wpmath0000015_5

Macroscopic_quantum_phenomena.html

  1. Ψ = Ψ 0 exp ( i φ ) \Psi=\Psi_{0}\exp(i\varphi)
  2. φ \varphi
  3. Ψ Ψ * d V = N s . \int\Psi\Psi^{*}\mathrm{d}V=N_{s}.
  4. Ψ Ψ * Δ V \Psi\Psi^{*}\Delta V
  5. J p = 1 2 m ( Ψ ( i h 2 π - q A ) Ψ * + c c ) \vec{J}_{p}=\frac{1}{2m}\left(\Psi(i\frac{h}{2\pi}\vec{\nabla}-q\vec{A})\Psi^{% *}+cc\right)
  6. A \vec{A}
  7. J p = Ψ 0 2 m ( h 2 π φ - q A ) . \vec{J}_{p}=\frac{\Psi_{0}^{2}}{m}\left(\frac{h}{2\pi}\vec{\nabla}\varphi-q% \vec{A}\right).
  8. m J p = ρ s v s . m\vec{J}_{p}=\rho_{s}\vec{v}_{s}.
  9. m Ψ 0 2 = ρ s m\Psi_{0}^{2}=\rho_{s}
  10. v s = 1 m ( h 2 π φ - q A ) . \vec{v}_{s}=\frac{1}{m}\left(\frac{h}{2\pi}\vec{\nabla}\varphi-q\vec{A}\right).
  11. v s = 1 m 4 h 2 π φ . \vec{v}_{s}=\frac{1}{m_{4}}\frac{h}{2\pi}\vec{\nabla}\varphi.
  12. v s d s = h 2 π m 4 φ d s . \oint\vec{v}_{s}\cdot\vec{\mathrm{d}s}=\frac{h}{2\pi m_{4}}\oint\vec{\nabla}% \varphi\cdot\vec{\mathrm{d}s}.
  13. φ d s = 2 π n \oint\vec{\nabla}\varphi\cdot\vec{\mathrm{d}s}=2\pi n
  14. v s d s = h m 4 n . \oint\vec{v}_{s}\cdot\vec{\mathrm{d}s}=\frac{h}{m_{4}}n.
  15. κ = h m 4 = 1.0 × 10 - 7 m 2 / s \kappa=\frac{h}{m_{4}}=1.0\times 10^{-7}m^{2}/s
  16. v s d s = 2 π v s r . \oint\vec{v}_{s}\cdot\vec{\mathrm{d}s}=2\pi v_{s}r.
  17. v s = 1 2 π r κ . v_{s}=\frac{1}{2\pi r}\kappa.
  18. 2 m e v s = h 2 π φ + 2 e A . 2m_{e}\vec{v}_{s}=\frac{h}{2\pi}\vec{\nabla}\varphi+2e\vec{A}.
  19. 2 m e v s d s = ( h 2 π φ + 2 e A ) d s 2m_{e}\oint\vec{v}_{s}\cdot\vec{\mathrm{d}s}=\oint(\frac{h}{2\pi}\vec{\nabla}% \varphi+2e\vec{A})\cdot\vec{\mathrm{d}s}
  20. v s d s = κ \oint\vec{v}_{s}\cdot\vec{\mathrm{d}s}=\kappa
  21. A d s = Φ \oint\vec{A}\cdot\vec{\mathrm{d}s}=\Phi
  22. Φ v = Φ - 2 m e 2 e κ . \Phi_{v}=\Phi-\frac{2m_{e}}{2e}\kappa.
  23. Φ v = n h 2 e . \Phi_{v}=n\frac{h}{2e}.
  24. Φ 0 = h 2 e = 2.067833758 ( 46 ) × 10 - 15 \Phi_{0}=\frac{h}{2e}=2.067833758(46)\times 10^{-15}
  25. Φ = Φ a + Φ s . \Phi=\Phi_{a}+\Phi_{s}.
  26. 0 = h 2 π φ + 2 e A . 0=\frac{h}{2\pi}\vec{\nabla}{\varphi}+2e\vec{A}.
  27. 0 = h 2 π × φ + 2 e × A . 0=\frac{h}{2\pi}\vec{\nabla}\times\vec{\nabla}\varphi+2e\vec{\nabla}\times\vec% {A}.
  28. × φ = 0 \vec{\nabla}\times\vec{\nabla}\varphi=0
  29. × A = B \vec{\nabla}\times\vec{A}=\vec{B}
  30. Φ = n Φ 0 . \Phi=n\Phi_{0}.
  31. Δ φ * = - 2 π h 2 m e δ v s d s . \Delta\varphi^{*}=-\frac{2\pi}{h}2m_{e}\int_{\delta}\vec{v}_{s}\cdot\vec{% \mathrm{d}s}.
  32. Φ a + Φ s + Φ 0 Δ φ * 2 π = n Φ 0 . \Phi_{a}+\Phi_{s}+\Phi_{0}\frac{\Delta\varphi^{*}}{2\pi}=n\Phi_{0}.
  33. i s = i 1 sin ( Δ φ * ) . i_{s}=i_{1}\sin(\Delta\varphi^{*}).
  34. V = 1 2 π h 2 e d Δ φ * d t . V=\frac{1}{2\pi}\frac{h}{2e}\frac{\mathrm{d}\Delta\varphi^{*}}{\mathrm{d}t}.
  35. Δ φ * \Delta\varphi^{*}
  36. Δ φ * = 2 π 2 e V h t . \Delta\varphi^{*}=2\pi\frac{2eV}{h}t.
  37. i s = i 1 sin ( 2 π 2 e V h t ) . i_{s}=i_{1}\sin(2\pi\frac{2eV}{h}t).
  38. ν = 2 e V h = V Φ 0 \nu=\frac{2eV}{h}=\frac{V}{\Phi_{0}}
  39. F = m d v s / d t . \vec{F}=m\mathrm{d}\vec{v}_{s}/\mathrm{d}t.
  40. F = q ( E + v s × B ) \vec{F}=q(\vec{E}+\vec{v}_{s}\times\vec{B})
  41. d v s / d t = v s / t + ( 1 / 2 ) v s 2 - v s × ( × v s ) \mathrm{d}\vec{v}_{s}/\mathrm{d}t=\partial\vec{v}_{s}/\partial t+(1/2)\vec{% \nabla}v_{s}^{2}-\vec{v}_{s}\times(\vec{\nabla}\times\vec{v}_{s})
  42. ( q / m ) ( E + v s × B ) = v s / t + ( 1 / 2 ) v s 2 - v s × ( × v s ) . (q/m)(\vec{E}+\vec{v}_{s}\times\vec{B})=\partial\vec{v}_{s}/\partial t+(1/2)% \vec{\nabla}v_{s}^{2}-\vec{v}_{s}\times(\vec{\nabla}\times\vec{v}_{s}).
  43. 0 = × v s + ( q / m ) × A = × v s + ( q / m ) B 0=\vec{\nabla}\times\vec{v}_{s}+(q/m)\vec{\nabla}\times\vec{A}=\vec{\nabla}% \times\vec{v}_{s}+(q/m)\vec{B}
  44. ( q / m ) E = v s / t + ( 1 / 2 ) v s 2 . (q/m)\vec{E}=\partial\vec{v}_{s}/\partial t+(1/2)\vec{\nabla}v_{s}^{2}.
  45. E d l = - V \int\vec{E}\cdot\mathrm{d}\vec{l}=-V
  46. Δ φ a * = Δ φ b * + 2 π Φ Φ 0 + 2 π n . \Delta\varphi_{a}^{*}=\Delta\varphi^{*}_{b}+2\pi\frac{\Phi}{\Phi_{0}}+2\pi n.
  47. Φ = Φ a = B A \Phi=\Phi_{a}=BA
  48. i s = i 1 sin ( Δ φ a * ) + i 1 sin ( Δ φ b * ) . i_{s}=i_{1}\sin(\Delta\varphi_{a}^{*})+i_{1}\sin(\Delta\varphi_{b}^{*}).
  49. i s = i 1 sin ( Δ φ b * + 2 π Φ Φ 0 ) + i 1 sin ( Δ φ b * ) . i_{s}=i_{1}\sin(\Delta\varphi_{b}^{*}+2\pi\frac{\Phi}{\Phi_{0}})+i_{1}\sin(% \Delta\varphi_{b}^{*}).
  50. i s = 2 i 1 sin ( Δ φ b * + π Φ Φ 0 ) cos ( π Φ a Φ 0 ) . i_{s}=2i_{1}\sin(\Delta\varphi_{b}^{*}+\pi\frac{\Phi}{\Phi_{0}})\cos(\pi\frac{% \Phi_{a}}{\Phi_{0}}).
  51. i c = 2 i 1 | cos ( π Φ a Φ 0 ) | . i_{c}=2i_{1}|\cos(\pi\frac{\Phi_{a}}{\Phi_{0}})|.
  52. × 10 1 2 \times 10^{1}2

Magnesium-protoporphyrin_IX_monomethyl_ester_(oxidative)_cyclase.html

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

Magnetic_Tower_of_Hanoi.html

  1. S R B B ( n ) S_{RBB}(n)
  2. S R R B ( n ) S_{RRB}(n)
  3. S R B B ( n + 1 ) = 2 S R B B ( n ) + S R R B ( n ) + 1 = 3 S R B B ( n ) + 1 S_{RBB}(n+1)=2S_{RBB}(n)+S_{RRB}(n)+1=3S_{RBB}(n)+1\,
  4. S R B B ( n ) = S R R B ( n ) S_{RBB}(n)=S_{RRB}(n)
  5. P R B B ( k ) P_{RBB}(k)
  6. P R R B ( k ) P_{RRB}(k)
  7. P ( k ) P(k)
  8. P R B B ( k ) = P R R B ( k ) P_{RBB}(k)=P_{RRB}(k)
  9. P R B B ( k + 1 ) = 2 P R B B ( k ) + P R R B ( k ) = 3 P R B B ( k ) P_{RBB}(k+1)=2P_{RBB}(k)+P_{RRB}(k)=3P_{RBB}(k)\,
  10. S R B B ( n ) S_{RBB}(n)
  11. P R B B ( k ) P_{RBB}(k)
  12. S R B B ( n ) = S R R B ( n ) = 1 2 3 n - 1 2 S_{RBB}(n)=S_{RRB}(n)=\tfrac{1}{2}3^{n}-\tfrac{1}{2}
  13. P R B B ( k ) = P R R B ( k ) = 3 k - 1 P_{RBB}(k)=P_{RRB}(k)=3^{k-1}\,
  14. lim n S ( n ) = s 3 n \lim_{n\to\infty}S(n)=s3^{n}\,
  15. lim k P ( k ) = p 3 k - 1 \lim_{k\to\infty}P(k)=p3^{k-1}\,
  16. S R R B ( n ) S_{RRB}(n)
  17. P R R B ( k ) P_{RRB}(k)
  18. S S I D ( n ) S_{SID}(n)
  19. S N N N ( n ) = S R N N ( n - 1 ) + 1 + S N N R ( n - 1 ) S_{NNN(n)}=S_{RNN(n-1)}+1+S_{NNR(n-1)}
  20. S R N N ( n ) = S R N N ( n - 1 ) + 1 + S N B R ( n - 1 ) S_{RNN(n)}=S_{RNN(n-1)}+1+S_{NBR(n-1)}
  21. S N N R ( n ) = 2 S R R B ( n - 1 ) + 1 + S R B N ( n - 1 ) + 1 + S R N N ( n - 1 ) S_{NNR(n)}=2\cdot S_{RRB(n-1)}+1+S_{RBN(n-1)}+1+S_{RNN(n-1)}
  22. S N B R ( n ) = 2 S R R B ( n - 1 ) + 1 + S R B N ( n - 1 ) + 1 + S R B N ( n - 1 ) S_{NBR(n)}=2\cdot S_{RRB(n-1)}+1+S_{RBN(n-1)}+1+S_{RBN(n-1)}
  23. S R N B ( n ) = S R B N ( n - 1 ) + 1 + S N B R ( n - 1 ) S_{RNB(n)}=S_{RBN(n-1)}+1+S_{NBR(n-1)}
  24. S R B N ( n ) = S R N B ( n - 1 ) + 2 S R R B ( n - 1 ) + 1 S_{RBN(n)}=S_{RNB(n-1)}+2\cdot S_{RRB(n-1)}+1
  25. S R R B ( n ) = 2 S R R B ( n - 1 ) + 1 + S R R B ( n - 1 ) S_{RRB(n)}=2\cdot S_{RRB(n-1)}+1+S_{RRB(n-1)}
  26. S ( n ) = [ S N N N ( n ) S R N N ( n ) S N N R ( n ) S N B R ( n ) S R N B ( n ) S R B N ( n ) S R R B ( n ) 1 ] \vec{S}(n)=\begin{bmatrix}S_{NNN}(n)\\ S_{RNN}(n)\\ S_{NNR}(n)\\ S_{NBR}(n)\\ S_{RNB}(n)\\ S_{RBN}(n)\\ S_{RRB}(n)\\ 1\end{bmatrix}
  27. S ( 1 ) = [ 1 1 2 2 1 1 1 1 ] , \vec{S}(1)=\begin{bmatrix}1\\ 1\\ 2\\ 2\\ 1\\ 1\\ 1\\ 1\end{bmatrix},
  28. S ( n ) = M S ( n - 1 ) , \vec{S}(n)=M\vec{S}(n-1),
  29. M M
  30. M [ 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 2 2 0 0 0 0 0 2 2 2 0 0 0 1 0 1 0 1 0 0 0 0 1 0 2 1 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 1 ] M\equiv\begin{bmatrix}0&1&1&0&0&0&0&1\\ 0&1&0&1&0&0&0&1\\ 0&1&0&0&0&1&2&2\\ 0&0&0&0&0&2&2&2\\ 0&0&0&1&0&1&0&1\\ 0&0&0&0&1&0&2&1\\ 0&0&0&0&0&0&3&1\\ 0&0&0&0&0&0&0&1\end{bmatrix}
  31. S ( n ) \vec{S}(n)
  32. S ( n ) = M n - 1 S ( 1 ) . \vec{S}(n)=M^{n-1}\vec{S}(1).
  33. M M
  34. ( K - 3 ) ( K 3 - K - 2 ) ( K - 1 ) 2 K 2 , (K-3)(K^{3}-K-2)(K-1)^{2}K^{2},
  35. K 1 , , K 8 = 3 , 1.52183 , - 0.7607 + 0.8579 i , - 0.7607 - 0.8579 i , 1 , 1 , 0 , 0 , {K_{1},\dots,K_{8}}={3,1.52183,-0.7607+0.8579i,-0.7607-0.8579i,1,1,0,0},
  36. ξ j \vec{\xi}_{j}
  37. M ξ j = K j ξ j . M\vec{\xi}_{j}=K_{j}\vec{\xi}_{j}.
  38. S ( 1 ) \vec{S}(1)
  39. S ( 1 ) = j = 1 8 b j K j ξ j , \vec{S}(1)=\sum_{j=1}^{8}b_{j}K_{j}\vec{\xi}_{j},
  40. S ( n ) = j = 1 8 b j K j n ξ j . \vec{S}(n)=\sum_{j=1}^{8}b_{j}K_{j}^{n}\vec{\xi}_{j}.
  41. | K 1 | > | K j | |K_{1}|>|K_{j}|
  42. j > 1 j>1
  43. lim n S ( n ) = b 1 K 1 n ξ 1 = b 1 3 n ξ 1 . \lim_{n\to\infty}\vec{S}(n)=b_{1}K_{1}^{n}\vec{\xi}_{1}=b_{1}3^{n}\vec{\xi}_{1}.
  44. lim n S S I D ( n ) = s 3 n \lim_{n\to\infty}S_{SID}(n)=s3^{n}\,

Magnussen_model.html

  1. R K _ i , k R_{K\_i^{\prime},k}
  2. i i^{\prime}
  3. k k
  4. R K _ i , k = - ν i , k M i A k T β k exp ( - E k R T ) j = 1 N [ C j ] η j , k = K i , k M i j = 1 N [ C j ] η j , k R_{K\_i^{\prime},k}=-\nu_{i^{\prime},k}M_{i}A_{k}T^{\beta_{k}}\exp{\left(-% \frac{E_{k}}{RT}\right)}\prod_{j^{\prime}=1}^{N}\left[C_{j^{\prime}}\right]^{% \eta_{j^{\prime},k}}=K_{i^{\prime},k}M_{i^{\prime}}\prod_{j^{\prime}=1}^{N}% \left[C_{j^{\prime}}\right]^{\eta_{j^{\prime},k}}
  5. i i^{\prime}
  6. k k
  7. A k A_{k}
  8. E k E_{k}
  9. i i^{\prime}
  10. k k
  11. ν i , k \nu_{i^{\prime},k}
  12. i i^{\prime}
  13. M i M_{i^{\prime}}
  14. T T
  15. β k \beta_{k}
  16. j j^{\prime}
  17. [ C j ] \left[C_{j^{\prime}}\right]
  18. K i , k K_{i^{\prime},k}

Mahler's_3::2_problem.html

  1. { x ( 3 2 ) n } \left\{x\left(\frac{3}{2}\right)^{n}\right\}
  2. Ω ( α ) = inf θ ( lim sup n { θ α n } - lim inf n { θ α n } ) . \Omega(\alpha)=\inf_{\theta}\left({\limsup_{n\rightarrow\infty}\left\{{\theta% \alpha^{n}}\right\}-\liminf_{n\rightarrow\infty}\left\{{\theta\alpha^{n}}% \right\}}\right).
  3. Ω ( p q ) > 1 p \Omega\left(\frac{p}{q}\right)>\frac{1}{p}

Malate_dehydrogenase_(NAD(P)+).html

  1. \rightleftharpoons

Maleamate_amidohydrolase.html

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Malonate_decarboxylase_holo-(acyl-carrier_protein)_synthase.html

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Malonyl-CoA_O-methyltransferase.html

  1. \rightleftharpoons

Malonyl-S-ACP:biotin-protein_carboxyltransferase.html

  1. \rightleftharpoons

Malonyl-S-ACP_decarboxylase.html

  1. \rightleftharpoons

Malonyl_CoA_reductase_(malonate_semialdehyde-forming).html

  1. \rightleftharpoons

Maltokinase.html

  1. \rightleftharpoons

Malware_research.html

  1. G = ( V , E ) G=(V,E)
  2. ( s , t ) E (s,t)\in E
  3. s s
  4. t t

Manin_matrix.html

  1. ( M i j M i l M k j M k l ) = ( a b c d ) \begin{pmatrix}\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&M_{ij}&\cdots&M_{il}&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&M_{kj}&\cdots&M_{kl}&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\end{pmatrix}=\begin{pmatrix}\cdots&\cdots&% \cdots&\cdots&\cdots\\ \cdots&a&\cdots&b&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&c&\cdots&d&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots\end{pmatrix}
  2. a c = c a , b d = d b , (entries in the same column commute) ac=ca,~{}~{}~{}bd=db,~{}~{}~{}\,\text{(entries in the same column commute) }
  3. a d - d a = c b - b c , (cross commutation relation) . ad-da=cb-bc,~{}~{}~{}\,\text{(cross commutation relation)}.
  4. M = ( a b c d ) . M=\begin{pmatrix}a&b\\ c&d\end{pmatrix}.
  5. ( y 1 y 2 ) = ( a b c d ) ( x 1 x 2 ) . \begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}.
  6. [ y 1 , y 2 ] = [ a x 1 + b x 2 , c x 1 + d x 2 ] = [ a , c ] x 1 2 + [ b , d ] x 2 2 + ( [ a , d ] + [ b , c ] ) x 1 x 2 . [y_{1},y_{2}]=[ax_{1}+bx_{2},cx_{1}+dx_{2}]=[a,c]x^{2}_{1}+[b,d]x^{2}_{2}+([a,% d]+[b,c])x_{1}x_{2}.
  7. ( ϕ 1 , ϕ 2 ) = ( ψ 1 , ψ 2 ) ( a b c d ) . \begin{pmatrix}\phi_{1},~{}\phi_{2}\end{pmatrix}=\begin{pmatrix}\psi_{1},~{}% \psi_{2}\end{pmatrix}\begin{pmatrix}a&b\\ c&d\end{pmatrix}.
  8. M - 1 = 1 a d - c b ( d - b - c a ) M^{-1}=\frac{1}{ad-cb}\begin{pmatrix}d&-b\\ -c&a\end{pmatrix}
  9. ( d - b - c a ) ( a b c d ) = ( d a - b c d b - b d - c a + a c - c b + a d ) = if and only if M is a Manin matrix = ( a d - c b 0 0 a d - c b ) . \begin{pmatrix}d&-b\\ -c&a\end{pmatrix}\begin{pmatrix}a&b\\ c&d\end{pmatrix}=\begin{pmatrix}da-bc&db-bd\\ -ca+ac&-cb+ad\end{pmatrix}=\,\text{if and only if }M\,\text{ is a Manin matrix% }=\begin{pmatrix}ad-cb&0\\ 0&ad-cb\end{pmatrix}.
  10. M 2 - ( a + d ) M + ( a d - c b ) 1 2 × 2 = 0 M^{2}-(a+d)M+(ad-cb)1_{2\times 2}=0
  11. \otimes
  12. y i = k M i k x k E n d P o l P o l y_{i}=\sum_{k}M_{ik}\otimes x_{k}\in EndPol\otimes Pol
  13. x i y i = k M i k x k x_{i}\mapsto y_{i}=\sum_{k}M_{ik}\otimes x_{k}
  14. \otimes
  15. Δ ( M i j ) = l M i l M l j \Delta(M_{ij})=\sum_{l}M_{il}\otimes M_{lj}
  16. i j \partial_{ij}
  17. z \partial_{z}
  18. ( z I d D t X z I d ) . \begin{pmatrix}zId&D^{t}\\ X&\partial_{z}Id\end{pmatrix}.
  19. M - 1 = 1 det c o l ( M ) M a d j , M^{-1}=\frac{1}{{\det}^{col}(M)}M^{adj},
  20. d e t c o l u m n ( t - M ) | t = M r i g h t s u b s t i t u t e = 0 , i . e . i = 0... n ( - 1 ) i σ i M n - i = 0. det^{column}(t-M)|_{t=M}^{right~{}substitute}=0,~{}~{}i.e.~{}~{}\sum_{i=0...n}% (-1)^{i}\sigma_{i}M^{n-i}=0.
  21. d e t c o l u m n ( t - M ) = i = 0... n ( - 1 ) i σ i t n - i det^{column}(t-M)=\sum_{i=0...n}(-1)^{i}\sigma_{i}t^{n-i}
  22. k 0 : - ( - 1 ) k k σ k = i = 0... k - 1 σ i T r ( M k - i ) \forall k\geq 0:-(-1)^{k}k\sigma_{k}=\sum_{i=0...k-1}\sigma_{i}Tr(M^{k-i})
  23. d e t c o l u m n ( t - M ) = i = 0... n ( - 1 ) i σ i t n - i det^{column}(t-M)=\sum_{i=0...n}(-1)^{i}\sigma_{i}t^{n-i}
  24. d e t c o l u m n ( A B C d ) = d e t c o l u m n ( A ) d e t c o l u m n ( D - C A - 1 B ) = d e t c o l u m n ( D ) d e t c o l u m n ( A - B D - 1 C ) . det^{column}\begin{pmatrix}A&B\\ C&d\\ \end{pmatrix}=det^{column}(A)det^{column}(D-CA^{-1}B)=det^{column}(D)det^{% column}(A-BD^{-1}C).
  25. ( D - C A - 1 B ) , ( A - B D - 1 C ) (D-CA^{-1}B),(A-BD^{-1}C)
  26. d / d z I d - E / z d/dzId-E/z
  27. M = ( d / d z - E 11 / z - E 12 / z - E 21 / z d / z - E 22 / z ) . M=\begin{pmatrix}d/dz-E_{11}/z&-E_{12}/z\\ -E_{21}/z&d/z-E_{22}/z\end{pmatrix}.
  28. [ d / d z - E 11 / z , - E 21 / z ] = [ d / d z , - E 21 / z ] + [ - E 11 / z , - E 21 / z ] = E 21 / z 2 - E 21 / z 2 = 0 [d/dz-E_{11}/z,-E_{21}/z]=[d/dz,-E_{21}/z]+[-E_{11}/z,-E_{21}/z]=E_{21}/z^{2}-% E_{21}/z^{2}=0
  29. e x p ( - d / d z ) ( I d + E / z ) exp(-d/dz)(Id+E/z)
  30. E i j = x i x j ; E i j = a = 1 n x i a x j a ; E i j = ψ i ψ j . E_{ij}=x_{i}\frac{\partial}{\partial x_{j}};~{}~{}~{}~{}~{}E_{ij}=\sum_{a=1}^{% n}x_{ia}\frac{\partial}{\partial x_{ja}};~{}~{}~{}~{}E_{ij}=\psi_{i}\frac{% \partial}{\partial\psi_{j}}.
  31. z n - 1 d e t c o l ( d / d z - E / z ) = d e t c o l ( z d / d z - E - d i a g ( n - 1 , n - 2 , , 1 , 0 ) ) z^{n-1}det^{col}(d/dz-E/z)=det^{col}(zd/dz-E-diag(n-1,n-2,...,1,0))
  32. d e t c o l ( d / d z - g E g - 1 / z ) = d e t c o l ( g ( d / d z - E / z ) g - 1 ) = d e t ( g ) d e t c o l ( d / d z - E / z ) d e t ( g - 1 ) = d e t c o l ( d / d z - E / z ) det^{col}(d/dz-gEg^{-1}/z)=det^{col}(g(d/dz-E/z)g^{-1})=det(g)det^{col}(d/dz-E% /z)det(g^{-1})=det^{col}(d/dz-E/z)
  33. d e t ( g M ) = d e t ( M g ) = d e t ( M ) d e t ( g ) det(gM)=det(Mg)=det(M)det(g)

Mannosylfructose-phosphate_phosphatase.html

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Mannosylfructose-phosphate_synthase.html

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Mannosylglucosyl-3-phosphoglycerate_synthase.html

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Mannosylglycerate_hydrolase.html

  1. \rightleftharpoons

Mannosylglycerate_synthase.html

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Marine_weather_forecasting.html

  1. t = 4 π X / ( g T ) t=4\pi X/(gT)

Markov–Kakutani_fixed-point_theorem.html

  1. x ( N ) = 1 N + 1 n = 0 N T n ( x ) . x(N)={1\over N+1}\sum_{n=0}^{N}T^{n}(x).
  2. x ( N i ) y . x(N_{i})\rightarrow y.\,
  3. | f ( T x ( N ) ) - f ( x ( N ) ) | = 1 N + 1 | f ( T N + 1 x ) - f ( x ) | 2 M N + 1 . |f(Tx(N))-f(x(N))|={1\over N+1}|f(T^{N+1}x)-f(x)|\leq{2M\over N+1}.
  4. f ( T y ) = f ( y ) . f(Ty)=f(y).\,
  5. T y = y . Ty=y.\,
  6. C S = { y C | T y = y , T S } = T S C T C^{S}=\{y\in C|Ty=y,\,T\in S\}=\bigcap_{T\in S}C^{T}\,

Mashreghi–Ransford_inequality.html

  1. ( a n ) n 0 (a_{n})_{n\geq 0}
  2. b n = k = 0 n ( n k ) a k , ( n 0 ) , b_{n}=\sum_{k=0}^{n}{n\choose k}a_{k},\qquad(n\geq 0),
  3. c n = k = 0 n ( - 1 ) k ( n k ) a k , ( n 0 ) . c_{n}=\sum_{k=0}^{n}(-1)^{k}{n\choose k}a_{k},\qquad(n\geq 0).
  4. ( n k ) = n ! k ! ( n - k ) ! . {n\choose k}=\frac{n!}{k!(n-k)!}.
  5. β > 1 \beta>1
  6. b n = O ( β n ) b_{n}=O(\beta^{n})
  7. c n = O ( β n ) c_{n}=O(\beta^{n})
  8. n n\to\infty
  9. a n = O ( α n ) a_{n}=O(\alpha^{n})
  10. n n\to\infty
  11. α = β 2 - 1 . \alpha=\sqrt{\beta^{2}-1}.
  12. κ \kappa
  13. ( lim sup n | a n | α n ) κ ( lim sup n | b n | β n ) 1 2 ( lim sup n | c n | β n ) 1 2 . \left(\limsup_{n\to\infty}\frac{|a_{n}|}{\alpha^{n}}\right)\leq\kappa\,\left(% \limsup_{n\to\infty}\frac{|b_{n}|}{\beta^{n}}\right)^{\frac{1}{2}}\left(% \limsup_{n\to\infty}\frac{|c_{n}|}{\beta^{n}}\right)^{\frac{1}{2}}.
  14. κ \kappa
  15. 2 3 κ 2. \frac{2}{\sqrt{3}}\leq\kappa\leq 2.

Mathematical_Q_models.html

  1. d U ( r , w ) d r - i k U ( r , w ) = 0 ( 1.1 ) \frac{dU(r,w)}{dr}-ikU(r,w)=0\quad(1.1)
  2. U ( r + r , w ) = U ( r , w ) exp ( i k r ) ( 1.2 ) U(r+\bigtriangleup r,w)=U(r,w)\exp(ik\bigtriangleup r)\quad(1.2)
  3. K ( i w ) = k ( w ) + i a ( w ) ( 1.3 ) K(iw)=k(w)+ia(w)\quad(1.3)
  4. c ( w ) = w k ( w ) ( 1.4 ) c(w)=\frac{w}{k(w)}\quad(1.4)
  5. α = | w | ( 2 c r Q r ) ( 1.5 ) \alpha=\frac{|w|}{(2c_{r}Q_{r})}\quad(1.5)
  6. 1 c ( w ) = 1 c r ( 1 - 1 π Q r l n | w w r | ) ( 1.6 ) \frac{1}{c(w)}=\frac{1}{c_{r}}(1-\frac{1}{\pi Q_{r}}ln|\frac{w}{w_{r}}|)\quad(% 1.6)
  7. 1 c ( w ) = 1 c r | w w r | - γ ( 1.7 ) \frac{1}{c(w)}=\frac{1}{c_{r}}|\frac{w}{w_{r}}|^{-\gamma}\quad(1.7)
  8. γ = ( π Q r ) - 1 \gamma=(\pi Q_{r})^{-1}

Matlis_duality.html

  1. D ( k ) D ( R ) : R Hom k ( R , - ) . D(k)\leftarrow D(R):R\operatorname{Hom}_{k}(R,-).
  2. E ( k ) E(k)
  3. D ( k ) D(k)

Matrix_analytic_method.html

  1. P = ( B 0 B 1 B 2 B 3 A 0 A 1 A 2 A 3 A 0 A 1 A 2 A 0 A 1 ) P=\begin{pmatrix}B_{0}&B_{1}&B_{2}&B_{3}&\cdots\\ A_{0}&A_{1}&A_{2}&A_{3}&\cdots\\ &A_{0}&A_{1}&A_{2}&\cdots\\ &&A_{0}&A_{1}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}
  2. P π = π and 𝐞 T π = 1 P\pi=\pi\quad\,\text{ and }\quad\mathbf{e}\text{T}\pi=1
  3. G = i = 0 G i A i . G=\sum_{i=0}^{\infty}G^{i}A_{i}.
  4. A ¯ i + 1 \displaystyle\overline{A}_{i+1}
  5. B ¯ 0 π 0 \displaystyle\overline{B}_{0}\pi_{0}
  6. π i = ( I - A ¯ 1 ) - 1 [ B ¯ i + 1 π 0 + j = 1 i - 1 A ¯ i + 1 - j π j ] , i 1. \pi_{i}=(I-\overline{A}_{1})^{-1}\left[\overline{B}_{i+1}\pi_{0}+\sum_{j=1}^{i% -1}\overline{A}_{i+1-j}\pi_{j}\right],i\geq 1.

Matrix_gamma_distribution.html

  1. Γ p \Gamma_{p}
  2. β = 2 , α = n 2 . \beta=2,\alpha=\frac{n}{2}.

Matrix_representation_of_Maxwell's_equations.html

  1. 𝐃 ( 𝐫 , t ) = ρ × 𝐇 ( 𝐫 , t ) - t 𝐃 ( 𝐫 , t ) = 𝐉 × 𝐄 ( 𝐫 , t ) + t 𝐁 ( 𝐫 , t ) = 0 𝐁 ( 𝐫 , t ) = 0 . \begin{aligned}&\displaystyle{\mathbf{\nabla}}\cdot{\mathbf{D}}\left({\mathbf{% r}},t\right)=\rho\\ &\displaystyle{\mathbf{\nabla}}\times{\mathbf{H}}\left({\mathbf{r}},t\right)-% \frac{\partial}{\partial t}{\mathbf{D}}\left({\mathbf{r}},t\right)={\mathbf{J}% }\\ &\displaystyle{\mathbf{\nabla}}\times{\mathbf{E}}\left({\mathbf{r}},t\right)+% \frac{\partial}{\partial t}{\mathbf{B}}\left({\mathbf{r}},t\right)=0\\ &\displaystyle{\mathbf{\nabla}}\cdot{\mathbf{B}}\left({\mathbf{r}},t\right)=0% \,.\end{aligned}
  2. 𝐃 = ϵ 𝐄 , 𝐁 = μ 𝐇 {\mathbf{D}}=\epsilon{\mathbf{E}}\,,\quad{\mathbf{B}}=\mu{\mathbf{H}}
  3. v ( 𝐫 , t ) = 1 ϵ ( 𝐫 , t ) μ ( 𝐫 , t ) v({\mathbf{r}},t)=\frac{1}{\sqrt{\epsilon({\mathbf{r}},t)\mu({\mathbf{r}},t)}}
  4. 𝐅 + ( 𝐫 , t ) = 1 2 ( ϵ ( 𝐫 , t ) 𝐄 ( 𝐫 , t ) + i 1 μ ( 𝐫 , t ) 𝐁 ( 𝐫 , t ) ) 𝐅 - ( 𝐫 , t ) = 1 2 ( ϵ ( 𝐫 , t ) 𝐄 ( 𝐫 , t ) - i 1 μ ( 𝐫 , t ) 𝐁 ( 𝐫 , t ) ) . \begin{aligned}\displaystyle{\mathbf{F}}^{+}\left({\mathbf{r}},t\right)&% \displaystyle=\frac{1}{\sqrt{2}}\left(\sqrt{\epsilon({\mathbf{r}},t)}{\mathbf{% E}}\left({\mathbf{r}},t\right)+{\rm i}\frac{1}{\sqrt{\mu({\mathbf{r}},t)}}{% \mathbf{B}}\left({\mathbf{r}},t\right)\right)\\ \displaystyle{\mathbf{F}}^{-}\left({\mathbf{r}},t\right)&\displaystyle=\frac{1% }{\sqrt{2}}\left(\sqrt{\epsilon({\mathbf{r}},t)}{\mathbf{E}}\left({\mathbf{r}}% ,t\right)-{\rm i}\frac{1}{\sqrt{\mu({\mathbf{r}},t)}}{\mathbf{B}}\left({% \mathbf{r}},t\right)\right)\,.\end{aligned}
  5. i t 𝐅 ± ( 𝐫 , t ) = ± v × 𝐅 ± ( 𝐫 , t ) - 1 2 ϵ ( i 𝐉 ) 𝐅 ± ( 𝐫 , t ) = 1 2 ϵ ( ρ ) . \begin{aligned}\displaystyle{\rm i}\frac{\partial}{\partial t}{\mathbf{F}}^{% \pm}\left({\mathbf{r}},t\right)&\displaystyle=\pm v{\mathbf{\nabla}}\times{% \mathbf{F}}^{\pm}\left({\mathbf{r}},t\right)-\frac{1}{\sqrt{2\epsilon}}({\rm i% }{\mathbf{J}})\\ \displaystyle{\mathbf{\nabla}}\cdot{\mathbf{F}}^{\pm}\left({\mathbf{r}},t% \right)&\displaystyle=\frac{1}{\sqrt{2\epsilon}}(\rho)\,.\end{aligned}
  6. Ψ + ( 𝐫 , t ) \displaystyle\Psi^{+}({\mathbf{r}},t)
  7. W + \displaystyle W^{+}
  8. t Ψ + \displaystyle\frac{\partial}{\partial t}\Psi^{+}
  9. Ω = ( 𝟎 - 𝐥 𝐥 𝟎 ) β = ( 𝐥 𝟎 𝟎 - 𝐥 ) 𝐥 = ( 1 0 0 1 ) . \Omega=\begin{pmatrix}{\mathbf{0}}&-{\mathbf{l}}\\ {\mathbf{l}}&{\mathbf{0}}\end{pmatrix}\,\qquad\beta=\begin{pmatrix}{\mathbf{l}% }&{\mathbf{0}}\\ {\mathbf{0}}&-{\mathbf{l}}\end{pmatrix}\,\qquad{\mathbf{l}}=\begin{pmatrix}1&0% \\ 0&1\end{pmatrix}\,.
  10. M x = [ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 ] = - β Ω M_{x}=\begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\end{bmatrix}=-\beta\Omega\,
  11. M y = [ 0 0 - i 0 0 0 0 - i i 0 0 0 0 i 0 0 ] = i Ω M_{y}=\begin{bmatrix}0&0&-{\rm i}&0\\ 0&0&0&-{\rm i}\\ {\rm i}&0&0&0\\ 0&{\rm i}&0&0\end{bmatrix}={\rm i}\Omega\,
  12. M z = [ 1 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 - 1 ] = β . M_{z}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{bmatrix}=\beta\,.
  13. M x β = - β M x \displaystyle M_{x}\beta=-\beta M_{x}
  14. Velocity function: v ( 𝐫 , t ) \displaystyle\,\text{ Velocity function:}\,v({\mathbf{r}},t)
  15. ε = 1 v h , μ = h v \varepsilon=\frac{1}{vh}\,,\quad\mu=\frac{h}{v}
  16. 𝐮 ( 𝐫 , t ) \displaystyle{\mathbf{u}}({\mathbf{r}},t)
  17. n ( 𝐫 , t ) = c v ( 𝐫 , t ) n({\mathbf{r}},t)=\frac{c}{v({\mathbf{r}},t)}
  18. 𝚺 = [ σ 𝟎 𝟎 σ ] α = [ 𝟎 σ σ 𝟎 ] 𝐈 = [ 𝟏 𝟎 𝟎 𝟏 ] \displaystyle{\mathbf{\Sigma}}=\left[\begin{array}[]{cc}{\mathbf{\sigma}}&{% \mathbf{0}}\\ {\mathbf{0}}&{\mathbf{\sigma}}\end{array}\right]\,\qquad{\mathbf{\alpha}}=% \left[\begin{array}[]{cc}{\mathbf{0}}&{\mathbf{\sigma}}\\ {\mathbf{\sigma}}&{\mathbf{0}}\end{array}\right]\,\qquad{\mathbf{I}}=\left[% \begin{array}[]{cc}{\mathbf{1}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{1}}\end{array}\right]
  19. σ = ( σ x , σ y , σ z ) = [ ( 0 1 1 0 ) , ( 0 - i i 0 ) , ( 1 0 0 - 1 ) ] {\mathbf{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})=\left[\begin{pmatrix}0&1% \\ 1&0\end{pmatrix},\begin{pmatrix}0&-{\rm i}\\ {\rm i}&0\end{pmatrix},\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\right]
  20. t [ 𝐈 𝟎 𝟎 𝐈 ] [ Ψ + Ψ - ] - v ˙ ( 𝐫 , t ) 2 v ( 𝐫 , t ) [ 𝐈 𝟎 𝟎 𝐈 ] [ Ψ + Ψ - ] + h ˙ ( 𝐫 , t ) 2 h ( 𝐫 , t ) [ 𝟎 i β α y i β α y 𝟎 ] [ Ψ + Ψ - ] \displaystyle\frac{\partial}{\partial t}\left[\begin{array}[]{cc}{\mathbf{I}}&% {\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{I}}\end{array}\right]\left[\begin{array}[]{cc}\Psi^{+}\\ \Psi^{-}\end{array}\right]-\frac{\dot{v}({\mathbf{r}},t)}{2v({\mathbf{r}},t)}% \left[\begin{array}[]{cc}{\mathbf{I}}&{\mathbf{0}}\\ {\mathbf{0}}&{\mathbf{I}}\end{array}\right]\left[\begin{array}[]{cc}\Psi^{+}\\ \Psi^{-}\end{array}\right]+\frac{\dot{h}({\mathbf{r}},t)}{2h({\mathbf{r}},t)}% \left[\begin{array}[]{cc}{\mathbf{0}}&{\rm i}\beta\alpha_{y}\\ {\rm i}\beta\alpha_{y}&{\mathbf{0}}\end{array}\right]\left[\begin{array}[]{cc}% \Psi^{+}\\ \Psi^{-}\end{array}\right]

