wpmath0000006_7

Lambdavacuum_solution.html

  1. G a b + Λ g a b = 8 π T a b , G^{ab}+\Lambda\,g^{ab}=8\pi\,T^{ab},
  2. Λ g a b \Lambda\,g^{ab}
  3. T a b T^{ab}
  4. G a b = - Λ g a b G^{ab}=-\Lambda\,g^{ab}
  5. R a b = ( R / 2 - Λ ) g a b . R^{ab}=\left(R/2-\Lambda\right)\,g^{ab}.
  6. Λ > 0 \Lambda>0
  7. Λ < 0 \Lambda<0
  8. e 0 , e 1 , e 2 , e 3 \vec{e}_{0},\;\vec{e}_{1},\;\vec{e}_{2},\;\vec{e}_{3}
  9. e 0 \vec{e}_{0}
  10. G a ^ b ^ = - Λ [ - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] G^{\hat{a}\hat{b}}=-\Lambda\,\left[\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}\right]
  11. χ ( ζ ) = ( ζ + Λ ) 4 \chi(\zeta)=\left(\zeta+\Lambda\right)^{4}
  12. t 2 = t 1 2 / 4 , t 3 = t 1 3 / 16 , t 4 = t 1 4 / 64 t_{2}=t_{1}^{2}/4,\;t_{3}=t_{1}^{3}/16,\;t_{4}=t_{1}^{4}/64
  13. t 1 = G a a , t 2 = G a b G b a , t 3 = G a b G b c G c a , t 4 = G a b G b c G c d G d a t_{1}={G^{a}}_{a},\;t_{2}={G^{a}}_{b}\,{G^{b}}_{a},\;t_{3}={G^{a}}_{b}\,{G^{b}% }_{c}\,{G^{c}}_{a},\;t_{4}={G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}

Lambert_conformal_conic_projection.html

  1. x = ρ sin [ n ( λ - λ 0 ) ] x=\rho\sin[n(\lambda-\lambda_{0})]
  2. y = ρ 0 - ρ cos [ n ( λ - λ 0 ) ] y=\rho_{0}-\rho\cos[n(\lambda-\lambda_{0})]
  3. n = ln ( cos ϕ 1 sec ϕ 2 ) ln [ tan ( 1 4 π + 1 2 ϕ 2 ) cot ( 1 4 π + 1 2 ϕ 1 ) ] n=\frac{\ln(\cos\phi_{1}\sec\phi_{2})}{\ln[\tan(\frac{1}{4}\pi+\frac{1}{2}\phi% _{2})\cot(\frac{1}{4}\pi+\frac{1}{2}\phi_{1})]}
  4. ρ = F cot n ( 1 4 π + 1 2 ϕ ) \rho=F\cot^{n}(\frac{1}{4}\pi+\frac{1}{2}\phi)
  5. ρ 0 = F cot n ( 1 4 π + 1 2 ϕ 0 ) \rho_{0}=F\cot^{n}(\frac{1}{4}\pi+\frac{1}{2}\phi_{0})
  6. F = cos ϕ 1 tan n ( 1 4 π + 1 2 ϕ 1 ) n F=\frac{\cos\phi_{1}\tan^{n}(\frac{1}{4}\pi+\frac{1}{2}\phi_{1})}{n}
  7. ϕ 1 = ϕ 2 \phi_{1}=\phi_{2}
  8. n = sin ( ϕ 1 ) n=\sin(\phi_{1})

Lamb–Oseen_vortex.html

  1. θ \theta
  2. V θ ( r , t ) = Γ 2 π r ( 1 - exp ( - r 2 r c 2 ( t ) ) ) , V_{\theta}(r,t)=\frac{\Gamma}{2\pi r}\left(1-\exp\left(-\frac{r^{2}}{r_{c}^{2}% (t)}\right)\right),
  3. r r
  4. r c ( t ) = 4 ν t r_{c}(t)=\sqrt{4\nu t}
  5. ν \nu
  6. Γ \Gamma
  7. V θ ( r ) = V θ max ( 1 + 1 2 α ) r max r [ 1 - exp ( - α r 2 r max 2 ) ] , V_{\theta}\left(r\right)=V_{\theta\max}\left(1+\frac{1}{2\alpha}\right)\frac{r% _{\max}}{r}\left[1-\exp\left(-\alpha\frac{r^{2}}{r_{\max}^{2}}\right)\right],
  8. r max ( t ) = α r c ( t ) r_{\max}(t)=\sqrt{\alpha}r_{c}(t)
  9. v max v_{\max}
  10. p r = ρ v 2 r , {\partial p\over\partial r}=\rho{v^{2}\over r},

Lamport_timestamps.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. a a
  6. b b
  7. C ( x ) C(x)
  8. x x
  9. C ( a ) C(a)
  10. C ( b ) C(b)
  11. a a
  12. b b
  13. a a
  14. b b
  15. a b a\rightarrow b
  16. C ( a ) < C ( b ) C(a)<C(b)
  17. \rightarrow\,
  18. C ( a ) < C ( b ) C(a)<C(b)
  19. a b a\rightarrow b
  20. C ( a ) C ( b ) C(a)\nless C(b)
  21. a b a\nrightarrow b
  22. C ( a ) C ( b ) C(a)\geq C(b)
  23. a a
  24. b b
  25. C ( a ) < C ( b ) C(a)<C(b)
  26. a a
  27. b b
  28. b b
  29. a a
  30. b b

Lanczos_algorithm.html

  1. n t h n^{th}
  2. m m
  3. m m
  4. n n
  5. m m
  6. A A\,
  7. x 0 x_{0}\,
  8. x n + 1 = A x n x_{n+1}=Ax_{n}\,
  9. n n
  10. x n / x n x_{n}/\|x_{n}\|
  11. A = U diag ( σ i ) U A=U\operatorname{diag}(\sigma_{i})U^{\prime}\,
  12. A A\,
  13. A n = U diag ( σ i n ) U A^{n}=U\operatorname{diag}(\sigma_{i}^{n})U^{\prime}
  14. n n\,
  15. x n * x n + 1 / x n * x n x_{n}^{*}x_{n+1}/{x_{n}^{*}x_{n}}\,
  16. x n / x n x_{n}/\|x_{n}\|\,
  17. x n x_{n}\,
  18. A n - 1 v A^{n-1}v
  19. A j v , j = 0 , 1 , , n - 2 A^{j}v,\,j=0,1,\cdots,n-2
  20. n n
  21. A A
  22. m m
  23. A A
  24. T m m T_{mm}
  25. m m
  26. A A
  27. T m m T_{mm}
  28. A A
  29. T m m = V m * A V m . T_{mm}=V_{m}^{*}AV_{m}.
  30. α j = t j j \alpha_{j}=t_{jj}
  31. β j = t j - 1 , j \beta_{j}=t_{j-1,j}
  32. t j - 1 , j = t j , j - 1 t_{j-1,j}=t_{j,j-1}
  33. v 1 v_{1}\leftarrow\,
  34. v 0 0 v_{0}\leftarrow 0\,
  35. β 1 0 \beta_{1}\leftarrow 0\,
  36. j = 1 , 2 , , m - 1 j=1,2,\cdots,m-1\,
  37. w j A v j w_{j}\leftarrow Av_{j}\,
  38. α j w j v j \alpha_{j}\leftarrow w_{j}\cdot v_{j}\,
  39. w j w j - α j v j - β j v j - 1 w_{j}\leftarrow w_{j}-\alpha_{j}v_{j}-\beta_{j}v_{j-1}\,
  40. β j + 1 w j \beta_{j+1}\leftarrow\left\|w_{j}\right\|\,
  41. v j + 1 w j / β j + 1 v_{j+1}\leftarrow w_{j}/\beta_{j+1}\,
  42. w m A v m w_{m}\leftarrow Av_{m}\,
  43. α m w m v m \alpha_{m}\leftarrow w_{m}\cdot v_{m}\,
  44. x y x\cdot y
  45. x x
  46. y y
  47. α j \alpha_{j}
  48. β j \beta_{j}
  49. T m m = ( α 1 β 2 0 β 2 α 2 β 3 β 3 α 3 β m - 1 β m - 1 α m - 1 β m 0 β m α m ) T_{mm}=\begin{pmatrix}\alpha_{1}&\beta_{2}&&&&0\\ \beta_{2}&\alpha_{2}&\beta_{3}&&&\\ &\beta_{3}&\alpha_{3}&\ddots&&\\ &&\ddots&\ddots&\beta_{m-1}&\\ &&&\beta_{m-1}&\alpha_{m-1}&\beta_{m}\\ 0&&&&\beta_{m}&\alpha_{m}\\ \end{pmatrix}
  50. v j v_{j}
  51. V m = ( v 1 , v 2 , , v m ) V_{m}=\left(v_{1},v_{2},\cdots,v_{m}\right)
  52. v 1 v_{1}
  53. T m m T_{mm}
  54. λ i ( m ) \lambda_{i}^{(m)}
  55. u i ( m ) u_{i}^{(m)}
  56. T T
  57. 𝒪 ( m 2 ) \mathcal{O}(m^{2})
  58. 𝒪 ( m 2 ) \mathcal{O}(m^{2})
  59. A A
  60. y i y_{i}
  61. A A
  62. y i = V m u i ( m ) y_{i}=V_{m}u_{i}^{(m)}
  63. V m V_{m}
  64. v 1 , v 2 , , v m v_{1},v_{2},\cdots,v_{m}
  65. v 1 , v 2 , , v m + 1 v_{1},v_{2},\cdots,v_{m+1}
  66. A A\,

Lanczos_resampling.html

  1. L ( x ) L(x)
  2. s i n c ( x ) sinc(x)
  3. s i n c ( x / a ) sinc(x/a)
  4. a x a −a≤x≤a
  5. L ( x ) = { sinc ( x ) sinc ( x / a ) if - a < x < a 0 otherwise L(x)=\begin{cases}\operatorname{sinc}(x)\,\operatorname{sinc}(x/a)&\,\text{if}% \;\;-a<x<a\\ 0&\,\text{otherwise}\end{cases}
  6. L ( x ) = { 1 if x = 0 a sin ( π x ) sin ( π x / a ) π 2 x 2 if 0 < | x | < a 0 otherwise L(x)=\begin{cases}1&\,\text{if}\;\;x=0\\ &\\ \displaystyle\frac{a\sin(\pi x)\sin(\pi x/a)}{\pi^{2}x^{2}}&\,\text{if}\;\;0<|% x|<a\\ &\\ 0&\,\text{otherwise}\end{cases}
  7. a a
  8. 2 a 1 2a−1
  9. a 1 a−1
  10. i i
  11. S ( x ) S(x)
  12. x x
  13. S ( x ) = i = x - a + 1 x + a s i L ( x - i ) , S(x)=\sum_{i=\lfloor x\rfloor-a+1}^{\lfloor x\rfloor+a}s_{i}L(x-i),
  14. a a
  15. x \lfloor x\rfloor
  16. a a
  17. x x
  18. ± a ±a
  19. S ( x ) S(x)
  20. x x
  21. x x
  22. S ( x ) S(x)
  23. x x
  24. i i
  25. L ( x , y ) = L ( x ) L ( y ) . L(x,y)=L(x)\cdot L(y).
  26. ( i , j ) (i,j)
  27. S ( x , y ) = i = x - a + 1 x + a j = y - a + 1 y + a s i j L ( x - i ) L ( y - j ) . S(x,y)=\sum_{i=\lfloor x\rfloor-a+1}^{\lfloor x\rfloor+a}\sum_{j=\lfloor y% \rfloor-a+1}^{\lfloor y\rfloor+a}s_{i\,j}L(x-i)L(y-j).
  28. ( x , y ) (x,y)
  29. 2 a 2a
  30. 2 a 2a
  31. a a
  32. a a
  33. a > 1 a>1
  34. U ( x ) = i L ( x - i ) U(x)=\sum_{i\in\mathbb{Z}}L(x-i)
  35. a a
  36. a a

Lanczos_tensor.html

  1. H a b c + H b a c = 0 , H_{abc}+H_{bac}=0,\,
  2. H a b c + H b c a + H c a b = 0. H_{abc}+H_{bca}+H_{cab}=0.
  3. C a b c d = H a b c ; d + H c d a ; b + H b a d ; c + H d c b ; a + ( H e + ( a c ) ; e H ( a | e | ) e ; c ) g b d + ( H e + ( b d ) ; e H ( b | e | ) e ; d ) g a c - ( H e + ( a d ) ; e H ( a | e | ) e ; d ) g b c - ( H e + ( b c ) ; e H ( b | e | ) e ; c ) g a d - 2 3 H e f ( g a c g b d - g a d g b c ) f ; e C_{abcd}=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a}+(H^{e}{}_{(ac);e}+H_{(a|e|}{}% ^{e}{}_{;c)})g_{bd}+(H^{e}{}_{(bd);e}+H_{(b|e|}{}^{e}{}_{;d)})g_{ac}-(H^{e}{}_% {(ad);e}+H_{(a|e|}{}^{e}{}_{;d)})g_{bc}-(H^{e}{}_{(bc);e}+H_{(b|e|}{}^{e}{}_{;% c)})g_{ad}-\frac{2}{3}H^{ef}{}_{f;e}(g_{ac}g_{bd}-g_{ad}g_{bc})
  4. C a b c d C_{abcd}
  5. Φ a \Phi^{a}
  6. H a b c = H a b c + Φ [ a g b ] c H^{\prime}_{abc}=H_{abc}+\Phi_{[a}g_{b]c}
  7. Φ a = - 2 3 H a b , b \Phi_{a}=-\frac{2}{3}H_{ab}{}^{b},
  8. H a b = b 0. H^{\prime}_{ab}{}^{b}=0.
  9. H a b = c ; c 0 H_{ab}{}^{c}{}_{;c}=0
  10. C a b c d = H a b c ; d + H c d a ; b + H b a d ; c + H d c b ; a + H e g b d a c ; e + H e g a c b d ; e - H e g b c a d ; e - H e g a d b c ; e . C_{abcd}=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a}+H^{e}{}_{ac;e}g_{bd}+H^{e}{}_% {bd;e}g_{ac}-H^{e}{}_{ad;e}g_{bc}-H^{e}{}_{bc;e}g_{ad}.
  11. H a b c = J a b c - 2 R c d H a b d + R a d H b c d + R b d H a c d + ( H d b e g a c - H d a e g b c ) R d e + 1 2 R H a b c , \begin{aligned}\displaystyle\Box H_{abc}=&\displaystyle J_{abc}\\ &\displaystyle{}-2{R_{c}}^{d}H_{abd}+{R_{a}}^{d}H_{bcd}+{R_{b}}^{d}H_{acd}\\ &\displaystyle{}+\left(H_{dbe}g_{ac}-H_{dae}g_{bc}\right)R^{de}+\frac{1}{2}RH_% {abc},\end{aligned}
  12. \Box
  13. J a b c = R c a ; b - R c b ; a - 1 6 ( g c a R ; b - g c b R ; a ) J_{abc}=R_{ca;b}-R_{cb;a}-\frac{1}{6}\left(g_{ca}R_{;b}-g_{cb}R_{;a}\right)
  14. H a b c = 0 , \Box H_{abc}=0,
  15. A a = 0 \Box A_{a}=0
  16. g a b = η a b + h a b g_{ab}=\eta_{ab}+h_{ab}
  17. 4 H a b c h a c , b - h b c , a - 1 6 ( η a c h d d , b - η b c h d d , a ) . 4H_{abc}\approx h_{ac,b}-h_{bc,a}-\frac{1}{6}(\eta_{ac}{h^{d}}_{d,b}-\eta_{bc}% {h^{d}}_{d,a}).
  18. H t r t = G M r 2 H_{trt}=\frac{GM}{r^{2}}
  19. H t r t = 2 G M 3 r 2 H_{trt}=\frac{2GM}{3r^{2}}
  20. H r θ θ = - G M 3 ( 1 - 2 G M / r ) H_{r\theta\theta}=\frac{-GM}{3(1-2GM/r)}
  21. H r ϕ ϕ = - G M sin 2 θ 3 ( 1 - 2 G M / r ) H_{r\phi\phi}=\frac{-GM\sin^{2}\theta}{3(1-2GM/r)}

Langmuir_(unit).html

  1. Φ = J N d t \Phi=\int{J_{N}}\,{\rm d}t
  2. J N = C u ¯ 4 J_{N}=\frac{C\bar{u}}{4}
  3. C = N V = p k T C=\frac{N}{V}=\frac{p}{kT}
  4. u ¯ = 8 k T π m \bar{u}=\sqrt{\frac{8kT}{\pi m}}
  5. J N = C u ¯ 4 = p 1 2 π k T m J_{N}=\frac{C\bar{u}}{4}=p\sqrt{\frac{1}{2\pi kTm}}

Langmuir–Blodgett_film.html

  1. Π \Pi
  2. Π \Pi
  3. Π = γ 0 - γ \Pi=\gamma_{0}-\gamma
  4. γ 0 \gamma_{0}
  5. γ \gamma
  6. γ 0 - γ \gamma_{0}-\gamma
  7. Γ \Gamma
  8. Π = R T Γ \Pi=RT\Gamma
  9. Π A = R T . \Pi A=RT.
  10. Π = - Δ γ = - [ Δ F 2 ( t p + w p ) ] - Δ F 2 w p \Pi=-\Delta\gamma=-\left[\frac{\Delta F}{2(t_{p}+w_{p})}\right]\approx-\frac{% \Delta F}{2w_{p}}
  11. w p t p w_{p}\gg t_{p}
  12. l p , w p l_{p},w_{p}
  13. t p t_{p}
  14. Δ F \Delta F
  15. Π \Pi

Laning_and_Zierler_system.html

  1. cos x \cos x
  2. x = 0 , 0.1 , , 1 x=0,0.1,...,1
  3. cos x \cos x

Large_set_(combinatorics).html

  1. S = { s 0 , s 1 , s 2 , s 3 , } S=\{s_{0},s_{1},s_{2},s_{3},\dots\}
  2. 1 s 0 + 1 s 1 + 1 s 2 + 1 s 3 + \frac{1}{s_{0}}+\frac{1}{s_{1}}+\frac{1}{s_{2}}+\frac{1}{s_{3}}+\cdots
  3. { 1 , 2 , 3 , 4 , 5 , } \{1,2,3,4,5,\dots\}
  4. { 1 , , 6 , 8 , , 16 , 18 , , 66 , 68 , 69 , 80 , } \{1,\dots,6,8,\dots,16,18,\dots,66,68,69,80,\dots\}
  5. S = { s 1 , s 2 , s 3 , } S=\{s_{1},s_{2},s_{3},\dots\}
  6. { 1 , x s 1 , x s 2 , x s 3 , } \{1,x^{s_{1}},x^{s_{2}},x^{s_{3}},\dots\}\,

Large_set_(Ramsey_theory).html

  1. S = { s 1 , s 2 , s 3 , } S=\{s_{1},s_{2},s_{3},\dots\}
  2. S = p ( ) S=p(\mathbb{N})\cap\mathbb{N}
  3. p p
  4. p ( 0 ) = 0 p(0)=0
  5. S S
  6. k = { k , 2 k , 3 k , } k\cdot\mathbb{N}=\{k,2k,3k,\dots\}
  7. k S k\cdot S
  8. S S
  9. m m
  10. S { x : x 0 ( mod m ) } S\cap\{x:x\equiv 0\;\;(\mathop{{\rm mod}}m)\}

Larmor_formula.html

  1. P = q 2 a 2 6 π ε 0 c 3 (SI units) P=\frac{q^{2}a^{2}}{6\pi\varepsilon_{0}c^{3}}\mbox{ (SI units)}~{}
  2. P = 2 3 q 2 a 2 c 3 (cgs units) P={2\over 3}\frac{q^{2}a^{2}}{c^{3}}\mbox{ (cgs units)}~{}
  3. a a
  4. q q
  5. c c
  6. P = 2 3 m e r e a 2 c P={2\over 3}\frac{m_{e}r_{e}a^{2}}{c}
  7. 𝐄 ( 𝐫 , t ) = q ( 𝐧 - s y m b o l β γ 2 ( 1 - s y m b o l β 𝐧 ) 3 R 2 ) ret + q c ( 𝐧 × [ ( 𝐧 - s y m b o l β ) × s y m b o l β ˙ ] ( 1 - s y m b o l β 𝐧 ) 3 R ) ret \mathbf{E}(\mathbf{r},t)=q\left(\frac{\mathbf{n}-symbol{\beta}}{\gamma^{2}(1-% symbol{\beta}\cdot\mathbf{n})^{3}R^{2}}\right)_{\rm{ret}}+\frac{q}{c}\left(% \frac{\mathbf{n}\times[(\mathbf{n}-symbol{\beta})\times\dot{symbol{\beta}}]}{(% 1-symbol{\beta}\cdot\mathbf{n})^{3}R}\right)_{\rm{ret}}
  8. 𝐁 = 𝐧 × 𝐄 , \mathbf{B}=\mathbf{n}\times\mathbf{E},
  9. s y m b o l β symbol{\beta}
  10. c c
  11. s y m b o l β ˙ \dot{symbol{\beta}}
  12. 𝐧 \mathbf{n}
  13. 𝐫 - 𝐫 0 \mathbf{r}-\mathbf{r}_{0}
  14. R R
  15. 𝐫 - 𝐫 0 \mathbf{r}-\mathbf{r}_{0}
  16. 𝐫 0 \mathbf{r}_{0}
  17. t r = t - R / c t\text{r}=t-R/c
  18. s y m b o l β symbol{\beta}
  19. s y m b o l β symbol{\beta}
  20. s y m b o l β ˙ \dot{symbol{\beta}}
  21. 1 / R 2 1/R^{2}
  22. 1 / R 1/R
  23. 𝐒 = c 4 π 𝐄 a × 𝐁 a , \mathbf{S}=\frac{c}{4\pi}\mathbf{E}\text{a}\times\mathbf{B}\text{a},
  24. t r t\text{r}
  25. 𝐒 = q 2 4 π c | 𝐧 × ( 𝐧 × s y m b o l β ˙ ) R | 2 . \mathbf{S}=\frac{q^{2}}{4\pi c}\left|\frac{\mathbf{n}\times(\mathbf{n}\times% \dot{symbol{\beta}})}{R}\right|^{2}.
  26. θ \theta
  27. 𝐚 = s y m b o l β ˙ c \mathbf{a}=\dot{symbol{\beta}}c
  28. d P d Ω = q 2 4 π c sin 2 ( θ ) a 2 c 2 . \frac{dP}{d\Omega}=\frac{q^{2}}{4\pi c}\frac{\sin^{2}(\theta)\,a^{2}}{c^{2}}.
  29. θ \theta
  30. ϕ \phi
  31. P = 2 3 q 2 a 2 c 3 , P=\frac{2}{3}\frac{q^{2}a^{2}}{c^{3}},
  32. E r E_{r}
  33. R R
  34. ( E t = 0 ) (E_{t}=0)
  35. E r E_{r}
  36. E t E_{t}
  37. 1 / R 1/R
  38. 1 / R 2 1/R^{2}
  39. 1 / R 2 1/R^{2}
  40. 1 / R 4 1/R^{4}
  41. E t = e a sin ( θ ) 4 π ε 0 c 2 R . E_{t}={{ea\sin(\theta)}\over{4\pi\varepsilon_{0}c^{2}R}}.
  42. R R
  43. E t E_{t}
  44. 𝐒 = < m t p l > E t 2 μ 0 c 𝐫 ^ = e 2 a 2 sin 2 ( θ ) 16 π 2 ε 0 c 3 R 2 𝐫 ^ \mathbf{S}=<mtpl>{{E_{t}^{2}\over\mu_{0}c}}\mathbf{\hat{r}}={{e^{2}a^{2}\sin^{% 2}(\theta)}\over{16\pi^{2}\varepsilon_{0}c^{3}R^{2}}}\mathbf{\hat{r}}
  45. P = e 2 a 2 6 π ε 0 c 3 . P={{e^{2}a^{2}}\over{6\pi\varepsilon_{0}c^{3}}}.
  46. P = μ 0 e 2 a 2 6 π c . P={{\mu_{0}e^{2}a^{2}}\over{6\pi c}}.
  47. 𝐩 \mathbf{p}
  48. P = 2 3 q 2 m 2 c 3 | 𝐩 ˙ | 2 . P=\frac{2}{3}\frac{q^{2}}{m^{2}c^{3}}|\dot{\mathbf{p}}|^{2}.
  49. P P
  50. P P
  51. | 𝐩 ˙ | 2 |\dot{\mathbf{p}}|^{2}
  52. d p μ d τ d p μ d τ = β 2 ( d p d τ ) 2 - ( d 𝐩 d τ ) 2 , \frac{dp_{\mu}}{d\tau}\frac{dp^{\mu}}{d\tau}=\beta^{2}\left(\frac{dp}{d\tau}% \right)^{2}-\left(\frac{d{\mathbf{p}}}{d\tau}\right)^{2},
  53. β 1 β≪1
  54. - | 𝐩 ˙ | 2 -|\dot{\mathbf{p}}|^{2}
  55. β \mathbf{β}
  56. γ 6 \gamma^{6}
  57. γ = 1 / 1 - β 2 \gamma=1/\sqrt{1-\beta^{2}}
  58. β 1 \beta\ll 1
  59. β 1 \beta\rightarrow 1
  60. γ 6 \gamma^{6}
  61. 1 - β 2 = 1 / γ 2 1-\beta^{2}=1/\gamma^{2}
  62. γ 6 \gamma^{6}
  63. γ 4 \gamma^{4}
  64. d P d Ω = q 2 4 π c | 𝐧 ^ × [ ( 𝐧 ^ - s y m b o l β ) × s y m b o l β ˙ ] | 2 ( 1 - 𝐧 ^ \cdotsymbol β ) 5 , \frac{dP}{d\Omega}=\frac{q^{2}}{4\pi c}\frac{|\mathbf{\hat{n}}\times[(\mathbf{% \hat{n}}-symbol{\beta})\times\dot{symbol{\beta}}]|^{2}}{(1-\mathbf{\hat{n}}% \cdotsymbol{\beta})^{5}},
  65. 𝐧 ^ \mathbf{\hat{n}}
  66. d P d Ω = q 2 a 2 4 π c 3 sin 2 θ ( 1 - β cos θ ) 5 , \frac{dP}{d\Omega}=\frac{q^{2}a^{2}}{4\pi c^{3}}\frac{\sin^{2}\theta}{(1-\beta% \cos\theta)^{5}},
  67. θ \theta
  68. β ( t r ) 0 \beta\left(t\text{r}\right)\neq 0

Larmor_precession.html

  1. Γ = μ × B = γ J × B \vec{\Gamma}=\vec{\mu}\times\vec{B}=\gamma\vec{J}\times\vec{B}
  2. Γ \vec{\Gamma}
  3. μ \vec{\mu}
  4. J \vec{J}
  5. B \vec{B}
  6. × \times
  7. γ \ \gamma
  8. J \vec{J}
  9. ω = - γ B \omega=-\gamma B
  10. ω \omega
  11. B B
  12. γ \gamma
  13. - e -e
  14. - e g 2 m -\frac{eg}{2m}
  15. m m
  16. g g
  17. ω s = g e B 2 m c + ( 1 - γ ) e B m c γ = e B 2 m c ( g - 2 + 2 γ ) \omega_{s}=\frac{geB}{2mc}+(1-\gamma)\frac{eB}{mc\gamma}=\frac{eB}{2mc}\left(g% -2+\frac{2}{\gamma}\right)
  18. γ \gamma
  19. ω s ( g = 2 ) = e B m c γ \omega_{s(g=2)}=\frac{eB}{mc\gamma}
  20. d a τ d s = e m u τ u σ F σ λ a λ + 2 μ ( F τ λ - u τ u σ F σ λ ) a λ , \frac{da^{\tau}}{ds}=\frac{e}{m}u^{\tau}u_{\sigma}F^{\sigma\lambda}a_{\lambda}% +2\mu(F^{\tau\lambda}-u^{\tau}u_{\sigma}F^{\sigma\lambda})a_{\lambda},
  21. a τ a^{\tau}
  22. e e
  23. m m
  24. μ \mu
  25. u τ u^{\tau}
  26. a τ a τ = - u τ u τ = - 1 a^{\tau}a_{\tau}=-u^{\tau}u_{\tau}=-1
  27. u τ a τ = 0 u^{\tau}a_{\tau}=0
  28. F τ σ F^{\tau\sigma}
  29. m d u τ d s = e F τ σ u σ , m\frac{du^{\tau}}{ds}=eF^{\tau\sigma}u_{\sigma},
  30. ( - u τ w λ + u λ w τ ) a λ (-u^{\tau}w^{\lambda}+u^{\lambda}w^{\tau})a_{\lambda}
  31. w τ = d u τ / d s w^{\tau}=du^{\tau}/ds
  32. ( s y m b o l μ s y m b o l B ) \nabla({symbol\mu}\cdot{symbolB})
  33. d u α d τ = e m F α β u β . {du^{\alpha}\over d\tau}={e\over m}F^{\alpha\beta}u_{\beta}\;.
  34. d S α d τ = e m [ g 2 F α β S β + ( g 2 - 1 ) u α ( S λ F λ μ U μ ) ] , {\;\,dS^{\alpha}\over d\tau}={e\over m}\bigg[{g\over 2}F^{\alpha\beta}S_{\beta% }+\left({g\over 2}-1\right)u^{\alpha}\left(S_{\lambda}F^{\lambda\mu}U_{\mu}% \right)\bigg]\;,

Law_of_comparative_judgment.html

  1. S i - S j = x i j σ i 2 + σ j 2 - 2 r i j σ i σ j , S_{i}-S_{j}=x_{ij}\sqrt{\sigma_{i}^{2}+\sigma_{j}^{2}-2r_{ij}\sigma_{i}\sigma_% {j}},
  2. S i S_{i}
  3. x i j x_{ij}
  4. σ i \sigma_{i}
  5. R i R_{i}
  6. r i j r_{ij}
  7. R i R_{i}
  8. S i S_{i}
  9. x i j = S i - S j σ x_{ij}=\frac{S_{i}-S_{j}}{\sigma}\,
  10. σ = σ i 2 + σ j 2 . {\sigma}=\sqrt{\sigma_{i}^{2}+\sigma_{j}^{2}}.\,
  11. S i - S j {S_{i}-S_{j}}
  12. x i j x_{ij}
  13. σ = 1 \sigma=1
  14. P i j P_{ij}
  15. P i j = 0.84 P_{ij}=0.84
  16. x i j x_{ij}
  17. S i - S j 1 S_{i}-S_{j}\cong 1

Lawrence_C._Evans.html

  1. C 2 , α C^{2,\alpha}

Lazarus_Fuchs.html

  1. p ( x ) y ′′ + q ( x ) y + r ( x ) y = 0 p(x)y^{\prime\prime}+q(x)y^{\prime}+r(x)y=0\,
  2. y = n = 0 a n ( x - x 0 ) n + σ , a 0 0 y=\sum_{n=0}^{\infty}a_{n}(x-x_{0})^{n+\sigma},\quad a_{0}\neq 0\,

Lazy_caterer's_sequence.html

  1. p = n 2 + n + 2 2 . p=\frac{n^{2}+n+2}{2}.
  2. p = ( n + 1 2 ) + 1 = ( n 2 ) + ( n 1 ) + ( n 0 ) . p={{\textstyle\left({{n+1}\atop{2}}\right)}}+1={{\textstyle\left({{n}\atop{2}}% \right)}}+{{\textstyle\left({{n}\atop{1}}\right)}}+{{\textstyle\left({{n}\atop% {0}}\right)}}.
  3. n = 0 n=0
  4. f ( n ) = n + f ( n - 1 ) . f(n)=n+f(n-1).\,
  5. f ( n ) = n + ( n - 1 ) + f ( n - 2 ) . f(n)=n+(n-1)+f(n-2).\,
  6. f ( n ) = n + ( n - 1 ) + ( n - 2 ) + + 1 + f ( 0 ) . f(n)=n+(n-1)+(n-2)+\cdots+1+f(0).\,
  7. f ( 0 ) = 1 f(0)=1
  8. f ( n ) = 1 + ( 1 + 2 + 3 + + n ) . f(n)=1+(1+2+3+\cdots+n).\,
  9. f ( n ) = 1 + n ( n + 1 ) 2 = n 2 + n + 2 2 . f(n)=1+\frac{n(n+1)}{2}=\frac{n^{2}+n+2}{2}.

Le_Cam's_theorem.html

  1. λ n = p 1 + + p n . \lambda_{n}=p_{1}+\cdots+p_{n}.\,
  2. S n = X 1 + + X n . S_{n}=X_{1}+\cdots+X_{n}.\,
  3. S n S_{n}
  4. k = 0 | Pr ( S n = k ) - λ n k e - λ n k ! | < 2 i = 1 n p i 2 . \sum_{k=0}^{\infty}\left|\Pr(S_{n}=k)-{\lambda_{n}^{k}e^{-\lambda_{n}}\over k!% }\right|<2\sum_{i=1}^{n}p_{i}^{2}.

