wpmath0000005_6

HVDC_converter_station.html

  1. 6 n + 1 6n+1
  2. 6 n - 1 6n-1
  3. 6 n 6n
  4. 12 n + 1 12n+1
  5. 12 n - 1 12n-1
  6. 12 n 12n

Hydraulic_analogy.html

  1. V V
  2. q q
  3. Q Q
  4. P P
  5. p p
  6. ϕ \phi
  7. T T
  8. k B k_{B}
  9. v v
  10. Φ V \Phi_{V}
  11. I I
  12. Q ˙ \dot{Q}
  13. F F
  14. v v
  15. j j
  16. Q ˙ ′′ \dot{Q}^{\prime\prime}
  17. σ \sigma
  18. Φ V = π r 4 8 η Δ p \Phi_{V}=\frac{\pi r^{4}}{8\eta}\frac{\Delta p^{\star}}{\ell}
  19. j = - σ ϕ j=-\sigma\nabla\phi
  20. Q ˙ ′′ = κ T \dot{Q}^{\prime\prime}=\kappa\nabla T
  21. σ = c Δ v \sigma=c\Delta v

Hydraulic_conductivity.html

  1. K K
  2. K = C ( D 10 ) 2 K=C(D_{10})^{2}
  3. C C
  4. D 10 D_{10}
  5. Q Q
  6. L L
  7. A A
  8. t t
  9. Q Q
  10. h h
  11. Q t = A v \frac{Q}{t}=Av\,
  12. v v
  13. v = K i v=Ki\,
  14. i i
  15. i = h L i=\frac{h}{L}
  16. h h
  17. L L
  18. Q t = A K h L \frac{Q}{t}=\frac{AKh}{L}
  19. K K
  20. K = Q L A h t K=\frac{QL}{Aht}
  21. K = 2.3 a L A t log ( h 1 h 2 ) K=\frac{2.3aL}{At}\log\left(\frac{h_{1}}{h_{2}}\right)
  22. r r
  23. h h^{\prime}
  24. r r
  25. h h^{\prime}
  26. r r
  27. h h^{\prime}
  28. h h^{\prime}
  29. n n
  30. T i T_{i}
  31. i - t h i-th
  32. d i d_{i}
  33. K i K_{i}
  34. T i = K i d i T_{i}=K_{i}d_{i}
  35. K i K_{i}
  36. d i d_{i}
  37. K i K_{i}
  38. d i d_{i}
  39. T i T_{i}
  40. T t T_{t}
  41. T t = T i T_{t}=\sum T_{i}
  42. \sum
  43. i = 1 , 2 , 3 , , n i=1,2,3,\cdots,n
  44. K A K_{A}
  45. K A = T t / D t K_{A}=T_{t}/D_{t}
  46. D t D_{t}
  47. D t = d i D_{t}=\sum d_{i}
  48. i = 1 , 2 , 3 , , n i=1,2,3,\cdots,n
  49. i - t h i-th
  50. d i d_{i}
  51. d i d_{i}
  52. d i d_{i}
  53. d i d_{i}
  54. i i
  55. n n
  56. d i d_{i}
  57. i i
  58. n n
  59. i - t h i-th

Hydraulic_head.html

  1. z > 0 z\,>\,0\,
  2. v = 2 g z , v=\sqrt{{2g}{z}},
  3. h = v 2 2 g h=\frac{v^{2}}{2g}
  4. g g
  5. v 2 2 g \frac{v^{2}}{2g}
  6. h = ψ + z h=\psi+z\,
  7. h h
  8. ψ \psi
  9. z z
  10. ψ = P γ = P ρ g \psi=\frac{P}{\gamma}=\frac{P}{\rho g}
  11. P P
  12. γ \gamma
  13. ρ \rho
  14. g g
  15. h f w = ψ ρ ρ f w + z h_{fw}=\psi\frac{\rho}{\rho_{fw}}+z
  16. h f w h_{fw}\,
  17. ρ f w \rho_{fw}\,
  18. i = d h d l = h 2 - h 1 length i=\frac{dh}{dl}=\frac{h_{2}-h_{1}}{\mathrm{length}}
  19. i i
  20. d h dh
  21. d l dl
  22. h = ( h x , h y , h z ) = h x 𝐢 + h y 𝐣 + h z 𝐤 \nabla h=\left({\frac{\partial h}{\partial x}},{\frac{\partial h}{\partial y}}% ,{\frac{\partial h}{\partial z}}\right)={\frac{\partial h}{\partial x}}\mathbf% {i}+{\frac{\partial h}{\partial y}}\mathbf{j}+{\frac{\partial h}{\partial z}}% \mathbf{k}

Hydraulic_machinery.html

  1. P o w e r l o s s = Δ p L S Q t o t Powerloss=\Delta p_{LS}\cdot Q_{tot}
  2. Δ p L S \Delta p_{LS}

Hydraulic_retention_time.html

  1. H R T = V o l u m e o f a e r a t i o n t a n k i n f l u e n t f l o w r a t e HRT=\frac{Volume\;of\;aeration\;tank}{influent\;flowrate}

Hydroquinone.html

  1. Fe ( CO ) 5 + 4 C 2 H 2 + 2 H 2 O 50 - 80 C 20 - 25 atm basic conditions 2 C 6 H 4 ( OH ) 2 + FeCO 3 \mathrm{Fe(CO)_{5}+4C_{2}H_{2}+2H_{2}O\xrightarrow[basic\ conditions]{\begin{% array}[]{c}50-80\ ^{\circ}\mathrm{C}\\ 20-25\ \mathrm{atm}\end{array}}2C_{6}H_{4}(OH)_{2}+FeCO_{3}}

Hyperbolic_coordinates.html

  1. { ( x , y ) : x > 0 , y > 0 } = Q \{(x,y)\ :\ x>0,\ y>0\ \}=Q\ \!
  2. H P = { ( u , v ) : u , v > 0 } HP=\{(u,v):u\in\mathbb{R},v>0\}
  3. ( x , y ) (x,y)
  4. Q Q
  5. u = ln x y u=\ln\sqrt{\frac{x}{y}}
  6. v = x y v=\sqrt{xy}
  7. u u
  8. v v
  9. x = v e u , y = v e - u x=ve^{u},\quad y=ve^{-u}
  10. Q H P Q\leftrightarrow HP
  11. x = 1 x=1
  12. y y
  13. 0 < y < 1 0<y<1
  14. u > 0 u>0
  15. 0 < z < y 0<z<y
  16. Δ u = ln y z \Delta u=\ln\sqrt{\frac{y}{z}}
  17. Δ u \Delta u

Hyperbolic_manifold.html

  1. n n
  2. n \mathbb{H}^{n}
  3. n \mathbb{H}^{n}
  4. n / Γ \mathbb{H}^{n}/\Gamma
  5. n \mathbb{H}^{n}
  6. S O 1 , n + SO^{+}_{1,n}\mathbb{R}

Hyperbolic_motion.html

  1. d ( a , b ) = | log ( b / a ) | d(a,b)=|\log(b/a)|

Hyperbolic_motion_(relativity).html

  1. α = 1 ( 1 - u 2 / c 2 ) 3 / 2 d u d t \alpha=\frac{1}{\left(1-u^{2}/c^{2}\right)^{3/2}}\frac{du}{dt}
  2. u u
  3. x 2 - c 2 t 2 = c 4 / α 2 x^{2}-c^{2}t^{2}=c^{4}/\alpha^{2}
  4. x x
  5. X = c 2 / α X=c^{2}/\alpha

Hyperbolic_partial_differential_equation.html

  1. u t t - u x x = 0. u_{tt}-u_{xx}=0.\,
  2. A u x x + 2 B u x y + C u y y + (lower order terms) = 0 Au_{xx}+2Bu_{xy}+Cu_{yy}+\,\text{(lower order terms)}=0\,
  3. B 2 - A C > 0 B^{2}-AC>0\,
  4. 2 u t 2 - c 2 2 u x 2 = 0 \frac{\partial^{2}u}{\partial t^{2}}-c^{2}\frac{\partial^{2}u}{\partial x^{2}}=0
  5. s s
  6. s s
  7. u = ( u 1 , , u s ) \vec{u}=(u_{1},\ldots,u_{s})
  8. u = u ( x , t ) \vec{u}=\vec{u}(\vec{x},t)
  9. x d \vec{x}\in\mathbb{R}^{d}
  10. ( * ) u t + j = 1 d x j f j ( u ) = 0 , (*)\quad\frac{\partial\vec{u}}{\partial t}+\sum_{j=1}^{d}\frac{\partial}{% \partial x_{j}}\vec{f^{j}}(\vec{u})=0,
  11. f j C 1 ( s , s ) , j = 1 , , d \vec{f^{j}}\in C^{1}(\mathbb{R}^{s},\mathbb{R}^{s}),j=1,\ldots,d
  12. f j \vec{f^{j}}
  13. s × s s\times s
  14. A j := ( f 1 j u 1 f 1 j u s f s j u 1 f s j u s ) , for j = 1 , , d . A^{j}:=\begin{pmatrix}\frac{\partial f_{1}^{j}}{\partial u_{1}}&\cdots&\frac{% \partial f_{1}^{j}}{\partial u_{s}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{s}^{j}}{\partial u_{1}}&\cdots&\frac{\partial f_{s}^{j}}{% \partial u_{s}}\end{pmatrix},\,\text{ for }j=1,\ldots,d.
  15. ( * ) (*)
  16. α 1 , , α d \alpha_{1},\ldots,\alpha_{d}\in\mathbb{R}
  17. A := α 1 A 1 + + α d A d A:=\alpha_{1}A^{1}+\cdots+\alpha_{d}A^{d}
  18. A A
  19. ( * ) (*)
  20. A A
  21. ( * ) (*)
  22. u = u ( x , t ) u=u(\vec{x},t)
  23. ( * ) (*)
  24. ( * * ) u t + j = 1 d x j f j ( u ) = 0 , (**)\quad\frac{\partial u}{\partial t}+\sum_{j=1}^{d}\frac{\partial}{\partial x% _{j}}{f^{j}}(u)=0,
  25. u u
  26. f = ( f 1 , , f d ) \vec{f}=(f^{1},\ldots,f^{d})
  27. u u
  28. ( * * ) (**)
  29. Ω \Omega
  30. Ω u t d Ω + Ω f ( u ) d Ω = 0. \int_{\Omega}\frac{\partial u}{\partial t}d\Omega+\int_{\Omega}\nabla\cdot\vec% {f}(u)d\Omega=0.
  31. u u
  32. f \vec{f}
  33. / t \partial/\partial t
  34. u u
  35. d d t Ω u d Ω + Ω f ( u ) n d Γ = 0 , \frac{d}{dt}\int_{\Omega}ud\Omega+\int_{\partial\Omega}\vec{f}(u)\cdot\vec{n}d% \Gamma=0,
  36. u u
  37. Ω \Omega
  38. u u
  39. Ω \partial\Omega
  40. u u
  41. Ω \Omega

Hyperbolic_secant_distribution.html

  1. f ( x ) = 1 2 sech ( π 2 x ) , f(x)=\frac{1}{2}\;\operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!,
  2. F ( x ) = 1 2 + 1 π arctan [ sinh ( π 2 x ) ] , F(x)=\frac{1}{2}+\frac{1}{\pi}\arctan\!\left[\operatorname{sinh}\!\left(\frac{% \pi}{2}\,x\right)\right]\!,
  3. = 2 π arctan [ exp ( π 2 x ) ] . =\frac{2}{\pi}\arctan\!\left[\exp\left(\frac{\pi}{2}\,x\right)\right]\!.
  4. F - 1 ( p ) = - 2 π arsinh [ cot ( π p ) ] , F^{-1}(p)=-\frac{2}{\pi}\,\operatorname{arsinh}\!\left[\cot(\pi\,p)\right]\!,
  5. = 2 π ln [ tan ( π 2 p ) ] . =\frac{2}{\pi}\,\ln\!\left[\tan\left(\frac{\pi}{2}\,p\right)\right]\!.

Hyperbolic_triangle.html

  1. K K
  2. R = 1 - K . R=\frac{1}{\sqrt{-K}}.
  3. R R
  4. ( π - A - B - C ) R 2 . (\pi-A-B-C)R^{2}{}{}.\!
  5. a a
  6. b b
  7. c c
  8. R R
  9. K K
  10. a a
  11. b b
  12. c c
  13. K K
  14. R R
  15. sin A = sinh(opposite) sinh(hypotenuse) = sinh a sinh c . \sin A=\frac{\textrm{sinh(opposite)}}{\textrm{sinh(hypotenuse)}}=\frac{\sinh a% }{\,\sinh c\,}.\,
  16. cos A = tanh(adjacent) tanh(hypotenuse) = tanh b tanh c . \cos A=\frac{\textrm{tanh(adjacent)}}{\textrm{tanh(hypotenuse)}}=\frac{\tanh b% }{\,\tanh c\,}.\,
  17. tan A = tanh(opposite) sinh(adjacent) = tanh a sinh b . \tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}}=\frac{\tanh a}{% \,\sinh b\,}.
  18. cosh(hypotenuse) = cosh(adjacent)cosh(opposite) . \textrm{cosh(hypotenuse)}=\textrm{cosh(adjacent)}\textrm{cosh(opposite)}.
  19. cosh(adjacent) = cos B sin A . \textrm{cosh(adjacent)}=\frac{\cos B}{\sin A}.
  20. cosh(hypotenuse) = cos A cos B sin A sin B . \textrm{cosh(hypotenuse)}=\frac{\cos A\cos B}{\sin A\sin B}.
  21. 0
  22. \infty
  23. tanh(c) = 1 \textrm{tanh(c)}=1
  24. cos A = tanh(adjacent) . \cos A=\textrm{tanh(adjacent)}.
  25. cosh c = cosh a cosh b - sinh a sinh b cos C , \cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C,
  26. cos C = - cos A cos B + sin A sin B cosh c , \cos C=-\cos A\cos B+\sin A\sin B\cosh c,
  27. sin A sinh a = sin B sinh b = sin C sinh c , \frac{\sin A}{\sinh a}=\frac{\sin B}{\sinh b}=\frac{\sin C}{\sinh c},
  28. cos C cosh a = sinh a coth b - sin C cot B . \cos C\cosh a=\sinh a\coth b-\sin C\cot B.

Hyperfunction.html

  1. 𝒪 \mathcal{O}
  2. ( 𝐑 ) = H 𝐑 1 ( 𝐂 , 𝒪 ) , \mathcal{B}(\mathbf{R})=H^{1}_{\mathbf{R}}(\mathbf{C},\mathcal{O}),
  3. 𝐂 + 𝐂 - = 𝐂 𝐑 . \mathbf{C}^{+}\cup\mathbf{C}^{-}=\mathbf{C}\setminus\mathbf{R}.\,
  4. H 𝐑 1 ( 𝐂 , 𝒪 ) = [ H 0 ( 𝐂 + , 𝒪 ) H 0 ( 𝐂 - , 𝒪 ) ] / H 0 ( 𝐂 , 𝒪 ) . H^{1}_{\mathbf{R}}(\mathbf{C},\mathcal{O})=\left[H^{0}(\mathbf{C}^{+},\mathcal% {O})\oplus H^{0}(\mathbf{C}^{-},\mathcal{O})\right]/H^{0}(\mathbf{C},\mathcal{% O}).
  5. ( U ) \mathcal{B}(U)
  6. U U\subseteq\mathbb{R}
  7. H 0 ( U ~ U , 𝒪 ) / H 0 ( U ~ , 𝒪 ) H^{0}(\tilde{U}\setminus U,\mathcal{O})/H^{0}(\tilde{U},\mathcal{O})
  8. U ~ \tilde{U}\subseteq\mathbb{C}
  9. U ~ = U \tilde{U}\cap\mathbb{R}=U
  10. U ~ \tilde{U}
  11. H ( x ) = ( 1 2 π i log ( z ) , 1 2 π i log ( z ) - 1 ) H(x)=\left(\frac{1}{2\pi i}\log(z),\frac{1}{2\pi i}\log(z)-1\right)
  12. ( 1 2 π i z , 1 2 π i z ) \left(\frac{1}{2\pi iz},\frac{1}{2\pi iz}\right)
  13. f ( z ) = 1 2 π i x I g ( x ) 1 z - x d x . f(z)={1\over 2\pi i}\int_{x\in I}g(x)\frac{1}{z-x}dx.
  14. 𝒟 ( ) ( ) \textstyle\mathcal{D}^{\prime}(\mathbb{R})\to\mathcal{B}(\mathcal{R})
  15. U \textstyle U\subseteq\mathbb{R}
  16. ( U ) \textstyle\mathcal{B}(U)
  17. a ( f + , f - ) + b ( g + , g - ) := ( a f + + a g + , a f - + b g - ) a(f_{+},f_{-})+b(g_{+},g_{-}):=(af_{+}+ag_{+},af_{-}+bg_{-})
  18. \textstyle\mathcal{B}
  19. h 𝒪 ( U ) \textstyle h\in\mathcal{O}(U)
  20. h ( f + , f - ) := ( h f + , h f - ) h(f_{+},f_{-}):=(hf_{+},hf_{-})
  21. d d z ( f + , f - ) := ( d f + d z , d f - d z ) \frac{d}{dz}(f_{+},f_{-}):=(\frac{df_{+}}{dz},\frac{df_{-}}{dz})
  22. \textstyle\mathcal{B}
  23. 𝒟 \textstyle\mathcal{D}^{\prime}\hookrightarrow\mathcal{B}
  24. a U \textstyle a\in U
  25. f ( U ) \textstyle f\in\mathcal{B}(U)
  26. a b \textstyle a\leq b
  27. a b f := - γ + f + ( z ) d z + γ - f - ( z ) d z \int_{a}^{b}f:=-\int_{\gamma_{+}}f_{+}(z)dz+\int_{\gamma_{-}}f_{-}(z)dz
  28. γ ± : [ 0 , 1 ] ± \textstyle\gamma_{\pm}:[0,1]\to\mathbb{C}^{\pm}
  29. γ ± ( 0 ) = a , γ ± ( 1 ) = b \textstyle\gamma_{\pm}(0)=a,\gamma_{\pm}(1)=b
  30. c ( U ) × 𝒪 ( U ) , ( f , φ ) f ϕ \mathcal{B}_{c}(U)\times\mathcal{O}(U)\to\mathbb{C},(f,\varphi)\mapsto\int f\cdot\phi
  31. 𝒪 ( U ) \mathcal{O}(U)
  32. 𝒪 ( U ) \mathcal{O}^{\prime}(U)
  33. C c 0 ( U ) C_{c}^{0}(U)
  34. ( U ) \mathcal{E}^{\prime}(U)
  35. ( U ) \mathcal{B}(U)
  36. u ( U ) \textstyle u\in\mathcal{E}^{\prime}(U)
  37. φ 𝒪 ( U ) \textstyle\varphi\in\mathcal{O}(U)
  38. supp ( u ) ( a , b ) \operatorname{supp}(u)\subset(a,b)
  39. a b u ϕ = u , ϕ \int_{a}^{b}u\cdot\phi=\langle u,\phi\rangle
  40. a b δ ( x ) d x \int_{a}^{b}\delta(x)dx
  41. ( U ) \mathcal{E}(U)
  42. ( U ) \mathcal{E}^{\prime}(U)
  43. 𝒪 ( U ) \mathcal{O}^{\prime}(U)
  44. \mathcal{E}^{\prime}\hookrightarrow\mathcal{B}
  45. Φ : U V \Phi:U\to V
  46. \mathbb{R}
  47. Φ \Phi
  48. ( V ) ( U ) \mathcal{B}(V)\to\mathcal{B}(U)
  49. f Φ := ( f + Φ , f - Φ ) f\circ\Phi:=(f_{+}\circ\Phi,f_{-}\circ\Phi)

Hyperstructure.html

  1. a A , b B ( a b ) \bigcup_{a\in A,b\in B}(a\star b)

Hypoexponential_distribution.html

  1. λ \lambda
  2. λ i \lambda_{i}
  3. i t h i^{th}
  4. s y m b o l X i symbol{X}_{i}
  5. s y m b o l X = i = 1 k s y m b o l X i symbol{X}=\sum^{k}_{i=1}symbol{X}_{i}
  6. 1 / k 1/k
  7. λ i \lambda_{i}
  8. λ k \lambda_{k}
  9. [ - λ 1 λ 1 0 0 0 0 - λ 2 λ 2 0 0 0 0 - λ k - 2 λ k - 2 0 0 0 0 - λ k - 1 λ k - 1 0 0 0 0 - λ k ] . \left[\begin{matrix}-\lambda_{1}&\lambda_{1}&0&\dots&0&0\\ 0&-\lambda_{2}&\lambda_{2}&\ddots&0&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&\ddots&-\lambda_{k-2}&\lambda_{k-2}&0\\ 0&0&\dots&0&-\lambda_{k-1}&\lambda_{k-1}\\ 0&0&\dots&0&0&-\lambda_{k}\end{matrix}\right]\;.
  10. Θ Θ ( λ 1 , , λ k ) \Theta\equiv\Theta(\lambda_{1},\dots,\lambda_{k})
  11. s y m b o l α = ( 1 , 0 , , 0 ) symbol{\alpha}=(1,0,\dots,0)
  12. H y p o ( λ 1 , , λ k ) = P H ( s y m b o l α , Θ ) . Hypo(\lambda_{1},\dots,\lambda_{k})=PH(symbol{\alpha},\Theta).
  13. λ 1 λ 2 \lambda_{1}\neq\lambda_{2}
  14. F ( x ) = 1 - λ 2 λ 2 - λ 1 e - λ 1 x + λ 1 λ 2 - λ 1 e - λ 2 x F(x)=1-\frac{\lambda_{2}}{\lambda_{2}-\lambda_{1}}e^{-\lambda_{1}x}+\frac{% \lambda_{1}}{\lambda_{2}-\lambda_{1}}e^{-\lambda_{2}x}
  15. f ( x ) = λ 1 λ 2 λ 1 - λ 2 ( e - x λ 2 - e - x λ 1 ) f(x)=\frac{\lambda_{1}\lambda_{2}}{\lambda_{1}-\lambda_{2}}(e^{-x\lambda_{2}}-% e^{-x\lambda_{1}})
  16. 1 λ 1 + 1 λ 2 \frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}
  17. 1 λ 1 2 + 1 λ 2 2 \frac{1}{\lambda_{1}^{2}}+\frac{1}{\lambda_{2}^{2}}
  18. λ 1 2 + λ 2 2 λ 1 + λ 2 \frac{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}}{\lambda_{1}+\lambda_{2}}
  19. c c
  20. λ 1 \lambda_{1}
  21. λ 2 \lambda_{2}
  22. λ 1 = 2 x ¯ [ 1 + 1 + 2 ( c 2 - 1 ) ] - 1 \lambda_{1}=\frac{2}{\bar{x}}\left[1+\sqrt{1+2(c^{2}-1)}\right]^{-1}
  23. λ 2 = 2 x ¯ [ 1 - 1 + 2 ( c 2 - 1 ) ] - 1 \lambda_{2}=\frac{2}{\bar{x}}\left[1-\sqrt{1+2(c^{2}-1)}\right]^{-1}
  24. s y m b o l X H y p o ( λ 1 , , λ k ) symbol{X}\sim Hypo(\lambda_{1},\dots,\lambda_{k})
  25. F ( x ) = 1 - s y m b o l α e x Θ s y m b o l 1 F(x)=1-symbol{\alpha}e^{x\Theta}symbol{1}
  26. f ( x ) = - s y m b o l α e x Θ \Thetasymbol 1 , f(x)=-symbol{\alpha}e^{x\Theta}\Thetasymbol{1}\;,
  27. s y m b o l 1 symbol{1}
  28. e A e^{A}
  29. λ i λ j \lambda_{i}\neq\lambda_{j}
  30. i j i\neq j
  31. f ( x ) = i = 1 k λ i e - x λ i ( j = 1 , j i k λ j λ j - λ i ) = i = 1 k i ( 0 ) λ i e - x λ i f(x)=\sum_{i=1}^{k}\lambda_{i}e^{-x\lambda_{i}}\left(\prod_{j=1,j\neq i}^{k}% \frac{\lambda_{j}}{\lambda_{j}-\lambda_{i}}\right)=\sum_{i=1}^{k}\ell_{i}(0)% \lambda_{i}e^{-x\lambda_{i}}
  32. 1 ( x ) , , k ( x ) \ell_{1}(x),\dots,\ell_{k}(x)
  33. λ 1 , , λ k \lambda_{1},\dots,\lambda_{k}
  34. { f ( x ) } = - s y m b o l α ( s I - Θ ) - 1 \Thetasymbol 1 \mathcal{L}\{f(x)\}=-symbol{\alpha}(sI-\Theta)^{-1}\Thetasymbol{1}
  35. E [ X n ] = ( - 1 ) n n ! s y m b o l α Θ - n s y m b o l 1 . E[X^{n}]=(-1)^{n}n!symbol{\alpha}\Theta^{-n}symbol{1}\;.
  36. a a
  37. λ 1 , λ 2 , , λ a \lambda_{1},\lambda_{2},\cdots,\lambda_{a}
  38. r 1 , r 2 , , r a r_{1},r_{2},\cdots,r_{a}
  39. t 0 t\geq 0
  40. F ( t ) = 1 - ( j = 1 a λ j r j ) k = 1 a l = 1 r k Ψ k , l ( - λ k ) t r k - l exp ( - λ k t ) ( r k - l ) ! ( l - 1 ) ! , F(t)=1-\left(\prod_{j=1}^{a}\lambda_{j}^{r_{j}}\right)\sum_{k=1}^{a}\sum_{l=1}% ^{r_{k}}\frac{\Psi_{k,l}(-\lambda_{k})t^{r_{k}-l}\exp(-\lambda_{k}t)}{(r_{k}-l% )!(l-1)!},
  41. Ψ k , l ( x ) = - l - 1 x l - 1 ( j = 0 , j k a ( λ j + x ) - r j ) . \Psi_{k,l}(x)=-\frac{\partial^{l-1}}{\partial x^{l-1}}\left(\prod_{j=0,j\neq k% }^{a}\left(\lambda_{j}+x\right)^{-r_{j}}\right).
  42. λ 0 = 0 , r 0 = 1 \lambda_{0}=0,r_{0}=1

Hypotrochoid.html

  1. x ( θ ) = ( R - r ) cos θ + d cos ( R - r r θ ) x(\theta)=(R-r)\cos\theta+d\cos\left({R-r\over r}\theta\right)
  2. y ( θ ) = ( R - r ) sin θ - d sin ( R - r r θ ) . y(\theta)=(R-r)\sin\theta-d\sin\left({R-r\over r}\theta\right).
  3. θ \theta
  4. θ \theta

IBEX_35.html

  1. I ( t ) = I ( t - 1 ) × i = 1 35 Cap i ( t ) [ i = 1 35 Cap i ( t - 1 ) ± J ] I(t)=I(t-1)\times\frac{\sum_{i=1}^{35}{\rm Cap}_{i}(t)\,}{[\,\sum_{i=1}^{35}{% \rm Cap}_{i}(t-1)\,\pm J\,]\,}

IC50.html

  1. K i = I C 50 1 + [ S ] K m K_{i}=\frac{IC_{50}}{1+\frac{[S]}{K_{m}}}
  2. K i = I C 50 [ A ] E C 50 + 1 K_{i}=\frac{IC_{50}}{\frac{[A]}{EC_{50}}+1}
  3. p I C 50 = - l o g 10 ( I C 50 ) pIC_{50}=-log_{10}(IC_{50})

Icy_Tower.html

  1. 100 [ S c o r e a S c o r e b + 2 ( F l o o r a F l o o r b ) + C o m b o a C o m b o b ] 4 \frac{100{\left[\sqrt{\frac{Score_{a}}{Score_{b}}}+2{\left(\frac{Floor_{a}}{% Floor_{b}}\right)}+\frac{Combo_{a}}{Combo_{b}}\right]}}{4}
  2. S c o r e Score
  3. F l o o r Floor
  4. C o m b o Combo
  5. a a
  6. b b

ID3_algorithm.html

  1. S S
  2. S S
  3. H ( S ) H(S)
  4. I G ( A ) IG(A)
  5. S S
  6. S S
  7. S S
  8. S S
  9. H ( S ) H(S)
  10. S S
  11. S S
  12. H ( S ) = - x X p ( x ) log 2 p ( x ) H(S)=-\sum_{x\in X}p(x)\log_{2}p(x)
  13. S S
  14. X X
  15. S S
  16. p ( x ) p(x)
  17. x x
  18. S S
  19. H ( S ) = 0 H(S)=0
  20. S S
  21. S S
  22. S S
  23. I G ( A ) IG(A)
  24. S S
  25. A A
  26. S S
  27. S S
  28. A A
  29. I G ( A , S ) = H ( S ) - t T p ( t ) H ( t ) IG(A,S)=H(S)-\sum_{t\in T}p(t)H(t)
  30. H ( S ) H(S)
  31. S S
  32. T T
  33. S S
  34. A A
  35. S = t T t S=\bigcup_{t\in T}t
  36. p ( t ) p(t)
  37. t t
  38. S S
  39. H ( t ) H(t)
  40. t t
  41. S S

Idempotency_of_entailment.html

  1. Γ , C , C B Γ , C B \frac{\Gamma,C,C\vdash B}{\Gamma,C\vdash B}

Ihara_zeta_function.html

  1. 1 ζ G ( u ) = p ( 1 - u L ( p ) ) \frac{1}{\zeta_{G}(u)}=\prod_{p}({1-u^{L(p)}})
  2. G = ( V , E ) G=(V,E)
  3. p = ( u 0 , , u L ( p ) - 1 , u 0 ) p=(u_{0},\cdots,u_{L(p)-1},u_{0})
  4. ( u i , u ( i + 1 ) mod L ( p ) ) E ; u i u ( i + 2 ) mod L ( p ) , (u_{i},u_{(i+1)\bmod L(p)})\in E~{};\quad u_{i}\neq u_{(i+2)\bmod L(p)~{}},
  5. L ( p ) L(p)
  6. ζ G ( u ) = 1 ( 1 - u 2 ) χ ( G ) - 1 det ( I - A u + ( k - 1 ) u 2 I ) \zeta_{G}(u)=\frac{1}{(1-u^{2})^{\chi(G)-1}\det(I-Au+(k-1)u^{2}I)}
  7. ζ G ( u ) = 1 det ( I - T u ) , \zeta_{G}(u)=\frac{1}{\det(I-Tu)}~{},

IJK.html

  1. 𝐬𝐲𝐦𝐛𝐨𝐥 ı ^ \mathbf{\hat{symbol{\imath}}}
  2. 𝐬𝐲𝐦𝐛𝐨𝐥 ȷ ^ \mathbf{\hat{symbol{\jmath}}}
  3. 𝐬𝐲𝐦𝐛𝐨𝐥𝐤 ^ \mathbf{\hat{symbol{k}}}
  4. 𝐬𝐲𝐦𝐛𝐨𝐥 ı ^ = [ 1 0 0 ] , 𝐬𝐲𝐦𝐛𝐨𝐥 ȷ ^ = [ 0 1 0 ] , 𝐬𝐲𝐦𝐛𝐨𝐥𝐤 ^ = [ 0 0 1 ] \mathbf{\hat{symbol{\imath}}}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix},\,\,\mathbf{\hat{symbol{\jmath}}}=\begin{bmatrix}0\\ 1\\ 0\end{bmatrix},\,\,\mathbf{\hat{symbol{k}}}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}

IMD3.html

  1. f 1 f_{1}
  2. f 2 f_{2}
  3. 2 × f 2 - f 1 2\times f_{2}-f_{1}
  4. 2 × f 1 - f 2 2\times f_{1}-f_{2}
  5. f 1 f_{1}
  6. f 2 f_{2}
  7. 2 × f 2 - f 1 2\times f_{2}-f_{1}
  8. 2 × f 1 - f 2 2\times f_{1}-f_{2}
  9. f 1 f_{1}
  10. f 2 f_{2}

