wpmath0000009_9

Papkovichโ€“Neuber_solution.html

  1. ๐Ÿ = 0 \mathbf{f}=0
  2. ๐ฎ = 1 2 ฮผ [ โˆ‡ ( ๐ฑ โ‹… ๐šฝ + ฯ‡ ) - 2 ๐šฝ ] \mathbf{u}={1\over{2\mu}}\left[\nabla(\mathbf{x}\cdot\mathbf{\Phi}+\chi)-2% \mathbf{\Phi}\right]
  3. p = โˆ‡ โ‹… ๐šฝ p=\nabla\cdot\mathbf{\Phi}
  4. ๐šฝ \mathbf{\Phi}
  5. ฯ‡ \chi

Papyrus_1.html

  1. ๐”“ \mathfrak{P}
  2. ๐”“ \mathfrak{P}
  3. ๐”“ \mathfrak{P}
  4. ๐”“ \mathfrak{P}

Papyrus_115.html

  1. ๐”“ \mathfrak{P}
  2. ๐”“ \mathfrak{P}
  3. ฮ™ ฮ— ฮ› ยฏ \overline{ฮ™ฮ—ฮ›}
  4. ฮ‘ ฮฅ ฮค ฮŸ ฮฅ ยฏ \overline{ฮ‘ฮฅฮคฮŸฮฅ}
  5. ฮ  ฮก ฮฃ ยฏ \overline{ฮ ฮกฮฃ}
  6. ฮ˜ ฮฉ ยฏ \overline{ฮ˜ฮฉ}
  7. ฮ˜ ฮฅ ยฏ \overline{ฮ˜ฮฅ}
  8. ฮ‘ ฮ ฮฉ ฮ ยฏ \overline{ฮ‘ฮฮฉฮ}
  9. ฮ  ฮ ฮ‘ ยฏ \overline{ฮ ฮฮ‘}
  10. ฮŸ ฮฅ ฮ ฮŸ ฮฅ ยฏ \overline{ฮŸฮฅฮฮŸฮฅ}
  11. ฮŸ ฮฅ ฮ ฮŸ ฮ ยฏ \overline{ฮŸฮฅฮฮŸฮ}
  12. ฮš ฮฅ ยฏ \overline{ฮšฮฅ}
  13. ฮ˜ ฮ ยฏ \overline{ฮ˜ฮ}
  14. ฮ‘ ฮ ฮŸ ฮฅ ยฏ \overline{ฮ‘ฮฮŸฮฅ}
  15. ฮŸ ฮฅ ฮ ฮฉ ยฏ \overline{ฮŸฮฅฮฮฉ}
  16. ๐”“ \mathfrak{P}
  17. ฯ‡ ฮพ ฯ› ยฏ \overline{ฯ‡ฮพฯ›}
  18. ฯ‡ ฮน ฯ› ยฏ \overline{ฯ‡ฮนฯ›}
  19. ๐”“ \mathfrak{P}
  20. ๐”“ \mathfrak{P}
  21. ๐”“ \mathfrak{P}
  22. ฮด ยฏ \overline{ฮด}
  23. ๐”“ \mathfrak{P}
  24. ๐” \mathfrak{M}
  25. ฮด ยฏ \overline{ฮด}
  26. ๐”“ \mathfrak{P}
  27. ๐”“ \mathfrak{P}
  28. ๐”“ \mathfrak{P}
  29. ๐” \mathfrak{M}
  30. ๐” \mathfrak{M}
  31. ๐”“ \mathfrak{P}
  32. ๐”“ \mathfrak{P}
  33. ๐”“ \mathfrak{P}
  34. ๐” \mathfrak{M}
  35. ๐”“ \mathfrak{P}
  36. ๐”“ \mathfrak{P}
  37. ๐”“ \mathfrak{P}
  38. ๐” \mathfrak{M}
  39. ๐”“ \mathfrak{P}
  40. ๐”“ \mathfrak{P}
  41. ฮฒ ฯ‡ ยฏ \overline{ฮฒฯ‡}
  42. ๐”“ \mathfrak{P}
  43. ฮฑ ฯ‡ ยฏ \overline{ฮฑฯ‡}
  44. ๐”“ \mathfrak{P}
  45. ๐” \mathfrak{M}

Papyrus_13.html

  1. ๐”“ \mathfrak{P}

Papyrus_2.html

  1. ๐”“ \mathfrak{P}

Papyrus_37.html

  1. ๐”“ \mathfrak{P}
  2. ๐”“ \mathfrak{P}

Papyrus_45.html

  1. ๐”“ \mathfrak{P}
  2. ๐”“ \mathfrak{P}
  3. ๐”“ \mathfrak{P}
  4. ๐”“ \mathfrak{P}
  5. ๐”“ \mathfrak{P}
  6. ๐”“ \mathfrak{P}
  7. ๐”“ \mathfrak{P}
  8. ๐”“ \mathfrak{P}
  9. ๐”“ \mathfrak{P}
  10. ๐”“ \mathfrak{P}
  11. ๐”“ \mathfrak{P}
  12. ๐”“ \mathfrak{P}
  13. ๐”“ \mathfrak{P}
  14. ๐”“ \mathfrak{P}
  15. ๐”“ \mathfrak{P}
  16. ๐”“ \mathfrak{P}
  17. ๐”“ \mathfrak{P}
  18. ๐”“ \mathfrak{P}
  19. ๐”“ \mathfrak{P}
  20. ๐”“ \mathfrak{P}
  21. ๐”“ \mathfrak{P}
  22. ๐”“ \mathfrak{P}
  23. ๐”“ \mathfrak{P}
  24. ๐”“ \mathfrak{P}
  25. ๐”“ \mathfrak{P}
  26. ๐”“ \mathfrak{P}
  27. ๐”“ \mathfrak{P}
  28. ๐”“ \mathfrak{P}
  29. ๐”“ \mathfrak{P}

Para-Nitrophenylphosphate.html

  1. P N P P + H 2 O A P โ†’ P N P + P i PNPP+H_{2}O\;\overrightarrow{{}_{AP}}\;PNP+P_{i}

Parabolic_induction.html

  1. P = M A N P=MAN
  2. M A MA

Parabolic_Lie_algebra.html

  1. ๐”ญ \mathfrak{p}
  2. ๐”ค \mathfrak{g}
  3. ๐”ญ \mathfrak{p}
  4. ๐”ค \mathfrak{g}
  5. ๐”ญ \mathfrak{p}
  6. ๐”ค \mathfrak{g}
  7. ๐”ญ \mathfrak{p}
  8. ๐”ฝ \mathbb{F}
  9. ๐”ญ โŠ— ๐”ฝ ๐”ฝ ยฏ \mathfrak{p}\otimes_{\mathbb{F}}\overline{\mathbb{F}}
  10. ๐”ค โŠ— ๐”ฝ ๐”ฝ ยฏ \mathfrak{g}\otimes_{\mathbb{F}}\overline{\mathbb{F}}
  11. ๐”ฝ ยฏ \overline{\mathbb{F}}
  12. ๐”ฝ \mathbb{F}

Paradoxes_of_set_theory.html

  1. โ„ต 0 \aleph_{0}
  2. ฯ• ( a , x ) \phi(a,x)

Parametricity.html

  1. โˆ€ \forall
  2. โˆ€ \forall
  3. โ†ฆ \mapsto

Parity-check_matrix.html

  1. H = [ 0 0 1 1 1 1 0 0 ] H=\left[\begin{array}[]{cccc}0&0&1&1\\ 1&1&0&0\end{array}\right]
  2. c 3 + c 4 = 0 c 1 + c 2 = 0 \begin{aligned}\displaystyle c_{3}+c_{4}&\displaystyle=0\\ \displaystyle c_{1}+c_{2}&\displaystyle=0\end{aligned}
  3. ( c 1 , c 2 , c 3 , c 4 ) (c_{1},c_{2},c_{3},c_{4})
  4. G = [ I k | P ] G=\begin{bmatrix}I_{k}|P\end{bmatrix}
  5. H = [ - P โŠค | I n - k ] H=\begin{bmatrix}-P^{\top}|I_{n-k}\end{bmatrix}
  6. G H โŠค = P - P = 0 GH^{\top}=P-P=0
  7. G = [ 1 0 1 0 1 0 1 1 1 0 ] G=\left[\begin{array}[]{cc|ccc}1&0&1&0&1\\ 0&1&1&1&0\\ \end{array}\right]
  8. H = [ 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 ] H=\left[\begin{array}[]{cc|ccc}1&1&1&0&0\\ 0&1&0&1&0\\ 1&0&0&0&1\\ \end{array}\right]

Parkland_formula.html

  1. V = 4 โ‹… m โ‹… ( A โ‹… 100 ) V=4\cdot m\cdot(A\cdot 100)

Partial_correlation.html

  1. ๐ฐ X * \mathbf{w}_{X}^{*}
  2. ๐ฐ Y * \mathbf{w}_{Y}^{*}
  3. ๐ฐ X * = arg min ๐ฐ { โˆ‘ i = 1 N ( x i - โŸจ ๐ฐ , ๐ณ i โŸฉ ) 2 } \mathbf{w}_{X}^{*}=\arg\min_{\mathbf{w}}\left\{\sum_{i=1}^{N}(x_{i}-\langle% \mathbf{w},\mathbf{z}_{i}\rangle)^{2}\right\}
  4. ๐ฐ Y * = arg min ๐ฐ { โˆ‘ i = 1 N ( y i - โŸจ ๐ฐ , ๐ณ i โŸฉ ) 2 } \mathbf{w}_{Y}^{*}=\arg\min_{\mathbf{w}}\left\{\sum_{i=1}^{N}(y_{i}-\langle% \mathbf{w},\mathbf{z}_{i}\rangle)^{2}\right\}
  5. โŸจ ๐ฏ , ๐ฐ โŸฉ \langle\mathbf{v},\mathbf{w}\rangle
  6. ๐ณ \mathbf{z}
  7. r X , i = x i - โŸจ ๐ฐ X * , ๐ณ i โŸฉ r_{X,i}=x_{i}-\langle\mathbf{w}_{X}^{*},\mathbf{z}_{i}\rangle
  8. r Y , i = y i - โŸจ ๐ฐ Y * , ๐ณ i โŸฉ r_{Y,i}=y_{i}-\langle\mathbf{w}_{Y}^{*},\mathbf{z}_{i}\rangle
  9. ฯ ^ X Y โ‹… ๐™ = N โˆ‘ i = 1 N r X , i r Y , i - โˆ‘ i = 1 N r X , i โˆ‘ i = 1 N r Y , i N โˆ‘ i = 1 N r X , i 2 - ( โˆ‘ i = 1 N r X , i ) 2 N โˆ‘ i = 1 N r Y , i 2 - ( โˆ‘ i = 1 N r Y , i ) 2 . \hat{\rho}_{XY\cdot\mathbf{Z}}=\frac{N\sum_{i=1}^{N}r_{X,i}r_{Y,i}-\sum_{i=1}^% {N}r_{X,i}\sum_{i=1}^{N}r_{Y,i}}{\sqrt{N\sum_{i=1}^{N}r_{X,i}^{2}-\left(\sum_{% i=1}^{N}r_{X,i}\right)^{2}}~{}\sqrt{N\sum_{i=1}^{N}r_{Y,i}^{2}-\left(\sum_{i=1% }^{N}r_{Y,i}\right)^{2}}}.
  10. Z 0 โˆˆ ๐™ Z_{0}\in\mathbf{Z}
  11. ฯ X Y โ‹… ๐™ = ฯ X Y โ‹… ๐™ โˆ– { Z 0 } - ฯ X Z 0 โ‹… ๐™ โˆ– { Z 0 } ฯ Z 0 Y โ‹… ๐™ โˆ– { Z 0 } 1 - ฯ X Z 0 โ‹… ๐™ โˆ– { Z 0 } 2 1 - ฯ Z 0 Y โ‹… ๐™ โˆ– { Z 0 } 2 . \rho_{XY\cdot\mathbf{Z}}=\frac{\rho_{XY\cdot\mathbf{Z}\setminus\{Z_{0}\}}-\rho% _{XZ_{0}\cdot\mathbf{Z}\setminus\{Z_{0}\}}\rho_{Z_{0}Y\cdot\mathbf{Z}\setminus% \{Z_{0}\}}}{\sqrt{1-\rho_{XZ_{0}\cdot\mathbf{Z}\setminus\{Z_{0}\}}^{2}}\sqrt{1% -\rho_{Z_{0}Y\cdot\mathbf{Z}\setminus\{Z_{0}\}}^{2}}}.
  12. ๐’ช ( n 3 ) \mathcal{O}(n^{3})
  13. ฯ X Y โ‹… Z = ฯ X Y - ฯ X Z ฯ Z Y 1 - ฯ X Z 2 1 - ฯ Z Y 2 . \rho_{XY\cdot Z}=\frac{\rho_{XY}-\rho_{XZ}\rho_{ZY}}{\sqrt{1-\rho_{XZ}^{2}}% \sqrt{1-\rho_{ZY}^{2}}}.
  14. ๐’ช ( n 3 ) \mathcal{O}(n^{3})
  15. ๐• โˆ– { X i , X j } \mathbf{V}\setminus\{X_{i},X_{j}\}
  16. ฯ X i X j โ‹… ๐• โˆ– { X i , X j } = - p i j p i i p j j . \rho_{X_{i}X_{j}\cdot\mathbf{V}\setminus\{X_{i},X_{j}\}}=-\frac{p_{ij}}{\sqrt{% p_{ii}p_{jj}}}.
  17. ฯ ^ X Y โ‹… ๐™ \hat{\rho}_{XY\cdot\mathbf{Z}}
  18. z ( ฯ ^ X Y โ‹… ๐™ ) = 1 2 ln ( 1 + ฯ ^ X Y โ‹… ๐™ 1 - ฯ ^ X Y โ‹… ๐™ ) . z(\hat{\rho}_{XY\cdot\mathbf{Z}})=\frac{1}{2}\ln\left(\frac{1+\hat{\rho}_{XY% \cdot\mathbf{Z}}}{1-\hat{\rho}_{XY\cdot\mathbf{Z}}}\right).
  19. H 0 : ฯ ^ X Y โ‹… ๐™ = 0 H_{0}:\hat{\rho}_{XY\cdot\mathbf{Z}}=0
  20. H A : ฯ ^ X Y โ‹… ๐™ โ‰  0 H_{A}:\hat{\rho}_{XY\cdot\mathbf{Z}}\neq 0
  21. N - | ๐™ | - 3 โ‹… | z ( ฯ ^ X Y โ‹… ๐™ ) | > ฮฆ - 1 ( 1 - ฮฑ / 2 ) , \sqrt{N-|\mathbf{Z}|-3}\cdot|z(\hat{\rho}_{XY\cdot\mathbf{Z}})|>\Phi^{-1}(1-% \alpha/2),
  22. ฯ• ( h ) = ฯ X 0 X h โ‹… { X 1 , โ€ฆ , X h - 1 } . \phi(h)=\rho_{X_{0}X_{h}\cdot\{X_{1},\dots,X_{h-1}\}}.

Partial_linear_space.html

  1. S = ( ๐’ซ , โ„ฌ , ๐ˆ ) S=({\mathcal{P}},{\mathcal{B}},\,\textbf{I})
  2. ๐’ซ {\mathcal{P}}
  3. โ„ฌ {\mathcal{B}}

Partial_oxidation.html

  1. C n H m + 2 n + m 4 O 2 โ†’ n CO + m 2 H 2 O \mathrm{C_{n}H_{m}+\frac{2n+m}{4}\ O_{2}\rightarrow n\ CO+\frac{m}{2}\ H_{2}O}
  2. C n H m + n 2 O 2 โ†’ n CO + m 2 H 2 \mathrm{C_{n}H_{m}+\frac{n}{2}\ O_{2}\rightarrow n\ CO+\frac{m}{2}\ H_{2}}
  3. C 12 H 24 + 6 O 2 โ†’ 12 CO + 12 H 2 \mathrm{C_{12}H_{24}+6\ O_{2}\rightarrow 12\ CO+12\ H_{2}}
  4. C 24 H 12 + 12 O 2 โ†’ 24 CO + 6 H 2 \mathrm{C_{24}H_{12}+12\ O_{2}\rightarrow 24\ CO+6\ H_{2}}

Particle-size_distribution.html

  1. f ( x ; P 80 , m ) = { 1 - e l n ( 0.2 ) ( x P 80 ) m x โ‰ฅ 0 , 0 x < 0 , f(x;P_{\rm{80}},m)=\begin{cases}1-e^{ln\left(0.2\right)\left(\frac{x}{P_{\rm{8% 0}}}\right)^{m}}&x\geq 0,\\ 0&x<0,\end{cases}
  2. x x
  3. P 80 P_{\rm{80}}
  4. m m
  5. f ( F ; P 80 , m ) = { P 80 l n ( 1 - F ) l n ( 0.2 ) m F > 0 , 0 F โ‰ค 0 , f(F;P_{\rm{80}},m)=\begin{cases}P_{\rm{80}}\sqrt[m]{\frac{ln(1-F)}{ln(0.2)}}&F% >0,\\ 0&F\leq 0,\end{cases}
  6. F F
  7. ln ( - ln ( 1 - F ) ) ) = m ln ( x ) + ln ( - ln ( 0.2 ) ( P 80 ) m ) \ln\left(-\ln\left(1-F)\right)\right)=m\ln(x)+\ln\left(\frac{-\ln(0.2)}{(P_{% \rm{80}})^{m}}\right)
  8. ln ( - ln ( 1 - F ) ) ) \ln\left(-\ln\left(1-F)\right)\right)
  9. ln ( x ) \ln(x)
  10. m m
  11. P 80 P_{\rm{80}}
  12. P 80 = ( - ln ( 0.2 ) e i n t e r c e p t ) 1 m P_{\rm{80}}=\left(\frac{-\ln(0.2)}{e^{intercept}}\right)^{\frac{1}{m}}

Particle_physics_in_cosmology.html

  1. 1 H \frac{1}{H}
  2. H H

Pauli_equation.html

  1. | ฯˆ โŸฉ = ( ฯˆ + ฯˆ - ) |\psi\rangle=\begin{pmatrix}\psi_{+}\\ \psi_{-}\end{pmatrix}
  2. H ^ = 1 2 m ( s y m b o l ฯƒ โ‹… ( ๐ฉ - q ๐€ ) ) 2 + q ฯ• \hat{H}=\frac{1}{2m}(symbol{\sigma}\cdot(\mathbf{p}-q\mathbf{A}))^{2}+q\phi
  3. ( s y m b o l ฯƒ โ‹… ๐š ) ( s y m b o l ฯƒ โ‹… ๐› ) = ๐š โ‹… ๐› + i s y m b o l ฯƒ โ‹… ( ๐š ร— ๐› ) (symbol{\sigma}\cdot\mathbf{a})(symbol{\sigma}\cdot\mathbf{b})=\mathbf{a}\cdot% \mathbf{b}+isymbol{\sigma}\cdot\left(\mathbf{a}\times\mathbf{b}\right)
  4. [ ๐ฉ 2 2 m + q ฯ• ] ( ฯˆ + ฯˆ - ) = i โ„ โˆ‚ โˆ‚ t ( ฯˆ + ฯˆ - ) . \left[\frac{\mathbf{p}^{2}}{2m}+q\phi\right]\begin{pmatrix}\psi_{+}\\ \psi_{-}\end{pmatrix}=i\hbar\frac{\partial}{\partial t}\begin{pmatrix}\psi_{+}% \\ \psi_{-}\end{pmatrix}.
  5. 1 ^ = ( 1 0 0 1 ) \hat{1}=\begin{pmatrix}1&0\\ 0&1\\ \end{pmatrix}

PDIFF.html

  1. Diff โ†’ PDiff โ†’ PL . \,\text{Diff}\to\,\text{PDiff}\to\,\text{PL}.

Peano-Russell_notation.html

  1. โˆผ p \sim p
  2. p โˆจ q p\lor q
  3. p โ‹… q p\cdot q
  4. p โŠƒ q p\supset q
  5. p โ‰ก q p\equiv q
  6. p โŠƒ q โ‹… q โŠƒ p p\supset q\cdot q\supset p
  7. โŠข p \vdash p
  8. p โ‹… p โŠƒ q . โŠƒ q p\cdot p\supset q.\supset q

Peano_existence_theorem.html

  1. f : D โ†’ โ„ f\colon D\to\mathbb{R}
  2. y โ€ฒ ( x ) = f ( x , y ( x ) ) y^{\prime}(x)=f\left(x,y(x)\right)
  3. y ( x 0 ) = y 0 y\left(x_{0}\right)=y_{0}
  4. ( x 0 , y 0 ) โˆˆ D (x_{0},y_{0})\in D
  5. z : I โ†’ โ„ z\colon I\to\mathbb{R}
  6. I I
  7. x 0 x_{0}
  8. โ„ \mathbb{R}
  9. z โ€ฒ ( x ) = f ( x , z ( x ) ) z^{\prime}(x)=f\left(x,z(x)\right)
  10. x โˆˆ I x\in I
  11. y โ€ฒ = | y | 1 2 y^{\prime}=\left|y\right|^{\frac{1}{2}}
  12. [ 0 , 1 ] . \left[0,1\right].
  13. y ( 0 ) = 0 y(0)=0
  14. y ( x ) = 0 y(x)=0
  15. y ( x ) = x 2 / 4 y(x)=x^{2}/4
  16. y = 0 y=0
  17. y = ( x - C ) 2 / 4 y=(x-C)^{2}/4

PEG_400.html

  1. P = - 4.8 P=-4.8

Peierls_stress.html

  1. ฯ„ PN = G e - 2 ฯ€ W / b \tau_{\mathrm{PN}}=Ge^{-2{\pi}W/b}
  2. W = a 1 - ฮฝ = W=\frac{a}{1-\nu}=
  3. G G
  4. ฮฝ \nu
  5. b b
  6. a a

Penney's_game.html

  1. k - 3 k-3
  2. ( 2 k - 1 + 1 ) : ( 2 k - 2 + 1 ) (2^{k-1}+1):(2^{k-2}+1)

Penrose_transform.html

  1. Z โ† ๐œ‚ Y โ†’ ๐œ X . Z\xleftarrow{\eta}Y\xrightarrow{\tau}X.
  2. G / H 1 โ† ๐œ‚ G / ( H 1 โˆฉ H 2 ) โ†’ ๐œ G / H 2 G/H_{1}\xleftarrow{\eta}G/(H_{1}\cap H_{2})\xrightarrow{\tau}G/H_{2}

Percolation_threshold.html

  1. z ยฏ \overline{z}
  2. z ยฏ \overline{z}
  3. b s [ 1 - ( t / ( 3 - t ) ) ( b - t ) ] = t bs[1-(\sqrt{t}/(3-t))(\sqrt{b}-\sqrt{t})]=t
  4. z ยฏ \overline{z}
  5. z ยฏ \overline{z}
  6. z ยฏ \overline{z}
  7. 1 / 3 {1}/{3}
  8. 2 / 7 {2}/{7}
  9. 2 / 7 {2}/{7}
  10. 2 / 3 {2}/{3}
  11. 1 / 4 {1}/{4}
  12. 1 / 7 {1}/{7}
  13. 1 / 3 {1}/{3}
  14. z ยฏ \overline{z}
  15. z ยฏ \overline{z}
  16. z ยฏ \overline{z}
  17. โˆž \infty
  18. โˆž \infty
  19. โˆž \infty
  20. ( 3 - 5 ) / 2 (3-\sqrt{5})/2
  21. โ„“ \ell
  22. โ„“ \ell
  23. ฮท c = ฯ€ r 2 N / L 2 \eta_{c}=\pi r^{2}N/L^{2}
  24. ฮท c = ฯ€ a b N / L 2 \eta_{c}=\pi abN/L^{2}
  25. ฯต = a / b \epsilon=a/b
  26. a > b a>b
  27. ฮท c = โ„“ m N / L 2 \eta_{c}=\ell mN/L^{2}
  28. โ„“ \ell
  29. m m
  30. ฯต = โ„“ / m \epsilon=\ell/m
  31. โ„“ > m \ell>m
  32. ฮท c = ฯ€ x N / ( 4 L 2 ( x - 2 ) ) \eta_{c}=\pi xN/(4L^{2}(x-2))
  33. Prob(radius โ‰ฅ R ) = R - x \hbox{Prob(radius}\geq R)=R^{-x}
  34. R โ‰ฅ 1 R\geq 1
  35. ฯ• c = 1 - e - ฮท c \phi_{c}=1-e^{-\eta_{c}}
  36. n c = โ„“ 2 N / L 2 n_{c}=\ell^{2}N/L^{2}
  37. โ„“ = 2 a \ell=2a
  38. n c = ( 4 ฯต / ฯ€ ) ฮท c n_{c}=(4\epsilon/\pi)\eta_{c}
  39. ฯ• c = e - ฮท c \phi_{c}=e^{-\eta_{c}}
  40. z ยฏ \overline{z}
  41. z ยฏ \overline{z}
  42. z ยฏ \overline{z}
  43. p C ( NN ) p_{C}(\,\text{NN})
  44. โ„“ = 2 a \ell=2a
  45. โ„“ = 2 a \ell=2a
  46. ฯ€ r \sqrt{\pi}r
  47. 2 ฯ€ / 3 1 / 4 r \sqrt{2\pi}/3^{1/4}r
  48. ฮท c = ( 4 / 3 ) ฯ€ r 3 N / L 3 \eta_{c}=(4/3)\pi r^{3}N/L^{3}
  49. ฯ• c = 1 - e - ฮท c \phi_{c}=1-e^{-\eta_{c}}
  50. ฯ• c = e - ฮท c \phi_{c}=e^{-\eta_{c}}
  51. z ยฏ \overline{z}
  52. ฮท c = ( ฯ€ d / 2 / ฮ“ [ d / 2 + 1 ] ) r d N / L d \eta_{c}=(\pi^{d/2}/\Gamma[d/2+1])r^{d}N/L^{d}
  53. ฮท c = ( 1 / 2 ) ฯ€ 2 r 4 N / L 4 \eta_{c}=(1/2)\pi^{2}r^{4}N/L^{4}
  54. ฮท c = ( 8 / 15 ) ฯ€ 2 r 5 N / L 5 \eta_{c}=(8/15)\pi^{2}r^{5}N/L^{5}
  55. ฮท c = ( 1 / 6 ) ฯ€ 3 r 6 N / L 6 \eta_{c}=(1/6)\pi^{3}r^{6}N/L^{6}
  56. ฯ• c = 1 - e - ฮท c \phi_{c}=1-e^{-\eta_{c}}
  57. ฯ• c = e - ฮท c \phi_{c}=e^{-\eta_{c}}
  58. ฮท c \eta_{c}
  59. z ยฏ \overline{z}
  60. z ยฏ \overline{z}
  61. z 2 ยฏ : \overline{z^{2}}:
  62. z ยฏ / ( z 2 ยฏ - z ยฏ ) \overline{z}/(\overline{z^{2}}-\overline{z})
  63. 1 - p 1 - p 2 - p 3 + p 1 p 2 p 3 = 0 1-p_{1}-p_{2}-p_{3}+p_{1}p_{2}p_{3}=0
  64. 1 - p 1 p 2 - p 1 p 3 - p 2 p 3 + p 1 p 2 p 3 = 0 1-p_{1}p_{2}-p_{1}p_{3}-p_{2}p_{3}+p_{1}p_{2}p_{3}=0
  65. 1 - 3 ( s 1 s 2 ) 2 + ( s 1 s 2 ) 3 = 0 , 1-3(s_{1}s_{2})^{2}+(s_{1}s_{2})^{3}=0,
  66. s 1 s 2 = 1 - 2 sin ( ฯ€ / 18 ) s_{1}s_{2}=1-2\sin(\pi/18)
  67. 1 - ( p 1 p 2 r 3 + p 2 p 3 r 1 + p 1 p 3 r 2 ) - ( p 1 p 2 r 1 r 2 + p 1 p 3 r 1 r 3 + p 2 p 3 r 2 r 3 ) + p 1 p 2 p 3 ( r 1 r 2 + r 1 r 3 + r 2 r 3 ) + 1-(p_{1}p_{2}r_{3}+p_{2}p_{3}r_{1}+p_{1}p_{3}r_{2})-(p_{1}p_{2}r_{1}r_{2}+p_{1% }p_{3}r_{1}r_{3}+p_{2}p_{3}r_{2}r_{3})+p_{1}p_{2}p_{3}(r_{1}r_{2}+r_{1}r_{3}+r% _{2}r_{3})+
  68. r 1 r 2 r 3 ( p 1 p 2 + p 1 p 3 + p 2 p 3 ) - 2 p 1 p 2 p 3 r 1 r 2 r 3 = 0 r_{1}r_{2}r_{3}(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})-2p_{1}p_{2}p_{3}r_{1}r_{2}r_% {3}=0
  69. 1 - r ( p 1 p 2 + p 1 p 3 + p 2 p 3 - p 1 p 2 p 3 ) = 0 1-r(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3}-p_{1}p_{2}p_{3})=0
  70. r 2 , p 1 r_{2},\ p_{1}
  71. r 1 , p 2 r_{1},\ p_{2}
  72. r 3 \ r_{3}
  73. 1 - p 1 r 2 - p 2 r 1 - p 1 p 2 r 3 - p 1 r 1 r 3 - p 2 r 2 r 3 + p 1 p 2 r 1 r 3 + p 1 p 2 r 2 r 3 + p 1 r 1 r 2 r 3 + p 2 r 1 r 2 r 3 - p 1 p 2 r 1 r 2 r 3 = 0 1-p_{1}r_{2}-p_{2}r_{1}-p_{1}p_{2}r_{3}-p_{1}r_{1}r_{3}-p_{2}r_{2}r_{3}+p_{1}p% _{2}r_{1}r_{3}+p_{1}p_{2}r_{2}r_{3}+p_{1}r_{1}r_{2}r_{3}+p_{2}r_{1}r_{2}r_{3}-% p_{1}p_{2}r_{1}r_{2}r_{3}=0
  74. y , x , z y,x,z
  75. 1 - 3 z + z 3 - ( 1 - z 2 ) [ 3 x 2 y ( 1 + y - y 2 ) ( 1 + z ) + x 3 y 2 ( 3 - 2 y ) ( 1 + 2 z ) ] = 0 1-3z+z^{3}-(1-z^{2})[3x^{2}y(1+y-y^{2})(1+z)+x^{3}y^{2}(3-2y)(1+2z)]=0
  76. 1 - ( p 1 p 2 + p 1 p 3 + p 1 p 4 + p 2 p 3 + p 2 p 4 + p 3 p 4 ) + p 1 p 2 p 3 + p 1 p 2 p 4 + p 1 p 3 p 4 + p 2 p 3 p 4 = 0 1-(p_{1}p_{2}+p_{1}p_{3}+p_{1}p_{4}+p_{2}p_{3}+p_{2}p_{4}+p_{3}p_{4})+p_{1}p_{% 2}p_{3}+p_{1}p_{2}p_{4}+p_{1}p_{3}p_{4}+p_{2}p_{3}p_{4}=0
  77. 1 - ( p 1 p 2 + p 1 p 3 + p 1 p 4 + p 2 p 3 + p 2 p 4 + p 3 p 4 ) + p 1 p 2 p 3 + p 1 p 2 p 4 + p 1 p 3 p 4 + p 2 p 3 p 4 + 1-(p_{1}p_{2}+p_{1}p_{3}+p_{1}p_{4}+p_{2}p_{3}+p_{2}p_{4}+p_{3}p_{4})+p_{1}p_{% 2}p_{3}+p_{1}p_{2}p_{4}+p_{1}p_{3}p_{4}+p_{2}p_{3}p_{4}+
  78. u ( 1 - p 1 p 2 - p 3 p 4 + p 1 p 2 p 3 p 4 ) = 0 u(1-p_{1}p_{2}-p_{3}p_{4}+p_{1}p_{2}p_{3}p_{4})=0
  79. p 1 , p 2 , p 3 , p 4 p_{1},p_{2},p_{3},p_{4}
  80. u u
  81. p 4 , p 1 p_{4},p_{1}
  82. p 2 , p 3 p_{2},p_{3}

Percusโ€“Yevick_approximation.html

  1. c ( r ) = g total ( r ) - g indirect ( r ) c(r)=g_{\rm total}(r)-g_{\rm indirect}(r)\,
  2. g total ( r ) g_{\rm total}(r)
  3. g ( r ) = exp [ - ฮฒ w ( r ) ] g(r)=\exp[-\beta w(r)]
  4. g indirect ( r ) g_{\rm indirect}(r)
  5. u ( r ) u(r)
  6. g indirect ( r ) = exp - ฮฒ [ w ( r ) - u ( r ) ] g_{\rm indirect}(r)=\exp^{-\beta[w(r)-u(r)]}
  7. c ( r ) = e - ฮฒ w ( r ) - e - ฮฒ [ w ( r ) - u ( r ) ] . c(r)=e^{-\beta w(r)}-e^{-\beta[w(r)-u(r)]}.\,
  8. y ( r ) = e ฮฒ u ( r ) g ( r ) y(r)=e^{\beta u(r)}g(r)
  9. c ( r ) = g ( r ) - y ( r ) = e - ฮฒ u y ( r ) - y ( r ) = f ( r ) y ( r ) . c(r)=g(r)-y(r)=e^{-\beta u}y(r)-y(r)=f(r)y(r).\,
  10. y ( r 12 ) = 1 + ฯ โˆซ f ( r 13 ) y ( r 13 ) h ( r 23 ) d ๐ซ ๐Ÿ‘ . y(r_{12})=1+\rho\int f(r_{13})y(r_{13})h(r_{23})d\mathbf{r_{3}}.\,

Perfect_spline.html

  1. m m
  2. m m
  3. + 1 +1
  4. - 1 -1

Perfect_totient_number.html

  1. n = โˆ‘ i = 1 c + 1 ฯ† i ( n ) , n=\sum_{i=1}^{c+1}\varphi^{i}(n),
  2. ฯ† i ( n ) = { ฯ† ( n ) if i = 1 ฯ† ( ฯ† i - 1 ( n ) ) otherwise \varphi^{i}(n)=\left\{\begin{matrix}\varphi(n)&\mbox{ if }~{}i=1\\ \varphi(\varphi^{i-1}(n))&\mbox{ otherwise}\end{matrix}\right.
  3. ฯ† c ( n ) = 2 , \displaystyle\varphi^{c}(n)=2,
  4. ฯ† ( 3 k ) = ฯ† ( 2 ร— 3 k ) = 2 ร— 3 k - 1 . \displaystyle\varphi(3^{k})=\varphi(2\times 3^{k})=2\times 3^{k-1}.

Performance_prediction.html

  1. T p r o g r a m = โˆ‘ i = 1 n ( T B B i * F B B i ) T_{program}=\sum_{i=1}^{n}{(T_{BB_{i}}*F_{BB_{i}})}

Periodic_boundary_conditions.html

  1. ฯ• : โ„ n โ†’ โ„ \phi:\mathbb{R}^{n}\to\mathbb{R}
  2. โˆ‚ m โˆ‚ x 1 m ฯ• ( a 1 , x 2 , โ€ฆ , x n ) = โˆ‚ m โˆ‚ x 1 m ฯ• ( b 1 , x 2 , โ€ฆ , x n ) , \frac{\partial^{m}}{\partial x_{1}^{m}}\phi(a_{1},x_{2},...,x_{n})=\frac{% \partial^{m}}{\partial x_{1}^{m}}\phi(b_{1},x_{2},...,x_{n}),
  3. โˆ‚ m โˆ‚ x 2 m ฯ• ( x 1 , a 2 , โ€ฆ , x n ) = โˆ‚ m โˆ‚ x 2 m ฯ• ( x 1 , b 2 , โ€ฆ , x n ) , \frac{\partial^{m}}{\partial x_{2}^{m}}\phi(x_{1},a_{2},...,x_{n})=\frac{% \partial^{m}}{\partial x_{2}^{m}}\phi(x_{1},b_{2},...,x_{n}),
  4. โ€ฆ , ...,
  5. โˆ‚ m โˆ‚ x n m ฯ• ( x 1 , x 2 , โ€ฆ , a n ) = โˆ‚ m โˆ‚ x n m ฯ• ( x 1 , x 2 , โ€ฆ , b n ) \frac{\partial^{m}}{\partial x_{n}^{m}}\phi(x_{1},x_{2},...,a_{n})=\frac{% \partial^{m}}{\partial x_{n}^{m}}\phi(x_{1},x_{2},...,b_{n})
  6. a i a_{i}
  7. b i b_{i}

Periodic_continued_fraction.html

  1. x = a 0 + 1 a 1 + 1 a 2 + โ‹ฑ โ‹ฑ a k + 1 a k + 1 + โ‹ฑ โ‹ฑ a k + m - 1 + 1 a k + m + 1 a k + 1 + 1 a k + 2 + 1 โ‹ฑ x=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{\ddots}{\quad\ddots\quad a_{k}+% \cfrac{1}{a_{k+1}+\cfrac{\ddots}{\quad\ddots\quad a_{k+m-1}+\cfrac{1}{a_{k+m}+% \cfrac{1}{a_{k+1}+\cfrac{1}{a_{k+2}+\cfrac{1}{\ddots}}}}}}}}}\,
  2. 2 \sqrt{2}
  3. x = [ a 0 ; a 1 , a 2 , โ€ฆ , a k , a k + 1 , a k + 2 , โ€ฆ , a k + m , a k + 1 , a k + 2 , โ€ฆ , a k + m , โ€ฆ ] = [ a 0 ; a 1 , a 2 , โ€ฆ , a k , a k + 1 , a k + 2 , โ€ฆ , a k + m ยฏ ] \begin{aligned}\displaystyle x&\displaystyle=[a_{0};a_{1},a_{2},\dots,a_{k},a_% {k+1},a_{k+2},\dots,a_{k+m},a_{k+1},a_{k+2},\dots,a_{k+m},\dots]\\ &\displaystyle=[a_{0};a_{1},a_{2},\dots,a_{k},\overline{a_{k+1},a_{k+2},\dots,% a_{k+m}}]\end{aligned}
  4. x = [ a 0 ; a 1 , a 2 , โ€ฆ , a k , a ห™ k + 1 , a k + 2 , โ€ฆ , a ห™ k + m ] \begin{aligned}\displaystyle x&\displaystyle=[a_{0};a_{1},a_{2},\dots,a_{k},% \dot{a}_{k+1},a_{k+2},\dots,\dot{a}_{k+m}]\end{aligned}
  5. x = [ a 0 ; a 1 , a 2 , โ€ฆ , a m ยฏ ] , x=[\overline{a_{0};a_{1},a_{2},\dots,a_{m}}],
  6. a x 2 + b x + c = 0 ax^{2}+bx+c=0\,
  7. ฮถ = P + D Q \zeta=\frac{P+\sqrt{D}}{Q}
  8. ฮถ = P + D Q \zeta=\frac{P+\sqrt{D}}{Q}
  9. ฮถ > 1 \zeta>1
  10. ฮท = P - D Q \eta=\frac{P-\sqrt{D}}{Q}
  11. - 1 < ฮท < 0 -1<\eta<0
  12. ฯ• = ( 1 + 5 ) / 2 = 1.618033... \phi=(1+\sqrt{5})/2=1.618033...
  13. ( 1 - 5 ) / 2 = - 0.618033... (1-\sqrt{5})/2=-0.618033...
  14. 2 = ( 0 + 8 ) / 2 \sqrt{2}=(0+\sqrt{8})/2
  15. - 2 = ( 0 - 8 ) / 2 -\sqrt{2}=(0-\sqrt{8})/2
  16. ฮถ = [ a 0 ; a 1 , a 2 , โ€ฆ , a m - 1 ยฏ ] - 1 ฮท = [ a m - 1 ; a m - 2 , a m - 3 , โ€ฆ , a 0 ยฏ ] \begin{aligned}\displaystyle\zeta&\displaystyle=[\overline{a_{0};a_{1},a_{2},% \dots,a_{m-1}}]\\ \displaystyle\frac{-1}{\eta}&\displaystyle=[\overline{a_{m-1};a_{m-2},a_{m-3},% \dots,a_{0}}]\end{aligned}
  17. r = [ a 0 ; a 1 , a 2 , โ€ฆ , a 2 , a 1 , 2 a 0 ยฏ ] . \sqrt{r}=[a_{0};\overline{a_{1},a_{2},\dots,a_{2},a_{1},2a_{0}}].\,
  18. P n + D Q n \frac{P_{n}+\sqrt{D}}{Q_{n}}
  19. L ( D ) = ๐’ช ( D ln D ) L(D)=\mathcal{O}(\sqrt{D}\ln{D})

Peripheral_cycle.html

  1. C C
  2. G G
  3. C C
  4. e 1 e_{1}
  5. e 2 e_{2}
  6. G โˆ– C G\setminus C
  7. G G
  8. e 1 e_{1}
  9. e 2 e_{2}
  10. C C
  11. C C
  12. G โˆ– C G\setminus C
  13. C C
  14. C C
  15. G G
  16. C C
  17. B B
  18. G G
  19. C C
  20. B B
  21. G โˆ– B G\setminus B
  22. C C
  23. C C
  24. G โˆ– C G\setminus C
  25. C C
  26. C C
  27. G G
  28. G G
  29. K 2 , 4 K_{2,4}
  30. K 2 , 3 K_{2,3}
  31. K 2 , 3 K_{2,3}
  32. G G
  33. C C

Peroxydisulfate.html

  1. โ† โ†’ \overrightarrow{\leftarrow}

Perpendicular_axis_theorem.html

  1. x x\,
  2. y y\,
  3. z z\,
  4. O O\,
  5. x y xy\,
  6. z z\,
  7. I z = I x + I y I_{z}=I_{x}+I_{y}\,
  8. I x I_{x}\,
  9. I y I_{y}\,
  10. I z = 2 I x = 2 I y I_{z}=2I_{x}=2I_{y}\,
  11. z z\,
  12. I z = โˆซ ( x 2 + y 2 ) d m = โˆซ x 2 d m + โˆซ y 2 d m = I y + I x I_{z}=\int\left(x^{2}+y^{2}\right)\,dm=\int x^{2}\,dm+\int y^{2}\,dm=I_{y}+I_{x}
  13. z = 0 z=0\,
  14. x x\,
  15. y y\,
  16. โˆซ x 2 d m = I y โ‰  I x \int x^{2}\,dm=I_{y}\neq I_{x}
  17. โˆซ r 2 d m \int r^{2}\,dm

Persymmetric_matrix.html

  1. a i j = a n - j + 1 , n - i + 1 a_{ij}=a_{n-j+1,n-i+1}
  2. A = [ a 11 a 12 a 13 a 14 a 15 a 21 a 22 a 23 a 24 a 14 a 31 a 32 a 33 a 23 a 13 a 41 a 42 a 32 a 22 a 12 a 51 a 41 a 31 a 21 a 11 ] . A=\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\ a_{21}&a_{22}&a_{23}&a_{24}&a_{14}\\ a_{31}&a_{32}&a_{33}&a_{23}&a_{13}\\ a_{41}&a_{42}&a_{32}&a_{22}&a_{12}\\ a_{51}&a_{41}&a_{31}&a_{21}&a_{11}\end{bmatrix}.
  3. A = [ r 1 r 2 r 3 โ‹ฏ r n r 2 r 3 r 4 โ‹ฏ r n + 1 r 3 r 4 r 5 โ‹ฏ r n + 2 โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฎ r n r n + 1 r n + 2 โ‹ฏ r 2 n - 1 ] . A=\begin{bmatrix}r_{1}&r_{2}&r_{3}&\cdots&r_{n}\\ r_{2}&r_{3}&r_{4}&\cdots&r_{n+1}\\ r_{3}&r_{4}&r_{5}&\cdots&r_{n+2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ r_{n}&r_{n+1}&r_{n+2}&\cdots&r_{2n-1}\end{bmatrix}.