Leadscrew.html

  1. π d m \pi d_{m}
  2. T r a i s e = F d m 2 ( l + π μ d m π d m - μ l ) = F d m 2 tan ( ϕ + λ ) T_{raise}=\frac{Fd_{m}}{2}\left(\frac{l+\pi\mu d_{m}}{\pi d_{m}-\mu l}\right)=% \frac{Fd_{m}}{2}\tan{\left(\phi+\lambda\right)}
  3. T l o w e r = F d m 2 ( π μ d m - l π d m + μ l ) = F d m 2 tan ( ϕ - λ ) T_{lower}=\frac{Fd_{m}}{2}\left(\frac{\pi\mu d_{m}-l}{\pi d_{m}+\mu l}\right)=% \frac{Fd_{m}}{2}\tan{\left(\phi-\lambda\right)}
  4. μ \mu\,
  5. ϕ \phi\,
  6. λ \lambda\,
  7. ϕ > λ \phi>\lambda
  8. efficiency = T 0 T r a i s e = F l 2 π T r a i s e = tan λ tan ( ϕ + λ ) \mbox{efficiency}~{}=\frac{T_{0}}{T_{raise}}=\frac{Fl}{2\pi T_{raise}}=\frac{% \tan{\lambda}}{\tan{\left(\phi+\lambda\right)}}
  9. T r a i s e = F d m 2 ( l + π μ d m sec α π d m - μ l sec α ) = F d m 2 ( μ sec α + tan λ 1 - μ sec α tan λ ) T_{raise}=\frac{Fd_{m}}{2}\left(\frac{l+\pi\mu d_{m}\sec{\alpha}}{\pi d_{m}-% \mu l\sec{\alpha}}\right)=\frac{Fd_{m}}{2}\left(\frac{\mu\sec{\alpha}+\tan{% \lambda}}{1-\mu\sec{\alpha}\tan{\lambda}}\right)
  10. T l o w e r = F d m 2 ( π μ d m sec α - l π d m + μ l sec α ) = F d m 2 ( μ sec α - tan λ 1 + μ sec α tan λ ) T_{lower}=\frac{Fd_{m}}{2}\left(\frac{\pi\mu d_{m}\sec{\alpha}-l}{\pi d_{m}+% \mu l\sec{\alpha}}\right)=\frac{Fd_{m}}{2}\left(\frac{\mu\sec{\alpha}-\tan{% \lambda}}{1+\mu\sec{\alpha}\tan{\lambda}}\right)
  11. α \alpha\,
  12. T c = F μ c d c 2 T_{c}=\frac{F\mu_{c}d_{c}}{2}
  13. μ c \mu_{c}
  14. μ c \mu_{c}
  15. μ c \mu_{c}
  16. N = ( 4.76 × 10 6 ) d r C L 2 N={(4.76\times 10^{6})d_{r}C\over L^{2}}

Leaf_area_index.html

  1. P = P max ( 1 - e - c L A I ) P=P_{\max}\left(1-e^{-c\cdot LAI}\right)
  2. c c

Least_slack_time_scheduling.html

  1. ( d - t ) - c (d-t)-c^{\prime}
  2. d d
  3. t t
  4. c c^{\prime}

Lebesgue's_lemma.html

  1. P P
  2. U U
  3. v - P v ( 1 + P ) inf u U v - u . \|v-Pv\|\leq(1+\|P\|)\inf_{u\in U}\|v-u\|.

Lebesgue_constant_(interpolation).html

  1. n n
  2. n + 1 n+1
  3. T T
  4. n n
  5. f - X ( f ) ( Λ n ( T ) + 1 ) f - p * . \|f-X(f)\|\leq(\Lambda_{n}(T)+1)\left\|f-p^{*}\right\|.
  6. n n
  7. [ u ! ! ] p f [ u ! ! ] [u^{\prime}!!^{\prime}]p−f[u^{\prime}!!^{\prime}]
  8. f - X ( f ) f - p * + p * - X ( f ) \|f-X(f)\|\leq\|f-p^{*}\|+\|p^{*}-X(f)\|
  9. l j ( x ) := i = 0 j i n x - x i x j - x i l_{j}(x):=\prod_{\begin{smallmatrix}i=0\\ j\neq i\end{smallmatrix}}^{n}\frac{x-x_{i}}{x_{j}-x_{i}}
  10. λ n ( x ) = j = 0 n | l j ( x ) | . \lambda_{n}(x)=\sum_{j=0}^{n}|l_{j}(x)|.
  11. Λ n ( T ) = max x [ a , b ] λ n ( x ) \Lambda_{n}(T)=\max_{x\in[a,b]}\lambda_{n}(x)
  12. Λ n ( T ) 2 n + 1 e n log n as n . \Lambda_{n}(T)\sim\frac{2^{n+1}}{e\,n\log n}\qquad\,\text{ as }n\to\infty.
  13. 2 π log ( n + 1 ) + a < Λ n ( T ) < 2 π log ( n + 1 ) + 1 , a = 0.9625 \tfrac{2}{\pi}\log(n+1)+a<\Lambda_{n}(T)<\tfrac{2}{\pi}\log(n+1)+1,\qquad a=0.% 9625\cdots
  14. i i
  15. s i = t i cos ( π 2 ( n + 1 ) ) . s_{i}=\frac{t_{i}}{\cos\left(\frac{\pi}{2(n+1)}\right)}.
  16. Λ n ( S ) < 2 π log ( n + 1 ) + b , b = 0.7219 \Lambda_{n}(S)<\tfrac{2}{\pi}\log(n+1)+b,\qquad b=0.7219\cdots
  17. C M n [ - 1 , 1 ] C_{M}^{n}[-1,1]
  18. n n
  19. n n
  20. M M
  21. 1 , 11 −1,11
  22. n n
  23. n n
  24. ( a , 0 , a ) (−a,0,a)
  25. 8 3 a 1 \frac{√8}{3}≤a≤1
  26. ( 1 , b , 1 ) (−1,b,1)
  27. n n
  28. p ( t i ) p(t_{i})
  29. p ^ ( x ) \hat{p}(x)
  30. u ^ \hat{u}
  31. p - p ^ p Λ n ( T ) u - u ^ u \frac{\|p-\hat{p}\|}{\|p\|}\leq\Lambda_{n}(T)\frac{\|u-\hat{u}\|}{\|u\|}
  32. p ^ ( x ) \hat{p}(x)

Lefschetz_pencil.html

  1. λ L + μ L = 0. \lambda L+\mu L^{\prime}=0.
  2. V P 1 V\rightarrow P^{1}

Lefschetz_zeta_function.html

  1. ζ f ( z ) = exp ( n = 1 L ( f n ) z n n ) , \zeta_{f}(z)=\exp\left(\sum_{n=1}^{\infty}L(f^{n})\frac{z^{n}}{n}\right),
  2. 1 ( 1 - t ) χ ( X ) , \frac{1}{(1-t)^{\chi(X)}},
  3. χ ( X ) \chi(X)
  4. ζ f ( t ) = exp ( n = 1 2 t 2 n + 1 2 n + 1 ) = exp ( { 2 n = 1 t n n } - { 2 n = 1 t 2 n 2 n } ) = exp ( - 2 log ( 1 - t ) + log ( 1 - t 2 ) ) = 1 - t 2 ( 1 - t ) 2 = 1 + t 1 - t \begin{aligned}\displaystyle\zeta_{f}(t)&\displaystyle=\exp\left(\sum_{n=1}^{% \infty}\frac{2t^{2n+1}}{2n+1}\right)\\ &\displaystyle=\exp\left(\left\{2\sum_{n=1}^{\infty}\frac{t^{n}}{n}\right\}-% \left\{2\sum_{n=1}^{\infty}\frac{t^{2n}}{2n}\right\}\right)\\ &\displaystyle=\exp\left(-2\log(1-t)+\log(1-t^{2})\right)\\ &\displaystyle=\frac{1-t^{2}}{(1-t)^{2}}\\ &\displaystyle=\frac{1+t}{1-t}\end{aligned}
  5. ζ f ( t ) = i = 0 n det ( 1 - t f | H i ( X , 𝐐 ) ) ( - 1 ) i + 1 . \zeta_{f}(t)=\prod_{i=0}^{n}\det(1-tf_{\ast}|H_{i}(X,\mathbf{Q}))^{(-1)^{i+1}}.

Left-right_planarity_test.html

  1. α \alpha
  2. β \beta

Legendre's_conjecture.html

  1. p \sqrt{p}
  2. O ( p ) O(\sqrt{p})
  3. ( log p ) 2 (\log p)^{2}
  4. O ( p log p ) O(\sqrt{p}\log p)
  5. [ x , x + O ( x 21 / 40 ) ] [x,\,x+O(x^{21/40})]
  6. x x

Lehmer_number.html

  1. a + b = R a+b=\sqrt{R}
  2. a b = Q ab=Q
  3. a / b a/b
  4. U n ( R , Q ) = a n - b n a - b U_{n}(\sqrt{R},Q)=\frac{a^{n}-b^{n}}{a-b}
  5. U n ( R , Q ) = a n - b n a 2 - b 2 U_{n}(\sqrt{R},Q)=\frac{a^{n}-b^{n}}{a^{2}-b^{2}}
  6. V n ( R , Q ) = a n + b n a + b V_{n}(\sqrt{R},Q)=\frac{a^{n}+b^{n}}{a+b}
  7. V n ( R , Q ) = a n + b n V_{n}(\sqrt{R},Q)=a^{n}+b^{n}
  8. U n = ( R - 2 Q ) U n - 2 - Q 2 U n - 4 = ( a 2 + b 2 ) U n - 2 - a 2 b 2 U n - 4 U_{n}=(R-2Q)U_{n-2}-Q^{2}U_{n-4}=(a^{2}+b^{2})U_{n-2}-a^{2}b^{2}U_{n-4}
  9. U 0 = 0 , U 1 = 1 , U 2 = 1 , U 3 = R - Q = a 2 + a b + b 2 U_{0}=0,U_{1}=1,U_{2}=1,U_{3}=R-Q=a^{2}+ab+b^{2}
  10. V n = ( R - 2 Q ) V n - 2 - Q 2 V n - 4 = ( a 2 + b 2 ) V n - 2 - a 2 b 2 V n - 4 V_{n}=(R-2Q)V_{n-2}-Q^{2}V_{n-4}=(a^{2}+b^{2})V_{n-2}-a^{2}b^{2}V_{n-4}
  11. V 0 = 2 , V 1 = 1 , V 2 = R - 2 Q = a 2 + b 2 , V 3 = R - 3 Q = a 2 - a b + b 2 V_{0}=2,V_{1}=1,V_{2}=R-2Q=a^{2}+b^{2},V_{3}=R-3Q=a^{2}-ab+b^{2}

Lehmer–Schur_algorithm.html

  1. p ( z ) p(z)
  2. D ( c , ρ ) D(c,\rho)
  3. p ( c + ρ z ) . p(c+\rho z).
  4. ρ | z | 2 ρ \rho\leq|z|\leq 2\rho
  5. c k = 5 3 ρ e i k π 4 c_{k}=\frac{5}{3}\rho e^{i\frac{k\pi}{4}}
  6. k = 0 , 1 , , 7 k=0,1,\dots,7
  7. ρ \rho
  8. 5 6 ρ \frac{5}{6}\rho
  9. p ( z ) = a n z n + + a 1 z + a 0 . p(z)=a_{n}z^{n}+\dots+a_{1}z+a_{0}.
  10. p * ( z ) = z n p ¯ ( z - 1 ) = z n p ¯ ( z ¯ - 1 ) p^{*}(z)=z^{n}\bar{p}(z^{-1})=z^{n}\overline{p}(\bar{z}^{-1})
  11. T p Tp
  12. p p
  13. ( T p ) ( z ) = a ¯ 0 p ( z ) - a n p * ( z ) . (Tp)(z)=\bar{a}_{0}p(z)-a_{n}p^{*}(z).
  14. deg T p < deg p \deg Tp<\deg p
  15. T p Tp
  16. δ 1 = ( T p ) ( 0 ) = | a 0 | 2 - | a n | 2 . \delta_{1}=(Tp)(0)=|a_{0}|^{2}-|a_{n}|^{2}.
  17. δ 1 > 0 \delta_{1}>0
  18. p p
  19. T p Tp
  20. δ 1 < 0 \delta_{1}<0
  21. p p
  22. T p Tp
  23. p * p^{*}
  24. T k + 1 p = T ( T k p ) T^{k+1}p=T(T^{k}p)
  25. T K + 1 p = 0 T^{K+1}p=0
  26. d k = deg T k p d_{k}=\deg T^{k}p
  27. d k + 1 < d k d_{k+1}<d_{k}
  28. δ k = ( T k p ) ( 0 ) \delta_{k}=(T^{k}p)(0)
  29. δ k > 0 \delta_{k}>0
  30. p p
  31. δ k ¯ < 0 \delta_{\bar{k}}<0
  32. δ k > 0 \delta_{k}>0
  33. k k ¯ k\neq\bar{k}
  34. d k ¯ d_{\bar{k}}
  35. T k ¯ p , , T K p T^{\bar{k}}p,\dots,T^{K}p
  36. T k ¯ p T^{\bar{k}}p
  37. d k ¯ = deg T k ¯ p d_{\bar{k}}=\deg T^{\bar{k}}p
  38. T k ¯ - 1 p T^{\bar{k}-1}p
  39. T k ¯ - 2 p , , T p , p T^{\bar{k}-2}p,\dots,Tp,p

Leibniz_integral_rule.html

  1. y 0 y 1 f ( x , y ) d y \int_{y_{0}}^{y_{1}}f(x,y)\,\mathrm{d}y
  2. d d x ( y 0 y 1 f ( x , y ) d y ) = y 0 y 1 f x ( x , y ) d y {\mathrm{d}\over\mathrm{d}x}\left(\int_{y_{0}}^{y_{1}}f(x,y)\,\mathrm{d}y% \right)=\int_{y_{0}}^{y_{1}}f_{x}(x,y)\,\mathrm{d}y
  3. d d θ ( a ( θ ) b ( θ ) f ( x , θ ) d x ) = a ( θ ) b ( θ ) θ f ( x , θ ) d x + f ( b ( θ ) , θ ) b ( θ ) - f ( a ( θ ) , θ ) a ( θ ) \frac{\mathrm{d}}{\mathrm{d}\theta}\left(\int_{a(\theta)}^{b(\theta)}f(x,% \theta)\,\mathrm{d}x\right)=\int_{a(\theta)}^{b(\theta)}\partial_{\theta}f(x,% \theta)\,\mathrm{d}x\,+\,f\big(b(\theta),\theta\big)\cdot b^{\prime}(\theta)\,% -\,f\big(a(\theta),\theta\big)\cdot a^{\prime}(\theta)
  4. d d t Σ ( t ) 𝐅 ( 𝐫 , t ) d 𝐀 = Σ ( t ) ( 𝐅 t ( 𝐫 , t ) + [ 𝐅 ( 𝐫 , t ) ] 𝐯 ) d 𝐀 - Σ ( t ) [ 𝐯 × 𝐅 ( 𝐫 , t ) ] d 𝐬 \frac{\mathrm{d}}{\mathrm{d}t}\iint_{\Sigma(t)}\mathbf{F}(\mathbf{r},t)\cdot% \mathrm{d}\mathbf{A}=\iint_{\Sigma(t)}\left(\mathbf{F}_{t}(\mathbf{r},t)+\left% [\mathrm{\nabla}\cdot\mathbf{F}(\mathbf{r},t)\right]\mathbf{v}\right)\cdot% \mathrm{d}\mathbf{A}\,-\,\oint_{\partial\Sigma(t)}\left[\mathbf{v}\times% \mathbf{F}(\mathbf{r},t)\right]\cdot\mathrm{d}\mathbf{s}
  5. X X
  6. \mathbb{R}
  7. Ω \Omega
  8. f : X × Ω f:X\times\Omega\rightarrow\mathbb{R}
  9. f ( x , ω ) f(x,\omega)
  10. ω \omega
  11. x X x\in X
  12. ω Ω \omega\in\Omega
  13. f x f_{x}
  14. x X x\in X
  15. θ : Ω \theta:\Omega\rightarrow\mathbb{R}
  16. | f x ( x , ω ) | θ ( ω ) |f_{x}(x,\omega)|\leq\theta(\omega)
  17. x X x\in X
  18. ω Ω \omega\in\Omega
  19. x X x\in X
  20. d d x Ω f ( x , ω ) d ω = Ω f x ( x , ω ) d ω \frac{\mathrm{d}}{\mathrm{d}x}\int_{\Omega}\,f(x,\omega)\mathrm{d}\omega=\int_% {\Omega}\,f_{x}(x,\omega)\mathrm{d}\omega
  21. u ( x ) \displaystyle u^{\prime}(x)
  22. u ( x ) = y 0 y 1 f x ( x , y ) d y u^{\prime}(x)=\int_{y_{0}}^{y_{1}}f_{x}(x,y)\,\mathrm{d}y
  23. f n ( y ) = f ( x + 1 n , y ) - f ( x , y ) 1 n . f_{n}(y)=\frac{f(x+\tfrac{1}{n},y)-f(x,y)}{\tfrac{1}{n}}.
  24. f x ( z , y ) = f ( x + 1 n , y ) - f ( x , y ) 1 n . f_{x}(z,y)=\frac{f(x+\tfrac{1}{n},y)-f(x,y)}{\tfrac{1}{n}}.
  25. d d x ( f 1 ( x ) f 2 ( x ) g ( t ) d t ) = g [ f 2 ( x ) ] f 2 ( x ) - g [ f 1 ( x ) ] f 1 ( x ) {\mathrm{d}\over\mathrm{d}x}\left(\int_{f_{1}(x)}^{f_{2}(x)}g(t)\,\mathrm{d}t% \right)=g[f_{2}(x)]{f_{2}^{\prime}(x)}-g[f_{1}(x)]{f_{1}^{\prime}(x)}
  26. φ ( α ) = a b f ( x , α ) d x , \varphi(\alpha)=\int_{a}^{b}f(x,\alpha)\,\mathrm{d}x,
  27. Δ φ \displaystyle\Delta\varphi
  28. a b f ( x ) d x = ( b - a ) f ( ξ ) \int_{a}^{b}f(x)\,\mathrm{d}x=(b-a)f(\xi)
  29. lim Δ α 0 a b f ( x , α + Δ α ) - f ( x , α ) Δ α d x = a b f α ( x , α ) d x \lim_{\Delta\alpha\to 0}\int_{a}^{b}\frac{f(x,\alpha+\Delta\alpha)-f(x,\alpha)% }{\Delta\alpha}\,\mathrm{d}x=\int_{a}^{b}f_{\alpha}(x,\alpha)\,\mathrm{d}x
  30. d φ d α = a b f α ( x , α ) d x + f ( b , α ) d b d α - f ( a , α ) d a d α \frac{\mathrm{d}\varphi}{\mathrm{d}\alpha}=\int_{a}^{b}f_{\alpha}(x,\alpha)\,% \mathrm{d}x+f(b,\alpha)\frac{\mathrm{d}b}{\mathrm{d}\alpha}-f(a,\alpha)\frac{% \mathrm{d}a}{\mathrm{d}\alpha}
  31. d d t ( Σ ( t ) d 𝐀 𝐫 𝐅 ( 𝐫 , t ) ) = Σ d 𝐀 𝐈 d d t 𝐅 ( 𝐂 ( t ) + 𝐈 , t ) \frac{\mathrm{d}}{\mathrm{d}t}\left(\iint_{\Sigma(t)}\mathrm{d}\mathbf{A}_{% \mathbf{r}}\cdot\mathbf{F}(\mathbf{r},t)\right)=\iint_{\Sigma}\mathrm{d}% \mathbf{A}_{\mathbf{I}}\cdot\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{F}(\mathbf{C% }(t)+\mathbf{I},t)
  32. d d t 𝐅 ( 𝐂 ( t ) + 𝐈 , t ) = 𝐅 t ( 𝐂 ( t ) + 𝐈 , t ) + 𝐯 𝐅 ( 𝐂 ( t ) + 𝐈 , t ) = 𝐅 t ( 𝐫 , t ) + 𝐯 𝐅 ( 𝐫 , t ) \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{F}(\mathbf{C}(t)+\mathbf{I},t)=\mathbf{F% }_{t}(\mathbf{C}(t)+\mathbf{I},t)+\mathbf{v\cdot\nabla F}(\mathbf{C}(t)+% \mathbf{I},t)=\mathbf{F}_{t}(\mathbf{r},t)+\mathbf{v}\cdot\nabla\mathbf{F}(% \mathbf{r},t)
  33. 𝐯 = d d t 𝐂 ( t ) \mathbf{v}=\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{C}(t)
  34. × ( 𝐯 × 𝐅 ) = ( 𝐅 + 𝐅 ) 𝐯 - ( 𝐯 + 𝐯 ) 𝐅 \mathbf{\nabla\times}\left(\mathbf{v\times F}\right)=(\nabla\cdot\mathbf{F}+% \mathbf{F}\cdot\nabla)\mathbf{v}-(\nabla\cdot\mathbf{v}+\mathbf{v}\cdot\nabla)% \mathbf{F}
  35. d d t ( Σ ( t ) 𝐅 ( 𝐫 , t ) d 𝐀 ) = Σ ( t ) ( 𝐅 t ( 𝐫 , t ) + ( 𝐅 ) 𝐯 + ( 𝐅 ) 𝐯 - ( 𝐯 ) 𝐅 ) d 𝐀 - Σ ( t ) ( 𝐯 × 𝐅 ) d 𝐬 . \frac{\mathrm{d}}{\mathrm{d}t}\left(\iint_{\Sigma(t)}\mathbf{F}(\mathbf{r},t)% \cdot\mathrm{d}\mathbf{A}\right)=\iint_{\Sigma(t)}\big(\mathbf{F}_{t}(\mathbf{% r},t)+\left(\mathbf{F\cdot\nabla}\right)\mathbf{v}+\left(\mathbf{\nabla\cdot F% }\right)\mathbf{v}-(\nabla\cdot\mathbf{v})\mathbf{F}\big)\,\cdot\,\mathrm{d}% \mathbf{A}\,-\,\oint_{\partial\Sigma(t)}\left(\mathbf{\mathbf{v}\times F}% \right)\mathbf{\cdot}\,\mathrm{d}\mathbf{s}.
  36. d d t Σ ( t ) 𝐅 ( 𝐫 , t ) d 𝐀 = Σ ( t ) ( 𝐅 t ( 𝐫 , t ) + ( 𝐅 ) 𝐯 ) d 𝐀 - Σ ( t ) ( 𝐯 × 𝐅 ) d 𝐬 \frac{\mathrm{d}}{\mathrm{d}t}\iint_{\Sigma(t)}\mathbf{F}(\mathbf{r},t)\cdot% \mathrm{d}\mathbf{A}=\iint_{\Sigma(t)}\big(\mathbf{F}_{t}(\mathbf{r},t)+\left(% \mathbf{\nabla\cdot F}\right)\mathbf{v}\big)\cdot\mathrm{d}\mathbf{A}-\oint_{% \partial\Sigma(t)}\left(\mathbf{\mathbf{v}\times F}\right)\mathbf{\cdot}\,% \mathrm{d}\mathbf{s}

Length_function.html

  1. L ( e ) \displaystyle L(e)
  2. S i := { g L ( g ) i } S_{i}:=\{g\mid L(g)\leq i\}
  3. L ( g h ) L ( g ) + L ( h ) L(gh)\leq L(g)+L(h)

Leontief_production_function.html

  1. q = Min ( z 1 a , z 2 b ) q=\,\text{Min}\left(\frac{z_{1}}{a},\frac{z_{2}}{b}\right)

Leray's_theorem.html

  1. \mathcal{F}
  2. X X
  3. 𝒰 \mathcal{U}
  4. X . X.
  5. \mathcal{F}
  6. 𝒰 \mathcal{U}
  7. H ˇ q ( 𝒰 , ) = H q ( X , ) , \check{H}^{q}(\mathcal{U},\mathcal{F})=H^{q}(X,\mathcal{F}),
  8. H ˇ q ( 𝒰 , ) \check{H}^{q}(\mathcal{U},\mathcal{F})
  9. q q
  10. \mathcal{F}
  11. 𝒰 . \mathcal{U}.

Leray_cover.html

  1. 𝔘 = { U i } \mathfrak{U}=\{U_{i}\}
  2. X X
  3. \mathcal{F}
  4. 𝔘 \mathfrak{U}
  5. \mathcal{F}
  6. i 1 , , i n i_{1},\cdots,i_{n}
  7. k > 0 k>0
  8. H k ( U i 1 U i n , ) = 0 H^{k}(U_{i_{1}}\cap\cdots\cap U_{i_{n}},\mathcal{F})=0
  9. \mathcal{F}

Levi_decomposition.html

  1. exp ( ad ( z ) ) \exp(\mathrm{ad}(z))

Lewis_number.html

  1. Le = α D \mathrm{Le}=\frac{\alpha}{D}
  2. α \alpha
  3. D D
  4. Le = Sc Pr \mathrm{Le}=\frac{\mathrm{Sc}}{\mathrm{Pr}}

LF-space.html

  1. ( V n , i n m ) (V_{n},i_{nm})
  2. ( V n , i n m ) (V_{n},i_{nm})
  3. V n V_{n}
  4. V n V_{n}
  5. V n + 1 V_{n+1}
  6. V n V_{n}
  7. U V n U\cap V_{n}
  8. V n V_{n}
  9. C c ( n ) C^{\infty}_{c}(\mathbb{R}^{n})
  10. n \mathbb{R}^{n}
  11. K 1 K 2 K i n K_{1}\subset K_{2}\subset\ldots\subset K_{i}\subset\ldots\subset\mathbb{R}^{n}
  12. i K i = n \bigcup_{i}K_{i}=\mathbb{R}^{n}
  13. K i K_{i}
  14. K i + 1 K_{i+1}
  15. C c ( K i ) C_{c}^{\infty}(K_{i})
  16. n \mathbb{R}^{n}
  17. K i K_{i}
  18. C c ( n ) C^{\infty}_{c}(\mathbb{R}^{n})
  19. K i K_{i}
  20. C c ( n ) C^{\infty}_{c}(\mathbb{R}^{n})

Library_sort.html

  1. k k
  2. A A

Lie_algebra-valued_differential_form.html

  1. [ ω η ] [\omega\wedge\eta]
  2. 𝔤 \mathfrak{g}
  3. ω \omega
  4. 𝔤 \mathfrak{g}
  5. η \eta
  6. [ ω η ] ( v 1 , , v p + q ) = 1 ( p + q ) ! σ sgn ( σ ) [ ω ( v σ ( 1 ) , , v σ ( p ) ) , η ( v σ ( p + 1 ) , , v σ ( p + q ) ) ] [\omega\wedge\eta](v_{1},\cdots,v_{p+q})={1\over(p+q)!}\sum_{\sigma}% \operatorname{sgn}(\sigma)[\omega(v_{\sigma(1)},\cdots,v_{\sigma(p)}),\eta(v_{% \sigma(p+1)},\cdots,v_{\sigma(p+q)})]
  7. ω \omega
  8. η \eta
  9. [ ω η ] ( v 1 , v 2 ) = 1 2 ( [ ω ( v 1 ) , η ( v 2 ) ] - [ ω ( v 2 ) , η ( v 1 ) ] ) . [\omega\wedge\eta](v_{1},v_{2})={1\over 2}([\omega(v_{1}),\eta(v_{2})]-[\omega% (v_{2}),\eta(v_{1})]).
  10. [ ω η ] [\omega\wedge\eta]
  11. Ω ( M , 𝔤 ) \Omega(M,\mathfrak{g})
  12. [ ( g α ) ( h β ) ] = [ g , h ] ( α β ) [(g\otimes\alpha)\wedge(h\otimes\beta)]=[g,h]\otimes(\alpha\wedge\beta)
  13. g , h 𝔤 g,h\in\mathfrak{g}
  14. α , β Ω ( M , ) \alpha,\beta\in\Omega(M,\mathbb{R})
  15. [ ω , η ] [\omega,\eta]
  16. [ ω η ] [\omega\wedge\eta]
  17. [ ω , η ] [\omega,\eta]
  18. 𝔤 \mathfrak{g}
  19. [ ω η ] [\omega\wedge\eta]
  20. ω \omega
  21. η \eta
  22. ω Ω p ( M , 𝔤 ) \omega\in\Omega^{p}(M,\mathfrak{g})
  23. η Ω q ( M , 𝔤 ) \eta\in\Omega^{q}(M,\mathfrak{g})
  24. [ ω η ] = ω η - ( - 1 ) p q η ω , [\omega\wedge\eta]=\omega\wedge\eta-(-1)^{pq}\eta\wedge\omega,
  25. ω η , η ω Ω p + q ( M , 𝔤 ) \omega\wedge\eta,\ \eta\wedge\omega\in\Omega^{p+q}(M,\mathfrak{g})
  26. 𝔤 \mathfrak{g}
  27. f : 𝔤 𝔥 f:\mathfrak{g}\to\mathfrak{h}
  28. 𝔤 \mathfrak{g}
  29. 𝔥 \mathfrak{h}
  30. f ( φ ) ( v 1 , , v k ) = f ( φ ( v 1 , , v k ) ) f(\varphi)(v_{1},\dots,v_{k})=f(\varphi(v_{1},\dots,v_{k}))
  31. 1 k 𝔤 \textstyle\prod_{1}^{k}\mathfrak{g}
  32. f ( φ 1 , , φ k ) ( v 1 , , v q ) = 1 q ! σ sgn ( σ ) f ( φ 1 ( v σ ( 1 ) , , v σ ( q 1 ) ) , , φ k ( v σ ( q - q k + 1 ) , , v σ ( q ) ) ) f(\varphi_{1},\dots,\varphi_{k})(v_{1},\dots,v_{q})={1\over q!}\sum_{\sigma}% \operatorname{sgn}(\sigma)f(\varphi_{1}(v_{\sigma(1)},\dots,v_{\sigma(q_{1})})% ,\dots,\varphi_{k}(v_{\sigma(q-q_{k}+1)},\dots,v_{\sigma(q)}))
  33. 𝔤 \mathfrak{g}
  34. f ( φ , η ) f(\varphi,\eta)
  35. f : 𝔤 × V V f:\mathfrak{g}\times V\to V
  36. 𝔤 \mathfrak{g}
  37. 𝔤 \mathfrak{g}
  38. ρ : 𝔤 V , ρ ( x ) y = f ( x , y ) \rho:\mathfrak{g}\to V,\rho(x)y=f(x,y)
  39. f ( x , y ) = [ x , y ] f(x,y)=[x,y]
  40. 𝔤 \mathfrak{g}
  41. [ ] [\cdot\wedge\cdot]
  42. 𝔤 \mathfrak{g}
  43. 𝔤 \mathfrak{g}
  44. f ( [ ω ω ] , φ ) = 2 f ( ω , f ( ω , φ ) ) . f([\omega\wedge\omega],\varphi)=2f(\omega,f(\omega,\varphi)).
  45. 𝔤 = Lie ( G ) \mathfrak{g}=\operatorname{Lie}(G)
  46. 𝔤 \mathfrak{g}
  47. 𝔤 P = P × Ad 𝔤 . \mathfrak{g}_{P}=P\times_{\operatorname{Ad}}\mathfrak{g}.
  48. 𝔤 P \mathfrak{g}_{P}

Lie_algebra_cohomology.html

  1. 𝔤 \mathfrak{g}
  2. U 𝔤 U\mathfrak{g}
  3. 𝔤 \mathfrak{g}
  4. U 𝔤 U\mathfrak{g}
  5. 𝔤 \mathfrak{g}
  6. H n ( 𝔤 ; M ) := Ext U 𝔤 n ( R , M ) \mathrm{H}^{n}(\mathfrak{g};M):=\mathrm{Ext}^{n}_{U\mathfrak{g}}(R,M)
  7. M M 𝔤 := { m M x m = 0 for all x 𝔤 } . M\mapsto M^{\mathfrak{g}}:=\{m\in M\mid xm=0\ \,\text{ for all }x\in\mathfrak{% g}\}.
  8. H n ( 𝔤 ; M ) := Tor n U 𝔤 ( R , M ) \mathrm{H}_{n}(\mathfrak{g};M):=\mathrm{Tor}_{n}^{U\mathfrak{g}}(R,M)
  9. M M 𝔤 := M / 𝔤 M . M\mapsto M_{\mathfrak{g}}:=M/\mathfrak{g}M.
  10. 𝔤 \mathfrak{g}
  11. k k
  12. 𝔤 \mathfrak{g}
  13. M M
  14. Hom k ( Λ 𝔤 , M ) \mathrm{Hom}_{k}(\Lambda^{\ast}\mathfrak{g},M)
  15. n n
  16. k k
  17. f : Λ n 𝔤 M f:\Lambda^{n}\mathfrak{g}\to M
  18. n n
  19. M M
  20. n n
  21. ( n + 1 ) (n+1)
  22. δ f \delta f
  23. ( δ f ) ( x 1 , , x n + 1 ) = i ( - 1 ) i + 1 x i f ( x 1 , , x ^ i , , x n + 1 ) + i < j ( - 1 ) i + j f ( [ x i , x j ] , x 1 , , x ^ i , , x ^ j , , x n + 1 ) , (\delta f)(x_{1},\ldots,x_{n+1})=\sum_{i}(-1)^{i+1}x_{i}\,f(x_{1},\ldots,\hat{% x}_{i},\ldots,x_{n+1})+\sum_{i<j}(-1)^{i+j}f([x_{i},x_{j}],x_{1},\ldots,\hat{x% }_{i},\ldots,\hat{x}_{j},\ldots,x_{n+1})\,,
  24. H 0 ( 𝔤 ; M ) = M 𝔤 = { m M x m = 0 for all x 𝔤 } . H^{0}(\mathfrak{g};M)=M^{\mathfrak{g}}=\{m\in M\mid xm=0\ \,\text{ for all }x% \in\mathfrak{g}\}.
  25. D e r Der
  26. I d e r Ider
  27. H 1 ( 𝔤 ; M ) = Der ( 𝔤 , M ) / Ider ( 𝔤 , M ) H^{1}(\mathfrak{g};M)=\mathrm{Der}(\mathfrak{g},M)/\mathrm{Ider}(\mathfrak{g},M)
  28. d d
  29. M M
  30. d [ x , y ] = x d y - y d x d[x,y]=xdy-ydx~{}
  31. d x = x a dx=xa~{}
  32. a a
  33. M M
  34. H 2 ( 𝔤 ; M ) H^{2}(\mathfrak{g};M)
  35. 0 M 𝔥 𝔤 0 0\rightarrow M\rightarrow\mathfrak{h}\rightarrow\mathfrak{g}\rightarrow 0
  36. M M

Lightface_analytic_game.html

  1. Σ 1 1 \Sigma^{1}_{1}
  2. ω × ω \omega\times\omega
  3. ( ω × ω ) < ω (\omega\times\omega)^{<\omega}

Limnor.html

  1. ( x 0 - x 1 ) 2 + ( y 0 - y 1 ) 2 \sqrt{(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}}

Line_spectral_pairs.html

  1. A ( z ) = 1 - k = 1 p a k z - k A(z)=1-\sum_{k=1}^{p}a_{k}z^{-k}
  2. A ( z ) = 0.5 [ P ( z ) + Q ( z ) ] A(z)=0.5[P(z)+Q(z)]
  3. P ( z ) = A ( z ) + z - ( p + 1 ) A ( z - 1 ) P(z)=A(z)+z^{-(p+1)}A(z^{-1})
  4. Q ( z ) = A ( z ) - z - ( p + 1 ) A ( z - 1 ) Q(z)=A(z)-z^{-(p+1)}A(z^{-1})
  5. ω \omega
  6. z = e i ω , P ( z ) = 0 z=e^{i\omega},P(z)=0
  7. π \pi
  8. a 0 = 1 a_{0}=1
  9. A ( z ) = 0.5 [ P ( z ) + Q ( z ) ] A(z)=0.5[P(z)+Q(z)]

Line_wrap_and_word_wrap.html

  1. 0 2 + 4 2 + 1 2 = 17 0^{2}+4^{2}+1^{2}=17
  2. 3 2 + 1 2 + 1 2 = 11 3^{2}+1^{2}+1^{2}=11
  3. O ( n 2 ) O(n^{2})
  4. n n

Linear_hashing.html

  1. N N
  2. L L
  3. 0
  4. S S
  5. H H
  6. H mod ( N × 2 L ) H\bmod(N\times 2^{L})
  7. S S
  8. H mod ( N × 2 L + 1 ) H\bmod(N\times 2^{L+1})
  9. H mod ( N × 2 L ) H\bmod(N\times 2^{L})
  10. S S
  11. S S
  12. H mod ( N × 2 L ) H\bmod(N\times 2^{L})
  13. S S
  14. N × 2 L N\times 2^{L}
  15. S S
  16. L L
  17. S S
  18. S S
  19. S S
  20. N × 2 L N\times 2^{L}
  21. N × 2 L N\times 2^{L}
  22. H mod ( N × 2 L + 1 ) H\bmod(N\times 2^{L+1})
  23. H mod ( N × 2 L ) H\bmod(N\times 2^{L})
  24. S S
  25. N × 2 L N\times 2^{L}