Implicit_surface.html

  1. F ( x , y , z ) = 0 F(x,y,z)=0
  2. z = f ( x , y ) z=f(x,y)
  3. ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) (x(s,t),y(s,t),z(s,t))
  4. x ( s , t ) , y ( s , t ) , z ( s , t ) x(s,t)\,,y(s,t)\,,z(s,t)
  5. s , t s,t
  6. z = f ( x , y ) z=f(x,y)
  7. z - f ( x , y ) = 0 z-f(x,y)=0
  8. ( s , t , f ( s , t ) ) (s,t,f(s,t))
  9. x + 2 y - 3 z + 1 = 0 x+2y-3z+1=0
  10. x 2 + y 2 + z 2 - 4 = 0 x^{2}+y^{2}+z^{2}-4=0
  11. ( x 2 + y 2 + z 2 + R 2 - a 2 ) 2 - 4 R 2 ( x 2 + y 2 ) = 0 (x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0
  12. 2 y ( y 2 - 3 x 2 ) ( 1 - z 2 ) + ( x 2 + y 2 ) 2 - ( 9 z 2 - 1 ) ( 1 - z 2 ) = 0 2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0
  13. x 2 + y 2 - ( ln ( z + 3.2 ) ) 2 - 0.02 = 0 x^{2}+y^{2}-(\ln(z+3.2))^{2}-0.02=0
  14. F ( x , y , z ) = 0 F(x,y,z)=0
  15. F ( x , y , z ) F(x,y,z)
  16. F ( x , y , z ) = 0 F(x,y,z)=0
  17. F F
  18. F F
  19. F x , , F x x , F_{x},...,F_{xx},...
  20. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  21. ( F x ( x 0 , y 0 , z 0 ) , F y ( x 0 , y 0 , z 0 ) , F z ( x 0 , y 0 , z 0 ) ) ( 0 , 0 , 0 ) (F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))% \neq(0,0,0)
  22. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  23. F x ( x 0 , y 0 , z 0 ) ( x - x 0 ) + F y ( x 0 , y 0 , z 0 ) ( y - y 0 ) + F z ( x 0 , y 0 , z 0 ) ( z - z 0 ) = 0 F_{x}(x_{0},y_{0},z_{0})(x-x_{0})+F_{y}(x_{0},y_{0},z_{0})(y-y_{0})+F_{z}(x_{0% },y_{0},z_{0})(z-z_{0})=0
  24. 𝐧 ( x 0 , y 0 , z 0 ) = ( F x ( x 0 , y 0 , z 0 ) , F y ( x 0 , y 0 , z 0 ) , F z ( x 0 , y 0 , z 0 ) ) T \mathbf{n}(x_{0},y_{0},z_{0})=(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0% }),F_{z}(x_{0},y_{0},z_{0}))^{T}
  25. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  26. κ n = 𝐯 H F 𝐯 grad F \kappa_{n}=\frac{\mathbf{v}^{\top}H_{F}\mathbf{v}}{\|\operatorname{grad}F\|}
  27. 𝐯 \mathbf{v}
  28. H F H_{F}
  29. F F
  30. q i q_{i}
  31. 𝐩 i = ( x i , y i , z i ) \mathbf{p}_{i}=(x_{i},y_{i},z_{i})
  32. 𝐩 = ( x , y , z ) \mathbf{p}=(x,y,z)
  33. F i ( x , y , z ) = q i 𝐩 - 𝐩 i F_{i}(x,y,z)=\frac{q_{i}}{\|\mathbf{p}-\mathbf{p}_{i}\|}
  34. c c
  35. F i ( x , y , z ) - c = 0 F_{i}(x,y,z)-c=0
  36. 𝐩 i \mathbf{p}_{i}
  37. 4 4
  38. F ( x , y , z ) = q 1 𝐩 - 𝐩 1 + q 2 𝐩 - 𝐩 2 + q 3 𝐩 - 𝐩 3 + q 4 𝐩 - 𝐩 4 F(x,y,z)=\frac{q_{1}}{\|\mathbf{p}-\mathbf{p}_{1}\|}+\frac{q_{2}}{\|\mathbf{p}% -\mathbf{p}_{2}\|}+\frac{q_{3}}{\|\mathbf{p}-\mathbf{p}_{3}\|}+\frac{q_{4}}{\|% \mathbf{p}-\mathbf{p}_{4}\|}
  39. ( ± 1 , ± 1 , 0 ) (\pm 1,\pm 1,0)
  40. F ( x , y , z ) - 2.8 = 0 F(x,y,z)-2.8=0
  41. F ( x , y , z ) = ( x - 1 ) 2 + y 2 + z 2 ( x + 1 ) 2 + y 2 + z 2 x 2 + ( y - 1 ) 2 + z 2 x 2 + ( y + 1 ) 2 + z 2 F(x,y,z)=\sqrt{(x-1)^{2}+y^{2}+z^{2}}\cdot\sqrt{(x+1)^{2}+y^{2}+z^{2}}\cdot% \sqrt{x^{2}+(y-1)^{2}+z^{2}}\cdot\sqrt{x^{2}+(y+1)^{2}+z^{2}}
  42. F ( x , y , z ) - 1.1 = 0 F(x,y,z)-1.1=0
  43. F 1 ( x , y , z ) = 0 , F 2 ( x , y , z ) = 0 F_{1}(x,y,z)=0,F_{2}(x,y,z)=0
  44. μ [ 0 , 1 ] \mu\in[0,1]
  45. F ( x , y , z ) = μ F 1 ( x , y , z ) + ( 1 - μ ) F 2 ( x , y , z ) = 0 F(x,y,z)=\mu F_{1}(x,y,z)+(1-\mu)F_{2}(x,y,z)=0
  46. μ = 0 , 0.33 , 0.66 , 1 \mu=0,\,0.33,\,0.66,\,1
  47. F ( x , y , z ) = F 1 ( x , y , z ) F 2 ( x , y , z ) F 3 ( x , y , z ) - c = 0 F(x,y,z)=F_{1}(x,y,z)\cdot F_{2}(x,y,z)\cdot F_{3}(x,y,z)-c=0
  48. c c
  49. F 1 = ( x 2 + y 2 + z 2 + R 2 - a 2 ) 2 - 4 R 2 ( x 2 + y 2 ) = 0 F_{1}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0
  50. F 2 = ( x 2 + y 2 + z 2 + R 2 - a 2 ) 2 - 4 R 2 ( x 2 + z 2 ) = 0 F_{2}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+z^{2})=0
  51. F 3 = ( x 2 + y 2 + z 2 + R 2 - a 2 ) 2 - 4 R 2 ( y 2 + z 2 ) = 0 F_{3}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(y^{2}+z^{2})=0
  52. R = 1 , a = 0.2 , c = 0.01 R=1,\,a=0.2,\,c=0.01

Incidence_geometry.html

  1. ( P , L , I ) (P,L,I)
  2. P P
  3. L L
  4. I I
  5. P × L P×L
  6. ( A , l ) (A,l)
  7. A A
  8. l l
  9. l l
  10. A A
  11. A I l AIl
  12. B B
  13. m m
  14. ( B , m ) (B,m)
  15. k k
  16. r r
  17. k > 1 k>1
  18. r > 1 r>1
  19. k , r > 1 k,r>1
  20. n n
  21. m m
  22. n r = m k nr=mk
  23. n = m n=m
  24. r = k r=k
  25. P P
  26. L L
  27. P P
  28. L L
  29. P P
  30. n = k 1 n=k–1
  31. n n
  32. A A
  33. l l
  34. m m
  35. A A
  36. A I m AIm
  37. l l
  38. l l
  39. m m
  40. k k
  41. n n
  42. α 1 α≥1
  43. ( B , m ) (B,m)
  44. α α
  45. ( A , l ) (A,l)
  46. B I l BIl
  47. A I m AIm
  48. s + 1 s+1
  49. t + 1 t+1
  50. p g ( s , t , α ) pg(s,t,α)
  51. α = 1 α=1
  52. α = s + 1 α=s+1
  53. n > 2 n>2
  54. n n
  55. Γ Γ
  56. Γ Γ
  57. Γ Γ
  58. n n
  59. n n
  60. m m
  61. n n
  62. n n
  63. n n
  64. d d
  65. 2 d 2d
  66. d d
  67. X X
  68. l l
  69. l l
  70. X X
  71. ( P , z ) (P,z)
  72. Q Q
  73. z z
  74. P P
  75. P P
  76. P P
  77. P P
  78. P P
  79. P P
  80. m m
  81. k = m + 1 k=m+1
  82. n n
  83. n n
  84. n n
  85. m m
  86. O ( n 2 3 m 2 3 + n + m ) , O\left(n^{\frac{2}{3}}m^{\frac{2}{3}}+n+m\right),
  87. C , K C,K
  88. n n
  89. n C \frac{n}{C}
  90. n < s u p > 2 K n\frac{<sup>2}{K}
  91. n n

Incomplete_polylogarithm.html

  1. Li s ( b , z ) = 1 Γ ( s ) b x s - 1 e x / z - 1 d x . \operatorname{Li}_{s}(b,z)=\frac{1}{\Gamma(s)}\int_{b}^{\infty}\frac{x^{s-1}}{% e^{x/z}-1}~{}dx.
  2. Li s ( b , z ) = k = 1 z k k s Γ ( s , k b ) Γ ( s ) \operatorname{Li}_{s}(b,z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{s}}~{}\frac{% \Gamma(s,kb)}{\Gamma(s)}

Indentation_hardness.html

  1. H V = 0.0018544 × L d 2 HV=0.0018544\times\tfrac{L}{d^{2}}

Independence-friendly_logic.html

  1. Σ 1 1 \Sigma^{1}_{1}
  2. Δ 2 1 \Delta_{2}^{1}
  3. Δ 2 1 \Delta_{2}^{1}
  4. Σ 1 1 \Sigma^{1}_{1}
  5. Σ n m \Sigma^{m}_{n}
  6. Σ 1 0 \Sigma_{1}^{0}
  7. Π 2 \Pi_{2}
  8. Π 1 0 \Pi_{1}^{0}
  9. Σ 2 \Sigma_{2}
  10. Π 1 0 \Pi_{1}^{0}
  11. Π 1 0 \Pi_{1}^{0}
  12. Σ 1 0 \Sigma_{1}^{0}
  13. Σ 1 0 \Sigma_{1}^{0}
  14. Π 1 1 \Pi_{1}^{1}

Indeterminate_(variable).html

  1. a 0 + a 1 X + a 2 X 2 + + a n X n a_{0}+a_{1}X+a_{2}X^{2}+\ldots+a_{n}X^{n}
  2. f ( x ) = 2 + 3 x , g ( x ) = 5 + 2 x f(x)=2+3x,g(x)=5+2x
  3. 2 + 3 X , 5 + 2 X 2+3X,5+2X
  4. 2 + 3 X = a + b X 2+3X=a+bX\,
  5. 0 - 0 2 = 0 , 1 - 1 2 = 0 , 0-0^{2}=0,\ 1-1^{2}=0,\,

Index_calculus_algorithm.html

  1. ( / q ) * (\mathbb{Z}/q\mathbb{Z})^{*}
  2. q q
  3. g x h ( mod n ) g^{x}\equiv h\;\;(\mathop{{\rm mod}}n)
  4. ( / q ) * (\mathbb{Z}/q\mathbb{Z})^{*}
  5. g k mod q g^{k}\mod q
  6. e i e_{i}
  7. g k mod q = ( - 1 ) e 0 2 e 1 3 e 2 p r e r g^{k}\mod q=(-1)^{e_{0}}2^{e_{1}}3^{e_{2}}\cdots p_{r}^{e_{r}}
  8. e i e_{i}
  9. ( e 0 , e 1 , e 2 , , e r , k ) (e_{0},e_{1},e_{2},\ldots,e_{r},k)
  10. g s h mod q = ( - 1 ) f 0 2 f 1 3 f 2 p r f r g^{s}h\mod q=(-1)^{f_{0}}2^{f_{1}}3^{f_{2}}\cdots p_{r}^{f_{r}}
  11. x = f 0 log g ( - 1 ) + f 1 log g 2 + + f r log g p r - s . x=f_{0}\log_{g}(-1)+f_{1}\log_{g}2+\cdots+f_{r}\log_{g}p_{r}-s.
  12. L n [ 1 / 2 , 2 + o ( 1 ) ] L_{n}[1/2,\sqrt{2}+o(1)]
  13. 𝔽 q \mathbb{F}_{q}
  14. q = p n q=p^{n}
  15. p p
  16. L q [ 1 / 3 , 64 / 9 3 ] L_{q}\left[1/3,\sqrt[3]{64/9}\right]
  17. p p
  18. q q
  19. L q [ 1 / 3 , 32 / 9 3 ] L_{q}\left[1/3,\sqrt[3]{32/9}\right]
  20. L q [ 1 / 4 + ϵ , c ] L_{q}\left[1/4+\epsilon,c\right]
  21. c > 0 c>0
  22. p p
  23. q q
  24. L q [ 1 / 3 , c ] L_{q}[1/3,c]
  25. c > 0 c>0
  26. p p
  27. L q [ 1 / 3 , c ] L_{q}\left[1/3,c\right]
  28. c > 0 c>0
  29. L ( 1 / 4 + o ( 1 ) ) L(1/4+o(1))

Index_of_dissimilarity.html

  1. 1 2 i = 1 N | b i B - w i W | \frac{1}{2}\sum_{i=1}^{N}\left|\frac{b_{i}}{B}-\frac{w_{i}}{W}\right|

Indian_mathematics.html

  1. 10 < s u p > 2 10<sup>2
  2. ( 3 , 4 , 5 ) (3,4,5)
  3. ( 5 , 12 , 13 ) (5,12,13)
  4. ( 8 , 15 , 17 ) (8,15,17)
  5. ( 7 , 24 , 25 ) (7,24,25)
  6. ( 12 , 35 , 37 ) (12,35,37)
  7. 2 = 1 + 1 3 + 1 3 4 - 1 3 4 34 1.4142156 \sqrt{2}=1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}\approx 1.% 4142156\ldots
  8. 2 = 1 + 24 60 + 51 60 2 + 10 60 3 = 1.41421297. \sqrt{2}=1+\frac{24}{60}+\frac{51}{60^{2}}+\frac{10}{60^{3}}=1.41421297.
  9. ( n 0 ) + ( n 1 ) + ( n 2 ) + + ( n n - 1 ) + ( n n ) = 2 n {n\choose 0}+{n\choose 1}+{n\choose 2}+\cdots+{n\choose n-1}+{n\choose n}=2^{n}
  10. sin 2 ( x ) + cos 2 ( x ) = 1 \sin^{2}(x)+\cos^{2}(x)=1
  11. sin ( x ) = cos ( π 2 - x ) \sin(x)=\cos\left(\frac{\pi}{2}-x\right)
  12. 1 - cos ( 2 x ) 2 = sin 2 ( x ) \frac{1-\cos(2x)}{2}=\sin^{2}(x)
  13. A = ( s - a ) ( s - b ) ( s - c ) ( s - d ) A=\sqrt{(s-a)(s-b)(s-c)(s-d)}\,
  14. s = a + b + c + d 2 . s=\frac{a+b+c+d}{2}.
  15. a , b , c a,b,c
  16. a = u 2 v + v , b = u 2 w + w , c = u 2 v + u 2 w - ( v + w ) a=\frac{u^{2}}{v}+v,\ \ b=\frac{u^{2}}{w}+w,\ \ c=\frac{u^{2}}{v}+\frac{u^{2}}% {w}-(v+w)
  17. u , v , u,v,
  18. w w
  19. a + 0 = a a+0=\ a
  20. a × 0 = 0 a\times 0=0
  21. 0 0 = 0 \frac{0}{0}=0
  22. a x 2 + b x = c \ ax^{2}+bx=c
  23. x = 4 a c + b 2 - b 2 a x=\frac{\sqrt{4ac+b^{2}}-b}{2a}
  24. x 2 - N y 2 = 1 , \ x^{2}-Ny^{2}=1,
  25. N N
  26. ( x 2 - N y 2 ) ( x 2 - N y 2 ) = ( x x + N y y ) 2 - N ( x y + x y ) 2 \ (x^{2}-Ny^{2})(x^{\prime 2}-Ny^{\prime 2})=(xx^{\prime}+Nyy^{\prime})^{2}-N(% xy^{\prime}+x^{\prime}y)^{2}
  27. x = x 1 , y = y 1 x=x_{1},\ \ y=y_{1}\
  28. x 2 - N y 2 = k 1 , \ \ x^{2}-Ny^{2}=k_{1},
  29. x = x 2 , y = y 2 x=x_{2},\ \ y=y_{2}\
  30. x 2 - N y 2 = k 2 , \ \ x^{2}-Ny^{2}=k_{2},
  31. x = x 1 x 2 + N y 1 y 2 , y = x 1 y 2 + x 2 y 1 x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\
  32. x 2 - N y 2 = k 1 k 2 \ x^{2}-Ny^{2}=k_{1}k_{2}
  33. x 2 - N y 2 = k \ x^{2}-Ny^{2}=k
  34. k = ± 4 , ± 2 , - 1 \ k=\pm 4,\pm 2,-1
  35. x 2 - N y 2 = 1 \ x^{2}-Ny^{2}=1
  36. x , y \ x,\ y
  37. x 2 - 92 y 2 = 1 \ x^{2}-92y^{2}=1
  38. x = 1151 , y = 120 \ x=1151,\ y=120
  39. a x n = q \ ax^{n}=q
  40. a x n - 1 x - 1 = p a\frac{x^{n}-1}{x-1}=p
  41. sin w - sin w \ \sin w^{\prime}-\sin w
  42. ( w - w ) cos w \ (w^{\prime}-w)\cos w
  43. sin ( a + b ) = sin ( a ) cos ( b ) + sin ( b ) cos ( a ) \ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)
  44. sin ( a - b ) = sin ( a ) cos ( b ) - sin ( b ) cos ( a ) \ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)
  45. 1 1 - x = 1 + x + x 2 + x 3 + x 4 + for | x | < 1 \frac{1}{1-x}=1+x+x^{2}+x^{3}+x^{4}+\cdots\,\text{ for }|x|<1
  46. 1 p + 2 p + + n p n p + 1 p + 1 1^{p}+2^{p}+\cdots+n^{p}\approx\frac{n^{p+1}}{p+1}
  47. sin x \sin x
  48. cos x \cos x
  49. arctan x \arctan x
  50. r arctan ( y x ) = 1 1 r y x - 1 3 r y 3 x 3 + 1 5 r y 5 x 5 - , where y / x 1. r\arctan\left(\frac{y}{x}\right)=\frac{1}{1}\cdot\frac{ry}{x}-\frac{1}{3}\cdot% \frac{ry^{3}}{x^{3}}+\frac{1}{5}\cdot\frac{ry^{5}}{x^{5}}-\cdots,\,\text{ % where }y/x\leq 1.
  51. sin x = x - x x 2 ( 2 2 + 2 ) r 2 + x x 2 ( 2 2 + 2 ) r 2 x 2 ( 4 2 + 4 ) r 2 - \sin x=x-x\frac{x^{2}}{(2^{2}+2)r^{2}}+x\frac{x^{2}}{(2^{2}+2)r^{2}}\cdot\frac% {x^{2}}{(4^{2}+4)r^{2}}-\cdots
  52. r - cos x = r x 2 ( 2 2 - 2 ) r 2 - r x 2 ( 2 2 - 2 ) r 2 x 2 ( 4 2 - 4 ) r 2 + , r-\cos x=r\frac{x^{2}}{(2^{2}-2)r^{2}}-r\frac{x^{2}}{(2^{2}-2)r^{2}}\frac{x^{2% }}{(4^{2}-4)r^{2}}+\cdots,
  53. sin x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + \sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots
  54. cos x = 1 - x 2 2 ! + x 4 4 ! - x 6 6 ! + \cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots
  55. arctan x \arctan x
  56. π \pi
  57. π 4 = 1 - 1 3 + 1 5 - 1 7 + \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots
  58. f i ( n + 1 ) f_{i}(n+1)
  59. π 4 1 - 1 3 + 1 5 - + ( - 1 ) ( n - 1 ) / 2 1 n + ( - 1 ) ( n + 1 ) / 2 f i ( n + 1 ) \frac{\pi}{4}\approx 1-\frac{1}{3}+\frac{1}{5}-\cdots+(-1)^{(n-1)/2}\frac{1}{n% }+(-1)^{(n+1)/2}f_{i}(n+1)
  60. where f 1 ( n ) = 1 2 n , f 2 ( n ) = n / 2 n 2 + 1 , f 3 ( n ) = ( n / 2 ) 2 + 1 ( n 2 + 5 ) n / 2 . \,\text{where }f_{1}(n)=\frac{1}{2n},\ f_{2}(n)=\frac{n/2}{n^{2}+1},\ f_{3}(n)% =\frac{(n/2)^{2}+1}{(n^{2}+5)n/2}.
  61. π \pi
  62. π 4 = 3 4 + 1 3 3 - 3 - 1 5 3 - 5 + 1 7 3 - 7 - \frac{\pi}{4}=\frac{3}{4}+\frac{1}{3^{3}-3}-\frac{1}{5^{3}-5}+\frac{1}{7^{3}-7% }-\cdots
  63. x + y = a , x - y = b , x y = c , x 2 + y 2 = d , \displaystyle x+y=a,\ x-y=b,\ xy=c,x^{2}+y^{2}=d,
  64. θ < π \theta<\pi
  65. 2 r sin ( θ / 2 ) 2r\sin\left(\theta/2\right)
  66. ( a , b , c ) (a,b,c)
  67. a < s u p > 2 + b 2 = c 2 a<sup>2+b^{2}=c^{2}
  68. ( 3 , 4 , 5 ) (3,4,5)
  69. ( 5 , 12 , 13 ) (5,12,13)
  70. ( 8 , 15 , 17 ) (8,15,17)
  71. ( 12 , 35 , 37 ) (12,35,37)
  72. π π
  73. ( a , b , c ) (a,b,c)
  74. c < s u p > 2 = a 2 + b 2 c<sup>2=a^{2}+b^{2}

Indirect_utility_function.html

  1. v ( p , w ) v(p,w)
  2. p p
  3. w w
  4. v ( p , w ) v(p,w)
  5. u ( x ) , u(x),
  6. x x
  7. x ( p , w ) x(p,w)
  8. u ( x ( p , w ) ) u(x(p,w))
  9. v ( p , w ) = u ( x ( p , w ) ) . v(p,w)=u(x(p,w)).
  10. ( p 0 , w 0 ) (p^{0},w^{0})
  11. v ( p , w ) w 0 \frac{\partial v(p,w)}{\partial w}\neq 0
  12. - v ( p 0 , w 0 ) / ( p i ) v ( p 0 , w 0 ) / w = x i ( p 0 , w 0 ) , i = 1 , , n . -\frac{\partial v(p^{0},w^{0})/(\partial p_{i})}{\partial v(p^{0},w^{0})/% \partial w}=x_{i}(p^{0},w^{0}),i=1,\dots,n.

Induced_metric.html

  1. g a b = a X μ b X ν g μ ν g_{ab}=\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}
  2. a , b a,b
  3. ξ a \xi^{a}
  4. X μ ( ξ a ) X^{\mu}(\xi^{a})
  5. μ , ν \mu,\nu
  6. Π : 𝒞 3 , τ { x 1 = ( a + b cos ( n τ ) ) cos ( m τ ) x 2 = ( a + b cos ( n τ ) ) sin ( m τ ) x 3 = b sin ( n τ ) . \Pi\colon\mathcal{C}\to\mathbb{R}^{3},\ \tau\mapsto\begin{cases}\begin{aligned% }\displaystyle x^{1}&\displaystyle=(a+b\cos(n\cdot\tau))\cos(m\cdot\tau)\\ \displaystyle x^{2}&\displaystyle=(a+b\cos(n\cdot\tau))\sin(m\cdot\tau)\\ \displaystyle x^{3}&\displaystyle=b\sin(n\cdot\tau).\end{aligned}\end{cases}
  7. 𝒞 \mathcal{C}
  8. τ \tau
  9. 3 \mathbb{R}^{3}
  10. a , b , m , n a,b,m,n\in\mathbb{R}
  11. 3 \mathbb{R}^{3}
  12. g = μ , ν g μ ν d x μ d x ν with g μ ν = ( 1 0 0 0 1 0 0 0 1 ) g=\sum\limits_{\mu,\nu}g_{\mu\nu}\mathrm{d}x^{\mu}\otimes\mathrm{d}x^{\nu}% \quad\,\text{with}\quad g_{\mu\nu}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}
  13. g τ τ = μ , ν x μ τ x ν τ g μ ν δ μ ν = μ ( x μ τ ) 2 = m 2 a 2 + 2 m 2 a b cos ( n τ ) + m 2 b 2 cos 2 ( n τ ) + b 2 n 2 g_{\tau\tau}=\sum\limits_{\mu,\nu}\frac{\partial x^{\mu}}{\partial\tau}\frac{% \partial x^{\nu}}{\partial\tau}\underbrace{g_{\mu\nu}}_{\delta_{\mu\nu}}=\sum% \limits_{\mu}\left(\frac{\partial x^{\mu}}{\partial\tau}\right)^{2}=m^{2}a^{2}% +2m^{2}ab\cos(n\cdot\tau)+m^{2}b^{2}\cos^{2}(n\cdot\tau)+b^{2}n^{2}
  14. g 𝒞 = ( m 2 a 2 + 2 m 2 a b cos ( n τ ) + m 2 b 2 cos 2 ( n τ ) + b 2 n 2 ) d τ d τ g_{\mathcal{C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot\tau)+m^{2}b^{2}\cos^{2}(n\cdot% \tau)+b^{2}n^{2})\mathrm{d}\tau\otimes\mathrm{d}\tau

Inequity_aversion.html

  1. U i ( { x i , x j } ) = x i - α i n - 1 × max ( x j - x i , 0 ) - β i n - 1 × max ( x i - x j , 0 ) , U_{i}(\{x_{i},x_{j}\})=x_{i}-\frac{\alpha_{i}}{n-1}\times\sum{\max(x_{j}-x_{i}% ,0)}-\frac{\beta_{i}}{n-1}\times\sum{\max(x_{i}-x_{j},0)},

Infinite-dimensional_holomorphy.html

  1. f ( z ) = lim ζ z f ( ζ ) - f ( z ) ζ - z . f^{\prime}(z)=\lim_{\zeta\to z}\frac{f(\zeta)-f(z)}{\zeta-z}.
  2. 1 k n f ( γ ( t k ) ) ( γ ( t k ) - γ ( t k - 1 ) ) \sum_{1\leq k\leq n}f(\gamma(t_{k}))(\gamma(t_{k})-\gamma(t_{k-1}))
  3. T ( γ f ( z ) d z ) = γ ( T f ) ( z ) d z . T\left(\int_{\gamma}f(z)\,dz\right)=\int_{\gamma}(T\circ f)(z)\,dz.
  4. γ f ( z ) d z = 0 \int_{\gamma}f(z)\,dz=0
  5. f φ ( z ) = φ f ( a + z b ) f_{\varphi}(z)=\varphi\circ f(a+zb)
  6. f ( x + y ) = n = 0 1 n ! D ^ n f ( x ) ( y ) f(x+y)=\sum_{n=0}^{\infty}\frac{1}{n!}\widehat{D}^{n}f(x)(y)
  7. D ^ n f ( x ) ( y ) \widehat{D}^{n}f(x)(y)
  8. f ( x + y ) = n = 0 1 n ! D ^ n f ( x ) ( y ) f(x+y)=\sum_{n=0}^{\infty}\frac{1}{n!}\widehat{D}^{n}f(x)(y)

Infinity_symbol.html

  1. s i z e = 150 % ∞size=150\%
  2. i = 0 1 2 i = lim x 2 x - 1 2 x - 1 = 2 , \sum_{i=0}^{\infty}\frac{1}{2^{i}}=\lim_{x\to\infty}\frac{2^{x}-1}{2^{x-1}}=2,
  3. \infty

Inflation_tax.html

  1. 1 + Interest 1 + Inflation = 1 + Real \frac{1+\mathrm{Interest}}{1+\mathrm{Inflation}}=1+\mathrm{Real}
  2. 1 + 0.06 1 + 0.04 - 1 = 1.92 % \frac{1+0.06}{1+0.04}-1=1.92\%
  3. 1 + 0.02 1 + 0.10 - 1 = - 7.27 % \frac{1+0.02}{1+0.10}-1=-7.27\%

Injective_sheaf.html

  1. \mathcal{F}
  2. X X
  3. U V X U\subset V\subset X
  4. r U V : Γ ( V , ) Γ ( U , ) r_{U\subset V}:\Gamma(V,\mathcal{F})\to\Gamma(U,\mathcal{F})

Instrumental_variable.html

  1. Y = β X + U Y=\beta X+U\,
  2. ( Z Y ) G X ¯ ( Z X ) G (Z\perp\!\!\!\perp Y)_{G_{\overline{X}}}\qquad(Z\not\!\!{\perp\!\!\!\perp}X)_{G}
  3. \perp\!\!\!\perp
  4. G X ¯ G_{\overline{X}}
  5. ( Z Y x ) ( Z X ) (Z\perp\!\!\!\perp Y_{x})\qquad(Z\not\!\!{\perp\!\!\!\perp}X)
  6. \perp\!\!\!\perp
  7. \rightarrow
  8. \rightarrow
  9. G X ¯ G_{\overline{X}}
  10. G X ¯ G_{\overline{X}}
  11. \rightarrow
  12. \leftrightarrow
  13. y i = β x i + ε i , y_{i}=\beta x_{i}+\varepsilon_{i},
  14. y i y_{i}
  15. x i x_{i}
  16. ε i \varepsilon_{i}
  17. y i y_{i}
  18. x i x_{i}
  19. β \beta
  20. β \beta
  21. y i y_{i}
  22. x i x_{i}
  23. y i y_{i}
  24. β \beta
  25. ε \varepsilon
  26. β ^ OLS = x T y x T x = x T ( x β + ε ) x T x = β + x T ε x T x . \widehat{\beta}_{\mathrm{OLS}}=\frac{x^{\mathrm{T}}y}{x^{\mathrm{T}}x}=\frac{x% ^{\mathrm{T}}(x\beta+\varepsilon)}{x^{\mathrm{T}}x}=\beta+\frac{x^{\mathrm{T}}% \varepsilon}{x^{\mathrm{T}}x}.
  27. ε \varepsilon
  28. ε \varepsilon
  29. ε \varepsilon
  30. E [ y z ] = β E [ x z ] + E [ ε z ] . E[y\mid z]=\beta E[x\mid z]+E[\varepsilon\mid z].\,
  31. β \beta
  32. β ^ IV = z T y z T x = β + z T ε z T x . \widehat{\beta}_{\mathrm{IV}}=\frac{z^{\mathrm{T}}y}{z^{\mathrm{T}}x}=\beta+% \frac{z^{\mathrm{T}}\varepsilon}{z^{\mathrm{T}}x}.\,
  33. ε \varepsilon
  34. β ^ IV = ( Z T X ) - 1 Z T y \widehat{\beta}_{\mathrm{IV}}=(Z^{\mathrm{T}}X)^{-1}Z^{\mathrm{T}}y\,
  35. β ^ GMM = ( X T P Z X ) - 1 X T P Z y , \widehat{\beta}_{\mathrm{GMM}}=(X^{\mathrm{T}}P_{Z}X)^{-1}X^{\mathrm{T}}P_{Z}y,
  36. P Z = Z ( Z T Z ) - 1 Z T P_{Z}=Z(Z^{\mathrm{T}}Z)^{-1}Z^{\mathrm{T}}
  37. β G M M \beta_{GMM}
  38. β ^ GMM = ( X T Z ( Z T Z ) - 1 Z T X ) - 1 X T Z ( Z T Z ) - 1 Z T y \widehat{\beta}_{\mathrm{GMM}}=(X^{\mathrm{T}}Z(Z^{\mathrm{T}}Z)^{-1}Z^{% \mathrm{T}}X)^{-1}X^{\mathrm{T}}Z(Z^{\mathrm{T}}Z)^{-1}Z^{\mathrm{T}}y
  39. X T Z , Z T Z X^{\mathrm{T}}Z,Z^{\mathrm{T}}Z
  40. Z T X Z^{\mathrm{T}}X
  41. β ^ GMM \displaystyle\widehat{\beta}_{\mathrm{GMM}}
  42. X = Z δ + errors X=Z\delta+\,\text{errors}
  43. δ ^ = ( Z T Z ) - 1 Z T X , \widehat{\delta}=(Z^{\mathrm{T}}Z)^{-1}Z^{\mathrm{T}}X,\,
  44. X ^ = Z δ ^ = Z ( Z T Z ) - 1 Z T X = P Z X . \widehat{X}=Z\widehat{\delta}=Z(Z^{\mathrm{T}}Z)^{-1}Z^{\mathrm{T}}X=P_{Z}X.\,
  45. Y = X ^ β + noise . Y=\widehat{X}\beta+\mathrm{noise}.\,
  46. β 2 S L S = ( X T P Z X ) - 1 X T P Z Y \beta_{2SLS}=\left(X^{\mathrm{T}}P_{Z}X\right)^{-1}X^{\mathrm{T}}P_{Z}Y
  47. ( X ^ T X ^ ) - 1 X ^ T Y (\widehat{X}^{\mathrm{T}}\widehat{X})^{-1}\widehat{X}^{\mathrm{T}}Y
  48. X ^ = P Z X \widehat{X}=P_{Z}X
  49. P Z P_{Z}
  50. P Z T P Z = P Z P Z = P Z P_{Z}^{\mathrm{T}}P_{Z}=P_{Z}P_{Z}=P_{Z}
  51. β 2 S L S = ( X ^ T X ^ ) - 1 X ^ T Y = ( X T P Z T P Z X ) - 1 X T P Z T Y = ( X T P Z X ) - 1 X T P Z Y \beta_{2SLS}=(\widehat{X}^{\mathrm{T}}\widehat{X})^{-1}\widehat{X}^{\mathrm{T}% }Y=\left(X^{\mathrm{T}}P_{Z}^{\mathrm{T}}P_{Z}X\right)^{-1}X^{\mathrm{T}}P_{Z}% ^{\mathrm{T}}Y=\left(X^{\mathrm{T}}P_{Z}X\right)^{-1}X^{\mathrm{T}}P_{Z}Y
  52. β \beta
  53. β \beta
  54. x 1 x k x_{1}\dots x_{k}
  55. z 1 z m z_{1}\dots z_{m}
  56. β 1 β k \beta_{1}\dots\beta_{k}
  57. x = g ( z , u ) x=g(z,u)\,
  58. y = f ( x , u ) y=f(x,u)\,
  59. f f
  60. g g
  61. Z Z
  62. U U
  63. Z , X Z,X
  64. Y Y
  65. X X
  66. Y Y
  67. ACE = Pr ( y do ( x ) ) = E u [ f ( x , u ) ] . \,\text{ACE}=\Pr(y\mid\,\text{do}(x))=\operatorname{E}_{u}[f(x,u)].
  68. Z Z
  69. ( X , Y ) (X,Y)
  70. X X
  71. f f
  72. g g
  73. Z Z
  74. max x y [ max z Pr ( y , x z ) ] 1. \max_{x}\sum_{y}[\max_{z}\Pr(y,x\mid z)]\leq 1.
  75. β \beta
  76. T R 2 TR^{2}