Matrix_splitting.html

  1. A x = k , ( 1 ) Ax=k,\quad(1)
  2. A = B - C , ( 2 ) A=B-C,\quad(2)
  3. B x = g , ( 3 ) Bx=g,\quad(3)
  4. B x ( m + 1 ) = C x ( m ) + k , m = 0 , 1 , 2 , , ( 4 ) Bx^{(m+1)}=Cx^{(m)}+k,\quad m=0,1,2,\ldots,\quad(4)
  5. x ( m + 1 ) = B - 1 C x ( m ) + B - 1 k , m = 0 , 1 , 2 , ( 5 ) x^{(m+1)}=B^{-1}Cx^{(m)}+B^{-1}k,\quad m=0,1,2,\ldots\quad(5)
  6. ρ ( D ) \rho(D)
  7. A = D - U - L , ( 6 ) A=D-U-L,\quad(6)
  8. x ( m + 1 ) = D - 1 ( U + L ) x ( m ) + D - 1 k . ( 7 ) x^{(m+1)}=D^{-1}(U+L)x^{(m)}+D^{-1}k.\quad(7)
  9. x ( m + 1 ) = ( D - L ) - 1 U x ( m ) + ( D - L ) - 1 k . ( 8 ) x^{(m+1)}=(D-L)^{-1}Ux^{(m)}+(D-L)^{-1}k.\quad(8)
  10. x ( m + 1 ) = ( D - ω L ) - 1 [ ( 1 - ω ) D + ω U ] x ( m ) + ω ( D - ω L ) - 1 k . ( 9 ) x^{(m+1)}=(D-\omega L)^{-1}[(1-\omega)D+\omega U]x^{(m)}+\omega(D-\omega L)^{-% 1}k.\quad(9)
  11. 𝐀 = ( 6 - 2 - 3 - 1 4 - 2 - 3 - 1 5 ) , 𝐤 = ( 5 - 12 10 ) . ( 10 ) \begin{aligned}&\displaystyle\mathbf{A}=\begin{pmatrix}6&-2&-3\\ -1&4&-2\\ -3&-1&5\end{pmatrix},\quad\mathbf{k}=\begin{pmatrix}5\\ -12\\ 10\end{pmatrix}.\quad(10)\end{aligned}
  12. 𝐁 = ( 6 0 0 0 4 0 0 0 5 ) , 𝐂 = ( 0 2 3 1 0 2 3 1 0 ) , ( 11 ) \begin{aligned}&\displaystyle\mathbf{B}=\begin{pmatrix}6&0&0\\ 0&4&0\\ 0&0&5\end{pmatrix},\quad\mathbf{C}=\begin{pmatrix}0&2&3\\ 1&0&2\\ 3&1&0\end{pmatrix},\quad(11)\end{aligned}
  13. 𝐀 - 𝟏 = 1 47 ( 18 13 16 11 21 15 13 12 22 ) , 𝐁 - 𝟏 = ( 1 6 0 0 0 1 4 0 0 0 1 5 ) , \begin{aligned}&\displaystyle\mathbf{A^{-1}}=\frac{1}{47}\begin{pmatrix}18&13&% 16\\ 11&21&15\\ 13&12&22\end{pmatrix},\quad\mathbf{B^{-1}}=\begin{pmatrix}\frac{1}{6}&0&0\\ 0&\frac{1}{4}&0\\ 0&0&\frac{1}{5}\end{pmatrix},\end{aligned}
  14. 𝐃 = 𝐁 - 𝟏 𝐂 = ( 0 1 3 1 2 1 4 0 1 2 3 5 1 5 0 ) , 𝐁 - 𝟏 𝐤 = ( 5 6 - 3 2 ) . \begin{aligned}\displaystyle\mathbf{D}=\mathbf{B^{-1}C}=\begin{pmatrix}0&\frac% {1}{3}&\frac{1}{2}\\ \frac{1}{4}&0&\frac{1}{2}\\ \frac{3}{5}&\frac{1}{5}&0\end{pmatrix},\quad\mathbf{B^{-1}k}=\begin{pmatrix}% \frac{5}{6}\\ -3\\ 2\end{pmatrix}.\end{aligned}
  15. ρ ( D ) \rho(D)
  16. x ( m + 1 ) = ( 0 1 3 1 2 1 4 0 1 2 3 5 1 5 0 ) x ( m ) + ( 5 6 - 3 2 ) , m = 0 , 1 , 2 , ( 12 ) x^{(m+1)}=\begin{aligned}\displaystyle\begin{pmatrix}0&\frac{1}{3}&\frac{1}{2}% \\ \frac{1}{4}&0&\frac{1}{2}\\ \frac{3}{5}&\frac{1}{5}&0\end{pmatrix}x^{(m)}+\begin{pmatrix}\frac{5}{6}\\ -3\\ 2\end{pmatrix}\end{aligned},\quad m=0,1,2,\ldots\quad(12)
  17. 𝐱 = ( 2 - 1 3 ) . ( 13 ) \begin{aligned}&\displaystyle\mathbf{x}=\begin{pmatrix}2\\ -1\\ 3\end{pmatrix}.\quad(13)\end{aligned}
  18. x 1 ( m ) x^{(m)}_{1}
  19. x 2 ( m ) x^{(m)}_{2}
  20. x 3 ( m ) x^{(m)}_{3}
  21. 0.0 0.0
  22. 0.0 0.0
  23. 0.0 0.0
  24. 0.83333 0.83333
  25. - 3.0000 -3.0000
  26. 2.0000 2.0000
  27. 0.83333 0.83333
  28. - 1.7917 -1.7917
  29. 1.9000 1.9000
  30. 1.1861 1.1861
  31. - 1.8417 -1.8417
  32. 2.1417 2.1417
  33. 1.2903 1.2903
  34. - 1.6326 -1.6326
  35. 2.3433 2.3433
  36. 1.4608 1.4608
  37. - 1.5058 -1.5058
  38. 2.4477 2.4477
  39. 1.5553 1.5553
  40. - 1.4110 -1.4110
  41. 2.5753 2.5753
  42. 1.6507 1.6507
  43. - 1.3235 -1.3235
  44. 2.6510 2.6510
  45. 1.7177 1.7177
  46. - 1.2618 -1.2618
  47. 2.7257 2.7257
  48. 1.7756 1.7756
  49. - 1.2077 -1.2077
  50. 2.7783 2.7783
  51. 1.8199 1.8199
  52. - 1.1670 -1.1670
  53. 2.8238 2.8238
  54. 𝐃 = ( 6 0 0 0 4 0 0 0 5 ) , 𝐔 = ( 0 2 3 0 0 2 0 0 0 ) , 𝐋 = ( 0 0 0 1 0 0 3 1 0 ) . ( 14 ) \begin{aligned}&\displaystyle\mathbf{D}=\begin{pmatrix}6&0&0\\ 0&4&0\\ 0&0&5\end{pmatrix},\quad\mathbf{U}=\begin{pmatrix}0&2&3\\ 0&0&2\\ 0&0&0\end{pmatrix},\quad\mathbf{L}=\begin{pmatrix}0&0&0\\ 1&0&0\\ 3&1&0\end{pmatrix}.\quad(14)\end{aligned}
  55. ( 𝐃 - 𝐋 ) - 𝟏 = 1 120 ( 20 0 0 5 30 0 13 6 24 ) , \begin{aligned}&\displaystyle\mathbf{(D-L)^{-1}}=\frac{1}{120}\begin{pmatrix}2% 0&0&0\\ 5&30&0\\ 13&6&24\end{pmatrix},\end{aligned}
  56. ( 𝐃 - 𝐋 ) - 𝟏 𝐔 = 1 120 ( 0 40 60 0 10 75 0 26 51 ) , ( 𝐃 - 𝐋 ) - 𝟏 𝐤 = 1 120 ( 100 - 335 233 ) . \begin{aligned}&\displaystyle\mathbf{(D-L)^{-1}U}=\frac{1}{120}\begin{pmatrix}% 0&40&60\\ 0&10&75\\ 0&26&51\end{pmatrix},\quad\mathbf{(D-L)^{-1}k}=\frac{1}{120}\begin{pmatrix}100% \\ -335\\ 233\end{pmatrix}.\end{aligned}
  57. x ( m + 1 ) = 1 120 ( 0 40 60 0 10 75 0 26 51 ) x ( m ) + 1 120 ( 100 - 335 233 ) , m = 0 , 1 , 2 , ( 15 ) x^{(m+1)}=\begin{aligned}&\displaystyle\frac{1}{120}\begin{pmatrix}0&40&60\\ 0&10&75\\ 0&26&51\end{pmatrix}x^{(m)}+\frac{1}{120}\begin{pmatrix}100\\ -335\\ 233\end{pmatrix},\end{aligned}\quad m=0,1,2,\ldots\quad(15)
  58. x 1 ( m ) x^{(m)}_{1}
  59. x 2 ( m ) x^{(m)}_{2}
  60. x 3 ( m ) x^{(m)}_{3}
  61. 0.0 0.0
  62. 0.0 0.0
  63. 0.0 0.0
  64. 0.8333 0.8333
  65. - 2.7917 -2.7917
  66. 1.9417 1.9417
  67. 0.8736 0.8736
  68. - 1.8107 -1.8107
  69. 2.1620 2.1620
  70. 1.3108 1.3108
  71. - 1.5913 -1.5913
  72. 2.4682 2.4682
  73. 1.5370 1.5370
  74. - 1.3817 -1.3817
  75. 2.6459 2.6459
  76. 1.6957 1.6957
  77. - 1.2531 -1.2531
  78. 2.7668 2.7668
  79. 1.7990 1.7990
  80. - 1.1668 -1.1668
  81. 2.8461 2.8461
  82. 1.8675 1.8675
  83. - 1.1101 -1.1101
  84. 2.8985 2.8985
  85. 1.9126 1.9126
  86. - 1.0726 -1.0726
  87. 2.9330 2.9330
  88. 1.9423 1.9423
  89. - 1.0479 -1.0479
  90. 2.9558 2.9558
  91. 1.9619 1.9619
  92. - 1.0316 -1.0316
  93. 2.9708 2.9708
  94. ( 𝐃 - ω 𝐋 ) - 𝟏 = 1 12 ( 2 0 0 0.55 3 0 1.441 0.66 2.4 ) , \begin{aligned}&\displaystyle\mathbf{(D-\omega L)^{-1}}=\frac{1}{12}\begin{% pmatrix}2&0&0\\ 0.55&3&0\\ 1.441&0.66&2.4\end{pmatrix},\end{aligned}
  95. ( 𝐃 - ω 𝐋 ) - 𝟏 [ ( 𝟏 - ω ) 𝐃 + ω 𝐔 ] = 1 12 ( - 1.2 4.4 6.6 - 0.33 0.01 8.415 - 0.8646 2.9062 5.0073 ) , \begin{aligned}&\displaystyle\mathbf{(D-\omega L)^{-1}[(1-\omega)D+\omega U]}=% \frac{1}{12}\begin{pmatrix}-1.2&4.4&6.6\\ -0.33&0.01&8.415\\ -0.8646&2.9062&5.0073\end{pmatrix},\end{aligned}
  96. ω ( 𝐃 - ω 𝐋 ) - 𝟏 𝐤 = 1 12 ( 11 - 36.575 25.6135 ) . \begin{aligned}&\displaystyle\mathbf{\omega(D-\omega L)^{-1}k}=\frac{1}{12}% \begin{pmatrix}11\\ -36.575\\ 25.6135\end{pmatrix}.\end{aligned}
  97. x ( m + 1 ) = 1 12 ( - 1.2 4.4 6.6 - 0.33 0.01 8.415 - 0.8646 2.9062 5.0073 ) x ( m ) + 1 12 ( 11 - 36.575 25.6135 ) , m = 0 , 1 , 2 , ( 16 ) x^{(m+1)}=\begin{aligned}&\displaystyle\frac{1}{12}\begin{pmatrix}-1.2&4.4&6.6% \\ -0.33&0.01&8.415\\ -0.8646&2.9062&5.0073\end{pmatrix}x^{(m)}+\frac{1}{12}\begin{pmatrix}11\\ -36.575\\ 25.6135\end{pmatrix},\end{aligned}\quad m=0,1,2,\ldots\quad(16)
  98. x 1 ( m ) x^{(m)}_{1}
  99. x 2 ( m ) x^{(m)}_{2}
  100. x 3 ( m ) x^{(m)}_{3}
  101. 0.0 0.0
  102. 0.0 0.0
  103. 0.0 0.0
  104. 0.9167 0.9167
  105. - 3.0479 -3.0479
  106. 2.1345 2.1345
  107. 0.8814 0.8814
  108. - 1.5788 -1.5788
  109. 2.2209 2.2209
  110. 1.4711 1.4711
  111. - 1.5161 -1.5161
  112. 2.6153 2.6153
  113. 1.6521 1.6521
  114. - 1.2557 -1.2557
  115. 2.7526 2.7526
  116. 1.8050 1.8050
  117. - 1.1641 -1.1641
  118. 2.8599 2.8599
  119. 1.8823 1.8823
  120. - 1.0930 -1.0930
  121. 2.9158 2.9158
  122. 1.9314 1.9314
  123. - 1.0559 -1.0559
  124. 2.9508 2.9508
  125. 1.9593 1.9593
  126. - 1.0327 -1.0327
  127. 2.9709 2.9709
  128. 1.9761 1.9761
  129. - 1.0185 -1.0185
  130. 2.9829 2.9829
  131. 1.9862 1.9862
  132. - 1.0113 -1.0113
  133. 2.9901 2.9901

Matrix_t-distribution.html

  1. × | 𝐈 n + s y m b o l Σ - 1 ( 𝐗 - 𝐌 ) s y m b o l Ω - 1 ( 𝐗 - 𝐌 ) T | - ν + n + p - 1 2 \times\left|\mathbf{I}_{n}+symbol\Sigma^{-1}(\mathbf{X}-\mathbf{M})symbol% \Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-\frac{\nu+n+p-1}{2}}
  2. 𝐌 \mathbf{M}
  3. ν + p - n > 1 \nu+p-n>1
  4. 𝐌 \mathbf{M}
  5. s y m b o l Σ s y m b o l Ω ν - 2 \frac{symbol\Sigma\otimes symbol\Omega}{\nu-2}
  6. ν > 2 \nu>2
  7. 𝐗 \mathbf{X}
  8. n × p n\times p
  9. f ( 𝐗 ; ν , 𝐌 , s y m b o l Σ , s y m b o l Ω ) = K × | 𝐈 n + s y m b o l Σ - 1 ( 𝐗 - 𝐌 ) s y m b o l Ω - 1 ( 𝐗 - 𝐌 ) T | - ν + n + p - 1 2 , f(\mathbf{X};\nu,\mathbf{M},symbol\Sigma,symbol\Omega)=K\times\left|\mathbf{I}% _{n}+symbol\Sigma^{-1}(\mathbf{X}-\mathbf{M})symbol\Omega^{-1}(\mathbf{X}-% \mathbf{M})^{\rm T}\right|^{-\frac{\nu+n+p-1}{2}},
  10. K = Γ p ( ν + n + p - 1 2 ) ( ν π ) n p 2 Γ p ( ν + p - 1 2 ) | s y m b o l Ω | - n 2 | s y m b o l Σ | - p 2 . K=\frac{\Gamma_{p}\left(\frac{\nu+n+p-1}{2}\right)}{(\nu\pi)^{\frac{np}{2}}% \Gamma_{p}\left(\frac{\nu+p-1}{2}\right)}|symbol\Omega|^{-\frac{n}{2}}|symbol% \Sigma|^{-\frac{p}{2}}.
  11. Γ p \Gamma_{p}
  12. × | 𝐈 n + β 2 s y m b o l Σ - 1 ( 𝐗 - 𝐌 ) s y m b o l Ω - 1 ( 𝐗 - 𝐌 ) T | - ( α + n / 2 ) \times\left|\mathbf{I}_{n}+\frac{\beta}{2}symbol\Sigma^{-1}(\mathbf{X}-\mathbf% {M})symbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-(\alpha+n/2)}
  13. Γ p \Gamma_{p}
  14. 𝐌 \mathbf{M}
  15. 2 ( s y m b o l Σ s y m b o l Ω ) β ( 2 α - n - 1 ) \frac{2(symbol\Sigma\otimes symbol\Omega)}{\beta(2\alpha-n-1)}
  16. β = 2 , α = ν + p - 1 2 . \beta=2,\alpha=\frac{\nu+p-1}{2}.
  17. 𝐗 T n , p ( α , β , 𝐌 , s y m b o l Σ , s y m b o l Ω ) \mathbf{X}\sim{\rm T}_{n,p}(\alpha,\beta,\mathbf{M},symbol\Sigma,symbol\Omega)
  18. 𝐗 T T p , n ( α , β , 𝐌 T , s y m b o l Ω , s y m b o l Σ ) . \mathbf{X}^{\rm T}\sim{\rm T}_{p,n}(\alpha,\beta,\mathbf{M}^{\rm T},symbol% \Omega,symbol\Sigma).
  19. det ( 𝐈 n + β 2 s y m b o l Σ - 1 ( 𝐗 - 𝐌 ) s y m b o l Ω - 1 ( 𝐗 - 𝐌 ) T ) = \det\left(\mathbf{I}_{n}+\frac{\beta}{2}symbol\Sigma^{-1}(\mathbf{X}-\mathbf{M% })symbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right)=
  20. det ( 𝐈 p + β 2 s y m b o l Ω - 1 ( 𝐗 T - 𝐌 T ) s y m b o l Σ - 1 ( 𝐗 T - 𝐌 T ) T ) . \det\left(\mathbf{I}_{p}+\frac{\beta}{2}symbol\Omega^{-1}(\mathbf{X}^{\rm T}-% \mathbf{M}^{\rm T})symbol\Sigma^{-1}(\mathbf{X}^{\rm T}-\mathbf{M}^{\rm T})^{% \rm T}\right).
  21. 𝐗 T n , p ( α , β , 𝐌 , s y m b o l Σ , s y m b o l Ω ) \mathbf{X}\sim{\rm T}_{n,p}(\alpha,\beta,\mathbf{M},symbol\Sigma,symbol\Omega)
  22. 𝐀 ( n × n ) \mathbf{A}(n\times n)
  23. 𝐁 ( p × p ) \mathbf{B}(p\times p)
  24. 𝐀𝐗𝐁 T n , p ( α , β , 𝐀𝐌𝐁 , 𝐀 s y m b o l Σ 𝐀 T , 𝐁 T s y m b o l Ω 𝐁 ) . \mathbf{AXB}\sim{\rm T}_{n,p}(\alpha,\beta,\mathbf{AMB},\mathbf{A}symbol\Sigma% \mathbf{A}^{\rm T},\mathbf{B}^{\rm T}symbol\Omega\mathbf{B}).
  25. ϕ T ( 𝐙 ) = exp ( tr ( i 𝐙 𝐌 ) ) | s y m b o l Ω | α Γ p ( α ) ( 2 β ) α p | 𝐙 s y m b o l Σ 𝐙 | α B α ( 1 2 β 𝐙 s y m b o l Σ 𝐙 s y m b o l Ω ) , \phi_{T}(\mathbf{Z})=\frac{\exp({\rm tr}(i\mathbf{Z}^{\prime}\mathbf{M}))|% symbol\Omega|^{\alpha}}{\Gamma_{p}(\alpha)(2\beta)^{\alpha p}}|\mathbf{Z}^{% \prime}symbol\Sigma\mathbf{Z}|^{\alpha}B_{\alpha}\left(\frac{1}{2\beta}\mathbf% {Z}^{\prime}symbol\Sigma\mathbf{Z}symbol\Omega\right),
  26. B δ ( 𝐖𝐙 ) = | 𝐖 | - δ 𝐒 > 0 exp ( tr ( - 𝐒𝐖 - 𝐒 - 𝟏 𝐙 ) ) | 𝐒 | - δ - 1 2 ( p + 1 ) d 𝐒 , B_{\delta}(\mathbf{WZ})=|\mathbf{W}|^{-\delta}\int_{\mathbf{S}>0}\exp\left({% \rm tr}(-\mathbf{SW}-\mathbf{S^{-1}Z})\right)|\mathbf{S}|^{-\delta-\frac{1}{2}% (p+1)}d\mathbf{S},
  27. B δ B_{\delta}

Matroid_minor.html

  1. r ( A ) = r ( A T ) - r ( T ) . r^{\prime}(A)=r(A\cup T)-r(T).
  2. U 4 2 U{}^{2}_{4}

Matroid_oracle.html

  1. n \scriptstyle n
  2. n \scriptstyle n
  3. n \scriptstyle n
  4. 2 2 n n - 3 / 2 + o ( 1 ) 2^{2^{n}n^{-3/2+o(1)}}
  5. I \scriptstyle I
  6. 1 \scriptstyle 1
  7. 0 \scriptstyle 0
  8. I \scriptstyle I
  9. x \scriptstyle x
  10. I { x } \scriptstyle I\cup\{x\}
  11. x \scriptstyle x
  12. + \scriptstyle+\infty
  13. x \scriptstyle x
  14. x \scriptstyle x
  15. X \scriptstyle X
  16. Y \scriptstyle Y
  17. X \scriptstyle X
  18. Y \scriptstyle Y
  19. X \scriptstyle X
  20. Y \scriptstyle Y
  21. X \scriptstyle X
  22. Y \scriptstyle Y
  23. n \scriptstyle n
  24. I { x } \scriptstyle I\cup\{x\}
  25. n \scriptstyle n
  26. y I \scriptstyle y\in I
  27. I { y } { x } \scriptstyle I\setminus\{y\}\cup\{x\}
  28. n \scriptstyle n
  29. O ( n ) \scriptstyle O(\sqrt{n})
  30. Ω ( ( n / log n ) 1 / 3 ) \scriptstyle\Omega((n/\log n)^{1/3})
  31. k \scriptstyle k
  32. k 3 \scriptstyle k\leq 3
  33. n \scriptstyle n
  34. r \scriptstyle r
  35. r \scriptstyle r
  36. M \scriptstyle M
  37. M \scriptstyle M^{\prime}
  38. M \scriptstyle M
  39. M \scriptstyle M^{\prime}
  40. M \scriptstyle M
  41. M \scriptstyle M
  42. M \scriptstyle M^{\prime}
  43. n \scriptstyle n
  44. M \scriptstyle M
  45. U n n / 2 \scriptstyle U{}^{n/2}_{n}
  46. M \scriptstyle M^{\prime}
  47. M \scriptstyle M
  48. n / 2 \scriptstyle n/2
  49. M \scriptstyle M
  50. M \scriptstyle M
  51. M \scriptstyle M^{\prime}
  52. n / 2 \scriptstyle n/2
  53. ( n n / 2 ) = Ω ( 2 n n ) {\left({{n}\atop{n/2}}\right)}=\Omega\left(\frac{2^{n}}{\sqrt{n}}\right)
  54. M \scriptstyle M
  55. M \scriptstyle M^{\prime}
  56. | aut ( M ) | i | fix ( M , Q i ) | \frac{|\operatorname{aut}(M)|}{\sum_{i}|\operatorname{fix}(M,Q_{i})|}
  57. aut ( M ) \scriptstyle\operatorname{aut}(M)
  58. M \scriptstyle M
  59. Q i \scriptstyle Q_{i}
  60. M \scriptstyle M
  61. M \scriptstyle M^{\prime}
  62. fix ( M , Q i ) \scriptstyle\operatorname{fix}(M,Q_{i})
  63. Q i \scriptstyle Q_{i}
  64. n ! \scriptstyle n!
  65. Q i \scriptstyle Q_{i}
  66. | fix ( M , Q i ) | = ( n / 2 ) ! 2 \scriptstyle|\operatorname{fix}(M,Q_{i})|=(n/2)!^{2}
  67. n ! \scriptstyle n!
  68. H \scriptstyle H
  69. H \scriptstyle H
  70. \scriptstyle\mathcal{H}
  71. \scriptstyle\mathcal{H}
  72. \scriptstyle\mathcal{H}
  73. n \scriptstyle n
  74. n \scriptstyle n
  75. k \scriptstyle k
  76. k \scriptstyle k
  77. k 4 \scriptstyle k\geq 4
  78. M \scriptstyle M
  79. U 4 2 \scriptstyle U{}^{2}_{4}
  80. U 4 2 \scriptstyle U{}^{2}_{4}

Matroid_partitioning.html

  1. H H
  2. G G
  3. | E ( H ) | | V ( H ) | - 1 \left\lceil\frac{|E(H)|}{|V(H)|-1}\right\rceil
  4. | V ( H ) | - 1 |V(H)|-1
  5. H H
  6. | E ( H ) | |E(H)|
  7. | E ( H ) | | V ( H ) | - 1 \frac{|E(H)|}{|V(H)|-1}
  8. x x
  9. y y
  10. x / y x/y
  11. G G
  12. M M
  13. H H
  14. G G
  15. M | S M|S
  16. M M
  17. S S
  18. H H
  19. | S | |S|
  20. | V ( H ) | - 1 |V(H)|-1
  21. H H
  22. r ( S ) r(S)
  23. M M
  24. k ( M ) = max S | S | r ( S ) , k(M)=\max_{S}\left\lceil\frac{|S|}{r(S)}\right\rceil,
  25. x x
  26. x x
  27. i \bot_{i}
  28. k k
  29. G x G_{x}
  30. i y \bot_{i}\rightarrow y
  31. y y
  32. i i
  33. z y z\rightarrow y
  34. ( y , z ) (y,z)
  35. z z
  36. y y
  37. i \bot_{i}
  38. x x
  39. x x
  40. S S
  41. x x
  42. D D
  43. M | S M|S
  44. y y
  45. S S
  46. i i
  47. i y \bot_{i}\rightarrow y
  48. i i
  49. M M
  50. z y z\rightarrow y
  51. z S z\notin S
  52. S S
  53. S S
  54. S S
  55. | S | r ( S ) = k r ( S ) + 1 r ( S ) = k + 1 \left\lceil\frac{|S|}{r(S)}\right\rceil=\left\lceil\frac{kr(S)+1}{r(S)}\right% \rceil=k+1
  56. x x
  57. x x
  58. G x G_{x}
  59. x x
  60. x x
  61. i M i \sum_{i}M_{i}
  62. M i M_{i}
  63. M 1 M_{1}
  64. M 2 M_{2}
  65. B B
  66. M 1 + M 2 * M_{1}+M_{2}^{*}
  67. M 2 * M_{2}^{*}
  68. M 2 M_{2}
  69. B B
  70. M 2 * M_{2}^{*}
  71. M 2 * M_{2}^{*}
  72. B B

Matroid_polytope.html

  1. M M
  2. P M P_{M}
  3. M M
  4. M M
  5. n n
  6. B { 1 , , n } B\subseteq\{1,\dots,n\}
  7. M M
  8. B B
  9. 𝐞 B := i B 𝐞 i , \mathbf{e}_{B}:=\sum_{i\in B}\mathbf{e}_{i},
  10. 𝐞 i \mathbf{e}_{i}
  11. i i
  12. n \mathbb{R}^{n}
  13. P M P_{M}
  14. { 𝐞 B B is a basis of M } n . \{\mathbf{e}_{B}\mid B\,\text{ is a basis of }M\}\subseteq\mathbb{R}^{n}.
  15. M M
  16. ( M ) = { { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } } . \mathcal{B}(M)=\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\}\}.
  17. { 1 , 2 , 3 , 4 } \{1,2,3,4\}
  18. { 3 , 4 } \{3,4\}
  19. ( M ) \mathcal{B}(M)
  20. { { 1 , 1 , 0 , 0 } , { 1 , 0 , 1 , 0 } , { 1 , 0 , 0 , 1 } , { 0 , 1 , 1 , 0 } , { 0 , 1 , 0 , 1 } } . \{\{1,1,0,0\},\{1,0,1,0\},\{1,0,0,1\},\{0,1,1,0\},\{0,1,0,1\}\}.
  21. M M
  22. P M = conv { { 1 , 1 , 0 , 0 } , { 1 , 0 , 1 , 0 } , { 1 , 0 , 0 , 1 } , { 0 , 1 , 1 , 0 } , { 0 , 1 , 0 , 1 } } , P_{M}=\,\text{conv}\{\{1,1,0,0\},\{1,0,1,0\},\{1,0,0,1\},\{0,1,1,0\},\{0,1,0,1% \}\},
  23. N N
  24. { 1 , 2 , 3 , 4 } \{1,2,3,4\}
  25. P N P_{N}
  26. P M P_{M}
  27. P N P_{N}
  28. M M
  29. r r
  30. n n
  31. P M P_{M}
  32. Δ n r \Delta_{n}^{r}
  33. Δ n r \Delta_{n}^{r}
  34. r r
  35. n n
  36. P M P_{M}
  37. e i - e j e_{i}-e_{j}
  38. i , j E i,j\in E
  39. P M P_{M}
  40. B , B B,B^{\prime}
  41. B = B i j B^{\prime}=B\setminus{i\cup j}
  42. i , j E . i,j\in E.
  43. r : 2 E r:2^{E}\rightarrow\mathbb{Z}
  44. M M
  45. P M P_{M}
  46. P M = A E β ~ ( M / A ) Δ E - A P_{M}=\sum_{A\subseteq E}\tilde{\beta}(M/A)\Delta_{E-A}
  47. E E
  48. M M
  49. β ( M ) \beta(M)
  50. M M
  51. β ~ ( M ) = ( - 1 ) r ( M ) + 1 β ( M ) , \tilde{\beta}(M)=(-1)^{r(M)+1}\beta(M),
  52. β ( M ) = ( - 1 ) r ( M ) X E ( - 1 ) | X | r ( X ) . \beta(M)=(-1)^{r(M)}\sum_{X\subseteq E}(-1)^{|X|}r(X).
  53. { 𝐞 I I is an independent set of M } n . \{\,\mathbf{e}_{I}\mid I\,\text{ is an independent set of }M\,\}\subseteq% \mathbb{R}^{n}.
  54. ψ \psi
  55. M M
  56. ψ \psi
  57. \mathcal{F}
  58. F 1 F 2 F m F^{1}\subset F^{2}\subset\cdots\subset F^{m}\,
  59. k i k_{i}
  60. F i F_{i}
  61. M M
  62. N N
  63. r M ( Y ) - r M ( X ) r N ( Y ) - r N ( X ) for all X Y E . r_{M}(Y)-r_{M}(X)\leq r_{N}(Y)-r_{N}(X)\,\text{ for all }X\subset Y\subseteq E.\,
  64. M 1 , , M m M_{1},\dots,M_{m}
  65. E E
  66. r 1 < < r m r_{1}<\cdots<r_{m}
  67. ( B 1 , , B m ) (B_{1},\dots,B_{m})
  68. B i B_{i}
  69. M i M_{i}
  70. B 1 B m B_{1}\subset\cdots\subset B_{m}
  71. \mathcal{F}
  72. M 1 , , M m M_{1},\dots,M_{m}
  73. \mathcal{F}
  74. B = ( B 1 , , B m ) B=(B_{1},\dots,B_{m})
  75. \mathcal{F}
  76. v B v_{B}
  77. B B
  78. v B = v B 1 + + v B m . v_{B}=v_{B_{1}}+\cdots+v_{B_{m}}.\,
  79. \mathcal{F}
  80. P P_{\mathcal{F}}
  81. { v B B is a flag in } . \{v_{B}\mid B\,\text{ is a flag in }\mathcal{F}\}.
  82. P = P M 1 + + P M k . P_{\mathcal{F}}=P_{M_{1}}+\cdots+P_{M_{k}}.\,

Matroid_rank.html

  1. A A
  2. B B
  3. E E
  4. r ( A B ) + r ( A B ) r ( A ) + r ( B ) r(A\cup B)+r(A\cap B)\leq r(A)+r(B)
  5. A A
  6. x x
  7. r ( A ) r ( A { x } ) r ( A ) + 1 r(A)\leq r(A\cup\{x\})\leq r(A)+1
  8. A B E A\subset B\subset E
  9. r ( A ) r ( B ) r ( E ) r(A)\leq r(B)\leq r(E)
  10. r ( A ) r ( A { x } ) r ( A ) + 1 r(A)\leq r(A\cup\{x\})\leq r(A)+1
  11. A A
  12. x x
  13. | A | - r ( A ) |A|-r(A)
  14. A A
  15. A A

Matroid_representation.html

  1. ( E , ) (E,\mathcal{I})
  2. E E
  3. \mathcal{I}
  4. E E
  5. A A
  6. B B
  7. x A B x\in A\setminus B
  8. B B
  9. E E
  10. \mathcal{I}
  11. E E
  12. ( E , ) (E,\mathcal{I})
  13. ( E , ) (E,\mathcal{I})
  14. ( E , ) (E,\mathcal{I})
  15. f f
  16. E E
  17. V V
  18. A A
  19. E E
  20. f ( A ) f(A)
  21. V V
  22. f f
  23. U 4 2 U{}^{2}_{4}
  24. U 4 2 U{}^{2}_{4}
  25. U n r U{}^{r}_{n}
  26. n n
  27. r r
  28. r r
  29. n n
  30. \mathbb{R}
  31. n n
  32. 2 n 2^{n}

Maximum_common_edge_subgraph_problem.html

  1. G G
  2. G G^{\prime}
  3. H H
  4. G G
  5. G G^{\prime}
  6. H H
  7. G G
  8. G G
  9. H H
  10. H H
  11. G G
  12. G G^{\prime}

McMullen_problem.html

  1. ν ( d ) \nu(d)
  2. ν ( d ) \nu(d)
  3. μ ( d ) \mu(d)
  4. μ ( d ) \mu(d)
  5. μ ( k ) \mu(k)
  6. ν ( d ) \nu(d)
  7. μ ( k ) = min { w w ν ( w - k - 1 ) } \mu(k)=\min\{w\mid w\leq\nu(w-k-1)\}\,
  8. ν ( d ) = max { w w μ ( w - d - 1 ) } \nu(d)=\max\{w\mid w\geq\mu(w-d-1)\}\,
  9. λ ( d ) \lambda(d)
  10. λ ( d ) \lambda(d)
  11. conv ( A \ { x } ) conv ( B \ { x } ) , x X . \operatorname{conv}(A\backslash\{x\})\cap\operatorname{conv}(B\backslash\{x\})% \not=\varnothing,\forall x\in X.\,
  12. μ \mu
  13. λ \lambda
  14. μ ( d + 1 ) = λ ( d ) , d 1 \mu(d+1)=\lambda(d),\qquad d\geq 1\,
  15. ν ( d ) \nu(d)
  16. ν ( d ) \nu(d)
  17. ν ( d ) \nu(d)
  18. 2 d + 1 ν ( d ) ( d + 1 ) 2 2d+1\leq\nu(d)\leq(d+1)^{2}
  19. ν ( d ) ( d + 1 ) ( d + 2 ) 2 \nu(d)\leq\frac{(d+1)(d+2)}{2}
  20. ν ( d ) 2 d + d + 1 2 \nu(d)\leq 2d+\lceil\frac{d+1}{2}\rceil
  21. ν ( d ) = 2 d + 1 \nu(d)=2d+1

Mean-periodic_function.html

  1. f ( x - y ) d μ ( y ) = 0 ( 1 ) \int f(x-y)\,d\mu(y)=0\qquad\qquad(1)
  2. μ \mu
  3. μ \mu

Mean_squared_displacement.html

  1. p ( x , t x 0 ) t = D 2 p ( x , t x 0 ) x 2 , \frac{\partial p(x,t\mid x_{0})}{\partial t}=D\frac{\partial^{2}p(x,t\mid x_{0% })}{\partial x^{2}},
  2. p ( x 0 , t = 0 x 0 ) = δ ( x - x 0 ) p(x_{0},t={0}\mid x_{0})=\delta(x-x_{0})
  3. x ( t ) x(t)
  4. x 0 x_{0}
  5. D D
  6. m 2 s - 1 m^{2}s^{-1}
  7. x ( t ) x(t)
  8. P ( x , t ) = 1 4 π D t exp ( - ( x - x 0 ) 2 4 D t ) . P(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{(x-x_{0})^{2}}{4Dt}\right).
  9. x ( t ) x(t)
  10. FWHM t . \rm{FWHM}\sim\sqrt{t}.
  11. L L
  12. t t
  13. L ( t ) - L ( x , t ) P ( x , t ) d x , \langle L(t)\rangle\equiv\int^{\infty}_{-\infty}L(x,t)P(x,t)dx,
  14. MSD ( x ( t ) - x 0 ) 2 , \rm{MSD}\equiv\langle\left(x(t)-x_{0}\right)^{2}\rangle,
  15. ( x - x 0 ) 2 = x 2 + x 0 2 - 2 x 0 x , \langle\left(x-x_{0}\right)^{2}\rangle=\langle x^{2}\rangle+x_{0}^{2}-2x_{0}% \langle x\rangle,
  16. x 2 \langle x^{2}\rangle
  17. x \langle x\rangle
  18. k th k^{\textrm{th}}
  19. x \langle x\rangle
  20. x 2 \langle x^{2}\rangle
  21. G ( k ) = e i k x I e i k x P ( x , t | x 0 ) d x , G(k)=\langle e^{ikx}\rangle\equiv\int_{I}e^{ikx}P(x,t|x_{0})dx,
  22. G ( k ) = m = 0 ( i k ) m m ! μ m . G(k)=\sum^{\infty}_{m=0}\frac{(ik)^{m}}{m!}\mu_{m}.
  23. ln ( G ( k ) ) = m = 1 ( i k ) m m ! κ m , \ln(G(k))=\sum^{\infty}_{m=1}\frac{(ik)^{m}}{m!}\kappa_{m},
  24. κ m \kappa_{m}
  25. m th m^{\rm{th}}
  26. x x
  27. μ \mu
  28. κ 1 = μ 1 ; \kappa_{1}=\mu_{1};
  29. κ 2 = μ 2 - μ 1 2 , \kappa_{2}=\mu_{2}-\mu_{1}^{2},
  30. σ 2 \sigma^{2}
  31. G ( k ) = 1 4 π D t I exp ( i k x ) exp ( - ( x - x 0 ) 2 4 D t ) d x ; G(k)=\frac{1}{\sqrt{4\pi Dt}}\int_{I}\exp(ikx)\exp\left(-\frac{(x-x_{0})^{2}}{% 4Dt}\right)dx;
  32. G ( k ) = exp ( i k x 0 - k 2 D t ) . G(k)=\exp(ikx_{0}-k^{2}Dt).
  33. i k ik
  34. κ 1 = x 0 , \kappa_{1}=x_{0},
  35. κ 2 = 2 D t , \kappa_{2}=2Dt,
  36. μ 2 = κ 2 + μ 1 2 = 2 D t + x 0 2 . \mu_{2}=\kappa_{2}+\mu_{1}^{2}=2Dt+x_{0}^{2}.
  37. ( x ( t ) - x 0 ) 2 = 2 D t . \langle\left(x(t)-x_{0}\right)^{2}\rangle=2Dt.

