wpmath0000007_8

Lossless_JPEG.html

  1. X = { min ( A , B ) if C max ( A , B ) max ( A , B ) if C min ( A , B ) A + B - C otherwise . X=\left\{\begin{aligned}&\displaystyle\min(A,B)\quad\,\mbox{if}~{}\,C\geq\max(% A,B)\\ &\displaystyle\max(A,B)\quad\mbox{if}~{}\,C\leq\min(A,B)\\ &\displaystyle A+B-C\quad\,\mbox{otherwise}~{}.\\ \end{aligned}\right.
  2. E { e | C t x } E\left\{e|Ctx\right\}
  3. e ¯ ( C ) \bar{e}(C)
  4. g 1 = D - B \displaystyle g_{1}=D-B
  5. P ( e C t x = [ q 1 , q 2 , q 3 ] ) = P ( - e C t x = [ - q 1 , - q 2 , - q 3 ] ) P(e\mid Ctx=[q_{1},q_{2},q_{3}])=P(-e\mid Ctx=[-q_{1},-q_{2},-q_{3}])
  6. ( ( 2 × 4 + 1 ) 3 + 1 ) / 2 = 365 ((2\times 4+1)^{3}+1)/2=365

Louvre_Pyramid.html

  1. 17 ( 17 + 1 ) 2 = 153 \textstyle\frac{17\cdot(17+1)}{2}=153
  2. 4 153 - 9 = 603 4\cdot 153-9=603
  3. 4 18 - 2 = 70 4\cdot 18-2=70

Lovász_conjecture.html

  1. G G
  2. S S
  3. G G
  4. S S
  5. G = S n G=S_{n}
  6. a = ( 1 , 2 , , n ) , b = ( 1 , 2 ) a=(1,2,\dots,n),b=(1,2)
  7. s 1 = ( 1 , 2 ) , s 2 = ( 2 , 3 ) , , s n - 1 = ( n - 1 , n ) s_{1}=(1,2),s_{2}=(2,3),\dots,s_{n-1}=(n-1,n)
  8. { 1 , 2 , . . , n } \{1,2,..,n\}
  9. a = ( 1 , 2 ) , b = ( 1 , 2 ) ( 3 , 4 ) , c = ( 2 , 3 ) ( 4 , 5 ) a=(1,2),b=(1,2)(3,4)\cdots,c=(2,3)(4,5)\cdots
  10. S = { a , b } , ( a b ) 2 = 1 S=\{a,b\},(ab)^{2}=1
  11. S = { a , b , c } , a 2 = b 2 = c 2 = [ a , b ] = 1 S=\{a,b,c\},a^{2}=b^{2}=c^{2}=[a,b]=1
  12. S = { a , b , c } , a 2 = 1 , c = a - 1 b a S=\{a,b,c\},a^{2}=1,c=a^{-1}ba
  13. S = { a , b } , a 2 = b s = ( a b ) 3 = 1 , S=\{a,b\},a^{2}=b^{s}=(ab)^{3}=1,
  14. | G | , s = 2 m o d 4 |G|,s=2~{}mod~{}4
  15. G G
  16. log 2 | G | \log_{2}|G|
  17. Ω ( log 5 | G | ) \Omega(\log^{5}|G|)

Low-energy_electron_diffraction.html

  1. I ( d ) = I 0 * e - d / Λ ( E ) \displaystyle I(d)=I_{0}*e^{-d/\Lambda(E)}
  2. Λ ( E ) \Lambda(E)
  3. λ = h 2 m E , λ [ nm ] 1.5 E [ eV ] \displaystyle\lambda=\frac{h}{\sqrt{2mE}},\qquad\lambda[\textrm{nm}]\approx% \sqrt{\frac{1.5}{E[\textrm{eV}]}}
  4. 𝐚 * = 2 π 𝐛 × 𝐜 𝐚 ( 𝐛 × 𝐜 ) , 𝐛 * = 2 π 𝐜 × 𝐚 𝐛 ( 𝐜 × 𝐚 ) , 𝐜 * = 2 π 𝐚 × 𝐛 𝐜 ( 𝐚 × 𝐛 ) \begin{aligned}\displaystyle\,\textbf{a}^{*}&\displaystyle=\frac{2\pi\,\textbf% {b}\times\,\textbf{c}}{\,\textbf{a}\cdot(\,\textbf{b}\times\,\textbf{c})},\\ \displaystyle\,\textbf{b}^{*}&\displaystyle=\frac{2\pi\,\textbf{c}\times\,% \textbf{a}}{\,\textbf{b}\cdot(\,\textbf{c}\times\,\textbf{a})},\\ \displaystyle\,\textbf{c}^{*}&\displaystyle=\frac{2\pi\,\textbf{a}\times\,% \textbf{b}}{\,\textbf{c}\cdot(\,\textbf{a}\times\,\textbf{b})}\end{aligned}
  5. 𝐤 0 = 2 π / λ 0 \,\textbf{k}_{0}=2\pi/\lambda_{0}
  6. 𝐤 = 2 π / λ \begin{aligned}\displaystyle\,\textbf{k}=2\pi/\lambda\end{aligned}
  7. 𝐤 - 𝐤 0 = 𝐆 hkl , ( 1 ) \begin{aligned}\displaystyle\,\textbf{k}-\,\textbf{k}_{0}=\,\textbf{G}_{% \textrm{hkl}},(1)\end{aligned}
  8. 𝐆 hkl = h 𝐚 * + k 𝐛 * + l 𝐜 * \begin{aligned}\displaystyle\,\textbf{G}_{\textrm{hkl}}=h\,\textbf{a}^{*}+k\,% \textbf{b}^{*}+l\,\textbf{c}^{*}\end{aligned}
  9. | 𝐤 0 | = | 𝐤 | |\,\textbf{k}_{0}|=|\,\textbf{k}|
  10. 𝐤 | | - 𝐤 0 | | = 𝐆 hk = h 𝐚 * + k 𝐛 * , ( 2 ) \begin{aligned}\displaystyle\,\textbf{k}^{||}-\,\textbf{k}_{0}^{||}=\,\textbf{% G}_{\textrm{hk}}=h\,\textbf{a}^{*}+k\,\textbf{b}^{*},(2)\end{aligned}
  11. 𝐚 * \,\textbf{a}^{*}
  12. 𝐛 * \,\textbf{b}^{*}
  13. 𝐤 | | , 𝐤 0 | | \,\textbf{k}^{||},\,\textbf{k}_{0}^{||}
  14. 𝐚 * \,\textbf{a}^{*}
  15. 𝐛 * \,\textbf{b}^{*}
  16. 𝐚 * \displaystyle\,\textbf{a}^{*}
  17. 𝐤 0 \,\textbf{k}_{0}
  18. | 𝐤 0 | |\,\textbf{k}_{0}|
  19. 𝐚 s = G 11 𝐚 + G 12 𝐛 , 𝐛 s = G 21 𝐚 + G 22 𝐛 . \begin{aligned}\displaystyle\,\textbf{a}_{s}&\displaystyle=G_{11}\,\textbf{a}+% G_{12}\,\textbf{b},\\ \displaystyle\,\textbf{b}_{s}&\displaystyle=G_{21}\,\textbf{a}+G_{22}\,\textbf% {b}.\end{aligned}
  20. G = ( G 11 G 12 G 21 G 22 ) . \begin{aligned}\displaystyle G=\left(\begin{array}[]{cc}G_{11}&G_{12}\\ G_{21}&G_{22}\end{array}\right).\end{aligned}
  21. 𝐚 s * = G 11 * 𝐚 * + G 12 * 𝐛 * , 𝐛 s * = G 21 * 𝐚 * + G 22 * 𝐛 * . \begin{aligned}\displaystyle\,\textbf{a}_{s}^{*}&\displaystyle=G_{11}^{*}\,% \textbf{a}^{*}+G_{12}^{*}\,\textbf{b}^{*},\\ \displaystyle\,\textbf{b}_{s}^{*}&\displaystyle=G_{21}^{*}\,\textbf{a}^{*}+G_{% 22}^{*}\,\textbf{b}^{*}.\end{aligned}
  22. G * = ( G - 1 ) T = 1 d e t ( G ) ( G 22 - G 21 - G 12 G 11 ) . \begin{aligned}\displaystyle G^{*}&\displaystyle=(G^{-1})^{T}\\ &\displaystyle=\frac{1}{det(G)}\left(\begin{array}[]{cc}G_{22}&-G_{21}\\ -G_{12}&G_{11}\end{array}\right).\end{aligned}
  23. L ( E ) \displaystyle L(E)
  24. R \displaystyle R
  25. Y ( E ) = L - 1 / ( L - 2 + V o i 2 ) Y(E)=L^{-1}/(L^{-2}+V^{2}_{oi})
  26. V o i V_{oi}
  27. R p 0.2 R_{p}\leq 0.2
  28. R p 0.3 R_{p}\simeq 0.3
  29. R p 0.5 R_{p}\simeq 0.5

Lowell_Schoenfeld.html

  1. | π ( x ) - li ( x ) | x ln x 8 π |\pi(x)-{\rm li}(x)|\leq\frac{\sqrt{x}\,\ln x}{8\pi}
  2. | ψ ( x ) - x | x ln 2 x 8 π |\psi(x)-x|\leq\frac{\sqrt{x}\,\ln^{2}x}{8\pi}

Lozanić's_triangle.html

  1. 2 n - 2 + 2 n / 2 - 1 2^{n-2}+2^{\lfloor n/2\rfloor-1}
  2. F 2 n - 1 + F n + 1 2 {F_{2n-1}+F_{n+1}}\over 2
  3. F 2 n + F n 2 {F_{2n}+F_{n}}\over 2

Lucas–Kanade_method.html

  1. ( V x , V y ) (V_{x},V_{y})
  2. I x ( q 1 ) V x + I y ( q 1 ) V y = - I t ( q 1 ) I_{x}(q_{1})V_{x}+I_{y}(q_{1})V_{y}=-I_{t}(q_{1})
  3. I x ( q 2 ) V x + I y ( q 2 ) V y = - I t ( q 2 ) I_{x}(q_{2})V_{x}+I_{y}(q_{2})V_{y}=-I_{t}(q_{2})
  4. \vdots
  5. I x ( q n ) V x + I y ( q n ) V y = - I t ( q n ) I_{x}(q_{n})V_{x}+I_{y}(q_{n})V_{y}=-I_{t}(q_{n})
  6. q 1 , q 2 , , q n q_{1},q_{2},\dots,q_{n}
  7. I x ( q i ) , I y ( q i ) , I t ( q i ) I_{x}(q_{i}),I_{y}(q_{i}),I_{t}(q_{i})
  8. I I
  9. q i q_{i}
  10. A v = b Av=b
  11. A = [ I x ( q 1 ) I y ( q 1 ) I x ( q 2 ) I y ( q 2 ) I x ( q n ) I y ( q n ) ] , v = [ V x V y ] , and b = [ - I t ( q 1 ) - I t ( q 2 ) - I t ( q n ) ] A=\begin{bmatrix}I_{x}(q_{1})&I_{y}(q_{1})\\ I_{x}(q_{2})&I_{y}(q_{2})\\ \vdots&\vdots\\ I_{x}(q_{n})&I_{y}(q_{n})\end{bmatrix},\quad\quad v=\begin{bmatrix}V_{x}\\ V_{y}\end{bmatrix},\quad\mbox{and}~{}\quad b=\begin{bmatrix}-I_{t}(q_{1})\\ -I_{t}(q_{2})\\ \vdots\\ -I_{t}(q_{n})\end{bmatrix}
  12. A T A v = A T b A^{T}Av=A^{T}b
  13. v = ( A T A ) - 1 A T b \mathrm{v}=(A^{T}A)^{-1}A^{T}b
  14. A T A^{T}
  15. A A
  16. [ V x V y ] = [ i I x ( q i ) 2 i I x ( q i ) I y ( q i ) i I y ( q i ) I x ( q i ) i I y ( q i ) 2 ] - 1 [ - i I x ( q i ) I t ( q i ) - i I y ( q i ) I t ( q i ) ] \begin{bmatrix}V_{x}\\ V_{y}\end{bmatrix}=\begin{bmatrix}\sum_{i}I_{x}(q_{i})^{2}&\sum_{i}I_{x}(q_{i}% )I_{y}(q_{i})\\ \sum_{i}I_{y}(q_{i})I_{x}(q_{i})&\sum_{i}I_{y}(q_{i})^{2}\end{bmatrix}^{-1}% \begin{bmatrix}-\sum_{i}I_{x}(q_{i})I_{t}(q_{i})\\ -\sum_{i}I_{y}(q_{i})I_{t}(q_{i})\end{bmatrix}
  17. A T A A^{T}A
  18. q i q_{i}
  19. A T W A v = A T W b A^{T}WAv=A^{T}Wb
  20. v = ( A T W A ) - 1 A T W b \mathrm{v}=(A^{T}WA)^{-1}A^{T}Wb
  21. W W
  22. W i i = w i W_{ii}=w_{i}
  23. q i q_{i}
  24. [ V x V y ] = [ i w i I x ( q i ) 2 i w i I x ( q i ) I y ( q i ) i w i I x ( q i ) I y ( q i ) i w i I y ( q i ) 2 ] - 1 [ - i w i I x ( q i ) I t ( q i ) - i w i I y ( q i ) I t ( q i ) ] \begin{bmatrix}V_{x}\\ V_{y}\end{bmatrix}=\begin{bmatrix}\sum_{i}w_{i}I_{x}(q_{i})^{2}&\sum_{i}w_{i}I% _{x}(q_{i})I_{y}(q_{i})\\ \sum_{i}w_{i}I_{x}(q_{i})I_{y}(q_{i})&\sum_{i}w_{i}I_{y}(q_{i})^{2}\end{% bmatrix}^{-1}\begin{bmatrix}-\sum_{i}w_{i}I_{x}(q_{i})I_{t}(q_{i})\\ -\sum_{i}w_{i}I_{y}(q_{i})I_{t}(q_{i})\end{bmatrix}
  25. w i w_{i}
  26. q i q_{i}
  27. A T A v = A T b A^{T}Av=A^{T}b
  28. A T A A^{T}A
  29. A T A A^{T}A
  30. λ 1 λ 2 > 0 \lambda_{1}\geq\lambda_{2}>0
  31. λ 2 \lambda_{2}
  32. λ 1 / λ 2 \lambda_{1}/\lambda_{2}
  33. λ 1 \lambda_{1}
  34. λ 2 \lambda_{2}
  35. V x , V y V_{x},V_{y}

M-estimator.html

  1. θ ^ \hat{\theta}
  2. θ ^ = arg max θ ( i = 1 n f ( x i , θ ) ) \widehat{\theta}=\arg\max_{\displaystyle\theta}{\left(\prod_{i=1}^{n}f(x_{i},% \theta)\right)}\,\!
  3. θ ^ = arg min θ ( - i = 1 n log ( f ( x i , θ ) ) ) . \widehat{\theta}=\arg\min_{\displaystyle\theta}{\left(-\sum_{i=1}^{n}\log{(f(x% _{i},\theta))}\right)}.\,\!
  4. i = 1 n ρ ( x i , θ ) , \sum_{i=1}^{n}\rho(x_{i},\theta),\,\!
  5. θ ^ = arg min θ ( i = 1 n ρ ( x i , θ ) ) \hat{\theta}=\arg\min_{\displaystyle\theta}\left(\sum_{i=1}^{n}\rho(x_{i},% \theta)\right)\,\!
  6. i = 1 n ρ ( x i , θ ) . \sum_{i=1}^{n}\rho(x_{i},\theta).\,\!
  7. ( 𝒳 , Σ ) (\mathcal{X},\Sigma)
  8. ( Θ r , S ) (\Theta\subset\mathbb{R}^{r},S)
  9. θ Θ \theta\in\Theta
  10. T T
  11. ρ : 𝒳 × Θ \rho:\mathcal{X}\times\Theta\rightarrow\mathbb{R}
  12. F F
  13. 𝒳 \mathcal{X}
  14. T ( F ) Θ T(F)\in\Theta
  15. 𝒳 ρ ( x , θ ) d F ( x ) \int_{\mathcal{X}}\rho(x,\theta)dF(x)
  16. T ( F ) := arg min θ Θ 𝒳 ρ ( x , θ ) d F ( x ) T(F):=\arg\min_{\theta\in\Theta}\int_{\mathcal{X}}\rho(x,\theta)dF(x)
  17. ρ ( x , θ ) = - log ( f ( x , θ ) ) \rho(x,\theta)=-\log(f(x,\theta))
  18. f ( x , θ ) = F ( x , θ ) x f(x,\theta)=\frac{\partial F(x,\theta)}{\partial x}
  19. ρ \rho
  20. θ ^ \widehat{\theta}
  21. ψ : 𝒳 × Θ r \psi:\mathcal{X}\times\Theta\rightarrow\mathbb{R}^{r}
  22. 𝒳 \mathcal{X}
  23. T ( F ) Θ T(F)\in\Theta
  24. 𝒳 ψ ( x , θ ) d F ( x ) = 0 \int_{\mathcal{X}}\psi(x,\theta)\,dF(x)=0
  25. 𝒳 ψ ( x , T ( F ) ) d F ( x ) = 0 \int_{\mathcal{X}}\psi(x,T(F))\,dF(x)=0
  26. ψ ( x , θ ) = ( log ( f ( x , θ ) ) θ 1 , , log ( f ( x , θ ) ) θ p ) T \psi(x,\theta)=\left(\frac{\partial\log(f(x,\theta))}{\partial\theta^{1}},% \dots,\frac{\partial\log(f(x,\theta))}{\partial\theta^{p}}\right)^{\mathrm{T}}
  27. u T u^{\mathrm{T}}
  28. f ( x , θ ) = F ( x , θ ) x f(x,\theta)=\frac{\partial F(x,\theta)}{\partial x}
  29. θ \theta
  30. ψ ( x , θ ) = θ ρ ( x , θ ) \psi(x,\theta)=\nabla_{\theta}\rho(x,\theta)
  31. x ± x\rightarrow\pm\infty
  32. ψ \psi
  33. ψ \psi
  34. T ( G ) T(G)
  35. IF ( x ; T , G ) = - ψ ( x , T ( G ) ) [ ψ ( y , θ ) θ ] f ( y ) d y \operatorname{IF}(x;T,G)=-\frac{\psi(x,T(G))}{\int\left[\frac{\partial\psi(y,% \theta)}{\partial\theta}\right]f(y)\mathrm{d}y}
  36. f ( y ) f(y)
  37. ρ ( x , θ ) = ( x - θ ) 2 2 , \rho(x,\theta)=\frac{(x-\theta)^{2}}{2},\,\!

Maekawa's_algorithm.html

  1. P i P_{i}
  2. request ( t s , i ) \,\text{request}(ts,i)
  3. R i R_{i}
  4. request ( t s , i ) \,\text{request}(ts,i)
  5. P j P_{j}
  6. P j P_{j}
  7. grant \,\text{grant}
  8. grant \,\text{grant}
  9. P j P_{j}
  10. grant ( j ) \,\text{grant}(j)
  11. P i P_{i}
  12. P j P_{j}
  13. grant \,\text{grant}
  14. P j P_{j}
  15. failed ( j ) \,\text{failed}(j)
  16. P i P_{i}
  17. P j P_{j}
  18. P i P_{i}
  19. P j P_{j}
  20. grant \,\text{grant}
  21. P j P_{j}
  22. inquire ( j ) \,\text{inquire}(j)
  23. P j P_{j}
  24. grant \,\text{grant}
  25. inquire ( j ) \,\text{inquire}(j)
  26. P k P_{k}
  27. yield ( k ) \,\text{yield}(k)
  28. P j P_{j}
  29. P k P_{k}
  30. failed \,\text{failed}
  31. P k P_{k}
  32. grant \,\text{grant}
  33. yield ( k ) \,\text{yield}(k)
  34. P j P_{j}
  35. grant ( j ) \,\text{grant}(j)
  36. P k P_{k}
  37. release ( i ) \,\text{release}(i)
  38. P j P_{j}
  39. P i P_{i}
  40. grant ( j ) \,\text{grant}(j)
  41. P i P_{i}
  42. grant \,\text{grant}
  43. R i R_{i}
  44. P i P_{i}
  45. release ( i ) \,\text{release}(i)
  46. R i R_{i}
  47. R x R_{x}
  48. i j [ R i R j ] \forall i\,\forall j\,[R_{i}\bigcap R_{j}\neq]
  49. i [ P i R i ] \forall i\,[P_{i}\in R_{i}]
  50. i [ | R i | = K ] \forall i\,[|R_{i}|=K]
  51. P i P_{i}
  52. K K
  53. | R i | N - 1 |R_{i}|\geq\sqrt{N-1}
  54. 3 N 3\sqrt{N}
  55. 6 N 6\sqrt{N}

Magnetic_pressure.html

  1. 𝐅 \mathbf{F}
  2. 𝐅 = I 2 c 2 R [ ln ( 8 R a ) - 1 + Y ] \mathbf{F}=\dfrac{I^{2}}{c^{2}R}[\ln\left(\dfrac{8R}{a}\right)-1+Y]
  3. j × B = 0 j\times B=0
  4. P B P_{B}
  5. P B = B 2 2 μ 0 P_{B}=\frac{B^{2}}{2\mu_{0}}
  6. P B = B 2 8 π P_{B}=\frac{B^{2}}{8\pi}
  7. P B [ bar ] = ( B [ T ] 0.501 ) 2 . P_{B}\mathrm{[bar]}=\left(\frac{B\mathrm{[T]}}{0.501}\right)^{2}.

Magnetic_Reynolds_number.html

  1. R m = U L η \mathrm{R}_{\mathrm{m}}=\frac{UL}{\eta}
  2. U U
  3. L L
  4. η \eta
  5. R m 1 \mathrm{R}_{\mathrm{m}}\ll 1
  6. R m 1 \mathrm{R}_{\mathrm{m}}\gg 1
  7. R m R_{m}
  8. R m = μ σ R_{m}=\mu\sigma
  9. μ \mu
  10. σ \sigma
  11. R m < 1 R_{m}<1
  12. R m > 30 R_{m}>30

Magnetic_tension_force.html

  1. ( 𝐁 ) 𝐁 μ 0 ( S.I. ) ( 𝐁 ) 𝐁 4 π ( c.g.s. ) \frac{\left(\mathbf{B}\cdot\nabla\right)\mathbf{B}}{\mu_{0}}\,(\,\text{S.I.})% \qquad\frac{(\mathbf{B}\cdot\nabla)\mathbf{B}}{4\pi}\,(\,\text{c.g.s.})
  2. 𝐉 \mathbf{J}
  3. 𝐁 \mathbf{B}
  4. 𝐉 \mathbf{J}
  5. ρ ( t + 𝐕 ) 𝐕 = 𝐉 × 𝐁 - p \rho\left(\frac{\partial}{\partial t}+\mathbf{V}\cdot\nabla\right)\mathbf{V}=% \mathbf{J}\times\mathbf{B}-\nabla p
  6. μ 0 𝐉 = × 𝐁 \mu_{0}\mathbf{J}=\nabla\times\mathbf{B}
  7. s y m b o l ( a. 𝐛 ) = ( a. s y m b o l ) 𝐛 + ( b. s y m b o l ) 𝐚 + 𝐚 × ( s y m b o l × 𝐛 ) + 𝐛 × ( s y m b o l × 𝐚 ) , symbol{\nabla}(\,\textbf{a.}\,\textbf{b})=(\,\textbf{a.}symbol{\nabla})\,% \textbf{b}+(\,\textbf{b.}symbol{\nabla})\,\textbf{a}+\,\textbf{a}\times(symbol% {\nabla}\times\,\textbf{b})+\,\textbf{b}\times(symbol{\nabla}\times\,\textbf{a% }),
  8. ρ ( t + 𝐕 ) 𝐕 = - s y m b o l ( B 2 / 2 μ 0 ) + B. s y m b o l 𝐁 μ 0 - p . \rho\left(\frac{\partial}{\partial t}+\mathbf{V}\cdot\nabla\right)\mathbf{V}=-% symbol{\nabla}(B^{2}/2\mu_{0})+{\,\textbf{B.}symbol{\nabla}\,\textbf{B}\over% \mu_{0}}-\nabla p.
  9. p + B 2 / 2 μ 0 p+B^{2}/2\mu_{0}
  10. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})
  11. 𝐟 = ρ 𝐄 + 𝐉 × 𝐁 \mathbf{f}=\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}
  12. 𝐟 = ϵ 0 [ ( s y m b o l 𝐄 ) 𝐄 + ( 𝐄 \cdotsymbol ) 𝐄 ] + 1 μ 0 [ ( s y m b o l 𝐁 ) 𝐁 + ( 𝐁 \cdotsymbol ) 𝐁 ] - 1 2 s y m b o l ( ϵ 0 E 2 + 1 μ 0 B 2 ) - ϵ 0 t ( 𝐄 × 𝐁 ) . \mathbf{f}=\epsilon_{0}\left[(symbol{\nabla}\cdot\mathbf{E})\mathbf{E}+(% \mathbf{E}\cdotsymbol{\nabla})\mathbf{E}\right]+\frac{1}{\mu_{0}}\left[(symbol% {\nabla}\cdot\mathbf{B})\mathbf{B}+(\mathbf{B}\cdotsymbol{\nabla})\mathbf{B}% \right]-\frac{1}{2}symbol{\nabla}\left(\epsilon_{0}E^{2}+\frac{1}{\mu_{0}}B^{2% }\right)-\epsilon_{0}\frac{\partial}{\partial t}\left(\mathbf{E}\times\mathbf{% B}\right).
  13. σ i j ϵ 0 ( E i E j - 1 2 δ i j E 2 ) + 1 μ 0 ( B i B j - 1 2 δ i j B 2 ) , \sigma_{ij}\equiv\epsilon_{0}\left(E_{i}E_{j}-\frac{1}{2}\delta_{ij}E^{2}% \right)+\frac{1}{\mu_{0}}\left(B_{i}B_{j}-\frac{1}{2}\delta_{ij}B^{2}\right),
  14. 𝐟 + ϵ 0 μ 0 𝐒 t = σ \mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla% \cdot\mathbf{\sigma}
  15. σ i j \sigma_{ij}
  16. 𝐒 = 𝐄 × 𝐁 / μ 0 \mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_{0}
  17. σ i j \sigma_{ij}
  18. σ \nabla\cdot\mathbf{\sigma}
  19. 𝐒 \mathbf{S}

Mainz_Microtron.html

  1. μ \mu
  2. Δ \Delta
  3. η \eta
  4. η \eta^{\prime}
  5. L = 10 38 cm 2 / s . L=10^{38}~{}\rm cm^{2}/s\ .

Major_seventh_chord.html

  1. 4 ^ \hat{4}

Majorana_fermion.html

  1. γ j \gamma^{\dagger}_{j}
  2. j j
  3. γ j \gamma_{j}
  4. γ j \gamma^{\dagger}_{j}
  5. γ j \gamma_{j}
  6. γ ( E ) \gamma(E)
  7. E E
  8. γ ( - E ) {\gamma^{\dagger}(-E)}
  9. - E -E

Majority_logic_decoding.html

  1. 0 , 1 0,1
  2. ( n , 1 ) (n,1)
  3. n n
  4. n = 2 t + 1 n=2t+1
  5. [ n / 2 ] [n/2]
  6. P e = k = n + 1 2 n ( n k ) ϵ k ( 1 - ϵ ) ( n - k ) P_{e}=\sum_{k=\frac{n+1}{2}}^{n}{n\choose k}\epsilon^{k}(1-\epsilon)^{(n-k)}
  7. ϵ \epsilon
  8. ( n , 1 ) (n,1)
  9. n = 2 t + 1 n=2t+1
  10. d H d_{H}
  11. d H t d_{H}\leq t
  12. d H t + 1 d_{H}\geq t+1
  13. ( n , 1 ) (n,1)
  14. n = 5 , t = 2 n=5,t=2
  15. d H = 3 d_{H}=3

Malnormal_subgroup.html

  1. H H
  2. G G
  3. x x
  4. G G
  5. H H
  6. H H
  7. x H x - 1 xHx^{-1}

Malthusian_growth_model.html

  1. P ( t ) = P 0 e r t P(t)=P_{0}e^{rt}\,

Manakov_system.html

  1. v 1 = - i ξ v 1 + q 1 v 2 + q 2 v 3 v_{1}^{\prime}=-i\,\xi\,v_{1}+q_{1}\,v_{2}+q_{2}\,v_{3}
  2. v 2 = - q 1 * v 1 + i ξ v 2 v_{2}^{\prime}=-q_{1}^{*}\,v_{1}+i\,\xi\,v_{2}
  3. v 3 = - q 2 * v 1 + i ξ v 3 . v_{3}^{\prime}=-q_{2}^{*}\,v_{1}+i\,\xi\,v_{3}.
  4. q 1 , q 2 q_{1},q_{2}
  5. lim x a e i ξ x v 1 - lim x b e i ξ x v 1 = a b [ e i ξ x q 1 v 2 + e i ξ x q 2 v 3 ] d x \lim_{x\to a}e^{i\xi x}v_{1}-\lim_{x\to b}e^{i\xi x}v_{1}=\int_{a}^{b}[e^{i\xi x% }\,q_{1}\,v_{2}+e^{i\xi x}\,q_{2}\,v_{3}]\,dx
  6. lim x a e - i ξ x v 2 - lim x b e - i ξ x v 2 = - a b e - i ξ x q 1 * v 1 d x \lim_{x\to a}e^{-i\xi x}v_{2}-\lim_{x\to b}e^{-i\xi x}v_{2}=-\int_{a}^{b}e^{-i% \xi x}\,q_{1}^{*}\,v_{1}\,dx
  7. lim x a e - i ξ x v 3 - lim x b e - i ξ x v 3 = - a b e - i ξ x q 2 * v 1 d x \lim_{x\to a}e^{-i\xi x}v_{3}-\lim_{x\to b}e^{-i\xi x}v_{3}=-\int_{a}^{b}e^{-i% \xi x}\,q_{2}^{*}\,v_{1}\,dx
  8. ξ \xi

Mandelate_racemase.html

  1. \rightleftharpoons

Manu_propria.html

  1. 792 ¯ \overline{792}

Marcus_Hutter.html

  1. n n
  2. n 7 n^{7}
  3. 5 n 7 + O ( 1 ) 5n^{7}+O(1)
  4. O ( ) O()