Integral_curve.html

  1. d x 1 d t \displaystyle\frac{dx_{1}}{dt}
  2. 𝐱 ( t ) = 𝐅 ( 𝐱 ( t ) ) . \mathbf{x}^{\prime}(t)=\mathbf{F}(\mathbf{x}(t)).\!\,
  3. π M : ( x , v ) x . \pi_{M}:(x,v)\mapsto x.
  4. α ( t 0 ) = p ; \alpha(t_{0})=p;\,
  5. α ( t ) = X ( α ( t ) ) for all t J . \alpha^{\prime}(t)=X(\alpha(t))\mbox{ for all }~{}t\in J.
  6. α ( t 0 ) = p ; \alpha(t_{0})=p;\,
  7. α ( t ) = X ( α ( t ) ) . \alpha^{\prime}(t)=X(\alpha(t)).\,
  8. ( d t f ) ( + 1 ) T α ( t ) M . (\mathrm{d}_{t}f)(+1)\in\mathrm{T}_{\alpha(t)}M.
  9. ( d α 1 d t , , d α n d t ) , \left(\frac{\mathrm{d}\alpha_{1}}{\mathrm{d}t},\dots,\frac{\mathrm{d}\alpha_{n% }}{\mathrm{d}t}\right),

Integrally_closed.html

  1. R R
  2. S S
  3. S S
  4. R R
  5. R R
  6. S S
  7. R R
  8. R R
  9. Spec \operatorname{Spec}

Integrating_factor.html

  1. y + P ( x ) y = Q ( x ) y^{\prime}+P(x)y=Q(x)
  2. M ( x ) M(x)
  3. M ( x ) = e s 0 x P ( s ) d s M(x)=e^{\int_{s_{0}}^{x}P(s)ds}
  4. M ( x ) M(x)
  5. M ( x ) M(x)
  6. ( 1 ) \displaystyle(1)
  7. M ( x ) P ( x ) = M ( x ) M(x)P(x)=M^{\prime}(x)
  8. M ( x ) M(x)
  9. P ( x ) P(x)
  10. ( 4 ) \displaystyle(4)
  11. M ( x ) M(x)
  12. y e s 0 x P ( s ) d s + P ( x ) y e s 0 x P ( s ) d s = Q ( x ) e s 0 x P ( s ) d s y^{\prime}e^{\int_{s_{0}}^{x}P(s)ds}+P(x)ye^{\int_{s_{0}}^{x}P(s)ds}=Q(x)e^{% \int_{s_{0}}^{x}P(s)ds}
  13. x x
  14. y e s 0 x P ( s ) d s + P ( x ) y e s 0 x P ( s ) d s = d d x ( y e s 0 x P ( s ) d s ) y^{\prime}e^{\int_{s_{0}}^{x}P(s)ds}+P(x)ye^{\int_{s_{0}}^{x}P(s)ds}=\frac{d}{% dx}(ye^{\int_{s_{0}}^{x}P(s)ds})
  15. d d x ( y e s 0 x P ( s ) d s ) = Q ( x ) e s 0 x P ( s ) d s \frac{d}{dx}(ye^{\int_{s_{0}}^{x}P(s)ds})=Q(x)e^{\int_{s_{0}}^{x}P(s)ds}
  16. x x
  17. x x
  18. t t
  19. y e s 0 x P ( s ) d s = t 0 x Q ( t ) e s 0 t P ( s ) d s d t + C ye^{\int_{s_{0}}^{x}P(s)ds}=\int_{t_{0}}^{x}Q(t)e^{\int_{s_{0}}^{t}P(s)ds}dt+C
  20. y = e - s 0 x P ( s ) d s t 0 x Q ( t ) e s 0 t P ( s ) d s d t + C e - s 0 x P ( s ) d s y=e^{-\int_{s_{0}}^{x}P(s)ds}\int_{t_{0}}^{x}Q(t)e^{\int_{s_{0}}^{t}P(s)ds}dt+% Ce^{-\int_{s_{0}}^{x}P(s)ds}
  21. Q ( x ) = 0 Q(x)=0
  22. y = C e s 0 x P ( s ) d s y=\frac{C}{e^{\int_{s_{0}}^{x}P(s)ds}}
  23. C C
  24. y - 2 y x = 0. y^{\prime}-\frac{2y}{x}=0.
  25. P ( x ) = - 2 x P(x)=\frac{-2}{x}
  26. M ( x ) = e P ( x ) d x M(x)=e^{\int P(x)\,dx}
  27. M ( x ) = e - 2 x d x = e - 2 ln x = ( e ln x ) - 2 = x - 2 M(x)=e^{\int\frac{-2}{x}\,dx}=e^{-2\ln x}={(e^{\ln x})}^{-2}=x^{-2}
  28. M ( x ) = 1 x 2 . M(x)=\frac{1}{x^{2}}.
  29. M ( x ) M(x)
  30. y x 2 - 2 y x 3 = 0 \frac{y^{\prime}}{x^{2}}-\frac{2y}{x^{3}}=0
  31. y x 3 - 2 x 2 y x 5 = 0 \frac{y^{\prime}x^{3}-2x^{2}y}{x^{5}}=0
  32. x ( y x 2 - 2 x y ) x 5 = 0 \frac{x(y^{\prime}x^{2}-2xy)}{x^{5}}=0
  33. y x 2 - 2 x y x 4 = 0. \frac{y^{\prime}x^{2}-2xy}{x^{4}}=0.
  34. ( y x 2 ) = 0 \left(\frac{y}{x^{2}}\right)^{\prime}=0
  35. y x 2 = C \frac{y}{x^{2}}=C\,
  36. y ( x ) = C x 2 . y\left(x\right)=Cx^{2}.
  37. d 2 y d t 2 = A y 2 / 3 \frac{d^{2}y}{dt^{2}}=Ay^{2/3}
  38. d y d t \tfrac{dy}{dt}
  39. d 2 y d t 2 d y d t = A y 2 / 3 d y d t . \frac{d^{2}y}{dt^{2}}\frac{dy}{dt}=Ay^{2/3}\frac{dy}{dt}.
  40. d d t ( 1 2 ( d y d t ) 2 ) = d d t ( A 3 5 y 5 / 3 ) . \frac{d}{dt}\left(\frac{1}{2}\left(\frac{dy}{dt}\right)^{2}\right)=\frac{d}{dt% }\left(A\frac{3}{5}y^{5/3}\right).
  41. ( d y d t ) 2 = 6 A 5 y 5 / 3 + C 0 . \left(\frac{dy}{dt}\right)^{2}=\frac{6A}{5}y^{5/3}+C_{0}.
  42. d y 6 A 5 y 5 / 3 + C 0 = t + C 1 ; \int\frac{dy}{\sqrt{\frac{6A}{5}y^{5/3}+C_{0}}}=t+C_{1};

Intel_8253.html

  1. f 𝑖𝑛𝑝𝑢𝑡 f 𝑜𝑢𝑡𝑝𝑢𝑡 {\it f_{input}}\over{\it f_{output}}
  2. n n
  3. n n
  4. n 2 n\over 2
  5. n 2 n\over 2
  6. n n
  7. n + 1 2 n+1\over 2
  8. n - 1 2 n-1\over 2
  9. n n

Interferometric_visibility.html

  1. Visibility ( real ) = amplitude of fringe oscillation average of fringe oscillation . \,\text{Visibility}(\,\text{real})=\frac{\,\text{amplitude of fringe % oscillation}}{\,\text{average of fringe oscillation}}.
  2. Visibility ( real ) = max - min max + min , \,\text{Visibility}(\,\text{real})=\frac{\max-\min}{\max+\min},
  3. Visibility ( ideal ) = 2 I 1 I 2 I 1 + I 2 , \,\text{Visibility}(\,\text{ideal})=\frac{2\sqrt{I_{1}I_{2}}}{I_{1}+I_{2}},
  4. I 1 I_{1}
  5. I 2 I_{2}

Interior_point_method.html

  1. f ( x ) f(x)~{}
  2. c i ( x ) 0 for i = 1 , , m , x n , c_{i}(x)\geq 0~{}~{}\,\text{for}~{}i=1,\ldots,m,~{}~{}x\in\mathbb{R}^{n},
  3. f : n , c i : n ( 1 ) f:\mathbb{R}^{n}\rightarrow\mathbb{R},c_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R% }~{}~{}~{}~{}~{}~{}(1)
  4. B ( x , μ ) = f ( x ) - μ i = 1 m ln ( c i ( x ) ) ( 2 ) B(x,\mu)=f(x)-\mu~{}\sum_{i=1}^{m}\ln(c_{i}(x))~{}~{}~{}~{}~{}(2)
  5. μ \mu
  6. μ \mu
  7. B ( x , μ ) B(x,\mu)
  8. g b = g - μ i = 1 m 1 c i ( x ) c i ( x ) ( 3 ) g_{b}=g-\mu\sum_{i=1}^{m}\frac{1}{c_{i}(x)}\nabla c_{i}(x)~{}~{}~{}~{}~{}~{}(3)
  9. g g
  10. f ( x ) f(x)
  11. c i \nabla c_{i}
  12. c i c_{i}
  13. x x
  14. λ m \lambda\in\mathbb{R}^{m}
  15. c i ( x ) λ i = μ , i = 1 , , m ( 4 ) c_{i}(x)\lambda_{i}=\mu,\forall i=1,\ldots,m~{}~{}~{}~{}~{}~{}~{}(4)
  16. ( x μ , λ μ ) (x_{\mu},\lambda_{\mu})
  17. g - A T λ = 0 ( 5 ) g-A^{T}\lambda=0~{}~{}~{}~{}~{}~{}(5)
  18. A A
  19. c ( x ) c(x)
  20. f ( x ) f(x)
  21. μ \mu
  22. c i ( x ) = 0 c_{i}(x)=0
  23. g g
  24. c i ( x ) c_{i}(x)
  25. ( x , λ ) (x,\lambda)
  26. ( p x , p λ ) (p_{x},p_{\lambda})
  27. ( W - A T Λ A C ) ( p x p λ ) = ( - g + A T λ μ 1 - C λ ) \begin{pmatrix}W&-A^{T}\\ \Lambda A&C\end{pmatrix}\begin{pmatrix}p_{x}\\ p_{\lambda}\end{pmatrix}=\begin{pmatrix}-g+A^{T}\lambda\\ \mu 1-C\lambda\end{pmatrix}
  28. W W
  29. f ( x ) f(x)
  30. Λ \Lambda
  31. λ \lambda
  32. C C
  33. C i i C_{ii}
  34. c i ( x ) c_{i}(x)
  35. λ 0 \lambda\geq 0
  36. α \alpha
  37. ( x , λ ) ( x + α p x , λ + α p λ ) (x,\lambda)\rightarrow(x+\alpha p_{x},\lambda+\alpha p_{\lambda})

International_America's_Cup_Class.html

  1. L + 1.25 × S - 9.8 × D S P 3 0.686 24.000 m e t r e s \frac{L+1.25\times\sqrt{S}-9.8\times\sqrt[3]{DSP}}{0.686}\leq 24.000\,metres

Interval_tree.html

  1. ( a i , b i ) \left(a_{i},b_{i}\right)
  2. i = 0 , 1 i=0,1
  3. a 0 a 1 < b 0 a_{0}\leqslant a_{1}<b_{0}
  4. a 0 < b 1 b 0 a_{0}<b_{1}\leqslant b_{0}
  5. a 1 a 0 < b 1 a_{1}\leqslant a_{0}<b_{1}
  6. a 1 < b 0 b 1 a_{1}<b_{0}\leqslant b_{1}
  7. m i = a i + b i 2 m_{i}=\frac{a_{i}+b_{i}}{2}
  8. d i = b i - a i 2 d_{i}=\frac{b_{i}-a_{i}}{2}
  9. | m 1 - m 0 | < d 0 + d 1 \left|m_{1}-m_{0}\right|<d_{0}+d_{1}
  10. d i d_{i}
  11. a q , b q , m q , d q a_{q},b_{q},m_{q},d_{q}
  12. M n M_{n}
  13. m i m_{i}
  14. min { d i } \min\left\{d_{i}\right\}
  15. m i = M n m_{i}=M_{n}
  16. min { d i } = | m q - M n | - d q \min\left\{d_{i}\right\}=\left|m_{q}-M_{n}\right|-d_{q}
  17. d i d_{i}
  18. min { d i } \min\left\{d_{i}\right\}

Intrinsic_equation.html

  1. s s
  2. θ \theta
  3. κ \kappa
  4. τ \tau
  5. κ ( s ) = 1 r \kappa(s)=\tfrac{1}{r}
  6. s s
  7. κ \kappa
  8. r r
  9. E = 0 L B κ 2 ( s ) d s E=\int_{0}^{L}B\kappa^{2}(s)ds
  10. B B
  11. E I EI
  12. κ ( s ) = d θ / d s \kappa(s)=d\theta/ds

Intrinsic_value_(finance).html

  1. I V out - of - the - money = 0 IV_{\mathrm{out-of-the-money}}=0
  2. I V in - the - money = | S - K | = | K - S | IV_{\mathrm{in-the-money}}=\left|S-K\right|=\left|K-S\right|

Introduction_to_general_relativity.html

  1. 𝐆 = 8 π G c 4 𝐓 , \mathbf{G}=\frac{8\pi G}{c^{4}}\mathbf{T},

Invariable_plane.html

  1. L = R M V L=RMV
  2. R R
  3. M M
  4. V V

Invariant_polynomial.html

  1. P P
  2. Γ \Gamma
  3. V V
  4. P P
  5. Γ \Gamma
  6. P ( γ x ) = P ( x ) P(\gamma x)=P(x)
  7. γ Γ \gamma\in\Gamma
  8. x V x\in V

Inverse-chi-squared_distribution.html

  1. Γ ( ν 2 , 1 2 x ) / Γ ( ν 2 ) \Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right)\bigg/\,\Gamma\!\left(\frac{\nu% }{2}\right)\!
  2. 1 ν - 2 \frac{1}{\nu-2}\!
  3. ν > 2 \nu>2\!
  4. 1 ν + 2 \frac{1}{\nu+2}\!
  5. 2 ( ν - 2 ) 2 ( ν - 4 ) \frac{2}{(\nu-2)^{2}(\nu-4)}\!
  6. ν > 4 \nu>4\!
  7. 4 ν - 6 2 ( ν - 4 ) \frac{4}{\nu-6}\sqrt{2(\nu-4)}\!
  8. ν > 6 \nu>6\!
  9. 12 ( 5 ν - 22 ) ( ν - 6 ) ( ν - 8 ) \frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\!
  10. ν > 8 \nu>8\!
  11. ν 2 + ln ( 1 2 Γ ( ν 2 ) ) \frac{\nu}{2}\!+\!\ln\!\left(\frac{1}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)
  12. - ( 1 + ν 2 ) ψ ( ν 2 ) \!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)
  13. 2 Γ ( ν 2 ) ( - t 2 i ) ν 4 K ν 2 ( - 2 t ) \frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4}}K% _{\frac{\nu}{2}}\!\left(\sqrt{-2t}\right)
  14. 2 Γ ( ν 2 ) ( - i t 2 ) ν 4 K ν 2 ( - 2 i t ) \frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4}}K% _{\frac{\nu}{2}}\!\left(\sqrt{-2it}\right)
  15. X X
  16. ν \nu
  17. 1 / X 1/X
  18. ν \nu
  19. ν / X \nu/X
  20. ν \nu
  21. f 1 ( x ; ν ) = 2 - ν / 2 Γ ( ν / 2 ) x - ν / 2 - 1 e - 1 / ( 2 x ) , f_{1}(x;\nu)=\frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1}e^{-1/(2x)},
  22. f 2 ( x ; ν ) = ( ν / 2 ) ν / 2 Γ ( ν / 2 ) x - ν / 2 - 1 e - ν / ( 2 x ) . f_{2}(x;\nu)=\frac{(\nu/2)^{\nu/2}}{\Gamma(\nu/2)}x^{-\nu/2-1}e^{-\nu/(2x)}.
  23. x > 0 x>0
  24. ν \nu
  25. Γ \Gamma
  26. σ 2 = 1 / ν , \sigma^{2}=1/\nu,
  27. σ 2 = 1 \sigma^{2}=1
  28. { 2 x 2 f 1 ( x ) + f 1 ( x ) ( ν x + 2 x - 1 ) = 0 , f 1 ( 1 ) = 2 - ν / 2 e Γ ( ν 2 ) } \left\{2x^{2}f_{1}^{\prime}(x)+f_{1}(x)(\nu x+2x-1)=0,f_{1}(1)=\frac{2^{-\nu/2% }}{\sqrt{e}\Gamma\left(\frac{\nu}{2}\right)}\right\}
  29. { 2 x 2 f 2 ( x ) + f 2 ( x ) ( - ν + ν x + 2 x ) = 0 , f 2 ( 1 ) = ( 2 e ) - ν / 2 v ν / 2 Γ ( ν 2 ) } \left\{2x^{2}f_{2}^{\prime}(x)+f_{2}(x)(-\nu+\nu x+2x)=0,f_{2}(1)=\frac{(2e)^{% -\nu/2}v^{\nu/2}}{\Gamma\left(\frac{\nu}{2}\right)}\right\}
  30. X χ 2 ( ν ) X\thicksim\chi^{2}(\nu)
  31. Y = 1 X Y=\frac{1}{X}
  32. Y Inv- χ 2 ( ν ) Y\thicksim\,\text{Inv-}\chi^{2}(\nu)
  33. X Scale-inv- χ 2 ( ν , 1 / ν ) X\thicksim\,\text{Scale-inv-}\chi^{2}(\nu,1/\nu)\,
  34. X inv- χ 2 ( ν ) X\thicksim\,\text{inv-}\chi^{2}(\nu)
  35. α = ν 2 \alpha=\frac{\nu}{2}
  36. β = 1 2 \beta=\frac{1}{2}

Inverse-gamma_distribution.html

  1. α > 3 \alpha>3
  2. 30 α - 66 ( α - 3 ) ( α - 4 ) \frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!
  3. α > 4 \alpha>4
  4. α + ln ( β Γ ( α ) ) - ( 1 + α ) Ψ ( α ) \alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)
  5. 2 ( - i β t ) α 2 Γ ( α ) K α ( - 4 i β t ) \frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha% }\left(\sqrt{-4i\beta t}\right)
  6. x > 0 x>0
  7. f ( x ; α , β ) = β α Γ ( α ) x - α - 1 exp ( - β x ) f(x;\alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{-\alpha-1}\exp\left(% -\frac{\beta}{x}\right)
  8. α \alpha
  9. β \beta
  10. Γ ( ) \Gamma(\cdot)
  11. β \beta
  12. f ( x ; α , β ) = f ( x β ; α , 1 ) β f(x;\alpha,\beta)=\frac{f(\frac{x}{\beta};\alpha,1)}{\beta}
  13. F ( x ; α , β ) = Γ ( α , β x ) Γ ( α ) = Q ( α , β x ) F(x;\alpha,\beta)=\frac{\Gamma\left(\alpha,\frac{\beta}{x}\right)}{\Gamma(% \alpha)}=Q\left(\alpha,\frac{\beta}{x}\right)\!
  14. K α ( ) K_{\alpha}(\cdot)
  15. α > 0 \alpha>0
  16. β > 0 \beta>0
  17. 𝔼 [ ln ( X ) ] = ln ( β ) - ψ ( α ) . \mathbb{E}[\ln(X)]=\ln(\beta)-\psi(\alpha).\,
  18. 𝔼 [ X - 1 ] = α β . \mathbb{E}[X^{-1}]=\frac{\alpha}{\beta}.\,
  19. ψ ( α ) \psi(\alpha)
  20. { x 2 f ( x ) + f ( x ) ( - β + α x + x ) = 0 , f ( 1 ) = e - β β α Γ ( α ) } \left\{x^{2}f^{\prime}(x)+f(x)(-\beta+\alpha x+x)=0,f(1)=\frac{e^{-\beta}\beta% ^{\alpha}}{\Gamma(\alpha)}\right\}
  21. X Inv-Gamma ( α , β ) X\sim\mbox{Inv-Gamma}~{}(\alpha,\beta)
  22. k X Inv-Gamma ( α , k β ) kX\sim\mbox{Inv-Gamma}~{}(\alpha,k\beta)\,
  23. X Inv-Gamma ( α , 1 2 ) X\sim\mbox{Inv-Gamma}~{}(\alpha,\tfrac{1}{2})
  24. X Inv- χ 2 ( 2 α ) X\sim\mbox{Inv-}~{}\chi^{2}(2\alpha)\,
  25. X Inv-Gamma ( α 2 , 1 2 ) X\sim\mbox{Inv-Gamma}~{}(\tfrac{\alpha}{2},\tfrac{1}{2})
  26. X Scaled Inv- χ 2 ( α , 1 α ) X\sim\mbox{Scaled Inv-}~{}\chi^{2}(\alpha,\tfrac{1}{\alpha})\,
  27. X Inv-Gamma ( 1 2 , c 2 ) X\sim\textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})
  28. X Levy ( 0 , c ) X\sim\textrm{Levy}(0,c)\,
  29. X Gamma ( k , θ ) X\sim\mbox{Gamma}~{}(k,\theta)\,
  30. 1 X Inv-Gamma ( k , θ - 1 ) \tfrac{1}{X}\sim\mbox{Inv-Gamma}~{}(k,\theta^{-1})\,
  31. f ( x ) = x k - 1 e - x / θ θ k Γ ( k ) f(x)=x^{k-1}\frac{e^{-x/\theta}}{\theta^{k}\,\Gamma(k)}
  32. Y = g ( X ) = 1 X Y=g(X)=\frac{1}{X}
  33. f Y ( y ) = f X ( g - 1 ( y ) ) | d d y g - 1 ( y ) | f_{Y}(y)=f_{X}\left(g^{-1}(y)\right)\left|\frac{d}{dy}g^{-1}(y)\right|
  34. = 1 θ k Γ ( k ) ( 1 y ) k - 1 exp ( - 1 θ y ) 1 y 2 =\frac{1}{\theta^{k}\Gamma(k)}\left(\frac{1}{y}\right)^{k-1}\exp\left(\frac{-1% }{\theta y}\right)\frac{1}{y^{2}}
  35. = 1 θ k Γ ( k ) ( 1 y ) k + 1 exp ( - 1 θ y ) =\frac{1}{\theta^{k}\Gamma(k)}\left(\frac{1}{y}\right)^{k+1}\exp\left(\frac{-1% }{\theta y}\right)
  36. = 1 θ k Γ ( k ) y - k - 1 exp ( - 1 θ y ) . =\frac{1}{\theta^{k}\Gamma(k)}y^{-k-1}\exp\left(\frac{-1}{\theta y}\right).
  37. k k
  38. α \alpha
  39. θ - 1 \theta^{-1}
  40. β \beta
  41. y y
  42. x x
  43. f ( x ) = β α Γ ( α ) x - α - 1 exp ( - β x ) . f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{-\alpha-1}\exp\left(\frac{-\beta}% {x}\right).

Inverse_distance_weighting.html

  1. u u
  2. u ( x ) : x , x 𝐃 \sub n u(x):x\rightarrow\mathbb{R},\quad x\in\mathbf{D}\sub\mathbb{R}^{n}
  3. 𝐃 \mathbf{D}
  4. N N
  5. [ ( x 1 , u 1 ) , ( x 2 , u 2 ) , , ( x N , u N ) ] [(x_{1},u_{1}),(x_{2},u_{2}),...,(x_{N},u_{N})]
  6. u ( x i ) = u i u(x_{i})=u_{i}
  7. z = exp ( - x 2 - y 2 ) z=\exp(-x^{2}-y^{2})
  8. u u
  9. x x
  10. u i = u ( x i ) u_{i}=u(x_{i})
  11. i = 1 , 2 , , N i=1,2,...,N
  12. u ( 𝐱 ) = { i = 1 N w i ( 𝐱 ) u i i = 1 N w i ( 𝐱 ) , if d ( 𝐱 , 𝐱 i ) 0 for all i u i , if d ( 𝐱 , 𝐱 i ) = 0 for some i u(\mathbf{x})=\begin{cases}\frac{\displaystyle\sum_{i=1}^{N}{w_{i}(\mathbf{x})% u_{i}}}{\displaystyle\sum_{i=1}^{N}{w_{i}(\mathbf{x})}},&\,\text{if }d(\mathbf% {x},\mathbf{x}_{i})\neq 0\,\text{ for all }i\\ u_{i},&\,\text{if }d(\mathbf{x},\mathbf{x}_{i})=0\,\text{ for some }i\end{cases}
  13. w i ( 𝐱 ) = 1 d ( 𝐱 , 𝐱 i ) p w_{i}(\mathbf{x})=\frac{1}{d(\mathbf{x},\mathbf{x}_{i})^{p}}
  14. d d
  15. p p
  16. p p
  17. p 2 p\leq 2
  18. ρ \rho
  19. r 0 r_{0}
  20. R R
  21. j w j r 0 R 2 π r ρ d r r p = 2 π ρ r 0 R r 1 - p d r , \sum_{j}w_{j}\approx\int_{r_{0}}^{R}\frac{2\pi r\rho dr}{r^{p}}=2\pi\rho\int_{% r_{0}}^{R}r^{1-p}dr,
  22. R R\rightarrow\infty
  23. p 2 p\leq 2
  24. p N p\leq N
  25. ϕ ( 𝐱 , u ) = ( i = 0 N ( u - u i ) 2 d ( 𝐱 , 𝐱 i ) p ) 1 p , \phi(\mathbf{x},u)=\left(\sum_{i=0}^{N}{\frac{(u-u_{i})^{2}}{d(\mathbf{x},% \mathbf{x}_{i})^{p}}}\right)^{\frac{1}{p}},
  26. ϕ ( 𝐱 , u ) u = 0. \frac{\partial\phi(\mathbf{x},u)}{\partial u}=0.
  27. w k ( 𝐱 ) = 1 ( D * * ( 𝐱 , 𝐱 k ) ) 1 2 , w_{k}(\mathbf{x})=\frac{1}{(D_{**}(\mathbf{x},\mathbf{x}_{k}))^{\frac{1}{2}}},
  28. D * * ( 𝐱 , 𝐱 k ) D_{**}(\mathbf{x},\mathbf{x}_{k})
  29. w k ( 𝐱 ) = ( max ( 0 , R - d ( 𝐱 , 𝐱 k ) ) R d ( 𝐱 , 𝐱 k ) ) 2 w_{k}(\mathbf{x})=\left(\frac{\max(0,R-d(\mathbf{x},\mathbf{x}_{k}))}{Rd(% \mathbf{x},\mathbf{x}_{k})}\right)^{2}

Inverse_quadratic_interpolation.html

  1. x n + 1 = f n - 1 f n ( f n - 2 - f n - 1 ) ( f n - 2 - f n ) x n - 2 + f n - 2 f n ( f n - 1 - f n - 2 ) ( f n - 1 - f n ) x n - 1 x_{n+1}=\frac{f_{n-1}f_{n}}{(f_{n-2}-f_{n-1})(f_{n-2}-f_{n})}x_{n-2}+\frac{f_{% n-2}f_{n}}{(f_{n-1}-f_{n-2})(f_{n-1}-f_{n})}x_{n-1}
  2. + f n - 2 f n - 1 ( f n - f n - 2 ) ( f n - f n - 1 ) x n , {}+\frac{f_{n-2}f_{n-1}}{(f_{n}-f_{n-2})(f_{n}-f_{n-1})}x_{n},
  3. f - 1 ( y ) = ( y - f n - 1 ) ( y - f n ) ( f n - 2 - f n - 1 ) ( f n - 2 - f n ) x n - 2 + ( y - f n - 2 ) ( y - f n ) ( f n - 1 - f n - 2 ) ( f n - 1 - f n ) x n - 1 f^{-1}(y)=\frac{(y-f_{n-1})(y-f_{n})}{(f_{n-2}-f_{n-1})(f_{n-2}-f_{n})}x_{n-2}% +\frac{(y-f_{n-2})(y-f_{n})}{(f_{n-1}-f_{n-2})(f_{n-1}-f_{n})}x_{n-1}
  4. + ( y - f n - 2 ) ( y - f n - 1 ) ( f n - f n - 2 ) ( f n - f n - 1 ) x n . {}+\frac{(y-f_{n-2})(y-f_{n-1})}{(f_{n}-f_{n-2})(f_{n}-f_{n-1})}x_{n}.

