wpmath0000007_11

Random_variate.html

  1. [ 0 ; 1 ] [0;1]

Range_(aeronautics).html

  1. t t
  2. W W
  3. W e + W f W_{e}+W_{f}
  4. W e W_{e}
  5. W f W_{f}
  6. F F
  7. - d W f d t = - d W d t -\frac{dW_{f}}{dt}=-\frac{dW}{dt}
  8. R R
  9. d W d R = d W d t d R d t = - F V \frac{dW}{dR}=\frac{\frac{dW}{dt}}{\frac{dR}{dt}}=-\frac{F}{V}
  10. V V
  11. d R d t = - V F d W d t \frac{dR}{dt}=-\frac{V}{F}{\frac{dW}{dt}}
  12. t 1 t_{1}
  13. t 2 t_{2}
  14. W 1 W_{1}
  15. W 2 W_{2}
  16. R = t 1 t 2 d R d t d t = W 1 W 2 - V F d W = W 2 W 1 V F d W R=\int_{t_{1}}^{t_{2}}\frac{dR}{dt}dt=\int_{W_{1}}^{W_{2}}-\frac{V}{F}dW=\int_% {W_{2}}^{W_{1}}\frac{V}{F}dW
  17. V F \frac{V}{F}
  18. P a = P r P_{a}=P_{r}
  19. η j \eta_{j}
  20. c p c_{p}
  21. P b r = P a η j P_{br}=\frac{P_{a}}{\eta_{j}}
  22. F = c p P b r F=c_{p}P_{br}
  23. P a = V C D C L W P_{a}=V\frac{C_{D}}{C_{L}}W
  24. R = η j c p C L C D W 2 W 1 d W W R=\frac{\eta_{j}}{c_{p}}\frac{C_{L}}{C_{D}}\int_{W_{2}}^{W_{1}}\frac{dW}{W}
  25. R = η j c p C L C D l n W 1 W 2 R=\frac{\eta_{j}}{c_{p}}\frac{C_{L}}{C_{D}}ln\frac{W_{1}}{W_{2}}
  26. D = C D C L W D=\frac{C_{D}}{C_{L}}W
  27. T = D = C D C L W T=D=\frac{C_{D}}{C_{L}}W
  28. F = - c T T = - c T C D C L W F=-c_{T}T=-c_{T}\frac{C_{D}}{C_{L}}W
  29. 1 2 ρ V 2 S C L = W \frac{1}{2}\rho V^{2}SC_{L}=W
  30. ρ \rho
  31. V F = 1 c T W W S 2 ρ C L C D 2 \frac{V}{F}=\frac{1}{c_{T}W}\sqrt{\frac{W}{S}\frac{2}{\rho}\frac{C_{L}}{C_{D}^% {2}}}
  32. R = W 2 W 1 1 c T W W S 2 ρ C L C D 2 d W R=\int_{W_{2}}^{W_{1}}\frac{1}{c_{T}W}\sqrt{\frac{W}{S}\frac{2}{\rho}\frac{C_{% L}}{C_{D}^{2}}}dW
  33. R = 2 c T 2 S ρ C L C D 2 ( W 1 - W 2 ) R=\frac{2}{c_{T}}\sqrt{\frac{2}{S\rho}\frac{C_{L}}{C_{D}^{2}}}\left(\sqrt{W_{1% }}-\sqrt{W_{2}}\right)
  34. V = a M V=aM
  35. M M
  36. a a
  37. R = a M c T C L C D W 2 W 1 d W W R=\frac{aM}{c_{T}}\frac{C_{L}}{C_{D}}\int_{W_{2}}^{W_{1}}\frac{dW}{W}
  38. R = a M c T C L C D l n W 1 W 2 R=\frac{aM}{c_{T}}\frac{C_{L}}{C_{D}}ln\frac{W_{1}}{W_{2}}

Rank_correlation.html

  1. τ \tau
  2. ρ \rho
  3. n n
  4. x x
  5. y y
  6. { x i } i n \{x_{i}\}_{i\leq n}
  7. { y i } i n \{y_{i}\}_{i\leq n}
  8. i i
  9. j j
  10. x x
  11. a i j a_{ij}
  12. y y
  13. b i j b_{ij}
  14. a i j = - a j i a_{ij}=-a_{ji}
  15. b i j = - b j i b_{ij}=-b_{ji}
  16. Γ \Gamma
  17. Γ = i , j = 1 n a i j b i j i , j = 1 n a i j 2 i , j = 1 n b i j 2 \Gamma=\frac{\sum_{i,j=1}^{n}a_{ij}b_{ij}}{\sqrt{\sum_{i,j=1}^{n}a_{ij}^{2}% \sum_{i,j=1}^{n}b_{ij}^{2}}}
  18. τ \tau
  19. r i r_{i}
  20. i i
  21. x x
  22. a i j = sgn ( r j - r i ) a_{ij}=\operatorname{sgn}(r_{j}-r_{i})
  23. b b
  24. a i j b i j \sum a_{ij}b_{ij}
  25. a i j 2 \sum a_{ij}^{2}
  26. a i j a_{ij}
  27. n ( n - 1 ) n(n-1)
  28. b i j 2 \sum b_{ij}^{2}
  29. Γ \Gamma
  30. τ \tau
  31. ρ \rho
  32. r i r_{i}
  33. s i s_{i}
  34. i i
  35. x x
  36. y y
  37. a i j = r j - r i a_{ij}=r_{j}-r_{i}
  38. b i j = s j - s i b_{ij}=s_{j}-s_{i}
  39. a i j 2 \sum a_{ij}^{2}
  40. b i j 2 \sum b_{ij}^{2}
  41. r i r_{i}
  42. s i s_{i}
  43. 1 1
  44. n n
  45. Γ = ( r j - r i ) ( s j - s i ) ( r j - r i ) 2 \Gamma=\frac{\sum(r_{j}-r_{i})(s_{j}-s_{i})}{\sum(r_{j}-r_{i})^{2}}
  46. i , j = 1 n ( r j - r i ) ( s j - s i ) = i = 1 n j = 1 n r i s i + i = 1 n j = 1 n r j s j - i = 1 n j = 1 n ( r i s j + r j s i ) \sum_{i,j=1}^{n}(r_{j}-r_{i})(s_{j}-s_{i})=\sum_{i=1}^{n}\sum_{j=1}^{n}r_{i}s_% {i}+\sum_{i=1}^{n}\sum_{j=1}^{n}r_{j}s_{j}-\sum_{i=1}^{n}\sum_{j=1}^{n}(r_{i}s% _{j}+r_{j}s_{i})
  47. = 2 n i = 1 n r i s i - 2 i = 1 n r i j = 1 n s j =2n\sum_{i=1}^{n}r_{i}s_{i}-2\sum_{i=1}^{n}r_{i}\sum_{j=1}^{n}s_{j}
  48. = 2 n i = 1 n r i s i - 1 2 n 2 ( n + 1 ) 2 =2n\sum_{i=1}^{n}r_{i}s_{i}-\frac{1}{2}n^{2}(n+1)^{2}
  49. r i \sum r_{i}
  50. s j \sum s_{j}
  51. n n
  52. 1 2 n ( n + 1 ) \frac{1}{2}n(n+1)
  53. S = i = 1 n ( r i - s i ) 2 = 2 r i 2 - 2 r i s i S=\sum_{i=1}^{n}(r_{i}-s_{i})^{2}=2\sum r_{i}^{2}-2\sum r_{i}s_{i}
  54. ( r j - r i ) ( s j - s i ) = 2 n r i 2 - 1 2 n 2 ( n + 1 ) 2 - n S \sum(r_{j}-r_{i})(s_{j}-s_{i})=2n\sum r_{i}^{2}-\frac{1}{2}n^{2}(n+1)^{2}-nS
  55. r i 2 \sum r_{i}^{2}
  56. n n
  57. 1 6 n ( n + 1 ) ( 2 n + 1 ) \frac{1}{6}n(n+1)(2n+1)
  58. ( r j - r i ) ( s j - s i ) = 1 6 n 2 ( n 2 - 1 ) - n S \sum(r_{j}-r_{i})(s_{j}-s_{i})=\frac{1}{6}n^{2}(n^{2}-1)-nS
  59. ( r j - r i ) 2 = 2 n r i 2 - 2 r i r j \sum(r_{j}-r_{i})^{2}=2n\sum r_{i}^{2}-2\sum r_{i}r_{j}
  60. = 2 n r i 2 - 2 ( r i ) 2 = 1 6 n 2 ( n 2 - 1 ) =2n\sum r_{i}^{2}-2(\sum r_{i})^{2}=\frac{1}{6}n^{2}(n^{2}-1)
  61. Γ R = 1 - 6 d i 2 n 3 - n \Gamma_{R}=1-\frac{6\sum d_{i}^{2}}{n^{3}-n}
  62. d i = x i - y i , d_{i}=x_{i}-y_{i},
  63. ρ \rho
  64. ρ \rho
  65. r = f - u r=f-u

Rank_mobility_index.html

  1. R M I = R 1 - R 2 R 1 + R 2 , RMI=\frac{R_{1}-R_{2}}{R_{1}+R_{2}},

Rapidity.html

  1. a r t a n h artanh
  2. c c
  3. v v
  4. v / c v/c
  5. 𝚲 ( φ ) = e 𝐙 φ \mathbf{\Lambda}(\varphi)=e^{\mathbf{Z}\varphi}
  6. 𝐙 \mathbf{Z}
  7. 𝐙 = ( 0 1 1 0 ) . \mathbf{Z}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}.
  8. 𝚲 ( φ 1 + φ 2 ) = 𝚲 ( φ 1 ) 𝚲 ( φ 2 ) \mathbf{\Lambda}(\varphi_{1}+\varphi_{2})=\mathbf{\Lambda}(\varphi_{1})\mathbf% {\Lambda}(\varphi_{2})
  9. A A
  10. B B
  11. C C
  12. φ AC = φ AB + φ BC \varphi_{\,\text{AC}}=\varphi_{\,\text{AB}}+\varphi_{\,\text{BC}}
  13. Q Q
  14. P P
  15. γ = 1 1 - v 2 / c 2 cosh φ \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\equiv\cosh\varphi
  16. u = ( u 1 + u 2 ) / ( 1 + u 1 u 2 / c 2 ) u=(u_{1}+u_{2})/(1+u_{1}u_{2}/c^{2})
  17. β i = u i c = tanh φ i \beta_{i}=\frac{u_{i}}{c}=\tanh{\varphi_{i}}
  18. tanh φ = tanh φ 1 + tanh φ 2 1 + tanh φ 1 tanh φ 2 = tanh ( φ 1 + φ 2 ) \begin{aligned}\displaystyle\tanh\varphi&\displaystyle=\frac{\tanh\varphi_{1}+% \tanh\varphi_{2}}{1+\tanh\varphi_{1}\tanh\varphi_{2}}\\ &\displaystyle=\tanh(\varphi_{1}+\varphi_{2})\end{aligned}
  19. k = e φ k=e^{\varphi}
  20. φ 1 2 + φ 2 2 \sqrt{\varphi_{1}^{2}+\varphi_{2}^{2}}
  21. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  22. | 𝐩 | |\mathbf{p}|
  23. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  24. E = m c 2 cosh φ E=mc^{2}\cosh\varphi
  25. | 𝐩 | = m c sinh φ |\mathbf{p}|=mc\,\sinh\varphi
  26. φ = artanh | 𝐩 | c E = 1 2 ln E + | 𝐩 | c E - | 𝐩 | c \varphi=\operatorname{artanh}\frac{|\mathbf{p}|c}{E}=\frac{1}{2}\ln\frac{E+|% \mathbf{p}|c}{E-|\mathbf{p}|c}
  27. y = 1 2 ln E + p z c E - p z c y=\frac{1}{2}\ln\frac{E+p_{z}c}{E-p_{z}c}
  28. < v a r > p z <var>p_{z}

Rate_function.html

  1. { x X | I ( x ) c } for c 0 \{x\in X|I(x)\leq c\}\mbox{ for }~{}c\geq 0
  2. lim sup δ 0 δ log μ δ ( F ) - inf x F I ( x ) , (U) \limsup_{\delta\downarrow 0}\delta\log\mu_{\delta}(F)\leq-\inf_{x\in F}I(x),% \quad\mbox{(U)}~{}
  3. lim inf δ 0 δ log μ δ ( G ) - inf x G I ( x ) . (L) \liminf_{\delta\downarrow 0}\delta\log\mu_{\delta}(G)\geq-\inf_{x\in G}I(x).% \quad\mbox{(L)}~{}
  4. lim sup δ 0 μ δ ( F ) μ ( F ) , \limsup_{\delta\downarrow 0}\mu_{\delta}(F)\leq\mu(F),
  5. lim inf δ 0 μ δ ( G ) μ ( G ) . \liminf_{\delta\downarrow 0}\mu_{\delta}(G)\geq\mu(G).
  6. lim δ 0 δ log μ δ ( S ) = - inf x S I ( x ) . (E) \lim_{\delta\downarrow 0}\delta\log\mu_{\delta}(S)=-\inf_{x\in S}I(x).\quad% \mbox{(E)}~{}
  7. I ( S ) = I ( S ¯ ) , I\big(\stackrel{\circ}{S}\big)=I\big(\bar{S}\big),
  8. S \stackrel{\circ}{S}
  9. S ¯ \bar{S}
  10. S S ¯ S\subseteq\bar{\stackrel{\circ}{S}}
  11. X n = 1 n i = 1 n Z i X_{n}=\frac{1}{n}\sum_{i=1}^{n}Z_{i}
  12. Ψ Z ( t ) = log 𝔼 e t Z . \Psi_{Z}(t)=\log\mathbb{E}e^{tZ}.

Rate_of_return.html

  1. r = V f - V i V i r=\frac{V_{f}-V_{i}}{V_{i}}
  2. V f V_{f}
  3. V i V_{i}
  4. R R
  5. t t
  6. R t \frac{R}{t}
  7. r r
  8. R R
  9. t t
  10. 1 + R = ( 1 + r ) t 1+R=(1+r)^{t}
  11. R R
  12. r r
  13. r = ( 1 + R ) 1 / t - 1 r=(1+R)^{1/t}-1
  14. 1.331 1 / 3 - 1 = 10 % 1.331^{1/3}-1=10\%
  15. R R
  16. r r
  17. t t
  18. r r
  19. R = ln ( V f V i ) R=\ln\left(\frac{V_{f}}{V_{i}}\right)
  20. r log = ln ( V f V i ) t r_{\mathrm{log}}=\frac{\ln\left(\frac{V_{f}}{V_{i}}\right)}{t}
  21. r r
  22. V f = V i e r t V_{f}=V_{i}e^{rt}
  23. r r
  24. t t
  25. R R
  26. r r
  27. t t
  28. R = r t R=rt
  29. r = R t r=\frac{R}{t}
  30. R R
  31. t t
  32. n n
  33. r 1 , r 2 , r 3 , , r n r_{1},r_{2},r_{3},\cdots,r_{n}
  34. ( 1 + r 1 ) ( 1 + r 2 ) ( 1 + r n ) - 1 (1+r_{1})(1+r_{2})\cdots(1+r_{n})-1
  35. i = 1 n r i = r 1 + r 2 + r 3 + + r n \sum_{i=1}^{n}{r_{i}}=r_{1}+r_{2}+r_{3}+\cdots+r_{n}
  36. n n
  37. r ¯ = 1 n i = 1 n r i = 1 n ( r 1 + + r n ) \bar{r}=\frac{1}{n}\sum_{i=1}^{n}{r_{i}}=\frac{1}{n}(r_{1}+\cdots+r_{n})
  38. r ¯ geometric = ( i = 1 n ( 1 + r i ) ) 1 / n - 1 \bar{r}_{\mathrm{geometric}}=\left({\prod_{i=1}^{n}(1+r_{i})}\right)^{1/n}-1
  39. ( 1 + 0.50 ) ( 1 - 0.20 ) ( 1 + 0.30 ) ( 1 - 0.40 ) - 1 = - 0.0640 = - 6.40 % (1+0.50)(1-0.20)(1+0.30)(1-0.40)-1=-0.0640=-6.40\%
  40. ( ( 1 + 0.50 ) ( 1 - 0.20 ) ( 1 + 0.30 ) ( 1 - 0.40 ) ) 1 / 4 - 1 = - 0.0164 = - 1.64 % ((1+0.50)(1-0.20)(1+0.30)(1-0.40))^{1/4}-1=-0.0164=-1.64\%
  41. ( 1 - 0.0640 ) 1 / 4 - 1 = - 0.0164 (1-0.0640)^{1/4}-1=-0.0164
  42. r r
  43. NPV = t = 0 n C t ( 1 + r ) t = 0 \mbox{NPV}~{}=\sum_{t=0}^{n}\frac{C_{t}}{(1+r)^{t}}=0
  44. C t {C_{t}}
  45. t {t}
  46. C 0 {C_{0}}
  47. C n {C_{n}}
  48. NPV = 0 \mbox{NPV}~{}=0
  49. r r
  50. r log r_{\mathrm{log}}
  51. r r
  52. V f / V i > 0 V_{f}/V_{i}>0
  53. V i V_{i}
  54. V f V_{f}
  55. V f - V i V_{f}-V_{i}
  56. r r
  57. r log r_{\mathrm{log}}
  58. 1.05 4 - 1 = 21.55 % 1.05^{4}-1=21.55\%
  59. ( 1 - 0.0164 ) 4 - 1 = - 6.4 % (1-0.0164)^{4}-1=-6.4\%
  60. ( 1 - 0.4274 ) 4 - 1 = - 89.25 % (1-0.4274)^{4}-1=-89.25\%
  61. 4.06 % = ( 1.01 ) 4 - 1 4.06\%=(1.01)^{4}-1
  62. ln ( 103.02 100 ) = 2.98 % \ln\left(\frac{103.02}{100}\right)=2.98\%
  63. P ( 1 + T ) n = ERV \mathrm{P\left(1+T\right)^{n}=ERV}
  64. T = ( ERV P ) 1 / n - 1 \mathrm{T=\left(\frac{ERV}{P}\right)^{1/n}-1}
  65. ( x + y ) ( x - y ) = x 2 - y 2 . (x+y)(x-y)=x^{2}-y^{2}.
  66. x = 100 % x=100\%
  67. x = 1 x=1

Ray_(optics).html

  1. θ i \theta_{\mathrm{i}}
  2. θ r \theta_{\mathrm{r}}
  3. θ R \theta_{\mathrm{R}}

Rayleigh_(unit).html

  1. I = 4 π 10 - 10 L I=4\pi{10^{-10}}L

Reactions_on_surfaces.html

  1. r = - d C A d t = k 2 C A S = k 2 θ C S r=-\frac{dC_{A}}{dt}=k_{2}C_{AS}=k_{2}\theta C_{S}
  2. C A S C_{AS}
  3. θ \theta
  4. C S C_{S}
  5. C S C_{S}
  6. d C A S d t = 0 = k 1 C A C S ( 1 - θ ) - k 2 θ C S - k - 1 θ C S \frac{dC_{AS}}{dt}=0=k_{1}C_{A}C_{S}(1-\theta)-k_{2}\theta C_{S}-k_{-1}\theta C% _{S}
  7. θ = k 1 C A k 1 C A + k - 1 + k 2 \theta=\frac{k_{1}C_{A}}{k_{1}C_{A}+k_{-1}+k_{2}}
  8. r = - d C A d t = k 1 k 2 C A C S k 1 C A + k - 1 + k 2 . r=-\frac{dC_{A}}{dt}=\frac{k_{1}k_{2}C_{A}C_{S}}{k_{1}C_{A}+k_{-1}+k_{2}}.
  9. K 1 = k 1 k - 1 K_{1}=\frac{k_{1}}{k_{-1}}
  10. k - 1 k_{-1}
  11. k 2 k 1 C A , k - 1 , so r k 1 C A C S . k_{2}\gg\ k_{1}C_{A},k_{-1},\,\text{ so }r\approx k_{1}C_{A}C_{S}.
  12. k 2 k 1 C A , k - 1 so θ = k 1 C A k 1 C A + k - 1 k_{2}\ll\ k_{1}C_{A},k_{-1}\,\text{ so }\theta=\frac{k_{1}C_{A}}{k_{1}C_{A}+k_% {-1}}
  13. r = K 1 k 2 C A C S K 1 C A + 1 r=\frac{K_{1}k_{2}C_{A}C_{S}}{K_{1}C_{A}+1}
  14. r = K 1 k 2 C A C S r=K_{1}k_{2}C_{A}C_{S}
  15. r = k 2 C S r=k_{2}C_{S}
  16. k 1 k_{1}
  17. k - 1 k_{-1}
  18. k 2 k_{2}
  19. k - 2 k_{-2}
  20. k k
  21. r = k θ A θ B C S 2 r=k\theta_{A}\theta_{B}C_{S}^{2}
  22. θ A = k 1 C A θ E k - 1 + k C S θ B \theta_{A}=\frac{k_{1}C_{A}\theta_{E}}{k_{-1}+kC_{S}\theta_{B}}
  23. θ E \theta_{E}
  24. θ A + θ B + θ E = 1 \theta_{A}+\theta_{B}+\theta_{E}=1
  25. θ A = K 1 C A θ E \theta_{A}=K_{1}C_{A}\theta_{E}
  26. K i = k i / k - i K_{i}=k_{i}/k_{-i}
  27. K 1 K_{1}
  28. K 2 K_{2}
  29. θ E \theta_{E}
  30. θ A \theta_{A}
  31. θ B \theta_{B}
  32. r = k C S 2 K 1 K 2 C A C B ( 1 + K 1 C A + K 2 C B ) 2 r=kC_{S}^{2}\frac{K_{1}K_{2}C_{A}C_{B}}{(1+K_{1}C_{A}+K_{2}C_{B})^{2}}
  33. 1 K 1 C A , K 2 C B 1\gg K_{1}C_{A},K_{2}C_{B}
  34. r = k C S 2 K 1 K 2 C A C B r=kC_{S}^{2}K_{1}K_{2}C_{A}C_{B}
  35. K 1 C A , 1 K 2 C B K_{1}C_{A},1\gg K_{2}C_{B}
  36. r = k C S 2 K 1 K 2 C A C B ( 1 + K 1 C A ) 2 r=kC_{S}^{2}\frac{K_{1}K_{2}C_{A}C_{B}}{(1+K_{1}C_{A})^{2}}
  37. r = k C S 2 K 1 K 2 C A C B r=kC_{S}^{2}K_{1}K_{2}C_{A}C_{B}
  38. r = k C S 2 K 2 C B K 1 C A r=kC_{S}^{2}\frac{K_{2}C_{B}}{K_{1}C_{A}}
  39. K 1 C A 1 , K 2 C B K_{1}C_{A}\gg 1,K_{2}C_{B}
  40. r = k C S 2 K 2 C B K 1 C A r=kC_{S}^{2}\frac{K_{2}C_{B}}{K_{1}C_{A}}
  41. k 1 , k - 1 k_{1},k_{-1}
  42. k k
  43. r = k C S θ A C B r=kC_{S}\theta_{A}C_{B}
  44. r = k C S C B K 1 C A K 1 C A + 1 r=kC_{S}C_{B}\frac{K_{1}C_{A}}{K_{1}C_{A}+1}
  45. r = k C S K 1 C A C B r=kC_{S}K_{1}C_{A}C_{B}
  46. r = k C S C B r=kC_{S}C_{B}