Linear_multistep_method.html

  1. y = f ( t , y ) , y ( t 0 ) = y 0 . y^{\prime}=f(t,y),\quad y(t_{0})=y_{0}.
  2. y ( t ) y(t)
  3. t i t_{i}
  4. y i y ( t i ) where t i = t 0 + i h , y_{i}\approx y(t_{i})\quad\,\text{where}\quad t_{i}=t_{0}+ih,
  5. h h
  6. Δ t \Delta t
  7. i i
  8. s s
  9. y i y_{i}
  10. f ( t i , y i ) f(t_{i},y_{i})
  11. y y
  12. y n + s + a s - 1 y n + s - 1 + a s - 2 y n + s - 2 + + a 0 y n \displaystyle y_{n+s}+a_{s-1}\cdot y_{n+s-1}+a_{s-2}\cdot y_{n+s-2}+\cdots+a_{% 0}\cdot y_{n}
  13. a 0 , , a s - 1 a_{0},\ldots,a_{s-1}
  14. b 0 , , b s b_{0},\ldots,b_{s}
  15. b s = 0 b_{s}=0
  16. y n + s y_{n+s}
  17. b s 0 b_{s}\neq 0
  18. y n + s y_{n+s}
  19. f ( t n + s , y n + s ) f(t_{n+s},y_{n+s})
  20. y n + s y_{n+s}
  21. y n + s y_{n+s}
  22. y = f ( t , y ) = y , y ( 0 ) = 1. y^{\prime}=f(t,y)=y,\quad y(0)=1.
  23. y ( t ) = e t y(t)=\mathrm{e}^{t}
  24. y n + 1 = y n + h f ( t n , y n ) . y_{n+1}=y_{n}+hf(t_{n},y_{n}).\,
  25. h = 1 2 h=\tfrac{1}{2}
  26. y = y y^{\prime}=y
  27. y 1 \displaystyle y_{1}
  28. y n + 2 = y n + 1 + 3 2 h f ( t n + 1 , y n + 1 ) - 1 2 h f ( t n , y n ) . y_{n+2}=y_{n+1}+\tfrac{3}{2}hf(t_{n+1},y_{n+1})-\tfrac{1}{2}hf(t_{n},y_{n}).
  29. y n + 1 y_{n+1}
  30. y n y_{n}
  31. y n + 2 y_{n+2}
  32. y 0 = 1 y_{0}=1
  33. y 1 y_{1}
  34. y 2 \displaystyle y_{2}
  35. t = t 4 = 2 t=t_{4}=2
  36. e 2 = 7.3891 \mathrm{e}^{2}=7.3891\ldots
  37. a s - 1 = - 1 a_{s-1}=-1
  38. a s - 2 = = a 0 = 0 a_{s-2}=\cdots=a_{0}=0
  39. b j b_{j}
  40. y n + 1 \displaystyle y_{n+1}
  41. b j b_{j}
  42. s - 1 s-1
  43. p ( t n + i ) = f ( t n + i , y n + i ) , for i = 0 , , s - 1. p(t_{n+i})=f(t_{n+i},y_{n+i}),\qquad\,\text{for }i=0,\ldots,s-1.
  44. p ( t ) = j = 0 s - 1 ( - 1 ) s - j - 1 f ( t n + j , y n + j ) j ! ( s - j - 1 ) ! h s - 1 i = 0 i j s - 1 ( t - t n + i ) . p(t)=\sum_{j=0}^{s-1}\frac{(-1)^{s-j-1}f(t_{n+j},y_{n+j})}{j!(s-j-1)!h^{s-1}}% \prod_{i=0\atop i\neq j}^{s-1}(t-t_{n+i}).
  45. y = f ( t , y ) y^{\prime}=f(t,y)
  46. y = p ( t ) y^{\prime}=p(t)
  47. y n + s = y n + s - 1 + t n + s - 1 t n + s p ( t ) d t . y_{n+s}=y_{n+s-1}+\int_{t_{n+s-1}}^{t_{n+s}}p(t)\,dt.
  48. b j b_{j}
  49. b s - j - 1 = ( - 1 ) j j ! ( s - j - 1 ) ! 0 1 i = 0 i j s - 1 ( u + i ) d u , for j = 0 , , s - 1. b_{s-j-1}=\frac{(-1)^{j}}{j!(s-j-1)!}\int_{0}^{1}\prod_{i=0\atop i\neq j}^{s-1% }(u+i)\,du,\qquad\,\text{for }j=0,\ldots,s-1.
  50. f ( t , y ) f(t,y)
  51. a s - 1 = - 1 a_{s-1}=-1
  52. a s - 2 = = a 0 = 0 a_{s-2}=\cdots=a_{0}=0
  53. b s = 0 b_{s}=0
  54. s + 1 s+1
  55. y n \displaystyle y_{n}
  56. t n - 1 , , t n - s t_{n-1},\dots,t_{n-s}
  57. t n t_{n}
  58. b s - j = ( - 1 ) j j ! ( s - j ) ! 0 1 i = 0 i j s ( u + i - 1 ) d u , for j = 0 , , s . b_{s-j}=\frac{(-1)^{j}}{j!(s-j)!}\int_{0}^{1}\prod_{i=0\atop i\neq j}^{s}(u+i-% 1)\,du,\qquad\,\text{for }j=0,\ldots,s.
  59. b s - 1 = = b 0 = 0 b_{s-1}=\cdots=b_{0}=0
  60. y n + s + a s - 1 y n + s - 1 + a s - 2 y n + s - 2 + + a 0 y n = h ( b s f ( t n + s , y n + s ) + b s - 1 f ( t n + s - 1 , y n + s - 1 ) + + b 0 f ( t n , y n ) ) , \begin{aligned}&\displaystyle y_{n+s}+a_{s-1}y_{n+s-1}+a_{s-2}y_{n+s-2}+\cdots% +a_{0}y_{n}\\ &\displaystyle\qquad{}=h\bigl(b_{s}f(t_{n+s},y_{n+s})+b_{s-1}f(t_{n+s-1},y_{n+% s-1})+\cdots+b_{0}f(t_{n},y_{n})\bigr),\end{aligned}
  61. y = f ( t , y ) y^{\prime}=f(t,y)
  62. y n + s y_{n+s}
  63. y n + s - 1 , , y n y_{n+s-1},\ldots,y_{n}
  64. t n + s t_{n+s}
  65. k = 0 s - 1 a k = - 1 and k = 0 s b k = s + k = 0 s - 1 k a k . \sum_{k=0}^{s-1}a_{k}=-1\quad\,\text{and}\quad\sum_{k=0}^{s}b_{k}=s+\sum_{k=0}% ^{s-1}ka_{k}.
  66. O ( h p + 1 ) O(h^{p+1})
  67. k = 0 s - 1 a k = - 1 and q k = 0 s k q - 1 b k = s q + k = 0 s - 1 k q a k for q = 1 , , p . \sum_{k=0}^{s-1}a_{k}=-1\quad\,\text{and}\quad q\sum_{k=0}^{s}k^{q-1}b_{k}=s^{% q}+\sum_{k=0}^{s-1}k^{q}a_{k}\,\text{ for }q=1,\ldots,p.
  68. s + 1 s+1
  69. ρ ( z ) = z s + k = 0 s - 1 a k z k and σ ( z ) = k = 0 s b k z k . \rho(z)=z^{s}+\sum_{k=0}^{s-1}a_{k}z^{k}\quad\,\text{and}\quad\sigma(z)=\sum_{% k=0}^{s}b_{k}z^{k}.
  70. ρ ( e h ) - h σ ( e h ) = O ( h p + 1 ) as h 0. \rho(\mathrm{e}^{h})-h\sigma(\mathrm{e}^{h})=O(h^{p+1})\quad\,\text{as }h\to 0.
  71. ρ ( 1 ) = 0 \rho(1)=0
  72. ρ ( 1 ) = σ ( 1 ) \rho^{\prime}(1)=\sigma(1)
  73. y 0 y_{0}
  74. y 0 , y 1 , , y s - 1 y_{0},y_{1},\ldots,y_{s-1}
  75. y = 0 y^{\prime}=0
  76. y 1 , , y s - 1 y_{1},\ldots,y_{s-1}
  77. y 0 y_{0}
  78. h 0 h\to 0
  79. h 0 h\to 0
  80. O ( h p ) O(h^{p})
  81. z = 1 z=1
  82. π ( z ; h λ ) = ( 1 - h λ β s ) z s + k = 0 s - 1 ( α k - h λ β k ) z k = ρ ( z ) - h λ σ ( z ) . \pi(z;h\lambda)=(1-h\lambda\beta_{s})z^{s}+\sum_{k=0}^{s-1}(\alpha_{k}-h% \lambda\beta_{k})z^{k}=\rho(z)-h\lambda\sigma(z).
  83. y n + 1 = y n + h ( 23 12 f ( t n , y n ) - 16 12 f ( t n - 1 , y n - 1 ) + 5 12 f ( t n - 2 , y n - 2 ) ) . y_{n+1}=y_{n}+h\left({23\over 12}f(t_{n},y_{n})-{16\over 12}f(t_{n-1},y_{n-1})% +{5\over 12}f(t_{n-2},y_{n-2})\right).
  84. ρ ( z ) = z 3 - z 2 = z 2 ( z - 1 ) \rho(z)=z^{3}-z^{2}=z^{2}(z-1)\,
  85. z = 0 , 1 z=0,1
  86. z = 1 z=1

Linear_network_coding.html

  1. P P
  2. S S
  3. K K
  4. G F ( 2 s ) GF(2^{s})
  5. p k p_{k}
  6. I n D e g ( p k ) = S InDeg(p_{k})=S
  7. X k X_{k}
  8. { M i } i = 1 S \{M_{i}\}_{i=1}^{S}
  9. X k = i = 1 S g k i M i X_{k}=\sum_{i=1}^{S}g_{k}^{i}\cdot M_{i}
  10. g k i g_{k}^{i}
  11. G F ( 2 s ) GF(2^{s})
  12. X k X_{k}
  13. g k i g_{k}^{i}
  14. k th k\text{th}
  15. g k i g_{k}^{i}
  16. e i = [ 0...010...0 ] e_{i}=[0...010...0]
  17. 𝒢 = ( V , E , C ) \mathcal{G}=(V,E,C)
  18. V V
  19. E E
  20. C C
  21. E E
  22. T ( s , t ) T(s,t)
  23. s s
  24. t t
  25. T ( s , t ) T(s,t)

Linearized_gravity.html

  1. g g
  2. h h
  3. g = η + h g\,=\eta+h
  4. η \eta
  5. h h
  6. g g
  7. h h
  8. h h
  9. δ ξ h = δ ξ g - δ ξ η = ξ g = ξ η + ξ h = [ ξ ν ; μ + ξ μ ; ν + ξ α h μ ν ; α + ξ ; μ α h α ν + ξ ; ν α h μ α ] d x μ d x ν \delta_{\vec{\xi}}h=\delta_{\vec{\xi}}g-\delta_{\vec{\xi}}\eta=\mathcal{L}_{% \vec{\xi}}g=\mathcal{L}_{\vec{\xi}}\eta+\mathcal{L}_{\vec{\xi}}h=\left[\xi_{% \nu;\mu}+\xi_{\mu;\nu}+\xi^{\alpha}h_{\mu\nu;\alpha}+\xi^{\alpha}_{;\mu}h_{% \alpha\nu}+\xi^{\alpha}_{;\nu}h_{\mu\alpha}\right]dx^{\mu}\otimes dx^{\nu}
  10. \mathcal{L}
  11. δ ξ h μ ν ( ξ η ) μ ν = ξ ν ; μ + ξ μ ; ν \delta_{\vec{\xi}}h_{\mu\nu}\approx\left(\mathcal{L}_{\vec{\xi}}\eta\right)_{% \mu\nu}=\xi_{\nu;\mu}+\xi_{\mu;\nu}
  12. g a b = η a b + h a b g_{ab}=\eta_{ab}+h_{ab}
  13. η a b \,\eta_{ab}
  14. h a b \,h_{ab}
  15. ϵ γ a b \epsilon\,\gamma_{ab}
  16. g a b \,g_{ab}
  17. h h
  18. η \eta
  19. | h μ ν | 1 \left|h_{\mu\nu}\right|\ll 1
  20. h h
  21. h h
  22. h h
  23. ϵ \epsilon
  24. η \eta
  25. g a b g b c = δ a c g_{ab}g^{bc}=\delta_{a}{}^{c}
  26. g a b = η a b - h a b g^{ab}\,=\eta^{ab}-h^{ab}
  27. 2 Γ b c a = ( h a + b , c h a - c , b h b c , ) a 2\Gamma^{a}_{bc}=(h^{a}{}_{b,c}+h^{a}{}_{c,b}-h_{bc,}{}^{a})
  28. h b c , = def a η a r h b c , r h_{bc,}{}^{a}\ \stackrel{\mathrm{def}}{=}\ \eta^{ar}h_{bc,r}
  29. 2 R a = b c d 2 ( Γ b d , c a - Γ b c , d a ) = η a e ( h e b , d c + h e d , b c - h b d , e c - h e b , c d - h e c , b d + h b c , e d ) = 2R^{a}{}_{bcd}=2(\Gamma^{a}_{bd,c}-\Gamma^{a}_{bc,d})=\eta^{ae}(h_{eb,dc}+h_{% ed,bc}-h_{bd,ec}-h_{eb,cd}-h_{ec,bd}+h_{bc,ed})=
  30. = η a e ( h e d , b c - h b d , e c - h e c , b d + h b c , e d ) = h d , b c a - h b d , + a c h b c , - a d h a c , b d =\eta^{ae}(h_{ed,bc}-h_{bd,ec}-h_{ec,bd}+h_{bc,ed})=h^{a}_{d,bc}-h_{bd,}{}^{a}% {}_{c}+h_{bc,}{}^{a}{}_{d}-h^{a}{}_{c,bd}
  31. R b d = δ c R a a b c d R_{bd}=\delta^{c}{}_{a}R^{a}{}_{bcd}
  32. 2 R b d = h d , b r r + h b , d r r - h , b d - h b d , r s η r s 2R_{bd}=h^{r}_{d,br}+h^{r}_{b,dr}-h_{,bd}-h_{bd,rs}\eta^{rs}
  33. R = R b d η b d = h , a b a b - h R=R_{bd}\eta^{bd}=h^{ab}_{,ab}-\square h
  34. 8 π T b d = R b d - R a c η a c η b d / 2 8\pi T_{bd}\,=R_{bd}-R_{ac}\eta^{ac}\eta_{bd}/2
  35. 8 π T b d = ( h d , b r r + h b , d r r - h , b d - h b d , r - r h s , r r η b d s ) / 2 + ( h , a η b d a + h a c , r η a c r η b d ) / 4 8\pi T_{bd}=(h^{r}_{d,br}+h^{r}_{b,dr}-h_{,bd}-h_{bd,r}{}^{r}-h^{r}_{s,r}{}^{s% }\eta_{bd})/2+(h_{,a}{}^{a}\eta_{bd}+h_{ac,r}{}^{r}\eta^{ac}\eta_{bd})/4
  36. 8 π ( T b d - T a c η a c η b d / 2 ) = R b d 8\pi(T_{bd}-T_{ac}\eta^{ac}\eta_{bd}/2)\,=R_{bd}
  37. 16 π ( T b d - T a c η a c η b d / 2 ) = h d , b r r + h b , d r r - h , b d - h b d , r s η r s 16\pi(T_{bd}-T_{ac}\eta^{ac}\eta_{bd}/2)\,=h^{r}_{d,br}+h^{r}_{b,dr}-h_{,bd}-h% _{bd,rs}\eta^{rs}
  38. h α β , γ η β γ = 1 2 h β γ , α η β γ , h_{\alpha\beta,\gamma}\eta^{\beta\gamma}=\frac{1}{2}h_{\beta\gamma,\alpha}\eta% ^{\beta\gamma}\,,
  39. 16 π ( T b d - T a c η a c η b d / 2 ) = - h b d , r s η r s . 16\pi(T_{bd}-T_{ac}\eta^{ac}\eta_{bd}/2)\,=\,-h_{bd,rs}\eta^{rs}\,.
  40. Δ h b d = - 16 π G c 4 ( T b d - T a c η a c η b d / 2 ) + 2 h b d c 2 t 2 \Delta h_{bd}={-16\pi G\over c^{4}}(T_{bd}-T_{ac}\eta^{ac}\eta_{bd}/2)+\frac{% \partial^{2}h_{bd}}{c^{2}{\partial t}^{2}}\,
  41. h b d ( r ) = - 1 4 π ( - 16 π G c 4 ( T b d ( s ) - T a c ( s ) η a c η b d / 2 ) + 2 h b d ( s ) c 2 t 2 ) 1 | r - s | d 3 s h_{bd}(r)=\frac{-1}{4\pi}\int\left({-16\pi G\over c^{4}}(T_{bd}(s)-T_{ac}(s)% \eta^{ac}\eta_{bd}/2)+\frac{\partial^{2}h_{bd}(s)}{c^{2}{\partial t}^{2}}% \right)\frac{1}{|r-s|}d^{3}s\,

Linearly_disjoint.html

  1. Ω \Omega
  2. A k B A B A\otimes_{k}B\to AB
  3. ( x , y ) x y (x,y)\mapsto xy
  4. u i , v j u_{i},v_{j}
  5. u i v j u_{i}v_{j}
  6. Ω \Omega
  7. A k B A\otimes_{k}B
  8. Ω \Omega
  9. A , B A,B
  10. A A A^{\prime}\subset A
  11. B B B^{\prime}\subset B
  12. A A^{\prime}
  13. B B^{\prime}

Linienzugbeeinflussung.html

  1. V permitted = 2 d e c e l ( X G - d i s t ) V_{\rm permitted}=\sqrt{2\cdot decel\cdot(XG-dist)}

Liquid–liquid_extraction.html

  1. D I + 2 D_{\mathrm{I}^{+2}}
  2. D I + 2 D_{\mathrm{I}^{+2}}

List_of_Arizona_Diamondbacks_team_records.html

  1. 9 × ( E R ÷ I P ) 9×(ER÷IP)
  2. E R ER
  3. I P IP

List_of_convex_uniform_tilings.html

  1. A ~ 2 {\tilde{A}}_{2}
  2. G ~ 2 {\tilde{G}}_{2}
  3. B C ~ 2 {\tilde{BC}}_{2}
  4. I ~ 1 2 {\tilde{I}}_{1}^{2}
  5. G ~ 2 {\tilde{G}}_{2}
  6. A ~ 2 {\tilde{A}}_{2}

List_of_Edmonton_Oilers_head_coaches.html

  1. W + 1 2 T G C \frac{W+\frac{1}{2}T}{GC}
  2. W + 1 2 T + 1 2 O T G C \frac{W+\frac{1}{2}T+\frac{1}{2}OT}{GC}

List_of_electromagnetic_projectile_devices_in_fiction.html

  1. m 1 - ( v / c ) 2 \frac{m}{\sqrt{1-(v/c)^{2}}}
  2. 1 2 m v 2 \frac{1}{2}mv^{2}

List_of_fractals_by_Hausdorff_dimension.html

  1. λ = 3.570 \scriptstyle{\lambda_{\infty}=3.570}
  2. log 3 ( 2 ) \log_{3}(2)
  3. log 2 ( φ ) = log 2 ( 1 + 5 ) - 1 \log_{2}(\varphi)=\log_{2}(1+\sqrt{5})-1
  4. log ( 2 ) log ( 8 / 3 ) \frac{\log(2)}{\log(8/3)}
  5. φ = ( 1 + 5 ) / 2 \scriptstyle\varphi=(1+\sqrt{5})/2
  6. log 10 ( 5 ) = 1 - log 10 ( 2 ) \log_{10}(5)=1-\log_{10}(2)
  7. log ( 1 + 2 ) \log(1+\sqrt{2})
  8. - log ( 2 ) log ( 1 - γ 2 ) \frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}
  9. γ l m - 1 \gamma\,l_{m-1}
  10. l m - 1 = ( 1 - γ ) m - 1 / 2 m - 1 l_{m-1}=(1-\gamma)^{m-1}/2^{m-1}
  11. γ = 1 / 3 \scriptstyle\gamma=1/3
  12. γ \scriptstyle\gamma
  13. 0 < D < 1 \scriptstyle 0\,<\,D\,<\,1
  14. 1 1
  15. 2 - 2 n 2^{-2n}
  16. 2 + log 2 ( 1 2 ) = 1 2+\log_{2}\left(\frac{1}{2}\right)=1
  17. f ( x ) = n = 0 2 - n s ( 2 n x ) f(x)=\sum_{n=0}^{\infty}2^{-n}s(2^{n}x)
  18. s ( x ) s(x)
  19. f ( x ) = n = 0 w n s ( 2 n x ) f(x)=\sum_{n=0}^{\infty}w^{n}s(2^{n}x)
  20. w = 1 / 2 w=1/2
  21. 2 + l o g 2 ( w ) 2+log_{2}(w)
  22. w w
  23. [ 1 / 2 , 1 ] \left[1/2,1\right]
  24. 2 | α | 3 s + | α | 4 s = 1 2|\alpha|^{3s}+|\alpha|^{4s}=1
  25. 1 12 1\mapsto 12
  26. 2 13 2\mapsto 13
  27. 3 1 3\mapsto 1
  28. α \alpha
  29. z 3 - z 2 - z - 1 = 0 z^{3}-z^{2}-z-1=0
  30. 2 log 7 ( 3 ) 2\log_{7}(3)
  31. 3 log ( φ ) log ( 3 + 13 2 ) 3\frac{\log(\varphi)}{\log\left(\displaystyle\frac{3+\sqrt{13}}{2}\right)}
  32. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  33. 2 log 2 ( 27 - 3 78 3 + 27 + 3 78 3 3 ) , or root of 2 x - 1 = 2 ( 2 - x ) / 2 \begin{aligned}&\displaystyle 2\log_{2}\left(\displaystyle\frac{\sqrt[3]{27-3% \sqrt{78}}+\sqrt[3]{27+3\sqrt{78}}}{3}\right),\\ &\displaystyle\,\text{or root of }2^{x}-1=2^{(2-x)/2}\end{aligned}
  34. log 3 ( 4 ) \log_{3}(4)
  35. log 3 ( 4 ) \log_{3}(4)
  36. log 3 ( 4 ) \log_{3}(4)
  37. log 3 ( 4 ) \log_{3}(4)
  38. log 3 ( 4 ) \log_{3}(4)
  39. log 3 ( 5 ) \log_{3}(5)
  40. log 3 ( 5 ) \log_{3}(5)
  41. log 5 ( 10 3 ) \log_{\sqrt{5}}\left(\frac{10}{3}\right)
  42. 2 - log 2 ( 2 ) = 3 2 2-\log_{2}(\sqrt{2})=\frac{3}{2}
  43. f ( x ) = k = 1 sin ( 2 k x ) 2 k \displaystyle f(x)=\sum_{k=1}^{\infty}\frac{\sin(2^{k}x)}{\sqrt{2}^{k}}
  44. f : [ 0 , 1 ] f:[0,1]\to\mathbb{R}
  45. f ( x ) = k = 1 a - k sin ( b k x ) f(x)=\sum_{k=1}^{\infty}a^{-k}\sin(b^{k}x)
  46. 1 < a < 2 1<a<2
  47. b > 1 b>1
  48. 2 - log b ( a ) 2-\log_{b}(a)
  49. log 4 ( 8 ) = 3 2 \log_{4}(8)=\frac{3}{2}
  50. log 2 ( 1 + 73 - 6 87 3 + 73 + 6 87 3 3 ) \log_{2}\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)
  51. log 2 ( 1 + 73 - 6 87 3 + 73 + 6 87 3 3 ) \log_{2}\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)
  52. log 2 ( 3 ) \log_{2}(3)
  53. log 2 ( 3 ) \log_{2}(3)
  54. log 2 ( 3 ) \log_{2}(3)
  55. log 2 ( 3 ) \log_{2}(3)
  56. log 2 ( 4 ) = 2 \log_{2}(4)=2
  57. log φ φ ( φ ) = φ \log_{\sqrt[\varphi]{\varphi}}(\varphi)=\varphi
  58. r r
  59. r 2 r^{2}
  60. r = 1 / φ 1 / φ r=1/\varphi^{1/\varphi}
  61. φ \varphi
  62. ( r 2 ) φ + r φ = 1 ({r^{2}})^{\varphi}+r^{\varphi}=1
  63. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  64. 1 + log 3 ( 2 ) 1+\log_{3}(2)
  65. 1 + log k ( k + 1 2 ) \scriptstyle{1+\log_{k}\left(\frac{k+1}{2}\right)}
  66. 1 + log 3 ( 2 ) 1+\log_{3}(2)
  67. 3 log ( φ ) log ( 1 + 2 ) 3\frac{\log(\varphi)}{\log(1+\sqrt{2})}
  68. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  69. ( 1 / 3 ) s + ( 1 / 2 ) s + ( 2 / 3 ) s = 1 (1/3)^{s}+(1/2)^{s}+(2/3)^{s}=1
  70. n n
  71. c n c_{n}
  72. s s
  73. k = 1 n c k s = 1 \sum_{k=1}^{n}c_{k}^{s}=1
  74. 1 + log 5 ( 3 ) 1+\log_{5}(3)
  75. 1 + log k ( k + 1 2 ) \scriptstyle{1+\log_{k}\left(\frac{k+1}{2}\right)}
  76. z n + 1 = a + b z n exp [ i [ k - p / ( 1 + z n 2 ) ] ] z_{n+1}=a+bz_{n}\exp\left[i\left[k-p/\left(1+\lfloor z_{n}\rfloor^{2}\right)% \right]\right]
  77. 1 + log 10 ( 5 ) 1+\log_{10}(5)
  78. 4 log 5 ( 2 ) 4\log_{5}(2)
  79. log 3 ( 7 ) \log_{3}(7)
  80. log ( 4 ) log ( 2 + 2 cos ( 85 ) ) \frac{\log(4)}{\log(2+2\cos(85^{\circ}))}
  81. log ( 4 ) log ( 2 + 2 cos ( a ) ) [ 1 , 2 ] \frac{\log(4)}{\log(2+2\cos(a))}\in[1,2]
  82. log 2 ( 3 0.63 + 2 0.63 ) \log_{2}\left(3^{0.63}+2^{0.63}\right)
  83. p × q p\times q
  84. p q p\leq q
  85. log p ( k = 1 p n k a ) \log_{p}\left(\sum_{k=1}^{p}n_{k}^{a}\right)
  86. a = log q ( p ) a=\log_{q}(p)
  87. n k n_{k}
  88. k k
  89. log ( 6 ) log ( 1 + φ ) \frac{\log(6)}{\log(1+\varphi)}
  90. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  91. 6 ( 1 / 3 ) s + 5 ( 1 / 3 3 ) s = 1 6(1/3)^{s}+5{(1/3\sqrt{3})}^{s}=1
  92. 1 / 3 1/3
  93. 1 / 3 3 1/3\sqrt{3}
  94. log 3 ( 8 ) \log_{3}(8)
  95. log 3 ( 8 ) \log_{3}(8)
  96. log 3 ( 4 ) + log 3 ( 2 ) = log ( 4 ) log ( 3 ) + log ( 2 ) log ( 3 ) = log ( 8 ) log ( 3 ) \log_{3}(4)+\log_{3}(2)=\frac{\log(4)}{\log(3)}+\frac{\log(2)}{\log(3)}=\frac{% \log(8)}{\log(3)}
  97. D i m H ( F × G ) = D i m H ( F ) + D i m H ( G ) Dim_{H}(F\times G)=Dim_{H}(F)+Dim_{H}(G)
  98. 2 2
  99. 2 2
  100. 2 2
  101. 2 2
  102. 2 2
  103. 2 2
  104. log 2 ( 2 ) = 2 \log_{\sqrt{2}}(2)=2
  105. log 2 ( 4 ) = 2 \log_{2}(4)=2
  106. 7 ( 1 / 3 ) s + 6 ( 1 / 3 3 ) s = 1 7({1/3})^{s}+6({1/3\sqrt{3}})^{s}=1
  107. 1 / 3 3 1/3\sqrt{3}
  108. log 2 ( 4 ) = 2 \log_{2}(4)=2
  109. log 2 ( 4 ) = 2 \log_{2}(4)=2
  110. log ( 2 ) log ( 2 / 2 ) = 2 \frac{\log(2)}{\log(2/\sqrt{2})}=2
  111. 1 / 2 1/\sqrt{2}
  112. log 2 ( 4 ) = 2 \log_{2}(4)=2
  113. σ \sigma
  114. log 2 ( 5 ) \log_{2}(5)
  115. log ( 20 ) log ( 2 + φ ) \frac{\log(20)}{\log(2+\varphi)}
  116. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  117. log 3 ( 13 ) \log_{3}(13)
  118. log 4 ( 32 ) = 5 2 \log_{4}(32)=\frac{5}{2}
  119. log 3 ( 16 ) \log_{3}(16)
  120. n log 3 ( 2 ) n\log_{3}(2)
  121. log ( 7 6 - 1 3 ) log ( 2 - 1 ) \frac{\log\left(\frac{\sqrt{7}}{6}-\frac{1}{3}\right)}{\log(\sqrt{2}-1)}
  122. 2 - 1 \sqrt{2}-1
  123. log ( 12 ) log ( 1 + φ ) \frac{\log(12)}{\log(1+\varphi)}
  124. φ = ( 1 + 5 ) / 2 \varphi=(1+\sqrt{5})/2
  125. 1 + log 2 ( 3 ) 1+\log_{2}(3)
  126. 1 + log 2 ( 3 ) 1+\log_{2}(3)
  127. 1 + log 2 ( 3 ) 1+\log_{2}(3)
  128. log 3 ( 20 ) \log_{3}(20)
  129. log 3 ( 20 ) \log_{3}(20)
  130. log 2 ( 8 ) = 3 \log_{2}(8)=3
  131. log 2 ( 8 ) = 3 \log_{2}(8)=3
  132. log 2 ( 8 ) = 3 \log_{2}(8)=3
  133. log 2 ( 8 ) = 3 \log_{2}(8)=3
  134. 3 3
  135. E ( C 1 s + C 2 s ) = 1 \scriptstyle{E(C_{1}^{s}+C_{2}^{s})=1}
  136. E ( C 1 ) = 0.5 \scriptstyle{E(C_{1})=0.5}
  137. E ( C 2 ) = 0.3 \scriptstyle{E(C_{2})=0.3}
  138. C 1 C_{1}
  139. C 2 C_{2}
  140. s s
  141. E ( C 1 s + C 2 s ) = 1 \scriptstyle{E(C_{1}^{s}+C_{2}^{s})=1}
  142. E ( X ) E(X)
  143. X X
  144. s + 1 = 12 * 2 - ( s + 1 ) - 6 * 3 - ( s + 1 ) s+1=12*2^{-(s+1)}-6*3^{-(s+1)}
  145. log ( 4 ) log ( 3 ) \textstyle{\frac{\log(4)}{\log(3)}}
  146. 4 3 \textstyle{\frac{4}{3}}
  147. 4 3 \textstyle{\frac{4}{3}}
  148. 4 3 \textstyle{\frac{4}{3}}
  149. 2 - 1 2 \textstyle{2-\frac{1}{2}}
  150. f ( x + h ) - f ( x ) f(x+h)-f(x)
  151. α \alpha
  152. = h 2 α =h^{2\alpha}
  153. 2 - α 2-\alpha
  154. 5 3 \textstyle{\frac{5}{3}}
  155. log ( 9 * 0.75 ) log ( 3 ) \textstyle{\frac{\log(9*0.75)}{\log(3)}}
  156. log ( 9 p ) log ( 3 ) \textstyle{\frac{\log(9p)}{\log(3)}}
  157. 91 48 \textstyle{\frac{91}{48}}
  158. log ( 2 ) log ( 2 ) = 2 \textstyle{\frac{\log(2)}{\log(\sqrt{2})}=2}
  159. log ( 13 ) log ( 3 ) \textstyle{\frac{\log(13)}{\log(3)}}
  160. 3 - 1 2 \textstyle{3-\frac{1}{2}}
  161. f : 2 - > \scriptstyle{f:\mathbb{R}^{2}->\mathbb{R}}
  162. ( x , y ) (x,y)
  163. h h
  164. k k
  165. f ( x + h , y + k ) - f ( x , y ) \scriptstyle{f(x+h,y+k)-f(x,y)}
  166. h 2 + k 2 \scriptstyle{\sqrt{h^{2}+k^{2}}}
  167. α \alpha
  168. ( h 2 + k 2 ) α (h^{2}+k^{2})^{\alpha}
  169. 3 - α 3-\alpha
  170. ( 0 , 2 ) \textstyle{\in(0,2)}

List_of_large_cardinal_properties.html

  1. 0 , 1 \aleph_{0},\aleph_{1}
  2. κ = κ \kappa=\aleph_{\kappa}

List_of_logic_symbols.html

  1. \Rightarrow
  2. \to
  3. \supset
  4. \implies
  5. \Leftrightarrow
  6. \equiv
  7. \leftrightarrow
  8. \iff
  9. ¬ \neg
  10. \sim
  11. \wedge
  12. \lor
  13. \oplus
  14. \veebar
  15. \top
  16. \bot
  17. \forall
  18. \exists
  19. ! \exists!
  20. : = :=
  21. \equiv
  22. \Leftrightarrow
  23. ( ) (~{})
  24. \vdash
  25. \vDash
  26. p p
  27. q ( p q ) q\equiv\Box(p\rightarrow q)
  28. \wedge
  29. \vee

List_of_mesons.html

  1. u u ¯ - d d ¯ 2 \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}\,
  2. u u ¯ + d d ¯ + s s ¯ 3 \mathrm{\tfrac{u\bar{u}+d\bar{d}+s\bar{s}}{\sqrt{3}}}\,
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. d s ¯ - s d ¯ 2 \mathrm{\tfrac{d\bar{s}-s\bar{d}}{\sqrt{2}}}\,
  6. 1 / 2 {1}/{2}
  7. d s ¯ + s d ¯ 2 \mathrm{\tfrac{d\bar{s}+s\bar{d}}{\sqrt{2}}}\,
  8. 1 / 2 {1}/{2}
  9. 1 / 2 {1}/{2}
  10. 1 / 2 {1}/{2}
  11. ħ / Γ {ħ}/{Γ}
  12. u u ¯ - d d ¯ 2 \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}\,
  13. u u ¯ + d d ¯ 2 \mathrm{\tfrac{u\bar{u}+d\bar{d}}{\sqrt{2}}}\,
  14. 1 / 2 {1}/{2}
  15. 1 / 2 {1}/{2}
  16. ħ / Γ {ħ}/{Γ}

List_of_New_Jersey_Devils_head_coaches.html

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

List_of_New_York_Yankees_team_records.html

  1. 9 × ( E R ÷ I P ) 9×(ER÷IP)
  2. E R ER
  3. I P IP

List_of_Philadelphia_Phillies_team_records.html

  1. 9 × ( E R ÷ I P ) 9×(ER÷IP)
  2. E R ER
  3. I P IP

List_of_Pittsburgh_Pirates_team_records.html

  1. 9 × ( E R ÷ I P ) 9×(ER÷IP)
  2. E R ER
  3. I P IP

List_of_probability_distributions.html

  1. μ - s , μ + s \mu-s,\mu+s

List_of_things_named_after_Leonhard_Euler.html

  1. v e + f = 2 v−e+f=2
  2. P cr = π 2 E I ( K L ) 2 P\text{cr}=\frac{\pi^{2}EI}{(KL)^{2}}
  3. e 2.71828 e≈2.71828
  4. a + b ω a+bω
  5. ω ω
  6. a a
  7. m m
  8. φ φ
  9. e 2.71828 e≈2.71828
  10. γ 0.577216 γ≈0.577216
  11. χ ( S 2 ) = F - E + V = 2 \scriptstyle\chi(S^{2})=F-E+V=2

