wpmath0000002_17

Re.html

  1. \Re

Reaction_rate.html

  1. r = - 1 a d [ A ] d t = - 1 b d [ B ] d t = 1 p d [ P ] d t = 1 q d [ Q ] d t r=-\frac{1}{a}\frac{d[A]}{dt}=-\frac{1}{b}\frac{d[B]}{dt}=\frac{1}{p}\frac{d[P% ]}{dt}=\frac{1}{q}\frac{d[Q]}{dt}
  2. F A 0 - F A + 0 V v d V = d N A d t F_{A0}-F_{A}+\int_{0}^{V}v\,dV=\frac{dN_{A}}{dt}
  3. F A 0 F_{A0}
  4. F A F_{A}
  5. v v
  6. V V
  7. r = d [ A ] d t r=\frac{d[A]}{dt}
  8. [ A ] [A]
  9. N A N_{A}
  10. [ A ] = N A N 0 V [A]=\frac{N_{A}}{N_{0}V}
  11. N 0 N_{0}
  12. r = d ξ d t = 1 ν i d n i d t = 1 ν i d ( C i V ) d t = 1 ν i ( V d C i d t + C i d V d t ) r=\frac{d\xi}{dt}=\frac{1}{\nu_{i}}\frac{dn_{i}}{dt}=\frac{1}{\nu_{i}}\frac{d(% C_{i}V)}{dt}=\frac{1}{\nu_{i}}\left(V\frac{dC_{i}}{dt}+C_{i}\frac{dV}{dt}\right)
  13. ν i \scriptstyle\nu_{i}
  14. i i
  15. V \scriptstyle V
  16. C i \scriptstyle C_{i}
  17. i i
  18. r = k ( T ) [ A ] n [ B ] m \,r=k(T)[A]^{n}[B]^{m}
  19. k ( T ) k(T)
  20. n n
  21. m m
  22. d [ P ] d t = k ( T ) [ A ] n [ B ] m \frac{d[P]}{dt}=k(T)[A]^{n}[B]^{m}
  23. 2 H 2 ( g ) + 2 N O ( g ) N 2 ( g ) + 2 H 2 O ( g ) 2H_{2}(g)+2NO(g)\rightarrow N_{2}(g)+2H_{2}O(g)
  24. r = k [ H 2 ] [ N O ] 2 r=k[H_{2}][NO]^{2}\,
  25. 2 N O ( g ) N 2 O 2 ( g ) 2NO(g)\ \overrightarrow{\longleftarrow}\ N_{2}O_{2}(g)
  26. N 2 O 2 + H 2 N 2 O + H 2 O N_{2}O_{2}+H_{2}\rightarrow N_{2}O+H_{2}O
  27. N 2 O + H 2 N 2 + H 2 O N_{2}O+H_{2}\rightarrow N_{2}+H_{2}O
  28. r = k 2 [ H 2 ] [ N 2 O 2 ] r=k_{2}[H_{2}][N_{2}O_{2}]\,
  29. [ N 2 O 2 ] = K 1 [ N O ] 2 [N_{2}O_{2}]=K_{1}[NO]^{2}\,
  30. r = k 2 K 1 [ H 2 ] [ N O ] 2 r=k_{2}K_{1}[H_{2}][NO]^{2}\,
  31. k = k 2 K 1 k=k_{2}K_{1}
  32. k = A e - E a R T k=Ae^{-\frac{E_{a}}{RT}}
  33. e - E a R T e^{-\frac{E_{a}}{RT}}
  34. A + B | A B | P A+B\rightleftharpoons|A\cdots B|^{\ddagger}\rightarrow P
  35. Δ V \Delta V^{\ddagger}
  36. Δ V = V ¯ - V ¯ A - V ¯ B \Delta V^{\ddagger}=\bar{V}_{\ddagger}-\bar{V}_{A}-\bar{V}_{B}
  37. V ¯ \bar{V}
  38. \ddagger
  39. - R T ( ln k x P ) T = Δ V -RT\left(\frac{\partial\ln k_{x}}{\partial P}\right)_{T}=\Delta V^{\ddagger}
  40. Δ V \Delta V^{\ddagger}
  41. Δ V \Delta V^{\ddagger}

Read–write_conflict.html

  1. S = [ T 1 T 2 R ( A ) R ( A ) W ( A ) C o m . R ( A ) W ( A ) C o m . ] S=\begin{bmatrix}T1&T2\\ R(A)&\\ &R(A)\\ &W(A)\\ &Com.\\ R(A)&\\ W(A)&\\ Com.&\end{bmatrix}

Reciprocity_law.html

  1. ( p q ) ( q p ) = ( - 1 ) p - 1 2 q - 1 2 . \left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}% {2}}.
  2. ( α β ) 3 = ( β α ) 3 . \Bigg(\frac{\alpha}{\beta}\Bigg)_{3}=\Bigg(\frac{\beta}{\alpha}\Bigg)_{3}.
  3. [ π θ ] [ θ π ] - 1 = ( - 1 ) N π - 1 4 N θ - 1 4 . \Bigg[\frac{\pi}{\theta}\Bigg]\left[\frac{\theta}{\pi}\right]^{-1}=(-1)^{\frac% {N\pi-1}{4}\frac{N\theta-1}{4}}.
  4. l l
  5. l l
  6. ( α 𝔭 ) l α N ( 𝔭 ) - 1 l ( mod 𝔭 ) \left(\frac{\alpha}{\mathfrak{p}}\right)_{l}\equiv\alpha^{\frac{N(\mathfrak{p}% )-1}{l}}\;\;(\mathop{{\rm mod}}\mathfrak{p})
  7. 𝔭 \mathfrak{p}
  8. ( a α ) l = ( α a ) l \left(\frac{a}{\alpha}\right)_{l}=\left(\frac{\alpha}{a}\right)_{l}
  9. l l
  10. l l
  11. { p q } n = { p n q } \left\{\frac{p}{q}\right\}^{n}=\left\{\frac{p^{n}}{q}\right\}
  12. { p q } = { q p } \left\{\frac{p}{q}\right\}=\left\{\frac{q}{p}\right\}
  13. v ( a , b ) v = 1 \prod_{v}(a,b)_{v}=1
  14. ( p , q ) ( p , q ) 2 ( p , q ) p ( p , q ) q = 1 (p,q)_{\infty}(p,q)_{2}(p,q)_{p}(p,q)_{q}=1
  15. θ : C K / N L / K ( C L ) Gal ( L / K ) ab . \theta:C_{K}/{N_{L/K}(C_{L})}\to\,\text{Gal}(L/K)^{\,\text{ab}}.
  16. K × / N L / K ( L × ) K^{\times}/N_{L/K}(L^{\times})
  17. G a l ( L / K ) Gal(L/K)
  18. ( α β ) n ( β α ) n - 1 = 𝔭 | n ( α , β ) 𝔭 . \left({\frac{\alpha}{\beta}}\right)_{n}\left({\frac{\beta}{\alpha}}\right)_{n}% ^{-1}=\prod_{\mathfrak{p}|n\infty}(\alpha,\beta)_{\mathfrak{p}}\ .

Recoil.html

  1. p f + p p = 0 p_{f}+p_{p}=0\,
  2. p f p_{f}\,
  3. p p p_{p}\,
  4. m f v f + m p v p = 0 m_{f}v_{f}+m_{p}v_{p}=0\,
  5. m f m_{f}\,
  6. v f v_{f}\,
  7. m p m_{p}\,
  8. v p v_{p}\,
  9. 0 t c r F c r ( t ) d t = - m f v f = m p v p \int_{0}^{t_{cr}}F_{cr}(t)\,dt=-m_{f}v_{f}=m_{p}v_{p}
  10. F c r ( t ) F_{cr}(t)\,
  11. t c r t_{cr}\,
  12. 0 t r F r ( t ) d t = m f v f = - m p v p \int_{0}^{t_{r}}F_{r}(t)\,dt=m_{f}v_{f}=-m_{p}v_{p}
  13. F r ( t ) F_{r}(t)\,
  14. t r t_{r}\,
  15. t r t c r t_{r}\ll t_{cr}
  16. F r ( t ) + F c r ( t ) = 0 F_{r}(t)+F_{cr}(t)=0
  17. τ \tau
  18. τ = I d 2 θ d t 2 = h F ( t ) \tau=I\frac{d^{2}\theta}{dt^{2}}=hF(t)
  19. θ \theta
  20. I d θ d t = h 0 t F ( t ) d t = h m g V g ( t ) = h m b V b ( t ) I\frac{d\theta}{dt}=h\int_{0}^{t}F(t)\,dt=hm_{g}V_{g}(t)=hm_{b}V_{b}(t)
  21. I θ f = h 0 t f m b V b d t = h m b L I\theta_{f}=h\int_{0}^{t_{f}}m_{b}V_{b}\,dt=hm_{b}L
  22. θ f \theta_{f}
  23. t f t_{f}
  24. θ f = h m b L I \theta_{f}=\frac{hm_{b}L}{I}
  25. E = 1 2 m v 2 = p 2 2 m E={\frac{1}{2}}mv^{2}=\frac{p^{2}}{2m}
  26. E E\,
  27. m m\,
  28. v v\,
  29. p p\,
  30. E t E_{t}\,
  31. E p = p p 2 / 2 m p E_{p}=p_{p}^{2}/2m_{p}
  32. E f = p f 2 / 2 m f E_{f}=p_{f}^{2}/2m_{f}
  33. p f + p p = 0 p_{f}+p_{p}=0
  34. E p E f = m f m p \frac{E_{p}}{E_{f}}=\frac{m_{f}}{m_{p}}
  35. m f m_{f}
  36. m p m_{p}
  37. α V 0 \alpha V_{0}
  38. V 0 V_{0}
  39. α \alpha
  40. p e p_{e}
  41. p e = m p V 0 + m g α V 0 p_{e}=m_{p}V_{0}+m_{g}\alpha V_{0}\,
  42. m g m_{g}\,

Reconstruction_conjecture.html

  1. G = ( V , E ) G=(V,E)
  2. G G
  3. G G
  4. G G
  5. G G
  6. D ( G ) D(G)
  7. G G
  8. D ( G ) D(G)
  9. G = ( V , E ) G=(V,E)
  10. G G
  11. G G
  12. G G
  13. E D ( G ) ED(G)
  14. G G
  15. E D ( G ) ED(G)
  16. n n
  17. n n
  18. n n
  19. G G
  20. D ( G ) D(G)
  21. D ( G ) D(G)
  22. G i G_{i}
  23. n n
  24. n n
  25. n - 1 n-1
  26. n n
  27. n - 1 n-1
  28. n n
  29. G G
  30. | V ( G ) | |V(G)|
  31. D ( G ) D(G)
  32. D ( G ) D(G)
  33. G G
  34. G G
  35. | V ( G ) | = | D ( G ) | |V(G)|=|D(G)|
  36. G G
  37. n n
  38. | E ( G ) | |E(G)|
  39. G G
  40. n - 2 n-2
  41. D ( G ) D(G)
  42. D ( G ) D(G)
  43. D ( G ) D(G)
  44. D ( G ) D(G)
  45. | E ( G ) | = q i n - 2 |E(G)|=\sum\frac{q_{i}}{n-2}
  46. q i q_{i}
  47. D ( G ) D(G)
  48. G G
  49. v i v_{i}
  50. G i G_{i}
  51. v i v_{i}
  52. q i q_{i}
  53. G i G_{i}
  54. q q
  55. G G
  56. q - q i = deg ( v i ) q-q_{i}=\deg(v_{i})

Recurrence_relation.html

  1. x n + 1 = r x n ( 1 - x n ) , x_{n+1}=rx_{n}(1-x_{n}),
  2. F n = F n - 1 + F n - 2 F_{n}=F_{n-1}+F_{n-2}
  3. F 0 = 0 F_{0}=0
  4. F 1 = 1 F_{1}=1
  5. F 2 = F 1 + F 0 F_{2}=F_{1}+F_{0}
  6. F 3 = F 2 + F 1 F_{3}=F_{2}+F_{1}
  7. F 4 = F 3 + F 2 F_{4}=F_{3}+F_{2}
  8. t 1 - t - t 2 . \frac{t}{1-t-t^{2}}.
  9. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  10. ( n k ) = ( n - 1 k - 1 ) + ( n - 1 k ) , {\left({{n}\atop{k}}\right)}={\left({{n-1}\atop{k-1}}\right)}+{\left({{n-1}% \atop{k}}\right)},
  11. ( n 0 ) = ( n n ) = 1 {\textstyle\left({{n}\atop{0}}\right)}={\textstyle\left({{n}\atop{n}}\right)}=1
  12. ( n k ) = n ! k ! ( n - k ) ! . {\left({{n}\atop{k}}\right)}=\frac{n!}{k!(n-k)!}.
  13. a n = c 1 a n - 1 + c 2 a n - 2 + + c d a n - d , a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+\cdots+c_{d}a_{n-d},
  14. a 0 , , a d - 1 a_{0},\dots,a_{d-1}
  15. p ( t ) = t d - c 1 t d - 1 - c 2 t d - 2 - - c d p(t)=t^{d}-c_{1}t^{d-1}-c_{2}t^{d-2}-\cdots-c_{d}
  16. a n = k 1 r 1 n + k 2 r 2 n + + k d r d n , a_{n}=k_{1}r_{1}^{n}+k_{2}r_{2}^{n}+\cdots+k_{d}r_{d}^{n},
  17. a n = k 1 r n + k 2 n r n + k 3 n 2 r n . a_{n}=k_{1}r^{n}+k_{2}nr^{n}+k_{3}n^{2}r^{n}.
  18. a n = a n - d , n d a_{n}=a_{n-d},n\geq d
  19. a 0 , a 1 , , a d - 1 , a 0 , a_{0},a_{1},\dots,a_{d-1},a_{0},\dots
  20. a 0 + a 1 x 1 + + a d - 1 x d - 1 1 - x d = \displaystyle\frac{a_{0}+a_{1}x^{1}+\cdots+a_{d-1}x^{d-1}}{1-x^{d}}=
  21. a n = c 1 a n - 1 + c 2 a n - 2 + + c d a n - d a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+\cdots+c_{d}a_{n-d}
  22. a 0 + a 1 x 1 + a 2 x 2 + , a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots,
  23. 1 - c 1 x 1 - c 2 x 2 - - c d x d . 1-c_{1}x^{1}-c_{2}x^{2}-\cdots-c_{d}x^{d}.
  24. b n = a n - c 1 a n - 1 - c 2 a n - 2 - - c d a n - d b_{n}=a_{n}-c_{1}a_{n-1}-c_{2}a_{n-2}-\cdots-c_{d}a_{n-d}
  25. x n x^{n}
  26. ( a 0 + a 1 x 1 + a 2 x 2 + ) ( 1 - c 1 x 1 - c 2 x 2 - - c d x d ) = ( b 0 + b 1 x 1 + b 2 x 2 + + b d - 1 x d - 1 ) \left(a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots\right)\left(1-c_{1}x^{1}-c_{2}x^{2}-% \cdots-c_{d}x^{d}\right)=\left(b_{0}+b_{1}x^{1}+b_{2}x^{2}+\cdots+b_{d-1}x^{d-% 1}\right)
  27. a 0 + a 1 x 1 + a 2 x 2 + = b 0 + b 1 x 1 + b 2 x 2 + + b d - 1 x d - 1 1 - c 1 x 1 - c 2 x 2 - - c d x d , a_{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots=\frac{b_{0}+b_{1}x^{1}+b_{2}x^{2}+\cdots+b_% {d-1}x^{d-1}}{1-c_{1}x^{1}-c_{2}x^{2}-\cdots-c_{d}x^{d}},
  28. x d p ( x - 1 ) , x^{d}p\left(x^{-1}\right),
  29. b 0 = a 0 b_{0}=a_{0}
  30. { a n } n = 1 \left\{a_{n}\right\}_{n=1}^{\infty}
  31. Δ ( a n ) \Delta(a_{n})
  32. Δ ( a n ) = a n + 1 - a n \Delta(a_{n})=a_{n+1}-a_{n}\,
  33. Δ 2 ( a n ) \Delta^{2}(a_{n})
  34. Δ 2 ( a n ) = Δ ( a n + 1 ) - Δ ( a n ) \Delta^{2}(a_{n})=\Delta(a_{n+1})-\Delta(a_{n})
  35. Δ 2 ( a n ) = a n + 2 - 2 a n + 1 + a n \Delta^{2}(a_{n})=a_{n+2}-2a_{n+1}+a_{n}
  36. Δ k ( a n ) \Delta^{k}(a_{n})
  37. Δ k ( a n ) = Δ k - 1 ( a n + 1 ) - Δ k - 1 ( a n ) = t = 0 k ( k t ) ( - 1 ) t a n + k - t \Delta^{k}(a_{n})=\Delta^{k-1}(a_{n+1})-\Delta^{k-1}(a_{n})=\sum_{t=0}^{k}{% \left({{k}\atop{t}}\right)}(-1)^{t}a_{n+k-t}
  38. a n + k = ( n 0 ) a n + ( n 1 ) Δ ( a n ) + + ( n k ) Δ k ( a n ) . a_{n+k}={n\choose 0}a_{n}+{n\choose 1}\Delta(a_{n})+\cdots+{n\choose k}\Delta^% {k}(a_{n}).
  39. 3 Δ 2 ( a n ) + 2 Δ ( a n ) + 7 a n = 0 3\Delta^{2}(a_{n})+2\Delta(a_{n})+7a_{n}=0
  40. 3 a n + 2 = 4 a n + 1 - 8 a n 3a_{n+2}=4a_{n+1}-8a_{n}
  41. a n = r a n - 1 a_{n}=ra_{n-1}
  42. a n = A a n - 1 + B a n - 2 . a_{n}=Aa_{n-1}+Ba_{n-2}.
  43. r n = A r n - 1 + B r n - 2 r^{n}=Ar^{n-1}+Br^{n-2}
  44. r 2 = A r + B , r^{2}=Ar+B,
  45. r 2 - A r - B = 0 , r^{2}-Ar-B=0,
  46. a n = C λ 1 n + D λ 2 n a_{n}=C\lambda_{1}^{n}+D\lambda_{2}^{n}
  47. a n = C λ n + D n λ n a_{n}=C\lambda^{n}+Dn\lambda^{n}
  48. λ 1 , λ 2 = α ± β i . \lambda_{1},\lambda_{2}=\alpha\pm\beta i.
  49. a n = C λ 1 n + D λ 2 n a_{n}=C\lambda_{1}^{n}+D\lambda_{2}^{n}
  50. a n = 2 M n ( E cos ( θ n ) + F sin ( θ n ) ) = 2 G M n cos ( θ n - δ ) , a_{n}=2M^{n}\left(E\cos(\theta n)+F\sin(\theta n)\right)=2GM^{n}\cos(\theta n-% \delta),
  51. M = α 2 + β 2 cos ( θ ) = α M sin ( θ ) = β M C , D = E F i G = E 2 + F 2 cos ( δ ) = E G sin ( δ ) = F G \begin{array}[]{lcl}M=\sqrt{\alpha^{2}+\beta^{2}}&\cos(\theta)=\tfrac{\alpha}{% M}&\sin(\theta)=\tfrac{\beta}{M}\\ C,D=E\mp Fi&&\\ G=\sqrt{E^{2}+F^{2}}&\cos(\delta)=\tfrac{E}{G}&\sin(\delta)=\tfrac{F}{G}\end{array}
  52. λ 1 + λ 2 = 2 α = A , \lambda_{1}+\lambda_{2}=2\alpha=A,
  53. λ 1 λ 2 = α 2 + β 2 = - B , \lambda_{1}\cdot\lambda_{2}=\alpha^{2}+\beta^{2}=-B,
  54. a n = ( - B ) n 2 ( E cos ( θ n ) + F sin ( θ n ) ) , a_{n}=(-B)^{\frac{n}{2}}\left(E\cos(\theta n)+F\sin(\theta n)\right),
  55. E = - A a 1 + a 2 B F = - i A 2 a 1 - A a 2 + 2 a 1 B B A 2 + 4 B θ = arccos ( A 2 - B ) \begin{aligned}\displaystyle E&\displaystyle=\frac{-Aa_{1}+a_{2}}{B}\\ \displaystyle F&\displaystyle=-i\frac{A^{2}a_{1}-Aa_{2}+2a_{1}B}{B\sqrt{A^{2}+% 4B}}\\ \displaystyle\theta&\displaystyle=\arccos\left(\frac{A}{2\sqrt{-B}}\right)\end% {aligned}
  56. b * = K 1 - A - B . b^{*}=\frac{K}{1-A-B}.
  57. [ b n - b * ] = A [ b n - 1 - b * ] + B [ b n - 2 - b * ] , [b_{n}-b^{*}]=A[b_{n-1}-b^{*}]+B[b_{n-2}-b^{*}],
  58. y n + k - c n - 1 y n - 1 + k - c n - 2 y n - 2 + k + - c 0 y k = 0 y_{n+k}-c_{n-1}y_{n-1+k}-c_{n-2}y_{n-2+k}+\cdots-c_{0}y_{k}=0
  59. y n = c n - 1 y n - 1 + c n - 2 y n - 2 + + c 0 y 0 . y_{n}=c_{n-1}y_{n-1}+c_{n-2}y_{n-2}+\cdots+c_{0}y_{0}.
  60. y n - k = y n - k , y_{n-k}=y_{n-k},
  61. y n = [ y n y n - 1 y 1 ] = [ c n - 1 c n - 2 c 0 1 0 0 0 0 0 1 0 ] [ y n - 1 y n - 2 y 0 ] = C y n - 1 = C n y 0 . \vec{y}_{n}=\begin{bmatrix}y_{n}\\ y_{n-1}\\ \vdots\\ \vdots\\ y_{1}\end{bmatrix}=\begin{bmatrix}c_{n-1}&c_{n-2}&\cdots&\cdots&c_{0}\\ 1&0&\cdots&\cdots&0\\ 0&\ddots&\ddots&&\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&0&1&0\end{bmatrix}\begin{bmatrix}y_{n-1}\\ y_{n-2}\\ \vdots\\ \vdots\\ y_{0}\end{bmatrix}=C\ \vec{y}_{n-1}=C^{n}\vec{y}_{0}.
  62. y n \vec{y}_{n}
  63. y 0 y_{0}
  64. y n , y n = y n [ n ] \vec{y}_{n},y_{n}=\vec{y}_{n}[n]
  65. y n = C n y 0 = c 1 λ 1 n e 1 + c 2 λ 2 n e 2 + + c n λ n n e n \vec{y}_{n}=\vec{C}^{n}\,\vec{y}_{0}=c_{1}\,\lambda_{1}^{n}\,\vec{e}_{1}+c_{2}% \,\lambda_{2}^{n}\,\vec{e}_{2}+\cdots+c_{n}\,\lambda_{n}^{n}\,\vec{e}_{n}
  66. λ 1 , λ 2 , , λ n \lambda_{1},\lambda_{2},\ldots,\lambda_{n}
  67. e 1 , e 2 , , e n \vec{e}_{1},\vec{e}_{2},\ldots,\vec{e}_{n}
  68. y n . \vec{y}_{n}.
  69. C e i = λ i e i = C [ e i , n e i , n - 1 e i , 1 ] = [ λ i e i , n λ i e i , n - 1 λ i e i , 1 ] C\,\vec{e}_{i}=\lambda_{i}\vec{e}_{i}=C\begin{bmatrix}e_{i,n}\\ e_{i,n-1}\\ \vdots\\ e_{i,1}\end{bmatrix}=\begin{bmatrix}\lambda_{i}\,e_{i,n}\\ \lambda_{i}\,e_{i,n-1}\\ \vdots\\ \lambda_{i}\,e_{i,1}\end{bmatrix}
  70. e i = [ λ i n - 1 λ i 2 λ i 1 ] T , \vec{e}_{i}=\begin{bmatrix}\lambda_{i}^{n-1}&\cdots&\lambda_{i}^{2}&\lambda_{i% }&1\end{bmatrix}^{T},
  71. y n = 1 n c i λ i n e i \vec{y}_{n}=\sum_{1}^{n}{c_{i}\,\lambda_{i}^{n}\,\vec{e}_{i}}
  72. c i c_{i}
  73. y 0 = [ y 0 y - 1 y - n + 1 ] = i = 1 n c i λ i 0 e i = [ e 1 e 2 e n ] [ c 1 c 2 c n ] = E [ c 1 c 2 c n ] \vec{y}_{0}=\begin{bmatrix}y_{0}\\ y_{-1}\\ \vdots\\ y_{-n+1}\end{bmatrix}=\sum_{i=1}^{n}{c_{i}\,\lambda_{i}^{0}\,\vec{e}_{i}}=% \begin{bmatrix}\vec{e}_{1}&\vec{e}_{2}&\cdots&\vec{e}_{n}\end{bmatrix}\,\begin% {bmatrix}c_{1}\\ c_{2}\\ \cdots\\ c_{n}\end{bmatrix}=E\,\begin{bmatrix}c_{1}\\ c_{2}\\ \cdots\\ c_{n}\end{bmatrix}
  74. [ c 1 c 2 c n ] = E - 1 y 0 = [ λ 1 n - 1 λ 2 n - 1 λ n n - 1 λ 1 λ 2 λ n 1 1 1 ] - 1 [ y 0 y - 1 y - n + 1 ] . \begin{bmatrix}c_{1}\\ c_{2}\\ \cdots\\ c_{n}\end{bmatrix}=E^{-1}\vec{y}_{0}=\begin{bmatrix}\lambda_{1}^{n-1}&\lambda_% {2}^{n-1}&\cdots&\lambda_{n}^{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ \lambda_{1}&\lambda_{2}&\cdots&\lambda_{n}\\ 1&1&\cdots&1\end{bmatrix}^{-1}\,\begin{bmatrix}y_{0}\\ y_{-1}\\ \vdots\\ y_{-n+1}\end{bmatrix}.
  75. y a , y b , n \underbrace{y_{a},y_{b},\ldots}\text{n}
  76. [ y a y b ] = [ y a [ n ] y b [ n ] ] = [ i = 1 n c i λ i a e i [ n ] i = 1 n c i λ i b e i [ n ] ] = [ i = 1 n c i λ i a λ i n - 1 i = 1 n c i λ i b λ i n - 1 ] = \begin{bmatrix}y_{a}\\ y_{b}\\ \vdots\end{bmatrix}=\begin{bmatrix}\vec{y}_{a}[n]\\ \vec{y}_{b}[n]\\ \vdots\end{bmatrix}=\begin{bmatrix}\sum_{i=1}^{n}{c_{i}\,\lambda_{i}^{a}\,\vec% {e}_{i}[n]}\\ \sum_{i=1}^{n}{c_{i}\,\lambda_{i}^{b}\,\vec{e}_{i}[n]}\\ \vdots\end{bmatrix}=\begin{bmatrix}\sum_{i=1}^{n}{c_{i}\,\lambda_{i}^{a}\,% \lambda_{i}^{n-1}}\\ \sum_{i=1}^{n}{c_{i}\,\lambda_{i}^{b}\,\lambda_{i}^{n-1}}\\ \vdots\end{bmatrix}=
  77. = [ c i λ i a + n - 1 c i λ i b + n - 1 ] = [ λ 1 a + n - 1 λ 2 a + n - 1 λ n a + n - 1 λ 1 b + n - 1 λ 2 b + n - 1 λ n b + n - 1 ] [ c 1 c 2 c n ] . =\begin{bmatrix}\sum{c_{i}\,\lambda_{i}^{a+n-1}}\\ \sum{c_{i}\,\lambda_{i}^{b+n-1}}\\ \vdots\end{bmatrix}=\begin{bmatrix}\lambda_{1}^{a+n-1}&\lambda_{2}^{a+n-1}&% \cdots&\lambda_{n}^{a+n-1}\\ \lambda_{1}^{b+n-1}&\lambda_{2}^{b+n-1}&\cdots&\lambda_{n}^{b+n-1}\\ \vdots&\vdots&\ddots&\vdots\end{bmatrix}\,\begin{bmatrix}c_{1}\\ c_{2}\\ \vdots\\ c_{n}\end{bmatrix}.
  78. { a n = a n - 1 - b n - 1 b n = 2 a n - 1 + b n - 1 . \begin{cases}a_{n}=a_{n-1}-b_{n-1}\\ b_{n}=2a_{n-1}+b_{n-1}.\end{cases}
  79. t d - c 1 t d - 1 - c 2 t d - 2 - - c d = 0 t^{d}-c_{1}t^{d-1}-c_{2}t^{d-2}-\cdots-c_{d}=0\,
  80. λ n , n λ n , n 2 λ n , , n r - 1 λ n \lambda^{n},n\lambda^{n},n^{2}\lambda^{n},\dots,n^{r-1}\lambda^{n}
  81. a n = ( b 1 λ 1 n + b 2 n λ 1 n + b 3 n 2 λ 1 n + + b r n r - 1 λ 1 n ) + + ( b d - q + 1 λ * n + + b d n q - 1 λ * n ) a_{n}=\left(b_{1}\lambda_{1}^{n}+b_{2}n\lambda_{1}^{n}+b_{3}n^{2}\lambda_{1}^{% n}+\cdots+b_{r}n^{r-1}\lambda_{1}^{n}\right)+\cdots+\left(b_{d-q+1}\lambda_{*}% ^{n}+\cdots+b_{d}n^{q-1}\lambda_{*}^{n}\right)
  82. a 0 , a 1 , , a d a_{0},a_{1},\dots,a_{d}
  83. a 0 , a 1 , , a d a_{0},a_{1},\dots,a_{d}
  84. b 1 , b 2 , , b d b_{1},b_{2},\dots,b_{d}
  85. a 0 , a 1 , a 2 , a_{0},a_{1},a_{2},\dots
  86. n = 0 f ( n ) ( a ) n ! ( x - a ) n \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}
  87. y [ k ] f [ n + k ] y^{[k]}\to f[n+k]
  88. x m * y [ k ] n ( n - 1 ) ( n - m + 1 ) f [ n + k - m ] x^{m}*y^{[k]}\to n(n-1)(n-m+1)f[n+k-m]
  89. ( x 2 + 3 x - 4 ) y [ 3 ] - ( 3 x + 1 ) y [ 2 ] + 2 y = 0 (x^{2}+3x-4)y^{[3]}-(3x+1)y^{[2]}+2y=0
  90. n ( n - 1 ) f [ n + 1 ] + 3 n f [ n + 2 ] - 4 f [ n + 3 ] - 3 n f [ n + 1 ] - f [ n + 2 ] + 2 f [ n ] = 0 n(n-1)f[n+1]+3nf[n+2]-4f[n+3]-3nf[n+1]-f[n+2]+2f[n]=0
  91. - 4 f [ n + 3 ] + 2 n f [ n + 2 ] + n ( n - 4 ) f [ n + 1 ] + 2 f [ n ] = 0. -4f[n+3]+2nf[n+2]+n(n-4)f[n+1]+2f[n]=0.
  92. a y ′′ + b y + c y = 0 ay^{\prime\prime}+by^{\prime}+cy=0
  93. y = e a x . y=e^{ax}.
  94. a f [ n + 2 ] + b f [ n + 1 ] + c f [ n ] = 0. af[n+2]+bf[n+1]+cf[n]=0.
  95. a n + 1 = a n + 1 a_{n+1}=a_{n}+1
  96. a n + 2 = a n + 1 + 1 a_{n+2}=a_{n+1}+1
  97. a n + 2 - a n + 1 = a n + 1 - a n a_{n+2}-a_{n+1}=a_{n+1}-a_{n}
  98. a n + 2 = 2 a n + 1 - a n a_{n+2}=2a_{n+1}-a_{n}
  99. a n + k = λ k - 1 a n + k - 1 + λ k - 2 a n + k - 2 + + λ 1 a n + 1 + λ 0 a n + p ( n ) a_{n+k}=\lambda_{k-1}a_{n+k-1}+\lambda_{k-2}a_{n+k-2}+\cdots+\lambda_{1}a_{n+1% }+\lambda_{0}a_{n}+p(n)
  100. λ 0 , λ 1 , , λ k - 1 \lambda_{0},\lambda_{1},\dots,\lambda_{k-1}
  101. P ( x ) = n = 0 p n x n P(x)=\sum_{n=0}^{\infty}p_{n}x^{n}
  102. A ( x ) = n = 0 a ( n ) x n A(x)=\sum_{n=0}^{\infty}a(n)x^{n}
  103. a n = i = 1 s c i a n - i + p n , n n r , a_{n}=\sum_{i=1}^{s}c_{i}a_{n-i}+p_{n},\quad n\geq n_{r},
  104. ( 1 - i = 1 s c i x i ) A ( x ) = P ( x ) + n = 0 n r - 1 [ a n - p n ] x n - i = 1 s c i x i n = 0 n r - i - 1 a n x n . \left(1-\sum_{i=1}^{s}c_{i}x^{i}\right)A(x)=P(x)+\sum_{n=0}^{n_{r}-1}[a_{n}-p_% {n}]x^{n}-\sum_{i=1}^{s}c_{i}x^{i}\sum_{n=0}^{n_{r}-i-1}a_{n}x^{n}.
  105. a n = 10 a n - 1 + n a_{n}=10a_{n-1}+n
  106. a n + 1 = f n a n + g n , f n 0 , a_{n+1}=f_{n}a_{n}+g_{n},\qquad f_{n}\neq 0,
  107. a n + 1 - f n a n = g n a_{n+1}-f_{n}a_{n}=g_{n}
  108. a n + 1 k = 0 n f k - f n a n k = 0 n f k = g n k = 0 n f k \frac{a_{n+1}}{\prod_{k=0}^{n}f_{k}}-\frac{f_{n}a_{n}}{\prod_{k=0}^{n}f_{k}}=% \frac{g_{n}}{\prod_{k=0}^{n}f_{k}}
  109. a n + 1 k = 0 n f k - a n k = 0 n - 1 f k = g n k = 0 n f k \frac{a_{n+1}}{\prod_{k=0}^{n}f_{k}}-\frac{a_{n}}{\prod_{k=0}^{n-1}f_{k}}=% \frac{g_{n}}{\prod_{k=0}^{n}f_{k}}
  110. A n = a n k = 0 n - 1 f k , A_{n}=\frac{a_{n}}{\prod_{k=0}^{n-1}f_{k}},
  111. A n + 1 - A n = g n k = 0 n f k A_{n+1}-A_{n}=\frac{g_{n}}{\prod_{k=0}^{n}f_{k}}
  112. m = 0 n - 1 ( A m + 1 - A m ) = A n - A 0 = m = 0 n - 1 g m k = 0 m f k \sum_{m=0}^{n-1}(A_{m+1}-A_{m})=A_{n}-A_{0}=\sum_{m=0}^{n-1}\frac{g_{m}}{\prod% _{k=0}^{m}f_{k}}
  113. a n k = 0 n - 1 f k = A 0 + m = 0 n - 1 g m k = 0 m f k \frac{a_{n}}{\prod_{k=0}^{n-1}f_{k}}=A_{0}+\sum_{m=0}^{n-1}\frac{g_{m}}{\prod_% {k=0}^{m}f_{k}}
  114. a n = ( k = 0 n - 1 f k ) ( A 0 + m = 0 n - 1 g m k = 0 m f k ) a_{n}=\left(\prod_{k=0}^{n-1}f_{k}\right)\left(A_{0}+\sum_{m=0}^{n-1}\frac{g_{% m}}{\prod_{k=0}^{m}f_{k}}\right)
  115. J n + 1 = 2 n z J n - J n - 1 J_{n+1}=\frac{2n}{z}J_{n}-J_{n-1}
  116. J n = J n ( z ) , J_{n}=J_{n}(z),\,
  117. ( b - n ) M n - 1 + ( 2 n - b - z ) M n - n M n + 1 = 0 (b-n)M_{n-1}+(2n-b-z)M_{n}-nM_{n+1}=0\,
  118. M n = M ( n , b ; z ) M_{n}=M(n,b;z)\,
  119. w t + 1 = a w t + b c w t + d w_{t+1}=\tfrac{aw_{t}+b}{cw_{t}+d}
  120. w t w_{t}
  121. x t x_{t}
  122. x t x_{t}
  123. a n = c 1 a n - 1 + c 2 a n - 2 + + c d a n - d , a_{n}=c_{1}a_{n-1}+c_{2}a_{n-2}+\dots+c_{d}a_{n-d},\,
  124. λ d - c 1 λ d - 1 - c 2 λ d - 2 - - c d λ 0 = 0. \lambda^{d}-c_{1}\lambda^{d-1}-c_{2}\lambda^{d-2}-\dots-c_{d}\lambda^{0}=0.\,
  125. [ x t - x * ] = A [ x t - 1 - x * ] [x_{t}-x^{*}]=A[x_{t-1}-x^{*}]\,
  126. x n = f ( x n - 1 ) . x_{n}=f(x_{n-1}).
  127. | f ( x * ) | < 1. |f^{\prime}(x^{*})|<1.\,
  128. g ( x ) := f f f ( x ) g(x):=f\circ f\circ\cdot\cdot\cdot\circ f(x)
  129. | g ( x * ) | < 1 , |g^{\prime}(x^{*})|<1,
  130. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , y^{\prime}(t)=f(t,y(t)),\ \ y(t_{0})=y_{0},
  131. y 0 = y ( t 0 ) , y 1 = y ( t 0 + h ) , y 2 = y ( t 0 + 2 h ) , y_{0}=y(t_{0}),\ \ y_{1}=y(t_{0}+h),\ \ y_{2}=y(t_{0}+2h),\ \dots
  132. y n + 1 = y n + h f ( t n , y n ) . \,y_{n+1}=y_{n}+hf(t_{n},y_{n}).
  133. N t + 1 = λ N t e - a P t N_{t+1}=\lambda N_{t}e^{-aP_{t}}\,
  134. P t + 1 = N t ( 1 - e - a P t ) , P_{t+1}=N_{t}(1-e^{-aP_{t}}),\,
  135. n n
  136. n n
  137. c 1 = 1 c_{1}=1
  138. c n = 1 + c n / 2 c_{n}=1+c_{n/2}
  139. log 2 ( n ) \log_{2}(n)
  140. y t = ( 1 - α ) x t + α y t - T y_{t}=(1-\alpha)x_{t}+\alpha y_{t-T}
  141. x t x_{t}
  142. y t y_{t}
  143. y t = ( 1 - α ) x t + α ( ( 1 - α ) x t - T + α y t - 2 T ) y_{t}=(1-\alpha)x_{t}+\alpha((1-\alpha)x_{t-T}+\alpha y_{t-2T})
  144. y t = ( 1 - α ) x t + ( α - α 2 ) x t - T + α 2 y t - 2 T ) y_{t}=(1-\alpha)x_{t}+(\alpha-\alpha^{2})x_{t-T}+\alpha^{2}y_{t-2T})

