1 i m ) } . \begin{aligned}\displaystyle\frac{1}{N_{e}^{(F)}}=\frac{1}{4T}\left\{\frac{1}{% N_{0}^{f}}+\frac{1}{N_{0}^{m}}+\sum_{i}\left(\ell_{i+1}^{f}\right)^{2}\left(v_% {i+1}^{f}\right)^{2}\left(\frac{1}{\ell_{i+1}^{f}}-\frac{1}{\ell_{i}^{f}}% \right)\right.&\\ \displaystyle\left.{}+\sum_{i}\left(\ell_{i+1}^{m}\right)^{2}\left(v_{i+1}^{m}% \right)^{2}\left(\frac{1}{\ell_{i+1}^{m}}-\frac{1}{\ell_{i}^{m}}\right)\right% \}.&\end{aligned}
  41. μ \mu
  42. μ \mu

Effective_rate_of_protection.html

  1. ( V A (VA
  2. V A VA
  3. 1 1
  4. ( T (T
  5. - T -T
  6. / V A /VA

Effective_temperature.html

  1. Bol \mathcal{F}_{\rm Bol}
  2. Bol = σ T eff 4 \mathcal{F}_{\rm Bol}=\sigma T_{\rm eff}^{4}
  3. L = 4 π R 2 σ T eff 4 L=4\pi R^{2}\sigma T_{\rm eff}^{4}
  4. R R
  5. P abs = L r 2 ( 1 - A ) 4 D 2 P_{\rm abs}=\frac{Lr^{2}(1-A)}{4D^{2}}
  6. P rad = 4 π r 2 σ T 4 P_{\rm rad}=4\pi r^{2}\sigma T^{4}
  7. T = ( L ( 1 - A ) 16 π σ D 2 ) 1 4 T=\left(\frac{L(1-A)}{16\pi\sigma D^{2}}\right)^{\tfrac{1}{4}}
  8. A total = 4 π r 2 A_{\rm total}=4\pi r^{2}
  9. P abs = L A abs ( 1 - a ) 4 π D 2 P_{\rm abs}=\frac{LA_{\rm abs}(1-a)}{4\pi D^{2}}
  10. P rad = A rad ε σ T 4 P_{\rm rad}=A_{\rm rad}\varepsilon\sigma T^{4}
  11. T = ( A abs A rad L ( 1 - a ) 4 π σ ε D 2 ) 1 4 T=\left(\frac{A_{\rm abs}}{A_{\rm rad}}\frac{L(1-a)}{4\pi\sigma\varepsilon D^{% 2}}\right)^{\tfrac{1}{4}}

Egorov's_theorem.html

  1. f n ( x ) = 1 [ n , n + 1 ] ( x ) , n , x , f_{n}(x)=1_{[n,n+1]}(x),\qquad n\in\mathbb{N},\ x\in\mathbb{R},
  2. n n
  3. E n , k = m n { x A | | f m ( x ) - f ( x ) | 1 k } . E_{n,k}=\bigcup_{m\geq n}\left\{x\in A\,\Big|\,|f_{m}(x)-f(x)|\geq\frac{1}{k}% \right\}.
  4. μ ( n E n , k ) = 0 \mu\biggl(\bigcap_{n\in\mathbb{N}}E_{n,k}\biggr)=0
  5. μ ( E n k , k ) < ε 2 k . \mu(E_{n_{k},k})<\frac{\varepsilon}{2^{k}}.
  6. B = k E n k , k B=\bigcup_{k\in\mathbb{N}}E_{n_{k},k}
  7. μ ( B ) k μ ( E n k , k ) < k ε 2 k = ε . \mu(B)\leq\sum_{k\in\mathbb{N}}\mu(E_{n_{k},k})<\sum_{k\in\mathbb{N}}\frac{% \varepsilon}{2^{k}}=\varepsilon.
  8. μ ( A k = 1 N A k ) 1 N \mu\left(A\setminus\bigcup_{k=1}^{N}A_{k}\right)\leq\frac{1}{N}
  9. H = A k = 1 A k H=A\setminus\bigcup_{k=1}^{\infty}A_{k}
  10. \scriptstyle\leq
  11. \scriptstyle\geq
  12. 𝔄 \scriptstyle\mathfrak{A}
  13. \scriptstyle\cap
  14. lim \scriptstyle\lim
  15. \scriptstyle\supset
  16. \scriptstyle\supset
  17. 𝔄 \scriptstyle\in\mathfrak{A}
  18. \scriptstyle\cup
  19. lim \scriptstyle\lim
  20. \scriptstyle\subset
  21. \scriptstyle\subset
  22. \scriptstyle\cup
  23. 𝔄 \scriptstyle\in\mathfrak{A}
  24. 𝔄 \scriptstyle\in\mathfrak{A}
  25. A 0 , m = { x A | d ( f n ( x ) , f ( x ) ) 1 n m } A_{0,m}=\left\{x\in A|d(f_{n}(x),f(x))\leq 1\ \forall n\geq m\right\}
  26. A 0 , 1 A 0 , 2 A 0 , 3 A_{0,1}\subseteq A_{0,2}\subseteq A_{0,3}\subseteq\dots
  27. A = m A 0 , m A=\bigcup_{m\in\mathbb{N}}A_{0,m}
  28. 0 μ ( A ) - μ ( A 0 ) ε 0\leq\mu(A)-\mu(A_{0})\leq\varepsilon
  29. A 1 , m = { x A 0 | d ( f m ( x ) , f ( x ) ) 1 2 n m } A_{1,m}=\left\{x\in A_{0}\left|d(f_{m}(x),f(x))\leq\frac{1}{2}\ \forall n\geq m% \right.\right\}
  30. A 1 , 1 A 1 , 2 A 1 , 3 A_{1,1}\subseteq A_{1,2}\subseteq A_{1,3}\subseteq\dots
  31. A 0 = m A 1 , m A_{0}=\bigcup_{m\in\mathbb{N}}A_{1,m}
  32. 0 μ ( A ) - μ ( A 1 ) ε 0\leq\mu(A)-\mu(A_{1})\leq\varepsilon
  33. A 0 A 1 A 2 \scriptstyle A_{0}\supseteq A_{1}\supseteq A_{2}\supseteq\dots
  34. 0 μ ( A ) - μ ( A m ) ε \scriptstyle 0\leq\mu(A)-\mu(A_{m})\leq\varepsilon
  35. \scriptstyle\in\mathbb{N}
  36. \scriptstyle\in\mathbb{N}
  37. \scriptstyle\geq
  38. d ( f n ( x ) , f ( x ) ) 2 - m \scriptstyle d(f_{n}(x),f(x))\leq 2^{-m}
  39. \scriptstyle\in
  40. A = n A n A^{\prime}=\bigcup_{n\in\mathbb{N}}A_{n}

