wpmath0000009_2

Compressed_sensing.html

  1. L 1 L^{1}
  2. L 1 L^{1}
  3. L 1 L^{1}
  4. L 1 L^{1}
  5. L 2 L^{2}
  6. L 0 L^{0}
  7. L 1 L^{1}
  8. L 0 L^{0}
  9. L 1 L^{1}
  10. L 0 L^{0}
  11. L 1 L^{1}
  12. l 1 l1
  13. l 1 l_{1}
  14. l 1 l_{1}
  15. l 1 l_{1}
  16. l 1 l_{1}
  17. l 1 l_{1}
  18. l 0 l_{0}
  19. f k - 1 = 0 f^{k-1}=0
  20. f k - 1 < 0 f^{k-1}<0
  21. f f
  22. σ \sigma
  23. σ \sigma
  24. σ \sigma
  25. I I
  26. d ^ \hat{d}
  27. J ρ ( I σ ) = G ρ * ( I σ I σ ) = ( J 11 J 12 J 12 J 22 ) J_{\rho}(\nabla I_{\sigma})=G_{\rho}*(\nabla I_{\sigma}\otimes\nabla I_{\sigma% })=\begin{pmatrix}J_{11}&J_{12}\\ J_{12}&J_{22}\end{pmatrix}
  28. J ρ J_{\rho}
  29. ρ \rho
  30. G G
  31. ( 0 , ρ 2 ) (0,\rho^{2})
  32. ρ \rho
  33. σ \sigma
  34. I I
  35. I σ \nabla I_{\sigma}
  36. I I
  37. ( I σ I σ ) (\nabla I_{\sigma}\otimes\nabla I_{\sigma})
  38. G G
  39. σ \sigma
  40. G G
  41. J J
  42. d ^ \hat{d}
  43. d ^ \hat{d}
  44. m i n \Chi \Chi d 1 + λ 2 Y - Φ \Chi 2 2 min_{\Chi}\lVert\nabla\Chi\bullet d\rVert_{1}+\frac{\lambda}{2}\ \lVert Y-\Phi% \Chi\rVert^{2}_{2}
  45. \Chi \Chi
  46. Φ \Phi
  47. d ^ \hat{d}
  48. X k - 1 X^{k-1}
  49. \Chi \Chi
  50. d d
  51. \Chi d \Chi\bullet d
  52. \Chi \Chi
  53. d d
  54. ( d h , d v ) (d_{h},d_{v})
  55. d h , d v d_{h},d_{v}
  56. d d
  57. m i n \Chi \Chi d 1 + λ 2 Y - Φ \Chi 2 2 min_{\Chi}\lVert\nabla\Chi\bullet d\rVert_{1}+\frac{\lambda}{2}\ \lVert Y-\Phi% \Chi\rVert^{2}_{2}
  58. d h , d v , H , V d_{h},d_{v},H,V
  59. L 1 L_{1}
  60. \Chi , P , Q , λ P , λ Q \Chi,P,Q,\lambda_{P},\lambda_{Q}
  61. H , V , P , Q H,V,P,Q
  62. H H
  63. d h \nabla d_{h}
  64. V V
  65. d v \nabla d_{v}
  66. P P
  67. \Chi \nabla\Chi
  68. Q Q
  69. P d P\bullet d
  70. λ H , λ V , λ P , λ Q \lambda_{H},\lambda_{V},\lambda_{P},\lambda_{Q}
  71. H , V , P , Q H,V,P,Q
  72. L 2 L_{2}
  73. \Chi , P , Q \Chi,P,Q
  74. ( λ H ) k = ( λ H ) k - 1 + γ H ( H k - ( d h ) k ) (\lambda_{H})^{k}=(\lambda_{H})^{k-1}+\gamma_{H}(H^{k}-\nabla(d_{h})^{k})
  75. ( λ V ) k = ( λ V ) k - 1 + γ V ( V k - ( d v ) k ) (\lambda_{V})^{k}=(\lambda_{V})^{k-1}+\gamma_{V}(V^{k}-\nabla(d_{v})^{k})
  76. ( λ P ) k = ( λ P ) k - 1 + γ P ( P k - ( \Chi ) k ) (\lambda_{P})^{k}=(\lambda_{P})^{k-1}+\gamma_{P}(P^{k}-\nabla(\Chi)^{k})
  77. ( λ Q ) k = ( λ Q ) k - 1 + γ Q ( Q k - P k d ) (\lambda_{Q})^{k}=(\lambda_{Q})^{k-1}+\gamma_{Q}(Q^{k}-P^{k}\bullet d)
  78. γ H , γ V , γ P , γ Q \gamma_{H},\gamma_{V},\gamma_{P},\gamma_{Q}
  79. l 1 l_{1}

Compressibility_equation.html

  1. k T ( ρ p ) = 1 + ρ V d 𝐫 [ g ( r ) - 1 ] kT\left(\frac{\partial\rho}{\partial p}\right)=1+\rho\int_{V}\mathrm{d}\mathbf% {r}[g(r)-1]
  2. ρ \rho
  3. k T ( ρ p ) kT\left(\frac{\partial\rho}{\partial p}\right)
  4. 1 k T ( p ρ ) = 1 1 + ρ h ( r ) d 𝐫 = 1 1 + ρ H ^ ( 0 ) = 1 - ρ C ^ ( 0 ) = 1 - ρ c ( r ) d 𝐫 \frac{1}{kT}\left(\frac{\partial p}{\partial\rho}\right)=\frac{1}{1+\rho\int h% (r)\mathrm{d}\mathbf{r}}=\frac{1}{1+\rho\hat{H}(0)}=1-\rho\hat{C}(0)=1-\rho% \int c(r)\mathrm{d}\mathbf{r}

Computational_magnetohydrodynamics.html

  1. 𝐁 = 0 \nabla\cdot{\mathbf{B}}=0

Computer_performance.html

  1. t = N * C / f t=N*C/f
  2. P = I * f / N P=I*f/N
  3. 1 / I 1/I
  4. 1 / C 1/C

Concatenated_error_correction_code.html

  1. R R
  2. C C
  3. N N
  4. N N
  5. C i n : A k A n C_{in}:A^{k}\rightarrow A^{n}
  6. C o u t : B K B N C_{out}:B^{K}\rightarrow B^{N}
  7. C o u t C i n : A k K A n N C_{out}\circ C_{in}:A^{kK}\rightarrow A^{nN}
  8. Δ ( C o u t ( m 1 ) , C o u t ( m 2 ) ) D . \Delta(C_{out}(m^{1}),C_{out}(m^{2}))\geq D.
  9. Δ ( C i n ( C o u t ( m 1 ) i ) , C i n ( C o u t ( m 2 ) i ) ) d . \Delta(C_{in}(C_{out}(m^{1})_{i}),C_{in}(C_{out}(m^{2})_{i}))\geq d.
  10. Δ ( C i n ( C o u t ( m 1 ) ) , C i n ( C o u t ( m 2 ) ) ) d D . \Delta(C_{in}(C_{out}(m^{1})),C_{in}(C_{out}(m^{2})))\geq dD.

Concatenation_(mathematics).html

  1. p | | q = p b l ( q ) + q p||q=pb^{l(q)}+q
  2. l ( q ) = l o g b ( q ) + 1 l(q)=\lfloor log_{b}(q)\rfloor+1
  3. x \lfloor x\rfloor
  4. n \mathbb{R}^{n}
  5. ( a 1 a 2 a n ) | | ( b 1 b 2 b n ) = ( a 1 | | b 1 a 2 | | b 2 a n | | b n ) \left(\begin{array}[]{c}a_{1}\\ a_{2}\\ \vdots\\ a_{n}\end{array}\right)||\left(\begin{array}[]{c}b_{1}\\ b_{2}\\ \vdots\\ b_{n}\end{array}\right)=\left(\begin{array}[]{c}a_{1}||b_{1}\\ a_{2}||b_{2}\\ \vdots\\ a_{n}||b_{n}\end{array}\right)
  6. 1 \mathbb{R}^{1}

Concordance_correlation_coefficient.html

  1. ρ c \rho_{c}
  2. ρ c = 2 ρ σ x σ y σ x 2 + σ y 2 + ( μ x - μ y ) 2 , \rho_{c}=\frac{2\rho\sigma_{x}\sigma_{y}}{\sigma_{x}^{2}+\sigma_{y}^{2}+(\mu_{% x}-\mu_{y})^{2}},
  3. μ x \mu_{x}
  4. μ y \mu_{y}
  5. σ x 2 \sigma^{2}_{x}
  6. σ y 2 \sigma^{2}_{y}
  7. ρ \rho
  8. ρ c = 1 - Expected orthogonal squared distance from the diagonal x = y Expected orthogonal squared distance from the diagonal x = y assuming independence . \rho_{c}=1-\frac{{\rm Expected\ orthogonal\ squared\ distance\ from\ the\ % diagonal\ }x=y}{{\rm Expected\ orthogonal\ squared\ distance\ from\ the\ % diagonal\ }x=y{\rm\ assuming\ independence}}.
  9. ρ ^ c = 2 s x y s x 2 + s y 2 + ( x ¯ - y ¯ ) 2 , \hat{\rho}_{c}=\frac{2s_{xy}}{s_{x}^{2}+s_{y}^{2}+(\bar{x}-\bar{y})^{2}},
  10. x ¯ = 1 N n = 1 N x n \bar{x}=\frac{1}{N}\sum_{n=1}^{N}x_{n}
  11. s x 2 = 1 N n = 1 N ( x n - x ¯ ) 2 s_{x}^{2}=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-\bar{x})^{2}
  12. s x y = 1 N n = 1 N ( x n - x ¯ ) ( y n - y ¯ ) . s_{xy}=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-\bar{x})(y_{n}-\bar{y}).

Conditional_variance.html

  1. Var ( Y | X = x ) = E ( ( Y - E ( Y X = x ) ) 2 X = x ) , \operatorname{Var}(Y|X=x)=\operatorname{E}((Y-\operatorname{E}(Y\mid X=x))^{2}% \mid X=x),
  2. Var Y X ( Y | x ) . \operatorname{Var}_{Y\mid X}(Y|x).
  3. Var ( Y ) = E X ( Var ( Y X ) ) + Var X ( E ( Y X ) ) , \operatorname{Var}(Y)=\operatorname{E}_{X}(\operatorname{Var}(Y\mid X))+% \operatorname{Var}_{X}(\operatorname{E}(Y\mid X)),
  4. Var ( Y | X ) \operatorname{Var}(Y|X)

Congruence_lattice_problem.html

  1. A ( B + C ) ( A B ) + ( A C ) A\cap(B+C)\neq(A\cap B)+(A\cap C)
  2. S = ( S i i I ) S=\bigcup(S_{i}\mid i\in I)
  3. 𝒮 \mathcal{S}
  4. ( Con c L i , Con c f i j i j in I ) (\mathrm{Con_{c}}\,L_{i},\mathrm{Con_{c}}\,f_{i}^{j}\mid i\leq j\,\text{ in }I)
  5. ( L i , f i j i j in I ) (L_{i},f_{i}^{j}\mid i\leq j\,\text{ in }I)
  6. 𝒮 \mathcal{S}
  7. L = lim i I L i L=\underrightarrow{\lim}_{i\in I}L_{i}
  8. Con c L S {\rm Con_{c}}\,L\cong S
  9. 𝒱 \mathcal{V}
  10. A 𝒱 A\in\mathcal{V}
  11. D ( S ) = ( R n ( S ) n < ω ) D(S)=\bigcup(R^{n}(S)\mid n<\omega)
  12. ( a i ) i I (a_{i})_{i\in I}
  13. ( b i ) i I (b_{i})_{i\in I}
  14. ( c i , j ( i , j ) I × I ) (c_{i,j}\mid(i,j)\in I\times I)
  15. U V = { u v ( u , v ) U × V } , for all U , V L , U\vee V=\{u\vee v\mid(u,v)\in U\times V\},\quad\,\text{for all }U,V\subseteq L,
  16. Z = { z 0 , z 1 , , z n } Z=\{z_{0},z_{1},\dots,z_{n}\}
  17. i < n z i z n \bigvee_{i<n}z_{i}\leq z_{n}
  18. α j = ( Θ L ( z i , z i + 1 ) i < n , ε ( i ) = j ) , for all j < 2. \alpha_{j}=\bigvee(\Theta_{L}(z_{i},z_{i+1})\mid i<n,\ \varepsilon(i)=j),\,% \text{ for all }j<2.
  19. θ j Con c { x j } Z L \theta_{j}\in\mathrm{Con_{c}}^{\{x_{j}\}\vee Z}L

Conic_optimization.html

  1. f : C f:C\to\mathbb{R}
  2. C X C\subset X
  3. \mathcal{H}
  4. h i ( x ) = 0 h_{i}(x)=0
  5. x x
  6. C C\cap\mathcal{H}
  7. f ( x ) f(x)
  8. C C
  9. + n = { x n : x 𝟎 } \mathbb{R}_{+}^{n}=\left\{x\in\mathbb{R}^{n}:\,x\geq\mathbf{0}\right\}
  10. 𝕊 + n \mathbb{S}^{n}_{+}
  11. { ( x , t ) n × : x t } \left\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}:\lVert x\rVert\leq t\right\}
  12. f f
  13. c T x c^{T}x
  14. A x = b , x C Ax=b,x\in C
  15. b T y b^{T}y
  16. A T y + s = c , s C * A^{T}y+s=c,s\in C^{*}
  17. C * C^{*}
  18. C C
  19. c T x c^{T}x
  20. x 1 F 1 + + x n F n + G 0 x_{1}F_{1}+\cdots+x_{n}F_{n}+G\leq 0
  21. tr ( G Z ) \mathrm{tr}\ (GZ)
  22. tr ( F i Z ) + c i = 0 , i = 1 , , n \mathrm{tr}\ (F_{i}Z)+c_{i}=0,\quad i=1,\dots,n
  23. Z 0 Z\geq 0

Consensus_theorem.html

  1. x y x ¯ z y z xy\vee\bar{x}z\vee yz
  2. x y x ¯ z xy\vee\bar{x}z
  3. x y x ¯ z y z = x y x ¯ z xy\vee\bar{x}z\vee yz=xy\vee\bar{x}z
  4. x y xy
  5. x ¯ z \bar{x}z
  6. y z yz
  7. ( x y ) ( x ¯ z ) ( y z ) = ( x y ) ( x ¯ z ) (x\vee y)(\bar{x}\vee z)(y\vee z)=(x\vee y)(\bar{x}\vee z)
  8. x y x ¯ z ( x x ¯ ) y z xy\vee\bar{x}z\vee(x\vee\bar{x})yz
  9. x y x ¯ z x y z x ¯ y z xy\vee\bar{x}z\vee xyz\vee\bar{x}yz
  10. x y x y z x ¯ z x ¯ y z xy\vee xyz\vee\bar{x}z\vee\bar{x}yz
  11. x y ( 1 z ) x ¯ z ( 1 y ) xy(1\vee z)\vee\bar{x}z(1\vee y)
  12. x y x ¯ z xy\vee\bar{x}z
  13. a a
  14. a ¯ \bar{a}
  15. a a
  16. a ¯ \bar{a}
  17. x ¯ y z \bar{x}yz
  18. w y ¯ z w\bar{y}z
  19. w x ¯ z w\bar{x}z
  20. ( x y ) (x\vee y)
  21. ( x ¯ z ) (\bar{x}\vee z)

Conservation_form.html

  1. d ξ d t + f ( ξ ) = 0 \frac{d\xi}{dt}+\nabla\cdot f(\xi)=0
  2. ξ \xi
  3. f f
  4. d d t V ξ d V + V f ( ξ ) ν d S \frac{d}{dt}\int_{V}\xi dV+\int_{\partial V}f(\xi)\cdot\nu~{}dS
  5. ξ \xi
  6. V V
  7. f ( ξ ) f(\xi)
  8. ν \nu
  9. ξ \xi
  10. V V
  11. f f
  12. f ( ξ ) = ξ u f(\xi)=\xi u
  13. u u
  14. ξ \xi
  15. ρ t + ( ρ u ) = 0 {\partial\rho\over\partial t}+\nabla\cdot(\rho u)=0
  16. ρ u t + ( ρ u u + p I ) = 0 {\partial\rho{u}\over\partial t}+\nabla\cdot(\rho u\otimes u+pI)=0
  17. E t + ( u ( E + p ) ) = 0 {\partial E\over\partial t}+\nabla\cdot(u(E+p))=0

Consistent_heuristic.html

  1. h ( N ) c ( N , P ) + h ( P ) h(N)\leq c(N,P)+h(P)
  2. h ( G ) = 0. h(G)=0.\,
  3. m m
  4. h ( N m ) h * ( N m ) h(N_{m})\leq h^{*}(N_{m})
  5. h * ( n ) h^{*}(n)
  6. h ( N m + 1 ) c ( N m + 1 , N m ) + h ( N m ) c ( N m + 1 , N m ) + h * ( N m ) = h * ( N m + 1 ) h(N_{m+1})\leq c(N_{m+1},N_{m})+h(N_{m})\leq c(N_{m+1},N_{m})+h^{*}(N_{m})=h^{% *}(N_{m+1})
  7. N m + 1 N_{m+1}
  8. N m N_{m}
  9. h h
  10. h h^{\prime}
  11. h ( P ) max ( h ( P ) , h ( N ) - c ( N , P ) ) h^{\prime}(P)\leftarrow\max(h(P),h(N)-c(N,P))
  12. f ( N j ) = g ( N j ) + h ( N j ) f(N_{j})=g(N_{j})+h(N_{j})
  13. g ( N j ) = i = 2 j c ( N i - 1 , N i ) g(N_{j})=\sum_{i=2}^{j}c(N_{i-1},N_{i})
  14. N 1 N_{1}
  15. N j N_{j}
  16. c ( N , P ) = c ( N , P ) + h ( P ) - h ( N ) c^{\prime}(N,P)=c(N,P)+h(P)-h(N)
  17. h h

Constant_phase_element.html

  1. Z C P E = 1 Y C P E = 1 Q 0 ω n e - π 2 n i Z_{CPE}=\frac{1}{Y_{CPE}}=\frac{1}{Q_{0}\omega^{n}}e^{-\frac{\pi}{2}ni}
  2. Y C P E = Q 0 ( ω i ) n Y_{CPE}=Q_{0}(\omega i)^{n}

Constant_Q_transform.html

  1. δ f k = 2 1 n * δ f k - 1 = ( 2 1 n ) k * δ f min \begin{aligned}\displaystyle\delta f_{k}&\displaystyle=2^{\frac{1}{n}}*\delta f% _{k-1}\\ &\displaystyle=\left({2^{\frac{1}{n}}}\right)^{k}*\delta f_{\mathrm{min}}\end{aligned}
  2. X [ k , m ] = n = 0 N - 1 W [ n - m ] x [ n ] e - j 2 π k n N X[k,m]=\sum_{n=0}^{N-1}W[n-m]x[n]e^{\frac{-j2\pi kn}{N}}
  3. Q = f k δ f k Q=\frac{f_{k}}{\delta f_{k}}
  4. N [ k ] = ( f s δ f k ) = ( S f k ) Q N[k]=\left(\frac{f_{s}}{\delta f_{k}}\right)=\left(\frac{S}{f_{k}}\right)Q
  5. N = N [ k ] = Q f s f k N=N[k]=Q\frac{f_{s}}{f_{k}}
  6. W [ k , n ] = α - ( 1 - α ) cos ( 2 π n N [ k ] ) , α = 25 / 46 , 0 n N [ k ] - 1 W[k,n]=\alpha-\left(1-\alpha\right)\cos\left(\frac{2\pi n}{N[k]}\right),\alpha% =25/46,0\leqslant n\leqslant N[k]-1
  7. 2 π k N \frac{2\pi k}{N}
  8. 2 π Q N [ k ] \frac{2\pi Q}{N[k]}
  9. X [ k ] = 1 N [ k ] n = 0 N [ k ] - 1 W [ k , n ] x [ n ] e - j 2 π Q n N [ k ] X[k]=\frac{1}{N[k]}\sum_{n=0}^{N[k]-1}W[k,n]x[n]e^{\frac{-j2\pi Qn}{N[k]}}

Constraint_algorithm.html

  1. 𝐌 d 2 𝐪 d t 2 = 𝐟 = - V 𝐪 \mathbf{M}\cdot\frac{d^{2}\mathbf{q}}{dt^{2}}=\mathbf{f}=-\frac{\partial V}{% \partial\mathbf{q}}
  2. g j ( 𝐪 ) = 0 g_{j}(\mathbf{q})=0
  3. σ k ( t ) := 𝐱 k α ( t ) - 𝐱 k β ( t ) 2 - d k 2 = 0 , k = 1 n \sigma_{k}(t):=\|\mathbf{x}_{k\alpha}(t)-\mathbf{x}_{k\beta}(t)\|^{2}-d_{k}^{2% }=0,\quad k=1\dots n
  4. 𝐱 k α ( t ) \scriptstyle\mathbf{x}_{k\alpha}(t)
  5. 𝐱 k β ( t ) \scriptstyle\mathbf{x}_{k\beta}(t)
  6. d k d_{k}
  7. 2 𝐱 i ( t ) t 2 m i = - 𝐱 i [ V ( 𝐱 i ( t ) ) + k = 1 n λ k σ k ( t ) ] , i = 1 N . \frac{\partial^{2}\mathbf{x}_{i}(t)}{\partial t^{2}}m_{i}=-\frac{\partial}{% \partial\mathbf{x}_{i}}\left[V(\mathbf{x}_{i}(t))+\sum_{k=1}^{n}\lambda_{k}% \sigma_{k}(t)\right],\quad i=1\dots N.
  8. t + Δ t t+\Delta t
  9. 𝐱 i ( t + Δ t ) = 𝐱 ^ i ( t + Δ t ) + k = 1 n λ k σ k ( t ) 𝐱 i ( Δ t ) 2 m i - 1 , i = 1 N \mathbf{x}_{i}(t+\Delta t)=\hat{\mathbf{x}}_{i}(t+\Delta t)+\sum_{k=1}^{n}% \lambda_{k}\frac{\partial\sigma_{k}(t)}{\partial\mathbf{x}_{i}}\left(\Delta t% \right)^{2}m_{i}^{-1},\quad i=1\dots N
  10. 𝐱 ^ i ( t + Δ t ) \hat{\mathbf{x}}_{i}(t+\Delta t)
  11. σ k ( t + Δ t ) \sigma_{k}(t+\Delta t)
  12. σ k ( t + Δ t ) := 𝐱 k α ( t + Δ t ) - 𝐱 k β ( t + Δ t ) 2 - d k 2 = 0. \sigma_{k}(t+\Delta t):=\left\|\mathbf{x}_{k\alpha}(t+\Delta t)-\mathbf{x}_{k% \beta}(t+\Delta t)\right\|^{2}-d_{k}^{2}=0.
  13. n n
  14. σ j ( t + Δ t ) := 𝐱 ^ j α ( t + Δ t ) - 𝐱 ^ j β ( t + Δ t ) + k = 1 n λ k ( Δ t ) 2 [ σ k ( t ) 𝐱 j α m j α - 1 - σ k ( t ) 𝐱 j β m j β - 1 ] 2 - d j 2 = 0 , j = 1 n \sigma_{j}(t+\Delta t):=\left\|\hat{\mathbf{x}}_{j\alpha}(t+\Delta t)-\hat{% \mathbf{x}}_{j\beta}(t+\Delta t)+\sum_{k=1}^{n}\lambda_{k}\left(\Delta t\right% )^{2}\left[\frac{\partial\sigma_{k}(t)}{\partial\mathbf{x}_{j\alpha}}m_{j% \alpha}^{-1}-\frac{\partial\sigma_{k}(t)}{\partial\mathbf{x}_{j\beta}}m_{j% \beta}^{-1}\right]\right\|^{2}-d_{j}^{2}=0,\quad j=1\dots n
  15. n n
  16. λ k \lambda_{k}
  17. n n
  18. n n
  19. λ ¯ \underline{\lambda}
  20. λ ¯ ( l + 1 ) λ ¯ ( l ) - 𝐉 σ - 1 σ ¯ ( t + Δ t ) \underline{\lambda}^{(l+1)}\leftarrow\underline{\lambda}^{(l)}-\mathbf{J}_{% \sigma}^{-1}\underline{\sigma}(t+\Delta t)
  21. 𝐉 σ \mathbf{J}_{\sigma}
  22. 𝐉 = ( σ 1 λ 1 σ 1 λ 2 σ 1 λ n σ 2 λ 1 σ 2 λ 2 σ 2 λ n σ n λ 1 σ n λ 2 σ n λ n ) . \mathbf{J}=\left(\begin{array}[]{cccc}\frac{\partial\sigma_{1}}{\partial% \lambda_{1}}&\frac{\partial\sigma_{1}}{\partial\lambda_{2}}&\dots&\frac{% \partial\sigma_{1}}{\partial\lambda_{n}}\\ \frac{\partial\sigma_{2}}{\partial\lambda_{1}}&\frac{\partial\sigma_{2}}{% \partial\lambda_{2}}&\dots&\frac{\partial\sigma_{2}}{\partial\lambda_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\sigma_{n}}{\partial\lambda_{1}}&\frac{\partial\sigma_{n}}{% \partial\lambda_{2}}&\dots&\frac{\partial\sigma_{n}}{\partial\lambda_{n}}\end{% array}\right).
  23. 𝐉 σ \mathbf{J}_{\sigma}
  24. 𝐉 σ \mathbf{J}_{\sigma}
  25. λ ¯ \underline{\lambda}
  26. λ ¯ ( 0 ) = 𝟎 \underline{\lambda}^{(0)}=\mathbf{0}
  27. σ k ( t ) \sigma_{k}(t)
  28. σ k ( t ) λ j \frac{\partial\sigma_{k}(t)}{\partial\lambda_{j}}
  29. J i j = σ j λ i | λ = 0 = 2 [ x ^ j α - x ^ j β ] [ σ i x j α m j α - 1 - σ i x j β m j β - 1 ] . J_{ij}=\left.\frac{\partial\sigma_{j}}{\partial\lambda_{i}}\right|_{\mathbf{% \lambda}=0}=2\left[\hat{x}_{j\alpha}-\hat{x}_{j\beta}\right]\left[\frac{% \partial\sigma_{i}}{\partial x_{j\alpha}}m_{j\alpha}^{-1}-\frac{\partial\sigma% _{i}}{\partial x_{j\beta}}m_{j\beta}^{-1}\right].
  30. λ \lambda
  31. λ j = - 𝐉 - 1 [ 𝐱 ^ j α ( t + Δ t ) - 𝐱 ^ j β ( t + Δ t ) 2 - d j 2 ] . \mathbf{\lambda}_{j}=-\mathbf{J}^{-1}\left[\left\|\hat{\mathbf{x}}_{j\alpha}(t% +\Delta t)-\hat{\mathbf{x}}_{j\beta}(t+\Delta t)\right\|^{2}-d_{j}^{2}\right].
  32. 𝐱 ^ i ( t + Δ t ) 𝐱 ^ i ( t + Δ t ) + k = 1 n λ k σ k 𝐱 i \hat{\mathbf{x}}_{i}(t+\Delta t)\leftarrow\hat{\mathbf{x}}_{i}(t+\Delta t)+% \sum_{k=1}^{n}\lambda_{k}\frac{\partial\sigma_{k}}{\partial\mathbf{x}_{i}}
  33. λ ¯ = 0. \underline{\lambda}=\mathbf{0}.
  34. σ k ( t + Δ t ) \sigma_{k}(t+\Delta t)
  35. n = 3 n=3
  36. λ ¯ = - 𝐉 σ - 1 σ ¯ \underline{\lambda}=-\mathbf{J}_{\sigma}^{-1}\underline{\sigma}
  37. 𝐉 σ \mathbf{J}_{\sigma}
  38. k k
  39. k k
  40. λ k \lambda_{k}
  41. \leftarrow
  42. σ k ( t ) σ k ( t ) / λ k \frac{\sigma_{k}(t)}{\partial\sigma_{k}(t)/\partial\lambda_{k}}
  43. 𝐱 k α \mathbf{x}_{k\alpha}
  44. \leftarrow
  45. 𝐱 k α + λ k σ k ( t ) 𝐱 k α \mathbf{x}_{k\alpha}+\lambda_{k}\frac{\partial\sigma_{k}(t)}{\partial\mathbf{x% }_{k\alpha}}
  46. 𝐱 k β \mathbf{x}_{k\beta}
  47. \leftarrow
  48. 𝐱 k β + λ k σ k ( t ) 𝐱 k β \mathbf{x}_{k\beta}+\lambda_{k}\frac{\partial\sigma_{k}(t)}{\partial\mathbf{x}% _{k\beta}}
  49. k = 1 n k=1\dots n
  50. σ k ( t + Δ t ) \sigma_{k}(t+\Delta t)
  51. 𝒪 ( n ) \mathcal{O}(n)
  52. λ k \lambda_{k}
  53. 𝐉 σ \mathbf{J}_{\sigma}
  54. 𝒪 ( n 2 ) \mathcal{O}(n^{2})
  55. λ ¯ = - 𝐉 σ - 1 σ ¯ \underline{\lambda}=-\mathbf{J}_{\sigma}^{-1}\underline{\sigma}
  56. 𝒪 ( n 3 ) \mathcal{O}(n^{3})
  57. 𝐉 σ \mathbf{J}_{\sigma}
  58. 𝒪 ( n 3 ) \mathcal{O}(n^{3})
  59. 𝒪 ( n 2 ) \mathcal{O}(n^{2})
  60. L r i g i d ( t + Δ t 2 ) = L n o n r i g i d ( t + Δ t 2 ) L^{rigid}\left(t+\frac{\Delta t}{2}\right)=L^{nonrigid}\left(t+\frac{\Delta t}% {2}\right)
  61. 𝐉 σ \mathbf{J}_{\sigma}
  62. ( 𝐈 - 𝐉 σ ) - 1 = 𝐈 + 𝐉 σ + 𝐉 σ 2 + 𝐉 σ 3 + (\mathbf{I}-\mathbf{J}_{\sigma})^{-1}=\mathbf{I}+\mathbf{J}_{\sigma}+\mathbf{J% }_{\sigma}^{2}+\mathbf{J}_{\sigma}^{3}+\dots