List_of_United_States_presidential_elections_by_Electoral_College_margin.html

  1. absolute margin of victory = { 0 ; w c 2 w - max { r , c 2 } ; w > c 2 \mbox{absolute margin of victory}~{}=\begin{cases}0;&w\leq\frac{c}{2}\\ w-\max\{r,\frac{c}{2}\};&w>\frac{c}{2}\end{cases}
  2. normalized margin of victory = { 0 ; w c 2 w - max { r , c 2 } c 2 ; w > c 2 \mbox{normalized margin of victory}~{}=\begin{cases}0;&w\leq\frac{c}{2}\\ \frac{w-\max\{r,\frac{c}{2}\}}{\frac{c}{2}};&w>\frac{c}{2}\end{cases}

Littrow_projection.html

  1. x = R sin ( λ - λ 0 ) cos φ x=R\frac{\sin\left(\lambda-\lambda_{0}\right)}{\cos\varphi}
  2. y = R cos ( λ - λ 0 ) tan φ y=R\cos\left(\lambda-\lambda_{0}\right)\tan\varphi

LMS_color_space.html

  1. [ L M S ] = [ 0.38971 0.68898 - 0.07868 - 0.22981 1.18340 0.04641 0.00000 0.00000 1.00000 ] [ X Y Z ] \begin{bmatrix}L\\ M\\ S\end{bmatrix}=\begin{bmatrix}0.38971&0.68898&-0.07868\\ -0.22981&1.18340&0.04641\\ 0.00000&0.00000&1.00000\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  2. [ L M S ] = [ 0.4002 0.7076 - 0.0808 - 0.2263 1.1653 0.0457 0 0 0.9182 ] [ X Y Z ] \begin{bmatrix}L\\ M\\ S\end{bmatrix}=\begin{bmatrix}0.4002&0.7076&-0.0808\\ -0.2263&1.1653&0.0457\\ 0&0&0.9182\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  3. [ L M S ] = [ 0.8951 0.2664 - 0.1614 - 0.7502 1.7135 0.0367 0.0389 - 0.0685 1.0296 ] [ X Y Z ] \begin{bmatrix}L\\ M\\ S\end{bmatrix}=\begin{bmatrix}0.8951&0.2664&-0.1614\\ -0.7502&1.7135&0.0367\\ 0.0389&-0.0685&1.0296\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  4. [ L M S ] = [ 0.8562 0.3372 - 0.1934 - 0.8360 1.8327 0.0033 0.0357 - 0.0469 1.0112 ] [ X Y Z ] \begin{bmatrix}L\\ M\\ S\end{bmatrix}=\begin{bmatrix}0.8562&0.3372&-0.1934\\ -0.8360&1.8327&0.0033\\ 0.0357&-0.0469&1.0112\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  5. [ L M S ] = [ 0.7328 0.4296 - 0.1624 - 0.7036 1.6975 0.0061 0.0030 0.0136 0.9834 ] [ X Y Z ] \begin{bmatrix}L\\ M\\ S\end{bmatrix}=\begin{bmatrix}0.7328&0.4296&-0.1624\\ -0.7036&1.6975&0.0061\\ 0.0030&0.0136&0.9834\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}

Local-density_approximation.html

  1. E x c LDA [ ρ ] = ρ ( 𝐫 ) ϵ x c ( ρ ) d 𝐫 , E_{xc}^{\mathrm{LDA}}[\rho]=\int\rho(\mathbf{r})\epsilon_{xc}(\rho)\ \mathrm{d% }\mathbf{r}\ ,
  2. E x c = E x + E c , E_{xc}=E_{x}+E_{c}\ ,
  3. E x LDA [ ρ ] = - 3 4 ( 3 π ) 1 / 3 ρ ( 𝐫 ) 4 / 3 d 𝐫 . E_{x}^{\mathrm{LDA}}[\rho]=-\frac{3}{4}\left(\frac{3}{\pi}\right)^{1/3}\int% \rho(\mathbf{r})^{4/3}\ \mathrm{d}\mathbf{r}\ .
  4. ϵ c = A ln ( r s ) + B + r s ( C ln ( r s ) + D ) , \epsilon_{c}=A\ln(r_{s})+B+r_{s}(C\ln(r_{s})+D)\ ,
  5. ϵ c = 1 2 ( g 0 r s + g 1 r s 3 / 2 + ) , \epsilon_{c}=\frac{1}{2}\left(\frac{g_{0}}{r_{s}}+\frac{g_{1}}{r_{s}^{3/2}}+% \dots\right)\ ,
  6. 4 3 π r s 3 = 1 ρ . \frac{4}{3}\pi r_{s}^{3}=\frac{1}{\rho}\ .
  7. E x c LSDA [ ρ α , ρ β ] = d 𝐫 ρ ( 𝐫 ) ϵ x c ( ρ α , ρ β ) . E_{xc}^{\mathrm{LSDA}}[\rho_{\alpha},\rho_{\beta}]=\int\mathrm{d}\mathbf{r}\ % \rho(\mathbf{r})\epsilon_{xc}(\rho_{\alpha},\rho_{\beta})\ .
  8. E x [ ρ α , ρ β ] = 1 2 ( E x [ 2 ρ α ] + E x [ 2 ρ β ] ) . E_{x}[\rho_{\alpha},\rho_{\beta}]=\frac{1}{2}\bigg(E_{x}[2\rho_{\alpha}]+E_{x}% [2\rho_{\beta}]\bigg)\ .
  9. ζ ( 𝐫 ) = ρ α ( 𝐫 ) - ρ β ( 𝐫 ) ρ α ( 𝐫 ) + ρ β ( 𝐫 ) . \zeta(\mathbf{r})=\frac{\rho_{\alpha}(\mathbf{r})-\rho_{\beta}(\mathbf{r})}{% \rho_{\alpha}(\mathbf{r})+\rho_{\beta}(\mathbf{r})}\ .
  10. ζ = 0 \zeta=0\,
  11. α \alpha\,
  12. β \beta\,
  13. ζ = ± 1 \zeta=\pm 1
  14. v x c LDA ( 𝐫 ) = δ E LDA δ ρ ( 𝐫 ) = ϵ x c ( ρ ( 𝐫 ) ) + ρ ( 𝐫 ) ϵ x c ( ρ ( 𝐫 ) ) ρ ( 𝐫 ) . v_{xc}^{\mathrm{LDA}}(\mathbf{r})=\frac{\delta E^{\mathrm{LDA}}}{\delta\rho(% \mathbf{r})}=\epsilon_{xc}(\rho(\mathbf{r}))+\rho(\mathbf{r})\frac{\partial% \epsilon_{xc}(\rho(\mathbf{r}))}{\partial\rho(\mathbf{r})}\ .

Locality_preserving_hashing.html

  1. | A - B | < | B - C | | f ( A ) - f ( B ) | < | f ( B ) - f ( C ) | . |A-B|<|B-C|\Rightarrow|f(A)-f(B)|<|f(B)-f(C)|.\,

Location-scale_family.html

  1. X X
  2. Y = d a + b X Y\stackrel{d}{=}a+bX
  3. = d \stackrel{d}{=}
  4. X X
  5. Y Y
  6. X X
  7. Y Y
  8. Y = d μ Y + σ Y X Y\stackrel{d}{=}\mu_{Y}+\sigma_{Y}X
  9. μ Y \mu_{Y}
  10. σ Y \sigma_{Y}
  11. Y Y
  12. Ω \Omega
  13. F Ω F\in\Omega
  14. a a\in\mathbb{R}
  15. b > 0 b>0
  16. G ( x ) = F ( a + b x ) G(x)=F(a+bx)
  17. Ω \Omega

Logarithm_of_a_matrix.html

  1. e B = A . e^{B}=A.\,
  2. A = ( cos ( α ) - sin ( α ) sin ( α ) cos ( α ) ) . A=\begin{pmatrix}\cos(\alpha)&-\sin(\alpha)\\ \sin(\alpha)&\cos(\alpha)\\ \end{pmatrix}.
  3. B n = ( α + 2 π n ) ( 0 - 1 1 0 ) , B_{n}=(\alpha+2\pi n)\begin{pmatrix}0&-1\\ 1&0\\ \end{pmatrix},
  4. ( 0 1 - 1 0 ) \begin{pmatrix}0&1\\ -1&0\\ \end{pmatrix}
  5. A B = e ln ( A ) + ln ( B ) . AB=e^{\ln(A)+\ln(B)}.\,
  6. A - 1 = e - ln ( A ) . A^{-1}=e^{-\ln(A)}.\,
  7. R R
  8. R R
  9. R R
  10. d g ( A , B ) := log ( A B ) F d_{g}(A,B):=\|\log(A^{\top}B)\|_{F}
  11. A = V - 1 A V . A^{\prime}=V^{-1}AV.\,
  12. ln A \ln A^{\prime}
  13. ln A = V ( ln A ) V - 1 . \ln A=V(\ln A^{\prime})V^{-1}.\,
  14. [ 1 1 0 1 ] . \begin{bmatrix}1&1\\ 0&1\end{bmatrix}.
  15. B = ( λ 1 0 0 0 0 λ 1 0 0 0 0 λ 1 0 0 0 0 0 λ 1 0 0 0 0 0 λ ) = λ ( 1 λ - 1 0 0 0 0 1 λ - 1 0 0 0 0 1 λ - 1 0 0 0 0 0 1 λ - 1 0 0 0 0 0 1 ) = λ ( I + K ) B=\begin{pmatrix}\lambda&1&0&0&\cdots&0\\ 0&\lambda&1&0&\cdots&0\\ 0&0&\lambda&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&0&0&\lambda&1\\ 0&0&0&0&0&\lambda\\ \end{pmatrix}=\lambda\begin{pmatrix}1&\lambda^{-1}&0&0&\cdots&0\\ 0&1&\lambda^{-1}&0&\cdots&0\\ 0&0&1&\lambda^{-1}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&0&0&1&\lambda^{-1}\\ 0&0&0&0&0&1\\ \end{pmatrix}=\lambda(I+K)
  16. 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
  17. ln B = ln ( λ ( I + K ) ) = ln ( λ I ) + ln ( I + K ) = ( ln λ ) I + K - K 2 2 + K 3 3 - K 4 4 + \ln B=\ln\big(\lambda(I+K)\big)=\ln(\lambda I)+\ln(I+K)=(\ln\lambda)I+K-\frac{% K^{2}}{2}+\frac{K^{3}}{3}-\frac{K^{4}}{4}+\cdots
  18. ln [ 1 1 0 1 ] = [ 0 1 0 0 ] . \ln\begin{bmatrix}1&1\\ 0&1\end{bmatrix}=\begin{bmatrix}0&1\\ 0&0\end{bmatrix}.
  19. exp : g G . \exp:g\rightarrow G.
  20. log = exp - 1 \log=\exp^{-1}
  21. 0 ¯ g \underline{0}\in g
  22. 1 ¯ G \underline{1}\in G
  23. log : V G U g . \log:V\subset G\rightarrow U\subset g.\,
  24. log ( det ( A ) ) = tr ( log A ) . \log(\det(A))=\mathrm{tr}(\log A)~{}.
  25. A = exp ( 0 a a 0 ) = ( cosh a sinh a sinh a cosh a ) = ( 1.25 .75 .75 1.25 ) A=\exp\begin{pmatrix}0&a\\ a&0\end{pmatrix}=\begin{pmatrix}\cosh a&\sinh a\\ \sinh a&\cosh a\end{pmatrix}=\begin{pmatrix}1.25&.75\\ .75&1.25\end{pmatrix}
  26. ln A = ( 0 ln 2 ln 2 0 ) \ln A=\begin{pmatrix}0&\ln 2\\ \ln 2&0\end{pmatrix}
  27. ( 3 / 4 5 / 4 5 / 4 3 / 4 ) , ( - 3 / 4 - 5 / 4 - 5 / 4 - 3 / 4 ) , ( - 5 / 4 - 3 / 4 - 3 / 4 - 5 / 4 ) \begin{pmatrix}3/4&5/4\\ 5/4&3/4\end{pmatrix},\ \begin{pmatrix}-3/4&-5/4\\ -5/4&-3/4\end{pmatrix},\ \begin{pmatrix}-5/4&-3/4\\ -3/4&-5/4\end{pmatrix}
  28. A = ( cosh ( ( ln 2 ) / 2 ) sinh ( ( ln 2 ) / 2 ) sinh ( ( ln 2 ) / 2 ) cosh ( ( ln 2 ) / 2 ) ) = ( 1.06 .35 .35 1.06 ) . \sqrt{A}=\begin{pmatrix}\cosh((\ln 2)/2)&\sinh((\ln 2)/2)\\ \sinh((\ln 2)/2)&\cosh((\ln 2)/2)\end{pmatrix}=\begin{pmatrix}1.06&.35\\ .35&1.06\end{pmatrix}~{}.
  29. a = l n ( p + r ) l n q a=ln(p+r)−lnq
  30. e a = p + r q = cosh a + sinh a e^{a}=\frac{p+r}{q}=\cosh a+\sinh a
  31. exp ( 0 a a 0 ) = ( r / q p / q p / q r / q ) \exp\begin{pmatrix}0&a\\ a&0\end{pmatrix}=\begin{pmatrix}r/q&p/q\\ p/q&r/q\end{pmatrix}
  32. 1 q ( r p p r ) \tfrac{1}{q}\begin{pmatrix}r&p\\ p&r\end{pmatrix}
  33. ( 0 a a 0 ) \begin{pmatrix}0&a\\ a&0\end{pmatrix}
  34. a = l n ( p + r ) l n q a=ln(p+r)−lnq

Logarithmic_mean_temperature_difference.html

  1. L M T D = Δ T A - Δ T B ln ( Δ T A Δ T B ) = Δ T A - Δ T B ln Δ T A - ln Δ T B LMTD=\frac{\Delta T_{A}-\Delta T_{B}}{\ln\left(\frac{\Delta T_{A}}{\Delta T_{B% }}\right)}=\frac{\Delta T_{A}-\Delta T_{B}}{\ln\Delta T_{A}-\ln\Delta T_{B}}
  2. Q = U × A r × L M T D Q=U\times Ar\times LMTD
  3. q ( z ) = U ( T 2 ( z ) - T 1 ( z ) ) / D = U ( Δ T ( z ) ) / D , q(z)=U(T_{2}(z)-T_{1}(z))/D=U(\Delta\;T(z))/D,
  4. d T 1 d z = k a ( T 1 ( z ) - T 2 ( z ) ) = - k a Δ T ( z ) \frac{\mathrm{d}\,T_{1}}{\mathrm{d}\,z}=k_{a}(T_{1}(z)-T_{2}(z))=-k_{a}\,% \Delta T(z)
  5. d T 2 d z = k b ( T 2 ( z ) - T 1 ( z ) ) = k b Δ T ( z ) \frac{\mathrm{d}\,T_{2}}{\mathrm{d}\,z}=k_{b}(T_{2}(z)-T_{1}(z))=k_{b}\,\Delta T% (z)
  6. d Δ T d z = d ( T 2 - T 1 ) d z = d T 2 d z - d T 1 d z = K Δ T ( z ) \frac{\mathrm{d}\,\Delta T}{\mathrm{d}\,z}=\frac{\mathrm{d}\,(T_{2}-T_{1})}{% \mathrm{d}\,z}=\frac{\mathrm{d}\,T_{2}}{\mathrm{d}\,z}-\frac{\mathrm{d}\,T_{1}% }{\mathrm{d}\,z}=K\Delta T(z)
  7. Q = A B q ( z ) d z = U D A B Δ T ( z ) d z = U D A B Δ T d z Q=\int^{B}_{A}q(z)dz=\frac{U}{D}\int^{B}_{A}\Delta T(z)dz=\frac{U}{D}\int^{B}_% {A}\Delta T\,dz
  8. Q = U A r ( B - A ) A B Δ T d z = U A r A B Δ T d z A B d z Q=\frac{UAr}{(B-A)}\int^{B}_{A}\Delta T\,dz=\frac{UAr\int^{B}_{A}\Delta T\,dz}% {\int^{B}_{A}\,dz}
  9. Q = U A r Δ T ( A ) Δ T ( B ) Δ T d z d Δ T d ( Δ T ) Δ T ( A ) Δ T ( B ) d z d Δ T d ( Δ T ) Q=\frac{UAr\int^{\Delta T(B)}_{\Delta T(A)}\Delta T\frac{\mathrm{d}\,z}{% \mathrm{d}\,\Delta T}\,d(\Delta T)}{\int^{\Delta T(B)}_{\Delta T(A)}\frac{% \mathrm{d}\,z}{\mathrm{d}\,\Delta T}\,d(\Delta T)}
  10. Q = U A r Δ T ( A ) Δ T ( B ) 1 K d ( Δ T ) Δ T ( A ) Δ T ( B ) 1 K Δ T d ( Δ T ) Q=\frac{UAr\int^{\Delta T(B)}_{\Delta T(A)}\frac{1}{K}\,d(\Delta T)}{\int^{% \Delta T(B)}_{\Delta T(A)}\frac{1}{K\Delta T}\,d(\Delta T)}
  11. Q = U × A r × Δ T ( B ) - Δ T ( A ) ln [ Δ T ( B ) / Δ T ( A ) ] Q=U\times Ar\times\frac{\Delta T(B)-\Delta T(A)}{\ln[\Delta T(B)/\Delta T(A)]}

Logical_equality.html

  1. x y x y E x y x EQ y x = y \begin{matrix}x\leftrightarrow y&&&x\Leftrightarrow y&&Exy\\ x\ \mbox{EQ}~{}\ y&&&x=y\end{matrix}
  2. x + y x y J x y x XOR y x y \begin{matrix}x+y&&&x\not\equiv y&&Jxy\\ x\ \mbox{XOR}~{}\ y&&&x\neq y\end{matrix}
  3. ( x = y ) = ¬ ( x y ) = ¬ x y = x ¬ y = ( x y ) ( ¬ x ¬ y ) = ( ¬ x y ) ( x ¬ y ) (x=y)=\lnot(x\oplus y)=\lnot x\oplus y=x\oplus\lnot y=(x\land y)\lor(\lnot x% \land\lnot y)=(\lnot x\lor y)\land(x\lor\lnot y)
  4. x y x\leftrightarrow y

Longest_common_substring_problem.html

  1. S S
  2. m m
  3. T T
  4. n n
  5. S S
  6. T T
  7. S = { S 1 , , S K } S=\{S_{1},...,S_{K}\}
  8. | S i | = n i |S_{i}|=n_{i}
  9. Σ n i = N \Sigma n_{i}=N
  10. 2 k K 2\leq k\leq K
  11. k k
  12. S S
  13. T T
  14. Θ ( n + m ) \Theta(n+m)
  15. Θ ( n m ) \Theta(nm)
  16. Θ ( n 1 + + n K ) \Theta(n_{1}+...+n_{K})
  17. Θ ( n 1 \Theta(n_{1}
  18. n K ) n_{K})
  19. Θ ( N * K ) \Theta(N*K)
  20. Θ ( N ) \Theta(N)
  21. Θ ( N K ) \Theta(NK)
  22. Θ ( N ) \Theta(N)
  23. 𝐿𝐶𝑆𝑢𝑓𝑓 ( S 1.. p , T 1.. q ) = { 𝐿𝐶𝑆𝑢𝑓𝑓 ( S 1.. p - 1 , T 1.. q - 1 ) + 1 if S [ p ] = T [ q ] 0 otherwise . \mathit{LCSuff}(S_{1..p},T_{1..q})=\begin{cases}\mathit{LCSuff}(S_{1..p-1},T_{% 1..q-1})+1&\mathrm{if}\;S[p]=T[q]\\ 0&\mathrm{otherwise}.\end{cases}
  24. 𝐿𝐶𝑆𝑢𝑏𝑠𝑡𝑟 ( S , T ) = max 1 i m , 1 j n 𝐿𝐶𝑆𝑢𝑓𝑓 ( S 1.. i , T 1.. j ) \mathit{LCSubstr}(S,T)=\max_{1\leq i\leq m,1\leq j\leq n}\mathit{LCSuff}(S_{1.% .i},T_{1..j})\;
  25. O ( n m ) O(nm)
  26. O ( min ( m , n ) ) O(\min(m,n))
  27. O ( n m ) O(nm)

Longitude_by_chronometer.html

  1. c o s L H A = ( s i n H o - s i n D e c * s i n B ) / ( c o s D e c * c o s B ) cosLHA=(sinHo-sinDec*sinB)/(cosDec*cosB)\,

Longitudinal_mode.html

  1. L = q λ 2 L=q\frac{\lambda}{2}
  2. Δ ν = c 2 n L \Delta\nu=\frac{c}{2nL}
  3. Δ ν = c 2 i n i L i = c 2 [ 1 n 1 L 1 + n 2 L 2 + n 3 L 3 + ] \Delta\nu=\frac{c}{2\sum_{i}n_{i}L_{i}}=\frac{c}{2}\left[\frac{1}{n_{1}L_{1}+n% _{2}L_{2}+n_{3}L_{3}+\ldots}\right]

Loop_space.html

  1. Map ( S 1 , X ) \mathrm{Map}(S^{1},X)
  2. X \mathcal{L}X
  3. π k ( X ) π k - 1 ( Ω X ) \pi_{k}(X)\approxeq\pi_{k-1}(\Omega X)
  4. [ Σ Z , X ] [ Z , Ω X ] [\Sigma Z,X]\approxeq[Z,\Omega X]
  5. [ A , B ] [A,B]
  6. A B A\rightarrow B
  7. Σ A \Sigma A
  8. [ A , B ] [A,B]
  9. A A
  10. B B
  11. [ Σ Z , X ] [\Sigma Z,X]
  12. [ Z , Ω X ] [Z,\Omega X]
  13. Z Z
  14. X X
  15. Z = S k - 1 Z=S^{k-1}
  16. k - 1 k-1

Lorentz_ether_theory.html

  1. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  2. v 2 / ( 2 c 2 ) v^{2}/(2c^{2})
  3. l = l 0 1 - v 2 / c 2 l=l_{0}\cdot\sqrt{1-v^{2}/c^{2}}
  4. t = t - v x / c 2 t^{\prime}=t-vx/c^{2}
  5. t = t - v x / c 2 t^{\prime}=t-vx/c^{2}
  6. ε - 1 / 2 \varepsilon^{-1/2}
  7. ( 1 - ( 1 / 2 ) v 2 / c 2 ) (1-(1/2)v^{2}/c^{2})
  8. k ε k\varepsilon
  9. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  10. ε \varepsilon
  11. \ell
  12. ε \varepsilon
  13. x = k ( x + ε t ) , y = y , z = z , t = k ( t + ε x ) x^{\prime}=k\ell\left(x+\varepsilon t\right),\qquad y^{\prime}=\ell y,\qquad z% ^{\prime}=\ell z,\qquad t^{\prime}=k\ell\left(t+\varepsilon x\right)
  14. k = 1 1 - ε 2 k=\frac{1}{\sqrt{1-\varepsilon^{2}}}
  15. x 2 + y 2 + z 2 - c 2 t 2 x^{2}+y^{2}+z^{2}-c^{2}t^{2}
  16. c t - 1 ct\sqrt{-1}
  17. m = ( 4 / 3 ) E / c 2 m=(4/3)E/c^{2}
  18. E e m / c 2 E_{em}/c^{2}
  19. k 3 ε k^{3}\varepsilon
  20. k ε k\varepsilon
  21. k = 1 - v 2 / c 2 k=\sqrt{1-v^{2}/c^{2}}
  22. ε \varepsilon
  23. ε = 1 \varepsilon=1
  24. m L = m 0 ( 1 - v 2 c 2 ) 3 , m T = m 0 1 - v 2 c 2 , m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=% \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},
  25. m 0 = 4 3 E e m c 2 m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
  26. - ( 1 / 3 ) E / c 2 -(1/3)E/c^{2}
  27. E / c 2 E/c^{2}
  28. E = m c 2 E=mc^{2}
  29. E / c 2 E/c^{2}
  30. E = m c 2 E=mc^{2}

Loschmidt_constant.html

  1. × 10 2 5 \times 10^{2}5
  2. × 10 2 5 \times 10^{2}5
  3. n 0 = p 0 k B T 0 n_{0}=\frac{p_{0}}{k_{\rm B}T_{0}}
  4. n 0 = p 0 N A R T 0 n_{0}=\frac{p_{0}N_{\rm A}}{RT_{0}}
  5. n 0 = N A R p 0 T 0 = A r ( e ) M u c α 2 2 R h R p 0 T 0 n_{0}=\frac{N_{\rm A}}{R}\frac{p_{0}}{T_{0}}=\frac{A_{\rm r}({\rm e})M_{\rm u}% c\alpha^{2}}{2R_{\infty}hR}\frac{p_{0}}{T_{0}}
  6. r {}_{r}
  7. u {}_{u}
  8. {}_{∞}
  9. = 3 4 n 0 π d 2 \ell=\frac{3}{4n_{0}\pi d^{2}}
  10. 0 {}_{0}
  11. 1 n 0 = 16 3 π d 2 4 \frac{1}{n_{0}}=\frac{16}{3}\frac{\pi\ell d^{2}}{4}
  12. 0 {}_{0}
  13. 2 {}^{2}
  14. 3 {}^{3}
  15. 0 {}_{0}
  16. 3 {}^{3}
  17. 0 {}_{0}
  18. 3 {}^{3}
  19. l i q u i d {}_{liquid}
  20. g a s {}_{gas}
  21. d = 8 V l V g d=8\frac{V_{\rm l}}{V_{\rm g}}\ell
  22. n 0 = ( V g V l ) 2 3 256 π 3 n_{0}=\left(\frac{V_{\rm g}}{V_{\rm l}}\right)^{2}\frac{3}{256\pi\ell^{3}}
  23. 0 {}_{0}
  24. × 10 2 4 \times 10^{2}4
  25. - 3 {}^{-3}
  26. 3 {}^{3}
  27. × 10 2 5 \times 10^{2}5
  28. - 3 {}^{-3}

Lottery_mathematics.html

  1. 49 ! ( 49 - 6 ) ! {49!\over(49-6)!}
  2. ( n k ) = n ! k ! ( n - k ) ! {n\choose k}={n!\over k!(n-k)!}
  3. ( n k ) {n\choose k}
  4. ( n k ) = ( 49 6 ) = 49 6 * 48 5 * 47 4 * 46 3 * 45 2 * 44 1 {n\choose k}={49\choose 6}={49\over 6}*{48\over 5}*{47\over 4}*{46\over 3}*{45% \over 2}*{44\over 1}
  5. ( 49 6 ) = 13 , 983 , 816 {49\choose 6}=13,983,816
  6. ( 6 n ) {6\choose n}
  7. ( 43 6 - n ) {43\choose 6-n}
  8. ( 6 n ) ( 43 6 - n ) ( 49 6 ) {6\choose n}{43\choose 6-n}\over{49\choose 6}
  9. ( K B ) ( N - K K - B ) ( N K ) {K\choose B}{N-K\choose K-B}\over{N\choose K}
  10. N N
  11. K K
  12. B B
  13. ( 6 0 ) ( 43 6 ) ( 49 6 ) {6\choose 0}{43\choose 6}\over{49\choose 6}
  14. ( 6 1 ) ( 43 5 ) ( 49 6 ) {6\choose 1}{43\choose 5}\over{49\choose 6}
  15. ( 6 2 ) ( 43 4 ) ( 49 6 ) {6\choose 2}{43\choose 4}\over{49\choose 6}
  16. ( 6 3 ) ( 43 3 ) ( 49 6 ) {6\choose 3}{43\choose 3}\over{49\choose 6}
  17. ( 6 4 ) ( 43 2 ) ( 49 6 ) {6\choose 4}{43\choose 2}\over{49\choose 6}
  18. ( 6 5 ) ( 43 1 ) ( 49 6 ) {6\choose 5}{43\choose 1}\over{49\choose 6}
  19. ( 6 6 ) ( 43 0 ) ( 49 6 ) {6\choose 6}{43\choose 0}\over{49\choose 6}
  20. 32 8 * 32 7 * 32 6 * 32 5 * 32 4 * 32 3 * 32 2 * 32 1 = 109951162776 40320 {32\over 8}*{32\over 7}*{32\over 6}*{32\over 5}*{32\over 4}*{32\over 3}*{32% \over 2}*{32\over 1}={109951162776\over 40320}
  21. ( 6 5 ) ( 43 1 ) {6\choose 5}{43\choose 1}
  22. 6 ( 49 6 ) {6}\over{49\choose 6}
  23. 258 42 43 ( 49 6 ) {258\cdot{{42}\over{43}}}\over{49\choose 6}
  24. ( 6 2 ) ( 43 4 ) {6\choose 2}{43\choose 4}
  25. 172 , 200 ( 49 6 ) {172,200}\over{49\choose 6}
  26. B B
  27. N N
  28. K K
  29. N N
  30. K - B N - K ( K B ) ( N - K K - B ) ( N K ) {K-B\over N-K}{K\choose B}{N-K\choose K-B}\over{N\choose K}
  31. B B
  32. N N
  33. K K
  34. N N
  35. N - K - K + B N - K ( K B ) ( N - K K - B ) ( N K ) {N-K-K+B\over N-K}{K\choose B}{N-K\choose K-B}\over{N\choose K}
  36. B B
  37. N N
  38. K K
  39. P P
  40. 1 P ( K B ) ( N - K K - B ) ( N K ) {1\over P}{K\choose B}{N-K\choose K-B}\over{N\choose K}
  41. B B
  42. N N
  43. K K
  44. P P
  45. P - 1 P ( K B ) ( N - K K - B ) ( N K ) {P-1\over P}{K\choose B}{N-K\choose K-B}\over{N\choose K}

Lovász_local_lemma.html

  1. 4 p d 1 4pd\leq 1
  2. e p ( d + 1 ) 1 , ep(d+1)\leq 1,
  3. { p < ( d - 1 ) d - 1 d d d > 1 p < 1 2 d = 1 \begin{cases}p<\frac{(d-1)^{d-1}}{d^{d}}&d>1\\ p<\tfrac{1}{2}&d=1\end{cases}
  4. e p d 1 epd\leq 1
  5. 𝒜 = { A 1 , , A n } \mathcal{A}=\{A_{1},\ldots,A_{n}\}
  6. A 𝒜 A\in\mathcal{A}
  7. Γ ( A ) \Gamma(A)
  8. 𝒜 \mathcal{A}
  9. 𝒜 ( { A } Γ ( A ) ) \mathcal{A}\setminus(\{A\}\cup\Gamma(A))
  10. x : 𝒜 ( 0 , 1 ) x:\mathcal{A}\to(0,1)
  11. A 𝒜 : Pr ( A ) x ( A ) B Γ ( A ) ( 1 - x ( B ) ) \forall A\in\mathcal{A}:\Pr(A)\leq x(A)\prod\nolimits_{B\in\Gamma(A)}(1-x(B))
  12. 𝒜 \mathcal{A}
  13. Pr ( A 1 ¯ A n ¯ ) A 𝒜 ( 1 - x ( A ) ) . \Pr\left(\overline{A_{1}}\wedge\ldots\wedge\overline{A_{n}}\right)\geq\prod% \nolimits_{A\in\mathcal{A}}(1-x(A)).
  14. A 𝒜 : x ( A ) = 1 d + 1 \forall A\in\mathcal{A}:x(A)=\frac{1}{d+1}
  15. p 1 d + 1 1 e p\leq\frac{1}{d+1}\cdot\frac{1}{e}
  16. 1 e ( 1 - 1 d + 1 ) d . \frac{1}{e}\leq\left(1-\frac{1}{d+1}\right)^{d}.
  17. Pr ( A | B S B ¯ ) x ( A ) S 𝒜 , A S \Pr\left(A|\wedge_{B\in S}\overline{B}\right)\leq x\left(A\right)\forall S% \subset\mathcal{A},A\not\in S
  18. S S
  19. S = S=\emptyset
  20. Pr ( A i ) x ( A i ) \Pr\left(A_{i}\right)\leq x\left(A_{i}\right)
  21. 𝒜 \mathcal{A}
  22. S 1 = S Γ ( A ) , S 2 = S S 1 S_{1}=S\cap\Gamma(A),S_{2}={{S\smallsetminus S_{1}}}
  23. Pr ( A | B S B ¯ ) = Pr ( A B S 1 B ¯ | B S 2 B ¯ ) Pr ( B S 1 B ¯ | B S 2 B ¯ ) \Pr\left(A|\wedge_{B\in S}\overline{B}\right)=\frac{\Pr\left(A\wedge\wedge_{B% \in S_{1}}\overline{B}|\wedge_{B\in S_{2}}\overline{B}\right)}{\Pr\left(\wedge% _{B\in S_{1}}\overline{B}|\wedge_{B\in S_{2}}\overline{B}\right)}
  24. S 1 = { B j 1 , B j 2 , , B j l } S_{1}=\{B_{j1},B_{j2},\ldots,B_{jl}\}
  25. A A
  26. S 2 S_{2}
  27. N u m e r a t o r Pr ( A | B S 2 B ¯ ) = Pr ( A ) x ( A ) B Γ ( A ) ( 1 - x ( B ) ) ( 1 ) Numerator\leq\Pr\left(A|\wedge_{B\in S_{2}}\overline{B}\right)=\Pr(A)\leq x(A)% \prod\limits_{B\in\Gamma(A)}\left(1-x(B)\right)\ldots\ldots(1)
  28. D e n o m i n a t o r = Pr ( B ¯ j 1 | t = 2 l B ¯ j t B S 2 B ¯ ) Pr ( B ¯ j 2 | t = 3 l B ¯ j t B S 2 B ¯ ) Pr ( B ¯ j l | B S 2 B ¯ ) B S 1 ( 1 - x ( B ) ) ( 2 ) Denominator=\Pr\left(\overline{B}_{j1}|\wedge_{t=2}^{l}\overline{B}_{jt}\wedge% _{B\in S_{2}}\overline{B}\right)\cdot\Pr\left(\overline{B}_{j2}|\wedge_{t=3}^{% l}\overline{B}_{jt}\wedge_{B\in S_{2}}\overline{B}\right)\cdot\ldots\cdot\Pr% \left(\overline{B}_{jl}|\wedge_{B\in S_{2}}\overline{B}\right)\geq\prod\limits% _{B\in S_{1}}\left(1-x(B)\right)\ldots\ldots(2)
  29. | S | |S|
  30. Pr ( A | B S B ¯ ) x ( A ) B Γ ( A ) - S 1 ( 1 - x ( B ) ) x ( A ) \Pr\left(A|\wedge_{B\in S}\overline{B}\right)\leq x(A)\prod\limits_{B\in\Gamma% (A)-S_{1}}(1-x(B))\leq x(A)
  31. Pr ( A ¯ | B S B ¯ ) 1 - x ( A ) \Pr\left(\overline{A}|\wedge_{B\in S}\overline{B}\right)\geq 1-x(A)
  32. Pr ( A 1 ¯ A n ¯ ) = Pr ( A 1 ¯ | A 2 ¯ A n ¯ ) Pr ( A 2 ¯ | A 3 ¯ A n ¯ ) Pr ( A n ¯ ) A 𝒜 ( 1 - x ( A ) ) \Pr\left(\overline{A_{1}}\wedge\ldots\wedge\overline{A_{n}}\right)=\Pr\left(% \overline{A_{1}}|\overline{A_{2}}\wedge\ldots\overline{A_{n}}\right)\cdot\Pr% \left(\overline{A_{2}}|\overline{A_{3}}\wedge\ldots\overline{A_{n}}\right)% \cdot\ldots\cdot\Pr\left(\overline{A_{n}}\right)\geq\prod\limits_{A\in\mathcal% {A}}(1-x(A))
  33. e p ( d + 1 ) 0.966 < 1. ep(d+1)\approx 0.966<1.