Pert.html

  1. P e r t Pe^{rt}

Pharmacokinetics.html

  1. V d F = V d T 1 + V d T 2 + V d T 3 + โ€ฆ + V d T n Vd_{F}=Vd_{T1}+Vd_{T2}+Vd_{T3}+...+Vd_{Tn}\,
  2. B A = [ A B C ] P . D I V [ A B C ] I V . D P B_{A}=\frac{[ABC]_{P}.D_{IV}}{[ABC]_{IV}.D_{P}}
  3. B R = [ A B C ] A . d o s e B [ A B C ] B . d o s e A \mathit{B}_{R}=\frac{[ABC]_{A}.dose_{B}}{[ABC]_{B}.dose_{A}}
  4. D e = B . D a De=B.Da\,
  5. D e = Q . D a . B De=Q.Da.B\,
  6. V a = D a . B . Q ฯ„ Va=\frac{Da.B.Q}{\tau}
  7. V a Va
  8. ฯ„ \tau
  9. p K a pKa\,
  10. p H = p K a + l o g B A pH=pKa+log\frac{B}{A}

Phase_retrieval.html

  1. F ( k ) F(k)
  2. | F ( k ) | |F(k)|
  3. ฯˆ ( k ) \psi(k)
  4. F ( k ) = | F ( k ) | e i ฯˆ ( k ) = โˆซ - โˆž โˆž f ( x ) e - 2 ฯ€ i k โ‹… x d x F(k)=|F(k)|e^{i\psi(k)}=\int_{-\infty}^{\infty}f(x)\ e^{-2\pi ik\cdot x}\,dx
  5. M = 2 M=2
  6. f ( x ) f(x)
  7. | F ( x ) | |F(x)|
  8. k k
  9. G k ( u ) G_{k}(u)
  10. ฯ• k \phi_{k}
  11. g k ( x ) g_{k}(x)
  12. F ( u ) F(u)
  13. ฯˆ \psi
  14. f ( x ) f(x)
  15. g k ( x ) g_{k}(x)
  16. G k ( u ) = | G k ( u ) | e i ฯ• k ( u ) = โ„ฑ ( g k ( x ) ) G_{k}(u)=|G_{k}(u)|e^{i\phi_{k}(u)}=\mathcal{F}(g_{k}(x))
  17. | F ( u ) | |F(u)|
  18. | G k ( u ) | |G_{k}(u)|
  19. G k โ€ฒ ( u ) = | F ( u ) | e i ฯ• k ( u ) G^{\prime}_{k}(u)=|F(u)|e^{i\phi_{k}(u)}
  20. G k โ€ฒ ( u ) G^{\prime}_{k}(u)
  21. g k โ€ฒ ( x ) = | g k โ€ฒ ( x ) | e i ฮธ k โ€ฒ ( x ) = โ„ฑ - 1 ( G k โ€ฒ ( u ) ) g^{\prime}_{k}(x)=|g^{\prime}_{k}(x)|e^{i\theta^{\prime}_{k}(x)}=\mathcal{F}^{% -1}(G^{\prime}_{k}(u))
  22. g k โ€ฒ ( x ) g^{\prime}_{k}(x)
  23. g k + 1 ( x ) g_{k+1}(x)
  24. g k + 1 ( x ) g_{k+1}(x)
  25. g k + 1 ( x ) โ‰ก { g k โ€ฒ ( x ) x โˆ‰ ฮณ 0 x โˆˆ ฮณ g_{k+1}(x)\equiv\begin{cases}g^{\prime}_{k}(x)&x\notin\gamma\\ 0&x\in\gamma\end{cases}
  26. ฮณ \gamma
  27. g k โ€ฒ ( x ) g^{\prime}_{k}(x)
  28. g k + 1 ( x ) g_{k+1}(x)
  29. g k ( x ) g_{k}(x)
  30. f ( x ) f(x)
  31. g k โ€ฒ ( x ) g^{\prime}_{k}(x)
  32. f ( x ) f(x)
  33. g k โ€ฒ ( x ) g^{\prime}_{k}(x)
  34. g k + 1 ( x ) g_{k+1}(x)
  35. g k ( x ) g_{k}(x)
  36. g k + 1 ( x ) โ‰ก { g k โ€ฒ ( u ) x โˆ‰ ฮณ g k ( x ) - ฮฒ g k โ€ฒ ( x ) x โˆˆ ฮณ g_{k+1}(x)\equiv\begin{cases}g^{\prime}_{k}(u)&x\notin\gamma\\ g_{k}(x)-\beta{g^{\prime}_{k}(x)}&x\in\gamma\end{cases}
  37. ฮฒ \beta
  38. ฮฒ โ‰ˆ 0.9 \beta\approx 0.9
  39. f ( x ) f(x)
  40. f * ( - x ) f^{*}(-x)

Philo_line.html

  1. 1 : 2 3 \scriptstyle 1:\sqrt[3]{2}

Phosphopentose_epimerase.html

  1. โ‡Œ \rightleftharpoons

Phosphoribosylaminoimidazole_carboxylase.html

  1. โ‡Œ \rightleftharpoons

Phred_quality_score.html

  1. Q Q
  2. P P
  3. Q = - 10 log 10 P Q=-10\ \log_{10}P
  4. P = 10 - Q 10 P=10^{\frac{-Q}{10}}
  5. P = 1 P=1

Physical_activity_level.html

  1. P A L = TEE/24h BMR PAL=\frac{\,\text{TEE/24h}}{\,\text{BMR}}

Physical_theories_modified_by_general_relativity.html

  1. F = m r ยจ F=m\ddot{r}
  2. d 2 r d t 2 = 0 \frac{\mathrm{d}^{2}r}{\mathrm{d}t^{2}}=0
  3. d 2 x a d ฯ„ 2 = 0 \frac{\mathrm{d}^{2}x^{a}}{\mathrm{d}\tau^{2}}=0
  4. d 2 x a d ฯ„ 2 + ฮ“ b c a d x b d ฯ„ d x c d ฯ„ = 0 \frac{\mathrm{d}^{2}x^{a}}{\mathrm{d}\tau^{2}}+\Gamma^{a}_{bc}\,\frac{\mathrm{% d}x^{b}}{\mathrm{d}\tau}\,\frac{\mathrm{d}x^{c}}{\mathrm{d}\tau}=0
  5. ฮ“ b c a \Gamma^{a}_{bc}
  6. ฮ“ b c a = 0 \Gamma^{a}_{bc}=0
  7. ๐ซ ยจ = G M ๐ซ ^ / r 2 \mathbf{\ddot{r}}=GM\mathbf{\hat{r}}/r^{2}
  8. ๐ซ ^ \mathbf{\hat{r}}
  9. T a b = ; b T a b + , b ฮ“ b c b T a c - ฮ“ c a b T c b = 0 {T_{a}}^{b}{}_{;b}={T_{a}}^{b}{}_{,b}+{\Gamma^{b}}_{cb}\,{T_{a}}^{c}-{\Gamma^{% c}}_{ab}\,{T_{c}}^{b}=0
  10. T a b {T_{a}}^{b}
  11. F a b = ; b 4 ฯ€ c J a F^{\,ab}{}_{;b}={4\pi\over c}\,J^{\,a}
  12. P a = ; ฯ„ ( q / m ) F a b P b P^{\,a}{}_{\,;\tau}=(q/m)\,F^{\,ab}P_{b}

Picone_identity.html

  1. ( p 1 ( x ) u โ€ฒ ) โ€ฒ + q 1 ( x ) u = 0 (p_{1}(x)u^{\prime})^{\prime}+q_{1}(x)u=0\,
  2. ( p 2 ( x ) v โ€ฒ ) โ€ฒ + q 2 ( x ) v = 0. (p_{2}(x)v^{\prime})^{\prime}+q_{2}(x)v=0.\,
  3. ( u v ( p 1 u โ€ฒ v - p 2 u v โ€ฒ ) ) โ€ฒ = ( q 2 - q 1 ) u 2 + ( p 1 - p 2 ) u โ€ฒ 2 + p 2 ( u โ€ฒ - v โ€ฒ u v ) 2 . \left(\frac{u}{v}(p_{1}u^{\prime}v-p_{2}uv^{\prime})\right)^{\prime}=\left(q_{% 2}-q_{1}\right)u^{2}+\left(p_{1}-p_{2}\right)u^{\prime 2}+p_{2}\left(u^{\prime% }-v^{\prime}\frac{u}{v}\right)^{2}.

Picosecond_ultrasonics.html

  1. ฮด r r = 4 i k n ~ 1 - n ~ 2 d n ~ d ฮท โˆซ 0 โˆž ฮท ( z , t ) e 2 i n ~ k z d z + 2 i k u ( t ) \frac{\delta r}{r}=\frac{4ik\tilde{n}}{1-{\tilde{n}}^{2}}\frac{d\tilde{n}}{d% \eta}\int_{0}^{\infty}\eta(z,t)e^{2i\tilde{n}kz}dz+2iku(t)
  2. n ~ = n + i ฮบ \tilde{n}=n+i\kappa
  3. d n ~ / d ฮท d\tilde{n}/d\eta
  4. u ( t ) = - โˆซ 0 โˆž ฮท ( z , t ) d z u(t)=-\int_{0}^{\infty}\eta(z,t)dz
  5. ฮด R / R = 2 R e ( ฮด r / r ) \delta R/R=2\rm{Re}(\it{\delta r/r})
  6. ฮด ฯ• = Im ( ฮด r / r ) \delta\it{\phi}=\rm{Im}(\it{\delta r/r})

Piecewise_linear_continuation.html

  1. f ( x , y ) = 0 f(x,y)=0\,
  2. f ( โ‹… ) f(\cdot)\,
  3. 0 โ‰ค x โ‰ค 1 , 0 โ‰ค y โ‰ค 1 0\leq x\leq 1,0\leq y\leq 1\,
  4. i h x โ‰ค x โ‰ค ( i + 1 ) h x ih_{x}\leq x\leq(i+1)h_{x}\,
  5. j h y โ‰ค y โ‰ค ( j + 1 ) h y jh_{y}\leq y\leq(j+1)h_{y}\,
  6. f ( x i , y j ) f(x_{i},y_{j})\,
  7. ( i , j ) (i,j)\,
  8. f ( x i , y j ) f(x_{i},y_{j})\,
  9. l f ( x , y ) lf(x,y)\,
  10. f ( โ‹… ) f(\cdot)\,
  11. ( x 0 , y 0 ) , ( x 1 , y 1 ) , ( x 2 , y 2 ) (x_{0},y_{0}),~{}(x_{1},y_{1}),~{}(x_{2},y_{2})\,
  12. ( x , y ) = ( x 0 , y 0 ) + ( x 1 - x 0 , y 1 - y 0 ) s + ( x 2 - x 0 , y 2 - y 0 ) t (x,y)=(x_{0},y_{0})+(x_{1}-x_{0},y_{1}-y_{0})s+(x_{2}-x_{0},y_{2}-y_{0})t\,
  13. 0 โ‰ค s 0\leq s\,
  14. 0 โ‰ค t 0\leq t\,
  15. s + t โ‰ค 1 s+t\leq 1\,
  16. l f ( x , y ) = f ( x 0 , y 0 ) + ( f ( x 1 , y 1 ) - f ( x 0 , y 0 ) ) s + ( f ( x 2 , y 2 ) - f ( x 0 , y 0 ) ) t lf(x,y)=f(x_{0},y_{0})+(f(x_{1},y_{1})-f(x_{0},y_{0}))s+(f(x_{2},y_{2})-f(x_{0% },y_{0}))t\,
  17. ( s , t ) (s,t)\,
  18. f ( โ‹… ) f(\cdot)\,
  19. ( x 0 , y 0 ) (x_{0},y_{0})\,
  20. ( x 1 , y 1 ) (x_{1},y_{1})\,
  21. ( x , y ) = ( x 0 , y 0 ) + t ( x 1 - x 0 , y 1 - y 0 ) , (x,y)=(x_{0},y_{0})+t(x_{1}-x_{0},y_{1}-y_{0}),\,
  22. t t\,
  23. ( 0 , 1 ) (0,1)\,
  24. f โˆผ f 0 + t ( f 1 - f 0 ) f\sim f_{0}+t(f_{1}-f_{0})\,
  25. f = 0 f=0\,
  26. t = - f 0 / ( f 1 - f 0 ) t=-f_{0}/(f_{1}-f_{0})\,
  27. ( x , y ) = ( x 0 , y 0 ) - f 0 * ( x 1 - x 0 , y 1 - y 0 ) / ( f 1 - f 0 ) (x,y)=(x_{0},y_{0})-f_{0}*(x_{1}-x_{0},y_{1}-y_{0})/(f_{1}-f_{0})\,
  28. โ„ n \mathbb{R}^{n}
  29. โ„ n - 1 \mathbb{R}^{n-1}
  30. v i v_{i}
  31. 0 < ฮฑ i 0<\alpha_{i}
  32. ๐ฑ = โˆ‘ i ฮฑ i ๐ฏ i \mathbf{x}=\sum_{i}\alpha_{i}\mathbf{v}_{i}
  33. โˆ‘ i ฮฑ i = 1. \sum_{i}\alpha_{i}=1.\,
  34. ฮฑ \alpha
  35. L F = โˆ‘ i ฮฑ i F ( ๐ฏ i ) LF=\sum_{i}\alpha_{i}F(\mathbf{v}_{i})
  36. P v i ( v 0 , v 1 , โ€ฆ , v n ) := ( v 0 , v 1 , โ€ฆ , v i - 1 , v ~ i , v i + 1 , โ€ฆ , v n ) P_{v_{i}}(v_{0},v_{1},...,v_{n}):=(v_{0},v_{1},...,v_{i-1},\tilde{v}_{i},v_{i+% 1},...,v_{n})
  37. v ~ i = { v 1 + v n - v 0 i = 0 v i + 1 + v i - 1 - v i 0 v n - 1 + v 0 - v n i = n \tilde{v}_{i}=\left\{\begin{array}[]{lcl}v_{1}+v_{n}-v_{0}&&i=0\\ v_{i+1}+v_{i-1}-v_{i}&&0\\ v_{n-1}+v_{0}-v_{n}&&i=n\\ \end{array}\right.

Pietro_Mengoli.html

  1. ฯ€ \pi

Pillai_prime.html

  1. n ! โ‰ก - 1 mod p n!\equiv-1\mod p
  2. p โ‰ข 1 mod n p\not\equiv 1\mod n

Pinhole_camera_model.html

  1. f f
  2. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  3. x 3 x_{3}
  4. ( y 1 , y 2 ) (y_{1},y_{2})
  5. ( y 1 , y 2 ) (y_{1},y_{2})
  6. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  7. - y 1 -y_{1}
  8. x 1 x_{1}
  9. x 3 x_{3}
  10. - y 1 f = x 1 x 3 \frac{-y_{1}}{f}=\frac{x_{1}}{x_{3}}
  11. y 1 = - f x 1 x 3 y_{1}=-\frac{f\,x_{1}}{x_{3}}
  12. - y 2 f = x 2 x 3 \frac{-y_{2}}{f}=\frac{x_{2}}{x_{3}}
  13. y 2 = - f x 2 x 3 y_{2}=-\frac{f\,x_{2}}{x_{3}}
  14. ( y 1 y 2 ) = - f x 3 ( x 1 x 2 ) \begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}=-\frac{f}{x_{3}}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}
  15. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  16. ( y 1 , y 2 ) (y_{1},y_{2})
  17. ( y 1 y 2 ) = f x 3 ( x 1 x 2 ) \begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}=\frac{f}{x_{3}}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}
  18. ๐ฑ \mathbf{x}
  19. ๐ฒ \mathbf{y}
  20. ๐ฒ โˆผ ๐‚ ๐ฑ \mathbf{y}\sim\mathbf{C}\,\mathbf{x}
  21. ๐‚ \mathbf{C}
  22. 3 ร— 4 3\times 4
  23. โˆผ \,\sim
  24. ๐‚ \mathbf{C}
  25. ๐‚ \mathbf{C}

Pinwheel_tiling.html

  1. T T
  2. 1 1
  3. 2 2
  4. 5 \sqrt{5}
  5. T T
  6. 1 / 5 1/\sqrt{5}
  7. T T
  8. T T
  9. T T
  10. arctan ( 1 / 2 ) \arctan(1/2)
  11. arctan ( 2 ) \arctan(2)
  12. T T
  13. ฯ€ \pi
  14. T T
  15. 2 2
  16. P P
  17. P โ€ฒ P^{\prime}
  18. 5 \sqrt{5}
  19. P P
  20. T T
  21. T T
  22. d = ln 4 ln 5 โ‰ˆ 1.7227 d=\frac{\ln 4}{\ln\sqrt{5}}\approx 1.7227

Pirincฬงlik_Air_Base.html

  1. 10 o {10^{o}}

Pivot_element.html

  1. [ 1 - 1 2 8 0 0 - 1 - 11 0 2 - 1 - 3 ] \left[\begin{array}[]{ccc|c}1&-1&2&8\\ 0&0&-1&-11\\ 0&2&-1&-3\end{array}\right]
  2. [ 1 - 1 2 8 0 2 - 1 - 3 0 0 - 1 - 11 ] \left[\begin{array}[]{ccc|c}1&-1&2&8\\ 0&2&-1&-3\\ 0&0&-1&-11\end{array}\right]
  3. [ 0.00300 59.14 59.17 5.291 - 6.130 46.78 ] \left[\begin{array}[]{cc|c}0.00300&59.14&59.17\\ 5.291&-6.130&46.78\\ \end{array}\right]
  4. [ 5.291 - 6.130 46.78 0.00300 59.14 59.17 ] . \left[\begin{array}[]{cc|c}5.291&-6.130&46.78\\ 0.00300&59.14&59.17\\ \end{array}\right].
  5. [ 30 591400 591700 5.291 - 6.130 46.78 ] \left[\begin{array}[]{cc|c}30&591400&591700\\ 5.291&-6.130&46.78\\ \end{array}\right]

Planck_momentum.html

  1. m P c m\text{P}c
  2. m P {m\text{P}}
  3. c c
  4. m P c = โ„ l P = โ„ c 3 G โ‰ˆ 6.52485 kg m/s , m\text{P}c=\frac{\hbar}{l\text{P}}=\sqrt{\frac{\hbar c^{3}}{G}}\approx 6.52485% \,\text{ kg m/s},
  5. โ„ \hbar
  6. l P {l\text{P}}
  7. G G
  8. m P c m\text{P}c
  9. m = m=

Plucker_matrix.html

  1. ๐ = A B T - B A T \mathbf{P}=AB^{T}-BA^{T}
  2. A A
  3. B B

Plus-end-directed_kinesin_ATPase.html

  1. โ‡Œ \rightleftharpoons

Pochhammer_k-symbol.html

  1. ( x ) n , k = x ( x + k ) ( x + 2 k ) โ‹ฏ ( x + ( n - 1 ) k ) , (x)_{n,k}=x(x+k)(x+2k)\cdots(x+(n-1)k),\,
  2. ฮ“ k ( x ) = lim n โ†’ โˆž n ! k n ( n k ) x / k - 1 ( x ) n , k . \Gamma_{k}(x)=\lim_{n\to\infty}\frac{n!k^{n}(nk)^{x/k-1}}{(x)_{n,k}}.

Poincareฬโ€“Lindstedt_method.html

  1. x ยจ + x + ฮต x 3 = 0 \ddot{x}+x+\varepsilon\,x^{3}=0\,
  2. x ( 0 ) = 1 , x(0)=1,\,
  3. x ห™ ( 0 ) = 0. \dot{x}(0)=0.\,
  4. x ( t ) = cos ( t ) + ฮต [ 1 32 ( cos ( 3 t ) - cos ( t ) ) - 3 8 t sin ( t ) ] + โ‹ฏ . x(t)=\cos(t)+\varepsilon\left[\tfrac{1}{32}\,\left(\cos(3t)-\cos(t)\right)-% \tfrac{3}{8}\,t\,\sin(t)\right]+\cdots.\,
  5. ฯ„ = ฯ‰ t , \tau=\omega t,\,
  6. ฯ‰ = ฯ‰ 0 + ฮต ฯ‰ 1 + โ‹ฏ . \omega=\omega_{0}+\varepsilon\omega_{1}+\cdots.\,
  7. ฯ‰ 2 x โ€ฒโ€ฒ ( ฯ„ ) + x ( ฯ„ ) + ฮต x 3 ( ฯ„ ) = 0 \omega^{2}\,x^{\prime\prime}(\tau)+x(\tau)+\varepsilon\,x^{3}(\tau)=0\,
  8. x 0 \displaystyle x_{0}
  9. 3 / 8 {3}/{8}
  10. x ( t ) โ‰ˆ cos ( ( 1 + 3 8 ฮต ) t ) + 1 32 ฮต [ cos ( 3 ( 1 + 3 8 ฮต ) t ) - cos ( ( 1 + 3 8 ฮต ) t ) ] . x(t)\approx\cos\Bigl(\left(1+\tfrac{3}{8}\,\varepsilon\right)\,t\Bigr)+\tfrac{% 1}{32}\,\varepsilon\,\left[\cos\Bigl(3\left(1+\tfrac{3}{8}\,\varepsilon\,% \right)\,t\Bigr)-\cos\Bigl(\left(1+\tfrac{3}{8}\,\varepsilon\,\right)\,t\Bigr)% \right].\,
  11. E = 1 2 x ห™ 2 + 1 2 x 2 + 1 4 ฮต x 4 \scriptstyle E=\tfrac{1}{2}\,\dot{x}^{2}+\tfrac{1}{2}\,x^{2}+\tfrac{1}{4}\,% \varepsilon\,x^{4}
  12. x ห™ \scriptstyle\dot{x}

Point-finite_collection.html

  1. ๐’ฐ \mathcal{U}
  2. X X
  3. X X
  4. ๐’ฐ \mathcal{U}

Point_distribution_model.html

  1. k k
  2. ๐— = ( x 1 , y 1 , โ€ฆ , x k , y k ) \mathbf{X}=(x_{1},y_{1},\ldots,x_{k},y_{k})
  3. i โˆˆ { 1 , โ€ฆ k } i\in\{1,\ldots k\}
  4. ( x 3 , y 3 ) (x_{3},y_{3})
  5. k k
  6. ๐— โˆˆ โ„ 2 k \mathbf{X}\in\mathbb{R}^{2k}
  7. d d
  8. ๐ โˆˆ โ„ 2 k ร— d \mathbf{P}\in\mathbb{R}^{2k\times d}
  9. ๐— โ€ฒ \mathbf{X}^{\prime}
  10. ๐— โ€ฒ = ๐— ยฏ + ๐๐› \mathbf{X}^{\prime}=\overline{\mathbf{X}}+\mathbf{P}\mathbf{b}
  11. ๐— ยฏ \overline{\mathbf{X}}
  12. ๐› \mathbf{b}
  13. ๐› \mathbf{b}
  14. ๐› \mathbf{b}
  15. ยฑ \pm
  16. โ„ 2 \mathbb{R}^{2}
  17. โ„ 3 \mathbb{R}^{3}
  18. k k

Point_reflection.html

  1. n - 1 n-1
  2. 1 โ‰ค k โ‰ค n - 1 1\leq k\leq n-1
  3. n - 1 n-1
  4. Ref ๐ฉ ( ๐š ) = 2 ๐ฉ - ๐š . \mathrm{Ref}_{\mathbf{p}}(\mathbf{a})=2\mathbf{p}-\mathbf{a}.
  5. 1 ยฏ \overline{1}
  6. - 1 -1
  7. - I -I
  8. I I
  9. ( x , y , z ) โ†ฆ ( - x , - y , - z ) (x,y,z)\mapsto(-x,-y,-z)
  10. - 1 -1
  11. O ( n ) O(n)
  12. 2 n 2n
  13. ( - 1 ) n (-1)^{n}
  14. O ( 2 n + 1 ) โ†’ ยฑ 1 O(2n+1)\to\pm 1
  15. O ( 2 n + 1 ) = S O ( 2 n + 1 ) ร— { ยฑ I } O(2n+1)=SO(2n+1)\times\{\pm I\}
  16. Q ( - v ) = Q ( v ) Q(-v)=Q(v)
  17. - 1 โˆˆ Spin ( n ) -1\in\mathrm{Spin}(n)
  18. - I โˆˆ S O ( 2 n ) -I\in SO(2n)
  19. Spin ( n ) \mathrm{Spin}(n)
  20. - 1 -1
  21. - I -I

Pointclass.html

  1. s y m b o l ฮฃ 1 0 symbol{\Sigma}^{0}_{1}
  2. s y m b o l ฮฃ 1 0 symbol{\Sigma}^{0}_{1}
  3. s y m b o l ฮฃ 1 0 symbol{\Sigma}^{0}_{1}
  4. s y m b o l ฮ  1 0 symbol{\Pi}^{0}_{1}
  5. s y m b o l ฮฃ 2 0 symbol{\Sigma}^{0}_{2}
  6. s y m b o l ฮ” 2 0 symbol{\Delta}^{0}_{2}
  7. s y m b o l ฮฃ 1 1 symbol{\Sigma}^{1}_{1}
  8. s y m b o l ฮ  1 0 symbol{\Pi}^{0}_{1}
  9. s y m b o l ฮ  1 0 symbol{\Pi}^{0}_{1}
  10. s y m b o l ฮฃ 1 0 symbol{\Sigma}^{0}_{1}
  11. ฮฃ 1 0 \Sigma^{0}_{1}
  12. ฮฃ \Sigma
  13. ฮฃ 1 0 \Sigma^{0}_{1}
  14. ฮ  1 0 \Pi^{0}_{1}
  15. ฮฃ 1 0 \Sigma^{0}_{1}
  16. ฮฃ 1 0 \Sigma^{0}_{1}
  17. ฮ  1 0 \Pi^{0}_{1}
  18. ฮฃ 2 0 \Sigma^{0}_{2}
  19. ฮ  1 0 \Pi^{0}_{1}
  20. ฮ  1 0 \Pi^{0}_{1}

Polar_homology.html

  1. C k C_{k}
  2. โ„‚ {\mathbb{C}}
  3. A k / R k A_{k}/R_{k}
  4. A k A_{k}
  5. R k R_{k}
  6. A k A_{k}
  7. A k A_{k}
  8. ( X , f , ฮฑ ) (X,f,\alpha)
  9. f : X โ†ฆ M f:\;X\mapsto M
  10. ฮฑ \alpha
  11. R k R_{k}
  12. R k R_{k}
  13. ฮป ( X , f , ฮฑ ) = ( X , f , ฮป ฮฑ ) \lambda(X,f,\alpha)=(X,f,\lambda\alpha)
  14. ( X , f , ฮฑ ) = 0 (X,f,\alpha)=0
  15. dim f ( X ) < k \dim f(X)<k
  16. โˆ‘ i ( X i , f i , ฮฑ i ) = 0 \ \sum_{i}(X_{i},f_{i},\alpha_{i})=0
  17. โˆ‘ i f i * ฮฑ i โ‰ก 0 , \sum_{i}f_{i*}\alpha_{i}\equiv 0,
  18. d i m f i ( X i ) = k dim\;f_{i}(X_{i})=k
  19. i i
  20. f i * ฮฑ i f_{i*}\alpha_{i}
  21. โˆช i f i ( X i ) \cup_{i}f_{i}(X_{i})
  22. โˆ‚ : C k โ†ฆ C k - 1 \partial:\;C_{k}\mapsto C_{k-1}
  23. โˆ‚ ( X , f , ฮฑ ) = 2 ฯ€ - 1 โˆ‘ i ( V i , f i , r e s V i ฮฑ ) \partial(X,f,\alpha)=2\pi\sqrt{-1}\sum_{i}(V_{i},f_{i},res_{V_{i}}\,\alpha)
  24. V i V_{i}
  25. ฮฑ \alpha
  26. f i = f | V i f_{i}=f|_{V_{i}}
  27. โˆ‚ 2 = 0 \partial^{2}=0
  28. ker โˆ‚ / im โˆ‚ \operatorname{ker}\;\partial/\operatorname{im}\;\partial

Polyconvex_function.html

  1. A โ†ฆ f ( A ) A\mapsto f(A)
  2. f ( A ) = { 1 det ( A ) , det ( A ) > 0 ; + โˆž , det ( A ) โ‰ค 0 ; f(A)=\begin{cases}\frac{1}{\det(A)},&\det(A)>0;\\ +\infty,&\det(A)\leq 0;\end{cases}

Polygonal_chain.html

  1. ( A 1 , A 2 , โ€ฆ , A n ) \scriptstyle(A_{1},A_{2},\dots,A_{n})
  2. n n
  3. โŒŠ n - 1 โŒ‹ \lfloor\sqrt{n-1}\rfloor

