wpmath0000014_3

Dependency_network.html

  1. C X , Y = ( X i ( n ) - μ i ) ( X j ( n ) - μ j ) σ i σ j C_{X,Y}=\frac{\left\langle(X_{i}(n)-\mu_{i})(X_{j}(n)-\mu_{j})\right\rangle}{% \sigma_{i}\sigma_{j}}
  2. X i ( n ) X_{i}(n)
  3. X j ( n ) X_{j}(n)
  4. ( i , j ) (i,j)
  5. j - P C ( i , k | j ) j-PC(i,k|j)
  6. P C ( i , k | j ) = C ( i , k ) - C ( i , j ) C ( k , j ) [ 1 - C 2 ( i , j ) ] [ 1 - C 2 ( k , j ) ] PC(i,k|j)=\frac{C(i,k)-C(i,j)C(k,j)}{\sqrt{[1-C^{2}(i,j)][1-C^{2}(k,j)]}}
  7. C ( i , j ) , C ( i , k ) C(i,j),C(i,k)
  8. C ( j , k ) C(j,k)
  9. C ( i , j ) C(i,j)
  10. C ( j , k ) C(j,k)
  11. d ( i , k | j ) C ( i , k ) - P C ( i , k | j ) d(i,k|j)\equiv C(i,k)-PC(i,k|j)
  12. C ( i , k ) C(i,k)
  13. C ( i , j ) , C ( i , k ) C(i,j),C(i,k)
  14. C ( j , k ) C(j,k)
  15. D ( i , j ) = 1 N - 1 k j N - 1 d ( i , k | j ) D(i,j)=\frac{1}{N-1}\sum_{k\neq j}^{N-1}d(i,k|j)
  16. D ( i , j ) D ( j , i ) D(i,j)\neq D(j,i)
  17. D P ( i k | j ) DP(i\rightarrow k|j)
  18. D P ( i k | j ) 1 t d ( i k | j + ) - 1 t d ( i k | j - ) DP(i\rightarrow k|j)\equiv\frac{1}{td(i\rightarrow k|j^{+})}-\frac{1}{td(i% \rightarrow k|j^{-})}
  19. t d ( i k | j + ) td(i\rightarrow k|j^{+})
  20. t d ( i k | j - ) td(i\rightarrow k|j^{-})
  21. D ( i , j ) = 1 N - 1 k = 1 N - 1 D P ( i k | j ) D(i,j)=\frac{1}{N-1}\sum_{k=1}^{N-1}DP(i\rightarrow k|j)
  22. D ( i , j ) D ( j , i ) D(i,j)\neq D(j,i)
  23. ( N - 1 ) (N-1)
  24. 3 ( N - 2 ) 3(N-2)

Dependent_source.html

  1. V = f a ( v x ) V={f_{a}}({v_{x}})
  2. I = f b ( v x ) I={f_{b}}({v_{x}})
  3. I = f c ( i x ) I={f_{c}}({i_{x}})
  4. V = f d ( i x ) V={f_{d}}({i_{x}})
  5. V D S > V G S - V T V_{DS}>V_{GS}-V_{T}
  6. V G S > V T V_{GS}>V_{T}
  7. V D S V_{DS}
  8. I D = μ n C o x 2 W L ( V G S - V t h ) 2 ( 1 + λ ( V D S - V D S s a t ) ) . I_{D}=\frac{\mu_{n}C_{ox}}{2}\frac{W}{L}(V_{GS}-V_{th})^{2}\left(1+\lambda(V_{% DS}-V_{DSsat})\right).
  9. V D S V_{DS}
  10. V D S - V T V_{DS}-V_{T}

Dephasing_rate_SP_formula.html

  1. Γ φ \Gamma_{\varphi}
  2. S ~ ( q , ω ) \tilde{S}(q,\omega)
  3. P ~ ( q , ω ) \tilde{P}(q,\omega)
  4. Γ φ = d q d ω 2 π S ~ ( q , ω ) P ~ ( - q , - ω ) \Gamma_{\varphi}\ =\ \int d{q}\int\frac{d\omega}{2\pi}\,\tilde{S}({q},\omega)% \,\tilde{P}(-{q},-\omega)
  5. S ~ ( q , ω ) \tilde{S}(q,\omega)
  6. P ~ ( q , ω ) \tilde{P}(q,\omega)
  7. P ( t ) = e - F ( t ) P(t)=e^{-F(t)}
  8. F ( t ) = Γ φ t F(t)=\Gamma_{\varphi}t
  9. P ( t ) P(t)
  10. T T
  11. η \eta
  12. S ~ ( q , ω ) = ( 2 π ) δ ( q ) q 2 [ 2 η ω 1 - e - ω / T ] \tilde{S}(q,\omega)\ =\ \frac{(2\pi)\delta(q)}{q^{2}}\,\left[\frac{2\eta\omega% }{1-e^{-\omega/T}}\right]
  13. q q
  14. S ~ ( q , ω ) = 1 ν D q 2 [ 2 ω 1 - e - ω / T ] . \tilde{S}(q,\omega)\ =\ \frac{1}{\nu Dq^{2}}\left[\frac{2\omega}{1-e^{-\omega/% T}}\right].
  15. P ~ ( q , ω ) = 2 D q 2 ω 2 + ( D q 2 ) 2 \tilde{P}(q,\omega)\ \ =\ \ \frac{2Dq^{2}}{\omega^{2}+(Dq^{2})^{2}}
  16. P ~ ( q , ω ) = d d ω [ ω 1 - e - ω / T ] × 2 D q 2 ω 2 + ( D q 2 ) 2 \tilde{P}(q,\omega)\ \ =\ \ \frac{d}{d\omega}\left[\frac{\omega}{1-e^{-\omega/% T}}\right]\times\frac{2Dq^{2}}{\omega^{2}+(Dq^{2})^{2}}
  17. Γ φ T 3 / 2 \Gamma_{\varphi}\propto T^{3/2}

Depreciation_(economics).html

  1. K t K_{t}
  2. t t
  3. I t I_{t}
  4. D t D_{t}
  5. K t + 1 K_{t+1}
  6. K t + I t - D t K_{t}+I_{t}-D_{t}

Depth_of_noncommutative_subrings.html

  1. B A B\subseteq A
  2. A n A B A A^{n}\rightarrow A\otimes_{B}A
  3. A n = A × × A A^{n}=A\times\ldots\times A
  4. g 1 , , g n g_{1},\cdots,g_{n}
  5. A n A B A A^{n}\rightarrow A\otimes_{B}A
  6. p ( ( a 1 , , a n ) ) = i = 1 n a i g i - 1 B g i p((a_{1},\cdots,a_{n}))=\sum_{i=1}^{n}a_{i}g_{i}^{-1}\otimes_{B}g_{i}
  7. q : A B A A n q:A\otimes_{B}A\rightarrow A^{n}
  8. q ( a B a ) = ( a γ 1 ( a ) , , a γ n ( a ) ) q(a\otimes_{B}a^{\prime})=(a\gamma_{1}(a^{\prime}),\cdots,a\gamma_{n}(a^{% \prime}))
  9. γ i ( g ) = δ i j g \gamma_{i}(g)=\delta_{ij}g
  10. g j H g_{j}H
  11. A B A A\otimes_{B}A
  12. f i : A B f_{i}:A\rightarrow B
  13. x i A x_{i}\in A
  14. i = 1 n x i f i ( a ) = a \sum_{i=1}^{n}x_{i}f_{i}(a)=a
  15. p ( a 1 , , a n ) = i = 1 n x i B a i p(a_{1},\cdots,a_{n})=\sum_{i=1}^{n}x_{i}\otimes_{B}a_{i}
  16. q ( a B a ) = ( f 1 ( a ) a , , f n ( a ) a ) q(a\otimes_{B}a^{\prime})=(f_{1}(a)a^{\prime},\cdots,f_{n}(a)a^{\prime})
  17. M M t M n M MM^{t}M\leq nM
  18. M 3 M^{3}
  19. m 1 m\geq 1
  20. M m + 1 n M m - 1 M^{m+1}\leq nM^{m-1}
  21. G = H C G ( X ) G=HC_{G}(X)
  22. B = C S 2 B=CS_{2}
  23. A = C S 3 A=CS_{3}
  24. A B B A A\otimes_{B}\cdots\otimes_{B}A
  25. i = 1 m A B B A \oplus_{i=1}^{m}A\otimes_{B}\cdots\otimes_{B}A
  26. d ( B , A ) d(B,A)
  27. d c ( H , G ) d_{c}(H,G)
  28. d c ( H , G ) d_{c}(H,G)
  29. End A B B \mbox{End}~{}\,{}_{B}A_{B}
  30. A B A A\otimes_{B}A
  31. End A B A R S \mbox{End}~{}\,A_{B}\cong A\otimes_{R}S
  32. End A B A A \mbox{End}~{}\,A\otimes_{B}A_{A}
  33. End A B B \mbox{End}~{}\,{}_{B}A_{B}
  34. λ : B End A B , λ ( b ) ( a ) = b a \lambda:B\rightarrow\mbox{End}~{}\,A_{B},\ \lambda(b)(a)=ba
  35. R = { r A : b B , b r = r b } R=\{r\in A:\forall b\in B,br=rb\}

Deriche_edge_detector.html

  1. f ( x ) = S ω e - α | x | s i n ω x f(x)=\frac{S}{\omega}e^{-\alpha|x|}sin\omega x
  2. ω \omega
  3. f ( x ) = S x e - α | x | f(x)=Sxe^{-\alpha|x|}
  4. y i j 1 = a 1 x i j + a 2 x i j - 1 + b 1 y i j - 1 1 + b 2 y i j - 2 1 y_{ij}^{1}=a_{1}x_{ij}+a_{2}x_{ij-1}+b_{1}y_{ij-1}^{1}+b_{2}y_{ij-2}^{1}
  5. y i j 2 = a 3 x i j + 1 + a 4 x i j + 2 + b 1 y i j + 1 2 + b 2 y i j + 2 2 y_{ij}^{2}=a_{3}x_{ij+1}+a_{4}x_{ij+2}+b_{1}y_{ij+1}^{2}+b_{2}y_{ij+2}^{2}
  6. θ i j = c 1 ( y i j 1 + y i j 2 ) \theta_{ij}=c_{1}(y_{ij}^{1}+y_{ij}^{2})
  7. y i j 1 = a 5 θ i j + a 6 θ i - 1 j + b 1 y i - 1 j 1 + b 2 y i - 2 j 1 y_{ij}^{1}=a_{5}\theta_{ij}+a_{6}\theta_{i-1j}+b_{1}y_{i-1j}^{1}+b_{2}y_{i-2j}% ^{1}
  8. y i j 2 = a 7 θ i + 1 j + a 8 θ i + 2 j + b 1 y i + 1 j 2 + b 2 y i + 2 j 2 y_{ij}^{2}=a_{7}\theta_{i+1j}+a_{8}\theta_{i+2j}+b_{1}y_{i+1j}^{2}+b_{2}y_{i+2% j}^{2}
  9. Θ i j = c 2 ( y i j 1 + y i j 2 ) \Theta_{ij}=c_{2}(y_{ij}^{1}+y_{ij}^{2})
  10. k k
  11. ( 1 - e - α ) 2 1 + 2 α e - α - e - 2 α \frac{{(1-e^{-\alpha})}^{2}}{1+2\alpha e^{-\alpha}-e^{-2\alpha}}
  12. ( 1 - e - α ) 2 1 + 2 α e - α - e - 2 α \frac{{(1-e^{-\alpha})}^{2}}{1+2\alpha e^{-\alpha}-e^{-2\alpha}}
  13. ( 1 - e - α ) 2 1 + 2 α e - α - e - 2 α \frac{{(1-e^{-\alpha})}^{2}}{1+2\alpha e^{-\alpha}-e^{-2\alpha}}
  14. a 1 a_{1}
  15. k k
  16. k k
  17. a 2 a_{2}
  18. k e - α ( α - 1 ) ke^{-\alpha}(\alpha-1)
  19. k e - α ( α - 1 ) ke^{-\alpha}(\alpha-1)
  20. a 3 a_{3}
  21. k e - α ( α + 1 ) ke^{-\alpha}(\alpha+1)
  22. k e - α ( α + 1 ) ke^{-\alpha}(\alpha+1)
  23. a 4 a_{4}
  24. - k e - 2 α -ke^{-2\alpha}
  25. - k e - 2 α -ke^{-2\alpha}
  26. a 5 a_{5}
  27. k k
  28. k k
  29. a 6 a_{6}
  30. k e - α ( α - 1 ) ke^{-\alpha}(\alpha-1)
  31. k e - α ( α - 1 ) ke^{-\alpha}(\alpha-1)
  32. a 7 a_{7}
  33. k e - α ( α + 1 ) ke^{-\alpha}(\alpha+1)
  34. k e - α ( α + 1 ) ke^{-\alpha}(\alpha+1)
  35. a 8 a_{8}
  36. - k e - 2 α -ke^{-2\alpha}
  37. - k e - 2 α -ke^{-2\alpha}
  38. b 1 b_{1}
  39. 2 e - α 2e^{-\alpha}
  40. 2 e - α 2e^{-\alpha}
  41. 2 e - α 2e^{-\alpha}
  42. b 2 b_{2}
  43. - e - 2 α -e^{-2\alpha}
  44. - e - 2 α -e^{-2\alpha}
  45. - e - 2 α -e^{-2\alpha}
  46. c 1 c_{1}
  47. - ( 1 - e - α ) 2 -{(1-e^{-\alpha})}^{2}
  48. c 2 c_{2}
  49. - ( 1 - e - α ) 2 -{(1-e^{-\alpha})}^{2}

Derrick's_theorem.html

  1. 2 θ - 2 θ t 2 = 1 2 f ( θ ) , θ ( x , t ) \R , x \R 3 , \nabla^{2}\theta-\frac{\partial^{2}\theta}{\partial t^{2}}=\frac{1}{2}f^{% \prime}(\theta),\qquad\theta(x,t)\in\R,\quad x\in\R^{3},
  2. f ( s ) f(s)
  3. f ( 0 ) = 0 f^{\prime}(0)=0
  4. θ ( x ) \theta(x)\,
  5. E = [ ( θ ) 2 + f ( θ ) ] d 3 x . E=\int\left[(\nabla\theta)^{2}+f(\theta)\right]\,d^{3}x.
  6. δ 2 E 0 \delta^{2}E\geq 0\,
  7. θ ( x ) \theta(x)\,
  8. δ E = 0 \delta E=0\,
  9. θ λ ( x ) = θ ( λ x ) \theta_{\lambda}(x)=\theta(\lambda x)\,
  10. λ \lambda
  11. I 1 = ( θ ) 2 d 3 x I_{1}=\int(\nabla\theta)^{2}d^{3}x
  12. I 2 = f ( θ ) d 3 x I_{2}=\int f(\theta)d^{3}x
  13. E λ = [ ( θ λ ) 2 + f ( θ λ ) ] d 3 x = I 1 / λ + I 2 / λ 3 . E_{\lambda}=\int\left[(\nabla\theta_{\lambda})^{2}+f(\theta_{\lambda})\right]d% ^{3}x=I_{1}/\lambda+I_{2}/\lambda^{3}.
  14. ( d E λ / d λ ) | λ = 1 = - I 1 - 3 I 2 = 0 (dE_{\lambda}/d\lambda)|_{\lambda=1}=-I_{1}-3I_{2}=0\,
  15. I 1 > 0 I_{1}>0\,
  16. ( d 2 E λ / d λ 2 ) | λ = 1 = 2 I 1 + 12 I 2 = - 2 I 1 < 0. (d^{2}E_{\lambda}/d\lambda^{2})|_{\lambda=1}=2I_{1}+12I_{2}=-2I_{1}\,<0.
  17. δ 2 E < 0 \delta^{2}E<0\,
  18. θ ( x ) \theta(x)\,
  19. x \R n x\in\R^{n}
  20. n > 3 n>3\,
  21. g g
  22. g ( 0 ) = 0 g(0)=0
  23. G ( t ) = 0 t g ( s ) d s G(t)=\int_{0}^{t}g(s)\,ds
  24. u L l o c ( \R n ) , u L 2 ( \R n ) , G ( u ) L 1 ( \R n ) , n 𝒩 , u\in L^{\infty}_{loc}(\R^{n}),\qquad\nabla u\in L^{2}(\R^{n}),\qquad G(u)\in L% ^{1}(\R^{n}),\qquad n\in\mathcal{N},
  25. - 2 u = g ( u ) -\nabla^{2}u=g(u)
  26. u u
  27. ( n - 2 ) \R n | u | 2 d x = n \R n G ( u ) d x , (n-2)\int_{\R^{n}}|\nabla u|^{2}\,dx=n\int_{\R^{n}}G(u)\,dx,
  28. t 2 u = 2 u - 1 2 f ( u ) \partial_{t}^{2}u=\nabla^{2}u-\frac{1}{2}f^{\prime}(u)
  29. t u = δ v H ( u , v ) \partial_{t}u=\delta_{v}H(u,v)
  30. t v = - δ u H ( u , v ) \partial_{t}v=-\delta_{u}H(u,v)
  31. u , v u,\,v
  32. x \R n , t \R x\in\R^{n},\,t\in\R
  33. H ( u , v ) = \R n ( 1 2 | v | 2 + 1 2 | u | 2 + 1 2 f ( u ) ) d x , H(u,v)=\int_{\R^{n}}\left(\frac{1}{2}|v|^{2}+\frac{1}{2}|\nabla u|^{2}+\frac{1% }{2}f(u)\right)\,dx,
  34. δ u H \delta_{u}H\,
  35. δ v H \delta_{v}H\,
  36. H ( u , v ) H(u,v)\,
  37. u ( x , t ) = θ ( x ) u(x,t)=\theta(x)\,
  38. H ( θ , 0 ) = \R n ( 1 2 | θ | 2 + 1 2 f ( θ ) ) d n x H(\theta,0)=\int_{\R^{n}}\left(\frac{1}{2}|\nabla\theta|^{2}+\frac{1}{2}f(% \theta)\right)\,d^{n}x
  39. 0 = t θ ( x ) = - u H ( θ , 0 ) = 1 2 E ( θ ) , 0=\partial_{t}\theta(x)=-\partial_{u}H(\theta,0)=\frac{1}{2}E^{\prime}(\theta),
  40. E E^{\prime}\,
  41. E = \R n [ | θ | 2 + f ( θ ) ] d n x E=\int_{\R^{n}}[|\nabla\theta|^{2}+f(\theta)]\,d^{n}x
  42. θ ( x ) \theta(x)\,
  43. E E\,
  44. E ( θ ) = 0 E^{\prime}(\theta)=0\,
  45. d 2 d λ 2 E ( θ ( λ x ) ) < 0 \frac{d^{2}}{d\lambda\,^{2}}E(\theta(\lambda x))<0
  46. λ = 1 \lambda=1\,
  47. u ( x , t ) = θ ( x ) u(x,t)=\theta(x)\,
  48. H H\,
  49. θ ( x ) \theta(x)\,
  50. u ( x , t ) = ϕ ω ( x ) e - i ω t u(x,t)=\phi_{\omega}(x)e^{-i\omega t}\,
  51. ω \omega\,
  52. Δ u + λ f ( u ) = 0 \Delta u+\lambda f(u)=0

Design_and_manufacturing_of_gears.html

  1. R i m t h i c k n e s s = M o d u l e N 2 N A , Rimthickness=Module\sqrt{\frac{N}{2N_{A}}},
  2. N N
  3. N A N_{A}

Determinantal_point_process.html

  1. Λ \Lambda
  2. μ \mu
  3. Λ \Lambda
  4. X X
  5. Λ \Lambda
  6. K K
  7. Λ \Lambda
  8. ρ n ( x 1 , , x n ) = det ( K ( x i , x j ) 1 i , j n ) \rho_{n}(x_{1},\ldots,x_{n})=\det(K(x_{i},x_{j})_{1\leq i,j\leq n})
  9. ρ k ( x σ ( 1 ) , , x σ ( k ) ) = ρ k ( x 1 , , x k ) σ S k , k \rho_{k}(x_{\sigma(1)},\ldots,x_{\sigma(k)})=\rho_{k}(x_{1},\ldots,x_{k})\quad% \forall\sigma\in S_{k},k
  10. φ 0 + k = 1 N i 1 i k φ k ( x i 1 x i k ) 0 for all k , ( x i ) i = 1 k \quad\varphi_{0}+\sum_{k=1}^{N}\sum_{i_{1}\neq\cdots\neq i_{k}}\varphi_{k}(x_{% i_{1}}\ldots x_{i_{k}})\geq 0\,\text{ for all }k,(x_{i})_{i=1}^{k}
  11. φ 0 + i = 1 N Λ k φ k ( x 1 , , x k ) ρ k ( x 1 , , x k ) d x 1 d x k 0 for all k , ( x i ) i = 1 k \quad\varphi_{0}+\sum_{i=1}^{N}\int_{\Lambda^{k}}\varphi_{k}(x_{1},\ldots,x_{k% })\rho_{k}(x_{1},\ldots,x_{k})\,\textrm{d}x_{1}\cdots\textrm{d}x_{k}\geq 0\,% \text{ for all }k,(x_{i})_{i=1}^{k}
  12. k = 0 ( 1 k ! A k ρ k ( x 1 , , x k ) d x 1 d x k ) - 1 k = \sum_{k=0}^{\infty}\left(\frac{1}{k!}\int_{A^{k}}\rho_{k}(x_{1},\ldots,x_{k})% \,\textrm{d}x_{1}\cdots\textrm{d}x_{k}\right)^{-\frac{1}{k}}=\infty
  13. \mathbb{R}
  14. K m ( x , y ) = k = 0 m - 1 ψ k ( x ) ψ k ( y ) K_{m}(x,y)=\sum_{k=0}^{m-1}\psi_{k}(x)\psi_{k}(y)
  15. ψ k ( x ) \psi_{k}(x)
  16. k k
  17. ψ k ( x ) = 1 2 n n ! H k ( x ) e - x 2 / 4 \psi_{k}(x)=\frac{1}{\sqrt{\sqrt{2n}n!}}H_{k}(x)e^{-x^{2}/4}
  18. H k ( x ) H_{k}(x)
  19. k k
  20. 1 / 2 {1}/{2}
  21. K ( x , y ) = { θ k + ( | x | , | y | ) | x | - | y | if x y > 0 , θ k - ( | x | , | y | ) x - y if x y < 0 , K(x,y)=\begin{cases}\sqrt{\theta}\,\dfrac{k_{+}(|x|,|y|)}{|x|-|y|}&\,\text{if % }xy>0,\\ \sqrt{\theta}\,\dfrac{k_{-}(|x|,|y|)}{x-y}&\,\text{if }xy<0,\end{cases}
  22. k + ( x , y ) = J x - 1 2 ( 2 θ ) J y + 1 2 ( 2 θ ) - J x + 1 2 ( 2 θ ) J y - 1 2 ( 2 θ ) , k_{+}(x,y)=J_{x-\frac{1}{2}}(2\sqrt{\theta})J_{y+\frac{1}{2}}(2\sqrt{\theta})-% J_{x+\frac{1}{2}}(2\sqrt{\theta})J_{y-\frac{1}{2}}(2\sqrt{\theta}),
  23. k - ( x , y ) = J x - 1 2 ( 2 θ ) J y - 1 2 ( 2 θ ) + J x + 1 2 ( 2 θ ) J y + 1 2 ( 2 θ ) k_{-}(x,y)=J_{x-\frac{1}{2}}(2\sqrt{\theta})J_{y-\frac{1}{2}}(2\sqrt{\theta})+% J_{x+\frac{1}{2}}(2\sqrt{\theta})J_{y+\frac{1}{2}}(2\sqrt{\theta})
  24. K ( e , f ) = I e , I f , e , f E K(e,f)=\langle I^{e},I^{f}\rangle,\quad e,f\in E

Deviation_risk_measure.html

  1. D : 2 [ 0 , + ] D:\mathcal{L}^{2}\to[0,+\infty]
  2. 2 \mathcal{L}^{2}
  3. D ( X + r ) = D ( X ) D(X+r)=D(X)
  4. r r\in\mathbb{R}
  5. D ( 0 ) = 0 D(0)=0
  6. D ( λ X ) = λ D ( X ) D(\lambda X)=\lambda D(X)
  7. X 2 X\in\mathcal{L}^{2}
  8. λ > 0 \lambda>0
  9. D ( X + Y ) D ( X ) + D ( Y ) D(X+Y)\leq D(X)+D(Y)
  10. X , Y 2 X,Y\in\mathcal{L}^{2}
  11. D ( X ) > 0 D(X)>0
  12. D ( X ) = 0 D(X)=0
  13. X 2 X\in\mathcal{L}^{2}
  14. D ( X ) = R ( X - 𝔼 [ X ] ) D(X)=R(X-\mathbb{E}[X])
  15. R ( X ) = D ( X ) - 𝔼 [ X ] R(X)=D(X)-\mathbb{E}[X]
  16. R ( X ) > 𝔼 [ - X ] R(X)>\mathbb{E}[-X]
  17. R ( X ) = 𝔼 [ - X ] R(X)=\mathbb{E}[-X]
  18. D ( X ) < 𝔼 [ X ] - ess inf X D(X)<\mathbb{E}[X]-\operatorname{ess\inf}X
  19. ess inf \operatorname{ess\inf}

Diagnostic_odds_ratio.html

  1. Diagnostic odds ratio, DOR = T P / F P F N / T N \,\text{Diagnostic odds ratio, DOR}=\frac{TP/FP}{FN/TN}
  2. T P TP
  3. F N FN
  4. F P FP
  5. T N TN
  6. SE ( log DOR ) = 1 T P + 1 F N + 1 F P + 1 T N \mathrm{SE}\left(\log{\,\text{DOR}}\right)=\sqrt{\frac{1}{TP}+\frac{1}{FN}+% \frac{1}{FP}+\frac{1}{TN}}
  7. log DOR ± 1.96 × SE ( log DOR ) \log{\,\text{DOR}}\pm 1.96\times\mathrm{SE}\left(\log{\,\text{DOR}}\right)
  8. DOR = sensitivity × specificity ( 1 - sensitivity ) × ( 1 - specificity ) \,\text{DOR}=\frac{\,\text{sensitivity}\times\,\text{specificity}}{\left(1-\,% \text{sensitivity}\right)\times\left(1-\,\text{specificity}\right)}
  9. DOR = PPV × NPV ( 1 - PPV ) × ( 1 - NPV ) \,\text{DOR}=\frac{\,\text{PPV}\times\,\text{NPV}}{\left(1-\,\text{PPV}\right)% \times\left(1-\,\text{NPV}\right)}
  10. L R + LR+
  11. L R - LR-
  12. DOR = L R + L R - \,\text{DOR}=\frac{LR+}{LR-}
  13. S S
  14. D = log DOR = log [ T P R ( 1 - T P R ) × ( 1 - F P R ) F P R ] = logit ( T P R ) - logit ( F P R ) D=\log{\,\text{DOR}}=\log{\left[\frac{TPR}{(1-TPR)}\times\frac{(1-FPR)}{FPR}% \right]}=\operatorname{logit}(TPR)-\operatorname{logit}(FPR)
  15. S = logit ( T P R ) + logit ( F P R ) S=\operatorname{logit}(TPR)+\operatorname{logit}(FPR)
  16. D = a + b S D=a+bS
  17. DOR = 26 / 3 12 / 48 = 34.666 35 \,\text{DOR}=\frac{26/3}{12/48}=34.666\ldots\approx 35
  18. DOR = 26 / 13 12 / 48 = 2 1 / 4 = 8 \,\text{DOR}^{\prime}=\frac{26/13}{12/48}=\frac{2}{1/4}=8

Diamond_coconut_model.html

  1. e ( t ) e(t)
  2. b ( e ( t ) ) b(e(t))
  3. y y
  4. b ( e ) > 0 b^{\prime}(e)>0
  5. f f
  6. f f
  7. e ( t ) e(t)
  8. 1 - e ( t ) 1-e(t)
  9. r V e = b ( e ) ( y + V u - V e ) + d V e d t rV_{e}=b(e)(y+V_{u}-V_{e})+\frac{dV_{e}}{dt}
  10. V e V_{e}
  11. V u V_{u}
  12. y y
  13. r r
  14. r V u = f ( - E ( c ) + V e - V u ) + d V u d t rV_{u}=f(-E(c)+V_{e}-V_{u})+\frac{dV_{u}}{dt}
  15. f f
  16. E ( c ) E(c)
  17. ( c l o w , c h i ) (c_{low},c_{hi})
  18. b ( e ) = e b(e)=e
  19. c c
  20. d e d t = f ( 1 - e ) - b ( e ) e = f ( 1 - e ) - e 2 \frac{de}{dt}=f(1-e)-b(e)e=f(1-e)-e^{2}
  21. d e d t = - e 2 \frac{de}{dt}=-e^{2}
  22. f ( 1 - e ) f(1-e)
  23. t t
  24. e 2 e^{2}
  25. d e d t = 0 \frac{de}{dt}=0
  26. e * = 0 e*=0
  27. e * = ( 1 / 2 ) ( - f + f 2 + 4 f ) e^{*}=(1/2)(-f+\sqrt{f^{2}+4f})
  28. d V e d t = d V u d t = 0 \frac{dV_{e}}{dt}=\frac{dV_{u}}{dt}=0
  29. V e V_{e}
  30. V u V_{u}
  31. V e - V u = e y + f c r + e + f V_{e}-V_{u}=\frac{ey+fc}{r+e+f}
  32. e * = 0 e^{*}=0
  33. f c / ( r + f ) > c fc/(r+f)>c
  34. e = 0 e=0
  35. e > r c y - c e>\frac{rc}{y-c}
  36. e e
  37. f f
  38. e * e^{*}
  39. f f
  40. e e
  41. r r
  42. e = 0 e=0
  43. c c
  44. r r

Dichromatic_reflectance_model.html

  1. L ( λ ) = m b c b ( λ ) + m s c s ( λ ) L(\lambda)=m_{\mathrm{b}}c_{\mathrm{b}}(\lambda)+m_{\mathrm{s}}c_{\mathrm{s}}(% \lambda)\,

Dieudonné's_theorem.html

  1. A , B X A,B\subset X
  2. A A
  3. B B
  4. recc ( A ) recc ( B ) \operatorname{recc}(A)\cap\operatorname{recc}(B)
  5. recc \operatorname{recc}
  6. A - B A-B

Dieudonné_determinant.html

  1. det ( a b c d ) = { - c b if a = 0 a d - a c a - 1 b if a 0 . \det\left({\begin{array}[]{*{20}c}a&b\\ c&d\end{array}}\right)=\left\{{\begin{array}[]{*{20}c}-cb&\,\text{if }a=0\\ ad-aca^{-1}b&\,\text{if }a\neq 0\end{array}}\right..

Differential_dynamic_microscopy.html

  1. Δ t \Delta t
  2. q q
  3. Δ t \Delta t
  4. D ( q ; Δ t ) D(q;\Delta t)
  5. f ( q ; Δ t ) f(q;\Delta t)
  6. I ( q ) I(q)
  7. B ( q ) B(q)
  8. T ( q ) T(q)
  9. f ( q ; Δ t ) = e - D q 2 Δ t , f(q;\Delta t)=e^{-Dq^{2}\Delta t},
  10. D D
  11. T ( q ) T(q)

Differential_dynamic_programming.html

  1. 𝐱 i + 1 = 𝐟 ( 𝐱 i , 𝐮 i ) \mathbf{x}_{i+1}=\mathbf{f}(\mathbf{x}_{i},\mathbf{u}_{i})
  2. 𝐱 \textstyle\mathbf{x}
  3. 𝐮 \mathbf{u}
  4. i i
  5. i + 1 i+1
  6. J 0 J_{0}
  7. \textstyle\ell
  8. f \ell_{f}
  9. 𝐱 \mathbf{x}
  10. 𝐔 { 𝐮 0 , 𝐮 1 , 𝐮 N - 1 } \mathbf{U}\equiv\{\mathbf{u}_{0},\mathbf{u}_{1}\dots,\mathbf{u}_{N-1}\}
  11. J 0 ( 𝐱 , 𝐔 ) = i = 0 N - 1 ( 𝐱 i , 𝐮 i ) + f ( 𝐱 N ) , J_{0}(\mathbf{x},\mathbf{U})=\sum_{i=0}^{N-1}\ell(\mathbf{x}_{i},\mathbf{u}_{i% })+\ell_{f}(\mathbf{x}_{N}),
  12. 𝐱 0 𝐱 \mathbf{x}_{0}\equiv\mathbf{x}
  13. 𝐱 i \mathbf{x}_{i}
  14. i > 0 i>0
  15. 𝐔 * ( 𝐱 ) argmin 𝐔 J 0 ( 𝐱 , 𝐔 ) . \mathbf{U}^{*}(\mathbf{x})\equiv\operatorname{argmin}_{\mathbf{U}}J_{0}(% \mathbf{x},\mathbf{U}).
  16. 𝐔 * ( 𝐱 ) \mathbf{U}^{*}(\mathbf{x})
  17. 𝐱 \mathbf{x}
  18. 𝐔 i \mathbf{U}_{i}
  19. 𝐔 i { 𝐮 i , 𝐮 i + 1 , 𝐮 N - 1 } \mathbf{U}_{i}\equiv\{\mathbf{u}_{i},\mathbf{u}_{i+1}\dots,\mathbf{u}_{N-1}\}
  20. J i J_{i}
  21. i i
  22. N N
  23. J i ( 𝐱 , 𝐔 i ) = j = i N - 1 ( 𝐱 j , 𝐮 j ) + f ( 𝐱 N ) . J_{i}(\mathbf{x},\mathbf{U}_{i})=\sum_{j=i}^{N-1}\ell(\mathbf{x}_{j},\mathbf{u% }_{j})+\ell_{f}(\mathbf{x}_{N}).
  24. i i
  25. V ( 𝐱 , i ) min 𝐔 i J i ( 𝐱 , 𝐔 i ) . V(\mathbf{x},i)\equiv\min_{\mathbf{U}_{i}}J_{i}(\mathbf{x},\mathbf{U}_{i}).
  26. V ( 𝐱 , N ) f ( 𝐱 N ) V(\mathbf{x},N)\equiv\ell_{f}(\mathbf{x}_{N})
  27. V ( 𝐱 , i ) = min 𝐮 [ ( 𝐱 , 𝐮 ) + V ( 𝐟 ( 𝐱 , 𝐮 ) , i + 1 ) ] . V(\mathbf{x},i)=\min_{\mathbf{u}}[\ell(\mathbf{x},\mathbf{u})+V(\mathbf{f}(% \mathbf{x},\mathbf{u}),i+1)].
  28. ( 𝐱 , 𝐮 ) + V ( 𝐟 ( 𝐱 , 𝐮 ) , i + 1 ) \ell(\mathbf{x},\mathbf{u})+V(\mathbf{f}(\mathbf{x},\mathbf{u}),i+1)
  29. min [ ] \min[]
  30. Q Q
  31. i i
  32. ( 𝐱 , 𝐮 ) (\mathbf{x},\mathbf{u})
  33. Q ( δ 𝐱 , δ 𝐮 ) \displaystyle Q(\delta\mathbf{x},\delta\mathbf{u})\equiv
  34. 1 2 [ 1 δ 𝐱 δ 𝐮 ] 𝖳 [ 0 Q 𝐱 𝖳 Q 𝐮 𝖳 Q 𝐱 Q 𝐱𝐱 Q 𝐱𝐮 Q 𝐮 Q 𝐮𝐱 Q 𝐮𝐮 ] [ 1 δ 𝐱 δ 𝐮 ] \approx\frac{1}{2}\begin{bmatrix}1\\ \delta\mathbf{x}\\ \delta\mathbf{u}\end{bmatrix}^{\mathsf{T}}\begin{bmatrix}0&Q_{\mathbf{x}}^{% \mathsf{T}}&Q_{\mathbf{u}}^{\mathsf{T}}\\ Q_{\mathbf{x}}&Q_{\mathbf{x}\mathbf{x}}&Q_{\mathbf{x}\mathbf{u}}\\ Q_{\mathbf{u}}&Q_{\mathbf{u}\mathbf{x}}&Q_{\mathbf{u}\mathbf{u}}\end{bmatrix}% \begin{bmatrix}1\\ \delta\mathbf{x}\\ \delta\mathbf{u}\end{bmatrix}
  35. Q Q
  36. i i
  37. V V ( i + 1 ) V^{\prime}\equiv V(i+1)
  38. Q 𝐱 = 𝐱 + 𝐟 𝐱 𝖳 V 𝐱 Q 𝐮 = 𝐮 + 𝐟 𝐮 𝖳 V 𝐱 Q 𝐱𝐱 = 𝐱𝐱 + 𝐟 𝐱 𝖳 V 𝐱𝐱 𝐟 𝐱 + V 𝐱 𝐟 𝐱𝐱 Q 𝐮𝐮 = 𝐮𝐮 + 𝐟 𝐮 𝖳 V 𝐱𝐱 𝐟 𝐮 + V 𝐱 𝐟 𝐮𝐮 Q 𝐮𝐱 = 𝐮𝐱 + 𝐟 𝐮 𝖳 V 𝐱𝐱 𝐟 𝐱 + V 𝐱 𝐟 𝐮𝐱 . \begin{aligned}\displaystyle Q_{\mathbf{x}}&\displaystyle=\ell_{\mathbf{x}}+% \mathbf{f}_{\mathbf{x}}^{\mathsf{T}}V^{\prime}_{\mathbf{x}}\\ \displaystyle Q_{\mathbf{u}}&\displaystyle=\ell_{\mathbf{u}}+\mathbf{f}_{% \mathbf{u}}^{\mathsf{T}}V^{\prime}_{\mathbf{x}}\\ \displaystyle Q_{\mathbf{x}\mathbf{x}}&\displaystyle=\ell_{\mathbf{x}\mathbf{x% }}+\mathbf{f}_{\mathbf{x}}^{\mathsf{T}}V^{\prime}_{\mathbf{x}\mathbf{x}}% \mathbf{f}_{\mathbf{x}}+V_{\mathbf{x}}^{\prime}\cdot\mathbf{f}_{\mathbf{x}% \mathbf{x}}\\ \displaystyle Q_{\mathbf{u}\mathbf{u}}&\displaystyle=\ell_{\mathbf{u}\mathbf{u% }}+\mathbf{f}_{\mathbf{u}}^{\mathsf{T}}V^{\prime}_{\mathbf{x}\mathbf{x}}% \mathbf{f}_{\mathbf{u}}+{V^{\prime}_{\mathbf{x}}}\cdot\mathbf{f}_{\mathbf{u}% \mathbf{u}}\\ \displaystyle Q_{\mathbf{u}\mathbf{x}}&\displaystyle=\ell_{\mathbf{u}\mathbf{x% }}+\mathbf{f}_{\mathbf{u}}^{\mathsf{T}}V^{\prime}_{\mathbf{x}\mathbf{x}}% \mathbf{f}_{\mathbf{x}}+{V^{\prime}_{\mathbf{x}}}\cdot\mathbf{f}_{\mathbf{u}% \mathbf{x}}.\end{aligned}
  39. δ 𝐮 \delta\mathbf{u}
  40. δ 𝐮 * = argmin δ 𝐮 Q ( δ 𝐱 , δ 𝐮 ) = - Q 𝐮𝐮 - 1 ( Q 𝐮 + Q 𝐮𝐱 δ 𝐱 ) , {\delta\mathbf{u}}^{*}=\operatorname{argmin}\limits_{\delta\mathbf{u}}Q(\delta% \mathbf{x},\delta\mathbf{u})=-Q_{\mathbf{u}\mathbf{u}}^{-1}(Q_{\mathbf{u}}+Q_{% \mathbf{u}\mathbf{x}}\delta\mathbf{x}),
  41. 𝐤 = - Q 𝐮𝐮 - 1 Q 𝐮 \mathbf{k}=-Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}}
  42. 𝐊 = - Q 𝐮𝐮 - 1 Q 𝐮𝐱 \mathbf{K}=-Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}\mathbf{x}}
  43. i i
  44. Δ V ( i ) \displaystyle\Delta V(i)
  45. V ( i ) V(i)
  46. { 𝐤 ( i ) , 𝐊 ( i ) } \{\mathbf{k}(i),\mathbf{K}(i)\}
  47. i = N - 1 i=N-1
  48. i = 1 i=1
  49. V ( 𝐱 , N ) f ( 𝐱 N ) V(\mathbf{x},N)\equiv\ell_{f}(\mathbf{x}_{N})
  50. 𝐱 ^ ( 1 ) \displaystyle\hat{\mathbf{x}}(1)
  51. Q 𝐮𝐮 Q_{\mathbf{u}\mathbf{u}}
  52. 𝐤 \mathbf{k}
  53. 0 < α < 1 0<\alpha<1