Marcus_theory.html

  1. Δ \Delta
  2. Δ \Delta
  3. D + A k 21 k 12 [ D A ] k 32 k 23 [ D + A - ] k 30 D + + A - \mathrm{D+A\ \overset{\xrightarrow{k_{12}}}{\xleftarrow[k_{21}]{}}\ [D{\cdots}% A]\ \overset{\xrightarrow{k_{23}}}{\xleftarrow[k_{32}]{}}\ [D^{+}{\cdots}A^{-}% ]\xrightarrow{k_{30}}\ D^{+}+A^{-}}
  4. Δ \Delta
  5. k a c t = A e - Δ G R T k_{act}=A\cdot e^{-\frac{\Delta G^{\ddagger}}{RT}}
  6. Δ G \Delta G^{\ddagger}
  7. Δ \Delta
  8. Δ \Delta
  9. Δ \Delta
  10. Δ \Delta
  11. Δ \Delta
  12. Δ \Delta
  13. G = ( 1 2 r 1 + 1 2 r 2 - 1 R ) ( 1 ϵ o p - 1 ϵ s ) ( Δ e ) 2 G=\left(\frac{1}{2r_{1}}+\frac{1}{2r_{2}}-\frac{1}{R}\right)\cdot\left(\frac{1% }{\epsilon_{op}}-\frac{1}{\epsilon_{s}}\right)\cdot(\Delta e)^{2}
  14. ϵ \epsilon
  15. ϵ \epsilon
  16. Δ \Delta
  17. Δ \Delta
  18. Δ \Delta
  19. λ \lambda
  20. Δ \Delta
  21. Δ \Delta
  22. Δ \Delta
  23. λ \lambda
  24. [ Fe II ( CN ) 6 ] 4 - + [ Ir IV Cl 6 ] 2 - [ Fe III ( CN ) 6 ] 3 - + [ Ir III Cl 6 ] 3 - \mathrm{[Fe^{II}(CN)_{6}]^{4-}}+\mathrm{[Ir^{IV}Cl_{6}]^{2-}}% \rightleftharpoons\mathrm{[Fe^{III}(CN)_{6}]^{3-}}+\mathrm{[Ir^{III}Cl_{6}]^{3% -}}
  25. Δ \Delta
  26. Δ \Delta
  27. λ \lambda
  28. Δ \Delta
  29. Δ \Delta
  30. Δ \Delta
  31. y = x 2 \mathrm{y=x^{2}}
  32. f ( 0 ) \mathrm{f^{(0)}}
  33. y = ( x - d ) 2 \mathrm{y=(x-d)^{2}}
  34. f 1 \mathrm{f_{1}}
  35. f 3 \mathrm{f_{3}}
  36. y = ( x - d ) 2 + c \mathrm{y=(x-d)^{2}+c}
  37. Δ G = ( λ o + Δ G 0 ) 2 4 λ o \Delta G^{\ddagger}=\frac{(\lambda_{o}+\Delta G^{0})^{2}}{4\lambda_{o}}
  38. Δ \Delta
  39. Δ \Delta
  40. Δ \Delta
  41. λ \lambda
  42. Δ \Delta
  43. Δ \Delta
  44. Δ \Delta
  45. Δ \Delta
  46. ν D \nu_{D}
  47. ν A \nu_{A}
  48. f = 4 π 2 ν 2 μ f=4\pi^{2}\nu^{2}\mu
  49. E D = E D ( q 0 , D ) + 3 f D ( Δ q D ) 2 E_{D}=E_{D}(q_{0,D})+3f_{D}(\Delta q_{D})^{2}
  50. E A = E A ( q 0 , A ) + 3 f A ( Δ q A ) 2 E_{A}=E_{A}(q_{0,A})+3f_{A}(\Delta q_{A})^{2}
  51. Δ q = ( q - q 0 ) \Delta q=(q-q_{0})
  52. \cdot
  53. λ \lambda
  54. q * = f D q 0 , D + f A q 0 , A f D + f A q^{*}=\frac{f_{D}q_{0,D}+f_{A}q_{0,A}}{f_{D}+f_{A}}
  55. λ i n = Δ E * = 3 f D f A f D + f A ( q 0 , D - q 0 , A ) 2 \lambda_{in}=\Delta E^{*}=\frac{3f_{D}f_{A}}{f_{D}+f_{A}}(q_{0,D}-q_{0,A})^{2}
  56. λ = λ i n + λ o \lambda=\lambda_{in}+\lambda_{o}
  57. k a c t = A e - Δ G i n + Δ G o k T k_{act}=A\cdot e^{-\frac{\Delta{G_{in}^{\ddagger}}+\Delta{G_{o}}^{\ddagger}}{% kT}}
  58. Δ \Delta
  59. {}^{\ddagger}
  60. Δ \Delta
  61. {}^{\ddagger}
  62. [ Fe ( H 2 O ) 6 ] 2 + + [ [ Co ( H 2 O ) 6 ] ] 3 + [ Fe ( H 2 O ) 6 ] 3 + + [ Co ( H 2 O ) 6 ] 2 + \mathrm{[Fe(H_{2}O)_{6}]^{2+}}+\mathrm{[[Co(H_{2}O)_{6}]]^{3+}}% \rightleftharpoons\mathrm{[Fe(H_{2}O)_{6}]^{3+}}+\mathrm{[Co(H_{2}O)_{6}]^{2+}}
  63. λ i n \lambda_{in}
  64. Δ \Delta
  65. Δ \Delta
  66. {}^{\ddagger}
  67. P i f = 1 - exp [ - 4 π 2 H i f 2 h v ( s i - s f ) ] P_{if}=1-\exp[-\frac{4\pi^{2}{H_{if}^{2}}}{hv\mid(s_{i}-s_{f})\mid}]
  68. k e t = 2 π | H A B | 2 1 4 π λ k b T exp ( - ( λ + Δ G ) 2 4 λ k b T ) k_{et}=\frac{2\pi}{\hbar}|H_{AB}|^{2}\frac{1}{\sqrt{4\pi\lambda k_{b}T}}\exp% \left(-\frac{(\lambda+\Delta G^{\circ})^{2}}{4\lambda k_{b}T}\right)
  69. k e t k_{et}
  70. | H A B | |H_{AB}|
  71. λ \lambda
  72. Δ G \Delta G^{\circ}
  73. k b k_{b}
  74. T T
  75. Δ \Delta
  76. Δ \Delta
  77. Δ \Delta
  78. Δ \Delta
  79. λ \lambda
  80. λ \lambda
  81. λ \lambda
  82. Δ \Delta
  83. Δ \Delta
  84. Δ G = Δ G 0 2 + Δ G ( 0 ) 2 + ( Δ G 0 2 ) 2 \Delta G^{\ddagger}=\frac{\Delta G^{0}}{2}+\sqrt{\Delta G^{\ddagger}(0)^{2}+% \left(\frac{\Delta G^{0}}{2}\right)^{2}}

Marginal_rate_of_technical_substitution.html

  1. - Δ x 2 -\Delta x_{2}
  2. Δ x 1 = 1 \Delta x_{1}=1
  3. y = y ¯ y=\bar{y}
  4. M R T S ( x 1 , x 2 ) = - Δ x 2 Δ x 1 = M P 1 M P 2 MRTS(x_{1},x_{2})=-\frac{\Delta x_{2}}{\Delta x_{1}}=\frac{MP_{1}}{MP_{2}}
  5. M P 1 MP_{1}
  6. M P 2 MP_{2}

Margolus–Levitin_theorem.html

  1. h 4 E \frac{h}{4E}

Marian_Rejewski.html

  1. \rightarrow
  2. \rightarrow
  3. \rightarrow

Market_concentration.html

  1. HK α ( x ) = { ( s i α ) 1 α - 1 if α > 0 , α 1 s i s i if α = 1 \,\text{HK}_{\alpha}(x)=\begin{cases}&\left(\sum s_{i}^{\alpha}\right)^{\frac{% 1}{\alpha-1}}\,\text{ if }\alpha>0,\alpha\neq 1\\ &\prod s_{i}^{s_{i}}\,\text{ if }\alpha=1\\ \end{cases}
  2. s i s i = exp ( s i log ( s i ) ) \prod s_{i}^{s_{i}}=\exp\left(\sum s_{i}\log(s_{i})\right)
  3. X ¯ 1 = x i 2 / x i = x i H \bar{X}_{1}=\sum x_{i}^{2}/x_{i}=\sum x_{i}H
  4. R = 1 2 i s i - 1 R=\frac{1}{2\sum is_{i}-1}
  5. C C I = s 1 + i = 2 N s i 2 ( 2 - s i ) CCI=s_{1}+\sum_{i=2}^{N}s_{i}^{2}(2-s_{i})
  6. 2 H - s i 3 2\,\text{H}-\sum s_{i}^{3}
  7. L = 1 N ( N - 1 ) i = 1 N - 1 Q i L=\frac{1}{N(N-1)}\sum_{i=1}^{N-1}Q_{i}
  8. i i
  9. N - i N-i
  10. U = I * a N - 1 U=I^{*a}N^{-1}
  11. I * I^{*}
  12. a a

Markov_additive_process.html

  1. 𝔼 [ f ( X t + s - X t ) g ( J t + s ) | t ] = 𝔼 J t , 0 [ f ( X s ) g ( J s ) ] \mathbb{E}[f(X_{t+s}-X_{t})g(J_{t+s})|\mathcal{F}_{t}]=\mathbb{E}_{J_{t},0}[f(% X_{s})g(J_{s})]

Markov_logic_network.html

  1. P ( A | B ) P(A|B)

Martingale_difference_sequence.html

  1. { X t , t } - \{X_{t},\mathcal{F}_{t}\}_{-\infty}^{\infty}
  2. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  3. X t X_{t}
  4. 𝔼 [ X t ] < \mathbb{E}\left[X_{t}\right]<\infty
  5. 𝔼 [ X t | t - 1 ] = 0 , a . s . \mathbb{E}\left[X_{t}|\mathcal{F}_{t-1}\right]=0,a.s.
  6. t t
  7. Y t Y_{t}
  8. X t = Y t - Y t - 1 X_{t}=Y_{t}-Y_{t-1}

Martingale_representation_theorem.html

  1. B t B_{t}
  2. ( Ω , , t , P ) (\Omega,\mathcal{F},\mathcal{F}_{t},P)
  3. 𝒢 t \mathcal{G}_{t}
  4. B B
  5. 𝒢 \mathcal{G}_{\infty}
  6. 𝒢 t \mathcal{G}_{t}
  7. X = E ( X ) + 0 C s d B s . X=E(X)+\int_{0}^{\infty}C_{s}\,dB_{s}.
  8. E ( X | 𝒢 t ) = E ( X ) + 0 t C s d B s . E(X|\mathcal{G}_{t})=E(X)+\int_{0}^{t}C_{s}\,dB_{s}.
  9. ( M t ) 0 t < \left(M_{t}\right)_{0\leq t<\infty}
  10. σ t \sigma_{t}
  11. ( N t ) 0 t < \left(N_{t}\right)_{0\leq t<\infty}
  12. \mathcal{F}
  13. ϕ \phi
  14. 0 T ϕ t 2 σ t 2 d t < \int_{0}^{T}\phi_{t}^{2}\sigma_{t}^{2}\,dt<\infty
  15. N t = N 0 + 0 t ϕ s d M s . N_{t}=N_{0}+\int_{0}^{t}\phi_{s}\,dM_{s}.
  16. ϕ t \phi_{t}
  17. ψ t B t = C t - ϕ t Z t \psi_{t}B_{t}=C_{t}-\phi_{t}Z_{t}
  18. Z t Z_{t}
  19. t t
  20. C t C_{t}
  21. t t
  22. V T = ϕ T S T + ψ T B T = C T = X V_{T}=\phi_{T}S_{T}+\psi_{T}B_{T}=C_{T}=X
  23. ( d V t = ϕ t d S t + ψ t d B t ) \left(dV_{t}=\phi_{t}dS_{t}+\psi_{t}\,dB_{t}\right)

Mason's_gain_formula.html

  1. G = y out y in = k = 1 N G k Δ k Δ G=\frac{y\text{out}}{y\text{in}}=\frac{\sum_{k=1}^{N}{G_{k}\Delta_{k}}}{\Delta\ }
  2. Δ = 1 - L i + L i L j - L i L j L k + + ( - 1 ) m + \Delta=1-\sum L_{i}+\sum L_{i}L_{j}-\sum L_{i}L_{j}L_{k}+\cdots+(-1)^{m}\sum% \cdots+\cdots
  3. G 1 = - y 21 R L G_{1}=-y_{21}R_{L}\,
  4. L 1 = - R i n y 11 L_{1}=-R_{in}y_{11}\,
  5. L 2 = - R L y 22 L_{2}=-R_{L}y_{22}\,
  6. L 3 = y 21 R L y 12 R i n L_{3}=y_{21}R_{L}y_{12}R_{in}\,
  7. Δ = 1 - ( L 1 + L 2 + L 3 ) + ( L 1 L 2 ) \Delta=1-(L_{1}+L_{2}+L_{3})+(L_{1}L_{2})\,
  8. Δ 1 = 1 \Delta_{1}=1\,
  9. G = G 1 Δ 1 Δ = - y 21 R L 1 + R i n y 11 + R L y 22 - y 21 R L y 12 R i n + R i n y 11 R L y 22 G=\frac{G_{1}\Delta_{1}}{\Delta}=\frac{-y_{21}R_{L}}{1+R_{in}y_{11}+R_{L}y_{22% }-y_{21}R_{L}y_{12}R_{in}+R_{in}y_{11}R_{L}y_{22}}\,
  10. L 1 = - a 1 Z - 1 L_{1}=-a_{1}Z^{-1}\,
  11. L 2 = - a 2 Z - 2 L_{2}=-a_{2}Z^{-2}\,
  12. Δ = 1 - ( L 1 + L 2 ) \Delta=1-(L_{1}+L_{2})\,
  13. G 0 = b 0 G_{0}=b_{0}\,
  14. G 1 = b 1 Z - 1 G_{1}=b_{1}Z^{-1}\,
  15. G 2 = b 2 Z - 2 G_{2}=b_{2}Z^{-2}\,
  16. Δ 0 = Δ 1 = Δ 2 = 1 \Delta_{0}=\Delta_{1}=\Delta_{2}=1\,
  17. G = G 0 Δ 0 + G 1 Δ 1 + G 2 Δ 2 Δ G=\frac{G_{0}\Delta_{0}+G_{1}\Delta_{1}+G_{2}\Delta_{2}}{\Delta}\,
  18. G = b 0 + b 1 Z - 1 + b 2 Z - 2 1 + a 1 Z - 1 + a 2 Z - 2 G=\frac{b_{0}+b_{1}Z^{-1}+b_{2}Z^{-2}}{1+a_{1}Z^{-1}+a_{2}Z^{-2}}\,
  19. L 0 = - β s M L_{0}=-\frac{\beta}{sM}\,
  20. L 1 = - ( R M + R S ) s L M L_{1}=\frac{-(R_{M}+R_{S})}{sL_{M}}\,
  21. L 2 = - G M K M s 2 L M M L_{2}=\,\frac{-G_{M}K_{M}}{s^{2}L_{M}M}
  22. L 3 = - K C R S s L M L_{3}=\frac{-K_{C}R_{S}}{sL_{M}}\,
  23. L 4 = - K V K C K M G T s 2 L M M L_{4}=\frac{-K_{V}K_{C}K_{M}G_{T}}{s^{2}L_{M}M}\,
  24. L 5 = - K P K V K C K M s 3 L M M L_{5}=\frac{-K_{P}K_{V}K_{C}K_{M}}{s^{3}L_{M}M}\,
  25. Δ = 1 - ( L 0 + L 1 + L 2 + L 3 + L 4 + L 5 ) + ( L 0 L 1 + L 0 L 3 ) \Delta=1-(L_{0}+L_{1}+L_{2}+L_{3}+L_{4}+L_{5})+(L_{0}L_{1}+L_{0}L_{3})\,
  26. g 0 = - K P K V K C K M s 3 L M M g_{0}=\frac{-K_{P}K_{V}K_{C}K_{M}}{s^{3}L_{M}M}\,
  27. Δ 0 = 1 \Delta_{0}=1
  28. θ L θ C = g 0 Δ 0 Δ \frac{\theta_{L}}{\theta_{C}}=\frac{g_{0}\Delta_{0}}{\Delta}\,
  29. 𝐓 \mathbf{T}
  30. t n m = [ 𝐓 ] n m t_{nm}=\left[\mathbf{T}\right]_{nm}
  31. u n m = [ 𝐔 ] n m u_{nm}=\left[\mathbf{U}\right]_{nm}
  32. 𝐔 = ( 𝐈 - 𝐓 ) - 1 \mathbf{U}=\left(\mathbf{I}-\mathbf{T}\right)^{-1}
  33. 𝐈 \mathbf{I}
  34. n n
  35. y i n y_{in}
  36. y o u t y_{out}
  37. ( n - 2 ) ! (n-2)!
  38. p 1 - q , \frac{p}{1-q},
  39. q q
  40. R ( 1 + L i ) - 1 R(1+\langle L_{i}\rangle)^{-1}

Mason–Weaver_equation.html

  1. c t = D 2 c z 2 + s g c z \frac{\partial c}{\partial t}=D\frac{\partial^{2}c}{\partial z^{2}}+sg\frac{% \partial c}{\partial z}
  2. D c z + s g c = 0 D\frac{\partial c}{\partial z}+sgc=0
  3. z a z_{a}
  4. z b z_{b}
  5. N t o t = z b z a d z c ( z , t ) N_{tot}=\int_{z_{b}}^{z_{a}}dz\ c(z,t)
  6. d N t o t / d t = 0 dN_{tot}/dt=0
  7. f v fv
  8. m g mg
  9. ρ V g \rho Vg
  10. ρ \rho
  11. v t e r m v_{term}
  12. ν ¯ \bar{\nu}
  13. f v t e r m = m ( 1 - ν ¯ ρ ) g = def m b g fv_{term}=m(1-\bar{\nu}\rho)g\ \stackrel{\mathrm{def}}{=}\ m_{b}g
  14. m b m_{b}
  15. s = def m b / f = v t e r m / g s\ \stackrel{\mathrm{def}}{=}\ m_{b}/f=v_{term}/g
  16. D = k B T f D=\frac{k_{B}T}{f}
  17. s D = m b k B T \frac{s}{D}=\frac{m_{b}}{k_{B}T}
  18. k B k_{B}
  19. J = - D c z - v t e r m c = - D c z - s g c . J=-D\frac{\partial c}{\partial z}-v_{term}c=-D\frac{\partial c}{\partial z}-sgc.
  20. v t e r m v_{term}
  21. c t = - J z . \frac{\partial c}{\partial t}=-\frac{\partial J}{\partial z}.
  22. c t = D 2 c z 2 + s g c z . \frac{\partial c}{\partial t}=D\frac{\partial^{2}c}{\partial z^{2}}+sg\frac{% \partial c}{\partial z}.
  23. z 0 z_{0}
  24. z 0 = def D s g z_{0}\ \stackrel{\mathrm{def}}{=}\ \frac{D}{sg}
  25. t 0 t_{0}
  26. t 0 = def D s 2 g 2 t_{0}\ \stackrel{\mathrm{def}}{=}\ \frac{D}{s^{2}g^{2}}
  27. ζ = def z / z 0 \zeta\ \stackrel{\mathrm{def}}{=}\ z/z_{0}
  28. τ = def t / t 0 \tau\ \stackrel{\mathrm{def}}{=}\ t/t_{0}
  29. c τ = 2 c ζ 2 + c ζ \frac{\partial c}{\partial\tau}=\frac{\partial^{2}c}{\partial\zeta^{2}}+\frac{% \partial c}{\partial\zeta}
  30. c ζ + c = 0 \frac{\partial c}{\partial\zeta}+c=0
  31. ζ a \zeta_{a}
  32. ζ b \zeta_{b}
  33. c ( ζ , τ ) = def e - ζ / 2 T ( τ ) P ( ζ ) c(\zeta,\tau)\ \stackrel{\mathrm{def}}{=}\ e^{-\zeta/2}T(\tau)P(\zeta)
  34. β \beta
  35. d T d τ + β T = 0 \frac{dT}{d\tau}+\beta T=0
  36. d 2 P d ζ 2 + [ β - 1 4 ] P = 0 \frac{d^{2}P}{d\zeta^{2}}+\left[\beta-\frac{1}{4}\right]P=0
  37. β \beta
  38. d P d ζ + 1 2 P = 0 \frac{dP}{d\zeta}+\frac{1}{2}P=0
  39. ζ a \zeta_{a}
  40. ζ b \zeta_{b}
  41. T ( τ ) = T 0 e - β τ T(\tau)=T_{0}e^{-\beta\tau}
  42. T 0 T_{0}
  43. P ( ζ ) P(\zeta)
  44. P k ( ζ ) P_{k}(\zeta)
  45. β k \beta_{k}
  46. β 0 \beta_{0}
  47. k 2 k^{2}
  48. c ( ζ , τ ) c(\zeta,\tau)
  49. c ( ζ , τ ) = k = 0 c k P k ( ζ ) e - β k τ c(\zeta,\tau)=\sum_{k=0}^{\infty}c_{k}P_{k}(\zeta)e^{-\beta_{k}\tau}
  50. c k c_{k}
  51. c ( ζ , τ = 0 ) c(\zeta,\tau=0)
  52. c k = ζ a ζ b d ζ c ( ζ , τ = 0 ) e ζ / 2 P k ( ζ ) c_{k}=\int_{\zeta_{a}}^{\zeta_{b}}d\zeta\ c(\zeta,\tau=0)e^{\zeta/2}P_{k}(\zeta)
  53. β = 0 \beta=0
  54. e - ζ / 2 P 0 ( ζ ) = B e - ζ = B e - m b g z / k B T e^{-\zeta/2}P_{0}(\zeta)=Be^{-\zeta}=Be^{-m_{b}gz/k_{B}T}
  55. P 0 ( ζ ) P_{0}(\zeta)
  56. ζ \zeta
  57. B = N t o t ( s g D ) ( 1 e - ζ b - e - ζ a ) B=N_{tot}\left(\frac{sg}{D}\right)\left(\frac{1}{e^{-\zeta_{b}}-e^{-\zeta_{a}}% }\right)
  58. β k \beta_{k}
  59. P ( ζ ) = e i ω k ζ P(\zeta)=e^{i\omega_{k}\zeta}
  60. ω k = ± β k - 1 4 \omega_{k}=\pm\sqrt{\beta_{k}-\frac{1}{4}}
  61. β k \beta_{k}
  62. ω k \omega_{k}
  63. β k 1 4 \beta_{k}\geq\frac{1}{4}
  64. β k < 1 4 \beta_{k}<\frac{1}{4}
  65. P ( ζ ) = A cos ω k ζ + B sin ω k ζ P(\zeta)=A\cos{\omega_{k}\zeta}+B\sin{\omega_{k}\zeta}
  66. ω \omega
  67. ρ \rho
  68. ϕ \phi
  69. u = def ρ sin ( ϕ ) = def P u\ \stackrel{\mathrm{def}}{=}\ \rho\sin(\phi)\ \stackrel{\mathrm{def}}{=}\ P
  70. v = def ρ cos ( ϕ ) = def - 1 ω ( d P d ζ ) v\ \stackrel{\mathrm{def}}{=}\ \rho\cos(\phi)\ \stackrel{\mathrm{def}}{=}\ -% \frac{1}{\omega}\left(\frac{dP}{d\zeta}\right)
  71. ρ = def u 2 + v 2 \rho\ \stackrel{\mathrm{def}}{=}\ u^{2}+v^{2}
  72. tan ( ϕ ) = def v / u \tan(\phi)\ \stackrel{\mathrm{def}}{=}\ v/u
  73. d ρ d ζ = 0 \frac{d\rho}{d\zeta}=0
  74. d ϕ d ζ = ω \frac{d\phi}{d\zeta}=\omega
  75. ρ \rho
  76. ζ a \zeta_{a}
  77. ζ b \zeta_{b}
  78. tan ( ϕ a ) = tan ( ϕ b ) = 1 2 ω k \tan(\phi_{a})=\tan(\phi_{b})=\frac{1}{2\omega_{k}}
  79. ϕ a - ϕ b + k π = k π = ζ b ζ a d ζ d ϕ d ζ = ω k ( ζ a - ζ b ) \phi_{a}-\phi_{b}+k\pi=k\pi=\int_{\zeta_{b}}^{\zeta_{a}}d\zeta\ \frac{d\phi}{d% \zeta}=\omega_{k}(\zeta_{a}-\zeta_{b})
  80. ω k \omega_{k}
  81. ω k = k π ζ a - ζ b \omega_{k}=\frac{k\pi}{\zeta_{a}-\zeta_{b}}
  82. ω k \omega_{k}
  83. ζ a > ζ b \zeta_{a}>\zeta_{b}
  84. ω 1 = def π / ( ζ a - ζ b ) \omega_{1}\ \stackrel{\mathrm{def}}{=}\ \pi/(\zeta_{a}-\zeta_{b})
  85. β k \beta_{k}
  86. ω k \omega_{k}
  87. β k = ω k 2 + 1 4 \beta_{k}=\omega_{k}^{2}+\frac{1}{4}
  88. c ( ζ , τ = 0 ) c(\zeta,\tau=0)
  89. e ζ / 2 e^{\zeta/2}
  90. e - β k τ e^{-\beta_{k}\tau}
  91. β k \beta_{k}
  92. ω k \omega_{k}

Mass-to-charge_ratio.html

  1. 𝐅 = Q ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{F}=Q(\mathbf{E}+\mathbf{v}\times\mathbf{B}),
  2. 𝐅 = m 𝐚 = m d 𝐯 d t \mathbf{F}=m\mathbf{a}=m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}
  3. ( m Q ) 𝐚 = 𝐄 + 𝐯 × 𝐁 \left(\frac{m}{Q}\right)\mathbf{a}=\mathbf{E}+\mathbf{v}\times\mathbf{B}
  4. e / m e e/m_{e}
  5. e / m e e/m_{e}
  6. e / m e e/m_{e}
  7. Δ E = e B 2 m ( m j , f g J , f - m j , i g J , i ) \Delta E=\frac{e\hbar B}{2m}(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})
  8. 𝐉 = 𝐋 + 𝐒 \mathbf{J}=\mathbf{L}+\mathbf{S}
  9. g J = 1 + j ( j + 1 ) + s ( s + 1 ) - l ( l + 1 ) 2 j ( j + 1 ) g_{J}=1+\frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}
  10. Δ E = h Δ ν = h c Δ ( 1 λ ) = h c Δ λ λ 2 \Delta E=h\Delta\nu=hc\Delta\left(\frac{1}{\lambda}\right)=hc\frac{\Delta% \lambda}{\lambda^{2}}
  11. Δ λ = λ 2 δ D 2 D Δ D \Delta\lambda=\lambda^{2}\frac{\delta D}{2D\Delta D}
  12. h c Δ λ λ 2 = h c δ D 2 D Δ D = e B 2 m ( m j , f g J , f - m j , i g J , i ) . hc\frac{\Delta\lambda}{\lambda^{2}}=hc\frac{\delta D}{2D\Delta D}=\frac{e\hbar B% }{2m}(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})\ .
  13. e m = 4 π c B ( m j , f g J , f - m j , i g J , i ) δ D D Δ D . \frac{e}{m}=\frac{4\pi c}{B(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})}\frac{\delta D}{D% \Delta D}\ .

Mass-to-light_ratio.html

  1. Υ \Upsilon
  2. Υ \Upsilon_{\odot}
  3. Υ \Upsilon_{\odot}
  4. Υ \Upsilon_{\odot}
  5. Υ \Upsilon_{\odot}

Mass_index.html

  1. M a s s = S u m [ 25 ] o f E M A [ 9 ] o f ( h i g h - l o w ) E M A [ 9 ] o f E M A [ 9 ] o f ( h i g h - l o w ) Mass=Sum[25]\;of\;{EMA[9]\;of\;(high-low)\over EMA[9]\,of\,EMA[9]\;of\;(high-% low)}

Mass_transfer_coefficient.html

  1. k c = n ˙ A A Δ c A k_{c}=\frac{\dot{n}_{A}}{A\Delta c_{A}}
  2. n ˙ A \dot{n}_{A}
  3. Δ c A {\Delta c_{A}}

Material_properties_(thermodynamics).html

  1. β T = - 1 V ( V P ) T = - 1 V 2 G P 2 \beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\quad=-% \frac{1}{V}\,\frac{\partial^{2}G}{\partial P^{2}}
  2. β S = - 1 V ( V P ) S = - 1 V 2 H P 2 \beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\quad=-% \frac{1}{V}\,\frac{\partial^{2}H}{\partial P^{2}}
  3. c P = T N ( S T ) P = - T N 2 G T 2 c_{P}=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)_{P}\quad=-\frac{T}% {N}\,\frac{\partial^{2}G}{\partial T^{2}}
  4. c V = T N ( S T ) V = - T N 2 A T 2 c_{V}=\frac{T}{N}\left(\frac{\partial S}{\partial T}\right)_{V}\quad=-\frac{T}% {N}\,\frac{\partial^{2}A}{\partial T^{2}}
  5. α = 1 V ( V T ) P = 1 V 2 G P T \alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\quad=\frac{1}% {V}\,\frac{\partial^{2}G}{\partial P\partial T}
  6. β T \beta_{T}
  7. c P c_{P}
  8. α \alpha
  9. c P = c V + T V α 2 N β T c_{P}=c_{V}+\frac{TV\alpha^{2}}{N\beta_{T}}
  10. β T = β S + T V α 2 N c P \beta_{T}=\beta_{S}+\frac{TV\alpha^{2}}{Nc_{P}}

Mathematics_in_medieval_Islam.html

  1. x 3 + a = b x \ x^{3}+a=bx
  2. y = b x - x 3 \ y=bx-x^{3}
  3. x = b 3 x=\textstyle\sqrt{\frac{b}{3}}

Matrix_grammar.html

  1. G = ( V N , V T , X 0 , M ) . G=(V_{N},V_{T},X_{0},M).
  2. V N V_{N}
  3. V T V_{T}
  4. X 0 X_{0}
  5. V N V_{N}
  6. M M
  7. ( P , Q ) , P W ( V ) V N W ( V ) , Q W ( V ) , V = V N V T . (P,Q),\quad P\in W(V)V_{N}W(V),\quad Q\in W(V),\quad V=V_{N}\cup V_{T}.
  8. P Q P\to Q
  9. m = [ P 1 Q 1 , , P r Q r ] . m=[P_{1}\to Q_{1},\ldots,P_{r}\to Q_{r}].
  10. F F
  11. m m
  12. G G
  13. G G
  14. i , i = 0 , 1 , 2 , 3 i,i=0,1,2,3
  15. λ \lambda
  16. G 1 = ( V N , V T , X 0 , F ) G_{1}=(V_{N},V_{T},X_{0},F)
  17. G G
  18. G \Rightarrow_{G}
  19. \Rightarrow
  20. P , Q W ( V ) P,Q\in W(V)
  21. P Q P\Rightarrow Q
  22. r 1 r\geq 1
  23. α 1 , , , α r + 1 , P 1 , , P r , R 1 , , R r , , R 1 , , R r \alpha_{1},,\ldots,\alpha_{r+1},\quad P_{1},\ldots,P_{r},\quad R_{1},\ldots,R_% {r},\quad,R^{1},\ldots,R^{r}
  24. α i = P \alpha_{i}=P
  25. α r + 1 = Q \alpha_{r+1}=Q
  26. m m
  27. G G
  28. α i = R i P i R i \alpha_{i}=R_{i}P_{i}R^{i}
  29. α i + 1 = R i Q i R i . \alpha_{i+1}=R_{i}Q_{i}R^{i}.
  30. P Q P\Rightarrow Q
  31. ( m , R 1 ) (m,R_{1})
  32. * \Rightarrow^{*}
  33. \Rightarrow
  34. G G
  35. L ( G ) = { P W ( V T ) | X 0 * P } . L(G)=\{P\in W(V_{T})|X_{0}\Rightarrow^{*}P\}.
  36. G = ( { S , X , Y } , { a , b , c } , S , M ) G=(\{S,X,Y\},\{a,b,c\},S,M)
  37. M M
  38. [ S X Y ] , [ X a X b , Y c Y ] , [ X a b , Y c ] [S\rightarrow XY],\quad[X\rightarrow aXb,Y\rightarrow cY],\quad[X\rightarrow ab% ,Y\rightarrow c]
  39. L = { a n b n c n | n 1 } . L=\{a^{n}b^{n}c^{n}|n\geq 1\}.
  40. M A T λ MAT^{\lambda}
  41. M A T MAT
  42. λ \lambda
  43. M A T MAT
  44. M A T λ MAT^{\lambda}
  45. M A T MAT
  46. M A T λ MAT^{\lambda}
  47. M A T MAT
  48. M A T MAT
  49. M A T λ MAT^{\lambda}
  50. M A T λ MAT^{\lambda}
  51. M A T MAT
  52. M A T λ MAT^{\lambda}

Matrix_group.html

  1. M 1 = [ 0 0 1 1 0 0 0 1 0 ] M_{1}=\begin{bmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{bmatrix}
  2. M 2 = [ 0 1 0 1 0 0 0 0 1 ] M_{2}=\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{bmatrix}

Maupertuis'_principle.html

  1. N N
  2. 𝐪 = ( q 1 , q 2 , , q N ) \mathbf{q}=\left(q_{1},q_{2},\ldots,q_{N}\right)
  3. 𝐪 1 \mathbf{q}_{1}
  4. 𝐪 2 \mathbf{q}_{2}
  5. 𝒮 0 [ 𝐪 ( t ) ] = def 𝐩 d 𝐪 \mathcal{S}_{0}[\mathbf{q}(t)]\ \stackrel{\mathrm{def}}{=}\ \int\mathbf{p}% \cdot d\mathbf{q}
  6. 𝐩 = ( p 1 , p 2 , , p N ) \mathbf{p}=\left(p_{1},p_{2},\ldots,p_{N}\right)
  7. p k = def L q ˙ k p_{k}\ \stackrel{\mathrm{def}}{=}\ \frac{\partial L}{\partial\dot{q}_{k}}
  8. L ( 𝐪 , 𝐪 ˙ , t ) L(\mathbf{q},\dot{\mathbf{q}},t)
  9. 𝒮 0 \mathcal{S}_{0}
  10. 𝒮 0 \mathcal{S}_{0}
  11. T T
  12. 𝐪 ˙ \dot{\mathbf{q}}
  13. T = 1 2 d 𝐪 d t 𝐌 d 𝐪 d t T=\frac{1}{2}\frac{d\mathbf{q}}{dt}\cdot\mathbf{M}\cdot\frac{d\mathbf{q}}{dt}
  14. 𝐌 \mathbf{M}
  15. 𝐪 \mathbf{q}
  16. 2 T = 𝐩 𝐪 ˙ 2T=\mathbf{p}\cdot\dot{\mathbf{q}}
  17. V ( 𝐪 ) V(\mathbf{q})
  18. d s ds
  19. d s 2 = d 𝐪 𝐌 d 𝐪 ds^{2}=d\mathbf{q}\cdot\mathbf{M}\cdot d\mathbf{q}
  20. T = 1 2 ( d s d t ) 2 T=\frac{1}{2}\left(\frac{ds}{dt}\right)^{2}
  21. 2 T d t = 𝐩 d 𝐪 = 2 m T d s . 2Tdt=\mathbf{p}\cdot d\mathbf{q}=\sqrt{2mT}\ ds.
  22. 𝒮 0 = def 𝐩 d 𝐪 = d s 2 m E t o t - V ( 𝐪 ) \mathcal{S}_{0}\ \stackrel{\mathrm{def}}{=}\ \int\mathbf{p}\cdot d\mathbf{q}=% \int ds\sqrt{2m}\sqrt{E_{tot}-V(\mathbf{q})}
  23. T = E t o t - V ( 𝐪 ) T=E_{tot}-V(\mathbf{q})
  24. E t o t E_{tot}
  25. V ( 𝐪 ) V(\mathbf{q})
  26. s = d s s=\int ds
  27. 𝒮 = def L d t \mathcal{S}\ \stackrel{\mathrm{def}}{=}\ \int L\,dt
  28. t 1 t_{1}
  29. t 2 t_{2}
  30. q 1 q_{1}
  31. q 2 q_{2}
  32. q 1 q_{1}
  33. q 2 q_{2}
  34. 𝐪 ( t ) \mathbf{q}(t)
  35. q 1 q_{1}
  36. q 2 q_{2}
  37. t 1 t_{1}
  38. t 2 t_{2}
  39. q 1 q_{1}
  40. q 2 q_{2}
  41. v d s \int v\,ds
  42. 𝒮 0 = def m v d s = def p d q \mathcal{S}_{0}\ \stackrel{\mathrm{def}}{=}\ \int mv\,ds\ \stackrel{\mathrm{% def}}{=}\ \int p\,dq

MAX-3SAT.html

  1. y R ¯ \bar{y_{R}}
  2. 1 2 1 q 2 q \frac{1}{2}\frac{1}{q2^{q}}
  3. z L π P r [ V π ( x ) = 1 ] 1 - ϵ z\in L\implies\exists\pi Pr[V^{\pi}(x)=1]\geq 1-\epsilon
  4. z L π P r [ V π ( x ) = 1 ] 1 2 + ϵ z\not\in L\implies\forall\pi Pr[V^{\pi}(x)=1]\leq\frac{1}{2}+\epsilon
  5. 1 - 1 4 ( 1 2 - ϵ ) 1 - ϵ = 7 8 + ϵ \frac{1-\frac{1}{4}(\frac{1}{2}-\epsilon)}{1-\epsilon}=\frac{7}{8}+\epsilon^{\prime}
  6. 7 / 8 + Ω ( 1 / B ) 7/8+\Omega(1/B)
  7. 7 / 8 + O ( 1 / B ) 7/8+O(1/\sqrt{B})
  8. k k
  9. 1 - ( 1 / 2 ) k 1-(1/2)^{k}

Max_Q.html

  1. q = 1 2 ρ v 2 , q=\tfrac{1}{2}\,\rho\,v^{2},

Maximum_satisfiability_problem.html

  1. ( x 0 x 1 ) ( x 0 ¬ x 1 ) ( ¬ x 0 x 1 ) ( ¬ x 0 ¬ x 1 ) (x_{0}\lor x_{1})\land(x_{0}\lor\lnot x_{1})\land(\lnot x_{0}\lor x_{1})\land(% \lnot x_{0}\lor\lnot x_{1})

Maximum_term_method.html

  1. n ! n!
  2. n 1 / 2 n n / e n n^{1/2}n^{n}/e^{n}
  3. lim M ( ln N = 1 M N ! ) = ln M ! \lim_{M\rightarrow\infty}(\ln{\sum_{N=1}^{M}N!})=\ln{M!}
  4. S = N = 1 M T N S=\sum_{N=1}^{M}T_{N}
  5. T N T_{N}
  6. T m a x T_{max}
  7. T m a x S M T m a x . T_{max}\leq S\leq MT_{max}.
  8. ln T m a x ln S ln T m a x + ln M . \ln T_{max}\leq\ln S\leq\ln T_{max}+\ln M.
  9. T m a x T_{max}
  10. O ( e M ) O(e^{M})
  11. O ( M ) ln S O ( M ) + ln M . O(M)\leq\ln S\leq O(M)+\ln M.
  12. ln T m a x \ln T_{max}
  13. ln S = ln T m a x . \ln S=\ln T_{max}.

