wpmath0000013_10

Peres_metric.html

  1. d τ 2 = d t 2 - 2 f ( t + z , x , y ) ( d t + d z ) 2 - d x 2 - d y 2 - d z 2 {d\tau}^{2}=dt^{2}-2f\,(t+z,\,x,\,y)(dt+dz)^{2}-dx^{2}-dy^{2}-dz^{2}

Perfect_digit-to-digit_invariant.html

  1. n = d k d k + d k - 1 d k - 1 + + d 2 d 2 + d 1 d 1 . n=d_{k}^{d_{k}}+d_{k-1}^{d_{k-1}}+\dots+d_{2}^{d_{2}}+d_{1}^{d_{1}}\,.
  2. 3 3 + 4 4 + 3 3 + 5 5 = 27 + 256 + 27 + 3125 = 3435 3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125=3435
  3. 4 4 + 3 3 + 8 8 + 5 5 + 7 7 + 9 9 + 0 0 + 8 8 + 8 8 4^{4}+3^{3}+8^{8}+5^{5}+7^{7}+9^{9}+0^{0}+8^{8}+8^{8}
  4. = 256 + 27 + 16777216 + 3125 + 823543 + 387420489 + 0 + 16777216 + 16777216 = 438579088 =256+27+16777216+3125+823543+387420489+0+16777216+16777216=438579088
  5. b b
  6. n n
  7. b b
  8. a ( b - 1 ) b - 1 a(b-1)^{b-1}
  9. a a
  10. n n
  11. b - 1 b-1
  12. b b
  13. a ( b - 1 ) b - 1 n b a - 1 . a(b-1)^{b-1}\geq n\geq b^{a-1}.
  14. a ( b - 1 ) b - 1 a(b-1)^{b-1}
  15. a a
  16. b a - 1 b^{a-1}
  17. a a
  18. k > 0 k>0
  19. a k , a ( b - 1 ) b - 1 < b a - 1 . \forall a\geq k,\,\,a(b-1)^{b-1}<b^{a-1}.
  20. n n
  21. n n
  22. b b

Perfect_ring.html

  1. I I\,
  2. R = { f I + j f F , j J } R=\{f\cdot I+j\mid f\in F,j\in J\}\,
  3. R R

Permanent_magnet_synchronous_generator.html

  1. P 120 \frac{\,\text{P}}{120}
  2. f ( Hz ) = R P M P 120 f\left(\,\text{Hz}\right)=RPM\frac{\,\text{P}}{120}

Permutation_pattern.html

  1. [ s , t ] = [s,t]=\emptyset
  2. s t s\nleq t
  3. μ ( x , y ) = { 1 if x = y - z : x z < y μ ( x , z ) for x < y 0 otherwise . \mu(x,y)=\begin{cases}{}\qquad 1&\textrm{if}\quad x=y\\ \displaystyle-\sum_{z:x\leq z<y}\mu(x,z)&\textrm{for}\quad x<y\\ {}\qquad 0&\textrm{otherwise}.\end{cases}
  4. lim n number of copies of β in a β -optimal permutation of length n ( n k ) . \lim_{n\rightarrow\infty}\frac{\,\text{number of copies of }\beta\,\text{ in a% }\beta\,\text{-optimal permutation of length }n}{\displaystyle{n\choose k}}.
  5. π \pi
  6. n n
  7. σ \sigma
  8. k n k\leq n
  9. 2 O ( k 2 log k ) n 2^{O(k^{2}\log k)}\cdot n

Perron_method.html

  1. φ ( x ) \varphi(x)
  2. S φ S_{\varphi}
  3. u ( x ) = sup v S φ v ( x ) u(x)=\sup_{v\in S_{\varphi}}v(x)
  4. S φ S_{\varphi}
  5. v ( x ) φ ( x ) v(x)\leq\varphi(x)
  6. φ ( x ) \varphi(x)
  7. w y ( x ) w_{y}(x)
  8. w y ( y ) = 0 w_{y}(y)=0
  9. w y ( x ) > 0 w_{y}(x)>0
  10. x y x\neq y
  11. x y , u ( x ) φ ( y ) x\rightarrow y,u(x)\rightarrow\varphi(y)
  12. Ω \Omega
  13. λ ( 0 , 1 ) \lambda\in(0,1)
  14. C j C_{j}
  15. B λ j ( x 0 ) Ω c B_{\lambda^{j}}(x_{0})\cap\Omega^{c}
  16. x 0 x_{0}
  17. j = 0 C j / λ j ( n - 2 ) \sum_{j=0}^{\infty}C_{j}/\lambda^{j(n-2)}

Persistent_current.html

  1. 𝐌 \mathbf{M}
  2. 𝐉 𝐦 = × 𝐌 \mathbf{J_{m}}=\nabla\times\mathbf{M}
  3. 𝐌 \mathbf{M}

Perspectivity.html

  1. A B C D A B C D , ABCD\doublebarwedge A^{\prime}B^{\prime}C^{\prime}D^{\prime},
  2. \ell
  3. m m
  4. \ell
  5. m m
  6. X Y . X\doublebarwedge Y.
  7. X 𝑃 Y . X\ \overset{P}{\doublebarwedge}\ Y.
  8. f P : [ ] [ m ] f_{P}\colon[\ell]\mapsto[m]
  9. f P ( X ) = Y whenever P X Y f_{P}(X)=Y\,\text{ whenever }P\in XY
  10. f P ( f P ( X ) ) = X for all X [ ] f_{P}(f_{P}(X))=X\,\text{ for all }X\in[\ell]

Phase_qubit.html

  1. δ \frac{}{}\delta
  2. I \frac{}{}I
  3. δ \frac{}{}\delta
  4. I = I 0 sin δ \frac{}{}I=I_{0}\sin\delta
  5. I 0 \frac{}{}I_{0}
  6. Δ \frac{}{}\Delta
  7. R n \frac{}{}R_{n}
  8. I 0 = π Δ 2 e R n I_{0}=\frac{\pi\Delta}{2eR_{n}}
  9. V \frac{}{}V
  10. V = 2 e d δ d t V=\frac{\hbar}{2e}\frac{d\delta}{dt}
  11. C \frac{}{}C
  12. R \frac{}{}R
  13. I \frac{}{}I
  14. C 2 e d 2 δ d t 2 + 2 e R d δ d t = I - I 0 sin δ \frac{\hbar C}{2e}\,\frac{d^{2}\delta}{dt^{2}}+\frac{\hbar}{2eR}\frac{d\delta}% {dt}=I-I_{0}\sin\delta
  15. δ \frac{}{}\delta
  16. C \frac{}{}C
  17. R \frac{}{}R
  18. U ( δ ) \frac{}{}U(\delta)
  19. U ( δ ) = 2 e ( - I 0 cos δ - I δ ) . U(\delta)=\frac{\hbar}{2e}\left(-I_{0}\cos\delta-I\,\delta\right).
  20. - I δ \frac{}{}-I\,\delta
  21. - I 0 cos δ -\frac{}{}I_{0}\,\cos\delta
  22. | I | < I 0 \frac{}{}|I|<I_{0}
  23. I 0 \frac{}{}I_{0}
  24. I \frac{}{}I
  25. | I | > I 0 \frac{}{}|I|>I_{0}
  26. R \frac{}{}R
  27. δ 0 \frac{}{}\delta_{0}
  28. I = I 0 sin δ 0 \frac{}{}I=I_{0}\sin\delta_{0}
  29. Δ δ \frac{}{}\Delta\delta
  30. δ 0 \frac{}{}\delta_{0}
  31. Δ I = ( I 0 cos δ 0 ) Δ δ . \frac{}{}\Delta I=\left(I_{0}\cos\delta_{0}\right)\Delta\delta.
  32. Δ V = 2 e d Δ δ d t = 2 e 1 I 0 cos δ 0 d Δ I d t = L d Δ I d t \Delta V=\frac{\hbar}{2e}\frac{d\Delta\delta}{dt}=\frac{\hbar}{2e}\frac{1}{I_{% 0}\cos\delta_{0}}\frac{d\Delta I}{dt}=L\frac{d\Delta I}{dt}
  33. L = 2 e 1 I 0 cos δ 0 . L=\frac{\hbar}{2e}\frac{1}{I_{0}\cos\delta_{0}}.
  34. δ 0 \frac{}{}\delta_{0}
  35. I \frac{}{}I
  36. L min = 2 e 1 I 0 = R n π Δ . L_{\rm min}=\frac{\hbar}{2e}\frac{1}{I_{0}}=\frac{\hbar R_{n}}{\pi\Delta}.
  37. I 0 \frac{}{}I_{0}
  38. δ 0 \frac{}{}\delta_{0}
  39. π / 2 \frac{}{}\pi/2
  40. L \frac{}{}L
  41. I \frac{}{}I
  42. I 0 \frac{}{}I_{0}
  43. L C \frac{}{}LC
  44. ω p = 1 L C = 2 e I 0 cos δ 0 C \omega_{p}=\frac{1}{\sqrt{LC}}=\sqrt{\frac{2eI_{0}\cos\delta_{0}}{\hbar C}}
  45. δ 0 1 - ( I / I 0 ) 2 \delta_{0}\approx\sqrt{1-(I/I_{0})^{2}}
  46. ω p 2 e I 0 C [ 1 - ( I / I 0 ) 2 ] 1 / 4 , \omega_{p}\approx\sqrt{\frac{2eI_{0}}{\hbar C}}\left[1-(I/I_{0})^{2}\right]^{1% /4},
  47. | I | < I 0 \frac{}{}|I|<I_{0}
  48. I \frac{}{}I
  49. ω p \frac{}{}\omega_{p}
  50. E 01 ω p \frac{}{}E_{01}\approx\hbar\omega_{p}
  51. | g \frac{}{}|g\rangle
  52. | e \frac{}{}|e\rangle
  53. I \frac{}{}I

Phi_coefficient.html

  1. ϕ 2 = χ 2 n \phi^{2}=\frac{\chi^{2}}{n}
  2. n 11 n_{11}
  3. n 10 n_{10}
  4. n 1 n_{1\bullet}
  5. n 01 n_{01}
  6. n 00 n_{00}
  7. n 0 n_{0\bullet}
  8. n 1 n_{\bullet 1}
  9. n 0 n_{\bullet 0}
  10. n n
  11. ϕ = n 11 n 00 - n 10 n 01 n 1 n 0 n 0 n 1 \phi=\frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{n_{1\bullet}n_{0\bullet}n_{\bullet 0% }n_{\bullet 1}}}

Phillips_relationship.html

  1. Δ m 15 \Delta{m}_{15}
  2. M max ( B ) = - 21.726 + 2.698 Δ m 15 ( B ) . M_{\mathrm{max}}(B)=-21.726+2.698\Delta m_{15}(B).
  3. Δ m 15 \Delta{m}_{15}

Phillips–Perron_test.html

  1. ρ = 0 \rho=0
  2. Δ y t = ρ y t - 1 + u t \Delta y_{t}=\rho y_{t-1}+u_{t}\,
  3. Δ \Delta
  4. y t y_{t}
  5. y t - 1 y_{t-1}
  6. Δ y t \Delta y_{t}

Phonon_scattering.html

  1. τ \tau
  2. τ C \tau_{C}
  3. 1 τ C = 1 τ U + 1 τ M + 1 τ B + 1 τ p h - e \frac{1}{\tau_{C}}=\frac{1}{\tau_{U}}+\frac{1}{\tau_{M}}+\frac{1}{\tau_{B}}+% \frac{1}{\tau_{ph-e}}
  4. τ U \tau_{U}
  5. τ M \tau_{M}
  6. τ B \tau_{B}
  7. τ p h - e \tau_{ph-e}
  8. ω \omega
  9. ω 2 \omega^{2}
  10. τ U \tau_{U}
  11. 1 τ U = 2 γ 2 k B T μ V 0 ω 2 ω D \frac{1}{\tau_{U}}=2\gamma^{2}\frac{k_{B}T}{\mu V_{0}}\frac{\omega^{2}}{\omega% _{D}}
  12. γ \gamma
  13. ω D \omega_{D}
  14. 1 τ M = V 0 Γ ω 4 4 π v g 3 \frac{1}{\tau_{M}}=\frac{V_{0}\Gamma\omega^{4}}{4\pi v_{g}^{3}}
  15. Γ \Gamma
  16. v g {v_{g}}
  17. 1 τ B = V D ( 1 - p ) \frac{1}{\tau_{B}}=\frac{V}{D}(1-p)
  18. 1 τ B = V D \frac{1}{\tau_{B}}=\frac{V}{D}
  19. 1 τ p h - e = n e ϵ 2 ω ρ V 2 k B T π m * V 2 2 k B T exp ( - m * V 2 2 k B T ) \frac{1}{\tau_{ph-e}}=\frac{n_{e}\epsilon^{2}\omega}{\rho V^{2}k_{B}T}\sqrt{% \frac{\pi m^{*}V^{2}}{2k_{B}T}}\exp\left(-\frac{m^{*}V^{2}}{2k_{B}T}\right)
  20. n e n_{e}

Phosphoribosylaminoimidazole_synthetase.html

  1. \rightleftharpoons

Phosphoribosylglycinamide_formyltransferase.html

  1. \rightleftharpoons

Photocatalytic_water_splitting.html

  1. 2 H 2 O photon energy > 1.23 e V 2 H 2 + O 2 2\,\text{ }H_{2}O\,\text{ }\stackrel{\mathrm{photon\,energy}\,>1.23eV}{% \rightleftharpoons}\,\text{ }2\,\text{ }H_{2}+\,\text{ }O_{2}

Photoelectrochemical_process.html

  1. m v 2 2 = h ν - 13.6 e V {mv^{2}\over 2}=h\nu-13.6eV
  2. Δ E m o l = N A h ν \Delta E_{mol}=N_{A}h\nu

Photon_Doppler_velocimetry.html

  1. v v
  2. ν o b s e r v e d \nu_{observed}
  3. ν o b s e r v e d = ν s o u r c e 1 + v / c 1 - v / c \nu_{observed}=\nu_{source}\sqrt{\frac{1+v/c}{1-v/c}}
  4. v * v^{*}
  5. λ 0 \lambda_{0}
  6. f ¯ \bar{f}
  7. v * = λ 0 2 f ¯ v^{*}=\frac{\lambda_{0}}{2}\bar{f}
  8. f ¯ \bar{f}

Photothermal_optical_microscopy.html

  1. Δ \Delta
  2. P d P_{d}
  3. P d , 0 P_{d,0}
  4. Δ P d \Delta P_{d}
  5. Φ \Phi
  6. Φ = Δ P d P d , 0 = P d ( heating beam on ) - P d ( heating beam off ) P d ( background, no particle ) \Phi=\frac{\Delta P_{d}}{P_{d,0}}=\frac{P_{d}\left(\,\text{heating beam on}% \right)-P_{d}\left(\,\text{heating beam off}\right)}{P_{d}\left(\,\text{% background, no particle}\right)}
  7. R R
  8. n 0 n_{0}
  9. d n / d T \mathrm{d}n/\mathrm{d}T
  10. n ( 𝐫 ) = n 0 + d n d T Δ T ( 𝐫 ) = n 0 + Δ n R r n\left(\mathbf{r}\right)=n_{0}+\frac{\mathrm{d}n}{\mathrm{d}T}\Delta T\left(% \mathbf{r}\right)=n_{0}+\Delta n\frac{R}{r}
  11. σ abs \sigma_{\rm abs}
  12. I h I_{h}
  13. κ \kappa
  14. Δ n = ( d n / d T ) σ abs I h / 4 π κ R \Delta n=\left(\mathrm{d}n/\mathrm{d}T\right)\sigma_{\rm abs}I_{h}/4\pi\kappa R

Physics_of_magnetic_resonance_imaging.html

  1. 1 / 2 {1}/{2}
  2. S ( t ) = ρ ~ eff ( k ( t ) ) - d x ρ ( x ) e 2 π ı k ( t ) x S(t)={\tilde{\rho}}_{\mathrm{eff}}\left(\vec{k}(t)\right)\equiv\int_{-\infty}^% {\infty}\mathrm{d}\vec{x}\ \rho(\vec{x})\cdot e^{2\pi\imath\ \vec{k}(t)\cdot% \vec{x}}
  3. k ( t ) 0 t G ( τ ) d τ \vec{k}(t)\equiv\int_{0}^{t}\vec{G}(\tau)\ \mathrm{d}\tau
  4. ρ ( x ) \rho(\vec{x})
  5. I ( x ) I(\vec{x})
  6. I ( x ) = - d k S ( k ( t ) ) e - 2 π ı k ( t ) x I\left(\vec{x}\right)=\int_{-\infty}^{\infty}\mathrm{d}\vec{k}\ S\left(\vec{k}% (t)\right)\cdot e^{-2\pi\imath\ \vec{k}(t)\cdot\vec{x}}
  7. x \vec{x}
  8. k \vec{k}
  9. F O V 1 Δ k Resolution | k max | . FOV\propto\frac{1}{\Delta k}\qquad\mathrm{Resolution}\propto|k_{\max}|\ .

Physics_of_roller_coasters.html

  1. a r = v 2 r a_{r}=\frac{v^{2}}{r}
  2. U g = m g h U_{g}=mgh
  3. K = 1 2 m v 2 K=\frac{1}{2}mv^{2}

Phytoene_synthase.html

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

Pierce–Birkhoff_conjecture.html

  1. f : R n R f:R^{n}\rightarrow R
  2. g i j R [ x 1 , , x n ] g_{ij}\in R[x_{1},\ldots,x_{n}]
  3. f = sup i inf j ( g i j ) f=\sup_{i}\inf_{j}(g_{ij})
  4. A = R [ x 1 , , x n ] A=R[x_{1},\ldots,x_{n}]
  5. Sper A \mathrm{Sper}A
  6. Sper A \mathrm{Sper}A
  7. g A g\in A
  8. g ( α ) 0 g(\alpha)\geq 0
  9. g ( β ) 0 g(\beta)\leq 0
  10. R n = i P i R^{n}=\cup_{i}P_{i}
  11. Sper A = i P ~ i \mathrm{Sper}A=\cup_{i}\tilde{P}_{i}
  12. f i f_{i}
  13. α Sper A \alpha\in\mathrm{Sper}A
  14. f | P i = f i | P i f|_{P_{i}}=f_{i}|_{P_{i}}
  15. α P ~ i \alpha\in\tilde{P}_{i}
  16. f i f_{i}
  17. Sper A \mathrm{Sper}A
  18. f α f_{\alpha}
  19. f β f_{\beta}
  20. f α - f β f_{\alpha}-f_{\beta}

Pieri's_formula.html

  1. s μ h r = λ s λ \displaystyle s_{\mu}h_{r}=\sum_{\lambda}s_{\lambda}

Piezoresponse_force_microscopy.html

  1. V ( s y m b o l ω ) = V ac cos ( s y m b o l ω t ) V(symbol\omega)=V_{\mathrm{ac}}\cos(symbol\omega t)\;
  2. d = d 0 + D cos ( s y m b o l ω t + s y m b o l φ ) d=d_{\mathrm{0}}+D\cos(symbol\omega t+symbol\varphi)\;
  3. X i = d ki E k X_{\mathrm{i}}=d_{\mathrm{ki}}E_{\mathrm{k}}\;
  4. [ X 1 X 2 X 3 X 4 X 5 X 6 ] = [ 0 0 d 31 0 0 d 31 0 0 d 33 0 d 15 0 d 15 0 0 0 0 0 ] [ E 1 E 2 E 3 ] \begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\\ X_{4}\\ X_{5}\\ X_{6}\end{bmatrix}=\begin{bmatrix}0&0&d_{31}\\ 0&0&d_{31}\\ 0&0&d_{33}\\ 0&d_{15}&0\\ d_{15}&0&0\\ 0&0&0\end{bmatrix}\begin{bmatrix}E_{1}\\ E_{2}\\ E_{3}\end{bmatrix}
  5. V ref = B cos ( s y m b o l ω t ) \scriptstyle V_{\mathrm{ref}}=B\cos(symbol\omega t)
  6. V in = A cos ( s y m b o l ω t + s y m b o l φ ) \scriptstyle V_{\mathrm{in}}=A\cos(symbol\omega t+symbol\varphi)
  7. V out = 1 2 A B cos ( s y m b o l φ ) + 1 2 A B cos ( 2 s y m b o l ω t + s y m b o l φ ) V_{\mathrm{out}}=\frac{1}{2}AB\cos(symbol\varphi)+\frac{1}{2}AB\cos(2symbol% \omega t+symbol\varphi)
  8. X = 1 2 A B cos ( s y m b o l θ ) X=\frac{1}{2}AB\cos(symbol\theta)
  9. Y = 1 2 A B cos ( s y m b o l θ + π 2 ) = 1 2 A B sin ( s y m b o l θ ) Y=\frac{1}{2}AB\cos(symbol\theta+\frac{\pi}{2})=\frac{1}{2}AB\sin(symbol\theta)
  10. s y m b o l θ = tan - 1 Y X symbol\theta=\tan^{-1}\frac{Y}{X}
  11. R = X 2 + Y 2 R=\sqrt{X^{2}+Y^{2}}

Pilling–Bedworth_ratio.html

  1. R PB = V oxide V metal = M oxide ρ metal n M metal ρ oxide \mathrm{R_{PB}=\frac{V_{oxide}}{V_{metal}}=\frac{M_{oxide}\cdot\rho_{metal}}{n% \cdot M_{metal}\cdot\rho_{oxide}}}

Pitzer_equations.html

  1. G e x W w R T = f ( I ) + i j b i b j λ i j ( I ) + i j k b i b j b k μ i j k + \frac{G^{ex}}{W_{w}RT}=f(I)+\sum_{i}\sum_{j}b_{i}b_{j}\lambda_{ij}(I)+\sum_{i}% \sum_{j}\sum_{k}b_{i}b_{j}b_{k}\mu_{ijk}+\cdots
  2. G = i μ i 0 + R T ln b i γ i G=\sum_{i}\mu^{0}_{i}+RT\ln b_{i}\gamma_{i}
  3. ln γ i = ( G e x W w R T ) b i = z i 2 2 f + 2 j λ i j b j + z i 2 2 j k λ j k b j b k + 3 j k μ i j k b j b k + \ln\gamma_{i}=\frac{\partial(\frac{G^{ex}}{W_{w}RT})}{\partial b_{i}}=\frac{z_% {i}^{2}}{2}f^{\prime}+2\sum_{j}\lambda_{ij}b_{j}+\frac{z_{i}^{2}}{2}\sum_{j}% \sum_{k}\lambda^{\prime}_{jk}b_{j}b_{k}+3\sum_{j}\sum_{k}\mu_{ijk}b_{j}b_{k}+\cdots
  4. ϕ - 1 = ( i b i ) - 1 [ I f - f + i j ( λ i j + I λ i j ) b i b j + 2 i j k μ i j k b i b j b k + ] \phi-1=\left(\sum_{i}b_{i}\right)^{-1}\left[If^{\prime}-f+\sum_{i}\sum_{j}% \left(\lambda_{ij}+I\lambda^{\prime}_{ij}\right)b_{i}b_{j}+2\sum_{i}\sum_{j}% \sum_{k}\mu_{ijk}b_{i}b_{j}b_{k}+\cdots\right]
  5. f ϕ f^{\phi}
  6. B M X ϕ B^{\phi}_{MX}
  7. C M X ϕ C^{\phi}_{MX}
  8. f ϕ = f - f I 2 f^{\phi}=\frac{f^{\prime}-\frac{f}{I}}{2}
  9. B M X ϕ = λ M X + I λ M X + ( p 2 q ) ( λ M M + I λ M M ) + ( q 2 p ) ( λ X X + I λ X X ) B^{\phi}_{MX}=\lambda_{MX}+I\lambda^{\prime}_{MX}+\left(\frac{p}{2q}\right)% \left(\lambda_{MM}+I\lambda^{\prime}_{MM}\right)+\left(\frac{q}{2p}\right)% \left(\lambda_{XX}+I\lambda^{\prime}_{XX}\right)
  10. C M X ϕ = [ 3 p q ] ( p μ M M X + q μ M X X ) . C^{\phi}_{MX}=\left[\frac{3}{\sqrt{pq}}\right]\left(p\mu_{MMX}+q\mu_{MXX}% \right).
  11. μ M M M \mu_{MMM}
  12. μ X X X \mu_{XXX}
  13. B M X ϕ = β M X ( 0 ) + β M X ( 1 ) e - α I . B^{\phi}_{MX}=\beta^{(0)}_{MX}+\beta^{(1)}_{MX}e^{-\alpha\sqrt{I}}.
  14. ϕ - 1 = | z + z - | f ϕ + b ( 2 p q p + q ) B M X ϕ + m 2 [ 2 ( p q ) 3 / 2 p + q ] C M X ϕ . \phi-1=|z^{+}z^{-}|f^{\phi}+b\left(\frac{2pq}{p+q}\right)B^{\phi}_{MX}+m^{2}% \left[2\frac{(pq)^{3/2}}{p+q}\right]C^{\phi}_{MX}.
  15. ln γ ± = p ln γ M + q ln γ X p + q \ln\gamma_{\pm}=\frac{p\ln\gamma_{M}+q\ln\gamma_{X}}{p+q}
  16. ln γ ± = | z + z - | f γ + m ( 2 p q p + q ) B M X γ + m 2 [ 2 ( p q ) 3 / 2 p + q ] C M X γ \ln\gamma_{\pm}=|z^{+}z^{-}|f^{\gamma}+m\left(\frac{2pq}{p+q}\right)B^{\gamma}% _{MX}+m^{2}\left[2\frac{(pq)^{3/2}}{p+q}\right]C^{\gamma}_{MX}

Pixel_connectivity.html

  1. ( x ± 1 , y ) \textstyle(x\pm 1,y)
  2. ( x , y ± 1 ) \textstyle(x,y\pm 1)
  3. ( x , y ) \textstyle(x,y)
  4. ( x + 1 , y + 1 ) \textstyle(x+1,y+1)
  5. ( x - 1 , y - 1 ) \textstyle(x-1,y-1)
  6. ( x , y ) \textstyle(x,y)
  7. ( x ± 1 , y ± 1 ) \textstyle(x\pm 1,y\pm 1)
  8. ( x ± 1 , y 1 ) \textstyle(x\pm 1,y\mp 1)
  9. ( x , y ) \textstyle(x,y)
  10. ( x ± 1 , y , z ) \textstyle(x\pm 1,y,z)
  11. ( x , y ± 1 , z ) \textstyle(x,y\pm 1,z)
  12. ( x , y , z ± 1 ) \textstyle(x,y,z\pm 1)
  13. ( x , y , z ) \textstyle(x,y,z)
  14. ( x ± 1 , y ± 1 , z ) \textstyle(x\pm 1,y\pm 1,z)
  15. ( x ± 1 , y 1 , z ) \textstyle(x\pm 1,y\mp 1,z)
  16. ( x ± 1 , y , z ± 1 ) \textstyle(x\pm 1,y,z\pm 1)
  17. ( x ± 1 , y , z 1 ) \textstyle(x\pm 1,y,z\mp 1)
  18. ( x , y ± 1 , z ± 1 ) \textstyle(x,y\pm 1,z\pm 1)
  19. ( x , y ± 1 , z 1 ) \textstyle(x,y\pm 1,z\mp 1)
  20. ( x , y , z ) \textstyle(x,y,z)
  21. ( x ± 1 , y ± 1 , z ± 1 ) \textstyle(x\pm 1,y\pm 1,z\pm 1)
  22. ( x ± 1 , y ± 1 , z 1 ) \textstyle(x\pm 1,y\pm 1,z\mp 1)
  23. ( x ± 1 , y 1 , z ± 1 ) \textstyle(x\pm 1,y\mp 1,z\pm 1)
  24. ( x 1 , y ± 1 , z ± 1 ) \textstyle(x\mp 1,y\pm 1,z\pm 1)
  25. ( x , y , z ) \textstyle(x,y,z)

Plactic_monoid.html

  1. Γ ( t ) = 1 ( 1 - t ) n 1 ( 1 - t 2 ) n ( n - 1 ) / 2 \Gamma(t)=\frac{1}{(1-t)^{n}}\frac{1}{(1-t^{2})^{n(n-1)/2}}
  2. n ( n + 1 ) 2 \frac{n(n+1)}{2}

Plane_of_rotation.html

  1. 𝐚 𝐛 0 , \mathbf{a}\wedge\mathbf{b}\neq 0,
  2. 𝐱 𝐁 = 0. \mathbf{x}\wedge\mathbf{B}=0.
  3. 𝐜 = λ 𝐚 + μ 𝐛 , \mathbf{c}=\lambda\mathbf{a}+\mu\mathbf{b},
  4. n 2 \left\lfloor\frac{n}{2}\right\rfloor
  5. ( - 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) , \begin{pmatrix}-1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix},
  6. e i θ = cos θ + i sin θ , e^{i\theta}=\cos{\theta}+i\sin{\theta},\,
  7. ( cos θ - sin θ sin θ cos θ ) . \begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}.
  8. ( cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ) . \begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{pmatrix}.
  9. ( 1 0 0 0 0 1 0 0 0 0 cos θ - sin θ 0 0 sin θ cos θ ) \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\theta&-\sin\theta\\ 0&0&\sin\theta&\cos\theta\end{pmatrix}
  10. ( cos α - sin α 0 0 sin α cos α 0 0 0 0 cos β - sin β 0 0 sin β cos β ) . \begin{pmatrix}\cos\alpha&-\sin\alpha&0&0\\ \sin\alpha&\cos\alpha&0&0\\ 0&0&\cos\beta&-\sin\beta\\ 0&0&\sin\beta&\cos\beta\end{pmatrix}.
  11. n 2 , \left\lfloor\frac{n}{2}\right\rfloor,
  12. 𝐱 = - 𝐦𝐱𝐦 \mathbf{x}^{\prime}=-\mathbf{mxm}\,
  13. 𝐱 ′′ = - 𝐧𝐱 𝐧 = - 𝐧 ( - 𝐦𝐱𝐦 ) 𝐧 = 𝐧𝐦𝐱𝐦𝐧 \mathbf{x}^{\prime\prime}=-\mathbf{nx^{\prime}n}=-\mathbf{n(-mxm)n}=\mathbf{nmxmn}
  14. ( 𝐦𝐧 ) ( 𝐧𝐦 ) = 𝐦𝐧𝐧𝐦 = 𝐦𝐦 = 1 \mathbf{(mn)(nm)}=\mathbf{mnnm}=\mathbf{mm}=1
  15. 𝐱 ′′ = R 𝐱 R - 1 \mathbf{x}^{\prime\prime}=R\mathbf{x}R^{-1}
  16. R 𝐁 = e 𝐁 2 . R_{\mathbf{B}}=e^{\frac{\mathbf{B}}{2}}.
  17. R 𝐁 2 = e 𝐁 2 e 𝐁 2 = e 𝐁 , {R_{\mathbf{B}}}^{2}=e^{\frac{\mathbf{B}}{2}}e^{\frac{\mathbf{B}}{2}}=e^{% \mathbf{B}},
  18. 𝐦𝐧 = e 𝐁 , \mathbf{mn}=e^{\mathbf{B}},
  19. 𝐁 = log ( 𝐦𝐧 ) . \mathbf{B}=\log{(\mathbf{mn})}.
  20. n 2 , \left\lfloor\frac{n}{2}\right\rfloor,

Plant_litter.html

  1. X X o = e - k \frac{X}{X_{o}}=e^{-k}
  2. X o X_{o}
  3. k k

Plasma_(physics).html

  1. α \alpha
  2. α = n i n i + n n \alpha=\frac{n_{i}}{n_{i}+n_{n}}
  3. n i n_{i}
  4. n n n_{n}
  5. Z \langle Z\rangle
  6. n e = Z n i n_{e}=\langle Z\rangle n_{i}
  7. n e n_{e}
  8. T e T n T_{e}\gg T_{n}
  9. n e = Z n i n_{e}=\langle Z\rangle n_{i}
  10. n e e e Φ / k B T e . n_{e}\propto e^{e\Phi/k_{B}T_{e}}.
  11. E = ( k B T e / e ) ( n e / n e ) . \vec{E}=(k_{B}T_{e}/e)(\nabla n_{e}/n_{e}).
  12. ω ce / v coll > 1 \omega_{\mathrm{ce}}/v_{\mathrm{coll}}>1
  13. ω ce \omega_{\mathrm{ce}}
  14. v coll v_{\mathrm{coll}}
  15. 𝐄 = - v × 𝐁 \mathbf{E}=-v\times\mathbf{B}
  16. 𝐄 \mathbf{E}
  17. 𝐯 \mathbf{v}
  18. 𝐁 \mathbf{B}
  19. T e = T i = T g a s T_{e}=T_{i}=T_{gas}
  20. T e T i = T g a s T_{e}\gg T_{i}=T_{gas}

Plasma_shaping.html

  1. κ = b a \kappa={b\over a}
  2. a a
  3. b b
  4. R R

Plasmonic_solar_cell.html

  1. C s c a t = 1 6 π ( 2 π λ ) 4 | α | 2 C_{scat}=\frac{1}{6\pi}\left(\frac{2\pi}{\lambda}\right)^{4}|\alpha|^{2}
  2. C a b s = 2 π λ I m [ α ] C_{abs}=\frac{2\pi}{\lambda}Im[\alpha]
  3. α = 3 V [ ϵ p / ϵ m - 1 ϵ p / ϵ m + 2 ] \alpha=3V\left[\frac{\epsilon_{p}/\epsilon_{m}-1}{\epsilon_{p}/\epsilon_{m}+2}\right]
  4. ϵ p \epsilon_{p}
  5. ϵ m \epsilon_{m}
  6. ϵ p = - 2 ϵ m \epsilon_{p}=-2\epsilon_{m}
  7. ϵ = 1 - ω p 2 ω 2 + i γ ω \epsilon=1-\frac{\omega_{p}^{2}}{\omega^{2}+i\gamma\omega}
  8. ω p \omega_{p}
  9. ω p 2 = N e 2 / m ϵ 0 \omega_{p}^{2}=Ne^{2}/m\epsilon_{0}
  10. ϵ 0 \epsilon_{0}
  11. α = 3 V ω p 2 ω p 2 - 3 ω 2 - i γ ω \alpha=3V\frac{\omega_{p}^{2}}{\omega_{p}^{2}-3\omega^{2}-i\gamma\omega}
  12. ω s p = 3 ω p \omega_{sp}=\sqrt{3}\omega_{p}
  13. C S s c a t C S p a r t i c l e \frac{CS_{scat}}{CS_{particle}}