Inverse_semigroup.html

  1. \cap
  2. X \mathcal{I}_{X}
  3. \mathcal{L}
  4. \mathcal{R}
  5. \mathcal{L}
  6. \mathcal{R}
  7. \mathcal{L}
  8. \mathcal{R}
  9. a b a - 1 a = b - 1 b , a b a a - 1 = b b - 1 a\,\mathcal{L}\,b\Longleftrightarrow a^{-1}a=b^{-1}b,\quad a\,\mathcal{R}\,b% \Longleftrightarrow aa^{-1}=bb^{-1}
  10. a b a = e b , a\leq b\Longleftrightarrow a=eb,
  11. a b a = b f , a\leq b\Longleftrightarrow a=bf,
  12. a b , c d a c b d a\leq b,c\leq d\Longrightarrow ac\leq bd
  13. a b a - 1 b - 1 . a\leq b\Longrightarrow a^{-1}\leq b^{-1}.
  14. \,\mathcal{L}\,
  15. \,\mathcal{R}\,
  16. e f e = e f , e\leq f\Longleftrightarrow e=ef,
  17. S \mathcal{I}_{S}
  18. a ρ b , c ρ d a c ρ b d . a\,\rho\,b,\quad c\,\rho\,d\Longrightarrow ac\,\rho\,bd.
  19. σ \sigma
  20. a σ b a\,\sigma\,b\Longleftrightarrow
  21. c S c\in S
  22. c a , b . c\leq a,b.
  23. a S , e E ( S ) , a ρ e a E ( S ) . a\in S,e\in E(S),a\,\rho\,e\Longrightarrow a\in E(S).
  24. e s E s E . es\in E\Longrightarrow s\in E.
  25. s e E s E . se\in E\Rightarrow s\in E.
  26. \sim
  27. \sim
  28. a b a b - 1 , a - 1 b a\sim b\Longleftrightarrow ab^{-1},a^{-1}b
  29. 𝒳 \mathcal{X}
  30. 𝒴 \mathcal{Y}
  31. 𝒳 \mathcal{X}
  32. 𝒴 \mathcal{Y}
  33. 𝒴 \mathcal{Y}
  34. \wedge
  35. 𝒴 \mathcal{Y}
  36. 𝒴 \mathcal{Y}
  37. 𝒳 \mathcal{X}
  38. 𝒳 \mathcal{X}
  39. 𝒴 \mathcal{Y}
  40. 𝒴 \mathcal{Y}
  41. 𝒳 \mathcal{X}
  42. 𝒳 \mathcal{X}
  43. 𝒳 \mathcal{X}
  44. 𝒳 \mathcal{X}
  45. 𝒳 \mathcal{X}
  46. 𝒳 \mathcal{X}
  47. ( G , 𝒳 , 𝒴 ) (G,\mathcal{X},\mathcal{Y})
  48. 𝒳 \mathcal{X}
  49. 𝒴 \mathcal{Y}
  50. 𝒴 \mathcal{Y}
  51. 𝒴 \mathcal{Y}
  52. ( G , 𝒳 , 𝒴 ) (G,\mathcal{X},\mathcal{Y})
  53. P ( G , 𝒳 , 𝒴 ) = { ( A , g ) 𝒴 × G : g - 1 A 𝒴 } P(G,\mathcal{X},\mathcal{Y})=\{(A,g)\in\mathcal{Y}\times G:g^{-1}A\in\mathcal{% Y}\}
  54. ( A , g ) ( B , h ) = ( A g B , g h ) (A,g)(B,h)=(A\wedge gB,gh)
  55. P ( G , 𝒳 , 𝒴 ) P(G,\mathcal{X},\mathcal{Y})
  56. ( G , 𝒳 , 𝒴 ) (G,\mathcal{X},\mathcal{Y})
  57. P ( G , 𝒳 , 𝒴 ) P(G,\mathcal{X},\mathcal{Y})
  58. ( G , 𝒳 , 𝒴 ) (G,\mathcal{X},\mathcal{Y})
  59. 𝒴 \mathcal{Y}
  60. 𝒳 \mathcal{X}
  61. 𝒳 \mathcal{X}
  62. { ( x x - 1 x , x ) , ( x x - 1 y y - 1 , y y - 1 x x - 1 ) | x , y ( X X - 1 ) + } . \{(xx^{-1}x,x),\;(xx^{-1}yy^{-1},yy^{-1}xx^{-1})\;|\;x,y\in(X\cup X^{-1})^{+}\}.

Inverse_synthetic_aperture_radar.html

  1. 2 D 2 λ \frac{2D^{2}}{\lambda}
  2. D D

Inversive_congruential_generator.html

  1. x 0 = x_{0}=
  2. x i + 1 ( a x i - 1 + c ) mod q x_{i+1}\equiv(ax_{i}^{-1}+c)\mod q
  3. x i 0 x_{i}\neq 0
  4. x i + 1 = c x_{i+1}=c
  5. x i = 0 x_{i}=0
  6. ( x n ) n 0 (x_{n})_{n\geq 0}
  7. x i = x j x_{i}=x_{j}
  8. x i + 1 = x j + 1 x_{i+1}=x_{j+1}
  9. f ( x ) = x 2 - c x - a 𝔽 q [ x ] f(x)=x^{2}-cx-a\in\mathbb{F}_{q}[x]
  10. 𝔽 q \mathbb{F}_{q}
  11. 𝔽 5 \in\mathbb{F}_{5}
  12. f ( x ) = x 2 - 3 x - 2 f(x)=x^{2}-3x-2
  13. 𝔽 5 [ x ] \mathbb{F}_{5}[x]
  14. 𝔽 5 [ x ] / ( f ) \mathbb{F}_{5}[x]/(f)
  15. p 1 , , p r p_{1},\dots,p_{r}
  16. p j 5 p_{j}\geq 5
  17. ( x n ) n 0 (x_{n})_{n\geq 0}
  18. 𝔽 p j \in\mathbb{F}_{p_{j}}
  19. p j p_{j}
  20. { x n ( j ) | 0 n p j } = 𝔽 p j \{x_{n}^{(j)}|0\leq n\leq p_{j}\}=\in\mathbb{F}_{p_{j}}
  21. T j = T / p j T_{j}=T/p_{j}
  22. T = p 1 p r T=p_{1}\cdots p_{r}
  23. ( x n ) n 0 (x_{n})_{n\geq 0}
  24. ( x n ) n 0 (x_{n})_{n\geq 0}
  25. x n = T 1 x n ( 1 ) + T 2 x n ( 2 ) + + T r x n ( r ) mod T x_{n}=T_{1}x_{n}^{(1)}+T_{2}x_{n}^{(2)}+\dots+T_{r}x_{n}^{(r)}\mod T
  26. p 1 = 5 p_{1}=5
  27. p 2 = 7 ( r = 2 ) p_{2}=7(r=2)
  28. ( x n ( 1 ) ) n 0 = ( 0 , 1 , 2 , 3 , 4 , ) (x_{n}^{(1)})_{n\geq 0}=(0,1,2,3,4,\dots)
  29. ( x n ( 2 ) ) n 0 = ( 0 , 1 , 2 , 3 , 4 , 5 , 6 , ) (x_{n}^{(2)})_{n\geq 0}=(0,1,2,3,4,5,6,\dots)
  30. x j = 7 x j ( 1 ) + 5 x j ( 2 ) mod 35 x_{j}=7x_{j}^{(1)}+5x_{j}^{(2)}\mod 35
  31. ( x n ) n 0 = ( 0 , 12 , 24 , 1 , 13 , 25 , 2 , 14 , 26 , 3 , 15 , 27 , 4 , 16 , 28 , 5 , 17 , 29 , 6 , 18 , 30 , 7 , 19 , 31 , 8 , 20 , 32 , 9 , 21 , 33 , 10 , 22 , 34 , 11 , 23 ) (x_{n})_{n\geq 0}=(0,12,24,1,13,25,2,14,26,3,15,27,4,16,28,5,17,29,6,18,30,7,1% 9,31,8,20,32,9,21,33,10,22,34,11,23)
  32. 5 × 7 = 35 5\times 7=35\,
  33. s = 1 s=1
  34. s = 2 s=2
  35. N N
  36. 𝐭 1 , , 𝐭 N - 1 [ 0 , 1 ) {\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1}\in[0,1)
  37. D N ( 𝐭 1 , , 𝐭 N - 1 ) = sup J | F N ( J ) - V ( J ) | D_{N}({\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1})={\rm sup}_{J}|F_{N}(J)-V(J)|
  38. J J
  39. [ 0 , 1 ) s [0,1)^{s}
  40. F N ( J ) F_{N}(J)
  41. N - 1 N^{-1}
  42. 𝐭 1 , , 𝐭 N - 1 {\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1}
  43. [ 0 , 1 ) s [0,1)^{s}
  44. 𝐭 1 , , 𝐭 N - 1 {\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1}
  45. J = [ 0 , 1 ) s J=[0,1)^{s}
  46. V ( J ) = 1 V(J)=1
  47. F N ( J ) = N / N = 1 F_{N}(J)=N/N=1
  48. D N ( 𝐭 1 , , 𝐭 N - 1 ) = 0 D_{N}({\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1})=0
  49. V ( j ) 0 V(j)\approx 0
  50. F N ( j ) N / N 1 F_{N}(j)\approx N/N\approx 1
  51. D N ( 𝐭 1 , , 𝐭 N - 1 ) = 1 D_{N}({\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1})=1
  52. 0 D N ( 𝐭 1 , , 𝐭 N - 1 ) 1 0\leq D_{N}({\mathbf{t}}_{1},\dots,{\mathbf{t}}_{N-1})\leq 1
  53. k 1 k\geq 1
  54. q 2 q\geq 2
  55. C k ( q ) C_{k}(q)
  56. ( h 1 , , h k ) Z k (h_{1},\dots,h_{k})\in Z^{k}
  57. - q / 2 < h j < q / 2 -q/2<h_{j}<q/2
  58. 1 j k 1\leq j\leq k
  59. r ( h , q ) = { q sin ( π | h | / q ) for h C 1 ( q ) 1 for h = 0 r(h,q)=\begin{cases}q\sin(\pi|h|/q)&\,\text{for }h\in C_{1}(q)\\ 1&\,\text{for }h=0\end{cases}
  60. r ( 𝐡 , q ) = j = 1 k r ( h j , q ) r(\mathbf{h},q)=\prod_{j=1}^{k}r(h_{j},q)
  61. 𝐡 = ( h 1 , , h k ) C k ( q ) {\mathbf{h}}=(h_{1},\dots,h_{k})\in C_{k}(q)
  62. t t
  63. e ( t ) = exp ( 2 π i t ) e(t)={\rm exp}(2\pi\cdot it)
  64. u v u\cdot v
  65. u , v i n R k u,vinR^{k}
  66. N 1 N\geq 1
  67. q 2 q\geq 2
  68. 𝐭 n = y n / q [ 0 , 1 ) k {\mathbf{t}}_{n}=y_{n}/q\in[0,1)^{k}
  69. y n { 0 , 1 , , q - 1 } k y_{n}\in\{0,1,\dots,q-1\}^{k}
  70. 0 n < N 0\leq n<N
  71. 𝐭 0 , , 𝐭 N - 1 {\mathbf{t}}_{0},\dots,{\mathbf{t}}_{N-1}
  72. D N ( 𝐭 0 , 𝐭 1 , , 𝐭 N - 1 ) D_{N}(\mathbf{t}_{0},\mathbf{t}_{1},\dots,\mathbf{t}_{N-1})
  73. k q \frac{k}{q}
  74. 1 N \frac{1}{N}
  75. h \C k ( q ) \sum_{h\in\C_{k}(q)}
  76. 1 r ( 𝐡 , q ) | n = 0 N - 1 e ( 𝐡 𝐭 n ) | \frac{1}{r(\mathbf{h},q)}\Bigg|\sum_{n=0}^{N-1}e(\mathbf{h}\cdot\mathbf{t}_{n}% )\Bigg|
  77. N N
  78. 𝐭 1 , , 𝐭 N - 1 [ 0 , 1 ) k \mathbf{t}_{1},\dots,\mathbf{t}_{N-1}\in[0,1)^{k}
  79. D N ( 𝐭 0 , 𝐭 1 , , 𝐭 N - 1 ) D_{N}(\mathbf{t}_{0},\mathbf{t}_{1},\dots,\mathbf{t}_{N-1})
  80. π 2 N ( ( π + 1 ) l - 1 ) j = 1 k max ( 1 , h j ) | n = 0 N - 1 e ( 𝐡 𝐭 n ) | \frac{\pi}{2N((\pi+1)^{l}-1)\prod_{j=1}^{k}{\rm max}(1,h_{j})}\Bigg|\sum_{n=0}% ^{N-1}e(\mathbf{h}\cdot\mathbf{t}_{n})\Bigg|
  81. 𝐡 = ( h 1 , , h k ) Z k {\mathbf{h}}=(h_{1},\dots,h_{k})\in Z^{k}
  82. l l
  83. 𝐡 {\mathbf{h}}

Iodine-131.html

  1. I 53 131 β + ν e ¯ + Xe * 54 131 {{}^{131}_{53}\mathrm{I}}\rightarrow\beta+\bar{\nu_{e}}+{{}^{131}_{54}\mathrm{% Xe}^{*}}
  2. Xe * 54 131 Xe 54 131 + γ {{}^{131}_{54}\mathrm{Xe}^{*}}\rightarrow{{}^{131}_{54}\mathrm{Xe}}+\gamma

Iodine_value.html

  1. R - CH = CH - R + I 2 R - CHI - CHI - R \mathrm{R{-}CH{=}CH{-}R^{\prime}+I_{2}\longrightarrow R{-}CHI{-}CHI{-}R^{% \prime}}

Ion_acoustic_wave.html

  1. ω = v s k \omega=v_{s}k
  2. v s = γ e Z K B T e + γ i K B T i M v_{s}=\sqrt{\frac{\gamma_{e}ZK_{B}T_{e}+\gamma_{i}K_{B}T_{i}}{M}}
  3. K B K_{B}
  4. M M
  5. Z Z
  6. T e T_{e}
  7. T i T_{i}
  8. p s 1 = γ s T s 0 n s 1 p_{s1}=\gamma_{s}T_{s0}n_{s1}
  9. γ s \gamma_{s}
  10. ( - m i t t + γ i T i 2 ) n i 1 = Z i e n i 0 E (-m_{i}\partial_{tt}+\gamma_{i}T_{i}\nabla^{2})n_{i1}=Z_{i}en_{i0}\nabla\cdot% \vec{E}
  11. E 1 \vec{E}_{1}
  12. n e 0 m e t v e 1 = - n e 0 e E 1 - γ e T e n e 1 n_{e0}m_{e}\partial_{t}\vec{v}_{e1}=-n_{e0}e\vec{E}_{1}-\gamma_{e}T_{e}\nabla n% _{e1}
  13. E 1 = - γ e T e n e 0 e n e 1 \vec{E}_{1}=-{\gamma_{e}T_{e}\over n_{e0}e}\nabla n_{e1}
  14. n i 1 n_{i1}
  15. n e 1 n_{e1}
  16. ( - m i t t + γ i T i 2 ) n i 1 = - γ e T e 2 n e 1 (-m_{i}\partial_{tt}+\gamma_{i}T_{i}\nabla^{2})n_{i1}=-\gamma_{e}T_{e}\nabla^{% 2}n_{e1}
  17. ϵ 0 e E = [ i = 1 N n i 0 Z i - n n e 0 ] + [ i = 1 N n i 1 Z i - n e 1 ] {\epsilon_{0}\over e}\nabla\cdot\vec{E}=\left[\sum_{i=1}^{N}n_{i0}Z_{i}-n_{ne0% }\right]+\left[\sum_{i=1}^{N}n_{i1}Z_{i}-n_{e1}\right]
  18. ( 1 - γ e λ D e 2 2 ) n e 1 = i = 1 N Z i n i 1 (1-\gamma_{e}\lambda_{De}^{2}\nabla^{2})n_{e1}=\sum_{i=1}^{N}Z_{i}n_{i1}
  19. λ D e 2 ϵ 0 T e / ( n e 0 e 2 ) \lambda_{De}^{2}\equiv\epsilon_{0}T_{e}/(n_{e0}e^{2})
  20. E \nabla\cdot\vec{E}
  21. k λ D e k\lambda_{De}
  22. n i 1 = γ e T e Z i n i 0 n e 0 [ m i v s 2 - γ i T i ] - 1 n e 1 n_{i1}=\gamma_{e}T_{e}Z_{i}{n_{i0}\over n_{e0}}[m_{i}v_{s}^{2}-\gamma_{i}T_{i}% ]^{-1}n_{e1}
  23. v s = ω / k v_{s}=\omega/k
  24. n e 1 n_{e1}
  25. n e 1 n_{e1}
  26. γ e T e i = 1 N Z i 2 f i [ m i v s 2 - γ i T i ] - 1 = Z ¯ ( 1 + γ e k 2 λ D e 2 ) \gamma_{e}T_{e}\sum_{i=1}^{N}Z_{i}^{2}f_{i}[m_{i}v_{s}^{2}-\gamma_{i}T_{i}]^{-% 1}=\bar{Z}(1+\gamma_{e}k^{2}\lambda_{De}^{2})
  27. n i 1 = f i n I 1 n_{i1}=f_{i}n_{I1}
  28. n I 1 = Σ i n i 1 n_{I1}=\Sigma_{i}n_{i1}
  29. Z ¯ = Σ i Z i f i \bar{Z}=\Sigma_{i}Z_{i}f_{i}
  30. i = 1 N F i u 2 - τ i = 1 + γ e k 2 λ D e 2 \sum_{i=1}^{N}{F_{i}\over u^{2}-\tau_{i}}=1+\gamma_{e}k^{2}\lambda_{De}^{2}
  31. A i = m i / m u A_{i}=m_{i}/m_{u}
  32. m u m_{u}
  33. u 2 = m u v s 2 / T e u^{2}=m_{u}v_{s}^{2}/T_{e}
  34. F i = Z i 2 f i γ e A i Z ¯ , τ i = γ i T i A i T e F_{i}={Z_{i}^{2}f_{i}\gamma_{e}\over A_{i}\bar{Z}},\quad\tau_{i}={\gamma_{i}T_% {i}\over A_{i}T_{e}}
  35. k λ D e k\lambda_{De}
  36. ω = v s k \omega=v_{s}k
  37. v s v_{s}
  38. u 2 u^{2}
  39. u u
  40. v s 2 = γ e Z i T e m i 1 1 + γ e ( k λ D e ) 2 + γ i T i m i = γ e Z i T e m i [ 1 1 + γ e ( k λ D e ) 2 + γ i T i Z i γ e T e ] v_{s}^{2}={\gamma_{e}Z_{i}T_{e}\over m_{i}}{1\over 1+\gamma_{e}(k\lambda_{De})% ^{2}}+{\gamma_{i}T_{i}\over m_{i}}={\gamma_{e}Z_{i}T_{e}\over m_{i}}\left[{1% \over 1+\gamma_{e}(k\lambda_{De})^{2}}+{\gamma_{i}T_{i}\over Z_{i}\gamma_{e}T_% {e}}\right]
  41. T i T e T_{i}\ll T_{e}
  42. T i = 0 T_{i}=0
  43. u 2 = 0 u^{2}=0
  44. v s 2 ( T i = 0 ) γ e T e 1 + γ e ( k λ D e ) 2 i Z i 2 f i Z ¯ m i v_{s}^{2}(T_{i}=0)\equiv{\gamma_{e}T_{e}\over 1+\gamma_{e}(k\lambda_{De})^{2}}% \sum_{i}{Z_{i}^{2}f_{i}\over\bar{Z}m_{i}}
  45. v s v_{s}
  46. T i T e T_{i}\ll T_{e}
  47. v s 2 v s 2 ( T i = 0 ) + i Z i 2 f i m i γ i T i m i j Z j 2 f j m j v_{s}^{2}\approx v_{s}^{2}(T_{i}=0)+{\sum_{i}{Z_{i}^{2}f_{i}\over m_{i}}{% \gamma_{i}T_{i}\over m_{i}}\over\sum_{j}{Z_{j}^{2}f_{j}\over m_{j}}}
  48. T i = 0 T_{i}=0
  49. v s v_{s}
  50. f D = f T = 1 / 2 f_{D}=f_{T}=1/2
  51. Z D = Z T = 1 Z_{D}=Z_{T}=1
  52. T e = T i T_{e}=T_{i}
  53. γ e = 1 , γ i = 3 \gamma_{e}=1,\gamma_{i}=3
  54. ( k λ D e ) 2 (k\lambda_{De})^{2}
  55. v s 2 v_{s}^{2}
  56. 2 A D A T u 4 - 7 ( A D + A T ) u 2 + 24 = 0 2A_{D}A_{T}u^{4}-7(A_{D}+A_{T})u^{2}+24=0
  57. ( A D , A T ) = ( 2.01 , 3.02 ) (A_{D},A_{T})=(2.01,3.02)
  58. u 2 = ( 1.10 , 1.81 ) u^{2}=(1.10,1.81)
  59. γ e = 1 \gamma_{e}=1
  60. γ i = 3 , T i = T e / 2 \gamma_{i}=3,T_{i}=T_{e}/2
  61. f B f_{B}
  62. f A u = 1 - f B f_{Au}=1-f_{B}
  63. Z ¯ = 50 - 45 f B , τ B = 0.139 , τ A u = 0.00761 , F B = 2.31 f B / Z ¯ , \bar{Z}=50-45f_{B},\tau_{B}=0.139,\tau_{Au}=0.00761,F_{B}=2.31f_{B}/\bar{Z},
  64. F A u = 12.69 ( 1 - f B ) / Z ¯ F_{Au}=12.69(1-f_{B})/\bar{Z}

Ion_chromatography.html

  1. R-X - C + + M + B - R-X - M + + C + + B - \,\text{R-X}^{-}\,\text{C}^{+}\,+\,\,\text{M}^{+}\,\,\text{B}^{-}% \rightleftarrows\,\,\text{R-X}^{-}\,\text{M}^{+}\,+\,\,\text{C}^{+}\,+\,\,% \text{B}^{-}
  2. R-X + A - + M + B - R-X + B - + M + + A - \,\text{R-X}^{+}\,\text{A}^{-}\,+\,\,\text{M}^{+}\,\,\text{B}^{-}% \rightleftarrows\,\,\text{R-X}^{+}\,\text{B}^{-}\,+\,\,\text{M}^{+}\,+\,\,% \text{A}^{-}

Ionocraft.html

  1. F = I d k F=\frac{Id}{k}

Isaac_Roberts.html

  1. 3 8 \begin{matrix}\frac{3}{8}\end{matrix}

ISO_31-0.html

  1. a b \frac{a}{b}

Isochron_dating.html

  1. D + Δ P t D i = Δ P t P - Δ P t ( P - Δ P t D i ) + D D i {D+\Delta{P}_{t}\over D_{i}}={\Delta{P}_{t}\over P-\Delta{P}_{t}}\left({P-% \Delta{P}_{t}\over D_{i}}\right)+{D\over D_{i}}
  2. D D
  3. D i D_{i}
  4. P P
  5. Δ P t \Delta{P}_{t}
  6. t t
  7. P - Δ P t P-\Delta{P}_{t}
  8. D + Δ P t D+\Delta{P}_{t}
  9. D i D_{i}
  10. D + Δ P t D i D+\Delta{P}_{t}\over D_{i}
  11. P - Δ P t D i {P-\Delta{P}_{t}\over D_{i}}
  12. Δ P t P - Δ P t \Delta{P}_{t}\over P-\Delta{P}_{t}

Isocitrate_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Isocost.html

  1. r K + w L = C rK+wL=C\,
  2. - w / r . -w/r.\,
  3. w / r w/r

Isotope_geochemistry.html

  1. δ 13 C = ( ( C 13 C 12 ) s a m p l e ( C 13 C 12 ) s t a n d a r d - 1 ) * 1000 o / o o \delta^{13}C=\Biggl(\frac{\bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{sample}}{% \bigl(\frac{{}^{13}C}{{}^{12}C}\bigr)_{standard}}-1\Biggr)*1000\ ^{o}\!/\!_{oo}

Isotropic_manifold.html

  1. ( M , g ) (M,g)
  2. p M p\in M
  3. v , w T p M v,w\in T_{p}M
  4. φ \varphi
  5. M M
  6. φ ( p ) = p \varphi(p)=p
  7. φ ( v ) = w \varphi_{\ast}(v)=w
  8. p , q M p,q\in M
  9. φ \varphi
  10. M M
  11. φ ( p ) = q . \varphi(p)=q.
  12. γ : [ 0 , 2 ] M \gamma:[0,2]\to M
  13. p p
  14. q q
  15. γ ( 1 ) \gamma(1)
  16. γ ( 1 ) \gamma^{\prime}(1)
  17. - γ ( 1 ) . -\gamma^{\prime}(1).
  18. n \mathbb{R}^{n}
  19. T = 2 / 2 T=\mathbb{R}^{2}/\mathbb{Z}^{2}
  20. T T
  21. p T p\in T
  22. 2 \mathbb{R}^{2}
  23. 2 \mathbb{Z}^{2}
  24. T T
  25. p p
  26. n \mathbb{CP}^{n}
  27. n > 1 n>1
  28. n \mathbb{RP}^{n}
  29. n \mathbb{CP}^{n}
  30. n \mathbb{HP}^{n}
  31. 𝕆 2 \mathbb{OP}^{2}
  32. × S 2 \mathbb{R}\times S^{2}

Iterated_function.html

  1. X X X → X
  2. X X
  3. f : X X f:X → X
  4. X X
  5. f : X X f:X→X
  6. f f
  7. f 0 = def id X f^{0}~{}\stackrel{\mathrm{def}}{=}~{}\operatorname{id}_{X}\,
  8. f n + 1 = def f f n , f^{n+1}~{}\stackrel{\mathrm{def}}{=}~{}f\circ f^{n},\,
  9. X X
  10. f g f○g
  11. ( f g ) ( x ) = f ( g ( x ) ) (f○g)(x)=f(g(x))
  12. f f
  13. f f
  14. f f
  15. m m
  16. n n
  17. f m f n = f n f m = f m + n . f^{m}\circ f^{n}=f^{n}\circ f^{m}=f^{m+n}~{}.\,
  18. f ( x ) = a x f(x)=ax
  19. m m
  20. n n
  21. x x
  22. X X
  23. x x
  24. m m
  25. m m
  26. x x
  27. x x
  28. f f
  29. g g
  30. g ( g ( x ) ) = f ( x ) g(g(x))=f(x)
  31. g ( x ) g(x)
  32. n n
  33. f f
  34. f ( a ) = a f(a)=a
  35. f n ( x ) = f n ( a ) + ( x - a ) d d x f n ( x ) | x = a + ( x - a ) 2 2 ! d 2 d x 2 f n ( x ) | x = a + f^{n}(x)=f^{n}(a)+(x-a)\frac{d}{dx}f^{n}(x)|_{x=a}+\frac{(x-a)^{2}}{2!}\frac{d% ^{2}}{dx^{2}}f^{n}(x)|_{x=a}+\cdots
  36. f n ( x ) = f n ( a ) + ( x - a ) f ( a ) f ( f ( a ) ) f ( f 2 ( a ) ) f ( f n - 1 ( a ) ) + f^{n}\left(x\right)=f^{n}(a)+(x-a)f^{\prime}(a)f^{\prime}(f(a))f^{\prime}(f^{2% }(a))\cdots f^{\prime}(f^{n-1}(a))+\cdots
  37. f n ( x ) = a + ( x - a ) f ( a ) n + ( x - a ) 2 2 ! ( f ′′ ( a ) f ( a ) n - 1 ) ( 1 + f ( a ) + + f ( a ) n - 1 ) + f^{n}\left(x\right)=a+(x-a)f^{\prime}(a)^{n}+\frac{(x-a)^{2}}{2!}(f^{\prime% \prime}(a)f^{\prime}(a)^{n-1})\left(1+f^{\prime}(a)+\cdots+f^{\prime}(a)^{n-1}% \right)+\cdots
  38. f n ( x ) = a + ( x - a ) f ( a ) n + ( x - a ) 2 2 ! ( f ′′ ( a ) f ( a ) n - 1 ) f ( a ) n - 1 f ( a ) - 1 + f^{n}\left(x\right)=a+(x-a)f^{\prime}(a)^{n}+\frac{(x-a)^{2}}{2!}(f^{\prime% \prime}(a)f^{\prime}(a)^{n-1})\frac{f^{\prime}(a)^{n}-1}{f^{\prime}(a)-1}+\cdots
  39. f ( a ) ) = 1 f(a))=1
  40. f n ( x ) = x + ( x - a ) 2 2 ! ( n f ′′ ( a ) ) + ( x - a ) 3 3 ! ( 3 2 n ( n - 1 ) f ′′ ( a ) 2 + n f ′′′ ( a ) ) + f^{n}\left(x\right)=x+\frac{(x-a)^{2}}{2!}(nf^{\prime\prime}(a))+\frac{(x-a)^{% 3}}{3!}\left(\frac{3}{2}n(n-1)f^{\prime\prime}(a)^{2}+nf^{\prime\prime\prime}(% a)\right)+\cdots
  41. f ( x ) = C x + D f(x)=Cx+D
  42. a = D / ( 1 - C ) a=D/(1-C)
  43. f n ( x ) = D 1 - C + ( x - D 1 - C ) C n = C n x + 1 - C n 1 - C D , f^{n}(x)=\frac{D}{1-C}+(x-\frac{D}{1-C})C^{n}=C^{n}x+\frac{1-C^{n}}{1-C}D~{},
  44. 2 2 2 \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdots}}}
  45. a = f ( 2 ) = 2 a=f(2)=2
  46. 2 2 2 = f n ( 1 ) = 2 - ( ln 2 ) n + ( ln 2 ) n + 1 ( ( ln 2 ) n - 1 ) 4 ( ln 2 - 1 ) - \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdots}}}=f^{n}(1)=2-(\ln 2)^{n}+\frac{(\ln 2)^{% n+1}((\ln 2)^{n}-1)}{4(\ln 2-1)}-\cdots
  47. a = f ( 4 ) = 4 a=f(4)=4
  48. n = 1 n=−1
  49. 2 l n x / l n 2 2lnx/ln2
  50. f n ( x ) = 1 + b n ( x - 1 ) + 1 2 ! b n ( b n - 1 ) ( x - 1 ) 2 + 1 3 ! b n ( b n - 1 ) ( b n - 2 ) ( x - 1 ) 3 + , f^{n}(x)=1+b^{n}(x-1)+\frac{1}{2!}b^{n}(b^{n}-1)(x-1)^{2}+\frac{1}{3!}b^{n}(b^% {n}-1)(b^{n}-2)(x-1)^{3}+\cdots~{},
  51. f f
  52. g g
  53. h h
  54. f f
  55. g g
  56. f ( x ) = x + 1 f(x)=x+1
  57. h h
  58. g ( ϕ ( y ) ) = ϕ ( y + 1 ) g(ϕ(y))=ϕ(y+1)
  59. x x
  60. f f
  61. Ψ Ψ
  62. f ( x ) f(x)
  63. g ( x ) = f ( 0 ) x g(x)=f(0)x
  64. f ( 0 ) 11 f(0)≠11
  65. n n
  66. n n
  67. Ψ Ψ
  68. f ( x ) = 4 x ( 1 x ) f(x)=4x(1−x)
  69. f ( x ) = 2 x ( 1 x ) f(x)=2x(1−x)
  70. Ψ ( x ) = ½ l n ( 1 2 x ) Ψ(x)=−½ln(1−2x)
  71. f f
  72. f ( x ) f(x)
  73. f n ( x ) f^{n}(x)
  74. x + b x+b
  75. x + n b x+nb
  76. a x + b ( a 1 ) ax+b\ (a\neq 1)
  77. a n x + a n - 1 a - 1 b a^{n}x+\frac{a^{n}-1}{a-1}b
  78. a x b ( b 1 ) ax^{b}\ (b\neq 1)
  79. a b n - 1 b - 1 x b n a^{\frac{b^{n}-1}{b-1}}x^{b^{n}}
  80. a x 2 + b x + b 2 - 2 b 4 a ax^{2}+bx+\frac{b^{2}-2b}{4a}
  81. 2 α 2 n - b 2 a \frac{2\alpha^{2^{n}}-b}{2a}
  82. α = 2 a x + b 2 \alpha=\frac{2ax+b}{2}
  83. a x 2 + b x + b 2 - 2 b - 8 4 a ax^{2}+bx+\frac{b^{2}-2b-8}{4a}
  84. 2 α 2 n + 2 α - 2 n - b 2 a \frac{2\alpha^{2^{n}}+2\alpha^{-2^{n}}-b}{2a}
  85. α = 2 a x + b ± ( 2 a x + b ) 2 - 16 4 \alpha=\frac{2ax+b\pm\sqrt{(2ax+b)^{2}-16}}{4}
  86. a x + b c x + d \frac{ax+b}{cx+d}
  87. a c + b c - a d c [ ( c x - a + α ) α n - 1 - ( c x - a + β ) β n - 1 ( c x - a + α ) α n - ( c x - a + β ) β n ] \frac{a}{c}+\frac{bc-ad}{c}\left[\frac{(cx-a+\alpha)\alpha^{n-1}-(cx-a+\beta)% \beta^{n-1}}{(cx-a+\alpha)\alpha^{n}-(cx-a+\beta)\beta^{n}}\right]
  88. α = a + d + ( a - d ) 2 + 4 b c 2 \alpha=\frac{a+d+\sqrt{(a-d)^{2}+4bc}}{2}
  89. β = a + d - ( a - d ) 2 + 4 b c 2 \beta=\frac{a+d-\sqrt{(a-d)^{2}+4bc}}{2}
  90. x 2 + b \sqrt{x^{2}+b}
  91. x 2 + b n \sqrt{x^{2}+bn}
  92. a x 2 + b ( a 1 ) \sqrt{ax^{2}+b}\ (a\neq 1)
  93. a n x 2 + a n - 1 a - 1 b \sqrt{a^{n}x^{2}+\frac{a^{n}-1}{a-1}b}
  94. g - 1 ( g ( x ) + b ) g^{-1}(g(x)+b)
  95. g - 1 ( g ( x ) + n b ) g^{-1}(g(x)+nb)
  96. g - 1 ( a g ( x ) + b ) ( a 1 ) g^{-1}(ag(x)+b)\ (a\neq 1)
  97. g - 1 ( a n g ( x ) + a n - 1 a - 1 b ) g^{-1}(a^{n}g(x)+\frac{a^{n}-1}{a-1}b)
  98. { b + 1 , i = a b g ( i ) } ( { i , x } { i + 1 , x + g ( i ) } ) b - a + 1 { a , 0 } \left\{b+1,\sum_{i=a}^{b}g(i)\right\}\equiv\left(\{i,x\}\rightarrow\{i+1,x+g(i% )\}\right)^{b-a+1}\{a,0\}
  99. { b + 1 , i = a b g ( i ) } ( { i , x } { i + 1 , x g ( i ) } ) b - a + 1 { a , 1 } \left\{b+1,\prod_{i=a}^{b}g(i)\right\}\equiv\left(\{i,x\}\rightarrow\{i+1,xg(i% )\}\right)^{b-a+1}\{a,1\}
  100. g ( f ( x ) ) g(f(x))
  101. v ( x ) = f n ( x ) / n | n = 0 v(x)=\partial f^{n}(x)/\partial n|_{n=0}
  102. n n
  103. f f
  104. g ( f ( x ) ) = exp [ v ( x ) x ] g ( x ) . g(f(x))=\exp\left[v(x)\dfrac{\partial}{\partial x}\right]g(x).
  105. f ( x ) = x + a f(x)=x+a
  106. v ( x ) = a v(x)=a
  107. g ( x + a ) = e x p ( a / x ) g ( x ) g(x+a)=exp(a∂/∂x)g(x)
  108. f ( x ) f(x)
  109. v ( x ) v(x)
  110. f ( x ) = h - 1 ( h ( x ) + 1 ) , f(x)=h^{-1}(h(x)+1),
  111. h ( x ) = 1 v ( x ) d x . h(x)=\int{\frac{1}{v(x)}dx}.
  112. f n ( x ) = h - 1 ( h ( x ) + n ) . f^{n}(x)=h^{-1}(h(x)+n)~{}.
  113. f < s u p > 1 / n f<sup>1/n
  114. f < s u p > n ( x ) = Ψ 1 ( ( l n 2 ) n Ψ ( x ) ) f<sup>n(x)=Ψ^{−1}((ln2)^{n}Ψ(x))