Construction_of_t-norms.html

  1. f ( - 1 ) ( y ) = { sup { x [ a , b ] f ( x ) < y } for f non-decreasing sup { x [ a , b ] f ( x ) > y } for f non-increasing. f^{(-1)}(y)=\begin{cases}\sup\{x\in[a,b]\mid f(x)<y\}&\,\text{for }f\,\text{ % non-decreasing}\\ \sup\{x\in[a,b]\mid f(x)>y\}&\,\text{for }f\,\text{ non-increasing.}\end{cases}
  2. lim p p 0 T p = T p 0 \lim_{p\to p_{0}}T_{p}=T_{p_{0}}
  3. T p SS ( x , y ) = { T min ( x , y ) if p = - ( x p + y p - 1 ) 1 / p if - < p < 0 T prod ( x , y ) if p = 0 ( max ( 0 , x p + y p - 1 ) ) 1 / p if 0 < p < + T D ( x , y ) if p = + . T^{\mathrm{SS}}_{p}(x,y)=\begin{cases}T_{\min}(x,y)&\,\text{if }p=-\infty\\ (x^{p}+y^{p}-1)^{1/p}&\,\text{if }-\infty<p<0\\ T_{\mathrm{prod}}(x,y)&\,\text{if }p=0\\ (\max(0,x^{p}+y^{p}-1))^{1/p}&\,\text{if }0<p<+\infty\\ T_{\mathrm{D}}(x,y)&\,\text{if }p=+\infty.\end{cases}
  4. T p SS T^{\mathrm{SS}}_{p}
  5. T p SS T^{\mathrm{SS}}_{p}
  6. f p SS ( x ) = { - log x if p = 0 1 - x p p otherwise. f^{\mathrm{SS}}_{p}(x)=\begin{cases}-\log x&\,\text{if }p=0\\ \frac{1-x^{p}}{p}&\,\text{otherwise.}\end{cases}
  7. T p H ( x , y ) = { T D ( x , y ) if p = + 0 if p = x = y = 0 x y p + ( 1 - p ) ( x + y - x y ) otherwise. T^{\mathrm{H}}_{p}(x,y)=\begin{cases}T_{\mathrm{D}}(x,y)&\,\text{if }p=+\infty% \\ 0&\,\text{if }p=x=y=0\\ \frac{xy}{p+(1-p)(x+y-xy)}&\,\text{otherwise.}\end{cases}
  8. T 0 H T^{\mathrm{H}}_{0}
  9. T p H T^{\mathrm{H}}_{p}
  10. T p H T^{\mathrm{H}}_{p}
  11. f p H ( x ) = { 1 - x x if p = 0 log p + ( 1 - p ) x x otherwise. f^{\mathrm{H}}_{p}(x)=\begin{cases}\frac{1-x}{x}&\,\text{if }p=0\\ \log\frac{p+(1-p)x}{x}&\,\text{otherwise.}\end{cases}
  12. T p F ( x , y ) = { T min ( x , y ) if p = 0 T prod ( x , y ) if p = 1 T Luk ( x , y ) if p = + log p ( 1 + ( p x - 1 ) ( p y - 1 ) p - 1 ) otherwise. T^{\mathrm{F}}_{p}(x,y)=\begin{cases}T_{\mathrm{min}}(x,y)&\,\text{if }p=0\\ T_{\mathrm{prod}}(x,y)&\,\text{if }p=1\\ T_{\mathrm{Luk}}(x,y)&\,\text{if }p=+\infty\\ \log_{p}\left(1+\frac{(p^{x}-1)(p^{y}-1)}{p-1}\right)&\,\text{otherwise.}\end{cases}
  13. T p F T^{\mathrm{F}}_{p}
  14. T p F T^{\mathrm{F}}_{p}
  15. f p F ( x ) = { - log x if p = 1 1 - x if p = + log p - 1 p x - 1 otherwise. f^{\mathrm{F}}_{p}(x)=\begin{cases}-\log x&\,\text{if }p=1\\ 1-x&\,\text{if }p=+\infty\\ \log\frac{p-1}{p^{x}-1}&\,\text{otherwise.}\end{cases}
  16. T p Y ( x , y ) = { T D ( x , y ) if p = 0 max ( 0 , 1 - ( ( 1 - x ) p + ( 1 - y ) p ) 1 / p ) if 0 < p < + T min ( x , y ) if p = + T^{\mathrm{Y}}_{p}(x,y)=\begin{cases}T_{\mathrm{D}}(x,y)&\,\text{if }p=0\\ \max\left(0,1-((1-x)^{p}+(1-y)^{p})^{1/p}\right)&\,\text{if }0<p<+\infty\\ T_{\mathrm{min}}(x,y)&\,\text{if }p=+\infty\end{cases}
  17. T p Y T^{\mathrm{Y}}_{p}
  18. T p Y T^{\mathrm{Y}}_{p}
  19. T p Y T^{\mathrm{Y}}_{p}
  20. f p Y ( x ) = ( 1 - x ) p . f^{\mathrm{Y}}_{p}(x)=(1-x)^{p}.
  21. T p AA ( x , y ) = { T D ( x , y ) if p = 0 e - ( | log x | p + | log y | p ) 1 / p if 0 < p < + T min ( x , y ) if p = + T^{\mathrm{AA}}_{p}(x,y)=\begin{cases}T_{\mathrm{D}}(x,y)&\,\text{if }p=0\\ e^{-\left(|\log x|^{p}+|\log y|^{p}\right)^{1/p}}&\,\text{if }0<p<+\infty\\ T_{\mathrm{min}}(x,y)&\,\text{if }p=+\infty\end{cases}
  22. T p AA T^{\mathrm{AA}}_{p}
  23. T p AA T^{\mathrm{AA}}_{p}
  24. T p AA T^{\mathrm{AA}}_{p}
  25. f p AA ( x ) = ( - log x ) p . f^{\mathrm{AA}}_{p}(x)=(-\log x)^{p}.
  26. T p D ( x , y ) = { 0 if x = 0 or y = 0 T D ( x , y ) if p = 0 T min ( x , y ) if p = + 1 1 + ( ( 1 - x x ) p + ( 1 - y y ) p ) 1 / p otherwise. T^{\mathrm{D}}_{p}(x,y)=\begin{cases}0&\,\text{if }x=0\,\text{ or }y=0\\ T_{\mathrm{D}}(x,y)&\,\text{if }p=0\\ T_{\mathrm{min}}(x,y)&\,\text{if }p=+\infty\\ \frac{1}{1+\left(\left(\frac{1-x}{x}\right)^{p}+\left(\frac{1-y}{y}\right)^{p}% \right)^{1/p}}&\,\text{otherwise.}\\ \end{cases}
  27. T p D T^{\mathrm{D}}_{p}
  28. T p D T^{\mathrm{D}}_{p}
  29. T p D T^{\mathrm{D}}_{p}
  30. f p D ( x ) = ( 1 - x x ) p . f^{\mathrm{D}}_{p}(x)=\left(\frac{1-x}{x}\right)^{p}.
  31. T p SW ( x , y ) = { T D ( x , y ) if p = - 1 max ( 0 , x + y - 1 + p x y 1 + p ) if - 1 < p < + T prod ( x , y ) if p = + T^{\mathrm{SW}}_{p}(x,y)=\begin{cases}T_{\mathrm{D}}(x,y)&\,\text{if }p=-1\\ \max\left(0,\frac{x+y-1+pxy}{1+p}\right)&\,\text{if }-1<p<+\infty\\ T_{\mathrm{prod}}(x,y)&\,\text{if }p=+\infty\end{cases}
  32. T p SW T^{\mathrm{SW}}_{p}
  33. T p SW T^{\mathrm{SW}}_{p}
  34. f p SW ( x ) = { 1 - x if p = 0 1 - log 1 + p ( 1 + p x ) otherwise. f^{\mathrm{SW}}_{p}(x)=\begin{cases}1-x&\,\text{if }p=0\\ 1-\log_{1+p}(1+px)&\,\text{otherwise.}\end{cases}
  35. T ( x , y ) = { a i + ( b i - a i ) T i ( x - a i b i - a i , y - a i b i - a i ) if x , y [ a i , b i ] 2 min ( x , y ) otherwise T(x,y)=\begin{cases}a_{i}+(b_{i}-a_{i})\cdot T_{i}\left(\frac{x-a_{i}}{b_{i}-a% _{i}},\frac{y-a_{i}}{b_{i}-a_{i}}\right)&\,\text{if }x,y\in[a_{i},b_{i}]^{2}\\ \min(x,y)&\,\text{otherwise}\end{cases}
  36. T = i I ( T i , a i , b i ) , T=\bigoplus\nolimits_{i\in I}(T_{i},a_{i},b_{i}),
  37. ( T 1 , a 1 , b 1 ) ( T n , a n , b n ) (T_{1},a_{1},b_{1})\oplus\dots\oplus(T_{n},a_{n},b_{n})
  38. T = i I ( T i , a i , b i ) T=\bigoplus\nolimits_{i\in I}(T_{i},a_{i},b_{i})
  39. R ( x , y ) = { 1 if x y a i + ( b i - a i ) R i ( x - a i b i - a i , y - a i b i - a i ) if a i < y < x b i y otherwise. R(x,y)=\begin{cases}1&\,\text{if }x\leq y\\ a_{i}+(b_{i}-a_{i})\cdot R_{i}\left(\frac{x-a_{i}}{b_{i}-a_{i}},\frac{y-a_{i}}% {b_{i}-a_{i}}\right)&\,\text{if }a_{i}<y<x\leq b_{i}\\ y&\,\text{otherwise.}\end{cases}
  40. R T 1 ( x , y ) = sup { z T 1 ( z , x ) y } . R_{T_{1}}(x,y)=\sup\{z\mid T_{1}(z,x)\leq y\}.
  41. T rot = { T 1 ( x , y ) if x , y ( t , 1 ] N ( R T 1 ( x , N ( y ) ) ) if x ( t , 1 ] and y [ 0 , t ] N ( R T 1 ( y , N ( x ) ) ) if x [ 0 , t ] and y ( t , 1 ] 0 if x , y [ 0 , t ] T_{\mathrm{rot}}=\begin{cases}T_{1}(x,y)&\,\text{if }x,y\in(t,1]\\ N(R_{T_{1}}(x,N(y)))&\,\text{if }x\in(t,1]\,\text{ and }y\in[0,t]\\ N(R_{T_{1}}(y,N(x)))&\,\text{if }x\in[0,t]\,\text{ and }y\in(t,1]\\ 0&\,\text{if }x,y\in[0,t]\end{cases}

Contact_mechanics.html

  1. R R
  2. d d
  3. a = R d a=\sqrt{Rd}
  4. F F
  5. d d
  6. F = 4 3 E * R 1 / 2 d 3 / 2 F=\tfrac{4}{3}E^{*}R^{1/2}d^{3/2}
  7. 1 E * = 1 - ν 1 2 E 1 + 1 - ν 2 2 E 2 \frac{1}{E^{*}}=\frac{1-\nu^{2}_{1}}{E_{1}}+\frac{1-\nu^{2}_{2}}{E_{2}}
  8. E 1 E_{1}
  9. E 2 E_{2}
  10. ν 1 \nu_{1}
  11. ν 2 \nu_{2}
  12. p ( r ) = p 0 ( 1 - r 2 a 2 ) 1 / 2 p(r)=p_{0}\left(1-\frac{r^{2}}{a^{2}}\right)^{1/2}
  13. p 0 p_{0}
  14. p 0 = 3 F 2 π a 2 = 1 π ( 6 F E * 2 R 2 ) 1 / 3 p_{0}=\cfrac{3F}{2\pi a^{2}}=\cfrac{1}{\pi}\left(\cfrac{6F{E^{*}}^{2}}{R^{2}}% \right)^{1/3}
  15. F F
  16. a 3 = 3 F R 4 E * a^{3}=\cfrac{3FR}{4E^{*}}
  17. d d
  18. d = a 2 R = ( 9 F 2 16 R E * 2 ) 1 / 3 d=\cfrac{a^{2}}{R}=\left(\cfrac{9F^{2}}{16R{E^{*}}^{2}}\right)^{1/3}
  19. z 0.49 a z\approx 0.49a
  20. ν = 0.33 \nu=0.33
  21. R 1 R_{1}
  22. R 2 R_{2}
  23. a a
  24. R R
  25. 1 R = 1 R 1 + 1 R 2 \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}
  26. R R
  27. R R
  28. p ( r ) = p 0 ( 1 - r 2 a 2 ) - 1 / 2 p(r)=p_{0}\left(1-\frac{r^{2}}{a^{2}}\right)^{-1/2}
  29. a a
  30. p 0 = 1 π E * d a p_{0}=\frac{1}{\pi}E^{*}\frac{d}{a}
  31. F = 2 a E * d F=2aE^{*}d\,
  32. E E
  33. ϵ \epsilon
  34. a a
  35. ϵ = a tan θ \epsilon=a\tan\theta
  36. θ \theta
  37. d d
  38. d = π 2 ϵ d=\frac{\pi}{2}\epsilon
  39. F = π E 2 ( 1 - ν 2 ) a 2 tan θ = 2 E π ( 1 - ν 2 ) d 2 tan θ F=\frac{\pi E}{2\left(1-\nu^{2}\right)}a^{2}\tan\theta=\frac{2E}{\pi\left(1-% \nu^{2}\right)}\frac{d^{2}}{\tan\theta}
  40. p ( r ) = E d π a ( 1 - ν 2 ) ln ( a r + ( a r ) 2 - 1 ) = E d π a ( 1 - ν 2 ) cosh - 1 ( a r ) p{\left(r\right)}=\frac{Ed}{\pi a\left(1-\nu^{2}\right)}\ln\left(\frac{a}{r}+% \sqrt{\left(\frac{a}{r}\right)^{2}-1}\right)=\frac{Ed}{\pi a\left(1-\nu^{2}% \right)}\cosh^{-1}\left(\frac{a}{r}\right)
  41. F = π 4 E * L d F=\frac{\pi}{4}E^{*}Ld
  42. a = 2 R d a=\sqrt{2Rd}
  43. 1 R = 1 R 1 + 1 R 2 \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}
  44. p 0 = ( E * F π L R ) 1 / 2 p_{0}=\left(\frac{E^{*}F}{\pi LR}\right)^{1/2}
  45. S S
  46. x x
  47. y y
  48. z z
  49. p z = p ( x , y ) = q z ( x , y ) p_{z}=p(x,y)=q_{z}(x,y)
  50. q x = q x ( x , y ) q_{x}=q_{x}(x,y)
  51. q y = q y ( x , y ) q_{y}=q_{y}(x,y)
  52. S S
  53. P z = S p ( x , y ) d A ; Q x = S q x ( x , y ) d A ; Q y = S q y ( x , y ) d A P_{z}=\int_{S}p(x,y)~{}\mathrm{d}A~{};~{}~{}Q_{x}=\int_{S}q_{x}(x,y)~{}\mathrm% {d}A~{};~{}~{}Q_{y}=\int_{S}q_{y}(x,y)~{}\mathrm{d}A
  54. M x = S y p ( x , y ) d A ; M y = S x p ( x , y ) d A ; M z = S [ x q y ( x , y ) - y q x ( x , y ) ] d A M_{x}=\int_{S}y~{}p(x,y)~{}\mathrm{d}A~{};~{}~{}M_{y}=\int_{S}x~{}p(x,y)~{}% \mathrm{d}A~{};~{}~{}M_{z}=\int_{S}[x~{}q_{y}(x,y)-y~{}q_{x}(x,y)]~{}\mathrm{d}A
  55. σ x z ( x , 0 ) = 0 ; σ z ( x , z ) = - P δ ( x , z ) \sigma_{xz}(x,0)=0~{};~{}~{}\sigma_{z}(x,z)=-P\delta(x,z)
  56. δ ( x , z ) \delta(x,z)
  57. σ x x = - 2 P π x 2 z ( x 2 + z 2 ) 2 σ z z = - 2 P π z 3 ( x 2 + z 2 ) 2 σ x z = - 2 P π x z 2 ( x 2 + z 2 ) 2 \begin{aligned}\displaystyle\sigma_{xx}&\displaystyle=-\frac{2P}{\pi}\frac{x^{% 2}z}{(x^{2}+z^{2})^{2}}\\ \displaystyle\sigma_{zz}&\displaystyle=-\frac{2P}{\pi}\frac{z^{3}}{(x^{2}+z^{2% })^{2}}\\ \displaystyle\sigma_{xz}&\displaystyle=-\frac{2P}{\pi}\frac{xz^{2}}{(x^{2}+z^{% 2})^{2}}\end{aligned}
  58. ( x , y ) (x,y)
  59. ( a , b ) (a,b)
  60. P P
  61. p ( x ) p(x)
  62. a < x < b a<x<b
  63. σ x x \displaystyle\sigma_{xx}
  64. ( a , b ) (a,b)
  65. Q Q
  66. q ( x ) q(x)
  67. σ x x \displaystyle\sigma_{xx}
  68. g N g_{N}
  69. g N 0 g_{N}\geq 0
  70. g N = 0 g_{N}=0
  71. p N = 𝐭 𝐧 p_{N}=\mathbf{t}\cdot\mathbf{n}
  72. p N 0 . p_{N}\leq 0\,.
  73. g N = 0 g_{N}=0
  74. p N < 0 p_{N}<0
  75. g N > 0 g_{N}>0
  76. p N = 0 p_{N}=0
  77. g N 0 , p N 0 , p N g N = 0 . g_{N}\geq 0\,,\quad p_{N}\leq 0\,,\quad p_{N}\,g_{N}=0\,.
  78. A A
  79. A 0 A_{0}
  80. F F
  81. A = κ E * h F A=\frac{\kappa}{E^{*}h^{\prime}}F
  82. h h^{\prime}
  83. κ 2 \kappa\approx 2
  84. p av = F A 1 2 E * h p_{\mathrm{av}}=\frac{F}{A}\approx\frac{1}{2}E^{*}h^{\prime}
  85. E * E^{*}
  86. h h^{\prime}
  87. p av = 1.1 σ y 0.39 σ 0 p_{\mathrm{av}}=1.1\sigma_{y}\approx 0.39\sigma_{0}
  88. σ y \sigma_{y}
  89. σ 0 \sigma_{0}
  90. Ψ \Psi
  91. σ 0 \sigma_{0}
  92. Ψ = E * h σ 0 > 2 3 . \Psi=\frac{E^{*}h^{\prime}}{\sigma_{0}}>\tfrac{2}{3}~{}.
  93. Ψ \Psi
  94. Ψ < 2 3 \Psi<\tfrac{2}{3}
  95. z z
  96. F ( z ) = 16 γ 3 z 0 [ ( z z 0 ) - 9 - ( z z 0 ) - 3 ] F(z)=\cfrac{16\gamma}{3z_{0}}\left[\left(\cfrac{z}{z_{0}}\right)^{-9}-\left(% \cfrac{z}{z_{0}}\right)^{-3}\right]
  97. F F
  98. 2 γ 2\gamma
  99. z 0 z_{0}
  100. F a ( z ) = 16 γ π R 3 [ 1 4 ( z z 0 ) - 8 - ( z z 0 ) - 2 ] ; 1 R = 1 R 1 + 1 R 2 F_{a}(z)=\cfrac{16\gamma\pi R}{3}\left[\cfrac{1}{4}\left(\cfrac{z}{z_{0}}% \right)^{-8}-\left(\cfrac{z}{z_{0}}\right)^{-2}\right]~{};~{}~{}\frac{1}{R}=% \frac{1}{R_{1}}+\frac{1}{R_{2}}
  101. R 1 , R 2 R_{1},R_{2}
  102. z = z 0 z=z_{0}
  103. F a = F c = - 4 γ π R . F_{a}=F_{c}=-4\gamma\pi R.
  104. p ( r ) = p 0 ( 1 - r 2 a 2 ) 1 / 2 + p 0 ( 1 - r 2 a 2 ) - 1 / 2 p(r)=p_{0}\left(1-\cfrac{r^{2}}{a^{2}}\right)^{1/2}+p_{0}^{\prime}\left(1-% \cfrac{r^{2}}{a^{2}}\right)^{-1/2}
  105. p 0 p_{0}^{\prime}
  106. p 0 = 2 a E * π R ; p 0 = - ( 4 γ E * π a ) 1 / 2 p_{0}=\cfrac{2aE^{*}}{\pi R}~{};~{}~{}p_{0}^{\prime}=-\left(\cfrac{4\gamma E^{% *}}{\pi a}\right)^{1/2}
  107. a a\,
  108. F F
  109. 2 γ 2\gamma
  110. R i , E i , ν i , i = 1 , 2 R_{i},E_{i},\nu_{i},~{}~{}i=1,2
  111. 1 R = 1 R 1 + 1 R 2 ; 1 E * = 1 - ν 1 2 E 1 + 1 - ν 2 2 E 2 \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}~{};~{}~{}\frac{1}{E^{*}}=\frac{1-% \nu_{1}^{2}}{E_{1}}+\frac{1-\nu_{2}^{2}}{E_{2}}
  112. d = π a 2 E * ( p 0 + 2 p 0 ) = a 2 R d=\cfrac{\pi a}{2E^{*}}(p_{0}+2p_{0}^{\prime})=\cfrac{a^{2}}{R}
  113. a 3 = 3 R 4 E * ( F + 6 γ π R + 12 γ π R F + ( 6 γ π R ) 2 ) a^{3}=\cfrac{3R}{4E^{*}}\left(F+6\gamma\pi R+\sqrt{12\gamma\pi RF+(6\gamma\pi R% )^{2}}\right)
  114. γ = 0 \gamma=0
  115. a 3 = 9 R 2 γ π E * a^{3}=\cfrac{9R^{2}\gamma\pi}{E^{*}}
  116. a = 0 a=0
  117. F c = - 3 γ π R F_{c}=-3\gamma\pi R\,
  118. a a
  119. a c a_{c}
  120. a c 3 = 9 R 2 γ π 4 E * a_{c}^{3}=\cfrac{9R^{2}\gamma\pi}{4E^{*}}
  121. Δ γ = γ 1 + γ 2 - γ 12 \Delta\gamma=\gamma_{1}+\gamma_{2}-\gamma_{12}
  122. γ 1 , γ 2 \gamma_{1},\gamma_{2}
  123. γ 12 \gamma_{12}
  124. a 3 = 3 R 4 E * ( F + 3 Δ γ π R + 6 Δ γ π R F + ( 3 Δ γ π R ) 2 ) a^{3}=\cfrac{3R}{4E^{*}}\left(F+3\Delta\gamma\pi R+\sqrt{6\Delta\gamma\pi RF+(% 3\Delta\gamma\pi R)^{2}}\right)
  125. F = - 3 2 Δ γ π R F=-\cfrac{3}{2}\Delta\gamma\pi R\,
  126. a c 3 = 9 R 2 Δ γ π 8 E * a_{c}^{3}=\cfrac{9R^{2}\Delta\gamma\pi}{8E^{*}}
  127. d c = a c 2 R = ( 9 4 ) 2 3 ( Δ γ ) 2 3 ( π 2 3 R 1 3 E * 2 3 ) d_{c}=\cfrac{a_{c}^{2}}{R}=\left(\cfrac{9}{4}\right)^{\tfrac{2}{3}}(\Delta% \gamma)^{\tfrac{2}{3}}\left(\cfrac{\pi^{\tfrac{2}{3}}~{}R^{\tfrac{1}{3}}}{{E^{% *}}^{\tfrac{2}{3}}}\right)
  128. a 3 = 3 R 4 E * ( F + 4 γ π R ) a^{3}=\cfrac{3R}{4E^{*}}\left(F+4\gamma\pi R\right)
  129. F c = - 4 γ π R F_{c}=-4\gamma\pi R\,
  130. Δ γ \Delta\gamma
  131. a 3 = 3 R 4 E * ( F + 2 Δ γ π R ) a^{3}=\cfrac{3R}{4E^{*}}\left(F+2\Delta\gamma\pi R\right)
  132. F c = - 2 Δ γ π R F_{c}=-2\Delta\gamma\pi R\,
  133. μ \mu
  134. μ := d c z 0 [ R ( Δ γ ) 2 E * 2 z 0 3 ] 1 / 3 \mu:=\cfrac{d_{c}}{z_{0}}\approx\left[\cfrac{R(\Delta\gamma)^{2}}{{E^{*}}^{2}z% _{0}^{3}}\right]^{1/3}
  135. z 0 z_{0}
  136. μ \mu
  137. μ \mu
  138. Δ γ = σ 0 h 0 \Delta\gamma=\sigma_{0}~{}h_{0}
  139. σ 0 \sigma_{0}
  140. h 0 h_{0}
  141. z 0 z z 0 + h 0 z_{0}\leq z\leq z_{0}+h_{0}
  142. a a
  143. σ 0 \sigma_{0}
  144. c > a c>a
  145. a < r < c a<r<c
  146. h ( r ) h(r)
  147. h ( a ) = 0 h(a)=0
  148. h ( c ) = h 0 h(c)=h_{0}
  149. m m
  150. m := c a m:=\cfrac{c}{a}
  151. - a < r < a -a<r<a
  152. p H ( r ) = ( 3 F H 2 π a 2 ) ( 1 - r 2 a 2 ) 1 / 2 p^{H}(r)=\left(\cfrac{3F^{H}}{2\pi a^{2}}\right)\left(1-\cfrac{r^{2}}{a^{2}}% \right)^{1/2}
  153. F H F^{H}
  154. F H = 4 E * a 3 3 R F^{H}=\cfrac{4E^{*}a^{3}}{3R}
  155. d H = a 2 R d^{H}=\cfrac{a^{2}}{R}
  156. r = c r=c
  157. u H ( c ) = 1 π R [ a 2 ( 2 - m 2 ) sin - 1 ( 1 m ) + a 2 m 2 - 1 ] u^{H}(c)=\cfrac{1}{\pi R}\left[a^{2}(2-m^{2})\sin^{-1}\left(\cfrac{1}{m}\right% )+a^{2}\sqrt{m^{2}-1}\right]
  158. r = c r=c
  159. h H ( c ) = c 2 2 R - d H + u H ( c ) h^{H}(c)=\cfrac{c^{2}}{2R}-d^{H}+u^{H}(c)
  160. p D ( r ) = { - σ 0 π cos - 1 [ 2 - m 2 - r 2 a 2 m 2 ( 1 - r 2 m 2 a 2 ) ] for r a - σ 0 for a r c p^{D}(r)=\begin{cases}-\cfrac{\sigma_{0}}{\pi}\cos^{-1}\left[\cfrac{2-m^{2}-% \cfrac{r^{2}}{a^{2}}}{m^{2}\left(1-\cfrac{r^{2}}{m^{2}a^{2}}\right)}\right]&% \quad\mathrm{for}\quad r\leq a\\ -\sigma_{0}&\quad\mathrm{for}\quad a\leq r\leq c\end{cases}
  161. F D = - 2 σ 0 m 2 a 2 [ cos - 1 ( 1 m ) + 1 m 2 m 2 - 1 ] F^{D}=-2\sigma_{0}m^{2}a^{2}\left[\cos^{-1}\left(\cfrac{1}{m}\right)+\frac{1}{% m^{2}}\sqrt{m^{2}-1}\right]
  162. d D = - ( 2 σ 0 a E * ) m 2 - 1 d^{D}=-\left(\cfrac{2\sigma_{0}a}{E^{*}}\right)\sqrt{m^{2}-1}
  163. r = c r=c
  164. h D ( c ) = ( 4 σ 0 a π E * ) [ m 2 - 1 cos - 1 ( 1 m ) + 1 - m ] h^{D}(c)=\left(\cfrac{4\sigma_{0}a}{\pi E^{*}}\right)\left[\sqrt{m^{2}-1}\cos^% {-1}\left(\cfrac{1}{m}\right)+1-m\right]
  165. p ( r ) = p H ( r ) + p D ( r ) p(r)=p^{H}(r)+p^{D}(r)
  166. F = F H + F D F=F^{H}+F^{D}
  167. h ( c ) = h H ( c ) + h D ( c ) = h 0 h(c)=h^{H}(c)+h^{D}(c)=h_{0}
  168. a , c , F , d a,c,F,d
  169. a ¯ = α a ; c ¯ := α c ; d ¯ := α 2 R d ; α := ( 4 E * 3 π Δ γ R 2 ) 1 / 3 ; A ¯ := π c 2 ; F ¯ = F π Δ γ R \bar{a}=\alpha a~{};~{}~{}\bar{c}:=\alpha c~{};~{}~{}\bar{d}:=\alpha^{2}Rd~{};% ~{}~{}\alpha:=\left(\cfrac{4E^{*}}{3\pi\Delta\gamma R^{2}}\right)^{1/3}~{};~{}% ~{}\bar{A}:=\pi c^{2}~{};~{}~{}\bar{F}=\cfrac{F}{\pi\Delta\gamma R}
  170. λ \lambda
  171. λ := σ 0 ( 9 R 2 π Δ γ E * 2 ) 1 / 3 = 1.16 μ \lambda:=\sigma_{0}\left(\cfrac{9R}{2\pi\Delta\gamma{E^{*}}^{2}}\right)^{1/3}=% 1.16\mu
  172. σ 0 \sigma_{0}
  173. σ t h = 16 Δ γ 9 3 z 0 \sigma_{th}=\cfrac{16\Delta\gamma}{9\sqrt{3}z_{0}}
  174. σ 0 = exp ( - 223 420 ) Δ γ z 0 0.588 Δ γ z 0 \sigma_{0}=\exp\left(-\cfrac{223}{420}\right)\cdot\cfrac{\Delta\gamma}{z_{0}}% \approx 0.588\cfrac{\Delta\gamma}{z_{0}}
  175. λ 0.663 μ \lambda\approx 0.663\mu
  176. F ¯ = a ¯ 3 - λ a ¯ 2 [ m 2 - 1 + m 2 sec - 1 m ] \bar{F}=\bar{a}^{3}-\lambda\bar{a}^{2}\left[\sqrt{m^{2}-1}+m^{2}\sec^{-1}m\right]
  177. d ¯ = a ¯ 2 - 4 3 λ a ¯ m 2 - 1 \bar{d}=\bar{a}^{2}-\cfrac{4}{3}~{}\lambda\bar{a}\sqrt{m^{2}-1}
  178. λ a ¯ 2 2 [ ( m 2 - 2 ) sec - 1 m + m 2 - 1 ] + 4 λ a ¯ 3 [ m 2 - 1 sec - 1 m - m + 1 ] = 1 \cfrac{\lambda\bar{a}^{2}}{2}\left[(m^{2}-2)\sec^{-1}m+\sqrt{m^{2}-1}\right]+% \cfrac{4\lambda\bar{a}}{3}\left[\sqrt{m^{2}-1}\sec^{-1}m-m+1\right]=1
  179. c c
  180. a a
  181. λ \lambda
  182. λ \lambda
  183. m 1 m\rightarrow 1
  184. λ \lambda
  185. λ \lambda
  186. a a
  187. a = a 0 ( β ) ( β + 1 - F / F c ( β ) 1 + β ) 2 / 3 a=a_{0}(\beta)\left(\cfrac{\beta+\sqrt{1-F/F_{c}(\beta)}}{1+\beta}\right)^{2/3}
  188. a 0 a_{0}
  189. β \beta
  190. λ \lambda
  191. λ = - 0.924 ln ( 1 - 1.02 β ) \lambda=-0.924\ln(1-1.02\beta)
  192. β = 1 \beta=1
  193. β = 0 \beta=0
  194. 0 < β < 1 0<\beta<1
  195. 0.1 < λ < 5 0.1<\lambda<5

Containment_order.html

  1. X ( x ) = { y X | y x } ; X_{\leq}(x)=\{y\in X|y\leq x\};
  2. X ( a ) X ( b ) precisely when a b . X_{\leq}(a)\subseteq X_{\leq}(b)\mbox{ precisely when }~{}a\leq b.
  3. S S
  4. | X | |X|

Content_(algebra).html

  1. 12 x 3 + 30 x - 20 12x^{3}+30x-20

Content_(measure_theory).html

  1. μ \mu
  2. 𝒜 \mathcal{A}
  3. μ ( A ) [ 0 , ] whenever A 𝒜 . \mu(A)\in\ [0,\infty]\mbox{ whenever }~{}A\in\mathcal{A}.
  4. μ ( ) = 0. \mu(\varnothing)=0.
  5. μ ( A 1 A 2 ) = μ ( A 1 ) + μ ( A 2 ) whenever A 1 , A 2 𝒜 and A 1 A 2 = . \mu(A_{1}\cup A_{2})=\mu(A_{1})+\mu(A_{2})\mbox{ whenever }~{}A_{1},A_{2}\in% \mathcal{A}\mbox{ and }~{}A_{1}\cap A_{2}=\varnothing.
  6. f d λ = lim i = 1 n f ( α i ) λ ( f - 1 ( A i ) ) \int fd\lambda=\lim\sum_{i=1}^{n}f(\alpha_{i})\lambda(f^{-1}(A_{i}))
  7. λ ( C ) [ 0 , ] \lambda(C)\in\ [0,\infty]
  8. λ ( ) = 0. \lambda(\varnothing)=0.
  9. λ ( C 1 ) λ ( C 2 ) whenever C 1 C 2 \lambda(C_{1})\leq\lambda(C_{2})\mbox{ whenever }~{}C_{1}\subset C_{2}
  10. λ ( C 1 C 2 ) λ ( C 1 ) + λ ( C 2 ) \lambda(C_{1}\cup C_{2})\leq\lambda(C_{1})+\lambda(C_{2})
  11. λ ( C 1 C 2 ) = λ ( C 1 ) + λ ( C 2 ) \lambda(C_{1}\cup C_{2})=\lambda(C_{1})+\lambda(C_{2})
  12. μ ( U ) = sup C U λ ( C ) \mu(U)=\sup_{C\subset U}\lambda(C)
  13. μ ( U ) [ 0 , ] \mu(U)\in\ [0,\infty]
  14. μ ( ) = 0. \mu(\varnothing)=0.
  15. μ ( U 1 ) μ ( U 2 ) whenever U 1 U 2 \mu(U_{1})\leq\mu(U_{2})\mbox{ whenever }~{}U_{1}\subset U_{2}
  16. μ ( n U n ) n λ ( U n ) \mu(\bigcup_{n}U_{n})\leq\oplus_{n}\lambda(U_{n})
  17. μ ( n U n ) = n λ ( U n ) \mu(\bigcup_{n}U_{n})=\oplus_{n}\lambda(U_{n})
  18. μ ( A ) = inf A U μ ( U ) \mu(A)=\inf_{A\subset U}\mu(U)
  19. μ ( A ) [ 0 , ] \mu(A)\in\ [0,\infty]
  20. μ ( ) = 0. \mu(\varnothing)=0.
  21. μ ( A 1 ) μ ( A 2 ) whenever A 1 A 2 \mu(A_{1})\leq\mu(A_{2})\mbox{ whenever }~{}A_{1}\subset A_{2}
  22. μ ( n A n ) n λ ( A n ) \mu(\bigcup_{n}A_{n})\leq\oplus_{n}\lambda(A_{n})

Continuous_function_(set_theory).html

  1. s := s α | α < γ s:=\langle s_{\alpha}|\alpha<\gamma\rangle
  2. s β = lim inf { s α | α < β } = sup { inf { s α | δ α < β } | δ < β } . s_{\beta}=\liminf\{s_{\alpha}|\alpha<\beta\}=\sup\{\inf\{s_{\alpha}|\delta\leq% \alpha<\beta\}|\delta<\beta\}\,.

Continuous_quantum_computation.html

  1. ε \scriptstyle\varepsilon
  2. ε - 1 \scriptstyle\varepsilon^{-1}
  3. ε - 2 log ε - 1 \scriptstyle\varepsilon^{-2}\log\varepsilon^{-1}
  4. ε - 1 \scriptstyle\varepsilon^{-1}
  5. log ε - 1 \scriptstyle\log\varepsilon^{-1}
  6. L p L_{p}

Continuous_stochastic_process.html

  1. 𝐏 ( { ω Ω | lim s t | X s ( ω ) - X t ( ω ) | = 0 } ) = 1. \mathbf{P}\left(\left\{\omega\in\Omega\left|\lim_{s\to t}\big|X_{s}(\omega)-X_% {t}(\omega)\big|=0\right.\right\}\right)=1.
  2. lim s t 𝐄 [ | X s - X t | 2 ] = 0. \lim_{s\to t}\mathbf{E}\left[\big|X_{s}-X_{t}\big|^{2}\right]=0.
  3. lim s t 𝐏 ( { ω Ω | | X s ( ω ) - X t ( ω ) | ε } ) = 0. \lim_{s\to t}\mathbf{P}\left(\left\{\omega\in\Omega\left|\big|X_{s}(\omega)-X_% {t}(\omega)\big|\geq\varepsilon\right.\right\}\right)=0.
  4. lim s t 𝐄 [ | X s - X t | 1 + | X s - X t | ] = 0. \lim_{s\to t}\mathbf{E}\left[\frac{\big|X_{s}-X_{t}\big|}{1+\big|X_{s}-X_{t}% \big|}\right]=0.
  5. lim s t F s ( x ) = F t ( x ) \lim_{s\to t}F_{s}(x)=F_{t}(x)
  6. A t = { ω Ω | lim s t | X s ( ω ) - X t ( ω ) | 0 } , A_{t}=\left\{\omega\in\Omega\left|\lim_{s\to t}\big|X_{s}(\omega)-X_{t}(\omega% )\big|\neq 0\right.\right\},
  7. A = t T A t . A=\bigcup_{t\in T}A_{t}.