Plate_theory.html

  1. x 1 x_{1}
  2. x 2 x_{2}
  3. x 3 x_{3}
  4. u 1 0 , u 2 0 u^{0}_{1},u^{0}_{2}
  5. w 0 w^{0}
  6. x 3 x_{3}
  7. φ α \varphi_{\alpha}
  8. φ α = w , α 0 . \varphi_{\alpha}=w^{0}_{,\alpha}\,.
  9. ε α β = 1 2 ( u α , β 0 + u β , α 0 ) - x 3 w , α β 0 ε α 3 = - w , α 0 + w , α 0 = 0 ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\tfrac{1}{% 2}(u^{0}_{\alpha,\beta}+u^{0}_{\beta,\alpha})-x_{3}~{}w^{0}_{,\alpha\beta}\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=-w^{0}_{,\alpha}+w^{0}_{,% \alpha}=0\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  10. ε α β = 1 2 ( u α , β 0 + u β , α 0 + w , α 0 w , β 0 ) - x 3 w , α β 0 ε α 3 = - w , α 0 + w , α 0 = 0 ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\frac{1}{2% }(u^{0}_{\alpha,\beta}+u^{0}_{\beta,\alpha}+w^{0}_{,\alpha}~{}w^{0}_{,\beta})-% x_{3}~{}w^{0}_{,\alpha\beta}\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=-w^{0}_{,\alpha}+w^{0}_{,% \alpha}=0\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  11. N α β , α = 0 M α β , α β = 0 \begin{aligned}\displaystyle N_{\alpha\beta,\alpha}&\displaystyle=0\\ \displaystyle M_{\alpha\beta,\alpha\beta}&\displaystyle=0\end{aligned}
  12. N α β := - h h σ α β d x 3 ; M α β := - h h x 3 σ α β d x 3 N_{\alpha\beta}:=\int_{-h}^{h}\sigma_{\alpha\beta}~{}dx_{3}~{};~{}~{}M_{\alpha% \beta}:=\int_{-h}^{h}x_{3}~{}\sigma_{\alpha\beta}~{}dx_{3}
  13. 2 h 2h
  14. σ α β \sigma_{\alpha\beta}
  15. q ( x ) q(x)
  16. x 3 x_{3}
  17. N α β , α = 0 M α β , α β + [ N α β w , β 0 ] , α - q = 0 \begin{aligned}\displaystyle N_{\alpha\beta,\alpha}&\displaystyle=0\\ \displaystyle M_{\alpha\beta,\alpha\beta}+[N_{\alpha\beta}~{}w^{0}_{,\beta}]_{% ,\alpha}-q&\displaystyle=0\end{aligned}
  18. n α N α β or u β 0 n α M α β , β or w 0 n β M α β or w , α 0 \begin{aligned}\displaystyle n_{\alpha}~{}N_{\alpha\beta}&\displaystyle\quad% \mathrm{or}\quad u^{0}_{\beta}\\ \displaystyle n_{\alpha}~{}M_{\alpha\beta,\beta}&\displaystyle\quad\mathrm{or}% \quad w^{0}\\ \displaystyle n_{\beta}~{}M_{\alpha\beta}&\displaystyle\quad\mathrm{or}\quad w% ^{0}_{,\alpha}\end{aligned}
  19. n α M α β , β n_{\alpha}~{}M_{\alpha\beta,\beta}
  20. [ σ 11 σ 22 σ 12 ] = [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] [ ε 11 ε 22 ε 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}
  21. σ α 3 \sigma_{\alpha 3}
  22. σ 33 \sigma_{33}
  23. [ N 11 N 22 N 12 ] = { - h h [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] d x 3 } [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\left\{\int_{-h}^{h}\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}
  24. [ M 11 M 22 M 12 ] = - { - h h x 3 2 [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] d x 3 } [ w , 11 0 w , 22 0 w , 12 0 ] . \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\left\{\int_{-h}^{h}x_{3}^{2}~{}\begin{bmatrix}C_{11}&C_{% 12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}w^{0}_{,11}\\ w^{0}_{,22}\\ w^{0}_{,12}\end{bmatrix}\,.
  25. A α β := - h h C α β d x 3 A_{\alpha\beta}:=\int_{-h}^{h}C_{\alpha\beta}~{}dx_{3}
  26. D α β := - h h x 3 2 C α β d x 3 D_{\alpha\beta}:=\int_{-h}^{h}x_{3}^{2}~{}C_{\alpha\beta}~{}dx_{3}
  27. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] . \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}\,.
  28. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 0 w , 22 0 w , 12 0 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w^{0}_{,11}\\ w^{0}_{,22}\\ w^{0}_{,12}\end{bmatrix}
  29. u 1 0 u^{0}_{1}
  30. u 2 0 u^{0}_{2}
  31. 4 w x 1 4 + 2 4 w x 1 2 x 2 2 + 4 w x 2 4 = 0 where w := w 0 . \frac{\partial^{4}w}{\partial x_{1}^{4}}+2\frac{\partial^{4}w}{\partial x_{1}^% {2}\partial x_{2}^{2}}+\frac{\partial^{4}w}{\partial x_{2}^{4}}=0\quad\,\text{% where}\quad w:=w^{0}\,.
  32. w , 1111 0 + 2 w , 1212 0 + w , 2222 0 = 0 . w^{0}_{,1111}+2~{}w^{0}_{,1212}+w^{0}_{,2222}=0\,.
  33. 4 w x 1 4 + 2 4 w x 1 2 x 2 2 + 4 w x 2 4 = - q D \frac{\partial^{4}w}{\partial x_{1}^{4}}+2\frac{\partial^{4}w}{\partial x_{1}^% {2}\partial x_{2}^{2}}+\frac{\partial^{4}w}{\partial x_{2}^{4}}=-\frac{q}{D}
  34. D := 2 h 3 E 3 ( 1 - ν 2 ) . D:=\cfrac{2h^{3}E}{3(1-\nu^{2})}\,.
  35. w , 1111 0 + 2 w , 1212 0 + w , 2222 0 = - q D w^{0}_{,1111}+2\,w^{0}_{,1212}+w^{0}_{,2222}=-\frac{q}{D}
  36. ( r , θ , z ) (r,\theta,z)
  37. 1 r d d r [ r d d r { 1 r d d r ( r d w d r ) } ] = - q D . \frac{1}{r}\cfrac{d}{dr}\left[r\cfrac{d}{dr}\left\{\frac{1}{r}\cfrac{d}{dr}% \left(r\cfrac{dw}{dr}\right)\right\}\right]=-\frac{q}{D}\,.
  38. [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] = 1 1 - ν 12 ν 21 [ E 1 ν 12 E 2 0 ν 21 E 1 E 2 0 0 0 2 G 12 ( 1 - ν 12 ν 21 ) ] . \begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}=\cfrac{1}{1-\nu_{12}\nu_{21}}\begin{bmatrix}% E_{1}&\nu_{12}E_{2}&0\\ \nu_{21}E_{1}&E_{2}&0\\ 0&0&2G_{12}(1-\nu_{12}\nu_{21})\end{bmatrix}\,.
  39. [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] = 2 h 1 - ν 12 ν 21 [ E 1 ν 12 E 2 0 ν 21 E 1 E 2 0 0 0 2 G 12 ( 1 - ν 12 ν 21 ) ] \begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}=\cfrac{2h}{1-\nu_{12}\nu_{21}}\begin{bmatrix% }E_{1}&\nu_{12}E_{2}&0\\ \nu_{21}E_{1}&E_{2}&0\\ 0&0&2G_{12}(1-\nu_{12}\nu_{21})\end{bmatrix}
  40. [ D 11 D 12 D 13 D 21 D 22 D 23 D 31 D 32 D 33 ] = 2 h 3 3 ( 1 - ν 12 ν 21 ) [ E 1 ν 12 E 2 0 ν 21 E 1 E 2 0 0 0 2 G 12 ( 1 - ν 12 ν 21 ) ] . \begin{bmatrix}D_{11}&D_{12}&D_{13}\\ D_{21}&D_{22}&D_{23}\\ D_{31}&D_{32}&D_{33}\end{bmatrix}=\cfrac{2h^{3}}{3(1-\nu_{12}\nu_{21})}\begin{% bmatrix}E_{1}&\nu_{12}E_{2}&0\\ \nu_{21}E_{1}&E_{2}&0\\ 0&0&2G_{12}(1-\nu_{12}\nu_{21})\end{bmatrix}\,.
  41. q q
  42. D x w , 1111 0 + 2 D x y w , 1122 0 + D y w , 2222 0 = - q D_{x}w^{0}_{,1111}+2D_{xy}w^{0}_{,1122}+D_{y}w^{0}_{,2222}=-q
  43. D x = D 11 = 2 h 3 E 1 3 ( 1 - ν 12 ν 21 ) D y = D 22 = 2 h 3 E 2 3 ( 1 - ν 12 ν 21 ) D x y = D 33 + 1 2 ( ν 21 D 11 + ν 12 D 22 ) = D 33 + ν 21 D 11 = 4 h 3 G 12 3 + 2 h 3 ν 21 E 1 3 ( 1 - ν 12 ν 21 ) . \begin{aligned}\displaystyle D_{x}&\displaystyle=D_{11}=\frac{2h^{3}E_{1}}{3(1% -\nu_{12}\nu_{21})}\\ \displaystyle D_{y}&\displaystyle=D_{22}=\frac{2h^{3}E_{2}}{3(1-\nu_{12}\nu_{2% 1})}\\ \displaystyle D_{xy}&\displaystyle=D_{33}+\tfrac{1}{2}(\nu_{21}D_{11}+\nu_{12}% D_{22})=D_{33}+\nu_{21}D_{11}=\frac{4h^{3}G_{12}}{3}+\frac{2h^{3}\nu_{21}E_{1}% }{3(1-\nu_{12}\nu_{21})}\,.\end{aligned}
  44. ρ = ρ ( x ) \rho=\rho(x)
  45. J 1 := - h h ρ d x 3 = 2 ρ h ; J 3 := - h h x 3 2 ρ d x 3 = 2 3 ρ h 3 J_{1}:=\int_{-h}^{h}\rho~{}dx_{3}=2~{}\rho~{}h~{};~{}~{}J_{3}:=\int_{-h}^{h}x_% {3}^{2}~{}\rho~{}dx_{3}=\frac{2}{3}~{}\rho~{}h^{3}
  46. u ˙ i = u i t ; u ¨ i = 2 u i t 2 ; u i , α = u i x α ; u i , α β = 2 u i x α x β \dot{u}_{i}=\frac{\partial u_{i}}{\partial t}~{};~{}~{}\ddot{u}_{i}=\frac{% \partial^{2}u_{i}}{\partial t^{2}}~{};~{}~{}u_{i,\alpha}=\frac{\partial u_{i}}% {\partial x_{\alpha}}~{};~{}~{}u_{i,\alpha\beta}=\frac{\partial^{2}u_{i}}{% \partial x_{\alpha}\partial x_{\beta}}
  47. D ( 4 w 0 x 1 4 + 2 4 w 0 x 1 2 x 2 2 + 4 w 0 x 2 4 ) = - q ( x 1 , x 2 , t ) - 2 ρ h 2 w 0 t 2 . D\,\left(\frac{\partial^{4}w^{0}}{\partial x_{1}^{4}}+2\frac{\partial^{4}w^{0}% }{\partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}w^{0}}{\partial x_{2% }^{4}}\right)=-q(x_{1},x_{2},t)-2\rho h\,\frac{\partial^{2}w^{0}}{\partial t^{% 2}}\,.
  48. D D
  49. 2 h 2h
  50. D := 2 h 3 E 3 ( 1 - ν 2 ) . D:=\cfrac{2h^{3}E}{3(1-\nu^{2})}\,.
  51. φ 1 \varphi_{1}
  52. φ 2 \varphi_{2}
  53. x 3 x_{3}
  54. φ 1 w , 1 ; φ 2 w , 2 \varphi_{1}\neq w_{,1}~{};~{}~{}\varphi_{2}\neq w_{,2}
  55. ε α β = 1 2 ( u α , β 0 + u β , α 0 ) - x 3 2 ( φ α , β + φ β , α ) ε α 3 = 1 2 ( w , α 0 - φ α ) ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\frac{1}{2% }(u^{0}_{\alpha,\beta}+u^{0}_{\beta,\alpha})-\frac{x_{3}}{2}~{}(\varphi_{% \alpha,\beta}+\varphi_{\beta,\alpha})\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=\cfrac{1}{2}\left(w^{0}_{,% \alpha}-\varphi_{\alpha}\right)\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  56. κ \kappa
  57. ε α 3 = 1 2 κ ( w , α 0 - φ α ) \varepsilon_{\alpha 3}=\cfrac{1}{2}~{}\kappa~{}\left(w^{0}_{,\alpha}-\varphi_{% \alpha}\right)
  58. Q α := κ - h h σ α 3 d x 3 . Q_{\alpha}:=\kappa~{}\int_{-h}^{h}\sigma_{\alpha 3}~{}dx_{3}\,.
  59. n α N α β or u β 0 n α M α β or φ α n α Q α or w 0 \begin{aligned}\displaystyle n_{\alpha}~{}N_{\alpha\beta}&\displaystyle\quad% \mathrm{or}\quad u^{0}_{\beta}\\ \displaystyle n_{\alpha}~{}M_{\alpha\beta}&\displaystyle\quad\mathrm{or}\quad% \varphi_{\alpha}\\ \displaystyle n_{\alpha}~{}Q_{\alpha}&\displaystyle\quad\mathrm{or}\quad w^{0}% \end{aligned}
  60. σ α β = C α β γ θ ε γ θ σ α 3 = C α 3 γ θ ε γ θ σ 33 = C 33 γ θ ε γ θ \begin{aligned}\displaystyle\sigma_{\alpha\beta}&\displaystyle=C_{\alpha\beta% \gamma\theta}~{}\varepsilon_{\gamma\theta}\\ \displaystyle\sigma_{\alpha 3}&\displaystyle=C_{\alpha 3\gamma\theta}~{}% \varepsilon_{\gamma\theta}\\ \displaystyle\sigma_{33}&\displaystyle=C_{33\gamma\theta}~{}\varepsilon_{% \gamma\theta}\end{aligned}
  61. σ 33 \sigma_{33}
  62. [ σ 11 σ 22 σ 23 σ 31 σ 12 ] = [ C 11 C 12 0 0 0 C 12 C 22 0 0 0 0 0 C 44 0 0 0 0 0 C 55 0 0 0 0 0 C 66 ] [ ε 11 ε 22 ε 23 ε 31 ε 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&0&0&0\\ C_{12}&C_{22}&0&0&0\\ 0&0&C_{44}&0&0\\ 0&0&0&C_{55}&0\\ 0&0&0&0&C_{66}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{23}\\ \varepsilon_{31}\\ \varepsilon_{12}\end{bmatrix}
  63. [ N 11 N 22 N 12 ] = { - h h [ C 11 C 12 0 C 12 C 22 0 0 0 C 66 ] d x 3 } [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\left\{\int_{-h}^{h}\begin{bmatrix}C_{11}&C_{12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}
  64. [ M 11 M 22 M 12 ] = - { - h h x 3 2 [ C 11 C 12 0 C 12 C 22 0 0 0 C 66 ] d x 3 } [ φ 1 , 1 φ 2 , 2 1 2 ( φ 1 , 2 + φ 2 , 1 ) ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\left\{\int_{-h}^{h}x_{3}^{2}~{}\begin{bmatrix}C_{11}&C_{% 12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}\varphi_{1,1}\\ \varphi_{2,2}\\ \frac{1}{2}~{}(\varphi_{1,2}+\varphi_{2,1})\end{bmatrix}
  65. [ Q 1 Q 2 ] = κ 2 { - h h [ C 55 0 0 C 44 ] d x 3 } [ w , 1 0 - φ 1 w , 2 0 - φ 2 ] \begin{bmatrix}Q_{1}\\ Q_{2}\end{bmatrix}=\cfrac{\kappa}{2}\left\{\int_{-h}^{h}\begin{bmatrix}C_{55}&% 0\\ 0&C_{44}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}w^{0}_{,1}-\varphi_{1}\\ w^{0}_{,2}-\varphi_{2}\end{bmatrix}
  66. A α β := - h h C α β d x 3 A_{\alpha\beta}:=\int_{-h}^{h}C_{\alpha\beta}~{}dx_{3}
  67. D α β := - h h x 3 2 C α β d x 3 D_{\alpha\beta}:=\int_{-h}^{h}x_{3}^{2}~{}C_{\alpha\beta}~{}dx_{3}
  68. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] . \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}\,.
  69. E E
  70. ν \nu
  71. ε α β \varepsilon_{\alpha\beta}
  72. σ 31 = 2 G ε 31 and σ 32 = 2 G ε 32 \sigma_{31}=2G\varepsilon_{31}\quad\,\text{and}\quad\sigma_{32}=2G\varepsilon_% {32}
  73. G = E / ( 2 ( 1 + ν ) ) G=E/(2(1+\nu))
  74. [ N 11 N 22 N 12 ] = 2 E h 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] , \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\cfrac{2Eh}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}\,,
  75. [ M 11 M 22 M 12 ] = - 2 E h 3 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ φ 1 , 1 φ 2 , 2 1 2 ( φ 1 , 2 + φ 2 , 1 ) ] , \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2Eh^{3}}{3(1-\nu^{2})}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varphi_{1,1}\\ \varphi_{2,2}\\ \frac{1}{2}(\varphi_{1,2}+\varphi_{2,1})\end{bmatrix}\,,
  76. [ Q 1 Q 2 ] = κ G h [ w , 1 0 - φ 1 w , 2 0 - φ 2 ] . \begin{bmatrix}Q_{1}\\ Q_{2}\end{bmatrix}=\kappa Gh\begin{bmatrix}w^{0}_{,1}-\varphi_{1}\\ w^{0}_{,2}-\varphi_{2}\end{bmatrix}\,.
  77. D = 2 E h 3 3 ( 1 - ν 2 ) . D=\cfrac{2Eh^{3}}{3(1-\nu^{2})}\,.
  78. h h
  79. D = E H 3 12 ( 1 - ν 2 ) . D=\cfrac{EH^{3}}{12(1-\nu^{2})}\,.
  80. M α β , β - Q α = 0 Q α , α + q = 0 . \begin{aligned}\displaystyle M_{\alpha\beta,\beta}-Q_{\alpha}&\displaystyle=0% \\ \displaystyle Q_{\alpha,\alpha}+q&\displaystyle=0\,.\end{aligned}
  81. w 0 , φ 1 , φ 2 w^{0},\varphi_{1},\varphi_{2}
  82. simply supported w 0 = 0 , M 11 = 0 ( or M 22 = 0 ) , φ 1 = 0 ( or φ 2 = 0 ) clamped w 0 = 0 , φ 1 = 0 , φ 2 = 0 . \begin{aligned}\displaystyle\,\text{simply supported}&\displaystyle\quad w^{0}% =0,M_{11}=0~{}(\,\text{or}~{}M_{22}=0),\varphi_{1}=0~{}(\,\text{or}~{}\varphi_% {2}=0)\\ \displaystyle\,\text{clamped}&\displaystyle\quad w^{0}=0,\varphi_{1}=0,\varphi% _{2}=0\,.\end{aligned}
  83. w ( x , y ) = w x ( x ) + y θ x ( x ) . w(x,y)=w_{x}(x)+y\,\theta_{x}(x)\,.
  84. q 1 ( x ) = - b / 2 b / 2 q ( x , y ) d y , q 2 ( x ) = - b / 2 b / 2 y q ( x , y ) d y , n 1 ( x ) = - b / 2 b / 2 n x ( x , y ) d y n 2 ( x ) = - b / 2 b / 2 y n x ( x , y ) d y , n 3 ( x ) = - b / 2 b / 2 y 2 n x ( x , y ) d y . \begin{aligned}\displaystyle q_{1}(x)&\displaystyle=\int_{-b/2}^{b/2}q(x,y)\,% \,\text{d}y~{},~{}~{}q_{2}(x)=\int_{-b/2}^{b/2}y\,q(x,y)\,\,\text{d}y~{},~{}~{% }n_{1}(x)=\int_{-b/2}^{b/2}n_{x}(x,y)\,\,\text{d}y\\ \displaystyle n_{2}(x)&\displaystyle=\int_{-b/2}^{b/2}y\,n_{x}(x,y)\,\,\text{d% }y~{},~{}~{}n_{3}(x)=\int_{-b/2}^{b/2}y^{2}\,n_{x}(x,y)\,\,\text{d}y\,.\end{aligned}
  85. x = 0 x=0
  86. w ( 0 , y ) = d w d x | x = 0 = 0 w x ( 0 ) = d w x d x | x = 0 = θ x ( 0 ) = d θ x d x | x = 0 = 0 . w(0,y)=\cfrac{dw}{dx}\Bigr|_{x=0}=0\qquad\implies\qquad w_{x}(0)=\cfrac{dw_{x}% }{dx}\Bigr|_{x=0}=\theta_{x}(0)=\cfrac{d\theta_{x}}{dx}\Bigr|_{x=0}=0\,.
  87. x = a x=a
  88. b D d 3 w x d x 3 + n 1 ( x ) d w x d x + n 2 ( x ) d θ x d x + q x 1 = 0 b 3 D 12 d 3 θ x d x 3 + [ n 3 ( x ) - 2 b D ( 1 - ν ) ] d θ x d x + n 2 ( x ) d w x d x + t = 0 b D d 2 w x d x 2 + m 1 = 0 , b 3 D 12 d 2 θ x d x 2 + m 2 = 0 \begin{aligned}&\displaystyle bD\cfrac{d^{3}w_{x}}{dx^{3}}+n_{1}(x)\cfrac{dw_{% x}}{dx}+n_{2}(x)\cfrac{d\theta_{x}}{dx}+q_{x1}=0\\ &\displaystyle\frac{b^{3}D}{12}\cfrac{d^{3}\theta_{x}}{dx^{3}}+\left[n_{3}(x)-% 2bD(1-\nu)\right]\cfrac{d\theta_{x}}{dx}+n_{2}(x)\cfrac{dw_{x}}{dx}+t=0\\ &\displaystyle bD\cfrac{d^{2}w_{x}}{dx^{2}}+m_{1}=0\quad,\quad\frac{b^{3}D}{12% }\cfrac{d^{2}\theta_{x}}{dx^{2}}+m_{2}=0\end{aligned}
  89. m 1 = - b / 2 b / 2 m x ( y ) d y , m 2 = - b / 2 b / 2 y m x ( y ) d y , q x 1 = - b / 2 b / 2 q x ( y ) d y t = q x 2 + m 3 = - b / 2 b / 2 y q x ( y ) d y + - b / 2 b / 2 m x y ( y ) d y . \begin{aligned}\displaystyle m_{1}&\displaystyle=\int_{-b/2}^{b/2}m_{x}(y)\,\,% \text{d}y~{},~{}~{}m_{2}=\int_{-b/2}^{b/2}y\,m_{x}(y)\,\,\text{d}y~{},~{}~{}q_% {x1}=\int_{-b/2}^{b/2}q_{x}(y)\,\,\text{d}y\\ \displaystyle t&\displaystyle=q_{x2}+m_{3}=\int_{-b/2}^{b/2}y\,q_{x}(y)\,\,% \text{d}y+\int_{-b/2}^{b/2}m_{xy}(y)\,\,\text{d}y\,.\end{aligned}
  90. h h
  91. U = 1 2 0 a - b / 2 b / 2 D { ( 2 w x 2 + 2 w y 2 ) 2 + 2 ( 1 - ν ) [ ( 2 w x y ) 2 - 2 w x 2 2 w y 2 ] } d x d y U=\frac{1}{2}\int_{0}^{a}\int_{-b/2}^{b/2}D\left\{\left(\frac{\partial^{2}w}{% \partial x^{2}}+\frac{\partial^{2}w}{\partial y^{2}}\right)^{2}+2(1-\nu)\left[% \left(\frac{\partial^{2}w}{\partial x\partial y}\right)^{2}-\frac{\partial^{2}% w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}}\right]\right\}\,\text{d% }x\,\text{d}y
  92. w w
  93. a a
  94. b b
  95. ν \nu
  96. E E
  97. D = E h 3 12 ( 1 - ν ) . D=\frac{Eh^{3}}{12(1-\nu)}.
  98. q ( x , y ) q(x,y)
  99. P q = 0 a - b / 2 b / 2 q ( x , y ) w ( x , y ) d x d y . P_{q}=\int_{0}^{a}\int_{-b/2}^{b/2}q(x,y)\,w(x,y)\,\,\text{d}x\,\text{d}y\,.
  100. n x ( x , y ) n_{x}(x,y)
  101. P n = 1 2 0 a - b / 2 b / 2 n x ( x , y ) ( w x ) 2 d x d y . P_{n}=\frac{1}{2}\int_{0}^{a}\int_{-b/2}^{b/2}n_{x}(x,y)\,\left(\frac{\partial w% }{\partial x}\right)^{2}\,\,\text{d}x\,\text{d}y\,.
  102. q x ( y ) q_{x}(y)
  103. m x ( y ) m_{x}(y)
  104. m x y ( y ) m_{xy}(y)
  105. P t = - b / 2 b / 2 ( q x ( y ) w ( x , y ) - m x ( y ) w x + m x y ( y ) w y ) d x d y . P_{t}=\int_{-b/2}^{b/2}\left(q_{x}(y)\,w(x,y)-m_{x}(y)\,\frac{\partial w}{% \partial x}+m_{xy}(y)\,\frac{\partial w}{\partial y}\right)\,\text{d}x\,\text{% d}y\,.
  106. W = U - ( P q + P n + P t ) . W=U-(P_{q}+P_{n}+P_{t})\,.
  107. U = 0 a b D 24 [ 12 ( d 2 w x d x 2 ) 2 + b 2 ( d 2 θ x d x 2 ) 2 + 24 ( 1 - ν ) ( d θ x d x ) 2 ] d x , U=\int_{0}^{a}\frac{bD}{24}\left[12\left(\cfrac{d^{2}w_{x}}{dx^{2}}\right)^{2}% +b^{2}\left(\cfrac{d^{2}\theta_{x}}{dx^{2}}\right)^{2}+24(1-\nu)\left(\cfrac{d% \theta_{x}}{dx}\right)^{2}\right]\,\,\text{d}x\,,
  108. P q = 0 a [ ( - b / 2 b / 2 q ( x , y ) d y ) w x + ( - b / 2 b / 2 y q ( x , y ) d y ) θ x ] d x , P_{q}=\int_{0}^{a}\left[\left(\int_{-b/2}^{b/2}q(x,y)\,\,\text{d}y\right)w_{x}% +\left(\int_{-b/2}^{b/2}yq(x,y)\,\,\text{d}y\right)\theta_{x}\right]\,dx\,,
  109. P n = 1 2 0 a [ ( - b / 2 b / 2 n x ( x , y ) d y ) ( d w x d x ) 2 + ( - b / 2 b / 2 y n x ( x , y ) d y ) d w x d x d θ x d x + ( - b / 2 b / 2 y 2 n x ( x , y ) d y ) ( d θ x d x ) 2 ] d x , \begin{aligned}\displaystyle P_{n}&\displaystyle=\frac{1}{2}\int_{0}^{a}\left[% \left(\int_{-b/2}^{b/2}n_{x}(x,y)\,\,\text{d}y\right)\left(\cfrac{dw_{x}}{dx}% \right)^{2}+\left(\int_{-b/2}^{b/2}yn_{x}(x,y)\,\,\text{d}y\right)\cfrac{dw_{x% }}{dx}\,\cfrac{d\theta_{x}}{dx}\right.\\ &\displaystyle\left.\qquad\qquad+\left(\int_{-b/2}^{b/2}y^{2}n_{x}(x,y)\,\,% \text{d}y\right)\left(\cfrac{d\theta_{x}}{dx}\right)^{2}\right]\,\text{d}x\,,% \end{aligned}
  110. P t = ( - b / 2 b / 2 q x ( y ) d y ) w x - ( - b / 2 b / 2 m x ( y ) d y ) d w x d x + [ - b / 2 b / 2 ( y q x ( y ) + m x y ( y ) ) d y ] θ x - ( - b / 2 b / 2 y m x ( y ) d y ) d θ x d x . \begin{aligned}\displaystyle P_{t}&\displaystyle=\left(\int_{-b/2}^{b/2}q_{x}(% y)\,\,\text{d}y\right)w_{x}-\left(\int_{-b/2}^{b/2}m_{x}(y)\,\,\text{d}y\right% )\cfrac{dw_{x}}{dx}+\left[\int_{-b/2}^{b/2}\left(yq_{x}(y)+m_{xy}(y)\right)\,% \,\text{d}y\right]\theta_{x}\\ &\displaystyle\qquad\qquad-\left(\int_{-b/2}^{b/2}ym_{x}(y)\,\,\text{d}y\right% )\cfrac{d\theta_{x}}{dx}\,.\end{aligned}
  111. W W
  112. ( w x , θ x , x ) (w_{x},\theta_{x},x)
  113. (1) b D d 4 w x d x 4 = q 1 ( x ) - n 1 ( x ) d 2 w x d x 2 - d n 1 d x d w x d x - 1 2 d n 2 d x d θ x d x - n 2 ( x ) 2 d 2 θ x d x 2 \,\text{(1)}\qquad bD\frac{\mathrm{d}^{4}w_{x}}{\mathrm{d}x^{4}}=q_{1}(x)-n_{1% }(x)\cfrac{d^{2}w_{x}}{dx^{2}}-\cfrac{dn_{1}}{dx}\,\cfrac{dw_{x}}{dx}-\frac{1}% {2}\cfrac{dn_{2}}{dx}\,\cfrac{d\theta_{x}}{dx}-\frac{n_{2}(x)}{2}\cfrac{d^{2}% \theta_{x}}{dx^{2}}
  114. (2) b 3 D 12 d 4 θ x d x 4 - 2 b D ( 1 - ν ) d 2 θ x d x 2 = q 2 ( x ) - n 3 ( x ) d 2 θ x d x 2 - d n 3 d x d θ x d x - n 2 ( x ) 2 d 2 w x d x 2 - 1 2 d n 2 d x d w x d x \,\text{(2)}\qquad\frac{b^{3}D}{12}\,\frac{\mathrm{d}^{4}\theta_{x}}{\mathrm{d% }x^{4}}-2bD(1-\nu)\cfrac{d^{2}\theta_{x}}{dx^{2}}=q_{2}(x)-n_{3}(x)\cfrac{d^{2% }\theta_{x}}{dx^{2}}-\cfrac{dn_{3}}{dx}\,\cfrac{d\theta_{x}}{dx}-\frac{n_{2}(x% )}{2}\,\cfrac{d^{2}w_{x}}{dx^{2}}-\frac{1}{2}\cfrac{dn_{2}}{dx}\,\cfrac{dw_{x}% }{dx}
  115. q 1 ( x ) = - b / 2 b / 2 q ( x , y ) d y , q 2 ( x ) = - b / 2 b / 2 y q ( x , y ) d y , n 1 ( x ) = - b / 2 b / 2 n x ( x , y ) d y n 2 ( x ) = - b / 2 b / 2 y n x ( x , y ) d y , n 3 ( x ) = - b / 2 b / 2 y 2 n x ( x , y ) d y . \begin{aligned}\displaystyle q_{1}(x)&\displaystyle=\int_{-b/2}^{b/2}q(x,y)\,% \,\text{d}y~{},~{}~{}q_{2}(x)=\int_{-b/2}^{b/2}y\,q(x,y)\,\,\text{d}y~{},~{}~{% }n_{1}(x)=\int_{-b/2}^{b/2}n_{x}(x,y)\,\,\text{d}y\\ \displaystyle n_{2}(x)&\displaystyle=\int_{-b/2}^{b/2}y\,n_{x}(x,y)\,\,\text{d% }y~{},~{}~{}n_{3}(x)=\int_{-b/2}^{b/2}y^{2}\,n_{x}(x,y)\,\,\text{d}y.\end{aligned}
  116. x = 0 x=0
  117. w ( 0 , y ) = d w d x | x = 0 = 0 w x ( 0 ) = d w x d x | x = 0 = θ x ( 0 ) = d θ x d x | x = 0 = 0 . w(0,y)=\cfrac{dw}{dx}\Bigr|_{x=0}=0\qquad\implies\qquad w_{x}(0)=\cfrac{dw_{x}% }{dx}\Bigr|_{x=0}=\theta_{x}(0)=\cfrac{d\theta_{x}}{dx}\Bigr|_{x=0}=0\,.
  118. x = a x=a
  119. b D d 3 w x d x 3 + n 1 ( x ) d w x d x + n 2 ( x ) d θ x d x + q x 1 = 0 \displaystyle bD\cfrac{d^{3}w_{x}}{dx^{3}}+n_{1}(x)\cfrac{dw_{x}}{dx}+n_{2}(x)% \cfrac{d\theta_{x}}{dx}+q_{x1}=0
  120. m 1 = - b / 2 b / 2 m x ( y ) d y , m 2 = - b / 2 b / 2 y m x ( y ) d y , q x 1 = - b / 2 b / 2 q x ( y ) d y t = q x 2 + m 3 = - b / 2 b / 2 y q x ( y ) d y + - b / 2 b / 2 m x y ( y ) d y . \begin{aligned}\displaystyle m_{1}&\displaystyle=\int_{-b/2}^{b/2}m_{x}(y)\,\,% \text{d}y~{},~{}~{}m_{2}=\int_{-b/2}^{b/2}y\,m_{x}(y)\,\,\text{d}y~{},~{}~{}q_% {x1}=\int_{-b/2}^{b/2}q_{x}(y)\,\,\text{d}y\\ \displaystyle t&\displaystyle=q_{x2}+m_{3}=\int_{-b/2}^{b/2}y\,q_{x}(y)\,\,% \text{d}y+\int_{-b/2}^{b/2}m_{xy}(y)\,\,\text{d}y.\end{aligned}

Plateau_principle.html

  1. C t = C 0 e - k e t C_{t}=C_{0}e^{-k_{e}t}\,
  2. k e = ln 2 t 1 / 2 k_{e}=\frac{\ln 2}{t_{1/2}}\,
  3. C t = C 0 + ( C s s - C 0 ) * ( 1 - e - k e t ) C_{t}=C_{0}+(C_{ss}-C_{0})*(1-e^{-k_{e}t})\,
  4. C s s = k s k e C_{ss}=\frac{k_{s}}{k_{e}}\,

Plessey_Code.html

  1. g ( x ) = x 8 + x 7 + x 6 + x 5 + x 3 + 1 g(x)=x^{8}+x^{7}+x^{6}+x^{5}+x^{3}+1
  2. a b < 1 \frac{a}{b}<1

Plücker's_conoid.html

  1. z = 2 x y x 2 + y 2 . z=\frac{2xy}{x^{2}+y^{2}}.
  2. x = v cos u , y = v sin u , z = sin 2 u . x=v\cos u,\quad y=v\sin u,\quad z=\sin 2u.
  3. x = v cos u , y = v sin u , z = sin n u . x=v\cos u,\quad y=v\sin u,\quad z=\sin nu.

Pochhammer_contour.html

  1. B ( α , β ) = 0 1 t α - 1 ( 1 - t ) β - 1 d t \displaystyle B(\alpha,\beta)=\int_{0}^{1}t^{\alpha-1}(1-t)^{\beta-1}\,dt
  2. ( 1 - e 2 π i α ) ( 1 - e 2 π i β ) B ( α , β ) = C t α - 1 ( 1 - t ) β - 1 d t . \displaystyle(1-e^{2\pi i\alpha})(1-e^{2\pi i\beta})B(\alpha,\beta)=\int_{C}t^% {\alpha-1}(1-t)^{\beta-1}\,dt.

Pocklington's_algorithm.html

  1. x 2 a ( mod p ) , x^{2}\equiv a\;\;(\mathop{{\rm mod}}p),\,
  2. \equiv
  3. ( mod p ) \;\;(\mathop{{\rm mod}}p)
  4. ( mod p ) \;\;(\mathop{{\rm mod}}p)
  5. x 2 a x^{2}\equiv a
  6. x - x x\neq-x
  7. p = 4 m + 3 p=4m+3
  8. m m\in\mathbb{N}
  9. x ± a m + 1 x\equiv\pm a^{m+1}
  10. p = 8 m + 5 p=8m+5
  11. m m\in\mathbb{N}
  12. a 2 m + 1 1 a^{2m+1}\equiv 1
  13. x ± a m + 1 x\equiv\pm a^{m+1}
  14. a 2 m + 1 - 1 a^{2m+1}\equiv-1
  15. 4 2 m + 1 - 1 4^{2m+1}\equiv-1
  16. ( 4 a ) 2 m + 1 1 (4a)^{2m+1}\equiv 1
  17. y ± ( 4 a ) m + 1 y\equiv\pm(4a)^{m+1}
  18. y 2 4 a y^{2}\equiv 4a
  19. x ± y / 2 x\equiv\pm y/2
  20. x ± ( p + y ) / 2 x\equiv\pm(p+y)/2
  21. p = 8 m + 1 p=8m+1
  22. D - a D\equiv-a
  23. x 2 + D 0 x^{2}+D\equiv 0
  24. t 1 t_{1}
  25. u 1 u_{1}
  26. N = t 1 2 - D u 1 2 N=t_{1}^{2}-Du_{1}^{2}
  27. t n = ( t 1 + u 1 D ) n + ( t 1 - u 1 D ) n 2 , u n = ( t 1 + u 1 D ) n - ( t 1 - u 1 D ) n 2 D t_{n}=\frac{(t_{1}+u_{1}\sqrt{D})^{n}+(t_{1}-u_{1}\sqrt{D})^{n}}{2},\qquad u_{% n}=\frac{(t_{1}+u_{1}\sqrt{D})^{n}-(t_{1}-u_{1}\sqrt{D})^{n}}{2\sqrt{D}}
  28. t m + n = t m t n + D u m u n , u m + n = t m u n + t n u m and t n 2 - D u n 2 = N n t_{m+n}=t_{m}t_{n}+Du_{m}u_{n},\quad u_{m+n}=t_{m}u_{n}+t_{n}u_{m}\quad\mbox{% and}~{}\quad t_{n}^{2}-Du_{n}^{2}=N^{n}
  29. 4 m + 1 4m+1
  30. 8 m + 1 8m+1
  31. t p t 1 p t 1 , u p u 1 p D ( p - 1 ) / 2 u 1 t_{p}\equiv t_{1}^{p}\equiv t_{1},\quad u_{p}\equiv u_{1}^{p}D^{(p-1)/2}\equiv u% _{1}
  32. t 1 t p - 1 t 1 + D u p - 1 u 1 and u 1 t p - 1 u 1 + t 1 u p - 1 t_{1}\equiv t_{p-1}t_{1}+Du_{p-1}u_{1}\quad\mbox{and}~{}\quad u_{1}\equiv t_{p% -1}u_{1}+t_{1}u_{p-1}
  33. t p - 1 1 , u p - 1 0 t_{p-1}\equiv 1,\quad u_{p-1}\equiv 0
  34. p - 1 = 2 r p-1=2r
  35. 0 u p - 1 2 t r u r 0\equiv u_{p-1}\equiv 2t_{r}u_{r}
  36. t r t_{r}
  37. u r u_{r}
  38. u r u_{r}
  39. r = 2 s r=2s
  40. 0 2 t s u s 0\equiv 2t_{s}u_{s}
  41. u i u_{i}
  42. u 1 u_{1}
  43. u m 0 u_{m}\equiv 0
  44. t m 2 - D u m 2 N m t_{m}^{2}-Du_{m}^{2}\equiv N^{m}
  45. t m 2 t_{m}^{2}
  46. t l 0 t_{l}\equiv 0
  47. - D u l 2 N l -Du_{l}^{2}\equiv N^{l}
  48. - D -D
  49. l = 2 k l=2k
  50. 0 t l t k 2 + D u k 2 0\equiv t_{l}\equiv t_{k}^{2}+Du_{k}^{2}
  51. x 2 + D 0 x^{2}+D\equiv 0
  52. u k x ± t k u_{k}x\equiv\pm t_{k}
  53. \equiv
  54. x 2 18 ( mod 23 ) . x^{2}\equiv 18\;\;(\mathop{{\rm mod}}23).
  55. 23 = 4 5 + 3 23=4\cdot 5+3
  56. m = 5 m=5
  57. x ± 18 6 ± 8 ( mod 23 ) x\equiv\pm 18^{6}\equiv\pm 8\;\;(\mathop{{\rm mod}}23)
  58. ( ± 8 ) 2 64 18 ( mod 23 ) (\pm 8)^{2}\equiv 64\equiv 18\;\;(\mathop{{\rm mod}}23)
  59. x 2 10 ( mod 13 ) . x^{2}\equiv 10\;\;(\mathop{{\rm mod}}13).
  60. 13 = 8 1 + 5 13=8\cdot 1+5
  61. m = 1 m=1
  62. 10 2 m + 1 10 3 - 1 ( mod 13 ) 10^{2m+1}\equiv 10^{3}\equiv-1\;\;(\mathop{{\rm mod}}13)
  63. x ± y / 2 ± ( 4 a ) 2 / 2 ± 800 ± 7 ( mod 13 ) x\equiv\pm y/2\equiv\pm(4a)^{2}/2\equiv\pm 800\equiv\pm 7\;\;(\mathop{{\rm mod% }}13)
  64. ( ± 7 ) 2 49 10 ( mod 13 ) (\pm 7)^{2}\equiv 49\equiv 10\;\;(\mathop{{\rm mod}}13)
  65. x 2 13 ( mod 17 ) x^{2}\equiv 13\;\;(\mathop{{\rm mod}}17)
  66. x 2 - 13 = 0 x^{2}-13=0
  67. t 1 t_{1}
  68. u 1 u_{1}
  69. t 1 2 + 13 u 1 2 t_{1}^{2}+13u_{1}^{2}
  70. t 1 = 3 , u 1 = 1 t_{1}=3,u_{1}=1
  71. t 8 t_{8}
  72. u 8 u_{8}
  73. t 2 = t 1 t 1 + 13 u 1 u 1 = 9 - 13 = - 4 13 ( mod 17 ) , t_{2}=t_{1}t_{1}+13u_{1}u_{1}=9-13=-4\equiv 13\;\;(\mathop{{\rm mod}}17),\,
  74. u 2 = t 1 u 1 + t 1 u 1 = 3 + 3 6 ( mod 17 ) . u_{2}=t_{1}u_{1}+t_{1}u_{1}=3+3\equiv 6\;\;(\mathop{{\rm mod}}17).\,
  75. t 4 = - 299 7 ( mod 17 ) u 4 = 156 3 ( mod 17 ) t_{4}=-299\equiv 7\;\;(\mathop{{\rm mod}}17)\,u_{4}=156\equiv 3\;\;(\mathop{{% \rm mod}}17)
  76. t 8 = - 68 0 ( mod 17 ) u 8 = 42 8 ( mod 17 ) . t_{8}=-68\equiv 0\;\;(\mathop{{\rm mod}}17)\,u_{8}=42\equiv 8\;\;(\mathop{{\rm mod% }}17).
  77. t 8 = 0 t_{8}=0
  78. 0 t 4 2 + 13 u 4 2 7 2 - 13 3 2 ( mod 17 ) 0\equiv t_{4}^{2}+13u_{4}^{2}\equiv 7^{2}-13\cdot 3^{2}\;\;(\mathop{{\rm mod}}% 17)
  79. 3 x ± 7 ( mod 17 ) 3x\equiv\pm 7\;\;(\mathop{{\rm mod}}17)
  80. x ± 8 ( mod 17 ) x\equiv\pm 8\;\;(\mathop{{\rm mod}}17)
  81. ( ± 8 ) 2 = 64 13 ( mod 17 ) (\pm 8)^{2}=64\equiv 13\;\;(\mathop{{\rm mod}}17)