Love_wave.html

  1. 1 r \frac{1}{\sqrt{r}}
  2. r r
  3. s y m b o l ( 𝖢 : s y m b o l 𝐮 ) = ρ 𝐮 ¨ symbol{\nabla}\cdot(\mathsf{C}:symbol{\nabla}\mathbf{u})=\rho~{}\ddot{\mathbf{% u}}
  4. 𝐮 \mathbf{u}
  5. 𝖢 \mathsf{C}
  6. 𝐮 \mathbf{u}
  7. x , y , z x,y,z
  8. z z
  9. λ ( z ) , μ ( z ) , ρ ( z ) \lambda(z),\mu(z),\rho(z)
  10. ( u , v , w ) (u,v,w)
  11. t t
  12. u ( x , y , z , t ) = 0 , v ( x , y , z , t ) = v ^ ( x , z , t ) , w ( x , y , z , t ) = 0 . u(x,y,z,t)=0~{},~{}~{}v(x,y,z,t)=\hat{v}(x,z,t)~{},~{}~{}w(x,y,z,t)=0\,.
  13. ( x , z ) (x,z)
  14. v ^ ( x , z , t ) \hat{v}(x,z,t)
  15. k k
  16. ω \omega
  17. v ^ ( x , z , t ) = V ( k , z , ω ) exp [ i ( k x - ω t ) ] \hat{v}(x,z,t)=V(k,z,\omega)\,\exp[i(kx-\omega t)]
  18. i = - 1 i=\sqrt{-1}
  19. σ x x = 0 , σ y y = 0 , σ z z = 0 , τ z x = 0 , τ y z = μ ( z ) d V d z exp [ i ( k x - ω t ) ] , τ x y = i k μ ( z ) V ( k , z , ω ) exp [ i ( k x - ω t ) ] . \sigma_{xx}=0~{},~{}~{}\sigma_{yy}=0~{},~{}~{}\sigma_{zz}=0~{},~{}~{}\tau_{zx}% =0~{},~{}~{}\tau_{yz}=\mu(z)\,\frac{dV}{dz}\,\exp[i(kx-\omega t)]~{},~{}~{}% \tau_{xy}=ik\mu(z)V(k,z,\omega)\,\exp[i(kx-\omega t)]\,.
  20. d d z [ μ ( z ) d V d z ] = [ k 2 μ ( z ) - ω 2 ρ ( z ) ] V ( k , z , ω ) . \frac{d}{dz}\left[\mu(z)\,\frac{dV}{dz}\right]=[k^{2}\,\mu(z)-\omega^{2}\,\rho% (z)]\,V(k,z,\omega)\,.
  21. ( z = 0 ) (z=0)
  22. τ y z \tau_{yz}
  23. V V
  24. τ y z = T ( k , z , ω ) exp [ i ( k x - ω t ) ] \tau_{yz}=T(k,z,\omega)\,\exp[i(kx-\omega t)]
  25. d d z [ V T ] = [ 0 1 / μ ( z ) k 2 μ ( z ) - ω 2 ρ ( z ) 0 ] [ V T ] . \frac{d}{dz}\begin{bmatrix}V\\ T\end{bmatrix}=\begin{bmatrix}0&1/\mu(z)\\ k^{2}\,\mu(z)-\omega^{2}\,\rho(z)&0\end{bmatrix}\begin{bmatrix}V\\ T\end{bmatrix}\,.

Low-dropout_regulator.html

  1. V o u t = ( 1 + R 1 R 2 ) V r e f V_{out}=\left(1+\frac{R_{1}}{R_{2}}\right)V_{ref}
  2. P L O S S P_{LOSS}
  3. P L O S S = ( V I N - V O U T ) × I O U T + ( V I N × I Q ) P_{LOSS}=(V_{IN}-V_{OUT})\times I_{OUT}+(V_{IN}\times I_{Q})
  4. I Q I_{Q}
  5. η = P I N - P L O S S P I N \eta=\frac{P_{IN}-P_{LOSS}}{P_{IN}}
  6. P I N = V I N × I O U T P_{IN}=V_{IN}\times I_{OUT}
  7. I O U T I Q I_{OUT}>>I_{Q}
  8. P L O S S P_{LOSS}
  9. P L O S S = ( V I N - V O U T ) × I O U T P_{LOSS}=(V_{IN}-V_{OUT})\times I_{OUT}
  10. η = V O U T V I N \eta=\frac{V_{OUT}}{V_{IN}}
  11. P L O S S P_{LOSS}
  12. P L O S S = V I N × I Q P_{LOSS}=V_{IN}\times I_{Q}
  13. P S R R = 20 × l o g R i p p l e I n p u t R i p p l e O u t p u t PSRR=20\times log\frac{Ripple_{Input}}{Ripple_{Output}}

Lower_convex_envelope.html

  1. f ˘ \breve{f}
  2. f f
  3. [ a , b ] [a,b]
  4. f ( x ) = sup { g ( x ) g is convex and g f over [ a , b ] } . f(x)=\sup\{g(x)\mid g\,\text{ is convex and }g\leq f\,\text{ over }[a,b]\}.

Luby_transform_code.html

  1. \oplus
  2. M i 1 M i 2 M i d M_{i_{1}}\oplus M_{i_{2}}\oplus\cdots\oplus M_{i_{d}}\,
  3. \oplus
  4. ( M i 1 M i d ) ( M i 1 M i k - 1 M i k + 1 M i d ) = M i 1 M i 1 M i k - 1 M i k - 1 M i k M i k + 1 M i k + 1 M i d M i d = 0 0 M i k 0 0 = M i k \begin{aligned}&\displaystyle{}\qquad(M_{i_{1}}\oplus\dots\oplus M_{i_{d}})% \oplus(M_{i_{1}}\oplus\dots\oplus M_{i_{k-1}}\oplus M_{i_{k+1}}\oplus\dots% \oplus M_{i_{d}})\\ &\displaystyle=M_{i_{1}}\oplus M_{i_{1}}\oplus\dots\oplus M_{i_{k-1}}\oplus M_% {i_{k-1}}\oplus M_{i_{k}}\oplus M_{i_{k+1}}\oplus M_{i_{k+1}}\oplus\dots\oplus M% _{i_{d}}\oplus M_{i_{d}}\\ &\displaystyle=0\oplus\dots\oplus 0\oplus M_{i_{k}}\oplus 0\oplus\dots\oplus 0% \\ &\displaystyle=M_{i_{k}}\end{aligned}
  5. P { d = 1 } = 1 n P { d = k } = 1 k ( k - 1 ) ( k = 2 , 3 , , n ) . \begin{aligned}\displaystyle\mathrm{P}\{d=1\}&\displaystyle=\frac{1}{n}\\ \displaystyle\mathrm{P}\{d=k\}&\displaystyle=\frac{1}{k(k-1)}\qquad(k=2,3,% \dots,n).\end{aligned}

Luminosity_distance.html

  1. M = m - 5 ( log 10 D L - 1 ) M=m-5(\log_{10}{D_{L}}-1)\!\,
  2. D L = 10 ( m - M ) 5 + 1 D_{L}=10^{\frac{(m-M)}{5}+1}
  3. F = L 4 π D L 2 F=\frac{L}{4\pi D_{L}^{2}}
  4. D L = L 4 π F D_{L}=\sqrt{\frac{L}{4\pi F}}
  5. D M D_{M}
  6. D L = ( 1 + z ) D M D_{L}=(1+z)D_{M}
  7. D M D_{M}
  8. δ θ \delta\theta
  9. D M δ θ D_{M}\delta\theta
  10. D M D_{M}
  11. D C D_{C}

Lunar_theory.html

  1. e e
  2. e e^{\prime}
  3. Γ \Gamma
  4. Γ \Gamma^{\prime}
  5. i i
  6. Ω \Omega
  7. Ω \Omega
  8. l l
  9. Γ \Gamma
  10. l l^{\prime}
  11. Γ \Gamma^{\prime}
  12. F F
  13. Ω \Omega
  14. D D
  15. + 22639 ′′ sin ( l ) + 769 ′′ sin ( 2 l ) + 36 ′′ sin ( 3 l ) +22639^{\prime\prime}\sin(l)+769^{\prime\prime}\sin(2l)+36^{\prime\prime}\sin(% 3l)
  16. + 4586 ′′ sin ( 2 D - l ) +4586^{\prime\prime}\sin(2D-l)
  17. + 2370 ′′ sin ( 2 D ) +2370^{\prime\prime}\sin(2D)
  18. - 668 ′′ sin ( l ) -668^{\prime\prime}\sin(l^{\prime})
  19. - 125 ′′ sin ( D ) -125^{\prime\prime}\sin(D)
  20. - 412 ′′ sin ( 2 F ) -412^{\prime\prime}\sin(2F)

Lusin's_theorem.html

  1. f : [ a , b ] f:[a,b]\rightarrow\mathbb{C}
  2. μ ( E ) > b - a - ε . \mu(E)>b-a-\varepsilon.\,
  3. ( X , Σ , μ ) (X,\Sigma,\mu)
  4. f : X Y f:X\rightarrow Y
  5. A Σ A\in\Sigma

Machin-like_formula.html

  1. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}
  2. c 0 π 4 = n = 1 N c n arctan a n b n c_{0}\frac{\pi}{4}=\sum_{n=1}^{N}c_{n}\arctan\frac{a_{n}}{b_{n}}
  3. a n a_{n}
  4. b n b_{n}
  5. a n < b n a_{n}<b_{n}
  6. c n c_{n}
  7. c 0 c_{0}
  8. arctan x = n = 0 ( - 1 ) n 2 n + 1 x 2 n + 1 = x - x 3 3 + x 5 5 - x 7 7 + \arctan x=\sum^{\infty}_{n=0}\frac{(-1)^{n}}{2n+1}x^{2n+1}=x-\frac{x^{3}}{3}+% \frac{x^{5}}{5}-\frac{x^{7}}{7}+...
  9. sin ( α + β ) = sin α cos β + cos α sin β \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
  10. cos ( α + β ) = cos α cos β - sin α sin β \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
  11. arctan a 1 b 1 + arctan a 2 b 2 = arctan a 1 b 2 + a 2 b 1 b 1 b 2 - a 1 a 2 , \arctan\frac{a_{1}}{b_{1}}+\arctan\frac{a_{2}}{b_{2}}=\arctan\frac{a_{1}b_{2}+% a_{2}b_{1}}{b_{1}b_{2}-a_{1}a_{2}},
  12. - π 2 < arctan a 1 b 1 + arctan a 2 b 2 < π 2 . -\frac{\pi}{2}<\arctan\frac{a_{1}}{b_{1}}+\arctan\frac{a_{2}}{b_{2}}<\frac{\pi% }{2}.
  13. 2 arctan 1 5 2\arctan\frac{1}{5}
  14. = arctan 1 5 + arctan 1 5 =\arctan\frac{1}{5}+\arctan\frac{1}{5}
  15. = arctan 1 * 5 + 1 * 5 5 * 5 - 1 * 1 =\arctan\frac{1*5+1*5}{5*5-1*1}
  16. = arctan 10 24 =\arctan\frac{10}{24}
  17. = arctan 5 12 =\arctan\frac{5}{12}
  18. 4 arctan 1 5 4\arctan\frac{1}{5}
  19. = 2 arctan 1 5 + 2 arctan 1 5 =2\arctan\frac{1}{5}+2\arctan\frac{1}{5}
  20. = arctan 5 12 + arctan 5 12 =\arctan\frac{5}{12}+\arctan\frac{5}{12}
  21. = arctan 5 * 12 + 5 * 12 12 * 12 - 5 * 5 =\arctan\frac{5*12+5*12}{12*12-5*5}
  22. = arctan 120 119 =\arctan\frac{120}{119}
  23. 4 arctan 1 5 - π 4 4\arctan\frac{1}{5}-\frac{\pi}{4}
  24. = 4 arctan 1 5 - arctan 1 1 =4\arctan\frac{1}{5}-\arctan\frac{1}{1}
  25. = 4 arctan 1 5 + arctan - 1 1 =4\arctan\frac{1}{5}+\arctan\frac{-1}{1}
  26. = arctan 120 119 + arctan - 1 1 =\arctan\frac{120}{119}+\arctan\frac{-1}{1}
  27. = arctan 120 * 1 + ( - 1 ) * 119 119 * 1 - 120 * ( - 1 ) =\arctan\frac{120*1+(-1)*119}{119*1-120*(-1)}
  28. = arctan 1 239 =\arctan\frac{1}{239}
  29. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}
  30. ( b 1 + a 1 i ) * ( b 2 + a 2 i ) (b_{1}+a_{1}i)*(b_{2}+a_{2}i)
  31. = b 1 b 2 + a 2 b 1 i + a 1 b 2 i - a 1 a 2 =b_{1}b_{2}+a_{2}b_{1}i+a_{1}b_{2}i-a_{1}a_{2}
  32. ( b n + a n i ) (b_{n}+a_{n}i)
  33. arctan a n b n \arctan\frac{a_{n}}{b_{n}}
  34. arctan a 1 b 2 + a 2 b 1 b 1 b 2 - a 1 a 2 \arctan\frac{a_{1}b_{2}+a_{2}b_{1}}{b_{1}b_{2}-a_{1}a_{2}}
  35. c n arctan a n b n c_{n}\arctan\frac{a_{n}}{b_{n}}
  36. ( b n + a n i ) c n (b_{n}+a_{n}i)^{c_{n}}
  37. k * ( 1 + i ) c 0 = n = 1 N ( b n + a n i ) c n k*(1+i)^{c_{0}}=\prod_{n=1}^{N}(b_{n}+a_{n}i)^{c_{n}}
  38. k k
  39. ( a + b i ) (a+bi)
  40. arctan b a \arctan\frac{b}{a}
  41. arctan b a \arctan\frac{b}{a}
  42. π 4 \frac{\pi}{4}
  43. π 4 \frac{\pi}{4}
  44. ( 2 + i ) (2+i)
  45. ( 3 + i ) (3+i)
  46. ( 5 + 5 i ) (5+5i)
  47. arctan 1 2 + arctan 1 3 = π 4 \arctan\frac{1}{2}+\arctan\frac{1}{3}=\frac{\pi}{4}
  48. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}
  49. ( 5 + i ) 4 ( - 239 + i ) = - 2 2 ( 13 4 ) ( 1 + i ) (5+i)^{4}(-239+i)=-2^{2}(13^{4})(1+i)
  50. 4 arctan 1 5 - arctan 1 239 = π 4 4\arctan\frac{1}{5}-\arctan\frac{1}{239}=\frac{\pi}{4}
  51. a n a_{n}
  52. π 4 = arctan 1 2 + arctan 1 3 \frac{\pi}{4}=\arctan\frac{1}{2}+\arctan\frac{1}{3}
  53. π 4 = 2 arctan 1 2 - arctan 1 7 \frac{\pi}{4}=2\arctan\frac{1}{2}-\arctan\frac{1}{7}
  54. π 4 = 2 arctan 1 3 + arctan 1 7 \frac{\pi}{4}=2\arctan\frac{1}{3}+\arctan\frac{1}{7}
  55. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}
  56. a n a_{n}
  57. π 4 = 22 arctan 24478 873121 + 17 arctan 685601 69049993 \frac{\pi}{4}=22\arctan\frac{24478}{873121}+17\arctan\frac{685601}{69049993}
  58. area ( P O N ) \displaystyle{\rm area}(PON)
  59. π 4 = 12 arctan 1 49 + 32 arctan 1 57 - 5 arctan 1 239 + 12 arctan 1 110443 \frac{\pi}{4}=12\arctan\frac{1}{49}+32\arctan\frac{1}{57}-5\arctan\frac{1}{239% }+12\arctan\frac{1}{110443}
  60. π 4 = 44 arctan 1 57 + 7 arctan 1 239 - 12 arctan 1 682 + 24 arctan 1 12943 \frac{\pi}{4}=44\arctan\frac{1}{57}+7\arctan\frac{1}{239}-12\arctan\frac{1}{68% 2}+24\arctan\frac{1}{12943}
  61. π 4 = \displaystyle\frac{\pi}{4}=
  62. π 4 = \displaystyle\frac{\pi}{4}=
  63. π 4 = \displaystyle\frac{\pi}{4}=
  64. N d N_{d}
  65. π \pi
  66. N t N_{t}
  67. u n u_{n}
  68. ( ( b n a n ) 2 ) N t = 10 N d \left(\left(\frac{b_{n}}{a_{n}}\right)^{2}\right)^{N_{t}}=10^{N_{d}}
  69. N t = N d ln 10 2 ln b n a n N_{t}=N_{d}\quad\frac{\ln 10}{2\ln\frac{b_{n}}{a_{n}}}
  70. N d N_{d}
  71. N d N t 2 \frac{N_{d}N_{t}}{2}
  72. t i m e = u n N d N t 2 time=\frac{u_{n}N_{d}N_{t}}{2}
  73. t i m e = u n N d 2 ln 10 4 ln b n a n = k u n ln b n a n time=\frac{u_{n}{N_{d}}^{2}\ln 10}{4\ln\frac{b_{n}}{a_{n}}}=\frac{ku_{n}}{\ln% \frac{b_{n}}{a_{n}}}
  74. k k
  75. k k
  76. t i m e = n = 1 N u n ln b n a n time=\sum_{n=1}^{N}\frac{u_{n}}{\ln\frac{b_{n}}{a_{n}}}
  77. u n u_{n}
  78. 1 : t e r m * = a n 2 1:\quad term\quad*=\quad{a_{n}}^{2}
  79. 2 : t e r m / = - b n 2 2:\quad term\quad/=\quad-{b_{n}}^{2}
  80. 3 : t m p = t e r m / ( 2 * n + 1 ) 3:\quad tmp\quad=\quad term\quad/\quad(2*n+1)
  81. 4 : s u m + = t m p 4:\quad sum\quad+=\quad tmp
  82. u n u_{n}
  83. a n a_{n}
  84. u n u_{n}
  85. π 4 = 44 arctan 74684 14967113 + 139 arctan 1 239 - 12 arctan 20138 15351991 \frac{\pi}{4}=44\arctan\frac{74684}{14967113}+139\arctan\frac{1}{239}-12% \arctan\frac{20138}{15351991}
  86. a n a_{n}
  87. b n b_{n}
  88. b n a n \frac{b_{n}}{a_{n}}
  89. ln b n a n \ln\frac{b_{n}}{a_{n}}
  90. u n u_{n}
  91. t i m e time
  92. a n a_{n}
  93. b n b_{n}
  94. b n a n \frac{b_{n}}{a_{n}}
  95. ln b n a n \ln\frac{b_{n}}{a_{n}}
  96. u n u_{n}
  97. t i m e time
  98. a n a_{n}
  99. b n b_{n}

Machine_epsilon.html

  1. ϵ \epsilon
  2. b b
  3. p p
  4. b - ( p - 1 ) / 2 b^{-(p-1)}/2
  5. b - ( p - 1 ) b^{-(p-1)}
  6. b b
  7. p p
  8. b b
  9. e e
  10. b e - ( p - 1 ) b^{e-(p-1)}
  11. b b
  12. b b
  13. e = 0 e=0
  14. b - ( p - 1 ) / 2 b^{-(p-1)}/2
  15. 1 + a 1+a
  16. a a
  17. 0
  18. b - ( p - 1 ) / 2 b^{-(p-1)}/2
  19. 1 1
  20. a / ( 1 + a ) a/(1+a)
  21. a a
  22. 1 + a 1+a
  23. b - ( p - 1 ) / 2 b^{-(p-1)}/2
  24. 1 1
  25. x x
  26. 2 ϵ 2\epsilon
  27. | x | |x|
  28. x x
  29. y y
  30. \bullet
  31. \circ
  32. x y = round ( x y ) x\bullet y=\mbox{round}(x\circ y)
  33. x y = ( x y ) ( 1 + z ) x\bullet y=(x\circ y)(1+z)
  34. z z
  35. ϵ \epsilon
  36. ϵ \epsilon
  37. b - ( p - 1 ) b^{-(p-1)}
  38. = ϵ / 2 =\epsilon/2

Madelung_constant.html

  1. V i = e 4 π ϵ 0 j i z j r i j V_{i}=\frac{e}{4\pi\epsilon_{0}}\sum_{j\neq i}\frac{z_{j}}{r_{ij}}\,\!
  2. × 10 19 \times 10^{−}19
  3. × 10 10 \times 10^{−}10
  4. V i = e 4 π ϵ 0 r 0 j z j r 0 r i j = e 4 π ϵ 0 r 0 M i V_{i}=\frac{e}{4\pi\epsilon_{0}r_{0}}\sum_{j}\frac{z_{j}r_{0}}{r_{ij}}=\frac{e% }{4\pi\epsilon_{0}r_{0}}M_{i}
  5. M i M_{i}
  6. M i = j z j r i j / r 0 . M_{i}=\sum_{j}\frac{z_{j}}{r_{ij}/r_{0}}.
  7. r i r_{i}
  8. E e l , i = z i e V i = e 2 4 π ϵ 0 r 0 z i M i . E_{el,i}=z_{i}eV_{i}=\frac{e^{2}}{4\pi\epsilon_{0}r_{0}}z_{i}M_{i}.
  9. M i M_{i}
  10. z N a = 1 z_{Na}=1
  11. z C l = - 1 z_{Cl}=-1
  12. r 0 = a / 2 r_{0}=a/2
  13. M Na = - M Cl = j , k , = - ( - 1 ) j + k + ( j 2 + k 2 + 2 ) 1 / 2 . M\text{Na}=-M\text{Cl}={\sum_{j,k,\ell=-\infty}^{\infty}}^{\prime}{{(-1)^{j+k+% \ell}}\over{(j^{2}+k^{2}+\ell^{2})^{1/2}}}.
  14. j = k = = 0 j=k=\ell=0
  15. M = - 6 + 12 / 2 - 8 / 3 + 6 / 2 - 24 / 5 + = - 1.74756 . M=-6+12/\sqrt{2}-8/\sqrt{3}+6/2-24/\sqrt{5}+\cdots=-1.74756\dots.
  16. M M
  17. r 0 r_{0}
  18. M ¯ \overline{M}
  19. w w
  20. r 0 r_{0}
  21. w = V 3 w=\sqrt[3]{V}
  22. M ¯ i = j z j r i j / w . \overline{M}_{i}=\sum_{j}\frac{z_{j}}{r_{ij}/w}.

Madhava_of_Sangamagrama.html

  1. r θ = r sin θ cos θ - ( 1 / 3 ) r ( sin θ ) 3 ( cos θ ) 3 + ( 1 / 5 ) r ( sin θ ) 5 ( cos θ ) 5 - ( 1 / 7 ) r ( sin θ ) 7 ( cos θ ) 7 + r\theta={\frac{r\sin\theta}{\cos\theta}}-(1/3)\,r\,{\frac{\left(\sin\theta% \right)^{3}}{\left(\cos\theta\right)^{3}}}+(1/5)\,r\,{\frac{\left(\sin\theta% \right)^{5}}{\left(\cos\theta\right)^{5}}}-(1/7)\,r\,{\frac{\left(\sin\theta% \right)^{7}}{\left(\cos\theta\right)^{7}}}+\cdots
  2. θ = tan θ - tan 3 θ 3 + tan 5 θ 5 - tan 7 θ 7 + \theta=\tan\theta-\frac{\tan^{3}\theta}{3}+\frac{\tan^{5}\theta}{5}-\frac{\tan% ^{7}\theta}{7}+\cdots
  3. π 4 = 1 - 1 3 + 1 5 - 1 7 + + ( - 1 ) n 2 n + 1 + \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots+\frac{(-1)^{n}}{2n+% 1}+\cdots
  4. π = 12 ( 1 - 1 3 3 + 1 5 3 2 - 1 7 3 3 + ) \pi=\sqrt{12}\left(1-{1\over 3\cdot 3}+{1\over 5\cdot 3^{2}}-{1\over 7\cdot 3^% {3}}+\cdots\right)

Magic_angle.html

  1. P 2 ( cos θ ) = 0 P_{2}(\cos\theta)=0\,
  2. θ m = arccos 1 3 = arctan 2 0.95532 rad 54.7 \theta_{m}=\rm{arccos}\frac{1}{\sqrt{3}}=\rm{arctan}\sqrt{2}\approx 0.95532rad% \approx 54.7^{\circ}
  3. 3 cos 2 θ - 1 = ( 3 cos 2 θ r - 1 ) ( 3 cos 2 β - 1 ) \left\langle 3{{\cos}^{2}}\theta-1\right\rangle=\left(3{{\cos}^{2}}{{\theta}_{% r}}-1\right)\left(3{{\cos}^{2}}\beta-1\right)
  4. θ \theta
  5. θ r \theta_{r}
  6. β \beta
  7. β \beta
  8. θ r \theta_{r}
  9. θ r = θ m 54.7 \theta_{r}=\theta_{m}\approx 54.7^{\circ}
  10. θ m \theta_{m}

Magnetic_capacitance.html

  1. x C = 1 ω C M x_{C}=\frac{1}{\omega C_{M}}
  2. C M C_{M}
  3. ω \omega
  4. - j x C = - j 1 ω C M = 1 j ω C M -jx_{C}=-j\frac{1}{\omega C_{M}}=\frac{1}{j\omega C_{M}}

Magnetic_capacitivity.html

  1. C M C_{M}
  2. C M = μ r μ 0 S l C_{M}=\mu_{r}\mu_{0}\frac{S}{l}
  3. μ r μ 0 = μ \mu_{r}\mu_{0}=\mu
  4. S S
  5. l l
  6. C M = Φ ϕ M 1 - ϕ M 2 C_{M}=\frac{\Phi}{\phi_{M1}-\phi_{M2}}
  7. ϕ M 1 - ϕ M 2 \phi_{M1}-\phi_{M2}

Magnetic_complex_impedance.html

  1. 1 Ω \frac{1}{\Omega}
  2. Z M = N ˙ I ˙ M = N ˙ m I ˙ M m = z M e j ϕ Z_{M}=\frac{\dot{N}}{\dot{I}_{M}}=\frac{\dot{N}_{m}}{\dot{I}_{M}m}=z_{M}e^{j\phi}
  3. z M = N I M = N m I M m z_{M}=\frac{N}{I_{M}}=\frac{N_{m}}{I_{Mm}}
  4. ϕ = β - α \phi=\beta-\alpha
  5. Z M = z M e j ϕ = z M cos ϕ + j z M sin ϕ = r M + j x M Z_{M}=z_{M}e^{j\phi}=z_{M}\cos\phi+jz_{M}\sin\phi=r_{M}+jx_{M}
  6. r M = z M cos ϕ r_{M}=z_{M}\cos\phi
  7. x M = z M sin ϕ x_{M}=z_{M}\sin\phi
  8. z M = r M 2 + x M 2 z_{M}=\sqrt{r_{M}^{2}+x_{M}^{2}}
  9. ϕ = arctan x M r M \phi=\arctan{\frac{x_{M}}{r_{M}}}

Magnetic_complex_reluctance.html

  1. Z μ = N ˙ Φ ˙ = N ˙ m Φ ˙ m = z μ e j ϕ Z_{\mu}=\frac{\dot{N}}{\dot{\Phi}}=\frac{\dot{N}_{m}}{\dot{\Phi}_{m}}=z_{\mu}e% ^{j\phi}
  2. N ˙ \dot{N}
  3. N ˙ m \dot{N}_{m}
  4. Φ ˙ \dot{\Phi}
  5. Φ ˙ m \dot{\Phi}_{m}
  6. z μ z_{\mu}
  7. e e
  8. j ϕ = j ( β - α ) j\phi=j\left(\beta-\alpha\right)
  9. j j
  10. β \beta
  11. α \alpha
  12. ϕ \phi
  13. Z μ = 1 μ ˙ μ 0 l S Z_{\mu}=\frac{1}{\dot{\mu}\mu_{0}}\frac{l}{S}
  14. l l
  15. S S
  16. μ ˙ μ 0 \dot{\mu}\mu_{0}

Magnetic_effective_resistance.html

  1. r M r_{M}
  2. I M 2 I_{M}^{2}
  3. P = r M I M 2 P=r_{M}I_{M}^{2}
  4. g M g_{M}
  5. g M = r M z M 2 g_{M}=\frac{r_{M}}{z_{M}^{2}}
  6. z M z_{M}

Magnetic_force_microscope.html

  1. F = μ o ( m ) H \vec{F}=\mu_{o}(\vec{m}\cdot\nabla)\vec{H}\,\!
  2. m \vec{m}\,\!
  3. H \vec{H}\,\!
  4. F z = F o cos ( ω t ) , z = z o cos ( ω t + θ ) F_{z}=F_{o}\cos(\omega t),\;z=z_{o}\cos(\omega t+\theta)\,\!
  5. z o = F o m ( ω n 2 - ω 2 ) + ( ω n ω Q ) 2 , θ = arctan [ ω n ω Q ( ω n 2 - ω 2 ) ] z_{o}=\frac{{\frac{F_{o}}{m}}}{\sqrt{(\omega_{n}^{2}-\omega^{2})+(\frac{\omega% _{n}\omega}{Q})^{2}}},\;\theta=\arctan\left[\frac{\omega_{n}\omega}{Q(\omega_{% n}^{2}-\omega^{2})}\right]\,\!
  6. Q = 2 π 1 2 k z o 2 π D z o 2 ω n = 1 2 δ , ω n = k m , δ = D 2 m k Q=2\pi\frac{\frac{1}{2}kz_{o}^{2}}{\pi Dz_{o}^{2}\omega_{n}}=\frac{1}{2\delta}% ,\;\omega_{n}=\sqrt{\frac{k}{m}},\;\delta=\frac{D}{2\sqrt{mk}}\,\!
  7. ω r = ω n 1 - 1 k F z z ω n ( 1 - 1 k F z z ) \omega_{r}=\omega_{n}\sqrt{1-\frac{1}{k}\frac{\partial F_{z}}{\partial z}}% \approx\omega_{n}\left(1-\frac{1}{k}\frac{\partial F_{z}}{\partial z}\right)\,\!
  8. Δ f = f r - f n - f n 2 k F z z \Delta f=f_{r}-f_{n}\approx-\frac{f_{n}}{2k}\frac{\partial F_{z}}{\partial z}\,\!
  9. f = ω 2 π f=\frac{\omega}{2\pi}\,\!
  10. U = - μ o V M H d V U=-\mu_{o}\int\limits_{V}{\vec{M}\cdot\vec{H}\,dV}\,\!
  11. F i = μ o V M H x i d V F_{i}=\mu_{o}\int\limits_{V}{\vec{M}\cdot\frac{\partial\vec{H}}{\partial x_{i}% }\,dV}\,\!