Maxwell's_equations_in_curved_spacetime.html

  1. F α β = α A β - β A α F_{\alpha\beta}\,=\,\partial_{\alpha}A_{\beta}\,-\,\partial_{\beta}A_{\alpha}\,
  2. 𝒟 μ ν = 1 μ 0 g μ α F α β g β ν - g \mathcal{D}^{\mu\nu}\,=\,\frac{1}{\mu_{0}}\,g^{\mu\alpha}\,F_{\alpha\beta}\,g^% {\beta\nu}\,\sqrt{-g}\,
  3. J μ = ν 𝒟 μ ν J^{\mu}\,=\,\partial_{\nu}\mathcal{D}^{\mu\nu}\,
  4. f μ = F μ ν J ν f_{\mu}\,=\,F_{\mu\nu}\,J^{\nu}\,
  5. 𝒟 μ ν \mathcal{D}^{\mu\nu}
  6. A ¯ β ( x ¯ ) = x γ x ¯ β A γ ( x ) . \bar{A}_{\beta}(\bar{x})=\frac{\partial x^{\gamma}}{\partial\bar{x}^{\beta}}A_% {\gamma}(x)\,.
  7. F α β = α A β - β A α . F_{\alpha\beta}\,=\,\partial_{\alpha}A_{\beta}\,-\,\partial_{\beta}A_{\alpha}\,.
  8. F ¯ α β = A ¯ β x ¯ α - A ¯ α x ¯ β = x ¯ α ( x γ x ¯ β A γ ) - x ¯ β ( x δ x ¯ α A δ ) = 2 x γ x ¯ α x ¯ β A γ + x γ x ¯ β A γ x ¯ α - 2 x δ x ¯ β x ¯ α A δ - x δ x ¯ α A δ x ¯ β = x γ x ¯ β x δ x ¯ α A γ x δ - x δ x ¯ α x γ x ¯ β A δ x γ = x δ x ¯ α x γ x ¯ β ( A γ x δ - A δ x γ ) = x δ x ¯ α x γ x ¯ β F δ γ . \begin{aligned}\displaystyle\bar{F}_{\alpha\beta}&\displaystyle=\frac{\partial% \bar{A}_{\beta}}{\partial\bar{x}^{\alpha}}\,-\,\frac{\partial\bar{A}_{\alpha}}% {\partial\bar{x}^{\beta}}\\ &\displaystyle=\,\frac{\partial}{\partial\bar{x}^{\alpha}}\left(\frac{\partial x% ^{\gamma}}{\partial\bar{x}^{\beta}}A_{\gamma}\right)\,-\,\frac{\partial}{% \partial\bar{x}^{\beta}}\left(\frac{\partial x^{\delta}}{\partial\bar{x}^{% \alpha}}A_{\delta}\right)\\ &\displaystyle=\,\frac{\partial^{2}x^{\gamma}}{\partial\bar{x}^{\alpha}\,% \partial\bar{x}^{\beta}}A_{\gamma}\,+\,\frac{\partial x^{\gamma}}{\partial\bar% {x}^{\beta}}\frac{\partial A_{\gamma}}{\partial\bar{x}^{\alpha}}\,-\,\frac{% \partial^{2}x^{\delta}}{\partial\bar{x}^{\beta}\,\partial\bar{x}^{\alpha}}A_{% \delta}\,-\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\frac{\partial A% _{\delta}}{\partial\bar{x}^{\beta}}\\ &\displaystyle=\,\frac{\partial x^{\gamma}}{\partial\bar{x}^{\beta}}\frac{% \partial x^{\delta}}{\partial\bar{x}^{\alpha}}\frac{\partial A_{\gamma}}{% \partial x^{\delta}}\,-\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}% \frac{\partial x^{\gamma}}{\partial\bar{x}^{\beta}}\frac{\partial A_{\delta}}{% \partial x^{\gamma}}\\ &\displaystyle=\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\frac{% \partial x^{\gamma}}{\partial\bar{x}^{\beta}}\,\left(\frac{\partial A_{\gamma}% }{\partial x^{\delta}}\,-\,\frac{\partial A_{\delta}}{\partial x^{\gamma}}% \right)\\ &\displaystyle=\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\frac{% \partial x^{\gamma}}{\partial\bar{x}^{\beta}}\,F_{\delta\gamma}\ .\end{aligned}
  9. λ F μ ν + μ F ν λ + ν F λ μ = 0 \partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{% \lambda\mu}=0\,
  10. λ F μ ν + μ F ν λ + ν F λ μ \partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{% \lambda\mu}\,
  11. = λ μ A ν - λ ν A μ + μ ν A λ - μ λ A ν + ν λ A μ - ν μ A λ = 0 . =\,\partial_{\lambda}\partial_{\mu}A_{\nu}-\partial_{\lambda}\partial_{\nu}A_{% \mu}+\partial_{\mu}\partial_{\nu}A_{\lambda}-\partial_{\mu}\partial_{\lambda}A% _{\nu}+\partial_{\nu}\partial_{\lambda}A_{\mu}-\partial_{\nu}\partial_{\mu}A_{% \lambda}\,=0\,.
  12. F [ μ ν ; λ ] = F [ μ ν , λ ] = 1 6 ( λ F μ ν + μ F ν λ + ν F λ μ - λ F ν μ - μ F λ ν - ν F μ λ ) F_{[\mu\nu;\lambda]}\,=\,F_{[\mu\nu,\lambda]}\,=\,\frac{1}{6}\left(\partial_{% \lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}-% \partial_{\lambda}F_{\nu\mu}-\partial_{\mu}F_{\lambda\nu}-\partial_{\nu}F_{\mu% \lambda}\right)\,
  13. = 1 3 ( λ F μ ν + μ F ν λ + ν F λ μ ) = 0 =\,\frac{1}{3}\left(\partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+% \partial_{\nu}F_{\lambda\mu}\right)=0\,
  14. F α β ; γ = F α β , γ - Γ μ α γ F μ β - Γ μ β γ F α μ F_{\alpha\beta;\gamma}\,=\,F_{\alpha\beta,\gamma}-{\Gamma^{\mu}}_{\alpha\gamma% }F_{\mu\beta}-{\Gamma^{\mu}}_{\beta\gamma}F_{\alpha\mu}\,
  15. 𝒟 μ ν = 1 μ 0 g μ α F α β g β ν - g . \mathcal{D}^{\mu\nu}\,=\,\frac{1}{\mu_{0}}\,g^{\mu\alpha}\,F_{\alpha\beta}\,g^% {\beta\nu}\,\sqrt{-g}\,.
  16. 𝒟 μ ν = 1 μ 0 g μ α F α β g β ν - g - μ ν . \mathcal{D}^{\mu\nu}\,=\,\frac{1}{\mu_{0}}\,g^{\mu\alpha}\,F_{\alpha\beta}\,g^% {\beta\nu}\,\sqrt{-g}\,-\,\mathcal{M}^{\mu\nu}\,.
  17. 𝒟 ¯ μ ν = x ¯ μ x α x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] \bar{\mathcal{D}}^{\mu\nu}\,=\,\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha% }}\,\frac{\partial\bar{x}^{\nu}}{\partial x^{\beta}}\,\mathcal{D}^{\alpha\beta% }\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\,
  18. J μ = ν 𝒟 μ ν . J^{\mu}\,=\,\partial_{\nu}\mathcal{D}^{\mu\nu}\,.
  19. J free μ = ν 𝒟 μ ν . J^{\mu}_{\,\text{free}}\,=\,\partial_{\nu}\mathcal{D}^{\mu\nu}\,.
  20. μ J μ = μ ν 𝒟 μ ν = 0 \partial_{\mu}J^{\mu}\,=\,\partial_{\mu}\partial_{\nu}\mathcal{D}^{\mu\nu}=0\,
  21. J ¯ μ = x ¯ μ x α J α det [ x σ x ¯ ρ ] . \bar{J}^{\mu}\,=\,\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,J^{\alpha% }\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\,.
  22. J ¯ μ = x ¯ ν ( 𝒟 ¯ μ ν ) = x ¯ ν ( x ¯ μ x α x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] ) = 2 x ¯ μ x ¯ ν x α x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] + x ¯ μ x α 2 x ¯ ν x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] + x ¯ μ x α x ¯ ν x β 𝒟 α β x ¯ ν det [ x σ x ¯ ρ ] + x ¯ μ x α x ¯ ν x β 𝒟 α β x ¯ ν det [ x σ x ¯ ρ ] = 2 x ¯ μ x β x α 𝒟 α β det [ x σ x ¯ ρ ] + x ¯ μ x α 2 x ¯ ν x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] + x ¯ μ x α 𝒟 α β x β det [ x σ x ¯ ρ ] + x ¯ μ x α x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] x ¯ ρ x σ 2 x σ x ¯ ν x ¯ ρ = 0 + x ¯ μ x α 2 x ¯ ν x ¯ ν x β 𝒟 α β det [ x σ x ¯ ρ ] + x ¯ μ x α J α det [ x σ x ¯ ρ ] + x ¯ μ x α 𝒟 α β det [ x σ x ¯ ρ ] x ¯ ρ x σ 2 x σ x β x ¯ ρ = x ¯ μ x α J α det [ x σ x ¯ ρ ] + x ¯ μ x α 𝒟 α β det [ x σ x ¯ ρ ] ( 2 x ¯ ν x ¯ ν x β + x ¯ ρ x σ 2 x σ x β x ¯ ρ ) \begin{aligned}\displaystyle\bar{J}^{\mu}&\displaystyle=\,\frac{\partial}{% \partial\bar{x}^{\nu}}\left(\bar{\mathcal{D}}^{\mu\nu}\right)\,=\,\frac{% \partial}{\partial\bar{x}^{\nu}}\left(\frac{\partial\bar{x}^{\mu}}{\partial x^% {\alpha}}\,\frac{\partial\bar{x}^{\nu}}{\partial x^{\beta}}\,\mathcal{D}^{% \alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}% \right]\right)\\ &\displaystyle=\,\frac{\partial^{2}\bar{x}^{\mu}}{\partial\bar{x}^{\nu}% \partial x^{\alpha}}\,\frac{\partial\bar{x}^{\nu}}{\partial x^{\beta}}\,% \mathcal{D}^{\alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x% }^{\rho}}\right]\,+\,\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,\frac{% \partial^{2}\bar{x}^{\nu}}{\partial\bar{x}^{\nu}\partial x^{\beta}}\,\mathcal{% D}^{\alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}% \right]\\ &\displaystyle+\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,\frac{% \partial\bar{x}^{\nu}}{\partial x^{\beta}}\,\frac{\partial\mathcal{D}^{\alpha% \beta}}{\partial\bar{x}^{\nu}}\,\det\left[\frac{\partial x^{\sigma}}{\partial% \bar{x}^{\rho}}\right]\,+\,\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,% \frac{\partial\bar{x}^{\nu}}{\partial x^{\beta}}\,\mathcal{D}^{\alpha\beta}\,% \frac{\partial}{\partial\bar{x}^{\nu}}\det\left[\frac{\partial x^{\sigma}}{% \partial\bar{x}^{\rho}}\right]\\ &\displaystyle=\,\frac{\partial^{2}\bar{x}^{\mu}}{\partial x^{\beta}\partial x% ^{\alpha}}\,\mathcal{D}^{\alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{% \partial\bar{x}^{\rho}}\right]\,+\,\frac{\partial\bar{x}^{\mu}}{\partial x^{% \alpha}}\,\frac{\partial^{2}\bar{x}^{\nu}}{\partial\bar{x}^{\nu}\partial x^{% \beta}}\,\mathcal{D}^{\alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{% \partial\bar{x}^{\rho}}\right]\\ &\displaystyle+\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,\frac{% \partial\mathcal{D}^{\alpha\beta}}{\partial x^{\beta}}\,\det\left[\frac{% \partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\,+\,\frac{\partial\bar{x}^% {\mu}}{\partial x^{\alpha}}\,\frac{\partial\bar{x}^{\nu}}{\partial x^{\beta}}% \,\mathcal{D}^{\alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar% {x}^{\rho}}\right]\frac{\partial\bar{x}^{\rho}}{\partial x^{\sigma}}\frac{% \partial^{2}x^{\sigma}}{\partial\bar{x}^{\nu}\partial\bar{x}^{\rho}}\\ &\displaystyle=\,0\,+\,\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,% \frac{\partial^{2}\bar{x}^{\nu}}{\partial\bar{x}^{\nu}\partial x^{\beta}}\,% \mathcal{D}^{\alpha\beta}\,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x% }^{\rho}}\right]\\ &\displaystyle+\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,J^{\alpha}\,% \det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\,+\,\frac{% \partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,\mathcal{D}^{\alpha\beta}\,\det% \left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\frac{\partial% \bar{x}^{\rho}}{\partial x^{\sigma}}\frac{\partial^{2}x^{\sigma}}{\partial x^{% \beta}\partial\bar{x}^{\rho}}\\ &\displaystyle=\,\frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,J^{\alpha}% \,\det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\,+\,% \frac{\partial\bar{x}^{\mu}}{\partial x^{\alpha}}\,\mathcal{D}^{\alpha\beta}\,% \det\left[\frac{\partial x^{\sigma}}{\partial\bar{x}^{\rho}}\right]\left(\frac% {\partial^{2}\bar{x}^{\nu}}{\partial\bar{x}^{\nu}\partial x^{\beta}}\,+\,\frac% {\partial\bar{x}^{\rho}}{\partial x^{\sigma}}\frac{\partial^{2}x^{\sigma}}{% \partial x^{\beta}\partial\bar{x}^{\rho}}\right)\end{aligned}
  23. 2 x ¯ ν x ¯ ν x β + x ¯ ρ x σ 2 x σ x β x ¯ ρ = 0 \frac{\partial^{2}\bar{x}^{\nu}}{\partial\bar{x}^{\nu}\partial x^{\beta}}\,+\,% \frac{\partial\bar{x}^{\rho}}{\partial x^{\sigma}}\frac{\partial^{2}x^{\sigma}% }{\partial x^{\beta}\partial\bar{x}^{\rho}}\,=\,0\,
  24. 2 x ¯ ν x ¯ ν x β + x ¯ ρ x σ 2 x σ x β x ¯ ρ = x σ x ¯ ν 2 x ¯ ν x σ x β + x ¯ ν x σ 2 x σ x β x ¯ ν = x σ x ¯ ν 2 x ¯ ν x β x σ + 2 x σ x β x ¯ ν x ¯ ν x σ = x β ( x σ x ¯ ν x ¯ ν x σ ) = x β ( x ¯ ν x ¯ ν ) = x β ( 𝟒 ) = 0 . \begin{aligned}\displaystyle\frac{\partial^{2}\bar{x}^{\nu}}{\partial\bar{x}^{% \nu}\partial x^{\beta}}\,+\,\frac{\partial\bar{x}^{\rho}}{\partial x^{\sigma}}% \frac{\partial^{2}x^{\sigma}}{\partial x^{\beta}\partial\bar{x}^{\rho}}&% \displaystyle=\,\frac{\partial x^{\sigma}}{\partial\bar{x}^{\nu}}\frac{% \partial^{2}\bar{x}^{\nu}}{\partial x^{\sigma}\partial x^{\beta}}\,+\,\frac{% \partial\bar{x}^{\nu}}{\partial x^{\sigma}}\frac{\partial^{2}x^{\sigma}}{% \partial x^{\beta}\partial\bar{x}^{\nu}}\\ &\displaystyle=\,\frac{\partial x^{\sigma}}{\partial\bar{x}^{\nu}}\frac{% \partial^{2}\bar{x}^{\nu}}{\partial x^{\beta}\partial x^{\sigma}}\,+\,\frac{% \partial^{2}x^{\sigma}}{\partial x^{\beta}\partial\bar{x}^{\nu}}\frac{\partial% \bar{x}^{\nu}}{\partial x^{\sigma}}\,=\,\frac{\partial}{\partial x^{\beta}}% \left(\frac{\partial x^{\sigma}}{\partial\bar{x}^{\nu}}\,\frac{\partial\bar{x}% ^{\nu}}{\partial x^{\sigma}}\right)\\ &\displaystyle=\,\frac{\partial}{\partial x^{\beta}}\left(\,\frac{\partial\bar% {x}^{\nu}}{\partial\bar{x}^{\nu}}\right)\,=\,\frac{\partial}{\partial x^{\beta% }}\left(\mathbf{4}\right)\,=\,0\,.\end{aligned}
  25. f μ = F μ ν J ν . f_{\mu}\,=\,F_{\mu\nu}\,J^{\nu}\,.
  26. d p α d t = Γ α γ β p β d x γ d t + q F α γ d x γ d t \frac{dp_{\alpha}}{dt}\,=\,\Gamma^{\beta}_{\alpha\gamma}\,p_{\beta}\,\frac{dx^% {\gamma}}{dt}\,+\,q\,F_{\alpha\gamma}\,\frac{dx^{\gamma}}{dt}\,
  27. d t / d t ¯ dt/d\bar{t}
  28. Γ ¯ α γ β = x ¯ β x ϵ x δ x ¯ α x ζ x ¯ γ Γ δ ζ ϵ + x ¯ β x η 2 x η x ¯ α x ¯ γ \bar{\Gamma}^{\beta}_{\alpha\gamma}\,=\,\frac{\partial\bar{x}^{\beta}}{% \partial x^{\epsilon}}\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\,% \frac{\partial x^{\zeta}}{\partial\bar{x}^{\gamma}}\,\Gamma^{\epsilon}_{\delta% \zeta}\,+\frac{\partial\bar{x}^{\beta}}{\partial x^{\eta}}\,\frac{\partial^{2}% x^{\eta}}{\partial\bar{x}^{\alpha}\partial\bar{x}^{\gamma}}\,
  29. d p ¯ α d t - Γ ¯ α γ β p ¯ β d x ¯ γ d t - q F ¯ α γ d x ¯ γ d t = d d t ( x δ x ¯ α p δ ) - ( x ¯ β x θ x δ x ¯ α x ι x ¯ γ Γ δ ι θ + x ¯ β x η 2 x η x ¯ α x ¯ γ ) x ϵ x ¯ β p ϵ x ¯ γ x ζ d x ζ d t - q x δ x ¯ α F δ ζ d x ζ d t = x δ x ¯ α ( d p δ d t - Γ δ ζ ϵ p ϵ d x ζ d t - q F δ ζ d x ζ d t ) + d d t ( x δ x ¯ α ) p δ - ( x ¯ β x η 2 x η x ¯ α x ¯ γ ) x ϵ x ¯ β p ϵ x ¯ γ x ζ d x ζ d t = 0 + d d t ( x δ x ¯ α ) p δ - 2 x ϵ x ¯ α x ¯ γ p ϵ d x ¯ γ d t = 0 . \begin{aligned}&\displaystyle\frac{d\bar{p}_{\alpha}}{dt}\,-\,\bar{\Gamma}^{% \beta}_{\alpha\gamma}\,\bar{p}_{\beta}\,\frac{d\bar{x}^{\gamma}}{dt}\,-\,q\,% \bar{F}_{\alpha\gamma}\,\frac{d\bar{x}^{\gamma}}{dt}\\ &\displaystyle=\,\frac{d}{dt}\left(\frac{\partial x^{\delta}}{\partial\bar{x}^% {\alpha}}\,p_{\delta}\right)\,-\,\left(\frac{\partial\bar{x}^{\beta}}{\partial x% ^{\theta}}\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\,\frac{% \partial x^{\iota}}{\partial\bar{x}^{\gamma}}\,\Gamma^{\theta}_{\delta\iota}+% \,\frac{\partial\bar{x}^{\beta}}{\partial x^{\eta}}\,\frac{\partial^{2}x^{\eta% }}{\partial\bar{x}^{\alpha}\partial\bar{x}^{\gamma}}\right)\,\frac{\partial x^% {\epsilon}}{\partial\bar{x}^{\beta}}\,p_{\epsilon}\,\frac{\partial\bar{x}^{% \gamma}}{\partial x^{\zeta}}\,\frac{dx^{\zeta}}{dt}\,-\,q\,\frac{\partial x^{% \delta}}{\partial\bar{x}^{\alpha}}\,F_{\delta\zeta}\,\frac{dx^{\zeta}}{dt}\\ &\displaystyle=\,\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\,\left(% \frac{dp_{\delta}}{dt}\,-\,\Gamma^{\epsilon}_{\delta\zeta}\,p_{\epsilon}\,% \frac{dx^{\zeta}}{dt}\,-\,q\,F_{\delta\zeta}\,\frac{dx^{\zeta}}{dt}\right)+% \frac{d}{dt}\left(\frac{\partial x^{\delta}}{\partial\bar{x}^{\alpha}}\right)% \,p_{\delta}\,-\,\left(\frac{\partial\bar{x}^{\beta}}{\partial x^{\eta}}\,% \frac{\partial^{2}x^{\eta}}{\partial\bar{x}^{\alpha}\partial\bar{x}^{\gamma}}% \right)\,\frac{\partial x^{\epsilon}}{\partial\bar{x}^{\beta}}\,p_{\epsilon}\,% \frac{\partial\bar{x}^{\gamma}}{\partial x^{\zeta}}\,\frac{dx^{\zeta}}{dt}\\ &\displaystyle=\,0\,+\,\frac{d}{dt}\left(\frac{\partial x^{\delta}}{\partial% \bar{x}^{\alpha}}\right)\,p_{\delta}\,-\,\frac{\partial^{2}x^{\epsilon}}{% \partial\bar{x}^{\alpha}\partial\bar{x}^{\gamma}}p_{\epsilon}\,\frac{d\bar{x}^% {\gamma}}{dt}\,=\,0\ .\end{aligned}
  30. = - 1 4 μ 0 F α β F α β - g + A α J α \mathcal{L}\,=\,-\frac{1}{4\mu_{0}}\,F_{\alpha\beta}\,F^{\alpha\beta}\,\sqrt{-% g}\,+\,A_{\alpha}\,J^{\alpha}\,
  31. F α β = g α γ F γ δ g δ β . F^{\alpha\beta}=g^{\alpha\gamma}F_{\gamma\delta}g^{\delta\beta}\,.
  32. = - 1 4 μ 0 F α β F α β - g + A α J free α + 1 2 F α β α β . \mathcal{L}\,=\,-\frac{1}{4\mu_{0}}\,F_{\alpha\beta}\,F^{\alpha\beta}\,\sqrt{-% g}\,+\,A_{\alpha}\,J^{\alpha}_{\,\text{free}}\,+\,\frac{1}{2}\,F_{\alpha\beta}% \,\mathcal{M}^{\alpha\beta}\,.
  33. T μ ν = - 1 μ 0 ( F μ α g α β F β ν - 1 4 g μ ν F σ α g α β F β ρ g ρ σ ) T_{\mu\nu}\,=\,-\frac{1}{\mu_{0}}(F_{\mu\alpha}g^{\alpha\beta}F_{\beta\nu}\,-% \,\frac{1}{4}g_{\mu\nu}\,F_{\sigma\alpha}g^{\alpha\beta}F_{\beta\rho}g^{\rho% \sigma})\,
  34. T μ ν T_{\mu\nu}
  35. T μ ν g μ ν = 0 T_{\mu\nu}g^{\mu\nu}\,=\,0\,
  36. 𝔗 μ ν = T μ γ g γ ν - g . \mathfrak{T}_{\mu}^{\nu}=T_{\mu\gamma}\,g^{\gamma\nu}\,\sqrt{-g}.
  37. 𝔗 μ ν ; ν + f μ = 0 {\mathfrak{T}_{\mu}^{\nu}}_{;\nu}\,+\,f_{\mu}\,=\,0\,
  38. - 𝔗 μ ν , ν = - Γ μ ν σ 𝔗 σ ν + f μ -{\mathfrak{T}_{\mu}^{\nu}}_{,\nu}\,=\,-\Gamma^{\sigma}_{\mu\nu}\mathfrak{T}_{% \sigma}^{\nu}\,+\,f_{\mu}\,
  39. 𝔗 μ ν ; ν + f μ = - 1 μ 0 ( F μ α ; ν g α β F β γ g γ ν + F μ α g α β F β γ ; ν g γ ν - 1 2 δ μ ν F σ α ; ν g α β F β ρ g ρ σ ) - g + 1 μ 0 F μ α g α β F β γ ; ν g γ ν - g = - 1 μ 0 ( F μ α ; ν F α ν - 1 2 F σ α ; μ F α σ ) - g = - 1 μ 0 ( ( - F ν μ ; α - F α ν ; μ ) F α ν - 1 2 F σ α ; μ F α σ ) - g = - 1 μ 0 ( F μ ν ; α F α ν - F α ν ; μ F α ν + 1 2 F σ α ; μ F σ α ) - g = - 1 μ 0 ( F μ α ; ν F ν α - 1 2 F α ν ; μ F α ν ) - g = - 1 μ 0 ( - F μ α ; ν F α ν + 1 2 F σ α ; μ F α σ ) - g , \begin{aligned}\displaystyle{\mathfrak{T}_{\mu}^{\nu}}_{;\nu}\,+\,f_{\mu}&% \displaystyle=\,-\frac{1}{\mu_{0}}(F_{\mu\alpha;\nu}g^{\alpha\beta}F_{\beta% \gamma}g^{\gamma\nu}\,+\,F_{\mu\alpha}g^{\alpha\beta}F_{\beta\gamma;\nu}g^{% \gamma\nu}\,-\,\frac{1}{2}\delta_{\mu}^{\nu}\,F_{\sigma\alpha;\nu}g^{\alpha% \beta}F_{\beta\rho}g^{\rho\sigma})\sqrt{-g}\\ &\displaystyle+\frac{1}{\mu_{0}}\,F_{\mu\alpha}\,g^{\alpha\beta}\,F_{\beta% \gamma;\nu}\,g^{\gamma\nu}\,\sqrt{-g}\\ &\displaystyle=\,-\frac{1}{\mu_{0}}(F_{\mu\alpha;\nu}F^{\alpha\nu}\,-\,\frac{1% }{2}F_{\sigma\alpha;\mu}F^{\alpha\sigma})\sqrt{-g}\\ &\displaystyle=\,-\frac{1}{\mu_{0}}((-F_{\nu\mu;\alpha}-F_{\alpha\nu;\mu})F^{% \alpha\nu}\,-\,\frac{1}{2}F_{\sigma\alpha;\mu}F^{\alpha\sigma})\sqrt{-g}\\ &\displaystyle=\,-\frac{1}{\mu_{0}}(F_{\mu\nu;\alpha}F^{\alpha\nu}-F_{\alpha% \nu;\mu}F^{\alpha\nu}\,+\,\frac{1}{2}F_{\sigma\alpha;\mu}F^{\sigma\alpha})% \sqrt{-g}\\ &\displaystyle=\,-\frac{1}{\mu_{0}}(F_{\mu\alpha;\nu}F^{\nu\alpha}-\frac{1}{2}% F_{\alpha\nu;\mu}F^{\alpha\nu})\sqrt{-g}\\ &\displaystyle=\,-\frac{1}{\mu_{0}}(-F_{\mu\alpha;\nu}F^{\alpha\nu}\,+\,\frac{% 1}{2}F_{\sigma\alpha;\mu}F^{\alpha\sigma})\sqrt{-g}\ ,\end{aligned}
  40. F a b = def F a b ; = d d - 2 R a c b d F c d + R a e F e - b R b e F e + a J a ; b - J b ; a \Box F_{ab}\ \stackrel{\mathrm{def}}{=}\ F_{ab;}{}^{d}{}_{d}=\,-2R_{acbd}F^{cd% }+R_{ae}F^{e}{}_{b}-R_{be}F^{e}{}_{a}+J_{a;b}-J_{b;a}
  41. \Box
  42. A a = A a ; b b \Box A^{a}={{A^{a;}}^{b}}_{b}
  43. A a - A b ; a b = - μ 0 J a \Box A^{a}-{A^{b;a}}_{b}=-\mu_{0}J^{a}
  44. A a ; a = 0 , {A^{a}}_{;a}=0\ ,
  45. A a = - μ 0 J a + R a b A b \Box A^{a}=-\mu_{0}J^{a}+{R^{a}}_{b}A^{b}
  46. R a b = def R s a s b R_{ab}\ \stackrel{\mathrm{def}}{=}\ {R^{s}}_{asb}
  47. G a b = 8 π G c 4 T a b G_{ab}=\frac{8\pi G}{c^{4}}T_{ab}
  48. G a b = def R a b - 1 2 R g a b {G}_{ab}\ \stackrel{\mathrm{def}}{=}\ {R}_{ab}-{1\over 2}{R}g_{ab}
  49. F ( ) F(\nabla)
  50. \nabla
  51. = 0 + i A \nabla=\nabla_{0}+iA
  52. 0 \nabla_{0}

Maxwell_bridge.html

  1. R 1 R_{1}
  2. R 4 R_{4}
  3. R 2 R_{2}
  4. C 2 C_{2}
  5. R 2 R_{2}
  6. C 2 C_{2}
  7. R 3 R_{3}
  8. L 3 L_{3}
  9. R 3 \displaystyle R_{3}