Polyiodide.html

  1. 1 12 \tfrac{1}{12}

Polynomial_chaos.html

  1. L 2 L_{2}
  2. L 2 L_{2}

Polynomial_greatest_common_divisor.html

  1. f = u d f=ud
  2. d = u - 1 f d=u^{-1}f
  3. gcd ( p , q ) = gcd ( q , p ) . \gcd(p,q)=\gcd(q,p).
  4. gcd ( p , q ) = gcd ( q , p + r q ) \gcd(p,q)=\gcd(q,p+rq)
  5. gcd ( p , q ) = gcd ( p , k q ) \gcd(p,q)=\gcd(p,kq)
  6. gcd ( p , q ) = gcd ( a 1 p + b 1 q , a 2 p + b 2 q ) \gcd(p,q)=\gcd(a_{1}p+b_{1}q,a_{2}p+b_{2}q)
  7. a 1 , b 1 , a 2 , b 2 a_{1},b_{1},a_{2},b_{2}
  8. a 1 b 2 - a 2 b 1 a_{1}b_{2}-a_{2}b_{1}
  9. gcd ( p , r ) = 1 \gcd(p,r)=1
  10. gcd ( p , q ) = gcd ( p , q r ) \gcd(p,q)=\gcd(p,qr)
  11. gcd ( q , r ) = 1 \gcd(q,r)=1
  12. gcd ( p , q r ) = gcd ( p , q ) gcd ( p , r ) \gcd(p,qr)=\gcd(p,q)\,\gcd(p,r)
  13. gcd ( p , q ) = a p + b q \gcd(p,q)=ap+bq
  14. gcd ( p , q ) \gcd(p,q)
  15. gcd ( p , q , r ) = gcd ( p , gcd ( q , r ) ) , \gcd(p,q,r)=\gcd(p,\gcd(q,r)),
  16. gcd ( p 1 , p 2 , โ€ฆ , p n ) = gcd ( p 1 , gcd ( p 2 , โ€ฆ , p n ) ) . \gcd(p_{1},p_{2},\dots,p_{n})=\gcd(p_{1},\gcd(p_{2},\dots,p_{n})).
  17. deg ( b ( x ) ) โ‰ค deg ( a ( x ) ) . \deg(b(x))\leq\deg(a(x))\,.
  18. a ( x ) = q 0 ( x ) b ( x ) + r 0 ( x ) a(x)=q_{0}(x)b(x)+r_{0}(x)
  19. deg ( r 0 ( x ) ) < deg ( b ( x ) ) \deg(r_{0}(x))<\deg(b(x))
  20. gcd ( a ( x ) , b ( x ) ) = gcd ( b ( x ) , r 0 ( x ) ) . \gcd(a(x),b(x))=\gcd(b(x),r_{0}(x))\,.
  21. a 1 ( x ) = b ( x ) , b 1 ( x ) = r 0 ( x ) . a_{1}(x)=b(x),b_{1}(x)=r_{0}(x)\,.
  22. deg ( a k + 1 ) + deg ( b k + 1 ) < deg ( a k ) + deg ( b k ) , \deg(a_{k+1})+\deg(b_{k+1})<\deg(a_{k})+\deg(b_{k})\ ,
  23. b N ( x ) = 0 b_{N}(x)=0
  24. gcd ( a , b ) = gcd ( a 1 , b 1 ) = โ‹ฏ = gcd ( a N , 0 ) = a N . \gcd(a,b)=\gcd(a_{1},b_{1})=\cdots=\gcd(a_{N},0)=a_{N}\,.
  25. a = b q + r a=bq+r
  26. deg ( r ) < deg ( b ) , \deg(r)<\deg(b),
  27. q := 0 ; r := a ; q:=0;\quad r:=a;
  28. d := deg ( b ) ; c := lc ( b ) ; d:=\deg(b);\quad c:=\,\text{lc}(b);
  29. deg ( r ) โ‰ฅ d \deg(r)\geq d
  30. s := lc ( r ) c x deg ( r ) - d ; s:=\frac{\,\text{lc}(r)}{c}x^{\deg(r)-d};
  31. q := q + s ; q:=q+s;
  32. r := r - s b ; r:=r-sb;
  33. r i = a s i + b t i r_{i}=as_{i}+bt_{i}
  34. s i t i + 1 - t i s i + 1 = s i t i - 1 - t i s i - 1 , s_{i}t_{i+1}-t_{i}s_{i+1}=s_{i}t_{i-1}-t_{i}s_{i-1},
  35. s i t i + 1 - t i s i + 1 = ( - 1 ) i . s_{i}t_{i+1}-t_{i}s_{i+1}=(-1)^{i}.
  36. a + L b = a + K [ X ] b . a+_{L}b=a+_{K[X]}b.
  37. a โ‹… L b = rem ( a . K [ X ] b ) , f ) . a\cdot_{L}b=\,\text{rem}(a._{K[X]}b),f).
  38. s 0 ( P , Q ) = โ‹ฏ = s d - 1 ( P , Q ) = 0 , s d ( P , Q ) โ‰  0 s_{0}(P,Q)=\cdots=s_{d-1}(P,Q)=0\ ,s_{d}(P,Q)\neq 0
  39. S 0 ( P , Q ) = โ‹ฏ = S d - 1 ( P , Q ) = 0. S_{0}(P,Q)=\cdots=S_{d-1}(P,Q)=0.
  40. deg ( P ) = deg ( ฯ† ( P ) ) \deg(P)=\deg(\varphi(P))
  41. deg ( Q ) = deg ( ฯ† ( Q ) ) , \deg(Q)=\deg(\varphi(Q)),
  42. P = p 0 + p 1 X + โ‹ฏ + p m X m , Q = q 0 + q 1 X + โ‹ฏ + q n X n . P=p_{0}+p_{1}X+\cdots+p_{m}X^{m},\quad Q=q_{0}+q_{1}X+\cdots+q_{n}X^{n}.
  43. ๐’ซ i \mathcal{P}_{i}
  44. ฯ† i : ๐’ซ n - i ร— ๐’ซ m - i โ†’ ๐’ซ m + n - i \varphi_{i}:\mathcal{P}_{n-i}\times\mathcal{P}_{m-i}\rightarrow\mathcal{P}_{m+% n-i}
  45. ฯ† i ( A , B ) = A P + B Q . \varphi_{i}(A,B)=AP+BQ.
  46. ฯ† 0 \varphi_{0}
  47. ฯ† i . \varphi_{i}.
  48. S = ( p m 0 โ‹ฏ 0 q n 0 โ‹ฏ 0 p m - 1 p m โ‹ฏ 0 q n - 1 q n โ‹ฏ 0 p m - 2 p m - 1 โ‹ฑ 0 q n - 2 q n - 1 โ‹ฑ 0 โ‹ฎ โ‹ฎ โ‹ฑ p m โ‹ฎ โ‹ฎ โ‹ฑ q n โ‹ฎ โ‹ฎ โ‹ฏ p m - 1 โ‹ฎ โ‹ฎ โ‹ฏ q n - 1 p 0 p 1 โ‹ฏ โ‹ฎ q 0 q 1 โ‹ฏ โ‹ฎ 0 p 0 โ‹ฑ โ‹ฎ 0 q 0 โ‹ฑ โ‹ฎ โ‹ฎ โ‹ฎ โ‹ฑ p 1 โ‹ฎ โ‹ฎ โ‹ฑ q 1 0 0 โ‹ฏ p 0 0 0 โ‹ฏ q 0 ) . S=\begin{pmatrix}p_{m}&0&\cdots&0&q_{n}&0&\cdots&0\\ p_{m-1}&p_{m}&\cdots&0&q_{n-1}&q_{n}&\cdots&0\\ p_{m-2}&p_{m-1}&\ddots&0&q_{n-2}&q_{n-1}&\ddots&0\\ \vdots&\vdots&\ddots&p_{m}&\vdots&\vdots&\ddots&q_{n}\\ \vdots&\vdots&\cdots&p_{m-1}&\vdots&\vdots&\cdots&q_{n-1}\\ p_{0}&p_{1}&\cdots&\vdots&q_{0}&q_{1}&\cdots&\vdots\\ 0&p_{0}&\ddots&\vdots&0&q_{0}&\ddots&\vdots&\\ \vdots&\vdots&\ddots&p_{1}&\vdots&\vdots&\ddots&q_{1}\\ 0&0&\cdots&p_{0}&0&0&\cdots&q_{0}\end{pmatrix}.
  49. ฯ† i \varphi_{i}
  50. V i = ( 1 0 โ‹ฏ 0 0 0 โ‹ฏ 0 0 1 โ‹ฏ 0 0 0 โ‹ฏ 0 โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฎ โ‹ฎ โ‹ฑ โ‹ฎ 0 0 0 โ‹ฏ 1 0 0 โ‹ฏ 0 0 0 โ‹ฏ 0 X i X i - 1 โ‹ฏ 1 ) . V_{i}=\begin{pmatrix}1&0&\cdots&0&0&0&\cdots&0\\ 0&1&\cdots&0&0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&0\\ 0&0&\cdots&1&0&0&\cdots&0\\ 0&0&\cdots&0&X^{i}&X^{i-1}&\cdots&1\end{pmatrix}.
  51. ฯ† i \varphi_{i}
  52. ฯ† i . \varphi_{i}.
  53. ฯ† i \varphi_{i}
  54. ฯ† i \varphi_{i}
  55. ฯ† i \varphi_{i}
  56. f = โˆ i = 1 deg ( f ) f i i f=\prod_{i=1}^{\deg(f)}f_{i}^{i}
  57. r i + 1 := rem ( r i - 1 , r i ) r_{i+1}:=\,\text{rem}(r_{i-1},r_{i})
  58. r i + 1 := - rem ( r i - 1 , r i ) . r_{i+1}:=-\,\text{rem}(r_{i-1},r_{i}).
  59. q = p c q=\frac{p}{c}
  60. cont ( q ) = cont ( p ) c . \,\text{cont}(q)=\frac{\,\text{cont}(p)}{c}.
  61. primpart ( p ) = p cont ( p ) . \,\text{primpart}(p)=\frac{p}{\,\text{cont}(p)}.
  62. p = cont ( p ) primpart ( p ) , p=\,\text{cont}(p)\,\,\text{primpart}(p),
  63. primpart ( p q ) = primpart ( p ) primpart ( q ) \,\text{primpart}(pq)=\,\text{primpart}(p)\,\,\text{primpart}(q)
  64. cont ( p q ) = cont ( p ) cont ( q ) . \,\text{cont}(pq)=\,\text{cont}(p)\,\,\text{cont}(q).
  65. primpart ( gcd F [ X ] ( q 1 , q 2 ) ) = gcd R [ X ] ( primpart ( q 1 ) , primpart ( q 2 ) ) . \,\text{primpart}(\,\text{gcd}_{F[X]}(q_{1},q_{2}))=\,\text{gcd}_{R[X]}(\,% \text{primpart}(q_{1}),\,\text{primpart}(q_{2})).
  66. gcd R [ X ] ( p 1 , p 2 ) = gcd R ( cont ( p 1 ) , cont ( p 2 ) ) gcd R [ X ] ( primpart ( p 1 ) , primpart ( p 2 ) ) , \,\text{gcd}_{R[X]}(p_{1},p_{2})=\,\text{gcd}_{R}(\,\text{cont}(p_{1}),\,\text% {cont}(p_{2}))\,\,\text{gcd}_{R[X]}(\,\text{primpart}(p_{1}),\,\text{primpart}% (p_{2})),
  67. gcd R [ X ] ( primpart ( p 1 ) , primpart ( p 2 ) ) = primpart ( gcd F [ X ] ( p 1 , p 2 ) ) . \,\text{gcd}_{R[X]}(\,\text{primpart}(p_{1}),\,\text{primpart}(p_{2}))=\,\text% {primpart}(\,\text{gcd}_{F[X]}(p_{1},p_{2})).
  68. X 8 + X 6 - 3 X 4 - 3 X 3 + 8 X 2 + 2 X - 5 X^{8}+X^{6}-3X^{4}-3X^{3}+8X^{2}+2X-5
  69. 3 X 6 + 5 X 4 - 4 X 2 - 9 X + 21 , 3X^{6}+5X^{4}-4X^{2}-9X+21,
  70. - 5 9 X 4 + 1 9 X 2 - 1 3 , -\frac{5}{9}X^{4}+\frac{1}{9}X^{2}-\frac{1}{3},
  71. - 117 25 X 2 - 9 X + 441 25 , -\frac{117}{25}X^{2}-9X+\frac{441}{25},
  72. 233150 19773 X - 102500 6591 , \frac{233150}{19773}X-\frac{102500}{6591},
  73. - 1288744821 543589225 . -\frac{1288744821}{543589225}.
  74. deg ( A ) = a \deg(A)=a
  75. deg ( B ) = b \deg(B)=b
  76. prem ( A , B ) = rem ( lc ( B ) a - b + 1 A , B ) , \,\text{prem}(A,B)=\,\text{rem}(\,\text{lc}(B)^{a-b+1}A,B),
  77. r i + 1 := rem ( r i - 1 , r i ) r_{i+1}:=\,\text{rem}(r_{i-1},r_{i})
  78. r i + 1 := prem ( r i - 1 , r i ) ฮฑ , r_{i+1}:=\frac{\,\text{prem}(r_{i-1},r_{i})}{\alpha},
  79. - 15 X 4 + 3 X 2 - 9 , -15\,X^{4}+3\,X^{2}-9,
  80. 15795 X 2 + 30375 X - 59535 , 15795\,X^{2}+30375\,X-59535,
  81. 1254542875143750 X - 1654608338437500 , 1254542875143750\,X-1654608338437500,
  82. 12593338795500743100931141992187500. 12593338795500743100931141992187500.
  83. - 5 X 4 + X 2 - 3 , -5\,X^{4}+X^{2}-3,
  84. 13 X 2 + 25 X - 49 , 13\,X^{2}+25\,X-49,
  85. 4663 X - 6150 , 4663\,X-6150,
  86. 1. 1.
  87. 15 X 4 - 3 X 2 + 9 , 15\,X^{4}-3\,X^{2}+9,
  88. 65 X 2 + 125 X - 245 , 65\,X^{2}+125\,X-245,
  89. 9326 X - 12300 , 9326\,X-12300,
  90. 260708. 260708.
  91. deg ( A ) = a \deg(A)=a
  92. deg ( B ) = b \deg(B)=b
  93. prem2 ( A , B ) = - rem ( | lc ( B ) | a - b + 1 A , B ) , \operatorname{prem2}(A,B)=-\operatorname{rem}(|\operatorname{lc}(B)|^{a-b+1}A,% B),
  94. - prem2 ( A , B ) \operatorname{-prem2}(A,B)
  95. prem ( A , B ) \operatorname{prem}(A,B)
  96. F = โ„š ( 3 ) F=\mathbb{Q}(\sqrt{3})
  97. D = โ„ค [ 3 ] D=\mathbb{Z}[\sqrt{3}]
  98. f = 3 x 3 - 5 x 2 + 4 x + 9 f=\sqrt{3}x^{3}-5x^{2}+4x+9
  99. g = x 4 + 4 x 2 + 3 3 x - 6 g=x^{4}+4x^{2}+3\sqrt{3}x-6
  100. I = ( 2 ) I=(2)
  101. D / I D/I
  102. I I
  103. D D
  104. f , g f,g
  105. ( D / I ) [ x ] (D/I)[x]
  106. f , g f,g
  107. F [ x ] F[x]
  108. I I

Pompeiu's_theorem.html

  1. P C = P โ€ฒ C \scriptstyle PC\ =\ P^{\prime}C
  2. โˆ  P C P โ€ฒ = 60 โˆ˜ \scriptstyle\angle PCP^{\prime}\ =\ 60^{\circ}
  3. P P โ€ฒ = P C \scriptstyle PP^{\prime}\ =\ PC
  4. P A = P โ€ฒ B \scriptstyle PA\ =\ P^{\prime}B

Ponderomotive_energy.html

  1. U p = e 2 E a 2 4 m ฯ‰ 0 2 U_{p}={e^{2}E_{a}^{2}\over 4m\omega_{0}^{2}}
  2. e e
  3. E a E_{a}
  4. ฯ‰ 0 2 \omega_{0}^{2}
  5. m m
  6. I I
  7. I = c ฯต 0 E a 2 / 2 I=c\epsilon_{0}E_{a}^{2}/2
  8. U p = e 2 I 2 c ฯต 0 m ฯ‰ 0 2 = 2 e 2 c ฯต 0 m ร— I 4 ฯ‰ 0 2 U_{p}={e^{2}I\over 2c\epsilon_{0}m\omega_{0}^{2}}={2e^{2}\over c\epsilon_{0}m}% \times{I\over 4\omega_{0}^{2}}
  9. e = m = 1 e=m=1
  10. ฯต 0 = 1 / 4 ฯ€ \epsilon_{0}=1/4\pi
  11. ฮฑ c = 1 \alpha c=1
  12. ฮฑ โ‰ˆ 1 / 137 \alpha\approx 1/137
  13. U p = I 4 ฯ‰ 0 2 . U_{p}=\frac{I}{4\omega_{0}^{2}}.
  14. e e
  15. E exp ( - i ฯ‰ t ) E\,\exp(-i\omega t)
  16. F = e E exp ( - i ฯ‰ t ) F=eE\,\exp(-i\omega t)
  17. a m = F m = e E m exp ( - i ฯ‰ t ) a_{m}={F\over m}={eE\over m}\exp(-i\omega t)
  18. x = - a ฯ‰ 2 = - e E m ฯ‰ 2 exp ( - i ฯ‰ t ) = - e m ฯ‰ 2 2 I 0 c ฯต 0 exp ( - i ฯ‰ t ) x={-a\over\omega^{2}}=-\frac{eE}{m\omega^{2}}\,\exp(-i\omega t)=-\frac{e}{m% \omega^{2}}\sqrt{\frac{2I_{0}}{c\epsilon_{0}}}\,\exp(-i\omega t)
  19. U = 1 2 m ฯ‰ 2 โŸจ x 2 โŸฉ = e 2 E 2 4 m ฯ‰ 2 U=\textstyle{\frac{1}{2}}m\omega^{2}\langle x^{2}\rangle={e^{2}E^{2}\over 4m% \omega^{2}}
  20. U p U_{p}

Pooled_variance.html

  1. i = 1 , โ€ฆ , k i=1,\ldots,k
  2. s p 2 s^{2}_{p}
  3. s i 2 s^{2}_{i}
  4. s p 2 = โˆ‘ i = 1 k ( n i - 1 ) s i 2 โˆ‘ i = 1 k ( n i - 1 ) = ( n 1 - 1 ) s 1 2 + ( n 2 - 1 ) s 2 2 + โ‹ฏ + ( n k - 1 ) s k 2 n 1 + n 2 + โ‹ฏ + n k - k s_{p}^{2}=\frac{\sum_{i=1}^{k}(n_{i}-1)s_{i}^{2}}{\sum_{i=1}^{k}(n_{i}-1)}=% \frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}+\cdots+(n_{k}-1)s_{k}^{2}}{n_{1}+n% _{2}+\cdots+n_{k}-k}
  5. n i n_{i}
  6. i i
  7. s i 2 s^{2}_{i}
  8. 1 n i - 1 โˆ‘ j = 1 n i ( y j - y j ยฏ ) 2 \frac{1}{n_{i}-1}\sum_{j=1}^{n_{i}}\left(y_{j}-\overline{y_{j}}\right)^{2}
  9. ( n i - 1 ) (n_{i}-1)
  10. n i n_{i}
  11. s p 2 = โˆ‘ i = 1 k ( n i - 1 ) s i 2 โˆ‘ i = 1 k ( n i - 1 ) s_{p}^{2}=\frac{\sum_{i=1}^{k}(n_{i}-1)s_{i}^{2}}{\sum_{i=1}^{k}(n_{i}-1)}
  12. s p 2 = โˆ‘ i = 1 k ( n i - 1 ) s i 2 โˆ‘ i = 1 k n i s_{p}^{2}=\frac{\sum_{i=1}^{k}(n_{i}-1)s_{i}^{2}}{\sum_{i=1}^{k}n_{i}}
  13. s p 2 s_{p}^{2}
  14. ฯƒ 2 \sigma^{2}
  15. s p 2 s_{p}^{2}
  16. ฯƒ 2 \sigma^{2}
  17. s i 2 s_{i}^{2}
  18. S P 2 = ( n 1 - 1 ) S 1 2 + ( n 2 - 1 ) S 2 2 + โ‹ฏ + ( n k - 1 ) S k 2 ( n 1 - 1 ) + ( n 2 - 1 ) + โ‹ฏ + ( n k - 1 ) S_{P}^{2}=\frac{(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}+\cdots+(n_{k}-1)S_{k}^{2% }}{(n_{1}-1)+(n_{2}-1)+\cdots+(n_{k}-1)}
  19. S P 2 = 2.765 S_{P}^{2}=2.765\,

Popper's_experiment.html

  1. ฮ” y \Delta y
  2. p y p_{y}
  3. ฯˆ ( y 1 , y 2 ) = โˆซ - โˆž โˆž e i k y 1 e - i k y 2 d k \psi(y_{1},y_{2})=\int_{-\infty}^{\infty}e^{iky_{1}}e^{-iky_{2}}\,dk
  4. | ฯˆ โŸฉ = โˆซ - โˆž โˆž | y , y โŸฉ d y = โˆซ - โˆž โˆž | p , - p โŸฉ d p |\psi\rangle=\int_{-\infty}^{\infty}|y,y\rangle\,dy=\int_{-\infty}^{\infty}|p,% -p\rangle\,dp
  5. x 0 x_{0}
  6. x 0 x_{0}
  7. p 0 p_{0}
  8. - p 0 -p_{0}
  9. ฯˆ ( y 1 , y 2 ) = A โˆซ - โˆž โˆž d p e - p 2 / 4 ฯƒ 2 e - i p y 2 / โ„ e i p y 1 / โ„ exp [ - ( y 1 + y 2 ) 2 / 16 ฮฉ 2 ] \psi(y_{1},y_{2})=A\!\int_{-\infty}^{\infty}dpe^{-p^{2}/4\sigma^{2}}e^{-ipy_{2% }/\hbar}e^{ipy_{1}/\hbar}\exp[-{(y_{1}+y_{2})^{2}/16\Omega^{2}}]
  10. ฯƒ \sigma
  11. ฮฉ \Omega
  12. ฮ” p 2 = ฮ” p 1 = ฯƒ 2 + โ„ 2 / 16 ฮฉ 2 , ฮ” y 1 = ฮ” y 2 = ฮฉ 2 + โ„ 2 / 16 ฯƒ 2 . \Delta p_{2}=\Delta p_{1}=\sqrt{\sigma^{2}+{\hbar^{2}/16\Omega^{2}}},~{}~{}~{}% ~{}\Delta y_{1}=\Delta y_{2}=\sqrt{\Omega^{2}+\hbar^{2}/16\sigma^{2}}.
  13. ฯ• 1 ( y 1 ) = 1 ( ฯต 2 2 ฯ€ ) 1 / 4 e - y 1 2 / 4 ฯต 2 \phi_{1}(y_{1})=\frac{1}{(\epsilon^{2}2\pi)^{1/4}}e^{-y_{1}^{2}/4\epsilon^{2}}
  14. ฯ• 2 ( y 2 ) = โˆซ - โˆž โˆž ฯˆ ( y 1 , y 2 ) ฯ• 1 * ( y 1 ) d y 1 \phi_{2}(y_{2})=\!\int_{-\infty}^{\infty}\psi(y_{1},y_{2})\phi_{1}^{*}(y_{1})% dy_{1}
  15. ฮ” p 2 = ฯƒ 2 ( 1 + ฯต 2 / ฮฉ 2 ) + โ„ 2 / 16 ฮฉ 2 1 + 4 ฯต 2 ( ฯƒ 2 / โ„ 2 + 1 / 16 ฮฉ 2 ) . \Delta p_{2}=\sqrt{\frac{\sigma^{2}(1+\epsilon^{2}/\Omega^{2})+\hbar^{2}/16% \Omega^{2}}{1+4\epsilon^{2}(\sigma^{2}/\hbar^{2}+1/16\Omega^{2})}}.
  16. ฯต โ†’ 0 \epsilon\to 0
  17. lim ฯต โ†’ 0 ฮ” p 2 = ฯƒ 2 + โ„ 2 / 16 ฮฉ 2 \lim_{\epsilon\to 0}\Delta p_{2}=\sqrt{\sigma^{2}+\hbar^{2}/16\Omega^{2}}
  18. ฯƒ 2 + โ„ 2 / 16 ฮฉ 2 \sqrt{\sigma^{2}+\hbar^{2}/16\Omega^{2}}
  19. ฯต , ฯƒ \epsilon,\sigma
  20. ฮฉ \Omega

Population_model.html

  1. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\,
  2. d N 1 d t = r 1 N 1 K 1 - N 1 - ฮฑ N 2 K 1 \frac{dN_{1}}{dt}=r_{1}N_{1}\frac{K_{1}-N_{1}-\alpha N_{2}}{K_{1}}\,
  3. S = I P I + E S=\frac{IP}{I+E}
  4. log ( S ) = log ( c ) + z log ( A ) \log(S)=\log(c)+z\log(A)\,

Population_vector.html

  1. F F
  2. F = โˆ‘ j m j F j โˆ‘ j m j F=\frac{\sum_{j}m_{j}F_{j}}{\sum_{j}m_{j}}
  3. m j m_{j}
  4. j j
  5. F j F_{j}
  6. j j
  7. F F
  8. F j F_{j}

Portal:Electronics::Laws.html

  1. I = V R I=\frac{V}{R}

Portal:Electronics::Selected_article::1.html

  1. I = V R I=\frac{V}{R}

Portal:Electronics::Selected_article::13.html

  1. F โ†’ = 1 4 ฯ€ ฯต 0 q 1 q 2 | r โ†’ | 3 r โ†’ = 1 4 ฯ€ ฯต 0 q 1 q 2 | r โ†’ | 2 r ^ \vec{F}=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{|\vec{r}|^{3}}\vec{r}=% \frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{|\vec{r}|^{2}}\hat{r}
  2. F โ†’ \vec{F}
  3. q 1 q_{1}
  4. q 2 q_{2}
  5. r โ†’ = r 1 โ†’ - r 2 โ†’ \vec{r}=\vec{r_{1}}-\vec{r_{2}}
  6. r 1 โ†’ \vec{r_{1}}
  7. q 1 q_{1}
  8. r 2 โ†’ \vec{r_{2}}
  9. q 2 q_{2}
  10. r ^ \hat{r}
  11. r โ†’ \vec{r}
  12. ฯต 0 \epsilon_{0}
  13. q 1 q 2 q_{1}q_{2}

Portal:Electronics::Selected_article::2.html

  1. Q = I 2 โ‹… R โ‹… t Q=I^{2}\cdot R\cdot t

Portal:Electronics::Selected_article::3.html

  1. R = V I R=\frac{V}{I}

Portal:Electronics::Selected_article::8.html

  1. C = Q V C=\frac{Q}{V}
  2. C = ฯต A d C=\epsilon\frac{A}{d}

Portal:Electronics::Selected_article::9.html

  1. L = ฮฆ i L=\frac{\Phi}{i}
  2. L = ฮป i = N ฮฆ i L=\frac{\lambda}{i}=N\frac{\Phi}{i}
  3. ฮป \lambda

Portal:Electronics::Selected_biography::3.html

  1. B B\,

Portal:Marine_life::Selected_Article::April,_2007.html

  1. f g f fgf

Portal:Mathematics::Featured_article::2005_6.html

  1. C = 2 ฯ€ r C=2\pi r\,\!
  2. A = ฯ€ r 2 A=\pi r^{2}\,\!

Portal:Mathematics::Featured_article::2007_24.html

  1. a + b a = a b = ฯ† . \frac{a+b}{a}=\frac{a}{b}=\varphi\,.
  2. ฯ† = 1 + 5 2 , \varphi=\frac{1+\sqrt{5}}{2},\,

Portal:Mathematics::Featured_article::2007_25.html

  1. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}\,
  2. a 2 + b 2 = c . \sqrt{a^{2}+b^{2}}=c.\,

Portal:Mathematics::Selected_article::31.html

  1. f ( x ) = a x 2 + b x + c . f(x)=ax^{2}+bx+c.\,

Portal:Physics::Selected_article::Week_17,_2007.html

  1. B B\,

Position_sensitive_device.html

  1. I a I_{a}
  2. I b I_{b}
  3. I c I_{c}
  4. I d I_{d}
  5. x = k x โ‹… I b - I d I b + I d x=k_{x}\cdot\frac{I_{b}-I_{d}}{I_{b}+I_{d}}
  6. y = k y โ‹… I a - I c I a + I c y=k_{y}\cdot\frac{I_{a}-I_{c}}{I_{a}+I_{c}}
  7. k x k_{x}
  8. k y k_{y}
  9. x = k x โ‹… I 4 - I 3 I 4 + I 3 x=k_{x}\cdot\frac{I_{4}-I_{3}}{I_{4}+I_{3}}
  10. y = k y โ‹… I 2 - I 1 I 2 + I 1 y=k_{y}\cdot\frac{I_{2}-I_{1}}{I_{2}+I_{1}}
  11. x = k x 1 โ‹… I 4 - I 3 I 0 - 1.02 ( I 2 - I 1 ) โ‹… 0.7 ( I 2 + I 1 ) + I 0 I 0 + 1.02 ( I 2 - I 1 ) x=k_{x1}\cdot\frac{I_{4}-I_{3}}{I_{0}-1.02(I_{2}-I_{1})}\cdot\frac{0.7(I_{2}+I% _{1})+I_{0}}{I_{0}+1.02(I_{2}-I_{1})}
  12. y = k y 1 โ‹… I 2 - I 1 I 0 - 1.02 ( I 4 - I 3 ) โ‹… 0.7 ( I 4 + I 3 ) + I 0 I 0 + 1.02 ( I 4 - I 3 ) y=k_{y1}\cdot\frac{I_{2}-I_{1}}{I_{0}-1.02(I_{4}-I_{3})}\cdot\frac{0.7(I_{4}+I% _{3})+I_{0}}{I_{0}+1.02(I_{4}-I_{3})}
  13. I 0 = I 1 + I 2 + I 3 + I 4 I_{0}=I_{1}+I_{2}+I_{3}+I_{4}
  14. k x 1 , k y 1 k_{x1},k_{y1}

Positive-definite_function_on_a_group.html

  1. โˆ‘ s , t โˆˆ G โŸจ F ( s - 1 t ) h ( t ) , h ( s ) โŸฉ โ‰ฅ 0 , \sum_{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle\geq 0,
  2. โˆ‘ s , t โˆˆ G โŸจ F ( s - 1 t ) h ( t ) , h ( s ) โŸฉ \displaystyle\sum_{s,t\in G}\langle F(s^{-1}t)h(t),h(s)\rangle
  3. V = โ‹ s โˆˆ G ฮฆ ( s ) H V=\bigvee_{s\in G}\Phi(s)H\,
  4. โ‹ \bigvee

Positive-definite_kernel.html

  1. { H n } n โˆˆ โ„ค \{H_{n}\}_{n\in{\mathbb{Z}}}
  2. โ„’ ( H i , H j ) \mathcal{L}(H_{i},H_{j})
  3. โ„ค ร— โ„ค {\mathbb{Z}}\times{\mathbb{Z}}
  4. A ( i , j ) โˆˆ โ„’ ( H i , H j ) A(i,j)\in\mathcal{L}(H_{i},H_{j})
  5. h i โˆˆ H i h_{i}\in H_{i}
  6. โˆ‘ - m โ‰ค i j โ‰ค m โŸจ A ( i , j ) h i , h j โŸฉ โ‰ฅ 0. \sum_{-m\leq i\quad\,\atop j\leq m}\langle A(i,j)h_{i},h_{j}\rangle\geq 0.
  7. ( โ„‹ , โŸจ โ‹… , โ‹… โŸฉ ) ({\mathcal{H}},\langle\cdot,\cdot\rangle)
  8. { H n } n โˆˆ โ„ค \{H_{n}\}_{n\in{\mathbb{Z}}}
  9. { โ„‚ } x โˆˆ X . \{{\mathbb{C}}\}_{x\in X}.
  10. X โŠ” Y = { ( x , ฮพ ) | x โˆˆ X } โˆช { ( ฮพ , y ) | y โˆˆ Y } . X\sqcup Y=\{(x,\xi)|x\in X\}\cup\{(\xi,y)|y\in Y\}.
  11. K โŠ• L K\oplus L
  12. H ( K โŠ• L , X โŠ” Y ) H(K\oplus L,X\sqcup Y)
  13. K โŠ— L K\otimes L
  14. ( K โŠ— L ) ( ( x , y ) , ( x โ€ฒ , y โ€ฒ ) ) = K ( x , x โ€ฒ ) L ( y , y โ€ฒ ) . (K\otimes L)((x,y),(x^{\prime},y^{\prime}))=K(x,x^{\prime})L(y,y^{\prime}).
  15. H ( K โŠ— L , X ร— Y ) H(K\otimes L,X\times Y)
  16. โŸจ x , B * B y โŸฉ = โŸจ B x , B y โŸฉ A = โŸจ [ x ] , [ y ] โŸฉ A = โŸจ x , A y โŸฉ \langle x,B^{*}By\rangle=\langle Bx,By\rangle_{A}=\langle[x],[y]\rangle_{A}=% \langle x,Ay\rangle
  17. โ„’ ( H n , H A ) , \mathcal{L}(H_{n},H_{A}),
  18. A ( i , j ) = B * ( i ) B ( j ) and H A = โ‹ n โˆˆ โ„ค B ( n ) H n , \quad A(i,j)=B^{*}(i)B(j)\quad\mbox{and}~{}\quad H_{A}=\bigvee_{n\in{\mathbb{Z% }}}B(n)H_{n}\;,
  19. U : G โ†’ H A such that U C ( n ) = B ( n ) for all n โˆˆ โ„ค . U:G\rightarrow H_{A}\quad\mbox{such that}~{}\quad UC(n)=B(n)\quad\mbox{for all% }~{}\quad n\in{\mathbb{Z}}.

Positive_current.html

  1. ฮท โ†ฆ โˆซ M ฮท โˆง ฯ \eta\mapsto\int_{M}\eta\wedge\rho
  2. ฮ› c * ( M ) \Lambda_{c}^{*}(M)
  3. ฮ› i ( M ) = โŠ• p + q = i ฮ› p , q ( M ) \Lambda^{i}(M)=\bigoplus_{p+q=i}\Lambda^{p,q}(M)
  4. ฮ› c p , q ( M ) \Lambda_{c}^{p,q}(M)
  5. ฮ˜ \Theta
  6. ฮ˜ \Theta
  7. ฮ˜ \Theta
  8. ฯ€ : M โ†ฆ X \pi:\;M\mapsto X
  9. ฯ€ \pi

Positive_form.html

  1. ฮ› p , p ( M ) โˆฉ ฮ› 2 p ( M , โ„ ) . \Lambda^{p,p}(M)\cap\Lambda^{2p}(M,{\mathbb{R}}).
  2. ฯ‰ \omega
  3. - ฯ‰ -\omega
  4. d z 1 , โ€ฆ d z n dz_{1},...dz_{n}
  5. ฮ› 1 , 0 M \Lambda^{1,0}M
  6. - 1 ฯ‰ \sqrt{-1}\omega
  7. - 1 ฯ‰ = โˆ‘ i ฮฑ i d z i โˆง d z ยฏ i , \sqrt{-1}\omega=\sum_{i}\alpha_{i}dz_{i}\wedge d\bar{z}_{i},
  8. ฮฑ i \alpha_{i}
  9. v โˆˆ T 1 , 0 M v\in T^{1,0}M
  10. - - 1 ฯ‰ ( v , v ยฏ ) โ‰ฅ 0 -\sqrt{-1}\omega(v,\bar{v})\geq 0
  11. v โˆˆ T M v\in TM
  12. ฯ‰ ( v , I ( v ) ) โ‰ฅ 0 \omega(v,I(v))\geq 0
  13. I : T M โ†ฆ T M I:\;TM\mapsto TM
  14. โˆ‚ ยฏ : L โ†ฆ L โŠ— ฮ› 0 , 1 ( M ) \bar{\partial}:\;L\mapsto L\otimes\Lambda^{0,1}(M)
  15. โˆ‡ 0 , 1 = โˆ‚ ยฏ \nabla^{0,1}=\bar{\partial}
  16. ฮ˜ \Theta
  17. - 1 ฮ˜ \sqrt{-1}\Theta
  18. - 1 ฮ˜ \sqrt{-1}\Theta
  19. d i m โ„‚ M = 2 dim_{\mathbb{C}}M=2
  20. ฮท , ฮถ โ†ฆ โˆซ M ฮท โˆง ฮถ \eta,\zeta\mapsto\int_{M}\eta\wedge\zeta
  21. 2 โ‰ค p โ‰ค d i m โ„‚ M - 2 2\leq p\leq dim_{\mathbb{C}}M-2
  22. ฮท \eta
  23. โˆซ M ฮท โˆง ฮถ โ‰ฅ 0 \int_{M}\eta\wedge\zeta\geq 0

Positive_invariant_set.html

  1. x ห™ = f ( x ) \dot{x}=f(x)
  2. x ( t , x 0 ) x(t,x_{0})\,
  3. x 0 x_{0}\,
  4. ๐’ช โ‰œ { x โˆˆ โ„ n | ฯ• ( x ) = 0 } \mathcal{O}\triangleq\left\{x\in\mathbb{R}^{n}|\phi(x)=0\right\}
  5. ฯ• \phi
  6. ๐’ช \mathcal{O}
  7. x 0 โˆˆ ๐’ช x_{0}\in\mathcal{O}
  8. x ( t , x 0 ) โˆˆ ๐’ช โˆ€ t โ‰ฅ 0 x(t,x_{0})\in\mathcal{O}\ \forall\ t\geq 0
  9. ๐’ช \mathcal{O}

Positively_separated_sets.html

  1. inf a โˆˆ A , b โˆˆ B d ( a , b ) \inf_{a\in A,b\in B}d(a,b)

Post-modern_portfolio_theory.html

  1. d = โˆซ - โˆž t ( t - r ) 2 f ( r ) d r d=\sqrt{\int_{-\infty}^{t}(t-r)^{2}f(r)\,dr}
  2. r - t d \frac{r-t}{d}

Posynomial.html

  1. f ( x 1 , x 2 , โ€ฆ , x n ) = โˆ‘ k = 1 K c k x 1 a 1 k โ‹ฏ x n a n k f(x_{1},x_{2},\dots,x_{n})=\sum_{k=1}^{K}c_{k}x_{1}^{a_{1k}}\cdots x_{n}^{a_{% nk}}
  2. x i x_{i}
  3. c k c_{k}
  4. a i k a_{ik}
  5. f ( x 1 , x 2 , x 3 ) = 2.7 x 1 2 x 2 - 1 / 3 x 3 0.7 + 2 x 1 - 4 x 3 2 / 5 f(x_{1},x_{2},x_{3})=2.7x_{1}^{2}x_{2}^{-1/3}x_{3}^{0.7}+2x_{1}^{-4}x_{3}^{2/5}

Potential_game.html

  1. N N
  2. A A
  3. A i A_{i}
  4. u u
  5. G = ( N , A = A 1 ร— โ€ฆ ร— A N , u : A โ†’ \reals N ) G=(N,A=A_{1}\times\ldots\times A_{N},u:A\rightarrow\reals^{N})
  6. ฮฆ : A โ†’ \reals \Phi:A\rightarrow\reals
  7. โˆ€ a - i โˆˆ A - i , โˆ€ a i โ€ฒ , a i โ€ฒโ€ฒ โˆˆ A i \forall{a_{-i}\in A_{-i}},\ \forall{a^{\prime}_{i},\ a^{\prime\prime}_{i}\in A% _{i}}
  8. ฮฆ ( a i โ€ฒ , a - i ) - ฮฆ ( a i โ€ฒโ€ฒ , a - i ) = u i ( a i โ€ฒ , a - i ) - u i ( a i โ€ฒโ€ฒ , a - i ) \Phi(a^{\prime}_{i},a_{-i})-\Phi(a^{\prime\prime}_{i},a_{-i})=u_{i}(a^{\prime}% _{i},a_{-i})-u_{i}(a^{\prime\prime}_{i},a_{-i})
  9. i i
  10. a โ€ฒ a^{\prime}
  11. a โ€ฒโ€ฒ a^{\prime\prime}
  12. ฮฆ : A โ†’ \reals \Phi:A\rightarrow\reals
  13. w โˆˆ \reals + + N w\in\reals_{++}^{N}
  14. โˆ€ a - i โˆˆ A - i , โˆ€ a i โ€ฒ , a i โ€ฒโ€ฒ โˆˆ A i \forall{a_{-i}\in A_{-i}},\ \forall{a^{\prime}_{i},\ a^{\prime\prime}_{i}\in A% _{i}}
  15. ฮฆ ( a i โ€ฒ , a - i ) - ฮฆ ( a i โ€ฒโ€ฒ , a - i ) = w i ( u i ( a i โ€ฒ , a - i ) - u i ( a i โ€ฒโ€ฒ , a - i ) ) \Phi(a^{\prime}_{i},a_{-i})-\Phi(a^{\prime\prime}_{i},a_{-i})=w_{i}(u_{i}(a^{% \prime}_{i},a_{-i})-u_{i}(a^{\prime\prime}_{i},a_{-i}))
  16. ฮฆ : A โ†’ \reals \Phi:A\rightarrow\reals
  17. โˆ€ a - i โˆˆ A - i , โˆ€ a i โ€ฒ , a i โ€ฒโ€ฒ โˆˆ A i \forall{a_{-i}\in A_{-i}},\ \forall{a^{\prime}_{i},\ a^{\prime\prime}_{i}\in A% _{i}}
  18. u i ( a i โ€ฒ , a - i ) - u i ( a i โ€ฒโ€ฒ , a - i ) > 0 โ‡” ฮฆ ( a i โ€ฒ , a - i ) - ฮฆ ( a i โ€ฒโ€ฒ , a - i ) > 0 u_{i}(a^{\prime}_{i},a_{-i})-u_{i}(a^{\prime\prime}_{i},a_{-i})>0% \Leftrightarrow\Phi(a^{\prime}_{i},a_{-i})-\Phi(a^{\prime\prime}_{i},a_{-i})>0
  19. ฮฆ : A โ†’ \reals \Phi:A\rightarrow\reals
  20. โˆ€ a - i โˆˆ A - i , โˆ€ a i โ€ฒ , a i โ€ฒโ€ฒ โˆˆ A i \forall{a_{-i}\in A_{-i}},\ \forall{a^{\prime}_{i},\ a^{\prime\prime}_{i}\in A% _{i}}
  21. u i ( a i โ€ฒ , a - i ) - u i ( a i โ€ฒโ€ฒ , a - i ) > 0 โ‡’ ฮฆ ( a i โ€ฒ , a - i ) - ฮฆ ( a i โ€ฒโ€ฒ , a - i ) > 0 u_{i}(a^{\prime}_{i},a_{-i})-u_{i}(a^{\prime\prime}_{i},a_{-i})>0\Rightarrow% \Phi(a^{\prime}_{i},a_{-i})-\Phi(a^{\prime\prime}_{i},a_{-i})>0
  22. ฮฆ : A โ†’ \reals \Phi:A\rightarrow\reals
  23. โˆ€ i โˆˆ N , โˆ€ a - i โˆˆ A - i \forall i\in N,\ \forall{a_{-i}\in A_{-i}}
  24. b i ( a - i ) = arg max a i โˆˆ A i ฮฆ ( a i , a - i ) b_{i}(a_{-i})=\arg\max_{a_{i}\in A_{i}}\Phi(a_{i},a_{-i})
  25. b i ( a - i ) b_{i}(a_{-i})
  26. i i
  27. a - i a_{-i}
  28. [ u 1 ( + 1 , - 1 ) + u 1 ( - 1 , + 1 ) ] - [ u 1 ( + 1 , + 1 ) + u 1 ( - 1 , - 1 ) ] = [ u 2 ( + 1 , - 1 ) + u 2 ( - 1 , + 1 ) ] - [ u 2 ( + 1 , + 1 ) + u 2 ( - 1 , - 1 ) ] [u_{1}(+1,-1)+u_{1}(-1,+1)]-[u_{1}(+1,+1)+u_{1}(-1,-1)]=[u_{2}(+1,-1)+u_{2}(-1% ,+1)]-[u_{2}(+1,+1)+u_{2}(-1,-1)]

Power_gain.html

  1. G P = P load P input G_{P}=\frac{P_{\mathrm{load}}}{P_{\mathrm{input}}}
  2. G T = P load P source , max G_{T}=\frac{P_{\mathrm{load}}}{P_{\mathrm{source,max}}}
  3. G T = 4 | y 21 | 2 โ„œ ( Y L ) โ„œ ( Y S ) | ( y 11 + Y S ) ( y 22 + Y L ) - y 12 y 21 | 2 G_{T}=\frac{4|y_{21}|^{2}\Re{(Y_{L})}\Re{(Y_{S})}}{|(y_{11}+Y_{S})(y_{22}+Y_{L% })-y_{12}y_{21}|^{2}}
  4. G T = 4 | k 21 | 2 โ„œ ( M L ) โ„œ ( M S ) | ( k 11 + M S ) ( k 22 + M L ) - k 12 k 21 | 2 G_{T}=\frac{4|k_{21}|^{2}\Re{(M_{L})}\Re{(M_{S})}}{|(k_{11}+M_{S})(k_{22}+M_{L% })-k_{12}k_{21}|^{2}}
  5. G A = P load , max P source , max G_{A}=\frac{P_{\mathrm{load,max}}}{P_{\mathrm{source,max}}}

Power_sum_symmetric_polynomial.html

  1. n n
  2. p k ( x 1 , x 2 , โ€ฆ , x n ) = โˆ‘ i = 1 n x i k . p_{k}(x_{1},x_{2},\dots,x_{n})=\sum_{i=1}^{n}x_{i}^{k}\,.
  3. p 0 ( x 1 , x 2 , โ€ฆ , x n ) = n , p_{0}(x_{1},x_{2},\dots,x_{n})=n,
  4. p 1 ( x 1 , x 2 , โ€ฆ , x n ) = x 1 + x 2 + โ‹ฏ + x n , p_{1}(x_{1},x_{2},\dots,x_{n})=x_{1}+x_{2}+\cdots+x_{n}\,,
  5. p 2 ( x 1 , x 2 , โ€ฆ , x n ) = x 1 2 + x 2 2 + โ‹ฏ + x n 2 , p_{2}(x_{1},x_{2},\dots,x_{n})=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\,,
  6. p 3 ( x 1 , x 2 , โ€ฆ , x n ) = x 1 3 + x 2 3 + โ‹ฏ + x n 3 . p_{3}(x_{1},x_{2},\dots,x_{n})=x_{1}^{3}+x_{2}^{3}+\cdots+x_{n}^{3}\,.
  7. k k
  8. k k
  9. n n
  10. n n
  11. n . n.
  12. p 0 = n p_{0}=n
  13. p 1 = x 1 . p_{1}=x_{1}\,.
  14. p 1 = x 1 + x 2 , p_{1}=x_{1}+x_{2}\,,
  15. p 2 = x 1 2 + x 2 2 . p_{2}=x_{1}^{2}+x_{2}^{2}\,.
  16. p 1 = x 1 + x 2 + x 3 , p_{1}=x_{1}+x_{2}+x_{3}\,,
  17. p 2 = x 1 2 + x 2 2 + x 3 2 , p_{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\,,
  18. p 3 = x 1 3 + x 2 3 + x 3 3 , p_{3}=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}\,,
  19. โ„š [ p 1 , โ€ฆ , p n ] . \mathbb{Q}[p_{1},\ldots,p_{n}].
  20. P ( x 1 , x 2 ) = x 1 2 x 2 + x 1 x 2 2 + x 1 x 2 P(x_{1},x_{2})=x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{1}x_{2}
  21. P ( x 1 , x 2 ) = p 1 3 - p 1 p 2 2 + p 1 2 - p 2 2 , P(x_{1},x_{2})=\frac{p_{1}^{3}-p_{1}p_{2}}{2}+\frac{p_{1}^{2}-p_{2}}{2}\,,
  22. P ( x 1 , x 2 ) P(x_{1},x_{2})
  23. โ„ค [ p 1 , โ€ฆ , p n ] . \mathbb{Z}[p_{1},\ldots,p_{n}].
  24. e 2 := โˆ‘ 1 โ‰ค i < j โ‰ค n x i x j = p 1 2 - p 2 2 . e_{2}:=\sum_{1\leq i<j\leq n}x_{i}x_{j}=\frac{p_{1}^{2}-p_{2}}{2}\,.
  25. p 2 = p 1 2 p_{2}=p_{1}^{2}
  26. p n = โˆ‘ j = 1 n ( - 1 ) j - 1 e j p n - j . p_{n}=\sum_{j=1}^{n}(-1)^{j-1}e_{j}p_{n-j}\,.
  27. e n = 1 n โˆ‘ j = 1 n ( - 1 ) j - 1 e n - j p j . e_{n}=\frac{1}{n}\sum_{j=1}^{n}(-1)^{j-1}e_{n-j}p_{j}\,.
  28. f ( p 1 , โ€ฆ , p n ) f(p_{1},\ldots,p_{n})
  29. โ„š [ p 1 , โ€ฆ , p n ] . \mathbb{Q}[p_{1},\ldots,p_{n}].