Measurements_of_neutrino_speed.html

  1. | v - c | / c |v-c|/c
  2. 10 - 4 \scriptstyle\lesssim 10^{-4}
  3. 10 - 8 \scriptstyle\lesssim 10^{-8}
  4. 10 - 14 \scriptstyle\lesssim 10^{-14}
  5. 10 - 20 \scriptstyle\lesssim 10^{-20}
  6. 10 - 26 \scriptstyle\lesssim 10^{-26}
  7. E = m c 2 1 - v 2 c 2 E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  8. | v - c | c < 4 × 10 - 5 \frac{|v-c|}{c}<4\times 10^{-5}
  9. | v - c | c < 2 × 10 - 9 \frac{|v-c|}{c}<2\times 10^{-9}
  10. ( 5.1 ± 2.9 ) × 10 - 5 \scriptstyle(5.1\pm 2.9)\times 10^{-5}
  11. - 2.4 × 10 - 5 < v - c c < 12.6 × 10 - 5 -2.4\times 10^{-5}<\frac{v-c}{c}<12.6\times 10^{-5}
  12. δ t = 6.5 ± 7.4 ( stat . ) + 8.3 - 8.0 ( sys . ) \delta t=6.5\pm 7.4\ (\mathrm{stat.}){\scriptstyle{+8.3\atop-8.0}}\ (\mathrm{% sys.})
  13. v - c c = ( 2.7 ± 3.1 ( stat . ) + 3.4 - 3.3 ( sys . ) ) × 10 - 6 \frac{v-c}{c}=(2.7\pm 3.1\ (\mathrm{stat.}){\scriptstyle{+3.4\atop-3.3}}\ (% \mathrm{sys.}))\times 10^{-6}
  14. δ t = - 1.9 ± 3.7 ( stat . ) \delta t=-1.9\pm 3.7\ (\mathrm{stat.})
  15. 10 - 6 10^{-6}
  16. δ t = 0.3 ± 4.9 ( stat . ) ± 9.0 ( sys . ) \delta t=0.3\pm 4.9\ (\mathrm{stat.})\pm 9.0\ (\mathrm{sys.})
  17. δ t = - 6.5 ± 7 ( stat . ) ± 6 ( sys . ) \delta t=-6.5\pm 7\ (\mathrm{stat.})\pm 6\ (\mathrm{sys.})
  18. δ t = 0.8 ± 0.7 ( stat . ) ± 2.9 ( sys . ) \delta t=0.8\pm 0.7\ (\mathrm{stat.})\pm 2.9\ (\mathrm{sys.})
  19. | v - c | c < 2.1 × 10 - 6 \frac{|v-c|}{c}<2.1\times 10^{-6}
  20. δ t = 3.1 ± 5.3 ( stat . ) ± 8 ( sys . ) \delta t=3.1\pm 5.3\ (\mathrm{stat.})\pm 8\ (\mathrm{sys.})
  21. δ t = 0.9 ± 0.6 ( stat . ) ± 3.2 ( sys . ) \delta t=0.9\pm 0.6\ (\mathrm{stat.})\pm 3.2\ (\mathrm{sys.})
  22. - 3.8 × 10 - 6 < v - c c < 3.1 × 10 - 6 -3.8\times 10^{-6}<\frac{v-c}{c}<3.1\times 10^{-6}
  23. δ t = 0.18 ± 0.69 ( stat . ) ± 2.17 ( sys . ) \delta t=0.18\pm 0.69\ (\mathrm{stat.})\pm 2.17\ (\mathrm{sys.})
  24. v - c c = ( 0.7 ± 2.8 ( stat . ) ± 8.9 ( sys . ) ) × 10 - 7 \frac{v-c}{c}=(0.7\pm 2.8\ (\mathrm{stat.})\pm 8.9\ (\mathrm{sys.}))\times 10^% {-7}
  25. 1.6 × 10 - 6 c 1.6\times 10^{-6}c
  26. δ t = 0.6 ± 0.4 ( stat . ) ± 3.0 ( sys . ) \delta t=0.6\pm 0.4\ (\mathrm{stat.})\pm 3.0\ (\mathrm{sys.})
  27. δ t = 1.7 ± 1.4 ( stat . ) ± 3.2 ( sys . ) \delta t=1.7\pm 1.4\ (\mathrm{stat.})\pm 3.2\ (\mathrm{sys.})
  28. - 1.8 × 10 - 6 < v - c c < 2.3 × 10 - 6 -1.8\times 10^{-6}<\frac{v-c}{c}<2.3\times 10^{-6}
  29. δ t = - 18 ± 11 ( stat . ) ± 29 ( sys . ) \delta t=-18\pm 11\ (\mathrm{stat.})\pm 29\ (\mathrm{sys.})
  30. δ t = - 11 ± 11 ( stat . ) ± 29 ( sys . ) \delta t=-11\pm 11\ (\mathrm{stat.})\pm 29\ (\mathrm{sys.})
  31. δ t = - 2.4 ± 0.1 ( stat . ) ± 2.6 ( sys . ) \delta t=-2.4\pm 0.1\ (\mathrm{stat.})\pm 2.6\ (\mathrm{sys.})
  32. v - c c = ( 1.0 ± 1.1 ) × 10 - 6 \frac{v-c}{c}=(1.0\pm 1.1)\times 10^{-6}
  33. | v - c | c < 5.6 × 10 - 19 \frac{|v-c|}{c}<5.6\times 10^{-19}

Medium-chain_acyl-CoA_dehydrogenase.html

  1. \rightleftharpoons

Meissner_equation.html

  1. d 2 y d t 2 + ( α 2 + ω 2 sgn cos ( t ) ) y = 0 \frac{d^{2}y}{dt^{2}}+(\alpha^{2}+\omega^{2}\operatorname{sgn}\cos(t))y=0
  2. d 2 y d t 2 + ( 1 + r f ( t ; a , b ) ) y = 0 \frac{d^{2}y}{dt^{2}}+(1+rf(t;a,b))y=0
  3. f ( t ; a , b ) = - 1 + 2 H a ( t mod ( a + b ) ) f(t;a,b)=-1+2H_{a}(t\mod(a+b))
  4. H c ( t ) H_{c}(t)
  5. c c
  6. d 2 y d t 2 + ( 1 + r sin ( ω t ) | sin ( ω t ) | ) y = 0. \frac{d^{2}y}{dt^{2}}+\left(1+r\frac{\sin(\omega t)}{|\sin(\omega t)|}\right)y% =0.
  7. a = b = 1 a=b=1
  8. λ 2 - 2 λ cosh ( r ) cos ( r ) + 1 = 0. \lambda^{2}-2\lambda\cosh(\sqrt{r})\cos(\sqrt{r})+1=0.
  9. | cosh ( r ) cos ( r ) | < 1 |\cosh(\sqrt{r})\cos(\sqrt{r})|<1

Menaquinol_oxidase_(H+-transporting).html

  1. \rightleftharpoons

Mercator_1569_world_map.html

  1. s = ( ϕ B - ϕ A ) sec α . s=(\phi_{B}-\phi_{A})\sec\alpha.

Metacube_–_Gigacube.html

  1. a i j a_{ij}
  2. a i j a_{ij}
  3. i , j i,j
  4. { a i j } \{a_{ij}\}
  5. ( { a i j } , k ) (\{a_{ij}\},k)
  6. { a i j } \{a_{ij}\}
  7. ( { a i j } , k ) (\{a_{ij}\},k)
  8. ( { a i j } , k ) (\{a_{ij}\},k)
  9. ( { a i j } , k , D ) (\{a_{ij}\},k,D)
  10. ( { a i j } , k , D ) (\{a_{ij}\},k,D)
  11. T \overrightarrow{T}
  12. ( { a i j } , k , D , T ) (\{a_{ij}\},k,D,\overrightarrow{T})

Methane_monooxygenase_(particulate).html

  1. \rightleftharpoons

Methanesulfonate_monooxygenase.html

  1. \rightleftharpoons

Methanogen_homoaconitase.html

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

Methanol_dehydrogenase_(cytochrome_c).html

  1. \rightleftharpoons

Methanol_dehydrogenase_(nicotinoprotein).html

  1. \rightleftharpoons

Methionine-S-oxide_reductase.html

  1. \rightleftharpoons

Methionine_transaminase.html

  1. \rightleftharpoons

Method_of_fundamental_solutions.html

  1. L u = f ( x , y ) , ( x , y ) Ω , Lu=f\left(x,y\right),\ \ \left(x,y\right)\in\Omega,
  2. u = g ( x , y ) , ( x , y ) Ω D , u=g\left(x,y\right),\ \ \left(x,y\right)\in\partial\Omega_{D},
  3. u n = h ( x , y ) , h ( x , y ) Ω N , \frac{\partial u}{\partial n}=h\left(x,y\right),\ \ h\left(x,y\right)\in% \partial\Omega_{N},
  4. L L
  5. Ω \Omega
  6. Ω D \partial\Omega_{D}
  7. Ω N \partial\Omega_{N}
  8. Ω D Ω N = Ω \partial\Omega_{D}\cup\partial\Omega_{N}=\partial\Omega
  9. Ω D Ω N = \partial\Omega_{D}\cap\partial\Omega_{N}=\varnothing
  10. u * ( x , y ) = i = 1 N α i ϕ ( r i ) {{u}^{*}}\left(x,y\right)=\sum\limits_{i=1}^{N}\alpha_{i}\phi\left(r_{i}\right)
  11. r i = ( x , y ) - ( s x i , s y i ) r_{i}=\left\|\left(x,y\right)-\left(sx_{i},sy_{i}\right)\right\|
  12. ( x , y ) \left(x,y\right)
  13. ( s x i , s y i ) \left(sx_{i},sy_{i}\right)
  14. ϕ ( ) \phi\left(\cdot\right)
  15. L ϕ = δ L\phi=\delta\,
  16. δ \delta
  17. α i {{\alpha}_{i}}
  18. [ ϕ ( r j | x i , y i ) ϕ ( r j | x k , y k ) n ] α = ( g ( x i , y i ) h ( x k , y k ) ) , \left[\begin{matrix}\phi\left(\left.r_{j}\right|_{x_{i},y_{i}}\right)\\ \frac{\partial\phi\left(\left.r_{j}\right|_{x_{k},y_{k}}\right)}{\partial n}\\ \end{matrix}\right]\ \cdot\ \alpha=\left(\begin{matrix}g\left(x_{i},y_{i}% \right)\\ h\left(x_{k},y_{k}\right)\\ \end{matrix}\right),
  19. ( x i , y i ) \left(x_{i},y_{i}\right)
  20. ( x k , y k ) \left(x_{k},y_{k}\right)
  21. α i \alpha_{i}

Methyl_halide_transferase.html

  1. \rightleftharpoons

Methylamine-corrinoid_protein_Co-methyltransferase.html

  1. \rightleftharpoons

Methylamine_dehydrogenase_(amicyanin).html

  1. \rightleftharpoons

Methylated-thiol-coenzyme_M_methyltransferase.html

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

Methylecgonone_reductase.html

  1. \rightleftharpoons

Methylenediurea_deaminase.html

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

Methylphenyltetrahydropyridine_N-monooxygenase.html

  1. \rightleftharpoons

Methylphosphonate_synthase.html

  1. \rightleftharpoons

Methylsterol_monooxygenase.html

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

Meyer_wavelet.html

  1. ν \nu
  2. Ψ ( ω ) := { 1 2 π sin ( π 2 ν ( 3 | ω | 2 π - 1 ) ) e j ω / 2 if 2 π / 3 < | ω | < 4 π / 3 , 1 2 π cos ( π 2 ν ( 3 | ω | 4 π - 1 ) ) e j ω / 2 if 4 π / 3 < | ω | < 8 π / 3 , 0 otherwise , \Psi(\omega):=\begin{cases}\frac{1}{\sqrt{2\pi}}\sin\left(\frac{\pi}{2}\nu% \left(\frac{3|\omega|}{2\pi}-1\right)\right)e^{j\omega/2}&\,\text{if }2\pi/3<|% \omega|<4\pi/3,\\ \frac{1}{\sqrt{2\pi}}\cos\left(\frac{\pi}{2}\nu\left(\frac{3|\omega|}{4\pi}-1% \right)\right)e^{j\omega/2}&\,\text{if }4\pi/3<|\omega|<8\pi/3,\\ 0&\,\text{otherwise},\end{cases}
  3. ν ( x ) := { 0 if x < 0 , x if 0 < x < 1 , 1 if x > 1. \nu(x):=\begin{cases}0&\,\text{if }x<0,\\ x&\,\text{if }0<x<1,\\ 1&\,\text{if }x>1.\end{cases}
  4. ν ( x ) := { x 4 ( 35 - 84 x + 70 x 2 - 20 x 3 ) if 0 < x < 1 , 0 otherwise . \nu(x):=\begin{cases}{x^{4}}(35-84x+70{x^{2}}-20{x^{3}})&\,\text{if }0<x<1,\\ 0&\,\text{otherwise}.\end{cases}
  5. Φ ( ω ) := { 1 2 π if | ω | < 2 π / 3 , 1 2 π cos ( π 2 ν ( 3 | ω | 2 π - 1 ) ) if 2 π / 3 < | ω | < 4 π / 3 , 0 otherwise . \Phi(\omega):=\begin{cases}\frac{1}{\sqrt{2\pi}}&\,\text{if }|\omega|<2\pi/3,% \\ \frac{1}{\sqrt{2\pi}}\cos\left(\frac{\pi}{2}\nu\left(\frac{3|\omega|}{2\pi}-1% \right)\right)&\,\text{if }2\pi/3<|\omega|<4\pi/3,\\ 0&\,\text{otherwise}.\end{cases}

Michael_McQuillan_(mathematician).html

  1. c 1 2 > c 2 c_{1}^{2}>c_{2}
  2. ¯ \partial\bar{\partial}

Miller's_Rule_(optics).html

  1. χ 2 \chi_{\,\text{2}}
  2. χ 1 \chi_{\,\text{1}}
  3. χ 2 \chi_{\,\text{2}}
  4. δ \delta
  5. χ 1 ( ω ) = N q 2 m ε 0 1 ω 0 2 - ω 2 - i ω τ \chi_{\,\text{1}}(\omega)=\frac{Nq^{2}}{m\varepsilon_{0}}\frac{1}{\omega_{% \mathrm{0}}^{2}-\omega^{2}-\tfrac{i\omega}{\tau}}
  6. ω \omega
  7. χ 1 \chi_{\textrm{1}}
  8. ω \omega
  9. ε 0 \varepsilon_{0}
  10. τ \tau
  11. D ( ω ) D(\omega)
  12. χ 1 \chi_{\,\text{1}}
  13. D ( ω ) = 1 ω 0 2 - ω 2 - i ω τ D(\omega)=\frac{1}{\omega_{\mathrm{0}}^{2}-\omega^{2}-\tfrac{i\omega}{\tau}}
  14. χ 1 ( ω ) = N q 2 ε 0 m 1 D ( ω ) \chi_{\,\text{1}}(\omega)=\frac{Nq^{2}}{\varepsilon_{0}m}\frac{1}{D(\omega)}
  15. χ 2 ( 2 ω ) = N q 3 ζ 2 ε 0 m 2 1 D ( 2 ω ) D ( ω ) 2 \chi_{\,\text{2}}(2\omega)=\frac{Nq^{3}\zeta_{2}}{\varepsilon_{0}m^{2}}\frac{1% }{D(2\omega)D(\omega)^{2}}
  16. ζ 2 \zeta_{2}
  17. χ 2 \chi_{\,\text{2}}
  18. χ 1 \chi_{\,\text{1}}
  19. χ 2 ( 2 ω ) = ε 0 2 m ζ 2 N 2 q 3 χ 1 ( ω ) χ 1 ( ω ) χ 1 ( 2 ω ) \chi_{\,\text{2}}(2\omega)=\frac{\varepsilon_{0}^{2}m\zeta_{2}}{N^{2}q^{3}}% \chi_{\,\text{1}}(\omega)\chi_{\,\text{1}}(\omega)\chi_{\,\text{1}}(2\omega)
  20. χ 2 \chi_{\,\text{2}}
  21. χ 1 \chi_{\,\text{1}}
  22. δ = ε 0 2 m ζ 2 N 2 q 3 \delta=\frac{\varepsilon_{0}^{2}m\zeta_{2}}{N^{2}q^{3}}

Mincer_earnings_function.html

  1. ln y = ln y 0 + r S + β 1 X + β 2 X 2 \ln y=\ln y_{0}+rS+\beta_{1}X+\beta_{2}X^{2}
  2. y y
  3. y 0 y_{0}
  4. S S
  5. X X

Mingarelli_identity.html

  1. n n
  2. ( p i ( t ) x i ) + q i ( t ) x i = 0 , x i ( a ) = 1 , x i ( a ) = R i (p_{i}(t)x_{i}^{\prime})^{\prime}+q_{i}(t)x_{i}=0,\,\,\,\,\,\,\,\,\,\,x_{i}(a)% =1,\,\,x_{i}^{\prime}(a)=R_{i}\,
  3. i = 1 , 2 , , n i=1,2,\ldots,n
  4. Δ \Delta
  5. Δ x i = x i + 1 - x i . \Delta x_{i}=x_{i+1}-x_{i}.
  6. Δ 2 ( x i ) = Δ ( Δ x i ) = x i + 2 - 2 x i + 1 + x i \Delta^{2}(x_{i})=\Delta(\Delta x_{i})=x_{i+2}-2x_{i+1}+x_{i}
  7. x i ( t ) 0 x_{i}(t)\neq 0
  8. x n - 1 2 Δ n - 1 ( p 1 r 1 ) ] a b \displaystyle x_{n-1}^{2}\Delta^{n-1}(p_{1}r_{1})]_{a}^{b}
  9. r i = x i / x i r_{i}=x^{\prime}_{i}/x_{i}
  10. W ( x i , x j ) = x i x j - x i x j W(x_{i},x_{j})=x^{\prime}_{i}x_{j}-x_{i}x^{\prime}_{j}
  11. C ( n - 1 , k ) C(n-1,k)
  12. n = 2 n=2
  13. p i , q i , p_{i},\,q_{i},\,
  14. ( p 1 ( t ) x 1 ) + q 1 ( t ) x 1 = 0 , x 1 ( a ) = 1 , x 1 ( a ) = R 1 (p_{1}(t)x_{1}^{\prime})^{\prime}+q_{1}(t)x_{1}=0,\,\,\,\,\,\,\,\,\,\,x_{1}(a)% =1,\,\,x_{1}^{\prime}(a)=R_{1}\,
  15. ( p 2 ( t ) x 2 ) + q 2 ( t ) x 2 = 0 , x 2 ( a ) = 1 , x 2 ( a ) = R 2 (p_{2}(t)x_{2}^{\prime})^{\prime}+q_{2}(t)x_{2}=0,\,\,\,\,\,\,\,\,\,\,x_{2}(a)% =1,\,\,x_{2}^{\prime}(a)=R_{2}\,
  16. ( p 3 ( t ) x 3 ) + q 3 ( t ) x 3 = 0 , x 3 ( a ) = 1 , x 3 ( a ) = R 3 (p_{3}(t)x_{3}^{\prime})^{\prime}+q_{3}(t)x_{3}=0,\,\,\,\,\,\,\,\,\,\,x_{3}(a)% =1,\,\,x_{3}^{\prime}(a)=R_{3}\,
  17. p i ( t ) > 0 p_{i}(t)>0\,
  18. R i R_{i}
  19. Δ 2 ( q 1 ) 0 \Delta^{2}(q_{1})\geq 0
  20. Δ 2 ( p 1 ) 0 \Delta^{2}(p_{1})\leq 0
  21. Δ 2 ( p 1 ( a ) R 1 ) 0 \Delta^{2}(p_{1}(a)R_{1})\leq 0
  22. x 1 ( t ) > 0 x_{1}(t)>0
  23. x 2 ( b ) = 0 x_{2}(b)=0
  24. x 3 ( t ) x_{3}(t)

Minimal-entropy_martingale_measure.html

  1. P P
  2. Q Q
  3. Q Q
  4. P P
  5. Q Q
  6. P P
  7. p i p_{i}
  8. q i q_{i}
  9. D K L ( Q P ) = i = 1 N q i ln ( q i p i ) D_{KL}(Q\|P)=\sum_{i=1}^{N}q_{i}\ln\left(\frac{q_{i}}{p_{i}}\right)
  10. r r
  11. r r

Minimal_surface_of_revolution.html

  1. ( x 1 , y 1 ) (x_{1},y_{1})
  2. ( x 2 , y 2 ) (x_{2},y_{2})
  3. f f
  4. 2 π x 1 x 2 f ( x ) 1 + f ( x ) 2 d x 2\pi\int_{x_{1}}^{x_{2}}f(x)\sqrt{1+f^{\prime}(x)^{2}}dx
  5. f ( x 1 ) = y 1 f(x_{1})=y_{1}
  6. f ( x 2 ) = y 2 f(x_{2})=y_{2}

Minimum_chi-square_estimation.html

  1. value frequency 0 1 1 2 2 4 3 5 4 3 5 3 6 1 7 0 8 1 > 8 0 \begin{array}[]{cc}\,\text{value}&\,\text{frequency}\\ \hline 0&1\\ 1&2\\ 2&4\\ 3&5\\ 4&3\\ 5&3\\ 6&1\\ 7&0\\ 8&1\\ >8&0\end{array}
  2. ( observed - expected ) 2 expected = k = 0 8 ( ( count in cell k ) - 20 ( λ k e - λ k ! ) ) 2 20 ( λ k e - λ k ! ) + ( 0 - a ) 2 a \sum\frac{(\,\text{observed}-\,\text{expected})^{2}}{\,\text{expected}}=\sum_{% k=0}^{8}\frac{\left((\,\text{count in cell }k)-20\left(\frac{\lambda^{k}e^{-% \lambda}}{k!}\right)\right)^{2}}{20\left(\frac{\lambda^{k}e^{-\lambda}}{k!}% \right)}+\frac{(0-a)^{2}}{a}

Minimum_rank_of_a_graph.html

  1. mr ( G ) \operatorname{mr}(G)
  2. G G
  3. mr ( G ) \operatorname{mr}(G)
  4. n 2 n\geq 2
  5. G G
  6. mr ( G ) = | G | - 2 \operatorname{mr}(G)=|G|-2
  7. G G
  8. G G
  9. mr ( G ) 2 \operatorname{mr}(G)\leq 2
  10. G G
  11. ( K s 1 K s 2 K p 1 , q 1 K p k , q k ) K r (K_{s_{1}}\cup K_{s_{2}}\cup K_{p_{1},q_{1}}\cup\cdots\cup K_{p_{k},q_{k}})% \vee K_{r}
  12. k , s 1 , s 2 , p 1 , q 1 , , p k , q k , r k,s_{1},s_{2},p_{1},q_{1},\ldots,p_{k},q_{k},r
  13. p i + q i > 0 p_{i}+q_{i}>0
  14. i = 1 , , k i=1,\ldots,k

Minkowski's_second_theorem.html

  1. g ( x ) = inf { λ : x λ K } . g(x)=\inf\{\lambda\in\mathbb{R}:x\in\lambda K\}.
  2. K = { x n : q ( x ) 1 } . K=\{x\in\mathbb{R}^{n}:q(x)\leq 1\}.
  3. 2 n n ! vol ( n / Γ ) λ 1 λ 2 λ n vol ( K ) 2 n vol ( n / Γ ) . \frac{2^{n}}{n!}\mathrm{vol}(\mathbb{R}^{n}/\Gamma)\leq\lambda_{1}\lambda_{2}% \cdots\lambda_{n}\mathrm{vol}(K)\leq 2^{n}\mathrm{vol}(\mathbb{R}^{n}/\Gamma).

Mirai_(song).html

  1. \cdot
  2. \cdot
  3. \cdot
  4. \cdot
  5. \cdot
  6. \cdot
  7. \cdot
  8. \cdot
  9. \cdot
  10. \cdot

Misleading_graph.html

  1. Lie factor = size of effect shown in graphic size of effect shown in data , \,\text{Lie factor}=\frac{\,\text{size of effect shown in graphic}}{\,\text{% size of effect shown in data}},
  2. size of effect = | second value - first value first value | . \,\text{size of effect}=\left|\frac{\,\text{second value}-\,\text{first value}% }{\,\text{first value}}\right|.
  3. graph discrepancy index = 100 ( a b - 1 ) , \,\text{graph discrepancy index}=100\left(\frac{a}{b}-1\right),
  4. a = percentage change depicted in graph , a=\,\text{percentage change depicted in graph},
  5. b = percentage change in data . b=\,\text{percentage change in data}.
  6. data-ink ratio = “ink” used to display the data total “ink” used to display the graphic . \,\text{data-ink ratio}=\frac{\,\text{``ink'' used to display the data}}{\,% \text{total ``ink'' used to display the graphic}}.
  7. data density = number of entries in data matrix area of data graphic . \,\text{data density}=\frac{\,\text{number of entries in data matrix}}{\,\text% {area of data graphic}}.

Missing_::_It's_You.html

  1. \cdot
  2. \cdot
  3. \cdot
  4. \cdot
  5. \cdot
  6. \cdot

Mittag-Leffler_summation.html

  1. y ( z ) = k = 0 y k z k y(z)=\sum_{k=0}^{\infty}y_{k}z^{k}
  2. α y \scriptstyle\mathcal{B}_{\alpha}y
  3. y \scriptstyle y
  4. α y ( t ) k = 0 y k Γ ( 1 + α k ) t k \mathcal{B}_{\alpha}y(t)\equiv\sum_{k=0}^{\infty}\frac{y_{k}}{\Gamma(1+\alpha k% )}t^{k}
  5. lim α 0 α y ( z ) \lim_{\alpha\rightarrow 0}\mathcal{B}_{\alpha}y(z)
  6. 0 e - t α y ( t α z ) d t \int_{0}^{\infty}e^{-t}\mathcal{B}_{\alpha}y(t^{\alpha}z)\,dt

Mixed_graph.html

  1. u , v V u,v\in V
  2. u v \overrightarrow{uv}
  3. ( u , v ) (u,v)
  4. u u
  5. v v
  6. u v uv
  7. [ u , v ] [u,v]
  8. { 1 , , k } \{1,...,k\}
  9. k k
  10. { 1 , 2 , 3 } \{1,2,3\}
  11. u u
  12. v v
  13. u u
  14. v v
  15. v v
  16. w w
  17. v v
  18. w w
  19. c : V [ k ] c:V\rightarrow[k]
  20. [ k ] := 1 , 2 , , k [k]:={1,2,\dots,k}
  21. c ( u ) c ( v ) c(u)\neq c(v)
  22. u v E uv\in E
  23. c ( u ) < c ( v ) c(u)<c(v)
  24. u v A \overrightarrow{uv}\in A
  25. c : V [ k ] c:V\rightarrow[k]
  26. [ k ] := 1 , 2 , , k [k]:={1,2,\dots,k}
  27. c ( u ) c ( v ) c(u)\neq c(v)
  28. u v E uv\in E
  29. c ( u ) c ( v ) c(u)\leq c(v)
  30. u v A \overrightarrow{uv}\in A
  31. ( v , w ) (v,w)
  32. χ ( G ) \chi(G)
  33. χ G ( k ) \chi_{G}(k)
  34. e e
  35. G = ( V , E , A ) G=(V,E,A)
  36. ( V , E - e , A ) (V,E-e,A)
  37. e e
  38. G - e G-e
  39. a a
  40. ( V , E , A - a ) (V,E,A-a)
  41. a a
  42. G - a G-a
  43. e e
  44. a a
  45. G / e G/e
  46. G / a G/a
  47. χ G ( k ) = χ G - e ( k ) - χ G / e ( k ) \chi_{G}(k)=\chi_{G-e}(k)-\chi_{G/e}(k)
  48. χ G ( k ) = χ G - a ( k ) + χ G / a ( k ) - χ G a ( k ) \chi_{G}(k)=\chi_{G-a}(k)+\chi_{G/a}(k)-\chi_{G_{a}}(k)

Mn2+-dependent_ADP-ribose::CDP-alcohol_diphosphatase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Molecular-scale_temperature.html

  1. T m ( z ) = M 0 M ( z ) T ( z ) T_{m}(z)=\frac{M_{0}}{M(z)}\cdot{T(z)}

Molybdenum_cofactor_cytidylyltransferase.html

  1. \rightleftharpoons

Molybdenum_cofactor_guanylyltransferase.html

  1. \rightleftharpoons

Molybdenum_cofactor_sulfurtransferase.html

  1. \rightleftharpoons

Molybdopterin-synthase_adenylyltransferase.html

  1. \rightleftharpoons

Molybdopterin_adenylyltransferase.html

  1. \rightleftharpoons

Molybdopterin_molybdotransferase.html

  1. \rightleftharpoons

Molybdopterin_synthase_sulfurtransferase.html

  1. \rightleftharpoons

Moment-area_theorem.html

  1. θ = ( M E I ) d x \theta=\int\left(\frac{M}{EI}\right)dx
  2. θ A B = A B M E I d x \theta_{AB}={\int_{A}}^{B}\frac{M}{EI}\;dx
  3. θ A B \theta_{AB}
  4. t A / B = A B M E I x d x t_{A/B}={\int_{A}}^{B}\frac{M}{EI}x\;dx
  5. t A / B t_{A/B}

Momentum-transfer_cross_section.html

  1. σ tr \sigma_{\mathrm{tr}}
  2. d σ d Ω ( θ ) \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}(\theta)
  3. σ tr = ( 1 - cos θ ) d σ d Ω ( θ ) d Ω \sigma_{\mathrm{tr}}=\int(1-\cos\theta)\frac{\mathrm{d}\sigma}{\mathrm{d}% \Omega}(\theta)\mathrm{d}\Omega
  4. = ( 1 - cos θ ) d σ d Ω ( θ ) sin θ d θ d ϕ =\int\int(1-\cos\theta)\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}(\theta)\sin% \theta\mathrm{d}\theta\mathrm{d}\phi
  5. σ tr = 4 π k 2 l = 0 ( l + 1 ) sin 2 [ δ l + 1 ( k ) - δ l ( k ) ] . \sigma_{\mathrm{tr}}=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(l+1)\sin^{2}[\delta% _{l+1}(k)-\delta_{l}(k)].
  6. 1 - cos θ 1-\cos\theta
  7. z z
  8. p in = q z ^ \vec{p}_{\mathrm{in}}=q\hat{z}
  9. θ \theta
  10. ϕ \phi
  11. p out = q cos θ z ^ + q sin θ cos ϕ x ^ + q sin θ cos ϕ y ^ \vec{p}_{\mathrm{out}}=q\cos\theta\hat{z}+q\sin\theta\cos\phi\hat{x}+q\sin% \theta\cos\phi\hat{y}
  12. Δ p = p in - p out = q ( 1 - cos θ ) z ^ - q sin θ cos ϕ x ^ - q sin θ cos ϕ y ^ \Delta\vec{p}=\vec{p}_{\mathrm{in}}-\vec{p}_{\mathrm{out}}=q(1-\cos\theta)\hat% {z}-q\sin\theta\cos\phi\hat{x}-q\sin\theta\cos\phi\hat{y}
  13. x x
  14. y y
  15. q ( 1 - cos θ ) z ^ q(1-\cos\theta)\hat{z}
  16. Δ p avg = Δ p Ω \Delta\vec{p}_{\mathrm{avg}}=\langle\Delta\vec{p}\rangle_{\Omega}
  17. = σ tot - 1 Δ p d σ d Ω ( θ ) d Ω =\sigma_{\mathrm{tot}}^{-1}\int\Delta\vec{p}\frac{\mathrm{d}\sigma}{\mathrm{d}% \Omega}(\theta)\mathrm{d}\Omega
  18. = σ tot - 1 [ q ( 1 - cos θ ) z ^ - q sin θ cos ϕ x ^ - q sin θ cos ϕ y ^ ] d σ d Ω ( θ ) d Ω =\sigma_{\mathrm{tot}}^{-1}\int\left[q(1-\cos\theta)\hat{z}-q\sin\theta\cos% \phi\hat{x}-q\sin\theta\cos\phi\hat{y}\right]\frac{\mathrm{d}\sigma}{\mathrm{d% }\Omega}(\theta)\mathrm{d}\Omega
  19. = q z ^ σ tot - 1 ( 1 - cos θ ) d σ d Ω ( θ ) d Ω =q\hat{z}\sigma_{\mathrm{tot}}^{-1}\int(1-\cos\theta)\frac{\mathrm{d}\sigma}{% \mathrm{d}\Omega}(\theta)\mathrm{d}\Omega
  20. = q z ^ σ tr / σ tot =q\hat{z}\sigma_{\mathrm{tr}}/\sigma_{\mathrm{tot}}
  21. σ tot = d σ d Ω ( θ ) d Ω \sigma_{\mathrm{tot}}=\int\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}(\theta)% \mathrm{d}\Omega
  22. σ tr \sigma_{\mathrm{tr}}

Momilactone-A_synthase.html

  1. \rightleftharpoons

Monocyclic_monoterpene_ketone_monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons
  5. \rightleftharpoons

Monodomain_model.html

  1. λ 1 + λ ( 𝚺 i v ) = χ ( C m v t + I ion ) , \frac{\lambda}{1+\lambda}\nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right)=% \chi\left(C_{m}\frac{\partial v}{\partial t}+I\text{ion}\right),
  2. 𝚺 i \mathbf{\Sigma}_{i}
  3. v v
  4. I ion I\text{ion}
  5. C m C_{m}
  6. λ \lambda
  7. χ \chi
  8. ( 𝚺 i v ) + ( 𝚺 i v e ) \displaystyle\nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right)+\nabla\cdot% \left(\mathbf{\Sigma}_{i}\nabla v_{e}\right)
  9. 𝚺 e = λ 𝚺 i \mathbf{\Sigma}_{e}=\lambda\mathbf{\Sigma}_{i}
  10. ( 𝚺 i v e ) = - 1 1 + λ ( 𝚺 i v ) . \nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v_{e}\right)=-\frac{1}{1+\lambda}% \nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right).
  11. λ 1 + λ ( 𝚺 i v ) = χ ( C m v t + I ion ) . \frac{\lambda}{1+\lambda}\nabla\cdot\left(\mathbf{\Sigma}_{i}\nabla v\right)=% \chi\left(C_{m}\frac{\partial v}{\partial t}+I\text{ion}\right).