Eine_Billion_Dollar.html

  1. 12 {}^{12}

Einstein's_constant.html

  1. G α γ = κ T α γ G^{\alpha\gamma}=\kappa\,T^{\alpha\gamma}~{}
  2. κ = - 8 π G c 2 \kappa\,=\,-{8\,\pi\,G\over c^{2}}~{}
  3. G α γ + Λ g α γ = ( const ) T α γ G^{\alpha\gamma}+\Lambda\mathrm{g}^{\alpha\gamma}=(\mathrm{const})T^{\alpha% \gamma}~{}
  4. G α γ = ( R α γ - 1 2 g α γ R ) = κ T α γ G^{\alpha\gamma}=\left(R^{\alpha\gamma}-\frac{1}{2}\mathrm{g}^{\alpha\gamma}\,% R\right)=\kappa\,T^{\alpha\gamma}~{}
  5. R α α - 1 2 g α α R = κ T α α {R^{\alpha}}_{\alpha}-\frac{1}{2}\,{\mathrm{g}^{\alpha}}_{\alpha}\,R={\kappa\,% T^{\alpha}}_{\alpha}~{}
  6. R = - κ T α α = - κ T R=-{\kappa\,T^{\alpha}}_{\alpha}=-\kappa\,T~{}
  7. R α γ = κ ( T α γ - 1 2 g α γ T ) R^{\alpha\gamma}=\kappa\left(T^{\alpha\gamma}-\frac{1}{2}\,\mathrm{g}^{\alpha% \gamma}\,T\right)
  8. | i |i
  9. | i | j |i|j
  10. x i \tfrac{\partial}{\partial x^{i}}
  11. 2 x i x j \tfrac{\partial^{2}}{\partial x^{i}\partial x^{j}}
  12. | i | i |i|i
  13. 2 ( x i ) 2 \tfrac{\partial^{2}}{(\partial{x^{i}})^{2}}
  14. T μ v = ρ ( 1 v x / c v y / c v z / c v x / c v x 2 / c 2 v x v y / c 2 v x v z / c 2 v y / c v y v x / c 2 v y 2 / c 2 v y v z / c 2 v z / c v z v x / c 2 v z v y / c 2 v z 2 / c 2 ) T_{{\mu}v}=\rho\begin{pmatrix}1&v_{x}/c&v_{y}/c&v_{z}/c\\ v_{x}/c&v^{2}_{x}/c^{2}&v_{x}v_{y}/c^{2}&v_{x}v_{z}/c^{2}\\ v_{y}/c&v_{y}v_{x}/c^{2}&v^{2}_{y}/c^{2}&v_{y}v_{z}/c^{2}\\ v_{z}/c&v_{z}v_{x}/c^{2}&v_{z}v_{y}/c^{2}&v^{2}_{z}/c^{2}\end{pmatrix}
  15. ( v c ) 2 (\tfrac{v}{c})^{2}
  16. ρ ( v c ) \rho(\tfrac{v}{c})
  17. T μ ν = ( ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) T_{\mu\nu}=\begin{pmatrix}\rho_{0}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}
  18. d s 2 = ( d x 0 ) 2 - ( d x 1 ) 2 - ( d x 2 ) 2 - ( d x 3 ) 2 \displaystyle\mathrm{d}s^{2}=(\mathrm{d}x^{0})^{2}-(\mathrm{d}x^{1})^{2}-(% \mathrm{d}x^{2})^{2}-(\mathrm{d}x^{3})^{2}
  19. ε γ μ ν \displaystyle\varepsilon\gamma_{\mu\nu}
  20. g μ ν = η μ ν + ε γ μ ν \displaystyle g_{\mu\nu}=\eta_{\mu\nu}+\varepsilon\gamma_{\mu\nu}
  21. d s 2 = ( d x 0 ) 2 - ( d x 1 ) 2 - ( d x 2 ) 2 - ( d x 3 ) 2 + ε γ μ ν d x μ d x ν \displaystyle\mathrm{d}s^{2}=(\mathrm{d}x^{0})^{2}-(\mathrm{d}x^{1})^{2}-(% \mathrm{d}x^{2})^{2}-(\mathrm{d}x^{3})^{2}+\varepsilon\gamma_{\mu\nu}\mathrm{d% }x^{\mu}\mathrm{d}x^{\nu}
  22. ε ρ 0 \displaystyle\varepsilon\rho_{0}
  23. T μ μ T^{\mu}_{\mu}
  24. T μ μ = Tr ( ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) = ρ 0 T^{\mu}_{\mu}=\operatorname{Tr}\begin{pmatrix}\rho_{0}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}=\rho_{0}
  25. ρ 0 \rho_{0}
  26. v c \tfrac{v}{c}
  27. ε γ μ ν \varepsilon\gamma_{\mu\nu}
  28. C ( T μ ν - 1 2 g μ ν T ) C ( T μ ν - 1 2 g μ ν T ) C [ ( ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) - 1 2 ( ρ 0 0 0 0 0 - ρ 0 0 0 0 0 - ρ 0 0 0 0 0 - ρ 0 ) ] C ρ 0 2 δ μ ν \begin{aligned}\displaystyle C\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)&% \displaystyle\simeq C\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\\ &\displaystyle\simeq C\left[\begin{pmatrix}\rho_{0}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}-\frac{1}{2}\begin{pmatrix}\rho_{0}&0&0&0\\ 0&-\rho_{0}&0&0\\ 0&0&-\rho_{0}&0\\ 0&0&0&-\rho_{0}\end{pmatrix}\right]\\ &\displaystyle\simeq\frac{C\rho_{0}}{2}\delta_{\mu\nu}\end{aligned}
  29. ε γ μ ν \varepsilon\gamma_{\mu\nu}
  30. R μ ν 1 2 [ ln ( - g ) ] | μ | ν | - [ μ ν , β ] | β R_{\mu\nu}\cong\frac{1}{2}\left[\ln(-g)\right]_{|\mu|\nu|}-[\mu\nu,\beta]_{|\beta}
  31. 1 2 [ ln ( - g ) ] | μ | ν | - [ μ ν , β ] | β = κ ρ 0 2 δ μ ν \frac{1}{2}\left[\ln(-g)\right]_{|\mu|\nu|}-[\mu\nu,\beta]_{|\beta}=\frac{% \kappa\,\rho_{0}}{2}\,\delta_{\mu\nu}
  32. [ 00 , β ] | β = ( g α β [ 00 , α ] ) | β = - κ ρ 0 2 ( * ) [00,\beta]_{|\beta}=\left(g^{\alpha\beta}\left[00,\alpha\right]\right)_{|\beta% }=-\frac{\kappa\,\rho_{0}}{2}\qquad\qquad(*)
  33. [ 00 , α ] = 1 2 ( g 0 α | 0 + g α 0 | 0 - g 00 | α ) \left[00,\alpha\right]=\frac{1}{2}\left(g_{0\alpha|0}+g_{\alpha 0|0}-g_{00|% \alpha}\right)
  34. [ 00 , α ] = - ε 2 γ 00 | α \left[00,\alpha\right]=-\frac{\varepsilon}{2}\gamma_{00|\alpha}
  35. γ μ ν \gamma_{\mu\nu}
  36. ε γ μ ν \varepsilon\gamma_{\mu\nu}
  37. g β α [ 00 , α ] = ε 2 γ 00 | β g^{\beta\alpha}\left[00,\alpha\right]=\frac{\varepsilon}{2}\gamma_{00|\beta}
  38. γ 00 \gamma_{00}
  39. ε β = 0 3 γ 00 | β | β = - κ ρ 0 \varepsilon\sum_{\beta=0}^{3}\gamma_{00|\beta|\beta}=-\kappa\,\rho_{0}
  40. ε β = 1 3 γ 00 | β | β = - κ ρ 0 \varepsilon\sum_{\beta=1}^{3}\gamma_{00|\beta|\beta}=-\kappa\,\rho_{0}
  41. β = 0 3 γ 00 | β | β = i = 1 3 γ 00 2 x β 2 = γ 00 2 x 1 2 + γ 00 2 x 2 2 + γ 00 2 x 3 2 = - κ ρ 0 \sum_{\beta=0}^{3}\gamma_{00|\beta|\beta}=\sum_{i=1}^{3}\frac{\partial{}^{2}% \gamma_{00}}{\partial{x_{\beta}}^{2}}=\frac{\partial{}^{2}\gamma_{00}}{% \partial{x_{1}}^{2}}+\frac{\partial{}^{2}\gamma_{00}}{\partial{x_{2}}^{2}}+% \frac{\partial{}^{2}\gamma_{00}}{\partial{x_{3}}^{2}}=-\kappa\,\rho_{0}
  42. - ε γ 00 κ = φ 4 π G -\frac{\varepsilon\gamma_{00}}{\kappa}=\frac{\varphi}{4\pi G}
  43. g μ ν = η μ ν + ε γ μ ν \displaystyle g_{\mu\nu}=\eta_{\mu\nu}+\varepsilon\gamma_{\mu\nu}
  44. d s 2 = ( d x 0 ) 2 - ( d x 1 ) 2 - ( d x 2 ) 2 - ( d x 3 ) 2 + ε γ μ ν d x μ d x ν \displaystyle\mathrm{d}s^{2}=(\mathrm{d}x^{0})^{2}-(\mathrm{d}x^{1})^{2}-(% \mathrm{d}x^{2})^{2}-(\mathrm{d}x^{3})^{2}+\varepsilon\gamma_{\mu\nu}\mathrm{d% }x^{\mu}\mathrm{d}x^{\nu}
  45. β = v c \beta=\tfrac{v}{c}
  46. x 0 = c t \displaystyle x^{0}=ct
  47. ( d s d t ) 2 = c 2 - v 2 + ε γ μ ν d x μ d t d x ν d t = c 2 ( 1 - β 2 + ε γ μ ν d x μ d x 0 d x ν d x 0 ) \begin{aligned}\displaystyle\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{2}&% \displaystyle=c^{2}-v^{2}+\varepsilon\gamma_{\mu\nu}\frac{\mathrm{d}x^{\mu}}{% \mathrm{d}t}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}t}\\ &\displaystyle=c^{2}\left(1-\beta^{2}+\varepsilon\gamma_{\mu\nu}\frac{\mathrm{% d}x^{\mu}}{\mathrm{d}x^{0}}\frac{\mathrm{d}x^{\nu}}{\mathrm{d}x^{0}}\right)% \end{aligned}
  48. ( d s d t ) 2 c 2 ( 1 + ε γ 00 ) \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^{2}\cong c^{2}(1+\varepsilon% \gamma_{00})
  49. d 2 x α d t 2 + [ 00 , α ] c 2 = 0 ( * * ) \frac{\mathrm{d}^{2}x^{\alpha}}{\mathrm{d}t^{2}}+[00,\alpha]c^{2}=0\qquad(**)
  50. [ 00 , i ] = 1 2 ε γ 00 | i [00,i]=\frac{1}{2}\varepsilon\gamma_{00|i}
  51. d 2 x i d t 2 = - c 2 2 ε γ 00 | i \displaystyle\frac{\mathrm{d}^{2}x^{i}}{\mathrm{d}t^{2}}=-\frac{c^{2}}{2}% \varepsilon\gamma_{00|i}
  52. d 2 X d t 2 = c 2 2 ε γ 00 \displaystyle\frac{\mathrm{d}^{2}X}{\mathrm{d}t^{2}}=\frac{c^{2}}{2}% \varepsilon\gamma_{00}
  53. φ = - c 2 2 ε γ 00 \varphi=-\frac{c^{2}}{2}\varepsilon\nabla\gamma_{00}
  54. g 00 = 1 + 2 φ c 2 g_{00}=1+\frac{2\varphi}{c^{2}}
  55. φ = c 2 2 ε γ 00 \varphi=\frac{c^{2}}{2}\varepsilon\gamma_{00}
  56. κ = - 8 π G c 2 \kappa\,=\,-{8\,\pi\,G\over c^{2}}~{}
  57. G α γ + Λ g α γ = - 8 π G c 2 T α γ G^{\alpha\gamma}+\Lambda g^{\alpha\gamma}=-\frac{8\pi G}{c^{2}}T^{\alpha\gamma% }~{}
  58. ( v c ) 2 (\tfrac{v}{c})^{2}
  59. ρ ( v c ) \rho(\tfrac{v}{c})
  60. T μ μ = Tr ( ρ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) = ρ 0 T^{\mu}_{\mu}=\operatorname{Tr}\begin{pmatrix}\rho_{0}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}=\rho_{0}
  61. κ = - 8 π G c 2 \kappa\,=\,-{8\,\pi\,G\over c^{2}}~{}
  62. T μ v = ρ ( c 2 v x c v y c v z c v x c v x 2 v x v y v x v z v y c v y v x v y 2 v y v z v z c v z v x v z v y v z 2 ) T_{{\mu}v}=\rho\begin{pmatrix}c^{2}&v_{x}c&v_{y}c&v_{z}c\\ v_{x}c&v^{2}_{x}&v_{x}v_{y}&v_{x}v_{z}\\ v_{y}c&v_{y}v_{x}&v^{2}_{y}&v_{y}v_{z}\\ v_{z}c&v_{z}v_{x}&v_{z}v_{y}&v^{2}_{z}\end{pmatrix}~{}
  63. T μ μ = Tr ( ρ 0 c 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) = ρ 0 c 2 T^{\mu}_{\mu}=\operatorname{Tr}\begin{pmatrix}\rho_{0}c^{2}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}=\rho_{0}c^{2}~{}
  64. κ = - 8 π G c 4 \kappa\,=\,-{8\,\pi\,G\over c^{4}}~{}
  65. G c 2 G\over c^{2}~{}