Itō_calculus.html

  1. Y t = 0 t H s d X s , Y_{t}=\int_{0}^{t}H_{s}\,dX_{s},
  2. Y t = 0 t H d X 0 t H s d X s , Y_{t}=\int_{0}^{t}H\,dX\equiv\int_{0}^{t}H_{s}\,dX_{s},
  3. ( Ω , , ( t ) t 0 , ) . (\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P}).
  4. 0 t H d B = lim n [ t i - 1 , t i ] π n H t i - 1 ( B t i - B t i - 1 ) . \int_{0}^{t}H\,dB=\lim_{n\rightarrow\infty}\sum_{[t_{i-1},t_{i}]\in\pi_{n}}H_{% t_{i-1}}(B_{t_{i}}-B_{t_{i-1}}).
  5. 0 t ( H - H n ) 2 d s 0 \int_{0}^{t}(H-H_{n})^{2}\,ds\to 0
  6. 0 t H d B = lim n 0 t H n d B \int_{0}^{t}H\,dB=\lim_{n\to\infty}\int_{0}^{t}H_{n}\,dB
  7. 𝔼 [ ( 0 t H s d B s ) 2 ] = 𝔼 [ 0 t H s 2 d s ] \mathbb{E}\left[\left(\int_{0}^{t}H_{s}\,dB_{s}\right)^{2}\right]=\mathbb{E}% \left[\int_{0}^{t}H_{s}^{2}\,ds\right]
  8. X t = X 0 + 0 t σ s d B s + 0 t μ s d s . X_{t}=X_{0}+\int_{0}^{t}\sigma_{s}\,dB_{s}+\int_{0}^{t}\mu_{s}\,ds.
  9. 0 t ( σ s 2 + | μ s | ) d s < \int_{0}^{t}(\sigma_{s}^{2}+|\mu_{s}|)\,ds<\infty
  10. 0 t H d X = 0 t H s σ s d B s + 0 t H s μ s d s . \int_{0}^{t}H\,dX=\int_{0}^{t}H_{s}\sigma_{s}\,dB_{s}+\int_{0}^{t}H_{s}\mu_{s}% \,ds.
  11. 0 t ( H 2 σ 2 + | H μ | ) d s < . \int_{0}^{t}(H^{2}\sigma^{2}+|H\mu|)ds<\infty.
  12. d f ( X t ) = f ( X t ) d X t + 1 2 f ′′ ( X t ) σ t 2 d t . df(X_{t})=f^{\prime}(X_{t})\,dX_{t}+\frac{1}{2}f^{\prime\prime}(X_{t})\sigma_{% t}^{2}\,dt.
  13. 0 t H d X = lim n t i - 1 , t i π n H t i - 1 ( X t i - X t i - 1 ) . \int_{0}^{t}H\,dX=\lim_{n\rightarrow\infty}\sum_{t_{i-1},t_{i}\in\pi_{n}}H_{t_% {i-1}}(X_{t_{i}}-X_{t_{i-1}}).
  14. 0 t H n d X 0 t H d X , \int_{0}^{t}H_{n}dX\to\int_{0}^{t}HdX,
  15. J ( K X ) = ( J K ) X J\cdot(K\cdot X)=(JK)\cdot X
  16. [ H X ] = H 2 [ X ] [H\cdot X]=H^{2}\cdot[X]
  17. X t Y t = X 0 Y 0 + 0 t X s - d Y s + 0 t Y s - d X s + [ X , Y ] t X_{t}Y_{t}=X_{0}Y_{0}+\int_{0}^{t}X_{s-}\,dY_{s}+\int_{0}^{t}Y_{s-}\,dX_{s}+[X% ,Y]_{t}
  18. d f ( X t ) = i = 1 d f i ( X t ) d X t i + 1 2 i , j = 1 d f i , j ( X t ) d [ X i , X j ] t . df(X_{t})=\sum_{i=1}^{d}f_{i}(X_{t})\,dX^{i}_{t}+\frac{1}{2}\sum_{i,j=1}^{d}f_% {i,j}(X_{t})\,d[X^{i},X^{j}]_{t}.
  19. 0 t H 2 d [ M ] < , \int_{0}^{t}H^{2}d[M]<\infty,
  20. 𝔼 [ ( H M t ) 2 ] = 𝔼 [ 0 t H 2 d [ M ] ] . \mathbb{E}\left[(H\cdot M_{t})^{2}\right]=\mathbb{E}\left[\int_{0}^{t}H^{2}\,d% [M]\right].
  21. c 𝔼 [ [ M ] t p 2 ] 𝔼 [ ( M t * ) p ] C 𝔼 [ [ M ] t p 2 ] c\mathbb{E}\left[[M]_{t}^{\frac{p}{2}}\right]\leq\mathbb{E}\left[(M^{*}_{t})^{% p}\right]\leq C\mathbb{E}\left[[M]_{t}^{\frac{p}{2}}\right]
  22. 𝔼 [ ( ( H M ) t * ) p ] C 𝔼 [ ( H 2 [ M ] t ) p 2 ] < \mathbb{E}\left[((H\cdot M)_{t}^{*})^{p}\right]\leq C\mathbb{E}\left[(H^{2}% \cdot[M]_{t})^{\frac{p}{2}}\right]<\infty
  23. H X t 𝟏 { t > T } A ( X t - X T ) . H\cdot X_{t}\equiv\mathbf{1}_{\{t>T\}}A(X_{t}-X_{T}).
  24. 𝔼 [ ( H B t ) 2 ] = 𝔼 [ 0 t H s 2 d s ] . \mathbb{E}\left[(H\cdot B_{t})^{2}\right]=\mathbb{E}\left[\int_{0}^{t}H_{s}^{2% }\,ds\right].
  25. 𝔼 [ 0 t H 2 d s ] < , \mathbb{E}\left[\int_{0}^{t}H^{2}ds\right]<\infty,
  26. 𝔼 [ ( H M t ) 2 ] = 𝔼 [ 0 t H s 2 d M s ] , \mathbb{E}\left[(H\cdot M_{t})^{2}\right]=\mathbb{E}\left[\int_{0}^{t}H^{2}_{s% }\,d\langle M\rangle_{s}\right],
  27. M t = M 0 + 0 t α s d B s M_{t}=M_{0}+\int_{0}^{t}\alpha_{s}\,\mathrm{d}B_{s}
  28. x ˙ k = h k + g k l ξ l , \dot{x}_{k}=h_{k}+g_{kl}\xi_{l},
  29. ξ j \xi_{j}
  30. ξ k ( t 1 ) ξ l ( t 2 ) = δ k l δ ( t 1 - t 2 ) \langle\xi_{k}(t_{1})\,\xi_{l}(t_{2})\rangle=\delta_{kl}\delta(t_{1}-t_{2})
  31. y = y ( x k ) y=y(x_{k})
  32. y ˙ = y x j x ˙ j + 1 2 2 y x k x l g k m g m l . \dot{y}=\frac{\partial y}{\partial x_{j}}\dot{x}_{j}+\frac{1}{2}\frac{\partial% ^{2}y}{\partial x_{k}\,\partial x_{l}}g_{km}g_{ml}.
  33. x ˙ k = h k + g k l ξ l - 1 2 g k l x m g m l . \dot{x}_{k}=h_{k}+g_{kl}\xi_{l}-\frac{1}{2}\frac{\partial g_{kl}}{\partial{x_{% m}}}g_{ml}.

Ivan_Mikheevich_Pervushin.html

  1. 2 2 12 + 1 2^{2^{12}}+1
  2. 7 * 2 14 + 1 = 114689 7*2^{14}+1=114689
  3. 2 2 23 + 1 2^{2^{23}}+1
  4. 5 * 2 25 + 1 = 167772161 5*2^{25}+1=167772161
  5. 2 61 - 1 2^{61}-1
  6. 2 127 - 1 2^{127}-1
  7. 2 89 - 1 2^{89}-1

Iverson_bracket.html

  1. [ P ] = { 1 if P is true; 0 otherwise. [P]=\begin{cases}1&\,\text{if }P\,\text{ is true;}\\ 0&\,\text{otherwise.}\end{cases}
  2. P P
  3. 𝐟𝐚𝐥𝐬𝐞 0 ; 𝐭𝐫𝐮𝐞 1 \,\textbf{false}\mapsto 0;\,\textbf{true}\mapsto 1
  4. ϕ ( n ) = i = 1 n [ gcd ( i , n ) = 1 ] , for n + . \phi(n)=\sum_{i=1}^{n}[\gcd(i,n)=1],\qquad\,\text{for }n\in\mathbb{N}^{+}.
  5. 1 i 10 i 2 = i i 2 [ 1 i 10 ] . \sum_{1\leq i\leq 10}i^{2}=\sum_{i}i^{2}[1\leq i\leq 10].
  6. i i
  7. 1 k n gcd ( k , n ) = 1 k = 1 2 n φ ( n ) \sum_{1\leq k\leq n\atop\gcd(k,n)=1}\!\!k=\frac{1}{2}n\varphi(n)
  8. n > 1 n>1
  9. 1 2 \frac{1}{2}
  10. n = 1 n=1
  11. ϕ ( n ) \phi(n)
  12. 1 k n gcd ( k , n ) = 1 k = 1 2 n ( φ ( n ) + [ n = 1 ] ) \sum_{1\leq k\leq n\atop\gcd(k,n)=1}\!\!k=\frac{1}{2}n(\varphi(n)+[n=1])
  13. δ i j = [ i = j ] . \delta_{ij}=[i=j].
  14. 𝟏 A ( x ) \mathbf{1}_{A}(x)
  15. 𝐈 A ( x ) \mathbf{I}_{A}(x)
  16. χ A ( x ) \chi_{A}(x)
  17. 𝐈 A ( x ) := [ x A ] \mathbf{I}_{A}(x):=[x\in A]
  18. sgn ( x ) = [ x > 0 ] - [ x < 0 ] \operatorname{sgn}(x)=[x>0]-[x<0]
  19. H ( x ) = [ x > 0 ] . H(x)=[x>0].
  20. max ( x , y ) = x [ x > y ] + y [ x y ] , \max(x,y)=x[x>y]+y[x\leq y],
  21. min ( x , y ) = x [ x y ] + y [ x > y ] , \min(x,y)=x[x\leq y]+y[x>y],
  22. | x | = x [ x 0 ] - x [ x < 0 ] . |x|=x[x\geq 0]-x[x<0].
  23. x = n = - n [ n x < n + 1 ] \lfloor x\rfloor=\sum_{n=-\infty}^{\infty}n[n\leq x<n+1]
  24. x = n = - n [ n - 1 < x n ] . \lceil x\rceil=\sum_{n=-\infty}^{\infty}n[n-1<x\leq n].
  25. { x } = x [ x 0 ] . \{x\}=x\cdot[x\geq 0].
  26. [ a < b ] + [ a = b ] + [ a > b ] = 1. [a<b]+[a=b]+[a>b]=1.

Jacobi_integral.html

  1. C J = n 2 ( x 2 + y 2 ) + 2 ( μ 1 r 1 + μ 2 r 2 ) - ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) C_{J}=n^{2}(x^{2}+y^{2})+2\left(\frac{\mu_{1}}{r_{1}}+\frac{\mu_{2}}{r_{2}}% \right)-\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)
  2. n = 2 π T n=\frac{2\pi}{T}
  3. μ 1 = G m 1 , μ 2 = G m 2 \mu_{1}=Gm_{1}\,\!,\mu_{2}=Gm_{2}\,\!
  4. r 1 , r 2 r_{1}\,\!,r_{2}\,\!
  5. C J = 2 ( μ 1 r 1 + μ 2 r 2 ) + 2 n ( ξ η ˙ - η ξ ˙ ) - ( ξ ˙ 2 + η ˙ 2 + ζ ˙ 2 ) . C_{J}=2\left(\frac{\mu_{1}}{r_{1}}+\frac{\mu_{2}}{r_{2}}\right)+2n\left(\xi% \dot{\eta}-\eta\dot{\xi}\right)-\left(\dot{\xi}^{2}+\dot{\eta}^{2}+\dot{\zeta}% ^{2}\right).
  6. U ( x , y , z ) = n 2 2 ( x 2 + y 2 ) + μ 1 r 1 + μ 2 r 2 U(x,y,z)=\frac{n^{2}}{2}(x^{2}+y^{2})+\frac{\mu_{1}}{r_{1}}+\frac{\mu_{2}}{r_{% 2}}
  7. x ¨ - 2 n y ˙ = δ U δ x \ddot{x}-2n\dot{y}=\frac{\delta U}{\delta x}
  8. y ¨ + 2 n x ˙ = δ U δ y \ddot{y}+2n\dot{x}=\frac{\delta U}{\delta y}
  9. z ¨ = δ U δ z \ddot{z}=\frac{\delta U}{\delta z}
  10. x ˙ , y ˙ \dot{x},\dot{y}
  11. z ˙ \dot{z}
  12. x ˙ x ¨ + y ˙ y ¨ + z ˙ z ¨ = δ U δ x x ˙ + δ U δ y y ˙ + δ U δ z z ˙ = d U d t \dot{x}\ddot{x}+\dot{y}\ddot{y}+\dot{z}\ddot{z}=\frac{\delta U}{\delta x}\dot{% x}+\frac{\delta U}{\delta y}\dot{y}+\frac{\delta U}{\delta z}\dot{z}=\frac{dU}% {dt}
  13. x ˙ 2 + y ˙ 2 + z ˙ 2 = 2 U - C J \dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}=2U-C_{J}

Jacobi_triple_product.html

  1. m = 1 ( 1 - x 2 m ) ( 1 + x 2 m - 1 y 2 ) ( 1 + x 2 m - 1 y 2 ) = n = - x n 2 y 2 n , \prod_{m=1}^{\infty}\left(1-x^{2m}\right)\left(1+x^{2m-1}y^{2}\right)\left(1+% \frac{x^{2m-1}}{y^{2}}\right)=\sum_{n=-\infty}^{\infty}x^{n^{2}}y^{2n},
  2. x = q q x=q\sqrt{q}
  3. y 2 = - q y^{2}=-\sqrt{q}
  4. ϕ ( q ) = m = 1 ( 1 - q m ) = n = - ( - 1 ) n q 3 n 2 - n 2 . \phi(q)=\prod_{m=1}^{\infty}\left(1-q^{m}\right)=\sum_{n=-\infty}^{\infty}(-1)% ^{n}q^{\frac{3n^{2}-n}{2}}.\,
  5. x = e i π τ x=e^{i\pi\tau}
  6. y = e i π z . y=e^{i\pi z}.
  7. ϑ ( z ; τ ) = n = - e π i n 2 τ + 2 π i n z \vartheta(z;\tau)=\sum_{n=-\infty}^{\infty}e^{\pi{\rm{i}}n^{2}\tau+2\pi{\rm{i}% }nz}
  8. n = - y 2 n x n 2 . \sum_{n=-\infty}^{\infty}y^{2n}x^{n^{2}}.
  9. ϑ ( z ; τ ) = m = 1 ( 1 - e 2 m π i τ ) [ 1 + e ( 2 m - 1 ) π i τ + 2 π i z ] [ 1 + e ( 2 m - 1 ) π i τ - 2 π i z ] . \vartheta(z;\tau)=\prod_{m=1}^{\infty}\left(1-e^{2m\pi{\rm{i}}\tau}\right)% \left[1+e^{(2m-1)\pi{\rm{i}}\tau+2\pi{\rm{i}}z}\right]\left[1+e^{(2m-1)\pi{\rm% {i}}\tau-2\pi{\rm{i}}z}\right].
  10. n = - q n ( n + 1 ) 2 z n = ( q ; q ) ( - 1 z ; q ) ( - z q ; q ) , \sum_{n=-\infty}^{\infty}q^{\frac{n(n+1)}{2}}z^{n}=(q;q)_{\infty}\;\left(-% \frac{1}{z};q\right)_{\infty}\;(-zq;q)_{\infty},
  11. ( a ; q ) (a;q)_{\infty}
  12. | a b | < 1 |ab|<1
  13. n = - a n ( n + 1 ) 2 b n ( n - 1 ) 2 = ( - a ; a b ) ( - b ; a b ) ( a b ; a b ) . \sum_{n=-\infty}^{\infty}a^{\frac{n(n+1)}{2}}\;b^{\frac{n(n-1)}{2}}=(-a;ab)_{% \infty}\;(-b;ab)_{\infty}\;(ab;ab)_{\infty}.
  14. n > 0 ( 1 + q n - 1 2 z ) ( 1 + q n - 1 2 z - 1 ) = ( l q l 2 / 2 z l ) ( n > 0 ( 1 - q n ) - 1 ) . \prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1})=\left(\sum_{l\in% \mathbb{Z}}q^{l^{2}/2}z^{l}\right)\left(\prod_{n>0}(1-q^{n})^{-1}\right).
  15. S S
  16. { v : v > 0 , v S } - { v : v < 0 , v S } \sum\{v\colon v>0,v\in S\}-\sum\{v\colon v<0,v\not\in S\}
  17. S S
  18. | { v : v > 0 , v S } | - | { v : v < 0 , v S } | . |\{v\colon v>0,v\in S\}|-|\{v\colon v<0,v\not\in S\}|.
  19. m , l s ( m , l ) q m z l \textstyle\sum_{m,l}s(m,l)q^{m}z^{l}
  20. s ( m , l ) s(m,l)
  21. m m
  22. l l
  23. n > 0 ( 1 + q n - 1 2 z ) ( 1 + q n - 1 2 z - 1 ) . \prod_{n>0}(1+q^{n-\frac{1}{2}}z)(1+q^{n-\frac{1}{2}}z^{-1}).
  24. l l
  25. l - l-
  26. { v : v < l } \{v\colon v<l\}
  27. λ 1 λ 2 λ j \lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{j}
  28. m m^{\prime}
  29. λ 1 \lambda_{1}
  30. λ 2 \lambda_{2}
  31. m + l 2 2 m^{\prime}+\frac{l^{2}}{2}
  32. ( l q l 2 / 2 z l ) ( n 0 p ( n ) q n ) = ( l q l 2 / 2 z l ) ( n > 0 ( 1 - q n ) - 1 ) \left(\sum_{l\in\mathbb{Z}}q^{l^{2}/2}z^{l}\right)\left(\sum_{n\geq 0}p(n)q^{n% }\right)=\left(\sum_{l\in\mathbb{Z}}q^{l^{2}/2}z^{l}\right)\left(\prod_{n>0}(1% -q^{n})^{-1}\right)
  33. p ( n ) p(n)

Jarzynski_equality.html

  1. Δ F = F B - F A \Delta F=F_{B}-F_{A}
  2. Δ F W \Delta F\leq W
  3. e - Δ F / k T = e - W / k T ¯ . e^{-\Delta F/kT}=\overline{e^{-W/kT}}.
  4. Δ F = W \Delta F=W
  5. Δ F \Delta F
  6. Δ F W ¯ , \Delta F\leq\overline{W},

Jean_Charles_Athanase_Peltier.html

  1. Q ˙ \dot{Q}
  2. Q ˙ = ( Π A - Π B ) I , \dot{Q}=\left(\Pi_{\mathrm{A}}-\Pi_{\mathrm{B}}\right)I,
  3. Π A \Pi_{A}
  4. Π B \Pi_{B}
  5. I I
  6. Π A \Pi_{A}
  7. Π B \Pi_{B}

Jeffreys_prior.html

  1. p ( θ ) det ( θ ) . p\left(\vec{\theta}\right)\propto\sqrt{\det\mathcal{I}\left(\vec{\theta}\right% )}.\,
  2. θ \vec{\theta}
  3. φ \varphi
  4. p ( φ ) I ( φ ) p(\varphi)\propto\sqrt{I(\varphi)}\,
  5. p ( θ ) I ( θ ) p(\theta)\propto\sqrt{I(\theta)}\,
  6. p ( φ ) \displaystyle p(\varphi)
  7. φ \vec{\varphi}
  8. p ( φ ) det I ( φ ) p(\vec{\varphi})\propto\sqrt{\det I(\vec{\varphi})}\,
  9. p ( θ ) det I ( θ ) p(\vec{\theta})\propto\sqrt{\det I(\vec{\theta})}\,
  10. p ( φ ) \displaystyle p(\vec{\varphi})
  11. θ \vec{\theta}
  12. θ \vec{\theta}
  13. θ \vec{\theta}
  14. x x
  15. f ( x μ ) = e - ( x - μ ) 2 / 2 σ 2 2 π σ 2 f(x\mid\mu)=\frac{e^{-(x-\mu)^{2}/2\sigma^{2}}}{\sqrt{2\pi\sigma^{2}}}
  16. σ \sigma
  17. μ \mu
  18. p ( μ ) \displaystyle p(\mu)
  19. μ \mu
  20. μ \mu
  21. x x
  22. f ( x σ ) = e - ( x - μ ) 2 / 2 σ 2 2 π σ 2 , f(x\mid\sigma)=\frac{e^{-(x-\mu)^{2}/2\sigma^{2}}}{\sqrt{2\pi\sigma^{2}}},
  23. μ \mu
  24. p ( σ ) \displaystyle p(\sigma)
  25. log σ = d σ / σ \log\sigma=\int d\sigma/\sigma
  26. n n
  27. f ( n λ ) = e - λ λ n n ! , f(n\mid\lambda)=e^{-\lambda}\frac{\lambda^{n}}{n!},
  28. p ( λ ) \displaystyle p(\lambda)
  29. λ = d λ / λ \sqrt{\lambda}=\int d\lambda/\sqrt{\lambda}
  30. γ H ( 1 - γ ) T \gamma^{H}(1-\gamma)^{T}
  31. γ \gamma
  32. p ( γ ) \displaystyle p(\gamma)
  33. α = β = 1 / 2 \alpha=\beta=1/2
  34. γ = sin 2 ( θ ) \gamma=\sin^{2}(\theta)
  35. θ \theta
  36. [ 0 , π / 2 ] [0,\pi/2]
  37. θ \theta
  38. [ 0 , 2 π ] [0,2\pi]
  39. N N
  40. γ = ( γ 1 , , γ N ) \vec{\gamma}=(\gamma_{1},\ldots,\gamma_{N})
  41. i = 1 N γ i = 1 \sum_{i=1}^{N}\gamma_{i}=1
  42. γ \vec{\gamma}
  43. γ i = ϕ i 2 \gamma_{i}={\phi_{i}}^{2}
  44. i i
  45. ϕ \vec{\phi}

Jellium.html

  1. H ^ = H ^ el + H ^ back + H ^ el - back , \hat{H}=\hat{H}_{\mathrm{el}}+\hat{H}_{\mathrm{back}}+\hat{H}_{\mathrm{el-back% }},\,
  2. H ^ el = i = 1 N p i 2 2 m + i < j N e 2 | 𝐫 i - 𝐫 j | \hat{H}_{\mathrm{el}}=\sum_{i=1}^{N}\frac{p_{i}^{2}}{2m}+\sum_{i<j}^{N}\frac{e% ^{2}}{|\mathbf{r}_{i}-\mathbf{r}_{j}|}
  3. H ^ back = e 2 2 Ω d 𝐑 Ω d 𝐑 n ( 𝐑 ) n ( 𝐑 ) | 𝐑 - 𝐑 | = e 2 2 ( N Ω ) 2 Ω d 𝐑 Ω d 𝐑 1 | 𝐑 - 𝐑 | \hat{H}_{\mathrm{back}}=\frac{e^{2}}{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{% \Omega}\mathrm{d}\mathbf{R}^{\prime}\ \frac{n(\mathbf{R})n(\mathbf{R}^{\prime}% )}{|\mathbf{R}-\mathbf{R}^{\prime}|}=\frac{e^{2}}{2}\left(\frac{N}{\Omega}% \right)^{2}\int_{\Omega}\mathrm{d}\mathbf{R}\int_{\Omega}\mathrm{d}\mathbf{R}^% {\prime}\ \frac{1}{|\mathbf{R}-\mathbf{R}^{\prime}|}
  4. H ^ el - back = Ω d 𝐫 Ω d 𝐑 ρ ( 𝐫 ) n ( 𝐑 ) | 𝐫 - 𝐑 | = - e 2 N Ω i = 1 N Ω d 𝐑 1 | 𝐫 i - 𝐑 | \hat{H}_{\mathrm{el-back}}=\int_{\Omega}\mathrm{d}\mathbf{r}\int_{\Omega}% \mathrm{d}\mathbf{R}\ \frac{\rho(\mathbf{r})n(\mathbf{R})}{|\mathbf{r}-\mathbf% {R}|}=-e^{2}\frac{N}{\Omega}\sum_{i=1}^{N}\int_{\Omega}\mathrm{d}\mathbf{R}\ % \frac{1}{|\mathbf{r}_{i}-\mathbf{R}|}
  5. K E = 3 5 E F = 3 5 2 k F 2 2 m e = 2.21 r s 2 Ryd KE=\frac{3}{5}E_{F}=\frac{3}{5}\frac{\hbar^{2}k_{F}^{2}}{2m_{e}}=\frac{2.21}{r% _{s}^{2}}\textrm{Ryd}
  6. E F E_{F}
  7. k F k_{F}
  8. r s r_{s}
  9. 1 / r 12 1/r_{12}
  10. r s r_{s}
  11. 1 / r s 2 1/r_{s}^{2}
  12. 1 / r s 1/r_{s}
  13. r s r_{s}
  14. E = 2.21 r s 2 - 0.916 r s E=\frac{2.21}{r_{s}^{2}}-\frac{0.916}{r_{s}}
  15. r s r_{s}
  16. E = 2.21 r s 2 - 0.916 r s + 0.0622 ln ( r s ) - 0.096 + O ( r s ) E=\frac{2.21}{r_{s}^{2}}-\frac{0.916}{r_{s}}+0.0622\ln(r_{s})-0.096+O(r_{s})
  17. r s r_{s}
  18. r s r_{s}