Reduced_mass.html

  1. μ \scriptstyle\mu
  2. μ \scriptstyle\mu
  3. μ = 1 1 m 1 + 1 m 2 = m 1 m 2 m 1 + m 2 , \mu=\cfrac{1}{\cfrac{1}{m_{1}}+\cfrac{1}{m_{2}}}=\cfrac{m_{1}m_{2}}{m_{1}+m_{2% }},\!\,
  4. μ m 1 , μ m 2 \mu\leq m_{1},\quad\mu\leq m_{2}\!\,
  5. 1 μ = 1 m 1 + 1 m 2 \frac{1}{\mu}=\frac{1}{m_{1}}+\frac{1}{m_{2}}\,\!
  6. F 12 = m 1 a 1 . {F}_{12}=m_{1}{a}_{1}.\!\,
  7. F 21 = m 2 a 2 . {F}_{21}=m_{2}{a}_{2}.\!\,
  8. F 12 = - F 21 . {F}_{12}=-{F}_{21}.\!\,
  9. m 1 a 1 = - m 2 a 2 . m_{1}{a}_{1}=-m_{2}{a}_{2}.\!\,
  10. a 2 = - m 1 m 2 a 1 . {a}_{2}=-{m_{1}\over m_{2}}{a}_{1}.\!\,
  11. a rel = a 1 - a 2 = ( 1 + m 1 m 2 ) a 1 = m 2 + m 1 m 1 m 2 m 1 a 1 = F 12 m red . {a}_{\rm rel}={a}_{1}-{a}_{2}=\left(1+\frac{m_{1}}{m_{2}}\right){a}_{1}=\frac{% m_{2}+m_{1}}{m_{1}m_{2}}m_{1}{a}_{1}=\frac{{F}_{12}}{m_{\rm red}}.
  12. L = 1 2 m 1 𝐫 ˙ 1 2 + 1 2 m 2 𝐫 ˙ 2 2 - V ( | 𝐫 1 - 𝐫 2 | ) L={1\over 2}m_{1}\mathbf{\dot{r}}_{1}^{2}+{1\over 2}m_{2}\mathbf{\dot{r}}_{2}^% {2}-V(|\mathbf{r}_{1}-\mathbf{r}_{2}|)\!\,
  13. 𝐫 i {\mathbf{r}}_{i}
  14. m i m_{i}
  15. i i
  16. 𝐫 = 𝐫 1 - 𝐫 2 \mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}
  17. m 1 𝐫 1 + m 2 𝐫 2 = 0 m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{2}=0
  18. 𝐫 1 = m 2 𝐫 m 1 + m 2 , 𝐫 2 = - m 1 𝐫 m 1 + m 2 . \mathbf{r}_{1}=\frac{m_{2}\mathbf{r}}{m_{1}+m_{2}},\mathbf{r}_{2}=\frac{-m_{1}% \mathbf{r}}{m_{1}+m_{2}}.
  19. L = 1 2 μ 𝐫 ˙ 2 - V ( r ) , L={1\over 2}\mu\mathbf{\dot{r}}^{2}-V(r),
  20. μ = m 1 m 2 m 1 + m 2 \mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}
  21. Δ K = 1 2 μ v rel 2 ( e 2 - 1 ) \Delta K=\frac{1}{2}\mu v^{2}_{\rm rel}(e^{2}-1)
  22. V ( | 𝐫 1 - 𝐫 2 | ) = - G m 1 m 2 | 𝐫 1 - 𝐫 2 | , V(|\mathbf{r}_{1}-\mathbf{r}_{2}|)=-\frac{Gm_{1}m_{2}}{|\mathbf{r}_{1}-\mathbf% {r}_{2}|}\,,
  23. m 1 m 2 = ( m 1 + m 2 ) μ m_{1}m_{2}=(m_{1}+m_{2})\mu\!\,
  24. m e m e m p m e + m p m_{e}\rightarrow\frac{m_{e}m_{p}}{m_{e}+m_{p}}
  25. m p m e + m p m_{p}\rightarrow m_{e}+m_{p}
  26. x * = 1 1 x 1 + 1 x 2 = x 1 x 2 x 1 + x 2 x^{*}={1\over{1\over x_{1}}+{1\over x_{2}}}={x_{1}x_{2}\over x_{1}+x_{2}}\!\,
  27. 1 x * = i = 1 n 1 x i = 1 x 1 + 1 x 2 + + 1 x n . \ {1\over x^{*}}=\sum_{i=1}^{n}{1\over x_{i}}={1\over x_{1}}+{1\over x_{2}}+% \cdots+{1\over x_{n}}.\!\,

Reference_range.html

  1. 95 % P I = m e a n ± t 0.975 , n - 1 n + 1 n s d , 95\%\ PI=mean\pm t_{0.975,n-1}\sqrt{\frac{n+1}{n}}sd,
  2. t 0.975 , n - 1 t_{0.975,n-1}
  3. t 0.975 , n - 1 2. t_{0.975,n-1}\simeq 2.
  4. 0.18 = 0.42 \sqrt{0.18}=0.42
  5. t 0.975 , 11 = 2.20 t_{0.975,11}=2.20
  6. L o w e r l i m i t = m - t 0.975 , 11 × n + 1 n × s . d . = 5.33 - 2.20 × 13 12 × 0.42 = 4.4 Lower~{}limit=m-t_{0.975,11}\times\sqrt{\frac{n+1}{n}}\times s.d.=5.33-2.20% \times\sqrt{\frac{13}{12}}\times 0.42=4.4
  7. U p p e r l i m i t = m + t 0.975 , 11 × n + 1 n × s . d . = 5.33 + 2.20 × 13 12 × 0.42 = 6.3. Upper~{}limit=m+t_{0.975,11}\times\sqrt{\frac{n+1}{n}}\times s.d.=5.33+2.20% \times\sqrt{\frac{13}{12}}\times 0.42=6.3.
  8. L l c i L l r r = L l r r - t 0.975 , n - 1 × n + 1 n × S D S R R L = 4.4 - 2.20 × 13 12 × 0.21 3.9 , LlciLlrr=Llrr-t_{0.975,n-1}\times\sqrt{\frac{n+1}{n}}\times SDSRRL=4.4-2.20% \times\sqrt{\frac{13}{12}}\times 0.21\approx 3.9,
  9. U l c i L l r r = L l r r + t 0.975 , n - 1 × n + 1 n × S D S R R L = 4.4 + 2.20 × 13 12 × 0.21 4.9 , UlciLlrr=Llrr+t_{0.975,n-1}\times\sqrt{\frac{n+1}{n}}\times SDSRRL=4.4+2.20% \times\sqrt{\frac{13}{12}}\times 0.21\approx 4.9,
  10. D i f f e r e n c e r a t i o = | L i m i t l o g - n o r m a l - L i m i t n o r m a l | L i m i t l o g - n o r m a l , Difference~{}ratio=\frac{|Limit_{log-normal}-Limit_{normal}|}{Limit_{log-% normal}},
  11. C o e f f i c i e n t o f v a r i a t i o n = s . d . m , Coefficient~{}of~{}variation=\frac{s.d.}{m},
  12. 0.0062 = 0.079 \sqrt{0.0062}=0.079
  13. L o g e ( l o w e r l i m i t ) \displaystyle Log_{e}(lower~{}limit)
  14. L o g e ( u p p e r l i m i t ) \displaystyle Log_{e}(upper~{}limit)
  15. L o w e r l i m i t = e L o g e ( l o w e r l i m i t ) = e 1.49 = 4.4 Lower~{}limit=e^{Log_{e}(lower~{}limit)}=e^{1.49}=4.4
  16. U p p e r l i m i t = e L o g e ( u p p e r l i m i t ) = e 1.85 = 6.4 Upper~{}limit=e^{Log_{e}(upper~{}limit)}=e^{1.85}=6.4
  17. μ l o g = ln ( m ) - 1 2 ln ( 1 + ( s . d . m ) 2 ) \mu_{log}=\ln(m)-\frac{1}{2}\ln\!\left(1+\!\left(\frac{s.d.}{m}\right)^{2}\right)
  18. σ l o g = ln ( 1 + ( s . d . m ) 2 ) \sigma_{log}=\sqrt{\ln\!\left(1+\!\left(\frac{s.d.}{m}\right)^{2}\right)}
  19. μ l o g = ln ( 5.33 ) - 1 2 ln ( 1 + ( 0.42 5.33 ) 2 ) = 1.67 \mu_{log}=\ln(5.33)-\frac{1}{2}\ln\!\left(1+\!\left(\frac{0.42}{5.33}\right)^{% 2}\right)=1.67
  20. σ l o g = ln ( 1 + ( 0.42 5.33 ) 2 ) = 0.079 \sigma_{log}=\sqrt{\ln\!\left(1+\!\left(\frac{0.42}{5.33}\right)^{2}\right)}=0% .079
  21. S t a n d a r d d e v i a t i o n ( s . d . ) | ( M e a n ) - ( U p p e r l i m i t ) | 2 . Standard~{}deviation~{}(s.d.)\approx\frac{|(Mean)-(Upper~{}limit)|}{2}.
  22. S t a n d a r d s c o r e ( z ) = | M e a n - ( i n d i v i d u a l m e a s u r e m e n t ) | s . d . . Standard~{}score~{}(z)=\frac{|Mean-(individual~{}measurement)|}{s.d.}.
  23. S t a n d a r d d e v i a t i o n ( s . d . ) \displaystyle Standard~{}deviation~{}(s.d.)
  24. S t a n d a r d s c o r e ( z ) \displaystyle Standard~{}score~{}(z)

Reflection_(mathematics).html

  1. Ref l ( v ) = 2 v l l l l - v \mathrm{Ref}_{l}(v)=2\frac{v\cdot l}{l\cdot l}l-v
  2. Ref l ( v ) = 2 P r o j l ( v ) - v \mathrm{Ref}_{l}(v)=2\mathrm{Proj}_{l}(v)-v\,
  3. Ref a ( v ) = v - 2 v a a a a \mathrm{Ref}_{a}(v)=v-2\frac{v\cdot a}{a\cdot a}a
  4. Ref a ( v ) = - a v a a 2 \mathrm{Ref}_{a}(v)=-\frac{ava}{a^{2}}
  5. R i j = δ i j - 2 a i a j a 2 R_{ij}=\delta_{ij}-2\frac{a_{i}a_{j}}{\|a\|^{2}}
  6. v a = c v\cdot a=c
  7. Ref a , c ( v ) = v - 2 v a - c a a a . \mathrm{Ref}_{a,c}(v)=v-2\frac{v\cdot a-c}{a\cdot a}a.

Reflexive_space.html

  1. X X
  2. 𝔽 = \mathbb{F}=\mathbb{R}
  3. 𝔽 = \mathbb{F}=\mathbb{C}
  4. \|\cdot\|
  5. X X^{\prime}
  6. f : X 𝔽 f:X\to{\mathbb{F}}
  7. \|\cdot\|^{\prime}
  8. f = sup { | f ( x ) | : x X , x 1 } . \|f\|^{\prime}=\sup\{|f(x)|\,:\,x\in X,\ \|x\|\leq 1\}.
  9. X X^{\prime}
  10. X ′′ = ( X ) X^{\prime\prime}=(X^{\prime})^{\prime}
  11. X X
  12. h : X 𝔽 h:X^{\prime}\to{\mathbb{F}}
  13. ′′ \|\cdot\|^{\prime\prime}
  14. \|\cdot\|^{\prime}
  15. x X x\in X
  16. J ( x ) : X 𝔽 J(x):X^{\prime}\to{\mathbb{F}}
  17. J ( x ) ( f ) = f ( x ) , f X , J(x)(f)=f(x),\qquad f\in X^{\prime},
  18. J ( x ) J(x)
  19. X X^{\prime}
  20. J ( x ) X ′′ J(x)\in X^{\prime\prime}
  21. J : X X ′′ J:X\to X^{\prime\prime}
  22. J J
  23. x X J ( x ) ′′ = x , \forall x\in X\qquad\|J(x)\|^{\prime\prime}=\|x\|,
  24. J J
  25. X X
  26. J ( X ) J(X)
  27. X ′′ X^{\prime\prime}
  28. J ( X ) J(X)
  29. X ′′ X^{\prime\prime}
  30. X ′′ X^{\prime\prime}
  31. X X
  32. J : X X ′′ J:X\to X^{\prime\prime}
  33. J : X X ′′ J:X\to X^{\prime\prime}
  34. J : X X ′′ J:X\to X^{\prime\prime}
  35. X X
  36. X X
  37. X ′′ X^{\prime\prime}
  38. x 1 - x - 1 t , x ε 1 , , ε k , 1 - x ε 1 , , ε k , - 1 t , 1 k < n . \|x_{1}-x_{-1}\|\geq t,\quad\|x_{\varepsilon_{1},\ldots,\varepsilon_{k},1}-x_{% \varepsilon_{1},\ldots,\varepsilon_{k},-1}\|\geq t,\quad 1\leq k<n.
  39. ( 1 - δ X ( t ) ) j , j = 1 , , n . (1-\delta_{X}(t))^{j},\ j=1,\ldots,n.
  40. ( 1 - δ X ( t ) ) n - 1 < t / 2 , (1-\delta_{X}(t))^{n-1}<t/2,
  41. δ X ( t ) c t q , t [ 0 , 2 ] . \delta_{X}(t)\geq c\,t^{q},\quad t\in[0,2].
  42. X X
  43. 𝔽 \mathbb{F}
  44. \mathbb{R}
  45. \mathbb{C}
  46. X β X^{\prime}_{\beta}
  47. f : X 𝔽 f:X\to{\mathbb{F}}
  48. β ( X , X ) \beta(X^{\prime},X)
  49. X X
  50. X β X^{\prime}_{\beta}
  51. ( X β ) β (X^{\prime}_{\beta})^{\prime}_{\beta}
  52. X X
  53. h : X β 𝔽 h:X^{\prime}_{\beta}\to{\mathbb{F}}
  54. β ( ( X β ) , X β ) \beta((X^{\prime}_{\beta})^{\prime},X^{\prime}_{\beta})
  55. x X x\in X
  56. J ( x ) : X β 𝔽 J(x):X^{\prime}_{\beta}\to{\mathbb{F}}
  57. J ( x ) ( f ) = f ( x ) , f X . J(x)(f)=f(x),\qquad f\in X^{\prime}.
  58. X β X^{\prime}_{\beta}
  59. J ( x ) ( X β ) β J(x)\in(X^{\prime}_{\beta})^{\prime}_{\beta}
  60. J : X ( X β ) β . J:X\to(X^{\prime}_{\beta})^{\prime}_{\beta}.
  61. X X
  62. J J
  63. U U
  64. X X
  65. V V
  66. ( X β ) β (X^{\prime}_{\beta})^{\prime}_{\beta}
  67. J ( U ) V J ( X ) J(U)\supseteq V\cap J(X)
  68. X X
  69. J : X ( X β ) β J:X\to(X^{\prime}_{\beta})^{\prime}_{\beta}
  70. J : X ( X β ) β J:X\to(X^{\prime}_{\beta})^{\prime}_{\beta}
  71. J J
  72. X X
  73. X X
  74. σ ( X , X * ) \sigma(X,X^{*})
  75. X X
  76. X X
  77. X X
  78. X X
  79. X X^{\prime}
  80. X β X^{\prime}_{\beta}
  81. J : X X ′′ J:X\to X^{\prime\prime}
  82. J : X ( X β ) β J:X\to(X^{\prime}_{\beta})^{\prime}_{\beta}
  83. X X
  84. J : X X ′′ J:X\to X^{\prime\prime}
  85. X X
  86. J : X ( X β ) β J:X\to(X^{\prime}_{\beta})^{\prime}_{\beta}
  87. X X
  88. J : X ( X β ) β J:X\to(X^{\prime}_{\beta})^{\prime}_{\beta}
  89. C ( M ) C^{\infty}(M)
  90. M M
  91. ( C ) ( M ) (C^{\infty})^{\prime}(M)
  92. M M
  93. 𝒟 ( M ) {\mathcal{D}}(M)
  94. M M
  95. 𝒟 ( M ) {\mathcal{D}}^{\prime}(M)
  96. M M
  97. 𝒪 ( M ) {\mathcal{O}}(M)
  98. M M
  99. 𝒪 ( M ) {\mathcal{O}}^{\prime}(M)
  100. M M
  101. 𝒮 ( n ) {\mathcal{S}}({\mathbb{R}}^{n})
  102. n {\mathbb{R}}^{n}
  103. 𝒮 ( n ) {\mathcal{S}}^{\prime}({\mathbb{R}}^{n})
  104. n {\mathbb{R}}^{n}
  105. X X
  106. J : X X , J ( x ) ( f ) = f ( x ) , x X , f X J:X\to X^{\star\star},\quad J(x)(f)=f(x),\quad x\in X,\quad f\in X^{\star}
  107. X X^{\star}
  108. X X^{\prime}
  109. X X
  110. X X^{\star\star}
  111. X X^{\star}

Refractory_period_(physiology).html

  1. u ˙ = 0 \dot{u}=0

Regenerative_brake.html

  1. E i n - E o u t = Δ E s y s t e m E_{in}-E_{out}=\Delta E_{system}
  2. E i n E_{in}
  3. E o u t E_{out}
  4. Δ E s y s t e m \Delta E_{system}
  5. m v 2 2 = Δ E f l y \frac{mv^{2}}{2}=\Delta E_{fly}
  6. m m
  7. v v
  8. η f l y \eta_{fly}
  9. K E f l y = η f l y m v 2 2 KE_{fly}=\frac{\eta_{fly}mv^{2}}{2}
  10. η g e n = W o u t W i n \eta_{gen}=\frac{W_{out}}{W_{in}}
  11. W i n W_{in}
  12. W o u t W_{out}
  13. P g e n = η g e n m v 2 2 Δ t P_{gen}=\frac{\eta_{gen}mv^{2}}{2\Delta t}
  14. Δ t \Delta t
  15. m m
  16. v v
  17. η b a t t = P o u t P i n \eta_{batt}=\frac{P_{out}}{P_{in}}
  18. P i n = P g e n P_{in}=P_{gen}
  19. P o u t = W o u t Δ t P_{out}=\frac{W_{out}}{\Delta t}
  20. W o u t = η b a t t η g e n m v 2 2 W_{out}=\frac{\eta_{batt}\eta_{gen}mv^{2}}{2}

Regression_toward_the_mean.html

  1. y = α + β x , y=\alpha+\beta x,\,
  2. min α , β Q ( α , β ) \min_{\alpha,\,\beta}Q(\alpha,\beta)
  3. Q ( α , β ) = i = 1 n ε ^ i 2 = i = 1 n ( y i - α - β x i ) 2 Q(\alpha,\beta)=\sum_{i=1}^{n}\hat{\varepsilon}_{i}^{\,2}=\sum_{i=1}^{n}(y_{i}% -\alpha-\beta x_{i})^{2}
  4. β ^ = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 = x y ¯ - x ¯ y ¯ x 2 ¯ - x ¯ 2 = Cov [ x , y ] Var [ x ] = r x y s y s x , \displaystyle\hat{\beta}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{% \sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}=\frac{\overline{xy}-\bar{x}\bar{y}}{% \overline{x^{2}}-\bar{x}^{2}}=\frac{\operatorname{Cov}[x,y]}{\operatorname{Var% }[x]}=r_{xy}\frac{s_{y}}{s_{x}},
  5. x y ¯ = 1 n i = 1 n x i y i . \overline{xy}=\tfrac{1}{n}\textstyle\sum_{i=1}^{n}x_{i}y_{i}\ .
  6. α ^ \hat{\alpha}
  7. β ^ \hat{\beta}
  8. y = α + β x , y=\alpha+\beta x,\,
  9. y ^ = α ^ + β ^ x , \hat{y}=\hat{\alpha}+\hat{\beta}x,\,
  10. y ^ - y ¯ s y = r x y x - x ¯ s x \frac{\hat{y}-\bar{y}}{s_{y}}=r_{xy}\frac{x-\bar{x}}{s_{x}}
  11. E ( Y X ) - E Y σ y = r X - E X σ x , \frac{E(Y\mid X)-EY}{\sigma_{y}}=r\frac{X-EX}{\sigma_{x}},

Regular_graph.html

  1. K m K_{m}
  2. m m
  3. k k
  4. n n
  5. n k + 1 n\geq k+1
  6. n k nk
  7. 𝐣 = ( 1 , , 1 ) \,\textbf{j}=(1,\dots,1)
  8. 𝐣 \,\textbf{j}
  9. v = ( v 1 , , v n ) v=(v_{1},\dots,v_{n})
  10. i = 1 n v i = 0 \sum_{i=1}^{n}v_{i}=0
  11. J i j = 1 J_{ij}=1
  12. k = λ 0 > λ 1 λ n - 1 k=\lambda_{0}>\lambda_{1}\geq\dots\geq\lambda_{n-1}
  13. D log ( n - 1 ) log ( k / λ ) + 1 D\leq\frac{\log{(n-1)}}{\log(k/\lambda)}+1
  14. λ = max i > 0 { λ i } \lambda=\max_{i>0}\{\mid\lambda_{i}\mid\}

Relational_algebra.html

  1. R × S := { ( r 1 , r 2 , , r n , s 1 , s 2 , , s m ) | ( r 1 , r 2 , , r n ) R , ( s 1 , s 2 , , s m ) S } R\times S:=\{(r_{1},r_{2},\dots,r_{n},s_{1},s_{2},\dots,s_{m})|(r_{1},r_{2},% \dots,r_{n})\in R,(s_{1},s_{2},\dots,s_{m})\in S\}
  2. Π a 1 , , a n ( R ) \Pi_{a_{1},\ldots,a_{n}}(R)
  3. a 1 , , a n a_{1},\ldots,a_{n}
  4. { a 1 , , a n } \{a_{1},\ldots,a_{n}\}
  5. Π contactName, contactPhoneNumber ( addressBook ) \Pi_{\,\text{contactName, contactPhoneNumber}}(\,\text{addressBook})
  6. σ φ ( R ) \sigma_{\varphi}(R)
  7. φ \varphi
  8. and \and
  9. $\or$
  10. ¬ \lnot
  11. φ \varphi
  12. σ isFriend = true isBusinessContact = true ( addressBook ) \sigma_{\,\text{isFriend = true}\,\text{isBusinessContact = true}}(\,\text{% addressBook})
  13. ρ a / b ( R ) \rho_{a/b}(R)
  14. ρ isBusinessContact / isFriend ( addressBook ) \rho_{\,\text{isBusinessContact / isFriend}}(\,\text{addressBook})
  15. \bowtie
  16. \bowtie
  17. \bowtie
  18. R S = { t s | t R s S 𝐹𝑢𝑛 ( t s ) } R\bowtie S=\left\{t\cup s\ |\ t\in R\ \land\ s\in S\ \land\ \mathit{Fun}(t\cup s% )\right\}
  19. T = ρ x 1 / c 1 , , x m / c m ( S ) = ρ x 1 / c 1 ( ρ x 2 / c 2 ( ρ x m / c m ( S ) ) ) T=\rho_{x_{1}/c_{1},\ldots,x_{m}/c_{m}}(S)=\rho_{x_{1}/c_{1}}(\rho_{x_{2}/c_{2% }}(\ldots\rho_{x_{m}/c_{m}}(S)\ldots))
  20. P = σ c 1 = x 1 , , c m = x m ( R × T ) = σ c 1 = x 1 ( σ c 2 = x 2 ( σ c m = x m ( R × T ) ) ) P=\sigma_{c_{1}=x_{1},\ldots,c_{m}=x_{m}}(R\times T)=\sigma_{c_{1}=x_{1}}(% \sigma_{c_{2}=x_{2}}(\ldots\sigma_{c_{m}=x_{m}}(R\times T)\ldots))
  21. U = π r 1 , , r n , c 1 , , c m , s 1 , , s k ( P ) U=\pi_{r_{1},\ldots,r_{n},c_{1},\ldots,c_{m},s_{1},\ldots,s_{k}}(P)
  22. \bowtie
  23. C a r B o a t C a r P r i c e B o a t P r i c e \begin{matrix}Car\bowtie Boat\\ \scriptstyle CarPrice\geq BoatPrice\end{matrix}
  24. R S a θ b \begin{matrix}R\ \bowtie\ S\\ a\ \theta\ b\end{matrix}
  25. R S a θ v \begin{matrix}R\ \bowtie\ S\\ a\ \theta\ v\end{matrix}
  26. \bowtie
  27. \ltimes
  28. \ltimes
  29. \in
  30. and \and
  31. \exists
  32. \in
  33. \cup
  34. \ltimes
  35. π \pi
  36. \bowtie
  37. \triangleright
  38. \triangleright
  39. \triangleright
  40. \in
  41. and \and
  42. ¬ \neg\exists
  43. \in
  44. \cup
  45. \triangleright
  46. \in
  47. \cup
  48. \cup
  49. \triangleright
  50. \ltimes
  51. \triangleright
  52. \in
  53. \wedge
  54. \forall
  55. \in
  56. \cup
  57. \in
  58. ( R S ) ( ( R - π r 1 , r 2 , , r n ( R S ) ) × { ( ω , ω ) } ) (R\bowtie S)\cup((R-\pi_{r_{1},r_{2},\dots,r_{n}}(R\bowtie S))\times\{(\omega,% \dots\omega)\})
  59. ( R S ) ( { ( ω , , ω ) } × ( S - π s 1 , s 2 , , s n ( R S ) ) ) (R\bowtie S)\cup(\{(\omega,\dots,\omega)\}\times(S-\pi_{s_{1},s_{2},\dots,s_{n% }}(R\bowtie S)))
  60. \cup
  61. x y z ( ( x , y ) R + ( y , z ) R + ( x , z ) R + ) \forall x\forall y\forall z\left((x,y)\in R^{+}\wedge(y,z)\in R^{+}\Rightarrow% (x,z)\in R^{+}\right)
  62. σ A ( R ) = σ A σ A ( R ) \sigma_{A}(R)=\sigma_{A}\sigma_{A}(R)\,\!
  63. σ A σ B ( R ) = σ B σ A ( R ) \sigma_{A}\sigma_{B}(R)=\sigma_{B}\sigma_{A}(R)\,\!
  64. σ A B ( R ) = σ A ( σ B ( R ) ) = σ B ( σ A ( R ) ) \sigma_{A\land B}(R)=\sigma_{A}(\sigma_{B}(R))=\sigma_{B}(\sigma_{A}(R))
  65. σ A B ( R ) = σ A ( R ) σ B ( R ) \sigma_{A\lor B}(R)=\sigma_{A}(R)\cup\sigma_{B}(R)
  66. N M NM
  67. σ A \sigma_{A}
  68. \wedge
  69. \wedge
  70. σ A ( R × P ) = σ B C D ( R × P ) = σ D ( σ B ( R ) × σ C ( P ) ) \sigma_{A}(R\times P)=\sigma_{B\wedge C\wedge D}(R\times P)=\sigma_{D}(\sigma_% {B}(R)\times\sigma_{C}(P))
  71. σ A ( R P ) = σ A ( R ) σ A ( P ) = σ A ( R ) P \sigma_{A}(R\setminus P)=\sigma_{A}(R)\setminus\sigma_{A}(P)=\sigma_{A}(R)\setminus P
  72. σ A ( R P ) = σ A ( R ) σ A ( P ) \sigma_{A}(R\cup P)=\sigma_{A}(R)\cup\sigma_{A}(P)
  73. σ A ( R P ) = σ A ( R ) σ A ( P ) = σ A ( R ) P = R σ A ( P ) \sigma_{A}(R\cap P)=\sigma_{A}(R)\cap\sigma_{A}(P)=\sigma_{A}(R)\cap P=R\cap% \sigma_{A}(P)
  74. π a 1 , , a n ( σ A ( R ) ) = σ A ( π a 1 , , a n ( R ) ) where fields in A { a 1 , , a n } \pi_{a_{1},\ldots,a_{n}}(\sigma_{A}(R))=\sigma_{A}(\pi_{a_{1},\ldots,a_{n}}(R)% )\,\text{ where fields in }A\subseteq\{a_{1},\ldots,a_{n}\}
  75. π a 1 , , a n ( π b 1 , , b m ( R ) ) = π a 1 , , a n ( R ) where { a 1 , , a n } { b 1 , , b m } \pi_{a_{1},\ldots,a_{n}}(\pi_{b_{1},\ldots,b_{m}}(R))=\pi_{a_{1},\ldots,a_{n}}% (R)\,\text{ where }\{a_{1},\ldots,a_{n}\}\subseteq\{b_{1},\ldots,b_{m}\}
  76. π a 1 , , a n ( R P ) = π a 1 , , a n ( R ) π a 1 , , a n ( P ) . \pi_{a_{1},\ldots,a_{n}}(R\cup P)=\pi_{a_{1},\ldots,a_{n}}(R)\cup\pi_{a_{1},% \ldots,a_{n}}(P).\,
  77. π A ( { A = a , B = b } { A = a , B = b } ) = \pi_{A}(\{\langle A=a,B=b\rangle\}\cap\{\langle A=a,B=b^{\prime}\rangle\})=\emptyset
  78. π A ( { A = a , B = b } ) π A ( { A = a , B = b } ) = { A = a } \pi_{A}(\{\langle A=a,B=b\rangle\})\cap\pi_{A}(\{\langle A=a,B=b^{\prime}% \rangle\})=\{\langle A=a\rangle\}
  79. π A ( { A = a , B = b } { A = a , B = b } ) = { A = a } \pi_{A}(\{\langle A=a,B=b\rangle\}\setminus\{\langle A=a,B=b^{\prime}\rangle\}% )=\{\langle A=a\}
  80. π A ( { A = a , B = b } ) π A ( { A = a , B = b } ) = , \pi_{A}(\{\langle A=a,B=b\rangle\})\setminus\pi_{A}(\{\langle A=a,B=b^{\prime}% \rangle\})=\emptyset\,,
  81. ρ a / b ( ρ b / c ( R ) ) = ρ a / c ( R ) \rho_{a/b}(\rho_{b/c}(R))=\rho_{a/c}(R)\,\!
  82. ρ a / b ( ρ c / d ( R ) ) = ρ c / d ( ρ a / b ( R ) ) \rho_{a/b}(\rho_{c/d}(R))=\rho_{c/d}(\rho_{a/b}(R))\,\!
  83. ρ a / b ( R P ) = ρ a / b ( R ) ρ a / b ( P ) \rho_{a/b}(R\setminus P)=\rho_{a/b}(R)\setminus\rho_{a/b}(P)
  84. ρ a / b ( R P ) = ρ a / b ( R ) ρ a / b ( P ) \rho_{a/b}(R\cup P)=\rho_{a/b}(R)\cup\rho_{a/b}(P)
  85. ρ a / b ( R P ) = ρ a / b ( R ) ρ a / b ( P ) \rho_{a/b}(R\cap P)=\rho_{a/b}(R)\cap\rho_{a/b}(P)

Relative_humidity.html

  1. ( ϕ ) \left(\phi\right)
  2. ( e w ) \left({e_{w}}\right)
  3. ( e * w ) \left({{e^{*}}_{w}}\right)
  4. ϕ = e w e * w \phi={{e_{w}}\over{{e^{*}}_{w}}}
  5. e * w = ( 1.0007 + 3.46 × 10 - 6 P ) × ( 6.1121 ) e ( 17.502 T 240.97 + T ) {{e^{*}}_{w}}=(1.0007+3.46\times 10^{-6}P)\times(6.1121)e^{\left(\frac{17.502T% }{240.97+T}\right)}
  6. T T
  7. P P
  8. e * w {{e^{*}}_{w}}
  9. ( f w ) \left(f_{w}\right)
  10. ( e w ) \left(e^{\prime}_{w}\right)
  11. f W = e w e w * f_{W}=\frac{e^{\prime}_{w}}{e^{*}_{w}}