Mazur–Ulam_theorem.html

  1. V V
  2. W W
  3. f : V W f\colon V\to W
  4. f f

McNemar's_test.html

  1. H 0 \displaystyle H_{0}
  2. χ 2 = ( b - c ) 2 b + c . \chi^{2}={(b-c)^{2}\over b+c}.
  3. χ 2 \chi^{2}
  4. χ 2 \chi^{2}
  5. 2 i = 0 b ( n i ) 0.5 i ( 1 - 0.5 ) n - i 2\sum_{i=0}^{b}{n\choose i}0.5^{i}(1-0.5)^{n-i}
  6. χ 2 = ( | b - c | - 1 ) 2 b + c . \chi^{2}={(|b-c|-1)^{2}\over b+c}.
  7. 2 ( i = 0 b ( n i ) 0.5 i ( 1 - 0.5 ) n - i - 0.5 ( n b ) 0.5 b ( 1 - 0.5 ) n - b ) 2(\sum_{i=0}^{b}{n\choose i}0.5^{i}(1-0.5)^{n-i}-0.5{n\choose b}0.5^{b}(1-0.5)% ^{n-b})
  8. - ( n b ) 0.5 b ( 1 - 0.5 ) n - b -{n\choose b}0.5^{b}(1-0.5)^{n-b}
  9. χ 2 = ( 121 - 59 ) 2 121 + 59 \chi^{2}={(121-59)^{2}\over{121+59}}
  10. χ 2 \chi^{2}
  11. χ 2 \chi^{2}
  12. Tonsillectomy No tonsillectomy Hodgkins 41 44 Control 33 52 \begin{array}[]{c|c|c}\hline&\,\text{Tonsillectomy}&\,\text{No tonsillectomy}% \\ \hline\,\text{Hodgkins}&41&44\\ \hline\,\text{Control}&33&52\end{array}
  13. Sibling Patient No tonsillectomy Tonsillectomy No tonsillectomy 37 7 Tonsillectomy 15 26 \begin{array}[]{cc}&\,\text{Sibling}\\ \,\text{Patient}&\begin{array}[]{c|c|c}\hline&\,\text{No tonsillectomy}&\,% \text{Tonsillectomy}\\ \hline\,\text{No tonsillectomy}&37&7\\ \hline\,\text{Tonsillectomy}&15&26\end{array}\end{array}

Mean_absolute_percentage_error.html

  1. M = 1 n t = 1 n | A t - F t A t | , \mbox{M}~{}=\frac{1}{n}\sum_{t=1}^{n}\left|\frac{A_{t}-F_{t}}{A_{t}}\right|,

Mean_squared_prediction_error.html

  1. g ^ \widehat{g}
  2. MSPE ( L ) = E [ ( g ( x i ) - g ^ ( x i ) ) 2 ] . \operatorname{MSPE}(L)=\operatorname{E}\left[\left(g(x_{i})-\widehat{g}(x_{i})% \right)^{2}\right].
  3. MSPE ( L ) = i = 1 n ( E [ g ^ ( x i ) ] - g ( x i ) ) 2 + i = 1 n var [ g ^ ( x i ) ] . \operatorname{MSPE}(L)=\sum_{i=1}^{n}\left(\operatorname{E}\left[\widehat{g}(x% _{i})\right]-g(x_{i})\right)^{2}+\sum_{i=1}^{n}\operatorname{var}\left[% \widehat{g}(x_{i})\right].
  4. y i = g ( x i ) + σ ε i y_{i}=g(x_{i})+\sigma\varepsilon_{i}
  5. ε i 𝒩 ( 0 , 1 ) \varepsilon_{i}\sim\mathcal{N}(0,1)
  6. MSPE ( L ) = g ( I - L ) ( I - L ) g + σ 2 tr [ L L ] . \operatorname{MSPE}(L)=g^{\prime}(I-L)^{\prime}(I-L)g+\sigma^{2}\operatorname{% tr}\left[L^{\prime}L\right].
  7. i = 1 n ( E [ g ^ ( x i ) ] - g ( x i ) ) 2 = E [ i = 1 n ( y i - g ^ ( x i ) ) 2 ] - σ 2 tr [ ( I - L ) ( I - L ) ] . \sum_{i=1}^{n}\left(\operatorname{E}\left[\widehat{g}(x_{i})\right]-g(x_{i})% \right)^{2}=\operatorname{E}\left[\sum_{i=1}^{n}\left(y_{i}-\widehat{g}(x_{i})% \right)^{2}\right]-\sigma^{2}\operatorname{tr}\left[\left(I-L\right)^{\prime}% \left(I-L\right)\right].
  8. MSPE ( L ) = E [ i = 1 n ( y i - g ^ ( x i ) ) 2 ] - σ 2 ( n - 2 tr [ L ] ) . \operatorname{MSPE}(L)=\operatorname{E}\left[\sum_{i=1}^{n}\left(y_{i}-% \widehat{g}(x_{i})\right)^{2}\right]-\sigma^{2}\left(n-2\operatorname{tr}\left% [L\right]\right).
  9. σ 2 \sigma^{2}
  10. σ ^ 2 \widehat{\sigma}^{2}
  11. MSPE ^ ( L ) = i = 1 n ( y i - g ^ ( x i ) ) 2 - σ ^ 2 ( n - 2 tr [ L ] ) . \operatorname{\widehat{MSPE}}(L)=\sum_{i=1}^{n}\left(y_{i}-\widehat{g}(x_{i})% \right)^{2}-\widehat{\sigma}^{2}\left(n-2\operatorname{tr}\left[L\right]\right).
  12. C p = i = 1 n ( y i - g ^ ( x i ) ) 2 σ ^ 2 - n + 2 tr [ L ] . C_{p}=\frac{\sum_{i=1}^{n}\left(y_{i}-\widehat{g}(x_{i})\right)^{2}}{\widehat{% \sigma}^{2}}-n+2\operatorname{tr}\left[L\right].
  13. p = tr [ L ] p=\operatorname{tr}\left[L\right]

Mean_value_theorem_(divided_differences).html

  1. ξ ( min { x 0 , , x n } , max { x 0 , , x n } ) \xi\in(\min\{x_{0},\dots,x_{n}\},\max\{x_{0},\dots,x_{n}\})\,
  2. f [ x 0 , , x n ] = f ( n ) ( ξ ) n ! . f[x_{0},\dots,x_{n}]=\frac{f^{(n)}(\xi)}{n!}.
  3. P P
  4. P P
  5. P P
  6. f [ x 0 , , x n ] ( x - x n - 1 ) ( x - x 1 ) ( x - x 0 ) f[x_{0},\dots,x_{n}](x-x_{n-1})\dots(x-x_{1})(x-x_{0})
  7. g g
  8. g = f - P g=f-P
  9. g g
  10. n + 1 n+1
  11. g g
  12. g g^{\prime}
  13. g ( n - 1 ) g^{(n-1)}
  14. g ( n ) g^{(n)}
  15. ξ \xi
  16. 0 = g ( n ) ( ξ ) = f ( n ) ( ξ ) - f [ x 0 , , x n ] n ! 0=g^{(n)}(\xi)=f^{(n)}(\xi)-f[x_{0},\dots,x_{n}]n!
  17. f [ x 0 , , x n ] = f ( n ) ( ξ ) n ! . f[x_{0},\dots,x_{n}]=\frac{f^{(n)}(\xi)}{n!}.

Mechanical_impedance.html

  1. ω \omega
  2. 𝐅 ( ω ) = 𝐙 ( ω ) 𝐯 ( ω ) \mathbf{F}(\omega)=\mathbf{Z}(\omega)\mathbf{v}(\omega)
  3. 𝐅 \mathbf{F}
  4. 𝐯 \mathbf{v}
  5. 𝐙 \mathbf{Z}
  6. ω \omega

Mechanical_resonance.html

  1. f = 1 2 π k m f={1\over 2\pi}\sqrt{k\over m}
  2. f = 1 2 π g L f={1\over 2\pi}\sqrt{g\over L}

Medial_axis.html

  1. γ : 𝐑 𝐑 2 \gamma:\mathbf{R}\to\mathbf{R}^{2}
  2. T ¯ ( t ) = d γ d t \underline{T}(t)={d\gamma\over dt}
  3. ( c - γ ( s ) ) T ¯ ( s ) = ( c - γ ( t ) ) T ¯ ( t ) = 0 , (c-\gamma(s))\cdot\underline{T}(s)=(c-\gamma(t))\cdot\underline{T}(t)=0,
  4. | c - γ ( s ) | = | c - γ ( t ) | = r . |c-\gamma(s)|=|c-\gamma(t)|=r.\,

Megamaser.html

  1. z 0.27 z\approx 0.27
  2. \propto
  3. \propto
  4. ± \pm

Meketaten.html

  1. γ \gamma
  2. α , β \alpha,\ \beta
  3. γ \gamma
  4. α \alpha
  5. γ \gamma
  6. α \alpha
  7. γ \gamma
  8. α \alpha

Memetic_algorithm.html

  1. Ω i l \Omega_{il}
  2. Ω i l \Omega_{il}
  3. f i l f_{il}
  4. t i l t_{il}
  5. t i l t_{il}

Mental_chronometry.html

  1. R T = a + b log 2 ( n + 1 ) RT=a+b\log_{2}(n+1)
  2. a a
  3. b b
  4. n n

Mercator_series.html

  1. ln ( 1 + x ) = x - x 2 2 + x 3 3 - x 4 4 + . \ln(1+x)\;=\;x\,-\,\frac{x^{2}}{2}\,+\,\frac{x^{3}}{3}\,-\,\frac{x^{4}}{4}\,+% \,\cdots.
  2. ln ( 1 + x ) = n = 1 ( - 1 ) n + 1 n x n . \ln(1+x)\;=\;\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^{n}.
  3. d d x ln x = 1 x . \frac{d}{dx}\ln x=\frac{1}{x}.
  4. 1 - t + t 2 - + ( - t ) n - 1 = 1 - ( - t ) n 1 + t 1-t+t^{2}-\cdots+(-t)^{n-1}=\frac{1-(-t)^{n}}{1+t}
  5. 1 1 + t = 1 - t + t 2 - + ( - t ) n - 1 + ( - t ) n 1 + t . \frac{1}{1+t}=1-t+t^{2}-\cdots+(-t)^{n-1}+\frac{(-t)^{n}}{1+t}.
  6. 0 x d t 1 + t = 0 x ( 1 - t + t 2 - + ( - t ) n - 1 + ( - t ) n 1 + t ) d t \int_{0}^{x}\frac{dt}{1+t}=\int_{0}^{x}\left(1-t+t^{2}-\cdots+(-t)^{n-1}+\frac% {(-t)^{n}}{1+t}\right)\,dt
  7. ln ( 1 + x ) = x - x 2 2 + x 3 3 - + ( - 1 ) n - 1 x n n + ( - 1 ) n 0 x t n 1 + t d t . \ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots+(-1)^{n-1}\frac{x^{n}}{n}+(-% 1)^{n}\int_{0}^{x}\frac{t^{n}}{1+t}\,dt.
  8. - x A k ( x ) + B k ( x ) ln ( 1 + x ) = n = 1 ( - 1 ) n - 1 x n + k n ( n + 1 ) ( n + k ) , -xA_{k}(x)+B_{k}(x)\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^{n+k}}{n(n+1)% \cdots(n+k)},
  9. A k ( x ) = 1 k ! m = 0 k ( k m ) x m l = 1 k - m ( - x ) l - 1 l A_{k}(x)=\frac{1}{k!}\sum_{m=0}^{k}{k\choose m}x^{m}\sum_{l=1}^{k-m}\frac{(-x)% ^{l-1}}{l}
  10. B k ( x ) = 1 k ! ( 1 + x ) k B_{k}(x)=\frac{1}{k!}(1+x)^{k}
  11. k = 1 ( - 1 ) k + 1 k = ln 2. \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}=\ln 2.
  12. n = 1 z n n = z + z 2 2 + z 3 3 + z 4 4 + \sum_{n=1}^{\infty}\frac{z^{n}}{n}=z\,+\,\frac{z^{2}}{2}\,+\,\frac{z^{3}}{3}\,% +\,\frac{z^{4}}{4}\,+\,\cdots
  13. ( 1 - z ) n = 1 m z n n = z - n = 2 m z n n ( n - 1 ) - z m + 1 m , (1-z)\sum_{n=1}^{m}\frac{z^{n}}{n}=z-\sum_{n=2}^{m}\frac{z^{n}}{n(n-1)}-\frac{% z^{m+1}}{m},

Mergelyan's_theorem.html

  1. \to

Merkle–Damgård_construction.html

  1. M M
  2. 𝖯𝖺𝖽 ( M ) . \mathsf{Pad}(M).
  3. | M 1 | = | M 2 | |M_{1}|=|M_{2}|
  4. | 𝖯𝖺𝖽 ( M 1 ) | = | 𝖯𝖺𝖽 ( M 2 ) | . |\mathsf{Pad}(M_{1})|=|\mathsf{Pad}(M_{2})|.
  5. | M 1 | | M 2 | |M_{1}|\neq|M_{2}|
  6. 𝖯𝖺𝖽 ( M 1 ) \mathsf{Pad}(M_{1})
  7. 𝖯𝖺𝖽 ( M 2 ) . \mathsf{Pad}(M_{2}).

Mermin–Wagner_theorem.html

  1. d 2 d≤2
  2. d 2 d≤2
  3. φ φ
  4. m m
  5. G ( x ) = φ ( x ) φ ( 0 ) = d 2 k ( 2 π ) 2 e i k x k 2 + m 2 . G(x)=\left\langle\varphi(x)\varphi(0)\right\rangle=\int\frac{d^{2}k}{(2\pi)^{2% }}\frac{e^{ik\cdot x}}{k^{2}+m^{2}}.
  6. m , G m,G
  7. 2 G = δ ( x ) . \nabla^{2}G=\delta(x).
  8. k k
  9. E = G E=∇G
  10. E = 1 2 π r . E={1\over 2\pi r}.
  11. G ( r ) = 1 2 π log ( r ) G(r)={1\over 2\pi}\log(r)
  12. θ θ
  13. θ θ
  14. θ θ
  15. O ( 2 ) O(2)
  16. d 2 d≤2
  17. ξ ξ
  18. G ( r ) exp ( - r / ξ ) G(r)\sim\exp(-r/\xi)
  19. r / ξ 1 r/\xi\gg 1
  20. G ( r ) G(r)
  21. r r
  22. a r ξ a≪r≪ξ
  23. a a
  24. n n
  25. d d
  26. J J
  27. H = - J i , j 𝐒 i 𝐒 j . H=-J\sum_{\left\langle{i,j}\right\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}.
  28. x x
  29. O ( n ) O(n)
  30. O ( n 1 ) O(n−1)
  31. 𝐒 = ( 1 - α σ α 2 , { σ α } ) , α = 1 , , n - 1. \mathbf{S}=\left(\sqrt{1-\sum_{\alpha}\sigma_{\alpha}^{2}},\left\{\sigma_{% \alpha}\right\}\right),\qquad\alpha=1,\cdots,n-1.
  32. 𝐒 i 𝐒 j = ( 1 - α σ i α 2 ) ( 1 - α σ j α 2 ) + α σ i α σ j α = 1 - 1 2 α ( σ i α 2 + σ j α 2 ) + α σ i α σ j α + 𝒪 ( σ 4 ) = 1 - 1 2 α ( σ i α - σ j α ) 2 + \begin{aligned}\displaystyle\mathbf{S}_{i}\cdot\mathbf{S}_{j}&\displaystyle=% \sqrt{\left(1-\sum_{\alpha}\sigma^{2}_{i\alpha}\right)\left(1-\sum_{\alpha}% \sigma^{2}_{j\alpha}\right)}+\sum_{\alpha}\sigma_{i\alpha}\sigma_{j\alpha}\\ &\displaystyle=1-\tfrac{1}{2}\sum_{\alpha}\left(\sigma^{2}_{i\alpha}+\sigma^{2% }_{j\alpha}\right)+\sum_{\alpha}\sigma_{i\alpha}\sigma_{j\alpha}+\mathcal{O}% \left(\sigma^{4}\right)\\ &\displaystyle=1-\tfrac{1}{2}\sum_{\alpha}\left(\sigma_{i\alpha}-\sigma_{j% \alpha}\right)^{2}+\ldots\end{aligned}
  33. H = H 0 + 1 2 J i , j α ( σ i α - σ j α ) 2 + H=H_{0}+\tfrac{1}{2}J\sum_{\left\langle i,j\right\rangle}\sum_{\alpha}\left(% \sigma_{i\alpha}-\sigma_{j\alpha}\right)^{2}+\cdots
  34. H = 1 2 J d d x α ( σ α ) 2 + . H=\tfrac{1}{2}J\int{d^{d}x\sum_{\alpha}{(\nabla\sigma_{\alpha})^{2}}}+\ldots.
  35. σ α ( r ) σ α ( 0 ) = 1 β J 1 a d d k ( 2 π ) d e i 𝐤 𝐫 k 2 \left\langle\sigma_{\alpha}(r)\sigma_{\alpha}(0)\right\rangle=\frac{1}{\beta J% }\int^{\frac{1}{a}}\frac{d^{d}k}{(2\pi)^{d}}\frac{e^{i\mathbf{k}\cdot\mathbf{r% }}}{k^{2}}
  36. S 1 = 1 - 1 2 α σ α 2 + \left\langle S_{1}\right\rangle=1-\tfrac{1}{2}\sum_{\alpha}\left\langle\sigma_% {\alpha}^{2}\right\rangle+\ldots
  37. α σ α 2 ( 0 ) = ( n - 1 ) 1 β J 1 a d d k ( 2 π ) d 1 k 2 . \sum_{\alpha}\left\langle\sigma_{\alpha}^{2}(0)\right\rangle=(n-1)\frac{1}{% \beta J}\int^{\frac{1}{a}}\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{k^{2}}.
  38. 1 a k d - 3 d k \int^{\frac{1}{a}}k^{d-3}dk
  39. d > 2 d>2
  40. d 2 d≤2
  41. d 2 d≤2
  42. T > 0 T>0
  43. T > 0 T>0
  44. d 2 d≤2
  45. G G
  46. G G

Mesh_analysis.html

  1. { Mesh 1: I 1 = I s Mesh 2: - V s + R 1 ( I 2 - I 1 ) + 1 s C ( I 2 - I 3 ) = 0 Mesh 3: 1 s C ( I 3 - I 2 ) + R 2 ( I 3 - I 1 ) + L s I 3 = 0 \begin{cases}\,\text{Mesh 1: }I_{1}=I_{s}\\ \,\text{Mesh 2: }-V_{s}+R_{1}(I_{2}-I_{1})+\frac{1}{sC}(I_{2}-I_{3})=0\\ \,\text{Mesh 3: }\frac{1}{sC}(I_{3}-I_{2})+R_{2}(I_{3}-I_{1})+LsI_{3}=0\\ \end{cases}\,
  2. { Mesh 1, 2: - V s + R 1 I 1 + R 2 I 2 = 0 Current source: I s = I 2 - I 1 \begin{cases}\,\text{Mesh 1, 2: }-V_{s}+R_{1}I_{1}+R_{2}I_{2}=0\\ \,\text{Current source: }I_{s}=I_{2}-I_{1}\end{cases}\,
  3. { Mesh 1: - V s + R 1 I 1 + R 3 ( I 1 - I 2 ) = 0 Mesh 2: R 2 I 2 + 3 I x + R 3 ( I 2 - I 1 ) = 0 Dependent variable: I x = I 1 - I 2 \begin{cases}\,\text{Mesh 1: }-V_{s}+R_{1}I_{1}+R_{3}(I_{1}-I_{2})=0\\ \,\text{Mesh 2: }R_{2}I_{2}+3I_{x}+R_{3}(I_{2}-I_{1})=0\\ \,\text{Dependent variable: }I_{x}=I_{1}-I_{2}\end{cases}\,

Mesopic_vision.html

  1. λ \lambda
  2. V m ( λ ) = ( 1 - x ) V ( λ ) + x V ( λ ) V_{m}(\lambda)=(1-x)V^{\prime}(\lambda)+xV(\lambda)
  3. V ( λ ) V(\lambda)
  4. V ( λ ) V^{\prime}(\lambda)
  5. x x
  6. L p L_{p}
  7. x x
  8. L p L_{p}

Metanilpotent_group.html

  1. G G
  2. N N
  3. N N
  4. G / N G/N

Method_of_averaging.html

  1. x ˙ = ϵ f ( t , x , ϵ ) \dot{x}=\epsilon f(t,x,\epsilon)
  2. f ( t , x ) f(t,x)
  3. t t
  4. T T
  5. t t
  6. f f
  7. ϵ \epsilon
  8. f f
  9. ϵ \epsilon
  10. x ˙ a = ϵ 1 T 0 T f ( τ , x , 0 ) d τ = f ~ ( x a ) . \dot{x}^{a}=\epsilon\frac{1}{T}\int_{0}^{T}f(\tau,x,0)d\tau=\tilde{f}(x^{a}).
  11. x a x^{a}
  12. m ( l θ ¨ - a k ω 2 sin ω t sin θ ) = - m g sin θ - k ( l θ ˙ + a ω cos ω t sin θ ) m(l\ddot{\theta}-ak\omega^{2}\sin\omega t\sin\theta)=-mg\sin\theta-k(l\dot{% \theta}+a\omega\cos\omega t\sin\theta)
  13. a sin ω t a\sin\omega t
  14. θ \theta

Method_of_image_charges.html

  1. ( 0 , 0 , a ) (0,0,a)
  2. V = 0 V=0
  3. ( 0 , 0 , - a ) (0,0,-a)
  4. z > 0 z>0
  5. V ( ρ , φ , z ) = 1 4 π ϵ 0 ( q ρ 2 + ( z - a ) 2 + - q ρ 2 + ( z + a ) 2 ) V\left(\rho,\varphi,z\right)=\frac{1}{4\pi\epsilon_{0}}\left(\frac{q}{\sqrt{% \rho^{2}+\left(z-a\right)^{2}}}+\frac{-q}{\sqrt{\rho^{2}+\left(z+a\right)^{2}}% }\right)\,
  6. σ = - ϵ 0 V z | z = 0 = - q a 2 π ( ρ 2 + a 2 ) 3 / 2 \sigma=-\epsilon_{0}\frac{\partial V}{\partial z}\Bigg|_{z=0}=\frac{-qa}{2\pi% \left(\rho^{2}+a^{2}\right)^{3/2}}
  7. Q t \displaystyle Q_{t}
  8. ( 0 , 0 , a ) (0,0,a)
  9. ( 0 , 0 , - a ) (0,0,-a)
  10. ( p sin θ cos ϕ , p sin θ sin ϕ , p cos θ ) (p\sin\theta\cos\phi,p\sin\theta\sin\phi,p\cos\theta)
  11. ( - p sin θ cos ϕ , - p sin θ sin ϕ , p cos θ ) (-p\sin\theta\cos\phi,-p\sin\theta\sin\phi,p\cos\theta)
  12. F = - 1 4 π ϵ 0 3 p 2 16 a 4 ( 1 + cos 2 θ ) F=-\frac{1}{4\pi\epsilon_{0}}\frac{3p^{2}}{16a^{4}}(1+\cos^{2}\theta)
  13. τ = - 1 4 π ϵ 0 p 2 16 a 3 sin 2 θ \tau=-\frac{1}{4\pi\epsilon_{0}}\frac{p^{2}}{16a^{3}}\sin 2\theta
  14. q q
  15. ϵ 1 \epsilon_{1}
  16. ϵ 2 \epsilon_{2}
  17. q q^{\prime}
  18. q = ϵ 1 - ϵ 2 ϵ 1 + ϵ 2 q q^{\prime}=\frac{\epsilon_{1}-\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}q
  19. 𝐩 \mathbf{p}
  20. ( R 2 / p 2 ) 𝐩 (R^{2}/p^{2})\mathbf{p}
  21. 𝐫 \mathbf{r}
  22. 4 π ϵ 0 V ( 𝐫 ) = q | 𝐫 1 | + ( - q R / p ) | 𝐫 2 | = q r 2 + p 2 - 2 𝐫 𝐩 + ( - q R / p ) r 2 + R 4 p 2 - 2 R 2 p 2 𝐫 𝐩 4\pi\epsilon_{0}V(\mathbf{r})=\frac{q}{|\mathbf{r}_{1}|}+\frac{(-qR/p)}{|% \mathbf{r}_{2}|}=\frac{q}{\sqrt{r^{2}+p^{2}-2\mathbf{r}\cdot\mathbf{p}}}+\frac% {(-qR/p)}{\sqrt{r^{2}+\frac{R^{4}}{p^{2}}-\frac{2R^{2}}{p^{2}}\mathbf{r}\cdot% \mathbf{p}}}
  23. V ( 𝐫 ) = 1 4 π ϵ 0 [ q r 2 + p 2 - 2 𝐫 𝐩 - q r 2 p 2 R 2 + R 2 - 2 𝐫 𝐩 ] V(\mathbf{r})=\frac{1}{4\pi\epsilon_{0}}\left[\frac{q}{\sqrt{r^{2}+p^{2}-2% \mathbf{r}\cdot\mathbf{p}}}-\frac{q}{\sqrt{\frac{r^{2}p^{2}}{R^{2}}+R^{2}-2% \mathbf{r}\cdot\mathbf{p}}}\right]
  24. 𝐩 \mathbf{p}
  25. σ ( θ ) = ϵ 0 V r | r = R = - q ( R 2 - p 2 ) 4 π R ( R 2 + p 2 - 2 p R cos θ ) 3 / 2 \sigma(\theta)=\epsilon_{0}\frac{\partial V}{\partial r}\Bigg|_{r=R}=\frac{-q(% R^{2}-p^{2})}{4\pi R(R^{2}+p^{2}-2pR\cos\theta)^{3/2}}
  26. Q t = 0 π d θ 0 2 π d ϕ σ ( θ ) R 2 sin θ = - q Q_{t}=\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\,\,\sigma(\theta)R^{2}\sin% \theta=-q
  27. 𝐩 \mathbf{p}
  28. ( R 2 / p 2 ) 𝐩 (R^{2}/p^{2})\mathbf{p}
  29. M M\,
  30. 𝐩 \mathbf{p}
  31. ( R 2 / p 2 ) 𝐩 (R^{2}/p^{2})\mathbf{p}
  32. q = R 𝐩 𝐌 p 3 q^{\prime}=\frac{R\mathbf{p}\cdot\mathbf{M}}{p^{3}}
  33. 𝐌 = R 3 [ - 𝐌 p 3 + 2 𝐩 ( 𝐩 𝐌 ) p 5 ] \mathbf{M}^{\prime}=R^{3}\left[-\frac{\mathbf{M}}{p^{3}}+\frac{2\mathbf{p}(% \mathbf{p}\cdot\mathbf{M})}{p^{5}}\right]
  34. Φ ( r , θ , ϕ ) \Phi(r,\theta,\phi)
  35. r , θ , ϕ r,\theta,\phi
  36. Φ ( r , θ , ϕ ) = R r Φ ( R 2 r , θ , ϕ ) \Phi^{\prime}(r,\theta,\phi)=\frac{R}{r}\Phi\left(\frac{R^{2}}{r},\theta,\phi\right)
  37. Φ \Phi
  38. q i q_{i}\,
  39. ( r i , θ i , ϕ i ) (r_{i},\theta_{i},\phi_{i})\,
  40. R q i / r i Rq_{i}/r_{i}\,
  41. ( R 2 / r i , θ i , ϕ i ) (R^{2}/r_{i},\theta_{i},\phi_{i})\,
  42. Φ \Phi
  43. ρ ( r , θ , ϕ ) \rho(r,\theta,\phi)\,
  44. ρ ( r , θ , ϕ ) = ( R / r ) 5 ρ ( R 2 / r , θ , ϕ ) \rho^{\prime}(r,\theta,\phi)=(R/r)^{5}\rho(R^{2}/r,\theta,\phi)\,

Method_of_matched_asymptotic_expansions.html

  1. ϵ y ′′ + ( 1 + ϵ ) y + y = 0 , \epsilon y^{\prime\prime}+(1+\epsilon)y^{\prime}+y=0,
  2. y y
  3. t t
  4. y ( 0 ) = 0 y(0)=0
  5. y ( 1 ) = 1 y(1)=1
  6. ϵ \epsilon
  7. 0 < ϵ 1 0<\epsilon\ll 1
  8. ϵ \epsilon
  9. ϵ = 0 \epsilon=0
  10. y + y = 0. y^{\prime}+y=0.\,
  11. y y
  12. t t
  13. ϵ \epsilon
  14. ϵ \epsilon
  15. ϵ 0 \epsilon\to 0
  16. y + y = 0. y^{\prime}+y=0.\,
  17. y = A e - t y=Ae^{-t}\,
  18. A A
  19. y ( 0 ) = 0 y(0)=0
  20. A = 0 A=0
  21. y ( 1 ) = 1 y(1)=1
  22. A = e A=e
  23. ϵ = 0 \epsilon=0
  24. ϵ \epsilon
  25. ϵ \epsilon
  26. t t
  27. t t
  28. ϵ \epsilon
  29. t = 0 t=0
  30. y ( 1 ) = 1 y(1)=1
  31. A = e A=e
  32. y O = e 1 - t y_{O}=e^{1-t}\,
  33. t t
  34. ϵ \epsilon
  35. τ = t / ϵ \tau=t/\epsilon
  36. t t
  37. τ ϵ \tau\epsilon
  38. 1 ϵ y ′′ ( τ ) + ( 1 + ϵ ) 1 ϵ y ( τ ) + y ( τ ) = 0 , \frac{1}{\epsilon}y^{\prime\prime}(\tau)+\left({1+\epsilon}\right)\frac{1}{% \epsilon}y^{\prime}(\tau)+y(\tau)=0,\,
  39. ϵ \epsilon
  40. ϵ = 0 \epsilon=0
  41. y ′′ + y = 0. y^{\prime\prime}+y^{\prime}=0.\,
  42. t t
  43. ϵ \epsilon
  44. y y
  45. y O y_{O}
  46. ϵ \epsilon
  47. ϵ \epsilon
  48. ϵ 0 \epsilon\to 0
  49. y ′′ + y = 0. y^{\prime\prime}+y^{\prime}=0.\,
  50. y = B - C e - τ y=B-Ce^{-\tau}\,
  51. B B
  52. C C
  53. y ( 0 ) = 0 y(0)=0
  54. B = C B=C
  55. y I = B ( 1 - e - τ ) = B ( 1 - e - t / ϵ ) . y_{I}=B\left({1-e^{-\tau}}\right)=B\left({1-e^{-t/\epsilon}}\right).\,
  56. B B
  57. t t
  58. ϵ t 1 \epsilon\ll t\ll 1
  59. lim τ y I = lim t 0 y O , \lim_{\tau\rightarrow\infty}y_{I}=\lim_{t\to 0}y_{O},\,
  60. B = e B=e
  61. y overlap \,y_{\mathrm{overlap}}
  62. e e
  63. y ( t ) = y I + y O - y overlap = e ( 1 - e - t / ϵ ) + e 1 - t - e = e ( e - t - e - t / ϵ ) . y(t)=y_{I}+y_{O}-y_{\mathrm{overlap}}=e\left({1-e^{-t/\epsilon}}\right)+e^{1-t% }-e=e\left({e^{-t}-e^{-t/\epsilon}}\right).\,
  64. y I y_{I}
  65. y O y_{O}
  66. t t
  67. ϵ \epsilon
  68. ϵ \epsilon
  69. ϵ \epsilon
  70. ϵ 0 \epsilon\rightarrow 0
  71. y ( t ) = e - t - e - t / ε e - 1 - e - 1 / ε , y(t)=\frac{{e^{-t}-e^{-t/\varepsilon}}}{{e^{-1}-e^{-1/\varepsilon}}},\,
  72. e 1 - 1 / ϵ e^{1-1/\epsilon}
  73. y y^{\prime}
  74. y ′′ y^{\prime\prime}
  75. t = 0 t=0
  76. τ = ( 1 - t ) / ϵ \tau=(1-t)/\epsilon
  77. ϵ \epsilon
  78. ϵ \epsilon
  79. ϵ log ϵ \epsilon\log\epsilon

Method_ringing.html

  1. n n
  2. n ! n!
  3. n n

Metric_connection.html

  1. \nabla
  2. X g = 0 \nabla_{X}g=0
  3. \nabla
  4. \nabla
  5. X g ( Y , Z ) = g ( X Y , Z ) + g ( Y , X Z ) Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}Z)
  6. X g ( Y , Z ) Xg(Y,Z)
  7. g ( Y , Z ) g(Y,Z)
  8. X X
  9. g a b g_{ab}
  10. c g a b = 0. \nabla_{c}\,g_{ab}=0.
  11. \nabla
  12. \nabla^{\prime}
  13. a x b = a x b - C a b c x c . \nabla_{a}x_{b}=\nabla_{a}^{\prime}x_{b}-{C_{ab}}^{c}x_{c}.
  14. C a b c {C_{ab}}^{c}

Metric_tree.html

  1. ( x , y ) (x,y)
  2. min x x max x \mbox{min}~{}_{x}\leq x\leq\mbox{max}~{}_{x}
  3. min y y max y \mbox{min}~{}_{y}\leq y\leq\mbox{max}~{}_{y}

Michel_parameters.html

  1. ρ , η , ξ \rho,\eta,\xi
  2. δ \delta
  3. l i - l j - ν i ν j ¯ l_{i}^{-}\rightarrow l_{j}^{-}\nu_{i}\bar{\nu_{j}}
  4. δ \delta
  5. ξ δ \xi\delta
  6. ρ = 3 4 , η = 0 , ξ = 1 , ξ δ = 3 4 . \rho={3\over 4},\quad\eta=0,\quad\xi=1,\quad\xi\delta={3\over 4}.
  7. μ + e + + ν e + ν ¯ μ . \mu^{+}\to e^{+}+\nu_{e}+\bar{\nu}_{\mu}.
  8. d 2 Γ x 2 d x d cos θ ( 3 - 3 x ) + 2 3 ρ ( 4 x - 3 ) + P μ ξ cos θ [ ( 1 - x ) + 2 3 δ ( 4 x - 3 ) ] , \frac{d^{2}\Gamma}{x^{2}dxd\cos\theta}\sim(3-3x)+\frac{2}{3}\rho(4x-3)+P_{\mu}% \xi\cos\theta[(1-x)+\frac{2}{3}\delta(4x-3)],
  9. P μ P_{\mu}
  10. x = E e / E e m a x x=E_{e}/E_{e}^{max}
  11. θ \theta
  12. c o s θ cos\theta
  13. d 2 Γ d x d cos θ x 2 [ ( 3 - 2 x ) - P μ cos θ ( 1 - 2 x ) ] . \frac{d^{2}\Gamma}{dxd\cos\theta}\sim x^{2}[(3-2x)-P_{\mu}\cos\theta(1-2x)].
  14. d Γ d cos θ 1 + 1 3 P μ cos θ . \frac{d\Gamma}{d\cos\theta}\sim 1+\frac{1}{3}P_{\mu}\cos\theta.
  15. d Γ d x ( 3 x 2 - 2 x 3 ) . \frac{d\Gamma}{dx}\sim(3x^{2}-2x^{3}).