Pocklington_primality_test.html

  1. N N
  2. N > 1 N>1
  3. q | N - 1 q|N-1
  4. q > N - 1 q>\sqrt{N}-1
  5. a N - 1 1 ( mod N ) a^{N-1}\equiv 1\;\;(\mathop{{\rm mod}}N)
  6. gcd ( a ( N - 1 ) / q - 1 , N ) = 1 \gcd{(a^{(N-1)/q}-1,N)}=1
  7. N N
  8. p N p\leq\sqrt{N}
  9. q > p - 1 q>p-1
  10. gcd ( q , p - 1 ) = 1 \gcd{(q,p-1)}=1
  11. u q 1 ( mod p - 1 ) uq\equiv 1\;\;(\mathop{{\rm mod}}p-1)
  12. 1 a N - 1 ( mod p ) 1\equiv a^{N-1}\;\;(\mathop{{\rm mod}}p)
  13. p | N p|N
  14. ( a N - 1 ) u a u ( N - 1 ) a u q ( ( N - 1 ) / q ) ( a u q ) ( N - 1 ) / q ( mod p ) \equiv(a^{N-1})^{u}\equiv a^{u(N-1)}\equiv a^{uq((N-1)/q)}\equiv(a^{uq})^{(N-1% )/q}\;\;(\mathop{{\rm mod}}p)
  15. a ( N - 1 ) / q ( mod p ) \equiv a^{(N-1)/q}\;\;(\mathop{{\rm mod}}p)
  16. p p
  17. gcd ( ) \gcd()
  18. gcd ( ) \gcd()
  19. 1 1
  20. N 10 , 000 N\leq 10,000
  21. 1 a < N 1\leq a<N
  22. a N - 1 1 ( mod N ) a^{N-1}\equiv 1\;\;(\mathop{{\rm mod}}N)
  23. ( N - 1 ) / q (N-1)/q
  24. A > N A>\sqrt{N}
  25. a p a_{p}
  26. a p N - 1 1 ( mod N ) a^{N-1}_{p}\equiv 1\;\;(\mathop{{\rm mod}}N)
  27. gcd ( a p ( N - 1 ) / p - 1 , N ) = 1 \gcd{(a^{(N-1)/p}_{p}-1,N)}=1
  28. p e p^{e}
  29. a p a_{p}
  30. b a p ( N - 1 ) / p e ( mod v ) b\equiv a^{(N-1)/p^{e}}_{p}\;\;(\mathop{{\rm mod}}v)
  31. b p e a p N - 1 1 ( mod v ) b^{p^{e}}\equiv a^{N-1}_{p}\equiv 1\;\;(\mathop{{\rm mod}}v)
  32. gcd ( a p ( N - 1 ) / p - 1 , N ) = 1 \gcd{(a^{(N-1)/p}_{p}-1,N)}=1
  33. b p e - 1 a p ( N - 1 ) / p 1 ( mod v ) b^{p^{e-1}}\equiv a^{(N-1)/p}_{p}\not\equiv 1\;\;(\mathop{{\rm mod}}v)
  34. b ( mod v ) b\;\;(\mathop{{\rm mod}}v)
  35. p e p^{e}
  36. p e | ( v - 1 ) p^{e}|(v-1)
  37. p e p^{e}
  38. A | ( v - 1 ) A|(v-1)
  39. v > A n . v>A\geq\sqrt{n}.
  40. N \sqrt{N}
  41. a p a_{p}
  42. a p a_{p}
  43. a p a_{p}
  44. a p a_{p}
  45. N = 11351 N=11351
  46. N - 1 = 2 5 2 227 N-1=2\cdot 5^{2}\cdot 227
  47. A = 227 5 2 A=227\cdot 5^{2}
  48. B = 2 B=2
  49. gcd ( A , B ) = 1 \gcd{(A,B)}=1
  50. A > N A>\sqrt{N}
  51. a p a_{p}
  52. a 5 = 2 a_{5}=2
  53. a p N - 1 2 11350 1 ( mod 11351 ) a^{N-1}_{p}\equiv 2^{11350}\equiv 1\;\;(\mathop{{\rm mod}}11351)
  54. gcd ( a p ( N - 1 ) / p - 1 , N ) = gcd ( 2 2 5 227 - 1 , 11351 ) = 1. \gcd{(a^{(N-1)/p}_{p}-1,N)}=\gcd{(2^{2\cdot 5\cdot 227}-1,11351)}=1.
  55. a 5 = 2 a_{5}=2
  56. a 227 = 7 a_{227}=7
  57. a p N - 1 7 11350 1 ( mod 11351 ) a^{N-1}_{p}\equiv 7^{11350}\equiv 1\;\;(\mathop{{\rm mod}}11351)
  58. gcd ( a p ( N - 1 ) / p - 1 , N ) = gcd ( 7 2 25 - 1 , 11351 ) = 1. \gcd{(a^{(N-1)/p}_{p}-1,N)}=\gcd(7^{2\cdot 25}-1,11351)=1.
  59. a p a_{p}
  60. a p a_{p}
  61. a p a_{p}
  62. a p a_{p}
  63. a p a_{p}
  64. a p a_{p}
  65. a p a_{p}
  66. N - 1 = m p N-1=mp
  67. 2 p + 1 N 2p+1\geq\sqrt{N}
  68. a ( N - 1 ) / 2 - 1 ( mod N ) a^{(N-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}N)
  69. a m / 2 - 1 ( mod N ) a^{m/2}\not\equiv-1\;\;(\mathop{{\rm mod}}N)
  70. ( N / 2 ) 1 / 3 (N/2)^{1/3}

Poincaré_plot.html

  1. x t , x t + 1 , x t + 2 , , x_{t},x_{t+1},x_{t+2},\ldots,\,

Poincaré_series_(modular_form).html

  1. γ Γ H ( γ ( z ) ) . \sum_{\gamma\in\Gamma}H(\gamma(z)).
  2. θ k ( z ) = γ Γ * ( J γ ( z ) ) k H ( γ ( z ) ) \theta_{k}(z)=\sum_{\gamma\in\Gamma^{*}}(J_{\gamma}(z))^{k}H(\gamma(z))
  3. θ k ( z ) = γ Γ * ( c z + d ) - 2 k H ( a z + b c z + d ) \theta_{k}(z)=\sum_{\gamma\in\Gamma^{*}}(cz+d)^{-2k}H\left(\frac{az+b}{cz+d}\right)
  4. γ = ( a b c d ) \gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  5. θ k , n ( z ) = γ Γ * ( c z + d ) - 2 k exp ( 2 π i n a z + b c z + d ) \theta_{k,n}(z)=\sum_{\gamma\in\Gamma^{*}}(cz+d)^{-2k}\exp\left(2\pi in\frac{% az+b}{cz+d}\right)

Poincaré–Steklov_operator.html

  1. u u
  2. Ω R n \Omega\subset R^{n}
  3. u u
  4. Ω \Omega
  5. u / n \partial u/\partial n
  6. Ω \Omega
  7. e i ω t e^{i\omega t}

Poisson_distribution.html

  1. f ( k ; λ ) = Pr ( X = k ) = λ k e - λ k ! , \!f(k;\lambda)=\Pr(X{=}k)=\frac{\lambda^{k}e^{-\lambda}}{k!},
  2. λ = E ( X ) = Var ( X ) . \lambda=\operatorname{E}(X)=\operatorname{Var}(X).
  3. λ - 1 / 2 \textstyle\lambda^{-1/2}
  4. E | X - λ | = 2 exp ( - λ ) λ λ + 1 λ ! . \operatorname{E}|X-\lambda|=2\exp(-\lambda)\frac{\lambda^{\lfloor\lambda% \rfloor+1}}{\lfloor\lambda\rfloor!}.
  5. λ \scriptstyle\lfloor\lambda\rfloor
  6. λ - ln 2 ν < λ + 1 3 . \lambda-\ln 2\leq\nu<\lambda+\frac{1}{3}.
  7. m k = i = 1 k λ i { k i } , m_{k}=\sum_{i=1}^{k}\lambda^{i}\left\{\begin{matrix}k\\ i\end{matrix}\right\},
  8. X i Pois ( λ i ) i = 1 , , n X_{i}\sim\mathrm{Pois}(\lambda_{i})\,i=1,\dots,n
  9. λ = i = 1 n λ i \lambda=\sum_{i=1}^{n}\lambda_{i}
  10. Y = ( i = 1 n X i ) Pois ( λ ) Y=\left(\sum_{i=1}^{n}X_{i}\right)\sim\mathrm{Pois}(\lambda)
  11. D KL ( λ λ 0 ) = λ 0 - λ + λ log λ λ 0 . D_{\mathrm{KL}}(\lambda\|\lambda_{0})=\lambda_{0}-\lambda+\lambda\log\frac{% \lambda}{\lambda_{0}}.
  12. X Pois ( λ ) X\sim\,\text{Pois}(\lambda)
  13. P ( X x ) e - λ ( e λ ) x x x , for x > λ , P(X\geq x)\leq\frac{e^{-\lambda}(e\lambda)^{x}}{x^{x}},\,\text{ for }x>\lambda,
  14. P ( X x ) e - λ ( e λ ) x x x , for x < λ . P(X\leq x)\leq\frac{e^{-\lambda}(e\lambda)^{x}}{x^{x}},\,\text{ for }x<\lambda.
  15. { ( k + 1 ) f ( k + 1 ) - λ f ( k ) = 0 , f ( 0 ) = e - λ } \left\{(k+1)f(k+1)-\lambda f(k)=0,f(0)=e^{-\lambda}\right\}
  16. X P o i ( λ ) X\sim Poi(\lambda)
  17. Y P o i ( μ ) Y\sim Poi(\mu)
  18. λ < μ \lambda<\mu
  19. e - ( μ - λ ) 2 ( λ + μ ) 2 - e - ( λ + μ ) 2 λ μ - e - ( λ + μ ) 4 λ μ P ( X - Y 0 ) e - ( μ - λ ) 2 \frac{e^{-(\sqrt{\mu}-\sqrt{\lambda})^{2}}}{(\lambda+\mu)^{2}}-\frac{e^{-(% \lambda+\mu)}}{2\sqrt{\lambda\mu}}-\frac{e^{-(\lambda+\mu)}}{4\lambda\mu}\leq P% (X-Y\geq 0)\leq e^{-(\sqrt{\mu}-\sqrt{\lambda})^{2}}
  20. P ( X - Y 0 | X + Y = i ) P(X-Y\geq 0|X+Y=i)
  21. Z i 2 Z\geq\frac{i}{2}
  22. Z B i n ( i , λ λ + μ ) Z\sim Bin\left(i,\frac{\lambda}{\lambda+\mu}\right)
  23. 1 ( i + 1 ) 2 e ( 2 D ( 0.5 | | λ λ + μ ) ) \frac{1}{(i+1)^{2}}e^{\left(2D\left(0.5||\frac{\lambda}{\lambda+\mu}\right)% \right)}
  24. D D
  25. X + Y P o i ( λ + μ ) X+Y\sim Poi(\lambda+\mu)
  26. X 1 Pois ( λ 1 ) X_{1}\sim\mathrm{Pois}(\lambda_{1})\,
  27. X 2 Pois ( λ 2 ) X_{2}\sim\mathrm{Pois}(\lambda_{2})\,
  28. Y = X 1 - X 2 Y=X_{1}-X_{2}
  29. X 1 Pois ( λ 1 ) X_{1}\sim\mathrm{Pois}(\lambda_{1})\,
  30. X 2 Pois ( λ 2 ) X_{2}\sim\mathrm{Pois}(\lambda_{2})\,
  31. X 1 X_{1}
  32. X 1 + X 2 X_{1}+X_{2}
  33. X 1 + X 2 = k X_{1}+X_{2}=k
  34. X 1 Binom ( k , λ 1 / ( λ 1 + λ 2 ) ) \!X_{1}\sim\mathrm{Binom}(k,\lambda_{1}/(\lambda_{1}+\lambda_{2}))
  35. j = 1 n X j = k , \sum_{j=1}^{n}X_{j}=k,
  36. X i Binom ( k , λ i j = 1 n λ j ) X_{i}\sim\mathrm{Binom}\left(k,\frac{\lambda_{i}}{\sum_{j=1}^{n}\lambda_{j}}\right)
  37. { X i } Multinom ( k , { λ i j = 1 n λ j } ) \{X_{i}\}\sim\mathrm{Multinom}\left(k,\left\{\frac{\lambda_{i}}{\sum_{j=1}^{n}% \lambda_{j}}\right\}\right)
  38. X Pois ( λ ) X\sim\mathrm{Pois}(\lambda)\,
  39. Y Y
  40. Y ( X = k ) Binom ( k , p ) Y\mid(X=k)\sim\mathrm{Binom}(k,p)
  41. Y Pois ( λ p ) Y\sim\mathrm{Pois}(\lambda\cdot p)\,
  42. { Y i } \{Y_{i}\}
  43. { Y i } ( X = k ) Multinom ( k , p i ) \{Y_{i}\}\mid(X=k)\sim\mathrm{Multinom}\left(k,p_{i}\right)
  44. Y i Y_{i}
  45. Y i Pois ( λ p i ) , ρ ( Y i , Y j ) = 0 Y_{i}\sim\mathrm{Pois}(\lambda\cdot p_{i}),\rho(Y_{i},Y_{j})=0
  46. F Binomial ( k ; n , p ) F Poisson ( k ; λ = n p ) F_{\mathrm{Binomial}}(k;n,p)\approx F_{\mathrm{Poisson}}(k;\lambda=np)\,
  47. λ \sqrt{\lambda}
  48. F Poisson ( x ; λ ) F normal ( x ; μ = λ , σ 2 = λ ) F_{\mathrm{Poisson}}(x;\lambda)\approx F_{\mathrm{normal}}(x;\mu=\lambda,% \sigma^{2}=\lambda)\,
  49. λ \sqrt{\lambda}
  50. F Poisson ( k ; λ ) = 1 - F χ 2 ( 2 λ ; 2 ( k + 1 ) ) integer k , F\text{Poisson}(k;\lambda)=1-F_{\chi^{2}}(2\lambda;2(k+1))\quad\quad\,\text{ % integer }k,
  51. Pr ( X = k ) = F χ 2 ( 2 λ ; 2 ( k + 1 ) ) - F χ 2 ( 2 λ ; 2 k ) . \Pr(X=k)=F_{\chi^{2}}(2\lambda;2(k+1))-F_{\chi^{2}}(2\lambda;2k).
  52. λ \lambda
  53. n n
  54. I 1 , , I n I_{1},\dots,I_{n}
  55. n n
  56. λ \lambda
  57. I k I_{k}
  58. k k
  59. λ / n \lambda/n
  60. i t h i^{th}
  61. I i I_{i}
  62. λ / n \lambda/n
  63. n n
  64. λ \lambda
  65. B ( n , λ / n ) \textrm{B}(n,\lambda/n)
  66. n n
  67. X B ( n , p ) . X\sim\textrm{B}(n,p).\,
  68. X Pois ( n p ) . X\sim\textrm{Pois}(np).\,
  69. P ( N ( D ) = k ) = ( λ | D | ) k e - λ | D | k ! . P(N(D)=k)=\frac{(\lambda|D|)^{k}e^{-\lambda|D|}}{k!}.
  70. σ k = λ \sigma_{k}=\sqrt{\lambda}
  71. I = e N / t I=eN/t
  72. σ I = e N / t \sigma_{I}=e\sqrt{N}/t
  73. e e
  74. t σ I 2 / I t\sigma_{I}^{2}/I
  75. Pr ( N t = k ) = f ( k ; λ t ) = e - λ t ( λ t ) k k ! . \Pr(N_{t}=k)=f(k;\lambda t)=\frac{e^{-\lambda t}(\lambda t)^{k}}{k!}.
  76. λ ^ MLE = 1 n i = 1 n k i . \widehat{\lambda}_{\mathrm{MLE}}=\frac{1}{n}\sum_{i=1}^{n}k_{i}.\!
  77. 𝐱 \mathbf{x}
  78. h ( 𝐱 ) h(\mathbf{x})
  79. λ \lambda
  80. 𝐱 \mathbf{x}
  81. T ( 𝐱 ) T(\mathbf{x})
  82. T ( 𝐱 ) T(\mathbf{x})
  83. λ \lambda
  84. P ( 𝐱 ) = i = 1 n λ x e - λ x ! = 1 i = 1 n x i ! × λ i = 1 n x i e - n λ P(\mathbf{x})=\prod_{i=1}^{n}\frac{\lambda^{x}e^{-\lambda}}{x!}=\frac{1}{\prod% _{i=1}^{n}x_{i}!}\times\lambda^{\sum_{i=1}^{n}x_{i}}e^{-n\lambda}
  85. h ( 𝐱 ) h(\mathbf{x})
  86. 𝐱 \mathbf{x}
  87. g ( T ( 𝐱 ) | λ ) g(T(\mathbf{x})|\lambda)
  88. T ( 𝐱 ) = i = 1 n x i T(\mathbf{x})=\sum_{i=1}^{n}x_{i}
  89. T ( 𝐱 ) T(\mathbf{x})
  90. L ( λ ) \displaystyle L(\lambda)
  91. d d λ L ( λ ) = 0 - n + ( i = 1 n k i ) 1 λ = 0. \frac{\mathrm{d}}{\mathrm{d}\lambda}L(\lambda)=0\iff-n+\left(\sum_{i=1}^{n}k_{% i}\right)\frac{1}{\lambda}=0.\!
  92. λ = i = 1 n k i n \lambda=\frac{\sum_{i=1}^{n}k_{i}}{n}
  93. 2 L λ 2 = - λ - 2 i = 1 n k i \frac{\partial^{2}L}{\partial\lambda^{2}}=-\lambda^{-2}\sum_{i=1}^{n}k_{i}
  94. 2 L λ 2 = - n 2 i = 1 n k i \frac{\partial^{2}L}{\partial\lambda^{2}}=-\frac{n^{2}}{\sum_{i=1}^{n}k_{i}}
  95. E ( g ( T ) ) = 0 E(g(T))=0
  96. P λ ( g ( T ) = 0 ) = 1 P_{\lambda}(g(T)=0)=1
  97. λ \lambda
  98. X i X_{i}
  99. Po ( λ ) \mathrm{Po}(\lambda)
  100. T ( 𝐱 ) = i = 1 n X i Po ( n λ ) T(\mathbf{x})=\sum_{i=1}^{n}X_{i}\sim\mathrm{Po}(n\lambda)
  101. E ( g ( T ) ) = t = 0 g ( t ) ( n λ ) t e - n λ t ! = 0 E(g(T))=\sum_{t=0}^{\infty}g(t)\frac{(n\lambda)^{t}e^{-n\lambda}}{t!}=0
  102. g ( t ) g(t)
  103. t t
  104. λ \lambda
  105. E ( g ( T ) ) = 0 E(g(T))=0
  106. λ \lambda
  107. P λ ( g ( T ) = 0 ) = 1 P_{\lambda}(g(T)=0)=1
  108. 1 α 1–α
  109. 1 2 χ 2 ( α / 2 ; 2 k ) μ 1 2 χ 2 ( 1 - α / 2 ; 2 k + 2 ) , \tfrac{1}{2}\chi^{2}(\alpha/2;2k)\leq\mu\leq\tfrac{1}{2}\chi^{2}(1-\alpha/2;2k% +2),
  110. F - 1 ( α / 2 ; k , 1 ) μ F - 1 ( 1 - α / 2 ; k + 1 , 1 ) , F^{-1}(\alpha/2;k,1)\leq\mu\leq F^{-1}(1-\alpha/2;k+1,1),
  111. χ 2 ( p ; n ) \chi^{2}(p;n)
  112. F - 1 ( p ; n , 1 ) F^{-1}(p;n,1)
  113. 1 α 1–α
  114. k ( 1 - 1 9 k - z α / 2 3 k ) 3 μ ( k + 1 ) ( 1 - 1 9 ( k + 1 ) + z α / 2 3 k + 1 ) 3 , k\left(1-\frac{1}{9k}-\frac{z_{\alpha/2}}{3\sqrt{k}}\right)^{3}\leq\mu\leq(k+1% )\left(1-\frac{1}{9(k+1)}+\frac{z_{\alpha/2}}{3\sqrt{k+1}}\right)^{3},
  115. z α / 2 z_{\alpha/2}
  116. α / 2 α/2
  117. k = i = 1 n k i , k=\sum_{i=1}^{n}k_{i},\!
  118. λ Gamma ( α , β ) \lambda\sim\mathrm{Gamma}(\alpha,\beta)\!
  119. g ( λ α , β ) = β α Γ ( α ) λ α - 1 e - β λ for λ > 0 . g(\lambda\mid\alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\;\lambda^{% \alpha-1}\;e^{-\beta\,\lambda}\qquad\,\text{ for }\lambda>0\,\!.
  120. λ Gamma ( α + i = 1 n k i , β + n ) . \lambda\sim\mathrm{Gamma}\left(\alpha+\sum_{i=1}^{n}k_{i},\beta+n\right).\!
  121. λ ^ MLE \widehat{\lambda}_{\mathrm{MLE}}
  122. α 0 , β 0 \alpha\to 0,\ \beta\to 0
  123. X 1 , X 2 , , X p X_{1},X_{2},\dots,X_{p}
  124. p p
  125. λ i \lambda_{i}
  126. i = 1 , , p i=1,\dots,p
  127. L ( λ , λ ^ ) = i = 1 p λ i - 1 ( λ ^ i - λ i ) 2 L(\lambda,{\hat{\lambda}})=\sum_{i=1}^{p}\lambda_{i}^{-1}({\hat{\lambda}}_{i}-% \lambda_{i})^{2}
  128. p > 1 p>1
  129. λ ^ i = X i {\hat{\lambda}}_{i}=X_{i}
  130. 0 < c 2 ( p - 1 ) 0<c\leq 2(p-1)
  131. b ( p - 2 + p - 1 ) b\geq(p-2+p^{-1})
  132. λ ^ i = ( 1 - c b + i = 1 p X i ) X i , i = 1 , , p . {\hat{\lambda}}_{i}=\left(1-\frac{c}{b+\sum_{i=1}^{p}X_{i}}\right)X_{i},\qquad i% =1,\dots,p.
  133. g ( u , v ) = exp [ ( θ 1 - θ 12 ) ( u - 1 ) + ( θ 2 - θ 12 ) ( v - 1 ) + θ 12 ( u v - 1 ) ] g(u,v)=\exp[(\theta_{1}-\theta_{12})(u-1)+(\theta_{2}-\theta_{12})(v-1)+\theta% _{12}(uv-1)]
  134. θ 1 , θ 2 > θ 12 > 0 \theta_{1},\theta_{2}>\theta_{12}>0\,
  135. 0 ρ min { θ 1 θ 2 , θ 2 θ 1 } 0\leq\rho\leq\min\left\{\frac{\theta_{1}}{\theta_{2}},\frac{\theta_{2}}{\theta% _{1}}\right\}
  136. X 1 , X 2 X_{1},X_{2}
  137. Y 1 , Y 2 , Y 3 Y_{1},Y_{2},Y_{3}
  138. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  139. X 1 = Y 1 + Y 3 , X 2 = Y 2 + Y 3 X_{1}=Y_{1}+Y_{3},X_{2}=Y_{2}+Y_{3}
  140. Pr ( X 1 = k 1 , X 2 = k 2 ) = exp ( - λ 1 - λ 2 - λ 3 ) λ 1 k 1 k 1 ! λ 2 k 2 k 2 ! k = 0 min ( k 1 , k 2 ) ( k 1 k ) ( k 2 k ) k ! ( λ 3 λ 1 λ 2 ) k \begin{aligned}&\displaystyle\Pr(X_{1}=k_{1},X_{2}=k_{2})\\ \displaystyle=&\displaystyle\exp\left(-\lambda_{1}-\lambda_{2}-\lambda_{3}% \right)\frac{\lambda_{1}^{k_{1}}}{k_{1}!}\frac{\lambda_{2}^{k_{2}}}{k_{2}!}% \sum_{k=0}^{\min(k_{1},k_{2})}{\left({{k_{1}}\atop{k}}\right)}{\left({{k_{2}}% \atop{k}}\right)}k!\left(\frac{\lambda_{3}}{\lambda_{1}\lambda_{2}}\right)^{k}% \end{aligned}

Poisson_games.html

  1. ( n , T , r , C , u ) (n,T,r,C,u)
  2. n n
  3. T T
  4. r r
  5. T T
  6. C C
  7. u u
  8. N N
  9. P ( N = k ) = e - n n k k ! P(N=k)=e^{-n}\frac{n^{k}}{k!}
  10. n n
  11. t t
  12. n r ( t ) nr(t)
  13. c c
  14. ...
  15. lim n P P \lim_{n\to\infty}\frac{P}{P}
  16. 2 ( x y - x + y 2 ) 2(\sqrt{xy}-\frac{x+y}{2})

Polar_hypersurface.html

  1. f ( x 0 , x 1 , x 2 , ) = 0 f(x_{0},x_{1},x_{2},\dots)=0\,
  2. a = ( a 0 : a 1 : a 2 : ) , a=(a_{0}:a_{1}:a_{2}:\dots),
  3. a 0 f 0 + a 1 f 1 + a 2 f 2 + = 0 , a_{0}f_{0}+a_{1}f_{1}+a_{2}f_{2}+\cdots=0,\,

Polarized_light_microscopy.html

  1. o . p . d . = Δ n t {o.p.d.}=\Delta\,n\cdot t
  2. δ = 2 π ( Δ n t / λ ) \delta\,=2\pi\,(\Delta\,n\cdot t/\lambda\,)
  3. λ / 2 \lambda\,/2
  4. π \pi
  5. n λ {n}\cdot\lambda\,
  6. 2 n π 2{n}\cdot\pi

Polder_tensor.html

  1. B = [ μ j κ 0 - j κ μ 0 0 0 μ 0 ] H B=\begin{bmatrix}\mu&j\kappa&0\\ -j\kappa&\mu&0\\ 0&0&\mu_{0}\end{bmatrix}H
  2. μ = μ 0 ( 1 + ω 0 ω m ω 0 2 - ω 2 ) \mu=\mu_{0}\left(1+\frac{\omega_{0}\omega_{m}}{\omega_{0}^{2}-\omega^{2}}\right)
  3. κ = μ 0 ω ω m ω 0 2 - ω 2 \kappa=\mu_{0}\frac{\omega\omega_{m}}{{\omega_{0}}^{2}-\omega^{2}}
  4. ω 0 = γ μ 0 H 0 \omega_{0}=\gamma\mu_{0}H_{0}
  5. ω m = γ μ 0 M \omega_{m}=\gamma\mu_{0}M
  6. γ = 17.6 g \gamma=17.6\cdot g
  7. ω = 2 π f \omega=2\pi f
  8. μ 0 \mu_{0}

Polynomial-time_algorithm_for_approximating_the_volume_of_convex_bodies.html

  1. ϵ \epsilon
  2. K K
  3. n n
  4. n n
  5. K K
  6. 1 / ϵ 1/\epsilon
  7. K K
  8. n n

Polynomial_(hyperelastic_model).html

  1. I 1 , I 2 I_{1},I_{2}
  2. W = i , j = 0 n C i j ( I 1 - 3 ) i ( I 2 - 3 ) j W=\sum_{i,j=0}^{n}C_{ij}(I_{1}-3)^{i}(I_{2}-3)^{j}
  3. C i j C_{ij}
  4. C 00 = 0 C_{00}=0
  5. W = i , j = 0 n C i j ( I ¯ 1 - 3 ) i ( I ¯ 2 - 3 ) j + k = 1 m D k ( J - 1 ) 2 k W=\sum_{i,j=0}^{n}C_{ij}(\bar{I}_{1}-3)^{i}(\bar{I}_{2}-3)^{j}+\sum_{k=1}^{m}D% _{k}(J-1)^{2k}
  6. I ¯ 1 = J - 2 / 3 I 1 ; I 1 = λ 1 2 + λ 2 2 + λ 3 2 ; J = det ( s y m b o l F ) I ¯ 2 = J - 4 / 3 I 2 ; I 2 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 \begin{aligned}\displaystyle\bar{I}_{1}&\displaystyle=J^{-2/3}~{}I_{1}~{};~{}~% {}I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}~{};~{}~{}J=\det(symbol% {F})\\ \displaystyle\bar{I}_{2}&\displaystyle=J^{-4/3}~{}I_{2}~{};~{}~{}I_{2}=\lambda% _{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_% {1}^{2}\end{aligned}
  7. C 01 = C 11 = 0 C_{01}=C{11}=0
  8. n = 1 , C 01 = C 2 , C 11 = 0 , C 10 = C 1 , m = 1 n=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},m=1
  9. W = C 01 ( I ¯ 2 - 3 ) + C 10 ( I ¯ 1 - 3 ) + D 1 ( J - 1 ) 2 W=C_{01}~{}(\bar{I}_{2}-3)+C_{10}~{}(\bar{I}_{1}-3)+D_{1}~{}(J-1)^{2}

Polynomial_identity_ring.html

  1. P ( r 1 , r 2 , , r N ) = 0. P(r_{1},r_{2},\ldots,r_{N})=0.
  2. P ( X 1 , X 2 ) = X 1 X 2 - X 2 X 1 = 0 P(X_{1},X_{2})=X_{1}X_{2}-X_{2}X_{1}=0~{}
  3. ( x y - y x ) 2 z = z ( x y - y x ) 2 (xy-yx)^{2}z=z(xy-yx)^{2}
  4. det ( A ) = σ S N sgn ( σ ) i = 1 N a i , σ ( i ) \det(A)=\sum_{\sigma\in S_{N}}\operatorname{sgn}(\sigma)\prod_{i=1}^{N}a_{i,% \sigma(i)}
  5. s N ( X 1 , , X N ) = σ S N sgn ( σ ) X σ ( 1 ) X σ ( N ) = 0 s_{N}(X_{1},\ldots,X_{N})=\sum_{\sigma\in S_{N}}\operatorname{sgn}(\sigma)X_{% \sigma(1)}\cdots X_{\sigma(N)}=0~{}
  6. R S R\otimes_{\mathbb{Z}}S
  7. p p
  8. P K P\cap K
  9. P Q P\subset Q
  10. P K Q K P\cap K\subset Q\cap K
  11. τ \tau
  12. \rightarrow
  13. f ( I ) I f(I)\subset I

Polynomial_regression.html

  1. y = a 0 + a 1 x + ε , y=a_{0}+a_{1}x+\varepsilon,\,
  2. y = a 0 + a 1 x + a 2 x 2 + ε . y=a_{0}+a_{1}x+a_{2}x^{2}+\varepsilon.\,
  3. y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + + a n x n + ε . y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots+a_{n}x^{n}+\varepsilon.\,
  4. y i = a 0 + a 1 x i + a 2 x i 2 + + a m x i m + ε i ( i = 1 , 2 , , n ) y_{i}\,=\,a_{0}+a_{1}x_{i}+a_{2}x_{i}^{2}+\cdots+a_{m}x_{i}^{m}+\varepsilon_{i% }\ (i=1,2,\dots,n)
  5. 𝐗 \scriptstyle\mathbf{X}
  6. y \scriptstyle\vec{y}
  7. a \scriptstyle\vec{a}
  8. ε \scriptstyle\vec{\varepsilon}
  9. 𝐗 \scriptstyle\mathbf{X}
  10. y \scriptstyle\vec{y}
  11. [ y 1 y 2 y 3 y n ] = [ 1 x 1 x 1 2 x 1 m 1 x 2 x 2 2 x 2 m 1 x 3 x 3 2 x 3 m 1 x n x n 2 x n m ] [ a 0 a 1 a 2 a m ] + [ ε 1 ε 2 ε 3 ε n ] \begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\\ \vdots\\ y_{n}\end{bmatrix}=\begin{bmatrix}1&x_{1}&x_{1}^{2}&\dots&x_{1}^{m}\\ 1&x_{2}&x_{2}^{2}&\dots&x_{2}^{m}\\ 1&x_{3}&x_{3}^{2}&\dots&x_{3}^{m}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&x_{n}&x_{n}^{2}&\dots&x_{n}^{m}\end{bmatrix}\begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ \vdots\\ a_{m}\end{bmatrix}+\begin{bmatrix}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \vdots\\ \varepsilon_{n}\end{bmatrix}
  12. y = 𝐗 a + ε . \vec{y}=\mathbf{X}\vec{a}+\vec{\varepsilon}.\,
  13. a ^ = ( 𝐗 T 𝐗 ) - 1 𝐗 T y . \widehat{\vec{a}}=(\mathbf{X}^{T}\mathbf{X})^{-1}\;\mathbf{X}^{T}\vec{y}.\,
  14. 𝐗 \scriptstyle\mathbf{X}
  15. 𝐗 \scriptstyle\mathbf{X}
  16. x R d x x\in R^{d_{x}}
  17. ϕ ( x ) R d ϕ \phi(x)\in R^{d_{\phi}}
  18. [ 1 , x ] ϕ [ 1 , x , x 2 , , x d ] [1,x]\stackrel{\phi}{\rightarrow}[1,x,x^{2},...,x^{d}]

Polyphase_sequence.html

  1. a n = e i 2 π q x n a_{n}=e^{i\frac{2\pi}{q}x_{n}}\,

Pompeiu_derivative.html

  1. x 3 \sqrt[3]{x}
  2. x . x.
  3. { q j } j 𝒩 \{q_{j}\}_{j\in\mathcal{N}}
  4. [ 0 , 1 ] . [0,\,1].
  5. { a j } j 𝒩 \{a_{j}\}_{j\in\mathcal{N}}
  6. j a j < . \textstyle\sum_{j}a_{j}<\infty.
  7. x [ 0 , 1 ] x\in[0,\,1]
  8. g ( x ) := j = 0 a j x - q j 3 . g(x):=\sum_{j=0}^{\infty}\,a_{j}\sqrt[3]{x-q_{j}}.
  9. x [ 0 , 1 ] x\in[0,\,1]
  10. g ( x ) := 1 3 j = 0 a j ( x - q j ) 2 3 > 0 , g^{\prime}(x):=\frac{1}{3}\sum_{j=0}^{\infty}\frac{a_{j}}{\sqrt[3]{(x-q_{j})^{% 2}}}>0,
  11. q j , q_{j},
  12. g ( x ) := + . \textstyle g^{\prime}(x):=+\infty.
  13. g g
  14. 0 = g ( 0 ) , 0=g(0),
  15. [ 0 , 1 ] [0,\,1]
  16. f := g - 1 f\,:=g^{-1}
  17. { g ( q j ) } j 𝒩 . \{g(q_{j})\}_{j\in\mathcal{N}}.
  18. [ 0 , 1 ] [0,\,1]

Popoviciu's_inequality.html

  1. I I\subseteq\mathbb{R}
  2. \mathbb{R}
  3. f ( x ) + f ( y ) + f ( z ) 3 + f ( x + y + z 3 ) 2 3 [ f ( x + y 2 ) + f ( y + z 2 ) + f ( z + x 2 ) ] . \frac{f(x)+f(y)+f(z)}{3}+f\left(\frac{x+y+z}{3}\right)\geq\frac{2}{3}\left[f% \left(\frac{x+y}{2}\right)+f\left(\frac{y+z}{2}\right)+f\left(\frac{z+x}{2}% \right)\right].
  4. I I
  5. I I\subseteq\mathbb{R}
  6. \mathbb{R}
  7. 2 k n - 1 2\leq k\leq n-1
  8. x 1 , , x n x_{1},\dots,x_{n}
  9. 1 k ( n - 2 k - 2 ) ( n - k k - 1 i = 1 n f ( x i ) + n f ( 1 n i = 1 n x i ) ) 1 i 1 < < i k n f ( 1 k j = 1 k x i j ) \frac{1}{k}{\left({{n-2}\atop{k-2}}\right)}\left(\frac{n-k}{k-1}\sum_{i=1}^{n}% f(x_{i})+nf\left(\frac{1}{n}\sum_{i=1}^{n}x_{i}\right)\right)\geq\sum_{1\leq i% _{1}<\dots<i_{k}\leq n}f\left(\frac{1}{k}\sum_{j=1}^{k}x_{i_{j}}\right)

Popoviciu's_inequality_on_variances.html

  1. variance 1 4 ( M - m ) 2 . \,\text{variance}\leq\frac{1}{4}(M-m)^{2}.
  2. variance + ( T h i r d c e n t r a l m o m e n t 2 variance ) 2 1 4 ( M - m ) 2 . {\text{variance}+(\frac{\text{}}{Thirdcentralmoment}\,\text{2 variance})^{2}}% \leq\frac{1}{4}(M-m)^{2}.