Monoid_factorisation.html

  1. w = x i 1 x i 2 x i n w=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}
  2. x i j X i j x_{i_{j}}\in X_{i_{j}}
  3. i 1 i 2 i n i_{1}\geq i_{2}\geq\ldots\geq i_{n}
  4. Y X A = X Y . YX\cup A=X\cup Y\ .

Monoterpene_epsilon-lactone_hydrolase.html

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

Moran's_theorem.html

  1. N 1 ( t + 1 ) = f ( N 1 ( t ) ) + ϵ 1 ( t ) N_{1}(t+1)=f(N_{1}(t))+\epsilon_{1}(t)
  2. N 2 ( t + 1 ) = f ( N 2 ( t ) ) + ϵ 2 ( t ) N_{2}(t+1)=f(N_{2}(t))+\epsilon_{2}(t)
  3. N i N_{i}
  4. i i
  5. f f
  6. ϵ i \epsilon_{i}
  7. ρ N 1 , N 2 = ρ ϵ 1 , ϵ 2 \rho_{N_{1},N_{2}}=\rho_{\epsilon_{1},\epsilon_{2}}

Morphic_word.html

  1. a s f ( s ) f ( f ( s ) ) f ( n ) ( s ) asf(s)f(f(s))\cdots f^{(n)}(s)\cdots

Motion_(geometry).html

  1. x 2 - y 2 \ x^{2}-y^{2}
  2. ϕ : R 1 , 3 R 1 , 3 \phi:R^{1,3}\mapsto R^{1,3}
  3. ϕ ( x ) - ϕ ( y ) , ϕ ( x ) - ϕ ( y ) = x - y , x - y \langle\phi(x)-\phi(y),\ \phi(x)-\phi(y)\rangle\ =\ \langle x-y,\ x-y\rangle
  4. z ω z z\mapsto\omega z
  5. ω = cos θ + i sin θ , i 2 = - 1 \ \omega=\cos\theta+i\sin\theta,\quad i^{2}=-1

Motor_constants.html

  1. τ P \frac{\tau}{\sqrt{P}}
  2. τ \scriptstyle\tau
  3. P \scriptstyle P

Moving_average_rate_procedure.html

  1. N e w a v e r a g e r a t e = Total balance in company currency + new funds in company currency Total balance in voucher currency + new funds in voucher currency Newaveragerate=\frac{\,\text{Total balance in company currency }+\,\text{new % funds in company currency}}{\,\text{Total balance in voucher currency }+\,% \text{new funds in voucher currency}}

Moving_crack_(metalworking).html

  1. β \beta
  2. β \beta
  3. β \beta
  4. A A
  5. β A \beta A
  6. β \beta
  7. T = β A ρ c d 3 T=\frac{\beta A}{\rho cd^{3}}
  8. ρ \rho
  9. c c
  10. d 3 d^{3}
  11. y > 0 y>0
  12. y = 0 y=0
  13. x x
  14. Δ T - 1 a 2 T t = - 1 λ Q v t \Delta T-\frac{1}{a^{2}}\frac{\partial T}{\partial t}=-\frac{1}{\lambda}\frac{% \partial Qv}{\partial t}
  15. T T
  16. t t
  17. λ \lambda
  18. a 2 a^{2}
  19. λ ρ c \frac{\lambda}{\rho c}
  20. Q v ( x , y , z , t ) Qv(x,y,z,t)
  21. T = Q v ρ c erf ( h 4 a t ) T=\frac{Qv}{\rho c}\operatorname{erf}\left(\frac{h}{4a\sqrt{t}}\right)
  22. erf ( z ) = 2 π 0 z e - t 2 d t . \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\,\mathrm{d}t.
  23. T T

Moving_horizon_estimation.html

  1. t t
  2. [ t - T , t ] [t-T,t]
  3. t t
  4. J = i = 1 N w y ( x i - y i ) 2 + i = 1 N w x ^ ( x i - x i ^ ) 2 + i = 1 N w p i Δ p i 2 J=\sum_{i=1}^{N}w_{y}(x_{i}-y_{i})^{2}+\sum_{i=1}^{N}w_{\hat{x}}(x_{i}-\hat{x_% {i}})^{2}+\sum_{i=1}^{N}w_{p_{i}}{\Delta p_{i}}^{2}
  5. x i x_{i}
  6. y i y_{i}
  7. p i p_{i}
  8. w y w_{y}
  9. y i y_{i}
  10. w x i ^ w_{\hat{x_{i}}}
  11. x i ^ \hat{x_{i}}
  12. w p i w_{p_{i}}
  13. p i p_{i}

Moving_load.html

  1. - N 2 w ( x , t ) x 2 + ρ A 2 w ( x , t ) t 2 = δ ( x - v t ) P . -N\frac{\partial^{2}w(x,t)}{\partial x^{2}}+\rho A\frac{\partial^{2}w(x,t)}{% \partial t^{2}}=\delta(x-vt)P\ .
  2. w ( x , t ) = 2 P ρ A l j = 1 1 ω ( j ) 2 - ω 2 ( sin ( ω t ) - ω ω ( j ) sin ( ω ( j ) t ) ) , w(x,t)=\frac{2P}{\rho Al}\sum_{j=1}^{\infty}\frac{1}{\omega_{(j)}^{2}-\omega^{% 2}}\left(\sin(\omega t)-\frac{\omega}{\omega_{(j)}}\sin(\omega_{(j)}t)\right)\ ,
  3. ω = j π v l , \omega=\frac{j\pi v}{l}\ ,
  4. ω ( j ) 2 = j 2 π 2 l 2 N ρ A . \omega_{(j)}^{2}=\frac{j^{2}\pi^{2}}{l^{2}}\frac{N}{\rho A}\ .
  5. - N 2 w ( x , t ) x 2 + ρ A 2 w ( x , t ) t 2 = δ ( x - v t ) P - δ ( x - v t ) m d w 2 ( v t , t ) d t 2 . -N\frac{\partial^{2}w(x,t)}{\partial x^{2}}+\rho A\frac{\partial^{2}w(x,t)}{% \partial t^{2}}=\delta(x-vt)P-\delta(x-vt)m\frac{\mbox{d}~{}^{2}w(vt,t)}{\mbox% {d}~{}t^{2}}\ .
  6. δ ( x - v t ) d d t [ m d w ( v t , t ) d t ] = δ ( x - v t ) m d w 2 ( v t , t ) d t 2 . \delta(x-vt)\frac{\mbox{d}~{}}{\mbox{d}~{}t}\left[m\frac{\mbox{d}~{}w(vt,t)}{% \mbox{d}~{}t}\right]=\delta(x-vt)m\frac{\mbox{d}~{}^{2}w(vt,t)}{\mbox{d}~{}t^{% 2}}\ .
  7. d d t [ δ ( x - v t ) m d w ( v t , t ) d t ] = - δ ( x - v t ) m v d w ( v t , t ) d t + δ ( x - v t ) m d w 2 ( v t , t ) d t 2 . \frac{\mbox{d}~{}}{\mbox{d}~{}t}\left[\delta(x-vt)m\frac{\mbox{d}~{}w(vt,t)}{% \mbox{d}~{}t}\right]=-\delta^{\prime}(x-vt)mv\frac{\mbox{d}~{}w(vt,t)}{\mbox{d% }~{}t}+\delta(x-vt)m\frac{\mbox{d}~{}^{2}w(vt,t)}{\mbox{d}~{}t^{2}}\ .
  8. ρ ρ
  9. - N 2 w ( x , t ) x 2 = δ ( x - v t ) P - δ ( x - v t ) m d w 2 ( v t , t ) d t 2 . -N\frac{\partial^{2}w(x,t)}{\partial x^{2}}=\delta(x-vt)P-\delta(x-vt)\,m\frac% {\mbox{d}~{}^{2}w(vt,t)}{\mbox{d}~{}t^{2}}\ .
  10. y y
  11. τ τ
  12. y ( τ ) = w ( v t , t ) w s t , τ = v t l , y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ ,
  13. w w
  14. y ( τ ) = 4 α α - 1 τ ( τ - 1 ) k = 1 i = 1 k ( a + i - 1 ) ( b + i - 1 ) c + i - 1 τ k k ! , y(\tau)=\frac{4\,\alpha}{\alpha\,-\,1}\,\tau\,(\tau-1)\,\sum_{k=1}^{\infty}\,% \prod_{i=1}^{k}\frac{(a+i-1)(b+i-1)}{c+i-1}\;\frac{\tau^{k}}{k!}\ ,
  15. α α
  16. α = N l 2 m v 2 > 0 α 1 . \alpha=\frac{Nl}{2mv2}\,>\,0\ \ \ \wedge\ \ \ \alpha\,\neq\,1\ .
  17. a a
  18. b b
  19. c c
  20. a 1 , 2 = 3 ± 1 + 8 α 2 , b 1 , 2 = 3 1 + 8 α 2 , c = 2 . a_{1,2}=\frac{3\,\pm\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ b_{1,2}=\frac{3\,\mp\,% \sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ c=2\ .
  21. α α
  22. y ( τ ) = [ 4 3 τ ( 1 - τ ) - 4 3 τ ( 1 + 2 τ ln ( 1 - τ ) + 2 ln ( 1 - τ ) ) ] . y(\tau)=\left[\frac{4}{3}\tau(1-\tau)-\frac{4}{3}\tau\left(1+2\tau\ln(1-\tau)+% 2\ln(1-\tau)\right)\right]\ .

Möbius_plane.html

  1. z z
  2. P P
  3. z z
  4. Q Q
  5. z z
  6. z z^{\prime}
  7. P , Q P,Q
  8. z z
  9. P P
  10. x 2 + y 2 = 1 x^{2}+y^{2}=1
  11. 𝔄 ( \R ) \mathfrak{A}(\R)
  12. ρ ( x , y ) = x 2 + y 2 \rho(x,y)=x^{2}+y^{2}
  13. \R 2 \R^{2}
  14. y = m x + b y=mx+b
  15. x = c x=c
  16. ρ ( x - x 0 , y - y 0 ) = ( x - x 0 ) 2 + ( y - y 0 ) 2 = r 2 , r > 0 \rho(x-x_{0},y-y_{0})=(x-x_{0})^{2}+(y-y_{0})^{2}=r^{2},\ r>0
  17. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  18. 𝒫 := \R 2 { } , \R {\mathcal{P}}:=\R^{2}\cup\{\infty\},\infty\notin\R
  19. 𝒵 := { g { } g line of 𝔄 ( \R ) } \mathcal{Z}:=\{g\cup\{\infty\}\mid g\,\text{ line of }{\mathfrak{A}}(\R)\}
  20. { k k circle of 𝔄 ( \R ) } \cup\{k\mid k\,\text{ circle of }{\mathfrak{A}}(\R)\}
  21. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  22. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  23. A , B , C A,B,C
  24. z z
  25. A , B , C A,B,C
  26. z z
  27. P z P\in z
  28. Q z Q\notin z
  29. z z^{\prime}
  30. P , Q z P,Q\in z^{\prime}
  31. z z = { P } z\cap z^{\prime}=\{P\}
  32. z z
  33. z z^{\prime}
  34. P P
  35. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  36. z = x + i y z=x+iy
  37. ( x , y ) \R 2 (x,y)\in\R^{2}
  38. 𝒫 := \C { } , \C {\mathcal{P}}:=\C\cup\{\infty\},\infty\notin\C
  39. 𝒵 := { { z \C a z + a z ¯ + b = 0 (line) } { } 0 a \C , b \R } {\mathcal{Z}}:=\{\{z\in\C\mid az+\overline{az}+b=0\ \,\text{(line)}\ \}\cup\{% \infty\}\mid\ 0\neq a\in\C,b\in\R\}
  40. { { z \C ( z - z 0 ) ( z - z 0 ) ¯ = d (circle) z 0 \C , d \R , d > 0 } \cup\{\{z\in\C\mid(z-z_{0})\overline{(z-z_{0})}=d\ \,\text{(circle)}\mid z_{0}% \in\C,d\in\R,d>0\}
  41. z ¯ = x - i y \overline{z}=x-iy
  42. z z
  43. 𝒫 {\mathcal{P}}
  44. z r z , , z\rightarrow rz,\ \ \infty\rightarrow\infty\quad,
  45. r \C r\in\C
  46. z z + s , , z\rightarrow z+s,\ \ \infty\rightarrow\infty\quad,
  47. s \C s\in\C
  48. z 1 z , z 0 , 0 , 0 , z\rightarrow\displaystyle\frac{1}{z},\ z\neq 0,\ \ 0\rightarrow\infty,\ \ % \infty\rightarrow 0\quad,
  49. ± 1 \pm 1
  50. z z ¯ , z\rightarrow\overline{z},\ \ \infty\rightarrow\infty\quad
  51. \C { } \C\cup\{\infty\}
  52. \C \C
  53. ( 1 ) - ( 3 ) (1)-(3)
  54. PGL ( 2 , \C ) \operatorname{PGL}(2,\C)
  55. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  56. z 1 z ¯ z\rightarrow\tfrac{1}{\overline{z}}
  57. z z ¯ = 1 z\overline{z}=1
  58. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  59. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  60. Φ : ( x , y ) ( x 1 + x 2 + y 2 , y 1 + x 2 + y 2 , x 2 + y 2 1 + x 2 + y 2 ) = ( u , v , w ) . \Phi:\ (x,y)\rightarrow(\frac{x}{1+x^{2}+y^{2}},\frac{y}{1+x^{2}+y^{2}},\frac{% x^{2}+y^{2}}{1+x^{2}+y^{2}})=(u,v,w)\ .
  61. Φ \Phi
  62. ( 0 , 0 , 1 ) (0,0,1)
  63. u 2 + v 2 + w 2 - w = 0 u^{2}+v^{2}+w^{2}-w=0
  64. ( 0 , 0 , 1 2 ) (0,0,\tfrac{1}{2})
  65. r = 1 2 r=\tfrac{1}{2}
  66. x 2 + y 2 - a x - b y - c = 0 x^{2}+y^{2}-ax-by-c=0
  67. a u + b v - ( 1 + c ) w + c = 0 au+bv-(1+c)w+c=0
  68. ( 0 , 0 , 1 ) (0,0,1)
  69. a x + b y + c = 0 ax+by+c=0
  70. a u + b v - c w + c = 0 au+bv-cw+c=0
  71. ( 0 , 0 , 1 ) (0,0,1)
  72. ( 0 , 0 , 1 ) (0,0,1)
  73. 𝔐 = ( 𝒫 , 𝒵 , ) \mathfrak{M}=({\mathcal{P}},{\mathcal{Z}},\in)
  74. 𝒫 {\mathcal{P}}
  75. 𝒵 {\mathcal{Z}}
  76. A , B , C A,B,C
  77. z z
  78. A , B , C A,B,C
  79. z z
  80. P z P\in z
  81. Q z Q\notin z
  82. z z^{\prime}
  83. P , Q z P,Q\in z^{\prime}
  84. z z = { P } z\cap z^{\prime}=\{P\}
  85. z z
  86. z z^{\prime}
  87. P P
  88. A , B , C , D A,B,C,D
  89. z z
  90. A , B , C , D z A,B,C,D\in z
  91. 5 5
  92. 𝒫 := { A , B , C , D , } , 𝒵 := { z z 𝒫 , | z | = 3 } {\mathcal{P}}:=\{A,B,C,D,\infty\},\quad{\mathcal{Z}}:=\{z\mid z\subset{% \mathcal{P}},|z|=3\}
  93. | 𝒵 | = ( 5 3 ) = 10 |\mathcal{Z}|={5\choose 3}=10
  94. 𝔐 = ( 𝒫 , 𝒵 , ) \mathfrak{M}=({\mathcal{P}},{\mathcal{Z}},\in)
  95. P 𝒫 P\in{\mathcal{P}}
  96. 𝔄 P := ( 𝒫 { P } , { z { P } | P z 𝒵 } , ) {\mathfrak{A}}_{P}:=({\mathcal{P}}\setminus\{P\},\{z\setminus\{P\}|P\in z\in{% \mathcal{Z}}\},\in)
  97. 𝔄 {\mathfrak{A}}_{\infty}
  98. \infty
  99. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  100. P 𝒫 P\in{\mathcal{P}}
  101. 𝔄 P {\mathfrak{A}}_{P}
  102. | 𝒫 | < |{\mathcal{P}}|<\infty
  103. 𝔐 = ( 𝒫 , 𝒵 , ) \mathfrak{M}=({\mathcal{P}},{\mathcal{Z}},\in)
  104. z 𝒵 z\in{\mathcal{Z}}
  105. n := | z | - 1 n:=|z|-1
  106. 𝔐 \mathfrak{M}
  107. 𝔐 = ( 𝒫 , 𝒵 , ) \mathfrak{M}=({\mathcal{P}},{\mathcal{Z}},\in)
  108. n n
  109. 𝔄 P {\mathfrak{A}}_{P}
  110. n n
  111. | 𝒫 | = n 2 + 1 |{\mathcal{P}}|=n^{2}+1
  112. | 𝒵 | = n ( n 2 + 1 ) . |{\mathcal{Z}}|=n(n^{2}+1).
  113. ρ \rho
  114. K K
  115. \R \R
  116. K K
  117. x 2 + y 2 x^{2}+y^{2}
  118. 𝔐 ( K , ρ ) \mathfrak{M}(K,\rho)
  119. 𝔐 ( K , ρ ) \mathfrak{M}(K,\rho)
  120. P 1 , , P 8 P_{1},...,P_{8}
  121. 𝔐 ( K , ρ ) \mathfrak{M}(K,\rho)
  122. 𝔐 ( K , ρ ) \mathfrak{M}(K,\rho)
  123. 𝔐 ( K , ρ ) \mathfrak{M}(K,\rho)
  124. K = G F ( 2 ) K=GF(2)
  125. { 0 , 1 } \{0,1\}
  126. ρ ( x , y ) = x 2 + x y + y 2 \rho(x,y)=x^{2}+xy+y^{2}
  127. x 2 + x y + y 2 = 1 x^{2}+xy+y^{2}=1
  128. { ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) } \{(0,1),(1,0),(1,1)\}
  129. K = \C K=\C
  130. K = K=\mathbb{Q}
  131. ρ ( x , y ) = x 2 + y 2 \rho(x,y)=x^{2}+y^{2}
  132. K = K=\mathbb{Q}
  133. ρ ( x , y ) = x 2 - 2 y 2 \rho(x,y)=x^{2}-2y^{2}
  134. 𝔐 ( K , ρ ) \mathfrak{M}(K,\rho)
  135. K K

Mu_to_E_Gamma.html

  1. B ( μ + e + γ ) < 5.7 × 10 - 13 B(\mu^{+}\to e^{+}\gamma)<5.7\times 10^{-13}

Mugineic-acid_3-dioxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Multicanonical_ensemble.html

  1. exp ( - β E ) \exp(-\beta E)
  2. Δ E \Delta E
  3. exp ( - β E ) \exp(-\beta E)
  4. Ω \Omega
  5. s y m b o l r Ω symbol{r}\in\Omega
  6. Γ \Gamma
  7. F ( Ω ) = Γ = [ Γ min , Γ max ] F(\Omega)=\Gamma=[\Gamma_{\min},\Gamma_{\max}]
  8. Q \langle Q\rangle
  9. Q = 1 V Ω Q ( s y m b o l r ) p ( s y m b o l r ) d s y m b o l r \langle Q\rangle=\frac{1}{V}\int_{\Omega}Q(symbol{r})p(symbol{r})\,dsymbol{r}
  10. p ( s y m b o l r ) p(symbol{r})
  11. p ( s y m b o l r ) = 1 V Ω p ( s y m b o l r ) d s y m b o l r . p(symbol{r})=\frac{1}{V}\int_{\Omega}p(symbol{r})\,dsymbol{r}.
  12. ρ ( f ) = 1 V Ω δ ( f - F ( s y m b o l r ) ) d s y m b o l r \rho(f)=\frac{1}{V}\int_{\Omega}\delta(f-F(symbol{r}))\,dsymbol{r}
  13. F ( s y m b o l r ) = F s y m b o l r F(symbol{r})=F_{s}ymbol{r}
  14. Q ( s y m b o l r ) = Q ( F s y m b o l r ) , p ( s y m b o l r ) = p ( F s y m b o l r ) , Q(symbol{r})=Q(F_{s}ymbol{r}),p(symbol{r})=p(F_{s}ymbol{r}),
  15. Q \langle Q\rangle
  16. Q \displaystyle\langle Q\rangle
  17. Ω \Omega
  18. Γ \Gamma
  19. Z = 1 V Ω e - β H ( s y m b o l r ) d s y m b o l r . Z=\frac{1}{V}\int_{\Omega}e^{-\beta H(symbol{r})}\,dsymbol{r}.
  20. ρ ( E ) \rho(E)
  21. E \langle E\rangle
  22. E = E min E max E e - β E Z ρ ( E ) d E \langle E\rangle=\int_{E_{\min}}^{E_{\max}}E\frac{e^{-\beta E}}{Z}\rho(E)\,dE
  23. Q \langle Q\rangle
  24. s y m b o l r i Ω symbol{r}_{i}\in\Omega
  25. Q ¯ N = 1 N i = 0 N Q ( s y m b o l r i ) p ( s y m b o l r i ) \overline{Q}_{N}=\frac{1}{N}\sum_{i=0}^{N}Q(symbol{r}_{i})p(symbol{r}_{i})
  26. Q \langle Q\rangle
  27. Q ¯ N \overline{Q}_{N}
  28. Q \langle Q\rangle
  29. lim N Q ¯ N = Q . \lim_{N\rightarrow\infty}\overline{Q}_{N}=\langle Q\rangle.
  30. Q ¯ N \overline{Q}_{N}
  31. P ( s y m b o l r ) P(symbol{r})
  32. Q ¯ N = 1 X i = 0 N Q ( s y m b o l r i ) P - 1 ( s y m b o l r i ) p ( s y m b o l r i ) \overline{Q}_{N}=\frac{1}{X}\sum_{i=0}^{N}Q(symbol{r}_{i})P^{-1}(symbol{r}_{i}% )p(symbol{r}_{i})
  33. X = i = 0 N P - 1 ( s y m b o l r i ) X=\sum_{i=0}^{N}P^{-1}(symbol{r}_{i})
  34. P ( s y m b o l r ) = p Boltzmann ( s y m b o l r ) = e - β F ( s y m b o l r ) Ω d s y m b o l r e - β F ( s y m b o l r ) P(symbol{r})=p_{\mathrm{Boltzmann}}(symbol{r})=\frac{e^{-\beta F(symbol{r})}}{% \int_{\Omega}\,dsymbol{r}e^{-\beta F(symbol{r})}}
  35. P ( s y m b o l r ) = 1 ρ ( F ( s y m b o l r ) ) P(symbol{r})=\frac{1}{\rho(F(symbol{r}))}
  36. ρ ( f ) = 1 V Ω δ ( F ( s y m b o l r ) - f ) d s y m b o l r \rho(f)=\frac{1}{V}\int_{\Omega}\delta(F(symbol{r})-f)\,dsymbol{r}
  37. P ( f ) = 1 f max - f min Ω δ ( f - F ( s y m b o l r ) ) P ( s y m b o l r ) d s y m b o l r = 1 f max - f min 1 V ρ ( f ) Ω δ ( f - F ( s y m b o l r ) ) d s y m b o l r = 1 f max - f min = constant P(f)=\frac{1}{f_{\max}-f_{\min}}\int_{\Omega}\delta(f-F(symbol{r}))P(symbol{r}% )\,dsymbol{r}=\frac{1}{f_{\max}-f_{\min}}\frac{1}{V}\rho(f)\int_{\Omega}\delta% (f-F(symbol{r}))\,dsymbol{r}=\frac{1}{f_{\max}-f_{\min}}=\,\text{constant}
  38. P ( s y m b o l r ) P(symbol{r})
  39. τ t t N 2 \tau_{tt}\propto N^{2}
  40. N 2 + z N^{2+z}
  41. z > 0 z>0

Multidimensional_sampling.html

  1. f ( ) f(\cdot)
  2. f ^ ( ξ ) = ( f ) ( ξ ) = n f ( x ) e - 2 π i x , ξ d x \hat{f}(\xi)=\mathcal{F}(f)(\xi)=\int_{\Re^{n}}f(x)e^{-2\pi i\langle x,\xi% \rangle}\,dx
  3. x , ξ \langle x,\xi\rangle
  4. f ( ) f(\cdot)
  5. Ω \Omega
  6. f ^ ( ξ ) = 0 \hat{f}(\xi)=0
  7. ξ Ω \xi\notin\Omega
  8. Λ n \Lambda\subset\Re^{n}
  9. Λ = { i = 1 n a i v i | a i } \Lambda=\left\{\sum_{i=1}^{n}a_{i}v_{i}\;|\;a_{i}\in\mathbb{Z}\right\}
  10. n \Re^{n}
  11. Γ \Gamma
  12. Λ \Lambda
  13. Γ = { i = 1 n a i u i | a i } \Gamma=\left\{\sum_{i=1}^{n}a_{i}u_{i}\;|\;a_{i}\in\mathbb{Z}\right\}
  14. u i u_{i}
  15. u i , v j = δ i j \langle u_{i},v_{j}\rangle=\delta_{ij}
  16. u i u_{i}
  17. A A
  18. v i v_{i}
  19. B B
  20. A = B - T A=B^{-T}
  21. Λ \Lambda
  22. n \Re^{n}
  23. Γ \Gamma
  24. f ( ) f(\cdot)
  25. Ω n \Omega\subset\Re^{n}
  26. Λ \Lambda
  27. Ω \Omega
  28. Ω + x \Omega+x
  29. Γ \Gamma
  30. f ( ) f(\cdot)
  31. Λ \Lambda
  32. Ω { x + y : y Ω } = ϕ \Omega\cap\{x+y:y\in\Omega\}=\phi
  33. x Γ { 0 } x\in\Gamma\setminus\{0\}
  34. { f ( x ) : x Λ } \{f(x):x\in\Lambda\}
  35. f ( ) f(\cdot)
  36. Λ \Lambda
  37. f ^ ( ) \hat{f}(\cdot)
  38. f ^ s ( ξ ) = def y Γ f ^ ( ξ - y ) = x Λ | Λ | f ( x ) e - i 2 π x , ξ , \hat{f}_{s}(\xi)\ \stackrel{\mathrm{def}}{=}\sum_{y\in\Gamma}\hat{f}\left(\xi-% y\right)=\sum_{x\in\Lambda}|\Lambda|f(x)\ e^{-i2\pi\langle x,\xi\rangle},
  39. | Λ | |\Lambda|
  40. f ^ ( ) \hat{f}(\cdot)
  41. Ω \Omega
  42. f ^ s ( ) \hat{f}_{s}(\cdot)
  43. Ω \Omega
  44. Γ \Gamma
  45. f ^ s ( ξ ) \hat{f}_{s}(\xi)
  46. f ^ ( ξ ) \hat{f}(\xi)
  47. ξ Ω \xi\in\Omega
  48. χ ˇ Ω ( ) \check{\chi}_{\Omega}(\cdot)
  49. Ω \Omega
  50. Ω \Omega
  51. f ^ s ( ) \hat{f}_{s}(\cdot)
  52. Ω \Omega
  53. n \Re^{n}
  54. Ω d \Omega\subset\Re^{d}
  55. 2 \Re^{2}
  56. 2 \Re^{2}
  57. 2 \Re^{2}
  58. Ω \Omega
  59. χ ˇ Ω ( ) \check{\chi}_{\Omega}(\cdot)

Multiplication_and_repeated_addition.html

  1. 8 + 8 + 8 = 24 , 8+8+8=24,
  2. 3 × 8 = 24. 3\times 8=24.
  3. 3 × 0 = 0 + 0 + 0 , 3\times 0=0+0+0,
  4. 0 × 3 = 0. 0\times 3=0.
  5. 7 / 4 × 5 / 6 7/4\times 5/6

Multipole_magnet.html

  1. z z
  2. B y + i B x = C n ( x + i y ) n - 1 B_{y}+iB_{x}=C_{n}\cdot(x+iy)^{n-1}
  3. x x
  4. y y
  5. C n C_{n}
  6. B x B_{x}
  7. B y B_{y}
  8. C n C_{n}
  9. C n C_{n}

Multisite-specific_tRNA:(cytosine-C5)-methyltransferase.html

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

Multivariate_Behrens–Fisher_problem.html

  1. X i j 𝒩 p ( μ i , Σ i ) ( j = 1 , , n i ; i = 1 , 2 ) X_{ij}\sim\mathcal{N}_{p}(\mu_{i},\,\Sigma_{i})\ \ (j=1,\dots,n_{i};\ \ i=1,2)
  2. p p
  3. μ i \mu_{i}
  4. Σ i \Sigma_{i}
  5. i i
  6. j j
  7. i i
  8. X i j X_{ij}
  9. H 0 H_{0}
  10. H 1 H_{1}
  11. H 0 : μ 1 = μ 2 vs H 1 : μ 1 μ 2 . H_{0}:\mu_{1}=\mu_{2}\ \ \,\text{vs}\ \ H_{1}:\mu_{1}\neq\mu_{2}.
  12. X i ¯ \displaystyle\bar{X_{i}}
  13. X i ¯ \bar{X_{i}}
  14. A i A_{i}
  15. μ i , Σ i , ( i = 1 , 2 ) \mu_{i},\Sigma_{i},\ (i=1,2)
  16. X i ¯ \bar{X_{i}}
  17. A i A_{i}
  18. X i ¯ \displaystyle\bar{X_{i}}
  19. T 2 T^{2}
  20. H 0 H_{0}
  21. T 2 T^{2}
  22. X i ¯ 𝒩 p ( μ i , Σ i / n i ) , \bar{X_{i}}\sim\mathcal{N}_{p}\left(\mu_{i},\Sigma_{i}/n_{i}\right),
  23. A 1 + A 2 A_{1}+A_{2}
  24. T 2 T^{2}
  25. F F
  26. tr \mathrm{tr}
  27. T 2 ν p ν - p + 1 F p , ν - p + 1 , T^{2}\sim\frac{\nu p}{\nu-p+1}F_{p,\nu-p+1},
  28. ν \displaystyle\nu
  29. T 2 q F p , ν , T^{2}\sim qF_{p,\nu},
  30. q \displaystyle q
  31. D = 1 2 i = 1 2 1 n i { \displaystyle D=\frac{1}{2}\sum_{i=1}^{2}\frac{1}{n_{i}}\Bigg\{
  32. T 2 ν p ν - p + 1 F p , ν - p + 1 , T^{2}\sim\frac{\nu p}{\nu-p+1}F_{p,\nu-p+1},
  33. ν = tr ( S ~ 2 ) + [ tr ( S ~ ) ] 2 1 n 1 { tr ( S 1 ~ 2 ) + [ tr ( S 1 ~ ) ] 2 } + 1 n 2 { tr ( S 2 ~ 2 ) + [ tr ( S 2 ~ ) ] 2 } . \nu=\frac{\mathrm{tr}(\tilde{S}^{2})+[\mathrm{tr}(\tilde{S})]^{2}}{\frac{1}{n_% {1}}\left\{\mathrm{tr}(\tilde{S_{1}}^{2})+[\mathrm{tr}(\tilde{S_{1}})]^{2}% \right\}+\frac{1}{n_{2}}\left\{\mathrm{tr}(\tilde{S_{2}}^{2})+[\mathrm{tr}(% \tilde{S_{2}})]^{2}\right\}}.
  34. T 2 T^{2}
  35. T 2 T^{2}
  36. [ min { n 1 , n 2 } , n 1 + n 2 ] \left[\min\{n_{1},n_{2}\},n_{1}+n_{2}\right]
  37. T 2 T^{2}
  38. T 2 ν p F p , ν - p + 1 / ( ν - p + 1 ) , T^{2}\sim\nu pF_{p,\nu-p+1}/(\nu-p+1),
  39. ν = p + p 2 1 n 1 { tr [ ( S ~ 1 S ~ - 1 ) 2 ] + [ tr ( S ~ 1 S ~ - 1 ) ] 2 } + 1 n 2 { tr [ ( S ~ 2 S ~ - 1 ) 2 ] + [ tr ( S ~ 2 S ~ - 1 ) ] 2 } . \nu=\frac{p+p^{2}}{\frac{1}{n_{1}}\{\mathrm{tr}[(\tilde{S}_{1}\tilde{S}^{-1})^% {2}]+[\mathrm{tr}(\tilde{S}_{1}\tilde{S}^{-1})]^{2}\}+\frac{1}{n_{2}}\{\mathrm% {tr}[(\tilde{S}_{2}\tilde{S}^{-1})^{2}]+[\mathrm{tr}(\tilde{S}_{2}\tilde{S}^{-% 1})]^{2}\}}.