Micromagnetics.html

  1. E = E exch + E anis + E Z + E demag + E m-e E=E\text{exch}+E\text{anis}+E\text{Z}+E\text{demag}+E\text{m-e}
  2. E exch = A V ( ( m x ) 2 + ( m y ) 2 + ( m z ) 2 ) d V E\text{exch}=A\int_{V}\left((\nabla m_{x})^{2}+(\nabla m_{y})^{2}+(\nabla m_{z% })^{2}\right)\mathrm{d}V
  3. E anis = V F anis ( 𝐦 ) d V E\text{anis}=\int_{V}F\text{anis}(\mathbf{m})\mathrm{d}V
  4. F anis ( 𝐦 ) = - K m z 2 F\text{anis}(\mathbf{m})=-Km_{z}^{2}
  5. E Z = - μ 0 V 𝐌 𝐇 a d V E\text{Z}=-\mu_{0}\int_{V}\mathbf{M}\cdot\mathbf{H}\text{a}\mathrm{d}V
  6. E demag = - μ 0 2 V 𝐌 𝐇 d d V E\text{demag}=-\frac{\mu_{0}}{2}\int_{V}\mathbf{M}\cdot\mathbf{H}\text{d}% \mathrm{d}V
  7. 𝐇 d = - 𝐌 \nabla\cdot\mathbf{H}\text{d}=-\nabla\cdot\mathbf{M}
  8. × 𝐇 d = 0 \nabla\times\mathbf{H}\text{d}=0
  9. 𝐇 d = - 1 4 π V 𝐌 𝐫 r 3 d V \mathbf{H}\text{d}=-\frac{1}{4\pi}\int_{V}\nabla\cdot\mathbf{M}\frac{\mathbf{r% }}{r^{3}}\mathrm{d}V
  10. ε 0 ( 𝐦 ) = 3 2 E [ 𝐦 𝐦 - 1 3 𝟏 ] \mathbf{\varepsilon}_{0}(\mathbf{m})=\frac{3}{2}E\,[\mathbf{m}\otimes\mathbf{m% }-\frac{1}{3}\mathbf{1}]
  11. ε e := ε - ε 0 \mathbf{\varepsilon}_{e}:=\mathbf{\varepsilon}-\mathbf{\varepsilon}_{0}
  12. E m-e = 1 2 [ ε - ε 0 ( 𝐦 ) ] : : [ ε - ε 0 ( 𝐦 ) ] E\text{m-e}=\frac{1}{2}[\mathbf{\varepsilon}-\mathbf{\varepsilon}_{0}(\mathbf{% m})]:\mathbb{C}:[\mathbf{\varepsilon}-\mathbf{\varepsilon}_{0}(\mathbf{m})]
  13. := λ 𝟏 𝟏 + 2 μ 𝕀 \mathbb{C}:=\lambda\mathbf{1}\otimes\mathbf{1}+2\mu\mathbb{I}
  14. E m-e = λ 2 tr [ ε ] 2 + μ tr [ ε 2 ] - 3 μ E { tr [ ε ( 𝐦 𝐦 ) ] - 1 3 tr [ ε ] } . E\text{m-e}=\frac{\lambda}{2}\mbox{tr}~{}^{2}[\mathbf{\varepsilon}]+\mu\,\mbox% {tr}~{}[\mathbf{\varepsilon}^{2}]-3\mu E\big\{\mbox{tr}~{}[\mathbf{\varepsilon% }(\mathbf{m}\otimes\mathbf{m})]-\frac{1}{3}\mbox{tr}~{}[\mathbf{\varepsilon}]% \big\}.
  15. 𝐇 eff = - 1 μ 0 M s d 2 E d 𝐦 d V \mathbf{H}_{\mathrm{eff}}=-\frac{1}{\mu_{0}M_{s}}\frac{\mathrm{d}^{2}E}{% \mathrm{d}\mathbf{m}\mathrm{d}V}
  16. d E = - μ 0 M s V ( d 𝐦 ) 𝐇 eff d V \mathrm{d}E=-\mu_{0}M_{s}\int_{V}(\mathrm{d}\mathbf{m})\cdot\mathbf{H}\text{% eff}\,\mathrm{d}V
  17. 𝐇 eff = 2 A μ 0 M s 2 𝐦 - 1 μ 0 M s F anis 𝐦 + 𝐇 a + 𝐇 d \mathbf{H}_{\mathrm{eff}}=\frac{2A}{\mu_{0}M_{s}}\nabla^{2}\mathbf{m}-\frac{1}% {\mu_{0}M_{s}}\frac{\partial F\text{anis}}{\partial\mathbf{m}}+\mathbf{H}\text% {a}+\mathbf{H}\text{d}
  18. 𝐦 t = - | γ | 𝐦 × 𝐇 eff + α 𝐦 × 𝐦 t \frac{\partial\mathbf{m}}{\partial t}=-|\gamma|\mathbf{m}\times\mathbf{H}_{% \mathrm{eff}}+\alpha\mathbf{m}\times\frac{\partial\mathbf{m}}{\partial t}
  19. 𝐦 t = - | γ | 1 + α 2 𝐦 × 𝐇 eff - α | γ | 1 + α 2 𝐦 × ( 𝐦 × 𝐇 eff ) \frac{\partial\mathbf{m}}{\partial t}=-\frac{|\gamma|}{1+\alpha^{2}}\mathbf{m}% \times\mathbf{H}_{\mathrm{eff}}-\frac{\alpha|\gamma|}{1+\alpha^{2}}\mathbf{m}% \times(\mathbf{m}\times\mathbf{H}\text{eff})

Microscopic_reversibility.html

  1. A B A\to B
  2. B A B\to A

MIDI_Tuning_Standard.html

  1. d = 69 + 12 log 2 ( f 440 Hz ) . d=69+12\log_{2}\left(\frac{f}{440\ \mathrm{Hz}}\right).\,
  2. f = 2 ( d - 69 ) / 12 440 Hz f=2^{(d-69)/12}\cdot 440\ \mathrm{Hz}\,

Midy's_theorem.html

  1. a p = 0. a 1 a 2 a 3 a n a n + 1 a 2 n ¯ \frac{a}{p}=0.\overline{a_{1}a_{2}a_{3}\dots a_{n}a_{n+1}\dots a_{2n}}
  2. a i + a i + n = 9 a_{i}+a_{i+n}=9\,
  3. a 1 a n + a n + 1 a 2 n = 10 n - 1. a_{1}\dots a_{n}+a_{n+1}\dots a_{2n}=10^{n}-1.\,
  4. 1 13 = 0. 076923 ¯ and 076 + 923 = 999. \frac{1}{13}=0.\overline{076923}\,\text{ and }076+923=999.\,
  5. 1 17 = 0. 0588235294117647 ¯ and 05882352 + 94117647 = 99999999. \frac{1}{17}=0.\overline{0588235294117647}\,\text{ and }05882352+94117647=9999% 9999.\,
  6. 1 19 = 0. 052631578947368421 ¯ \frac{1}{19}=0.\overline{052631578947368421}\,
  7. 052631 + 578947 + 368421 = 999999. 052631+578947+368421=999999.
  8. 052 + 631 + 578 + 947 + 368 + 421 = 2997 = 3 × 999. 052+631+578+947+368+421=2997=3\times 999.
  9. 1 19 = 0. 032745 ¯ 8 032 8 + 745 8 = 777 8 03 8 + 27 8 + 45 8 = 77 8 . \begin{aligned}&\displaystyle\frac{1}{19}=0.\overline{032745}_{8}\\ &\displaystyle 032_{8}+745_{8}=777_{8}\\ &\displaystyle 03_{8}+27_{8}+45_{8}=77_{8}.\end{aligned}
  10. 1 19 = 0. 076 45 ¯ 12 \displaystyle\frac{1}{19}=0.\overline{076\mathcal{E}45}_{12}
  11. a p = [ 0. a 1 a 2 a ¯ ] b a p b = [ a 1 a 2 a . a 1 a 2 a ¯ ] b a p b = N + [ 0. a 1 a 2 a ¯ ] b = N + a p a p = N b - 1 \begin{aligned}&\displaystyle\frac{a}{p}=[0.\overline{a_{1}a_{2}\dots a_{\ell}% }]_{b}\\ &\displaystyle\Rightarrow\frac{a}{p}b^{\ell}=[a_{1}a_{2}\dots a_{\ell}.% \overline{a_{1}a_{2}\dots a_{\ell}}]_{b}\\ &\displaystyle\Rightarrow\frac{a}{p}b^{\ell}=N+[0.\overline{a_{1}a_{2}\dots a_% {\ell}}]_{b}=N+\frac{a}{p}\\ &\displaystyle\Rightarrow\frac{a}{p}=\frac{N}{b^{\ell}-1}\end{aligned}
  12. a p = N m ( b k - 1 ) . \frac{a}{p}=\frac{N}{m(b^{k}-1)}.
  13. a m p = N b k - 1 \frac{am}{p}=\frac{N}{b^{k}-1}
  14. N 0 ( mod b k - 1 ) . N\equiv 0\;\;(\mathop{{\rm mod}}b^{k}-1).\,
  15. N h - 1 = [ a 1 a k ] b N h - 2 = [ a k + 1 a 2 k ] b N 0 = [ a l - k + 1 a l ] b \begin{aligned}\displaystyle N_{h-1}&\displaystyle=[a_{1}\dots a_{k}]_{b}\\ \displaystyle N_{h-2}&\displaystyle=[a_{k+1}\dots a_{2k}]_{b}\\ &\displaystyle{}\ \ \vdots\\ \displaystyle N_{0}&\displaystyle=[a_{l-k+1}\dots a_{l}]_{b}\end{aligned}
  16. N = i = 0 h - 1 N i b i k = i = 0 h - 1 N i ( b k ) i N=\sum_{i=0}^{h-1}N_{i}b^{ik}=\sum_{i=0}^{h-1}N_{i}(b^{k})^{i}
  17. N i = 0 h - 1 N i ( mod b k - 1 ) \Rightarrow N\equiv\sum_{i=0}^{h-1}N_{i}\;\;(\mathop{{\rm mod}}b^{k}-1)
  18. i = 0 h - 1 N i 0 ( mod b k - 1 ) \Rightarrow\sum_{i=0}^{h-1}N_{i}\equiv 0\;\;(\mathop{{\rm mod}}b^{k}-1)
  19. 0 N i b k - 1. 0\leq N_{i}\leq b^{k}-1.\,
  20. 0 < N 0 + N 1 < 2 ( b k - 1 ) 0<N_{0}+N_{1}<2(b^{k}-1)\,
  21. N 0 + N 1 = b k - 1. N_{0}+N_{1}=b^{k}-1.\,
  22. a m p \frac{am}{p}
  23. m = 0 ( mod p ) m=0\;\;(\mathop{{\rm mod}}p)
  24. k = l 2 k=\frac{l}{2}
  25. b l 2 + 1 = 0 ( mod p ) b^{\frac{l}{2}}+1=0\;\;(\mathop{{\rm mod}}p)
  26. k = l 3 k=\frac{l}{3}
  27. b 2 3 l + b l 3 + 1 = 0 ( mod p ) b^{\frac{2}{3}l}+b^{\frac{l}{3}}+1=0\;\;(\mathop{{\rm mod}}p)

Mikhail_Molodenskii.html

  1. h = H + N h=H+N
  2. h = H * + ζ h=H^{*}+\zeta
  3. ζ \zeta
  4. H * H^{*}

Miller_cylindrical_projection.html

  1. x = λ x=\lambda
  2. y = 5 4 ln [ tan ( π 4 + 2 φ 5 ) ] = 5 4 sinh - 1 ( tan 4 φ 5 ) y=\frac{5}{4}\ln\left[\tan\left(\frac{\pi}{4}+\frac{2\varphi}{5}\right)\right]% =\frac{5}{4}\sinh^{-1}\left(\tan\frac{4\varphi}{5}\right)
  3. λ = x \lambda=x
  4. φ = 5 2 tan - 1 ( e 4 y 5 ) - 5 π 8 = 5 4 tan - 1 ( sinh 4 y 5 ) \varphi=\frac{5}{2}\tan^{-1}\left(e^{\frac{4y}{5}}\right)-\frac{5\pi}{8}=\frac% {5}{4}\tan^{-1}\left(\sinh\frac{4y}{5}\right)
  5. φ \varphi\,

Miller_effect.html

  1. C M = C ( 1 + A v ) C_{M}=C(1+A_{v})\,
  2. - A v -A_{v}
  3. - A v -A_{v}
  4. Z Z
  5. V o = - A v V i V_{o}=-A_{v}V_{i}
  6. Z Z
  7. I i = V i - V o Z = V i ( 1 + A v ) Z I_{i}=\frac{V_{i}-V_{o}}{Z}=\frac{V_{i}(1+A_{v})}{Z}
  8. Z i n = V i I i = Z 1 + A v . Z_{in}=\frac{V_{i}}{I_{i}}=\frac{Z}{1+A_{v}}.
  9. Z = 1 s C Z=\frac{1}{sC}
  10. Z i n = 1 s C M where C M = C ( 1 + A v ) . Z_{in}=\frac{1}{sC_{M}}\quad\mathrm{where}\quad C_{M}=C(1+A_{v}).
  11. ( 1 + A v ) (1+A_{v})
  12. A v A_{v}
  13. C ( 1 + 1 / A v ) {C}({1+1/A_{v}})
  14. A v A_{v}
  15. R C RC
  16. j ω C C ( V i - V O ) = j ω C M V i , \ j\omega C_{C}(V_{i}-V_{O})=j\omega C_{M}V_{i},
  17. C M = C C ( 1 - V o V i ) = C C ( 1 + A v ) . C_{M}=C_{C}\left(1-\frac{V_{o}}{V_{i}}\right)=C_{C}(1+A_{v}).
  18. V o = - A v V i = - A v V A 1 + j ω C M R A , V_{o}=-A_{v}V_{i}=-A_{v}\frac{V_{A}}{1+j\omega C_{M}R_{A}},

Milliken's_tree_theorem.html

  1. 𝕊 T n \mathbb{S}^{n}_{T}
  2. 𝕊 T n = C 1 C r \mathbb{S}^{n}_{T}=C_{1}\cup...\cup C_{r}
  3. 𝕊 R n C i \mathbb{S}^{n}_{R}\subset C_{i}
  4. 𝕊 n = T 𝕊 T n \mathbb{S}^{n}=\bigcup_{T}\mathbb{S}^{n}_{T}
  5. 𝕊 n \mathbb{S}^{n}
  6. { q P : q p } \{q\in P:q\geq p\}
  7. I S ( p , P ) IS(p,P)
  8. S T S\subset T
  9. s S s\in S
  10. t I S ( s , T ) t\in IS(s,T)
  11. | S u c c ( t , T ) I S ( s , S ) | = 1 |Succ(t,T)\cap IS(s,S)|=1
  12. α \alpha
  13. β \beta
  14. S ( n ) T ( f ( n ) ) . S(n)\subset T(f(n)).

Milliken–Taylor_theorem.html

  1. 𝒫 f ( ) \mathcal{P}_{f}(\mathbb{N})
  2. \mathbb{N}
  3. 𝒫 f ( ) \mathcal{P}_{f}(\mathbb{N})
  4. [ F S ( a n n = 0 ) ] < k = { { α 1 , , α k } : α 1 , , α k 𝒫 f ( ) and α 1 < < α k } . [FS(\langle a_{n}\rangle_{n=0}^{\infty})]^{k}_{<}=\left\{\left\{\sum\alpha_{1}% ,\ldots,\sum\alpha_{k}\right\}:\alpha_{1},\cdots,\alpha_{k}\in\mathcal{P}_{f}(% \mathbb{N})\,\text{ and }\alpha_{1}<\cdots<\alpha_{k}\right\}.
  5. [ S ] k [S]^{k}
  6. [ ] k = C 1 C 2 C r [\mathbb{N}]^{k}=C_{1}\cup C_{2}\cup\cdots\cup C_{r}
  7. a n n = 0 \langle a_{n}\rangle_{n=0}^{\infty}\subset\mathbb{N}
  8. [ F S ( a n n = 0 ) ] < k C i [FS(\langle a_{n}\rangle_{n=0}^{\infty})]^{k}_{<}\subset C_{i}
  9. a n n = 0 \langle a_{n}\rangle_{n=0}^{\infty}\subset\mathbb{N}
  10. [ F S ( a n n = 0 ) ] < k [FS(\langle a_{n}\rangle_{n=0}^{\infty})]^{k}_{<}

MIMIC.html

  1. f ˙ \dot{f}
  2. s ˙ \dot{s}
  3. α \alpha
  4. β \beta
  5. γ \gamma
  6. ϵ \epsilon
  7. f ˙ = α f - β f s \dot{f}=\alpha f-\beta fs
  8. s ˙ = ϵ β f s - γ s \dot{s}=\epsilon\beta fs-\gamma s
  9. f ( 0 ) = f o f(0)=f_{o}
  10. s ( 0 ) = s o s(0)=s_{o}
  11. f o f_{o}
  12. s o s_{o}
  13. α \alpha
  14. β \beta
  15. γ \gamma
  16. ϵ \epsilon

Minakshisundaram–Pleijel_zeta_function.html

  1. λ 1 , λ 2 , \lambda_{1},\lambda_{2},\ldots
  2. Re ( s ) \operatorname{Re}(s)
  3. Z ( s ) = Tr ( Δ - s ) = n = 1 | λ n | - s . Z(s)=\operatorname{Tr}(\Delta^{-s})=\sum_{n=1}^{\infty}|\lambda_{n}|^{-s}.
  4. Z ( P , Q , s ) = n = 1 f n ( P ) f n ( Q ) λ n s Z(P,Q,s)=\sum_{n=1}^{\infty}\frac{f_{n}(P)f_{n}(Q)}{\lambda_{n}^{s}}
  5. Z ( P , P , s ) T N / 2 ( 2 π ) N Γ ( N / 2 + 1 ) \displaystyle Z(P,P,s)\sim\frac{T^{N/2}}{(2\sqrt{\pi})^{N}\Gamma(N/2+1)}
  6. Z ( s ) = M Z ( P , P , s ) d P \displaystyle Z(s)=\int_{M}Z(P,P,s)dP
  7. K ( P , Q , t ) = n = 1 f n ( P ) f n ( Q ) e - λ n t K(P,Q,t)=\sum_{n=1}^{\infty}f_{n}(P)f_{n}(Q)e^{-\lambda_{n}t}
  8. Z ( P , Q , s ) = 1 Γ ( s ) 0 K ( P , Q , t ) t s - 1 d t Z(P,Q,s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}K(P,Q,t)t^{s-1}dt
  9. Z ( s ) = i = 1 e - λ i s . Z(s)=\sum^{\infty}_{i=1}e^{-\lambda_{i}s}.
  10. Z ( s ) = n 0 1 ( n 2 ) s = 2 ζ ( 2 s ) Z(s)=\sum_{n\neq 0}\frac{1}{(n^{2})^{s}}=2\zeta(2s)
  11. Z ( s ) ( 4 π s ) - n / 2 m = 0 a m s m . Z(s)\sim(4\pi s)^{-n/2}\sum^{\infty}_{m=0}a_{m}s^{m}.
  12. a 0 = V o l ( M , g ) , a 1 = 1 6 M S ( x ) d V a_{0}=Vol(M,g),\ \ \ \ a_{1}=\frac{1}{6}\int_{M}S(x)dV
  13. 0 = λ 0 λ 1 λ 2 , 0=\lambda_{0}\leq\lambda_{1}\leq\lambda_{2}\cdots,
  14. λ \lambda
  15. ω n \omega_{n}
  16. n \mathbb{R}^{n}
  17. N ( λ ) ω n V o l ( M ) λ n / 2 ( 2 π ) n , N(\lambda)\sim\frac{\omega_{n}Vol(M)\lambda^{n/2}}{(2\pi)^{n}},
  18. ( λ k ) n / 2 ( 2 π ) n k ω n V o l ( M ) . (\lambda_{k})^{n/2}\sim\frac{(2\pi)^{n}k}{\omega_{n}Vol(M)}.

Minimax_approximation_algorithm.html

  1. f f
  2. [ a , b ] [a,b]
  3. n n
  4. p p
  5. n n
  6. max a x b | f ( x ) - p ( x ) | . \max_{a\leq x\leq b}|f(x)-p(x)|.

Minimum_energy_control.html

  1. u ( t ) u(t)
  2. 𝐱 ˙ ( t ) = A 𝐱 ( t ) + B 𝐮 ( t ) \dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{u}(t)
  3. 𝐲 ( t ) = C 𝐱 ( t ) + D 𝐮 ( t ) \mathbf{y}(t)=C\mathbf{x}(t)+D\mathbf{u}(t)
  4. x ( t 0 ) = x 0 x(t_{0})=x_{0}
  5. u ( t ) u(t)
  6. x 1 x_{1}
  7. t 1 t_{1}
  8. u ¯ ( t ) \bar{u}(t)
  9. x 0 x_{0}
  10. x 1 x_{1}
  11. t 1 t_{1}
  12. t 0 t 1 u ¯ * ( t ) u ¯ ( t ) d t t 0 t 1 u * ( t ) u ( t ) d t . \int_{t_{0}}^{t_{1}}\bar{u}^{*}(t)\bar{u}(t)dt\ \geq\ \int_{t_{0}}^{t_{1}}u^{*% }(t)u(t)dt.
  13. W c ( t ) = t 0 t e A ( t - τ ) B B * e A * ( t - τ ) d τ . W_{c}(t)=\int_{t_{0}}^{t}e^{A(t-\tau)}BB^{*}e^{A^{*}(t-\tau)}d\tau.
  14. W c W_{c}
  15. u ( t ) = - B * e A * ( t 1 - t ) W c - 1 ( t 1 ) [ e A ( t 1 - t 0 ) x 0 - x 1 ] . u(t)=-B^{*}e^{A^{*}(t_{1}-t)}W_{c}^{-1}(t_{1})[e^{A(t_{1}-t_{0})}x_{0}-x_{1}].
  16. x ( t ) = e A ( t - t 0 ) x 0 + t 0 t e A ( t - τ ) B u ( τ ) d τ x(t)=e^{A(t-t_{0})}x_{0}+\int_{t_{0}}^{t}e^{A(t-\tau)}Bu(\tau)d\tau
  17. x 1 x_{1}
  18. t 1 t_{1}

Minimum_resolvable_temperature_difference.html

  1. F ( x ) = Δ t ( i ) f s ( i ) F(x)=\frac{\Delta\,t(i)}{f_{s}(i)}
  2. Δ t ( i ) \Delta\,t(i)
  3. f s ( i ) f_{s}(i)\,\!
  4. Δ \Delta\,

Minimum_total_potential_energy_principle.html

  1. s y m b o l Π symbol{\Pi}
  2. s y m b o l Π = 𝐔 + 𝐕 ( 1 ) symbol{\Pi}=\mathbf{U}+\mathbf{V}\qquad\mathrm{(1)}
  3. \deltasymbol Π = δ ( 𝐔 + 𝐕 ) = 0 ( 2 ) \deltasymbol{\Pi}=\delta(\mathbf{U}+\mathbf{V})=0\qquad\mathrm{(2)}
  4. S t δ 𝐮 T 𝐓 d S + V δ 𝐮 T 𝐟 d V = V \deltasymbol ϵ T s y m b o l σ d V ( 3 ) \int_{S_{t}}\delta\ \mathbf{u}^{T}\mathbf{T}dS+\int_{V}\delta\ \mathbf{u}^{T}% \mathbf{f}dV=\int_{V}\deltasymbol{\epsilon}^{T}symbol{\sigma}dV\qquad\mathrm{(% 3)}
  5. 𝐮 \mathbf{u}
  6. 𝐓 \mathbf{T}
  7. S t S_{t}
  8. 𝐟 \mathbf{f}
  9. δ 𝐔 \delta\mathbf{U}
  10. 𝐕 = - S t 𝐮 T 𝐓 d S - V 𝐮 T 𝐟 d V \mathbf{V}=-\int_{S_{t}}\mathbf{u}^{T}\mathbf{T}dS-\int_{V}\mathbf{u}^{T}% \mathbf{f}dV
  11. - δ 𝐕 = δ 𝐔 -\delta\ \mathbf{V}=\delta\ \mathbf{U}

Minkowski_functional.html

  1. p ( x ) = inf { λ > 0 : x λ K } p(x)=\inf\{\lambda\in\mathbb{R}_{>0}:x\in\lambda K\}
  2. λ \lambda
  3. p ( x ) = inf { r > 0 : x r K } . p(x)=\inf\left\{r>0:x\in rK\right\}.
  4. p ( x ) = x p(x)=\|x\|
  5. K = { x X : | ϕ ( x ) | a } . K=\{x\in X:|\phi(x)|\leq a\}.
  6. p ( x ) = inf { r > 0 : x r K } . p(x)=\inf\left\{r>0:x\in rK\right\}.
  7. p ( x ) = 1 a | ϕ ( x ) | . p(x)=\frac{1}{a}|\phi(x)|.
  8. p K : X [ 0 , ) p_{K}:X\rightarrow[0,\infty)
  9. p K ( x ) = inf { r > 0 : x r K } , p_{K}(x)=\inf\left\{r>0:x\in rK\right\},
  10. K K
  11. p K ( 1 2 x + 1 2 y ) r + ϵ = 1 2 p K ( x ) + 1 2 p K ( y ) + ϵ . p_{K}\left(\frac{1}{2}x+\frac{1}{2}y\right)\leq r+\epsilon=\frac{1}{2}p_{K}(x)% +\frac{1}{2}p_{K}(y)+\epsilon.
  12. p K ( x + y ) p K ( x ) + p K ( y ) + ϵ , for all ϵ > 0. p_{K}(x+y)\leq p_{K}(x)+p_{K}(y)+\epsilon,\quad\mbox{for all}~{}\quad\epsilon>0.
  13. λ x r K if and only if x r | λ | K . \lambda x\in rK\quad\mbox{if and only if}~{}\quad x\in\frac{r}{|\lambda|}K.
  14. p K ( λ x ) = inf { r > 0 : λ x r K } = inf { r > 0 : x r | λ | K } = inf { | λ | r | λ | > 0 : x r | λ | K } = | λ | p K ( x ) . p_{K}(\lambda x)=\inf\left\{r>0:\lambda x\in rK\right\}=\inf\left\{r>0:x\in% \frac{r}{|\lambda|}K\right\}=\inf\left\{|\lambda|\frac{r}{|\lambda|}>0:x\in% \frac{r}{|\lambda|}K\right\}=|\lambda|p_{K}(x).

Minkowski_plane.html

  1. d P ( P 1 , P 2 ) = ( x 1 - x 2 ) 2 - ( y 1 - y 2 ) 2 d_{P}(P_{1},P_{2})=(x_{1}-x_{2})^{2}-(y_{1}-y_{2})^{2}
  2. P i = ( x i . y i ) P_{i}=(x_{i}.y_{i})
  3. { P \R 2 | d P ( P , M ) = r } \{P\in\R^{2}\ |\ d_{P}(P,M)=r\}
  4. M M
  5. d P ( P 1 , P 2 ) = ( x 1 - x 2 ) ( y 1 - y 2 ) d^{\prime}_{P}(P_{1},P_{2})=(x_{1}-x_{2})(y_{1}-y_{2})
  6. 𝒫 := ( \R { } ) 2 = \R 2 ( { } × \R ) ( \R × { } ) { ( , ) } , \R \mathcal{P}:=(\R\cup\{\infty\})^{2}=\R^{2}\cup(\{\infty\}\times\R)\cup(\R% \times\{\infty\})\ \cup\{(\infty,\infty)\}\ ,\ \infty\notin\R
  7. 𝒵 := { { ( x , y ) \R 2 | y = a x + b } { ( , ) } | a , b \R , a 0 } \mathcal{Z}:=\{\{(x,y)\in\R^{2}\ |\ y=ax+b\}\cup\{(\infty,\infty)\}\ |\ a,b\in% \R,a\neq 0\}
  8. { { ( x , y ) \R 2 | y = a x - b + c , x b } { ( b , ) , ( , c ) } | a , b , c \R , a 0 } , \cup\{\{(x,y)\in\R^{2}\ |y=\frac{a}{x-b}+c,x\neq b\}\cup\{(b,\infty),(\infty,c% )\}\ |\ a,b,c\in\R,a\neq 0\},
  9. ( 𝒫 , 𝒵 , ) ({\mathcal{P}},{\mathcal{Z}},\in)
  10. \R 2 \R^{2}
  11. \R \R
  12. ( , ) (\infty,\infty)
  13. y = a x + b , a 0 y=ax+b,a\neq 0
  14. ( , ) (\infty,\infty)
  15. y = a x - b + c , a 0 y=\frac{a}{x-b}+c,a\neq 0
  16. ( b , ) , ( , c ) (b,\infty),(\infty,c)
  17. ( x 1 , y 1 ) ( x 2 , y 2 ) (x_{1},y_{1})\neq(x_{2},y_{2})
  18. x 1 = x 2 x_{1}=x_{2}
  19. y 1 = y 2 y_{1}=y_{2}
  20. P 1 , P 2 P_{1},P_{2}
  21. P 1 | + P 2 P_{1}\parallel_{+}P_{2}
  22. x 1 = x 2 x_{1}=x_{2}
  23. P 1 | - P 2 P_{1}\parallel_{-}P_{2}
  24. y 1 = y 2 y_{1}=y_{2}
  25. P 1 , P 2 P_{1},P_{2}
  26. P 1 P 2 P_{1}\parallel P_{2}
  27. P 1 | + P 2 P_{1}\parallel_{+}P_{2}
  28. P 1 | - P 2 P_{1}\parallel_{-}P_{2}
  29. A , B A,B
  30. C C
  31. A | + C | - B A\parallel_{+}C\parallel_{-}B
  32. P P
  33. z z
  34. A , B z A,B\in z
  35. A | + P | - B A\parallel_{+}P\parallel_{-}B
  36. A , B , C A,B,C
  37. z z
  38. A , B , C A,B,C
  39. z z
  40. P z P\in z
  41. Q , P ∦ Q Q,P\not\parallel Q
  42. Q z Q\notin z
  43. z z^{\prime}
  44. z z = { P } z\cap z^{\prime}=\{P\}
  45. z z
  46. z z^{\prime}
  47. ( 𝒫 , 𝒵 ; + , - , ) \left({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in\right)
  48. 𝒫 \mathcal{P}
  49. 𝒵 \mathcal{Z}
  50. + \parallel_{+}
  51. - \parallel_{-}
  52. 𝒫 \mathcal{P}
  53. P 𝒫 P\in\mathcal{P}
  54. P ¯ + := { Q 𝒫 | Q + P } \overline{P}_{+}:=\left\{\left.Q\in\mathcal{P}\ \right|\ Q\parallel_{+}P\right\}
  55. P ¯ - := { Q 𝒫 | Q - P } \overline{P}_{-}:=\left\{\left.Q\in\mathcal{P}\ \right|\ Q\parallel_{-}P\right\}
  56. P ¯ + \overline{P}_{+}
  57. P ¯ - \overline{P}_{-}
  58. A , B A,B
  59. A B A\parallel B
  60. A | + B A\parallel_{+}B
  61. A | - B A\parallel_{-}B
  62. 𝔐 := ( 𝒫 , 𝒵 ; + , - , ) {\mathfrak{M}}:=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in)
  63. A , B A,B
  64. C C
  65. A | + C | - B A\parallel_{+}C\parallel_{-}B
  66. P P
  67. z z
  68. A , B z A,B\in z
  69. A | + P | - B A\parallel_{+}P\parallel_{-}B
  70. A , B , C A,B,C
  71. z z
  72. A , B , C A,B,C
  73. z z
  74. P z P\in z
  75. Q , P ∦ Q Q,P\not\parallel Q
  76. Q z Q\notin z
  77. z z^{\prime}
  78. z z = { P } z\cap z^{\prime}=\{P\}
  79. z z
  80. z z^{\prime}
  81. P P
  82. z z
  83. P P
  84. z z
  85. A , B A,B
  86. | A ¯ + B ¯ - | = 1 \left|\overline{A}_{+}\cap\overline{B}_{-}\right|=1
  87. P P
  88. z z
  89. | P ¯ + z | = 1 = | P ¯ - z | \left|\overline{P}_{+}\cap z\right|=1=\left|\overline{P}_{-}\cap z\right|
  90. 𝔐 {\mathfrak{M}}
  91. 𝔐 = ( 𝒫 , 𝒵 ; + , - , ) {\mathfrak{M}}=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in)
  92. P 𝒫 P\in\mathcal{P}
  93. 𝔄 P := ( 𝒫 P ¯ , { z { P ¯ } | P z 𝒵 } { E P ¯ | E { P ¯ + , P ¯ - } } , ) \mathfrak{A}_{P}:=(\mathcal{P}\setminus\overline{P},\{z\setminus\{\overline{P}% \}\ |\ P\in z\in\mathcal{Z}\}\cup\{E\setminus\overline{P}\ |\ E\in{\mathcal{E}% }\setminus\{\overline{P}_{+},\overline{P}_{-}\}\},\in)
  94. 𝔄 ( , ) \mathfrak{A}_{(\infty,\infty)}
  95. \R 2 \R^{2}
  96. 𝔐 = ( 𝒫 , 𝒵 ; + , , ) {\mathfrak{M}}=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel,\in)
  97. 𝔐 = ( 𝒫 , 𝒵 ; + , - , ) {\mathfrak{M}}=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in)
  98. + \parallel_{+}
  99. - \parallel_{-}
  100. 𝒫 \mathcal{P}
  101. 𝔐 {\mathfrak{M}}
  102. P P
  103. 𝔄 P \mathfrak{A}_{P}
  104. K ¯ := { 0 , 1 , } \overline{K}:=\{0,1,\infty\}
  105. 𝒫 := K ¯ 2 \mathcal{P}:={\overline{K}}^{2}\qquad
  106. 𝒵 := { { ( a 1 , b 1 ) , ( a 2 , b 2 ) , ( a 3 , b 3 ) } | \mathcal{Z}:=\{\{(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\}|
  107. | { a 1 , a 2 , a 3 } = { b 1 , b 2 , b 3 } = K ¯ } = |\{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}=\overline{K}\}=
  108. { { ( 0 , 0 ) , ( 1 , 1 ) , ( , ) } , \{\{(0,0),(1,1),(\infty,\infty)\},
  109. { ( 0 , 0 ) , ( 1 , ) , ( , 1 ) } , \{(0,0),(1,\infty),(\infty,1)\},
  110. { ( 0 , 1 ) , ( 1 , 0 ) , ( , ) } , \{(0,1),(1,0),(\infty,\infty)\},
  111. { ( 0 , 1 ) , ( 1 , ) , ( , 0 ) } , \{(0,1),(1,\infty),(\infty,0)\},
  112. { ( 0 , ) , ( 1 , 1 ) , ( , 0 ) } , \{(0,\infty),(1,1),(\infty,0)\},
  113. { ( 0 , ) , ( 1 , 0 ) , ( , 1 ) } } \{(0,\infty),(1,0),(\infty,1)\}\}
  114. ( x 1 , y 1 ) | + ( x 2 , y 2 ) (x_{1},y_{1})\parallel_{+}(x_{2},y_{2})
  115. x 1 = x 2 x_{1}=x_{2}
  116. ( x 1 , y 1 ) | - ( x 2 , y 2 ) (x_{1},y_{1})\parallel_{-}(x_{2},y_{2})
  117. y 1 = y 2 y_{1}=y_{2}
  118. | 𝒫 | = 9 \left|\mathcal{P}\right|=9
  119. | 𝒵 | = 6 \left|\mathcal{Z}\right|=6
  120. 𝔐 = ( 𝒫 , 𝒵 ; + , - , ) {\mathfrak{M}}=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in)
  121. | 𝒫 | < \left|\mathcal{P}\right|<\infty
  122. z 1 , z 2 z_{1},z_{2}
  123. e 1 , e 2 e_{1},e_{2}
  124. | z 1 | = | z 2 | = | e 1 | = | e 2 | \left|z_{1}\right|=\left|z_{2}\right|=\left|e_{1}\right|=\left|e_{2}\right|
  125. 𝔐 {\mathfrak{M}}
  126. z z
  127. 𝔐 {\mathfrak{M}}
  128. n = | z | - 1 n=\left|z\right|-1
  129. 𝔐 {\mathfrak{M}}
  130. 𝔐 = ( 𝒫 , 𝒵 ; + , - , ) {\mathfrak{M}}=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in)
  131. n n
  132. | 𝒫 | = ( n + 1 ) 2 \left|\mathcal{P}\right|=(n+1)^{2}
  133. | 𝒵 | = ( n + 1 ) n ( n - 1 ) \left|\mathcal{Z}\right|=(n+1)n(n-1)
  134. \R \R
  135. K K
  136. 𝔐 ( K ) = ( 𝒫 , 𝒵 ; + , - , ) {\mathfrak{M}}(K)=({\mathcal{P}},{\mathcal{Z}};\parallel_{+},\parallel_{-},\in)
  137. 𝔐 ( K ) \mathfrak{M}(K)
  138. 𝔐 ( K ) \mathfrak{M}(K)
  139. P 1 , , P 8 P_{1},...,P_{8}
  140. 𝔐 ( K ) \mathfrak{M}(K)
  141. 𝔐 ( K ) \mathfrak{M}(K)
  142. 𝔐 ( K ) \mathfrak{M}(K)
  143. K = G F ( 2 ) K=GF(2)
  144. { 0 , 1 } \{0,1\}
  145. 𝔐 ( K ) \mathfrak{M}(K)
  146. K K