Population_equivalent.html

  1. P E = B O D l o a d f r o m i n d u s t r y [ k g d a y ] 0.054 [ k g i n h a b d a y ] PE=\dfrac{BOD\ load\ from\ industry\ \left[\dfrac{kg}{day}\right]}{0.054\ % \left[\dfrac{kg}{inhab\cdot day}\right]}

Pore_pressure_gradient.html

  1. 1 psi ft × 1 ft 12 in × 1 lb / in 2 1 psi × 231 in 3 1 US Gal = 19.25000000 lb / gal \mathrm{\frac{1\;psi}{ft}\times\frac{1\;ft}{12\;in}\times\frac{1\;lb/in^{2}}{1% \;psi}\times\frac{231\;in^{3}}{1\;US\;Gal}=19.25000000\;lb/gal}

Portal:Featured_content::Lists::List_of_New_Jersey_Devils_head_coaches.html

  1. W i n s + 1 2 T i e s G a m e s \frac{Wins+\frac{1}{2}Ties}{Games}

Positional_game.html

  1. ( X , ) (X,\mathcal{F})
  2. X X
  3. \mathcal{F}
  4. X X
  5. X X
  6. F F

Positive-real_function.html

  1. [ Z ( s ) ] > 0 if ( s ) > 0 [ Z ( s ) ] = 0 if ( s ) = 0 \begin{aligned}&\displaystyle\Re[Z(s)]>0\quad\,\text{if}\quad\Re(s)>0\\ &\displaystyle\Im[Z(s)]=0\quad\,\text{if}\quad\Im(s)=0\end{aligned}
  2. s = σ + i ω s=\sigma+i\omega\,\!
  3. [ Z ( s ) ] > 0 if σ > 0 [ Z ( s ) ] = 0 if ω = 0 \begin{aligned}&\displaystyle\Re[Z(s)]>0\quad\,\text{if}\quad\sigma>0\\ &\displaystyle\Im[Z(s)]=0\quad\,\text{if}\quad\omega=0\end{aligned}

Post-quantum_cryptography.html

  1. mod ( 2 10 ) \bmod{\left(2^{10}\right)}
  2. 𝔽 31 \mathbb{F}_{31}

Potential_flow_around_a_circular_cylinder.html

  1. R R
  2. V \vec{V}
  3. p p
  4. V = U i ^ + 0 j ^ , \vec{V}=U\widehat{i}+0\widehat{j},
  5. U U
  6. V n ^ = 0 , \vec{V}\cdot\widehat{n}=0,
  7. n ^ \widehat{n}
  8. ρ \rho
  9. × V = 0 \nabla\times\vec{V}=0
  10. ϕ \phi
  11. V = ϕ . \vec{V}=\nabla\phi.
  12. V = 0 \nabla\cdot\vec{V}=0
  13. ϕ \phi
  14. 2 ϕ = 0. \nabla^{2}\phi=0.
  15. ϕ \phi
  16. r r
  17. θ \theta
  18. x = r cos θ x=r\cos\theta
  19. y = r sin θ y=r\sin\theta
  20. 2 ϕ r 2 + 1 r ϕ r + 1 r 2 2 ϕ θ 2 = 0. \frac{\partial^{2}\phi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\phi}{% \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}\phi}{\partial\theta^{2}}=0.
  21. ϕ ( r , θ ) = U ( r + R 2 r ) cos θ . \phi(r,\theta)=U\left(r+\frac{R^{2}}{r}\right)\cos\theta.
  22. ϕ \nabla\phi
  23. V r = ϕ r = U ( 1 - R 2 r 2 ) cos θ V_{r}=\frac{\partial\phi}{\partial r}=U\left(1-\frac{R^{2}}{r^{2}}\right)\cos\theta
  24. V θ = 1 r ϕ θ = - U ( 1 + R 2 r 2 ) sin θ . V_{\theta}=\frac{1}{r}\frac{\partial\phi}{\partial\theta}=-U\left(1+\frac{R^{2% }}{r^{2}}\right)\sin\theta.
  25. p = 1 2 ρ ( U 2 - V 2 ) + p , p=\frac{1}{2}\rho\left(U^{2}-V^{2}\right)+p_{\infty},
  26. U U
  27. p p_{\infty}
  28. p p p\rightarrow p_{\infty}
  29. V = U V=U
  30. V 2 = V r 2 + V θ 2 , V^{2}=V_{r}^{2}+V_{\theta}^{2},
  31. p = 1 2 ρ U 2 ( 2 R 2 r 2 cos ( 2 θ ) - R 4 r 4 ) + p . p=\frac{1}{2}\rho U^{2}\left(2\frac{R^{2}}{r^{2}}\cos(2\theta)-\frac{R^{4}}{r^% {4}}\right)+p_{\infty}.
  32. 2 p - p ρ U 2 = 2 R 2 r 2 cos ( 2 θ ) - R 4 r 4 . 2\frac{p-p_{\infty}}{\rho U^{2}}=2\frac{R^{2}}{r^{2}}\cos(2\theta)-\frac{R^{4}% }{r^{4}}.
  33. r = R r=R
  34. θ = 0 \theta=0
  35. θ = π \theta=\pi
  36. θ = 1 2 π \theta=\tfrac{1}{2}\pi
  37. θ = 3 2 π . \theta=\tfrac{3}{2}\pi.
  38. V V
  39. V = 2 U V=2U
  40. V = ψ × k ^ . \vec{V}=\nabla\psi\times\widehat{k}.
  41. V ψ = 0. \vec{V}\cdot\nabla{\psi}=0.
  42. ψ \psi
  43. V . \vec{V}.
  44. ψ = U ( r - R 2 r ) sin θ . \psi=U\left(r-\frac{R^{2}}{r}\right)\sin\theta.
  45. V \vec{V}
  46. p p
  47. V \vec{V}
  48. p , p,
  49. ρ U 2 / 2 , \rho U^{2}/2,
  50. U . U.
  51. p r = ρ V 2 L , \frac{\partial p}{\partial r}=\frac{\rho V^{2}}{L},
  52. L L
  53. L R , L\approx R,
  54. V U . V\approx U.
  55. Δ r R \Delta r\approx R
  56. p - p - ρ U 2 . p-p_{\infty}\approx-\rho U^{2}.
  57. p - p = - 3 2 ρ U 2 . p-p_{\infty}=-\frac{3}{2}\rho U^{2}.
  58. V = 2 U , V=2U,
  59. V > U V>U
  60. V V
  61. U U

Pounds_per_square_inch.html

  1. 1 lbf ( 1 in ) 2 \frac{1\,\text{ lbf}}{(1\,\text{ in})^{2}}
  2. 4.4482216152605 N ( 0.0254 m ) 2 \frac{4.4482216152605\,\text{ N}}{(0.0254\,\text{ m})^{2}}
  3. 101325 Pa 101325\,\text{ Pa}
  4. 101325 6894.757293168 \frac{101325}{6894.757293168}

Power_optimizer.html

  1. V o c V_{oc}

Power_usage_effectiveness.html

  1. PUE = Total Facility Energy IT Equipment Energy \mathrm{PUE}={\mbox{Total Facility Energy}~{}\over\mbox{IT Equipment Energy}~{}}

Power–speed_number.html

  1. P S N = 2 * H R * S B H R + S B PSN=\frac{2*HR*SB}{HR+SB}

Pöschl–Teller_potential.html

  1. U ( x ) = - λ ( λ + 1 ) 2 sech 2 ( x ) U(x)=-\frac{\lambda(\lambda+1)}{2}\mathrm{sech}^{2}(x)
  2. - 1 2 ψ ′′ ( x ) + U ( x ) ψ ( x ) = E ψ ( x ) -\frac{1}{2}\psi^{\prime\prime}(x)+U(x)\psi(x)=E\psi(x)
  3. u = tanh ( x ) u=\mathrm{tanh(x)}
  4. [ ( 1 - u 2 ) ψ ( u ) ] + λ ( λ + 1 ) ψ ( u ) + 2 E 1 - u 2 ψ ( u ) = 0 \left[(1-u^{2})\psi^{\prime}(u)\right]^{\prime}+\lambda(\lambda+1)\psi(u)+% \frac{2E}{1-u^{2}}\psi(u)=0
  5. ψ ( u ) \psi(u)
  6. P λ μ ( u ) P_{\lambda}^{\mu}(u)
  7. E = - μ 2 2 E=\frac{-\mu^{2}}{2}
  8. λ \lambda

PPA_(complexity).html

  1. n n

PPP_(complexity).html

  1. n n
  2. 2 n 2^{n}

Precision_(statistics).html

  1. 2 \scriptstyle\sqrt{2}
  2. φ Δ = h π e - h h Δ Δ . \varphi\Delta=\tfrac{h}{\surd\pi}\,e^{-hh\Delta\Delta}.

Precursor_(physics).html

  1. f ( x , t ) = 1 2 π ζ ^ 0 ( ω ) exp [ - i ( k ( ω ) x - ω t ) ] d ω f(x,t)=\frac{1}{2\pi}\int\hat{\zeta}_{0}(\omega)\exp\left[-i\left(k(\omega)x-% \omega t\right)\right]d\omega
  2. ζ ^ 0 ( ω ) \hat{\zeta}_{0}(\omega)
  3. exp [ - i ( k ( ω ) x - ω t ) ] \exp\left[-i\left(k(\omega)x-\omega t\right)\right]
  4. k ( ω ) k(\omega)
  5. t = 0 t=0
  6. f ( t ) = { 0 t < 0 sin 2 π t τ t 0 , f(t)=\left\{\begin{array}[]{rl}0&t<0\\ \sin\frac{2\pi t}{\tau}&t\geq 0\end{array}\right.,
  7. f ( x , t ) = - 1 τ e - i ( k ( ω ) x - ω t ) d ω ω 2 - ( 2 π / τ ) 2 . f(x,t)=-\frac{1}{\tau}\int e^{-i(k(\omega)x-\omega t)}\frac{d\omega}{\omega^{2% }-(2\pi/\tau)^{2}}.
  8. k ( ω ) = ω c 1 + a 2 ω 0 2 ω 0 2 - ω 2 k(\omega)=\frac{\omega}{c}\sqrt{1+\frac{a^{2}\omega_{0}^{2}}{\omega_{0}^{2}-% \omega^{2}}}
  9. a 2 = N q 2 m ϵ 0 ω 0 2 a^{2}=\frac{Nq^{2}}{m\epsilon_{0}\omega_{0}^{2}}
  10. N N
  11. q q
  12. m m
  13. ω 0 \omega_{0}
  14. ϵ 0 \epsilon_{0}
  15. f ( x , t ) = - 1 τ exp [ - i ( x ω c 1 + a 2 ω 0 2 ω 0 2 - ω 2 - ω t ) ] d ω ω 2 - ( 2 π / τ ) 2 . f(x,t)=-\frac{1}{\tau}\int\exp\left[-i\left(x\frac{\omega}{c}\sqrt{1+\frac{a^{% 2}\omega_{0}^{2}}{\omega_{0}^{2}-\omega^{2}}}-\omega t\right)\right]\frac{d% \omega}{\omega^{2}-(2\pi/\tau)^{2}}.
  16. t = t - x c t^{\prime}=t-\frac{x}{c}
  17. c c
  18. | ω | |\omega|
  19. 2 π τ \frac{2\pi}{\tau}
  20. ω \omega
  21. ξ = a 2 ω 0 2 2 c x \xi=\frac{a^{2}\omega_{0}^{2}}{2c}x
  22. f ( ξ , t ) = - 1 τ exp [ - i ( ξ ω + ω t ) ] d ω ω 2 f(\xi,t^{\prime})=-\frac{1}{\tau}\int\exp\left[-i\left(\frac{\xi}{\omega}+% \omega t^{\prime}\right)\right]\frac{d\omega}{\omega^{2}}
  23. f ( ξ , t ) = - 1 τ exp [ - i ξ t ( 1 ω ξ t + ω t ξ ) ] d ω ω 2 f(\xi,t^{\prime})=-\frac{1}{\tau}\int\exp\left[-i\sqrt{\xi t^{\prime}}\left(% \frac{1}{\omega}\sqrt{\frac{\xi}{t^{\prime}}}+\omega\sqrt{\frac{t^{\prime}}{% \xi}}\right)\right]\frac{d\omega}{\omega^{2}}
  24. ω t ξ = e i k , d ω ω = i d k , d ω ω 2 = i t ξ e - i k d k \omega\sqrt{\frac{t^{\prime}}{\xi}}=e^{ik},\qquad\frac{d\omega}{\omega}=idk,% \qquad\frac{d\omega}{\omega^{2}}=i\sqrt{\frac{t^{\prime}}{\xi}}e^{-ik}dk
  25. f ( ξ , t ) = - i τ t ξ exp [ - 2 i ξ t cos k ] e - i k d k , f(\xi,t^{\prime})=-\frac{i}{\tau}\sqrt{\frac{t^{\prime}}{\xi}}\int\exp\left[-2% i\sqrt{\xi t^{\prime}}\cos k\right]e^{-ik}dk,
  26. k k
  27. f ( ξ , t ) = 2 π τ t ξ J 1 ( 2 ξ t ) , f(\xi,t^{\prime})=\frac{2\pi}{\tau}\sqrt{\frac{t^{\prime}}{\xi}}J_{1}\left(2% \sqrt{\xi t^{\prime}}\right),
  28. J 1 J_{1}
  29. x t \frac{x}{t}
  30. x x
  31. t t
  32. ω D \omega_{D}
  33. x t \frac{x}{t}
  34. v g ( ω D ) = d ω d k | ω D = x t . v_{g}(\omega_{D})=\left.\frac{d\omega}{dk}\right|_{\omega_{D}}=\frac{x}{t}.

Pregaussian_class.html

  1. L P 2 ( S ) L^{2}_{P}(S)
  2. f : S R f:S\to R
  3. f 2 d P < \int f^{2}\,dP<\infty
  4. L P 2 ( S ) \mathcal{F}\subset L^{2}_{P}(S)
  5. G P G_{P}
  6. \mathcal{F}
  7. Cov ( G P ( f ) , G P ( g ) ) = E G P ( f ) G P ( g ) = f g d P - f d P g d P for f , g \operatorname{Cov}(G_{P}(f),G_{P}(g))=EG_{P}(f)G_{P}(g)=\int fg\,dP-\int f\,dP% \int g\,dP\,\text{ for }f,g\in\mathcal{F}
  8. L P 2 ( S ) L^{2}_{P}(S)
  9. ϱ P ( f , g ) = ( E ( G P ( f ) - G P ( g ) ) 2 ) 1 / 2 \varrho_{P}(f,g)=(E(G_{P}(f)-G_{P}(g))^{2})^{1/2}
  10. L P 2 ( S ) \mathcal{F}\subset L^{2}_{P}(S)
  11. ω S , \omega\in S,
  12. f G P ( f ) ( ω ) f\mapsto G_{P}(f)(\omega)
  13. \mathcal{F}
  14. ϱ P \varrho_{P}
  15. G P G_{P}
  16. S = [ 0 , 1 ] , S=[0,1],
  17. G P G_{P}
  18. I [ 0 , x ] I_{[0,x]}
  19. x [ 0 , 1 ] , x\in[0,1],

Pressure-volume_curves.html

  1. ψ 0 ( V ) \psi_{0}\mathit{(}V)\!
  2. ψ 0 \psi_{0}\!
  3. ( V ) \mathit{(}V)\!
  4. ψ 0 \psi_{0}\!
  5. 1 V × \dfrac{1}{V}\!\times

Press–Schechter_formalism.html

  1. M M
  2. M + d M M+dM
  3. N ( M ) d M = 1 π ( 1 + n 3 ) ρ ¯ M 2 ( M M * ) ( 3 + n ) / 6 exp ( - ( M M * ) ( 3 + n ) / 3 ) d M N(M)dM=\frac{1}{\sqrt{\pi}}\left(1+\frac{n}{3}\right)\frac{\bar{\rho}}{M^{2}}% \left(\frac{M}{M^{*}}\right)^{\left(3+n\right)/6}\exp\left(-\left(\frac{M}{M^{% *}}\right)^{\left(3+n\right)/3}\right)dM
  4. ρ ¯ \bar{\rho}
  5. n n
  6. P ( k ) k n P(k)\propto k^{n}
  7. M * M^{*}

Price_of_stability.html

  1. PoS = value of best Nash equilibrium value of optimal solution , PoS 0. \,\text{PoS}=\frac{\,\text{value of best Nash equilibrium}}{\,\text{value of % optimal solution}},\ \,\text{PoS}\geq 0.
  2. O ( log n / log log n ) O(\log n/\log\log n)
  3. n n
  4. n n
  5. n n
  6. i i
  7. s i s_{i}
  8. t i t_{i}
  9. G = ( V , E ) G=(V,E)
  10. P i P_{i}
  11. s i s_{i}
  12. t i t_{i}
  13. G G
  14. c i c_{i}
  15. n e n_{e}
  16. e e
  17. d e ( n e ) = c e n e \textstyle d_{e}(n_{e})=\frac{c_{e}}{n_{e}}
  18. C i ( S ) = e P i c e n e \textstyle C_{i}(S)=\sum_{e\in P_{i}}\frac{c_{e}}{n_{e}}
  19. S C ( S ) = i C i ( S ) = e S n e c e n e = e S c e \textstyle SC(S)=\sum_{i}C_{i}(S)=\sum_{e\in S}n_{e}\frac{c_{e}}{n_{e}}=\sum_{% e\in S}c_{e}
  20. Ω ( n ) \Omega(n)
  21. Ω ( n ) \Omega(n)
  22. 1 + ε 1+\varepsilon
  23. 1 + ε 1+\varepsilon
  24. n n
  25. 1 1
  26. 1 + ε 1+\varepsilon
  27. n n
  28. s i s_{i}
  29. t t
  30. 1 + ε 1+\varepsilon
  31. 1 + ε 1+\varepsilon
  32. 1 + ε n \textstyle\frac{1+\varepsilon}{n}
  33. 1 n \textstyle\frac{1}{n}
  34. 1 n - 1 \textstyle\frac{1}{n-1}
  35. 1 + 1 2 + + 1 n = H n \textstyle 1+\frac{1}{2}+\cdots+\frac{1}{n}=H_{n}
  36. H n H_{n}
  37. n n
  38. Θ ( log n ) \Theta(\log n)
  39. Φ = e i = 1 n e c e i \textstyle\Phi=\sum_{e}\sum_{i=1}^{n_{e}}\frac{c_{e}}{i}
  40. A A
  41. B B
  42. S S
  43. A S C ( S ) Φ ( S ) B S C ( S ) . A\cdot SC(S)\leq\Phi(S)\leq B\cdot SC(S).
  44. B / A B/A
  45. N E NE
  46. Φ \Phi
  47. S C ( N E ) 1 / A Φ ( N E ) 1 / A Φ ( O P T ) B / A S C ( O P T ) . SC(NE)\leq 1/A\cdot\Phi(NE)\leq 1/A\cdot\Phi(OPT)\leq B/A\cdot SC(OPT).
  48. Φ ( S ) = e S i = 1 n e c e i = e S c e H n e e S c e H n = H n S C ( S ) . \Phi(S)=\sum_{e\in S}\sum_{i=1}^{n_{e}}\frac{c_{e}}{i}=\sum_{e\in S}c_{e}H_{n_% {e}}\leq\sum_{e\in S}c_{e}H_{n}=H_{n}\cdot SC(S).
  49. A = 1 A=1
  50. B = H n B=H_{n}
  51. O ( log n / log log n ) O(\log n/\log\log n)

Priestley_space.html

  1. ( X , τ , ) (X,τ,≤)
  2. X X
  3. τ τ
  4. ( X , τ ) (X,τ)
  5. x y \scriptstyle x\,\not\leq\,y
  6. U U
  7. X X
  8. y U y∉U
  9. x , y x,y
  10. ( X , τ , ) (X,τ,≤)
  11. x y x≠y
  12. x y \scriptstyle x\,\not\leq\,y
  13. y x \scriptstyle y\,\not\leq\,x
  14. x y \scriptstyle x\,\not\leq\,y
  15. U U
  16. X X
  17. y U y∉U
  18. U U
  19. V = X U V=X−U
  20. X X
  21. x x
  22. y y
  23. U U
  24. x x
  25. ( X , τ , ) (X,τ,≤)
  26. C C
  27. x x
  28. x y \scriptstyle x\,\not\leq\,y
  29. y x \scriptstyle y\,\not\leq\,x
  30. x x
  31. y y
  32. x x
  33. X U X−U
  34. X X
  35. x x
  36. X U X−U
  37. C C
  38. x x
  39. U U
  40. ( X , τ , ) (X,τ,≤)
  41. ( X , τ ) (X,τ)
  42. ( X , τ , ) (X,τ,≤)
  43. F F
  44. X X
  45. X X
  46. X X
  47. X X
  48. X X
  49. X X
  50. X X
  51. X X
  52. X X
  53. X X
  54. X X
  55. ( X , τ ) (X,τ)
  56. F F
  57. G G
  58. X X
  59. F G = ↑F∩↓G=∅
  60. U U
  61. F U F⊆U
  62. U G = U∩G=∅
  63. ( X , τ , ) (X,τ,≤)
  64. ( X , τ , ) (X′,τ′,≤′)
  65. f : X X f:X→X′
  66. ( X , τ , ) (X,τ,≤)
  67. X X
  68. X X
  69. ( X , τ , ) (X,τ,≤)
  70. ( X , τ ) (X,τ)
  71. X X
  72. ( X , τ ) (X,τ)
  73. ( X , τ ) (X,τ)
  74. ( X , τ ) (X,τ)
  75. ( X , τ , ) (X,τ,≤)
  76. ( X , τ , ) (X,τ,≤)
  77. τ τ
  78. 𝐒𝐩𝐞𝐜 𝐏𝐫𝐢𝐞𝐬 𝐏𝐒𝐭𝐨𝐧𝐞 \mathbf{Spec}\cong\mathbf{Pries}\cong\mathbf{PStone}

Primary_line_constants.html

  1. Z 0 2 = L / C \scriptstyle{Z_{0}}^{2}=L/C
  2. d x \scriptstyle dx
  3. d L = lim δ x 0 ( L δ x ) = L d x dL=\lim_{\delta x\to 0}(L\delta x)=Ldx
  4. d R = lim δ x 0 ( R δ x ) = R d x dR=\lim_{\delta x\to 0}(R\delta x)=Rdx
  5. d C = lim δ x 0 ( C δ x ) = C d x dC=\lim_{\delta x\to 0}(C\delta x)=Cdx
  6. d G = lim δ x 0 ( G δ x ) = G d x dG=\lim_{\delta x\to 0}(G\delta x)=Gdx
  7. d Z = ( R + i ω L ) d x = Z d x , dZ=(R+i\omega L)dx=Zdx\,,
  8. d Y = ( G + i ω C ) d x = Y d x . dY=(G+i\omega C)dx=Ydx\,.
  9. γ \scriptstyle\gamma
  10. α \scriptstyle\alpha
  11. β \scriptstyle\beta
  12. Z 0 \scriptstyle Z_{0}
  13. R 0 \scriptstyle R_{0}
  14. X 0 \scriptstyle X_{0}
  15. ω L \scriptstyle\omega L
  16. 1 / ( ω C ) \scriptstyle 1/(\omega C)
  17. ω \scriptstyle\omega
  18. α \scriptstyle\alpha
  19. Z 0 \scriptstyle Z_{0}
  20. β \scriptstyle\beta
  21. ω \scriptstyle\omega
  22. Z 0 \scriptstyle Z_{0}
  23. Z 0 \scriptstyle Z_{0}
  24. d Z \scriptstyle dZ
  25. d Y \scriptstyle dY
  26. Z 0 \scriptstyle Z_{0}
  27. Z 0 \scriptstyle Z_{0}
  28. Z 0 = δ Z + Z 0 1 + Z 0 δ Y Z_{0}=\delta Z+\frac{Z_{0}}{1+Z_{0}\delta Y}
  29. Z 0 2 - Z 0 δ Z = δ Z δ Y {Z_{0}}^{2}-Z_{0}\delta Z=\frac{\delta Z}{\delta Y}
  30. lim δ x 0 ( Z 0 2 - Z 0 δ Z ) = Z 0 2 = d Z d Y \lim_{\delta x\to 0}({Z_{0}}^{2}-Z_{0}\delta Z)={Z_{0}}^{2}=\frac{dZ}{dY}
  31. Z 0 2 = Z Y {Z_{0}}^{2}=\frac{Z}{Y}
  32. δ x \scriptstyle\delta x
  33. Z 0 \scriptstyle Z_{0}
  34. V i V x 1 = δ Z + Z 0 / δ Y Z 0 + 1 / δ Y Z 0 / δ Y Z 0 + 1 / δ Y = 1 + δ Z Z 0 + δ Z δ Y \frac{V_{\mathrm{i}}}{V_{x1}}=\frac{\delta Z+\frac{Z_{0}/\delta Y}{Z_{0}+1/% \delta Y}}{\frac{Z_{0}/\delta Y}{Z_{0}+1/\delta Y}}=1+\frac{\delta Z}{Z_{0}}+% \delta Z\delta Y
  35. n \scriptstyle n
  36. V i V x n = ( 1 + δ Z Z 0 + δ Z δ Y ) n \frac{V_{\mathrm{i}}}{V_{xn}}=\left(1+\frac{\delta Z}{Z_{0}}+\delta Z\delta Y% \right)^{n}
  37. x \scriptstyle x
  38. x / δ x \scriptstyle x/\delta x
  39. V i V x n = ( 1 + δ Z Z 0 + δ Z δ Y ) x δ x \frac{V_{\mathrm{i}}}{V_{xn}}=\left(1+\frac{\delta Z}{Z_{0}}+\delta Z\delta Y% \right)^{\frac{x}{\delta x}}
  40. δ x 0 \scriptstyle\delta x\to 0
  41. V i V x = lim δ x 0 V i V x n = lim δ x 0 ( 1 + δ Z Z 0 + δ Z δ Y ) x δ x \frac{V_{\mathrm{i}}}{V_{x}}=\lim_{\delta x\to 0}\frac{V_{\mathrm{i}}}{V_{xn}}% =\lim_{\delta x\to 0}\left(1+\frac{\delta Z}{Z_{0}}+\delta Z\delta Y\right)^{% \frac{x}{\delta x}}
  42. δ Z δ Y \scriptstyle\delta Z\delta Y
  43. V i V x = lim δ x 0 ( 1 + δ Z Z 0 ) x δ x \frac{V_{\mathrm{i}}}{V_{x}}=\lim_{\delta x\to 0}\left(1+\frac{\delta Z}{Z_{0}% }\right)^{\frac{x}{\delta x}}
  44. e x lim p ( 1 + 1 / p ) p x e^{x}\equiv\lim_{p\to\infty}(1+1/p)^{px}
  45. V i = V x e Z Z 0 x V_{\mathrm{i}}=V_{x}e^{\frac{Z}{Z_{0}}x}
  46. V i = V x e γ x V_{\mathrm{i}}=V_{x}e^{\gamma x}\,\!
  47. γ = Z Z 0 = Z Y \gamma=\frac{Z}{Z_{0}}=\sqrt{ZY}
  48. γ = i ω L C \gamma=i\omega\sqrt{LC}
  49. α = 0 \alpha=0\,
  50. β = ω L C \beta=\omega\sqrt{LC}
  51. Z 0 = L C Z_{0}=\sqrt{\frac{L}{C}}
  52. R ω L \scriptstyle R\gg\omega L
  53. γ i ω C R \gamma\approx\sqrt{i\omega CR}
  54. α ω C R 2 \alpha\approx\sqrt{\frac{\omega CR}{2}}
  55. β ω C R 2 \beta\approx\sqrt{\frac{\omega CR}{2}}
  56. Z 0 R i ω C = R 2 ω C - i R 2 ω C Z_{0}\approx\sqrt{\frac{R}{i\omega C}}=\sqrt{\frac{R}{2\omega C}}-i\sqrt{\frac% {R}{2\omega C}}
  57. β \scriptstyle\beta
  58. β \scriptstyle\beta
  59. ω \scriptstyle\omega
  60. ω \scriptstyle\sqrt{\omega}
  61. Z 0 \scriptstyle Z_{0}
  62. R ω L \scriptstyle R\ll\omega L
  63. G ω C \scriptstyle G\ll\omega C
  64. γ i ω L C \gamma\approx i\omega\sqrt{LC}
  65. α L G + R C 2 L C = 1 2 ( Z 0 G + R Z 0 ) R 2 Z 0 \alpha\approx\frac{LG+RC}{2\sqrt{LC}}=\tfrac{1}{2}\left(Z_{0}G+\frac{R}{Z_{0}}% \right)\approx\frac{R}{2Z_{0}}
  66. β ω L C \beta\approx\omega\sqrt{LC}
  67. Z 0 L C Z_{0}\approx\sqrt{\frac{L}{C}}
  68. Z 0 \scriptstyle Z_{0}
  69. γ = R G + i ω L C \gamma=\sqrt{RG}+i\omega\sqrt{LC}
  70. α = R G \alpha=\sqrt{RG}
  71. β = ω L C \beta=\omega\sqrt{LC}
  72. Z 0 = L C = R G Z_{0}=\sqrt{\frac{L}{C}}=\sqrt{\frac{R}{G}}
  73. v = λ f . v=\lambda f.
  74. ω = 2 π f \omega=2\pi f
  75. β = 2 π λ \beta=\frac{2\pi}{\lambda}
  76. v = ω β . v=\frac{\omega}{\beta}.
  77. β β
  78. β = ω L C \beta=\omega\sqrt{LC}
  79. v = 1 L C . v={1\over\sqrt{LC}}.

Primitive_abundant_number.html

  1. o ( n log 2 ( n ) ) . o\left(\frac{n}{\log^{2}(n)}\right)\,.

Prince_Rupert's_cube.html

  1. 3 2 4 1.0606601. \frac{3\sqrt{2}}{4}\approx 1.0606601.
  2. 3 2 4 \tfrac{3\sqrt{2}}{4}
  3. 6 - 2 1.03527 \sqrt{6}-\sqrt{2}\approx 1.03527
  4. m m
  5. n n
  6. ( m , n ) = ( 3 , 4 ) (m,n)=(3,4)
  7. 4 x 4 - 28 x 3 - 7 x 2 + 16 x + 16 4x^{4}-28x^{3}-7x^{2}+16x+16
  8. m = 2 m=2
  9. n n
  10. n / 2 \sqrt{n/2}
  11. n / 2 - 3 / 8 \sqrt{n/2-3/8}
  12. n n

Principal_curvature-based_region_detector.html

  1. H ( 𝐱 ) = [ L x x ( 𝐱 ) L x y ( 𝐱 ) L x y ( 𝐱 ) L y y ( 𝐱 ) ] H(\mathbf{x})=\begin{bmatrix}L_{xx}(\mathbf{x})&L_{xy}(\mathbf{x})\\ L_{xy}(\mathbf{x})&L_{yy}(\mathbf{x})\\ \end{bmatrix}
  2. L a a ( 𝐱 ) L_{aa}(\mathbf{x})
  3. a a
  4. L a b ( 𝐱 ) L_{ab}(\mathbf{x})
  5. a a
  6. b b

Principles_of_Hindu_Reckoning.html

  1. 1 / 2 {1}/{2}
  2. ( 63342 ) = 255 371 511 \sqrt{(}63342)=255\frac{371}{511}

Prism_correction.html

  1. P = 100 tan d P=100\tan d\!
  2. P P
  3. d d
  4. a a
  5. n n
  6. d = ( n - 1 ) a d=(n-1)a
  7. P = c f P=cf

Probabilistic_relevance_model.html

  1. s i m ( d j , q ) = P ( R | d j ) P ( R ¯ | d j ) sim(d_{j},q)=\frac{P(R|\vec{d}_{j})}{P(\bar{R}|\vec{d}_{j})}

Probability_of_precipitation.html

  1. PoP = C × A \,\text{PoP}=C\times A

Product_state.html

  1. ρ = ρ 1 ρ 2 , \rho=\rho_{1}\otimes\rho_{2},
  2. ρ 1 \rho_{1}
  3. ρ 2 \rho_{2}
  4. ρ = k p k ρ 1 k ρ 2 k . \rho=\sum_{k}p_{k}\rho^{k}_{1}\otimes\rho^{k}_{2}.

Project_Mathematics!.html

  1. π \pi
  2. 2 π r 2\pi r
  3. π r 2 \pi r^{2}
  4. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  5. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  6. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  7. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  8. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  9. 6 π 2 \tfrac{6}{\pi^{2}}

Projection_pursuit_regression.html

  1. Y = β 0 + j = 1 r f j ( β j x ) + ε , Y=\beta_{0}+\sum_{j=1}^{r}f_{j}(\beta_{j}^{\prime}x)+\varepsilon,
  2. β j x \beta_{j}^{\prime}x
  3. β j X \beta_{j}^{\prime}X
  4. ( y i , x i ) (y_{i},x_{i})
  5. S = i = 1 n [ y i - j = 1 r f j ( β j x i ) ] 2 , S=\sum_{i=1}^{n}\left[y_{i}-\sum_{j=1}^{r}f_{j}(\beta_{j}^{\prime}x_{i})\right% ]^{2},
  6. f j f_{j}
  7. β j \beta_{j}
  8. f j f_{j}
  9. β j \beta_{j}
  10. f j f_{j}
  11. β j \beta_{j}
  12. f j f_{j}
  13. β j \beta_{j}
  14. n 1 2 n^{\frac{1}{2}}
  15. β j \beta_{j}
  16. β j \beta_{j}
  17. f j f_{j}
  18. f j f_{j}

Projective_polyhedron.html

  1. S 2 𝐑𝐏 2 S^{2}\to\mathbf{RP}^{2}
  2. 𝐑𝐏 n = 𝐏 ( 𝐑 n + 1 ) , \mathbf{RP}^{n}=\mathbf{P}(\mathbf{R}^{n+1}),
  3. P O ( 2 k + 1 ) = P S O ( 2 k + 1 ) S O ( 2 k + 1 ) PO(2k+1)=PSO(2k+1)\cong SO(2k+1)
  4. O P O O\to PO
  5. S O P S O SO\to PSO
  6. P S O ( 2 k ) P O ( 2 k ) PSO(2k)\neq PO(2k)
  7. S n 𝐑𝐏 n , S^{n}\to\mathbf{RP}^{n},
  8. S n S^{n}
  9. n 2 n\geq 2

Projective_superspace.html

  1. 𝒩 = 2 \mathcal{N}=2

Projectivization.html

  1. ( V ) {\mathbb{P}}(V)
  2. ( V ) {\mathbb{P}}(V)
  3. ( V ) {\mathbb{P}}(V)
  4. ( V ) {\mathbb{P}}(V)
  5. f : V W f:V\to W
  6. ( f ) : ( V ) ( W ) . \mathbb{P}(f):\mathbb{P}(V)\to\mathbb{P}(W).
  7. ( V ) {\mathbb{P}}(V)
  8. ( V K ) {\mathbb{P}}(V\oplus K)
  9. ( V ) . {\mathbb{P}}(V).

Proofs_involving_ordinary_least_squares.html

  1. S ( b ) = ( y - X b ) ( y - X b ) S(b)=(y-Xb)^{\prime}(y-Xb)\,
  2. {}^{\prime}
  3. 0 = d S d b ( β ^ ) = d d b ( y y - b X y - y X b + b X X b ) | b = β ^ = - 2 X y + 2 X X β ^ 0=\frac{dS}{db^{\prime}}(\hat{\beta})=\frac{d}{db^{\prime}}\bigg(y^{\prime}y-b% ^{\prime}X^{\prime}y-y^{\prime}Xb+b^{\prime}X^{\prime}Xb\bigg)\bigg|_{b=\hat{% \beta}}=-2X^{\prime}y+2X^{\prime}X\hat{\beta}
  4. β ^ = ( X X ) - 1 X y \hat{\beta}=(X^{\prime}X)^{-1}X^{\prime}y\,
  5. β ^ \hat{\beta}
  6. β ^ \hat{\beta}
  7. E [ β ^ ] \displaystyle\operatorname{E}[\,\hat{\beta}]
  8. σ 2 I \sigma^{2}I
  9. ε \varepsilon
  10. E [ ( β ^ - β ) ( β ^ - β ) T ] = E [ ( ( X X ) - 1 X ε ) ( ( X X ) - 1 X ε ) T ] = σ 2 ( X X ) - 1 , \begin{aligned}\displaystyle\operatorname{E}[\,(\hat{\beta}-\beta)(\hat{\beta}% -\beta)^{T}]&\displaystyle=\operatorname{E}\Big[((X^{\prime}X)^{-1}X^{\prime}% \varepsilon)((X^{\prime}X)^{-1}X^{\prime}\varepsilon)^{T}\Big]\\ &\displaystyle=\sigma^{2}(X^{\prime}X)^{-1},\\ \end{aligned}
  11. β ^ - β \hat{\beta}-\beta
  12. ε \varepsilon
  13. ( X X ) - 1 X (X^{\prime}X)^{-1}X^{\prime}
  14. β = [ β 0 , β 1 ] T \beta=[\beta_{0},\beta_{1}]^{T}
  15. β 0 \beta_{0}
  16. β 1 \beta_{1}
  17. σ 2 ( X X ) - 1 = σ 2 ( x i x i ) - 1 = σ 2 ( ( 1 , x i ) ( 1 , x i ) ) - 1 = σ 2 ( ( 1 x i x i x i 2 ) ) - 1 = σ 2 ( n x i x i x i 2 ) - 1 = σ 2 1 n x i 2 - ( x i ) 2 ( x i 2 - x i - x i n ) = σ 2 1 n i = 1 n ( x i - x ¯ ) 2 ( x i 2 - x i - x i n ) \begin{aligned}\displaystyle\sigma^{2}(X^{\prime}X)^{-1}&\displaystyle=\sigma^% {2}\left(\sum x_{i}x_{i}^{\prime}\right)^{-1}\\ &\displaystyle=\sigma^{2}\left(\sum(1,x_{i})^{\prime}(1,x_{i})\right)^{-1}\\ &\displaystyle=\sigma^{2}\left(\sum\begin{pmatrix}1&x_{i}\\ x_{i}&x_{i}^{2}\end{pmatrix}\right)^{-1}\\ &\displaystyle=\sigma^{2}\begin{pmatrix}n&\sum x_{i}\\ \sum x_{i}&\sum x_{i}^{2}\end{pmatrix}^{-1}\\ &\displaystyle=\sigma^{2}\cdot\frac{1}{n\sum x_{i}^{2}-(\sum x_{i})^{2}}\begin% {pmatrix}\sum x_{i}^{2}&-\sum x_{i}\\ -\sum x_{i}&n\end{pmatrix}\\ &\displaystyle=\sigma^{2}\cdot\frac{1}{n\sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}}% \begin{pmatrix}\sum x_{i}^{2}&-\sum x_{i}\\ -\sum x_{i}&n\end{pmatrix}\\ \end{aligned}
  18. V a r ( β 1 ) = σ 2 i = 1 n ( x i - x ¯ ) 2 . \begin{aligned}\displaystyle Var(\beta_{1})&\displaystyle=\frac{\sigma^{2}}{% \sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}}.\end{aligned}
  19. σ ^ 2 \hat{\sigma}^{2}
  20. σ ^ 2 = 1 n y M y = 1 n ( X β + ε ) M ( X β + ε ) = 1 n ε M ε \hat{\sigma}^{2}=\tfrac{1}{n}y^{\prime}My=\tfrac{1}{n}(X\beta+\varepsilon)^{% \prime}M(X\beta+\varepsilon)=\tfrac{1}{n}\varepsilon^{\prime}M\varepsilon
  21. E σ ^ 2 = 1 n E [ tr ( ε M ε ) ] = 1 n tr ( E [ M ε ε ] ) \operatorname{E}\,\hat{\sigma}^{2}=\tfrac{1}{n}\operatorname{E}\big[% \operatorname{tr}(\varepsilon^{\prime}M\varepsilon)\big]=\tfrac{1}{n}% \operatorname{tr}\big(\operatorname{E}[M\varepsilon\varepsilon^{\prime}]\big)
  22. E σ ^ 2 = 1 n tr ( E [ M E [ ε ε | X ] ] ) = 1 n tr ( E [ σ 2 M I ] ) = 1 n σ 2 E [ tr M ] \operatorname{E}\,\hat{\sigma}^{2}=\tfrac{1}{n}\operatorname{tr}\Big(% \operatorname{E}\big[M\,\operatorname{E}[\varepsilon\varepsilon^{\prime}|X]% \big]\Big)=\tfrac{1}{n}\operatorname{tr}\big(\operatorname{E}[\sigma^{2}MI]% \big)=\tfrac{1}{n}\sigma^{2}\operatorname{E}\big[\operatorname{tr}\,M\big]
  23. E σ ^ 2 = n - p n σ 2 \operatorname{E}\,\hat{\sigma}^{2}=\frac{n-p}{n}\sigma^{2}
  24. σ ^ 2 \hat{\sigma}^{2}
  25. β ^ \hat{\beta}
  26. β ^ \hat{\beta}
  27. β ^ = ( 1 n X X ) - 1 1 n X y = β + ( 1 n X X ) - 1 1 n X ε = β + ( 1 n i = 1 n x i x i ) - 1 ( 1 n i = 1 n x i ε i ) \hat{\beta}=\big(\tfrac{1}{n}X^{\prime}X\big)^{-1}\tfrac{1}{n}X^{\prime}y=% \beta+\big(\tfrac{1}{n}X^{\prime}X\big)^{-1}\tfrac{1}{n}X^{\prime}\varepsilon=% \beta\;+\;\bigg(\frac{1}{n}\sum_{i=1}^{n}x_{i}x^{\prime}_{i}\bigg)^{\!\!-1}% \bigg(\frac{1}{n}\sum_{i=1}^{n}x_{i}\varepsilon_{i}\bigg)
  28. 1 n i = 1 n x i x i 𝑝 E [ x i x i ] = Q x x n , 1 n i = 1 n x i ε i 𝑝 E [ x i ε i ] = 0 \frac{1}{n}\sum_{i=1}^{n}x_{i}x^{\prime}_{i}\ \xrightarrow{p}\ \operatorname{E% }[x_{i}x_{i}^{\prime}]=\frac{Q_{xx}}{n},\qquad\frac{1}{n}\sum_{i=1}^{n}x_{i}% \varepsilon_{i}\ \xrightarrow{p}\ \operatorname{E}[x_{i}\varepsilon_{i}]=0
  29. β ^ \hat{\beta}
  30. β ^ 𝑝 β + Q x x - 1 0 = β \hat{\beta}\ \xrightarrow{p}\ \beta+Q_{xx}^{-1}\cdot 0=\beta
  31. 1 n i = 1 n x i ε i 𝑑 𝒩 ( 0 , V ) , \frac{1}{\sqrt{n}}\sum_{i=1}^{n}x_{i}\varepsilon_{i}\ \xrightarrow{d}\ % \mathcal{N}\big(0,\,V\big),
  32. V = Var [ x i ε i ] = E [ ε i 2 x i x i ] = E [ E [ ε i 2 | x i ] x i x i ] = σ 2 Q x x n V=\operatorname{Var}[x_{i}\varepsilon_{i}]=\operatorname{E}[\,\varepsilon_{i}^% {2}x_{i}x^{\prime}_{i}\,]=\operatorname{E}\big[\,\operatorname{E}[\varepsilon_% {i}^{2}|x_{i}]\;x_{i}x^{\prime}_{i}\,\big]=\sigma^{2}\frac{Q_{xx}}{n}
  33. n ( β ^ - β ) = ( 1 n i = 1 n x i x i ) - 1 ( 1 n i = 1 n x i ε i ) 𝑑 Q x x - 1 n 𝒩 ( 0 , σ 2 Q x x n ) = 𝒩 ( 0 , σ 2 Q x x - 1 n ) \sqrt{n}(\hat{\beta}-\beta)=\bigg(\frac{1}{n}\sum_{i=1}^{n}x_{i}x^{\prime}_{i}% \bigg)^{\!\!-1}\bigg(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}x_{i}\varepsilon_{i}\bigg% )\ \xrightarrow{d}\ Q_{xx}^{-1}n\cdot\mathcal{N}\big(0,\sigma^{2}\frac{Q_{xx}}% {n}\big)=\mathcal{N}\big(0,\sigma^{2}Q_{xx}^{-1}n\big)
  34. y | X 𝒩 ( X β , σ 2 I ) y|X\ \sim\ \mathcal{N}(X\beta,\,\sigma^{2}I)
  35. ( β , σ 2 | X ) \displaystyle\mathcal{L}(\beta,\sigma^{2}|X)
  36. β = - 1 2 σ 2 ( - 2 X y + 2 X X β ) = 0 β ^ = ( X X ) - 1 X y \displaystyle\frac{\partial\mathcal{L}}{\partial\beta^{\prime}}=-\frac{1}{2% \sigma^{2}}\Big(-2X^{\prime}y+2X^{\prime}X\beta\Big)=0\quad\Rightarrow\quad% \hat{\beta}=(X^{\prime}X)^{-1}X^{\prime}y
  37. β ^ \hat{\beta}
  38. σ ^ 2 \hat{\sigma}^{2}
  39. β ^ = ( X X ) - 1 X y = ( X X ) - 1 X ( X β + ε ) = β + ( X X ) - 1 X 𝒩 ( 0 , σ 2 I ) \hat{\beta}=(X^{\prime}X)^{-1}X^{\prime}y=(X^{\prime}X)^{-1}X^{\prime}(X\beta+% \varepsilon)=\beta+(X^{\prime}X)^{-1}X^{\prime}\mathcal{N}(0,\sigma^{2}I)
  40. β ^ | X 𝒩 ( β , σ 2 ( X X ) - 1 ) . \hat{\beta}|X\ \sim\ \mathcal{N}(\beta,\,\sigma^{2}(X^{\prime}X)^{-1}).
  41. σ ^ 2 \hat{\sigma}^{2}
  42. σ ^ 2 \displaystyle\hat{\sigma}^{2}
  43. M = I - X ( X X ) - 1 X M=I-X(X^{\prime}X)^{-1}X^{\prime}
  44. n σ 2 σ ^ 2 | X = ( ε / σ ) M ( ε / σ ) χ n - p 2 \tfrac{n}{\sigma^{2}}\hat{\sigma}^{2}|X=(\varepsilon/\sigma)^{\prime}M(% \varepsilon/\sigma)\ \sim\ \chi^{2}_{n-p}
  45. β ^ \hat{\beta}
  46. σ ^ 2 \hat{\sigma}^{2}
  47. β ^ \hat{\beta}
  48. y ^ = X β ^ = P y = X β + P ε \hat{y}=X\hat{\beta}=Py=X\beta+P\varepsilon
  49. β ^ \hat{\beta}
  50. σ ^ 2 \hat{\sigma}^{2}
  51. β ^ \hat{\beta}
  52. σ ^ 2 \hat{\sigma}^{2}