Differential_optical_absorption_spectroscopy.html

  1. I = I 0 exp ( i ρ i β i d s ) I=I_{0}\exp\left(\sum_{i}\int\rho_{i}\beta_{i}\,ds\right)
  2. ρ \rho
  3. β \beta
  4. σ = ρ d s \sigma=\int\rho\,ds
  5. I = I 0 exp ( i β i σ i ) = I 0 i e β i σ i I=I_{0}\exp\left(\sum_{i}\beta_{i}\sigma_{i}\right)=I_{0}\prod_{i}e^{\beta_{i}% \sigma_{i}}
  6. δ = ln ( I 1 I 2 ) = i β i ( σ i 2 - σ i 1 ) = σ i β i Δ σ i \delta=\ln\left(\frac{I_{1}}{I_{2}}\right)=\sum_{i}\beta_{i}\left(\sigma_{i2}-% \sigma_{i1}\right)=\sigma_{i}\beta_{i}\,\Delta\sigma_{i}
  7. I = I 0 exp [ i ( β i * + α i ) σ i ] I=I_{0}\exp\left[\sum_{i}\left(\beta_{i}^{*}+\alpha_{i}\right)\sigma_{i}\right]
  8. α \alpha
  9. β * \beta^{*}
  10. δ d + δ c = ln ( I 1 d I 2 d ) + ln ( I 1 c I 2 c ) = ( β i * + α i ) ( σ i 2 - σ i 1 ) = i β i * ( σ i 2 - σ i 1 ) + i α i ( σ i 2 - σ i 1 ) \delta_{d}+\delta_{c}=\ln\left(\frac{I_{1d}}{I_{2d}}\right)+\ln\left(\frac{I_{% 1c}}{I_{2c}}\right)=\sum\left(\beta_{i}^{*}+\alpha_{i}\right)\left(\sigma_{i2}% -\sigma_{i1}\right)=\sum_{i}\beta_{i}^{*}\left(\sigma_{i2}-\sigma_{i1}\right)+% \sum_{i}\alpha_{i}\left(\sigma_{i2}-\sigma_{i1}\right)
  11. δ d \delta_{d}
  12. δ d ( λ ) = σ i β i * ( λ ) Δ σ i \delta_{d}(\lambda)=\sigma_{i}\beta_{i}^{*}(\lambda)\,\Delta\sigma_{i}
  13. { Δ σ i } \{\Delta\sigma_{i}\}
  14. σ i 0 = amf i ( θ ) σ i θ \sigma_{i0}=\mathrm{amf}_{i}(\theta)\sigma_{i\theta}
  15. σ i 0 \sigma_{i0}
  16. σ i θ \sigma_{i\theta}
  17. θ \theta
  18. σ i 0 = Δ σ i amf i ( θ 2 ) - amf i ( θ 1 ) \sigma_{i0}=\frac{\Delta\sigma_{i}}{\mathrm{amf}_{i}(\theta_{2})-\mathrm{amf}_% {i}(\theta_{1})}
  19. θ 1 \theta_{1}
  20. θ 2 \theta_{2}

Diffractive_beam_splitter.html

  1. d sin θ m = m λ d\sin\theta_{m}=m\lambda

Diffusion_map.html

  1. ( X , 𝒜 , μ ) (X,\mathcal{A},\mu)
  2. X X
  3. μ \mu
  4. X X
  5. k k
  6. x x
  7. y y
  8. x x
  9. y y
  10. k : X × X k:X\times X\rightarrow\mathbb{R}
  11. k ( x , y ) = e - || x - y || 2 ϵ k(x,y)=e^{-\frac{||x-y||^{2}}{\epsilon}}
  12. k ( x , y ) = k ( y , x ) k(x,y)=k(y,x)
  13. k k
  14. k ( x , y ) 0 x , y k(x,y)\geq 0\,\,\forall x,y
  15. k k
  16. ( X , k ) (X,k)
  17. X X
  18. d ( x ) = X k ( x , y ) d μ ( y ) d(x)=\int_{X}k(x,y)d\mu(y)
  19. p ( x , y ) = k ( x , y ) d ( x ) p(x,y)=\frac{k(x,y)}{d(x)}
  20. X p ( x , y ) d μ ( y ) = 1 \int_{X}p(x,y)d\mu(y)=1
  21. p ( x , y ) p(x,y)
  22. M M
  23. X X
  24. p ( x , y ) p(x,y)
  25. x x
  26. y y
  27. M t M^{t}
  28. L L
  29. L i , j = k ( x i , x j ) L_{i,j}=k(x_{i},x_{j})\,
  30. L i , j ( α ) = k ( α ) ( x i , x j ) = L i , j ( d ( x i ) d ( x j ) ) α L^{(\alpha)}_{i,j}=k^{(\alpha)}(x_{i},x_{j})=\frac{L_{i,j}}{(d(x_{i})d(x_{j}))% ^{\alpha}}\,
  31. L ( α ) = D - α L D - α L^{(\alpha)}=D^{-\alpha}LD^{-\alpha}\,
  32. D i , i = j L i , j . D_{i,i}=\sum_{j}L_{i,j}.
  33. M = ( D ( α ) ) - 1 L ( α ) , M=({D}^{(\alpha)})^{-1}L^{(\alpha)},\,
  34. D ( α ) D^{(\alpha)}
  35. D i , i ( α ) = j L i , j ( α ) . {D}^{(\alpha)}_{i,i}=\sum_{j}L^{(\alpha)}_{i,j}.
  36. p ( x j , t | x i ) = M i , j t p(x_{j},t|x_{i})=M^{t}_{i,j}\,
  37. M M
  38. X X
  39. M t M^{t}
  40. M i , j t = l λ l t ψ l ( x i ) ϕ l ( x j ) M^{t}_{i,j}=\sum_{l}\lambda_{l}^{t}\psi_{l}(x_{i})\phi_{l}(x_{j})\,
  41. { λ l } \{\lambda_{l}\}
  42. M t M^{t}
  43. { ψ l } \{\psi_{l}\}
  44. { ϕ l } \{\phi_{l}\}
  45. α \alpha
  46. α \alpha
  47. α = 1 \alpha=1
  48. α = 0.5 \alpha=0.5
  49. α = 0 \alpha=0
  50. t t
  51. D t ( x i , x j ) 2 = y ( p ( y , t | x i ) - p ( y , t | x j ) ) 2 ϕ 0 ( y ) D_{t}(x_{i},x_{j})^{2}=\sum_{y}\frac{(p(y,t|x_{i})-p(y,t|x_{j}))^{2}}{\phi_{0}% (y)}
  52. ϕ 0 ( y ) \phi_{0}(y)
  53. M M
  54. ϕ 0 ( y ) = d ( y ) z X d ( z ) \phi_{0}(y)=\frac{d(y)}{\sum_{z\in X}d(z)}
  55. D t ( x i , x j ) D_{t}(x_{i},x_{j})
  56. x i x_{i}
  57. x j x_{j}
  58. t t
  59. D t ( x i , x j ) D_{t}(x_{i},x_{j})
  60. t t
  61. x i x_{i}
  62. x j x_{j}
  63. D t ( x i , x j ) 2 = l λ l 2 t ( ψ l ( x i ) - ψ l ( x j ) ) 2 D_{t}(x_{i},x_{j})^{2}=\sum_{l}\lambda_{l}^{2t}(\psi_{l}(x_{i})-\psi_{l}(x_{j}% ))^{2}\,
  64. Ψ t ( x ) = ( λ 1 t ψ 1 ( x ) , λ 2 t ψ 2 ( x ) , , λ k t ψ k ( x ) ) \Psi_{t}(x)=(\lambda_{1}^{t}\psi_{1}(x),\lambda_{2}^{t}\psi_{2}(x),\ldots,% \lambda_{k}^{t}\psi_{k}(x))
  65. D t ( x i , x j ) 2 = || Ψ t ( x i ) - Ψ t ( x j ) || 2 D_{t}(x_{i},x_{j})^{2}=||\Psi_{t}(x_{i})-\Psi_{t}(x_{j})||^{2}\,
  66. α \alpha
  67. L ( α ) = D - α L D - α L^{(\alpha)}=D^{-\alpha}LD^{-\alpha}
  68. M = ( D ( α ) ) - 1 L ( α ) M=({D}^{(\alpha)})^{-1}L^{(\alpha)}
  69. M t M^{t}
  70. Ψ t \Psi_{t}

Diffusion_wavelets.html

  1. T T
  2. T T
  3. T T
  4. T T
  5. T T
  6. T T
  7. Φ a \Phi_{a}
  8. Ψ b \Psi_{b}
  9. a a
  10. b b
  11. [ Φ b ] Φ a [\Phi_{b}]_{\Phi_{a}}
  12. Φ b \Phi_{b}
  13. Φ a \Phi_{a}
  14. [ T ] Φ a Φ b [T]_{\Phi_{a}}^{\Phi_{b}}
  15. T T
  16. T T
  17. Φ a \Phi_{a}
  18. T T
  19. Φ b \Phi_{b}
  20. T T
  21. Φ a \Phi_{a}
  22. Φ b \Phi_{b}
  23. Q R QR
  24. ϵ \epsilon

Dimensionless_momentum-depth_relationship_in_open-channel_flow.html

  1. M = Q 2 g A + y ¯ A M=\frac{Q^{2}}{gA}+\overline{y}A
  2. M 1 = Q 2 g A 1 + y ¯ 1 A 1 = Q 2 g A 2 + y ¯ 2 A 2 = M 2 M_{1}=\frac{Q^{2}}{gA_{1}}+\overline{y}_{1}A_{1}=\frac{Q^{2}}{gA_{2}}+% \overline{y}_{2}A_{2}=M_{2}
  3. M 1 u = q 2 g y 1 + y 1 2 2 = q 2 g y 2 + y 2 2 2 = M 2 u {{}_{u}M_{1}}=\frac{q^{2}}{gy_{1}}+\frac{y_{1}^{2}}{2}=\frac{q^{2}}{gy_{2}}+% \frac{y_{2}^{2}}{2}={{}_{u}M_{2}}
  4. d M d y = y c - q 2 g y c 2 \frac{dM}{dy}=y_{c}-\frac{q^{2}}{gy_{c}^{2}}
  5. y c = 2 3 M c y_{c}=\frac{2}{3}\sqrt{M_{c}}
  6. M 1 u = q 2 g y 1 + y 1 2 2 = q 2 g y 2 + y 2 2 2 = M 2 u {{}_{u}M_{1}}=\frac{q^{2}}{gy_{1}}+\frac{y_{1}^{2}}{2}=\frac{q^{2}}{gy_{2}}+% \frac{y_{2}^{2}}{2}={{}_{u}M_{2}}
  7. q 2 = g y c 3 {q^{2}}={gy_{c}^{3}}
  8. M u y c 2 = g y c 3 g y y c 2 + y 2 2 y c 2 \frac{{}_{u}M}{y_{c}^{2}}=\frac{gy_{c}^{3}}{gyy_{c}^{2}}+\frac{y^{2}}{2y_{c}^{% 2}}
  9. M = 1 y + y 2 2 M^{\prime}=\frac{1}{y^{\prime}}+\frac{y^{\prime 2}}{2}
  10. q = Q b q=\frac{Q}{b}
  11. q = 100 f t 3 / s 10 f t = 10 f t 2 / s q=\frac{100ft^{3}/s}{10ft}=10ft^{2}/s
  12. y c = q 2 g 3 y_{c}=\sqrt[3]{q^{2}\over g}
  13. y c = ( 10 f t 2 / s ) 2 32.2 f t / s 2 3 = 1.459 f t y_{c}=\sqrt[3]{{(10ft^{2}/s)}^{2}\over 32.2ft/s^{2}}=1.459ft
  14. y = y d y c y^{\prime}=\frac{y_{d}}{y_{c}}
  15. y = 6 f t 1.46 f t = 4.112 y^{\prime}=\frac{6ft}{1.46ft}=4.112
  16. y = y u y c y^{\prime}=\frac{y_{u}}{y_{c}}
  17. y u = y y c = 0.115 ( 1.459 f t ) = 0.168 f t {y_{u}}={y^{\prime}}{y_{c}}=0.115(1.459ft)=0.168ft
  18. M u u = q 2 g y 1 + y 1 2 2 {{}_{u}M_{u}}=\frac{q^{2}}{gy_{1}}+\frac{y_{1}^{2}}{2}
  19. M u u = 10 2 ( 32.2 ) 0.168 + 0.168 2 2 = 18.500 f t 2 {{}_{u}M_{u}}=\frac{10^{2}}{(32.2)0.168}+\frac{0.168^{2}}{2}=18.500ft^{2}
  20. M d u = q 2 g y 2 + y 2 2 2 {{}_{u}M_{d}}=\frac{q^{2}}{gy_{2}}+\frac{y_{2}^{2}}{2}
  21. M d u = 10 2 ( 32.2 ) 6 + 6 2 2 = 18.518 f t 2 {{}_{u}M_{d}}=\frac{10^{2}}{(32.2)6}+\frac{6^{2}}{2}=18.518ft^{2}
  22. E 1 = E 2 E_{1}=E_{2}
  23. y 1 + q 2 2 g y 1 = y 2 + q 2 2 g y 2 = 8.02 f t y_{1}+\frac{q^{2}}{2gy_{1}}=y_{2}+\frac{q^{2}}{2gy_{2}}=8.02ft
  24. y 2 = 2 y 1 - 1 + 1 + 8 g y 1 3 q 2 = 0.45 f t y_{2}=\frac{2y_{1}}{-1+\sqrt{1+8\frac{gy_{1}^{3}}{q^{2}}}}=0.45ft
  25. M 2 = M 3 M_{2}=M_{3}
  26. y 2 2 + q 2 g y 2 = y 3 2 + q 2 g y 3 = 32.4 f t \frac{y_{2}}{2}+\frac{q^{2}}{gy_{2}}=\frac{y_{3}}{2}+\frac{q^{2}}{gy_{3}}=32.4ft
  27. y 3 = y 2 2 ( - 1 + 1 + 8 q 2 g y 2 3 ) = 3.46 f t y_{3}=\frac{y_{2}}{2}(-1+\sqrt{1+8\frac{q^{2}}{gy_{2}^{3}}})=3.46ft
  28. M = M 1 - M 2 = 25.5 \triangle{M}=M_{1}-M_{2}=25.5
  29. T h r u s t = γ M = ρ g M = 1590 l b s / f f Thrust=\gamma\triangle{M}=\rho g\triangle{M}=1590lbs/ff

Dimensionless_Specific_Energy_Diagrams_for_Open_Channel_Flow.html

  1. E = v 2 2 g + y + p γ E=\frac{v^{2}}{2g}+y+\frac{p}{\gamma}
  2. γ \gamma
  3. E 1 = v 1 2 2 g + y 1 + p 1 γ = E 2 = v 2 2 2 g + y 2 + p 2 γ E_{1}=\frac{v_{1}^{2}}{2g}+y_{1}+\frac{p_{1}}{\gamma}=E_{2}=\frac{v_{2}^{2}}{2% g}+y_{2}+\frac{p_{2}}{\gamma}
  4. E 1 = v 1 2 2 g + y 1 = E 2 = v 2 2 2 g + y 2 E_{1}=\frac{v_{1}^{2}}{2g}+y_{1}=E_{2}=\frac{v_{2}^{2}}{2g}+y_{2}
  5. q = Q b q=\frac{Q}{b}
  6. v = q y v=\frac{q}{y}
  7. E = q 2 2 g y 2 + y E=\frac{q^{2}}{2gy^{2}}+y
  8. y 2 = 2 y 1 - 1 + 1 + 8 g y 1 3 q 2 y_{2}=\cfrac{2y_{1}}{-1+\sqrt{1+\cfrac{8gy_{1}^{3}}{q^{2}}}}
  9. d E d y = 1 - q 2 g y c 3 = 0 \frac{\operatorname{d}E}{\operatorname{d}y}=1-\frac{q^{2}}{gy_{c}^{3}}=0
  10. y c = q 2 g 3 y_{c}=\sqrt[3]{q^{2}\over g}
  11. E c = g y c 3 2 g y c 2 + y c = y c 2 + y c = 3 y c 2 E_{c}=\frac{gy_{c}^{3}}{2gy_{c}^{2}}+y_{c}=\frac{y_{c}}{2}+y_{c}=\frac{3y_{c}}% {2}
  12. F r = v g y Fr=\frac{v}{\sqrt{gy}}
  13. E = ( 20 f t 2 s ) 2 2 ( 32.2 f t s 2 ) ( 4.4 f t ) 2 + 4.4 f t = 4.7 f t E=\frac{(20\frac{ft^{2}}{s})^{2}}{2(32.2\frac{ft}{s^{2}})(4.4ft)^{2}}+4.4ft=4.% 7ft
  14. y c = ( 20 f t 2 s ) 2 32.2 f t s 2 3 = 2.3 f t , y_{c}=\sqrt[3]{(20\frac{ft^{2}}{s})^{2}\over 32.2\frac{ft}{s^{2}}}=2.3ft,
  15. y 2 = 2 ( 4.4 ) - 1 + 1 + 8 ( 32.2 ) ( 4.4 ) 3 ( 20 ) 2 = 1.4 f t y_{2}=\frac{2(4.4)}{-1+\sqrt{1+\frac{8(32.2)(4.4)^{3}}{(20)^{2}}}}=1.4ft
  16. E = ( 20 f t 2 s ) 2 2 ( 32.2 f t s 2 ) ( 1 f t ) 2 + 1 f t = 7.2 f t E=\frac{(20\frac{ft^{2}}{s})^{2}}{2(32.2\frac{ft}{s^{2}})(1ft)^{2}}+1ft=7.2ft
  17. q = 2 g y 2 E - 2 g y 3 = ( 2 ) ( 32.2 f t s 2 ) ( 1 f t ) 2 ( 4.7 f t ) - ( 2 ) ( 32.2 f t s 2 ) ( 1 f t ) 3 = 15.4 f t 2 s q=\sqrt{2gy^{2}E-2gy^{3}}=\sqrt{(2)(32.2\frac{ft}{s^{2}})(1ft)^{2}(4.7ft)-(2)(% 32.2\frac{ft}{s^{2}})(1ft)^{3}}=15.4\frac{ft^{2}}{s}
  18. y 1 = ( 2 ) ( 1 f t ) - 1 + 1 + 8 ( 32.2 f t s 2 ) ( 1 f t ) 3 ( 20 f t 2 s ) 2 = 7.1 f t y_{1}=\frac{(2)(1ft)}{-1+\sqrt{1+\frac{8(32.2\frac{ft}{s^{2}})(1ft)^{3}}{(20% \frac{ft^{2}}{s})^{2}}}}=7.1ft
  19. E y c = y y c + q 2 2 g y 2 y c o r \frac{E}{y_{c}}=\frac{y}{y_{c}}+\frac{q^{2}}{2gy^{2}y_{c}}\;or
  20. E = y + 1 2 y 2 E^{\prime}=y^{\prime}+\frac{1}{2y^{\prime 2}}
  21. w h e r e y = y y c where\;y^{\prime}=\frac{y}{y_{c}}
  22. a n d E = E y c and\;E^{\prime}=\frac{E}{y_{c}}
  23. d E d y = 1 - 1 y 3 = 0 \ \frac{\operatorname{d}E^{\prime}}{\operatorname{d}y^{\prime}}=1-\frac{1}{y^{% \prime 3}}=0
  24. E = y + 1 2 y 2 E^{\prime}=y^{\prime}+\frac{1}{2y^{\prime 2}}
  25. = 1 + 1 2 ( 1 ) 2 =1+\frac{1}{2(1)^{2}}
  26. = 3 2 =\frac{3}{2}
  27. y = y y c = 4.4 f t 2.3 f t = 1.9 y^{\prime}=\frac{y}{y_{c}}=\frac{4.4ft}{2.3ft}=1.9
  28. E = y + 1 2 y 2 = 1.9 + 1 2 ( 1.9 ) 2 = 2.0 E^{\prime}=y^{\prime}+\frac{1}{2y^{\prime 2}}=1.9+\frac{1}{2(1.9)^{2}}=2.0
  29. 2.0 = y + 1 2 y 2 2.0=y^{\prime}+\frac{1}{2y^{\prime 2}}
  30. o r \ or
  31. 4 y 2 - 2 y 3 - 1 = 0 \ 4y^{\prime 2}-2y^{\prime 3}-1=0
  32. w h e r e \ where
  33. y = 0.60 \ y^{\prime}=0.60
  34. y 2 = y y c = ( 0.60 ) ( 2.3 f t ) = 1.4 f t \ y_{2}=y^{\prime}y_{c}=(0.60)(2.3ft)=1.4ft
  35. y = y y c = 7.1 f t 2.3 f t = 3.1 y^{\prime}=\frac{y}{y_{c}}=\frac{7.1ft}{2.3ft}=3.1
  36. E = y + 1 2 y 2 = 3.1 + 1 2 ( 3.1 ) 2 = 3.1 E^{\prime}=y^{\prime}+\frac{1}{2y^{\prime 2}}=3.1+\frac{1}{2(3.1)^{2}}=3.1
  37. 3.1 = y + 1 2 y 2 3.1=y^{\prime}+\frac{1}{2y^{\prime 2}}
  38. o r \ or
  39. 6.3 y 2 - 2 y 3 - 1 = 0 \ 6.3y^{\prime 2}-2y^{\prime 3}-1=0
  40. w h e r e \ where
  41. y = 0.93 \ y^{\prime}=0.93
  42. y 2 = y y c = ( 0.93 ) ( 2.3 f t ) = 1.0 f t \ y_{2}=y^{\prime}y_{c}=(0.93)(2.3ft)=1.0ft

Dini–Lipschitz_criterion.html

  1. lim δ 0 + ω ( δ , f ) log ( δ ) = 0 \lim_{\delta\rightarrow 0^{+}}\omega(\delta,f)\log(\delta)=0

Dinostratus'_theorem.html

  1. | A E | | A B | = 2 π \frac{|AE|}{|AB|}=\frac{2}{\pi}
  2. 2 : π 2:\pi

Direct_bonding.html

  1. Si - OH + OH - Si polymerization Si - O - Si + H O 2 \mathrm{Si-OH+OH-Si\ \xrightarrow{polymerization}\ Si-O-Si+H{{}_{2}}O}
  2. Si - OH + OH - Si slowfracture Si - O - Si + H O 2 \mathrm{Si-OH+OH-Si\ \xleftarrow[slowfracture]{\ }Si-O-Si+H{{}_{2}}O}
  3. Si - OH + Si - OH Si - O - Si + HOH \mathrm{Si-OH+Si-OH\rightleftharpoons Si-O-Si+HOH}
  4. J / m < s u p > 2 {J}/{m<sup>2}

Direct_labor_cost.html

  1. Direct labor cost = job time × wage \,\text{Direct labor cost}=\,\text{job time}\times\,\text{wage}

Dirichlet_space.html

  1. Ω , 𝒟 ( Ω ) \Omega\subseteq\mathbb{C},\,\mathcal{D}(\Omega)
  2. H 2 ( Ω ) H^{2}(\Omega)
  3. 𝒟 ( f ) := 1 π Ω | f ( z ) | 2 d A = 1 4 π Ω | x f | 2 + | y f | 2 d x d y \mathcal{D}(f):={1\over\pi}\iint_{\Omega}|f^{\prime}(z)|^{2}\,dA={1\over 4\pi}% \iint_{\Omega}|\partial_{x}f|^{2}+|\partial_{y}f|^{2}\,dx\,dy
  4. \mathbb{C}
  5. 𝒟 ( Ω ) \mathcal{D}(\Omega)
  6. 𝒟 ( f ) = 0 \mathcal{D}(f)=0
  7. f , g 𝒟 ( Ω ) f,\,g\in\mathcal{D}(\Omega)
  8. 𝒟 ( f , g ) := 1 π Ω f ( z ) g ( z ) ¯ d A ( z ) . \mathcal{D}(f,\,g):={1\over\pi}\iint_{\Omega}f^{\prime}(z)\overline{g^{\prime}% (z)}\,dA(z).
  9. 𝒟 ( f , f ) = 𝒟 ( f ) \mathcal{D}(f,\,f)=\mathcal{D}(f)
  10. 𝒟 ( Ω ) \mathcal{D}(\Omega)
  11. f , g 𝒟 ( Ω ) := f , g H 2 ( Ω ) + 𝒟 ( f , g ) ( f , g 𝒟 ( Ω ) ) , \langle f,g\rangle_{\mathcal{D}(\Omega)}:=\langle f,\,g\rangle_{H^{2}(\Omega)}% +\mathcal{D}(f,\,g)\;\;\;\;\;(f,\,g\in\mathcal{D}(\Omega)),
  12. , H 2 ( Ω ) \langle\cdot,\,\cdot\rangle_{H^{2}(\Omega)}
  13. H 2 ( Ω ) . H^{2}(\Omega).
  14. 𝒟 ( Ω ) \|\cdot\|_{\mathcal{D}(\Omega)}
  15. f 𝒟 ( Ω ) 2 := f H 2 ( Ω ) 2 + 𝒟 ( f ) ( f 𝒟 ( Ω ) ) . \|f\|^{2}_{\mathcal{D}(\Omega)}:=\|f\|^{2}_{H^{2}(\Omega)}+\mathcal{D}(f)\;\;% \;\;\;(f\in\mathcal{D}(\Omega)).
  16. f 2 = | f ( c ) | 2 + 𝒟 ( f ) \|f\|^{2}=|f(c)|^{2}+\mathcal{D}(f)
  17. c Ω c\in\Omega
  18. 𝒟 ( Ω ) H ( Ω ) \mathcal{D}(\Omega)\cap H^{\infty}(\Omega)
  19. f 𝒟 ( Ω ) H ( Ω ) := f H ( Ω ) + 𝒟 ( f ) 1 / 2 ( f 𝒟 ( Ω ) H ( Ω ) ) . \|f\|_{\mathcal{D}(\Omega)\cap H^{\infty}(\Omega)}:=\|f\|_{H^{\infty}(\Omega)}% +\mathcal{D}(f)^{1/2}\;\;\;\;\;(f\in\mathcal{D}(\Omega)\cap H^{\infty}(\Omega)).
  20. Ω = 𝔻 \Omega=\mathbb{D}
  21. \mathbb{C}
  22. 𝒟 ( 𝔻 ) := 𝒟 \mathcal{D}(\mathbb{D}):=\mathcal{D}
  23. f ( z ) = n 0 a n z n ( f 𝒟 ) , f(z)=\sum_{n\geq 0}a_{n}z^{n}\;\;\;\;\;(f\in\mathcal{D}),
  24. D ( f ) = n 1 n | a n | 2 , D(f)=\sum_{n\geq 1}n|a_{n}|^{2},
  25. f 𝒟 2 = n 0 ( n + 1 ) | a n | 2 . \|f\|^{2}_{\mathcal{D}}=\sum_{n\geq 0}(n+1)|a_{n}|^{2}.
  26. 𝒟 \mathcal{D}
  27. f f
  28. 𝔻 \mathbb{D}
  29. f f^{\prime}
  30. 𝔻 \mathbb{D}
  31. 𝒟 \mathcal{D}
  32. w { 0 } w\in\mathbb{C}\setminus\{0\}
  33. k w ( z ) = 1 z w ¯ log ( 1 1 - z w ¯ ) ( z { 0 } ) . k_{w}(z)=\frac{1}{z\overline{w}}\log\left(\frac{1}{1-z\overline{w}}\right)\;\;% \;\;\;(z\in\mathbb{C}\setminus\{0\}).