Power_supply_rejection_ratio.html

  1. P S R R ( d B ) = 20 log 10 ( ฮ” V supply ฮ” V out โ‹… A v ) dB PSRR(dB)=20\log_{10}\left({\Delta V_{\mathrm{supply}}\over{\Delta V_{\mathrm{% out}}}}\cdot A_{v}\right)\mbox{dB}~{}
  2. 100 - 40 = 60 dB 100-40=60\ \mathrm{dB}
  3. 1 V โ‹… 10 - 60 20 = .001 V = 1 mV 1\ \mathrm{V}\cdot 10^{\frac{-60}{20}}=.001\ \mathrm{V}=1\ \mathrm{mV}

Power_transform.html

  1. y i ( ฮป ) = { y i ฮป - 1 ฮป ( GM ( y ) ) ฮป - 1 , if ฮป โ‰  0 GM ( y ) ln y i , if ฮป = 0 y_{i}^{(\lambda)}=\begin{cases}\dfrac{y_{i}^{\lambda}-1}{\lambda(\operatorname% {GM}(y))^{\lambda-1}},&\,\text{if }\lambda\neq 0\\ \operatorname{GM}(y)\ln{y_{i}},&\,\text{if }\lambda=0\end{cases}
  2. GM ( y ) = ( y 1 โ‹ฏ y n ) 1 / n \operatorname{GM}(y)=(y_{1}\cdots y_{n})^{1/n}\,
  3. y i ( ฮป ) y_{i}^{(\lambda)}
  4. y ฮป - 1 ฮป \dfrac{y^{\lambda}-1}{\lambda}
  5. J ( ฮป ; y 1 , โ€ฆ , y n ) = โˆ i = 1 n | d y i ( ฮป ) / d y | = โˆ i = 1 n y i ฮป - 1 = GM ( y ) n ( ฮป - 1 ) J(\lambda;y_{1},...,y_{n})=\prod_{i=1}^{n}|dy_{i}^{(\lambda)}/dy|=\prod_{i=1}^% {n}y_{i}^{\lambda-1}=\operatorname{GM}(y)^{n(\lambda-1)}
  6. log ( โ„’ ( ฮผ ^ , ฯƒ ^ ) ) = ( - n / 2 ) ( log ( 2 ฯ€ ฯƒ ^ 2 ) + 1 ) + n ( ฮป - 1 ) log ( GM ( y ) ) = ( - n / 2 ) ( log ( 2 ฯ€ ฯƒ ^ 2 / GM ( y ) 2 ( ฮป - 1 ) ) + 1 ) . \log(\mathcal{L}(\hat{\mu},\hat{\sigma}))=(-n/2)(\log(2\pi\hat{\sigma}^{2})+1)% +n(\lambda-1)\log(\operatorname{GM}(y))=(-n/2)(\log(2\pi\hat{\sigma}^{2}/% \operatorname{GM}(y)^{2(\lambda-1)})+1).
  7. GM ( y ) 2 ( ฮป - 1 ) \operatorname{GM}(y)^{2(\lambda-1)}
  8. ฯƒ ^ 2 \hat{\sigma}^{2}
  9. y i ( ฮป ) y_{i}^{(\lambda)}
  10. ( y ฮป - 1 ) / ฮป (y^{\lambda}-1)/\lambda
  11. ฯ„ ( y i ; ฮป , ฮฑ ) = { ( y i + ฮฑ ) ฮป - 1 ฮป ( GM ( y + ฮฑ ) ) ฮป - 1 if ฮป โ‰  0 , GM ( y + ฮฑ ) ln ( y i + ฮฑ ) if ฮป = 0 , \tau(y_{i};\lambda,\alpha)=\begin{cases}\dfrac{(y_{i}+\alpha)^{\lambda}-1}{% \lambda(\operatorname{GM}(y+\alpha))^{\lambda-1}}&\,\text{if }\lambda\neq 0,\\ \\ \operatorname{GM}(y+\alpha)\ln(y_{i}+\alpha)&\,\text{if }\lambda=0,\end{cases}
  12. ฯ„ ( y i ; ฮป , ฮฑ ) = { sgn ( y i + ฮฑ ) | y i + ฮฑ | ฮป - 1 ฮป ( GM ( y + ฮฑ ) ) ฮป - 1 if ฮป โ‰  0 , GM ( y + ฮฑ ) sgn ( y + ฮฑ ) ln ( y i + ฮฑ ) if ฮป = 0 , \tau(y_{i};\lambda,\alpha)=\begin{cases}\dfrac{\operatorname{sgn}(y_{i}+\alpha% )|y_{i}+\alpha|^{\lambda}-1}{\lambda(\operatorname{GM}(y+\alpha))^{\lambda-1}}% &\,\text{if }\lambda\neq 0,\\ \\ \operatorname{GM}(y+\alpha)\operatorname{sgn}(y+\alpha)\ln(y_{i}+\alpha)&\,% \text{if }\lambda=0,\end{cases}
  13. ฮฑ \alpha
  14. min ( y i ) \operatorname{min}(y_{i})
  15. y i ( ฮป ) = { y i ฮป - 1 ฮป if ฮป โ‰  0 , ln ( y i ) if ฮป = 0 , y_{i}^{(\lambda)}=\begin{cases}\dfrac{y_{i}^{\lambda}-1}{\lambda}&\,\text{if }% \lambda\neq 0,\\ \ln{(y_{i})}&\,\text{if }\lambda=0,\end{cases}
  16. y i ( s y m b o l ฮป ) = { ( y i + ฮป 2 ) ฮป 1 - 1 ฮป 1 if ฮป 1 โ‰  0 , ln ( y i + ฮป 2 ) if ฮป 1 = 0 , y_{i}^{(symbol{\lambda})}=\begin{cases}\dfrac{(y_{i}+\lambda_{2})^{\lambda_{1}% }-1}{\lambda_{1}}&\,\text{if }\lambda_{1}\neq 0,\\ \ln{(y_{i}+\lambda_{2})}&\,\text{if }\lambda_{1}=0,\end{cases}
  17. y i > 0 y_{i}>0
  18. y i > - ฮป 2 y_{i}>-\lambda_{2}
  19. ฮป \lambda
  20. ฯ„ ( Q ) = ฮฑ ฯ„ ( K ) + ( 1 - ฮฑ ) ฯ„ ( N ) . \tau(Q)=\alpha\tau(K)+(1-\alpha)\tau(N).\,
  21. Q = ( ฮฑ K ฮป + ( 1 - ฮฑ ) N ฮป ) 1 / ฮป , Q=\big(\alpha K^{\lambda}+(1-\alpha)N^{\lambda}\big)^{1/\lambda},\,
  22. Q = ฮฑ K + ( 1 - ฮฑ ) N . Q=\alpha K+(1-\alpha)N.\,
  23. Q = K ฮฑ N 1 - ฮฑ . Q=K^{\alpha}N^{1-\alpha}.\,

Poynting_effect.html

  1. ln p v p v , o = v l i q R T ( P - p v , o ) \ln\frac{p_{v}}{p_{v,o}}=\frac{v_{liq}}{RT}(P-p_{v,o})\!

PPAD_(complexity).html

  1. โ‰  \neq

Prandtlโ€“Meyer_function.html

  1. ฮฝ \nu
  2. M M
  3. ฮณ \gamma
  4. ฮฝ max \nu\text{max}
  5. ฮฝ ( M ) \displaystyle\nu(M)
  6. ฮฝ \nu\,
  7. M M
  8. ฮณ \gamma
  9. ฮฝ ( 1 ) = 0. \nu(1)=0.\,
  10. โˆž \infty
  11. ฮฝ \nu\,
  12. ฮฝ max \nu\text{max}\,
  13. ฮฝ max = ฯ€ 2 ( ฮณ + 1 ฮณ - 1 - 1 ) \nu\text{max}=\frac{\pi}{2}\bigg(\sqrt{\frac{\gamma+1}{\gamma-1}}-1\bigg)
  14. ฮฝ ( M 2 ) = ฮฝ ( M 1 ) + ฮธ \nu(M_{2})=\nu(M_{1})+\theta\,
  15. ฮฝ ( M 2 ) = ฮฝ ( M 1 ) - ฮธ \nu(M_{2})=\nu(M_{1})-\theta\,
  16. ฮธ \theta
  17. M M

Precedence_graph.html

  1. D = [ T 1 T 2 T 3 R ( A ) W ( A ) W ( A ) W ( A ) ] D=\begin{bmatrix}T1&T2&T3\\ R(A)&&\\ &W(A)&\\ &\\ W(A)&&\\ &&\\ &&W(A)\\ &\\ \end{bmatrix}
  2. D = R 1 ( A ) D=R1(A)
  3. W 2 ( A ) W2(A)
  4. C o m .2 Com.2
  5. W 1 ( A ) W1(A)
  6. C o m .1 Com.1
  7. W 3 ( A ) W3(A)
  8. C o m .3 Com.3

Precision_tests_of_QED.html

  1. R โˆž = ฮฑ 2 m e c 4 ฯ€ โ„ . R_{\infty}=\frac{\alpha^{2}m_{e}c}{4\pi\hbar}.
  2. e + e - โ†’ e + e - e + e - e^{+}e^{-}\to e^{+}e^{-}e^{+}e^{-}
  3. e + e - โ†’ e + e - ฮผ + ฮผ - e^{+}e^{-}\to e^{+}e^{-}\mu^{+}\mu^{-}

Precoding.html

  1. s s
  2. h h
  3. n n
  4. r = s h + n r=sh+n
  5. h h
  6. n n
  7. n n
  8. h h
  9. h h
  10. h est h_{\,\text{est}}
  11. s h est {s\over h_{\,\text{est}}}
  12. r = ( h h est ) s + n r=\left(\frac{h}{h_{\,\text{est}}}\right)s+n
  13. h est = h h_{\,\text{est}}=h
  14. r = s + n r=s+n
  15. N N
  16. K K
  17. k k
  18. N ร— 1 N\times 1
  19. ๐ก k \mathbf{h}_{k}
  20. i i
  21. i i
  22. y k = ๐ก k H ๐ฑ + n k , k = 1 , 2 , โ€ฆ , K y_{k}=\mathbf{h}_{k}^{H}\mathbf{x}+n_{k},\quad k=1,2,\ldots,K
  23. ๐ฑ \mathbf{x}
  24. N ร— 1 N\times 1
  25. y k y_{k}
  26. n k n_{k}
  27. ๐ฑ = โˆ‘ i = 1 K ๐ฐ i s i , \mathbf{x}=\sum_{i=1}^{K}\mathbf{w}_{i}s_{i},
  28. s i s_{i}
  29. ๐ฐ i \mathbf{w}_{i}
  30. N ร— 1 N\times 1
  31. k k
  32. SINR k = | ๐ก k H ๐ฐ k | 2 1 + โˆ‘ i โ‰  k | ๐ก k H ๐ฐ i | 2 \textrm{SINR}_{k}=\frac{|\mathbf{h}_{k}^{H}\mathbf{w}_{k}|^{2}}{1+\sum_{i\neq k% }|\mathbf{h}_{k}^{H}\mathbf{w}_{i}|^{2}}
  33. log 2 ( 1 + SINR k ) \log_{2}(1+\textrm{SINR}_{k})
  34. โˆ‘ i = 1 K โˆฅ ๐ฐ i โˆฅ 2 โ‰ค P \sum_{i=1}^{K}\|\mathbf{w}_{i}\|^{2}\leq P
  35. P P
  36. maximize { ๐ฐ k } : โˆ‘ i โˆฅ ๐ฐ i โˆฅ 2 โ‰ค P โˆ‘ k = 1 K a k log 2 ( 1 + SINR k ) \underset{\{\mathbf{w}_{k}\}:\sum_{i}\|\mathbf{w}_{i}\|^{2}\leq P}{\mathrm{% maximize}}\sum_{k=1}^{K}a_{k}\log_{2}(1+\textrm{SINR}_{k})
  37. a k a_{k}
  38. ๐ฐ k W-MMSE = p k ( ๐ˆ + โˆ‘ i โ‰  k q i ๐ก i ๐ก i H ) - 1 ๐ก k โˆฅ ( ๐ˆ + โˆ‘ i โ‰  k q i ๐ก i ๐ก i H ) - 1 ๐ก k โˆฅ \mathbf{w}^{\textrm{W-MMSE}}_{k}=\sqrt{p_{k}}\frac{(\mathbf{I}+\sum_{i\neq k}q% _{i}\mathbf{h}_{i}\mathbf{h}_{i}^{H})^{-1}\mathbf{h}_{k}}{\|(\mathbf{I}+\sum_{% i\neq k}q_{i}\mathbf{h}_{i}\mathbf{h}_{i}^{H})^{-1}\mathbf{h}_{k}\|}
  39. q 1 , โ€ฆ , q K q_{1},\ldots,q_{K}
  40. โˆ‘ i = 1 K q i = P \sum_{i=1}^{K}q_{i}=P
  41. p i p_{i}
  42. ๐ฐ k MRT = p k ๐ก k โˆฅ ๐ก k โˆฅ , \mathbf{w}^{\mathrm{MRT}}_{k}=\sqrt{p_{k}}\frac{\mathbf{h}_{k}}{\|\mathbf{h}_{% k}\|},
  43. ๐ก i H ๐ฐ k ZF = 0 \mathbf{h}_{i}^{H}\mathbf{w}^{\mathrm{ZF}}_{k}=0
  44. SINR k ZF = | ๐ก k H ๐ฐ k ZF | 2 . \textrm{SINR}^{\mathrm{ZF}}_{k}=|\mathbf{h}_{k}^{H}\mathbf{w}^{\mathrm{ZF}}_{k% }|^{2}.
  45. N N
  46. ๐ฒ = โˆ‘ k = 1 K ๐ก k q k s k + ๐ง \mathbf{y}=\sum_{k=1}^{K}\mathbf{h}_{k}\sqrt{q_{k}}s_{k}+\mathbf{n}
  47. s k s_{k}
  48. k k
  49. q k q_{k}
  50. ๐ฒ \mathbf{y}
  51. ๐ง \mathbf{n}
  52. N ร— 1 N\times 1
  53. ๐ก k \mathbf{h}_{k}
  54. N ร— 1 N\times 1
  55. N N
  56. k k
  57. SINR k uplink = q k | ๐ก k H ๐ฏ k | 2 1 + โˆ‘ i โ‰  k q i | ๐ก i H ๐ฏ k | 2 \textrm{SINR}^{\mathrm{uplink}}_{k}=\frac{q_{k}|\mathbf{h}_{k}^{H}\mathbf{v}_{% k}|^{2}}{1+\sum_{i\neq k}q_{i}|\mathbf{h}_{i}^{H}\mathbf{v}_{k}|^{2}}
  58. ๐ฏ k \mathbf{v}_{k}
  59. ๐ฏ k MMSE = ( ๐ˆ + โˆ‘ i โ‰  k q i ๐ก i ๐ก i H ) - 1 ๐ก k โˆฅ ( ๐ˆ + โˆ‘ i โ‰  k q i ๐ก i ๐ก i H ) - 1 ๐ก k โˆฅ \mathbf{v}^{\textrm{MMSE}}_{k}=\frac{(\mathbf{I}+\sum_{i\neq k}q_{i}\mathbf{h}% _{i}\mathbf{h}_{i}^{H})^{-1}\mathbf{h}_{k}}{\|(\mathbf{I}+\sum_{i\neq k}q_{i}% \mathbf{h}_{i}\mathbf{h}_{i}^{H})^{-1}\mathbf{h}_{k}\|}
  60. q 1 , โ€ฆ , q K q_{1},\ldots,q_{K}
  61. y k = ๐ก k H โˆ‘ i = 1 K ๐ฐ ^ i s i + n k , k = 1 , 2 , โ€ฆ , K . y_{k}=\mathbf{h}_{k}^{H}\sum_{i=1}^{K}\hat{\mathbf{w}}_{i}s_{i}+n_{k},\quad k=% 1,2,\ldots,K.
  62. ๐ฐ ^ i = ๐ฐ i + ๐ž i \hat{\mathbf{w}}_{i}=\mathbf{w}_{i}+\mathbf{e}_{i}
  63. ๐ฐ i \mathbf{w}_{i}
  64. ๐ž i \mathbf{e}_{i}
  65. y k = ๐ก k H โˆ‘ i = 1 K ๐ฐ i s i + ๐ก k H โˆ‘ i = 1 K ๐ž i s i + n k , k = 1 , 2 , โ€ฆ , K y_{k}=\mathbf{h}_{k}^{H}\sum_{i=1}^{K}\mathbf{w}_{i}s_{i}+\mathbf{h}_{k}^{H}% \sum_{i=1}^{K}\mathbf{e}_{i}s_{i}+n_{k},\quad k=1,2,\ldots,K
  66. ๐ก k H โˆ‘ i โ‰  k ๐ž i s i \mathbf{h}_{k}^{H}\sum_{i\neq k}\mathbf{e}_{i}s_{i}
  67. k k

Preimage_theorem.html

  1. f : X โ†’ Y f:X\to Y\,\!
  2. y โˆˆ Y y\in Y
  3. x โˆˆ f - 1 ( y ) x\in f^{-1}(y)
  4. d f x : T x X โ†’ T y Y df_{x}:T_{x}X\to T_{y}Y\,\!
  5. T x X T_{x}X\,\!
  6. T y Y T_{y}Y\,\!
  7. f : X โ†’ Y f:X\to Y\,\!
  8. y โˆˆ Y y\in Y
  9. f - 1 ( y ) f^{-1}(y)
  10. y โˆˆ im ( f ) y\in\,\text{im}(f)
  11. f - 1 ( y ) f^{-1}(y)
  12. f - 1 ( y ) f^{-1}(y)
  13. x x
  14. ker ( d f x ) \ker(df_{x})

Pressure-correction_method.html

  1. ฯ ( โˆ‚ ๐ฏ โˆ‚ t โŸ Unsteady acceleration + ( ๐ฏ โ‹… โˆ‡ ) ๐ฏ โŸ Convective acceleration ) โž Inertia = - โˆ‡ p โŸ Pressure gradient + ฮผ โˆ‡ 2 ๐ฏ โŸ Viscosity + ๐Ÿ โŸ Other forces \overbrace{\rho\Big(\underbrace{\frac{\partial\mathbf{v}}{\partial t}}_{\begin% {smallmatrix}\,\text{Unsteady}\\ \,\text{acceleration}\end{smallmatrix}}+\underbrace{\left(\mathbf{v}\cdot% \nabla\right)\mathbf{v}}_{\begin{smallmatrix}\,\text{Convective}\\ \,\text{acceleration}\end{smallmatrix}}\Big)}^{\,\text{Inertia}}=\underbrace{-% \nabla p}_{\begin{smallmatrix}\,\text{Pressure}\\ \,\text{gradient}\end{smallmatrix}}+\underbrace{\mu\nabla^{2}\mathbf{v}}_{\,% \text{Viscosity}}+\underbrace{\mathbf{f}}_{\begin{smallmatrix}\,\text{Other}\\ \,\text{forces}\end{smallmatrix}}
  2. ๐ฏ \mathbf{v}
  3. p p
  4. div ๐ฏ = 0 \,\text{div}\,\mathbf{v}=0
  5. m ห™ \dot{m}
  6. โˆ‡ โ‹… ( ฯ ( ๐ฑ ) ๐ฏ ( ๐ฑ ) ) = d d t p ( ๐ฑ ) c 2 \nabla\cdot\left(\rho(\mathbf{x})\mathbf{v}(\mathbf{x})\right)=\frac{\frac{d}{% dt}p(\mathbf{x})}{c^{2}}
  7. c 2 c^{2}
  8. c c
  9. div ๐ฏ \displaystyle\,\text{div}\,\mathbf{v}
  10. โˆ‡ โ‹… โˆ‚ t ๐ฏ \displaystyle\nabla\cdot\partial_{t}\mathbf{v}
  11. ( โˆ— ) (\ast)
  12. โˆ‚ t ฯ \displaystyle\partial_{t}\rho
  13. p p
  14. ฯ = constant . \rho=\,\text{constant}.

Prevalent_and_shy_sets.html

  1. โˆซ 0 1 f ( x ) d x โ‰  0. \int_{0}^{1}f(x)\,\mathrm{d}x\neq 0.
  2. โˆ‘ n โˆˆ โ„• a n \sum_{n\in\mathbb{N}}a_{n}

Preฬkopaโ€“Leindler_inequality.html

  1. h ( ( 1 - ฮป ) x + ฮป y ) โ‰ฅ f ( x ) 1 - ฮป g ( y ) ฮป h\left((1-\lambda)x+\lambda y\right)\geq f(x)^{1-\lambda}g(y)^{\lambda}
  2. โˆฅ h โˆฅ 1 := โˆซ โ„ n h ( x ) d x โ‰ฅ ( โˆซ โ„ n f ( x ) d x ) 1 - ฮป ( โˆซ โ„ n g ( x ) d x ) ฮป = : โˆฅ f โˆฅ 1 1 - ฮป โˆฅ g โˆฅ 1 ฮป . \|h\|_{1}:=\int_{\mathbb{R}^{n}}h(x)\,\mathrm{d}x\geq\left(\int_{\mathbb{R}^{n% }}f(x)\,\mathrm{d}x\right)^{1-\lambda}\left(\int_{\mathbb{R}^{n}}g(x)\,\mathrm% {d}x\right)^{\lambda}=:\|f\|_{1}^{1-\lambda}\|g\|_{1}^{\lambda}.\,
  3. ess sup x โˆˆ โ„ n f ( x ) = inf { t โˆˆ [ - โˆž , + โˆž ] | f ( x ) โ‰ค t for almost all x โˆˆ โ„ n } . \mathop{\mathrm{ess\,sup}}_{x\in\mathbb{R}^{n}}f(x)=\inf\left\{t\in[-\infty,+% \infty]|f(x)\leq t\mbox{ for almost all }~{}x\in\mathbb{R}^{n}\right\}.
  4. s ( x ) = ess sup y โˆˆ โ„ n f ( x - y 1 - ฮป ) 1 - ฮป g ( y ฮป ) ฮป . s(x)=\mathop{\mathrm{ess\,sup}}_{y\in\mathbb{R}^{n}}f\left(\frac{x-y}{1-% \lambda}\right)^{1-\lambda}g\left(\frac{y}{\lambda}\right)^{\lambda}.
  5. โˆฅ s โˆฅ 1 โ‰ฅ โˆฅ f โˆฅ 1 1 - ฮป โˆฅ g โˆฅ 1 ฮป . \|s\|_{1}\geq\|f\|_{1}^{1-\lambda}\|g\|_{1}^{\lambda}.
  6. ฮผ ( ( 1 - ฮป ) A + ฮป B ) โ‰ฅ ฮผ ( A ) 1 - ฮป ฮผ ( B ) ฮป , \mu\left((1-\lambda)A+\lambda B\right)\geq\mu(A)^{1-\lambda}\mu(B)^{\lambda},
  7. ฮผ ( ( 1 - ฮป ) A + ฮป B ) 1 / n โ‰ฅ ( 1 - ฮป ) ฮผ ( A ) 1 / n + ฮป ฮผ ( B ) 1 / n . \mu\left((1-\lambda)A+\lambda B\right)^{1/n}\geq(1-\lambda)\mu(A)^{1/n}+% \lambda\mu(B)^{1/n}.
  8. H ( ( 1 - ฮป ) ( x 1 , y 1 ) + ฮป ( x 2 , y 2 ) ) โ‰ฅ H ( x 1 , y 1 ) 1 - ฮป H ( x 2 , y 2 ) ฮป , H\left((1-\lambda)(x_{1},y_{1})+\lambda(x_{2},y_{2})\right)\geq H(x_{1},y_{1})% ^{1-\lambda}H(x_{2},y_{2})^{\lambda},
  9. M ( y ) = โˆซ โ„ m H ( x , y ) d x . M(y)=\int_{\mathbb{R}^{m}}H(x,y)dx.
  10. M ( ( 1 - ฮป ) y 1 + ฮป y 2 ) โ‰ฅ M ( y 1 ) 1 - ฮป M ( y 2 ) ฮป , M((1-\lambda)y_{1}+\lambda y_{2})\geq M(y_{1})^{1-\lambda}M(y_{2})^{\lambda},

Prime_zeta_function.html

  1. โ„œ ( s ) > 1 \Re(s)>1
  2. P ( s ) = โˆ‘ p โˆˆ primes 1 p s P(s)=\sum_{p\,\in\mathrm{\,primes}}\frac{1}{p^{s}}
  3. log ฮถ ( s ) = โˆ‘ n > 0 P ( n s ) n \log\zeta(s)=\sum_{n>0}\frac{P(ns)}{n}
  4. P ( s ) = โˆ‘ n > 0 ฮผ ( n ) log ฮถ ( n s ) n P(s)=\sum_{n>0}\mu(n)\frac{\log\zeta(ns)}{n}
  5. P ( s ) โˆผ log ฮถ ( s ) โˆผ log ( 1 s - 1 ) P(s)\sim\log\zeta(s)\sim\log\left(\frac{1}{s-1}\right)
  6. โ„œ ( s ) > 0 \Re(s)>0
  7. โ„œ ( s ) = 0 \Re(s)=0
  8. a n = โˆ p k โˆฃ n 1 k = โˆ p k โˆฃ โˆฃ n 1 k ! a_{n}=\prod_{p^{k}\mid n}\frac{1}{k}=\prod_{p^{k}\mid\mid n}\frac{1}{k!}
  9. P ( s ) = log โˆ‘ n = 1 โˆž a n n s . P(s)=\log\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.
  10. ln C Artin = - โˆ‘ n = 2 โˆž ( L n - 1 ) P ( n ) n \ln C_{\mathrm{Artin}}=-\sum_{n=2}^{\infty}\frac{(L_{n}-1)P(n)}{n}
  11. 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + โ‹ฏ โ†’ โˆž . \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{5}+\tfrac{1}{7}+\tfrac{1}{11}+\cdots\to\infty.
  12. 0.45224 74200 41065 49850 โ€ฆ 0{.}45224\,\text{ }74200\,\text{ }41065\,\text{ }49850\ldots
  13. 0.17476 26392 99443 53642 โ€ฆ 0{.}17476\,\text{ }26392\,\text{ }99443\,\text{ }53642\ldots
  14. 0.07699 31397 64246 84494 โ€ฆ 0{.}07699\,\text{ }31397\,\text{ }64246\,\text{ }84494\ldots
  15. 0.03575 50174 83924 25713 โ€ฆ 0{.}03575\,\text{ }50174\,\text{ }83924\,\text{ }25713\ldots
  16. 0.00200 44675 74962 45066 โ€ฆ 0{.}00200\,\text{ }44675\,\text{ }74962\,\text{ }45066\ldots
  17. s = 1 s=1
  18. โˆซ s โˆž P ( t ) d t = โˆ‘ p 1 p s log p \int_{s}^{\infty}P(t)dt=\sum_{p}\frac{1}{p^{s}\log p}
  19. โˆ‘ p 1 / ( p s log p ) \sum_{p}1/(p^{s}\log p)
  20. 1.63661632 โ€ฆ 1.63661632\ldots
  21. 0.50778218 โ€ฆ 0.50778218\ldots
  22. 0.22120334 โ€ฆ 0.22120334\ldots
  23. 0.10266547 โ€ฆ 0.10266547\ldots
  24. P โ€ฒ ( s ) โ‰ก d d s P ( s ) = - โˆ‘ p log p p s P^{\prime}(s)\equiv\frac{d}{ds}P(s)=-\sum_{p}\frac{\log p}{p^{s}}
  25. P โ€ฒ ( s ) P^{\prime}(s)
  26. - 0.493091109 โ€ฆ -0.493091109\ldots
  27. - 0.150757555 โ€ฆ -0.150757555\ldots
  28. - 0.060607633 โ€ฆ -0.060607633\ldots
  29. - 0.026838601 โ€ฆ -0.026838601\ldots
  30. k k
  31. P k ( s ) โ‰ก โˆ‘ n : ฮฉ ( n ) = k 1 n s P_{k}(s)\equiv\sum_{n:\Omega(n)=k}\frac{1}{n^{s}}
  32. ฮฉ \Omega
  33. P k ( s ) P_{k}(s)
  34. 0.14076043434 โ€ฆ 0.14076043434\ldots
  35. 0.02380603347 โ€ฆ 0.02380603347\ldots
  36. 0.03851619298 โ€ฆ 0.03851619298\ldots
  37. 0.00304936208 โ€ฆ 0.00304936208\ldots
  38. ฮถ \zeta
  39. k k
  40. P k P_{k}
  41. ฮถ ( s ) = 1 + โˆ‘ k = 1 , 2 , โ€ฆ P k ( s ) \zeta(s)=1+\sum_{k=1,2,\ldots}P_{k}(s)

Primes_in_arithmetic_progression.html

  1. a n = 3 + 4 * n a_{n}=3+4*n
  2. 0 โ‰ค n โ‰ค 2 0\leq n\leq 2
  3. a * n + b a*n+b

Principal_axis_theorem.html

  1. x 2 9 + y 2 25 = 1 \frac{x^{2}}{9}+\frac{y^{2}}{25}=1
  2. x 2 9 - y 2 25 = 1 {}\frac{x^{2}}{9}-\frac{y^{2}}{25}=1
  3. 5 x 2 + 8 x y + 5 y 2 = 1. 5x^{2}+8xy+5y^{2}=1.
  4. u ( x , y ) 2 + v ( x , y ) 2 = 1 (ellipse) u(x,y)^{2}+v(x,y)^{2}=1\qquad\,\text{(ellipse)}
  5. u ( x , y ) 2 - v ( x , y ) 2 = 1 (hyperbola) . u(x,y)^{2}-v(x,y)^{2}=1\qquad\,\text{(hyperbola)}.
  6. 5 x 2 + 8 x y + 5 y 2 = [ x y ] [ 5 4 4 5 ] [ x y ] = ๐ฑ T A ๐ฑ 5x^{2}+8xy+5y^{2}=\begin{bmatrix}x&y\end{bmatrix}\begin{bmatrix}5&4\\ 4&5\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}=\mathbf{x}^{T}A\mathbf{x}
  7. ฮป 1 = 1 , ฮป 2 = 9 \lambda_{1}=1,\quad\lambda_{2}=9
  8. ๐ฏ 1 = [ 1 - 1 ] , ๐ฏ 2 = [ 1 1 ] . \mathbf{v}_{1}=\begin{bmatrix}1\\ -1\end{bmatrix},\quad\mathbf{v}_{2}=\begin{bmatrix}1\\ 1\end{bmatrix}.
  9. ๐ฎ 1 = [ 1 / 2 - 1 / 2 ] , ๐ฎ 2 = [ 1 / 2 1 / 2 ] . \mathbf{u}_{1}=\begin{bmatrix}1/\sqrt{2}\\ -1/\sqrt{2}\end{bmatrix},\quad\mathbf{u}_{2}=\begin{bmatrix}1/\sqrt{2}\\ 1/\sqrt{2}\end{bmatrix}.
  10. A = S D S - 1 = S D S T = [ 1 / 2 1 / 2 - 1 / 2 1 / 2 ] [ 1 0 0 9 ] [ 1 / 2 - 1 / 2 1 / 2 1 / 2 ] . A=SDS^{-1}=SDS^{T}=\begin{bmatrix}1/\sqrt{2}&1/\sqrt{2}\\ -1/\sqrt{2}&1/\sqrt{2}\end{bmatrix}\begin{bmatrix}1&0\\ 0&9\end{bmatrix}\begin{bmatrix}1/\sqrt{2}&-1/\sqrt{2}\\ 1/\sqrt{2}&1/\sqrt{2}\end{bmatrix}.
  11. 5 x 2 + 8 x y + 5 y 2 = ๐ฑ T A ๐ฑ = ( S T ๐ฑ ) T D ( S T ๐ฑ ) = 1 ( x - y 2 ) 2 + 9 ( x + y 2 ) 2 . 5x^{2}+8xy+5y^{2}=\mathbf{x}^{T}A\mathbf{x}=(S^{T}\mathbf{x})^{T}D(S^{T}% \mathbf{x})=1\left(\frac{x-y}{\sqrt{2}}\right)^{2}+9\left(\frac{x+y}{\sqrt{2}}% \right)^{2}.
  12. c 1 = x - y 2 , c 2 = x + y 2 c_{1}=\frac{x-y}{\sqrt{2}},\quad c_{2}=\frac{x+y}{\sqrt{2}}
  13. E 1 = span ( [ 1 / 2 - 1 / 2 ] ) , E 2 = span ( [ 1 / 2 1 / 2 ] ) . E_{1}=\,\text{span}\left(\begin{bmatrix}1/\sqrt{2}\\ -1/\sqrt{2}\end{bmatrix}\right),\quad E_{2}=\,\text{span}\left(\begin{bmatrix}% 1/\sqrt{2}\\ 1/\sqrt{2}\end{bmatrix}\right).
  14. Q ( ๐ฑ ) = ๐ฑ T A ๐ฑ Q(\mathbf{x})=\mathbf{x}^{T}A\mathbf{x}
  15. Q ( ๐ฑ ) = ฮป 1 c 1 2 + ฮป 2 c 2 2 + โ€ฆ + ฮป n c n 2 , Q(\mathbf{x})=\lambda_{1}c_{1}^{2}+\lambda_{2}c_{2}^{2}+\dots+\lambda_{n}c_{n}% ^{2},

Probabilistic_automaton.html

  1. ( Q , ฮฃ , ฮด , q 0 , F ) (Q,\Sigma,\delta,q_{0},F)
  2. P P
  3. q 0 q_{0}
  4. Q Q
  5. ฮฃ \Sigma
  6. ฮด : Q ร— ฮฃ โ†’ P ( Q ) \delta:Q\times\Sigma\to P(Q)
  7. F F
  8. F โŠ‚ Q F\subset Q
  9. P ( Q ) P(Q)
  10. Q Q
  11. ฮด : Q ร— ฮฃ โ†’ P ( Q ) \delta:Q\times\Sigma\to P(Q)
  12. ฮด : Q ร— ฮฃ ร— Q โ†’ { 0 , 1 } \delta:Q\times\Sigma\times Q\to\{0,1\}
  13. ฮด ( q , a , q โ€ฒ ) = 1 \delta(q,a,q^{\prime})=1
  14. q โ€ฒ โˆˆ ฮด ( q , a ) q^{\prime}\in\delta(q,a)
  15. ฮด ( q , a , q โ€ฒ ) = 0 \delta(q,a,q^{\prime})=0
  16. q โ€ฒ โˆ‰ ฮด ( q , a ) q^{\prime}\notin\delta(q,a)
  17. [ ฮธ a ] q q โ€ฒ = ฮด ( q , a , q โ€ฒ ) \left[\theta_{a}\right]_{qq^{\prime}}=\delta(q,a,q^{\prime})
  18. ฮธ a \theta_{a}
  19. q โ†’ a q โ€ฒ q\stackrel{a}{\rightarrow}q^{\prime}
  20. P P
  21. [ P a ] q q โ€ฒ \left[P_{a}\right]_{qq^{\prime}}
  22. โˆ‘ q โ€ฒ [ P a ] q q โ€ฒ = 1 \sum_{q^{\prime}}\left[P_{a}\right]_{qq^{\prime}}=1
  23. a a
  24. q q
  25. v v
  26. โˆ‘ q [ v ] q = 1 \sum_{q}\left[v\right]_{q}=1
  27. a b c abc
  28. v P a P b P c vP_{a}P_{b}P_{c}
  29. ( Q , ฮฃ , P , v , F ) (Q,\Sigma,P,v,F)
  30. v v
  31. F = Q accept โŠ‚ Q F=Q\text{accept}\subset Q
  32. Q accept Q\text{accept}
  33. Q accept Q\text{accept}
  34. Q accept Q\text{accept}
  35. L ฮท = { s โˆˆ ฮฃ * | v P s Q accept > ฮท } L_{\eta}=\{s\in\Sigma^{*}|vP_{s}Q\text{accept}>\eta\}
  36. ฮฃ * \Sigma^{*}
  37. ฮฃ \Sigma
  38. ฮท \eta
  39. 0 โ‰ค ฮท < 1 0\leq\eta<1
  40. ฮท \eta
  41. 0 โ‰ค ฮท < 1 0\leq\eta<1
  42. L ฮท L_{\eta}
  43. ฮด > 0 \delta>0
  44. | v P ( s ) Q accept - ฮท | โ‰ฅ ฮด |vP(s)Q\text{accept}-\eta|\geq\delta
  45. s โˆˆ ฮฃ * s\in\Sigma^{*}
  46. 0 < ฮท < 1 0<\eta<1
  47. ฮท \eta
  48. L ฮท L_{\eta}
  49. 0 , 1 , 2 , โ€ฆ , ( p - 1 ) 0,1,2,\ldots,(p-1)
  50. L ฮท ( p ) = { 0. n 1 n 2 n 3 โ€ฆ | 0 โ‰ค n k < p and 0. n 1 n 2 n 3 โ€ฆ > ฮท } L_{\eta}(p)=\{0.n_{1}n_{2}n_{3}\ldots|0\leq n_{k}<p\,\text{ and }0.n_{1}n_{2}n% _{3}\ldots>\eta\}
  51. ฮท \eta
  52. ฮท \eta

Probabilistic_bisimulation.html

  1. S = ( St , Act , ฯ„ : St ร— Act ร— St โ†’ [ 0 , 1 ] ) S=(\operatorname{St},\operatorname{Act},\tau:\operatorname{St}\times% \operatorname{Act}\times\operatorname{St}\rightarrow[0,1])
  2. ฯ„ ( s , a , t ) \tau(s,a,t)
  3. ฯ„ ( s , a , C ) = ฯ„ ( t , a , C ) . \tau(s,a,C)=\tau(t,a,C).