Relaxation_oscillator.html

  1. V D D V_{DD}
  2. V + \,\!V_{+}
  3. V o u t \,\!V_{out}
  4. V + = V o u t 2 V_{+}=\frac{V_{out}}{2}
  5. V - \,\!V_{-}
  6. V o u t - V - R = C d V - d t \frac{V_{out}-V_{-}}{R}=C\frac{dV_{-}}{dt}
  7. V - \,\!V_{-}
  8. d V - d t + V - R C = V o u t R C \frac{dV_{-}}{dt}+\frac{V_{-}}{RC}=\frac{V_{out}}{RC}
  9. V - = A \,\!V_{-}=A
  10. d V - d t = 0 \frac{dV_{-}}{dt}=0
  11. A R C = V o u t R C \frac{A}{RC}=\frac{V_{out}}{RC}
  12. A = V o u t \,\!A=V_{out}
  13. d V - d t + V - R C = 0 \frac{dV_{-}}{dt}+\frac{V_{-}}{RC}=0
  14. V - = B e - 1 R C t V_{-}=Be^{\frac{-1}{RC}t}
  15. V - \,\!V_{-}
  16. V - = A + B e - 1 R C t V_{-}=A+Be^{\frac{-1}{RC}t}
  17. V - = V o u t + B e - 1 R C t V_{-}=V_{out}+Be^{\frac{-1}{RC}t}
  18. V o u t = V d d V_{out}=V_{dd}
  19. V - = 0 \,\!V_{-}=0
  20. 0 = V d d + B \,\!0=V_{dd}+B
  21. B = - V d d \,\!B=-V_{dd}
  22. V d d = - V s s V_{dd}=-V_{ss}
  23. V d d 2 \frac{V_{dd}}{2}
  24. V s s 2 \frac{V_{ss}}{2}
  25. V o u t V_{out}
  26. - V d d 2 -\frac{V_{dd}}{2}
  27. V d d 2 \frac{V_{dd}}{2}
  28. V - = A + B e - 1 R C t V_{-}=A+Be^{\frac{-1}{RC}t}
  29. V d d 2 = V d d ( 1 - 3 2 e - 1 R C T 2 ) \frac{V_{dd}}{2}=V_{dd}(1-\frac{3}{2}e^{\frac{-1}{RC}\frac{T}{2}})
  30. 1 3 = e - 1 R C T 2 \frac{1}{3}=e^{\frac{-1}{RC}\frac{T}{2}}
  31. ln ( 1 3 ) = - 1 R C T 2 \ln\left(\frac{1}{3}\right)=\frac{-1}{RC}\frac{T}{2}
  32. T = 2 ln ( 3 ) R C \,\!T=2\ln(3)RC
  33. f = 1 2 ln ( 3 ) R C \,\!f=\frac{1}{2\ln(3)RC}
  34. T = ( R C ) [ ln ( 2 V s s - V d d V s s ) + ln ( 2 V d d - V s s V d d ) ] T=(RC)\left[\ln\left(\frac{2V_{ss}-V_{dd}}{V_{ss}}\right)+\ln\left(\frac{2V_{% dd}-V_{ss}}{V_{dd}}\right)\right]
  35. V d d = - V s s V_{dd}=-V_{ss}

Relevance_logic.html

  1. M , a ϕ ψ b , c ( ( R a b c M , b ϕ ) M , c ψ ) M,a\models\phi\to\psi\iff\forall b,c((Rabc\land M,b\models\phi)\Rightarrow M,c% \models\psi)
  2. M , a ¬ ϕ M , a * ⊧̸ ϕ M,a\models\lnot\phi\iff M,a^{*}\not\models\phi

Reliability_(psychometrics).html

  1. σ X 2 = σ T 2 + σ E 2 \sigma^{2}_{X}=\sigma^{2}_{T}+\sigma^{2}_{E}
  2. ρ x x \rho_{xx^{\prime}}
  3. ρ x x = σ T 2 σ X 2 = 1 - σ E 2 σ X 2 \rho_{xx^{\prime}}=\frac{\sigma^{2}_{T}}{\sigma^{2}_{X}}=1-\frac{\sigma^{2}_{E% }}{\sigma^{2}_{X}}
  4. R ( t ) = 1 - F ( t ) . R(t)=1-F(t).
  5. R ( t ) = exp ( - λ t ) . R(t)=\exp(-\lambda t).
  6. λ \lambda

Removable_singularity.html

  1. sinc ( z ) = sin z z \,\text{sinc}(z)=\frac{\sin z}{z}
  2. sin ( z ) z \frac{\sin(z)}{z}
  3. sinc ( z ) = 1 z ( k = 0 ( - 1 ) k z 2 k + 1 ( 2 k + 1 ) ! ) = k = 0 ( - 1 ) k z 2 k ( 2 k + 1 ) ! = 1 - z 2 3 ! + z 4 5 ! - z 6 7 ! + . \,\text{sinc}(z)=\frac{1}{z}\left(\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k+1}}{(% 2k+1)!}\right)=\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k}}{(2k+1)!}=1-\frac{z^{2}% }{3!}+\frac{z^{4}}{5!}-\frac{z^{6}}{7!}+\cdots.
  4. U U\subset\mathbb{C}
  5. \mathbb{C}
  6. a U a\in U
  7. U U
  8. f : U { a } f:U\setminus\{a\}\rightarrow\mathbb{C}
  9. a a
  10. f f
  11. g : U g:U\rightarrow\mathbb{C}
  12. f f
  13. U { a } U\setminus\{a\}
  14. f f
  15. U U
  16. g g
  17. D C D\subset C
  18. a D a\in D
  19. D D
  20. f f
  21. D { a } D\setminus\{a\}
  22. f f
  23. a a
  24. f f
  25. a a
  26. a a
  27. f f
  28. lim z a ( z - a ) f ( z ) = 0 \lim_{z\to a}(z-a)f(z)=0
  29. a a
  30. a a
  31. h ( z ) = { ( z - a ) 2 f ( z ) z a , 0 z = a . h(z)=\begin{cases}(z-a)^{2}f(z)&z\neq a,\\ 0&z=a.\end{cases}
  32. h ( a ) = lim z a ( z - a ) 2 f ( z ) - 0 z - a = lim z a ( z - a ) f ( z ) = 0 h^{\prime}(a)=\lim_{z\to a}\frac{(z-a)^{2}f(z)-0}{z-a}=\lim_{z\to a}(z-a)f(z)=0
  33. h ( z ) = c 0 + c 1 ( z - a ) + c 2 ( z - a ) 2 + c 3 ( z - a ) 3 + . h(z)=c_{0}+c_{1}(z-a)+c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots\,.
  34. h ( z ) = c 2 ( z - a ) 2 + c 3 ( z - a ) 3 + . h(z)=c_{2}(z-a)^{2}+c_{3}(z-a)^{3}+\cdots\,.
  35. f ( z ) = h ( z ) / ( z - a ) 2 = c 2 + c 3 ( z - a ) + . f(z)=h(z)/(z-a)^{2}=c_{2}+c_{3}(z-a)+\cdots\,.
  36. g ( z ) = c 2 + c 3 ( z - a ) + . g(z)=c_{2}+c_{3}(z-a)+\cdots\,.
  37. m m
  38. lim z a ( z - a ) m + 1 f ( z ) = 0 \lim_{z\rightarrow a}(z-a)^{m+1}f(z)=0
  39. a a
  40. f f
  41. m m
  42. a a
  43. a a
  44. f f
  45. f f
  46. U { a } U\setminus\{a\}

Renormalization.html

  1. m em = 1 2 E 2 d V = r e 1 2 ( q 4 π r 2 ) 2 4 π r 2 d r = q 2 8 π r e , m_{\mathrm{em}}=\int{1\over 2}E^{2}\,dV=\int_{r_{e}}^{\infty}\frac{1}{2}\left(% {q\over 4\pi r^{2}}\right)^{2}4\pi r^{2}\,dr={q^{2}\over 8\pi r_{e}},
  2. m em m_{\mathrm{em}}
  3. q = e q=e
  4. c c
  5. ε 0 \varepsilon_{0}
  6. r e = e 2 4 π ε 0 m e c 2 = α m e c 2.8 × 10 - 15 m . r_{e}={e^{2}\over 4\pi\varepsilon_{0}m_{\mathrm{e}}c^{2}}=\alpha{\hbar\over m_% {\mathrm{e}}c}\approx 2.8\times 10^{-15}\ \mathrm{m}.
  7. α 1 / 137 \alpha\approx 1/137
  8. / m e c \hbar/m_{\mathrm{e}}c
  9. E 2 - p 2 E^{2}-p^{2}
  10. ( p 2 - a 2 ) 1 2 \left(p^{2}-a^{2}\right)^{\frac{1}{2}}
  11. p = a p=a
  12. 1 p 2 - a 2 , \frac{1}{p^{2}-a^{2}},
  13. - i e 3 d 4 q ( 2 π ) 4 γ μ i ( γ α ( r - q ) α + m ) ( r - q ) 2 - m 2 + i ϵ γ ρ i ( γ β ( p - q ) β + m ) ( p - q ) 2 - m 2 + i ϵ γ ν - i g μ ν q 2 + i ϵ -ie^{3}\int\frac{d^{4}q}{(2\pi)^{4}}\gamma^{\mu}\frac{i(\gamma^{\alpha}(r-q)_{% \alpha}+m)}{(r-q)^{2}-m^{2}+i\epsilon}\gamma^{\rho}\frac{i(\gamma^{\beta}(p-q)% _{\beta}+m)}{(p-q)^{2}-m^{2}+i\epsilon}\gamma^{\nu}\frac{-ig_{\mu\nu}}{q^{2}+i\epsilon}
  14. e e
  15. i ϵ i\epsilon
  16. e 3 γ μ γ α γ ρ γ β γ μ d 4 q ( 2 π ) 4 q α q β ( r - q ) 2 ( p - q ) 2 q 2 e^{3}\gamma^{\mu}\gamma^{\alpha}\gamma^{\rho}\gamma^{\beta}\gamma_{\mu}\int% \frac{d^{4}q}{(2\pi)^{4}}\frac{q_{\alpha}q_{\beta}}{(r-q)^{2}(p-q)^{2}q^{2}}
  17. = ψ ¯ B [ i γ μ ( μ + i e B A B μ ) - m B ] ψ B - 1 4 F B μ ν F B μ ν \mathcal{L}=\bar{\psi}_{B}\left[i\gamma_{\mu}\left(\partial^{\mu}+ie_{B}A_{B}^% {\mu}\right)-m_{B}\right]\psi_{B}-\frac{1}{4}F_{B\mu\nu}F_{B}^{\mu\nu}
  18. B B
  19. ( ψ ¯ m ψ ) B = Z 0 ψ ¯ m ψ \left(\bar{\psi}m\psi\right)_{B}=Z_{0}\bar{\psi}m\psi
  20. ( ψ ¯ ( μ + i e A μ ) ψ ) B = Z 1 ψ ¯ ( μ + i e A μ ) ψ \left(\bar{\psi}\left(\partial^{\mu}+ieA^{\mu}\right)\psi\right)_{B}=Z_{1}\bar% {\psi}\left(\partial^{\mu}+ieA^{\mu}\right)\psi
  21. ( F μ ν F μ ν ) B = Z 3 F μ ν F μ ν . \left(F_{\mu\nu}F^{\mu\nu}\right)_{B}=Z_{3}\,F_{\mu\nu}F^{\mu\nu}.
  22. ψ ¯ ( + i e A ) ψ \bar{\psi}(\partial+ieA)\psi
  23. I = - e ψ ¯ γ μ A μ ψ - ( Z 1 - 1 ) e ψ ¯ γ μ A μ ψ \mathcal{L}_{I}=-e\bar{\psi}\gamma_{\mu}A^{\mu}\psi-(Z_{1}-1)e\bar{\psi}\gamma% _{\mu}A^{\mu}\psi
  24. e e
  25. I = 0 a 1 z d z - 0 b 1 z d z = ln a - ln b - ln 0 + ln 0 I=\int_{0}^{a}\frac{1}{z}\,dz-\int_{0}^{b}\frac{1}{z}\,dz=\ln a-\ln b-\ln 0+\ln 0
  26. I = ln a - ln b - ln ε a + ln ε b = ln a b - ln ε b ε a I=\ln a-\ln b-\ln{\varepsilon_{a}}+\ln{\varepsilon_{b}}=\ln\tfrac{a}{b}-\ln% \tfrac{\varepsilon_{b}}{\varepsilon_{a}}
  27. I = l n a b . I=ln\frac{a}{b}.
  28. ∞−∞
  29. I ( n , Λ ) = 0 Λ d p p n 1 + 2 n + 3 n + + Λ n ζ ( - n ) I(n,\Lambda)=\int_{0}^{\Lambda}dp\,p^{n}\sim 1+2^{n}+3^{n}+\cdots+\Lambda^{n}% \to\zeta(-n)
  30. Λ Λ→∞
  31. ζ ( n ) ζ(−n)
  32. I ( n , Λ ) = n 2 I ( n - 1 , Λ ) + ζ ( - n ) - r = 1 B 2 r ( 2 r ) ! a n , r ( n - 2 r + 1 ) I ( n - 2 r , Λ ) , I(n,\Lambda)=\frac{n}{2}I(n-1,\Lambda)+\zeta(-n)-\sum_{r=1}^{\infty}\frac{B_{2% r}}{(2r)!}a_{n,r}(n-2r+1)I(n-2r,\Lambda),
  33. a n , r = Γ ( n + 1 ) Γ ( n - 2 r + 2 ) . a_{n,r}=\frac{\Gamma(n+1)}{\Gamma(n-2r+2)}.
  34. I ( m , Λ ) I(m,Λ)
  35. ζ ( 1 ) , ζ ( 3 ) , ζ ( 5 ) , , ζ ( m ) ζ(−1),ζ(−3),ζ(−5),...,ζ(−m)
  36. ζ ( - m , β ) - β m 2 - i 0 d t ( i t + β ) m - ( - i t + β ) m e 2 π t - 1 = 0 d p ( p + β ) m \zeta(-m,\beta)-\frac{\beta^{m}}{2}-i\int_{0}^{\infty}dt\frac{(it+\beta)^{m}-(% -it+\beta)^{m}}{e^{2\pi t}-1}=\int_{0}^{\infty}dp\,(p+\beta)^{m}
  37. m > 0 m>0
  38. 0 d x ( β + x ) m n = 0 h m + 1 ζ ( β h - 1 , - m ) \int_{0}^{\infty}dx\,(\beta+x)^{m}\approx\sum_{n=0}^{\infty}h^{m+1}\zeta\left(% \beta h^{-1},-m\right)
  39. n = 0 1 n + a = - ψ ( a ) \sum_{n=0}^{\infty}\frac{1}{n+a}=-\psi(a)
  40. k 1 , , k n k_{1},\cdots,k_{n}
  41. d Ω \int d\Omega
  42. r 2 = k 1 2 + + k n 2 r^{2}=k_{1}^{2}+\cdots+k_{n}^{2}
  43. F ( q 1 , , q n ) F(q_{1},\cdots,q_{n})
  44. F ( r , Ω ) F(r,Ω)
  45. r r
  46. ( 1 + q i q i ) - s \left(1+\sqrt{q}_{i}q^{i}\right)^{-s}
  47. s s
  48. s s
  49. s = 0 s=0
  50. ζ ( m ) ζ(−m)

Renormalization_group.html

  1. ψ ( g ) = G d / ( G / g ) ψ(g)=Gd/(∂G/∂g)
  2. T T
  3. J J
  4. H ( T , J ) H(T,J)
  5. 2 × 2 2\times 2
  6. T T
  7. J J
  8. H ( T , J ) H(T^{\prime},J^{\prime})
  9. H ( T ′′ , J ′′ ) H(T^{\prime\prime},J^{\prime\prime})
  10. ( T , J ) ( T , J ) (T,J)\to(T^{\prime},J^{\prime})
  11. ( T , J ) ( T ′′ , J ′′ ) (T^{\prime},J^{\prime})\to(T^{\prime\prime},J^{\prime\prime})
  12. T = 0 T=0
  13. J J\to\infty
  14. T T\to\infty
  15. J 0 J\to 0
  16. T = T c T=T_{c}
  17. J = J c J=J_{c}
  18. Z Z
  19. { s i } \{s_{i}\}
  20. { J k } \{J_{k}\}
  21. { s i } { s ~ i } \{s_{i}\}\to\{\tilde{s}_{i}\}
  22. s ~ i \tilde{s}_{i}
  23. s i s_{i}
  24. Z Z
  25. s ~ i \tilde{s}_{i}
  26. { J k } { J ~ k } \{J_{k}\}\to\{\tilde{J}_{k}\}
  27. { J ~ k } = β ( { J k } ) \{\tilde{J}_{k}\}=\beta(\{J_{k}\})
  28. J J
  29. J J
  30. A A
  31. p 2 Λ 2 p^{2}\leq\Lambda^{2}
  32. Z = p 2 Λ 2 𝒟 ϕ exp [ - S Λ [ ϕ ] ] . Z=\int_{p^{2}\leq\Lambda^{2}}\mathcal{D}\phi\exp\left[-S_{\Lambda}[\phi]\right].
  33. p 2 Λ 2 p^{2}\leq\Lambda^{\prime 2}
  34. exp ( - S Λ [ ϕ ] ) = def Λ p Λ 𝒟 ϕ exp [ - S Λ [ ϕ ] ] . \exp\left(-S_{\Lambda^{\prime}}[\phi]\right)\ \stackrel{\mathrm{def}}{=}\ \int% _{\Lambda^{\prime}\leq p\leq\Lambda}\mathcal{D}\phi\exp\left[-S_{\Lambda}[\phi% ]\right].
  35. Z = p 2 Λ 2 𝒟 ϕ exp [ - S Λ [ ϕ ] ] . Z=\int_{p^{2}\leq{\Lambda^{\prime}}^{2}}\mathcal{D}\phi\exp\left[-S_{\Lambda^{% \prime}}[\phi]\right].
  36. Z Λ [ J ] = 𝒟 ϕ exp ( - S Λ [ ϕ ] + J ϕ ) = 𝒟 ϕ exp ( - 1 2 ϕ R Λ ϕ - S int Λ [ ϕ ] + J ϕ ) Z_{\Lambda}[J]=\int\mathcal{D}\phi\exp\left(-S_{\Lambda}[\phi]+J\cdot\phi% \right)=\int\mathcal{D}\phi\exp\left(-\tfrac{1}{2}\phi\cdot R_{\Lambda}\cdot% \phi-S_{\,\text{int}\,\Lambda}[\phi]+J\cdot\phi\right)
  37. 1 2 d d p ( 2 π ) d ϕ ~ * ( p ) R Λ ( p ) ϕ ~ ( p ) \frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}\tilde{\phi}^{*}(p)R_{\Lambda}(p)% \tilde{\phi}(p)
  38. p Λ p\ll\Lambda
  39. p Λ p\gg\Lambda
  40. d d Λ Z Λ = 0 \frac{d}{d\Lambda}Z_{\Lambda}=0
  41. d d Λ S int Λ = 1 2 δ S int Λ δ ϕ ( d d Λ R Λ - 1 ) δ S int Λ δ ϕ - 1 2 Tr [ δ 2 S int Λ δ ϕ δ ϕ R Λ - 1 ] . \frac{d}{d\Lambda}S_{\,\text{int}\,\Lambda}=\frac{1}{2}\frac{\delta S_{\,\text% {int}\,\Lambda}}{\delta\phi}\cdot\left(\frac{d}{d\Lambda}R_{\Lambda}^{-1}% \right)\cdot\frac{\delta S_{\,\text{int}\,\Lambda}}{\delta\phi}-\frac{1}{2}% \operatorname{Tr}\left[\frac{\delta^{2}S_{\,\text{int}\,\Lambda}}{\delta\phi\,% \delta\phi}\cdot R_{\Lambda}^{-1}\right].
  42. 1 2 d d p ( 2 π ) d ϕ ~ * ( p ) R k ( p ) ϕ ~ ( p ) \frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}\tilde{\phi}^{*}(p)R_{k}(p)\tilde{\phi% }(p)
  43. p k p\gg k
  44. p k p\ll k
  45. R k ( p ) k 2 R_{k}(p)\gtrsim k^{2}
  46. 1 2 ϕ R k ϕ \frac{1}{2}\phi\cdot R_{k}\cdot\phi
  47. exp ( W k [ J ] ) = Z k [ J ] = 𝒟 ϕ exp ( - S [ ϕ ] - 1 2 ϕ R k ϕ + J ϕ ) \exp\left(W_{k}[J]\right)=Z_{k}[J]=\int\mathcal{D}\phi\exp\left(-S[\phi]-\frac% {1}{2}\phi\cdot R_{k}\cdot\phi+J\cdot\phi\right)
  48. ϕ [ J ; k ] = δ W k δ J [ J ] \phi[J;k]=\frac{\delta W_{k}}{\delta J}[J]
  49. Γ k [ ϕ ] = def ( - W [ J k [ ϕ ] ] + J k [ ϕ ] ϕ ) - 1 2 ϕ R k ϕ . \Gamma_{k}[\phi]\ \stackrel{\mathrm{def}}{=}\ \left(-W\left[J_{k}[\phi]\right]% +J_{k}[\phi]\cdot\phi\right)-\tfrac{1}{2}\phi\cdot R_{k}\cdot\phi.
  50. d d k Γ k [ ϕ ] = - d d k W k [ J k [ ϕ ] ] - δ W k δ J d d k J k [ ϕ ] + d d k J k [ ϕ ] ϕ - 1 2 ϕ d d k R k ϕ = - d d k W k [ J k [ ϕ ] ] - 1 2 ϕ d d k R k ϕ = 1 2 ϕ d d k R k ϕ J k [ ϕ ] ; k - 1 2 ϕ d d k R k ϕ = 1 2 Tr [ ( δ J k δ ϕ ) - 1 d d k R k ] = 1 2 Tr [ ( δ 2 Γ k δ ϕ δ ϕ + R k ) - 1 d d k R k ] \begin{aligned}\displaystyle\frac{d}{dk}\Gamma_{k}[\phi]&\displaystyle=-\frac{% d}{dk}W_{k}[J_{k}[\phi]]-\frac{\delta W_{k}}{\delta J}\cdot\frac{d}{dk}J_{k}[% \phi]+\frac{d}{dk}J_{k}[\phi]\cdot\phi-\tfrac{1}{2}\phi\cdot\frac{d}{dk}R_{k}% \cdot\phi\\ &\displaystyle=-\frac{d}{dk}W_{k}[J_{k}[\phi]]-\tfrac{1}{2}\phi\cdot\frac{d}{% dk}R_{k}\cdot\phi\\ &\displaystyle=\tfrac{1}{2}\left\langle\phi\cdot\frac{d}{dk}R_{k}\cdot\phi% \right\rangle_{J_{k}[\phi];k}-\tfrac{1}{2}\phi\cdot\frac{d}{dk}R_{k}\cdot\phi% \\ &\displaystyle=\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta J_{k}}{% \delta\phi}\right)^{-1}\cdot\frac{d}{dk}R_{k}\right]\\ &\displaystyle=\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{\delta^{2}\Gamma% _{k}}{\delta\phi\delta\phi}+R_{k}\right)^{-1}\cdot\frac{d}{dk}R_{k}\right]\end% {aligned}
  51. d d k Γ k [ ϕ ] = 1 2 Tr [ ( δ 2 Γ k δ ϕ δ ϕ + R k ) - 1 d d k R k ] \frac{d}{dk}\Gamma_{k}[\phi]=\tfrac{1}{2}\operatorname{Tr}\left[\left(\frac{% \delta^{2}\Gamma_{k}}{\delta\phi\delta\phi}+R_{k}\right)^{-1}\cdot\frac{d}{dk}% R_{k}\right]
  52. φ φ
  53. T c T_{c}

Repdigit.html

  1. B B
  2. x B y - 1 B - 1 x\frac{B^{y}-1}{B-1}
  3. 0 < x < B 0<x<B
  4. y y
  5. 7 × 10 2 - 1 10 - 1 7\times\frac{10^{2}-1}{10-1}

Representation_of_a_Lie_group.html

  1. J = [ 0 I n - I n 0 ] J=\left[\begin{smallmatrix}0&I_{n}\\ -I_{n}&0\end{smallmatrix}\right]
  2. S p 2 ( 𝔽 q ) = { g G L 2 n ( 𝔽 q ) t g J g = J } . Sp_{2}(\mathbb{F}_{q})=\left\{g\in GL_{2n}(\mathbb{F}_{q})\mid^{t}gJg=J\right\}.
  3. n j n_{j}\,\!
  4. n 1 + + n r = n n_{1}+\ldots+n_{r}=n\,\!
  5. P = P ( n 1 , , n r ) = M × N P=P_{(n_{1},\ldots,n_{r})}=M\times N
  6. M G L n 1 ( 𝔽 q ) × × G L n r ( 𝔽 q ) M\simeq GL_{n_{1}}(\mathbb{F}_{q})\times\ldots\times GL_{n_{r}}(\mathbb{F}_{q})
  7. M = { ( A 1 0 0 0 A 2 0 0 0 A r ) | A j G L n j ( 𝔽 q ) , 1 j r } , M=\left\{\left.\begin{pmatrix}A_{1}&0&\cdots&0\\ 0&A_{2}&\cdots&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&\cdots&0&A_{r}\end{pmatrix}\right|A_{j}\in GL_{n_{j}}(\mathbb{F}_{q}),1\leq j% \leq r\right\},
  8. N = { ( I n 1 * * 0 I n 2 * 0 0 I n r ) } , N=\left\{\begin{pmatrix}I_{n_{1}}&*&\cdots&*\\ 0&I_{n_{2}}&\cdots&*\\ \vdots&\ddots&\ddots&\vdots\\ 0&\cdots&0&I_{n_{r}}\end{pmatrix}\right\},
  9. * *\,\!
  10. 𝔽 q \mathbb{F}_{q}

Repunit.html

  1. R n ( b ) = b n - 1 b - 1 for | b | 2 , n 1. R_{n}^{(b)}={b^{n}-1\over{b-1}}\qquad\mbox{for }~{}|b|\geq 2,n\geq 1.
  2. R 1 ( b ) = b - 1 b - 1 = 1 and R 2 ( b ) = b 2 - 1 b - 1 = b + 1 for | b | 2. R_{1}^{(b)}={b-1\over{b-1}}=1\qquad\,\text{and}\qquad R_{2}^{(b)}={b^{2}-1% \over{b-1}}=b+1\qquad\,\text{for}\ |b|\geq 2.
  3. R n = R n ( 10 ) = 10 n - 1 10 - 1 = 10 n - 1 9 for n 1. R_{n}=R_{n}^{(10)}={10^{n}-1\over{10-1}}={10^{n}-1\over 9}\qquad\mbox{for }~{}% n\geq 1.
  4. R n ( 2 ) = 2 n - 1 2 - 1 = 2 n - 1 for n 1. R_{n}^{(2)}={2^{n}-1\over{2-1}}={2^{n}-1}\qquad\mbox{for }~{}n\geq 1.
  5. R n ( b ) = 1 b - 1 d | n Φ d ( b ) R_{n}^{(b)}=\frac{1}{b-1}\prod_{d|n}\Phi_{d}(b)
  6. Φ d ( x ) \Phi_{d}(x)
  7. d th d^{\mathrm{th}}
  8. Φ p ( x ) = i = 0 p - 1 x i \Phi_{p}(x)=\sum_{i=0}^{p-1}x^{i}
  9. Φ 3 ( x ) \Phi_{3}(x)
  10. Φ 9 ( x ) \Phi_{9}(x)
  11. x 2 + x + 1 x^{2}+x+1
  12. x 6 + x 3 + 1 x^{6}+x^{3}+1
  13. n n
  14. 11 4 11_{4}
  15. 4 n - 1 = ( 2 n + 1 ) ( 2 n - 1 ) 4^{n}-1=\left(2^{n}+1\right)\left(2^{n}-1\right)
  16. 2 n + 1 2^{n}+1
  17. 2 n - 1 2^{n}-1
  18. 2 n + 1 2^{n}+1
  19. 2 n - 1 2^{n}-1
  20. n n
  21. n n
  22. n n
  23. 111 8 111_{8}
  24. 8 n - 1 = ( 4 n + 2 n + 1 ) ( 2 n - 1 ) 8^{n}-1=\left(4^{n}+2^{n}+1\right)\left(2^{n}-1\right)
  25. 4 n + 2 n + 1 4^{n}+2^{n}+1
  26. 2 n - 1 2^{n}-1
  27. 9 n - 1 = ( 3 n + 1 ) ( 3 n - 1 ) 9^{n}-1=\left(3^{n}+1\right)\left(3^{n}-1\right)
  28. 3 n + 1 3^{n}+1
  29. 3 n - 1 3^{n}-1
  30. n n
  31. n n
  32. R p R_{p}
  33. R n ( b ) R_{n}(b)
  34. b = - 2 b=-2

Residue_(complex_analysis).html

  1. f : { a k } k f:\mathbb{C}\setminus\{a_{k}\}_{k}\rightarrow\mathbb{C}
  2. f f
  3. a a
  4. Res ( f , a ) \operatorname{Res}(f,a)
  5. Res a ( f ) \operatorname{Res}_{a}(f)
  6. R R
  7. f ( z ) - R / ( z - a ) f(z)-R/(z-a)
  8. 0 < | z - a | < δ 0<|z-a|<\delta
  9. ω \omega
  10. ω \omega
  11. x x
  12. ω \omega
  13. f ( z ) d z f(z)\;dz
  14. ω \omega
  15. x x
  16. f ( z ) f(z)
  17. x x
  18. C e z z 5 d z \oint_{C}{e^{z}\over z^{5}}\,dz
  19. e z e^{z}
  20. C 1 z 5 ( 1 + z + z 2 2 ! + z 3 3 ! + z 4 4 ! + z 5 5 ! + z 6 6 ! + ) d z . \oint_{C}{1\over z^{5}}\left(1+z+{z^{2}\over 2!}+{z^{3}\over 3!}+{z^{4}\over 4% !}+{z^{5}\over 5!}+{z^{6}\over 6!}+\cdots\right)\,dz.
  21. C ( 1 z 5 + z z 5 + z 2 2 ! z 5 + z 3 3 ! z 5 + z 4 4 ! z 5 + z 5 5 ! z 5 + z 6 6 ! z 5 + ) d z \oint_{C}\left({1\over z^{5}}+{z\over z^{5}}+{z^{2}\over 2!\;z^{5}}+{z^{3}% \over 3!\;z^{5}}+{z^{4}\over 4!\;z^{5}}+{z^{5}\over 5!\;z^{5}}+{z^{6}\over 6!% \;z^{5}}+\cdots\right)\,dz
  22. = C ( 1 z 5 + 1 z 4 + 1 2 ! z 3 + 1 3 ! z 2 + 1 4 ! z + 1 5 ! + z 6 ! + ) d z . =\oint_{C}\left({1\over\;z^{5}}+{1\over\;z^{4}}+{1\over 2!\;z^{3}}+{1\over 3!% \;z^{2}}+{1\over 4!\;z}+{1\over\;5!}+{z\over 6!}+\cdots\right)\,dz.
  23. C 1 z n d z = 0 , n , for n 1. \oint_{C}{1\over z^{n}}\,dz=0,\quad n\in\mathbb{Z},\mbox{ for }~{}n\neq 1.
  24. C 1 4 ! z d z = 1 4 ! C 1 z d z = 1 4 ! ( 2 π i ) = π i 12 . \oint_{C}{1\over 4!\;z}\,dz={1\over 4!}\oint_{C}{1\over z}\,dz={1\over 4!}(2% \pi i)={\pi i\over 12}.
  25. Res 0 e z z 5 , or Res z = 0 e z z 5 , or Res ( f , 0 ) . \mathrm{Res}_{0}{e^{z}\over z^{5}},\ \mathrm{or}\ \mathrm{Res}_{z=0}{e^{z}% \over z^{5}},\ \mathrm{or}\ \mathrm{Res}(f,0).
  26. Res ( f , c ) = 1 2 π i γ f ( z ) d z \operatorname{Res}(f,c)={1\over 2\pi i}\oint_{\gamma}f(z)\,dz
  27. Res ( f , c ) = g ( c ) h ( c ) . \operatorname{Res}(f,c)=\frac{g(c)}{h^{\prime}(c)}.
  28. Res ( f , c ) = 1 ( n - 1 ) ! lim z c d n - 1 d z n - 1 ( ( z - c ) n f ( z ) ) . \mathrm{Res}(f,c)=\frac{1}{(n-1)!}\lim_{z\to c}\frac{d^{n-1}}{dz^{n-1}}\left((% z-c)^{n}f(z)\right).
  29. Res ( f ( z ) , ) = - Res ( 1 z 2 f ( 1 z ) , 0 ) \mathrm{Res}(f(z),\infty)=-\mathrm{Res}\left(\frac{1}{z^{2}}f\left(\frac{1}{z}% \right),0\right)
  30. lim | z | f ( z ) = 0 \lim_{|z|\to\infty}f(z)=0
  31. Res ( f , ) = - lim | z | z f ( z ) \mathrm{Res}(f,\infty)=-\lim_{|z|\to\infty}z\cdot f(z)
  32. lim | z | f ( z ) = c 0 \lim_{|z|\to\infty}f(z)=c\neq 0
  33. Res ( f , ) = - lim | z | z 2 f ( z ) \mathrm{Res}(f,\infty)=-\lim_{|z|\to\infty}z^{2}\cdot f^{\prime}(z)
  34. f ( z ) = sin z z 2 - z f(z)={\sin{z}\over z^{2}-z}
  35. f ( z ) = sin z z ( z - 1 ) f(z)={\sin{z}\over z(z-1)}
  36. g ( z ) = g ( a ) + g ( a ) ( z - a ) + g ′′ ( a ) ( z - a ) 2 2 ! + g ′′′ ( a ) ( z - a ) 3 3 ! + g(z)=g(a)+g^{\prime}(a)(z-a)+{g^{\prime\prime}(a)(z-a)^{2}\over 2!}+{g^{\prime% \prime\prime}(a)(z-a)^{3}\over 3!}+\cdots
  37. sin z = sin 1 + cos 1 ( z - 1 ) + - sin 1 ( z - 1 ) 2 2 ! + - cos 1 ( z - 1 ) 3 3 ! + . \sin{z}=\sin{1}+\cos{1}(z-1)+{-\sin{1}(z-1)^{2}\over 2!}+{-\cos{1}(z-1)^{3}% \over 3!}+\cdots.
  38. 1 z = 1 ( z - 1 ) + 1 = 1 - ( z - 1 ) + ( z - 1 ) 2 - ( z - 1 ) 3 + . \frac{1}{z}=\frac{1}{(z-1)+1}=1-(z-1)+(z-1)^{2}-(z-1)^{3}+\cdots.
  39. sin z z ( z - 1 ) = sin 1 z - 1 + ( cos 1 - sin 1 ) + ( z - 1 ) ( - sin 1 2 ! - cos 1 + sin 1 ) + . \frac{\sin{z}}{z(z-1)}={\sin{1}\over z-1}+(\cos{1}-\sin 1)+(z-1)\left(-\frac{% \sin{1}}{2!}-\cos 1+\sin 1\right)+\cdots.
  40. u ( z ) := k 1 u k z k u(z):=\sum_{k\geq 1}u_{k}z^{k}
  41. v ( z ) := k 1 v k z k v(z):=\sum_{k\geq 1}v_{k}z^{k}
  42. v 1 0 \textstyle v_{1}\neq 0
  43. v ( z ) \textstyle v(z)
  44. V ( z ) \textstyle V(z)
  45. u ( 1 / V ( z ) ) \textstyle u(1/V(z))
  46. Res 0 ( u ( 1 / V ( z ) ) ) = k = 0 k u k v k \mathrm{Res_{0}}\big(u(1/V(z))\big)=\sum_{k=0}^{\infty}ku_{k}v_{k}
  47. Res 0 ( u ( 1 / V ( z ) ) ) = Res 0 ( k 1 u k V ( z ) - k ) = k 1 u k Res 0 ( V ( z ) - k ) \mathrm{Res_{0}}\big(u(1/V(z))\big)=\mathrm{Res_{0}}\Big(\sum_{k\geq 1}u_{k}V(% z)^{-k}\Big)=\sum_{k\geq 1}u_{k}\mathrm{Res_{0}}\big(V(z)^{-k}\big)
  48. Res 0 ( V ( z ) - k ) = k v k \mathrm{Res_{0}}\big(V(z)^{-k}\big)=kv_{k}
  49. u ( z ) = z + z 2 u(z)=z+z^{2}
  50. v ( z ) = z + z 2 v(z)=z+z^{2}
  51. V ( z ) = 2 z 1 + 1 + 4 z V(z)=\frac{2z}{1+\sqrt{1+4z}}
  52. u ( 1 / V ( z ) ) = 1 + 1 + 4 z 2 z + 1 + 2 z + 1 + 4 z 2 z 2 u(1/V(z))=\frac{1+\sqrt{1+4z}}{2z}+\frac{1+2z+\sqrt{1+4z}}{2z^{2}}
  53. 1 / z 2 + 2 / z 1/z^{2}+2/z
  54. u ( z ) \textstyle u(z)
  55. v ( z ) \textstyle v(z)
  56. Res 0 ( u ( 1 / V ) ) = Res 0 ( v ( 1 / U ) ) \mathrm{Res_{0}}\big(u(1/V)\big)=\mathrm{Res_{0}}\big(v(1/U)\big)
  57. U ( z ) \textstyle U(z)
  58. u ( z ) \textstyle u(z)