Mumford–Shah_functional.html

  1. E [ J , B ] = C ( I ( x ) - J ( x ) ) 2 d x + A D / B J ( x ) J ( x ) d x + B B d s E[J,B]=C\int(I(\vec{x})-J(\vec{x}))^{2}d\vec{x}+A\int_{D/B}\vec{\nabla}J(\vec{% x})\cdot\vec{\nabla}J(\vec{x})d\vec{x}+B\int_{B}ds
  2. E [ J , z ; ϵ ] = C ( I ( x ) - J ( x ) ) 2 d x + A z ( x ) | J ( x ) | 2 d x + B { ϵ | ϕ ( x ) | 2 + ϵ - 1 ϕ 2 ( z ( x ) ) } d x E[J,z;\epsilon]=C\int(I(\vec{x})-J(\vec{x}))^{2}d\vec{x}+A\int z(\vec{x})|\vec% {\nabla}J(\vec{x})|^{2}d\vec{x}+B\int\{\epsilon|\vec{\nabla}\phi(\vec{x})|^{2}% +\epsilon^{-1}\phi^{2}(z(\vec{x}))\}d\vec{x}
  3. ϕ 1 ( z ) = ( 1 - z ) / 2 z [ 0 , 1 ] . \phi_{1}(z)=(1-z)/2\quad z\in[0,1].
  4. ϕ 2 ( z ) = 3 z ( 1 - z ) z [ 0 , 1 ] . \phi_{2}(z)=3z(1-z)\quad z\in[0,1].
  5. ϵ 0 \epsilon\to 0

Muscle_architecture.html

  1. C S A = V l CSA=\frac{V}{l}
  2. V \textstyle V
  3. l \textstyle l
  4. P C S A = m cos θ l ρ PCSA=\frac{m\cdot\cos\theta}{l\cdot\rho}
  5. m \textstyle m
  6. θ \textstyle\theta
  7. l \textstyle l
  8. ρ \textstyle\rho

Mutation_(Jordan_algebra).html

  1. L ( a ) b = a b \displaystyle{L(a)b=ab}
  2. Q ( a ) = 2 L ( a ) 2 - L ( a 2 ) . Q(a)=2L(a)^{2}-L(a^{2}).\,
  3. Q ( 1 ) = I . Q(1)=I.\,
  4. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) \displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a)}
  5. Q ( a ) R ( b , a ) = R ( a , b ) Q ( a ) = 2 Q ( Q ( a ) b , a ) , \displaystyle{Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a),}
  6. R ( a , b ) c = 2 Q ( a , c ) b , Q ( x , y ) = 1 2 ( Q ( x + y ) - Q ( x ) - Q ( y ) ) . R(a,b)c=2Q(a,c)b,\,\,\,Q(x,y)=\frac{1}{2}(Q(x+y)-Q(x)-Q(y)).
  7. R ( a , b ) = 2 Q ( a ) Q ( a - 1 , b ) = 2 Q ( a , b - 1 ) Q ( b ) . \displaystyle{R(a,b)=2Q(a)Q(a^{-1},b)=2Q(a,b^{-1})Q(b).}
  8. { a , b , c } = ( a b ) c + ( c b ) a - ( a c ) b , \{a,b,c\}=(ab)c+(cb)a-(ac)b,\,
  9. Q ( a ) b = { a , b , a } , Q ( a , c ) b = { a , b , c } , R ( a , b ) c = { a , b , c } . Q(a)b=\{a,b,a\},\,\,\,Q(a,c)b=\{a,b,c\},\,\,\,R(a,b)c=\{a,b,c\}.\,
  10. Q ( a , b ) = L ( a ) L ( b ) + L ( b ) L ( a ) - L ( a b ) , R ( a , b ) = [ L ( a ) , L ( b ) ] + L ( a b ) . Q(a,b)=L(a)L(b)+L(b)L(a)-L(ab),\,\,\,R(a,b)=[L(a),L(b)]+L(ab).\,
  11. a b = { a , y , b } . a\circ b=\{a,y,b\}.\,
  12. Q ( a ) R ( b , a ) = R ( a , b ) Q ( a ) = 2 Q ( Q ( a ) b , a ) . \displaystyle{Q(a)R(b,a)=R(a,b)Q(a)=2Q(Q(a)b,a).}
  13. 2 Q ( a ) Q ( b , c ) a = 2 Q ( Q ( a ) c , a ) b . \displaystyle{2Q(a)Q(b,c)a=2Q(Q(a)c,a)b.}
  14. Q ( a ) R ( b , a ) c = 2 Q ( Q ( a ) b , a ) c . \displaystyle{Q(a)R(b,a)c=2Q(Q(a)b,a)c.}
  15. L ( a ) = 1 2 R ( a , 1 ) . \displaystyle{L(a)=\frac{1}{2}R(a,1).}
  16. R ( a , 1 ) = R ( 1 , a ) = 2 Q ( a , 1 ) . \displaystyle{R(a,1)=R(1,a)=2Q(a,1).}
  17. L ( a ) = Q ( a , 1 ) , L ( 1 ) = Q ( 1 , 1 ) = I . \displaystyle{L(a)=Q(a,1),\,\,\,L(1)=Q(1,1)=I.}
  18. a b = L ( a ) b = 1 2 R ( a , 1 ) b = Q ( a , b ) 1 , \displaystyle{a\circ b=L(a)b=\frac{1}{2}R(a,1)b=Q(a,b)1,}
  19. a b = b a . \displaystyle{a\circ b=b\circ a.}
  20. [ L ( a ) , L ( a 2 ) ] = 0. \displaystyle{[L(a),L(a^{2})]=0.}
  21. Q ( Q ( a ) b ) = Q ( a ) Q ( b ) Q ( a ) , \displaystyle{Q(Q(a)b)=Q(a)Q(b)Q(a),}
  22. Q ( a ) = 2 Q ( a , 1 ) 2 - Q ( a 2 , 1 ) = 2 L ( a ) 2 - L ( a 2 ) . \displaystyle{Q(a)=2Q(a,1)^{2}-Q(a^{2},1)=2L(a)^{2}-L(a^{2}).}
  23. Q ( a ) L ( a ) = L ( a ) Q ( a ) , \displaystyle{Q(a)L(a)=L(a)Q(a),}
  24. [ L ( x 2 ) , L ( y ) ] + 2 [ L ( x y ) , L ( x ) ] = 0. \displaystyle{[L(x^{2}),L(y)]+2[L(xy),L(x)]=0.}
  25. a b = { a , y , b } , \displaystyle{a\circ b=\{a,y,b\},}
  26. a b = L y ( a ) b , \displaystyle{a\circ b=L_{y}(a)b,}
  27. L y ( a ) = [ L ( a ) , L ( y ) ] + L ( a y ) . \displaystyle{L_{y}(a)=[L(a),L(y)]+L(ay).}
  28. y e = 1. \displaystyle{ye=1.}
  29. e ( y a ) = y ( e a ) \displaystyle{e(ya)=y(ea)}
  30. Q y ( a ) = Q ( a ) Q ( y ) , R y ( a , b ) = R ( a , Q ( y ) b ) . \displaystyle{Q_{y}(a)=Q(a)Q(y),\,\,\,R_{y}(a,b)=R(a,Q(y)b).}
  31. Q y ( e ) = Q y ( y - 1 ) = Q ( y - 1 ) Q ( y ) = I . \displaystyle{Q_{y}(e)=Q_{y}(y^{-1})=Q(y^{-1})Q(y)=I.}
  32. Q y ( a ) Q y ( b ) Q y ( a ) = Q ( a ) Q ( y ) Q ( b ) Q ( y ) Q ( a ) Q ( y ) = Q ( a ) Q ( Q ( y ) b ) Q ( a ) Q ( y ) = Q ( Q ( a ) Q ( y ) b ) Q ( y ) = Q y ( Q y ( a ) b ) . \displaystyle{Q_{y}(a)Q_{y}(b)Q_{y}(a)=Q(a)Q(y)Q(b)Q(y)Q(a)Q(y)=Q(a)Q(Q(y)b)Q(% a)Q(y)=Q(Q(a)Q(y)b)Q(y)=Q_{y}(Q_{y}(a)b).}
  33. Q ( y ) R ( c , Q ( y ) d ) Q ( y ) - 1 = R ( Q ( y ) c , d ) , \displaystyle{Q(y)R(c,Q(y)d)Q(y)^{-1}=R(Q(y)c,d),}
  34. Q ( y ) R ( c , Q ( y ) d ) x = 2 Q ( y ) Q ( c , x ) Q ( y ) d = 2 Q ( Q ( y ) c , Q ( y ) x ) d = R ( Q ( y ) c , d ) Q ( y ) x . \displaystyle{Q(y)R(c,Q(y)d)x=2Q(y)Q(c,x)Q(y)d=2Q(Q(y)c,Q(y)x)d=R(Q(y)c,d)Q(y)% x.}
  35. Q y ( a ) R y ( b , a ) = Q ( a ) Q ( y ) R ( b , Q ( y ) a ) = Q ( y - 1 ) Q ( Q ( y ) a ) R ( b , Q ( y ) a ) = Q ( y ) - 1 R ( Q ( y ) a , b ) Q ( Q ( y ) a ) = R y ( a , b ) Q y ( a ) . \displaystyle{Q_{y}(a)R_{y}(b,a)=Q(a)Q(y)R(b,Q(y)a)=Q(y^{-1})Q(Q(y)a)R(b,Q(y)a% )=Q(y)^{-1}R(Q(y)a,b)Q(Q(y)a)=R_{y}(a,b)Q_{y}(a).}
  36. M ( a ) b = 1 2 R y ( a , e ) b = 1 2 R ( a , Q ( y ) e ) b = 1 2 R ( a , y ) b = { a , y , b } = L y ( a ) b . \displaystyle{M(a)b=\frac{1}{2}R_{y}(a,e)b=\frac{1}{2}R(a,Q(y)e)b=\frac{1}{2}R% (a,y)b=\{a,y,b\}=L_{y}(a)b.}
  37. Q 1 ( a α 1 ) ( b β 1 ) = α 2 β 1 [ α 2 a + α 2 b + 2 α β a + α { a , y , b } + β Q ( a ) y + Q ( a ) Q ( y ) b ] . \displaystyle{Q_{1}(a\oplus\alpha 1)(b\oplus\beta 1)=\alpha^{2}\beta 1\oplus[% \alpha^{2}a+\alpha^{2}b+2\alpha\beta a+\alpha\{a,y,b\}+\beta Q(a)y+Q(a)Q(y)b].}
  38. ( 1 - x ) - 1 + ( 1 - x - 1 ) - 1 = 1. \displaystyle{(1-x)^{-1}+(1-x^{-1})^{-1}=1.}
  39. a - 1 = Q ( a ) - 1 ( a - 1 - Q ( a ) - 1 b ) - 1 + Q ( a ) - 1 ( a - 1 - b - 1 ) - 1 = ( a - b ) - 1 + ( a - Q ( a ) b - 1 ) - 1 . \displaystyle{a^{-1}=Q(a)^{-1}(a^{-1}-Q(a)^{-1}b)^{-1}+Q(a)^{-1}(a^{-1}-b^{-1}% )^{-1}=(a-b)^{-1}+(a-Q(a)b^{-1})^{-1}.}
  40. B ( a , b ) = I - R ( a , b ) + Q ( a ) Q ( b ) . \displaystyle{B(a,b)=I-R(a,b)+Q(a)Q(b).}
  41. B ( a , b ) = Q ( a ) Q ( a - 1 - b ) ; \displaystyle{B(a,b)=Q(a)Q(a^{-1}-b);}
  42. B ( a , b ) = Q ( a - b - 1 ) Q ( b ) . \displaystyle{B(a,b)=Q(a-b^{-1})Q(b).}
  43. B ( a , b ) - 1 ( a - Q ( a ) b ) = Q ( a b ) Q ( a - 1 ) ( a - Q ( a ) b ) = Q ( a b ) ( a b ) - 1 = a b . \displaystyle{B(a,b)^{-1}(a-Q(a)b)=Q(a^{b})Q(a^{-1})(a-Q(a)b)=Q(a^{b})(a^{b})^% {-1}=a^{b}.}
  44. a b + c = ( ( a - 1 - b ) - c ) - 1 = ( ( a b ) - 1 - c ) - 1 = ( a b ) c . \displaystyle{a^{b+c}=((a^{-1}-b)-c)^{-1}=((a^{b})^{-1}-c)^{-1}=(a^{b})^{c}.}
  45. a b = B ( a , b ) - 1 ( a - Q ( a ) b ) . \displaystyle{a^{b}=B(a,b)^{-1}(a-Q(a)b).}
  46. a b + c = ( a b ) c . \displaystyle{a^{b+c}=(a^{b})^{c}.}
  47. B ( a , b ) Q ( a b ) B ( b , a ) = B ( a , b ) Q ( a ) = Q ( a ) B ( b , a ) . \displaystyle{B(a,b)Q(a^{b})B(b,a)=B(a,b)Q(a)=Q(a)B(b,a).}
  48. a 1 - 1 - b 1 + b 3 = ( a 1 - 1 - b 1 + b 2 ) - b 2 + b 3 = a 2 - 1 - b 2 + b 3 \displaystyle{a_{1}^{-1}-b_{1}+b_{3}=(a_{1}^{-1}-b_{1}+b_{2})-b_{2}+b_{3}=a_{2% }^{-1}-b_{2}+b_{3}}
  49. a 3 = a 2 b 2 - b 3 = ( a 1 b 1 - b 2 ) b 2 - b 3 = a 1 b 1 - b 3 . \displaystyle{a_{3}=a_{2}^{b_{2}-b_{3}}=(a_{1}^{b_{1}-b_{2}})^{b_{2}-b_{3}}=a_% {1}^{b_{1}-b_{3}}.}
  50. B ( a 1 , b 1 - b 3 ) = B ( a 1 , b 1 - b 2 ) B ( a 2 , b 2 - b 3 ) , \displaystyle{B(a_{1},b_{1}-b_{3})=B(a_{1},b_{1}-b_{2})B(a_{2},b_{2}-b_{3}),}
  51. a 3 = a 2 b 2 - b 3 = ( a 1 b 1 - b 2 ) b 2 - b 3 = a 1 b 1 - b 3 . \displaystyle{a_{3}=a_{2}^{b_{2}-b_{3}}=(a_{1}^{b_{1}-b_{2}})^{b_{2}-b_{3}}=a_% {1}^{b_{1}-b_{3}}.}
  52. Q ( g a ) = g Q ( a ) g t . \displaystyle{Q(ga)=gQ(a)g^{t}.}
  53. g ( a , b ) = ( g a , ( g t ) - 1 b ) . \displaystyle{g(a,b)=(ga,(g^{t})^{-1}b).}
  54. g ( x y ) = ( g x ) ( g t ) - 1 y . \displaystyle{g(x^{y})=(gx)^{(g^{t})^{-1}y}.}
  55. g B ( x , y ) g - 1 = B ( g x , ( g t ) - 1 y ) \displaystyle{gB(x,y)g^{-1}=B(gx,(g^{t})^{-1}y)}
  56. 𝔤 0 \mathfrak{g}_{0}
  57. 𝔤 0 \mathfrak{g}_{0}
  58. φ c b = φ c φ b - 1 : φ b ( X b X c ) φ c ( X b X c ) . \displaystyle{\varphi_{cb}=\varphi_{c}\circ\varphi_{b}^{-1}:\varphi_{b}(X_{b}% \cap X_{c})\rightarrow\varphi_{c}(X_{b}\cap X_{c}).}
  59. φ c b ( a ) = a b - c \displaystyle{\varphi_{cb}(a)=a^{b-c}}
  60. φ c b ( a ) = B ( a , b - c ) - 1 . \displaystyle{\varphi_{cb}^{\prime}(a)=B(a,b-c)^{-1}.}
  61. g = ( α β γ δ ) , \displaystyle{g=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix},}
  62. g ( z ) = ( α z + β ) ( γ z + δ ) - 1 . \displaystyle{g(z)=(\alpha z+\beta)(\gamma z+\delta)^{-1}.}
  63. J = ( 0 1 - 1 0 ) . \displaystyle{J=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}.}
  64. J = ( 1 0 - 1 1 ) ( 1 1 0 1 ) ( 1 0 - 1 1 ) . \displaystyle{J=\begin{pmatrix}1&0\\ -1&1\end{pmatrix}\begin{pmatrix}1&1\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -1&1\end{pmatrix}.}
  65. g ( a ) = ( α a + β 1 ) ( γ a + δ 1 ) - 1 . \displaystyle{g(a)=(\alpha a+\beta 1)(\gamma a+\delta 1)^{-1}.}
  66. g ( a : 0 ) = ( a : - γ ) = ( a - γ : 0 ) = ( a ( γ a + 1 ) - 1 : 0 ) \displaystyle{g(a:0)=(a:-\gamma)=(a^{-\gamma}:0)=(a(\gamma a+1)^{-1}:0)}
  67. Q ( a ) = T a j T a - 1 j T a j . \displaystyle{Q(a)=T_{a}\circ j\circ T_{a^{-1}}\circ j\circ T_{a}\circ j.}
  68. j T a j T b j = j T c T λ j T λ - 1 T d j = λ 2 j T c j T - λ j T d j = λ 2 T - c - 1 j Q ( c - 1 ) T - c - 1 - λ - d - 1 j Q ( d - 1 ) j T - d - 1 , \displaystyle{jT_{a}jT_{b}j=jT_{c}T_{\lambda}jT_{\lambda^{-1}}T_{d}\circ j=% \lambda^{2}jT_{c}jT_{-\lambda}jT_{d}\ j=\lambda^{2}T_{-c^{-1}}jQ(c^{-1})T_{-c^% {-1}-\lambda-d^{-1}}jQ(d^{-1})jT_{-d^{-1}},}
  69. Q ( a ) = T a j T a - 1 j T a j . \displaystyle{Q(a)=T_{a}\circ j\circ T_{a^{-1}}\circ j\circ T_{a}\circ j.}
  70. B ( c + b , a ) = B ( c , a b ) B ( b , a ) \displaystyle{B(c+b,a)=B(c,a^{b})B(b,a)}
  71. R ( a , b ) = R ( a b , b - Q ( b ) a ) = R ( a - Q ( a ) b , b a ) \displaystyle{R(a,b)=R(a^{b},b-Q(b)a)=R(a-Q(a)b,b^{a})}
  72. 𝔤 0 \mathfrak{g}_{0}
  73. 𝔤 - 1 \mathfrak{g}_{-1}
  74. 𝔤 1 \mathfrak{g}_{1}
  75. 𝔤 = 𝔤 - 1 𝔤 0 𝔤 1 . \displaystyle{\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\oplus% \mathfrak{g}_{1}.}
  76. 𝔤 i = ( 0 ) \mathfrak{g}_{i}=(0)
  77. 𝔤 \mathfrak{g}
  78. [ 𝔤 p , 𝔤 q ] 𝔤 p + q . \displaystyle{[\mathfrak{g}_{p},\mathfrak{g}_{q}]\subseteq\mathfrak{g}_{p+q}}.
  79. 𝔤 \mathfrak{g}
  80. [ ( a 1 , T 1 , b 1 ) , ( a 2 , T 2 , b 2 ) ] = ( T 1 a 2 - T 2 a 1 , [ T 1 , T 2 ] + R ( a 1 , b 2 ) - R ( a 2 , b 1 ) , T 2 t b 1 - T 1 t b 2 ) \displaystyle{[(a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2})]=(T_{1}a_{2}-T_{2}a_{1}% ,[T_{1},T_{2}]+R(a_{1},b_{2})-R(a_{2},b_{1}),T_{2}^{t}b_{1}-T_{1}^{t}b_{2})}
  81. 𝔤 \mathfrak{g}
  82. σ ( a , T , b ) = ( b , - T t , a ) . \displaystyle{\sigma(a,T,b)=(b,-T^{t},a).}
  83. g = ( α β γ δ ) \displaystyle{g=\begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}}
  84. ( α 0 0 α - 1 ) ( a , T , b ) = ( α 2 a , T , α - 2 b ) , \displaystyle{\begin{pmatrix}\alpha&0\\ 0&\alpha^{-1}\end{pmatrix}(a,T,b)=(\alpha^{2}a,T,\alpha^{-2}b),}
  85. ( 1 β 0 1 ) ( a , T , b ) = ( a + β T ( 1 ) - β 2 b , T - β L ( a ) , b ) , \displaystyle{\begin{pmatrix}1&\beta\\ 0&1\end{pmatrix}(a,T,b)=(a+\beta T(1)-\beta^{2}b,T-\beta L(a),b),}
  86. ( 1 0 γ 1 ) ( a , T , b ) = ( a , T - γ L ( b ) , b - γ T t ( 1 ) - γ 2 a ) . \displaystyle{\begin{pmatrix}1&0\\ \gamma&1\end{pmatrix}(a,T,b)=(a,T-\gamma L(b),b-\gamma T^{t}(1)-\gamma^{2}a).}
  87. ( g ( T ) g ( a ) g ( b ) g ( T ) t ) = g ( T a b T t ) g - 1 . \displaystyle{\begin{pmatrix}g(T)&g(a)\\ g(b)&g(T)^{t}\end{pmatrix}=g\begin{pmatrix}T&a\\ b&T^{t}\end{pmatrix}g^{-1}.}
  88. < m t p l > B ( ( a 1 , T 1 , b 1 ) , ( a 2 , T 2 , b 2 ) ) = ( a 1 , b 2 ) + ( b 1 , a 2 ) + β ( T 1 , T 2 ) , \displaystyle{\mathbf{<}mtpl>{{B}}((a_{1},T_{1},b_{1}),(a_{2},T_{2},b_{2}))=(a% _{1},b_{2})+(b_{1},a_{2})+\beta(T_{1},T_{2}),}
  89. β ( R ( a , b ) , R ( c , d ) ) = ( R ( a , b ) c , d ) = ( R ( c , d ) a , b ) , \displaystyle{\beta(R(a,b),R(c,d))=(R(a,b)c,d)=(R(c,d)a,b),}
  90. 𝔤 \mathfrak{g}
  91. W ( a , T , b ) = ( W a , W T W - 1 , ( W t ) - 1 b ) , \displaystyle{W(a,T,b)=(Wa,WTW^{-1},(W^{t})^{-1}b),}
  92. J ( a , T , b ) = ( - b , - T t , - a ) , \displaystyle{J(a,T,b)=(-b,-T^{t},-a),}
  93. T x ( a , T , b ) = ( a + T x - Q ( x ) b , T - R ( x , b ) , b ) , \displaystyle{T_{x}(a,T,b)=(a+Tx-Q(x)b,T-R(x,b),b),}
  94. S y ( a , T , b ) = ( a , T - R ( a , y ) , b - T t y - Q ( y ) a ) . \displaystyle{S_{y}(a,T,b)=(a,T-R(a,y),b-T^{t}y-Q(y)a).}
  95. 𝔤 \mathfrak{g}
  96. 𝔤 \mathfrak{g}
  97. 𝔤 ± 1 \mathfrak{g}_{\pm 1}
  98. e t M ( 0 , 0 , b ) = J e t N J ( 0 , 0 , b ) = J e t N ( b , 0 , 0 ) = ( 0 , 0 , e t N b ) . \displaystyle{e^{tM}(0,0,b)=Je^{tN}J(0,0,b)=Je^{tN}(b,0,0)=(0,0,e^{tN}b).}
  99. Q ( U t a ) b = U t Q ( a ) V - t b . \displaystyle{Q(U_{t}a)b=U_{t}Q(a)V_{-t}b.}
  100. 𝔤 0 \mathfrak{g}_{0}
  101. 𝔤 \mathfrak{g}
  102. 𝔤 \mathfrak{g}
  103. 𝔤 \mathfrak{g}
  104. 𝔤 0 \mathfrak{g}_{0}
  105. 𝔤 \mathfrak{g}
  106. θ ( a , T , b ) = ( b * , - T * , a * ) . \displaystyle{\theta(a,T,b)=(b^{*},-T^{*},a^{*}).}
  107. θ ( S a ) = T a * , θ ( j ) = j , θ ( T b ) = S b * , θ ( W ) = ( W * ) - 1 . \displaystyle{\theta(S_{a})=T_{a^{*}},\,\,\,\theta(j)=j,\,\,\,\theta(T_{b})=S_% {b^{*}},\,\,\,\theta(W)=(W^{*})^{-1}.}
  108. 𝔥 \mathfrak{h}
  109. 𝔤 \mathfrak{g}
  110. 𝔤 \mathfrak{g}
  111. 𝔤 \mathfrak{g}
  112. 𝔥 \mathfrak{h}
  113. ( α β γ δ ) \begin{pmatrix}\alpha&\beta\\ \gamma&\delta\end{pmatrix}
  114. ( δ ¯ - γ ¯ - β ¯ α ¯ ) \begin{pmatrix}\overline{\delta}&-\overline{\gamma}\\ -\overline{\beta}&\overline{\alpha}\end{pmatrix}
  115. M = ( ζ 1 0 0 ζ 1 - 1 ) ( cos φ sin φ - sin φ cos φ ) ( ζ 2 0 0 ζ 2 - 1 ) . \displaystyle{M=\begin{pmatrix}\zeta_{1}&0\\ 0&\zeta_{1}^{-1}\end{pmatrix}\begin{pmatrix}\cos\varphi&\sin\varphi\\ -\sin\varphi&\cos\varphi\end{pmatrix}\begin{pmatrix}\zeta_{2}&0\\ 0&\zeta_{2}^{-1}\end{pmatrix}.}
  116. 𝔤 \mathfrak{g}
  117. 𝔥 1 \mathfrak{h}_{1}
  118. 𝔥 \mathfrak{h}
  119. 𝔥 \mathfrak{h}
  120. 𝔥 \mathfrak{h}
  121. 𝔥 1 \mathfrak{h}_{1}
  122. 𝔡 \mathfrak{d}
  123. 𝔥 \mathfrak{h}
  124. 𝔥 \mathfrak{h}
  125. 𝔥 \mathfrak{h}
  126. 𝔥 1 \mathfrak{h}_{1}
  127. 𝔡 \mathfrak{d}
  128. 𝔥 \mathfrak{h}
  129. 𝔥 \mathfrak{h}
  130. 𝔡 \mathfrak{d}
  131. 𝔥 \mathfrak{h}
  132. 𝔡 \mathfrak{d}
  133. 𝔡 \mathfrak{d}
  134. 𝔥 \mathfrak{h}
  135. 𝔥 \mathfrak{h}
  136. 𝔤 \mathfrak{g}
  137. 𝔤 \mathfrak{g}
  138. 𝔤 \mathfrak{g}
  139. 𝔤 \mathfrak{g}
  140. 𝔤 \mathfrak{g}
  141. 𝔥 \mathfrak{h}
  142. 𝔥 \mathfrak{h}
  143. 𝔤 \mathfrak{g}
  144. 𝔤 \mathfrak{g}
  145. 𝔤 \mathfrak{g}
  146. 𝔤 \mathfrak{g}
  147. 𝔥 \mathfrak{h}
  148. 𝔤 \mathfrak{g}
  149. τ ( a , T , b ) = ( - a * , - T * , - b * ) . \displaystyle{\tau(a,T,b)=(-a^{*},-T^{*},-b^{*}).}
  150. 𝔥 * \mathfrak{h}^{*}
  151. 𝔤 τ \mathfrak{g}_{\tau}
  152. 𝔤 τ = 𝔤 τ , + 1 𝔤 τ , 0 𝔤 τ , - 1 . \displaystyle{\mathfrak{g}_{\tau}=\mathfrak{g}_{\tau,+1}\oplus\mathfrak{g}_{% \tau,0}\oplus\mathfrak{g}_{\tau,-1}.}
  153. 𝔤 τ \mathfrak{g}_{\tau}
  154. ρ ( a , T , b ) = ( - a * , - T * , - b * ) . \displaystyle{\rho(a,T,b)=(-a^{*},-T^{*},-b^{*}).}
  155. 𝔥 τ * \mathfrak{h}^{*}_{\tau}
  156. g = ( a b c d ) g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  157. J = ( 0 I - I 0 ) J=\begin{pmatrix}0&I\\ -I&0\end{pmatrix}
  158. ω ( z 1 , z 2 ) = z J z t . \displaystyle{\omega(z_{1},z_{2})=zJz^{t}.}
  159. g = ( a b c d ) g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  160. g - 1 = ( d t - c t - b t a t ) . \displaystyle{g^{-1}=\begin{pmatrix}d^{t}&-c^{t}\\ -b^{t}&a^{t}\end{pmatrix}.}
  161. J = ( 0 I - I 0 ) J=\begin{pmatrix}0&I\\ -I&0\end{pmatrix}

Mycophenolic_acid_acyl-glucuronide_esterase.html

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Mycoredoxin.html

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Mycothiol_synthase.html

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Mylar_balloon_(geometry).html

  1. z ( r ) = 0 z(r)=0
  2. 0 r 1 + z ( x ) 2 d x = a \int_{0}^{r}\!\sqrt{1+z^{\prime}(x)^{2}}\,dx\,=a
  3. 0 r 4 π x z ( x ) d x \int_{0}^{r}\!4\pi xz(x)\,dx
  4. x ( u ) = r cos u ; z ( u ) = r 2 [ E ( u , 1 2 ) - 1 2 F ( u , 1 2 ) ] for u [ 0 , π 2 ] x(u)=r\cos u;\qquad z(u)=r\sqrt{2}\left[E(u,\frac{1}{\sqrt{2}})-\frac{1}{2}F(u% ,\frac{1}{\sqrt{2}})\right]\,\text{ for }u\in[0,\frac{\pi}{2}]\,
  5. 2 z ( π 2 ) 2z({\frac{\pi}{2}})
  6. V = 2 3 π a r 2 , V=\frac{2}{3}\pi ar^{2},
  7. V = 4 3 τ a 2 , V=\frac{4}{3}\tau a^{2},

N,N'-diacetylbacillosaminyl-diphospho-undecaprenol_alpha-1,3-N-acetylgalactosaminyltransferase.html

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N,N'-diacetylchitobiose_phosphorylase.html

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N,N'-diacetyllegionaminate_synthase.html

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N-acetyl-1-D-myo-inositol-2-amino-2-deoxy-alpha-D-glucopyranoside_deacetylase.html

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N-acetyl-beta-glucosaminyl-glycoprotein_4-beta-N-acetylgalactosaminyltransferase.html

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N-acetylgalactosamine-N,N'-diacetylbacillosaminyl-diphospho-undecaprenol_4-alpha-N-acetylgalactosaminyltransferase.html

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N-acetylgalactosamine_4-sulfate_6-O-sulfotransferase.html

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N-acetylglucosaminyl-diphospho-decaprenol_L-rhamnosyltransferase.html

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N-acetylglucosaminyldiphosphodolichol_N-acetylglucosaminyltransferase.html

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N-acetylglucosaminyldiphosphoundecaprenol_N-acetyl-beta-D-mannosaminyltransferase.html

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N-acetylhexosamine_1-kinase.html

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N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysyl-(N6-glycyl)-D-alanyl-D-alanine-diphosphoundecaprenyl-N-acetylglucosamine:glycine_glycyltransferase.html

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N-acetylmuramoyl-L-alanyl-D-glutamyl-L-lysyl-(N6-triglycine)-D-alanyl-D-alanine-diphosphoundecaprenyl-N-acetylglucosamine:glycine_glycyltransferase.html

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N-acetylneuraminylgalactosylglucosylceramide_b-1,4-N-acetylgalactosaminyltransferase.html

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N-acetylphosphatidylethanolamine-hydrolysing_phospholipase_D.html

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N-alkylglycine_oxidase.html

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N-formylmaleamate_deformylase.html

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N-succinylornithine_carbamoyltransferase.html

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N1-acetylpolyamine_oxidase.html

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N8-acetylspermidine_oxidase_(propane-1,3-diamine-forming).html

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N_=_4_supersymmetric_Yang–Mills_theory.html

  1. L = tr { - 1 2 g 2 F μ ν F μ ν + θ I 8 π 2 F μ ν F ¯ μ ν - i λ a σ μ D μ λ a - D μ X i D μ X i + g C i a b λ a [ X i , λ b ] + C i a b λ a [ X i , λ b ] + g 2 2 [ X i , X j ] 2 } L=\operatorname{tr}\left\{-\frac{1}{2g^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{\theta_{% I}}{8\pi^{2}}F_{\mu\nu}\bar{F}^{\mu\nu}-i\lambda^{a}\sigma^{\mu}D_{\mu}\lambda% _{a}-D_{\mu}X^{i}D^{\mu}X^{i}+gC^{ab}_{i}\lambda_{a}[X^{i},\lambda_{b}]+C_{iab% }\lambda^{a}[X^{i},\lambda^{b}]+\frac{g^{2}}{2}[X^{i},X^{j}]^{2}\right\}
  2. F μ ν k = μ A ν k - ν A μ k + g f k l m A μ l A ν m F^{k}_{\mu\nu}=\partial_{\mu}A^{k}_{\nu}-\partial_{\nu}A^{k}_{\mu}+gf^{klm}A^{% l}_{\mu}A^{m}_{\nu}
  3. L = tr { 1 g 2 F I J F I J - i λ ¯ Γ I D I λ } L=\operatorname{tr}\left\{\frac{1}{g^{2}}F_{IJ}F^{IJ}-i\bar{\lambda}\Gamma^{I}% D_{I}\lambda\right\}
  4. Γ I \Gamma^{I}
  5. ( 32 = 2 10 / 2 ) (32=2^{10/2})
  6. θ I \theta_{I}
  7. A i A_{i}
  8. θ I \theta_{I}
  9. τ = θ 2 π + 4 π i g 2 . \tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^{2}}.
  10. τ - 1 n G τ \tau\mapsto\frac{-1}{n_{G}\tau}
  11. N 2 - 2 g N^{2-2g}
  12. N N\rightarrow\infty

NADH:ubiquinone_reductase_(Na+-transporting).html

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NADH:ubiquinone_reductase_(non-electrogenic).html

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NADH_dehydrogenase.html

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NADP-retinol_dehydrogenase.html

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Nagao's_theorem.html

  1. B ( R ) = { ( a b 0 d ) : a , d R * , b R } . B(R)=\left\{{\left({\begin{array}[]{*{20}c}a&b\\ 0&d\end{array}}\right):a,d\in R^{*},~{}b\in R}\right\}.

Nakajima–Zwanzig_equation.html

  1. t ρ = i [ ρ , H ] = L ρ , \partial_{t}\rho=\frac{i}{\hbar}[\rho,H]=L\rho,
  2. L L
  3. L A = i [ A , H ] LA=\frac{i}{\hbar}[A,H]
  4. ρ \rho
  5. 𝒫 \mathcal{P}
  6. ρ = ( 𝒫 + 𝒬 ) ρ \rho=\left(\mathcal{P}+\mathcal{Q}\right)\rho
  7. 𝒬 1 - 𝒫 \mathcal{Q}\equiv 1-\mathcal{P}
  8. 𝒫 \mathcal{P}
  9. t ( 𝒫 𝒬 ) ρ = ( 𝒫 𝒬 ) L ( 𝒫 𝒬 ) ρ + ( 𝒫 𝒬 ) L ( 𝒬 𝒫 ) ρ . {\partial_{t}}\left(\begin{matrix}\mathcal{P}\\ \mathcal{Q}\\ \end{matrix}\right)\rho=\left(\begin{matrix}\mathcal{P}\\ \mathcal{Q}\\ \end{matrix}\right)L\left(\begin{matrix}\mathcal{P}\\ \mathcal{Q}\\ \end{matrix}\right)\rho+\left(\begin{matrix}\mathcal{P}\\ \mathcal{Q}\\ \end{matrix}\right)L\left(\begin{matrix}\mathcal{Q}\\ \mathcal{P}\\ \end{matrix}\right)\rho.
  10. 𝒬 ρ = e 𝒬 L t Q ρ ( t = 0 ) + 0 t d t e 𝒬 L t 𝒬 L 𝒫 ρ ( t - t ) . \mathcal{Q}\rho={{e}^{\mathcal{Q}Lt}}Q\rho(t=0)+\int_{0}^{t}dt^{\prime}{e}^{% \mathcal{Q}Lt^{\prime}}\mathcal{Q}L\mathcal{P}\rho(t-{t}^{\prime}).
  11. t 𝒫 ρ = 𝒫 L 𝒫 ρ + 𝒫 L e 𝒬 L t Q ρ ( t = 0 ) = 0 + 𝒫 L 0 t d t e 𝒬 L t 𝒬 L 𝒫 ρ ( t - t ) . \partial_{t}\mathcal{P}\rho=\mathcal{P}L\mathcal{P}\rho+\underbrace{\mathcal{P% }L{{e}^{\mathcal{Q}Lt}}Q\rho(t=0)}_{=0}+\mathcal{P}L\int_{0}^{t}{dt^{\prime}{{% e}^{\mathcal{Q}Lt^{\prime}}}\mathcal{Q}L\mathcal{P}\rho(t-{t}^{\prime})}.
  12. 𝒦 ( t ) 𝒫 L e 𝒬 L t 𝒬 L 𝒫 , \mathcal{K}\left(t\right)\equiv\mathcal{P}L{{e}^{\mathcal{Q}Lt}}\mathcal{Q}L% \mathcal{P},
  13. 𝒫 ρ ρ rel , \mathcal{P}\rho\equiv{{\rho}_{\mathrm{rel}}},
  14. 𝒫 2 = 𝒫 , \mathcal{P}^{2}=\mathcal{P},
  15. t ρ rel = 𝒫 L ρ rel + 0 t d t 𝒦 ( t ) ρ rel ( t - t ) . \partial_{t}{\rho}_{\mathrm{rel}}=\mathcal{P}L{{\rho}_{\mathrm{rel}}}+\int_{0}% ^{t}{dt^{\prime}\mathcal{K}({t}^{\prime}){{\rho}_{\mathrm{rel}}}(t-{t}^{\prime% })}.

Nanothermometry.html

  1. S r = d Q / d T Q . S_{r}=\frac{dQ/dT}{Q}.

Napoleon_points.html

  1. ( csc ( A + π 6 ) , csc ( B + π 6 ) , csc ( C + π 6 ) ) \displaystyle\left(\csc\left(A+\frac{\pi}{6}\right),\csc\left(B+\frac{\pi}{6}% \right),\csc\left(C+\frac{\pi}{6}\right)\right)
  2. ( a csc ( A + π 6 ) , b csc ( B + π 6 ) , c csc ( C + π 6 ) ) \left(a\csc\left(A+\frac{\pi}{6}\right),b\csc\left(B+\frac{\pi}{6}\right),c% \csc\left(C+\frac{\pi}{6}\right)\right)
  3. ( csc ( A - π 6 ) , csc ( B - π 6 ) , csc ( C - π 6 ) ) \displaystyle\left(\csc\left(A-\frac{\pi}{6}\right),\csc\left(B-\frac{\pi}{6}% \right),\csc\left(C-\frac{\pi}{6}\right)\right)
  4. ( a csc ( A - π 6 ) , b csc ( B - π 6 ) , c csc ( C - π 6 ) ) \left(a\csc\left(A-\frac{\pi}{6}\right),b\csc\left(B-\frac{\pi}{6}\right),c% \csc\left(C-\frac{\pi}{6}\right)\right)
  5. θ \theta
  6. X ( - sin θ , sin ( C + θ ) , sin ( B + θ ) ) X(-\sin\theta,\sin(C+\theta),\sin(B+\theta))
  7. Y ( sin ( C + θ ) , - sin θ , sin ( A + θ ) ) Y(\sin(C+\theta),-\sin\theta,\sin(A+\theta))
  8. Z ( sin ( B + θ ) , sin ( A + θ ) , - sin θ ) Z(\sin(B+\theta),\sin(A+\theta),-\sin\theta)
  9. ( csc ( A + θ ) , csc ( B + θ ) , csc ( C + θ ) ) (\csc(A+\theta),\csc(B+\theta),\csc(C+\theta))
  10. θ \theta
  11. N N
  12. θ \theta
  13. sin ( B - C ) x + sin ( C - A ) y + sin ( A - B ) z = 0. \frac{\sin(B-C)}{x}+\frac{\sin(C-A)}{y}+\frac{\sin(A-B)}{z}=0.