Contraction_(operator_theory).html

  1. \mathcal{H}
  2. 𝒟 T \mathcal{D}_{T}
  3. 𝒟 T * \mathcal{D}_{T*}
  4. \mathcal{H}
  5. 𝒟 T * \mathcal{D}_{T*}
  6. ( dim 𝒟 T , dim 𝒟 T * ) . (\dim\mathcal{D}_{T},\dim\mathcal{D}_{T^{*}}).
  7. T = Γ U T=\Gamma\oplus U
  8. = H H H , \displaystyle{\mathcal{H}=H\oplus H\oplus H\oplus\cdots,}
  9. \mathcal{H}
  10. V ( ξ 1 , ξ 2 , ξ 3 , ) = ( T ξ 1 , I - T * T ξ 1 , ξ 2 , ξ 3 , ) . \displaystyle{V(\xi_{1},\xi_{2},\xi_{3},\dots)=(T\xi_{1},\sqrt{I-T^{*}T}\xi_{1% },\xi_{2},\xi_{3},\dots).}
  11. 𝒦 = . \displaystyle{\mathcal{K}=\mathcal{H}\oplus\mathcal{H}.}
  12. 𝒦 \mathcal{K}
  13. W ( x , y ) = ( V x + ( I - V V * ) y , - V * y ) . \displaystyle{W(x,y)=(Vx+(I-VV^{*})y,-V^{*}y).}
  14. 𝒦 \mathcal{H}\subset\mathcal{K}
  15. Φ ( g ) = P U ( g ) P , \displaystyle{\Phi(g)=PU(g)P,}
  16. λ i λ j ¯ Φ ( g j - 1 g i ) = P T * T P 0 , \sum\lambda_{i}\overline{\lambda_{j}}\Phi(g_{j}^{-1}g_{i})=PT^{*}TP\geq 0,
  17. T = λ i U ( g i ) . \displaystyle{T=\sum\lambda_{i}U(g_{i}).}
  18. Φ ( 1 ) = P . \displaystyle{\Phi(1)=P.}
  19. \mathcal{H}
  20. ( f 1 , f 2 ) = g , h ( Φ ( h - 1 g ) f 1 ( g ) , f 2 ( h ) ) . \displaystyle{(f_{1},f_{2})=\sum_{g,h}(\Phi(h^{-1}g)f_{1}(g),f_{2}(h)).}
  21. \mathcal{H}
  22. U ( g ) f ( x ) = f ( g - 1 x ) . \displaystyle{U(g)f(x)=f(g^{-1}x).}
  23. \mathcal{H}
  24. f v ( g ) = δ g , 1 v . f_{v}(g)=\delta_{g,1}v.\,
  25. \mathcal{H}
  26. P U ( g ) P = Φ ( g ) , \displaystyle{PU(g)P=\Phi(g),}
  27. \mathcal{H}
  28. \mathcal{H}
  29. Φ ( 0 ) = I , Φ ( n ) = T n , Φ ( - n ) = ( T * ) n , \displaystyle\Phi(0)=I,\,\,\,\Phi(n)=T^{n},\,\,\,\Phi(-n)=(T^{*})^{n},
  30. T ( t ) = P U ( t ) P \displaystyle{T(t)=PU(t)P}
  31. Φ ( 0 ) = I , Φ ( t ) = T ( t ) , Φ ( - t ) = T ( t ) * , \displaystyle{\Phi(0)=I,\,\,\,\Phi(t)=T(t),\,\,\,\Phi(-t)=T(t)^{*},}
  32. A ξ = lim t 0 1 t ( T ( t ) - I ) ξ , \displaystyle{A\xi=\lim_{t\downarrow 0}{1\over t}(T(t)-I)\xi,}
  33. - ( A ξ , ξ ) 0 \displaystyle{-\Re(A\xi,\xi)\geq 0}
  34. T ( t ) = e A t , \displaystyle{T(t)=e^{At},}
  35. T = ( A + I ) ( A - I ) - 1 . \displaystyle{T=(A+I)(A-I)^{-1}.}
  36. A = ( T + I ) ( T - I ) - 1 . \displaystyle{A=(T+I)(T-I)^{-1}.}
  37. f ( T ) ξ = P f ( U ) ξ . \displaystyle{f(T)\xi=Pf(U)\xi.}
  38. f ( z ) = n 0 a n z n \displaystyle{f(z)=\sum_{n\geq 0}a_{n}z^{n}}
  39. f ( T ) ξ = lim r 1 f r ( T ) ξ . \displaystyle{f(T)\xi=\lim_{r\rightarrow 1}f_{r}(T)\xi.}
  40. f ( z ) = n 0 a n z ¯ n , \displaystyle{f^{\sim}(z)=\sum_{n\geq 0}a_{n}\overline{z}^{n},}
  41. f ( T ) = f ( T * ) * . \displaystyle{f^{\sim}(T)=f(T^{*})^{*}.}
  42. e t ( z ) = exp t z + 1 z - 1 . \displaystyle{e_{t}(z)=\exp t{z+1\over z-1}.}
  43. T ( t ) = e t ( T ) \displaystyle{T(t)=e_{t}(T)}
  44. T = 1 2 I - 1 2 0 e - t T ( t ) d t . \displaystyle{T={1\over 2}I-{1\over 2}\int_{0}^{\infty}e^{-t}T(t)\,dt.}
  45. φ ( z ) = c B ( z ) e - P ( z ) , \displaystyle{\varphi(z)=cB(z)e^{-P(z)},}
  46. B ( z ) = [ | λ i | λ i λ i - z 1 - λ ¯ i ] m i , \displaystyle{B(z)=\prod\left[{|\lambda_{i}|\over\lambda_{i}}{\lambda_{i}-z% \over 1-\overline{\lambda}_{i}}\right]^{m_{i}},}
  47. m i ( 1 - | λ i | ) < , \displaystyle{\sum m_{i}(1-|\lambda_{i}|)<\infty,}
  48. P ( z ) = 0 2 π 1 + e - i θ z 1 - e - i θ z d μ ( θ ) \displaystyle{P(z)=\int_{0}^{2\pi}{1+e^{-i\theta}z\over 1-e^{-i\theta}z}\,d\mu% (\theta)}
  49. H i = { ξ : ( T - λ i I ) m i ξ = 0 } . \displaystyle{H_{i}=\{\xi:(T-\lambda_{i}I)^{m_{i}}\xi=0\}.}
  50. A T 1 = T 2 A , B T 2 = T 1 B . \displaystyle{AT_{1}=T_{2}A,\,\,\,BT_{2}=T_{1}B.}
  51. H = H 2 φ H 2 , \displaystyle{H=H^{2}\ominus\varphi H^{2},}
  52. S e i = λ i e i \displaystyle{Se_{i}=\lambda_{i}e_{i}}
  53. ( 1 - | λ i | ) < 1 \displaystyle{\sum(1-|\lambda_{i}|)<1}
  54. S = S ( φ 1 ) S ( φ 1 φ 2 ) S ( φ 1 φ 2 φ 3 ) \displaystyle{S=S(\varphi_{1})\oplus S(\varphi_{1}\varphi_{2})\oplus S(\varphi% _{1}\varphi_{2}\varphi_{3})\oplus\cdots}

Contraharmonic_mean.html

  1. L p L_{p}
  2. C ( x 1 , x 2 , , x n ) = ( x 1 2 + x 2 2 + + x n 2 n ) ( x 1 + x 2 + + x n n ) , C(x_{1},x_{2},\dots,x_{n})={\left({x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\over n% }\right)\over\left({x_{1}+x_{2}+\cdots+x_{n}\over n}\right)},
  3. C ( x 1 , x 2 , , x n ) = x 1 2 + x 2 2 + + x n 2 x 1 + x 2 + + x n . C(x_{1},x_{2},\dots,x_{n})={{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}\over{x_{1}+% x_{2}+\cdots+x_{n}}}.
  4. C ( x 1 , x 2 , , x n ) [ min ( x 1 , x 2 , , x n ) , max ( x 1 , x 2 , , x n ) ] C(x_{1},x_{2},\dots,x_{n})\in[\min(x_{1},x_{2},\dots,x_{n}),\max(x_{1},x_{2},% \dots,x_{n})]
  5. C ( t x 1 , t x 2 , , t x n ) = t C ( x 1 , x 2 , , x n ) for t > 0 C(t\cdot x_{1},t\cdot x_{2},\dots,t\cdot x_{n})=t\cdot C(x_{1},x_{2},\dots,x_{% n})\,\text{ for }t>0
  6. min ( 𝐱 ) H ( 𝐱 ) G ( 𝐱 ) L ( 𝐱 ) A ( 𝐱 ) R ( 𝐱 ) C ( 𝐱 ) max ( 𝐱 ) \min(\mathbf{x})\leq H(\mathbf{x})\leq G(\mathbf{x})\leq L(\mathbf{x})\leq A(% \mathbf{x})\leq R(\mathbf{x})\leq C(\mathbf{x})\leq\max(\mathbf{x})
  7. G ( A ( a , b ) , H ( a , b ) ) = G ( a + b 2 , 2 a b a + b ) = a + b 2 2 a b a + b = a b = G ( a , b ) G(A(a,b),H(a,b))=G\left({{a+b}\over 2},{{2ab}\over{a+b}}\right)=\sqrt{{{a+b}% \over 2}\cdot{{2ab}\over{a+b}}}=\sqrt{ab}=G(a,b)
  8. G ( A ( a , b ) , C ( a , b ) ) = G ( a + b 2 , a 2 + b 2 a + b ) G(A(a,b),C(a,b))=G\left({{a+b}\over 2},{{a^{2}+b^{2}}\over{a+b}}\right)
  9. = a + b 2 a 2 + b 2 a + b = a 2 + b 2 2 = R ( a , b ) =\sqrt{{{a+b}\over 2}\cdot{{a^{2}+b^{2}}\over{a+b}}}=\sqrt{{{a^{2}+b^{2}}\over 2% }}=R(a,b)
  10. g ( x ) = x f ( x ) m g(x)=\frac{xf(x)}{m}
  11. E [ 1 x ] = 1 m \operatorname{E}\left[\frac{1}{x}\right]=\frac{1}{m}
  12. Var ( 1 x ) = m ( E [ 1 / x - 1 ] ) n m 2 \operatorname{Var}\left(\frac{1}{x}\right)=\frac{m(E[1/x-1])}{nm^{2}}

Contraposition.html

  1. P Q P\rightarrow Q
  2. ¬ Q ¬ P \neg Q\rightarrow\neg P
  3. ¬ P ¬ Q \neg P\rightarrow\neg Q
  4. Q P Q\rightarrow P
  5. ¬ ( P Q ) \neg(P\rightarrow Q)
  6. P Q P\rightarrow Q
  7. ¬ Q \neg Q
  8. ¬ P \neg P
  9. A B A\to B
  10. ¬ B ¬ A \neg B\to\neg A
  11. ( A B ) ( ¬ B ¬ A ) (A\to B)\to(\neg B\to\neg A)
  12. A B A\to B
  13. ¬ B ¬ A \neg B\to\neg A
  14. ¬ B ¬ A \neg B\to\neg A
  15. A B A\to B
  16. ( P Q ) (P\to Q)
  17. ( ¬ Q ¬ P ) (\neg Q\to\neg P)
  18. x ( P x Q x ) \forall{x}(P{x}\to Q{x})
  19. x ( ¬ Q x ¬ P x ) \forall{x}(\neg Q{x}\to\neg P{x})
  20. A B ¬ A B A\to B\iff\neg AB
  21. ¬ A B \displaystyle\neg AB
  22. ( A B ) and ¬ B (A\to B)\and\neg B
  23. ( A B ) ( ¬ B ¬ A ) (A\to B)\to(\neg B\to\neg A)
  24. ( ¬ B ¬ A ) and A (\neg B\to\neg A)\and A
  25. ( ¬ B ¬ A ) ( A B ) (\neg B\to\neg A)\to(A\to B)
  26. ( A B ) ( ¬ B ¬ A ) (A\to B)\iff(\neg B\to\neg A)
  27. ( P Q ) (P\to Q)
  28. ¬ ( P and ¬ Q ) \neg(P\and\neg Q)
  29. ¬ ( ¬ Q and P ) \neg(\neg Q\and P)
  30. R R
  31. ¬ Q \neg Q
  32. S S
  33. ¬ P \neg P
  34. ¬ S \neg S
  35. ¬ ¬ P \neg\neg P
  36. P P
  37. ¬ ( R and ¬ S ) \neg(R\and\neg S)
  38. ( R S ) (R\to S)
  39. ( ¬ Q ¬ P ) (\neg Q\to\neg P)
  40. 2 \sqrt{2}
  41. 2 \sqrt{2}
  42. 2 \sqrt{2}
  43. 2 \sqrt{2}

Convergence_problem.html

  1. x = b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + . x=b_{0}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\cfrac{a_{% 4}}{b_{4}+\ddots}}}}.\,
  2. x = a 1 b 1 + a 2 b 2 + b k - 1 + a k b k + a 1 b 1 + a 2 b 2 + x=\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{\ddots}{\quad\ddots\quad b_{k% -1}+\cfrac{a_{k}}{b_{k}+\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\ddots}}}}}}\,
  3. s ( w ) = A k - 1 w + A k B k - 1 w + B k s(w)=\frac{A_{k-1}w+A_{k}}{B_{k-1}w+B_{k}}\,
  4. B k - 1 w 2 + ( B k - A k - 1 ) w - A k = 0. B_{k-1}w^{2}+(B_{k}-A_{k-1})w-A_{k}=0.\,
  5. x = K 1 a b , x=\underset{1}{\overset{\infty}{\mathrm{K}}}\frac{a}{b},\,
  6. y = 1 + K 1 z 1 ( z = a b 2 ) y=1+\underset{1}{\overset{\infty}{\mathrm{K}}}\frac{z}{1}\qquad\left(z=\frac{a% }{b^{2}}\right)\,
  7. y = 1 + z 1 + z 1 + z 1 + y=1+\cfrac{z}{1+\cfrac{z}{1+\cfrac{z}{1+\ddots}}}\,
  8. y = 1 2 ( 1 ± 4 z + 1 ) y=\frac{1}{2}\left(1\pm\sqrt{4z+1}\right)\,
  9. x = K 1 1 z = 1 z + 1 z + 1 z + x=\underset{1}{\overset{\infty}{\mathrm{K}}}\frac{1}{z}=\cfrac{1}{z+\cfrac{1}{% z+\cfrac{1}{z+\ddots}}}\,
  10. x = z - 1 1 + z - 2 1 + z - 2 1 + x=\cfrac{z^{-1}}{1+\cfrac{z^{-2}}{1+\cfrac{z^{-2}}{1+\ddots}}}\,
  11. x = K 1 1 z x=\underset{1}{\overset{\infty}{\mathrm{K}}}\frac{1}{z}
  12. x = 1 1 + a 2 1 + a 3 1 + a 4 1 + x=\cfrac{1}{1+\cfrac{a_{2}}{1+\cfrac{a_{3}}{1+\cfrac{a_{4}}{1+\ddots}}}}\,
  13. Ω = { z : | z - 4 / 3 | 2 / 3 } . \Omega=\{z:|z-4/3|\leq 2/3\}.\,
  14. f ( z ) = 1 1 + c 2 z 1 + c 3 z 1 + c 4 z 1 + f(z)=\cfrac{1}{1+\cfrac{c_{2}z}{1+\cfrac{c_{3}z}{1+\cfrac{c_{4}z}{1+\ddots}}}}\,
  15. | z | < 1 4 M |z|<\frac{1}{4M}\,
  16. | b n | | a n | + 1 |b_{n}|\geq|a_{n}|+1
  17. n 1. n\geq 1.
  18. - π / 2 + ϵ < arg ( b i ) < π / 2 - ϵ , i 1 , -\pi/2+\epsilon<\arg(b_{i})<\pi/2-\epsilon,i\geq 1,
  19. π / 2 \pi/2
  20. π - 2 ϵ \pi-2\epsilon
  21. - π / 2 + ϵ < arg ( f i ) < π / 2 - ϵ , i 1. -\pi/2+\epsilon<\arg(f_{i})<\pi/2-\epsilon,i\geq 1.

Converse_nonimplication.html

  1. p q {}_{p\not\subset q}\!
  2. ( p q ) {}_{\sim(p\subset q)}\!
  3. p q {}_{p\not\subset q}\!
  4. {}_{\not\subset}\!
  5. p q {}_{p\not\subset q}\!
  6. p ~ q {}_{p\tilde{\leftarrow}q}\!
  7. ~ {}_{\tilde{\leftarrow}}\!
  8. {}_{\leftarrow}\!
  9. {}_{\sim}\!
  10. M p q {}_{Mpq}\!
  11. p q {}_{p\nleftarrow q}\!
  12. {}_{\nleftarrow}\!
  13. {}_{\leftarrow}\!
  14. / {}_{/}\!
  15. q p = q p {}_{q\nleftarrow p=q^{\prime}p}\!
  16. {}_{\sim}\!
  17. {}_{{}_{\vee}}\!
  18. {}_{{}_{\wedge}}\!
  19. x {}_{\sim x}\!
  20. 1 {}_{1}\!
  21. 0 {}_{0}\!
  22. x {}_{x}\!
  23. 0 {}_{0}\!
  24. 1 {}_{1}\!
  25. y {}_{y}\!
  26. 1 {}_{1}\!
  27. 1 {}_{1}\!
  28. 1 {}_{1}\!
  29. 0 {}_{0}\!
  30. 0 {}_{0}\!
  31. 1 {}_{1}\!
  32. y x {}_{y_{\vee}x}\!
  33. 0 {}_{0}\!
  34. 1 {}_{1}\!
  35. x {}_{x}\!
  36. y {}_{y}\!
  37. 1 {}_{1}\!
  38. 0 {}_{0}\!
  39. 1 {}_{1}\!
  40. 0 {}_{0}\!
  41. 0 {}_{0}\!
  42. 0 {}_{0}\!
  43. y x {}_{y_{\wedge}x}\!
  44. 0 {}_{0}\!
  45. 1 {}_{1}\!
  46. x {}_{x}\!
  47. y x {}_{y\nleftarrow x}\!
  48. y {}_{y}\!
  49. 1 {}_{1}\!
  50. 0 {}_{0}\!
  51. 0 {}_{0}\!
  52. 0 {}_{0}\!
  53. 0 {}_{0}\!
  54. 1 {}_{1}\!
  55. y x {}_{y\nleftarrow x}\!
  56. 0 {}_{0}\!
  57. 1 {}_{1}\!
  58. x {}_{x}\!
  59. c {}_{{}^{c}}\!
  60. {}_{{}_{\vee}}\!
  61. {}_{{}_{\wedge}}\!
  62. x c {}_{x^{c}}\!
  63. 6 {}_{6}\!
  64. 3 {}_{3}\!
  65. 2 {}_{2}\!
  66. 1 {}_{1}\!
  67. x {}_{x}\!
  68. 1 {}_{1}\!
  69. 2 {}_{2}\!
  70. 3 {}_{3}\!
  71. 6 {}_{6}\!
  72. y {}_{y}\!
  73. 6 {}_{6}\!
  74. 6 {}_{6}\!
  75. 6 {}_{6}\!
  76. 6 {}_{6}\!
  77. 6 {}_{6}\!
  78. 3 {}_{3}\!
  79. 3 {}_{3}\!
  80. 6 {}_{6}\!
  81. 3 {}_{3}\!
  82. 6 {}_{6}\!
  83. 2 {}_{2}\!
  84. 2 {}_{2}\!
  85. 2 {}_{2}\!
  86. 6 {}_{6}\!
  87. 6 {}_{6}\!
  88. 1 {}_{1}\!
  89. 1 {}_{1}\!
  90. 2 {}_{2}\!
  91. 3 {}_{3}\!
  92. 6 {}_{6}\!
  93. y x {}_{y_{\vee}x}\!
  94. 1 {}_{1}\!
  95. 2 {}_{2}\!
  96. 3 {}_{3}\!
  97. 6 {}_{6}\!
  98. x {}_{x}\!
  99. y {}_{y}\!
  100. 6 {}_{6}\!
  101. 1 {}_{1}\!
  102. 2 {}_{2}\!
  103. 3 {}_{3}\!
  104. 6 {}_{6}\!
  105. 3 {}_{3}\!
  106. 1 {}_{1}\!
  107. 1 {}_{1}\!
  108. 3 {}_{3}\!
  109. 3 {}_{3}\!
  110. 2 {}_{2}\!
  111. 1 {}_{1}\!
  112. 2 {}_{2}\!
  113. 1 {}_{1}\!
  114. 2 {}_{2}\!
  115. 1 {}_{1}\!
  116. 1 {}_{1}\!
  117. 1 {}_{1}\!
  118. 1 {}_{1}\!
  119. 1 {}_{1}\!
  120. y x {}_{y_{\wedge}x}\!
  121. 1 {}_{1}\!
  122. 2 {}_{2}\!
  123. 3 {}_{3}\!
  124. 6 {}_{6}\!
  125. x {}_{x}\!
  126. y x {}_{y\nleftarrow x}\!
  127. y {}_{y}\!
  128. 6 {}_{6}\!
  129. 1 {}_{1}\!
  130. 1 {}_{1}\!
  131. 1 {}_{1}\!
  132. 1 {}_{1}\!
  133. 3 {}_{3}\!
  134. 1 {}_{1}\!
  135. 2 {}_{2}\!
  136. 1 {}_{1}\!
  137. 2 {}_{2}\!
  138. 2 {}_{2}\!
  139. 1 {}_{1}\!
  140. 1 {}_{1}\!
  141. 3 {}_{3}\!
  142. 3 {}_{3}\!
  143. 1 {}_{1}\!
  144. 1 {}_{1}\!
  145. 2 {}_{2}\!
  146. 3 {}_{3}\!
  147. 6 {}_{6}\!
  148. y x {}_{y\nleftarrow x}\!
  149. 1 {}_{1}\!
  150. 2 {}_{2}\!
  151. 3 {}_{3}\!
  152. 6 {}_{6}\!
  153. x {}_{x}\!
  154. r ( q p ) = ( r q ) p {}_{r\nleftarrow(q\nleftarrow p)=(r\nleftarrow q)\nleftarrow p}\!
  155. r p = 0 {}_{rp=0}\!
  156. r = 0 {}_{r=0}\!
  157. p = 0 {}_{p=0}\!
  158. : : ( r q ) p = r q p (by definition) = ( r q ) p (by definition) = ( r + q ) p (De Morgan’s laws) = ( r + r q ) p (Absorption law) = r p + r q p = r p + r ( q p ) (by definition) = r p + r ( q p ) (by definition) ::\begin{aligned}\displaystyle(r\nleftarrow q)\nleftarrow p&\displaystyle=r^{% \prime}q\nleftarrow p\qquad\qquad\qquad~{}~{}~{}~{}\,\text{(by definition)}\\ &\displaystyle=(r^{\prime}q)^{\prime}p\qquad\qquad\qquad~{}~{}~{}~{}~{}~{}\,% \text{(by definition)}\\ &\displaystyle=(r+q^{\prime})p\qquad\qquad~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{(% De Morgan's laws)}\\ &\displaystyle=(r+r^{\prime}q^{\prime})p\qquad\qquad~{}~{}~{}~{}~{}~{}~{}\,% \text{(Absorption law)}\\ &\displaystyle=rp+r^{\prime}q^{\prime}p\\ &\displaystyle=rp+r^{\prime}(q\nleftarrow p)\qquad~{}~{}~{}~{}~{}~{}~{}~{}\,% \text{(by definition)}\\ &\displaystyle=rp+r\nleftarrow(q\nleftarrow p)\qquad~{}~{}~{}~{}\,\text{(by % definition)}\\ \end{aligned}
  159. r p = 0 {}_{rp=0}\!
  160. q p = p q {}_{q\nleftarrow p=p\nleftarrow q\,}\!
  161. q = p {}_{q=p\,}\!
  162. 0 {}_{0}\!
  163. 0 p = p {}_{0\nleftarrow p=p}\!
  164. p 0 = 0 {}_{p\nleftarrow 0=0}\!
  165. 1 p = 0 {}_{1\nleftarrow p=0}\!
  166. p 1 = p {}_{p\nleftarrow 1=p^{\prime}}\!
  167. p p = 0 {}_{p\nleftarrow p=0}\!
  168. q p {}_{q\rightarrow p}\!
  169. q p {}_{q\nleftarrow p}\!
  170. s .1 {}_{s.1\,}\!
  171. q ~ p = q p {}_{q\tilde{\leftarrow}p=q^{\prime}p\,}\!
  172. s .2 {}_{s.2\,}\!
  173. p ~ q = p q {}_{p\tilde{\leftarrow}q=p^{\prime}q\,}\!
  174. s .3 {}_{s.3\,}\!
  175. s .1 s .2 {}_{s.1\ s.2\,}\!
  176. q ~ p = p ~ q q p = q p {}_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\ \Leftrightarrow\ q^{\prime}p=qp% ^{\prime}\,}\!
  177. s .4 {}_{s.4\,}\!
  178. q {}_{q\,}\!
  179. = {}_{=\,}\!
  180. q .1 {}_{q.1\,}\!
  181. s .5 {}_{s.5\,}\!
  182. s .4. r i g h t {}_{s.4.right\,}\!
  183. = {}_{=\,}\!
  184. q . ( p + p ) {}_{q.(p+p^{\prime})\,}\!
  185. s .6 {}_{s.6\,}\!
  186. s .5. r i g h t {}_{s.5.right\,}\!
  187. = {}_{=\,}\!
  188. q p + q p {}_{qp+qp^{\prime}\,}\!
  189. s .7 {}_{s.7\,}\!
  190. s .4. l e f t = s .6. r i g h t {}_{s.4.left=s.6.right\,}\!
  191. q = q p + q p {}_{q=qp+qp^{\prime}\,}\!
  192. s .8 {}_{s.8\,}\!
  193. q p = q p {}_{q^{\prime}p=qp^{\prime}\,}\!
  194. {}_{\Rightarrow\,}\!
  195. q p + q p = q p + q p {}_{qp+qp^{\prime}=qp+q^{\prime}p\,}\!
  196. s .9 {}_{s.9\,}\!
  197. s .8 {}_{s.8\,}\!
  198. {}_{\Rightarrow\,}\!
  199. q . ( p + p ) = ( q + q ) . p {}_{q.(p+p^{\prime})=(q+q^{\prime}).p\,}\!
  200. s .10 {}_{s.10\,}\!
  201. s .9 {}_{s.9\,}\!
  202. {}_{\Rightarrow\,}\!
  203. q .1 = 1. p {}_{q.1=1.p\,}\!
  204. s .11 {}_{s.11\,}\!
  205. s .10. r i g h t {}_{s.10.right\,}\!
  206. {}_{\Rightarrow\,}\!
  207. q = p {}_{q=p\,}\!
  208. s .12 {}_{s.12\,}\!
  209. s .8 s .11 {}_{s.8\ s.11\,}\!
  210. q p = q p q = p {}_{q^{\prime}p=qp^{\prime}\ \Rightarrow\ q=p\,}\!
  211. s .13 {}_{s.13\,}\!
  212. q = p q p = q p {}_{q=p\ \Rightarrow\ q^{\prime}p=qp^{\prime}\,}\!
  213. s .14 {}_{s.14\,}\!
  214. s .12 s .13 {}_{s.12\ s.13\,}\!
  215. q = p q p = q p {}_{q=p\ \Leftrightarrow\ q^{\prime}p=qp^{\prime}\,}\!
  216. s .15 {}_{s.15\,}\!
  217. s .3 s .14 {}_{s.3\ s.14\,}\!
  218. q ~ p = p ~ q q = p {}_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\ \Leftrightarrow\ q=p\,}\!
  219. s .1 {}_{s.1\,}\!
  220. d u a l ( q ~ p ) {}_{dual(q\tilde{\leftarrow}p)\,}\!
  221. = {}_{=\,}\!
  222. d u a l ( q p ) {}_{dual(q^{\prime}p)\,}\!
  223. s .2 {}_{s.2\,}\!
  224. s .1. r i g h t {}_{s.1.right\,}\!
  225. = {}_{=\,}\!
  226. q + p {}_{q^{\prime}+p\,}\!
  227. s .3 {}_{s.3\,}\!
  228. s .2. r i g h t {}_{s.2.right\,}\!
  229. = {}_{=\,}\!
  230. ( q + p ) ′′ {}_{(q^{\prime}+p)^{\prime\prime}\,}\!
  231. s .4 {}_{s.4\,}\!
  232. s .3. r i g h t {}_{s.3.right\,}\!
  233. = {}_{=\,}\!
  234. ( q p ) {}_{(qp^{\prime})^{\prime}\,}\!
  235. s .5 {}_{s.5\,}\!
  236. s .4. r i g h t {}_{s.4.right\,}\!
  237. = {}_{=\,}\!
  238. ( p q ) {}_{(p^{\prime}q)^{\prime}\,}\!
  239. s .6 {}_{s.6\,}\!
  240. s .5. r i g h t {}_{s.5.right\,}\!
  241. = {}_{=\,}\!
  242. ( p ~ q ) {}_{(p\tilde{\leftarrow}q)^{\prime}\,}\!
  243. s .7 {}_{s.7\,}\!
  244. s .6. r i g h t {}_{s.6.right\,}\!
  245. = {}_{=\,}\!
  246. p q {}_{p\leftarrow q\,}\!
  247. s .8 {}_{s.8\,}\!
  248. s .7. r i g h t {}_{s.7.right\,}\!
  249. = {}_{=\,}\!
  250. q p {}_{q\rightarrow p\,}\!
  251. s .9 {}_{s.9\,}\!
  252. s .1. l e f t = s .8. r i g h t {}_{s.1.left=s.8.right\,}\!
  253. d u a l ( q ~ p ) = q p {}_{dual(q\tilde{\leftarrow}p)=q\rightarrow p\,}\!

Convex_body.html

  1. \R n \R^{n}

Convex_hull_algorithms.html

  1. x 1 , , x n x_{1},\dots,x_{n}
  2. ( x 1 , x 1 2 ) , , ( x n , x n 2 ) (x_{1},x^{2}_{1}),\dots,(x_{n},x^{2}_{n})
  3. x 1 , , x n x_{1},\dots,x_{n}
  4. v 1 , , v n v_{1},...,v_{n}
  5. O ( n ) \ O(n)
  6. h 1 h_{1}
  7. h k - 1 , h k , v i h_{k-1},h_{k},v_{i}
  8. h k + 1 = v i h_{k+1}=v_{i}
  9. h k h_{k}
  10. h k - 2 , h k - 1 , v i h_{k-2},h_{k-1},v_{i}
  11. h 1 h_{1}

Convex_lattice_polytope.html

  1. A A
  2. a x y + b x 2 + c y 5 + d axy+bx^{2}+cy^{5}+d
  3. a , b , c , d 0 a,b,c,d\neq 0
  4. conv ( { ( 1 , 1 ) , ( 2 , 0 ) , ( 0 , 5 ) , ( 0 , 0 ) } ) . {\rm conv}(\{(1,1),(2,0),(0,5),(0,0)\}).

Convex_metric_space.html

  1. d ( x , z ) + d ( z , y ) = d ( x , y ) , d(x,z)+d(z,y)=d(x,y),\,
  2. x x
  3. y y
  4. z z
  5. x x
  6. y , y,
  7. x x
  8. y , y,
  9. ( X , d ) (X,d)
  10. S S
  11. X X
  12. x x
  13. y y
  14. X , X,
  15. [ a , b ] [a,b]
  16. γ : [ a , b ] X , \gamma:[a,b]\to X,\,
  17. γ ( [ a , b ] ) = S , \gamma([a,b])=S,
  18. γ ( a ) = x \gamma(a)=x
  19. γ ( b ) = y . \gamma(b)=y.
  20. S S
  21. x x
  22. y y
  23. x x
  24. y . y.
  25. ( X , d ) (X,d)
  26. ( X , d ) (X,d)
  27. x y x\neq y
  28. X X
  29. x x
  30. y y
  31. { x , y } \{x,y\}

Convolution_power.html

  1. x x
  2. n n
  3. x * n = x * x * x * * x * x n , x * 0 = δ 0 x^{*n}=\underbrace{x*x*x*\cdots*x*x}_{n},\quad x^{*0}=\delta_{0}
  4. P ( x * n σ n < β ) Φ ( β ) as n P\left(\frac{x^{*n}}{\sigma\sqrt{n}}<\beta\right)\to\Phi(\beta)\quad\rm{as}\ n\to\infty
  5. x * n / σ n x^{*n}/\sigma\sqrt{n}
  6. μ 1 / n * n = μ . \mu_{1/n}^{*n}=\mu.
  7. π α , μ = e - α n = 0 α n n ! μ * n . \pi_{\alpha,\mu}=e^{-\alpha}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}\mu^{*n}.
  8. F ( z ) = n = 0 a n z n \textstyle{F(z)=\sum_{n=0}^{\infty}a_{n}z^{n}}
  9. F * ( x ) = a 0 δ 0 + n = 1 a n x * n . F^{*}(x)=a_{0}\delta_{0}+\sum_{n=1}^{\infty}a_{n}x^{*n}.
  10. exp * ( x ) = δ 0 + n = 1 x * n n ! . \exp^{*}(x)=\delta_{0}+\sum_{n=1}^{\infty}\frac{x^{*n}}{n!}.
  11. x = log * ( exp * x ) = exp * ( log * x ) x=\log^{*}(\exp^{*}x)=\exp^{*}(\log^{*}x)
  12. 𝒟 { x * n } = ( 𝒟 x ) * x * ( n - 1 ) = x * 𝒟 { x * ( n - 1 ) } \mathcal{D}\big\{x^{*n}\big\}=(\mathcal{D}x)*x^{*(n-1)}=x*\mathcal{D}\big\{x^{% *(n-1)}\big\}
  13. 𝒟 \mathcal{D}

Convolution_random_number_generator.html

  1. X Erlang ( k , θ ) X\ \sim\operatorname{Erlang}(k,\theta)
  2. Exp ( k θ ) \operatorname{Exp}(k\theta)\,
  3. E [ X ] = 1 k θ + 1 k θ + + 1 k θ = 1 θ . \operatorname{E}[X]=\frac{1}{k\theta}+\frac{1}{k\theta}+\cdots+\frac{1}{k% \theta}=\frac{1}{\theta}.
  4. Erlang ( k , θ ) \operatorname{Erlang}(k,\theta)
  5. X i Exp ( k θ ) X_{i}\ \sim\operatorname{Exp}(k\theta)
  6. X = i = 1 k X i Erlang ( k , θ ) . X=\sum_{i=1}^{k}X_{i}\sim\operatorname{Erlang}(k,\theta).

