wpmath0000001_17

Phase_(waves).html

  1. x ( t ) \displaystyle x(t)
  2. A \scriptstyle A\,
  3. f \scriptstyle f\,
  4. φ \scriptstyle\varphi\,
  5. T = 1 f \scriptstyle T=\frac{1}{f}\,
  6. T 4 \scriptstyle\frac{T}{4}\,
  7. t \scriptstyle t\,
  8. cos ( 2 π f t ) \scriptstyle\cos(2\pi ft)\,
  9. x ( t ) \scriptstyle x(t)\,
  10. φ \scriptstyle\varphi\,
  11. y ( t ) \scriptstyle y(t)\,
  12. φ - π 2 \scriptstyle\varphi\,-\,\frac{\pi}{2}\,
  13. φ \scriptstyle\varphi\,
  14. x ( t ) \scriptstyle x(t)\,
  15. y ( t ) \scriptstyle y(t)\,
  16. 2 π f t + φ \scriptstyle 2\pi ft\,+\,\varphi
  17. φ \scriptstyle\varphi\,
  18. φ \scriptstyle\varphi\,
  19. x ( t ) \scriptstyle x(t)\,
  20. 1 4 \scriptstyle\frac{1}{4}\,
  21. x ( t - 1 4 T ) \displaystyle x\left(t-\tfrac{1}{4}T\right)
  22. φ - π 2 \scriptstyle\varphi\,-\,\frac{\pi}{2}
  23. π 2 \scriptstyle\frac{\pi}{2}

Phase_angle.html

  1. A \ang θ , A\ang\!\ \theta,

Phase_modulation.html

  1. m ( t ) m(t)
  2. c ( t ) = A c sin ( ω c t + ϕ c ) . c(t)=A_{c}\sin\left(\omega_{\mathrm{c}}t+\phi_{\mathrm{c}}\right).
  3. y ( t ) = A c sin ( ω c t + m ( t ) + ϕ c ) . y(t)=A_{c}\sin\left(\omega_{\mathrm{c}}t+m(t)+\phi_{\mathrm{c}}\right).
  4. m ( t ) m(t)
  5. m ( t ) = cos ( ω c t + k ω m ( t ) ) m(t)=\cos\left(\omega_{\mathrm{c}}t+k\omega_{\mathrm{m}}(t)\right)
  6. 2 ( h + 1 ) f M 2\left(h+1\right)f_{\mathrm{M}}
  7. f M = ω m / 2 π f_{\mathrm{M}}=\omega_{\mathrm{m}}/2\pi
  8. h h
  9. h = Δ θ h\,=\Delta\theta\,
  10. Δ θ \Delta\theta

Phase_noise.html

  1. ( f ) ℒ(f)
  2. Jitter (seconds ) = Phase error (degrees) / ( 360 × Frequency (Hertz) ) \,\text{Jitter (seconds})=\,\text{Phase error (degrees)}/(360\times\,\text{% Frequency (Hertz)})
  3. σ c 2 = f 2 ( f ) f o s c 3 \sigma^{2}_{c}=\frac{f^{2}\mathcal{L}\left(f\right)}{f_{osc}^{3}}
  4. ( f ) = f o s c 3 σ c 2 f 2 \mathcal{L}\left(f\right)=\frac{f_{osc}^{3}\sigma^{2}_{c}}{f^{2}}

Phase_transition.html

  1. T T
  2. T T
  3. C C
  4. C | T c - T | - α . C\propto|T_{c}-T|^{-\alpha}.
  5. ν ν
  6. α α
  7. ν ν
  8. α α
  9. α α
  10. α α
  11. α α
  12. α α
  13. β , γ , δ , ν β,γ,δ,ν
  14. η η
  15. β = γ / ( δ - 1 ) , ν = γ / ( 2 - η ) \beta=\gamma/(\delta-1),\qquad\nu=\gamma/(2-\eta)
  16. ν ν
  17. η η

Phase_velocity.html

  1. v p = λ T . v_{\mathrm{p}}=\frac{\lambda}{T}.
  2. v p = ω k . v_{\mathrm{p}}=\frac{\omega}{k}.
  3. cos [ ( k - Δ k ) x - ( ω - Δ ω ) t ] + cos [ ( k + Δ k ) x - ( ω + Δ ω ) t ] = 2 cos ( Δ k x - Δ ω t ) cos ( k x - ω t ) , \cos[(k-\Delta k)x-(\omega-\Delta\omega)t]\;+\;\cos[(k+\Delta k)x-(\omega+% \Delta\omega)t]=2\;\cos(\Delta kx-\Delta\omega t)\;\cos(kx-\omega t),
  4. d ω d k = c n - c k n 2 d n d k . \frac{\,\text{d}\omega}{\,\text{d}k}=\frac{c}{n}-\frac{ck}{n^{2}}\cdot\frac{\,% \text{d}n}{\,\text{d}k}.

Phased_array.html

  1. ψ = ψ 0 ( sin ( < m t p l > π a λ sin θ ) π a λ sin θ ) ( sin ( N 2 k d sin θ ) sin ( k d 2 sin θ ) ) \psi={{\psi}_{0}}\left(\frac{\sin\left(\frac{<}{m}tpl>{{\pi a}}{\lambda}\sin% \theta\right)}{\frac{{\pi a}}{\lambda}\sin\theta}\right)\left(\frac{\sin\left(% \frac{N}{2}{kd}\sin\theta\right)}{\sin\left(\frac{{kd}}{2}\sin\theta\right)}\right)
  2. k d sin θ \begin{matrix}kd\sin\theta\end{matrix}
  3. ψ = ψ 0 ( sin ( < m t p l > π a λ sin θ ) π a λ sin θ ) ( sin ( N 2 ( 2 π d λ sin θ + ϕ ) ) sin ( π d λ sin θ + ϕ ) ) \psi={{\psi}_{0}}\left(\frac{\sin\left(\frac{<}{m}tpl>{{\pi a}}{\lambda}\sin% \theta\right)}{\frac{{\pi a}}{\lambda}\sin\theta}\right)\left(\frac{\sin\left(% \frac{N}{2}\big(\frac{{2\pi d}}{\lambda}\sin\theta+\phi\big)\right)}{\sin\left% (\frac{{\pi d}}{\lambda}\sin\theta+\phi\right)}\right)
  4. I = I 0 ( sin ( π a λ sin θ ) < m t p l > π a λ sin θ ) 2 ( sin ( N 2 ( 2 π d λ sin θ + ϕ ) ) sin ( π d λ sin θ + ϕ ) ) 2 I=I_{0}{{\left(\frac{\sin\left(\frac{\pi a}{\lambda}\sin\theta\right)}{\frac{<% }{m}tpl>{{\pi a}}{\lambda}\sin\theta}\right)}^{2}}{{\left(\frac{\sin\left(% \frac{N}{2}(\frac{2\pi d}{\lambda}\sin\theta+\phi)\right)}{\sin\left(\frac{{% \pi d}}{\lambda}\sin\theta+\phi\right)}\right)}^{2}}
  5. I = I 0 ( sin ( < m t p l > π a λ sin θ ) π a λ sin θ ) 2 ( sin ( π λ N d sin θ + N 2 ϕ ) sin ( π d λ sin θ + ϕ ) ) 2 I=I_{0}{{\left(\frac{\sin\left(\frac{<}{m}tpl>{{\pi a}}{\lambda}\sin\theta% \right)}{\frac{{\pi a}}{\lambda}\sin\theta}\right)}^{2}}{{\left(\frac{\sin% \left(\frac{\pi}{\lambda}Nd\sin\theta+\frac{N}{2}\phi\right)}{\sin\left(\frac{% {\pi d}}{\lambda}\sin\theta+\phi\right)}\right)}^{2}}
  6. d = λ 4 d=\begin{matrix}\frac{\lambda}{4}\end{matrix}
  7. I = I 0 ( sin ( π a λ sin θ ) π a λ sin θ ) 2 ( sin ( π 4 N sin θ + N 2 ϕ ) sin ( π 4 sin θ + ϕ ) ) 2 I=I_{0}{{\left(\frac{\sin\left(\frac{\pi a}{\lambda}\sin\theta\right)}{\frac{% \pi a}{\lambda}\sin\theta}\right)}^{2}}{{\left(\frac{\sin\left(\frac{\pi}{4}N% \sin\theta+\frac{N}{2}\phi\right)}{\sin\left(\frac{\pi}{4}\sin\theta+\phi% \right)}\right)}^{2}}
  8. π 2 \begin{matrix}\frac{\pi}{2}\end{matrix}
  9. π 4 N sin θ + N 2 ϕ = π 2 \frac{\pi}{4}N\sin\theta+\frac{N}{2}\phi=\frac{\pi}{2}
  10. sin θ = ( π 2 - N 2 ϕ ) 4 N π \sin\theta=\left(\frac{\pi}{2}-\frac{N}{2}\phi\right)\frac{4}{N\pi}
  11. sin θ = 2 N - 2 ϕ π \sin\theta=\frac{2}{N}-\frac{2\phi}{\pi}
  12. 2 ϕ π \begin{matrix}\frac{2\phi}{\pi}\end{matrix}
  13. ϕ = - π 2 \phi=-\begin{matrix}\frac{\pi}{2}\end{matrix}
  14. θ = sin - 1 ( 1 ) = π 2 = 90 \theta=\sin^{-1}(1)=\begin{matrix}\frac{\pi}{2}\end{matrix}=90^{\circ}

Phidias.html

  1. φ \varphi

Phosphate.html

  1. K a 1 = [ H ] + [ H PO 2 ] 4 - [ H PO 3 ] 4 7.5 × 10 - 3 K_{a1}=\frac{[\mbox{H}~{}^{+}][\mbox{H}~{}_{2}\mbox{PO}~{}_{4}^{-}]}{[\mbox{H}% ~{}_{3}\mbox{PO}~{}_{4}]}\simeq 7.5\times 10^{-3}
  2. K a 2 = [ H ] + [ HPO ] 4 2 - [ H PO 2 ] 4 - 6.2 × 10 - 8 K_{a2}=\frac{[\mbox{H}~{}^{+}][\mbox{HPO}~{}_{4}^{2-}]}{[\mbox{H}~{}_{2}\mbox{% PO}~{}_{4}^{-}]}\simeq 6.2\times 10^{-8}
  3. K a 3 = [ H ] + [ PO ] 4 3 - [ HPO ] 4 2 - 2.14 × 10 - 13 K_{a3}=\frac{[\mbox{H}~{}^{+}][\mbox{PO}~{}_{4}^{3-}]}{[\mbox{HPO}~{}_{4}^{2-}% ]}\simeq 2.14\times 10^{-13}
  4. [ H PO 2 ] 4 - [ H PO 3 ] 4 7.5 × 10 4 , [ HPO ] 4 2 - [ H PO 2 ] 4 - 0.62 , [ PO ] 4 3 - [ HPO ] 4 2 - 2.14 × 10 - 6 \frac{[\mbox{H}~{}_{2}\mbox{PO}~{}_{4}^{-}]}{[\mbox{H}~{}_{3}\mbox{PO}~{}_{4}]% }\simeq 7.5\times 10^{4}\mbox{ , }~{}\frac{[\mbox{HPO}~{}_{4}^{2-}]}{[\mbox{H}% ~{}_{2}\mbox{PO}~{}_{4}^{-}]}\simeq 0.62\mbox{ , }~{}\frac{[\mbox{PO}~{}_{4}^{% 3-}]}{[\mbox{HPO}~{}_{4}^{2-}]}\simeq 2.14\times 10^{-6}

Photodiode.html

  1. τ = R C \tau=RC
  2. D D
  3. D D^{\star}
  4. A A
  5. D = D A D^{\star}=D\sqrt{A}
  6. D D^{\star}

Photoelectric_effect.html

  1. K max K_{\max}
  2. K max = h f - φ , K_{\max}=h\,f-\varphi,
  3. h h
  4. f f
  5. φ \varphi
  6. W W
  7. ϕ \phi
  8. φ = h f 0 , \varphi=h\,f_{0},
  9. f 0 f_{0}
  10. K max = h ( f - f 0 ) . K_{\max}=h\left(f-f_{0}\right).
  11. f > f 0 f>f_{0}
  12. K max K_{\max}
  13. V 0 V_{0}
  14. q e V 0 q_{e}V_{0}
  15. q e V 0 = K max . q_{e}V_{0}=K_{\max}.
  16. K max = h ( f - f 0 ) , K_{\max}=h\left(f-f_{0}\right),
  17. σ = constant Z n E 3 \sigma=\mathrm{constant}\cdot\frac{Z^{n}}{E^{3}}

Photolithography.html

  1. C D = k 1 λ N A CD=k_{1}\cdot\frac{\lambda}{NA}
  2. C D \,CD
  3. k 1 \,k_{1}
  4. λ \,\lambda
  5. N A \,NA
  6. D F = k 2 λ N A 2 D_{F}=k_{2}\cdot\frac{\lambda}{{NA}^{2}}
  7. k 2 \,k_{2}

Photoluminescence.html

  1. 𝐤 \mathbf{k}

Photon.html

  1. A i j A_{ij}
  2. B j i B_{ji}
  3. B i j B_{ij}
  4. B i j B_{ij}
  5. h ν h\nu
  6. ν \nu
  7. E = n h ν E=nh\nu
  8. ν \nu
  9. E = n h ν E=nh\nu
  10. n n
  11. h ν h\nu
  12. A i j A_{ij}
  13. B i j B_{ij}
  14. E = p c E=pc
  15. | n k 0 | n k 1 | n k n |n_{k_{0}}\rangle\otimes|n_{k_{1}}\rangle\otimes\dots\otimes|n_{k_{n}}\rangle\dots
  16. | n k i |n_{k_{i}}\rangle
  17. n k i \,n_{k_{i}}
  18. k i k_{i}
  19. k i k_{i}
  20. | n k i | n k i + 1 |n_{k_{i}}\rangle\rightarrow|n_{k_{i}}+1\rangle
  21. ± \pm\hbar
  22. E E
  23. E / c 2 {E}/{c^{2}}
  24. E / c 2 {E}/{c^{2}}
  25. E = h ν E=h\nu

Phrase_structure_rules.html

  1. A B C A\to B\quad C
  2. A A
  3. B B
  4. C C
  5. S N P V P S\to NP\quad VP
  6. N P ( D e t ) N 1 NP\to(Det)\quad N1
  7. N 1 ( A P ) N 1 ( P P ) N1\to(AP)\quad N1\quad(PP)

Physical_constant.html

  1. K J - 90 K_{\mathrm{J-90}}\,
  2. R K - 90 R_{\mathrm{K-90}}\,
  3. M u = M ( C 12 ) / 12 M_{\mathrm{u}}=M({}^{12}\mathrm{C})/12\,
  4. M ( C 12 ) = N A m ( C 12 ) M({}^{12}\mathrm{C})=N_{\mathrm{A}}m({}^{12}\mathrm{C})\,
  5. g n g_{\mathrm{n}}\,\!
  6. atm \mathrm{atm}\,

Physical_cosmology.html

  1. 1 / H 1/H
  2. H H
  3. 1 / H 1/H

Physical_quantity.html

  1. u \vec{u}\,\!
  2. 𝐀 A 𝐧 ^ , 𝐒 S 𝐧 ^ \mathbf{A}\equiv A\mathbf{\hat{n}},\quad\mathbf{S}\equiv S\mathbf{\hat{n}}\,\!
  3. 𝐭 ^ \mathbf{\hat{t}}
  4. X X ( x 1 , x 2 x n ) X\equiv X\left(x_{1},x_{2}\cdots x_{n}\right)
  5. d n x d V n d x 1 d x 2 d x n \mathrm{d}^{n}x\equiv\mathrm{d}V_{n}\equiv\mathrm{d}x_{1}\mathrm{d}x_{2}\cdots% \mathrm{d}x_{n}
  6. X d n x X d V n X d x 1 d x 2 d x n \int X\mathrm{d}^{n}x\equiv\int X\mathrm{d}V_{n}\equiv\int\cdots\int\int X% \mathrm{d}x_{1}\mathrm{d}x_{2}\cdots\mathrm{d}x_{n}\,\!
  7. q ˙ \dot{q}\,\!
  8. q ˙ d q d t \dot{q}\equiv\frac{\mathrm{d}q}{\mathrm{d}t}
  9. q = ρ n d V n q=\int\rho_{n}\mathrm{d}V_{n}
  10. q m = d q d m q_{m}=\frac{\mathrm{d}q}{\mathrm{d}m}\,\!
  11. q n = d q d n q_{n}=\frac{\mathrm{d}q}{\mathrm{d}n}\,\!
  12. q \nabla q
  13. q = q λ d λ q=\int q_{\lambda}\mathrm{d}\lambda
  14. q = q ν d ν q=\int q_{\nu}\mathrm{d}\nu
  15. q = F d A d t q=\iiint F\mathrm{d}A\mathrm{d}t
  16. Φ F = S 𝐅 d 𝐀 \Phi_{F}=\iint_{S}\mathbf{F}\cdot\mathrm{d}\mathbf{A}
  17. 𝐅 𝐧 ^ = d Φ F d A \mathbf{F}\cdot\mathbf{\hat{n}}=\frac{\mathrm{d}\Phi_{F}}{\mathrm{d}A}\,\!
  18. I = d q d t I=\frac{\mathrm{d}q}{\mathrm{d}t}
  19. I = 𝐉 d 𝐒 I=\iint\mathbf{J}\cdot\mathrm{d}\mathbf{S}
  20. 𝐦 = 𝐫 q \mathbf{m}=\mathbf{r}q
  21. 𝐦 = 𝐫 × 𝐪 \mathbf{m}=\mathbf{r}\times\mathbf{q}

Physics.html

  1. Ξ b - \Xi_{b}^{-}

Physisorption.html

  1. V = e 2 4 π ε 0 ( - 1 | 2 𝐑 | + - 1 | 2 𝐑 + 𝐫 - 𝐫 | + 1 | 2 𝐑 - 𝐫 | + 1 | 2 𝐑 + 𝐫 | ) . V={e^{2}\over 4\pi\varepsilon_{0}}\left(\frac{-1}{|2\mathbf{R}|}+\frac{-1}{|2% \mathbf{R}+\mathbf{r}-\mathbf{r}^{\prime}|}+\frac{1}{|2\mathbf{R}-\mathbf{r}^{% \prime}|}+\frac{1}{|2\mathbf{R}+\mathbf{r}|}\right).
  2. V = - e 2 16 π ε 0 Z 3 ( x 2 + y 2 2 + z 2 ) + 3 e 2 32 π ε 0 Z 4 ( x 2 + y 2 2 z + z 3 ) + O ( 1 Z 5 ) . V={-e^{2}\over 16\pi\varepsilon_{0}Z^{3}}\left(\frac{x^{2}+y^{2}}{2}+z^{2}% \right)+{3e^{2}\over 32\pi\varepsilon_{0}Z^{4}}\left(\frac{x^{2}+y^{2}}{2}{z}+% z^{3}\right)+O\left(\frac{1}{Z^{5}}\right).
  3. V a = m e 2 ω 2 ( x 2 + y 2 + z 2 ) , V_{a}=\frac{m_{e}}{2}{\omega^{2}}(x^{2}+y^{2}+z^{2}),
  4. V a = m e 2 ω 2 ( x 2 + y 2 + z 2 ) - e 2 16 π ε 0 Z 3 ( x 2 + y 2 2 + z 2 ) + V_{a}=\frac{m_{e}}{2}{\omega^{2}}(x^{2}+y^{2}+z^{2})-{e^{2}\over 16\pi% \varepsilon_{0}Z^{3}}\left(\frac{x^{2}+y^{2}}{2}+z^{2}\right)+\ldots
  5. m e ω 2 e 2 16 π ε 0 Z 3 , m_{e}\omega^{2}>>{e^{2}\over 16\pi\varepsilon_{0}Z^{3}},
  6. V a m e 2 ω 1 2 ( x 2 + y 2 ) + m e 2 ω 2 2 z 2 V_{a}\sim\frac{m_{e}}{2}{\omega_{1}^{2}}(x^{2}+y^{2})+\frac{m_{e}}{2}{\omega_{% 2}^{2}}z^{2}
  7. ω 1 = ω - e 2 32 π ε 0 m e ω Z 3 , ω 2 = ω - e 2 16 π ε 0 m e ω Z 3 . \begin{aligned}\displaystyle\omega_{1}&\displaystyle=\omega-{e^{2}\over 32\pi% \varepsilon_{0}m_{e}\omega Z^{3}},\\ \displaystyle\omega_{2}&\displaystyle=\omega-{e^{2}\over 16\pi\varepsilon_{0}m% _{e}\omega Z^{3}}.\end{aligned}
  8. V v = 2 ( 2 ω 1 + ω 2 - 3 ω ) = - e 2 16 π ε 0 m e ω Z 3 . V_{v}=\frac{\hbar}{2}(2\omega_{1}+\omega_{2}-3\omega)=-{\hbar e^{2}\over 16\pi% \varepsilon_{0}m_{e}\omega Z^{3}}.
  9. α = e 2 m e ω 2 , \alpha=\frac{e^{2}}{m_{e}\omega^{2}},
  10. V v = - α ω 16 π ε 0 Z 3 = - C v Z 3 , V_{v}=-{\hbar\alpha\omega\over 16\pi\varepsilon_{0}Z^{3}}=-\frac{C_{v}}{Z^{3}},
  11. C v = α ω 16 π ε 0 , C_{v}={\hbar\alpha\omega\over 16\pi\varepsilon_{0}},
  12. V v = - C v ( Z - Z 0 ) 3 + O ( 1 Z 5 ) . V_{v}=-\frac{C_{v}}{(Z-Z_{0})^{3}}+O\left(\frac{1}{Z^{5}}\right).

Pi.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi
  6. π \pi
  7. π \pi
  8. π \pi
  9. π \pi
  10. π \pi
  11. π \pi
  12. π \pi
  13. π \pi
  14. π \pi
  15. π \pi
  16. π \pi
  17. π \pi
  18. \PI \PI
  19. π \pi
  20. π \pi
  21. C C
  22. d d
  23. π = C d \pi=\frac{C}{d}
  24. C / d C/d
  25. C / d C/d
  26. π \pi
  27. π = C / d \pi=C/d
  28. x 2 + y 2 = 1 x^{2}+y^{2}=1
  29. π = - 1 1 d x 1 - x 2 . \pi=\int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}}.
  30. π \pi
  31. π \pi
  32. π \pi
  33. π \pi
  34. π \pi
  35. e x p ( z ) exp(z)
  36. z z
  37. e x p ( z ) exp(z)
  38. { , - 2 π i , 0 , 2 π i , 4 π i , } = { 2 π k i | k } \{\dots,-2\pi i,0,2\pi i,4\pi i,\dots\}=\{2\pi ki|k\in\mathbb{Z}\}
  39. π \pi
  40. / \mathbb{R}/\mathbb{Z}
  41. π \pi
  42. π \pi
  43. 22 7 \frac{22}{7}
  44. π \pi
  45. π \pi
  46. π \pi
  47. π \pi
  48. e e
  49. l n ( 2 ) ln(2)
  50. π \pi
  51. π \pi
  52. π \pi
  53. 313 \sqrt{313}
  54. 10 \sqrt{10}
  55. π \pi
  56. π \pi
  57. π \pi
  58. π \pi
  59. π \pi
  60. π \pi
  61. π \pi
  62. π \pi
  63. π \pi
  64. π = 3 + 1 7 + 1 15 + 1 1 + 1 292 + 1 1 + 1 1 + 1 1 + \pi=3+\textstyle\frac{1}{7+\textstyle\frac{1}{15+\textstyle\frac{1}{1+% \textstyle\frac{1}{292+\textstyle\frac{1}{1+\textstyle\frac{1}{1+\textstyle% \frac{1}{1+\ddots}}}}}}}
  65. π \pi
  66. π \pi
  67. π \pi
  68. π \pi
  69. π \pi
  70. π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + = 3 + 1 2 6 + 3 2 6 + 5 2 6 + 7 2 6 + 9 2 6 + = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + 4 2 9 + \pi=\textstyle\cfrac{4}{1+\textstyle\frac{1^{2}}{2+\textstyle\frac{3^{2}}{2+% \textstyle\frac{5^{2}}{2+\textstyle\frac{7^{2}}{2+\textstyle\frac{9^{2}}{2+% \ddots}}}}}}=3+\textstyle\frac{1^{2}}{6+\textstyle\frac{3^{2}}{6+\textstyle% \frac{5^{2}}{6+\textstyle\frac{7^{2}}{6+\textstyle\frac{9^{2}}{6+\ddots}}}}}=% \textstyle\cfrac{4}{1+\textstyle\frac{1^{2}}{3+\textstyle\frac{2^{2}}{5+% \textstyle\frac{3^{2}}{7+\textstyle\frac{4^{2}}{9+\ddots}}}}}
  71. 22 7 \frac{22}{7}
  72. 333 106 \frac{333}{106}
  73. 355 113 \frac{355}{113}
  74. 52163 16604 \frac{52163}{16604}
  75. 103993 33102 \frac{103993}{33102}
  76. 245850922 78256779 \frac{245850922}{78256779}
  77. π \pi
  78. π \pi
  79. 22 / 7 {22}/{7}
  80. π \pi
  81. π \pi
  82. 25 / 8 {25}/{8}
  83. π \pi
  84. 16 / 9 {16}/{9}
  85. 339 / 108 {339}/{108}
  86. π \pi
  87. 10 \sqrt{10}
  88. π \pi
  89. π \pi
  90. π \pi
  91. 22 7 \frac{22}{7}
  92. π \pi
  93. π \pi
  94. π \pi
  95. π \pi
  96. π \pi
  97. π \pi
  98. 2 π = 2 2 2 + 2 2 2 + 2 + 2 2 \frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{% \sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots
  99. π \pi
  100. π \pi
  101. arctan z = z - z 3 3 + z 5 5 - z 7 7 + \arctan z=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdots
  102. π / 4 π/4
  103. z z
  104. π \pi
  105. π \pi
  106. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\,\arctan\frac{1}{5}-\arctan\frac{1}{239}
  107. π \pi
  108. π \pi
  109. π \pi
  110. π \pi
  111. π \pi
  112. π \pi
  113. π \pi
  114. π \pi
  115. π \pi
  116. π = 4 1 - 4 3 + 4 5 - 4 7 + 4 9 - 4 11 + 4 13 - \pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+% \frac{4}{13}-\cdots
  117. π \pi
  118. π \pi
  119. π \pi
  120. π \pi
  121. π = 3 + 4 2 × 3 × 4 - 4 4 × 5 × 6 + 4 6 × 7 × 8 - 4 8 × 9 × 10 + \pi=3+\frac{4}{2\times 3\times 4}-\frac{4}{4\times 5\times 6}+\frac{4}{6\times 7% \times 8}-\frac{4}{8\times 9\times 10}+\cdots
  122. π \pi
  123. π = 4 1 - 4 3 + 4 5 - 4 7 + 4 9 - 4 11 + 4 13 . \pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+% \frac{4}{13}\cdots.
  124. π \pi
  125. π = < m t p l > 3 + 4 2 × 3 × 4 - 4 4 × 5 × 6 + 4 6 × 7 × 8 . \pi=<mtpl>{{3}}+\frac{{4}}{2\times 3\times 4}-\frac{{4}}{4\times 5\times 6}+% \frac{{4}}{6\times 7\times 8}\cdots.
  126. π \pi
  127. π \pi
  128. π \pi
  129. π \pi
  130. π \pi
  131. π 2 6 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + \frac{\pi^{2}}{6}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{% 2}}+\cdots
  132. π \pi
  133. π \pi
  134. π \pi
  135. π \pi
  136. π \pi
  137. π \pi
  138. π \pi
  139. π \pi
  140. π \pi
  141. π \pi
  142. π \pi
  143. π \pi
  144. π \pi
  145. π \pi
  146. π \pi
  147. π \pi
  148. π \pi
  149. π \pi
  150. π \pi
  151. π \pi
  152. π \pi
  153. π \pi
  154. π \pi
  155. π \pi
  156. 1 π = 2 2 9801 k = 0 ( 4 k ) ! ( 1103 + 26390 k ) k ! 4 ( 396 4 k ) . \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{k=0}^{\infty}\frac{(4k)!(1103+26390k% )}{k!^{4}(396^{4k})}.
  157. π \pi
  158. 1 π = 12 640320 3 / 2 k = 0 ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 ( - 640320 ) 3 k . \frac{1}{\pi}=\frac{12}{640320^{3/2}}\sum_{k=0}^{\infty}\frac{(6k)!(13591409+5% 45140134k)}{(3k)!(k!)^{3}(-640320)^{3k}}.
  159. π \pi
  160. π \pi
  161. π \pi
  162. π k = n = 1 1 n k ( a q n - 1 + b q 2 n - 1 + c q 4 n - 1 ) , \pi^{k}=\sum_{n=1}^{\infty}\frac{1}{n^{k}}\left(\frac{a}{q^{n}-1}+\frac{b}{q^{% 2n}-1}+\frac{c}{q^{4n}-1}\right),
  163. q q
  164. e e
  165. π \pi
  166. k k
  167. a , b , c a,b,c
  168. π \pi
  169. π \pi
  170. π = k = 0 1 16 k ( 4 8 k + 1 - 2 8 k + 4 - 1 8 k + 5 - 1 8 k + 6 ) \pi=\sum_{k=0}^{\infty}\frac{1}{16^{k}}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-% \frac{1}{8k+5}-\frac{1}{8k+6}\right)
  171. π \pi
  172. π \pi
  173. π \pi
  174. π \pi
  175. π \pi
  176. π \pi
  177. π \pi
  178. π \pi
  179. r r
  180. 2 π r 2πr
  181. r r
  182. r r
  183. r r
  184. S n ( r ) = n π n / 2 Γ ( n 2 + 1 ) r n - 1 S_{n}(r)=\frac{n\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^{n-1}
  185. V n ( r ) = π n / 2 Γ ( n 2 + 1 ) r n V_{n}(r)=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}r^{n}
  186. π \pi
  187. - 1 1 1 - x 2 d x = π 2 . \int_{-1}^{1}\sqrt{1-x^{2}}\,dx=\frac{\pi}{2}.
  188. [ u i n t m a t h , u i n t , u - 1 , u 1 ] [u^{\prime}intmath^{\prime},u^{\prime}int^{\prime},u^{\prime}-1^{\prime},u^{% \prime}1^{\prime}]
  189. x x
  190. π \pi
  191. π \pi
  192. π \pi
  193. π \pi
  194. π \pi
  195. π \pi
  196. θ θ
  197. k k
  198. sin θ = sin ( θ + 2 π k ) \sin\theta=\sin\left(\theta+2\pi k\right)
  199. cos θ = cos ( θ + 2 π k ) . \cos\theta=\cos\left(\theta+2\pi k\right).
  200. π \pi
  201. n n
  202. t t
  203. x x
  204. x x
  205. π \pi
  206. π 2 n x t \pi\approx\frac{2n\ell}{xt}
  207. π \pi
  208. π / 4 π/4
  209. π \pi
  210. π \pi
  211. e e
  212. z z
  213. z z
  214. φ φ
  215. z = r ( cos φ + i sin φ ) , z=r\cdot(\cos\varphi+i\sin\varphi),
  216. i i
  217. π \pi
  218. e i φ = cos φ + i sin φ , e^{i\varphi}=\cos\varphi+i\sin\varphi,
  219. e e
  220. e e
  221. φ φ
  222. π \pi
  223. e i π + 1 = 0. e^{i\pi}+1=0.
  224. n n
  225. z z
  226. n n
  227. e 2 π i k / n ( k = 0 , 1 , 2 , , n - 1 ) . e^{2\pi ik/n}\qquad(k=0,1,2,\dots,n-1).
  228. f ( z 0 ) = 1 2 π i γ f ( z ) z - z 0 d z f(z_{0})=\frac{1}{2\pi i}\oint_{\gamma}{f(z)\over z-z_{0}}\,dz
  229. π \pi
  230. π \pi
  231. π \pi
  232. π \pi
  233. Γ ( 1 / 2 ) = π \Gamma(1/2)=\sqrt{\pi}
  234. Γ ( 5 / 2 ) = 3 π 4 \Gamma(5/2)=\frac{3\sqrt{\pi}}{4}
  235. n ! n!
  236. n n
  237. n ! 2 π n ( n e ) n n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}
  238. ζ ( s ) ζ(s)
  239. s = 2 s=2
  240. ζ ( 2 ) = 1 1 2 + 1 2 2 + 1 3 2 + \zeta(2)=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots
  241. p p
  242. 1 / p 1/p
  243. p ( 1 - 1 p 2 ) = ( p 1 1 - p - 2 ) - 1 = 1 1 + 1 2 2 + 1 3 2 + = 1 ζ ( 2 ) = 6 π 2 61 % \prod_{p}^{\infty}\left(1-\frac{1}{p^{2}}\right)=\left(\prod_{p}^{\infty}\frac% {1}{1-p^{-2}}\right)^{-1}=\frac{1}{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots}=% \frac{1}{\zeta(2)}=\frac{6}{\pi^{2}}\approx 61\%
  244. π \pi
  245. π \sqrt{\pi}
  246. π \pi
  247. μ μ
  248. σ σ
  249. f ( x ) = 1 σ 2 π e - ( x - μ ) 2 / ( 2 σ 2 ) f(x)={1\over\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^{2}/(2\sigma^{2})}
  250. - e - x 2 d x = π , \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi},
  251. - 1 x 2 + 1 d x = π . \int_{-\infty}^{\infty}\frac{1}{x^{2}+1}\,dx=\pi.
  252. π \pi
  253. π \pi
  254. T T
  255. L L
  256. g g
  257. T 2 π L g . T\approx 2\pi\sqrt{\frac{L}{g}}.
  258. x x
  259. p p
  260. h h
  261. Δ x Δ p h 4 π . \Delta x\,\Delta p\geq\frac{h}{4\pi}.
  262. π \pi
  263. R i k - g i k R 2 + Λ g i k = 8 π G c 4 T i k , R_{ik}-{g_{ik}R\over 2}+\Lambda g_{ik}={8\pi G\over c^{4}}T_{ik},
  264. R R
  265. Λ Λ
  266. G G
  267. c c
  268. r r
  269. F = | q 1 q 2 | 4 π ε 0 r 2 . F=\frac{\left|q_{1}q_{2}\right|}{4\pi\varepsilon_{0}r^{2}}.
  270. π \pi
  271. α α
  272. 1 τ = 2 π 2 - 9 9 π m α 6 , \frac{1}{\tau}=2\frac{\pi^{2}-9}{9\pi}m\alpha^{6},
  273. m m
  274. π \pi
  275. F F
  276. L L
  277. E E
  278. I I
  279. F = π 2 E I L 2 . F=\frac{\pi^{2}EI}{L^{2}}.
  280. π \pi
  281. F F
  282. R R
  283. v v
  284. η η
  285. F = 6 π η R v . F=6\,\pi\,\eta\,R\,v.
  286. f ^ ( ξ ) = - f ( x ) e - 2 π i x ξ d x \hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)\ e^{-2\pi ix\xi}\,dx
  287. π \pi
  288. π \pi
  289. π \pi
  290. π \pi
  291. π \pi
  292. π \pi
  293. π \pi
  294. π \pi
  295. π \pi
  296. π \pi
  297. π \pi
  298. π \pi
  299. π \pi
  300. π \pi
  301. π \pi
  302. π \pi
  303. π \pi
  304. π \pi
  305. τ \tau
  306. π \pi
  307. τ \tau
  308. π \pi
  309. τ \tau
  310. π \pi
  311. τ \tau
  312. π \pi
  313. τ τ
  314. π \pi
  315. π \pi
  316. 2 \sqrt{2}
  317. e e
  318. π = - d x 1 + x 2 . \pi=\int_{-\infty}^{\infty}\frac{dx}{1+x^{2}}.
  319. π \pi
  320. π \pi
  321. π \pi
  322. π \pi