Probability_derivations_for_making_rank-based_hands_in_Omaha_hold_'em.html

  1. ( 48 3 ) = 17 , 296 \begin{matrix}{48\choose 3}=17,296\end{matrix}
  2. ( 48 4 ) = 194 , 580 \begin{matrix}{48\choose 4}=194,580\end{matrix}
  3. ( 48 5 ) = 1 , 712 , 304 \begin{matrix}{48\choose 5}=1,712,304\end{matrix}
  4. ( 12 1 ) ( 4 3 ) \begin{matrix}{12\choose 1}{4\choose 3}\end{matrix}
  5. ( 12 1 ) ( 4 2 ) ( 44 1 ) \begin{matrix}{12\choose 1}{4\choose 2}{44\choose 1}\end{matrix}
  6. ( 12 3 ) ( 4 1 ) 3 \begin{matrix}{12\choose 3}{4\choose 1}^{3}\end{matrix}
  7. ( 12 1 ) ( 4 3 ) ( 44 1 ) \begin{matrix}{12\choose 1}{4\choose 3}{44\choose 1}\end{matrix}
  8. ( 12 1 ) ( 4 4 ) \begin{matrix}{12\choose 1}{4\choose 4}\end{matrix}
  9. ( 12 2 ) ( 4 2 ) 2 \begin{matrix}{12\choose 2}{4\choose 2}^{2}\end{matrix}
  10. ( 12 1 ) ( 4 2 ) ( 11 2 ) ( 4 1 ) 2 \begin{matrix}{12\choose 1}{4\choose 2}{11\choose 2}{4\choose 1}^{2}\end{matrix}
  11. ( 12 4 ) ( 4 1 ) 4 \begin{matrix}{12\choose 4}{4\choose 1}^{4}\end{matrix}
  12. ( 12 1 ) ( 4 3 ) ( 44 2 ) \begin{matrix}{12\choose 1}{4\choose 3}{44\choose 2}\end{matrix}
  13. ( 12 1 ) ( 4 4 ) ( 44 1 ) \begin{matrix}{12\choose 1}{4\choose 4}{44\choose 1}\end{matrix}
  14. ( 12 2 ) ( 4 2 ) 2 ( 40 1 ) \begin{matrix}{12\choose 2}{4\choose 2}^{2}{40\choose 1}\end{matrix}
  15. ( 12 1 ) ( 4 2 ) ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{12\choose 1}{4\choose 2}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  16. ( 12 5 ) ( 4 1 ) 5 \begin{matrix}{12\choose 5}{4\choose 1}^{5}\end{matrix}
  17. ( 3 3 ) \begin{matrix}{3\choose 3}\end{matrix}
  18. ( 1 1 ) ( 3 2 ) \begin{matrix}{1\choose 1}{3\choose 2}\end{matrix}
  19. ( 1 1 ) ( 11 1 ) ( 4 2 ) \begin{matrix}{1\choose 1}{11\choose 1}{4\choose 2}\end{matrix}
  20. ( 11 1 ) ( 4 3 ) \begin{matrix}{11\choose 1}{4\choose 3}\end{matrix}
  21. ( 1 1 ) ( 3 1 ) ( 44 1 ) \begin{matrix}{1\choose 1}{3\choose 1}{44\choose 1}\end{matrix}
  22. ( 1 1 ) ( 11 2 ) ( 4 1 ) 2 \begin{matrix}{1\choose 1}{11\choose 2}{4\choose 1}^{2}\end{matrix}
  23. ( 3 2 ) ( 44 1 ) \begin{matrix}{3\choose 2}{44\choose 1}\end{matrix}
  24. ( 3 1 ) ( 11 1 ) ( 4 2 ) \begin{matrix}{3\choose 1}{11\choose 1}{4\choose 2}\end{matrix}
  25. ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{11\choose 1}{4\choose 2}{40\choose 1}\end{matrix}
  26. ( 3 1 ) ( 11 2 ) ( 4 1 ) 2 \begin{matrix}{3\choose 1}{11\choose 2}{4\choose 1}^{2}\end{matrix}
  27. ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  28. ( 3 3 ) ( 45 1 ) \begin{matrix}{3\choose 3}{45\choose 1}\end{matrix}
  29. ( 1 1 ) ( 3 2 ) ( 44 1 ) \begin{matrix}{1\choose 1}{3\choose 2}{44\choose 1}\end{matrix}
  30. ( 1 1 ) ( 3 1 ) ( 11 1 ) ( 4 2 ) \begin{matrix}{1\choose 1}{3\choose 1}{11\choose 1}{4\choose 2}\end{matrix}
  31. ( 1 1 ) ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{1\choose 1}{11\choose 1}{4\choose 2}{40\choose 1}\end{matrix}
  32. ( 11 1 ) ( 4 3 ) ( 44 1 ) \begin{matrix}{11\choose 1}{4\choose 3}{44\choose 1}\end{matrix}
  33. ( 11 1 ) ( 4 4 ) \begin{matrix}{11\choose 1}{4\choose 4}\end{matrix}
  34. ( 1 1 ) ( 3 1 ) ( 11 2 ) ( 4 1 ) 2 \begin{matrix}{1\choose 1}{3\choose 1}{11\choose 2}{4\choose 1}^{2}\end{matrix}
  35. ( 1 1 ) ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{1\choose 1}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  36. ( 3 2 ) ( 44 2 ) \begin{matrix}{3\choose 2}{44\choose 2}\end{matrix}
  37. ( 3 1 ) ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{3\choose 1}{11\choose 1}{4\choose 2}{40\choose 1}\end{matrix}
  38. ( 11 2 ) ( 4 2 ) 2 \begin{matrix}{11\choose 2}{4\choose 2}^{2}\end{matrix}
  39. ( 11 1 ) ( 4 2 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{11\choose 1}{4\choose 2}{10\choose 2}{4\choose 1}^{2}\end{matrix}
  40. ( 3 1 ) ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{3\choose 1}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  41. ( 11 4 ) ( 4 1 ) 4 \begin{matrix}{11\choose 4}{4\choose 1}^{4}\end{matrix}
  42. ( 3 3 ) ( 45 2 ) \begin{matrix}{3\choose 3}{45\choose 2}\end{matrix}
  43. ( 1 1 ) ( 3 2 ) ( 44 2 ) \begin{matrix}{1\choose 1}{3\choose 2}{44\choose 2}\end{matrix}
  44. ( 1 1 ) ( 3 1 ) ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{1\choose 1}{3\choose 1}{11\choose 1}{4\choose 2}{40\choose 1}% \end{matrix}
  45. ( 1 1 ) ( 11 2 ) ( 4 2 ) 2 \begin{matrix}{1\choose 1}{11\choose 2}{4\choose 2}^{2}\end{matrix}
  46. ( 1 1 ) ( 11 1 ) ( 4 2 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{1\choose 1}{11\choose 1}{4\choose 2}{10\choose 2}{4\choose 1}^{% 2}\end{matrix}
  47. ( 11 1 ) ( 4 3 ) ( 44 2 ) \begin{matrix}{11\choose 1}{4\choose 3}{44\choose 2}\end{matrix}
  48. ( 11 1 ) ( 4 4 ) ( 44 1 ) \begin{matrix}{11\choose 1}{4\choose 4}{44\choose 1}\end{matrix}
  49. ( 1 1 ) ( 3 1 ) ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{1\choose 1}{3\choose 1}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  50. ( 1 1 ) ( 11 4 ) ( 4 1 ) 4 \begin{matrix}{1\choose 1}{11\choose 4}{4\choose 1}^{4}\end{matrix}
  51. ( 3 2 ) ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{3\choose 2}{11\choose 1}{4\choose 2}{40\choose 1}\end{matrix}
  52. ( 3 2 ) ( 11 3 ) ( 4 2 ) 3 \begin{matrix}{3\choose 2}{11\choose 3}{4\choose 2}^{3}\end{matrix}
  53. ( 3 1 ) ( 11 2 ) ( 4 2 ) 2 \begin{matrix}{3\choose 1}{11\choose 2}{4\choose 2}^{2}\end{matrix}
  54. ( 3 1 ) ( 11 1 ) ( 4 2 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{3\choose 1}{11\choose 1}{4\choose 2}{10\choose 2}{4\choose 1}^{% 2}\end{matrix}
  55. ( 11 2 ) ( 4 2 ) 2 ( 36 1 ) \begin{matrix}{11\choose 2}{4\choose 2}^{2}{36\choose 1}\end{matrix}
  56. ( 11 1 ) ( 4 2 ) ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{11\choose 1}{4\choose 2}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  57. ( 3 1 ) ( 11 4 ) ( 4 1 ) 4 \begin{matrix}{3\choose 1}{11\choose 4}{4\choose 1}^{4}\end{matrix}
  58. ( 11 5 ) ( 4 1 ) 5 \begin{matrix}{11\choose 5}{4\choose 1}^{5}\end{matrix}
  59. ( 2 2 ) ( 2 1 ) ( 46 1 ) \begin{matrix}{2\choose 2}{2\choose 1}{46\choose 1}\end{matrix}
  60. ( 2 1 ) ( 2 1 ) ( 11 1 ) ( 4 2 ) \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 1}{4\choose 2}\end{matrix}
  61. ( 11 1 ) ( 4 3 ) \begin{matrix}{11\choose 1}{4\choose 3}\end{matrix}
  62. ( 2 1 ) 2 ( 44 1 ) \begin{matrix}{2\choose 1}^{2}{44\choose 1}\end{matrix}
  63. ( 2 1 ) ( 2 1 ) ( 11 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 2}{4\choose 1}^{2}\end{matrix}
  64. ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{11\choose 1}{4\choose 2}{40\choose 1}\end{matrix}
  65. ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  66. ( 2 2 ) ( 2 2 ) \begin{matrix}{2\choose 2}{2\choose 2}\end{matrix}
  67. ( 2 2 ) ( 2 1 ) ( 2 1 ) ( 44 1 ) \begin{matrix}{2\choose 2}{2\choose 1}{2\choose 1}{44\choose 1}\end{matrix}
  68. ( 2 2 ) ( 2 1 ) ( 44 2 ) \begin{matrix}{2\choose 2}{2\choose 1}{44\choose 2}\end{matrix}
  69. ( 2 1 ) 2 ( 11 1 ) ( 4 2 ) \begin{matrix}{2\choose 1}^{2}{11\choose 1}{4\choose 2}\end{matrix}
  70. ( 2 1 ) ( 2 1 ) ( 11 1 ) ( 4 3 ) \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 1}{4\choose 3}\end{matrix}
  71. ( 2 1 ) ( 2 1 ) ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 1}{4\choose 2}{40\choose 1}% \end{matrix}
  72. ( 11 1 ) ( 4 4 ) \begin{matrix}{11\choose 1}{4\choose 4}\end{matrix}
  73. ( 11 1 ) ( 4 3 ) ( 40 1 ) \begin{matrix}{11\choose 1}{4\choose 3}{40\choose 1}\end{matrix}
  74. ( 2 1 ) 2 ( 11 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}^{2}{11\choose 2}{4\choose 1}^{2}\end{matrix}
  75. ( 2 1 ) ( 2 1 ) ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  76. ( 11 2 ) ( 4 2 ) 2 \begin{matrix}{11\choose 2}{4\choose 2}^{2}\end{matrix}
  77. ( 11 1 ) ( 4 2 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{11\choose 1}{4\choose 2}{10\choose 2}{4\choose 1}^{2}\end{matrix}
  78. ( 11 4 ) ( 4 1 ) 4 \begin{matrix}{11\choose 4}{4\choose 1}^{4}\end{matrix}
  79. ( 2 2 ) ( 2 2 ) ( 44 1 ) \begin{matrix}{2\choose 2}{2\choose 2}{44\choose 1}\end{matrix}
  80. ( 2 1 ) ( 2 2 ) ( 2 1 ) ( 44 2 ) \begin{matrix}{2\choose 1}{2\choose 2}{2\choose 1}{44\choose 2}\end{matrix}
  81. ( 2 1 ) ( 2 2 ) ( 44 3 ) \begin{matrix}{2\choose 1}{2\choose 2}{44\choose 3}\end{matrix}
  82. ( 2 1 ) 2 ( 11 1 ) ( 4 3 ) \begin{matrix}{2\choose 1}^{2}{11\choose 1}{4\choose 3}\end{matrix}
  83. ( 2 1 ) 2 ( 11 1 ) ( 4 2 ) ( 40 1 ) \begin{matrix}{2\choose 1}^{2}{11\choose 1}{4\choose 2}{40\choose 1}\end{matrix}
  84. ( 2 1 ) ( 2 1 ) ( 11 1 ) ( 4 3 ) ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 1}{4\choose 3}{40\choose 1}% \end{matrix}
  85. ( 2 1 ) ( 2 1 ) ( 11 2 ) ( 4 2 ) 2 \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 2}{4\choose 2}^{2}\end{matrix}
  86. ( 2 1 ) ( 2 1 ) ( 11 1 ) ( 4 2 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 1}{4\choose 2}{10\choose 2}{4% \choose 1}^{2}\end{matrix}
  87. ( 11 1 ) ( 4 3 ) ( 40 2 ) \begin{matrix}{11\choose 1}{4\choose 3}{40\choose 2}\end{matrix}
  88. ( 11 1 ) ( 4 4 ) ( 44 1 ) \begin{matrix}{11\choose 1}{4\choose 4}{44\choose 1}\end{matrix}
  89. ( 2 1 ) 2 ( 11 3 ) ( 4 1 ) 3 \begin{matrix}{2\choose 1}^{2}{11\choose 3}{4\choose 1}^{3}\end{matrix}
  90. ( 2 1 ) ( 2 1 ) ( 11 4 ) ( 4 1 ) 4 \begin{matrix}{2\choose 1}{2\choose 1}{11\choose 4}{4\choose 1}^{4}\end{matrix}
  91. ( 11 2 ) ( 4 2 ) 2 ( 36 1 ) \begin{matrix}{11\choose 2}{4\choose 2}^{2}{36\choose 1}\end{matrix}
  92. ( 11 1 ) ( 4 2 ) ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{11\choose 1}{4\choose 2}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  93. ( 11 5 ) ( 4 1 ) 5 \begin{matrix}{11\choose 5}{4\choose 1}^{5}\end{matrix}
  94. ( 2 2 ) ( 46 1 ) \begin{matrix}{2\choose 2}{46\choose 1}\end{matrix}
  95. ( 2 1 ) ( 3 3 ) \begin{matrix}{2\choose 1}{3\choose 3}\end{matrix}
  96. ( 2 1 ) ( 2 1 ) ( 3 2 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 2}\end{matrix}
  97. ( 2 1 ) ( 10 1 ) ( 4 2 ) \begin{matrix}{2\choose 1}{10\choose 1}{4\choose 2}\end{matrix}
  98. ( 2 1 ) ( 3 2 ) ( 3 1 ) \begin{matrix}{2\choose 1}{3\choose 2}{3\choose 1}\end{matrix}
  99. ( 10 1 ) ( 4 3 ) \begin{matrix}{10\choose 1}{4\choose 3}\end{matrix}
  100. ( 2 1 ) ( 3 1 ) 2 \begin{matrix}{2\choose 1}{3\choose 1}^{2}\end{matrix}
  101. ( 2 1 ) ( 2 1 ) ( 3 1 ) ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 1}{40\choose 1}\end{matrix}
  102. ( 2 1 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{10\choose 2}{4\choose 1}^{2}\end{matrix}
  103. ( 2 1 ) ( 3 2 ) ( 40 1 ) \begin{matrix}{2\choose 1}{3\choose 2}{40\choose 1}\end{matrix}
  104. ( 3 1 ) 2 ( 40 1 ) \begin{matrix}{3\choose 1}^{2}{40\choose 1}\end{matrix}
  105. ( 2 1 ) ( 3 1 ) ( 10 1 ) ( 4 2 ) \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 1}{4\choose 2}\end{matrix}
  106. ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{10\choose 1}{4\choose 2}{36\choose 1}\end{matrix}
  107. ( 2 1 ) ( 3 1 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 2}{4\choose 1}^{2}\end{matrix}
  108. ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  109. ( 2 2 ) ( 46 2 ) \begin{matrix}{2\choose 2}{46\choose 2}\end{matrix}
  110. ( 2 1 ) ( 3 3 ) ( 45 1 ) \begin{matrix}{2\choose 1}{3\choose 3}{45\choose 1}\end{matrix}
  111. ( 2 1 ) ( 2 1 ) ( 3 2 ) ( 3 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 2}{3\choose 1}\end{matrix}
  112. ( 2 1 ) ( 2 1 ) ( 3 2 ) ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 2}{40\choose 1}\end{matrix}
  113. ( 2 1 ) ( 2 1 ) ( 3 1 ) ( 10 1 ) ( 4 2 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 1}{10\choose 1}{4\choose 2}% \end{matrix}
  114. ( 2 1 ) ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{2\choose 1}{10\choose 1}{4\choose 2}{36\choose 1}\end{matrix}
  115. ( 2 2 ) ( 3 2 ) 2 \begin{matrix}{2\choose 2}{3\choose 2}^{2}\end{matrix}
  116. ( 2 1 ) ( 2 2 ) ( 3 2 ) ( 3 1 ) ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 2}{3\choose 2}{3\choose 1}{40\choose 1}% \end{matrix}
  117. ( 10 1 ) ( 4 3 ) ( 44 1 ) \begin{matrix}{10\choose 1}{4\choose 3}{44\choose 1}\end{matrix}
  118. ( 10 1 ) ( 4 4 ) \begin{matrix}{10\choose 1}{4\choose 4}\end{matrix}
  119. ( 2 1 ) ( 2 2 ) ( 3 1 ) 2 ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 2}{3\choose 1}^{2}{40\choose 1}\end{matrix}
  120. ( 2 1 ) ( 2 1 ) ( 3 1 ) ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 1}{10\choose 2}{4\choose 1}^{2% }\end{matrix}
  121. ( 2 1 ) ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{2\choose 1}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  122. ( 2 1 ) ( 3 2 ) ( 40 2 ) \begin{matrix}{2\choose 1}{3\choose 2}{40\choose 2}\end{matrix}
  123. ( 3 1 ) 2 ( 40 2 ) \begin{matrix}{3\choose 1}^{2}{40\choose 2}\end{matrix}
  124. ( 2 1 ) ( 3 1 ) ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 1}{4\choose 2}{36\choose 1}% \end{matrix}
  125. ( 10 2 ) ( 4 2 ) 2 \begin{matrix}{10\choose 2}{4\choose 2}^{2}\end{matrix}
  126. ( 10 1 ) ( 4 2 ) ( 9 2 ) ( 4 1 ) 2 \begin{matrix}{10\choose 1}{4\choose 2}{9\choose 2}{4\choose 1}^{2}\end{matrix}
  127. ( 2 1 ) ( 3 1 ) ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  128. ( 10 4 ) ( 4 1 ) 4 \begin{matrix}{10\choose 4}{4\choose 1}^{4}\end{matrix}
  129. ( 2 2 ) ( 46 3 ) \begin{matrix}{2\choose 2}{46\choose 3}\end{matrix}
  130. ( 2 1 ) ( 3 3 ) ( 45 2 ) \begin{matrix}{2\choose 1}{3\choose 3}{45\choose 2}\end{matrix}
  131. ( 2 2 ) ( 2 1 ) ( 3 3 ) \begin{matrix}{2\choose 2}{2\choose 1}{3\choose 3}\end{matrix}
  132. ( 2 1 ) ( 2 2 ) ( 3 2 ) ( 3 2 ) \begin{matrix}{2\choose 1}{2\choose 2}{3\choose 2}{3\choose 2}\end{matrix}
  133. ( 2 1 ) ( 2 1 ) ( 3 2 ) ( 3 1 ) ( 40 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 2}{3\choose 1}{40\choose 1}% \end{matrix}
  134. ( 2 1 ) ( 2 1 ) ( 3 2 ) ( 40 2 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 2}{40\choose 2}\end{matrix}
  135. ( 2 1 ) ( 2 2 ) ( 3 1 ) 2 ( 10 1 ) ( 4 2 ) \begin{matrix}{2\choose 1}{2\choose 2}{3\choose 1}^{2}{10\choose 1}{4\choose 2% }\end{matrix}
  136. ( 2 1 ) ( 2 1 ) ( 3 1 ) ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 1}{10\choose 1}{4\choose 2}{36% \choose 1}\end{matrix}
  137. ( 2 1 ) ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{2\choose 1}{10\choose 1}{4\choose 2}{36\choose 1}\end{matrix}
  138. ( 2 1 ) ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{2\choose 1}{10\choose 1}{4\choose 2}{36\choose 1}\end{matrix}
  139. ( 2 2 ) ( 3 2 ) 2 ( 40 1 ) \begin{matrix}{2\choose 2}{3\choose 2}^{2}{40\choose 1}\end{matrix}
  140. ( 2 1 ) ( 2 2 ) ( 3 2 ) ( 3 1 ) ( 40 2 ) \begin{matrix}{2\choose 1}{2\choose 2}{3\choose 2}{3\choose 1}{40\choose 2}% \end{matrix}
  141. ( 10 1 ) ( 4 3 ) ( 44 2 ) \begin{matrix}{10\choose 1}{4\choose 3}{44\choose 2}\end{matrix}
  142. ( 10 1 ) ( 4 3 ) ( 2 2 ) \begin{matrix}{10\choose 1}{4\choose 3}{2\choose 2}\end{matrix}
  143. ( 10 1 ) ( 4 4 ) ( 44 1 ) \begin{matrix}{10\choose 1}{4\choose 4}{44\choose 1}\end{matrix}
  144. ( 2 1 ) ( 2 2 ) ( 3 1 ) 2 ( 10 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{2\choose 2}{3\choose 1}^{2}{10\choose 2}{4\choose 1% }^{2}\end{matrix}
  145. ( 2 1 ) ( 2 1 ) ( 3 1 ) ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{2\choose 1}{2\choose 1}{3\choose 1}{10\choose 3}{4\choose 1}^{3% }\end{matrix}
  146. ( 2 1 ) ( 10 4 ) ( 4 1 ) 4 \begin{matrix}{2\choose 1}{10\choose 4}{4\choose 1}^{4}\end{matrix}
  147. ( 2 1 ) ( 3 2 ) ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{2\choose 1}{3\choose 2}{10\choose 1}{4\choose 2}{36\choose 1}% \end{matrix}
  148. ( 2 1 ) ( 3 2 ) ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{2\choose 1}{3\choose 2}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  149. ( 3 1 ) 2 ( 10 1 ) ( 4 2 ) ( 36 1 ) \begin{matrix}{3\choose 1}^{2}{10\choose 1}{4\choose 2}{36\choose 1}\end{matrix}
  150. ( 3 1 ) 2 ( 10 3 ) ( 4 1 ) 3 \begin{matrix}{3\choose 1}^{2}{10\choose 3}{4\choose 1}^{3}\end{matrix}
  151. ( 2 1 ) ( 3 1 ) ( 10 2 ) ( 4 2 ) 2 \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 2}{4\choose 2}^{2}\end{matrix}
  152. ( 2 1 ) ( 3 1 ) ( 10 1 ) ( 4 2 ) ( 9 2 ) ( 4 1 ) 2 \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 1}{4\choose 2}{9\choose 2}{4% \choose 1}^{2}\end{matrix}
  153. ( 10 2 ) ( 4 2 ) 2 ( 32 1 ) \begin{matrix}{10\choose 2}{4\choose 2}^{2}{32\choose 1}\end{matrix}
  154. ( 10 1 ) ( 4 2 ) ( 9 3 ) ( 4 1 ) 3 \begin{matrix}{10\choose 1}{4\choose 2}{9\choose 3}{4\choose 1}^{3}\end{matrix}
  155. ( 2 1 ) ( 3 1 ) ( 10 4 ) ( 4 1 ) 4 \begin{matrix}{2\choose 1}{3\choose 1}{10\choose 4}{4\choose 1}^{4}\end{matrix}
  156. ( 10 5 ) ( 4 1 ) 5 \begin{matrix}{10\choose 5}{4\choose 1}^{5}\end{matrix}
  157. ( 4 1 ) ( 3 3 ) \begin{matrix}{4\choose 1}{3\choose 3}\end{matrix}
  158. ( 4 1 ) ( 3 2 ) ( 3 1 ) ( 3 1 ) \begin{matrix}{4\choose 1}{3\choose 2}{3\choose 1}{3\choose 1}\end{matrix}
  159. ( 4 1 ) ( 3 2 ) ( 36 1 ) \begin{matrix}{4\choose 1}{3\choose 2}{36\choose 1}\end{matrix}
  160. ( 9 1 ) ( 4 3 ) \begin{matrix}{9\choose 1}{4\choose 3}\end{matrix}
  161. ( 4 3 ) ( 3 1 ) 3 \begin{matrix}{4\choose 3}{3\choose 1}^{3}\end{matrix}
  162. ( 4 2 ) ( 3 1 ) 2 ( 36 1 ) \begin{matrix}{4\choose 2}{3\choose 1}^{2}{36\choose 1}\end{matrix}
  163. ( 4 1 ) ( 3 1 ) ( 36 2 ) \begin{matrix}{4\choose 1}{3\choose 1}{36\choose 2}\end{matrix}
  164. ( 9 1 ) ( 4 2 ) ( 32 1 ) \begin{matrix}{9\choose 1}{4\choose 2}{32\choose 1}\end{matrix}
  165. ( 9 3 ) ( 4 1 ) 3 \begin{matrix}{9\choose 3}{4\choose 1}^{3}\end{matrix}
  166. ( 4 1 ) ( 3 3 ) ( 45 1 ) \begin{matrix}{4\choose 1}{3\choose 3}{45\choose 1}\end{matrix}
  167. ( 4 2 ) ( 3 2 ) 2 \begin{matrix}{4\choose 2}{3\choose 2}^{2}\end{matrix}
  168. ( 4 1 ) ( 3 2 ) ( 3 2 ) ( 3 1 ) 2 \begin{matrix}{4\choose 1}{3\choose 2}{3\choose 2}{3\choose 1}^{2}\end{matrix}
  169. ( 4 1 ) ( 3 2 ) ( 3 1 ) ( 3 1 ) ( 36 1 ) \begin{matrix}{4\choose 1}{3\choose 2}{3\choose 1}{3\choose 1}{36\choose 1}% \end{matrix}
  170. ( 4 1 ) ( 3 2 ) ( 36 2 ) \begin{matrix}{4\choose 1}{3\choose 2}{36\choose 2}\end{matrix}
  171. ( 4 1 ) ( 3 1 ) ( 9 1 ) ( 4 3 ) \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 1}{4\choose 3}\end{matrix}
  172. ( 9 1 ) ( 4 4 ) \begin{matrix}{9\choose 1}{4\choose 4}\end{matrix}
  173. ( 9 1 ) ( 4 3 ) ( 32 1 ) \begin{matrix}{9\choose 1}{4\choose 3}{32\choose 1}\end{matrix}
  174. ( 4 4 ) ( 3 1 ) 4 \begin{matrix}{4\choose 4}{3\choose 1}^{4}\end{matrix}
  175. ( 4 3 ) ( 3 1 ) 3 ( 36 1 ) \begin{matrix}{4\choose 3}{3\choose 1}^{3}{36\choose 1}\end{matrix}
  176. ( 4 2 ) ( 3 1 ) 2 ( 36 2 ) \begin{matrix}{4\choose 2}{3\choose 1}^{2}{36\choose 2}\end{matrix}
  177. ( 4 1 ) ( 3 1 ) ( 9 1 ) ( 4 2 ) ( 32 1 ) \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 1}{4\choose 2}{32\choose 1}% \end{matrix}
  178. ( 4 1 ) ( 3 1 ) ( 9 3 ) ( 4 1 ) 3 \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 3}{4\choose 1}^{3}\end{matrix}
  179. ( 9 2 ) ( 4 2 ) 2 \begin{matrix}{9\choose 2}{4\choose 2}^{2}\end{matrix}
  180. ( 9 1 ) ( 4 2 ) ( 8 2 ) ( 4 1 ) 2 \begin{matrix}{9\choose 1}{4\choose 2}{8\choose 2}{4\choose 1}^{2}\end{matrix}
  181. ( 9 4 ) ( 4 1 ) 4 \begin{matrix}{9\choose 4}{4\choose 1}^{4}\end{matrix}
  182. ( 4 1 ) ( 3 3 ) ( 45 2 ) \begin{matrix}{4\choose 1}{3\choose 3}{45\choose 2}\end{matrix}
  183. ( 4 2 ) ( 3 2 ) 2 ( 2 1 ) ( 3 1 ) \begin{matrix}{4\choose 2}{3\choose 2}^{2}{2\choose 1}{3\choose 1}\end{matrix}
  184. ( 4 2 ) ( 3 2 ) 2 ( 36 1 ) \begin{matrix}{4\choose 2}{3\choose 2}^{2}{36\choose 1}\end{matrix}
  185. ( 4 1 ) ( 3 2 ) ( 3 3 ) ( 3 1 ) 3 \begin{matrix}{4\choose 1}{3\choose 2}{3\choose 3}{3\choose 1}^{3}\end{matrix}
  186. ( 4 1 ) ( 3 2 ) ( 3 2 ) ( 3 1 ) 2 ( 36 1 ) \begin{matrix}{4\choose 1}{3\choose 2}{3\choose 2}{3\choose 1}^{2}{36\choose 1% }\end{matrix}
  187. ( 4 1 ) ( 3 2 ) ( 3 1 ) ( 3 1 ) ( 36 2 ) \begin{matrix}{4\choose 1}{3\choose 2}{3\choose 1}{3\choose 1}{36\choose 2}% \end{matrix}
  188. ( 4 1 ) ( 3 2 ) ( 36 3 ) \begin{matrix}{4\choose 1}{3\choose 2}{36\choose 3}\end{matrix}
  189. ( 4 2 ) ( 3 1 ) 2 ( 9 1 ) ( 4 3 ) \begin{matrix}{4\choose 2}{3\choose 1}^{2}{9\choose 1}{4\choose 3}\end{matrix}
  190. ( 4 1 ) ( 3 1 ) ( 9 1 ) ( 4 3 ) ( 32 1 ) \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 1}{4\choose 3}{32\choose 1}% \end{matrix}
  191. ( 9 1 ) ( 4 4 ) ( 44 1 ) \begin{matrix}{9\choose 1}{4\choose 4}{44\choose 1}\end{matrix}
  192. ( 9 1 ) ( 4 3 ) ( 32 2 ) \begin{matrix}{9\choose 1}{4\choose 3}{32\choose 2}\end{matrix}
  193. ( 4 4 ) ( 3 1 ) 4 ( 36 1 ) \begin{matrix}{4\choose 4}{3\choose 1}^{4}{36\choose 1}\end{matrix}
  194. ( 4 3 ) ( 3 1 ) 3 ( 36 2 ) \begin{matrix}{4\choose 3}{3\choose 1}^{3}{36\choose 2}\end{matrix}
  195. ( 4 2 ) ( 3 1 ) 2 ( 9 1 ) ( 4 2 ) ( 32 1 ) \begin{matrix}{4\choose 2}{3\choose 1}^{2}{9\choose 1}{4\choose 2}{32\choose 1% }\end{matrix}
  196. ( 4 2 ) ( 3 1 ) 2 ( 9 3 ) ( 4 1 ) 3 \begin{matrix}{4\choose 2}{3\choose 1}^{2}{9\choose 3}{4\choose 1}^{3}\end{matrix}
  197. ( 4 1 ) ( 3 1 ) ( 9 2 ) ( 4 2 ) 2 \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 2}{4\choose 2}^{2}\end{matrix}
  198. ( 4 1 ) ( 3 1 ) ( 9 1 ) ( 4 2 ) ( 8 2 ) ( 4 1 ) 2 \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 1}{4\choose 2}{8\choose 2}{4% \choose 1}^{2}\end{matrix}
  199. ( 4 1 ) ( 3 1 ) ( 9 4 ) ( 4 1 ) 4 \begin{matrix}{4\choose 1}{3\choose 1}{9\choose 4}{4\choose 1}^{4}\end{matrix}
  200. ( 9 2 ) ( 4 2 ) 2 ( 28 1 ) \begin{matrix}{9\choose 2}{4\choose 2}^{2}{28\choose 1}\end{matrix}
  201. ( 9 1 ) ( 4 2 ) ( 8 3 ) ( 4 1 ) 3 \begin{matrix}{9\choose 1}{4\choose 2}{8\choose 3}{4\choose 1}^{3}\end{matrix}
  202. ( 9 5 ) ( 4 1 ) 5 \begin{matrix}{9\choose 5}{4\choose 1}^{5}\end{matrix}

Probability_integral_transform.html

  1. Y = F X ( X ) , Y=F_{X}(X)\,,
  2. ฮฆ ( x ) = 1 2 ฯ€ โˆซ - โˆž x e - t 2 / 2 d t = 1 2 [ 1 + erf ( x 2 ) ] , x โˆˆ โ„ , \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt=\frac{1}{2}% \Big[\,1+\operatorname{erf}\Big(\frac{x}{\sqrt{2}}\Big)\,\Big],\quad x\in% \mathbb{R},\,
  3. erf ( ) , \operatorname{erf}(),
  4. F ( x ) = 1 - exp ( - x ) , F(x)=1-\exp(-x),
  5. Y = 1 - exp ( - X ) Y=1-\exp(-X)
  6. Y โ€ฒ = exp ( - X ) Y^{\prime}=\exp(-X)

Probability_matching.html

  1. p 2 + ( 1 - p ) 2 p^{2}+(1-p)^{2}
  2. .6 ร— .6 + .4 ร— .4 .6\times.6+.4\times.4

Problem_of_points.html

  1. r + s - 1 r+s-1
  2. r + s - 1 r+s-1
  3. 2 r + s - 1 2^{r+s-1}
  4. r + s - 1 r+s-1
  5. 2 r + s - 1 2^{r+s-1}
  6. 2 r + s - 1 2^{r+s-1}
  7. r + s - 1 r+s-1
  8. โˆ‘ k = 0 s - 1 ( r + s - 1 k ) to โˆ‘ k = s r + s - 1 ( r + s - 1 k ) . \sum_{k=0}^{s-1}{\left({{r+s-1}\atop{k}}\right)}\mbox{ to }~{}\sum_{k=s}^{r+s-% 1}{\left({{r+s-1}\atop{k}}\right)}.
  9. r + s - 1 r+s-1

Problems_in_Latin_squares.html

  1. L n L_{n}
  2. p ( n ) p(n)
  3. 2 p ( n ) 2^{p(n)}
  4. L n L_{n}
  5. p ( n ) p(n)
  6. L n = n ! ( n - 1 ) ! R n L_{n}=n!(n-1)!R_{n}
  7. R n R_{n}
  8. R n R_{n}
  9. R n R_{n}

Process_Window_Index.html

  1. C ^ p k \hat{C}_{pk}
  2. PWI = 100 ร— max i = 1 โ€ฆ N j = 1 โ€ฆ M { | measured value [ i , j ] - average limits [ i , j ] range [ i , j ] / 2 | } \,\text{PWI}=100\times\max_{i=1\dots N\atop j=1\dots M}\left\{\left|\frac{\,% \text{measured value}_{[i,j]}-\,\text{average limits}_{[i,j]}}{\,\text{range}_% {[i,j]}/2}\right|\right\}

Product_integral.html

  1. โˆ \prod
  2. โˆซ \int
  3. f : [ a , b ] โ†’ โ„ f:[a,b]\to\mathbb{R}
  4. โˆซ a b f ( x ) d x = lim ฮ” x โ†’ 0 โˆ‘ f ( x i ) ฮ” x , \int_{a}^{b}f(x)\,dx=\lim_{\Delta x\to 0}\sum f(x_{i})\,\Delta x,
  5. [ a , b ] [a,b]
  6. โˆ a b f ( x ) d x = lim ฮ” x โ†’ 0 โˆ f ( x i ) ฮ” x = exp ( โˆซ a b ln f ( x ) d x ) , \prod_{a}^{b}f(x)^{dx}=\lim_{\Delta x\to 0}\prod{f(x_{i})^{\Delta x}}=\exp% \left(\int_{a}^{b}\ln f(x)\,dx\right),
  7. โˆ i = a b \prod_{i=a}^{b}
  8. i , a , b โˆˆ โ„ค i,a,b\in\mathbb{Z}
  9. โˆซ a b d x \int_{a}^{b}dx
  10. x โˆˆ [ a , b ] x\in[a,b]
  11. โˆ‘ i = a b f ( i ) \sum_{i=a}^{b}f(i)
  12. โˆ i = a b f ( i ) \prod_{i=a}^{b}f(i)
  13. โˆซ a b f ( x ) d x \int_{a}^{b}f(x)dx
  14. โˆ a b f ( x ) d x \prod_{a}^{b}f(x){}^{dx}
  15. ln โˆ a b p ( x ) d x = โˆซ a b ln p ( x ) d x \ln\prod_{a}^{b}p(x)^{dx}=\int_{a}^{b}\ln p(x)\,dx
  16. โˆ a b ( 1 + f ( x ) d x ) = lim ฮ” x โ†’ 0 โˆ ( 1 + f ( x i ) ฮ” x ) \prod_{a}^{b}(1+f(x)\,dx)=\lim_{\Delta x\to 0}\prod(1+f(x_{i})\,\Delta x)
  17. f : [ a , b ] โ†’ โ„ f:[a,b]\to\mathbb{R}
  18. โˆ a b ( 1 + f ( x ) d x ) = exp ( โˆซ a b f ( x ) d x ) , \prod_{a}^{b}(1+f(x)\,dx)=\exp\left(\int_{a}^{b}f(x)\,dx\right),
  19. โˆ a b c d x = c b - a \;\prod_{a}^{b}{c^{dx}}=c^{b-a}
  20. โˆ a b ( f ( x ) k ) d x = ( โˆ a b f ( x ) d x ) k \;\prod_{a}^{b}{(f(x)^{k})^{dx}}=(\prod_{a}^{b}{f(x)^{dx}})^{k}
  21. โˆ a b ( c f ( x ) ) d x = c โˆซ a b f ( x ) d x \;\prod_{a}^{b}{(c^{f(x)})^{dx}}=c^{\int_{a}^{b}\!f(x)dx}
  22. โˆ a b f โ€ฒ * ( x ) d x = โˆ a b exp ( f โ€ฒ ( x ) f ( x ) d x ) = f ( b ) f ( a ) \;\prod_{a}^{b}{f^{\prime*}(x)^{dx}}=\prod_{a}^{b}\exp\left(\frac{f^{\prime}(x% )}{f(x)}\,dx\right)=\frac{f(b)}{f(a)}
  23. f โ€ฒ * ( x ) f^{\prime*}(x)
  24. ( f g ) * = f * g * \;(fg)^{*}=f^{*}g^{*}
  25. ( f / g ) * = f * / g * \;(f/g)^{*}=f^{*}/g^{*}
  26. X 1 X 2 โ‹ฏ X n n โ†’ โˆ x X d F ( x ) as n โ†’ โˆž \;\sqrt[n]{X_{1}X_{2}\cdots X_{n}}\to\sideset{}{}{\prod}_{x}X^{dF(x)}\,\text{ % as }n\to\infty
  27. X 1 + X 2 + โ‹ฏ + X n n โ†’ โˆซ X d F ( x ) as n โ†’ โˆž \;\frac{X_{1}+X_{2}+\cdots+X_{n}}{n}\;\to\;\int X\,dF(x)\,\text{ as }n\to\infty