Reactivity–selectivity_principle.html

  1. δ \delta\,
  2. E a = E o + α Δ H r + β δ 2 E_{a}=E_{o}+\alpha\Delta H_{r}+\beta\delta^{2}\,
  3. E a E_{a}\,
  4. Δ H r \Delta H_{r}\,

Recessional_velocity.html

  1. v = H 0 D v=H_{0}D
  2. H 0 H_{0}
  3. D D
  4. v v

Reciprocal_difference.html

  1. ( x 0 , x 1 , , x n ) (x_{0},x_{1},...,x_{n})
  2. f ( x ) f(x)
  3. ρ 1 ( x 0 , x 1 ) = x 0 - x 1 f ( x 0 ) - f ( x 1 ) \rho_{1}(x_{0},x_{1})=\frac{x_{0}-x_{1}}{f(x_{0})-f(x_{1})}
  4. ρ 2 ( x 0 , x 1 , x 2 ) = x 0 - x 2 ρ 1 ( x 0 , x 1 ) - ρ 1 ( x 1 , x 2 ) + f ( x 1 ) \rho_{2}(x_{0},x_{1},x_{2})=\frac{x_{0}-x_{2}}{\rho_{1}(x_{0},x_{1})-\rho_{1}(% x_{1},x_{2})}+f(x_{1})
  5. ρ n ( x 0 , x 1 , , x n ) = x 0 - x n ρ n - 1 ( x 0 , x 1 , , x n - 1 ) - ρ n - 1 ( x 1 , x 2 , , x n ) + ρ n - 2 ( x 1 , , x n - 1 ) \rho_{n}(x_{0},x_{1},\ldots,x_{n})=\frac{x_{0}-x_{n}}{\rho_{n-1}(x_{0},x_{1},% \ldots,x_{n-1})-\rho_{n-1}(x_{1},x_{2},\ldots,x_{n})}+\rho_{n-2}(x_{1},\ldots,% x_{n-1})

Reciprocal_gamma_function.html

  1. f ( z ) = 1 Γ ( z ) , f(z)=\frac{1}{\Gamma(z)},
  2. log ( log | 1 / Γ ( z ) | ) \log(\log|1/\Gamma(z)|)
  3. log | z | \log|z|
  4. log | 1 / Γ ( z ) | \log|1/\Gamma(z)|
  5. | z | log | z | |z|\log|z|
  6. 1 Γ ( z ) = z + γ z 2 + ( γ 2 2 - π 2 12 ) z 3 + \frac{1}{\Gamma(z)}=z+\gamma z^{2}+\left(\frac{\gamma^{2}}{2}-\frac{\pi^{2}}{1% 2}\right)z^{3}+\cdots
  7. a k = - a 2 a k - 1 + j = 2 k - 1 ( - 1 ) j ζ ( j ) a k - j 1 - k a_{k}=\frac{{-a_{2}a_{k-1}+\sum_{j=2}^{k-1}(-1)^{j}\,\zeta(j)\,a_{k-j}}}{1-k}
  8. a k a_{k}
  9. ln ( 1 / Γ ( z ) ) - z ln ( z ) + z + 1 2 ln ( z 2 π ) - 1 12 z + 1 360 z 3 - 1 1260 z 5 for | arg ( z ) | < π \ln(1/\Gamma(z))\sim-z\ln(z)+z+\tfrac{1}{2}\ln\left(\frac{z}{2\pi}\right)-% \frac{1}{12z}+\frac{1}{360z^{3}}-\frac{1}{1260z^{5}}\qquad\qquad\,\text{for}% \quad|\arg(z)|<\pi
  10. 1 Γ ( z ) = i 2 π C ( - t ) - z e - t d t , \frac{1}{\Gamma(z)}=\frac{i}{2\pi}\oint_{C}(-t)^{-z}e^{-t}\,dt,
  11. 0 1 Γ ( x ) d x 2.80777024 , \int_{0}^{\infty}\frac{1}{\Gamma(x)}\,dx\approx 2.80777024,

Reciprocal_rule.html

  1. d d x ( 1 g ( x ) ) = - g ( x ) ( g ( x ) ) 2 \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{g(x)}\right)=\frac{-g^{\prime}(x)% }{(g(x))^{2}}
  2. d d x ( 1 g ( x ) ) = d d x ( f ( x ) g ( x ) ) \displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{g(x)}\right)=\frac{% \mathrm{d}}{\mathrm{d}x}\left(\frac{f(x)}{g(x)}\right)
  3. d d x ( 1 g ( x ) ) = d d x ( g ( x ) ) - 1 = - 1 ( g ( x ) ) - 2 g ( x ) = - g ( x ) ( g ( x ) ) 2 \frac{d}{dx}\left(\frac{1}{g(x)}\right)=\frac{d}{dx}\left({g(x)}\right)^{-1}=-% 1\cdot\left({g(x)}\right)^{-2}\cdot g^{\prime}(x)=-\frac{g^{\prime}(x)}{\left(% {g(x)}\right)^{2}}
  4. d d x ( 1 x 3 + 4 x ) = - 3 x 2 - 4 ( x 3 + 4 x ) 2 . \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{x^{3}+4x}\right)=\frac{-3x^{2}-4}% {(x^{3}+4x)^{2}}.
  5. d d x ( 1 cos ( x ) ) = sin ( x ) cos 2 ( x ) = 1 cos ( x ) sin ( x ) cos ( x ) = sec ( x ) tan ( x ) . \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{\cos(x)}\right)=\frac{\sin(x)}{% \cos^{2}(x)}=\frac{1}{\cos(x)}\frac{\sin(x)}{\cos(x)}=\sec(x)\tan(x).

Recovery_(metallurgy).html

  1. P = - α R d d R ( γ s R ) P=-\alpha\;R\frac{d}{dR}\left(\frac{\gamma\;_{s}}{R}\right)\,\!

Recrystallization_(metallurgy).html

  1. Δ E ρ G b 2 o r 3 γ s / d s \Delta\ E\approx\;\rho\;Gb^{2}or\approx\;3\gamma\;_{s}/d_{s}\,\!
  2. R = G ( t - t 0 ) R=G\left(t-t_{0}\right)\,\!
  3. f = 4 3 π N ˙ G 3 0 t ( t - t 0 ) 3 d t = π 3 N ˙ G 3 t 4 f=\frac{4}{3}\pi\ \dot{N}G^{3}\int_{0}^{t}(t-t_{0})^{3}\,dt=\frac{\pi\ }{3}% \dot{N}G^{3}t^{4}\,\!
  4. F v / r F_{v}/r
  5. F v F_{v}
  6. F v / r F_{v}/r
  7. F v / r F_{v}/r

Rectified_tesseract.html

  1. ( 0 , ± 2 , ± 2 , ± 2 ) (0,\ \pm\sqrt{2},\ \pm\sqrt{2},\ \pm\sqrt{2})

Recursion_(computer_science).html

  1. 0 ! = 1 0!=1
  2. n > 0 n>0
  3. n ! = n ( n 1 ) ! n!=n(n−1)!
  4. e = 1 / 0 ! + 1 / 1 ! + 1 / 2 ! + 1 / 3 ! + e=1/0!+1/1!+1/2!+1/3!+...
  5. fact ( n ) = { 1 if n = 0 n fact ( n - 1 ) if n > 0 \operatorname{fact}(n)=\begin{cases}1&\mbox{if }~{}n=0\\ n\cdot\operatorname{fact}(n-1)&\mbox{if }~{}n>0\\ \end{cases}
  6. b n = n b n - 1 b_{n}=nb_{n-1}
  7. b 0 = 1 b_{0}=1
  8. fact ( n ) = fact acc ( n , 1 ) fact acc ( n , t ) = { t if n = 0 fact acc ( n - 1 , n t ) if n > 0 \begin{array}[]{rcl}\operatorname{fact}(n)&=&\operatorname{fact_{acc}}(n,1)\\ \operatorname{fact_{acc}}(n,t)&=&\begin{cases}t&\mbox{if }~{}n=0\\ \operatorname{fact_{acc}}(n-1,nt)&\mbox{if }~{}n>0\\ \end{cases}\end{array}
  9. gcd ( x , y ) = { x if y = 0 gcd ( y , remainder ( x , y ) ) if y > 0 \gcd(x,y)=\begin{cases}x&\mbox{if }~{}y=0\\ \gcd(y,\operatorname{remainder}(x,y))&\mbox{if }~{}y>0\\ \end{cases}
  10. x % y x\%y
  11. x / y x/y
  12. gcd ( x , y ) = gcd ( y , x % y ) \gcd(x,y)=\gcd(y,x\%y)
  13. y 0 y\neq 0
  14. gcd ( x , 0 ) = x \gcd(x,0)=x
  15. hanoi ( n ) = { 1 if n = 1 2 hanoi ( n - 1 ) + 1 if n > 1 \operatorname{hanoi}(n)=\begin{cases}1&\mbox{if }~{}n=1\\ 2\cdot\operatorname{hanoi}(n-1)+1&\mbox{if }~{}n>1\\ \end{cases}
  16. h n = 2 h n - 1 + 1 h_{n}=2h_{n-1}+1
  17. h 1 = 1 h_{1}=1

Reduction_of_order.html

  1. y 1 ( x ) y_{1}(x)
  2. y 2 ( x ) y_{2}(x)
  3. v v
  4. a y ′′ ( x ) + b y ( x ) + c y ( x ) = 0 , ay^{\prime\prime}(x)+by^{\prime}(x)+cy(x)=0,\;
  5. a , b , c a,b,c
  6. a λ 2 + b λ + c = 0 a\lambda^{2}+b\lambda+c=0\;
  7. b 2 - 4 a c b^{2}-4ac
  8. λ 1 = λ 2 = - b 2 a . \lambda_{1}=\lambda_{2}=-\frac{b}{2a}.
  9. y 1 ( x ) = e - b 2 a x . y_{1}(x)=e^{-\frac{b}{2a}x}.
  10. y 2 ( x ) = v ( x ) y 1 ( x ) y_{2}(x)=v(x)y_{1}(x)\;
  11. v ( x ) v(x)
  12. y 2 ( x ) y_{2}(x)
  13. a ( v ′′ y 1 + 2 v y 1 + v y 1 ′′ ) + b ( v y 1 + v y 1 ) + c v y 1 = 0. a\left(v^{\prime\prime}y_{1}+2v^{\prime}y_{1}^{\prime}+vy_{1}^{\prime\prime}% \right)+b\left(v^{\prime}y_{1}+vy_{1}^{\prime}\right)+cvy_{1}=0.
  14. v ( x ) v(x)
  15. ( a y 1 ) v ′′ + ( 2 a y 1 + b y 1 ) v + ( a y 1 ′′ + b y 1 + c y 1 ) v = 0. \left(ay_{1}\right)v^{\prime\prime}+\left(2ay_{1}^{\prime}+by_{1}\right)v^{% \prime}+\left(ay_{1}^{\prime\prime}+by_{1}^{\prime}+cy_{1}\right)v=0.
  16. y 1 ( x ) y_{1}(x)
  17. y 1 ( x ) y_{1}(x)
  18. 2 a ( - b 2 a e - b 2 a x ) + b e - b 2 a x = ( - b + b ) e - b 2 a x = 0. 2a\left(-\frac{b}{2a}e^{-\frac{b}{2a}x}\right)+be^{-\frac{b}{2a}x}=\left(-b+b% \right)e^{-\frac{b}{2a}x}=0.
  19. a y 1 v ′′ = 0. ay_{1}v^{\prime\prime}=0.\;
  20. a a
  21. y 1 ( x ) y_{1}(x)
  22. v ′′ = 0. v^{\prime\prime}=0.\;
  23. v ( x ) = c 1 x + c 2 v(x)=c_{1}x+c_{2}\;
  24. c 1 , c 2 c_{1},c_{2}
  25. y 2 ( x ) = ( c 1 x + c 2 ) y 1 ( x ) = c 1 x y 1 ( x ) + c 2 y 1 ( x ) . y_{2}(x)=(c_{1}x+c_{2})y_{1}(x)=c_{1}xy_{1}(x)+c_{2}y_{1}(x).\;
  26. y 2 ( x ) y_{2}(x)
  27. y 2 ( x ) = x y 1 ( x ) = x e - b 2 a x . y_{2}(x)=xy_{1}(x)=xe^{-\frac{b}{2a}x}.
  28. y 2 ( x ) y_{2}(x)
  29. W ( y 1 , y 2 ) ( x ) = | y 1 x y 1 y 1 y 1 + x y 1 | = y 1 ( y 1 + x y 1 ) - x y 1 y 1 = y 1 2 + x y 1 y 1 - x y 1 y 1 = y 1 2 = e - b a x 0. W(y_{1},y_{2})(x)=\begin{vmatrix}y_{1}&xy_{1}\\ y_{1}^{\prime}&y_{1}+xy_{1}^{\prime}\end{vmatrix}=y_{1}(y_{1}+xy_{1}^{\prime})% -xy_{1}y_{1}^{\prime}=y_{1}^{2}+xy_{1}y_{1}^{\prime}-xy_{1}y_{1}^{\prime}=y_{1% }^{2}=e^{-\frac{b}{a}x}\neq 0.
  30. y 2 ( x ) y_{2}(x)
  31. y ′′ + p ( t ) y + q ( t ) y = r ( t ) y^{\prime\prime}+p(t)y^{\prime}+q(t)y=r(t)\,
  32. y 1 ( t ) y_{1}(t)
  33. r ( t ) = 0 r(t)=0
  34. y 2 = v ( t ) y 1 ( t ) y_{2}=v(t)y_{1}(t)\,
  35. v ( t ) v(t)
  36. y 2 = v ( t ) y 1 ( t ) + v ( t ) y 1 ( t ) y_{2}^{\prime}=v^{\prime}(t)y_{1}(t)+v(t)y_{1}^{\prime}(t)\,
  37. y 2 ′′ = v ′′ ( t ) y 1 ( t ) + 2 v ( t ) y 1 ( t ) + v ( t ) y 1 ′′ ( t ) . y_{2}^{\prime\prime}=v^{\prime\prime}(t)y_{1}(t)+2v^{\prime}(t)y_{1}^{\prime}(% t)+v(t)y_{1}^{\prime\prime}(t).\,
  38. y y
  39. y y^{\prime}
  40. y ′′ y^{\prime\prime}
  41. y 1 ( t ) v ′′ + ( 2 y 1 ( t ) + p ( t ) y 1 ( t ) ) v + ( y 1 ′′ ( t ) + p ( t ) y 1 ( t ) + q ( t ) y 1 ( t ) ) v = r ( t ) . y_{1}(t)\,v^{\prime\prime}+(2y_{1}^{\prime}(t)+p(t)y_{1}(t))\,v^{\prime}+(y_{1% }^{\prime\prime}(t)+p(t)y_{1}^{\prime}(t)+q(t)y_{1}(t))\,v=r(t).
  42. y 1 ( t ) y_{1}(t)
  43. y 1 ′′ ( t ) + p ( t ) y 1 ( t ) + q ( t ) y 1 ( t ) = 0 y_{1}^{\prime\prime}(t)+p(t)y_{1}^{\prime}(t)+q(t)y_{1}(t)=0
  44. y 1 ( t ) v ′′ + ( 2 y 1 ( t ) + p ( t ) y 1 ( t ) ) v = r ( t ) y_{1}(t)\,v^{\prime\prime}+(2y_{1}^{\prime}(t)+p(t)y_{1}(t))\,v^{\prime}=r(t)
  45. v ( t ) v^{\prime}(t)
  46. y 1 ( t ) y_{1}(t)
  47. v ′′ + ( 2 y 1 ( t ) y 1 ( t ) + p ( t ) ) v = r ( t ) y 1 ( t ) v^{\prime\prime}+\left(\frac{2y_{1}^{\prime}(t)}{y_{1}(t)}+p(t)\right)\,v^{% \prime}=\frac{r(t)}{y_{1}(t)}
  48. μ ( t ) = e ( 2 y 1 ( t ) y 1 ( t ) + p ( t ) ) d t = y 1 2 ( t ) e p ( t ) d t \mu(t)=e^{\int(\frac{2y_{1}^{\prime}(t)}{y_{1}(t)}+p(t))dt}=y_{1}^{2}(t)e^{% \int p(t)dt}
  49. μ ( t ) \mu(t)
  50. v ( t ) v(t)
  51. d d t ( v ( t ) y 1 2 ( t ) e p ( t ) d t ) = y 1 ( t ) r ( t ) e p ( t ) d t \frac{d}{dt}(v^{\prime}(t)y_{1}^{2}(t)e^{\int p(t)dt})=y_{1}(t)r(t)e^{\int p(t% )dt}
  52. v ( t ) v^{\prime}(t)
  53. v ( t ) v^{\prime}(t)
  54. y 2 ( t ) = v ( t ) y 1 ( t ) y_{2}(t)=v(t)y_{1}(t)\,