Magnetic_impedance.html

  1. N m N_{m}
  2. I M m I_{Mm}
  3. z M = N I M = N m I M m z_{M}=\frac{N}{I_{M}}=\frac{N_{m}}{I_{Mm}}
  4. r M = z M cos ϕ r_{M}=z_{M}\cos\phi
  5. x M = z M sin ϕ x_{M}=z_{M}\sin\phi
  6. ϕ \phi
  7. ϕ = arctan x M r M \phi=\arctan{\frac{x_{M}}{r_{M}}}

Magnetic_inductance.html

  1. x L = ω L M x_{L}=\omega L_{M}
  2. L M L_{M}
  3. ω \omega
  4. j x L = j ω L M jx_{L}=j\omega L_{M}

Magnetic_reactance.html

  1. 1 Ω \tfrac{1}{\Omega}
  2. x x
  3. X X
  4. x L = ω L M x_{L}=\omega L_{M}
  5. x C = 1 ω C M x_{C}=\tfrac{1}{\omega C_{M}}
  6. ω \omega
  7. L M L_{M}
  8. C M C_{M}
  9. x = x L - x C = ω L M - 1 ω C M x=x_{L}-x_{C}=\omega L_{M}-\frac{1}{\omega C_{M}}
  10. x L = x C x_{L}=x_{C}
  11. x = 0 x=0
  12. x = z 2 - r 2 x=\sqrt{z^{2}-r^{2}}
  13. r = 0 r=0
  14. x = z x=z
  15. ϕ = arctan x r \phi=\arctan{\frac{x}{r}}

Magnetic_reluctance.html

  1. \scriptstyle\mathcal{R}
  2. = Φ \mathcal{R}=\frac{\mathcal{F}}{\Phi}
  3. \scriptstyle\mathcal{R}
  4. \scriptstyle\mathcal{F}
  5. = l μ 0 μ r A \mathcal{R}=\frac{l}{\mu_{0}\mu_{r}A}
  6. = l μ A \mathcal{R}=\frac{l}{\mu A}
  7. μ 0 \scriptstyle\mu_{0}
  8. 7 {}^{−7}
  9. μ r \scriptstyle\mu_{r}
  10. μ \scriptstyle\mu
  11. μ = μ 0 μ r \scriptstyle\mu\;=\;\mu_{0}\mu_{r}
  12. 𝒫 = 1 \mathcal{P}=\frac{1}{\mathcal{R}}

Magnetic_resonance_angiography.html

  1. G b i p G_{bip}
  2. G b i p d t = 0 \int G_{bip}\,dt=0
  3. Δ Φ \Delta\Phi
  4. v x v_{x}
  5. Δ Φ = γ v x Δ m 1 \Delta\Phi=\gamma v_{x}\Delta m_{1}
  6. v x v_{x}
  7. Δ m 1 \Delta m_{1}
  8. v x v_{x}
  9. γ \gamma
  10. Δ Φ \Delta\Phi

Magnetization.html

  1. 𝐌 = d 𝐦 d V \mathbf{M}=\frac{d\mathbf{m}}{dV}
  2. 𝐦 = 𝐌 d V \mathbf{m}=\iiint\mathbf{M}\,dV
  3. 𝐏 = d 𝐩 d V , 𝐩 = 𝐏 d V \mathbf{P}={d\mathbf{p}\over dV},\quad\mathbf{p}=\iiint\mathbf{P}\,dV
  4. m o m e n t = M V moment=MV
  5. M = B r / μ 0 M=B_{r}/\mu_{0}
  6. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) \mathbf{B}=\mu_{0}\mathbf{(H+M)}
  7. 𝐁 = ( 𝐇 + 4 π 𝐌 ) \mathbf{B}=(\mathbf{H}+4\pi\mathbf{M})
  8. 𝐌 = χ m 𝐇 \mathbf{M}=\chi_{m}\mathbf{H}
  9. 𝐉 𝐦 = × 𝐌 \mathbf{J_{m}}=\nabla\times\mathbf{M}
  10. 𝐊 𝐦 = 𝐌 × 𝐧 ^ \mathbf{K_{m}}=\mathbf{M}\times\mathbf{\hat{n}}
  11. 𝐉 = 𝐉 𝐟 + × 𝐌 + 𝐏 t \mathbf{J}=\mathbf{J_{f}}+\nabla\times\mathbf{M}+\frac{\partial\mathbf{P}}{% \partial t}
  12. × 𝐇 = 0 𝐇 = - 𝐌 \begin{aligned}\displaystyle\mathbf{\nabla\times H}&\displaystyle=0\\ \displaystyle\mathbf{\nabla\cdot H}&\displaystyle=-\nabla\cdot\mathbf{M}\end{aligned}
  13. 𝐄 = ρ ϵ 0 × 𝐄 = 0 \begin{aligned}\displaystyle\mathbf{\nabla\cdot E}&\displaystyle=\frac{\rho}{% \epsilon_{0}}\\ \displaystyle\mathbf{\nabla\times E}&\displaystyle=0\end{aligned}

Magnitude_(astronomy).html

  1. 100 5 \sqrt[5]{100}\approx
  2. m - M = 5 ( log 10 d - 1 ) . m-M=5\left(\log_{10}d-1\right).
  3. m 1 - m ref = - 2.5 log 10 ( I 1 I ref ) . m_{1}-m_{\rm ref}=-2.5\log_{10}\left(\frac{I_{1}}{I_{\rm ref}}\right).
  4. × 10 1 0 \times 10^{1}0
  5. × 10 6 \times 10^{−}6
  6. × 10 1 0 \times 10^{1}0
  7. × 10 6 \times 10^{−}6
  8. × 10 1 0 \times 10^{1}0
  9. × 10 6 \times 10^{−}6
  10. × 10 9 \times 10^{9}
  11. × 10 7 \times 10^{−}7
  12. × 10 9 \times 10^{9}
  13. × 10 7 \times 10^{−}7
  14. × 10 8 \times 10^{8}
  15. × 10 8 \times 10^{−}8
  16. × 10 8 \times 10^{8}
  17. × 10 8 \times 10^{−}8
  18. × 10 8 \times 10^{8}
  19. × 10 8 \times 10^{−}8
  20. × 10 7 \times 10^{7}
  21. × 10 9 \times 10^{−}9
  22. × 10 7 \times 10^{7}
  23. × 10 9 \times 10^{−}9
  24. × 10 6 \times 10^{6}
  25. × 10 10 \times 10^{−}10
  26. × 10 6 \times 10^{6}
  27. × 10 10 \times 10^{−}10
  28. × 10 6 \times 10^{6}
  29. × 10 10 \times 10^{−}10
  30. × 10 5 \times 10^{5}
  31. × 10 3 \times 10^{−}3
  32. × 10 11 \times 10^{−}11
  33. × 10 5 \times 10^{5}
  34. × 10 3 \times 10^{−}3
  35. × 10 11 \times 10^{−}11
  36. × 10 4 \times 10^{4}
  37. × 10 4 \times 10^{−}4
  38. × 10 12 \times 10^{−}12
  39. × 10 4 \times 10^{4}
  40. × 10 4 \times 10^{−}4
  41. × 10 12 \times 10^{−}12
  42. × 10 4 \times 10^{4}
  43. × 10 4 \times 10^{−}4
  44. × 10 12 \times 10^{−}12
  45. × 10 3 \times 10^{3}
  46. × 10 5 \times 10^{−}5
  47. × 10 13 \times 10^{−}13
  48. × 10 3 \times 10^{3}
  49. × 10 5 \times 10^{−}5
  50. × 10 13 \times 10^{−}13

Majorana_equation.html

  1. - i / ψ + m ψ c = 0 ( 1 ) -i{\partial\!\!\!\big/}\psi+m\psi_{c}=0\qquad\qquad(1)
  2. / {\partial\!\!\!\big/}
  3. ψ c := i ψ * . \psi_{c}:=i\psi^{*}.
  4. i / ψ c + m ψ = 0 ( 2 ) i{\partial\!\!\!\big/}\psi_{c}+m\psi=0\qquad\qquad(2)
  5. m m
  6. ψ = ψ < s u b > c ψ=ψ<sub>c

Majorization.html

  1. 𝐚 d \mathbf{a}\in\mathbb{R}^{d}
  2. 𝐚 d \mathbf{a}^{\downarrow}\in\mathbb{R}^{d}
  3. 𝐚 , 𝐛 d \mathbf{a},\mathbf{b}\in\mathbb{R}^{d}
  4. 𝐚 \mathbf{a}
  5. 𝐛 \mathbf{b}
  6. 𝐚 w 𝐛 \mathbf{a}\succ_{w}\mathbf{b}
  7. i = 1 k a i i = 1 k b i for k = 1 , , d , \sum_{i=1}^{k}a_{i}^{\downarrow}\geq\sum_{i=1}^{k}b_{i}^{\downarrow}\quad\,% \text{for }k=1,\dots,d,
  8. a i a^{\downarrow}_{i}
  9. b i b^{\downarrow}_{i}
  10. 𝐚 \mathbf{a}
  11. 𝐛 \mathbf{b}
  12. 𝐛 \mathbf{b}
  13. 𝐚 \mathbf{a}
  14. 𝐛 w 𝐚 \mathbf{b}\prec_{w}\mathbf{a}
  15. 𝐚 \mathbf{a}
  16. 𝐛 \mathbf{b}
  17. 𝐚 w 𝐛 \mathbf{a}\succ^{w}\mathbf{b}
  18. i = k d a i i = k d b i for k = 1 , , d , \sum_{i=k}^{d}a_{i}^{\downarrow}\leq\sum_{i=k}^{d}b_{i}^{\downarrow}\quad\,% \text{for }k=1,\dots,d,
  19. 𝐛 \mathbf{b}
  20. 𝐚 \mathbf{a}
  21. 𝐛 w 𝐚 \mathbf{b}\prec^{w}\mathbf{a}
  22. 𝐚 w 𝐛 \mathbf{a}\succ_{w}\mathbf{b}
  23. i = 1 d a i = i = 1 d b i \sum_{i=1}^{d}a_{i}=\sum_{i=1}^{d}b_{i}
  24. 𝐚 \mathbf{a}
  25. 𝐛 \mathbf{b}
  26. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}
  27. 𝐛 \mathbf{b}
  28. 𝐚 \mathbf{a}
  29. 𝐛 𝐚 \mathbf{b}\prec\mathbf{a}
  30. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}
  31. 𝐚 w 𝐛 \mathbf{a}\succ_{w}\mathbf{b}
  32. 𝐚 w 𝐛 \mathbf{a}\succ^{w}\mathbf{b}
  33. 𝐚 \mathbf{a}
  34. 𝐛 \mathbf{b}
  35. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}
  36. 𝐛 𝐚 \mathbf{b}\succ\mathbf{a}
  37. 𝐚 = 𝐛 \mathbf{a}=\mathbf{b}
  38. \succ
  39. \prec
  40. f : d f:\mathbb{R}^{d}\to\mathbb{R}
  41. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}
  42. f ( 𝐚 ) f ( 𝐛 ) f(\mathbf{a})\geq f(\mathbf{b})
  43. f ( 𝐚 ) f(\mathbf{a})
  44. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}
  45. f ( 𝐚 ) f ( 𝐛 ) . f(\mathbf{a})\leq f(\mathbf{b}).
  46. ( 1 , 2 ) ( 0 , 3 ) (1,2)\prec(0,3)
  47. ( 2 , 1 ) ( 3 , 0 ) (2,1)\prec(3,0)
  48. ( 1 , 2 , 3 ) ( 0 , 3 , 3 ) ( 0 , 0 , 6 ) (1,2,3)\prec(0,3,3)\prec(0,0,6)
  49. ( 1 n , , 1 n ) ( 1 n - 1 , , 1 n - 1 , 0 ) ( 1 2 , 1 2 , 0 , , 0 ) ( 1 , 0 , , 0 ) . \left(\frac{1}{n},\ldots,\frac{1}{n}\right)\prec\left(\frac{1}{n-1},\ldots,% \frac{1}{n-1},0\right)\prec\cdots\prec\left(\frac{1}{2},\frac{1}{2},0,\ldots,0% \right)\prec\left(1,0,\ldots,0\right).
  50. ( 1 , 2 , 3 ) w ( 1 , 3 , 3 ) w ( 1 , 3 , 4 ) (1,2,3)\prec_{w}(1,3,3)\prec_{w}(1,3,4)
  51. ( 1 n , , 1 n ) w ( 1 n - 1 , , 1 n - 1 , 1 ) . \left(\frac{1}{n},\ldots,\frac{1}{n}\right)\prec_{w}\left(\frac{1}{n-1},\ldots% ,\frac{1}{n-1},1\right).
  52. 𝐱 , 𝐲 n , \mathbf{x},\mathbf{y}\in\mathbb{R}^{n},
  53. 𝐱 𝐲 \mathbf{x}\prec\mathbf{y}
  54. 𝐱 \mathbf{x}
  55. 𝐲 \mathbf{y}
  56. 𝐲 = ( 3 , 1 ) \mathbf{y}=(3,\,1)
  57. 𝐱 = ( 2 , 2 ) \mathbf{x}=(2,\,2)
  58. 𝐱 𝐲 \mathbf{x}\prec\mathbf{y}
  59. 𝐲 \mathbf{y}
  60. 𝐱 \mathbf{x}
  61. 𝐱 𝐲 \mathbf{x}\prec\mathbf{y}
  62. 𝐲 \mathbf{y}
  63. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}
  64. 𝐛 = D 𝐚 \mathbf{b}=D\mathbf{a}
  65. D D
  66. a a
  67. 𝐚 \mathbf{a}
  68. 𝐛 \mathbf{b}
  69. a i a_{i}
  70. a j < a i a_{j}<a_{i}
  71. a i - ε a_{i}-\varepsilon
  72. a j + ε a_{j}+\varepsilon
  73. ε ( 0 , a i - a j ) \varepsilon\in(0,a_{i}-a_{j})
  74. h : h:\mathbb{R}\to\mathbb{R}
  75. i = 1 d h ( a i ) i = 1 d h ( b i ) \sum_{i=1}^{d}h(a_{i})\geq\sum_{i=1}^{d}h(b_{i})
  76. t j = 1 d | a j - t | j = 1 d | b j - t | \forall t\in\mathbb{R}\quad\sum_{j=1}^{d}|a_{j}-t|\geq\sum_{j=1}^{d}|b_{j}-t|
  77. v , v d v,v^{\prime}\in\mathbb{R}^{d}
  78. v v
  79. v v^{\prime}
  80. ( p 1 , p 2 , , p d ) , i = 1 d p i = 1 (p_{1},p_{2},\ldots,p_{d}),\sum_{i=1}^{d}p_{i}=1
  81. ( P 1 , P 2 , , P d ) (P_{1},P_{2},\ldots,P_{d})
  82. v = i = 1 d p i P i v v^{\prime}=\sum_{i=1}^{d}p_{i}P_{i}v
  83. D D
  84. v D = v vD=v^{\prime}
  85. H H
  86. H H^{\prime}
  87. H H
  88. H H^{\prime}
  89. f , g : f,g:\mathbb{N}\to\mathbb{N}\,\!
  90. f f\,\!
  91. g g\,\!
  92. x x\,\!
  93. f ( x ) g ( x ) f(x)\geq g(x)\,\!
  94. n n\,\!
  95. f ( x ) g ( x ) f(x)\geq g(x)\,\!
  96. x > n x>n\,\!
  97. f f\,\!
  98. g g\,\!
  99. f ( x ) > g ( x ) f(x)>g(x)\,\!
  100. f ( x ) g ( x ) f(x)\geq g(x)\,\!

Malecot's_method_of_coancestry.html

  1. f f
  2. f f
  3. f f
  4. f f
  5. f 1 f\rightarrow 1
  6. N N
  7. f i f_{i}
  8. i i
  9. f f
  10. k 1 k\gg 1
  11. N N
  12. n n
  13. f n = k - 1 k N + k ( N - 1 ) k N f n - 1 f_{n}=\frac{k-1}{kN}+\frac{k(N-1)}{kN}f_{n-1}
  14. 1 N + ( 1 - 1 N ) f n - 1 . \approx\frac{1}{N}+(1-\frac{1}{N})f_{n-1}.
  15. f 0 = 0 f_{0}=0
  16. f n = 1 - ( 1 - 1 N ) n . f_{n}=1-(1-\frac{1}{N})^{n}.
  17. n ¯ = - 1 / log ( 1 - 1 / N ) N . \bar{n}=-1/\log(1-1/N)\approx N.
  18. N N
  19. 2 N 2N

Malmquist_bias.html

  1. M ¯ \overline{M}
  2. Δ M ¯ = M ¯ - M 0 \overline{\Delta M}=\overline{M}-M_{0}
  3. Δ M ¯ = - σ 2 d ln A ( m lim ) d m lim \overline{\Delta M}=-\sigma^{2}\frac{\operatorname{d}\ln A(m_{\lim})}{% \operatorname{d}m_{\lim}}
  4. Δ M ¯ = - 1.382 σ 2 \overline{\Delta M}=-1.382\sigma^{2}
  5. D C = D A ( 1 + z ) D_{C}=D_{A}(1+z)
  6. V C = V A ( 1 + z ) 3 V_{C}=V_{A}(1+z)^{3}
  7. M ¯ = α P + β \overline{M}=\alpha P+\beta

Manifold.html

  1. χ top ( x , y ) = x . \chi_{\mathrm{top}}(x,y)=x.\,
  2. χ bottom ( x , y ) = x \chi_{\mathrm{bottom}}(x,y)=x
  3. χ left ( x , y ) = y \chi_{\mathrm{left}}(x,y)=y
  4. χ right ( x , y ) = y . \chi_{\mathrm{right}}(x,y)=y.
  5. T ( a ) \displaystyle T(a)
  6. χ minus ( x , y ) = s = y 1 + x \chi_{\mathrm{minus}}(x,y)=s=\frac{y}{1+x}
  7. χ plus ( x , y ) = t = y 1 - x \chi_{\mathrm{plus}}(x,y)=t=\frac{y}{1-x}{}
  8. x \displaystyle x
  9. t = 1 s t=\frac{1}{s}
  10. z = 0 z=0
  11. z > 0 z>0
  12. x y xy
  13. V - E + F = 2. V-E+F=2.
  14. θ ( y ) \theta(y)
  15. θ ( y ) \theta^{\prime}(y^{\prime})
  16. y y
  17. y y^{\prime}
  18. y y
  19. y y^{\prime}
  20. y = 1 - x 2 y=\sqrt{1-x^{2}}
  21. y = - 1 - x 2 y=-\sqrt{1-x^{2}}
  22. x = 1 - y 2 x=\sqrt{1-y^{2}}
  23. x = - 1 - y 2 x=-\sqrt{1-y^{2}}
  24. x 2 + y 2 - 1 = 0 x^{2}+y^{2}-1=0
  25. 𝐁 n = { ( x 1 , x 2 , , x n ) n x 1 2 + x 2 2 + + x n 2 < 1 } . \mathbf{B}^{n}=\{(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}\mid x_{1}^{2}+x_{2% }^{2}+\cdots+x_{n}^{2}<1\}.
  26. S = { ( x , y , z ) 𝐑 3 | x 2 + y 2 + z 2 = 1 } . S=\{(x,y,z)\in\mathbf{R}^{3}|x^{2}+y^{2}+z^{2}=1\}.
  27. χ ( x , y , z ) = ( x , y ) , \chi(x,y,z)=(x,y),
  28. 𝐑 n { 0 } 𝐑 n { 0 } : x x / x 2 . \mathbf{R}^{n}\setminus\{0\}\to\mathbf{R}^{n}\setminus\{0\}:x\mapsto x/\|x\|^{% 2}.
  29. \C n \C^{n}
  30. \C n \C^{n}
  31. f : M 𝐑 f\colon M\to\mathbf{R}
  32. f : M 𝐂 , f\colon M\to\mathbf{C},
  33. M M
  34. M M
  35. C \R M C\subseteq\R^{M}
  36. M M
  37. C C
  38. H C ( \R i ) H\in C^{\infty}(\R^{i})
  39. i 𝒩 i\in\mathcal{N}
  40. f 1 , , f n C f_{1},\dots,f_{n}\in C
  41. H ( f 1 , , f n ) C H\circ(f_{1},\dots,f_{n})\in C
  42. M M
  43. C C
  44. C C
  45. ( M , C ) (M,C)
  46. ϵ \epsilon

Manning_formula.html

  1. V = k n R h 2 / 3 S 1 / 2 V=\frac{k}{n}{R_{h}}^{2/3}\,S^{1/2}
  2. R h = A P R_{h}=\frac{A}{P}

Margin_(finance).html

  1. P = P 0 ( 1 - initial margin requirement ) ( 1 - maintenance margin requirement ) \textstyle P=P_{0}\frac{(1-\,\text{initial margin requirement})}{(1-\,\text{% maintenance margin requirement})}
  2. P = $ 50 ( 1 - 0.6 ) ( 1 - 0.25 ) = $ 26.66. P=\$50\frac{(1-0.6)}{(1-0.25)}=\$26.66.

Marginal_propensity_to_import.html

  1. M P M = d I d Y MPM=\frac{dI}{dY}

Marginal_value.html

  1. y = f ( x 1 , x 2 , , x n ) y=f\left(x_{1},x_{2},\ldots,x_{n}\right)
  2. x i x_{i}
  3. x i , 0 x_{i,0}
  4. x i , 1 x_{i,1}
  5. x i x_{i}
  6. Δ x i = x i , 1 - x i , 0 \Delta x_{i}=x_{i,1}-x_{i,0}
  7. y y
  8. Δ y = f ( x 1 , x 2 , , x i , 1 , , x n ) - f ( x 1 , x 2 , , x i , 0 , , x n ) \Delta y=f\left(x_{1},x_{2},\ldots,x_{i,1},\ldots,x_{n}\right)-f\left(x_{1},x_% {2},\ldots,x_{i,0},\ldots,x_{n}\right)
  9. Δ y Δ x = f ( x 1 , x 2 , , x i , 1 , , x n ) - f ( x 1 , x 2 , , x i , 0 , , x n ) x i , 1 - x i , 0 \frac{\Delta y}{\Delta x}=\frac{f\left(x_{1},x_{2},\ldots,x_{i,1},\ldots,x_{n}% \right)-f\left(x_{1},x_{2},\ldots,x_{i,0},\ldots,x_{n}\right)}{x_{i,1}-x_{i,0}}
  10. x i x_{i}
  11. d x i dx_{i}
  12. y y
  13. y x i = f ( x 1 , x 2 , , x n ) x i \frac{\partial y}{\partial x_{i}}=\frac{\partial f\left(x_{1},x_{2},\ldots,x_{% n}\right)}{\partial x_{i}}
  14. y = a + b x y=a+b\cdot x
  15. y y
  16. x x
  17. b b
  18. x x
  19. y = a b x y=a\cdot b^{x}
  20. y y
  21. x x
  22. C = C ( Y ) C=C\left(Y\right)
  23. Y Y
  24. M P C = d C d Y MPC=\frac{dC}{dY}

Mark_Six.html

  1. 1 < m t p l > ( 49 6 ) = 1 13 , 983 , 816 \frac{1}{<}mtpl>{{49\choose 6}}=\frac{1}{13,983,816}
  2. < m t p l > ( 6 5 ) ( 49 6 ) = 1 2 , 330 , 636 \frac{<}{m}tpl>{{6\choose 5}}{{49\choose 6}}=\frac{1}{2,330,636}
  3. ( 6 5 ) ( 42 1 ) < m t p l > ( 49 6 ) 1 55 , 491.33 \frac{{6\choose 5}{42\choose 1}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{55,491.% 33}
  4. ( 6 4 ) ( 42 1 ) < m t p l > ( 49 6 ) 1 22 , 196.53 \frac{{6\choose 4}{42\choose 1}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{22,196.% 53}
  5. ( 6 4 ) ( 42 2 ) < m t p l > ( 49 6 ) 1 1 , 082.76 \frac{{6\choose 4}{42\choose 2}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{1,082.76}
  6. ( 6 3 ) ( 42 2 ) < m t p l > ( 49 6 ) 1 812.07 \frac{{6\choose 3}{42\choose 2}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{812.07}
  7. ( 6 3 ) ( 42 3 ) < m t p l > ( 49 6 ) 1 60.9 \frac{{6\choose 3}{42\choose 3}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{60.9}
  8. ( 6 2 ) ( 42 3 ) < m t p l > ( 49 6 ) 1 81.207 \frac{{6\choose 2}{42\choose 3}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{81.207}
  9. ( 6 2 ) ( 42 4 ) < m t p l > ( 49 6 ) 1 8.33 \frac{{6\choose 2}{42\choose 4}}{<}mtpl>{{49\choose 6}}\approx\frac{1}{8.33}

Markov_number.html

  1. x 2 + y 2 + z 2 = 3 x y z , x^{2}+y^{2}+z^{2}=3xyz,\,
  2. ( 1 , F 2 n - 1 , F 2 n + 1 ) , (1,F_{2n-1},F_{2n+1}),\,
  3. ( 2 , P 2 n - 1 , P 2 n + 1 ) , (2,P_{2n-1},P_{2n+1}),\,
  4. m n = 1 3 e C n + o ( 1 ) with C = 2.3523418721 . m_{n}=\tfrac{1}{3}e^{C\sqrt{n}+o(1)}\quad\,\text{with }C=2.3523418721\ldots.
  5. x 2 + y 2 + z 2 = 3 x y z + 4 / 9 x^{2}+y^{2}+z^{2}=3xyz+4/9
  6. f ( x ) + f ( y ) = f ( z ) f(x)+f(y)=f(z)
  7. L n = 9 - 4 m n 2 . L_{n}=\sqrt{9-{4\over{m_{n}}^{2}}}.\,
  8. f ( x , y ) = a x 2 + b x y + c y 2 f(x,y)=ax^{2}+bxy+cy^{2}\,
  9. D = b 2 - 4 a c D=b^{2}-4ac
  10. D 3 \frac{\sqrt{D}}{3}
  11. p x 2 + ( 3 p - 2 a ) x y + ( b - 3 a ) y 2 px^{2}+(3p-2a)xy+(b-3a)y^{2}\,
  12. 0 < a < p / 2 , a q ± r mod p , b p - a 2 = 1 0<a<p/2,aq\equiv\pm r\bmod p,bp-a^{2}=1\,

Martingale_central_limit_theorem.html

  1. X 1 , X 2 , X_{1},X_{2},\dots\,
  2. E [ X t + 1 - X t | X 1 , , X t ] = 0 , \operatorname{E}[X_{t+1}-X_{t}|X_{1},\dots,X_{t}]=0\,,
  3. | X t + 1 - X t | k |X_{t+1}-X_{t}|\leq k
  4. | X 1 | k |X_{1}|\leq k
  5. σ t 2 = E [ ( X t + 1 - X t ) 2 | X 1 , , X t ] , \sigma_{t}^{2}=\operatorname{E}[(X_{t+1}-X_{t})^{2}|X_{1},\ldots,X_{t}],
  6. τ ν = min { t : i = 1 t σ i 2 ν } . \tau_{\nu}=\min\left\{t:\sum_{i=1}^{t}\sigma_{i}^{2}\geq\nu\right\}.
  7. X τ ν ν \frac{X_{\tau_{\nu}}}{\sqrt{\nu}}
  8. ν + \nu\to+\infty\!
  9. lim ν + P ( X τ ν ν < x ) = Φ ( x ) = 1 2 π - x exp ( - u 2 2 ) d u , x . \lim_{\nu\to+\infty}\operatorname{P}\left(\frac{X_{\tau_{\nu}}}{\sqrt{\nu}}<x% \right)=\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}\exp\left(-\frac{u^{2}}% {2}\right)\,du,\quad x\in\mathbb{R}.

Mass_balance.html

  1. Input = Output + Accumulation \mathrm{Input}=\mathrm{Output}+\mathrm{Accumulation}\,
  2. Input + Generation = Output + Accumulation + Consumption \,\text{Input}+\,\text{Generation}=\,\text{Output}+\,\text{Accumulation}\ +\,% \text{Consumption}
  3. IN + PROD = OUT + ACC \mathrm{IN}+\mathrm{PROD}=\mathrm{OUT}+\mathrm{ACC}
  4. 0 + r A V = 0 + d n A d t 0+r_{\mathrm{A}}V=0+\frac{dn_{\mathrm{A}}}{dt}
  5. Carbon : mass of air mass of C = 4.773 × 28.96 12.01 = 11.51 \mathrm{Carbon:}\frac{\mathrm{mass\ of\ air}}{\mathrm{mass\ of\ C}}=\frac{4.77% 3\times 28.96}{12.01}=11.51
  6. Hydrogen : mass of air mass of H = 1 4 ( 4.773 ) × 28.96 1.008 = 34.28 \mathrm{Hydrogen:}\frac{\mathrm{mass\ of\ air}}{\mathrm{mass\ of\ H}}=\frac{% \frac{1}{4}(4.773)\times 28.96}{1.008}=34.28
  7. Sulfur : mass of air mass of S = 4.773 × 28.96 32.06 = 4.31 \mathrm{Sulfur:}\frac{\mathrm{mass\ of\ air}}{\mathrm{mass\ of\ S}}=\frac{4.77% 3\times 28.96}{32.06}=4.31
  8. mass of air mass of fuel = A F R m a s s = 11.5 ( w C ) + 34.3 ( w H ) + ( w S - w O ) \frac{\mathrm{mass\ of\ air}}{\mathrm{mass\ of\ fuel}}=AFR_{mass}=11.5(wC)+34.% 3(wH)+(wS-wO)
  9. % excess air = 1.2804 × ( % O 2 in combustion gas ) 2 + 4.49 × ( % O 2 in combustion gas ) \mathrm{\%\ excess\ air}=1.2804\times(\mathrm{\%O_{2}\ in\ combustion\ gas})^{% 2}+4.49\times(\mathrm{\%O_{2}\ in\ combustion\ gas})
  10. % O 2 in combustion gas = - 0.00138 × ( % excess air ) 2 + 0.210 × ( % excess air ) \mathrm{\%O_{2}\ in\ combustion\ gas}=-0.00138\times(\mathrm{\%\ excess\ air})% ^{2}+0.210\times(\mathrm{\%\ excess\ air})
  11. a A + b B c C + d D a\mathrm{A}+b\mathrm{B}\leftrightarrow c\mathrm{C}+d\mathrm{D}
  12. IN + PROD = OUT + ACC \mathrm{IN}+\mathrm{PROD}=\mathrm{OUT}+\mathrm{ACC}
  13. 0 + r A V = 0 + d n A d t 0+r_{\mathrm{A}}V=0+\frac{dn_{\mathrm{A}}}{dt}
  14. n A = V * C A n_{\mathrm{A}}=V*C_{\mathrm{A}}
  15. r A V = V d C A d t r_{\mathrm{A}}V=V\frac{dC_{\mathrm{A}}}{dt}
  16. r A = d C A d t r_{\mathrm{A}}=\frac{dC_{\mathrm{A}}}{dt}
  17. r 1 = k 1 [ A ] a [ B ] b r_{1}=k_{1}[\mathrm{A}]^{a}[\mathrm{B}]^{b}
  18. r - 1 = k - 1 1 [ C ] c [ D ] d r_{-1}=k_{-1}1[\mathrm{C}]^{c}[\mathrm{D}]^{d}
  19. r A = a ( r - 1 - r 1 ) r_{\mathrm{A}}=a(r_{-1}-r_{1})
  20. r A = a ( r - 1 - r 1 ) = d C A d t = 0 r_{\mathrm{A}}=a(r_{-1}-r_{1})=\frac{dC_{\mathrm{A}}}{dt}=0
  21. k 1 k - 1 = [ C ] c [ D ] d [ A ] a [ B ] b = K e q \frac{k_{1}}{k_{-1}}=\frac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[% \mathrm{B}]^{b}}=K_{eq}
  22. IN + PROD = OUT + ACC \mathrm{IN}+\mathrm{PROD}=\mathrm{OUT}+\mathrm{ACC}
  23. Q 0 C A , 0 + r A V = Q C A + d n A d t Q_{0}\cdot C_{\mathrm{A},0}+r_{A}\cdot V=Q\cdot C_{\mathrm{A}}+\frac{dn_{% \mathrm{A}}}{dt}
  24. ACC = 0 \mathrm{ACC}=0
  25. PROD = 0 \mathrm{PROD}=0
  26. Q = 0 Q=0
  27. IN + PROD = OUT + ACC \mathrm{IN}+\mathrm{PROD}=\mathrm{OUT}+\mathrm{ACC}
  28. Q 0 C A , 0 + 0 = 0 C A + d n A d t Q_{0}\cdot C_{\mathrm{A},0}+0=0\cdot C_{\mathrm{A}}+\frac{dn_{\mathrm{A}}}{dt}
  29. Q 0 C A , 0 = d C A V d t = V d C A d t + C A d V d t Q_{0}\cdot C_{\mathrm{A},0}=\frac{dC_{\mathrm{A}}V}{dt}=V\frac{dC_{\mathrm{A}}% }{dt}+C_{\mathrm{A}}\frac{dV}{dt}
  30. d V d t = Q 0 \frac{dV}{dt}=Q_{0}
  31. V = V t = 0 + Q 0 t V=V_{t=0}+Q_{0}t
  32. d C A d t = Q 0 ( V t = 0 + Q 0 t ) ( C A , 0 - C A ) \frac{dC_{\mathrm{A}}}{dt}=\frac{Q_{0}}{(V_{t=0}+Q_{0}t)}\left(C_{\mathrm{A},0% }-C_{\mathrm{A}}\right)
  33. d C A d t 0 \frac{dC_{\mathrm{A}}}{dt}\neq 0
  34. d C d t \frac{dC}{dt}
  35. d C d t \frac{dC}{dt}
  36. v = d C A d t v=\frac{dC_{\mathrm{A}}}{dt}
  37. d ( Q C A ) d V = r A \frac{d(Q\cdot C_{A})}{dV}=r_{A}

Mass_flow_rate.html

  1. m ˙ \dot{m}
  2. m ˙ = lim Δ t 0 Δ m Δ t = d m d t \dot{m}=\lim\limits_{\Delta t\rightarrow 0}\frac{\Delta m}{\Delta t}=\frac{{% \rm d}m}{{\rm d}t}
  3. m ˙ = ρ V ˙ = ρ v A = j m A \dot{m}=\rho\cdot\dot{V}=\rho\cdot{v}\cdot{A}={j}_{\rm m}\cdot{A}
  4. V ˙ \dot{V}
  5. m ˙ = A ρ v d A = A j m d A \dot{m}=\iint_{A}\rho{v}\cdot{\rm d}{A}=\iint_{A}{j}_{\rm m}\cdot{\rm d}{A}
  6. n ^ {\hat{n}}
  7. A = A n ^ {A}=A{\hat{n}}
  8. m ˙ = ρ v A cos θ \dot{m}=\rho vA\cos\theta
  9. n ^ {\hat{n}}
  10. cos θ \cos\theta
  11. m ˙ = ρ v A cos ( π / 2 ) = 0 \dot{m}=\rho vA\cos(\pi/2)=0
  12. m ˙ s = v s ρ = m ˙ / A \dot{m}_{s}=v_{s}\cdot\rho=\dot{m}/A
  13. ρ 1 v 1 A 1 = ρ 2 v 2 A 2 \rho_{1}{v}_{1}\cdot{A}_{1}=\rho_{2}{v}_{2}\cdot{A}_{2}

Mass_ratio.html

  1. Δ v \Delta v
  2. Δ v = v e ln m 0 m 1 \Delta v=v_{e}\ln\frac{m_{0}}{m_{1}}
  3. m 0 m 1 = e Δ v / v e \frac{m_{0}}{m_{1}}=e^{\Delta v/v_{e}}
  4. n n
  5. e n e^{n}
  6. Δ v \Delta v
  7. e 2.5 e^{2.5}
  8. n n
  9. e n e^{n}
  10. M R = m 1 m 0 M_{R}=\frac{m_{1}}{m_{0}}

Massey_product.html

  1. Γ \Gamma
  2. d d
  3. H * ( Γ ) H^{*}(\Gamma)
  4. u ¯ \bar{u}
  5. ( - 1 ) d e g ( u ) + 1 u (-1)^{deg(u)+1}u
  6. u u
  7. Γ \Gamma
  8. [ u ] , [ v ] , [ w ] = { [ s ¯ w + u ¯ t ] d s = u ¯ v , d t = v ¯ w } . \langle[u],[v],[w]\rangle=\{[\bar{s}w+\bar{u}t]\mid ds=\bar{u}v,dt=\bar{v}w\}.
  9. H * ( Γ ) H^{*}(\Gamma)
  10. H * ( Γ ) H^{*}(\Gamma)
  11. u , v , w u,v,w
  12. i , j , k , i,j,k,
  13. i + j + k - 1 , i+j+k-1,
  14. u v uv
  15. v w vw
  16. H ( Γ ) / ( [ u ] H ( Γ ) + H ( Γ ) [ w ] ) . \displaystyle H(\Gamma)/([u]H(\Gamma)+H(\Gamma)[w]).
  17. [ u ] [ v ] [u][v]
  18. [ v ] [ w ] [v][w]
  19. [ u ] [ v ] = [ v ] [ w ] = 0 [u][v]=[v][w]=0
  20. u v = d s uv=ds
  21. v w = d t vw=dt
  22. s s
  23. t t
  24. [ u ] [ v ] [ w ] [u][v][w]
  25. s w sw
  26. u t ut
  27. d ( s w ) = d s w + s d w , d(sw)=ds\cdot w+s\cdot dw,
  28. [ d w ] = 0 [dw]=0
  29. s s
  30. t t
  31. s w sw
  32. u t ut
  33. a ¯ 1 , 1 a 2 , n + a ¯ 1 , 2 a 3 , n + + a ¯ 1 , n - 1 a n , n \bar{a}_{1,1}a_{2,n}+\bar{a}_{1,2}a_{3,n}+\cdots+\bar{a}_{1,n-1}a_{n,n}
  34. d a i , j = a ¯ i , i a i + 1 , j + a ¯ i , i + 1 a i + 1 , j + + a ¯ i , j - 1 a j , j da_{i,j}=\bar{a}_{i,i}a_{i+1,j}+\bar{a}_{i,i+1}a_{i+1,j}+\cdots+\bar{a}_{i,j-1% }a_{j,j}
  35. d 2 p + 1 d_{2p+1}