Residue_theorem.html

  1. γ f ( z ) d z = 2 π i k = 1 n I ( γ , a k ) Res ( f , a k ) . \oint_{\gamma}f(z)\,dz=2\pi i\sum_{k=1}^{n}\operatorname{I}(\gamma,a_{k})% \operatorname{Res}(f,a_{k}).
  2. γ f ( z ) d z = 2 π i Res ( f , a k ) \oint_{\gamma}f(z)\,dz=2\pi i\sum\operatorname{Res}(f,a_{k})
  3. V \ W d ( f d z ) - W \ V d ( f d z ) \scriptstyle\int_{V\backslash W}d(f\,dz)-\int_{W\backslash V}d(f\,dz)
  4. - e i t x x 2 + 1 d x \int_{-\infty}^{\infty}{e^{itx}\over x^{2}+1}\,dx
  5. C f ( z ) d z = C e i t z z 2 + 1 d z . \int_{C}{f(z)}\,dz=\int_{C}{e^{itz}\over z^{2}+1}\,dz.
  6. e i t z z 2 + 1 \displaystyle\frac{e^{itz}}{z^{2}+1}
  7. Res z = i f ( z ) = e - t 2 i . \operatorname{Res}\limits_{z=i}f(z)={e^{-t}\over 2i}.
  8. C f ( z ) d z = 2 π i Res z = i f ( z ) = 2 π i e - t 2 i = π e - t . \int_{C}f(z)\,dz=2\pi i\cdot\operatorname{Res}\limits_{z=i}f(z)=2\pi i{e^{-t}% \over 2i}=\pi e^{-t}.
  9. straight f ( z ) d z + arc f ( z ) d z = π e - t \int_{\mathrm{straight}}f(z)\,dz+\int_{\mathrm{arc}}f(z)\,dz=\pi e^{-t}\,
  10. - a a f ( z ) d z = π e - t - arc f ( z ) d z . \int_{-a}^{a}f(z)\,dz=\pi e^{-t}-\int_{\mathrm{arc}}f(z)\,dz.
  11. | arc e i t z z 2 + 1 d z | arc | e i t z z 2 + 1 | d z arc 1 | z 2 + 1 | d z arc 1 a 2 - 1 d z = π a a 2 - 1 . \left|\int_{\mathrm{arc}}{e^{itz}\over z^{2}+1}\,dz\right|\leq\int_{\mathrm{% arc}}\left|{e^{itz}\over z^{2}+1}\right|dz\leq\int_{\mathrm{arc}}{1\over|z^{2}% +1|}dz\leq\int_{\mathrm{arc}}{1\over a^{2}-1}dz=\frac{\pi a}{a^{2}-1}.
  12. lim a π a a 2 - 1 = 0. \lim_{a\to\infty}\frac{\pi a}{a^{2}-1}=0.
  13. | e i t z | = | e i t | z | ( cos ϕ + i sin ϕ ) | = | e - t | z | sin ϕ + i t | z | cos ϕ | = e - t | z | sin ϕ 1. \left|e^{itz}\right|=\left|e^{it|z|(\cos\phi+i\sin\phi)}\right|=\left|e^{-t|z|% \sin\phi+it|z|\cos\phi}\right|=e^{-t|z|\sin\phi}\leq 1.
  14. - e i t z z 2 + 1 d z = π e - t . \int_{-\infty}^{\infty}{e^{itz}\over z^{2}+1}\,dz=\pi e^{-t}.
  15. - e i t z z 2 + 1 d z = π e - | t | . \int_{-\infty}^{\infty}{e^{itz}\over z^{2}+1}\,dz=\pi e^{-\left|t\right|}.
  16. π cot ( π z ) \pi\operatorname{cot}(\pi z)
  17. - f ( n ) \displaystyle\sum_{-\infty}^{\infty}f(n)
  18. f ( z ) = z - 2 f(z)=z^{-2}
  19. Γ N \Gamma_{N}
  20. [ - N - 1 / 2 , N + 1 / 2 ] 2 [-N-1/2,N+1/2]^{2}
  21. 1 2 π i Γ N f ( z ) π cot ( π z ) d z = Res z = 0 + n = - N , n 0 N n - 2 {1\over 2\pi i}\int_{\Gamma_{N}}f(z)\pi\operatorname{cot}(\pi z)\,dz=% \operatorname{Res}_{z=0}+\sum_{n=-N,n\neq 0}^{N}n^{-2}
  22. N N\to\infty
  23. O ( N - 2 ) O(N^{-2})
  24. z / 2 cot ( z / 2 ) = 1 - B 2 z 2 2 ! + , B 2 = 1 6 {z/2}\operatorname{cot}(z/2)=1-B_{2}{z^{2}\over 2!}+\cdots,\,B_{2}={1\over 6}
  25. z / 2 cot ( z / 2 ) = i z / ( 1 - e - i z ) - i z / 2 {z/2}\operatorname{cot}(z/2)={iz/(1-e^{-iz}})-iz/2
  26. Res z = 0 \operatorname{Res}_{z=0}
  27. z = 0 z=0
  28. - π 2 / 3 -{\pi^{2}/3}
  29. n = 1 1 n 2 = π 2 6 \sum_{n=1}^{\infty}{1\over n^{2}}={\pi^{2}\over 6}
  30. π cot ( π z ) = lim N n = - N N ( z - n ) - 1 \pi\operatorname{cot}(\pi z)=\lim_{N\to\infty}\sum_{n=-N}^{N}(z-n)^{-1}
  31. f ( z ) = ( w - z ) - 1 f(z)=(w-z)^{-1}
  32. Γ N π cot ( π z ) z d z = 0 \int_{\Gamma_{N}}{\pi\operatorname{cot}(\pi z)\over z}\,dz=0
  33. Γ N f ( z ) π cot ( π z ) d z = Γ N ( 1 w - z + 1 z ) π cot ( π z ) d z \int_{\Gamma_{N}}f(z)\pi\operatorname{cot}(\pi z)\,dz=\int_{\Gamma_{N}}\left({% 1\over w-z}+{1\over z}\right)\pi\operatorname{cot}(\pi z)\,dz
  34. N N\to\infty

Restricted_product.html

  1. I I
  2. S S
  3. I I
  4. i I i\in I
  5. G i G_{i}
  6. i I \ S i\in I\backslash S
  7. K i G i K_{i}\subset G_{i}
  8. i G i {\prod_{i}}^{\prime}G_{i}\,
  9. G i G_{i}
  10. ( g i ) i I (g_{i})_{i\in I}
  11. g i K i g_{i}\in K_{i}
  12. i I \ S i\in I\backslash S
  13. i A i , \prod_{i}A_{i}\,,
  14. A i A_{i}
  15. G i G_{i}
  16. A i = K i A_{i}=K_{i}
  17. i i

Reverberation.html

  1. R T 60 = 24 ln 10 1 c 20 V S a 0.1611 sm - 1 V S a RT_{60}=\frac{24\ln 10^{1}}{c_{20}}\frac{V}{Sa}\approx 0.1611\,\mathrm{s}% \mathrm{m}^{-1}\frac{V}{Sa}
  2. c 20 c_{20}
  3. V V
  4. S S
  5. a a
  6. S a Sa
  7. d c 0.057 V R T 60 d_{\mathrm{c}}\approx 0{.}057\cdot\sqrt{\frac{V}{RT_{60}}}
  8. d c d_{c}
  9. V V
  10. R T 60 RT_{60}

Rhombus.html

  1. 4 a 2 = p 2 + q 2 . \displaystyle 4a^{2}=p^{2}+q^{2}.
  2. A = a h . A=a\cdot h.
  3. A = a 2 sin α = a 2 sin β , A=a^{2}\cdot\sin\alpha=a^{2}\cdot\sin\beta,
  4. A = p q 2 , A=\frac{p\cdot q}{2},
  5. A = 2 a r . A=2a\cdot r.
  6. r r
  7. p p
  8. q q
  9. r = p q 2 p 2 + q 2 . r=\frac{p\cdot q}{2\sqrt{p^{2}+q^{2}}}.

Rhumb_line.html

  1. λ \lambda
  2. - π / 2 ϕ π / 2 -\pi/2\leq\phi\leq\pi/2
  3. ı ^ \hat{\imath}
  4. ȷ ^ \hat{\jmath}
  5. k ^ \hat{k}
  6. r ( λ , ϕ ) = cos λ cos ϕ ı ^ + sin λ cos ϕ ȷ ^ + sin ϕ k ^ . \vec{r}(\lambda,\phi)=\cos{\lambda}\cos{\phi}\,\hat{\imath}+\sin{\lambda}\cos{% \phi}\,\hat{\jmath}+\sin{\phi}\,\hat{k}\ .
  7. λ ^ ( λ , ϕ ) = sec ϕ r λ = - sin λ ı ^ + cos λ ȷ ^ \hat{\lambda}(\lambda,\phi)=\sec{\phi}\frac{\partial\vec{r}}{\partial\lambda}=% -\sin{\lambda}\,\hat{\imath}+\cos{\lambda}\,\hat{\jmath}
  8. ϕ ^ ( λ , ϕ ) = r ϕ = - cos λ sin ϕ ı ^ - sin λ sin ϕ ȷ ^ + cos ϕ k ^ \hat{\phi}(\lambda,\phi)=\frac{\partial\vec{r}}{\partial\phi}=-\cos{\lambda}% \sin{\phi}\,\hat{\imath}-\sin{\lambda}\sin{\phi}\,\hat{\jmath}+\cos{\phi}\,% \hat{k}
  9. λ ^ ϕ ^ = λ ^ r = ϕ ^ r = 0 . \hat{\lambda}\cdot\hat{\phi}=\hat{\lambda}\cdot\vec{r}=\hat{\phi}\cdot\vec{r}=% 0\ .
  10. λ ^ \hat{\lambda}
  11. ϕ \phi
  12. ϕ ^ \hat{\phi}
  13. λ \lambda
  14. β ^ ( λ , ϕ ) = sin β λ ^ + cos β ϕ ^ \hat{\beta}(\lambda,\phi)=\sin{\beta}\,\hat{\lambda}+\cos{\beta}\,\hat{\phi}
  15. β \beta
  16. ϕ ^ \hat{\phi}
  17. λ \lambda
  18. ϕ \phi
  19. β ^ ϕ ^ = cos β . \hat{\beta}\cdot\hat{\phi}=\cos{\beta}\ .
  20. β \beta
  21. β ^ \hat{\beta}
  22. d s ds
  23. d r = β ^ d s r λ d λ + r ϕ d ϕ = ( sin β λ ^ + cos β ϕ ^ ) d s cos ϕ d λ λ ^ + d ϕ ϕ ^ = sin β d s λ ^ + cos β d s ϕ ^ d s = cos ϕ sin β d λ = d ϕ cos β d λ < m t p l > d ϕ = tan β sec ϕ λ ( ϕ ) = tan β tanh - 1 sin ϕ + λ 0 ϕ ( λ ) = sin - 1 tanh ( ( λ - λ 0 ) cot β ) \begin{aligned}\displaystyle d\vec{r}&\displaystyle=\hat{\beta}\,ds\\ \displaystyle\frac{\partial\vec{r}}{\partial\lambda}\,d\lambda+\frac{\partial% \vec{r}}{\partial\phi}\,d\phi&\displaystyle=(\sin{\beta}\,\hat{\lambda}+\cos{% \beta}\,\hat{\phi})ds\\ \displaystyle\cos{\phi}\,d\lambda\,\hat{\lambda}+d\phi\,\hat{\phi}&% \displaystyle=\sin{\beta}\,ds\,\hat{\lambda}+\cos{\beta}\,ds\,\hat{\phi}\\ \displaystyle ds&\displaystyle=\frac{\cos{\phi}}{\sin{\beta}}\,d\lambda=\frac{% d\phi}{\cos{\beta}}\\ \displaystyle\frac{d\lambda}{<}mtpl>{{d\phi}}&\displaystyle=\tan{\beta}\,\sec{% \phi}\\ \displaystyle\lambda(\phi)&\displaystyle=\tan\beta\,\tanh^{-1}\sin\phi+\lambda% _{0}\\ \displaystyle\phi(\lambda)&\displaystyle=\sin^{-1}\tanh((\lambda-\lambda_{0})% \cot\beta)\end{aligned}
  24. λ \lambda
  25. ϕ \phi
  26. r ( λ , β ) = cos λ sech ( ( λ - λ 0 ) cot β ) ı ^ + sin λ sech ( ( λ - λ 0 ) cot β ) ȷ ^ + tanh ( ( λ - λ 0 ) cot β ) k ^ . \vec{r}(\lambda,\beta)=\cos{\lambda}\,\textrm{sech}((\lambda-\lambda_{0})\cot% \beta)\,\hat{\imath}\;+\;\sin{\lambda}\,\textrm{sech}((\lambda-\lambda_{0})% \cot\beta)\,\hat{\jmath}\;+\;\tanh((\lambda-\lambda_{0})\cot\beta)\,\hat{k}\ .
  27. ϕ π 2 , sin ϕ 1 , tanh - 1 sin ϕ , \phi\,\to\,\frac{\pi}{2}\,,\sin\phi\,\to\,1\,,\tanh^{-1}\sin\phi\,\to\,\infty\,,
  28. λ \lambda
  29. ψ = tanh - 1 sin ϕ \psi=\tanh^{-1}\sin\phi
  30. ϕ = gd ( ( λ - λ 0 ) cot β ) . \phi=\rm{gd}((\lambda-\lambda_{0})\cot\beta)\,.
  31. λ \lambda\,\!
  32. ϕ \phi
  33. x = λ - λ 0 x=\lambda-\lambda_{0}
  34. y = tanh - 1 sin ϕ y=\tanh^{-1}\sin\phi\,
  35. β \beta
  36. y = m x y=mx
  37. m = cot β m=\cot\beta\,
  38. m = cot β m=\cot\beta
  39. λ 0 \lambda_{0}

Riccati_equation.html

  1. y ( x ) = q 0 ( x ) + q 1 ( x ) y ( x ) + q 2 ( x ) y 2 ( x ) y^{\prime}(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)
  2. q 0 ( x ) 0 q_{0}(x)\neq 0
  3. q 2 ( x ) 0 q_{2}(x)\neq 0
  4. q 0 ( x ) = 0 q_{0}(x)=0
  5. q 2 ( x ) = 0 q_{2}(x)=0
  6. y = q 0 ( x ) + q 1 ( x ) y + q 2 ( x ) y 2 y^{\prime}=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}\!
  7. q 2 q_{2}
  8. v = y q 2 v=yq_{2}
  9. v = v 2 + R ( x ) v + S ( x ) , v^{\prime}=v^{2}+R(x)v+S(x),\!
  10. S = q 2 q 0 S=q_{2}q_{0}
  11. R = q 1 + ( q 2 q 2 ) R=q_{1}+\left(\frac{q_{2}^{\prime}}{q_{2}}\right)
  12. v = ( y q 2 ) = y q 2 + y q 2 = ( q 0 + q 1 y + q 2 y 2 ) q 2 + v q 2 q 2 = q 0 q 2 + ( q 1 + q 2 q 2 ) v + v 2 . v^{\prime}=(yq_{2})^{\prime}=y^{\prime}q_{2}+yq_{2}^{\prime}=(q_{0}+q_{1}y+q_{% 2}y^{2})q_{2}+v\frac{q_{2}^{\prime}}{q_{2}}=q_{0}q_{2}+\left(q_{1}+\frac{q_{2}% ^{\prime}}{q_{2}}\right)v+v^{2}.\!
  13. v = - u / u v=-u^{\prime}/u
  14. u u
  15. u ′′ - R ( x ) u + S ( x ) u = 0 u^{\prime\prime}-R(x)u^{\prime}+S(x)u=0\!
  16. v = - ( u / u ) = - ( u ′′ / u ) + ( u / u ) 2 = - ( u ′′ / u ) + v 2 v^{\prime}=-(u^{\prime}/u)^{\prime}=-(u^{\prime\prime}/u)+(u^{\prime}/u)^{2}=-% (u^{\prime\prime}/u)+v^{2}\!
  17. u ′′ / u = v 2 - v = - S - R v = - S + R u / u u^{\prime\prime}/u=v^{2}-v^{\prime}=-S-Rv=-S+Ru^{\prime}/u\!
  18. u ′′ - R u + S u = 0. u^{\prime\prime}-Ru^{\prime}+Su=0.\!
  19. y = - u / ( q 2 u ) y=-u^{\prime}/(q_{2}u)
  20. S ( w ) := ( w ′′ / w ) - ( w ′′ / w ) 2 / 2 = f S(w):=(w^{\prime\prime}/w^{\prime})^{\prime}-(w^{\prime\prime}/w^{\prime})^{2}% /2=f
  21. S ( w ) S(w)
  22. S ( ( a w + b ) / ( c w + d ) ) = S ( w ) S((aw+b)/(cw+d))=S(w)
  23. a d - b c ad-bc
  24. y = w ′′ / w y=w^{\prime\prime}/w^{\prime}
  25. y = y 2 / 2 + f . y^{\prime}=y^{2}/2+f.
  26. y = - 2 u / u y=-2u^{\prime}/u
  27. u u
  28. u ′′ + ( 1 / 2 ) f u = 0. u^{\prime\prime}+(1/2)fu=0.
  29. w ′′ / w = - 2 u / u w^{\prime\prime}/w^{\prime}=-2u^{\prime}/u
  30. w = C / u 2 w^{\prime}=C/u^{2}
  31. C C
  32. U U
  33. U u - U u U^{\prime}u-Uu^{\prime}
  34. C C
  35. w = ( U u - U u ) / u 2 = ( U / u ) w^{\prime}=(U^{\prime}u-Uu^{\prime})/u^{2}=(U/u)^{\prime}
  36. w = U / u . w=U/u.
  37. y 1 y_{1}
  38. y = y 1 + u y=y_{1}+u
  39. y 1 + u y_{1}+u
  40. y 1 + u = q 0 + q 1 ( y 1 + u ) + q 2 ( y 1 + u ) 2 , y_{1}^{\prime}+u^{\prime}=q_{0}+q_{1}\cdot(y_{1}+u)+q_{2}\cdot(y_{1}+u)^{2},
  41. y 1 = q 0 + q 1 y 1 + q 2 y 1 2 y_{1}^{\prime}=q_{0}+q_{1}\,y_{1}+q_{2}\,y_{1}^{2}
  42. u = q 1 u + 2 q 2 y 1 u + q 2 u 2 u^{\prime}=q_{1}\,u+2\,q_{2}\,y_{1}\,u+q_{2}\,u^{2}
  43. u - ( q 1 + 2 q 2 y 1 ) u = q 2 u 2 , u^{\prime}-(q_{1}+2\,q_{2}\,y_{1})\,u=q_{2}\,u^{2},
  44. z = 1 u z=\frac{1}{u}
  45. y = y 1 + 1 z y=y_{1}+\frac{1}{z}
  46. z + ( q 1 + 2 q 2 y 1 ) z = - q 2 z^{\prime}+(q_{1}+2\,q_{2}\,y_{1})\,z=-q_{2}
  47. y = y 1 + 1 z y=y_{1}+\frac{1}{z}

Ricci_curvature.html

  1. ( M , g ) (M,g)
  2. \nabla
  3. M M
  4. ( 1 , 3 ) (1,3)
  5. R ( X , Y ) Z = X Y Z - Y X Z - [ X , Y ] Z R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z
  6. X , Y , Z X,Y,Z
  7. T p M T_{p}M
  8. ξ \xi
  9. η \eta
  10. T p M T_{p}M
  11. Ric \mathrm{Ric}
  12. ( ξ , η ) (\xi,\eta)
  13. T p M T p M T_{p}M\to T_{p}M
  14. ζ R ( ζ , η ) ξ . \zeta\mapsto R(\zeta,\eta)\xi.
  15. Ric = R i j d x i d x j \operatorname{Ric}=R_{ij}\,dx^{i}\otimes dx^{j}
  16. R i j = R k i k j . R_{ij}={R^{k}}_{ikj}.
  17. R α β = R ρ α ρ β = ρ Γ β α ρ - β Γ ρ α ρ + Γ ρ λ ρ Γ β α λ - Γ β λ ρ Γ ρ α λ = 2 Γ < ρ m t p l > α [ β , ρ ] + 2 Γ λ [ ρ ρ Γ β ] α λ . R_{\alpha\beta}={R^{\rho}}_{\alpha\rho\beta}=\partial_{\rho}{\Gamma^{\rho}_{% \beta\alpha}}-\partial_{\beta}\Gamma^{\rho}_{\rho\alpha}+\Gamma^{\rho}_{\rho% \lambda}\Gamma^{\lambda}_{\beta\alpha}-\Gamma^{\rho}_{\beta\lambda}\Gamma^{% \lambda}_{\rho\alpha}=2\Gamma^{\rho}_{<}mtpl>{{\alpha[\beta,\rho]}}+2\Gamma^{% \rho}_{\lambda[\rho}\Gamma^{\lambda}_{\beta]\alpha}.
  18. Ric ( ξ , η ) = Ric ( η , ξ ) . \operatorname{Ric}(\xi,\eta)=\operatorname{Ric}(\eta,\xi).
  19. Ric ( ξ , ξ ) \operatorname{Ric}(\xi,\xi)
  20. ξ \xi
  21. ξ \xi
  22. ξ \xi
  23. R i j = - 1 2 Δ ( g i j ) + lower order terms R_{ij}=-\frac{1}{2}\Delta(g_{ij})+\,\text{lower order terms}
  24. R i j = - 3 2 Δ ( g i j ) . R_{ij}=-\frac{3}{2}\Delta(g_{ij}).
  25. g i j = δ i j + O ( | x | 2 ) . g_{ij}=\delta_{ij}+O(|x|^{2}).\,
  26. g i j = δ i j - 1 3 R i k j x k x + O ( | x | 3 ) . g_{ij}=\delta_{ij}-\frac{1}{3}R_{ikj\ell}x^{k}x^{\ell}+O(|x|^{3}).
  27. d μ g = [ 1 - 1 6 R j k x j x k + O ( | x | 3 ) ] d μ < m t p l > Euclidean d\mu_{g}=\Big[1-\frac{1}{6}R_{jk}x^{j}x^{k}+O(|x|^{3})\Big]d\mu_{<}mtpl>{{\rm Euclidean}}
  28. ( n - 1 ) k > 0 \left(n-1\right)k>0\,\!
  29. π / k \leq\pi/\sqrt{k}
  30. v p ( R ) v_{p}(R)
  31. R R
  32. V ( R ) = c m R m V(R)=c_{m}R^{m}
  33. v p ( R ) / V ( R ) v_{p}(R)/V(R)
  34. Ric 0 \operatorname{Ric}\geq 0
  35. d ( γ ( u ) , γ ( v ) ) = | u - v | d(\gamma(u),\gamma(v))=|u-v|
  36. v , u v,u\in\mathbb{R}
  37. × L \mathbb{R}\times L
  38. e 2 f e^{2f}
  39. g ~ = e 2 f g \tilde{g}=e^{2f}g
  40. Ric ~ = Ric + ( 2 - n ) [ d f - d f d f ] + [ Δ f - ( n - 2 ) d f 2 ] g , \widetilde{\operatorname{Ric}}=\operatorname{Ric}+(2-n)[\nabla df-df\otimes df% ]+[\Delta f-(n-2)\|df\|^{2}]g,
  41. \mapsto
  42. ( M , g ) (M,g)
  43. Z = Ric - S n g Z=\operatorname{Ric}-\frac{S}{n}g
  44. Ric \operatorname{Ric}
  45. S S
  46. g g
  47. n n
  48. M M
  49. Z a b g a b = 0. Z_{ab}g^{ab}=\,0.
  50. n 3 n\geq 3
  51. Ric = λ g \operatorname{Ric}=\lambda g
  52. λ \lambda
  53. ( M , g ) (M,g)
  54. ( M , g ) (M,g)
  55. κ = n Ω X . \kappa=\wedge^{n}\Omega_{X}.
  56. ρ ( X , Y ) = def Ric ( J X , Y ) \rho(X,Y)\,\stackrel{\,\text{def}}{=}\,\operatorname{Ric}(JX,Y)
  57. Ric ( X , Y ) = ρ ( X , J Y ) . \operatorname{Ric}(X,Y)=\rho(X,JY).
  58. ρ = - i ¯ log det ( g α β ¯ ) \rho=-i\partial\overline{\partial}\log\det(g_{\alpha\overline{\beta}})
  59. \partial
  60. g α β ¯ = g ( z α , z ¯ β ) . g_{\alpha\overline{\beta}}=g\left(\frac{\partial}{\partial z^{\alpha}},\frac{% \partial}{\partial\overline{z}^{\beta}}\right).
  61. \nabla
  62. R R
  63. ( 1 , 3 ) (1,3)
  64. R ( X , Y ) Z = X Y Z - Y X Z - [ X , Y ] Z R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z
  65. X , Y , Z X,Y,Z
  66. ric ( X , Y ) = tr ( Z R ( Z , X ) Y ) . \operatorname{ric}(X,Y)=\operatorname{tr}(Z\mapsto R(Z,X)Y).

Ricci_flow.html

  1. g i j g_{ij}
  2. R i j R_{ij}
  3. t g i j = - 2 R i j . \partial_{t}g_{ij}=-2R_{ij}.
  4. t g i j = - 2 R i j + 2 n R avg g i j \partial_{t}g_{ij}=-2R_{ij}+\frac{2}{n}R_{\mathrm{avg}}g_{ij}
  5. R avg R_{\mathrm{avg}}
  6. n n
  7. t t
  8. ( 1 - 2 t ( n - 1 ) ) (1-2t(n-1))
  9. 1 / 2 ( n - 1 ) 1/2(n-1)
  10. d s 2 = exp ( 2 p ( x , y ) ) ( d x 2 + d y 2 ) . ds^{2}=\exp(2\,p(x,y))\,\left(dx^{2}+dy^{2}\right).
  11. σ 1 = exp ( p ) d x , σ 2 = exp ( p ) d y \sigma^{1}=\exp(p)\,dx,\;\;\sigma^{2}=\exp(p)\,dy
  12. σ 1 σ 1 + σ 2 σ 2 = exp ( 2 p ) ( d x d x + d y d y ) . \sigma^{1}\otimes\sigma^{1}+\sigma^{2}\otimes\sigma^{2}=\exp(2p)\,\left(dx% \otimes dx+dy\otimes dy\right).
  13. h ( x , y ) h(x,y)
  14. d h = h x d x + h y d y = exp ( - p ) h x σ 1 + exp ( - p ) h y σ 2 . dh=h_{x}dx+h_{y}dy=\exp(-p)h_{x}\,\sigma^{1}+\exp(-p)h_{y}\,\sigma^{2}.
  15. d h = - exp ( - p ) h y σ 1 + exp ( - p ) h x σ 2 = - h y d x + h x d y . \star dh=-\exp(-p)h_{y}\,\sigma^{1}+\exp(-p)h_{x}\,\sigma^{2}=-h_{y}\,dx+h_{x}% \,dy.
  16. d d h = - h y y d y d x + h x x d x d y = ( h x x + h y y ) d x d y d\star dh=-h_{yy}\,dy\wedge dx+h_{xx}\,dx\wedge dy=\left(h_{xx}+h_{yy}\right)% \,dx\wedge dy
  17. d d h = exp ( - 2 p ) ( h x x + h y y ) σ 1 σ 2 . d\star dh=\exp(-2p)\,\left(h_{xx}+h_{yy}\right)\,\sigma^{1}\wedge\sigma^{2}.
  18. Δ h = d d h = exp ( - 2 p ) ( h x x + h y y ) \Delta h=\star d\star dh=\exp(-2p)\,\left(h_{xx}+h_{yy}\right)
  19. Δ = exp ( - 2 p ( x , y ) ) ( D x 2 + D y 2 ) . \Delta=\exp(-2\,p(x,y))\left(D_{x}^{2}+D_{y}^{2}\right).
  20. d σ 1 = p y exp ( p ) d y d x = - ( p y d x ) σ 2 = - ω 1 2 σ 2 d\sigma^{1}=p_{y}\exp(p)dy\wedge dx=-\left(p_{y}dx\right)\wedge\sigma^{2}=-{% \omega^{1}}_{2}\wedge\sigma^{2}
  21. d σ 2 = p x exp ( p ) d x d y = - ( p x d y ) σ 1 = - ω 2 1 σ 1 . d\sigma^{2}=p_{x}\exp(p)dx\wedge dy=-\left(p_{x}dy\right)\wedge\sigma^{1}=-{% \omega^{2}}_{1}\wedge\sigma^{1}.
  22. ω 1 2 = p y d x - p x d y . {\omega^{1}}_{2}=p_{y}dx-p_{x}dy.
  23. d ω 1 2 = p y y d y d x - p x x d x d y = - ( p x x + p y y ) d x d y . d{\omega^{1}}_{2}=p_{yy}dy\wedge dx-p_{xx}dx\wedge dy=-\left(p_{xx}+p_{yy}% \right)\,dx\wedge dy.
  24. Ω 1 2 = - exp ( - 2 p ) ( p x x + p y y ) σ 1 σ 2 = - Δ p σ 1 σ 2 {\Omega^{1}}_{2}=-\exp(-2p)\left(p_{xx}+p_{yy}\right)\,\sigma^{1}\wedge\sigma^% {2}=-\Delta p\,\sigma^{1}\wedge\sigma^{2}
  25. Ω 1 2 = R 1 212 σ 1 σ 2 . {\Omega^{1}}_{2}={R^{1}}_{212}\,\sigma^{1}\wedge\sigma^{2}.
  26. R 1 212 = - Δ p {R^{1}}_{212}=-\Delta p
  27. R 22 = R 11 = - Δ p . R_{22}=R_{11}=-\Delta p.
  28. R x x = R y y = - ( p x x + p y y ) . R_{xx}=R_{yy}=-\left(p_{xx}+p_{yy}\right).
  29. g x x = g y y = exp ( 2 p ) g_{xx}=g_{yy}=\exp(2p)
  30. p t = Δ p . \frac{\partial p}{\partial t}=\Delta p.
  31. u t = Δ u \frac{\partial u}{\partial t}=\Delta u
  32. Δ = D x 2 + D y 2 \Delta=D_{x}^{2}+D_{y}^{2}
  33. p ( x , y ) = 0 p(x,y)=0
  34. p p
  35. t 0 t_{0}

Richard_Herrnstein.html

  1. R 1 R 1 + R 2 = r 1 r 1 + r 2 , \frac{R_{1}}{R_{1}+R_{2}}=\frac{r_{1}}{r_{1}+r_{2}},

Rician_fading.html

  1. K K
  2. Ω \Omega
  3. K K
  4. Ω \Omega
  5. Ω = ν 2 + 2 σ 2 \Omega=\nu^{2}+2\sigma^{2}
  6. R R
  7. ν 2 = K 1 + K Ω \nu^{2}=\frac{K}{1+K}\Omega
  8. σ 2 = Ω 2 ( 1 + K ) \sigma^{2}=\frac{\Omega}{2(1+K)}
  9. f ( x ) = 2 ( K + 1 ) x Ω exp ( - K - ( K + 1 ) x 2 Ω ) I 0 ( 2 K ( K + 1 ) Ω x ) , f(x)=\frac{2(K+1)x}{\Omega}\exp\left(-K-\frac{(K+1)x^{2}}{\Omega}\right)I_{0}% \left(2\sqrt{\frac{K(K+1)}{\Omega}}x\right),
  10. I 0 ( ) I_{0}(\cdot)

Riemann_curvature_tensor.html

  1. \nabla
  2. R ( u , v ) w = u v w - v u w - [ u , v ] w R(u,v)w=\nabla_{u}\nabla_{v}w-\nabla_{v}\nabla_{u}w-\nabla_{[u,v]}w
  3. u = / x i u=\partial/\partial x^{i}
  4. v = / x j v=\partial/\partial x^{j}
  5. [ u , v ] = 0 [u,v]=0
  6. R ( u , v ) w = u v w - v u w . R(u,v)w=\nabla_{u}\nabla_{v}w-\nabla_{v}\nabla_{u}w.
  7. w R ( u , v ) w w\mapsto R(u,v)w
  8. u , v 2 w = u v w - u v w \nabla^{2}_{u,v}w=\nabla_{u}\nabla_{v}w-\nabla_{\nabla_{u}v}w
  9. R ( u , v ) = u , v 2 - v , u 2 R(u,v)=\nabla^{2}_{u,v}-\nabla^{2}_{v,u}
  10. x ˙ 0 Y = lim h 0 1 h ( Y x 0 - τ x h - 1 ( Y x h ) ) = d d t ( τ x t Y ) | t = 0 \nabla_{\dot{x}_{0}}Y=\lim_{h\to 0}\frac{1}{h}\left(Y_{x_{0}}-\tau^{-1}_{x_{h}% }(Y_{x_{h}})\right)=\left.\frac{d}{dt}(\tau_{x_{t}}Y)\right|_{t=0}
  11. τ s X - 1 τ t Y - 1 τ s X τ t Y Z . \tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z.
  12. d d s d d t τ s X - 1 τ t Y - 1 τ s X τ t Y Z | s = t = 0 = ( X Y - Y X - [ X , Y ] ) Z = R ( X , Y ) Z \left.\frac{d}{ds}\frac{d}{dt}\tau_{sX}^{-1}\tau_{tY}^{-1}\tau_{sX}\tau_{tY}Z% \right|_{s=t=0}=(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]})Z=R(% X,Y)Z
  13. R ρ = σ μ ν d x ρ ( R ( μ , ν ) σ ) R^{\rho}{}_{\sigma\mu\nu}=dx^{\rho}(R(\partial_{\mu},\partial_{\nu})\partial_{% \sigma})
  14. μ = / x μ \partial_{\mu}=\partial/\partial x^{\mu}
  15. R ρ = σ μ ν μ Γ ρ - ν σ ν Γ ρ + μ σ Γ ρ Γ λ μ λ - ν σ Γ ρ Γ λ ν λ μ σ R^{\rho}{}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma}-\partial_{% \nu}\Gamma^{\rho}{}_{\mu\sigma}+\Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}% _{\nu\sigma}-\Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}
  16. A ν A_{\nu}\,
  17. A ν ; ρ σ - A ν ; σ ρ = A β R β , ν ρ σ A_{\nu;\rho\sigma}-A_{\nu;\sigma\rho}=A_{\beta}R^{\beta}{}_{\nu\rho\sigma}\,,
  18. Γ α β μ \Gamma^{\alpha}{}_{\beta\mu}\,
  19. Γ λ - μ ν Γ λ ν μ \Gamma^{\lambda}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\mu}\,
  20. R β ν ρ σ R^{\beta}{}_{\nu\rho\sigma}\,
  21. T α 1 α r - β 1 β s ; γ δ T α 1 α r = β 1 β s ; δ γ - R α 1 T ρ α 2 α r ρ γ δ - β 1 β s - R α r T α 1 α r - 1 ρ ρ γ δ β 1 β s + R σ T α 1 α r β 1 γ δ + σ β 2 β s + R σ T α 1 α r β s γ δ . β 1 β s - 1 σ \begin{aligned}\displaystyle T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots% \beta_{s};\gamma\delta}-T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta% _{s};\delta\gamma}=&\displaystyle-R^{\alpha_{1}}{}_{\rho\gamma\delta}T^{\rho% \alpha_{2}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s}}-\cdots-R^{\alpha_{r}}% {}_{\rho\gamma\delta}T^{\alpha_{1}\cdots\alpha_{r-1}\rho}{}_{\beta_{1}\cdots% \beta_{s}}\\ &\displaystyle+\,R^{\sigma}{}_{\beta_{1}\gamma\delta}T^{\alpha_{1}\cdots\alpha% _{r}}{}_{\sigma\beta_{2}\cdots\beta_{s}}+\cdots+R^{\sigma}{}_{\beta_{s}\gamma% \delta}T^{\alpha_{1}\cdots\alpha_{r}}{}_{\beta_{1}\cdots\beta_{s-1}\sigma}\,.% \end{aligned}
  22. μ ( g ) ( g ) ; μ = 0 , where g = | det ( g μ ν ) | . \nabla_{\mu}(\sqrt{g}\,)\equiv(\sqrt{g}\,)_{;\mu}=0\,,\quad{\mathrm{where}}% \quad{g}=|{\mathrm{det}}(g_{\mu\nu})|\,.
  23. R ρ σ μ ν = g ρ ζ R ζ . σ μ ν R_{\rho\sigma\mu\nu}=g_{\rho\zeta}R^{\zeta}{}_{\sigma\mu\nu}\,.
  24. R ( u , v ) = - R ( v , u ) R(u,v)=-R(v,u)
  25. R ( u , v ) w , z = - R ( u , v ) z , w \langle R(u,v)w,z\rangle=-\langle R(u,v)z,w\rangle
  26. R ( u , v ) w + R ( v , w ) u + R ( w , u ) v = 0. R(u,v)w+R(v,w)u+R(w,u)v=0.
  27. , \langle,\rangle
  28. n 2 ( n 2 - 1 ) / 12 n^{2}(n^{2}-1)/12
  29. R ( u , v ) w , z = R ( w , z ) u , v . \langle R(u,v)w,z\rangle=\langle R(w,z)u,v\rangle.
  30. u R \nabla_{u}R
  31. ( u R ) ( v , w ) + ( v R ) ( w , u ) + ( w R ) ( u , v ) = 0. (\nabla_{u}R)(v,w)+(\nabla_{v}R)(w,u)+(\nabla_{w}R)(u,v)=0.
  32. R a b c d = - R b a c d = - R a b d c R_{abcd}=-R_{bacd}=-R_{abdc}
  33. R a b c d = R c d a b R_{abcd}=R_{cdab}
  34. R a b c d + R a c d b + R a d b c = 0 R_{abcd}+R_{acdb}+R_{adbc}=0
  35. R a [ b c d ] = 0 , R_{a[bcd]}=0,
  36. R a b c d ; e + R a b d e ; c + R a b e c ; d = 0 R_{abcd;e}+R_{abde;c}+R_{abec;d}=0
  37. R a b [ c d ; e ] = 0 R_{ab[cd;e]}=0
  38. R a b c d = K ( g a c g d b - g a d g c b ) R_{abcd}=K(g_{ac}g_{db}-g_{ad}g_{cb})\,
  39. g a b g_{ab}
  40. K K
  41. Ric a b = K g a b . \operatorname{Ric}_{ab}=Kg_{ab}.\,
  42. R a b c d = K ( g a c g d b - g a d g c b ) . R_{abcd}=K(g_{ac}g_{db}-g_{ad}g_{cb}).