Conway_base_13_function.html

  1. x x\in\mathbb{R}
  2. x x
  3. 0 ¯ \underline{0}
  4. 1 ¯ \underline{1}
  5. 2 ¯ \underline{2}
  6. 8 ¯ \underline{8}
  7. 9 ¯ \underline{9}
  8. + ¯ \underline{+}
  9. - ¯ \underline{-}
  10. ¯ \underline{\cdot}
  11. ¯ \underline{\cdot}
  12. x x
  13. r r
  14. f ( x ) = r f(x)=r
  15. f ( x ) = 0 f(x)=0
  16. f ( 7 + 1 ¯ . 4 + 3 14159 ¯ ) = f ( 7 + 14 + 3 141 ¯ . 59 ¯ ) = π f(\underline{7{+}{\cdot}1}\,.\,\underline{4{+}3{\cdot}14159\ldots})=f(% \underline{7{+}{\cdot}14{+}3{\cdot}141}\,.\,\underline{59\ldots})=\pi
  17. + ¯ \underline{+}
  18. ¯ \underline{\cdot}
  19. f ( x ) = r f(x)=r
  20. + ¯ \underline{+}
  21. - ¯ \underline{-}
  22. ¯ \underline{\cdot}
  23. f f
  24. [ a , b ] [a,b]
  25. f f
  26. f ( a ) f(a)
  27. f ( b ) f(b)
  28. f f
  29. ( a , b ) (a,b)
  30. c ( a , b ) c\in(a,b)
  31. r r
  32. c c
  33. r ¯ \underline{r}
  34. r r
  35. r r
  36. c c^{\prime}
  37. c c
  38. c c^{\prime}
  39. ( a , b ) (a,b)
  40. f ( c ) = r f(c^{\prime})=r
  41. f f
  42. f f
  43. f f
  44. f f

Cooperative_diversity.html

  1. x s x_{s}
  2. r d , s = h d , s x s + n d , s r_{d,s}=h_{d,s}x_{s}+n_{d,s}\quad
  3. r r , s = h r , s x s + n r , s r_{r,s}=h_{r,s}x_{s}+n_{r,s}\quad
  4. h d , s h_{d,s}
  5. h r , s h_{r,s}
  6. n r , s n_{r,s}
  7. h r , s h_{r,s}
  8. n d , s n_{d,s}
  9. h d , s h_{d,s}
  10. r d , s = h d , s x s + n d , s r_{d,s}=h_{d,s}x_{s}+n_{d,s}\quad
  11. r d , r = h d , r r r , s + n d , r = h d , r h r , s x s + h d , r n r , s + n d , r r_{d,r}=h_{d,r}r_{r,s}+n_{d,r}=h_{d,r}h_{r,s}x_{s}+h_{d,r}n_{r,s}+n_{d,r}\quad
  12. h d , r h_{d,r}
  13. n r , s n_{r,s}
  14. h d , r h_{d,r}
  15. 𝐫 = [ r d , s r d , r ] T = [ h d , s h d , r h r , s ] T x s + [ 1 | h d , r | 2 + 1 ] T n d = 𝐡 x s + 𝐪 n d \mathbf{r}=[r_{d,s}\quad r_{d,r}]^{T}=[h_{d,s}\quad h_{d,r}h_{r,s}]^{T}x_{s}+% \left[1\quad\sqrt{|h_{d,r}|^{2}+1}\right]^{T}n_{d}=\mathbf{h}x_{s}+\mathbf{q}n% _{d}
  16. r d , s r_{d,s}
  17. r d , r r_{d,r}
  18. y = 𝐰 H 𝐫 y=\mathbf{w}^{H}\mathbf{r}
  19. 𝐰 \mathbf{w}
  20. c 21 e j φ 21 , c 31 e j φ 31 , c 32 e j φ 32 c_{21}e^{j\varphi_{21}},c_{31}e^{j\varphi_{31}},c_{32}e^{j\varphi_{32}}
  21. C + = max f ( X 1 , X 2 ) min { I ( X 1 ; Y 2 , Y 3 | X 2 ) , I ( X 1 , X 2 ; Y 3 ) } C^{+}=\max_{f(X_{1},X_{2})}\min\{I(X_{1};Y_{2},Y_{3}|X_{2}),I(X_{1},X_{2};Y_{3% })\}
  22. X 1 X_{1}
  23. X 2 X_{2}
  24. Y 2 Y_{2}
  25. Y 3 Y_{3}
  26. X 1 X_{1}
  27. Y 2 Y_{2}
  28. Y 3 Y_{3}
  29. X 2 X_{2}
  30. max f ( X 1 , X 2 ) I ( X 1 ; Y 2 , Y 3 | X 2 ) = 1 2 log ( 1 + ( 1 - β ) ( c 21 2 + c 31 2 ) P 1 ) \max_{f(X_{1},X_{2})}I(X_{1};Y_{2},Y_{3}|X_{2})=\frac{1}{2}\log(1+(1-\beta)(c^% {2}_{21}+c^{2}_{31})P_{1})
  31. X 1 X_{1}
  32. X 2 X_{2}
  33. Y 3 Y_{3}
  34. max f ( X 1 , X 2 ) I ( X 2 , X 2 ; Y 3 ) = 1 2 log ( 1 + c 31 2 P 1 + c 32 2 P 2 + 2 β c 31 2 c 32 2 P 1 P 2 ) \max_{f(X_{1},X_{2})}I(X_{2},X_{2};Y_{3})=\frac{1}{2}\log(1+c^{2}_{31}P_{1}+c^% {2}_{32}P_{2}+2\sqrt{\beta c^{2}_{31}c^{2}_{32}P_{1}P_{2}})
  35. β \beta
  36. X 1 X_{1}
  37. X 2 X_{2}
  38. X 2 X_{2}
  39. X 1 X_{1}
  40. C + = max 0 β 1 min { 1 2 log ( 1 + ( 1 - β ) ( c 21 2 + c 31 2 ) P 1 ) , 1 2 log ( 1 + c 31 2 P 1 + c 32 2 P 2 + 2 β c 31 2 c 32 2 P 1 P 2 ) } C^{+}=\max_{0\leq\beta\leq 1}\min\left\{\frac{1}{2}\log(1+(1-\beta)(c^{2}_{21}% +c^{2}_{31})P_{1}),\frac{1}{2}\log(1+c^{2}_{31}P_{1}+c^{2}_{32}P_{2}+2\sqrt{% \beta c^{2}_{31}c^{2}_{32}P_{1}P_{2}})\right\}
  41. R 1 = max f ( X 1 , X 2 ) min { I ( X 1 ; Y 2 | X 2 ) , I ( X 1 , X 2 ; Y 3 ) } R_{1}=\max_{f(X_{1},X_{2})}\min\{I(X_{1};Y_{2}|X_{2}),I(X_{1},X_{2};Y_{3})\}
  42. I ( X 1 ; Y 2 , Y 3 | X 2 ) I(X_{1};Y_{2},Y_{3}|X_{2})
  43. I ( X 1 ; Y 2 | X 2 ) I(X_{1};Y_{2}|X_{2})
  44. max f ( X 1 , X 2 ) I ( X 1 ; Y 2 | X 2 ) = 1 2 log ( 1 + ( 1 - β ) c 21 2 P 1 ) . \max_{f(X_{1},X_{2})}I(X_{1};Y_{2}|X_{2})=\frac{1}{2}\log(1+(1-\beta)c^{2}_{21% }P_{1}).
  45. R 1 = max 0 β 1 min { 1 2 log ( 1 + ( 1 - β ) c 21 2 P 1 ) , 1 2 log ( 1 + c 31 2 P 1 + c 32 2 P 2 + 2 β c 31 2 c 32 2 P 1 P 2 ) } R_{1}=\max_{0\leq\beta\leq 1}\min\left\{\frac{1}{2}\log(1+(1-\beta)c^{2}_{21}P% _{1}),\frac{1}{2}\log(1+c^{2}_{31}P_{1}+c^{2}_{32}P_{2}+2\sqrt{\beta c^{2}_{31% }c^{2}_{32}P_{1}P_{2}})\right\}
  46. C + = max 0 β 1 min { C 1 + ( β ) , C 2 + ( β ) } C^{+}=\max_{0\leq\beta\leq 1}\min\{C_{1}^{+}(\beta),C_{2}^{+}(\beta)\}
  47. C 1 + ( β ) = α 2 log ( 1 + ( c 31 2 + c 21 2 ) P 1 ( 1 ) ) + 1 - α 2 log ( 1 + ( 1 - β ) c 31 2 P 1 ( 2 ) ) C_{1}^{+}(\beta)=\frac{\alpha}{2}\log\left(1+(c_{31}^{2}+c_{21}^{2})P_{1}^{(1)% }\right)+\frac{1-\alpha}{2}\log\left(1+(1-\beta)c_{31}^{2}P_{1}^{(2)}\right)
  48. C 2 + ( β ) = α 2 log ( 1 + c 31 2 P 1 ( 1 ) ) + 1 - α 2 log ( 1 + c 31 2 P 1 ( 2 ) + c 32 2 P 2 + 2 β C 31 2 P 1 ( 2 ) C 32 2 P 2 ) C_{2}^{+}(\beta)=\frac{\alpha}{2}\log\left(1+c_{31}^{2}P_{1}^{(1)}\right)+% \frac{1-\alpha}{2}\log\left(1+c_{31}^{2}P_{1}^{(2)}+c_{32}^{2}P_{2}+2\sqrt{% \beta C_{31}^{2}P_{1}^{(2)}C_{32}^{2}P_{2}}\right)

Coordination_number.html

  1. g ( r ) g(r)
  2. n 1 = 4 π r 0 r 1 r 2 g ( r ) ρ d r , n_{1}=4\pi\int_{r_{0}}^{r_{1}}r^{2}g(r)\rho\,dr,
  3. r 0 r_{0}
  4. r = 0 r=0
  5. g ( r ) g(r)
  6. r 1 r_{1}
  7. g ( r ) g(r)
  8. n 2 = 4 π r 1 r 2 r 2 g ( r ) ρ d r . n_{2}=4\pi\int_{r_{1}}^{r_{2}}r^{2}g(r)\rho\,dr.
  9. r p r_{p}
  10. n 1 = 4 π × 2 r 0 r p r 2 g ( r ) ρ d r . n_{1}^{\prime}=4\pi\times 2\int_{r_{0}}^{r_{p}}r^{2}g(r)\rho\,dr.
  11. r 0 r_{0}
  12. r 1 r_{1}

Cophasing.html

  1. λ / 40 \lambda/40

Cophenetic_correlation.html

  1. x ¯ \bar{x}
  2. t ¯ \bar{t}
  3. c = i < j ( x ( i , j ) - x ¯ ) ( t ( i , j ) - t ¯ ) [ i < j ( x ( i , j ) - x ¯ ) 2 ] [ i < j ( t ( i , j ) - t ¯ ) 2 ] . c=\frac{\sum_{i<j}(x(i,j)-\bar{x})(t(i,j)-\bar{t})}{\sqrt{[\sum_{i<j}(x(i,j)-% \bar{x})^{2}][\sum_{i<j}(t(i,j)-\bar{t})^{2}]}}.

Copositive_matrix.html

  1. x T A x 0 x^{T}Ax\geq 0
  2. x 0 x\geq 0

Copper_cable_certification.html

  1. 2 2
  2. 2 \sqrt{2}

Coprecipitation.html

  1. ln a a - x = λ ln b b - y \ln{a\over{a-x}}=\lambda\ln{b\over{b-y}}
  2. x a - x = D y b - y {x\over{a-x}}=D{y\over{b-y}}

Coptic_versions_of_the_Bible.html

  1. 𝔓 \mathfrak{P}

Copying_mechanism.html

  1. p p
  2. 1 - p 1-p
  3. k k
  4. ( 1 - p ) k (1-p)k
  5. P ( k i n ) = k - ( 2 - p ) / ( 1 - p ) P(k_{in})=k^{-(2-p)/(1-p)}
  6. P ( k ) P(k)
  7. p 0 p\rightarrow 0
  8. \infty
  9. p 1 p\rightarrow 1
  10. L N L_{N}

Core_(graph_theory).html

  1. G G
  2. f : G G f:G\to G
  3. G G
  4. H H
  5. G G
  6. H H
  7. G G
  8. G G
  9. H H
  10. G G
  11. G H G\to H
  12. H G H\to G
  13. G G
  14. H H

CORR.html

  1. corr ( X , Y ) = ρ X , Y \mathrm{corr}(X,Y)=\rho_{X,Y}\,

Correlate_summation_analysis.html

  1. t = | r | 1 - r 2 n - 2 t=\frac{|r|}{\sqrt{\frac{1-r^{2}}{n-2}}}
  2. y = m x a y=mx^{a}

Correspondence_analysis.html

  1. χ 2 \chi^{2}
  2. w m = ( 1 C 1 ) - 1 C 1 w_{m}=(1C1)^{-1}C1
  3. w n = ( 1 C 1 ) - 1 1 C w_{n}=(1C1)^{-1}1C
  4. S = ( 1 C 1 ) - 1 C S=(1C1)^{-1}C
  5. M = S - w m w n * M=S-w_{m}w_{n}^{*}
  6. w n * w_{n}^{*}
  7. w n w_{n}
  8. W m = d i a g { w m } W_{m}=diag\{w_{m}\}
  9. W n = d i a g { w n } W_{n}=diag\{w_{n}\}
  10. W n W_{n}
  11. w n w_{n}
  12. M = U Σ V * M=U\Sigma V^{*}\,
  13. U * W m U = V * W n V = I . U^{*}W_{m}U=V^{*}W_{n}V=I.
  14. F m = W m U Σ F_{m}=W_{m}U\Sigma
  15. F n = W n V Σ F_{n}=W_{n}V\Sigma

Cosine_similarity.html

  1. D C ( A , B ) = 1 - S C ( A , B ) D_{C}(A,B)=1-S_{C}(A,B)
  2. 𝐚 𝐛 = 𝐚 𝐛 cos θ \mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta
  3. similarity = cos ( θ ) = A B A B = i = 1 n A i × B i i = 1 n ( A i ) 2 × i = 1 n ( B i ) 2 \,\text{similarity}=\cos(\theta)={A\cdot B\over\|A\|\|B\|}=\frac{\sum\limits_{% i=1}^{n}{A_{i}\times B_{i}}}{\sqrt{\sum\limits_{i=1}^{n}{(A_{i})^{2}}}\times% \sqrt{\sum\limits_{i=1}^{n}{(B_{i})^{2}}}}
  4. A - A ¯ A-\bar{A}
  5. 1 - cos - 1 ( similarity ) π 1-\frac{\cos^{-1}(\,\text{similarity})}{\pi}
  6. 1 - 2 cos - 1 ( similarity ) π 1-\frac{2\cdot\cos^{-1}(\,\text{similarity})}{\pi}
  7. T ( A , B ) = A B A 2 + B 2 - A B T(A,B)={A\cdot B\over\|A\|^{2}+\|B\|^{2}-A\cdot B}
  8. K = n ( A B ) n ( A ) × n ( B ) K=\frac{n(A\cap B)}{\sqrt{n(A)\times n(B)}}
  9. A A
  10. B B
  11. n ( A ) n(A)
  12. A A
  13. A - B \|A-B\|
  14. A - B 2 = ( A - B ) ( A - B ) = A 2 + B 2 - 2 A B \|A-B\|^{2}=(A-B)^{\top}(A-B)=\|A\|^{2}+\|B\|^{2}-2A^{\top}B
  15. A A
  16. B B
  17. A 2 = B 2 = 1 \|A\|^{2}=\|B\|^{2}=1
  18. 2 ( 1 - cos ( A , B ) ) 2(1-\cos(A,B))
  19. 1 / n 1/n
  20. n n
  21. n n
  22. 𝐬 \mathbf{s}
  23. N N
  24. soft _ cosine 1 ( a , b ) = i , j N s i j a i b j i , j N s i j a i a j i , j N s i j b i b j , \displaystyle\operatorname{soft\_cosine}_{1}(a,b)=\frac{\sum\nolimits_{i,j}^{N% }s_{ij}a_{i}b_{j}}{\sqrt{\sum\nolimits_{i,j}^{N}s_{ij}a_{i}a_{j}}\sqrt{\sum% \nolimits_{i,j}^{N}s_{ij}b_{i}b_{j}}},
  25. s < s u b > i j = s i m i l a r i t y ( f e a t u r e i , f e a t u r e j ) s<sub>ij=similarity(feature_{i},feature_{j})

COSMO_Solvation_Model.html

  1. q = f ( ε ) q * . q=f(\varepsilon)q^{*}.
  2. f ( ε ) = ε - 1 ε + x , f(\varepsilon)=\frac{\varepsilon-1}{\varepsilon+x},

Cottrell_equation.html

  1. i = n F A c j 0 D j π t i=\frac{nFAc_{j}^{0}\sqrt{D_{j}}}{\sqrt{\pi t}}

Coulomb_damping.html

  1. F s = μ s N F_{s}=\mu_{s}N\,
  2. F k = μ k N F_{k}=\mu_{k}N\,
  3. F = k x F=kx
  4. m x ¨ = - k x - F m\ddot{x}\ =-kx-F
  5. m x ¨ = - k x + F m\ddot{x}\ =-kx+F
  6. x ¨ \ddot{x}

Counterpart_theory.html

  1. \Box
  2. \Box
  3. \Box
  4. \Box

Courant_algebroid.html

  1. T M T * M TM\oplus T^{*}M
  2. E M E\to M
  3. [ . , . ] : Γ E × Γ E Γ E [.,.]:\Gamma E\times\Gamma E\to\Gamma E
  4. . , . : E × E M × \R \langle.,.\rangle:E\times E\to M\times\R
  5. ρ : E T M \rho:E\to TM
  6. [ ϕ , [ χ , ψ ] ] = [ [ ϕ , χ ] , ψ ] + [ χ , [ ϕ , ψ ] ] [\phi,[\chi,\psi]]=[[\phi,\chi],\psi]+[\chi,[\phi,\psi]]
  7. [ ϕ , f ψ ] = ρ ( ϕ ) f ψ + f [ ϕ , ψ ] [\phi,f\psi]=\rho(\phi)f\psi+f[\phi,\psi]
  8. [ ϕ , ϕ ] = 1 2 D ϕ , ϕ [\phi,\phi]=\tfrac{1}{2}D\langle\phi,\phi\rangle
  9. ρ ( ϕ ) ψ , ψ = 2 [ ϕ , ψ ] , ψ \rho(\phi)\langle\psi,\psi\rangle=2\langle[\phi,\psi],\psi\rangle
  10. κ - 1 ρ T d \kappa^{-1}\rho^{T}d
  11. ρ T \rho^{T}
  12. ρ \rho
  13. E * E^{*}
  14. ρ [ ϕ , ψ ] = [ ρ ( ϕ ) , ρ ( ψ ) ] . \rho[\phi,\psi]=[\rho(\phi),\rho(\psi)].
  15. ρ ( ϕ ) χ , ψ = [ ϕ , χ ] , ψ + χ , [ ϕ , ψ ] . \rho(\phi)\langle\chi,\psi\rangle=\langle[\phi,\chi],\psi\rangle+\langle\chi,[% \phi,\psi]\rangle.
  16. T M T * M TM\oplus T^{*}M
  17. [ X + ξ , Y + η ] = [ X , Y ] + ( X η - i ( Y ) d ξ + i ( X ) i ( Y ) H ) [X+\xi,Y+\eta]=[X,Y]+(\mathcal{L}_{X}\,\eta-i(Y)d\xi+i(X)i(Y)H)
  18. A * A^{*}
  19. ρ A \rho_{A}
  20. [ . , . ] A [.,.]_{A}
  21. A * A^{*}
  22. d A * d_{A^{*}}
  23. * A \wedge^{*}A
  24. d A * [ X , Y ] A = [ d A * X , Y ] A + [ X , d A * Y ] A d_{A^{*}}[X,Y]_{A}=[d_{A^{*}}X,Y]_{A}+[X,d_{A^{*}}Y]_{A}
  25. * A \wedge^{*}A
  26. A * A^{*}
  27. E = A A * E=A\oplus A^{*}
  28. ρ ( X + α ) = ρ A ( X ) + ρ A * ( α ) \rho(X+\alpha)=\rho_{A}(X)+\rho_{A^{*}}(\alpha)
  29. [ X + α , Y + β ] = ( [ X , Y ] A + α A * Y - i β d A * X ) + ( [ α , β ] A * + X A β - i Y d A α ) [X+\alpha,Y+\beta]=([X,Y]_{A}+\mathcal{L}^{A^{*}}_{\alpha}Y-i_{\beta}d_{A^{*}}% X)+([\alpha,\beta]_{A^{*}}+\mathcal{L}^{A}_{X}\beta-i_{Y}d_{A}\alpha)
  30. [ [ ϕ , ψ ] ] = 1 2 ( [ ϕ , ψ ] - [ ψ , ϕ ] . ) [[\phi,\psi]]=\tfrac{1}{2}\big([\phi,\psi]-[\psi,\phi]\big.)
  31. [ [ ϕ , [ [ ψ , χ ] ] ] ] + cycl. = D T ( ϕ , ψ , χ ) [[\phi,[[\psi,\chi]]\,]]+\,\text{cycl.}=DT(\phi,\psi,\chi)
  32. T ( ϕ , ψ , χ ) = 1 3 [ ϕ , ψ ] , χ + cycl. T(\phi,\psi,\chi)=\frac{1}{3}\langle[\phi,\psi],\chi\rangle+\,\text{cycl.}
  33. [ [ ϕ , ψ ] ] = [ ϕ , ψ ] - 1 2 D ϕ , ψ [[\phi,\psi]]=[\phi,\psi]-\tfrac{1}{2}D\langle\phi,\psi\rangle
  34. ρ D = 0 \rho\circ D=0
  35. . , . \langle.,.\rangle
  36. T M T * M TM\oplus T^{*}M
  37. L , L 0 \langle L,L\rangle\equiv 0
  38. rk L = 1 2 rk E \mathrm{rk}\,L=\tfrac{1}{2}\mathrm{rk}\,E
  39. [ Γ L , Γ L ] Γ L [\Gamma L,\Gamma L]\subset\Gamma L
  40. Π Γ ( 2 T M ) \Pi\in\Gamma(\wedge^{2}TM)
  41. L L ¯ = 0 L\cap\bar{L}=0
  42. ¯ \bar{\ }

Covariance_function.html

  1. C ( x , y ) := cov ( Z ( x ) , Z ( y ) ) . C(x,y):=\operatorname{cov}(Z(x),Z(y)).\,
  2. X = i = 1 N w i Z ( x i ) X=\sum_{i=1}^{N}w_{i}Z(x_{i})
  3. var ( X ) = i = 1 N j = 1 N w i C ( x i , x j ) w j . \operatorname{var}(X)=\sum_{i=1}^{N}\sum_{j=1}^{N}w_{i}C(x_{i},x_{j})w_{j}.
  4. C ( x i , x j ) = C ( x i + h , x j + h ) C(x_{i},x_{j})=C(x_{i}+h,x_{j}+h)\,
  5. C s ( h ) = C ( 0 , h ) = C ( x , x + h ) C_{s}(h)=C(0,h)=C(x,x+h)\,
  6. C ( x , y ) = C s ( y - x ) . C(x,y)=C_{s}(y-x).\,
  7. C ( d ) = exp ( - d / V ) C(d)=\exp(-d/V)
  8. C ( d ) = exp ( - d 2 / V ) C(d)=\exp(-d^{2}/V)

Cover_tree.html

  1. C i C i - 1 C_{i}\subseteq C_{i-1}
  2. p C i - 1 p\in C_{i-1}
  3. q C i q\in C_{i}
  4. p p
  5. q q
  6. 2 i 2^{i}
  7. q q
  8. p p
  9. p , q C i p,q\in C_{i}
  10. p p
  11. q q
  12. 2 i 2^{i}
  13. O ( η * log n ) O(\eta*\log{n})
  14. η \eta
  15. O ( n ) O(n)
  16. n n
  17. η \eta
  18. O ( c 12 log n ) O(c^{12}\log{n})
  19. c c
  20. O ( c 6 log n ) O(c^{6}\log{n})

Craig's_theorem.html

  1. A 1 , A 2 , A_{1},A_{2},\dots
  2. A i A i i \underbrace{A_{i}\land\dots\land A_{i}}_{i}
  3. A 1 A_{1}
  4. B j B j j . \underbrace{B_{j}\land\dots\land B_{j}}_{j}.
  5. A 1 A_{1}
  6. A j A_{j}
  7. A n A_{n}
  8. A i A_{i}
  9. A i A i i \underbrace{A_{i}\land\dots\land A_{i}}_{i}
  10. A i A i f ( i ) \underbrace{A_{i}\land\dots\land A_{i}}_{f(i)}
  11. A i A_{i}
  12. A i A_{i}

Criticism_of_sport_utility_vehicles.html

  1. P a c c e l = m v e h i c l e a v {P_{accel}=m_{vehicle}\cdot a\cdot v}
  2. P a c c e l P_{accel}\,\!
  3. m v e h i c l e m_{vehicle}\,\!
  4. a {a}\,\!
  5. v {v}\,\!
  6. P d r a g = A c r o s s c w v e h i c l e v a i r 3 ρ a i r 2 {P_{drag}=A_{cross}\cdot cw_{vehicle}\cdot\frac{v_{air}^{3}\rho_{air}}{2}}
  7. P d r a g P_{drag}\,\!
  8. A c r o s s {A_{cross}}\,\!
  9. ρ a i r {\rho_{air}}\,\!
  10. v a i r v_{air}\,\!
  11. P r o l l = μ r o l l m v e h i c l e v {P_{roll}=\mu_{roll}\cdot m_{vehicle}\cdot v}
  12. μ r o l l \mu_{roll}\,\!
  13. m v e h i c l e m_{vehicle}\,\!

Cross_slope.html

  1. = R i s e R u n * 100 % =\frac{Rise}{Run}*100\%

Crystal_base.html

  1. U q ( G ) U_{q}(G)
  2. U q ( G ) U_{q}(G)
  3. ( q ) {\mathbb{Q}}(q)
  4. \mathbb{Q}
  5. α i \alpha_{i}
  6. n n
  7. e i ( n ) = e i n / [ n ] q i ! e_{i}^{(n)}=e_{i}^{n}/[n]_{q_{i}}!
  8. f i ( n ) = f i n / [ n ] q i ! f_{i}^{(n)}=f_{i}^{n}/[n]_{q_{i}}!
  9. e i ( 0 ) = f i ( 0 ) = 1 e_{i}^{(0)}=f_{i}^{(0)}=1
  10. M M
  11. λ \lambda
  12. u M λ u\in M_{\lambda}
  13. u u
  14. M M
  15. λ \lambda
  16. u = n = 0 f i ( n ) u n = n = 0 e i ( n ) v n , u=\sum_{n=0}^{\infty}f_{i}^{(n)}u_{n}=\sum_{n=0}^{\infty}e_{i}^{(n)}v_{n},
  17. u n ker ( e i ) M λ + n α i u_{n}\in\mathrm{ker}(e_{i})\cap M_{\lambda+n\alpha_{i}}
  18. v n ker ( f i ) M λ - n α i v_{n}\in\mathrm{ker}(f_{i})\cap M_{\lambda-n\alpha_{i}}
  19. u n 0 u_{n}\neq 0
  20. n + 2 ( λ , α i ) ( α i , α i ) 0 n+\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}\geq 0
  21. v n 0 v_{n}\neq 0
  22. n - 2 ( λ , α i ) ( α i , α i ) 0 n-\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}\geq 0
  23. e ~ i : M M \tilde{e}_{i}:M\to M
  24. f ~ i : M M \tilde{f}_{i}:M\to M
  25. M λ M_{\lambda}
  26. e ~ i u = n = 1 f i ( n - 1 ) u n = n = 0 e i ( n + 1 ) v n , \tilde{e}_{i}u=\sum_{n=1}^{\infty}f_{i}^{(n-1)}u_{n}=\sum_{n=0}^{\infty}e_{i}^% {(n+1)}v_{n},
  27. f ~ i u = n = 0 f i ( n + 1 ) u n = n = 1 e i ( n - 1 ) . v n \tilde{f}_{i}u=\sum_{n=0}^{\infty}f_{i}^{(n+1)}u_{n}=\sum_{n=1}^{\infty}e_{i}^% {(n-1)}.v_{n}
  28. A A
  29. ( q ) {\mathbb{Q}}(q)
  30. q = 0 q=0
  31. f ( q ) f(q)
  32. A A
  33. g ( q ) g(q)
  34. h ( q ) h(q)
  35. [ q ] {\mathbb{Q}}[q]
  36. h ( 0 ) 0 h(0)\neq 0
  37. f ( q ) = g ( q ) / h ( q ) f(q)=g(q)/h(q)
  38. M M
  39. ( L , B ) (L,B)
  40. L L
  41. A A
  42. M M
  43. M = ( q ) A L ; M={\mathbb{Q}}(q)\otimes_{A}L;
  44. B B
  45. \mathbb{Q}
  46. L / q L L/qL
  47. , \mathbb{Q},
  48. L = λ L λ L=\oplus_{\lambda}L_{\lambda}
  49. B = λ B λ B=\sqcup_{\lambda}B_{\lambda}
  50. L λ = L M λ L_{\lambda}=L\cap M_{\lambda}
  51. B λ = B ( L λ / q L λ ) , B_{\lambda}=B\cap(L_{\lambda}/qL_{\lambda}),
  52. e ~ i L L \tilde{e}_{i}L\subset L
  53. f ~ i L L for all i , \tilde{f}_{i}L\subset L\,\text{ for all }i,
  54. e ~ i B B { 0 } \tilde{e}_{i}B\subset B\cup\{0\}
  55. f ~ i B B { 0 } for all i , \tilde{f}_{i}B\subset B\cup\{0\}\,\text{ for all }i,
  56. for all b B and b B , and for all i , e ~ i b = b if and only if f ~ i b = b . \,\text{for all }b\in B\,\text{ and }b^{\prime}\in B,\,\text{ and for all }i,% \quad\tilde{e}_{i}b=b^{\prime}\,\text{ if and only if }\tilde{f}_{i}b^{\prime}% =b.
  57. e i f i e_{i}f_{i}
  58. f i e i f_{i}e_{i}
  59. q = 0 q=0
  60. M M
  61. e ~ i \tilde{e}_{i}
  62. f ~ i \tilde{f}_{i}
  63. e ~ i f ~ i \tilde{e}_{i}\tilde{f}_{i}
  64. f ~ i e ~ i \tilde{f}_{i}\tilde{e}_{i}
  65. q = 0 q=0
  66. ( q ) {\mathbb{Q}}(q)
  67. B ~ \tilde{B}
  68. M M
  69. e ~ i \tilde{e}_{i}
  70. f ~ i \tilde{f}_{i}
  71. q = 0 q=0
  72. A A
  73. A A
  74. e ~ i \tilde{e}_{i}
  75. f ~ i \tilde{f}_{i}
  76. q = 0 q=0
  77. q = 0 q=0
  78. i i
  79. e ~ i \tilde{e}_{i}
  80. f ~ i \tilde{f}_{i}
  81. \mathbb{Q}
  82. B B
  83. L / q L L/qL
  84. v 1 v_{1}
  85. v 2 v_{2}
  86. b 2 = f ~ i b 1 b_{2}=\tilde{f}_{i}b_{1}
  87. b 1 = e ~ i b 2 b_{1}=\tilde{e}_{i}b_{2}
  88. b 1 b_{1}
  89. v 1 v_{1}
  90. b 2 b_{2}
  91. v 2 v_{2}
  92. e ~ i \tilde{e}_{i}
  93. f ~ i \tilde{f}_{i}
  94. q = 0 q=0
  95. V 1 V_{1}
  96. V 2 V_{2}
  97. V 1 V_{1}
  98. V 2 V_{2}
  99. M M
  100. ( L , B ) (L,B)
  101. M M^{\prime}
  102. ( L , B ) (L^{\prime},B^{\prime})
  103. Δ \Delta
  104. Δ ( k λ ) = k λ k λ , Δ ( e i ) = e i k i - 1 + 1 e i , Δ ( f i ) = f i 1 + k i f i \Delta(k_{\lambda})=k_{\lambda}\otimes k_{\lambda},\ \Delta(e_{i})=e_{i}% \otimes k_{i}^{-1}+1\otimes e_{i},\ \Delta(f_{i})=f_{i}\otimes 1+k_{i}\otimes f% _{i}
  105. M ( q ) M M\otimes_{{\mathbb{Q}}(q)}M^{\prime}
  106. ( L A L , B B ) (L\otimes_{A}L^{\prime},B\otimes B^{\prime})
  107. B B = { b b : b B , b B } B\otimes B^{\prime}=\{b\otimes_{\mathbb{Q}}b^{\prime}:b\in B,\ b^{\prime}\in B% ^{\prime}\}
  108. b B b\in B
  109. ϵ i ( b ) = max { n 0 : e ~ i n b 0 } \epsilon_{i}(b)=\max\{n\geq 0:\tilde{e}_{i}^{n}b\neq 0\}
  110. ϕ i ( b ) = max { n 0 : f ~ i n b 0 } \phi_{i}(b)=\max\{n\geq 0:\tilde{f}_{i}^{n}b\neq 0\}
  111. e ~ i \tilde{e}_{i}
  112. f ~ i \tilde{f}_{i}
  113. b b b\otimes b^{\prime}
  114. e ~ i ( b b ) = { e ~ i b b , if ϕ i ( b ) ϵ i ( b ) , b e ~ i b , if ϕ i ( b ) < ϵ i ( b ) , \tilde{e}_{i}(b\otimes b^{\prime})=\begin{cases}\tilde{e}_{i}b\otimes b^{% \prime},&\,\text{if }\phi_{i}(b)\geq\epsilon_{i}(b^{\prime}),\\ b\otimes\tilde{e}_{i}b^{\prime},&\,\text{if }\phi_{i}(b)<\epsilon_{i}(b^{% \prime}),\end{cases}
  115. f ~ i ( b b ) = { f ~ i b b , if ϕ i ( b ) > ϵ i ( b ) , b f ~ i b , if ϕ i ( b ) ϵ i ( b ) . \tilde{f}_{i}(b\otimes b^{\prime})=\begin{cases}\tilde{f}_{i}b\otimes b^{% \prime},&\,\text{if }\phi_{i}(b)>\epsilon_{i}(b^{\prime}),\\ b\otimes\tilde{f}_{i}b^{\prime},&\,\text{if }\phi_{i}(b)\leq\epsilon_{i}(b^{% \prime}).\end{cases}