Einstein_solid.html

  1. T 3 T^{3}
  2. C V = ( U T ) V . C_{V}=\left({\partial U\over\partial T}\right)_{V}.
  3. T T
  4. 1 T = S U . {1\over T}={\partial S\over\partial U}.
  5. N N
  6. 3 N 3N
  7. N = 3 N N^{\prime}=3N
  8. E n = ω ( n + 1 2 ) E_{n}=\hbar\omega\left(n+{1\over 2}\right)
  9. ε = ω \varepsilon=\hbar\omega
  10. q q
  11. N N^{\prime}
  12. q q
  13. N N^{\prime}
  14. N - 1 N^{\prime}-1
  15. q q
  16. N - 1 N^{\prime}-1
  17. n n
  18. n ! n!
  19. q q
  20. N - 1 N^{\prime}-1
  21. ( q + N - 1 ) ! \left(q+N^{\prime}-1\right)!
  22. q ! q!
  23. ( N - 1 ) ! (N^{\prime}-1)!
  24. Ω = ( q + N - 1 ) ! q ! ( N - 1 ) ! \Omega={\left(q+N^{\prime}-1\right)!\over q!(N^{\prime}-1)!}
  25. q q
  26. N - 1 N^{\prime}-1
  27. S / k = ln Ω = ln ( q + N - 1 ) ! q ! ( N - 1 ) ! . S/k=\ln\Omega=\ln{\left(q+N^{\prime}-1\right)!\over q!(N^{\prime}-1)!}.
  28. N N^{\prime}
  29. S / k ln ( q + N ) ! q ! N ! S/k\approx\ln{\left(q+N^{\prime}\right)!\over q!N^{\prime}!}
  30. S / k ( q + N ) ln ( q + N ) - N ln N - q ln q . S/k\approx\left(q+N^{\prime}\right)\ln\left(q+N^{\prime}\right)-N^{\prime}\ln N% ^{\prime}-q\ln q.
  31. U = N ε 2 + q ε , U={N^{\prime}\varepsilon\over 2}+q\varepsilon,
  32. 1 T = S U = S q d q d U = 1 ε S q = k ε ln ( 1 + N / q ) {1\over T}={\partial S\over\partial U}={\partial S\over\partial q}{dq\over dU}% ={1\over\varepsilon}{\partial S\over\partial q}={k\over\varepsilon}\ln\left(1+% N^{\prime}/q\right)
  33. U = N ε 2 + N ε e ε / k T - 1 . U={N^{\prime}\varepsilon\over 2}+{N^{\prime}\varepsilon\over e^{\varepsilon/kT% }-1}.
  34. C V C_{V}
  35. C V = U T = N ε 2 k T 2 e ε / k T ( e ε / k T - 1 ) 2 C_{V}={\partial U\over\partial T}={N^{\prime}\varepsilon^{2}\over kT^{2}}{e^{% \varepsilon/kT}\over\left(e^{\varepsilon/kT}-1\right)^{2}}
  36. C V = 3 N k ( ε k T ) 2 e ε / k T ( e ε / k T - 1 ) 2 . C_{V}=3Nk\left({\varepsilon\over kT}\right)^{2}{e^{\varepsilon/kT}\over\left(e% ^{\varepsilon/kT}-1\right)^{2}}.
  37. Z = n = 0 e - E n / k T Z=\sum_{n=0}^{\infty}e^{-E_{n}/kT}
  38. E n = ε ( n + 1 2 ) E_{n}=\varepsilon\left(n+{1\over 2}\right)
  39. Z = n = 0 e - ε ( n + 1 / 2 ) / k T = e - ε / 2 k T n = 0 e - n ε / k T = e - ε / 2 k T n = 0 ( e - ε / k T ) n = e - ε / 2 k T 1 - e - ε / k T = 1 e ε / 2 k T - e - ε / 2 k T = 1 2 sinh ( ε 2 k T ) . \begin{aligned}\displaystyle Z=\sum_{n=0}^{\infty}e^{-\varepsilon\left(n+1/2% \right)/kT}=e^{-\varepsilon/2kT}\sum_{n=0}^{\infty}e^{-n\varepsilon/kT}=e^{-% \varepsilon/2kT}\sum_{n=0}^{\infty}\left(e^{-\varepsilon/kT}\right)^{n}\\ \displaystyle={e^{-\varepsilon/2kT}\over 1-e^{-\varepsilon/kT}}={1\over e^{% \varepsilon/2kT}-e^{-\varepsilon/2kT}}={1\over 2\sinh\left({\varepsilon\over 2% kT}\right)}.\end{aligned}
  40. N N^{\prime}
  41. E = u = - 1 Z β Z \langle E\rangle=u=-{1\over Z}\partial_{\beta}Z
  42. β = 1 k T . \beta={1\over kT}.
  43. u = - 2 sinh ( ε 2 k T ) - cosh ( ε 2 k T ) 2 sinh 2 ( ε 2 k T ) ε 2 = ε 2 coth ( ε 2 k T ) . u=-2\sinh\left({\varepsilon\over 2kT}\right){-\cosh\left({\varepsilon\over 2kT% }\right)\over 2\sinh^{2}\left({\varepsilon\over 2kT}\right)}{\varepsilon\over 2% }={\varepsilon\over 2}\coth\left({\varepsilon\over 2kT}\right).
  44. c V = u T = - ε 2 1 sinh 2 ( ε 2 k T ) ( - ε 2 k T 2 ) = k ( ε 2 k T ) 2 1 sinh 2 ( ε 2 k T ) . c_{V}={\partial u\over\partial T}=-{\varepsilon\over 2}{1\over\sinh^{2}\left({% \varepsilon\over 2kT}\right)}\left(-{\varepsilon\over 2kT^{2}}\right)=k\left({% \varepsilon\over 2kT}\right)^{2}{1\over\sinh^{2}\left({\varepsilon\over 2kT}% \right)}.
  45. C V = 3 N c V C_{V}=3Nc_{V}
  46. N N
  47. C V = 3 N k ( ε 2 k T ) 2 1 sinh 2 ( ε 2 k T ) . C_{V}=3Nk\left({\varepsilon\over 2kT}\right)^{2}{1\over\sinh^{2}\left({% \varepsilon\over 2kT}\right)}.
  48. T E = ε / k T_{E}=\varepsilon/k
  49. T / T E T/T_{E}
  50. T / T D T/T_{D}