Refinable_function.html

  1. φ \varphi
  2. h h
  3. φ ( x ) = 2 k = 0 N - 1 h k φ ( 2 x - k ) \varphi(x)=2\cdot\sum_{k=0}^{N-1}h_{k}\cdot\varphi(2\cdot x-k)
  4. D D
  5. φ = 2 D 1 / 2 ( h * φ ) \varphi=2\cdot D_{1/2}(h*\varphi)
  6. φ 2 D 1 / 2 ( h * φ ) \varphi\mapsto 2\cdot D_{1/2}(h*\varphi)
  7. φ \varphi
  8. c c
  9. c φ c\cdot\varphi
  10. φ \varphi
  11. a a
  12. b b
  13. h h
  14. ( φ ( a ) φ ( a + 1 ) φ ( b ) ) = ( h a h a + 2 h a + 1 h a h a + 4 h a + 3 h a + 2 h a + 1 h a h b h b - 1 h b - 2 h b - 3 h b - 4 h b h b - 1 h b - 2 h b ) ( φ ( a ) φ ( a + 1 ) φ ( b ) ) . \begin{pmatrix}\varphi(a)\\ \varphi(a+1)\\ \vdots\\ \varphi(b)\end{pmatrix}=\begin{pmatrix}h_{a}&&&&&\\ h_{a+2}&h_{a+1}&h_{a}&&&\\ h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ &h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\ &&&h_{b}&h_{b-1}&h_{b-2}\\ &&&&&h_{b}\end{pmatrix}\cdot\begin{pmatrix}\varphi(a)\\ \varphi(a+1)\\ \vdots\\ \varphi(b)\end{pmatrix}.
  15. Q Q
  16. h h
  17. T h T_{h}
  18. Q φ = T h Q φ . Q\varphi=T_{h}\cdot Q\varphi.\,
  19. T h T_{h}
  20. k 2 - j k\cdot 2^{-j}
  21. k k\in\mathbb{Z}
  22. j j\in\mathbb{N}
  23. φ = D 1 / 2 ( 2 ( h * φ ) ) \varphi=D_{1/2}(2\cdot(h*\varphi))
  24. D 2 φ = 2 ( h * φ ) D_{2}\varphi=2\cdot(h*\varphi)
  25. Q ( D 2 φ ) = Q ( 2 ( h * φ ) ) = 2 ( h * Q φ ) Q(D_{2}\varphi)=Q(2\cdot(h*\varphi))=2\cdot(h*Q\varphi)
  26. k 2 \frac{k}{2}
  27. φ \varphi
  28. D 2 φ D_{2}\varphi
  29. Q ( D 2 j + 1 φ ) = 2 ( h * Q ( D 2 j φ ) ) Q(D_{2^{j+1}}\varphi)=2\cdot(h*Q(D_{2^{j}}\varphi))
  30. φ \varphi
  31. h h
  32. ψ \psi
  33. g g
  34. φ * ψ \varphi*\psi
  35. h * g h*g
  36. φ \varphi
  37. h h
  38. φ \varphi^{\prime}
  39. φ \varphi^{\prime}
  40. 2 h 2\cdot h
  41. φ \varphi
  42. h h
  43. Φ \Phi
  44. Φ ( t ) = 0 t φ ( τ ) d τ \Phi(t)=\int_{0}^{t}\varphi(\tau)\mathrm{d}\tau
  45. t Φ ( t ) + c t\mapsto\Phi(t)+c
  46. 1 2 h \frac{1}{2}\cdot h
  47. c c
  48. c ( 1 - j h j ) = j h j Φ ( - j ) c\cdot(1-\sum_{j}h_{j})=\sum_{j}h_{j}\cdot\Phi(-j)
  49. φ \varphi
  50. T T
  51. φ , T k ψ = φ * ψ * , T k δ = ( φ * ψ * ) ( k ) \langle\varphi,T_{k}\psi\rangle=\langle\varphi*\psi^{*},T_{k}\delta\rangle=(% \varphi*\psi^{*})(k)
  52. ψ * \psi^{*}
  53. ψ \psi
  54. ψ * \psi^{*}
  55. ψ \psi
  56. ψ * ( t ) = ψ ( - t ) ¯ \psi^{*}(t)=\overline{\psi(-t)}
  57. φ * ψ * \varphi*\psi^{*}
  58. h * g * h*g^{*}
  59. h h
  60. b b
  61. ( 1 , 1 ) (1,1)
  62. q q
  63. b b
  64. q q
  65. b b
  66. T q * q * T_{q*q^{*}}
  67. \R d \R \R^{d}\to\R
  68. φ \varphi
  69. ψ \psi
  70. h h
  71. g g
  72. φ ψ \varphi\otimes\psi
  73. h g h\otimes g
  74. M M
  75. M M
  76. | det M | > 1 |\det M|>1
  77. φ ( x ) = | det M | k \Z d h k φ ( M x - k ) \varphi(x)=|\det M|\cdot\sum_{k\in\Z^{d}}h_{k}\cdot\varphi(M\cdot x-k)
  78. φ = | det M | D M - 1 ( h * φ ) \varphi=|\det M|\cdot D_{M^{-1}}(h*\varphi)
  79. δ \delta
  80. n n
  81. 2 n δ 2^{n}\cdot\delta
  82. 1 2 δ \frac{1}{2}\cdot\delta
  83. n n
  84. 1 2 n + 1 δ \frac{1}{2^{n+1}}\cdot\delta
  85. [ 0 , 1 ) [0,1)
  86. n n
  87. v * ( 1 , - 1 ) n + 1 v*(1,-1)^{n+1}
  88. v v
  89. ( 1 , - 1 ) n + 1 (1,-1)^{n+1}
  90. φ \varphi
  91. φ ( x ) = i s i ( x - i ) k \varphi(x)=\sum_{i\in\mathbb{Z}}\frac{s_{i}}{(x-i)^{k}}
  92. k k
  93. s s
  94. s | ( s 2 ) s|(s\uparrow 2)
  95. h ( z ) [ z , z - 1 ] h ( z ) s ( z ) = s ( z 2 ) \exists h(z)\in\mathbb{R}[z,z^{-1}]\ h(z)\cdot s(z)=s(z^{2})
  96. 2 k - 1 h 2^{k-1}\cdot h

Reflection_formula.html

  1. f ( - x ) = f ( x ) , f(-x)=f(x),\,\!
  2. f ( - x ) = - f ( x ) . f(-x)=-f(x).\,\!
  3. Γ ( z ) Γ ( 1 - z ) = π sin ( π z ) \Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin{(\pi z)}}\!
  4. ψ ( n ) ( 1 - z ) + ( - 1 ) n + 1 ψ ( n ) ( z ) = ( - 1 ) n π d n d z n cot ( π z ) \psi^{(n)}(1-z)+(-1)^{n+1}\psi^{(n)}(z)=(-1)^{n}\pi\frac{d^{n}}{dz^{n}}\cot{(% \pi z)}\,
  5. ln Γ \ln\Gamma
  6. ζ ( 1 - z ) ζ ( z ) = 2 Γ ( z ) ( 2 π ) z cos ( π z 2 ) , \frac{\zeta(1-z)}{\zeta(z)}=\frac{2\,\Gamma(z)}{(2\pi)^{z}}\cos\left(\frac{\pi z% }{2}\right),\,\!
  7. ξ ( z ) = ξ ( 1 - z ) . \xi(z)=\xi(1-z).\,\!

Reflection_theorem.html

  1. ( ζ p ) \mathbb{Q}\left(\zeta_{p}\right)
  2. ( ζ p ) + \mathbb{Q}\left(\zeta_{p}\right)^{+}
  3. ( d ) \mathbb{Q}\left(\sqrt{d}\right)
  4. ( - 3 d ) \mathbb{Q}\left(\sqrt{-3d}\right)
  5. [ A ϕ : A ϕ p ] = q e ϕ . [A_{\phi}:A_{\phi}^{p}]=q^{e_{\phi}}.
  6. ζ g = ζ ω ( g ) for ζ μ p . \zeta^{g}=\zeta^{\omega(g)}\,\text{ for }\zeta\in\mu_{p}.
  7. ϕ * ( g ) = ω ( g ) ϕ ( g - 1 ) . \phi^{*}(g)=\omega(g)\phi(g^{-1}).
  8. K ( ϵ p ) / K K(\sqrt[p]{\epsilon})/K
  9. | e ϕ * - e ϕ | δ ϕ . |e_{\phi^{*}}-e_{\phi}|\leq\delta_{\phi}.

Reflective_subcategory.html

  1. A B A_{B}
  2. r B : B A B r_{B}\colon B\to A_{B}
  3. f : B A f\colon B\to A
  4. f ¯ : A B A \overline{f}\colon A_{B}\to A
  5. f ¯ r B = f \overline{f}\circ r_{B}=f
  6. ( A B , r B ) (A_{B},r_{B})
  7. r B r_{B}
  8. A B A_{B}
  9. E : 𝐀 𝐁 E\colon\mathbf{A}\hookrightarrow\mathbf{B}
  10. R : 𝐁 𝐀 R\colon\mathbf{B}\to\mathbf{A}
  11. r B r_{B}
  12. B B
  13. A B A_{B}
  14. R f Rf
  15. f f
  16. E E
  17. E E
  18. E E
  19. E E

Registered_Retirement_Income_Fund.html

  1. 500000 × 0.0748 = 37400 500000\times 0.0748=37400

Regular_grid.html

  1. ( i d x , j d y ) (i\cdot dx,j\cdot dy)
  2. ( i d x , j d y , k d z ) (i\cdot dx,j\cdot dy,k\cdot dz)

Regular_measure.html

  1. μ ( A ) = sup { μ ( F ) | F A , F compact and measurable } \mu(A)=\sup\{\mu(F)|F\subseteq A,F\mbox{ compact and measurable}~{}\}
  2. μ ( A ) = inf { μ ( G ) | G A , G open and measurable } \mu(A)=\inf\{\mu(G)|G\supseteq A,G\mbox{ open and measurable}~{}\}
  3. μ ( ) = 0 \mu(\emptyset)=0
  4. μ ( { 1 } ) = 0 \mu\left(\{1\}\right)=0\,\,
  5. μ ( A ) = \mu(A)=\infty\,\,
  6. A A

Reinhardt_cardinal.html

  1. { x | ϕ ( x , a ) } \{x|\phi(x,a)\}
  2. Π 2 V \Pi^{V}_{2}

Rejection_rate.html

  1. R R = ( R e j e c t e d p i e c e s / P r o c e s s e d P i e c e s ) * 100 RR=(Rejectedpieces/ProcessedPieces)*100

Relation_algebra.html

  1. ˘ \breve{\ }
  2. ˘ \breve{\ }
  3. ˘ \breve{}
  4. ˘ \breve{\ }
  5. ˘ \breve{\ }
  6. \vee
  7. ˘ \breve{\ }
  8. ˘ ˘ \breve{\ }\breve{\ }
  9. ˘ \breve{\ }
  10. ˘ \breve{\ }
  11. ˘ \breve{\ }
  12. ˘ \breve{\ }
  13. ˘ \breve{\ }
  14. ˘ \breve{\ }
  15. ˘ \breve{\ }
  16. ˘ \breve{\ }
  17. ˘ \breve{\ }
  18. ˘ {}^{\breve{\ }}
  19. ˘ {}^{\breve{\ }}
  20. ˘ {}^{\breve{\ }}
  21. ˘ {}^{\breve{\ }}
  22. ˘ {}^{\breve{\ }}
  23. ˘ {}^{\breve{\ }}
  24. ˘ {}^{\breve{\ }}
  25. ˘ {}^{\breve{\ }}
  26. ˘ {}^{\breve{\ }}
  27. ˘ {}^{\breve{\ }}
  28. ˘ {}^{\breve{\ }}
  29. ˘ {}^{\breve{\ }}
  30. ˘ {}^{\breve{\ }}
  31. ˘ {}^{\breve{\ }}
  32. ˘ \breve{\ }
  33. ˘ \breve{\ }
  34. ˘ \breve{\ }
  35. ˘ \breve{\ }
  36. ˘ {}^{\breve{\ }}
  37. ˘ {}^{\breve{\ }}

Relative_velocity.html

  1. v B | A \vec{v}_{\mathrm{B|A}}
  2. v BA \vec{v}_{\mathrm{BA}}
  3. v BrelA \vec{v}_{\mathrm{BrelA}}
  4. v M | E 50 k m / h r = v M | T 10 k m / h r + v T | E 40 k m / h r , \underbrace{\vec{v}_{M|E}}_{50\;km/hr}=\underbrace{\vec{v}_{M|T}}_{10\;km/hr}+% \underbrace{\vec{v}_{T|E}}_{40\;km/hr}\,,
  5. v M | E \vec{v}_{M|E}
  6. v M | T \vec{v}_{M|T}
  7. v T | E \vec{v}_{T|E}
  8. r A = r A i + v A t \vec{r}_{A}=\vec{r}_{Ai}+\vec{v}_{A}t
  9. r B = r B i + v B t \vec{r}_{B}=\vec{r}_{Bi}+\vec{v}_{B}t
  10. r B - r A \vec{r}_{B}-\vec{r}_{A}
  11. r B - r A = r B i - r A i i n i t i a l s e p a r a t i o n + ( v B - v A ) t r e l a t i v e v e l o c i t y \vec{r}_{B}-\vec{r}_{A}=\underbrace{\vec{r}_{Bi}-\vec{r}_{Ai}}_{initial\;% separation}+\underbrace{(\vec{v}_{B}-\vec{v}_{A})t}_{relative\;velocity}
  12. v B | A = v B - v A \vec{v}_{B|A}=\vec{v}_{B}-\vec{v}_{A}
  13. v A | C = v A \vec{v}_{A|C}=\vec{v}_{A}
  14. v B | C = v B \vec{v}_{B|C}=\vec{v}_{B}
  15. v B | A = v B | C - v A | C \vec{v}_{B|A}=\vec{v}_{B|C}-\vec{v}_{A|C}\Rightarrow
  16. v B | C = v B | A + v A | C \vec{v}_{B|C}=\vec{v}_{B|A}+\vec{v}_{A|C}
  17. x = x - v t x^{\prime}=x-vt
  18. t = t t^{\prime}=t
  19. d x = d x - v d t dx^{\prime}=dx-vdt
  20. d t = d t dt^{\prime}=dt
  21. d x d t = d x d t - v \frac{dx^{\prime}}{dt^{\prime}}=\frac{dx}{dt}-v
  22. d x / d t = v A | O dx/dt=v_{A|O}
  23. d x / d t = v A | O dx^{\prime}/dt=v_{A|O^{\prime}}
  24. O O
  25. O O^{\prime}
  26. v = v O | O v=v_{O^{\prime}|O}
  27. v A | O = v A | O - v O | O v A | O = v A | O + v O | O v_{A|O^{\prime}}=v_{A|O}-v_{O^{\prime}|O}\Rightarrow v_{A|O}=v_{A|O^{\prime}}+% v_{O^{\prime}|O}
  28. v B | A \vec{v}_{\mathrm{B|A}}
  29. v B | A = - v A | B \vec{v}_{\mathrm{B|A}}=-\vec{v}_{\mathrm{A|B}}
  30. v B | A = v A | B = v B | A = v A | B \|\vec{v}_{\mathrm{B|A}}\|=\|\vec{v}_{\mathrm{A|B}}\|=v_{\mathrm{B|A}}=v_{% \mathrm{A|B}}
  31. v B | A = v B - v A 1 - v A v B c 2 \vec{v}_{\mathrm{B|A}}=\frac{\vec{v}_{\mathrm{B}}-\vec{v}_{\mathrm{A}}}{1-% \frac{\vec{v}_{\mathrm{A}}\vec{v}_{\mathrm{B}}}{c^{2}}}
  32. v B | A = | v B - v A | 1 - v A v B c 2 v_{\mathrm{B|A}}=\frac{\left|v_{\mathrm{B}}-v_{\mathrm{A}}\right|}{1-\frac{v_{% \mathrm{A}}v_{\mathrm{B}}}{c^{2}}}
  33. v B | A \vec{v}_{\mathrm{B|A}}
  34. v B | A = v B γ A - v A \vec{v}_{\mathrm{B|A}}={\frac{\vec{v}_{\mathrm{B}}}{\gamma_{\mathrm{A}}}}-\vec% {v}_{\mathrm{A}}
  35. γ A = 1 1 - ( v A c ) 2 \gamma_{\mathrm{A}}=\frac{1}{\sqrt{1-(\frac{v_{\mathrm{A}}}{c})^{2}}}
  36. v B | A = c 4 - ( c 2 - v A 2 ) ( c 2 - v B 2 ) c v_{\mathrm{B|A}}=\frac{\sqrt{c^{4}-(c^{2}-v_{\mathrm{A}}^{2})(c^{2}-v_{\mathrm% {B}}^{2})}}{c}
  37. v B | A \vec{v}_{\mathrm{B|A}}
  38. v B | A = 1 γ A ( 1 - v A v B c 2 ) [ v B - v A + v A ( γ A - 1 ) ( v A v B v A 2 - 1 ) ] \vec{v}_{\mathrm{B|A}}={\frac{1}{\gamma_{\mathrm{A}}(1-\frac{\vec{v}_{\mathrm{% A}}\vec{v}_{\mathrm{B}}}{c^{2}})}}[\vec{v}_{\mathrm{B}}-\vec{v}_{\mathrm{A}}+% \vec{v}_{\mathrm{A}}(\gamma_{\mathrm{A}}-1)(\frac{\vec{v}_{\mathrm{A}}\cdot% \vec{v}_{\mathrm{B}}}{v_{\mathrm{A}}^{2}}-1)]
  39. γ A = 1 1 - ( v A c ) 2 \gamma_{\mathrm{A}}=\frac{1}{\sqrt{1-(\frac{v_{\mathrm{A}}}{c})^{2}}}
  40. v B | A = 1 - ( c 2 - v A 2 ) ( c 2 - v B 2 ) ( c 2 - v A v B ) 2 * c v_{\mathrm{B|A}}=\sqrt{1-\frac{(c^{2}-v_{\mathrm{A}}^{2})(c^{2}-v_{\mathrm{B}}% ^{2})}{(c^{2}-\vec{v}_{\mathrm{A}}\cdot\vec{v}_{\mathrm{B}})^{2}}}*c
  41. r = r - v t \vec{r}\,^{\prime}=\vec{r}-\vec{v}t

Relative_viscosity.html

  1. η r \eta_{r}
  2. η \eta
  3. η s \eta_{s}
  4. η r = η η s \eta_{r}=\frac{\eta}{\eta_{s}}

Relativistic_particle.html

  1. m c 2 mc^{2}

Relativity_of_simultaneity.html

  1. t = t - v x / c 2 1 - v 2 / c 2 , t^{\prime}=\frac{t-{v\,x/c^{2}}}{\sqrt{1-v^{2}/c^{2}}}\ ,
  2. x = x - v t 1 - v 2 / c 2 , x^{\prime}=\frac{x-v\,t}{\sqrt{1-v^{2}/c^{2}}}\ ,
  3. y = y , y^{\prime}=y\ ,
  4. z = z , z^{\prime}=z\ ,

Relativity_priority_dispute.html

  1. m r = E / c 2 m_{r}=E/c^{2}
  2. Δ E \Delta E
  3. Δ E / c 2 \Delta E/c^{2}
  4. g μ ν g^{\mu\nu}
  5. g μ ν g^{\mu\nu}
  6. q s q_{s}
  7. g μ ν g_{\mu\nu}
  8. g g
  9. g μ ν g^{\mu\nu}
  10. q s q_{s}
  11. E = m c 2 E=mc^{2}

Relaxation_(NMR).html

  1. M z , eq M_{z,\mathrm{eq}}
  2. M z ( t ) = M z , eq - [ M z , eq - M z ( 0 ) ] e - t / T 1 M_{z}(t)=M_{z,\mathrm{eq}}-[M_{z,\mathrm{eq}}-M_{z}(0)]e^{-t/T_{1}}
  3. M z ( 0 ) = 0 M_{z}(0)=0
  4. M z ( t ) = M z , eq ( 1 - e - t / T 1 ) M_{z}(t)=M_{z,\mathrm{eq}}\left(1-e^{-t/T_{1}}\right)
  5. M z ( 0 ) = - M z , eq M_{z}(0)=-M_{z,\mathrm{eq}}
  6. M z ( t ) = M z , eq ( 1 - 2 e - t / T 1 ) M_{z}(t)=M_{z,\mathrm{eq}}\left(1-2e^{-t/T_{1}}\right)
  7. T 1 T_{1}
  8. T 2 T_{2}
  9. M M_{\perp}
  10. M x y ( t ) = M x y ( 0 ) e - t / T 2 M_{xy}(t)=M_{xy}(0)e^{-t/T_{2}}\,
  11. T 2 T_{2}
  12. 1 T 2 * = 1 T 2 + 1 T i n h o m = 1 T 2 + γ Δ B 0 \frac{1}{T_{2}^{*}}=\frac{1}{T_{2}}+\frac{1}{T_{inhom}}=\frac{1}{T_{2}}+\gamma% \Delta B_{0}
  13. 2 T 1 T 2 T 2 * 2T_{1}\geq T_{2}\geq T_{2}^{*}
  14. T 1 T_{1}
  15. T 2 T_{2}
  16. M x ( t ) t = γ ( M ( t ) × B ( t ) ) x - M x ( t ) T 2 \frac{\partial M_{x}(t)}{\partial t}=\gamma({M}(t)\times{B}(t))_{x}-\frac{M_{x% }(t)}{T_{2}}
  17. M y ( t ) t = γ ( M ( t ) × B ( t ) ) y - M y ( t ) T 2 \frac{\partial M_{y}(t)}{\partial t}=\gamma({M}(t)\times{B}(t))_{y}-\frac{M_{y% }(t)}{T_{2}}
  18. M z ( t ) t = γ ( M ( t ) × B ( t ) ) z - M z ( t ) - M 0 T 1 \frac{\partial M_{z}(t)}{\partial t}=\gamma({M}(t)\times{B}(t))_{z}-\frac{M_{z% }(t)-M_{0}}{T_{1}}
  19. e - t / τ c e^{-t/\tau_{c}}
  20. τ c \tau_{c}
  21. 1 T 1 = K [ τ c 1 + ω 0 2 τ c 2 + 4 τ c 1 + 4 ω 0 2 τ c 2 ] \frac{1}{T_{1}}=K[\frac{\tau_{c}}{1+\omega_{0}^{2}\tau_{c}^{2}}+\frac{4\tau_{c% }}{1+4\omega_{0}^{2}\tau_{c}^{2}}]
  22. 1 T 2 = K 2 [ 3 τ c + 5 τ c 1 + ω 0 2 τ c 2 + 2 τ c 1 + 4 ω 0 2 τ c 2 ] \frac{1}{T_{2}}=\frac{K}{2}[3\tau_{c}+\frac{5\tau_{c}}{1+\omega_{0}^{2}\tau_{c% }^{2}}+\frac{2\tau_{c}}{1+4\omega_{0}^{2}\tau_{c}^{2}}]
  23. ω 0 \omega_{0}
  24. B 0 B_{0}
  25. τ c \tau_{c}
  26. K = 3 μ 0 2 160 π 2 2 γ 4 r 6 K=\frac{3\mu_{0}^{2}}{160\pi^{2}}\frac{\hbar^{2}\gamma^{4}}{r^{6}}
  27. μ 0 \mu_{0}
  28. = h 2 π \hbar=\frac{h}{2\pi}
  29. τ c \tau_{c}
  30. 10 - 12 10^{-12}
  31. ω 0 τ c = 3.2 × 10 - 5 \omega_{0}\tau_{c}=3.2\times 10^{-5}
  32. T 1 = ( 1.02 × 10 10 [ 5 × 10 - 12 1 + ( 3.2 × 10 - 5 ) 2 + 4 5 × 10 - 12 1 + 4 ( 3.2 × 10 - 5 ) 2 ] ) - 1 T_{1}=\left(1.02\times 10^{10}\left[\frac{5\times 10^{-12}}{1+(3.2\times 10^{-% 5})^{2}}+\frac{4\cdot 5\times 10^{-12}}{1+4\cdot(3.2\times 10^{-5})^{2}}\right% ]\right)^{-1}
  33. T 2 = ( 1.02 × 10 10 2 [ 3 5 × 10 - 12 + 5 5 × 10 - 12 1 + ( 3.2 × 10 - 5 ) 2 + 2 5 × 10 - 12 1 + 4 ( 3.2 × 10 - 5 ) 2 ] ) - 1 T_{2}=\left(\frac{1.02\times 10^{10}}{2}\left[3\cdot 5\times 10^{-12}+\frac{5% \cdot 5\times 10^{-12}}{1+\left(3.2\times 10^{-5}\right)^{2}}+\frac{2\cdot 5% \times 10^{-12}}{1+4\cdot(3.2\times 10^{-5})^{2}}\right]\right)^{-1}