Discrete_Chebyshev_transform.html

  1. T n ( x ) T_{n}(x)
  2. x n {x_{n}}
  3. a m = p m N n = 0 N - 1 u ( x n ) T m ( x n ) a_{m}=\frac{p_{m}}{N}\sum_{n=0}^{N-1}u(x_{n})T_{m}(x_{n})
  4. x n = - cos ( π N ( n + 1 2 ) ) x_{n}=-\cos\left(\frac{\pi}{N}(n+\frac{1}{2})\right)
  5. a m = p m N n = 0 N - 1 u ( x n ) cos ( m cos - 1 ( x n ) ) a_{m}=\frac{p_{m}}{N}\sum_{n=0}^{N-1}u(x_{n})\cos\left(m\cos^{-1}(x_{n})\right)
  6. p m = 1 m = 0 p_{m}=1\Leftrightarrow m=0
  7. p m = 2 p_{m}=2
  8. x n x_{n}
  9. a m = p m N n = 0 N - 1 u ( x n ) cos ( m π N ( N + n + 1 2 ) ) a_{m}=\frac{p_{m}}{N}\sum_{n=0}^{N-1}u(x_{n})\cos\left(\frac{m\pi}{N}(N+n+% \frac{1}{2})\right)
  10. a m = p m N n = 0 N - 1 u ( x n ) ( - 1 ) m cos ( m π N ( n + 1 2 ) ) a_{m}=\frac{p_{m}}{N}\sum_{n=0}^{N-1}u(x_{n})(-1)^{m}\cos\left(\frac{m\pi}{N}(% n+\frac{1}{2})\right)
  11. u n = m = 0 N - 1 a m T m ( x n ) u_{n}=\sum_{m=0}^{N-1}a_{m}T_{m}(x_{n})
  12. u n = m = 0 N - 1 a m cos ( m π N ( N + n + 1 2 ) ) u_{n}=\sum_{m=0}^{N-1}a_{m}\cos\left(\frac{m\pi}{N}(N+n+\frac{1}{2})\right)
  13. u n = m = 0 N - 1 a m ( - 1 ) m cos ( m π N ( n + 1 2 ) ) \therefore u_{n}=\sum_{m=0}^{N-1}a_{m}(-1)^{m}\cos\left(\frac{m\pi}{N}(n+\frac% {1}{2})\right)
  14. x n = - cos ( n π N ) x_{n}=-\cos\left(\frac{n\pi}{N}\right)
  15. T n ( x m ) = cos ( π m n N + n π ) = ( - 1 ) n cos ( π m n N ) T_{n}(x_{m})=\cos\left(\frac{\pi mn}{N}+n\pi\right)=(-1)^{n}\cos\left(\frac{% \pi mn}{N}\right)
  16. u ( x n ) = u n = m = 0 N a m T m ( x n ) u(x_{n})=u_{n}=\sum_{m=0}^{N}a_{m}T_{m}(x_{n})
  17. a m = p m N [ 1 2 ( u 0 ( - 1 ) m + u N ) + n = 1 N - 1 u n T m ( x n ) ] a_{m}=\frac{p_{m}}{N}\left[\frac{1}{2}(u_{0}(-1)^{m}+u_{N})+\sum_{n=1}^{N-1}u_% {n}T_{m}(x_{n})\right]
  18. p m = 1 m = 0 p_{m}=1\Leftrightarrow m=0
  19. p m = 2 p_{m}=2

Discrete_logarithm_records.html

  1. 𝔽 2 4 1223 \mathbb{F}_{2^{4\cdot 1223}}
  2. 𝔽 2 1971 \mathbb{F}_{2^{1971}}
  3. 𝔽 2 4 1223 {\mathbb{F}}_{2^{4\cdot 1223}}
  4. 𝔽 2 12 367 {\mathbb{F}}_{2^{12\cdot 367}}

Discrete_q-Hermite_polynomials.html

  1. h n ( x ; q ) = q ( n 2 ) ϕ 1 2 ( q - n , x - 1 ; 0 ; q , - q x ) = x n ϕ 0 2 ( q - n , q - n + 1 ; ; q 2 , q 2 n - 1 / x 2 ) = U n ( - 1 ) ( x ; q ) \displaystyle h_{n}(x;q)=q^{{\left({{n}\atop{2}}\right)}}{}_{2}\phi_{1}(q^{-n}% ,x^{-1};0;q,-qx)=x^{n}{}_{2}\phi_{0}(q^{-n},q^{-n+1};;q^{2},q^{2n-1}/x^{2})=U_% {n}^{(-1)}(x;q)
  2. h ^ n ( x ; q ) = i - n q - ( n 2 ) ϕ 0 2 ( q - n , i x ; ; q , - q n ) = x n ϕ 1 2 ( q - n , q - n + 1 ; 0 ; q 2 , - q 2 / x 2 ) = i - n V n ( - 1 ) ( i x ; q ) \displaystyle\hat{h}_{n}(x;q)=i^{-n}q^{-{\left({{n}\atop{2}}\right)}}{}_{2}% \phi_{0}(q^{-n},ix;;q,-q^{n})=x^{n}{}_{2}\phi_{1}(q^{-n},q^{-n+1};0;q^{2},-q^{% 2}/x^{2})=i^{-n}V_{n}^{(-1)}(ix;q)
  3. h n ( i x ; q - 1 ) = i n h ^ n ( x ; q ) h_{n}(ix;q^{-1})=i^{n}\hat{h}_{n}(x;q)

Dissipation_model_for_extended_environment.html

  1. = p 2 2 m + V ( x ) + i n t + b a t h \mathcal{H}=\frac{p^{2}}{2m}+V(x)+\mathcal{H}_{int}+\mathcal{H}_{bath}
  2. b a t h = α ( P α 2 2 m α + 1 2 m ω α 2 Q α 2 ) \mathcal{H}_{bath}=\sum_{\alpha}\left(\frac{P_{\alpha}^{2}}{2m_{\alpha}}+\frac% {1}{2}m\omega_{\alpha}^{2}Q_{\alpha}^{2}\right)
  3. i n t = - α c α Q α u ( x - x α ) \mathcal{H}_{int}=-\sum_{\alpha}c_{\alpha}Q_{\alpha}u(x-x_{\alpha})
  4. Q α Q_{\alpha}
  5. α \alpha
  6. u ( x - x α ) u(x-x_{\alpha})
  7. c α c_{\alpha}
  8. π 2 α c α 2 m α ω α δ ( ω - ω α ) δ ( x - x α ) = J ( ω ) \frac{\pi}{2}\sum_{\alpha}\frac{c^{2}_{\alpha}}{m_{\alpha}\omega_{\alpha}}% \delta(\omega-\omega_{\alpha})\ \delta(x-x_{\alpha})\ =\ J(\omega)
  9. J ( ω ) J(\omega)
  10. S ~ ( q , ω ) \tilde{S}(q,\omega)
  11. J ( ω ) J(\omega)
  12. u ( r ) u(r)

Distance_correlation.html

  1. a j , k = X j - X k , j , k = 1 , 2 , , n , b j , k = Y j - Y k , j , k = 1 , 2 , , n , \begin{aligned}\displaystyle a_{j,k}&\displaystyle=\|X_{j}-X_{k}\|,\qquad j,k=% 1,2,\ldots,n,\\ \displaystyle b_{j,k}&\displaystyle=\|Y_{j}-Y_{k}\|,\qquad j,k=1,2,\ldots,n,% \end{aligned}
  2. A j , k := a j , k - a ¯ j . - a ¯ . k + a ¯ . . , B j , k := b j , k - b ¯ j . - b ¯ . k + b ¯ . . , A_{j,k}:=a_{j,k}-\overline{a}_{j.}-\overline{a}_{.k}+\overline{a}_{..},\qquad B% _{j,k}:=b_{j,k}-\overline{b}_{j.}-\overline{b}_{.k}+\overline{b}_{..},
  3. a ¯ j . \textstyle\overline{a}_{j.}
  4. j j
  5. a ¯ . k \textstyle\overline{a}_{.k}
  6. k k
  7. a ¯ . . \textstyle\overline{a}_{..}
  8. dCov n 2 ( X , Y ) := 1 n 2 j , k = 1 n A j , k B j , k . \operatorname{dCov}^{2}_{n}(X,Y):=\frac{1}{n^{2}}\sum_{j,k=1}^{n}A_{j,k}\,B_{j% ,k}.
  9. μ μ
  10. ν ν
  11. a μ ( x ) := E [ X - x ] , D ( μ ) := E [ a μ ( X ) ] , d μ ( x , x ) := x - x - a μ ( x ) - a μ ( x ) + D ( μ ) . a_{\mu}(x):=\operatorname{E}[\|X-x\|],\quad D(\mu):=\operatorname{E}[a_{\mu}(X% )],\quad d_{\mu}(x,x^{\prime}):=\|x-x^{\prime}\|-a_{\mu}(x)-a_{\mu}(x^{\prime}% )+D(\mu).
  12. dCov 2 ( X , Y ) := E [ d μ ( X , X ) d ν ( Y , Y ) ] . \operatorname{dCov}^{2}(X,Y):=\operatorname{E}\big[d_{\mu}(X,X^{\prime})d_{\nu% }(Y,Y^{\prime})\big].
  13. dCov 2 ( X , Y ) \displaystyle\operatorname{dCov}^{2}(X,Y)
  14. ( X , Y ) , \textstyle(X,Y),
  15. ( X , Y ) , \textstyle(X^{\prime},Y^{\prime}),
  16. ( X ′′ , Y ′′ ) \textstyle(X^{\prime\prime},Y^{\prime\prime})
  17. dCov 2 ( X , Y ) = cov ( X - X , Y - Y ) - 2 cov ( X - X , Y - Y ′′ ) . \operatorname{dCov}^{2}(X,Y)=\operatorname{cov}(\|X-X^{\prime}\|,\|Y-Y^{\prime% }\|)-2\operatorname{cov}(\|X-X^{\prime}\|,\|Y-Y^{\prime\prime}\|).
  18. dCov 2 ( X , Y ) = 1 c p c q p + q | ϕ X , Y ( s , t ) - ϕ X ( s ) ϕ Y ( t ) | 2 | s | p 1 + p | t | q 1 + q d t d s \operatorname{dCov}^{2}(X,Y)=\frac{1}{c_{p}c_{q}}\int_{\mathbb{R}^{p+q}}\frac{% \left|\phi_{X,Y}(s,t)-\phi_{X}(s)\phi_{Y}(t)\right|^{2}}{|s|_{p}^{1+p}|t|_{q}^% {1+q}}dt\,ds
  19. ( c p c q | s | p 1 + p | t | q 1 + q ) - 1 ({c_{p}c_{q}}{|s|_{p}^{1+p}|t|_{q}^{1+q}})^{-1}
  20. dVar 2 ( X ) := E [ X - X 2 ] + E 2 [ X - X ] - 2 E [ X - X X - X ′′ ] , \operatorname{dVar}^{2}(X):=\operatorname{E}[\|X-X^{\prime}\|^{2}]+% \operatorname{E}^{2}[\|X-X^{\prime}\|]-2\operatorname{E}[\|X-X^{\prime}\|\,\|X% -X^{\prime\prime}\|],
  21. E \operatorname{E}
  22. X X^{\prime}
  23. X X
  24. X ′′ X^{\prime\prime}
  25. X X
  26. X X^{\prime}
  27. X X
  28. X X^{\prime}
  29. dVar n 2 ( X ) := dCov n 2 ( X , X ) = 1 n 2 k , A k , 2 , \operatorname{dVar}^{2}_{n}(X):=\operatorname{dCov}^{2}_{n}(X,X)=\tfrac{1}{n^{% 2}}\sum_{k,\ell}A_{k,\ell}^{2},
  30. dCor ( X , Y ) = dCov ( X , Y ) dVar ( X ) dVar ( Y ) , \operatorname{dCor}(X,Y)=\frac{\operatorname{dCov}(X,Y)}{\sqrt{\operatorname{% dVar}(X)\,\operatorname{dVar}(Y)}},
  31. 0 dCor n ( X , Y ) 1 0\leq\operatorname{dCor}_{n}(X,Y)\leq 1
  32. 0 dCor ( X , Y ) 1 0\leq\operatorname{dCor}(X,Y)\leq 1
  33. dCor ( X , Y ) = 0 \operatorname{dCor}(X,Y)=0
  34. X X
  35. Y Y
  36. dCor n ( X , Y ) = 1 \operatorname{dCor}_{n}(X,Y)=1
  37. X X
  38. Y Y
  39. Y = A + b 𝐂 X Y=A+b\,\mathbf{C}X
  40. A A
  41. b b
  42. 𝐂 \mathbf{C}
  43. dCov ( X , Y ) 0 \operatorname{dCov}(X,Y)\geq 0
  44. dCov n ( X , Y ) 0 \operatorname{dCov}_{n}(X,Y)\geq 0
  45. dCov 2 ( a 1 + b 1 𝐂 1 X , a 2 + b 2 𝐂 2 Y ) = | b 1 b 2 | dCov 2 ( X , Y ) \operatorname{dCov}^{2}(a_{1}+b_{1}\,\mathbf{C}_{1}\,X,a_{2}+b_{2}\,\mathbf{C}% _{2}\,Y)=|b_{1}\,b_{2}|\operatorname{dCov}^{2}(X,Y)
  46. a 1 , a 2 a_{1},a_{2}
  47. b 1 , b 2 b_{1},b_{2}
  48. 𝐂 1 , 𝐂 2 \mathbf{C}_{1},\mathbf{C}_{2}
  49. ( X 1 , Y 1 ) (X_{1},Y_{1})
  50. ( X 2 , Y 2 ) (X_{2},Y_{2})
  51. dCov ( X 1 + X 2 , Y 1 + Y 2 ) dCov ( X 1 , Y 1 ) + dCov ( X 2 , Y 2 ) . \operatorname{dCov}(X_{1}+X_{2},Y_{1}+Y_{2})\leq\operatorname{dCov}(X_{1},Y_{1% })+\operatorname{dCov}(X_{2},Y_{2}).
  52. X 1 X_{1}
  53. Y 1 Y_{1}
  54. X 2 X_{2}
  55. Y 2 Y_{2}
  56. X 1 , X 2 , Y 1 , Y 2 X_{1},X_{2},Y_{1},Y_{2}
  57. dCov ( X , Y ) = 0 \operatorname{dCov}(X,Y)=0
  58. X X
  59. Y Y
  60. dCov n 2 ( X , Y ) \operatorname{dCov}^{2}_{n}(X,Y)
  61. dCov 2 ( X , Y ) \operatorname{dCov}^{2}(X,Y)
  62. E [ dCov n 2 ( X , Y ) ] = n - 1 n 2 { ( n - 2 ) dCov 2 ( X , Y ) + E [ X - X ] E [ Y - Y ] } = n - 1 n 2 E [ X - X ] E [ Y - Y ] . \operatorname{E}[\operatorname{dCov}^{2}_{n}(X,Y)]=\frac{n-1}{n^{2}}\left\{(n-% 2)\operatorname{dCov}^{2}(X,Y)+\operatorname{E}[\|X-X^{\prime}\|]\,% \operatorname{E}[\|Y-Y^{\prime}\|]\right\}=\frac{n-1}{n^{2}}\operatorname{E}[% \|X-X^{\prime}\|]\,\operatorname{E}[\|Y-Y^{\prime}\|].
  63. dCov 2 ( X , Y ) \operatorname{dCov}^{2}(X,Y)
  64. dVar ( X ) = 0 \operatorname{dVar}(X)=0
  65. X = E [ X ] X=\operatorname{E}[X]
  66. dVar n ( X ) = 0 \operatorname{dVar}_{n}(X)=0
  67. dVar ( A + b 𝐂 X ) = | b | dVar ( X ) \operatorname{dVar}(A+b\,\mathbf{C}\,X)=|b|\operatorname{dVar}(X)
  68. A A
  69. b b
  70. 𝐂 \mathbf{C}
  71. X X
  72. Y Y
  73. dVar ( X + Y ) dVar ( X ) + dVar ( Y ) \operatorname{dVar}(X+Y)\leq\operatorname{dVar}(X)+\operatorname{dVar}(Y)
  74. X X
  75. Y Y
  76. dCov 2 ( X , Y ; α ) : = E [ X - X α Y - Y α ] + E [ X - X α ] E [ Y - Y α ] - 2 E [ X - X α Y - Y ′′ α ] . \begin{aligned}\displaystyle\operatorname{dCov}^{2}(X,Y;\alpha)&\displaystyle:% =\operatorname{E}[\|X-X^{\prime}\|^{\alpha}\,\|Y-Y^{\prime}\|^{\alpha}]+% \operatorname{E}[\|X-X^{\prime}\|^{\alpha}]\,\operatorname{E}[\|Y-Y^{\prime}\|% ^{\alpha}]\\ &\displaystyle\qquad-2\operatorname{E}[\|X-X^{\prime}\|^{\alpha}\,\|Y-Y^{% \prime\prime}\|^{\alpha}].\end{aligned}
  77. 0 < α < 2 0<\alpha<2
  78. X X
  79. Y Y
  80. dCov 2 ( X , Y ; α ) = 0 \operatorname{dCov}^{2}(X,Y;\alpha)=0
  81. α = 2 \alpha=2
  82. ( X , Y ) (X,Y)
  83. dCor ( X , Y ; α = 2 ) \operatorname{dCor}(X,Y;\alpha=2)
  84. a k , a_{k,\ell}
  85. b k , b_{k,\ell}
  86. α \alpha
  87. 0 < α 2 0<\alpha\leq 2
  88. α \alpha
  89. dCov n 2 ( X , Y ; α ) := 1 n 2 k , A k , B k , . \operatorname{dCov}^{2}_{n}(X,Y;\alpha):=\frac{1}{n^{2}}\sum_{k,\ell}A_{k,\ell% }\,B_{k,\ell}.
  90. dCov \operatorname{dCov}
  91. X X
  92. Y Y
  93. X X
  94. μ \mu
  95. d d
  96. a μ ( x ) := E [ d ( X , x ) ] a_{\mu}(x):=\operatorname{E}[d(X,x)]
  97. D ( μ ) := E [ a μ ( X ) ] D(\mu):=\operatorname{E}[a_{\mu}(X)]
  98. a μ a_{\mu}
  99. X X
  100. d μ ( x , x ) := d ( x , x ) - a μ ( x ) - a μ ( x ) + D ( μ ) d_{\mu}(x,x^{\prime}):=d(x,x^{\prime})-a_{\mu}(x)-a_{\mu}(x^{\prime})+D(\mu)
  101. Y Y
  102. ν \nu
  103. dCov 2 ( X , Y ) := E [ d μ ( X , X ) d ν ( Y , Y ) ] . \operatorname{dCov}^{2}(X,Y):=\operatorname{E}\big[d_{\mu}(X,X^{\prime})d_{\nu% }(Y,Y^{\prime})\big].
  104. X , Y X,Y
  105. ( M , d ) (M,d)
  106. ( M , d 1 / 2 ) (M,d^{1/2})
  107. dCov 2 ( X , Y ) = 0 \operatorname{dCov}^{2}(X,Y)=0
  108. X , Y X,Y
  109. dCov 2 ( X , Y ) \operatorname{dCov}^{2}(X,Y)
  110. dCov ( X , Y ) \operatorname{dCov}(X,Y)
  111. X , Y \operatorname{X},Y
  112. X \operatorname{X}
  113. dCov 2 ( X , Y ) . \operatorname{dCov}^{2}(X,Y).
  114. X X
  115. X X
  116. dCor 2 ( X , Y ) \operatorname{dCor}^{2}(X,Y)
  117. cov ( X , Y ) 2 = E [ ( X - E ( X ) ) ( X - E ( X ) ) ( Y - E ( Y ) ) ( Y - E ( Y ) ) ] \operatorname{cov}(X,Y)^{2}=\operatorname{E}\left[\big(X-\operatorname{E}(X)% \big)\big(X^{\mathrm{{}^{\prime}}}-\operatorname{E}(X^{\mathrm{{}^{\prime}}})% \big)\big(Y-\operatorname{E}(Y)\big)\big(Y^{\mathrm{{}^{\prime}}}-% \operatorname{E}(Y^{\mathrm{{}^{\prime}}})\big)\right]
  118. X U := U ( X ) - E X [ U ( X ) { U ( t ) } ] X_{U}:=U(X)-\operatorname{E}_{X}\left[U(X)\mid\left\{U(t)\right\}\right]
  119. cov U , V 2 ( X , Y ) := E [ X U X U Y V Y V ] \operatorname{cov}_{U,V}^{2}(X,Y):=\operatorname{E}\left[X_{U}X_{U}^{\mathrm{{% }^{\prime}}}Y_{V}Y_{V}^{\mathrm{{}^{\prime}}}\right]
  120. cov W ( X , Y ) . \operatorname{cov}_{W}(X,Y).
  121. cov W ( X , Y ) = dCov ( X , Y ) , \operatorname{cov}_{\mathrm{W}}(X,Y)=\operatorname{dCov}(X,Y),
  122. cov id ( X , Y ) = | cov ( X , Y ) | . \operatorname{cov}_{\mathrm{id}}(X,Y)=\left|\operatorname{cov}(X,Y)\right|.

Distortion_function.html

  1. g : [ 0 , 1 ] [ 0 , 1 ] g:[0,1]\to[0,1]
  2. g ( 0 ) = 0 g(0)=0
  3. g ( 1 ) = 1 g(1)=1
  4. g ~ ( x ) = 1 - g ( 1 - x ) \tilde{g}(x)=1-g(1-x)
  5. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  6. X X
  7. g g
  8. \mathbb{Q}
  9. A A\in\mathcal{F}
  10. ( A ) = g ( ( X A ) ) . \mathbb{Q}(A)=g(\mathbb{P}(X\in A)).

Distortion_problem.html

  1. sup y 1 , y 2 Y , y i = 1 | y 1 | | y 2 | λ \sup_{y_{1},y_{2}\in Y,\|y_{i}\|=1}\frac{|y_{1}|}{|y_{2}|}\geq\lambda

Distortion_risk_measure.html

  1. ρ g : L p \rho_{g}:L^{p}\to\mathbb{R}
  2. g : [ 0 , 1 ] [ 0 , 1 ] g:[0,1]\to[0,1]
  3. X L p X\in L^{p}
  4. L p L^{p}
  5. ρ g ( X ) = - 0 1 F - X - 1 ( p ) d g ~ ( p ) = - 0 g ~ ( F - X ( x ) ) d x - 0 g ( 1 - F - X ( x ) ) d x \rho_{g}(X)=-\int_{0}^{1}F_{-X}^{-1}(p)d\tilde{g}(p)=\int_{-\infty}^{0}\tilde{% g}(F_{-X}(x))dx-\int_{0}^{\infty}g(1-F_{-X}(x))dx
  6. F - X F_{-X}
  7. - X -X
  8. g ~ \tilde{g}
  9. g ~ ( u ) = 1 - g ( 1 - u ) \tilde{g}(u)=1-g(1-u)
  10. X 0 X\leq 0
  11. ρ g \rho_{g}
  12. ρ g ( X ) = - 0 g ( 1 - F - X ( x ) ) d x . \rho_{g}(X)=-\int_{0}^{\infty}g(1-F_{-X}(x))dx.
  13. ρ g ( X ) = 𝔼 [ - X ] \rho_{g}(X)=\mathbb{E}^{\mathbb{Q}}[-X]
  14. \mathbb{Q}
  15. g g
  16. A A\in\mathcal{F}
  17. ( A ) = g ( ( A ) ) \mathbb{Q}(A)=g(\mathbb{P}(A))
  18. X X
  19. Y Y
  20. ρ g ( X ) = ρ g ( Y ) \rho_{g}(X)=\rho_{g}(Y)
  21. g g
  22. ρ g \rho_{g}
  23. g g
  24. ρ g \rho_{g}
  25. g ( x ) = { 0 if 0 x < 1 - α 1 if 1 - α x 1 . g(x)=\begin{cases}0&\,\text{if }0\leq x<1-\alpha\\ 1&\,\text{if }1-\alpha\leq x\leq 1\end{cases}.
  26. g ( x ) = { x 1 - α if 0 x < 1 - α 1 if 1 - α x 1 . g(x)=\begin{cases}\frac{x}{1-\alpha}&\,\text{if }0\leq x<1-\alpha\\ 1&\,\text{if }1-\alpha\leq x\leq 1\end{cases}.
  27. g ( x ) = x g(x)=x

Distortionmeter.html

  1. f i = a 1 s i n ( ω t ) {f_{i}}=a_{1}\cdot sin(\omega t)
  2. f o = a 1 s i n ( ω t ) + a 2 s i n ( 2 ω t ) + a 3 s i n ( 3 ω t ) + . . {f_{o}}=a_{1}\cdot sin(\omega t)+a_{2}\cdot sin(2\omega t)+a_{3}\cdot sin(3% \omega t)+..
  3. T H D = a 2 2 + a 3 2 + . . a 1 THD=\frac{\sqrt{{a_{2}}^{2}+{a_{3}}^{2}+..}}{a_{1}}

Disulfur_difluoride.html

  1. FS - SF KF S = SF 2 \rm\ FS-SF\xrightarrow{KF}S=SF_{2}
  2. S 8 + 8 A g F 2 398 K 4 S 2 F 2 + 8 A g F \rm\ S_{8}+8AgF_{2}\xrightarrow{398K}4S_{2}F_{2}+8AgF
  3. 2 KSO 2 F + S 2 Cl 2 S = SF 2 + 2 K C l + 2 S O 2 \rm\ 2KSO_{2}F+S_{2}Cl_{2}\rightarrow S=SF_{2}+2KCl+2SO_{2}
  4. 2 S 2 F 2 180 o C SF 4 + 3 S \mathrm{2S_{2}F_{2}\ \xrightarrow{180^{o}C}\ SF_{4}+3S}
  5. 2 S 2 F 2 + 2 H 2 O SO 2 + 3 S + 4 H F \mathrm{2S_{2}F_{2}+2H_{2}O\ \xrightarrow{}\ SO_{2}+3S+4HF}
  6. S 2 F 2 + 3 H 2 SO 4 80 o C 5 SO 2 + 2 H F + 2 H 2 O \mathrm{S_{2}F_{2}+3H_{2}SO_{4}\ \xrightarrow{80^{o}C}\ 5SO_{2}+2HF+2H_{2}O}
  7. 2 S 2 F 2 + 6 N a O H Na 2 SO 3 + 3 S + 4 N a F + 3 H 2 O \mathrm{2S_{2}F_{2}+6NaOH\ \xrightarrow{}\ Na_{2}SO_{3}+3S+4NaF+3H_{2}O}
  8. 2 S 2 F 2 + 5 O 2 NO 2 SOF 4 + 3 S O 3 \mathrm{2S_{2}F_{2}+5O_{2}\ \xrightarrow{NO_{2}}\ SOF_{4}+3SO_{3}}

Dividend_policy.html

  1. P = D + ( r ) ( ( E - D ) / k e ) k e P={D+(r)((E-D)/k_{e})\over{k_{e}}}\,
  2. P = E ( 1 - b ) k e - b r P={E(1-b)\over{k_{e}-br}}\,
  3. D E q > 1 - T C 1 - T D - 1 \frac{D}{E_{\,\text{q}}}>\frac{1-T_{\,\text{C}}}{1-T_{\,\text{D}}}-1
  4. E q E_{\,\text{q}}
  5. T C T_{\,\text{C}}
  6. T D T_{\,\text{D}}

Divisibility_(ring_theory).html

  1. a b a\mid b
  2. a b a\mid b
  3. b a b\mid a
  4. R R
  5. a b a\mid b
  6. ( b ) ( a ) (b)\subseteq(a)
  7. ( a ) = ( b ) (a)=(b)
  8. ( u ) = R (u)=R
  9. a = b u a=bu
  10. ( a ) (a)
  11. R R
  12. a a

Divisibility_sequence.html

  1. ( a n ) n 𝒩 {(a_{n})}_{n\in\mathcal{N}}
  2. if m n then a m a n , \,\text{if }m\mid n\,\text{ then }a_{m}\mid a_{n},
  3. ( a n ) n 𝒩 {(a_{n})}_{n\in\mathcal{N}}
  4. gcd ( a m , a n ) = a gcd ( m , n ) . \gcd(a_{m},a_{n})=a_{\gcd(m,n)}.
  5. m n m\mid n
  6. g c d ( m , n ) = m gcd(m,n)=m
  7. g c d ( a m , a n ) = a m gcd(a_{m},a_{n})=a_{m}
  8. a m a n a_{m}\mid a_{n}
  9. a n = k n a_{n}=kn
  10. a n = A n - B n a_{n}=A^{n}-B^{n}
  11. A > B > 0 A>B>0

Divisor_topology.html

  1. S n = { x Z + : x | n } S_{n}=\{x\in{Z}^{+}:x|n\}

Document_mosaicing.html

  1. p i y = u = 1 i v = 1 j b u v p_{iy}=\sum_{u=1}^{i}\sum_{v=1}^{j}b_{uv}
  2. u = u 1 u 2 v = v 1 v 2 b u v = p u 2 v 2 + p u 1 v 1 - p u 1 v 2 - p u 2 v 1 \sum_{u=u_{1}}^{u_{2}}\sum_{v=v_{1}}^{v_{2}}b_{uv}=p_{u_{2}v_{2}}+p_{u_{1}v_{1% }}-p_{u_{1}v_{2}}-p_{u_{2}v_{1}}
  3. [ s u s v s ] = [ p 11 p 12 p 13 p 21 p 22 p 23 p 31 p 32 1 ] [ u v 1 ] E q .1 \left[\begin{array}[]{c}su^{\prime}\\ sv^{\prime}\\ s\end{array}\right]=\left[\begin{array}[]{ccc}p_{11}&p_{12}&p_{13}\\ p_{21}&p_{22}&p_{23}\\ p_{31}&p_{32}&1\end{array}\right]\left[\begin{array}[]{c}u\\ v\\ 1\end{array}\right]\qquad Eq.1

Dold–Thom_theorem.html

  1. H ~ i ( X ; ) \widetilde{H}_{i}(X;\mathbb{Z})

Double_centralizer_theorem.html

  1. C R ( S ) = { r R r s = s r for all s S } . \mathrm{C}_{R}(S)=\{r\in R\mid rs=sr\,\text{ for all }s\in S\}.\,
  2. dim F ( B ) dim F ( C A ( B ) ) = dim F ( A ) . \mathrm{dim}_{F}(B)\cdot\mathrm{dim}_{F}(\mathrm{C}_{A}(B))=\mathrm{dim}_{F}(A% ).\,
  3. R U = C E ( C E ( R U ) ) R_{U}=\mathrm{C}_{E}(\mathrm{C}_{E}(R_{U}))\,

Double_Chooz.html

  1. e , μ , τ e,\mu,\tau
  2. θ 13 \theta_{13}
  3. θ 13 \theta_{13}
  4. sin 2 ( 2 θ 13 ) < 0.2 \sin^{2}(2\theta_{13})<0.2
  5. θ 13 \theta_{13}
  6. 0.03 < sin 2 ( 2 θ 13 ) < 0.2 0.03<\sin^{2}(2\theta_{13})<0.2
  7. ν e ¯ \bar{\nu_{e}}
  8. ν e ¯ \bar{\nu_{e}}
  9. P = 1 - sin 2 ( 2 θ 13 ) sin 2 ( 1.27 Δ m 31 2 L E ν ) ( natural units ) . P=1-\sin^{2}(2\theta_{13})\,\sin^{2}\left(\frac{1.27\Delta m^{2}_{31}L}{E_{\nu% }}\right)\,\mathrm{(natural\,units)}.
  10. L L
  11. E ν E_{\nu}
  12. ν e ¯ \bar{\nu_{e}}
  13. ν ¯ e + p e + + n . \bar{\nu}_{e}+p\to e^{+}+n.
  14. θ 13 \theta_{13}
  15. sin 2 ( 2 θ 13 ) = 0.097 ± 0.034 ( stat ) ± 0.034 ( syst ) . \sin^{2}(2\theta_{13})=0.097\pm 0.034\,\mathrm{(stat)}\pm 0.034\,\mathrm{(syst% )}.
  16. sin 2 ( 2 θ 13 ) = 0.102 ± 0.028 ( stat ) ± 0.033 ( syst ) . \sin^{2}(2\theta_{13})=0.102\pm 0.028\,\mathrm{(stat)}\pm 0.033\,\mathrm{(syst% )}.
  17. sin 2 ( 2 θ 13 ) = 0.090 - 0.029 + 0.032 . \sin^{2}(2\theta_{13})=0.090^{+0.032}_{-0.029}.

Double_origin_topology.html

  1. N ( 0 , n ) = { ( x , y ) R 2 : x 2 + y 2 < 1 / n 2 , y > 0 } { 0 } . \ N(0,n)=\{(x,y)\in{R}^{2}:x^{2}+y^{2}<1/n^{2},\ y>0\}\cup\{0\}.
  2. N ( 0 * , n ) = { ( x , y ) R 2 : x 2 + y 2 < 1 / n 2 , y < 0 } { 0 * } . N(0^{*},n)=\{(x,y)\in{R}^{2}:x^{2}+y^{2}<1/n^{2},\ y<0\}\cup\{0^{*}\}.

Doubling_space.html

  1. μ ( B ( x , 2 r ) ) C μ ( B ( x , r ) ) \mu(B(x,2r))\leq C\mu(B(x,r))\,
  2. d μ n = i = 1 n ( 1 + a cos ( 3 i 2 π x ) ) d x , | a | < 1. d\mu_{n}=\prod_{i=1}^{n}(1+a\cos(3^{i}2\pi x))\,dx,\;\;\;|a|<1.

Downlink_CNR.html

  1. f = g t f=\frac{g}{t}
  2. t = t a + ( n - 1 ) t 0 t=t_{a}+(n-1)\cdot t_{0}
  3. f = g t a + ( n - 1 ) t 0 f=\frac{g}{t_{a}+(n-1)\cdot t_{0}}
  4. F = 10 log 10 ( f ) F=10\ \log_{10}(f)
  5. l = ( 4 π d λ ) 2 l=(\frac{4\cdot\pi\cdot d}{\lambda})^{2}
  6. λ \lambda
  7. L = 20 log 10 ( 4 π d λ ) L=20\ \log_{10}\left(\frac{4\cdot\pi\cdot d}{\lambda}\right)
  8. L = 20 log 10 ( 4 π d f c ) L=20\ \log_{10}\left(\frac{4\cdot\pi\cdot d\cdot f}{c}\right)
  9. L = - 147.56 + 20 log 10 ( d ) + 20 log 10 ( f ) L=-147.56+20\ \log_{10}(d)+20\ \log_{10}(f)
  10. L = 92.45 + 20 log 10 ( d ) + 20 log 10 ( f ) L=92.45+20\ \log_{10}(d)+20\ \log_{10}(f)
  11. L = 96.58 + 20 log 10 ( d ) + 20 log 10 ( f ) L=96.58+20\ \log_{10}(d)+20\ \log_{10}(f)
  12. P e = 10 log 10 ( p ) + 10 log 10 ( g t ) P_{e}=10\ \log_{10}(p)+10\ \log_{10}(g_{t})
  13. B = 10 log 10 ( b ) B=10\ \log_{10}(b)
  14. B = 10 log 10 ( b ) + 60 B=10\ \log_{10}(b)+60
  15. K = 10 log 10 ( 1.380 10 - 23 ) = - 228.6 K=10\ \log_{10}(1.380\cdot 10^{-23})=-228.6
  16. CNR = F + P - B - K - L \mbox{CNR}~{}=F+P-B-K-L

Downside_risk.html

  1. S D ( X ) = ( 𝔼 [ ( X - 𝔼 [ X ] ) 2 1 { X 𝔼 [ X ] } ] ) 1 2 SD(X)=\left(\mathbb{E}[(X-\mathbb{E}[X])^{2}1_{\{X\leq\mathbb{E}[X]\}}]\right)% ^{\frac{1}{2}}
  2. 1 { X 𝔼 [ X ] } 1_{\{X\leq\mathbb{E}[X]\}}
  3. 1 { X 𝔼 [ X ] } = { 1 if X 𝔼 [ X ] 0 else 1_{\{X\leq\mathbb{E}[X]\}}=\begin{cases}1&\,\text{if }X\leq\mathbb{E}[X]\\ 0&\,\text{else}\end{cases}
  4. t t
  5. T S V ( X , t ) = ( 𝔼 [ ( X - t ) 2 1 { X t } ] ) 1 2 TSV(X,t)=\left(\mathbb{E}[(X-t)^{2}1_{\{X\leq t\}}]\right)^{\frac{1}{2}}

DSRP.html

  1. S T n = i n f o j n { : D o i S w p R r a P v ρ : } j {ST}_{n}=\underset{info}{\oplus}\underset{j\leq n}{\otimes}{\left\{:{D}_{o}^{i% }\circ{S}_{w}^{p}\circ{R}_{r}^{a}\circ{P}_{v}^{\rho}:\right\}}_{j}
  2. P := ( e 1 e 2 ) P:=\left({e}_{1}\leftrightarrow{e}_{2}\right)
  3. e 1 {e}_{1}
  4. e 2 {e}_{2}
  5. ( e 1 e 2 ) \left({e}_{1}\leftrightarrow{e}_{2}\right)
  6. : = P :=P
  7. D := ( i o ) D:=\left({i}\leftrightarrow{o}\right)
  8. i {i}
  9. o {o}
  10. ( i o ) \left({i}\leftrightarrow{o}\right)
  11. : = D :=D
  12. S := ( p w ) S:=\left({p}\leftrightarrow{w}\right)
  13. w {w}
  14. p {p}
  15. ( w p ) \left({w}\leftrightarrow{p}\right)
  16. : = S :=S
  17. R := ( a r ) R:=\left({a}\leftrightarrow{r}\right)
  18. a {a}
  19. r {r}
  20. ( a r ) \left({a}\leftrightarrow{r}\right)
  21. : = R :=R
  22. P := ( ρ v ) P:=\left(\rho\leftrightarrow{v}\right)
  23. ρ {\rho}
  24. v {v}
  25. ( ρ v ) \left(\rho\leftrightarrow{v}\right)
  26. : = P :=P
  27. ( D S R P ) ( D S R P ) \intercal\left(D\vee S\vee R\vee P\right)\rightarrow\left(D\wedge S\wedge R% \wedge P\right)
  28. ( D S R P ) \left(D\vee S\vee R\vee P\right)
  29. ( D S R P ) \left(D\wedge S\wedge R\wedge P\right)
  30. ( D S R P ) ( D S R P ) \intercal\left(D\vee S\vee R\vee P\right)\rightarrow\left(D\wedge S\wedge R% \wedge P\right)

Du_Noüy–Padday_method.html

  1. 2 π τ p γ cos θ = m m g 2\pi\tau_{p}\gamma\cos\theta=m_{m}g
  2. τ p \tau_{p}
  3. m m m_{m}
  4. θ \theta
  5. F p = m m g + F b u o a y a n c y = 2 τ p + F b u o y a n c y F_{p}=m_{m}g+F_{buoayancy}=2\tau_{p}+F_{buoyancy}
  6. F p F_{p}
  7. F b u o y a n c y F_{buoyancy}
  8. γ = F m a x 2 τ p \gamma=\frac{F_{max}}{2\tau_{p}}
  9. μ \mu

Dual_Hahn_polynomials.html

  1. R n ( λ ( x ) ; γ , δ , N ) = F 2 3 ( - n , - x , x + γ + δ + 1 ; γ + 1 , - N ; 1 ) . R_{n}(\lambda(x);\gamma,\delta,N)={}_{3}F_{2}(-n,-x,x+\gamma+\delta+1;\gamma+1% ,-N;1).
  2. a ¯ \overline{a}
  3. b ¯ \overline{b}
  4. R n ( λ ( x ) ; γ , δ , N ) = Q x ( n ; γ , δ , N ) R_{n}(\lambda(x);\gamma,\delta,N)=Q_{x}(n;\gamma,\delta,N)

Dual_q-Krawtchouk_polynomials.html

  1. K n ( λ ( x ) ; c , N | q ) = 3 Φ 2 ( q - n , q - x , c q x - N ; q - N , 0 | q ; q ) K_{n}(\lambda(x);c,N|q)=_{3}\Phi_{2}(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)

Dual_total_correlation.html

  1. { X 1 , , X n } \{X_{1},\ldots,X_{n}\}
  2. D ( X 1 , , X n ) D(X_{1},\ldots,X_{n})
  3. D ( X 1 , , X n ) = H ( X 1 , , X n ) - i = 1 n H ( X i | X 1 , , X i - 1 , X i + 1 , , X n ) , D(X_{1},\ldots,X_{n})=H\left(X_{1},\ldots,X_{n}\right)-\sum_{i=1}^{n}H\left(X_% {i}|X_{1},\ldots,X_{i-1},X_{i+1},\ldots,X_{n}\right),
  4. H ( X 1 , , X n ) H(X_{1},\ldots,X_{n})
  5. { X 1 , , X n } \{X_{1},\ldots,X_{n}\}
  6. H ( X i | ) H(X_{i}|...)
  7. X i X_{i}
  8. H ( X 1 , , X n ) H(X_{1},\ldots,X_{n})
  9. N D ( X 1 , , X n ) = D ( X 1 , , X n ) H ( X 1 , , X n ) . ND(X_{1},\ldots,X_{n})=\frac{D(X_{1},\ldots,X_{n})}{H(X_{1},\ldots,X_{n})}.
  10. H ( X 1 , , X n ) H(X_{1},\ldots,X_{n})
  11. 0 D ( X 1 , , X n ) H ( X 1 , , X n ) . 0\leq D(X_{1},\ldots,X_{n})\leq H(X_{1},\ldots,X_{n}).
  12. C ( X 1 , , X n ) C(X_{1},\ldots,X_{n})
  13. C ( X 1 , , X n ) n - 1 D ( X 1 , , X n ) ( n - 1 ) C ( X 1 , , X n ) . \frac{C(X_{1},\ldots,X_{n})}{n-1}\leq D(X_{1},\ldots,X_{n})\leq(n-1)\;C(X_{1},% \ldots,X_{n}).
  14. D ( X 1 , , X n ) = μ ( i X ~ i \ ( j X ~ j \ k j X ~ k ) ) ) D(X_{1},\ldots,X_{n})=\mu\left(\bigcup_{i}\tilde{X}_{i}\backslash\left(\bigcup% _{j}\tilde{X}_{j}\backslash\bigcup_{k\neq j}\tilde{X}_{k})\right)\right)
  15. D ( X 1 , , X n ) = μ ( i j i ( X ~ i X ~ j ) ) D(X_{1},\ldots,X_{n})=\mu\left(\bigcup_{i}\bigcup_{j\neq i}\left(\tilde{X}_{i}% \cap\tilde{X}_{j}\right)\right)
  16. D ( X 1 , , X n ) \displaystyle{}\qquad D(X_{1},\ldots,X_{n})
  17. D ( X 1 , , X n ) \displaystyle{}\qquad D(X_{1},\ldots,X_{n})