Eisenstein's_theorem.html

  1. a n t n \sum a_{n}t^{n}

Eisenstein_series.html

  1. G 2 k ( τ ) = ( m , n ) 𝐙 2 \ ( 0 , 0 ) 1 ( m + n τ ) 2 k . G_{2k}(\tau)=\sum_{(m,n)\in\mathbf{Z}^{2}\backslash(0,0)}\frac{1}{(m+n\tau)^{2% k}}.
  2. G 2 k ( a τ + b c τ + d ) = ( c τ + d ) 2 k G 2 k ( τ ) G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2k}G_{2k}(\tau)
  3. g 2 = 60 G 4 g_{2}=60G_{4}
  4. g 3 = 140 G 6 g_{3}=140G_{6}
  5. k = 0 n ( n k ) d k d n - k = 2 n + 9 3 n + 6 d n + 2 \sum_{k=0}^{n}{n\choose k}d_{k}d_{n-k}=\frac{2n+9}{3n+6}d_{n+2}
  6. ( n k ) {n\choose k}
  7. d 0 = 3 G 4 d_{0}=3G_{4}
  8. d 1 = 5 G 6 d_{1}=5G_{6}
  9. ( z ) = 1 z 2 + z 2 k = 0 d k z 2 k k ! = 1 z 2 + k = 1 ( 2 k + 1 ) G 2 k + 2 z 2 k . \wp(z)=\frac{1}{z^{2}}+z^{2}\sum_{k=0}^{\infty}\frac{d_{k}z^{2k}}{k!}=\frac{1}% {z^{2}}+\sum_{k=1}^{\infty}(2k+1)G_{2k+2}z^{2k}.
  10. q = e 2 π i τ q=e^{2\pi i\tau}
  11. q = e i π τ q=e^{i\pi\tau}
  12. q = e 2 π i τ q=e^{2\pi i\tau}
  13. G 2 k ( τ ) = 2 ζ ( 2 k ) ( 1 + c 2 k n = 1 σ 2 k - 1 ( n ) q n ) G_{2k}(\tau)=2\zeta(2k)\left(1+c_{2k}\sum_{n=1}^{\infty}\sigma_{2k-1}(n)q^{n}\right)
  14. c 2 k = ( 2 π i ) 2 k ( 2 k - 1 ) ! ζ ( 2 k ) = - 4 k B 2 k = 2 ζ ( 1 - 2 k ) . c_{2k}=\frac{(2\pi i)^{2k}}{(2k-1)!\zeta(2k)}=\frac{-4k}{B_{2k}}=\frac{2}{% \zeta(1-2k)}.
  15. G 4 ( τ ) = π 4 45 [ 1 + 240 n = 1 σ 3 ( n ) q n ] G 6 ( τ ) = 2 π 6 945 [ 1 - 504 n = 1 σ 5 ( n ) q n ] . \begin{aligned}\displaystyle G_{4}(\tau)&\displaystyle=\frac{\pi^{4}}{45}\left% [1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}\right]\\ \displaystyle G_{6}(\tau)&\displaystyle=\frac{2\pi^{6}}{945}\left[1-504\sum_{n% =1}^{\infty}\sigma_{5}(n)q^{n}\right].\end{aligned}
  16. n = 1 q n σ a ( n ) = n = 1 n a q n 1 - q n \sum_{n=1}^{\infty}q^{n}\sigma_{a}(n)=\sum_{n=1}^{\infty}\frac{n^{a}q^{n}}{1-q% ^{n}}
  17. E 2 k ( τ ) = G 2 k ( τ ) 2 ζ ( 2 k ) = 1 + 2 ζ ( 1 - 2 k ) n = 1 n 2 k - 1 q n 1 - q n = 1 - 4 k B 2 k d , n 1 n 2 k - 1 q n d E_{2k}(\tau)=\frac{G_{2k}(\tau)}{2\zeta(2k)}=1+\frac{2}{\zeta(1-2k)}\sum_{n=1}% ^{\infty}\frac{n^{2k-1}q^{n}}{1-q^{n}}=1-\frac{4k}{B_{2k}}\sum_{d,n\geq 1}n^{2% k-1}q^{nd}
  18. q = e 2 π i τ q=e^{2\pi i\tau}
  19. E 4 ( τ ) = 1 + 240 n = 1 n 3 q n 1 - q n E_{4}(\tau)=1+240\sum_{n=1}^{\infty}\frac{n^{3}q^{n}}{1-q^{n}}
  20. E 6 ( τ ) = 1 - 504 n = 1 n 5 q n 1 - q n E_{6}(\tau)=1-504\sum_{n=1}^{\infty}\frac{n^{5}q^{n}}{1-q^{n}}
  21. E 8 ( τ ) = 1 + 480 n = 1 n 7 q n 1 - q n E_{8}(\tau)=1+480\sum_{n=1}^{\infty}\frac{n^{7}q^{n}}{1-q^{n}}
  22. a = θ 2 ( 0 ; e π i τ ) = ϑ 10 ( 0 ; τ ) a=\theta_{2}(0;e^{\pi i\tau})=\vartheta_{10}(0;\tau)
  23. b = θ 3 ( 0 ; e π i τ ) = ϑ 00 ( 0 ; τ ) b=\theta_{3}(0;e^{\pi i\tau})=\vartheta_{00}(0;\tau)
  24. c = θ 4 ( 0 ; e π i τ ) = ϑ 01 ( 0 ; τ ) c=\theta_{4}(0;e^{\pi i\tau})=\vartheta_{01}(0;\tau)
  25. θ m \theta_{m}
  26. ϑ i j \vartheta_{ij}
  27. E 4 ( τ ) = 1 2 ( a 8 + b 8 + c 8 ) E_{4}(\tau)=\tfrac{1}{2}(a^{8}+b^{8}+c^{8})
  28. E 6 ( τ ) = 1 2 ( - 3 a 8 ( b 4 + c 4 ) + b 12 + c 12 ) = 1 2 ( a 8 + b 8 + c 8 ) 3 - 54 ( a b c ) 8 2 \begin{aligned}\displaystyle E_{6}(\tau)&\displaystyle=\tfrac{1}{2}\big(-3a^{8% }(b^{4}+c^{4})+b^{12}+c^{12}\big)\\ &\displaystyle=\tfrac{1}{2}\sqrt{\frac{(a^{8}+b^{8}+c^{8})^{3}-54(abc)^{8}}{2}% }\end{aligned}
  29. E 4 3 - E 6 2 = 27 4 ( a b c ) 8 E_{4}^{3}-E_{6}^{2}=\tfrac{27}{4}(abc)^{8}
  30. Δ = g 2 3 - 27 g 3 2 = ( 2 π ) 12 ( 1 2 a b c ) 8 \Delta=g_{2}^{3}-27g_{3}^{2}=(2\pi)^{12}\left(\tfrac{1}{2}abc\right)^{8}
  31. E 8 = E 4 2 E_{8}=E_{4}^{2}
  32. a 4 - b 4 + c 4 = 0 a^{4}-b^{4}+c^{4}=0
  33. E 8 ( τ ) = 1 2 ( a 16 + b 16 + c 16 ) E_{8}(\tau)=\tfrac{1}{2}(a^{16}+b^{16}+c^{16})
  34. E 4 2 = E 8 , E 4 E 6 = E 10 , E 4 E 10 = E 14 , E 6 E 8 = E 14 . E_{4}^{2}=E_{8},\quad E_{4}E_{6}=E_{10},\quad E_{4}E_{10}=E_{14},\quad E_{6}E_% {8}=E_{14}.
  35. ( 1 + 240 n = 1 σ 3 ( n ) q n ) 2 = 1 + 480 n = 1 σ 7 ( n ) q n , (1+240\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n})^{2}=1+480\sum_{n=1}^{\infty}% \sigma_{7}(n)q^{n},
  36. σ 7 ( n ) = σ 3 ( n ) + 120 m = 1 n - 1 σ 3 ( m ) σ 3 ( n - m ) , \sigma_{7}(n)=\sigma_{3}(n)+120\sum_{m=1}^{n-1}\sigma_{3}(m)\sigma_{3}(n-m),
  37. θ Γ ( τ ) = 1 + n = 1 r Γ ( 2 n ) q n = E 4 ( τ ) , r Γ ( n ) = 240 σ 3 ( n ) \theta_{\Gamma}(\tau)=1+\sum_{n=1}^{\infty}r_{\Gamma}(2n)q^{n}=E_{4}(\tau),% \quad r_{\Gamma}(n)=240\sigma_{3}(n)
  38. E 8 \displaystyle E_{8}
  39. Δ 2 = - 691 1728 2 250 det | E 4 E 6 E 8 E 6 E 8 E 10 E 8 E 10 E 12 | \Delta^{2}=-\frac{691}{1728^{2}\cdot 250}\det\begin{vmatrix}E_{4}&E_{6}&E_{8}% \\ E_{6}&E_{8}&E_{10}\\ E_{8}&E_{10}&E_{12}\end{vmatrix}
  40. Δ = E 4 3 - E 6 2 1728 \Delta=\frac{E_{4}^{3}-E_{6}^{2}}{1728}
  41. L ( q ) = 1 - 24 n = 1 n q n 1 - q n = E 2 ( τ ) L(q)=1-24\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}=E_{2}(\tau)
  42. M ( q ) = 1 + 240 n = 1 n 3 q n 1 - q n = E 4 ( τ ) M(q)=1+240\sum_{n=1}^{\infty}\frac{n^{3}q^{n}}{1-q^{n}}=E_{4}(\tau)
  43. N ( q ) = 1 - 504 n = 1 n 5 q n 1 - q n = E 6 ( τ ) , N(q)=1-504\sum_{n=1}^{\infty}\frac{n^{5}q^{n}}{1-q^{n}}=E_{6}(\tau),
  44. q d L d q = L 2 - M 12 q\frac{dL}{dq}=\frac{L^{2}-M}{12}
  45. q d M d q = L M - N 3 q\frac{dM}{dq}=\frac{LM-N}{3}
  46. q d N d q = L N - M 2 2 . q\frac{dN}{dq}=\frac{LN-M^{2}}{2}.
  47. σ p ( 0 ) = 1 2 ζ ( - p ) . \sigma_{p}(0)=\frac{1}{2}\zeta(-p).\;
  48. σ ( 0 ) \displaystyle\sigma(0)
  49. k = 0 n σ ( k ) σ ( n - k ) = 5 12 σ 3 ( n ) - 1 2 n σ ( n ) . \sum_{k=0}^{n}\sigma(k)\sigma(n-k)=\frac{5}{12}\sigma_{3}(n)-\frac{1}{2}n% \sigma(n).
  50. k = 0 n σ 3 ( k ) σ 3 ( n - k ) = 1 120 σ 7 ( n ) \sum_{k=0}^{n}\sigma_{3}(k)\sigma_{3}(n-k)=\frac{1}{120}\sigma_{7}(n)
  51. k = 0 n σ ( 2 k + 1 ) σ 3 ( n - k ) = 1 240 σ 5 ( 2 n + 1 ) \sum_{k=0}^{n}\sigma(2k+1)\sigma_{3}(n-k)=\frac{1}{240}\sigma_{5}(2n+1)
  52. k = 0 n σ ( 3 k + 1 ) σ ( 3 n - 3 k + 1 ) = 1 9 σ 3 ( 3 n + 2 ) . \sum_{k=0}^{n}\sigma(3k+1)\sigma(3n-3k+1)=\frac{1}{9}\sigma_{3}(3n+2).