Relay_channel.html

  1. X 1 , X 2 , Y 1 , X_{1},X_{2},Y_{1},
  2. Y Y
  3. p ( y , y 1 | x 1 , x 2 ) p(y,y_{1}|x_{1},x_{2})
  4. p ( x 1 , x 2 ) p(x_{1},x_{2})
  5. max p ( x 1 , x 2 ) min ( I ( x 1 ; y 1 | x 2 ) , I ( x 1 , x 2 ; y ) ) \max_{p(x_{1},x_{2})}\min\left(I\left(x_{1};y_{1}|x_{2}\right),I\left(x_{1},x_% {2};y\right)\right)
  6. max p ( x 1 ) p ( y ^ 1 | y 1 ) p ( x 2 ) I ( x 1 ; y 1 ^ , y | x 2 ) \max_{p(x_{1})p(\hat{y}_{1}|y_{1})p(x_{2})}I\left(x_{1};\hat{y_{1}},y|x_{2}\right)
  7. I ( x 2 ; y ) I ( y 1 ; y ^ 1 | y ) I(x_{2};y)\geq I(y_{1};\hat{y}_{1}|y)
  8. C max p ( x 1 , x 2 ) min ( I ( x 1 ; y 1 , y | x 2 ) , I ( x 1 , x 2 ; y ) ) C\leq\max_{p(x_{1},x_{2})}\min\left(I\left(x_{1};y_{1},y|x_{2}\right),I\left(x% _{1},x_{2};y\right)\right)
  9. x 1 x_{1}
  10. y 1 y_{1}
  11. x 2 x_{2}
  12. p ( y | x 1 , x 2 , y 1 ) = p ( y | x 2 , y 1 ) p(y|x_{1},x_{2},y_{1})=p(y|x_{2},y_{1})
  13. p ( y , y 1 | x 1 , x 2 ) = p ( y | x 1 , x 2 ) p ( y 1 | y , x 2 ) p(y,y_{1}|x_{1},x_{2})=p(y|x_{1},x_{2})p(y_{1}|y,x_{2})

Relevance_vector_machine.html

  1. k ( 𝐱 , 𝐱 ) = j = 1 N 1 α j φ ( 𝐱 , 𝐱 j ) φ ( 𝐱 , 𝐱 j ) k(\mathbf{x},\mathbf{x^{\prime}})=\sum_{j=1}^{N}\frac{1}{\alpha_{j}}\varphi(% \mathbf{x},\mathbf{x}_{j})\varphi(\mathbf{x}^{\prime},\mathbf{x}_{j})
  2. φ \varphi
  3. α j \alpha_{j}
  4. w N ( 0 , α - 1 I ) w\sim N(0,\alpha^{-1}I)
  5. 𝐱 1 , , 𝐱 N \mathbf{x}_{1},\ldots,\mathbf{x}_{N}

Remez_algorithm.html

  1. n + 2 n+2
  2. x 1 , x 2 , , x n + 2 x_{1},x_{2},...,x_{n+2}
  3. b 0 + b 1 x i + + b n x i n + ( - 1 ) i E = f ( x i ) b_{0}+b_{1}x_{i}+...+b_{n}x_{i}^{n}+(-1)^{i}E=f(x_{i})
  4. i = 1 , 2 , n + 2 i=1,2,...n+2
  5. b 0 , b 1 b n b_{0},b_{1}...b_{n}
  6. b i b_{i}
  7. P n P_{n}
  8. | P n ( x ) - f ( x ) | |P_{n}(x)-f(x)|
  9. m M m\in M
  10. P n P_{n}
  11. f - L n ( f ) ( 1 + L n ) inf p P n f - p \lVert f-L_{n}(f)\rVert_{\infty}\leq(1+\lVert L_{n}\rVert_{\infty})\inf_{p\in P% _{n}}\lVert f-p\rVert
  12. L n = Λ ¯ n ( T ) = max - 1 x 1 λ n ( T ; x ) , \lVert L_{n}\rVert_{\infty}=\overline{\Lambda}_{n}(T)=\max_{-1\leq x\leq 1}% \lambda_{n}(T;x),
  13. λ n ( T ; x ) = j = 1 n + 1 | l j ( x ) | , l j ( x ) = i j i = 1 n + 1 ( x - t i ) ( t j - t i ) . \lambda_{n}(T;x)=\sum_{j=1}^{n+1}\left|l_{j}(x)\right|,\quad l_{j}(x)=\prod_{% \stackrel{i=1}{i\neq j}}^{n+1}\frac{(x-t_{i})}{(t_{j}-t_{i})}.
  14. Λ ¯ n ( T ) = min - 1 x 1 λ n ( T ; x ) \underline{\Lambda}_{n}(T)=\min_{-1\leq x\leq 1}\lambda_{n}(T;x)
  15. Λ ¯ n - Λ ¯ n 0. \overline{\Lambda}_{n}-\underline{\Lambda}_{n}\geq 0.
  16. Λ ¯ n ( T ) = 2 π log ( n + 1 ) + 2 π ( γ + log 8 π ) + α n + 1 \overline{\Lambda}_{n}(T)=\frac{2}{\pi}\log(n+1)+\frac{2}{\pi}\left(\gamma+% \log\frac{8}{\pi}\right)+\alpha_{n+1}
  17. 0 < α n < π 72 n 2 0<\alpha_{n}<\frac{\pi}{72n^{2}}
  18. n 1 , n\geq 1,
  19. Λ ¯ n ( T ) 2 π log ( n + 1 ) + 1 \overline{\Lambda}_{n}(T)\leq\frac{2}{\pi}\log(n+1)+1
  20. n 3 n\geq 3
  21. T ^ \hat{T}
  22. Λ ¯ n ( T ^ ) - Λ ¯ n ( T ^ ) < Λ ¯ 3 - 1 6 cot π 8 + π 64 1 sin 2 ( 3 π / 16 ) - 2 π ( γ - log π ) 0.201. \overline{\Lambda}_{n}(\hat{T})-\underline{\Lambda}_{n}(\hat{T})<\overline{% \Lambda}_{3}-\frac{1}{6}\cot\frac{\pi}{8}+\frac{\pi}{64}\frac{1}{\sin^{2}(3\pi% /16)}-\frac{2}{\pi}(\gamma-\log\pi)\approx 0.201.
  23. n 40 n\geq 40
  24. Λ ¯ n ( T ^ ) - Λ ¯ n ( T ^ ) < 0.0196. \overline{\Lambda}_{n}(\hat{T})-\underline{\Lambda}_{n}(\hat{T})<0.0196.
  25. x 0 , x 1 , x n + 1 x_{0},x_{1},...x_{n+1}
  26. b 0 + b 1 x i + + b n x i n + ( - 1 ) i E = f ( x i ) b_{0}+b_{1}x_{i}+...+b_{n}x_{i}^{n}+(-1)^{i}E=f(x_{i})
  27. i = 0 , 1 , n + 1 i=0,1,...n+1
  28. b 0 , b 1 , b n b_{0},b_{1},...b_{n}
  29. ( - 1 ) i E (-1)^{i}E
  30. x 0 , , x n + 1 x_{0},...,x_{n+1}
  31. O ( n 2 ) O(n^{2})
  32. O ( n 3 ) O(n^{3})
  33. p 1 ( x ) p_{1}(x)
  34. f ( x ) f(x)
  35. p 2 ( x ) p_{2}(x)
  36. ( - 1 ) i (-1)^{i}
  37. p 1 ( x i ) = f ( x i ) , p 2 ( x i ) = ( - 1 ) i , i = 0 , , n . p_{1}(x_{i})=f(x_{i}),p_{2}(x_{i})=(-1)^{i},i=0,...,n.
  38. 0 , , n 0,...,n
  39. O ( n 2 ) O(n^{2})
  40. p 2 ( x ) p_{2}(x)
  41. x i - 1 x_{i-1}
  42. x i , i = 1 , , n x_{i},\ i=1,...,n
  43. x n x_{n}
  44. x n + 1 x_{n+1}
  45. p 2 ( x n ) p_{2}(x_{n})
  46. p 2 ( x n + 1 ) p_{2}(x_{n+1})
  47. ( - 1 ) n (-1)^{n}
  48. p ( x ) := p 1 ( x ) - p 2 ( x ) E p(x):=p_{1}(x)-p_{2}(x)\!\cdot\!E
  49. p ( x i ) = p 1 ( x i ) - p 2 ( x i ) E = f ( x i ) - ( - 1 ) i E , i = 0 , , n . p(x_{i})=p_{1}(x_{i})-p_{2}(x_{i})\!\cdot\!E\ =\ f(x_{i})-(-1)^{i}E,\ \ \ \ i=% 0,\ldots,n.
  50. i = 0 , , n i=0,...,n
  51. p ( x n + 1 ) = p 1 ( x n + 1 ) - p 2 ( x n + 1 ) E = f ( x n + 1 ) - ( - 1 ) n + 1 E p(x_{n+1})\ =\ p_{1}(x_{n+1})-p_{2}(x_{n+1})\!\cdot\!E\ =\ f(x_{n+1})-(-1)^{n+% 1}E
  52. E := p 1 ( x n + 1 ) - f ( x n + 1 ) p 2 ( x n + 1 ) + ( - 1 ) n . E\ :=\ \frac{p_{1}(x_{n+1})-f(x_{n+1})}{p_{2}(x_{n+1})+(-1)^{n}}.
  53. p ( x ) b 0 + b 1 x + + b n x n p(x)\equiv b_{0}+b_{1}x+\ldots+b_{n}x^{n}
  54. p ( x i ) - f ( x i ) = - ( - 1 ) i E , i = 0 , , n + 1. p(x_{i})-f(x_{i})\ =\ -(-1)^{i}E,\ \ i=0,...,n\!+\!1.
  55. p ~ ( x ) \tilde{p}(x)
  56. p ( x ) - p ~ ( x ) = ( p ( x ) - f ( x ) ) - ( p ~ ( x ) - f ( x ) ) p(x)-\tilde{p}(x)=(p(x)-f(x))-(\tilde{p}(x)-f(x))
  57. x i x_{i}
  58. b 0 + b 1 x + + b n x n b_{0}+b_{1}x+...+b_{n}x^{n}
  59. p ( x ) p(x)
  60. x 0 , , x n + 1 x_{0},...,x_{n+1}
  61. ± E \pm E
  62. x i x_{i}
  63. x ¯ i \bar{x}_{i}
  64. x i x_{i}
  65. x ¯ 0 \bar{x}_{0}
  66. x ¯ i \bar{x}_{i}
  67. x i x_{i}
  68. x ¯ n + 1 \bar{x}_{n+1}
  69. z i := p ( x ¯ i ) - f ( x ¯ i ) z_{i}:=p(\bar{x}_{i})-f(\bar{x}_{i})
  70. | z i | |z_{i}|
  71. z 0 , , z n + 1 z_{0},...,z_{n+1}
  72. min { | z i | } E \min\{|z_{i}|\}\geq E
  73. max { | z i | } \max\{|z_{i}|\}
  74. min { | z i | } \min\,\{|z_{i}|\}
  75. max { | z i | } \max\,\{|z_{i}|\}
  76. max { | z i | } - min { | z i | } \max\{|z_{i}|\}-\min\{|z_{i}|\}

Remineralisation.html

  1. \rightarrow

Renal_blood_flow.html

  1. R P F = U x V P a - P v RPF=\frac{U_{x}V}{P_{a}-P_{v}}
  2. R P F a × P a = R P F v × P v + U x × V RPF_{a}\times P_{a}=RPF_{v}\times P_{v}+U_{x}\times V
  3. R P F × P a = R P F × P v + U x V RPF\times P_{a}=RPF\times P_{v}+U_{x}V
  4. R P F = U x V P a - P v RPF=\frac{U_{x}V}{P_{a}-P_{v}}
  5. e R P F = U x P a V eRPF=\frac{U_{x}}{P_{a}}V
  6. e R P F = U P A H P P A H V eRPF=\frac{U_{PAH}}{P_{PAH}}V
  7. R B F = R P F 1 - H c t RBF=\frac{RPF}{1-Hct}

Renato_Caccioppoli.html

  1. n n
  2. C 2 C^{2}

Rencontres_numbers.html

  1. k n {}_{n}\!\!\diagdown\!\!^{k}
  2. D 0 , 0 = 1 , D_{0,0}=1,\!
  3. D 1 , 0 = 0 , D_{1,0}=0,\!
  4. D n + 2 , 0 = ( n + 1 ) ( D n + 1 , 0 + D n , 0 ) D_{n+2,0}=(n+1)(D_{n+1,0}+D_{n,0})\!
  5. D n , 0 = [ n ! e ] D_{n,0}=\left[{n!\over e}\right]
  6. D n , k = ( n k ) D n - k , 0 . D_{n,k}={n\choose k}\cdot D_{n-k,0}.
  7. D n , m = n ! m ! [ z n - m ] e - z 1 - z = n ! m ! k = 0 n - m ( - 1 ) k k ! . D_{n,m}=\frac{n!}{m!}[z^{n-m}]\frac{e^{-z}}{1-z}=\frac{n!}{m!}\sum_{k=0}^{n-m}% \frac{(-1)^{k}}{k!}.
  8. D n , m = ( n m ) D n - m , 0 and D n , m n ! e - 1 m ! D_{n,m}={n\choose m}D_{n-m,0}\;\;\mbox{ and }~{}\;\;\frac{D_{n,m}}{n!}\approx% \frac{e^{-1}}{m!}
  9. D n , k n ! . {D_{n,k}\over n!}.
  10. lim n D n , k n ! = e - 1 k ! . \lim_{n\to\infty}{D_{n,k}\over n!}={e^{-1}\over k!}.

Repetition_code.html

  1. r r
  2. ( 3 , 1 ) (3,1)
  3. m = 101001 m=101001
  4. c = 111000111000000111 c=111000111000000111
  5. ( 3 , 1 ) (3,1)
  6. c = 110001111 c=110001111
  7. m = 101 m=101
  8. d m i n d_{min}
  9. r r
  10. ( r , 1 ) (r,1)
  11. w m i n w_{min}
  12. r 2 \tfrac{r}{2}
  13. r 2 \tfrac{r}{2}

Replica_trick.html

  1. lim n 0 Z n - 1 n = ln Z . \lim_{n\to 0}{Z^{n}-1\over n}=\ln Z.
  2. f ( z ) f(z)
  3. z z
  4. z z
  5. f ( z ) f(z)
  6. z z
  7. - ln Z [ J i j ] -\ln Z[J_{ij}]
  8. J i j J_{ij}
  9. Z [ J i j ] e - β J i j Z[J_{ij}]\sim e^{-\beta J_{ij}}
  10. Z [ J i j ] Z[J_{ij}]
  11. J i j J_{ij}
  12. d J i j e - β J - α J 2 \int dJ_{ij}e^{-\beta J-\alpha J^{2}}
  13. ln Z i j = lim n 0 Z n - 1 n \ln Z_{ij}=\lim_{n\to 0}\dfrac{Z^{n}-1}{n}
  14. n n
  15. Z n Z^{n}
  16. n n
  17. n 0 n\to 0
  18. n 0 n\to 0
  19. F = F [ J i j ] ¯ = - k B T ln Z [ J ] ¯ F=\overline{F[J_{ij}]}=-k_{B}T\overline{\ln Z[J]}
  20. J i j J_{ij}
  21. i i
  22. j j
  23. [ ] [\cdots]
  24. J J
  25. Z n Z^{n}
  26. n n
  27. lim n 0 x n - 1 n = ln x . \lim_{n\to 0}{x^{n}-1\over n}=\ln x.
  28. exp y = 1 + y + 1 2 ! y 2 + 1 3 ! y 3 + = lim N r = 0 N N ! r ! ( N - r ) ! ( y N ) r = lim N ( 1 + y N ) N . \exp y=1+y+{1\over 2!}y^{2}+{1\over 3!}y^{3}+\dots=\lim_{N\to\infty}\sum_{r=0}% ^{N}{N!\over r!(N-r)!}({y\over N})^{r}=\lim_{N\to\infty}(1+{y\over N})^{N}.
  29. y = ln x y=\ln x
  30. x = lim N ( 1 + ln x N ) N x=\lim_{N\to\infty}(1+{\ln x\over N})^{N}
  31. ln x = lim N ( x 1 / N - 1 ) 1 / N = lim n 0 x n - 1 n . \ln x=\lim_{N\to\infty}{(x^{1/N}-1)\over 1/N}=\lim_{n\to 0}{x^{n}-1\over n}.
  32. lim n 0 x n - 1 n = d x n d n | n = 0 = x n ln x | n = 0 = ln x . \lim_{n\to 0}{x^{n}-1\over n}=\left.\frac{\mathrm{d}x^{n}}{\mathrm{d}n}\right|% _{n=0}=\left.x^{n}\ln x\right|_{n=0}=\ln x.

Reproductive_value_(population_genetics).html

  1. x \ell_{x}
  2. m x m_{x}
  3. x \ell_{x}
  4. x x
  5. m x m_{x}
  6. x . x.
  7. v x = y = x λ - ( y - x + 1 ) y x m y v_{x}=\sum_{y=x}^{\infty}\lambda^{-(y-x+1)}\frac{\ell_{y}}{\ell_{x}}m_{y}
  8. λ \lambda
  9. v x = x e - r ( y - x ) y x m y d y v_{x}=\int_{x}^{\infty}e^{-r(y-x)}\frac{\ell_{y}}{\ell_{x}}m_{y}dy
  10. r r

Resampling_(statistics).html

  1. A A
  2. B B
  3. x ¯ A \bar{x}_{A}
  4. x ¯ B \bar{x}_{B}
  5. n A n_{A}
  6. n B n_{B}
  7. 0 {}_{0}
  8. A A
  9. B B
  10. n A n_{A}
  11. n B n_{B}
  12. N \scriptstyle\ N
  13. N = 10000 \scriptstyle\ N=10000
  14. p ^ = 0.05 \scriptstyle\ \hat{p}=0.05
  15. p \scriptstyle\ p
  16. [ 0.044 , 0.056 ] \scriptstyle\ [0.044,0.056]
  17. p α \scriptstyle\ p\leq\alpha
  18. α \scriptstyle\ \alpha
  19. α = 0.05 \scriptstyle\ \alpha=0.05
  20. α = 0.05 \scriptstyle\ \alpha=0.05
  21. p α \scriptstyle\ p\leq\alpha
  22. α \scriptstyle\ \alpha
  23. p α \scriptstyle\ p\leq\alpha
  24. ϵ \scriptstyle\ \epsilon
  25. p ^ > α \scriptstyle\ \hat{p}>\alpha
  26. p α \scriptstyle\ p\leq\alpha
  27. p α \scriptstyle\ p\leq\alpha
  28. p > α \scriptstyle\ p>\alpha
  29. 1 - ϵ \scriptstyle\ 1-\epsilon
  30. ϵ \scriptstyle\ \epsilon

Residue-class-wise_affine_group.html

  1. \mathbb{Z}
  2. f : f:\mathbb{Z}\rightarrow\mathbb{Z}
  3. m m
  4. f f
  5. m m
  6. r ( m ) / m r(m)\in\mathbb{Z}/m\mathbb{Z}
  7. a r ( m ) , b r ( m ) , c r ( m ) a_{r(m)},b_{r(m)},c_{r(m)}\in\mathbb{Z}
  8. f f
  9. r ( m ) = { r + k m k } r(m)=\{r+km\mid k\in\mathbb{Z}\}
  10. f | r ( m ) : r ( m ) , n a r ( m ) n + b r ( m ) c r ( m ) f|_{r(m)}:r(m)\rightarrow\mathbb{Z},\ n\mapsto\frac{a_{r(m)}\cdot n+b_{r(m)}}{% c_{r(m)}}
  11. \mathbb{Z}
  12. r 1 ( m 1 ) r_{1}(m_{1})
  13. r 2 ( m 2 ) r_{2}(m_{2})
  14. \mathbb{Z}
  15. r 1 + k m 1 r_{1}+km_{1}
  16. r 2 + k m 2 r_{2}+km_{2}
  17. k k\in\mathbb{Z}
  18. 0 r 1 < m 1 0\leq r_{1}<m_{1}
  19. 0 r 2 < m 2 0\leq r_{2}<m_{2}
  20. \mathbb{Z}
  21. \mathbb{Z}

Resistor_ladder.html

  1. 2 n - 1 \scriptstyle 2^{n}-1
  2. 2 n \scriptstyle 2n

RESOLFT.html

  1. Δ d = λ π n I I s a t \Delta d=\frac{\lambda}{\pi\cdot n\cdot\sqrt{\frac{I}{Isat}}}
  2. I s a t I_{sat}
  3. I I
  4. N A = n sin α NA=n\sin\alpha
  5. Δ d = λ 2 n sin α 1 + I I s a t \Delta d=\frac{\lambda}{2n\cdot\sin\alpha\cdot\sqrt{1+\frac{I}{Isat}}}
  6. Δ d \Delta d
  7. ς = I I s a t \varsigma=\frac{I}{Isat}
  8. I I
  9. I s a t I_{sat}