Proofs_of_convergence_of_random_variables.html

  1. X n a s X X n 𝑝 X X_{n}\ \xrightarrow{as}\ X\quad\Rightarrow\quad X_{n}\ \xrightarrow{p}\ X
  2. A n = m n { | X m - X | > ε } A_{n}=\bigcup_{m\geq n}\left\{\left|X_{m}-X\right|>\varepsilon\right\}
  3. A = n 1 A n . A_{\infty}=\bigcap_{n\geq 1}A_{n}.
  4. Pr ( | X n - X | > ε ) Pr ( A n ) n 0 , \operatorname{Pr}\left(|X_{n}-X|>\varepsilon\right)\leq\operatorname{Pr}(A_{n}% )\ \underset{n\to\infty}{\rightarrow}0,
  5. X n 𝑝 X X n 𝑑 X , X_{n}\ \xrightarrow{p}\ X\quad\Rightarrow\quad X_{n}\ \xrightarrow{d}\ X,
  6. Pr ( Y a ) Pr ( X a + ε ) + Pr ( | Y - X | > ε ) . \operatorname{Pr}(Y\leq a)\leq\operatorname{Pr}(X\leq a+\varepsilon)+% \operatorname{Pr}(|Y-X|>\varepsilon).
  7. { Y a } { X a + ε } { | Y - X | > ε } . \{Y\leq a\}\subset\{X\leq a+\varepsilon\}\cup\{|Y-X|>\varepsilon\}.
  8. Pr ( Y a ) \displaystyle\operatorname{Pr}(Y\leq a)
  9. Pr ( X n a ) \displaystyle\operatorname{Pr}(X_{n}\leq a)
  10. Pr ( X a - ε ) - Pr ( | X n - X | > ε ) Pr ( X n a ) Pr ( X a + ε ) + Pr ( | X n - X | > ε ) . \operatorname{Pr}(X\leq a-\varepsilon)-\operatorname{Pr}\left(\left|X_{n}-X% \right|>\varepsilon\right)\leq\operatorname{Pr}\left(X_{n}\leq a\right)\leq% \operatorname{Pr}(X\leq a+\varepsilon)+\operatorname{Pr}\left(\left|X_{n}-X% \right|>\varepsilon\right).
  11. F X ( a - ε ) lim n Pr ( X n a ) F X ( a + ε ) , F_{X}(a-\varepsilon)\leq\lim_{n\to\infty}\operatorname{Pr}(X_{n}\leq a)\leq F_% {X}(a+\varepsilon),
  12. lim n Pr ( X n a ) = Pr ( X a ) , \lim_{n\to\infty}\operatorname{Pr}(X_{n}\leq a)=\operatorname{Pr}(X\leq a),
  13. X n 𝑑 c X n 𝑝 c , X_{n}\ \xrightarrow{d}\ c\quad\Rightarrow\quad X_{n}\ \xrightarrow{p}\ c,
  14. Pr ( | X n - c | ε ) = Pr ( X n B ε c ( c ) ) . \operatorname{Pr}\left(|X_{n}-c|\geq\varepsilon\right)=\operatorname{Pr}\left(% X_{n}\in B_{\varepsilon}^{c}(c)\right).
  15. lim n Pr ( | X n - c | ε ) \displaystyle\lim_{n\to\infty}\operatorname{Pr}\left(\left|X_{n}-c\right|\geq% \varepsilon\right)
  16. | Y n - X n | 𝑝 0 , X n 𝑑 X Y n 𝑑 X |Y_{n}-X_{n}|\ \xrightarrow{p}\ 0,\ \ X_{n}\ \xrightarrow{d}\ X\ \quad% \Rightarrow\quad Y_{n}\ \xrightarrow{d}\ X
  17. K > 0 , x , y : | f ( x ) - f ( y ) | K | x - y | . \exists K>0,\forall x,y:\quad|f(x)-f(y)|\leq K|x-y|.
  18. | E [ f ( Y n ) ] - E [ f ( X n ) ] | \displaystyle\left|\operatorname{E}\left[f(Y_{n})\right]-\operatorname{E}\left% [f(X_{n})\right]\right|
  19. | E [ f ( Y n ) ] - E [ f ( X ) ] | \displaystyle\left|\operatorname{E}\left[f(Y_{n})\right]-\operatorname{E}\left% [f(X)\right]\right|
  20. lim n | E [ f ( Y n ) ] - E [ f ( X ) ] | K ε . \lim_{n\to\infty}\left|\operatorname{E}\left[f(Y_{n})\right]-\operatorname{E}% \left[f(X)\right]\right|\leq K\varepsilon.
  21. X n 𝑑 X , Y n 𝑑 c ( X n , Y n ) 𝑑 ( X , c ) X_{n}\ \xrightarrow{d}\ X,\ \ Y_{n}\ \xrightarrow{d}\ c\ \quad\Rightarrow\quad% (X_{n},Y_{n})\ \xrightarrow{d}\ (X,c)
  22. { | ( X n , Y n ) - ( X n , c ) | 𝑝 0 , ( X n , c ) 𝑑 ( X , c ) . \begin{cases}\left|(X_{n},Y_{n})-(X_{n},c)\right|\ \xrightarrow{p}\ 0,\\ (X_{n},c)\ \xrightarrow{d}\ (X,c).\end{cases}
  23. X n 𝑝 X , Y n 𝑝 Y ( X n , Y n ) 𝑝 ( X , Y ) X_{n}\ \xrightarrow{p}\ X,\ \ Y_{n}\ \xrightarrow{p}\ Y\ \quad\Rightarrow\quad% (X_{n},Y_{n})\ \xrightarrow{p}\ (X,Y)
  24. Pr ( | ( X n , Y n ) - ( X , Y ) | ε ) \displaystyle\operatorname{Pr}\left(\left|(X_{n},Y_{n})-(X,Y)\right|\geq% \varepsilon\right)

Proofs_related_to_chi-squared_distribution.html

  1. for y < 0 , P ( Y < y ) = 0 and for y 0 , P ( Y < y ) = P ( X 2 < y ) = P ( | X | < y ) = P ( - y < X < y ) = F X ( y ) - F X ( - y ) = F X ( y ) - ( 1 - F X ( y ) ) = 2 F X ( y ) - 1 \begin{aligned}\displaystyle\,\text{for}~{}y<0,&\displaystyle~{}~{}P(Y<y)=0~{}% ~{}\,\text{and}\\ \displaystyle\,\text{for}~{}y\geq 0,&\displaystyle~{}~{}P(Y<y)=P(X^{2}<y)=P(|X% |<\sqrt{y})=P(-\sqrt{y}<X<\sqrt{y})\\ &\displaystyle=F_{X}(\sqrt{y})-F_{X}(-\sqrt{y})=F_{X}(\sqrt{y})-(1-F_{X}(\sqrt% {y}))=2F_{X}(\sqrt{y})-1\end{aligned}
  2. f Y ( y ) \displaystyle f_{Y}(y)
  3. F F
  4. f f
  5. Y = X 2 χ 1 2 . Y=X^{2}\sim\chi^{2}_{1}.
  6. f Y ( y ) = i f X ( g i - 1 ( y ) ) | d g i - 1 ( y ) d y | . f_{Y}(y)=\sum_{i}f_{X}(g_{i}^{-1}(y))\left|\frac{dg_{i}^{-1}(y)}{dy}\right|.
  7. Y \scriptstyle Y
  8. X \scriptstyle X
  9. f Y ( y ) = 2 f X ( g - 1 ( y ) ) | d g - 1 ( y ) d y | . f_{Y}(y)=2f_{X}(g^{-1}(y))\left|\frac{dg^{-1}(y)}{dy}\right|.
  10. x = g - 1 ( y ) = y x=g^{-1}(y)=\sqrt{y}
  11. d g - 1 ( y ) d y = 1 2 y . \frac{dg^{-1}(y)}{dy}=\frac{1}{2\sqrt{y}}.
  12. f Y ( y ) = 2 1 2 π e - y / 2 1 2 y = 1 2 π y e - y / 2 . f_{Y}(y)=2\frac{1}{\sqrt{2\pi}}e^{-y/2}\frac{1}{2\sqrt{y}}=\frac{1}{\sqrt{2\pi y% }}e^{-y/2}.
  13. Γ ( 1 2 ) = ( π ) \Gamma(\frac{1}{2})=\sqrt{(}\pi)
  14. x x
  15. y y
  16. x χ 1 2 x\sim\chi^{2}_{1}
  17. y χ 1 2 y\sim\chi^{2}_{1}
  18. x x
  19. y y
  20. f ( x ) = 1 2 1 2 Γ ( 1 2 ) x - 1 2 e - x 2 f(x)=\frac{1}{2^{\frac{1}{2}}\Gamma(\frac{1}{2})}x^{-\frac{1}{2}}e^{-\frac{x}{% 2}}
  21. f ( y ) = 1 2 1 2 Γ ( 1 2 ) y - 1 2 e - y 2 f(y)=\frac{1}{2^{\frac{1}{2}}\Gamma(\frac{1}{2})}y^{-\frac{1}{2}}e^{-\frac{y}{% 2}}
  22. x x
  23. y y
  24. f ( x , y ) = 1 2 π ( x y ) - 1 2 e - x + y 2 f(x,y)=\frac{1}{2\pi}(xy)^{-\frac{1}{2}}e^{-\frac{x+y}{2}}
  25. Γ ( 1 2 ) 2 \Gamma(\frac{1}{2})^{2}
  26. π \pi
  27. A = x y A=xy
  28. B = x + y B=x+y
  29. x = B + B 2 - 4 A 2 x=\frac{B+\sqrt{B^{2}-4A}}{2}
  30. y = B - B 2 - 4 A 2 y=\frac{B-\sqrt{B^{2}-4A}}{2}
  31. x = B - B 2 - 4 A 2 x=\frac{B-\sqrt{B^{2}-4A}}{2}
  32. y = B + B 2 - 4 A 2 y=\frac{B+\sqrt{B^{2}-4A}}{2}
  33. Jacobian ( x , y A , B ) = | - ( B 2 - 4 A ) - 1 2 1 + B ( B 2 - 4 A ) - 1 2 2 ( B 2 - 4 A ) - 1 2 1 - B ( B 2 - 4 A ) - 1 2 2 | = ( B 2 - 4 A ) - 1 2 \operatorname{Jacobian}\left(\frac{x,y}{A,B}\right)=\begin{vmatrix}-(B^{2}-4A)% ^{-\frac{1}{2}}&\frac{1+B(B^{2}-4A)^{-\frac{1}{2}}}{2}\\ (B^{2}-4A)^{-\frac{1}{2}}&\frac{1-B(B^{2}-4A)^{-\frac{1}{2}}}{2}\\ \end{vmatrix}=(B^{2}-4A)^{-\frac{1}{2}}
  34. f ( x , y ) f(x,y)
  35. f ( A , B ) f(A,B)
  36. f ( A , B ) = 2 × 1 2 π A - 1 2 e - B 2 ( B 2 - 4 A ) - 1 2 f(A,B)=2\times\frac{1}{2\pi}A^{-\frac{1}{2}}e^{-\frac{B}{2}}(B^{2}-4A)^{-\frac% {1}{2}}
  37. A A
  38. B B
  39. x + y x+y
  40. f ( B ) = 2 × e - B 2 2 π 0 B 2 4 A - 1 2 ( B 2 - 4 A ) - 1 2 d A f(B)=2\times\frac{e^{-\frac{B}{2}}}{2\pi}\int_{0}^{\frac{B^{2}}{4}}A^{-\frac{1% }{2}}(B^{2}-4A)^{-\frac{1}{2}}dA
  41. A = B 2 4 sin 2 ( t ) A=\frac{B^{2}}{4}\sin^{2}(t)
  42. f ( B ) = 2 × e - B 2 2 π 0 π 2 d t f(B)=2\times\frac{e^{-\frac{B}{2}}}{2\pi}\int_{0}^{\frac{\pi}{2}}\,dt
  43. f ( B ) = e - B 2 2 f(B)=\frac{e^{-\frac{B}{2}}}{2}
  44. x i x_{i}
  45. P ( Q ) d Q = 𝒱 i = 1 k ( N ( x i ) d x i ) = 𝒱 e - ( x 1 2 + x 2 2 + + x k 2 ) / 2 ( 2 π ) k / 2 d x 1 d x 2 d x k P(Q)\,dQ=\int_{\mathcal{V}}\prod_{i=1}^{k}(N(x_{i})\,dx_{i})=\int_{\mathcal{V}% }\frac{e^{-(x_{1}^{2}+x_{2}^{2}+\cdots+x_{k}^{2})/2}}{(2\pi)^{k/2}}\,dx_{1}\,% dx_{2}\cdots dx_{k}
  46. N ( x ) N(x)
  47. 𝒱 \mathcal{V}
  48. Q = i = 1 k x i 2 Q=\sum_{i=1}^{k}x_{i}^{2}
  49. R = Q R=\sqrt{Q}
  50. P ( Q ) d Q = e - Q / 2 ( 2 π ) k / 2 𝒱 d x 1 d x 2 d x k P(Q)\,dQ=\frac{e^{-Q/2}}{(2\pi)^{k/2}}\int_{\mathcal{V}}dx_{1}\,dx_{2}\cdots dx% _{k}
  51. d R = d Q 2 Q 1 / 2 . dR=\frac{dQ}{2Q^{1/2}}.
  52. A = k R k - 1 π k / 2 Γ ( k / 2 + 1 ) A=\frac{kR^{k-1}\pi^{k/2}}{\Gamma(k/2+1)}
  53. Γ ( z + 1 ) = z Γ ( z ) \Gamma(z+1)=z\Gamma(z)
  54. P ( Q ) d Q = e - Q / 2 ( 2 π ) k / 2 A d R = 1 2 k / 2 Γ ( k / 2 ) Q k / 2 - 1 e - Q / 2 d Q P(Q)\,dQ=\frac{e^{-Q/2}}{(2\pi)^{k/2}}A\,dR=\frac{1}{2^{k/2}\Gamma(k/2)}Q^{k/2% -1}e^{-Q/2}\,dQ

Properties_of_concrete.html

  1. E c = 33 w c 1.5 f c E_{c}=33w_{c}^{1.5}\sqrt{f^{\prime}_{c}}
  2. w c = w_{c}=
  3. 90 lb ft 3 w c 155 lb ft 3 90\frac{\textrm{lb}}{\textrm{ft}^{3}}\leq w_{c}\leq 155\frac{\textrm{lb}}{% \textrm{ft}^{3}}
  4. f c = f^{\prime}_{c}=
  5. 57000 f c 57000\sqrt{f^{\prime}_{c}}
  6. E c = 33000 K 1 w c 1.5 f c E_{c}=33000K_{1}w_{c}^{1.5}\sqrt{f^{\prime}_{c}}
  7. K 1 = K_{1}=
  8. w c = w_{c}=
  9. 0.090 kip ft 3 w c 0.155 kip ft 3 0.090\frac{\textrm{kip}}{\textrm{ft}^{3}}\leq w_{c}\leq 0.155\frac{\textrm{kip% }}{\textrm{ft}^{3}}
  10. f y 15.0 kip in 2 f_{y}\leq 15.0\frac{\textrm{kip}}{\textrm{in}^{2}}
  11. f c = f^{\prime}_{c}=
  12. E c = 1820 f c E_{c}=1820\sqrt{f^{\prime}_{c}}

Prosolvable_group.html

  1. 𝐐 p \mathbf{Q}_{p}
  2. Gal ( 𝐐 ¯ p / 𝐐 p ) \,\text{Gal}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})
  3. 𝐐 ¯ p \overline{\mathbf{Q}}_{p}
  4. 𝐐 p \mathbf{Q}_{p}
  5. L L
  6. 𝐐 p \mathbf{Q}_{p}
  7. Gal ( L / 𝐐 p ) \,\text{Gal}(L/\mathbf{Q}_{p})
  8. Gal ( L / 𝐐 p ) = ( R Q ) P \,\text{Gal}(L/\mathbf{Q}_{p})=(R\rtimes Q)\rtimes P
  9. P P
  10. f f
  11. f 𝐍 f\in\mathbf{N}
  12. Q Q
  13. p f - 1 p^{f}-1
  14. R R
  15. p p
  16. Gal ( L / 𝐐 p ) \,\text{Gal}(L/\mathbf{Q}_{p})

Protective_index.html

  1. Protective index = TD 50 ED 50 \mbox{Protective index}~{}=\frac{\mathrm{TD}_{50}}{\mathrm{ED}_{50}}

Protein_I-sites.html

  1. D p q = i j l o g [ P i j ( p ) + α F i ( 1 + α ) F i ] l o g [ k ϵ q P i j ( k ) + α F i ( N q + α ) F i ] D_{pq}=\sum_{ij}log\left[\dfrac{P_{ij}(p)+\alpha F_{i}}{(1+\alpha)F_{i}}\right% ]log\left[\dfrac{\sum_{k\epsilon q}P_{ij}(k)+\alpha^{\prime}F_{i}}{(N_{q}+% \alpha^{\prime})F_{i}}\right]
  2. d m e = i = 1 L j = i - 5 i + 5 ( α i j s 1 - α i j s 2 ) 2 N dme=\sqrt{\dfrac{\sum\limits_{i=1}^{L}\sum\limits_{j=i-5}^{i+5}(\alpha_{i% \rightarrow j}^{s1}-\alpha_{i\rightarrow j}^{s2})^{2}}{N}}
  3. m d a ( L ) = m a x i = 1 , L - 1 ( Δ Φ i + 1 , Δ Ψ i ) mda(L)=max_{i=1,L-1}(\Delta\Phi_{i+1},\Delta\Psi_{i})

Prune_and_search.html

  1. 1 / ( 1 - ( 1 - p ) ) = 1 / p . 1/(1-(1-p))=1/p.

Pseudo-determinant.html

  1. | 𝐀 | + = lim α 0 | 𝐀 + α 𝐈 | α n - rank ( 𝐀 ) |\mathbf{A}|_{+}=\lim_{\alpha\to 0}\frac{|\mathbf{A}+\alpha\mathbf{I}|}{\alpha% ^{n-\operatorname{rank}(\mathbf{A})}}
  2. ( a x + b ) ( c x + d ) - 1 (ax+b)(cx+d)^{-1}
  3. a , b , c , d 𝒢 ( p , q ) a,b,c,d\in\mathcal{G}(p,q)
  4. [ f ] = : : [ a b c d ] [f]=::\begin{bmatrix}a&b\\ c&d\end{bmatrix}
  5. p d e t : : [ a b c d ] = a d - b c pdet::\begin{bmatrix}a&b\\ c&d\end{bmatrix}=ad^{\dagger}-bc^{\dagger}
  6. p d e t [ f ] > 0 pdet[f]>0
  7. p d e t [ f ] < 0 pdet[f]<0
  8. A A
  9. A A
  10. | 𝐀 | + |\mathbf{A}|_{+}

Pseudoideal.html

  1. \subseteq
  2. \subseteq
  3. \subseteq
  4. \subseteq

Pseudoisotopy_theorem.html

  1. M × { 0 } M × [ 0 , 1 ] M\times\{0\}\cup\partial M\times[0,1]
  2. f : M × [ 0 , 1 ] M × [ 0 , 1 ] f:M\times[0,1]\to M\times[0,1]
  3. M × { 1 } M\times\{1\}
  4. g g
  5. M × { t } M\times\{t\}
  6. t [ 0 , 1 ] t\in[0,1]
  7. π [ 0 , 1 ] f t \pi_{[0,1]}\circ f_{t}

Pulse-code_modulation.html

  1. f s / 2 f_{s}/2

Pulse_wave_velocity.html

  1. F = m a F=ma
  2. PWV = E inc h 2 r ρ \mathrm{PWV}=\sqrt{\dfrac{E_{\mathrm{inc}}\cdot h}{2r\rho}}
  3. h h
  4. r r
  5. d v ( d r - 1 ) d x d t \operatorname{d}\!v(\operatorname{d}\!r^{-1})\operatorname{d}\!x\cdot% \operatorname{d}\!t
  6. d V / d P \operatorname{d}\!V/\operatorname{d}\!P
  7. V V
  8. d V / d P \operatorname{d}\!V/\operatorname{d}\!P
  9. V V
  10. Δ x \Delta x
  11. Δ t \Delta t
  12. PWV = Δ x Δ t \mathrm{PWV}=\dfrac{\Delta x}{\Delta t}
  13. δ P \delta P
  14. δ y = y 2 δ P / ( E c ) \delta y=y^{2}\delta P/(Ec)
  15. y y
  16. δ V = 2 π y 3 δ P / ( E c ) \delta V=2\pi y^{3}\delta P/(Ec)
  17. V V
  18. 2 y / E c = d V / ( V d P ) 2y/Ec=\operatorname{d}\!V/(V\operatorname{d}\!P)
  19. c c
  20. h h
  21. y y
  22. r r
  23. d V / ( V d P ) \operatorname{d}\!V/(V\operatorname{d}\!P)
  24. PWV = d P V ρ d V \mathrm{PWV}=\sqrt{\dfrac{\operatorname{d}\!P\cdot V}{\rho\cdot\operatorname{d% }\!V}}
  25. Z c Z_{\mathrm{c}}
  26. P P
  27. v v
  28. PWV = P i / ( v i ρ ) = Z c / ρ \mathrm{PWV}=P_{\mathrm{i}}/\left(v_{\mathrm{i}}\cdot\rho\right)=Z_{\mathrm{c}% }/\rho
  29. ρ \rho
  30. h h
  31. E inc E_{\mathrm{inc}}
  32. P P
  33. P W V PWV
  34. r r
  35. t t
  36. V V
  37. v v
  38. Z c Z_{\mathrm{c}}

Purity_(quantum_mechanics).html

  1. γ Tr ( ρ 2 ) \gamma\,\equiv\,\mbox{Tr}~{}(\rho^{2})\,
  2. ρ \rho\,
  3. 1 / d 1/d\,
  4. d d\,
  5. S L S_{L}\,
  6. γ = 1 - S L . \gamma=1-S_{L}\,.

Purnell_equation.html

  1. R s = N 2 4 ( α - 1 α ) ( k 2 1 + k 2 ) R_{s}=\frac{\sqrt{N_{2}}}{4}\left(\frac{\alpha-1}{\alpha}\right)\left(\frac{k^% {\prime}_{2}}{1+k^{\prime}_{2}}\right)

PVIFA.html

  1. W W
  2. X X
  3. Y Y
  4. Z Z
  5. A A

Pythagorean_theorem.html

  1. a 2 + b 2 = c 2 , a^{2}+b^{2}=c^{2},
  2. a 2 + b 2 = c 2 . a^{2}+b^{2}=c^{2}.\,
  3. c = a 2 + b 2 . c=\sqrt{a^{2}+b^{2}}.\,
  4. a = c 2 - b 2 a=\sqrt{c^{2}-b^{2}}\,
  5. b = c 2 - a 2 . b=\sqrt{c^{2}-a^{2}}.\,
  6. B C A B = B H B C and A C A B = A H A C . \frac{BC}{AB}=\frac{BH}{BC}\,\text{ and }\frac{AC}{AB}=\frac{AH}{AC}.\,
  7. B C 2 = A B × B H and A C 2 = A B × A H . {BC}^{2}={AB}\times{BH}\,\text{ and }{AC}^{2}={AB}\times{AH}.\,
  8. B C 2 + A C 2 = A B × B H + A B × A H = A B × ( A H + B H ) = A B 2 , {BC}^{2}+{AC}^{2}={AB}\times{BH}+{AB}\times{AH}={AB}\times({AH}+{BH})={AB}^{2}% ,\,\!
  9. B C 2 + A C 2 = A B 2 . {BC}^{2}+{AC}^{2}={AB}^{2}\ .\,\!
  10. 1 2 a b \tfrac{1}{2}ab
  11. ( b - a ) 2 + 4 a b 2 = ( b - a ) 2 + 2 a b = a 2 + b 2 . (b-a)^{2}+4\frac{ab}{2}=(b-a)^{2}+2ab=a^{2}+b^{2}.\,
  12. c 2 = a 2 + b 2 . c^{2}=a^{2}+b^{2}.\,
  13. ( b + a ) 2 = c 2 + 4 a b 2 = c 2 + 2 a b , (b+a)^{2}=c^{2}+4\frac{ab}{2}=c^{2}+2ab,\,
  14. c 2 = ( b + a ) 2 - 2 a b = a 2 + b 2 . c^{2}=(b+a)^{2}-2ab=a^{2}+b^{2}.\,
  15. 1 2 ( b + a ) 2 . \frac{1}{2}(b+a)^{2}.
  16. 1 2 \frac{1}{2}
  17. d y d x = x y . \frac{dy}{dx}=\frac{x}{y}.
  18. y d y - x d x = 0. y\cdot dy-x\cdot dx=0.\,
  19. y 2 - x 2 = C , y^{2}-x^{2}=C,\,
  20. y 2 = x 2 + a 2 . y^{2}=x^{2}+a^{2}.\,
  21. z = x + i y , z=x+iy,\,
  22. r = | z | = x 2 + y 2 . r=|z|=\sqrt{x^{2}+y^{2}}.\,
  23. r 2 = x 2 + y 2 . r^{2}=x^{2}+y^{2}.\,
  24. | z 1 - z 2 | = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 , |z_{1}-z_{2}|=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}},\,
  25. | z 1 - z 2 | 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 . |z_{1}-z_{2}|^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}.\,
  26. ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 . \sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}.
  27. A = ( a 1 , a 2 , , a n ) A\,=\,(a_{1},a_{2},\dots,a_{n})
  28. B = ( b 1 , b 2 , , b n ) B\,=\,(b_{1},b_{2},\dots,b_{n})
  29. ( a 1 - b 1 ) 2 + ( a 2 - b 2 ) 2 + + ( a n - b n ) 2 = i = 1 n ( a i - b i ) 2 . \sqrt{(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots+(a_{n}-b_{n})^{2}}=\sqrt{\sum% _{i=1}^{n}(a_{i}-b_{i})^{2}}.
  30. x = r cos θ , y = r sin θ . x=r\cos\theta,\ y=r\sin\theta.\,
  31. s 2 = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 = ( r 1 cos θ 1 - r 2 cos θ 2 ) 2 + ( r 1 sin θ 1 - r 2 sin θ 2 ) 2 . s^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}=(r_{1}\cos\theta_{1}-r_{2}\cos\theta% _{2})^{2}+(r_{1}\sin\theta_{1}-r_{2}\sin\theta_{2})^{2}.\,
  32. s 2 = r 1 2 + r 2 2 - 2 r 1 r 2 ( cos θ 1 cos θ 2 + sin θ 1 sin θ 2 ) = r 1 2 + r 2 2 - 2 r 1 r 2 cos ( θ 1 - θ 2 ) = r 1 2 + r 2 2 - 2 r 1 r 2 cos Δ θ , \begin{aligned}\displaystyle s^{2}&\displaystyle=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{% 2}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}\right)\\ &\displaystyle=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos\left(\theta_{1}-\theta_{2}% \right)\\ &\displaystyle=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos\Delta\theta,\end{aligned}\,
  33. s 2 = r 1 2 + r 2 2 . s^{2}=r_{1}^{2}+r_{2}^{2}.\,
  34. sin θ = b c , cos θ = a c . \sin\theta=\frac{b}{c},\quad\cos\theta=\frac{a}{c}.
  35. cos 2 θ + sin 2 θ = a 2 + b 2 c 2 = 1 , {\cos}^{2}\theta+{\sin}^{2}\theta=\frac{a^{2}+b^{2}}{c^{2}}=1,
  36. 𝐚 × 𝐛 2 + ( 𝐚 𝐛 ) 2 = 𝐚 2 𝐛 2 . \|\mathbf{a}\times\mathbf{b}\|^{2}+(\mathbf{a}\cdot\mathbf{b})^{2}=\|\mathbf{a% }\|^{2}\|\mathbf{b}\|^{2}.\,
  37. 𝐚 × 𝐛 = a b 𝐧 sin θ 𝐚 𝐛 = a b cos θ , \begin{aligned}\displaystyle\mathbf{a}\times\mathbf{b}&\displaystyle=ab\mathbf% {n}\sin{\theta}\\ \displaystyle\mathbf{a}\cdot\mathbf{b}&\displaystyle=ab\cos{\theta},\end{aligned}
  38. 𝐚 × 𝐛 2 = 𝐚 2 𝐛 2 - ( 𝐚 𝐛 ) 2 . \|\mathbf{a}\times\mathbf{b}\|^{2}=\|\mathbf{a}\|^{2}\|\mathbf{b}\|^{2}-(% \mathbf{a}\cdot\mathbf{b})^{2}.\,
  39. A a 2 = B b 2 = C c 2 , \frac{A}{a^{2}}=\frac{B}{b^{2}}=\frac{C}{c^{2}}\,,
  40. A + B = a 2 c 2 C + b 2 c 2 C . \Rightarrow A+B=\frac{a^{2}}{c^{2}}C+\frac{b^{2}}{c^{2}}C\,.
  41. a 2 + b 2 - 2 a b cos θ = c 2 , a^{2}+b^{2}-2ab\cos{\theta}=c^{2},\,
  42. a 2 + b 2 = c ( r + s ) . a^{2}+b^{2}=c(r+s)\ .
  43. c a = a r . \frac{c}{a}=\frac{a}{r}\ .
  44. c b = b s . \frac{c}{b}=\frac{b}{s}\ .
  45. c r + c s = a 2 + b 2 , cr+cs=a^{2}+b^{2}\ ,
  46. θ \theta
  47. B D ¯ 2 = B C ¯ 2 + C D ¯ 2 , \overline{BD}^{\,2}=\overline{BC}^{\,2}+\overline{CD}^{\,2}\ ,
  48. A D ¯ 2 = A B ¯ 2 + B D ¯ 2 , \overline{AD}^{\,2}=\overline{AB}^{\,2}+\overline{BD}^{\,2}\ ,
  49. A D ¯ 2 = A B ¯ 2 + B C ¯ 2 + C D ¯ 2 . \overline{AD}^{\,2}=\overline{AB}^{\,2}+\overline{BC}^{\,2}+\overline{CD}^{\,2% }\ .
  50. 𝐯 2 = k = 1 3 𝐯 k 2 . \|\mathbf{v}\|^{2}=\sum_{k=1}^{3}\|\mathbf{v}_{k}\|^{2}.
  51. 𝐯 , 𝐰 \langle\mathbf{v},\mathbf{w}\rangle
  52. 𝐯 𝐯 , 𝐯 . \lVert\mathbf{v}\rVert\equiv\sqrt{\langle\mathbf{v},\mathbf{v}\rangle}\,.
  53. 𝐯 + 𝐰 2 = 𝐯 2 + 𝐰 2 . \left\|\mathbf{v}+\mathbf{w}\right\|^{2}=\left\|\mathbf{v}\right\|^{2}+\left\|% \mathbf{w}\right\|^{2}.
  54. 𝐯 + 𝐰 2 = 𝐯 + 𝐰 , 𝐯 + 𝐰 = 𝐯 , 𝐯 + 𝐰 , 𝐰 + 𝐯 , 𝐰 + 𝐰 , 𝐯 = 𝐯 2 + 𝐰 2 , \left\|\mathbf{v}+\mathbf{w}\right\|^{2}=\langle\mathbf{v+w},\ \mathbf{v+w}% \rangle=\langle\mathbf{v},\ \mathbf{v}\rangle+\langle\mathbf{w},\ \mathbf{w}% \rangle+\langle\mathbf{v,\ w}\rangle+\langle\mathbf{w,\ v}\rangle\ =\left\|% \mathbf{v}\right\|^{2}+\left\|\mathbf{w}\right\|^{2},
  55. 2 𝐯 2 + 2 𝐰 2 = 𝐯 + 𝐰 2 + 𝐯 - 𝐰 2 , 2\|\mathbf{v}\|^{2}+2\|\mathbf{w}\|^{2}=\|\mathbf{v+w}\|^{2}+\|\mathbf{v-w}\|^% {2}\ ,
  56. k = 1 n 𝐯 k 2 = k = 1 n 𝐯 k 2 . \|\sum_{k=1}^{n}\mathbf{v}_{k}\|^{2}=\sum_{k=1}^{n}\|\mathbf{v}_{k}\|^{2}.
  57. cos ( c R ) = cos ( a R ) cos ( b R ) . \cos\left(\frac{c}{R}\right)=\cos\left(\frac{a}{R}\right)\cos\left(\frac{b}{R}% \right).
  58. cos ( c R ) = cos ( a R ) cos ( b R ) + sin ( a R ) sin ( b R ) cos γ . \cos\left(\frac{c}{R}\right)=\cos\left(\frac{a}{R}\right)\cos\left(\frac{b}{R}% \right)+\sin\left(\frac{a}{R}\right)\sin\left(\frac{b}{R}\right)\cos\gamma\ .
  59. cos x = 1 - x 2 2 + O ( x 4 ) as x 0 , \cos x=1-\frac{x^{2}}{2}+O(x^{4})\,\text{ as }x\to 0\ ,
  60. 1 - 1 2 ( c R ) 2 + O ( 1 R 4 ) = [ 1 - 1 2 ( a R ) 2 + O ( 1 R 4 ) ] [ 1 - 1 2 ( b R ) 2 + O ( 1 R 4 ) ] as R . 1-\frac{1}{2}\left(\frac{c}{R}\right)^{2}+O\left(\frac{1}{R^{4}}\right)=\left[% 1-\frac{1}{2}\left(\frac{a}{R}\right)^{2}+O\left(\frac{1}{R^{4}}\right)\right]% \left[1-\frac{1}{2}\left(\frac{b}{R}\right)^{2}+O\left(\frac{1}{R^{4}}\right)% \right]\,\text{ as }R\to\infty\ .
  61. ( c R ) 2 = ( a R ) 2 + ( b R ) 2 + O ( 1 R 4 ) as R . \left(\frac{c}{R}\right)^{2}=\left(\frac{a}{R}\right)^{2}+\left(\frac{b}{R}% \right)^{2}+O\left(\frac{1}{R^{4}}\right)\,\text{ as }R\to\infty\ .
  62. c 2 = a 2 + b 2 + O ( 1 R 2 ) as R . c^{2}=a^{2}+b^{2}+O\left(\frac{1}{R^{2}}\right)\,\text{ as }R\to\infty\ .
  63. cosh c = cosh a cosh b \cosh c=\cosh a\,\cosh b
  64. cosh c = cosh a cosh b - sinh a sinh b cos γ , \cosh c=\cosh a\ \cosh b-\sinh a\ \sinh b\ \cos\gamma\ ,
  65. d s 2 = d x 2 + d y 2 + d z 2 , ds^{2}=dx^{2}+dy^{2}+dz^{2},\,
  66. d s 2 = i , j n g i j d x i d x j ds^{2}=\sum_{i,j}^{n}g_{ij}\,dx_{i}\,dx_{j}
  67. d s 2 = d r 2 + r 2 d θ 2 . ds^{2}=dr^{2}+r^{2}d\theta^{2}\ .