Profitability_index.html

  1. Profitability index = PV of future cash flows Initial investment \,\text{Profitability index}=\frac{\,\text{PV of future cash flows}}{\,\text{% Initial investment}}

Program_structure_tree.html

  1. O ( | V | + | E | ) O(|V|+|E|)
  2. | V | |V|
  3. | E | |E|
  4. O ( | E | ) O(|E|)
  5. E E

Progressive_Graphics_File.html

  1. [ Y r U r V r ] = [ 1 4 1 2 1 4 1 - 1 0 0 - 1 1 ] [ R G B ] ; [ R G B ] = [ 1 3 4 - 1 4 1 - 1 4 - 1 4 1 - 1 4 3 4 ] [ Y r U r V r ] \begin{bmatrix}Y_{r}\\ U_{r}\\ V_{r}\end{bmatrix}=\begin{bmatrix}\frac{1}{4}&\frac{1}{2}&\frac{1}{4}\\ 1&-1&0\\ 0&-1&1\end{bmatrix}\begin{bmatrix}R\\ G\\ B\end{bmatrix};\qquad\qquad\begin{bmatrix}R\\ G\\ B\end{bmatrix}=\begin{bmatrix}1&\frac{3}{4}&-\frac{1}{4}\\ 1&-\frac{1}{4}&-\frac{1}{4}\\ 1&-\frac{1}{4}&\frac{3}{4}\end{bmatrix}\begin{bmatrix}Y_{r}\\ U_{r}\\ V_{r}\end{bmatrix}

Projective_cover.html

  1. ๐’ž \mathcal{C}
  2. ๐’ž \mathcal{C}
  3. ๐’ž \mathcal{C}
  4. p : P โ†’ X p:P\to X
  5. p ( N ) โ‰  M p(N)\neq M
  6. p โ€ฒ : P โ€ฒ โ†’ M p^{\prime}:P^{\prime}\rightarrow M
  7. p ฮฑ = p โ€ฒ p\alpha=p^{\prime}

Proof_sketch_for_Goฬˆdel's_first_incompleteness_theorem.html

  1. 0
  2. S S
  3. โˆง โˆง
  4. โˆจ โˆจ
  5. ยฌ ยฌ
  6. โˆ€ โˆ€
  7. โˆƒ โˆƒ
  8. x x
  9. x * , x * * , โ€ฆ x*,x**,โ€ฆ
  10. x = S S 0 x=SS0
  11. x = โˆ€ + x=โˆ€+
  12. F F
  13. F F
  14. F ( x ) F(x)
  15. x x
  16. โˆƒ m F ( m ) โˆƒmF(m)
  17. ยฌ F ( n ) ยฌF(n)
  18. n n
  19. P F ( x , y ) PF(x,y)
  20. n n
  21. m , P F ( n , m ) m,PF(n,m)
  22. n n
  23. m m
  24. S S
  25. x x
  26. 0
  27. G ( F ) G(F)
  28. F F
  29. โˆ€ x โˆ€ x * ( x + x * = x * + x ) โˆ€xโˆ€x*(x+x*=x*+x)
  30. 626026206260262016303620262011202620163011102620163011202620323 626026206260262016303620262011202620163011102620163011202620323
  31. 111062601120262 111062601120262
  32. = โˆ€ + x =โˆ€+x
  33. S S
  34. 0
  35. 4 , S S S S 0 4,SSSS0
  36. 1230123012301230666 1230123012301230666
  37. F ( x ) F(x)
  38. x x
  39. m m
  40. F ( m ) F(m)
  41. G ( x ) G(x)
  42. G ( F ( x ) ) G(F(x))
  43. G ( m ) G(m)
  44. F F
  45. m m
  46. F F
  47. S S
  48. n n
  49. m m
  50. n n
  51. m m
  52. n n
  53. S S
  54. n n
  55. m m
  56. P P
  57. P P
  58. A X AX
  59. n n
  60. P P
  61. m m
  62. i i
  63. m m
  64. P P
  65. P P
  66. P P
  67. S S
  68. S S
  69. P F ( x , y ) PF(x,y)
  70. x x
  71. y y
  72. x x
  73. S S
  74. y = G ( S ) y=G(S)
  75. P F ( x , y ) PF(x,y)
  76. x + y = 6 x+y=6
  77. R ( x , y ) R(x,y)
  78. n n
  79. m m
  80. R ( m , n ) R(m,n)
  81. ยฌ R ( m , n ) ยฌR(m,n)
  82. P F PF
  83. n n
  84. F ( y ) F(y)
  85. y y
  86. q ( n , G ( F ) ) q(n,G(F))
  87. n n
  88. G ( F ) G(F)
  89. n n
  90. F ( G ( F ) ) F(G(F))
  91. F ( G ( F ) ) F(G(F))
  92. F F
  93. q q
  94. G ( F ) G(F)
  95. F F
  96. q ( n , G ( F ) ) q(n,G(F))
  97. ยฌ q ( n , G ( F ) ) ยฌq(n,G(F))
  98. G ( F ) G(F)
  99. G ( F ) G(F)
  100. F F
  101. y y
  102. F F
  103. G ( F ) G(F)
  104. F ( G ( F ) ) F(G(F))
  105. n n
  106. F ( y ) , q ( n , G ( F ) ) F(y),q(n,G(F))
  107. n n
  108. G ( F ) G(F)
  109. P F PF
  110. q ( n , G ( F ) ) q(n,G(F))
  111. n n
  112. F ( G ( F ) ) F(G(F))
  113. ยฌ q ( n , G ( F ) ) ยฌq(n,G(F))
  114. n n
  115. F ( G ( F ) ) F(G(F))
  116. n n
  117. G ( F ) G(F)
  118. q ( n , G ( F ) ) q(n,G(F))
  119. ยฌ q ( n , G ( F ) ) ยฌq(n,G(F))
  120. F ( G ( F ) ) F(G(F))
  121. n n
  122. q ( n , G ( F ) ) q(n,G(F))
  123. q ( n , G ( F ) ) q(n,G(F))
  124. n n
  125. F ( G ( F ) ) F(G(F))
  126. โˆ€ y q ( y , G ( F ) ) โˆ€yq(y,G(F))
  127. F ( G ( F ) ) F(G(F))
  128. P ( x ) = โˆ€ y q ( y , x ) P(x)=โˆ€yq(y,x)
  129. x x
  130. P P
  131. G ( P ) G(P)
  132. x x
  133. G ( F ) G(F)
  134. F ( z ) F(z)
  135. z z
  136. P ( G ( F ) ) = โˆ€ y q ( y , G ( F ) ) P(G(F))=โˆ€yq(y,G(F))
  137. F ( G ( F ) ) F(G(F))
  138. P ( G ( P ) ) = โˆ€ y , q ( y , G ( P ) ) P(G(P))=โˆ€y,q(y,G(P))
  139. G ( P ) G(P)
  140. P ( G ( P ) ) P(G(P))
  141. P ( G ( P ) ) P(G(P))
  142. P ( G ( P ) ) P(G(P))
  143. ยฌ P ( G ( P ) ) ยฌP(G(P))
  144. P ( G ( P ) ) = โˆ€ y , q ( y , G ( P ) ) P(G(P))=โˆ€y,q(y,G(P))
  145. n n
  146. P ( G ( P ) ) P(G(P))
  147. ยฌ q ( n , G ( P ) ) ยฌq(n,G(P))
  148. ยฌ q ( n , G ( P ) ) ยฌq(n,G(P))
  149. โˆ€ y q ( y , G ( P ) ) โˆ€yq(y,G(P))
  150. P ( G ( P ) ) P(G(P))
  151. n n
  152. ยฌ q ( n , G ( P ) ) ยฌq(n,G(P))
  153. n n
  154. P ( G ( P ) ) P(G(P))
  155. P ( G ( P ) ) P(G(P))
  156. q ( n , G ( P ) ) q(n,G(P))
  157. ยฌ q ( n , G ( P ) ) ยฌq(n,G(P))
  158. n , q ( n , G ( P ) ) n,q(n,G(P))
  159. P ( G ( P ) ) P(G(P))
  160. ยฌ P ( G ( P ) ) = โˆƒ x ยฌ q ( x , G ( P ) ) ยฌP(G(P))=โˆƒxยฌq(x,G(P))
  161. โˆƒ x ยฌ q ( x , G ( P ) ) โˆƒxยฌq(x,G(P))
  162. q ( n , G ( P ) ) q(n,G(P))
  163. n n
  164. ยฌ P ( G ( P ) ) ยฌP(G(P))
  165. P ( G ( P ) ) P(G(P))
  166. P ( G ( P ) ) P(G(P))
  167. P ( G ( P ) ) P(G(P))
  168. q ( n , G ( P ) ) q(n,G(P))
  169. n n
  170. P ( G ( P ) ) = โˆ€ y q ( y , G ( P ) ) P(G(P))=\forall y\,q(y,G(P))
  171. P ( G ( P ) ) P(G(P))
  172. M M
  173. n n
  174. 0
  175. C x z Cxz
  176. z z
  177. x x
  178. F F
  179. n n
  180. โˆ€ x ( F ( x ) โ†” x = n ) \forall x(F(x)\leftrightarrow x=n)
  181. B x y โ†” โˆƒ z ( z < y โˆง C x z ) Bxyโ†”โˆƒz(z<yโˆงCxz)

Proofs_involving_the_Mooreโ€“Penrose_pseudoinverse.html

  1. A A
  2. ๐•‚ \mathbb{K}
  3. ๐•‚ \mathbb{K}
  4. โ„ \mathbb{R}
  5. โ„‚ \mathbb{C}
  6. A + A^{+}
  7. ๐•‚ \mathbb{K}
  8. 0 = Tr ( A * A ) = โˆ‘ j = 1 n ( A * A ) j j = โˆ‘ j = 1 n โˆ‘ i = 1 m ( A * ) j i A i j = โˆ‘ i = 1 m โˆ‘ j = 1 n | A i j | 2 0=\operatorname{Tr}(A^{*}A)=\sum_{j=1}^{n}(A^{*}A)_{jj}=\sum_{j=1}^{n}\sum_{i=% 1}^{m}(A^{*})_{ji}A_{ij}=\sum_{i=1}^{m}\sum_{j=1}^{n}|A_{ij}|^{2}
  9. A i j A_{ij}
  10. ๐•‚ \mathbb{K}
  11. x โˆˆ ๐•‚ x\in\mathbb{K}
  12. x + := { x - 1 , if x โ‰  0 0 , if x = 0 x^{+}:=\begin{cases}x^{-1},&\mbox{if }~{}x\neq 0\\ 0,&\mbox{if }~{}x=0\end{cases}
  13. x + x^{+}
  14. x x
  15. D D
  16. D + D^{+}
  17. ( D + ) i j := ( D i j ) + (D^{+})_{ij}:=(D_{ij})^{+}
  18. D i j + D^{+}_{ij}
  19. ( D + ) i j = ( D i j ) + (D^{+})_{ij}=(D_{ij})^{+}
  20. D + D^{+}
  21. D + D^{+}
  22. D D
  23. ( D D + D ) i j = D i j D i j + D i j = D i j โ‡’ D D + D = D (DD^{+}D)_{ij}=D_{ij}D^{+}_{ij}D_{ij}=D_{ij}\Rightarrow DD^{+}D=D
  24. ( D + D D + ) i j = D i j + D i j D i j + = D i j + โ‡’ D + D D + = D + (D^{+}DD^{+})_{ij}=D^{+}_{ij}D_{ij}D^{+}_{ij}=D^{+}_{ij}\Rightarrow D^{+}DD^{+% }=D^{+}
  25. ( D D + ) i j * = ( D D + ) j i ยฏ = D j i D j i + ยฏ = ( D j i D j i + ) * = D j i D j i + = D i j D i j + โ‡’ ( D D + ) * = D D + (DD^{+})^{*}_{ij}=\overline{(DD^{+})_{ji}}=\overline{D_{ji}D^{+}_{ji}}=(D_{ji}% D^{+}_{ji})^{*}=D_{ji}D^{+}_{ji}=D_{ij}D^{+}_{ij}\Rightarrow(DD^{+})^{*}=DD^{+}
  26. ( D + D ) i j * = ( D + D ) j i ยฏ = D j i + D j i ยฏ = ( D j i + D j i ) * = D j i + D j i = D i j + D i j โ‡’ ( D + D ) * = D + D (D^{+}D)^{*}_{ij}=\overline{(D^{+}D)_{ji}}=\overline{D^{+}_{ji}D_{ji}}=(D^{+}_% {ji}D_{ji})^{*}=D^{+}_{ji}D_{ji}=D^{+}_{ij}D_{ij}\Rightarrow(D^{+}D)^{*}=D^{+}D
  27. A = U ฮฃ V * A=U\Sigma V^{*}
  28. A + A^{+}
  29. V ฮฃ + U * V\Sigma^{+}U^{*}
  30. A + A^{+}
  31. A A
  32. A A + A = U ฮฃ V * V ฮฃ + U * U ฮฃ V * = U ฮฃ ฮฃ + ฮฃ V * = U ฮฃ V * = A AA^{+}A=U\Sigma V^{*}V\Sigma^{+}U^{*}U\Sigma V^{*}=U\Sigma\Sigma^{+}\Sigma V^{% *}=U\Sigma V^{*}=A
  33. A + A A + = V ฮฃ + U * U ฮฃ V * V ฮฃ + U * = V ฮฃ + ฮฃ ฮฃ + U * = V ฮฃ + U * = A + A^{+}AA^{+}=V\Sigma^{+}U^{*}U\Sigma V^{*}V\Sigma^{+}U^{*}=V\Sigma^{+}\Sigma% \Sigma^{+}U^{*}=V\Sigma^{+}U^{*}=A^{+}
  34. ( A A + ) * = ( U ฮฃ V * V ฮฃ + U * ) * = ( U ฮฃ ฮฃ + U * ) * = U ( ฮฃ ฮฃ + ) * U * = U ( ฮฃ ฮฃ + ) U * = U ฮฃ V * V ฮฃ + U * = A A + (AA^{+})^{*}=(U\Sigma V^{*}V\Sigma^{+}U^{*})^{*}=(U\Sigma\Sigma^{+}U^{*})^{*}=% U(\Sigma\Sigma^{+})^{*}U^{*}=U(\Sigma\Sigma^{+})U^{*}=U\Sigma V^{*}V\Sigma^{+}% U^{*}=AA^{+}
  35. ( A + A ) * = ( V ฮฃ + U * U ฮฃ V * ) * = ( V ฮฃ + ฮฃ V * ) * = V ( ฮฃ + ฮฃ ) * V * = V ( ฮฃ + ฮฃ ) V * = V ฮฃ + U * U ฮฃ V * = A + A (A^{+}A)^{*}=(V\Sigma^{+}U^{*}U\Sigma V^{*})^{*}=(V\Sigma^{+}\Sigma V^{*})^{*}% =V(\Sigma^{+}\Sigma)^{*}V^{*}=V(\Sigma^{+}\Sigma)V^{*}=V\Sigma^{+}U^{*}U\Sigma V% ^{*}=A^{+}A
  36. B = A * B=A^{*}
  37. C = A A * C=AA^{*}
  38. D = A + * A + D=A^{+*}A^{+}
  39. D D
  40. C D C = A A * A + * A + A A * = A ( A + A ) * A + A A * = A A + A A + A A * = A A + A A * = A A * = C CDC=AA^{*}A^{+*}A^{+}AA^{*}=A(A^{+}A)^{*}A^{+}AA^{*}=AA^{+}AA^{+}AA^{*}=AA^{+}% AA^{*}=AA^{*}=C
  41. D C D = A + * A + A A * A + * A + = A + * A + A ( A + A ) * A + = A + * A + A A + A A + = A + * A + A A + = A + * A + A A + = A + * A + = D DCD=A^{+*}A^{+}AA^{*}A^{+*}A^{+}=A^{+*}A^{+}A(A^{+}A)^{*}A^{+}=A^{+*}A^{+}AA^{% +}AA^{+}=A^{+*}A^{+}AA^{+}=A^{+*}A^{+}AA^{+}=A^{+*}A^{+}=D
  42. ( C D ) * = ( A A * A + * A + ) * = A + * A + A A * = A + * ( A + A ) * A * = A + * A * A + * A * = ( A A + ) * ( A A + ) * = A A + A A + = A ( A + A ) * A + = (CD)^{*}=(AA^{*}A^{+*}A^{+})^{*}=A^{+*}A^{+}AA^{*}=A^{+*}(A^{+}A)^{*}A^{*}=A^{% +*}A^{*}A^{+*}A^{*}=(AA^{+})^{*}(AA^{+})^{*}=AA^{+}AA^{+}=A(A^{+}A)^{*}A^{+}=
  43. = A A * A + * A + = C D =AA^{*}A^{+*}A^{+}=CD
  44. ( D C ) * = ( A + * A + A A * ) * = A A * A + * A + = A ( A + A ) * A + = A A + A A + = ( A A + ) * ( A A + ) * = A + * A * A + * A * = A + * ( A + A ) * A * = (DC)^{*}=(A^{+*}A^{+}AA^{*})^{*}=AA^{*}A^{+*}A^{+}=A(A^{+}A)^{*}A^{+}=AA^{+}AA% ^{+}=(AA^{+})^{*}(AA^{+})^{*}=A^{+*}A^{*}A^{+*}A^{*}=A^{+*}(A^{+}A)^{*}A^{*}=
  45. = A + * A + A A * = D C =A^{+*}A^{+}AA^{*}=DC
  46. D = C + D=C^{+}
  47. ( A A * ) + = A + * A + (AA^{*})^{+}=A^{+*}A^{+}
  48. ( A * ) * = A (A^{*})^{*}=A
  49. ( A * A ) + = A + A + * (A^{*}A)^{+}=A^{+}A^{+*}
  50. Ker ( A + ) = Ker ( A * ) . Im ( A + ) = Im ( A * ) . \begin{aligned}\displaystyle\operatorname{Ker}(A^{+})&\displaystyle=% \operatorname{Ker}(A^{*}).\\ \displaystyle\operatorname{Im}(A^{+})&\displaystyle=\operatorname{Im}(A^{*}).% \\ \end{aligned}
  51. m ร— n m\times n
  52. A A
  53. โˆฅ A x - b โˆฅ 2 โ‰ฅ โˆฅ A z - b โˆฅ 2 \|Ax-b\|_{2}\geq\|Az-b\|_{2}
  54. z = A + b z=A^{+}b
  55. A x = b Ax=b
  56. P = A A + P=AA^{+}
  57. P A = A PA=A
  58. P = P * P=P^{*}
  59. A * ( A z - b ) = A * ( A A + b - b ) = A * ( P b - b ) = A * P * b - A * b = ( P A ) * b - A * b = 0 \begin{aligned}\displaystyle A^{*}(Az-b)&\displaystyle=A^{*}(AA^{+}b-b)\\ &\displaystyle=A^{*}(Pb-b)\\ &\displaystyle=A^{*}P^{*}b-A^{*}b\\ &\displaystyle=(PA)^{*}b-A^{*}b\\ &\displaystyle=0\end{aligned}
  60. โˆฅ A x - b โˆฅ 2 2 = โˆฅ A z - b โˆฅ 2 2 + ( A ( x - z ) ) * ( A z - b ) + c.c. + โˆฅ A ( x - z ) โˆฅ 2 2 = โˆฅ A z - b โˆฅ 2 2 + ( x - z ) * A * ( A z - b ) + c.c. + โˆฅ A ( x - z ) โˆฅ 2 2 = โˆฅ A z - b โˆฅ 2 2 + โˆฅ A ( x - z ) โˆฅ 2 2 โ‰ฅ โˆฅ A z - b โˆฅ 2 2 \begin{aligned}\displaystyle\|Ax-b\|_{2}^{2}&\displaystyle=\|Az-b\|_{2}^{2}+(A% (x-z))^{*}(Az-b)+\,\text{c.c.}+\|A(x-z)\|_{2}^{2}\\ &\displaystyle=\|Az-b\|_{2}^{2}+(x-z)^{*}A^{*}(Az-b)+\,\text{c.c.}+\|A(x-z)\|_% {2}^{2}\\ &\displaystyle=\|Az-b\|_{2}^{2}+\|A(x-z)\|_{2}^{2}\\ &\displaystyle\geq\|Az-b\|_{2}^{2}\end{aligned}
  61. A A
  62. m โ‰ฅ n m\geq n
  63. z z
  64. A x = b Ax=b
  65. z = A + b z=A^{+}b
  66. z z
  67. Q = A + A Q=A^{+}A
  68. Q z = A + A A + b = A + b = z Qz=A^{+}AA^{+}b=A^{+}b=z
  69. Q * = Q Q^{*}=Q
  70. A x = b Ax=b
  71. z * ( x - z ) = ( Q z ) * ( x - z ) = z * Q ( x - z ) = z * ( A + A x - z ) = z * ( A + b - z ) = 0. \begin{aligned}\displaystyle z^{*}(x-z)&\displaystyle=(Qz)^{*}(x-z)\\ &\displaystyle=z^{*}Q(x-z)\\ &\displaystyle=z^{*}(A^{+}Ax-z)\\ &\displaystyle=z^{*}(A^{+}b-z)\\ &\displaystyle=0.\end{aligned}
  72. โˆฅ x โˆฅ 2 2 = โˆฅ z โˆฅ 2 2 + 2 z * ( x - z ) + โˆฅ x - z โˆฅ 2 2 = โˆฅ z โˆฅ 2 2 + โˆฅ x - z โˆฅ 2 2 โ‰ฅ โˆฅ z โˆฅ 2 2 \begin{aligned}\displaystyle\|x\|_{2}^{2}&\displaystyle=\|z\|_{2}^{2}+2z^{*}(x% -z)+\|x-z\|_{2}^{2}\\ &\displaystyle=\|z\|_{2}^{2}+\|x-z\|_{2}^{2}\\ &\displaystyle\geq\|z\|_{2}^{2}\end{aligned}
  73. x = z x=z

Proper_acceleration.html

  1. a โ†’ a c c = a โ†’ o - a โ†’ f r a m e \vec{a}_{acc}=\vec{a}_{o}-\vec{a}_{frame}
  2. a โ†’ r o t = a โ†’ o - ฯ‰ โ†’ ร— ( ฯ‰ โ†’ ร— r โ†’ ) - 2 ฯ‰ โ†’ ร— v โ†’ r o t - d ฯ‰ โ†’ d t ร— r โ†’ \vec{a}_{rot}=\vec{a}_{o}-\vec{\omega}\times(\vec{\omega}\times\vec{r})-2\vec{% \omega}\times\vec{v}_{rot}-\frac{d\vec{\omega}}{dt}\times\vec{r}
  3. - T c e n t r i p e t a l = ฮฃ F r a d i a l = m a r a d i a l = - m v 2 r . -T_{centripetal}=\Sigma F_{radial}=ma_{radial}=-m\frac{v^{2}}{r}.
  4. m v 2 r - T c e n t r i p e t a l = ฮฃ F r o t = m a r o t = 0. m\frac{v^{2}}{r}-T_{centripetal}=\Sigma F_{rot}=ma_{rot}=0.
  5. ฮฑ = ฮ” w ฮ” t = c ฮ” ฮท ฮ” ฯ„ = c 2 ฮ” ฮณ ฮ” x \alpha=\frac{\Delta w}{\Delta t}=c\frac{\Delta\eta}{\Delta\tau}=c^{2}\frac{% \Delta\gamma}{\Delta x}
  6. ฮท = sinh - 1 ( w c ) = tanh - 1 ( v c ) = ยฑ cosh - 1 ( ฮณ ) \eta=\sinh^{-1}\left(\frac{w}{c}\right)=\tanh^{-1}\left(\frac{v}{c}\right)=\pm% \cosh^{-1}\left(\gamma\right)
  7. A ฮป := D U ฮป d ฯ„ = d U ฮป d ฯ„ + ฮ“ ฮป U ฮผ ฮผ ฮฝ U ฮฝ A^{\lambda}:=\frac{DU^{\lambda}}{d\tau}=\frac{dU^{\lambda}}{d\tau}+\Gamma^{% \lambda}{}_{\mu\nu}U^{\mu}U^{\nu}
  8. d U ฮป d ฯ„ = A ฮป - ฮ“ ฮป U ฮผ ฮผ ฮฝ U ฮฝ \frac{dU^{\lambda}}{d\tau}=A^{\lambda}-\Gamma^{\lambda}{}_{\mu\nu}U^{\mu}U^{\nu}
  9. d E d t = v โ†’ โ‹… d p โ†’ d t \frac{dE}{dt}=\vec{v}\cdot\frac{d\vec{p}}{dt}
  10. d p โ†’ d t = ฮฃ f o โ†’ + ฮฃ f g โ†’ = m ( a o โ†’ + a g โ†’ ) \frac{d\vec{p}}{dt}=\Sigma\vec{f_{o}}+\Sigma\vec{f_{g}}=m(\vec{a_{o}}+\vec{a_{% g}})
  11. a โ†’ s h e l l = a โ†’ o - r r - r s G M r 2 r ^ \vec{a}_{shell}=\vec{a}_{o}-\sqrt{\frac{r}{r-r_{s}}}\frac{GM}{r^{2}}\hat{r}
  12. a โ†’ s h e l l = a โ†’ o - g r ^ \vec{a}_{shell}=\vec{a}_{o}-g\hat{r}
  13. r ^ \hat{r}
  14. ( { ฮ“ t t t , ฮ“ t r t , ฮ“ t ฮธ t , ฮ“ t ฯ• t } { ฮ“ r t t , ฮ“ r r t , ฮ“ r ฮธ t , ฮ“ r ฯ• t } { ฮ“ ฮธ t t , ฮ“ ฮธ r t , ฮ“ ฮธ ฮธ t , ฮ“ ฮธ ฯ• t } { ฮ“ ฯ• t t , ฮ“ ฯ• r t , ฮ“ ฯ• ฮธ t , ฮ“ ฯ• ฯ• t } { ฮ“ t t r , ฮ“ t r r , ฮ“ t ฮธ r , ฮ“ t ฯ• r } { ฮ“ r t r , ฮ“ r r r , ฮ“ r ฮธ r , ฮ“ r ฯ• r } { ฮ“ ฮธ t r , ฮ“ ฮธ r r , ฮ“ ฮธ ฮธ r , ฮ“ ฮธ ฯ• r } { ฮ“ ฯ• t r , ฮ“ ฯ• r r , ฮ“ ฯ• ฮธ r , ฮ“ ฯ• ฯ• r } { ฮ“ t t ฮธ , ฮ“ t r ฮธ , ฮ“ t ฮธ ฮธ , ฮ“ t ฯ• ฮธ } { ฮ“ r t ฮธ , ฮ“ r r ฮธ , ฮ“ r ฮธ ฮธ , ฮ“ r ฯ• ฮธ } { ฮ“ ฮธ t ฮธ , ฮ“ ฮธ r ฮธ , ฮ“ ฮธ ฮธ ฮธ , ฮ“ ฮธ ฯ• ฮธ } { ฮ“ ฯ• t ฮธ , ฮ“ ฯ• r ฮธ , ฮ“ ฯ• ฮธ ฮธ , ฮ“ ฯ• ฯ• ฮธ } { ฮ“ t t ฯ• , ฮ“ t r ฯ• , ฮ“ t ฮธ ฯ• , ฮ“ t ฯ• ฯ• } { ฮ“ r t ฯ• , ฮ“ r r ฯ• , ฮ“ r ฮธ ฯ• , ฮ“ r ฯ• ฯ• } { ฮ“ ฮธ t ฯ• , ฮ“ ฮธ r ฯ• , ฮ“ ฮธ ฮธ ฯ• , ฮ“ ฮธ ฯ• ฯ• } { ฮ“ ฯ• t ฯ• , ฮ“ ฯ• r ฯ• , ฮ“ ฯ• ฮธ ฯ• , ฮ“ ฯ• ฯ• ฯ• } ) \left(\begin{array}[]{llll}\left\{\Gamma_{tt}^{t},\Gamma_{tr}^{t},\Gamma_{t% \theta}^{t},\Gamma_{t\phi}^{t}\right\}&\left\{\Gamma_{rt}^{t},\Gamma_{rr}^{t},% \Gamma_{r\theta}^{t},\Gamma_{r\phi}^{t}\right\}&\left\{\Gamma_{\theta t}^{t},% \Gamma_{\theta r}^{t},\Gamma_{\theta\theta}^{t},\Gamma_{\theta\phi}^{t}\right% \}&\left\{\Gamma_{\phi t}^{t},\Gamma_{\phi r}^{t},\Gamma_{\phi\theta}^{t},% \Gamma_{\phi\phi}^{t}\right\}\\ \left\{\Gamma_{tt}^{r},\Gamma_{tr}^{r},\Gamma_{t\theta}^{r},\Gamma_{t\phi}^{r}% \right\}&\left\{\Gamma_{rt}^{r},\Gamma_{rr}^{r},\Gamma_{r\theta}^{r},\Gamma_{r% \phi}^{r}\right\}&\left\{\Gamma_{\theta t}^{r},\Gamma_{\theta r}^{r},\Gamma_{% \theta\theta}^{r},\Gamma_{\theta\phi}^{r}\right\}&\left\{\Gamma_{\phi t}^{r},% \Gamma_{\phi r}^{r},\Gamma_{\phi\theta}^{r},\Gamma_{\phi\phi}^{r}\right\}\\ \left\{\Gamma_{tt}^{\theta},\Gamma_{tr}^{\theta},\Gamma_{t\theta}^{\theta},% \Gamma_{t\phi}^{\theta}\right\}&\left\{\Gamma_{rt}^{\theta},\Gamma_{rr}^{% \theta},\Gamma_{r\theta}^{\theta},\Gamma_{r\phi}^{\theta}\right\}&\left\{% \Gamma_{\theta t}^{\theta},\Gamma_{\theta r}^{\theta},\Gamma_{\theta\theta}^{% \theta},\Gamma_{\theta\phi}^{\theta}\right\}&\left\{\Gamma_{\phi t}^{\theta},% \Gamma_{\phi r}^{\theta},\Gamma_{\phi\theta}^{\theta},\Gamma_{\phi\phi}^{% \theta}\right\}\\ \left\{\Gamma_{tt}^{\phi},\Gamma_{tr}^{\phi},\Gamma_{t\theta}^{\phi},\Gamma_{t% \phi}^{\phi}\right\}&\left\{\Gamma_{rt}^{\phi},\Gamma_{rr}^{\phi},\Gamma_{r% \theta}^{\phi},\Gamma_{r\phi}^{\phi}\right\}&\left\{\Gamma_{\theta t}^{\phi},% \Gamma_{\theta r}^{\phi},\Gamma_{\theta\theta}^{\phi},\Gamma_{\theta\phi}^{% \phi}\right\}&\left\{\Gamma_{\phi t}^{\phi},\Gamma_{\phi r}^{\phi},\Gamma_{% \phi\theta}^{\phi},\Gamma_{\phi\phi}^{\phi}\right\}\end{array}\right)
  15. ( { 0 , r s 2 r ( r - r s ) , 0 , 0 } { r s 2 r ( r - r s ) , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { r s c 2 ( r - r s ) 2 r 3 , 0 , 0 , 0 } { 0 , r s 2 r ( r s - r ) , 0 , 0 } { 0 , 0 , r s - r , 0 } { 0 , 0 , 0 , ( r s - r ) sin 2 ฮธ } { 0 , 0 , 0 , 0 } { 0 , 0 , 1 r , 0 } { 0 , 1 r , 0 , 0 } { 0 , 0 , 0 , - cos ฮธ sin ฮธ } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 1 r } { 0 , 0 , 0 , cot ( ฮธ ) } { 0 , 1 r , cot ฮธ , 0 } ) \left(\begin{array}[]{llll}\left\{0,\frac{r_{s}}{2r(r-r_{s})},0,0\right\}&% \left\{\frac{r_{s}}{2r(r-r_{s})},0,0,0\right\}&\{0,0,0,0\}&\{0,0,0,0\}\\ \left\{\frac{r_{s}c^{2}(r-r_{s})}{2r^{3}},0,0,0\right\}&\left\{0,\frac{r_{s}}{% 2r(r_{s}-r)},0,0\right\}&\{0,0,r_{s}-r,0\}&\left\{0,0,0,(r_{s}-r)\sin^{2}% \theta\right\}\\ \{0,0,0,0\}&\left\{0,0,\frac{1}{r},0\right\}&\left\{0,\frac{1}{r},0,0\right\}&% \{0,0,0,-\cos\theta\sin\theta\}\\ \{0,0,0,0\}&\left\{0,0,0,\frac{1}{r}\right\}&\{0,0,0,\cot(\theta)\}&\left\{0,% \frac{1}{r},\cot\theta,0\right\}\end{array}\right)
  16. A ฮป = ฮ“ ฮป U ฮผ ฮผ ฮฝ U ฮฝ = { 0 , G M / r 2 , 0 , 0 } A^{\lambda}=\Gamma^{\lambda}{}_{\mu\nu}U^{\mu}U^{\nu}=\{0,GM/r^{2},0,0\}
  17. ฮฑ = 1 / ( 1 - r s / r ) G M / r 2 \alpha=\sqrt{1/(1-r_{s}/r)}GM/r^{2}
  18. ( { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 0 } { 0 , 0 , - r , 0 } { 0 , 0 , 0 , - r sin 2 ฮธ } { 0 , 0 , 0 , 0 } { 0 , 0 , 1 r , 0 } { 0 , 1 r , 0 , 0 } { 0 , 0 , 0 , - cos ฮธ sin ฮธ } { 0 , 0 , 0 , 0 } { 0 , 0 , 0 , 1 r } { 0 , 0 , 0 , cot ฮธ } { 0 , 1 r , cot ฮธ , 0 } ) \left(\begin{array}[]{llll}\left\{0,0,0,0\right\}&\left\{0,0,0,0\right\}&\{0,0% ,0,0\}&\{0,0,0,0\}\\ \left\{0,0,0,0\right\}&\left\{0,0,0,0\right\}&\{0,0,-r,0\}&\left\{0,0,0,-r\sin% ^{2}\theta\right\}\\ \{0,0,0,0\}&\left\{0,0,\frac{1}{r},0\right\}&\left\{0,\frac{1}{r},0,0\right\}&% \{0,0,0,-\cos\theta\sin\theta\}\\ \{0,0,0,0\}&\left\{0,0,0,\frac{1}{r}\right\}&\{0,0,0,\cot\theta\}&\left\{0,% \frac{1}{r},\cot\theta,0\right\}\end{array}\right)
  19. A ฮป = ฮ“ ฮป U ฮผ ฮผ ฮฝ U ฮฝ = { 0 , - r ( d ฯ• / d ฯ„ ) 2 , 0 , 0 } A^{\lambda}=\Gamma^{\lambda}{}_{\mu\nu}U^{\mu}U^{\nu}=\{0,-r(d\phi/d\tau)^{2},% 0,0\}

Proper_equilibrium.html

  1. ฯต > 0 \epsilon>0
  2. ฯƒ \sigma
  3. ฯต \epsilon
  4. u ( s , ฯƒ - i ) < u ( s โ€ฒ , ฯƒ - i ) u(s,\sigma_{-i})<u(s^{\prime},\sigma_{-i})
  5. ฯต \epsilon
  6. ฯต \epsilon
  7. ฯต \epsilon