Riemann_sum.html

  1. P = { [ x 0 , x 1 ] , [ x 1 , x 2 ] , , [ x n - 1 , x n ] } , P=\left\{[x_{0},x_{1}],[x_{1},x_{2}],\dots,[x_{n-1},x_{n}]\right\},
  2. a = x 0 < x 1 < x 2 < < x n = b . a=x_{0}<x_{1}<x_{2}<\cdots<x_{n}=b.
  3. S = i = 1 n f ( x i * ) ( x i - x i - 1 ) , x i - 1 x i * x i . S=\sum_{i=1}^{n}f(x_{i}^{*})(x_{i}-x_{i-1}),\quad x_{i-1}\leq x_{i}^{*}\leq x_% {i}.
  4. x i * x_{i}^{*}
  5. [ x i - 1 , x i ] [x_{i-1},x_{i}]
  6. x i * x_{i}^{*}
  7. x i - 1 x i * x i x_{i-1}\leq x_{i}^{*}\leq x_{i}
  8. x i * x_{i}^{*}
  9. x i * = x i - 1 x_{i}^{*}=x_{i-1}
  10. x i * = x i x_{i}^{*}=x_{i}
  11. x i * = 1 2 ( x i + x i - 1 ) x_{i}^{*}=\tfrac{1}{2}(x_{i}+x_{i-1})
  12. S = i = 1 n v i ( x i - x i - 1 ) , S=\sum_{i=1}^{n}v_{i}(x_{i}-x_{i-1}),
  13. v i v_{i}
  14. [ x i - 1 , x i ] [x_{i-1},x_{i}]
  15. v i v_{i}
  16. [ x i - 1 , x i ] [x_{i-1},x_{i}]
  17. x i * x_{i}^{*}
  18. x i - 1 x_{i-1}
  19. x i x_{i}
  20. Δ x = b - a n . \Delta x=\frac{b-a}{n}.
  21. a , a + Δ x , a + 2 Δ x , , a + ( n - 2 ) Δ x , a + ( n - 1 ) Δ x , b . a,a+\Delta x,a+2\Delta x,\ldots,a+(n-2)\Delta x,a+(n-1)\Delta x,b.
  22. Δ x [ f ( a ) + f ( a + Δ x ) + f ( a + 2 Δ x ) + + f ( b - Δ x ) ] . \Delta x\left[f(a)+f(a+\Delta x)+f(a+2\Delta x)+\cdots+f(b-\Delta x)\right].
  23. Δ x [ f ( a + Δ x ) + f ( a + 2 Δ x ) + + f ( b ) ] . \Delta x\left[f(a+\Delta x)+f(a+2\Delta x)+\cdots+f(b)\right].
  24. | a b f ( x ) d x - A right | M 1 ( b - a ) 2 2 n , \left|\int_{a}^{b}f(x)\,dx-A_{\mathrm{right}}\right|\leq\frac{M_{1}(b-a)^{2}}{% 2n},
  25. M 1 M_{1}
  26. f ( x ) f^{\prime}(x)
  27. Δ x [ f ( a + Δ x 2 ) + f ( a + 3 Δ x 2 ) + + f ( b - Δ x 2 ) ] . \Delta x\left[f(a+\tfrac{\Delta x}{2})+f(a+\tfrac{3\Delta x}{2})+\cdots+f(b-% \tfrac{\Delta x}{2})\right].
  28. | a b f ( x ) d x - A mid | M 2 ( b - a ) 3 24 n 2 , \left|\int_{a}^{b}f(x)\,dx-A_{\mathrm{mid}}\right|\leq\frac{M_{2}(b-a)^{3}}{24% n^{2}},
  29. M 2 M_{2}
  30. f ′′ ( x ) f^{\prime\prime}(x)
  31. A = 1 2 h ( b 1 + b 2 ) A=\tfrac{1}{2}h(b_{1}+b_{2})
  32. 1 2 Δ x [ f ( a ) + 2 f ( a + Δ x ) + 2 f ( a + 2 Δ x ) + 2 f ( a + 3 Δ x ) + + f ( b ) ] . \tfrac{1}{2}\Delta x\left[f(a)+2f(a+\Delta x)+2f(a+2\Delta x)+2f(a+3\Delta x)+% \cdots+f(b)\right].
  33. | a b f ( x ) d x - A trap | M 2 ( b - a ) 3 12 n 2 , \left|\int_{a}^{b}f(x)\,dx-A_{\mathrm{trap}}\right|\leq\frac{M_{2}(b-a)^{3}}{1% 2n^{2}},
  34. M 2 M_{2}
  35. f ′′ ( x ) . f^{\prime\prime}(x).
  36. 2 n \tfrac{2}{n}
  37. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  38. x 1 2 , x 2 2 , , x n 2 x_{1}^{2},x_{2}^{2},\ldots,x_{n}^{2}
  39. x i = 2 i n x_{i}=\tfrac{2i}{n}
  40. x n = 2 x_{n}=2
  41. 2 n × x i 2 \tfrac{2}{n}\times x_{i}^{2}
  42. S \displaystyle S
  43. lim n S = lim n ( 8 3 + 4 n + 4 3 n 2 ) = 8 3 \lim_{n\to\infty}S=\lim_{n\to\infty}\left(\frac{8}{3}+\frac{4}{n}+\frac{4}{3n^% {2}}\right)=\frac{8}{3}
  44. 0 2 x 2 d x = 8 3 \int_{0}^{2}x^{2}\,dx=\frac{8}{3}
  45. y = x 2 y=x^{2}

Riemann_surface.html

  1. τ ( z ) \wp_{\tau}(z)
  2. τ ( z ) \wp_{\tau}^{\prime}(z)
  3. [ ( z ) ] 2 = 4 [ ( z ) ] 3 - g 2 ( z ) - g 3 , [\wp^{\prime}(z)]^{2}=4[\wp(z)]^{3}-g_{2}\wp(z)-g_{3},
  4. 𝐂 ^ := 𝐂 { } \widehat{\mathbf{C}}:=\mathbf{C}\cup\{\infty\}
  5. Δ 𝐂 𝐂 ^ , \Delta\subset\mathbf{C}\subset\widehat{\mathbf{C}},
  6. 𝐂 ^ \widehat{\mathbf{C}}
  7. z z n , z\mapsto z^{n},
  8. 84 ( g - 1 ) , 84(g-1),

Riemannian_manifold.html

  1. g p g_{p}
  2. T p M T_{p}M
  3. p p
  4. p g p ( X ( p ) , Y ( p ) ) p\mapsto g_{p}(X(p),Y(p))
  5. g p g_{p}
  6. L ( α ) = 0 1 α ( t ) d t . L(\alpha)=\int_{0}^{1}{\|\alpha^{\prime}(t)\|\,\mathrm{d}t}.
  7. L ( γ ) = a b γ ( t ) d t . L(\gamma)=\int_{a}^{b}\|\gamma^{\prime}(t)\|\,\mathrm{d}t.
  8. g p : T p M × T p M 𝐑 , p M g_{p}\colon T_{p}M\times T_{p}M\longrightarrow\mathbf{R},\qquad p\in M
  9. p g p ( X ( p ) , Y ( p ) ) p\mapsto g_{p}(X(p),Y(p))
  10. { x 1 , , x n } \left\{\frac{\partial}{\partial x^{1}},\ldots,\frac{\partial}{\partial x^{n}}\right\}
  11. g i j ( p ) := g p ( ( x i ) p , ( x j ) p ) . g_{ij}(p):=g_{p}\Biggl(\left(\frac{\partial}{\partial x^{i}}\right)_{p},\left(% \frac{\partial}{\partial x^{j}}\right)_{p}\Biggr).
  12. g = i , j g i j d x i d x j . g=\sum_{i,j}g_{ij}\mathrm{d}x^{i}\otimes\mathrm{d}x^{j}.
  13. x i \frac{\partial}{\partial x^{i}}
  14. g p can : T p U × T p U 𝐑 , ( i a i x i , j b j x j ) i a i b i . g^{\mathrm{can}}_{p}\colon T_{p}U\times T_{p}U\longrightarrow\mathbf{R},\qquad% \left(\sum_{i}a_{i}\frac{\partial}{\partial x^{i}},\sum_{j}b_{j}\frac{\partial% }{\partial x^{j}}\right)\longmapsto\sum_{i}a_{i}b_{i}.
  15. g i j can = e i , e j = δ i j . g^{\mathrm{can}}_{ij}=\langle e_{i},e_{j}\rangle=\delta_{ij}.
  16. g p M : T p M × T p M 𝐑 , g^{M}_{p}\colon T_{p}M\times T_{p}M\longrightarrow\mathbf{R},
  17. ( u , v ) g p M ( u , v ) := g f ( p ) N ( T p f ( u ) , T p f ( v ) ) . (u,v)\longmapsto g^{M}_{p}(u,v):=g^{N}_{f(p)}(T_{p}f(u),T_{p}f(v)).
  18. h : 𝐑 n 𝐑 , ( x 1 , , x n ) i = 1 n ( x i ) 2 - 1. h\colon\mathbf{R}^{n}\longrightarrow\mathbf{R},\qquad(x^{1},\ldots,x^{n})% \longmapsto\sum_{i=1}^{n}(x^{i})^{2}-1.
  19. h - 1 ( 0 ) = { x 𝐑 n | i = 1 n ( x i ) 2 = 1 } = 𝐒 n - 1 h^{-1}(0)=\left\{x\in\mathbf{R}^{n}|\sum_{i=1}^{n}(x^{i})^{2}=1\right\}=% \mathbf{S}^{n-1}
  20. g ( p , q ) M 1 × M 2 : T ( p , q ) ( M 1 × M 2 ) × T ( p , q ) ( M 1 × M 2 ) 𝐑 , g^{M_{1}\times M_{2}}_{(p,q)}\colon T_{(p,q)}(M_{1}\times M_{2})\times T_{(p,q% )}(M_{1}\times M_{2})\longrightarrow\mathbf{R},
  21. ( u , v ) g p M 1 ( T ( p , q ) π 1 ( u ) , T ( p , q ) π 1 ( v ) ) + g q M 2 ( T ( p , q ) π 2 ( u ) , T ( p , q ) π 2 ( v ) ) . (u,v)\longmapsto g^{M_{1}}_{p}(T_{(p,q)}\pi_{1}(u),T_{(p,q)}\pi_{1}(v))+g^{M_{% 2}}_{q}(T_{(p,q)}\pi_{2}(u),T_{(p,q)}\pi_{2}(v)).
  22. T ( p , q ) ( M 1 × M 2 ) T p M 1 T q M 2 T_{(p,q)}(M_{1}\times M_{2})\cong T_{p}M_{1}\oplus T_{q}M_{2}
  23. g ~ := λ g 0 + ( 1 - λ ) g 1 , λ [ 0 , 1 ] , \tilde{g}:=\lambda g_{0}+(1-\lambda)g_{1},\qquad\lambda\in[0,1],
  24. ( f * g N ) ( v , w ) = g N ( d f ( v ) , d f ( w ) ) . (f^{*}g^{N})(v,w)=g^{N}(df(v),df(w))\,.
  25. ϕ : U α ϕ ( U α ) 𝐑 n . \phi\colon U_{\alpha}\to\phi(U_{\alpha})\subseteq\mathbf{R}^{n}.
  26. g := β τ β g ~ β , with g ~ β := ϕ ~ β * g can . g:=\sum_{\beta}\tau_{\beta}\cdot\tilde{g}_{\beta},\qquad\,\text{with}\qquad% \tilde{g}_{\beta}:=\tilde{\phi}_{\beta}^{*}g^{\mathrm{can}}.
  27. g M = f * g N , g^{M}=f^{*}g^{N}\,,
  28. g p M ( u , v ) = g f ( p ) N ( d f ( u ) , d f ( v ) ) p M , u , v T p M . g^{M}_{p}(u,v)=g^{N}_{f(p)}(df(u),df(v))\qquad\forall p\in M,\forall u,v\in T_% {p}M.
  29. L a b ( c ) := a b g ( c ( t ) , c ( t ) ) d t = a b c ( t ) d t . L_{a}^{b}(c):=\int_{a}^{b}\sqrt{g(c^{\prime}(t),c^{\prime}(t))}\,\mathrm{d}t=% \int_{a}^{b}\|c^{\prime}(t)\|\,\mathrm{d}t.
  30. c ( t ) = 1 \|c^{\prime}(t)\|=1
  31. t [ a , b ] t\in[a,b]
  32. d ( p , q ) = inf L ( γ ) d(p,q)=\inf L(\gamma)
  33. diam ( M ) := sup p , q M d ( p , q ) 𝐑 0 { + } . \mathrm{diam}(M):=\sup_{p,q\in M}d(p,q)\in\mathbf{R}_{\geq 0}\cup\{+\infty\}.
  34. exp p \exp_{p}
  35. v T p M v\in T_{p}M
  36. γ ( t ) \gamma(t)

Riemann–Roch_theorem.html

  1. ( f ) := z ν R ( f ) s ν z ν (f):=\sum_{z_{\nu}\in R(f)}s_{\nu}z_{\nu}
  2. s ν := { a if z ν is a zero of order a - a if z ν is a pole of order a . s_{\nu}:=\begin{cases}a&\,\text{if }z_{\nu}\,\text{ is a zero of order }a\\ -a&\,\text{if }z_{\nu}\,\text{ is a pole of order }a.\end{cases}
  3. l ( D ) - l ( K - D ) = deg ( D ) - g + 1. l(D)-l(K-D)=\textrm{deg}(D)-g+1.
  4. l ( D ) deg ( D ) - g + 1. l(D)\geq\textrm{deg}(D)-g+1.
  5. l ( D ) = deg ( D ) - g + 1. l(D)=\textrm{deg}(D)-g+1.
  6. l ( n P ) , n 0 l(n\cdot P),n\geq 0
  7. 𝐂 × z 1 / z . \mathbf{C}^{\times}\ni z\mapsto 1/z.
  8. d ( 1 / z ) = - 1 z 2 d z . d(1/z)=-\frac{1}{z^{2}}dz.
  9. H 0 ( X , L ) H^{0}(X,L)
  10. h 0 ( X , L ) h^{0}(X,L)
  11. h 0 ( X , L ) - h 0 ( X , L - 1 K ) = deg ( L ) + 1 - g . h^{0}(X,L)-h^{0}(X,L^{-1}\otimes K)=\textrm{deg}(L)+1-g.
  12. h 0 ( X , L ) = 1 h^{0}(X,L)=1
  13. L - 1 L^{-1}
  14. 1 - h 0 ( X , K ) = 1 - g . 1-h^{0}(X,K)=1-g.
  15. h 0 ( X , K ) = g h^{0}(X,K)=g
  16. g ( C ) := d i m k Γ ( C , Ω C 1 ) g(C):=dim_{k}\Gamma(C,\Omega^{1}_{C})
  17. l ( D ) - l ( K - D ) = deg ( D ) - g + 1. l(D)-l(K-D)=\textrm{deg}(D)-g+1.
  18. 𝒪 \mathcal{O}
  19. g a := d i m k H 1 ( C , 𝒪 C ) . g_{a}:=dim_{k}H^{1}(C,\mathcal{O}_{C}).
  20. ( D ) \mathcal{L}(D)
  21. I ( D ) = dim H 0 ( X , ( D ) ) I(D)=\mathrm{dim}H^{0}(X,\mathcal{L}(D))
  22. I ( 𝒦 X - D ) = dim H 0 ( X , ω X ( D ) ) I(\mathcal{K}_{X}-D)=\mathrm{dim}H^{0}(X,\omega_{X}\otimes\mathcal{L}(D)^{\vee})
  23. H 0 ( X , ω X ( D ) ) H^{0}(X,\omega_{X}\otimes\mathcal{L}(D)^{\vee})
  24. H 1 ( X , ( D ) ) \simeq H^{1}(X,\mathcal{L}(D))^{\vee}
  25. 1 - g 1-g
  26. l ( D ) d e g D 2 + 1. l(D)\leq\frac{degD}{2}+1.

Rifling.html

  1. T w i s t = L D b o r e {Twist}=\frac{L}{D_{bore}}
  2. T w i s t = C D 2 L × S G 10.9 Twist=\frac{CD^{2}}{L}\times\sqrt{\frac{SG}{10.9}}
  3. S S
  4. S = υ C S=\frac{\upsilon}{C}
  5. υ \upsilon
  6. C C
  7. S = υ 0 L S=\frac{\upsilon_{0}}{L}
  8. υ 0 \upsilon_{0}
  9. L L

Right-hand_rule.html

  1. 𝐅 = q 𝐯 × 𝐁 \mathbf{F}=q\mathbf{v}\times\mathbf{B}

Right_quotient.html

  1. L 1 L_{1}
  2. L 2 L_{2}
  3. L 1 L_{1}
  4. L 2 L_{2}
  5. L 1 / L 2 = { w | x ( ( x L 2 ) ( w x L 1 ) ) } L_{1}/L_{2}=\{w\ |\ \exists x((x\in L_{2})\land(wx\in L_{1}))\}
  6. L 1 / L 2 L_{1}/L_{2}
  7. w x wx
  8. L 1 L_{1}
  9. L 2 L_{2}
  10. L 1 = { a n b n c n | n 0 } L_{1}=\{a^{n}b^{n}c^{n}\ \ |\ \ n\geq 0\}
  11. L 2 = { b i c j | i , j 0 } L_{2}=\{b^{i}c^{j}\ \ |\ \ i,j\geq 0\}
  12. L 1 L_{1}
  13. L 2 L_{2}
  14. a n b n - i a^{n}b^{n-i}
  15. a n b n c n - j a^{n}b^{n}c^{n-j}
  16. L 1 / L 2 L_{1}/L_{2}
  17. { a p b q c r | p = q r p q and r = 0 } \{a^{p}b^{q}c^{r}\ \ |\ \ p=q\geq r\ \ \ \ p\geq q\and r=0\}
  18. L 1 L_{1}
  19. L 2 L_{2}

Ring-opening_polymerization.html

  1. Δ G p ( x y ) = Δ H p ( x y ) - T Δ S p ( x y ) \Delta G_{p}(xy)=\Delta H_{p}(xy)-T\Delta S_{p}(xy)
  2. Δ G p = Δ G p + R T l n [ - ( m ) i + 1 m ] [ M ] [ - ( m ) i m ] \Delta G_{p}=\Delta G^{\circ}_{p}+RTln\frac{[...-(m)_{i+1}m^{\ast}]}{[M][...-(% m)_{i}m^{\ast}]}
  3. Δ G p = Δ H p - T ( Δ S p + R l n [ M ] ) \Delta G_{p}=\Delta H^{\circ}_{p}-T(\Delta S^{\circ}_{p}+Rln[M])
  4. [ M ] e q = e ( Δ H p R T - Δ S p R ) [M]_{eq}=e^{(}\frac{\Delta H^{\circ}_{p}}{RT}-\frac{\Delta S^{\circ}_{p}}{R})
  5. l n ( D P n D P n - 1 [ M ] e q ) = Δ H p R T - Δ S p R ln(\frac{DP_{n}}{DP_{n}-1}[M]_{eq})=\frac{\Delta H^{\circ}_{p}}{RT}-\frac{% \Delta S^{\circ}_{p}}{R}
  6. [ M ] e q = D P n - 1 D P n e ( Δ H p R T - Δ S p R ) [M]_{eq}=\frac{DP_{n}-1}{DP_{n}}e^{(}\frac{\Delta H^{\circ}_{p}}{RT}-\frac{% \Delta S^{\circ}_{p}}{R})
  7. T c = Δ H p Δ S p + R l n [ M ] 0 ; ( Δ H p < 0 , Δ S p < 0 ) T_{c}=\frac{\Delta H^{\circ}_{p}}{\Delta S^{\circ}_{p}+Rln[M]_{0}};(\Delta H^{% \circ}_{p}<0,\Delta S^{\circ}_{p}<0)
  8. T f = Δ H p Δ S p + R l n [ M ] 0 ; ( Δ H p > 0 , Δ S p > 0 ) T_{f}=\frac{\Delta H^{\circ}_{p}}{\Delta S^{\circ}_{p}+Rln[M]_{0}};(\Delta H^{% \circ}_{p}>0,\Delta S^{\circ}_{p}>0)

Ring_theory.html

  1. 𝔭 0 𝔭 1 𝔭 n \mathfrak{p}_{0}\subsetneq\mathfrak{p}_{1}\subsetneq\cdots\subsetneq\mathfrak{% p}_{n}
  2. k [ t 1 , , t n ] k[t_{1},\cdots,t_{n}]
  3. ( R , 𝔪 ) (R,\mathfrak{m})
  4. 𝔪 \mathfrak{m}
  5. gr 𝔪 ( R ) = k 0 𝔪 k / 𝔪 k + 1 \operatorname{gr}_{\mathfrak{m}}(R)=\oplus_{k\geq 0}\mathfrak{m}^{k}/{% \mathfrak{m}^{k+1}}
  6. 𝔭 𝔭 \mathfrak{p}\subset\mathfrak{p}^{\prime}
  7. 𝔭 = 𝔭 0 𝔭 n = 𝔭 \mathfrak{p}=\mathfrak{p}_{0}\subsetneq\cdots\subsetneq\mathfrak{p}_{n}=% \mathfrak{p}^{\prime}
  8. ( R , 𝔪 ) (R,\mathfrak{m})
  9. dim R = ht 𝔭 + dim R / 𝔭 \operatorname{dim}R=\operatorname{ht}\mathfrak{p}+\operatorname{dim}R/% \mathfrak{p}
  10. ht 𝔭 = dim R 𝔭 \operatorname{ht}\mathfrak{p}=\operatorname{dim}R_{\mathfrak{p}}
  11. 𝔭 \mathfrak{p}
  12. 𝐏 ( R ) \mathbf{P}(R)
  13. 𝐏 n ( R ) \mathbf{P}_{n}(R)
  14. Spec R , 𝔭 dim M R k ( 𝔭 ) \operatorname{Spec}R\to\mathbb{Z},\,\mathfrak{p}\mapsto\dim M\otimes_{R}k(% \mathfrak{p})
  15. 𝐏 1 ( R ) \mathbf{P}_{1}(R)
  16. 1 R * F * f f R Cart ( R ) Pic ( R ) 1 1\to R^{*}\to F^{*}\overset{f\mapsto fR}{\to}\operatorname{Cart}(R)\to% \operatorname{Pic}(R)\to 1
  17. Cart ( R ) \operatorname{Cart}(R)
  18. 𝐏 ( R ) \mathbf{P}(R)
  19. k [ V ] k[V]
  20. R [ σ 1 , , σ n ] R[\sigma_{1},\ldots,\sigma_{n}]
  21. σ i \sigma_{i}

Risch_algorithm.html

  1. g = v + i < n α i ln ( u i ) g=v+\sum_{i<n}\alpha_{i}\,\ln(u_{i})
  2. ( f e g ) = ( f + f g ) e g , \left(f\cdot e^{g}\right)^{\prime}=\left(f^{\prime}+f\cdot g^{\prime}\right)% \cdot e^{g},\,
  3. ( f ( ln g ) n ) = f ( ln g ) n + n f g g ( ln g ) n - 1 \left(f\cdot(\ln g)^{n}\right)^{\prime}=f^{\prime}(\ln{g})^{n}+nf\frac{g^{% \prime}}{g}(\ln{g})^{n-1}
  4. f ( x ) = x x 4 + 10 x 2 - 96 x - 71 , f(x)=\frac{x}{\sqrt{x^{4}+10x^{2}-96x-71}},
  5. F ( x ) = - 1 8 ln \displaystyle F(x)=-\frac{1}{8}\ln
  6. f ( x ) = x 2 + 2 x + 1 + ( 3 x + 1 ) x + ln x x x + ln x ( x + x + ln x ) . f(x)=\frac{x^{2}+2x+1+(3x+1)\sqrt{x+\ln x}}{x\,\sqrt{x+\ln x}(x+\sqrt{x+\ln x}% )}.
  7. F ( x ) = 2 ( x + ln x + ln ( x + x + ln x ) ) + C . F(x)=2(\sqrt{x+\ln x}+\ln(x+\sqrt{x+\ln x}))+C.
  8. \mathbb{Q}
  9. ( y ) \mathbb{Q}(y)

Risk_assessment.html

  1. R i = L i p ( L i ) R_{i}=L_{i}p(L_{i})\,\!
  2. R t o t a l = i L i p ( L i ) R_{total}=\sum_{i}L_{i}p(L_{i})\,\!
  3. R i = p ( L i ) R_{i}=p(L_{i})\,\!

Risk_aversion.html

  1. E ( u ) = ( u ( 0 ) + u ( 100 ) ) / 2 E(u)=(u(0)+u(100))/2
  2. ( $ 50 - $ 40 ) / $ 40 (\$50-\$40)/\$40
  3. 1 2 0 + 1 2 100 \tfrac{1}{2}0+\tfrac{1}{2}100
  4. u ( c ) u(c)
  5. A ( c ) = - u ′′ ( c ) u ( c ) A(c)=-\frac{u^{\prime\prime}(c)}{u^{\prime}(c)}
  6. u ( c ) = 1 - e - α c u(c)=1-e^{-\alpha c}
  7. A ( c ) = α A(c)=\alpha
  8. A ( c ) = - u ′′ ( c ) u ( c ) = 1 a c + b A(c)=-\frac{u^{\prime\prime}(c)}{u^{\prime}(c)}=\frac{1}{ac+b}
  9. u ( c ) = ( c - c s ) 1 - R 1 - R u(c)=\frac{(c-c_{s})^{1-R}}{1-R}
  10. R = 1 / a R=1/a
  11. c s = - b / a c_{s}=-b/a
  12. a = 0 a=0
  13. A ( c ) = 1 / b = c o n s t A(c)=1/b=const
  14. b = 0 b=0
  15. c A ( c ) = 1 / a = c o n s t cA(c)=1/a=const
  16. A ( c ) A(c)
  17. A ( c ) c = - u ( c ) u ′′′ ( c ) - [ u ′′ ( c ) ] 2 [ u ( c ) ] 2 < 0 \frac{\partial A(c)}{\partial c}=-\frac{u^{\prime}(c)u^{\prime\prime\prime}(c)% -[u^{\prime\prime}(c)]^{2}}{[u^{\prime}(c)]^{2}}<0
  18. u ′′′ ( c ) > 0 u^{\prime\prime\prime}(c)>0
  19. u ′′′ ( c ) > 0 u^{\prime\prime\prime}(c)>0
  20. u ′′′ ( c ) < 0 u^{\prime\prime\prime}(c)<0
  21. u ( c ) = log ( c ) u(c)=\log(c)
  22. A ( c ) = 1 / c A(c)=1/c
  23. u ( c ) = c - α c 2 , u(c)=c-\alpha c^{2},
  24. α > 0 \alpha>0
  25. A ( c ) = 2 α / ( 1 - 2 α c ) A(c)=2\alpha/(1-2\alpha c)
  26. A ( c ) = - u ′′ ( c ) u ( c ) A(c)=-\frac{u^{\prime\prime}(c)}{u^{\prime}(c)}
  27. R ( c ) = c A ( c ) = - c u ′′ ( c ) u ( c ) R(c)=cA(c)=\frac{-cu^{\prime\prime}(c)}{u^{\prime}(c)}
  28. u ( c ) = log ( c ) u(c)=\log(c)
  29. u ( c ) = c 1 - ρ - 1 1 - ρ u(c)=\frac{c^{1-\rho}-1}{1-\rho}
  30. R ( c ) = ρ R(c)=\rho
  31. ε u ( c ) = 1 / ρ \varepsilon_{u(c)}=1/\rho
  32. ρ = 1 , \rho=1,
  33. A = d E ( r ) d σ A=\frac{dE(r)}{d\sigma}
  34. A n = d E ( r ) d μ n n = 1 n d E ( r ) d μ n A_{n}=\frac{dE(r)}{d\sqrt[n]{\mu_{n}}}=\frac{1}{n}\frac{dE(r)}{d\mu_{n}}

Ritchey–Chrétien_telescope.html

  1. R 1 = - 2 D F F - B R_{1}=-\frac{2DF}{F-B}
  2. R 2 = - 2 D B F - B - D R_{2}=-\frac{2DB}{F-B-D}
  3. F F
  4. B B
  5. D D
  6. B B
  7. D D
  8. f 1 f_{1}
  9. b b
  10. D = f 1 ( F - b ) / ( F + f 1 ) D=f_{1}(F-b)/(F+f_{1})
  11. B = D + b B=D+b
  12. K 1 K_{1}
  13. K 2 K_{2}
  14. K 1 = - 1 - 2 M 3 B D K_{1}=-1-\frac{2}{M^{3}}\cdot\frac{B}{D}
  15. K 2 = - 1 - 2 ( M - 1 ) 3 [ M ( 2 M - 1 ) + B D ] K_{2}=-1-\frac{2}{(M-1)^{3}}\left[M(2M-1)+\frac{B}{D}\right]
  16. M = F / f 1 = ( F - B ) / D M=F/f_{1}=(F-B)/D
  17. K 1 K_{1}
  18. K 2 K_{2}
  19. - 1 -1
  20. M > 1 M>1

Robert_Rosen_(theoretical_biologist).html

  1. ( M , R ) (M{,}R)
  2. ( M , R ) (M{,}R)
  3. M M
  4. R R
  5. ( M , R ) (M{,}R)
  6. M M
  7. R R
  8. β \beta
  9. ( M , R ) (M{,}R)
  10. ( M , R ) (M{,}R)
  11. ( M , R ) (M{,}R)
  12. ( M , R ) (M{,}R)
  13. ( M , R ) (M{,}R)
  14. ( M , R ) (M{,}R)

Roche_lobe.html

  1. r 1 A = 0.38 + 0.2 log M 1 M 2 \frac{r_{1}}{A}=0.38+0.2\log\frac{M_{1}}{M_{2}}
  2. 0.3 < M 1 M 2 < 20 0.3<\frac{M_{1}}{M_{2}}<20
  3. r 1 A = 0.46224 ( M 1 M 1 + M 2 ) 1 / 3 \frac{r_{1}}{A}=0.46224\left(\frac{M_{1}}{M_{1}+M_{2}}\right)^{1/3}
  4. M 1 M 2 < 0.8 \frac{M_{1}}{M_{2}}<0.8
  5. r 1 r_{1}
  6. M 1 M_{1}
  7. r 1 A = 0.49 q 2 / 3 0.6 q 2 / 3 + ln ( 1 + q 1 / 3 ) \frac{r_{1}}{A}=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}
  8. q = M 1 / M 2 q=M_{1}/M_{2}
  9. q q

Rocket_engine.html

  1. L * L^{*}
  2. L * = V c A t L^{*}=\frac{V_{c}}{A_{t}}
  3. V c V_{c}
  4. A t A_{t}
  5. I s p I_{sp}
  6. v e v_{e}
  7. F n = m ˙ v e = m ˙ v e - a c t + A e ( p e - p a m b ) F_{n}=\dot{m}\;v_{e}=\dot{m}\;v_{e-act}+A_{e}(p_{e}-p_{amb})
  8. m ˙ \dot{m}
  9. v e v_{e}
  10. v e - a c t v_{e-act}
  11. A e A_{e}
  12. p e p_{e}
  13. p a m b p_{amb}
  14. m ˙ v e - a c t \dot{m}\;v_{e-act}\,
  15. A e ( p e - p a m b ) A_{e}(p_{e}-p_{amb})\,
  16. p e = p a m b p_{e}=p_{amb}
  17. m ˙ \dot{m}
  18. F v a c = C f m ˙ c * F_{vac}=C_{f}\,\dot{m}\,c^{*}
  19. v e v a c = C f c * v_{evac}=C_{f}\,c^{*}\,
  20. c * c^{*}
  21. C f C_{f}
  22. F n = m ˙ v e v a c - A e p a m b F_{n}=\dot{m}\,v_{evac}-A_{e}\,p_{amb}
  23. m ˙ \dot{m}
  24. P = F * V P=F*V

Roger_Cotes.html

  1. E X + X C - 1 EX+XC\sqrt{-1}
  2. - 1 \sqrt{-1}
  3. E X + X C - 1 EX+XC\sqrt{-1}
  4. - 1 \sqrt{-1}
  5. E X + X C - 1 EX+XC\sqrt{-1}
  6. cos θ + - 1 sin θ \cos\theta+\sqrt{-1}\sin\theta
  7. - 1 \sqrt{-1}
  8. - 1 C E ln ( cos θ + - 1 sin θ ) = ( C E ) θ \sqrt{-1}CE\ln{\left(\cos\theta+\sqrt{-1}\sin\theta\right)\ }=(CE)\theta
  9. - 1 \sqrt{-1}
  10. cos θ + - 1 sin θ = e - 1 θ \cos\theta+\sqrt{-1}\sin\theta=e^{\sqrt{-1}\theta}