Electric-field_integral_equation.html

  1. e + j w t e^{+jwt}\,
  2. ϵ \epsilon\,
  3. μ \mu\,
  4. × 𝐄 = - j ω μ 𝐇 \nabla\times\,\textbf{E}=-j\omega\mu\,\textbf{H}\,
  5. × 𝐇 = j ω ϵ 𝐄 + 𝐉 \nabla\times\,\textbf{H}=j\omega\epsilon\,\textbf{E}+\,\textbf{J}\,
  6. 𝐇 = 0 \nabla\cdot\,\textbf{H}=0\,
  7. × 𝐀 = 𝐇 \nabla\times\,\textbf{A}=\,\textbf{H}\,
  8. × ( 𝐄 + j ω μ 𝐀 ) = 0 \nabla\times(\,\textbf{E}+j\omega\mu\,\textbf{A})=0\,
  9. 𝐄 + j ω μ 𝐀 = - Φ \,\textbf{E}+j\omega\mu\,\textbf{A}=-\nabla\Phi
  10. Φ \Phi
  11. × × 𝐀 - k 2 𝐀 = 𝐉 - j ω ϵ Φ \nabla\times\nabla\times\,\textbf{A}-k^{2}\,\textbf{A}=\,\textbf{J}-j\omega% \epsilon\nabla\Phi\,
  12. k = ω μ ϵ k=\omega\sqrt{\mu\epsilon}
  13. ( 𝐀 ) - 2 𝐀 - k 2 𝐀 = 𝐉 - j ω ϵ Φ \nabla(\nabla\cdot\,\textbf{A})-\nabla^{2}\,\textbf{A}-k^{2}\,\textbf{A}=\,% \textbf{J}-j\omega\epsilon\nabla\Phi\,
  14. 𝐀 = - j ω ϵ Φ \nabla\cdot\,\textbf{A}=-j\omega\epsilon\Phi\,
  15. 2 𝐀 + k 2 𝐀 = - 𝐉 \nabla^{2}\,\textbf{A}+k^{2}\,\textbf{A}=-\,\textbf{J}\,
  16. 𝐀 ( 𝐫 ) = 1 4 π 𝐉 ( 𝐫 ) G ( 𝐫 , 𝐫 ) d 𝐫 \,\textbf{A}(\,\textbf{r})=\frac{1}{4\pi}\iiint\,\textbf{J}(\,\textbf{r}^{% \prime})\ G(\,\textbf{r},\,\textbf{r}^{\prime})\ d\,\textbf{r}^{\prime}\,
  17. G ( 𝐫 , 𝐫 ) G(\,\textbf{r},\,\textbf{r}^{\prime})\,
  18. G ( 𝐫 , 𝐫 ) = e - j k | 𝐫 - 𝐫 | | 𝐫 - 𝐫 | G(\,\textbf{r},\,\textbf{r}^{\prime})=\frac{e^{-jk|\,\textbf{r}-\,\textbf{r}^{% \prime}|}}{|\,\textbf{r}-\,\textbf{r}^{\prime}|}\,
  19. 𝐄 = - j ω μ 𝐀 + 1 j ω ϵ ( 𝐀 ) \,\textbf{E}=-j\omega\mu\,\textbf{A}+\frac{1}{j\omega\epsilon}\nabla(\nabla% \cdot\,\textbf{A})\,
  20. 𝐄 = - j ω μ V d 𝐫 𝐆 ( 𝐫 , 𝐫 ) 𝐉 ( 𝐫 ) \,\textbf{E}=-j\omega\mu\int_{V}d\,\textbf{r}^{\prime}\,\textbf{G}(\,\textbf{r% },\,\textbf{r}^{\prime})\cdot\,\textbf{J}(\,\textbf{r}^{\prime})\,
  21. 𝐆 ( 𝐫 , 𝐫 ) \,\textbf{G}(\,\textbf{r},\,\textbf{r}^{\prime})\,
  22. 𝐆 ( 𝐫 , 𝐫 ) = 1 4 π [ 𝐈 + k 2 ] G ( 𝐫 , 𝐫 ) \,\textbf{G}(\,\textbf{r},\,\textbf{r}^{\prime})=\frac{1}{4\pi}\left[\,\textbf% {I}+\frac{\nabla\nabla}{k^{2}}\right]G(\,\textbf{r},\,\textbf{r}^{\prime})\,
  23. E s E_{s}
  24. E i E_{i}
  25. E i E_{i}
  26. 𝐀 \nabla\cdot\mathbf{A}
  27. 𝐀 \nabla\cdot\mathbf{A}
  28. 𝐀 \nabla\cdot\mathbf{A}
  29. 𝐀 \mathbf{A}
  30. 𝐀 \mathbf{A}
  31. 𝐀 \mathbf{A}
  32. 𝐀 \mathbf{A}
  33. 𝐄 \mathbf{E}
  34. 𝐁 \mathbf{B}
  35. 𝐄 \mathbf{E}
  36. 𝐁 \mathbf{B}
  37. 𝐀 \mathbf{A}
  38. × 𝐀 \nabla\times\mathbf{A}

Electric_flux.html

  1. 𝐒 \mathbf{S}
  2. Φ E = 𝐄 𝐒 = E S cos θ , \Phi_{E}=\mathbf{E}\cdot\mathbf{S}=ES\cos\theta,
  3. 𝐄 \mathbf{E}
  4. V / m V/m
  5. E E
  6. S S
  7. θ θ
  8. S S
  9. d 𝐒 d\mathbf{S}
  10. d Φ E = 𝐄 d 𝐒 d\Phi_{E}=\mathbf{E}\cdot d\mathbf{S}
  11. 𝐄 \mathbf{E}
  12. S S
  13. Φ E = S 𝐄 d 𝐒 \Phi_{E}=\iint_{S}\mathbf{E}\cdot d\mathbf{S}
  14. 𝐄 \mathbf{E}
  15. d 𝐒 d\mathbf{S}
  16. S S
  17. 𝐄 \mathbf{E}
  18. S S
  19. Q Q
  20. S S
  21. ε < s u b > 0 ε<sub>0

Electric_potential_energy.html

  1. Φ \scriptstyle\Phi
  2. Φ \scriptstyle\Phi
  3. k e = 1 4 π ε 0 k_{e}=\frac{1}{4\pi\varepsilon_{0}}
  4. 𝐅 = q 𝐄 \mathbf{F}=q\mathbf{E}
  5. U E ( r ) - U E ( r ref ) = - W r ref r = - r ref r q 𝐄 d 𝐬 U_{E}(r)-U_{E}(r_{\rm ref})=-W_{r_{\rm ref}\rightarrow r}=-\int_{{r}_{\rm ref}% }^{r}q\mathbf{E}\cdot\mathrm{d}\mathbf{s}
  6. W r ref r \scriptstyle W_{r_{\rm ref}\rightarrow r}
  7. U E ( r ref = ) = 0 U_{E}(r_{\rm ref}=\infty)=0
  8. U E ( r ) = - r q 𝐄 d 𝐬 U_{E}(r)=-\int_{\infty}^{r}q\mathbf{E}\cdot\mathrm{d}\mathbf{s}
  9. 𝐄 d 𝐬 = | 𝐄 | | d 𝐬 | cos ( 0 ) = E d s \mathbf{E}\cdot\mathrm{d}\mathbf{s}=|\mathbf{E}|\cdot|\mathrm{d}\mathbf{s}|% \cos(0)=E\mathrm{d}s
  10. | 𝐄 | = E = 1 4 π ε 0 Q s 2 |\mathbf{E}|=E=\frac{1}{4\pi\varepsilon_{0}}\frac{Q}{s^{2}}
  11. U E ( r ) = - r q 𝐄 d 𝐬 = - r 1 4 π ε 0 q Q s 2 d s = 1 4 π ε 0 q Q r = k e q Q r U_{E}(r)=-\int_{\infty}^{r}q\mathbf{E}\cdot\mathrm{d}\mathbf{s}=-\int_{\infty}% ^{r}\frac{1}{4\pi\varepsilon_{0}}\frac{qQ}{s^{2}}{\rm d}s=\frac{1}{4\pi% \varepsilon_{0}}\frac{qQ}{r}=k_{e}\frac{qQ}{r}
  12. k e = 1 4 π ε 0 k_{e}=\frac{1}{4\pi\varepsilon_{0}}
  13. Φ ( 𝐫 i ) = j = 1 N ( j i ) k e q j 𝐫 i j \Phi(\mathbf{r}_{i})=\sum_{j=1}^{N(j\neq i)}k_{e}\frac{q_{j}}{\mathbf{r}_{ij}}
  14. U E = q 2 Φ 1 ( 𝐫 2 ) . U_{\mathrm{E}}=q_{2}\Phi_{1}(\mathbf{r}_{2}).
  15. U E = q 1 Φ 2 ( 𝐫 1 ) . U_{\mathrm{E}}=q_{1}\Phi_{2}(\mathbf{r}_{1}).
  16. U E = 1 2 i = 1 N q i Φ ( 𝐫 i ) U_{\mathrm{E}}=\frac{1}{2}\sum_{i=1}^{N}q_{i}\Phi(\mathbf{r}_{i})
  17. Φ ( r ) = k e Q 1 r \Phi(r)=k_{e}\frac{Q_{1}}{r}
  18. U E = 1 4 π ε 0 q Q 1 r 1 U_{E}=\frac{1}{4\pi\varepsilon_{0}}\frac{qQ_{1}}{r_{1}}
  19. U E = 1 4 π ε 0 ( Q 1 Q 2 r 12 + Q 1 Q 3 r 13 + Q 2 Q 3 r 23 ) U_{\mathrm{E}}=\frac{1}{4\pi\varepsilon_{0}}\left(\frac{Q_{1}Q_{2}}{r_{12}}+% \frac{Q_{1}Q_{3}}{r_{13}}+\frac{Q_{2}Q_{3}}{r_{23}}\right)
  20. U E = 1 2 ( Q 1 Φ ( 𝐫 1 ) + Q 2 Φ ( 𝐫 2 ) + Q 3 Φ ( 𝐫 3 ) ) U_{\mathrm{E}}=\frac{1}{2}(Q_{1}\Phi(\mathbf{r}_{1})+Q_{2}\Phi(\mathbf{r}_{2})% +Q_{3}\Phi(\mathbf{r}_{3}))
  21. Φ ( 𝐫 1 ) \Phi(\mathbf{r}_{1})
  22. Φ ( 𝐫 2 ) \Phi(\mathbf{r}_{2})
  23. Φ ( 𝐫 3 ) \Phi(\mathbf{r}_{3})
  24. Φ ( 𝐫 1 ) = Φ 2 ( 𝐫 1 ) + Φ 3 ( 𝐫 1 ) = 1 4 π ε 0 Q 2 r 12 + 1 4 π ε 0 Q 3 r 13 \Phi(\mathbf{r}_{1})=\Phi_{2}(\mathbf{r}_{1})+\Phi_{3}(\mathbf{r}_{1})=\frac{1% }{4\pi\varepsilon_{0}}\frac{Q_{2}}{r_{12}}+\frac{1}{4\pi\varepsilon_{0}}\frac{% Q_{3}}{r_{13}}
  25. Φ ( 𝐫 2 ) = Φ 1 ( 𝐫 2 ) + Φ 3 ( 𝐫 2 ) = 1 4 π ε 0 Q 1 r 21 + 1 4 π ε 0 Q 3 r 23 \Phi(\mathbf{r}_{2})=\Phi_{1}(\mathbf{r}_{2})+\Phi_{3}(\mathbf{r}_{2})=\frac{1% }{4\pi\varepsilon_{0}}\frac{Q_{1}}{r_{21}}+\frac{1}{4\pi\varepsilon_{0}}\frac{% Q_{3}}{r_{23}}
  26. Φ ( 𝐫 3 ) = Φ 1 ( 𝐫 3 ) + Φ 2 ( 𝐫 3 ) = 1 4 π ε 0 Q 1 r 31 + 1 4 π ε 0 Q 2 r 32 \Phi(\mathbf{r}_{3})=\Phi_{1}(\mathbf{r}_{3})+\Phi_{2}(\mathbf{r}_{3})=\frac{1% }{4\pi\varepsilon_{0}}\frac{Q_{1}}{r_{31}}+\frac{1}{4\pi\varepsilon_{0}}\frac{% Q_{2}}{r_{32}}
  27. U E = 1 2 1 4 π ε 0 ( Q 1 Q 2 r 12 + Q 1 Q 3 r 13 + Q 2 Q 1 r 21 + Q 2 Q 3 r 23 + Q 3 Q 1 r 31 + Q 3 Q 2 r 32 ) U_{\mathrm{E}}=\frac{1}{2}\frac{1}{4\pi\varepsilon_{0}}(\frac{Q_{1}Q_{2}}{r_{1% 2}}+\frac{Q_{1}Q_{3}}{r_{13}}+\frac{Q_{2}Q_{1}}{r_{21}}+\frac{Q_{2}Q_{3}}{r_{2% 3}}+\frac{Q_{3}Q_{1}}{r_{31}}+\frac{Q_{3}Q_{2}}{r_{32}})
  28. U E = 1 4 π ε 0 ( Q 1 Q 2 r 12 + Q 1 Q 3 r 13 + Q 2 Q 3 r 23 ) U_{\mathrm{E}}=\frac{1}{4\pi\varepsilon_{0}}(\frac{Q_{1}Q_{2}}{r_{12}}+\frac{Q% _{1}Q_{3}}{r_{13}}+\frac{Q_{2}Q_{3}}{r_{23}})
  29. d U d V \frac{dU}{dV}
  30. u e = d U d V = 1 2 ε 0 | 𝐄 | 2 . u_{e}=\frac{dU}{dV}=\frac{1}{2}\varepsilon_{0}\left|{\mathbf{E}}\right|^{2}.
  31. 𝐄 = ρ ε 0 \mathbf{\nabla}\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  32. 𝐄 \mathbf{E}
  33. ρ \rho
  34. ε 0 \varepsilon_{0}
  35. U \displaystyle U
  36. ( A B ) = ( A ) B + A ( B ) ( A ) B = ( A B ) - A ( B ) \nabla\cdot({A}{B})=(\nabla\cdot{A}){B}+{A}\cdot(\nabla{B})\Rightarrow(\nabla% \cdot{A}){B}=\nabla\cdot({A}{B})-{A}\cdot(\nabla{B})
  37. U = ε 0 2 all space ( 𝐄 Φ ) d V - ε 0 2 all space ( Φ ) 𝐄 d V U=\frac{\varepsilon_{0}}{2}\int\limits_{\,\text{all space}}\mathbf{\nabla}% \cdot(\mathbf{E}\Phi)dV-\frac{\varepsilon_{0}}{2}\int\limits_{\,\text{all % space}}(\mathbf{\nabla}\Phi)\cdot\mathbf{E}dV
  38. Φ ( ) = 0 \Phi(\infty)=0
  39. U \displaystyle U
  40. d U d V \frac{dU}{dV}
  41. u e = 1 2 ε 0 | 𝐄 | 2 . u_{e}=\frac{1}{2}\varepsilon_{0}\left|{\mathbf{E}}\right|^{2}.
  42. U E = 1 2 Q V = 1 2 C V 2 = Q 2 2 C U_{E}=\frac{1}{2}QV=\frac{1}{2}CV^{2}=\frac{Q^{2}}{2C}