Properties_of_polynomial_roots.html

  1. a 0 + a 1 x + โ‹ฏ + a n x n , a n โ‰  0 , a_{0}+a_{1}x+\cdots+a_{n}x^{n},\quad a_{n}\not=0,
  2. โ„‚ \mathbb{C}
  3. n n
  4. p p
  5. p = a 0 + a 1 x + โ‹ฏ + a n x n . p=a_{0}+a_{1}x+\cdots+a_{n}x^{n}.
  6. p p
  7. p = 0 p=0
  8. a a
  9. p ( a ) = 0 p(a)=0
  10. p p
  11. b \sqrt{b}
  12. ( x - [ a + b ] ) ( x - [ a - b ] ) = ( x - a ) 2 - b . \left(x-\left[a+\sqrt{b}\right]\right)\left(x-\left[a-\sqrt{b}\right]\right)=(% x-a)^{2}-b.
  13. P ( x ) = D ( x ) Q ( x ) + c x + d = ( ( x - a ) 2 - b ) Q ( x ) + c x + d , P(x)=D(x)Q(x)+cx+d=((x-a)^{2}-b)Q(x)+cx+d,\,\!
  14. P ( a + b ) = c ( a + b ) + d = ( a c + d ) + c b = 0. P\left(a+\sqrt{b}\right)=c\left(a+\sqrt{b}\right)+d=(ac+d)+c\sqrt{b}=0.
  15. | a k | R k > | a 0 | + โ‹ฏ + | a k - 1 | R k - 1 + | a k + 1 | R k + 1 + โ‹ฏ + | a n | R n |a_{k}|\,R^{k}>|a_{0}|+\cdots+|a_{k-1}|\,R^{k-1}+|a_{k+1}|\,R^{k+1}+\cdots+|a_% {n}|\,R^{n}
  16. R = 1 + 1 | a n | max { | a 0 | , | a 1 | , โ€ฆ , | a n - 1 | } R=1+\frac{1}{|a_{n}|}\max\{|a_{0}|,|a_{1}|,\dots,|a_{n-1}|\}
  17. R = max ( 1 , 1 | a n | ( | a 0 | + | a 1 | + โ‹ฏ + | a n - 1 | ) ) R=\max\left(1,\,\frac{1}{|a_{n}|}\left(|a_{0}|+|a_{1}|+\cdots+|a_{n-1}|\right)\right)
  18. R = | a 0 | | a 0 | + max { | a 1 | , | a 2 | , โ€ฆ , | a n | } R=\frac{|a_{0}|}{|a_{0}|+\max\{|a_{1}|,|a_{2}|,\dots,|a_{n}|\}}
  19. R = | a 0 | max ( | a 0 | , | a 1 | + | a 2 | + โ‹ฏ + | a n | ) R=\frac{|a_{0}|}{\max(|a_{0}|,\,|a_{1}|+|a_{2}|+\cdots+|a_{n}|)}
  20. h ( R ) = | a 0 | R - k + โ‹ฏ + | a k - 1 | R - 1 - | a k | + | a k + 1 | R + โ‹ฏ + | a n | R n - k h(R)=|a_{0}|\,R^{-k}+\cdots+|a_{k-1}|\,R^{-1}-|a_{k}|+|a_{k+1}|\,R+\cdots+|a_{% n}|\,R^{n-k}
  21. 2 max { | a n - 1 a n | , | a n - 2 a n | 1 2 , โ‹ฏ , | a 0 2 a n | 1 n } 2\max\left\{\left|\frac{a_{n-1}}{a_{n}}\right|,\left|\frac{a_{n-2}}{a_{n}}% \right|^{\frac{1}{2}},\cdots,\left|\frac{a_{0}}{2a_{n}}\right|^{\frac{1}{n}}\right\}
  22. 2 max { | a n - 1 a n | , | a n - 2 a n - 1 | , โ‹ฏ , | a 0 2 a 1 | } 2\max\left\{\left|\frac{a_{n-1}}{a_{n}}\right|,\left|\frac{a_{n-2}}{a_{n-1}}% \right|,\cdots,\left|\frac{a_{0}}{2a_{1}}\right|\right\}
  23. 1 + max { | a 0 a n | , | a 1 a n | , โ‹ฏ , | a n - 1 a n | } 1+\max\left\{\left|\frac{a_{0}}{a_{n}}\right|,\left|\frac{a_{1}}{a_{n}}\right|% ,\cdots,\left|\frac{a_{n-1}}{a_{n}}\right|\right\}
  24. max { 1 , โˆ‘ i = 0 n - 1 | a i a n | } \max\left\{1,\sum_{i=0}^{n-1}\left|\frac{a_{i}}{a_{n}}\right|\right\}
  25. โˆ‘ i = 0 n - 1 | a i a i + 1 | \sum_{i=0}^{n-1}\left|\frac{a_{i}}{a_{i+1}}\right|
  26. max { 1 + | a n - 1 | | a n | , | a n - 2 | | a n | , โ€ฆ , | a 0 | | a n | } . \max\left\{1+\frac{|a_{n-1}|}{|a_{n}|},\frac{|a_{n-2}|}{|a_{n}|},\dots,\frac{|% a_{0}|}{|a_{n}|}\right\}.
  27. 0 โ‰ค i โ‰ค m 0โ‰คiโ‰คm
  28. d d
  29. d 1 = 1 2 ( ( | a n - 1 | - 1 ) + ( | a n - 1 | - 1 ) 2 + 4 a ) , a = max { | a i | } . d_{1}=\tfrac{1}{2}\left((|a_{n-1}|-1)+\sqrt{(|a_{n-1}|-1)^{2}+4a}\right),% \qquad a=\max\{|a_{i}|\}.
  30. Q ( x ) = x 3 + ( 2 - | a n - 1 | ) x 2 + ( 1 - | a n - 1 | - | a n - 2 | ) x - a , a = max { | a i | } Q(x)=x^{3}+(2-|a_{n-1}|)x^{2}+(1-|a_{n-1}|-|a_{n-2}|)x-a,\qquad a=\max\{|a_{i}|\}
  31. ฮถ ฮถ
  32. z n + a n - 1 z n - 1 + โ‹ฏ + a 1 z + a 0 ; z^{n}+a_{n-1}z^{n-1}+\cdots+a_{1}z+a_{0};
  33. | ฮถ | > 1 |ฮถ|>1
  34. - ฮถ n = a n - 1 ฮถ n - 1 + โ‹ฏ + a 1 ฮถ + a 0 , -\zeta^{n}=a_{n-1}\zeta^{n-1}+\cdots+a_{1}\zeta+a_{0},
  35. | ฮถ | n โ‰ค โˆฅ a โˆฅ p โˆฅ ( ฮถ n - 1 , โ‹ฏ , ฮถ , 1 ) โˆฅ q . |\zeta|^{n}\leq\|a\|_{p}\left\|(\zeta^{n-1},\cdots,\zeta,1)\right\|_{q}.
  36. p = 1 p=1
  37. | ฮถ | n โ‰ค โˆฅ a โˆฅ 1 max { | ฮถ | n - 1 , โ‹ฏ , | ฮถ | , 1 } = โˆฅ a โˆฅ 1 | ฮถ | n - 1 , |\zeta|^{n}\leq\|a\|_{1}\max\left\{|\zeta|^{n-1},\cdots,|\zeta|,1\right\}=\|a% \|_{1}|\zeta|^{n-1},
  38. | ฮถ | โ‰ค max { 1 , โˆฅ a โˆฅ 1 } . |\zeta|\leq\max\{1,\|a\|_{1}\}.
  39. | ฮถ | n โ‰ค โˆฅ a โˆฅ p ( | ฮถ | q ( n - 1 ) + โ‹ฏ + | ฮถ | q + 1 ) 1 q = โˆฅ a โˆฅ p ( | ฮถ | q n - 1 | ฮถ | q - 1 ) 1 q โ‰ค โˆฅ a โˆฅ p ( | ฮถ | q n | ฮถ | q - 1 ) 1 q , |\zeta|^{n}\leq\|a\|_{p}\left(|\zeta|^{q(n-1)}+\cdots+|\zeta|^{q}+1\right)^{% \frac{1}{q}}=\|a\|_{p}\left(\frac{|\zeta|^{qn}-1}{|\zeta|^{q}-1}\right)^{\frac% {1}{q}}\leq\|a\|_{p}\left(\frac{|\zeta|^{qn}}{|\zeta|^{q}-1}\right)^{\frac{1}{% q}},
  40. | ฮถ | n q โ‰ค โˆฅ a โˆฅ p q | ฮถ | q n | ฮถ | q - 1 |\zeta|^{nq}\leq\|a\|_{p}^{q}\frac{|\zeta|^{qn}}{|\zeta|^{q}-1}
  41. | ฮถ | q โ‰ค 1 + โˆฅ a โˆฅ p q . |\zeta|^{q}\leq 1+\|a\|_{p}^{q}.
  42. | ฮถ | โ‰ค โˆฅ ( 1 , โˆฅ a โˆฅ p ) โˆฅ q = R p , |\zeta|\leq\left\|\left(1,\|a\|_{p}\right)\right\|_{q}=R_{p},
  43. 1 โ‰ค p โ‰ค โˆž 1โ‰คpโ‰คโˆž
  44. z 1 , โ€ฆ , z n z_{1},\ldots,z_{n}
  45. n n
  46. p p
  47. M ( p ) = | a n | โˆ j = 1 n max ( 1 , | z j | ) . M(p)=|a_{n}|\prod_{j=1}^{n}\max(1,|z_{j}|).
  48. M ( p ) โ‰ค | a 0 | 2 + | a 1 | 2 + โ‹ฏ | a n | 2 . M(p)\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+\cdots|a_{n}|^{2}}\,.
  49. q = b m x m + โ‹ฏ + b 0 q=b_{m}x^{m}+\cdots+b_{0}
  50. p p
  51. | b 0 | + | b 1 | + โ‹ฏ | b m | โ‰ค 2 m | b m a n | M ( p ) . |b_{0}|+|b_{1}|+\cdots|b_{m}|\leq 2^{m}\,\left|\frac{b_{m}}{a_{n}}\right|\,M(p% )\,.
  52. max | z | โ‰ค 1 | P โ€ฒ ( z ) | โ‰ค n max | z | โ‰ค 1 | P ( z ) | . \max_{|z|\leq 1}\big|P^{\prime}(z)\big|\leq n\max_{|z|\leq 1}\big|P(z)\big|.
  53. a n x n + a n - 1 x n - 1 + โ€ฆ + a 1 x + a 0 a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{1}x+a_{0}
  54. x ยฑ = - a n - 1 n a n ยฑ n - 1 n a n a n - 1 2 - 2 n n - 1 a n a n - 2 . x_{\pm}=-\frac{a_{n-1}}{na_{n}}\pm\frac{n-1}{na_{n}}\sqrt{a^{2}_{n-1}-\frac{2n% }{n-1}a_{n}a_{n-2}}.
  55. x 4 + 5 x 3 + 5 x 2 - 5 x - 6 x^{4}+5x^{3}+5x^{2}-5x-6
  56. x ยฑ = - 5 4 ยฑ 3 4 35 3 , x_{\pm}=-\frac{5}{4}\pm\frac{3}{4}\sqrt{\frac{35}{3}},
  57. I = [ - 3.8117 , 1.3117 ] I=[-3.8117,1.3117]
  58. p ( x ) = a n x n + a n - 1 x n - 1 + โ‹ฏ + a 2 x 2 + a 1 x + a 0 p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}
  59. m ( x ) = A ( x ) C ( x ) - B ( x ) 2 ฯ€ A ( x ) m(x)=\frac{\sqrt{A(x)C(x)-B(x)^{2}}}{\pi A(x)}
  60. A ( x ) \displaystyle A(x)
  61. m ( x ) = 1 ฯ€ | 1 - x 2 | m(x)=\frac{1}{\pi|1-x^{2}|}
  62. m ( ยฑ 1 ) = 1 ฯ€ n 2 - 1 12 m(\pm 1)=\frac{1}{\pi}\sqrt{\frac{n^{2}-1}{12}}
  63. N n = 2 ฯ€ ln n + C + O ( n - 2 ) N_{n}=\frac{2}{\pi}\ln n+C+O(n^{-2})

Proto-Indo-European_root.html

  1. root + suffix โŸ stem + ending โŸ word \underbrace{\underbrace{\mathrm{root+suffix}}_{\mathrm{stem}}+\mathrm{ending}}% _{\mathrm{word}}

Proximity_effect_(electromagnetism).html

  1. R A C = R D C ( R e ( M ) + ( m 2 - 1 ) R e ( D ) 3 ) R_{AC}=R_{DC}\bigg(Re(M)+\frac{(m^{2}-1)Re(D)}{3}\bigg)
  2. M = ฮฑ h coth ( ฮฑ h ) M=\alpha h\coth(\alpha h)\,
  3. D = 2 ฮฑ h tanh ( ฮฑ h / 2 ) D=2\alpha h\tanh(\alpha h/2)\,
  4. ฮฑ = j ฯ‰ ฮผ 0 ฮท ฯ \alpha=\sqrt{\frac{j\omega\mu_{0}\eta}{\rho}}
  5. ฯ‰ \omega
  6. ฯ \rho
  7. ฮท = N l a b \eta=N_{l}\frac{a}{b}
  8. ๐ƒ = ฮณ 1 โŸจ [ | B โ†’ 1 ^ | 2 B โ†’ 1 ^ โ‹… B โ†’ 2 ^ B โ†’ 2 ^ โ‹… B โ†’ 1 ^ | B โ†’ 2 ^ | 2 ] โŸฉ 1 + ฮณ 2 โŸจ [ | B โ†’ 1 ^ | 2 B โ†’ 1 ^ โ‹… B โ†’ 2 ^ B โ†’ 2 ^ โ‹… B โ†’ 1 ^ | B โ†’ 2 ^ | 2 ] โŸฉ 2 \mathbf{D}=\gamma_{1}\left\langle\begin{bmatrix}\left|\hat{\vec{B}_{1}}\right|% ^{2}&\hat{\vec{B}_{1}}\cdot\hat{\vec{B}_{2}}\\ \hat{\vec{B}_{2}}\cdot\hat{\vec{B}_{1}}&\left|\hat{\vec{B}_{2}}\right|^{2}\end% {bmatrix}\right\rangle_{1}+\gamma_{2}\left\langle\begin{bmatrix}\left|\hat{% \vec{B}_{1}}\right|^{2}&\hat{\vec{B}_{1}}\cdot\hat{\vec{B}_{2}}\\ \hat{\vec{B}_{2}}\cdot\hat{\vec{B}_{1}}&\left|\hat{\vec{B}_{2}}\right|^{2}\end% {bmatrix}\right\rangle_{2}
  9. B โ†’ j ^ \hat{\vec{B}_{j}}
  10. ฮณ j = ฯ€ N j l t , j d c , j 4 64 ฯ c \gamma_{j}=\frac{\pi N_{j}l_{t,j}d_{c,j}^{4}}{64\rho_{c}}
  11. N j N_{j}
  12. l t , j l_{t,j}
  13. d c , j d_{c,j}
  14. ฯ c \rho_{c}
  15. P = [ d i 1 d t d i 2 d t ] ๐ƒ [ d i 1 d t d i 2 d t ] ยฏ P=\overline{\begin{bmatrix}\frac{di_{1}}{dt}\frac{di_{2}}{dt}\end{bmatrix}% \mathbf{D}\begin{bmatrix}\frac{di_{1}}{dt}\\ \frac{di_{2}}{dt}\end{bmatrix}}
  16. I r m s 2 ร— R D C I_{rms}^{2}\times R_{DC}

Proximity_effect_(electron_beam_lithography).html

  1. ฮฑ \alpha
  2. ฮฒ \beta
  3. P S F ( r ) = 1 ฯ€ ( 1 + ฮท ) [ 1 ฮฑ 2 e - r 2 ฮฑ 2 + ฮท ฮฒ 2 e - r 2 ฮฒ 2 ] PSF(r)=\frac{1}{\pi(1+\eta)}\left[\frac{1}{\alpha^{2}}e^{-\frac{r^{2}}{\alpha^% {2}}}+\frac{\eta}{\beta^{2}}e^{-\frac{r^{2}}{\beta^{2}}}\right]
  4. ฮท \eta
  5. ฮฑ \alpha
  6. ฮฒ \beta
  7. ฮท \eta

Proximity_effect_(superconductivity).html

  1. T c T_{c}
  2. T c T_{c}
  3. T c T_{c}

Pruฬˆfer_domain.html

  1. I โ‹… I - 1 = R \ I\cdot I^{-1}=R
  2. I - 1 = { r โˆˆ q ( R ) : r I โІ R } I^{-1}=\{r\in q(R):rI\subseteq R\}
  3. q ( R ) \ q(R)
  4. I โˆฉ ( J + K ) = ( I โˆฉ J ) + ( I โˆฉ K ) . I\cap(J+K)=(I\cap J)+(I\cap K).
  5. I ( J โˆฉ K ) = I J โˆฉ I K . I(J\cap K)=IJ\cap IK.
  6. ( I + J ) ( I โˆฉ J ) = I J . (I+J)(I\cap J)=IJ.

Pseudo-Euclidean_space.html

  1. n n
  2. q q
  3. q ( x ) = ( x 1 2 + โ‹ฏ + x k 2 ) - ( x k + 1 2 + โ‹ฏ + x n 2 ) q(x)=\left(x_{1}^{2}+\cdots+x_{k}^{2}\right)-\left(x_{k+1}^{2}+\cdots+x_{n}^{2% }\right)
  4. x x
  5. k = n k=n
  6. q q
  7. i โ‰ค k iโ‰คk
  8. j > k j>k
  9. k > 1 k>1
  10. k = 1 k=1
  11. U U
  12. q q
  13. U U
  14. d i m U โ‰ฅ 2 dimUโ‰ฅ2
  15. d i m U = 2 dimU=2
  16. U U
  17. ฮฝ \mathbf{ฮฝ}
  18. N = โŸจ ฮฝ โŸฉ N=\langle\mathbf{ฮฝ}\rangle
  19. N N
  20. U + U < s u p > โŠฅ = U+U<sup>โŠฅ=
  21. U U
  22. N N
  23. q q
  24. N N
  25. m i n ( k , n โˆ’ k ) min(k,nโˆ’k)
  26. k k
  27. ( n โˆ’ k ) (nโˆ’k)
  28. U U
  29. q ( x ) + q ( y ) = 1 2 ( q ( x + y ) + q ( x - y ) ) . q(x)+q(y)=\frac{1}{2}(q(x+y)+q(x-y)).
  30. q ( x + y ) = q ( x ) + q ( y ) + 2 โŸจ x , y โŸฉ . q(x+y)=q(x)+q(y)+2\langle x,y\rangle.
  31. โŸจ x , y โŸฉ = 0 โ‡’ q ( x ) + q ( y ) = q ( x + y ) . \langle x,y\rangle=0\Rightarrow q(x)+q(y)=q(x+y).
  32. | โŸจ x , y โŸฉ | |\langle x,y\rangle|
  33. | q ( x ) q ( y ) | \sqrt{|}{q}{(}{x}{)}{q}{(}{y}{)}{|}
  34. k = 1 k=1
  35. q q
  36. | โŸจ x , y โŸฉ | โ‰ฅ q ( x ) q ( y ) , |\langle x,y\rangle|\geq\sqrt{q(x)q(y)}\,,
  37. arcosh | โŸจ x , y โŸฉ | q ( x ) q ( y ) . \operatorname{arcosh}\frac{|\langle x,y\rangle|}{\sqrt{q(x)q(y)}}\,.
  38. ( n โˆ’ 1 ) (nโˆ’1)
  39. [ 0 , + โˆž ) [0,โ€‰+โˆž)
  40. q q
  41. โˆ’ q โˆ’q
  42. v ฮฑ = q ฮฑ ฮฒ v ฮฒ , v_{\alpha}=q_{\alpha\beta}v^{\beta}\,,
  43. q ฮฑ ฮฒ = ( I k ร— k 0 0 - I ( n - k ) ร— ( n - k ) ) q_{\alpha\beta}=\begin{pmatrix}I_{k\times k}&0\\ 0&-I_{(n-k)\times(n-k)}\end{pmatrix}
  44. k k
  45. n โˆ’ k nโˆ’k
  46. n = 4 n=4
  47. k = 3 k=3
  48. q ( x ) = x 1 2 + x 2 2 + x 3 2 - x 4 2 , q(x)=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2},
  49. z = x + y j z=x+yโ€‰j
  50. โˆฅ z โˆฅ = z z * = z * z = x 2 - y 2 . \lVert z\rVert=zz^{*}=z^{*}z=x^{2}-y^{2}.
  51. n = 2 n=2
  52. k = 1 k=1
  53. S O < s u p > + ( 1 , 1 ) SO<sup>+(1,โ€‰1)
  54. ๐‘ < s u p > n \mathbf{R}<sup>n
  55. U โˆฉ U < s u p > โŠฅ UโˆฉU<sup>โŠฅ
  56. c o s ( i โ‹… a r c o s h s ) = s cos(iโ‹…arcoshs)=s
  57. s > 0 s>0
  58. k = 1 k=1
  59. 0
  60. q ( x ) = x < s u b > 0 2 โˆ’ x 1 2 โˆ’ x 2 2 โˆ’ x 3 2 q(x)=x<sub>0^{2}โˆ’x_{1}^{2}โˆ’x_{2}^{2}โˆ’x_{3}^{2}

Pseudo-monotone_operator.html

  1. u j โ‡€ u in X as j โ†’ โˆž u_{j}\rightharpoonup u\mbox{ in }~{}X\mbox{ as }~{}j\to\infty
  2. lim sup j โ†’ โˆž โŸจ T ( u j ) , u j - u โŸฉ โ‰ค 0 , \limsup_{j\to\infty}\langle T(u_{j}),u_{j}-u\rangle\leq 0,
  3. lim inf j โ†’ โˆž โŸจ T ( u j ) , u j - v โŸฉ โ‰ฅ โŸจ T ( u ) , u - v โŸฉ . \liminf_{j\to\infty}\langle T(u_{j}),u_{j}-v\rangle\geq\langle T(u),u-v\rangle.

PTAS_reduction.html

  1. A โ‰ค PTAS B \,\text{A}\leq_{\,\text{PTAS}}\,\text{B}
  2. f ( x ) f(x)
  3. f ( x ) f(x)
  4. 1 + ฮฑ ( ฯต ) 1+\alpha(\epsilon)
  5. g ( x , y , ฯต ) g(x,y,\epsilon)
  6. 1 + ฯต 1+\epsilon
  7. A โ‰ค PTAS B \,\text{A}\leq_{\,\text{PTAS}}\,\text{B}
  8. B โˆˆ PTAS โŸน A โˆˆ PTAS \,\text{B}\in\,\text{PTAS}\implies\,\text{A}\in\,\text{PTAS}
  9. A โ‰ค PTAS B \,\text{A}\leq_{\,\text{PTAS}}\,\text{B}
  10. A โˆ‰ PTAS โŸน B โˆ‰ PTAS \,\text{A}\not\in\,\text{PTAS}\implies\,\text{B}\not\in\,\text{PTAS}

Pu's_inequality.html

  1. โ„ โ„™ 2 \mathbb{RP}^{2}
  2. sys 2 โ‰ค ฯ€ 2 area ( โ„ โ„™ 2 ) , \operatorname{sys}^{2}\leq\frac{\pi}{2}\operatorname{area}(\mathbb{RP}^{2}),
  3. S 2 S^{2}
  4. p , q โˆˆ S 2 p,q\in S^{2}
  5. d = d ( p , q ) d=d(p,q)
  6. d 2 โ‰ค ฯ€ 4 area ( S 2 ) . d^{2}\leq\frac{\pi}{4}\operatorname{area}(S^{2}).
  7. 2 ฯ€ 2\pi
  8. 2 2
  9. 2 2
  10. S 2 โŠ‚ โ„ 3 S^{2}\subset\mathbb{R}^{3}
  11. S 1 S^{1}
  12. 2 ฯ€ 2\pi
  13. S 1 S^{1}
  14. 2 2
  15. ฯ€ \pi
  16. S 1 S^{1}
  17. 2 2
  18. 2 ฯ€ 2\pi
  19. L 2 โ‰ฅ 4 ฯ€ A L^{2}\geq 4\pi A
  20. L L
  21. A A

Publicly_Verifiable_Secret_Sharing.html

  1. s s
  2. D D
  3. s 1 , s 2 โ€ฆ s n s_{1},s_{2}...s_{n}
  4. P 1 , P 2 โ€ฆ P n P_{1},P_{2}...P_{n}
  5. E i ( s i ) E_{i}(s_{i})
  6. P i P_{i}
  7. proof D \mathrm{proof}_{D}
  8. E i E_{i}
  9. s i s_{i}
  10. proof D \mathrm{proof}_{D}
  11. s s
  12. E i E_{i}
  13. P i P_{i}
  14. s i s_{i}
  15. E i ( s i ) E_{i}(s_{i})
  16. E i ( s i ) E_{i}(s_{i})
  17. s i s_{i}
  18. s i s_{i}
  19. proof P i \mathrm{proof}_{P_{i}}
  20. proof P i \mathrm{proof}_{P_{i}}
  21. E i ( s i ) E_{i}(s_{i})
  22. s s
  23. log g 1 h 1 = log g 2 h 2 \log_{{}_{g1}}h_{1}=\log_{{}_{g2}}h_{2}
  24. r โˆˆ s y m b o l \Zeta q * r\in symbol{\Zeta}_{q^{*}}
  25. c โˆˆ R s y m b o l \Zeta q c\in_{R}symbol{\Zeta}_{q}
  26. s = r - c x ( mod q ) s=r-cx(\mathrm{mod}\,q)
  27. ฮฑ 1 = g 1 s h 1 c \alpha_{1}=g_{1}^{s}h_{1}^{c}
  28. ฮฑ 2 = g 2 s h 2 c \alpha_{2}=g_{2}^{s}h_{2}^{c}
  29. dleq ( g 1 , h 1 , g 2 , h 2 ) \mathrm{dleq}(g_{1},h_{1},g_{2},h_{2})
  30. dleq ( g 1 , h 1 , g 2 , h 2 ) \mathrm{dleq}(g_{1},h_{1},g_{2},h_{2})
  31. dleq ( X , Y , g 1 , h 1 , g 2 , h 2 ) \,\text{dleq}(X,Y,g_{1},h_{1},g_{2},h_{2})
  32. X = g 1 x 1 g 2 x 2 X=g_{1}^{x_{1}}g_{2}^{x_{2}}
  33. Y = h 1 x 1 h 2 x 2 Y=h_{1}^{x_{1}}h_{2}^{x_{2}}
  34. r 1 , r 2 โˆˆ Z q * r_{1},r_{2}\in Z_{q}^{*}
  35. t 1 = g 1 r 1 g 2 r 2 t_{1}=g_{1}^{r_{1}}g_{2}^{r_{2}}
  36. t 2 = h 1 r 1 h 2 r 2 t_{2}=h_{1}^{r_{1}}h_{2}^{r_{2}}
  37. c โˆˆ R s y m b o l \Zeta q c\in_{R}symbol{\Zeta}_{q}
  38. s 1 = r 1 - c x 1 ( mod q ) s_{1}=r_{1}-cx_{1}(\mathrm{mod}\,q)
  39. s 2 = r 2 - c x 2 ( mod q ) s_{2}=r_{2}-cx_{2}(\mathrm{mod}\,q)
  40. t 1 = X c g 1 s 1 g 2 s 2 t_{1}=X^{c}g_{1}^{s_{1}}g_{2}^{s_{2}}
  41. t 2 = Y c h 1 s 1 h 2 s 2 t_{2}=Y^{c}h_{1}^{s_{1}}h_{2}^{s_{2}}
  42. c c
  43. m m

Publius_Publishing_System.html

  1. n a m e i = w r a p ( H ( M * s h a r e i ) ) name_{i}\ =wrap(H(M*share_{i}))
  2. M * s h a r e i M*share_{i}
  3. s h a r e i share_{i}
  4. l o c a t i o n i = ( n a m e i m o d m ) + 1 location_{i}=(name_{i}\ mod\ m)+1
  5. d โ‰ฅ k d>=k
  6. l o c a t i o n i location_{i}
  7. n a m e i name_{i}
  8. E ( M , K ) E(M,K)
  9. s h a r e i share_{i}
  10. n a m e i name_{i}
  11. l o c a t i o n i location_{i}
  12. n a m e i name_{i}
  13. s h a r e i share_{i}
  14. n a m e i name_{i}
  15. n a m e i name_{i}
  16. l o c a t i o n i = ( n a m e i m o d m ) + 1 location_{i}=(name_{i}\ mod\ m)+1
  17. n a m e i name_{i}
  18. n a m e i name_{i}
  19. s h a r e i share_{i}
  20. w r a p ( H ( M * s h a r e i ) ) wrap(H(M*share_{i}))
  21. n a m e i name_{i}
  22. H ( S D N * P A S S W D ) H(SDN*PASSWD)
  23. n a m e i name_{i}
  24. n a m e i name_{i}
  25. n a m e i name_{i}
  26. n a m e i name_{i}
  27. h t t p : / / ! a n o n ! / o p t i o n s n a m e 1 n a m e 2 โ€ฆ n a m e n http://!anon!/\ options\ name_{1}\ name_{2}\ ...\ name_{n}
  28. n a m e 1 n a m e 2 โ€ฆ n a m e n name_{1}name_{2}...name_{n}

Pullback.html

  1. f ( y ( x ) ) โ‰ก g ( x ) f(y(x))\equiv g(x)\,
  2. f * s = s โˆ˜ f f^{*}s=s\circ f

Pulmonary_compliance.html

  1. C o m p l i a n c e = ฮ” V ฮ” P Compliance=\frac{\Delta V}{\Delta P}
  2. C o m p l i a n c e = ฮ” V ฮ” P = .5 L ( - 5 c m H 2 O - ( - 10 c m H 2 O ) ) = .5 L 5 c m H 2 O = 0.1 L ร— c m H 2 O - 1 Compliance=\frac{\Delta V}{\Delta P}=\frac{.5\;L}{(-5\;cmH_{2}O-(-10\;cmH_{2}O% ))}=\frac{.5\;L}{5\;cmH_{2}O}=0.1\;L\;\times\;cmH_{2}O^{-1}
  3. C s t a t = < m t p l > V T P p l a t - P E E P C_{stat}=\frac{<}{m}tpl>{{V_{T}}}{{P_{plat}-PEEP}}
  4. C d y n = < m t p l > V T P I P - P E E P C_{dyn}=\frac{<}{m}tpl>{{V_{T}}}{{PIP-PEEP}}

Pulsating_white_dwarf.html

  1. โ‰ฒ 0.2 M โŠ™ \lesssim 0.2M_{\odot}

Pulse_compression.html

  1. A A
  2. f 0 f_{0}
  3. T \scriptstyle T
  4. s s
  5. t = 0 t\,=\,0
  6. s ( t ) = { A e 2 i ฯ€ f 0 t if 0 โ‰ค t < T 0 otherwise s(t)=\left\{\begin{array}[]{ll}Ae^{2i\pi f_{0}t}&\,\text{if}\;0\leq t<T\\ 0&\,\text{otherwise}\end{array}\right.
  7. r ( t ) \scriptstyle r(t)
  8. B ( t ) \scriptstyle B(t)
  9. < s , r > ( t ) \scriptstyle<s,\,r>(t)
  10. โŸจ s , r โŸฉ ( t ) = โˆซ t โ€ฒ = 0 + โˆž s โ‹† ( t โ€ฒ ) r ( t + t โ€ฒ ) d t โ€ฒ \langle s,r\rangle(t)=\int_{t^{\prime}\,=\,0}^{+\infty}s^{\star}(t^{\prime})r(% t+t^{\prime})dt^{\prime}
  11. t r \scriptstyle t_{r}
  12. K \scriptstyle K
  13. r ( t ) = { K A e 2 i ฯ€ f 0 ( t - t r ) + B ( t ) if t r โ‰ค t < t r + T B ( t ) otherwise r(t)=\left\{\begin{array}[]{ll}KAe^{2i\pi f_{0}(t\,-\,t_{r})}+B(t)&\mbox{if}~{% }\;t_{r}\leq t<t_{r}+T\\ B(t)&\mbox{otherwise}\end{array}\right.
  14. โŸจ s , r โŸฉ ( t ) = K A 2 ฮ› ( t - t r T ) e 2 i ฯ€ f 0 ( t - t r ) + B โ€ฒ ( t ) \langle s,r\rangle(t)=KA^{2}\Lambda\left(\frac{t-t_{r}}{T}\right)e^{2i\pi f_{0% }(t\,-\,t_{r})}+B^{\prime}(t)
  15. B โ€ฒ ( t ) \scriptstyle B^{\prime}(t)
  16. ฮ› \Lambda
  17. [ - โˆž , - 1 2 ] โˆช [ 1 2 , + โˆž ] \scriptstyle[-\infty,\,-\frac{1}{2}]\,\cup\,[\frac{1}{2},\,+\infty]
  18. [ - 1 2 , 0 ] \scriptstyle[-\frac{1}{2},\,0]
  19. [ 0 , 1 2 ] \scriptstyle[0,\,\frac{1}{2}]
  20. T = 1 \scriptstyle T\,=\,1
  21. f 0 = 10 \scriptstyle f_{0}\,=\,10
  22. 1 / 2 {1}/{2}
  23. T \scriptstyle T
  24. T \scriptstyle T
  25. c T \scriptstyle cT
  26. 1 2 c T \scriptstyle\frac{1}{2}cT
  27. T \scriptstyle T
  28. c \scriptstyle c
  29. P ( t ) = | s | 2 ( t ) \scriptstyle P(t)\,=\,|s|^{2}(t)
  30. E = โˆซ 0 T P ( t ) d t = A 2 T E=\int_{0}^{T}P(t)dt=A^{2}T
  31. E r = K 2 A 2 T \scriptstyle E_{r}\,=\,K^{2}A^{2}T
  32. ฯƒ \scriptstyle\sigma
  33. S N R = E r ฯƒ 2 = K 2 A 2 T ฯƒ 2 SNR=\frac{E_{r}}{\sigma^{2}}=\frac{K^{2}A^{2}T}{\sigma^{2}}
  34. T T
  35. T T
  36. T \scriptstyle T
  37. t = 0 \scriptstyle t\,=\,0
  38. ฮ” f \scriptstyle\Delta f
  39. f 0 \scriptstyle f_{0}
  40. s c ( t ) = { A e i 2 ฯ€ ( ( f 0 - ฮ” f 2 ) t + ฮ” f 2 T t 2 ) if 0 โ‰ค t < T 0 otherwise s_{c}(t)=\left\{\begin{array}[]{ll}Ae^{i2\pi\left(\left(f_{0}\,-\,\frac{\Delta f% }{2}\right)t\,+\,\frac{\Delta f}{2T}t^{2}\,\right)}&\mbox{if}~{}\;0\leq t<T\\ 0&\mbox{otherwise}\end{array}\right.
  41. ฯ• ( t ) = 2 ฯ€ ( ( f 0 - ฮ” f 2 ) t + ฮ” f 2 T t 2 ) \phi(t)=2\pi\left(\left(f_{0}\,-\,\frac{\Delta f}{2}\right)t\,+\,\frac{\Delta f% }{2T}t^{2}\,\right)
  42. f ( t ) = 1 2 ฯ€ [ d ฯ• d t ] t = f 0 - ฮ” f 2 + ฮ” f T t f(t)=\frac{1}{2\pi}\left[\frac{d\phi}{dt}\right]_{t}=f_{0}-\frac{\Delta f}{2}+% \frac{\Delta f}{T}t
  43. f 0 - ฮ” f 2 \scriptstyle f_{0}\,-\,\frac{\Delta f}{2}
  44. t = 0 \scriptstyle t\,=\,0
  45. f 0 + ฮ” f 2 \scriptstyle f_{0}\,+\,\frac{\Delta f}{2}
  46. t = T \scriptstyle t\,=\,T
  47. f ( t ) f(t)
  48. ฯ• ( t ) = 2 ฯ€ โˆซ 0 t f ( u ) d u \phi(t)=2\pi\int_{0}^{t}f(u)\,du
  49. s c โ€ฒ ( t ) = { A e 2 i ฯ€ ( f 0 + ฮ” f 2 T t ) t if - T 2 โ‰ค t < T 2 0 otherwise s_{c^{\prime}}(t)=\left\{\begin{array}[]{ll}Ae^{2i\pi\left(f_{0}\,+\,\frac{% \Delta f}{2T}t\right)t}&\mbox{if}~{}\;-\frac{T}{2}\leq t<\frac{T}{2}\\ 0&\mbox{otherwise}\end{array}\right.
  50. K K
  51. s c โ€ฒ \scriptstyle s_{c^{\prime}}
  52. โŸจ s c โ€ฒ , s c โ€ฒ โŸฉ ( t ) = โˆซ - โˆž + โˆž s c โ€ฒ โ‹† ( - t โ€ฒ ) s c โ€ฒ ( t - t โ€ฒ ) d t โ€ฒ \langle s_{c^{\prime}},s_{c^{\prime}}\rangle(t)=\int_{-\infty}^{+\infty}s_{c^{% \prime}}^{\star}(-t^{\prime})s_{c^{\prime}}(t-t^{\prime})dt^{\prime}
  53. s c โ€ฒ s_{c^{\prime}}
  54. โŸจ s c โ€ฒ , s c โ€ฒ โŸฉ ( t ) = A 2 T ฮ› ( t T ) sinc [ ฯ€ ฮ” f t ฮ› ( t T ) ] e 2 i ฯ€ f 0 t \langle s_{c^{\prime}},s_{c^{\prime}}\rangle(t)=A^{2}T\Lambda\left(\frac{t}{T}% \right)\mathrm{sinc}\left[\pi\Delta ft\Lambda\left(\frac{t}{T}\right)\right]e^% {2i\pi f_{0}t}
  55. s c โ€ฒ \scriptstyle s_{c^{\prime}}
  56. T โ€ฒ = 1 ฮ” f \scriptstyle T^{\prime}\,=\,\frac{1}{\Delta f}
  57. T โ€ฒ \scriptstyle T^{\prime}
  58. ฮ” f \scriptstyle\Delta f
  59. T โ€ฒ \scriptstyle T^{\prime}
  60. T \scriptstyle T
  61. ฮ” f \scriptstyle\Delta f
  62. c 2 ฮ” f \scriptstyle\frac{c}{2\Delta f}
  63. c \scriptstyle c
  64. T T โ€ฒ = T ฮ” f \scriptstyle\frac{T}{T^{\prime}}\,=\,T\Delta f
  65. T โ€ฒ โ‰ˆ 1 ฮ” f \scriptstyle T^{\prime}\,\approx\,\frac{1}{\Delta f}
  66. P \scriptstyle P
  67. P โ€ฒ \scriptstyle P^{\prime}
  68. P ร— T = P โ€ฒ ร— T โ€ฒ P\times T=P^{\prime}\times T^{\prime}
  69. P โ€ฒ = P ร— T T โ€ฒ P^{\prime}=P\times\frac{T}{T^{\prime}}
  70. T ฮ” f \scriptstyle T\Delta f
  71. ฯƒ = 0.5 \scriptstyle\sigma\,=\,0.5
  72. N \scriptstyle N
  73. T N \scriptstyle\frac{T}{N}
  74. ฯ€ \scriptstyle\pi
  75. { 0 , ฯ€ } \scriptstyle\{0,\,\pi\}
  76. ฯ€ \scriptstyle\pi
  77. 1 T \scriptstyle\frac{1}{T}
  78. 1 / 2 {1}/{2}

Pumping_(oil_well).html

  1. ฮ” V = P ร— V ร— k \Delta V=P\times V\times k
  2. ฮ” V = P V k \Delta V=PVk
  3. ฮ” V = 3000 p s i ร— 300 b b l ร— 3.5 ร— 10 - 6 p s i - 1 \Delta V=3000psi\times 300bbl\times 3.5\times 10^{-6}psi^{-1}
  4. ฮ” V = 3.15 b b l \Delta V=3.15bbl

Punchscan.html

  1. P 1 P_{1}
  2. P 1 P_{1}
  3. P 1 P_{1}
  4. P 1 P_{1}
  5. P 1 โˆˆ { 0 , 1 } = { AB , BA } P_{1}\in\{0,1\}=\{\mbox{AB}~{},\mbox{BA}~{}\}\,
  6. P 2 P_{2}
  7. P 2 โˆˆ { 0 , 1 } = { AB , BA } P_{2}\in\{0,1\}=\{\mbox{AB}~{},\mbox{BA}~{}\}\,
  8. P 3 P_{3}
  9. P 3 โˆˆ { 0 , 1 } = { 1st , 2nd } P_{3}\in\{0,1\}=\{\mbox{1st}~{},\mbox{2nd}~{}\}\,
  10. R R
  11. R โˆˆ { 0 , 1 } = { Coke , Pepsi } R\in\{0,1\}=\{\mbox{Coke}~{},\mbox{Pepsi}~{}\}\,
  12. R = P 1 + P 2 + P 3 mod 2 R=P_{1}+P_{2}+P_{3}\bmod 2\,
  13. P 1 P_{1}
  14. P 2 P_{2}
  15. R R
  16. D 1 D_{1}
  17. D 2 D_{2}
  18. D 3 D_{3}
  19. D 4 D_{4}
  20. D 2 + D 4 = P 1 + P 2 mod 2 D_{2}+D_{4}=P_{1}+P_{2}\bmod 2\,
  21. R R
  22. D 5 D_{5}
  23. R R
  24. P 3 P_{3}
  25. D 3 D_{3}
  26. D 3 = P 3 + D 2 mod 2 D_{3}=P_{3}+D_{2}\bmod 2\,
  27. R = D 3 + D 4 mod 2 R=D_{3}+D_{4}\bmod 2\,
  28. R = ( D 3 ) + D 4 mod 2 = ( P 3 + D 2 ) + D 4 mod 2 = P 3 + ( D 2 + D 4 ) mod 2 = P 3 + ( P 1 + P 2 ) mod 2 \begin{aligned}\displaystyle R&\displaystyle=(D_{3})+D_{4}\bmod 2\\ &\displaystyle=(P_{3}+D_{2})+D_{4}\bmod 2\\ &\displaystyle=P_{3}+(D_{2}+D_{4})\bmod 2\\ &\displaystyle=P_{3}+(P_{1}+P_{2})\bmod 2\end{aligned}
  29. n n
  30. { D 1 , D 2 , D 3 } \{D_{1},D_{2},D_{3}\}
  31. { D 3 , D 4 , D 5 } \{D_{3},D_{4},D_{5}\}
  32. D 1 D_{1}
  33. D 2 D_{2}
  34. D 5 D_{5}

Pushforward_measure.html

  1. ( f * ( ฮผ ) ) ( B ) = ฮผ ( f - 1 ( B ) ) for B โˆˆ ฮฃ 2 . (f_{*}(\mu))(B)=\mu\left(f^{-1}(B)\right)\mbox{ for }~{}B\in\Sigma_{2}.
  2. g โˆ˜ f g\circ f
  3. โˆซ X 2 g d ( f * ฮผ ) = โˆซ X 1 g โˆ˜ f d ฮผ . \int_{X_{2}}g\,d(f_{*}\mu)=\int_{X_{1}}g\circ f\,d\mu.
  4. f ( n ) = f โˆ˜ f โˆ˜ โ€ฆ โˆ˜ f โŸ n times : X โ†’ X . f^{(n)}=\underbrace{f\circ f\circ\dots\circ f}_{n\mathrm{\,times}}:X\to X.

Pyramorphix.html

  1. 8 ! ร— 3 4 24 = 136080 \frac{8!\times 3^{4}}{24}=136080
  2. 8 ! ร— 3 7 ร— 12 ! ร— 2 9 ร— 4 6 โ‰ˆ 8.86 ร— 10 22 {8!\times 3^{7}\times 12!\times 2^{9}\times 4^{6}}\approx 8.86\times 10^{22}
  3. 8 ! ร— 3 4 ร— 12 ! ร— 2 10 ร— 4 6 6 4 โ‰ˆ 5.06 ร— 10 18 \frac{8!\times 3^{4}\times 12!\times 2^{10}\times 4^{6}}{6^{4}}\approx 5.06% \times 10^{18}

Pโ€“P_plot.html

  1. ( F ( z ) , G ( z ) ) (F(z),G(z))
  2. - โˆž -\infty
  3. โˆž . \infty.
  4. ( - โˆž , โˆž ) (-\infty,\infty)
  5. [ 0 , 1 ] ร— [ 0 , 1 ] . [0,1]\times[0,1].