Rogowski_coil.html

  1. V = - A N μ 0 l d I d t V=\frac{-AN\mu_{0}}{l}\frac{dI}{dt}
  2. A = π r 2 A=\pi r^{2}
  3. N N
  4. l = 2 π R l=2\pi R
  5. d I d t \frac{dI}{dt}
  6. μ 0 = 4 π × 10 - 7 \mu_{0}=4\pi\times 10^{-7}
  7. R R
  8. r r
  9. L = μ 0 N 2 ( R - R 2 - a 2 ) L=\mu_{0}N^{2}(R-\sqrt{R^{2}-a^{2}})

Rolle's_theorem.html

  1. f ( c ) = 0 f^{\prime}(c)=0
  2. f ( x ) = r 2 - x 2 , x [ - r , r ] . f(x)=\sqrt{r^{2}-x^{2}},\quad x\in[-r,r].
  3. f ( x ) = | x | , x [ - 1 , 1 ] . f(x)=|x|,\qquad x\in[-1,1].
  4. f ( x + ) := lim h 0 + f ( x + h ) - f ( x ) h f^{\prime}(x+):=\lim_{h\to 0^{+}}\frac{f(x+h)-f(x)}{h}
  5. f ( x - ) := lim h 0 - f ( x + h ) - f ( x ) h f^{\prime}(x-):=\lim_{h\to 0^{-}}\frac{f(x+h)-f(x)}{h}
  6. f ( c + ) and f ( c - ) f^{\prime}(c+)\quad\,\text{and}\quad f^{\prime}(c-)
  7. f ( x - ) f ( x + ) f ( y - ) , x < y . f^{\prime}(x-)\leq f^{\prime}(x+)\leq f^{\prime}(y-),\qquad x<y.
  8. f ( c + h ) - f ( c ) h 0 , \frac{f(c+h)-f(c)}{h}\leq 0,
  9. f ( c + ) := lim h 0 + f ( c + h ) - f ( c ) h 0 , f^{\prime}(c+):=\lim_{h\to 0^{+}}\frac{f(c+h)-f(c)}{h}\leq 0,
  10. f ( c - ) := lim h 0 - f ( c + h ) - f ( c ) h 0 , f^{\prime}(c-):=\lim_{h\to 0^{-}}\frac{f(c+h)-f(c)}{h}\geq 0,
  11. x 3 - x = x ( x - 1 ) ( x + 1 ) x^{3}-x=x(x-1)(x+1)
  12. 3 x 2 - 1 = 3 ( x - 1 / 3 ) ( x + 1 / 3 ) 3x^{2}-1=3(x-1/\sqrt{3})(x+1/\sqrt{3})
  13. 𝐅 2 \mathbf{F}_{2}
  14. 𝐅 4 \mathbf{F}_{4}

Root-finding_algorithm.html

  1. 1 + Φ 2.6 1+\Phi\approx 2.6
  2. p = p 1 p 2 2 p k k p=p_{1}p_{2}^{2}\cdots p_{k}^{k}
  3. p i p_{i}
  4. p i p_{i}

Root_of_unity.html

  1. n n
  2. n n
  3. n n
  4. n n
  5. n n
  6. n n
  7. n = 1 , 2 , 3 , n=1, 2, 3, …
  8. z z
  9. z n = 1. z^{n}=1.
  10. z z
  11. z R z∈R
  12. R R
  13. n n
  14. n n
  15. n n
  16. k k
  17. k k
  18. z k 1 ( k = 1 , 2 , 3 , , n - 1 ) . z^{k}\neq 1\qquad(k=1,2,3,\dots,n-1).
  19. n n
  20. z z
  21. a a
  22. a a
  23. 1 a n 1≤a≤n
  24. z z
  25. z z
  26. n n
  27. z z
  28. n n
  29. a b ( m o d n ) a≡b(modn)
  30. a = b + k n a=b+kn
  31. k k
  32. z a = z b + k n = z b z k n = z b ( z n ) k = z b 1 k = z b . z^{a}=z^{b+kn}=z^{b}z^{kn}=z^{b}(z^{n})^{k}=z^{b}1^{k}=z^{b}.
  33. z z
  34. 1 a n 1≤a≤n
  35. n n
  36. n n
  37. ( z k ) n = z k n = ( z n ) k = 1 k = 1. (z^{k})^{n}=z^{kn}=(z^{n})^{k}=1^{k}=1.
  38. k k
  39. n n
  40. n n
  41. 1 z = z - 1 = z n - 1 = z ¯ . \frac{1}{z}=z^{-1}=z^{n-1}=\bar{z}.
  42. z z
  43. n n
  44. z z
  45. n n
  46. n n
  47. z z
  48. n n
  49. z z
  50. n n
  51. z a = z b a b ( mod n ) . z^{a}=z^{b}\iff a\equiv b\;\;(\mathop{{\rm mod}}n).
  52. z z
  53. a b ( mod n ) z a = z b . a\equiv b\;\;(\mathop{{\rm mod}}n)\implies z^{a}=z^{b}.
  54. n = 4 , z = - 1 , z 2 = z 4 = 1 , 2 4 ( mod 4 ) . n=4,\;\;z=-1,\;\;z^{2}=z^{4}=1,\;\;2\not\equiv 4\;\;(\mathop{{\rm mod}}4).
  55. z z
  56. n n
  57. k k
  58. a a
  59. k a ka
  60. n n
  61. n n
  62. k k
  63. l c m ( n , k ) lcm(n,k)
  64. g c d ( n , k ) gcd(n,k)
  65. k n = gcd ( k , n ) lcm ( k , n ) , k\,n=\gcd(k,n)\,\operatorname{lcm}(k,n),
  66. lcm ( k , n ) = k n gcd ( k , n ) . \operatorname{lcm}(k,n)=k\frac{n}{\gcd(k,n)\,}.
  67. a a
  68. a = n gcd ( k , n ) . a=\frac{n}{\gcd(k,n)}.
  69. k k
  70. n n
  71. n n
  72. φ ( n ) φ(n)
  73. φ φ
  74. n n
  75. n n
  76. R ( n ) R(n)
  77. n n
  78. P ( n ) P(n)
  79. R ( n ) R(n)
  80. P ( n ) P(n)
  81. R ( n ) = d | n P ( d ) , \operatorname{R}(n)=\bigcup_{d\,|\,n}\operatorname{P}(d),
  82. d d
  83. n n
  84. n n
  85. R ( n ) R(n)
  86. n n
  87. P ( n ) P(n)
  88. φ ( n ) φ(n)
  89. d | n ϕ ( d ) = n . \sum_{d\,|\,n}\phi(d)=n.
  90. x x
  91. n n
  92. ( cos x + i sin x ) n = cos n x + i sin n x . (\cos x+i\sin x)^{n}=\cos nx+i\sin nx.
  93. x = 2 π / n x=2π/n
  94. n n
  95. ( cos 2 π n + i sin 2 π n ) n = cos 2 π + i sin 2 π = 1 , \left(\cos\tfrac{2\pi}{n}+i\sin\tfrac{2\pi}{n}\right)^{n}=\cos 2\pi+i\sin 2\pi% =1,
  96. k = 1 , 2 , , n 1 k=1,2,⋯ ,n−1
  97. ( cos 2 π n + i sin 2 π n ) k = cos 2 k π n + i sin 2 k π n 1 \left(\cos\tfrac{2\pi}{n}+i\sin\tfrac{2\pi}{n}\right)^{k}=\cos\tfrac{2k\pi}{n}% +i\sin\tfrac{2k\pi}{n}\neq 1
  98. n n
  99. n n
  100. n = 3 n=3
  101. n = 5 n=5
  102. e i x = cos x + i sin x , e^{ix}=\cos x+i\sin x,
  103. x x
  104. n n
  105. e 2 π i k n 0 k < n . e^{2\pi i\frac{k}{n}}\qquad 0\leq k<n.
  106. n n
  107. k / n k/n
  108. k k
  109. n n
  110. { e 2 π i 3 , e - 2 π i 3 } = { - 1 + i 3 2 , - 1 - i 3 2 } \left\{e^{\frac{2\pi i}{3}},e^{-\frac{2\pi i}{3}}\right\}=\left\{\frac{-1+i% \sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2}\right\}
  111. { e 2 π i 4 , e - 2 π i 4 } = { ± - 1 } = { + i , - i } . \left\{e^{\frac{2\pi i}{4}},e^{-\frac{2\pi i}{4}}\right\}=\left\{\pm\sqrt{-1}% \right\}=\left\{+i,-i\right\}.
  112. { e 2 π i k 5 | 1 k 4 } = { u 5 - 1 4 + v i 5 + u 5 8 | u , v { - 1 , 1 } } . \left\{\left.e^{\frac{2\pi ik}{5}}\right|1\leq k\leq 4\right\}=\left\{\left.% \frac{u\sqrt{5}-1}{4}+v\,i\,\sqrt{\frac{5+u\sqrt{5}}{8}}\;\right|u,v\in\{-1,1% \}\right\}.
  113. { e 2 π i 6 , e - 2 π i 6 } = { 1 + i 3 2 , 1 - i 3 2 } . \left\{e^{\frac{2\pi i}{6}},e^{-\frac{2\pi i}{6}}\right\}=\left\{\frac{1+i% \sqrt{3}}{2},\frac{1-i\sqrt{3}}{2}\right\}.
  114. n n
  115. n n
  116. e 2 π i 7 = - 1 + 7 + 21 - 3 2 3 + 7 - 21 - 3 2 3 6 + i 2 7 - ω 2 7 + 21 - 3 2 3 - ω 7 - 21 - 3 2 3 3 e^{\frac{2\pi i}{7}}=\frac{-1+\sqrt[3]{\frac{7+21\sqrt{-3}}{2}}+\sqrt[3]{\frac% {7-21\sqrt{-3}}{2}}}{6}+\frac{i}{2}\sqrt{\frac{7-\omega^{2}\sqrt[3]{\frac{7+21% \sqrt{-3}}{2}}-\omega\sqrt[3]{\frac{7-21\sqrt{-3}}{2}}}{3}}
  117. ω ω
  118. e x p ( 2 π i / 3 ) exp(2πi/3)
  119. e x p ( 4 π i / 3 ) exp(4πi/3)
  120. ± i ±i
  121. e 2 π i 8 = i = 2 2 + i 2 2 . e^{\frac{2\pi i}{8}}=\sqrt{i}=\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}.
  122. z z
  123. n n
  124. n n
  125. j j
  126. n n
  127. k = 1 , , n k=1, … ,n
  128. n n
  129. n n
  130. n n
  131. n n
  132. x j = k X k z k j = X 1 z 1 j + + X n z n j x_{j}=\sum_{k}X_{k}\cdot z^{k\cdot j}=X_{1}z^{1\cdot j}+\cdots+X_{n}\cdot z^{n% \cdot j}
  133. j j
  134. j j
  135. k k
  136. n n
  137. c o s cos
  138. s i n sin
  139. S R ( n ) SR(n)
  140. n n
  141. SR ( n ) = { 1 , n = 1 0 , n > 1. \operatorname{SR}(n)=\begin{cases}1,&n=1\\ 0,&n>1.\end{cases}
  142. n = 1 n=1
  143. n > 1 n>1
  144. z z
  145. n n
  146. k = 0 n - 1 z k = z n - 1 z - 1 = 0. \sum_{k=0}^{n-1}z^{k}=\frac{z^{n}-1}{z-1}=0.
  147. S P ( n ) SP(n)
  148. n n
  149. SP ( n ) = μ ( n ) , \operatorname{SP}(n)=\mu(n),
  150. μ ( n ) μ(n)
  151. R ( n ) R(n)
  152. n n
  153. P ( n ) P(n)
  154. R ( n ) R(n)
  155. P ( n ) P(n)
  156. R ( n ) = d | n P ( d ) , \operatorname{R}(n)=\bigcup_{d\,|\,n}\operatorname{P}(d),
  157. SR ( n ) = d | n SP ( d ) . \operatorname{SR}(n)=\sum_{d\,|\,n}\operatorname{SP}(d).
  158. SP ( n ) = d | n μ ( d ) SR ( n d ) . \operatorname{SP}(n)=\sum_{d\,|\,n}\mu(d)\operatorname{SR}\left(\frac{n}{d}% \right).
  159. S R ( n / d ) = 0 SR(n/d)=0
  160. d = n d=n
  161. S R ( n / d ) = 1 SR(n/d)=1
  162. S P ( n ) = μ ( n ) SP(n)=μ(n)
  163. s s
  164. n n
  165. c n ( s ) = a = 1 gcd ( a , n ) = 1 n e 2 π i a n s . c_{n}(s)=\sum_{a=1\atop\gcd(a,n)=1}^{n}e^{2\pi i\tfrac{a}{n}s}.
  166. j = 1 , , n j=1, … ,n
  167. j = 1 , , n j′=1, … ,n
  168. k = 1 n z j k ¯ z j k = n δ j , j \sum_{k=1}^{n}\overline{z^{j\cdot k}}\cdot z^{j^{\prime}\cdot k}=n\cdot\delta_% {j,j^{\prime}}
  169. δ δ
  170. z z
  171. n n
  172. n × n n×n
  173. U U
  174. ( j , k ) (j,k)
  175. U j , k = n - 1 2 z j k U_{j,k}=n^{-\frac{1}{2}}\cdot z^{j\cdot k}
  176. U U
  177. k = 1 n U j , k ¯ U k , j = δ j , j , \sum_{k=1}^{n}\overline{U_{j,k}}\cdot U_{k,j^{\prime}}=\delta_{j,j^{\prime}},
  178. U U
  179. U U
  180. O ( n l o g n ) O(nlogn)
  181. p ( z ) = z n - 1 p(z)=z^{n}-1
  182. n n
  183. n n
  184. n n
  185. Φ n ( z ) = k = 1 φ ( n ) ( z - z k ) \Phi_{n}(z)=\prod_{k=1}^{\varphi(n)}(z-z_{k})
  186. n n
  187. φ ( n ) φ(n)
  188. n n
  189. ( z + 1 ) n - 1 ( ( z + 1 ) - 1 ) , \frac{(z+1)^{n}-1}{((z+1)-1)},
  190. n n
  191. d d
  192. d d
  193. n n
  194. z n - 1 = d n Φ d ( z ) . z^{n}-1=\prod_{d\,\mid\,n}\Phi_{d}(z).
  195. Φ n ( z ) = d n ( z n / d - 1 ) μ ( d ) = d n ( z d - 1 ) μ ( n / d ) , \Phi_{n}(z)=\prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)}=\prod_{d\,\mid n}(z^{d}-1)^{% \mu(n/d)},
  196. μ μ
  197. p p
  198. p p
  199. p p
  200. Φ p ( z ) = z p - 1 z - 1 = k = 0 p - 1 z k . \Phi_{p}(z)=\frac{z^{p}-1}{z-1}=\sum_{k=0}^{p-1}z^{k}.
  201. z z
  202. z z
  203. Φ < s u b > [ u n u m , u 105 ] Φ<sub>[u^{\prime}num^{\prime},u^{\prime}105^{\prime}]

Root_system.html

  1. ( , ) (\cdot,\cdot)
  2. , : Φ × Φ \langle\cdot,\cdot\rangle\colon\Phi\times\Phi\to\mathbb{Z}
  3. A 1 × A 1 A_{1}\times A_{1}
  4. D 2 D_{2}
  5. A 2 A_{2}
  6. G 2 G_{2}
  7. B 2 B_{2}
  8. C 2 C_{2}
  9. x , y \langle x,y\rangle
  10. { α , - α } \{\alpha,-\alpha\}
  11. A 1 A_{1}
  12. σ α ( β ) = β + n α \sigma_{\alpha}(\beta)=\beta+n\alpha
  13. n = 0 , 1 , 2 , 3 n=0,1,2,3
  14. A 1 × A 1 A_{1}\times A_{1}
  15. B 2 B_{2}
  16. A 2 A_{2}
  17. G 2 G_{2}
  18. L L
  19. 𝔥 \mathfrak{h}
  20. det ( ad L x - t ) \det(\mathrm{ad}_{L}x-t)
  21. x 𝔥 x\in\mathfrak{h}
  22. 𝔥 \mathfrak{h}
  23. 𝔥 * \mathfrak{h}^{*}
  24. 𝔥 * \mathfrak{h}^{*}
  25. β , α \langle\beta,\alpha\rangle
  26. α , β \langle\alpha,\beta\rangle
  27. β , α α , β = 2 ( α , β ) ( α , α ) 2 ( α , β ) ( β , β ) = 4 ( α , β ) 2 | α | 2 | β | 2 = 4 cos 2 ( θ ) = ( 2 cos ( θ ) ) 2 . \langle\beta,\alpha\rangle\langle\alpha,\beta\rangle=2\frac{(\alpha,\beta)}{(% \alpha,\alpha)}\cdot 2\frac{(\alpha,\beta)}{(\beta,\beta)}=4\frac{(\alpha,% \beta)^{2}}{|\alpha|^{2}|\beta|^{2}}=4\cos^{2}(\theta)=(2\cos(\theta))^{2}\in% \mathbb{Z}.
  28. 2 cos ( θ ) [ - 2 , 2 ] 2\cos(\theta)\in[-2,2]
  29. cos ( θ ) \cos(\theta)
  30. 0 , ± 1 2 , ± 2 2 , ± 3 2 , ± 4 2 = ± 1 0,\pm\tfrac{1}{2},\pm\tfrac{\sqrt{2}}{2},\pm\tfrac{\sqrt{3}}{2},\pm\tfrac{% \sqrt{4}}{2}=\pm 1
  31. Φ + \Phi^{+}
  32. α Φ \alpha\in\Phi
  33. α \alpha
  34. α \alpha
  35. Φ + \Phi^{+}
  36. α , β Φ + \alpha,\beta\in\Phi^{+}
  37. α + β \alpha+\beta
  38. α + β Φ + \alpha+\beta\in\Phi^{+}
  39. Φ + \Phi^{+}
  40. - Φ + -\Phi^{+}
  41. Φ + \Phi^{+}
  42. Φ + \Phi^{+}
  43. Δ \Delta
  44. V V
  45. Φ \Phi
  46. Δ \Delta
  47. α β \alpha\leq\beta
  48. β - α \beta-\alpha
  49. deg ( α Δ λ α α ) = α Δ λ α \operatorname{deg}\big(\sum_{\alpha\in\Delta}\lambda_{\alpha}\alpha\big)=\sum_% {\alpha\in\Delta}\lambda_{\alpha}
  50. α = 2 ( α , α ) α . \alpha^{\vee}={2\over(\alpha,\alpha)}\,\alpha.
  51. ( s α ) = s ( α ) . (s\alpha)^{\vee}=s(\alpha^{\vee}).
  52. Φ = Φ 1 Φ 2 \Phi=\Phi_{1}\cup\Phi_{2}
  53. ( α , β ) = 0 (\alpha,\beta)=0
  54. α Φ 1 \alpha\in\Phi_{1}
  55. β Φ 2 \beta\in\Phi_{2}
  56. 2 π / 3 2\pi/3
  57. 3 π / 4 3\pi/4
  58. 5 π / 6 5\pi/6
  59. Φ \Phi
  60. | Φ | |\Phi|
  61. | Φ < | |\Phi^{<}|
  62. | W | |W|
  63. - 1 2 ( i = 1 8 e i ) -\textstyle\frac{1}{2}(\textstyle\sum_{i=1}^{8}e_{i})
  64. 1 2 ( - i = 1 j e i + i = j + 1 8 e i ) \textstyle\frac{1}{2}(-\textstyle\sum_{i=1}^{j}e_{i}+\textstyle\sum_{i=j+1}^{8% }e_{i})
  65. 1 2 i = 1 4 e i \textstyle\frac{1}{2}\sum_{i=1}^{4}e_{i}

Rotating_furnace.html

  1. 2 f ω 2 = g 2f\omega^{2}=g
  2. f f
  3. ω \omega
  4. g g
  5. f f
  6. ω \omega
  7. g g
  8. ω \omega
  9. g g
  10. f s 2 447 fs^{2}\approx 447
  11. f f
  12. s s