PID_controller.html

  1. u ( t ) u(t)
  2. u ( t ) = MV ( t ) = K p e ( t ) + K i 0 t e ( τ ) d τ + K d d d t e ( t ) \mathrm{u}(t)=\mathrm{MV}(t)=K_{p}{e(t)}+K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau}+K% _{d}\frac{d}{dt}e(t)
  3. K p K_{p}
  4. K i K_{i}
  5. K d K_{d}
  6. e e
  7. = S P - P V =SP-PV
  8. t t
  9. τ \tau
  10. t t
  11. L ( s ) = K p + K i / s + K d s L(s)=K_{p}+K_{i}/s+K_{d}s
  12. s s
  13. P out = K p e ( t ) P_{\mathrm{out}}=K_{p}\,{e(t)}
  14. K i K_{i}
  15. I out = K i 0 t e ( τ ) d τ I_{\mathrm{out}}=K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau}
  16. D out = K d d d t e ( t ) D_{\mathrm{out}}=K_{d}\frac{d}{dt}e(t)
  17. H ( s ) = K ( s ) G ( s ) 1 + K ( s ) G ( s ) H(s)=\frac{K(s)G(s)}{1+K(s)G(s)}
  18. K ( s ) K(s)
  19. G ( s ) G(s)
  20. s s
  21. K ( s ) G ( s ) = - 1 K(s)G(s)=-1
  22. | K ( s ) G ( s ) | = 1 |K(s)G(s)|=1
  23. K ( s ) G ( s ) < 1 K(s)G(s)<1
  24. K i K_{i}
  25. K d K_{d}
  26. K p K_{p}
  27. K p K_{p}
  28. K i K_{i}
  29. K i K_{i}
  30. K d K_{d}
  31. K d K_{d}
  32. K p K_{p}
  33. K p K_{p}
  34. K p K_{p}
  35. K i K_{i}
  36. K d K_{d}
  37. K d K_{d}
  38. K i K_{i}
  39. K d K_{d}
  40. K u K_{u}
  41. K u K_{u}
  42. T u T_{u}
  43. K p K_{p}
  44. K i K_{i}
  45. K d K_{d}
  46. 0.50 K u 0.50{K_{u}}
  47. 0.45 K u 0.45{K_{u}}
  48. 1.2 K p / T u 1.2{K_{p}}/T_{u}
  49. 0.60 K u 0.60{K_{u}}
  50. 2 K p / T u 2{K_{p}}/T_{u}
  51. K p T u / 8 {K_{p}}{T_{u}}/8
  52. T i T_{i}
  53. T d T_{d}
  54. T u T_{u}
  55. K P Δ + K I Δ d t K_{P}\Delta+K_{I}\int\Delta\,dt
  56. Δ \Delta
  57. Δ = S P - P V \Delta=SP-PV
  58. C = G ( 1 + τ s ) τ s C=\frac{G(1+\tau s)}{\tau s}
  59. G = K P G=K_{P}
  60. G / τ = K I G/\tau=K_{I}
  61. G G
  62. K p K_{p}
  63. I out I_{\mathrm{out}}
  64. D out D_{\mathrm{out}}
  65. MV ( t ) = K p ( e ( t ) + 1 T i 0 t e ( τ ) d τ + T d d d t e ( t ) ) \mathrm{MV(t)}=K_{p}\left(\,{e(t)}+\frac{1}{T_{i}}\int_{0}^{t}{e(\tau)}\,{d% \tau}+T_{d}\frac{d}{dt}e(t)\right)
  66. T i T_{i}
  67. T d T_{d}
  68. T d T_{d}
  69. T i T_{i}
  70. K p K_{p}
  71. MV ( t ) = K p e ( t ) + K i 0 t e ( τ ) d τ + K d d d t e ( t ) \mathrm{MV(t)}=K_{p}{e(t)}+K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau}+K_{d}\frac{d}{% dt}e(t)
  72. K i = K p T i K_{i}=\frac{K_{p}}{T_{i}}
  73. K d = K p T d K_{d}=K_{p}T_{d}\,
  74. K p K_{p}
  75. 1 / K p 1/K_{p}
  76. MV ( t ) = K p ( e ( t ) + 1 T i 0 t e ( τ ) d τ - T d d d t P V ( t ) ) \mathrm{MV(t)}=K_{p}\left(\,{e(t)}+\frac{1}{T_{i}}\int_{0}^{t}{e(\tau)}\,{d% \tau}-T_{d}\frac{d}{dt}PV(t)\right)
  77. MV ( t ) = K p ( - P V ( t ) + 1 T i 0 t e ( τ ) d τ - T d d d t P V ( t ) ) \mathrm{MV(t)}=K_{p}\left(\,{-PV(t)}+\frac{1}{T_{i}}\int_{0}^{t}{e(\tau)}\,{d% \tau}-T_{d}\frac{d}{dt}PV(t)\right)
  78. G ( s ) = K p + K i s + K d s = K d s 2 + K p s + K i s G(s)=K_{p}+\frac{K_{i}}{s}+K_{d}{s}=\frac{K_{d}{s^{2}}+K_{p}{s}+K_{i}}{s}
  79. G ( s ) = K d s 2 + K p K d s + K i K d s G(s)=K_{d}\frac{s^{2}+\frac{K_{p}}{K_{d}}s+\frac{K_{i}}{K_{d}}}{s}
  80. H ( s ) = 1 s 2 + 2 ζ ω 0 s + ω 0 2 H(s)=\frac{1}{s^{2}+2\zeta\omega_{0}s+\omega_{0}^{2}}
  81. K i K d = ω 0 2 \frac{K_{i}}{K_{d}}=\omega_{0}^{2}
  82. K p K d = 2 ζ ω 0 \frac{K_{p}}{K_{d}}=2\zeta\omega_{0}
  83. G ( s ) H ( s ) = K d s G(s)H(s)=\frac{K_{d}}{s}
  84. G ( s ) = K c ( τ i s + 1 ) τ i s ( τ d s + 1 ) G(s)=K_{c}\frac{(\tau_{i}{s}+1)}{\tau_{i}{s}}(\tau_{d}{s}+1)
  85. K p = K c α K_{p}=K_{c}\cdot\alpha
  86. T i = τ i α T_{i}=\tau_{i}\cdot\alpha
  87. T d = τ d α T_{d}=\frac{\tau_{d}}{\alpha}
  88. α = 1 + τ d τ i \alpha=1+\frac{\tau_{d}}{\tau_{i}}
  89. Δ t \Delta t
  90. 0 t k e ( τ ) d τ = i = 1 k e ( t i ) Δ t \int_{0}^{t_{k}}{e(\tau)}\,{d\tau}=\sum_{i=1}^{k}e(t_{i})\Delta t
  91. d e ( t k ) d t = e ( t k ) - e ( t k - 1 ) Δ t \dfrac{de(t_{k})}{dt}=\dfrac{e(t_{k})-e(t_{k-1})}{\Delta t}
  92. u ( t ) u(t)
  93. u ( t k ) u(t_{k})
  94. u ( t k ) = u ( t k - 1 ) + K p [ ( 1 + Δ t T i + T d Δ t ) e ( t k ) + ( - 1 - 2 T d Δ t ) e ( t k - 1 ) + T d Δ t e ( t k - 2 ) ] u(t_{k})=u(t_{k-1})+K_{p}\left[\left(1+\dfrac{\Delta t}{T_{i}}+\dfrac{T_{d}}{% \Delta t}\right)e(t_{k})+\left(-1-\dfrac{2T_{d}}{\Delta t}\right)e(t_{k-1})+% \dfrac{T_{d}}{\Delta t}e(t_{k-2})\right]
  95. T i = K p / K i , T d = K d / K p T_{i}=K_{p}/K_{i},T_{d}=K_{d}/K_{p}

Piezoelectricity.html

  1. 𝐃 = s y m b o l ε 𝐄 D i = ε i j E j \mathbf{D}=symbol{\varepsilon}\,\mathbf{E}\quad\implies\quad D_{i}=\varepsilon% _{ij}\,E_{j}\;
  2. s y m b o l S = 𝗌 s y m b o l T S i j = s i j k l T k l symbol{S}=\mathsf{s}\,symbol{T}\quad\implies\quad S_{ij}=s_{ijkl}\,T_{kl}\;
  3. s y m b o l S \displaystyle symbol{S}
  4. { S } = [ s E ] { T } + [ d t ] { E } { D } = [ d ] { T } + [ ε T ] { E } , \begin{aligned}\displaystyle\{S\}&\displaystyle=\left[s^{E}\right]\{T\}+[d^{t}% ]\{E\}\\ \displaystyle\{D\}&\displaystyle=[d]\{T\}+\left[\varepsilon^{T}\right]\{E\}\,,% \end{aligned}
  5. [ d ] [d]
  6. [ d t ] [d^{t}]
  7. [ S 1 S 2 S 3 S 4 S 5 S 6 ] = [ s 11 E s 12 E s 13 E 0 0 0 s 21 E s 22 E s 23 E 0 0 0 s 31 E s 32 E s 33 E 0 0 0 0 0 0 s 44 E 0 0 0 0 0 0 s 55 E 0 0 0 0 0 0 s 66 E = 2 ( s 11 E - s 12 E ) ] [ T 1 T 2 T 3 T 4 T 5 T 6 ] + [ 0 0 d 31 0 0 d 32 0 0 d 33 0 d 24 0 d 15 0 0 0 0 0 ] [ E 1 E 2 E 3 ] \begin{bmatrix}S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6}\end{bmatrix}=\begin{bmatrix}s_{11}^{E}&s_{12}^{E}&s_{13}^{E}&0&0&0\\ s_{21}^{E}&s_{22}^{E}&s_{23}^{E}&0&0&0\\ s_{31}^{E}&s_{32}^{E}&s_{33}^{E}&0&0&0\\ 0&0&0&s_{44}^{E}&0&0\\ 0&0&0&0&s_{55}^{E}&0\\ 0&0&0&0&0&s_{66}^{E}=2\left(s_{11}^{E}-s_{12}^{E}\right)\end{bmatrix}\begin{% bmatrix}T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\\ T_{5}\\ T_{6}\end{bmatrix}+\begin{bmatrix}0&0&d_{31}\\ 0&0&d_{32}\\ 0&0&d_{33}\\ 0&d_{24}&0\\ d_{15}&0&0\\ 0&0&0\end{bmatrix}\begin{bmatrix}E_{1}\\ E_{2}\\ E_{3}\end{bmatrix}
  8. [ D 1 D 2 D 3 ] = [ 0 0 0 0 d 15 0 0 0 0 d 24 0 0 d 31 d 32 d 33 0 0 0 ] [ T 1 T 2 T 3 T 4 T 5 T 6 ] + [ ε 11 0 0 0 ε 22 0 0 0 ε 33 ] [ E 1 E 2 E 3 ] \begin{bmatrix}D_{1}\\ D_{2}\\ D_{3}\end{bmatrix}=\begin{bmatrix}0&0&0&0&d_{15}&0\\ 0&0&0&d_{24}&0&0\\ d_{31}&d_{32}&d_{33}&0&0&0\end{bmatrix}\begin{bmatrix}T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\\ T_{5}\\ T_{6}\end{bmatrix}+\begin{bmatrix}{\varepsilon}_{11}&0&0\\ 0&{\varepsilon}_{22}&0\\ 0&0&{\varepsilon}_{33}\end{bmatrix}\begin{bmatrix}E_{1}\\ E_{2}\\ E_{3}\end{bmatrix}
  9. S 4 , S 5 , S 6 S_{4},S_{5},S_{6}
  10. 2 ( s 11 E - s 12 E ) 2(s_{11}^{E}-s_{12}^{E})
  11. S 6 = 2 S 12 S_{6}=2S_{12}
  12. s 66 = 1 / G 12 s_{66}=1/G_{12}
  13. G 12 G_{12}
  14. d i j d_{ij}
  15. e i j e_{ij}
  16. g i j g_{ij}
  17. h i j h_{ij}
  18. d i j = ( D i T j ) E = ( S j E i ) T d_{ij}=\left(\frac{\partial D_{i}}{\partial T_{j}}\right)^{E}=\left(\frac{% \partial S_{j}}{\partial E_{i}}\right)^{T}
  19. e i j = ( D i S j ) E = - ( T j E i ) S e_{ij}=\left(\frac{\partial D_{i}}{\partial S_{j}}\right)^{E}=-\left(\frac{% \partial T_{j}}{\partial E_{i}}\right)^{S}
  20. g i j = - ( E i T j ) D = ( S j D i ) T g_{ij}=-\left(\frac{\partial E_{i}}{\partial T_{j}}\right)^{D}=\left(\frac{% \partial S_{j}}{\partial D_{i}}\right)^{T}
  21. h i j = - ( E i S j ) D = - ( T j D i ) S h_{ij}=-\left(\frac{\partial E_{i}}{\partial S_{j}}\right)^{D}=-\left(\frac{% \partial T_{j}}{\partial D_{i}}\right)^{S}
  22. d i j d_{ij}
  23. 4 ¯ \overline{4}
  24. 4 ¯ \overline{4}
  25. 6 ¯ \overline{6}
  26. 6 ¯ \overline{6}
  27. 4 ¯ \overline{4}
  28. T C T_{C}
  29. Q m 900 Q_{m}\approx 900

Pigeonhole_principle.html

  1. n n
  2. m m
  3. n > m n>m
  4. k k
  5. m m
  6. n = k m + 1 n=km+1
  7. m m
  8. k + 1 k+1
  9. n n
  10. m m
  11. k + 1 = ( n - 1 ) / m + 1 k+1=⌊(n-1)/m⌋+1
  12. ⌊...⌋
  13. ( n - 1 ) m + 1 = ( 7 - 1 ) 4 + 1 = 6 4 + 1 = 1 + 1 = 2 \left\lfloor\frac{(n-1)}{m}\right\rfloor+1=\left\lfloor\frac{(7-1)}{4}\right% \rfloor+1=\left\lfloor\frac{6}{4}\right\rfloor+1=1+1=2
  14. q 1 + q 2 + + q n - n + 1 q_{1}+q_{2}+\cdots+q_{n}-n+1
  15. n ( r - 1 ) + 1 n(r-1)+1
  16. k / n \lceil k/n\rceil
  17. x \lceil x\rceil
  18. k / n \lfloor k/n\rfloor
  19. x \lfloor x\rfloor
  20. 1 - ( m ) n m n , 1-\frac{(m)_{n}}{m^{n}},\!

Pink_noise.html

  1. 1 / f {1}/{f}
  2. S ( f ) 1 f α S(f)\propto\frac{1}{f^{\alpha}}
  3. 1 / f {1}/{f}
  4. d d f ln f = 1 f . \textstyle\frac{d}{df}\ln f=\frac{1}{f}.

Pion.html

  1. v m ¯ \overline{vm}
  2. m ¯ \overline{m}
  3. R π = ( m e / m μ ) 2 ( m π 2 - m e 2 m π 2 - m μ 2 ) 2 = 1.283 × 10 - 4 R_{\pi}=(m_{e}/m_{\mu})^{2}\left(\frac{m_{\pi}^{2}-m_{e}^{2}}{m_{\pi}^{2}-m_{% \mu}^{2}}\right)^{2}=1.283\times 10^{-4}
  4. u u ¯ - d d ¯ 2 \tfrac{\mathrm{u\bar{u}}-\mathrm{d\bar{d}}}{\sqrt{2}}

Pitot_tube.html

  1. p t = p s + ( ρ u 2 2 ) p_{t}=p_{s}+\left(\frac{\rho u^{2}}{2}\right)
  2. u = 2 ( p t - p s ) ρ u=\sqrt{\frac{2(p_{t}-p_{s})}{\rho}}
  3. u u
  4. p t p_{t}
  5. p s p_{s}
  6. ρ \rho
  7. k g / m 3 kg/m^{3}
  8. p t p_{t}
  9. p s p_{s}
  10. Δ p \Delta p
  11. Δ h = Δ p ρ l g \Delta h=\frac{\Delta p}{\rho_{l}g}
  12. Δ h \Delta h
  13. ρ l \rho_{l}
  14. m / s 2 m/s^{2}
  15. V = 2 ( Δ h * ( ρ l g ) ) ρ V=\sqrt{\frac{2(\Delta h*(\rho_{l}g))}{\rho}}

Planar_graph.html

  1. 2 e 3 f 2e\geq 3f
  2. n n
  3. g n - 7 / 2 γ n n ! g\cdot n^{-7/2}\cdot\gamma^{n}\cdot n!
  4. γ 27.22687 \gamma\approx 27.22687
  5. g 0.43 × 10 - 5 g\approx 0.43\times 10^{-5}
  6. n n
  7. 27.2 n 27.2^{n}
  8. 30.06 n 30.06^{n}

Plane_wave.html

  1. A ( x , t ) = A o cos ( k x - ω t + φ ) A(x,t)=A_{o}\cos(kx-\omega t+\varphi)
  2. A ( x , t ) A(x,t)\,
  3. A ( x , t ) A(x,t)\,
  4. A o A_{o}\,
  5. k k\,
  6. 2 π / λ 2π/λ
  7. λ λ
  8. k k\,
  9. x x\,
  10. y y\,
  11. z z\,
  12. y - z y-z
  13. ω \omega\,
  14. 2 π / T 2π/T
  15. T T
  16. ω \omega\,
  17. t t\,
  18. φ \varphi\,
  19. 2 π
  20. λ \lambda\,
  21. T T\,
  22. f f\,
  23. c c\,
  24. A = A o cos [ 2 π ( x / λ - t / T ) + φ ] A=A_{o}\cos[2\pi(x/\lambda-t/T)+\varphi]\,
  25. A = A o cos [ 2 π ( x / λ - f t ) + φ ] A=A_{o}\cos[2\pi(x/\lambda-ft)+\varphi]\,
  26. A = A o cos [ ( 2 π / λ ) ( x - c t ) + φ ] A=A_{o}\cos[(2\pi/\lambda)(x-ct)+\varphi]\,
  27. f = 1 / T f=1/T\,\!
  28. c = λ / T = ω / k c=\lambda/T=\omega/k\,\!
  29. A ( 𝐫 , t ) = A o cos ( 𝐤 𝐫 - ω t + φ ) A(\mathbf{r},t)=A_{o}\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi)
  30. 𝐤 \mathbf{k}
  31. | 𝐤 | = k = 2 π / λ |\mathbf{k}|=k=2\pi/\lambda
  32. \cdot
  33. 𝐫 \mathbf{r}
  34. e e\,
  35. i i\,
  36. U ( 𝐫 , t ) = A o e i ( 𝐤 𝐫 - ω t + φ ) U(\mathbf{r},t)=A_{o}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi)}
  37. U ( 𝐫 , t ) = A o cos ( 𝐤 𝐫 - ω t + φ ) + i A o sin ( 𝐤 𝐫 - ω t + φ ) U(\mathbf{r},t)=A_{o}\cos(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi)+iA_{o}% \sin(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi)
  38. U ( 𝐫 , t ) = A ( 𝐫 , t ) + i A o sin ( 𝐤 𝐫 - ω t + φ ) U(\mathbf{r},t)=\qquad\ \ A(\mathbf{r},t)\qquad\qquad+iA_{o}\sin(\mathbf{k}% \cdot\mathbf{r}-\omega t+\varphi)
  39. U o U_{o}\,
  40. A o A_{o}\,
  41. U ( 𝐫 , t ) = A o e i ( 𝐤 𝐫 - ω t + φ ) U(\mathbf{r},t)=A_{o}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t+\varphi)}
  42. U ( 𝐫 , t ) = A o e i ( 𝐤 𝐫 - ω t ) e i φ U(\mathbf{r},t)=A_{o}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}e^{i\varphi}
  43. e i φ e^{i\varphi}
  44. U o = A o e i φ U_{o}=A_{o}e^{i\varphi}
  45. U ( 𝐫 , t ) = U o e i ( 𝐤 𝐫 - ω t ) U(\mathbf{r},t)=U_{o}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}
  46. R e [ U ( 𝐫 , t ) ] = A ( 𝐫 , t ) = A o cos ( 𝐤 𝐫 - ω t + φ ) Re[U(\mathbf{r},t)]=A(\mathbf{r},t)=A_{o}\cos(\mathbf{k}\cdot\mathbf{r}-\omega t% +\varphi)

Plasma_stability.html

  1. β = p p m a g = n k B T ( B 2 / 2 μ 0 ) \beta=\frac{p}{p_{mag}}=\frac{nk_{B}T}{(B^{2}/2\mu_{0})}
  2. β 2 \beta^{2}
  3. β N 4 \beta_{N}^{4}
  4. β N = β / ( I / a B ) \beta_{N}=\beta/(I/aB)