Crystalline_cohomology.html

  1. H i ( X / W ) = lim H i ( X / W n ) H^{i}(X/W)=\lim_{\leftarrow}H^{i}(X/W_{n})
  2. H i ( X / W n ) = H i ( C r i s ( X / W n ) , O ) H^{i}(X/W_{n})=H^{i}(Cris(X/W_{n}),O)
  3. H i ( X / W ) = H D R i ( Z / W ) ( = H i ( Z , Ω Z / W * ) = lim H i ( Z , Ω Z / W n * ) ) H^{i}(X/W)=H^{i}_{DR}(Z/W)\quad(=H^{i}(Z,\Omega_{Z/W}^{*})=\lim_{\leftarrow}H^% {i}(Z,\Omega_{Z/W_{n}}^{*}))

Császár_polyhedron.html

  1. K 7 K_{7}
  2. h = ( v - 3 ) ( v - 4 ) 12 . h=\frac{(v-3)(v-4)}{12}.

Cu2+-exporting_ATPase.html

  1. \rightleftharpoons

Cunningham_correction_factor.html

  1. C = 1 + 2 λ d ( A 1 + A 2 e - A 3 d λ ) C=1+\frac{2\lambda}{d}\cdot(A_{1}+A_{2}\cdot e^{\frac{-A_{3}\cdot d}{\lambda}})

Cupriavidus_necator.html

  1. \rightleftharpoons

Current_differencing_transconductance_amplifier.html

  1. g m gm\,
  2. V p = V n = 0 Vp=Vn=0\,
  3. I z = I p - I n Iz=Ip-In\,
  4. I x + = g m . V z Ix+=gm.Vz\,
  5. I x - = - g m . V z Ix-=-gm.Vz\,
  6. V z - = I z . Z z Vz-=Iz.Zz\,
  7. Z z Zz\,
  8. I x = g m . ( V + - V - ) Ix=gm.(V+-V-)\,
  9. V + V+\,
  10. V - V-\,
  11. V - = 0 V V-=0\;V\,
  12. I x + = - I x - Ix+=-Ix-\,

Curvature_of_a_measure.html

  1. c ( x , y , z ) = 1 R ( x , y , z ) , c(x,y,z)=\frac{1}{R(x,y,z)},
  2. c 2 ( μ ) = 2 c ( x , y , z ) 2 d μ ( x ) d μ ( y ) d μ ( z ) . c^{2}(\mu)=\iiint_{\mathbb{R}^{2}}c(x,y,z)^{2}\,\mathrm{d}\mu(x)\mathrm{d}\mu(% y)\mathrm{d}\mu(z).
  3. c 2 α ( μ ) = 2 c ( x , y , z ) 2 α d μ ( x ) d μ ( y ) d μ ( z ) . c^{2\alpha}(\mu)=\iiint_{\mathbb{R}^{2}}c(x,y,z)^{2\alpha}\,\mathrm{d}\mu(x)% \mathrm{d}\mu(y)\mathrm{d}\mu(z).
  4. c 2 ( μ ; x ) = 2 c ( x , y , z ) 2 d μ ( y ) d μ ( z ) , c^{2}(\mu;x)=\iint_{\mathbb{R}^{2}}c(x,y,z)^{2}\,\mathrm{d}\mu(y)\mathrm{d}\mu% (z),
  5. c 2 ( μ ) = 2 c 2 ( μ ; x ) d μ ( x ) . c^{2}(\mu)=\int_{\mathbb{R}^{2}}c^{2}(\mu;x)\,\mathrm{d}\mu(x).
  6. μ ( B r ( x ) ) C 0 r \mu(B_{r}(x))\leq C_{0}r
  7. | 6 | 𝒞 ε ( μ ) ( z ) | 2 d μ ( z ) - c ε 2 ( μ ) | C μ \left|6\int_{\mathbb{C}}|\mathcal{C}_{\varepsilon}(\mu)(z)|^{2}\,\mathrm{d}\mu% (z)-c_{\varepsilon}^{2}(\mu)\right|\leq C\|\mu\|
  8. | x - y | > ε ; |x-y|>\varepsilon;
  9. | y - z | > ε ; |y-z|>\varepsilon;
  10. | z - x | > ε . |z-x|>\varepsilon.
  11. 𝒞 ε \mathcal{C}_{\varepsilon}
  12. 𝒞 ε ( μ ) ( z ) = 1 ξ - z d μ ( ξ ) , \mathcal{C}_{\varepsilon}(\mu)(z)=\int\frac{1}{\xi-z}\,\mathrm{d}\mu(\xi),
  13. | ξ - z | > ε . |\xi-z|>\varepsilon.

Cutler's_bar_notation.html

  1. a b = a × a × × a b copies of a \begin{matrix}a^{b}&=&\underbrace{a\times a\times\dots\times a}\\ &&b\mbox{ copies of }~{}a\end{matrix}
  2. a a . . . a b copies of a \begin{matrix}&\underbrace{a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}}&\\ &b\mbox{ copies of }~{}a\end{matrix}
  3. a ¯ b {{}^{b}}\bar{a}
  4. a ¯ b = a a . . . a b copies of a \begin{matrix}{{}^{b}}\bar{a}=&\underbrace{a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}}&% \\ &b\mbox{ copies of }~{}a\end{matrix}
  5. b b a ¯ = a a . . . a a ¯ b copies of a \begin{matrix}^{{}^{b}{b}}\bar{a}=&\underbrace{a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}% }}&\\ &{{{}^{b}}\bar{a}}\mbox{ copies of }~{}a\end{matrix}
  6. b b . . . b a ¯ = a ¯ c c copies of b \begin{matrix}\underbrace{b^{b^{{}^{.\,^{.\,^{.\,^{b}}}}}}}\bar{a}={{}_{c}}% \bar{a}\\ c\mbox{ copies of }~{}b\end{matrix}
  7. a ¯ d \bar{a}_{d}
  8. 10 ¯ 10 \bar{10}_{10}
  9. a b b c {}^{a^{b^{b^{c}}}}

Cyril_Berry.html

  1. A B V = ( S t a r t i n g S G - F i n a l S G ) / 7.36 ABV=(StartingSG-FinalSG)/7.36

Cystathionine_beta_synthase.html

  1. \rightleftharpoons

D'Agostino's_K-squared_test.html

  1. x ¯ \bar{x}
  2. g 1 = m 3 m 2 3 / 2 = 1 n i = 1 n ( x i - x ¯ ) 3 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 3 / 2 , \displaystyle g_{1}=\frac{m_{3}}{m_{2}^{3/2}}=\frac{\frac{1}{n}\sum_{i=1}^{n}% \left(x_{i}-\bar{x}\right)^{3}}{\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-% \bar{x}\right)^{2}\right)^{3/2}}\ ,
  3. μ 1 ( g 1 ) = 0 , \displaystyle\mu_{1}(g_{1})=0,
  4. μ 1 ( g 2 ) = - 6 n + 1 , \displaystyle\mu_{1}(g_{2})=-\frac{6}{n+1},
  5. Z 1 ( g 1 ) = δ ln ( g 1 α μ 2 + g 1 2 α 2 μ 2 + 1 ) , Z_{1}(g_{1})=\delta\cdot\ln\!\left(\frac{g_{1}}{\alpha\sqrt{\mu_{2}}}+\sqrt{% \frac{g_{1}^{2}}{\alpha^{2}\mu_{2}}+1}\right),
  6. W 2 = 2 γ 2 + 4 - 1 , \displaystyle W^{2}=\sqrt{2\gamma_{2}+4}-1,
  7. Z 2 ( g 2 ) = 9 A 2 { 1 - 2 9 A - ( 1 - 2 / A 1 + g 2 - μ 1 μ 2 2 / ( A - 4 ) ) 1 / 3 } , Z_{2}(g_{2})=\sqrt{\frac{9A}{2}}\left\{1-\frac{2}{9A}-\left(\frac{1-2/A}{1+% \frac{g_{2}-\mu_{1}}{\sqrt{\mu_{2}}}\sqrt{2/(A-4)}}\right)^{\!1/3}\right\},
  8. A = 6 + 8 γ 1 ( 2 γ 1 + 1 + 4 / γ 1 2 ) , A=6+\frac{8}{\gamma_{1}}\left(\frac{2}{\gamma_{1}}+\sqrt{1+4/\gamma_{1}^{2}}% \right),
  9. K 2 = Z 1 ( g 1 ) 2 + Z 2 ( g 2 ) 2 K^{2}=Z_{1}(g_{1})^{2}+Z_{2}(g_{2})^{2}\,

D-ary_heap.html

  1. d d
  2. d d
  3. d d
  4. d d
  5. d d
  6. d d
  7. d d
  8. n n
  9. d d
  10. d d
  11. i i
  12. i > 0 i>0
  13. f l o o r ( ( i 1 ) / d ) floor((i−1)/d)
  14. d i + 1 di+1
  15. d i + d di+d
  16. n n
  17. n 1 n−1
  18. d d
  19. n n
  20. O ( l o g n / l o g d ) O(logn/logd)
  21. d d
  22. O ( d ) O(d)
  23. d 1 d−1
  24. O ( d l o g n / l o g d ) O(dlogn/logd)
  25. d d
  26. d d
  27. n / d + 1 n/d+1
  28. O ( d ) O(d)
  29. O ( d ) O(d)
  30. i = 1 log d n ( n d i + 1 ) O ( d ) = O ( n ) . \sum_{i=1}^{\log_{d}n}\left(\frac{n}{d^{i}}+1\right)O(d)=O(n).
  31. d d - 1 ( n - s d ( n ) ) - ( d - 1 - ( n mod d ) ) ( e d ( n d ) + 1 ) \frac{d}{d-1}(n-s_{d}(n))-(d-1-(n\mod d))(e_{d}(\lfloor\frac{n}{d}\rfloor)+1)
  32. 2 n - 2 s 2 ( n ) - e 2 ( n ) 2n-2s_{2}(n)-e_{2}(n)
  33. 3 2 ( n - s 3 ( n ) ) - 2 e 3 ( n ) - e 3 ( n - 1 ) \frac{3}{2}(n-s_{3}(n))-2e_{3}(n)-e_{3}(n-1)
  34. d - a r y d-ary
  35. n n
  36. m m
  37. n n
  38. d d
  39. d = m / n d=m/n
  40. O ( m l o g < s u b > m / n n ) O(mlog<sub>m/nn)

Danskin's_theorem.html

  1. f ( x ) = max z Z ϕ ( x , z ) . f(x)=\max_{z\in Z}\phi(x,z).
  2. ϕ ( x , z ) \phi(x,z)
  3. ϕ : n × Z \phi:{\mathbb{R}}^{n}\times Z\rightarrow{\mathbb{R}}
  4. Z m Z\subset{\mathbb{R}}^{m}
  5. ϕ ( x , z ) \phi(x,z)
  6. x x
  7. z Z z\in Z
  8. f ( x ) = max z Z ϕ ( x , z ) . f(x)=\max_{z\in Z}\phi(x,z).
  9. Z 0 ( x ) Z_{0}(x)
  10. Z 0 ( x ) = { z ¯ : ϕ ( x , z ¯ ) = max z Z ϕ ( x , z ) } . Z_{0}(x)=\left\{\overline{z}:\phi(x,\overline{z})=\max_{z\in Z}\phi(x,z)\right\}.
  11. f ( x ) f(x)
  12. f ( x ) f(x)
  13. y y
  14. D y f ( x ) D_{y}\ f(x)
  15. D y f ( x ) = max z Z 0 ( x ) ϕ ( x , z ; y ) , D_{y}f(x)=\max_{z\in Z_{0}(x)}\phi^{\prime}(x,z;y),
  16. ϕ ( x , z ; y ) \phi^{\prime}(x,z;y)
  17. ϕ ( , z ) \phi(\cdot,z)
  18. x x
  19. y y
  20. f ( x ) f(x)
  21. x x
  22. Z 0 ( x ) Z_{0}(x)
  23. z ¯ \overline{z}
  24. f ( x ) f(x)
  25. f ( x ) f(x)
  26. x x
  27. f x = ϕ ( x , z ¯ ) x . \frac{\partial f}{\partial x}=\frac{\partial\phi(x,\overline{z})}{\partial x}.
  28. ϕ ( x , z ) \phi(x,z)
  29. x x
  30. z Z z\in Z
  31. ϕ / x \partial\phi/\partial x
  32. z z
  33. x x
  34. f ( x ) f(x)
  35. f ( x ) = conv { ϕ ( x , z ) x : z Z 0 ( x ) } \partial f(x)=\mathrm{conv}\left\{\frac{\partial\phi(x,z)}{\partial x}:z\in Z_% {0}(x)\right\}
  36. conv \mathrm{conv}
  37. ϕ ( , z ) \phi(\cdot,z)
  38. ϕ ( , z ) \phi(\cdot,z)
  39. z z
  40. Z Z
  41. i n t ( d o m ( f ) ) int(dom(f))
  42. f f
  43. ϕ \phi
  44. i n t ( d o m ( f ) ) × Z int(dom(f))\times Z
  45. x x
  46. i n t ( d o m ( f ) ) int(dom(f))
  47. f f
  48. x x
  49. f ( x ) = conv { ϕ ( x , z ) : z Z 0 ( x ) } \partial f(x)=\mathrm{conv}\left\{\partial\phi(x,z):z\in Z_{0}(x)\right\}
  50. ϕ ( x , z ) \partial\phi(x,z)
  51. ϕ ( , z ) \phi(\cdot,z)
  52. x x
  53. z z
  54. Z Z

Darboux_frame.html

  1. 𝐓 ( s ) = γ ( s ) , \mathbf{T}(s)=\gamma^{\prime}(s),
  2. 𝐮 ( s ) = 𝐮 ( γ ( s ) ) , \mathbf{u}(s)=\mathbf{u}(\gamma(s)),
  3. 𝐭 ( s ) = 𝐮 ( s ) × 𝐓 ( s ) , \mathbf{t}(s)=\mathbf{u}(s)\times\mathbf{T}(s),
  4. 𝐓 ( s ) = γ ( s ) , \mathbf{T}(s)=\gamma^{\prime}(s),
  5. 𝐍 ( s ) = 𝐓 ( s ) 𝐓 ( s ) , \mathbf{N}(s)=\frac{\mathbf{T}^{\prime}(s)}{\|\mathbf{T}^{\prime}(s)\|},
  6. 𝐁 ( s ) = 𝐓 ( s ) × 𝐍 ( s ) , \mathbf{B}(s)=\mathbf{T}(s)\times\mathbf{N}(s),
  7. [ 𝐓 𝐭 𝐮 ] = [ 1 0 0 0 cos α sin α 0 - sin α cos α ] [ 𝐓 𝐍 𝐁 ] . \begin{bmatrix}\mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&\cos\alpha&\sin\alpha\\ 0&-\sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}\mathbf{T}\\ \mathbf{N}\\ \mathbf{B}\end{bmatrix}.
  8. d [ 𝐓 𝐭 𝐮 ] = [ 0 κ cos α d s - κ sin α d s - κ cos α d s 0 τ d s + d α κ sin α d s - τ d s - d α 0 ] [ 𝐓 𝐭 𝐮 ] \mathrm{d}\begin{bmatrix}\mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}=\begin{bmatrix}0&\kappa\cos\alpha\,\mathrm{d}s&-\kappa% \sin\alpha\,\mathrm{d}s\\ -\kappa\cos\alpha\,\mathrm{d}s&0&\tau\,\mathrm{d}s+\mathrm{d}\alpha\\ \kappa\sin\alpha\,\mathrm{d}s&-\tau\,\mathrm{d}s-\mathrm{d}\alpha&0\end{% bmatrix}\begin{bmatrix}\mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}
  9. = [ 0 κ g d s κ n d s - κ g d s 0 τ r d s - κ n d s - τ r d s 0 ] [ 𝐓 𝐭 𝐮 ] =\begin{bmatrix}0&\kappa_{g}\,\mathrm{d}s&\kappa_{n}\,\mathrm{d}s\\ -\kappa_{g}\,\mathrm{d}s&0&\tau_{r}\,\mathrm{d}s\\ -\kappa_{n}\,\mathrm{d}s&-\tau_{r}\,\mathrm{d}s&0\end{bmatrix}\begin{bmatrix}% \mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}
  10. d [ 𝐏 𝐓 𝐭 𝐮 ] = [ 0 d s 0 0 0 0 κ g d s κ n d s 0 - κ g d s 0 τ r d s 0 - κ n d s - τ r d s 0 ] [ 𝐏 𝐓 𝐭 𝐮 ] . \mathrm{d}\begin{bmatrix}\mathbf{P}\\ \mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}=\begin{bmatrix}0&\mathrm{d}s&0&0\\ 0&0&\kappa_{g}\,\mathrm{d}s&\kappa_{n}\,\mathrm{d}s\\ 0&-\kappa_{g}\,\mathrm{d}s&0&\tau_{r}\,\mathrm{d}s\\ 0&-\kappa_{n}\,\mathrm{d}s&-\tau_{r}\,\mathrm{d}s&0\end{bmatrix}\begin{bmatrix% }\mathbf{P}\\ \mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}.
  11. [ 𝐏 𝐞 1 𝐞 2 𝐞 3 ] = [ 1 0 0 0 0 cos θ sin θ 0 0 - sin θ cos θ 0 0 0 0 1 ] [ 𝐏 𝐓 𝐭 𝐮 ] \begin{bmatrix}\mathbf{P}\\ \mathbf{e}_{1}\\ \mathbf{e}_{2}\\ \mathbf{e}_{3}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&\cos\theta&\sin\theta&0\\ 0&-\sin\theta&\cos\theta&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}\mathbf{P}\\ \mathbf{T}\\ \mathbf{t}\\ \mathbf{u}\end{bmatrix}
  12. d 𝐏 = 𝐓 d s = ω 1 𝐞 1 + ω 2 𝐞 2 d 𝐞 i = j ω i j 𝐞 j \begin{aligned}\displaystyle\mathrm{d}\mathbf{P}&\displaystyle=\mathbf{T}% \mathrm{d}s=\omega^{1}\mathbf{e}_{1}+\omega^{2}\mathbf{e}_{2}\\ \displaystyle\mathrm{d}\mathbf{e}_{i}&\displaystyle=\sum_{j}\omega^{j}_{i}% \mathbf{e}_{j}\end{aligned}
  13. ω 1 = cos θ d s , ω 2 = - sin θ d s ω i j = - ω j i ω 1 2 = κ g d s + d θ ω 1 3 = ( κ n cos θ + τ r sin θ ) d s ω 2 3 = - ( κ n sin θ + τ r cos θ ) d s \begin{aligned}\displaystyle\omega^{1}&\displaystyle=\cos\theta\,\mathrm{d}s,% \quad\omega^{2}=-\sin\theta\,\mathrm{d}s\\ \displaystyle\omega_{i}^{j}&\displaystyle=-\omega_{j}^{i}\\ \displaystyle\omega_{1}^{2}&\displaystyle=\kappa_{g}\,\mathrm{d}s+\mathrm{d}% \theta\\ \displaystyle\omega_{1}^{3}&\displaystyle=(\kappa_{n}\cos\theta+\tau_{r}\sin% \theta)\,\mathrm{d}s\\ \displaystyle\omega_{2}^{3}&\displaystyle=-(\kappa_{n}\sin\theta+\tau_{r}\cos% \theta)\,\mathrm{d}s\end{aligned}
  14. d ω 1 = ω 2 ω 2 1 d ω 2 = ω 1 ω 1 2 0 = ω 1 ω 1 3 + ω 2 ω 2 3 \begin{aligned}\displaystyle\mathrm{d}\omega^{1}&\displaystyle=\omega^{2}% \wedge\omega_{2}^{1}\\ \displaystyle\mathrm{d}\omega^{2}&\displaystyle=\omega^{1}\wedge\omega_{1}^{2}% \\ \displaystyle 0&\displaystyle=\omega^{1}\wedge\omega_{1}^{3}+\omega^{2}\wedge% \omega_{2}^{3}\end{aligned}
  15. d ω 1 2 = ω 1 3 ω 3 2 d ω 1 3 = ω 1 2 ω 2 3 d ω 2 3 = ω 2 1 ω 1 3 \begin{aligned}\displaystyle\mathrm{d}\omega_{1}^{2}&\displaystyle=\omega_{1}^% {3}\wedge\omega_{3}^{2}\\ \displaystyle\mathrm{d}\omega_{1}^{3}&\displaystyle=\omega_{1}^{2}\wedge\omega% _{2}^{3}\\ \displaystyle\mathrm{d}\omega_{2}^{3}&\displaystyle=\omega_{2}^{1}\wedge\omega% _{1}^{3}\end{aligned}
  16. I I = - d 𝐍 d 𝐏 = ω 1 3 ω 1 + ω 2 3 ω 2 = ( ω 1 ω 2 ) ( i i 11 i i 12 i i 21 i i 22 ) ( ω 1 ω 2 ) . II=-\mathrm{d}\mathbf{N}\cdot\mathrm{d}\mathbf{P}=\omega_{1}^{3}\odot\omega^{1% }+\omega_{2}^{3}\odot\omega^{2}=\begin{pmatrix}\omega^{1}\omega^{2}\end{% pmatrix}\begin{pmatrix}ii_{11}&ii_{12}\\ ii_{21}&ii_{22}\end{pmatrix}\begin{pmatrix}\omega^{1}\\ \omega^{2}\end{pmatrix}.
  17. ϕ ( x ) = A x + x 0 \phi(x)=Ax+x_{0}
  18. ϕ ( v ; f 1 , , f n ) := ( ϕ ( v ) ; A f 1 , , A f n ) . \phi(v;f_{1},\dots,f_{n}):=(\phi(v);Af_{1},\dots,Af_{n}).
  19. P ( v ; f 1 , , f n ) = v e i ( v ; f 1 , , f n ) = f i , i = 1 , 2 , , n . \begin{aligned}\displaystyle P(v;f_{1},\dots,f_{n})&\displaystyle=v\\ \displaystyle e_{i}(v;f_{1},\dots,f_{n})&\displaystyle=f_{i},\qquad i=1,2,% \dots,n.\end{aligned}
  20. d P = i ω i e i , \mathrm{d}P=\sum_{i}\omega^{i}e_{i},\,
  21. d e i = j ω i j e j . \mathrm{d}e_{i}=\sum_{j}\omega_{i}^{j}e_{j}.
  22. ϕ * ( ω i ) = ( A - 1 ) j i ω j \phi^{*}(\omega^{i})=(A^{-1})_{j}^{i}\omega^{j}
  23. ϕ * ( ω j i ) = ( A - 1 ) p i ω q p A j q . \phi^{*}(\omega_{j}^{i})=(A^{-1})_{p}^{i}\,\omega_{q}^{p}\,A_{j}^{q}.
  24. d ω i = - ω j i ω j \mathrm{d}\omega^{i}=-\omega_{j}^{i}\wedge\omega^{j}
  25. d ω j i = - ω k i ω j k . \mathrm{d}\omega_{j}^{i}=-\omega^{i}_{k}\wedge\omega^{k}_{j}.
  26. θ i = ϕ * ω i , θ j i = ϕ * ω j i . \theta^{i}=\phi^{*}\omega^{i},\quad\theta_{j}^{i}=\phi^{*}\omega_{j}^{i}.
  27. d θ i = - θ j i θ j , d θ j i = - θ k i θ j k . \mathrm{d}\theta^{i}=-\theta_{j}^{i}\wedge\theta^{j},\quad\mathrm{d}\theta_{j}% ^{i}=-\theta_{k}^{i}\wedge\theta_{j}^{k}.
  28. θ μ = 0 , μ = p + 1 , , n \theta^{\mu}=0,\quad\mu=p+1,\dots,n
  29. d θ a = - b = 1 p θ b a θ b 0 = d θ μ = - b = 1 p θ b μ θ b } ( 1 ) \left.\begin{array}[]{l}\mathrm{d}\theta^{a}=-\sum_{b=1}^{p}\theta_{b}^{a}% \wedge\theta^{b}\\ \\ 0=\mathrm{d}\theta^{\mu}=-\sum_{b=1}^{p}\theta_{b}^{\mu}\wedge\theta^{b}\end{% array}\right\}\,\,\,(1)
  30. θ b μ = s a b μ θ a \theta_{b}^{\mu}=s^{\mu}_{ab}\theta^{a}
  31. d θ b a + c = 1 p θ c a θ b c = Ω b a = - μ = p + 1 n θ μ a θ b μ d θ b γ = - c = 1 p θ c γ θ b c - μ = p + 1 n θ μ γ θ b μ d θ μ γ = - c = 1 p θ c γ θ μ c - δ = p + 1 n θ δ γ θ μ δ } ( 2 ) \left.\begin{array}[]{l}\mathrm{d}\theta_{b}^{a}+\sum_{c=1}^{p}\theta_{c}^{a}% \wedge\theta_{b}^{c}=\Omega_{b}^{a}=-\sum_{\mu=p+1}^{n}\theta_{\mu}^{a}\wedge% \theta^{\mu}_{b}\\ \\ \mathrm{d}\theta_{b}^{\gamma}=-\sum_{c=1}^{p}\theta_{c}^{\gamma}\wedge\theta_{% b}^{c}-\sum_{\mu=p+1}^{n}\theta_{\mu}^{\gamma}\wedge\theta_{b}^{\mu}\\ \\ \mathrm{d}\theta_{\mu}^{\gamma}=-\sum_{c=1}^{p}\theta_{c}^{\gamma}\wedge\theta% _{\mu}^{c}-\sum_{\delta=p+1}^{n}\theta_{\delta}^{\gamma}\wedge\theta_{\mu}^{% \delta}\end{array}\right\}\,\,\,(2)

Darwin–Radau_equation.html

  1. C M R e 2 = 2 3 λ = 2 3 ( 1 - 2 5 1 + η ) \frac{C}{MR_{e}^{2}}=\frac{2}{3\lambda}=\frac{2}{3}\left(1-\frac{2}{5}\sqrt{1+% \eta}\right)
  2. η = 5 q 2 ϵ - 2 \eta=\frac{5q}{2\epsilon}-2
  3. q = ω 2 R e 3 G M q=\frac{\omega^{2}R_{e}^{3}}{GM}
  4. ϵ = R p - R e R e \epsilon=\frac{R_{p}-R_{e}}{R_{e}}
  5. q 3.461391 × 10 - 3 q\approx 3.461391\times 10^{-3}
  6. ϵ 1 / 298.257 \epsilon\approx 1/298.257
  7. C M R e 2 0.3313 \frac{C}{MR_{e}^{2}}\approx 0.3313

Data_transformation_(statistics).html

  1. Y = a + b X Y=a+bX
  2. log ( Y ) = a + b X \log(Y)=a+bX
  3. Y = e a e b X Y=e^{a}e^{bX}
  4. Y = a + b log ( X ) Y=a+b\log(X)
  5. log ( Y ) = a + b log ( X ) \log(Y)=a+b\log(X)
  6. Y = e a X b Y=e^{a}X^{b}

Database_storage_structures.html

  1. O ( 1 ) O\left(1\right)
  2. O ( n ) O\left(n\right)
  3. O ( log n ) O\left(\log n\right)
  4. O ( 1 ) O\left(1\right)
  5. O ( log n ) O\left(\log n\right)

Davidon–Fletcher–Powell_formula.html

  1. f ( x ) f(x)
  2. f \nabla f
  3. B B
  4. f ( x k + s k ) = f ( x k ) + f ( x k ) T s k + 1 2 s k T B s k , f(x_{k}+s_{k})=f(x_{k})+\nabla f(x_{k})^{T}s_{k}+\frac{1}{2}s^{T}_{k}{B}s_{k},
  5. f ( x k + s k ) = f ( x k ) + B s k , \nabla f(x_{k}+s_{k})=\nabla f(x_{k})+Bs_{k},
  6. B B
  7. B k B_{k}
  8. B k + 1 = ( I - γ k y k s k T ) B k ( I - γ k s k y k T ) + γ k y k y k T , B_{k+1}=(I-\gamma_{k}y_{k}s_{k}^{T})B_{k}(I-\gamma_{k}s_{k}y_{k}^{T})+\gamma_{% k}y_{k}y_{k}^{T},
  9. y k = f ( x k + s k ) - f ( x k ) , y_{k}=\nabla f(x_{k}+s_{k})-\nabla f(x_{k}),
  10. γ k = 1 y k T s k . \gamma_{k}=\frac{1}{y_{k}^{T}s_{k}}.
  11. B k B_{k}
  12. H k = B k - 1 H_{k}=B_{k}^{-1}
  13. H k + 1 = H k - H k y k y k T H k y k T H k y k + s k s k T y k T s k . H_{k+1}=H_{k}-\frac{H_{k}y_{k}y_{k}^{T}H_{k}}{y_{k}^{T}H_{k}y_{k}}+\frac{s_{k}% s_{k}^{T}}{y_{k}^{T}s_{k}}.
  14. B B
  15. s k T s_{k}^{T}
  16. y y
  17. s k T y k = s k T B s k > 0. s_{k}^{T}y_{k}=s_{k}^{T}Bs_{k}>0.\,

Davydov_soliton.html

  1. H ^ \hat{H}
  2. H ^ = H ^ qp + H ^ ph + H ^ int \hat{H}=\hat{H}_{\textrm{qp}}+\hat{H}_{\textrm{ph}}+\hat{H}_{\textrm{int}}
  3. H ^ qp \hat{H}_{\textrm{qp}}
  4. H ^ ph \hat{H}_{\textrm{ph}}
  5. H ^ int \hat{H}_{\textrm{int}}
  6. H ^ qp \hat{H}_{\textrm{qp}}
  7. H ^ qp = \hat{H}_{\textrm{qp}}=
  8. n , α E 0 A ^ n , α A ^ n , α \sum_{n,\alpha}E_{0}\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha}
  9. - J n , α ( A ^ n , α A ^ n + 1 , α + A ^ n , α A ^ n - 1 , α ) -J\sum_{n,\alpha}\left(\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n+1,\alpha}+\hat{A% }_{n,\alpha}^{\dagger}\hat{A}_{n-1,\alpha}\right)
  10. + L n , α ( A ^ n , α A ^ n , α + 1 + A ^ n , α A ^ n , α - 1 ) +L\sum_{n,\alpha}\left(\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha+1}+\hat{A% }_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha-1}\right)
  11. n = 1 , 2 , , N n=1,2,\cdots,N
  12. α = 1 , 2 , 3 \alpha=1,2,3
  13. E 0 = 3.28 × 10 - 20 E_{0}=3.28\times 10^{-20}
  14. J = 2.46 × 10 - 22 J=2.46\times 10^{-22}
  15. L = 1.55 × 10 - 22 L=1.55\times 10^{-22}
  16. A ^ n , α \hat{A}_{n,\alpha}^{\dagger}
  17. A ^ n , α \hat{A}_{n,\alpha}
  18. n , α n,\alpha
  19. H ^ ph \hat{H}_{\textrm{ph}}
  20. H ^ ph = 1 2 n , α [ w ( u ^ n + 1 , α - u ^ n , α ) 2 + p ^ n , α 2 M ] \hat{H}_{\textrm{ph}}=\frac{1}{2}\sum_{n,\alpha}\left[w(\hat{u}_{n+1,\alpha}-% \hat{u}_{n,\alpha})^{2}+\frac{\hat{p}_{n,\alpha}^{2}}{M}\right]
  21. u ^ n , α \hat{u}_{n,\alpha}
  22. n , α n,\alpha
  23. p ^ n , α \hat{p}_{n,\alpha}
  24. n , α n,\alpha
  25. w = 19.5 w=19.5
  26. - 1 {}^{-1}
  27. H ^ int \hat{H}_{\textrm{int}}
  28. H ^ int = χ n , α [ ( u ^ n + 1 , α - u ^ n , α ) A ^ n , α A ^ n , α ] \hat{H}_{\textrm{int}}=\chi\sum_{n,\alpha}\left[(\hat{u}_{n+1,\alpha}-\hat{u}_% {n,\alpha})\hat{A}_{n,\alpha}^{\dagger}\hat{A}_{n,\alpha}\right]
  29. χ = - 30 \chi=-30
  30. Δ v = 133 \Delta v=133

De_Bruijn_index.html

  1. n [ N 1 N n ] = N n ( M 1 M 2 ) [ s ] = ( M 1 [ s ] ) ( M 2 [ s ] ) ( λ M ) [ s ] = λ ( M [ 1.1 [ s ] .2 [ s ] .3 [ s ] ] ) where s = s 1 \begin{aligned}\displaystyle n[N_{1}\ldots N_{n}\ldots]=&\displaystyle N_{n}\\ \displaystyle(M_{1}\;M_{2})[s]=&\displaystyle(M_{1}[s])(M_{2}[s])\\ \displaystyle(\lambda\;M)[s]=&\displaystyle\lambda\;(M[1.1[s^{\prime}].2[s^{% \prime}].3[s^{\prime}]\ldots])\\ &\displaystyle\,\text{where }s^{\prime}=s\uparrow^{1}\end{aligned}

De_Bruijn_notation.html

  1. M , N , M,N,\ldots
  2. v v
  3. [ v ] [v]
  4. ( M ) (M)
  5. M , N , : := v | [ v ] M | ( M ) N M,N,...::=\ v\ |\ [v]\;M\ |\ (M)\;N
  6. \mathcal{I}
  7. ( v ) = v ( λ v . M ) = [ v ] ( M ) ( M N ) = ( ( N ) ) ( M ) . \begin{aligned}\displaystyle\mathcal{I}(v)&\displaystyle=v\\ \displaystyle\mathcal{I}(\lambda v.\ M)&\displaystyle=[v]\;\mathcal{I}(M)\\ \displaystyle\mathcal{I}(M\;N)&\displaystyle=(\mathcal{I}(N))\mathcal{I}(M).% \end{aligned}
  8. \mathcal{I}
  9. ( λ v . M ) N β M [ v := N ] (\lambda v.\ M)\;N\ \ \longrightarrow_{\beta}\ \ M[v:=N]
  10. ( N ) [ v ] M β M [ v := N ] . (N)\;[v]\;M\ \ \longrightarrow_{\beta}\ \ M[v:=N].
  11. ( M ) ( N ) [ u ] ( P ) [ v ] [ w ] ( Q ) z (M)\;(N)\;[u]\;(P)\;[v]\;[w]\;(Q)\;z
  12. ( M ) ( N ) [ u ] ¯ ( P ) [ v ] [ w ] ( Q ) z β ( M ) ( P [ u := N ] ) [ v ] ¯ [ w ] ( Q [ u := N ] ) z β ( M ) [ w ] ¯ ( Q [ u := N , v := P [ u := N ] ] ) z β ( Q [ u := N , v := P [ u := N ] , w := M ] ) z . \begin{aligned}\displaystyle(M)\;\underline{(N)\;[u]}\;(P)\;[v]\;[w]\;(Q)\;z&% \displaystyle{\ \longrightarrow_{\beta}\ }(M)\;\underline{(P[u:=N])\;[v]}\;[w]% \;(Q[u:=N])\;z\\ &\displaystyle{\ \longrightarrow_{\beta}\ }\underline{(M)\;[w]}\;(Q[u:=N,v:=P[% u:=N]])\;z\\ &\displaystyle{\ \longrightarrow_{\beta}\ }(Q[u:=N,v:=P[u:=N],w:=M])\;z.\end{aligned}
  13. ( M ) (M)
  14. [ w ] [w]

De_Vaucouleurs'_law.html

  1. I I
  2. R R
  3. ln I ( R ) = ln I 0 - k R 1 / 4 . \ln I(R)=\ln I_{0}-kR^{1/4}.
  4. ln I ( R ) = ln I e + 7.669 [ 1 - ( R R e ) 1 / 4 ] \ln I(R)=\ln I_{e}+7.669\left[1-\left(\frac{R}{R_{e}}\right)^{1/4}\right]
  5. I ( R ) = I e e - 7.669 [ ( R R e ) 1 / 4 - 1 ] I(R)=I_{e}e^{-7.669\left[(\frac{R}{R_{e}})^{1/4}-1\right]}
  6. 0 R e I ( R ) r d r = 1 2 0 I ( R ) r d r . \int^{R_{e}}_{0}I(R)rdr=\frac{1}{2}\int^{\infty}_{0}I(R)rdr.