Electrical_mobility.html

  1. v d = μ E \,v_{d}=\mu E
  2. v d \,v_{d}
  3. E \,E
  4. μ \,\mu
  5. μ = v d E \,\mu=\frac{v_{d}}{E}
  6. μ = q m ν m \mu=\frac{q}{m\,\nu_{m}}
  7. q \,q
  8. ν m \,\nu_{m}
  9. m \,m
  10. D \,D
  11. μ = q k T D \mu=\frac{q}{k\,T}D
  12. k \,k
  13. T \,T
  14. D \,D
  15. D = π 8 λ 2 ν m D=\frac{\pi}{8}\lambda^{2}\nu_{m}
  16. λ \,\lambda
  17. D = 1 2 λ v D=\frac{1}{2}\lambda\,v
  18. v \,v
  19. v = 3 k T m v=\sqrt{{3\,k\,T}\over{m}}
  20. m \,m

Electrohydrodynamics.html

  1. F = I d k F=\frac{Id}{k}
  2. F F
  3. I I
  4. d d
  5. k k

Electromagnetic_shielding.html

  1. μ \mu
  2. a a
  3. b b
  4. H 0 = H 0 z ^ = H 0 cos θ r ^ \vec{H}_{0}=H_{0}\hat{z}=H_{0}\cos\theta\hat{r}
  5. H = - Φ M \vec{H}=-\nabla\Phi_{M}
  6. 2 Φ M = 0 \nabla^{2}\Phi_{M}=0
  7. B = μ H \vec{B}=\mu\vec{H}
  8. Φ M = l = 0 ( A l r l + B l r l + 1 ) P l ( cos θ ) \Phi_{M}=\sum_{l=0}^{\infty}\left(A_{l}r^{l}+\frac{B_{l}}{r^{l+1}}\right)P_{l}% (\cos\theta)
  9. ( H 2 - H 1 ) × n ^ = 0 (\vec{H_{2}}-\vec{H_{1}})\times\hat{n}=0
  10. ( B 2 - B 1 ) n ^ = 0 (\vec{B_{2}}-\vec{B_{1}})\cdot\hat{n}=0
  11. n ^ \hat{n}
  12. H i n = η H 0 \vec{H_{in}}=\eta\vec{H_{0}}
  13. η \eta
  14. η = 9 μ ( 2 μ + 1 ) ( μ + 2 ) - 2 ( a b ) 3 ( μ - 1 ) 2 \eta=\frac{9\mu}{(2\mu+1)(\mu+2)-2\left(\frac{a}{b}\right)^{3}(\mu-1)^{2}}
  15. μ 1 \mu\rightarrow 1
  16. μ 0 , \mu\rightarrow 0,\infty
  17. η = 9 2 1 ( 1 - a 3 b 3 ) μ \eta=\frac{9}{2}\frac{1}{(1-\frac{a^{3}}{b^{3}})\mu}
  18. μ - 1 \mu^{-1}