Platonic_solid.html

  1. 1 + 5 2 \tfrac{1+\sqrt{5}}{2}
  2. p F = 2 E = q V . pF=2E=qV.\,
  3. V - E + F = 2. V-E+F=2.\,
  4. V = 4 p 4 - ( p - 2 ) ( q - 2 ) , E = 2 p q 4 - ( p - 2 ) ( q - 2 ) , F = 4 q 4 - ( p - 2 ) ( q - 2 ) . V=\frac{4p}{4-(p-2)(q-2)},\quad E=\frac{2pq}{4-(p-2)(q-2)},\quad F=\frac{4q}{4% -(p-2)(q-2)}.
  5. V - E + F = 2 V-E+F=2
  6. p F = 2 E = q V pF=2E=qV
  7. 2 E q - E + 2 E p = 2. \frac{2E}{q}-E+\frac{2E}{p}=2.
  8. 1 q + 1 p = 1 2 + 1 E . {1\over q}+{1\over p}={1\over 2}+{1\over E}.
  9. E E
  10. 1 q + 1 p > 1 2 . \frac{1}{q}+\frac{1}{p}>\frac{1}{2}.
  11. sin θ 2 = cos ( π / q ) sin ( π / p ) . \sin{\theta\over 2}=\frac{\cos(\pi/q)}{\sin(\pi/p)}.
  12. tan θ 2 = cos ( π / q ) sin ( π / h ) . \tan{\theta\over 2}=\frac{\cos(\pi/q)}{\sin(\pi/h)}.
  13. δ = 2 π - q π ( 1 - 2 p ) . \delta=2\pi-q\pi\left(1-{2\over p}\right).
  14. Ω = q θ - ( q - 2 ) π . \Omega=q\theta-(q-2)\pi.\,
  15. θ \theta
  16. tan θ 2 \tan\frac{\theta}{2}
  17. δ \delta
  18. Ω \Omega
  19. 1 2 1\over{\sqrt{2}}
  20. π \pi
  21. cos - 1 ( 23 27 ) \cos^{-1}\left(\frac{23}{27}\right)
  22. 0.551286 \approx 0.551286
  23. π \pi
  24. 1 1
  25. π 2 \pi\over 2
  26. π 2 \frac{\pi}{2}
  27. 1.57080 \approx 1.57080
  28. 2 π 3 2\pi\over 3
  29. 2 \sqrt{2}
  30. 2 π 3 {2\pi}\over 3
  31. 4 sin - 1 ( 1 3 ) 4\sin^{-1}\left({1\over 3}\right)
  32. 1.35935 \approx 1.35935
  33. π 2 \pi\over 2
  34. φ \varphi
  35. π 5 \pi\over 5
  36. π - tan - 1 ( 2 11 ) \pi-\tan^{-1}\left(\frac{2}{11}\right)
  37. 2.96174 \approx 2.96174
  38. π 3 \pi\over 3
  39. φ 2 \varphi^{2}
  40. π 3 \pi\over 3
  41. 2 π - 5 sin - 1 ( 2 3 ) 2\pi-5\sin^{-1}\left({2\over 3}\right)
  42. 2.63455 \approx 2.63455
  43. π 5 \pi\over 5
  44. R = ( a 2 ) tan π q tan θ 2 R=\left({a\over 2}\right)\tan\frac{\pi}{q}\tan\frac{\theta}{2}
  45. r = ( a 2 ) cot π p tan θ 2 r=\left({a\over 2}\right)\cot\frac{\pi}{p}\tan\frac{\theta}{2}
  46. ρ = ( a 2 ) cos ( π / p ) sin ( π / h ) \rho=\left({a\over 2}\right)\frac{\cos(\pi/p)}{\sin(\pi/h)}
  47. R r = tan π p tan π q = sin - 2 ( θ / 2 ) - cos 2 ( α / 2 ) sin ( α / 2 ) . {R\over r}=\tan\frac{\pi}{p}\tan\frac{\pi}{q}=\frac{{\sqrt{{\sin^{-2}{(\theta/% 2)}}-{\cos^{2}{(\alpha/2)}}}}}{\sin{(\alpha/2)}}.
  48. A = ( a 2 ) 2 F p cot π p . A=\left({a\over 2}\right)^{2}Fp\cot\frac{\pi}{p}.
  49. V = 1 3 r A . V={1\over 3}rA.
  50. 1 6 1\over{\sqrt{6}}
  51. 1 2 1\over{\sqrt{2}}
  52. 3 2 \sqrt{3\over 2}
  53. 4 3 4\sqrt{3}
  54. 8 3 \frac{\sqrt{8}}{3}
  55. 1 1\,
  56. 2 \sqrt{2}
  57. 3 \sqrt{3}
  58. 24 24\,
  59. 8 8\,
  60. 2 3 \sqrt{2\over 3}
  61. 1 1\,
  62. 2 \sqrt{2}
  63. 8 3 8\sqrt{3}
  64. 128 3 \frac{\sqrt{128}}{3}
  65. φ 2 ξ \frac{\varphi^{2}}{\xi}
  66. φ 2 \varphi^{2}
  67. 3 φ \sqrt{3}\,\varphi
  68. 12 25 + 10 5 12{\sqrt{25+10\sqrt{5}}}
  69. 20 φ 3 ξ 2 20\frac{\varphi^{3}}{\xi^{2}}
  70. φ 2 3 \frac{\varphi^{2}}{\sqrt{3}}
  71. φ \varphi
  72. ξ φ \xi\varphi
  73. 20 3 20\sqrt{3}
  74. 20 φ 2 3 \frac{20\varphi^{2}}{3}
  75. φ = 2 cos π 5 = 1 + 5 2 ξ = 2 sin π 5 = 5 - 5 2 = 5 1 / 4 φ - 1 / 2 . \varphi=2\cos{\pi\over 5}=\frac{1+\sqrt{5}}{2}\qquad\xi=2\sin{\pi\over 5}=% \sqrt{\frac{5-\sqrt{5}}{2}}=5^{1/4}\varphi^{-1/2}.
  76. d 2 = R r = r R = ρ ρ . d^{2}=R^{\ast}r=r^{\ast}R=\rho^{\ast}\rho.
  77. | |
  78. | |
  79. | |
  80. | |
  81. | |

Poincaré_conjecture.html

  1. t g i j = - 2 R i j \partial_{t}g_{ij}=-2R_{ij}

Pointless_topology.html

  1. b ( a i ) = ( a i b ) b\wedge\left(\bigvee a_{i}\right)=\bigvee\left(a_{i}\wedge b\right)

Polar_coordinate_system.html

  1. x = r cos φ x=r\cos\varphi\,
  2. y = r sin φ y=r\sin\varphi\,
  3. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}\quad
  4. φ = atan2 ( y , x ) \varphi=\operatorname{atan2}(y,x)\quad
  5. atan2 ( y , x ) = { arctan ( y x ) if x > 0 arctan ( y x ) + π if x < 0 and y 0 arctan ( y x ) - π if x < 0 and y < 0 π 2 if x = 0 and y > 0 - π 2 if x = 0 and y < 0 undefined if x = 0 and y = 0 \operatorname{atan2}(y,x)=\begin{cases}\arctan(\frac{y}{x})&\mbox{if }~{}x>0\\ \arctan(\frac{y}{x})+\pi&\mbox{if }~{}x<0\mbox{ and }~{}y\geq 0\\ \arctan(\frac{y}{x})-\pi&\mbox{if }~{}x<0\mbox{ and }~{}y<0\\ \frac{\pi}{2}&\mbox{if }~{}x=0\mbox{ and }~{}y>0\\ -\frac{\pi}{2}&\mbox{if }~{}x=0\mbox{ and }~{}y<0\\ \,\text{undefined}&\mbox{if }~{}x=0\mbox{ and }~{}y=0\end{cases}
  6. r 2 - 2 r r 0 cos ( φ - γ ) + r 0 2 = a 2 . r^{2}-2rr_{0}\cos(\varphi-\gamma)+r_{0}^{2}=a^{2}.\,
  7. r ( φ ) = a r(\varphi)=a\,
  8. r r
  9. a a
  10. r = 2 a cos ( φ - γ ) r=2a\cos(\varphi-\gamma)
  11. r r
  12. r = r 0 cos ( φ - γ ) + a 2 - r 0 2 sin 2 ( φ - γ ) r=r_{0}\cos(\varphi-\gamma)+\sqrt{a^{2}-r_{0}^{2}\sin^{2}(\varphi-\gamma)}
  13. φ = γ \varphi=\gamma\,
  14. r ( φ ) = r 0 sec ( φ - γ ) . r(\varphi)={r_{0}}\sec(\varphi-\gamma).\,
  15. r ( φ ) = a cos ( k φ + γ 0 ) r(\varphi)=a\cos(k\varphi+\gamma_{0})\,
  16. r ( φ ) = a + b φ . r(\varphi)=a+b\varphi.\,
  17. r = 1 - e cos φ r={\ell\over{1-e\cos\varphi}}
  18. \ell
  19. \ell
  20. r = f ( θ ) r=f(\theta)
  21. r = g ( θ ) r=g(\theta)
  22. f ( θ ) = 0 f(\theta)=0
  23. g ( θ ) = 0 g(\theta)=0
  24. [ g ( θ i ) , θ i ] [g(\theta_{i}),\theta_{i}]
  25. θ i \theta_{i}
  26. f ( θ ) = g ( θ ) f(\theta)=g(\theta)
  27. [ g ( θ i ) , θ i ] [g(\theta_{i}),\theta_{i}]
  28. θ i \theta_{i}
  29. f ( θ + ( 2 k + 1 ) π ) = - g ( θ ) f(\theta+(2k+1)\pi)=-g(\theta)
  30. k k
  31. z = x + i y z=x+iy\,
  32. z = r ( cos φ + i sin φ ) z=r\cdot(\cos\varphi+i\sin\varphi)
  33. z = r e i φ z=re^{i\varphi}\,
  34. r 0 e i φ 0 r 1 e i φ 1 = r 0 r 1 e i ( φ 0 + φ 1 ) r_{0}e^{i\varphi_{0}}\cdot r_{1}e^{i\varphi_{1}}=r_{0}r_{1}e^{i(\varphi_{0}+% \varphi_{1})}\,
  35. r 0 e i φ 0 r 1 e i φ 1 = r 0 r 1 e i ( φ 0 - φ 1 ) \frac{r_{0}e^{i\varphi_{0}}}{r_{1}e^{i\varphi_{1}}}=\frac{r_{0}}{r_{1}}e^{i(% \varphi_{0}-\varphi_{1})}\,
  36. ( r e i φ ) n = r n e i n φ (re^{i\varphi})^{n}=r^{n}e^{in\varphi}\,
  37. r u r = r u x x r + r u y y r , r\frac{\partial u}{\partial r}=r\frac{\partial u}{\partial x}\frac{\partial x}% {\partial r}+r\frac{\partial u}{\partial y}\frac{\partial y}{\partial r},
  38. u φ = u x x φ + u y y φ , \frac{\partial u}{\partial\varphi}=\frac{\partial u}{\partial x}\frac{\partial x% }{\partial\varphi}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial% \varphi},
  39. r u r = r u x cos φ + r u y sin φ = x u x + y u y , r\frac{\partial u}{\partial r}=r\frac{\partial u}{\partial x}\cos\varphi+r% \frac{\partial u}{\partial y}\sin\varphi=x\frac{\partial u}{\partial x}+y\frac% {\partial u}{\partial y},
  40. u φ = - u x r sin φ + u y r cos φ = - y u x + x u y . \frac{\partial u}{\partial\varphi}=-\frac{\partial u}{\partial x}r\sin\varphi+% \frac{\partial u}{\partial y}r\cos\varphi=-y\frac{\partial u}{\partial x}+x% \frac{\partial u}{\partial y}.
  41. r r = x x + y y r\frac{\partial}{\partial r}=x\frac{\partial}{\partial x}+y\frac{\partial}{% \partial y}\,
  42. φ = - y x + x y . \frac{\partial}{\partial\varphi}=-y\frac{\partial}{\partial x}+x\frac{\partial% }{\partial y}.
  43. u x = u r r x + u φ φ x , \frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{\partial r}{% \partial x}+\frac{\partial u}{\partial\varphi}\frac{\partial\varphi}{\partial x},
  44. u y = u r r y + u φ φ y , \frac{\partial u}{\partial y}=\frac{\partial u}{\partial r}\frac{\partial r}{% \partial y}+\frac{\partial u}{\partial\varphi}\frac{\partial\varphi}{\partial y},
  45. u x = u r x x 2 + y 2 - u φ y x 2 + y 2 = cos φ u r - 1 r sin φ u φ , \frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{x}{\sqrt{x^{2% }+y^{2}}}-\frac{\partial u}{\partial\varphi}\frac{y}{x^{2}+y^{2}}=\cos\varphi% \frac{\partial u}{\partial r}-\frac{1}{r}\sin\varphi\frac{\partial u}{\partial% \varphi},
  46. u y = u r y x 2 + y 2 + u φ x x 2 + y 2 = sin φ u r + 1 r cos φ u φ . \frac{\partial u}{\partial y}=\frac{\partial u}{\partial r}\frac{y}{\sqrt{x^{2% }+y^{2}}}+\frac{\partial u}{\partial\varphi}\frac{x}{x^{2}+y^{2}}=\sin\varphi% \frac{\partial u}{\partial r}+\frac{1}{r}\cos\varphi\frac{\partial u}{\partial% \varphi}.
  47. x = cos φ r - 1 r sin φ φ \frac{\partial}{\partial x}=\cos\varphi\frac{\partial}{\partial r}-\frac{1}{r}% \sin\varphi\frac{\partial}{\partial\varphi}\,
  48. y = sin φ r + 1 r cos φ φ . \frac{\partial}{\partial y}=\sin\varphi\frac{\partial}{\partial r}+\frac{1}{r}% \cos\varphi\frac{\partial}{\partial\varphi}.
  49. x = r ( φ ) cos φ x=r(\varphi)\cos\varphi\,
  50. y = r ( φ ) sin φ y=r(\varphi)\sin\varphi\,
  51. d x d φ = r ( φ ) cos φ - r ( φ ) sin φ \frac{dx}{d\varphi}=r^{\prime}(\varphi)\cos\varphi-r(\varphi)\sin\varphi\,
  52. d y d φ = r ( φ ) sin φ + r ( φ ) cos φ . \frac{dy}{d\varphi}=r^{\prime}(\varphi)\sin\varphi+r(\varphi)\cos\varphi.\,
  53. d y d x = r ( φ ) sin φ + r ( φ ) cos φ r ( φ ) cos φ - r ( φ ) sin φ . \frac{dy}{dx}=\frac{r^{\prime}(\varphi)\sin\varphi+r(\varphi)\cos\varphi}{r^{% \prime}(\varphi)\cos\varphi-r(\varphi)\sin\varphi}.
  54. 1 2 a b [ r ( φ ) ] 2 d φ . \frac{1}{2}\int_{a}^{b}\left[r(\varphi)\right]^{2}\,d\varphi.
  55. [ r ( φ i ) ] 2 π Δ φ 2 π = 1 2 [ r ( φ i ) ] 2 Δ φ . \left[r(\varphi_{i})\right]^{2}\pi\cdot\frac{\Delta\varphi}{2\pi}=\frac{1}{2}% \left[r(\varphi_{i})\right]^{2}\Delta\varphi.
  56. i = 1 n 1 2 r ( φ i ) 2 Δ φ . \sum_{i=1}^{n}\tfrac{1}{2}r(\varphi_{i})^{2}\,\Delta\varphi.
  57. J = det ( x , y ) ( r , φ ) = | x r x φ y r y φ | = | cos φ - r sin φ sin φ r cos φ | = r cos 2 φ + r sin 2 φ = r . J=\det\frac{\partial(x,y)}{\partial(r,\varphi)}=\begin{vmatrix}\frac{\partial x% }{\partial r}&\frac{\partial x}{\partial\varphi}\\ \frac{\partial y}{\partial r}&\frac{\partial y}{\partial\varphi}\end{vmatrix}=% \begin{vmatrix}\cos\varphi&-r\sin\varphi\\ \sin\varphi&r\cos\varphi\end{vmatrix}=r\cos^{2}\varphi+r\sin^{2}\varphi=r.
  58. d A = d x d y = J d r d φ = r d r d φ . dA=dx\,dy\ =J\,dr\,d\varphi=r\,dr\,d\varphi.
  59. R f ( x , y ) d A = a b 0 r ( φ ) f ( r , φ ) r d r d φ . \iint_{R}f(x,y)\,dA=\int_{a}^{b}\int_{0}^{r(\varphi)}f(r,\varphi)\,r\,dr\,d\varphi.
  60. - e - x 2 d x = π . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}.
  61. 𝐫 \mathbf{r}
  62. 𝐫 ^ = ( cos ( φ ) , sin ( φ ) ) \hat{\mathbf{r}}=(\cos(\varphi),\sin(\varphi))
  63. s y m b o l φ ^ = ( - sin ( φ ) , cos ( φ ) ) = 𝐤 ^ × 𝐫 ^ , \hat{symbol\varphi}=(-\sin(\varphi),\cos(\varphi))=\hat{\mathbf{k}}\times\hat{% \mathbf{r}}\ ,
  64. 𝐤 ^ \hat{\mathbf{k}}
  65. 𝐫 = ( x , y ) = r ( cos φ , sin φ ) = r 𝐫 ^ , \mathbf{r}=(x,\ y)=r(\cos\varphi,\ \sin\varphi)=r\hat{\mathbf{r}}\ ,
  66. 𝐫 ˙ = ( x ˙ , y ˙ ) = r ˙ ( cos φ , sin φ ) + r φ ˙ ( - sin φ , cos φ ) = r ˙ 𝐫 ^ + r φ ˙ s y m b o l φ ^ , \dot{\mathbf{r}}=(\dot{x},\ \dot{y})=\dot{r}(\cos\varphi,\ \sin\varphi)+r\dot{% \varphi}(-\sin\varphi,\ \cos\varphi)=\dot{r}\hat{\mathbf{r}}+r\dot{\varphi}% \hat{symbol{\varphi}}\ ,
  67. 𝐫 ¨ = ( x ¨ , y ¨ ) = r ¨ ( cos φ , sin φ ) + 2 r ˙ φ ˙ ( - sin φ , cos φ ) + r φ ¨ ( - sin φ , cos φ ) - r φ ˙ 2 ( cos φ , sin φ ) = \ddot{\mathbf{r}}=(\ddot{x},\ \ddot{y})=\ddot{r}(\cos\varphi,\ \sin\varphi)+2% \dot{r}\dot{\varphi}(-\sin\varphi,\ \cos\varphi)+r\ddot{\varphi}(-\sin\varphi,% \ \cos\varphi)-r{\dot{\varphi}}^{2}(\cos\varphi,\ \sin\varphi)\ =
  68. ( r ¨ - r φ ˙ 2 ) 𝐫 ^ + ( r φ ¨ + 2 r ˙ φ ˙ ) s y m b o l φ ^ = ( r ¨ - r φ ˙ 2 ) 𝐫 ^ + 1 r d d t ( r 2 φ ˙ ) s y m b o l φ ^ \left(\ddot{r}-r\dot{\varphi}^{2}\right)\hat{\mathbf{r}}+\left(r\ddot{\varphi}% +2\dot{r}\dot{\varphi}\right)\hat{symbol{\varphi}}\ =(\ddot{r}-r\dot{\varphi}^% {2})\hat{\mathbf{r}}+\frac{1}{r}\;\frac{d}{dt}\left(r^{2}\dot{\varphi}\right)% \hat{symbol{\varphi}}
  69. r φ ˙ 2 r\dot{\varphi}^{2}
  70. 2 r ˙ φ ˙ 2\dot{r}\dot{\varphi}
  71. s y m b o l F + s y m b o l F c f + s y m b o l F C o r = m s y m b o l r ¨ , symbol{F}+symbol{F_{cf}}+symbol{F_{Cor}}=m\ddot{symbol{r}}\ ,
  72. F r + m r Ω 2 = m r ¨ F_{r}+mr\Omega^{2}=m\ddot{r}
  73. F φ - 2 m r ˙ Ω = m r φ ¨ , F_{\varphi}-2m\dot{r}\Omega=mr\ddot{\varphi}\ ,
  74. F r = m r ¨ - m r φ ˙ 2 F_{r}=m\ddot{r}-mr\dot{\varphi}^{2}
  75. F φ = m r φ ¨ + 2 m r ˙ φ ˙ . F_{\varphi}=mr\ddot{\varphi}+2m\dot{r}\dot{\varphi}\ .
  76. φ ˙ \dot{\varphi}

Polarization_(waves).html

  1. E ( z , t ) = [ e x e y 0 ] e i 2 π ( z / ( λ / n ) - t / T ) = [ e x e y 0 ] e i ( k z - ω t ) \vec{E}(z,t)=\begin{bmatrix}e_{x}\\ e_{y}\\ 0\end{bmatrix}\;e^{i2\pi(z/(\lambda/n)-t/T)}=\begin{bmatrix}e_{x}\\ e_{y}\\ 0\end{bmatrix}\;e^{i(kz-\omega t)}
  2. H ( z , t ) = [ h x h y 0 ] e i 2 π ( z / ( λ / n ) - t / T ) = [ h x h y 0 ] e i ( k z - ω t ) \vec{H}(z,t)=\begin{bmatrix}h_{x}\\ h_{y}\\ 0\end{bmatrix}\;e^{i2\pi(z/(\lambda/n)-t/T)}=\begin{bmatrix}h_{x}\\ h_{y}\\ 0\end{bmatrix}\;e^{i(kz-\omega t)}
  3. k = 2 π n / λ k=2\pi n/\lambda
  4. ω = 2 π f \omega=2\pi f
  5. k r \vec{k}\cdot\vec{r}
  6. k \vec{k}
  7. η \eta
  8. h y = e x / η h_{y}=e_{x}/\eta
  9. h x = - e y / η h_{x}=-e_{y}/\eta
  10. E ( r , t ) H ( r , t ) = e x h x + e y h y + e z h z = e x ( - e y / η ) + e y ( e x / η ) + 0 0 = 0 \vec{E}(\vec{r},t)\cdot\vec{H}(\vec{r},t)=e_{x}h_{x}+e_{y}h_{y}+e_{z}h_{z}=e_{% x}(-e_{y}/\eta)+e_{y}(e_{x}/\eta)+0\cdot 0=0
  11. I = ( | e x | 2 + | e y | 2 ) 1 2 η I=(|e_{x}|^{2}+|e_{y}|^{2})\,\frac{1}{2\eta}
  12. η = \eta=
  13. η 0 \eta_{0}
  14. e x = 1 + Q 2 e_{x}=\sqrt{\frac{1+Q}{2}}
  15. e y = 1 - Q 2 e i ϕ e_{y}=\sqrt{\frac{1-Q}{2}}\,e^{i\phi}
  16. e = 1 - b 2 / a 2 e=\sqrt{1-b^{2}/a^{2}}
  17. 𝐞 = [ a 1 e i θ 1 a 2 e i θ 2 ] . \mathbf{e}=\begin{bmatrix}a_{1}e^{i\theta_{1}}\\ a_{2}e^{i\theta_{2}}\end{bmatrix}.
  18. a 1 a_{1}
  19. a 2 a_{2}
  20. θ 1 \theta_{1}
  21. θ 2 \theta_{2}
  22. 𝚿 = 𝐞𝐞 \mathbf{\Psi}=\left\langle\mathbf{e}\mathbf{e}^{\dagger}\right\rangle\,
  23. = [ e 1 e 1 * e 1 e 2 * e 2 e 1 * e 2 e 2 * ] =\left\langle\begin{bmatrix}e_{1}e_{1}^{*}&e_{1}e_{2}^{*}\\ e_{2}e_{1}^{*}&e_{2}e_{2}^{*}\end{bmatrix}\right\rangle
  24. = [ a 1 2 a 1 a 2 e i ( θ 1 - θ 2 ) a 1 a 2 e - i ( θ 1 - θ 2 ) a 2 2 ] =\left\langle\begin{bmatrix}a_{1}^{2}&a_{1}a_{2}e^{i(\theta_{1}-\theta_{2})}\\ a_{1}a_{2}e^{-i(\theta_{1}-\theta_{2})}&a_{2}^{2}\end{bmatrix}\right\rangle
  25. S 0 = I S_{0}=I\,
  26. S 1 = I p cos 2 ψ cos 2 χ S_{1}=Ip\cos 2\psi\cos 2\chi\,
  27. S 2 = I p sin 2 ψ cos 2 χ S_{2}=Ip\sin 2\psi\cos 2\chi\,
  28. S 3 = I p sin 2 χ S_{3}=Ip\sin 2\chi\,
  29. P = S 1 2 + S 2 2 + S 3 2 P=\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}
  30. 𝐒 = 1 S 0 [ S 0 S 1 S 2 S 3 ] . \mathbf{S^{\prime}}=\frac{1}{S_{0}}\begin{bmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{bmatrix}.
  31. 𝐒 \mathbf{S^{\prime}}
  32. S 0 = 1 S^{\prime}_{0}=1
  33. P 0 = 1 P^{\prime}_{0}=1
  34. P = S 1 2 + S 2 2 + S 3 2 P^{\prime}=\sqrt{S_{1}^{\prime 2}+S_{2}^{\prime 2}+S_{3}^{\prime 2}}
  35. 𝐒 ′′ = 1 P [ 1 S 1 S 2 S 3 ] = 1 P [ S 0 S 1 S 2 S 3 ] . \mathbf{S^{\prime\prime}}=\frac{1}{P^{\prime}}\begin{bmatrix}1\\ S^{\prime}_{1}\\ S^{\prime}_{2}\\ S^{\prime}_{3}\end{bmatrix}=\frac{1}{P}\begin{bmatrix}S_{0}\\ S_{1}\\ S_{2}\\ S_{3}\end{bmatrix}.
  36. 𝐞 ( z + Δ z , t + Δ t ) = 𝐞 ( z , t ) e i k ( c Δ t - Δ z ) , \mathbf{e}(z+\Delta z,t+\Delta t)=\mathbf{e}(z,t)e^{ik(c\Delta t-\Delta z)},
  37. e - i ω t e^{-i\omega t}
  38. 𝐞 = 𝐉𝐞 . \mathbf{e^{\prime}}=\mathbf{J}\mathbf{e}.
  39. 𝐉 = 𝐓 [ g 1 0 0 g 2 ] 𝐓 - 1 , \mathbf{J}=\mathbf{T}\begin{bmatrix}g_{1}&0\\ 0&g_{2}\end{bmatrix}\mathbf{T}^{-1},

Polish_notation.html

  1. ¬ φ \neg\varphi
  2. N φ \mathrm{N}\varphi
  3. φ ψ \varphi\land\psi
  4. K φ ψ \mathrm{K}\varphi\psi
  5. φ ψ \varphi\lor\psi
  6. A φ ψ \mathrm{A}\varphi\psi
  7. φ ψ \varphi\to\psi
  8. C φ ψ \mathrm{C}\varphi\psi
  9. φ ψ \varphi\leftrightarrow\psi
  10. E φ ψ \mathrm{E}\varphi\psi
  11. \bot
  12. O \mathrm{O}
  13. φ ψ \varphi\mid\psi
  14. D φ ψ \mathrm{D}\varphi\psi
  15. φ \Diamond\varphi
  16. M φ \mathrm{M}\varphi
  17. φ \Box\varphi
  18. L φ \mathrm{L}\varphi
  19. p φ \forall p\,\varphi
  20. Π p φ \Pi p\,\varphi
  21. p φ \exists p\,\varphi
  22. Σ p φ \Sigma p\,\varphi

Polygon.html

  1. ( 1 - 2 n ) π \left(1-\tfrac{2}{n}\right)\pi
  2. 180 - 360 n 180-\tfrac{360}{n}
  3. p q \tfrac{p}{q}
  4. π ( p - 2 q ) p \tfrac{\pi(p-2q)}{p}
  5. 180 ( p - 2 q ) p \tfrac{180(p-2q)}{p}
  6. A = 1 2 | i = 0 n - 1 ( x i y i + 1 - x i + 1 y i ) | A=\frac{1}{2}\left|\sum_{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\right|\,
  7. C x = 1 6 A i = 0 n - 1 ( x i + x i + 1 ) ( x i y i + 1 - x i + 1 y i ) C_{x}=\frac{1}{6A}\sum_{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})\,
  8. C y = 1 6 A i = 0 n - 1 ( y i + y i + 1 ) ( x i y i + 1 - x i + 1 y i ) . C_{y}=\frac{1}{6A}\sum_{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}).\,
  9. C x C_{x}
  10. C y C_{y}
  11. A A
  12. A = 1 2 ( a 1 [ a 2 sin ( θ 1 ) + a 3 sin ( θ 1 + θ 2 ) + + a n - 1 sin ( θ 1 + θ 2 + + θ n - 2 ) ] \displaystyle A=\frac{1}{2}(a_{1}[a_{2}\sin(\theta_{1})+a_{3}\sin(\theta_{1}+% \theta_{2})+\cdots+a_{n-1}\sin(\theta_{1}+\theta_{2}+\cdots+\theta_{n-2})]
  13. p 2 > 4 π A p^{2}>4\pi A
  14. A = 1 2 p r . A=\tfrac{1}{2}\cdot p\cdot r.
  15. A = n s 4 4 - s 2 . A=\frac{ns}{4}\sqrt{4-s^{2}}.
  16. A = r 2 p 1 - p 2 4 n 2 r 2 . A=\frac{r}{2}\cdot p\cdot\sqrt{1-\tfrac{p^{2}}{4n^{2}r^{2}}}.
  17. A = n s 2 4 cot π n = n s 2 4 cot θ n - 2 = n sin π n cos π n = n sin θ n - 2 cos θ n - 2 . A=\frac{ns^{2}}{4}\cot\frac{\pi}{n}=\frac{ns^{2}}{4}\cot\frac{\theta}{n-2}=n% \cdot\sin\frac{\pi}{n}\cdot\cos\frac{\pi}{n}=n\cdot\sin\frac{\theta}{n-2}\cdot% \cos\frac{\theta}{n-2}.