Deal–Grove_model.html

  1. J g a s = h g ( C g - C s ) J_{gas}=h_{g}(C_{g}-C_{s})
  2. J o x i d e = D o x C s - C i x J_{oxide}=D_{ox}\frac{C_{s}-C_{i}}{x}
  3. J r e a c t i n g = k i C i J_{reacting}=k_{i}C_{i}
  4. J g a s = J o x i d e = J r e a c t i n g = C g 1 k i + x D o x + 1 h g J_{gas}=J_{oxide}=J_{reacting}=\frac{C_{g}}{\frac{1}{k_{i}}+\frac{x}{D_{ox}}+% \frac{1}{h_{g}}}
  5. t = X o 2 B + X o B / A t=\frac{X_{o}^{2}}{B}+\frac{X_{o}}{B/A}
  6. A = 2 D o x ( 1 k i + 1 h g ) A=2D_{ox}(\frac{1}{k_{i}}+\frac{1}{h_{g}})
  7. B = 2 D o x C s N i B=\frac{2D_{ox}C_{s}}{N_{i}}
  8. τ = X o 2 + A X o B \tau=\frac{X_{o}^{2}+AX_{o}}{B}
  9. C s = H P g C_{s}=HP_{g}
  10. H H
  11. P g P_{g}
  12. N i N_{i}
  13. X o ( t ) = - A + A 2 + 4 ( B ) ( t + τ ) 2 X_{o}(t)=\frac{-A+\sqrt{{A^{2}}+4(B)(t+\tau)}}{2}
  14. t + τ A 2 4 B X o ( t ) = B A ( t + τ ) t+\tau\ll\frac{A^{2}}{4B}\Rightarrow X_{o}(t)=\frac{B}{A}(t+\tau)
  15. t + τ A 2 4 B X o ( t ) = B ( t + τ ) t+\tau\gg\frac{A^{2}}{4B}\Rightarrow X_{o}(t)=\sqrt{B(t+\tau)}
  16. B = B 0 e - E A / k T ; B / A = ( B / A ) 0 e - E A / k T B=B_{0}e^{-E_{A}/kT};\quad B/A=(B/A)_{0}e^{-E_{A}/kT}
  17. E A E_{A}
  18. k k
  19. E A E_{A}
  20. H 2 O H_{2}O
  21. O 2 O_{2}
  22. ( B / A ) 0 ( μ m h r ) (B/A)_{0}\ \left(\frac{\mu m}{hr}\right)
  23. E A E_{A}
  24. B 0 ( ( μ m ) 2 h r ) B_{0}\ \left(\frac{(\mu m)^{2}}{hr}\right)
  25. E A E_{A}
  26. O 2 O_{2}

Dean_number.html

  1. 𝐷𝑒 = ρ V d μ ( d R ) 1 / 2 \mathit{De}=\frac{\rho V\!d}{\mu}\left(\frac{d}{R}\right)^{1/2}
  2. ρ \rho
  3. μ \mu
  4. V V
  5. d d
  6. R R
  7. V V
  8. d d
  9. a / r 1 a/r\ll 1
  10. ( x , y , z ) (x,y,z)
  11. ( s y m b o l x ^ , s y m b o l y ^ , s y m b o l z ^ ) (\hat{symbol{x}},\hat{symbol{y}},\hat{symbol{z}})
  12. s y m b o l z ^ \hat{symbol{z}}
  13. s y m b o l x ^ \hat{symbol{x}}
  14. s y m b o l y ^ \hat{symbol{y}}
  15. G G
  16. u z u_{z}
  17. U = G a 2 / μ U=Ga^{2}/\mu
  18. u x , u y u_{x},u_{y}
  19. ( a / R ) 1 / 2 U (a/R)^{1/2}U
  20. ρ a U 2 / L \rho aU^{2}/L
  21. a a
  22. D ( D u x D t + u z 2 ) = - D p x + 2 u x D\left(\frac{\mathrm{D}u_{x}}{\mathrm{D}t}+u_{z}^{2}\right)=-D\frac{\partial p% }{\partial x}+\nabla^{2}u_{x}
  23. D D u y D t = - D p y + 2 u y D\frac{\mathrm{D}u_{y}}{\mathrm{D}t}=-D\frac{\partial p}{\partial y}+\nabla^{2% }u_{y}
  24. D D u z D t = 1 + 2 u z D\frac{\mathrm{D}u_{z}}{\mathrm{D}t}=1+\nabla^{2}u_{z}
  25. u x x + u y y = 0 \frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}=0
  26. D D t = u x x + u y y \frac{\mathrm{D}}{\mathrm{D}t}=u_{x}\frac{\partial}{\partial x}+u_{y}\frac{% \partial}{\partial y}
  27. D c 956 D_{c}\approx 956

Defense-Independent_Component_ERA.html

  1. D I C E = 3.00 + 13 H R + 3 ( B B + H B P ) - 2 K I P DICE=3.00+\frac{13HR+3(BB+HBP)-2K}{IP}

Dehn–Sommerville_equations.html

  1. f ( P ) = ( f 0 , f 1 , , f d - 1 ) f(P)=(f_{0},f_{1},\ldots,f_{d-1})
  2. f - 1 = 1 , f d = 1. f_{-1}=1,f_{d}=1.
  3. j = k d - 1 ( - 1 ) j ( j + 1 k + 1 ) f j = ( - 1 ) d - 1 f k . \sum_{j=k}^{d-1}(-1)^{j}{\left({{j+1}\atop{k+1}}\right)}f_{j}=(-1)^{d-1}f_{k}.
  4. [ d + 1 2 ] \left[\frac{d+1}{2}\right]
  5. i = - 1 k - 1 ( - 1 ) d + i ( d - i - 1 d - k ) f i = i = - 1 d - k - 1 ( - 1 ) i ( d - i - 1 k ) f i , \sum_{i=-1}^{k-1}(-1)^{d+i}{\left({{d-i-1}\atop{d-k}}\right)}f_{i}=\sum_{i=-1}% ^{d-k-1}(-1)^{i}{\left({{d-i-1}\atop{k}}\right)}f_{i},
  6. h k = i = 0 k ( - 1 ) k - i ( d - i k - i ) f i - 1 . h_{k}=\sum_{i=0}^{k}(-1)^{k-i}{\left({{d-i}\atop{k-i}}\right)}f_{i-1}.
  7. h ( P ) = ( h 0 , h 1 , , h d ) h(P)=(h_{0},h_{1},\ldots,h_{d})
  8. i = 0 d f i - 1 ( t - 1 ) d - i = k = 0 d h k t d - k . \sum_{i=0}^{d}f_{i-1}(t-1)^{d-i}=\sum_{k=0}^{d}h_{k}t^{d-k}.
  9. h k = h d - k for 0 k d . h_{k}=h_{d-k}\quad\textrm{for}\quad 0\leq k\leq d.
  10. h k = dim IH 2 k ( X , ) h_{k}=\operatorname{dim}_{\mathbb{Q}}\operatorname{IH}^{2k}(X,\mathbb{Q})

Deligne–Lusztig_theory.html

  1. R θ ( w ) = i ( - 1 ) i H c i ( X ( w ) , F θ ) R^{\theta}(w)=\sum_{i}(-1)^{i}H_{c}^{i}(X(w),F_{\theta})
  2. dim ( R T θ ) = ± | G F | | T F | | U F | \dim(R^{\theta}_{T})={\pm|G^{F}|\over|T^{F}||U^{F}|}
  3. R T θ ( x ) = 1 | G s F | g G F , g - 1 s g T Q g T g - 1 , G s ( u ) θ ( g - 1 s g ) R^{\theta}_{T}(x)={1\over|G_{s}^{F}|}\sum_{g\in G^{F},g^{-1}sg\in T}Q_{gTg^{-1% },G_{s}}(u)\theta(g^{-1}sg)
  4. ( T , θ ) κ , mod G F R T θ ( R T θ , R T θ ) \sum_{(T,\theta)\in\kappa,\bmod G^{F}}{R_{T}^{\theta}\over(R_{T}^{\theta},R_{T% }^{\theta})}
  5. ( T , θ ) κ , mod G F ϵ G ϵ T R T θ ( R T θ , R T θ ) \sum_{(T,\theta)\in\kappa,\bmod G^{F}}{\epsilon_{G}\epsilon_{T}R_{T}^{\theta}% \over(R_{T}^{\theta},R_{T}^{\theta})}
  6. x y q - y x q 0 xy^{q}-yx^{q}\neq 0
  7. x y q - y x q = 1 xy^{q}-yx^{q}=1

Delimited_continuation.html

  1. \mathcal{F}
  2. # \#

Delta-cadinene_synthase.html

  1. \rightleftharpoons

Demand_for_money.html

  1. M d = P * L ( R , Y ) M^{d}=P*L(R,Y)\,
  2. M d M^{d}
  3. L ( R , Y ) L(R,Y)
  4. M d = P Y V M^{d}=P\frac{Y}{V}\,
  5. M d P = Y V \frac{M^{d}}{P}=\frac{Y}{V}\,
  6. M d P = t Y 2 R \frac{M^{d}}{P}=\sqrt{\frac{tY}{2R}}\,
  7. g m + g v = π + g y g_{m}+g_{v}=\pi+g_{y}\,
  8. g v = 0 g_{v}=0
  9. π = - g y + g m \pi=-g_{y}+g_{m}\,

Dense-in-itself.html

  1. A A
  2. A A
  3. x x
  4. y x y\neq x
  5. \mathbb{R}
  6. [ 0 , 1 ] \mathbb{Q}\cap[0,1]
  7. \mathbb{R}

Density_logging.html

  1. ρ bulk \rho\text{bulk}
  2. ρ e = 2 ρ bulk Z A \rho\text{e}=2\rho\text{bulk}\ \frac{Z}{A}
  3. Z Z
  4. A A
  5. Z / A Z/A
  6. ρ e \rho\text{e}
  7. ρ bulk \rho\text{bulk}
  8. ρ matrix \rho\text{matrix}
  9. ρ fluid \rho\text{fluid}
  10. ϕ \phi
  11. ϕ = ρ matrix - ρ bulk ρ matrix - ρ fluid \phi=\frac{\rho\text{matrix}-\rho\text{bulk}}{\rho\text{matrix}-\rho\text{% fluid}}
  12. ρ matrix \rho\text{matrix}
  13. ρ fluid \rho\text{fluid}

Density_wave_theory.html

  1. Ω g p \Omega_{gp}
  2. Ω g p \Omega_{gp}
  3. R c R_{c}
  4. Ω > Ω g p \Omega>\Omega_{gp}
  5. Ω < Ω g p \Omega<\Omega_{gp}
  6. m ( Ω g p - Ω ( R ) ) m(\Omega_{gp}-\Omega(R))
  7. κ ( R ) \kappa(R)
  8. Ω ( R ) = Ω g p + κ / m \Omega(R)=\Omega_{gp}+\kappa/m
  9. Ω ( R ) = Ω g p - κ / m \Omega(R)=\Omega_{gp}-\kappa/m

Deoxidization.html

  1. n D + m O D n K m nD+mO\longrightarrow D_{n}K_{m}
  2. K e q = a o x / ( a D n * a O m ) K_{eq}=a_{ox}/(a_{D}^{n}*a_{O}^{m})
  3. l o g K e q = A D / T - B D logK_{eq}=A_{D}/T-B_{D}
  4. C + O C O C+O\longrightarrow CO
  5. K C O = P C O / ( a C * a O ) K_{CO}=P_{CO}/(a_{C}*a_{O})
  6. K F e O = a [ O ] / a ( O ) K_{FeO}=a_{[O]}/a_{(O)}

Dependency_relation.html

  1. D D
  2. ( a , b ) D (a,b)\in D
  3. ( b , a ) D (b,a)\in D
  4. a a
  5. ( a , a ) D (a,a)\in D
  6. Σ \Sigma
  7. D D
  8. D D
  9. I I
  10. I = Σ × Σ D I=\Sigma\times\Sigma\setminus D
  11. D D
  12. ( Σ , D ) (\Sigma,D)
  13. ( Σ , I ) (\Sigma,I)
  14. ( Σ , D , I ) (\Sigma,D,I)
  15. I I
  16. D D
  17. Σ = { a , b , c } \Sigma=\{a,b,c\}
  18. D \displaystyle D
  19. I D = { ( b , c ) , ( c , b ) } I_{D}=\{(b,c)\,,\,(c,b)\}
  20. b , c b,c

Depth_(ring_theory).html

  1. depth ( M ) dim ( M ) , \mathrm{depth}(M)\leq\dim(M),
  2. depth I ( M ) = min { i : Ext i ( R / I , M ) 0 } . \mathrm{depth}_{I}(M)=\min\{i:\operatorname{Ext}^{i}(R/I,M)\neq 0\}.
  3. 𝔪 \mathfrak{m}
  4. 𝔪 \mathfrak{m}
  5. 𝔪 \mathfrak{m}
  6. 𝔪 \mathfrak{m}
  7. pd R ( M ) + depth ( M ) = depth ( R ) . \mathrm{pd}_{R}(M)+\mathrm{depth}(M)=\mathrm{depth}(R).
  8. 𝔪 \mathfrak{m}
  9. x 𝔪 = 0 x\mathfrak{m}=0
  10. 𝔪 \mathfrak{m}
  11. k [ x , y ] / ( x 2 , x y ) k[x,y]/(x^{2},xy)
  12. x = 0 x=0

Derivation_of_the_Navier–Stokes_equations.html

  1. Ω \Omega
  2. Ω \partial\Omega
  3. D D t = def t + 𝐮 \frac{D}{Dt}\ \stackrel{\mathrm{def}}{=}\ \frac{\partial}{\partial t}+\mathbf{% u}\cdot\nabla
  4. 𝐮 \mathbf{u}
  5. Ω \Omega
  6. d d t Ω ϕ d V = - Ω ϕ 𝐮 𝐧 d A - Ω s d V \frac{d}{dt}\int_{\Omega}\phi\ dV=-\int_{\partial\Omega}\phi\mathbf{u\cdot n}% \ dA-\int_{\Omega}s\ dV
  7. s s
  8. Ω \Omega
  9. Ω \partial\Omega
  10. d d t Ω ϕ d V = - Ω ( ϕ 𝐮 ) d V - Ω s d V . \frac{d}{dt}\int_{\Omega}\phi\ dV=-\int_{\Omega}\nabla\cdot(\phi\mathbf{u})\ % dV-\int_{\Omega}s\ dV.
  11. Ω ϕ t d V = - Ω ( ϕ 𝐮 ) d V - Ω s d V Ω ( ϕ t + ( ϕ 𝐮 ) + s ) d V = 0. \int_{\Omega}\frac{\partial\phi}{\partial t}\ dV=-\int_{\Omega}\nabla\cdot(% \phi\mathbf{u})\ dV-\int_{\Omega}s\ dV\qquad\Rightarrow\qquad\int_{\Omega}% \left(\frac{\partial\phi}{\partial t}+\nabla\cdot(\phi\mathbf{u})+s\ \right)dV% =0.
  12. ϕ t + ( ϕ 𝐮 ) + s = 0. \frac{\partial\phi}{\partial t}+\nabla\cdot(\phi\mathbf{u})+s=0.
  13. ρ 𝐮 \rho\mathbf{u}
  14. t ( ρ 𝐮 ) + ( ρ 𝐮𝐮 ) + 𝐬 = 0 \frac{\partial}{\partial t}(\rho\mathbf{u})+\nabla\cdot(\rho\mathbf{u}\mathbf{% u})+\mathbf{s}=0
  15. 𝐮𝐮 \mathbf{u}\mathbf{u}
  16. 𝐮 ρ t + ρ 𝐮 t + 𝐮𝐮 ρ + ρ 𝐮 𝐮 + ρ 𝐮 𝐮 = 𝐬 \mathbf{u}\frac{\partial\rho}{\partial t}+\rho\frac{\partial\mathbf{u}}{% \partial t}+\mathbf{u}\mathbf{u}\cdot\nabla\rho+\rho\mathbf{u}\cdot\nabla% \mathbf{u}+\rho\mathbf{u}\nabla\cdot\mathbf{u}=\mathbf{s}
  17. 𝐮 ρ + ρ 𝐮 = ( ρ 𝐮 ) \mathbf{u}\cdot\nabla\rho+\rho\nabla\cdot\mathbf{u}=\nabla\cdot(\rho\mathbf{u})
  18. 𝐮 ( ρ t + 𝐮 ρ + ρ 𝐮 ) + ρ ( 𝐮 t + 𝐮 𝐮 ) = 𝐬 \mathbf{u}\left(\frac{\partial\rho}{\partial t}+\mathbf{u}\cdot\nabla\rho+\rho% \nabla\cdot\mathbf{u}\right)+\rho\left(\frac{\partial\mathbf{u}}{\partial t}+% \mathbf{u}\cdot\nabla\mathbf{u}\right)=\mathbf{s}
  19. 𝐮 ( ρ t + ( ρ 𝐮 ) ) + ρ ( 𝐮 t + 𝐮 𝐮 ) = 𝐬 \mathbf{u}\left(\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})% \right)+\rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla% \mathbf{u}\right)=\mathbf{s}
  20. ρ ( 𝐮 t + 𝐮 𝐮 ) = 𝐬 \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=\mathbf{s}
  21. ρ D 𝐮 D t = 𝐬 \qquad\rho\frac{D\mathbf{u}}{Dt}=\mathbf{s}
  22. ρ d d t ( 𝐮 ( x , y , z , t ) ) = 𝐬 \displaystyle\rho\frac{d}{dt}(\mathbf{u}(x,y,z,t))=\mathbf{s}
  23. 𝐮 = ( u , v , w ) \mathbf{u}=(u,v,w)
  24. 𝐮 = ( d x d t , d y d t , d z d t ) \mathbf{u}=\left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right)
  25. Q = 0 Q=0
  26. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  27. ρ \rho
  28. 𝐮 \mathbf{u}
  29. ρ \rho
  30. 𝐮 = 0 \nabla\cdot\mathbf{u}=0
  31. 𝐬 \mathbf{s}
  32. ρ D 𝐮 D t = s y m b o l σ + 𝐟 \rho\frac{D\mathbf{u}}{Dt}=\nabla\cdot symbol{\sigma}+\mathbf{f}
  33. s y m b o l σ symbol{\sigma}
  34. 𝐟 \mathbf{f}
  35. s y m b o l σ symbol{\sigma}
  36. σ i j = ( σ x x τ x y τ x z τ y x σ y y τ y z τ z x τ z y σ z z ) \sigma_{ij}=\begin{pmatrix}\sigma_{xx}&\tau_{xy}&\tau_{xz}\\ \tau_{yx}&\sigma_{yy}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{zz}\end{pmatrix}
  37. σ \sigma
  38. τ \tau
  39. σ i j = ( σ x x τ x y τ x z τ y x σ y y τ y z τ z x τ z y σ z z ) = - ( p 0 0 0 p 0 0 0 p ) + ( σ x x + p τ x y τ x z τ y x σ y y + p τ y z τ z x τ z y σ z z + p ) = - p I + s y m b o l τ \sigma_{ij}=\begin{pmatrix}\sigma_{xx}&\tau_{xy}&\tau_{xz}\\ \tau_{yx}&\sigma_{yy}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{zz}\end{pmatrix}=-\begin{pmatrix}p&0&0\\ 0&p&0\\ 0&0&p\end{pmatrix}+\begin{pmatrix}\sigma_{xx}+p&\tau_{xy}&\tau_{xz}\\ \tau_{yx}&\sigma_{yy}+p&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{zz}+p\end{pmatrix}=-pI+symbol\tau
  40. I I
  41. s y m b o l τ symbol\tau
  42. p = - 1 3 ( σ x x + σ y y + σ z z ) . p=-\frac{1}{3}\left(\sigma_{xx}+\sigma_{yy}+\sigma_{zz}\right).
  43. s y m b o l τ symbol\tau
  44. s y m b o l τ symbol\tau
  45. ρ D 𝐮 D t = - p + \cdotsymbol τ + 𝐟 \rho\frac{D\mathbf{u}}{Dt}=-\nabla p+\nabla\cdotsymbol\tau+\mathbf{f}
  46. s y m b o l τ symbol\tau
  47. p p
  48. τ u y \tau\propto\frac{\partial u}{\partial y}
  49. s y m b o l τ \nabla\cdot symbol\tau
  50. τ i j = μ ( u i x j + u j x i - 2 3 δ i j u k x k ) + δ i j λ u k x k \tau_{ij}=\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}% {\partial x_{i}}-\tfrac{2}{3}\delta_{ij}\frac{\partial u_{k}}{\partial x_{k}}% \right)+\delta_{ij}\lambda\frac{\partial u_{k}}{\partial x_{k}}
  51. δ i j \delta_{ij}
  52. τ i j \tau_{ij}
  53. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p + ( μ ( 𝐮 + ( 𝐮 ) T ) ) + ( - 2 μ 3 𝐮 ) + ρ 𝐠 \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p+\nabla\cdot\left(\mu(\nabla\mathbf{u}+(\nabla\mathbf{u})^{T% })\right)+\nabla\left(-\frac{2\mu}{3}\nabla\cdot\mathbf{u}\right)+\rho\mathbf{g}
  54. 𝐟 = ρ 𝐠 \mathbf{f}=\rho\mathbf{g}
  55. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  56. ρ D h D t = D p D t + ( k T ) + Φ \rho\frac{Dh}{Dt}=\frac{Dp}{Dt}+\nabla\cdot(k\nabla T)+\Phi
  57. h h
  58. T T
  59. Φ \Phi
  60. Φ = μ ( 2 ( u x ) 2 + 2 ( v y ) 2 + 2 ( w z ) 2 + ( v x + u y ) 2 + ( w y + v z ) 2 + ( u z + w x ) 2 ) + λ ( 𝐮 ) 2 \Phi=\mu\left(2\left(\frac{\partial u}{\partial x}\right)^{2}+2\left(\frac{% \partial v}{\partial y}\right)^{2}+2\left(\frac{\partial w}{\partial z}\right)% ^{2}+\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)^% {2}+\left(\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}\right)^{% 2}+\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)^{2% }\right)+\lambda(\nabla\cdot\mathbf{u})^{2}
  61. μ \mu
  62. λ = 0 \lambda=0
  63. 𝐮 = 0 \nabla\cdot\mathbf{u}=0
  64. x x
  65. x ( 2 μ u x + λ 𝐮 ) + y ( μ ( u y + v x ) ) + z ( μ ( u z + w x ) ) = 2 μ 2 u x 2 + μ 2 u y 2 + μ 2 v y x + μ 2 u z 2 + μ 2 w z x = μ 2 u x 2 + μ 2 u y 2 + μ 2 u z 2 + μ 2 u x 2 + μ 2 v y x + μ 2 w z x = μ 2 u + μ x \cancelto 0 ( u x + v y + w z ) = μ 2 u \begin{aligned}&\displaystyle\frac{\partial}{\partial x}\left(2\mu\frac{% \partial u}{\partial x}+\lambda\nabla\cdot\mathbf{u}\right)+\frac{\partial}{% \partial y}\left(\mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{% \partial x}\right)\right)+\frac{\partial}{\partial z}\left(\mu\left(\frac{% \partial u}{\partial z}+\frac{\partial w}{\partial x}\right)\right)\\ \\ &\displaystyle=2\mu\frac{\partial^{2}u}{\partial x^{2}}+\mu\frac{\partial^{2}u% }{\partial y^{2}}+\mu\frac{\partial^{2}v}{\partial y\,\partial x}+\mu\frac{% \partial^{2}u}{\partial z^{2}}+\mu\frac{\partial^{2}w}{\partial z\,\partial x}% \\ \\ &\displaystyle=\mu\frac{\partial^{2}u}{\partial x^{2}}+\mu\frac{\partial^{2}u}% {\partial y^{2}}+\mu\frac{\partial^{2}u}{\partial z^{2}}+\mu\frac{\partial^{2}% u}{\partial x^{2}}+\mu\frac{\partial^{2}v}{\partial y\,\partial x}+\mu\frac{% \partial^{2}w}{\partial z\,\partial x}\\ \\ &\displaystyle=\mu\nabla^{2}u+\mu\frac{\partial}{\partial x}\cancelto{0}{\left% (\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}% {\partial z}\right)}=\mu\nabla^{2}u\end{aligned}\,
  66. y y
  67. z z
  68. μ 2 v \mu\nabla^{2}v
  69. μ 2 w \mu\nabla^{2}w
  70. u y = { 0 , τ < τ 0 ( τ - τ 0 ) / μ , τ τ 0 \frac{\partial u}{\partial y}=\left\{\begin{matrix}0&,\quad\tau<\tau_{0}\\ (\tau-\tau_{0})/{\mu}&,\quad\tau\geq\tau_{0}\end{matrix}\right.
  71. τ \tau
  72. τ = K ( u y ) n \tau=K\left(\frac{\partial u}{\partial y}\right)^{n}
  73. × ( ϕ ) = 0 \nabla\times(\nabla\phi)=0
  74. ( × 𝐀 ) = 0 \nabla\cdot(\nabla\times\mathbf{A})=0
  75. ϕ \phi
  76. 𝐀 \mathbf{A}
  77. × ( 𝐮 t + 𝐮 𝐮 ) = ν × ( 2 𝐮 ) \nabla\times\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla% \mathbf{u}\right)=\nu\nabla\times(\nabla^{2}\mathbf{u})
  78. ψ \vec{\psi}
  79. 𝐮 = 0 ( × ψ ) = 0 0 = 0 \nabla\cdot\mathbf{u}=0\quad\Rightarrow\quad\nabla\cdot(\nabla\times\vec{\psi}% )=0\quad\Rightarrow\quad 0=0
  80. 𝐮 = × ψ \mathbf{u}=\nabla\times\vec{\psi}
  81. × ( t ( × ψ ) + ( × ψ ) ( × ψ ) ) = ν × ( 2 ( × ψ ) ) \nabla\times\left(\frac{\partial}{\partial t}(\nabla\times\vec{\psi})+(\nabla% \times\vec{\psi})\cdot\nabla(\nabla\times\vec{\psi})\right)=\nu\nabla\times(% \nabla^{2}(\nabla\times\vec{\psi}))
  82. 𝐮 = u 1 𝐞 1 + u 2 𝐞 2 + u 3 𝐞 3 \mathbf{u}=u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}+u_{3}\mathbf{e}_{3}
  83. 𝐞 i \mathbf{e}_{i}
  84. u i u_{i}
  85. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  86. 𝐮 = u 1 𝐞 1 + u 2 𝐞 2 \mathbf{u}=u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}
  87. u 1 x 3 = u 2 x 3 = 0 \frac{\partial u_{1}}{\partial x_{3}}=\frac{\partial u_{2}}{\partial x_{3}}=0
  88. ψ \vec{\psi}
  89. 𝐮 = × ψ \mathbf{u}=\nabla\times\vec{\psi}
  90. u 1 𝐞 1 + u 2 𝐞 2 = 𝐞 1 h 2 h 3 [ x 2 ( h 3 ψ 3 ) - x 3 ( h 2 ψ 2 ) ] + 𝐞 2 h 3 h 1 [ x 3 ( h 1 ψ 1 ) - x 1 ( h 3 ψ 3 ) ] + 𝐞 3 h 1 h 2 [ x 1 ( h 2 ψ 2 ) - x 2 ( h 1 ψ 1 ) ] u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}=\frac{\mathbf{e}_{1}}{h_{2}h_{3}}\left% [\frac{\partial}{\partial x_{2}}\left(h_{3}\psi_{3}\right)-\frac{\partial}{% \partial x_{3}}\left(h_{2}\psi_{2}\right)\right]+\frac{\mathbf{e}_{2}}{h_{3}h_% {1}}\left[\frac{\partial}{\partial x_{3}}\left(h_{1}\psi_{1}\right)-\frac{% \partial}{\partial x_{1}}\left(h_{3}\psi_{3}\right)\right]+\frac{\mathbf{e}_{3% }}{h_{1}h_{2}}\left[\frac{\partial}{\partial x_{1}}\left(h_{2}\psi_{2}\right)-% \frac{\partial}{\partial x_{2}}\left(h_{1}\psi_{1}\right)\right]
  91. ψ 1 = ψ 2 = 0 \psi_{1}=\psi_{2}=0
  92. u 1 𝐞 1 + u 2 𝐞 2 = 𝐞 1 h 2 h 3 x 2 ( h 3 ψ 3 ) - 𝐞 2 h 3 h 1 x 1 ( h 3 ψ 3 ) u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}=\frac{\mathbf{e}_{1}}{h_{2}h_{3}}\frac% {\partial}{\partial x_{2}}\left(h_{3}\psi_{3}\right)-\frac{\mathbf{e}_{2}}{h_{% 3}h_{1}}\frac{\partial}{\partial x_{1}}\left(h_{3}\psi_{3}\right)
  93. ψ \vec{\psi}
  94. ψ 3 = ψ \psi_{3}=\psi
  95. ψ \psi
  96. ψ = 1 h 1 h 2 h 3 x 3 ( ψ h 1 h 2 ) = 0 \nabla\cdot\vec{\psi}=\frac{1}{h_{1}h_{2}h_{3}}\frac{\partial}{\partial x_{3}}% \left(\psi h_{1}h_{2}\right)=0
  97. h 1 h_{1}
  98. h 2 h_{2}
  99. x 3 x_{3}
  100. × ( × ψ ) = ( ψ ) - 2 ψ = - 2 ψ \nabla\times(\nabla\times\vec{\psi})=\nabla(\nabla\cdot\vec{\psi})-\nabla^{2}% \vec{\psi}=-\nabla^{2}\vec{\psi}
  101. t ( 2 ψ ) + ( × ψ ) ( 2 ψ ) = ν 4 ψ \frac{\partial}{\partial t}(\nabla^{2}\psi)+(\nabla\times\vec{\psi})\cdot% \nabla(\nabla^{2}\psi)=\nu\nabla^{4}\psi
  102. 4 \nabla^{4}
  103. t ( × ( × ψ ) ) + × ( ( × ψ ) ( × ψ ) ) = ν × ( 2 ( × ψ ) ) \frac{\partial}{\partial t}(\nabla\times(\nabla\times\vec{\psi}))+\nabla\times% \left((\nabla\times\vec{\psi})\cdot\nabla(\nabla\times\vec{\psi})\right)=\nu% \nabla\times(\nabla^{2}(\nabla\times\vec{\psi}))
  104. - t ( 2 ψ ) + × ( ( ( × ψ ) ( × ψ ) 2 ) + ( × ( × ψ ) ) × ( × ψ ) ) = ν ( 2 ( ( ψ ) ) - 4 ψ ) -\frac{\partial}{\partial t}(\nabla^{2}\vec{\psi})+\nabla\times\left(\nabla% \left(\frac{(\nabla\times\vec{\psi})\cdot(\nabla\times\vec{\psi})}{2}\right)+% \left(\nabla\times(\nabla\times\vec{\psi})\right)\times(\nabla\times\vec{\psi}% )\right)=\nu(\nabla^{2}(\nabla(\nabla\cdot\vec{\psi}))-\nabla^{4}\vec{\psi})
  105. ψ \vec{\psi}
  106. t ( 2 ψ ) + × ( 2 ψ × ( × ψ ) ) = ν 4 ψ \frac{\partial}{\partial t}(\nabla^{2}\vec{\psi})+\nabla\times\left(\nabla^{2}% \vec{\psi}\times(\nabla\times\vec{\psi})\right)=\nu\nabla^{4}\vec{\psi}
  107. t ( 2 ψ ) + ( × ψ ) ( 2 ψ ) - ( 2 ψ ) ( × ψ ) + ( 2 ψ ) ( ( × ψ ) ) - ( × ψ ) ( ( 2 ψ ) ) = ν 4 ψ \frac{\partial}{\partial t}(\nabla^{2}\vec{\psi})+(\nabla\times\vec{\psi})% \cdot\nabla(\nabla^{2}\vec{\psi})-(\nabla^{2}\vec{\psi})\cdot\nabla(\nabla% \times\vec{\psi})+(\nabla^{2}\vec{\psi})(\nabla\cdot(\nabla\times\vec{\psi}))-% (\nabla\times\vec{\psi})(\nabla\cdot(\nabla^{2}\vec{\psi}))=\nu\nabla^{4}\vec{\psi}
  108. h 3 h_{3}
  109. t ( 2 ψ ) + ( × ψ ) ( 2 ψ ) = ν 4 ψ \frac{\partial}{\partial t}(\nabla^{2}\vec{\psi})+(\nabla\times\vec{\psi})% \cdot\nabla(\nabla^{2}\vec{\psi})=\nu\nabla^{4}\vec{\psi}
  110. u 3 = u 1 x 3 = u 2 x 3 = 0 u_{3}=\frac{\partial u_{1}}{\partial x_{3}}=\frac{\partial u_{2}}{\partial x_{% 3}}=0
  111. h 1 x 3 = h 2 x 3 = 0 \frac{\partial h_{1}}{\partial x_{3}}=\frac{\partial h_{2}}{\partial x_{3}}=0
  112. - 2 ψ = × ( × ψ ) = × 𝐮 -\nabla^{2}\vec{\psi}=\nabla\times(\nabla\times\vec{\psi})=\nabla\times\mathbf% {u}
  113. σ = - π I + s y m b o l τ \sigma=-\pi I+symbol\tau
  114. σ i j = - p δ i j + μ ( u i x j + u j x i ) + δ i j λ 𝐮 . \sigma_{ij}=-p\delta_{ij}+\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac% {\partial u_{j}}{\partial x_{i}}\right)+\delta_{ij}\lambda\nabla\cdot\mathbf{u}.
  115. σ \displaystyle\sigma
  116. e e
  117. e i j = 1 2 ( u i x j + u j x i ) . e_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u% _{j}}{\partial x_{i}}\right).