Jet_(mathematics).html

  1. f : f:{\mathbb{R}}\rightarrow{\mathbb{R}}
  2. x 0 x_{0}
  3. f ( x ) = f ( x 0 ) + f ( x 0 ) ( x - x 0 ) + + f ( k ) ( x 0 ) k ! ( x - x 0 ) k + R k + 1 ( x ) ( k + 1 ) ! ( x - x 0 ) k + 1 f(x)=f(x_{0})+f^{\prime}(x_{0})(x-x_{0})+\cdots+\frac{f^{(k)}(x_{0})}{k!}(x-x_% {0})^{k}+\frac{R_{k+1}(x)}{(k+1)!}(x-x_{0})^{k+1}
  4. | R k + 1 ( x ) | sup x U | f ( k + 1 ) ( x ) | . |R_{k+1}(x)|\leq\sup_{x\in U}|f^{(k+1)}(x)|.
  5. x 0 x_{0}
  6. ( J x 0 k f ) ( z ) = f ( x 0 ) + f ( x 0 ) z + + f ( k ) ( x 0 ) k ! z k . (J^{k}_{x_{0}}f)(z)=f(x_{0})+f^{\prime}(x_{0})z+\cdots+\frac{f^{(k)}(x_{0})}{k% !}z^{k}.
  7. x 0 x_{0}
  8. f : n m f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}
  9. f ( x ) = f ( x 0 ) + ( D f ( x 0 ) ) ( x - x 0 ) + 1 2 ( D 2 f ( x 0 ) ) ( x - x 0 ) 2 + + D k f ( x 0 ) k ! ( x - x 0 ) k + R k + 1 ( x ) ( k + 1 ) ! ( x - x 0 ) ( k + 1 ) . f(x)=f(x_{0})+(Df(x_{0}))\cdot(x-x_{0})+\frac{1}{2}(D^{2}f(x_{0}))\cdot(x-x_{0% })^{\otimes 2}+\cdots+\frac{D^{k}f(x_{0})}{k!}\cdot(x-x_{0})^{\otimes k}+\frac% {R_{k+1}(x)}{(k+1)!}\cdot(x-x_{0})^{\otimes(k+1)}.
  10. ( J x 0 k f ) ( z ) = f ( x 0 ) + ( D f ( x 0 ) ) z + 1 2 ( D 2 f ( x 0 ) ) z 2 + + D k f ( x 0 ) k ! z k (J^{k}_{x_{0}}f)(z)=f(x_{0})+(Df(x_{0}))\cdot z+\frac{1}{2}(D^{2}f(x_{0}))% \cdot z^{\otimes 2}+\cdots+\frac{D^{k}f(x_{0})}{k!}\cdot z^{\otimes k}
  11. [ z ] {\mathbb{R}}[z]
  12. z = ( z 1 , , z n ) z=(z_{1},\ldots,z_{n})
  13. f , g : n f,g:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}
  14. J x 0 k f J x 0 k g = J x 0 k ( f g ) J^{k}_{x_{0}}f\cdot J^{k}_{x_{0}}g=J^{k}_{x_{0}}(f\cdot g)
  15. z k + 1 z^{k+1}
  16. [ z ] / ( z k + 1 ) {\mathbb{R}}[z]/(z^{k+1})
  17. ( z k + 1 ) (z^{k+1})
  18. f : m f:{\mathbb{R}}^{m}\rightarrow{\mathbb{R}}^{\ell}
  19. g : n m g:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}
  20. f g : n f\circ g:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{\ell}
  21. J 0 k f J 0 k g = J 0 k ( f g ) . J^{k}_{0}f\circ J^{k}_{0}g=J^{k}_{0}(f\circ g).
  22. > k >k
  23. f ( x ) = log ( 1 - x ) f(x)=\log(1-x)
  24. g ( x ) = sin x g(x)=\sin\,x
  25. ( J 0 3 f ) ( x ) = - x - x 2 2 - x 3 3 (J^{3}_{0}f)(x)=-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}
  26. ( J 0 3 g ) ( x ) = x - x 3 6 (J^{3}_{0}g)(x)=x-\frac{x^{3}}{6}
  27. ( J 0 3 f ) ( J 0 3 g ) = - ( x - x 3 6 ) - 1 2 ( x - x 3 6 ) 2 - 1 3 ( x - x 3 6 ) 3 ( mod x 4 ) (J^{3}_{0}f)\circ(J^{3}_{0}g)=-\left(x-\frac{x^{3}}{6}\right)-\frac{1}{2}\left% (x-\frac{x^{3}}{6}\right)^{2}-\frac{1}{3}\left(x-\frac{x^{3}}{6}\right)^{3}\ % \ (\hbox{mod}\ x^{4})
  28. = - x - x 2 2 - x 3 6 =-x-\frac{x^{2}}{2}-\frac{x^{3}}{6}
  29. C ( n , m ) C^{\infty}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  30. f : n m f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}
  31. n {\mathbb{R}}^{n}
  32. E p k E_{p}^{k}
  33. f g f\sim g\,\!
  34. f - g = 0 f-g=0
  35. C ( n , m ) C^{\infty}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  36. E p k E^{k}_{p}
  37. J p k ( n , m ) J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  38. f C ( n , m ) f\in C^{\infty}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  39. J p k ( n , m ) J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  40. C p ( n , m ) C_{p}^{\infty}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  41. f : n m f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}
  42. n {\mathbb{R}}^{n}
  43. 𝔪 p {\mathfrak{m}}_{p}
  44. C p ( n , m ) C_{p}^{\infty}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  45. 𝔪 p k + 1 {\mathfrak{m}}_{p}^{k+1}
  46. J p k ( n , m ) = C p ( n , m ) / 𝔪 p k + 1 J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})=C_{p}^{\infty}({\mathbb{R}}^{n},{% \mathbb{R}}^{m})/{\mathfrak{m}}_{p}^{k+1}
  47. f : n m f:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{m}
  48. J p k ( n , m ) J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  49. J p k f = f ( mod 𝔪 p k + 1 ) J^{k}_{p}f=f\ (\hbox{mod}\ {\mathfrak{m}}_{p}^{k+1})
  50. J p k ( n , m ) J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  51. m [ z ] / ( z k + 1 ) {\mathbb{R}}^{m}[z]/(z^{k+1})
  52. J p k ( n , m ) J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})
  53. p n p\in{\mathbb{R}}^{n}
  54. J p k ( n , m ) q = { J k f J p k ( n , m ) | f ( p ) = q } J^{k}_{p}({\mathbb{R}}^{n},{\mathbb{R}}^{m})_{q}=\left\{J^{k}f\in J^{k}_{p}({% \mathbb{R}}^{n},{\mathbb{R}}^{m})|f(p)=q\right\}
  55. f : M N f:M\rightarrow N
  56. f : M f:{\mathbb{R}}\rightarrow M
  57. E p k E_{p}^{k}
  58. φ : U \varphi:U\rightarrow{\mathbb{R}}
  59. J 0 k ( φ f ) = J 0 k ( φ g ) J^{k}_{0}(\varphi\circ f)=J^{k}_{0}(\varphi\circ g)
  60. φ f \varphi\circ f
  61. φ g \varphi\circ g
  62. E p k E^{k}_{p}
  63. J k f J^{k}\!f\,
  64. J 0 k f J^{k}_{0}f
  65. J 0 k ( , M ) p J^{k}_{0}({\mathbb{R}},M)_{p}
  66. J 0 k ( , M ) p J^{k}_{0}({\mathbb{R}},M)_{p}
  67. J 0 k ( , M ) p J^{k}_{0}({\mathbb{R}},M)_{p}
  68. ( x i ) : M \R n (x^{i}):M\rightarrow\R^{n}
  69. E p k E_{p}^{k}
  70. J 0 k ( ( x i ) f ) = J 0 k ( ( x i ) g ) J^{k}_{0}\left((x^{i})\circ f\right)=J^{k}_{0}\left((x^{i})\circ g\right)
  71. {\mathbb{R}}
  72. E p k E_{p}^{k}
  73. J 0 k ( x i f ) = J 0 k ( x i g ) J^{k}_{0}(x^{i}\circ f)=J^{k}_{0}(x^{i}\circ g)
  74. φ ( Q ) = ψ ( x 1 ( Q ) , , x n ( Q ) ) \varphi(Q)=\psi(x^{1}(Q),\dots,x^{n}(Q))
  75. φ f = ψ ( x 1 f , , x n f ) \varphi\circ f=\psi(x^{1}\circ f,\dots,x^{n}\circ f)
  76. φ g = ψ ( x 1 g , , x n g ) \varphi\circ g=\psi(x^{1}\circ g,\dots,x^{n}\circ g)
  77. d d t ( ψ f ) ( t ) | t = 0 = i = 1 n d d t ( x i f ) ( t ) | t = 0 ( D i ψ ) f ( 0 ) \left.\frac{d}{dt}\left(\psi\circ f\right)(t)\right|_{t=0}=\sum_{i=1}^{n}\left% .\frac{d}{dt}(x^{i}\circ f)(t)\right|_{t=0}\ (D_{i}\psi)\circ f(0)
  78. ( y i ) : M n (y^{i}):M\rightarrow{\mathbb{R}}^{n}
  79. ρ = ( x i ) ( y i ) - 1 : n n \rho=(x^{i})\circ(y^{i})^{-1}:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{n}
  80. n {\mathbb{R}}^{n}
  81. J 0 k ρ : J 0 k ( n , n ) J 0 k ( n , n ) J^{k}_{0}\rho:J^{k}_{0}({\mathbb{R}}^{n},{\mathbb{R}}^{n})\rightarrow J^{k}_{0% }({\mathbb{R}}^{n},{\mathbb{R}}^{n})
  82. ρ - 1 \rho^{-1}
  83. I = J 0 k I = J 0 k ( ρ ρ - 1 ) = J 0 k ( ρ ) J 0 k ( ρ - 1 ) I=J^{k}_{0}I=J^{k}_{0}(\rho\circ\rho^{-1})=J^{k}_{0}(\rho)\circ J^{k}_{0}(\rho% ^{-1})
  84. J 0 k ρ J^{k}_{0}\rho
  85. v = i v i x i v=\sum_{i}v^{i}\frac{\partial}{\partial x^{i}}
  86. x i f ( t ) = t v i x^{i}\circ f(t)=tv^{i}
  87. φ f : \varphi\circ f:{\mathbb{R}}\rightarrow{\mathbb{R}}
  88. J 0 1 ( φ f ) ( t ) = t v i f x i ( p ) J^{1}_{0}(\varphi\circ f)(t)=tv^{i}\frac{\partial f}{\partial x^{i}}(p)
  89. x i ( t ) = t d x i d t ( 0 ) + t 2 2 d 2 x i d t 2 . x^{i}(t)=t\frac{dx^{i}}{dt}(0)+\frac{t^{2}}{2}\frac{d^{2}x^{i}}{dt^{2}}.
  90. ( x ˙ i , x ¨ i ) (\dot{x}^{i},\ddot{x}^{i})
  91. d d t y i ( x ( t ) ) = y i x j ( x ( t ) ) d x j d t ( t ) \frac{d}{dt}y^{i}(x(t))=\frac{\partial y^{i}}{\partial x^{j}}(x(t))\frac{dx^{j% }}{dt}(t)
  92. d 2 d t 2 y i ( x ( t ) ) = 2 y i x j x k ( x ( t ) ) d x j d t ( t ) d x k d t ( t ) + y i x j ( x ( t ) ) d 2 x j d t 2 ( t ) \frac{d^{2}}{dt^{2}}y^{i}(x(t))=\frac{\partial^{2}y^{i}}{\partial x^{j}% \partial x^{k}}(x(t))\frac{dx^{j}}{dt}(t)\frac{dx^{k}}{dt}(t)+\frac{\partial y% ^{i}}{\partial x^{j}}(x(t))\frac{d^{2}x^{j}}{dt^{2}}(t)
  93. y ˙ i = y i x j ( 0 ) x ˙ j \dot{y}^{i}=\frac{\partial y^{i}}{\partial x^{j}}(0)\dot{x}^{j}
  94. y ¨ i = 2 y i x j x k ( 0 ) x ˙ j x ˙ k + y i x j ( 0 ) x ¨ k . \ddot{y}^{i}=\frac{\partial^{2}y^{i}}{\partial x^{j}\partial x^{k}}(0)\dot{x}^% {j}\dot{x}^{k}+\frac{\partial y^{i}}{\partial x^{j}}(0)\ddot{x}^{k}.
  95. C p ( M , N ) C^{\infty}_{p}(M,N)
  96. f : M N f:M\rightarrow N
  97. E p k E^{k}_{p}
  98. C p ( M , N ) C^{\infty}_{p}(M,N)
  99. γ : M \gamma:{\mathbb{R}}\rightarrow M
  100. γ ( 0 ) = p \gamma(0)=p
  101. J 0 k ( f γ ) = J 0 k ( g γ ) J^{k}_{0}(f\circ\gamma)=J^{k}_{0}(g\circ\gamma)
  102. J p k ( M , N ) J^{k}_{p}(M,N)
  103. C p ( M , N ) C^{\infty}_{p}(M,N)
  104. E p k E^{k}_{p}
  105. J p k ( M , N ) J^{k}_{p}(M,N)
  106. f : M N f:M\rightarrow N
  107. J p k f J^{k}_{p}f
  108. E p k E^{k}_{p}
  109. π : E M \pi:E\rightarrow M
  110. s : M E s:M\rightarrow E
  111. π s \pi\circ s
  112. J p k ( M , E ) J^{k}_{p}(M,E)
  113. J p k ( M , E ) J^{k}_{p}(M,E)
  114. v = v i ( x ) / x i v=v^{i}(x)\partial/\partial x^{i}
  115. v i ( x ) = v i ( 0 ) + x j v i x j ( 0 ) = v i + v j i x j . v^{i}(x)=v^{i}(0)+x^{j}\frac{\partial v^{i}}{\partial x^{j}}(0)=v^{i}+v^{i}_{j% }x^{j}.
  116. ( v i , v j i ) (v^{i},v^{i}_{j})
  117. ( v i , v j i ) (v^{i},v^{i}_{j})
  118. ( w i , w j i ) (w^{i},w^{i}_{j})
  119. v = w k ( y ) / y k = v i ( x ) / x i , v=w^{k}(y)\partial/\partial y^{k}=v^{i}(x)\partial/\partial x^{i},
  120. w k ( y ) = v i ( x ) y k x i ( x ) . w^{k}(y)=v^{i}(x)\frac{\partial y^{k}}{\partial x^{i}}(x).
  121. w k ( 0 ) + y j w k y j ( 0 ) = ( v i ( 0 ) + x j v i x j ) y k x i ( x ) w^{k}(0)+y^{j}\frac{\partial w^{k}}{\partial y^{j}}(0)=\left(v^{i}(0)+x^{j}% \frac{\partial v^{i}}{\partial x^{j}}\right)\frac{\partial y^{k}}{\partial x^{% i}}(x)
  122. w k = y k x i ( 0 ) v i w^{k}=\frac{\partial y^{k}}{\partial x^{i}}(0)v^{i}
  123. w j k = v i 2 y k x i x j + v j i y k x i . w^{k}_{j}=v^{i}\frac{\partial^{2}y^{k}}{\partial x^{i}\partial x^{j}}+v_{j}^{i% }\frac{\partial y^{k}}{\partial x^{i}}.

John_Earman.html

  1. d d
  2. M M
  3. H H
  4. M M
  5. H H
  6. d d
  7. d d
  8. d d
  9. t = 0 t=0
  10. t t
  11. t = 0 t=0

Johnson's_algorithm.html

  1. q q
  2. q q
  3. v v
  4. h ( v ) h(v)
  5. q q
  6. v v
  7. u u
  8. v v
  9. w ( u , v ) w(u,v)
  10. w ( u , v ) + h ( u ) h ( v ) w(u,v)+h(u)−h(v)
  11. q q
  12. s s
  13. q q
  14. q q
  15. h ( v ) h(v)
  16. q q
  17. q q
  18. w ( u , v ) w(u,v)
  19. w ( u , v ) + h ( u ) h ( v ) w(u,v)+h(u)−h(v)
  20. s s
  21. t t
  22. h ( s ) h ( t ) h(s)−h(t)
  23. ( w ( s , p 1 ) + h ( s ) - h ( p 1 ) ) + ( w ( p 1 , p 2 ) + h ( p 1 ) - h ( p 2 ) ) + + ( w ( p n , t ) + h ( p n ) - h ( t ) ) . \bigl(w(s,p1)+h(s)-h(p1)\bigr)+\bigl(w(p1,p2)+h(p1)-h(p2)\bigr)+...+\bigl(w(p_% {n},t)+h(p_{n})-h(t)\bigr).
  24. + h ( p i ) +h(p_{i})
  25. - h ( p i ) -h(p_{i})
  26. ( w ( s , p 1 ) + w ( p 1 , p 2 ) + + w ( p n , t ) ) + h ( s ) - h ( t ) \bigl(w(s,p1)+w(p1,p2)+...+w(p_{n},t)\bigr)+h(s)-h(t)
  27. O ( V < s u p > 2 l o g V + V E ) O(V<sup>2logV+VE)

Josephus_problem.html

  1. n n
  2. k k
  3. k - 1 k-1
  4. k k
  5. 1 1
  6. n n
  7. k = 2 k=2
  8. k 2 k\neq 2
  9. f ( n ) f(n)
  10. n n
  11. k = 2 k=2
  12. x x
  13. 2 x - 1 2x-1
  14. x x
  15. n = 2 j n=2j
  16. f ( j ) f(j)
  17. 2 f ( j ) - 1 2f(j)-1
  18. f ( 2 j ) = 2 f ( j ) - 1 . f(2j)=2f(j)-1\;.
  19. x x
  20. 2 x + 1 2x+1
  21. f ( 2 j + 1 ) = 2 f ( j ) + 1 . f(2j+1)=2f(j)+1\;.
  22. n n
  23. f ( n ) f(n)
  24. n n
  25. f ( n ) f(n)
  26. f ( n ) f(n)
  27. f ( n ) = 1 f(n)=1
  28. n = 2 m + l n=2^{m}+l
  29. 0 l < 2 m 0\leq l<2^{m}
  30. f ( n ) = 2 l + 1 f(n)=2\cdot l+1
  31. l l
  32. 2 m 2^{m}
  33. 2 l + 1 2l+1
  34. f ( n ) = 2 l + 1 f(n)=2l+1
  35. n = 2 m + l n=2^{m}+l
  36. 0 l < 2 m 0\leq l<2^{m}
  37. f ( n ) = 2 l + 1 f(n)=2l+1
  38. n n
  39. n = 1 n=1
  40. n n
  41. n n
  42. n n
  43. l 1 l_{1}
  44. m 1 m_{1}
  45. n / 2 = 2 m 1 + l 1 n/2=2^{m_{1}}+l_{1}
  46. 0 l 1 < 2 m 1 0\leq l_{1}<2^{m_{1}}
  47. l 1 = l / 2 l_{1}=l/2
  48. f ( n ) = 2 f ( n / 2 ) - 1 = 2 ( ( 2 l 1 ) + 1 ) - 1 = 2 l + 1 f(n)=2f(n/2)-1=2((2l_{1})+1)-1=2l+1
  49. n n
  50. l 1 l_{1}
  51. m 1 m_{1}
  52. ( n - 1 ) / 2 = 2 m 1 + l 1 (n-1)/2=2^{m_{1}}+l_{1}
  53. 0 l 1 < 2 m 1 0\leq l_{1}<2^{m_{1}}
  54. l 1 = ( l - 1 ) / 2 l_{1}=(l-1)/2
  55. f ( n ) = 2 f ( ( n - 1 ) / 2 ) + 1 = 2 ( ( 2 l 1 ) + 1 ) + 1 = 2 l + 1 f(n)=2f((n-1)/2)+1=2((2l_{1})+1)+1=2l+1
  56. l l
  57. f ( n ) f(n)
  58. f ( n ) = 2 ( n - 2 log 2 ( n ) ) + 1 f(n)=2(n-2^{\lfloor\log_{2}(n)\rfloor})+1
  59. n n
  60. f ( n ) f(n)
  61. n n
  62. n n
  63. n = 1 b 1 b 2 b 3 b m n=1b_{1}b_{2}b_{3}\dots b_{m}
  64. f ( n ) = b 1 b 2 b 3 b m 1 f(n)=b_{1}b_{2}b_{3}\dots b_{m}1
  65. n n
  66. 2 m + l 2^{m}+l
  67. f ( n ) f(n)
  68. s s
  69. ( ( s - 1 ) mod n ) + 1 ((s-1)\bmod n)+1
  70. f ( n , k ) f(n,k)
  71. k k
  72. n - 1 n-1
  73. ( k mod n ) + 1 (k\bmod n)+1
  74. f ( n - 1 , k ) f(n-1,k)
  75. 1 1
  76. ( k mod n ) + 1 (k\bmod n)+1
  77. f ( n , k ) = ( ( f ( n - 1 , k ) + k - 1 ) mod n ) + 1 , with f ( 1 , k ) = 1 , f(n,k)=((f(n-1,k)+k-1)\bmod n)+1,\,\text{ with }f(1,k)=1\,,
  78. g ( n , k ) = ( g ( n - 1 , k ) + k ) mod n , with g ( 1 , k ) = 0 g(n,k)=(g(n-1,k)+k)\bmod n,\,\text{ with }g(1,k)=0
  79. 0
  80. n - 1 n-1
  81. O ( n ) O(n)
  82. k k
  83. n n
  84. O ( k log n ) O(k\log n)
  85. ( n / k k ) (\lfloor n/k\rfloor k)
  86. m m
  87. n n
  88. p p
  89. x x
  90. p + m x p+mx
  91. n + x n+x
  92. x x
  93. ( p + m x ) > ( n + x ) (p+mx)>(n+x)
  94. ( p + m x ) - ( n + x ) (p+mx)-(n+x)

Joule_expansion.html

  1. S = n R ln [ ( V N ) ( 4 π m 3 h 2 U N ) 3 2 ] + 5 2 n R . S=nR\ln\left[\left(\frac{V}{N}\right)\left(\frac{4\pi m}{3h^{2}}\frac{U}{N}% \right)^{\frac{3}{2}}\right]+{\frac{5}{2}}nR.
  2. S ( V , T ) = S 0 + n R ln ( V V 0 ) + n C V ln ( T T 0 ) S(V,T)=S_{0}+nR\ln\left(\frac{V}{V_{0}}\right)+nC_{V}\ln\left(\frac{T}{T_{0}}\right)
  3. d U = T d S - P d V . \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V.
  4. Δ S = i f d S = V 0 2 V 0 P d V T = V 0 2 V 0 n R d V V = n R ln 2. \Delta S=\int_{i}^{f}\mathrm{d}S=\int_{V_{0}}^{2V_{0}}\frac{P\,\mathrm{d}V}{T}% =\int_{V_{0}}^{2V_{0}}\frac{nR\,\mathrm{d}V}{V}=nR\ln 2.
  5. T = T i 2 - R / C V = T i 2 - 2 / 3 T=T_{i}2^{-R/C_{V}}=T_{i}2^{-2/3}
  6. Δ S = n T T i C V d T T = n R ln 2. \Delta S=n\int_{T}^{T_{i}}C_{V}\frac{\mathrm{d}T^{\prime}}{T^{\prime}}=nR\ln 2.
  7. W = - 2 V 0 V 0 P d V = - 2 V 0 V 0 n R T V d V = n R T ln 2 = T Δ S g a s . W=-\int_{2V_{0}}^{V_{0}}P\,\mathrm{d}V=-\int_{2V_{0}}^{V_{0}}\frac{nRT}{V}% \mathrm{d}V=nRT\ln 2=T\Delta S_{gas}.

Jounce.html

  1. s = d j d t = d 2 a d t 2 = d 3 v d t 3 = d 4 r d t 4 \vec{s}=\frac{d\vec{j}}{dt}=\frac{d^{2}\vec{a}}{dt^{2}}=\frac{d^{3}\vec{v}}{dt% ^{3}}=\frac{d^{4}\vec{r}}{dt^{4}}
  2. j = j 0 + s t \vec{j}=\vec{j}_{0}+\vec{s}\,t
  3. a = a 0 + j 0 t + 1 2 s t 2 \vec{a}=\vec{a}_{0}+\vec{j}_{0}\,t+\frac{1}{2}\vec{s}\,t^{2}
  4. v = v 0 + a 0 t + 1 2 j 0 t 2 + 1 6 s t 3 \vec{v}=\vec{v}_{0}+\vec{a}_{0}\,t+\frac{1}{2}\vec{j}_{0}\,t^{2}+\frac{1}{6}% \vec{s}\,t^{3}
  5. r = r 0 + v 0 t + 1 2 a 0 t 2 + 1 6 j 0 t 3 + 1 24 s t 4 \vec{r}=\vec{r}_{0}+\vec{v}_{0}\,t+\frac{1}{2}\vec{a}_{0}\,t^{2}+\frac{1}{6}% \vec{j}_{0}\,t^{3}+\frac{1}{24}\vec{s}\,t^{4}
  6. s \vec{s}
  7. j 0 \vec{j}_{0}
  8. j \vec{j}
  9. a 0 \vec{a}_{0}
  10. a \vec{a}
  11. v 0 \vec{v}_{0}
  12. v \vec{v}
  13. r 0 \vec{r}_{0}
  14. r \vec{r}
  15. t t
  16. s \vec{s}

K-d_tree.html

  1. t worst = O ( k N 1 - 1 k ) t\text{worst}=O(k\cdot N^{1-\frac{1}{k}})
  2. O ( 1 ϵ d log n ) O\left(\frac{1}{{\epsilon\ }^{d}}\log n\right)
  3. O ( log n + ( 1 ϵ ) d ) O\left(\log n+{\left(\frac{1}{\epsilon\ }\right)}^{d}\right)

K-means_clustering.html

  1. arg min 𝐒 i = 1 k 𝐱 S i 𝐱 - s y m b o l μ i 2 \underset{\mathbf{S}}{\operatorname{arg\,min}}\sum_{i=1}^{k}\sum_{\mathbf{x}% \in S_{i}}\left\|\mathbf{x}-symbol\mu_{i}\right\|^{2}
  2. S i ( t ) = { x p : x p - m i ( t ) 2 x p - m j ( t ) 2 j , 1 j k } , S_{i}^{(t)}=\big\{x_{p}:\big\|x_{p}-m^{(t)}_{i}\big\|^{2}\leq\big\|x_{p}-m^{(t% )}_{j}\big\|^{2}\ \forall j,1\leq j\leq k\big\},
  3. x p x_{p}
  4. S ( t ) S^{(t)}
  5. m i ( t + 1 ) = 1 | S i ( t ) | x j S i ( t ) x j m^{(t+1)}_{i}=\frac{1}{|S^{(t)}_{i}|}\sum_{x_{j}\in S^{(t)}_{i}}x_{j}
  6. O ( n d k + 1 log n ) O(n^{dk+1}\log{n})
  7. O ( n k d i ) O(nkdi)
  8. [ 0 , 1 ] d [0,1]^{d}
  9. 0
  10. σ 2 \sigma^{2}
  11. k k
  12. O ( n 34 k 34 d 8 l o g 4 ( n ) / σ 6 ) O(n^{34}k^{34}d^{8}log^{4}(n)/\sigma^{6})
  13. n n
  14. k k
  15. d d
  16. 1 / σ 1/\sigma
  17. O ( d n 4 M 2 ) O(dn^{4}M^{2})
  18. n n
  19. { 1 , , M } d \{1,\dots,M\}^{d}
  20. L 1 L_{1}
  21. k = 3 k=3
  22. k = 2 k=2
  23. k = 3 k=3
  24. k = 2 k=2
  25. < v a r > k <var>k

Kazhdan's_property_(T).html

  1. g K : π ( g ) ξ - ξ < ε . \forall g\in K\ :\ \left\|\pi(g)\xi-\xi\right\|<\varepsilon.

KEKB_(accelerator).html

  1. β γ \beta\gamma
  2. 2.11 × 10 34 cm - 2 s - 1 2.11\times 10^{34}\mathrm{cm}^{-2}\mathrm{s}^{-1}

Kerr–Newman_metric.html

  1. g μ ν g_{\mu\nu}\!
  2. c 2 d τ 2 = - ( d r 2 Δ + d θ 2 ) ρ 2 + ( c d t - α sin 2 θ d ϕ ) 2 Δ ρ 2 - ( ( r 2 + α 2 ) d ϕ - α c d t ) 2 sin 2 θ ρ 2 c^{2}d\tau^{2}=-\left(\frac{dr^{2}}{\Delta}+d\theta^{2}\right)\rho^{2}+\left(c% \,dt-\alpha\sin^{2}\theta\,d\phi\right)^{2}\frac{\Delta}{\rho^{2}}-\left(\left% (r^{2}+\alpha^{2}\right)d\phi-\alpha c\,dt\right)^{2}\frac{\sin^{2}\theta}{% \rho^{2}}
  3. α = J M c , \alpha=\frac{J}{Mc}\,,
  4. ρ 2 = r 2 + α 2 cos 2 θ , \ \rho^{2}=r^{2}+\alpha^{2}\cos^{2}\theta\,,
  5. Δ = r 2 - r s r + α 2 + r Q 2 , \ \Delta=r^{2}-r_{s}r+\alpha^{2}+r_{Q}^{2}\,,
  6. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}
  7. r Q 2 = Q 2 G 4 π ϵ 0 c 4 r_{Q}^{2}=\frac{Q^{2}G}{4\pi\epsilon_{0}c^{4}}
  8. c 2 d τ 2 \displaystyle c^{2}d\tau^{2}
  9. g μ ν = η μ ν + f k μ k ν g_{\mu\nu}=\eta_{\mu\nu}+fk_{\mu}k_{\nu}\!
  10. f = G r 2 r 4 + a 2 z 2 [ 2 M r - Q 2 ] f=\frac{Gr^{2}}{r^{4}+a^{2}z^{2}}\left[2Mr-Q^{2}\right]
  11. 𝐤 = ( k x , k y , k z ) = ( r x + a y r 2 + a 2 , r y - a x r 2 + a 2 , z r ) \mathbf{k}=(k_{x},k_{y},k_{z})=\left(\frac{rx+ay}{r^{2}+a^{2}},\frac{ry-ax}{r^% {2}+a^{2}},\frac{z}{r}\right)
  12. k 0 = 1. k_{0}=1.\!
  13. a \vec{a}
  14. 1 = x 2 + y 2 r 2 + a 2 + z 2 r 2 1=\frac{x^{2}+y^{2}}{r^{2}+a^{2}}+\frac{z^{2}}{r^{2}}
  15. R = x 2 + y 2 + z 2 R=\sqrt{x^{2}+y^{2}+z^{2}}
  16. A μ = Q r 3 r 4 + a 2 z 2 k μ A_{\mu}=\frac{Qr^{3}}{r^{4}+a^{2}z^{2}}k_{\mu}
  17. A μ = Q R k μ A_{\mu}=\frac{Q}{R}k_{\mu}
  18. a 2 + Q 2 M 2 . a^{2}+Q^{2}\leq M^{2}.
  19. A μ = ( - ϕ , A x , A y , A z ) A_{\mu}=\left(-\phi,A_{x},A_{y},A_{z}\right)\,
  20. E = - ϕ \vec{E}=-\vec{\nabla}\phi\,
  21. B = × A \vec{B}=\vec{\nabla}\times\vec{A}\,
  22. E + i B = - Ω \vec{E}+i\vec{B}=-\vec{\nabla}\Omega\,
  23. Ω = Q ( R - i a ) 2 \Omega=\frac{Q}{\sqrt{(\vec{R}-i\vec{a})^{2}}}\,
  24. Ω \Omega

Killing_horizon.html

  1. κ \kappa
  2. r = r + := M + M 2 - Q 2 - J 2 / M 2 r=r_{+}:=M+\sqrt{M^{2}-Q^{2}-J^{2}/M^{2}}
  3. / t \partial/\partial t
  4. T H = κ / 2 π T_{H}=\kappa/2\pi
  5. r = 3 / Λ r=\sqrt{3/\Lambda}
  6. T = ( 1 / 2 π ) Λ / 3 T=(1/2\pi)\sqrt{\Lambda/3}

Kirchhoff's_theorem.html

  1. t ( G ) = 1 n λ 1 λ 2 λ n - 1 . t(G)=\frac{1}{n}\lambda_{1}\lambda_{2}\cdots\lambda_{n-1}\,.
  2. Q = [ 2 - 1 - 1 0 - 1 3 - 1 - 1 - 1 - 1 3 - 1 0 - 1 - 1 2 ] . Q=\left[\begin{array}[]{rrrr}2&-1&-1&0\\ -1&3&-1&-1\\ -1&-1&3&-1\\ 0&-1&-1&2\end{array}\right].
  3. Q = [ 3 - 1 - 1 - 1 3 - 1 - 1 - 1 2 ] . Q^{\ast}=\left[\begin{array}[]{rrr}3&-1&-1\\ -1&3&-1\\ -1&-1&2\end{array}\right].
  4. E E
  5. E = [ 1 1 0 0 0 - 1 0 1 1 0 0 - 1 - 1 0 1 0 0 0 - 1 - 1 ] . E=\begin{bmatrix}1&1&0&0&0\\ -1&0&1&1&0\\ 0&-1&-1&0&1\\ 0&0&0&-1&-1\\ \end{bmatrix}.
  6. det ( M 11 ) = S det ( F S ) det ( F S T ) = S det ( F S ) 2 \det(M_{11})=\sum_{S}\det(F_{S})\det(F^{T}_{S})=\sum_{S}\det(F_{S})^{2}
  7. [ n - 1 - 1 - 1 - 1 n - 1 - 1 - 1 - 1 n - 1 ] . \begin{bmatrix}n-1&-1&\cdots&-1\\ -1&n-1&\cdots&-1\\ \vdots&\vdots&\ddots&\vdots\\ -1&-1&\cdots&n-1\\ \end{bmatrix}.

Kleene_fixed-point_theorem.html

  1. f ( ) f ( f ( ) ) f n ( ) \bot\;\sqsubseteq\;f(\bot)\;\sqsubseteq\;f\left(f(\bot)\right)\;\sqsubseteq\;% \dots\;\sqsubseteq\;f^{n}(\bot)\;\sqsubseteq\;\dots
  2. lfp ( f ) = sup ( { f n ( ) n } ) \textrm{lfp}(f)=\sup\left(\left\{f^{n}(\bot)\mid n\in\mathbb{N}\right\}\right)
  3. $\textrm{lfp}$
  4. f n ( ) f n + 1 ( ) , n 0 f^{n}(\bot)\sqsubseteq f^{n+1}(\bot),n\in\mathbb{N}_{0}
  5. f 0 ( ) = f 1 ( ) f^{0}(\bot)=\bot\sqsubseteq f^{1}(\bot)
  6. f n ( ) f n + 1 ( ) f^{n}(\bot)\sqsubseteq f^{n+1}(\bot)
  7. f ( f n - 1 ( ) ) f ( f n ( ) ) f(f^{n-1}(\bot))\sqsubseteq f(f^{n}(\bot))
  8. f n - 1 ( ) f n ( ) f^{n-1}(\bot)\sqsubseteq f^{n}(\bot)
  9. 𝕄 \mathbb{M}
  10. 𝕄 = { , f ( ) , f ( f ( ) ) , } \mathbb{M}=\{\bot,f(\bot),f(f(\bot)),\ldots\}
  11. m m
  12. m m
  13. m m
  14. f ( m ) = m f(m)=m
  15. f f
  16. f ( sup ( 𝕄 ) ) = sup ( f ( 𝕄 ) ) f(\sup(\mathbb{M}))=\sup(f(\mathbb{M}))
  17. f ( m ) = sup ( f ( 𝕄 ) ) f(m)=\sup(f(\mathbb{M}))
  18. f ( 𝕄 ) = 𝕄 { } f(\mathbb{M})=\mathbb{M}\setminus\{\bot\}
  19. \bot
  20. sup \sup
  21. sup ( f ( 𝕄 ) ) = sup ( 𝕄 ) \sup(f(\mathbb{M}))=\sup(\mathbb{M})
  22. f ( m ) = m f(m)=m
  23. m m
  24. f f
  25. m m
  26. 𝕄 \mathbb{M}
  27. f f
  28. D L D\subseteq L
  29. L L
  30. sup ( D ) \sup(D)
  31. L L
  32. k k
  33. f f
  34. i i
  35. i : f i ( ) k \forall i\in\mathbb{N}\colon f^{i}(\bot)\sqsubseteq k
  36. i = 0 i=0
  37. f 0 ( ) = k f^{0}(\bot)=\bot\sqsubseteq k
  38. \bot
  39. L L
  40. f i ( ) k f^{i}(\bot)\sqsubseteq k
  41. f f
  42. f f
  43. f i ( ) k f i + 1 ( ) f ( k ) f^{i}(\bot)\sqsubseteq k~{}\implies~{}f^{i+1}(\bot)\sqsubseteq f(k)
  44. k k
  45. f f
  46. f ( k ) = k f(k)=k
  47. f i + 1 ( ) k f^{i+1}(\bot)\sqsubseteq k

Klein_transformation.html

  1. [ ϕ i ( x ) , ϕ j ( y ) ] = [ χ i ( x ) , χ j ( y ) ] = { ϕ i ( x ) , χ j ( y ) } = 0. [\phi^{i}(x),\phi^{j}(y)]=[\chi^{i}(x),\chi^{j}(y)]=\{\phi^{i}(x),\chi^{j}(y)% \}=0.
  2. ϕ = i K χ ϕ \phi^{\prime}=iK_{\chi}\phi\,
  3. χ = K χ χ . \chi^{\prime}=K_{\chi}\chi.\,
  4. [ ϕ i ( x ) , ϕ j ( y ) ] = [ χ i ( x ) , χ j ( y ) ] = [ ϕ i ( x ) , χ j ( y ) ] = 0. [\phi^{\prime i}(x),\phi^{\prime j}(y)]=[\chi^{\prime i}(x),\chi^{\prime j}(y)% ]=[\phi^{\prime i}(x),\chi^{\prime j}(y)]=0.\,
  5. { ϕ i ( x ) , ϕ j ( y ) } = { χ i ( x ) , χ j ( y ) } = [ ϕ i ( x ) , χ j ( y ) ] = 0 \{\phi^{i}(x),\phi^{j}(y)\}=\{\chi^{i}(x),\chi^{j}(y)\}=[\phi^{i}(x),\chi^{j}(% y)]=0
  6. ϕ = i K χ ϕ \phi^{\prime}=iK_{\chi}\phi\,
  7. χ = K χ χ . \chi^{\prime}=K_{\chi}\chi.\,
  8. { ϕ i ( x ) , ϕ j ( y ) } = { χ i ( x ) , χ j ( y ) } = { ϕ i ( x ) , χ j ( y ) } = 0. \{\phi^{\prime i}(x),\phi^{\prime j}(y)\}=\{\chi^{\prime i}(x),\chi^{\prime j}% (y)\}=\{\phi^{\prime i}(x),\chi^{\prime j}(y)\}=0.