Mitotic_index.html

  1. c e l l s o b s e r v e d w i t h v i s i b l e c h r o m o s o m e s ÷ t o t a l n u m b e r o f c e l l s v i s i b l e cellsobservedwithvisiblechromosomes÷totalnumberofcellsvisible
  2. Mitotic I = ( P + M + A + T ) N * 100 % \mbox{Mitotic I}~{}=\frac{\mbox{}}{(P+M+A+T)}~{}\mbox{N}~{}*100\%

Mitsuhiro_Shishikura.html

  1. d d\,
  2. 2 d - 2 2d-2\,

MODFLOW.html

  1. x [ K x x h x ] + y [ K y y h y ] + z [ K z z h z ] + W = S S h t \frac{\partial}{\partial x}\left[K_{xx}\frac{\partial h}{\partial x}\right]+% \frac{\partial}{\partial y}\left[K_{yy}\frac{\partial h}{\partial y}\right]+% \frac{\partial}{\partial z}\left[K_{zz}\frac{\partial h}{\partial z}\right]+W=% S_{S}\frac{\partial h}{\partial t}
  2. K x x K_{xx}
  3. K y y K_{yy}
  4. K z z K_{zz}
  5. h h
  6. W W
  7. S S S_{S}
  8. t t\,
  9. 𝐶𝑅 i , j - 1 2 , k ( h i , j - 1 , k m - h i , j , k m ) + 𝐶𝑅 i , j + 1 2 , k ( h i , j + 1 , k m - h i , j , k m ) + \displaystyle\mathit{CR}_{i,j-\tfrac{1}{2},k}\left(h^{m}_{i,j-1,k}-h^{m}_{i,j,% k}\right)+\mathit{CR}_{i,j+\tfrac{1}{2},k}\left(h^{m}_{i,j+1,k}-h^{m}_{i,j,k}% \right)+
  10. h i , j , k m h^{m}_{i,j,k}\,
  11. P i , j , k P_{i,j,k}\,
  12. Q i , j , k Q_{i,j,k}\,
  13. Q i , j , k < 0.0 Q_{i,j,k}<0.0\,
  14. Q i , j , k > 0.0 Q_{i,j,k}>0.0\,
  15. 𝑆𝑆 i , j , k \mathit{SS}_{i,j,k}\,
  16. Δ r j Δ c i Δ v k \Delta r_{j}\Delta c_{i}\Delta v_{k}\,
  17. t m t^{m}\,
  18. 𝐶𝑉 i , j , k - 1 2 h i , j , k - 1 m + 𝐶𝐶 i - 1 2 , j , k h i - 1 , j , k m + 𝐶𝑅 i , j - 1 2 , k h i , j - 1 , k m \displaystyle\mathit{CV}_{i,j,k-\tfrac{1}{2}}h^{m}_{i,j,k-1}+\mathit{CC}_{i-% \tfrac{1}{2},j,k}h^{m}_{i-1,j,k}+\mathit{CR}_{i,j-\tfrac{1}{2},k}h^{m}_{i,j-1,k}
  19. 𝐻𝐶𝑂𝐹 i , j , k = P i , j , k - 𝑆𝑆 i , j , k Δ r j Δ c i Δ k t m - t m - 1 𝑅𝐻𝑆 i , j , k = - Q i , j , k - 𝑆𝑆 i , j , k Δ r j Δ c i Δ v k h i , j , k m - 1 t m - t m - 1 \begin{aligned}\displaystyle\mathit{HCOF}_{i,j,k}&\displaystyle=P_{i,j,k}-% \frac{\mathit{SS}_{i,j,k}\Delta r_{j}\Delta c_{i}\Delta_{k}}{t^{m}-t^{m-1}}\\ \displaystyle\mathit{RHS}_{i,j,k}&\displaystyle=-Q_{i,j,k}-\mathit{SS}_{i,j,k}% \Delta r_{j}\Delta c_{i}\Delta v_{k}\frac{h^{m-1}_{i,j,k}}{t^{m}-t^{m-1}}\end{aligned}
  20. A h = q A{h}={q}
  21. h {h}
  22. q {q}
  23. 𝐊 = [ K x x 0 0 0 K y y 0 0 0 K z z ] \mathbf{K}=\begin{bmatrix}K_{xx}&0&0\\ 0&K_{yy}&0\\ 0&0&K_{zz}\end{bmatrix}

Modified_internal_rate_of_return.html

  1. MIRR = F V ( positive cash flows, reinvestment rate ) - P V ( negative cash flows, finance rate ) n - 1 \mbox{MIRR}~{}=\sqrt[n]{\frac{FV(\,\text{positive cash flows, reinvestment % rate})}{-PV(\,\text{negative cash flows, finance rate})}}-1
  2. r r
  3. NPV = - 1000 + - 4000 ( 1 + r ) 1 + 5000 ( 1 + r ) 2 + 2000 ( 1 + r ) 3 = 0 \mbox{NPV}~{}=-1000+\frac{-4000}{(1+r)^{1}}+\frac{5000}{(1+r)^{2}}+\frac{2000}% {(1+r)^{3}}=0
  4. P V ( negative cash flows, finance rate ) = - 1000 + - 4000 ( 1 + 10 % ) 1 = - 4636.36 PV(\,\text{negative cash flows, finance rate})=-1000+\frac{-4000}{(1+10\%)^{1}% }=-4636.36
  5. F V ( positive cash flows, reinvestment rate ) = 5000 ( 1 + 12 % ) 1 + 2000 = 7600 FV(\,\text{positive cash flows, reinvestment rate})=5000\cdot(1+12\%)^{1}+2000% =7600
  6. MIRR = 7600 4636.36 3 - 1 = 17.91 % \mbox{MIRR}~{}=\sqrt[3]{\frac{7600}{4636.36}}-1=17.91\%

Modified_Morlet_wavelet.html

  1. ψ 2 ( t ) = C ψ 2 cos ( ω 0 t ) sech ( t ) \psi_{2}(t)=C_{\psi_{2}}\cos(\omega_{0}t){\rm sech}(t)

Modular_elliptic_curve.html

  1. y 2 = x ( x - a n ) ( x + b n ) . y^{2}=x(x-a^{n})(x+b^{n}).\,
  2. X 0 ( N ) X_{0}(N)
  3. L ( s , E ) = n = 1 a n n s . L(s,E)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.
  4. a n a_{n}
  5. f ( q , E ) = n = 1 a n q n . f(q,E)=\sum_{n=1}^{\infty}a_{n}q^{n}.
  6. q = e 2 π i τ q=e^{2\pi i\tau}
  7. f ( τ , E ) f(\tau,E)
  8. f f

Moisture_recycling.html

  1. P L P_{L}
  2. P P
  3. ρ \rho
  4. ρ = P L / P \rho=P_{L}/P

Molar_solubility.html

  1. A x B y ( s ) x A ( a q ) + y B ( a q ) {\,\text{A}_{x}\,\text{B}_{y}}_{(s)}\Longleftrightarrow x\,\text{A}_{(aq)}+y\,% \text{B}_{(aq)}\,
  2. - N A x B y ( Δ ) 1 = N A ( Δ ) x = N B ( Δ ) y -\frac{N_{AxBy(\Delta)}}{1}=\frac{N_{A(\Delta)}}{x}=\frac{N_{B(\Delta)}}{y}\,
  3. N A ( Δ ) = - x N A x B y ( Δ ) N_{A(\Delta)}=-xN_{AxBy(\Delta)}\,
  4. N B ( Δ ) = - y N A x B y ( Δ ) N_{B(\Delta)}=-yN_{AxBy(\Delta)}\,
  5. N ( i ) + N ( Δ ) = N ( f ) N_{(i)}+N_{(\Delta)}=N_{(f)}\,
  6. N A ( i ) = 0 N_{A(i)}=0\,
  7. N B ( i ) = 0 N_{B(i)}=0\,
  8. N A ( f ) = N A ( Δ ) N_{A(f)}=N_{A(\Delta)}\,
  9. N B ( f ) = N B ( Δ ) N_{B(f)}=N_{B(\Delta)}\,
  10. S 0 = - N A x B y ( Δ ) V S_{0}=-\frac{N_{AxBy(\Delta)}}{V}\,
  11. K s p = [ A ] x [ B ] y K_{sp}=[A]^{x}[B]^{y}\,
  12. K s p = ( N A ( f ) V ) x ( N B ( f ) V ) y K_{sp}={\left(\frac{N_{A(f)}}{V}\right)}^{x}{\left(\frac{N_{B(f)}}{V}\right)}^% {y}\,
  13. K s p = ( N A ( Δ ) V ) x ( N B ( Δ ) V ) y K_{sp}={\left(\frac{N_{A(\Delta)}}{V}\right)}^{x}{\left(\frac{N_{B(\Delta)}}{V% }\right)}^{y}\,
  14. K s p = ( N A ( Δ ) ) x ( N B ( Δ ) ) y V ( x + y ) K_{sp}=\frac{{(N_{A(\Delta)})}^{x}{(N_{B(\Delta)})}^{y}}{V^{(x+y)}}\,
  15. K s p = ( - x N A x B y ( Δ ) ) x ( - y N A x B y ( Δ ) ) y V ( x + y ) K_{sp}=\frac{{(-xN_{AxBy(\Delta)})}^{x}{(-yN_{AxBy(\Delta)})}^{y}}{V^{(x+y)}}\,
  16. K s p = ( - 1 ) x ( x ) x ( N A x B y ( Δ ) ) x ( - 1 ) y ( y ) y ( N A x B y ( Δ ) ) y V ( x + y ) K_{sp}=\frac{{(-1)}^{x}{(x)}^{x}{(N_{AxBy(\Delta)})}^{x}{(-1)}^{y}{(y)}^{y}{(N% _{AxBy(\Delta)})}^{y}}{V^{(x+y)}}\,
  17. K s p = x x y y ( - 1 ) x ( - 1 ) y ( N A x B y ( Δ ) ) x ( N A x B y ( Δ ) ) y V ( x + y ) K_{sp}=x^{x}y^{y}\frac{{(-1)}^{x}{(-1)}^{y}{(N_{AxBy(\Delta)})}^{x}{(N_{AxBy(% \Delta)})}^{y}}{V^{(x+y)}}\,
  18. K s p = x x y y ( - 1 ) ( x + y ) ( N A x B y ( Δ ) ) ( x + y ) V ( x + y ) K_{sp}=x^{x}y^{y}\frac{{(-1)}^{(x+y)}{(N_{AxBy(\Delta)})}^{(x+y)}}{V^{(x+y)}}\,
  19. K s p x x y y = ( - N A x B y ( Δ ) V ) ( x + y ) \frac{K_{sp}}{x^{x}y^{y}}={\left(-\frac{N_{AxBy(\Delta)}}{V}\right)}^{(x+y)}\,
  20. K s p x x y y = ( S 0 ) ( x + y ) \frac{K_{sp}}{x^{x}y^{y}}={\left(S_{0}\right)}^{(x+y)}\,
  21. S 0 = K s p x x y y ( x + y ) S_{0}=\sqrt[(x+y)]{\frac{K_{sp}}{x^{x}y^{y}}}\,
  22. A B 2 ( s ) A a q + 2 B a q AB_{2(s)}\Longleftrightarrow A_{aq}+2B_{aq}
  23. K s p = [ A ] [ B ] 2 K_{sp}=[A][B]^{2}
  24. K s p = ( x ) ( 2 x ) 2 K_{sp}=(x)(2x)^{2}
  25. K s p = ( x ) ( 4 x 2 ) K_{sp}=(x)(4x^{2})
  26. K s p = 4 x 3 K_{sp}=4x^{3}

Molecular_tagging_velocimetry.html

  1. 10 - 7 10^{-7}
  2. 10 - 9 10^{-9}

Molecule_mining.html

  1. \neq

Molien_series.html

  1. M ( t ) = d n d t d . M(t)=\sum_{d}n_{d}t^{d}.
  2. M ( t ) = 1 | G | g G 1 det ( I - t g ) M(t)=\frac{1}{|G|}\sum_{g\in G}\frac{1}{\det(I-tg)}
  3. | G | M ( t ) = g G 1 det ( I - t g ) |G|\cdot M(t)=\sum_{g\in G}\frac{1}{\det(I-tg)}
  4. S 3 S_{3}
  5. det ( I - t g ) \det(I-tg)
  6. S 3 S_{3}
  7. det ( I - t e ) = ( 1 - t ) 3 , det ( I - t σ 2 ) = ( 1 - t ) ( 1 - t 2 ) \det(I-te)=(1-t)^{3},\det(I-t\sigma_{2})=(1-t)(1-t^{2})
  8. det ( 1 - t σ 3 ) = ( 1 - t 3 ) \det(1-t\sigma_{3})=(1-t^{3})
  9. σ 2 = ( 1 , 2 ) \sigma_{2}=(1,2)
  10. σ 3 = ( 1 , 2 , 3 ) \sigma_{3}=(1,2,3)
  11. M ( t ) = 1 6 ( 1 ( 1 - t ) 3 + 3 ( 1 - t ) ( 1 - t 2 ) + 2 1 - t 3 ) = 1 ( 1 - t ) ( 1 - t 2 ) ( 1 - t 3 ) M(t)=\frac{1}{6}\left(\frac{1}{(1-t)^{3}}+\frac{3}{(1-t)(1-t^{2})}+\frac{2}{1-% t^{3}}\right)=\frac{1}{(1-t)(1-t^{2})(1-t^{3})}

Moment_map.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 * \mathfrak{g}^{*}
  3. , : 𝔤 * × 𝔤 𝐑 \langle,\rangle:\mathfrak{g}^{*}\times\mathfrak{g}\to\mathbf{R}
  4. 𝔤 \mathfrak{g}
  5. ρ ( ξ ) x \rho(\xi)_{x}
  6. d d t | t = 0 exp ( t ξ ) x , \left.\frac{d}{dt}\right|_{t=0}\exp(t\xi)\cdot x,
  7. exp : 𝔤 G \exp:\mathfrak{g}\to G
  8. \cdot
  9. ι ρ ( ξ ) ω \iota_{\rho(\xi)}\omega\,
  10. ι ρ ( ξ ) ω \iota_{\rho(\xi)}\omega\,
  11. 𝔤 \mathfrak{g}
  12. μ : M 𝔤 * \mu:M\to\mathfrak{g}^{*}
  13. d ( μ , ξ ) = ι ρ ( ξ ) ω d(\langle\mu,\xi\rangle)=\iota_{\rho(\xi)}\omega
  14. 𝔤 \mathfrak{g}
  15. μ , ξ \langle\mu,\xi\rangle
  16. μ , ξ ( x ) = μ ( x ) , ξ \langle\mu,\xi\rangle(x)=\langle\mu(x),\xi\rangle
  17. 𝔤 * \mathfrak{g}^{*}
  18. 𝔤 * \mathfrak{g}^{*}
  19. ι ρ ( ξ ) ω \iota_{\rho(\xi)}\omega
  20. 𝔤 \mathfrak{g}
  21. ι ρ ( ξ ) ω \iota_{\rho(\xi)}\omega
  22. d H ξ dH_{\xi}
  23. H ξ : M 𝐑 . H_{\xi}:M\to\mathbf{R}.
  24. H ξ H_{\xi}
  25. ξ H ξ \xi\mapsto H_{\xi}
  26. ξ H ξ \xi\mapsto H_{\xi}
  27. 𝔤 \mathfrak{g}
  28. μ : M 𝔤 * \mu:M\to\mathfrak{g}^{*}
  29. H ξ = μ , ξ H_{\xi}=\langle\mu,\xi\rangle
  30. ξ H ξ \xi\mapsto H_{\xi}
  31. ρ ( ξ ) = X H ξ \rho(\xi)=X_{H_{\xi}}
  32. X H ξ X_{H_{\xi}}
  33. H ξ H_{\xi}
  34. ι X H ξ ω = d H ξ . \iota_{X_{H_{\xi}}}\omega=dH_{\xi}.
  35. G = 𝒰 ( 1 ) G=\mathcal{U}(1)
  36. 𝔤 * \mathfrak{g}^{*}
  37. \mathbb{R}
  38. M M
  39. 3 \mathbb{R}^{3}
  40. G G
  41. G G
  42. S O ( 3 ) SO(3)
  43. 3 \mathbb{R}^{3}
  44. N N
  45. T * N T^{*}N
  46. π : T * N N \pi:T^{*}N\rightarrow N
  47. τ \tau
  48. T * N T^{*}N
  49. G G
  50. N N
  51. G G
  52. ( T * N , d τ ) (T^{*}N,\mathrm{d}\tau)
  53. g η := ( T π ( η ) g - 1 ) * η g\cdot\eta:=(T_{\pi(\eta)}g^{-1})^{*}\eta
  54. g G , η T * N g\in G,\eta\in T^{*}N
  55. - ι ρ ( ξ ) τ -\iota_{\rho(\xi)}\tau
  56. ξ 𝔤 \xi\in\mathfrak{g}
  57. ι ρ ( ξ ) τ \iota_{\rho(\xi)}\tau
  58. ρ ( ξ ) \rho(\xi)
  59. ξ \xi
  60. τ \tau
  61. G , H G,H
  62. 𝔤 , 𝔥 \mathfrak{g},\mathfrak{h}
  63. 𝒪 ( F ) , F 𝔤 * \mathcal{O}(F),F\in\mathfrak{g}^{*}
  64. 𝒪 ( F ) \mathcal{O}(F)
  65. 𝒪 ( F ) 𝔤 * \mathcal{O}(F)\hookrightarrow\mathfrak{g}^{*}
  66. G G
  67. ( M , ω ) (M,\omega)
  68. Φ G : M 𝔤 * \Phi_{G}:M\rightarrow\mathfrak{g}^{*}
  69. ψ : H G \psi:H\rightarrow G
  70. H H
  71. M M
  72. H H
  73. M M
  74. ( d ψ ) e * Φ G (\mathrm{d}\psi)_{e}^{*}\circ\Phi_{G}
  75. ( d ψ ) e * : 𝔤 * 𝔥 * (\mathrm{d}\psi)_{e}^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{h}^{*}
  76. ( d ψ ) e : 𝔥 𝔤 (\mathrm{d}\psi)_{e}:\mathfrak{h}\rightarrow\mathfrak{g}
  77. e e
  78. H H
  79. H H
  80. G G
  81. ψ \psi
  82. ( M 1 , ω 1 ) (M_{1},\omega_{1})
  83. G G
  84. ( M 2 , ω 2 ) (M_{2},\omega_{2})
  85. H H
  86. G × H G\times H
  87. ( M 1 × M 2 , ω 1 × ω 2 ) (M_{1}\times M_{2},\omega_{1}\times\omega_{2})
  88. Φ G \Phi_{G}
  89. Φ H \Phi_{H}
  90. ω 1 × ω 2 := π 1 * ω 1 + π 2 * ω 2 \omega_{1}\times\omega_{2}:=\pi_{1}^{*}\omega_{1}+\pi_{2}^{*}\omega_{2}
  91. π i : M 1 × M 2 M i \pi_{i}:M_{1}\times M_{2}\rightarrow M_{i}
  92. M M
  93. G G
  94. N N
  95. M M
  96. G G
  97. M M
  98. N N
  99. N N
  100. G G
  101. N N
  102. M M
  103. μ : M 𝔤 * \mu:M\to\mathfrak{g}^{*}
  104. μ - 1 ( 0 ) \mu^{-1}(0)
  105. μ - 1 ( 0 ) \mu^{-1}(0)
  106. μ - 1 ( 0 ) \mu^{-1}(0)
  107. μ - 1 ( 0 ) / G \mu^{-1}(0)/G
  108. μ - 1 ( 0 ) \mu^{-1}(0)
  109. μ - 1 ( 0 ) \mu^{-1}(0)
  110. M / / G M/\!\!/G

Monetary_policy_reaction_function.html

  1. u = u 0 + Φ ( π - π t ) u=u_{0}+\Phi(\pi-\pi_{t})
  2. Φ \Phi
  3. r r
  4. 1 Φ \frac{1}{\Phi}

Monge–Ampère_equation.html

  1. L [ u ] = A ( u x x u y y - u x y 2 ) + B u x x + 2 C u x y + D u y y + E = 0 L[u]=A(u_{xx}u_{yy}-u_{xy}^{2})+Bu_{xx}+2Cu_{xy}+Du_{yy}+E=0\,
  2. L [ u ] = 0 , on Ω L[u]=0,\quad\,\text{on}\ \Omega
  3. u | Ω = g . u|_{\partial\Omega}=g.
  4. B D - C 2 - A E > 0 , BD-C^{2}-AE>0,
  5. L [ u ] = det D 2 u - f ( 𝐱 , u , D u ) = 0 ( 1 ) L[u]=\det D^{2}u-f(\mathbf{x},u,Du)=0\qquad\qquad(1)
  6. det D 2 u - K ( 𝐱 ) ( 1 + | D u | 2 ) ( n + 2 ) / 2 = 0. \det D^{2}u-K(\mathbf{x})(1+|Du|^{2})^{(n+2)/2}=0.

Monoidal_adjunction.html

  1. ( 𝒞 , , I ) (\mathcal{C},\otimes,I)
  2. ( 𝒟 , , J ) (\mathcal{D},\bullet,J)
  3. ( F , m ) : ( 𝒞 , , I ) ( 𝒟 , , J ) (F,m):(\mathcal{C},\otimes,I)\to(\mathcal{D},\bullet,J)
  4. ( G , n ) : ( 𝒟 , , J ) ( 𝒞 , , I ) (G,n):(\mathcal{D},\bullet,J)\to(\mathcal{C},\otimes,I)
  5. ( F , G , η , ε ) (F,G,\eta,\varepsilon)
  6. η : 1 𝒞 G F \eta:1_{\mathcal{C}}\Rightarrow G\circ F
  7. ε : F G 1 𝒟 \varepsilon:F\circ G\Rightarrow 1_{\mathcal{D}}
  8. ( F , m ) : ( 𝒞 , , I ) ( 𝒟 , , J ) (F,m):(\mathcal{C},\otimes,I)\to(\mathcal{D},\bullet,J)
  9. F : 𝒞 𝒟 F:\mathcal{C}\to\mathcal{D}
  10. G : 𝒟 𝒞 G:\mathcal{D}\to\mathcal{C}
  11. ( F , m ) (F,m)
  12. ( G , n ) (G,n)
  13. ( F , m ) (F,m)
  14. ( F , m ) (F,m)
  15. ( G , n ) (G,n)
  16. G F G\circ F

Monoidal_functor.html

  1. ( 𝒞 , , I 𝒞 ) (\mathcal{C},\otimes,I_{\mathcal{C}})
  2. ( 𝒟 , , I 𝒟 ) (\mathcal{D},\bullet,I_{\mathcal{D}})
  3. 𝒞 \mathcal{C}
  4. 𝒟 \mathcal{D}
  5. F : 𝒞 𝒟 F:\mathcal{C}\to\mathcal{D}
  6. ϕ A , B : F A F B F ( A B ) \phi_{A,B}:FA\bullet FB\to F(A\otimes B)
  7. ϕ : I 𝒟 F I 𝒞 \phi:I_{\mathcal{D}}\to FI_{\mathcal{C}}
  8. A A
  9. B B
  10. C C
  11. 𝒞 \mathcal{C}
  12. 𝒟 \mathcal{D}
  13. α , ρ , λ \alpha,\rho,\lambda
  14. 𝒞 \mathcal{C}
  15. 𝒟 \mathcal{D}
  16. ϕ A , B , ϕ \phi_{A,B},\phi
  17. 𝒞 \mathcal{C}
  18. U : ( 𝐀𝐛 , 𝐙 , 𝐙 ) ( 𝐒𝐞𝐭 , × , { * } ) U\colon(\mathbf{Ab},\otimes_{\mathbf{Z}},\mathbf{Z})\rightarrow(\mathbf{Set},% \times,\{*\})
  19. ϕ A , B : U ( A ) × U ( B ) U ( A B ) \phi_{A,B}\colon U(A)\times U(B)\to U(A\otimes B)
  20. a b a\otimes b
  21. ϕ : { * } \phi\colon\{*\}\to\mathbb{Z}
  22. 𝐁𝐨𝐫𝐝 n - 1 , n \mathbf{Bord}_{\langle n-1,n\rangle}
  23. F : ( 𝐁𝐨𝐫𝐝 n - 1 , n , , ) ( 𝐤𝐕𝐞𝐜𝐭 , k , k ) . F\colon(\mathbf{Bord}_{\langle n-1,n\rangle},\sqcup,\emptyset)\rightarrow(% \mathbf{kVect},\otimes_{k},k).
  24. F : 𝒞 𝒟 F:\mathcal{C}\to\mathcal{D}
  25. ( G , n ) : ( 𝒟 , , I 𝒟 ) ( 𝒞 , , I 𝒞 ) (G,n):(\mathcal{D},\bullet,I_{\mathcal{D}})\to(\mathcal{C},\otimes,I_{\mathcal% {C}})
  26. F F
  27. ( F , m ) (F,m)
  28. ( G , n ) (G,n)
  29. m A , B = ε F A F B F n F A , F B F ( η A η B ) : F ( A B ) F A F B m_{A,B}=\varepsilon_{FA\bullet FB}\circ Fn_{FA,FB}\circ F(\eta_{A}\otimes\eta_% {B}):F(A\otimes B)\to FA\bullet FB
  30. m = ε I 𝒟 F n : F I 𝒞 I 𝒟 m=\varepsilon_{I_{\mathcal{D}}}\circ Fn:FI_{\mathcal{C}}\to I_{\mathcal{D}}
  31. F F

Monoidal_monad.html

  1. ( T , η , μ , m ) (T,\eta,\mu,m)
  2. ( T , η , μ ) (T,\eta,\mu)
  3. ( C , , I ) (C,\otimes,I)
  4. T : ( C , , I ) ( C , , I ) T:(C,\otimes,I)\to(C,\otimes,I)
  5. η , μ \eta,\mu
  6. T T
  7. m A , B : T A T B T ( A B ) m_{A,B}:TA\otimes TB\to T(A\otimes B)
  8. m : I T I m:I\to TI
  9. η : i d T \eta:id\Rightarrow T
  10. μ : T 2 T \mu:T^{2}\Rightarrow T
  11. ( C , , I ) (C,\otimes,I)
  12. η \eta
  13. m m
  14. η I \eta_{I}
  15. m A , B : T ( A B ) T A T B m^{A,B}:T(A\otimes B)\to TA\otimes TB
  16. m 0 : T I I m^{0}:TI\to I
  17. Vect \operatorname{Vect}
  18. - A -\otimes A
  19. A A
  20. A A
  21. A A
  22. C C
  23. M M
  24. X X × M X\mapsto X\times M
  25. M M

Monoidal_natural_transformation.html

  1. ( 𝒞 , , I ) (\mathcal{C},\otimes,I)
  2. ( 𝒟 , , J ) (\mathcal{D},\bullet,J)
  3. ( F , m ) : ( 𝒞 , , I ) ( 𝒟 , , J ) (F,m):(\mathcal{C},\otimes,I)\to(\mathcal{D},\bullet,J)
  4. ( G , n ) : ( 𝒞 , , I ) ( 𝒟 , , J ) (G,n):(\mathcal{C},\otimes,I)\to(\mathcal{D},\bullet,J)
  5. θ : ( F , m ) ( G , n ) \theta:(F,m)\to(G,n)
  6. θ : F G \theta:F\to G
  7. A A
  8. B B
  9. 𝒞 \mathcal{C}

Monopulse_radar.html

  1. Σ \scriptstyle\Sigma
  2. E r r o r = ( ( R e a l D e l t a ) + i ( I m a g i n a r y D e l t a ) ( R e a l S u m ) + i ( I m a g i n a r y S u m ) ) ÷ ( ( R e a l D e l t a C a l ) + i ( I m a g i n a r y D e l t a C a l ) ( R e a l S u m C a l ) + i ( I m a g i n a r y S u m C a l ) ) Error=\left(\frac{(RealDelta)+i(ImaginaryDelta)}{(RealSum)+i(ImaginarySum)}% \right)\div\left(\frac{(RealDeltaCal)+i(ImaginaryDeltaCal)}{(RealSumCal)+i(% ImaginarySumCal)}\right)