Determination_of_equilibrium_constants.html

  1. K = [ S ] σ [ T ] τ [ A ] α [ B ] β K=\frac{{[S]}^{\sigma}{[T]}^{\tau}\cdots}{{[A]}^{\alpha}{[B]}^{\beta}\cdots}
  2. E = E 0 + R T n F ln A E=E^{0}+\frac{RT}{nF}\ln{A}
  3. p H = n F R T ( E 0 - E ) pH=\frac{nF}{RT}(E^{0}-E)
  4. E = E 0 + s log 10 [ A ] E=E^{0}+s\log_{10}[A]
  5. A = ϵ c A=\ell\sum{\epsilon c}
  6. I = ϕ c I=\sum\phi c
  7. δ ¯ = c i δ i c i \bar{\delta}=\frac{\sum c_{i}\delta_{i}}{\sum c_{i}}
  8. p A + q B A p B q : β p q = [ A p B q ] [ A ] p [ B ] q pA+qB...\rightleftharpoons A_{p}B_{q}\cdots:\beta_{pq\cdots}=\frac{[A_{p}B_{q}% \cdots]}{[A]^{p}[B]^{q}\cdots}
  9. H 2 O H + + O H - : K W = [ H + ] [ O H - ] [ H 2 O ] H_{2}O\rightleftharpoons H^{+}+OH^{-}:K_{W}^{\prime}=\frac{[H^{+}][OH^{-}]}{[H% _{2}O]}
  10. K W = [ H + ] [ O H - ] K_{W}=[H^{+}][OH^{-}]\,
  11. [ O H - ] = K W [ H + ] - 1 [OH^{-}]=K_{W}[H^{+}]^{-1}\,
  12. T A = [ A ] + p β p q [ A ] p [ B ] q T B = [ B ] + q β p q [ A ] p [ B ] q \begin{aligned}\displaystyle T_{A}&\displaystyle=[A]+\sum p\beta_{pq\cdots}[A]% ^{p}[B]^{q}\cdots\\ \displaystyle T_{B}&\displaystyle=[B]+\sum q\beta_{pq\cdots}[A]^{p}[B]^{q}% \cdots\\ &\displaystyle{}\ \ \vdots\end{aligned}
  13. β \beta
  14. [ A ] = β 10 [ A ] , [ B ] = β 01 [ B ] [A]=\beta_{10\ldots}[A],[B]=\beta_{01\ldots}[B]\ldots\,
  15. β 10 = β 01 = 1 \beta_{10\ldots}=\beta_{01\ldots}\ldots=1\,
  16. T A = p β p q [ A ] p [ B ] q T B = q β p q [ A ] p [ B ] q \begin{aligned}\displaystyle T_{A}&\displaystyle=\sum p\beta_{pq\cdots}[A]^{p}% [B]^{q}\cdots\\ \displaystyle T_{B}&\displaystyle=\sum q\beta_{pq\cdots}[A]^{p}[B]^{q}\cdots\\ &\displaystyle{}\ \ \vdots\end{aligned}
  17. T R = R 0 + v i [ R ] v 0 + v i T_{R}=\frac{R_{0}+v_{i}[R]}{v_{0}+v_{i}}
  18. R 0 R_{0}
  19. v 0 v_{0}
  20. [ R ] [R]
  21. v i v_{i}
  22. r i = y i obs - y i calc r_{i}=y_{i}\text{obs}-y_{i}\text{calc}
  23. U = i j r i W i j r j U=\sum_{i}\sum_{j}r_{i}W_{ij}r_{j}\,
  24. U = i W i i r i 2 U=\sum_{i}W_{ii}r_{i}^{2}
  25. W i j = 0 W_{ij}=0
  26. W i i = 1 W_{ii}=1
  27. U = U 0 + i U p i δ p i U=U^{0}+\sum_{i}\frac{\partial U}{\partial p_{i}}\delta p_{i}
  28. δ p i \delta p_{i}
  29. U p i \frac{\partial U}{\partial p_{i}}
  30. J j k = y j c a l c p k J_{jk}=\frac{\partial y_{j}^{calc}}{\partial p_{k}}
  31. δ 𝐩 \mathbf{\delta p}
  32. 𝐔 𝐩 = 𝟎 \mathbf{\frac{\partial U}{\partial p}=0}
  33. ( 𝐉 𝐓 𝐖𝐉 ) δ 𝐩 = 𝐉 𝐓 𝐖𝐫 \mathbf{\left(J^{T}WJ\right)\delta p=J^{T}Wr}
  34. δ p \delta p
  35. 𝐩 𝐧 + 𝟏 = 𝐩 𝐧 + δ 𝐩 \mathbf{p^{n+1}=p^{n}+\delta p}
  36. y c a l c y^{calc}
  37. 𝐩 𝐧 + 𝟏 = 𝐩 𝐧 + f δ 𝐩 \mathbf{p^{n+1}=p^{n}}+f\mathbf{\delta p}
  38. δ 𝐩 \mathbf{\delta p}
  39. ( 𝐉 𝐓 𝐖𝐉 + λ 𝐈 ) δ 𝐩 = 𝐉 𝐓 𝐖𝐫 \mathbf{\left(J^{T}WJ+\lambda I\right)\delta p=J^{T}Wr}
  40. λ \lambda
  41. A i = ϵ p q c p q , i A_{i}=\sum\epsilon_{pq\cdots}c_{pq\cdots,i}
  42. ϵ \epsilon
  43. 𝐀 = \Epsilon 𝐂 \mathbf{A=\Epsilon C}\,
  44. ϵ \epsilon
  45. ϵ \epsilon
  46. \Epsilon = ( 𝐂 𝐓 𝐂 ) - 𝟏 𝐂 𝐓 𝐀 \mathbf{\Epsilon=\left(C^{T}C\right)^{-1}C^{T}A}
  47. ( 𝐂 𝐓 𝐂 ) - 𝟏 𝐂 𝐓 \mathbf{\left(C^{T}C\right)^{-1}C^{T}}
  48. 𝐀 = ( ( 𝐂 𝐓 𝐂 ) - 𝟏 𝐂 𝐓 𝐀 ) 𝐂 \mathbf{A=\left(\left(C^{T}C\right)^{-1}C^{T}A\right)C}
  49. 𝐩 = ( 𝐉 𝐓 𝐖𝐉 ) - 𝟏 𝐉 𝐓 𝐖𝐲 𝐨𝐛𝐬 \mathbf{p=(J^{T}WJ)^{-1}J^{T}Wy^{obs}}
  50. Σ y \Sigma^{y}
  51. Σ p \Sigma^{p}
  52. 𝚺 𝐩 = ( 𝐉 𝐓 𝐖𝐉 ) - 𝟏 𝐉 𝐓 𝐖 𝚺 𝐲 𝐖 𝐓 𝐉 ( 𝐉 𝐓 𝐖𝐉 ) - 𝟏 \mathbf{\Sigma^{p}=(J^{T}WJ)^{-1}J^{T}W\Sigma^{y}W^{T}J(J^{T}WJ)^{-1}}
  53. 𝐖 = ( 𝚺 𝐲 ) - 𝟏 \mathbf{W=(\Sigma^{y})^{-1}}
  54. 𝚺 𝐩 = ( 𝐉 𝐓 𝐖𝐉 ) - 𝟏 \mathbf{\Sigma^{p}=(J^{T}WJ)^{-1}}
  55. Σ y \Sigma^{y}
  56. W k = 1 σ E 2 + ( E v ) k 2 σ v 2 W_{k}=\frac{1}{\sigma^{2}_{E}+\left(\frac{\partial E}{\partial v}\right)^{2}_{% k}\sigma^{2}_{v}}
  57. σ E \sigma_{E}\,
  58. ( E v ) k \left(\frac{\partial E}{\partial v}\right)_{k}
  59. σ v \sigma_{v}\,
  60. 𝐖 = 𝐈 , 𝐩 = ( 𝐉 𝐓 𝐉 ) - 𝟏 𝐉 𝐓 𝐲 \mathbf{W=I,p=(J^{T}J)^{-1}J^{T}y}
  61. Σ y = σ 2 𝐈 \Sigma^{y}=\sigma^{2}\mathbf{I}
  62. σ 2 \sigma^{2}\,
  63. 𝐈 \mathbf{I}
  64. σ 2 \sigma^{2}\,
  65. U n d - n p \frac{U}{n_{d}-n_{p}}
  66. 𝚺 𝐩 = 𝐔 𝐧 𝐝 - 𝐧 𝐩 ( 𝐉 𝐓 𝐉 ) - 𝟏 \mathbf{\Sigma^{p}=\frac{U}{n_{d}-n_{p}}(J^{T}J)^{-1}}
  67. Σ i i p \Sigma^{p}_{ii}
  68. Σ i j p \Sigma^{p}_{ij}
  69. L 3 - + 2 H + L H 2 - : [ L H 2 - ] = β 12 [ L 3 - ] [ H + ] 2 L^{3-}+2H^{+}\leftrightharpoons LH_{2}^{-}:[LH_{2}^{-}]=\beta_{12}[L^{3-}][H^{% +}]^{2}
  70. L 3 - + 3 H + L H 3 : [ L H 3 ] = β 13 [ L 3 - ] [ H + ] 3 L^{3-}+3H^{+}\leftrightharpoons LH_{3}:[LH_{3}]=\beta_{13}[L^{3-}][H^{+}]^{3}
  71. L H 2 - + H + L H 3 : [ L H 3 ] = K [ L H 2 - ] [ H + ] LH_{2}^{-}+H^{+}\leftrightharpoons LH_{3}:[LH_{3}]=K[LH_{2}^{-}][H^{+}]
  72. β 13 [ L 3 - ] [ H + ] 3 = K β 12 [ L 3 - ] [ H + ] 2 [ H + ] \beta_{13}[L^{3-}][H^{+}]^{3}=K\beta_{12}[L^{3-}][H^{+}]^{2}[H^{+}]\,
  73. β 13 = K β 12 : K = β 13 β 12 \beta_{13}=K\beta_{12}:K=\frac{\beta_{13}}{\beta_{12}}\,
  74. p K a = - log ( 1 / K ) pK_{a}=-\log(1/K)\,
  75. p K a 1 = log β 13 - log β 12 pKa_{1}=\log\beta_{13}-\log\beta_{12}\,
  76. σ K 2 = σ β 12 2 + σ β 13 2 - 2 σ β 12 σ β 13 ρ 12 , 13 \sigma^{2}_{K}=\sigma^{2}_{\beta_{12}}+\sigma^{2}_{\beta_{13}}-2\sigma_{\beta_% {12}}\sigma_{\beta_{13}}\rho_{12,13}\,
  77. σ log K = σ K K \sigma_{\log K}=\frac{\sigma_{K}}{K}
  78. U n d - n p \frac{U}{n_{d}-n_{p}}
  79. U n d - n p \frac{U}{n_{d}-n_{p}}
  80. β \beta
  81. σ p H \sigma_{pH}
  82. 𝐫 = 𝐲 obs - 𝐉 ( 𝐉 𝐓 𝐓 ) - 𝟏 𝐉 𝐓 𝐲 obs \mathbf{r=y\text{obs}-J\left(J^{T}T\right)^{-1}J^{T}y\text{obs}}
  83. 𝐉 ( 𝐉 𝐓 𝐓 ) - 𝟏 𝐉 \mathbf{J\left(J^{T}T\right)^{-1}J}
  84. 𝐇 \mathbf{H}
  85. 𝐫 = ( 𝐈 - 𝐇 ) 𝐲 obs \mathbf{r=\left(I-H\right)y\text{obs}}
  86. 𝐌 𝐫 = ( 𝐈 - 𝐇 ) 𝐌 𝐲 ( 𝐈 - 𝐇 ) \mathbf{M^{r}=\left(I-H\right)M^{y}\left(I-H\right)}

Detrended_fluctuation_analysis.html

  1. x t x_{t}
  2. t t\in\mathbb{N}
  3. X t X_{t}
  4. X t = i = 1 t ( x i - x ) X_{t}=\sum_{i=1}^{t}(x_{i}-\langle x\rangle)
  5. X t X_{t}
  6. X t X_{t}
  7. Y j Y_{j}
  8. L L
  9. E 2 E^{2}
  10. a , b a,b
  11. E 2 = j = 1 L ( Y j - j a - b ) 2 . E^{2}=\sum_{j=1}^{L}\left(Y_{j}-ja-b\right)^{2}.
  12. j a + b ja+b
  13. n n
  14. F ( L ) = [ 1 L j = 1 L ( Y j - j a - b ) 2 ] 1 2 . F(L)=\left[\frac{1}{L}\sum_{j=1}^{L}\left(Y_{j}-ja-b\right)^{2}\right]^{\frac{% 1}{2}}.
  15. L L
  16. L L
  17. F ( L ) F(L)
  18. F ( L ) L α F(L)\propto L^{\alpha}
  19. α \alpha
  20. L L
  21. F ( L ) F(L)
  22. L \sqrt{L}
  23. 1 2 \tfrac{1}{2}
  24. α < 1 / 2 \alpha<1/2
  25. α 1 / 2 \alpha\simeq 1/2
  26. α > 1 / 2 \alpha>1/2
  27. α 1 \alpha\simeq 1
  28. α > 1 \alpha>1
  29. α 3 / 2 \alpha\simeq 3/2
  30. n = 1 n=1
  31. n n
  32. n n
  33. x i x_{i}
  34. X t X_{t}
  35. x i x_{i}
  36. n n
  37. n - 1 n-1
  38. x i x_{i}
  39. F ( L ) F(L)
  40. α = α ( 2 ) \alpha=\alpha(2)
  41. q q
  42. α ( q ) \alpha(q)
  43. H = α ( 2 ) H=\alpha(2)
  44. H = α ( 2 ) - 1 H=\alpha(2)-1
  45. γ \gamma
  46. C ( L ) L - γ C(L)\sim L^{-\gamma}\!
  47. P ( f ) f - β P(f)\sim f^{-\beta}\!
  48. γ = 2 - 2 α \gamma=2-2\alpha
  49. β = 2 α - 1 \beta=2\alpha-1
  50. γ = 1 - β \gamma=1-\beta
  51. α \alpha
  52. β \beta
  53. α = ( β + 1 ) / 2 \alpha=(\beta+1)/2
  54. β [ - 1 , 1 ] \beta\in[-1,1]
  55. α = [ 0 , 1 ] \alpha=[0,1]
  56. β = 2 H - 1 \beta=2H-1
  57. H H
  58. α \alpha
  59. H H
  60. β [ 1 , 3 ] \beta\in[1,3]
  61. α = [ 1 , 2 ] \alpha=[1,2]
  62. β = 2 H + 1 \beta=2H+1
  63. H H
  64. α \alpha
  65. H + 1 H+1
  66. α \alpha
  67. L L
  68. α \alpha

Deviation_(statistics).html

  1. D i = | x i - m ( X ) | D_{i}=|x_{i}-m(X)|
  2. x ¯ \overline{x}

Devil's_curve.html

  1. y 2 ( y 2 - a 2 ) = x 2 ( x 2 - b 2 ) . y^{2}(y^{2}-a^{2})=x^{2}(x^{2}-b^{2}).

Dielectric_barrier_discharge.html

  1. C 1 C_{1}
  2. C g C_{g}
  3. C p C_{p}
  4. R p R_{p}
  5. S S
  6. C 1 C_{1}
  7. C 2 C_{2}
  8. C 1 C_{1}
  9. C 2 C_{2}
  10. C s = C 1 C 2 C 1 + C 2 C_{s}=\frac{C_{1}C_{2}}{C_{1}+C_{2}}
  11. I ( t ) I(t)
  12. I ( t ) = C s d U d t I(t)=C_{s}\frac{dU}{dt}
  13. U ( t ) U(t)
  14. U ( t ) U(t)
  15. I ( t ) I(t)
  16. S S
  17. R p R_{p}
  18. U g ( t ) U_{g}(t)
  19. R p R_{p}
  20. C p C_{p}
  21. U g ( t ) U_{g}(t)
  22. U g ( t ) = U ( t ) - 1 C 1 t 0 I ( t 1 ) d t 1 + U 0 U_{g}(t)=U(t)-\frac{1}{C_{1}}\int\limits_{t}^{0}I(t_{1})dt_{1}+U_{0}
  23. C 1 C_{1}
  24. U 0 U_{0}
  25. I ( t ) I(t)
  26. U g ( t ) U_{g}(t)
  27. R p R_{p}
  28. C p C_{p}
  29. U g ( t ) U_{g}(t)
  30. U g ( t ) = U R p + U C p U_{g}(t)=U_{Rp}+U_{Cp}
  31. U R p ( t ) U_{Rp}(t)
  32. U C p ( t ) U_{Cp}(t)
  33. R p R_{p}
  34. C p C_{p}
  35. I ( t ) = C p ( d U C p / d t ) I(t)=C_{p}(dU_{Cp}/dt)
  36. U R p ( t ) = R p I ( t ) = R p C p ( d U C p / d t ) U_{Rp}(t)=R_{p}I(t)=R_{p}C_{p}(dU_{Cp}/dt)
  37. d U C p ( t ) d t + 1 R p C p U C p ( t ) = 1 R p C p U g ( t ) \frac{dU_{Cp}(t)}{dt}+\frac{1}{R_{p}C_{p}}U_{Cp}(t)=\frac{1}{R_{p}C_{p}}U_{g}(t)
  38. U C p ( t ) = 1 R p C p exp ( - t / R p C p ) ( 0 t U g ( t 1 ) exp ( t 1 / R p C p ) d t 1 + C i n t ) U_{Cp}(t)=\frac{1}{R_{p}C_{p}}\exp(-t/R_{p}C_{p})\left(\int\limits_{0}^{t}U_{g% }(t_{1})\exp(t_{1}/R_{p}C_{p})dt_{1}+C_{int}\right)
  39. C i n t C_{int}
  40. t t
  41. I ( t ) I(t)
  42. U g ( t ) U_{g}(t)
  43. R p R_{p}
  44. C p C_{p}
  45. I ( t ) = 1 R p [ U g ( t ) - exp ( - t / R p C p ) R p C p ( 0 t U g ( t 1 ) exp ( t 1 / R p C p ) d t 1 + C i n t ) ] I(t)=\frac{1}{R_{p}}\left[U_{g}(t)-\frac{\exp(-t/R_{p}C_{p})}{R_{p}C_{p}}\left% (\int\limits_{0}^{t}U_{g}(t_{1})\exp(t_{1}/R_{p}C_{p})dt_{1}+C_{int}\right)\right]
  46. U g = U p + Δ U U_{g}=U_{p}+\Delta U
  47. U p U_{p}
  48. Δ U \Delta U
  49. R p R_{p}
  50. τ = R p C p \tau=R_{p}C_{p}
  51. Δ U \Delta U
  52. C i n t C_{int}
  53. I ( t ) I(t)
  54. U g ( t ) U_{g}(t)
  55. I ( t ) I(t)

Dielectric_elastomers.html

  1. U U
  2. p e l p_{el}
  3. U U
  4. p e l p_{el}
  5. p e q p_{eq}
  6. p e l p_{el}
  7. p e q = ε 0 ε r U 2 z 2 p_{eq}=\varepsilon_{0}\varepsilon_{r}\frac{U^{2}}{z^{2}}
  8. ε 0 \varepsilon_{0}
  9. ε r \varepsilon_{r}
  10. z z

Difference-map_algorithm.html

  1. x x
  2. A A
  3. B B
  4. P A P_{A}
  5. P B P_{B}
  6. x x
  7. A A
  8. B B
  9. x x
  10. x D ( x ) \displaystyle x\mapsto D(x)
  11. β \beta
  12. β = 1 \beta=1
  13. β = - 1 \beta=-1
  14. D ( x ) = x + P A ( 2 P B ( x ) - x ) - P B ( x ) D(x)=x+P_{A}\left(2P_{B}(x)-x\right)-P_{B}(x)
  15. x x
  16. x D ( x ) x\mapsto D(x)
  17. P A ( f B ( x ) ) = P B ( f A ( x ) ) P_{A}\left(f_{B}(x)\right)=P_{B}\left(f_{A}(x)\right)
  18. A A
  19. B B
  20. x x
  21. A A
  22. B B
  23. Δ = | P A ( f B ( x ) ) - P B ( f A ( x ) ) | \Delta=\left|P_{A}\left(f_{B}(x)\right)-P_{B}\left(f_{A}(x)\right)\right|

Differential_equations_of_addition.html

  1. ( x + y ) ( ( x a ) + ( y b ) ) = c (x+y)\oplus((x\oplus a)+(y\oplus b))=c
  2. x x
  3. y y
  4. n n
  5. a a
  6. b b
  7. c c
  8. + +
  9. \oplus
  10. 2 n 2^{n}
  11. ( a , b , c ) (a,b,c)
  12. S = { ( a i , b i , c i ) | i S=\{(a_{i},b_{i},c_{i})|i
  13. k } k\}
  14. k k
  15. k k
  16. n n

Differential_graded_category.html

  1. H o m ( A , B ) Hom(A,B)
  2. n 𝐙 H o m n ( A , B ) \oplus_{n\in\mathbf{Z}}Hom_{n}(A,B)
  3. d : H o m n ( A , B ) H o m n + 1 ( A , B ) d:Hom_{n}(A,B)\rightarrow Hom_{n+1}(A,B)
  4. d d = 0 d\circ d=0
  5. H o m ( A , B ) Hom(A,B)
  6. H o m ( A , B ) H o m ( B , C ) H o m ( A , C ) Hom(A,B)\otimes Hom(B,C)\rightarrow Hom(A,C)
  7. d ( i d A ) = 0 d(id_{A})=0
  8. Hom n ( - , - ) \mathrm{Hom}_{n}(-,-)
  9. C ( 𝒜 ) C(\mathcal{A})
  10. 𝒜 \mathcal{A}
  11. Hom C ( 𝒜 ) , n ( A , B ) \mathrm{Hom}_{C(\mathcal{A}),n}(A,B)
  12. A B [ n ] A\rightarrow B[n]
  13. Hom C ( 𝒜 ) , n ( A , B ) = Π l 𝐙 Hom ( A l , B l + n ) \mathrm{Hom}_{C(\mathcal{A}),n}(A,B)=\Pi_{l\in\mathbf{Z}}\mathrm{Hom}(A_{l},B_% {l+n})
  14. f = ( f l : A l B l + n ) f=(f_{l}:A_{l}\rightarrow B_{l+n})
  15. f l + 1 d A + ( - 1 ) n + 1 d B f l f_{l+1}\circ d_{A}+(-1)^{n+1}d_{B}\circ f_{l}
  16. d A , d B d_{A},d_{B}

Differentiation_rules.html

  1. h ( x ) = a f ( x ) + b g ( x ) . h^{\prime}(x)=af^{\prime}(x)+bg^{\prime}(x).\,
  2. d ( a f + b g ) d x = a d f d x + b d g d x . \frac{d(af+bg)}{dx}=a\frac{df}{dx}+b\frac{dg}{dx}.
  3. ( a f ) = a f (af)^{\prime}=af^{\prime}\,
  4. ( f + g ) = f + g (f+g)^{\prime}=f^{\prime}+g^{\prime}\,
  5. ( f - g ) = f - g . (f-g)^{\prime}=f^{\prime}-g^{\prime}.\,
  6. h ( x ) = f ( x ) g ( x ) + f ( x ) g ( x ) . h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x).\,
  7. d ( f g ) d x = d f d x g + f d g d x . \frac{d(fg)}{dx}=\frac{df}{dx}g+f\frac{dg}{dx}.
  8. h ( x ) = f ( g ( x ) ) g ( x ) . h^{\prime}(x)=f^{\prime}(g(x))g^{\prime}(x).\,
  9. d h d x = d f ( g ( x ) ) d g ( x ) d g ( x ) d x . \frac{dh}{dx}=\frac{df(g(x))}{dg(x)}\frac{dg(x)}{dx}.\,
  10. d h d x = d h d g d g d x . \frac{dh}{dx}=\frac{dh}{dg}\frac{dg}{dx}.\,
  11. g = 1 f g . g^{\prime}=\frac{1}{f^{\prime}\circ g}.
  12. d x d y = 1 d y / d x . \frac{dx}{dy}=\frac{1}{dy/dx}.
  13. f ( x ) = x n f(x)=x^{n}
  14. f ( x ) = n x n - 1 . f^{\prime}(x)=nx^{n-1}.\,
  15. h ( x ) = - f ( x ) ( f ( x ) ) 2 . h^{\prime}(x)=-\frac{f^{\prime}(x)}{(f(x))^{2}}.
  16. d ( 1 / f ) d x = - 1 f 2 d f d x . \frac{d(1/f)}{dx}=-\frac{1}{f^{2}}\frac{df}{dx}.\,
  17. ( f g ) = f g - g f g 2 \left(\frac{f}{g}\right)^{\prime}=\frac{f^{\prime}g-g^{\prime}f}{g^{2}}\quad
  18. ( f g ) = ( e g ln f ) = f g ( f g f + g ln f ) , (f^{g})^{\prime}=\left(e^{g\ln f}\right)^{\prime}=f^{g}\left(f^{\prime}{g\over f% }+g^{\prime}\ln f\right),\quad
  19. d d x ( c a x ) = c a x ln c a , c > 0 \frac{d}{dx}\left(c^{ax}\right)={c^{ax}\ln c\cdot a},\qquad c>0
  20. d d x ( log c x ) = 1 x ln c , c > 0 , c 1 \frac{d}{dx}\left(\log_{c}x\right)={1\over x\ln c},\qquad c>0,c\neq 1
  21. d d x ( ln | x | ) = 1 x \frac{d}{dx}\left(\ln|x|\right)={1\over x}
  22. d d x ( x x ) = x x ( 1 + ln x ) . \frac{d}{dx}\left(x^{x}\right)=x^{x}(1+\ln x).
  23. ( ln f ) = f f (\ln f)^{\prime}=\frac{f^{\prime}}{f}\quad
  24. ( sin x ) = cos x (\sin x)^{\prime}=\cos x\,
  25. ( arcsin x ) = 1 1 - x 2 (\arcsin x)^{\prime}={1\over\sqrt{1-x^{2}}}\,
  26. ( cos x ) = - sin x (\cos x)^{\prime}=-\sin x\,
  27. ( arccos x ) = - 1 1 - x 2 (\arccos x)^{\prime}=-{1\over\sqrt{1-x^{2}}}\,
  28. ( tan x ) = sec 2 x = 1 cos 2 x = 1 + tan 2 x (\tan x)^{\prime}=\sec^{2}x={1\over\cos^{2}x}=1+\tan^{2}x\,
  29. ( arctan x ) = 1 1 + x 2 (\arctan x)^{\prime}={1\over 1+x^{2}}\,
  30. ( sec x ) = sec x tan x (\sec x)^{\prime}=\sec x\tan x\,
  31. ( arcsec x ) = 1 | x | x 2 - 1 (\operatorname{arcsec}x)^{\prime}={1\over|x|\sqrt{x^{2}-1}}\,
  32. ( csc x ) = - csc x cot x (\csc x)^{\prime}=-\csc x\cot x\,
  33. ( arccsc x ) = - 1 | x | x 2 - 1 (\operatorname{arccsc}x)^{\prime}=-{1\over|x|\sqrt{x^{2}-1}}\,
  34. ( cot x ) = - csc 2 x = - 1 sin 2 x = - ( 1 + cot 2 x ) (\cot x)^{\prime}=-\csc^{2}x={-1\over\sin^{2}x}=-(1+\cot^{2}x)\,
  35. ( arccot x ) = - 1 1 + x 2 (\operatorname{arccot}x)^{\prime}=-{1\over 1+x^{2}}\,
  36. arctan ( y , x ) \arctan(y,x)
  37. [ - π , π ] [-\pi,\pi]
  38. ( x , y ) (x,y)
  39. x > 0 x>0
  40. arctan ( y , x > 0 ) = arctan ( y / x ) \arctan(y,x>0)=\arctan(y/x)
  41. arctan ( y , x ) y = x x 2 + y 2 \frac{\partial\arctan(y,x)}{\partial y}=\frac{x}{x^{2}+y^{2}}
  42. arctan ( y , x ) x = - y x 2 + y 2 . \frac{\partial\arctan(y,x)}{\partial x}=\frac{-y}{x^{2}+y^{2}}.
  43. ( sinh x ) = cosh x = e x + e - x 2 (\sinh x)^{\prime}=\cosh x=\frac{e^{x}+e^{-x}}{2}
  44. ( arsinh x ) = 1 x 2 + 1 (\operatorname{arsinh}\,x)^{\prime}={1\over\sqrt{x^{2}+1}}
  45. ( cosh x ) = sinh x = e x - e - x 2 (\cosh x)^{\prime}=\sinh x=\frac{e^{x}-e^{-x}}{2}
  46. ( arcosh x ) = 1 x 2 - 1 (\operatorname{arcosh}\,x)^{\prime}={\frac{1}{\sqrt{x^{2}-1}}}
  47. ( tanh x ) = sech 2 x (\tanh x)^{\prime}={\operatorname{sech}^{2}\,x}
  48. ( artanh x ) = 1 1 - x 2 (\operatorname{artanh}\,x)^{\prime}={1\over 1-x^{2}}
  49. ( sech x ) = - tanh x sech x (\operatorname{sech}\,x)^{\prime}=-\tanh x\,\operatorname{sech}\,x
  50. ( arsech x ) = - 1 x 1 - x 2 (\operatorname{arsech}\,x)^{\prime}=-{1\over x\sqrt{1-x^{2}}}
  51. ( csch x ) = - coth x csch x (\operatorname{csch}\,x)^{\prime}=-\,\operatorname{coth}\,x\,\operatorname{% csch}\,x
  52. ( arcsch x ) = - 1 | x | 1 + x 2 (\operatorname{arcsch}\,x)^{\prime}=-{1\over|x|\sqrt{1+x^{2}}}
  53. ( coth x ) = - csch 2 x (\operatorname{coth}\,x)^{\prime}=-\,\operatorname{csch}^{2}\,x
  54. ( arcoth x ) = - 1 1 - x 2 (\operatorname{arcoth}\,x)^{\prime}=-{1\over 1-x^{2}}
  55. Γ ( x ) = 0 t x - 1 e - t ln t d t \Gamma^{\prime}(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\ln t\,dt
  56. = Γ ( x ) ( n = 1 ( ln ( 1 + 1 n ) - 1 x + n ) - 1 x ) = Γ ( x ) ψ ( x ) =\Gamma(x)\left(\sum_{n=1}^{\infty}\left(\ln\left(1+\dfrac{1}{n}\right)-\dfrac% {1}{x+n}\right)-\dfrac{1}{x}\right)=\Gamma(x)\psi(x)
  57. ζ ( x ) = - n = 1 ln n n x = - ln 2 2 x - ln 3 3 x - ln 4 4 x - \zeta^{\prime}(x)=-\sum_{n=1}^{\infty}\frac{\ln n}{n^{x}}=-\frac{\ln 2}{2^{x}}% -\frac{\ln 3}{3^{x}}-\frac{\ln 4}{4^{x}}-\cdots\!
  58. = - p prime p - x ln p ( 1 - p - x ) 2 q prime , q p 1 1 - q - x =-\sum_{p\,\text{ prime}}\frac{p^{-x}\ln p}{(1-p^{-x})^{2}}\prod_{q\,\text{ % prime},q\neq p}\frac{1}{1-q^{-x}}\!
  59. F ( x ) = a ( x ) b ( x ) f ( x , t ) d t , F(x)=\int_{a(x)}^{b(x)}f(x,t)\,dt,
  60. f ( x , t ) f(x,t)\,
  61. x f ( x , t ) \frac{\partial}{\partial x}\,f(x,t)\,
  62. t t\,
  63. x x\,
  64. ( t , x ) (t,x)\,
  65. a ( x ) t b ( x ) , a(x)\leq t\leq b(x),
  66. x 0 x x 1 x_{0}\leq x\leq x_{1}\,
  67. a ( x ) a(x)\,
  68. b ( x ) b(x)\,
  69. x 0 x x 1 x_{0}\leq x\leq x_{1}\,
  70. x 0 x x 1 \,x_{0}\leq x\leq x_{1}\,\,
  71. F ( x ) = f ( x , b ( x ) ) b ( x ) - f ( x , a ( x ) ) a ( x ) + a ( x ) b ( x ) x f ( x , t ) d t . F^{\prime}(x)=f(x,b(x))\,b^{\prime}(x)-f(x,a(x))\,a^{\prime}(x)+\int_{a(x)}^{b% (x)}\frac{\partial}{\partial x}\,f(x,t)\;dt\,.
  72. d n d x n [ f ( g ( x ) ) ] = n ! { k m } f ( r ) ( g ( x ) ) m = 1 n 1 k m ! ( g ( m ) ( x ) ) k m \frac{d^{n}}{dx^{n}}[f(g(x))]=n!\sum_{\{k_{m}\}}f^{(r)}(g(x))\prod_{m=1}^{n}% \frac{1}{k_{m}!}\left(g^{(m)}(x)\right)^{k_{m}}
  73. r = m = 1 n - 1 k m r=\sum_{m=1}^{n-1}k_{m}
  74. { k m } \{k_{m}\}
  75. m = 1 n m k m = n \sum_{m=1}^{n}mk_{m}=n
  76. d n d x n [ f ( x ) g ( x ) ] = k = 0 n ( n k ) d n - k d x n - k f ( x ) d k d x k g ( x ) \frac{d^{n}}{dx^{n}}[f(x)g(x)]=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\frac% {d^{n-k}}{dx^{n-k}}f(x)\frac{d^{k}}{dx^{k}}g(x)