Naringenin_7-O-methyltransferase.html

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Neamine_transaminase.html

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Near-horizon_metric.html

  1. r r
  2. d s 2 = - ( 1 - M r ) 2 d t 2 + ( 1 - M r ) - 2 d r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) . ds^{2}\,=\,-\Big(1-\frac{M}{r}\Big)^{2}\,dt^{2}+\Big(1-\frac{M}{r}\Big)^{-2}dr% ^{2}+r^{2}\,\big(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\big)\,.
  3. t t ~ ϵ , r M + ϵ r ~ , ϵ 0 , t\mapsto\frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,% \quad\epsilon\to 0\,,
  4. d s 2 = - r 2 M 2 d t 2 + M 2 r 2 d r 2 + M 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=-\frac{r^{2}}{M^{2}}\,dt^{2}+\frac{M^{2}}{r^{2}}\,dr^{2}+M^{2}\,\big(d% \theta^{2}+\sin^{2}\theta\,d\phi^{2}\big)
  5. M = a = J / M M=a=J/M
  6. d s 2 = - ρ K 2 Δ K Σ 2 d t 2 + ρ K 2 Δ K d r 2 + ρ K 2 d θ 2 + Σ 2 sin 2 θ ρ K 2 ( d ϕ - ω K d t ) 2 , ds^{2}\,=\,-\frac{\rho_{K}^{2}\Delta_{K}}{\Sigma^{2}}\,dt^{2}+\frac{\rho_{K}^{% 2}}{\Delta_{K}}\,dr^{2}+\rho_{K}^{2}d\theta^{2}+\frac{\Sigma^{2}\sin^{2}\theta% }{\rho_{K}^{2}}\big(d\phi-\omega_{K}\,dt\big)^{2}\,,
  7. d s 2 = - Δ K ρ K 2 ( d t - M sin 2 θ d ϕ ) 2 + ρ K 2 Δ K d r 2 + ρ K 2 d θ 2 + sin 2 θ ρ K 2 ( M d t - ( r 2 + M 2 ) d ϕ ) 2 , ds^{2}\,=\,-\frac{\Delta_{K}}{\rho_{K}^{2}}\,\big(dt-M\sin^{2}\theta d\phi\big% )^{2}+\frac{\rho_{K}^{2}}{\Delta_{K}}\,dr^{2}+\rho_{K}^{2}d\theta^{2}+\frac{% \sin^{2}\theta}{\rho_{K}^{2}}\Big(Mdt-(r^{2}+M^{2})d\phi\Big)^{2}\,,
  8. ρ K 2 := r 2 + M 2 cos 2 θ , Δ K := ( r - M ) 2 , Σ 2 := ( r + M 2 ) 2 - M 2 Δ K sin 2 θ , ω K := 2 M 2 r Σ 2 . \rho_{K}^{2}:=r^{2}+M^{2}\cos^{2}\theta\,,\;\;\Delta_{K}:=\big(r-M\big)^{2}\,,% \;\;\Sigma^{2}:=\big(r+M^{2}\big)^{2}-M^{2}\Delta_{K}\sin^{2}\theta\,,\;\;% \omega_{K}:=\frac{2M^{2}r}{\Sigma^{2}}\,.
  9. t t ~ ϵ , r M + ϵ r ~ , ϕ ϕ ~ + 1 2 M ϵ t ~ , ϵ 0 , t\mapsto\frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,% \quad\phi\mapsto\tilde{\phi}+\frac{1}{2M\epsilon}\tilde{t}\,,\quad\epsilon\to 0\,,
  10. d s 2 1 + cos 2 θ 2 ( - r 2 2 M 2 d t 2 + 2 M 2 r 2 d r 2 + 2 M 2 d θ 2 ) + 4 M 2 sin 2 θ 1 + cos 2 θ ( d ϕ + r d t 2 M 2 ) 2 . ds^{2}\simeq\frac{1+\cos^{2}\theta}{2}\,\Big(-\frac{r^{2}}{2M^{2}}\,dt^{2}+% \frac{2M^{2}}{r^{2}}\,dr^{2}+2M^{2}d\theta^{2}\Big)+\frac{4M^{2}\sin^{2}\theta% }{1+\cos^{2}\theta}\,\Big(d\phi+\frac{rdt}{2M^{2}}\Big)^{2}\,.
  11. r + 2 = M 2 + Q 2 r_{+}^{2}=M^{2}+Q^{2}
  12. d s 2 = - ( 1 - 2 M r - Q 2 ρ K N ) d t 2 - 2 a sin 2 θ ( 2 M r - Q 2 ) ρ K N d t d ϕ + ρ K N ( d r 2 Δ K N + d θ 2 ) + Σ 2 ρ K N d ϕ 2 , ds^{2}=-\Big(1-\frac{2Mr-Q^{2}}{\rho_{KN}}\!\Big)dt^{2}-\frac{2a\sin^{2}\!% \theta\,(2Mr-Q^{2})}{\rho_{KN}}dtd\phi+\rho_{KN}\Big(\frac{dr^{2}}{\Delta_{KN}% }+d\theta^{2}\Big)+\frac{\Sigma^{2}}{\rho_{KN}}d\phi^{2},
  13. Δ K N := r 2 - 2 M r + a 2 + Q 2 , ρ K N := r 2 + a 2 cos 2 θ , Σ 2 := ( r 2 + a 2 ) 2 - Δ K N a 2 sin 2 θ . \Delta_{KN}\,:=\,r^{2}-2Mr+a^{2}+Q^{2}\,,\;\;\rho_{KN}\,:=\,r^{2}+a^{2}\cos^{2% }\!\theta\,,\;\;\Sigma^{2}\,:=\,(r^{2}+a^{2})^{2}-\Delta_{KN}a^{2}\sin^{2}% \theta\,.
  14. t t ~ ϵ , r M + ϵ r ~ , ϕ ϕ ~ + a r 0 2 ϵ t ~ , ϵ 0 , ( r 0 2 := M 2 + a 2 ) t\mapsto\frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,% \quad\phi\mapsto\tilde{\phi}+\frac{a}{r^{2}_{0}\epsilon}\tilde{t}\,,\quad% \epsilon\to 0\,,\quad\Big(r^{2}_{0}\,:=\,M^{2}+a^{2}\Big)
  15. d s 2 ( 1 - a 2 r 0 2 sin 2 θ ) ( - r 2 r 0 2 d t 2 + r 0 2 r 2 d r 2 + r 0 2 d θ 2 ) + r 0 2 sin 2 θ ( 1 - a 2 r 0 2 sin 2 θ ) - 1 ( d ϕ + 2 a r M r 0 4 d t ) - 1 . ds^{2}\simeq\Big(1-\frac{a^{2}}{r_{0}^{2}}\sin^{2}\!\theta\Big)\left(-\frac{r^% {2}}{r^{2}_{0}}dt^{2}+\frac{r^{2}_{0}}{r^{2}}dr^{2}+r^{2}_{0}d\theta^{2}\right% )+r^{2}_{0}\sin^{2}\!\theta\,\Big(1-\frac{a^{2}}{r_{0}^{2}}\sin^{2}\!\theta% \Big)^{-1}\left(d\phi+\frac{2arM}{r^{4}_{0}}dt\right)^{-1}\,.
  16. d s 2 = ( h ^ A B G A G B - F ) r 2 d v 2 + 2 d v d r - h ^ A B G B r d v d y A - h ^ A B G A r d v d y B + h ^ A B d y A d y B ds^{2}=(\hat{h}_{AB}G^{A}G^{B}-F)r^{2}dv^{2}+2dvdr-\hat{h}_{AB}G^{B}rdvdy^{A}-% \hat{h}_{AB}G^{A}rdvdy^{B}+\hat{h}_{AB}dy^{A}dy^{B}
  17. = - F r 2 d v 2 + 2 d v d r + h ^ A B ( d y A - G A r d v ) ( d y B - G B r d v ) , =-F\,r^{2}dv^{2}+2dvdr+\hat{h}_{AB}\big(dy^{A}-G^{A}\,rdv\big)\big(dy^{B}-G^{B% }\,rdv\big)\,,
  18. { F , G A } \{F,G^{A}\}
  19. h ^ A B \hat{h}_{AB}
  20. y A y^{A}
  21. r = 0 r=0

Nearest_centroid_classifier.html

  1. { ( x 1 , y 1 ) , , ( x n , y n ) } \textstyle\{(\vec{x}_{1},y_{1}),\dots,(\vec{x}_{n},y_{n})\}
  2. y i 𝐘 y_{i}\in\mathbf{Y}
  3. μ l = 1 | C l | i C l x i \textstyle\vec{\mu_{l}}=\frac{1}{|C_{l}|}\underset{i\in C_{l}}{\sum}\vec{x}_{i}
  4. C l C_{l}
  5. l 𝐘 l\in\mathbf{Y}
  6. x \vec{x}
  7. y ^ = arg min l 𝐘 μ l - x \hat{y}={\arg\min}_{l\in\mathbf{Y}}\|\vec{\mu}_{l}-\vec{x}\|

Nearest_neighbour_classifiers.html

  1. ( X , Y ) , ( X 1 , Y 1 ) , , ( X n , Y n ) (X,Y),(X_{1},Y_{1}),\dots,(X_{n},Y_{n})
  2. d × { 1 , 2 } \mathbb{R}^{d}\times\{1,2\}
  3. Y Y
  4. X X
  5. X | Y = r P r X|Y=r\sim P_{r}
  6. r = 1 , 2 r=1,2
  7. P r P_{r}
  8. \|\cdot\|
  9. d \mathbb{R}^{d}
  10. x d x\in\mathbb{R}^{d}
  11. ( X ( 1 ) , Y ( 1 ) ) , , ( X ( n ) , Y ( n ) ) (X_{(1)},Y_{(1)}),\dots,(X_{(n)},Y_{(n)})
  12. X ( 1 ) - x X ( n ) - x \|X_{(1)}-x\|\leq\dots\leq\|X_{(n)}-x\|
  13. 1 1
  14. x x
  15. C n 1 n n ( x ) = Y ( 1 ) C_{n}^{1nn}(x)=Y_{(1)}
  16. k k
  17. k k
  18. x x
  19. k k
  20. x x
  21. k k
  22. k k
  23. ( X , Y ) (X,Y)
  24. k := k n k:=k_{n}
  25. k n / n k_{n}/n
  26. n n\to\infty
  27. C n k n n C_{n}^{knn}
  28. k k
  29. n n
  30. ( C n k n n ) - ( C B a y e s ) = { B 1 1 k + B 2 ( k n ) 4 / d } { 1 + o ( 1 ) } , \mathcal{R}_{\mathcal{R}}(C^{knn}_{n})-\mathcal{R}_{\mathcal{R}}(C^{Bayes})=% \left\{B_{1}\frac{1}{k}+B_{2}\left(\frac{k}{n}\right)^{4/d}\right\}\{1+o(1)\},
  31. B 1 B_{1}
  32. B 2 B_{2}
  33. k * = B n 4 d + 4 k^{*}=\lfloor Bn^{\frac{4}{d+4}}\rfloor
  34. k * k^{*}
  35. 𝒪 ( n - 4 d + 4 ) \mathcal{O}(n^{-\frac{4}{d+4}})
  36. k k
  37. k k
  38. 1 / k 1/k
  39. 0
  40. i i
  41. w n i w_{ni}
  42. i = 1 n w n i = 1 \sum_{i=1}^{n}w_{ni}=1
  43. C n w n n C^{wnn}_{n}
  44. { w n i } i = 1 n \{w_{ni}\}_{i=1}^{n}
  45. ( C n w n n ) - ( C B a y e s ) = ( B 1 s n 2 + B 2 t n 2 ) { 1 + o ( 1 ) } , \mathcal{R}_{\mathcal{R}}(C^{wnn}_{n})-\mathcal{R}_{\mathcal{R}}(C^{Bayes})=% \left(B_{1}s_{n}^{2}+B_{2}t_{n}^{2}\right)\{1+o(1)\},
  46. B 1 B_{1}
  47. B 2 B_{2}
  48. s n 2 = i = 1 n w n i 2 s_{n}^{2}=\sum_{i=1}^{n}w_{ni}^{2}
  49. t n = n - 2 / d i = 1 n w n i { i 1 + 2 / d - ( i - 1 ) 1 + 2 / d } t_{n}=n^{-2/d}\sum_{i=1}^{n}w_{ni}\{i^{1+2/d}-(i-1)^{1+2/d}\}
  50. { w n i * } i = 1 n \{w_{ni}^{*}\}_{i=1}^{n}
  51. k * = B n 4 d + 4 k^{*}=\lfloor Bn^{\frac{4}{d+4}}\rfloor
  52. w n i * = 1 k * [ 1 + d 2 - d 2 k * 2 / d { i 1 + 2 / d - ( i - 1 ) 1 + 2 / d } ] w_{ni}^{*}=\frac{1}{k^{*}}\left[1+\frac{d}{2}-\frac{d}{2{k^{*}}^{2/d}}\{i^{1+2% /d}-(i-1)^{1+2/d}\}\right]
  53. i = 1 , 2 , , k * i=1,2,\dots,k^{*}
  54. w n i * = 0 w^{*}_{ni}=0
  55. i = k * + 1 , , n i=k^{*}+1,\dots,n
  56. 𝒪 ( n - 4 d + 4 ) \mathcal{O}(n^{-\frac{4}{d+4}})

Negativity_(quantum_mechanics).html

  1. A A
  2. ρ \rho
  3. 𝒩 ( ρ ) || ρ Γ A || 1 - 1 2 \mathcal{N}(\rho)\equiv\frac{||\rho^{\Gamma_{A}}||_{1}-1}{2}
  4. ρ Γ A \rho^{\Gamma_{A}}
  5. ρ \rho
  6. A A
  7. || X || 1 = Tr | X | = Tr X X ||X||_{1}=\,\text{Tr}|X|=\,\text{Tr}\sqrt{X^{\dagger}X}
  8. X X
  9. ρ Γ A \rho^{\Gamma_{A}}
  10. 𝒩 ( ρ ) = i | λ i | - λ i 2 \mathcal{N}(\rho)=\sum_{i}\frac{|\lambda_{i}|-\lambda_{i}}{2}
  11. λ i \lambda_{i}
  12. ρ \rho
  13. 𝒩 ( i p i ρ i ) i p i 𝒩 ( ρ i ) \mathcal{N}(\sum_{i}p_{i}\rho_{i})\leq\sum_{i}p_{i}\mathcal{N}(\rho_{i})
  14. 𝒩 ( P ( ρ i ) ) 𝒩 ( ρ i ) \mathcal{N}(P(\rho_{i}))\leq\mathcal{N}(\rho_{i})
  15. P ( ρ ) P(\rho)
  16. ρ \rho
  17. E N ( ρ ) log 2 || ρ Γ A || 1 E_{N}(\rho)\equiv\log_{2}||\rho^{\Gamma_{A}}||_{1}
  18. Γ A \Gamma_{A}
  19. | | | | 1 ||\cdot||_{1}
  20. E N ( ρ ) := log 2 ( 2 𝒩 + 1 ) E_{N}(\rho):=\log_{2}(2\mathcal{N}+1)
  21. E N ( ρ σ ) = E N ( ρ ) + E N ( σ ) E_{N}(\rho\otimes\sigma)=E_{N}(\rho)+E_{N}(\sigma)
  22. H 1 , H 2 , H_{1},H_{2},\ldots
  23. ρ 1 , ρ 2 , \rho_{1},\rho_{2},\ldots
  24. ρ n 1 , ρ n 2 , \rho^{\otimes n_{1}},\rho^{\otimes n_{2}},\ldots
  25. n i n_{i}
  26. E N ( ρ 1 ) / n 1 , E N ( ρ 2 ) / n 2 , E_{N}(\rho_{1})/n_{1},E_{N}(\rho_{2})/n_{2},\ldots
  27. E N ( ρ ) E_{N}(\rho)

Neofunctionalization.html

  1. ρ S - 1 1 - e s \frac{\rho\,\!S-1}{1-e^{s}}

Neomycin_C_transaminase.html

  1. \rightleftharpoons

Neopentalenolactone_D_synthase.html

  1. \rightleftharpoons

Neovius_surface.html

  1. 3 [ cos ( x ) + cos ( y ) + cos ( z ) ] + 4 cos ( x ) cos ( y ) cos ( z ) = 0 3[\cos(x)+\cos(y)+\cos(z)]+4\cos(x)\cos(y)\cos(z)=0

Networks_in_marketing.html

  1. N t + 1 = r N t [ ( K - N t ) / K ] N_{t+1}=rN_{t}[(K-N_{t})/K]
  2. r N t rN_{t}
  3. N t N_{t}
  4. K K
  5. r r
  6. r r
  7. r r

Neumann–Poincaré_operator.html

  1. Ω A d x + B d y = Ω ( B x - A y ) d x d y . \displaystyle{\int_{\partial\Omega}A\,dx+B\,dy=\iint_{\Omega}(B_{x}-A_{y})\,dx% \,dy.}
  2. Ω u n v = Ω u x v x + u y v y - u Δ v , \displaystyle{\int_{\partial\Omega}u\,\partial_{n}v=\iint_{\Omega}u_{x}v_{x}+u% _{y}v_{y}-u\,\Delta v,}
  3. Δ = - x 2 - y 2 . \displaystyle\Delta=-\partial^{2}_{x}-\partial_{y}^{2}.
  4. Ω u n v - n u v = Ω Δ u v - u Δ v . \displaystyle{\int_{\partial\Omega}u\,\partial_{n}v-\partial_{n}u\,v=\iint_{% \Omega}\,\Delta u\,v-u\,\Delta v.}
  5. Ω n u = 0 , \displaystyle{\int_{\partial\Omega}\partial_{n}u=0,}
  6. Ω f = 0. \displaystyle{\int_{\partial\Omega}f\,\,=\,\,0.}
  7. F ( z ) = a 0 + a 1 z - 1 + a 2 z - 2 + \displaystyle{F(z)=a_{0}+a_{1}z^{-1}+a_{2}z^{-2}+\cdots}
  8. | r F ( z ) | = | F ( z ) | C R - 2 . \displaystyle{|\partial_{r}F(z)|=|F^{\prime}(z)|\leq CR^{-2}.}
  9. | | z | = R n u | 2 π C R - 1 , \displaystyle{\left|\int_{|z|=R}\partial_{n}u\right|\leq 2\pi CR^{-1},}
  10. E ( z ) = - 1 2 π log | z | . \displaystyle{E(z)=-{1\over 2\pi}\log|z|.}
  11. E Δ φ = φ ( 0 ) . \displaystyle{\iint E\cdot\Delta\varphi=\varphi(0).}
  12. u ( z ) = Ω K ( z , w ) u ( w ) - N ( z - w ) n u ( w ) | d w | , \displaystyle{u(z)=\int_{\partial\Omega}K(z,w)u(w)-N(z-w)\partial_{n}u(w)\,|dw% |,}
  13. K ( z , w ) = n , w N ( z - w ) . \displaystyle{K(z,w)=\partial_{n,w}N(z-w).}
  14. u ( 0 ) = lim r 0 1 2 π 0 2 π u ( r e i θ ) - r log r r u ( r e i θ ) d θ , \displaystyle{u(0)=\lim_{r\rightarrow 0}{1\over 2\pi}\int_{0}^{2\pi}u(re^{i% \theta})-r\cdot\log r\cdot\partial_{r}u(re^{i\theta})\,d\theta,}
  15. u ( z ) = Ω - K ( z , w ) u ( w ) + N ( z - w ) n u ( w ) | d w | . \displaystyle{u(z)=\int_{\partial\Omega}-K(z,w)u(w)+N(z-w)\partial_{n}u(w)\,|% dw|.}
  16. 𝐯 ˙ 𝐯 ˙ = 1 , 𝐯 ¨ 𝐯 ˙ = 0. \displaystyle{\dot{\mathbf{v}}\cdot\dot{\mathbf{v}}=1,\,\,\,\ddot{\mathbf{v}}% \cdot\dot{\mathbf{v}}=0.}
  17. 𝐭 = 𝐯 ˙ , \displaystyle{\mathbf{t}=\dot{\mathbf{v}},}
  18. 𝐧 = ( - y ˙ , x ˙ ) . \displaystyle{\mathbf{n}=(-\dot{y},\dot{x}).}
  19. 𝐯 ¨ = κ ( t ) 𝐧 ( t ) . \displaystyle{\ddot{\mathbf{v}}=\kappa(t)\,\mathbf{n}(t).}
  20. κ = 𝐯 ¨ 𝐧 = y ¨ x ˙ - x ¨ y ˙ . \displaystyle{\kappa=\ddot{\mathbf{v}}\cdot\mathbf{n}=\ddot{y}\dot{x}-\ddot{x}% \dot{y}.}
  21. 𝐧 ˙ = - κ 𝐭 , 𝐧 ¨ = κ ˙ 𝐧 - κ 2 𝐭 . \displaystyle{\dot{\mathbf{n}}=-\kappa\mathbf{t},\,\,\,\ddot{\mathbf{n}}=\dot{% \kappa}\mathbf{n}-\kappa^{2}\mathbf{t}.}
  22. K ( 𝐮 , 𝐯 ( t ) ) = - 1 2 π ( 𝐮 - 𝐯 ( t ) ) 𝐧 ( t ) | 𝐮 - 𝐯 | 2 . \displaystyle{K(\mathbf{u},\mathbf{v}(t))=-{1\over 2\pi}{(\mathbf{u}-\mathbf{v% }(t))\cdot\mathbf{n}(t)\over|\mathbf{u}-\mathbf{v}|^{2}}.}
  23. k ( s , t ) = K ( v ( s ) , v ( t ) ) . \displaystyle{k(s,t)=K(v(s),v(t)).}
  24. a ( s , t ) = | v ( s ) - v ( t ) | 2 | e i s / L - e i t / L | 2 \displaystyle{a(s,t)={|v(s)-v(t)|^{2}\over|e^{is/L}-e^{it/L}|^{2}}}
  25. π \pi
  26. b ( s , t ) = ( 𝐯 ( s ) - 𝐯 ( t ) ) 𝐧 ( t ) | e i s / L - e i t / L | 2 \displaystyle{b(s,t)={(\mathbf{v}(s)-\mathbf{v}(t))\cdot\mathbf{n}(t)\over|e^{% is/L}-e^{it/L}|^{2}}}
  27. 𝐯 ( t + h ) - 𝐯 ( t ) = h 𝐭 ( t ) + h 2 2 κ ( t ) 𝐧 ( t ) - h 3 6 κ ( t ) 𝐭 ( t ) + h 4 24 ( κ ˙ 𝐧 - κ 2 𝐭 ) + , \displaystyle{\mathbf{v}(t+h)-\mathbf{v}(t)=h\mathbf{t}(t)+{h^{2}\over 2}% \kappa(t)\mathbf{n}(t)-{h^{3}\over 6}\kappa(t)\mathbf{t}(t)+{h^{4}\over 24}(% \dot{\kappa}\mathbf{n}-\kappa^{2}\mathbf{t})+\cdots,}
  28. h - 2 ( 𝐯 ( t + h ) - 𝐯 ( t ) ) 𝐧 ( t ) = κ ( t ) / 2 + h 2 κ ˙ ( t ) / 24 + \displaystyle{h^{-2}(\mathbf{v}(t+h)-\mathbf{v}(t))\cdot\mathbf{n}(t)=\kappa(t% )/2+h^{2}\dot{\kappa}(t)/24+\cdots}
  29. k ( t , t ) = - κ ( t ) 4 π . \displaystyle{k(t,t)=-{\kappa(t)\over 4\pi}.}
  30. k ( s , t ) = 1 2 π t arg ( z ( s ) - z ( t ) ) , \displaystyle{k(s,t)={1\over 2\pi}\partial_{t}\arg(z(s)-z(t)),}
  31. log z = log | z | + i arg z \displaystyle{\log z=\log|z|+i\arg z}
  32. K ( 𝐯 ( t ) + λ 𝐧 ( t ) , 𝐯 ( t ) ) = - 1 2 π λ , \displaystyle{K(\mathbf{v}(t)+\lambda\mathbf{n}(t),\mathbf{v}(t))=-{1\over 2% \pi\lambda},}
  33. D ( φ ) ( z ) = Ω K ( z , w ) φ ( w ) | d w | . \displaystyle{D(\varphi)(z)=\int_{\partial\Omega}K(z,w)\varphi(w)\,|dw|.}
  34. D ( φ ) ( z ) = 1 2 π i Ω φ ( w ) w - z d w . \displaystyle{D(\varphi)(z)=\Re{1\over 2\pi i}\int_{\partial\Omega}{\varphi(w)% \over w-z}\,dw.}
  35. Ω K ( z , w ) | d w | = 1 2 π s - π L s + π L t arg ( z ( s ) - z ( t ) ) d t = 1 / 2. \displaystyle{\int_{\partial\Omega}K(z,w)|dw|={1\over 2\pi}\int_{s-\pi L}^{s+% \pi L}\partial_{t}\arg(z(s)-z(t))\,dt=1/2.}
  36. Ω | K ( z , w ) | | d w | C \displaystyle{\int_{\partial\Omega}|K(z,w)|\,|dw|\leq C}
  37. 𝐮 = 𝐯 ( 0 ) + λ 𝐧 ( 0 ) , \displaystyle{\mathbf{u}=\mathbf{v}(0)+\lambda\mathbf{n}(0),}
  38. 𝐯 ( t ) - 𝐮 = - λ 𝐧 ( 0 ) + t 𝐭 ( 0 ) + t 2 2 κ ( 0 ) 𝐧 ( 0 ) + O ( t 3 ) . \displaystyle{\mathbf{v}(t)-\mathbf{u}=-\lambda\mathbf{n}(0)+t\mathbf{t}(0)+{t% ^{2}\over 2}\kappa(0)\mathbf{n}(0)+O(t^{3}).}
  39. | 𝐯 ( t ) - 𝐮 | 2 = λ 2 + t 2 ( 1 - λ κ ( 0 ) ) + O ( t 3 ) ( λ 2 + t 2 ) / 2 , | ( 𝐯 ( t ) - 𝐮 ) 𝐧 ( 0 ) | = | - λ + t 2 2 κ ( 0 ) + O ( t 3 ) | | λ | + C 1 t 2 \displaystyle{|\mathbf{v}(t)-\mathbf{u}|^{2}=\lambda^{2}+t^{2}(1-\lambda\kappa% (0))+O(t^{3})\geq(\lambda^{2}+t^{2})/2,\,\,\,\,\,\,|(\mathbf{v}(t)-\mathbf{u})% \cdot\mathbf{n}(0)|=|-\lambda+{t^{2}\over 2}\kappa(0)+O(t^{3})|\leq|\lambda|+C% _{1}t^{2}}
  40. 2 π | K ( 𝐮 , 𝐯 ( t ) ) | = | ( 𝐯 ( t ) - 𝐮 ) 𝐧 ( 0 ) | | 𝐯 ( t ) - 𝐮 | 2 2 | λ | + C 1 t 2 λ 2 + t 2 2 | λ | λ 2 + t 2 + C 1 . \displaystyle{2\pi\cdot|K(\mathbf{u},\mathbf{v}(t))|={|(\mathbf{v}(t)-\mathbf{% u})\cdot\mathbf{n}(0)|\over|\mathbf{v}(t)-\mathbf{u}|^{2}}\leq{2|\lambda|+C_{1% }t^{2}\over\lambda^{2}+t^{2}}\leq{2|\lambda|\over\lambda^{2}+t^{2}}+C_{1}.}
  41. - | λ | d t t 2 + λ 2 = - d t t 2 + 1 = π < . \displaystyle{\int_{-\infty}^{\infty}{|\lambda|\,dt\over t^{2}+\lambda^{2}}=% \int_{-\infty}^{\infty}{dt\over t^{2}+1}=\pi<\infty.}
  42. | D ( φ n ) ( z n ) - D ( φ ) ( z ) | | w - z | δ | K ( z n , w ) - K ( z , w ) | | φ ( w ) | | d w | + | w - z | δ ( | K ( z n , w ) | + | K ( z , w ) | ) | φ ( w ) | | d w | |D(\varphi_{n})(z_{n})-D(\varphi)(z)|\leq\int_{|w-z|\geq\delta}|K(z_{n},w)-K(z% ,w)||\varphi(w)|\,|dw|+\int_{|w-z|\leq\delta}(|K(z_{n},w)|+|K(z,w)|)\cdot|% \varphi(w)|\,|dw|
  43. + Ω | K ( z n , w ) | | φ n ( w ) - φ ( w ) | | d w | . +\int_{\partial\Omega}|K(z_{n},w)|\cdot|\varphi_{n}(w)-\varphi(w)|\,|dw|.
  44. u - ( z ) = 1 2 φ ( z ) + T K φ ( z ) , u + ( z ) = - 1 2 φ ( z ) + T K φ ( z ) . \displaystyle{u_{-}(z)={1\over 2}\varphi(z)+T_{K}\varphi(z),\,\,\,\,\,u_{+}(z)% =-{1\over 2}\varphi(z)+T_{K}\varphi(z).}
  45. φ = u - - u + . \displaystyle{\varphi=u_{-}-u_{+}.}
  46. u ( z n ) 1 2 φ ( z ) - T K φ ( z ) = Ω ( K ( z n , w ) - K ( z , w ) ) ( φ ( w ) - φ ( z ) ) | d w | = D ( ψ ) ( z n ) - D ( ψ ) ( z ) , \displaystyle{u(z_{n})\mp{1\over 2}\varphi(z)-T_{K}\varphi(z)=\int_{\partial% \Omega}(K(z_{n},w)-K(z,w))(\varphi(w)-\varphi(z))\,|dw|=D(\psi)(z_{n})-D(\psi)% (z),}
  47. S ( φ ) ( z ) = Ω N ( z - w ) φ ( w ) | d w | , \displaystyle{S(\varphi)(z)=\int_{\partial\Omega}N(z-w)\varphi(w)\,|dw|,}
  48. N ( x ) = 1 2 π log | z | . \displaystyle{N(x)={1\over 2\pi}\log|z|.}
  49. S ( φ ) ( z ) = 1 2 π ( log | z - w | - log | z | ) φ ( w ) | d w | + log | z | 2 π φ ( w ) | d w | , \displaystyle{S(\varphi)(z)={1\over 2\pi}\int_{\partial}(\log|z-w|-\log|z|)% \varphi(w)\,|dw|+{\log|z|\over 2\pi}\int\varphi(w)\,|dw|,}
  50. | S ( φ ) ( z ) - S ( φ ) ( z n ) | 1 2 π | w - z | ε | log | z - w | - log | z n - w | | | φ ( w ) | | d w | + φ | w - z | ε ( | log | z - w | | + | log | z n - w | | ) | d w | . |S(\varphi)(z)-S(\varphi)(z_{n})|\leq{1\over 2\pi}\int_{|w-z|\geq\varepsilon}|% \log|z-w|-\log|z_{n}-w||\,|\varphi(w)|\,|dw|+\|\varphi\|_{\infty}\int_{|w-z|% \leq\varepsilon}(|\log|z-w||+|\log|z_{n}-w||)\,|dw|.
  51. 2 | w - z | 2 ε | log | z - w | | | d w | , \displaystyle{2\int_{|w-z|\leq 2\varepsilon}|\log|z-w||\,\,|dw|,}
  52. S φ C φ , \displaystyle{\|S\varphi\|_{\infty}\leq C^{\prime}\|\varphi\|_{\infty},}
  53. n u ( z + a 𝐧 z ) = d d t a ( z + t 𝐧 z ) | t = a . \displaystyle{\partial_{n}u(z+a\mathbf{n}_{z})={d\over dt}a(z+t\mathbf{n}_{z})% |_{t=a}.}
  54. n S ( φ ) ( z ) = Ω K ( w , z ) φ ( w ) | d w | , \displaystyle{\partial_{n}S(\varphi)(z)=\int_{\partial\Omega}K(w,z)\varphi(w)% \,|dw|,}
  55. K * ( z , w ) = K ( w , z ) . \displaystyle{K^{*}(z,w)=K(w,z).}
  56. n - u ( z ) = - 1 2 φ ( z ) + T K * φ ( z ) , n + u ( z ) = 1 2 φ ( z ) + T K * φ ( z ) . \displaystyle{\partial_{n-}u(z)=-{1\over 2}\varphi(z)+T_{K}^{*}\varphi(z),\,\,% \,\,\,\partial_{n+}u(z)={1\over 2}\varphi(z)+T_{K}^{*}\varphi(z).}
  57. φ = n + u - n - u . \displaystyle{\varphi=\partial_{n+}u-\partial_{n-}u.}
  58. f = T K φ + T K * φ \displaystyle{f=T_{K}\varphi+T_{K}^{*}\varphi}
  59. f = v + n u . \displaystyle{f=v+\partial_{n}u.}
  60. f ( z n ) - f ( z ) = Ω ( K ( w , z n ) + K ( z n , w ) - K ( w , z ) - K ( z , w ) ) φ ( w ) | d w | = | w - z | δ + | w - z | δ . \displaystyle{f(z_{n})-f(z)=\int_{\partial\Omega}(K(w,z_{n})+K(z_{n},w)-K(w,z)% -K(z,w))\varphi(w)\,|dw|=\int_{|w-z|\geq\delta}+\int_{|w-z|\leq\delta}.}
  61. 2 π | K ( z n , w ) + K ( w , z n ) | = | ( z n - w ) ( 𝐧 ζ n - 𝐧 w ) | | z n - w | 2 | 𝐧 ζ n - 𝐧 w | | z n - w | = | 𝐧 ζ n - 𝐧 w | | ζ n - w | | ζ n - w | | z n - w | . \displaystyle{2\pi|K(z_{n},w)+K(w,z_{n})|={|(z_{n}-w)\cdot(\mathbf{n}_{\zeta_{% n}}-\mathbf{n}_{w})|\over|z_{n}-w|^{2}}\leq{|\mathbf{n}_{\zeta_{n}}-\mathbf{n}% _{w}|\over|z_{n}-w|}={|\mathbf{n}_{\zeta_{n}}-\mathbf{n}_{w}|\over|\zeta_{n}-w% |}\cdot{|\zeta_{n}-w|\over|z_{n}-w|}.}
  62. | z n - w | 2 ( | z n - ζ n | 2 + | ζ n - w | 2 ) / 2. \displaystyle{|z_{n}-w|^{2}\geq(|z_{n}-\zeta_{n}|^{2}+|\zeta_{n}-w|^{2})/2.}
  63. T K φ + T K * φ = v ± + n ± u = φ / 2 + T K φ + n ± u , \displaystyle{T_{K}\varphi+T^{*}_{K}\varphi=v_{\pm}+\partial_{n\pm}u=\mp% \varphi/2+T_{K}\varphi+\partial_{n\pm}u,}
  64. f = ( x f , y f ) . \displaystyle{\nabla f=(\partial_{x}f,\partial_{y}f).}
  65. ~ f = ( y f , - x f ) . \displaystyle{\widetilde{\nabla}f=(\partial_{y}f,-\partial_{x}f).}
  66. S ( φ ) = - D ( φ 𝐧 ) + S ( t ( φ 𝐭 ) ) , D ( φ ) = ~ S ( φ ˙ ) . \displaystyle{\nabla S(\varphi)=-D(\varphi\mathbf{n})+S(\partial_{t}(\varphi% \mathbf{t})),\,\,\,\nabla D(\varphi)=\widetilde{\nabla}S(\dot{\varphi}).}
  67. z N ( z - w ) = - w N ( z - w ) = - n , w N ( z - w ) 𝐧 w - t N ( z - w ) 𝐭 w , \displaystyle{\nabla_{z}N(z-w)=-\nabla_{w}N(z-w)=-\partial_{n,w}N(z-w)\mathbf{% n}_{w}-\partial_{t}N(z-w)\mathbf{t}_{w},}
  68. S ( φ ) = - D ( φ 𝐧 ) - Ω ( t N ( z - 𝐯 ( t ) ) ) φ ( t ) 𝐭 ( t ) d t = - D ( φ 𝐧 ) + S ( t ( φ 𝐭 ) ) . \displaystyle{\nabla S(\varphi)=-D(\varphi\mathbf{n})-\int_{\partial\Omega}(% \partial_{t}N(z-\mathbf{v}(t)))\varphi(t)\mathbf{t}(t)\,dt=-D(\varphi\mathbf{n% })+S(\partial_{t}(\varphi\mathbf{t})).}
  69. D ( φ ) = Ω Δ z N ( z - w ) 𝐧 φ - ~ Ω t N ( z - 𝐯 ( t ) ) φ d t = ~ S ( φ ˙ ) . \displaystyle{\nabla D(\varphi)=\int_{\partial\Omega}\Delta_{z}N(z-w)\mathbf{n% }\varphi-\widetilde{\nabla}\int_{\partial\Omega}\partial_{t}N(z-\mathbf{v}(t))% \,\varphi\,dt=\widetilde{\nabla}S(\dot{\varphi}).}
  70. D ( φ ) = D ( φ ˙ 𝐭 ) + S ( t ( φ ˙ 𝐧 ) ) , \displaystyle{\nabla D(\varphi)=D(\dot{\varphi}\mathbf{t})+S(\partial_{t}(\dot% {\varphi}\mathbf{n})),}
  71. n D ( φ ) ( 𝐯 ( s ) + λ 𝐧 ( s ) ) = D ( φ ˙ 𝐭 𝐧 ( s ) ) + S ( t ( φ ˙ 𝐧 ) 𝐧 ( s ) ) . \displaystyle{\partial_{n}D(\varphi)(\mathbf{v}(s)+\lambda\mathbf{n}(s))=D(% \dot{\varphi}\mathbf{t}\cdot\mathbf{n}(s))+S(\partial_{t}(\dot{\varphi}\mathbf% {n})\cdot\mathbf{n}(s)).}
  72. T K φ - 1 2 φ = 1 \displaystyle{T_{K}\varphi-{1\over 2}\varphi=1}
  73. T K * φ + 1 2 φ = f , \displaystyle{T_{K}^{*}\varphi+{1\over 2}\varphi=f,}
  74. φ = f , \displaystyle{\int\varphi=\int f,}
  75. f = ( f , 1 ) = ( ( T K * + 1 2 ) φ , 1 ) = ( φ , ( T K + 1 2 ) 1 ) = ( φ , 1 ) = φ . \int f=(f,1)=((T_{K}^{*}+{1\over 2})\varphi,1)=(\varphi,(T_{K}+{1\over 2})1)=(% \varphi,1)=\int\varphi.
  76. T K * φ + 1 2 φ = 0 , \displaystyle{T_{K}^{*}\varphi+{1\over 2}\varphi=0,}
  77. H ( φ ) = - n D ( φ ) | Ω = - t ( S ( t φ ) | Ω ) . \displaystyle{H(\varphi)=-\partial_{n}D(\varphi)|_{\partial\Omega}=-\partial_{% t}(S(\partial_{t}\varphi)|_{\partial\Omega}).}
  78. C = ( 1 2 I + T K S H 1 2 I - T K * ) . \displaystyle{C=\begin{pmatrix}{1\over 2}I+T_{K}&S\\ H&{1\over 2}I-T_{K}^{*}\end{pmatrix}.}
  79. S T * = T S , S H = 1 4 I - T 2 , H S = 1 4 I - ( T * ) 2 , H T = T * H . \displaystyle{ST^{*}=TS,\,\,\,SH={1\over 4}I-T^{2},\,\,\,HS={1\over 4}I-(T^{*}% )^{2},\,\,\,HT=T^{*}H.}
  80. u = D ( S φ ) - S ( T * φ ) + S ( φ ) / 2. \displaystyle{u=D(S\varphi)-S(T^{*}\varphi)+S(\varphi)/2.}
  81. n ± u = ( λ 1 2 ) φ , \displaystyle{\partial_{n\pm}u=(\lambda\mp{1\over 2})\varphi,}
  82. ( λ + 1 2 ) Ω n + u u ¯ = ( λ - 1 2 ) Ω n - u u ¯ . \displaystyle{(\lambda+{1\over 2})\int_{\partial\Omega}\partial_{n+}u\,% \overline{u}=(\lambda-{1\over 2})\int_{\partial\Omega}\partial_{n-}u\,% \overline{u}.}
  83. ( λ + 1 2 ) Ω | u x | 2 + | u y | 2 = ( λ - 1 2 ) Ω c | u x | 2 + | u y | 2 . \displaystyle{(\lambda+{1\over 2})\iint_{\Omega}|u_{x}|^{2}+|u_{y}|^{2}=(% \lambda-{1\over 2})\iint_{\Omega^{c}}|u_{x}|^{2}+|u_{y}|^{2}.}
  84. n + u λ + 1 2 = n - u λ - 1 2 = φ . \displaystyle{{\partial_{n+}u\over\lambda+{1\over 2}}={\partial_{n_{-}}u\over% \lambda-{1\over 2}}=\varphi.}
  85. u | Ω = D ( ψ ) - ( λ - 1 2 ) S ( φ ) , u | Ω c = - D ( ψ ) + ( λ + 1 2 ) S ( φ ) . \displaystyle{u|_{\Omega}=D(\psi)-(\lambda-{1\over 2})S(\varphi),\,\,\,u|_{% \Omega^{c}}=-D(\psi)+(\lambda+{1\over 2})S(\varphi).}
  86. ψ = T ψ + 1 2 ψ - ( λ - 1 2 ) S φ = - ( T ψ - 1 2 ψ ) + ( λ + 1 2 ) S φ \displaystyle{\psi=T\psi+{1\over 2}\psi-(\lambda-{1\over 2})S\varphi=-(T\psi-{% 1\over 2}\psi)+(\lambda+{1\over 2})S\varphi}
  87. ( λ - 1 2 ) φ = n D ( ψ ) | Ω - ( λ - 1 2 ) ( T * φ - 1 2 φ ) , ( λ + 1 2 ) φ = - n D ( ψ ) | Ω + ( λ + 1 2 ) ( T * φ + 1 2 φ ) . \displaystyle{(\lambda-{1\over 2})\varphi=\partial_{n}D(\psi)|_{\partial\Omega% }-(\lambda-{1\over 2})(T^{*}\varphi-{1\over 2}\varphi),\,\,\,\,(\lambda+{1% \over 2})\varphi=-\partial_{n}D(\psi)|_{\partial\Omega}+(\lambda+{1\over 2})(T% ^{*}\varphi+{1\over 2}\varphi).}
  88. T ψ = λ ψ , T * φ = λ φ , S φ = ψ , n D ( ψ ) | Ω = ( λ 2 - 1 4 ) φ , \displaystyle{T\psi=\lambda\psi,\,\,\,T^{*}\varphi=\lambda\varphi,\,\,\,S% \varphi=\psi,\,\,\,\partial_{n}D(\psi)|_{\partial\Omega}=(\lambda^{2}-{1\over 4% })\varphi,}
  89. u x = - v y , u y = v x . \displaystyle{u_{x}=-v_{y},\,\,u_{y}=v_{x}.}
  90. v ( z ) = a z - u y d x + u x d y , \displaystyle{v(z)=\int_{a}^{z}-u_{y}dx+u_{x}dy,}
  91. u + = u - , n + u λ - 1 2 = n - u λ + 1 2 = φ , \displaystyle{u_{+}=u_{-},\,\,\,{\partial_{n+}u\over\lambda-{1\over 2}}={% \partial_{n_{-}}u\over\lambda+{1\over 2}}=\varphi,}
  92. n + v = n - v , v + = λ + 1 2 λ - 1 2 v - . \displaystyle{\partial_{n+}v=\partial_{n_{-}}v,\,\,\,\,v_{+}={\lambda+{1\over 2% }\over\lambda-{1\over 2}}\cdot v_{-}.}
  93. U - = v - , U + = λ + 1 2 λ - 1 2 v + . \displaystyle{U_{-}=v_{-},\,\,\,\,U_{+}={\lambda+{1\over 2}\over\lambda-{1% \over 2}}\cdot v_{+}.}
  94. n + U = λ - 1 2 λ + 1 2 n - U . \displaystyle{\partial_{n+}U={\lambda-{1\over 2}\over\lambda+{1\over 2}}\cdot% \partial_{n_{-}}U.}
  95. Ω S ( φ 1 ) x S ( φ 2 ) x + S ( φ 1 ) y S ( φ 2 ) y = Ω S ( φ 1 ) n - S ( φ 2 ) , Ω c S ( φ 1 ) x S ( φ 2 ) x + S ( φ 1 ) y S ( φ 2 ) y = Ω S ( φ 1 ) n - S ( φ 2 ) . \displaystyle{\iint_{\Omega}S(\varphi_{1})_{x}S(\varphi_{2})_{x}+S(\varphi_{1}% )_{y}S(\varphi_{2})_{y}=\int_{\partial\Omega}S(\varphi_{1})\partial_{n-}S(% \varphi_{2}),\,\,\,\iint_{\Omega^{c}}S(\varphi_{1})_{x}S(\varphi_{2})_{x}+S(% \varphi_{1})_{y}S(\varphi_{2})_{y}=\int_{\partial\Omega}S(\varphi_{1})\partial% _{n-}S(\varphi_{2}).}
  96. S ( φ 1 ) x S ( φ 2 ) x + S ( φ 1 ) y S ( φ 2 ) y = Ω S ( φ 1 ) φ 2 . \displaystyle{\iint S(\varphi_{1})_{x}S(\varphi_{2})_{x}+S(\varphi_{1})_{y}S(% \varphi_{2})_{y}=\int_{\partial\Omega}S(\varphi_{1})\varphi_{2}.}
  97. ( S φ , φ ) = S ( ϕ ) 2 , ( S φ 1 , φ 2 ) = S ( φ 1 ) S ( φ 2 ) ¯ . \displaystyle{(S\varphi,\varphi)=\int\|\nabla S(\phi)\|^{2},\,\,\,(S\varphi_{1% },\varphi_{2})=\int\nabla S(\varphi_{1})\cdot\overline{\nabla S(\varphi_{2})}.}
  98. ( f , g ) S = ( S f , g ) . \displaystyle{(f,g)_{S}=(Sf,g).}
  99. Δ ( z ) = det ( I - z T 2 ) . \displaystyle{\Delta(z)=\det(I-zT^{2}).}
  100. Δ ( z ) = ( 1 - z / 4 ) n 1 ( 1 - z λ n 2 ) . \displaystyle{\Delta(z)=(1-z/4)\cdot\prod_{n\geq 1}(1-z\lambda_{n}^{2}).}
  101. T c f ( w ) = lim ε 0 1 π i | z - w | ε f ( z ) ¯ ( z - w ) 2 d x d y . \displaystyle{T_{c}f(w)=\lim_{\varepsilon\rightarrow 0}{1\over\pi i}\iint_{|z-% w|\geq\varepsilon}{\overline{f(z)}\over(z-w)^{2}}\,dx\,dy.}
  102. z ¯ \overline{z}
  103. K F ( z , w ) = F ( z ) F ( w ) ( F ( z ) - F ( w ) ) 2 . K_{F}(z,w)={F^{\prime}(z)F^{\prime}(w)\over(F(z)-F(w))^{2}}.
  104. z ¯ \overline{z}
  105. F ( z ) F ( w ) ( F ( z ) - F ( w ) ) 2 - 1 ( z - w ) 2 \displaystyle{{F^{\prime}(z)F^{\prime}(w)\over(F(z)-F(w))^{2}}\,-\,{1\over(z-w% )^{2}}}
  106. ( T u , v ) = ( T v , u ) \displaystyle{(Tu,v)=(Tv,u)}
  107. ( A u , u ) = ( T u , T u ) = T u 2 0 , \displaystyle{(Au,u)=(Tu,Tu)=\|Tu\|^{2}\geq 0,}
  108. A u n = μ n u n , \displaystyle{Au_{n}=\mu_{n}u_{n},}
  109. μ n = λ n 2 \displaystyle{\mu_{n}=\lambda_{n}^{2}}
  110. T u n = λ n u n . \displaystyle{Tu_{n}=\lambda_{n}u_{n}.}
  111. T ( i u n ) = - λ n i u n \displaystyle{T(iu_{n})=-\lambda_{n}iu_{n}}
  112. T f ( w ) = 1 2 π Ω n ( log | z - w | ) f ( z ) = 1 2 ( H f ) ( w ) , \displaystyle{Tf(w)={1\over 2\pi}\int_{\partial\Omega}\partial_{n}(\log|z-w|)f% (z)={1\over 2}\Re(Hf)(w),}
  113. 2 T h = ( H f ) + i ( H g ) = 1 2 ( H f + J H f + i H g + i J H g ) = 1 2 ( H + J H J ) h \displaystyle{2Th=\Re(Hf)+i\Re(Hg)={1\over 2}(Hf+JHf+iHg+iJHg)={1\over 2}(H+% JHJ)h}
  114. T = 1 4 ( H + J H J ) , \displaystyle{T={1\over 4}(H+JHJ),}
  115. A = 1 2 ( H + J H J ) , B = 1 2 i ( H - J H J ) , \displaystyle{A={1\over 2}(H+JHJ),\,\,\,\,B={1\over 2i}(H-JHJ),}
  116. H = A + i B . \displaystyle{H=A+iB.}
  117. A B = - B A , A 2 - B 2 = I . \displaystyle{AB=-BA,\,\,\,\,A^{2}-B^{2}=I.}
  118. A 1 2 - I = B 1 2 \displaystyle{A_{1}^{2}-I=B_{1}^{2}}
  119. = ( Ω ¯ ) ( Ω c ¯ ) , \displaystyle{\mathfrak{H}=\mathfrak{H}(\overline{\Omega})\oplus\mathfrak{H}(% \overline{\Omega^{c}}),}
  120. f - f + 2 = Ω | f - | 2 + Ω c | f + | 2 . \displaystyle{\|f_{-}\oplus f_{+}\|_{\mathfrak{H}}^{2}=\iint_{\Omega}|\nabla f% _{-}|^{2}+\iint_{\Omega^{c}}|\nabla f_{+}|^{2}.}
  121. f - = D ( φ ) | Ω + S ( ψ ) | Ω , f + = D ( φ ) | Ω c + S ( ψ ) | Ω c . \displaystyle{f_{-}=D(\varphi)|_{\Omega}+S(\psi)|_{\Omega},\,\,\,\,\,f_{+}=D(% \varphi)|_{\Omega^{c}}+S(\psi)|_{\Omega^{c}}.}
  122. φ = f - | Ω - f + | Ω , ψ = n f - | Ω - n f + | Ω . \displaystyle{\varphi=f_{-}|_{\partial\Omega}-f_{+}|_{\partial\Omega},\,\,\,\,% \psi=\partial_{n}f_{-}|_{\partial\Omega}-\partial_{n}f_{+}|_{\partial\Omega}.}
  123. D ( φ ) = D ( φ ) | Ω D ( φ ) | Ω c , S ( ψ ) = S ( ψ ) | Ω S ( ψ ) Ω c . \displaystyle{D(\varphi)=D(\varphi)|_{\Omega}\oplus D(\varphi)|_{\Omega^{c}},% \,\,\,\,S(\psi)=S(\psi)|_{\Omega}\oplus S(\psi)_{\Omega^{c}}.}
  124. ( S , D ) = Ω Ω c S D = - Ω S n D + Ω S n D = 0. \displaystyle{(S,D)=\iint_{\Omega\cup\Omega^{c}}\nabla S\cdot\nabla D=-\int_{% \partial\Omega}S\partial_{n}D+\int_{\partial\Omega}S\partial_{n}D=0.}
  125. U ( f - f + ) = ( z f - ) χ Ω + ( z f - ) χ Ω c . \displaystyle{U(f_{-}\oplus f_{+})=(\partial_{z}f_{-})\chi_{\Omega}+(\partial_% {z}f_{-})\chi_{\Omega^{c}}.}
  126. T c ( U ( D ( φ ) + S ( ψ ) ) ) = U ( D ( φ ) - S ( ψ ) ) . \displaystyle{T_{c}(U(D(\varphi)+S(\psi)))=U(D(\varphi)-S(\psi)).}
  127. Ω Ω c , | z - w | > ε N ( w - z ) S ( ψ ) ( z ) d x d y = - | z - w | = ε n N ( z - w ) S ( ψ ) ( z ) = - S ( ψ ) ( z ) , \displaystyle{\iint_{\Omega\cup\Omega^{c},\,\,|z-w|>\varepsilon}\nabla N(w-z)% \cdot\nabla S(\psi)(z)\,dx\,dy=-\int_{|z-w|=\varepsilon}\partial_{n}N(z-w)\,S(% \psi)(z)=-S(\psi)(z),}
  128. π ¯ z \overline{π}{z}
  129. 1 π Ω Ω c z S ( ψ ) ( z ) ¯ z - w d x d y = - S ( ψ ) ( w ) . \displaystyle{{1\over\pi}\,\iint_{\Omega\cup\Omega^{c}}{\overline{\partial_{z}% S(\psi)(z)}\over z-w}\,dx\,dy=-S(\psi)(w).}
  130. T c ( z S ( ψ ) ) = - z S ( ψ ) . \displaystyle{T_{c}(\partial_{z}S(\psi))=-\partial_{z}S(\psi).}
  131. Ω Ω c , | z - w | > ε N ( w - z ) D ( φ ) ( z ) d x d y = | z - w | = ε N ( z - w ) n D ( φ ) ( z ) = 0 , \displaystyle{\iint_{\Omega\cup\Omega^{c},\,\,|z-w|>\varepsilon}\nabla N(w-z)% \cdot\nabla D(\varphi)(z)\,dx\,dy=\int_{|z-w|=\varepsilon}N(z-w)\,\partial_{n}% D(\varphi)(z)=0,}
  132. π ¯ z \overline{π}{z}
  133. 1 π Ω Ω c z D ( φ ) ( z ) ¯ z - w d x d y = D ( φ ) ( w ) . \displaystyle{{1\over\pi}\,\iint_{\Omega\cup\Omega^{c}}{\overline{\partial_{z}% D(\varphi)(z)}\over z-w}\,dx\,dy=D(\varphi)(w).}
  134. T c ( z D ( φ ) ) = z D ( φ ) . \displaystyle{T_{c}(\partial_{z}D(\varphi))=\partial_{z}D(\varphi).}
  135. T K , 0 = P 0 T K P 0 . \displaystyle{T_{K,0}=P_{0}T_{K}P_{0}.}
  136. ( f , g ) 0 = ( ( 1 / 2 - T K ) - 1 S f , g ) \displaystyle{(f,g)_{0}=((1/2-T_{K})^{-1}Sf,g)}
  137. V ( ψ ) = U ( D ( φ ) + S ( ψ ) ) | Ω , \displaystyle{V(\psi)=U(D(\varphi)+S(\psi))|_{\Omega},}
  138. ( 1 2 I - T K ) φ = - S ψ . \displaystyle{({1\over 2}I-T_{K})\varphi=-S\psi.}
  139. V T K , 0 V * = T Ω . \displaystyle{VT_{K,0}V^{*}=T_{\Omega}.}
  140. Φ - = z D ( φ ) | Ω A 2 ( Ω ) , Φ + = z D ( φ ) | Ω c A 2 ( Ω c ) . \displaystyle{\Phi_{-}=\partial_{z}D(\varphi)|_{\Omega}\in A^{2}(\Omega),\,\,% \,\Phi_{+}=\partial_{z}D(\varphi)|_{\Omega^{c}}\in A^{2}(\Omega^{c}).}
  141. Φ ± ( w ) = 1 2 π i Ω φ ( z ) ( z - w ) 2 d z . \displaystyle{\Phi_{\pm}(w)={1\over 2\pi i}\int_{\partial\Omega}{\varphi(z)% \over(z-w)^{2}}\,dz.}
  142. T Ω Φ - = λ Φ - , T Ω c Φ + = λ Φ + . \displaystyle{T_{\Omega}\Phi_{-}=\lambda\Phi_{-},\,\,\,T_{\Omega^{c}}\Phi_{+}=% \lambda\Phi_{+}.}
  143. ( T c Φ - ) | Ω c = ( λ + 1 2 ) Φ + , ( T c Φ + ) | Ω = ( λ - 1 2 ) Φ - . \displaystyle{(T_{c}\Phi_{-})|_{\Omega^{c}}=(\lambda+{1\over 2})\Phi_{+},\,\,% \,(T_{c}\Phi_{+})|_{\Omega}=(\lambda-{1\over 2})\Phi_{-}.}
  144. F ( w ) = 1 2 π i Ω f ( z ) z - w d z \displaystyle{F(w)={1\over 2\pi i}\int_{\partial\Omega}{f(z)\over z-w}\,dz}
  145. C f ( w ) = 1 2 π i Ω f ( z ) ¯ z - w d z ¯ . \displaystyle{Cf(w)={1\over 2\pi i}\int_{\partial\Omega}{\overline{f(z)}\over z% -w}\,d\overline{z}.}
  146. C f ( w ) = T Ω F ( w ) , \displaystyle{Cf(w)=T_{\Omega}F(w),}
  147. 1 π Ω , | z - w | > ε F ( z ) ¯ ( z - w ) 2 d x d y = C f ( w ) - 1 2 π i | z - w | = ε F ( z ) ¯ z - w d z ¯ = C f ( w ) . \displaystyle{{1\over\pi}\iint_{\Omega,\,\,\,|z-w|>\varepsilon}{\overline{F(z)% }\over(z-w)^{2}}\,dx\,dy=Cf(w)-{1\over 2\pi i}\int_{|z-w|=\varepsilon}{% \overline{F(z)}\over z-w}\,d\overline{z}=Cf(w).}