Electromagnetic_tensor.html

  1. F = def d A . F\ \stackrel{\mathrm{def}}{=}\ \mathrm{d}A.
  2. F μ ν = μ A ν - ν A μ . F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.
  3. E i = c F 0 i , E_{i}=cF_{0i},
  4. B i = - 1 2 ϵ i j k F j k , B_{i}=-\frac{1}{2}\epsilon_{ijk}F^{jk},
  5. ϵ i j k \epsilon_{ijk}
  6. [ 0 - E x / c - E y / c - E z / c E x / c 0 - B z B y E y / c B z 0 - B x E z / c - B y B x 0 ] = F μ ν . \begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\ E_{x}/c&0&-B_{z}&B_{y}\\ E_{y}/c&B_{z}&0&-B_{x}\\ E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}=F^{\mu\nu}.
  7. F μ ν = η μ α F α β η β ν = [ 0 E x / c E y / c E z / c - E x / c 0 - B z B y - E y / c B z 0 - B x - E z / c - B y B x 0 ] . F_{\mu\nu}=\eta_{\mu\alpha}F^{\alpha\beta}\eta_{\beta\nu}=\begin{bmatrix}0&E_{% x}/c&E_{y}/c&E_{z}/c\\ -E_{x}/c&0&-B_{z}&B_{y}\\ -E_{y}/c&B_{z}&0&-B_{x}\\ -E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}.
  8. d p μ d τ = q F μ u ν ν \frac{dp^{\mu}}{d\tau}=qF^{\mu}{}_{\nu}u^{\nu}
  9. F μ = ν F μ β η β ν = [ 0 E x / c E y / c E z / c E x / c 0 B z - B y E y / c - B z 0 B x E z / c B y - B x 0 ] . F^{\mu}{}_{\nu}=F^{\mu\beta}\eta_{\beta\nu}=\begin{bmatrix}0&E_{x}/c&E_{y}/c&E% _{z}/c\\ E_{x}/c&0&B_{z}&-B_{y}\\ E_{y}/c&-B_{z}&0&B_{x}\\ E_{z}/c&B_{y}&-B_{x}&0\end{bmatrix}.
  10. 𝐄 = ρ ϵ 0 , × 𝐁 - 1 c 2 𝐄 t = μ 0 𝐉 \nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}},\quad\nabla\times\mathbf{B}-% \frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=\mu_{0}\mathbf{J}
  11. α F α β = μ 0 J β \partial_{\alpha}F^{\alpha\beta}=\mu_{0}J^{\beta}
  12. J α = ( c ρ , 𝐉 ) J^{\alpha}=(c\rho,\mathbf{J})
  13. 𝐁 = 0 , 𝐁 t + × 𝐄 = 0 \nabla\cdot\mathbf{B}=0,\quad\frac{\partial\mathbf{B}}{\partial t}+\nabla% \times\mathbf{E}=0
  14. γ F α β + α F β γ + β F γ α = 0 \partial_{\gamma}F_{\alpha\beta}+\partial_{\alpha}F_{\beta\gamma}+\partial_{% \beta}F_{\gamma\alpha}=0
  15. [ α F β γ ] = 0 \partial_{[\alpha}F_{\beta\gamma]}=0
  16. α J α = J α = , α 0 \partial_{\alpha}J^{\alpha}{}=J^{\alpha}{}_{,\alpha}=0
  17. F [ α β ; γ ] = 0 F_{[\alpha\beta;\gamma]}=0
  18. F α β = ; α μ 0 J β F^{\alpha\beta}{}_{;\alpha}\,=\mu_{0}J^{\beta}
  19. J α = ; α 0 J^{\alpha}{}_{;\alpha}\,=0
  20. 𝒮 = ( - 1 4 μ 0 F μ ν F μ ν - J μ A μ ) d 4 x \mathcal{S}=\int\left(-\begin{matrix}\frac{1}{4\mu_{0}}\end{matrix}F_{\mu\nu}F% ^{\mu\nu}-J^{\mu}A_{\mu}\right)\mathrm{d}^{4}x\,
  21. d 4 x \mathrm{d}^{4}x\;
  22. = - 1 4 μ 0 F μ ν F μ ν - J μ A μ = - 1 4 μ 0 ( μ A ν - ν A μ ) ( μ A ν - ν A μ ) - J μ A μ = - 1 4 μ 0 ( μ A ν μ A ν - ν A μ μ A ν - μ A ν ν A μ + ν A μ ν A μ ) - J μ A μ \begin{aligned}\displaystyle\mathcal{L}&\displaystyle=-\frac{1}{4\mu_{0}}F_{% \mu\nu}F^{\mu\nu}-J^{\mu}A_{\mu}\\ &\displaystyle=-\frac{1}{4\mu_{0}}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_% {\mu}\right)\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)-J^{\mu}A_% {\mu}\\ &\displaystyle=-\frac{1}{4\mu_{0}}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{% \nu}-\partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu}-\partial_{\mu}A_{\nu}\partial^% {\nu}A^{\mu}+\partial_{\nu}A_{\mu}\partial^{\nu}A^{\mu}\right)-J^{\mu}A_{\mu}% \\ \end{aligned}
  23. = - 1 2 μ 0 ( μ A ν μ A ν - ν A μ μ A ν ) - J μ A μ . \mathcal{L}=-\frac{1}{2\mu_{0}}\left(\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu% }-\partial_{\nu}A_{\mu}\partial^{\mu}A^{\nu}\right)-J^{\mu}A_{\mu}.
  24. μ ( ( μ A ν ) ) - A ν = 0 \partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}A_{\nu})% }\right)-\frac{\partial\mathcal{L}}{\partial A_{\nu}}=0
  25. - μ 1 μ 0 ( μ A ν - ν A μ ) + J ν = 0. -\partial_{\mu}\frac{1}{\mu_{0}}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{% \mu}\right)+J^{\nu}=0.\,
  26. μ F μ ν = μ 0 J ν \partial_{\mu}F^{\mu\nu}=\mu_{0}J^{\nu}
  27. E i / c = - F 0 i ~{}E^{i}/c=-F^{0i}\,
  28. ϵ i j k B k = - F i j \epsilon^{ijk}B_{k}=-F^{ij}\,
  29. = ψ ¯ ( i c γ α D α - m c 2 ) ψ - 1 4 μ 0 F α β F α β , \mathcal{L}=\bar{\psi}(i\hbar c\,\gamma^{\alpha}D_{\alpha}-mc^{2})\psi-\frac{1% }{4\mu_{0}}F_{\alpha\beta}F^{\alpha\beta},

Electromagnetically_induced_transparency.html

  1. | 3 |3\rangle
  2. | 3 |3\rangle
  3. | 1 |1\rangle
  4. | 2 |2\rangle
  5. | 1 |1\rangle
  6. | 2 |2\rangle
  7. | 1 |1\rangle
  8. | 2 |2\rangle
  9. | 3 |3\rangle
  10. | 2 |2\rangle
  11. | 3 |3\rangle
  12. | 1 |1\rangle
  13. | 2 |2\rangle

Electron_magnetic_moment.html

  1. s y m b o l μ = - e 2 m e 𝐋 . symbol{\mu}=\frac{-e}{2m\text{e}}\,\mathbf{L}.
  2. s y m b o l μ = g - e 2 m e 𝐋 . symbol{\mu}=g\frac{-e}{2m\text{e}}\mathbf{L}.
  3. s y m b o l μ = - g μ B 𝐋 . symbol{\mu}=-g\mu\text{B}\frac{\mathbf{L}}{\hbar}.
  4. s y m b o l μ s = - g s μ B 𝐒 . symbol{\mu}\text{s}=-g\text{s}\mu\text{B}\frac{\mathbf{S}}{\hbar}.
  5. μ S 2 e 2 m e 2 = μ B . \mu\text{S}\approx 2\frac{e\hbar}{2m\text{e}}\frac{\frac{\hbar}{2}}{\hbar}=\mu% \text{B}.
  6. ( s y m b o l μ s ) z = - g s μ B m s (symbol{\mu}\text{s})_{z}=-g\text{s}\mu\text{B}m\text{s}
  7. ρ e ( r ) = e N e e - r 2 r e 2 \rho\text{e}(r)=eN\text{e}e^{-\frac{r^{2}}{r\text{e}^{2}}}
  8. ρ m ( r ) = m e N m e - r 2 r m 2 \rho\text{m}(r)=m\text{e}N\text{m}e^{-\frac{r^{2}}{r\text{m}^{2}}}
  9. r m r\text{m}
  10. r e r\text{e}
  11. g = ( r e r m ) 8 g=\left(\frac{r\text{e}}{r\text{m}}\right)^{8}
  12. g = 2 g=2
  13. ( r e r m ) 1.09051 \left(\frac{r\text{e}}{r\text{m}}\right)\approx 1.09051
  14. s y m b o l μ L = - g L μ B 𝐋 . symbol{\mu}\text{L}=-g\text{L}\mu\text{B}\frac{\mathbf{L}}{\hbar}.
  15. s y m b o l μ J = g J μ B 𝐉 . symbol{\mu}\text{J}=g\text{J}\mu\text{B}\frac{\mathbf{J}}{\hbar}.
  16. μ L = - g L μ B Ψ n , , m | L | Ψ n , , m = - μ B ( + 1 ) . \mu\text{L}=-g\text{L}\frac{\mu\text{B}}{\hbar}\langle\Psi_{n,\ell,m}|L|\Psi_{% n,\ell,m}\rangle=-\mu\text{B}\sqrt{\ell(\ell+1)}.
  17. ( μ L ) z = - μ B m . (\mathbf{\mu\text{L}})_{z}=-\mu\text{B}m_{\ell}.
  18. 1 / 2 {1}/{2}
  19. H = 1 2 m [ s y m b o l σ ( 𝐩 - e c 𝐀 ) ] 2 + e ϕ . H=\frac{1}{2m}\left[symbol{\sigma}\cdot\left(\mathbf{p}-\frac{e}{c}\mathbf{A}% \right)\right]^{2}+e\phi.
  20. H = 1 2 m ( 𝐩 - e c 𝐀 ) 2 + e ϕ - e 2 m c s y m b o l σ 𝐁 . H=\frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)^{2}+e\phi-\frac{e% \hbar}{2mc}symbol{\sigma}\cdot\mathbf{B}.
  21. [ - i γ μ ( μ + i e A μ ) + m ] ψ = 0 \left[-i\gamma^{\mu}\left(\partial_{\mu}+ieA_{\mu}\right)+m\right]\psi=0\,
  22. γ μ \scriptstyle\gamma^{\mu}
  23. ( ( m c 2 - E + e ϕ ) c σ ( 𝐩 - e c 𝐀 ) - c s y m b o l σ ( 𝐩 - e c 𝐀 ) ( m c 2 + E - e ϕ ) ) ( ψ + ψ - ) = ( 0 0 ) . \begin{pmatrix}(mc^{2}-E+e\phi)&c\sigma\cdot\left(\mathbf{p}-\frac{e}{c}% \mathbf{A}\right)\\ -csymbol{\sigma}\cdot\left(\mathbf{p}-\frac{e}{c}\mathbf{A}\right)&\left(mc^{2% }+E-e\phi\right)\end{pmatrix}\begin{pmatrix}\psi_{+}\\ \psi_{-}\end{pmatrix}=\begin{pmatrix}0\\ 0\end{pmatrix}.
  24. ( E - e ϕ ) ψ + - c s y m b o l σ ( 𝐩 - e c 𝐀 ) ψ - = m c 2 ψ + - ( E - e ϕ ) ψ - + c s y m b o l σ ( 𝐩 - e c 𝐀 ) ψ + = m c 2 ψ - \begin{aligned}\displaystyle(E-e\phi)\psi_{+}-csymbol{\sigma}\cdot\left(% \mathbf{p}-\frac{e}{c}\mathbf{A}\right)\psi_{-}&\displaystyle=mc^{2}\psi_{+}\\ \displaystyle-(E-e\phi)\psi_{-}+csymbol{\sigma}\cdot\left(\mathbf{p}-\frac{e}{% c}\mathbf{A}\right)\psi_{+}&\displaystyle=mc^{2}\psi_{-}\end{aligned}
  25. E - e ϕ m c 2 p m v \begin{aligned}\displaystyle E-e\phi&\displaystyle\approx mc^{2}\\ \displaystyle p&\displaystyle\approx mv\end{aligned}
  26. ψ - 1 2 m c s y m b o l σ ( 𝐩 - e c 𝐀 ) ψ + \psi_{-}\approx\frac{1}{2mc}symbol{\sigma}\cdot\left(\mathbf{p}-\frac{e}{c}% \mathbf{A}\right)\psi_{+}
  27. ( E - m c 2 ) ψ + = 1 2 m [ s y m b o l σ ( 𝐩 - e c 𝐀 ) ] 2 ψ + + e ϕ ψ + \left(E-mc^{2}\right)\psi_{+}=\frac{1}{2m}\left[symbol{\sigma}\cdot\left(% \mathbf{p}-\frac{e}{c}\mathbf{A}\right)\right]^{2}\psi_{+}+e\phi\psi_{+}