Q-guidance.html

  1. V T B G = V c - V m V_{TBG}=V_{c}-V_{m}
  2. d V T B G d t = - a T - Q V T B G \frac{dV_{TBG}}{dt}=-a_{T}-QV_{TBG}
  3. Q = V c r | r T , t f Q=\left.\frac{\partial V_{c}}{\partial r}\right|_{r_{T},t_{f}}
  4. ω = κ ( a T × V T B G ) \omega=\kappa(a_{T}\times V_{TBG})
  5. κ \kappa
  6. ω = κ ( V T B G × d V T B G d t ) \omega=\kappa(V_{TBG}\times\frac{dV_{TBG}}{dt})

QI_(G_series).html

  1. Ax. 1. P ( φ ) x [ φ ( x ) ψ ( x ) ] P ( ψ ) Ax. 2. P ( ¬ φ ) ¬ P ( φ ) Th. 1. P ( φ ) x [ φ ( x ) ] Df. 1. G ( x ) φ [ P ( φ ) φ ( x ) ] Ax. 3. P ( G ) Th. 2. x G ( x ) Df. 2. φ ess x φ ( x ) ψ { ψ ( x ) x [ φ ( x ) ψ ( x ) ] } Ax. 4. P ( φ ) P ( φ ) Th. 3. G ( x ) G ess x Df. 3. E ( x ) φ [ φ ess x x φ ( x ) ] Ax. 5. P ( E ) Th. 4. x G ( x ) \begin{array}[]{rl}\mbox{Ax. 1.}&P(\varphi)\land\Box\;\forall x[\varphi(x)% \rightarrow\psi(x)]\rightarrow P(\psi)\\ \mbox{Ax. 2.}&P(\neg\varphi)\leftrightarrow\neg P(\varphi)\\ \mbox{Th. 1.}&P(\varphi)\rightarrow\Diamond\;\exists x\;[\varphi(x)]\\ \mbox{Df. 1.}&G(x)\iff\forall\varphi[P(\varphi)\rightarrow\varphi(x)]\\ \mbox{Ax. 3.}&P(G)\\ \mbox{Th. 2.}&\Diamond\;\exists x\;G(x)\\ \mbox{Df. 2.}&\varphi\;\operatorname{ess}\;x\iff\varphi(x)\land\forall\psi\{% \psi(x)\rightarrow\Box\;\forall x[\varphi(x)\rightarrow\psi(x)]\}\\ \mbox{Ax. 4.}&P(\varphi)\rightarrow\Box\;P(\varphi)\\ \mbox{Th. 3.}&G(x)\rightarrow G\;\operatorname{ess}\;x\\ \mbox{Df. 3.}&E(x)\iff\forall\varphi[\varphi\;\operatorname{ess}\;x\rightarrow% \Box\;\exists x\;\varphi(x)]\\ \mbox{Ax. 5.}&P(E)\\ \mbox{Th. 4.}&\Box\;\exists x\;G(x)\end{array}

QMA.html

  1. QMA ( c , s ) \mbox{QMA}~{}(c,s)
  2. x L \forall x\in L
  3. | ψ |\psi\rangle
  4. ( | x , | ψ ) (|x\rangle,|\psi\rangle)
  5. c c
  6. x L \forall x\notin L
  7. | ψ |\psi\rangle
  8. ( | x , | ψ ) (|x\rangle,|\psi\rangle)
  9. s s
  10. | ψ |\psi\rangle
  11. QMA \mbox{QMA}~{}
  12. QMA ( 2 / 3 , 1 / 3 ) \mbox{QMA}~{}({2}/{3},1/3)
  13. c c
  14. s s
  15. c c
  16. s s
  17. q ( n ) q(n)
  18. r ( n ) r(n)
  19. QMA ( 2 3 , 1 3 ) = QMA ( 1 2 + 1 q ( n ) , 1 2 - 1 q ( n ) ) = QMA ( 1 - 2 - r ( n ) , 2 - r ( n ) ) \mbox{QMA}~{}\left(\frac{2}{3},\frac{1}{3}\right)=\mbox{QMA}~{}\left(\frac{1}{% 2}+\frac{1}{q(n)},\frac{1}{2}-\frac{1}{q(n)}\right)=\mbox{QMA}~{}(1-2^{-r(n)},% 2^{-r(n)})
  20. 2 n × 2 n 2^{n}\times 2^{n}
  21. a < b [ 0 , 1 ] a<b\in[0,1]
  22. 1 b - a = O ( n c ) \frac{1}{b-a}=O(n^{c})
  23. H = i h i Z i + i < j J i j Z i Z i + i < j K i j X i X i H=\sum_{i}h_{i}Z_{i}+\sum_{i<j}J^{ij}Z_{i}Z_{i}+\sum_{i<j}K^{ij}X_{i}X_{i}
  24. Z , X Z,X
  25. σ z , σ x \sigma_{z},\sigma_{x}
  26. P NP MA QCMA QMA PP PSPACE \mbox{P}~{}\subseteq\mbox{NP}~{}\subseteq\mbox{MA}~{}\subseteq\mbox{QCMA}~{}% \subseteq\mbox{QMA}~{}\subseteq\mbox{PP}~{}\subseteq\mbox{PSPACE}~{}

Quadratic_algebra.html

  1. A = T ( V ) / S A=T(V)/\langle S\rangle

Quadratic_integer.html

  1. B B
  2. C C
  3. 2 \sqrt{2}
  4. i = 1 i=\sqrt{–1}
  5. - 1 + 3 2 -1+\frac{\sqrt{–3}}{2}
  6. D D
  7. \mathbb{Q}
  8. ( D ) , \mathbb{Q}(\sqrt{D}),
  9. D D
  10. E E
  11. ( D ) , \mathbb{Q}(\sqrt{D}),
  12. ( D ) . \mathbb{Q}(\sqrt{D}).
  13. D D
  14. a a
  15. ( D ) = ( a 2 D ) . \mathbb{Q}(\sqrt{D})=\mathbb{Q}(\sqrt{a^{2}D}).
  16. a + ω b a+ωb
  17. a a
  18. b b
  19. ω ω
  20. ω = { D if D 2 , 3 ( mod 4 ) 1 + D 2 if D 1 ( mod 4 ) \omega=\begin{cases}\sqrt{D}&\mbox{if }~{}D\equiv 2,3\;\;(\mathop{{\rm mod}}4)% \\ {{1+\sqrt{D}}\over 2}&\mbox{if }~{}D\equiv 1\;\;(\mathop{{\rm mod}}4)\end{cases}
  21. D D
  22. D 0 ( mod 4 ) D\equiv 0\;\;(\mathop{{\rm mod}}4)
  23. 2 + 3 \sqrt{2}+\sqrt{3}
  24. ( D ) \mathbb{Q}(\sqrt{D})
  25. a + b D a+b\sqrt{D}
  26. a a
  27. b b
  28. D 1 ( m o d 4 ) D≡1(mod4)
  29. D > 0 D>0
  30. a + b D a+b\sqrt{D}
  31. a + b D ¯ = a - b D . \overline{a+b\sqrt{D}}=a-b\sqrt{D}.
  32. ( D ) . \mathbb{Q}(\sqrt{D}).
  33. ( D ) \mathbb{Q}(\sqrt{D})
  34. 1 1
  35. 1 –1
  36. ( D ) \mathbb{Q}(\sqrt{D})
  37. D = 1 D=–1
  38. 1 , 1 , 1 , 1 1,–1,\sqrt{–1},–\sqrt{–1}
  39. D = 3 D=–3
  40. ± 1 , ± 1 ± 3 2 ±1,±1±\frac{\sqrt{–3}}{2}
  41. D D
  42. 1 1
  43. 1 –1
  44. D > 0 D>0
  45. ( D ) \mathbb{Q}(\sqrt{D})
  46. i i
  47. u u
  48. u u
  49. u ¯ , \overline{u},
  50. - u -u
  51. - u ¯ . -\overline{u}.
  52. a + b D a+b\sqrt{D}
  53. a a
  54. b b
  55. D D
  56. 1 + 2 1+\sqrt{2}
  57. 2 + 3 2+\sqrt{3}
  58. 1 + 5 2 1+\frac{\sqrt{5}}{2}
  59. 5 + 2 6 5+2\sqrt{6}
  60. 8 + 3 7 8+3\sqrt{7}
  61. 3 + 10 3+\sqrt{10}
  62. 10 + 3 11 10+3\sqrt{11}
  63. 3 + 13 2 3+\frac{\sqrt{13}}{2}
  64. 15 + 4 14 15+4\sqrt{14}
  65. 4 + 15 4+\sqrt{15}
  66. D D
  67. D = 19 , 31 , 43 D=19,31,43
  68. 170 + 39 19 170+39\sqrt{19}
  69. 1520 + 273 31 1520+273\sqrt{31}
  70. 3482 + 531 43 3482+531\sqrt{43}
  71. D D
  72. 𝐐 ( D ) . \mathbf{Q}(\sqrt{D}).
  73. 𝐙 ω ω = = a + ω b : a , b 𝐙 \mathbf{Z}ωω==a+ωb:a,b∈\mathbf{Z}
  74. ω ω
  75. D \sqrt{D}
  76. 𝒪 𝐐 ( D ) . \mathcal{O}_{\mathbf{Q}(\sqrt{D})}.
  77. 𝐙 \mathbf{Z}
  78. 𝐐 ( D ) . \mathbf{Q}(\sqrt{D}).
  79. D D
  80. D > 0 D>0
  81. 𝒪 𝐐 ( D ) \mathcal{O}_{\mathbf{Q}(\sqrt{D})}
  82. 𝒪 𝐐 ( D ) \mathcal{O}_{\mathbf{Q}(\sqrt{D})}
  83. D D
  84. 𝐙 [ - 1 ] \mathbf{Z}[\sqrt{-1}]
  85. 𝒪 𝐐 ( - 3 ) = 𝐙 [ 1 + - 3 2 ] \mathcal{O}_{\mathbf{Q}(\sqrt{-3})}=\mathbf{Z}\left[{{1+\sqrt{-3}}\over 2}\right]
  86. 3 \sqrt{−3}
  87. 𝒪 𝐐 ( - 5 ) = 𝐙 [ - 5 ] , \mathcal{O}_{\mathbf{Q}(\sqrt{-5})}=\mathbf{Z}\left[\sqrt{-5}\right],
  88. 𝒪 𝐐 ( - 5 ) , \mathcal{O}_{\mathbf{Q}(\sqrt{-5})},
  89. 9 = 3 3 = ( 2 + - 5 ) ( 2 - - 5 ) . 9=3\cdot 3=(2+\sqrt{-5})(2-\sqrt{-5}).
  90. 2 + - 5 2+\sqrt{-5}
  91. 2 - - 5 2-\sqrt{-5}
  92. ± 1 ±1
  93. 3 , 1 + - 5 \langle 3,1+\sqrt{-5}\rangle
  94. 3 , 1 - - 5 \langle 3,1-\sqrt{-5}\rangle
  95. D > 0 D>0
  96. ω ω
  97. D 2 , 3 ( m o d 4 ) D≡2, 3(mod4)
  98. D = 5 D=5
  99. ω = 1 + 5 2 ω=1+\frac{\sqrt{5}}{2}
  100. n n
  101. 𝐙 s q r t q r t 61 \mathbf{Z}sqrtqrt61
  102. 𝐙 s q r t q r t 5 \mathbf{Z}sqrtqrt−5
  103. 𝒪 𝐐 ( D ) \mathcal{O}_{\mathbf{Q}(\sqrt{D})}
  104. D = 1 , 2 , 3 , 7 , 11 , 19 , 43 , 67 , 163 D=−1,−2,−3,−7,−11,−19,−43,−67,−163
  105. D > 0 D>0
  106. N ( a + b D ) = a 2 - D b 2 , N(a+b\sqrt{D})=a^{2}-Db^{2},
  107. 𝒪 𝐐 ( D ) \mathcal{O}_{\mathbf{Q}(\sqrt{D})}
  108. D D
  109. D = 1 , 2 , 3 , 7 , 11 D=−1, −2, −3, −7, −11
  110. D D
  111. D = 2 , 3 , 5 , 6 , 7 , 11 , 13 , 17 , 19 , 21 , 29 , 33 , 37 , 41 , 57 , 73 D=2,3,5,6,7,11,13,17,19,21,29,33,37,41,57,73
  112. D D
  113. D = 19 , 43 , 67 , 163 D=−19,−43,−67,−163

Quadratic_Lie_algebra.html

  1. ( . , . ) : 𝔤 𝔤 (.,.)\colon\mathfrak{g}\otimes\mathfrak{g}\to\mathbb{R}
  2. ( ( x 1 , , x n ) , ( y 1 , , y n ) ) := j x j y j ((x_{1},\dots,x_{n}),(y_{1},\dots,y_{n})):=\sum_{j}x_{j}y_{j}
  3. [ X , Y ] = Z , [ Y , Z ] = X , [ Z , X ] = Y [X,Y]=Z,\quad[Y,Z]=X,\quad[Z,X]=Y
  4. ad : 𝔤 End ( 𝔤 ) : X ( ad X : Y [ X , Y ] ) \mathrm{ad}\colon\mathfrak{g}\to\mathrm{End}(\mathfrak{g}):X\mapsto(\mathrm{ad% }_{X}\colon Y\mapsto[X,Y])
  5. k : 𝔤 𝔤 : X Y - tr ( ad X ad Y ) k\colon\mathfrak{g}\otimes\mathfrak{g}\to\mathbb{R}:X\otimes Y\mapsto-\mathrm{% tr}(\mathrm{ad}_{X}\circ\mathrm{ad}_{Y})

Quantization_of_the_electromagnetic_field.html

  1. m photon = 0 H | 𝐤 , μ = h ν | 𝐤 , μ with ν = c | 𝐤 | P EM | 𝐤 , μ = 𝐤 | 𝐤 , μ S z | 𝐤 , μ = μ | 𝐤 , μ , μ = 1 , - 1. \begin{aligned}\displaystyle m_{\textrm{photon}}&\displaystyle=0\\ \displaystyle H\,|\,\mathbf{k},\mu\,\rangle&\displaystyle=h\nu\,|\,\mathbf{k},% \mu\,\rangle\quad\hbox{with}\quad\nu=c|\mathbf{k}|\\ \displaystyle P_{\textrm{EM}}\,|\,\mathbf{k},\mu\,\rangle&\displaystyle=\hbar% \mathbf{k}|\,\mathbf{k},\mu\,\rangle\\ \displaystyle S_{z}|\,\mathbf{k},\mu\,\rangle&\displaystyle=\mu|\,\mathbf{k},% \mu\,\rangle,\quad\mu=1,-1.\\ \end{aligned}
  2. 𝐁 ( 𝐫 , t ) = s y m b o l × 𝐀 ( 𝐫 , t ) 𝐄 ( 𝐫 , t ) = - s y m b o l ϕ ( 𝐫 , t ) - 𝐀 ( 𝐫 , t ) t , \begin{aligned}\displaystyle\mathbf{B}(\mathbf{r},t)&\displaystyle=symbol{% \nabla}\times\mathbf{A}(\mathbf{r},t)\\ \displaystyle\mathbf{E}(\mathbf{r},t)&\displaystyle=-symbol{\nabla}\phi(% \mathbf{r},t)-\frac{\partial\mathbf{A}(\mathbf{r},t)}{\partial t},\\ \end{aligned}
  3. 𝐀 ( 𝐫 , t ) = 𝐤 μ = - 1 , 1 ( 𝐞 ( μ ) ( 𝐤 ) a 𝐤 ( μ ) ( t ) e i 𝐤 𝐫 + 𝐞 ¯ ( μ ) ( 𝐤 ) a ¯ 𝐤 ( μ ) ( t ) e - i 𝐤 𝐫 ) , \mathbf{A}(\mathbf{r},t)=\sum_{\mathbf{k}}\sum_{\mu=-1,1}\left(\mathbf{e}^{(% \mu)}(\mathbf{k})\,a^{(\mu)}_{\mathbf{k}}(t)\,e^{i\mathbf{k}\cdot\mathbf{r}}+% \bar{\mathbf{e}}^{(\mu)}(\mathbf{k})\,\bar{a}^{(\mu)}_{\mathbf{k}}(t)\,e^{-i% \mathbf{k}\cdot\mathbf{r}}\right),
  4. a 𝐤 ( μ ) ¯ = a - 𝐤 ( μ ) \overline{a_{\mathbf{k}}^{(\mu)}}=a_{-\mathbf{k}}^{(\mu)}
  5. k x = 2 π n x L , k y = 2 π n y L , k z = 2 π n z L , n x , n y , n z = 0 , ± 1 , ± 2 , . k_{x}=\frac{2\pi n_{x}}{L},\quad k_{y}=\frac{2\pi n_{y}}{L},\quad k_{z}=\frac{% 2\pi n_{z}}{L},\qquad n_{x},\;n_{y},\;n_{z}=0,\,\pm 1,\,\pm 2,\,\ldots\,.
  6. 𝐞 ( 1 ) - 1 2 ( 𝐞 x + i 𝐞 y ) and 𝐞 ( - 1 ) 1 2 ( 𝐞 x - i 𝐞 y ) with 𝐞 x 𝐤 = 𝐞 y 𝐤 = 0. \mathbf{e}^{(1)}\equiv\frac{-1}{\sqrt{2}}(\mathbf{e}_{x}+i\mathbf{e}_{y})\quad% \hbox{and}\quad\mathbf{e}^{(-1)}\equiv\frac{1}{\sqrt{2}}(\mathbf{e}_{x}-i% \mathbf{e}_{y})\quad\hbox{with}\quad\mathbf{e}_{x}\cdot\mathbf{k}=\mathbf{e}_{% y}\cdot\mathbf{k}=0.
  7. a 𝐤 ( μ ) ( t ) a^{(\mu)}_{\mathbf{k}}(t)
  8. a ¯ 𝐤 ( μ ) ( t ) \bar{a}^{(\mu)}_{\mathbf{k}}(t)
  9. 𝐩 ( t ) - i \hbarsymbol \mathbf{p}(t)\rightarrow-i\hbarsymbol{\nabla}
  10. a 𝐤 ( μ ) ( t ) \displaystyle a^{(\mu)}_{\mathbf{k}}(t)
  11. [ a ( μ ) ( 𝐤 ) , a ( μ ) ( 𝐤 ) ] = 0 [ a ( μ ) ( 𝐤 ) , a ( μ ) ( 𝐤 ) ] = 0 [ a ( μ ) ( 𝐤 ) , a ( μ ) ( 𝐤 ) ] = δ 𝐤 , 𝐤 δ μ , μ . \begin{aligned}\displaystyle\big[a^{(\mu)}(\mathbf{k}),\,a^{(\mu^{\prime})}(% \mathbf{k}^{\prime})\big]&\displaystyle=0\\ \displaystyle\big[{a^{\dagger}}^{(\mu)}(\mathbf{k}),\,{a^{\dagger}}^{(\mu^{% \prime})}(\mathbf{k}^{\prime})\big]&\displaystyle=0\\ \displaystyle\big[a^{(\mu)}(\mathbf{k}),\,{a^{\dagger}}^{(\mu^{\prime})}(% \mathbf{k}^{\prime})\big]&\displaystyle=\delta_{\mathbf{k},\mathbf{k}^{\prime}% }\delta_{\mu,\mu^{\prime}}.\end{aligned}
  12. [ A , B ] A B - B A \big[A,B\big]\equiv AB-BA
  13. 𝐀 ( 𝐫 ) = 𝐤 , μ 2 ω V ϵ 0 ( 𝐞 ( μ ) a ( μ ) ( 𝐤 ) e i 𝐤 𝐫 + 𝐞 ¯ ( μ ) a ( μ ) ( 𝐤 ) e - i 𝐤 𝐫 ) 𝐄 ( 𝐫 ) = i 𝐤 , μ ω 2 V ϵ 0 ( 𝐞 ( μ ) a ( μ ) ( 𝐤 ) e i 𝐤 𝐫 - 𝐞 ¯ ( μ ) a ( μ ) ( 𝐤 ) e - i 𝐤 𝐫 ) 𝐁 ( 𝐫 ) = i 𝐤 , μ 2 ω V ϵ 0 ( ( 𝐤 × 𝐞 ( μ ) ) a ( μ ) ( 𝐤 ) e i 𝐤 𝐫 - ( 𝐤 × 𝐞 ¯ ( μ ) ) a ( μ ) ( 𝐤 ) e - i 𝐤 𝐫 ) , \begin{aligned}\displaystyle\mathbf{A}(\mathbf{r})&\displaystyle=\sum_{\mathbf% {k},\mu}\sqrt{\frac{\hbar}{2\omega V\epsilon_{0}}}\left(\mathbf{e}^{(\mu)}a^{(% \mu)}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}}+\bar{\mathbf{e}}^{(\mu)}{a^{% \dagger}}^{(\mu)}(\mathbf{k})e^{-i\mathbf{k}\cdot\mathbf{r}}\right)\\ \displaystyle\mathbf{E}(\mathbf{r})&\displaystyle=i\sum_{\mathbf{k},\mu}\sqrt{% \frac{\hbar\omega}{2V\epsilon_{0}}}\left(\mathbf{e}^{(\mu)}a^{(\mu)}(\mathbf{k% })e^{i\mathbf{k}\cdot\mathbf{r}}-\bar{\mathbf{e}}^{(\mu)}{a^{\dagger}}^{(\mu)}% (\mathbf{k})e^{-i\mathbf{k}\cdot\mathbf{r}}\right)\\ \displaystyle\mathbf{B}(\mathbf{r})&\displaystyle=i\sum_{\mathbf{k},\mu}\sqrt{% \frac{\hbar}{2\omega V\epsilon_{0}}}\left((\mathbf{k}\times\mathbf{e}^{(\mu)})% a^{(\mu)}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}}-(\mathbf{k}\times\bar{% \mathbf{e}}^{(\mu)}){a^{\dagger}}^{(\mu)}(\mathbf{k})e^{-i\mathbf{k}\cdot% \mathbf{r}}\right),\\ \end{aligned}
  14. H = 1 2 ϵ 0 V ( E ( 𝐫 , t ) 2 + c 2 B ( 𝐫 , t ) 2 ) d 3 𝐫 = V ϵ 0 𝐤 μ = 1 , - 1 ω 2 ( a ¯ 𝐤 ( μ ) ( t ) a 𝐤 ( μ ) ( t ) + a 𝐤 ( μ ) ( t ) a ¯ 𝐤 ( μ ) ( t ) ) . H=\frac{1}{2}\epsilon_{0}\iiint_{V}\left(E(\mathbf{r},t)^{2}+c^{2}B(\mathbf{r}% ,t)^{2}\right)\mathrm{d}^{3}\mathbf{r}=V\epsilon_{0}\sum_{\mathbf{k}}\sum_{\mu% =1,-1}\omega^{2}\big(\bar{a}^{(\mu)}_{\mathbf{k}}(t)a^{(\mu)}_{\mathbf{k}}(t)+% a^{(\mu)}_{\mathbf{k}}(t)\bar{a}^{(\mu)}_{\mathbf{k}}(t)\big).
  15. H = 1 2 𝐤 , μ = - 1 , 1 ω ( a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) + a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) ) = 𝐤 , μ ω ( a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) + 1 2 ) \begin{aligned}\displaystyle H&\displaystyle=\frac{1}{2}\sum_{\mathbf{k},\mu=-% 1,1}\hbar\omega\Big({a^{\dagger}}^{(\mu)}(\mathbf{k})\,a^{(\mu)}(\mathbf{k})+a% ^{(\mu)}(\mathbf{k})\,{a^{\dagger}}^{(\mu)}(\mathbf{k})\Big)\\ &\displaystyle=\sum_{\mathbf{k},\mu}\hbar\omega\Big({a^{\dagger}}^{(\mu)}(% \mathbf{k})a^{(\mu)}(\mathbf{k})+\frac{1}{2}\Big)\end{aligned}
  16. H = ω ( a a + 1 2 ) H=\hbar\omega\big(a^{\dagger}a+\tfrac{1}{2}\big)
  17. a a^{\dagger}
  18. a | n = | n + 1 n + 1 in particular a | 0 = | 1 and ( a ) n | 0 | n . a^{\dagger}|n\rangle=|n+1\rangle\sqrt{n+1}\quad\hbox{in particular}\quad a^{% \dagger}|0\rangle=|1\rangle\quad\hbox{and}\quad(a^{\dagger})^{n}|0\rangle% \propto|n\rangle.
  19. a a^{\dagger}
  20. a \,a
  21. a | n = | n - 1 n in particular a | 0 0 , a|n\rangle=|n-1\rangle\sqrt{n}\quad\hbox{in particular}\quad a|0\rangle\propto 0,
  22. a | 0 = 0. a|0\rangle=0.
  23. a \,a
  24. [ a , ( a ) n ] = n ( a ) n - 1 with ( a ) 0 = 1. [a,(a^{\dagger})^{n}]=n(a^{\dagger})^{n-1}\quad\hbox{with}\quad(a^{\dagger})^{% 0}=1.
  25. H = i ω i ( a ( i ) a ( i ) + 1 2 ) . H=\sum_{i}\hbar\omega_{i}\Big(a^{\dagger}(i)a(i)+\tfrac{1}{2}\Big).
  26. i ( 𝐤 , μ ) i\rightarrow(\mathbf{k},\mu)
  27. ( a ( μ ) ( 𝐤 ) ) m ( a ( μ ) ( 𝐤 ) ) n | 0 | ( 𝐤 , μ ) m ; ( 𝐤 , μ ) n , \big({a^{\dagger}}^{(\mu)}(\mathbf{k})\big)^{m}\,\big({a^{\dagger}}^{(\mu^{% \prime})}(\mathbf{k}^{\prime})\big)^{n}\,\big|\,0\,\big\rangle\propto\big|(% \mathbf{k},\mu)^{m};\,(\mathbf{k}^{\prime},\mu^{\prime})^{n}\,\big\rangle,
  28. N ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) N^{(\mu)}(\mathbf{k})\equiv{a^{\dagger}}^{(\mu)}(\mathbf{k})\,a^{(\mu)}(% \mathbf{k})
  29. N ( μ ) ( 𝐤 ) | 𝐤 , μ = a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) | 0 = a ( μ ) ( 𝐤 ) ( δ 𝐤 , 𝐤 δ μ , μ + a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) ) | 0 = δ 𝐤 , 𝐤 δ μ , μ | 𝐤 , μ , \begin{aligned}\displaystyle N^{(\mu)}(\mathbf{k})\;|\,\mathbf{k}^{\prime},\mu% ^{\prime}\,\rangle&\displaystyle={a^{\dagger}}^{(\mu)}(\mathbf{k})\,a^{(\mu)}(% \mathbf{k})\;{a^{\dagger}}^{(\mu^{\prime})}(\mathbf{k^{\prime}})\,|\,0\,% \rangle={a^{\dagger}}^{(\mu)}(\mathbf{k})\,\left(\delta_{\mathbf{k},\mathbf{k^% {\prime}}}\delta_{\mu,\mu^{\prime}}+{a^{\dagger}}^{(\mu^{\prime})}(\mathbf{k^{% \prime}})\,a^{(\mu)}(\mathbf{k})\right)\,|\,0\,\rangle\\ &\displaystyle=\delta_{\mathbf{k},\mathbf{k^{\prime}}}\delta_{\mu,\mu^{\prime}% }\,|\,\mathbf{k},\mu\rangle,\end{aligned}
  30. N ( a ) n | 0 = a ( [ a , ( a ) n ] + ( a ) n a ) | 0 = a [ a , ( a ) n ] | 0 . N(a^{\dagger})^{n}|\,0\,\rangle=a^{\dagger}\left([a,(a^{\dagger})^{n}]+(a^{% \dagger})^{n}a\right)|0\rangle=a^{\dagger}\,[a,(a^{\dagger})^{n}]\,|0\,\rangle.
  31. N ( a ) n | 0 = n ( a ) n | 0 . N(a^{\dagger})^{n}|\,0\,\rangle=n(a^{\dagger})^{n}|\,0\,\rangle.
  32. H = 𝐤 , μ ω ( a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) + 1 2 ) H=\sum_{\mathbf{k},\mu}\hbar\omega\Big({a^{\dagger}}^{(\mu)}(\mathbf{k})a^{(% \mu)}(\mathbf{k})+\frac{1}{2}\Big)
  33. H = 𝐤 , μ ω N ( μ ) ( 𝐤 ) H=\sum_{\mathbf{k},\mu}\hbar\omega N^{(\mu)}(\mathbf{k})
  34. H | 𝐤 , μ H ( a ( μ ) ( 𝐤 ) | 0 ) = 𝐤 , μ ω N ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) | 0 = ω ( a ( μ ) ( 𝐤 ) | 0 ) = ω | 𝐤 , μ . H|\mathbf{k},\mu\rangle\equiv H\left({a^{\dagger}}^{(\mu)}(\mathbf{k})\,|0% \rangle\right)=\sum_{\mathbf{k^{\prime}},\mu^{\prime}}\hbar\omega^{\prime}N^{(% \mu^{\prime})}(\mathbf{k}^{\prime}){a^{\dagger}}^{(\mu)}(\mathbf{k})\,|\,0\,% \rangle=\hbar\omega\left({a^{\dagger}}^{(\mu)}(\mathbf{k})\,|0\rangle\right)=% \hbar\omega|\mathbf{k},\mu\rangle.
  35. H | ( 𝐤 , μ ) m ; ( 𝐤 , μ ) n = [ m ( ω ) + n ( ω ) ] | ( 𝐤 , μ ) m ; ( 𝐤 , μ ) n , H\big|(\mathbf{k},\mu)^{m};\,(\mathbf{k}^{\prime},\mu^{\prime})^{n}\,\big% \rangle=\left[m(\hbar\omega)+n(\hbar\omega^{\prime})\right]\big|(\mathbf{k},% \mu)^{m};\,(\mathbf{k}^{\prime},\mu^{\prime})^{n}\,\big\rangle,
  36. ω = c | 𝐤 | and ω = c | 𝐤 | . \omega=c|\mathbf{k}|\quad\hbox{and}\quad\omega^{\prime}=c|\mathbf{k}^{\prime}|.
  37. ( λ 2 π ) 3 , \left(\frac{\lambda}{2\pi}\right)^{3},
  38. ( λ 2 π ) 3 \left(\frac{\lambda}{2\pi}\right)^{3}
  39. 𝐏 EM = ϵ 0 V 𝐄 ( 𝐫 , t ) × 𝐁 ( 𝐫 , t ) d 3 𝐫 , \mathbf{P}_{\textrm{EM}}=\epsilon_{0}\iiint_{V}\mathbf{E}(\mathbf{r},t)\times% \mathbf{B}(\mathbf{r},t)\,\textrm{d}^{3}\mathbf{r},
  40. 𝐏 EM = V ϵ 0 𝐤 μ = 1 , - 1 ω 𝐤 ( a 𝐤 ( μ ) ( t ) a ¯ 𝐤 ( μ ) ( t ) + a ¯ 𝐤 ( μ ) ( t ) a 𝐤 ( μ ) ( t ) ) . \mathbf{P}_{\textrm{EM}}=V\epsilon_{0}\sum_{\mathbf{k}}\sum_{\mu=1,-1}\omega% \mathbf{k}\left(a^{(\mu)}_{\mathbf{k}}(t)\bar{a}^{(\mu)}_{\mathbf{k}}(t)+\bar{% a}^{(\mu)}_{\mathbf{k}}(t)a^{(\mu)}_{\mathbf{k}}(t)\right).
  41. 𝐏 EM = 𝐤 , μ 𝐤 ( a ( μ ) ( 𝐤 ) a ( μ ) ( 𝐤 ) + 1 2 ) = 𝐤 , μ 𝐤 N ( μ ) ( 𝐤 ) . \mathbf{P}_{\textrm{EM}}=\sum_{\mathbf{k},\mu}\hbar\mathbf{k}\Big({a^{\dagger}% }^{(\mu)}(\mathbf{k})a^{(\mu)}(\mathbf{k})+\frac{1}{2}\Big)=\sum_{\mathbf{k},% \mu}\hbar\mathbf{k}N^{(\mu)}(\mathbf{k}).
  42. 𝐏 EM | 𝐤 , μ = 𝐏 EM ( a ( μ ) ( 𝐤 ) | 0 ) = 𝐤 ( a ( μ ) ( 𝐤 ) | 0 ) = 𝐤 | 𝐤 , μ . \mathbf{P}_{\textrm{EM}}\,|\,\mathbf{k},\mu\,\rangle=\mathbf{P}_{\textrm{EM}}% \left({a^{\dagger}}^{(\mu)}(\mathbf{k})\,|0\rangle\right)=\hbar\mathbf{k}\left% ({a^{\dagger}}^{(\mu)}(\mathbf{k})\,|0\rangle\right)=\hbar\mathbf{k}\,|\,% \mathbf{k},\mu\,\rangle.
  43. E 2 = m 0 2 c 4 1 - v 2 / c 2 , p 2 = m 0 2 v 2 1 - v 2 / c 2 . E^{2}=\frac{m_{0}^{2}c^{4}}{1-v^{2}/c^{2}},\quad p^{2}=\frac{m_{0}^{2}v^{2}}{1% -v^{2}/c^{2}}.
  44. v 2 c 2 = c 2 p 2 E 2 E 2 = m 0 2 c 4 1 - c 2 p 2 / E 2 m 0 2 c 4 = E 2 - c 2 p 2 . \frac{v^{2}}{c^{2}}=\frac{c^{2}p^{2}}{E^{2}}\quad\Longrightarrow\quad E^{2}=% \frac{m_{0}^{2}c^{4}}{1-c^{2}p^{2}/E^{2}}\quad\Longrightarrow\quad m_{0}^{2}c^% {4}=E^{2}-c^{2}p^{2}.
  45. E 2 = 2 ω 2 and p 2 = 2 k 2 = 2 ω 2 c 2 E^{2}=\hbar^{2}\omega^{2}\quad\mathrm{and}\quad p^{2}=\hbar^{2}k^{2}=\frac{% \hbar^{2}\omega^{2}}{c^{2}}
  46. m 0 2 c 4 = E 2 - c 2 p 2 = 2 ω 2 - c 2 2 ω 2 c 2 = 0 , m_{0}^{2}c^{4}=E^{2}-c^{2}p^{2}=\hbar^{2}\omega^{2}-c^{2}\frac{\hbar^{2}\omega% ^{2}}{c^{2}}=0,
  47. S z - i ( 𝐞 x 𝐞 y - 𝐞 y 𝐞 x ) and cyclically x y z x . S_{z}\equiv-i\hbar\Big(\mathbf{e}_{x}\otimes\mathbf{e}_{y}-\mathbf{e}_{y}% \otimes\mathbf{e}_{x}\Big)\quad\hbox{and cyclically}\quad x\rightarrow y% \rightarrow z\rightarrow x.
  48. [ S x , S y ] = i S z and cyclically x y z x . [S_{x},\,S_{y}]=i\hbar S_{z}\quad\hbox{and cyclically}\quad x\rightarrow y% \rightarrow z\rightarrow x.
  49. ( 𝐞 y 𝐞 z ) ( 𝐞 z 𝐞 x ) = ( 𝐞 y 𝐞 x ) ( 𝐞 z 𝐞 z ) = 𝐞 y 𝐞 x \big(\mathbf{e}_{y}\otimes\mathbf{e}_{z}\big)\;\big(\mathbf{e}_{z}\otimes% \mathbf{e}_{x}\big)=(\mathbf{e}_{y}\otimes\mathbf{e}_{x})(\mathbf{e}_{z}\cdot% \mathbf{e}_{z})=\mathbf{e}_{y}\otimes\mathbf{e}_{x}
  50. [ S x , S y ] = - 2 ( 𝐞 y 𝐞 z - 𝐞 z 𝐞 y ) ( 𝐞 z 𝐞 x - 𝐞 x 𝐞 z ) + 2 ( 𝐞 z 𝐞 x - 𝐞 x 𝐞 z ) ( 𝐞 y 𝐞 z - 𝐞 z 𝐞 y ) = i [ - i ( 𝐞 x 𝐞 y - 𝐞 y 𝐞 x ) ] = i S z . \begin{aligned}\displaystyle\left[S_{x},\,S_{y}\right]&\displaystyle=-\hbar^{2% }\Big(\mathbf{e}_{y}\otimes\mathbf{e}_{z}-\mathbf{e}_{z}\otimes\mathbf{e}_{y}% \Big)\;\Big(\mathbf{e}_{z}\otimes\mathbf{e}_{x}-\mathbf{e}_{x}\otimes\mathbf{e% }_{z}\Big)+\hbar^{2}\Big(\mathbf{e}_{z}\otimes\mathbf{e}_{x}-\mathbf{e}_{x}% \otimes\mathbf{e}_{z}\Big)\;\Big(\mathbf{e}_{y}\otimes\mathbf{e}_{z}-\mathbf{e% }_{z}\otimes\mathbf{e}_{y}\Big)\\ &\displaystyle=i\hbar\Big[-i\hbar\big(\mathbf{e}_{x}\otimes\mathbf{e}_{y}-% \mathbf{e}_{y}\otimes\mathbf{e}_{x}\big)\Big]=i\hbar S_{z}.\\ \end{aligned}
  51. - i ( 𝐞 x 𝐞 y - 𝐞 y 𝐞 x ) 𝐞 ( μ ) = μ 𝐞 ( μ ) , μ = 1 , - 1 , -i\hbar\Big(\mathbf{e}_{x}\otimes\mathbf{e}_{y}-\mathbf{e}_{y}\otimes\mathbf{e% }_{x}\Big)\cdot\mathbf{e^{(\mu)}}=\mu\mathbf{e}^{(\mu)},\quad\mu=1,-1,
  52. S z | 𝐤 , μ = μ | 𝐤 , μ , μ = 1 , - 1. S_{z}|\mathbf{k},\mu\rangle=\mu|\mathbf{k},\mu\rangle,\quad\mu=1,-1.