Polyhedron.html

  1. χ = V - E + F . \chi=V-E+F.
  2. A A
  3. r r
  4. A r / 3 Ar/3
  5. n n
  6. volume = n A r / 3 \,\text{volume}=nAr/3
  7. L L
  8. A = L 2 A=L^{2}
  9. r = L / 2 r=L/2
  10. volume = 6 L 2 L 2 3 = L 3 , \,\text{volume}=\frac{6\cdot L^{2}\cdot\frac{L}{2}}{3}=L^{3},
  11. F ( x ) = 1 3 x = ( x 1 3 , x 2 3 , x 3 3 ) \vec{F}(\vec{x})=\frac{1}{3}\vec{x}=\left(\frac{x_{1}}{3},\frac{x_{2}}{3},% \frac{x_{3}}{3}\right)
  12. F ( x ) F(x)
  13. volume ( Ω ) = Ω F d Ω = S F n ^ d S . \,\text{volume}(\Omega)=\int_{\Omega}\nabla\cdot\vec{F}d\Omega=\oint_{S}\vec{F% }\cdot\hat{n}dS.
  14. volume = 1 3 face i x i n ^ i A i \,\text{volume}=\frac{1}{3}\sum_{\,\text{face }i}\vec{x}_{i}\cdot\hat{n}_{i}A_% {i}
  15. x i \vec{x}_{i}
  16. n ^ i \hat{n}_{i}
  17. A i A_{i}

Polymer.html

  1. η \eta\,
  2. η \eta\,

Polymerization.html

  1. A + A + A A A A A+A+A...\rightarrow AAA...
  2. A + B + A A B A A+B+A...\rightarrow ABA...
  3. σ \sigma

Polynomial.html

  1. x x
  2. x P ( x ) x\mapsto P(x)
  3. P = P ( X ) . P=P(X).
  4. a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 , a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0},
  5. a 0 , , a n a_{0},\ldots,a_{n}
  6. x x
  7. x x
  8. i = 0 n a i x i \sum_{i=0}^{n}a_{i}x^{i}
  9. - 5 x 2 y -5x^{2}y\,
  10. 5 −5
  11. x x
  12. y y
  13. x x
  14. y y
  15. 2 + 1 = 3 2+1=3
  16. 3 x 2 term 1 - 5 x term 2 + 4 term 3 . \underbrace{3x^{2}}_{\begin{smallmatrix}\mathrm{term}\\ \mathrm{1}\end{smallmatrix}}\underbrace{-5x}_{\begin{smallmatrix}\mathrm{term}% \\ \mathrm{2}\end{smallmatrix}}\underbrace{+4}_{\begin{smallmatrix}\mathrm{term}% \\ \mathrm{3}\end{smallmatrix}}.
  17. 2 x 2x
  18. x x
  19. x x
  20. x x
  21. 3 3
  22. x x
  23. 2 2
  24. x , y x,y
  25. z z
  26. ( ( ( ( ( a n x + a n - 1 ) x + a n - 2 ) x + + a 3 ) x + a 2 ) x + a 1 ) x + a 0 . (((\cdots((a_{n}x+a_{n-1})x+a_{n-2})x+\cdots+a_{3})x+a_{2})x+a_{1})x+a_{0}.
  27. P = 3 x 2 - 2 x + 5 x y - 2 Q = - 3 x 2 + 3 x + 4 y 2 + 8 \begin{aligned}\displaystyle P&\displaystyle=3x^{2}-2x+5xy-2\\ \displaystyle Q&\displaystyle=-3x^{2}+3x+4y^{2}+8\end{aligned}
  28. P + Q = 3 x 2 - 2 x + 5 x y - 2 - 3 x 2 + 3 x + 4 y 2 + 8 P+Q=3x^{2}-2x+5xy-2-3x^{2}+3x+4y^{2}+8
  29. P + Q = x + 5 x y + 4 y 2 + 6 P+Q=x+5xy+4y^{2}+6
  30. P = 2 x + 3 y + 5 Q = 2 x + 5 y + x y + 1 \begin{aligned}\displaystyle P&\displaystyle{=2x+3y+5}\\ \displaystyle Q&\displaystyle{=2x+5y+xy+1}\end{aligned}
  31. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  32. P Q = 4 x 2 + 21 x y + 2 x 2 y + 12 x + 15 y 2 + 3 x y 2 + 28 y + 5 PQ=4x^{2}+21xy+2x^{2}y+12x+15y^{2}+3xy^{2}+28y+5
  33. f ( x ) f(x)
  34. ( x a ) (x−a)
  35. f ( a ) f(a)
  36. p p
  37. k k
  38. p p
  39. 0
  40. c c
  41. a = b q + r a=bq+r
  42. 5 x 3 - 5 5x^{3}-5
  43. 5 ( x - 1 ) ( x 2 + x + 1 ) 5(x-1)\left(x^{2}+x+1\right)
  44. 5 ( x - 1 ) ( x + 1 + i 3 2 ) ( x + 1 - i 3 2 ) 5(x-1)\left(x+\frac{1+i\sqrt{3}}{2}\right)\left(x+\frac{1-i\sqrt{3}}{2}\right)
  45. 1 / 12 1/12
  46. 1 / 12 1/12
  47. 2 + 3 i 2+3i
  48. f f
  49. f ( x ) = a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}\,
  50. x x
  51. n n
  52. f f
  53. f ( x ) = x 3 - x f(x)=x^{3}-x\,
  54. f ( x , y ) = 2 x 3 + 4 x 2 y + x y 5 + y 2 - 7. f(x,y)=2x^{3}+4x^{2}y+xy^{5}+y^{2}-7.\,
  55. f ( x ) = cos ( 2 arccos ( x ) ) f(x)=\cos(2\arccos(x))
  56. [ - 1 , 1 ] [-1,1]
  57. x x
  58. [ - 1 , 1 ] [-1,1]
  59. f ( x ) = 2 x 2 - 1 f(x)=2x^{2}-1
  60. f ( x ) = x < s u p > 2 x 2 f(x)=x<sup>2−x−2
  61. f ( x ) = x < s u p > 3 / 4 + 3 x 2 / 4 3 x / 2 2 f(x)=x<sup>3/4+3x^{2}/4−3x/2−2
  62. f ( x ) = 1 / 14 ( x + 4 ) ( x + 1 ) ( x 1 ) ( x 3 ) + 0.5 f(x)=1/14(x+4)(x+1)(x−1)(x−3)+0.5
  63. f ( x ) = 1 / 20 ( x + 4 ) ( x + 2 ) ( x + 1 ) ( x 1 ) ( x 3 ) f(x)=1/20(x+4)(x+2)(x+1)(x−1)(x−3)
  64. + 2 +2
  65. f ( x ) = 1 / 100 ( x < s u p > 6 2 x 5 26 x 4 + 28 x 3 f(x)=1/100(x<sup>6−2x^{5}−26x^{4}+28x^{3}
  66. f ( x ) = ( x 3 ) ( x 2 ) ( x 1 ) ( x ) ( x + 1 ) ( x + 2 ) f(x)=(x−3)(x−2)(x−1)(x)(x+1)(x+2)
  67. ( x + 3 ) (x+3)
  68. f ( x ) = 0 f(x)=0
  69. x x
  70. a n x n + a n - 1 x n - 1 + + a 2 x 2 + a 1 x + a 0 = 0 a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}=0
  71. 3 x 2 + 4 x - 5 = 0 3x^{2}+4x-5=0\,
  72. P P
  73. x x
  74. f ( x ) = P f(x)=P
  75. x x
  76. P P
  77. f f
  78. P P
  79. x x
  80. f ( x ) = 0 f(x)=0
  81. P P
  82. f f
  83. f f
  84. x x
  85. a a
  86. P P
  87. x a x−a
  88. x x
  89. P P
  90. x a x−a
  91. P P
  92. P P
  93. a a
  94. P P
  95. a a
  96. P P
  97. P P
  98. m m
  99. P P
  100. a a
  101. P P
  102. P P
  103. P P
  104. P P
  105. x a x−a
  106. P ( x ) = i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + + a n x n , P(x)=\sum_{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n},
  107. P ( A ) = i = 0 n a i A i = a 0 I + a 1 A + a 2 A 2 + + a n A n , P(A)=\sum_{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots+a_{n}A^{n},
  108. i = 0 n a i x i \sum_{i=0}^{n}a_{i}x^{i}
  109. i = 1 n a i i x i - 1 \sum_{i=1}^{n}a_{i}ix^{i-1}
  110. i = 0 n a i i + 1 x i + 1 + c . \sum_{i=0}^{n}{a_{i}\over i+1}x^{i+1}+c.
  111. f f
  112. X X
  113. R R
  114. f = a n X n + a n - 1 X n - 1 + + a 1 X 1 + a 0 X 0 f=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots+a_{1}X^{1}+a_{0}X^{0}
  115. n n
  116. R R
  117. X X
  118. n n
  119. i > n i>n
  120. n n
  121. X X
  122. a X k b X l = a b X k + l aX^{k}\;bX^{l}=abX^{k+l}
  123. R R
  124. R R
  125. R X X RXX
  126. R R
  127. R X X RXX
  128. r r
  129. R R
  130. R X X RXX
  131. R R
  132. R X X RXX
  133. R R
  134. R X X RXX
  135. R R
  136. X X
  137. R R
  138. X r = r X Xr=rX
  139. X X
  140. R X X RXX
  141. R R
  142. R R
  143. P P
  144. R X X RXX
  145. f f
  146. R R
  147. R R
  148. f ( r ) f(r)
  149. R R
  150. X X
  151. P P
  152. R R
  153. p p
  154. R R
  155. X X
  156. R R
  157. f f
  158. g g
  159. R X X RXX
  160. f f
  161. g g
  162. f f
  163. g g
  164. q q
  165. R X X RXX
  166. f q = g fq=g
  167. f f
  168. R X X RXX
  169. r r
  170. R R
  171. f ( r ) = 0 f(r)=0
  172. X r X−r
  173. f f
  174. F F
  175. f f
  176. g g
  177. F X X FXX
  178. g 0 g≠0
  179. q q
  180. r r
  181. F X X FXX
  182. f = q g + r f=q\,g+r
  183. r r
  184. g g
  185. q q
  186. r r
  187. f f
  188. g g
  189. F X X FXX
  190. 3 x + 2 y + z = 29 3x+ 2y+z=29
  191. a a
  192. x x
  193. p p

Polytope.html

  1. \vdots
  2. \vdots
  3. \vdots
  4. \vdots
  5. { ( x , y ) 2 x 0 } \{(x,y)\in\mathbb{R}^{2}\mid x\geq 0\}

Population_inversion.html

  1. N 1 + N 2 = N N_{1}+N_{2}=N
  2. Δ E 12 = E 2 - E 1 , \Delta E_{12}=E_{2}-E_{1},
  3. ν 12 \nu_{12}
  4. E 2 - E 1 = Δ E = h ν 12 , E_{2}-E_{1}=\Delta E=h\nu_{12},
  5. N 2 N 1 = exp - ( E 2 - E 1 ) k T , \frac{N_{2}}{N_{1}}=\exp{\frac{-(E_{2}-E_{1})}{kT}},
  6. N 2 ( t ) = N 2 ( 0 ) exp - t τ 21 , N_{2}(t)=N_{2}(0)\exp{\frac{-t}{\tau_{21}}},
  7. ν 13 = 1 h ( E 3 - E 1 ) \scriptstyle\nu_{13}\,=\,\frac{1}{h}\left(E_{3}-E_{1}\right)

Positive-definite_matrix.html

  1. M M
  2. z T M z z^{T}Mz
  3. z z
  4. n n
  5. z T z^{T}
  6. z z
  7. M M
  8. z * M z z^{*}Mz
  9. z z
  10. n n
  11. z * z^{*}
  12. z z
  13. z T M z z^{T}Mz
  14. z * M z z^{*}Mz
  15. I = [ 1 0 0 1 ] I=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  16. z T I z = [ a b ] [ 1 0 0 1 ] [ a b ] = a 2 + b 2 z^{\mathrm{T}}Iz=\begin{bmatrix}a&b\end{bmatrix}\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix}=a^{2}+b^{2}
  17. z * I z = [ a * b * ] [ 1 0 0 1 ] [ a b ] = a * a + b * b = | a | 2 + | b | 2 z^{*}Iz=\begin{bmatrix}a^{*}&b^{*}\end{bmatrix}\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix}=a^{*}a+b^{*}b=|a|^{2}+|b|^{2}
  18. M = [ 2 - 1 0 - 1 2 - 1 0 - 1 2 ] M=\begin{bmatrix}2&-1&0\\ -1&2&-1\\ 0&-1&2\end{bmatrix}
  19. z T M z = ( z T M ) z = [ ( 2 a - b ) ( - a + 2 b - c ) ( - b + 2 c ) ] [ a b c ] = 2 a 2 - 2 a b + 2 b 2 - 2 b c + 2 c 2 = a 2 + ( a - b ) 2 + ( b - c ) 2 + c 2 \begin{aligned}\displaystyle z^{\mathrm{T}}Mz=(z^{\mathrm{T}}M)z&\displaystyle% =\begin{bmatrix}(2a-b)&(-a+2b-c)&(-b+2c)\end{bmatrix}\begin{bmatrix}a\\ b\\ c\end{bmatrix}\\ &\displaystyle=2{a}^{2}-2ab+2{b}^{2}-2bc+2{c}^{2}\\ &\displaystyle={a}^{2}+(a-b)^{2}+(b-c)^{2}+{c}^{2}\end{aligned}
  20. N = [ 1 2 2 1 ] N=\begin{bmatrix}1&2\\ 2&1\end{bmatrix}
  21. [ 1 - 1 ] \begin{bmatrix}1\\ -1\end{bmatrix}
  22. z T N z = [ 1 - 1 ] [ 1 2 2 1 ] [ 1 - 1 ] = [ - 1 1 ] [ 1 - 1 ] = - 2 0. z^{\mathrm{T}}Nz=\begin{bmatrix}1&-1\end{bmatrix}\begin{bmatrix}1&2\\ 2&1\end{bmatrix}\begin{bmatrix}1\\ -1\end{bmatrix}=\begin{bmatrix}-1&1\end{bmatrix}\begin{bmatrix}1\\ -1\end{bmatrix}=-2\not>0.
  23. A A
  24. A T A A^{T}A
  25. z z
  26. z T A T A z = A z 2 2 > 0 , z^{T}A^{T}Az=\|Az\|_{2}^{2}>0,
  27. A A
  28. A z 0. Az\neq 0.
  29. , \langle\cdot,\cdot\rangle
  30. x , y := y * M x \langle x,y\rangle:=y^{*}Mx
  31. z , z \langle z,z\rangle
  32. x 1 , , x n x_{1},\ldots,x_{n}
  33. , \langle\cdot,\cdot\rangle
  34. M i j = x i , x j M_{ij}=\langle x_{i},x_{j}\rangle
  35. E i i = λ i E_{ii}=\sqrt{\lambda_{i}}
  36. E 2 = D E^{2}=D
  37. P - 1 D P = P * D P = P * E E P = ( E P ) * E P . P^{-1}DP=P^{*}DP=P^{*}EEP=(EP)^{*}EP.
  38. x 1 , , x n x_{1},\ldots,x_{n}
  39. x i , x j = x i T x j \langle x_{i},x_{j}\rangle=x_{i}^{T}x_{j}
  40. x * M x < 0 x^{*}Mx<0\,
  41. x * M x 0 x^{*}Mx\geq 0
  42. x * M x 0 x^{*}Mx\leq 0
  43. M \sqrt{M}
  44. N \sqrt{N}
  45. | m i j | m i i m j j m i i + m j j 2 |m_{ij}|\leq\sqrt{m_{ii}m_{jj}}\leq\frac{m_{ii}+m_{jj}}{2}
  46. max | m i j | max | m i i | \max|m_{ij}|\leq\max|m_{ii}|
  47. j 0 | m ( j ) | < m ( 0 ) \sum\nolimits_{j\neq 0}|m(j)|<m(0)
  48. det ( M k ) / det ( M k - 1 ) \det(M_{k})/\det(M_{k-1})
  49. x ( α M + ( 1 - α ) N ) x = α x M x + ( 1 - α ) x N x 0. x^{\top}(\alpha M+(1-\alpha)N)x=\alpha x^{\top}Mx+(1-\alpha)x^{\top}Nx\geq 0.
  50. det ( M N ) det ( N ) i m i i . \det(M\circ N)\geq\det(N)\prod\nolimits_{i}m_{ii}.
  51. M = [ A B C D ] M=\begin{bmatrix}A&B\\ C&D\end{bmatrix}
  52. [ v * 0 ] [ A B B * D ] [ v 0 ] = v * A v 0. \begin{bmatrix}v^{*}&0\end{bmatrix}\begin{bmatrix}A&B\\ B^{*}&D\end{bmatrix}\begin{bmatrix}v\\ 0\end{bmatrix}=v^{*}Av\geq 0.
  53. M = [ 1 1 - 1 1 ] , M=\begin{bmatrix}1&1\\ -1&1\end{bmatrix},
  54. [ 1 1 - 1 1 ] \bigl[\begin{smallmatrix}1&1\\ -1&1\end{smallmatrix}\bigr]
  55. M 0 M\succeq 0
  56. M 0 M\succ 0

Positron_emission_tomography.html

  1. 1 \ell_{1}

Positronium.html

  1. t 0 = 2 m e c 2 α 5 = 1.244 × 10 - 10 s . t_{0}=\frac{2\hbar}{m_{\mathrm{e}}c^{2}\alpha^{5}}=1.244\times 10^{-10}~{}% \mathrm{s}.
  2. t 1 = 1 2 9 h 2 m e c 2 α 6 ( π 2 - 9 ) = 1.386 × 10 - 7 s . t_{1}=\frac{\frac{1}{2}9h}{2m_{\mathrm{e}}c^{2}\alpha^{6}(\pi^{2}-9)}=1.386% \times 10^{-7}~{}\mathrm{s}.
  3. E n = - μ q e 4 8 h 2 ε 0 2 1 n 2 , E_{n}=-\frac{\mu q_{\mathrm{e}}^{4}}{8h^{2}\varepsilon_{0}^{2}}\frac{1}{n^{2}},
  4. h h
  5. μ μ
  6. μ = m e m p m e + m p = m e 2 2 m e = m e 2 , \mu=\frac{m_{\mathrm{e}}m_{\mathrm{p}}}{m_{\mathrm{e}}+m_{\mathrm{p}}}=\frac{m% _{\mathrm{e}}^{2}}{2m_{\mathrm{e}}}=\frac{m_{\mathrm{e}}}{2},
  7. E n = - 1 2 m e q e 4 8 h 2 ε 0 2 1 n 2 = - 6.8 eV n 2 . E_{n}=-\frac{1}{2}\frac{m_{\mathrm{e}}q_{\mathrm{e}}^{4}}{8h^{2}\varepsilon_{0% }^{2}}\frac{1}{n^{2}}=\frac{-6.8~{}\mathrm{eV}}{n^{2}}.
  8. n = 1 n=1
  9. c < s u p > 2 n c\frac{<sup>2}{n}

Post_correspondence_problem.html

  1. α 1 , , α N \alpha_{1},\ldots,\alpha_{N}
  2. β 1 , , β N \beta_{1},\ldots,\beta_{N}
  3. A A
  4. ( i k ) 1 k K (i_{k})_{1\leq k\leq K}
  5. K 1 K\geq 1
  6. 1 i k N 1\leq i_{k}\leq N
  7. k k
  8. α i 1 α i K = β i 1 β i K . \alpha_{i_{1}}\ldots\alpha_{i_{K}}=\beta_{i_{1}}\ldots\beta_{i_{K}}.
  9. α 3 α 2 α 3 α 1 = b b a + a b + b b a + a = b b a a b b b a a = b b + a a + b b + b a a = β 3 β 2 β 3 β 1 . \alpha_{3}\alpha_{2}\alpha_{3}\alpha_{1}=bba+ab+bba+a=bbaabbbaa=bb+aa+bb+baa=% \beta_{3}\beta_{2}\beta_{3}\beta_{1}.
  10. α 2 , α 3 \alpha_{2},\alpha_{3}
  11. β 2 , β 3 \beta_{2},\beta_{3}
  12. A A
  13. A A
  14. i 1 , i 2 , i_{1},i_{2},\ldots
  15. α i 1 α i k \alpha_{i_{1}}\cdots\alpha_{i_{k}}
  16. β i 1 β i k \beta_{i_{1}}\cdots\beta_{i_{k}}
  17. i 1 , i 2 , i_{1},i_{2},\ldots
  18. α i 1 α i k \alpha_{i_{1}}\cdots\alpha_{i_{k}}
  19. β i 1 β i k \beta_{i_{1}}\cdots\beta_{i_{k}}
  20. { 1 , , N } \{1,\ldots,N\}

Potential_energy.html

  1. W = - Δ U \,W=-\Delta U
  2. Δ U \Delta U
  3. W = C F d x = U ( 𝐱 A ) - U ( 𝐱 B ) W=\int_{C}{F}\cdot\mathrm{d}{x}=U(\mathbf{x}_{A})-U(\mathbf{x}_{B})
  4. 𝐅 = φ = ( φ x , φ y , φ z ) . \mathbf{F}={\nabla\varphi}=\left(\frac{\partial\varphi}{\partial x},\frac{% \partial\varphi}{\partial y},\frac{\partial\varphi}{\partial z}\right).
  5. W = C F d x = C φ d x , W=\int_{C}{F}\cdot\mathrm{d}{x}=\int_{C}\nabla\varphi\cdot\mathrm{d}{x},
  6. W = φ ( 𝐱 B ) - φ ( 𝐱 A ) . W=\varphi(\mathbf{x}_{B})-\varphi(\mathbf{x}_{A}).
  7. W = U ( 𝐱 A ) - U ( 𝐱 B ) . W=U(\mathbf{x}_{A})-U(\mathbf{x}_{B}).
  8. W = - U = - ( U x , U y , U z ) = 𝐅 , {\nabla W}=-{\nabla U}=-\left(\frac{\partial U}{\partial x},\frac{\partial U}{% \partial y},\frac{\partial U}{\partial z}\right)=\mathbf{F},
  9. γ φ ( 𝐫 ) d 𝐫 = a b φ ( 𝐫 ( t ) ) 𝐫 ( t ) d t , = a b d d t φ ( 𝐫 ( t ) ) d t = φ ( 𝐫 ( b ) ) - φ ( 𝐫 ( a ) ) = φ ( 𝐱 B ) - φ ( 𝐱 A ) . \begin{aligned}\displaystyle\int_{\gamma}\nabla\varphi(\mathbf{r})\cdot d% \mathbf{r}&\displaystyle=\int_{a}^{b}\nabla\varphi(\mathbf{r}(t))\cdot\mathbf{% r}^{\prime}(t)dt,\\ &\displaystyle=\int_{a}^{b}\frac{d}{dt}\varphi(\mathbf{r}(t))dt=\varphi(% \mathbf{r}(b))-\varphi(\mathbf{r}(a))=\varphi\left(\mathbf{x}_{B}\right)-% \varphi\left(\mathbf{x}_{A}\right).\end{aligned}
  10. γ 𝐅 d 𝐫 = a b 𝐅 𝐯 d t , = - a b d d t U ( 𝐫 ( t ) ) d t = U ( 𝐱 A ) - U ( 𝐱 B ) . \begin{aligned}\displaystyle\int_{\gamma}\mathbf{F}\cdot d\mathbf{r}&% \displaystyle=\int_{a}^{b}\mathbf{F}\cdot\mathbf{v}dt,\\ &\displaystyle=-\int_{a}^{b}\frac{d}{dt}U(\mathbf{r}(t))dt=U\left(\mathbf{x}_{% A}\right)-U\left(\mathbf{x}_{B}\right).\end{aligned}
  11. P ( t ) = - U 𝐯 = 𝐅 𝐯 . P(t)=-{\nabla U}\cdot\mathbf{v}=\mathbf{F}\cdot\mathbf{v}.
  12. W = t 1 t 2 s y m b o l F \cdotsymbol v d t = t 1 t 2 F z v z d t = F z Δ z . W=\int_{t_{1}}^{t_{2}}symbol{F}\cdotsymbol{v}\mathrm{d}t=\int_{t_{1}}^{t_{2}}F% _{z}v_{z}\mathrm{d}t=F_{z}\Delta z.
  13. U ( 𝐫 ) = m g h ( 𝐫 ) , U(\mathbf{r})=mg\,h(\mathbf{r}),
  14. W = 0 t 𝐅 𝐯 d t = - 0 t k x v x d t = - 1 2 k x 2 . W=\int_{0}^{t}\mathbf{F}\cdot\mathbf{v}\mathrm{\,}{d}t=-\int_{0}^{t}kxv_{x}% \mathrm{\,}{d}t=-\frac{1}{2}kx^{2}.
  15. U ( x ) = 1 2 k x 2 , U(x)=\frac{1}{2}kx^{2},
  16. 𝐅 = - G M m r 3 𝐫 , \mathbf{F}=-\frac{GMm}{r^{3}}\mathbf{r},
  17. 𝐅 = - G M m r 2 𝐫 ^ , \mathbf{F}=-\frac{GMm}{r^{2}}\mathbf{\hat{r}},
  18. 𝐫 ^ \mathbf{\hat{r}}
  19. W = - 𝐫 ( t 1 ) 𝐫 ( t 2 ) G M m r 3 𝐫 d 𝐫 = - t 1 t 2 G M m r 3 𝐫 𝐯 d t . W=-\int^{\mathbf{r}(t_{2})}_{\mathbf{r}(t_{1})}\frac{GMm}{r^{3}}\mathbf{r}% \cdot d\mathbf{r}=-\int^{t_{2}}_{t_{1}}\frac{GMm}{r^{3}}\mathbf{r}\cdot\mathbf% {v}\mathrm{d}t.
  20. 𝐫 = r 𝐞 r , 𝐯 = r ˙ 𝐞 r + r θ ˙ 𝐞 t , \mathbf{r}=r\mathbf{e}_{r},\qquad\mathbf{v}=\dot{r}\mathbf{e}_{r}+r\dot{\theta% }\mathbf{e}_{t},
  21. W = - t 1 t 2 G m M r 3 ( r 𝐞 r ) ( r ˙ 𝐞 r + r θ ˙ 𝐞 t ) d t = - t 1 t 2 G m M r 3 r r ˙ d t = G M m r ( t 2 ) - G M m r ( t 1 ) . W=-\int^{t_{2}}_{t_{1}}\frac{GmM}{r^{3}}(r\mathbf{e}_{r})\cdot(\dot{r}\mathbf{% e}_{r}+r\dot{\theta}\mathbf{e}_{t})\mathrm{d}t=-\int^{t_{2}}_{t_{1}}\frac{GmM}% {r^{3}}r\dot{r}\mathrm{d}t=\frac{GMm}{r(t_{2})}-\frac{GMm}{r(t_{1})}.
  22. d d t r - 1 = - r - 2 r ˙ = - r ˙ r 2 . \frac{\mathrm{d}}{\mathrm{d}t}r^{-1}=-r^{-2}\dot{r}=-\frac{\dot{r}}{r^{2}}.
  23. U = - G M m r , U=-\frac{GMm}{r},
  24. 𝐅 = 1 4 π ε 0 Q q r 3 𝐫 , \mathbf{F}=\frac{1}{4\pi\varepsilon_{0}}\frac{Qq}{r^{3}}\mathbf{r},
  25. U ( r ) = 1 4 π ε 0 Q q r . U({r})=\frac{1}{4\pi\varepsilon_{0}}\frac{Qq}{r}.
  26. W F = - Δ U F . W_{F}=-\Delta U_{F}.\!
  27. U = m g h U=mgh\!
  28. Δ U = m g Δ h . \,\Delta U=mg\Delta h.
  29. U = - G m 1 M 2 r + K U=-G\frac{m_{1}M_{2}}{r}\ +K
  30. n ( n - 1 ) 2 \frac{n(n-1)}{2}
  31. U = - m ( G M 1 r 1 + G M 2 r 2 ) U=-m\left(G\frac{M_{1}}{r_{1}}+G\frac{M_{2}}{r_{2}}\right)
  32. U = - m G M r U=-m\sum G\frac{M}{r}
  33. r = 0 r=0
  34. r = r=\infty
  35. U = 0 U=0
  36. r = 0 r=0
  37. U = 0 U=0
  38. r = 0 r=0
  39. U = 0 U=0
  40. 𝐅 = - 1 4 π ε 0 Q q r 3 𝐫 , \mathbf{F}=-\frac{1}{4\pi\varepsilon_{0}}\frac{Qq}{r^{3}}\mathbf{r},
  41. U ( r ) = 1 4 π ε 0 Q q r . U({r})=\frac{1}{4\pi\varepsilon_{0}}\frac{Qq}{r}.
  42. 𝐦 \mathbf{m}
  43. 𝐁 \mathbf{B}
  44. U = - 𝐦 𝐁 . U=-\mathbf{m}\cdot\mathbf{B}.
  45. 𝐌 \mathbf{M}
  46. U = - 1 2 𝐌 𝐁 d V , U=-\frac{1}{2}\int\mathbf{M}\cdot\mathbf{B}\mathrm{d}V,
  47. 𝐌 \mathbf{M}
  48. ϕ \phi
  49. V V
  50. U = - G M m r , U=-\frac{GMm}{r},
  51. ϕ = - ( G M r + G m r ) = - G ( M + m ) r = - G M m μ r = U μ . \phi=-\left(\frac{GM}{r}+\frac{Gm}{r}\right)=-\frac{G(M+m)}{r}=-\frac{GMm}{\mu r% }=\frac{U}{\mu}.
  52. μ \mu
  53. U = a U=a
  54. U = b U=b
  55. ( b - a ) (b-a)
  56. ( a - b ) (a-b)
  57. U A B A = ( b - a ) + ( a - b ) = 0. U_{A\to B\to A}=(b-a)+(a-b)=0.\,
  58. a + c a+c
  59. b + c b+c
  60. c c
  61. c c
  62. U A B = ( b + c ) - ( a + c ) = b - a U_{A\to B}=(b+c)-(a+c)=b-a\,
  63. U U
  64. ϕ \phi