Resolvent_formalism.html

  1. R ( z ; A ) = ( A - z I ) - 1 . R(z;A)=(A-zI)^{-1}.
  2. λ \lambda
  3. C λ C_{\lambda}
  4. λ \lambda
  5. - 1 2 π i C λ ( A - z I ) - 1 d z \frac{-1}{2\pi i}\oint_{C_{\lambda}}(A-zI)^{-1}~{}dz
  6. λ \lambda
  7. U ( t ) = exp ( i t A ) U(t)=\exp(itA)
  8. R ( z ; A ) = 0 e - z t U ( t ) d t . R(z;A)=\int_{0}^{\infty}e^{-zt}U(t)dt.
  9. z , w z,w
  10. ρ ( A ) \rho(A)
  11. A A
  12. R ( z ; A ) - R ( w ; A ) = ( z - w ) R ( z ; A ) R ( w ; A ) . R(z;A)-R(w;A)=(z-w)R(z;A)R(w;A)\,.
  13. ( z I - A ) - 1 (zI-A)^{-1}
  14. A A
  15. B B
  16. z z
  17. ρ ( A ) ρ ( B ) \rho(A)\cap\rho(B)
  18. R ( z ; A ) - R ( z ; B ) = R ( z ; A ) ( B - A ) R ( z ; B ) . R(z;A)-R(z;B)=R(z;A)(B-A)R(z;B)\,.
  19. A : H H A:H\to H
  20. H H
  21. z ρ ( A ) z\in\rho(A)
  22. R ( z ; A ) R(z;A)
  23. A A
  24. σ ( A ) \sigma(A)
  25. A A
  26. \mathbb{C}
  27. A A
  28. σ ( A ) \sigma(A)\subset\mathbb{R}
  29. { v i } i \{v_{i}\}_{i\in\mathbb{N}}
  30. A A
  31. { λ i } i \{\lambda_{i}\}_{i\in\mathbb{N}}
  32. { λ i } \{\lambda_{i}\}

Resource_bounded_measure.html

  1. \R n \R^{n}
  2. { 0 , 1 } \{0,1\}^{\infty}
  3. d : { 0 , 1 } * [ 0 , ) d:\{0,1\}^{*}\to[0,\infty)
  4. d ( w ) = d ( w 0 ) + d ( w 1 ) 2 d(w)=\frac{d(w0)+d(w1)}{2}
  5. S { 0 , 1 } S\in\{0,1\}^{\infty}
  6. lim sup n d ( S n ) = , \limsup_{n\to\infty}d(S\upharpoonright n)=\infty,
  7. S n S\upharpoonright n
  8. X { 0 , 1 } X\subseteq\{0,1\}^{\infty}
  9. w { 0 , 1 } * w\in\{0,1\}^{*}
  10. X { 0 , 1 } X\subseteq\{0,1\}^{\infty}

Response_amplitude_operator.html

  1. [ M + A ( ω ) ] x ¨ + B ( ω ) x ˙ + C x = F ( ω ) \big[M+A(\omega)\big]\ddot{x}+B(\omega)\dot{x}+Cx=F(\omega)
  2. x x
  3. ω \omega
  4. M M
  5. A ( ω ) A(\omega)
  6. B ( ω ) B(\omega)
  7. C C
  8. F ( ω ) F(\omega)
  9. x x
  10. ζ a \zeta_{a}
  11. x x
  12. RAO ( ω ) = x ζ a = F 0 C - ( M + A ( ω ) ) ω 2 + i B ( ω ) ω \mathrm{RAO}(\omega)=\frac{x}{\zeta_{a}}=\frac{F_{0}}{C-(M+A(\omega))\omega^{2% }+iB(\omega)\omega}
  13. F 0 F_{0}
  14. i i
  15. B v B_{v}
  16. B v = ξ B crit B_{v}=\xi\ B\text{crit}
  17. [ M + A ( ω ) ] x ¨ + [ B ( ω ) + B v ] x ˙ + C x = F ( ω ) \big[M+A(\omega)\big]\ddot{x}+\big[B(\omega)+B_{v}\big]\dot{x}+Cx=F(\omega)
  18. B v = ξ B crit = ξ 2 ( A ( ω ) + M ) C B_{v}=\xi\ B\text{crit}=\xi\ 2\sqrt{(A(\omega)+M)C}
  19. A = lim ω A ( ω ) A_{\infty}=\lim_{\omega\to\infty}A(\omega)

Restricted_open-shell_Hartree–Fock.html

  1. 𝐅𝐂 = 𝐒𝐂 ϵ \mathbf{F}\mathbf{C}=\mathbf{S}\mathbf{C}\mathbf{\epsilon}
  2. ϵ \epsilon
  3. 𝐒 2 \mathbf{S}^{2}

Restriction_of_scalars.html

  1. R R
  2. S S
  3. f : R S f:R\to S
  4. R R
  5. S S
  6. f : R S f:R\to S
  7. M M
  8. S S
  9. R R
  10. R R
  11. r m = f ( r ) m r\cdot m=f(r)\cdot m
  12. r R r\in R
  13. m M m\in M
  14. S S
  15. R R
  16. S S
  17. u : M N u:M\to N
  18. R R
  19. M M
  20. N N
  21. m M m\in M
  22. r R r\in R
  23. u ( r m ) = u ( f ( r ) m ) = f ( r ) u ( m ) = r u ( m ) u(r\cdot m)=u(f(r)\cdot m)=f(r)\cdot u(m)=r\cdot u(m)\,
  24. R R
  25. R R
  26. S S
  27. f f
  28. R R
  29. S S
  30. S S
  31. S S
  32. R S R\subset S

Retract_(group_theory).html

  1. H H
  2. G G
  3. σ : G G \sigma:G\to G
  4. σ ( h ) = h \sigma(h)=h
  5. h H h\in H
  6. σ ( g ) H \sigma(g)\in H
  7. g G g\in G

Reynolds_transport_theorem.html

  1. 𝐟 = 𝐟 ( 𝐱 , t ) \mathbf{f}=\mathbf{f}(\mathbf{x},t)
  2. Ω ( t ) \Omega(t)
  3. Ω ( t ) \partial\Omega(t)
  4. d d t Ω ( t ) 𝐟 dV . \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega(t)}\mathbf{f}~{}\,\text{dV}~{}.
  5. 𝐟 \mathbf{f}
  6. Ω \Omega
  7. d d t Ω ( t ) 𝐟 dV = Ω ( t ) 𝐟 t dV + Ω ( t ) ( 𝐯 b 𝐧 ) 𝐟 dA \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega(t)}\mathbf{f}~{}\,\text{dV}=\int_{% \Omega(t)}\frac{\partial\mathbf{f}}{\partial t}~{}\,\text{dV}+\int_{\partial% \Omega(t)}(\mathbf{v}^{b}\cdot\mathbf{n})\mathbf{f}~{}\,\text{dA}~{}
  8. 𝐧 ( 𝐱 , t ) \mathbf{n}(\mathbf{x},t)
  9. 𝐱 \mathbf{x}
  10. dV \,\text{dV}
  11. dA \,\text{dA}
  12. 𝐱 \mathbf{x}
  13. 𝐯 b ( 𝐱 , t ) \mathbf{v}^{b}(\mathbf{x},t)
  14. 𝐟 \mathbf{f}
  15. Ω ( t ) \Omega(t)
  16. 𝐯 = 𝐯 ( 𝐱 , t ) \mathbf{v}=\mathbf{v}(\mathbf{x},t)
  17. 𝐯 b 𝐧 = 𝐯 𝐧 . \mathbf{v}^{b}\cdot\mathbf{n}=\mathbf{v}\cdot\mathbf{n}.
  18. d d t ( Ω ( t ) 𝐟 dV ) = Ω ( t ) 𝐟 t dV + Ω ( t ) ( 𝐯 𝐧 ) 𝐟 dA . \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}~{}\,\text{dV}% \right)=\int_{\Omega(t)}\frac{\partial\mathbf{f}}{\partial t}~{}\,\text{dV}+% \int_{\partial\Omega(t)}(\mathbf{v}\cdot\mathbf{n})\mathbf{f}~{}\,\text{dA}~{}.
  19. Ω 0 \Omega_{0}
  20. Ω ( t ) \Omega(t)
  21. 𝐱 = s y m b o l φ ( 𝐗 , t ) ; s y m b o l F ( 𝐗 , t ) = s y m b o l s y m b o l φ . \mathbf{x}=symbol{\varphi}(\mathbf{X},t)~{};\qquad\implies\qquad symbol{F}(% \mathbf{X},t)=symbol{\nabla}_{\circ}symbol{\varphi}~{}.
  22. J ( 𝐗 , t ) = det [ s y m b o l F ( 𝐗 , t ) ] J(\mathbf{X},t)=\det[symbol{F}(\mathbf{X},t)]
  23. Ω ( t ) 𝐟 ( 𝐱 , t ) dV = Ω 0 𝐟 [ s y m b o l φ ( 𝐗 , t ) , t ] J ( 𝐗 , t ) dV 0 = Ω 0 𝐟 ^ ( 𝐗 , t ) J ( 𝐗 , t ) dV 0 . \int_{\Omega(t)}\mathbf{f}(\mathbf{x},t)~{}\,\text{dV}=\int_{\Omega_{0}}% \mathbf{f}[symbol{\varphi}(\mathbf{X},t),t]~{}J(\mathbf{X},t)~{}\,\text{dV}_{0% }=\int_{\Omega_{0}}\hat{\mathbf{f}}(\mathbf{X},t)~{}J(\mathbf{X},t)~{}\,\text{% dV}_{0}~{}.
  24. d d t ( Ω ( t ) 𝐟 ( 𝐱 , t ) dV ) = lim Δ t 0 1 Δ t ( Ω ( t + Δ t ) 𝐟 ( 𝐱 , t + Δ t ) dV - Ω ( t ) 𝐟 ( 𝐱 , t ) dV ) . \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}(\mathbf{x},t)~% {}\,\text{dV}\right)=\lim_{\Delta t\rightarrow 0}\cfrac{1}{\Delta t}\left(\int% _{\Omega(t+\Delta t)}\mathbf{f}(\mathbf{x},t+\Delta t)~{}\,\text{dV}-\int_{% \Omega(t)}\mathbf{f}(\mathbf{x},t)~{}\,\text{dV}\right)~{}.
  25. d d t ( Ω ( t ) 𝐟 ( 𝐱 , t ) dV ) = lim Δ t 0 1 Δ t ( Ω 0 𝐟 ^ ( 𝐗 , t + Δ t ) J ( 𝐗 , t + Δ t ) dV 0 - Ω 0 𝐟 ^ ( 𝐗 , t ) J ( 𝐗 , t ) dV 0 ) . \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}(\mathbf{x},t)~% {}\,\text{dV}\right)=\lim_{\Delta t\rightarrow 0}\cfrac{1}{\Delta t}\left(\int% _{\Omega_{0}}\hat{\mathbf{f}}(\mathbf{X},t+\Delta t)~{}J(\mathbf{X},t+\Delta t% )~{}\,\text{dV}_{0}-\int_{\Omega_{0}}\hat{\mathbf{f}}(\mathbf{X},t)~{}J(% \mathbf{X},t)~{}\,\text{dV}_{0}\right)~{}.
  26. Ω 0 \Omega_{0}
  27. d d t ( Ω ( t ) 𝐟 ( 𝐱 , t ) dV ) = Ω 0 [ lim Δ t 0 𝐟 ^ ( 𝐗 , t + Δ t ) J ( 𝐗 , t + Δ t ) - 𝐟 ^ ( 𝐗 , t ) J ( 𝐗 , t ) Δ t ] dV 0 = Ω 0 t [ 𝐟 ^ ( 𝐗 , t ) J ( 𝐗 , t ) ] dV 0 = Ω 0 ( t [ 𝐟 ^ ( 𝐗 , t ) ] J ( 𝐗 , t ) + 𝐟 ^ ( 𝐗 , t ) t [ J ( 𝐗 , t ) ] ) dV 0 \begin{aligned}\displaystyle\cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(% t)}\mathbf{f}(\mathbf{x},t)~{}\,\text{dV}\right)&\displaystyle=\int_{\Omega_{0% }}\left[\lim_{\Delta t\rightarrow 0}\cfrac{\hat{\mathbf{f}}(\mathbf{X},t+% \Delta t)~{}J(\mathbf{X},t+\Delta t)-\hat{\mathbf{f}}(\mathbf{X},t)~{}J(% \mathbf{X},t)}{\Delta t}\right]~{}\,\text{dV}_{0}\\ &\displaystyle=\int_{\Omega_{0}}\frac{\partial}{\partial t}[\hat{\mathbf{f}}(% \mathbf{X},t)~{}J(\mathbf{X},t)]~{}\,\text{dV}_{0}\\ &\displaystyle=\int_{\Omega_{0}}\left(\frac{\partial}{\partial t}[\hat{\mathbf% {f}}(\mathbf{X},t)]~{}J(\mathbf{X},t)+\hat{\mathbf{f}}(\mathbf{X},t)~{}\frac{% \partial}{\partial t}[J(\mathbf{X},t)]\right)~{}\,\text{dV}_{0}\end{aligned}
  28. \detsymbol F \detsymbol{F}
  29. J ( 𝐗 , t ) t = t ( \detsymbol F ) = ( \detsymbol F ) ( s y m b o l 𝐯 ) = J ( 𝐗 , t ) s y m b o l 𝐯 ( s y m b o l φ ( 𝐗 , t ) , t ) = J ( 𝐗 , t ) s y m b o l 𝐯 ( 𝐱 , t ) . \frac{\partial J(\mathbf{X},t)}{\partial t}=\frac{\partial}{\partial t}(% \detsymbol{F})=(\detsymbol{F})(symbol{\nabla}\cdot\mathbf{v})=J(\mathbf{X},t)~% {}symbol{\nabla}\cdot\mathbf{v}(symbol{\varphi}(\mathbf{X},t),t)=J(\mathbf{X},% t)~{}symbol{\nabla}\cdot\mathbf{v}(\mathbf{x},t)~{}.
  30. d d t ( Ω ( t ) 𝐟 ( 𝐱 , t ) dV ) = Ω 0 ( t [ 𝐟 ^ ( 𝐗 , t ) ] J ( 𝐗 , t ) + 𝐟 ^ ( 𝐗 , t ) J ( 𝐗 , t ) s y m b o l 𝐯 ( 𝐱 , t ) ) dV 0 = Ω 0 ( t [ 𝐟 ^ ( 𝐗 , t ) ] + 𝐟 ^ ( 𝐗 , t ) s y m b o l 𝐯 ( 𝐱 , t ) ) J ( 𝐗 , t ) dV 0 = Ω ( t ) ( 𝐟 ˙ ( 𝐱 , t ) + 𝐟 ( 𝐱 , t ) s y m b o l 𝐯 ( 𝐱 , t ) ) dV \begin{aligned}\displaystyle\cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(% t)}\mathbf{f}(\mathbf{x},t)~{}\,\text{dV}\right)&\displaystyle=\int_{\Omega_{0% }}\left(\frac{\partial}{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~{}J(% \mathbf{X},t)+\hat{\mathbf{f}}(\mathbf{X},t)~{}J(\mathbf{X},t)~{}symbol{\nabla% }\cdot\mathbf{v}(\mathbf{x},t)\right)~{}\,\text{dV}_{0}\\ &\displaystyle=\int_{\Omega_{0}}\left(\frac{\partial}{\partial t}[\hat{\mathbf% {f}}(\mathbf{X},t)]+\hat{\mathbf{f}}(\mathbf{X},t)~{}symbol{\nabla}\cdot% \mathbf{v}(\mathbf{x},t)\right)~{}J(\mathbf{X},t)~{}\,\text{dV}_{0}\\ &\displaystyle=\int_{\Omega(t)}\left(\dot{\mathbf{f}}(\mathbf{x},t)+\mathbf{f}% (\mathbf{x},t)~{}symbol{\nabla}\cdot\mathbf{v}(\mathbf{x},t)\right)~{}\,\text{% dV}\end{aligned}
  31. 𝐟 ˙ \dot{\mathbf{f}}
  32. 𝐟 \mathbf{f}
  33. 𝐟 ˙ ( 𝐱 , t ) = 𝐟 ( 𝐱 , t ) t + [ s y m b o l 𝐟 ( 𝐱 , t ) ] 𝐯 ( 𝐱 , t ) . \dot{\mathbf{f}}(\mathbf{x},t)=\frac{\partial\mathbf{f}(\mathbf{x},t)}{% \partial t}+[symbol{\nabla}\mathbf{f}(\mathbf{x},t)]\cdot\mathbf{v}(\mathbf{x}% ,t)~{}.
  34. d d t ( Ω ( t ) 𝐟 ( 𝐱 , t ) dV ) = Ω ( t ) ( 𝐟 ( 𝐱 , t ) t + [ s y m b o l 𝐟 ( 𝐱 , t ) ] 𝐯 ( 𝐱 , t ) + 𝐟 ( 𝐱 , t ) s y m b o l 𝐯 ( 𝐱 , t ) ) dV \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}(\mathbf{x},t)~% {}\,\text{dV}\right)=\int_{\Omega(t)}\left(\frac{\partial\mathbf{f}(\mathbf{x}% ,t)}{\partial t}+[symbol{\nabla}\mathbf{f}(\mathbf{x},t)]\cdot\mathbf{v}(% \mathbf{x},t)+\mathbf{f}(\mathbf{x},t)~{}symbol{\nabla}\cdot\mathbf{v}(\mathbf% {x},t)\right)~{}\,\text{dV}
  35. d d t ( Ω ( t ) 𝐟 dV ) = Ω ( t ) ( 𝐟 t + s y m b o l 𝐟 𝐯 + 𝐟 s y m b o l 𝐯 ) dV . \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}~{}\,\text{dV}% \right)=\int_{\Omega(t)}\left(\frac{\partial\mathbf{f}}{\partial t}+symbol{% \nabla}\mathbf{f}\cdot\mathbf{v}+\mathbf{f}~{}symbol{\nabla}\cdot\mathbf{v}% \right)~{}\,\text{dV}~{}.
  36. s y m b o l ( 𝐯 𝐰 ) = 𝐯 ( s y m b o l 𝐰 ) + s y m b o l 𝐯 𝐰 symbol{\nabla}\cdot(\mathbf{v}\otimes\mathbf{w})=\mathbf{v}(symbol{\nabla}% \cdot\mathbf{w})+symbol{\nabla}\mathbf{v}\cdot\mathbf{w}
  37. d d t ( Ω ( t ) 𝐟 dV ) = Ω ( t ) ( 𝐟 t + s y m b o l ( 𝐟 𝐯 ) ) dV . \cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}~{}\,\text{dV}% \right)=\int_{\Omega(t)}\left(\frac{\partial\mathbf{f}}{\partial t}+symbol{% \nabla}\cdot(\mathbf{f}\otimes\mathbf{v})\right)~{}\,\text{dV}~{}.
  38. ( 𝐚 𝐛 ) 𝐧 = ( 𝐛 𝐧 ) 𝐚 (\mathbf{a}\otimes\mathbf{b})\cdot\mathbf{n}=(\mathbf{b}\cdot\mathbf{n})% \mathbf{a}
  39. d d t ( Ω ( t ) 𝐟 dV ) = Ω ( t ) 𝐟 t dV + Ω ( t ) ( 𝐟 𝐯 ) 𝐧 dA = Ω ( t ) 𝐟 t dV + Ω ( t ) ( 𝐯 𝐧 ) 𝐟 dA {\cfrac{\mathrm{d}}{\mathrm{d}t}\left(\int_{\Omega(t)}\mathbf{f}~{}\,\text{dV}% \right)=\int_{\Omega(t)}\frac{\partial\mathbf{f}}{\partial t}~{}\,\text{dV}+% \int_{\partial\Omega(t)}(\mathbf{f}\otimes\mathbf{v})\cdot\mathbf{n}~{}\,\text% {dA}=\int_{\Omega(t)}\frac{\partial\mathbf{f}}{\partial t}~{}\,\text{dV}+\int_% {\partial\Omega(t)}(\mathbf{v}\cdot\mathbf{n})\mathbf{f}~{}\,\text{dA}\qquad\square}
  40. Ω \Omega
  41. 𝐯 b = 0 \mathbf{v}_{b}=0
  42. d d t Ω f dV = Ω f t dV . \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}f~{}\,\text{dV}=\int_{\Omega}\frac% {\partial f}{\partial t}~{}\,\text{dV}~{}.
  43. f f
  44. y y
  45. z z
  46. Ω ( t ) \Omega(t)
  47. y - z y-z
  48. x x
  49. a ( t ) a(t)
  50. b ( t ) b(t)
  51. d d t a ( t ) b ( t ) f dx = a ( t ) b ( t ) f t dx + b ( t ) t f ( b ( t ) , t ) - a ( t ) t f ( a ( t ) , t ) , \cfrac{\mathrm{d}}{\mathrm{d}t}\int_{a(t)}^{b(t)}f~{}\,\text{dx}=\int_{a(t)}^{% b(t)}\frac{\partial f}{\partial t}~{}\,\text{dx}+\frac{\partial b(t)}{\partial t% }f(b(t),t)-\frac{\partial a(t)}{\partial t}f(a(t),t)~{},
  52. 𝐯 b = 𝐯 . \mathbf{v}^{b}=\mathbf{v}.