Neutral_Density.html

  1. γ n \gamma^{n}\,
  2. γ n \gamma^{n}\,
  3. β S - α θ \beta\nabla S-\alpha\nabla\theta
  4. θ \theta\,
  5. α \alpha\,
  6. β \beta\,
  7. ρ ( β S - α θ ) \rho(\beta\nabla S-\alpha\nabla\theta)
  8. γ n \gamma^{n}\,
  9. γ n = b ρ ( β S - α θ ) ; \nabla\gamma^{n}\ =b\rho(\beta\nabla S-\alpha\nabla\theta);
  10. γ n \gamma^{n}\,
  11. γ n \gamma^{n}\,
  12. γ n \gamma^{n}\,
  13. γ n \gamma^{n}\,
  14. γ n \gamma^{n}\,
  15. γ n \gamma^{n}\,
  16. γ n \gamma^{n}\,
  17. γ n \gamma^{n}\,
  18. γ n \gamma^{n}\,
  19. γ n \gamma^{n}\,
  20. γ n \gamma^{n}\,
  21. γ n \gamma^{n}\,
  22. γ n \gamma^{n}\,
  23. γ n \gamma^{n}\,
  24. γ n \gamma^{n}\,
  25. γ n \gamma^{n}\,
  26. γ n \gamma^{n}\,

Nevanlinna_invariant.html

  1. K X + r D K_{X}+rD

Néel_effect.html

  1. M ( H ) = χ 0 H + N e H 3 + ε ( H 3 ) M(H)=\chi_{0}H+N_{e}H^{3}+\varepsilon(H^{3})
  2. χ 0 \chi_{0}
  3. N e N_{e}
  4. N N
  5. S S
  6. I e x c I_{exc}
  7. H e x t H_{ext}
  8. e e
  9. e = - d ϕ / d t = - S d B / d t e=-d\phi/dt=-SdB/dt
  10. B B
  11. B = μ 0 μ r ( H + M ) B=\mu_{0}\mu_{r}(H+M)
  12. M = 0 M=0
  13. B = μ 0 μ r ( H e x t + H e x c ) B=\mu_{0}\mu_{r}(H_{ext}+H_{exc})
  14. i e x c i_{exc}
  15. H e x t H_{ext}
  16. B = μ 0 μ r ( ( 1 + χ 0 ) ( H e x t + H e x c ) + N e ( H e x t + H e x c ) 3 ) B=\mu_{0}\mu_{r}((1+\chi_{0})(H_{ext}+H_{exc})+N_{e}(H_{ext}+H_{exc})^{3})
  17. μ 0 μ r ( 1 + χ 0 ) ( H e x t + H e x c ) \mu_{0}\mu_{r}(1+\chi_{0})(H_{ext}+H_{exc})
  18. i e x c i_{exc}
  19. H e x t H_{ext}
  20. ( H e x t + H e x c ) 3 = H e x t 3 + 3 H e x t 2 H e x c + 3 H e x t H e x c 2 + H e x c 3 (H_{ext}+H_{exc})^{3}=H_{ext}^{3}+3H_{ext}^{2}H_{exc}+3H_{ext}H_{exc}^{2}+H_{% exc}^{3}
  21. H ( l ) H(l)
  22. L p L_{p}
  23. t t
  24. u ( t ) = L d I ( t ) d t + F R o g d d t [ 0 H L p ( l ) d l ] + F N e e l [ 0 H L p ( l ) d l ] I ( t ) d I ( t ) d t u(t)=L\frac{dI(t)}{dt}+F_{Rog}\frac{d}{dt}\left[\int_{0}^{H}Lp(l)dl\right]+F_{% Neel}\left[\int_{0}^{H}Lp(l)dl\right]I(t)\frac{dI(t)}{dt}
  25. u ( t ) = L I e x w e x cos ( w e x t ) + F R o g d d t [ 0 L p H ( l ) d l ] + F N e e l [ 0 L p H ( l ) d l ] I e x 2 2 w e x sin ( 2 w e x t ) u(t)=LI_{ex}w_{ex}\cos(w_{ex}t)+F_{Rog}\frac{d}{dt}\left[\int_{0}^{Lp}H(l)dl% \right]+F_{Neel}\left[\int_{0}^{Lp}H(l)dl\right]\frac{I_{ex}^{2}}{2}w_{ex}\sin% (2w_{ex}t)
  26. I ( t ) = I e x sin ( w e x t ) I(t)=I_{ex}\sin(w_{ex}t)
  27. i e x c ( t ) i_{exc}(t)
  28. H e x t ( t ) H_{ext}(t)
  29. f e x c f_{exc}
  30. H e x t ( t ) H_{ext}(t)
  31. f e m ( t ) = F N e e l i e x c 2 ( t ) H ( t ) fem(t)=F_{Neel}i_{exc}^{2}(t)H(t)
  32. I c r = I p / N c r I_{cr}=I_{p}/N_{cr}

Néron_differential.html

  1. y 2 + a 1 x y + a 3 = x 3 + a 2 x 2 2 + a 4 x + a 6 y^{2}+a_{1}xy+a_{3}=x^{3}+a_{2}x^{2}_{2}+a_{4}x+a_{6}
  2. d x 2 y + a 1 x + a 3 \frac{dx}{2y+a_{1}x+a_{3}}

NGC_4845.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

NGC_6166.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Nicotinate_dehydrogenase_(cytochrome).html

  1. \rightleftharpoons

Nicotinate_riboside_kinase.html

  1. \rightleftharpoons

Nicotine_blue_oxidoreductase.html

  1. \rightleftharpoons

Nielsen–Ninomiya_theorem.html

  1. S [ ψ ] S[\psi]
  2. ψ \psi

Nigerose_phosphorylase.html

  1. \rightleftharpoons

Nitrate_reductase_(quinone).html

  1. \rightleftharpoons

Nitric-oxide_synthase_(NAD(P)H-dependent).html

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

Nitric_oxide_reductase_(cytochrome_c).html

  1. \rightleftharpoons

Nitric_oxide_reductase_(menaquinol).html

  1. \rightleftharpoons

Nitric_oxide_reductase_(NAD(P),_nitrous_oxide-forming).html

  1. \rightleftharpoons

Nitrilotriacetate_monooxygenase.html

  1. \rightleftharpoons

Nitrite_dismutase.html

  1. \rightleftharpoons

Nitronate_monooxygenase.html

  1. \rightleftharpoons

Nodal_period.html

  1. T n T_{n}
  2. T n = 2 π a 3 2 μ 1 2 [ 1 - 3 J 2 ( 4 - 5 sin 2 i ) 4 ( a R ) 2 ( 1 - ϵ 2 ( 1 + ϵ cos ω ) 2 - 3 J 2 ( 1 - ϵ cos ω ) 3 2 ( a R ) 2 ( 1 - ϵ 2 ) 3 ] T_{n}=\frac{2\pi a^{\frac{3}{2}}}{\mu^{\frac{1}{2}}}\left[1-\frac{3J_{2}(4-5% \sin^{2}i)}{4(\frac{a}{R})^{2}\sqrt{(1-\epsilon^{2}}(1+\epsilon\cos\omega)^{2}% }-\frac{3J_{2}(1-\epsilon\cos\omega)^{3}}{2(\frac{a}{R})^{2}(1-\epsilon^{2})^{% 3}}\right]

Nodary.html

  1. y 2 + 2 a y 1 + y 2 = b 2 y^{2}+\frac{2ay}{\sqrt{1+y^{\prime 2}}}=b^{2}
  2. x ( u ) = a sn ( u , k ) + ( a / k ) ( ( 1 - k 2 ) u - E ( u , k ) ) x(u)=a\operatorname{sn}(u,k)+(a/k)\big((1-k^{2})u-E(u,k)\big)
  3. y ( u ) = - a cn ( u , k ) + ( a / k ) dn ( u , k ) y(u)=-a\operatorname{cn}(u,k)+(a/k)\operatorname{dn}(u,k)
  4. k = cos ( tan - 1 ( b / a ) ) k=\cos(\tan^{-1}(b/a))
  5. E ( u , k ) E(u,k)

Non-autonomous_system_(mathematics).html

  1. Q Q\to\mathbb{R}
  2. \mathbb{R}
  3. Q Q\to\mathbb{R}
  4. J r Q J^{r}Q
  5. Q Q\to\mathbb{R}
  6. Q Q\to\mathbb{R}
  7. Q Q\to\mathbb{R}
  8. Γ \Gamma
  9. Q Q\to\mathbb{R}
  10. ( t , q i ) (t,q^{i})
  11. Q Q
  12. ( t , q i , q t i ) (t,q^{i},q^{i}_{t})
  13. J 1 Q J^{1}Q
  14. q t i = Γ ( t , q i ) . q^{i}_{t}=\Gamma(t,q^{i}).
  15. q t t i = ξ i ( t , q j , q t j ) q^{i}_{tt}=\xi^{i}(t,q^{j},q^{j}_{t})
  16. Q Q\to\mathbb{R}
  17. ξ \xi
  18. J 1 Q J^{1}Q\to\mathbb{R}
  19. J 1 Q Q J^{1}Q\to Q
  20. J 1 Q T Q J^{1}Q\to TQ
  21. T Q TQ
  22. Q Q

Non-dimensionalization_and_scaling_of_the_Navier–Stokes_equations.html

  1. 𝐮 t + 𝐮 𝐮 = - 1 ρ p + ν 2 𝐮 + 𝐠 . \frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u}=-\frac{1% }{\rho}\nabla p+\nu\nabla^{2}\mathbf{u}+\mathbf{g}.
  2. 𝐫 * = 𝐫 L \mathbf{r}^{*}\ =\frac{\mathbf{r}}{L}
  3. 𝐮 * = 𝐮 U \mathbf{u}^{*}\ =\frac{\mathbf{u}}{U}\,
  4. t * = t L / U t^{*}\ =\frac{t}{L/U}\,
  5. p * = p ρ U 2 p^{*}=\frac{p}{\rho U^{2}}
  6. p * = p L μ U p^{*}=\frac{pL}{\mu U}
  7. 𝐮 * t = - p * + 1 R e 2 𝐮 * . \frac{\partial\mathbf{u^{*}}}{\partial t}=-\nabla p^{*}+\frac{1}{Re}\nabla^{2}% \mathbf{u^{*}}.
  8. 𝐮 * t - 2 𝐮 * = 0. \frac{\partial\mathbf{u^{*}}}{\partial t}-\nabla^{2}\mathbf{u^{*}}=\mathbf{0}.
  9. 𝐮 * t + ( 𝐮 * ) 𝐮 * = - p * . \frac{\partial\mathbf{u^{*}}}{\partial t}+(\mathbf{u^{*}}\cdot\nabla)\mathbf{u% ^{*}}\ =-\nabla p^{*}.
  10. W t + U W X + W W Z = - 1 ρ o p d Z + v ( 2 W X 2 + 2 W Z 2 ) - g ( β s S - β T T ) \frac{\partial W}{\partial t}+U\frac{\partial W}{\partial X}+W\frac{\partial W% }{\partial Z}\ =-\frac{1}{\rho_{o}}\frac{\partial p_{d}}{\partial Z}+v\left(% \frac{\partial^{2}W}{\partial X^{2}}+\frac{\partial^{2}W}{\partial Z^{2}}% \right)\ -g\left(\beta_{s}\nabla{S}-\beta_{T}\nabla{T}\right)
  11. S * = S - S B S T - S B S^{*}=\frac{S-S_{B}}{S_{T}-S_{B}}
  12. T * = T - T B T T - T B T^{*}=\frac{T-T_{B}}{T_{T}-T_{B}}
  13. W * t * + U * W * X * + W * W * Z * = - p d Z * + P r ( 2 W * X * 2 + 2 W * Z * 2 ) - R a s P r s S + R a T P r T T \frac{\partial W^{*}}{\partial t^{*}}+U^{*}\frac{\partial W^{*}}{\partial X^{*% }}+W^{*}\frac{\partial W^{*}}{\partial Z^{*}}\ =-\frac{\partial p_{d}}{% \partial Z^{*}}+Pr\left(\frac{\partial^{2}W^{*}}{\partial X^{*2}}+\frac{% \partial^{2}W^{*}}{\partial Z^{*2}}\right)\ -{Ra_{s}Pr_{s}S}+{Ra_{T}Pr_{T}T}