Potential_flow.html

  1. φ φ
  2. 𝐯 \mathbf{v}
  3. φ φ
  4. 𝐯 = φ . \mathbf{v}=\nabla\varphi.
  5. 𝐯 = φ \mathbf{v}=−∇φ
  6. × φ = 𝟎 , \nabla\times\nabla\varphi=\mathbf{0},
  7. 𝐯 \mathbf{v}
  8. × 𝐯 = 0. \nabla\times\mathbf{v}=\mathbf{0}.
  9. 𝐯 \mathbf{v}
  10. 𝐯 = 0 , \nabla\cdot\mathbf{v}=0,
  11. φ φ
  12. 2 φ = 0 , \nabla^{2}\varphi=0,
  13. Δ Δ
  14. ( 1 - M x 2 ) 2 Φ x 2 + ( 1 - M y 2 ) 2 Φ y 2 + ( 1 - M z 2 ) 2 Φ z 2 - 2 M x M y 2 Φ x y - 2 M y M z 2 Φ y z - 2 M z M x 2 Φ z x = 0 , \left(1-M_{x}^{2}\right)\frac{\partial^{2}\Phi}{\partial x^{2}}+\left(1-M_{y}^% {2}\right)\frac{\partial^{2}\Phi}{\partial y^{2}}+\left(1-M_{z}^{2}\right)% \frac{\partial^{2}\Phi}{\partial z^{2}}-2M_{x}M_{y}\frac{\partial^{2}\Phi}{% \partial x\,\partial y}-2M_{y}M_{z}\frac{\partial^{2}\Phi}{\partial y\,% \partial z}-2M_{z}M_{x}\frac{\partial^{2}\Phi}{\partial z\,\partial x}=0,
  15. M x = 1 a Φ x , M y = 1 a Φ y , and M z = 1 a Φ z , M_{x}=\frac{1}{a}\frac{\partial\Phi}{\partial x},\qquad M_{y}=\frac{1}{a}\frac% {\partial\Phi}{\partial y},\qquad\,\text{and}\qquad M_{z}=\frac{1}{a}\frac{% \partial\Phi}{\partial z},
  16. 𝐯 \mathbf{v}
  17. Φ ∇Φ
  18. Φ Φ
  19. φ ∇φ
  20. Φ = V x + φ . \nabla\Phi=V_{\infty}x+\nabla\varphi.
  21. ( 1 - M 2 ) 2 φ x 2 + 2 φ y 2 + 2 φ z 2 = 0 , \left(1-M_{\infty}^{2}\right)\frac{\partial^{2}\varphi}{\partial x^{2}}+\frac{% \partial^{2}\varphi}{\partial y^{2}}+\frac{\partial^{2}\varphi}{\partial z^{2}% }=0,
  22. x i ( ρ v i ) = 0 , ρ v j v i x j = - p x i , \begin{aligned}\displaystyle\frac{\partial}{\partial x_{i}}\left(\rho\,v_{i}% \right)&\displaystyle=0,\\ \displaystyle\rho\,v_{j}\,\frac{\partial v_{i}}{\partial x_{j}}&\displaystyle=% -\frac{\partial p}{\partial x_{i}},\end{aligned}
  23. j j
  24. ρ ρ
  25. p p
  26. 𝐯 \mathbf{v}
  27. ρ ρ
  28. a 2 = [ p ρ ] S . a^{2}=\left[\frac{\partial p}{\partial\rho}\right]_{S}.
  29. v i ρ x i + ρ v i x i = 0 and ρ v j v i x j = - a 2 ρ x i . v_{i}\,\frac{\partial\rho}{\partial x_{i}}+\rho\,\frac{\partial v_{i}}{% \partial x_{i}}=0\qquad\,\text{ and }\qquad\rho\,v_{j}\,\frac{\partial v_{i}}{% \partial x_{j}}=-a^{2}\,\frac{\partial\rho}{\partial x_{i}}.
  30. ρ v i v j v i x j = ρ a 2 v i x i . \rho\,v_{i}\,v_{j}\,\frac{\partial v_{i}}{\partial x_{j}}=\rho\,a^{2}\,\frac{% \partial v_{i}}{\partial x_{i}}.
  31. ρ ρ
  32. v i x i - v i v j a 2 v i x j = 0. \frac{\partial v_{i}}{\partial x_{i}}-\frac{v_{i}\,v_{j}}{a^{2}}\frac{\partial v% _{i}}{\partial x_{j}}=0.
  33. 𝐯 \mathbf{v}
  34. Φ Φ
  35. v i = Φ x i and M i = v i a = 1 a Φ x i . v_{i}=\frac{\partial\Phi}{\partial x_{i}}\qquad\,\text{ and }\qquad M_{i}=% \frac{v_{i}}{a}=\frac{1}{a}\frac{\partial\Phi}{\partial x_{i}}.
  36. 2 Φ x i x i - M i M j 2 Φ x i x j = 0. \frac{\partial^{2}\Phi}{\partial x_{i}\,\partial x_{i}}-M_{i}\,M_{j}\,\frac{% \partial^{2}\Phi}{\partial x_{i}\,\partial x_{j}}=0.
  37. p p
  38. ρ ρ
  39. 2 φ t 2 = a ¯ 2 Δ φ , \frac{\partial^{2}\varphi}{\partial t^{2}}=\overline{a}^{2}\Delta\varphi,
  40. φ φ
  41. 𝐯 \mathbf{v}
  42. 𝐯 = φ \mathbf{v}=∇φ
  43. Δ Δ
  44. p p
  45. ρ ρ
  46. f f
  47. ( x , y ) (x,y)
  48. ( φ , ψ ) (φ,ψ)
  49. x , y , φ x,y,φ
  50. ψ ψ
  51. z = x + i y and w = φ + i ψ . z=x+iy\qquad\,\text{and}\qquad w=\varphi+i\psi.
  52. f f
  53. f ( x + i y ) = φ + i ψ or f ( z ) = w . f(x+iy)=\varphi+i\psi\qquad\,\text{or}\qquad f(z)=w.
  54. f f
  55. φ x = ψ y , φ y = - ψ x . \frac{\partial\varphi}{\partial x}=\frac{\partial\psi}{\partial y},\qquad\frac% {\partial\varphi}{\partial y}=-\frac{\partial\psi}{\partial x}.
  56. ( u , v ) (u,v)
  57. ( x , y ) (x,y)
  58. f f
  59. z z
  60. d f d z = u - i v \frac{df}{dz}=u-iv
  61. 𝐯 = ( u , v ) \mathbf{v}=(u,v)
  62. u = φ x = ψ y , v = φ y = - ψ x . u=\frac{\partial\varphi}{\partial x}=\frac{\partial\psi}{\partial y},\qquad v=% \frac{\partial\varphi}{\partial y}=-\frac{\partial\psi}{\partial x}.
  63. φ φ
  64. ψ ψ
  65. Δ φ = 2 φ x 2 + 2 φ y 2 = 0 and Δ ψ = 2 ψ x 2 + 2 ψ y 2 = 0. \Delta\varphi=\frac{\partial^{2}\varphi}{\partial x^{2}}+\frac{\partial^{2}% \varphi}{\partial y^{2}}=0\qquad\,\text{and}\qquad\Delta\psi=\frac{\partial^{2% }\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}}=0.
  66. φ φ
  67. ψ ψ
  68. ψ ψ
  69. φ φ
  70. ϕ ψ = ϕ x ψ x + ϕ y ψ y = ψ y ψ x - ψ x ψ y = 0. \nabla\phi\cdot\nabla\psi=\frac{\partial\phi}{\partial x}\frac{\partial\psi}{% \partial x}+\frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}=% \frac{\partial\psi}{\partial y}\frac{\partial\psi}{\partial x}-\frac{\partial% \psi}{\partial x}\frac{\partial\psi}{\partial y}=0.
  71. ψ ψ
  72. φ φ
  73. Δ ψ = 0 Δψ=0
  74. × 𝐯 = 𝟎 ∇×\mathbf{v}=\mathbf{0}
  75. · 𝐯 = 0 ∇·\mathbf{v}=0
  76. f f
  77. w = A z < s u p > n w=Az<sup>n
  78. z = x + i y z=x+iy
  79. w = φ + i ψ w=φ+iψ
  80. w = A z n , w=Az^{n},
  81. z z
  82. φ = A r n cos n θ and ψ = A r n sin n θ . \varphi=Ar^{n}\cos n\theta\qquad\,\text{and}\qquad\psi=Ar^{n}\sin n\theta.
  83. n n
  84. n n
  85. n = 1 2 n=\frac{1}{2}
  86. n = 2 3 n=\frac{2}{3}
  87. n = 1 n=1
  88. n = 2 n=2
  89. n = 1 n=−1
  90. A A
  91. | A | |A|
  92. a r g ( A ) arg(A)
  93. n = 1 n=1
  94. n = 1 n=1
  95. ψ ψ
  96. x x
  97. f ( x + i y ) = A × ( x + i y ) = A x + i A y f(x+iy)=A\times(x+iy)=Ax+i\cdot Ay
  98. φ = A x φ=Ax
  99. ψ = A y ψ=Ay
  100. x x
  101. n = 2 n=2
  102. n = 2 n=2
  103. ψ ψ
  104. ψ = A r 2 sin 2 θ , \psi=Ar^{2}\sin 2\theta,
  105. sin 2 θ = 2 sin θ cos θ \sin 2\theta=2\sin\theta\,\cos\theta
  106. sin θ = y / r \sin\theta=y/r
  107. cos θ = x / r \cos\theta=x/r
  108. ψ = 2 A x y . \psi=2Axy.
  109. φ ∇φ
  110. ( u v ) = ( φ x φ y ) = ( + ψ y - ψ x ) = ( + 2 A x - 2 A y ) . \begin{pmatrix}u\\ v\end{pmatrix}=\begin{pmatrix}\frac{\partial\varphi}{\partial x}\\ \frac{\partial\varphi}{\partial y}\end{pmatrix}=\begin{pmatrix}+{\partial\psi% \over\partial y}\\ -{\partial\psi\over\partial x}\end{pmatrix}=\begin{pmatrix}+2Ax\\ -2Ay\end{pmatrix}.
  111. z = 0 z=0
  112. ψ = 0 ψ=0
  113. x = 0 x=0
  114. y = 0 y=0
  115. x x
  116. x x
  117. n = 3 n=3
  118. n = 3 n=3
  119. n = 2 n=2
  120. n = 1 n=−1
  121. n = 1 n=−1
  122. ψ = - A r sin θ . \psi=-\frac{A}{r}\sin\theta.
  123. ψ = - A y r 2 = - A y x 2 + y 2 , \psi={-Ay\over r^{2}}={-Ay\over x^{2}+y^{2}},
  124. x 2 + y 2 + A y ψ = 0 , x^{2}+y^{2}+{Ay\over\psi}=0,
  125. x 2 + ( y + A 2 ψ ) 2 = ( A 2 ψ ) 2 . x^{2}+\left(y+\frac{A}{2\psi}\right)^{2}=\left(\frac{A}{2\psi}\right)^{2}.
  126. ( u , v ) = ( ψ y , - ψ x ) = ( A y 2 - x 2 ( x 2 + y 2 ) 2 , - A 2 x y ( x 2 + y 2 ) 2 ) . (u,v)=\left({\partial\psi\over\partial y},-{\partial\psi\over\partial x}\right% )=\left(A\frac{y^{2}-x^{2}}{(x^{2}+y^{2})^{2}},-A\frac{2xy}{(x^{2}+y^{2})^{2}}% \right).
  127. ( u r , u θ ) = ( 1 r ψ θ , - ψ r ) = ( - A r 2 cos θ , - A r 2 sin θ ) . (u_{r},u_{\theta})=\left(\frac{1}{r}{\partial\psi\over\partial\theta},-{% \partial\psi\over\partial r}\right)=\left(-\frac{A}{r^{2}}\cos\theta,-\frac{A}% {r^{2}}\sin\theta\right).
  128. n = 2 n=−2
  129. n = 2 n=−2
  130. ψ = - A r 2 sin ( 2 θ ) . \psi=-\frac{A}{r^{2}}\sin(2\theta).

Power-law_index_profile.html

  1. n ( r ) = { n 1 1 - 2 Δ ( r α ) g r α n 1 1 - 2 Δ r α n(r)=\begin{cases}n_{1}\sqrt{1-2\Delta\left({r\over\alpha}\right)^{g}}&r\leq% \alpha\\ n_{1}\sqrt{1-2\Delta}&r\geq\alpha\end{cases}
  2. Δ = n 1 2 - n 2 2 2 n 1 2 , \Delta={n_{1}^{2}-n_{2}^{2}\over 2n_{1}^{2}},
  3. n ( r ) n(r)
  4. n 1 n_{1}
  5. n 2 n_{2}
  6. n ( r ) = n 2 for r α n(r)=n_{2}\mathrm{\ for\ }r\geq\alpha
  7. α \alpha
  8. g g
  9. α \alpha
  10. g g
  11. g g
  12. g g

Power_(physics).html

  1. P avg = Δ W Δ t . P_{\mathrm{avg}}=\frac{\Delta W}{\Delta t}\,.
  2. P = lim Δ t 0 P avg = lim Δ t 0 Δ W Δ t = d W d t . P=\lim_{\Delta t\rightarrow 0}P_{\mathrm{avg}}=\lim_{\Delta t\rightarrow 0}% \frac{\Delta W}{\Delta t}=\frac{\mathrm{d}W}{\mathrm{d}t}\,.
  3. W = P T . W=PT\,.
  4. W C = C F v d t = C F d x , W_{C}=\int_{C}{F}\cdot{v}\,\mathrm{d}t=\int_{C}{F}\cdot\mathrm{d}{x},
  5. W C = U ( B ) - U ( A ) , W_{C}=U(B)-U(A),
  6. P ( t ) = d W d t = F v = - d U d t . P(t)=\frac{\mathrm{d}W}{\mathrm{d}t}={F}\cdot{v}=-\frac{\mathrm{d}U}{\mathrm{d% }t}.
  7. P ( t ) = F v . P(t)=F\cdot v.
  8. P ( t ) = s y m b o l τ s y m b o l ω , P(t)=symbol{\tau}\cdot symbol{\omega},\,
  9. \cdot
  10. P ( t ) = p Q , P(t)=pQ,\!
  11. P = F B v B = F A v A , P=F_{B}v_{B}=F_{A}v_{A},\!
  12. MA = F B F A = v A v B . \mathrm{MA}=\frac{F_{B}}{F_{A}}=\frac{v_{A}}{v_{B}}.
  13. P = T A ω A = T B ω B , P=T_{A}\omega_{A}=T_{B}\omega_{B},\!
  14. MA = T B T A = ω A ω B . \mathrm{MA}=\frac{T_{B}}{T_{A}}=\frac{\omega_{A}}{\omega_{B}}.
  15. P ( t ) = I ( t ) V ( t ) P(t)=I(t)\cdot V(t)\,
  16. P = I V = I 2 R = V 2 R P=I\cdot V=I^{2}\cdot R=\frac{V^{2}}{R}\,
  17. R = V I R=\frac{V}{I}\,
  18. s ( t ) s(t)
  19. T T
  20. p ( t ) = | s ( t ) | 2 p(t)=|s(t)|^{2}
  21. T T
  22. P 0 = max [ p ( t ) ] P_{0}=\max[p(t)]
  23. P avg P_{\mathrm{avg}}
  24. ϵ pulse = 0 T p ( t ) d t \epsilon_{\mathrm{pulse}}=\int_{0}^{T}p(t)\mathrm{d}t\,
  25. P avg = 1 T 0 T p ( t ) d t = ϵ pulse T P_{\mathrm{avg}}=\frac{1}{T}\int_{0}^{T}p(t)\mathrm{d}t=\frac{\epsilon_{% \mathrm{pulse}}}{T}\,
  26. τ \tau
  27. P 0 τ = ϵ pulse P_{0}\tau=\epsilon_{\mathrm{pulse}}
  28. P avg P 0 = τ T \frac{P_{\mathrm{avg}}}{P_{0}}=\frac{\tau}{T}\,

Power_factor.html

  1. φ = 90 \varphi=90^{\circ}
  2. cos φ = 0 \cos\varphi=0
  3. φ = 45 \varphi=45^{\circ}
  4. cos φ 0.71 \cos\varphi\approx 0.71
  5. φ \varphi
  6. P S \frac{P}{S}
  7. S 2 = P 2 + Q 2 S^{2}=P^{2}+Q^{2}
  8. φ \varphi
  9. cos φ \cos\varphi
  10. | P | = | S | cos φ |P|=|S|\cos\varphi
  11. distortion power factor = 1 1 + THD i 2 = I 1, rms I rms \mbox{distortion power factor}~{}={1\over\sqrt{1+\mbox{THD}~{}_{i}^{2}}}={I_{% \mbox{1, rms}~{}}\over I_{\mbox{rms}~{}}}
  12. THD i \mbox{THD}~{}_{i}
  13. I 1 , rms I_{1,\mbox{rms}~{}}
  14. I rms I_{\mbox{rms}~{}}
  15. PF = I 1, rms I rms cos φ \mbox{PF}~{}={I_{\mbox{1, rms}~{}}\over I_{\mbox{rms}~{}}}\cos\varphi

Power_law.html

  1. f ( x ) = a x k f(x)=ax^{k}
  2. x x
  3. c c
  4. f ( c x ) = a ( c x ) k = c k f ( x ) f ( x ) . f(cx)=a(cx)^{k}=c^{k}f(x)\propto f(x).\!
  5. c c
  6. c k c^{k}
  7. f ( x ) f(x)
  8. x x
  9. o ( x k ) o(x^{k})
  10. ε \varepsilon
  11. y = a x k + ε . y=ax^{k}+\varepsilon.\!
  12. p ( x ) = C x - α p(x)=Cx^{-\alpha}
  13. x > x min x>x\text{min}
  14. α \alpha
  15. x min x\text{min}
  16. 2 < α < 3 2<\alpha<3
  17. f ( x ) x α 1 f(x)\propto x^{\alpha_{1}}
  18. x < x th , x<x\text{th},
  19. f ( x ) x α 1 - α 2 th x α 2 for x > x th f(x)\propto x^{\alpha_{1}-\alpha_{2}}\text{th}x^{\alpha_{2}}\,\text{ for }x>x% \text{th}
  20. f ( x ) x α e β x . f(x)\propto x^{\alpha}e^{\beta x}.
  21. f ( x ) x α + β x f(x)\propto x^{\alpha+\beta x}
  22. p ( x ) L ( x ) x - α p(x)\propto L(x)x^{-\alpha}
  23. α > 1 \alpha>1
  24. L ( x ) L(x)
  25. lim x L ( r x ) / L ( x ) = 1 \lim_{x\rightarrow\infty}L(r\,x)/L(x)=1
  26. r r
  27. L ( x ) L(x)
  28. p ( x ) p(x)
  29. L ( x ) L(x)
  30. L ( x ) L(x)
  31. x x
  32. x min x_{\mathrm{min}}
  33. x x
  34. p ( x ) = α - 1 x min ( x x min ) - α , p(x)=\frac{\alpha-1}{x_{\min}}\left(\frac{x}{x_{\min}}\right)^{-\alpha},
  35. α - 1 x min \frac{\alpha-1}{x_{\min}}
  36. x m = x min x m p ( x ) d x = α - 1 α - 1 - m x min m \langle x^{m}\rangle=\int_{x_{\min}}^{\infty}x^{m}p(x)\,\mathrm{d}x=\frac{% \alpha-1}{\alpha-1-m}x_{\min}^{m}
  37. m < α - 1 m<\alpha-1
  38. m α - 1 m\geq\alpha-1
  39. α < 2 \alpha<2
  40. 2 < α < 3 2<\alpha<3
  41. p ( x ) L ( x ) x - α e - λ x . p(x)\propto L(x)x^{-\alpha}\mathrm{e}^{-\lambda x}.
  42. e - λ x \mathrm{e}^{-\lambda x}
  43. x x
  44. λ = 0 \lambda=0
  45. x x\rightarrow\infty
  46. α \alpha
  47. α \alpha
  48. P ( x ) = Pr ( X > x ) P(x)=\mathrm{Pr}(X>x)
  49. P ( x ) = Pr ( X > x ) = C x p ( X ) d X = α - 1 x min - α + 1 x X - α d X = ( x x min ) - α + 1 . P(x)=\Pr(X>x)=C\int_{x}^{\infty}p(X)\,\mathrm{d}X=\frac{\alpha-1}{x_{\min}^{-% \alpha+1}}\int_{x}^{\infty}X^{-\alpha}\,\mathrm{d}X=\left(\frac{x}{x_{\min}}% \right)^{-\alpha+1}.
  50. n n
  51. [ 1 , n - 1 n , n - 2 n , , 1 n ] \left[1,\frac{n-1}{n},\frac{n-2}{n},\dots,\frac{1}{n}\right]
  52. p ( x ) = α - 1 x min ( x x min ) - α p(x)=\frac{\alpha-1}{x_{\min}}\left(\frac{x}{x_{\min}}\right)^{-\alpha}
  53. x x min x\geq x_{\min}
  54. α - 1 x min \frac{\alpha-1}{x_{\min}}
  55. x min x_{\min}
  56. ( α ) = log i = 1 n α - 1 x min ( x x min ) - α \mathcal{L}(\alpha)=\log\prod_{i=1}^{n}\frac{\alpha-1}{x_{\min}}\left(\frac{x}% {x_{\min}}\right)^{-\alpha}
  57. α \alpha
  58. α ^ = 1 + n [ i = 1 n ln x i x min ] - 1 \hat{\alpha}=1+n\left[\sum_{i=1}^{n}\ln\frac{x_{i}}{x_{\min}}\right]^{-1}
  59. { x i } \{x_{i}\}
  60. n n
  61. x i x min x_{i}\geq x_{\min}
  62. O ( n - 1 ) O(n^{-1})
  63. σ = α ^ - 1 n + O ( n - 1 ) \sigma=\frac{\hat{\alpha}-1}{\sqrt{n}}+O(n^{-1})
  64. { x i } \{x_{i}\}
  65. x i x min x_{i}\geq x_{\min}
  66. ζ ( α ^ , x min ) ζ ( α ^ , x min ) = - 1 n i = 1 n ln x i x min \frac{\zeta^{\prime}(\hat{\alpha},x_{\min})}{\zeta(\hat{\alpha},x_{\min})}=-% \frac{1}{n}\sum_{i=1}^{n}\ln\frac{x_{i}}{x_{\min}}
  67. ζ ( α , x min ) \zeta(\alpha,x_{\mathrm{min}})
  68. α ^ \hat{\alpha}
  69. x min x_{\min}
  70. L ( x ) L(x)
  71. x min x_{\min}
  72. α ^ \hat{\alpha}
  73. α ^ \hat{\alpha}
  74. x min x_{\min}
  75. L ( x ) L(x)
  76. D D
  77. α ^ = arg min 𝛼 D α \hat{\alpha}=\underset{\alpha}{\operatorname{arg\,min}}\,D_{\alpha}
  78. D α = max x | P emp ( x ) - P α ( x ) | D_{\alpha}=\max_{x}|P_{\mathrm{emp}}(x)-P_{\alpha}(x)|
  79. P emp ( x ) P_{\mathrm{emp}}(x)
  80. P α ( x ) P_{\alpha}(x)
  81. α \alpha
  82. l o g ( < v a r > x ) log(<var>x)

Power_number.html

  1. N p = P ρ n 3 d 5 N_{\mathrm{p}}={P\over\rho n^{3}d^{5}}

Power_series.html

  1. f ( x ) = n = 0 a n ( x - c ) n = a 0 + a 1 ( x - c ) 1 f(x)=\sum_{n=0}^{\infty}a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}
  2. + a 2 ( x - c ) 2 + a 3 ( x - c ) 3 + +a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots
  3. f ( x ) = n = 0 a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + . f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots.
  4. f ( x ) = x 2 + 2 x + 3 f(x)=x^{2}+2x+3
  5. c = 0 c=0
  6. f ( x ) = 3 + 2 x + 1 x 2 + 0 x 3 + 0 x 4 + f(x)=3+2x+1x^{2}+0x^{3}+0x^{4}+\cdots\,
  7. c = 1 c=1
  8. f ( x ) = 6 + 4 ( x - 1 ) + 1 ( x - 1 ) 2 f(x)=6+4(x-1)+1(x-1)^{2}
  9. + 0 ( x - 1 ) 3 + 0 ( x - 1 ) 4 + +0(x-1)^{3}+0(x-1)^{4}+\cdots\,
  10. 1 1 - x = n = 0 x n = 1 + x + x 2 + x 3 + , \frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}=1+x+x^{2}+x^{3}+\cdots,
  11. | x | < 1 |x|<1
  12. e x = n = 0 x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + , e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!% }+\cdots,
  13. sin ( x ) = n = 0 ( - 1 ) n x 2 n + 1 ( 2 n + 1 ) ! = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + , \sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}=x-\frac{x^{3}}{3!}% +\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots,
  14. 1 + x - 1 + x - 2 + 1+x^{-1}+x^{-2}+\cdots
  15. x 1 / 2 x^{1/2}
  16. a n a_{n}
  17. x x
  18. sin ( x ) x + sin ( 2 x ) x 2 + sin ( 3 x ) x 3 + \sin(x)x+\sin(2x)x^{2}+\sin(3x)x^{3}+\cdots\,
  19. r = lim inf n | a n | - 1 n r=\liminf_{n\to\infty}\left|a_{n}\right|^{-\frac{1}{n}}
  20. r - 1 = lim sup n | a n | 1 n r^{-1}=\limsup_{n\to\infty}\left|a_{n}\right|^{\frac{1}{n}}
  21. r - 1 = lim n | a n + 1 a n | r^{-1}=\lim_{n\to\infty}\left|{a_{n+1}\over a_{n}}\right|
  22. g ( x ) = n = 0 b n ( x - c ) n g(x)=\sum_{n=0}^{\infty}b_{n}(x-c)^{n}
  23. f ( x ) ± g ( x ) = n = 0 ( a n ± b n ) ( x - c ) n . f(x)\pm g(x)=\sum_{n=0}^{\infty}(a_{n}\pm b_{n})(x-c)^{n}.
  24. f ( x ) g ( x ) = ( n = 0 a n ( x - c ) n ) ( n = 0 b n ( x - c ) n ) f(x)g(x)=\left(\sum_{n=0}^{\infty}a_{n}(x-c)^{n}\right)\left(\sum_{n=0}^{% \infty}b_{n}(x-c)^{n}\right)
  25. = i = 0 j = 0 a i b j ( x - c ) i + j =\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{i}b_{j}(x-c)^{i+j}
  26. = n = 0 ( i = 0 n a i b n - i ) ( x - c ) n . =\sum_{n=0}^{\infty}\left(\sum_{i=0}^{n}a_{i}b_{n-i}\right)(x-c)^{n}.
  27. m n = i = 0 n a i b n - i m_{n}=\sum_{i=0}^{n}a_{i}b_{n-i}
  28. a n a_{n}
  29. b n b_{n}
  30. f ( x ) g ( x ) = n = 0 a n ( x - c ) n n = 0 b n ( x - c ) n = n = 0 d n ( x - c ) n {f(x)\over g(x)}={\sum_{n=0}^{\infty}a_{n}(x-c)^{n}\over\sum_{n=0}^{\infty}b_{% n}(x-c)^{n}}=\sum_{n=0}^{\infty}d_{n}(x-c)^{n}
  31. f ( x ) = ( n = 0 b n ( x - c ) n ) ( n = 0 d n ( x - c ) n ) f(x)=\left(\sum_{n=0}^{\infty}b_{n}(x-c)^{n}\right)\left(\sum_{n=0}^{\infty}d_% {n}(x-c)^{n}\right)
  32. f ( x ) = n = 1 a n n ( x - c ) n - 1 = n = 0 a n + 1 ( n + 1 ) ( x - c ) n f^{\prime}(x)=\sum_{n=1}^{\infty}a_{n}n\left(x-c\right)^{n-1}=\sum_{n=0}^{% \infty}a_{n+1}\left(n+1\right)\left(x-c\right)^{n}
  33. f ( x ) d x = n = 0 a n ( x - c ) n + 1 n + 1 + k = n = 1 a n - 1 ( x - c ) n n + k . \int f(x)\,dx=\sum_{n=0}^{\infty}\frac{a_{n}\left(x-c\right)^{n+1}}{n+1}+k=% \sum_{n=1}^{\infty}\frac{a_{n-1}\left(x-c\right)^{n}}{n}+k.
  34. a n = f ( n ) ( c ) n ! a_{n}=\frac{f^{\left(n\right)}\left(c\right)}{n!}
  35. f ( n ) ( c ) f^{(n)}(c)
  36. f ( 0 ) ( c ) = f ( c ) f^{(0)}(c)=f(c)
  37. f ( x 1 , , x n ) = j 1 , , j n = 0 a j 1 , , j n k = 1 n ( x k - c k ) j k , f(x_{1},\dots,x_{n})=\sum_{j_{1},\dots,j_{n}=0}^{\infty}a_{j_{1},\dots,j_{n}}% \prod_{k=1}^{n}\left(x_{k}-c_{k}\right)^{j_{k}},
  38. f ( x ) = α n a α ( x - c ) α . f(x)=\sum_{\alpha\in\mathbb{N}^{n}}a_{\alpha}\left(x-c\right)^{\alpha}.
  39. n = 0 x 1 n x 2 n \sum_{n=0}^{\infty}x_{1}^{n}x_{2}^{n}
  40. { ( x 1 , x 2 ) : | x 1 x 2 | < 1 } \{(x_{1},x_{2}):|x_{1}x_{2}|<1\}
  41. ( log | x 1 | , log | x 2 | ) (\log|x_{1}|,\log|x_{2}|)
  42. ( x 1 , x 2 ) (x_{1},x_{2})

Power_set.html

  1. 𝒫 ( S ) \mathcal{P}(S)
  2. 𝒫 ( S ) \mathcal{P}(S)
  3. \varnothing
  4. | 𝒫 ( S ) | = 2 n |\mathcal{P}(S)|=2^{n}
  5. { ω 1 , ω 2 , , ω n } \{\omega_{1},\omega_{2},\ldots,\omega_{n}\}
  6. ω i , 1 i n \omega_{i},1\leq i\leq n
  7. 0
  8. 1 1
  9. ω i = 1 \omega_{i}=1
  10. i i
  11. i i
  12. 2 n 2^{n}
  13. 𝒫 ( S ) \mathcal{P}(S)
  14. 𝒫 ( S ) \mathcal{P}(S)
  15. 𝒫 ( S ) \mathcal{P}(S)
  16. S = { x , y , z } S=\{x,y,z\}
  17. k k
  18. n n
  19. C ( n , k ) , C(n,k),
  20. C ( 3 , 0 ) = 1 C(3,0)=1
  21. C ( 3 , 1 ) = 3 C(3,1)=3
  22. C ( 3 , 2 ) = 3 C(3,2)=3
  23. C ( 3 , 3 ) = 1 C(3,3)=1
  24. S S\!
  25. 𝒫 ( S ) \mathcal{P}(S)
  26. ( e , T ) = { X { e } | X T } \mathcal{F}(e,T)=\{X\cup\{e\}|X\in T\}
  27. e e\!
  28. X X\!
  29. T T\!
  30. S = { } S=\{\}\!
  31. 𝒫 ( S ) = { { } } \mathcal{P}(S)=\{\{\}\}
  32. e e\!
  33. S S\!
  34. T = S { e } T=S\setminus\{e\}\!
  35. S { e } S\setminus\{e\}\!
  36. { e } \{e\}\!
  37. S S\!
  38. 𝒫 ( S ) = 𝒫 ( T ) ( e , 𝒫 ( T ) ) \mathcal{P}(S)=\mathcal{P}(T)\cup\mathcal{F}(e,\mathcal{P}(T))
  39. 𝒫 κ ( S ) \mathcal{P}_{\kappa}(S)
  40. 𝒫 < κ ( S ) . \mathcal{P}_{<\kappa}(S)\,.
  41. 𝒫 1 ( S ) . \mathcal{P}_{\geq 1}(S)\,.