Rhombic_dodecahedral_honeycomb.html

  1. 3 ¯ \overline{3}
  2. C ~ 3 {\tilde{C}}_{3}
  3. B ~ 3 {\tilde{B}}_{3}
  4. A ~ 3 {\tilde{A}}_{3}
  5. B ~ 3 {\tilde{B}}_{3}
  6. A ~ 3 {\tilde{A}}_{3}
  7. 3 ¯ \overline{3}

Ricart–Agrawala_algorithm.html

  1. r e l e a s e release
  2. P i P_{i}
  3. r e p l y reply
  4. P j P_{j}
  5. P i P_{i}
  6. P j P_{j}
  7. P i P_{i}
  8. r e p l y reply
  9. P j P_{j}

RICE_chart.html

  1. H A A - + H + HA\rightleftharpoons A^{-}+H^{+}
  2. K a = [ H + ] [ A - ] [ H A ] K_{a}=\frac{[H^{+}][A^{-}]}{[HA]}
  3. K a = x 2 C a - x K_{a}=\frac{x^{2}}{C_{a}-x}
  4. x 2 + K a x - K a C a = 0 x^{2}+K_{a}x-K_{a}C_{a}=0
  5. K a = x 2 C a K_{a}=\frac{x^{2}}{C_{a}}

Ridged_mirror.html

  1. C ~{}C~{}
  2. r r 0 ( L C , K sin ( θ ) ) \displaystyle r\approx r_{0}\!\left(\frac{\ell}{L}C,\!~{}K\sin(\theta)\right)
  3. ~{}\ell~{}
  4. L ~{}L~{}
  5. θ \displaystyle~{}\theta~{}
  6. K = m V / ~{}K=mV/\hbar~{}
  7. r 0 ( C , k ) ~{}r_{0}(C,k)~{}
  8. k ~{}k~{}
  9. L ~{}L~{}
  10. K L θ 2 1 KL\!~{}\theta^{2}\ll 1
  11. r 0 ~{}r_{0}~{}
  12. L L
  13. r exp ( - 8 K L θ ) ~{}\displaystyle r\approx\exp\!\left(-\sqrt{8\!~{}K\!~{}L}~{}\theta\right)~{}
  14. θ \displaystyle~{}\theta~{}
  15. L ~{}L~{}
  16. / L 1 ~{}\ell/L\ll 1~{}
  17. \ell
  18. L L

Riesz_sequence.html

  1. ( H , , ) (H,\langle\cdot,\cdot\rangle)
  2. 0 < c C < + 0<c\leq C<+\infty
  3. c ( n | a n | 2 ) n a n x n 2 C ( n | a n | 2 ) c\left(\sum_{n}|a_{n}|^{2}\right)\leq\left\|\sum_{n}a_{n}x_{n}\right\|^{2}\leq C% \left(\sum_{n}|a_{n}|^{2}\right)
  4. span ( x n ) ¯ = H \overline{\mathop{\rm span}(x_{n})}=H
  5. ϕ n ( x ) = ϕ ( x - n ) \phi_{n}(x)=\phi(x-n)
  6. φ ^ \hat{\varphi}
  7. 0 < c C < + 0<c\leq C<+\infty
  8. 1. ( a n ) 2 , c ( n | a n | 2 ) n a n φ n 2 C ( n | a n | 2 ) 1.\quad\forall(a_{n})\in\ell^{2},\ \ c\left(\sum_{n}|a_{n}|^{2}\right)\leq% \left\|\sum_{n}a_{n}\varphi_{n}\right\|^{2}\leq C\left(\sum_{n}|a_{n}|^{2}\right)
  9. 2. c n | φ ^ ( ω + 2 π n ) | 2 C 2.\quad c\leq\sum_{n}\left|\hat{\varphi}(\omega+2\pi n)\right|^{2}\leq C

Rifleman's_rule.html

  1. R S R_{S}
  2. α \alpha
  3. R H = R S cos ( α ) R_{H}=R_{S}\cos(\alpha)
  4. R H R_{H}
  5. R H = R S cos ( α ) R_{H}=R_{S}\cos(\alpha)
  6. angle correction = - bullet drop R H \mbox{angle correction}~{}=-\frac{\mbox{bullet drop}~{}}{R_{H}}
  7. 20 20^{\circ}
  8. R H = 300 meters cos ( 20 ) = 282 meters R_{H}=300\mbox{ meters}~{}\cos(20^{\circ})=282\mbox{ meters}~{}
  9. bullet drop = - 11.0 ( 282 - 200 ) + - 2.9 ( 300 - 282 ) 300 - 200 = - 9.5 cm \mbox{bullet drop}~{}=\frac{-11.0\cdot(282-200)+-2.9\cdot(300-282)}{300-200}=-% 9.5\mbox{ cm}~{}
  10. angle correction = - - 9.5 cm 282 meters = 0.00094 radians = 3.2 (minutes of angle) \mbox{angle correction}~{}=-\frac{-9.5\mbox{ cm}~{}}{282\mbox{ meters}~{}}=0.0% 0094\mbox{ radians}~{}=3.2^{\prime}\mbox{ (minutes of angle)}~{}
  11. δ θ \delta\theta
  12. δ θ \delta\theta
  13. δ θ \delta\theta
  14. x ( t ) = v b u l l e t cos ( δ θ ) t x(t)=v_{bullet}\cos(\delta\theta)t\,
  15. y ( t ) = v b u l l e t sin ( δ θ ) t - 1 2 g t 2 y(t)=v_{bullet}\sin(\delta\theta)t-\frac{1}{2}gt^{2}\,
  16. t = x v b u l l e t cos ( δ θ ) t=\frac{x}{v_{bullet}\cos(\delta\theta)}
  17. y = 0 y=0
  18. t 0 t\neq 0
  19. y ( t ) = 0 = ( v b u l l e t sin ( δ θ ) - 1 2 g t ) t y(t)=0=\left(v_{bullet}\sin(\delta\theta)-\frac{1}{2}gt\right)t
  20. 0 = v b u l l e t sin ( δ θ ) t - 1 2 g t 2 0=v_{bullet}\sin(\delta\theta)t-\frac{1}{2}gt^{2}
  21. v b u l l e t sin ( δ θ ) = 1 2 g t = 1 2 g x v b u l l e t cos ( δ θ ) v_{bullet}\sin(\delta\theta)=\frac{1}{2}gt=\frac{1}{2}g\frac{x}{v_{bullet}\cos% (\delta\theta)}
  22. x = v b u l l e t 2 2 sin ( δ θ ) cos ( δ θ ) g x=\frac{v^{2}_{bullet}2\sin(\delta\theta)\cos(\delta\theta)}{g}\,
  23. v b u l l e t v_{bullet}
  24. x = R H x=R_{H}
  25. y = 0 y=0
  26. x = R H x=R_{H}
  27. R H = v b u l l e t 2 2 sin ( δ θ ) cos ( δ θ ) g = v b u l l e t 2 sin ( 2 δ θ ) g R_{H}=\frac{v_{bullet}^{2}\;2\,\sin(\delta\theta)\,\cos(\delta\theta)}{g}=% \frac{v_{bullet}^{2}\sin(2\delta\theta)}{g}\,
  28. R H R_{H}
  29. δ θ \delta\theta
  30. δ θ = 0.039 \delta\theta=0.039^{\circ}
  31. δ θ \delta\theta
  32. δ θ \delta\theta
  33. R H R_{H}
  34. R S R_{S}
  35. R S = R H ( 1 - tan ( δ θ ) tan ( α ) ) sec ( α ) R_{S}=R_{H}\,(1-\tan(\delta\theta)\tan(\alpha))\sec(\alpha)\,
  36. δ θ \delta\theta
  37. α \alpha
  38. R H R_{H}
  39. δ θ \delta\theta
  40. α \alpha
  41. 1 - tan ( δ θ ) tan ( α ) 1 1-\tan(\delta\theta)\tan(\alpha)\approx 1
  42. R S R_{S}
  43. R S R H sec ( α ) R_{S}\approx R_{H}\sec(\alpha)\,
  44. sec ( α ) 1 \sec(\alpha)\geq 1\,
  45. R H R_{H}
  46. R S > R H R_{S}>R_{H}
  47. R H R_{H}
  48. R S R_{S}
  49. R H R_{H}
  50. R Z e r o = R H cos ( α ) R_{Zero}=R_{H}\cos(\alpha)
  51. R S = R Z e r o sec ( α ) = ( R H cos ( α ) ) sec ( α ) = R H R_{S}=R_{Zero}\sec(\alpha)=\left(R_{H}\cos(\alpha)\right)\sec(\alpha)=R_{H}
  52. R S = v B u l l e t 2 sin ( 2 θ ) g ( 1 - cot ( θ ) tan ( α ) ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}\sin(2\theta)}{g}\,(1-\cot(\theta)\tan(\alpha))\sec(% \alpha)\,
  53. R S = v B u l l e t 2 g 2 sin ( θ ) cos ( θ ) ( 1 - cos ( θ ) sin ( θ ) sin ( α ) cos ( α ) ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}}{g}\,2\sin(\theta)\cos(\theta)\left(1-\frac{\cos(% \theta)}{\sin(\theta)}\frac{\sin(\alpha)}{\cos(\alpha)}\right)\sec(\alpha)\,
  54. R S = v B u l l e t 2 g 2 ( sin ( θ ) cos ( θ ) - cos ( θ ) 2 sin ( α ) cos ( α ) ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}}{g}\,\,2\left(\sin(\theta)\cos(\theta)-\frac{\cos(% \theta)^{2}\sin(\alpha)}{\cos(\alpha)}\right)\sec(\alpha)\,
  55. cos ( θ ) / cos ( α ) \cos(\theta)/\cos(\alpha)
  56. R S = v B u l l e t 2 g 2 cos ( θ ) cos ( α ) ( sin ( θ ) cos ( α ) - sin ( α ) cos ( θ ) ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}}{g}\,2\frac{\cos(\theta)}{\cos(\alpha)}\left(\sin(% \theta)\cos(\alpha)-\sin(\alpha)\cos(\theta)\right)\sec(\alpha)\,
  57. cos ( θ - α ) / cos ( θ - α ) \cos(\theta-\alpha)/\cos(\theta-\alpha)
  58. R S = v B u l l e t 2 g ( 2 sin ( θ - α ) cos ( θ ) - α ) ) cos ( θ ) cos ( α ) cos ( θ - α ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}}{g}\,\left(2\sin(\theta-\alpha)\cos(\theta)-\alpha)% \right)\frac{\cos(\theta)}{\cos(\alpha)\cos(\theta-\alpha)}\sec(\alpha)\,
  59. sin ( 2 ( θ - α ) ) \sin(2(\theta-\alpha))
  60. cos ( α ) cos ( θ - α ) \cos(\alpha)\cos(\theta-\alpha)
  61. R S = v B u l l e t 2 g sin ( 2 ( θ - α ) ) ( cos ( α ) cos ( θ - α ) + cos ( θ ) - cos ( α ) cos ( θ - α ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}}{g}\,\sin(2\left(\theta-\alpha)\right)\left(\frac{% \cos(\alpha)\cos(\theta-\alpha)+\cos(\theta)-\cos(\alpha)\cos(\theta-\alpha)}{% \cos(\alpha)\cos(\theta-\alpha)}\right)\sec(\alpha)\,
  62. δ θ = θ - α \delta\theta=\theta-\alpha
  63. R S = v B u l l e t 2 g sin ( 2 δ θ ) ( cos ( α ) cos ( θ - α ) + cos ( θ ) - cos ( α ) cos ( θ - α ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=\frac{v_{Bullet}^{2}}{g}\,\sin(2\delta\theta)\left(\frac{\cos(\alpha)% \cos(\theta-\alpha)+\cos(\theta)-\cos(\alpha)\cos(\theta-\alpha)}{\cos(\alpha)% \cos(\theta-\alpha)}\right)\sec(\alpha)\,
  64. R H = v B u l l e t 2 sin ( 2 δ θ ) g R_{H}=\frac{v_{Bullet}^{2}\sin(2\delta\theta)}{g}
  65. R S = R H ( 1 + cos ( θ ) - cos ( α ) cos ( θ - α ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=R_{H}\left(1+\frac{\cos(\theta)-\cos(\alpha)\cos(\theta-\alpha)}{\cos(% \alpha)\cos(\theta-\alpha)}\right)\sec(\alpha)\,
  66. cos ( θ - α ) = cos ( θ ) cos ( α ) + sin ( θ ) sin ( α ) \cos(\theta-\alpha)=\cos(\theta)\cos(\alpha)+\sin(\theta)\sin(\alpha)
  67. R S = R H ( 1 + cos ( θ ) - cos ( α ) ( cos ( α ) cos ( θ ) + sin ( α ) sin ( θ ) ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=R_{H}\left(1+\frac{\cos(\theta)-\cos(\alpha)\left(\cos(\alpha)\cos(% \theta)+\sin(\alpha)\sin(\theta)\right)}{\cos(\alpha)\cos(\theta-\alpha)}% \right)\sec(\alpha)\,
  68. cos ( α ) \cos(\alpha)
  69. R S = R H ( 1 + cos ( θ ) - cos ( α ) 2 cos ( θ ) - cos ( α ) sin ( θ ) sin ( α ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=R_{H}\left(1+\frac{\cos(\theta)-\cos(\alpha)^{2}\cos(\theta)-\cos(\alpha% )\sin(\theta)\sin(\alpha)}{\cos(\alpha)\cos(\theta-\alpha)}\right)\sec(\alpha)\,
  70. cos ( α ) \cos(\alpha)
  71. sin ( α ) 2 = 1 - cos ( α ) 2 \sin(\alpha)^{2}=1-\cos(\alpha)^{2}
  72. R S = R H ( 1 + cos ( θ ) sin ( α ) 2 - cos ( α ) sin ( θ ) sin ( α ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=R_{H}\left(1+\frac{\cos(\theta)\sin(\alpha)^{2}-\cos(\alpha)\sin(\theta)% \sin(\alpha)}{\cos(\alpha)\cos(\theta-\alpha)}\right)\sec(\alpha)\,
  73. sin ( α ) \sin(\alpha)
  74. R S = R H ( 1 + sin ( α ) ( cos ( θ ) sin ( α ) - cos ( α ) sin ( θ ) ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=R_{H}\left(1+\frac{\sin(\alpha)\left(\cos(\theta)\sin(\alpha)-\cos(% \alpha)\sin(\theta)\right)}{\cos(\alpha)\cos(\theta-\alpha)}\right)\sec(\alpha)\,
  75. - sin ( θ - α ) = cos ( θ ) sin ( α ) - sin ( θ ) cos ( α ) -\sin(\theta-\alpha)=\cos(\theta)\sin(\alpha)-\sin(\theta)\cos(\alpha)
  76. R S = R H ( 1 - sin ( α ) sin ( θ - α ) cos ( α ) cos ( θ - α ) ) sec ( α ) R_{S}=R_{H}\left(1-\frac{\sin(\alpha)\sin(\theta-\alpha)}{\cos(\alpha)\cos(% \theta-\alpha)}\right)\sec(\alpha)\,
  77. tan ( δ θ ) \tan(\delta\theta)
  78. tan ( α ) \tan(\alpha)
  79. δ θ = θ - α \delta\theta=\theta-\alpha\,
  80. R S = R H ( 1 - tan ( α ) tan ( δ θ ) ) sec ( α ) R_{S}=R_{H}\left(1-\tan(\alpha)\tan(\delta\theta)\right)\sec(\alpha)

Right-to-left_shunt.html

  1. Q s / Q t = ( C c O 2 - C a O 2 ) / ( C c O 2 - C v O 2 ) Q_{s}/Q_{t}=(CcO_{2}-CaO_{2})/(CcO_{2}-CvO_{2})

Ripple_(electrical).html

  1. V pp = I 2 f C V_{\mathrm{pp}}=\frac{I}{2fC}
  2. V pp = I f C V_{\mathrm{pp}}=\frac{I}{fC}
  3. V pp V_{\mathrm{pp}}
  4. I I
  5. f f
  6. C C
  7. γ = 1 4 3 f C R \gamma=\frac{1}{4\sqrt{3}fCR}
  8. γ \gamma
  9. R R
  10. γ = 0.236 R ω L \gamma=\frac{0.236R}{\omega L}
  11. ω \omega
  12. 2 π f 2\pi f
  13. L L

Risk-seeking.html

  1. x < 0 x<0
  2. x < 0 x<0
  3. x > 0 x>0

Roadway_air_dispersion_modeling.html

  1. χ = 0 q π ( u c d x 2 ) ( c o s α ) ( exp y 2 2 c 2 x 2 ) d x \chi\ =\int_{0}^{\infty}\frac{q}{\pi\left(ucdx^{2}\right)\left(cos\alpha\ % \right)}\left(\exp\frac{y^{2}}{2c^{2}x^{2}}\right)dx

Rocket_engine_nozzle.html

  1. v e = T R M 2 γ γ - 1 [ 1 - ( p e / p ) ( γ - 1 ) / γ ] v_{e}=\sqrt{\;\frac{T\;R}{M}\cdot\frac{2\;\gamma}{\gamma-1}\cdot\bigg[1-(p_{e}% /p)^{(\gamma-1)/\gamma}\bigg]}
  2. v e v_{e}
  3. T T
  4. R R
  5. M M
  6. γ \gamma
  7. c p / c v c_{p}/c_{v}
  8. c p c_{p}
  9. c v c_{v}
  10. p e p_{e}
  11. p p
  12. F F
  13. = m ˙ v e + ( p e - p o ) A e =\,\dot{m}\,v_{e}+(p_{e}-p_{o})\,A_{e}
  14. = m ˙ [ v e + ( p e - p o m ˙ ) A e ] =\,\dot{m}\,\bigg[v_{e}+\bigg(\frac{p_{e}-p_{o}}{\dot{m}}\bigg)A_{e}\bigg]
  15. F F
  16. = m ˙ v e q =\,\dot{m}\,v_{eq}
  17. I s p I_{sp}
  18. I s p = F m ˙ g o = m ˙ v e q m ˙ g o = v e q g o I_{sp}=\,\frac{F}{\dot{m}\,g_{o}}\,=\,\frac{\dot{m}\,v_{eq}}{\dot{m}\,g_{o}}\,% =\,\frac{v_{eq}}{g_{o}}
  19. F F
  20. m ˙ \dot{m}
  21. v e v_{e}
  22. p e p_{e}
  23. p o p_{o}
  24. A e A_{e}
  25. v e q v_{eq}
  26. I s p I_{sp}
  27. g o g_{o}
  28. p e p_{e}
  29. p o p_{o}
  30. I s p = F m ˙ g o = m ˙ v e m ˙ g o = v e g o I_{sp}=\,\frac{F}{\dot{m}\,g_{o}}\,=\,\frac{\dot{m}\,v_{e}}{\dot{m}\,g_{o}}\,=% \,\frac{v_{e}}{g_{o}}
  31. p e p_{e}
  32. m ˙ \dot{m}
  33. I s p ( v a c ) I_{sp}(vac)
  34. I s p ( v a c ) = v e g o + p e A e m ˙ g o I_{sp}(vac)=\,\frac{v_{e}}{g_{o}}+\frac{p_{e}\,A_{e}}{\dot{m}\,g_{o}}
  35. F = I s p ( v a c ) g o m ˙ - A e p o F=I_{sp}(vac)\,g_{o}\,\dot{m}-A_{e}p_{o}

Rolling_hash.html

  1. H = c 1 a k - 1 + c 2 a k - 2 + c 3 a k - 3 + + c k a 0 H=c_{1}a^{k-1}+c_{2}a^{k-2}+c_{3}a^{k-3}+...+c_{k}a^{0}
  2. a a
  3. c 1 , , c k c_{1},...,c_{k}
  4. H H
  5. n n
  6. a a
  7. n n
  8. H H
  9. a a
  10. H H
  11. a a
  12. a a
  13. a - 1 a^{-1}
  14. H H
  15. h h
  16. [ 0 , 2 L ) [0,2^{L})
  17. s s
  18. s ( 10011 ) = 00111 s(10011)=00111
  19. \oplus
  20. H = s k - 1 ( h ( c 1 ) ) s k - 2 ( h ( c 2 ) ) s ( h ( c k - 1 ) ) h ( c k ) H=s^{k-1}(h(c_{1}))\oplus s^{k-2}(h(c_{2}))\oplus\ldots\oplus s(h(c_{k-1}))% \oplus h(c_{k})
  21. [ 0 , 2 L ) [0,2^{L})
  22. H H
  23. H H
  24. H s ( H ) H\leftarrow s(H)
  25. c 1 c_{1}
  26. k k
  27. s k ( h ( c 1 ) ) s^{k}(h(c_{1}))
  28. H s ( H ) s k ( h ( c 1 ) ) h ( c k + 1 ) H\leftarrow s(H)\oplus s^{k}(h(c_{1}))\oplus h(c_{k+1})
  29. c k + 1 c_{k+1}
  30. L - k + 1 L-k+1
  31. H H
  32. k - 1 k-1
  33. H H ÷ 2 k - 1 H\rightarrow H\div 2^{k-1}
  34. S ( n ) = i = n - 8196 n c i S(n)=\sum_{i=n-8196}^{n}c_{i}
  35. A ( n ) = S ( n ) 8196 A(n)=\frac{S(n)}{8196}
  36. H ( n ) = S ( n ) mod 4096 H(n)=S(n)\mod 4096
  37. S ( n ) S(n)
  38. n n
  39. c i c_{i}
  40. i i
  41. A ( n ) A(n)
  42. n n
  43. H ( n ) H(n)
  44. S ( n ) S(n)
  45. n n
  46. H ( n ) = = 0 H(n)==0
  47. n n
  48. n + 1 n+1
  49. k k
  50. k k
  51. O ( k log k 2 O ( log * k ) ) O(k\log k2^{O(\log^{*}k)})