Kleinian_group.html

  1. π 1 \pi_{1}
  2. B ¯ 3 \bar{B}^{3}
  3. S 2 S^{2}_{\infty}
  4. S 2 S^{2}_{\infty}
  5. Λ ( G ) \Lambda(G)
  6. Ω ( G ) = S 2 - Λ ( G ) \Omega(G)=S^{2}_{\infty}-\Lambda(G)
  7. Ω ( G ) / G \Omega(G)/G
  8. d s 2 = 4 | d x | 2 ( 1 - | x | 2 ) 2 ds^{2}=\frac{4\left|dx\right|^{2}}{\left(1-|x|^{2}\right)^{2}}
  9. S 2 S^{2}_{\infty}
  10. Mob ( S 2 ) Conf ( B 3 ) Isom ( 𝐇 3 ) . \mbox{Mob}~{}(S^{2}_{\infty})\cong\mbox{Conf}~{}(B^{3})\cong\mbox{Isom}~{}(% \mathbf{H}^{3}).

Kleinian_model.html

  1. 3 / Γ \mathbb{H}^{3}/\Gamma
  2. π 1 ( N ) \pi_{1}(N)

Knoop_hardness_test.html

  1. H K = load ( kgf ) impression area ( mm ) 2 = P C p L 2 HK={{\textrm{load}(\mbox{kgf}~{})}\over{\textrm{impression\ area}(\mbox{mm}~{}% ^{2})}}={P\over{C_{p}L^{2}}}

Knot_complement.html

  1. X K = M - interior ( N ) . X_{K}=M-\mbox{interior}~{}(N).

Kodaira_dimension.html

  1. K X = n Ω X 1 , \,\!K_{X}=\bigwedge^{n}\Omega^{1}_{X},
  2. P d = h 0 ( X , K X d ) = dim H 0 ( X , K X d ) . P_{d}=h^{0}(X,K_{X}^{d})=\operatorname{dim}\ H^{0}(X,K_{X}^{d}).
  3. 𝐏 ( H 0 ( X , K X d ) ) = 𝐏 P d - 1 \mathbf{P}(H^{0}(X,K_{X}^{d}))=\mathbf{P}^{P_{d}-1}
  4. R ( K X ) := d 0 H 0 ( X , K X d ) . R(K_{X}):=\bigoplus_{d\geq 0}H^{0}(X,K_{X}^{d}).
  5. 1 1
  6. 2 \geq 2
  7. 0
  8. 1 1
  9. - -\infty
  10. 0
  11. 1 \mathbb{P}^{1}
  12. 2 2
  13. 1 1
  14. 0
  15. 1 1
  16. 2 2
  17. 0
  18. 1 1
  19. 1 1
  20. 0
  21. 0
  22. 0
  23. - -\infty
  24. 0
  25. 1 \geq 1
  26. 0
  27. 0
  28. 3 3
  29. 2 2
  30. 1 1
  31. 0
  32. 1 1
  33. 3 3
  34. 0
  35. 2 2
  36. 0
  37. 1 1
  38. 1 1
  39. 0
  40. 1 1
  41. 0
  42. - -\infty
  43. 0
  44. 1 \geq 1
  45. 0
  46. 0
  47. κ ( X ) = dim X . \kappa(X)=\operatorname{dim}\ X.
  48. κ ( V ) = κ ( F ) + κ ( W ) . \kappa(V)=\kappa(F)+\kappa(W).

Kolchuga_passive_sensor.html

  1. d ( k m ) = 130 ( h r ( k m ) + h t ( k m ) ) d(km)=130(\sqrt{hr(km)}+\sqrt{ht(km)})

Kosterlitz–Thouless_transition.html

  1. T c T_{c}
  2. F = E - T S F=E-TS
  3. T c T_{c}
  4. T c T_{c}
  5. κ ln ( R / a ) \kappa\ln(R/a)
  6. κ \kappa
  7. R R
  8. a a
  9. R a R\gg a
  10. ( R / a ) 2 (R/a)^{2}
  11. S = 2 k B ln ( R / a ) S=2k_{B}\ln(R/a)
  12. k B k_{B}
  13. F = E - T S = ( κ - 2 k B T ) ln ( R / a ) . F=E-TS=(\kappa-2k_{B}T)\ln(R/a).
  14. F > 0 F>0
  15. F < 0 F<0
  16. F = 0 F=0
  17. T c T_{c}
  18. T c = κ 2 k B . T_{c}=\frac{\kappa}{2k_{B}}.
  19. T c T_{c}
  20. V I V\sim I
  21. T c T_{c}
  22. V I 3 V\sim I^{3}
  23. I 2 I^{2}
  24. T c T_{c}
  25. E = 1 2 ϕ ϕ d 2 x E=\int\frac{1}{2}\nabla\phi\cdot\nabla\phi d^{2}x
  26. γ d ϕ \oint_{\gamma}d\phi
  27. γ d ϕ \oint_{\gamma}d\phi
  28. i = 1 n n i arg ( z - z i ) \sum_{i=1}^{n}n_{i}\arg(z-z_{i})
  29. E = 1 2 ϕ 0 ϕ 0 d 2 x + 1 2 i = 1 n n i arg ( z - z i ) j = 1 n n j arg ( z - z j ) d 2 x E=\int\frac{1}{2}\nabla\phi_{0}\cdot\nabla\phi_{0}d^{2}x+\int\frac{1}{2}\nabla% \sum_{i=1}^{n}n_{i}\arg(z-z_{i})\cdot\nabla\sum_{j=1}^{n}n_{j}\arg(z-z_{j})d^{% 2}x
  30. i = 1 n n i = 0 \sum_{i=1}^{n}n_{i}=0
  31. i = 1 n n i = 0 \sum_{i=1}^{n}n_{i}=0
  32. 1 i < j n - 2 π n i n j ln ( | x j - x i | / L ) \sum_{1\leq i<j\leq n}-2\pi n_{i}n_{j}\ln(|x_{j}-x_{i}|/L)

Kraft's_inequality.html

  1. leaves 2 - depth ( ) 1. \sum_{\ell\in\mathrm{leaves}}2^{-\mathrm{depth}(\ell)}\leq 1.
  2. 1 4 + 4 ( 1 8 ) = 3 4 1. \frac{1}{4}+4\left(\frac{1}{8}\right)=\frac{3}{4}\leq 1.
  3. Ω = p P 2 - | p | . \Omega=\sum_{p\in P}2^{-|p|}.
  4. Ω 1 \Omega\leq 1
  5. S = { s 1 , s 2 , , s n } S=\{\,s_{1},s_{2},\ldots,s_{n}\,\}\,
  6. r r
  7. 1 , 2 , , n . \ell_{1},\ell_{2},\ldots,\ell_{n}.\,
  8. i = 1 n ( 1 r ) i 1. \sum_{i=1}^{n}\left(\frac{1}{r}\right)^{\ell_{i}}\leq 1.
  9. 1 , 2 , , n \ell_{1},\ell_{2},\ldots,\ell_{n}\,
  10. r r
  11. 1 2 n \ell_{1}\leq\ell_{2}\leq...\leq\ell_{n}
  12. A A
  13. r r
  14. n \ell_{n}
  15. n \ell\leq\ell_{n}
  16. r r
  17. \ell
  18. i i
  19. v i v_{i}
  20. A i A_{i}
  21. A A
  22. v i v_{i}
  23. | A i | = r n - i . |A_{i}|=r^{\ell_{n}-\ell_{i}}.
  24. A i A j = , i j A_{i}\cap A_{j}=\varnothing,\quad i\neq j
  25. n \ell_{n}
  26. r n r^{\ell_{n}}
  27. | i = 1 n A i | = i = 1 n r n - i r n |\bigcup_{i=1}^{n}A_{i}|=\sum_{i=1}^{n}r^{\ell_{n}-\ell_{i}}\leq r^{\ell_{n}}
  28. n n
  29. 1 2 n \ell_{1}\leq\ell_{2}\leq\dots\leq\ell_{n}
  30. i \ell_{i}
  31. r r
  32. n \ell_{n}
  33. 1 \ell_{1}
  34. r - 1 r^{-\ell_{1}}
  35. r - 2 r^{-\ell_{2}}
  36. r - 1 + r - 2 r^{-\ell_{1}}+r^{-\ell_{2}}
  37. m m
  38. i = 1 m r - i \sum_{i=1}^{m}r^{-\ell_{i}}
  39. m < n m<n
  40. i \ell_{i}
  41. n n
  42. r - l i = P ( E i ) = P ( E i ) 1. \sum r^{-li}=\sum P(E_{i})=P(\cup E_{i})\leq 1.
  43. F ( x ) = i = 1 n x - | s i | = = min max p x - F(x)=\sum_{i=1}^{n}x^{-|s_{i}|}=\sum_{\ell=\min}^{\max}p_{\ell}\,x^{-\ell}
  44. p p_{\ell}
  45. x - x^{-\ell}
  46. \ell
  47. s i 1 s i 2 s i m s_{i_{1}}s_{i_{2}}\dots s_{i_{m}}
  48. i 1 , i 2 , , i m i_{1},i_{2},\dots,i_{m}
  49. s i 1 s i 2 s i m = s j 1 s j 2 s j m s_{i_{1}}s_{i_{2}}\dots s_{i_{m}}=s_{j_{1}}s_{j_{2}}\dots s_{j_{m}}
  50. i 1 = j 1 , i 2 = j 2 , , i m = j m i_{1}=j_{1},i_{2}=j_{2},\dots,i_{m}=j_{m}
  51. G ( x ) G(x)
  52. S m S^{m}
  53. F ( x ) F(x)
  54. G ( x ) = ( F ( x ) ) m = ( i = 1 n x - | s i | ) m = G(x)=\left(F(x)\right)^{m}=\left(\sum_{i=1}^{n}x^{-|s_{i}|}\right)^{m}=
  55. = i 1 = 1 n i 2 = 1 n i m = 1 n x - ( | s i 1 | + | s i 2 | + + | s i m | ) = =\sum_{i_{1}=1}^{n}\sum_{i_{2}=1}^{n}\cdots\sum_{i_{m}=1}^{n}x^{-\left(|s_{i_{% 1}}|+|s_{i_{2}}|+\cdots+|s_{i_{m}}|\right)}=
  56. = i 1 = 1 n i 2 = 1 n i m = 1 n x - | s i 1 s i 2 s i m | = = m min m max q x - . =\sum_{i_{1}=1}^{n}\sum_{i_{2}=1}^{n}\cdots\sum_{i_{m}=1}^{n}x^{-|s_{i_{1}}s_{% i_{2}}\cdots s_{i_{m}}|}=\sum_{\ell=m\cdot\min}^{m\cdot\max}q_{\ell}\,x^{-\ell% }\;.
  57. q q_{\ell}
  58. x - x^{-\ell}
  59. G ( x ) G(x)
  60. \ell
  61. S m S^{m}
  62. q q_{\ell}
  63. r r^{\ell}
  64. ( F ( x ) ) m = m min m max r x - . \left(F(x)\right)^{m}\leq\sum_{\ell=m\cdot\min}^{m\cdot\max}r^{\ell}\,x^{-\ell% }\;.
  65. ( F ( r ) ) m m ( max - min ) + 1 \left(F(r)\right)^{m}\leq m\cdot(\max-\min)+1
  66. m m
  67. m m
  68. m m
  69. F ( r ) 1 F(r)\leq 1
  70. F ( x ) F(x)
  71. i = 1 n r - i = i = 1 n r - | s i | = F ( r ) 1 . \sum_{i=1}^{n}r^{-\ell_{i}}=\sum_{i=1}^{n}r^{-|s_{i}|}=F(r)\leq 1\;.
  72. n n
  73. 1 2 n \ell_{1}\leq\ell_{2}\leq\dots\leq\ell_{n}
  74. j = 1 i - 1 r - l j . \sum_{j=1}^{i-1}r^{-l_{j}}.

Kripke–Platek_set_theory_with_urelements.html

  1. L * L^{*}
  2. \in
  3. p , q , r , p,q,r,...
  4. a , b , c , a,b,c,...
  5. x , y , z , x,y,z,...
  6. \in
  7. p a p\in a
  8. b a b\in a
  9. Δ 0 \Delta_{0}
  10. Δ 0 \Delta_{0}
  11. \in
  12. ¬ \neg
  13. \wedge
  14. \vee
  15. x a \forall x\in a
  16. x a \exists x\in a
  17. a a
  18. x ( x a x b ) a = b \forall x(x\in a\leftrightarrow x\in b)\rightarrow a=b
  19. ϕ ( x ) \phi(x)
  20. a ϕ ( a ) a ( ϕ ( a ) x a ( ¬ ϕ ( x ) ) ) \exists a\phi(a)\rightarrow\exists a\,(\phi(a)\wedge\forall x\in a\,(\neg\phi(% x)))
  21. a ( x a y a ) \exists a\,(x\in a\land y\in a)
  22. a c b y c ( y a ) \exists a\forall c\in b\forall y\in c\,(y\in a)
  23. Δ 0 \Delta_{0}
  24. ϕ ( x ) \phi(x)
  25. a x ( x a x b ϕ ( x ) ) \exists a\forall x\,(x\in a\leftrightarrow x\in b\wedge\phi(x))
  26. Δ 0 \Delta_{0}
  27. Δ 0 \Delta_{0}
  28. ϕ ( x , y ) \phi(x,y)
  29. x a y ϕ ( x , y ) b x a y b ϕ ( x , y ) \forall x\in a\exists y\,\phi(x,y)\rightarrow\exists b\forall x\in a\exists y% \in b\,\phi(x,y)
  30. a ( a = a ) \exists a\,(a=a)
  31. p a ( p a ) \forall p\forall a\,(p\neq a)
  32. p x ( x p ) \forall p\forall x\,(x\notin p)

Kruskal–Katona_theorem.html

  1. N = ( n i i ) + ( n i - 1 i - 1 ) + + ( n j j ) , n i > n i - 1 > > n j j 1. N={\left({{n_{i}}\atop{i}}\right)}+{\left({{n_{i-1}}\atop{i-1}}\right)}+\ldots% +{\left({{n_{j}}\atop{j}}\right)},\quad n_{i}>n_{i-1}>\ldots>n_{j}\geq j\geq 1.
  2. N ( n i ) , N\geq{\left({{n}\atop{i}}\right)},
  3. N ( i ) = ( n i i + 1 ) + ( n i - 1 i ) + + ( n j j + 1 ) . N^{(i)}={\left({{n_{i}}\atop{i+1}}\right)}+{\left({{n_{i-1}}\atop{i}}\right)}+% \ldots+{\left({{n_{j}}\atop{j+1}}\right)}.
  4. ( f 0 , f 1 , , f d - 1 ) (f_{0},f_{1},...,f_{d-1})
  5. ( d - 1 ) (d-1)
  6. 0 f i f i - 1 ( i ) , 1 i d - 1. 0\leq f_{i}\leq f_{i-1}^{(i)},\quad 1\leq i\leq d-1.
  7. ( i - r ) (i-r)
  8. | B | ( n i i - r ) + ( n i - 1 i - r - 1 ) + + ( n j j - r ) . |B|\geq{\left({{n_{i}}\atop{i-r}}\right)}+{\left({{n_{i-1}}\atop{i-r-1}}\right% )}+\ldots+{\left({{n_{j}}\atop{j-r}}\right)}.
  9. 123 , 124 , 134 , 234 , 125 , 135 , 235 , 145 , 245 , 345 , . 123,124,134,234,125,135,235,145,245,345,\ldots.
  10. f = ( f 0 , f 1 , , f d - 1 ) f=(f_{0},f_{1},...,f_{d-1})
  11. f i - 1 f_{i-1}
  12. f i f i - 1 ( i ) , 1 i d - 1. f_{i}\leq f_{i-1}^{(i)},\quad 1\leq i\leq d-1.

Kuder–Richardson_Formula_20.html

  1. r = K K - 1 [ 1 - i = 1 K p i q i σ X 2 ] r=\frac{K}{K-1}\left[1-\frac{\sum_{i=1}^{K}p_{i}q_{i}}{\sigma^{2}_{X}}\right]
  2. σ X 2 = i = 1 n ( X i - X ¯ ) 2 n . \sigma^{2}_{X}=\frac{\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\,{}}{n}.
  3. n n - 1 \frac{n}{n-1}

Kummer's_function.html

  1. Λ n ( z ) = 0 z log n - 1 | t | 1 + t d t . \Lambda_{n}(z)=\int_{0}^{z}\frac{\log^{n-1}|t|}{1+t}\;dt.
  2. Λ n ( z ) + Λ n ( - z ) = 2 1 - n Λ n ( - z 2 ) \Lambda_{n}(z)+\Lambda_{n}(-z)=2^{1-n}\Lambda_{n}(-z^{2})
  3. Li n ( z ) + Li n ( - z ) = 2 1 - n Li n ( z 2 ) . \operatorname{Li}_{n}(z)+\operatorname{Li}_{n}(-z)=2^{1-n}\operatorname{Li}_{n% }(z^{2}).
  4. Li n ( z ) = Li n ( 1 ) + k = 1 n - 1 ( - ) k - 1 log k | z | k ! Li n - k ( z ) + ( - ) n - 1 ( n - 1 ) ! [ Λ n ( - 1 ) - Λ n ( - z ) ] . \operatorname{Li}_{n}(z)=\operatorname{Li}_{n}(1)\;\;+\;\;\sum_{k=1}^{n-1}(-)^% {k-1}\;\frac{\log^{k}|z|}{k!}\;\operatorname{Li}_{n-k}(z)\;\;+\;\;\frac{(-)^{n% -1}}{(n-1)!}\;\left[\Lambda_{n}(-1)-\Lambda_{n}(-z)\right].

Lag_operator.html

  1. X = { X 1 , X 2 , } X=\{X_{1},X_{2},\dots\}\,
  2. L X t = X t - 1 \,LX_{t}=X_{t-1}
  3. t > 1 \;t>1\,
  4. X t = L X t + 1 \,X_{t}=LX_{t+1}
  5. t 1 \;t\geq 1\,
  6. L - 1 X t = X t + 1 \,L^{-1}X_{t}=X_{t+1}\,
  7. L k X t = X t - k . \,L^{k}X_{t}=X_{t-k}.\,
  8. ε t = X t - i = 1 p φ i X t - i = ( 1 - i = 1 p φ i L i ) X t \varepsilon_{t}=X_{t}-\sum_{i=1}^{p}\varphi_{i}X_{t-i}=\left(1-\sum_{i=1}^{p}% \varphi_{i}L^{i}\right)X_{t}\,
  9. φ ( L ) X t = θ ( L ) ε t \varphi(L)X_{t}=\theta(L)\varepsilon_{t}\,
  10. φ ( L ) \varphi(L)
  11. θ ( L ) \theta(L)
  12. φ ( L ) = 1 - i = 1 p φ i L i \varphi(L)=1-\sum_{i=1}^{p}\varphi_{i}L^{i}\,
  13. θ ( L ) = 1 + i = 1 q θ i L i . \theta(L)=1+\sum_{i=1}^{q}\theta_{i}L^{i}.\,
  14. X t = θ ( L ) φ ( L ) ε t , X_{t}=\frac{\theta(L)}{\varphi(L)}\varepsilon_{t},
  15. φ ( L ) X t = θ ( L ) ε t . \varphi(L)X_{t}=\theta(L)\varepsilon_{t}\,.
  16. [ ] + [\ ]_{+}
  17. Δ X t = X t - X t - 1 Δ X t = ( 1 - L ) X t . \begin{array}[]{lcr}\Delta X_{t}&=X_{t}-X_{t-1}\\ \Delta X_{t}&=(1-L)X_{t}~{}.\end{array}
  18. Δ ( Δ X t ) \displaystyle\Delta(\Delta X_{t})
  19. Δ i X t = ( 1 - L ) i X t . \Delta^{i}X_{t}=(1-L)^{i}X_{t}\ .
  20. Ω t \Omega_{t}
  21. E [ X t + j | Ω t ] = E t [ X t + j ] . E[X_{t+j}|\Omega_{t}]=E_{t}[X_{t+j}]\,.
  22. L n E t [ X t + j ] = E t - n [ X t + j - n ] , L^{n}E_{t}[X_{t+j}]=E_{t-n}[X_{t+j-n}]\,,
  23. B n E t [ X t + j ] = E t [ X t + j - n ] . B^{n}E_{t}[X_{t+j}]=E_{t}[X_{t+j-n}]\,.

Lambert_series.html

  1. S ( q ) = n = 1 a n q n 1 - q n . S(q)=\sum_{n=1}^{\infty}a_{n}\frac{q^{n}}{1-q^{n}}.
  2. S ( q ) = n = 1 a n k = 1 q n k = m = 1 b m q m S(q)=\sum_{n=1}^{\infty}a_{n}\sum_{k=1}^{\infty}q^{nk}=\sum_{m=1}^{\infty}b_{m% }q^{m}
  3. b m = ( a * 1 ) ( m ) = n m a n . b_{m}=(a*1)(m)=\sum_{n\mid m}a_{n}.\,
  4. n = 1 q n σ 0 ( n ) = n = 1 q n 1 - q n \sum_{n=1}^{\infty}q^{n}\sigma_{0}(n)=\sum_{n=1}^{\infty}\frac{q^{n}}{1-q^{n}}
  5. σ 0 ( n ) = d ( n ) \sigma_{0}(n)=d(n)
  6. n = 1 q n σ α ( n ) = n = 1 n α q n 1 - q n \sum_{n=1}^{\infty}q^{n}\sigma_{\alpha}(n)=\sum_{n=1}^{\infty}\frac{n^{\alpha}% q^{n}}{1-q^{n}}
  7. α \alpha
  8. σ α ( n ) = ( Id α * 1 ) ( n ) = d n d α \sigma_{\alpha}(n)=(\textrm{Id}_{\alpha}*1)(n)=\sum_{d\mid n}d^{\alpha}\,
  9. μ ( n ) \mu(n)
  10. n = 1 μ ( n ) q n 1 - q n = q . \sum_{n=1}^{\infty}\mu(n)\,\frac{q^{n}}{1-q^{n}}=q.
  11. ϕ ( n ) \phi(n)
  12. n = 1 φ ( n ) q n 1 - q n = q ( 1 - q ) 2 . \sum_{n=1}^{\infty}\varphi(n)\,\frac{q^{n}}{1-q^{n}}=\frac{q}{(1-q)^{2}}.
  13. λ ( n ) \lambda(n)
  14. n = 1 λ ( n ) q n 1 - q n = n = 1 q n 2 \sum_{n=1}^{\infty}\lambda(n)\,\frac{q^{n}}{1-q^{n}}=\sum_{n=1}^{\infty}q^{n^{% 2}}
  15. q = e - z q=e^{-z}
  16. n = 1 a n e z n - 1 = m = 1 b m e - m z \sum_{n=1}^{\infty}\frac{a_{n}}{e^{zn}-1}=\sum_{m=1}^{\infty}b_{m}e^{-mz}
  17. b m = ( a * 1 ) ( m ) = d m a d b_{m}=(a*1)(m)=\sum_{d\mid m}a_{d}\,
  18. z = 2 π z=2\pi
  19. q n / ( 1 - q n ) = Li 0 ( q n ) q^{n}/(1-q^{n})=\mathrm{Li}_{0}(q^{n})
  20. n = 1 ξ n Li u ( α q n ) n s = n = 1 α n Li s ( ξ q n ) n u \sum_{n=1}^{\infty}\frac{\xi^{n}\,\mathrm{Li}_{u}(\alpha q^{n})}{n^{s}}=\sum_{% n=1}^{\infty}\frac{\alpha^{n}\,\mathrm{Li}_{s}(\xi q^{n})}{n^{u}}
  21. 12 ( n = 1 n 2 Li - 1 ( q n ) ) 2 = n = 1 n 2 Li - 5 ( q n ) - n = 1 n 4 Li - 3 ( q n ) , 12\left(\sum_{n=1}^{\infty}n^{2}\,\mathrm{Li}_{-1}(q^{n})\right)^{\!2}=\sum_{n% =1}^{\infty}n^{2}\,\mathrm{Li}_{-5}(q^{n})-\sum_{n=1}^{\infty}n^{4}\,\mathrm{% Li}_{-3}(q^{n}),

Lanczos_approximation.html

  1. Γ ( z + 1 ) = 2 π ( z + g + 1 2 ) z + 1 2 e - ( z + g + 1 2 ) A g ( z ) \Gamma(z+1)=\sqrt{2\pi}{\left(z+g+\frac{1}{2}\right)}^{z+\frac{1}{2}}e^{-\left% (z+g+\frac{1}{2}\right)}A_{g}(z)
  2. A g ( z ) = 1 2 p 0 ( g ) + p 1 ( g ) z z + 1 + p 2 ( g ) z ( z - 1 ) ( z + 1 ) ( z + 2 ) + . A_{g}(z)=\frac{1}{2}p_{0}(g)+p_{1}(g)\frac{z}{z+1}+p_{2}(g)\frac{z(z-1)}{(z+1)% (z+2)}+\cdots.
  3. Γ ( 1 - z ) Γ ( z ) = π sin π z . \Gamma(1-z)\;\Gamma(z)={\pi\over\sin\pi z}.
  4. A g ( z ) = c 0 + k = 1 N c k z + k A_{g}(z)=c_{0}+\sum_{k=1}^{N}\frac{c_{k}}{z+k}
  5. p k ( g ) = a = 0 k C ( 2 k + 1 , 2 a + 1 ) 2 π ( a - 1 2 ) ! ( a + g + 1 2 ) - ( a + 1 2 ) e a + g + 1 2 p_{k}(g)=\sum_{a=0}^{k}C(2k+1,2a+1)\frac{\sqrt{2}}{\pi}\left(a-\begin{matrix}% \frac{1}{2}\end{matrix}\right)!{\left(a+g+\begin{matrix}\frac{1}{2}\end{matrix% }\right)}^{-\left(a+\frac{1}{2}\right)}e^{a+g+\frac{1}{2}}
  6. C ( i , j ) C(i,j)
  7. C ( 1 , 1 ) = 1 C(1,1)=1\,
  8. C ( 2 , 2 ) = 1 C(2,2)=1\,
  9. C ( i , 1 ) = - C ( i - 2 , 1 ) C(i,1)=-C(i-2,1)\,
  10. i = 3 , 4 , i=3,4,\dots\,
  11. C ( i , j ) = 2 C ( i - 1 , j - 1 ) C(i,j)=2C(i-1,j-1)\,
  12. i = j = 3 , 4 , i=j=3,4,\dots\,
  13. C ( i , j ) = 2 C ( i - 1 , j - 1 ) - C ( i - 2 , j ) C(i,j)=2C(i-1,j-1)-C(i-2,j)\,
  14. i > j = 2 , 3 , . i>j=2,3,\dots.
  15. Γ ( z + 1 ) = 0 t z e - t d t , \Gamma(z+1)=\int_{0}^{\infty}t^{z}\,e^{-t}\,dt,
  16. Γ ( z + 1 ) = ( z + g + 1 ) z + 1 e - ( z + g + 1 ) 0 e [ v ( 1 - log v ) ] z - 1 2 v g d v , \Gamma(z+1)=(z+g+1)^{z+1}e^{-(z+g+1)}\int_{0}^{e}[v(1-\log v)]^{z-\frac{1}{2}}% v^{g}\,dv,

Land-use_forecasting.html

  1. R = Y ( P - c ) - Y t d R=Y\left({P-c}\right)-Ytd
  2. max Z = k = 1 u i = 1 n h = 1 m x i h k ( b i h - c i h k ) x i h k 0 \max Z=\sum_{k=1}^{u}{\sum_{i=1}^{n}{\sum_{h=1}^{m}{x_{ih}^{k}\left({b_{ih}-c_% {ih}^{k}}\right)}}}\quad x_{ih}^{k}\geq 0
  3. i = 1 n h = 1 m s i h x i h k L k \sum_{i=1}^{n}{\sum_{h=1}^{m}{s_{ih}x_{ih}^{k}}}\leq L^{k}
  4. k = 1 u h = 1 m x i h k = N i \sum_{k=1}^{u}{\sum_{h=1}^{m}{x_{ih}^{k}}}=N_{i}
  5. min Z = k = 1 u r k L k + i = 1 n v i ( - N i ) \min Z^{\prime}=\sum_{k=1}^{u}{r^{k}L^{k}+\sum_{i=1}^{n}{v_{i}\left({-N_{i}}% \right)}}
  6. s i h r k - v i b i h - c i h k s_{ih}r^{k}-v_{i}\geq b_{ih}-c_{ih}^{k}
  7. r k 0 r^{k}\geq 0