Quantum-confined_Stark_effect.html

  1. V ( z ) = { 0 ; | z | < L / 2 V 0 ; otherwise V(z)=\begin{cases}0;&|z|<L/2\\ V_{0};&\mbox{otherwise}\end{cases}
  2. L L
  3. V 0 V_{0}
  4. E n E_{n}
  5. ψ ( 𝐫 ) = ϕ n ( z ) 1 A e i ( k x x + k y y ) u ( 𝐫 ) . \psi(\mathbf{r})=\phi_{n}(z)\frac{1}{\sqrt{A}}e^{i(k_{x}\cdot{x}+k_{y}\cdot{y}% )}u(\mathbf{r}).
  6. A A
  7. u ( 𝐫 ) u(\mathbf{r})
  8. ϕ n ( z ) \phi_{n}(z)
  9. V 0 V_{0}\to\infty
  10. ϕ n ( z ) = 2 L × { cos ( n π z L ) n odd sin ( n π z L ) n even . \phi_{n}(z)=\sqrt{\frac{2}{L}}\times\begin{cases}\cos\left(\frac{n\pi z}{L}% \right)&n\,\,\text{odd}\\ \sin\left(\frac{n\pi z}{L}\right)&n\,\,\text{even}\end{cases}.
  11. E n = 2 n 2 π 2 2 m * L 2 , E_{n}=\frac{\hbar^{2}n^{2}\pi^{2}}{2m^{*}L^{2}},
  12. m * m^{*}
  13. 𝐄 = E 𝐳 , \mathbf{E}=E\mathbf{z},
  14. H = e E z . H^{\prime}=eEz.
  15. E n ( 1 ) = n ( 0 ) | e E z | n ( 0 ) = 0 E_{n}^{(1)}=\langle n^{(0)}|eEz|n^{(0)}\rangle=0
  16. E 1 ( 2 ) = k 1 | k ( 0 ) | e E z | 1 ( 0 ) | 2 E 1 ( 0 ) - E k ( 0 ) | 2 ( 0 ) | e E z | 1 ( 0 ) | 2 E 1 ( 0 ) - E 2 ( 0 ) = - 24 ( 2 3 π ) 6 e 2 E 2 m e * L 4 2 E_{1}^{(2)}=\sum_{k\neq 1}\frac{|\langle k^{(0)}|eEz|1^{(0)}\rangle|^{2}}{E_{1% }^{(0)}-E_{k}^{(0)}}\approx\frac{|\langle 2^{(0)}|eEz|1^{(0)}\rangle|^{2}}{E_{% 1}^{(0)}-E_{2}^{(0)}}=-24\left(\frac{2}{3\pi}\right)^{6}\frac{e^{2}E^{2}m_{e}^% {*}L^{4}}{\hbar^{2}}

Quantum-mechanical_explanation_of_intermolecular_interactions.html

  1. H ( 0 ) H A + H B H^{(0)}\equiv H^{A}+H^{B}
  2. Φ n A Φ m B \Phi_{n}^{A}\Phi_{m}^{B}\quad
  3. H A Φ n A = E n A Φ n A \quad H^{A}\Phi_{n}^{A}=E_{n}^{A}\Phi_{n}^{A}\quad
  4. H B Φ m B = E m B Φ m B \quad H^{B}\Phi_{m}^{B}=E_{m}^{B}\Phi_{m}^{B}
  5. 𝒜 ~ A B \tilde{\mathcal{A}}^{AB}
  6. [ 𝒜 ~ A B , H ( 0 ) ] 0. \big[\tilde{\mathcal{A}}^{AB},H^{(0)}\big]\neq 0.
  7. 𝒜 ~ A B Φ n A Φ m B \tilde{\mathcal{A}}^{AB}\Phi^{A}_{n}\Phi^{B}_{m}
  8. E antisymmetric ( 1 ) = Φ 0 A Φ 0 B | V A B 𝒜 ~ A B | Φ 0 A Φ 0 B Φ 0 A Φ 0 B | 𝒜 ~ A B | Φ 0 A Φ 0 B . E^{(1)}_{\mathrm{antisymmetric}}=\frac{\langle\Phi_{0}^{A}\Phi_{0}^{B}|V^{AB}% \tilde{\mathcal{A}}^{AB}|\Phi_{0}^{A}\Phi_{0}^{B}\rangle}{\langle\Phi_{0}^{A}% \Phi_{0}^{B}|\tilde{\mathcal{A}}^{AB}|\Phi_{0}^{A}\Phi_{0}^{B}\rangle}.
  9. E electrostatic ( 1 ) = Φ 0 A Φ 0 B | V A B | Φ 0 A Φ 0 B . E^{(1)}_{\mathrm{electrostatic}}=\langle\Phi_{0}^{A}\Phi_{0}^{B}|V^{AB}|\Phi_{% 0}^{A}\Phi_{0}^{B}\rangle.
  10. ρ tot A ( 𝐫 ) = α Z α δ ( 𝐫 - 𝐑 α ) - ρ el A ( 𝐫 ) \rho^{A}_{\mathrm{tot}}(\mathbf{r})=\sum_{\alpha}Z_{\alpha}\delta(\mathbf{r}-% \mathbf{R}_{\alpha})-\rho^{A}_{\mathrm{el}}(\mathbf{r})
  11. ρ el A ( 𝐫 ) = n A | Φ 0 A ( 𝐫 , 𝐫 2 , , 𝐫 n A ) | 2 d 𝐫 2 d 𝐫 n A . \rho^{A}_{\mathrm{el}}(\mathbf{r})=n_{A}\int|\Phi^{A}_{0}(\mathbf{r},\mathbf{r% }^{\prime}_{2},\ldots,\mathbf{r}^{\prime}_{n_{A}})|^{2}d\mathbf{r}^{\prime}_{2% }\cdots d\mathbf{r}^{\prime}_{n_{A}}.
  12. E electrostatic ( 1 ) = ρ tot A ( 𝐫 1 ) 1 r 12 ρ tot B ( 𝐫 2 ) d 𝐫 1 d 𝐫 2 , E^{(1)}_{\mathrm{electrostatic}}=\int\int\rho^{A}_{\mathrm{tot}}(\mathbf{r}_{1% })\frac{1}{r_{12}}\rho^{B}_{\mathrm{tot}}(\mathbf{r}_{2})d\mathbf{r}_{1}d% \mathbf{r}_{2},
  13. V A B = A = 0 B = 0 ( - 1 ) B ( 2 A + 2 B 2 A ) 1 / 2 M = - A - B A + B ( - 1 ) M I A + B , - M ( 𝐑 A B ) [ 𝐐 A 𝐐 B ] M A + B V^{AB}=\sum_{\ell_{A}=0}^{\infty}\sum_{\ell_{B}=0}^{\infty}(-1)^{\ell_{B}}{% \left({{2\ell_{A}+2\ell_{B}}\atop{2\ell_{A}}}\right)}^{1/2}\sum_{M=-\ell_{A}-% \ell_{B}}^{\ell_{A}+\ell_{B}}(-1)^{M}I_{\ell_{A}+\ell_{B},-M}(\mathbf{R}_{AB})% \;\left[\mathbf{Q}^{\ell_{A}}\otimes\mathbf{Q}^{\ell_{B}}\right]^{\ell_{A}+% \ell_{B}}_{M}
  14. [ 𝐐 A 𝐐 B ] M A + B m A = - A A m B = - B B Q m A A Q m B B A , m A ; B , m B | A + B , M . \left[\mathbf{Q}^{\ell_{A}}\otimes\mathbf{Q}^{\ell_{B}}\right]^{\ell_{A}+\ell_% {B}}_{M}\equiv\sum_{m_{A}=-\ell_{A}}^{\ell_{A}}\sum_{m_{B}=-\ell_{B}}^{\ell_{B% }}\;Q_{m_{A}}^{\ell_{A}}Q_{m_{B}}^{\ell_{B}}\;\langle\ell_{A},m_{A};\ell_{B},m% _{B}|\ell_{A}+\ell_{B},M\rangle.
  15. I L , M ( 𝐑 A B ) [ 4 π 2 L + 1 ] 1 / 2 Y L , M ( 𝐑 ^ A B ) R A B L + 1 . I_{L,M}(\mathbf{R}_{AB})\equiv\left[\frac{4\pi}{2L+1}\right]^{1/2}\;\frac{Y_{L% ,M}(\widehat{\mathbf{R}}_{AB})}{R_{AB}^{L+1}}.
  16. Q m A A Q^{\ell_{A}}_{m_{A}}
  17. Q m B B Q^{\ell_{B}}_{m_{B}}
  18. E electrostatic ( 1 ) = A = 0 B = 0 ( - 1 ) B ( 2 A + 2 B 2 A ) 1 / 2 M = - A - B A + B ( - 1 ) M I A + B , - M ( 𝐑 A B ) [ 𝐌 A 𝐌 B ] M A + B , \begin{aligned}\displaystyle E^{(1)}_{\mathrm{electrostatic}}=&\displaystyle% \sum_{\ell_{A}=0}^{\infty}\sum_{\ell_{B}=0}^{\infty}(-1)^{\ell_{B}}{\left({{2% \ell_{A}+2\ell_{B}}\atop{2\ell_{A}}}\right)}^{1/2}\\ &\displaystyle\sum_{M=-\ell_{A}-\ell_{B}}^{\ell_{A}+\ell_{B}}(-1)^{M}I_{\ell_{% A}+\ell_{B},-M}(\mathbf{R}_{AB})\;\left[\mathbf{M}^{\ell_{A}}\otimes\mathbf{M}% ^{\ell_{B}}\right]^{\ell_{A}+\ell_{B}}_{M},\end{aligned}
  19. M m A A Φ 0 A | Q m A A | Φ 0 A and M m B B Φ 0 B | Q m B B | Φ 0 B . M^{\ell_{A}}_{m_{A}}\equiv\langle\Phi_{0}^{A}|Q^{\ell_{A}}_{m_{A}}|\Phi_{0}^{A% }\rangle\quad\hbox{and}\quad M^{\ell_{B}}_{m_{B}}\equiv\langle\Phi_{0}^{B}|Q^{% \ell_{B}}_{m_{B}}|\Phi_{0}^{B}\rangle.
  20. I 0 , 0 ( 𝐑 A B ) = 1 R A B , I_{0,0}(\mathbf{R}_{AB})=\frac{1}{R_{AB}},
  21. M 0 0 A = q A , M 0 0 B = q B and [ 𝐌 0 A 𝐌 0 A ] 0 0 = q A q B , M^{0_{A}}_{0}=q_{A},\quad M^{0_{B}}_{0}=q_{B}\quad\hbox{and}\quad[\mathbf{M}^{% 0_{A}}\otimes\mathbf{M}^{0_{A}}]^{0}_{0}=q_{A}q_{B},
  22. E electrostatic ( 1 ) = q A q B R A B + higher terms . E^{(1)}_{\mathrm{electrostatic}}=\frac{q_{A}q_{B}}{R_{AB}}+\hbox{higher terms}.
  23. s y m b o l μ A = ( μ x A , μ y A , μ z A ) and μ z A = s y m b o l μ A 𝐑 ^ A B s y m b o l μ A 𝐑 A B R A B symbol{\mu}^{A}=(\mu_{x}^{A},\mu_{y}^{A},\mu_{z}^{A})\quad\hbox{and}\quad\mu_{% z}^{A}=symbol{\mu}^{A}\cdot\hat{\mathbf{R}}_{AB}\equiv symbol{\mu}^{A}\cdot% \frac{\mathbf{R}_{AB}}{R_{AB}}
  24. E dip - dip = 1 R A B 3 [ s y m b o l μ A \cdotsymbol μ B - 3 ( s y m b o l μ A 𝐑 ^ A B ) ( 𝐑 ^ A B s y m b o l μ B ) ] . E_{\mathrm{dip-dip}}=\frac{1}{R^{3}_{AB}}\left[symbol{\mu}^{A}\cdotsymbol{\mu}% ^{B}-3(symbol{\mu}^{A}\cdot\hat{\mathbf{R}}_{AB})(\hat{\mathbf{R}}_{AB}\cdot symbol% {\mu}^{B})\right].
  25. s y m b o l μ A = s y m b o l μ B = μ HCl ( 0 0 - 1 ) and E dip - dip = - 2 μ HCl 2 R A B 3 . symbol{\mu}^{A}=symbol{\mu}^{B}=\mu_{\mathrm{HCl}}\begin{pmatrix}0\\ 0\\ -1\end{pmatrix}\quad\hbox{and}\quad E_{\mathrm{dip-dip}}=\frac{-2\mu^{2}_{% \mathrm{HCl}}}{R^{3}_{AB}}.
  26. E ¯ dip - dip = - 2 k T 3 R A B 6 | μ A | 2 | μ B | 2 . \overline{E}_{\mathrm{dip-dip}}=-\frac{2kT}{3R_{AB}^{6}}|\mu^{A}|^{2}|\mu^{B}|% ^{2}.
  27. r 1 \vec{r}_{1}
  28. r 2 \vec{r}_{2}
  29. | r 1 - r 2 | |\vec{r}_{1}-\vec{r}_{2}|
  30. U = u ( r 12 ) + u ( r 13 ) + u ( r 23 ) . U=u(r_{12})+u(r_{13})+u(r_{23}).
  31. U = u ( r 12 ) + u ( r 13 ) + u ( r 23 ) + u ( r 12 , r 13 , r 23 ) , U=u(r_{12})+u(r_{13})+u(r_{23})+u(r_{12},r_{13},r_{23}),
  32. u ( r 12 , r 13 , r 23 ) u(r_{12},r_{13},r_{23})

Quantum_beats.html

  1. Λ \Lambda
  2. Λ \Lambda
  3. Λ \Lambda
  4. Λ \Lambda
  5. | a |a\rangle
  6. | b |b\rangle
  7. | c |c\rangle
  8. | a |a\rangle
  9. | b |b\rangle
  10. | c |c\rangle
  11. | a |a\rangle
  12. | b |b\rangle
  13. | c |c\rangle
  14. Λ \Lambda
  15. | a |a\rangle
  16. | b |b\rangle
  17. | c |c\rangle
  18. | a |a\rangle
  19. | b |b\rangle
  20. | c |c\rangle
  21. | a |a\rangle
  22. | b |b\rangle
  23. | c |c\rangle
  24. | ψ ( t ) = c a e x p ( - i ω a t ) | a + c b e x p ( - i ω b t ) | b + c c e x p ( - i ω c t ) | c |\psi(t)\rangle=c_{a}exp(-i\omega_{a}t)|a\rangle+c_{b}exp(-i\omega_{b}t)|b% \rangle+c_{c}exp(-i\omega_{c}t)|c\rangle
  25. 𝒫 a c = e a | r | c , 𝒫 b c = e b | r | c \mathcal{P}_{ac}=e\langle a|r|c\rangle,\mathcal{P}_{bc}=e\langle b|r|c\rangle
  26. 𝒫 a b = e a | r | b , 𝒫 a c = e a | r | c \mathcal{P}_{ab}=e\langle a|r|b\rangle,\mathcal{P}_{ac}=e\langle a|r|c\rangle
  27. Λ \Lambda
  28. P ( t ) = 𝒫 a c ( c a * c c ) e x p ( i ν 1 t ) + 𝒫 b c ( c b * c c ) e x p ( i ν 2 t ) + c . c . P(t)=\mathcal{P}_{ac}(c_{a}^{*}c_{c})exp(i\nu_{1}t)+\mathcal{P}_{bc}(c_{b}^{*}% c_{c})exp(i\nu_{2}t)+c.c.
  29. ν 1 = ω a - ω c , ν 2 = ω b - ω c \nu_{1}=\omega_{a}-\omega_{c},\nu_{2}=\omega_{b}-\omega_{c}
  30. P ( t ) = 𝒫 a b ( c a * c b ) e x p ( i ν 1 t ) + 𝒫 a c ( c a * c c ) e x p ( i ν 2 t ) + c . c . P(t)=\mathcal{P}_{ab}(c_{a}^{*}c_{b})exp(i\nu_{1}t)+\mathcal{P}_{ac}(c_{a}^{*}% c_{c})exp(i\nu_{2}t)+c.c.
  31. Λ \Lambda
  32. ν 1 = ω a - ω b , ν 2 = ω a - ω c \nu_{1}=\omega_{a}-\omega_{b},\nu_{2}=\omega_{a}-\omega_{c}
  33. E ( + ) = 1 e x p ( - i ν 1 t ) + 2 e x p ( - i ν 2 t ) E^{(+)}=\mathcal{E}_{1}exp(-i\nu_{1}t)+\mathcal{E}_{2}exp(-i\nu_{2}t)
  34. | E ( + ) | 2 = | 1 | 2 + | 2 | 2 + { 1 * 2 e x p [ i ( ν 1 - ν 2 ) t ] + c . c . } |E^{(+)}|^{2}=|\mathcal{E}_{1}|^{2}+|\mathcal{E}_{2}|^{2}+\{\mathcal{E}_{1}^{*% }\mathcal{E}_{2}exp[i(\nu_{1}-\nu_{2})t]+c.c.\}
  35. E n ( + ) = a n e x p ( - i ν n t ) E_{n}^{(+)}=a_{n}exp(-i\nu_{n}t)
  36. E n ( - ) = a n e x p ( i ν n t ) E_{n}^{(-)}=a_{n}^{\dagger}exp(i\nu_{n}t)
  37. ψ V ( t ) | E 1 ( - ) ( t ) E 2 ( + ) ( t ) | ψ V ( t ) \langle\psi_{V}(t)|E_{1}^{(-)}(t)E_{2}^{(+)}(t)|\psi_{V}(t)\rangle
  38. ψ Λ ( t ) | E 1 ( - ) ( t ) E 2 ( + ) ( t ) | ψ Λ ( t ) \langle\psi_{\Lambda}(t)|E_{1}^{(-)}(t)E_{2}^{(+)}(t)|\psi_{\Lambda}(t)\rangle
  39. Λ \Lambda
  40. | ψ V ( t ) = i = a , b , c c i | i , 0 + c 1 | c , 1 ν 1 + c 2 | c , 1 ν 2 |\psi_{V}(t)\rangle=\sum_{i=a,b,c}c_{i}|i,0\rangle+c_{1}|c,1_{\nu_{1}}\rangle+% c_{2}|c,1_{\nu_{2}}\rangle
  41. | ψ Λ ( t ) = i = a , b , c c i | i , 0 + c 1 | b , 1 ν 1 + c 2 | c , 1 ν 2 |\psi_{\Lambda}(t)\rangle=\sum_{i=a,b,c}c_{i}^{\prime}|i,0\rangle+c_{1}^{% \prime}|b,1_{\nu_{1}}\rangle+c_{2}^{\prime}|c,1_{\nu_{2}}\rangle
  42. ψ V ( t ) | E 1 ( - ) ( t ) E 2 ( + ) ( t ) | ψ V ( t ) = κ 1 ν 1 0 ν 2 | a 1 a 2 | 0 ν 1 1 ν 2 e x p [ i ( ν 1 - ν 2 ) t ] c | c = κ e x p [ i ( ν 1 - ν 2 ) t ] c | c \langle\psi_{V}(t)|E_{1}^{(-)}(t)E_{2}^{(+)}(t)|\psi_{V}(t)\rangle=\kappa% \langle 1_{\nu_{1}}0_{\nu_{2}}|a_{1}^{\dagger}a_{2}|0_{\nu_{1}}1_{\nu_{2}}% \rangle exp[i(\nu_{1}-\nu_{2})t]\langle c|c\rangle=\kappa exp[i(\nu_{1}-\nu_{2% })t]\langle c|c\rangle
  43. ψ Λ ( t ) | E 1 ( - ) ( t ) E 2 ( + ) ( t ) | ψ Λ ( t ) = κ 1 ν 1 0 ν 2 | a 1 a 2 | 0 ν 1 1 ν 2 e x p [ i ( ν 1 - ν 2 ) t ] b | c = κ e x p [ i ( ν 1 - ν 2 ) t ] b | c \langle\psi_{\Lambda}(t)|E_{1}^{(-)}(t)E_{2}^{(+)}(t)|\psi_{\Lambda}(t)\rangle% =\kappa^{\prime}\langle 1_{\nu_{1}}0_{\nu_{2}}|a_{1}^{\dagger}a_{2}|0_{\nu_{1}% }1_{\nu_{2}}\rangle exp[i(\nu_{1}-\nu_{2})t]\langle b|c\rangle=\kappa^{\prime}% exp[i(\nu_{1}-\nu_{2})t]\langle b|c\rangle
  44. Λ \Lambda
  45. c | c = 1 \langle c|c\rangle=1
  46. b | c = o \langle b|c\rangle=o
  47. Λ \Lambda
  48. Λ \Lambda
  49. | c |c\rangle
  50. ν 1 \nu_{1}
  51. ν 2 \nu_{2}
  52. Λ \Lambda

Quantum_capacitance.html

  1. Q = C V Q=CV
  2. ρ \rho
  3. C geom C\text{geom}
  4. Q e Qe
  5. Δ V galvani = Q / C g e o m \Delta V\text{galvani}=Q/C_{geom}
  6. Δ μ internal = N / ρ = Q / ( ρ e ) \Delta\mu\text{internal}=N/\rho=Q/(\rho e)
  7. Δ V quantum = ( Δ μ internal ) / e = Q / ( ρ e 2 ) \Delta V\text{quantum}=(\Delta\mu\text{internal})/e=Q/(\rho e^{2})
  8. C quantum = g v m * e 2 π 2 C\text{quantum}=\frac{g_{v}m^{*}e^{2}}{\pi\hbar^{2}}
  9. g v g_{v}

Quantum_convolutional_code.html

  1. , \mathcal{H},
  2. { i } i + \left\{\mathcal{H}_{i}\right\}_{i\in\mathbb{Z}^{+}}
  3. = i = 0 i . \mathcal{H}={\displaystyle\bigotimes\limits_{i=0}^{\infty}}\ \mathcal{H}_{i}.
  4. 𝐀 \mathbf{A}
  5. { A i } i + \left\{A_{i}\right\}_{i\in\mathbb{Z}^{+}}
  6. 𝐀 = i = 0 A i , \mathbf{A}={\displaystyle\bigotimes\limits_{i=0}^{\infty}}\ A_{i},
  7. \mathcal{H}
  8. Π + \Pi^{\mathbb{Z}^{+}}
  9. ( 𝐀 ) \left(\mathbf{A}\right)
  10. 𝐀 \mathbf{A}
  11. 𝐀 \mathbf{A}
  12. 𝐀 \mathbf{A}
  13. | supp ( 𝐀 ) | \left|\,\text{supp}\left(\mathbf{A}\right)\right|
  14. ( 𝐀 ) \left(\mathbf{A}\right)
  15. 𝐀 \mathbf{A}
  16. ( 𝐀 ) \left(\mathbf{A}\right)
  17. 𝐀 \mathbf{A}
  18. I X I Y Z I I , \begin{array}[c]{cccccccc}I&X&I&Y&Z&I&I&\cdots\end{array},
  19. { 1 , 3 , 4 } \left\{1,3,4\right\}
  20. F ( Π + ) F(\Pi^{\mathbb{Z}^{+}})
  21. F ( Π + ) F(\Pi^{\mathbb{Z}^{+}})
  22. k / n k/n
  23. 0 k n 0\leq k\leq n
  24. 𝒢 \mathcal{G}
  25. n n
  26. 𝒢 0 \mathcal{G}_{0}
  27. 𝒢 0 \mathcal{G}_{0}
  28. n - k n-k
  29. 𝒢 0 = { 𝐆 i F ( Π + ) : 1 i n - k } . \mathcal{G}_{0}=\left\{\mathbf{G}_{i}\in F(\Pi^{\mathbb{Z}^{+}}):1\leq i\leq n% -k\right\}.
  30. ν \nu
  31. 𝒢 0 \mathcal{G}_{0}
  32. n n
  33. D D
  34. D D
  35. 𝒢 0 \mathcal{G}_{0}
  36. n n
  37. D D
  38. 𝐀 Π + \mathbf{A}\in\Pi^{\mathbb{Z}^{+}}
  39. D ( 𝐀 ) = I n 𝐀 . D\left(\mathbf{A}\right)=I^{\otimes n}\otimes\mathbf{A.}
  40. j j
  41. D D
  42. D D
  43. D j ( 𝐀 ) = I j n 𝐀 . D^{j}\left(\mathbf{A}\right)=I^{\otimes jn}\otimes\mathbf{A.}
  44. D j ( 𝒢 0 ) D^{j}\left(\mathcal{G}_{0}\right)
  45. 𝒢 0 \mathcal{G}_{0}
  46. j j
  47. 𝒢 \mathcal{G}
  48. 𝒢 = j + D j ( 𝒢 0 ) . \mathcal{G}={\textstyle\bigcup\limits_{j\in\mathbb{Z}^{+}}}D^{j}\left(\mathcal% {G}_{0}\right).
  49. 𝒢 \mathcal{G}
  50. 𝒢 \mathcal{G}
  51. | I I I I I I I I I I I I | X X X Z Z Z I I I I I I | X Z Y Z Y X X X X Z Z Z | I I I I I I X Z Y Z Y X | I I I I I I I I I I I I | \cdots\left|\begin{array}[c]{c}III\\ III\\ III\\ III\end{array}\right|\begin{array}[c]{c}XXX\\ ZZZ\\ III\\ III\end{array}\left|\begin{array}[c]{c}XZY\\ ZYX\\ XXX\\ ZZZ\end{array}\right|\begin{array}[c]{c}III\\ III\\ XZY\\ ZYX\end{array}\left|\begin{array}[c]{c}III\\ III\\ III\\ III\end{array}\right|\cdots

Quantum_differential_calculus.html

  1. A A
  2. k k
  3. A A
  4. A A
  5. A A
  6. Ω 1 \Omega^{1}
  7. A A
  8. Ω 1 \Omega^{1}
  9. A A
  10. a ( ω b ) = ( a ω ) b , a , b A , ω Ω 1 a(\omega b)=(a\omega)b,\ \forall a,b\in A,\ \omega\in\Omega^{1}
  11. d : A Ω 1 {\rm d}:A\to\Omega^{1}
  12. d ( a b ) = a ( d b ) + ( d a ) b , a , b A {\rm d}(ab)=a({\rm d}b)+({\rm d}a)b,\ \forall a,b\in A
  13. Ω 1 = { a ( d b ) | a , b A } \Omega^{1}=\{a({\rm d}b)\ |\ a,b\in A\}
  14. ker d = k 1 \ker\ {\rm d}=k1
  15. d {\rm d}
  16. A A
  17. Ω 1 \Omega^{1}
  18. Ω = n Ω n , d : Ω n Ω n + 1 \Omega=\oplus_{n}\Omega^{n},\ {\rm d}:\Omega^{n}\to\Omega^{n+1}
  19. Ω \Omega
  20. d 2 = 0 {\rm d}^{2}=0
  21. Ω 0 = A \Omega^{0}=A
  22. Ω \Omega
  23. A , Ω 1 A,\Omega^{1}
  24. \wedge
  25. A A
  26. a ( d b ) = ( db ) a , a , b A a({\rm d}b)=({\rm db})a,\ \forall a,b\in A
  27. d ( a b - b a ) = 0 , a , b A {\rm d}(ab-ba)=0,\ \forall a,b\in A
  28. A = [ x ] A={\mathbb{C}}[x]
  29. λ \lambda\in\mathbb{C}
  30. Ω 1 = . d x , ( d x ) f ( x ) = f ( x + λ ) ( d x ) , d f = f ( x + λ ) - f ( x ) λ d x \Omega^{1}={\mathbb{C}}.{\rm d}x,\quad({\rm d}x)f(x)=f(x+\lambda)({\rm d}x),% \quad{\rm d}f={f(x+\lambda)-f(x)\over\lambda}{\rm d}x
  31. λ 0 \lambda\to 0
  32. A = [ t , t - 1 ] A={\mathbb{C}}[t,t^{-1}]
  33. q 0 q\neq 0\in\mathbb{C}
  34. Ω 1 = . d t , ( d t ) f ( t ) = f ( q t ) ( d t ) , d f = f ( q t ) - f ( t ) q ( t - 1 ) dt \Omega^{1}={\mathbb{C}}.{\rm d}t,\quad({\rm d}t)f(t)=f(qt)({\rm d}t),\quad{\rm d% }f={f(qt)-f(t)\over q(t-1)}\,{\rm dt}
  35. q q
  36. A A
  37. Ω 1 = ker ( m : A A A ) , d a = 1 a - a 1 , a A \Omega^{1}=\ker(m:A\otimes A\to A),\quad{\rm d}a=1\otimes a-a\otimes 1,\quad% \forall a\in A
  38. m m