Power_transmission.html

  1. watt = joule second = newton × meter second \,\text{watt}=\frac{\,\text{joule}}{\,\text{second}}=\frac{\,\text{newton}% \times\,\text{meter}}{\,\text{second}}

Poynting_vector.html

  1. 𝐒 = 𝐄 × 𝐇 , \mathbf{S}=\mathbf{E}\times\mathbf{H},
  2. u t = - 𝐒 - 𝐉 f 𝐄 , \frac{\partial u}{\partial t}=-\mathbf{\nabla}\cdot\mathbf{S}-\mathbf{J_{% \mathrm{f}}}\cdot\mathbf{E},
  3. u = 1 2 ( 𝐄 𝐃 + 𝐁 𝐇 ) , u=\frac{1}{2}\!\left(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H}\right% )\!,
  4. 𝐃 = ε 𝐄 , 𝐇 = 1 μ 𝐁 , \mathbf{D}=\varepsilon\mathbf{E},\quad\mathbf{H}=\frac{1}{\mu}\mathbf{B},
  5. 𝐒 = 𝐒 + × 𝐅 𝐒 = 𝐒 , \mathbf{S}^{\prime}=\mathbf{S}+\nabla\times\mathbf{F}\Rightarrow\nabla\cdot% \mathbf{S}^{\prime}=\nabla\cdot\mathbf{S},
  6. 𝐒 = 1 μ 0 𝐄 × 𝐁 , \mathbf{S}=\frac{1}{\mu_{0}}\mathbf{E}\times\mathbf{B},
  7. u t = - 𝐒 - 𝐉 𝐄 , \frac{\partial u}{\partial t}=-\nabla\cdot\mathbf{S}-\mathbf{J}\cdot\mathbf{E},
  8. u = 1 2 ( ε 0 𝐄 2 + 1 μ 0 𝐁 2 ) , u=\frac{1}{2}\!\left(\varepsilon_{0}\mathbf{E}^{2}+\frac{1}{\mu_{0}}\mathbf{B}% ^{2}\right)\!,
  9. 𝐒 \displaystyle\mathbf{S}
  10. 𝐒 = 1 T 0 T 𝐒 ( t ) d t = 1 T 0 T [ 1 2 Re ( 𝐄 𝐦 ¯ × 𝐇 𝐦 * ¯ ) + 1 2 Re ( 𝐄 𝐦 ¯ × 𝐇 𝐦 ¯ e 2 j ω t ) ] d t . \langle\mathbf{S}\rangle=\frac{1}{T}\int_{0}^{T}\mathbf{S}(t)\,dt=\frac{1}{T}% \int_{0}^{T}\!\left[\frac{1}{2}\operatorname{Re}\!\left(\underline{\mathbf{E_{% m}}}\times\underline{\mathbf{H_{m}^{*}}}\right)+\frac{1}{2}\operatorname{Re}\!% \left(\underline{\mathbf{E_{m}}}\times\underline{\mathbf{H_{m}}}e^{2j\omega t}% \right)\right]dt.
  11. Re ( e 2 j ω t ) = cos ( 2 ω t ) \operatorname{Re}\!\left(e^{2j\omega t}\right)=\cos(2\omega t)
  12. 𝐒 = 1 2 Re ( 𝐄 𝐦 ¯ × 𝐇 𝐦 * ¯ ) = 1 2 Re ( 𝐄 𝐦 ¯ e j ω t × 𝐇 𝐦 * ¯ e - j ω t ) = 1 2 Re ( 𝐄 a × 𝐇 a * ) . \langle\mathbf{S}\rangle=\frac{1}{2}\operatorname{Re}\!\left(\underline{% \mathbf{E_{m}}}\times\underline{\mathbf{H_{m}^{*}}}\right)=\frac{1}{2}% \operatorname{Re}\!\left(\underline{\mathbf{E_{m}}}e^{j\omega t}\times% \underline{\mathbf{H_{m}^{*}}}e^{-j\omega t}\right)=\frac{1}{2}\operatorname{% Re}\!\left(\mathbf{E_{\mathrm{a}}}\times\mathbf{H_{\mathrm{a}}^{*}}\right)\!.
  13. S = 1 2 μ 0 c E m 2 = ε 0 c 2 E m 2 \langle S\rangle=\frac{1}{2\mu_{0}\mathrm{c}}E_{\mathrm{m}}^{2}=\frac{% \varepsilon_{0}\mathrm{c}}{2}E_{\mathrm{m}}^{2}
  14. B m = 1 c E m B_{\mathrm{m}}=\frac{1}{\mathrm{c}}E_{\mathrm{m}}
  15. E ( 𝐫 , t ) = E m cos ( ω t - 𝐤 𝐫 ) E(\mathbf{r},t)=E_{\mathrm{m}}\cos(\omega t-\mathbf{k}\cdot\mathbf{r})
  16. B ( 𝐫 , t ) = B m cos ( ω t - 𝐤 𝐫 ) B(\mathbf{r},t)=B_{\mathrm{m}}\cos(\omega t-\mathbf{k}\cdot\mathbf{r})
  17. S ( 𝐫 , t ) = 1 μ 0 E m B m cos 2 ( ω t - 𝐤 𝐫 ) = 1 μ 0 c E m 2 cos 2 ( ω t - 𝐤 𝐫 ) = ε 0 c E m 2 cos 2 ( ω t - 𝐤 𝐫 ) . S(\mathbf{r},t)=\frac{1}{\mu_{0}}E_{\mathrm{m}}B_{\mathrm{m}}\cos^{2}(\omega t% -\mathbf{k}\cdot\mathbf{r})=\frac{1}{\mu_{0}c}E_{\mathrm{m}}^{2}\cos^{2}(% \omega t-\mathbf{k}\cdot\mathbf{r})=\varepsilon_{0}\mathrm{c}E_{\mathrm{m}}^{2% }\cos^{2}(\omega t-\mathbf{k}\cdot\mathbf{r}).
  18. S = 1 2 μ 0 c E m 2 = ε 0 c 2 E m 2 . \langle S\rangle=\frac{1}{2\mu_{0}\mathrm{c}}E_{\mathrm{m}}^{2}=\frac{% \varepsilon_{0}\mathrm{c}}{2}E_{\mathrm{m}}^{2}.
  19. P rad = S c . P_{\mathrm{rad}}=\frac{\langle S\rangle}{\mathrm{c}}.

Prandtl_number.html

  1. Pr \mathrm{Pr}
  2. Pr = ν α = viscous diffusion rate thermal diffusion rate = c p μ k \mathrm{Pr}=\frac{\nu}{\alpha}=\frac{\mbox{viscous diffusion rate}~{}}{\mbox{% thermal diffusion rate}~{}}=\frac{c_{p}\mu}{k}
  3. ν \nu
  4. ν = μ / ρ \nu=\mu/\rho
  5. α \alpha
  6. α = k / ( ρ c p ) \alpha=k/(\rho c_{p})
  7. μ \mu
  8. k k
  9. c p c_{p}
  10. ρ \rho
  11. Pr \mathrm{Pr}
  12. × 10 2 5 \times 10^{2}5
  13. Pr 1 \mathrm{Pr}\ll 1
  14. Pr 1 \mathrm{Pr}\gg 1

Precession.html

  1. s y m b o l ω p = s y m b o l I s s y m b o l ω s s y m b o l I p cos ( s y m b o l α ) symbol\omega_{p}=\frac{symbolI_{s}symbol\omega_{s}}{symbolI_{p}\cos(symbol% \alpha)}
  2. s y m b o l ω p \scriptstyle symbol\omega_{p}
  3. s y m b o l ω s \scriptstyle symbol\omega_{s}
  4. s y m b o l I s \scriptstyle symbolI_{s}
  5. s y m b o l I p \scriptstyle symbolI_{p}
  6. s y m b o l α symbol\alpha
  7. s y m b o l R \scriptstyle symbolR
  8. s y m b o l I 0 \scriptstyle symbolI_{0}
  9. s y m b o l L \scriptstyle symbolL
  10. s y m b o l ω ( s y m b o l R ) = s y m b o l R s y m b o l I 0 - 1 s y m b o l R T s y m b o l L \scriptstyle symbol\omega(symbolR)\;=\;symbolRsymbolI_{0}^{-1}symbolR^{T}symbolL
  11. s y m b o l ω symbol\omega
  12. s y m b o l ω d t \scriptstyle symbol\omega dt
  13. d t \scriptstyle dt
  14. s y m b o l R new = exp ( [ s y m b o l ω ( s y m b o l R old ) ] × d t ) s y m b o l R old \scriptstyle symbolR\text{new}\;=\;\exp([symbol\omega(symbolR\text{old})]_{% \times}dt)symbolR\text{old}
  15. [ s y m b o l ω ] × \scriptstyle[symbol\omega]_{\times}
  16. E ( s y m b o l R ) = s y m b o l ω ( s y m b o l R ) s y m b o l L / 2 \scriptstyle E(symbolR)\;=\;symbol\omega(symbolR)\cdot symbolL/2
  17. s y m b o l v \scriptstyle symbolv
  18. s y m b o l ω \scriptstyle symbol\omega
  19. s y m b o l L \scriptstyle symbolL
  20. E ( exp ( [ s y m b o l v ] × ) s y m b o l R ) E ( s y m b o l R ) + ( s y m b o l ω ( s y m b o l R ) × s y m b o l L ) s y m b o l v \scriptstyle E(\exp([symbolv]_{\times})symbolR)\;\approx\;E(symbolR)\,+\,(% symbol\omega(symbolR)\,\times\,symbolL)\cdot symbolv
  21. s y m b o l ω p = m g r I s s y m b o l ω s symbol\omega_{p}=\frac{\ mgr}{I_{s}symbol\omega_{s}}
  22. s y m b o l ω s \scriptstyle symbol\omega_{s}
  23. s y m b o l ω \scriptstyle symbol\omega
  24. 2 π T \scriptstyle\frac{2\pi}{T}
  25. T p = 4 π 2 I s m g r T s T_{p}=\frac{4\pi^{2}I_{s}}{\ mgrT_{s}}
  26. s y m b o l τ \scriptstyle symbol\tau

Prefix_code.html

  1. 2 k 2^{k}

Preorder.html

  1. \lesssim
  2. x y x\leq y
  3. f ( x ) f ( y ) f(x)\leq f(y)
  4. x y x\leq y
  5. x x
  6. y y
  7. \lesssim
  8. \lesssim
  9. \lesssim
  10. \lesssim
  11. \lesssim
  12. \lesssim
  13. \lesssim
  14. \lesssim
  15. \lesssim
  16. \lesssim
  17. \lesssim
  18. \lesssim
  19. \lesssim
  20. \lesssim

Presburger_arithmetic.html

  1. 2 2 c n 2^{2^{cn}}

Present_value.html

  1. P V = C ( 1 + i ) n PV=\frac{C}{(1+i)^{n}}\,
  2. C \,C\,
  3. n \,n\,
  4. C \,C\,
  5. i \,i\,
  6. i \,i\,
  7. v n = ( 1 + i ) - n v^{n}=\,(1+i)^{-n}
  8. P V = $ 1000 ( 1 + 0.10 ) 5 = $ 620.92 PV=\frac{\$1000}{(1+0.10)^{5}}=\$620.92\,
  9. C \,C\,
  10. n \,n\,
  11. i \,i\,
  12. N P V \,NPV\,
  13. P V 1 = $ 100 ( 1.05 ) 1 = $ 95.24 PV_{1}=\frac{\$100}{(1.05)^{1}}=\$95.24\,
  14. P V 2 = - $ 50 ( 1.05 ) 2 = - $ 45.35 PV_{2}=\frac{-\$50}{(1.05)^{2}}=-\$45.35\,
  15. P V 3 = $ 35 ( 1.05 ) 3 = $ 30.23 PV_{3}=\frac{\$35}{(1.05)^{3}}=\$30.23\,
  16. N P V = P V 1 + P V 2 + P V 3 = 100 ( 1.05 ) 1 + - 50 ( 1.05 ) 2 + 35 ( 1.05 ) 3 = 95.24 - 45.35 + 30.23 = 80.12 , NPV=PV_{1}+PV_{2}+PV_{3}=\frac{100}{(1.05)^{1}}+\frac{-50}{(1.05)^{2}}+\frac{3% 5}{(1.05)^{3}}=95.24-45.35+30.23=80.12,
  17. N P V = 100 ( 1.05 ) - 1 + 200 ( 1.10 ) - 1 ( 1.05 ) - 1 = 100 ( 1.05 ) 1 + 200 ( 1.10 ) 1 ( 1.05 ) 1 = $ 95.24 + $ 173.16 = $ 268.40 NPV=100\,(1.05)^{-1}+200\,(1.10)^{-1}\,(1.05)^{-1}=\frac{100}{(1.05)^{1}}+% \frac{200}{(1.10)^{1}(1.05)^{1}}=\$95.24+\$173.16=\$268.40
  18. i \,i\,
  19. ( 1 + i ) = ( 1 + i 4 4 ) 4 (1+i)=\left(1+\frac{i^{4}}{4}\right)^{4}
  20. i 4 i^{4}
  21. i 4 4 \frac{i^{4}}{4}
  22. n \,n\,
  23. n \,n\,
  24. n \,n\,
  25. n - 1 \,n-1\,
  26. ( 1 + i ) (1+i)
  27. P V annuity due = P V annuity immediate ( 1 + i ) PV\text{annuity due}=PV\text{annuity immediate}(1+i)\,\!
  28. P V = k = 1 n C ( 1 + i ) k = C [ 1 - ( 1 + i ) - n i ] , ( 1 ) PV=\sum_{k=1}^{n}\frac{C}{(1+i)^{k}}=C\left[\frac{1-(1+i)^{-n}}{i}\right],% \qquad(1)
  29. n \,n\,
  30. C \,C\,
  31. i \,i\,
  32. C P V ( 1 n + 2 3 i ) C\approx PV\left(\frac{1}{n}+\frac{2}{3}i\right)
  33. C P V i C\approx PVi
  34. P V = C i . ( 2 ) PV\,=\,\frac{C}{i}.\qquad(2)
  35. P V = k = 1 C ( 1 + i ) k = C i , i > 0 , PV=\sum_{k=1}^{\infty}\frac{C}{(1+i)^{k}}=\frac{C}{i},\qquad i>0,
  36. ( 1 + i ) (1+i)
  37. P V perpetuity due = P V perpetuity immediate ( 1 + i ) PV\text{perpetuity due}=PV\text{perpetuity immediate}(1+i)\,\!
  38. F F
  39. r r
  40. F r Fr
  41. F ( 1 + r ) F(1+r)
  42. P V = [ k = 1 n F r ( 1 + i ) - k ] PV=\left[\sum_{k=1}^{n}Fr(1+i)^{-k}\right]
  43. + F ( 1 + i ) - n +F(1+i)^{-n}

Pressure.html

  1. 1 / 760 {1}/{760}
  2. p = F A p=\frac{F}{A}
  3. p p
  4. F F
  5. A A
  6. d 𝐅 n = - p d 𝐀 = - p 𝐧 d A d\mathbf{F}_{n}=-p\,d\mathbf{A}=-p\,\mathbf{n}\,dA
  7. F \vec{F}
  8. A \vec{A}
  9. F = σ A \vec{F}=\sigma\vec{A}\,
  10. p γ + v 2 2 g + z = const \frac{p}{\gamma}+\frac{v^{2}}{2g}+z=\mbox{const}~{}
  11. p γ \frac{p}{\gamma}
  12. v 2 2 g \frac{v^{2}}{2g}
  13. p 0 = 1 2 ρ v 2 + p p_{0}=\frac{1}{2}\rho v^{2}+p
  14. p 0 p_{0}
  15. v v
  16. p p
  17. π = F l \pi=\frac{F}{l}
  18. p = n R T V p=\frac{nRT}{V}
  19. p = ρ g h p=\rho gh
  20. p = weight density × depth p=\,\text{weight density}\times\!\,\,\text{depth}
  21. Weight density = weight volume \,\text{Weight density}=\frac{\,\text{weight}}{\,\text{volume}}
  22. Weight = weight density × volume \,\text{Weight}=\,\text{weight density}\times\!\,\,\text{volume}
  23. Pressure = force area = weight area = weight density × volume area \,\text{Pressure}=\frac{\,\text{force}}{\,\text{area}}=\frac{\,\text{weight}}{% \,\text{area}}=\frac{\,\text{weight density}\times\!\,\,\text{volume}}{\,\text% {area}}
  24. Pressure = weight density × (area × depth) area \,\text{Pressure}=\frac{\,\text{weight density}\times\!\,\,\text{(area}\times% \!\,\,\text{depth)}}{\,\text{area}}
  25. Pressure = weight density × depth \,\text{Pressure}=\,\text{weight density}\times\!\,\,\text{depth}
  26. p = ρ g h p=\rho gh
  27. p γ + z = const \frac{p}{\gamma}+z=\mbox{const}~{}
  28. 2 g h \scriptstyle\sqrt{2gh}
  29. P = p / ρ 0 P=p/\rho_{0}
  30. p p
  31. ρ 0 \rho_{0}
  32. ν \nu
  33. ρ 0 \rho_{0}
  34. u t + ( u ) u = - P + ν 2 u \frac{\partial u}{\partial t}+(u\nabla)u=-\nabla P+\nu\nabla^{2}u

Prim's_algorithm.html

  1. w ( f ) w ( e ) . w(f)\geq w(e).

Prime_number.html

  1. n \sqrt{n}
  2. n n
  3. n n
  4. n n
  5. n n
  6. n n
  7. n n
  8. n n
  9. 2 , 3 , , n 1 2,3,...,n−1
  10. n n
  11. n > 1 n>1
  12. a a
  13. b b
  14. n = a · b n=a·b
  15. n n
  16. n n
  17. 𝐏 \mathbf{P}
  18. n n
  19. n n
  20. p p
  21. p p
  22. a b ab
  23. p p
  24. a a
  25. p p
  26. b b
  27. N = 1 + p S p . N=1+\prod_{p\in S}p.
  28. S ( p ) = 1 2 + 1 3 + 1 5 + 1 7 + + 1 p . S(p)=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots+\frac{1}{p}.
  29. 1 1 2 + 1 2 2 + 1 3 2 + + 1 n 2 = i = 1 n 1 i 2 \frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\cdots+\frac{1}{n^{2}}=\sum_{i% =1}^{n}\frac{1}{i^{2}}
  30. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  31. n = a b n=ab
  32. n \sqrt{n}
  33. n = 37 n=37
  34. n \sqrt{n}
  35. n \sqrt{n}
  36. n / ln ( n ) \sqrt{n}/\ln(\sqrt{n})
  37. n n
  38. n n
  39. ( n + 1 ) ! + 2 (n+1)!+2
  40. ( n + 1 ) ! + n + 1 (n+1)!+n+1
  41. ( n + 1 ) ! + k (n+1)!+k
  42. k k
  43. k k
  44. 2 2
  45. n + 1 n+1
  46. p ( n ) = a + b n p(n)=a+bn\,
  47. A 3 n and 2 2 2 μ \left\lfloor A^{3^{n}}\right\rfloor\,\text{ and }\left\lfloor 2^{\dots^{2^{2^{% \mu}}}}\right\rfloor
  48. - \lfloor-\rfloor
  49. π ( n ) n ln n , \pi(n)\approx\frac{n}{\ln n},
  50. Li ( n ) = 2 n d t ln t . \operatorname{Li}(n)=\int_{2}^{n}\frac{dt}{\ln t}.
  51. n 2 + n + 41 n^{2}+n+41\,
  52. f ( n ) = a x 2 + b x + c f(n)=ax^{2}+bx+c\,
  53. ζ ( s ) = n = 1 1 n s , \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},
  54. p prime 1 1 - p - s . \prod_{p\,\text{ prime}}\frac{1}{1-p^{-s}}.
  55. ζ ( 2 ) = p 1 1 - p - 2 = π 2 6 . \zeta(2)=\prod_{p}\frac{1}{1-p^{-2}}=\frac{\pi^{2}}{6}.
  56. { 0 , 1 , 2 , , n - 1 } , \{0,1,2,\dots,n-1\},\,
  57. 3 + 5 1 ( mod 7 ) . 3+5\equiv 1\;\;(\mathop{{\rm mod}}7).
  58. 3 x 2 ( mod 6 ) , 3\cdot x\equiv 2\;\;(\mathop{{\rm mod}}6),
  59. 3 x 2 ( mod 7 ) , 3\cdot x\equiv 2\ \ (\operatorname{mod}\ 7),
  60. a p - 1 1 ( mod p ) a^{p-1}\equiv 1(\operatorname{mod}\ p)
  61. a = 1 p - 1 a p - 1 ( p - 1 ) 1 - 1 ( mod p ) . \sum_{a=1}^{p-1}a^{p-1}\equiv(p-1)\cdot 1\equiv-1\;\;(\mathop{{\rm mod}}p).
  62. { , - 11 , - 7 , - 5 , - 3 , - 2 , 2 , 3 , 5 , 7 , 11 , } . \{\dots,-11,-7,-5,-3,-2,2,3,5,7,11,\dots\}\,.
  63. x 2 p ( mod q ) , x^{2}\equiv p\ \ (\,\text{mod }q),\,
  64. a 0 + a 1 ζ + + a p - 1 ζ p - 1 , a_{0}+a_{1}\zeta+\cdots+a_{p-1}\zeta^{p-1}\,,
  65. q = p v p ( q ) r s , q=p^{v_{p}(q)}\frac{r}{s},
  66. | q | p := p - v p ( q ) . \left|q\right|_{p}:=p^{-v_{p}(q)}.\,
  67. | q | p := e - v p ( q ) . \left|q\right|_{p}:=e^{-v_{p}(q)}.\,