Duality_gap.html

  1. d * d^{*}
  2. p * p^{*}
  3. p * - d * p^{*}-d^{*}
  4. ( X , X * ) \left(X,X^{*}\right)
  5. ( Y , Y * ) \left(Y,Y^{*}\right)
  6. f : X { + } f:X\to\mathbb{R}\cup\{+\infty\}
  7. inf x X f ( x ) . \inf_{x\in X}f(x).\,
  8. f f
  9. f = f + I constraints f=f+I_{\mathrm{constraints}}
  10. I I
  11. F : X × Y { + } F:X\times Y\to\mathbb{R}\cup\{+\infty\}
  12. F ( x , 0 ) = f ( x ) F(x,0)=f(x)
  13. inf x X [ F ( x , 0 ) ] - sup y * Y * [ - F * ( 0 , y * ) ] \inf_{x\in X}[F(x,0)]-\sup_{y^{*}\in Y^{*}}[-F^{*}(0,y^{*})]
  14. F * F^{*}

Duffin–Kemmer–Petiau_algebra.html

  1. β a β b β c + β c β b β a = β a η b c + β c η b a \beta^{a}\beta^{b}\beta^{c}+\beta^{c}\beta^{b}\beta^{a}=\beta^{a}\eta^{bc}+% \beta^{c}\eta^{ba}
  2. η a b \eta^{ab}
  3. β \beta
  4. η a b \eta^{ab}
  5. β 0 = ( 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) \beta^{0}=\begin{pmatrix}0&1&0&0&0\\ 1&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{pmatrix}
  6. β 1 = ( 0 0 - 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) \quad\beta^{1}=\begin{pmatrix}0&0&-1&0&0\\ 0&0&0&0&0\\ 1&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{pmatrix}
  7. β 2 = ( 0 0 0 - 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 ) \quad\beta^{2}=\begin{pmatrix}0&0&0&-1&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 1&0&0&0&0\\ 0&0&0&0&0\end{pmatrix}
  8. β 3 = ( 0 0 0 0 - 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ) \quad\beta^{3}=\begin{pmatrix}0&0&0&0&-1\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 1&0&0&0&0\end{pmatrix}
  9. ( i β a a - m c ) ψ = 0 (i\hbar\beta^{a}\partial_{a}-mc)\psi=0
  10. β a \beta^{a}
  11. m m
  12. ψ \psi
  13. \hbar
  14. c c
  15. m c mc
  16. γ \gamma
  17. β a γ + γ β a = β a \beta^{a}\gamma+\gamma\beta^{a}=\beta^{a}
  18. γ 2 = γ \gamma^{2}=\gamma

Duffin–Schaeffer_conjecture.html

  1. f : + f:\mathbb{N}\rightarrow\mathbb{R}^{+}
  2. α \alpha
  3. | α - p q | < f ( q ) q \left|\alpha-\frac{p}{q}\right|<\frac{f(q)}{q}
  4. p , q p,q
  5. q > 0 q>0
  6. q = 1 f ( q ) φ ( q ) q = . \sum_{q=1}^{\infty}f(q)\frac{\varphi(q)}{q}=\infty.
  7. φ ( q ) \varphi(q)
  8. c > 0 c>0
  9. n n
  10. f ( n ) = c / n f(n)=c/n
  11. f ( n ) = 0 f(n)=0
  12. f ( n ) = O ( n - 1 ) f(n)=O(n^{-1})
  13. ϵ > 0 \epsilon>0
  14. n = 1 ( f ( n ) n ) 1 + ϵ φ ( n ) = \sum_{n=1}^{\infty}\left(\frac{f(n)}{n}\right)^{1+\epsilon}\varphi(n)=\infty

Dunn_index.html

  1. Δ i = max x , y C i d ( x , y ) \Delta_{i}=\underset{x,y\in C_{i}}{\,\text{max}}d(x,y)
  2. Δ i = 1 | C i | | C i - 1 | x , y C i , x y d ( x , y ) \Delta_{i}=\dfrac{1}{|C_{i}||C_{i}-1|}\underset{x,y\in C_{i},x\neq y}{\sum}d(x% ,y)
  3. Δ i = x C i d ( x , μ ) | C i | , μ = x C i x | C i | \Delta_{i}=\dfrac{\underset{x\in C_{i}}{\sum}d(x,\mu)}{|C_{i}|},\mu=\dfrac{% \underset{x\in C_{i}}{\sum}x}{|C_{i}|}
  4. δ ( C i , C j ) \delta(C_{i},C_{j})
  5. 𝐷𝐼 m = min 1 i < j m δ ( C i , C j ) max 1 k m Δ k \mathit{DI}_{m}=\frac{\underset{1\leqslant i<j\leqslant m}{\,\text{min}}\left.% \delta(C_{i},C_{j})\right.}{\underset{1\leqslant k\leqslant m}{\,\text{max}}% \left.\Delta_{k}\right.}

Durfee_square.html

  1. n \lfloor\sqrt{n}\rfloor

Dwork_family.html

  1. x 1 n + x 2 n + + x n n = - n λ x 1 x 2 x n x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=-n\lambda x_{1}x_{2}\cdots x_{n}\,

Dyadic_cubes.html

  1. 2 - k 𝐙 n = { 2 - k ( v 1 , , v n ) : v j 𝐙 } 2^{-k}\mathbf{Z}^{n}=\left\{2^{-k}(v_{1},\dots,v_{n}):v_{j}\in\mathbf{Z}\right\}
  2. Δ k α = { Q + α : Q Δ k } . \Delta_{k}^{\alpha}=\{Q+\alpha:Q\in\Delta_{k}\}.
  3. M f ( x ) = sup r > 0 1 | B ( x , r ) | B ( x , r ) | f ( x ) | d x Mf(x)=\sup_{r>0}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(x)|dx
  4. | { x 𝐑 n : M f ( x ) > λ } | C n λ f 1 \left|\{x\in\mathbf{R}^{n}:Mf(x)>\lambda\}\right|\leq\frac{C_{n}}{\lambda}\|f% \|_{1}
  5. M Δ α f ( x ) = sup x Q Δ α 1 | Q | Q | f ( x ) | d x . M_{\Delta^{\alpha}}f(x)=\sup_{x\in Q\in\Delta^{\alpha}}\frac{1}{|Q|}\int_{Q}|f% (x)|dx.
  6. μ ( B ( x , 2 r ) ) C μ ( B ( x , r ) ) \mu(B(x,2r))\leq C\mu(B(x,r))
  7. μ ( X \ Q Δ k Q ) = 0. \mu\left(X\backslash\bigcup\nolimits_{Q\in\Delta_{k}}Q\right)=0.
  8. B ( z Q , c 1 δ k ) Q B ( z Q , c 2 δ k ) B(z_{Q},c_{1}\delta^{k})\subseteq Q\subseteq B(z_{Q},c_{2}\delta^{k})
  9. μ ( { x Q : d ( x , X \ Q ) t δ k } ) C 3 t η μ ( Q ) . \mu\left(\left\{x\in Q:d(x,X\backslash Q)\leq t\delta^{k}\right\}\right)\leq C% _{3}t^{\eta}\mu(Q).

Dykstra's_projection_algorithm.html

  1. r r
  2. x ¯ C D \bar{x}\in C\cap D
  3. x ¯ - r 2 x - r 2 , for all x C D , \|\bar{x}-r\|^{2}\leq\|x-r\|^{2},\,\text{for all }x\in C\cap D,
  4. C , D C,D
  5. r r
  6. C D C\cap D
  7. 𝒫 C D \mathcal{P}_{C\cap D}
  8. C C
  9. D D
  10. C , D C,D
  11. x 0 = r x_{0}=r
  12. x k + 1 = 𝒫 C ( 𝒫 D ( x k ) ) x_{k+1}=\mathcal{P}_{C}\left(\mathcal{P}_{D}(x_{k})\right)
  13. x 0 = r , p 0 = q 0 = 0 x_{0}=r,p_{0}=q_{0}=0
  14. y k = 𝒫 D ( x k + p k ) y_{k}=\mathcal{P}_{D}(x_{k}+p_{k})
  15. p k + 1 = x k + p k - y k p_{k+1}=x_{k}+p_{k}-y_{k}
  16. x k + 1 = 𝒫 C ( y k + q k ) x_{k+1}=\mathcal{P}_{C}(y_{k}+q_{k})
  17. q k + 1 = y k + q k - x k + 1 . q_{k+1}=y_{k}+q_{k}-x_{k+1}.
  18. ( x k ) (x_{k})

Dynamic_fluid_film_equations.html

  1. ρ \rho
  2. S S
  3. C C
  4. V α V^{\alpha}
  5. α \nabla_{\alpha}
  6. B α β B_{\alpha\beta}
  7. B β α B^{\alpha}_{\beta}
  8. B α α B^{\alpha}_{\alpha}
  9. e ( ρ ) e\left(\rho\right)
  10. E E
  11. E = S ρ e ( ρ ) d S . E=\int_{S}\rho e\left(\rho\right)\,dS.
  12. e ( ρ ) e\left(\rho\right)
  13. e ( ρ ) = σ ρ e\left(\rho\right)=\frac{\sigma}{\rho}
  14. σ \sigma
  15. E = σ A E=\sigma A\,
  16. δ ρ δ t + α ( ρ V α ) \displaystyle\frac{\delta\rho}{\delta t}+\nabla_{\alpha}\left(\rho V^{\alpha}\right)
  17. δ / δ t {\delta}/{\delta}t
  18. ρ 2 e ρ \rho^{2}e_{\rho}
  19. p p
  20. ( e ( ρ ) = σ / ρ ) \left(e\left(\rho\right)=\sigma/\rho\right)
  21. δ ρ δ t + α ( ρ V α ) \displaystyle\frac{\delta\rho}{\delta t}+\nabla_{\alpha}\left(\rho V^{\alpha}\right)
  22. B α β = 0 B_{\alpha\beta}=0
  23. C = 0 C=0
  24. ρ t + α ( ρ V α ) \displaystyle\frac{\partial\rho}{\partial t}+\nabla_{\alpha}\left(\rho V^{% \alpha}\right)
  25. ρ \rho
  26. C C
  27. δ ρ δ t \displaystyle{\frac{\delta\rho}{\delta t}}

Dynamic_insulation.html

  1. λ d 2 T ( x ) d x 2 - u ρ a c a d T ( x ) d x = 0 \lambda\frac{d^{2}T\left(x\right)}{dx^{2}}-u\rho_{a}c_{a}\frac{dT\left(x\right% )}{dx}=0
  2. T ( x ) - T o T L - T o = e ( A x ) - 1 e ( A L ) - 1 \frac{T\left(x\right)-T_{o}}{T_{L}-T_{o}}=\frac{e^{\left(}Ax\right)-1}{e^{% \left(}AL\right)-1}
  3. A = u ρ a c a λ A=\frac{u\rho_{a}c_{a}}{\lambda}
  4. d T d x = ( T L - T o ) A e ( A x ) e ( A L ) - 1 \frac{dT}{dx}=\frac{\left(T_{L}-T_{o}\right)Ae^{\left(}Ax\right)}{e^{\left(}AL% \right)-1}
  5. d T d x | x = 0 = ( T L - T o ) A e ( A L ) - 1 \frac{dT}{dx}\Bigg|_{x=0}=\frac{\left(T_{L}-T_{o}\right)A}{e^{\left(}AL\right)% -1}
  6. d T d x | x = L = d T d x | x = 0 e ( A L ) \frac{dT}{dx}\Bigg|_{x=L}=\frac{dT}{dx}\Bigg|_{x=0}e^{\left(}AL\right)
  7. U d y n = λ A e ( A L ) - 1 U_{dyn}=\frac{\lambda\,A}{e^{\left(}AL\right)-1}
  8. U d y n U s t a t i c = A L e ( A L ) - 1 \frac{U_{dyn}}{U_{static}}=\frac{A\,L}{e^{\left(}AL\right)-1}
  9. P w - P o = C p ( 1 2 ρ a v w 2 ) P_{w}-P_{o}=C_{p}\left(\tfrac{1}{2}\rho_{a}v_{w}^{2}\right)
  10. Φ = L V A Δ P \Phi=\frac{L\ V^{\prime}}{A\ \Delta\!P}

Dynamic_logic_(modal_logic).html

  1. p \Box p
  2. p p\,\!
  3. p \Diamond p
  4. p p\,\!
  5. a a\,\!
  6. [ a ] [a]\,\!
  7. a \langle a\rangle\,\!
  8. [ a ] p [a]p\,\!
  9. a a\,\!
  10. p p\,\!
  11. a a\,\!
  12. p p\,\!
  13. a p \langle a\rangle p\,\!
  14. a a\,\!
  15. p p\,\!
  16. a a\,\!
  17. p p\,\!
  18. [ a ] p ¬ a ¬ p [a]p\equiv\neg\langle a\rangle\neg p\,\!
  19. a p ¬ [ a ] ¬ p \langle a\rangle p\equiv\neg[a]\neg p\,\!
  20. \forall\,\!
  21. \exists\,\!
  22. a a\,\!
  23. b b\,\!
  24. a b a\cup b\,\!
  25. a + b a+b\,\!
  26. a | b a|b\,\!
  27. a a\,\!
  28. b b\,\!
  29. a ; b a;b\,\!
  30. a a\,\!
  31. b b\,\!
  32. a * a*\,\!
  33. a a\,\!
  34. 0 0\,\!
  35. 1 1\,\!
  36. 0 * 0*\,\!
  37. [ a ] p ¬ a ¬ p [a]p\equiv\neg\langle a\rangle\neg p\,\!
  38. p \vdash p\!
  39. p q \vdash p\to q\!
  40. q \vdash q\,\!
  41. p \vdash p\!
  42. [ a ] p \vdash[a]p\,\!
  43. [ 0 ] p [0]p\,\!
  44. [ 1 ] p p [1]p\equiv p\,\!
  45. [ a b ] p [ a ] p [ b ] p [a\cup b]p\equiv[a]p\land[b]p\,\!
  46. [ a ; b ] p [ a ] [ b ] p [a;b]p\equiv[a][b]p\,\!
  47. [ a * ] p p [ a ] [ a * ] p [a*]p\equiv p\land[a][a*]p\,\!
  48. p [ a * ] ( p [ a ] p ) [ a * ] p p\land[a*](p\to[a]p)\to[a*]p\,\!
  49. p p\,\!
  50. p p\,\!
  51. p p\,\!
  52. a a\,\!
  53. b b\,\!
  54. p p\,\!
  55. a a\,\!
  56. p p\,\!
  57. b b\,\!
  58. a a\,\!
  59. b b\,\!
  60. p p\,\!
  61. a a\,\!
  62. b b\,\!
  63. p p\,\!
  64. a * = 1 a ; a * a*=1\cup a;a*\,\!
  65. p p\,\!
  66. a a\,\!
  67. p p\,\!
  68. a a\,\!
  69. p p\,\!
  70. a a\,\!
  71. a a\,\!
  72. [ a ] p ¬ a ¬ p [a]p\equiv\neg\langle a\rangle\neg p\,\!
  73. ¬ 0 p \neg\langle 0\rangle p\,\!
  74. 1 p p \langle 1\rangle p\equiv p\,\!
  75. a b p a p b p \langle a\cup b\rangle p\equiv\langle a\rangle p\lor\langle b\rangle p\,\!
  76. a ; b p a b p \langle a;b\rangle p\equiv\langle a\rangle\langle b\rangle p\,\!
  77. a * p p a a * p \langle a*\rangle p\equiv p\lor\langle a\rangle\langle a*\rangle p\,\!
  78. a * p p a * ( ¬ p a p ) \langle a*\rangle p\to p\lor\langle a*\rangle(\neg p\land\langle a\rangle p)\,\!
  79. [ 1 ] [1]\,\!
  80. 1 \langle 1\rangle\,\!
  81. a a\,\!
  82. b b\,\!
  83. p p\,\!
  84. a a\,\!
  85. b b\,\!
  86. p p\,\!
  87. p p\,\!
  88. a a\,\!
  89. p p\,\!
  90. a a\,\!
  91. p p\,\!
  92. a a\,\!
  93. p p\,\!
  94. p q p\to q\,\!
  95. p p\,\!
  96. q q\,\!
  97. p q p\vdash q\,\!
  98. p p\,\!
  99. q q\,\!
  100. p [ a ] p p\vdash[a]p\,\!
  101. p p\,\!
  102. a a\,\!
  103. p p\,\!
  104. p [ a ] p p\to[a]p\,\!
  105. p p\,\!
  106. a a\,\!
  107. p [ a ] p p\to[a]p\,\!
  108. p p\,\!
  109. [ a ] p [a]p\,\!
  110. ( x = 1 ) [ x := x + 1 ] ( x = 1 ) (x=1)\to[x:=x+1](x=1)\,\!
  111. x x\,\!
  112. k k\,\!
  113. b b\,\!
  114. b [ k ] b b [ k * ] b b\to[k]b\vdash b\to[k*]b\,\!
  115. b [ k ] b b\to[k]b\,\!
  116. b [ k * ] b b\to[k*]b\,\!
  117. [ k ] b [k]b\,\!
  118. b [ k ] b b\to[k]b\,\!
  119. k * k*\,\!
  120. b [ k * ] b b\to[k*]b\,\!
  121. b [ k ] b b [ k * ] b b\to[k]b\vdash b\to[k*]b\,\!
  122. ( b [ k ] b ) ( b [ k * ] b ) (b\to[k]b)\to(b\to[k*]b)\,\!
  123. b [ k ] b b\to[k]b\,\!
  124. b [ k * ] b b\to[k*]b\,\!
  125. b b\,\!
  126. [ k * ] b [k*]b\,\!
  127. [ k ] b [k]b\,\!
  128. b [ k ] b b\to[k]b\,\!
  129. b [ k ] b b\to[k]b\,\!
  130. ( x 3 ) [ x := x + 1 ] ( x 4 ) (x\geq 3)\to[x:=x+1](x\geq 4)\,\!
  131. x x\,\!
  132. x x\,\!
  133. x x\,\!
  134. a a\,\!
  135. x := x + 1 x:=x+1\,\!
  136. [ a ] [a]\,\!
  137. a \langle a\rangle\,\!
  138. ( x 3 ) x := x + 1 ( x 4 ) (x\geq 3)\to\langle x:=x+1\rangle(x\geq 4)\,\!
  139. x x\,\!
  140. x := x + 1 x:=x+1\,\!
  141. x x\,\!
  142. x := e x:=e\,\!
  143. x x\,\!
  144. e e\,\!
  145. [ x := e ] Φ ( x ) Φ ( e ) [x:=e]\Phi(x)\equiv\Phi(e)\,\!
  146. Φ ( x ) \Phi(x)\,\!
  147. Φ \Phi\,\!
  148. x x\,\!
  149. Φ ( e ) \Phi(e)\,\!
  150. Φ \Phi\,\!
  151. x x\,\!
  152. Φ \Phi\,\!
  153. x \forall x\,\!
  154. e e\,\!
  155. [ x := e ] ( x = y 2 ) e = y 2 [x:=e](x=y^{2})\equiv e=y^{2}\,\!
  156. [ x := e ] ( b = c + x ) b = c + e [x:=e](b=c+x)\equiv b=c+e\,\!
  157. [ x := x + 1 ] ( x 4 ) ( x + 1 ) 4 [x:=x+1](x\geq 4)\equiv(x+1)\geq 4\,\!
  158. [ x := x + 1 ] x 4 [x:=x+1]x\geq 4\,\!
  159. ( x + 1 ) 4 (x+1)\geq 4\,\!
  160. x 3 x\geq 3\,\!
  161. * *\,\!
  162. ( x := x + 1 ) * x = 7 \langle(x:=x+1)*\rangle x=7\,\!
  163. x x\,\!
  164. x x\,\!
  165. x x\,\!
  166. x x\,\!
  167. x x\,\!
  168. x 7 x\leq 7\,\!
  169. p p\,\!
  170. Φ ( n ) \Phi(n)\,\!
  171. a a\,\!
  172. n := n + 1 n:=n+1\,\!
  173. n n\,\!
  174. 0 0\,\!
  175. ( Φ ( n ) [ ( n := n + 1 ) * ] ( Φ ( n ) [ n := n + 1 ] Φ ( n ) ) ) [ ( n := n + 1 ) * ] Φ ( n ) (\Phi(n)\land[(n:=n+1)*](\Phi(n)\to[n:=n+1]\Phi(n)))\to[(n:=n+1)*]\Phi(n)\,\!
  176. 0 0\,\!
  177. n n\,\!
  178. Φ ( n ) \Phi(n)\,\!
  179. p p\,\!
  180. p p\,\!
  181. [ n := n + 1 ] [n:=n+1]\,\!
  182. n n\,\!
  183. n n\,\!
  184. a a\,\!
  185. n n\,\!
  186. n n\,\!
  187. n n\,\!
  188. [ n := n + 1 ] [n:=n+1]\,\!
  189. n n\,\!
  190. 0 0\,\!
  191. n n\,\!
  192. [ ( n := n + 1 ) * ] Φ ( n ) [(n:=n+1)*]\Phi(n)\,\!
  193. [ ( n := n + 1 ) 0 ] Φ ( n ) [ ( n := n + 1 ) 1 ] Φ ( n ) [ ( n := n + 1 ) 2 ] Φ ( n ) [(n:=n+1)^{0}]\Phi(n)\land[(n:=n+1)^{1}]\Phi(n)\land[(n:=n+1)^{2}]\Phi(n)\land% \ldots\,\!
  194. i i\,\!
  195. [ ( n := n + 1 ) i ] Φ ( n ) [(n:=n+1)^{i}]\Phi(n)\,\!
  196. [ ( n := n + 1 ) i ] Φ ( n ) [(n:=n+1)^{i}]\Phi(n)\,\!
  197. [ n := n + 1 ] [ n := n + 1 ] Φ ( n ) [n:=n+1][n:=n+1]\ldots\Phi(n)\,\!
  198. i i\,\!
  199. i i\,\!
  200. Φ ( n + i ) \Phi(n+i)\,\!
  201. i Φ ( n + i ) \forall i\Phi(n+i)\,\!
  202. [ ( n := n + 1 ) * ] [(n:=n+1)*]\,\!
  203. ( Φ ( n ) i ( Φ ( n + i ) [ n := n + 1 ] Φ ( n + i ) ) ) i Φ ( n + i ) (\Phi(n)\land\forall i(\Phi(n+i)\to[n:=n+1]\Phi(n+i)))\to\forall i\Phi(n+i)\,\!
  204. ( Φ ( n ) i ( Φ ( n + i ) Φ ( n + i + 1 ) ) ) i Φ ( n + i ) (\Phi(n)\land\forall i(\Phi(n+i)\to\Phi(n+i+1)))\to\forall i\Phi(n+i)\,\!
  205. 0 0\,\!
  206. n n\,\!
  207. ( Φ ( 0 ) i ( Φ ( i ) Φ ( i + 1 ) ) ) i Φ ( i ) (\Phi(0)\land\forall i(\Phi(i)\to\Phi(i+1)))\to\forall i\Phi(i)\,\!
  208. i \forall i\,\!
  209. i i\,\!
  210. a * a*\,\!
  211. a i a^{i}\,\!
  212. i i\,\!
  213. n n\,\!
  214. n n\,\!
  215. Φ ( n ) \Phi(n)\,\!
  216. n n\,\!
  217. ( Φ ( n ) i ( Φ ( n + i ) Φ ( n + i + 1 ) ) ) i Φ ( n + i ) (\Phi(n)\land\forall i(\Phi(n+i)\to\Phi(n+i+1)))\to\forall i\Phi(n+i)\,\!
  218. n n\,\!
  219. n n\,\!
  220. n n\,\!
  221. n + i n+i\,\!
  222. n n\,\!
  223. p [ a * ] ( p [ a ] p ) [ a * ] p p\land[a*](p\to[a]p)\equiv[a*]p\,\!
  224. p p\,\!
  225. p ? p?\,\!
  226. p p\,\!
  227. p ? p?\,\!
  228. p p\,\!
  229. p ? p?\,\!
  230. [ p ? ] q p q [p?]q\equiv p\to q\,\!
  231. p ? \langle p?\rangle\,\!
  232. p ? q p q \langle p?\rangle q\equiv p\land q\,\!
  233. ( p ? ; a ) ( ¬ p ? ; b ) (p?;a)\cup(\neg p?;b)\,\!
  234. p p\,\!
  235. p ? ; a p?;a\,\!
  236. a a\,\!
  237. ¬ p ? ; b \neg p?;b\,\!
  238. a 0 a\cup 0\,\!
  239. a a\,\!
  240. p p\,\!
  241. p p\,\!
  242. ( p ? ; a ) * ; ¬ p ? (p?;a)*;\neg p?\,\!
  243. p ? ; a p?;a\,\!
  244. ¬ p ? \neg p?\,\!
  245. p p\,\!
  246. ¬ p ? \neg p?\,\!
  247. p p\,\!
  248. ¬ p ? \neg p?\,\!
  249. x := ? x:=?\,\!
  250. x x\,\!
  251. [ x := ? ] p [x:=?]p\,\!
  252. p p\,\!
  253. x x\,\!
  254. x := ? p \langle x:=?\rangle p\,\!
  255. x x\,\!
  256. p p\,\!
  257. [ x := ? ] [x:=?]\,\!
  258. x \forall x\,\!
  259. x := ? \langle x:=?\rangle\,\!
  260. x \exists x\,\!
  261. x := ? x:=?\,\!
  262. [ x := x + 1 ] ( x 4 ) [x:=x+1](x\geq 4)\,\!
  263. x x\,\!
  264. x + 1 x+1\,\!
  265. 4 4\,\!
  266. x := x + 1 x:=x+1\,\!
  267. x 4 x\geq 4\,\!
  268. [ x := x + 1 ] ( x 4 ) [x:=x+1](x\geq 4)\,\!
  269. x := x + 1 x:=x+1\,\!
  270. p { a } q p\{a\}q\,\!
  271. p [ a ] q p\to[a]q\,\!
  272. w p ( a , p ) wp(a,p)\,\!
  273. [ a ] p [a]p\,\!
  274. w l p ( a , p ) wlp(a,p)\,\!

Dynamic_method.html

  1. sin ( Θ 2 ) = 1 ϵ \sin\left(\frac{\Theta}{2}\right)=\frac{1}{\epsilon}
  2. Θ \Theta
  3. ϵ \epsilon

Dynamic_risk_measure.html

  1. ρ t : L ( T ) L t = L ( t ) \rho_{t}:L^{\infty}\left(\mathcal{F}_{T}\right)\rightarrow L^{\infty}_{t}=L^{% \infty}\left(\mathcal{F}_{t}\right)
  2. m t L t : ρ t ( X + m t ) = ρ t ( X ) - m t \forall m_{t}\in L^{\infty}_{t}:\;\rho_{t}(X+m_{t})=\rho_{t}(X)-m_{t}
  3. If X Y then ρ t ( X ) ρ t ( Y ) \mathrm{If}\;X\leq Y\;\mathrm{then}\;\rho_{t}(X)\geq\rho_{t}(Y)
  4. ρ t ( 0 ) = 0 \rho_{t}(0)=0
  5. λ L t , 0 λ 1 : ρ t ( λ X + ( 1 - λ ) Y ) λ ρ t ( X ) + ( 1 - λ ) ρ t ( Y ) \forall\lambda\in L^{\infty}_{t},0\leq\lambda\leq 1:\rho_{t}(\lambda X+(1-% \lambda)Y)\leq\lambda\rho_{t}(X)+(1-\lambda)\rho_{t}(Y)
  6. λ L t , λ 0 : ρ t ( λ X ) = λ ρ t ( X ) \forall\lambda\in L^{\infty}_{t},\lambda\geq 0:\rho_{t}(\lambda X)=\lambda\rho% _{t}(X)
  7. t t
  8. A t = { X L T : ρ ( X ) 0 a.s. } A_{t}=\{X\in L^{\infty}_{T}:\rho(X)\leq 0\,\text{ a.s.}\}
  9. t t
  10. ρ t = ess inf { Y L t : X + Y A t } \rho_{t}=\,\text{ess}\inf\{Y\in L^{\infty}_{t}:X+Y\in A_{t}\}
  11. ess inf \,\text{ess}\inf
  12. ρ t \rho_{t}
  13. X L T X\in L^{\infty}_{T}
  14. A t A\in\mathcal{F}_{t}
  15. ρ t ( 1 A X ) = 1 A ρ t ( X ) \rho_{t}(1_{A}X)=1_{A}\rho_{t}(X)
  16. 1 A 1_{A}
  17. A A
  18. ρ t ( X ) [ ω ] \rho_{t}(X)[\omega]
  19. ρ t + 1 ( X ) ρ t + 1 ( Y ) ρ t ( X ) ρ t ( Y ) X , Y L 0 ( T ) \rho_{t+1}(X)\leq\rho_{t+1}(Y)\Rightarrow\rho_{t}(X)\leq\rho_{t}(Y)\;\forall X% ,Y\in L^{0}(\mathcal{F}_{T})
  20. ρ t ( - X ) = ess sup Q E M M 𝔼 Q [ X | t ] \rho_{t}(-X)=\operatorname*{ess\sup}_{Q\in EMM}\mathbb{E}^{Q}[X|\mathcal{F}_{t}]

Dynamic_similarity_(Reynolds_and_Womersley_numbers).html

  1. N R = V L ρ μ N_{R}=\tfrac{VL\rho}{\mu}\,\!
  2. N R = Convective Inertial Force Shear Force N_{R}={\,\text{Convective Inertial Force}\over\,\text{Shear Force}}\,\!
  3. N W = L ω ρ μ = N S N_{W}=L\sqrt{\tfrac{\omega\rho}{\mu}}=\sqrt{N_{S}}\,\!
  4. N W = Transient Inertial Force Shear Force N_{W}={\,\text{Transient Inertial Force}\over\,\text{Shear Force}}\,\!
  5. ρ ( u t + u u x + v u y + w u z ) = ρ g - P x + μ ( 2 u x 2 + 2 v y 2 + 2 w z 2 ) {\rho}\left(\frac{{\partial}u}{{\partial}t}+u\frac{{\partial}u}{{\partial}x}+v% \frac{{\partial}u}{{\partial}y}+w\frac{{\partial}u}{{\partial}z}\right)={\rho}% g-\frac{{\partial}P}{{\partial}x}+{\mu}\left(\frac{{\partial^{2}}u}{{\partial}% x^{2}}+\frac{{\partial^{2}}v}{{\partial}y^{2}}+\frac{{\partial^{2}}w}{{% \partial}z^{2}}\right)\,\!
  6. transient inertial forces + convective inertial forces = gravitational force + Pressure force + viscous forces \,\text{transient inertial forces + convective inertial forces}=\,\text{% gravitational force + Pressure force + viscous forces}\,\!
  7. ρ \rho
  8. ( u t + u u x + v u y + w u z ) = - 1 ρ P x + ν ( 2 u x 2 + 2 v y 2 + 2 w z 2 ) \left(\frac{{\partial}u}{{\partial}t}+u\frac{{\partial}u}{{\partial}x}+v\frac{% {\partial}u}{{\partial}y}+w\frac{{\partial}u}{{\partial}z}\right)=-\frac{1}{% \rho}\frac{{\partial}P}{{\partial}x}+{\nu}\left(\frac{{\partial^{2}}u}{{% \partial}x^{2}}+\frac{{\partial^{2}}v}{{\partial}y^{2}}+\frac{{\partial^{2}}w}% {{\partial}z^{2}}\right)\,\!
  9. ν = μ / ρ \nu={\mu}/{\rho}
  10. V V
  11. ω \omega
  12. L L
  13. x = x / L x^{\prime}={x/L}\,\!
  14. u = u / V u^{\prime}={u/V}\,\!
  15. P = P / ρ V 2 P^{\prime}={P/{{\rho}V^{2}}}\,\!
  16. t = t ω t^{\prime}=t{\omega}\,\!
  17. V 2 L \tfrac{V^{2}}{L}\,\!
  18. N W 2 N R ( u t ) + ( u u x + v u y + w u z ) = - P x + 1 N R ( 2 u x 2 + 2 v y 2 + 2 w z 2 ) \frac{{N_{W}}^{2}}{N_{R}}\left(\frac{{\partial}u^{\prime}}{{\partial}t^{\prime% }}\right)+\left(u^{\prime}\frac{{\partial}u^{\prime}}{{\partial}x^{\prime}}+v^% {\prime}\frac{{\partial}u^{\prime}}{{\partial}y^{\prime}}+w^{\prime}\frac{{% \partial}u^{\prime}}{{\partial}z^{\prime}}\right)=-\frac{{\partial}P^{\prime}}% {{\partial}x^{\prime}}+\frac{1}{N_{R}}\left(\frac{{\partial^{2}}u^{\prime}}{{% \partial}x^{\prime 2}}+\frac{{\partial^{2}}v^{\prime}}{{\partial}y^{\prime 2}}% +\frac{{\partial^{2}}w^{\prime}}{{\partial}z^{\prime 2}}\right)\,\!
  19. u x + v y + w z = 0 \frac{{\partial}u^{\prime}}{{\partial}x^{\prime}}+\frac{{\partial}v^{\prime}}{% {\partial}y^{\prime}}+\frac{{\partial}w^{\prime}}{{\partial}z^{\prime}}=0\,\!
  20. ρ ω V {\rho}{\omega}V\,\!
  21. μ V δ 1 2 \tfrac{\mu V}{\delta_{1}^{2}}\,\!
  22. ρ ω V = μ V δ 1 2 {\rho}{\omega}V=\tfrac{\mu V}{\delta_{1}^{2}}\,\!
  23. δ 1 {\delta}_{1}\,\!
  24. δ 1 = μ ρ ω {\delta}_{1}=\sqrt{\tfrac{\mu}{\rho\omega}}\,\!
  25. L δ 1 = L ρ ω μ = L ω ν = N W \tfrac{L}{\delta_{1}}=L\sqrt{\tfrac{\rho\omega}{\mu}}=L\sqrt{\tfrac{\omega}{% \nu}}=N_{W}\,\!
  26. ρ V 2 L \tfrac{{\rho}{V^{2}}}{L}\,\!
  27. ρ V 2 L = μ V < m t p l > δ 2 2 \tfrac{{\rho}{V^{2}}}{L}=\tfrac{{\mu}{V}}{<}mtpl>{{\delta_{2}^{2}}}\,\!
  28. δ 2 = μ L ρ V {\delta}_{2}=\sqrt{\tfrac{\mu L}{\rho V}}\,\!
  29. L δ 2 = ρ V L μ = V L ν = N R \tfrac{L}{\delta_{2}}=\sqrt{\tfrac{\rho VL}{\mu}}=\sqrt{\tfrac{VL}{\nu}}=\sqrt% {N_{R}}\,\!