Massive_gravity.html

  1. h μ ν h_{\mu\nu}
  2. g μ ν = η μ ν + M Pl - 1 h μ ν g_{\mu\nu}=\eta_{\mu\nu}+M_{\mathrm{Pl}}^{-1}h_{\mu\nu}
  3. M Pl = ( 8 π G ) - 1 / 2 M_{\mathrm{Pl}}=(8\pi G)^{-1/2}
  4. G G
  5. h μ ν h_{\mu\nu}
  6. h μ ν T μ ν h^{\mu\nu}T_{\mu\nu}
  7. T μ ν T_{\mu\nu}
  8. h μ ν h_{\mu\nu}
  9. h μ ν h_{\mu\nu}
  10. int = a h μ ν h μ ν + b ( η μ ν h μ ν ) 2 . \mathcal{L}_{\mathrm{int}}=ah^{\mu\nu}h_{\mu\nu}+b\left(\eta^{\mu\nu}h_{\mu\nu% }\right)^{2}.
  11. a = - b a=-b
  12. FP = m 2 ( h μ ν h μ ν - ( η μ ν h μ ν ) 2 ) \mathcal{L}_{\mathrm{FP}}=m^{2}\left(h^{\mu\nu}h_{\mu\nu}-\left(\eta^{\mu\nu}h% _{\mu\nu}\right)^{2}\right)
  13. m 0 m\to 0
  14. h μ ν h_{\mu\nu}
  15. a = - b a=-b
  16. h μ ν T μ ν h^{\mu\nu}T_{\mu\nu}
  17. S = d 4 x - g ( - M Pl 2 2 R + m 2 M Pl 2 n = 0 4 α n e n ( 𝕂 ) + m ( g , Φ i ) ) , S=\int d^{4}x\sqrt{-g}\left(-\frac{M_{\mathrm{Pl}}^{2}}{2}R+m^{2}M_{\mathrm{Pl% }}^{2}\displaystyle\sum_{n=0}^{4}\alpha_{n}e_{n}(\mathbb{K})+\mathcal{L}_{% \mathrm{m}}(g,\Phi_{i})\right),
  18. S = d 4 x - g ( - M Pl 2 2 R + m 2 M Pl 2 n = 0 4 β n e n ( 𝕏 ) + m ( g , Φ i ) ) . S=\int d^{4}x\sqrt{-g}\left(-\frac{M_{\mathrm{Pl}}^{2}}{2}R+m^{2}M_{\mathrm{Pl% }}^{2}\displaystyle\sum_{n=0}^{4}\beta_{n}e_{n}(\mathbb{X})+\mathcal{L}_{% \mathrm{m}}(g,\Phi_{i})\right).
  19. R R
  20. m \mathcal{L}_{\mathrm{m}}
  21. Φ i \Phi_{i}
  22. m m
  23. β i \beta_{i}
  24. 𝒪 ( 1 ) \mathcal{O}(1)
  25. e n e_{n}
  26. 𝕂 = 𝕀 - g - 1 f \mathbb{K}=\mathbb{I}-\sqrt{g^{-1}f}
  27. 𝕏 = g - 1 f \mathbb{X}=\sqrt{g^{-1}f}
  28. α i \alpha_{i}
  29. β i \beta_{i}
  30. g - 1 f \sqrt{g^{-1}f}
  31. g - 1 f g^{-1}f
  32. 𝕏 \mathbb{X}
  33. X μ X α α = ν g μ α f ν α . X^{\mu}{}_{\alpha}X^{\alpha}{}_{\nu}=g^{\mu\alpha}f_{\nu\alpha}.
  34. f μ ν f_{\mu\nu}
  35. g μ ν g_{\mu\nu}
  36. g μ α g α ν = δ ν μ g^{\mu\alpha}g_{\alpha\nu}=\delta^{\mu}_{\nu}
  37. det g \operatorname{det}g
  38. f μ ν f_{\mu\nu}
  39. f μ ν = η μ ν f_{\mu\nu}=\eta_{\mu\nu}
  40. f μ ν f_{\mu\nu}
  41. f μ ν f_{\mu\nu}
  42. m m
  43. M Pl M_{\mathrm{Pl}}
  44. f μ ν f_{\mu\nu}
  45. 𝕏 \mathbb{X}
  46. 𝕂 \mathbb{K}
  47. e n e_{n}
  48. 𝕏 \mathbb{X}
  49. e 0 ( 𝕏 ) = 1 , e 1 ( 𝕏 ) = [ 𝕏 ] , e 2 ( 𝕏 ) = 1 2 ( [ 𝕏 ] 2 - [ 𝕏 2 ] ) , e 3 ( 𝕏 ) = 1 6 ( [ 𝕏 ] 3 - 3 [ 𝕏 ] [ 𝕏 2 ] + 2 [ 𝕏 3 ] ) , e 4 ( 𝕏 ) = det 𝕏 , \begin{aligned}\displaystyle e_{0}(\mathbb{X})&\displaystyle=1,\\ \displaystyle e_{1}(\mathbb{X})&\displaystyle=[\mathbb{X}],\\ \displaystyle e_{2}(\mathbb{X})&\displaystyle=\frac{1}{2}\left([\mathbb{X}]^{2% }-[\mathbb{X}^{2}]\right),\\ \displaystyle e_{3}(\mathbb{X})&\displaystyle=\frac{1}{6}\left([\mathbb{X}]^{3% }-3[\mathbb{X}][\mathbb{X}^{2}]+2[\mathbb{X}^{3}]\right),\\ \displaystyle e_{4}(\mathbb{X})&\displaystyle=\operatorname{det}\mathbb{X},% \end{aligned}
  50. [ 𝕏 ] X μ μ [\mathbb{X}]\equiv X^{\mu}{}_{\mu}
  51. e n e_{n}
  52. 𝕏 \mathbb{X}
  53. 𝕂 = 𝕀 - 𝕏 \mathbb{K}=\mathbb{I}-\mathbb{X}
  54. 𝕀 \mathbb{I}
  55. β n = ( 4 - n ) ! i = n 4 ( - 1 ) i + n ( 4 - i ) ! ( i - n ) ! α i . \beta_{n}=(4-n)!\displaystyle\sum_{i=n}^{4}\frac{(-1)^{i+n}}{(4-i)!(i-n)!}% \alpha_{i}.
  56. g μ ν = η a b e a , μ f μ ν = η a b f a , μ \begin{aligned}\displaystyle g_{\mu\nu}=\eta_{ab}e^{a}{}_{\mu},\\ \displaystyle f_{\mu\nu}=\eta_{ab}f^{a}{}_{\mu},\end{aligned}
  57. 𝐞 a = e a d μ x μ , 𝐟 a = f a d μ x μ , \begin{aligned}\displaystyle\mathbf{e}^{a}=e^{a}{}_{\mu}dx^{\mu},\\ \displaystyle\mathbf{f}^{a}=f^{a}{}_{\mu}dx^{\mu},\end{aligned}
  58. e 0 ( 𝕏 ) ϵ a b c d 𝐞 a 𝐞 b 𝐞 c 𝐞 d e 1 ( 𝕏 ) ϵ a b c d 𝐞 a 𝐞 b 𝐞 c 𝐟 d e 2 ( 𝕏 ) ϵ a b c d 𝐞 a 𝐞 b 𝐟 c 𝐟 d e 3 ( 𝕏 ) ϵ a b c d 𝐞 a 𝐟 b 𝐟 c 𝐟 d e 4 ( 𝕏 ) ϵ a b c d 𝐟 a 𝐟 b 𝐟 c 𝐟 d \begin{aligned}\displaystyle e_{0}(\mathbb{X})\propto\epsilon_{abcd}\mathbf{e}% ^{a}\wedge\mathbf{e}^{b}\wedge\mathbf{e}^{c}\wedge\mathbf{e}^{d}\\ \displaystyle e_{1}(\mathbb{X})\propto\epsilon_{abcd}\mathbf{e}^{a}\wedge% \mathbf{e}^{b}\wedge\mathbf{e}^{c}\wedge\mathbf{f}^{d}\\ \displaystyle e_{2}(\mathbb{X})\propto\epsilon_{abcd}\mathbf{e}^{a}\wedge% \mathbf{e}^{b}\wedge\mathbf{f}^{c}\wedge\mathbf{f}^{d}\\ \displaystyle e_{3}(\mathbb{X})\propto\epsilon_{abcd}\mathbf{e}^{a}\wedge% \mathbf{f}^{b}\wedge\mathbf{f}^{c}\wedge\mathbf{f}^{d}\\ \displaystyle e_{4}(\mathbb{X})\propto\epsilon_{abcd}\mathbf{f}^{a}\wedge% \mathbf{f}^{b}\wedge\mathbf{f}^{c}\wedge\mathbf{f}^{d}\\ \end{aligned}
  59. ( e - 1 ) a f b ν μ = ( e - 1 ) b f a ν μ (e^{-1})_{a}{}^{\mu}f_{b\nu}=(e^{-1})_{b}{}^{\mu}f_{a\nu}
  60. m m
  61. H 0 H_{0}
  62. m = 0 m=0
  63. m H 0 m\sim H_{0}
  64. f μ ν f_{\mu\nu}
  65. S = M 3 2 d 3 x - g ( R - 2 Λ ) + 1 4 μ ϵ λ μ ν Γ λ σ ρ ( μ Γ ρ ν σ + 2 3 Γ μ α σ Γ ν ρ α ) , S=\frac{M_{3}}{2}\int d^{3}x\sqrt{-g}(R-2\Lambda)+\frac{1}{4\mu}\epsilon^{% \lambda\mu\nu}\Gamma^{\rho}_{\lambda\sigma}\left(\partial_{\mu}\Gamma^{\sigma}% _{\rho\nu}+\frac{2}{3}\Gamma^{\sigma}_{\mu\alpha}\Gamma^{\alpha}_{\nu\rho}% \right),
  66. M 3 M_{3}
  67. S = M 3 d 3 x - g [ ± R + 1 m 2 ( R μ ν R μ ν - 3 8 R 2 ) ] . S=M_{3}\int d^{3}x\sqrt{-g}\left[\pm R+\frac{1}{m^{2}}\left(R_{\mu\nu}R^{\mu% \nu}-\frac{3}{8}R^{2}\right)\right].

Mathematical_joke.html

  1. 31 8 = 25 10 31_{8}=25_{10}
  2. i 2 = - 1 i^{2}=-1\,
  3. ( lim x 8 + 1 x - 8 = ) ( lim x 3 + 1 x - 3 = ω ) \left(\lim_{x\to 8^{+}}\frac{1}{x-8}=\infty\right)\Rightarrow\left(\lim_{x\to 3% ^{+}}\frac{1}{x-3}=\omega\right)
  4. d d x ( x ) = 1 x x = 1 \frac{d}{dx}(x)=\frac{1}{x}x=1
  5. 64 16 = / 6 4 1 / 6 = 4 1 = 4 \frac{64}{16}=\frac{/\!\!\!{6}\;{4}}{{1}\;/\!\!\!{6}}=\frac{4}{1}=4
  6. b \displaystyle b
  7. 1 = 2 1=2
  8. J = ( 7 e - 1 / e - 9 ) π 2 = 867.5309 . J=(7^{e-1/e}-9)\cdot\pi^{2}=867.5309\ldots.
  9. 12 + 144 + 20 + 3 4 7 + ( 5 × 11 ) = 9 2 + 0 \frac{12+144+20+3\sqrt{4}}{7}+(5\times 11)=9^{2}+0

Mathematics_of_Sudoku.html

  1. n n \mathbb{Z}_{n}\oplus\mathbb{Z}_{n}
  2. n n
  3. n \mathbb{Z}_{n}
  4. 0
  5. n n \mathbb{Z}_{n}\oplus\mathbb{Z}_{n}
  6. n = 3 n=3
  7. 3 \mathbb{Z}_{3}
  8. 3 \mathbb{Z}_{3}
  9. 9 9
  10. 1 , 4 , 7 , 2 , 5 , 8 , 3 , 6 , 9 1,4,7,2,5,8,3,6,9
  11. 3 3
  12. 3 \mathbb{Z}_{3}
  13. 4 \mathbb{Z}_{4}
  14. 2 \mathbb{Z}_{2}
  15. Number of Grids b R , C C × b C , R R ( R C ) ! R C \mbox{Number of Grids}~{}\simeq\frac{b_{R,C}^{C}\times b_{C,R}^{R}}{(RC)!^{RC}}
  16. ( 3 C ) ! ( C ! ) 6 k = 0 C ( C k ) 3 (3C)!(C!)^{6}\sum_{k=0\ldots C}{C\choose k}^{3}
  17. ( 4 C ) ! ( C ! ) 12 a , b , c ( C ! 2 a ! b ! c ! k 12 , k 13 , k 14 , k 23 , k 24 , k 34 ( a k 12 ) ( b k 13 ) ( c k 14 ) ( c k 23 ) ( b k 24 ) ( a k 34 ) ) 2 (4C)!(C!)^{12}\sum_{a,b,c}{\left(\frac{C!^{2}}{a!b!c!}\cdot\sum_{k_{12},k_{13}% ,k_{14},\atop k_{23},k_{24},k_{34}}{{a\choose k_{12}}{b\choose k_{13}}{c% \choose k_{14}}{c\choose k_{23}}{b\choose k_{24}}{a\choose k_{34}}}\right)^{2}}
  18. N combinations for B2 = k = 0..3 ( 3 k ) 3 \mbox{N combinations for B2}~{}=\sum_{k=0..3}{{3\choose k}^{3}}
  19. N = { S b } Sb.z × Sb.n N=\sum_{\{Sb\}}\mbox{Sb.z}~{}\times\mbox{Sb.n}~{}

Matrix_population_models.html

  1. N t + 1 = N t + B - D + I - E , N_{t+1}=N_{t}+B-D+I-E,
  2. N t + 1 = N t , a × S a + N t , i × R i × S i N_{t+1}=N_{t,a}\times S_{a}+N_{t,i}\times R_{i}\times S_{i}
  3. ( N t + l i N t + l a ) = ( S i R i S a R i S i S a ) ( N t i N t a ) . \begin{aligned}\displaystyle\begin{pmatrix}N_{t+l_{i}}\\ N_{t+l_{a}}\end{pmatrix}&\displaystyle=\begin{pmatrix}S_{i}R_{i}&S_{a}R_{i}\\ S_{i}&S_{a}\end{pmatrix}\begin{pmatrix}N_{t_{i}}\\ N_{t_{a}}\end{pmatrix}\end{aligned}.
  4. ( N t + l 1 N t + l 2 N t + l 3 ) = ( F 1 F 2 F 3 S 1 0 0 0 S 2 0 ) ( N t 1 N t 2 N t 3 ) . \begin{aligned}\displaystyle\begin{pmatrix}N_{t+l_{1}}\\ N_{t+l_{2}}\\ N_{t+l_{3}}\end{pmatrix}&\displaystyle=\begin{pmatrix}F_{1}&F_{2}&F_{3}\\ S_{1}&0&0\\ 0&S_{2}&0\end{pmatrix}\begin{pmatrix}N_{t_{1}}\\ N_{t_{2}}\\ N_{t_{3}}\end{pmatrix}\end{aligned}.

Matrix_scheme.html

  1. ρ = λ / μ \rho=\lambda/\mu
  2. N = ρ / ( 1 - ρ ) N=\rho/(1-\rho)
  3. T = 1 / ( μ - λ ) T=1/(\mu-\lambda)

Maxime_Bôcher.html

  1. r ( z ) r^{\prime}(z)
  2. r ( z ) r(z)
  3. r ( z ) r(z)
  4. r ( z ) r(z)
  5. r ( z ) r(z)
  6. y ′′ + 1 2 [ m 1 x - a 1 + + m n - 1 x - a n - 1 ] y + 1 4 [ A 0 + A 1 x + + A x ( x - a 1 ) 1 m ( x - a 2 ) 2 m ( x - a n - 1 ) n - 1 m ] y = 0. y^{\prime\prime}+\frac{1}{2}\left[\frac{m_{1}}{x-a_{1}}+\cdots+\frac{m_{n-1}}{% x-a_{n-1}}\right]y^{\prime}+\frac{1}{4}\left[\frac{A_{0}+A_{1}x+\cdots+A_{\ell% }x^{\ell}}{(x-a_{1})^{m}_{1}(x-a_{2})^{m}_{2}\cdots(x-a_{n-1})^{m}_{n-1}}% \right]y=0.

Maximilien_Toepler.html

  1. R ( t ) = k T D 0 t I ( t ) d t R(t)={\frac{k_{T}D}{\int_{0}^{t}I(t)\,dt}}
  2. k T k_{T}
  3. 4 × 10 - 3 V s m - 1 4\times 10^{-3}\,V\cdot s\cdot m^{-1}

Maximum_entropy_thermodynamics.html

  1. S I = - p i ln p i S_{I}=-\sum p_{i}\ln p_{i}
  2. S T h ( P , V , T , ) ( e q m ) = k B S I ( P , V , T , ) S_{Th}(P,V,T,...)_{(eqm)}=k_{B}\,S_{I}(P,V,T,...)
  3. S I = - p Γ ln p Γ S_{I}=-\sum p_{\Gamma}\ln p_{\Gamma}
  4. H c = - p ( x ) log p ( x ) m ( x ) d x . H_{c}=-\int p(x)\log\frac{p(x)}{m(x)}\,dx.
  5. Δ S I = 0 \Delta S_{I}=0\,
  6. S T h ( 2 ) S T h ( 1 ) {S_{Th}}^{(2)}\geq{S_{Th}}^{(1)}

Maxwell_stress_tensor.html

  1. 𝐄 = ρ ϵ 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}}
  2. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  3. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  4. × 𝐁 = μ 0 𝐉 + μ 0 ϵ 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\epsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}
  5. t ( 𝐄 × 𝐁 ) = 𝐄 t × 𝐁 + 𝐄 × 𝐁 t = 𝐄 t × 𝐁 - 𝐄 × ( s y m b o l × 𝐄 ) \frac{\partial}{\partial t}(\mathbf{E}\times\mathbf{B})=\frac{\partial\mathbf{% E}}{\partial t}\times\mathbf{B}+\mathbf{E}\times\frac{\partial\mathbf{B}}{% \partial t}=\frac{\partial\mathbf{E}}{\partial t}\times\mathbf{B}-\mathbf{E}% \times(symbol{\nabla}\times\mathbf{E})\,
  6. 𝐟 = ϵ 0 ( s y m b o l 𝐄 ) 𝐄 + 1 μ 0 ( s y m b o l × 𝐁 ) × 𝐁 - ϵ 0 t ( 𝐄 × 𝐁 ) - ϵ 0 𝐄 × ( s y m b o l × 𝐄 ) , \mathbf{f}=\epsilon_{0}\left(symbol{\nabla}\cdot\mathbf{E}\right)\mathbf{E}+% \frac{1}{\mu_{0}}\left(symbol{\nabla}\times\mathbf{B}\right)\times\mathbf{B}-% \epsilon_{0}\frac{\partial}{\partial t}\left(\mathbf{E}\times\mathbf{B}\right)% -\epsilon_{0}\mathbf{E}\times(symbol{\nabla}\times\mathbf{E}),
  7. 𝐟 = ϵ 0 [ ( s y m b o l 𝐄 ) 𝐄 - 𝐄 × ( s y m b o l × 𝐄 ) ] + 1 μ 0 [ - 𝐁 × ( s y m b o l × 𝐁 ) ] - ϵ 0 t ( 𝐄 × 𝐁 ) . \mathbf{f}=\epsilon_{0}\left[(symbol{\nabla}\cdot\mathbf{E})\mathbf{E}-\mathbf% {E}\times(symbol{\nabla}\times\mathbf{E})\right]+\frac{1}{\mu_{0}}\left[-% \mathbf{B}\times\left(symbol{\nabla}\times\mathbf{B}\right)\right]-\epsilon_{0% }\frac{\partial}{\partial t}\left(\mathbf{E}\times\mathbf{B}\right).
  8. 𝐟 = ϵ 0 [ ( s y m b o l 𝐄 ) 𝐄 - 𝐄 × ( s y m b o l × 𝐄 ) ] + 1 μ 0 [ ( s y m b o l 𝐁 ) 𝐁 - 𝐁 × ( s y m b o l × 𝐁 ) ] - ϵ 0 t ( 𝐄 × 𝐁 ) . \mathbf{f}=\epsilon_{0}\left[(symbol{\nabla}\cdot\mathbf{E})\mathbf{E}-\mathbf% {E}\times(symbol{\nabla}\times\mathbf{E})\right]+\frac{1}{\mu_{0}}\left[(% symbol{\nabla}\cdot\mathbf{B})\mathbf{B}-\mathbf{B}\times\left(symbol{\nabla}% \times\mathbf{B}\right)\right]-\epsilon_{0}\frac{\partial}{\partial t}\left(% \mathbf{E}\times\mathbf{B}\right).
  9. 1 2 s y m b o l ( 𝐀 𝐀 ) = 𝐀 × ( s y m b o l × 𝐀 ) + ( 𝐀 s y m b o l ) 𝐀 , \tfrac{1}{2}symbol{\nabla}(\mathbf{A}\cdot\mathbf{A})=\mathbf{A}\times(symbol{% \nabla}\times\mathbf{A})+(\mathbf{A}\cdot symbol{\nabla})\mathbf{A},
  10. 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).
  11. 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),
  12. 𝐟 + ϵ 0 μ 0 𝐒 t = σ \mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla% \cdot\mathbf{\sigma}
  13. 𝐒 = 1 μ 0 𝐄 × 𝐁 . \mathbf{S}=\frac{1}{\mu_{0}}\mathbf{E}\times\mathbf{B}.
  14. σ \nabla\cdot\mathbf{\sigma}
  15. 𝐒 \mathbf{S}
  16. σ i j = ϵ 0 E i E j + 1 < m t p l > μ 0 B i B j - 1 2 ( ϵ 0 E 2 + 1 μ 0 B 2 ) δ i j \sigma_{ij}=\epsilon_{0}E_{i}E_{j}+\frac{1}{<}mtpl>{{\mu_{0}}}B_{i}B_{j}-\frac% {1}{2}\bigl({\epsilon_{0}E^{2}+\tfrac{1}{{\mu_{0}}}B^{2}}\bigr)\delta_{ij}
  17. σ i j = 1 4 π ( E i E j + H i H j - 1 2 ( E 2 + H 2 ) δ i j ) \sigma_{ij}=\frac{1}{4\pi}\left(E_{i}E_{j}+H_{i}H_{j}-\frac{1}{2}(E^{2}+H^{2})% \delta_{ij}\right)
  18. σ = 1 4 π [ 𝐄 𝐄 + 𝐇 𝐇 - E 2 + H 2 2 𝕀 ] \overset{\leftrightarrow}{\mathbf{\sigma}}=\frac{1}{4\pi}\left[\mathbf{E}% \otimes\mathbf{E}+\mathbf{H}\otimes\mathbf{H}-\frac{E^{2}+H^{2}}{2}\mathbb{I}\right]
  19. 𝕀 ( 1 0 0 0 1 0 0 0 1 ) = ( 𝐱 ^ 𝐱 ^ + 𝐲 ^ 𝐲 ^ + 𝐳 ^ 𝐳 ^ ) \mathbb{I}\equiv\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)=(\mathbf{\hat{x}}\otimes\mathbf{\hat{x}}+\mathbf{\hat{% y}}\otimes\mathbf{\hat{y}}+\mathbf{\hat{z}}\otimes\mathbf{\hat{z}})
  20. σ i j = 1 μ 0 B i B j - 1 2 μ 0 B 2 δ i j . \sigma_{ij}=\frac{1}{\mu_{0}}B_{i}B_{j}-\frac{1}{2\mu_{0}}B^{2}\delta_{ij}\,.
  21. σ r t = 1 μ 0 B r B t - 1 2 μ 0 B 2 δ r t . \sigma_{rt}=\frac{1}{\mu_{0}}B_{r}B_{t}-\frac{1}{2\mu_{0}}B^{2}\delta_{rt}\,.
  22. { λ } = { - ϵ 0 E 2 + B 2 / μ 0 2 , ± ( ϵ 0 E 2 - B 2 / μ 0 2 ) 2 + ( ϵ 0 μ 0 s y m b o l E s y m b o l B ) 2 } \{\lambda\}=\left\{-\frac{\epsilon_{0}E^{2}+B^{2}/\mu_{0}}{2},~{}\pm\sqrt{% \left(\frac{\epsilon_{0}E^{2}-B^{2}/\mu_{0}}{2}\right)^{2}+\left(\frac{% \epsilon_{0}}{\mu_{0}}symbol{E}\cdot symbol{B}\right)^{2}}\right\}

McCullagh's_parametrization_of_the_Cauchy_distributions.html

  1. f ( x ) = 1 π ( 1 + x 2 ) f(x)={1\over\pi(1+x^{2})}
  2. f ( x ) = | θ | π | x - θ | 2 , f(x)={\left|\Im{\theta}\right|\over\pi\left|x-\theta\right|^{2}}\,,
  3. θ = σ \Im{\theta}=\sigma
  4. { f ( x ) ( μ 2 + σ 2 + x 2 - 2 μ x ) + f ( x ) ( 2 x - 2 μ ) = 0 , f ( 0 ) = 1 π | σ | ( μ 2 σ 2 + 1 ) } \left\{\begin{array}[]{l}f^{\prime}(x)\left(\mu^{2}+\sigma^{2}+x^{2}-2\mu x% \right)+f(x)(2x-2\mu)=0,\\ f(0)=\frac{1}{\pi\left|\sigma\right|\left(\frac{\mu^{2}}{\sigma^{2}}+1\right)}% \end{array}\right\}

MDC-2.html

  1. M M
  2. E E
  3. n n
  4. A 1 , B 1 A_{1},B_{1}
  5. n n
  6. M = M 1 | | . . | | M m M=M_{1}||..||M_{m}
  7. M i M_{i}
  8. n n
  9. V m | | W m V_{m}||W_{m}
  10. i = 1 i=1
  11. m m
  12. V i = M i E ( M i , A i ) V_{i}=M_{i}\oplus E(M_{i},A_{i})
  13. W i = M i E ( M i , B i ) W_{i}=M_{i}\oplus E(M_{i},B_{i})
  14. V i L | | V i R = V i V_{i}^{L}||V_{i}^{R}=V_{i}
  15. W i L | | W i R = W i W_{i}^{L}||W_{i}^{R}=W_{i}
  16. A i + 1 = V i R | | W i L A_{i+1}=V_{i}^{R}||W_{i}^{L}
  17. B i + 1 = W i R | | V i L B_{i+1}=W_{i}^{R}||V_{i}^{L}
  18. A m + 1 | | B m + 1 A_{m+1}||B_{m+1}

Mean_effective_pressure.html

  1. W W
  2. P P
  3. p m e p_{me}
  4. V d V_{d}
  5. n c n_{c}
  6. n c = 2 n_{c}=2
  7. N N
  8. T T
  9. n c n_{c}
  10. W = P n c N W={Pn_{c}\over N}
  11. W = p m e V d W=p_{me}V_{d}
  12. p m e = P n c V d N p_{me}={Pn_{c}\over V_{d}N}
  13. P = T N 2 π P=TN{2\pi}
  14. p m e = T n c V d 2 π p_{me}={Tn_{c}\over V_{d}}{2\pi}

Mean_time_between_coincidences.html

  1. M T B C = 1 λ d = 2 λ μ τ MTBC=\frac{1}{\lambda\ _{d}}=\frac{2}{\lambda\ \mu\ \tau\ }

Measurement_uncertainty.html

  1. Y Y
  2. X 1 , , X N X_{1},\ldots,X_{N}
  3. Y = f ( X 1 , , X N ) , Y=f(X_{1},\ldots,X_{N}),
  4. f f
  5. h ( Y , h(Y,
  6. X 1 , , X N ) = 0. X_{1},\ldots,X_{N})=0.
  7. Y Y
  8. X 1 , , X N X_{1},\ldots,X_{N}
  9. Y Y
  10. X 1 , , X N X_{1},\ldots,X_{N}
  11. X 1 , , X N X_{1},\ldots,X_{N}
  12. X 1 , , X N X_{1},\ldots,X_{N}
  13. x 1 , , x N x_{1},\ldots,x_{N}
  14. X 1 , , X N X_{1},\ldots,X_{N}
  15. X 1 , , X N X_{1},\ldots,X_{N}
  16. x 1 , , x N x_{1},\ldots,x_{N}
  17. X 1 , , X N X_{1},\ldots,X_{N}
  18. i i
  19. u ( x i ) u(x_{i})
  20. X i X_{i}
  21. x i x_{i}
  22. X i X_{i}
  23. Y Y
  24. Y Y
  25. X i X_{i}
  26. Y Y
  27. Y = X 1 + X 2 Y=X_{1}+X_{2}
  28. X 1 X_{1}
  29. X 2 X_{2}
  30. Y Y
  31. X 1 X_{1}
  32. X 2 X_{2}
  33. X 1 , , X N X_{1},\ldots,X_{N}
  34. Y Y
  35. Y Y
  36. Y Y
  37. Y Y
  38. Y Y
  39. Y Y
  40. Y Y
  41. Y Y
  42. Y Y
  43. Y Y
  44. X i X_{i}
  45. X X
  46. X X
  47. t t
  48. X X
  49. a , b a,b
  50. a a
  51. b b
  52. c 1 , , c N c_{1},\ldots,c_{N}
  53. y y
  54. Y Y
  55. x 1 , , x N x_{1},\ldots,x_{N}
  56. X 1 , , X N X_{1},\ldots,X_{N}
  57. Y = f ( X 1 , , X N ) Y=f(X_{1},\ldots,X_{N})
  58. c i c_{i}
  59. f f
  60. X i X_{i}
  61. X 1 = x 1 X_{1}=x_{1}
  62. X 2 = x 2 X_{2}=x_{2}
  63. Y = c 1 X 1 + + c N X N , Y=c_{1}X_{1}+\cdots+c_{N}X_{N},
  64. X 1 , , X N X_{1},\ldots,X_{N}
  65. x i x_{i}
  66. u ( x i ) u(x_{i})
  67. c i u ( x i ) c_{i}u(x_{i})
  68. y y
  69. Y = f ( X 1 , , X N ) Y=f(X_{1},\ldots,X_{N})
  70. | c i | u ( x i ) |c_{i}|u(x_{i})
  71. u ( y ) u(y)
  72. y y
  73. u ( y ) u(y)
  74. y y
  75. Y Y
  76. | c i | u ( x i ) |c_{i}|u(x_{i})
  77. Y = f ( X 1 , , X N ) Y=f(X_{1},\ldots,X_{N})
  78. u 2 ( y ) = c 1 2 u 2 ( x 1 ) + + c N 2 u 2 ( x N ) , u^{2}(y)=c_{1}^{2}u^{2}(x_{1})+\cdots+c_{N}^{2}u^{2}(x_{N}),
  79. X i X_{i}
  80. u ( y ) u(y)
  81. Y Y
  82. Y Y
  83. Y Y
  84. Y Y
  85. Y Y
  86. y y
  87. Y Y
  88. Y Y
  89. u ( y ) u(y)
  90. y y
  91. Y Y
  92. Y Y
  93. t t
  94. Y Y
  95. Y Y

Mediated_reference_theory.html

  1. x ( ( K ( x ) B ( x ) ) y ( ( K ( y ) B ( y ) ) y = x ) z ( K ( z ) B ( z ) ) ) \exists\;x\Bigg(\Big(K(x)\bullet B(x)\Big)\bullet\forall\;y\Big(\big(K(y)% \bullet B(y)\big)\rightarrow y=x\Big)\bullet\forall\;z\Big(K(z)\rightarrow B(z% )\Big)\Bigg)

Memory_refresh.html

  1. refresh cycle interval = refresh time / number of rows \,\text{refresh cycle interval}=\,\text{refresh time}\,/\,\,\text{number of % rows}\,
  2. refresh overhead = time required for refresh, ms refresh interval, ms \,\text{refresh overhead}=\frac{\,\text{time required for refresh, ms}}{\,% \text{refresh interval, ms}}\,
  3. length of refresh cycle = 4 / f = 4 1.33 ( 10 8 ) Hz = 30 ns \,\text{length of refresh cycle}=4/f=\frac{4}{1.33(10^{8})\,\,\text{Hz}}=30\,% \,\text{ns}\,
  4. time required for refresh = ( length of refresh cycle ) ( rows ) = ( 30 ns ) ( 8192 ) = 0.246 ms \,\text{time required for refresh}=(\,\text{length of refresh cycle})(\,\text{% rows})=(30\,\,\text{ns})(8192)=0.246\,\,\text{ms}\,
  5. refresh overhead = 0.246 ms 64 ms = .0038 \,\text{refresh overhead}=\frac{0.246\,\,\text{ms}}{64\,\,\text{ms}}=.0038\,

Mental_poker.html

  1. Π E ( r i ) = E ( Σ r i ) \Pi E(r_{i})=E(\Sigma r_{i})
  2. r * = Σ r i r*=\Sigma r_{i}
  3. r r^{\prime}
  4. r = r * + r r=r*+r^{\prime}

Mercer's_condition.html

  1. K ( x , y ) g ( x ) g ( y ) d x d y 0. \iint K(x,y)g(x)g(y)\,dxdy\geq 0.
  2. K ( x , y ) = 1 K(x,y)=1\,
  3. g ( x ) g ( y ) d x d y = g ( x ) d x g ( y ) d y = ( g ( x ) d x ) 2 \iint g(x)g(y)\,dxdy=\int\!g(x)\,dx\int\!g(y)\,dy=\left(\int\!g(x)\,dx\right)^% {2}

Mercury_cadmium_telluride.html

  1. n i ( t , x ) = ( 5.585 - 3.82 x + ( 1.753 10 - 3 ) t - 1.364 10 - 3 t x ) 10 14 E g ( t , x ) 0.75 T 1.5 e - E g ( t , x ) q 2 k t n_{i}(t,x)=(5.585-3.82x+(1.753\cdot 10^{-3})t-1.364\cdot 10^{-3}t\cdot x)\cdot 1% 0^{14}\cdot E_{g}(t,x)^{0.75}\cdot T^{1.5}\cdot e^{\frac{-E_{g}(t,x)\cdot q}{2% \cdot k\cdot t}}
  2. E g ( t , x ) = - 0.302 + 1.93 x + ( 5.35 10 - 4 ) t ( 1 - 2 x ) - 0.81 x 2 + 0.832 x 3 E_{g}(t,x)=-0.302+1.93\cdot x+(5.35\cdot 10^{-4})\cdot t\cdot(1-2\cdot x)-0.81% \cdot x^{2}+0.832\cdot x^{3}
  3. λ p = 1.24 E g \lambda_{p}=\frac{1.24}{E_{g}}
  4. λ p = ( - 0.244 + 1.556 x + ( 4.31 10 - 4 ) t ( 1 - 2 x ) - 0.65 x 2 + 0.671 x 3 ) - 1 \lambda_{p}=(-0.244+1.556\cdot x+(4.31\cdot 10^{-4})\cdot t\cdot(1-2\cdot x)-0% .65\cdot x^{2}+0.671\cdot x^{3})^{-1}
  5. τ A u g e r 1 ( t , x ) = 2.12 10 - 14 E g ( t , x ) e q E g ( t , x ) k t F F 2 ( k t q ) 1.5 \tau_{Auger1}(t,x)=\frac{2.12\cdot 10^{-14}\cdot\sqrt{E_{g}(t,x)}\cdot e^{% \frac{q\cdot E_{g}(t,x)}{k\cdot t}}}{FF^{2}\cdot(\frac{k\cdot t}{q})^{1.5}}
  6. τ A u g e r 1 d o p e d ( t , x , n ) = 2 τ A u g e r 1 ( t , x ) 1 + ( n n i ( t , x ) ) 2 \tau_{Auger1_{doped}}(t,x,n)=\frac{2\cdot\tau_{Auger1(t,x)}}{1+(\frac{n}{n_{i}% (t,x)})^{2}}
  7. τ A u g e r 7 ( t , x ) = 10 τ A u g e r 1 ( t , x ) \tau_{Auger7}(t,x)=10\cdot\tau_{Auger1}(t,x)
  8. τ A u g e r 7 d o p e d ( t , x , n ) = 2 τ A u g e r 7 ( t , x ) 1 + ( n i ( t , x ) n ) 2 \tau_{Auger7_{doped}}(t,x,n)=\frac{2\cdot\tau_{Auger7(t,x)}}{1+(\frac{n_{i}(t,% x)}{n})^{2}}
  9. τ A u g e r ( t , x ) = τ A u g e r 1 ( t , x ) τ A u g e r 7 ( t , x ) τ A u g e r 1 ( t , x ) + τ A u g e r 7 ( t , x ) \tau_{Auger}(t,x)=\frac{\tau_{Auger1}(t,x)\cdot\tau_{Auger7}(t,x)}{\tau_{Auger% 1}(t,x)+\tau_{Auger7}(t,x)}

Metaplectic_group.html

  1. g z = a z + b c z + d g\cdot z=\frac{az+b}{cz+d}
  2. g = ( a b c d ) SL 2 ( ) g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R})
  3. g SL 2 ( ) , ϵ 2 = c z + d = j ( g z ) , g\in\mathrm{SL}_{2}(\mathbb{R}),\epsilon^{2}=cz+d=j(g\cdot z),
  4. ( g 1 , ϵ 1 ) ( g 2 , ϵ 2 ) = ( g 1 g 2 , ϵ ) , (g_{1},\epsilon_{1})\cdot(g_{2},\epsilon_{2})=(g_{1}g_{2},\epsilon),
  5. ϵ ( z ) = ϵ 1 ( g 2 z ) ϵ 2 ( z ) . \epsilon(z)=\epsilon_{1}(g_{2}\cdot z)\epsilon_{2}(z).
  6. ( g , ϵ ) g (g,\epsilon)\mapsto g
  7. \mathcal{H}
  8. ρ : ( V ) U ( ) \rho:\mathbb{H}(V)\longrightarrow U(\mathcal{H})
  9. ρ \rho^{\prime}
  10. ψ U ( ) \psi\in U(\mathcal{H})
  11. ρ = Ad ψ ( ρ ) \rho^{\prime}=\mathrm{Ad}_{\psi}(\rho)
  12. \mathcal{H}
  13. \mathcal{H}
  14. [ ψ ] P U ( ) [\psi]\in PU(\mathcal{H})

Methanol_(data_page).html

  1. P m m H g = 10 7.87863 - 1473.11 230.0 + T \scriptstyle P_{mmHg}=10^{7.87863-\frac{1473.11}{230.0+T}}
  2. log 10 P m m H g = 7.87863 - 1473.11 230.0 + T \scriptstyle\log_{10}{P_{mmHg}}=7.87863-\frac{1473.11}{230.0+T}

Method_of_moments_(statistics).html

  1. k k
  2. θ 1 , θ 2 , , θ k \theta_{1},\theta_{2},\dots,\theta_{k}
  3. f W ( w ; θ ) f_{W}(w;\theta)
  4. W W
  5. k k
  6. θ \theta
  7. μ 1 E [ W ] = g 1 ( θ 1 , θ 2 , , θ k ) , \mu_{1}\equiv E[W]=g_{1}(\theta_{1},\theta_{2},\dots,\theta_{k}),
  8. μ 2 E [ W 2 ] = g 2 ( θ 1 , θ 2 , , θ k ) , \mu_{2}\equiv E[W^{2}]=g_{2}(\theta_{1},\theta_{2},\dots,\theta_{k}),
  9. \vdots
  10. μ k E [ W k ] = g k ( θ 1 , θ 2 , , θ k ) . \mu_{k}\equiv E[W^{k}]=g_{k}(\theta_{1},\theta_{2},\dots,\theta_{k}).
  11. n n
  12. w 1 , , w n w_{1},\dots,w_{n}
  13. j = 1 , , k j=1,\dots,k
  14. μ ^ j = 1 n i = 1 n w i j \hat{\mu}_{j}=\frac{1}{n}\sum_{i=1}^{n}w_{i}^{j}
  15. μ j \mu_{j}
  16. θ 1 , θ 2 , , θ k \theta_{1},\theta_{2},\dots,\theta_{k}
  17. θ ^ 1 , θ ^ 2 , , θ ^ k \hat{\theta}_{1},\hat{\theta}_{2},\dots,\hat{\theta}_{k}
  18. μ ^ 1 = g 1 ( θ ^ 1 , θ ^ 2 , , θ ^ k ) , \hat{\mu}_{1}=g_{1}(\hat{\theta}_{1},\hat{\theta}_{2},\dots,\hat{\theta}_{k}),
  19. μ ^ 2 = g 2 ( θ ^ 1 , θ ^ 2 , , θ ^ k ) , \hat{\mu}_{2}=g_{2}(\hat{\theta}_{1},\hat{\theta}_{2},\dots,\hat{\theta}_{k}),
  20. \vdots
  21. μ ^ k = g k ( θ ^ 1 , θ ^ 2 , , θ ^ k ) . \hat{\mu}_{k}=g_{k}(\hat{\theta}_{1},\hat{\theta}_{2},\dots,\hat{\theta}_{k}).