Prime_number_theorem.html

  1. lim x π ( x ) x / ln ( x ) = 1 , \lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1,
  2. π ( x ) x ln x . \pi(x)\sim\frac{x}{\ln x}.\!
  3. p n n ln ( n ) p_{n}\sim n\ln(n)
  4. lim x ϑ ( x ) x = 1 \lim_{x\to\infty}\frac{\vartheta\left(x\right)}{x}=1
  5. lim x ψ ( x ) x = 1 \lim_{x\to\infty}\frac{\psi(x)}{x}=1
  6. ϑ \vartheta
  7. ψ \psi
  8. ψ ( x ) \psi(x)
  9. ψ ( x ) = p k x , p is prime log p . \psi(x)=\sum_{p^{k}\leq x,\atop p\,\,\text{is prime}}\log p.
  10. ψ ( x ) = n x Λ ( n ) \psi(x)=\sum_{n\leq x}\Lambda(n)
  11. Λ ( n ) \Lambda(n)
  12. Λ ( n ) = { log p if n = p k for some prime p and integer k 1 , 0 otherwise. \Lambda(n)=\begin{cases}\log p&\,\text{if }n=p^{k}\,\text{ for some prime }p\,% \text{ and integer }k\geq 1,\\ 0&\,\text{otherwise.}\end{cases}
  13. lim x ψ ( x ) / x = 1 \lim_{x\to\infty}\psi(x)/x=1
  14. ψ ( x ) = p x log p log x log p p x log x = π ( x ) log x \psi(x)=\sum_{p\leq x}\log p\left\lfloor\frac{\log x}{\log p}\right\rfloor\leq% \sum_{p\leq x}\log x=\pi(x)\log x
  15. ϵ > 0 \epsilon>0
  16. ψ ( x ) x 1 - ϵ p x log p x 1 - ϵ p x ( 1 - ϵ ) log x = ( 1 - ϵ ) ( π ( x ) + O ( x 1 - ϵ ) ) log x . \psi(x)\geq\sum_{x^{1-\epsilon}\leq p\leq x}\log p\geq\sum_{x^{1-\epsilon}\leq p% \leq x}(1-\epsilon)\log x=(1-\epsilon)(\pi(x)+O(x^{1-\epsilon}))\log x.
  17. ψ ( x ) \psi(x)
  18. ζ ( s ) \zeta(s)
  19. ζ ( s ) \zeta(s)
  20. Λ ( n ) \Lambda(n)
  21. ψ ( x ) \psi(x)
  22. - ζ ( s ) ζ ( s ) = n = 1 Λ ( n ) n - s . -\frac{\zeta^{\prime}(s)}{\zeta(s)}=\sum_{n=1}^{\infty}\Lambda(n)n^{-s}.
  23. ψ ( x ) = x - ρ x ρ ρ - log ( 2 π ) \psi(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\log(2\pi)
  24. ψ ( x ) \psi(x)
  25. n = 1 1 2 n x 2 n = - 1 2 ln ( 1 - 1 x 2 ) , \sum_{n=1}^{\infty}\frac{1}{2n\,x^{2n}}=-\frac{1}{2}\ln\left(1-\frac{1}{x^{2}}% \right),
  26. 0 ( s ) 1 0\leq\Re(s)\leq 1
  27. ( ρ ) = 1 \Re(\rho)=1
  28. ζ ( s ) \zeta(s)
  29. ( s ) > 0 \Re(s)>0
  30. s = 1 s=1
  31. ζ ( s ) = p ( 1 - p - s ) - 1 \zeta(s)=\prod_{p}(1-p^{-s})^{-1}
  32. ( s ) > 1. \Re(s)>1.
  33. ζ ( s ) \zeta(s)
  34. log ζ ( s ) = - p log ( 1 - p - s ) = p , n p - n s / n . \log\zeta(s)=-\sum_{p}\log(1-p^{-s})=\sum_{p,n}p^{-ns}/n.
  35. s = x + i y s=x+iy
  36. | ζ ( x + i y ) | = exp ( n , p cos n y log p n p n x ) . |\zeta(x+iy)|=\exp(\sum_{n,p}\frac{\cos ny\log p}{np^{nx}}).
  37. 3 + 4 cos ϕ + cos 2 ϕ = 2 ( 1 + cos ϕ ) 2 0 , 3+4\cos\phi+\cos 2\phi=2(1+\cos\phi)^{2}\geq 0,
  38. | ζ ( x ) 3 ζ ( x + i y ) 4 ζ ( x + 2 i y ) | = exp n , p 3 + 4 cos ( n y log p ) + cos ( 2 n y log p ) n p n x 1 |\zeta(x)^{3}\zeta(x+iy)^{4}\zeta(x+2iy)|=\exp\sum_{n,p}\frac{3+4\cos(ny\log p% )+\cos(2ny\log p)}{np^{nx}}\geq 1
  39. x > 1 x>1
  40. ζ ( 1 + i y ) = 0 \zeta(1+iy)=0
  41. y y
  42. ζ ( s ) \zeta(s)
  43. s = 1 s=1
  44. x > 1 x>1
  45. x x
  46. 1 1
  47. ζ ( s ) \zeta(s)
  48. s = 1 s=1
  49. ζ ( x + 2 i y ) \zeta(x+2iy)
  50. 0
  51. ψ ( x ) \psi(x)
  52. Li ( x ) = 2 x 1 ln t d t = li ( x ) - li ( 2 ) . \mathrm{Li}(x)=\int_{2}^{x}\frac{1}{\ln t}\,\mathrm{d}t=\mathrm{li}(x)-\mathrm% {li}(2).
  53. Li ( x ) x ln x k = 0 k ! ( ln x ) k = x ln x + x ( ln x ) 2 + 2 x ( ln x ) 3 + . \mathrm{Li}(x)\sim\frac{x}{\ln x}\sum_{k=0}^{\infty}\frac{k!}{(\ln x)^{k}}=% \frac{x}{\ln x}+\frac{x}{(\ln x)^{2}}+\frac{2x}{(\ln x)^{3}}+\cdots.
  54. π ( x ) = Li ( x ) + O ( x e - a ln x ) as x \pi(x)={\rm Li}(x)+O\left(x\mathrm{e}^{-a\sqrt{\ln x}}\right)\quad\,\text{as }% x\to\infty
  55. π ( x ) = Li ( x ) + O ( x exp ( - A ( ln x ) 3 / 5 ( ln ln x ) 1 / 5 ) ) . \pi(x)={\rm Li}(x)+O\left(x\,\exp\left(-\frac{A(\ln x)^{3/5}}{(\ln\ln x)^{1/5}% }\right)\right).
  56. π ( x ) = Li ( x ) + O ( x ln x ) . \pi(x)={\rm Li}(x)+O\left(\sqrt{x}\ln x\right).
  57. | π ( x ) - li ( x ) | < x ln x 8 π |\pi(x)-{\rm li}(x)|<\frac{\sqrt{x}\,\ln x}{8\pi}
  58. | ψ ( x ) - x | < x ln 2 x 8 π |\psi(x)-x|<\frac{\sqrt{x}\,\ln^{2}x}{8\pi}
  59. ϑ ( x ) log ( x ) + p x log ( p ) ϑ ( x p ) = 2 x log ( x ) + O ( x ) \vartheta\left(x\right)\log\left(x\right)+\sum\limits_{p\leq x}{\log\left(p% \right)}\ \vartheta\left({\frac{x}{p}}\right)=2x\log\left(x\right)+O\left(x\right)
  60. ϑ ( x ) = p x log ( p ) \vartheta\left(x\right)=\sum\limits_{p\leq x}{\log\left(p\right)}
  61. p p
  62. I Δ 0 + exp I\Delta_{0}+\exp
  63. π n , a ( x ) \pi_{n,a}(x)
  64. π n , a ( x ) 1 φ ( n ) Li ( x ) , \pi_{n,a}(x)\sim\frac{1}{\varphi(n)}\mathrm{Li}(x),
  65. π 4 , 1 ( x ) π 4 , 3 ( x ) , \pi_{4,1}(x)\sim\pi_{4,3}(x),\,
  66. π 4 , 1 ( x ) - π 4 , 3 ( x ) , \pi_{4,1}(x)-\pi_{4,3}(x),\,
  67. ( 1 - ε ) x ln x < π ( x ) < ( 1 + ε ) x ln x . (1-\varepsilon)\frac{x}{\ln x}<\pi(x)<(1+\varepsilon)\frac{x}{\ln x}.
  68. x ln x ( 1 + 1 ln x ) < π ( x ) < x ln x ( 1 + 1 ln x + 2.51 ( ln x ) 2 ) . \frac{x}{\ln x}\left(1+\frac{1}{\ln x}\right)<\pi(x)<\frac{x}{\ln x}\left(1+% \frac{1}{\ln x}+\frac{2.51}{(\ln x)^{2}}\right).
  69. x ln x + 2 < π ( x ) < x ln x - 4 \frac{x}{\ln x+2}<\pi(x)<\frac{x}{\ln x-4}
  70. x ln x - ( 1 - ε ) < π ( x ) < x ln x - ( 1 + ε ) . \frac{x}{\ln x-(1-\varepsilon)}<\pi(x)<\frac{x}{\ln x-(1+\varepsilon)}.
  71. p n n ln n . p_{n}\sim n\ln n.
  72. p n n = ln n + ln ln n - 1 + ln ln n - 2 ln n - ( ln ln n ) 2 - 6 ln ln n + 11 2 ( ln n ) 2 + o ( 1 ( ln n ) 2 ) . {\frac{p_{n}}{n}=\ln n+\ln\ln n-1+\frac{\ln\ln n-2}{\ln n}-\frac{(\ln\ln n)^{2% }-6\ln\ln n+11}{2(\ln n)^{2}}+o\left(\frac{1}{(\ln n)^{2}}\right).}
  73. ln n + ln ln n - 1 < p n n < ln n + ln ln n for n 6. \ln n+\ln\ln n-1<\frac{p_{n}}{n}<\ln n+\ln\ln n\quad\,\text{for }n\geq 6.
  74. N n q n n . N_{n}\sim\frac{q^{n}}{n}.
  75. x log q x , \frac{x}{\log_{q}x},
  76. N n = q n n + O ( q n / 2 n ) . N_{n}=\frac{q^{n}}{n}+O\left(\frac{q^{n/2}}{n}\right).
  77. q n = d n d N d , q^{n}=\sum_{d\mid n}dN_{d},
  78. N n = 1 n d n μ ( n / d ) q d , N_{n}=\frac{1}{n}\sum_{d\mid n}\mu(n/d)q^{d},

Primitive_recursive_function.html

  1. h ( x 1 , , x m ) = f ( g 1 ( x 1 , , x m ) , , g k ( x 1 , , x m ) ) h(x_{1},\ldots,x_{m})=f(g_{1}(x_{1},\ldots,x_{m}),\ldots,g_{k}(x_{1},\ldots,x_% {m}))\,
  2. h ( 0 , x 1 , , x k ) = f ( x 1 , , x k ) h(0,x_{1},\ldots,x_{k})=f(x_{1},\ldots,x_{k})\,
  3. h ( S ( y ) , x 1 , , x k ) = g ( y , h ( y , x 1 , , x k ) , x 1 , , x k ) . h(S(y),x_{1},\ldots,x_{k})=g(y,h(y,x_{1},\ldots,x_{k}),x_{1},\ldots,x_{k})\,.
  4. f ( a , b , c ) = g ( h ( c , a ) , h ( a , b ) ) f(a,b,c)=g(h(c,a),h(a,b))\!
  5. f ( a , b , c ) = g ( h ( P 3 3 ( a , b , c ) , P 1 3 ( a , b , c ) ) , h ( P 1 3 ( a , b , c ) , P 2 3 ( a , b , c ) ) ) . f(a,b,c)=g(h(P^{3}_{3}(a,b,c),P^{3}_{1}(a,b,c)),h(P^{3}_{1}(a,b,c),P^{3}_{2}(a% ,b,c))).
  6. n n
  7. n n
  8. n n
  9. n n
  10. f < s u b > n f<sub>n

Principal_ideal_domain.html

  1. ( x k ) (x^{k})
  2. M M
  3. R / x R R/xR
  4. x R x\in R
  5. x x
  6. 0
  7. R / x R R/xR
  8. R R
  9. ( 2 , X ) [ X ] (2,X)\subseteq\mathbb{Z}[X]
  10. [ X ] \mathbb{Z}[X]
  11. [ ( 1 + - 19 ) / 2 ] . \mathbb{Z}\left[(1+\sqrt{-19})/2\right].
  12. 1 + - 19 1+\sqrt{-19}
  13. X , Y . \left\langle X,Y\right\rangle.

Principle_of_bivalence.html

  1. P x ( x P x P ) \forall P\,\forall x(x\in P\lor x\notin P)

Priority_queue.html

  1. C C
  2. t o p top
  3. C C
  4. k k
  5. k k
  6. t o p m i n ( t o p , k ) top←min(top,k)
  7. t o p top
  8. t o p top
  9. O ( C ) O(C)
  10. O ( log n ) O(\sqrt{\log n})
  11. n log ( n ) n\log(n)
  12. n log ( n ) n\log(n)
  13. n log ( n ) n\log(n)
  14. n n
  15. n log ( n ) n\log(n)
  16. n log ( n ) n\log(n)
  17. n 2 n^{2}
  18. n 2 n^{2}
  19. n 2 n^{2}
  20. n n
  21. n 2 n^{2}
  22. n 2 n^{2}
  23. n log ( n ) n\log(n)
  24. n log ( n ) n\log(n)
  25. n log ( n ) n\log(n)

Prismatoid.html

  1. V = h ( A 1 + 4 A 2 + A 3 ) 6 V=\frac{h(A_{1}+4A_{2}+A_{3})}{6}
  2. V = n h ( a 2 + 4 b 2 + c 2 ) 24 tan ( 180 / n ) V=\frac{nh(a^{2}+4b^{2}+c^{2})}{24\tan(180/n)}

Prisoner's_dilemma.html

  1. P = { P c c , P c d , P d c , P d d } P=\{P_{cc},P_{cd},P_{dc},P_{dd}\}
  2. P a b P_{ab}
  3. P c d P_{cd}
  4. Q = { Q c c , Q c d , Q d c , Q d d } Q=\{Q_{cc},Q_{cd},Q_{dc},Q_{dd}\}
  5. M c d , c d = P c d ( 1 - Q d c ) M_{cd,cd}=P_{cd}(1-Q_{dc})
  6. v M = v v\cdot M=v
  7. M n M^{n}
  8. M M^{\infty}
  9. M n M^{n}
  10. M M^{\infty}
  11. M M^{\infty}
  12. S x = { R , S , T , P } S_{x}=\{R,S,T,P\}
  13. S y = { R , T , S , P } S_{y}=\{R,T,S,P\}
  14. s x = v S x s_{x}=v\cdot S_{x}
  15. s y = v S y s_{y}=v\cdot S_{y}
  16. s x = D ( P , Q , S x ) s_{x}=D(P,Q,S_{x})
  17. s y = D ( P , Q , S y ) s_{y}=D(P,Q,S_{y})
  18. s y = D ( P , Q , f ) s_{y}=D(P,Q,f)
  19. α s x + β s y + γ = D ( P , Q , α S x + β S y + γ U ) \alpha s_{x}+\beta s_{y}+\gamma=D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)
  20. D ( P , Q , α S x + β S y + γ U ) = 0 D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)=0
  21. α s x + β s y + γ = 0 \alpha s_{x}+\beta s_{y}+\gamma=0
  22. D ( P , Q , β S y + γ U ) = 0 D(P,Q,\beta S_{y}+\gamma U)=0
  23. s y s_{y}
  24. s x s_{x}

Probability.html

  1. ϕ ( x ) = c e - h 2 x 2 , \phi(x)=ce^{-h^{2}x^{2}},
  2. h h
  3. c c
  4. = 1 - 1 6 = 5 6 =1-\tfrac{1}{6}=\tfrac{5}{6}
  5. P ( A B ) P(A\cap B)
  6. P ( A and B ) = P ( A B ) = P ( A ) P ( B ) , P(A\mbox{ and }~{}B)=P(A\cap B)=P(A)P(B),\,
  7. 1 2 × 1 2 = 1 4 \tfrac{1}{2}\times\tfrac{1}{2}=\tfrac{1}{4}
  8. P ( A B ) P(A\cup B)
  9. P ( A or B ) = P ( A B ) = P ( A ) + P ( B ) . P(A\mbox{ or }~{}B)=P(A\cup B)=P(A)+P(B).
  10. P ( 1 or 2 ) = P ( 1 ) + P ( 2 ) = 1 6 + 1 6 = 1 3 . P(1\mbox{ or }~{}2)=P(1)+P(2)=\tfrac{1}{6}+\tfrac{1}{6}=\tfrac{1}{3}.
  11. P ( A or B ) = P ( A ) + P ( B ) - P ( A and B ) . P\left(A\hbox{ or }B\right)=P\left(A\right)+P\left(B\right)-P\left(A\mbox{ and% }~{}B\right).
  12. 13 52 + 12 52 - 3 52 = 11 26 \tfrac{13}{52}+\tfrac{12}{52}-\tfrac{3}{52}=\tfrac{11}{26}
  13. P ( A B ) P(A\mid B)
  14. P ( A B ) = P ( A B ) P ( B ) . P(A\mid B)=\frac{P(A\cap B)}{P(B)}.\,
  15. P ( B ) = 0 P(B)=0
  16. P ( A B ) P(A\mid B)
  17. 1 / 2 1/2
  18. 1 / 3 1/3
  19. A 1 A_{1}
  20. A 2 A_{2}
  21. B B
  22. A 1 A_{1}
  23. A 2 A_{2}
  24. A A
  25. P ( A | B ) P ( A ) P ( B | A ) P(A|B)\propto P(A)P(B|A)
  26. A A
  27. B B
  28. P ( A ) [ 0 , 1 ] P(A)\in[0,1]\,
  29. P ( A c ) = 1 - P ( A ) P(A^{c})=1-P(A)\,
  30. P ( A B ) = P ( A ) + P ( B ) - P ( A B ) P ( A B ) = P ( A ) + P ( B ) if A and B are mutually exclusive \begin{aligned}\displaystyle P(A\cup B)&\displaystyle=P(A)+P(B)-P(A\cap B)\\ \displaystyle P(A\cup B)&\displaystyle=P(A)+P(B)\qquad\mbox{if A and B are % mutually exclusive}\\ \end{aligned}
  31. P ( A B ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) P ( A B ) = P ( A ) P ( B ) if A and B are independent \begin{aligned}\displaystyle P(A\cap B)&\displaystyle=P(A|B)P(B)=P(B|A)P(A)\\ \displaystyle P(A\cap B)&\displaystyle=P(A)P(B)\qquad\mbox{if A and B are % independent}\\ \end{aligned}
  32. P ( A B ) = P ( A B ) P ( B ) = P ( B | A ) P ( A ) P ( B ) P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{P(B|A)P(A)}{P(B)}\,

Probability_axioms.html

  1. P ( E ) P(E)
  2. P ( E ) , P ( E ) 0 E F P(E)\in\mathbb{R},P(E)\geq 0\qquad\forall E\in F
  3. F F
  4. P ( E ) P(E)
  5. P ( Ω ) = 1. P(\Omega)=1.
  6. E 1 , E 2 , E_{1},E_{2},...
  7. P ( i = 1 E i ) = i = 1 P ( E i ) . P\left(\bigcup_{i=1}^{\infty}E_{i}\right)=\sum_{i=1}^{\infty}P(E_{i}).
  8. P ( ) = 0. P(\varnothing)=0.
  9. if A B then P ( A ) P ( B ) . \quad\,\text{if}\quad A\subseteq B\quad\,\text{then}\quad P(A)\leq P(B).
  10. 0 P ( E ) 1 E F . 0\leq P(E)\leq 1\qquad\forall E\in F.
  11. E 1 = A E_{1}=A
  12. E 2 = B \ A E_{2}=B\backslash A
  13. A B and E i = \quad A\subseteq B\,\text{ and }E_{i}=\varnothing
  14. i 3 i\geq 3
  15. E i E_{i}
  16. E 1 E 2 = B E_{1}\cup E_{2}\cup\ldots=B
  17. P ( A ) + P ( B \ A ) + i = 3 P ( ) = P ( B ) . P(A)+P(B\backslash A)+\sum_{i=3}^{\infty}P(\varnothing)=P(B).
  18. P ( B ) P(B)
  19. P ( A ) P ( B ) P(A)\leq P(B)
  20. P ( ) = 0 P(\varnothing)=0
  21. P ( ) = a P(\varnothing)=a
  22. i = 3 P ( E i ) = i = 3 P ( ) = i = 3 a = { 0 if a = 0 , if a > 0. \sum_{i=3}^{\infty}P(E_{i})=\sum_{i=3}^{\infty}P(\varnothing)=\sum_{i=3}^{% \infty}a=\begin{cases}0&\,\text{if }a=0,\\ \infty&\,\text{if }a>0.\end{cases}
  23. a > 0 a>0
  24. P ( B ) P(B)
  25. a = 0 a=0
  26. P ( ) = 0 P(\varnothing)=0
  27. P ( A B ) = P ( A ) + P ( B ) - P ( A B ) . P(A\cup B)=P(A)+P(B)-P(A\cap B).
  28. P ( A B ) = P ( A ) + P ( B ( A B ) ) P(A\cup B)=P(A)+P(B\setminus(A\cap B))\,\,
  29. P ( B ) = P ( B ( A B ) ) + P ( A B ) P(B)=P(B\setminus(A\cap B))+P(A\cap B)
  30. P ( B ( A B ) ) P(B\setminus(A\cap B))
  31. P ( E c ) = P ( Ω E ) = 1 - P ( E ) P\left(E^{c}\right)=P(\Omega\setminus E)=1-P(E)
  32. Ω = { H , T } \Omega=\{H,T\}
  33. F = { , { H } , { T } , { H , T } } F=\{\varnothing,\{H\},\{T\},\{H,T\}\}
  34. P ( ) = 0 P(\varnothing)=0
  35. P ( { H , T } ) = 1 P(\{H,T\})=1
  36. P ( { H } ) + P ( { T } ) = 1 P(\{H\})+P(\{T\})=1

Probability_density_function.html

  1. Pr [ a X b ] = a b f X ( x ) d x . \Pr[a\leq X\leq b]=\int_{a}^{b}f_{X}(x)\,dx.
  2. F X ( x ) = - x f X ( u ) d u , F_{X}(x)=\int_{-\infty}^{x}f_{X}(u)\,du,
  3. f X ( x ) = d d x F X ( x ) . f_{X}(x)=\frac{d}{dx}F_{X}(x).
  4. ( 𝒳 , 𝒜 ) (\mathcal{X},\mathcal{A})
  5. ( 𝒳 , 𝒜 ) (\mathcal{X},\mathcal{A})
  6. ( 𝒳 , 𝒜 ) (\mathcal{X},\mathcal{A})
  7. f = d X * P d μ . f=\frac{dX_{*}P}{d\mu}.
  8. Pr [ X A ] = X - 1 A d P = A f d μ \Pr[X\in A]=\int_{X^{-1}A}\,dP=\int_{A}f\,d\mu
  9. A 𝒜 A\in\mathcal{A}
  10. f ( x ) = 1 2 π e - x 2 / 2 . f(x)=\frac{1}{\sqrt{2\pi}}\;e^{-x^{2}/2}.
  11. E [ X ] = - x f ( x ) d x . \operatorname{E}[X]=\int_{-\infty}^{\infty}x\,f(x)\,dx.
  12. d d x F ( x ) = f ( x ) . \frac{d}{dx}F(x)=f(x).
  13. Pr ( t < X < t + d t ) = f ( t ) d t . \Pr(t<X<t+dt)=f(t)\,dt.
  14. f ( t ) = 1 2 ( δ ( t + 1 ) + δ ( t - 1 ) ) . f(t)=\frac{1}{2}(\delta(t+1)+\delta(t-1)).
  15. f ( t ) = i = 1 n p i δ ( t - x i ) , f(t)=\sum_{i=1}^{n}p_{i}\,\delta(t-x_{i}),
  16. μ \mu
  17. σ 2 \sigma^{2}
  18. f ( x ; μ , σ 2 ) = 1 σ 2 π e - 1 2 ( x - μ σ ) 2 . f(x;\mu,\sigma^{2})=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-% \mu}{\sigma}\right)^{2}}.
  19. Pr ( X 1 , , X N \isin D ) = D f X 1 , , X N ( x 1 , , x N ) d x 1 d x N . \Pr\left(X_{1},\cdots,X_{N}\isin D\right)=\int_{D}f_{X_{1},\cdots,X_{N}}(x_{1}% ,\cdots,x_{N})\,dx_{1}\cdots dx_{N}.
  20. f ( x ) = n F x 1 x n | x f(x)=\frac{\partial^{n}F}{\partial x_{1}\cdots\partial x_{n}}\bigg|_{x}
  21. f X i ( x i ) = f ( x 1 , , x n ) d x 1 d x i - 1 d x i + 1 d x n . f_{X_{i}}(x_{i})=\int f(x_{1},\cdots,x_{n})\,dx_{1}\cdots dx_{i-1}\,dx_{i+1}% \cdots dx_{n}.
  22. f X 1 , , X n ( x 1 , , x n ) = f X 1 ( x 1 ) f X n ( x n ) . f_{X_{1},\cdots,X_{n}}(x_{1},\cdots,x_{n})=f_{X_{1}}(x_{1})\cdots f_{X_{n}}(x_% {n}).
  23. f X 1 , , X n ( x 1 , , x n ) = f 1 ( x 1 ) f n ( x n ) , f_{X_{1},\cdots,X_{n}}(x_{1},\cdots,x_{n})=f_{1}(x_{1})\cdots f_{n}(x_{n}),
  24. f X i ( x i ) = f i ( x i ) f i ( x ) d x . f_{X_{i}}(x_{i})=\frac{f_{i}(x_{i})}{\int f_{i}(x)\,dx}.
  25. R \vec{R}
  26. R \vec{R}
  27. Pr ( X > 0 , Y > 0 ) = 0 0 f X , Y ( x , y ) d x d y . \Pr\left(X>0,Y>0\right)=\int_{0}^{\infty}\int_{0}^{\infty}f_{X,Y}(x,y)\,dx\,dy.
  28. f Y ( y ) = | d d y ( g - 1 ( y ) ) | f X ( g - 1 ( y ) ) . f_{Y}(y)=\left|\frac{d}{dy}(g^{-1}(y))\right|\cdot f_{X}(g^{-1}(y)).
  29. | f Y ( y ) d y | = | f X ( x ) d x | , \left|f_{Y}(y)\,dy\right|=\left|f_{X}(x)\,dx\right|,
  30. f Y ( y ) = | d x d y | f X ( x ) = | d d y ( x ) | f X ( x ) = | d d y ( g - 1 ( y ) ) | f X ( g - 1 ( y ) ) = f X ( g - 1 ( y ) ) | g ( g - 1 ( y ) ) | . f_{Y}(y)=\left|\frac{dx}{dy}\right|f_{X}(x)=\left|\frac{d}{dy}(x)\right|f_{X}(% x)=\left|\frac{d}{dy}(g^{-1}(y))\right|f_{X}(g^{-1}(y))=\frac{f_{X}(g^{-1}(y))% }{|g^{\prime}(g^{-1}(y))|}.
  31. k = 1 n ( y ) | d d y g k - 1 ( y ) | f X ( g k - 1 ( y ) ) \sum_{k=1}^{n(y)}\left|\frac{d}{dy}g^{-1}_{k}(y)\right|\cdot f_{X}(g^{-1}_{k}(% y))
  32. E ( g ( X ) ) = - y f g ( X ) ( y ) d y , \operatorname{E}(g(X))=\int_{-\infty}^{\infty}yf_{g(X)}(y)\,dy,
  33. E ( g ( X ) ) = - g ( x ) f X ( x ) d x . \operatorname{E}(g(X))=\int_{-\infty}^{\infty}g(x)f_{X}(x)\,dx.
  34. y = g ( x 1 , , x n ) f ( x 1 , , x n ) j = 1 n g x j ( x 1 , , x n ) 2 d V \int\limits_{y=g(x_{1},\cdots,x_{n})}\frac{f(x_{1},\cdots,x_{n})}{\sqrt{\sum_{% j=1}^{n}\frac{\partial g}{\partial x_{j}}(x_{1},\cdots,x_{n})^{2}}}\;dV
  35. g ( 𝐲 ) = f ( 𝐱 ) | det ( d 𝐱 d 𝐲 ) | g(\mathbf{y})=f(\mathbf{x})\left|\det\left(\frac{\mathrm{d}\mathbf{x}}{\mathrm% {d}\mathbf{y}}\right)\right|
  36. f Y ( y ) = - - - f X 1 ( x 1 ) f X 2 ( x 2 ) f X n ( x n ) δ ( y - G ( x 1 , x 2 , , x n ) ) d x 1 d x 2 d x n f_{Y}(y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{% \infty}f_{X_{1}}(x_{1})f_{X_{2}}(x_{2})\cdots f_{X_{n}}(x_{n})\delta(y-G(x_{1}% ,x_{2},\cdots,x_{n}))\,dx_{1}\,dx_{2}\,\cdots dx_{n}
  37. f U + V ( x ) = - f U ( y ) f V ( x - y ) d y = ( f U * f V ) ( x ) f_{U+V}(x)=\int_{-\infty}^{\infty}f_{U}(y)f_{V}(x-y)\,dy=\left(f_{U}*f_{V}% \right)(x)
  38. f U 1 + + U N ( x ) = ( f U 1 * * f U N ) ( x ) f_{U_{1}+\cdots+U_{N}}(x)=\left(f_{U_{1}}*\cdots*f_{U_{N}}\right)(x)
  39. Y = U / V Y=U/V
  40. Z = V Z=V
  41. U = Y Z U=YZ
  42. V = Z V=Z
  43. J ( U , V | Y , Z ) J(U,V|Y,Z)
  44. | U Y U Z V Y V Z | = | Z Y 0 1 | = | Z | . \begin{vmatrix}\frac{\partial U}{\partial Y}&\frac{\partial U}{\partial Z}\\ \frac{\partial V}{\partial Y}&\frac{\partial V}{\partial Z}\\ \end{vmatrix}=\begin{vmatrix}Z&Y\\ 0&1\\ \end{vmatrix}=|Z|.
  45. p ( Y , Z ) = p ( U , V ) J ( U , V | Y , Z ) = p ( U ) p ( V ) J ( U , V | Y , Z ) = p U ( Y Z ) p V ( Z ) | Z | . p(Y,Z)=p(U,V)\,J(U,V|Y,Z)=p(U)\,p(V)\,J(U,V|Y,Z)=p_{U}(YZ)\,p_{V}(Z)\,|Z|.
  46. p ( Y ) = - p U ( Y Z ) p V ( Z ) | Z | d Z p(Y)=\int_{-\infty}^{\infty}p_{U}(YZ)\,p_{V}(Z)\,|Z|\,dZ
  47. p ( U ) = 1 2 π e - U 2 2 p(U)=\frac{1}{\sqrt{2\pi}}e^{-\frac{U^{2}}{2}}
  48. p ( V ) = 1 2 π e - V 2 2 p(V)=\frac{1}{\sqrt{2\pi}}e^{-\frac{V^{2}}{2}}
  49. Y = U / V Y=U/V
  50. Z = V Z=V
  51. p ( Y ) \displaystyle p(Y)

Probability_distribution.html

  1. F ( x ) = Pr [ X x ] for all x . F(x)=\Pr\left[X\leq x\right]\qquad\,\text{ for all }x\in\mathbb{R}.
  2. u Pr ( X = u ) = 1 \sum_{u}\Pr(X=u)=1
  3. Pr ( X = n ) = 1 2 n \Pr(X=n)=\tfrac{1}{2^{n}}
  4. X : A B X\colon A\to B
  5. ( A , 𝒜 , P ) (A,\mathcal{A},P)
  6. ( B , ) (B,\mathcal{B})
  7. X - 1 ( b ) 𝒜 X^{-1}(b)\in\mathcal{A}
  8. b B b\in B
  9. f X : X ( A ) f_{X}\colon X(A)\to\mathbb{R}
  10. f X ( b ) := P ( X - 1 ( b ) ) f_{X}(b):=P(X^{-1}(b))
  11. b X ( A ) f X ( b ) = b X ( A ) P ( X - 1 ( b ) ) = P ( b X ( A ) X - 1 ( b ) ) = P ( A ) = 1. \sum_{b\in X(A)}f_{X}(b)=\sum_{b\in X(A)}P(X^{-1}(b))=P\left(\bigcup_{b\in X(A% )}X^{-1}(b)\right)=P(A)=1.
  12. Ω i = X - 1 ( u i ) = { ω : X ( ω ) = u i } , i = 0 , 1 , 2 , \Omega_{i}=X^{-1}(u_{i})=\{\omega:X(\omega)=u_{i}\},\,i=0,1,2,\dots
  13. Pr ( i Ω i ) = i Pr ( Ω i ) = i Pr ( X = u i ) = 1. \Pr\left(\bigcup_{i}\Omega_{i}\right)=\sum_{i}\Pr(\Omega_{i})=\sum_{i}\Pr(X=u_% {i})=1.
  14. X = i u i 1 Ω i X=\sum_{i}u_{i}1_{\Omega_{i}}
  15. 1 A 1_{A}
  16. 31 / 2 3{1}/{2}
  17. Pr [ a X b ] = a b f ( x ) d x \Pr[a\leq X\leq b]=\int_{a}^{b}f(x)\,dx
  18. μ \,\mu
  19. F ( x ) = μ ( - , x ] F(x)=\mu(-\infty,x]
  20. μ { x } = 0 \mu\{x\}\,=\,0
  21. x \,x
  22. f \,f
  23. F ( x ) = μ ( - , x ] = - x f ( t ) d t . F(x)=\mu(-\infty,x]=\int_{-\infty}^{x}f(t)\,dt.
  24. ( Ω , , P ) \scriptstyle(\Omega,\mathcal{F},\operatorname{P})
  25. ( 𝒳 , 𝒜 ) \scriptstyle(\mathcal{X},\mathcal{A})
  26. ( 𝒳 , 𝒜 ) \scriptstyle(\mathcal{X},\mathcal{A})