Rotation_group_SO(3).html

  1. 𝐮 𝐯 = 1 2 ( 𝐮 + 𝐯 2 - 𝐮 2 - 𝐯 2 ) . \mathbf{u}\cdot\mathbf{v}=\tfrac{1}{2}\left(\|\mathbf{u}+\mathbf{v}\|^{2}-\|% \mathbf{u}\|^{2}-\|\mathbf{v}\|^{2}\right).
  2. S O ( 3 ) SO(3)
  3. R R
  4. R R
  5. R R
  6. R R
  7. R 𝖳 R = I , R^{\mathsf{T}}R=I,
  8. R R
  9. I I
  10. 3 × 3 3 × 3
  11. 3 × 3 3 × 3
  12. O ( 3 ) O(3)
  13. R R
  14. d e t R = ± 1 detR=±1
  15. + 1 +1
  16. S O ( 3 ) SO(3)
  17. S O ( 3 ) SO(3)
  18. 1 −1
  19. R z ( φ ) = [ cos φ - sin φ 0 sin φ cos φ 0 0 0 1 ] . R_{z}(\varphi)=\begin{bmatrix}\cos\varphi&-\sin\varphi&0\\ \sin\varphi&\cos\varphi&0\\ 0&0&1\end{bmatrix}.
  20. 1 2 \frac{1}{2}
  21. ( x , y , z ) = ( 0 , 0 , 1 2 ) (x,y,z)=(0,0,\frac{1}{2})
  22. M M
  23. z = 1 2 z=−\frac{1}{2}
  24. ( ξ , η ) (ξ,η)
  25. P P
  26. N N
  27. S ( P ) = P ´ S(P)=P´
  28. M M
  29. z = 1 2 z=−\frac{1}{2}
  30. S S
  31. M M
  32. ( ξ , η ) (ξ,η)
  33. L L
  34. N N
  35. P P
  36. L = N + t ( N - P ) = ( 0 , 0 , 1 / 2 ) + t ( ( 0 , 0 , 1 / 2 ) - ( x , y , z ) ) , t . L=N+t(N-P)=(0,0,1/2)+t((0,0,1/2)-(x,y,z)),\quad t\in\mathbb{R}.
  37. 1 2 −\frac{1}{2}
  38. t = 1 z t=1\frac{z}{−}
  39. S : 𝐒 M ; P P ; ( x , y , z ) ( ξ , η ) = ( x 1 2 - z , y 1 2 - z ) ζ = ξ + i η , S:\mathbf{S}\rightarrow M;P\mapsto P^{\prime};(x,y,z)\mapsto(\xi,\eta)=\left(% \frac{x}{\frac{1}{2}-z},\frac{y}{\frac{1}{2}-z}\right)\equiv\zeta=\xi+i\eta,
  40. M M
  41. L L
  42. L = N + s ( P - N ) = ( 0 , 0 , 1 2 ) + s ( ( ξ , η , - 1 2 ) - ( 0 , 0 , 1 2 ) ) , L=N+s(P^{\prime}-N)=(0,0,\frac{1}{2})+s\left((\xi,\eta,-\frac{1}{2})-(0,0,% \frac{1}{2})\right),
  43. S - 1 : M 𝐒 ; P P ; ( ξ , η ) ( x , y , z ) = ( ξ 1 + ξ 2 + η 2 , η 1 + ξ 2 + η 2 , - 1 + ξ 2 + η 2 2 + 2 ξ 2 + 2 η 2 ) . S^{-1}:M\rightarrow\mathbf{S};P^{\prime}\mapsto P;(\xi,\eta)\mapsto(x,y,z)=% \left(\frac{\xi}{1+\xi^{2}+\eta^{2}},\frac{\eta}{1+\xi^{2}+\eta^{2}},\frac{-1+% \xi^{2}+\eta^{2}}{2+2\xi^{2}+2\eta^{2}}\right).
  44. g S O ( 3 ) g∈SO(3)
  45. 𝐒 \mathbf{S}
  46. 𝐒 \mathbf{S}
  47. S S
  48. M M
  49. g S O ( 3 ) g∈SO(3)
  50. φ φ
  51. x = x cos φ - y sin φ , y = x sin φ + y cos φ , z = z . \begin{aligned}\displaystyle x^{\prime}&\displaystyle=x\cos\varphi-y\sin% \varphi,\\ \displaystyle y^{\prime}&\displaystyle=x\sin\varphi+y\cos\varphi,\\ \displaystyle z^{\prime}&\displaystyle=z.\end{aligned}
  52. ζ = x + i y 1 2 - z = e i φ ( x + i y ) 1 2 - z = e i φ ζ = cos φ ζ + i sin φ 0 ζ + 1 , \zeta^{\prime}=\frac{x^{\prime}+iy^{\prime}}{\frac{1}{2}-z^{\prime}}=\frac{e^{% i\varphi}(x+iy)}{\frac{1}{2}-z}=e^{i\varphi}\zeta=\frac{\cos\varphi\zeta+i\sin% \varphi}{0\zeta+1},
  53. θ θ
  54. w = e i θ w , w = y + i z 1 2 - x , w^{\prime}=e^{i\theta}w,\quad w=\frac{y+iz}{\frac{1}{2}-x},
  55. ζ = cos θ 2 ζ + i sin θ 2 i sin θ 2 ζ + cos θ 2 . \zeta^{\prime}=\frac{\cos\frac{\theta}{2}\zeta+i\sin\frac{\theta}{2}}{i\sin% \frac{\theta}{2}\zeta+\cos\frac{\theta}{2}}.
  56. ζ = α ζ + β γ ζ + δ , α δ - β γ 0. \zeta^{\prime}=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta},\quad\alpha\delta-% \beta\gamma\neq 0.
  57. S O ( 3 ) SO(3)
  58. ( α β γ δ ) , α δ - β γ = 1 , \left(\begin{matrix}\alpha&\beta\\ \gamma&\delta\end{matrix}\right),\quad\quad\alpha\delta-\beta\gamma=1,
  59. α , β , γ , δ α,β,γ,δ
  60. I −I
  61. g , g S L ( 2 , ) g,−g∈SL(2,ℂ)
  62. Π u ( g φ ) = Π u [ ( cos φ - sin φ 0 sin φ cos φ 0 0 0 0 ) ] = ± ( e i φ 2 0 0 e - i φ 2 ) , Π u ( g θ ) = Π u [ ( 0 0 0 0 cos θ - sin θ 0 sin θ cos θ ) ] = ± ( cos θ 2 i sin θ 2 i sin θ 2 cos θ 2 ) . \begin{aligned}\displaystyle\Pi_{u}(g_{\varphi})&\displaystyle=\Pi_{u}\left[% \left(\begin{matrix}\cos\varphi&-\sin\varphi&0\\ \sin\varphi&\cos\varphi&0\\ 0&0&0\end{matrix}\right)\right]=\pm\left(\begin{matrix}e^{i\frac{\varphi}{2}}&% 0\\ 0&e^{-i\frac{\varphi}{2}}\end{matrix}\right),\\ \displaystyle\Pi_{u}(g_{\theta})&\displaystyle=\Pi_{u}\left[\left(\begin{% matrix}0&0&0\\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta\end{matrix}\right)\right]=\pm\left(\begin{matrix}\cos% \frac{\theta}{2}&i\sin\frac{\theta}{2}\\ i\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{matrix}\right).\end{aligned}
  63. g ( φ , θ , ψ ) = g φ g θ g ψ = ( cos φ - sin φ 0 sin φ cos φ 0 0 0 1 ) ( 1 0 0 0 cos θ - sin θ 0 sin θ cos θ ) ( cos ψ - sin ψ 0 sin ψ cos ψ 0 0 0 1 ) = ( cos φ cos ψ - cos θ sin φ sin ψ - cos φ sin ψ - cos θ sin φ cos ψ sin φ sin θ sin φ cos ψ + cos θ cos φ sin ψ - sin φ sin ψ + cos θ cos φ cos ψ - cos φ sin θ sin ψ sin θ cos ψ sin θ cos θ ) , \begin{aligned}\displaystyle g(\varphi,\theta,\psi)&\displaystyle=g_{\varphi}g% _{\theta}g_{\psi}=\left(\begin{matrix}\cos\varphi&-\sin\varphi&0\\ \sin\varphi&\cos\varphi&0\\ 0&0&1\end{matrix}\right)\left(\begin{matrix}1&0&0\\ 0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta\end{matrix}\right)\left(\begin{matrix}\cos\psi&-\sin% \psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{matrix}\right)\\ &\displaystyle=\left(\begin{matrix}\cos\varphi\cos\psi-\cos\theta\sin\varphi% \sin\psi&-\cos\varphi\sin\psi-\cos\theta\sin\varphi\cos\psi&\sin\varphi\sin% \theta\\ \sin\varphi\cos\psi+\cos\theta\cos\varphi\sin\psi&-\sin\varphi\sin\psi+\cos% \theta\cos\varphi\cos\psi&-\cos\varphi\sin\theta\\ \sin\psi\sin\theta&\cos\psi\sin\theta&\cos\theta\end{matrix}\right),\end{aligned}
  64. Π u ( g ( φ , θ , ψ ) ) = ± ( e i φ 2 0 0 e - i φ 2 ) ( cos θ 2 i sin θ 2 i sin θ 2 cos θ 2 ) ( e i ψ 2 0 0 e - i ψ 2 ) = ± ( cos θ 2 e i φ + ψ 2 i sin θ 2 e i φ - ψ 2 i sin θ 2 e - i φ - ψ 2 cos θ 2 e - i φ + ψ 2 ) . \begin{aligned}\displaystyle\Pi_{u}(g(\varphi,\theta,\psi))&\displaystyle=\pm% \left(\begin{matrix}e^{i\frac{\varphi}{2}}&0\\ 0&e^{-i\frac{\varphi}{2}}\end{matrix}\right)\left(\begin{matrix}\cos\frac{% \theta}{2}&i\sin\frac{\theta}{2}\\ i\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{matrix}\right)\left(\begin{% matrix}e^{i\frac{\psi}{2}}&0\\ 0&e^{-i\frac{\psi}{2}}\end{matrix}\right)\\ &\displaystyle=\pm\left(\begin{matrix}\cos\frac{\theta}{2}e^{i\frac{\varphi+% \psi}{2}}&i\sin\frac{\theta}{2}e^{i\frac{\varphi-\psi}{2}}\\ i\sin\frac{\theta}{2}e^{-i\frac{\varphi-\psi}{2}}&\cos\frac{\theta}{2}e^{-i% \frac{\varphi+\psi}{2}}\end{matrix}\right).\end{aligned}
  65. ± Π u ( g α , β ) = ± ( α β - β ¯ α ¯ ) SU ( 2 ) . \pm\Pi_{u}(g_{\alpha,\beta})=\pm\left(\begin{matrix}\alpha&\beta\\ -\overline{\beta}&\overline{\alpha}\end{matrix}\right)\in\mathrm{SU}(2).
  66. cos θ 2 = | α | , sin θ 2 = | β | , ( 0 θ π ) , φ + ψ 2 = arg α , ψ - φ 2 = arg β . \begin{aligned}\displaystyle\cos\frac{\theta}{2}&\displaystyle=|\alpha|,\quad% \sin\frac{\theta}{2}=|\beta|,\quad(0\leq\theta\leq\pi),\\ \displaystyle\frac{\varphi+\psi}{2}&\displaystyle=\arg\alpha,\quad\frac{\psi-% \varphi}{2}=\arg\beta.\end{aligned}
  67. φ , θ , ψ φ,θ,ψ
  68. α , β α,β
  69. g α , β = ( 1 2 ( α 2 - β 2 + α 2 ¯ - β 2 ¯ ) i 2 ( - α 2 - β 2 + α 2 ¯ + β 2 ¯ ) - α β - α ¯ β ¯ i 2 ( α 2 - β 2 - α 2 ¯ + β 2 ¯ ) 1 2 ( α 2 + β 2 + α 2 ¯ + β 2 ¯ ) - i ( + α β - α ¯ β ¯ ) α β ¯ + α ¯ β i ( - α β ¯ + α ¯ β ) α α ¯ - β β ¯ ) . g_{\alpha,\beta}=\left(\begin{matrix}\frac{1}{2}(\alpha^{2}-\beta^{2}+% \overline{\alpha^{2}}-\overline{\beta^{2}})&\frac{i}{2}(-\alpha^{2}-\beta^{2}+% \overline{\alpha^{2}}+\overline{\beta^{2}})&-\alpha\beta-\overline{\alpha}% \overline{\beta}\\ \frac{i}{2}(\alpha^{2}-\beta^{2}-\overline{\alpha^{2}}+\overline{\beta^{2}})&% \frac{1}{2}(\alpha^{2}+\beta^{2}+\overline{\alpha^{2}}+\overline{\beta^{2}})&-% i(+\alpha\beta-\overline{\alpha}\overline{\beta})\\ \alpha\overline{\beta}+\overline{\alpha}\beta&i(-\alpha\overline{\beta}+% \overline{\alpha}\beta)&\alpha\overline{\alpha}-\beta\overline{\beta}\end{% matrix}\right).
  70. α . β α.β
  71. 2 : 1 2:1
  72. S O ( 3 ) SO(3)
  73. S U ( 2 ) SU(2)
  74. S U ( 2 ) SU(2)
  75. q = a 1 + b i + c j + d k = α + j β [ α - β ¯ β α ¯ ] = U , q , a , b , c , d , α , β , U SU ( 2 ) . q=a\mathrm{1}+b\mathrm{i}+c\mathrm{j}+d\mathrm{k}=\alpha+j\beta\leftrightarrow% \begin{bmatrix}\alpha&-\overline{\beta}\\ \beta&\overline{\alpha}\end{bmatrix}=U,\quad q\in\mathbb{H},\quad a,b,c,d\in% \mathbb{R},\quad\alpha,\beta\in\mathbb{C},\quad U\in\mathrm{SU}(2).
  76. 2 : 1 2:1
  77. S O ( 3 ) SO(3)
  78. q q
  79. q = w + i x + j y + k z , 1 = w 2 + x 2 + y 2 + z 2 , \begin{aligned}\displaystyle q&\displaystyle{}=w+{i}x+{j}y+{k}z,\\ \displaystyle 1&\displaystyle{}=w^{2}+x^{2}+y^{2}+z^{2},\end{aligned}
  80. Q = [ 1 - 2 y 2 - 2 z 2 2 x y - 2 z w 2 x z + 2 y w 2 x y + 2 z w 1 - 2 x 2 - 2 z 2 2 y z - 2 x w 2 x z - 2 y w 2 y z + 2 x w 1 - 2 x 2 - 2 y 2 ] . Q=\begin{bmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\ 2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\ 2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{bmatrix}.
  81. ( x , y , z ) (x,y,z)
  82. 2 θ
  83. c o s θ = w cosθ=w
  84. | s i n θ | = || ( x , y , z ) || |sinθ|=||(x,y,z)||
  85. s i n θ sinθ
  86. q q
  87. q −q
  88. Q Q
  89. S O ( 3 ) SO(3)
  90. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  91. 3 × 3 3×3
  92. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  93. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  94. d d ϕ | ϕ = 0 R ( ϕ , s y m b o l n ) s y m b o l x = s y m b o l n × s y m b o l x \left.{\operatorname{d}\over\operatorname{d}\phi}\right|_{\phi=0}R(\phi,symbol% {n})symbol{x}=symbol{n}\times symbol{x}
  95. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  96. s y m b o l ω 3 symbol{\omega}\in\mathbb{R}^{3}
  97. ω ~ {\tilde{\omega}}
  98. ω ~ ( s y m b o l x ) = s y m b o l ω \timessymbol x {\tilde{\omega}}(symbol{x})=symbol{\omega}\timessymbol{x}
  99. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  100. L x = [ 0 0 0 0 0 - 1 0 1 0 ] , L y = [ 0 0 1 0 0 0 - 1 0 0 ] , L z = [ 0 - 1 0 1 0 0 0 0 0 ] . L_{{x}}=\begin{bmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{bmatrix},\quad L_{{y}}=\begin{bmatrix}0&0&1\\ 0&0&0\\ -1&0&0\end{bmatrix},\quad L_{{z}}=\begin{bmatrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{bmatrix}.
  101. [ L x , L y ] = L z , [ L z , L x ] = L y , [ L y , L z ] = L x [L_{{x}},L_{{y}}]=L_{{z}},\quad[L_{{z}},L_{{x}}]=L_{{y}},\quad[L_{{y}},L_{{z}}% ]=L_{{x}}
  102. s y m b o l ω = ( x , y , z ) 3 , s y m b o l ω ~ = s y m b o l ω L = x L x + y L y + z L z = [ 0 - z y z 0 - x - y x 0 ] 𝔰 𝔬 ( 3 ) . \begin{aligned}\displaystyle symbol{\omega}&\displaystyle=(x,y,z)\in\mathbb{R}% ^{3},\\ \displaystyle symbol{\tilde{\omega}}&\displaystyle=symbol{\omega\cdot L}=xL_{{% x}}+yL_{{y}}+zL_{{z}}=\begin{bmatrix}0&-z&y\\ z&0&-x\\ -y&x&0\end{bmatrix}\in\mathfrak{so}(3).\end{aligned}
  103. [ u ~ , v ~ ] = u × v ~ . [\tilde{{u}},\tilde{{v}}]=\widetilde{{u}\!\times\!{v}}.
  104. 𝐮 \mathbf{u}
  105. u ~ v = u × v , \tilde{{u}}{v}={u}\times{v},
  106. 𝐮 \mathbf{u}
  107. 𝐮 × 𝐮 = 0 \mathbf{u}×\mathbf{u}=0
  108. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  109. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  110. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  111. t 1 = 1 2 [ 0 - i - i 0 ] , t 2 = 1 2 [ 0 - 1 1 0 ] , t 3 = 1 2 [ - i 0 0 i ] . t_{1}=\frac{1}{2}\begin{bmatrix}0&-i\\ -i&0\end{bmatrix},\quad t_{2}=\frac{1}{2}\begin{bmatrix}0&-1\\ 1&0\end{bmatrix},\quad t_{3}=\frac{1}{2}\begin{bmatrix}-i&0\\ 0&i\end{bmatrix}.
  112. i i
  113. i i
  114. i i
  115. i i
  116. [ t i , t j ] = ϵ i j k t k , [t_{i},t_{j}]=\epsilon_{ijk}t_{k},
  117. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  118. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  119. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  120. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  121. L x t 1 , L y t 2 , L z t 3 , L_{x}\leftrightarrow t_{1},\quad L_{y}\leftrightarrow t_{2},\quad L_{z}% \leftrightarrow t_{3},
  122. S O ( 3 ) SO(3)
  123. S O ( 3 ) SO(3)
  124. exp : 𝔰 𝔬 ( 3 ) S O ( 3 ) ; A e A = k = 0 1 k ! A k = I + A + 1 2 A 2 + + 1 k ! A k + . \exp\colon\mathfrak{so}(3)\to SO(3);A\mapsto e^{A}=\sum_{k=0}^{\infty}\frac{1}% {k!}A^{k}=I+A+\tfrac{1}{2}A^{2}+\cdots+\tfrac{1}{k!}A^{k}+\cdots.
  125. A 𝐬𝐨 ( 3 ) A∈\mathbf{so}(3)
  126. S O ( 3 ) SO(3)
  127. S O ( 3 ) SO(3)
  128. A 0 A≠0
  129. B e θ L z B - 1 = e B θ L z B - 1 , Be^{\theta L_{z}}B^{-1}=e^{B\theta L_{z}B^{-1}},
  130. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  131. S O ( 3 ) SO(3)
  132. e - π L x / 2 e θ L z e π L x / 2 = e θ L y e^{-\pi L_{x}/2}~{}e^{\theta L_{z}}~{}e^{\pi L_{x}/2}=e^{\theta L_{y}}
  133. A 𝐬𝐨 ( 3 ) A∈\mathbf{so}(3)
  134. ω = θ 𝐮 \mathbf{ω}=θ\mathbf{u}
  135. 𝐮 = ( x , y , z ) \mathbf{u}=(x,y,z)
  136. 𝐮 \mathbf{u}
  137. A A
  138. O O
  139. 𝐮 \mathbf{u}
  140. z z
  141. 𝐮 \mathbf{u}
  142. 𝐮 \mathbf{u}
  143. exp ( s y m b o l ω ~ ) = exp ( [ 0 - z θ y θ z θ 0 - x θ - y θ x θ 0 ] ) = s y m b o l I + 2 c s s y m b o l u L + 2 s 2 ( s y m b o l u L ) 2 = [ 2 ( x 2 - 1 ) s 2 + 1 2 x y s 2 - 2 z c s 2 x z s 2 + 2 y c s 2 x y s 2 + 2 z c s 2 ( y 2 - 1 ) s 2 + 1 2 y z s 2 - 2 x c s 2 x z s 2 - 2 y c s 2 y z s 2 + 2 x c s 2 ( z 2 - 1 ) s 2 + 1 ] , \begin{aligned}\displaystyle\exp(\tilde{symbol{\omega}})&\displaystyle{}=\exp% \left(\begin{bmatrix}0&-z\theta&y\theta\\ z\theta&0&-x\theta\\ -y\theta&x\theta&0\end{bmatrix}\right)\\ &\displaystyle{}=symbol{I}+2cs~{}symbol{u\cdot L}+2s^{2}~{}(symbol{u\cdot L})^% {2}\\ &\displaystyle{}=\begin{bmatrix}2(x^{2}-1)s^{2}+1&2xys^{2}-2zcs&2xzs^{2}+2ycs% \\ 2xys^{2}+2zcs&2(y^{2}-1)s^{2}+1&2yzs^{2}-2xcs\\ 2xzs^{2}-2ycs&2yzs^{2}+2xcs&2(z^{2}-1)s^{2}+1\end{bmatrix},\end{aligned}
  144. c = c o s θ / 2 , s = s i n θ / 2 c=cos{θ}/{2},s=sin{θ}/{2}
  145. 𝐮 \mathbf{u}
  146. θ θ
  147. R S O ( 3 ) R∈SO(3)
  148. A = R - R T 2 A=\frac{R-R^{\mathrm{T}}}{2}
  149. A A
  150. log R = sin - 1 | | A | | || A || A . \log R=\frac{\sin^{-1}||A||}{||A||}A.
  151. e X = I + sin θ θ X + 2 sin 2 θ 2 θ 2 X 2 , θ = || X || , e^{X}=I+\frac{\sin\theta}{\theta}X+2\frac{\sin^{2}\frac{\theta}{2}}{\theta^{2}% }X^{2},\quad\theta=||X||,
  152. X X
  153. Y Y
  154. e x p ( X ) exp(X)
  155. e x p ( Y ) exp(Y)
  156. Z Z
  157. e x p ( Z ) = e x p ( X ) e x p ( Y ) exp(Z)=exp(X)exp(Y)
  158. Z = C ( X , Y ) , Z=C(X,Y),
  159. C C
  160. X X
  161. Y Y
  162. e x p ( X ) exp(X)
  163. e x p ( Y ) exp(Y)
  164. Z = X + Y Z=X+Y
  165. Z = C ( X , Y ) = X + Y + 1 2 [ X , Y ] + 1 12 [ X , [ X , Y ] ] - 1 12 [ Y , [ X , Y ] ] + . Z=C(X,Y)=X+Y+\tfrac{1}{2}[X,Y]+\tfrac{1}{12}[X,[X,Y]]-\tfrac{1}{12}[Y,[X,Y]]+% \cdots~{}.
  166. S O ( 3 ) SO(3)
  167. Z = α X + β Y + γ [ X , Y ] , Z=\alpha X+\beta Y+\gamma[X,Y],
  168. ( α , β , γ ) (α,β,γ)
  169. ( α , β , γ ) (α,β,γ)
  170. α = ϕ cot ( ϕ / 2 ) γ , β = θ cot ( θ / 2 ) γ , γ = s i n - 1 d d c θ ϕ , \alpha=\phi\cot(\phi/2)~{}\gamma,\qquad\beta=\theta\cot(\theta/2)~{}\gamma,% \qquad\gamma=\frac{sin^{-1}d}{d}\frac{c}{\theta\phi}~{}~{},
  171. c = 1 2 sin θ sin ϕ - 2 sin 2 θ 2 sin 2 ϕ 2 cos ( ( u , v ) ) , a = c cot ( ϕ / 2 ) , b = c cot ( θ / 2 ) , d = a 2 + b 2 + 2 a b cos ( ( u , v ) ) + c 2 sin 2 ( ( u , v ) ) , \begin{aligned}\displaystyle c&\displaystyle=\frac{1}{2}\sin\theta\sin\phi-2% \sin^{2}\frac{\theta}{2}\sin^{2}\frac{\phi}{2}\cos(\angle(u,v)),\quad a=c\cot(% \phi/2),\quad b=c\cot(\theta/2),\\ \displaystyle d&\displaystyle=\sqrt{a^{2}+b^{2}+2ab\cos(\angle(u,v))+c^{2}\sin% ^{2}(\angle(u,v))}~{}~{},\end{aligned}
  172. θ = 1 2 || X || , ϕ = 1 2 || Y || , ( u , v ) = cos - 1 X , Y || X || || Y || . \theta=\frac{1}{\sqrt{2}}||X||~{},\quad\phi=\frac{1}{\sqrt{2}}||Y||~{},\quad% \angle(u,v)=\cos^{-1}\frac{\langle X,Y\rangle}{||X||||Y||}~{}.
  173. u , v = 1 2 Tr X T Y , \langle u,v\rangle=\frac{1}{2}\operatorname{Tr}X^{\mathrm{T}}Y,
  174. θ θ
  175. φ φ
  176. α X + β Y + γ [ X , Y ] = 𝔰 𝔬 ( 3 ) X + Y + 1 2 [ X , Y ] + 1 12 [ X , [ X , Y ] ] - 1 12 [ Y , [ X , Y ] ] + , \alpha X+\beta Y+\gamma[X,Y]~{}\underset{\mathfrak{so}(3)}{=}~{}X+Y+\tfrac{1}{% 2}[X,Y]+\tfrac{1}{12}[X,[X,Y]]-\tfrac{1}{12}[Y,[X,Y]]+\cdots,
  177. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  178. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  179. e i a ( u ^ σ ) e i b ( v ^ σ ) = exp ( c sin c sin a sin b ( ( i cot b u ^ + i cot a v ^ ) σ + 1 2 [ i u ^ σ , i v ^ σ ] ) ) , e^{ia^{\prime}(\hat{u}\cdot\vec{\sigma})}e^{ib^{\prime}(\hat{v}\cdot\vec{% \sigma})}=\exp\left(\frac{c^{\prime}}{\sin c^{\prime}}\sin a^{\prime}\sin b^{% \prime}~{}\left((i\cot b^{\prime}\hat{u}+i\cot a^{\prime}\hat{v})\cdot\vec{% \sigma}+\frac{1}{2}[i\hat{u}\cdot\vec{\sigma},i\hat{v}\cdot\vec{\sigma}]\right% )\right)~{},
  180. cos c = cos a cos b - u ^ v ^ sin a sin b , \cos c^{\prime}=\cos a^{\prime}\cos b^{\prime}-\hat{u}\cdot\hat{v}\sin a^{% \prime}\sin b^{\prime}~{},
  181. a , b b , c c a,bb,cc
  182. a , b , c a,b,c
  183. Z = α X + β Y + γ [ X , Y ] , Z=\alpha^{\prime}X+\beta^{\prime}Y+\gamma^{\prime}[X,Y],
  184. X = i a u ^ σ , Y = i b v ^ σ 𝔰 𝔲 ( 2 ) , X=ia^{\prime}\hat{u}\cdot\mathbf{\sigma},\quad Y=ib^{\prime}\hat{v}\cdot% \mathbf{\sigma}~{}\in\mathfrak{su}(2),
  185. α = c sin c sin a a cos b β = c sin c sin b b cos a γ = 1 2 c sin c sin a a sin b b . \begin{aligned}\displaystyle\alpha^{\prime}&\displaystyle=\frac{c^{\prime}}{% \sin c^{\prime}}\frac{\sin a^{\prime}}{a^{\prime}}\cos b^{\prime}\\ \displaystyle\beta^{\prime}&\displaystyle=\frac{c^{\prime}}{\sin c^{\prime}}% \frac{\sin b^{\prime}}{b^{\prime}}\cos a^{\prime}\\ \displaystyle\gamma^{\prime}&\displaystyle=\frac{1}{2}\frac{c^{\prime}}{\sin c% ^{\prime}}\frac{\sin a^{\prime}}{a^{\prime}}\frac{\sin b^{\prime}}{b^{\prime}}% ~{}.\end{aligned}
  186. t t
  187. σ 2 i 𝐭 \mathbf{σ}→2i\mathbf{t}
  188. a - θ 2 , b - ϕ 2 . a^{\prime}\mapsto-\frac{\theta}{2},\quad b^{\prime}\mapsto-\frac{\phi}{2}.
  189. α γ = θ cot θ 2 = α γ β γ = ϕ cot ϕ 2 = β γ . \begin{aligned}\displaystyle\frac{\alpha^{\prime}}{\gamma^{\prime}}&% \displaystyle={\theta}\cot\frac{\theta}{2}&\displaystyle=\frac{\alpha}{\gamma}% \\ \displaystyle\frac{\beta^{\prime}}{\gamma^{\prime}}&\displaystyle=\phi\cot% \frac{\phi}{2}&\displaystyle=\frac{\beta}{\gamma}~{}.\end{aligned}
  190. γ = γ γ=γ
  191. $\mathbf{ }$
  192. n × n n×n
  193. I + A d θ , I+A\,d\theta~{},
  194. d θ
  195. A 𝐬𝐨 ( 3 ) A∈\mathbf{so}(3)
  196. d A x = [ 1 0 0 0 1 - d θ 0 d θ 1 ] . dA_{{x}}=\begin{bmatrix}1&0&0\\ 0&1&-d\theta\\ 0&d\theta&1\end{bmatrix}~{}.
  197. d A x T d A x = [ 1 0 0 0 1 + d θ 2 0 0 0 1 + d θ 2 ] , dA_{{x}}^{T}\,dA_{{x}}=\begin{bmatrix}1&0&0\\ 0&1+d\theta^{2}&0\\ 0&0&1+d\theta^{2}\end{bmatrix},
  198. d A x 2 = [ 1 0 0 0 1 - d θ 2 - 2 d θ 0 2 d θ 1 - d θ 2 ] . dA_{{x}}^{2}=\begin{bmatrix}1&0&0\\ 0&1-d\theta^{2}&-2d\theta\\ 0&2d\theta&1-d\theta^{2}\end{bmatrix}~{}.
  199. d A y = [ 1 0 d ϕ 0 1 0 - d ϕ 0 1 ] . dA_{{y}}=\begin{bmatrix}1&0&d\phi\\ 0&1&0\\ -d\phi&0&1\end{bmatrix}.
  200. d A x d A y = [ 1 0 d ϕ d θ d ϕ 1 - d θ - d ϕ d θ 1 ] d A y d A x = [ 1 d θ d ϕ d ϕ 0 1 - d θ - d ϕ d θ 1 ] . \begin{aligned}\displaystyle dA_{{x}}\,dA_{{y}}&\displaystyle{}=\begin{bmatrix% }1&0&d\phi\\ d\theta\,d\phi&1&-d\theta\\ -d\phi&d\theta&1\end{bmatrix}\\ \displaystyle dA_{{y}}\,dA_{{x}}&\displaystyle{}=\begin{bmatrix}1&d\theta\,d% \phi&d\phi\\ 0&1&-d\theta\\ -d\phi&d\theta&1\end{bmatrix}.\\ \end{aligned}
  201. d θ d φ dθdφ
  202. d A x d A y = d A y d A x , dA_{{x}}\,dA_{{y}}=dA_{{y}}\,dA_{{x}},\,\!
  203. J x , J y , J z J_{x},\,J_{y},\,J_{z}
  204. [ J x , J y ] = J z , [ J z , J x ] = J y , [ J y , J z ] = J x . [J_{{x}},J_{{y}}]=J_{{z}},\quad[J_{{z}},J_{{x}}]=J_{{y}},\quad[J_{{y}},J_{{z}}% ]=J_{{x}}.
  205. J 2 s y m b o l J J = J x 2 + J y 2 + J z 2 I . J^{2}\equiv symbol{J\cdot J}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\propto I~{}.
  206. j j
  207. J 2 = - j ( j + 1 ) I 2 j + 1 , J^{2}=-j(j+1)~{}I_{2j+1}~{},
  208. j j
  209. j j
  210. j j
  211. i i
  212. [ J x , J y ] = i J z , [ J z , J x ] = i J y , [ J y , J z ] = i J x . [J_{{x}},J_{{y}}]=iJ_{{z}},\quad[J_{{z}},J_{{x}}]=iJ_{{y}},\quad[J_{{y}},J_{{z% }}]=iJ_{{x}}.
  213. J 2 = j ( j + 1 ) I 2 j + 1 . J^{2}=j(j+1)~{}I_{2j+1}~{}.
  214. ( J z ( j ) ) b a = ( j + 1 - a ) δ a b , a ( J x ( j ) ) b a = 1 2 ( δ b , a + 1 + δ b + 1 , a ) ( j + 1 ) ( a + b - 1 ) - a b ( J y ( j ) ) b a = 1 2 i ( δ b , a + 1 - δ b + 1 , a ) ( j + 1 ) ( a + b - 1 ) - a b 1 a , b 2 j + 1 , \begin{aligned}\displaystyle\left(J_{z}^{(j)}\right)_{ba}&\displaystyle=(j+1-a% )~{}\delta_{ab,a}\\ \displaystyle\left(J_{x}^{(j)}\right)_{ba}&\displaystyle=\frac{1}{2}(\delta_{b% ,a+1}+\delta_{b+1,a})\sqrt{(j+1)(a+b-1)-ab}\\ \displaystyle\left(J_{y}^{(j)}\right)_{ba}&\displaystyle=\frac{1}{2i}(\delta_{% b,a+1}-\delta_{b+1,a})\sqrt{(j+1)(a+b-1)-ab}\\ &\displaystyle 1\leq a,b\leq 2j+1~{},\end{aligned}
  215. j j
  216. 3 2 \frac{3}{2}
  217. 5 2 \frac{5}{2}
  218. j = 1 j=1
  219. J x = 1 2 ( 0 1 0 1 0 1 0 1 0 ) J y = 1 2 ( 0 - i 0 i 0 - i 0 i 0 ) J z = ( 1 0 0 0 0 0 0 0 - 1 ) \begin{aligned}\displaystyle J_{x}&\displaystyle=\frac{1}{\sqrt{2}}\begin{% pmatrix}0&1&0\\ 1&0&1\\ 0&1&0\end{pmatrix}\\ \displaystyle J_{y}&\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i&0\\ i&0&-i\\ 0&i&0\end{pmatrix}\\ \displaystyle J_{z}&\displaystyle=\begin{pmatrix}1&0&0\\ 0&0&0\\ 0&0&-1\end{pmatrix}\end{aligned}
  220. i 𝐋 i\mathbf{L}
  221. j = 3 2 j=\textstyle\frac{3}{2}
  222. J x \displaystyle J_{x}
  223. j = 5 2 j=\textstyle\frac{5}{2}
  224. J x \displaystyle J_{x}
  225. S O ( 3 ) SO(3)
  226. f , g = 𝕊 2 f ¯ g d Ω = 0 2 π 0 π f ¯ g sin θ d θ d φ . \langle f,g\rangle=\int_{\mathbb{S}^{2}}\overline{f}gd\Omega=\int_{0}^{2\pi}% \int_{0}^{\pi}\overline{f}g\sin\theta d\theta d\varphi.
  227. f f
  228. | f = l = 1 m = - l m = l | Y m l Y m l | f , f ( θ , φ ) = l = 1 m = - l m = l f l m Y m l ( θ , φ ) , |f\rangle=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}|Y_{m}^{l}\rangle\langle Y_{m}^{% l}|f\rangle,\quad f(\theta,\varphi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}f_{lm}% Y^{l}_{m}(\theta,\varphi),
  229. f l m = Y m l , f = 𝕊 2 < ¯ m t p l > Y m l f_{lm}=\langle Y_{m}^{l},f\rangle=\int_{\mathbb{S}^{2}}\overline{<}mtpl>{{Y^{l% }_{m}}}
  230. S O ( 3 ) SO(3)
  231. ( Π ( R ) f ) ( θ ( x ) , φ ( x ) ) = l = 1 m = - l m = l m = - l m = l D m m ( l ) ( R ) f l m Y m l ( θ ( R - 1 x ) , φ ( R - 1 x ) ) , R SO ( 3 ) , x 𝕊 2 . (\Pi(R)f)(\theta(x),\varphi(x))=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}\sum_{m^{% \prime}=-l}^{m^{\prime}=l}D^{(l)}_{mm^{\prime}}(R)f_{lm^{\prime}}Y^{l}_{m}(% \theta(R^{-1}x),\varphi(R^{-1}x)),\qquad R\in\mathrm{SO}(3),\quad x\in\mathbb{% S}^{2}.
  232. Π ( R ) f , Π ( R ) g = f , g f , g 𝕊 2 , R SO ( 3 ) . \langle\Pi(R)f,\Pi(R)g\rangle=\langle f,g\rangle\qquad\forall f,g\in\mathbb{S}% ^{2},\quad\forall R\in\mathrm{SO}(3).
  233. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  234. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  235. S O ( 3 ) SO(3)
  236. Π Π
  237. ( Π , V ) (Π,V)
  238. f , g U SO ( 3 ) Π ( R ) f , Π ( R ) g d g = 1 8 π 2 0 2 π 0 π 0 2 π Π ( R ) f , Π ( R ) g sin θ d φ d θ d ψ , f , g V , \langle f,g\rangle_{U}\equiv\int_{\mathrm{SO}(3)}\langle\Pi(R)f,\Pi(R)g\rangle dg% =\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\langle\Pi(R)f,% \Pi(R)g\rangle\sin\theta d\varphi d\theta d\psi,\quad f,g\in V,
  239. S O ( 3 ) SO(3)
  240. 1 1
  241. V V
  242. g < s u b > φ g<sub>φ
  243. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  244. || X || + || Y || < l o g 2 ||X||+||Y||<log2
  245. L < s u p > 2 ( 𝐒 2 ) L<sup>2(\mathbf{S}^{2})
  246. D < s u p > ( [ u e l l ] ) D<sup>([u^{\prime}ell^{\prime}])

Rounding.html

  1. z = round ( x , m ) = round ( x / m ) m z=\mathrm{round}(x,m)=\mathrm{round}(x/m)\cdot m\,
  2. q = floor ( y ) = y = - - y q=\mathrm{floor}(y)=\left\lfloor y\right\rfloor=-\left\lceil-y\right\rceil\,
  3. q = ceil ( y ) = y = - - y q=\mathrm{ceil}(y)=\left\lceil y\right\rceil=-\left\lfloor-y\right\rfloor\,
  4. q = truncate ( y ) = sgn ( y ) | y | = - sgn ( y ) - | y | q=\mathrm{truncate}(y)=\operatorname{sgn}(y)\left\lfloor\left|y\right|\right% \rfloor=-\operatorname{sgn}(y)\left\lceil-\left|y\right|\right\rceil\,
  5. q = sgn ( y ) | y | = - sgn ( y ) - | y | q=\operatorname{sgn}(y)\left\lceil\left|y\right|\right\rceil=-\operatorname{% sgn}(y)\left\lfloor-\left|y\right|\right\rfloor\,
  6. q = y + 0.5 = - - y - 0.5 q=\left\lfloor y+0.5\right\rfloor=-\left\lceil-y-0.5\right\rceil
  7. q = y - 0.5 = - - y + 0.5 q=\left\lceil y-0.5\right\rceil=-\left\lfloor-y+0.5\right\rfloor\,
  8. q = sgn ( y ) | y | - 0.5 = - sgn ( y ) - | y | + 0.5 q=\operatorname{sgn}(y)\left\lceil\left|y\right|-0.5\right\rceil=-% \operatorname{sgn}(y)\left\lfloor-\left|y\right|+0.5\right\rfloor\,
  9. q = sgn ( y ) | y | + 0.5 = - sgn ( y ) - | y | - 0.5 q=\operatorname{sgn}(y)\left\lfloor\left|y\right|+0.5\right\rfloor=-% \operatorname{sgn}(y)\left\lceil-\left|y\right|-0.5\right\rceil\,

Route_inspection_problem.html

  1. 1 2 \tfrac{1}{2}

Row_and_column_spaces.html

  1. J = [ 2 4 1 3 2 - 1 - 2 1 0 5 1 6 2 2 2 3 6 2 5 1 ] J=\begin{bmatrix}2&4&1&3&2\\ -1&-2&1&0&5\\ 1&6&2&2&2\\ 3&6&2&5&1\end{bmatrix}

Rubik's_Revenge.html

  1. 8 ! × 3 7 × 24 ! 2 4 ! 6 × 24 7.40 × 10 45 . \frac{8!\times 3^{7}\times 24!^{2}}{4!^{6}\times 24}\approx 7.40\times 10^{45}.

Rule_of_inference.html

  1. A B A\to B
  2. A ¯ \underline{A\quad\quad\quad}\,\!
  3. B B\!
  4. \vdash
  5. p \vdash p
  6. Provable ( p ) \vdash\,\text{Provable}(p)
  7. p p
  8. p p
  9. p Provable ( p ) p\to\,\text{Provable}(p)
  10. n 𝗇𝖺𝗍 n\,\,\mathsf{nat}
  11. n n
  12. 𝟎 𝗇𝖺𝗍 n 𝗇𝖺𝗍 𝐬 ( n ) 𝗇𝖺𝗍 \begin{matrix}\frac{}{\mathbf{0}\,\,\mathsf{nat}}&\frac{n\,\,\mathsf{nat}}{% \mathbf{s(}n\mathbf{)}\,\,\mathsf{nat}}\\ \end{matrix}
  13. n 𝗇𝖺𝗍 𝐬 ( 𝐬 ( n ) ) 𝗇𝖺𝗍 \frac{n\,\,\mathsf{nat}}{\mathbf{s(s(}n\mathbf{))}\,\,\mathsf{nat}}
  14. 𝐬 ( n ) 𝗇𝖺𝗍 n 𝗇𝖺𝗍 \frac{\mathbf{s(}n\mathbf{)}\,\,\mathsf{nat}}{n\,\,\mathsf{nat}}
  15. n 𝗇𝖺𝗍 n\,\,\mathsf{nat}
  16. 𝐬 ( - 𝟑 ) 𝗇𝖺𝗍 \frac{}{\mathbf{s(-3)}\,\,\mathsf{nat}}
  17. - 𝟑 𝗇𝖺𝗍 \mathbf{-3}\,\,\mathsf{nat}

Runge's_phenomenon.html

  1. lim n ( max a x b | f ( x ) - P n ( x ) | ) = 0. \lim_{n\rightarrow\infty}\left(\max_{a\leq x\leq b}\left|f(x)-P_{n}(x)\right|% \right)=0.
  2. f ( x ) = 1 1 + 25 x 2 . f(x)=\frac{1}{1+25x^{2}}.\,
  3. x i = 2 i n - 1 , i { 0 , 1 , , n } x_{i}=\frac{2i}{n}-1,\quad i\in\left\{0,1,\dots,n\right\}
  4. lim n ( max - 1 x 1 | f ( x ) - P n ( x ) | ) = + . \lim_{n\rightarrow\infty}\left(\max_{-1\leq x\leq 1}|f(x)-P_{n}(x)|\right)=+\infty.
  5. f ( x ) - P n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! i = 1 n + 1 ( x - x i ) f(x)-P_{n}(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}\prod_{i=1}^{n+1}(x-x_{i})
  6. ξ \xi
  7. max - 1 x 1 | f ( x ) - P n ( x ) | max - 1 x 1 | f ( n + 1 ) ( x ) | ( n + 1 ) ! max - 1 x 1 i = 0 n | x - x i | . \max_{-1\leq x\leq 1}|f(x)-P_{n}(x)|\leq\max_{-1\leq x\leq 1}\frac{\left|f^{(n% +1)}(x)\right|}{(n+1)!}\max_{-1\leq x\leq 1}\prod_{i=0}^{n}|x-x_{i}|.
  8. w n ( x ) w_{n}(x)
  9. w n ( x ) = ( x - x 0 ) ( x - x 1 ) ( x - x n ) . w_{n}(x)=(x-x_{0})(x-x_{1})\cdots(x-x_{n}).
  10. W n W_{n}
  11. w n w_{n}
  12. W n = max - 1 x 1 w n ( x ) . W_{n}=\max_{-1\leq x\leq 1}w_{n}(x).
  13. W n h n ( n - 1 ) ! 4 W_{n}\leq h^{n}\frac{(n-1)!}{4}
  14. h = 2 / ( n - 1 ) h=2/(n-1)
  15. f f
  16. max - 1 x 1 f ( n ) ( x ) M n \max_{-1\leq x\leq 1}f^{(n)}(x)\leq M_{n}
  17. max - 1 x 1 | f ( x ) - P n ( x ) | M n h n 4 n \max_{-1\leq x\leq 1}|f(x)-P_{n}(x)|\leq M_{n}\frac{h^{n}}{4n}
  18. 1 / 1 - x 2 1/\sqrt{1-x^{2}}
  19. n 2 n^{2}
  20. n + 1 n+1
  21. L 2 L^{2}
  22. N < 2 m N<2\sqrt{m}
  23. P N ( x ) P_{N}(x)