Landé_g-factor.html

  1. g J = g L J ( J + 1 ) - S ( S + 1 ) + L ( L + 1 ) 2 J ( J + 1 ) + g S J ( J + 1 ) + S ( S + 1 ) - L ( L + 1 ) 2 J ( J + 1 ) . g_{J}=g_{L}\frac{J(J+1)-S(S+1)+L(L+1)}{2J(J+1)}+g_{S}\frac{J(J+1)+S(S+1)-L(L+1% )}{2J(J+1)}.
  2. g S = 2 g_{S}=2
  3. g J 3 2 + S ( S + 1 ) - L ( L + 1 ) 2 J ( J + 1 ) . g_{J}\approx\frac{3}{2}+\frac{S(S+1)-L(L+1)}{2J(J+1)}.
  4. g F = g J F ( F + 1 ) - I ( I + 1 ) + J ( J + 1 ) 2 F ( F + 1 ) + g I F ( F + 1 ) + I ( I + 1 ) - J ( J + 1 ) 2 F ( F + 1 ) g_{F}=g_{J}\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}+g_{I}\frac{F(F+1)+I(I+1)-J(J+1% )}{2F(F+1)}
  5. g J F ( F + 1 ) - I ( I + 1 ) + J ( J + 1 ) 2 F ( F + 1 ) \approx g_{J}\frac{F(F+1)-I(I+1)+J(J+1)}{2F(F+1)}
  6. g I g_{I}
  7. g J g_{J}
  8. μ L = L g L μ B \vec{\mu}_{L}=\vec{L}g_{L}\mu_{B}
  9. μ S = S g S μ B \vec{\mu}_{S}=\vec{S}g_{S}\mu_{B}
  10. μ J = μ L + μ S \vec{\mu}_{J}=\vec{\mu}_{L}+\vec{\mu}_{S}
  11. g L = - 1 g_{L}=-1
  12. g S = - 2 g_{S}=-2
  13. g S g_{S}
  14. μ J \vec{\mu}_{J}
  15. J = L + S \vec{J}=\vec{L}+\vec{S}
  16. J \vec{J}
  17. J , J z | μ J | J , J < m t p l > z = g J μ B J , J z | J | J , J z \langle J,J_{z}|\vec{\mu}_{J}|J,J_{<}mtpl>{{z^{\prime}}}\rangle=g_{J}\mu_{B}% \langle J,J_{z}|\vec{J}|J,J_{z^{\prime}}\rangle
  18. J , J z | μ J | J , J z J , J z | J | J , J z = g J μ B J , J z | J | J , J z J , J z | J | J , J z \langle J,J_{z}|\vec{\mu}_{J}|J,J_{z^{\prime}}\rangle\cdot\langle J,J_{z^{% \prime}}|\vec{J}|J,J_{z}\rangle=g_{J}\mu_{B}\langle J,J_{z}|\vec{J}|J,J_{z^{% \prime}}\rangle\cdot\langle J,J_{z^{\prime}}|\vec{J}|J,J_{z}\rangle
  19. J z J , J z | μ J | J , J z J , J z | J | J , J z = J z g J μ B J , J z | J | J , J z J , J z | J | J , J z \sum_{J_{z^{\prime}}}\langle J,J_{z}|\vec{\mu}_{J}|J,J_{z^{\prime}}\rangle% \cdot\langle J,J_{z^{\prime}}|\vec{J}|J,J_{z}\rangle=\sum_{J_{z^{\prime}}}g_{J% }\mu_{B}\langle J,J_{z}|\vec{J}|J,J_{z^{\prime}}\rangle\cdot\langle J,J_{z^{% \prime}}|\vec{J}|J,J_{z}\rangle
  20. J , J z | μ J J | J , J z = g J μ B J , J z | J J | J , J z \langle J,J_{z}|\vec{\mu}_{J}\cdot\vec{J}|J,J_{z}\rangle=g_{J}\mu_{B}\langle J% ,J_{z}|\vec{J}\cdot\vec{J}|J,J_{z}\rangle
  21. g J J , J z | J J | J , J z = g L L J + g S S J g_{J}\langle J,J_{z}|\vec{J}\cdot\vec{J}|J,J_{z}\rangle=g_{L}{{\vec{L}}\cdot{% \vec{J}}}+g_{S}{{\vec{S}}\cdot{\vec{J}}}
  22. = g L ( L 2 + 1 2 ( J 2 - L 2 - S 2 ) ) + g S ( S 2 + 1 2 ( J 2 - L 2 - S 2 ) ) =g_{L}{(\vec{L}^{2}+\frac{1}{2}(\vec{J}^{2}-\vec{L}^{2}-\vec{S}^{2}))}+g_{S}{(% \vec{S}^{2}+\frac{1}{2}(\vec{J}^{2}-\vec{L}^{2}-\vec{S}^{2}))}
  23. g J = g L J ( J + 1 ) + L ( L + 1 ) - S ( S + 1 ) < m t p l > 2 J ( J + 1 ) + g S J ( J + 1 ) - L ( L + 1 ) + S ( S + 1 ) 2 J ( J + 1 ) g_{J}=g_{L}\frac{J(J+1)+L(L+1)-S(S+1)}{<}mtpl>{{2J(J+1)}}+g_{S}\frac{J(J+1)-L(% L+1)+S(S+1)}{{2J(J+1)}}

Langmuir–Blodgett_trough.html

  1. [ F o r c e o n P l a t e ] = [ W e i g h t o f p l a t e ] + [ S u r f a c e T e n s i o n F o r c e ] - [ B u o y a n t F o r c e ] [Force\ on\ Plate]=[Weight\ of\ plate]+[Surface\ Tension\ Force]-[Buoyant\ Force]
  2. F = ( m p g ) + 2 ( t p + w p ) γ L V cos ( θ ) - ρ l V p g F=\left(m_{p}g\right)+2(t_{p}+w_{p})\gamma_{LV}\cos(\theta)-\rho_{l}V_{p}g
  3. m p = mass of plate m_{p}=\mathrm{mass\ of\ plate}
  4. g = gravitational acceleration g=\mathrm{gravitational\ acceleration}
  5. t p = thickness of plate t_{p}=\mathrm{thickness\ of\ plate}
  6. w p = width of plate w_{p}=\mathrm{width\ of\ plate}
  7. γ L V = surface tension of liquid \gamma_{LV}=\mathrm{surface\ tension\ of\ liquid}
  8. θ = contact angle \theta=\mathrm{contact\ angle}
  9. ρ l = density of liquid \rho_{l}=\mathrm{density\ of\ liquid}
  10. V p = Volume of the proportion of plate immersed in liquid V_{p}=\mathrm{Volume\ of\ the\ proportion\ of\ plate\ immersed\ in\ liquid}
  11. F = 2 ( t p + w p ) γ L V cos ( θ ) F=2(t_{p}+w_{p})\gamma_{LV}\cos(\theta)\,
  12. Π = γ o - γ \Pi=\gamma_{o}-\gamma\,
  13. Π = surface pressure \Pi=\mathrm{surface\ pressure}
  14. γ = surface tension of subphase with monolayer \gamma=\mathrm{surface\ tension\ of\ subphase\ with\ monolayer}
  15. γ o = surface tension of pure subphase \gamma_{o}=\mathrm{surface\ tension\ of\ pure\ subphase}

Language_identification_in_the_limit.html

  1. s 0 , s 1 , s_{0},s_{1},...
  2. L 1 , L 2 , L_{1},L_{2},...
  3. s n L n s_{n}\not\in L_{n}
  4. L n L_{n}
  5. { s 1 , , s n - 1 } \{s_{1},...,s_{n-1}\}
  6. \mathcal{L}
  7. L i L_{i}
  8. \mathcal{L}
  9. T i T_{i}
  10. T 1 \sub T 2 \sub T_{1}\sub T_{2}\sub...
  11. T i L i T_{i}\in L_{i}
  12. T i + 1 L i T_{i+1}\not\in L_{i}
  13. lim n = T i = L \lim_{n=\infty}T_{i}=L

Language_model.html

  1. m m
  2. P ( w 1 , , w m ) P(w_{1},\ldots,w_{m})
  3. P ( Q M d ) P(Q\mid M_{d})
  4. P ( t 1 t 2 t 3 ) = P ( t 1 ) P ( t 2 t 1 ) P ( t 3 t 1 t 2 ) P(t_{1}t_{2}t_{3})=P(t_{1})P(t_{2}\mid t_{1})P(t_{3}\mid t_{1}t_{2})
  5. P uni ( t 1 t 2 t 3 ) = P ( t 1 ) P ( t 2 ) P ( t 3 ) P\text{uni}(t_{1}t_{2}t_{3})=P(t_{1})P(t_{2})P(t_{3})
  6. term in doc P ( term ) = 1 \sum_{\,\text{term in doc}}P(\,\text{term})=1\,
  7. P ( query ) = term in query P ( term ) P(\,\text{query})=\prod_{\,\text{term in query}}P(\,\text{term})
  8. P ( w 1 , , w m ) P(w_{1},\ldots,w_{m})
  9. w 1 , , w m w_{1},\ldots,w_{m}
  10. P ( w 1 , , w m ) = i = 1 m P ( w i w 1 , , w i - 1 ) i = 1 m P ( w i w i - ( n - 1 ) , , w i - 1 ) P(w_{1},\ldots,w_{m})=\prod^{m}_{i=1}P(w_{i}\mid w_{1},\ldots,w_{i-1})\approx% \prod^{m}_{i=1}P(w_{i}\mid w_{i-(n-1)},\ldots,w_{i-1})
  11. P ( w i w i - ( n - 1 ) , , w i - 1 ) = count ( w i - ( n - 1 ) , , w i - 1 , w i ) count ( w i - ( n - 1 ) , , w i - 1 ) P(w_{i}\mid w_{i-(n-1)},\ldots,w_{i-1})=\frac{\mathrm{count}(w_{i-(n-1)},% \ldots,w_{i-1},w_{i})}{\mathrm{count}(w_{i-(n-1)},\ldots,w_{i-1})}
  12. P ( I, saw, the, red, house ) \displaystyle P(\,\text{I, saw, the, red, house})
  13. P ( I, saw, the, red, house ) \displaystyle P(\,\text{I, saw, the, red, house})
  14. P ( w t | context ) t V P(w_{t}|\mathrm{context})\,\forall t\in V
  15. P ( w t | w t - k , , w t - 1 ) P(w_{t}|w_{t-k},\dots,w_{t-1})
  16. k k
  17. P ( w t | w t - k , , w t - 1 , w t + 1 , , w t + k ) P(w_{t}|w_{t-k},\dots,w_{t-1},w_{t+1},\dots,w_{t+k})
  18. - k j - 1 , j k log P ( w t + j | w t ) \sum_{-k\leq j-1,\,j\leq k}\log P(w_{t+j}|w_{t})
  19. n n
  20. n n
  21. v v
  22. w w
  23. n n
  24. v ( king ) - v ( male ) + v ( female ) v ( queen ) v(\mathrm{king})-v(\mathrm{male})+v(\mathrm{female})\approx v(\mathrm{queen})

Laplace–Beltrami_operator.html

  1. 2 f = f . \nabla^{2}f=\nabla\cdot\nabla f.
  2. vol n := | g | d x 1 d x n \operatorname{vol}_{n}:=\sqrt{|g|}\;dx^{1}\wedge\cdots\wedge dx^{n}
  3. i := x i \partial_{i}:=\frac{\partial}{\partial x^{i}}
  4. \wedge
  5. ( X ) vol n := L X vol n (\nabla\cdot X)\operatorname{vol}_{n}:=L_{X}\operatorname{vol}_{n}
  6. X = 1 | g | i ( | g | X i ) \nabla\cdot X=\frac{1}{\sqrt{|g|}}\partial_{i}\left(\sqrt{|g|}X^{i}\right)
  7. , \langle\cdot,\cdot\rangle
  8. grad f ( x ) , v x = d f ( x ) ( v x ) \langle\operatorname{grad}f(x),v_{x}\rangle=df(x)(v_{x})
  9. ( grad f ) i = i f = g i j j f \left(\operatorname{grad}f\right)^{i}=\partial^{i}f=g^{ij}\partial_{j}f
  10. 2 f = 1 | g | i ( | g | g i j j f ) . \nabla^{2}f=\frac{1}{\sqrt{|g|}}\partial_{i}\left(\sqrt{|g|}g^{ij}\partial_{j}% f\right).
  11. M d f ( X ) vol n = - M f X vol n \int_{M}df(X)\operatorname{vol}_{n}=-\int_{M}f\nabla\cdot X\operatorname{vol}_% {n}
  12. M f 2 h vol n = - M d f , d h vol n \int_{M}f\,\nabla^{2}h\,\operatorname{vol}_{n}=-\int_{M}\langle df,dh\rangle\,% \operatorname{vol}_{n}
  13. M f 2 h vol n = - M d f , d h vol n = M h 2 f vol n . \int_{M}f\,\nabla^{2}h\operatorname{vol}_{n}=-\int_{M}\langle df,dh\rangle% \operatorname{vol}_{n}=\int_{M}h\,\nabla^{2}f\operatorname{vol}_{n}.
  14. - 2 u = λ u , -\nabla^{2}u=\lambda u,
  15. u u
  16. λ \lambda
  17. λ \lambda
  18. λ \lambda
  19. λ = 0 \lambda=0
  20. - 2 -\nabla^{2}
  21. λ 0 \lambda\geq 0
  22. u u
  23. M M
  24. d V = vol n dV=\operatorname{vol}_{n}
  25. - M 2 u u d V = λ M u 2 d V -\int_{M}\nabla^{2}u\ u\ dV=\lambda\int_{M}u^{2}\ dV
  26. M M
  27. - M 2 u u d V = M | u | 2 d V -\int_{M}\nabla^{2}u\ u\ dV=\int_{M}|\nabla u|^{2}\ dV
  28. M | u | 2 d V = λ M u 2 d V \int_{M}|\nabla u|^{2}\ dV=\lambda\int_{M}u^{2}\ dV
  29. λ 0 \lambda\geq 0
  30. n 2 n\geq 2
  31. R i c ( X , X ) κ g ( X , X ) , κ > 0 , Ric(X,X)\geq\kappa g(X,X),\kappa>0,
  32. g ( , ) g(\cdot,\cdot)
  33. X X
  34. M M
  35. λ 1 \lambda_{1}
  36. λ 1 n n - 1 κ . \lambda_{1}\geq\frac{n}{n-1}\kappa.
  37. 𝕊 n \mathbb{S}^{n}
  38. 𝕊 2 \mathbb{S}^{2}
  39. λ 1 \lambda_{1}
  40. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  41. 3 \mathbb{R}^{3}
  42. 𝕊 2 \mathbb{S}^{2}
  43. ( θ , ϕ ) (\theta,\phi)
  44. 𝕊 2 \mathbb{S}^{2}
  45. x 3 = cos ϕ = u 1 , x_{3}=\cos\phi=u_{1},
  46. - 𝕊 2 2 u 1 = 2 u 1 -\nabla^{2}_{\mathbb{S}^{2}}u_{1}=2u_{1}
  47. λ 1 \lambda_{1}
  48. λ 1 = n n - 1 κ , \lambda_{1}=\frac{n}{n-1}\kappa,
  49. 𝕊 n ( 1 / κ ) \mathbb{S}^{n}(1/\sqrt{\kappa})
  50. 1 / κ 1/\sqrt{\kappa}
  51. n . \mathbb{C}^{n}.
  52. H ( f ) i j = H f ( X i , X j ) = X i X j f - X i X j f H(f)_{ij}=H_{f}(X_{i},X_{j})=\nabla_{X_{i}}\nabla_{X_{j}}f-\nabla_{\nabla_{X_{% i}}X_{j}}f
  53. 2 f = i j g i j H ( f ) i j . \nabla^{2}f=\sum_{ij}g^{ij}H(f)_{ij}.
  54. 2 f = a a f \nabla^{2}f=\nabla^{a}\nabla_{a}f
  55. 2 T = g i j ( X i X j T - X i X j T ) \nabla^{2}T=g^{ij}\left(\nabla_{X_{i}}\nabla_{X_{j}}T-\nabla_{\nabla_{X_{i}}X_% {j}}T\right)
  56. 2 = d δ + δ d = ( d + δ ) 2 , \nabla^{2}=\mathrm{d}\delta+\delta\mathrm{d}=(\mathrm{d}+\delta)^{2},\;
  57. 2 f = δ d f . \nabla^{2}f=\delta\mathrm{d}f.
  58. | g | = 1 |g|=1
  59. 2 f = 1 | g | i | g | i f = i i f \nabla^{2}f=\frac{1}{\sqrt{|g|}}\partial_{i}\sqrt{|g|}\partial^{i}f=\partial_{% i}\partial^{i}f
  60. S n - 1 2 f ( x ) = 2 f ( x / | x | ) \nabla^{2}_{S^{n-1}}f(x)=\nabla^{2}f(x/|x|)
  61. 2 f = r 1 - n r ( r n - 1 f r ) + r - 2 S n - 1 2 f . \nabla^{2}f=r^{1-n}\frac{\partial}{\partial r}\left(r^{n-1}\frac{\partial f}{% \partial r}\right)+r^{-2}\nabla^{2}_{S^{n-1}}f.
  62. ϕ \phi
  63. ϕ \phi
  64. S n - 1 2 f ( ξ , ϕ ) = ( sin ϕ ) 2 - n ϕ ( ( sin ϕ ) n - 2 f ϕ ) + ( sin ϕ ) - 2 ξ 2 f \nabla^{2}_{S^{n-1}}f(\xi,\phi)=(\sin\phi)^{2-n}\frac{\partial}{\partial\phi}% \left((\sin\phi)^{n-2}\frac{\partial f}{\partial\phi}\right)+(\sin\phi)^{-2}% \nabla^{2}_{\xi}f
  65. ξ 2 \nabla^{2}_{\xi}
  66. S 2 2 f ( θ , ϕ ) = ( sin ϕ ) - 1 ϕ ( sin ϕ f ϕ ) + ( sin ϕ ) - 2 2 θ 2 f \nabla^{2}_{S^{2}}f(\theta,\phi)=(\sin\phi)^{-1}\frac{\partial}{\partial\phi}% \left(\sin\phi\frac{\partial f}{\partial\phi}\right)+(\sin\phi)^{-2}\frac{% \partial^{2}}{\partial\theta^{2}}f
  67. q ( x ) = x 1 2 - x 2 2 - - x n 2 . q(x)=x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}.
  68. H n = { x q ( x ) = 1 , x 1 > 1 } . H^{n}=\{x\mid q(x)=1,x_{1}>1\}.\,
  69. H n - 1 2 f = f ( x / q ( x ) 1 / 2 ) | H n - 1 \nabla^{2}_{H^{n-1}}f=\Box f\left(x/q(x)^{1/2}\right)|_{H^{n-1}}
  70. f ( x / q ( x ) 1 / 2 ) f(x/q(x)^{1/2})
  71. = 2 x 1 2 - - 2 x n 2 . \Box=\frac{\partial^{2}}{\partial x_{1}^{2}}-\cdots-\frac{\partial^{2}}{% \partial x_{n}^{2}}.
  72. H n - 1 2 f ( t , ξ ) = sinh ( t ) 2 - n t ( sinh ( t ) n - 2 f t ) + sinh ( t ) - 2 ξ 2 f \nabla^{2}_{H^{n-1}}f(t,\xi)=\sinh(t)^{2-n}\frac{\partial}{\partial t}\left(% \sinh(t)^{n-2}\frac{\partial f}{\partial t}\right)+\sinh(t)^{-2}\nabla^{2}_{% \xi}f
  73. ξ 2 \nabla^{2}_{\xi}
  74. H 2 2 f ( r , θ ) = sinh ( r ) - 1 r ( sinh ( r ) f r ) + sinh ( r ) - 2 2 θ 2 f \nabla^{2}_{H^{2}}f(r,\theta)=\sinh(r)^{-1}\frac{\partial}{\partial r}\left(% \sinh(r)\frac{\partial f}{\partial r}\right)+\sinh(r)^{-2}\frac{\partial^{2}}{% \partial\theta^{2}}f

Laplacian_matrix.html

  1. L n × n L_{n\times n}
  2. L = D - A , L=D-A,
  3. L L
  4. L i , j := { deg ( v i ) if i = j - 1 if i j and v i is adjacent to v j 0 otherwise L_{i,j}:=\begin{cases}\deg(v_{i})&\mbox{if}~{}\ i=j\\ -1&\mbox{if}~{}\ i\neq j\ \mbox{and}~{}\ v_{i}\mbox{ is adjacent to }~{}v_{j}% \\ 0&\mbox{otherwise}\end{cases}
  5. L sym := D - 1 / 2 L D - 1 / 2 = I - D - 1 / 2 A D - 1 / 2 L^{\,\text{sym}}:=D^{-1/2}LD^{-1/2}=I-D^{-1/2}AD^{-1/2}
  6. L sym L^{\,\text{sym}}
  7. L i , j sym := { 1 if i = j and deg ( v i ) 0 - 1 deg ( v i ) deg ( v j ) if i j and v i is adjacent to v j 0 otherwise . L^{\,\text{sym}}_{i,j}:=\begin{cases}1&\mbox{if}~{}\ i=j\ \mbox{and}~{}\ \deg(% v_{i})\neq 0\\ -\frac{1}{\sqrt{\deg(v_{i})\deg(v_{j})}}&\mbox{if}~{}\ i\neq j\ \mbox{and}~{}% \ v_{i}\mbox{ is adjacent to }~{}v_{j}\\ 0&\mbox{otherwise}~{}.\end{cases}
  8. L rw := D - 1 L = I - D - 1 A L^{\,\text{rw}}:=D^{-1}L=I-D^{-1}A
  9. L rw L^{\,\text{rw}}
  10. L i , j rw := { 1 if i = j and deg ( v i ) 0 - 1 deg ( v i ) if i j and v i is adjacent to v j 0 otherwise . L^{\,\text{rw}}_{i,j}:=\begin{cases}1&\mbox{if}~{}\ i=j\ \mbox{and}~{}\ \deg(v% _{i})\neq 0\\ -\frac{1}{\deg(v_{i})}&\mbox{if}~{}\ i\neq j\ \mbox{and}~{}\ v_{i}\mbox{ is % adjacent to }~{}v_{j}\\ 0&\mbox{otherwise}~{}.\end{cases}
  11. ( 2 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 ) \left(\begin{array}[]{rrrrrr}2&0&0&0&0&0\\ 0&3&0&0&0&0\\ 0&0&2&0&0&0\\ 0&0&0&3&0&0\\ 0&0&0&0&3&0\\ 0&0&0&0&0&1\\ \end{array}\right)
  12. ( 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 0 1 0 0 ) \left(\begin{array}[]{rrrrrr}0&1&0&0&1&0\\ 1&0&1&0&1&0\\ 0&1&0&1&0&0\\ 0&0&1&0&1&1\\ 1&1&0&1&0&0\\ 0&0&0&1&0&0\\ \end{array}\right)
  13. ( 2 - 1 0 0 - 1 0 - 1 3 - 1 0 - 1 0 0 - 1 2 - 1 0 0 0 0 - 1 3 - 1 - 1 - 1 - 1 0 - 1 3 0 0 0 0 - 1 0 1 ) \left(\begin{array}[]{rrrrrr}2&-1&0&0&-1&0\\ -1&3&-1&0&-1&0\\ 0&-1&2&-1&0&0\\ 0&0&-1&3&-1&-1\\ -1&-1&0&-1&3&0\\ 0&0&0&-1&0&1\\ \end{array}\right)
  14. λ 0 λ 1 λ n - 1 \lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{n-1}
  15. λ i 0 \lambda_{i}\geq 0
  16. λ 0 = 0 \lambda_{0}=0
  17. 𝐯 0 = ( 1 , 1 , , 1 ) \mathbf{v}_{0}=(1,1,\dots,1)
  18. L 𝐯 0 = 0. L\mathbf{v}_{0}=\mathbf{0}.
  19. \mathbb{R}
  20. = 1 k L = I - 1 k A \mathcal{L}=\tfrac{1}{k}L=I-\tfrac{1}{k}A
  21. | e | |e|
  22. | v | |v|
  23. M e v = { 1 , if v = i - 1 , if v = j 0 , otherwise . M_{ev}=\left\{\begin{array}[]{rl}1,&\,\text{if}\,v=i\\ -1,&\,\text{if}\,v=j\\ 0,&\,\text{otherwise}.\end{array}\right.
  24. L = M T M , L=M\text{T}M\,,
  25. M T M\text{T}
  26. L L
  27. 𝐯 i \mathbf{v}_{i}
  28. λ i \lambda_{i}
  29. λ i \displaystyle\lambda_{i}
  30. λ i \lambda_{i}
  31. M 𝐯 i M\mathbf{v}_{i}
  32. λ i 0 \lambda_{i}\geq 0
  33. L L
  34. Δ ( s ) = I - s A + s 2 ( D - I ) \Delta(s)=I-sA+s^{2}(D-I)
  35. Δ ( 1 ) \Delta(1)
  36. L sym := D - 1 / 2 L D - 1 / 2 = I - D - 1 / 2 A D - 1 / 2 L^{\,\text{sym}}:=D^{-1/2}LD^{-1/2}=I-D^{-1/2}AD^{-1/2}
  37. D - 1 / 2 D^{-1/2}
  38. L sym = S S * L^{\,\text{sym}}=SS^{*}
  39. 1 d u \frac{1}{\sqrt{d}_{u}}
  40. - 1 d v -\frac{1}{\sqrt{d}_{v}}
  41. S * S^{*}
  42. L sym L^{\,\text{sym}}
  43. L sym L^{\,\text{sym}}
  44. g = D 1 / 2 f g=D^{1/2}f
  45. λ = g , L sym g g , g = g , D - 1 / 2 L D - 1 / 2 g g , g = f , L f D 1 / 2 f , D 1 / 2 f = u v ( f ( u ) - f ( v ) ) 2 v f ( v ) 2 d v > 0 , \lambda\ {}=\ {}\frac{\langle g,L^{\,\text{sym}}g\rangle}{\langle g,g\rangle}% \ {}=\ {}\frac{\langle g,D^{-1/2}LD^{-1/2}g\rangle}{\langle g,g\rangle}\ {}=\ % {}\frac{\langle f,Lf\rangle}{\langle D^{1/2}f,D^{1/2}f\rangle}\ {}=\ {}\frac{% \sum_{u\sim v}(f(u)-f(v))^{2}}{\sum_{v}f(v)^{2}d_{v}}\ >\ 0,
  46. f , g = v f ( v ) g ( v ) \langle f,g\rangle=\sum_{v}f(v)g(v)
  47. u v \sum_{u\sim v}
  48. u , v ( f ( u ) - f ( v ) ) 2 \sum_{u,v}(f(u)-f(v))^{2}
  49. g , L sym g g , g \frac{\langle g,L^{\,\text{sym}}g\rangle}{\langle g,g\rangle}
  50. D 1 / 2 1 D^{1/2}1
  51. L sym L^{\,\text{sym}}
  52. L rw := D - 1 A L^{\,\text{rw}}:=D^{-1}A
  53. D - 1 D^{-1}
  54. L i , i rw L^{\,\text{rw}}_{i,i}
  55. L rw L^{\,\text{rw}}
  56. L i , j rw := { 1 if i = j and deg ( v i ) 0 - 1 deg ( v i ) if i j and v i is adjacent to v j 0 otherwise . L^{\,\text{rw}}_{i,j}:=\begin{cases}1&\mbox{if}~{}\ i=j\ \mbox{and}~{}\ \deg(v% _{i})\neq 0\\ -\frac{1}{\deg(v_{i})}&\mbox{if}~{}\ i\neq j\ \mbox{and}~{}\ v_{i}\mbox{ is % adjacent to }~{}v_{j}\\ 0&\mbox{otherwise}~{}.\end{cases}
  57. e i e_{i}
  58. x = e i L rw x=e_{i}L^{\,\text{rw}}
  59. i i
  60. x j = ( v i v j ) x_{j}=\mathbb{P}(v_{i}\to v_{j})
  61. x x
  62. x = x ( L rw ) t x^{\prime}=x(L^{\,\text{rw}})^{t}
  63. t t
  64. L rw = D - 1 2 ( I - L sym ) D 1 2 L^{\,\text{rw}}=D^{-\frac{1}{2}}\left(I-L^{\,\text{sym}}\right)D^{\frac{1}{2}}
  65. L rw L^{\,\text{rw}}
  66. L sym L^{\,\text{sym}}
  67. L rw L^{\,\text{rw}}
  68. L sym L^{\,\text{sym}}
  69. I - D - 1 A I-D^{-1}A
  70. p i ( t ) = j A i j d e g ( v j ) p j ( t - 1 ) , p_{i}(t)=\sum_{j}\frac{A_{ij}}{deg(v_{j})}p_{j}(t-1),
  71. p ( t ) = A D - 1 p ( t - 1 ) . p(t)=AD^{-1}p(t-1).
  72. t t\rightarrow\infty
  73. p = A D - 1 p p=AD^{-1}p
  74. D - 1 2 p ( t ) = [ D - 1 2 A D - 1 2 ] D - 1 2 p ( t - 1 ) . \begin{aligned}\displaystyle D^{-\frac{1}{2}}p(t)&\displaystyle=\left[D^{-% \frac{1}{2}}AD^{-\frac{1}{2}}\right]D^{-\frac{1}{2}}p(t-1).\end{aligned}
  75. A r e d u c e d D - 1 2 A D - 1 2 A_{reduced}\equiv D^{-\frac{1}{2}}AD^{-\frac{1}{2}}
  76. A r e d u c e d A_{reduced}
  77. A r e d u c e d A_{reduced}
  78. ϕ i \phi_{i}
  79. d ϕ i d t = - k j A i j ( ϕ i - ϕ j ) = - k ϕ i j A i j + k j A i j ϕ j = - k ϕ i d e g ( v i ) + k j A i j ϕ j = - k j ( δ i j d e g ( v i ) - A i j ) ϕ j = - k j ( i j ) ϕ j . \begin{aligned}\displaystyle\frac{d\phi_{i}}{dt}&\displaystyle=-k\sum_{j}A_{ij% }(\phi_{i}-\phi_{j})\\ &\displaystyle=-k\phi_{i}\sum_{j}A_{ij}+k\sum_{j}A_{ij}\phi_{j}\\ &\displaystyle=-k\phi_{i}\ deg(v_{i})+k\sum_{j}A_{ij}\phi_{j}\\ &\displaystyle=-k\sum_{j}(\delta_{ij}\ deg(v_{i})-A_{ij})\phi_{j}\\ &\displaystyle=-k\sum_{j}(\ell_{ij})\phi_{j}.\end{aligned}
  80. d ϕ d t = - k ( D - A ) ϕ = - k L ϕ , \begin{aligned}\displaystyle\frac{d\phi}{dt}&\displaystyle=-k(D-A)\phi\\ &\displaystyle=-kL\phi,\end{aligned}
  81. d ϕ d t + k L ϕ = 0. \begin{aligned}\displaystyle\frac{d\phi}{dt}+kL\phi=0.\end{aligned}
  82. 2 \nabla^{2}
  83. ϕ \phi
  84. 𝐯 i \mathbf{v}_{i}
  85. L 𝐯 i = λ i 𝐯 i L\mathbf{v}_{i}=\lambda_{i}\mathbf{v}_{i}
  86. ϕ = i c i 𝐯 i . \begin{aligned}\displaystyle\phi=\sum_{i}c_{i}\mathbf{v}_{i}.\end{aligned}
  87. 𝐯 i \mathbf{v}_{i}
  88. d ( i c i 𝐯 i ) d t + k L ( i c i 𝐯 i ) \displaystyle\frac{d(\sum_{i}c_{i}\mathbf{v}_{i})}{dt}+kL(\sum_{i}c_{i}\mathbf% {v}_{i})
  89. c i ( t ) = c i ( 0 ) exp ( - k λ i t ) . \begin{aligned}\displaystyle c_{i}(t)=c_{i}(0)\exp(-k\lambda_{i}t).\end{aligned}
  90. λ i \lambda_{i}
  91. λ i \lambda_{i}
  92. c i ( 0 ) c_{i}(0)
  93. c i ( 0 ) c_{i}(0)
  94. i i
  95. ϕ ( 0 ) \phi(0)
  96. ϕ ( 0 ) \phi(0)
  97. 𝐯 i \mathbf{v}_{i}
  98. c i ( 0 ) = ϕ ( 0 ) , 𝐯 i c_{i}(0)=\langle\phi(0),\mathbf{v}_{i}\rangle
  99. L L
  100. L L
  101. lim t ϕ ( t ) \lim_{t\to\infty}\phi(t)
  102. c i ( t ) = c i ( 0 ) exp ( - k λ i t ) c_{i}(t)=c_{i}(0)\exp(-k\lambda_{i}t)
  103. λ i = 0 \lambda_{i}=0
  104. lim t exp ( - k λ i t ) = { 0 if λ i > 0 1 if λ i = 0 } \lim_{t\to\infty}\exp(-k\lambda_{i}t)=\left\{\begin{array}[]{rlr}0&\,\text{if}% &\lambda_{i}>0\\ 1&\,\text{if}&\lambda_{i}=0\end{array}\right\}
  105. L L
  106. j L i j = 0 \sum_{j}L_{ij}=0
  107. 𝐯 1 \mathbf{v}^{1}
  108. k k
  109. k k
  110. λ = 0 \lambda=0
  111. c ( 0 ) c(0)
  112. N N
  113. lim t ϕ ( t ) = c ( 0 ) , 𝐯 𝟏 𝐯 𝟏 \lim_{t\to\infty}\phi(t)=\langle c(0),\mathbf{v^{1}}\rangle\mathbf{v^{1}}
  114. 𝐯 𝟏 = 1 N [ 1 , 1 , , 1 ] \mathbf{v^{1}}=\frac{1}{\sqrt{N}}[1,1,...,1]
  115. ϕ j \phi_{j}
  116. ϕ \phi
  117. j j
  118. lim t ϕ j ( t ) = 1 N i = 1 N c i ( 0 ) \lim_{t\to\infty}\phi_{j}(t)=\frac{1}{N}\sum_{i=1}^{N}c_{i}(0)
  119. ϕ \phi
  120. ϕ \phi