Dynamic_software_updating.html

  1. ( δ , P ) (\delta,P)
  2. δ \delta
  3. P P
  4. ( δ , P ) (\delta,P)
  5. ( δ , P ) (\delta^{\prime},P^{\prime})
  6. P P^{\prime}
  7. ( δ , P ) (\delta,P)
  8. ( S ( δ ) , P ) (S(\delta),P^{\prime})
  9. ( S ( δ ) , P ) (S(\delta),P^{\prime})
  10. ( δ , P ) (\delta,P^{\prime})
  11. ( δ i n i t , P ) (\delta_{init},P^{\prime})

Dynamic_topic_model.html

  1. t t
  2. t + 1 t+1
  3. α t \alpha_{t}
  4. β t , k \beta_{t,k}
  5. θ t , d \theta_{t,d}
  6. z t , d , n z_{t,d,n}
  7. w t , d , n w_{t,d,n}
  8. α t + 1 \alpha_{t+1}
  9. β t + 1 , k \beta_{t+1,k}
  10. α t \alpha_{t}
  11. β t , k \beta_{t,k}
  12. β t , k | β t - 1 , k N ( β t - 1 , k , σ 2 I ) \beta_{t,k}|\beta_{t-1,k}\sim N(\beta_{t-1,k},\sigma^{2}I)
  13. α t | α t - 1 N ( α t - 1 , δ 2 I ) \alpha_{t}|\alpha_{t-1}\sim N(\alpha_{t-1},\delta^{2}I)
  14. β t , k | β t - 1 , k N ( β t - 1 , k , σ 2 I ) k \beta_{t,k}|\beta_{t-1,k}\sim N(\beta_{t-1,k},\sigma^{2}I)\forall k
  15. α t | α t - 1 N ( α t - 1 , δ 2 I ) \alpha_{t}|\alpha_{t-1}\sim N(\alpha_{t-1},\delta^{2}I)
  16. η t , d N ( α t , a 2 I ) \eta_{t,d}\sim N(\alpha_{t},a^{2}I)
  17. Z t , d , n Mult ( π ( η t , d ) ) Z_{t,d,n}\sim\textrm{Mult}(\pi(\eta_{t,d}))
  18. W t , d , n Mult ( π ( β t , Z t , d , n ) ) W_{t,d,n}\sim\textrm{Mult}(\pi(\beta_{t,Z_{t,d,n}}))
  19. π ( x ) \pi(x)
  20. π ( x i ) = exp ( x i ) i exp ( x i ) \pi(x_{i})=\frac{\exp(x_{i})}{\sum_{i}\exp(x_{i})}
  21. W t , d , n W_{t,d,n}

Dynamical_mean_field_theory.html

  1. Δ ( τ ) \Delta(\tau)
  2. Δ ( τ ) \Delta(\tau)
  3. U U
  4. H Hubbard = t i j σ c i σ c j σ + U i n i n i H_{\,\text{Hubbard}}=t\sum_{\langle ij\rangle\sigma}c_{i\sigma}^{\dagger}c_{j% \sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow}
  5. c i , c i c_{i}^{\dagger},c_{i}
  6. i i
  7. n i = c i c i n_{i}=c_{i}^{\dagger}c_{i}
  8. d d
  9. t t
  10. a p a_{p}
  11. a p a_{p}^{\dagger}
  12. H AIM = p ϵ p a p a p H bath + p σ ( V p σ c σ a p σ + h . c . ) H mix + U n n - μ ( n + n ) H loc H_{\,\text{AIM}}=\underbrace{\sum_{p}\epsilon_{p}a_{p}^{\dagger}a_{p}}_{H_{\,% \text{bath}}}+\underbrace{\sum_{p\sigma}\left(V_{p}^{\sigma}c_{\sigma}^{% \dagger}a_{p\sigma}+h.c.\right)}_{H_{\,\text{mix}}}+\underbrace{Un_{\uparrow}n% _{\downarrow}-\mu\left(n_{\uparrow}+n_{\downarrow}\right)}_{H_{\,\text{loc}}}
  13. H bath H_{\,\text{bath}}
  14. ϵ p \epsilon_{p}
  15. H loc H_{\,\text{loc}}
  16. U U
  17. H mix H_{\,\text{mix}}
  18. V p σ V_{p}^{\sigma}
  19. G imp ( τ ) = - T c ( τ ) c ( 0 ) G_{\,\text{imp}}(\tau)=-\langle Tc(\tau)c^{\dagger}(0)\rangle
  20. U , μ U,\mu
  21. Δ σ ( i ω n ) = p | V p σ | 2 i ω n - ϵ p \Delta_{\sigma}(i\omega_{n})=\sum_{p}\frac{|V_{p}^{\sigma}|^{2}}{i\omega_{n}-% \epsilon_{p}}
  22. Δ σ ( τ ) \Delta_{\sigma}(\tau)
  23. ( G 0 ) - 1 ( i ω n ) = i ω n + μ - Δ ( i ω n ) (G_{0})^{-1}(i\omega_{n})=i\omega_{n}+\mu-\Delta(i\omega_{n})
  24. G ( i ω n ) G(i\omega_{n})
  25. Δ ( i ω n ) \Delta(i\omega_{n})
  26. U , μ U,\mu
  27. G i m p ( τ ) G_{imp}(\tau)
  28. G i i ( τ ) = - T c i ( τ ) c i ( 0 ) G_{ii}(\tau)=-\langle Tc_{i}(\tau)c_{i}^{\dagger}(0)\rangle
  29. G ( i ω n ) = G i i ( i ω n ) = k 1 i ω n + μ - ϵ ( k ) - Σ ( k , i ω n ) G(i\omega_{n})=G_{ii}(i\omega_{n})=\sum_{k}\frac{1}{i\omega_{n}+\mu-\epsilon(k% )-\Sigma(k,i\omega_{n})}
  30. Σ ( k , i ω n ) \Sigma(k,i\omega_{n})
  31. Σ ( k , i ω n ) Σ i m p ( i ω n ) \Sigma(k,i\omega_{n})\approx\Sigma_{imp}(i\omega_{n})
  32. U U
  33. μ \mu
  34. T T
  35. G 0 ( k , i ω n ) G_{0}(k,i\omega_{n})
  36. G 0 , l o c G_{0,loc}
  37. 𝒢 0 ( τ ) = G 0 , l o c \mathcal{G}^{0}(\tau)=G_{0,loc}
  38. G i m p 0 ( τ ) G_{imp}^{0}(\tau)
  39. Σ i m p ( i ω n ) = ( 𝒢 0 ) - 1 ( i ω n ) - ( G i m p 0 ) - 1 ( i ω n ) \Sigma_{imp}(i\omega_{n})=(\mathcal{G}^{0})^{-1}(i\omega_{n})-(G_{imp}^{0})^{-% 1}(i\omega_{n})
  40. Σ ( k , i ω n ) Σ i m p ( i ω n ) \Sigma(k,i\omega_{n})\approx\Sigma_{imp}(i\omega_{n})
  41. 𝒢 1 \mathcal{G}^{1}
  42. G i m p n = G i m p n + 1 G_{imp}^{n}=G_{imp}^{n+1}
  43. U U
  44. μ \mu
  45. T T
  46. U U
  47. U α β n α n β U_{\alpha\beta}n_{\alpha}n_{\beta}
  48. α \alpha
  49. β \beta
  50. d d

E9_honeycomb.html

  1. T ¯ 9 {\bar{T}}_{9}
  2. T ¯ 9 {\bar{T}}_{9}
  3. T ¯ 9 {\bar{T}}_{9}
  4. T ¯ 9 {\bar{T}}_{9}

Eady_Model.html

  1. ψ ¯ = - Λ z y {\overline{\psi}}=-\Lambda zy
  2. q t + U ¯ q x = 0 {\partial q^{\prime}\over\partial t}+{\overline{U}}{\partial q^{\prime}\over% \partial x}=0
  3. T t - v f 0 Λ = 0 {\partial T^{\prime}\over\partial t}-v^{\prime}f_{0}\Lambda=0
  4. T t + Λ H T x - v f 0 Λ = 0 {\partial T^{\prime}\over\partial t}+\Lambda H{\partial T^{\prime}\over% \partial x}-v^{\prime}f_{0}\Lambda=0
  5. U ¯ \overline{U}
  6. f 0 f_{0}
  7. κ L R = 1.61 \kappa L_{R}=1.61
  8. L R L_{R}
  9. κ L R = 2.4 \kappa L_{R}=2.4

Ear_decomposition.html

  1. G = ( V , E ) G=(V,E)
  2. | V | 2 |V|\geq 2
  3. G = ( V , E ) G=(V,E)
  4. | V | 2 |V|\geq 2
  5. P i P_{i}
  6. P j P_{j}
  7. i > j i>j
  8. P i P_{i}
  9. P j P_{j}
  10. P j P_{j}
  11. P i P_{i}
  12. P j P_{j}

Earliest_deadline_first_scheduling.html

  1. U = i = 1 n C i T i 1 , U=\sum_{i=1}^{n}\frac{C_{i}}{T_{i}}\leq 1,
  2. { C i } \left\{C_{i}\right\}
  3. n n
  4. { T i } \left\{T_{i}\right\}
  5. ( 1 8 + 2 5 + 4 10 ) = ( 37 40 ) = 0.925 = 92.5 % \left(\frac{1}{8}+\frac{2}{5}+\frac{4}{10}\right)=\left(\frac{37}{40}\right)=0% .925={\mathbf{92.5\%}}

Earth_Similarity_Index.html

  1. E S I = i = 1 n ( 1 - | x i - x i 0 x i + x i 0 | ) w i n ESI=\prod_{i=1}^{n}\left(1-\left|\frac{x_{i}-x_{i_{0}}}{x_{i}+x_{i_{0}}}\right% |\right)^{\frac{w_{i}}{n}}
  2. x i x_{i}
  3. x i 0 x_{i_{0}}
  4. w i w_{i}

Eckert_IV_projection.html

  1. x = 2 4 π + π 2 R ( λ - λ 0 ) ( 1 + cos θ ) 0.4222382 R ( λ - λ 0 ) ( 1 + cos θ ) x=\frac{2}{\sqrt{4\pi+\pi^{2}}}R\,(\lambda-\lambda_{0})(1+\cos\theta)\approx 0% .4222382\,R\,(\lambda-\lambda_{0})(1+\cos\theta)
  2. y = 2 π 4 + π R sin θ 1.3265004 R sin θ y=2\sqrt{\frac{\pi}{4+\pi}}R\sin\theta\approx 1.3265004\,R\sin\theta
  3. θ + sin θ cos θ + 2 sin θ = ( 2 + π 2 ) sin φ \theta+\sin\theta\cos\theta+2\sin\theta=\left(2+\frac{\pi}{2}\right)\sin\varphi
  4. θ = arcsin [ y 4 + π 2 π R ] arcsin [ y 1.3265004 R ] \theta=\arcsin\left[y\frac{\sqrt{4+\pi}}{2\sqrt{\pi}R}\right]\approx\arcsin% \left[\frac{y}{1.3265004\,R}\right]
  5. φ = arcsin [ θ + sin θ cos θ + 2 sin θ 2 + π 2 ] \varphi=\arcsin\left[\frac{\theta+\sin\theta\cos\theta+2\sin\theta}{2+\frac{% \pi}{2}}\right]
  6. λ = λ 0 + x 4 π + π 2 2 R ( 1 + cos θ ) λ 0 + x 0.4222382 R ( 1 + cos θ ) \lambda=\lambda_{0}+x\frac{\sqrt{4\pi+\pi^{2}}}{2R(1+\cos\theta)}\approx% \lambda_{0}+\frac{x}{0.4222382\,R\,(1+\cos\theta)}

Ecometrics.html

  1. T o t a l . C o s t = M o r t g a g e ( r e a l . c o s t - R e b a t e / C o m p a n y f u n d i n g ) + ( P L C u r r e n t . F u e l . p r i c e ( c o s t / g a l ) ) * a v g . w e e k l y . f u e l . c o n s ( g a l / w e e k ) + c u r r e n t . f u e l . p r i c e ( c o s t ) ) + i n s u r a n c e ( c o s t ) Total.Cost=Mortgage(real.cost-Rebate/Companyfunding)+(\int_{P}^{L}\!Current.% Fuel.price(cost/gal))*avg.weekly.fuel.cons(gal/week)\ +current.fuel.price(cost% ))+insurance(cost)
  2. Efficiency of Conversion for ingested food(ECI) * Food Calories = Net Production(Calories) \,\text{Efficiency of Conversion for ingested food(ECI)}*\,\text{Food Calories% }=\,\text{Net Production(Calories)}
  3. 100 M C a l o r i e s ( f e e d ) * 10 % ( E C I ) ( m e e t / f e e d ) = 10 M c a l o r i e s ( m e a t ) 100MCalories(feed)*10\%(ECI)(meet/feed)=10Mcalories(meat)
  4. 10 M C a l o r i e s = corresponds to = 1 p e s t i c i d e / t r a n s p o r t a t i o n a l . i m p a c t . v e g e t a r i a n . c i t y 10MCalories=\,\text{corresponds to}=1pesticide/transportational.impact.% vegetarian.city
  5. 100 M C a l o r i e s = 10 * p e s t i c i d e / t r a n s p o r a t i o n a l . i m p a c t . v e g e t a r i a n . c i t y 100MCalories=10*pesticide/transporational.impact.vegetarian.city
  6. G e n e r a l F o r m u l a . t r u c k l o a d s : N e c . i n t a k e ( C a l o r i e s ) / M a x . t r a n s . c a p a c i t y ( C a l o r i e s / T r u c k l o a d ) = T r u c k l o a d s . N e c . I n t a k e GeneralFormula.truckloads:Nec.intake(Calories)/Max.trans.capacity(Calories/% Truckload)=Truckloads.Nec.Intake
  7. G e n e r a l F o r m u l a . t r u c k l o a d . E m i s s i o n s : T r u c k l o a d s . N e c . I n t a k e ( s c a l a r ) * d i s t a n c e ( m i l e s ) * e m i s s i o n s ( e m i s s i o n s / m i l e s ) = E m i s s i o n s GeneralFormula.truckload.Emissions:Truckloads.Nec.Intake(scalar)*distance(% miles)*emissions(emissions/miles)=Emissions
  8. 1 t r u c k l o a d = 200 k c a l o r i e s ( 200 , 000 c a l o r i e s ) ; t h e r e a r e s e p a r a t e t r u c k s f o r m e a t a n d f e e d ; 1truckload=200kcalories(200,000calories);thereareseparatetrucksformeatandfeed;
  9. t h a t t h e m e a t f a c i l i t y i s 600 m i l e s f r o m t h e c i t y a n d t h e f a r m i s 300 m i l e s f r o m t h e c i t y ; thatthemeatfacilityis600milesfromthecityandthefarmis300milesfromthecity;
  10. e a c h t r u c k t y p e m u s t g o b a c k a n d f o r t h ( m i l e s a r e a l r e a d y X 2 ) ; eachtrucktypemustgobackandforth(milesarealreadyX2);
  11. 10 M c a l o r i e s ( c i t y . v e g . i n t a k e ) / 200 k c a l o r i e s ( c a l o r i e s / t r u c k l o a d ) = 50 t r u c k l o a d s 10Mcalories(city.veg.intake)/200kcalories(calories/truckload)=50truckloads
  12. 50 t r u c k l o a d s 300 m i l e s 0.654 l b s ( C O 2 / M i l e ) = 9 , 810 l b s CO2 emitted 50truckloads300miles0.654lbs(CO2/Mile)=9,810lbs\,\text{ CO2 emitted}
  13. 100 M c a l o r i e s ( m e a t . f a c . i n t a k e ) / 200 k c a l o r i e s ( c a l o r i e s / t r u c k l o a d ) = 500 t r u c k l o a d s 100Mcalories(meat.fac.intake)/200kcalories(calories/truckload)=500truckloads
  14. 500 t r u c k l o a d s * .10 ( E C I ) = 50 m e a t r u c k l o a d s 500truckloads*.10(ECI)=50meatruckloads
  15. ( ( 500 t r u c k l o a d s 300 m i l e s ) + ( 50 t r u c k l o a d s 600 m i l e s ) ) 0.654 l b s ( C O 2 / m i l e s ) = 180 , 000 l b s CO2 emitted ((500truckloads300miles)+(50truckloads600miles))0.654lbs(CO2/miles)=180,000lbs% \,\text{ CO2 emitted}
  16. U s a g e ( t i m e ) / L i f e t i m e ( t i m e ) Usage(time)/Lifetime(time)
  17. V a l u e = b e n e f i t . v a l u e / m o n e y Value=benefit.value/money
  18. D e c r e a s e . s p a c e ( u n i t s . s p a c e ) = u n i t . s p a c e ( u n i t . s p a c e ) * ( n u m . u s e r s . i n d i v i d u a l . c o n s ( s c a l a r ) - n u m . m a x . r e n t . u n i t s . s t o c k e d ( s c a l a r ) ) Decrease.space(units.space)=unit.space(unit.space)*(num.users.individual.cons(% scalar)-num.max.rent.units.stocked(scalar))
  19. R a t i o . o f . i n d v . s p a c e / r e n t s p a c e = u n i t . s p a c e ( n u m . u s e r s . i n d i v i d u a l . c o n s u m e d / n u m . m a x . r e n t . u n i t s . s t o c k e d ) Ratio.of.indv.space/rentspace=unit.space(num.users.individual.consumed/num.max% .rent.units.stocked)
  20. n u m . m a x . r e n t . u n i t s . s t o c k e d = m a x . u s e r s . p e r . d a y * m a x . r e n t . p e r i o d * n u m . s t o c k e d / r e n t . p e r i o d num.max.rent.units.stocked=max.users.per.day*max.rent.period*num.stocked/rent.period
  21. D e c r e a s e . w a s t e . u s i n g . c o m . l i b = w a s t e . u n i t * ( n u m . u s e r s . i n d i v i d u a l . c o n s ( s c a l a r ) - n u m . m a x . r e n t . u n i t s . s t o c k e d ( s c a l a r ) ) Decrease.waste.using.com.lib=waste.unit*(num.users.individual.cons(scalar)-num% .max.rent.units.stocked(scalar))
  22. R a t i o . o f . i n d v . w a s t e . t o . c o l l . r e n t . w a s t e = ( n u m u s e r s . i n d i v i d u a l . c o n s u m p / n u m . m a x . r e n t . u n i t s . s t o c k e d ) Ratio.of.indv.waste.to.coll.rent.waste=(numusers.individual.consump/num.max.% rent.units.stocked)
  23. e x t . i m p a c t . o f . p r o d u c t i o n . u n i t = e m i s s i o n s . e n e r g y . p r o d . u n i t + p o l l u t i o n . e n e r g y . p r o d . u n i t + e m i s s i o n s . m a t e r i a l . p r o d . u n i t + p o l l u t i o n . m a t e r i a l . p r o d . u n i t ext.impact.of.production.unit=emissions.energy.prod.unit+pollution.energy.prod% .unit+emissions.material.prod.unit+pollution.material.prod.unit
  24. l o c a l . i m p a c t . o f . p r o d = e x t . i m p a c t . o f . p r o d local.impact.of.prod=ext.impact.of.prod
  25. n u m . m a x . r e n t . u n i t s . s t o c k e d = m a x . u s e r s . p e r . d a y * m a x . r e n t . p e r i o d * n u m . s t o c k e d / r e n t . p e r i o d num.max.rent.units.stocked=max.users.per.day*max.rent.period*num.stocked/rent.period

Effective_domain.html

  1. f : X { ± } f:X\to\mathbb{R}\cup\{\pm\infty\}
  2. dom f = { x X : f ( x ) < + } . \operatorname{dom}f=\{x\in X:f(x)<+\infty\}.\,
  3. dom f = { x X : f ( x ) > - } . \operatorname{dom}f=\{x\in X:f(x)>-\infty\}.\,
  4. f : X { ± } f:X\to\mathbb{R}\cup\{\pm\infty\}
  5. dom f = { x X : y : ( x , y ) epi f } . \operatorname{dom}f=\{x\in X:\exists y\in\mathbb{R}:(x,y)\in\operatorname{epi}% f\}.\,
  6. f : X f:X\to\mathbb{R}
  7. f : X { ± } f:X\to\mathbb{R}\cup\{\pm\infty\}
  8. f ( x ) > - f(x)>-\infty
  9. x X x\in X

Effective_dose_(radiation).html

  1. E = T W T H T = T W T R W R D ¯ T , R E=\sum_{T}W_{T}\cdot H_{T}=\sum_{T}W_{T}\sum_{R}W_{R}\cdot\bar{D}_{T,R}
  2. E = T W T R W R T D R ( x , y , z ) ρ ( x , y , z ) d V T ρ ( x , y , z ) d V E=\sum_{T}W_{T}\sum_{R}W_{R}\cdot\frac{\int_{T}D_{R}(x,y,z)\rho(x,y,z)dV}{\int% _{T}\rho(x,y,z)dV}
  3. E E
  4. H T H_{T}
  5. W T W_{T}
  6. W R W_{R}
  7. D ¯ T , R \bar{D}_{T,R}
  8. D R ( x , y , z ) D_{R}(x,y,z)
  9. ρ ( x , y , z ) \rho(x,y,z)
  10. V V
  11. T T

Egyptian_algebra.html

  1. 3 / 2 × x + 4 = 10. 3/2\times x+4=10.
  2. pefsu = number loaves of bread (or jugs of beer) number of heqats of grain . \mbox{pefsu}~{}=\frac{\mbox{number loaves of bread (or jugs of beer)}~{}}{% \mbox{number of heqats of grain}~{}}.
  3. 1 / 2 k 1/2^{k}

Egyptian_geometry.html

  1. A = 1 2 b h A=\frac{1}{2}bh
  2. A = b h A=bh
  3. A = 1 4 ( 256 81 ) d 2 A=\frac{1}{4}(\frac{256}{81})d^{2}
  4. π = 3.14159... \pi=3.14159...
  5. A = 1 2 b h A=\frac{1}{2}bh
  6. 9 2 - 4 1 2 ( 3 ) ( 3 ) = 63 9^{2}-4\frac{1}{2}(3)(3)=63
  7. 64 = 8 2 64=8^{2}
  8. 4 ( 8 9 ) 2 = 3.16049... 4(\frac{8}{9})^{2}=3.16049...
  9. V = 256 81 r 2 h V=\frac{256}{81}r^{2}\ h
  10. V = 32 27 d 2 h = 128 27 r 2 h V=\frac{32}{27}d^{2}\ h=\frac{128}{27}r^{2}\ h
  11. V = w l h V=w\ l\ h
  12. V = 1 3 ( a 2 + a b + b 2 ) h V=\frac{1}{3}(a^{2}+ab+b^{2})h

Ehrenfest–Tolman_effect.html

  1. ξ \xi
  2. T T
  3. T ξ = constant T\,\xi=\mathrm{constant}

Eigenform.html

  1. f = a 0 + q + i = 2 a i q i f=a_{0}+q+\sum_{i=2}^{\infty}a_{i}q^{i}
  2. f , f = 1 \langle f,f\rangle=1\,

EigenMoments.html

  1. s n s\in\mathcal{R}^{n}
  2. C n × n C\in\mathcal{R}^{n\times n}
  3. C = E [ s s T ] C=E[ss^{T}]
  4. q = W T X T s , q=W^{T}X^{T}s,
  5. q = [ q 1 , , q n ] T k q=[q_{1},...,q_{n}]^{T}\in\mathcal{R}^{k}
  6. X = [ x 1 , , x n ] T n × m X=[x_{1},...,x_{n}]^{T}\in\mathcal{R}^{n\times m}
  7. W = [ w 1 , , w n ] T m × k W=[w_{1},...,w_{n}]^{T}\in\mathcal{R}^{m\times k}
  8. q q
  9. m = k = n m=k=n
  10. m n m\leq n
  11. k m k\leq m
  12. n n
  13. W W
  14. S N R t r a n s f o r m = w T X T C X w w T X T N X w , SNR_{transform}=\frac{w^{T}X^{T}CXw}{w^{T}X^{T}NXw},
  15. w 1 , , w k = a r g m a x w w T X T C X w w T X T N X w {w_{1},...,w_{k}}=argmax_{w}\frac{w^{T}X^{T}CXw}{w^{T}X^{T}NXw}
  16. w i T X T N X w j = δ i j , w_{i}^{T}X^{T}NXw_{j}=\delta_{ij},
  17. δ i j \delta_{ij}
  18. A = X T C X A=X^{T}CX
  19. B = X T N X B=X^{T}NX
  20. w 1 , , w k = arg max 𝑥 w T A w w T B w {w_{1},...,w_{k}}=\underset{x}{\operatorname{arg\,max}}\frac{w^{T}Aw}{w^{T}Bw}
  21. w i T B w j = δ i j w_{i}^{T}Bw_{j}=\delta_{ij}
  22. max w R ( w ) = max w w T A w w T B w \max_{w}R(w)=\max_{w}\frac{w^{T}Aw}{w^{T}Bw}
  23. A A
  24. B B
  25. B B
  26. w w
  27. w T B w = 1 w^{T}Bw=1
  28. w w
  29. max w w T A w \max_{w}{w^{T}Aw}
  30. w T B w = 1 {w^{T}Bw}=1
  31. max w ( w ) = max w ( w T A w - λ w T B w ) \max_{w}\mathcal{L}(w)=\max_{w}(w{T}Aw-\lambda w^{T}Bw)
  32. A w = λ B w Aw=\lambda Bw
  33. A w = λ B w Aw=\lambda Bw
  34. ( w , λ ) (w,\lambda)
  35. w w
  36. λ \lambda
  37. w w
  38. λ \lambda
  39. w i w_{i}
  40. i = 1 , , k i=1,...,k
  41. R ( w ) R(w)
  42. m m
  43. k k
  44. k k
  45. A = X T C X A=X^{T}CX
  46. B = X T N X B=X^{T}NX
  47. W W
  48. W = W 1 W 2 . W=W_{1}W_{2}.
  49. W 1 W_{1}
  50. P T B P = D B P^{T}BP=D_{B}
  51. D B D_{B}
  52. B B
  53. D B > 0 D_{B}>0
  54. P ^ \hat{P}
  55. k k
  56. P P
  57. P T ^ B P ^ = D B ^ \hat{P^{T}}B\hat{P}=\hat{D_{B}}
  58. D B ^ \hat{D_{B}}
  59. k × k k\times k
  60. D B D_{B}
  61. W 1 = P ^ D B ^ - 1 / 2 W_{1}=\hat{P}\hat{D_{B}}^{-1/2}
  62. W 1 T B W 1 = ( P ^ D B ^ - 1 / 2 ) T B ( P ^ D B ^ - 1 / 2 ) = I W_{1}^{T}BW_{1}=(\hat{P}\hat{D_{B}}^{-1/2})^{T}B(\hat{P}\hat{D_{B}}^{-1/2})=I
  63. W 1 W_{1}
  64. B B
  65. m m
  66. k k
  67. q = W 1 T X T s q^{\prime}=W_{1}^{T}X^{T}s
  68. W 1 T A W 1 W_{1}^{T}AW_{1}
  69. W 2 T W 1 T A W 1 W 2 = D A W_{2}^{T}W_{1}^{T}AW_{1}W_{2}=D_{A}
  70. W 2 T W 2 = I W_{2}^{T}W_{2}=I
  71. D A D_{A}
  72. W 1 T A W 1 W_{1}^{T}AW_{1}
  73. W = W 1 W 2 W=W_{1}W_{2}
  74. W W
  75. W T A W = D A W^{T}AW=D_{A}
  76. W T B W = I W^{T}BW=I
  77. s s
  78. q = W T X T s = W 2 T W 1 T X T s q=W^{T}X^{T}s=W_{2}^{T}W_{1}^{T}X^{T}s
  79. η = 1 - t r a c e ( W 1 T A W 1 ) t r a c e ( D B - 1 / 2 P T A P D B - 1 / 2 ) = 1 - t r a c e ( D B ^ - 1 / 2 P ^ T A P ^ D B ^ - 1 / 2 ) t r a c e ( D B - 1 / 2 P T A P D B - 1 / 2 ) \begin{array}[]{lll}\eta&=&1-\frac{trace(W_{1}^{T}AW_{1})}{trace(D_{B}^{-1/2}P% ^{T}APD_{B}^{-1/2})}\\ &=&1-\frac{trace(\hat{D_{B}}^{-1/2}\hat{P}^{T}A\hat{P}\hat{D_{B}}^{-1/2})}{% trace(D_{B}^{-1/2}P^{T}APD_{B}^{-1/2})}\end{array}
  80. X = [ 1 , x , x 2 , , x m - 1 ] X=[1,x,x^{2},...,x^{m-1}]
  81. X T X^{T}
  82. M M
  83. s = [ s ( x ) ] s=[s(x)]
  84. M = X T s M=X^{T}s
  85. r ( x 1 , x 2 ) = r ( 0 , 0 ) e - c ( x 1 - x 2 ) 2 r(x_{1},x_{2})=r(0,0)e^{-c(x_{1}-x_{2})^{2}}
  86. r ( x 1 , x 2 ) = r ( 0 , 0 ) e - c ( x 1 - x 2 ) 2 r(x_{1},x_{2})=r(0,0)e^{-c(x_{1}-x_{2})^{2}}
  87. r ( 0 , 0 ) = E [ t r ( s s T ) ] r(0,0)=E[tr(ss^{T})]
  88. r ( 0 , 0 ) r(0,0)
  89. A = X T C X = - 1 1 - 1 1 [ x 1 j x 2 i e - c ( x 1 - x 2 ) 2 ] i , j = 0 i , j = m - 1 d x 1 d x 2 A=X^{T}CX=\int_{-1}^{1}\int_{-1}^{1}[x_{1}^{j}x_{2}^{i}e^{-c(x_{1}-x_{2})^{2}}% ]_{i,j=0}^{i,j=m-1}dx_{1}dx_{2}
  90. σ n 2 δ ( x 1 , x 2 ) \sigma_{n}^{2}\delta(x_{1},x_{2})
  91. σ n 2 \sigma_{n}^{2}
  92. σ n 2 \sigma_{n}^{2}
  93. B = X T N X = - 1 1 - 1 1 [ x 1 j x 2 i δ ( x 1 , x 2 ) ] i , j = 0 i , j = m - 1 d x 1 d x 2 B=X^{T}NX=\int_{-1}^{1}\int_{-1}^{1}[x_{1}^{j}x_{2}^{i}\delta(x_{1},x_{2})]_{i% ,j=0}^{i,j=m-1}dx_{1}dx_{2}
  94. B = X T N X = - 1 1 [ x 1 j + i ] i , j = 0 i , j = m - 1 d x 1 = X T X B=X^{T}NX=\int_{-1}^{1}[x_{1}^{j+i}]_{i,j=0}^{i,j=m-1}dx_{1}=X^{T}X
  95. W W
  96. Φ = [ ϕ 1 , , ϕ k ] = X W \Phi=[\phi_{1},...,\phi_{k}]=XW
  97. Φ T C Φ = ( X W ) T C ( X W ) = D C \Phi^{T}C\Phi=(XW)^{T}C(XW)=D_{C}
  98. Φ T Φ = ( X W ) T ( X W ) = W T X T X = W T X T N X W = W T B W = I \begin{array}[]{lll}\Phi^{T}\Phi&=&(XW)^{T}(XW)\\ &=&W^{T}X^{T}X\\ &=&W^{T}X^{T}NXW\\ &=&W^{T}BW\\ &=&I\\ \end{array}
  99. c = 0.5 c=0.5
  100. m = 6 m=6
  101. k = 4 k=4
  102. W = ( 0.0 0 - 0.7745 - 0.8960 2.8669 - 4.4622 0.0 0.0 0.0 0.0 7.9272 2.4523 - 4.0225 20.6505 0.0 0.0 0.0 0.0 - 9.2789 - 0.1239 - 0.5092 - 18.4582 0.0 0.0 ) W=\left(\begin{array}[]{cccc}0.0&0&-0.7745&-0.8960\\ 2.8669&-4.4622&0.0&0.0\\ 0.0&0.0&7.9272&2.4523\\ -4.0225&20.6505&0.0&0.0\\ 0.0&0.0&-9.2789&-0.1239\\ -0.5092&-18.4582&0.0&0.0\end{array}\right)
  103. ϕ 1 = 2.8669 x - 4.0225 x 3 - 0.5092 x 5 ϕ 2 = - 4.4622 x + 20.6505 x 3 - 18.4582 x 5 ϕ 3 = - 0.7745 + 7.9272 x 2 - 9.2789 x 4 ϕ 4 = - 0.8960 + 2.4523 x 2 - 0.1239 x 4 \begin{array}[]{lll}\phi_{1}&=&2.8669x-4.0225x^{3}-0.5092x^{5}\\ \phi_{2}&=&-4.4622x+20.6505x^{3}-18.4582x^{5}\\ \phi_{3}&=&-0.7745+7.9272x^{2}-9.2789x^{4}\\ \phi_{4}&=&-0.8960+2.4523x^{2}-0.1239x^{4}\\ \end{array}
  104. ( p + q ) (p+q)
  105. m p q = - 1 1 - 1 1 x p y q f ( x , y ) d x d y m_{pq}=\int_{-1}^{1}\int_{-1}^{1}x^{p}y^{q}f(x,y)dxdy
  106. M = { m j , i } i , j = 0 i , j = m - 1 M=\{m_{j,i}\}_{i,j=0}^{i,j=m-1}
  107. Ω = W T M W \Omega=W^{T}MW
  108. Ω = { Ω j , i } i , j = 0 i , j = k - 1 \Omega=\{\Omega_{j,i}\}_{i,j=0}^{i,j=k-1}
  109. Ω j , i = Σ r = 0 m - 1 Σ s = 0 m - 1 w r , j w s , i m r , s \Omega_{j,i}=\Sigma_{r=0}^{m-1}\Sigma_{s=0}^{m-1}w_{r,j}w_{s,i}m_{r,s}
  110. M ^ \hat{M}
  111. M M
  112. m ^ p q = α p + q + 2 - 1 1 - 1 1 [ ( x - x c ) c o s ( θ ) + ( y - y c ) s i n ( θ ) ] p = × [ - ( x - x c ) s i n ( θ ) + ( y - y c ) c o s ( θ ) ] q = × f ( x , y ) d x d y , \begin{array}[]{lll}\hat{m}_{pq}&=&\alpha^{p}+q+2\int_{-1}^{1}\int_{-1}^{1}[(x% -x^{c})cos(\theta)+(y-y^{c})sin(\theta)]^{p}\\ &=&\times[-(x-x^{c})sin(\theta)+(y-y^{c})cos(\theta)]^{q}\\ &=&\times f(x,y)dxdy,\\ \end{array}
  113. ( x c , y c ) = ( m 10 / m 00 , m 01 / m 00 ) (x^{c},y^{c})=(m_{10}/m_{00},m_{01}/m_{00})
  114. f ( x , y ) f(x,y)
  115. α = [ m 00 S / m 00 ] 1 / 2 θ = 1 2 t a n - 1 2 m 11 m 20 - m 02 \begin{array}[]{lll}\alpha&=&[m_{00}^{S}/m_{00}]^{1/2}\\ \theta&=&\frac{1}{2}tan^{-1}\frac{2m_{11}}{m_{20}-m_{02}}\end{array}
  116. m 00 S m_{00}^{S}
  117. m 00 S m_{00}^{S}

Eigensystem_realization_algorithm.html

  1. H ( k - 1 ) = [ Y ( k ) Y ( k + 1 ) Y ( k + p ) Y ( k + 1 ) Y ( k + r ) Y ( k + p + r ) ] H(k-1)=\begin{bmatrix}Y(k)&Y(k+1)&\cdots&Y(k+p)\\ Y(k+1)&\ddots&&\vdots\\ \vdots&&&\\ Y(k+r)&\cdots&&Y(k+p+r)\end{bmatrix}
  2. Y ( k ) Y(k)
  3. m × n m\times n
  4. k k
  5. H ( 0 ) H(0)
  6. H ( 0 ) = P D Q T H(0)=PDQ^{T}
  7. D n , P n , and Q n D_{n},P_{n},\,\text{ and }Q_{n}
  8. A ^ = D n - 1 2 P n T H ( 1 ) Q n D n - 1 2 \hat{A}=D_{n}^{-\frac{1}{2}}P_{n}^{T}H(1)Q_{n}D_{n}^{-\frac{1}{2}}
  9. B ^ = D n 1 2 Q n T E m \hat{B}=D_{n}^{\frac{1}{2}}Q_{n}^{T}E_{m}
  10. C ^ = E n T P n D n 1 2 \hat{C}=E_{n}^{T}P_{n}D_{n}^{\frac{1}{2}}
  11. Λ = C ^ Φ ^ \Lambda=\hat{C}\hat{\Phi}
  12. Φ ^ \hat{\Phi}
  13. A ^ \hat{A}

Einstein's_unsuccessful_investigations.html

  1. m / ( 1 - v 2 / c 2 ) \scriptstyle m/(1-v^{2}/c^{2})
  2. m / 1 - v 2 / c 2 \scriptstyle m/\sqrt{1-v^{2}/c^{2}}

Eisenbud–Levine–Khimshiashvili_signature_formula.html

  1. X = f 1 ( x ) x 1 + + f n ( x ) x n . X=f_{1}({x})\,\frac{\partial}{\partial x_{1}}+\cdots+f_{n}({x})\,\frac{% \partial}{\partial x_{n}}.
  2. X = f 1 ( x ) x 1 + + f n ( x ) x n X=f_{1}({x})\,\frac{\partial}{\partial x_{1}}+\cdots+f_{n}({x})\,\frac{% \partial}{\partial x_{n}}
  3. B X := A n , 0 / I X . B_{X}:=A_{n,0}/I_{X}\,.
  4. F ( x ) := ( f 1 ( x ) , , f n ( x ) ) , F({x}):=(f_{1}({x}),\ldots,f_{n}({x})),
  5. β : B X × B X * B X \R ; β ( g , h ) = ( g * h ) , \beta:B_{X}\times B_{X}\stackrel{*}{\longrightarrow}B_{X}\stackrel{\ell}{% \longrightarrow}\R;\ \ \beta(g,h)=\ell(g*h),
  6. \scriptstyle\ell
  7. ( [ J F ] ) > 0. \ell\left(\left[J_{F}\right]\right)>0.
  8. X := ( x 3 - 3 x y 2 ) x + ( 3 x 2 y - y 3 ) y . X:=(x^{3}-3xy^{2})\,\frac{\partial}{\partial x}+(3x^{2}y-y^{3})\,\frac{% \partial}{\partial y}.
  9. B X = A 2 , 0 / ( x 3 - 3 x y 2 , 3 x 2 y - y 3 ) \R 1 , x , y , x 2 , x y , y 2 , x y 2 , y 3 , y 4 . B_{X}=A_{2,0}/(x^{3}-3xy^{2},3x^{2}y-y^{3})\cong\R\langle 1,x,y,x^{2},xy,y^{2}% ,xy^{2},y^{3},y^{4}\rangle.
  10. \scriptstyle\ell
  11. ( 1 ) = ( x ) = ( y ) = ( x 2 ) = ( x y ) = ( y 2 ) = ( x y 2 ) = ( y 3 ) = 0 , and ( y 4 ) = 3. \ell(1)=\ell(x)=\ell(y)=\ell(x^{2})=\ell(xy)=\ell(y^{2})=\ell(xy^{2})=\ell(y^{% 3})=0,\ \,\text{and}\ \ell(y^{4})=3.
  12. ( [ J F ] ) > 0 \scriptstyle\ell\left(\left[J_{F}\right]\right)\,>\,0
  13. [ 0 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 3 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 0 0 0 0 0 ] \left[\begin{array}[]{ccccccccc}0&0&0&0&0&0&0&3\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&3&0\\ 0&0&0&3&0&1&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&1&0&3&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&3&0&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ \end{array}\right]
  14. G : X X || X || . G:X\longmapsto\frac{X}{||X||}.
  15. G ( θ ) = ( cos ( 3 θ ) , sin ( 3 θ ) ) , G(\theta)=(\cos(3\theta),\sin(3\theta)),\,

Eisenstein_integral.html

  1. E ( P : ψ : ν : x ) = K ψ ( x k ) τ ( k - 1 ) exp ( ( i ν - ρ P ) H P ( x k ) ) d k \displaystyle E(P:\psi:\nu:x)=\int_{K}\psi(xk)\tau(k^{-1})\exp((i\nu-\rho_{P})% H_{P}(xk))\,dk

Ekman_velocity.html

  1. f s y m b o l z ^ × s y m b o l u = - ϕ + s y m b o l τ z , f\hat{symbol{z}}\times symbol{u}=-\nabla\phi+\frac{\partial symbol{\tau}}{% \partial z},
  2. ϕ = p / ρ 0 , \phi=p/\rho_{0},\,
  3. s y m b o l τ symbol{\tau}
  4. ρ 0 \rho_{0}\,
  5. s y m b o l z ^ \hat{symbol{z}}
  6. s y m b o l u symbol{u}
  7. s y m b o l u g symbol{u}_{g}
  8. s y m b o l u e = s y m b o l u - s y m b o l u g . symbol{u}_{e}=symbol{u}-symbol{u}_{g}.
  9. s y m b o l u g symbol{u}_{g}
  10. s y m b o l u g = 1 f s y m b o l z ^ × ϕ , symbol{u}_{g}=\frac{1}{f}\hat{symbol{z}}\times\nabla\phi,
  11. f s y m b o l z ^ \timessymbol u e = s y m b o l τ z f\hat{symbol{z}}\timessymbol{u}_{e}=\frac{\partial symbol{\tau}}{\partial z}
  12. s y m b o l u e = - s y m b o l z ^ × ( 1 f s y m b o l τ z ) . symbol{u}_{e}=-\hat{symbol{z}}\times\Big(\frac{1}{f}\frac{\partial symbol{\tau% }}{\partial z}\Big).
  13. z = z b z=z_{b}\,
  14. z = z t z=z_{t}\,
  15. s y m b o l U e = z b z t s y m b o l u e d z = - s y m b o l z ^ × ( s y m b o l τ t - s y m b o l τ b f ) . symbol{U}_{e}=\int_{z_{b}}^{z_{t}}symbol{u}_{e}\;dz=-\hat{symbol{z}}\times\Big% (\frac{symbol{\tau}_{t}-symbol{\tau}_{b}}{f}\Big).