Diffraction_formalism.html

  1. a \ a
  2. L \ L
  3. S = L 2 + ( x + a / 2 ) 2 S=\sqrt{L^{2}+(x+a/2)^{2}}
  4. L ( x + a / 2 ) \ L>>(x+a/2)
  5. S ( L + ( x + a / 2 ) 2 2 L ) = L + x 2 2 L + x a 2 L + a 2 8 L S\approx\left(L+\frac{(x+a/2)^{2}}{2L}\right)=L+\frac{x^{2}}{2L}+\frac{xa}{2L}% +\frac{a^{2}}{8L}
  6. L \ L
  7. a 2 L λ \frac{a^{2}}{L}<<\lambda
  8. S L + x 2 2 L + x a 2 L S\approx L+\frac{x^{2}}{2L}+\frac{xa}{2L}
  9. a \ a
  10. Δ S = a sin θ \ \Delta S={a}\sin\theta
  11. a sin θ = n λ \ {a}\sin\theta=n\lambda
  12. n \ n
  13. λ \ \lambda
  14. a \ a
  15. θ \ \theta
  16. a sin θ = λ ( n + 1 / 2 ) {a}\sin\theta=\lambda(n+1/2)\,
  17. E ( r ) = A cos ( k r - ω t + ϕ ) / r \ E(r)=A\cos(kr-\omega t+\phi)/r
  18. k = 2 π λ \ k=\frac{2\pi}{\lambda}
  19. λ \ \lambda
  20. ω \ \omega
  21. ϕ \ \phi
  22. Ψ \ \Psi
  23. E \ E
  24. Ψ ( r ) = A e i ( k r - ω t + ϕ ) / r \ \Psi(r)=Ae^{i(kr-\omega t+\phi)}/r
  25. E ( r ) = R e ( Ψ ( r ) ) \ E(r)=Re(\Psi(r))
  26. Ψ \ \Psi
  27. N \ N
  28. x \ x
  29. Ψ t o t a l = A e i ( - ω t + ϕ ) n = 0 N - 1 e i k ( x - n a ) 2 + L 2 ( x - n a ) 2 + L 2 \Psi_{total}=Ae^{i(-\omega t+\phi)}\sum_{n=0}^{N-1}\frac{e^{ik\sqrt{(x-na)^{2}% +L^{2}}}}{\sqrt{(x-na)^{2}+L^{2}}}
  30. x \ x
  31. n \ n
  32. ( x - n a ) 2 + L 2 L + ( x - n a ) 2 / 2 L \sqrt{(x-na)^{2}+L^{2}}\approx L+(x-na)^{2}/2L
  33. a 2 2 L \frac{a^{2}}{2L}
  34. a / L \ a/L
  35. x / L \ x/L
  36. Ψ = A e i ( k ( x 2 2 L + L ) - ω t + ϕ ) L n = 0 N - 1 e - i k x n a L \Psi=A\frac{e^{i\left(k(\frac{x^{2}}{2L}+L)-\omega t+\phi\right)}}{L}\sum_{n=0% }^{N-1}e^{-ik\frac{xna}{L}}
  37. Ψ = A e i ( k ( x 2 - ( N - 1 ) a x 2 L + L ) - ω t + ϕ ) L sin ( N k a x 2 L ) sin ( k a x 2 L ) \Psi=A\frac{e^{i\left(k(\frac{x^{2}-(N-1)ax}{2L}+L)-\omega t+\phi\right)}}{L}% \frac{\sin\left(\frac{Nkax}{2L}\right)}{\sin\left(\frac{kax}{2L}\right)}
  38. I ( x ) = Ψ Ψ * = | Ψ | 2 = I 0 ( sin ( N k a x 2 L ) sin ( k a x 2 L ) ) 2 I(x)=\Psi\Psi^{*}=|\Psi|^{2}=I_{0}\left(\frac{\sin\left(\frac{Nkax}{2L}\right)% }{\sin\left(\frac{kax}{2L}\right)}\right)^{2}
  39. Ψ * \Psi^{*}
  40. Ψ \Psi
  41. Ψ \Psi^{\prime}
  42. Ψ = slit i r λ Ψ e - i k r d slit \Psi=\int_{\mathrm{slit}}\frac{i}{r\lambda}\Psi^{\prime}e^{-ikr}\,d\mathrm{slit}
  43. x = - a / 2 x^{\prime}=-a/2
  44. + a / 2 +a/2\,
  45. y = - y^{\prime}=-\infty
  46. \infty
  47. r = ( x - x ) 2 + y 2 + z 2 r=\sqrt{\left(x-x^{\prime}\right)^{2}+y^{\prime 2}+z^{2}}
  48. r = z ( 1 + ( x - x ) 2 + y 2 z 2 ) 1 2 r=z\left(1+\frac{\left(x-x^{\prime}\right)^{2}+y^{\prime 2}}{z^{2}}\right)^{% \frac{1}{2}}
  49. z | ( x - x ) | z\gg\big|\left(x-x^{\prime}\right)\big|
  50. r z ( 1 + 1 2 ( x - x ) 2 + y 2 z 2 ) r\approx z\left(1+\frac{1}{2}\frac{\left(x-x^{\prime}\right)^{2}+y^{\prime 2}}% {z^{2}}\right)
  51. r z + ( x - x ) 2 + y 2 2 z r\approx z+\frac{\left(x-x^{\prime}\right)^{2}+y^{\prime 2}}{2z}
  52. Ψ \Psi\,
  53. = i Ψ z λ - a 2 a 2 - e - i k [ z + ( x - x ) 2 + y 2 2 z ] d y d x =\frac{i\Psi^{\prime}}{z\lambda}\int_{-\frac{a}{2}}^{\frac{a}{2}}\int_{-\infty% }^{\infty}e^{-ik\left[z+\frac{\left(x-x^{\prime}\right)^{2}+y^{\prime 2}}{2z}% \right]}\,dy^{\prime}\,dx^{\prime}
  54. = i Ψ z λ e - i k z - a 2 a 2 e - i k [ ( x - x ) 2 2 z ] d x - e - i k [ y 2 2 z ] d y =\frac{i\Psi^{\prime}}{z\lambda}e^{-ikz}\int_{-\frac{a}{2}}^{\frac{a}{2}}e^{-% ik\left[\frac{\left(x-x^{\prime}\right)^{2}}{2z}\right]}\,dx^{\prime}\int_{-% \infty}^{\infty}e^{-ik\left[\frac{y^{\prime 2}}{2z}\right]}\,dy^{\prime}
  55. = Ψ i z λ e - i k x 2 2 z - a 2 a 2 e i k x x z e - i k x 2 2 z d x =\Psi^{\prime}\sqrt{\frac{i}{z\lambda}}e^{\frac{-ikx^{2}}{2z}}\int_{-\frac{a}{% 2}}^{\frac{a}{2}}e^{\frac{ikxx^{\prime}}{z}}e^{\frac{-ikx^{\prime 2}}{2z}}\,dx% ^{\prime}
  56. k x 2 / z kx^{\prime 2}/z
  57. e - i k x 2 2 z 1 e^{\frac{-ikx^{\prime 2}}{2z}}\approx 1
  58. x x^{\prime}
  59. e - i k x 2 2 z e^{\frac{-ikx^{2}}{2z}}
  60. e - i k x 2 2 z | e - i k x 2 2 z = e - i k x 2 2 z ( e - i k x 2 2 z ) * = e - i k x 2 2 z e + i k x 2 2 z = e 0 = 1 \langle e^{\frac{-ikx^{2}}{2z}}|e^{\frac{-ikx^{2}}{2z}}\rangle=e^{\frac{-ikx^{% 2}}{2z}}(e^{\frac{-ikx^{2}}{2z}})^{*}=e^{\frac{-ikx^{2}}{2z}}e^{\frac{+ikx^{2}% }{2z}}=e^{0}=1
  61. e - i k z e^{-ikz}
  62. C = Ψ i z λ C=\Psi^{\prime}\sqrt{\frac{i}{z\lambda}}
  63. Ψ \Psi\,
  64. = C - a 2 a 2 e i k x x z d x =C\int_{-\frac{a}{2}}^{\frac{a}{2}}e^{\frac{ikxx^{\prime}}{z}}\,dx^{\prime}
  65. = C ( e i k a x 2 z - e - i k a x 2 z ) i k x z =C\frac{\left(e^{\frac{ikax}{2z}}-e^{\frac{-ikax}{2z}}\right)}{\frac{ikx}{z}}
  66. sin x = e i x - e - i x 2 i \sin x=\frac{e^{ix}-e^{-ix}}{2i}
  67. sin θ = x z \sin\theta=\frac{x}{z}
  68. Ψ = a C sin k a sin θ 2 k a sin θ 2 = a C [ sinc ( k a sin θ 2 ) ] \Psi=aC\frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}=aC\left[% \operatorname{sinc}\left(\frac{ka\sin\theta}{2}\right)\right]
  69. sinc ( x ) = def sin ( x ) x \operatorname{sinc}(x)\ \stackrel{\mathrm{def}}{=}\ \frac{\operatorname{sin}(x% )}{x}
  70. 2 π λ = k \frac{2\pi}{\lambda}=k
  71. I I
  72. I ( θ ) I(\theta)\,
  73. = I 0 [ sinc ( π a λ sin θ ) ] 2 =I_{0}{\left[\operatorname{sinc}\left(\frac{\pi a}{\lambda}\sin\theta\right)% \right]}^{2}
  74. Ψ = slit i r λ Ψ e - i k r d slit \Psi=\int_{\mathrm{slit}}\frac{i}{r\lambda}\Psi^{\prime}e^{-ikr}\,d\mathrm{slit}
  75. r = z ( 1 + ( x - x ) 2 + y 2 z 2 ) 1 2 r=z\left(1+\frac{\left(x-x^{\prime}\right)^{2}+y^{\prime 2}}{z^{2}}\right)^{% \frac{1}{2}}
  76. x j = 0 n - 1 = x 0 - j d x_{j=0\cdots n-1}^{\prime}=x_{0}^{\prime}-jd
  77. r j = z ( 1 + ( x - x - j d ) 2 + y 2 z 2 ) 1 2 r_{j}=z\left(1+\frac{\left(x-x^{\prime}-jd\right)^{2}+y^{\prime 2}}{z^{2}}% \right)^{\frac{1}{2}}
  78. Ψ = j = 0 N - 1 C - a 2 a 2 e i k x ( x - j d ) z e - i k ( x - j d ) 2 2 z d x \Psi=\sum_{j=0}^{N-1}C\int_{-\frac{a}{2}}^{\frac{a}{2}}e^{\frac{ikx\left(x^{% \prime}-jd\right)}{z}}e^{\frac{-ik\left(x^{\prime}-jd\right)^{2}}{2z}}\,dx^{\prime}
  79. k ( x - j d ) 2 z \frac{k\left(x^{\prime}-jd\right)^{2}}{z}
  80. e - i k ( x - j d ) 2 2 z 1 e^{\frac{-ik\left(x^{\prime}-jd\right)^{2}}{2z}}\approx 1
  81. Ψ \Psi\,
  82. = C j = 0 N - 1 - a 2 a 2 e i k x ( x - j d ) z d x =C\sum_{j=0}^{N-1}\int_{-\frac{a}{2}}^{\frac{a}{2}}e^{\frac{ikx\left(x^{\prime% }-jd\right)}{z}}\,dx^{\prime}
  83. = a C j = 0 N - 1 ( e i k a x 2 z - i j k x d z - e - i k a x 2 z - i j k x d z ) 2 i k a x 2 z =aC\sum_{j=0}^{N-1}\frac{\left(e^{\frac{ikax}{2z}-\frac{ijkxd}{z}}-e^{\frac{-% ikax}{2z}-\frac{ijkxd}{z}}\right)}{\frac{2ikax}{2z}}
  84. = a C j = 0 N - 1 e i j k x d z ( e i k a x 2 z - e - i k a x 2 z ) 2 i k a x 2 z =aC\sum_{j=0}^{N-1}e^{\frac{ijkxd}{z}}\frac{\left(e^{\frac{ikax}{2z}}-e^{\frac% {-ikax}{2z}}\right)}{\frac{2ikax}{2z}}
  85. = a C sin k a sin θ 2 k a sin θ 2 j = 1 N - 1 e i j k d sin θ =aC\frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\sum_{j=1}^{N-1}e^% {ijkd\sin\theta}
  86. j = 0 N - 1 e x j = 1 - e N x 1 - e x . \sum_{j=0}^{N-1}e^{xj}=\frac{1-e^{Nx}}{1-e^{x}}.
  87. Ψ \Psi\,
  88. = a C sin k a sin θ 2 k a sin θ 2 ( 1 - e i N k d sin θ 1 - e i k d sin θ ) =aC\frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\left(\frac{1-e^{% iNkd\sin\theta}}{1-e^{ikd\sin\theta}}\right)
  89. = a C sin k a sin θ 2 k a sin θ 2 ( e - i N k d sin θ 2 - e i N k d sin θ 2 e - i k d sin θ 2 - e i k d sin θ 2 ) ( e i N k d sin θ 2 e i k d sin θ 2 ) =aC\frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\left(\frac{e^{-% iNkd\frac{\sin\theta}{2}}-e^{iNkd\frac{\sin\theta}{2}}}{e^{-ikd\frac{\sin% \theta}{2}}-e^{ikd\frac{\sin\theta}{2}}}\right)\left(\frac{e^{iNkd\frac{\sin% \theta}{2}}}{e^{ikd\frac{\sin\theta}{2}}}\right)
  90. = a C sin k a sin θ 2 k a sin θ 2 e - i N k d sin θ 2 - e i N k d sin θ 2 2 i e - i k d sin θ 2 - e i k d sin θ 2 2 i ( e i ( N - 1 ) k d sin θ 2 ) =aC\frac{\sin\frac{ka\sin\theta}{2}}{\frac{ka\sin\theta}{2}}\frac{\frac{e^{-% iNkd\frac{\sin\theta}{2}}-e^{iNkd\frac{\sin\theta}{2}}}{2i}}{\frac{e^{-ikd% \frac{\sin\theta}{2}}-e^{ikd\frac{\sin\theta}{2}}}{2i}}\left(e^{i(N-1)kd\frac{% \sin\theta}{2}}\right)
  91. = a C sin ( k a sin θ 2 ) k a sin θ 2 sin ( N k d sin θ 2 ) sin ( k d sin θ 2 ) e i ( N - 1 ) k d sin θ 2 =aC\frac{\sin\left(\frac{ka\sin\theta}{2}\right)}{\frac{ka\sin\theta}{2}}\frac% {\sin\left(\frac{Nkd\sin\theta}{2}\right)}{\sin\left(\frac{kd\sin\theta}{2}% \right)}e^{i\left(N-1\right)kd\frac{\sin\theta}{2}}
  92. I 0 I_{0}
  93. e i x | e i x = e 0 = 1 \langle e^{ix}\Big|e^{ix}\rangle\ =e^{0}=1
  94. I ( θ ) = I 0 [ sinc ( π a λ sin θ ) ] 2 [ sin ( N π d λ sin θ ) sin ( π d λ sin θ ) ] 2 I\left(\theta\right)=I_{0}\left[\operatorname{sinc}\left(\frac{\pi a}{\lambda}% \sin\theta\right)\right]^{2}\cdot\left[\frac{\sin\left(\frac{N\pi d}{\lambda}% \sin\theta\right)}{\sin\left(\frac{\pi d}{\lambda}\sin\theta\right)}\right]^{2}
  95. Ψ = slit i r λ Ψ e - i k r d slit \Psi=\int_{\mathrm{slit}}\frac{i}{r\lambda}\Psi^{\prime}e^{-ikr}\,d\mathrm{slit}

Diffraction_topography.html

  1. 2 ϑ B 2\vartheta_{B}
  2. Δ ϑ ( r ) = 1 h cos ϑ B s h [ h u ( r ) ] \Delta\vartheta(\vec{r})=\frac{1}{\vec{h}\cdot\cos\vartheta_{B}}\frac{\partial% }{\partial s_{\vec{h}}}\left[\vec{h}\cdot\vec{u}(\vec{r})\right]
  3. u ( r ) \vec{u}(\vec{r})
  4. s 0 \vec{s}_{0}
  5. s h \vec{s}_{h}
  6. Δ ϑ ( r ) = - tan ϑ B Δ d d ( r ) ± Δ φ ( r ) \Delta\vartheta(\vec{r})=-\tan\vartheta_{B}\frac{\Delta d}{d}(\vec{r})\pm% \Delta\varphi(\vec{r})
  7. 𝐠 𝐛 \mathbf{g}\cdot\mathbf{b}
  8. l l
  9. Δ x = S d D = S D d \Delta x=S\cdot\frac{d}{D}=\frac{S}{D}\cdot d

Diffusion_Monte_Carlo.html

  1. i d Ψ ( x , t ) d t = - 1 2 d 2 Ψ ( x , t ) d x 2 + V ( x ) Ψ ( x , t ) . i\frac{d\Psi(x,t)}{dt}=-\frac{1}{2}\frac{d^{2}\Psi(x,t)}{dx^{2}}+V(x)\Psi(x,t).
  2. H = - 1 2 d 2 d x 2 + V ( x ) H=-\frac{1}{2}\frac{d^{2}}{dx^{2}}+V(x)
  3. i d Ψ ( x , t ) d t = H Ψ ( x , t ) , i\frac{d\Psi(x,t)}{dt}=H\Psi(x,t),
  4. H Ψ ( x ) = E Ψ ( x ) H\Psi(x)=E\Psi(x)
  5. - d Ψ ( x , t ) d t = ( H - E 0 ) Ψ ( x , t ) -\frac{d\Psi(x,t)}{dt}=(H-E_{0})\Psi(x,t)
  6. E 0 E_{0}
  7. H Φ 0 ( x ) = E 0 Φ 0 ( x ) H\Phi_{0}(x)=E_{0}\Phi_{0}(x)
  8. Ψ \Psi
  9. Ψ = c 0 Φ 0 + i = 1 c i Φ i \Psi=c_{0}\Phi_{0}+\sum_{i=1}^{\infty}c_{i}\Phi_{i}
  10. Φ 0 \Phi_{0}
  11. Φ 1 \Phi_{1}
  12. Φ 0 \Phi_{0}
  13. Φ 1 \Phi_{1}
  14. E 1 > E 0 E_{1}>E_{0}
  15. c 1 c_{1}
  16. E 0 E_{0}
  17. E 0 E_{0}
  18. x ( t + τ ) = x ( t ) + τ v ( t ) + 0.5 F ( t ) τ 2 x(t+\tau)=x(t)+\tau v(t)+0.5F(t)\tau^{2}
  19. τ \tau
  20. Ψ ( x , t + τ ) = G ( x , x , τ ) Ψ ( x , t ) d x \Psi(x,t+\tau)=\int G(x,x^{\prime},\tau)\Psi(x^{\prime},t)dx^{\prime}

Diffusionless_transformation.html

  1. y = S x y=Sx

Digital_biquad_filter.html

  1. H ( z ) = b 0 + b 1 z - 1 + b 2 z - 2 a 0 + a 1 z - 1 + a 2 z - 2 \ H(z)=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}}{a_{0}+a_{1}z^{-1}+a_{2}z^{-2}}
  2. H ( z ) = b 0 + b 1 z - 1 + b 2 z - 2 1 + a 1 z - 1 + a 2 z - 2 \ H(z)=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}}{1+a_{1}z^{-1}+a_{2}z^{-2}}
  3. y [ n ] = 1 a 0 ( b 0 x [ n ] + b 1 x [ n - 1 ] + b 2 x [ n - 2 ] - a 1 y [ n - 1 ] - a 2 y [ n - 2 ] ) \ y[n]=\frac{1}{a_{0}}\left(b_{0}x[n]+b_{1}x[n-1]+b_{2}x[n-2]-a_{1}y[n-1]-a_{2% }y[n-2]\right)
  4. y [ n ] = b 0 x [ n ] + b 1 x [ n - 1 ] + b 2 x [ n - 2 ] - a 1 y [ n - 1 ] - a 2 y [ n - 2 ] \ y[n]=b_{0}x[n]+b_{1}x[n-1]+b_{2}x[n-2]-a_{1}y[n-1]-a_{2}y[n-2]
  5. b 0 b_{0}
  6. b 1 b_{1}
  7. b 2 b_{2}
  8. a 1 a_{1}
  9. a 2 a_{2}
  10. y [ n ] = b 0 w [ n ] + b 1 w [ n - 1 ] + b 2 w [ n - 2 ] , \ y[n]=b_{0}w[n]+b_{1}w[n-1]+b_{2}w[n-2],
  11. w [ n ] = x [ n ] - a 1 w [ n - 1 ] - a 2 w [ n - 2 ] . \ w[n]=x[n]-a_{1}w[n-1]-a_{2}w[n-2].

Digital_probabilistic_physics.html

  1. X ( A n ) X(A_{n})
  2. A n A_{n}
  3. p p
  4. X ( A n ) = log 2 ( 1 p ( A n ) ) = - log 2 ( p ( A n ) ) X(A_{n})=\log_{2}\left(\frac{1}{p(A_{n})}\right)=-\log_{2}(p(A_{n}))

Digital_root.html

  1. n n
  2. x \lfloor x\rfloor
  3. d r ( n ) = n - 9 n - 1 9 . dr(n)=n-9\left\lfloor\frac{n-1}{9}\right\rfloor.
  4. S ( n ) S(n)
  5. n n
  6. S ( n ) S(n)
  7. S 1 ( n ) = S ( n ) , S m ( n ) = S ( S m - 1 ( n ) ) , for m 2. S^{1}(n)=S(n),\ \ S^{m}(n)=S(S^{m-1}(n)),\ \,\text{for}\ m\geq 2.
  8. S 1 ( n ) , S 2 ( n ) , S 3 ( n ) , S^{1}(n),S^{2}(n),S^{3}(n),\cdots
  9. S σ ( n ) S^{\sigma}(n)
  10. n n
  11. 1853 1853
  12. S ( 1853 ) = 17 S(1853)=17\,
  13. S ( 17 ) = 8 S(17)=8\,
  14. S 2 ( 1853 ) = 8. S^{2}(1853)=8.\,
  15. S σ ( 1853 ) = d r ( 1853 ) = 8. S^{\sigma}(1853)=dr(1853)=8.\,
  16. S 1 ( n ) , S 2 ( n ) , S 3 ( n ) , S^{1}(n),S^{2}(n),S^{3}(n),\cdots
  17. n = d 1 + 10 d 2 + + 10 m - 1 d m n=d_{1}+10d_{2}+\cdots+10^{m-1}d_{m}
  18. i i
  19. d i d_{i}
  20. S ( n ) = d 1 + d 2 + + d m S(n)=d_{1}+d_{2}+\cdots+d_{m}
  21. S ( n ) < n S(n)<n
  22. d 2 , d 3 , , d m = 0 d_{2},d_{3},\cdots,d_{m}=0
  23. n n
  24. S ( n ) S(n)
  25. n n
  26. S ( d 1 ) = d 1 S(d_{1})=d_{1}
  27. dr ( n ) = { 0 if n = 0 , 9 if n 0 , n 0 ( mod 9 ) , n mod 9 if n 0 ( mod 9 ) . \operatorname{dr}(n)=\begin{cases}0&\mbox{if}~{}\ n=0,\\ 9&\mbox{if}~{}\ n\neq 0,\ n\ \equiv 0\;\;(\mathop{{\rm mod}}9),\\ n\ {\rm mod}\ 9&\mbox{if}~{}\ n\not\equiv 0\;\;(\mathop{{\rm mod}}9).\end{cases}
  28. dr ( n ) = 1 + ( ( n - 1 ) mod 9 ) . \mbox{dr}~{}(n)=1\ +\ ((n-1)\ {\rm mod}\ 9).
  29. 10 1 ( mod 9 ) , 10\equiv 1\;\;(\mathop{{\rm mod}}9),
  30. 10 k 1 k 1 ( mod 9 ) , 10^{k}\equiv 1^{k}\equiv 1\;\;(\mathop{{\rm mod}}9),
  31. a 100 a 10 a ( mod 9 ) a\cdot 100\equiv a\cdot 10\equiv a\;\;(\mathop{{\rm mod}}9)
  32. dr ( a b c ) a 10 2 + b 10 + c 1 a 1 + b 1 + c 1 a + b + c ( mod 9 ) \mbox{dr}~{}(abc)\equiv a\cdot 10^{2}+b\cdot 10+c\cdot 1\equiv a\cdot 1+b\cdot 1% +c\cdot 1\equiv a+b+c\;\;(\mathop{{\rm mod}}9)
  33. 10 k 10^{k}
  34. b k b^{k}
  35. 10 - 1 ( mod 11 ) , 10\equiv-1\;\;(\mathop{{\rm mod}}11),
  36. 10 2 ( - 1 ) 2 1 ( mod 11 ) , 10^{2}\equiv(-1)^{2}\equiv 1\;\;(\mathop{{\rm mod}}11),
  37. 𝑑𝑟 ( n ) = 0 n = 0. \mathit{dr}(n)=0\Leftrightarrow n=0.
  38. 𝑑𝑟 ( n ) > 0 n > 0. \mathit{dr}(n)>0\Leftrightarrow n>0.
  39. 𝑑𝑟 ( n ) = n n { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } . \mathit{dr}(n)=n\Leftrightarrow n\in\{0,1,2,3,4,5,6,7,8,9\}.
  40. 𝑑𝑟 ( n ) < n n 10. \mathit{dr}(n)<n\Leftrightarrow n\geq 10.
  41. 𝑑𝑟 ( a + b ) 𝑑𝑟 ( a ) + 𝑑𝑟 ( b ) ( mod 9 ) . \mathit{dr}(a+b)\equiv\mathit{dr}(a)+\mathit{dr}(b)\;\;(\mathop{{\rm mod}}9).
  42. 𝑑𝑟 ( a - b ) 𝑑𝑟 ( a ) - 𝑑𝑟 ( b ) ( mod 9 ) . \mathit{dr}(a-b)\equiv\mathit{dr}(a)-\mathit{dr}(b)\;\;(\mathop{{\rm mod}}9).
  43. 𝑑𝑟 ( a × b ) 𝑑𝑟 ( a ) × 𝑑𝑟 ( b ) ( mod 9 ) . \mathit{dr}(a\times b)\equiv\mathit{dr}(a)\times\mathit{dr}(b)\;\;(\mathop{{% \rm mod}}9).
  44. 𝑑𝑟 ( n ) = 9 n = 9 m for m = 1 , 2 , 3 , . \mathit{dr}(n)=9\Leftrightarrow n=9m\ \ \ \,\text{for}\ m=1,2,3,\cdots.
  45. 𝑑𝑟 ( n ) = 3 n = 9 m + 3 for m = 0 , 1 , 2 , , 𝑑𝑟 ( n ) = 6 n = 9 m + 6 for m = 0 , 1 , 2 , , 𝑑𝑟 ( n ) = 9 n = 9 m for m = 1 , 2 , 3 , . \begin{aligned}\displaystyle\mathit{dr}(n)&\displaystyle=3\Leftrightarrow n=9m% +3&\displaystyle\ \,\text{for}\ m=0,1,2,\cdots,\\ \displaystyle\mathit{dr}(n)&\displaystyle=6\Leftrightarrow n=9m+6&% \displaystyle\ \,\text{for}\ m=0,1,2,\cdots,\\ \displaystyle\mathit{dr}(n)&\displaystyle=9\Leftrightarrow n=9m&\displaystyle% \ \,\text{for}\ m=1,2,3,\cdots.\end{aligned}
  46. 𝑑𝑟 ( n ! ) = 9 n 6. \mathit{dr}(n!)=9\Leftrightarrow n\geq 6.
  47. 𝑑𝑟 ( a n ) 𝑑𝑟 n ( a ) ( mod 9 ) . \mathit{dr}(a^{n})\equiv\mathit{dr}^{n}(a)\;\;(\mathop{{\rm mod}}9).

Digital_signature_forgery.html

  1. m m
  2. σ \sigma
  3. m m
  4. m m
  5. ( m , σ ) (m,\sigma)
  6. σ \sigma
  7. m m
  8. m m
  9. ( m , σ ) (m,\sigma)
  10. e e
  11. S i g Sig
  12. S i g e ( m o d n ) | | S i g ( m o d n ) Sig^{e}(modn)||Sig(modn)
  13. S i g e = = S i g e Sig^{e}==Sig^{e}
  14. x 1 = S k ( y 1 ) x_{1}=S_{k}(y_{1})
  15. y 1 y_{1}
  16. k k
  17. x 2 = S k ( y 2 ) x_{2}=S_{k}(y_{2})
  18. x 1 x 2 ( mod n ) x_{1}\cdot x_{2}\;\;(\mathop{{\rm mod}}n)
  19. y 1 y 2 ( mod n ) y_{1}\cdot y_{2}\;\;(\mathop{{\rm mod}}n)
  20. k k
  21. ( m , σ ) (m,\sigma)
  22. m m
  23. m m
  24. m m
  25. σ \sigma
  26. m m