Electron_paramagnetic_resonance.html

  1. s = 1 2 s=\tfrac{1}{2}
  2. m s = + 1 2 m_{\mathrm{s}}=+\tfrac{1}{2}
  3. m s = - 1 2 m_{\mathrm{s}}=-\tfrac{1}{2}
  4. B 0 B_{\mathrm{0}}
  5. m s = - 1 2 m_{\mathrm{s}}=-\tfrac{1}{2}
  6. m s = + 1 2 m_{\mathrm{s}}=+\tfrac{1}{2}
  7. E = m s g e μ B B 0 E=m_{\mathrm{s}}g_{\mathrm{e}}\mu_{\mathrm{B}}B_{\mathrm{0}}
  8. g e g_{\mathrm{e}}
  9. g e = 2.0023 g_{\mathrm{e}}=2.0023
  10. μ B \mu_{\mathrm{B}}
  11. Δ E = g e μ B B 0 \Delta E=g_{\mathrm{e}}\mu_{\mathrm{B}}B_{\mathrm{0}}
  12. h ν h\nu
  13. h ν = Δ E h\nu=\Delta E
  14. h ν = g e μ B B 0 h\nu=g_{\mathrm{e}}\mu_{\mathrm{B}}B_{\mathrm{0}}
  15. m s = + 1 2 m_{\mathrm{s}}=+\tfrac{1}{2}
  16. m s = - 1 2 m_{\mathrm{s}}=-\tfrac{1}{2}
  17. B 0 = h ν / g e μ B B_{\mathrm{0}}=h\nu/g_{\mathrm{e}}\mu_{\mathrm{B}}
  18. h ν = g N μ N B 0 h\nu=g_{\mathrm{N}}\mu_{\mathrm{N}}B_{\mathrm{0}}
  19. g N g_{\mathrm{N}}
  20. μ N \mu_{\mathrm{N}}
  21. n upper n lower = exp ( - E upper - E lower k T ) = exp ( - Δ E k T ) = exp ( - ϵ k T ) = exp ( - h ν k T ) ( E q .1 ) \frac{n\text{upper}}{n\text{lower}}=\exp{\left(-\frac{E\text{upper}-E\text{% lower}}{kT}\right)}=\exp{\left(-\frac{\Delta E}{kT}\right)}=\exp{\left(-\frac{% \epsilon}{kT}\right)}=\exp{\left(-\frac{h\nu}{kT}\right)}(Eq.1)
  22. n upper n\text{upper}
  23. k k
  24. T T
  25. ν \nu
  26. n upper / n lower n\text{upper}/n\text{lower}
  27. N min N\text{min}
  28. ν \nu
  29. N min = k 1 V Q 0 k f ν 2 P 1 / 2 ( E q .2 ) N\text{min}=\frac{k_{1}V}{Q_{0}k_{f}\nu^{2}P^{1/2}}(Eq.2)
  30. k 1 k_{1}
  31. V V
  32. Q 0 Q_{0}
  33. k f k_{f}
  34. P P
  35. k f k_{f}
  36. P P
  37. N min N\text{min}
  38. ( Q 0 ν 2 ) - 1 (Q_{0}\nu^{2})^{-1}
  39. N min N\text{min}
  40. ν - α \nu^{-\alpha}
  41. α \alpha
  42. α \alpha
  43. N min N\text{min}
  44. g e g_{\mathrm{e}}
  45. B 0 B_{\mathrm{0}}
  46. B eff B_{\mathrm{eff}}
  47. B eff = B 0 ( 1 - σ ) B_{\mathrm{eff}}=B_{0}(1-\sigma)\,
  48. σ \sigma
  49. σ \sigma
  50. h ν = g e μ B B eff h\nu=g_{\mathrm{e}}\mu_{\mathrm{B}}B_{\mathrm{eff}}
  51. h ν = g e μ B B eff = g e μ B B 0 ( 1 - σ ) h\nu=g_{\mathrm{e}}\mu_{B}B_{\mathrm{eff}}=g_{\mathrm{e}}\mu_{B}B_{0}(1-\sigma)\,
  52. g e ( 1 - σ ) g_{\mathrm{e}}(1-\sigma)
  53. g g
  54. g g
  55. h ν = g μ B B 0 h\nu=g\mu_{\mathrm{B}}B_{\mathrm{0}}\,
  56. g g
  57. g g
  58. g e g_{\mathrm{e}}
  59. π \pi
  60. Δ B h \Delta B_{h}
  61. Δ B 1 / 2 \Delta B_{1/2}
  62. Δ B 1 / 2 = 2 Δ B h \Delta B_{1/2}=2\Delta B_{h}
  63. Δ B m a x = 2 Δ B 1 s \Delta B_{max}=2\Delta B_{1s}
  64. π \pi
  65. T 2 T_{2}
  66. λ / mm \lambda/\,\text{mm}
  67. ν / GHz \nu/\,\text{GHz}
  68. B 0 / T B_{0}/\,\text{T}
  69. ν \nu
  70. ν \nu
  71. ν \nu
  72. ν \nu

Electronic_correlation.html

  1. Φ I \Phi_{I}
  2. c I c_{I}
  3. ρ ( 𝐫 a , 𝐫 b ) ρ ( 𝐫 a ) ρ ( 𝐫 b ) , \rho(\mathbf{r}_{a},\mathbf{r}_{b})\sim\rho(\mathbf{r}_{a})\rho(\mathbf{r}_{b}% ),\,

Electronic_filter.html

  1. H ( s ) \ H(s)
  2. Y ( s ) \ Y(s)
  3. X ( s ) \ X(s)
  4. s \ s
  5. H ( s ) = Y ( s ) X ( s ) \ H(s)=\frac{Y(s)}{X(s)}
  6. s = σ + j ω \ s=\sigma+j\omega
  7. s \ s
  8. s \ s
  9. s \ s

Electrostatic_ion_cyclotron_wave.html

  1. m e m i \sqrt{\frac{m_{e}}{m_{i}}}
  2. ω 2 = Ω c 2 + k 2 v s 2 \omega^{2}=\Omega_{c}^{2}+k^{2}v_{s}^{2}

Electrostatic_precipitator.html

  1. E = ρ j \vec{E}={\rho}\,\vec{j}
  2. ρ = A V I l \rho=\frac{AV}{Il}

Electrostriction.html

  1. Q i j k l Q_{ijkl}
  2. x i j x_{ij}
  3. P k P_{k}
  4. P l P_{l}
  5. x i j = Q i j k l × P k × P l x_{ij}=Q_{ijkl}\times P_{k}\times P_{l}

Elementary_arithmetic.html

  1. 5 × 3 = 15. 5\times 3=15.\,
  2. 3 × 729 = 2187 3\times 729=2187
  3. 789 × 345 = 272205 789\times 345=272205
  4. c × b = a c\times b=a\,
  5. a b = c \frac{a}{b}=c
  6. 6 3 = 2 \frac{6}{3}=2
  7. 2 × 3 = 6 2\times 3=6\,
  8. a b . \frac{a}{b}.
  9. a / b . a/b.\,
  10. a / b {a}/{b}
  11. a ÷ b . a\div b.
  12. 5 ÷ 1 2 = 5 × 2 1 = 5 × 2 = 10 \textstyle{5\div{1\over 2}=5\times{2\over 1}=5\times 2=10}
  13. 2 3 ÷ 2 5 = 2 3 × 5 2 = 10 6 = 5 3 \textstyle{{2\over 3}\div{2\over 5}={2\over 3}\times{5\over 2}={10\over 6}={5% \over 3}}

Elementary_class.html

  1. Δ {}_{\Delta}
  2. K K^{\prime}
  3. K K^{\prime}
  4. x y ( ( f ( x ) = f ( y ) ) ( x = y ) ) \forall x\forall y((f(x)=f(y))\to(x=y))
  5. ρ 2 = \rho_{2}={}
  6. \exist x 1 \exist x 2 ( x 1 x 2 ) \exist x_{1}\exist x_{2}(x_{1}\not=x_{2})
  7. ρ 3 = \rho_{3}={}
  8. \exist x 1 \exist x 2 \exist x 3 ( ( x 1 x 2 ) and ( x 1 x 3 ) and ( x 2 x 3 ) ) \exist x_{1}\exist x_{2}\exist x_{3}((x_{1}\not=x_{2})\and(x_{1}\not=x_{3})% \and(x_{2}\not=x_{3}))
  9. ρ n \rho_{n}
  10. T = { ρ 2 , ρ 3 , ρ 4 , } T_{\infty}=\{\rho_{2},\rho_{3},\rho_{4},\dots\}
  11. { ¬ τ , ρ 2 , ρ 3 , ρ 4 , } \{\neg\tau,\rho_{2},\rho_{3},\rho_{4},\dots\}
  12. { ¬ τ , ρ 2 , ρ 3 , ρ 4 , , ρ n } \{\neg\tau,\rho_{2},\rho_{3},\rho_{4},\dots,\rho_{n}\}
  13. n + 1 n+1
  14. \cup
  15. ( x y ( f ( x ) = f ( y ) x = y ) y ¬ x ( y = f ( x ) ) ) , (\forall x\forall y(f(x)=f(y)\rightarrow x=y)\land\exists y\neg\exists x(y=f(x% ))),
  16. σ \sigma^{\prime}
  17. K K^{\prime}
  18. σ \sigma^{\prime}
  19. K K^{\prime}