Elastic_map.html

  1. S S
  2. W j W_{j}
  3. s S s\in S
  4. W j W_{j}
  5. S S
  6. K j = { s | W j is a host of s } K_{j}=\{s\ |\ W_{j}\mbox{ is a host of }~{}s\}
  7. D = 1 2 j = 1 k s K j s - W j 2 D=\frac{1}{2}\sum_{j=1}^{k}\sum_{s\in K_{j}}\|s-W_{j}\|^{2}
  8. { s i } \{s_{i}\}
  9. ( W i , W j ) (W_{i},W_{j})
  10. E E
  11. ( W i , W j , W k ) (W_{i},W_{j},W_{k})
  12. G G
  13. U E = 1 2 λ ( W i , W j ) E W i - W j 2 U_{E}=\frac{1}{2}\lambda\sum_{(W_{i},W_{j})\in E}\|W_{i}-W_{j}\|^{2}
  14. U G = 1 2 μ ( W i , W j , W l ) G W i - 2 W j + W l 2 U_{G}=\frac{1}{2}\mu\sum_{(W_{i},W_{j},W_{l})\in G}\|W_{i}-2W_{j}+W_{l}\|^{2}
  15. λ \lambda
  16. μ \mu
  17. U = D + U E + U G . U=D+U_{E}+U_{G}.
  18. { W j } \{W_{j}\}
  19. U U
  20. S S
  21. K j K_{j}
  22. U U
  23. { W j } \{W_{j}\}
  24. { K j } \{K_{j}\}
  25. { K j } \{K_{j}\}
  26. U U
  27. { W j } \{W_{j}\}
  28. U U
  29. λ \lambda
  30. μ \mu
  31. λ \lambda
  32. μ \mu

Elastic_net_regularization.html

  1. β 1 = j = 1 p | β j | . \|\beta\|_{1}=\textstyle\sum_{j=1}^{p}|\beta_{j}|.
  2. β 2 \|\beta\|^{2}
  3. β ^ = argmin 𝛽 ( y - X β 2 + λ 2 β 2 + λ 1 β 1 ) . \hat{\beta}=\underset{\beta}{\operatorname{argmin}}(\|y-X\beta\|^{2}+\lambda_{% 2}\|\beta\|^{2}+\lambda_{1}\|\beta\|_{1}).
  4. λ 1 = λ , λ 2 = 0 \lambda_{1}=\lambda,\lambda_{2}=0
  5. λ 1 = 0 , λ 2 = λ \lambda_{1}=0,\lambda_{2}=\lambda
  6. λ 2 \lambda_{2}
  7. ( 1 + λ 2 ) (1+\lambda_{2})

Electoral_College_(India).html

  1. Value of an MLA vote = Total population of the state Total number of elected members × 1000 \textrm{Value\;of\;an\;MLA\;vote}=\cfrac{\mbox{Total population of the state}~% {}}{\mbox{Total number of elected members}~{}\times{1000}}
  2. Value of an MP vote = The sum of vote value of elected members of all the Legislative Assemblies The sum of elected members of both the houses of Parliament \textrm{Value\;of\;an\;MP\;vote}=\cfrac{\mbox{The sum of vote value of elected% members of all the Legislative Assemblies}~{}}{\mbox{The sum of elected % members of both the houses of Parliament}~{}}

Electric_dipole_moment.html

  1. 𝐩 = q 𝐝 \mathbf{p}=q\mathbf{d}
  2. s y m b o l τ = p × E , symbol{\tau}={p}\times{E}\ ,
  3. 𝐩 ( 𝐫 ) = V ρ ( 𝐫 0 ) ( 𝐫 0 - 𝐫 ) d 3 𝐫 0 , \mathbf{p}(\mathbf{r})=\int\limits_{V}\rho(\mathbf{r}_{0})\,(\mathbf{r}_{0}-% \mathbf{r})\ d^{3}\mathbf{r}_{0},
  4. ρ ( 𝐫 ) = i = 1 N q i δ ( 𝐫 - 𝐫 i ) , \rho(\mathbf{r})=\sum_{i=1}^{N}\,q_{i}\,\delta(\mathbf{r}-\mathbf{r}_{i}),
  5. 𝐩 ( 𝐫 ) = i = 1 N q i V δ ( 𝐫 0 - 𝐫 i ) ( 𝐫 0 - 𝐫 ) d 3 𝐫 0 = i = 1 N q i ( 𝐫 i - 𝐫 ) . \mathbf{p}(\mathbf{r})=\sum_{i=1}^{N}\,q_{i}\int\limits_{V}\delta(\mathbf{r}_{% 0}-\mathbf{r}_{i})\,(\mathbf{r}_{0}-\mathbf{r})\ d^{3}\mathbf{r}_{0}=\sum_{i=1% }^{N}\,q_{i}(\mathbf{r}_{i}-\mathbf{r}).
  6. 𝐩 ( 𝐫 ) \mathbf{p}(\mathbf{r})
  7. = q 1 ( 𝐫 1 - 𝐫 ) + q 2 ( 𝐫 2 - 𝐫 ) = q ( 𝐫 + - 𝐫 ) - q ( 𝐫 - - 𝐫 ) = q ( 𝐫 + - 𝐫 - ) = q 𝐝 , =q_{1}(\mathbf{r}_{1}-\mathbf{r})+q_{2}(\mathbf{r}_{2}-\mathbf{r})=q(\mathbf{r% }_{+}-\mathbf{r})-q(\mathbf{r}_{-}-\mathbf{r})=q(\mathbf{r}_{+}-\mathbf{r}_{-}% )=q\mathbf{d},
  8. 𝐩 ( 𝐫 ) \displaystyle\mathbf{p}(\mathbf{r})
  9. ϕ ( 𝐫 ) = q 4 π ε 0 | 𝐫 - 𝐫 + | - q 4 π ε 0 | 𝐫 - 𝐫 - | , \phi(\mathbf{r})=\frac{q}{4\pi\varepsilon_{0}|\mathbf{r}-\mathbf{r}_{+}|}-% \frac{q}{4\pi\varepsilon_{0}|\mathbf{r}-\mathbf{r}_{-}|}\ ,
  10. 𝐝 = 𝐫 + - 𝐫 - , \mathbf{d}=\mathbf{r}_{+}-\mathbf{r}_{-}\ ,
  11. 𝐑 = 𝐫 - 𝐫 + + 𝐫 - 2 , 𝐑 ^ = 𝐑 R , {\mathbf{R}}=\mathbf{r}-\frac{\mathbf{r}_{+}+\mathbf{r}_{-}}{2},\quad\hat{% \mathbf{R}}=\frac{\mathbf{R}}{R}\ ,
  12. ϕ ( 𝐑 ) = 1 4 π ε 0 q 𝐝 𝐑 ^ R 2 + O ( d 2 R 2 ) 1 4 π ε 0 𝐩 𝐑 ^ R 2 , \phi(\mathbf{R})=\frac{1}{4\pi\varepsilon_{0}}\frac{q\mathbf{d}\cdot\hat{% \mathbf{R}}}{R^{2}}+O\left(\frac{d^{2}}{R^{2}}\right)\approx\frac{1}{4\pi% \varepsilon_{0}}\frac{\mathbf{p}\cdot\hat{\mathbf{R}}}{R^{2}}\ ,
  13. Q i j = d 3 𝐫 0 ( 3 x i x j - r 0 2 δ i j ) ρ ( 𝐫 0 ) , Q_{ij}=\int d^{3}\mathbf{r}_{0}\left(3x_{i}x_{j}-r_{0}^{2}\delta_{ij}\right)% \rho(\mathbf{r}_{0})\ ,
  14. 𝐩 = q 𝐝 . \mathbf{p}=q\mathbf{d}\ .
  15. ϕ ( 𝐑 ) = - 𝐩 1 4 π ε 0 R , \phi(\mathbf{R})=-\mathbf{p}\cdot\mathbf{\nabla}\frac{1}{4\pi\varepsilon_{0}R}\ ,
  16. 𝐄 ( 𝐑 ) = 3 ( 𝐩 𝐑 ^ ) 𝐑 ^ - 𝐩 4 π ε 0 R 3 . \mathbf{E}\left(\mathbf{R}\right)=\frac{3(\mathbf{p}\cdot\hat{\mathbf{R}})\hat% {\mathbf{R}}-\mathbf{p}}{4\pi\varepsilon_{0}R^{3}}\ .
  17. p = i = 1 N q i d i , p=\sum_{i=1}^{N}\ q_{i}{d_{i}}\ ,
  18. D = ε 0 E + P , {D}=\varepsilon_{0}{E}+{P}\ ,
  19. D = ρ f = ε 0 E + P , \nabla\cdot{D}=\rho_{f}=\varepsilon_{0}\nabla\cdot{E}+\nabla\cdot{P}\ ,
  20. P = - ρ b , \nabla\cdot{P}=-\rho_{b}\ ,
  21. × E = s y m b o l 0 , \nabla\times{E}=symbol{0}\ ,
  22. × ( D - P ) = s y m b o l 0 , \nabla\times\left({D}-{P}\right)=symbol{0}\ ,
  23. D - P = - φ , {D-P=-\nabla}\varphi\ ,
  24. ( D - P ) = ε 0 E = ρ f + ρ b = - 2 φ . {\nabla\cdot(D-P)}=\varepsilon_{0}{\nabla\cdot E}=\rho_{f}+\rho_{b}=-\nabla^{2% }\varphi\ .
  25. φ = φ f + φ b . \varphi=\varphi_{f}+\varphi_{b}\ .
  26. 2 φ f = 0 . \nabla^{2}\varphi_{f}=0\ .
  27. P ( r ) = p ( r ) {P}({r})={p}({r})\,
  28. p ( r ) = ρ b , \nabla\cdot{p}({r})=\rho_{b},
  29. ϕ ( r ) = 1 4 π ε 0 ρ ( r 0 ) | r - r 0 | d 3 r 0 + 1 4 π ε 0 p ( r 0 ) ( r - r 0 ) | r - r 0 | 3 d 3 r 0 , \phi({r})=\frac{1}{4\pi\varepsilon_{0}}\int\frac{\rho({r}_{0})}{|{r}-{r}_{0}|}% d^{3}{r}_{0}\ +\frac{1}{4\pi\varepsilon_{0}}\int\frac{{p}({r}_{0}){\cdot(r-r_{% 0})}}{|{r}-{r}_{0}|^{3}}d^{3}{r}_{0},
  30. r 0 1 | r - r 0 | = r - r 0 | r - r 0 | 3 \nabla_{{r}_{0}}\frac{1}{|r-{r}_{0}|}=\frac{r-{r}_{0}}{|r-{r}_{0}|^{3}}
  31. 1 4 π ε 0 p ( r 0 ) ( r - r 0 ) | r - r 0 | 3 d 3 r 0 = 1 4 π ε 0 p ( r 0 ) r 0 1 | r - r 0 | d 3 r 0 , \frac{1}{4\pi\varepsilon_{0}}\int\frac{{p}({r}_{0}){\cdot(r-r_{0})}}{|{r}-{r}_% {0}|^{3}}d^{3}{r}_{0}=\frac{1}{4\pi\varepsilon_{0}}\int{p}({r}_{0}){\cdot% \nabla}_{{r}_{0}}\frac{1}{|r-{r}_{0}|}d^{3}{r}_{0},
  32. = 1 4 π ε 0 r 0 ( p ( r 0 ) 1 | r - r 0 | ) d 3 r 0 - 1 4 π ε 0 r 0 p ( r 0 ) | r - r 0 | d 3 r 0 , =\frac{1}{4\pi\varepsilon_{0}}\int{\nabla_{{r_{0}}}\cdot}\left({p}({r}_{0})% \frac{1}{|r-{r}_{0}|}\right)d^{3}{r}_{0}-\frac{1}{4\pi\varepsilon_{0}}\int% \frac{{\nabla_{{r_{0}}}\cdot}{p}({r}_{0})}{|r-{r}_{0}|}d^{3}{r}_{0},
  33. ϕ ( r ) = 1 4 π ε 0 ρ ( r 0 ) - r 0 p ( r 0 ) | r - r 0 | d 3 r 0 , \phi({r})=\frac{1}{4\pi\varepsilon_{0}}\int\frac{\rho({r}_{0})-{\nabla_{{r_{0}% }}\cdot}{p}({r}_{0})}{|{r}-{r}_{0}|}d^{3}{r}_{0}\ ,
  34. ρ total ( r 0 ) = ρ ( r 0 ) - r 0 p ( r 0 ) , \rho_{\rm total}({r}_{0})=\rho({r}_{0})-{\nabla_{{r_{0}}}\cdot}{p}({r}_{0})\ ,
  35. - r 0 p ( r 0 ) = ρ b . -{\nabla_{{r_{0}}}\cdot}{p}({r}_{0})=\rho_{b}\ .
  36. D = ρ f , \nabla\cdot{D}=\rho_{f}\ ,
  37. P ( r ) = p dip - p quad + , {P}({r})={p}_{\rm dip}-\nabla\cdot{p}_{\rm quad}+\ldots\ ,
  38. ϕ ( r ) = 1 4 π ε 0 r 0 ( p ( r 0 ) 1 | r - r 0 | ) d 3 r 0 - 1 4 π ε 0 r 0 p ( r 0 ) | r - r 0 | d 3 r 0 . \phi({r})=\frac{1}{4\pi\varepsilon_{0}}\int{\nabla_{{r_{0}}}\cdot}\left({p}({r% }_{0})\frac{1}{|r-{r}_{0}|}\right)d^{3}{r}_{0}-\frac{1}{4\pi\varepsilon_{0}}% \int\frac{{\nabla_{{r_{0}}}\cdot}{p}({r}_{0})}{|r-{r}_{0}|}d^{3}{r}_{0}\ .
  39. 1 4 π ε 0 r 0 ( p ( r 0 ) 1 | r - r 0 | ) d 3 r 0 \frac{1}{4\pi\varepsilon_{0}}\int{\nabla_{{r_{0}}}\cdot}\left({p}({r}_{0})% \frac{1}{|r-{r}_{0}|}\right)d^{3}{r}_{0}
  40. = 1 4 π ε 0 p ( r 0 ) d A 0 | r - r 0 | , =\frac{1}{4\pi\varepsilon_{0}}\int\frac{{p}({r}_{0}){\cdot}d{A_{0}}}{|r-{r}_{0% }|}\ ,
  41. ϕ ( r ) = 1 4 π ε 0 1 | r - r 0 | p d A 0 , \phi({r})=\frac{1}{4\pi\varepsilon_{0}}\int\frac{1}{|r-{r}_{0}|}\ {p}\cdot d{A% _{0}}\ ,
  42. σ = p d A \sigma={p}\cdot d{A}\,
  43. 1 | r - r 0 | \frac{1}{|{r-r_{0}}|}
  44. , m 4 π 2 + 1 \sum_{\ell,\ m}\frac{4\pi}{2\ell+1}
  45. 1 r ( r 0 r ) \frac{1}{r}\left({\frac{r_{0}}{r}}\right)^{\ell}
  46. Y * m ( θ 0 , ϕ 0 ) Y m ( θ , ϕ ) \ {Y^{*}}_{\ell}^{m}(\theta_{0},\ \phi_{0})Y_{\ell}^{m}(\theta,\ \phi)
  47. E ( r ) = - 1 4 π ε 0 r 1 | r - r 0 | p d A 0 , E({r})=-\frac{1}{4\pi\varepsilon_{0}}\nabla_{{r}}\int\frac{1}{|r-{r}_{0}|}\ {p% }\cdot d{A_{0}}\ ,
  48. E = - p 3 ε 0 . E=-\frac{p}{3\varepsilon_{0}}\ .
  49. p ( r ) = ε 0 χ ( r ) E ( r ) , {p(r)}=\varepsilon_{0}\chi(r){E(r)}\ ,
  50. p ( r ) = ( χ ( r ) ε 0 E ( r ) ) = - ρ b . {\nabla\cdot p(r)}={\nabla\cdot}\left(\chi{(r)}\varepsilon_{0}{E(r)}\right)=-% \rho_{b}\ .
  51. ε 0 n ^ [ χ ( r + ) E ( r + ) - χ ( r - ) E ( r - ) ] = 1 A n d Ω n ρ b = 0 , \varepsilon_{0}\hat{n}\cdot\left[\chi{(r_{+})}{E(r_{+})}-\chi{(r_{-})}{E(r_{-}% )}\right]=\frac{1}{A_{n}}\int d\Omega_{n}\ \rho_{b}=0\ ,
  52. n ^ \hat{n}
  53. p ( r ) = χ ( r ) E ( r ) {p}({r})=\chi({r}){E}({r})\,
  54. ϕ = - E z = - E r cos θ . \phi_{\infty}=-E_{\infty}z=-E_{\infty}r\cos\theta\ .
  55. D = κ ϵ 0 E , {D}=\kappa\epsilon_{0}{E}\ ,
  56. ϕ < = A r cos θ , \phi_{<}=Ar\cos\theta\ ,
  57. ϕ > = ( B r + C r 2 ) cos θ . \phi_{>}=\left(Br+\frac{C}{r^{2}}\right)\cos\theta\ .
  58. A = - 3 κ + 2 E ; C = κ - 1 κ + 2 E R 3 , A=-\frac{3}{\kappa+2}E_{\infty}\ ;\ C=\frac{\kappa-1}{\kappa+2}E_{\infty}R^{3}\ ,
  59. ϕ > = ( - r + κ - 1 κ + 2 < m t p l > R 3 r 2 ) E cos θ , \phi_{>}=\left({-r}+\frac{\kappa-1}{\kappa+2}\frac{<}{m}tpl>{{R^{3}}}{r^{2}}% \right)E_{\infty}\cos\theta\ ,
  60. p = 4 π ε 0 ( κ - 1 κ + 2 R 3 ) E , p=4\pi\varepsilon_{0}\left(\frac{\kappa-1}{\kappa+2}{R^{3}}\right){E_{\infty}}\ ,
  61. p V = 3 ε 0 ( κ - 1 κ + 2 ) E . \frac{p}{V}={3}\varepsilon_{0}\left(\frac{\kappa-1}{\kappa+2}\right){E_{\infty% }}\ .
  62. - ϕ < = 3 κ + 2 E = ( 1 - κ - 1 κ + 2 ) E , {-\nabla}\phi_{<}=\frac{3}{\kappa+2}{E_{\infty}}=\left(1-\frac{\kappa-1}{% \kappa+2}\right){E_{\infty}}\ ,
  63. σ = 3 ε 0 κ - 1 κ + 2 E cos θ = 1 V p R ^ . \sigma={3}\varepsilon_{0}\frac{\kappa-1}{\kappa+2}E_{\infty}\cos\theta=\frac{1% }{V}{p\cdot\hat{R}}\ .
  64. Tot \mathcal{M}_{\rm Tot}\,
  65. ϵ = 1 + k Tot 2 \epsilon=1+k\langle\mathcal{M}_{\rm Tot}^{2}\rangle
  66. Tot 2 = Tot ( t = 0 ) Tot ( t = 0 ) \langle\mathcal{M}_{\rm Tot}^{2}\rangle=\langle\mathcal{M}_{\rm Tot}(t=0)% \mathcal{M}_{\rm Tot}(t=0)\rangle
  67. Tot = Trans + Rot . \mathcal{M}_{\rm Tot}=\mathcal{M}_{\rm Trans}+\mathcal{M}_{\rm Rot}.

Electrical_resistivity_measurement_of_concrete.html

  1. ρ = R A , \rho=R\frac{A}{\ell},\,\!
  2. \ell
  3. ρ = 2 π a V I \rho=2\pi a\frac{V}{I}
  4. ρ = 2 R D \rho=2RD

Electrocommunication.html

  1. 1 / r 3 1/r^{3}

Electromigrated_nanogaps.html

  1. G = 2 e 2 / h G=2e^{2}/h
  2. I = v e N / L I=veN/L
  3. G = v e N / L V G=veN/LV
  4. G = v e 2 N / L E G=ve^{2}N/LE
  5. G = 2 e 2 / h G=2e^{2}/h

Electrovibration.html

  1. F e = ε 0 ε r A V 2 2 d 2 F_{\mathrm{e}}=\frac{\varepsilon_{0}\varepsilon_{r}AV^{2}}{2d^{2}}
  2. ε 0 \varepsilon_{0}
  3. ε r \varepsilon_{r}
  4. A A
  5. V V
  6. d d
  7. f = μ F e f=\mu F_{\mathrm{e}}
  8. μ \mu

Elementary_group.html

  1. 1 C G P 1 1\longrightarrow C\longrightarrow G\longrightarrow P\longrightarrow 1
  2. C C

Ellipse_Law.html

  1. m ˙ 0 \scriptstyle\dot{m}_{0}\,
  2. T 0 \scriptstyle T_{0}\,
  3. p 0 \scriptstyle p_{0}\,
  4. p 2 \scriptstyle p_{2}\,
  5. p 1 \scriptstyle p_{1}\,
  6. m ˙ 01 \scriptstyle\dot{m}_{01}\,
  7. T 01 \scriptstyle T_{01}\,
  8. p 01 \scriptstyle p_{01}\,
  9. p 21 \scriptstyle p_{21}\,
  10. p 01 \scriptstyle p_{01}\,
  11. p 21 \scriptstyle p_{21}\,
  12. m ˙ 01 0 p 21 \scriptstyle\dot{m}_{01}\,0\,p_{21}\,
  13. p 21 \scriptstyle p_{21}\,
  14. p 21 \scriptstyle p_{21}\,
  15. p 01 \scriptstyle p_{01}\,
  16. m ˙ 01 0 p 01 \scriptstyle\dot{m}_{01}\,0\,p_{01}\,
  17. m ˙ 0 m \scriptstyle\dot{m}_{0m}\,
  18. p 0 m \scriptstyle p_{0m}\,
  19. p 2 m \scriptstyle p_{2m}\,
  20. ϵ 0 = p 0 / p 0 m \scriptstyle\epsilon_{0}=p_{0}/p_{0m}\,
  21. ϵ 2 = p 2 / p 2 m \scriptstyle\epsilon_{2}=p_{2}/p_{2m}\,
  22. ϵ 01 = p 01 / p 0 m \scriptstyle\epsilon_{01}=p_{01}/p_{0m}\,
  23. ϵ 21 = p 21 / p 2 m \scriptstyle\epsilon_{21}=p_{21}/p_{2m}\,
  24. 0 p 02 \scriptstyle 0\,p_{02}\,
  25. ϵ c = p c / p 01 \scriptstyle\epsilon_{c}=p_{c}/p_{01}\,
  26. p c \scriptstyle p_{c}\,
  27. m ˙ 0 m ˙ 01 = T 01 T 0 ϵ 0 2 ( 1 - ϵ c ) 2 - ( ϵ 2 - ϵ c ϵ 0 ) 2 ϵ 01 2 ( 1 - ϵ c ) 2 - ( ϵ 21 - ϵ c ϵ 01 ) 2 \frac{\dot{m}_{0}}{\dot{m}_{01}}=\sqrt{\frac{T_{01}}{T_{0}}}\sqrt{\frac{% \epsilon_{0}^{2}(1-\epsilon_{c})^{2}-(\epsilon_{2}-\epsilon_{c}\epsilon_{0})^{% 2}}{\epsilon_{01}^{2}(1-\epsilon_{c})^{2}-(\epsilon_{21}-\epsilon_{c}\epsilon_% {01})^{2}}}
  28. ϵ c \scriptstyle\epsilon_{c}\,
  29. m ˙ 0 m ˙ 01 = T 01 T 0 ϵ 0 2 - ϵ 2 2 ϵ 01 2 - ϵ 21 2 \frac{\dot{m}_{0}}{\dot{m}_{01}}=\sqrt{\frac{T_{01}}{T_{0}}}\sqrt{\frac{% \epsilon_{0}^{2}-\epsilon_{2}^{2}}{\epsilon_{01}^{2}-\epsilon_{21}^{2}}}
  30. m ˙ 0 m ˙ 01 = ϵ 0 2 - ϵ 2 2 ϵ 01 2 - ϵ 21 2 \frac{\dot{m}_{0}}{\dot{m}_{01}}=\sqrt{\frac{\epsilon_{0}^{2}-\epsilon_{2}^{2}% }{\epsilon_{01}^{2}-\epsilon_{21}^{2}}}
  31. ϵ 2 ϵ 21 0 \scriptstyle\epsilon_{2}\,\approx\,\epsilon_{21}\,\approx\,0
  32. m ˙ 0 m ˙ 01 = ϵ 0 ϵ 01 = p 01 p 0 \frac{\dot{m}_{0}}{\dot{m}_{01}}=\frac{\epsilon_{0}}{\epsilon_{01}}=\frac{p_{0% 1}}{p_{0}}

Elliptic_curve_point_multiplication.html

  1. Q = n P \scriptstyle Q~{}=~{}nP
  2. P + Q = R ( x p , y p ) + ( x q , y q ) = ( x r , y r ) \begin{aligned}\displaystyle P+Q&\displaystyle=R\\ \displaystyle(x_{p},y_{p})+(x_{q},y_{q})&\displaystyle=(x_{r},y_{r})\end{aligned}
  3. x r \displaystyle x_{r}
  4. λ = 3 x p 2 + a 2 y p \lambda=\frac{3x_{p}^{2}+a}{2y_{p}}
  5. d = d 0 + 2 d 1 + 2 2 d 2 + + 2 m d m d=d_{0}+2d_{1}+2^{2}d_{2}+\cdots+2^{m}d_{m}
  6. d 0 . . d m d_{0}..d_{m}
  7. 2 w 2^{w}
  8. d P dP
  9. d = 0 , 1 , 2 , , 2 w - 1 d=0,1,2,\dots,2^{w}-1
  10. d = d 0 + 2 w d 1 + 2 2 w d 2 + + 2 m w d m d=d_{0}+2^{w}d_{1}+2^{2w}d_{2}+\cdots+2^{mw}d_{m}
  11. w = 4 w=4
  12. n + 1 n+1
  13. 2 w - 2 + n w 2^{w}-2+{n\over w}
  14. d P dP
  15. d = 2 w - 1 , 2 w - 1 + 1 , , 2 w - 1 d=2^{w-1},2^{w-1}+1,\dots,2^{w}-1
  16. d = d 0 + 2 d 1 + 2 2 d 2 + + 2 m d m d=d_{0}+2d_{1}+2^{2}d_{2}+\cdots+2^{m}d_{m}
  17. w - 1 + n w-1+n
  18. 2 w - 1 - 1 + n w 2^{w-1}-1+{n\over w}
  19. d d
  20. { 1 , 3 , 5 , , 2 w - 1 - 1 } P \{1,3,5,\dots,2^{w-1}-1\}P
  21. P P
  22. P = { x , y } P=\{x,y\}
  23. - P = { x , - y } -P=\{x,-y\}
  24. d P dP
  25. 1 w + 1 1\over{w+1}
  26. 2 w - 2 - 1 2^{w-2}-1
  27. n n
  28. n w + 1 n\over{w+1}
  29. d i d_{i}
  30. w - 1 w-1
  31. w w
  32. d d
  33. 2 w 2^{w}
  34. d i d_{i}

Empty_lattice_approximation.html

  1. E n ( k ) = 2 ( k + G n ) 2 2 m E_{n}({k})=\frac{\hbar^{2}({k}+{G_{n}})^{2}}{2m}
  2. G n {G}_{n}
  3. E n ( k ) E_{n}({k})
  4. G n {G}_{n}
  5. E n ( k ) E_{n}({k})
  6. G n {G}_{n}
  7. [ k , k + d k ] [{k},{k}+d{k}]
  8. [ E , E + d E ] [E,E+dE]
  9. [ k , k + d k ] [{k},{k}+d{k}]
  10. E n ( k ) E_{n}({k})
  11. D 3 ( E ) D_{3}\left(E\right)
  12. D 3 ( E ) = 2 π E - E 0 c k 3 . D_{3}\left(E\right)=2\pi\sqrt{\frac{E-E_{0}}{c_{k}^{3}}}\ .
  13. k {k}
  14. V ( r ) = Z e r e - q r V(r)=\frac{Ze}{r}e^{-qr}
  15. U G U_{{G}}
  16. V ( r ) V({r})
  17. U G = 4 π Z e q 2 + G 2 U_{{G}}=\frac{4\pi Ze}{q^{2}+{G}^{2}}
  18. U G U_{{G}}
  19. 2 | U G | 2|U_{{G}}|

End_(graph_theory).html

  1. κ 0 κ 1 κ 2 \kappa_{0}\subset\kappa_{1}\subset\kappa_{2}\dots
  2. U 0 U 1 U 2 U_{0}\supset U_{1}\supset U_{2}\dots
  3. U i = β ( κ i ) U_{i}=\beta(\kappa_{i})
  4. U 0 U 1 U 2 U_{0}\supset U_{1}\supset U_{2}\dots
  5. h = inf { | V | | V | } , h=\inf\left\{\frac{|\partial V|}{|V|}\right\},
  6. V \partial V
  7. S = X ( β ( X ) X ) S=\bigcap_{X}\left(\beta(X)\cup X\right)
  8. β T ( X ) β G ( X ) \beta_{T}(X)\subset\beta_{G}(X)

Energy_distance.html

  1. D 2 ( F , G ) = 2 𝔼 X - Y - 𝔼 X - X - 𝔼 Y - Y 0 , D^{2}(F,G)=2\mathbb{E}\|X-Y\|-\mathbb{E}\|X-X^{\prime}\|-\mathbb{E}\|Y-Y^{% \prime}\|\geq 0,
  2. 𝔼 \mathbb{E}
  3. - ( F ( x ) - G ( x ) ) 2 d x \int_{-\infty}^{\infty}(F(x)-G(x))^{2}\,dx
  4. ( M , d ) (M,d)
  5. ( M ) \mathcal{B}(M)
  6. 𝒫 ( M ) \mathcal{P}(M)
  7. ( M , ( M ) ) (M,\mathcal{B}(M))
  8. 𝒫 ( M ) \mathcal{P}(M)
  9. D D
  10. D 2 ( μ , ν ) = 2 𝔼 [ d ( X , Y ) ] - 𝔼 [ d ( X , X ) ] - 𝔼 [ d ( Y , Y ) ] . D^{2}(\mu,\nu)=2\mathbb{E}[d(X,Y)]-\mathbb{E}[d(X,X^{\prime})]-\mathbb{E}[d(Y,% Y^{\prime})].
  11. ( M , d ) (M,d)
  12. D D
  13. ( M , d ) (M,d)
  14. D D
  15. ( M , d ) (M,d)
  16. T = n m n + m E n , m ( X , Y ) T=\frac{nm}{n+m}E_{n,m}(X,Y)
  17. H = D 2 ( F X , F Y ) 2 𝔼 X - Y = 2 𝔼 X - Y - 𝔼 X - X - 𝔼 Y - Y 2 𝔼 X - Y , H=\frac{D^{2}(F_{X},F_{Y})}{2\operatorname{\mathbb{E}}\|X-Y\|}=\frac{2\mathbb{% E}\|X-Y\|-\mathbb{E}\|X-X^{\prime}\|-\mathbb{E}\|Y-Y^{\prime}\|}{2% \operatorname{\mathbb{E}}\|X-Y\|},
  18. 𝔼 \mathbb{E}
  19. Q n = n ( 2 n i = 1 n 𝔼 x i - X α - 𝔼 X - X α - 1 n 2 i = 1 n j = 1 n x i - x j α ) , Q_{n}=n\left(\frac{2}{n}\sum_{i=1}^{n}\mathbb{E}\|x_{i}-X\|^{\alpha}-\mathbb{E% }\|X-X^{\prime}\|^{\alpha}-\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\|x_{i}-% x_{j}\|^{\alpha}\right),
  20. α ( 0 , 2 ) \alpha\in(0,2)
  21. α \alpha
  22. 𝔼 Q n = 𝔼 X - X α \mathbb{E}Q_{n}=\mathbb{E}\|X-X^{\prime}\|^{\alpha}