Ronald_Jensen.html

  1. \diamondsuit

Root-mean-square_deviation_of_atomic_positions.html

  1. RMSD = 1 N i = 1 N δ i 2 \mathrm{RMSD}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\delta_{i}^{2}}
  2. n n
  3. 𝐯 \mathbf{v}
  4. 𝐰 \mathbf{w}
  5. RMSD ( 𝐯 , 𝐰 ) \displaystyle\mathrm{RMSD}(\mathbf{v},\mathbf{w})

Rosenbrock_function.html

  1. f ( x , y ) = ( a - x ) 2 + b ( y - x 2 ) 2 f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}
  2. ( x , y ) = ( a , a 2 ) (x,y)=(a,a^{2})
  3. f ( x , y ) = 0 f(x,y)=0
  4. a = 1 a=1
  5. b = 100 b=100
  6. N / 2 N/2
  7. f ( 𝐱 ) = f ( x 1 , x 2 , , x N ) = i = 1 N / 2 [ 100 ( x 2 i - 1 2 - x 2 i ) 2 + ( x 2 i - 1 - 1 ) 2 ] . f(\mathbf{x})=f(x_{1},x_{2},\dots,x_{N})=\sum_{i=1}^{N/2}\left[100(x_{2i-1}^{2% }-x_{2i})^{2}+(x_{2i-1}-1)^{2}\right].
  8. N N
  9. f ( 𝐱 ) = i = 1 N - 1 [ ( 1 - x i ) 2 + 100 ( x i + 1 - x i 2 ) 2 ] x N . f(\mathbf{x})=\sum_{i=1}^{N-1}\left[(1-x_{i})^{2}+100(x_{i+1}-x_{i}^{2})^{2}% \right]\quad\forall x\in\mathbb{R}^{N}.
  10. N = 3 N=3
  11. ( 1 , 1 , 1 ) (1,1,1)
  12. 4 N 7 4\leq N\leq 7
  13. ( x 1 , x 2 , , x N ) = ( - 1 , 1 , , 1 ) (x_{1},x_{2},\dots,x_{N})=(-1,1,\dots,1)
  14. x x
  15. N N
  16. | x i | < 2.4 |x_{i}|<2.4
  17. N N
  18. x 0 = ( - 3 , - 4 ) x_{0}=(-3,-4)
  19. 10 - 10 10^{-10}

Rotation_number.html

  1. F ( x + m ) = F ( x ) + m F(x+m)=F(x)+m
  2. ω ( f ) = lim n F n ( x ) - x n . \omega(f)=\lim_{n\to\infty}\frac{F^{n}(x)-x}{n}.
  3. F ( x ) = x + θ , F(x)=x+\theta,
  4. h f = g h h\circ f=g\circ h
  5. C 0 C^{0}

Rotational_temperature.html

  1. θ R = h c B ¯ k B = 2 2 k B I \theta_{R}=\frac{hc\bar{B}}{k_{B}}=\frac{\hbar^{2}}{2k_{B}I}
  2. B ¯ = B / h c \bar{B}=B/hc
  3. I I
  4. θ R \theta_{R}

Rotational–vibrational_coupling.html

  1. x = a cos ( ω t ) x=a\cos(\omega t)
  2. y = b sin ( ω t ) y=b\sin(\omega t)
  3. a a
  4. b b
  5. ω \omega
  6. x = ( a + b 2 ) cos ( ω t ) + ( a - b 2 ) cos ( ω t ) x=\left(\begin{matrix}\frac{a+b}{2}\end{matrix}\right)\cos(\omega t)+\left(% \begin{matrix}\frac{a-b}{2}\end{matrix}\right)\cos(\omega t)
  7. y = ( a + b 2 ) sin ( ω t ) - ( a - b 2 ) sin ( ω t ) y=\left(\begin{matrix}\frac{a+b}{2}\end{matrix}\right)\sin(\omega t)-\left(% \begin{matrix}\frac{a-b}{2}\end{matrix}\right)\sin(\omega t)
  8. ( a + b ) / 2 (a+b)/2
  9. x = ( a - b 2 ) cos ( 2 ω t ) x=\left(\begin{matrix}\frac{a-b}{2}\end{matrix}\right)\cos(2\omega t)
  10. y = - ( a - b 2 ) sin ( 2 ω t ) y=-\left(\begin{matrix}\frac{a-b}{2}\end{matrix}\right)\sin(2\omega t)
  11. F = - C r \vec{F}=-\ C\vec{r}
  12. a = Ω 2 r + 2 ( Ω × v ) a\ =\ \Omega^{2}\vec{r}\ +\ 2(\vec{\Omega}\times\vec{v})
  13. F = - C r + m Ω 2 r + 2 m ( Ω × v ) \vec{F}=-\ C\vec{r}+m\Omega^{2}\vec{r}+2m(\vec{\Omega}\times\vec{v})
  14. F = 2 m ( Ω × v ) \vec{F}=2m(\vec{\Omega}\times\vec{v})

Rotor–stator_interaction.html

  1. Ω \Omega
  2. B Ω B\Omega
  3. m B Ω mB\Omega
  4. F m F_{m}
  5. m B Ω mB\Omega
  6. P m P_{m}
  7. B = 13 B=13
  8. Ω = 12000 \Omega=12000
  9. F m P m {{F_{m}}\over{P_{m}}}
  10. s = - s = + e - i ( m B - s V ) π 2 J m B - s V ( m B M ) \sum\limits_{s=-\infty}^{s=+\infty}{e^{-{{i(mB-sV)\pi}\over 2}}J_{mB-sV}}(mBM)
  11. J m B - s V J_{mB-sV}
  12. F m P m {{F_{m}}\over{P_{m}}}
  13. F m F_{m}
  14. θ = 0 \theta=0
  15. θ = P i / 2 \theta=Pi/2

Round_function.html

  1. M M\to{\mathbb{R}}
  2. M M
  3. S 1 S^{1}
  4. M M
  5. K = ( 0 , 2 π ) × ( 0 , 2 π ) . K=(0,2\pi)\times(0,2\pi).\,
  6. X : K 3 X\colon K\to{\mathbb{R}}^{3}\,
  7. X ( θ , ϕ ) = ( ( 2 + cos θ ) cos ϕ , ( 2 + cos θ ) sin ϕ , sin θ ) X(\theta,\phi)=((2+\cos\theta)\cos\phi,(2+\cos\theta)\sin\phi,\sin\theta)\,
  8. M M
  9. π 3 : 3 \pi_{3}\colon{\mathbb{R}}^{3}\to{\mathbb{R}}
  10. G = π 3 | M : M , ( θ , ϕ ) sin θ G=\pi_{3}|_{M}\colon M\to{\mathbb{R}},(\theta,\phi)\mapsto\sin\theta\,
  11. G = G ( θ , ϕ ) = sin θ G=G(\theta,\phi)=\sin\theta
  12. grad G ( θ , ϕ ) = ( G θ , G ϕ ) ( θ , ϕ ) = ( 0 , 0 ) , {\rm grad}\ G(\theta,\phi)=\left({{\partial}G\over{\partial}\theta},{{\partial% }G\over{\partial}\phi}\right)\!\left(\theta,\phi\right)=(0,0),\,
  13. θ = π 2 , 3 π 2 \theta={\pi\over 2},\ {3\pi\over 2}
  14. θ \theta
  15. X ( π / 2 , ϕ ) = ( 2 cos ϕ , 2 sin ϕ , 1 ) X({\pi/2},\phi)=(2\cos\phi,2\sin\phi,1)\,
  16. X ( 3 π / 2 , ϕ ) = ( 2 cos ϕ , 2 sin ϕ , - 1 ) X({3\pi/2},\phi)=(2\cos\phi,2\sin\phi,-1)\,
  17. M M
  18. hess ( G ) = [ - sin θ 0 0 0 ] {\rm hess}(G)=\begin{bmatrix}-\sin\theta&0\\ 0&0\end{bmatrix}
  19. hess ( G ) {\rm hess}(G)

Routh's_theorem.html

  1. A B C ABC
  2. D D
  3. E E
  4. F F
  5. B C BC
  6. C A CA
  7. A B AB
  8. C D B D \tfrac{CD}{BD}
  9. = x =x
  10. A E C E \tfrac{AE}{CE}
  11. = y =y
  12. B F A F \tfrac{BF}{AF}
  13. = z =z
  14. A D AD
  15. B E BE
  16. C F CF
  17. A B C ABC
  18. ( x y z - 1 ) 2 ( x y + y + 1 ) ( y z + z + 1 ) ( z x + x + 1 ) . \frac{(xyz-1)^{2}}{(xy+y+1)(yz+z+1)(zx+x+1)}.
  19. x = y = z = 2 x=y=z=2
  20. x = y = z = 1 x=y=z=1
  21. A B C ABC
  22. A B D ABD
  23. F R C FRC
  24. A F F B × B C C D × D R R A = 1 \frac{AF}{FB}\times\frac{BC}{CD}\times\frac{DR}{RA}=1
  25. D R R A = B F F A × D C C B = z x x + 1 \frac{DR}{RA}=\frac{BF}{FA}\times\frac{DC}{CB}=\frac{zx}{x+1}
  26. A R C ARC
  27. S A R C = A R A D S A D C = A R A D × D C B C S A B C = x z x + x + 1 S_{ARC}=\frac{AR}{AD}S_{ADC}=\frac{AR}{AD}\times\frac{DC}{BC}S_{ABC}=\frac{x}{% zx+x+1}
  28. S B P A = y x y + y + 1 S_{BPA}=\frac{y}{xy+y+1}
  29. S C Q B = z y z + z + 1 S_{CQB}=\frac{z}{yz+z+1}
  30. P Q R PQR
  31. S P Q R = S A B C - S A R C - S B P A - S C Q B \displaystyle S_{PQR}=S_{ABC}-S_{ARC}-S_{BPA}-S_{CQB}
  32. = 1 - x z x + x + 1 - y x y + y + 1 - z y z + z + 1 =1-\frac{x}{zx+x+1}-\frac{y}{xy+y+1}-\frac{z}{yz+z+1}
  33. = ( x y z - 1 ) 2 ( x z + x + 1 ) ( y x + y + 1 ) ( z y + z + 1 ) . =\frac{(xyz-1)^{2}}{(xz+x+1)(yx+y+1)(zy+z+1)}.