Probability_interpretations.html

  1. P ( A ) = N A N P(A)={N_{A}\over N}
  2. n a \textstyle n_{a}
  3. 𝒜 \mathcal{A}
  4. n \textstyle n
  5. lim n n a n = p \lim_{n\to\infty}{n_{a}\over n}=p
  6. P ( 𝒜 ) = p \textstyle P(\mathcal{A})=p

Probability_measure.html

  1. μ μ
  2. μ μ
  3. μ μ
  4. { E i } \{E_{i}\}
  5. μ ( i I E i ) = i I μ ( E i ) . \mu\Bigl(\bigcup_{i\in I}E_{i}\Bigr)=\sum_{i\in I}\mu(E_{i}).
  6. P ( B A ) = P ( A B ) P ( A ) . P(B\mid A)=\frac{P(A\cap B)}{P(A)}.
  7. P ( A ) P(A)

Probability_space.html

  1. \scriptstyle\mathcal{F}
  2. \scriptstyle\mathcal{F}
  3. \scriptstyle\mathcal{F}
  4. \scriptstyle\mathcal{F}
  5. \scriptstyle\mathcal{F}
  6. \scriptstyle\mathcal{F}
  7. P : [ 0 , 1 ] \scriptstyle P:\ \mathcal{F}\rightarrow[0,1]
  8. ( Ω , , P ) \scriptstyle(\Omega,\;\mathcal{F},\;P)
  9. \scriptstyle\mathcal{F}
  10. \scriptstyle\mathcal{F}
  11. Ω \scriptstyle\Omega\in\mathcal{F}
  12. \scriptstyle\mathcal{F}
  13. \scriptstyle\mathcal{F}
  14. \scriptstyle\mathcal{F}
  15. \scriptstyle\mathcal{F}
  16. \scriptstyle\mathcal{F}
  17. \scriptstyle\mathcal{F}
  18. \scriptstyle\mathcal{F}
  19. \scriptstyle\mathcal{F}
  20. \scriptstyle\mathcal{F}
  21. \scriptstyle\mathcal{F}
  22. \scriptstyle\mathcal{F}
  23. \scriptstyle\mathcal{F}
  24. \scriptstyle\mathcal{F}
  25. ( * ) P ( A ) = ω A p ( ω ) for all A Ω . (*)\qquad P(A)=\sum_{\omega\in A}p(\omega)\quad\,\text{for all }A\subseteq% \Omega\,.
  26. \scriptstyle\mathcal{F}
  27. \scriptstyle\mathcal{F}
  28. \scriptstyle\mathcal{F}
  29. \scriptstyle\mathcal{F}
  30. \scriptstyle\mathcal{F}
  31. ( Ω , , P ) \scriptstyle(\Omega,\;\mathcal{F},\;P)
  32. B \scriptstyle B\,\in\,\mathcal{F}
  33. P ( B ) = 0 \scriptstyle P(B)\,=\;0
  34. A B \scriptstyle A\;\subset\;B
  35. A \scriptstyle A\;\in\;\mathcal{F}
  36. \scriptstyle\mathcal{F}
  37. \scriptstyle\mathcal{F}
  38. \scriptstyle\mathcal{F}
  39. \scriptstyle\mathcal{F}
  40. \scriptstyle\mathcal{F}
  41. \scriptstyle\mathcal{F}
  42. \scriptstyle\mathcal{F}
  43. \scriptstyle\mathcal{F}
  44. \scriptstyle\mathcal{F}
  45. \scriptstyle\mathcal{F}
  46. \scriptstyle\mathcal{F}
  47. \scriptstyle\mathcal{F}
  48. \scriptstyle\mathcal{F}
  49. \scriptstyle\mathcal{F}
  50. \scriptstyle\mathcal{F}
  51. \scriptstyle\mathcal{F}
  52. \scriptstyle\mathcal{F}
  53. \scriptstyle\mathcal{F}
  54. \scriptstyle\mathcal{F}
  55. P ( B | A ) = P ( B A ) P ( A ) P(B|A)={P(B\cap A)\over P(A)}

Probability_theory.html

  1. 3 6 = 1 2 \tfrac{3}{6}=\tfrac{1}{2}
  2. Ω \Omega
  3. x Ω x\in\Omega\,
  4. f ( x ) f(x)\,
  5. f ( x ) [ 0 , 1 ] for all x Ω ; f(x)\in[0,1]\mbox{ for all }~{}x\in\Omega\,;
  6. x Ω f ( x ) = 1 . \sum_{x\in\Omega}f(x)=1\,.
  7. E E\,
  8. Ω \Omega\,
  9. E E\,
  10. P ( E ) = x E f ( x ) . P(E)=\sum_{x\in E}f(x)\,.
  11. f ( x ) f(x)\,
  12. \mathbb{R}
  13. F F\,
  14. F ( x ) = P ( X x ) F(x)=P(X\leq x)\,
  15. F F\,
  16. lim x - F ( x ) = 0 ; \lim_{x\rightarrow-\infty}F(x)=0\,;
  17. lim x F ( x ) = 1 . \lim_{x\rightarrow\infty}F(x)=1\,.
  18. F F\,
  19. f ( x ) = d F ( x ) d x . f(x)=\frac{dF(x)}{dx}\,.
  20. E E\subseteq\mathbb{R}
  21. E E\,
  22. P ( X E ) = x E d F ( x ) . P(X\in E)=\int_{x\in E}dF(x)\,.
  23. P ( X E ) = x E f ( x ) d x . P(X\in E)=\int_{x\in E}f(x)\,dx\,.
  24. . \mathbb{R}\,.
  25. n \mathbb{R}^{n}
  26. ( δ [ x ] + φ ( x ) ) / 2 (\delta[x]+\varphi(x))/2
  27. δ [ x ] \delta[x]
  28. Ω \Omega\,
  29. \mathcal{F}\,
  30. P P\,
  31. \mathcal{F}\,
  32. P ( Ω ) = 1. P(\Omega)=1.\,
  33. \mathcal{F}\,
  34. \mathcal{F}\,
  35. E E\,
  36. \mathcal{F}\,
  37. P ( E ) = ω E μ F ( d ω ) P(E)=\int_{\omega\in E}\mu_{F}(d\omega)\,
  38. μ F \mu_{F}\,
  39. F . F\,.
  40. n \mathbb{R}^{n}
  41. X 1 , X 2 , , X_{1},X_{2},\dots,\,
  42. X X\,
  43. F 1 , F 2 , F_{1},F_{2},\dots\,
  44. F F\,
  45. X X\,
  46. F F\,
  47. X n 𝒟 X . X_{n}\,\xrightarrow{\mathcal{D}}\,X\,.
  48. X 1 , X 2 , X_{1},X_{2},\dots\,
  49. X X\,
  50. lim n P ( | X n - X | ε ) = 0 \lim_{n\rightarrow\infty}P\left(\left|X_{n}-X\right|\geq\varepsilon\right)=0
  51. X n 𝑃 X . X_{n}\,\xrightarrow{P}\,X\,.
  52. X 1 , X 2 , X_{1},X_{2},\dots\,
  53. X X\,
  54. P ( lim n X n = X ) = 1 P(\lim_{n\rightarrow\infty}X_{n}=X)=1
  55. X n a . s . X . X_{n}\,\xrightarrow{\mathrm{a.s.}}\,X\,.
  56. X ¯ n = 1 n k = 1 n X k \overline{X}_{n}=\frac{1}{n}{\sum_{k=1}^{n}X_{k}}
  57. X k X_{k}
  58. μ \mu
  59. | X k | |X_{k}|
  60. Weak law: X ¯ n 𝑃 μ for n Strong law: X ¯ n a . s . μ for n . \begin{array}[]{lll}\,\text{Weak law:}&\overline{X}_{n}\,\xrightarrow{P}\,\mu&% \,\text{for }n\to\infty\\ \,\text{Strong law:}&\overline{X}_{n}\,\xrightarrow{\mathrm{a.\,s.}}\,\mu&\,% \text{for }n\to\infty.\end{array}
  61. Y 1 , Y 2 , Y_{1},Y_{2},...\,
  62. E ( Y i ) = p \textrm{E}(Y_{i})=p
  63. Y ¯ n \bar{Y}_{n}
  64. X 1 , X 2 , X_{1},X_{2},\dots\,
  65. μ \mu
  66. σ 2 > 0. \sigma^{2}>0.\,
  67. Z n = i = 1 n ( X i - μ ) σ n Z_{n}=\frac{\sum_{i=1}^{n}(X_{i}-\mu)}{\sigma\sqrt{n}}\,

Product_(mathematics).html

  1. x ( 2 + x ) x\cdot(2+x)
  2. x x
  3. ( 2 + x ) (2+x)
  4. r r
  5. s s
  6. r s = i = 1 s r = j = 1 r s r\cdot s=\sum_{i=1}^{s}r=\sum_{j=1}^{r}s
  7. - + - + - + - + \begin{array}[]{|c|c c|}\hline\cdot&-&+\\ \hline-&+&-\\ +&-&+\\ \hline\end{array}
  8. z n z n = z z n n \frac{z}{n}\cdot\frac{z^{\prime}}{n^{\prime}}=\frac{z\cdot z^{\prime}}{n\cdot n% ^{\prime}}
  9. i 2 = - 1 \mathrm{i}^{2}=-1
  10. ( a + b i ) ( c + d i ) \displaystyle(a+b\,\mathrm{i})\cdot(c+d\,\mathrm{i})
  11. a + b i = r ( cos ( φ ) + i sin ( φ ) ) = r e i φ a+b\,\mathrm{i}=r\cdot(\cos(\varphi)+\mathrm{i}\sin(\varphi))=r\cdot\mathrm{e}% ^{\mathrm{i}\varphi}
  12. c + d i = s ( cos ( ψ ) + i sin ( ψ ) ) = s e i ψ c+d\,\mathrm{i}=s\cdot(\cos(\psi)+\mathrm{i}\sin(\psi))=s\cdot\mathrm{e}^{% \mathrm{i}\psi}
  13. ( a c - b d ) + ( a d + b c ) i = r s ( cos ( φ + ψ ) + i sin ( φ + ψ ) ) = r s e i ( φ + ψ ) (a\cdot c-b\cdot d)+(a\cdot d+b\cdot c)\,\mathrm{i}=r\cdot s\cdot(\cos(\varphi% +\psi)+\mathrm{i}\sin(\varphi+\psi))=r\cdot s\cdot\mathrm{e}^{\mathrm{i}(% \varphi+\psi)}
  14. a b a\cdot b
  15. b a b\cdot a
  16. \Z / N \Z \Z/N\Z
  17. ( a + N \Z ) + ( b + N \Z ) = a + b + N \Z (a+N\Z)+(b+N\Z)=a+b+N\Z
  18. ( a + N \Z ) ( b + N \Z ) = a b + N \Z (a+N\Z)\cdot(b+N\Z)=a\cdot b+N\Z
  19. ( f + g ) ( m ) := f ( m ) + g ( m ) (f+g)(m):=f(m)+g(m)
  20. ( f g ) ( m ) := f ( m ) g ( m ) (f\cdot g)(m):=f(m)\cdot g(m)
  21. - | f ( t ) | d t < and - | g ( t ) | d t < \int\limits_{-\infty}^{\infty}|f(t)|\,\mathrm{d}\,t\;<\;\infty\quad\mbox{and }% ~{}\int\limits_{-\infty}^{\infty}|g(t)|\,\mathrm{d}\,t\;<\;\infty
  22. ( f * g ) ( t ) := - f ( τ ) g ( t - τ ) d τ (f*g)(t)\;:=\int\limits_{-\infty}^{\infty}f(\tau)\cdot g(t-\tau)\,\mathrm{d}\tau
  23. ( i = 0 n a i X i ) ( j = 0 m b j X j ) = k = 0 n + m c k X k \left(\sum_{i=0}^{n}a_{i}X^{i}\right)\cdot\left(\sum_{j=0}^{m}b_{j}X^{j}\right% )=\sum_{k=0}^{n+m}c_{k}X^{k}
  24. c k = i + j = k a i b j c_{k}=\sum_{i+j=k}a_{i}\cdot b_{j}
  25. \R × V V \R\times V\rightarrow V
  26. : V × V \R \cdot:V\times V\rightarrow\R
  27. v v > 0 v\cdot v>0
  28. 0 v V 0\not=v\in V
  29. v := v v \|v\|:=\sqrt{v\cdot v}
  30. cos ( v , w ) = v w v w \cos\angle(v,w)=\frac{v\cdot w}{\|v\|\cdot\|w\|}
  31. n n
  32. ( i = 1 n α i e i ) ( i = 1 n β i e i ) = i = 1 n α i β i \left(\sum_{i=1}^{n}\alpha_{i}e_{i}\right)\cdot\left(\sum_{i=1}^{n}\beta_{i}e_% {i}\right)=\sum_{i=1}^{n}\alpha_{i}\,\beta_{i}
  33. 𝐮 × 𝐯 = | 𝐢 𝐣 𝐤 u 1 u 2 u 3 v 1 v 2 v 3 | \mathbf{u\times v}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ u_{1}&u_{2}&u_{3}\\ v_{1}&v_{2}&v_{3}\\ \end{vmatrix}
  34. f ( t 1 x 1 + t 2 x 2 ) = t 1 f ( x 1 ) + t 2 f ( x 2 ) , x 1 , x 2 V , t 1 , t 2 𝔽 . f(t_{1}x_{1}+t_{2}x_{2})=t_{1}f(x_{1})+t_{2}f(x_{2}),\forall x_{1},x_{2}\in V,% \forall t_{1},t_{2}\in\mathbb{F}.
  35. f ( 𝐯 ) = f ( v i 𝐛 𝐕 i ) = v i f ( 𝐛 𝐕 i ) = f i j v i 𝐛 𝐖 j , f(\mathbf{v})=f(v_{i}\mathbf{b_{V}}^{i})=v_{i}f(\mathbf{b_{V}}^{i})={f^{i}}_{j% }v_{i}\mathbf{b_{W}}^{j},
  36. g f ( 𝐯 ) = g ( f i j v i 𝐛 𝐖 j ) = g j k f i j v i 𝐛 𝐔 k . g\circ f(\mathbf{v})=g({f^{i}}_{j}v_{i}\mathbf{b_{W}}^{j})={g^{j}}_{k}{f^{i}}_% {j}v_{i}\mathbf{b_{U}}^{k}.
  37. g f ( 𝐯 ) = 𝐆𝐅𝐯 , g\circ f(\mathbf{v})=\mathbf{G}\mathbf{F}\mathbf{v},
  38. A = ( a i , j ) i = 1 s ; j = 1 r \R s × r A=(a_{i,j})_{i=1\ldots s;j=1\ldots r}\in\R^{s\times r}
  39. B = ( b j , k ) j = 1 r ; k = 1 t \R r × t B=(b_{j,k})_{j=1\ldots r;k=1\ldots t}\in\R^{r\times t}
  40. B A = ( j = 1 r a i , j b j , k ) i = 1 s ; k = 1 t \R s × t B\cdot A=\left(\sum_{j=1}^{r}a_{i,j}\cdot b_{j,k}\right)_{i=1\ldots s;k=1% \ldots t}\;\in\R^{s\times t}
  41. 𝒰 = { u 1 , u r } \mathcal{U}=\{u_{1},\ldots u_{r}\}
  42. 𝒱 = { v 1 , v s } \mathcal{V}=\{v_{1},\ldots v_{s}\}
  43. 𝒲 = { w 1 , w t } \mathcal{W}=\{w_{1},\ldots w_{t}\}
  44. A = M 𝒱 𝒰 ( f ) \R s × r A=M^{\mathcal{U}}_{\mathcal{V}}(f)\in\R^{s\times r}
  45. B = M 𝒲 𝒱 ( g ) \R r × t B=M^{\mathcal{V}}_{\mathcal{W}}(g)\in\R^{r\times t}
  46. B A = M 𝒲 𝒰 ( g f ) \R s × t B\cdot A=M^{\mathcal{U}}_{\mathcal{W}}(g\circ f)\in\R^{s\times t}
  47. g f : U W g\circ f:U\rightarrow W
  48. V W ( v , m ) = V ( v ) W ( w ) , v V * , w W * , V\otimes W(v,m)=V(v)W(w),\forall v\in V^{*},\forall w\in W^{*},

Product_of_group_subsets.html

  1. S T = { s t : s S and t T } . ST=\{st:s\in S\,\text{ and }t\in T\}.
  2. | S T | = | S | | T | | S T | |ST|=\frac{|S||T|}{|S\cap T|}

Product_of_rings.html

  1. n = p 1 n 1 p 2 n 2 p k n k n=p_{1}^{n_{1}}\ p_{2}^{n_{2}}\ \cdots\ p_{k}^{n_{k}}
  2. 𝐙 / p 1 n 1 𝐙 × 𝐙 / p 2 n 2 𝐙 × × 𝐙 / p k n k 𝐙 \mathbf{Z}/p_{1}^{n_{1}}\mathbf{Z}\ \times\ \mathbf{Z}/p_{2}^{n_{2}}\mathbf{Z}% \ \times\ \cdots\ \times\ \mathbf{Z}/p_{k}^{n_{k}}\mathbf{Z}

Product_topology.html

  1. X := i I X i X:=\prod_{i\in I}X_{i}
  2. i I i\in I
  3. i I U i \prod_{i\in I}U_{i}
  4. i I X i \prod_{i\in I}X_{i}
  5. { ( x , y ) 2 x y = 1 } , \{(x,y)\in\mathbb{R}^{2}\mid xy=1\},

Profinite_group.html

  1. N = 1 \cap N=1

Projective_plane.html

  1. { k x : k K } \{kx:k\in K\}
  2. { k x + l y : k , l K } \{kx+ly:k,l\in K\}
  3. N \sqrt{N}
  4. ( x 1 , x 2 ) ( 1 , x 1 , x 2 ) . (x_{1},x_{2})\to(1,x_{1},x_{2}).
  5. { k ( 0 , 0 , 1 ) + l ( 0 , 1 , 0 ) : k , l K } \{k(0,0,1)+l(0,1,0):k,l\in K\}
  6. u = { ( x , 0 ) : x K } u=\{(x,0):x\in K\}
  7. y = { ( x , 1 ) : x K } y=\{(x,1):x\in K\}
  8. u ¯ = { ( 1 , x , 0 ) : x K } { ( 0 , 1 , 0 ) } \bar{u}=\{(1,x,0):x\in K\}\cup\{(0,1,0)\}
  9. y ¯ = { ( 1 , x , 1 ) : x K } { ( 0 , 1 , 0 ) } \bar{y}=\{(1,x,1):x\in K\}\cup\{(0,1,0)\}
  10. { k ( 1 , 0 , 0 ) + l ( 0 , 1 , 0 ) : k , l K } \{k(1,0,0)+l(0,1,0):k,l\in K\}
  11. k ( 1 , 0 , 1 ) + l ( 0 , 1 , 0 ) : k , l K {k(1,0,1)+l(0,1,0):k,l\in K}
  12. ( x 1 , x 2 ) ( x 2 , 1 , x 1 ) , (x_{1},x_{2})\to(x_{2},1,x_{1}),

Proof_by_contradiction.html

  1. 2 ¯ \overline{2}
  2. 2 ¯ \overline{2}
  3. S { P } 𝔽 S\cup\{P\}\vdash\mathbb{F}
  4. S ¬ P . S\vdash\neg P.
  5. S { ¬ P } 𝔽 S\cup\{\neg P\}\vdash\mathbb{F}
  6. S P . S\vdash P.
  7. \rightarrow\!\leftarrow
  8. \Rightarrow\!\Leftarrow
  9. \nleftrightarrow

Propagation_constant.html

  1. A 0 A x = e γ x \frac{A_{0}}{A_{x}}=e^{\gamma x}
  2. γ = α + i β \gamma=\alpha+i\beta\,
  3. e i θ = cos θ + i sin θ e^{i\theta}=\cos{\theta}+i\sin{\theta}\,\!
  4. | e i θ | = cos 2 θ + sin 2 θ = 1 \left|e^{i\theta}\right|=\sqrt{\cos^{2}{\theta}+\sin^{2}{\theta}}=1
  5. γ = Z Y \gamma=\sqrt{ZY}
  6. Z = R + i ω L Z=R+i\omega L\,\!
  7. Y = G + i ω C Y=G+i\omega C\,\!
  8. | A 0 A x | = e α x \left|\frac{A_{0}}{A_{x}}\right|=e^{\alpha x}
  9. α = R G \alpha=\sqrt{RG}\,\!
  10. R ω R\propto\sqrt{\omega}
  11. tan δ {\tan\delta}
  12. α d = π ε r λ tan δ \alpha_{d}={{\pi}\sqrt{\varepsilon_{r}}\over{\lambda}}{\tan\delta}
  13. k = 2 π λ = β k=\frac{2\pi}{\lambda}=\beta
  14. v p = λ T = f ν ~ = ω β v_{p}=\frac{\lambda}{T}=\frac{f}{\tilde{\nu}}=\frac{\omega}{\beta}
  15. β = ω L C \beta=\omega\sqrt{LC}
  16. V 1 V 2 = Z I 1 Z I 2 e γ 1 \frac{V_{1}}{V_{2}}=\sqrt{\frac{Z_{I1}}{Z_{I2}}}e^{\gamma_{1}}
  17. V 2 V 3 = Z I 2 Z I 3 e γ 2 \frac{V_{2}}{V_{3}}=\sqrt{\frac{Z_{I2}}{Z_{I3}}}e^{\gamma_{2}}
  18. V 3 V 4 = Z I 3 Z I 4 e γ 3 \frac{V_{3}}{V_{4}}=\sqrt{\frac{Z_{I3}}{Z_{I4}}}e^{\gamma_{3}}
  19. Z I n Z I m \sqrt{\frac{Z_{In}}{Z_{Im}}}
  20. V 1 V 4 = V 1 V 2 V 2 V 3 V 3 V 4 = Z I 1 Z I 4 e γ 1 + γ 2 + γ 3 \frac{V_{1}}{V_{4}}=\frac{V_{1}}{V_{2}}\cdot\frac{V_{2}}{V_{3}}\cdot\frac{V_{3% }}{V_{4}}=\sqrt{\frac{Z_{I1}}{Z_{I4}}}e^{\gamma_{1}+\gamma_{2}+\gamma_{3}}
  21. γ T o t = γ 1 + γ 2 + γ 3 + + γ n \gamma_{Tot}=\gamma_{1}+\gamma_{2}+\gamma_{3}+\cdots+\gamma_{n}

Propeller.html

  1. α \alpha
  2. F ρ A V 2 = f ( R n , α ) \frac{F}{\rho AV^{2}}=f(R_{n},\alpha)
  3. L = C L 1 2 ρ V 2 A L=C_{L}\tfrac{1}{2}\rho V^{2}A
  4. D = C D 1 2 ρ V 2 A D=C_{D}{\tfrac{1}{2}\rho V^{2}A}
  5. τ \tau
  6. Lift = L = ρ V τ \mbox{Lift}~{}=L=\rho V\tau
  7. 2 π N r \scriptstyle 2\pi Nr
  8. V a \scriptstyle V_{a}
  9. Slip = N P - V a N P = 1 - J p \mbox{Slip}~{}=\frac{NP-V_{a}}{NP}=1-\frac{J}{p}
  10. J = V a N D \scriptstyle J=\frac{V_{a}}{ND}
  11. p = P D \scriptstyle p=\frac{P}{D}
  12. d L = 1 2 ρ V 1 2 C L d A = 1 2 ρ C L [ V a 2 ( 1 + a ) 2 + 4 π 2 r 2 ( 1 - a ) 2 ] b d r \mbox{d}~{}L=\frac{1}{2}\rho V_{1}^{2}C_{L}dA=\frac{1}{2}\rho C_{L}[V_{a}^{2}(% 1+a)^{2}+4\pi^{2}r^{2}(1-a^{\prime})^{2}]b\mbox{d}~{}r
  13. V 1 2 \displaystyle V_{1}^{2}
  14. d T = d L cos φ - d D sin φ = d L ( cos φ - d D d L sin φ ) \mbox{d}~{}T=\mbox{d}~{}L\cos\varphi-\mbox{d}~{}D\sin\varphi=\mbox{d}~{}L(\cos% \varphi-\frac{\mbox{d}~{}D}{\mbox{d}~{}L}\sin\varphi)
  15. t a n β \displaystyle tan\beta
  16. V 1 = V a ( 1 + a ) sin φ \scriptstyle V_{1}=\frac{V_{a}(1+a)}{\sin\varphi}
  17. d T = 1 2 ρ C L V a 2 ( 1 + a ) 2 cos ( φ + β ) sin 2 φ cos β b d r \mbox{d}~{}T=\frac{1}{2}\rho C_{L}\frac{V_{a}^{2}(1+a)^{2}\cos(\varphi+\beta)}% {\sin^{2}\varphi\cos\beta}b\mbox{d}~{}r
  18. d M \displaystyle\mbox{d}~{}M
  19. V 1 \scriptstyle V_{1}
  20. d Q = r d M = 1 2 ρ C L V a 2 ( 1 + a ) 2 sin ( φ + β ) sin 2 φ cos β b r d r \mbox{d}~{}Q=r\mbox{d}~{}M=\frac{1}{2}\rho C_{L}\frac{V_{a}^{2}(1+a)^{2}\sin(% \varphi+\beta)}{\sin^{2}\varphi\cos\beta}br\mbox{d}~{}r
  21. T V a \scriptstyle TV_{a}
  22. 2 π N Q \scriptstyle 2\pi NQ
  23. T V a 2 π N Q \scriptstyle\frac{TV_{a}}{2\pi NQ}
  24. blade element efficiency = V a 2 π N r 1 tan ( φ + β ) \mbox{blade element efficiency}~{}=\frac{V_{a}}{2\pi Nr}\cdot\frac{1}{\tan(% \varphi+\beta)}
  25. φ \scriptstyle\varphi
  26. β \scriptstyle\beta
  27. β \scriptstyle\beta
  28. V a V_{a}
  29. T = ρ V 2 D 2 [ f 1 ( N D V a ) , f 2 ( v V a D ) , f 3 ( g D V a 2 ) ] T=\rho V^{2}D^{2}\left[f_{1}\left(\frac{ND}{V_{a}}\right),f_{2}\left(\frac{v}{% V_{a}D}\right),f_{3}\left(\frac{gD}{V_{a}^{2}}\right)\right]
  30. f 1 f_{1}
  31. f 2 f_{2}
  32. f 3 f_{3}
  33. f 2 f_{2}
  34. f 3 f_{3}
  35. f 1 f_{1}
  36. T = ρ V a 2 D 2 × f r ( N D V a ) T=\rho V_{a}^{2}D^{2}\times f_{r}\left(\frac{ND}{V_{a}}\right)
  37. T 1 , T 2 T_{1},T_{2}
  38. T 1 T 2 = ρ 1 ρ 2 × V a 1 2 V a 2 2 × D 1 2 D 2 2 \frac{T_{1}}{T_{2}}=\frac{\rho_{1}}{\rho_{2}}\times\frac{V_{a1}^{2}}{V_{a2}^{2% }}\times\frac{D_{1}^{2}}{D_{2}^{2}}
  39. T 1 T 2 = ρ 1 ρ 2 × D 1 3 D 2 3 = ρ 1 ρ 2 λ 3 \frac{T_{1}}{T_{2}}=\frac{\rho_{1}}{\rho_{2}}\times\frac{D_{1}^{3}}{D_{2}^{3}}% =\frac{\rho_{1}}{\rho_{2}}\lambda^{3}
  40. λ \lambda
  41. P T 1 P T 2 = ρ 1 ρ 2 λ 3.5 \frac{P_{T1}}{P_{T2}}=\frac{\rho_{1}}{\rho_{2}}\lambda^{3.5}
  42. Q = ρ V a 2 D 3 × f q ( N D V a ) Q=\rho V_{a}^{2}D^{3}\times f_{q}\left(\frac{ND}{V_{a}}\right)
  43. ...
  44. P D \scriptstyle P_{D}
  45. P D \scriptstyle P^{\prime}_{D}
  46. P E \scriptstyle P_{E}
  47. P C \scriptstyle PC
  48. P S \scriptstyle P_{S}
  49. P E \scriptstyle P^{\prime}_{E}
  50. P T \scriptstyle P_{T}
  51. P E P^{\prime}_{E}
  52. P T P_{T}
  53. η H \eta_{H}
  54. P T P_{T}
  55. P D P^{\prime}_{D}
  56. η O \eta_{O}
  57. P D P^{\prime}_{D}
  58. P D P_{D}
  59. η R \eta_{R}
  60. P D P_{D}
  61. P S P_{S}
  62. P C = ( η H η O η R appendage coefficient ) transmission efficiency PC=\left(\frac{\eta_{H}\cdot\eta_{O}\cdot\eta_{R}}{\mbox{appendage coefficient% }~{}}\right)\cdot\mbox{transmission efficiency}~{}
  63. Q P C \scriptstyle QPC
  64. η D \scriptstyle\eta_{D}
  65. Q P C \scriptstyle QPC