Energy–depth_relationship_in_a_rectangular_channel.html

  1. 𝐾𝐸 Weight = 1 2 m v 2 γ V = 1 2 ( ρ V ) v 2 ρ g V \frac{\mathit{KE}}{\textrm{Weight}}=\frac{\frac{1}{2}mv^{2}}{\gamma V}=\frac{% \frac{1}{2}\bigl(\rho V\bigr)v^{2}}{\rho gV}
  2. 𝐾𝐸 = v 2 2 g \mathit{KE}=\frac{v^{2}}{2g}
  3. 𝑃𝐸 ρ g = y \frac{\mathit{PE}}{\rho g}=y
  4. v 1 2 2 g + y 1 + P 1 γ - h f = v 2 2 2 g + y 2 + P 2 γ \frac{v_{1}^{2}}{2g}+y_{1}+\frac{P_{1}}{\gamma}-h_{f}=\frac{v_{2}^{2}}{2g}+y_{% 2}+\frac{P_{2}}{\gamma}
  5. v 1 2 2 g + y 1 = v 2 2 2 g + y 2 \frac{v_{1}^{2}}{2g}+y_{1}=\frac{v_{2}^{2}}{2g}+y_{2}
  6. E = v 2 2 g + y E=\frac{v^{2}}{2g}+y
  7. q = Q b q=\frac{Q}{b}
  8. E = q 2 2 g y 2 + y E=\frac{q^{2}}{2gy^{2}}+y
  9. 𝐹𝑟 = v g y \mathit{Fr}=\frac{v}{\sqrt{gy}}
  10. 𝐹𝑟 = 1 ; 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 \mathit{Fr}=1;\;\,\textit{Critical}
  11. 𝐹𝑟 < 1 ; 𝑆𝑢𝑏𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 \mathit{Fr}<1;\;\,\textit{Subcritical}
  12. 𝐹𝑟 > 1 ; 𝑆𝑢𝑝𝑒𝑟𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 \mathit{Fr}>1;\;\,\textit{Supercritical}
  13. y = 2 / 3 E y=2/3E
  14. y c = 2 / 3 E c y_{c}=2/3E_{c}
  15. d E d y = d d y ( q 2 2 g y 2 + y ) = 0 \frac{dE}{dy}=\frac{d}{dy}\biggl(\frac{q^{2}}{2gy^{2}}+y\biggr)=0
  16. y c = ( q 2 g ) 1 3 y_{c}=\biggl(\frac{q^{2}}{g}\biggr)^{\frac{1}{3}}
  17. E c = 3 2 y c E_{c}=\frac{3}{2}y_{c}
  18. y 2 = 2 y 1 - 1 + 1 + 8 g y 1 3 q 2 = 2 y 1 - 1 + 1 + 8 F r 1 2 y_{2}=\frac{2y_{1}}{-1+\sqrt{1+\frac{8gy_{1}^{3}}{q^{2}}}}=\frac{2y_{1}}{-1+% \sqrt{1+\frac{8}{{Fr_{1}}^{2}}}}
  19. y 2 = y 1 2 ( 1 + 8 F r 1 2 - 1 ) y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8Fr_{1}^{2}}-1\right)
  20. M = q 2 g y + y 2 2 {M}=\frac{q^{2}}{gy}+\frac{y^{2}}{2}
  21. M y c 2 = q 2 g y y c 2 + y 2 2 y c 2 \frac{M}{y_{c}^{2}}=\frac{q^{2}}{gyy_{c}^{2}}+\frac{y^{2}}{2y_{c}^{2}}
  22. M = M y c 2 M^{\prime}=\frac{M}{y_{c}^{2}}
  23. y = y y c y^{\prime}=\frac{y}{y_{c}}
  24. q 2 = g y c 3 q^{2}=gy_{c}^{3}
  25. M = 1 y + y 2 2 M^{\prime}=\frac{1}{y^{\prime}}+\frac{y^{\prime 2}}{2}
  26. E = q 2 2 g y 2 + y E=\frac{q^{2}}{2gy^{2}}+y
  27. E y c = y y c + q 2 2 g y 2 y c \frac{E}{y_{c}}=\frac{y}{y_{c}}+\frac{q^{2}}{2gy^{2}y_{c}}
  28. E = E y c E^{\prime}=\frac{E}{y_{c}}
  29. y = y y c y^{\prime}=\frac{y}{y_{c}}
  30. q 2 = g y c 3 q^{2}=gy_{c}^{3}
  31. E = y + 1 2 y 2 E^{\prime}=y^{\prime}+\frac{1}{2y^{\prime 2}}
  32. y ′′ = 1 y = y c y y^{\prime\prime}=\frac{1}{y^{\prime}}=\frac{y_{c}}{y}
  33. E ′′ = 1 y ′′ + y ′′ 2 2 E^{\prime\prime}=\frac{1}{y^{\prime\prime}}+\frac{y^{\prime\prime 2}}{2}
  34. M = 1 y + y 2 2 M^{\prime}=\frac{1}{y^{\prime}}+\frac{y^{\prime 2}}{2}
  35. E ′′ = 1 y ′′ + y ′′ 2 2 E^{\prime\prime}=\frac{1}{y^{\prime\prime}}+\frac{y^{\prime\prime 2}}{2}
  36. y 2 = y 1 2 ( 1 + 8 𝐹𝑟 1 2 - 1 ) y_{2}=\frac{y_{1}}{2}\left(\sqrt{1+8\mathit{Fr}_{1}^{2}}-1\right)
  37. q = g q=\sqrt{g}
  38. 𝐹𝑟 ( q = g ) = v g y = q g y 3 = g g y 3 = 1 y 3 \mathit{Fr}_{\left(q=\sqrt{g}\right)}=\frac{v}{\sqrt{gy}}=\frac{q}{\sqrt{gy^{3% }}}=\frac{\sqrt{g}}{\sqrt{gy^{3}}}=\frac{1}{\sqrt{y^{3}}}
  39. 𝐹𝑟 1 ~ = 1 y ′′ 3 \tilde{\mathit{Fr}_{1}}=\frac{1}{\sqrt{y^{\prime\prime 3}}}
  40. 𝐹𝑟 1 ~ \tilde{\mathit{Fr}_{1}}
  41. 𝐹𝑟 1 ~ \tilde{\mathit{Fr}_{1}}
  42. y ′′ = 1 y = y c y y^{\prime\prime}=\frac{1}{y^{\prime}}=\frac{y_{c}}{y}
  43. y 2 ′′ = y 1 ′′ 2 ( - 1 + 1 + 8 ( y 1 ′′ ) 3 ) y^{\prime\prime}_{2}=\frac{y^{\prime\prime}_{1}}{2}\left(-1+\sqrt{1+\frac{8}{(% y_{1}^{\prime\prime})^{3}}}\right)
  44. y 2 = y c y 2 ′′ y_{2}=\frac{y_{c}}{y_{2}^{\prime\prime}}
  45. ( y 1 ′′ ) 3 = ( y c y 1 ) 3 = q 2 g y 1 3 \left(y_{1}^{\prime\prime}\right)^{3}=\left(\frac{y_{c}}{y_{1}}\right)^{3}=% \frac{q^{2}}{gy_{1}^{3}}
  46. y 2 = 2 y 1 - 1 + 1 + 8 g y 1 3 q 2 y_{2}=\frac{2y_{1}}{-1+\sqrt{1+\frac{8gy_{1}^{3}}{q^{2}}}}
  47. y c = ( q 2 g ) 1 3 y_{c}=\left(\frac{q^{2}}{g}\right)^{\frac{1}{3}}
  48. F r = v g y = q y g y Fr=\frac{v}{\sqrt{gy}}=\frac{q}{y\sqrt{gy}}
  49. 𝐹𝑟 2 = ( y c y ) 3 = q 2 g y 3 \mathit{Fr}^{2}=\left(\frac{y_{c}}{y}\right)^{3}=\frac{q^{2}}{gy^{3}}
  50. y 2 = 2 y 1 - 1 + 1 + 8 g y 1 3 q 2 = 2 y 1 - 1 + 1 + 8 𝐹𝑟 1 2 y_{2}=\frac{2y_{1}}{-1+\sqrt{1+\frac{8gy_{1}^{3}}{q^{2}}}}=\frac{2y_{1}}{-1+% \sqrt{1+\frac{8}{{\mathit{Fr}_{1}}^{2}}}}
  51. q = 10. ft 2 s q=10.\frac{\textrm{ft}^{2}}{\textrm{s}}
  52. E 1 = E 2 E_{1}=E_{2}
  53. y 2 = 2 y 1 - 1 + 1 + 8 g y 1 3 q 2 = 2 ( 5.0 ft ) - 1 + 1 + 8 ( 32.2 ft s 2 ) ( 5.0 ft ) 3 ( 10 ft 2 s ) 2 = 0.59 ft y_{2}=\frac{2y_{1}}{-1+\sqrt{1+\frac{8gy_{1}^{3}}{q^{2}}}}=\frac{2(5.0\,% \mathrm{ft})}{-1+\sqrt{1+\frac{8(32.2\frac{\mathrm{ft}}{\mathrm{s}^{2}})(5.0\,% \mathrm{ft})^{3}}{\left(10\frac{\mathrm{ft}^{2}}{\mathrm{s}}\right)^{2}}}}=0.5% 9\,\mathrm{ft}
  54. E 1 = E 2 E_{1}=E_{2}
  55. E 1 = y 1 + q 2 2 g y 1 2 = 5.0 + ft + ( 10 ft 2 s ) 2 2 ( 32.2 ft s 2 ) ( 5.0 ft ) 2 = 5.06 ft E_{1}=y_{1}+\frac{q^{2}}{2gy_{1}^{2}}=5.0+\,\mathrm{ft}+\frac{\left(10\frac{% \mathrm{ft}^{2}}{\mathrm{s}}\right)^{2}}{2\left(32.2\frac{\mathrm{ft}}{\mathrm% {s}^{2}}\right)(5.0\,\mathrm{ft})^{2}}=5.06\,\mathrm{ft}
  56. E 2 = y 2 + q 2 2 g y 2 2 = 0.59 ft + ( 10 ft 2 s ) 2 2 ( 32.2 ft s 2 ) ( 0.59 ft ) 2 = 5.06 ft E_{2}=y_{2}+\frac{q^{2}}{2gy_{2}^{2}}=0.59\,\mathrm{ft}+\frac{\left(10\frac{% \mathrm{ft}^{2}}{\mathrm{s}}\right)^{2}}{2\left(32.2\frac{\mathrm{ft}}{\mathrm% {s}^{2}}\right)(0.59\,\mathrm{ft})^{2}}=5.06\,\mathrm{ft}
  57. E 1 = E 2 = 5.06 ft E_{1}=E_{2}=5.06\,\mathrm{ft}

Enneacontagon.html

  1. A = 45 2 t 2 cot π 90 A=\frac{45}{2}t^{2}\cot\frac{\pi}{90}
  2. r = 1 2 t cot π 90 r=\frac{1}{2}t\cot\frac{\pi}{90}
  3. R = 1 2 t csc π 90 R=\frac{1}{2}t\csc\frac{\pi}{90}

Ensemble_averaging.html

  1. y i y_{i}
  2. y ~ \tilde{y}
  3. y ~ ( x ; α ) = j = 1 p α j y j ( x ) \tilde{y}({x};{\alpha})=\sum_{j=1}^{p}\alpha_{j}y_{j}({x})
  4. α {\alpha}
  5. α j = 0 \alpha_{j}=0
  6. α k = 1 \alpha_{k}=1
  7. α j \alpha_{j}

Enthalpy–entropy_chart.html

  1. h h
  2. u u
  3. P P
  4. v v
  5. h = u + P v h=u+Pv\,\!

Entropic_risk_measure.html

  1. θ > 0 \theta>0
  2. ρ ent ( X ) = 1 θ log ( 𝔼 [ e - θ X ] ) = sup Q 1 { E Q [ - X ] - 1 θ H ( Q | P ) } \rho^{\mathrm{ent}}(X)=\frac{1}{\theta}\log\left(\mathbb{E}[e^{-\theta X}]% \right)=\sup_{Q\in\mathcal{M}_{1}}\left\{E^{Q}[-X]-\frac{1}{\theta}H(Q|P)% \right\}\,
  3. H ( Q | P ) = E [ d Q d P log d Q d P ] H(Q|P)=E\left[\frac{dQ}{dP}\log\frac{dQ}{dP}\right]
  4. A = { X L p ( ) : E [ u ( X ) ] 0 } = { X L p ( ) : E [ e - θ X ] 1 } A=\{X\in L^{p}(\mathcal{F}):E[u(X)]\geq 0\}=\{X\in L^{p}(\mathcal{F}):E\left[e% ^{-\theta X}\right]\leq 1\}
  5. u ( X ) u(X)
  6. θ \theta
  7. ρ t ent ( X ) = 1 θ log ( 𝔼 [ e - θ X | t ] ) . \rho^{\mathrm{ent}}_{t}(X)=\frac{1}{\theta}\log\left(\mathbb{E}[e^{-\theta X}|% \mathcal{F}_{t}]\right).
  8. θ \theta

Envelope_(radar).html

  1. B l i n d R a n g e = 0.5 × C × ( T r a n s m i t P u l s e W i d t h + S e t u p T i m e ) Blind\ Range=0.5\times C\times(Transmit\ Pulse\ Width+Setup\ Time)
  2. H e i g h t < C 2 × ( T r a n s m i t P u l s e W i d t h + S e t u p T i m e ) 2 8 × E a r t h R a d i u s Height<\frac{C^{2}\times(Transmit\ Pulse\ Width+Setup\ Time)^{2}}{8\times Earth% \ Radius}

Enzo_Martinelli.html

  1. n n
  2. n n
  3. ( n + 1 ) (n+1)
  4. n n
  5. n n
  6. 2 n 2n
  7. n n
  8. 2 n - 1 2n-1
  9. n n

EPSP_synthase.html

  1. \rightleftharpoons

Equalization_(audio).html

  1. | H ( f ) | 2 = 1 + ( f / f z ) 2 1 + ( f / f p ) 2 |H(f)|^{2}={{1+(f/f_{z})^{2}}\over{1+(f/f_{p})^{2}}}
  2. | H ( f ) | 2 = ( f z / f p ) 2 1 + ( f / f z ) 2 1 + ( f / f p ) 2 |H(f)|^{2}=(f_{z}/f_{p})^{2}\;{{1+(f/f_{z})^{2}}\over{1+(f/f_{p})^{2}}}
  3. B W = F 0 / Q BW\ =\ F_{0}/Q
  4. Q = 2 N 2 N - 1 = 1 2 sinh ( ln ( 2 ) 2 N ) Q\ =\ \frac{\sqrt{2^{N}}}{2^{N}-1}\ =\ \frac{1}{2\sinh\left(\frac{\ln(2)}{2}N% \right)}
  5. N N
  6. Q Q

Equalization_(proof).html

  1. α \alpha
  2. β \beta
  3. F x 1 = F x 2 F_{x1}=F_{x2}\,
  4. F 1 S i n ( α ) = F 2 S i n ( β ) F_{1}Sin(\alpha)=F_{2}Sin(\beta)\,
  5. F 1 = F 2 S i n ( β ) S i n ( α ) F_{1}=F_{2}\frac{Sin(\beta)}{Sin(\alpha)}\,
  6. F y 1 + F y 2 = F l o a d F_{y1}+F_{y2}=F_{load}\,
  7. F 1 C o s ( α ) + F 2 C o s ( β ) = F l o a d F_{1}Cos(\alpha)+F_{2}Cos(\beta)=F_{load}\,
  8. F 1 F_{1}
  9. F 2 F_{2}
  10. F 2 S i n ( β ) S i n ( α ) C o s ( α ) + F 2 C o s ( β ) = F l o a d F_{2}\frac{Sin(\beta)}{Sin(\alpha)}Cos(\alpha)+F_{2}Cos(\beta)=F_{load}\,
  11. F 2 [ S i n ( β ) S i n ( α ) C o s ( α ) + C o s ( β ) ] = F l o a d F_{2}\left[\frac{Sin(\beta)}{Sin(\alpha)}Cos(\alpha)+Cos(\beta)\right]=F_{load}\,
  12. F 2 F_{2}
  13. F 2 = F l o a d [ S i n ( β ) S i n ( α ) C o s ( α ) + C o s ( β ) ] F_{2}=\frac{F_{load}}{\left[\frac{Sin(\beta)}{Sin(\alpha)}Cos(\alpha)+Cos(% \beta)\right]}\,
  14. F 2 = F l o a d [ S i n ( β ) C o s ( α ) S i n ( α ) + C o s ( β ) S i n ( α ) S i n ( α ) ] F_{2}=\frac{F_{load}}{\left[\frac{Sin(\beta)Cos(\alpha)}{Sin(\alpha)}+\frac{% Cos(\beta)Sin(\alpha)}{Sin(\alpha)}\right]}\,
  15. F 2 = F l o a d [ C o s ( α ) S i n ( β ) + S i n ( α ) C o s ( β ) S i n ( α ) ] F_{2}=\frac{F_{load}}{\left[\frac{Cos(\alpha)Sin(\beta)+Sin(\alpha)Cos(\beta)}% {Sin(\alpha)}\right]}\,
  16. F 2 = F l o a d S i n ( α ) < m t p l > C o s ( α ) S i n ( β ) + S i n ( α ) C o s ( β ) F_{2}=F_{load}\frac{Sin(\alpha)}{<}mtpl>{{Cos(\alpha)Sin(\beta)+Sin(\alpha)Cos% (\beta)}}\,
  17. F 2 F_{2}
  18. F 2 = F l o a d S i n ( α ) S i n ( α + β ) F_{2}=F_{load}\frac{Sin(\alpha)}{Sin(\alpha+\beta)}\,
  19. F 2 F_{2}
  20. F 1 F_{1}
  21. F 1 = F l o a d S i n ( α ) S i n ( α + β ) S i n ( β ) S i n ( α ) F_{1}=F_{load}\frac{Sin(\alpha)}{Sin(\alpha+\beta)}\frac{Sin(\beta)}{Sin(% \alpha)}\,
  22. F 1 = F l o a d S i n ( β ) S i n ( α + β ) F_{1}=F_{load}\frac{Sin(\beta)}{Sin(\alpha+\beta)}\,
  23. 2 α = θ 2\alpha=\theta
  24. α \alpha
  25. β \beta
  26. β \beta
  27. α \alpha
  28. F e a c h A n c h o r = F l o a d S i n ( α ) S i n ( α + α ) F_{eachAnchor}=F_{load}\frac{Sin(\alpha)}{Sin(\alpha+\alpha)}\,
  29. F e a c h A n c h o r = F l o a d S i n ( α ) S i n ( 2 α ) F_{eachAnchor}=F_{load}\frac{Sin(\alpha)}{Sin(2\alpha)}\,
  30. F e a c h A n c h o r = F l o a d S i n ( α ) 2 S i n ( α ) C o s ( α ) F_{eachAnchor}=F_{load}\frac{Sin(\alpha)}{2Sin(\alpha)Cos(\alpha)}\,
  31. F e a c h A n c h o r = F l o a d 2 C o s ( α ) F_{eachAnchor}=\frac{F_{load}}{2Cos(\alpha)}\,
  32. α \alpha
  33. θ \theta
  34. θ \theta
  35. α = θ / 2 \alpha=\theta/2
  36. F e a c h A n c h o r = F l o a d 2 C o s ( θ 2 ) F_{eachAnchor}=\frac{F_{load}}{2Cos(\frac{\theta}{2})}\,

Equichordal_point_problem.html

  1. C C
  2. O 1 O_{1}
  3. O 2 O_{2}
  4. C C
  5. C C
  6. O 1 O_{1}
  7. O 2 O_{2}
  8. L L
  9. C C
  10. O 1 O_{1}
  11. O 2 O_{2}
  12. a = O 1 - O 2 L a=\frac{\|O_{1}-O_{2}\|}{L}
  13. O 1 - O 2 \|O_{1}-O_{2}\|
  14. O 1 O_{1}
  15. O 2 O_{2}
  16. F : 2 2 F:\mathbb{C}^{2}\to\mathbb{C}^{2}

Equidiagonal_quadrilateral.html

  1. p p
  2. q q
  3. m m
  4. n n
  5. p q = m 2 + n 2 . pq=m^{2}+n^{2}.
  6. K = m n . \displaystyle K=mn.
  7. K = 1 4 ( 2 ( a 2 + c 2 ) - 4 x 2 ) ( 2 ( b 2 + d 2 ) - 4 x 2 ) . K=\tfrac{1}{4}\sqrt{(2(a^{2}+c^{2})-4x^{2})(2(b^{2}+d^{2})-4x^{2})}.
  8. K = a 2 + c 2 + 4 ( a 2 c 2 + b 2 d 2 ) - ( a 2 + c 2 ) 2 4 . K=\frac{a^{2}+c^{2}+\sqrt{4(a^{2}c^{2}+b^{2}d^{2})-(a^{2}+c^{2})^{2}}}{4}.

Equilateral_dimension.html

  1. x p = ( | x 1 | p + | x 2 | p + + | x d | p ) 1 / p . \ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots+|x_{d}|^{p}\right)^{1/p}.
  2. k exp ( c log d ) k\geq\exp(c\sqrt{\log d})

Equioscillation_theorem.html

  1. f f
  2. [ a , b ] [a,b]
  3. 𝐑 \mathbf{R}
  4. n \leq n
  5. g g
  6. || f - g || ||f-g||_{\infty}
  7. n + 2 n+2
  8. a x 0 < x 1 < < x n + 1 b a\leq x_{0}<x_{1}<\cdots<x_{n+1}\leq b
  9. f ( x i ) - g ( x i ) = σ ( - 1 ) i || f - g || f(x_{i})-g(x_{i})=\sigma(-1)^{i}||f-g||_{\infty}
  10. σ = ± 1 \sigma=\pm 1

Equivalent_latitude.html

  1. ϕ = sin - 1 ( A 2 π R 2 - 1 ) \phi=\sin^{-1}\left(\frac{A}{2\pi R^{2}}-1\right)

Equivariant_index_theorem.html

  1. π : E M \pi:E\to M
  2. π \pi
  3. str ( g ker D ) = tr ( g ker D + ) - tr ( g ker D - ) . \operatorname{str}(g\mid\ker D)=\operatorname{tr}(g\mid\ker D^{+})-% \operatorname{tr}(g\mid\ker D^{-}).

Erdelyi–Kober_operator.html

  1. x - ν - α + 1 Γ ( α ) 0 x ( t - x ) α - 1 t - α - ν f ( t ) d t \frac{x^{-\nu-\alpha+1}}{\Gamma(\alpha)}\int_{0}^{x}(t-x)^{\alpha-1}t^{-\alpha% -\nu}f(t)dt

Erdős_distinct_distances_problem.html

  1. n n
  2. g ( n ) g(n)
  3. n n
  4. n - 3 / 4 - 1 / 2 g ( n ) c n / log n \sqrt{n-3/4}-1/2\leq g(n)\leq cn/\sqrt{\log n}
  5. c c
  6. n × n \sqrt{n}\times\sqrt{n}
  7. O ( n / log n ) O(n/\sqrt{\log n})
  8. g ( n ) = Ω ( n c ) g(n)=\Omega(n^{c})
  9. c < 1 c<1

Erdős–Gallai_theorem.html

  1. d 1 d n d_{1}\geq\cdots\geq d_{n}
  2. d 1 + + d n d_{1}+\cdots+d_{n}
  3. i = 1 k d i k ( k - 1 ) + i = k + 1 n min ( d i , k ) \sum^{k}_{i=1}d_{i}\leq k(k-1)+\sum^{n}_{i=k+1}\min(d_{i},k)
  4. 1 k n 1\leq k\leq n
  5. k k
  6. k k
  7. k ( k - 1 ) k(k-1)
  8. t t
  9. d t > d t + 1 d_{t}>d_{t+1}
  10. d t d_{t}
  11. m = d i m=\sum d_{i}
  12. m m
  13. m m
  14. d i d_{i}
  15. d i + 1 d_{i+1}
  16. k k
  17. 1 k < n 1\leq k<n
  18. a k > a k + 1 a_{k}>a_{k+1}
  19. k = n k=n
  20. l l
  21. l = max { k d k k } l=\max\{k\mid d_{k}\geq k\}
  22. ( d 1 , , d n ) (d_{1},\cdots,d_{n})
  23. d 1 d n d_{1}\geq\cdots\geq d_{n}
  24. i = 1 n d i \sum_{i=1}^{n}d_{i}
  25. ( c 1 , , c n ) (c_{1},\cdots,c_{n})
  26. ( d 1 , , d n ) (d_{1},\cdots,d_{n})

Erdős–Szemerédi_theorem.html

  1. ε \varepsilon
  2. max ( | A + A | , | A A | ) c | A | 1 + ε \max(|A+A|,|A\cdot A|)\geq c|A|^{1+\varepsilon}
  3. A + A = { a + b : a , b A } A+A=\{a+b:a,b\in A\}
  4. A A = { a b : a , b A } A\cdot A=\{ab:a,b\in A\}
  5. ε \varepsilon
  6. ε \varepsilon

Erdős–Turán_conjecture_on_additive_bases.html

  1. f ( n ) > 0 f(n)>0
  2. n > n 0 n>n_{0}
  3. lim ¯ n f ( n ) = \overline{\lim}_{n\rightarrow\infty}f(n)=\infty
  4. f ( n ) f(n)
  5. n n
  6. B B
  7. f ( n ) f(n)
  8. n n
  9. B B
  10. B B\subset\mathbb{N}
  11. r B ( n ) = # { ( a 1 , a 2 ) B 2 | a 1 + a 2 = n } r_{B}(n)=\#\{(a_{1},a_{2})\in B^{2}|a_{1}+a_{2}=n\}
  12. r B ( n ) > 0 r_{B}(n)>0
  13. n n
  14. lim sup n r B ( n ) = \limsup_{n\rightarrow\infty}r_{B}(n)=\infty
  15. h h\in\mathbb{N}
  16. B B\subset\mathbb{N}
  17. h h
  18. r B , h ( n ) = # { ( a 1 , , a h ) B h | a 1 + + a h = n } r_{B,h}(n)=\#\{(a_{1},\cdots,a_{h})\in B^{h}|a_{1}+\cdots+a_{h}=n\}
  19. B B
  20. h h
  21. r B , h ( n ) > 0 r_{B,h}(n)>0
  22. n n
  23. B B
  24. h h
  25. n m = 1 n r B , h ( m ) | B [ 1 , n ] | h \displaystyle n\leq\sum_{m=1}^{n}r_{B,h}(m)\leq|B\cap[1,n]|^{h}
  26. n 1 / h | B [ 1 , n ] | n^{1/h}\leq|B\cap[1,n]|
  27. | B [ 1 , n ] | n 1 / h |B\cap[1,n]|\geq n^{1/h}
  28. B B
  29. h h
  30. h = 2 h=2
  31. B B
  32. c 1 , c 2 > 0 c_{1},c_{2}>0
  33. c 1 log n r B ( n ) c 2 log n c_{1}\log n\leq r_{B}(n)\leq c_{2}\log n
  34. n n
  35. B B
  36. r B ( n ) = n 1 / 2 + o ( 1 ) r_{B}(n)=n^{1/2+o(1)}
  37. B B
  38. h h
  39. lim sup n r B ( n ) / log n > 0. \limsup_{n\rightarrow\infty}r_{B}(n)/\log n>0.
  40. B B
  41. f ( n ) f(n)

Erdős–Turán_inequality.html

  1. sup A | μ ( A ) - mes A | C ( 1 n + k = 1 n | μ ^ ( k ) | k ) , \sup_{A}\left|\mu(A)-\mathrm{mes}\,A\right|\leq C\left(\frac{1}{n}+\sum_{k=1}^% {n}\frac{|\hat{\mu}(k)|}{k}\right),
  2. μ ^ ( k ) = exp ( 2 π i k θ ) d μ ( θ ) \hat{\mu}(k)=\int\exp(2\pi ik\theta)\,d\mu(\theta)
  3. μ m ( S ) = 1 m # { 1 j m | s j mod 1 S } , S [ 0 , 1 ) , \mu_{m}(S)=\frac{1}{m}\#\{1\leq j\leq m\,|\,s_{j}\,\mathrm{mod}\,1\in S\},% \quad S\subset[0,1),
  4. D ( m ) ( = sup 0 a b 1 | m - 1 # { 1 j m | a s j mod 1 b } - ( b - a ) | ) C ( 1 n + 1 m k = 1 n 1 k | j = 1 m e 2 π i s j k | ) . ( 1 ) \begin{aligned}\displaystyle D(m)&\displaystyle\left(=\sup_{0\leq a\leq b\leq 1% }\Big|m^{-1}\#\{1\leq j\leq m\,|\,a\leq s_{j}\,\mathrm{mod}\,1\leq b\}-(b-a)% \Big|\right)\\ &\displaystyle\leq C\left(\frac{1}{n}+\frac{1}{m}\sum_{k=1}^{n}\frac{1}{k}% \left|\sum_{j=1}^{m}e^{2\pi is_{j}k}\right|\right).\end{aligned}\qquad(1)