Runge–Kutta_methods.html

  1. y ˙ = f ( t , y ) , y ( t 0 ) = y 0 . \dot{y}=f(t,y),\quad y(t_{0})=y_{0}.
  2. y ˙ \dot{y}
  3. t 0 t_{0}
  4. y 0 y_{0}
  5. t 0 t_{0}
  6. y 0 y_{0}
  7. y n + 1 = y n + h 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) t n + 1 = t n + h \begin{aligned}\displaystyle y_{n+1}&\displaystyle=y_{n}+\tfrac{h}{6}\left(k_{% 1}+2k_{2}+2k_{3}+k_{4}\right)\\ \displaystyle t_{n+1}&\displaystyle=t_{n}+h\\ \end{aligned}
  8. k 1 = f ( t n , y n ) , k 2 = f ( t n + h 2 , y n + h 2 k 1 ) , k 3 = f ( t n + h 2 , y n + h 2 k 2 ) , k 4 = f ( t n + h , y n + h k 3 ) . \begin{aligned}\displaystyle k_{1}&\displaystyle=f(t_{n},y_{n}),\\ \displaystyle k_{2}&\displaystyle=f(t_{n}+\tfrac{h}{2},y_{n}+\tfrac{h}{2}k_{1}% ),\\ \displaystyle k_{3}&\displaystyle=f(t_{n}+\tfrac{h}{2},y_{n}+\tfrac{h}{2}k_{2}% ),\\ \displaystyle k_{4}&\displaystyle=f(t_{n}+h,y_{n}+hk_{3}).\end{aligned}
  9. y n + 1 y_{n+1}
  10. y ( t n + 1 ) y(t_{n+1})
  11. y n + 1 y_{n+1}
  12. y n y_{n}
  13. k 1 k_{1}
  14. y {y}
  15. k 2 k_{2}
  16. y + h 2 k 1 {y}+\tfrac{h}{2}k_{1}
  17. k 3 k_{3}
  18. y + h 2 k 2 {y}+\tfrac{h}{2}k_{2}
  19. k 4 k_{4}
  20. y + h k 3 {y}+hk_{3}
  21. f f
  22. y y
  23. O ( h 5 ) O(h^{5})
  24. O ( h 4 ) O(h^{4})
  25. y n + 1 = y n + h i = 1 s b i k i , y_{n+1}=y_{n}+h\sum_{i=1}^{s}b_{i}k_{i},
  26. k 1 = f ( t n , y n ) , k_{1}=f(t_{n},y_{n}),\,
  27. k 2 = f ( t n + c 2 h , y n + h ( a 21 k 1 ) ) , k_{2}=f(t_{n}+c_{2}h,y_{n}+h(a_{21}k_{1})),\,
  28. k 3 = f ( t n + c 3 h , y n + h ( a 31 k 1 + a 32 k 2 ) ) , k_{3}=f(t_{n}+c_{3}h,y_{n}+h(a_{31}k_{1}+a_{32}k_{2})),\,
  29. \vdots
  30. k s = f ( t n + c s h , y n + h ( a s 1 k 1 + a s 2 k 2 + + a s , s - 1 k s - 1 ) ) . k_{s}=f(t_{n}+c_{s}h,y_{n}+h(a_{s1}k_{1}+a_{s2}k_{2}+\cdots+a_{s,s-1}k_{s-1})).
  31. c 2 c_{2}
  32. a 21 a_{21}
  33. c 3 c_{3}
  34. a 31 a_{31}
  35. a 32 a_{32}
  36. \vdots
  37. \vdots
  38. \ddots
  39. c s c_{s}
  40. a s 1 a_{s1}
  41. a s 2 a_{s2}
  42. \cdots
  43. a s , s - 1 a_{s,s-1}
  44. b 1 b_{1}
  45. b 2 b_{2}
  46. \cdots
  47. b s - 1 b_{s-1}
  48. b s b_{s}
  49. j = 1 i - 1 a i j = c i for i = 2 , , s . \sum_{j=1}^{i-1}a_{ij}=c_{i}\ \mathrm{for}\ i=2,\ldots,s.
  50. s s
  51. p p
  52. s p s\geq p
  53. p 5 p\geq 5
  54. s > p s>p
  55. s s
  56. s s
  57. p p
  58. p 1 2 3 4 5 6 7 8 min s 1 2 3 4 6 7 9 11 \begin{array}[]{c|cccccccc}p&1&2&3&4&5&6&7&8\\ \hline\min s&1&2&3&4&6&7&9&11\end{array}
  59. y n + 1 = y n + h f ( t n , y n ) y_{n+1}=y_{n}+hf(t_{n},y_{n})
  60. y n + 1 = y n + h f ( t n + 1 2 h , y n + 1 2 h f ( t n , y n ) ) . y_{n+1}=y_{n}+hf\left(t_{n}+\frac{1}{2}h,y_{n}+\frac{1}{2}hf(t_{n},y_{n})% \right).
  61. y n + 1 = y n + h ( ( 1 - 1 2 α ) f ( t n , y n ) + 1 2 α f ( t n + α h , y n + α h f ( t n , y n ) ) ) . y_{n+1}=y_{n}+h\bigl((1-\tfrac{1}{2\alpha})f(t_{n},y_{n})+\tfrac{1}{2\alpha}f(% t_{n}+\alpha h,y_{n}+\alpha hf(t_{n},y_{n}))\bigr).
  62. α \alpha
  63. α \alpha
  64. ( 1 - 1 2 α ) (1-\tfrac{1}{2\alpha})
  65. 1 2 α \tfrac{1}{2\alpha}
  66. α = 1 2 \alpha=\tfrac{1}{2}
  67. α = 1 \alpha=1
  68. k 1 = f ( t n , y n ) , k 2 = f ( t n + 2 3 h , y n + 2 3 h k 1 ) , y n + 1 = y n + h ( 1 4 k 1 + 3 4 k 2 ) . \begin{aligned}\displaystyle k_{1}&\displaystyle=f(t_{n},y_{n}),\\ \displaystyle k_{2}&\displaystyle=f(t_{n}+\tfrac{2}{3}h,y_{n}+\tfrac{2}{3}hk_{% 1}),\\ \displaystyle y_{n+1}&\displaystyle=y_{n}+h\left(\tfrac{1}{4}k_{1}+\tfrac{3}{4% }k_{2}\right).\end{aligned}
  69. y = tan ( y ) + 1 , y 0 = 1 , t [ 1 , 1.1 ] y^{\prime}=\tan(y)+1,\quad y_{0}=1,\ t\in[1,1.1]
  70. t 0 = 1 : t_{0}=1\colon
  71. y 0 = 1 y_{0}=1
  72. t 1 = 1.025 : t_{1}=1.025\colon
  73. y 0 = 1 y_{0}=1
  74. k 1 = 2.557407725 k_{1}=2.557407725
  75. k 2 = f ( t 0 + 2 3 h , y 0 + 2 3 h k 1 ) = 2.7138981184 k_{2}=f(t_{0}+\tfrac{2}{3}h,y_{0}+\tfrac{2}{3}hk_{1})=2.7138981184
  76. y 1 = y 0 + h ( 1 4 k 1 + 3 4 k 2 ) = 1.066869388 ¯ y_{1}=y_{0}+h(\tfrac{1}{4}k_{1}+\tfrac{3}{4}k_{2})=\underline{1.066869388}
  77. t 2 = 1.05 : t_{2}=1.05\colon
  78. y 1 = 1.066869388 y_{1}=1.066869388
  79. k 1 = 2.813524695 k_{1}=2.813524695
  80. k 2 = f ( t 1 + 2 3 h , y 1 + 2 3 h k 1 ) k_{2}=f(t_{1}+\tfrac{2}{3}h,y_{1}+\tfrac{2}{3}hk_{1})
  81. y 2 = y 1 + h ( 1 4 k 1 + 3 4 k 2 ) = 1.141332181 ¯ y_{2}=y_{1}+h(\tfrac{1}{4}k_{1}+\tfrac{3}{4}k_{2})=\underline{1.141332181}
  82. t 3 = 1.075 : t_{3}=1.075\colon
  83. y 2 = 1.141332181 y_{2}=1.141332181
  84. k 1 = 3.183536647 k_{1}=3.183536647
  85. k 2 = f ( t 2 + 2 3 h , y 2 + 2 3 h k 1 ) k_{2}=f(t_{2}+\tfrac{2}{3}h,y_{2}+\tfrac{2}{3}hk_{1})
  86. y 3 = y 2 + h ( 1 4 k 1 + 3 4 k 2 ) = 1.227417567 ¯ y_{3}=y_{2}+h(\tfrac{1}{4}k_{1}+\tfrac{3}{4}k_{2})=\underline{1.227417567}
  87. t 4 = 1.1 : t_{4}=1.1\colon
  88. y 3 = 1.227417567 y_{3}=1.227417567
  89. k 1 = 3.796866512 k_{1}=3.796866512
  90. k 2 = f ( t 3 + 2 3 h , y 3 + 2 3 h k 1 ) k_{2}=f(t_{3}+\tfrac{2}{3}h,y_{3}+\tfrac{2}{3}hk_{1})
  91. y 4 = y 3 + h ( 1 4 k 1 + 3 4 k 2 ) = 1.335079087 ¯ . y_{4}=y_{3}+h(\tfrac{1}{4}k_{1}+\tfrac{3}{4}k_{2})=\underline{1.335079087}.
  92. p p
  93. p - 1 p-1
  94. y n + 1 * = y n + h i = 1 s b i * k i , y^{*}_{n+1}=y_{n}+h\sum_{i=1}^{s}b^{*}_{i}k_{i},
  95. k i k_{i}
  96. e n + 1 = y n + 1 - y n + 1 * = h i = 1 s ( b i - b i * ) k i , e_{n+1}=y_{n+1}-y^{*}_{n+1}=h\sum_{i=1}^{s}(b_{i}-b^{*}_{i})k_{i},
  97. O ( h p ) O(h^{p})
  98. b i * b^{*}_{i}
  99. c 2 c_{2}
  100. a 21 a_{21}
  101. c 3 c_{3}
  102. a 31 a_{31}
  103. a 32 a_{32}
  104. \vdots
  105. \vdots
  106. \ddots
  107. c s c_{s}
  108. a s 1 a_{s1}
  109. a s 2 a_{s2}
  110. \cdots
  111. a s , s - 1 a_{s,s-1}
  112. b 1 b_{1}
  113. b 2 b_{2}
  114. \cdots
  115. b s - 1 b_{s-1}
  116. b s b_{s}
  117. b 1 * b^{*}_{1}
  118. b 2 * b^{*}_{2}
  119. \cdots
  120. b s - 1 * b^{*}_{s-1}
  121. b s * b^{*}_{s}
  122. c i , i = 1 , 2 , , s c_{i},\,i=1,2,\ldots,s
  123. y n + 1 = y n + h i = 1 s b i k i , y_{n+1}=y_{n}+h\sum_{i=1}^{s}b_{i}k_{i},
  124. k i = f ( t n + c i h , y n + h j = 1 s a i j k j ) , i = 1 , , s . k_{i}=f\left(t_{n}+c_{i}h,y_{n}+h\sum_{j=1}^{s}a_{ij}k_{j}\right),\quad i=1,% \ldots,s.
  125. a i j a_{ij}
  126. c 1 a 11 a 12 a 1 s c 2 a 21 a 22 a 2 s c s a s 1 a s 2 a s s b 1 b 2 b s = 𝐜 A 𝐛 𝐓 \begin{array}[]{c|cccc}c_{1}&a_{11}&a_{12}&\dots&a_{1s}\\ c_{2}&a_{21}&a_{22}&\dots&a_{2s}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ c_{s}&a_{s1}&a_{s2}&\dots&a_{ss}\\ \hline&b_{1}&b_{2}&\dots&b_{s}\\ \end{array}=\begin{array}[]{c|c}\mathbf{c}&A\\ \hline&\mathbf{b^{T}}\\ \end{array}
  127. y n + 1 = y n + h f ( t n + h , y n + 1 ) . y_{n+1}=y_{n}+hf(t_{n}+h,y_{n+1}).\,
  128. 1 1 1 \begin{array}[]{c|c}1&1\\ \hline&1\\ \end{array}
  129. k 1 = f ( t n + h , y n + h k 1 ) and y n + 1 = y n + h k 1 , k_{1}=f(t_{n}+h,y_{n}+hk_{1})\quad\,\text{and}\quad y_{n+1}=y_{n}+hk_{1},
  130. 0 0 0 1 1 2 1 2 1 2 1 2 \begin{array}[]{c|cc}0&0&0\\ 1&\frac{1}{2}&\frac{1}{2}\\ \hline&\frac{1}{2}&\frac{1}{2}\\ \end{array}
  131. 1 2 - 1 6 3 1 4 1 4 - 1 6 3 1 2 + 1 6 3 1 4 + 1 6 3 1 4 1 2 1 2 \begin{array}[]{c|cc}\frac{1}{2}-\frac{1}{6}\sqrt{3}&\frac{1}{4}&\frac{1}{4}-% \frac{1}{6}\sqrt{3}\\ \frac{1}{2}+\frac{1}{6}\sqrt{3}&\frac{1}{4}+\frac{1}{6}\sqrt{3}&\frac{1}{4}\\ \hline&\frac{1}{2}&\frac{1}{2}\end{array}
  132. y n + 1 = r ( h λ ) y n y_{n+1}=r(h\lambda)\,y_{n}
  133. r ( z ) = 1 + z b T ( I - z A ) - 1 e = det ( I - z A + z e b T ) det ( I - z A ) , r(z)=1+zb^{T}(I-zA)^{-1}e=\frac{\det(I-zA+zeb^{T})}{\det(I-zA)},
  134. r ( z ) = e z + O ( z p + 1 ) r(z)=\textrm{e}^{z}+O(z^{p+1})
  135. z 0 z\to 0
  136. y = λ y y^{\prime}=\lambda y
  137. y = f ( y ) y^{\prime}=f(y)
  138. f ( y ) - f ( z ) , y - z < 0 \langle f(y)-f(z),y-z\rangle<0
  139. y n + 1 - z n + 1 y n - z n \|y_{n+1}-z_{n+1}\|\leq\|y_{n}-z_{n}\|
  140. B B
  141. M M
  142. Q Q
  143. s × s s\times s
  144. B = diag ( b 1 , b 2 , , b s ) , M = B A + A T B - b b T , Q = B A - 1 + A - T B - A - T b b T A - 1 . B=\operatorname{diag}(b_{1},b_{2},\ldots,b_{s}),\,M=BA+A^{T}B-bb^{T},\,Q=BA^{-% 1}+A^{-T}B-A^{-T}bb^{T}A^{-1}.
  145. B B
  146. M M
  147. B B
  148. Q Q
  149. s s
  150. y t + h = y t + h i = 1 s a i k i + 𝒪 ( h s + 1 ) , y_{t+h}=y_{t}+h\cdot\sum_{i=1}^{s}a_{i}k_{i}+\mathcal{O}(h^{s+1}),
  151. k i = f ( y t + h j = 1 s β i j k j , t n + α i h ) k_{i}=f\left(y_{t}+h\cdot\sum_{j=1}^{s}\beta_{ij}k_{j},t_{n}+\alpha_{i}h\right)
  152. y t y_{t}
  153. i i
  154. s = 4 s=4
  155. ( t , t + h ) (t,t+h)
  156. α i β i j α 1 = 0 β 21 = 1 2 α 2 = 1 2 β 32 = 1 2 α 3 = 1 2 β 43 = 1 α 4 = 1 \begin{array}[]{|l|l|}\hline\alpha_{i}&\beta_{ij}\\ \hline\alpha_{1}=0&\beta_{21}=\frac{1}{2}\\ \alpha_{2}=\frac{1}{2}&\beta_{32}=\frac{1}{2}\\ \alpha_{3}=\frac{1}{2}&\beta_{43}=1\\ \alpha_{4}=1&\\ \hline\end{array}
  157. β i j = 0 \beta_{ij}=0
  158. y t + h 1 \displaystyle y^{1}_{t+h}
  159. y t + h / 2 1 = y t + y t + h 1 2 y^{1}_{t+h/2}=\dfrac{y_{t}+y^{1}_{t+h}}{2}
  160. y t + h / 2 2 = y t + y t + h 2 2 y^{2}_{t+h/2}=\dfrac{y_{t}+y^{2}_{t+h}}{2}
  161. k 1 \displaystyle k_{1}
  162. 𝒪 ( h 2 ) \mathcal{O}(h^{2})
  163. k 2 \displaystyle k_{2}
  164. d d t f ( y t , t ) = y f ( y t , t ) y ˙ t + t f ( y t , t ) = f y ( y t , t ) y ˙ + f t ( y t , t ) := y ¨ t \frac{d}{dt}f(y_{t},t)=\frac{\partial}{\partial y}f(y_{t},t)\dot{y}_{t}+\frac{% \partial}{\partial t}f(y_{t},t)=f_{y}(y_{t},t)\dot{y}+f_{t}(y_{t},t):=\ddot{y}% _{t}
  165. f f
  166. y t + h \displaystyle y_{t+h}
  167. y t + h y_{t+h}
  168. y t y_{t}
  169. y t + h \displaystyle y_{t+h}
  170. { a + b + c + d = 1 1 2 b + 1 2 c + d = 1 2 1 4 c + 1 2 d = 1 6 1 4 d = 1 24 \begin{cases}&a+b+c+d=1\\ &\frac{1}{2}b+\frac{1}{2}c+d=\frac{1}{2}\\ &\frac{1}{4}c+\frac{1}{2}d=\frac{1}{6}\\ &\frac{1}{4}d=\frac{1}{24}\end{cases}
  171. a = 1 6 , b = 1 3 , c = 1 3 , d = 1 6 a=\frac{1}{6},b=\frac{1}{3},c=\frac{1}{3},d=\frac{1}{6}

Runs_created.html

  1. R C = A × B C RC=\frac{A\;\times\;B}{C}
  2. R C = ( H + B B ) × T B A B + B B RC=\frac{(H+BB)\times TB}{AB+BB}
  3. R C = ( H + B B - C S ) × ( T B + ( .55 × S B ) ) A B + B B RC=\frac{(H+BB-CS)\times(TB+(.55\times SB))}{AB+BB}
  4. R C = ( H + B B - C S + H B P - G I D P ) × ( T B + ( .26 × ( B B - I B B + H B P ) ) + ( .52 × ( S H + S F + S B ) ) ) A B + B B + H B P + S H + S F RC=\frac{(H+BB-CS+HBP-GIDP)\times(TB+(.26\times(BB-IBB+HBP))+(.52\times(SH+SF+% SB)))}{AB+BB+HBP+SH+SF}
  5. H + B B - C S + H B P - G I D P H+BB-CS+HBP-GIDP
  6. ( 1.125 × Singles ) + ( 1.69 × Doubles ) + ( 3.02 × Triples ) + ( 3.73 × HR ) + .29 × ( BB - IBB + HBP ) + .492 × ( SH + SF + SB ) - ( .04 × K ) (1.125\times\ \rm{Singles})+(1.69\times\rm{Doubles})+(3.02\times\rm{Triples})+% (3.73\times HR)+.29\times(BB-IBB+HBP)+.492\times(SH+SF+SB)-(.04\times K)
  7. A B + B B + H B P + S H + S F AB+BB+HBP+SH+SF
  8. R C = ( ( 2.4 C + A ) ( 3 C + B ) 9 C ) - .9 C RC=\left(\frac{(2.4C+A)\;(3C+B)}{9C}\right)-.9C
  9. H R I S P - ( A B R I S P × B A ) + H R R O B - A B R O B × H R A B H_{RISP}-(AB_{RISP}\times BA)+HR_{ROB}-\frac{AB_{ROB}\times HR}{AB}
  10. H R I S P H_{RISP}
  11. R C 27 \frac{RC}{27}

Rydberg_constant.html

  1. R = m e e 4 8 ε 0 2 h 3 c = 1.097 373 156 8539 ( 55 ) × 10 7 m - 1 , R_{\infty}=\frac{m\text{e}e^{4}}{8{\varepsilon_{0}}^{2}h^{3}c}=1.097\;373\;156% \;8539(55)\times 10^{7}\,\,\text{m}^{-1},
  2. m e m\text{e}
  3. e e
  4. ε 0 \varepsilon_{0}
  5. h h
  6. c c
  7. 1 Ry h c R = 13.605 692 53 ( 30 ) eV . 1\ \,\text{Ry}\equiv hcR_{\infty}=13.605\;692\;53(30)\,\,\text{eV}.
  8. \infty
  9. 1 λ = R ( 1 n 1 2 - 1 n 2 2 ) = m e e 4 8 ε 0 2 h 3 c ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda}=R_{\infty}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}% \right)=\frac{m\text{e}e^{4}}{8\varepsilon_{0}^{2}h^{3}c}\left(\frac{1}{n_{1}^% {2}}-\frac{1}{n_{2}^{2}}\right)
  10. λ \lambda
  11. 1 λ = R M ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda}=R_{M}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)
  12. R M = R / ( 1 + m e / M ) , R_{M}=R_{\infty}/(1+m_{\,\text{e}}/M),
  13. R R_{\infty}
  14. R R_{\infty}
  15. R = α 2 m e c 4 π = α 2 2 λ e = α 4 π a 0 R_{\infty}=\frac{\alpha^{2}m\text{e}c}{4\pi\hbar}=\frac{\alpha^{2}}{2\lambda_{% \,\text{e}}}=\frac{\alpha}{4\pi a_{0}}
  16. h c R = m e c 2 α 2 2 = m e e 4 32 π 2 ε 0 2 2 = m e c 2 r e 2 a 0 = h c α 2 2 λ e = h f C α 2 2 = ω C 2 α 2 = 2 2 m e a 0 2 = e 2 ( 4 π ε 0 ) 2 a 0 . hcR_{\infty}=m_{\,\text{e}}c^{2}\frac{\alpha^{2}}{2}=\frac{m_{\,\text{e}}e^{4}% }{32\pi^{2}\varepsilon_{0}^{2}\hbar^{2}}=\frac{m_{\,\text{e}}c^{2}r_{e}}{2a_{0% }}=\frac{hc\alpha^{2}}{2\lambda_{\,\text{e}}}=\frac{hf_{\,\text{C}}\alpha^{2}}% {2}=\frac{\hbar\omega_{\,\text{C}}}{2}\alpha^{2}=\dfrac{\hbar^{2}}{2m_{\,\text% {e}}a_{0}^{2}}=\frac{e^{2}}{(4\pi\varepsilon_{0})2a_{0}}.
  17. m e m\text{e}
  18. e e
  19. h h
  20. = h / 2 π \hbar=h/2\pi
  21. c c
  22. ε 0 \varepsilon_{0}
  23. α \alpha
  24. λ e = h / m e c \lambda_{\,\text{e}}=h/m\text{e}c
  25. f C = m e c 2 / h f_{\,\text{C}}=m_{\,\text{e}}c^{2}/h
  26. ω C = 2 π f C \omega_{\,\text{C}}=2\pi f_{\,\text{C}}
  27. a 0 = 4 π ε 0 2 e 2 m e a_{0}=\frac{4\pi\varepsilon_{0}\hbar^{2}}{e^{2}m_{\,\text{e}}}
  28. r e = 1 4 π ε 0 e 2 m e c 2 r_{\mathrm{e}}=\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{m_{\mathrm{e}}c^{2}}
  29. E n = - h c R / n 2 E_{n}=-hcR_{\infty}/n^{2}

Rydberg_formula.html

  1. n = n 0 - C 0 m + m \scriptstyle n=n_{0}-\frac{C_{0}}{m+m^{\prime}}
  2. n = n 0 - C 0 ( m + m ) 2 \scriptstyle n=n_{0}-\frac{C_{0}}{\left(m+m^{\prime}\right)^{2}}
  3. λ = h m 2 m 2 - 4 \scriptstyle\lambda={hm^{2}\over m^{2}-4}
  4. n = n 0 - 4 n 0 m 2 \scriptstyle n=n_{0}-{4n_{0}\over m^{2}}
  5. m = 0 \scriptstyle m^{\prime}=0\!
  6. C 0 = 4 n 0 \scriptstyle C_{0}=4n_{0}\!
  7. n 0 = 1 h \scriptstyle n_{0}=\frac{1}{h}
  8. 1 λ = f c \scriptstyle\frac{1}{\lambda}=\frac{f}{c}
  9. 1 λ = E h c \scriptstyle\frac{1}{\lambda}=\frac{E}{hc}
  10. 1 λ vac = R ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda_{\mathrm{vac}}}=R\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}% }\right)
  11. λ vac \lambda_{\mathrm{vac}}\!
  12. R R\!
  13. n 1 n_{1}\!
  14. n 2 n_{2}\!
  15. n 1 < n 2 n_{1}<n_{2}\!
  16. n 1 n_{1}
  17. n 2 n_{2}
  18. 1 λ vac = R Z 2 ( 1 n 1 2 - 1 n 2 2 ) \frac{1}{\lambda_{\mathrm{vac}}}=RZ^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2% }^{2}}\right)
  19. λ vac \lambda_{\mathrm{vac}}\!
  20. R R\!
  21. Z Z\!
  22. n 1 n_{1}\!
  23. n 2 n_{2}\!
  24. n 1 < n 2 n_{1}<n_{2}\!
  25. 3 / 4 {3}/{4}

Sainte-Laguë_method.html

  1. q u o t = V 2 s + 1 quot=\frac{V}{2s+1}
  2. ( q u o t = V s + 1 ) \left(quot=\frac{V}{s+1}\right)

Sampling_(signal_processing).html

  1. , s ^ ( t ) , ,\hat{s}(t),
  2. , s ( t ) , ,s(t),\,
  3. s a ( t ) = def s ( t ) + j s ^ ( t ) , s_{a}(t)\ \stackrel{\,\text{def}}{=}\ s(t)+j\cdot\hat{s}(t),
  4. s a ( t ) e - j 2 π B 2 t , s_{a}(t)\cdot e^{-j2\pi\frac{B}{2}t},
  5. s ^ ( t ) , \hat{s}(t),
  6. , [ s ( n T ) e - j 2 π B 2 T n ] , ,\left[s(nT)\cdot e^{-j2\pi\frac{B}{2}Tn}\right],
  7. [ s ( n T ) ( - j ) n ] . \left[s(nT)\cdot(-j)^{n}\right].

Satellite_modem.html

  1. 1 / 2 1/2
  2. 2 / 3 2/3
  3. 3 / 4 3/4
  4. 5 / 6 5/6
  5. 7 / 8 7/8

Satisficing.html

  1. max s X I S ( s ) \max_{s\in X}I_{S}(s)
  2. I S ( s ) := { 1 , s S 0 , s S , s X I_{S}(s):=\begin{cases}\begin{array}[]{ccc}1&,&s\in S\\ 0&,&s\notin S\end{array}\end{cases}\ ,\ s\in X
  3. max s X U ( s ) \max_{s\in X}U(s)
  4. < v a r > O <var>O

Scalability.html

  1. n n
  2. n n
  3. α \alpha
  4. 1 - α 1-\alpha
  5. 1 α + 1 - α P \frac{1}{\alpha+\frac{1-\alpha}{P}}
  6. 1 0.3 + 1 - 0.3 4 = 2.105 \frac{1}{0.3+\frac{1-0.3}{4}}=2.105
  7. 1 0.3 + 1 - 0.3 8 = 2.581 \frac{1}{0.3+\frac{1-0.3}{8}}=2.581

Scalar_curvature.html

  1. S = tr Ric g . S=\mbox{tr}~{}_{g}\,\operatorname{Ric}.
  2. S = g i j R i j = R j j S=g^{ij}R_{ij}=R^{j}_{j}
  3. Ric = R i j d x i d x j . \operatorname{Ric}=R_{ij}\,dx^{i}\otimes dx^{j}.
  4. S = g a b ( Γ a b , c c - Γ a c , b c + Γ a b d Γ c d c - Γ a c d Γ b d c ) = 2 g a b ( Γ a [ b , c ] c + Γ a [ b d Γ c ] d c ) S=g^{ab}(\Gamma^{c}_{ab,c}-\Gamma^{c}_{ac,b}+\Gamma^{d}_{ab}\Gamma^{c}_{cd}-% \Gamma^{d}_{ac}\Gamma^{c}_{bd})=2g^{ab}(\Gamma^{c}_{a[b,c]}+\Gamma^{d}_{a[b}% \Gamma^{c}_{c]d})
  5. Γ b c a \Gamma^{a}_{bc}
  6. Γ j k , i l \Gamma_{jk,i}^{l}
  7. Γ j k l \Gamma_{jk}^{l}
  8. i i
  9. ( M , g ) (M,g)
  10. Vol ( B ε ( p ) M ) Vol ( B ε ( 0 ) n ) = 1 - S 6 ( n + 2 ) ε 2 + O ( ε 4 ) . \frac{\operatorname{Vol}(B_{\varepsilon}(p)\subset M)}{\operatorname{Vol}(B_{% \varepsilon}(0)\subset{\mathbb{R}}^{n})}=1-\frac{S}{6(n+2)}\varepsilon^{2}+O(% \varepsilon^{4}).
  11. ϵ \epsilon
  12. Area ( B ε ( p ) M ) Area ( B ε ( 0 ) n ) = 1 - S 6 n ε 2 + O ( ε 4 ) . \frac{\operatorname{Area}(\partial B_{\varepsilon}(p)\subset M)}{\operatorname% {Area}(\partial B_{\varepsilon}(0)\subset{\mathbb{R}}^{n})}=1-\frac{S}{6n}% \varepsilon^{2}+O(\varepsilon^{4}).
  13. S = 2 ρ 1 ρ 2 S=\frac{2}{\rho_{1}\rho_{2}}\,
  14. ρ 1 , ρ 2 \rho_{1},\,\rho_{2}
  15. 2 R 1212 = S det ( g i j ) = S [ g 11 g 22 - ( g 12 ) 2 ] . 2R_{1212}\,=S\det(g_{ij})=S[g_{11}g_{22}-(g_{12})^{2}].
  16. x 0 2 - x 1 2 - - x n 2 = r 2 , x 0 > 0. x_{0}^{2}-x_{1}^{2}-\cdots-x_{n}^{2}=r^{2},\quad x_{0}>0.
  17. R i j k l R_{ijk}^{l}
  18. R a b c d R_{abcd}
  19. R i j R_{ij}

Scalar_field.html

  1. sin ( 2 π ( x y + σ ) ) \sin(2\pi(xy+\sigma))
  2. σ \sigma

Scale-free_network.html

  1. P ( k ) k s y m b o l - γ P(k)\ \sim\ k^{s}ymbol{-\gamma}
  2. γ \gamma
  3. v v
  4. v v
  5. deg ( v ) \deg(v)
  6. s ( G ) = ( u , v ) E deg ( u ) deg ( v ) . s(G)=\sum_{(u,v)\in E}\deg(u)\cdot\deg(v).
  7. S ( G ) = s ( G ) s max , S(G)=\frac{s(G)}{s_{\max}},
  8. Π \Pi
  9. Π ( k i ) = k i j k j \Pi(k_{i})=\frac{k_{i}}{\sum_{j}k_{j}}
  10. Π ( k ) \Pi(k)
  11. Π ( 0 ) 0 \Pi(0)\neq 0
  12. Π ( k ) \Pi(k)
  13. Π ( k ) = A + k α \Pi(k)=A+k^{\alpha}
  14. A A
  15. Π ( k i ) = k i + C ( i , j ) k j j k j + C j k j 2 \Pi(k_{i})=\frac{k_{i}+C\sum_{(i,j)}k_{j}}{\sum_{j}k_{j}+C\sum_{j}k_{j}^{2}}
  16. Π ( k ) \Pi(k)
  17. i i
  18. k k
  19. i i
  20. Π ( k ) \Pi(k)
  21. k k
  22. Π ( k ) = p \Pi(k)=p
  23. Π ( k ) \Pi(k)
  24. k k
  25. Π ( k ) \Pi(k)
  26. Π ( k ) \Pi(k)
  27. Π ( k i ) a k i \Pi(k_{i})\sim a_{\infty}k_{i}
  28. k i k_{i}\to\infty
  29. P ( k ) k - γ with γ = 1 + μ a . P(k)\sim k^{-\gamma}\,\text{ with }\gamma=1+\frac{\mu}{a_{\infty}}.
  30. \infty
  31. p ( x i , x j ) p(x_{i},x_{j})
  32. p ( x i , x j ) = δ x i x j 1 + δ x i x j p(x_{i},x_{j})=\frac{\delta x_{i}x_{j}}{1+\delta x_{i}x_{j}}

Scale_model.html

  1. Re m = ρ m 𝐯 m L m μ m = ρ p 𝐯 p L p μ p = Re p \mathrm{Re}_{m}={{\rho_{m}{\mathbf{v}_{m}}L_{m}}\over{\mu_{m}}}={{\rho_{p}{% \mathbf{v}_{p}}L_{p}}\over{\mu_{p}}}=\mathrm{Re}_{p}
  2. 𝐯 {\mathbf{v}}
  3. L {L}
  4. μ {\mu}
  5. ρ {\rho}\,
  6. S L \mathrm{S}_{L}
  7. S i = i p i m \mathrm{S}_{i}={{i_{p}}\over{i_{m}}}
  8. i p i_{p}
  9. i m i_{m}

Scattering.html

  1. α = π D p / λ , \alpha=\pi D\text{p}/\lambda,

Schumann_resonances.html

  1. n n
  2. f n f_{n}
  3. a a
  4. c c
  5. f n = c 2 π a n ( n + 1 ) f_{n}=\frac{c}{2\pi a}\sqrt{n(n+1)}

Schwarzschild_metric.html

  1. c 2 d τ 2 = ( 1 - r s r ) c 2 d t 2 - ( 1 - r s r ) - 1 d r 2 - r 2 ( d θ 2 + sin 2 θ d φ 2 ) , c^{2}{d\tau}^{2}=\left(1-\frac{r_{s}}{r}\right)c^{2}dt^{2}-\left(1-\frac{r_{s}% }{r}\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}% \right),
  2. τ \tau
  3. r r
  4. r s r_{s}
  5. R α β γ δ R α β γ δ = 12 r s 2 r 6 = 48 G 2 M 2 c 4 r 6 . R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}=\frac{12{r_{s}}^{2}}{r^% {6}}=\frac{48G^{2}M^{2}}{c^{4}r^{6}}\,.
  6. ( 1 - r s r ) d v 2 - 2 d v d r - r 2 d Ω 2 \left(1-\frac{r_{s}}{r}\right)dv^{2}-2dvdr-r^{2}d\Omega^{2}
  7. ( 1 - r s r ) d u 2 + 2 d u d r - r 2 d Ω 2 \left(1-\frac{r_{s}}{r}\right)du^{2}+2dudr-r^{2}d\Omega^{2}
  8. ( 1 - r s r ) d T 2 - 2 r s r d T d r - d r 2 - r 2 d Ω 2 \left(1-\frac{r_{s}}{r}\right)dT^{2}-2\sqrt{\frac{r_{s}}{r}}dTdr-dr^{2}-r^{2}d% \Omega^{2}
  9. ( 1 - r s 4 R ) 2 ( 1 + r s 4 R ) 2 d t 2 - ( 1 + r s 4 R ) 4 ( d x 2 + d y 2 + d z 2 ) \frac{(1-\frac{r_{s}}{4R})^{2}}{(1+\frac{r_{s}}{4R})^{2}}{dt}^{2}-\left(1+% \frac{r_{s}}{4R}\right)^{4}(dx^{2}+dy^{2}+dz^{2})
  10. R = x 2 + y 2 + z 2 R=\sqrt{x^{2}+y^{2}+z^{2}}
  11. 4 r s 3 r e - r / r s ( d T 2 - d R 2 ) - r 2 d Ω 2 , \frac{4r_{s}^{3}}{r}e^{-r/r_{s}}(dT^{2}-dR^{2})-r^{2}d\Omega^{2},
  12. T 2 - R 2 = ( 1 - r r s ) e r / r s T^{2}-R^{2}=\left(1-\frac{r}{r_{s}}\right)e^{r/r_{s}}
  13. d T 2 - r s r d R 2 - r 2 d Ω 2 dT^{2}-\frac{r_{s}}{r}dR^{2}-r^{2}d\Omega^{2}
  14. r = [ 3 2 ( R - T ) ] 2 / 3 r s 1 / 3 r=\left[\frac{3}{2}(R-T)\right]^{2/3}r_{s}^{1/3}
  15. d Ω 2 = d θ 2 + sin ( θ ) 2 d ϕ 2 d\Omega^{2}=d\theta^{2}+\sin(\theta)^{2}d\phi^{2}
  16. r > r s r>r_{s}
  17. w = 2 r s ( r - r s ) . w=2\sqrt{r_{s}\left(r-r_{s}\right)}.
  18. d w 2 + d r 2 + r 2 d φ 2 = - c 2 d τ 2 = d r 2 1 - r s r + r 2 d φ 2 dw^{2}+dr^{2}+r^{2}d\varphi^{2}=-c^{2}d\tau^{2}=\frac{dr^{2}}{1-\frac{r_{s}}{r% }}+r^{2}d\varphi^{2}
  19. d s 2 = d w 2 + d r 2 + r 2 d ϕ 2 . \mathrm{d}s^{2}=\mathrm{d}w^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2}.\,
  20. w = w ( r ) w=w(r)
  21. d s 2 = [ 1 + ( d w d r ) 2 ] d r 2 + r 2 d ϕ 2 , \mathrm{d}s^{2}=\left[1+\left(\frac{\mathrm{d}w}{\mathrm{d}r}\right)^{2}\right% ]\mathrm{d}r^{2}+r^{2}\mathrm{d}\phi^{2},
  22. d s 2 = ( 1 - r s r ) - 1 d r 2 + r 2 d ϕ 2 , \mathrm{d}s^{2}=\left(1-\frac{r_{s}}{r}\right)^{-1}\mathrm{d}r^{2}+r^{2}% \mathrm{d}\phi^{2},
  23. w ( r ) = d r r r s - 1 = 2 r s r r s - 1 + constant w(r)=\int\frac{\mathrm{d}r}{\sqrt{\frac{r}{r_{s}}-1}}=2r_{s}\sqrt{\frac{r}{r_{% s}}-1}+\mbox{constant}~{}
  24. r > 3 r s r>3r_{s}
  25. r r
  26. 3 r s / 2 3r_{s}/2
  27. 3 r s 3r_{s}
  28. r < 3 r s / 2 r<3r_{s}/2
  29. 3 r s / 2 3r_{s}/2
  30. r r
  31. r s r_{s}
  32. 3 r s / 2 3r_{s}/2

Schwarzschild_radius.html

  1. M = G m c 2 M=\frac{Gm}{c^{2}}
  2. 1 2 M - r \frac{1}{2M-r}
  3. r r
  4. radius s \,\text{radius}_{s}
  5. density s \,\text{density}_{s}
  6. × 10 1 5 \times 10^{1}5
  7. × 10 - 8 \times 10^{-}8
  8. × 10 3 \times 10^{3}
  9. × 10 1 6 \times 10^{1}6
  10. × 10 - 3 \times 10^{-}3
  11. × 10 2 7 \times 10^{2}7
  12. × 10 1 0 \times 10^{1}0
  13. × 10 1 1 \times 10^{1}1
  14. × 10 1 3 \times 10^{1}3
  15. r s = 2 G M c 2 , r_{\mathrm{s}}=\frac{2GM}{c^{2}},
  16. G G
  17. M M
  18. c c
  19. V s ρ - 3 / 2 , V_{\mathrm{s}}\propto\rho^{-3/2},
  20. V s = 4 π 3 r s 3 V_{\mathrm{s}}\!=\frac{4\pi}{3}r_{\mathrm{s}}^{3}
  21. ρ = m V s \rho\!=\frac{m}{V_{\mathrm{s}}}
  22. t r t = 1 - r s r \frac{t_{r}}{t}=\sqrt{1-\frac{r_{\mathrm{s}}}{r}}
  23. t t
  24. r r
  25. g r s ( r c ) 2 = 1 2 \frac{g}{r_{\mathrm{s}}}\left(\frac{r}{c}\right)^{2}=\frac{1}{2}
  26. g g
  27. r r
  28. c c
  29. 9.80665 m / s 2 8.870056 mm ( 6375416 m 299792458 m / s ) 2 = ( 1105.59 s - 2 ) ( 0.0212661 s ) 2 = 1 2 . \frac{9.80665\ \mathrm{m}/\mathrm{s}^{2}}{8.870056\ \mathrm{mm}}\left(\frac{63% 75416\ \mathrm{m}}{299792458\ \mathrm{m}/\mathrm{s}}\right)^{2}=\left(1105.59% \ \mathrm{s}^{-2}\right)\left(0.0212661\ \mathrm{s}\right)^{2}=\frac{1}{2}.
  30. r r s ( v c ) 2 = 1 2 \frac{r}{r_{\mathrm{s}}}\left(\frac{v}{c}\right)^{2}=\frac{1}{2}
  31. r r
  32. v v
  33. c c
  34. a r s ( 2 π a c T ) 2 = 1 2 \frac{a}{r_{\mathrm{s}}}\left(\frac{2\pi a}{cT}\right)^{2}=\frac{1}{2}
  35. a a
  36. T T
  37. 1 AU 2953.25 m ( 2 π AU light year ) 2 = ( 50655379.7 ) ( 9.8714403 × 10 - 9 ) = 1 2 . \frac{1\,\mathrm{AU}}{2953.25\,\mathrm{m}}\left(\frac{2\pi\,\mathrm{AU}}{% \mathrm{light\,year}}\right)^{2}=\left(50655379.7\right)\left(9.8714403\times 1% 0^{-9}\right)=\frac{1}{2}.
  38. r r s ( v c 1 - r s r ) 2 = 1 2 \frac{r}{r_{s}}\left(\frac{v}{c}\sqrt{1-\frac{r_{s}}{r}}\right)^{2}=\frac{1}{2}
  39. r r s ( v c ) 2 ( 1 - r s r ) = 1 2 \frac{r}{r_{s}}\left(\frac{v}{c}\right)^{2}\left(1-\frac{r_{s}}{r}\right)=% \frac{1}{2}
  40. ( v c ) 2 ( r r s - 1 ) = 1 2 . \left(\frac{v}{c}\right)^{2}\left(\frac{r}{r_{s}}-1\right)=\frac{1}{2}.