Monotone_class_theorem.html

  1. R R
  2. M M
  3. R R
  4. R R
  5. A i M A_{i}\in M
  6. A 1 A 2 A_{1}\subset A_{2}\subset\ldots
  7. i = 1 A i M \cup_{i=1}^{\infty}A_{i}\in M
  8. 𝒜 \mathcal{A}
  9. Ω \Omega\,
  10. \mathcal{H}
  11. Ω \Omega
  12. A 𝒜 A\in\mathcal{A}
  13. 𝟏 A \mathbf{1}_{A}\in\mathcal{H}
  14. f , g f,g\in\mathcal{H}
  15. f + g f+g
  16. c f cf\in\mathcal{H}
  17. c c
  18. f n f_{n}\in\mathcal{H}
  19. f f
  20. f f\in\mathcal{H}
  21. \mathcal{H}
  22. σ ( 𝒜 ) \sigma(\mathcal{A})
  23. 𝒜 \mathcal{A}
  24. Ω 𝒜 \Omega\,\in\mathcal{A}
  25. 𝒢 = { A : 𝟏 A } \mathcal{G}=\{A:\mathbf{1}_{A}\in\mathcal{H}\}
  26. σ ( 𝒜 ) 𝒢 \sigma(\mathcal{A})\subset\mathcal{G}
  27. \mathcal{H}
  28. \mathcal{H}
  29. σ ( 𝒜 ) \sigma(\mathcal{A})

Monotone_preferences.html

  1. x x
  2. y y
  3. y x y\gg x
  4. y x y\succ x
  5. x x
  6. y y
  7. y x y\geq x
  8. y x y\neq x
  9. y x y\succ x

Monte_Carlo_localization.html

  1. ( x , y , θ ) (x,y,\theta)
  2. x , y x,y
  3. θ \theta
  4. ( θ 1 , θ 2 , , θ 10 ) (\theta_{1},\theta_{2},...,\theta_{10})
  5. t t
  6. M M
  7. X t = { x t [ 1 ] , x t [ 2 ] , , x t [ M ] } X_{t}=\{x_{t}^{[1]},x_{t}^{[2]},\ldots,x_{t}^{[M]}\}
  8. X t X_{t}
  9. X t - 1 X_{t-1}
  10. t t
  11. X t - 1 = { x t - 1 [ 1 ] , x t - 1 [ 2 ] , , x t - 1 [ M ] } X_{t-1}=\{x_{t-1}^{[1]},x_{t-1}^{[2]},\ldots,x_{t-1}^{[M]}\}
  12. u t u_{t}
  13. z t z_{t}
  14. X t X_{t}
  15. ( X t - 1 , u t , z t ) (X_{t-1},u_{t},z_{t})
  16. X t ¯ = X t = \bar{X_{t}}=X_{t}=\emptyset
  17. m = 1 m=1
  18. M M
  19. x t [ m ] = x_{t}^{[m]}=
  20. ( u t , x t - 1 [ m ] ) (u_{t},x_{t-1}^{[m]})
  21. w t [ m ] = w_{t}^{[m]}=
  22. ( z t , x t [ m ] ) (z_{t},x_{t}^{[m]})
  23. X t ¯ = X t ¯ + x t [ m ] , w t [ m ] \bar{X_{t}}=\bar{X_{t}}+\langle x_{t}^{[m]},w_{t}^{[m]}\rangle
  24. m = 1 m=1
  25. M M
  26. x t [ m ] x_{t}^{[m]}
  27. X t ¯ \bar{X_{t}}
  28. w t [ m ] \propto w_{t}^{[m]}
  29. X t = X t + x t [ m ] X_{t}=X_{t}+x_{t}^{[m]}
  30. X t X_{t}
  31. t = 0 t=0
  32. t = 1 t=1
  33. t = 2 t=2
  34. w t [ i ] w_{t}^{[i]}
  35. M M
  36. w t [ i ] w_{t}^{[i]}
  37. M M
  38. M M
  39. u t u_{t}
  40. z t z_{t}
  41. M 50 M\leq 50
  42. M M
  43. M x M_{x}
  44. M x M_{x}
  45. 1 - δ 1-\delta
  46. ϵ \epsilon
  47. δ \delta
  48. ϵ \epsilon
  49. k k
  50. M x M_{x}
  51. k k
  52. M M
  53. M x M_{x}
  54. M x M_{x}

Moore_graph.html

  1. 1 + d i = 0 k - 1 ( d - 1 ) i . 1+d\sum_{i=0}^{k-1}(d-1)^{i}.
  2. n g ( m - n + 1 ) \frac{n}{g}(m-n+1)
  3. 1 + d i = 0 k - 1 ( d - 1 ) i . 1+d\sum_{i=0}^{k-1}(d-1)^{i}.
  4. 1 + d i = 0 k - 1 ( d - 1 ) i . 1+d\sum_{i=0}^{k-1}(d-1)^{i}.
  5. 2 i = 0 k - 1 ( d - 1 ) i = 1 + ( d - 1 ) k - 1 + d i = 0 k - 2 ( d - 1 ) i . 2\sum_{i=0}^{k-1}(d-1)^{i}=1+(d-1)^{k-1}+d\sum_{i=0}^{k-2}(d-1)^{i}.
  6. K n K_{n}
  7. C 2 n + 1 C_{2n+1}
  8. C 2 n C_{2n}
  9. K n , n K_{n,n}

Morita_equivalence.html

  1. \to
  2. \to
  3. F ( - ) P R - \operatorname{F}(-)\cong P\otimes_{R}-
  4. G ( - ) Hom ( S P , - ) . \operatorname{G}(-)\cong\operatorname{Hom}(_{S}P,-).
  5. F ( - ) \operatorname{F}(-)
  6. E R - E\otimes_{R}-
  7. M S N R M\otimes_{S}N\cong R
  8. N R M S N\otimes_{R}M\cong S
  9. N Hom ( M S , S S ) N\cong\operatorname{Hom}(M_{S},S_{S})
  10. S End ( P R ) S\cong\operatorname{End}(P_{R})
  11. S e 𝕄 n ( R ) e S\cong e\mathbb{M}_{n}(R)e
  12. R = C ( R ) C ( S ) = S R=\operatorname{C}(R)\cong\operatorname{C}(S)=S
  13. 𝒫 \mathcal{P}
  14. 𝒫 \mathcal{P}
  15. 𝒫 \mathcal{P}
  16. 𝒫 \mathcal{P}
  17. 𝒫 \mathcal{P}
  18. 𝒫 \mathcal{P}

Moscow_Mathematical_Papyrus.html

  1. 3 / 2 × x + 4 = 10 3/2\times x+4=10
  2. pefsu = number loaves of bread (or jugs of beer) number of heqats of grain \mbox{pefsu}~{}=\frac{\mbox{number loaves of bread (or jugs of beer)}~{}}{% \mbox{number of heqats of grain}~{}}
  3. Area = 2 × ( 8 9 ) 2 × ( diameter ) 2 = 2 × 256 81 ( radius ) 2 \,\text{Area}=2\times\left(\frac{8}{9}\right)^{2}\times(\,\text{diameter})^{2}% =2\times\frac{256}{81}(\,\text{radius})^{2}
  4. 256 81 3.16049 \frac{256}{81}\approx 3.16049
  5. V = 1 3 h ( a 2 + a b + b 2 ) . V=\frac{1}{3}h(a^{2}+ab+b^{2}).

Motion_graphs_and_derivatives.html

  1. v = Δ y Δ x = Δ s Δ t . v=\frac{\Delta y}{\Delta x}=\frac{\Delta s}{\Delta t}.
  2. s s
  3. t t
  4. ( m s ) (\begin{matrix}\frac{m}{s}\end{matrix})
  5. Δ s {\Delta s}
  6. d s {ds}
  7. Δ t {\Delta t}
  8. d t {dt}
  9. t t
  10. y y
  11. x x
  12. a = Δ y Δ x = Δ v Δ t . a=\frac{\Delta y}{\Delta x}=\frac{\Delta v}{\Delta t}.
  13. v v
  14. m s \begin{matrix}\frac{m}{s}\end{matrix}
  15. t t
  16. d v d t \begin{matrix}\frac{dv}{dt}\end{matrix}
  17. t t
  18. m s 2 \begin{matrix}\frac{m}{s^{2}}\end{matrix}
  19. m s s = m \begin{matrix}\frac{m}{s}\end{matrix}s=m
  20. ( m s 2 ) (\begin{matrix}\frac{m}{s^{2}}\end{matrix})
  21. m s 2 s = m s \begin{matrix}\frac{m}{s^{2}}\end{matrix}s=\begin{matrix}\frac{m}{s}\end{matrix}
  22. v = d s d t , v=\frac{ds}{dt},
  23. a = d v d t . a=\frac{dv}{dt}.
  24. a = d 2 s d t 2 . a=\frac{d^{2}s}{dt^{2}}.
  25. s ( t 2 ) - s ( t 1 ) = t 1 t 2 v d t , s(t_{2})-s(t_{1})=\int_{t_{1}}^{t_{2}}{v}\,dt,
  26. v ( t 2 ) - v ( t 1 ) = t 1 t 2 a d t . v(t_{2})-v(t_{1})=\int_{t_{1}}^{t_{2}}{a}\,dt.

Motion_planning.html

  1. × \times
  2. × \times

Motive_power.html

  1. P = W t = ( m g ) h t P=\frac{W}{t}=\frac{(mg)h}{t}

Mountain_pass_theorem.html

  1. I C 1 ( H , ) I\in C^{1}(H,\mathbb{R})
  2. I I^{\prime}
  3. I [ 0 ] = 0 I[0]=0
  4. I [ u ] a I[u]\geq a
  5. u = r \|u\|=r
  6. v H v\in H
  7. v > r \|v\|>r
  8. I [ v ] 0 I[v]\leq 0
  9. Γ = { 𝐠 C ( [ 0 , 1 ] ; H ) | 𝐠 ( 0 ) = 0 , 𝐠 ( 1 ) = v } \Gamma=\{\mathbf{g}\in C([0,1];H)\,|\,\mathbf{g}(0)=0,\mathbf{g}(1)=v\}
  10. c = inf 𝐠 Γ max 0 t 1 I [ 𝐠 ( t ) ] , c=\inf_{\mathbf{g}\in\Gamma}\max_{0\leq t\leq 1}I[\mathbf{g}(t)],
  11. I [ 0 ] = 0 I[0]=0
  12. I [ v ] 0 I[v]\leq 0
  13. u = r \|u\|=r
  14. X X
  15. Φ C ( X , 𝐑 ) \Phi\in C(X,\mathbf{R})
  16. Φ : X X * \Phi^{\prime}\colon X\to X^{*}
  17. X X
  18. X * X^{*}
  19. r > 0 r>0
  20. x > r \|x^{\prime}\|>r
  21. max ( Φ ( 0 ) , Φ ( x ) ) < inf x = r Φ ( x ) = : m ( r ) \max\,(\Phi(0),\Phi(x^{\prime}))<\inf\limits_{\|x\|=r}\Phi(x)=:m(r)
  22. Φ \Phi
  23. { x X m ( r ) Φ ( x ) } \{x\in X\mid m(r)\leq\Phi(x)\}
  24. x ¯ X \overline{x}\in X
  25. Φ \Phi
  26. m ( r ) Φ ( x ¯ ) m(r)\leq\Phi(\overline{x})
  27. Γ = { c C ( [ 0 , 1 ] , X ) c ( 0 ) = 0 , c ( 1 ) = x } \Gamma=\{c\in C([0,1],X)\mid c\,(0)=0,\,c\,(1)=x^{\prime}\}
  28. Φ ( x ¯ ) = inf c Γ max 0 t 1 Φ ( c ( t ) ) . \Phi(\overline{x})=\inf_{c\,\in\,\Gamma}\max_{0\leq t\leq 1}\Phi(c\,(t)).

MU_puzzle.html

  1. n n
  2. n 2 a 0 ( mod 3 ) . n\equiv 2^{a}\not\equiv 0\;\;(\mathop{{\rm mod}}3).\,
  3. a a

Multilateration.html

  1. E = ( x , y , z ) \vec{E}=(x,\,y,\,z)
  2. P 0 , P 1 , , P m , , P N . \vec{P}_{0},\,\vec{P}_{1},\,\cdots,\,\vec{P}_{m},\,\cdots,\,\vec{P}_{N}.
  3. P m = ( x m , y m , z m ) \vec{P}_{m}=(x_{m},\,y_{m},\,z_{m})
  4. cos σ v m = sin ϕ v sin ϕ m + cos ϕ v cos ϕ m cos λ v m . \cos\sigma_{vm}=\sin\phi_{v}\sin\phi_{m}+\cos\phi_{v}\cos\phi_{m}\cos\lambda_{% vm}.
  5. R m = R E σ v m R_{m}=R_{E}\sigma_{vm}
  6. R m R_{m}
  7. v v
  8. T m T_{m}
  9. τ m \tau_{m}
  10. P 0 P_{0}
  11. P 1 P_{1}
  12. E E
  13. P 1 P_{1}
  14. P 0 P_{0}
  15. P 1 P_{1}
  16. P 0 P_{0}
  17. ( P 1 P 0 ) (P_{1}\star P_{0})
  18. τ \tau
  19. τ 1 = 5 \tau_{1}=5
  20. θ \theta
  21. θ = 2 π f T \theta=2\pi f\cdot T
  22. v τ m v\tau_{m}
  23. A m , B m , C m , D m A_{m},B_{m},C_{m},D_{m}

Multilevel_model.html

  1. Y i j = β 0 j + β 1 j X i j + e i j Y_{ij}=\beta_{0j}+\beta_{1j}X_{ij}+e_{ij}
  2. Y i j Y_{ij}
  3. X i j X_{ij}
  4. β 0 j \beta_{0j}
  5. β 1 j \beta_{1j}
  6. e i j e_{ij}
  7. r i j r_{ij}
  8. β 0 j = γ 00 + γ 01 W j + u 0 j \beta_{0j}=\gamma_{00}+\gamma_{01}W_{j}+u_{0j}
  9. β 1 j = γ 10 + u 1 j \beta_{1j}=\gamma_{10}+u_{1j}
  10. γ 00 \gamma_{00}
  11. W j W_{j}
  12. γ 01 \gamma_{01}
  13. u 0 j u_{0j}
  14. γ 10 \gamma_{10}
  15. u 1 j u_{1j}

Multinomial_logistic_regression.html

  1. score ( 𝐗 i , k ) = s y m b o l β k 𝐗 i , \operatorname{score}(\mathbf{X}_{i},k)=symbol\beta_{k}\cdot\mathbf{X}_{i},
  2. f ( k , i ) f(k,i)
  3. f ( k , i ) = β 0 , k + β 1 , k x 1 , i + β 2 , k x 2 , i + + β M , k x M , i , f(k,i)=\beta_{0,k}+\beta_{1,k}x_{1,i}+\beta_{2,k}x_{2,i}+\cdots+\beta_{M,k}x_{% M,i},
  4. β m , k \beta_{m,k}
  5. f ( k , i ) = s y m b o l β k 𝐱 i , f(k,i)=symbol\beta_{k}\cdot\mathbf{x}_{i},
  6. s y m b o l β k symbol\beta_{k}
  7. 𝐱 i \mathbf{x}_{i}
  8. ln Pr ( Y i = 1 ) Pr ( Y i = K ) \displaystyle\ln\frac{\Pr(Y_{i}=1)}{\Pr(Y_{i}=K)}
  9. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  10. Pr ( Y i = K ) = 1 1 + k = 1 K - 1 e s y m b o l β k 𝐗 i \Pr(Y_{i}=K)=\frac{1}{1+\sum_{k=1}^{K-1}e^{symbol\beta_{k}\cdot\mathbf{X}_{i}}}
  11. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  12. ln Pr ( Y i = 1 ) \displaystyle\ln\Pr(Y_{i}=1)
  13. - ln Z -\ln Z
  14. k = 1 K Pr ( Y i = k ) = 1 \sum_{k=1}^{K}\Pr(Y_{i}=k)=1
  15. - ln Z -\ln Z
  16. + Z +Z
  17. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  18. 1 = k = 1 K Pr ( Y i = k ) \displaystyle 1=\sum_{k=1}^{K}\Pr(Y_{i}=k)
  19. Z = k = 1 K e s y m b o l β k 𝐗 i Z=\sum_{k=1}^{K}e^{symbol\beta_{k}\cdot\mathbf{X}_{i}}
  20. Pr ( Y i = 1 ) = e s y m b o l β 1 𝐗 i k = 1 K e s y m b o l β k 𝐗 i Pr ( Y i = 2 ) = e s y m b o l β 2 𝐗 i k = 1 K e s y m b o l β k 𝐗 i Pr ( Y i = K ) = e s y m b o l β K 𝐗 i k = 1 K e s y m b o l β k 𝐗 i \begin{aligned}\displaystyle\Pr(Y_{i}=1)&\displaystyle=\frac{e^{symbol\beta_{1% }\cdot\mathbf{X}_{i}}}{\sum_{k=1}^{K}e^{symbol\beta_{k}\cdot\mathbf{X}_{i}}}\\ \displaystyle\Pr(Y_{i}=2)&\displaystyle=\frac{e^{symbol\beta_{2}\cdot\mathbf{X% }_{i}}}{\sum_{k=1}^{K}e^{symbol\beta_{k}\cdot\mathbf{X}_{i}}}\\ \displaystyle\cdots&\displaystyle\cdots\\ \displaystyle\Pr(Y_{i}=K)&\displaystyle=\frac{e^{symbol\beta_{K}\cdot\mathbf{X% }_{i}}}{\sum_{k=1}^{K}e^{symbol\beta_{k}\cdot\mathbf{X}_{i}}}\\ \end{aligned}
  21. Pr ( Y i = c ) = e s y m b o l β c 𝐗 i k = 1 K e s y m b o l β k 𝐗 i \Pr(Y_{i}=c)=\frac{e^{symbol\beta_{c}\cdot\mathbf{X}_{i}}}{\sum_{k=1}^{K}e^{% symbol\beta_{k}\cdot\mathbf{X}_{i}}}
  22. softmax ( k , x 1 , , x n ) = e x k i = 1 n e x i \operatorname{softmax}(k,x_{1},\ldots,x_{n})=\frac{e^{x_{k}}}{\sum_{i=1}^{n}e^% {x_{i}}}
  23. x 1 , , x n x_{1},\ldots,x_{n}
  24. softmax ( k , x 1 , , x n ) \operatorname{softmax}(k,x_{1},\ldots,x_{n})
  25. x k x_{k}
  26. f ( k ) = { 1 if k = arg max ( x 1 , , x n ) , 0 otherwise . f(k)=\begin{cases}1\;\textrm{ if }\;k=\operatorname{\arg\max}(x_{1},\ldots,x_{% n}),\\ 0\;\textrm{ otherwise}.\end{cases}
  27. Pr ( Y i = c ) = softmax ( c , s y m b o l β 1 𝐗 i , , s y m b o l β K 𝐗 i ) \Pr(Y_{i}=c)=\operatorname{softmax}(c,symbol\beta_{1}\cdot\mathbf{X}_{i},% \ldots,symbol\beta_{K}\cdot\mathbf{X}_{i})
  28. β k \beta_{k}
  29. k - 1 k-1
  30. k - 1 k-1
  31. e ( s y m b o l β c + C ) 𝐗 i k = 1 K e ( s y m b o l β k + C ) 𝐗 i \displaystyle\frac{e^{(symbol\beta_{c}+C)\cdot\mathbf{X}_{i}}}{\sum_{k=1}^{K}e% ^{(symbol\beta_{k}+C)\cdot\mathbf{X}_{i}}}
  32. C = - s y m b o l β K C=-symbol\beta_{K}
  33. s y m b o l β 1 \displaystyle symbol\beta^{\prime}_{1}
  34. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  35. Y i , 1 \displaystyle Y_{i,1}^{\ast}
  36. ε k EV 1 ( 0 , 1 ) , \varepsilon_{k}\sim\operatorname{EV}_{1}(0,1),
  37. Y i Y_{i}
  38. Y i , k Y_{i,k}^{\ast}
  39. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  40. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  41. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  42. X EV 1 ( a , b ) X\sim\operatorname{EV}_{1}(a,b)
  43. Y EV 1 ( a , b ) Y\sim\operatorname{EV}_{1}(a,b)
  44. X - Y Logistic ( 0 , b ) . X-Y\sim\operatorname{Logistic}(0,b).
  45. X Logistic ( 0 , 1 ) X\sim\operatorname{Logistic}(0,1)
  46. b X Logistic ( 0 , b ) . bX\sim\operatorname{Logistic}(0,b).

Multiple-conclusion_logic.html

  1. \vdash
  2. Γ Δ \Gamma\vdash\Delta
  3. Γ \Gamma
  4. Δ \Delta
  5. Δ \Delta
  6. Γ \Gamma

Multiple_(mathematics).html

  1. 14 = 7 × 2 14=7\times 2
  2. 49 = 7 × 7 49=7\times 7
  3. - 21 = 7 × ( - 3 ) -21=7\times(-3)
  4. 0 = 7 × 0 0=7\times 0
  5. 3 = 7 × ( 3 / 7 ) 3=7\times(3/7)
  6. 3 / 7 3/7
  7. - 6 = 7 × ( - 6 / 7 ) -6=7\times(-6/7)
  8. - 6 / 7 -6/7
  9. 0 = 0 b 0=0\cdot b
  10. n n
  11. n n
  12. n n
  13. n × 1 n\times 1
  14. n n
  15. a a
  16. b b
  17. x x
  18. a + b a+b
  19. a - b a-b
  20. x x

Multiplicity_function_for_N_noninteracting_spins.html

  1. W ( n , N ) = ( N n ) = N ! n ! ( N - n ) ! W(n,N)={N\choose n}={{N!}\over{n!(N-n)!}}
  2. S = k ln W S=k\ln{W}\,

Multiplicity_of_infection.html

  1. n n
  2. m m
  3. P ( n ) = m n e - m n ! P(n)=\frac{m^{n}\cdot e^{-m}}{n!}
  4. m m
  5. n n
  6. P ( n ) P(n)
  7. n n
  8. P ( 0 ) = 36.79 % P(0)=36.79\%
  9. P ( 1 ) = 36.79 % P(1)=36.79\%
  10. P ( 2 ) = 18.39 % P(2)=18.39\%
  11. P ( 3 ) = 6.13 % P(3)=6.13\%
  12. P ( n > 0 ) = 1 - P ( 0 ) P(n>0)=1-P(0)
  13. m m
  14. P ( n > 0 ) = 1 - P ( n = 0 ) = 1 - m 0 e - m 0 ! = 1 - e - m P(n>0)=1-P(n=0)=1-\frac{m^{0}\cdot e^{-m}}{0!}=1-e^{-m}
  15. m m
  16. m 1 m\ll 1

Multitape_Turing_machine.html

  1. M = Q , Γ , s , b , F , δ M=\langle Q,\Gamma,s,b,F,\delta\rangle
  2. Q Q
  3. Γ \Gamma
  4. s Q s\in Q
  5. b Γ b\in\Gamma
  6. F Q F\subseteq Q
  7. δ : Q × Γ k Q × ( Γ × { L , R , S } ) k \delta:Q\times\Gamma^{k}\rightarrow Q\times(\Gamma\times\{L,R,S\})^{k}

Musical_isomorphism.html

  1. T M TM
  2. ( M , g ) (M,g)
  3. T M TM
  4. 2 2
  5. X := g i j X i d x j = X j d x j . X^{\flat}:=g_{ij}X^{i}\,dx^{j}=X_{j}\,dx^{j}.
  6. g g
  7. X ( Y ) = X , Y X^{\flat}(Y)=\langle X,Y\rangle
  8. X X
  9. Y Y
  10. ω := g i j ω i j = ω j j \omega^{\sharp}:=g^{ij}\omega_{i}\partial_{j}=\omega^{j}\partial_{j}
  11. ω , Y = ω ( Y ) , \left\langle\omega^{\sharp},Y\right\rangle=\omega(Y),
  12. ω ω
  13. Y Y
  14. : T M T * M , : T * M T M . \flat:TM\to T^{*}M,\qquad\sharp:T^{*}M\to TM.
  15. p p
  16. M M
  17. T [ u s u , u p = 009 217 , u b = , u p ] M T[u^{\prime}su^{\prime},u^{\prime}p=\u{2}009\u{2}217^{\prime},u^{\prime}b=^{% \prime},u^{\prime}p^{\prime}]M
  18. k T M , k T * M . \bigotimes^{k}TM,\qquad\bigotimes^{k}T^{*}M.
  19. ( 2 , 0 ) (2,0)
  20. ( 1 , 1 ) (1,1)
  21. X = g j k X i j d x i k . X^{\sharp}=g^{jk}X_{ij}\,dx^{i}\otimes\partial_{k}.
  22. ( 0 , 2 ) (0,2)
  23. X X
  24. g g
  25. tr g ( X ) := tr ( X ) = tr ( g j k X i j ) = g j i X i j = g i j X i j . \operatorname{tr}_{g}(X):=\operatorname{tr}(X^{\sharp})=\operatorname{tr}(g^{% jk}X_{ij})=g^{ji}X_{ij}=g^{ij}X_{ij}.

Müntz–Szász_theorem.html

  1. x n x^{n}

Møller_scattering.html

  1. e - e - e - e - e^{-}e^{-}\longrightarrow e^{-}e^{-}
  2. A P V = - m E G F 2 π α 16 sin 2 Θ cm ( 3 + cos 2 Θ cm ) 2 ( 1 4 - sin 2 θ W ) A_{PV}=-mE\frac{G_{F}}{\sqrt{2}\pi\alpha}\frac{16\sin^{2}\Theta_{\textrm{cm}}}% {\left(3+\cos^{2}\Theta_{\textrm{cm}}\right)^{2}}\left(\frac{1}{4}-\sin^{2}% \theta_{W}\right)
  3. G F G_{F}
  4. α \alpha
  5. Θ cm \Theta_{\textrm{cm}}
  6. θ W \theta_{W}

N-body_simulation.html

  1. 2 Φ = 4 π G ρ , \nabla^{2}\Phi=4\pi G{\rho},\,
  2. ρ {\rho}
  3. Φ ^ = - 4 π G ρ ^ k 2 , \hat{\Phi}=-4\pi G\frac{\hat{\rho}}{k^{2}},\,
  4. k \vec{k}
  5. k \vec{k}

N-electron_valence_state_perturbation_theory.html

  1. Ψ m ( 0 ) \Psi_{m}^{(0)}
  2. Ψ m ( 0 ) = I CAS C I , m | I \Psi_{m}^{(0)}=\sum_{I\in{\rm CAS}}C_{I,m}\left|I\right\rangle
  3. ^ \hat{\mathcal{H}}
  4. 𝒫 ^ CAS ^ 𝒫 ^ CAS | Ψ m ( 0 ) = E m ( 0 ) | Ψ m ( 0 ) \hat{\mathcal{P}}_{\rm CAS}\hat{\mathcal{H}}\hat{\mathcal{P}}_{\rm CAS}\left|% \Psi_{m}^{(0)}\right\rangle=E_{m}^{(0)}\left|\Psi_{m}^{(0)}\right\rangle
  5. 𝒫 ^ CAS \hat{\mathcal{P}}_{\rm CAS}
  6. k k
  7. - 2 k 2 -2\leq k\leq 2
  8. Φ c \Phi_{c}
  9. Ψ m v \Psi_{m}^{v}
  10. | Ψ m ( 0 ) = | Φ c Ψ m v \left|\Psi_{m}^{(0)}\right\rangle=\left|\Phi_{c}\Psi_{m}^{v}\right\rangle
  11. | Ψ l , μ k = | Φ l - k Ψ μ v + k \left|\Psi_{l,\mu}^{k}\right\rangle=\left|\Phi_{l}^{-k}\Psi_{\mu}^{v+k}\right\rangle
  12. l l
  13. Ψ l , μ k \Psi_{l,\mu}^{k}
  14. μ \mu
  15. i i
  16. j j
  17. a a
  18. b b
  19. r r
  20. s s
  21. k = 0 k=0
  22. k = + 1 k=+1
  23. k = - 1 k=-1
  24. k = + 2 k=+2
  25. k = - 2 k=-2
  26. k = 0 k=0
  27. k = + 1 k=+1
  28. k = - 1 k=-1
  29. S l k S_{l}^{k}
  30. Φ l - k \Phi_{l}^{-k}
  31. Ψ I k \Psi_{I}^{k}
  32. S l k = def { Φ l - k Ψ I k } S_{l}^{k}\ \stackrel{\mathrm{def}}{=}\ \{\Phi_{l}^{-k}\Psi_{I}^{k}\}
  33. 𝒫 ^ S l k ^ 𝒫 ^ S l k | Φ l - k Ψ μ v + k = E l , μ | Φ l - k Ψ μ v + k \hat{\mathcal{P}}_{S_{l}^{k}}\hat{\mathcal{H}}\hat{\mathcal{P}}_{S_{l}^{k}}% \left|\Phi_{l}^{-k}\Psi_{\mu}^{v+k}\right\rangle=E_{l,\mu}\left|\Phi_{l}^{-k}% \Psi_{\mu}^{v+k}\right\rangle
  34. S l k S_{l}^{k}
  35. ^ D \hat{\mathcal{H}}^{D}
  36. ^ D | Φ l - k Ψ μ v + k = E l , μ k | Φ l - k Ψ μ v + k \hat{\mathcal{H}}^{D}\left|\Phi_{l}^{-k}\Psi_{\mu}^{v+k}\right\rangle=E_{l,\mu% }^{k}\left|\Phi_{l}^{-k}\Psi_{\mu}^{v+k}\right\rangle
  37. ^ v D | Ψ μ v + k = E μ k | Ψ μ v + k \hat{\mathcal{H}}^{D}_{v}\left|\Psi_{\mu}^{v+k}\right\rangle=E_{\mu}^{k}\left|% \Psi_{\mu}^{v+k}\right\rangle
  38. E l , μ k E_{l,\mu}^{k}
  39. E μ k E_{\mu}^{k}
  40. Φ l - k \Phi_{l}^{-k}
  41. S l k S_{l}^{k}
  42. 𝒫 ^ S l k \hat{\mathcal{P}}_{S_{l}^{k}}
  43. Ψ l k = 𝒫 ^ S l k ^ Ψ m ( 0 ) \Psi_{l}^{k}=\hat{\mathcal{P}}_{S_{l}^{k}}\hat{\mathcal{H}}\Psi_{m}^{(0)}
  44. 𝒫 ^ S l k \hat{\mathcal{P}}_{S_{l}^{k}}
  45. Ψ l k = V l k Ψ m ( 0 ) \Psi_{l}^{k}=V_{l}^{k}\Psi_{m}^{(0)}
  46. N l k = Ψ l k | Ψ l k = Ψ m ( 0 ) | ( V l k ) + V l k | Ψ m ( 0 ) N_{l}^{k}=\left\langle\Psi_{l}^{k}\left.\right|\Psi_{l}^{k}\right\rangle=\left% \langle\Psi_{m}^{(0)}\left|\left(V_{l}^{k}\right)^{+}V_{l}^{k}\right|\Psi_{m}^% {(0)}\right\rangle
  47. Ψ m ( 0 ) \Psi_{m}^{(0)}
  48. Ψ l k \Psi_{l}^{k}
  49. S l k S_{l}^{k}
  50. Ψ l k \Psi_{l}^{k}
  51. Ψ l k \Psi_{l}^{k}
  52. ^ 0 = l k | Ψ l k E l k Ψ l k + m | Ψ m ( 0 ) E m ( 0 ) Ψ m ( 0 ) | \hat{\mathcal{H}}_{0}=\sum_{lk}\left|\Psi_{l}^{k}{}^{\prime}\right\rangle E_{l% }^{k}\left\langle\Psi_{l}^{k}{}^{\prime}\right\rangle+\sum_{m}\left|\Psi_{m}^{% (0)}\right\rangle E_{m}^{(0)}\left\langle\Psi_{m}^{(0)}\right|
  53. | Ψ l k \left|\Psi_{l}^{k}{}^{\prime}\right\rangle
  54. | Ψ l k \left|\Psi_{l}^{k}\right\rangle
  55. Ψ m ( 1 ) = k l | Ψ l k Ψ l k | ^ | Ψ m ( 0 ) E m ( 0 ) - E l k = k l | Ψ l k N l k E m ( 0 ) - E l k \Psi_{m}^{(1)}=\sum_{kl}\left|\Psi_{l}^{k}{}^{\prime}\right\rangle\frac{\left% \langle\Psi_{l}^{k}{}^{\prime}\left|\hat{\mathcal{H}}\right|\Psi_{m}^{(0)}% \right\rangle}{E_{m}^{(0)}-E_{l}^{k}}=\sum_{kl}\left|\Psi_{l}^{k}{}^{\prime}% \right\rangle\frac{\sqrt{N_{l}^{k}}}{E_{m}^{(0)}-E_{l}^{k}}
  56. E m ( 2 ) = k l | Ψ l k | ^ | Ψ m ( 0 ) | 2 E m ( 0 ) - E l k = k l N l k E m ( 0 ) - E l k E_{m}^{(2)}=\sum_{kl}\frac{\left|\left\langle\Psi_{l}^{k}{}^{\prime}\left|\hat% {\mathcal{H}}\right|\Psi_{m}^{(0)}\right\rangle\right|^{2}}{E_{m}^{(0)}-E_{l}^% {k}}=\sum_{kl}\frac{N_{l}^{k}}{E_{m}^{(0)}-E_{l}^{k}}
  57. E l k E_{l}^{k}
  58. E l k = 1 N l k Ψ l k | ^ D | Ψ l k E_{l}^{k}=\frac{1}{N_{l}^{k}}\left\langle\Psi_{l}^{k}\left|\hat{\mathcal{H}}^{% D}\right|\Psi_{l}^{k}\right\rangle
  59. N l k E l k = Ψ m ( 0 ) | ( V l k ) + ^ D V l k | Ψ m ( 0 ) = Ψ m ( 0 ) | ( V l k ) + V l k ^ D | Ψ m ( 0 ) + Ψ m ( 0 ) | ( V l k ) + [ ^ D , V l k ] | Ψ m ( 0 ) N_{l}^{k}E_{l}^{k}=\left\langle\Psi_{m}^{(0)}\left|\left(V_{l}^{k}\right)^{+}% \hat{\mathcal{H}}^{D}V_{l}^{k}\right|\Psi_{m}^{(0)}\right\rangle=\left\langle% \Psi_{m}^{(0)}\left|\left(V_{l}^{k}\right)^{+}V_{l}^{k}\hat{\mathcal{H}}^{D}% \right|\Psi_{m}^{(0)}\right\rangle+\left\langle\Psi_{m}^{(0)}\left|\left(V_{l}% ^{k}\right)^{+}\left[\hat{\mathcal{H}}^{D},V_{l}^{k}\right]\right|\Psi_{m}^{(0% )}\right\rangle
  60. E l k = E m ( 0 ) + Δ ϵ l + 1 N l k Ψ m ( 0 ) | ( V l k ) + [ ^ v , V l k ] | Ψ m ( 0 ) E_{l}^{k}=E_{m}^{(0)}+\Delta\epsilon_{l}+\frac{1}{N_{l}^{k}}\left\langle\Psi_{% m}^{(0)}\left|\left(V_{l}^{k}\right)^{+}\left[\hat{\mathcal{H}}_{v},V_{l}^{k}% \right]\right|\Psi_{m}^{(0)}\right\rangle
  61. Δ ϵ l \Delta\epsilon_{l}
  62. V V
  63. V i j r s ( 0 ) V_{ijrs}^{(0)}
  64. E m ( 2 ) ( S r s i j 0 ) = - N r s i j 0 ϵ r + ϵ s - ϵ i - ϵ j E_{m}^{(2)}\left(S_{rsij}^{0}\right)=-\frac{N_{rsij}^{0}}{\epsilon_{r}+% \epsilon_{s}-\epsilon_{i}-\epsilon_{j}}
  65. S ¯ l k \overline{S}_{l}^{k}
  66. S l k S_{l}^{k}
  67. Φ \Phi
  68. V l k V_{l}^{k}
  69. V r s i - 1 V_{rsi}^{-1}
  70. V r s i - 1 = γ r s a ( r s | i a E r i E s a + s r | i a E s i E r a ) r s V_{rsi}^{-1}=\gamma_{rs}\sum_{a}\left(\left\langle rs\left.\right|ia\right% \rangle E_{ri}E_{sa}+\left\langle sr\left.\right|ia\right\rangle E_{si}E_{ra}% \right)\quad r\leq s
  71. Φ r i s a = E r i E s a Ψ m ( 0 ) \Phi_{risa}=E_{ri}E_{sa}\Psi_{m}^{(0)}
  72. Φ r i s a = E s i E r a Ψ m ( 0 ) \Phi_{risa}=E_{si}E_{ra}\Psi_{m}^{(0)}
  73. S ¯ r s i - 1 \overline{S}_{rsi}^{-1}
  74. S ¯ l k \overline{S}_{l}^{k}
  75. 𝒫 ^ S ¯ l k ^ 𝒫 ^ S ¯ l k | Ψ l μ k = E l , μ k | Ψ l μ k \hat{\mathcal{P}}_{\overline{S}_{l}^{k}}\hat{\mathcal{H}}\hat{\mathcal{P}}_{% \overline{S}_{l}^{k}}\left|\Psi_{l\mu}^{k}\right\rangle=E_{l,\mu}^{k}\left|% \Psi_{l\mu}^{k}\right\rangle
  76. E ( A - B ) = E ( A ) + E ( B ) E(A-B)=E(A)+E(B)