Non-expanding_horizon.html

  1. S 2 × S^{2}\times\mathbb{R}
  2. l l
  3. θ ( l ) := h ^ a b ^ a l b \displaystyle\theta_{(l)}:=\hat{h}^{ab}\hat{\nabla}_{a}l_{b}
  4. T a b T_{ab}
  5. V a := - T b a l b V^{a}:=-T^{a}_{b}l^{b}
  6. V a V a 0 V^{a}V_{a}\leq 0
  7. l a l^{a}
  8. g = 0 g=0
  9. θ ( l ) = 0 \theta_{(l)}=0
  10. = ^ \hat{=}
  11. h ^ a b \hat{h}^{ab}
  12. ^ \hat{\nabla}
  13. : = n a a :=n^{a}\nabla_{a}
  14. { ( - , + , + , + ) ; l a n a = - 1 , m a m ¯ a = 1 } \{(-,+,+,+);l^{a}n_{a}=-1,m^{a}\bar{m}_{a}=1\}
  15. { ( + , - , - , - ) ; l a n a = 1 , m a m ¯ a = - 1 } \{(+,-,-,-);l^{a}n_{a}=1,m^{a}\bar{m}_{a}=-1\}
  16. l a l^{a}
  17. κ := - m a l b b l a = ^ 0 \kappa:=-m^{a}l^{b}\nabla_{b}l_{a}\,\hat{=}\,0
  18. Im ( ρ ) = Im ( - m a m ¯ b b l a ) = ^ 0 \,\text{Im}(\rho)=\,\text{Im}(-m^{a}\bar{m}^{b}\nabla_{b}l_{a})\,\hat{=}\,0
  19. θ ( l ) \theta_{(l)}
  20. l a l^{a}
  21. θ ( l ) = ^ 0 \theta_{(l)}\,\hat{=}\,0
  22. Re ( ρ ) = Re ( - m a m ¯ b b l a ) = - 1 2 θ ( l ) = ^ 0 \,\text{Re}(\rho)=\,\text{Re}(-m^{a}\bar{m}^{b}\nabla_{b}l_{a})=-\frac{1}{2}% \theta_{(l)}\,\hat{=}\,0
  23. ( 1 ) D ρ = ρ 2 + σ σ ¯ + 1 2 R a b l a l b = ^ 0 , (1)\qquad D\rho=\rho^{2}+\sigma\bar{\sigma}+\frac{1}{2}R_{ab}l^{a}l^{b}\,\hat{% =}\,0\,,
  24. ( 2 ) σ σ ¯ + 1 2 R a b l a l b = ^ 0 , (2)\qquad\sigma\bar{\sigma}+\frac{1}{2}R_{ab}l^{a}l^{b}\,\hat{=}\,0\,,
  25. σ := - m b m a a l b \sigma:=-m^{b}m^{a}\nabla_{a}l_{b}
  26. R a b l a l b = R a b l a l b - 1 2 R g a b l a l b = 8 π T a b l a l b R_{ab}l^{a}l^{b}=R_{ab}l^{a}l^{b}-\frac{1}{2}Rg_{ab}l^{a}l^{b}=8\pi\,T_{ab}l^{% a}l^{b}
  27. c = G = 1 c=G=1
  28. R a b l a l b R_{ab}l^{a}l^{b}
  29. σ σ ¯ \sigma\bar{\sigma}
  30. σ σ ¯ \sigma\bar{\sigma}
  31. R a b l a l b R_{ab}l^{a}l^{b}
  32. σ = ^ 0 \sigma\,\hat{=}\,0
  33. R a b l a l b = ^ 0 R_{ab}l^{a}l^{b}\,\hat{=}\,0
  34. ( 3 ) κ = ^ 0 , Im ( ρ ) = ^ 0 , Re ( ρ ) = ^ 0 , σ = ^ 0 , R a b l a l b = ^ 0. (3)\qquad\kappa\,\hat{=}\,0\,,\quad\,\text{Im}(\rho)\,\hat{=}\,0\,,\quad\,% \text{Re}(\rho)\,\hat{=}\,0\,,\quad\sigma\,\hat{=}\,0\,,\quad R_{ab}l^{a}l^{b}% \,\hat{=}\,0.
  35. R a b l a l b = 8 π T a b l a l b = 8 π T b a l b l a = ^ 0 R_{ab}l^{a}l^{b}=8\pi\cdot T_{ab}l^{a}l^{b}=8\pi\cdot T^{a}_{b}l^{b}\cdot l_{a% }\,\hat{=}\,0
  36. - T b a l b -T^{a}_{b}l^{b}
  37. l a l^{a}
  38. R a b l b R_{ab}l^{b}
  39. l a l_{a}
  40. - T b a l b = ^ c l a -T^{a}_{b}l^{b}\,\hat{=}\,cl^{a}
  41. R a b l b = ^ c l a R_{ab}l^{b}\,\hat{=}\,cl_{a}
  42. c c\in\mathbb{R}
  43. Φ 00 := 1 2 R a b l a l b = ^ c 2 l b l b = ^ 0 \Phi_{00}:=\frac{1}{2}R_{ab}l^{a}l^{b}\,\hat{=}\,\frac{c}{2}\,l_{b}l^{b}\,\hat% {=}\,0
  44. Φ 01 = Φ 10 ¯ := 1 2 R a b l a m b = ^ c 2 l b m b = ^ 0 \Phi_{01}=\overline{\Phi_{10}}:=\frac{1}{2}R_{ab}l^{a}m^{b}\,\hat{=}\,\frac{c}% {2}\,l_{b}m^{b}\,\hat{=}\,0
  45. ( 4 ) R a b l b = ^ c l a , Φ 00 = ^ 0 , Φ 10 = Φ 01 ¯ = ^ 0 . (4)\qquad R_{ab}l^{b}\,\hat{=}\,cl_{a}\,,\quad\Phi_{00}\,\hat{=}\,0\,,\quad% \Phi_{10}=\overline{\Phi_{01}}\,\hat{=}\,0\,.
  46. { Φ 00 , Φ 01 , Φ 10 } \{\Phi_{00}\,,\Phi_{01}\,,\Phi_{10}\}
  47. Ψ i ( i = 0 , 1 , 3 , 4 ) \Psi_{i}\;(i=0,1,3,4)
  48. Ψ 2 \Psi_{2}
  49. Φ i j \Phi_{ij}
  50. ( 5 ) D σ = σ ( ρ + ρ ¯ ) + Ψ 0 = - 2 σ θ ( l ) + Ψ 0 , (5)\qquad D\sigma=\sigma(\rho+\bar{\rho})+\Psi_{0}=-2\sigma\theta_{(l)}+\Psi_{% 0}\,,
  51. ( 6 ) D σ - δ κ = ( ρ + ρ ¯ ) σ + ( 3 ε - ε ¯ ) σ - ( τ - π ¯ + α ¯ + 3 β ) κ + Ψ 0 = ^ 0 , (6)\qquad D\sigma-\delta\kappa=(\rho+\bar{\rho})\sigma+(3\varepsilon-\bar{% \varepsilon})\sigma-(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa+\Psi_{0}\,\hat{% =}\,0\,,
  52. Ψ 0 := C a b c d l a m b l c m d = ^ 0 \Psi_{0}:=C_{abcd}l^{a}m^{b}l^{c}m^{d}\,\hat{=}\,0
  53. ( 7 ) δ ρ - δ ¯ σ = ρ ( α ¯ + β ) - σ ( 3 α - β ¯ ) + ( ρ - ρ ¯ ) τ + ( μ - μ ¯ ) κ - Ψ 1 + Φ 01 = ^ 0 (7)\qquad\delta\rho-\bar{\delta}\sigma=\rho(\bar{\alpha}+\beta)-\sigma(3\alpha% -\bar{\beta})+(\rho-\bar{\rho})\tau+(\mu-\bar{\mu})\kappa-\Psi_{1}+\Phi_{01}\,% \hat{=}\,0
  54. Ψ 1 := C a b c d l a n b l c m d = ^ 0 \Psi_{1}:=C_{abcd}l^{a}n^{b}l^{c}m^{d}\,\hat{=}\,0
  55. ( 8 ) Ψ 0 = ^ 0 , Ψ 1 = ^ 0 , (8)\qquad\Psi_{0}\,\hat{=}\,0\,,\quad\Psi_{1}\,\hat{=}\,0\,,
  56. l a l^{a}
  57. Ψ 0 \Psi_{0}
  58. Ψ 1 \Psi_{1}
  59. ( 9 ) θ ( l ) = - 1 2 θ ( l ) 2 + κ ~ ( l ) θ ( l ) - σ a b σ a b + ω ~ a b ω ~ a b - R a b l a l b , (9)\qquad\mathcal{L}_{\ell}\theta_{(l)}=-\frac{1}{2}\theta_{(l)}^{2}+\tilde{% \kappa}_{(l)}\theta_{(l)}-\sigma_{ab}\sigma^{ab}+\tilde{\omega}_{ab}\tilde{% \omega}^{ab}-R_{ab}l^{a}l^{b}\,,
  60. κ ~ ( l ) \tilde{\kappa}_{(l)}
  61. κ ~ ( l ) l b := l a a l b \tilde{\kappa}_{(l)}l^{b}:=l^{a}\nabla_{a}l^{b}
  62. ( 10 ) θ ( l ) = - ( ρ + ρ ¯ ) = - 2 Re ( ρ ) , θ ( n ) = μ + μ ¯ = 2 Re ( μ ) , (10)\qquad\theta_{(l)}=-(\rho+\bar{\rho})=-2\,\text{Re}(\rho)\,,\quad\theta_{(% n)}=\mu+\bar{\mu}=2\,\text{Re}(\mu)\,,
  63. ( 11 ) σ a b = - σ m ¯ a m ¯ b - σ ¯ m a m b , (11)\qquad\sigma_{ab}=-\sigma\bar{m}_{a}\bar{m}_{b}-\bar{\sigma}m_{a}m_{b}\,,
  64. ( 12 ) ω ~ a b = 1 2 ( ρ - ρ ¯ ) ( m a m ¯ b - m ¯ a m b ) = Im ( ρ ) ( m a m ¯ b - m ¯ a m b ) , (12)\qquad\tilde{\omega}_{ab}=\frac{1}{2}\,\Big(\rho-\bar{\rho}\Big)\,\Big(m_{% a}\bar{m}_{b}-\bar{m}_{a}m_{b}\Big)=\,\text{Im}(\rho)\cdot\Big(m_{a}\bar{m}_{b% }-\bar{m}_{a}m_{b}\Big)\,,
  65. h ^ a b = h ^ b a = m b m ¯ a + m ¯ b m a \hat{h}^{ab}=\hat{h}^{ba}=m^{b}\bar{m}^{a}+\bar{m}^{b}m^{a}
  66. ( 13 ) θ ( l ) = h ^ b a a l b = m b m ¯ a a l b + m ¯ b m a a l b = m b δ ¯ l b + m ¯ b δ l b = - ( ρ + ρ ¯ ) , (13)\qquad\theta_{(l)}=\hat{h}^{ba}\nabla_{a}l_{b}=m^{b}\bar{m}^{a}\nabla_{a}l% _{b}+\bar{m}^{b}m^{a}\nabla_{a}l_{b}=m^{b}\bar{\delta}l_{b}+\bar{m}^{b}\delta l% _{b}=-(\rho+\bar{\rho})\,,
  67. ( 14 ) θ ( n ) = h ^ b a a n b = m ¯ b m a a n b + m b m ¯ a a n b = m ¯ b δ n b + m b δ ¯ n b = μ + μ ¯ . (14)\qquad\theta_{(n)}=\hat{h}^{ba}\nabla_{a}n_{b}=\bar{m}^{b}m^{a}\nabla_{a}n% _{b}+m^{b}\bar{m}^{a}\nabla_{a}n_{b}=\bar{m}^{b}\delta n_{b}+m^{b}\bar{\delta}% n_{b}=\mu+\bar{\mu}\,.
  68. Im ( ρ ) = 0 \,\text{Im}(\rho)=0
  69. { Φ i j = ^ 0 , Λ = ^ 0 } \{\Phi_{ij}\hat{=}0\,,\Lambda\hat{=}0\}
  70. Λ = ^ 0 \Lambda\hat{=}0
  71. ( 15 ) T a b = 1 4 π ( F a c F b c - 1 4 g a b F c d F c d ) , (15)\qquad T_{ab}=\frac{1}{4\pi}\,\Big(\,F_{ac}F_{b}^{c}-\frac{1}{4}g_{ab}F_{% cd}F^{cd}\Big)\,,
  72. F a b F_{ab}
  73. F a b = - F b a F_{ab}=-F_{ba}
  74. F a a = 0 F^{a}_{a}=0
  75. T a b T_{ab}
  76. T a a = 0 T^{a}_{a}=0
  77. F a b F_{ab}
  78. ϕ i \phi_{i}
  79. Φ i j \Phi_{ij}
  80. ( 16 ) Φ i j = 2 ϕ i ϕ j ¯ , i , j { 0 , 1 , 2 } , (16)\qquad\Phi_{ij}=\,2\,\phi_{i}\,\overline{\phi_{j}}\,,\quad i,j\in\{0,1,2\}\,,
  81. ϕ i \phi_{i}
  82. Φ 00 = 0 \Phi_{00}=0
  83. ( 17 ) D ρ - δ ¯ κ = ( ρ 2 + σ σ ¯ ) + ( ε + ε ¯ ) ρ - κ ¯ τ - ( 3 α + β ¯ - π ) κ + Φ 00 = ^ 0 (17)\qquad D\rho-\bar{\delta}\kappa=(\rho^{2}+\sigma\bar{\sigma})+(\varepsilon% +\bar{\varepsilon})\rho-\bar{\kappa}\tau-(3\alpha+\bar{\beta}-\pi)\,\kappa+% \Phi_{00}\,\hat{=}\,0\,
  84. κ = ^ ρ = ^ σ = 0 \kappa\,\hat{=}\,\rho\,\hat{=}\,\sigma=0
  85. ( 18 ) Φ 00 = ^ 0 2 ϕ 0 ϕ 0 ¯ = ^ 0 ϕ 0 = ϕ 0 ¯ = ^ 0 . (18)\qquad\Phi_{00}\,\hat{=}\,0\;\;\Leftrightarrow\;\;2\,\phi_{0}\,\overline{% \phi_{0}}\,\hat{=}\,0\;\;\Rightarrow\;\;\phi_{0}=\overline{\phi_{0}}\,\hat{=}% \,0\,.
  86. ( 19 ) Φ 01 = Φ 10 ¯ = 2 ϕ 0 ϕ 1 ¯ = ^ 0 , Φ 02 = Φ 20 ¯ = 2 ϕ 0 ϕ 2 ¯ = ^ 0 . (19)\qquad\Phi_{01}=\overline{\Phi_{10}}=\,2\,\phi_{0}\,\overline{\phi_{1}}\,% \hat{=}\,0\,,\quad\Phi_{02}=\overline{\Phi_{20}}=\,2\,\phi_{0}\,\overline{\phi% _{2}}\,\hat{=}\,0\,.
  87. Φ 00 \Phi_{00}
  88. Φ 01 \Phi_{01}
  89. Φ i j = 2 ϕ i ϕ j ¯ \Phi_{ij}=2\,\phi_{i}\,\overline{\phi_{j}}
  90. Φ i j = Tr ( ϝ i ϝ ¯ j ) \Phi_{ij}=\,\,\text{Tr}\,\big(\,\digamma_{i}\,\bar{\digamma}_{j}\,\big)
  91. ϝ i ( i { 0 , 1 , 2 } \digamma_{i}(i\in\{0,1,2\}
  92. v v
  93. v v
  94. S v 2 S^{2}_{v}
  95. v = constant v=\,\text{constant}
  96. v v
  97. n a n_{a}
  98. n a = - d v n_{a}=-dv
  99. l a l^{a}
  100. S v 2 S^{2}_{v}
  101. l a n a = - 1 l^{a}n_{a}=-1
  102. D v = 1 Dv=1
  103. { l a , n a } \{l^{a},n^{a}\}
  104. { l a , n a } \{l^{a}\,,n^{a}\}
  105. { m a , m ¯ a } \{m^{a},\bar{m}^{a}\}
  106. S v 2 S^{2}_{v}
  107. { l a , n a } \{l^{a}\,,n^{a}\}
  108. m = ^ m ¯ = ^ 0 \mathcal{L}_{\ell}m\,\hat{=}\,\mathcal{L}_{\ell}\bar{m}\hat{=}0
  109. ( 20 ) m = [ , m ] = ^ 0 δ D - D δ = ( α ¯ + β - π ¯ ) D + κ Δ - ( ρ ¯ + ε - ε ¯ ) δ - σ δ ¯ = ^ 0 , (20)\qquad\mathcal{L}_{\ell}m=[\ell,m]\,\hat{=}\,0\;\Rightarrow\;\delta D-D% \delta=(\bar{\alpha}+\beta-\bar{\pi})D+\kappa\Delta-(\bar{\rho}+\varepsilon-% \bar{\varepsilon})\delta-\sigma\bar{\delta}\,\hat{=}\,0\,,
  110. κ = ^ ρ = ^ σ = ^ 0 \kappa\,\hat{=}\,\rho\,\hat{=}\,\sigma\,\hat{=}\,0
  111. ( 21 ) π = ^ α + β ¯ , ε = ^ ε ¯ . (21)\qquad\pi\,\hat{=}\,\alpha+\bar{\beta}\,,\quad\varepsilon\,\hat{=}\,\bar{% \varepsilon}\,.
  112. m ¯ m \mathcal{L}_{\bar{m}}m
  113. ( 22 ) m ¯ m = [ m ¯ , m ] = δ ¯ δ - δ δ ¯ = ( μ ¯ - μ ) D + ( ρ ¯ - ρ ) Δ - ( β ¯ - α ) δ - ( α ¯ - β ) δ ¯ , (22)\qquad\mathcal{L}_{\bar{m}}m=[\bar{m},m]=\bar{\delta}\delta-\delta\bar{% \delta}=(\bar{\mu}-\mu)D+(\bar{\rho}-\rho)\Delta-(\bar{\beta}-\alpha)\delta-(% \bar{\alpha}-\beta)\bar{\delta}\,,
  114. D D
  115. ( 23 ) μ ¯ = ^ μ , m ¯ m = ^ ( α - β ¯ ) δ - ( α ¯ - β ) δ ¯ , (23)\qquad\bar{\mu}\,\hat{=}\,\mu\,,\quad\mathcal{L}_{\bar{m}}m\,\hat{=}\,(% \alpha-\bar{\beta})\delta-(\bar{\alpha}-\beta)\bar{\delta}\,,
  116. n a n^{a}
  117. Im ( μ ) = Im ( m ¯ a m b b n a ) = 0 \,\text{Im}(\mu)=\,\text{Im}(\bar{m}^{a}m^{b}\nabla_{b}n_{a})=0
  118. 2 μ = 2 Re ( μ ) 2\mu=2\,\text{Re}(\mu)
  119. θ ( n ) \theta_{(n)}
  120. [ ] [\ell]
  121. 𝒟 \mathcal{D}

Non-extensive_self-consistent_thermodynamical_theory.html

  1. ln [ 1 + Z q ( V o , T ) ] = V o 2 π 2 n = 1 1 n 0 d m 0 d p p 2 ρ ( n ; m ) [ 1 + ( q - 1 ) β p 2 + m 2 ] - n q ( q - 1 ) . \ln[1+Z_{q}(V_{o},T)]=\frac{V_{o}}{2\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n}\int% _{0}^{\infty}dm\int_{0}^{\infty}dp\,p^{2}\rho(n;m)[1+(q-1)\beta\sqrt{p^{2}+m^{% 2}}]^{-\frac{nq}{(q-1)}}\,.
  2. q > 1 q>1
  3. Z q ( V o , T ) = 0 σ ( E ) [ 1 + ( q - 1 ) β E ] - q ( q - 1 ) d E , Z_{q}(V_{o},T)=\int_{0}^{\infty}\sigma(E)[1+(q-1)\beta E]^{-\frac{q}{(q-1)}}dE\,,
  4. σ ( E ) \sigma(E)
  5. l o g [ ρ ( m ) ] = l o g [ σ ( E ) ] log[\rho(m)]=log[\sigma(E)]
  6. m , E m,E
  7. m 3 / 2 ρ ( m ) = γ m [ 1 + ( q o - 1 ) β o m ] 1 q o - 1 = γ m [ 1 + ( q o - 1 ) m ] β o q o - 1 m^{3/2}\rho(m)=\frac{\gamma}{m}\big[1+(q_{o}-1)\beta_{o}m\big]^{\frac{1}{q_{o}% -1}}=\frac{\gamma}{m}[1+(q^{\prime}_{o}-1)m]^{\frac{\beta_{o}}{q^{\prime}_{o}-% 1}}
  8. σ ( E ) = b E a [ 1 + ( q o - 1 ) E ] β o q o - 1 , \sigma(E)=bE^{a}\big[1+(q^{\prime}_{o}-1)E\big]^{\frac{\beta_{o}}{q^{\prime}_{% o}-1}}\,,
  9. γ \gamma
  10. q o - 1 = β o ( q o - 1 ) q^{\prime}_{o}-1=\beta_{o}(q_{o}-1)
  11. a , b , γ a,b,\gamma
  12. q 1 q^{\prime}\rightarrow 1
  13. Z q ( V o , T ) ( 1 β - β o ) α Z_{q}(V_{o},T)\rightarrow\bigg(\frac{1}{\beta-\beta_{o}}\bigg)^{\alpha}
  14. α = γ V o 2 π 2 β 3 / 2 , \alpha=\frac{\gamma V_{o}}{2\pi^{2}\beta^{3/2}}\,,
  15. a + 1 = α = γ V o 2 π 2 β 3 / 2 . a+1=\alpha=\frac{\gamma V_{o}}{2\pi^{2}\beta^{3/2}}\,.
  16. T o = 1 / β o T_{o}=1/\beta_{o}

Non-linear_coherent_states.html

  1. a | α = α | α a|\alpha\rangle=\alpha|\alpha\rangle
  2. | α = D ( α ) | 0 |\alpha\rangle=D(\alpha)|0\rangle
  3. D ( α ) = exp ( α a - α * a ) D(\alpha)=\exp(\alpha a^{\dagger}-\alpha^{*}a)
  4. A = a f ( a a ) A=af(a^{\dagger}a)
  5. a a
  6. A A
  7. f f

Non-neutral_plasmas.html

  1. 0 = q E r + q v θ B + m v θ 2 / r , 0=qE_{r}+qv_{\theta}B+m{v_{\theta}}^{2}/r,
  2. v θ v_{\theta}
  3. ω = - v θ / r \omega=-v_{\theta}/r
  4. ω = Ω c 2 ± Ω c 2 / 4 - q E r / m r \omega=\frac{\Omega_{c}}{2}\pm\sqrt{{\Omega_{c}}^{2}/4-qE_{r}/{mr}}
  5. Ω c = q B / m \Omega_{c}=qB/m
  6. 0 ω / Ω c 1 0\leq\omega/\Omega_{c}\leq 1
  7. q E r / m r = Ω c 2 / 4 qE_{r}/{mr}={\Omega_{c}}^{2}/4
  8. 1 r r ( r E r ) = q n / ϵ 0 , \frac{1}{r}\frac{\partial}{\partial r}(rE_{r})=qn/\epsilon_{0},
  9. ω \omega
  10. n = 2 ϵ 0 m ω ( Ω c - ω ) q 2 . n=\frac{2\epsilon_{0}m\omega(\Omega_{c}-\omega)}{q^{2}}.
  11. n B = ϵ 0 m Ω c 2 2 q 2 = B 2 / ( 2 μ 0 ) m c 2 n_{B}=\frac{\epsilon_{0}m{\Omega_{c}}^{2}}{2q^{2}}=\frac{B^{2}/(2\mu_{0})}{mc^% {2}}
  12. c = 1 / μ 0 ϵ 0 c=1/\sqrt{\mu_{0}\epsilon_{0}}
  13. B 2 / ( 2 μ 0 ) B^{2}/(2\mu_{0})
  14. Γ = q 2 4 π ϵ 0 k B T a , \Gamma=\frac{q^{2}}{4\pi\epsilon_{0}k_{B}Ta},
  15. T T
  16. a a
  17. n n
  18. 4 π a 3 n / 3 = 1 4\pi a^{3}n/3=1
  19. q 2 / ( 4 π ϵ 0 a ) q^{2}/(4\pi\epsilon_{0}a)
  20. k b T k_{b}T
  21. Γ > 1 \Gamma>1
  22. Γ \Gamma
  23. Γ 2 \Gamma\approx 2
  24. Γ 175 \Gamma\simeq 175
  25. Γ \Gamma

Non-specific_polyamine_oxidase.html

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

Non-wellfounded_mereology.html

  1. < x < < x < \cdots<x<\cdots<x<\cdots
  2. < x < < x < \cdots<x<\cdots<x<\cdots

Noncommutative_signal-flow_graph.html

  1. x i = j = 0 n a i j x j , 1 i n , x_{i}=\sum_{j=0}^{n}a_{ij}x_{j},\;\;\;1\leq i\leq n,
  2. \in
  3. x m = T x 0 , x_{m}=Tx_{0},
  4. \in
  5. p j = ( w k j ( j ) , , w 2 ( j ) , w 1 ( j ) ) . p_{j}=(w^{(j)}_{k_{j}},\ldots,w^{(j)}_{2},w^{(j)}_{1}).
  6. T j = i = k j 1 ( 1 - S i ( j ) ) - 1 w i ( j ) , T_{j}=\prod_{i=k_{j}}^{1}(1-S^{(j)}_{i})^{-1}w^{(j)}_{i},
  7. T = j J T j . T=\sum_{j\in J}T_{j}.
  8. f + e ( 1 - b ) - 1 c , f+e(1-b)^{-1}c,
  9. T d = [ 1 - f - e ( 1 - b ) - 1 c ] - 1 d . T_{d}=\left[1-f-e(1-b)^{-1}c\right]^{-1}d.
  10. T ( F R L ) = [ 1 - f - e ( 1 - b ) - 1 c ] - 1 d + [ 1 - f - e ( 1 - b ) - 1 c ] - 1 e ( 1 - b ) - 1 a , T^{(FRL)}=\left[1-f-e(1-b)^{-1}c\right]^{-1}d+\left[1-f-e(1-b)^{-1}c\right]^{-% 1}e(1-b)^{-1}a,
  11. T ( B R L ) = [ 1 - f - e ( 1 - b ) - 1 c ] - 1 d + ( 1 - f ) - 1 e [ 1 - b - c ( 1 - f ) e ] - 1 a , T^{(BRL)}=\left[1-f-e(1-b)^{-1}c\right]^{-1}d+(1-f)^{-1}e\left[1-b-c(1-f)^{e}% \right]^{-1}a,
  12. y i = j = 1 2 a i j x j + j = 1 2 b i j y j y_{i}=\sum_{j=1}^{2}a_{ij}x_{j}+\sum_{j=1}^{2}b_{ij}y_{j}
  13. z i = j = 1 2 c i j y j , z_{i}=\sum_{j=1}^{2}c_{ij}y_{j},
  14. T = C ( 1 - B ) - 1 A . T=C(1-B)^{-1}A.
  15. Σ \Sigma
  16. Σ * \Sigma^{*}
  17. Σ * \subseteq\Sigma^{*}
  18. A B = { a b a A , b B } . A\cdot B=\{ab\mid a\in A,b\in B\}.
  19. A * = { λ } + A + A A + A A A + , A^{*}=\{\lambda\}+A+AA+AAA+\cdots,
  20. λ \lambda
  21. ( 1 - x ) - 1 = 1 + x + x 2 + x 3 , (1-x)^{-1}=1+x+x^{2}+x^{3}\cdots,
  22. Σ * \Sigma^{*}
  23. Σ * \Sigma^{*}
  24. λ \lambda
  25. a c * b , ac^{*}b,
  26. λ \lambda
  27. a c * d c * a , ac^{*}dc^{*}a,
  28. λ \lambda
  29. b c * a . bc^{*}a.
  30. L = a c * b + a c * d c * a + b c * a . L=ac^{*}b+ac^{*}dc^{*}a+bc^{*}a.

Noncommutative_torus.html

  1. B ( L 2 ( 𝕋 ) ) B(L^{2}(\mathbb{T}))
  2. \mathbb{C}
  3. U U
  4. V V
  5. U ( f ) ( z ) = z f ( z ) U(f)(z)=zf(z)
  6. V ( f ) ( z ) = f ( e - 2 π i θ z ) V(f)(z)=f(e^{-2\pi i\theta}z)
  7. V U = e - 2 π i θ U V VU=e^{-2\pi i\theta}UV
  8. V U = e 2 π i θ U V . VU=e^{2\pi i\theta}UV.
  9. π \pi
  10. A θ A_{\theta}
  11. A η A_{\eta}

Nonlinear_Dirac_equation.html

  1. = ψ ¯ ( i / - m ) ψ - g 2 ( ψ ¯ γ μ ψ ) ( ψ ¯ γ μ ψ ) , \mathcal{L}=\overline{\psi}(i\partial\!\!\!/-m)\psi-\frac{g}{2}\left(\overline% {\psi}\gamma^{\mu}\psi\right)\left(\overline{\psi}\gamma_{\mu}\psi\right),
  2. / = μ = 0 , 1 γ μ x μ , \partial\!\!\!/=\sum_{\mu=0,1}\gamma^{\mu}\frac{\partial}{\partial x^{\mu}}\,,
  3. g g
  4. m m
  5. μ = 0 , 1 μ=0,1
  6. = ψ ¯ ( i / - m ) ψ + g 2 ( ψ ¯ ψ ) 2 , \mathcal{L}=\overline{\psi}\left(i\partial\!\!\!/-m\right)\psi+\frac{g}{2}% \left(\overline{\psi}\psi\right)^{2},
  7. / = μ = 0 3 γ μ x μ , \partial\!\!\!/=\sum_{\mu=0}^{3}\gamma^{\mu}\frac{\partial}{\partial x^{\mu}}\,,
  8. μ = 0 , 1 , 2 , 3 μ=0,1,2,3
  9. c = = 1 c=\hbar=1
  10. = - g ( ψ ¯ ( i γ μ D μ - m ) ψ ) , \mathcal{L}=\sqrt{-g}\bigl(\overline{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)% \psi\bigr),
  11. D μ = μ + 1 4 ω ν ρ μ γ ν γ ρ D_{\mu}=\partial_{\mu}+\frac{1}{4}\omega_{\nu\rho\mu}\gamma^{\nu}\gamma^{\rho}
  12. ω μ ν ρ \omega_{\mu\nu\rho}
  13. g g
  14. g μ ν g_{\mu\nu}
  15. γ μ γ ν + γ ν γ μ = 2 g μ ν I . \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}I.
  16. i γ μ D μ ψ - m ψ = i γ μ μ ψ + 3 κ 8 ( ψ ¯ γ μ γ 5 ψ ) γ μ γ 5 ψ - m ψ = 0 , i\gamma^{\mu}D_{\mu}\psi-m\psi=i\gamma^{\mu}\nabla_{\mu}\psi+\frac{3\kappa}{8}% (\overline{\psi}\gamma_{\mu}\gamma^{5}\psi)\gamma^{\mu}\gamma^{5}\psi-m\psi=0,
  17. μ \nabla_{\mu}
  18. m 2 κ \frac{m^{2}}{\kappa}

Nonlinear_expectation.html

  1. 𝔼 : \mathbb{E}:\mathcal{H}\to\mathbb{R}
  2. \mathcal{H}
  3. X , Y X,Y\in\mathcal{H}
  4. X Y X\geq Y
  5. 𝔼 [ X ] 𝔼 [ Y ] \mathbb{E}[X]\geq\mathbb{E}[Y]
  6. c c\in\mathbb{R}
  7. 𝔼 [ c ] = c \mathbb{E}[c]=c
  8. ρ \rho
  9. 𝔼 [ X ] := ρ ( - X ) \mathbb{E}[X]:=\rho(-X)

Nonlinear_realization.html

  1. G G
  2. H H
  3. G G
  4. 𝔤 \mathfrak{g}
  5. G G
  6. G G
  7. H H
  8. V V
  9. 𝔤 \mathfrak{g}
  10. G G
  11. 𝔤 = 𝔥 + 𝔣 \mathfrak{g}=\mathfrak{h}+\mathfrak{f}
  12. 𝔥 \mathfrak{h}
  13. H H
  14. 𝔣 \mathfrak{f}
  15. [ 𝔣 , 𝔣 ] 𝔥 , [ 𝔣 , 𝔥 ] 𝔣 . [\mathfrak{f},\mathfrak{f}]\subset\mathfrak{h},\qquad[\mathfrak{f},\mathfrak{h% }]\subset\mathfrak{f}.
  16. U U
  17. G G
  18. g U g\in U
  19. g = exp ( F ) exp ( I ) , F 𝔣 , I 𝔥 . g=\exp(F)\exp(I),\qquad F\in\mathfrak{f},\qquad I\in\mathfrak{h}.
  20. U G U_{G}
  21. G G
  22. U G 2 U U_{G}^{2}\subset U
  23. U 0 U_{0}
  24. H H
  25. σ 0 \sigma_{0}
  26. G / H G/H
  27. σ = g σ 0 = exp ( F ) σ 0 , g U G . \sigma=g\sigma_{0}=\exp(F)\sigma_{0},\qquad g\in U_{G}.
  28. s ( g σ 0 ) = exp ( F ) s(g\sigma_{0})=\exp(F)
  29. G G / H G\to G/H
  30. U 0 U_{0}
  31. g U G G g\in U_{G}\subset G
  32. U 0 × V U_{0}\times V
  33. g exp ( F ) = exp ( F ) exp ( I ) , g : ( exp ( F ) σ 0 , v ) ( exp ( F ) σ 0 , exp ( I ) v ) . g\exp(F)=\exp(F^{\prime})\exp(I^{\prime}),\qquad g:(\exp(F)\sigma_{0},v)\to(% \exp(F^{\prime})\sigma_{0},\exp(I^{\prime})v).
  34. 𝔤 \mathfrak{g}
  35. G G
  36. { F α } \{F_{\alpha}\}
  37. { I a } \{I_{a}\}
  38. 𝔣 \mathfrak{f}
  39. 𝔥 \mathfrak{h}
  40. [ I a , I b ] = c a b d I d , [ F α , F β ] = c α β d I d , [ F α , I b ] = c α b β F β . [I_{a},I_{b}]=c^{d}_{ab}I_{d},\qquad[F_{\alpha},F_{\beta}]=c^{d}_{\alpha\beta}% I_{d},\qquad[F_{\alpha},I_{b}]=c^{\beta}_{\alpha b}F_{\beta}.
  41. 𝔤 \mathfrak{g}
  42. 𝔣 × V \mathfrak{f}\times V
  43. F α : ( σ γ F γ , v ) ( F α ( σ γ ) F γ , F α ( v ) ) , I a : ( σ γ F γ , v ) ( I a ( σ γ ) F γ , I a v ) , F_{\alpha}:(\sigma^{\gamma}F_{\gamma},v)\to(F_{\alpha}(\sigma^{\gamma})F_{% \gamma},F_{\alpha}(v)),\qquad I_{a}:(\sigma^{\gamma}F_{\gamma},v)\to(I_{a}(% \sigma^{\gamma})F_{\gamma},I_{a}v),
  44. F α ( σ γ ) = δ α γ + 1 12 ( c α μ β c β ν γ - 3 c α μ b c ν b γ ) σ μ σ ν , I a ( σ γ ) = c a ν γ σ ν , F_{\alpha}(\sigma^{\gamma})=\delta^{\gamma}_{\alpha}+\frac{1}{12}(c^{\beta}_{% \alpha\mu}c^{\gamma}_{\beta\nu}-3c^{b}_{\alpha\mu}c^{\gamma}_{\nu b})\sigma^{% \mu}\sigma^{\nu},\qquad I_{a}(\sigma^{\gamma})=c^{\gamma}_{a\nu}\sigma^{\nu},
  45. σ α \sigma^{\alpha}
  46. σ α \sigma^{\alpha}

Nonuniform_sampling.html

  1. z 0 , z 1 , , z n z_{0},z_{1},\ldots,z_{n}
  2. w 0 , w 1 , , w n w_{0},w_{1},\ldots,w_{n}
  3. p n ( z ) p_{n}(z)
  4. p n ( z i ) = w i , where i = 0 , 1 , , n . p_{n}(z_{i})=w_{i},\,\text{ where }i=0,1,\ldots,n.
  5. p n ( z ) p_{n}(z)
  6. I k ( z ) = ( z - z 0 ) ( z - z 1 ) ( z - z k - 1 ) ( z - z k + 1 ) ( z - z n ) ( z k - z 0 ) ( z k - z 1 ) ( z k - z k - 1 ) ( z k - z k + 1 ) ( z k - z n ) I_{k}(z)=\frac{(z-z_{0})(z-z_{1})\cdots(z-z_{k-1})(z-z_{k+1})\cdots(z-z_{n})}{% (z_{k}-z_{0})(z_{k}-z_{1})\cdots(z_{k}-z_{k-1})(z_{k}-z_{k+1})\cdots(z_{k}-z_{% n})}
  7. I k ( z j ) = δ k , j = { 0 , if k j 1 , if k = j I_{k}(z_{j})=\delta_{k,j}=\begin{cases}0,&\,\text{if }k\neq j\\ 1,&\,\text{if }k=j\end{cases}
  8. p n ( z ) = k = 0 n w k I k ( z ) p_{n}(z)=\sum_{k=0}^{n}w_{k}I_{k}(z)
  9. p n ( z j ) = w j , j = 0 , 1 , , n p_{n}(z_{j})=w_{j},j=0,1,\ldots,n
  10. G n ( z ) = ( z - z 0 ) ( z - z 1 ) ( z - z n ) G_{n}(z)=(z-z_{0})(z-z_{1})\cdots(z-z_{n})
  11. p n ( z ) = k = 0 n w k G n ( z ) ( z - z k ) G n ( z k ) p_{n}(z)=\sum_{k=0}^{n}w_{k}\frac{G_{n}(z)}{(z-z_{k})G^{\prime}_{n}(z_{k})}
  12. f ( z j ) = p n ( z j ) , j = 0 , 1 , , n , f(z_{j})=p_{n}(z_{j}),j=0,1,\ldots,n,
  13. f ( z ) = k = 0 n f ( z k ) G n ( z ) ( z - z k ) G n ( z k ) f(z)=\sum_{k=0}^{n}f(z_{k})\frac{G_{n}(z)}{(z-z_{k})G^{\prime}_{n}(z_{k})}
  14. C f ( z ) = n = - f ( a + n W ) sin [ π ( z - a - n W / W ) ] [ π ( z - a - n W / W ) ] C_{f}(z)=\sum_{n=-\infty}^{\infty}f(a+nW)\frac{\sin[\pi(z-a-nW/W)]}{[\pi(z-a-% nW/W)]}
  15. f ( z ) f(z)
  16. z n = a + n W z_{n}=a+nW
  17. C f ( z ) C_{f}(z)
  18. C f ( z ) = n = - f ( z n ) G ( z ) G ( z n ) ( z - z n ) , where G ( z ) = sin [ π ( z - a ) / W ] and z n = a + n W C_{f}(z)=\sum_{n=-\infty}^{\infty}f(z_{n})\frac{G(z)}{G^{\prime}(z_{n})(z-z_{n% })},\,\text{ where }G(z)=\sin[\pi(z-a)/W]\,\text{ and }z_{n}=a+nW
  19. f ( t ) = - σ σ e j x t g ( x ) d x ( t ) , g L 2 ( - σ , σ ) , f(t)=\int_{-\sigma}^{\sigma}e^{jxt}g(x)\,dx\qquad(t\in\mathbb{R}),\qquad% \forall g\in L^{2}(-\sigma,\sigma),
  20. f ( t ) = k = - f ( k π σ ) sin ( σ t - k π ) σ t - k π ( t ) f(t)=\sum_{k=-\infty}^{\infty}f\left(\frac{k\pi}{\sigma}\right)\frac{\sin(% \sigma t-k\pi)}{\sigma t-k\pi}\qquad(t\in\mathbb{R})
  21. { t k } k \{t_{k}\}_{k\in\mathbb{Z}}
  22. D = sup k | t k - k | < 1 4 , D=\sup_{k\in\mathbb{Z}}|t_{k}-k|<\frac{1}{4},
  23. f ( t ) = k = - f ( t k ) G ( t ) G ( t k ) ( t - t k ) , f B π 2 , ( t ) , f(t)=\sum_{k=-\infty}^{\infty}f(t_{k})\frac{G(t)}{G^{\prime}(t_{k})(t-t_{k})},% \qquad\forall f\in B^{2}_{\pi},\qquad(t\in\mathbb{R}),
  24. where G ( t ) = ( t - t 0 ) k = 1 ( 1 - t t k ) ( 1 - t t - k ) , \,\text{ where }G(t)=(t-t_{0})\prod_{k=1}^{\infty}\left(1-\frac{t}{t_{k}}% \right)\left(1-\frac{t}{t_{-k}}\right),
  25. B σ 2 . B^{2}_{\sigma}.
  26. and f ( t ) \,\text{and }f(t)

Normal_form_for_free_groups_and_free_product_of_groups.html

  1. G G
  2. S S
  3. G G
  4. w w
  5. w = a 1 a 2 a n w=a_{1}a_{2}\ldots\,a_{n}
  6. a j S ± 1 j n . a_{j}\in\,S^{\pm}\,\forall\,1\leq\,j\leq\,n.
  7. w w
  8. a a - 1 , a S ± aa^{-1},\,a\in\,S^{\pm}
  9. G G
  10. S S
  11. S S
  12. G G
  13. G G
  14. w G w\in\,G
  15. a a - 1 aa^{-1}
  16. a S ± a\in S^{\pm}
  17. w 1 w_{1}
  18. w 2 w_{2}
  19. w 1 w 2 w_{1}\equiv\,w_{2}
  20. w 1 w_{1}
  21. w 2 w_{2}
  22. G G
  23. G 0 G_{0}
  24. a a - 1 aa^{-1}
  25. w w
  26. u u
  27. v v
  28. x S x\in\,S
  29. x Δ x\Delta
  30. G 0 G_{0}
  31. w ( x Δ ) = w x w(x\Delta)=wx
  32. w x wx
  33. w ( x Δ ) = u w(x\Delta)=u
  34. w = u x - 1 w=ux^{-1}
  35. P P
  36. G 0 G_{0}
  37. x Δ , x S x\Delta,\,x\in\,S
  38. Δ * {\Delta}^{*}
  39. Δ \Delta
  40. Δ * : W P {\Delta}^{*}:W\mapsto\,P
  41. u 1 u 2 , u_{1}\equiv\,u_{2},
  42. u 1 Δ * = u 2 Δ * u_{1}{\Delta}^{*}=u_{2}{\Delta}^{*}
  43. 1 ( u Δ * ) = u 0 1(u{\Delta}^{*})=u_{0}
  44. u 0 u . u_{0}\equiv\,u.
  45. u 1 u 2 u_{1}\equiv\,u_{2}
  46. u 1 , u 2 u_{1},\,u_{2}
  47. u 1 = u 2 u_{1}=u_{2}
  48. G = A * B G\,=\,A\,*\,B
  49. A A
  50. B B
  51. w G w\,\in\,G
  52. w = g 1 g 2 g n w\,=\,g_{1}g_{2}...g_{n}
  53. g j A or B g_{j}\in\,A\,\,\,\text{or}\,\,B
  54. 1 j n 1\leq\,j\leq\,n
  55. g 1 , g 2 g n g_{1},\,g_{2}\ldots\,g_{n}
  56. g j A or B 1 j n g_{j}\,\in\,A\,\,\text{or}\,B\,\forall\,1\leq\,j\leq\,n
  57. g j e j , g_{j}\,\neq\,e\,\forall\,j,
  58. g j , g j + 1 g_{j},\,g_{j+1}
  59. A A
  60. B B
  61. A * B A*B
  62. A A
  63. B B
  64. w = g 1 g 2 g n , n > 0 w=g_{1}g_{2}\cdots g_{n}\,\,,n>0
  65. g 1 , g 2 , , g n g_{1},g_{2},\dots,g_{n}
  66. w 1 w\neq 1
  67. A * B A*B
  68. w w
  69. A * B A*B
  70. w = g 1 g 2 g n w=g_{1}g_{2}\cdots g_{n}
  71. g 1 , g 2 , , g n g_{1},g_{2},\dots,g_{n}
  72. w = 1 w=1
  73. w = g 1 g 2 g m w=g_{1}g_{2}\cdots g_{m}
  74. w = h 1 h 2 h n w=h_{1}h_{2}\cdots h_{n}
  75. g 1 g 2 g m = h 1 h 2 h n . g_{1}g_{2}\cdots g_{m}\,=\,h_{1}h_{2}\cdots h_{n}.
  76. h n - 1 h n - 1 - 1 h 1 - 1 g 1 g 2 g m = 1. h_{n}^{-1}h_{n-1}^{-1}\cdots h_{1}^{-1}g_{1}g_{2}\cdots g_{m}\,=\,1.
  77. h 1 - 1 g 1 = 1 h_{1}^{-1}g_{1}\,=\,1
  78. g 1 = h 1 . g_{1}\,=\,h_{1}.
  79. m = n m\,=\,n
  80. g i = h i g_{i}\,=\,h_{i}
  81. i = 1 , 2 , , n . i\,=\,1,2,\dots,n.
  82. W W
  83. A * B A*B
  84. S ( W ) S(W)
  85. W W
  86. ϕ : A S ( W ) \phi:A\rightarrow S(W)
  87. a = id a=\operatorname{id}
  88. ϕ ( a ) = id \phi(a)=\operatorname{id}
  89. ϕ \phi
  90. ϕ ( x ) ( g 1 , g 2 , , g m ) = { ( a , g 1 , g 2 , , g m ) if g 1 B . ( a g 1 , g 2 , , g n ) if g 1 A and a g 1 1. ( g 2 , g 3 , , g n ) if a g 1 = 1. \phi(x)(g_{1},g_{2},\dots,g_{m})=\begin{cases}(a,g_{1},g_{2},\cdots,g_{m})&\,% \text{if }g_{1}\in B.\\ (ag_{1},g_{2},\dots,g_{n})&\,\text{if }g_{1}\in A\,\text{ and }ag_{1}\neq 1.\\ (g_{2},g_{3},\dots,g_{n})&\,\text{if }ag_{1}=1.\end{cases}
  91. ψ : B S ( W ) \psi:B\rightarrow S(W)
  92. ϕ \phi
  93. ψ \psi
  94. ϕ * ψ : A * B S ( W ) \phi*\psi:A*B\rightarrow S(W)
  95. ϕ * ψ ( id ) ( 1 ) = id ( 1 ) = 1 \phi*\psi(\operatorname{id})(1)=\operatorname{id}(1)=1
  96. w = g 1 g 2 g n , n > 0 w=g_{1}g_{2}\cdots g_{n}\,\,,n>0
  97. g 1 , g 2 , , g n g_{1},g_{2},\dots,g_{n}
  98. ϕ * ψ ( w ) ( 1 ) = ( g 1 , g 2 , , g n ) . \phi*\psi(w)(1)=(g_{1},g_{2},\dots,g_{n}).
  99. w = 1 w=1
  100. A * B A*B
  101. n = 0 n=0