Methods_of_computing_square_roots.html

  1. S \sqrt{S}
  2. f ( x ) = x 2 - S = 0 f(x)=x^{2}-S=0\,\!
  3. x n + 1 = x n - f ( x n ) f ( x n ) = x n - x n 2 - S 2 x n = 1 2 ( x n + S x n ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}=x_{n}-\frac{x_{n}^{2}-S}{2x_{% n}}=\frac{1}{2}\left(x_{n}+\frac{S}{x_{n}}\right)
  4. S S
  5. a × 10 2 n a\times 10^{2n}
  6. 1 a < 100 1\leq a<100
  7. S = a × 10 n \sqrt{S}=\sqrt{a}\times 10^{n}
  8. S { 2 10 n if a < 10 , 6 10 n if a 10. \sqrt{S}\approx\begin{cases}2\cdot 10^{n}&\,\text{if }a<10,\\ 6\cdot 10^{n}&\,\text{if }a\geq 10.\end{cases}
  9. 1 10 = 10 4 2 \sqrt{\sqrt{1}\cdot\sqrt{10}}=\sqrt[4]{10}\approx 2\,
  10. 10 100 = 1000 4 6 \sqrt{\sqrt{10}\cdot\sqrt{100}}=\sqrt[4]{1000}\approx 6\,
  11. S = 125348 = 12.5348 × 10 4 S=125348=12.5348\times 10^{4}
  12. S 6 10 2 = 600 \sqrt{S}\approx 6\cdot 10^{2}=600
  13. S S
  14. a × 2 2 n a\times 2^{2n}
  15. 0.1 2 a < 10 2 0.1_{2}\leq a<10_{2}
  16. S = a × 2 n \sqrt{S}=\sqrt{a}\times 2^{n}
  17. S 2 n \sqrt{S}\approx 2^{n}
  18. 0.1 2 10 2 = 1 4 = 1 \sqrt{\sqrt{0.1_{2}}\cdot\sqrt{10_{2}}}=\sqrt[4]{1}=1
  19. S = 125348 = 1 1110 1001 1010 0100 2 = 1.1110 1001 1010 0100 2 × 2 16 S=125348=1\;1110\;1001\;1010\;0100_{2}=1.1110\;1001\;1010\;0100_{2}\times 2^{1% 6}\,
  20. S 2 8 = 1 0000 0000 2 = 256 . \sqrt{S}\approx 2^{8}=1\;0000\;0000_{2}=256\,.
  21. S \sqrt{S}
  22. S / x \scriptstyle S/x\,
  23. x x
  24. S \sqrt{S}
  25. e e
  26. S = ( x + e ) 2 S=(x+e)^{2}
  27. e = S - x 2 2 x + e S - x 2 2 x e=\frac{S-x^{2}}{2x+e}\approx\frac{S-x^{2}}{2x}
  28. e x e\ll x
  29. x := x + e = S + x 2 2 x = x + S x 2 x:=x+e=\frac{S+x^{2}}{2x}=\frac{x+\frac{S}{x}}{2}
  30. x 0 S . x_{0}\approx\sqrt{S}.
  31. x n + 1 = 1 2 ( x n + S x n ) , x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{S}{x_{n}}\right),
  32. S = lim n x n . \sqrt{S}=\lim_{n\to\infty}x_{n}.
  33. S \sqrt{S}
  34. x 0 = 6 10 2 = 600.000 x_{0}=6\cdot 10^{2}=600.000\,
  35. x 1 = 1 2 ( x 0 + S x 0 ) = 1 2 ( 600.000 + 125348 600.000 ) = 404.457 x_{1}=\frac{1}{2}\left(x_{0}+\frac{S}{x_{0}}\right)=\frac{1}{2}\left(600.000+% \frac{125348}{600.000}\right)=404.457
  36. x 2 = 1 2 ( x 1 + S x 1 ) = 1 2 ( 404.457 + 125348 404.457 ) = 357.187 x_{2}=\frac{1}{2}\left(x_{1}+\frac{S}{x_{1}}\right)=\frac{1}{2}\left(404.457+% \frac{125348}{404.457}\right)=357.187
  37. x 3 = 1 2 ( x 2 + S x 2 ) = 1 2 ( 357.187 + 125348 357.187 ) = 354.059 x_{3}=\frac{1}{2}\left(x_{2}+\frac{S}{x_{2}}\right)=\frac{1}{2}\left(357.187+% \frac{125348}{357.187}\right)=354.059
  38. x 4 = 1 2 ( x 3 + S x 3 ) = 1 2 ( 354.059 + 125348 354.059 ) = 354.045 x_{4}=\frac{1}{2}\left(x_{3}+\frac{S}{x_{3}}\right)=\frac{1}{2}\left(354.059+% \frac{125348}{354.059}\right)=354.045
  39. x 5 = 1 2 ( x 4 + S x 4 ) = 1 2 ( 354.045 + 125348 354.045 ) = 354.045 . x_{5}=\frac{1}{2}\left(x_{4}+\frac{S}{x_{4}}\right)=\frac{1}{2}\left(354.045+% \frac{125348}{354.045}\right)=354.045\,.
  40. 125348 354.045 . \sqrt{125348}\approx 354.045\,.
  41. ε n = x n S - 1 \varepsilon_{n}=\frac{x_{n}}{\sqrt{S}}-1
  42. x n = S ( 1 + ε n ) . x_{n}=\sqrt{S}\cdot(1+\varepsilon_{n}).
  43. ε n + 1 = ε n 2 2 ( 1 + ε n ) \varepsilon_{n+1}=\frac{\varepsilon_{n}^{2}}{2(1+\varepsilon_{n})}
  44. 0 ε n + 2 min { ε n + 1 2 2 , ε n + 1 2 } 0\leq\varepsilon_{n+2}\leq\min\left\{\frac{\varepsilon_{n+1}^{2}}{2},\frac{% \varepsilon_{n+1}}{2}\right\}
  45. S \displaystyle S
  46. ε 1 2 - 2 . \varepsilon_{1}\leq 2^{-2}.\,
  47. ε 2 < 2 - 5 < 10 - 1 . \varepsilon_{2}<2^{-5}<10^{-1}.\,
  48. ε 3 < 2 - 11 < 10 - 3 . \varepsilon_{3}<2^{-11}<10^{-3}.\,
  49. ε 4 < 2 - 23 < 10 - 6 . \varepsilon_{4}<2^{-23}<10^{-6}.\,
  50. ε 5 < 2 - 47 < 10 - 14 . \varepsilon_{5}<2^{-47}<10^{-14}.\,
  51. ε 6 < 2 - 95 < 10 - 28 . \varepsilon_{6}<2^{-95}<10^{-28}.\,
  52. ε 7 < 2 - 191 < 10 - 57 . \varepsilon_{7}<2^{-191}<10^{-57}.\,
  53. ε 8 < 2 - 383 < 10 - 115 . \varepsilon_{8}<2^{-383}<10^{-115}.\,
  54. N = ( a 1 + a 2 + a 3 + + a n ) 2 . N=(a_{1}+a_{2}+a_{3}+\cdots+a_{n})^{2}.
  55. ( x + y ) 2 = x 2 + 2 x y + y 2 , (x+y)^{2}=x^{2}+2xy+y^{2},
  56. ( a 1 + a 2 + a 3 + + a n ) 2 = a 1 2 + 2 a 1 a 2 + a 2 2 + 2 ( a 1 + a 2 ) a 3 + a 3 2 + + a n - 1 2 + 2 ( i = 1 n - 1 a i ) a n + a n 2 = a 1 2 + [ 2 a 1 + a 2 ] a 2 + [ 2 ( a 1 + a 2 ) + a 3 ] a 3 + + [ 2 ( i = 1 n - 1 a i ) + a n ] a n . \begin{aligned}&\displaystyle(a_{1}+a_{2}+a_{3}+\cdots+a_{n})^{2}\\ \displaystyle=&\displaystyle\,a_{1}^{2}+2a_{1}a_{2}+a_{2}^{2}+2(a_{1}+a_{2})a_% {3}+a_{3}^{2}+\cdots+a_{n-1}^{2}+2\left(\sum_{i=1}^{n-1}a_{i}\right)a_{n}+a_{n% }^{2}\\ \displaystyle=&\displaystyle\,a_{1}^{2}+[2a_{1}+a_{2}]a_{2}+[2(a_{1}+a_{2})+a_% {3}]a_{3}+\cdots+\left[2\left(\sum_{i=1}^{n-1}a_{i}\right)+a_{n}\right]a_{n}.% \end{aligned}
  57. a i a_{i}
  58. a 1 , , a m - 1 a_{1},\ldots,a_{m-1}
  59. Y m = [ 2 P m - 1 + a m ] a m , Y_{m}=[2P_{m-1}+a_{m}]a_{m},
  60. P m - 1 = i = 1 m - 1 a i P_{m-1}=\sum_{i=1}^{m-1}a_{i}
  61. a m a_{m}
  62. X m = X m - 1 - Y m , X_{m}=X_{m-1}-Y_{m},
  63. X m 0 X_{m}\geq 0
  64. 1 m n , 1\leq m\leq n,
  65. X 0 = N . X_{0}=N.
  66. X n = 0 , X_{n}=0,
  67. a i a_{i}
  68. X n X_{n}
  69. N = ( a 1 10 n - 1 + a 2 10 n - 2 + + a n - 1 10 + a n ) 2 , N=(a_{1}\cdot 10^{n-1}+a_{2}\cdot 10^{n-2}+\cdots+a_{n-1}\cdot 10+a_{n})^{2},
  70. 10 n - i 10^{n-i}
  71. a i { 0 , 1 , 2 , , 9 } a_{i}\in\{0,1,2,\ldots,9\}
  72. P m - 1 P_{m-1}
  73. Y m Y_{m}
  74. P m - 1 = i = 1 m - 1 a i 10 n - i = 10 n - m + 1 i = 1 m - 1 a i 10 m - i - 1 , P_{m-1}=\sum_{i=1}^{m-1}a_{i}\cdot 10^{n-i}=10^{n-m+1}\sum_{i=1}^{m-1}a_{i}% \cdot 10^{m-i-1},
  75. Y m = [ 2 P m - 1 + a m 10 n - m ] a m 10 n - m = [ 20 i = 1 m - 1 a i 10 m - i - 1 + a m ] a m 10 2 ( n - m ) . Y_{m}=[2P_{m-1}+a_{m}\cdot 10^{n-m}]a_{m}\cdot 10^{n-m}=[20\sum_{i=1}^{m-1}a_{% i}\cdot 10^{m-i-1}+a_{m}]a_{m}\cdot 10^{2(n-m)}.
  76. Y m Y_{m}
  77. X m - 1 X_{m-1}
  78. a i a_{i}
  79. Y m Y_{m}
  80. Y m = 0 Y_{m}=0
  81. a m = 0 a_{m}=0
  82. Y m = 2 P m - 1 + 1 Y_{m}=2P_{m-1}+1
  83. a m = 1 a_{m}=1
  84. a m a_{m}
  85. a m a_{m}
  86. Y m X m - 1 Y_{m}\leq X_{m-1}
  87. a m = 1. a_{m}=1.
  88. a m = 1 a_{m}=1
  89. a m = 0. a_{m}=0.
  90. x ( 20 p + x ) c x(20p+x)\leq c
  91. x x
  92. r \,r
  93. ( r + e ) ( r + e ) x \,(r+e)\cdot(r+e)\leq x
  94. x \,x
  95. r r + 2 r e + e e x \,r\cdot r+2re+e\cdot e\leq x
  96. r r \,r\cdot r
  97. e \,e
  98. 2 r e \,2\cdot r\cdot e
  99. e e \,e\cdot e
  100. e m 2 e_{m}^{2}
  101. ( 2 m ) 2 = 4 m (2^{m})^{2}=4^{m}
  102. 2 r e m 2re_{m}
  103. X m X_{m}
  104. 2 m + 1 2^{m+1}
  105. e e
  106. X m X_{m}
  107. 2 r e m + e m 2 2re_{m}+e_{m}^{2}
  108. e m e_{m}
  109. r r
  110. 2 e m 2 2e^{2}_{m}
  111. 2 r e m 2re_{m}
  112. e m - 1 e_{m-1}
  113. X m X_{m}
  114. 2 r e m + e m 2 2re_{m}+e_{m}^{2}
  115. e m e_{m}
  116. e m - 1 e_{m-1}
  117. e m e_{m}
  118. e m - 1 e_{m-1}
  119. e e
  120. ln x n = n ln x \ln x^{n}=n\ln x
  121. e ln x = x e^{\ln x}=x
  122. S = e 1 2 ln S . \sqrt{S}=e^{\frac{1}{2}\ln S}.
  123. S \sqrt{S}
  124. d = S - N 2 d=S-N^{2}\,\!
  125. P = d 2 N P=\frac{d}{2N}
  126. A = N + P A=N+P\,\!
  127. S A - P 2 2 A \sqrt{S}\approx A-\frac{P^{2}}{2A}
  128. S N + d 2 N - d 2 8 N 3 + 4 N d = 8 N 4 + 8 N 2 d + d 2 8 N 3 + 4 N d = N 4 + 6 N 2 S + S 2 4 N 3 + 4 N S = N 2 ( N 2 + 6 S ) + S 2 4 N ( N 2 + S ) \sqrt{S}\approx N+\frac{d}{2N}-\frac{d^{2}}{8N^{3}+4Nd}=\frac{8N^{4}+8N^{2}d+d% ^{2}}{8N^{3}+4Nd}=\frac{N^{4}+6N^{2}S+S^{2}}{4N^{3}+4NS}=\frac{N^{2}(N^{2}+6S)% +S^{2}}{4N(N^{2}+S)}
  129. 9.2345 \sqrt{9.2345}
  130. N = 3 N=3\,\!
  131. d = 9.2345 - 3 2 = 0.2345 d=9.2345-3^{2}=0.2345\,\!
  132. P = 0.2345 2 × 3 = 0.0391 P=\frac{0.2345}{2\times 3}=0.0391
  133. A = 3 + 0.0391 = 3.0391 A=3+0.0391=3.0391\,\!
  134. 9.2345 3.0391 - 0.0391 2 2 × 3.0391 3.0388 \sqrt{9.2345}\approx 3.0391-\frac{0.0391^{2}}{2\times 3.0391}\approx 3.0388
  135. N = ( a 0 + a 1 + + a n - 1 ) 2 N=(a_{0}+a_{1}+\ldots+a_{n-1})^{2}
  136. = a 0 2 + 2 a 0 i = 1 n - 1 a i + a 1 2 + 2 a 1 i = 2 n - 1 a i + + a n - 1 2 . =a_{0}^{2}+2a_{0}\sum_{i=1}^{n-1}a_{i}+a_{1}^{2}+2a_{1}\sum_{i=2}^{n-1}a_{i}+% \cdots+a_{n-1}^{2}.
  137. q = 2 a 0 q=2a_{0}
  138. d m = { a m / 2 2 + i = 1 m / 2 2 a i a m - i + 1 for m odd i = 1 m / 2 2 a i a m - i + 1 for m even . d_{m}=\begin{cases}a_{\lceil m/2\rceil}^{2}+\sum_{i=1}^{\lfloor m/2\rfloor}2a_% {i}a_{m-i+1}&\mbox{for}~{}\;m\;\mbox{odd}\\ \sum_{i=1}^{m/2}2a_{i}a_{m-i+1}&\mbox{for}~{}\;m\;\mbox{even}~{}.\\ \end{cases}
  139. N - a 0 2 = i = 1 n - 1 ( q a i + d i ) . N-a_{0}^{2}=\sum_{i=1}^{n-1}(qa_{i}+d_{i}).
  140. a m a_{m}
  141. X m = X m - 1 - q a m - d m , X_{m}=X_{m-1}-qa_{m}-d_{m},
  142. X m 0 X_{m}\geq 0
  143. 1 m n - 1 1\leq m\leq n-1
  144. X 0 = N - a 0 2 . X_{0}=N-a_{0}^{2}.
  145. X m = 0 X_{m}=0
  146. a i a_{i}
  147. X m - 1 X_{m-1}
  148. X m X_{m}
  149. N = ( a 0 10 n - 1 + a 1 10 n - 2 + + a n - 2 10 + a n - 1 ) 2 N=(a_{0}\cdot 10^{n-1}+a_{1}\cdot 10^{n-2}+\cdots+a_{n-2}\cdot 10+a_{n-1})^{2}
  150. a i { 0 , 1 , 2 , , 9 } a_{i}\in\{0,1,2,\ldots,9\}
  151. X 0 = N - a 0 2 10 2 ( n - 1 ) X_{0}=N-a_{0}^{2}\cdot 10^{2(n-1)}
  152. q = 2 a 0 10 n - 1 q=2a_{0}\cdot 10^{n-1}
  153. q a m 10 n - m - 1 = 2 a 0 a m 10 2 n - m - 2 qa_{m}\cdot 10^{n-m-1}=2a_{0}a_{m}\cdot 10^{2n-m-2}
  154. d m = d ~ m 10 2 n - m - 3 d_{m}=\tilde{d}_{m}\cdot 10^{2n-m-3}
  155. d ~ m \tilde{d}_{m}
  156. a 1 , a 2 , , a m a_{1},a_{2},\ldots,a_{m}
  157. d ~ m \tilde{d}_{m}
  158. 2 a 0 a m 2a_{0}a_{m}
  159. X m - 1 X_{m-1}
  160. X m - 1 X_{m-1}
  161. 0 < S < 3 0<S<3\,\!
  162. S 1 S\approx 1
  163. S S\,\!
  164. S \sqrt{S}
  165. S S\,\!
  166. 1 2 S < 2 \frac{1}{2}\leq S<2
  167. a 0 = S a_{0}=S\,\!
  168. c 0 = S - 1 c_{0}=S-1\,\!
  169. a n + 1 = a n - a n c n / 2 a_{n+1}=a_{n}-a_{n}c_{n}/2\,\!
  170. c n + 1 = c n 2 ( c n - 3 ) / 4 c_{n+1}=c_{n}^{2}(c_{n}-3)/4\,\!
  171. a n S a_{n}\rightarrow\sqrt{S}
  172. c n 0 c_{n}\rightarrow 0
  173. c n c_{n}\,\!
  174. a n a_{n}\,\!
  175. c n c_{n}\,\!
  176. 1 + c n + 1 = ( 1 + c n ) ( 1 - c n / 2 ) 2 1+c_{n+1}=(1+c_{n})(1-c_{n}/2)^{2}\,\!
  177. S ( 1 + c n ) = a n 2 S(1+c_{n})=a_{n}^{2}
  178. a n a_{n}\,\!
  179. S \sqrt{S}
  180. c n c_{n}\,\!
  181. - 1 < c 0 < 2 -1<c_{0}<2\,\!
  182. 1 / S 1/\sqrt{S}
  183. S \sqrt{S}
  184. S = S ( 1 / S ) \sqrt{S}=S\cdot(1/\sqrt{S})
  185. ( 1 / x 2 ) - S = 0 (1/x^{2})-S=0
  186. x n + 1 = x n 2 ( 3 - S x n 2 ) . x_{n+1}=\frac{x_{n}}{2}\cdot(3-S\cdot x_{n}^{2}).
  187. y n = S x n 2 , y_{n}=S\cdot x_{n}^{2}\,,\!
  188. x n + 1 = x n 8 ( 15 - y n ( 10 - 3 y n ) ) . x_{n+1}=\frac{x_{n}}{8}\cdot(15-y_{n}\cdot(10-3\cdot y_{n})).
  189. S \sqrt{S}
  190. 1 / S 1/\sqrt{S}
  191. S \sqrt{S}
  192. b 0 = S b_{0}=S
  193. Y 0 1 / S Y_{0}\approx 1/\sqrt{S}
  194. y 0 = Y 0 y_{0}=Y_{0}
  195. x 0 = S y 0 x_{0}=Sy_{0}
  196. b n + 1 = b n Y n 2 b_{n+1}=b_{n}Y_{n}^{2}
  197. Y n + 1 = ( 3 - b n + 1 ) / 2 Y_{n+1}=(3-b_{n+1})/2
  198. x n + 1 = x n Y n + 1 x_{n+1}=x_{n}Y_{n+1}
  199. y n + 1 = y n Y n + 1 y_{n+1}=y_{n}Y_{n+1}
  200. b i b_{i}
  201. S = lim n x n . \sqrt{S}=\lim_{n\to\infty}x_{n}.
  202. 1 / S = lim n y n . 1/\sqrt{S}=\lim_{n\to\infty}y_{n}.
  203. y 0 1 / S y_{0}\approx 1/\sqrt{S}
  204. x 0 = S y 0 x_{0}=Sy_{0}
  205. h 0 = y 0 / 2 h_{0}=y_{0}/2
  206. r n = ( 1 / 2 ) - x n h n r_{n}=(1/2)-x_{n}h_{n}
  207. x n + 1 = x n + x n r n x_{n+1}=x_{n}+x_{n}r_{n}
  208. h n + 1 = h n + h n r n h_{n+1}=h_{n}+h_{n}r_{n}
  209. r i r_{i}
  210. S = lim n x n . \sqrt{S}=\lim_{n\to\infty}x_{n}.
  211. 1 / S = lim n 2 h n . 1/\sqrt{S}=\lim_{n\to\infty}2h_{n}.
  212. S \sqrt{S}
  213. N 2 + d = n = 0 ( - 1 ) n ( 2 n ) ! d n ( 1 - 2 n ) n ! 2 4 n N 2 n - 1 = N + d 2 N - d 2 8 N 3 + d 3 16 N 5 - 5 d 4 128 N 7 + \sqrt{N^{2}+d}=\sum_{n=0}^{\infty}\frac{(-1)^{n}(2n)!d^{n}}{(1-2n)n!^{2}4^{n}N% ^{2n-1}}=N+\frac{d}{2N}-\frac{d^{2}}{8N^{3}}+\frac{d^{3}}{16N^{5}}-\frac{5d^{4% }}{128N^{7}}+\cdots
  214. | d | N 2 \frac{|d|}{N^{2}}\,
  215. x n x_{n}
  216. x 0 = S + 1 x_{0}=S+1
  217. y 0 = S - 1 y_{0}=S-1
  218. ω 0 = 0 \omega_{0}=0
  219. a + b c \frac{a+\sqrt{b}}{c}
  220. m 0 = 0 m_{0}=0\,\!
  221. d 0 = 1 d_{0}=1\,\!
  222. a 0 = S a_{0}=\left\lfloor\sqrt{S}\right\rfloor\,\!
  223. m n + 1 = d n a n - m n m_{n+1}=d_{n}a_{n}-m_{n}\,\!
  224. d n + 1 = S - m n + 1 2 d n d_{n+1}=\frac{S-m_{n+1}^{2}}{d_{n}}\,\!
  225. a n + 1 = S + m n + 1 d n + 1 = a 0 + m n + 1 d n + 1 . a_{n+1}=\left\lfloor\frac{\sqrt{S}+m_{n+1}}{d_{n+1}}\right\rfloor=\left\lfloor% \frac{a_{0}+m_{n+1}}{d_{n+1}}\right\rfloor\!.
  226. S = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + \sqrt{S}=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\,\ddots}}}
  227. 114 \displaystyle\sqrt{114}
  228. m 1 = d 0 a 0 - m 0 = 1 10 - 0 = 10 . m_{1}=d_{0}\cdot a_{0}-m_{0}=1\cdot 10-0=10\,.
  229. d 1 = S - m 1 2 d 0 = 114 - 10 2 1 = 14 . d_{1}=\frac{S-m_{1}^{2}}{d_{0}}=\frac{114-10^{2}}{1}=14\,.
  230. a 1 = a 0 + m 1 d 1 = 10 + 10 14 = 20 14 = 1 . a_{1}=\left\lfloor\frac{a_{0}+m_{1}}{d_{1}}\right\rfloor=\left\lfloor\frac{10+% 10}{14}\right\rfloor=\left\lfloor\frac{20}{14}\right\rfloor=1\,.
  231. 114 + 10 14 = 1 + 114 - 4 14 = 1 + 114 - 16 14 ( 114 + 4 ) = 1 + 1 114 + 4 7 . \frac{\sqrt{114}+10}{14}=1+\frac{\sqrt{114}-4}{14}=1+\frac{114-16}{14(\sqrt{11% 4}+4)}=1+\frac{1}{\frac{\sqrt{114}+4}{7}}.
  232. 114 + 4 7 = 2 + 114 - 10 7 = 2 + 14 7 ( 114 + 10 ) = 2 + 1 114 + 10 2 . \frac{\sqrt{114}+4}{7}=2+\frac{\sqrt{114}-10}{7}=2+\frac{14}{7(\sqrt{114}+10)}% =2+\frac{1}{\frac{\sqrt{114}+10}{2}}.
  233. 114 + 10 2 = 10 + 114 - 10 2 = 10 + 14 2 ( 114 + 10 ) = 10 + 1 114 + 10 7 . \frac{\sqrt{114}+10}{2}=10+\frac{\sqrt{114}-10}{2}=10+\frac{14}{2(\sqrt{114}+1% 0)}=10+\frac{1}{\frac{\sqrt{114}+10}{7}}.
  234. 114 + 10 7 = 2 + 114 - 4 7 = 2 + 98 7 ( 114 + 4 ) = 2 + 1 114 + 4 14 . \frac{\sqrt{114}+10}{7}=2+\frac{\sqrt{114}-4}{7}=2+\frac{98}{7(\sqrt{114}+4)}=% 2+\frac{1}{\frac{\sqrt{114}+4}{14}}.
  235. 114 + 4 14 = 1 + 114 - 10 14 = 1 + 14 14 ( 114 + 10 ) = 1 + 1 114 + 10 1 . \frac{\sqrt{114}+4}{14}=1+\frac{\sqrt{114}-10}{14}=1+\frac{14}{14(\sqrt{114}+1% 0)}=1+\frac{1}{\frac{\sqrt{114}+10}{1}}.
  236. 114 + 10 1 = 20 + 114 - 10 1 = 20 + 14 114 + 10 = 20 + 1 114 + 10 14 . \frac{\sqrt{114}+10}{1}=20+\frac{\sqrt{114}-10}{1}=20+\frac{14}{\sqrt{114}+10}% =20+\frac{1}{\frac{\sqrt{114}+10}{14}}.
  237. 114 = [ 10 ; 1 , 2 , 10 , 2 , 1 , 20 , 1 , 2 , 10 , 2 , 1 , 20 , 1 , 2 , 10 , 2 , 1 , 20 , ] . \sqrt{114}=[10;1,2,10,2,1,20,1,2,10,2,1,20,1,2,10,2,1,20,\dots].\,
  238. z = x 2 + y = x + y 2 x + y 2 x + y 2 x + = x + 2 x y 2 ( 2 z - y ) - y - y 2 2 ( 2 z - y ) - y 2 2 ( 2 z - y ) - \sqrt{z}=\sqrt{x^{2}+y}=x+\cfrac{y}{2x+\cfrac{y}{2x+\cfrac{y}{2x+\ddots}}}=x+% \cfrac{2x\cdot y}{2(2z-y)-y-\cfrac{y^{2}}{2(2z-y)-\cfrac{y^{2}}{2(2z-y)-\ddots% }}}
  239. 114 = 1026 3 = 32 2 + 2 3 = 32 3 + 2 / 3 64 + 2 64 + 2 64 + 2 64 + = 32 3 + 2 192 + 18 192 + 18 192 + , \sqrt{114}=\cfrac{\sqrt{1026}}{3}=\cfrac{\sqrt{32^{2}+2}}{3}=\cfrac{32}{3}+% \cfrac{2/3}{64+\cfrac{2}{64+\cfrac{2}{64+\cfrac{2}{64+\ddots}}}}=\cfrac{32}{3}% +\cfrac{2}{192+\cfrac{18}{192+\cfrac{18}{192+\ddots}}},
  240. 114 = 32 2 + 2 3 = 32 3 + 64 / 3 2050 - 1 - 1 2050 - 1 2050 - = 32 3 + 64 6150 - 3 - 9 6150 - 9 6150 - , \sqrt{114}=\cfrac{\sqrt{32^{2}+2}}{3}=\cfrac{32}{3}+\cfrac{64/3}{2050-1-\cfrac% {1}{2050-\cfrac{1}{2050-\ddots}}}=\cfrac{32}{3}+\cfrac{64}{6150-3-\cfrac{9}{61% 50-\cfrac{9}{6150-\ddots}}},
  241. p 2 = S q 2 ± 1 p^{2}=S\cdot q^{2}\pm 1\!
  242. p q \frac{p}{q}
  243. S \sqrt{S}
  244. p = p a p b + S q a q b p=p_{a}p_{b}+S\cdot q_{a}q_{b}\,\!
  245. q = p a q b + p b q a q=p_{a}q_{b}+p_{b}q_{a}\,\!
  246. p m + n = p m p n + S q m q n p_{m+n}=p_{m}p_{n}+S\cdot q_{m}q_{n}\,\!
  247. q m + n = p m q n + p n q m q_{m+n}=p_{m}q_{n}+p_{n}q_{m}\,\!
  248. p 1 2 = S q 1 2 ± 1 p_{1}^{2}=S\cdot q_{1}^{2}\pm 1
  249. p n + 1 = p 1 p n + S q 1 q n p_{n+1}=p_{1}p_{n}+S\cdot q_{1}q_{n}\,\!
  250. q n + 1 = p 1 q n + p n q 1 q_{n+1}=p_{1}q_{n}+p_{n}q_{1}\,\!
  251. p 2 n = p n 2 + S q n 2 p_{2n}=p_{n}^{2}+S\cdot q_{n}^{2}\,\!
  252. q 2 n = 2 p n q n q_{2n}=2p_{n}q_{n}\,\!
  253. p 1 q 1 \textstyle\frac{p_{1}}{q_{1}}
  254. p n q n \frac{p_{n}}{q_{n}}
  255. | p n q n - S | < 1 q n 2 S . \left|\frac{p_{n}}{q_{n}}-\sqrt{S}\right|<\frac{1}{q_{n}^{2}\cdot\sqrt{S}}.
  256. m × b p m\times b^{p}
  257. m × b p / 2 \sqrt{m}\times b^{p/2}
  258. b p / 2 b^{p/2}
  259. log 2 ( m × 2 p ) = p + log 2 ( m ) \log_{2}(m\times 2^{p})=p+\log_{2}(m)
  260. 2 - 23 2^{-23}
  261. x int 2 - 23 - 127 log 2 ( x ) . x\text{int}\cdot 2^{-23}-127\approx\log_{2}(x).
  262. 1065353216 = 127 2 23 1065353216=127\cdot 2^{23}
  263. 1065353216 2 - 23 - 127 = 0 1065353216\cdot 2^{-23}-127=0
  264. log 2 ( 1.0 ) \log_{2}(1.0)
  265. b b
  266. n n
  267. ( ( ( x int / 2 n - b ) / 2 ) + b ) 2 n = ( x int - 2 n ) / 2 + ( ( b + 1 ) / 2 ) 2 n . (((x\text{int}/2^{n}-b)/2)+b)\cdot 2^{n}=(x\text{int}-2^{n})/2+((b+1)/2)\cdot 2% ^{n}.
  268. S = | S | + a 2 + sgn ( b ) | S | - a 2 i . \sqrt{S}=\sqrt{\frac{|S|+a}{2}}\,+\,\operatorname{sgn}(b)\sqrt{\frac{|S|-a}{2}% }\,\,i\,.
  269. | S | = a 2 + b 2 |S|=\sqrt{a^{2}+b^{2}}