Quantum_ergodicity.html

  1. U t = exp ( i t Δ ) U_{t}=\exp(it\sqrt{\Delta})
  2. Δ \sqrt{\Delta}

Quantum_LC_circuit.html

  1. ω = 1 L C \omega=\sqrt{1\over LC}
  2. ω \omega\,
  3. U C = 1 2 C V 2 = Q 2 2 C U_{C}=\frac{1}{2}CV^{2}=\frac{Q^{2}}{2C}
  4. Q ( t ) = - t I ( τ ) d τ Q(t)=\int_{-\infty}^{t}I(\tau)d\tau
  5. U L = 1 2 L I 2 = ϕ 2 2 L U_{L}=\frac{1}{2}LI^{2}=\frac{\phi^{2}}{2L}
  6. ϕ \phi
  7. ϕ ( t ) - t V ( τ ) d τ \phi(t)\equiv\int_{-\infty}^{t}V(\tau)d\tau
  8. ϕ ϕ ^ \phi\rightarrow\hat{\phi}
  9. q q ^ q\rightarrow\hat{q}
  10. H H ^ = ϕ ^ 2 2 L + q ^ 2 2 C H\rightarrow\hat{H}=\frac{\hat{\phi}^{2}}{2L}+\frac{\hat{q}^{2}}{2C}
  11. [ ϕ , q ] = i \left[\phi,q\right]=i\hbar
  12. H = ϕ 2 2 L + 1 2 L ω 2 Q 2 H=\frac{\phi^{2}}{2L}+\frac{1}{2}L\omega^{2}Q^{2}
  13. ϕ \phi
  14. H | ψ E | ψ > H|\psi>=E|\psi>
  15. E ψ = - 2 2 L 2 ψ + 1 2 L ω 2 Q 2 ψ E\psi=-\frac{\hbar^{2}}{2L}\nabla^{2}\psi+\frac{1}{2}L\omega^{2}Q^{2}\psi
  16. Q | ψ n = 1 2 n n ! ( L ω π ) 1 / 4 exp ( - L ω Q 2 2 ) H n ( L ω Q ) \left\langle Q|\psi_{n}\right\rangle=\sqrt{\frac{1}{2^{n}\,n!}}\cdot\left(% \frac{L\omega}{\pi\hbar}\right)^{1/4}\cdot\exp\left(-\frac{L\omega Q^{2}}{2% \hbar}\right)\cdot H_{n}\left(\sqrt{\frac{L\omega}{\hbar}}Q\right)
  17. n = 0 , 1 , 2 , n=0,1,2,\ldots
  18. π = C d ϕ d t \pi=C\frac{d\phi}{dt}
  19. C d V d t + 1 L 0 t V d t = 0 C\frac{dV}{dt}+\frac{1}{L}\int_{0}^{t}V\,dt\,=0
  20. V = d ϕ d t V=\frac{d\phi}{dt}
  21. C d 2 ϕ d t 2 + 1 L ϕ = 0 C\frac{d^{2}\phi}{dt^{2}}+\frac{1}{L}\phi\,=0
  22. i d ψ d t = - 2 2 C 2 ψ + ϕ 2 2 L ψ i\hbar\frac{d\psi}{dt}=-\frac{\hbar^{2}}{2C}\nabla^{2}\psi+\frac{\phi^{2}}{2L}\psi
  23. ψ \psi
  24. L = 1 2 L 1 d Q 1 d t 2 + 1 2 L 2 d Q 2 d t 2 + m d Q 1 d t d Q 2 d t - Q 1 2 2 C 1 - Q 2 2 2 C 2 L=\frac{1}{2}L_{1}\frac{dQ_{1}}{dt}^{2}+\frac{1}{2}L_{2}\frac{dQ_{2}}{dt}^{2}+% m\frac{dQ_{1}}{dt}\frac{dQ_{2}}{dt}-\frac{Q_{1}^{2}}{2C_{1}}-\frac{Q_{2}^{2}}{% 2C_{2}}
  25. H = 1 2 L 1 d Q 1 d t 2 + 1 2 L 2 d Q 2 d t 2 + m d Q 1 d t d Q 2 d t + Q 1 2 2 C 1 + Q 2 2 2 C 2 H=\frac{1}{2}L_{1}\frac{dQ_{1}}{dt}^{2}+\frac{1}{2}L_{2}\frac{dQ_{2}}{dt}^{2}+% m\frac{dQ_{1}}{dt}\frac{dQ_{2}}{dt}+\frac{Q_{1}^{2}}{2C_{1}}+\frac{Q_{2}^{2}}{% 2C_{2}}
  26. E ψ = - 2 2 L 1 d 2 ψ d Q 1 2 - 2 2 L 2 d 2 ψ d Q 2 2 - 2 m d 2 ψ d Q 1 d Q 2 + 1 2 L 1 ω 2 Q 1 2 ψ + 1 2 L 2 ω 2 Q 2 2 ψ E\psi=-\frac{\hbar^{2}}{2L_{1}}\frac{d^{2}\psi}{dQ_{1}^{2}}-\frac{\hbar^{2}}{2% L_{2}}\frac{d^{2}\psi}{dQ_{2}^{2}}-\frac{\hbar^{2}}{m}\frac{d^{2}\psi}{dQ_{1}% dQ_{2}}+\frac{1}{2}L_{1}\omega^{2}Q_{1}^{2}\psi+\frac{1}{2}L_{2}\omega^{2}Q_{2% }^{2}\psi
  27. Q d Q_{d}
  28. Q d = Q 1 - Q 2 Q_{d}=Q_{1}-Q_{2}
  29. Q c Q_{c}
  30. Q c = L 1 Q 1 + L 2 Q 2 L 1 + L 2 Q_{c}=\frac{L_{1}Q_{1}+L_{2}Q_{2}}{L_{1}+L_{2}}
  31. E ψ = - 2 ( 1 - λ ) 2 ( L 1 + L 2 ) d 2 ψ d Q c 2 - 2 ( 1 - λ ) 2 μ d 2 ψ d Q d 2 + 1 2 μ ω 2 Q d 2 ψ E\psi=-\frac{\hbar^{2}(1-\lambda)}{2(L_{1}+L_{2})}\frac{d^{2}\psi}{dQ_{c}^{2}}% -\frac{\hbar^{2}(1-\lambda)}{2\mu}\frac{d^{2}\psi}{dQ_{d}^{2}}+\frac{1}{2}\mu% \omega^{2}Q_{d}^{2}\psi
  32. λ \lambda
  33. 2 m L 1 + L 2 \frac{2m}{L_{1}+L_{2}}
  34. μ \mu
  35. E ψ c = - 2 ( 1 - λ ) 2 ( L 1 + L 2 ) d 2 ψ c d Q c 2 E\psi_{c}=-\frac{\hbar^{2}(1-\lambda)}{2(L_{1}+L_{2})}\frac{d^{2}\psi_{c}}{dQ_% {c}^{2}}
  36. E ψ d = - 2 ( 1 - λ ) 2 μ d 2 ψ d d Q d 2 + 1 2 μ ω 2 Q d 2 ψ d E\psi_{d}=-\frac{\hbar^{2}(1-\lambda)}{2\mu}\frac{d^{2}\psi_{d}}{dQ_{d}^{2}}+% \frac{1}{2}\mu\omega^{2}Q_{d}^{2}\psi_{d}
  37. = q 2 ( t ) 2 C + p 2 ( t ) 2 L \mathcal{H}=\frac{q^{2}(t)}{2C}+\frac{p^{2}(t)}{2L}
  38. ( q , p ) q = q ( t ) C = - p ˙ ( t ) \frac{\partial\mathcal{H}(q,p)}{\partial q}=\frac{q(t)}{C}=-\dot{p}(t)
  39. ( q , p ) p = p ( t ) L = - q ˙ ( t ) \frac{\partial\mathcal{H}(q,p)}{\partial p}=\frac{p(t)}{L}=-\dot{q}(t)
  40. q ( t ) = C v ( t ) q(t)=Cv(t)
  41. p ( t ) = L i ( t ) p(t)=Li(t)
  42. v ( t ) - v(t)-
  43. i ( t ) - i(t)-
  44. t - t-
  45. q ( 0 ) , p ( 0 ) q(0),p(0)
  46. ω = 1 L C \omega=\frac{1}{\sqrt{LC}}
  47. ρ = L C \rho=\sqrt{\frac{L}{C}}
  48. t 0 t\geq 0
  49. 𝐪 = q ( 0 ) + j p ( 0 ) ω L \mathbf{q}=q(0)+j\frac{p(0)}{\omega L}
  50. < q ( t ) = R e [ 𝐪 e - j ω t ] <q(t)=Re[\mathbf{q}e^{-j\omega t}]
  51. p ( t ) = I m [ ω L 𝐪 e - j ω t ] p(t)=Im[\omega L\mathbf{q}e^{-j\omega t}]
  52. = | 𝐪 | 2 2 C = c o n s t a n t \mathcal{H}=\frac{|\mathbf{q}|^{2}}{2C}=constant
  53. a ( t ) = a 1 ( t ) + j a 2 ( t ) a(t)=a_{1}(t)+ja_{2}(t)
  54. j = - 1 j=\sqrt{-1}
  55. a 1 ( t ) = q ( t ) q ( 0 ) a_{1}(t)=\frac{q(t)}{q(0)}
  56. q ( 0 ) = D ( 0 ) S C = 2 ρ q(0)=D(0)S_{C}=\sqrt{\frac{2\hbar}{\rho}}
  57. S C - S_{C}-
  58. a 2 ( t ) = p ( t ) p ( 0 ) a_{2}(t)=\frac{p(t)}{p(0)}
  59. p ( 0 ) = 2 ρ p(0)=\sqrt{2\hbar\rho}
  60. S L - S_{L}-
  61. S C = S L = S q S_{C}=S_{L}=S_{q}
  62. ρ = L C = q ( 0 ) p ( 0 ) \rho=\sqrt{\frac{L}{C}}=\frac{q(0)}{p(0)}
  63. a ( t ) = a e - j ω t a(t)=ae^{-j\omega t}
  64. = ω [ a 1 2 ( t ) + a 2 2 ( t ) ] = ω | a | 2 \mathcal{H}=\hbar\omega[a_{1}^{2}(t)+a_{2}^{2}(t)]=\hbar\omega|a|^{2}
  65. p ^ = - j q \hat{p}=-j\hbar\frac{\partial}{\partial q}
  66. [ q ^ , p ^ ] = j [\hat{q},\hat{p}]=j\hbar
  67. a ^ = a 1 ^ + j a 2 ^ = q ^ q 0 + j p ^ p 0 \hat{a}=\hat{a_{1}}+j\hat{a_{2}}=\frac{\hat{q}}{q_{0}}+j\frac{\hat{p}}{p_{0}}
  68. a ( t ) ^ = a ^ e - j ω t \hat{a(t)}=\hat{a}e^{-j\omega t}
  69. ^ ω [ a ^ 1 2 ( t ) + a ^ 2 2 ( t ) ] = ω [ a ^ a ^ + 1 / 2 ] \hat{\mathcal{H}}\hbar\omega[\hat{a}_{1}^{2}(t)+\hat{a}_{2}^{2}(t)]=\hbar% \omega[\hat{a}\hat{a}^{\dagger}+1/2]
  70. [ a ^ 1 ( t ) , a ^ 2 ( t ) ] = j / 2 [\hat{a}_{1}(t),\hat{a}_{2}(t)]=j/2
  71. [ a ^ ( t ) , a ^ ( t ) ] = 1 [\hat{a}(t),\hat{a}^{\dagger}(t)]=1
  72. Δ a ^ 1 2 ( t ) Δ a ^ 2 2 ( t ) 1 16 \langle\Delta\hat{a}_{1}^{2}(t)\rangle\langle\Delta\hat{a}_{2}^{2}(t)\rangle% \geq\frac{1}{16}
  73. ρ 0 = L 0 C 0 = μ 0 ϵ 0 = 2 α h e 2 \rho_{0}=\sqrt{\frac{L_{0}}{C_{0}}}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}=2% \alpha\frac{h}{e^{2}}
  74. e - e-
  75. α - \alpha-
  76. q 0 = 2 ρ 0 = e 2 π α q_{0}=\sqrt{\frac{2\hbar}{\rho_{0}}}=\frac{e}{\sqrt{2\pi\alpha}}
  77. p 0 = 2 ρ 0 = ϕ 0 2 α π p_{0}=\sqrt{2\hbar\rho_{0}}=\phi_{0}\sqrt{\frac{2\alpha}{\pi}}
  78. ϕ 0 = h e - \phi_{0}=\frac{h}{e}-
  79. W L C = W C + W L , W_{LC}=W_{C}+W_{L},
  80. W C = 0.5 C V C 2 - W_{C}=0.5CV_{C}^{2}-
  81. W L = 0.5 L I L 2 - W_{L}=0.5LI_{L}^{2}-
  82. Q C = C V C Q_{C}=CV_{C}
  83. Φ L = L I L . \Phi_{L}=LI_{L}.
  84. W m a x = W L C = Q C 0 2 2 C + Φ L 0 2 2 L . W_{max}=W_{LC}=\frac{Q_{C0}^{2}}{2C}+\frac{\Phi_{L0}^{2}}{2L}.
  85. ω 0 = 1 / L C \omega_{0}=1/\sqrt{LC}
  86. W 0 = ω 0 2 = 2 L C . W_{0}=\frac{\hbar\omega_{0}}{2}=\frac{\hbar}{2\sqrt{LC}}.
  87. Q C 0 = e = C V C Q_{C0}=e=CV_{C}
  88. W C = e 2 2 C . W_{C}=\frac{e^{2}}{2C}.
  89. ξ C = W C W 0 = 2 π R H L C = 2 π ρ q R H . \xi_{C}=\frac{W_{C}}{W_{0}}=\frac{2\pi}{R_{H}}\sqrt{\frac{L}{C}}=2\pi\cdot% \frac{\rho_{q}}{R_{H}}.
  90. ρ q = L / C \rho_{q}=\sqrt{L/C}
  91. ρ q = L C = { ρ w = 2 α R H , - wave impedance ρ D O S = R H , - DOS impedance \rho_{q}=\sqrt{\frac{L}{C}}=\begin{cases}\rho_{w}=2\alpha R_{H},&\mbox{ - wave% impedance }\\ \rho_{DOS}=R_{H},&\mbox{ - DOS impedance }\end{cases}
  92. ξ C = W C W 0 = 2 π ρ q R H = { 4 π α , at ρ w 2 π , at ρ D O S \xi_{C}=\frac{W_{C}}{W_{0}}=2\pi\frac{\rho_{q}}{R_{H}}=\begin{cases}4\pi\alpha% ,&\mbox{at }~{}\rho_{w}\\ 2\pi,&\mbox{at }~{}\rho_{DOS}\end{cases}
  93. ρ D O S i j = Δ Φ j Δ Q i = i j R H , \rho_{DOS}^{ij}=\frac{\Delta\Phi_{j}}{\Delta Q_{i}}=\frac{i}{j}R_{H},
  94. Δ Φ j = j Φ 0 - \Delta\Phi_{j}=j\Phi_{0}-
  95. Δ Q i = i e - \Delta Q_{i}=ie-
  96. i , j = i n t e g e r . i,j=integer.
  97. ω Q = 1 L Q A C Q A = ω B 2 π , \omega_{Q}=\sqrt{\frac{1}{L_{QA}C_{QA}}}=\frac{\omega_{B}}{2\pi},
  98. ω B = e B / m - \omega_{B}=eB/m-
  99. L Q A = 4 π R H ω B L_{QA}=\frac{4\pi R_{H}}{\omega_{B}}
  100. C Q A = 4 π R H ω B . C_{QA}=\frac{4\pi}{R_{H}\omega_{B}}.
  101. I B = e ω B 4 π . I_{B}=\frac{e\omega_{B}}{4\pi}.
  102. W L = L Q A I B 2 2 = ω B 4 . W_{L}=\frac{L_{QA}I_{B}^{2}}{2}=\frac{\hbar\omega_{B}}{4}.
  103. Φ 0 = h / e \Phi_{0}=h/e
  104. W 0 = 1 2 W C 2 π α = γ B Y W C 2 W_{0}=\frac{1}{2}\cdot\frac{W_{C}}{2\pi\alpha}=\frac{\gamma_{BY}W_{C}}{2}
  105. γ B Y = 1 2 π α \gamma_{BY}=\frac{1}{2\pi\alpha}
  106. a B = γ B Y λ 0 , a_{B}=\gamma_{BY}\cdot\lambda_{0},
  107. λ 0 = h / m 0 c - \lambda_{0}=h/m_{0}c-
  108. γ B Y - \gamma_{BY}-
  109. W C = Q C 2 2 C = 2 π α W L C . W_{C}=\frac{Q_{C}^{2}}{2C}=2\pi\alpha W_{LC}.
  110. W L = Φ L 2 2 L = 2 π α W L C . W_{L}=\frac{\Phi_{L}^{2}}{2L}=2\pi\alpha W_{LC}.
  111. W L C = ω L C = L C . W_{LC}=\hbar\omega_{LC}=\frac{\hbar}{\sqrt{LC}}.
  112. W t o t = W L C + W C + W L . W_{tot}=W_{LC}+W_{C}+W_{L}.
  113. W L C W_{LC}
  114. W C W_{C}
  115. e e
  116. W L W_{L}
  117. μ B = e 2 m 0 = 0.5 e ν B S B = e a B 2 L B C B . \mu_{B}=\frac{e\hbar}{2m_{0}}=0.5e\nu_{B}S_{B}=\frac{ea_{B}^{2}}{\sqrt{L_{B}C_% {B}}}.
  118. μ e = 0.5 e ν e S e = 0.5 e m 0 c 2 h λ 0 2 2 π = e 2 m 0 , \mu_{e}=0.5e\nu_{e}S_{e}=0.5e\frac{m_{0}c^{2}}{h}\frac{\lambda_{0}^{2}}{2\pi}=% \frac{e\hbar}{2m_{0}},
  119. C e = 4 π ϵ 0 1 r e - 1 r e + λ 0 = ϵ 0 λ 0 2 π , C_{e}=\frac{4\pi\epsilon_{0}}{\frac{1}{r_{e}}-\frac{1}{r_{e}+\lambda_{0}}}=% \frac{\epsilon_{0}\lambda_{0}}{2\pi},
  120. r e = λ 0 2 2 π - r_{e}=\frac{\lambda_{0}}{2\sqrt{2}\pi}-
  121. λ 0 - \lambda_{0}-
  122. l e = m 0 ω e r e 2 = / 2 , l_{e}=m_{0}\omega_{e}r_{e}^{2}=\hbar/2,
  123. ω e = m 0 c 2 / \omega_{e}=m_{0}c^{2}/\hbar
  124. L e = μ 0 λ 0 2 π . L_{e}=\frac{\mu_{0}\lambda_{0}}{2\pi}.
  125. ρ e = L e C e = μ 0 ϵ 0 = ρ 0 = 2 α R H . \rho_{e}=\sqrt{\frac{L_{e}}{C_{e}}}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}=\rho_{% 0}=2\alpha R_{H}.
  126. ω e = 1 L e C e = 2 π c λ 0 = m 0 c 2 . \omega_{e}=\sqrt{\frac{1}{L_{e}C_{e}}}=\frac{2\pi c}{\lambda_{0}}=\frac{m_{0}c% ^{2}}{\hbar}.
  127. Q e = C e V e . Q_{e}=C_{e}V_{e}.
  128. W C e = C e V e 2 2 = Q e 2 2 C e = 2 π α W 0 , W_{Ce}=\frac{C_{e}V_{e}^{2}}{2}=\frac{Q_{e}^{2}}{2C_{e}}=2\pi\alpha W_{0},
  129. W 0 = m 0 c 2 - W_{0}=m_{0}c^{2}-
  130. Q e = 2 α m 0 c 2 ϵ 0 λ 0 = e . Q_{e}=\sqrt{2\alpha m_{0}c^{2}\epsilon_{0}\lambda_{0}}=e.
  131. Φ e = L e I e . \Phi_{e}=L_{e}I_{e}.
  132. W L e = L e I e 2 2 = Φ e 2 2 L e = 2 π α W 0 . W_{Le}=\frac{L_{e}I_{e}^{2}}{2}=\frac{\Phi_{e}^{2}}{2L_{e}}=2\pi\alpha W_{0}.
  133. Φ e = 2 μ 0 h c α = 2 α Φ 0 . \Phi_{e}=\sqrt{2\mu_{0}hc\alpha}=2\alpha\Phi_{0}.
  134. Φ 0 = h / e - \Phi_{0}=h/e-
  135. a B = λ 0 2 π α a_{B}=\frac{\lambda_{0}}{2\pi\alpha}
  136. λ 0 = h m 0 c - \lambda_{0}=\frac{h}{m_{0}c}-
  137. α - \alpha-
  138. S B = 4 π a B 2 S_{B}=4\pi a_{B}^{2}
  139. L B = μ 0 λ 0 S B L_{B}=\frac{\mu_{0}}{\lambda_{0}}\cdot S_{B}
  140. C B = ϵ 0 λ 0 S B C_{B}=\frac{\epsilon_{0}}{\lambda_{0}}\cdot S_{B}
  141. ρ B = L B C B = μ 0 ϵ 0 = ρ 0 . \rho_{B}=\sqrt{\frac{L_{B}}{C_{B}}}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}=\rho_{% 0}.
  142. ω B = 1 L B C B = α 2 2 m 0 c 2 = 2 π c λ B , \omega_{B}=\sqrt{\frac{1}{L_{B}C_{B}}}=\frac{\alpha^{2}}{2}\cdot\frac{m_{0}c^{% 2}}{\hbar}=\frac{2\pi c}{\lambda_{B}},
  143. λ B = 4 π a B α - \lambda_{B}=\frac{4\pi a_{B}}{\alpha}-
  144. Q B = C B V B . Q_{B}=C_{B}V_{B}.
  145. W C B = C B V B 2 2 = Q B 2 2 C B = 2 π α W B , W_{C}B=\frac{C_{B}V_{B}^{2}}{2}=\frac{Q_{B}^{2}}{2C_{B}}=2\pi\alpha W_{B},
  146. W B = ω B - W_{B}=\hbar\omega_{B}-
  147. Q B = 2 π α 3 m 0 c 2 C B = e . Q_{B}=\sqrt{2\pi\alpha^{3}m_{0}c^{2}C_{B}}=e.
  148. W L B = L B I B 2 2 = Φ B 2 2 L B = 2 π α W 0 . W_{L}B=\frac{L_{B}I_{B}^{2}}{2}=\frac{\Phi_{B}^{2}}{2L_{B}}=2\pi\alpha W_{0}.
  149. Φ B = π α 2 e c λ 0 L B = 2 α Φ 0 . \Phi_{B}=\frac{\pi\alpha^{2}ec}{\lambda_{0}}L_{B}=2\alpha\Phi_{0}.
  150. ω w = 1 L w C w . \omega_{w}=\sqrt{\frac{1}{L_{w}C_{w}}}.
  151. ρ w = L w C w = μ 0 ϵ 0 = ρ 0 . \rho_{w}=\sqrt{\frac{L_{w}}{C_{w}}}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}=\rho_{% 0}.
  152. L w = ρ 0 ω w . L_{w}=\frac{\rho_{0}}{\omega_{w}}.
  153. C w = 1 ρ 0 ω w . C_{w}=\frac{1}{\rho_{0}\omega_{w}}.
  154. ϕ w = L w I w = ϕ 0 = h e . \phi_{w}=L_{w}I_{w}=\phi_{0}=\frac{h}{e}.
  155. I w = h e L w = e ω w 2 α . I_{w}=\frac{h}{eL_{w}}=\frac{e\omega_{w}}{2\alpha}.
  156. D 2 D = m * π 2 D_{2D}=\frac{m^{*}}{\pi\hbar^{2}}
  157. m * = ξ m 0 - m^{*}=\xi m_{0}-
  158. m 0 - m_{0}-
  159. ξ - \xi-
  160. L Q L = ϕ 0 2 D 2 D = ξ L Q 0 L_{QL}=\phi_{0}^{2}\cdot D_{2D}=\xi\cdot L_{Q0}
  161. L Q 0 8 π β L Q Y L_{Q0}8\pi\beta\cdot L_{QY}
  162. ξ = 1 \xi=1
  163. L Q Y = μ 0 λ 0 = H / m 2 L_{QY}=\frac{\mu_{0}}{\lambda_{0}}=H/m^{2}
  164. μ 0 - \mu_{0}-
  165. β = 1 4 α - \beta=\frac{1}{4\alpha}-
  166. α - \alpha-
  167. λ 0 - \lambda_{0}-
  168. L Q A = L Q L n B L_{QA}=\frac{L_{QL}}{n_{B}}
  169. n B = e B h n_{B}=\frac{eB}{h}
  170. h - h-
  171. C Q A = C Q L n B C_{QA}=\frac{C_{QL}}{n_{B}}
  172. C Q L = e 2 D 2 D = ξ C Q 0 C_{QL}=e^{2}\cdot D_{2D}=\xi\cdot C_{Q0}
  173. C Q 0 = 8 π α C Q Y C_{Q0}=8\pi\alpha\cdot C_{QY}
  174. ξ = 1 \xi=1
  175. C Q Y = ϵ 0 λ 0 = 3.6492417 F / m 2 C_{QY}=\frac{\epsilon_{0}}{\lambda_{0}}=3.6492417F/m^{2}
  176. ϵ 0 - \epsilon_{0}-
  177. ρ Q = L Q A C Q A = ϕ 0 2 e 2 = h e 2 = R H \rho_{Q}=\sqrt{\frac{L_{QA}}{C_{QA}}}=\sqrt{\frac{\phi_{0}^{2}}{e^{2}}}=\frac{% h}{e^{2}}=R_{H}
  178. R H = h e 2 = 25.812813 k Ω R_{H}=\frac{h}{e^{2}}=25.812813k\Omega
  179. ω Q = 1 L Q A C Q A = ω c ϕ 0 e = ω c 2 π \omega_{Q}=\frac{1}{\sqrt{L_{QA}C_{QA}}}=\frac{\hbar\omega_{c}}{\phi_{0}e}=% \frac{\omega_{c}}{2\pi}
  180. ω c = e B m * - \omega_{c}=\frac{eB}{m^{*}}-
  181. I H = h e L Q A = e ω B 4 π I_{H}=\frac{h}{eL_{QA}}=\frac{e\omega_{B}}{4\pi}
  182. ω B = e B m * - \omega_{B}=\frac{eB}{m^{*}}-
  183. V i n d = Φ t = - L I t , V_{ind}=\frac{\partial\Phi}{\partial t}=-L\frac{\partial I}{\partial t},
  184. Φ - \Phi-
  185. L - L-
  186. I - I-
  187. I = I J sin ϕ , I=I_{J}\cdot\sin\phi,
  188. I J - I_{J}-
  189. ϕ - \phi-
  190. I t = I J cos ϕ ϕ t . \frac{\partial I}{\partial t}=I_{J}\cos\phi\cdot\frac{\partial\phi}{\partial t}.
  191. ϕ t = q V = 2 π Φ 0 V , \frac{\partial\phi}{\partial t}=\frac{q}{\hbar}V=\frac{2\pi}{\Phi_{0}}V,
  192. - \hbar-
  193. Φ 0 = h / 2 e - \Phi_{0}=h/2e-
  194. q = 2 e q=2e
  195. e - e-
  196. V = Φ 0 2 π I J 1 cos ϕ I t = L J I t , V=\frac{\Phi_{0}}{2\pi I_{J}}\cdot\frac{1}{\cos\phi}\cdot\frac{\partial I}{% \partial t}=L_{J}\cdot\frac{\partial I}{\partial t},
  197. L J = Φ 0 2 π I J 1 cos ϕ L_{J}=\frac{\Phi_{0}}{2\pi I_{J}}\cdot\frac{1}{\cos\phi}
  198. ω J = q V . \omega_{J}=\frac{q}{\hbar}\cdot V.
  199. ω J = 1 L J C J . \omega_{J}=\sqrt{\frac{1}{L_{J}C_{J}}}.
  200. C J - C_{J}-
  201. C J = 1 L J ω J 2 = Φ 0 I J V 0 2 cos ϕ 2 π . C_{J}=\frac{1}{L_{J}\omega_{J}^{2}}=\frac{\Phi_{0}I_{J}}{V_{0}^{2}}\cdot\frac{% \cos\phi}{2\pi}.
  202. ρ J = L J C J = V 0 I J 1 cos ϕ . \rho_{J}=\sqrt{\frac{L_{J}}{C_{J}}}=\frac{V_{0}}{I_{J}}\cdot\frac{1}{\sqrt{% \cos\phi}}.
  203. V 0 = 0 , 1 V_{0}=0,1
  204. I J = 0 , 2 μ I_{J}=0,2\mu
  205. ρ J = 500 Ω . \rho_{J}=500\Omega.
  206. C F 0 = ϵ 0 λ 0 S F 0 = 5.1805 10 - 7 C_{F0}=\frac{\epsilon_{0}}{\lambda_{0}}\cdot S_{F0}=5.1805\cdot 10^{-7}
  207. λ 0 = h m 0 c \lambda_{0}=\frac{h}{m_{0}c}
  208. L F 0 = μ 0 λ 0 S F 0 = 7.3524 10 - 2 L_{F0}=\frac{\mu_{0}}{\lambda_{0}}\cdot S_{F0}=7.3524\cdot 10^{-2}
  209. S F 0 = λ 0 c ω F 0 = h m 0 ω F 0 = 1.4196 10 - 7 S_{F0}=\frac{\lambda_{0}c}{\omega_{F0}}=\frac{h}{m_{0}\omega_{F0}}=1.4196\cdot 1% 0^{-7}
  210. ω F 0 = 1 L F 0 C F 0 = 5123.9 \omega_{F0}=\sqrt{\frac{1}{L_{F0}C_{F0}}}=5123.9
  211. ρ F 0 = L F 0 C F 0 = ρ 0 = 2 α R H , \rho_{F0}=\sqrt{\frac{L_{F0}}{C_{F0}}}=\rho_{0}=2\alpha R_{H},
  212. ρ 0 \rho_{0}
  213. Q F 1 = e S F 0 S B = 2 10 6 e Q_{F1}=e\sqrt{\frac{S_{F0}}{S_{B}}}=2\cdot 10^{6}e
  214. S B = 4 π a B 2 - S_{B}=4\pi a_{B}^{2}-
  215. ω F 0 n = ω F 0 n . \omega_{F0n}=\frac{\omega_{F0}}{n}.

Quantum_limit.html

  1. x ^ \hat{x}
  2. x ^ \hat{x}
  3. 𝒪 ^ \hat{\mathcal{O}}
  4. ^ \hat{\mathcal{F}}
  5. 𝒪 ^ \hat{\mathcal{O}}
  6. x ^ \hat{x}
  7. δ 𝒪 ^ \delta\hat{\mathcal{O}}
  8. x ^ \hat{x}
  9. Δ 𝒪 Δ / 2 , \Delta{\mathcal{O}}\Delta{\mathcal{F}}\geqslant\hbar/2\,,
  10. Δ a = a ^ 2 - a ^ 2 \Delta a=\sqrt{\langle\hat{a}^{2}\rangle-\langle\hat{a}\rangle^{2}}
  11. a a
  12. a ^ \langle\hat{a}\rangle
  13. a a
  14. Δ δ 𝒪 \Delta\mathcal{\delta O}
  15. x ^ \hat{x}
  16. 𝒪 ^ = x ^ f r e e + δ 𝒪 ^ + δ x ^ B A [ ^ ] , \hat{\mathcal{O}}=\hat{x}_{free}+\delta\hat{\mathcal{O}}+\delta\hat{x}_{BA}[% \hat{\mathcal{F}}]\,,
  17. x ^ f r e e \hat{x}_{free}
  18. x ^ \hat{x}
  19. δ x B A ^ [ ^ ] \delta\hat{x_{BA}}[\hat{\mathcal{F}}]
  20. x ^ \hat{x}
  21. ^ \hat{\mathcal{F}}
  22. Δ Δ 𝒪 - 1 \Delta\mathcal{F}\propto\Delta\mathcal{O}^{-1}
  23. δ 𝒪 ^ \delta\hat{\mathcal{O}}
  24. ^ \hat{\mathcal{F}}
  25. M M
  26. x x
  27. M M
  28. ϑ \vartheta
  29. M M
  30. j j
  31. x ( t j ) x(t_{j})
  32. ϕ ^ j refl = ϕ ^ j - 2 k p x ^ ( t j ) , \hat{\phi}_{j}^{\mathrm{refl}}=\hat{\phi}_{j}-2k_{p}\hat{x}(t_{j})\,,
  33. k p = ω p / c k_{p}=\omega_{p}/c
  34. ω p \omega_{p}
  35. j = , - 1 , 0 , 1 , j=\dots,-1,0,1,\dots
  36. ϕ ^ j \hat{\phi}_{j}
  37. j j
  38. ϕ ^ j = 0 \langle\hat{\phi}_{j}\rangle=0
  39. ( ϕ 2 ^ - ϕ ^ 2 ) 1 / 2 \langle(\hat{\phi^{2}}\rangle-\langle\hat{\phi}\rangle^{2})^{1/2}
  40. Δ ϕ \Delta\phi
  41. ϕ ^ j \hat{\phi}_{j}
  42. 𝒪 \mathcal{O}
  43. ϕ ^ j refl \hat{\phi}_{j}^{\mathrm{refl}}
  44. Δ ϕ \Delta\phi
  45. Δ x meas = Δ ϕ 2 k p . \Delta x_{\mathrm{meas}}=\frac{\Delta\phi}{2k_{p}}\,.
  46. x ~ j - ϕ ^ j refl 2 k p = x ^ ( t j ) + x ^ fl ( t j ) , \tilde{x}_{j}\equiv-\frac{\hat{\phi}_{j}^{\mathrm{refl}}}{2k_{p}}=\hat{x}(t_{j% })+\hat{x}_{\mathrm{fl}}(t_{j})\,,
  47. x ^ fl ( t j ) = - ϕ ^ j 2 k p \hat{x}_{\mathrm{fl}}(t_{j})=-\frac{\hat{\phi}_{j}}{2k_{p}}
  48. p ^ j after - p ^ j before = p ^ j b . a . = 2 c 𝒲 ^ j , \hat{p}_{j}^{\mathrm{after}}-\hat{p}_{j}^{\mathrm{before}}=\hat{p}_{j}^{% \mathrm{b.a.}}=\frac{2}{c}\hat{\mathcal{W}}_{j}\,,
  49. p ^ j before \hat{p}_{j}^{\mathrm{before}}
  50. p ^ j after \hat{p}_{j}^{\mathrm{after}}
  51. 𝒲 j \mathcal{W}_{j}
  52. j j
  53. ^ \hat{\mathcal{F}}
  54. p ^ j b . a . = 2 c 𝒲 , \langle\hat{p}_{j}^{\mathrm{b.a.}}\rangle=\frac{2}{c}\mathcal{W}\,,
  55. 𝒲 \mathcal{W}
  56. p ^ b . a . ( t j ) = p ^ j b . a . - p ^ j b . a . = 2 c ( 𝒲 ^ j - 𝒲 ) , \hat{p}^{\mathrm{b.a.}}(t_{j})=\hat{p}_{j}^{\mathrm{b.a.}}-\langle\hat{p}_{j}^% {\mathrm{b.a.}}\rangle=\frac{2}{c}\bigl(\hat{\mathcal{W}}_{j}-\mathcal{W}\bigr% )\,,
  57. Δ p b . a . = 2 Δ 𝒲 c , \Delta p_{\mathrm{b.a.}}=\frac{2\Delta\mathcal{W}}{c}\,,
  58. Δ 𝒲 \Delta\mathcal{W}
  59. ϑ T \vartheta\ll T
  60. j j
  61. j + 1 j+1
  62. ϑ \vartheta
  63. x ^ b . a . ( t j ) = p ^ b . a . ( t j ) ϑ M . \hat{x}_{\mathrm{b.a.}}(t_{j})=\frac{\hat{p}^{\mathrm{b.a.}}(t_{j})\vartheta}{% M}\,.
  64. Δ x b . a . ( t j ) = Δ p b . a . ( t j ) ϑ M . \Delta x_{\mathrm{b.a.}}(t_{j})=\frac{\Delta{p}_{\mathrm{b.a.}}(t_{j})% \vartheta}{M}\,.
  65. j j
  66. j + 1 j+1
  67. δ x ~ j + 1 , j = x ~ ( t j + 1 ) - x ~ ( t j + 1 ) \delta\tilde{x}_{j+1,j}=\tilde{x}(t_{j+1})-\tilde{x}(t_{j+1})
  68. Δ x ~ j + 1 , j = [ ( Δ x meas ( t j + 1 ) ) 2 + ( Δ x meas ( t j ) ) 2 + ( Δ x b . a . ( t j ) ) 2 ] 1 / 2 , \Delta\tilde{x}_{j+1,j}=\Bigl[(\Delta x_{\rm meas}(t_{j+1}))^{2}+(\Delta x_{% \rm meas}(t_{j}))^{2}+(\Delta x_{\rm b.a.}(t_{j}))^{2}\Bigr]^{1/2}\,,
  69. Δ x meas ( t j + 1 ) = Δ x meas ( t j ) Δ x meas = Δ ϕ / ( 2 k p ) \Delta x_{\rm meas}(t_{j+1})=\Delta x_{\rm meas}(t_{j})\equiv\Delta x_{\rm meas% }=\Delta\phi/(2k_{p})
  70. Δ 𝒲 Δ ϕ ω p 2 . \Delta\mathcal{W}\Delta\phi\geq\frac{\hbar\omega_{p}}{2}\,.
  71. Δ x meas \Delta x_{\mathrm{meas}}
  72. Δ p b . a . \Delta p_{\mathrm{b.a.}}
  73. Δ x meas Δ p b . a . 2 . \Delta x_{\mathrm{meas}}\Delta p_{\mathrm{b.a.}}\geq\frac{\hbar}{2}\,.
  74. Δ x meas \Delta x_{\mathrm{meas}}
  75. Δ x b . a . \Delta x_{\mathrm{b.a.}}
  76. Δ x min = ϑ 2 M \Delta x_{\mathrm{min}}=\sqrt{\frac{\hbar\vartheta}{2M}}
  77. Δ x ~ j + 1 , j [ 2 ( Δ x meas ) 2 + ( ϑ 2 M Δ x meas ) 2 ] 1 / 2 3 ϑ 2 M , \Delta\tilde{x}_{j+1,j}\geqslant\Bigl[2(\Delta x_{\rm meas})^{2}+\Bigl(\frac{% \hbar\vartheta}{2M\Delta x_{\rm meas}}\Bigr)^{2}\Bigr]^{1/2}\geqslant\sqrt{% \frac{3\hbar\vartheta}{2M}}\,,
  78. Δ x meas ( t j + 1 ) \Delta x_{\rm meas}(t_{j+1})
  79. Δ p b . a . ( t j + 1 ) \Delta p_{\rm b.a.}(t_{j+1})\to\infty
  80. Δ x ~ S Q L = ϑ M , \Delta\tilde{x}_{SQL}=\sqrt{\frac{\hbar\vartheta}{M}}\,,
  81. x ^ fl ( t j ) \hat{x}_{\mathrm{fl}}(t_{j})
  82. p ^ b . a . ( t j ) \hat{p}^{\mathrm{b.a.}}(t_{j})

Quantum_money.html

  1. 3 / 4 3/4
  2. N N
  3. ( 3 / 4 ) N (3/4)^{N}
  4. N N

Quantum_spacetime.html

  1. x , p x,p
  2. λ \lambda
  3. λ 0 \lambda\to 0
  4. λ 0 \lambda\to 0
  5. [ x i , x j ] = 0 , [ x i , t ] = i λ x i [x_{i},x_{j}]=0,\quad[x_{i},t]=i\lambda x_{i}
  6. x i x_{i}
  7. t t
  8. λ \lambda
  9. λ - 1 \lambda^{-1}
  10. p i p_{i}
  11. p 0 p_{0}
  12. p i p_{i}
  13. p 1 2 + p 2 2 + p 3 2 < λ - 1 \sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}<\lambda^{-1}\,
  14. p 0 , p i p_{0},p_{i}
  15. e i i p i x i e i p 0 t e^{i\sum_{i}p_{i}x_{i}}e^{ip_{0}t}\,
  16. κ \kappa
  17. κ \kappa
  18. κ = λ - 1 \kappa=\lambda^{-1}
  19. q q
  20. ( α β γ δ ) = ( α β γ δ ) , β α = q 2 α β , [ α , δ ] = 0 , [ β , γ ] = ( 1 - q - 2 ) α ( δ - α ) , [ δ , β ] = ( 1 - q - 2 ) α β \begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}^{\dagger},\quad\beta\alpha=q^{2}\alpha\beta,\ [% \alpha,\delta]=0,\ [\beta,\gamma]=(1-q^{-2})\alpha(\delta-\alpha),\ [\delta,% \beta]=(1-q^{-2})\alpha\beta
  21. q 1 q\to 1
  22. x , y , z , t x,y,z,t
  23. t = Trace q ( α β γ δ ) = q δ + q - 1 α t=\,\text{Trace}_{q}\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}=q\delta+q^{-1}\alpha
  24. det q ( α β γ δ ) = α δ - q 2 γ β {\det}_{q}\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}=\alpha\delta-q^{2}\gamma\beta
  25. q 1 q\to 1
  26. q = e λ q=e^{\lambda}
  27. q = e i λ q=e^{i\lambda}
  28. λ \lambda
  29. q q
  30. q q
  31. q q
  32. [ x 1 , x 2 ] = 2 i λ x 3 , [ x 2 , x 3 ] = 2 i λ x 1 , [ x 3 , x 1 ] = 2 i λ x 2 [x_{1},x_{2}]=2i\lambda x_{3},\ [x_{2},x_{3}]=2i\lambda x_{1},\ [x_{3},x_{1}]=% 2i\lambda x_{2}
  33. [ x μ , x ν ] = ı M μ ν [x_{\mu},x_{\nu}]=\imath M_{\mu\nu}
  34. M μ ν M_{\mu\nu}
  35. M μ ν M_{\mu\nu}
  36. x μ x_{\mu}
  37. x μ x_{\mu}
  38. M M
  39. θ \theta
  40. [ x μ , x ν ] = i θ μ ν [x_{\mu},x_{\nu}]=i\theta_{\mu\nu}
  41. D D
  42. θ \theta
  43. 2 {}^{2}

Quantum_triviality.html

  1. g g
  2. g g
  3. g o b s = g 0 1 + β 2 g 0 ln Λ / m , g_{obs}=\frac{g_{0}}{1+\beta_{2}g_{0}\ln\Lambda/m}~{},
  4. m m
  5. Λ Λ
  6. g g
  7. g g
  8. Λ Λ
  9. g g
  10. Λ Λ
  11. g g
  12. g 0 = g o b s 1 - β 2 g o b s ln Λ / m . g_{0}=\frac{g_{obs}}{1-\beta_{2}g_{obs}\ln\Lambda/m}~{}.
  13. g g
  14. Λ Λ
  15. g g
  16. g g
  17. g ( μ ) g(μ)
  18. μ μ
  19. d g d ln μ = β ( g ) = β 2 g 2 + β 3 g 3 + , \frac{dg}{d\ln\mu}=\beta(g)=\beta_{2}g^{2}+\beta_{3}g^{3}+\ldots~{},
  20. μ μ
  21. m m
  22. g ( μ ) g(μ)
  23. g g
  24. μ μ
  25. Λ Λ
  26. β 2 \beta_{2}
  27. g ( μ ) g(\mu)
  28. β ( g ) β(g)
  29. g g
  30. μ 0 \mu_{0}
  31. g g
  32. Λ Λ
  33. g g
  34. g g
  35. m m
  36. g g
  37. β β

Quantum_well_infrared_photodetector.html

  1. ϕ \phi
  2. I p h I_{ph}
  3. I p h = e ϕ η g p h I_{ph}=e\phi\eta g_{ph}
  4. e e
  5. η \eta
  6. g p h g_{ph}
  7. η \eta
  8. g p h g_{ph}
  9. η \eta
  10. g p h g_{ph}
  11. g p h g_{ph}
  12. g p h g_{ph}
  13. p c 1 p_{c}\leq 1
  14. g p h g_{ph}
  15. g p h g_{ph}
  16. p c p_{c}
  17. p e p_{e}
  18. g p h = p e N p c g_{ph}=\frac{p_{e}}{N\,p_{c}}
  19. N N
  20. p c p_{c}
  21. p e p_{e}

Quartet_distance.html

  1. O ( N 4 ) O(N^{4})
  2. N N
  3. O ( N 2 ) O(N^{2})
  4. O ( N log 2 N ) O(N\log^{2}N)
  5. O ( N log N ) O(N\log N)
  6. O ( N 2 D 2 ) O(N^{2}D^{2})
  7. D D

Quasi-analytic_function.html

  1. M = { M k } k = 0 M=\{M_{k}\}_{k=0}^{\infty}
  2. | d k f d x k ( x ) | A k + 1 M k \left|\frac{d^{k}f}{dx^{k}}(x)\right|\leq A^{k+1}M_{k}
  3. d k f d x k ( x ) = 0 \frac{d^{k}f}{dx^{k}}(x)=0
  4. 1 / L j = \sum 1/L_{j}=\infty
  5. L j = inf k j M k 1 / k L_{j}=\inf_{k\geq j}M_{k}^{1/k}
  6. j ( M j * ) - 1 / j = \sum_{j}(M_{j}^{*})^{-1/j}=\infty
  7. j M j - 1 * / M j * = . \sum_{j}M_{j-1}^{*}/M_{j}^{*}=\infty.
  8. n n , ( n log n ) n , ( n log n log log n ) n , ( n log n log log n log log log n ) n n^{n},\,(n\log n)^{n},\,(n\log n\log\log n)^{n},\,(n\log n\log\log n\log\log% \log n)^{n}\dots

Quasi-homogeneous_polynomial.html

  1. f ( x ) = α a α x α , where α = ( i 1 , , i r ) r , and x α = x 1 i 1 x r i r , f(x)=\sum_{\alpha}a_{\alpha}x^{\alpha}\,\text{, where }\alpha=(i_{1},\dots,i_{% r})\in\mathbb{N}^{r}\,\text{, and }x^{\alpha}=x_{1}^{i_{1}}\cdots x_{r}^{i_{r}},
  2. w 1 , , w r w_{1},\ldots,w_{r}
  3. w = w 1 i 1 + + w r i r w=w_{1}i_{1}+\cdots+w_{r}i_{r}
  4. f ( λ w 1 x 1 , , λ w r x r ) = λ w f ( x 1 , , x r ) f(\lambda^{w_{1}}x_{1},\ldots,\lambda^{w_{r}}x_{r})=\lambda^{w}f(x_{1},\ldots,% x_{r})
  5. λ \lambda
  6. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  7. w 1 , , w r w_{1},\ldots,w_{r}
  8. f ( y 1 w 1 , , y n w n ) f(y_{1}^{w_{1}},\ldots,y_{n}^{w_{n}})
  9. y i y_{i}
  10. α \alpha
  11. { α | a α 0 } , \{\alpha|a_{\alpha}\neq 0\},
  12. f ( x , y ) = 5 x 3 y 3 + x y 9 - 2 y 12 f(x,y)=5x^{3}y^{3}+xy^{9}-2y^{12}
  13. f ( λ x , λ y ) f(\lambda x,\lambda y)
  14. ( λ 3 , λ ) (\lambda^{3},\lambda)
  15. f ( λ 3 x , λ y ) = 5 ( λ 3 x ) 3 ( λ y ) 3 + ( λ 3 x ) ( λ y ) 9 - 2 ( λ y ) 12 = λ 12 f ( x , y ) . f(\lambda^{3}x,\lambda y)=5(\lambda^{3}x)^{3}(\lambda y)^{3}+(\lambda^{3}x)(% \lambda y)^{9}-2(\lambda y)^{12}=\lambda^{12}f(x,y).\,
  16. f ( x , y ) f(x,y)
  17. 3 i 1 + 1 i 2 = 12 3i_{1}+1i_{2}=12
  18. f ( x , y ) f(x,y)
  19. 3 x + y = 12 3x+y=12
  20. 2 \mathbb{R}^{2}
  21. 1 4 x + 1 12 y = 1 \tfrac{1}{4}x+\tfrac{1}{12}y=1
  22. 1 4 , 1 12 \tfrac{1}{4},\tfrac{1}{12}
  23. g ( x , y ) g(x,y)
  24. 1 i 1 + 1 i 2 = d 1i_{1}+1i_{2}=d
  25. f ( x ) f(x)
  26. x = x 1 x r x=x_{1}\ldots x_{r}
  27. f ( x ) = α r a α x α , α = ( i 1 , , i r ) , a α . f(x)=\sum_{\alpha\in\mathbb{N}^{r}}a_{\alpha}x^{\alpha},\alpha=(i_{1},\ldots,i% _{r}),a_{\alpha}\in\mathbb{R}.
  28. φ = ( φ 1 , , φ r ) \varphi=(\varphi_{1},\ldots,\varphi_{r})
  29. φ i \varphi_{i}\in\mathbb{N}
  30. a a\in\mathbb{R}
  31. α , φ = k r i k φ k = a , \langle\alpha,\varphi\rangle=\sum_{k}^{r}i_{k}\varphi_{k}=a,
  32. a α 0 a_{\alpha}\neq 0

Quasiregular_element.html

  1. x + y - x y = 0 x+y-xy=0
  2. x y = x + y - x y x\cdot y=x+y-xy
  3. \cdot
  4. ( R , ) ( R , × ) ; x 1 - x (R,\cdot)\to(R,\times);x\mapsto 1-x
  5. x + y + x y = 0 x+y+xy=0
  6. x y = x + y + x y x\circ y=x+y+xy
  7. ( - x ) ( - y ) = - ( x y ) (-x)\circ(-y)=-(x\cdot y)
  8. x 2 x^{2}
  9. x x
  10. x n + 1 = 0 x^{n+1}=0
  11. ( 1 - x ) ( 1 + x + x 2 + + x n ) = 1 (1-x)(1+x+x^{2}+\cdots+x^{n})=1
  12. ( 1 + x ) ( 1 - x + x 2 - + ( - x ) n ) = 1 (1+x)(1-x+x^{2}-\cdots+(-x)^{n})=1
  13. - x - x 2 - - x n -x-x^{2}-\cdots-x^{n}
  14. - x + x 2 - + ( - x ) n -x+x^{2}-\cdots+(-x)^{n}
  15. x < 1 \|x\|<1
  16. 0 x n \sum_{0}^{\infty}x^{n}
  17. μ a ( r ) = r a + 1 \mu_{a}(r)=ra+1
  18. μ a \mu_{a}
  19. μ a \mu_{a}
  20. 1 1 - a \frac{1}{1-a}
  21. x y = 0 = y x x\cdot y=0=y^{\prime}\cdot x
  22. y = ( y x ) y = y ( x y ) = y y=(y^{\prime}\cdot x)\cdot y=y^{\prime}\cdot(x\cdot y)=y^{\prime}