NACA_airfoil.html

  1. y t = 5 t c [ 0.2969 x c + ( - 0.1260 ) ( x c ) + ( - 0.3516 ) ( x c ) 2 + 0.2843 ( x c ) 3 + ( - 0.1015 ) ( x c ) 4 ] , y_{t}=5tc\,\left[0.2969\sqrt{\frac{x}{c}}+(-0.1260)\left(\frac{x}{c}\right)+(-% 0.3516)\left(\frac{x}{c}\right)^{2}+0.2843\left(\frac{x}{c}\right)^{3}+(-0.101% 5)\left(\frac{x}{c}\right)^{4}\right],
  2. y t y_{t}
  3. r = 1.1019 t 2 . r=1.1019t^{2}.\,
  4. ( x U , y U ) (x_{U},y_{U})
  5. ( x L , y L ) (x_{L},y_{L})
  6. x U = x L = x , y U = + y t , and y L = - y t . x_{U}=x_{L}=x,\qquad y_{U}=+y_{t},\quad\,\text{and}\quad y_{L}=-y_{t}.
  7. y c = { m x p 2 ( 2 p - x c ) , 0 x p c m c - x ( 1 - p ) 2 ( 1 + x c - 2 p ) , p c x c y_{c}=\left\{\begin{array}[]{ll}\displaystyle{m\,\frac{x}{p^{2}}\left(2\,p\,-% \frac{x}{c}\right)},&0\leq x\leq pc\\ \\ \displaystyle{m\,\frac{c-x}{(1-p)^{2}}\left(1+\frac{x}{c}-2\,p\right)},&pc\leq x% \leq c\end{array}\right.
  8. ( x U , y U ) (x_{U},y_{U})
  9. ( x L , y L ) (x_{L},y_{L})
  10. x U = x - y t sin θ , y U = y c + y t cos θ , x L = x + y t sin θ , y L = y c - y t cos θ , \begin{aligned}\displaystyle x_{U}&\displaystyle=x-y_{t}\,\sin\theta,&% \displaystyle y_{U}&\displaystyle=y_{c}+y_{t}\,\cos\theta,\\ \displaystyle x_{L}&\displaystyle=x+y_{t}\,\sin\theta,&\displaystyle y_{L}&% \displaystyle=y_{c}-y_{t}\,\cos\theta,\end{aligned}
  11. θ = arctan ( d y c d x ) , \theta=\arctan{\left(\frac{dy_{c}}{dx}\right)},
  12. d y c d x = { 2 m p 2 ( p - x c ) , 0 x p c 2 m ( 1 - p ) 2 ( p - x c ) , p c x c \frac{dy_{c}}{dx}=\left\{\begin{array}[]{ll}\displaystyle{\frac{2m}{p^{2}}% \left(p-\frac{x}{c}\right)},&0\leq x\leq pc\\ \\ \displaystyle{\frac{2m}{(1-p)^{2}}\left(p-\frac{x}{c}\right)},&pc\leq x\leq c% \end{array}\right.
  13. y c = { k 1 6 { x 3 - 3 m x 2 + m 2 ( 3 - m ) x } , 0 < x < p k 1 m 3 6 ( 1 - x ) , p < x < 1 y_{c}=\begin{cases}\frac{k_{1}}{6}\left\{x^{3}-3mx^{2}+m^{2}(3-m)x\right\},&0<% x<p\\ \frac{k_{1}m^{3}}{6}(1-x),&p<x<1\end{cases}
  14. x x
  15. y y
  16. m m
  17. x = p x=p
  18. p = 0.3 / 2 = 0.15 p=0.3/2=0.15
  19. m = 0.2025 m=0.2025
  20. k 1 k_{1}
  21. k 1 = 15.957 k_{1}=15.957
  22. y c = { p m 2 ( 2 m x c ) - ( x c ) 2 , 0 < x < m p ( 1 - m ) 2 [ ( 1 - 2 m ) + 2 m x c - ( x c ) 2 ] , m < x < 1 y_{c}=\begin{cases}\frac{p}{m^{2}}\left(2m\frac{x}{c}\right)-\left(\frac{x}{c}% \right)^{2},&0<x<m\\ \frac{p}{(1-m)^{2}\left[(1-2m)+2m\frac{x}{c}-\left(\frac{x}{c}\right)^{2}% \right]},&m<x<1\end{cases}
  23. p p
  24. m m
  25. k 1 k_{1}
  26. x c r \frac{x}{c}<=r
  27. y c = k 1 6 [ ( x c - r ) 3 - k 2 k 1 ( 1 - r ) 3 x c - r 3 x c + r 3 ] \frac{y}{c}=\frac{k_{1}}{6}\left[\left(\frac{x}{c}-r\right)^{3}-\frac{k_{2}}{k% _{1}}(1-r)^{3}\frac{x}{c}-r^{3}\frac{x}{c}+r^{3}\right]
  28. r < x c 1.0 r<\frac{x}{c}<=1.0
  29. y c = k 1 6 [ k 2 k 1 ( x c - r ) 3 - k 2 k 1 ( 1 - r ) 3 x c - r 3 x c + r 3 ] \frac{y}{c}=\frac{k_{1}}{6}\left[\frac{k_{2}}{k_{1}}\left(\frac{x}{c}-r\right)% ^{3}-\frac{k_{2}}{k_{1}}(1-r)^{3}\frac{x}{c}-r^{3}\frac{x}{c}+r^{3}\right]
  30. p p
  31. m m
  32. k 1 k_{1}
  33. k 2 / k 1 k_{2}/k_{1}

Nachbin's_theorem.html

  1. | f ( r e i θ ) | M e τ r |f(re^{i\theta})|\leq Me^{\tau r}
  2. r r\to\infty
  3. z = r e i θ z=re^{i\theta}
  4. f ( z ) = sin ( π z ) f(z)=\sin(\pi z)
  5. sin ( π z ) \sin(\pi z)
  6. sin ( π z ) \sin(\pi z)
  7. Ψ ( t ) \Psi(t)
  8. Ψ ( t ) = n = 0 Ψ n t n \Psi(t)=\sum_{n=0}^{\infty}\Psi_{n}t^{n}
  9. Ψ n > 0 \Psi_{n}>0
  10. lim n Ψ n + 1 Ψ n = 0. \lim_{n\to\infty}\frac{\Psi_{n+1}}{\Psi_{n}}=0.
  11. Ψ ( t ) \Psi(t)
  12. | f ( r e i θ ) | M Ψ ( τ r ) \left|f\left(re^{i\theta}\right)\right|\leq M\Psi(\tau r)
  13. r r\to\infty
  14. f ( z ) = n = 0 f n z n f(z)=\sum_{n=0}^{\infty}f_{n}z^{n}
  15. lim sup n | f n Ψ n | 1 / n = τ . \limsup_{n\to\infty}\left|\frac{f_{n}}{\Psi_{n}}\right|^{1/n}=\tau.
  16. F ( w ) = n = 0 f n Ψ n w n + 1 . F(w)=\sum_{n=0}^{\infty}\frac{f_{n}}{\Psi_{n}w^{n+1}}.
  17. F ( w ) F(w)
  18. | w | τ . |w|\leq\tau.
  19. f ( z ) = 1 2 π i γ Ψ ( z w ) F ( w ) d w f(z)=\frac{1}{2\pi i}\oint_{\gamma}\Psi(zw)F(w)\,dw
  20. | w | τ |w|\leq\tau
  21. Ψ ( t ) = e t \Psi(t)=e^{t}
  22. α ( t ) \alpha(t)
  23. [ 0 , ) [0,\infty)
  24. 1 Ψ n = 0 t n d α ( t ) \frac{1}{\Psi_{n}}=\int_{0}^{\infty}t^{n}\,d\alpha(t)
  25. d α ( t ) = α ( t ) d t d\alpha(t)=\alpha^{\prime}(t)\,dt
  26. F ( w ) = 1 w 0 f ( t w ) d α ( t ) . F(w)=\frac{1}{w}\int_{0}^{\infty}f\left(\frac{t}{w}\right)\,d\alpha(t).
  27. α ( t ) = e - t \alpha(t)=e^{-t}
  28. g ( s ) = s 0 K ( s t ) f ( t ) d t g(s)=s\int_{0}^{\infty}K(st)f(t)\,dt
  29. f ( x ) = n = 0 a n M ( n + 1 ) x n f(x)=\sum_{n=0}^{\infty}\frac{a_{n}}{M(n+1)}x^{n}
  30. g ( s ) = n = 0 a n s - n g(s)=\sum_{n=0}^{\infty}a_{n}s^{-n}
  31. π ( x ) n = 1 log n ( x ) n n ! ζ ( n + 1 ) . \pi(x)\approx\sum_{n=1}^{\infty}\frac{\log^{n}(x)}{n\cdot n!\zeta(n+1)}.
  32. τ \tau
  33. f n = sup z exp [ - ( τ + 1 n ) | z | ] | f ( z ) | . \|f\|_{n}=\sup_{z\in\mathbb{C}}\exp\left[-\left(\tau+\frac{1}{n}\right)|z|% \right]|f(z)|.

Nanoindentation.html

  1. H H
  2. P m a x P_{max}
  3. A r A_{r}
  4. H = P max A r . H=\frac{P\text{max}}{A\text{r}}.
  5. d P / d h dP/dh
  6. S S
  7. E r E_{r}
  8. E r = 1 β π 2 S A p ( h c ) , E_{r}=\frac{1}{\beta}\frac{\sqrt{\pi}}{2}\frac{S}{\sqrt{A_{p}(h_{c})}},
  9. A p ( h c ) A_{p}(h_{c})
  10. h c h_{c}
  11. β \beta
  12. A p ( h c ) A_{p}(h_{c})
  13. A p ( h c ) = C 0 h c 2 + C 1 h c 1 + C 2 h c 1 / 2 + C 3 h c 1 / 4 + + C 8 h c 1 / 128 A_{p}(h_{c})=C_{0}h_{c}^{2}+C_{1}h_{c}^{1}+C_{2}h_{c}^{1/2}+C_{3}h_{c}^{1/4}+% \ldots+C_{8}h_{c}^{1/128}
  14. C 0 C_{0}
  15. E r E_{r}
  16. E s E_{s}
  17. 1 / E r = ( 1 - ν i 2 ) / E i + ( 1 - ν s 2 ) / E s . 1/E_{r}=(1-\nu_{i}^{2})/E_{i}+(1-\nu_{s}^{2})/E_{s}.
  18. i i
  19. ν \nu
  20. E i E_{i}
  21. ν i \nu_{i}
  22. ν s \nu_{s}
  23. H = P m a x A r . H=\frac{P_{max}}{A_{r}}.
  24. A p ( h c ) A_{p}(h_{c})
  25. h h
  26. m m
  27. m = ln σ ln ε ˙ , m=\frac{\partial\ln{\sigma}}{\partial\ln{\dot{\varepsilon}}},
  28. σ = σ ( ε ˙ ) \sigma=\sigma(\dot{\varepsilon})
  29. ε ˙ \dot{\varepsilon}
  30. m m
  31. d ln H = m d ln ε p ˙ + n d ln h p . d\ln{H}=md\ln{\dot{\varepsilon_{p}}}+nd\ln{h_{p}}.
  32. p p
  33. V * V^{*}
  34. V * = 9 k B T ln ε ˙ H , V^{*}=9k_{B}T\frac{\partial\ln{\dot{\varepsilon}}}{\partial H},
  35. T T
  36. m m
  37. V * ( H m ) - 1 V^{*}\propto(Hm)^{-1}
  38. ( - h ( P 0 ) ) (-h(P_{0}))
  39. X Y XY
  40. H M HM
  41. h m a x h_{max}
  42. P m a x P_{max}
  43. H M = P m a x A s . HM=\frac{P_{max}}{A_{s}}.
  44. A s A_{s}
  45. A s = 24.5 h m a x 2 A_{s}=24.5h_{max}^{2}
  46. H I T H_{IT}
  47. H I T = P m a x A p . H_{IT}=\frac{P_{max}}{A_{p}}.
  48. A p A_{p}
  49. C 1 C_{1}
  50. C 0 C_{0}
  51. C 1 C_{1}
  52. A p = C 0 h m a x 2 + C 1 h m a x . A_{p}=C_{0}h_{max}^{2}+C_{1}h_{max}.
  53. h h
  54. h f h_{f}
  55. k k
  56. m m
  57. k k
  58. h f h_{f}
  59. m m
  60. d P / d h dP/dh
  61. P = k ( h - h f ) m . P=k\left(h-h_{f}\right)^{m}.
  62. σ \sigma

Nanoparticle_tracking_analysis.html

  1. ( x , y ) 2 4 = D t {(x,y)^{2}\over 4}=Dt
  2. D t = T K b 3 π η d = D t Dt={TK_{b}\over 3\pi\eta d}=Dt
  3. K b K_{b}

Narcissistic_number.html

  1. 153 = 1 3 + 5 3 + 3 3 153=1^{3}+5^{3}+3^{3}
  2. 370 = 3 3 + 7 3 + 0 3 370=3^{3}+7^{3}+0^{3}
  3. 371 = 3 3 + 7 3 + 1 3 371=3^{3}+7^{3}+1^{3}
  4. 407 = 4 3 + 0 3 + 7 3 407=4^{3}+0^{3}+7^{3}
  5. k ( b - 1 ) k , k(b-1)^{k}\,,
  6. k ( b - 1 ) k < b k - 1 , k(b-1)^{k}<b^{k-1}\,,
  7. 72 , 90 , 108 , 153 , 270 , 423 , 450 , 531 , 558 , 630 , 648 , 738 , 72,90,108,153,270,423,450,531,558,630,648,738,...
  8. n = m d k + m d k - 1 + + m d 2 + m d 1 n=m^{d_{k}}+m^{d_{k-1}}+\dots+m^{d_{2}}+m^{d_{1}}
  9. n = i = 0 k d i d i , e.g. 3435 = 3 3 + 4 4 + 3 3 + 5 5 . \textstyle n=\sum_{i=0}^{k}d_{i}^{d_{i}}\,,\,\text{ e.g. }3435=3^{3}+4^{4}+3^{% 3}+5^{5}\,.
  10. n = d k 1 + d k - 1 2 + + d 2 k - 1 + d 1 k , e.g. 135 = 1 1 + 3 2 + 5 3 . n=d_{k}^{1}+d_{k-1}^{2}+\dots+d_{2}^{k-1}+d_{1}^{k}\,,\,\text{ e.g. }135=1^{1}% +3^{2}+5^{3}\,.
  11. e . g .729 = ( 7 + 2 ) 9 , 4096 = 4 + 0 96 {e.g.}729=(7+2)^{\sqrt{9}},\,\text{ }4096=\sqrt{\sqrt{\sqrt{\sqrt{4}+0}}}^{96}
  12. n = ( i = 1 k d i ) ( i = 1 k d i ) , e.g. 144 = ( 1 + 4 + 4 ) × ( 1 × 4 × 4 ) . n=\left(\sum_{i=1}^{k}{d_{i}}\right)\left(\prod_{i=1}^{k}{d_{i}}\right)\,,\,% \text{ e.g. }144=(1+4+4)\times(1\times 4\times 4)\,.
  13. n = ( i = 1 k d i ) 3 , e.g. 512 = ( 5 + 1 + 2 ) 3 . n=\left(\sum_{i=1}^{k}{d_{i}}\right)^{3}\,,\,\text{ e.g. }512=(5+1+2)^{3}\,.
  14. n = i = 1 k d i ! , e.g. 145 = 1 ! + 4 ! + 5 ! . n=\sum_{i=1}^{k}{d_{i}}!\,,\,\text{ e.g. }145=1!+4!+5!\,.

Nash–Sutcliffe_model_efficiency_coefficient.html

  1. E = 1 - t = 1 T ( Q o t - Q m t ) 2 t = 1 T ( Q o t - Q o ¯ ) 2 E=1-\frac{\sum_{t=1}^{T}\left(Q_{o}^{t}-Q_{m}^{t}\right)^{2}}{\sum_{t=1}^{T}% \left(Q_{o}^{t}-\overline{Q_{o}}\right)^{2}}

Natural_frequency.html

  1. K 1 e - s 1 t K_{1}e^{-s_{1}t}
  2. K 1 0 K_{1}\neq 0
  3. ω 0 = 1 L C \omega_{0}=\frac{1}{\sqrt{LC}}

Natural_process_variation.html

  1. x ¯ = i = 1 n x i / n \bar{x}=\sum_{i=1}^{n}x_{i}/n
  2. σ x ¯ = σ / n \sigma_{\bar{x}}=\sigma/\sqrt{n}
  3. x ¯ \bar{x}
  4. σ x ¯ \sigma_{\bar{x}}
  5. x ¯ = 1 \bar{x}=1
  6. σ x ¯ = 0.1 / 1 0 = 0.0316 \sigma_{\bar{x}}=0.1/\sqrt{1}0=0.0316

Naturalness_(physics).html

  1. Λ Λ
  2. h = c Λ 4 - d , h=c\Lambda^{4-d},
  3. d d
  4. c c
  5. c c
  6. E E
  7. l n ( E / Λ ) ln(E/Λ)

Near-field_(mathematics).html

  1. Q Q
  2. + +
  3. \cdot
  4. ( Q , + ) (Q,+)
  5. ( a b ) c (a\cdot b)\cdot c
  6. a ( b c ) a\cdot(b\cdot c)
  7. a a
  8. b b
  9. c c
  10. Q Q
  11. ( a + b ) c = a c + b c (a+b)\cdot c=a\cdot c+b\cdot c
  12. a a
  13. b b
  14. c c
  15. Q Q
  16. Q Q
  17. 1 a = a 1 = a 1\cdot a=a\cdot 1=a
  18. a a
  19. Q Q
  20. Q Q
  21. a - 1 a^{-1}
  22. a a - 1 = 1 = a - 1 a a\cdot a^{-1}=1=a^{-1}\cdot a
  23. c ( a + b ) = c a + c b c\cdot(a+b)=c\cdot a+c\cdot b
  24. x 0 = 0 x\cdot 0=0
  25. x x
  26. x y = x x\cdot y=x
  27. x x
  28. y y
  29. K K
  30. K K
  31. * *
  32. b b
  33. K K
  34. a a
  35. K K
  36. a b = a * b a\cdot b=a*b
  37. b b
  38. K K
  39. a a
  40. K K
  41. a b = a 3 * b a\cdot b=a^{3}*b
  42. K K
  43. K K
  44. K m K_{m}
  45. K a K_{a}
  46. c K m c\in K_{m}
  47. b K a b\in K_{a}
  48. b b c b\mapsto b\cdot c
  49. K a K_{a}
  50. K a K_{a}
  51. A A
  52. M M
  53. Aut ( A ) \mathrm{Aut}(A)
  54. A A
  55. A A
  56. M M
  57. A A
  58. 1 1
  59. ϕ : M A { 0 } \phi:M\to A\setminus\{0\}
  60. m 1 m m\mapsto 1\ast m
  61. A A
  62. A A
  63. a b = 1 ϕ - 1 ( a ) ϕ - 1 ( b ) a\cdot b=1\ast\phi^{-1}(a)\phi^{-1}(b)
  64. A M A\rtimes M
  65. M M
  66. A A
  67. | M | = | A | - 1 |M|=|A|-1
  68. ( A , M ) (A,M)
  69. A A
  70. M M
  71. A A
  72. A A
  73. q q
  74. n n
  75. n n
  76. q - 1 q-1
  77. q 3 mod 4 q\equiv 3\bmod 4
  78. n n
  79. 4 4
  80. F F
  81. q n q^{n}
  82. A A
  83. F F
  84. F F
  85. x x q x\mapsto x^{q}
  86. F F
  87. C n C q n - 1 C_{n}\ltimes C_{q^{n}-1}
  88. C k C_{k}
  89. k k
  90. n n
  91. C n C q n - 1 C_{n}\ltimes C_{q^{n}-1}
  92. q n - 1 q^{n}-1
  93. A A
  94. n = 1 n=1
  95. q = 3 q=3
  96. n = 2 n=2
  97. A A
  98. C p 2 C_{p}^{2}
  99. A = C p 2 A=C_{p}^{2}
  100. M M
  101. M M
  102. p = 5 p=5
  103. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  104. ( 1 - 2 - 1 - 2 ) \left(\begin{smallmatrix}1&-2\\ -1&-2\\ \end{smallmatrix}\right)
  105. 2 T 2T
  106. p = 11 p=11
  107. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  108. ( 1 5 - 5 - 2 ) \left(\begin{smallmatrix}1&5\\ -5&-2\\ \end{smallmatrix}\right)
  109. ( 4 0 0 4 ) \left(\begin{smallmatrix}4&0\\ 0&4\\ \end{smallmatrix}\right)
  110. 2 T × C 5 2T\times C_{5}
  111. p = 7 p=7
  112. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  113. ( 1 3 - 1 - 2 ) \left(\begin{smallmatrix}1&3\\ -1&-2\\ \end{smallmatrix}\right)
  114. 2 O 2O
  115. p = 23 p=23
  116. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  117. ( 1 - 6 12 - 2 ) \left(\begin{smallmatrix}1&-6\\ 12&-2\\ \end{smallmatrix}\right)
  118. ( 2 0 0 2 ) \left(\begin{smallmatrix}2&0\\ 0&2\\ \end{smallmatrix}\right)
  119. 2 O × C 11 2O\times C_{11}
  120. p = 11 p=11
  121. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  122. ( 2 4 1 - 3 ) \left(\begin{smallmatrix}2&4\\ 1&-3\\ \end{smallmatrix}\right)
  123. 2 I 2I
  124. p = 29 p=29
  125. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  126. ( 1 - 7 - 12 - 2 ) \left(\begin{smallmatrix}1&-7\\ -12&-2\\ \end{smallmatrix}\right)
  127. ( 16 0 0 16 ) \left(\begin{smallmatrix}16&0\\ 0&16\\ \end{smallmatrix}\right)
  128. 2 I × C 7 2I\times C_{7}
  129. p = 59 p=59
  130. ( 0 - 1 1 0 ) \left(\begin{smallmatrix}0&-1\\ 1&0\\ \end{smallmatrix}\right)
  131. ( 9 15 - 10 - 10 ) \left(\begin{smallmatrix}9&15\\ -10&-10\\ \end{smallmatrix}\right)
  132. ( 4 0 0 4 ) \left(\begin{smallmatrix}4&0\\ 0&4\\ \end{smallmatrix}\right)
  133. 2 I × C 29 2I\times C_{29}
  134. A 4 A_{4}
  135. S 4 S_{4}
  136. A 5 A_{5}
  137. 2 T 2T
  138. 2 I 2I
  139. S L ( 2 , 𝔽 3 ) SL(2,\mathbb{F}_{3})
  140. S L ( 2 , 𝔽 5 ) SL(2,\mathbb{F}_{5})
  141. ( Q ) , \mathcal{(}Q),

Near-field_scanning_optical_microscope.html

  1. d = 0.61 λ 0 N A d=0.61\frac{\lambda_{0}}{N\!A}\;\!

Nearly_free_electron_model.html

  1. ψ k ( r ) = 1 Ω r e i k r \psi_{{k}}({r})=\frac{1}{\sqrt{\Omega_{r}}}e^{i{k}\cdot{r}}
  2. E k = 2 k 2 2 m E_{k}=\frac{\hbar^{2}k^{2}}{2m}
  3. ψ k ( r ) = 1 Ω r [ cos ( k r ) + i sin ( k r ) ] \psi_{{k}}({r})=\frac{1}{\sqrt{\Omega_{r}}}\left[\cos({k}\cdot{r})+i\sin({k}% \cdot{r})\right]
  4. cos ( k r ) = 1 2 [ e i k r + e - i k r ] \cos({k}\cdot{r})=\frac{1}{2}[e^{i{k}\cdot{r}}+e^{-i{k}\cdot{r}}]
  5. H = T + V = - 2 2 m 2 + V ( r ) H=T+V=-\frac{\hbar^{2}}{2m}\nabla^{2}+V({r})
  6. T T
  7. V V
  8. E = H = Ω r ψ k * ( r ) [ T + V ] ψ k ( r ) d r E=\langle H\rangle=\int_{\Omega_{r}}\psi_{{k}}^{*}({r})[T+V]\psi_{{k}}({r})d{r}
  9. r {r}
  10. k {k}
  11. V V
  12. E k = 1 Ω r Ω r e - i k r [ 2 k 2 2 m + V ( r ) ] e i k r d r = 2 k 2 2 m + V E_{k}=\frac{1}{\Omega_{r}}\int_{\Omega_{r}}e^{-i{k}\cdot{r}}\left[\frac{\hbar^% {2}k^{2}}{2m}+V({r})\right]e^{i{k}\cdot{r}}d{r}=\frac{\hbar^{2}k^{2}}{2m}+% \langle V\rangle
  13. k {k}
  14. V V
  15. k {k}
  16. r = 0 {r}=0
  17. k = 0 {k}=0
  18. \infty
  19. k {k}
  20. r = 0 {r}=0
  21. r = ± π / ( 2 k ) {r}=\pm\pi/(2{k})
  22. k {k}
  23. k {k}
  24. cos ( k r ) \cos({k}\cdot{r})
  25. cos ( k r ) \cos({k}\cdot{r})
  26. sin ( k r ) \sin({k}\cdot{r})
  27. ρ k \rho_{{k}}
  28. | ψ k ( r ) | 2 |\psi_{{k}}({r})|^{2}
  29. ρ k = ρ k c ( r ) + ρ k s ( r ) \rho_{{k}}=\rho_{{k}}^{c}({r})+\rho_{{k}}^{s}({r})
  30. cos ( k r ) \cos({k}\cdot{r})
  31. sin ( k r ) \sin({k}\cdot{r})
  32. ρ k c ( r ) = 1 2 Ω [ 1 + cos ( 2 k r ) ] \rho_{{k}}^{c}({r})=\frac{1}{2\Omega}\left[1+\cos(2{k}\cdot{r})\right]
  33. sin ( k r ) \sin({k}\cdot{r})
  34. ρ k s ( r ) = 1 2 Ω [ 1 - cos ( 2 k r ) ] \rho_{{k}}^{s}({r})=\frac{1}{2\Omega}\left[1-\cos(2{k}\cdot{r})\right]
  35. k {k}
  36. cos ( k r ) \cos({k}\cdot{r})
  37. sin ( k r ) \sin({k}\cdot{r})
  38. ψ k ( r ) = u k ( r ) e i k r \psi_{{k}}({r})=u_{{k}}({r})e^{i{k}\cdot{r}}
  39. u k ( r ) = u k ( r + T ) u_{{k}}({r})=u_{{k}}({r}+{T})
  40. u k ( r ) 1 Ω r u_{{k}}({r})\approx\frac{1}{\sqrt{\Omega_{r}}}
  41. ( λ k - ϵ ) C k + G U G C k - G = 0 (\lambda_{{k}}-\epsilon)C_{{k}}+\sum_{{G}}U_{{G}}C_{{k}-{G}}=0
  42. λ k \lambda_{{k}}
  43. λ k ψ k ( r ) = - 2 2 m 2 ψ k ( r ) = - 2 2 m 2 ( u k ( r ) e i k r ) \lambda_{{k}}\psi_{{k}}({r})=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{{k}}({r})=-% \frac{\hbar^{2}}{2m}\nabla^{2}(u_{{k}}({r})e^{i{k}\cdot{r}})
  44. ψ k ( r ) \psi_{{k}}({r})
  45. λ k = 2 k 2 2 m \lambda_{{k}}=\frac{\hbar^{2}k^{2}}{2m}
  46. u k ( r ) u_{{k}}({r})
  47. 2 u k ( r ) k 2 . \nabla^{2}u_{{k}}({r})\ll k^{2}.
  48. U ( r ) = G U G e i G r U({r})=\sum_{{G}}U_{{G}}e^{i{G}\cdot{r}}
  49. ψ ( r ) = k C k e i k r \psi({r})=\sum_{{k}}C_{{k}}e^{i{k}\cdot{r}}
  50. ( λ k - ϵ ) C k = 0 (\lambda_{{k}}-\epsilon)C_{{k}}=0
  51. C k = 0 C_{{k}}=0
  52. λ k = ϵ \lambda_{{k}}=\epsilon
  53. λ k \lambda_{k}
  54. C k C_{k}
  55. ψ k e i k r \psi_{k}\propto e^{i{k}\cdot{r}}
  56. ϵ \epsilon
  57. ψ i = 1 m A i e i k i r \psi\propto\sum_{i=1}^{m}A_{i}e^{i{k}_{i}\cdot{r}}
  58. ϵ = λ k ± | U k | \epsilon=\lambda_{{k}}\pm|U_{{k}}|
  59. 2 | U K | 2|U_{K}|