Q-Pochhammer_symbol.html

  1. ( a ; q ) n = โˆ k = 0 n - 1 ( 1 - a q k ) = ( 1 - a ) ( 1 - a q ) ( 1 - a q 2 ) โ‹ฏ ( 1 - a q n - 1 ) (a;q)_{n}=\prod_{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots(1-aq^{n-1})
  2. ( a ; q ) 0 = 1 (a;q)_{0}=1
  3. ( a ; q ) โˆž = โˆ k = 0 โˆž ( 1 - a q k ) . (a;q)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}).
  4. ฯ• ( q ) = ( q ; q ) โˆž = โˆ k = 1 โˆž ( 1 - q k ) \phi(q)=(q;q)_{\infty}=\prod_{k=1}^{\infty}(1-q^{k})
  5. ( a ; q ) n = ( a ; q ) โˆž ( a q n ; q ) โˆž , (a;q)_{n}=\frac{(a;q)_{\infty}}{(aq^{n};q)_{\infty}},
  6. ( a ; q ) - n = 1 ( a q - n ; q ) n = โˆ k = 1 n 1 ( 1 - a / q k ) (a;q)_{-n}=\frac{1}{(aq^{-n};q)_{n}}=\prod_{k=1}^{n}\frac{1}{(1-a/q^{k})}
  7. ( a ; q ) - n = ( - q / a ) n q n ( n - 1 ) / 2 ( q / a ; q ) n . (a;q)_{-n}=\frac{(-q/a)^{n}q^{n(n-1)/2}}{(q/a;q)_{n}}.
  8. ( x ; q ) โˆž = โˆ‘ n = 0 โˆž ( - 1 ) n q n ( n - 1 ) / 2 ( q ; q ) n x n (x;q)_{\infty}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n(n-1)/2}}{(q;q)_{n}}x^{n}
  9. 1 ( x ; q ) โˆž = โˆ‘ n = 0 โˆž x n ( q ; q ) n \frac{1}{(x;q)_{\infty}}=\sum_{n=0}^{\infty}\frac{x^{n}}{(q;q)_{n}}
  10. ( a x ; q ) โˆž ( x ; q ) โˆž = โˆ‘ n = 0 โˆž ( a ; q ) n ( q ; q ) n x n . \frac{(ax;q)_{\infty}}{(x;q)_{\infty}}=\sum_{n=0}^{\infty}\frac{(a;q)_{n}}{(q;% q)_{n}}x^{n}.
  11. q m a n q^{m}a^{n}
  12. ( a ; q ) โˆž - 1 = โˆ k = 0 โˆž ( 1 - a q k ) - 1 (a;q)_{\infty}^{-1}=\prod_{k=0}^{\infty}(1-aq^{k})^{-1}
  13. ( a ; q ) โˆž - 1 = โˆ‘ k = 0 โˆž ( โˆ j = 1 k 1 1 - q j ) a k = โˆ‘ k = 0 โˆž a k ( q ; q ) k (a;q)_{\infty}^{-1}=\sum_{k=0}^{\infty}\left(\prod_{j=1}^{k}\frac{1}{1-q^{j}}% \right)a^{k}=\sum_{k=0}^{\infty}\frac{a^{k}}{(q;q)_{k}}
  14. q m a n q^{m}a^{n}
  15. ( - a ; q ) โˆž = โˆ k = 0 โˆž ( 1 + a q k ) (-a;q)_{\infty}=\prod_{k=0}^{\infty}(1+aq^{k})
  16. ( - a ; q ) โˆž = โˆ k = 0 โˆž ( 1 + a q k ) = โˆ‘ k = 0 โˆž ( q ( k 2 ) โˆ j = 1 k 1 1 - q j ) a k = โˆ‘ k = 0 โˆž q ( k 2 ) ( q ; q ) k a k (-a;q)_{\infty}=\prod_{k=0}^{\infty}(1+aq^{k})=\sum_{k=0}^{\infty}\left(q^{k% \choose 2}\prod_{j=1}^{k}\frac{1}{1-q^{j}}\right)a^{k}=\sum_{k=0}^{\infty}% \frac{q^{k\choose 2}}{(q;q)_{k}}a^{k}
  17. ( a 1 , a 2 , โ€ฆ , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n โ€ฆ ( a m ; q ) n . (a_{1},a_{2},\ldots,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots(a_{m};q)_{n}.
  18. ( a ; q ) n (a;q)_{n}
  19. lim q โ†’ 1 1 - q n 1 - q = n , \lim_{q\rightarrow 1}\frac{1-q^{n}}{1-q}=n,
  20. [ n ] q = 1 - q n 1 - q . [n]_{q}=\frac{1-q^{n}}{1-q}.
  21. [ n ] q ! \big[n]_{q}!
  22. = โˆ k = 1 n [ k ] q =\prod_{k=1}^{n}[k]_{q}
  23. = [ 1 ] q [ 2 ] q โ‹ฏ [ n - 1 ] q [ n ] q =[1]_{q}[2]_{q}\cdots[n-1]_{q}[n]_{q}
  24. = 1 - q 1 - q 1 - q 2 1 - q โ‹ฏ 1 - q n - 1 1 - q 1 - q n 1 - q =\frac{1-q}{1-q}\frac{1-q^{2}}{1-q}\cdots\frac{1-q^{n-1}}{1-q}\frac{1-q^{n}}{1% -q}
  25. = 1 ( 1 + q ) โ‹ฏ ( 1 + q + โ‹ฏ + q n - 2 ) ( 1 + q + โ‹ฏ + q n - 1 ) =1(1+q)\cdots(1+q+\cdots+q^{n-2})(1+q+\cdots+q^{n-1})
  26. = ( q ; q ) n ( 1 - q ) n . =\frac{(q;q)_{n}}{(1-q)^{n}}.
  27. โˆ k = 1 n [ - k ] q = ( - 1 ) n [ n ] q ! q n ( n + 1 ) / 2 \prod_{k=1}^{n}[-k]_{q}=\frac{(-1)^{n}\,[n]_{q}!}{q^{n(n+1)/2}}
  28. [ n k ] q = [ n ] q ! [ n - k ] q ! [ k ] q ! . \begin{bmatrix}n\\ k\end{bmatrix}_{q}=\frac{[n]_{q}!}{[n-k]_{q}![k]_{q}!}.
  29. [ n + 1 k ] q = [ n k ] q + q n - k + 1 [ n k - 1 ] q . \begin{bmatrix}n+1\\ k\end{bmatrix}_{q}=\begin{bmatrix}n\\ k\end{bmatrix}_{q}+q^{n-k+1}\begin{bmatrix}n\\ k-1\end{bmatrix}_{q}.
  30. ฮ“ q ( x ) = ( 1 - q ) 1 - x ( q ; q ) โˆž ( q x ; q ) โˆž \Gamma_{q}(x)=\frac{(1-q)^{1-x}(q;q)_{\infty}}{(q^{x};q)_{\infty}}
  31. ฮ“ q ( x + 1 ) = [ x ] q ฮ“ q ( x ) \Gamma_{q}(x+1)=[x]_{q}\Gamma_{q}(x)\,
  32. ฮ“ q ( n + 1 ) = [ n ] q ! . \Gamma_{q}(n+1)=[n]_{q}!\frac{}{}.

Quadratically_constrained_quadratic_program.html

  1. minimize 1 2 x T P 0 x + q 0 T x subject to 1 2 x T P i x + q i T x + r i โ‰ค 0 for i = 1 , โ€ฆ , m , A x = b , \begin{aligned}&\displaystyle\,\text{minimize}&&\displaystyle\tfrac{1}{2}x^{% \mathrm{T}}P_{0}x+q_{0}^{\mathrm{T}}x\\ &\displaystyle\,\text{subject to}&&\displaystyle\tfrac{1}{2}x^{\mathrm{T}}P_{i% }x+q_{i}^{\mathrm{T}}x+r_{i}\leq 0\quad\,\text{for }i=1,\dots,m,\\ &&&\displaystyle Ax=b,\end{aligned}

Quadrature_domains.html

  1. โˆฌ D u d x d y = โˆ‘ j = 1 k c j u ( z j ) , \iint_{D}u\,dxdy=\sum_{j=1}^{k}c_{j}u(z_{j}),

Qualitative_variation.html

  1. M = โˆ‘ i = 1 K ( f m - f i ) M=\sum_{i=1}^{K}(f_{m}-f_{i})
  2. M = K f m - N M=Kf_{m}-N
  3. v = 1 - f m N v=1-\frac{f_{m}}{N}
  4. ( f m N ) - 1 K N K ( K - 1 ) N = M N ( K - 1 ) \frac{(\frac{f_{m}}{N})-\frac{1}{K}}{\frac{N}{K}\frac{(K-1)}{N}}=\frac{M}{N(K-% 1)}
  5. M o d V R = 1 - K f m - N N ( K - 1 ) = K ( N - f m ) N ( K - 1 ) = K v K - 1 ModVR=1-\frac{Kf_{m}-N}{N(K-1)}=\frac{K(N-f_{m})}{N(K-1)}=\frac{Kv}{K-1}
  6. R a n V R = 1 - f m - f l f m = f l f m RanVR=1-\frac{f_{m}-f_{l}}{f_{m}}=\frac{f_{l}}{f_{m}}
  7. A v D e v = 1 - 1 2 N K K - 1 โˆ‘ i = 1 K | f i - N K | AvDev=1-\frac{1}{2N}\frac{K}{K-1}\sum^{K}_{i=1}|f_{i}-\frac{N}{K}|
  8. M N D i f = 1 - 1 N ( K - 1 ) โˆ‘ i = 1 K - 1 โˆ‘ j = i + 1 K | f i - f j | MNDif=1-\frac{1}{N(K-1)}\sum_{i=1}^{K-1}\sum_{j=i+1}^{K}|f_{i}-f_{j}|
  9. V a r N C = 1 - 1 N 2 K ( K - 1 ) โˆ‘ ( f i - N K ) 2 VarNC=1-\frac{1}{N^{2}}\frac{K}{(K-1)}\sum(f_{i}-\frac{N}{K})^{2}
  10. S t D e v 1 = 1 - โˆ‘ i = 1 K ( f i - N K ) 2 ( N - N K ) 2 + ( K - 1 ) ( N K ) 2 StDev_{1}=1-\sqrt{\frac{\sum_{i=1}^{K}(f_{i}-\frac{N}{K})^{2}}{(N-\frac{N}{K})% ^{2}+(K-1)(\frac{N}{K})^{2}}}
  11. S t D e v 2 = 1 - โˆ‘ i = 1 K - 1 โˆ‘ j = i + 1 K ( f i - f j ) N 2 ( K - 1 ) StDev_{2}=1-\sqrt{\frac{\sum^{K-1}_{i=1}\sum^{K}_{j=i+1}(f_{i}-f_{j})}{N^{2}(K% -1)}}
  12. H R e l = - โˆ‘ p i l o g 2 p i log 2 K HRel=\frac{-\sum p_{i}log_{2}p_{i}}{\log_{2}K}
  13. M 1 = 1 - โˆ‘ i = 1 K p i 2 M1=1-\sum_{i=1}^{K}p_{i}^{2}
  14. p i = f i / N p_{i}=f_{i}/N
  15. M 2 = K K - 1 ( 1 - โˆ‘ i = 1 K p i 2 ) M2=\frac{K}{K-1}\left(1-\sum_{i=1}^{K}p_{i}^{2}\right)
  16. p i = f i / N p_{i}=f_{i}/N
  17. K K - 1 \frac{K}{K-1}
  18. M 4 = โˆ‘ i = 1 K | X i - m | 2 โˆ‘ i = 1 K X i M4=\frac{\sum_{i=1}^{K}|X_{i}-m|}{2\sum_{i=1}^{K}X_{i}}
  19. M 6 = K [ 1 - โˆ‘ i = 1 K | X i - m | 2 N ] M6=K\left[1-\frac{\sum_{i=1}^{K}|X_{i}-m|}{2N}\right]
  20. m = โˆ‘ i = 1 K X i N m=\frac{\sum_{i=1}^{K}X_{i}}{N}
  21. M 6 = K [ 1 - โˆ‘ i = 1 K | p i - 1 N | 2 ] M6=K\left[1-\frac{\sum_{i=1}^{K}|p_{i}-\frac{1}{N}|}{2}\right]
  22. โˆ‘ i = 1 K p i 2 \sum_{i=1}^{K}p_{i}^{2}
  23. I C = โˆ‘ f i ( f i - 1 ) n ( n - 1 ) IC=\sum\frac{f_{i}(f_{i}-1)}{n(n-1)}
  24. D = 1 - โˆ‘ i = 1 K n i ( n i - 1 ) n ( n - 1 ) D=1-\sum_{i=1}^{K}{\frac{n_{i}(n_{i}-1)}{n(n-1)}}
  25. u โˆผ 1 - โˆ‘ i = 1 K p i 2 u\sim 1-\sum_{i=1}^{K}p_{i}^{2}
  26. u = c ( x , y ) n 2 - n u=\frac{c(x,y)}{n^{2}-n}
  27. u โˆผ 1 - โˆ‘ i = 1 K p i 2 u\sim 1-\sum_{i=1}^{K}p_{i}^{2}
  28. H 2 = 2 ( 1 - โˆ‘ i = 1 K p i 2 ) H^{2}=2\left(1-\sum_{i=1}^{K}p_{i}^{2}\right)
  29. M 7 = โˆ‘ i = 1 K โˆ‘ j = 1 L | R i - R | 2 โˆ‘ R i M7=\frac{\sum_{i=1}^{K}\sum_{j=1}^{L}|R_{i}-R|}{2\sum R_{i}}
  30. R i j = O i j E i j = O i j n i p j R_{ij}=\frac{O_{ij}}{E_{ij}}=\frac{O_{ij}}{n_{i}p_{j}}
  31. R = โˆ‘ i = 1 K โˆ‘ j = 1 L R i j โˆ‘ i = 1 K n i R=\frac{\sum_{i=1}^{K}\sum_{j=1}^{L}R_{ij}}{\sum_{i=1}^{K}n_{i}}
  32. p i p_{i}
  33. p i p_{i}
  34. I B = log ( N ! ) - โˆ‘ i = 1 K ( log ( n i ! ) ) N I_{B}=\frac{\log(N!)-\sum_{i=1}^{K}(\log(n_{i}!))}{N}
  35. E B = I B / I B ( max ) E_{B}=I_{B}/I_{B(\max)}
  36. N a = 1 [ โˆ‘ i = 1 K p i a ] a - 1 N_{a}=\frac{1}{\left[\sum_{i=1}^{K}p_{i}^{a}\right]^{a-1}}
  37. E a , b = N a N b E_{a,b}=\frac{N_{a}}{N_{b}}
  38. E 4 = N 2 N 1 E_{4}=\frac{N_{2}}{N_{1}}
  39. E 5 = N 2 - 1 N 1 - 1 E_{5}=\frac{N_{2}-1}{N_{1}-1}
  40. I M a r g = S - 1 l o g e N I_{Marg}=\frac{S-1}{log_{e}N}
  41. I Men = S N I_{\mathrm{Men}}=\frac{S}{\sqrt{N}}
  42. Q = 1 2 ( n R 1 + n R 2 ) + โˆ‘ j = R 1 + 1 R 2 - 1 n j l o g ( R 2 / R 1 ) Q=\frac{\frac{1}{2}(n_{R1}+n_{R2})+\sum_{j=R_{1}+1}^{R_{2}-1}n_{j}}{log(R_{2}/% R_{1})}
  43. H = log e N - 1 N โˆ‘ n i p i log ( p i ) H=\log_{e}N-\frac{1}{N}\sum n_{i}p_{i}\log(p_{i})
  44. S D ( H ) = 1 N [ โˆ‘ p i [ log e ( p i ) ] 2 - H 2 ] SD(H)=\frac{1}{N}\left[\sum p_{i}[\log_{e}(p_{i})]^{2}-H^{2}\right]
  45. var ( H ) = โˆ‘ p i [ log ( p i ) ] 2 - [ โˆ‘ p i log ( p i ) ] 2 N + K - 1 2 N 2 + - 1 + โˆ‘ p i 2 - โˆ‘ p i - 1 log ( p i ) + โˆ‘ p i - 1 โˆ‘ p i log ( p i ) 6 N 3 \operatorname{var}(H)=\frac{\sum p_{i}[\log(p_{i})]^{2}-\left[\sum p_{i}\log(p% _{i})\right]^{2}}{N}+\frac{K-1}{2N^{2}}+\frac{-1+\sum p_{i}^{2}-\sum p_{i}^{-1% }\log(p_{i})+\sum p_{i}^{-1}\sum p_{i}\log(p_{i})}{6N^{3}}
  46. J = H log e ( S ) J=\frac{H}{\log_{e}(S)}
  47. H q = 1 1 - q ln ( โˆ‘ i = 1 K p i q ) {}^{q}H=\frac{1}{1-q}\;\ln\left(\sum_{i=1}^{K}p_{i}^{q}\right)
  48. H q = ln ( 1 โˆ‘ i = 1 K p i p i q - 1 q - 1 ) = ln ( D q ) {}^{q}H=\ln\left({1\over\sqrt[q-1]{{\sum_{i=1}^{K}p_{i}p_{i}^{q-1}}}}\right)=% \ln({}^{q}\!D)
  49. D q {}^{q}\!D
  50. D = N - โˆ‘ i = 1 K n i N - N D=\frac{N-\sqrt{\sum_{i=1}^{K}n_{i}}}{N-\sqrt{N}}
  51. E = N - โˆ‘ i = 1 K n i N - N K E=\frac{N-\sqrt{\sum_{i=1}^{K}n_{i}}}{N-\frac{N}{\sqrt{K}}}
  52. K = ฮฑ ln ( 1 + N ฮฑ ) K=\alpha\ln(1+\frac{N}{\alpha})
  53. E ( n r ) = ฮฑ X r r E(n_{r})=\alpha\frac{X^{r}}{r}
  54. N = ฮฑ X 1 - X N=\frac{\alpha X}{1-X}
  55. K = - ฮฑ ln ( 1 - X ) K=-\alpha\ln(1-X)
  56. var ( ฮฑ ) = ฮฑ ln ( X ) ( 1 - X ) \operatorname{var}(\alpha)=\frac{\alpha}{\ln(X)(1-X)}
  57. D w = m a x [ c i K - i N ] D_{w}=max[\frac{c_{i}}{K}-\frac{i}{N}]
  58. E = 1 D / K E=\frac{1}{D}/K
  59. E 1 = 1 - D 1 - 1 K E_{1}=\frac{1-D}{1-\frac{1}{K}}
  60. E 2 = log e ( D ) log e ( K ) E_{2}=\frac{\log_{e}(D)}{\log_{e}(K)}
  61. E = e H - 1 K - 1 E=\frac{e^{H}-1}{K-1}
  62. E = e H K E=\frac{e^{H}}{K}
  63. E = 1 - โˆ‘ i = 1 K โˆ‘ j = i + 1 K p i - p j K E=1-\sum_{i=1}^{K}\sum_{j=i+1}^{K}\frac{p_{i}-p_{j}}{K}
  64. B = 1 - 2 ฯ€ a r c t a n ( ฮธ ) B=1-\frac{2}{\pi}arctan(\theta)
  65. E c = O - 1 K 1 - 1 K E_{c}=\frac{O-\frac{1}{K}}{1-\frac{1}{K}}
  66. E d = O - 1 K - K - 1 N 1 - 1 K - K - 1 N E_{d}=\frac{O-\frac{1}{K}-\frac{K-1}{N}}{1-\frac{1}{K}-\frac{K-1}{N}}
  67. O = 1 - 1 2 | p i - 1 K | O=1-\frac{1}{2}|p_{i}-\frac{1}{K}|
  68. R i k = H max - H obs H max - H min R_{ik}=\frac{H_{\max}-H_{\mathrm{obs}}}{H_{\max}-H_{\min}}
  69. X = โˆ‘ x i j X=\sum x_{ij}
  70. X = โˆ‘ x k j X=\sum x_{kj}
  71. H ( X ) = โˆ‘ x i j X log X x i j H(X)=\sum\frac{x_{ij}}{X}\log\frac{X}{x_{ij}}
  72. H ( Y ) = โˆ‘ x k j Y log Y x k j H(Y)=\sum\frac{x_{kj}}{Y}\log\frac{Y}{x_{kj}}
  73. H min = X X + Y H ( X ) + Y X + Y H ( Y ) H_{\min}=\frac{X}{X+Y}H(X)+\frac{Y}{X+Y}H(Y)
  74. H max = โˆ‘ ( x i j X + Y log X + Y x i j + x k j X + Y log X + Y x k j ) H_{\max}=\sum\left(\frac{x_{ij}}{X+Y}\log\frac{X+Y}{x_{ij}}+\frac{x_{kj}}{X+Y}% \log\frac{X+Y}{x_{kj}}\right)
  75. H obs = โˆ‘ x i j + x k j X + Y log X + Y x i j + x k j H_{\mathrm{obs}}=\sum\frac{x_{ij}+x_{kj}}{X+Y}\log\frac{X+Y}{x_{ij}+x_{kj}}
  76. X n X_{n}
  77. X n X_{n}
  78. f n f_{n}
  79. f n = E [ X n ] = K - ( N n ) - 1 โˆ‘ i = 1 K ( N - N i n ) f_{n}=E[X_{n}]=K-{\left({{N}\atop{n}}\right)}^{-1}\sum_{i=1}^{K}{\left({{N-N_{% i}}\atop{n}}\right)}
  80. f ( 0 ) = 0 , f ( 1 ) = 1 , f ( N ) = K f(0)=0,\ f(1)=1,\ f(N)=K
  81. V = H - E ( H ) S D ( H ) V=\frac{H-E(H)}{SD(H)}
  82. S D ( H ) = 1 N [ โˆ‘ p i [ log e ( p i ) ] 2 - H 2 ] SD(H)=\frac{1}{N}\left[\sum p_{i}[\log_{e}(p_{i})]^{2}-H^{2}\right]
  83. I L G = K K โ€ฒ I_{LG}=\frac{K}{K^{\prime}}
  84. S T D = 2 โˆ‘ โˆ‘ i < j ฯ‰ i j s ( s - 1 ) S_{TD}=2\frac{\sum\sum_{i<j}\omega_{ij}}{s(s-1)}
  85. D = 1 2 โˆ‘ i = 1 K | A i A - B i B | D=\frac{1}{2}\sum_{i=1}^{K}\left|\frac{A_{i}}{A}-\frac{B_{i}}{B}\right|
  86. A = โˆ‘ i = 1 K A i A=\sum_{i=1}^{K}A_{i}
  87. B = โˆ‘ i = 1 K B i B=\sum_{i=1}^{K}B_{i}
  88. G T = D ( 1 - A A + B ) GT=D\left(1-\frac{A}{A+B}\right)
  89. S I = 1 2 โˆ‘ i = 1 K | A i A - t i - A i T - A | SI=\frac{1}{2}\sum_{i=1}^{K}|\frac{A_{i}}{A}-\frac{t_{i}-A_{i}}{T-A}|
  90. A = โˆ‘ i = 1 K A i A=\sum_{i=1}^{K}A_{i}
  91. T = โˆ‘ i = 1 K t i T=\sum_{i=1}^{K}t_{i}
  92. H = 1 - โˆ‘ i = 1 K โˆ‘ j = 1 i p i p j H=1-\sum_{i=1}^{K}\sum_{j=1}^{i}\sqrt{p_{i}p_{j}}
  93. L x y = 1 N โˆ‘ i = 1 K X i Y i X tot L_{xy}=\frac{1}{N}\sum_{i=1}^{K}\frac{X_{i}Y_{i}}{X_{\mathrm{tot}}}
  94. I R = p x x - p x 1 - p x I_{R}=\frac{p_{xx}-p_{x}}{1-p_{x}}
  95. p x x = โˆ‘ i = 1 K x i p i N x p_{xx}=\frac{\sum_{i=1}^{K}x_{i}p_{i}}{N_{x}}
  96. I I = โˆ‘ i = 1 K A i A A i t i II=\sum_{i=1}^{K}\frac{A_{i}}{A}\frac{A_{i}}{t_{i}}
  97. M I I = I I - A T 1 - A T MII=\frac{II-\frac{A}{T}}{1-\frac{A}{T}}
  98. G S = 1 2 โˆ‘ i = 1 K | A i A - t i T | GS=\frac{1}{2}\sum_{i=1}^{K}|\frac{A_{i}}{A}-\frac{t_{i}}{T}|
  99. A = โˆ‘ i = 1 K A i A=\sum_{i=1}^{K}A_{i}
  100. T = โˆ‘ i = 1 K t i T=\sum_{i=1}^{K}t_{i}
  101. I E = โˆ‘ i = 1 K A i A B i t i IE=\sum_{i=1}^{K}\frac{A_{i}}{A}\frac{B_{i}}{t_{i}}
  102. A = โˆ‘ i = 1 K A i A=\sum_{i=1}^{K}A_{i}
  103. O = a ( a + b ) ( a + c ) O=\frac{a}{\sqrt{(a+b)(a+c)}}
  104. K = a 2 ( 1 a + b + 1 a + c ) K=\frac{a}{2}(\frac{1}{a+b}+\frac{1}{a+c})
  105. Q = a d - b c a d + b c Q=\frac{ad-bc}{ad+bc}
  106. Y = a d - b c a d + b c Y=\frac{\sqrt{ad}-\sqrt{bc}}{\sqrt{ad}+\sqrt{bc}}
  107. B U B = a d + a a d + a + b + c BUB=\frac{\sqrt{ad}+a}{\sqrt{ad}+a+b+c}
  108. H = ( a + d ) - ( b + c ) a + b + c + d H=\frac{(a+d)-(b+c)}{a+b+c+d}
  109. R T = ( a + d ) a + 2 ( b + c ) + d RT=\frac{(a+d)}{a+2(b+c)+d}
  110. S S = 2 ( a + d ) 2 ( a + d ) + b + c SS=\frac{2(a+d)}{2(a+d)+b+c}
  111. S B D = b + c a + b + c + d SBD=\sqrt{\frac{b+c}{a+b+c+d}}
  112. R R = a a + b + c + d RR=\frac{a}{a+b+c+d}
  113. ฯ• = a d - b c ( a + b ) ( a + c ) ( b + c ) ( c + d ) \phi=\frac{ad-bc}{\sqrt{(a+b)(a+c)(b+c)(c+d)}}
  114. S = b + c b + c + d S=\frac{b+c}{b+c+d}
  115. S = a a + m i n ( b , c ) S=\frac{a}{a+min(b,c)}
  116. D = a d - b c ( a + b + c + d ) ( a + b ) ( a + c ) D=\frac{ad-bc}{\sqrt{(a+b+c+d)(a+b)(a+c)}}
  117. F = a ( a + b + c + d ) ( a + b ) ( a + c ) F=\frac{a(a+b+c+d)}{(a+b)(a+c)}
  118. S M = a + d ( a + b + c + d ) SM=\frac{a+d}{(a+b+c+d)}
  119. F = ( a + b + c + d ) ( a - 0.5 ) 2 ( a + b ) ( a + c ) F=\frac{(a+b+c+d)(a-0.5)^{2}}{(a+b)(a+c)}
  120. S = l o g [ n ( | a d - b c | - n 2 ) 2 ( a + b ) ( a + c ) ( b + d ) ( c + d ) ] S=log[\frac{n(|ad-bc|-\frac{n}{2})^{2}}{(a+b)(a+c)(b+d)(c+d)}]
  121. M = 4 ( a d - b c ) ( a + d ) 2 + ( b + c ) 2 M=\frac{4(ad-bc)}{(a+d)^{2}+(b+c)^{2}}
  122. P = a b + b c a b + 2 b c + c d P=\frac{ab+bc}{ab+2bc+cd}
  123. H D = 1 2 ( a a + b + c + d b + c + d ) HD=\frac{1}{2}(\frac{a}{a+b+c}+\frac{d}{b+c+d})
  124. C Z I = โˆ‘ min ( x i , x j ) โˆ‘ ( x i + x j ) CZI=\frac{\sum\min(x_{i},x_{j})}{\sum(x_{i}+x_{j})}
  125. d ( ๐ฉ , ๐ช ) = โˆ‘ i = 1 n | p i - q i | | p i | + | q i | d(\mathbf{p},\mathbf{q})=\sum_{i=1}^{n}\frac{|p_{i}-q_{i}|}{|p_{i}|+|q_{i}|}
  126. C C = 2 c s 1 + s 2 CC=\frac{2c}{s_{1}+s_{2}}
  127. J = A A + B + C J=\frac{A}{A+B+C}
  128. J = 1 A ( 1 A + B + C ) J=\frac{1}{A}(\frac{1}{A+B+C})
  129. S E ( J ) = A ( B + C ) N ( A + B + C ) 3 SE(J)=\sqrt{\frac{A(B+C)}{N(A+B+C)^{3}}}
  130. D = 2 A 2 A + B + C D=\frac{2A}{2A+B+C}
  131. M = N - B - C N M=\frac{N-B-C}{N}
  132. I m = โˆ‘ x ( x - 1 ) n m ( m - 1 ) I_{m}=\frac{\sum x(x-1)}{nm(m-1)}
  133. I m = n โˆ‘ x 2 - โˆ‘ x ( โˆ‘ x ) 2 - โˆ‘ x I_{m}=n\frac{\sum x^{2}-\sum x}{\left(\sum x\right)^{2}-\sum x}
  134. I m = n I M C n m - 1 I_{m}=\frac{n\ IMC}{nm-1}
  135. I m ( โˆ‘ x - 1 ) + n - โˆ‘ x I_{m}\left(\sum x-1\right)+n-\sum x
  136. z = I m - 1 2 / n m 2 z=\frac{I_{m}-1}{2/nm^{2}}
  137. M u = ฯ‡ 0.975 2 - k + โˆ‘ x โˆ‘ x - 1 M_{u}=\frac{\chi^{2}_{0.975}-k+\sum x}{\sum x-1}
  138. M c = ฯ‡ 0.025 2 - k + โˆ‘ x โˆ‘ x - 1 M_{c}=\frac{\chi^{2}_{0.025}-k+\sum x}{\sum x-1}
  139. I p = 0.5 + 0.5 ( I d - M c k - M c ) I_{p}=0.5+0.5\left(\frac{I_{d}-M_{c}}{k-M_{c}}\right)
  140. I p = 0.5 ( I d - 1 M u - 1 ) I_{p}=0.5\left(\frac{I_{d}-1}{M_{u}-1}\right)
  141. I p = - 0.5 ( I d - 1 M u - 1 ) I_{p}=-0.5\left(\frac{I_{d}-1}{M_{u}-1}\right)
  142. I p = - 0.5 + 0.5 ( I d - M u M u ) I_{p}=-0.5+0.5\left(\frac{I_{d}-M_{u}}{M_{u}}\right)
  143. E 1 = I - I min I max - I min E_{1}=\frac{I-I_{\min}}{I_{\max}-I_{\min}}
  144. E 2 = I I max E_{2}=\frac{I}{I_{\max}}
  145. H = p m a x ( 1 - p m i n ) p m i n ( 1 - p m a x ) H=\sqrt{\frac{p_{max}(1-p_{min})}{p_{min}(1-p_{max})}}
  146. d j k = โˆ‘ i = 1 N ( x i j - x i k ) 2 d_{jk}=\sqrt{\sum_{i=1}^{N}(x_{ij}-x_{ik})^{2}}
  147. d j k = โˆ‘ i = 1 N | x i j - x i k | d_{jk}=\sum_{i=1}^{N}|x_{ij}-x_{ik}|
  148. D P Q = 1 r โˆ‘ j = 1 r โˆ‘ i = 1 k | p j i - q j i | D_{PQ}=\frac{1}{r}\sum_{j=1}^{r}\sum_{i=1}^{k}|p_{ji}-q_{ji}|
  149. A = โˆ‘ x i j A=\sum x_{ij}
  150. B = โˆ‘ x i k B=\sum x_{ik}
  151. J = โˆ‘ min ( x i j , x j k ) J=\sum\min(x_{ij},x_{jk})
  152. d j k = A + B - 2 J d_{jk}=A+B-2J
  153. d j k = A + B - 2 J A + B d_{jk}=\frac{A+B-2J}{A+B}
  154. d j k = A + B - 2 J A + B - J d_{jk}=\frac{A+B-2J}{A+B-J}
  155. d j k = 1 - 1 2 ( J A + J B ) d_{jk}=1-\frac{1}{2}\left(\frac{J}{A}+\frac{J}{B}\right)
  156. c a = โˆ‘ i = 1 a p j c_{a}=\sum^{a}_{i=1}p_{j}
  157. D = 2 โˆ‘ a = i K d a K - 1 D=2\sum_{a=i}^{K}\frac{d_{a}}{K-1}
  158. H = 1 N - 1 s 2 m 2 H=\frac{1}{N-1}\frac{s^{2}}{m^{2}}
  159. P C I = X t Z [ 1 - | โˆ‘ i = 1 r + X + X t - โˆ‘ i = 1 r - X - X t | ] PCI=\frac{X_{t}}{Z}\left[1-\left|\frac{\sum_{i=1}^{r_{+}}X_{+}}{X_{t}}-\frac{% \sum_{i=1}^{r_{-}}X_{-}}{X_{t}}\right|\right]
  160. X t = โˆ‘ i = 1 r + | X + | + โˆ‘ i = 1 r - | X - | X_{t}=\sum_{i=1}^{r_{+}}|X_{+}|+\sum_{i=1}^{r_{-}}|X_{-}|
  161. P C I 2 = โˆ‘ i = 1 K โˆ‘ j = 1 i k i k j d i j ฮด PCI_{2}=\frac{\sum_{i=1}^{K}\sum_{j=1}^{i}k_{i}k_{j}d_{ij}}{\delta}
  162. ฮด = N 2 2 d max \delta=\frac{N^{2}}{2}d_{\max}
  163. ฮด = N 2 - 1 2 d max \delta=\frac{N^{2}-1}{2}d_{\max}
  164. D 1 : d i j = | r i - r j | - 1 D_{1}:d_{ij}=|r_{i}-r_{j}|-1
  165. D 2 : d i j = | r i - r j | D_{2}:d_{ij}=|r_{i}-r_{j}|
  166. D 3 : d i j = | r i - r j | p D_{3}:d_{ij}=|r_{i}-r_{j}|^{p}
  167. D p i j : d i j = [ | r i - r j | - ( m - 1 ) ] p Dp_{ij}:d_{ij}=[|r_{i}-r_{j}|-(m-1)]^{p}
  168. A = U ( 1 - S - 1 K - 1 ) A=U\left(1-\frac{S-1}{K-1}\right)
  169. A overall = โˆ‘ w i A i A_{\mathrm{overall}}=\sum w_{i}A_{i}
  170. p = โˆ i = 1 n ( 1 - i k ) p=\prod_{i=1}^{n}\left(1-\frac{i}{k}\right)
  171. p = exp ( - n 2 2 c ) p=\exp\left(\frac{-n^{2}}{2c}\right)
  172. log e ( 1 - i c ) โ‰ˆ - i c \log_{e}\left(1-\frac{i}{c}\right)\approx-\frac{i}{c}
  173. n = 1.2 c n=1.2\sqrt{c}
  174. n = 2.448 c โ‰ˆ 2.5 c n=2.448\sqrt{c}\approx 2.5\sqrt{c}
  175. n = 1.2 1 โˆ‘ i = 1 k 1 c i n=1.2\sqrt{\frac{1}{\sum_{i=1}^{k}\frac{1}{c_{i}}}}
  176. n โ‰ˆ 2.5 1 โˆ‘ i = 1 k 1 c i n\approx 2.5\sqrt{\frac{1}{\sum_{i=1}^{k}\frac{1}{c_{i}}}}
  177. n = 1.2 c 2 j + 1 n=1.2\sqrt{\frac{c}{2j+1}}
  178. n โ‰ˆ 2.5 c 2 j + 1 n\approx 2.5\sqrt{\frac{c}{2j+1}}
  179. - log 10 ( 1 + 2 d 365 ) -\log_{10}\left(\frac{1+2d}{365}\right)

Quality_control_and_genetic_algorithms.html

  1. โŠ‚ \subset
  2. โŠ‚ \subset
  3. โŠ‚ \subset

Quantile_function.html

  1. F X : R โ†’ [ 0 , 1 ] \scriptstyle F_{X}\colon R\to[0,1]
  2. F X ( x ) := Pr ( X โ‰ค x ) = p . F_{X}(x):=\Pr(X\leq x)=p.\,
  3. Q ( p ) = inf { x โˆˆ โ„ : p โ‰ค F ( x ) } Q(p)\,=\,\inf\left\{x\in\mathbb{R}:p\leq F(x)\right\}
  4. F ( x ; ฮป ) = { 1 - e - ฮป x x โ‰ฅ 0 , 0 x < 0. F(x;\lambda)=\begin{cases}1-e^{-\lambda x}&x\geq 0,\\ 0&x<0.\end{cases}
  5. 1 - e - ฮป Q = p 1-e^{-\lambda Q}=p
  6. Q ( p ; ฮป ) = - ln ( 1 - p ) ฮป , Q(p;\lambda)=\frac{-\ln(1-p)}{\lambda},\!
  7. ln ( 2 ) / ฮป \ln(2)/\lambda\,
  8. ln ( 4 ) / ฮป . \ln(4)/\lambda.\,
  9. d 2 w d p 2 = w ( d w d p ) 2 \frac{d^{2}w}{dp^{2}}=w\left(\frac{dw}{dp}\right)^{2}
  10. w ( 1 / 2 ) = 0 , w\left(1/2\right)=0,\,
  11. w โ€ฒ ( 1 / 2 ) = 2 ฯ€ . w^{\prime}\left(1/2\right)=\sqrt{2\pi}.\,
  12. Q ( p ) = tan ( ฯ€ ( p - 1 / 2 ) ) Q(p)=\tan(\pi(p-1/2))\!
  13. Q ( p ) = 2 ( p - 1 / 2 ) 2 ฮฑ Q(p)=2(p-1/2)\sqrt{\frac{2}{\alpha}}\!
  14. Q ( p ) = sign ( p - 1 / 2 ) 2 q - 1 Q(p)=\operatorname{sign}(p-1/2)\,2\,\sqrt{q-1}\!
  15. q = cos ( 1 3 arccos ( ฮฑ ) ) ฮฑ q=\frac{\cos\left(\frac{1}{3}\arccos\left(\sqrt{\alpha}\,\right)\right)}{\sqrt% {\alpha}}\!
  16. ฮฑ = 4 p ( 1 - p ) . \alpha=4p(1-p).\!
  17. Q ( p ) = โˆ‘ i = 1 m a i Q i ( p ) Q(p)=\sum_{i=1}^{m}a_{i}Q_{i}(p)
  18. Q i ( p ) Q_{i}(p)
  19. i = 1 , โ€ฆ , m i=1,\ldots,m
  20. a i a_{i}
  21. i = 1 , โ€ฆ , m i=1,\ldots,m
  22. a i a_{i}
  23. Q ( p ) Q(p)
  24. d 2 Q d p 2 = H ( Q ) ( d Q d p ) 2 \frac{d^{2}Q}{dp^{2}}=H(Q)\left(\frac{dQ}{dp}\right)^{2}
  25. H ( x ) = - d log [ f ( x ) ] d x H(x)=-\frac{d\log[f(x)]}{dx}