Error_analysis_for_the_Global_Positioning_System.html

  1. 3 σ R \ 3\sigma_{R}
  2. 3 σ R \ 3\sigma_{R}
  3. σ n u m \ \sigma_{num}
  4. σ R \ \sigma_{R}
  5. σ r c \ \sigma_{rc}
  6. σ R \ \sigma_{R}
  7. σ R \ \sigma_{R}
  8. σ R \ \sigma_{R}
  9. 3 σ R = 3 2 + 5 2 + 2.5 2 + 2 2 + 1 2 + 0.5 2 m = 6.7 m 3\sigma_{R}=\sqrt{3^{2}+5^{2}+2.5^{2}+2^{2}+1^{2}+0.5^{2}}\,\mathrm{m}\,=\,6.7% \,\mathrm{m}
  10. σ r c \ \sigma_{rc}
  11. σ r c = P D O P 2 × σ R 2 + σ n u m 2 = P D O P 2 × 2.2 2 + 1 2 m \ \sigma_{rc}=\sqrt{PDOP^{2}\times\sigma_{R}^{2}+\sigma_{num}^{2}}=\sqrt{PDOP^% {2}\times 2.2^{2}+1^{2}}\,\mathrm{m}
  12. 0.01 ( 1.023 × 10 6 / s ) \frac{0.01}{(1.023\times 10^{6}/\mathrm{s})}
  13. ( 0.01 × 300 , 000 , 000 m / s ) ( 10.23 × 10 6 / s ) \frac{(0.01\times 300,000,000\ \mathrm{m/s})}{(10.23\times 10^{6}/\mathrm{s})}
  14. ( x i - x ) R i \frac{(x_{i}-x)}{R_{i}}
  15. ( y i - y ) R i \frac{(y_{i}-y)}{R_{i}}
  16. ( z i - z ) R i \frac{(z_{i}-z)}{R_{i}}
  17. R i \ R_{i}
  18. R i = ( x i - x ) 2 + ( y i - y ) 2 + ( z i - z ) 2 R_{i}\,=\,\sqrt{(x_{i}-x)^{2}+(y_{i}-y)^{2}+(z_{i}-z)^{2}}
  19. x , y , \ x,y,
  20. z \ z
  21. x i , y i , \ x_{i},y_{i},
  22. z i \ z_{i}
  23. A = [ ( x 1 - x ) R 1 ( y 1 - y ) R 1 ( z 1 - z ) R 1 c ( x 2 - x ) R 2 ( y 2 - y ) R 2 ( z 2 - z ) R 2 c ( x 3 - x ) R 3 ( y 3 - y ) R 3 ( z 3 - z ) R 3 c ( x 4 - x ) R 4 ( y 4 - y ) R 4 ( z 4 - z ) R 4 c ] A=\begin{bmatrix}\frac{(x_{1}-x)}{R_{1}}&\frac{(y_{1}-y)}{R_{1}}&\frac{(z_{1}-% z)}{R_{1}}&c\\ \frac{(x_{2}-x)}{R_{2}}&\frac{(y_{2}-y)}{R_{2}}&\frac{(z_{2}-z)}{R_{2}}&c\\ \frac{(x_{3}-x)}{R_{3}}&\frac{(y_{3}-y)}{R_{3}}&\frac{(z_{3}-z)}{R_{3}}&c\\ \frac{(x_{4}-x)}{R_{4}}&\frac{(y_{4}-y)}{R_{4}}&\frac{(z_{4}-z)}{R_{4}}&c\end{bmatrix}
  24. Q = ( A T A ) - 1 Q=\left(A^{T}A\right)^{-1}
  25. Q = [ d x 2 d x y 2 d x z 2 d x t 2 d x y 2 d y 2 d y z 2 d y t 2 d x z 2 d y z 2 d z 2 d z t 2 d x t 2 d y t 2 d z t 2 d t 2 ] Q=\begin{bmatrix}d_{x}^{2}&d_{xy}^{2}&d_{xz}^{2}&d_{xt}^{2}\\ d_{xy}^{2}&d_{y}^{2}&d_{yz}^{2}&d_{yt}^{2}\\ d_{xz}^{2}&d_{yz}^{2}&d_{z}^{2}&d_{zt}^{2}\\ d_{xt}^{2}&d_{yt}^{2}&d_{zt}^{2}&d_{t}^{2}\end{bmatrix}
  26. σ \ \sigma
  27. P D O P = d x 2 + d y 2 + d z 2 T D O P = d t 2 = | d t | G D O P = P D O P 2 + T D O P 2 \begin{aligned}\displaystyle PDOP&\displaystyle=\sqrt{d_{x}^{2}+d_{y}^{2}+d_{z% }^{2}}\\ \displaystyle TDOP&\displaystyle=\sqrt{d_{t}^{2}}=|d_{t}|\\ \displaystyle GDOP&\displaystyle=\sqrt{PDOP^{2}+TDOP^{2}}\end{aligned}
  28. H D O P = d x 2 + d y 2 HDOP=\sqrt{d_{x}^{2}+d_{y}^{2}}
  29. V D O P = d z 2 = | d z | \ VDOP=\sqrt{d_{z}^{2}}=|d_{z}|
  30. 𝐞 \mathbf{e}
  31. 𝐞 = e x x ^ + e y y ^ + e z z ^ \mathbf{e}=e_{x}\hat{x}+e_{y}\hat{y}+e_{z}\hat{z}
  32. x ^ \hat{x}
  33. y ^ \hat{y}
  34. z ^ \hat{z}
  35. e t \ e_{t}
  36. 𝐞 \mathbf{e}
  37. e t \ e_{t}
  38. A [ e x e y e z e t ] = [ ( x 1 - x ) R 1 ( y 1 - y ) R 1 ( z 1 - z ) R 1 c ( x 2 - x ) R 2 ( y 2 - y ) R 2 ( z 2 - z ) R 2 c ( x 3 - x ) R 3 ( y 3 - y ) R 3 ( z 3 - z ) R 3 c ( x 4 - x ) R 4 ( y 4 - y ) R 4 ( z 4 - z ) R 4 c ] [ e x e y e z e t ] = [ e 1 e 2 e 3 e 4 ] ( 1 ) A\begin{bmatrix}e_{x}\\ e_{y}\\ e_{z}\\ e_{t}\end{bmatrix}=\begin{bmatrix}\frac{(x_{1}-x)}{R_{1}}&\frac{(y_{1}-y)}{R_{% 1}}&\frac{(z_{1}-z)}{R_{1}}&c\\ \frac{(x_{2}-x)}{R_{2}}&\frac{(y_{2}-y)}{R_{2}}&\frac{(z_{2}-z)}{R_{2}}&c\\ \frac{(x_{3}-x)}{R_{3}}&\frac{(y_{3}-y)}{R_{3}}&\frac{(z_{3}-z)}{R_{3}}&c\\ \frac{(x_{4}-x)}{R_{4}}&\frac{(y_{4}-y)}{R_{4}}&\frac{(z_{4}-z)}{R_{4}}&c\end{% bmatrix}\begin{bmatrix}e_{x}\\ e_{y}\\ e_{z}\\ e_{t}\end{bmatrix}=\begin{bmatrix}e_{1}\\ e_{2}\\ e_{3}\\ e_{4}\end{bmatrix}\ (1)
  39. e 1 e_{1}
  40. e 2 e_{2}
  41. e 3 e_{3}
  42. e 4 e_{4}
  43. A - 1 A^{-1}
  44. [ e x e y e z e t ] = A - 1 [ e 1 e 2 e 3 e 4 ] ( 2 ) \begin{bmatrix}e_{x}\\ e_{y}\\ e_{z}\\ e_{t}\end{bmatrix}=A^{-1}\begin{bmatrix}e_{1}\\ e_{2}\\ e_{3}\\ e_{4}\end{bmatrix}\ (2)
  45. [ e x e y e z e t ] = [ e 1 e 2 e 3 e 4 ] ( A - 1 ) T ( 3 ) \begin{bmatrix}e_{x}&e_{y}&e_{z}&e_{t}\end{bmatrix}=\begin{bmatrix}e_{1}&e_{2}% &e_{3}&e_{4}\end{bmatrix}\left(A^{-1}\right)^{T}\ (3)
  46. [ e x e y e z e t ] [ e x e y e z e t ] = A - 1 [ e 1 e 2 e 3 e 4 ] [ e 1 e 2 e 3 e 4 ] ( A - 1 ) T ( 4 ) \begin{bmatrix}e_{x}\\ e_{y}\\ e_{z}\\ e_{t}\end{bmatrix}\begin{bmatrix}e_{x}&e_{y}&e_{z}&e_{t}\end{bmatrix}=A^{-1}% \begin{bmatrix}e_{1}\\ e_{2}\\ e_{3}\\ e_{4}\end{bmatrix}\begin{bmatrix}e_{1}&e_{2}&e_{3}&e_{4}\end{bmatrix}\left(A^{% -1}\right)^{T}\ (4)
  47. E ( [ e x e y e z e t ] [ e x e y e z e t ] ) = A - 1 E ( [ e 1 e 2 e 3 e 4 ] [ e 1 e 2 e 3 e 4 ] ) ( A - 1 ) T ( 5 ) E\left(\begin{bmatrix}e_{x}\\ e_{y}\\ e_{z}\\ e_{t}\end{bmatrix}\begin{bmatrix}e_{x}&e_{y}&e_{z}&e_{t}\end{bmatrix}\right)=A% ^{-1}E\left(\begin{bmatrix}e_{1}\\ e_{2}\\ e_{3}\\ e_{4}\end{bmatrix}\begin{bmatrix}e_{1}&e_{2}&e_{3}&e_{4}\end{bmatrix}\right)% \left(A^{-1}\right)^{T}\ (5)
  48. [ σ x 2 σ x y 2 σ x z 2 σ x t 2 σ x y 2 σ y 2 σ y z 2 σ y t 2 σ x z 2 σ y z 2 σ z 2 σ z t 2 σ x t 2 σ y t 2 σ z t 2 σ t 2 ] = σ R 2 A - 1 ( A - 1 ) T = σ R 2 ( A T A ) - 1 ( 6 ) \begin{bmatrix}\sigma_{x}^{2}&\sigma_{xy}^{2}&\sigma_{xz}^{2}&\sigma_{xt}^{2}% \\ \sigma_{xy}^{2}&\sigma_{y}^{2}&\sigma_{yz}^{2}&\sigma_{yt}^{2}\\ \sigma_{xz}^{2}&\sigma_{yz}^{2}&\sigma_{z}^{2}&\sigma_{zt}^{2}\\ \sigma_{xt}^{2}&\sigma_{yt}^{2}&\sigma_{zt}^{2}&\sigma_{t}^{2}\end{bmatrix}=% \sigma_{R}^{2}\ A^{-1}\left(A^{-1}\right)^{T}=\sigma_{R}^{2}\ \left(A^{T}A% \right)^{-1}\ (6)
  49. A - 1 ( A - 1 ) T ( A T A ) = I \ A^{-1}\left(A^{-1}\right)^{T}\left(A^{T}A\right)=I
  50. ( A - 1 ) T = ( A T ) - 1 , \left(A^{-1}\right)^{T}=\left(A^{T}\right)^{-1},
  51. I = ( A A - 1 ) T = ( A - 1 ) T A T I=\left(AA^{-1}\right)^{T}=\left(A^{-1}\right)^{T}A^{T}
  52. ( A T A ) - 1 = Q \left(A^{T}A\right)^{-1}=Q
  53. [ σ x 2 σ x y 2 σ x z 2 σ x t 2 σ x y 2 σ y 2 σ y z 2 σ y t 2 σ x z 2 σ y z 2 σ z 2 σ z t 2 σ x t 2 σ y t 2 σ z t 2 σ t 2 ] = σ R 2 [ d x 2 d x y 2 d x z 2 d x t 2 d x y 2 d y 2 d y z 2 d y t 2 d x z 2 d y z 2 d z 2 d z t 2 d x t 2 d y t 2 d z t 2 d t 2 ] ( 7 ) \begin{bmatrix}\sigma_{x}^{2}&\sigma_{xy}^{2}&\sigma_{xz}^{2}&\sigma_{xt}^{2}% \\ \sigma_{xy}^{2}&\sigma_{y}^{2}&\sigma_{yz}^{2}&\sigma_{yt}^{2}\\ \sigma_{xz}^{2}&\sigma_{yz}^{2}&\sigma_{z}^{2}&\sigma_{zt}^{2}\\ \sigma_{xt}^{2}&\sigma_{yt}^{2}&\sigma_{zt}^{2}&\sigma_{t}^{2}\end{bmatrix}=% \sigma_{R}^{2}\begin{bmatrix}d_{x}^{2}&d_{xy}^{2}&d_{xz}^{2}&d_{xt}^{2}\\ d_{xy}^{2}&d_{y}^{2}&d_{yz}^{2}&d_{yt}^{2}\\ d_{xz}^{2}&d_{yz}^{2}&d_{z}^{2}&d_{zt}^{2}\\ d_{xt}^{2}&d_{yt}^{2}&d_{zt}^{2}&d_{t}^{2}\end{bmatrix}\ (7)
  54. σ r c 2 = σ x 2 + σ y 2 + σ z 2 = σ R 2 ( d x 2 + d y 2 + d z 2 ) = P D O P 2 σ R 2 \sigma_{rc}^{2}=\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}=\sigma_{R}^{2}% \left(d_{x}^{2}+d_{y}^{2}+d_{z}^{2}\right)=PDOP^{2}\sigma_{R}^{2}
  55. σ t 2 = σ R 2 d t 2 = T D O P 2 σ R 2 \sigma_{t}^{2}=\sigma_{R}^{2}d_{t}^{2}=TDOP^{2}\sigma_{R}^{2}
  56. v 2 2 c 2 10 - 10 \frac{v^{2}}{2c^{2}}\approx 10^{-10}
  57. 1 γ = 1 - v 2 c 2 \frac{1}{\gamma}=\sqrt{1-\frac{v^{2}}{c^{2}}}
  58. 1 γ 1 - v 2 2 c 2 \frac{1}{\gamma}\approx 1-\frac{v^{2}}{2c^{2}}
  59. 1 γ 1 - 3874 2 2 ( 2.998 × 10 8 ) 2 1 - 8.349 × 10 - 11 \frac{1}{\gamma}\approx 1-\frac{3874^{2}}{2\left(2.998\times 10^{8}\right)^{2}% }\approx 1-8.349\times 10^{-11}
  60. - 8.349 × 10 - 11 × 60 × 60 × 24 × 10 9 - 7214 ns -8.349\times 10^{-11}\times 60\times 60\times 24\times 10^{9}\approx-7214\,% \text{ ns}
  61. 1 γ = 1 - 2 G M r c 2 \frac{1}{\gamma}=\sqrt{1-\frac{2GM}{rc^{2}}}
  62. 1 γ 1 - G M r c 2 \frac{1}{\gamma}\approx 1-\frac{GM}{rc^{2}}
  63. Δ ( 1 γ ) G M earth R earth c 2 - G M earth R gps c 2 \Delta\left(\frac{1}{\gamma}\right)\approx\frac{GM_{\,\text{earth}}}{R_{\,% \text{earth}}c^{2}}-\frac{GM_{\,\text{earth}}}{R_{\,\text{gps}}c^{2}}
  64. Δ ( 1 γ ) 5.307 × 10 - 10 \Delta\left(\frac{1}{\gamma}\right)\approx 5.307\times 10^{-10}
  65. 5.307 × 10 - 10 × 60 × 60 × 24 × 10 9 45850 ns 5.307\times 10^{-10}\times 60\times 60\times 24\times 10^{9}\approx 45850\,% \text{ ns}

Ershove_Number.html

  1. n = m a x ( c h i l d 1 , c h i l d 2 ) , i f c h i l d 1 c h i l d 2 n=max(child_{1},child_{2}),ifchild_{1}\neq child_{2}
  2. n = c h i l d 1 + 1 , i f c h i l d 1 = c h i l d 2 n=child_{1}+1,ifchild_{1}=child_{2}

Escaping_set.html

  1. z 0 z_{0}\in\mathbb{C}
  2. z n + 1 := f ( z n ) z_{n+1}:=f(z_{n})
  3. n n
  4. f f
  5. I ( f ) I(f)
  6. f ( z ) = e z f(z)=e^{z}
  7. 0 , 1 , e , e e , e e e , 0,1,e,e^{e},e^{e^{e}},\dots
  8. f ( z ) = z + 1 + exp ( - z ) f(z)=z+1+\exp(-z)
  9. f ( z ) = c sin ( z ) f(z)=c\sin(z)
  10. f ( z ) = a z + b f(z)=az+b
  11. f f
  12. I ( f ) { } I(f)\cup\{\infty\}
  13. I ( f ) I(f)
  14. f ( z ) = z 2 f(z)=z^{2}
  15. I ( f ) = { z : a b s ( z ) > 1 } I(f)=\{z:abs(z)>1\}

Espejos.html

  1. B ′′′ o n u s t r a c k ′′′ {}^{\prime\prime\prime}Bonustrack^{\prime\prime\prime}
  2. C l o t h Cloth

Esscher_transform.html

  1. f ( x ; h ) = e h x f ( x ) - e h x f ( x ) d x . f(x;h)=\frac{e^{hx}f(x)}{\int_{-\infty}^{\infty}e^{hx}f(x)dx}.\,
  2. e h x - e h x d μ ( x ) \frac{e^{hx}}{\int_{-\infty}^{\infty}e^{hx}d\mu(x)}
  3. E h ( 𝒩 ( μ , σ 2 ) ) = 𝒩 ( μ + h σ 2 , σ 2 ) . E_{h}(\mathcal{N}(\mu,\,\sigma^{2}))=\mathcal{N}(\mu+h\sigma^{2},\,\sigma^{2}).\,
  4. e h k p k ( 1 - p ) n - k 1 - p + p e h \,\frac{e^{hk}p^{k}(1-p)^{n-k}}{1-p+pe^{h}}
  5. ( n k ) e h k p k ( 1 - p ) n - k ( 1 - p + p e h ) n \,\frac{{n\choose k}e^{hk}p^{k}(1-p)^{n-k}}{(1-p+pe^{h})^{n}}
  6. 1 2 π σ 2 e - ( x - μ - σ 2 h ) 2 2 σ 2 \,\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu-\sigma^{2}h)^{2}}{2\sigma^{2% }}}
  7. e h k - λ e h λ k k ! \,\frac{e^{hk-\lambda e^{h}}\lambda^{k}}{k!}

Essentially_finite_vector_bundle.html

  1. X X
  2. k k
  3. x X ( k ) x\in X(k)
  4. V V
  5. X X
  6. k k
  7. G G
  8. G G
  9. p : P X p:P\to X
  10. V V
  11. P P
  12. p * ( V ) O P r p^{*}(V)\cong O_{P}^{\oplus r}
  13. r = r k ( V ) r=rk(V)

Eta_invariant.html

  1. η ( s ) = λ 0 sign ( λ ) | λ | s \eta(s)=\sum_{\lambda\neq 0}\frac{\operatorname{sign}(\lambda)}{|\lambda|^{s}}

Euler's_pump_and_turbine_equation.html

  1. M + M r = ρ . Q ( c 2 u . r 2 - c 1 u . r 1 ) M+Mr=\rho.Q(c_{2}u.r_{2}-c_{1}u.r_{1})
  2. c 2 u = c 2 c o s α 2 c_{2}u=c_{2}cos\alpha_{2}\,
  3. c 1 u = c 1 c o s α 1 . c_{1}u=c_{1}cos\alpha_{1}.\,
  4. c 1 c_{1}\,
  5. c 2 c_{2}\,
  6. w 1 w_{1}\,
  7. w 2 w_{2}\,
  8. u 1 u_{1}\,
  9. u 2 u_{2}\,
  10. ω \omega
  11. Y t h . g = H t = c 2 u . u 2 - c 1 u . u 1 Yth.g=H_{t}=c_{2}u.u_{2}-c_{1}u.u_{1}
  12. Y t h = 1 / 2 ( u 2 2 - u 1 2 + w 1 2 - w 2 2 + c 2 2 - c 1 2 ) Yth=1/2(u_{2}^{2}-u_{1}^{2}+w_{1}^{2}-w_{2}^{2}+c_{2}^{2}-c_{1}^{2})
  13. H = 1 2 g ( V 1 2 - V 2 2 ) . H={1\over{2g}}(V_{1}^{2}-V_{2}^{2}).\,

Euler_tour_technique.html

  1. succ ( u , v ) = { next ( v , u ) next ( v , u ) nil first ( v ) otherwise . \mathrm{succ}(u,v)=\begin{cases}\mathrm{next}(v,u)&\mathrm{next}(v,u)\neq% \mathrm{nil}\\ \mathrm{first}(v)&\,\text{otherwise}.\end{cases}

Euler–Heisenberg_Lagrangian.html

  1. = - - 1 8 π 2 0 d s s 3 exp ( - m 2 s ) [ ( e s ) 2 Re cosh ( e s 2 ( + i 𝒢 ) ) Im cosh ( e s 2 ( + i 𝒢 ) ) 𝒢 - 2 3 ( e s ) 2 - 1 ] \mathcal{L}=-\mathcal{F}-\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}% \exp\left(-m^{2}s\right)\left[(es)^{2}\frac{\operatorname{Re}\cosh\left(es% \sqrt{2\left(\mathcal{F}+i\mathcal{G}\right)}\right)}{\operatorname{Im}\cosh% \left(es\sqrt{2\left(\mathcal{F}+i\mathcal{G}\right)}\right)}\mathcal{G}-\frac% {2}{3}(es)^{2}\mathcal{F}-1\right]
  2. = 1 2 ( 𝐁 2 - 𝐄 2 ) \mathcal{F}=\frac{1}{2}\left(\mathbf{B}^{2}-\mathbf{E}^{2}\right)
  3. 𝒢 = 𝐄 𝐁 \mathcal{G}=\mathbf{E}\cdot\mathbf{B}
  4. = 1 2 ( 𝐄 2 - 𝐁 2 ) + 2 α 2 45 m 4 [ ( 𝐄 2 - 𝐁 2 ) 2 + 7 ( 𝐄 𝐁 ) 2 ] \mathcal{L}=\frac{1}{2}\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)+\frac{2% \alpha^{2}}{45m^{4}}\left[\left(\mathbf{E}^{2}-\mathbf{B}^{2}\right)^{2}+7% \left(\mathbf{E}\cdot\mathbf{B}\right)^{2}\right]

Euler–Rodrigues_formula.html

  1. a 2 + b 2 + c 2 + d 2 = 1. a^{2}+b^{2}+c^{2}+d^{2}=1.
  2. x \vec{x}
  3. x = ( a 2 + b 2 - c 2 - d 2 2 ( b c - a d ) 2 ( b d + a c ) 2 ( b c + a d ) a 2 + c 2 - b 2 - d 2 2 ( c d - a b ) 2 ( b d - a c ) 2 ( c d + a b ) a 2 + d 2 - b 2 - c 2 ) x . \vec{x}^{\prime}=\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\ 2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\ 2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}\vec{x}.
  4. a a
  5. [ u v e c , u 3 c 9 ] = ( b , c , d ) [u^{\prime}vec^{\prime},u^{\prime}\u{0}3c9^{\prime}]=(b,c,d)
  6. ( a 1 , b 1 , c 1 , d 1 ) (a_{1},b_{1},c_{1},d_{1})
  7. ( a 2 , b 2 , c 2 , d 2 ) (a_{2},b_{2},c_{2},d_{2})
  8. a \displaystyle a
  9. a 2 + b 2 + c 2 + d 2 = 1 a^{2}+b^{2}+c^{2}+d^{2}=1
  10. k = ( k x , k y , k z ) \vec{k}=(k_{x},k_{y},k_{z})
  11. ϕ \phi
  12. a \displaystyle a
  13. ϕ \phi
  14. ϕ = 0 \phi=0
  15. ( a , b , c , d ) = ( ± 1 , 0 , 0 , 0 ) (a,b,c,d)=(\pm 1,0,0,0)
  16. a = 0 a=0
  17. a a
  18. b , c , d b,c,d
  19. q = a + b i + c j + d k , q=a+bi+cj+dk,
  20. q 2 = a 2 + b 2 + c 2 + d 2 = 1. \left\|q\right\|^{2}=a^{2}+b^{2}+c^{2}+d^{2}=1.
  21. U = ( a + d i b + c i - b + c i a - d i ) . U=\begin{pmatrix}\ \ \,a+di&b+ci\\ -b+ci&a-di\end{pmatrix}.
  22. U = a ( 1 0 0 1 ) + b ( 0 1 - 1 0 ) + c ( 0 i i 0 ) + d ( i 0 0 - i ) = a I + i c σ x + i b σ y + i d σ z , \begin{aligned}\displaystyle U&\displaystyle=a\ \begin{pmatrix}1&0\\ 0&1\end{pmatrix}+b\ \begin{pmatrix}0&1\\ -1&0\end{pmatrix}+c\ \begin{pmatrix}0&i\\ i&0\end{pmatrix}+d\ \begin{pmatrix}i&0\\ 0&-i\end{pmatrix}\\ &\displaystyle=a\,I+ic\,\sigma_{x}+ib\,\sigma_{y}+id\,\sigma_{z},\end{aligned}
  23. σ i \sigma_{i}

European_Fiscal_Compact.html

  1. \scriptscriptstyle\leq
  2. \scriptscriptstyle\leq
  3. \scriptscriptstyle\leq
  4. \scriptscriptstyle\leq
  5. \scriptscriptstyle\nleq
  6. \scriptscriptstyle\nleq
  7. \scriptscriptstyle\nleq
  8. \scriptscriptstyle\nleq
  9. \scriptscriptstyle\nleq
  10. \scriptscriptstyle\nleq

Evans_balance.html

  1. 3 {}_{3}
  2. 2 {}_{2}
  3. 3 {}_{3}
  4. χ g = C L ( R s - R 0 ) m \chi_{g}=\frac{CL(R_{s}-R_{0})}{m}

Evenly_spaced_integer_topology.html

  1. a + k Z := { a + k λ : λ Z } . a+k{Z}:=\{a+k\lambda:\lambda\in{Z}\}.

Event_horizon.html

  1. d p d_{p}
  2. d p = 0 t 0 c a ( t ) d t . d_{p}=\int_{0}^{t_{0}}\frac{c}{a(t)}dt\ .
  3. d p d_{p}\rightarrow\infty
  4. d p d_{p}\neq\infty

Event_structure.html

  1. ( E , , # ) (E,\leq,\#)
  2. E E
  3. E E
  4. # \#
  5. e E e\in E
  6. [ e ] = { f E | f e } [e]=\{f\in E|f\leq e\}
  7. e e
  8. E E
  9. d , e , f E d,e,f\in E
  10. d e d\leq e
  11. d # f d\#f
  12. e # f e\#f

Evolving_networks.html

  1. P ( k ) k - γ P\left(k\right)\sim k^{-\gamma}
  2. p i = k i j k j , p_{i}=\frac{k_{i}}{\displaystyle\sum_{j}k_{j}},
  3. Π ( k i ) = η i k i j η j k j , \Pi(k_{i})=\frac{\eta_{i}k_{i}}{\displaystyle\sum_{j}\eta_{j}k_{j}},
  4. η \eta
  5. Π ( k i ) k i ( t - t i ) - ν , \Pi(k_{i})\propto k_{i}(t-t_{i})^{-\nu},
  6. γ \gamma
  7. ν \nu

Ex-tangential_quadrilateral.html

  1. a + b = c + d a+b=c+d
  2. a + d = b + c . a+d=b+c.
  3. | a - c | = | b - d | . |a-c|=|b-d|.
  4. A B + B C = A D + D C A E + E C = A F + F C . AB+BC=AD+DC\quad\Leftrightarrow\quad AE+EC=AF+FC.
  5. R 1 + R 3 = R 2 + R 4 R_{1}+R_{3}=R_{2}+R_{4}
  6. R 1 + R 2 = R 3 + R 4 R_{1}+R_{2}=R_{3}+R_{4}
  7. R 1 + R 4 = R 2 + R 3 R_{1}+R_{4}=R_{2}+R_{3}
  8. a g h + c e f = b e h + d f g agh+cef=beh+dfg
  9. a g h + b e h = c e f + d f g agh+beh=cef+dfg
  10. a g h + d f g = b e h + c e f agh+dfg=beh+cef
  11. 1 h 1 + 1 h 3 = 1 h 2 + 1 h 4 \frac{1}{h_{1}}+\frac{1}{h_{3}}=\frac{1}{h_{2}}+\frac{1}{h_{4}}
  12. 1 h 1 + 1 h 2 = 1 h 3 + 1 h 4 \frac{1}{h_{1}}+\frac{1}{h_{2}}=\frac{1}{h_{3}}+\frac{1}{h_{4}}
  13. 1 h 1 + 1 h 4 = 1 h 2 + 1 h 3 \frac{1}{h_{1}}+\frac{1}{h_{4}}=\frac{1}{h_{2}}+\frac{1}{h_{3}}
  14. tan x 2 tan z 2 = tan y 2 tan w 2 \tan{\frac{x}{2}}\tan{\frac{z}{2}}=\tan{\frac{y}{2}}\tan{\frac{w}{2}}
  15. tan x 2 tan w 2 = tan y 2 tan z 2 \tan{\frac{x}{2}}\tan{\frac{w}{2}}=\tan{\frac{y}{2}}\tan{\frac{z}{2}}
  16. tan x 2 tan y 2 = tan z 2 tan w 2 \tan{\frac{x}{2}}\tan{\frac{y}{2}}=\tan{\frac{z}{2}}\tan{\frac{w}{2}}
  17. R a R c = R b R d R_{a}R_{c}=R_{b}R_{d}
  18. R a R b = R c R d R_{a}R_{b}=R_{c}R_{d}
  19. R a R d = R b R c R_{a}R_{d}=R_{b}R_{c}
  20. K = a b c d sin B + D 2 . \displaystyle K=\sqrt{abcd}\sin{\frac{B+D}{2}}.
  21. r = K | a - c | = K | b - d | r=\frac{K}{|a-c|}=\frac{K}{|b-d|}
  22. K = a b c d \displaystyle K=\sqrt{abcd}
  23. 1 ( R - x ) 2 + 1 ( R + x ) 2 = 1 r 2 , \frac{1}{(R-x)^{2}}+\frac{1}{(R+x)^{2}}=\frac{1}{r^{2}},
  24. x = R 2 + r 2 + r 4 R 2 + r 2 . x=\sqrt{R^{2}+r^{2}+r\sqrt{4R^{2}+r^{2}}}.
  25. x > R + r , \displaystyle x>R+r,

Exact_solutions_of_classical_central-force_problems.html

  1. φ = φ 0 + L 2 m u d u E tot - V ( 1 / u ) - L 2 u 2 2 m \varphi=\varphi_{0}+\frac{L}{\sqrt{2m}}\int^{u}\frac{du}{\sqrt{E_{\mathrm{tot}% }-V(1/u)-\frac{L^{2}u^{2}}{2m}}}

Excess_molar_quantity.html

  1. z E = z - i x i z i i d . z^{E}=z-\sum_{i}x_{i}z^{id}_{i}.
  2. * *
  3. E E
  4. z z
  5. z = i x i Z i ¯ , z=\sum_{i}x_{i}\bar{Z_{i}},
  6. z E = i x i ( Z i ¯ - z i i d ) . z^{E}=\sum_{i}x_{i}(\bar{Z_{i}}-z_{i}^{id}).
  7. V E ¯ i = V ¯ i - V ¯ i i d \bar{V^{E}}_{i}=\bar{V}_{i}-\bar{V}^{id}_{i}
  8. H E ¯ i = H ¯ i - H ¯ i i d \bar{H^{E}}_{i}=\bar{H}_{i}-\bar{H}^{id}_{i}
  9. S E ¯ i = S ¯ i - S ¯ i i d \bar{S^{E}}_{i}=\bar{S}_{i}-\bar{S}^{id}_{i}
  10. G E ¯ i = G ¯ i - G ¯ i i d \bar{G^{E}}_{i}=\bar{G}_{i}-\bar{G}^{id}_{i}
  11. V = i V i + i V i E {V}=\sum_{i}V_{i}+\sum_{i}V_{i}^{E}
  12. V E ¯ i = R T ( l n ( γ i ) ) P \bar{V^{E}}_{i}=RT\frac{\partial(ln(\gamma_{i}))}{\partial P}
  13. V T = i V i T + i V i E T \frac{\partial V}{\partial T}=\sum_{i}\frac{\partial V_{i}}{\partial T}+\sum_{% i}\frac{\partial V_{i}^{E}}{\partial T}
  14. : α V V = i α V , i V i + i V i E T :\alpha_{V}V=\sum_{i}\alpha_{V,i}V_{i}+\sum_{i}\frac{\partial V_{i}^{E}}{% \partial T}
  15. V E ¯ i T = R ( l n ( γ i ) ) P + R T 2 T P l n ( γ i ) \frac{\partial\bar{V^{E}}_{i}}{\partial T}=R\frac{\partial(ln(\gamma_{i}))}{% \partial P}+RT{\partial^{2}\over\partial T\partial P}ln(\gamma_{i})

Existential_generalization.html

  1. Q ( a ) x Q ( x ) Q(a)\to\ \exists{x}\,Q(x)

Existential_instantiation.html

  1. ( x ) ϕ ( x ) (\exists x)\phi(x)
  2. ϕ ( c ) \phi(c)
  3. ( x ) x : : a , (\exists x)\mathcal{F}x::\mathcal{F}a,

Existential_theory_of_the_reals.html

  1. X 1 X n F ( X 1 , , X n ) , \exists X_{1}\cdots\exists X_{n}\,F(X_{1},\dots,X_{n}),\,
  2. F ( X 1 , X n ) F(X_{1},\dots X_{n})
  3. \exists\mathbb{R}
  4. X i X_{i}
  5. X 1 X n F ( X 1 , , X n ) , \exists X_{1}\cdots\exists X_{n}\,F(X_{1},\dots,X_{n}),\,
  6. F ( X 1 , X n ) F(X_{1},\dots X_{n})
  7. ( X 1 , X n ) (X_{1},\dots X_{n})
  8. F ( X 1 , X n ) F(X_{1},\dots X_{n})
  9. φ \varphi
  10. x 2 - x - 1 x^{2}-x-1
  11. X 1 X 1 > 1 X 1 × X 1 - X 1 - 1 = 0. \exists X_{1}\ X_{1}>1\wedge X_{1}\times X_{1}-X_{1}-1=0.
  12. x x
  13. y y
  14. x + y 2 x y . \frac{x+y}{2}\geq\sqrt{xy}.
  15. X 1 X 2 X 1 0 X 2 0 ( X 1 + X 2 ) × ( X 1 + X 2 ) < ( 1 + 1 + 1 + 1 ) × X 1 × X 2 . \exists X_{1}\exists X_{2}\ X_{1}\geq 0\wedge X_{2}\geq 0\wedge(X_{1}+X_{2})% \times(X_{1}+X_{2})<(1+1+1+1)\times X_{1}\times X_{2}.
  16. L 3 ( m d ) 2 O ( n ) L^{3}(md)^{2^{O(n)}}
  17. L L
  18. m m
  19. d d
  20. n n
  21. n n
  22. L ( m d ) n 2 L(md)^{n^{2}}
  23. n n
  24. L log L log log L ( m d ) O ( n ) . L\log L\log\log L(md)^{O(n)}.
  25. \exists\mathbb{R}

Expander_code.html

  1. [ n , n - m ] 2 [n,n-m]_{2}\,
  2. 2 ( 1 - ε ) γ 2(1-\varepsilon)\gamma\,
  3. ε \varepsilon\,
  4. γ \gamma\,
  5. ( 1 - m n ) \left(1-\tfrac{m}{n}\right)\,
  6. O ( n ) O(n)\,
  7. G ( L , R , E ) G(L,R,E)\,
  8. L L\,
  9. R R\,
  10. E E\,
  11. L L\,
  12. R R\,
  13. L L\,
  14. d d\,
  15. d d\,
  16. | L | = n |L|=n\,
  17. | R | = m |R|=m\,
  18. m < n m<n\,
  19. G G\,
  20. ( N , M , d , γ , α ) (N,M,d,\gamma,\alpha)\,
  21. S L S\subset L\,
  22. | S | γ n |S|\leq\gamma n\,
  23. S S\,
  24. d α | S | d\alpha|S|\,
  25. R R\,
  26. γ 1 n \gamma\leq\tfrac{1}{n}\,
  27. 1 n < γ 1 \tfrac{1}{n}<\gamma\leq 1\,
  28. α = 1 - ε \alpha=1-\varepsilon\,
  29. ε \varepsilon\,
  30. G G\,
  31. G G\,
  32. n × m n\times m\,
  33. C C\,
  34. C C\,
  35. m × n m\times n\,
  36. C C\,
  37. ( n - m ) / n = 1 - m n (n-m)/n=1-\tfrac{m}{n}\,
  38. ε < 1 2 \varepsilon<\tfrac{1}{2}\,
  39. ( n , m , d , γ , 1 - ε ) (n,m,d,\gamma,1-\varepsilon)\,
  40. C C\,
  41. 2 ( 1 - ε ) γ n 2(1-\varepsilon)\gamma n\,
  42. c c\,
  43. C C\,
  44. S L S\subset L\,
  45. v i S v_{i}\in S\,
  46. i i\,
  47. c c\,
  48. v R v\in R\,
  49. S S\,
  50. c P = 0 cP=0\,
  51. P P\,
  52. R R\,
  53. P P\,
  54. GF ( 2 ) = { 0 , 1 } \,\text{GF}(2)=\{0,1\}\,
  55. v R v\in R\,
  56. S S\,
  57. c c\,
  58. N ( S ) N(S)\,
  59. R R\,
  60. S S\,
  61. U ( S ) U(S)\,
  62. S S\,
  63. S S\,
  64. S L S\subset L\,
  65. | S | γ n |S|\leq\gamma n\,
  66. d | S | | N ( S ) | | U ( S ) | d ( 1 - 2 ε ) | S | d|S|\geq|N(S)|\geq|U(S)|\geq d(1-2\varepsilon)|S|\,
  67. | N ( S ) | | U ( S ) | |N(S)|\geq|U(S)|\,
  68. v U ( S ) v\in U(S)\,
  69. v N ( S ) v\in N(S)\,
  70. | N ( S ) | d | S | |N(S)|\leq d|S|\,
  71. S S\,
  72. d d\,
  73. d ( 1 - ε ) | S | d(1-\varepsilon)|S|\,
  74. d ε | S | d\varepsilon|S|\,
  75. d ε | S | d\varepsilon|S|\,
  76. U ( S ) d ( 1 - ε ) | S | - d ε | S | = d ( 1 - 2 ε ) | S | U(S)\geq d(1-\varepsilon)|S|-d\varepsilon|S|=d(1-2\varepsilon)|S|\,
  77. S S\,
  78. ε < 1 2 \varepsilon<\tfrac{1}{2}\,
  79. T L T\subset L\,
  80. | T | < 2 ( 1 - ε ) γ n |T|<2(1-\varepsilon)\gamma n\,
  81. | T | γ n |T|\leq\gamma n\,
  82. 2 ( 1 - ε ) γ n > | T | > γ n 2(1-\varepsilon)\gamma n>|T|>\gamma n\,
  83. S T S\subset T\,
  84. | S | = γ n |S|=\gamma n\,
  85. | U ( S ) | d ( 1 - 2 ε ) | S | |U(S)|\geq d(1-2\varepsilon)|S|\,
  86. v U ( S ) v\in U(S)\,
  87. U ( T ) U(T)\,
  88. v N ( T S ) v\notin N(T\setminus S)\,
  89. | T S | 2 ( 1 - ε ) γ n - γ n = ( 1 - 2 ε ) γ n |T\setminus S|\leq 2(1-\varepsilon)\gamma n-\gamma n=(1-2\varepsilon)\gamma n\,
  90. | N ( T S ) | d ( 1 - 2 ε ) γ n |N(T\setminus S)|\leq d(1-2\varepsilon)\gamma n\,
  91. ε < 1 2 \varepsilon<\tfrac{1}{2}\,
  92. | U ( T ) | | U ( S ) N ( T S ) | | U ( S ) | - | N ( T S ) | > 0 |U(T)|\geq|U(S)\setminus N(T\setminus S)|\geq|U(S)|-|N(T\setminus S)|>0\,
  93. U ( T ) U(T)\,
  94. T L T\subset L\,
  95. | U ( T ) | > 0 |U(T)|>0\,
  96. c c\,
  97. T T\,
  98. c C w t ( c ) 2 ( 1 - ε ) γ n c\in C\implies wt(c)\geq 2(1-\varepsilon)\gamma n\,
  99. C C\,
  100. C C\,
  101. 2 ( 1 - ε ) γ n 2(1-\varepsilon)\gamma n\,
  102. O ( n 2 ) O(n^{2})\,
  103. O ( n ) O(n)\,
  104. O ( n ) O(n)\,
  105. ε < 1 4 \varepsilon<\tfrac{1}{4}\,
  106. v i v_{i}\,
  107. L L\,
  108. i i\,
  109. C C\,
  110. y { 0 , 1 } n y\in\{0,1\}^{n}\,
  111. V ( y ) = { v i | the i th position of y is a 1 } V(y)=\{v_{i}|\,\text{ the }i^{\,\text{th}}\,\text{ position of }y\,\text{ is a% }1\}\,
  112. e ( i ) e(i)\,
  113. | { v R | N ( v ) V ( y ) |\{v\in R|N(v)\cap V(y)\,
  114. } | \}|\,
  115. o ( i ) o(i)\,
  116. | { v R | N ( v ) V ( y ) |\{v\in R|N(v)\cap V(y)\,
  117. } | \}|\,
  118. y y\,
  119. y y^{\prime}\,
  120. S S\,
  121. s = | S | s=|S|\,
  122. R R\,
  123. c c\,
  124. 0 < s < γ n 0<s<\gamma n\,
  125. v i v_{i}\,
  126. o ( i ) > e ( i ) o(i)>e(i)\,
  127. U ( S ) d ( 1 - 2 ε ) s U(S)\geq d(1-2\varepsilon)s\,
  128. d ( 1 - 2 ε ) > d / 2 d(1-2\varepsilon)>d/2\,
  129. o ( i ) o(i)\,
  130. ε < 1 4 \varepsilon<\tfrac{1}{4}\,
  131. v i v_{i}\,
  132. o ( i ) > e ( i ) o(i)>e(i)\,
  133. γ n \gamma n\,
  134. s < γ ( 1 - 2 ε ) n s<\gamma(1-2\varepsilon)n\,
  135. s = γ n s=\gamma n\,
  136. v i v_{i}\,
  137. o ( i ) o(i)\,
  138. e ( i ) e(i)\,
  139. o ( i ) > e ( i ) o(i)>e(i)\,
  140. s < γ ( 1 - 2 ε ) n s<\gamma(1-2\varepsilon)n\,
  141. d γ ( 1 - 2 ε ) n d\gamma(1-2\varepsilon)n\,
  142. d d\,
  143. γ n \gamma n\,
  144. d γ ( 1 - 2 ε ) n d\gamma(1-2\varepsilon)n\,
  145. d γ ( 1 - 2 ε ) n d\gamma(1-2\varepsilon)n\,
  146. s < γ ( 1 - 2 ε ) n s<\gamma(1-2\varepsilon)n\,
  147. C C\,
  148. v i v_{i}\,
  149. R R\,
  150. m m\,
  151. γ n \gamma n\,
  152. 2 ( 1 - ε ) γ n > γ n 2(1-\varepsilon)\gamma n>\gamma n\,
  153. ε < 1 2 \varepsilon<\tfrac{1}{2}\,
  154. n m \tfrac{n}{m}\,
  155. r r\,
  156. R R\,
  157. r r\,
  158. O ( m r ) O(mr)\,
  159. R R\,
  160. O ( d n ) = O ( d m r ) O(dn)=O(dmr)\,
  161. v i v_{i}\,
  162. L L\,
  163. o ( i ) > e ( i ) o(i)>e(i)\,
  164. R R\,
  165. O ( d ) O(d)\,
  166. O ( d r ) O(dr)\,
  167. L L\,
  168. O ( d r ) O(dr)\,
  169. m m\,
  170. O ( m d r ) = O ( n ) O(mdr)=O(n)\,
  171. d d\,
  172. r r\,