Routhian_mechanics.html

  1. t t
  2. L ( q 1 , q 2 , , q ˙ 1 , q ˙ 2 , , t ) , q ˙ i = d q i d t , L(q_{1},q_{2},\ldots,\dot{q}_{1},\dot{q}_{2},\ldots,t)\,,\quad\dot{q}_{i}=% \frac{dq_{i}}{dt}\,,
  3. H ( q 1 , q 2 , , p 1 , p 2 , , t ) = i q ˙ i p i - L ( q 1 , q 2 , , q ˙ 1 ( p 1 ) , q ˙ 2 ( p 2 ) , , t ) , p i = L q ˙ i , H(q_{1},q_{2},\ldots,p_{1},p_{2},\ldots,t)=\sum_{i}\dot{q}_{i}p_{i}-L(q_{1},q_% {2},\ldots,\dot{q}_{1}(p_{1}),\dot{q}_{2}(p_{2}),\ldots,t)\,,\quad p_{i}=\frac% {\partial L}{\partial\dot{q}_{i}}\,,
  4. L L
  5. H H
  6. n n
  7. n + s n+s
  8. R ( 𝐪 , s y m b o l ζ , 𝐩 , s y m b o l ζ ˙ , t ) = 𝐩 𝐪 ˙ - L ( 𝐪 , s y m b o l ζ , 𝐪 ˙ , s y m b o l ζ ˙ , t ) , R(\mathbf{q},symbol{\zeta},\mathbf{p},\dot{symbol{\zeta}},t)=\mathbf{p}\cdot% \dot{{\mathbf{q}}}-L(\mathbf{q},symbol{\zeta},\dot{\mathbf{q}},\dot{symbol{% \zeta}},t)\,,
  9. 𝐩 𝐪 ˙ = i = 1 n p i q ˙ i . \mathbf{p}\cdot\dot{{\mathbf{q}}}=\sum_{i=1}^{n}p_{i}\dot{q}_{i}\,.
  10. s s
  11. s s
  12. d d t L q ˙ j = L q j \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{j}}=\frac{\partial L}{\partial q% _{j}}
  13. j = 1 , 2 , , s j=1,2,...,s
  14. n n
  15. 2 n 2n
  16. q ˙ i = H p i , p ˙ i = - H q i \dot{q}_{i}=\frac{\partial H}{\partial p_{i}}\,,\quad\dot{p}_{i}=-\frac{% \partial H}{\partial q_{i}}
  17. q q
  18. ζ ζ
  19. d q / d t dq/dt
  20. d ζ / d t dζ/dt
  21. L ( q , ζ , q ˙ , ζ ˙ , t ) L(q,\zeta,\dot{q},\dot{\zeta},t)
  22. L L
  23. d L = L q d q + L ζ d ζ + L q ˙ d q ˙ + L ζ ˙ d ζ ˙ + L t d t . dL=\frac{\partial L}{\partial q}dq+\frac{\partial L}{\partial\zeta}d\zeta+% \frac{\partial L}{\partial\dot{q}}d\dot{q}+\frac{\partial L}{\partial\dot{% \zeta}}d\dot{\zeta}+\frac{\partial L}{\partial t}dt\,.
  24. q q
  25. ζ ζ
  26. d q / d t dq/dt
  27. d ζ / d t dζ/dt
  28. q q
  29. ζ ζ
  30. p p
  31. d ζ / d t dζ/dt
  32. d q / d t dq/dt
  33. p p
  34. L L
  35. d q dq
  36. d ζ
  37. d p dp
  38. d ( d ζ / d t ) d(dζ/dt)
  39. d t dt
  40. q q
  41. p = L q ˙ , p ˙ = d d t L q ˙ = L q p=\frac{\partial L}{\partial\dot{q}}\,,\quad\dot{p}=\frac{d}{dt}\frac{\partial L% }{\partial\dot{q}}=\frac{\partial L}{\partial q}
  42. d L = p ˙ d q + L ζ d ζ + p d q ˙ + L ζ ˙ d ζ ˙ + L t d t dL=\dot{p}dq+\frac{\partial L}{\partial\zeta}d\zeta+pd\dot{q}+\frac{\partial L% }{\partial\dot{\zeta}}d\dot{\zeta}+\frac{\partial L}{\partial t}dt
  43. p d ( d q / d t ) pd(dq/dt)
  44. ( d q / d t ) d p (dq/dt)dp
  45. p d q ˙ = d ( q ˙ p ) - q ˙ d p pd\dot{q}=d(\dot{q}p)-\dot{q}dp
  46. d ( L - p q ˙ ) = p ˙ d q + L ζ d ζ - q ˙ d p + L ζ ˙ d ζ ˙ + L t d t . d(L-p\dot{q})=\dot{p}dq+\frac{\partial L}{\partial\zeta}d\zeta-\dot{q}dp+\frac% {\partial L}{\partial\dot{\zeta}}d\dot{\zeta}+\frac{\partial L}{\partial t}dt\,.
  47. R ( q , ζ , p , ζ ˙ , t ) = p q ˙ ( p ) - L R(q,\zeta,p,\dot{\zeta},t)=p\dot{q}(p)-L
  48. d q / d t dq/dt
  49. p p
  50. d R = - p ˙ d q - L ζ d ζ + q ˙ d p - L ζ ˙ d ζ ˙ - L t d t , dR=-\dot{p}dq-\frac{\partial L}{\partial\zeta}d\zeta+\dot{q}dp-\frac{\partial L% }{\partial\dot{\zeta}}d\dot{\zeta}-\frac{\partial L}{\partial t}dt\,,
  51. d R = R q d q + R ζ d ζ + R p d p + R ζ ˙ d ζ ˙ + R t d t . dR=\frac{\partial R}{\partial q}dq+\frac{\partial R}{\partial\zeta}d\zeta+% \frac{\partial R}{\partial p}dp+\frac{\partial R}{\partial\dot{\zeta}}d\dot{% \zeta}+\frac{\partial R}{\partial t}dt\,.
  52. d q dq
  53. d ζ
  54. d p dp
  55. d ( d ζ / d t ) d(dζ/dt)
  56. d t dt
  57. q q
  58. q ˙ = R p , p ˙ = - R q , \dot{q}=\frac{\partial R}{\partial p}\,,\quad\dot{p}=-\frac{\partial R}{% \partial q}\,,
  59. ζ ζ
  60. d d t R ζ ˙ = R ζ \frac{d}{dt}\frac{\partial R}{\partial\dot{\zeta}}=\frac{\partial R}{\partial\zeta}
  61. L ζ = - R ζ , L ζ ˙ = - R ζ ˙ , \frac{\partial L}{\partial\zeta}=-\frac{\partial R}{\partial\zeta}\,,\quad% \frac{\partial L}{\partial\dot{\zeta}}=-\frac{\partial R}{\partial\dot{\zeta}}\,,
  62. L L
  63. R R
  64. L t = - R t . \frac{\partial L}{\partial t}=-\frac{\partial R}{\partial t}\,.
  65. n + s n+s
  66. R ( q 1 , , q n , ζ 1 , , ζ s , p 1 , , p n , ζ ˙ 1 , , ζ ˙ s , t ) = i = 1 n p i q ˙ i ( p i ) - L R(q_{1},\ldots,q_{n},\zeta_{1},\ldots,\zeta_{s},p_{1},\ldots,p_{n},\dot{\zeta}% _{1},\ldots,\dot{\zeta}_{s},t)=\sum_{i=1}^{n}p_{i}\dot{q}_{i}(p_{i})-L
  67. R R
  68. i = 1 , 2 , , n i=1,2,...,n
  69. j = 1 , 2 , , s j=1,2,...,s
  70. R q i = - L q i = - d d t L q ˙ i = - p ˙ i \frac{\partial R}{\partial q_{i}}=-\frac{\partial L}{\partial q_{i}}=-\frac{d}% {dt}\frac{\partial L}{\partial\dot{q}_{i}}=-\dot{p}_{i}\,
  71. R p i = q ˙ i \frac{\partial R}{\partial p_{i}}=\dot{q}_{i}\,
  72. R ζ j = - L ζ j , \frac{\partial R}{\partial\zeta_{j}}=-\frac{\partial L}{\partial\zeta_{j}}\,,
  73. R ζ ˙ j = - L ζ ˙ j , \frac{\partial R}{\partial\dot{\zeta}_{j}}=-\frac{\partial L}{\partial\dot{% \zeta}_{j}}\,,
  74. R t = - L t . \frac{\partial R}{\partial t}=-\frac{\partial L}{\partial t}\,.
  75. j j
  76. 2 n + s 2n+s
  77. 2 n 2n
  78. s s
  79. L L
  80. R R
  81. L L
  82. R R
  83. L q i = p ˙ i = - R q i = 0 p i = α i , \frac{\partial L}{\partial q_{i}}=\dot{p}_{i}=-\frac{\partial R}{\partial q_{i% }}=0\quad\Rightarrow\quad p_{i}=\alpha_{i}\,,
  84. R R
  85. R ( ζ 1 , , ζ s , α 1 , , α n , ζ ˙ 1 , , ζ ˙ s , t ) = i = 1 n α i q ˙ i ( α i ) - L ( ζ 1 , , ζ s , q ˙ 1 ( α 1 ) , , q ˙ n ( α n ) , ζ ˙ 1 , , ζ ˙ s , t ) , R(\zeta_{1},\ldots,\zeta_{s},\alpha_{1},\ldots,\alpha_{n},\dot{\zeta}_{1},% \ldots,\dot{\zeta}_{s},t)=\sum_{i=1}^{n}\alpha_{i}\dot{q}_{i}(\alpha_{i})-L(% \zeta_{1},\ldots,\zeta_{s},\dot{q}_{1}(\alpha_{1}),\ldots,\dot{q}_{n}(\alpha_{% n}),\dot{\zeta}_{1},\ldots,\dot{\zeta}_{s},t)\,,
  86. 2 n 2n
  87. q ˙ i = R p i = 0 , p ˙ i = - R q i = 0 , \dot{q}_{i}=\frac{\partial R}{\partial p_{i}}=0\,,\quad\dot{p}_{i}=-\frac{% \partial R}{\partial q_{i}}=0\,,
  88. s s
  89. d d t R ζ ˙ j = R ζ j . \frac{d}{dt}\frac{\partial R}{\partial\dot{\zeta}_{j}}=\frac{\partial R}{% \partial\zeta_{j}}\,.
  90. m m
  91. V ( r ) V(r)
  92. ( r , θ , z ) (r,θ,z)
  93. L ( r , r ˙ , θ ˙ , z ˙ ) = m 2 ( r ˙ 2 + r 2 θ 2 ˙ + z ˙ 2 ) - V ( r ) . L(r,\dot{r},\dot{\theta},\dot{z})=\frac{m}{2}(\dot{r}^{2}+r^{2}\dot{\theta^{2}% }+\dot{z}^{2})-V(r)\,.
  94. V ( r ) V(r)
  95. θ θ
  96. z z
  97. θ θ
  98. z z
  99. p θ = L θ ˙ = m r 2 θ ˙ , p z = L z ˙ = m z ˙ . p_{\theta}=\frac{\partial L}{\partial\dot{\theta}}=mr^{2}\dot{\theta}\,,\quad p% _{z}=\frac{\partial L}{\partial\dot{z}}=m\dot{z}\,.
  100. r r
  101. d θ / d t dθ/dt
  102. d z / d t dz/dt
  103. R ( r , r ˙ ) = p θ θ ˙ + p z z ˙ - L = p θ θ ˙ + p z z ˙ - m 2 r ˙ 2 - p θ θ ˙ 2 - p z z ˙ 2 + V ( r ) = p θ θ ˙ 2 + p z z ˙ 2 - m 2 r ˙ 2 + V ( r ) = p θ 2 2 m r 2 + p z 2 2 m - m 2 r ˙ 2 + V ( r ) . \begin{aligned}\displaystyle R(r,\dot{r})&\displaystyle=p_{\theta}\dot{\theta}% +p_{z}\dot{z}-L\\ &\displaystyle=p_{\theta}\dot{\theta}+p_{z}\dot{z}-\frac{m}{2}\dot{r}^{2}-% \frac{p_{\theta}\dot{\theta}}{2}-\frac{p_{z}\dot{z}}{2}+V(r)\\ &\displaystyle=\frac{p_{\theta}\dot{\theta}}{2}+\frac{p_{z}\dot{z}}{2}-\frac{m% }{2}\dot{r}^{2}+V(r)\\ &\displaystyle=\frac{p_{\theta}^{2}}{2mr^{2}}+\frac{p_{z}^{2}}{2m}-\frac{m}{2}% \dot{r}^{2}+V(r)\,.\end{aligned}
  104. θ θ
  105. z z
  106. r r
  107. d d t R r ˙ = R r \frac{d}{dt}\frac{\partial R}{\partial\dot{r}}=\frac{\partial R}{\partial r}
  108. - m r ¨ = - p θ 2 m r 3 + V r -m\ddot{r}=-\frac{p_{\theta}^{2}}{mr^{3}}+\frac{\partial V}{\partial r}
  109. z z
  110. z = p z m t + c z , z=\frac{p_{z}}{m}t+c_{z}\,,
  111. z z
  112. r r
  113. d θ / d t dθ/dt
  114. θ θ
  115. z z
  116. r r
  117. d d t L r ˙ = L r m r ¨ = m r θ ˙ 2 - V r , \frac{d}{dt}\frac{\partial L}{\partial\dot{r}}=\frac{\partial L}{\partial r}% \quad\Rightarrow\quad m\ddot{r}=mr\dot{\theta}^{2}-\frac{\partial V}{\partial r% }\,,
  118. θ θ
  119. d d t L θ ˙ = L θ m ( 2 r r ˙ θ ˙ + r 2 θ ¨ ) = 0 , \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial% \theta}\quad\Rightarrow\quad m(2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta})=0\,,
  120. z z
  121. d d t L z ˙ = L z m z ¨ = 0 . \frac{d}{dt}\frac{\partial L}{\partial\dot{z}}=\frac{\partial L}{\partial z}% \quad\Rightarrow\quad m\ddot{z}=0\,.
  122. z z
  123. θ θ
  124. m m
  125. V ( r ) V(r)
  126. ( r , θ , φ ) (r,θ,φ)
  127. L ( r , r ˙ , θ , θ ˙ , ϕ ˙ ) = m 2 ( r ˙ 2 + r 2 θ ˙ 2 + r 2 sin 2 θ ϕ ˙ 2 ) - V ( r ) . L(r,\dot{r},\theta,\dot{\theta},\dot{\phi})=\frac{m}{2}(\dot{r}^{2}+{r}^{2}% \dot{\theta}^{2}+r^{2}\sin^{2}\theta\dot{\phi}^{2})-V(r)\,.
  128. φ φ
  129. φ φ
  130. p ϕ = L ϕ ˙ = m r 2 sin 2 θ ϕ ˙ , p_{\phi}=\frac{\partial L}{\partial\dot{\phi}}=mr^{2}\sin^{2}\theta\dot{\phi}\,,
  131. r r
  132. d φ / d t dφ/dt
  133. R ( r , r ˙ , θ , θ ˙ ) = p ϕ ϕ ˙ - L = p ϕ ϕ ˙ - m 2 r ˙ 2 - m 2 r 2 θ ˙ 2 - p ϕ ϕ ˙ 2 + V ( r ) = p ϕ ϕ ˙ 2 - m 2 r ˙ 2 - m 2 r 2 θ ˙ 2 + V ( r ) = p ϕ 2 2 m r 2 sin 2 θ - m 2 r ˙ 2 - m 2 r 2 θ ˙ 2 + V ( r ) . \begin{aligned}\displaystyle R(r,\dot{r},\theta,\dot{\theta})&\displaystyle=p_% {\phi}\dot{\phi}-L\\ &\displaystyle=p_{\phi}\dot{\phi}-\frac{m}{2}\dot{r}^{2}-\frac{m}{2}r^{2}\dot{% \theta}^{2}-\frac{p_{\phi}\dot{\phi}}{2}+V(r)\\ &\displaystyle=\frac{p_{\phi}\dot{\phi}}{2}-\frac{m}{2}\dot{r}^{2}-\frac{m}{2}% r^{2}\dot{\theta}^{2}+V(r)\\ &\displaystyle=\frac{p_{\phi}^{2}}{2mr^{2}\sin^{2}\theta}-\frac{m}{2}\dot{r}^{% 2}-\frac{m}{2}r^{2}\dot{\theta}^{2}+V(r)\,.\end{aligned}
  134. r r
  135. θ θ
  136. φ φ
  137. r r
  138. d d t R r ˙ = R r - m r ¨ = - p ϕ 2 m r 3 sin 2 θ - m r θ ˙ 2 + V r , \frac{d}{dt}\frac{\partial R}{\partial\dot{r}}=\frac{\partial R}{\partial r}% \quad\Rightarrow\quad-m\ddot{r}=-\frac{p_{\phi}^{2}}{mr^{3}\sin^{2}\theta}-mr% \dot{\theta}^{2}+\frac{\partial V}{\partial r}\,,
  139. θ θ
  140. d d t R θ ˙ = R θ - m ( 2 r r ˙ θ ˙ + r 2 θ ¨ ) = - p ϕ 2 cos θ m r 2 sin 3 θ . \frac{d}{dt}\frac{\partial R}{\partial\dot{\theta}}=\frac{\partial R}{\partial% \theta}\quad\Rightarrow\quad-m(2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta})=-% \frac{p_{\phi}^{2}\cos\theta}{mr^{2}\sin^{3}\theta}\,.
  141. φ φ
  142. r r
  143. d d t L r ˙ = L r m r ¨ = m r θ ˙ 2 + m r sin 2 θ ϕ ˙ 2 - V r , \frac{d}{dt}\frac{\partial L}{\partial\dot{r}}=\frac{\partial L}{\partial r}% \quad\Rightarrow\quad m\ddot{r}=mr\dot{\theta}^{2}+mr\sin^{2}\theta\dot{\phi}^% {2}-\frac{\partial V}{\partial r}\,,
  144. θ θ
  145. d d t L θ ˙ = L θ 2 r r ˙ θ ˙ + r 2 θ ¨ = r 2 sin θ cos θ ϕ ˙ 2 , \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial% \theta}\quad\Rightarrow\quad 2r\dot{r}\dot{\theta}+r^{2}\ddot{\theta}=r^{2}% \sin\theta\cos\theta\dot{\phi}^{2}\,,
  146. φ φ
  147. d d t L ϕ ˙ = L ϕ 2 r r ˙ sin 2 θ ϕ ˙ + 2 r 2 sin θ cos θ θ ˙ ϕ ˙ + r 2 sin 2 θ ϕ ¨ = 0 . \frac{d}{dt}\frac{\partial L}{\partial\dot{\phi}}=\frac{\partial L}{\partial% \phi}\quad\Rightarrow\quad 2r\dot{r}\sin^{2}\theta\dot{\phi}+2r^{2}\sin\theta% \cos\theta\dot{\theta}\dot{\phi}+r^{2}\sin^{2}\theta\ddot{\phi}=0\,.
  148. m m
  149. l l
  150. g g
  151. d φ / d t dφ/dt
  152. θ θ
  153. V = m g ( 1 - cos θ ) , V=mg\ell(1-\cos\theta)\,,
  154. L ( θ , θ ˙ , ϕ ˙ ) = m 2 2 ( θ ˙ 2 + sin 2 θ ϕ ˙ 2 ) + m g cos θ , L(\theta,\dot{\theta},\dot{\phi})=\frac{m\ell^{2}}{2}(\dot{\theta}^{2}+\sin^{2% }\theta\dot{\phi}^{2})+mg\ell\cos\theta\,,
  155. φ φ
  156. p ϕ = L ϕ ˙ = m 2 sin 2 θ ϕ ˙ . p_{\phi}=\frac{\partial L}{\partial\dot{\phi}}=m\ell^{2}\sin^{2}\theta\dot{% \phi}\,.
  157. θ θ
  158. d φ / d t dφ/dt
  159. R ( θ , θ ˙ ) = p ϕ ϕ ˙ - L = p ϕ ϕ ˙ - m 2 2 θ ˙ 2 - p ϕ ϕ ˙ 2 - m g cos θ = p ϕ ϕ ˙ 2 - m 2 2 θ ˙ 2 - m g cos θ = p ϕ 2 2 m 2 sin 2 θ - m 2 2 θ ˙ 2 - m g cos θ \begin{aligned}\displaystyle R(\theta,\dot{\theta})&\displaystyle=p_{\phi}\dot% {\phi}-L\\ &\displaystyle=p_{\phi}\dot{\phi}-\frac{m\ell^{2}}{2}\dot{\theta}^{2}-\frac{p_% {\phi}\dot{\phi}}{2}-mg\ell\cos\theta\\ &\displaystyle=\frac{p_{\phi}\dot{\phi}}{2}-\frac{m\ell^{2}}{2}\dot{\theta}^{2% }-mg\ell\cos\theta\\ &\displaystyle=\frac{p_{\phi}^{2}}{2m\ell^{2}\sin^{2}\theta}-\frac{m\ell^{2}}{% 2}\dot{\theta}^{2}-mg\ell\cos\theta\end{aligned}
  160. θ θ
  161. d d t R θ ˙ = R θ - m 2 θ ¨ = - p ϕ 2 cos θ m 2 sin 3 θ + m g sin θ , \frac{d}{dt}\frac{\partial R}{\partial\dot{\theta}}=\frac{\partial R}{\partial% \theta}\quad\Rightarrow\quad-m\ell^{2}\ddot{\theta}=-\frac{p_{\phi}^{2}\cos% \theta}{m\ell^{2}\sin^{3}\theta}+mg\ell\sin\theta\,,
  162. a = p ϕ 2 m 2 4 , b = g , a=\frac{p_{\phi}^{2}}{m^{2}\ell^{4}}\,,\quad b=\frac{g}{\ell}\,,
  163. θ ¨ = a cos θ sin 3 θ - b sin θ . \ddot{\theta}=a\frac{\cos\theta}{\sin^{3}\theta}-b\sin\theta\,.
  164. a a
  165. θ θ
  166. s i n θ θ sinθ~{}θ
  167. c o s θ 1 cosθ~{}1
  168. φ φ
  169. θ θ
  170. d d t L θ ˙ = L θ m 2 θ ¨ = m 2 sin θ cos θ ϕ ˙ 2 - m g sin θ , \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial% \theta}\quad\Rightarrow\quad m\ell^{2}\ddot{\theta}=m\ell^{2}\sin\theta\cos% \theta\dot{\phi}^{2}-mg\ell\sin\theta\,,
  171. φ φ
  172. d d t L ϕ ˙ = L ϕ 2 sin θ cos θ θ ˙ ϕ ˙ + sin 2 θ ϕ ¨ = 0 . \frac{d}{dt}\frac{\partial L}{\partial\dot{\phi}}=\frac{\partial L}{\partial% \phi}\quad\Rightarrow\quad 2\sin\theta\cos\theta\dot{\theta}\dot{\phi}+\sin^{2% }\theta\ddot{\phi}=0\,.
  173. M M
  174. L ( θ , θ ˙ , ψ ˙ , ϕ ˙ ) = I 1 2 ( θ ˙ 2 + ϕ ˙ 2 sin 2 θ ) + I 3 2 ( ψ ˙ 2 + ϕ ˙ 2 cos 2 θ ) + I 3 ψ ˙ ϕ ˙ cos θ - M g cos θ L(\theta,\dot{\theta},\dot{\psi},\dot{\phi})=\frac{I_{1}}{2}(\dot{\theta}^{2}+% \dot{\phi}^{2}\sin^{2}\theta)+\frac{I_{3}}{2}(\dot{\psi}^{2}+\dot{\phi}^{2}% \cos^{2}\theta)+I_{3}\dot{\psi}\dot{\phi}\cos\theta-Mg\ell\cos\theta
  175. ψ , φ , θ ψ,φ,θ
  176. θ θ
  177. z z
  178. z z′
  179. ψ ψ
  180. z z′
  181. φ φ
  182. z z′
  183. z z
  184. x x′
  185. y y′
  186. z z′
  187. z z′
  188. V = M g l c o s θ V=Mglcosθ
  189. g g
  190. l l
  191. z z′
  192. ψ , φ ψ,φ
  193. p ψ = L ψ ˙ = I 3 ψ ˙ + I 3 ϕ ˙ cos θ p_{\psi}=\frac{\partial L}{\partial\dot{\psi}}=I_{3}\dot{\psi}+I_{3}\dot{\phi}\cos\theta
  194. p ϕ = L ϕ ˙ = ϕ ˙ ( I 1 sin 2 θ + I 3 cos 2 θ ) + I 3 ψ ˙ cos θ p_{\phi}=\frac{\partial L}{\partial\dot{\phi}}=\dot{\phi}(I_{1}\sin^{2}\theta+% I_{3}\cos^{2}\theta)+I_{3}\dot{\psi}\cos\theta
  195. d ψ / d t dψ/dt
  196. p ϕ - p ψ cos θ = I 1 ϕ ˙ sin 2 θ p_{\phi}-p_{\psi}\cos\theta=I_{1}\dot{\phi}\sin^{2}\theta
  197. ϕ ˙ = p ϕ - p ψ cos θ I 1 sin 2 θ , \dot{\phi}=\frac{p_{\phi}-p_{\psi}\cos\theta}{I_{1}\sin^{2}\theta}\,,
  198. d φ / d t dφ/dt
  199. d ψ / d t dψ/dt
  200. ψ ˙ = p ψ I 3 - cos θ ( p ϕ - p ψ cos θ I 1 sin 2 θ ) . \dot{\psi}=\frac{p_{\psi}}{I_{3}}-\cos\theta\left(\frac{p_{\phi}-p_{\psi}\cos% \theta}{I_{1}\sin^{2}\theta}\right)\,.
  201. R ( θ , θ ˙ ) = p ψ ψ ˙ + p ϕ ϕ ˙ - L = 1 2 ( p ψ ψ ˙ + p ϕ ϕ ˙ ) - I 1 θ ˙ 2 2 + M g cos θ R(\theta,\dot{\theta})=p_{\psi}\dot{\psi}+p_{\phi}\dot{\phi}-L=\frac{1}{2}(p_{% \psi}\dot{\psi}+p_{\phi}\dot{\phi})-\frac{I_{1}\dot{\theta}^{2}}{2}+Mg\ell\cos\theta
  202. p ϕ ϕ ˙ 2 = p ϕ 2 2 I 1 sin 2 θ - p ψ p ϕ cos θ 2 I 1 sin 2 θ , \frac{p_{\phi}\dot{\phi}}{2}=\frac{p_{\phi}^{2}}{2I_{1}\sin^{2}\theta}-\frac{p% _{\psi}p_{\phi}\cos\theta}{2I_{1}\sin^{2}\theta}\,,
  203. p ψ ψ ˙ 2 = p ψ 2 2 I 3 - p ψ p ϕ cos θ 2 I 1 sin 2 θ + p ψ 2 cos 2 θ 2 I 1 sin 2 θ \frac{p_{\psi}\dot{\psi}}{2}=\frac{p_{\psi}^{2}}{2I_{3}}-\frac{p_{\psi}p_{\phi% }\cos\theta}{2I_{1}\sin^{2}\theta}+\frac{p_{\psi}^{2}\cos^{2}\theta}{2I_{1}% \sin^{2}\theta}
  204. R = p ψ 2 2 I 3 + p ψ 2 cos 2 θ 2 I 1 sin 2 θ + p ϕ 2 2 I 1 sin 2 θ - p ψ p ϕ cos θ 4 I 1 sin 2 θ - I 1 θ ˙ 2 2 + M g cos θ . R=\frac{p_{\psi}^{2}}{2I_{3}}+\frac{p_{\psi}^{2}\cos^{2}\theta}{2I_{1}\sin^{2}% \theta}+\frac{p_{\phi}^{2}}{2I_{1}\sin^{2}\theta}-\frac{p_{\psi}p_{\phi}\cos% \theta}{4I_{1}\sin^{2}\theta}-\frac{I_{1}\dot{\theta}^{2}}{2}+Mg\ell\cos\theta\,.
  205. R = 1 2 I 1 sin 2 θ [ p ψ 2 cos 2 θ + p ϕ 2 - p ψ p ϕ 2 cos θ ] - I 1 θ ˙ 2 2 + M g cos θ R=\frac{1}{2I_{1}\sin^{2}\theta}\left[p_{\psi}^{2}\cos^{2}\theta+p_{\phi}^{2}-% \frac{p_{\psi}p_{\phi}}{2}\cos\theta\right]-\frac{I_{1}\dot{\theta}^{2}}{2}+Mg% \ell\cos\theta
  206. θ θ
  207. d d t R θ ˙ = R θ \frac{d}{dt}\frac{\partial R}{\partial\dot{\theta}}=\frac{\partial R}{\partial% \theta}\quad\Rightarrow\quad
  208. - I 1 θ ¨ = - cos θ I 1 sin 3 θ [ p ψ 2 cos 2 θ + p ϕ 2 - p ψ p ϕ 2 cos θ ] + 1 2 I 1 sin 2 θ [ - 2 p ψ 2 cos θ sin θ + p ψ p ϕ 2 sin θ ] - M g sin θ , -I_{1}\ddot{\theta}=-\frac{\cos\theta}{I_{1}\sin^{3}\theta}\left[p_{\psi}^{2}% \cos^{2}\theta+p_{\phi}^{2}-\frac{p_{\psi}p_{\phi}}{2}\cos\theta\right]+\frac{% 1}{2I_{1}\sin^{2}\theta}\left[-2p_{\psi}^{2}\cos\theta\sin\theta+\frac{p_{\psi% }p_{\phi}}{2}\sin\theta\right]-Mg\ell\sin\theta\,,
  209. a = p ψ 2 I 1 2 , b = p ϕ 2 I 1 2 , c = p ψ p ϕ 2 I 1 2 , k = M g I 1 , a=\frac{p_{\psi}^{2}}{I_{1}^{2}}\,,\quad b=\frac{p_{\phi}^{2}}{I_{1}^{2}}\,,% \quad c=\frac{p_{\psi}p_{\phi}}{2I_{1}^{2}}\,,\quad k=\frac{Mg\ell}{I_{1}}\,,
  210. θ ¨ = cos θ sin 3 θ ( a cos 2 θ + b - c cos θ ) + 1 2 sin θ ( 2 a cos θ - c ) + k sin θ . \ddot{\theta}=\frac{\cos\theta}{\sin^{3}\theta}(a\cos^{2}\theta+b-c\cos\theta)% +\frac{1}{2\sin\theta}(2a\cos\theta-c)+k\sin\theta\,.
  211. ψ ψ
  212. φ φ
  213. θ θ
  214. d d t L θ ˙ = L θ I 1 θ ¨ = ( I 1 - I 3 ) ϕ ˙ 2 sin θ cos θ - I 3 ψ ˙ ϕ ˙ sin θ + M g sin θ , \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}=\frac{\partial L}{\partial% \theta}\quad\Rightarrow\quad I_{1}\ddot{\theta}=(I_{1}-I_{3})\dot{\phi}^{2}% \sin\theta\cos\theta-I_{3}\dot{\psi}\dot{\phi}\sin\theta+Mg\ell\sin\theta\,,
  215. ψ ψ
  216. d d t L ψ ˙ = L ψ ψ ¨ + ϕ ¨ cos θ - ϕ ˙ θ ˙ sin θ = 0 , \frac{d}{dt}\frac{\partial L}{\partial\dot{\psi}}=\frac{\partial L}{\partial% \psi}\quad\Rightarrow\quad\ddot{\psi}+\ddot{\phi}\cos\theta-\dot{\phi}\dot{% \theta}\sin\theta=0\,,
  217. φ φ
  218. d d t L ϕ ˙ = L ϕ ϕ ¨ ( I 1 sin 2 θ + I 3 cos 2 θ ) + ϕ ˙ ( I 1 - I 3 ) 2 sin θ cos θ θ ˙ + I 3 ψ ¨ cos θ - I 3 ψ ˙ sin θ θ ˙ = 0 , \frac{d}{dt}\frac{\partial L}{\partial\dot{\phi}}=\frac{\partial L}{\partial% \phi}\quad\Rightarrow\quad\ddot{\phi}(I_{1}\sin^{2}\theta+I_{3}\cos^{2}\theta)% +\dot{\phi}(I_{1}-I_{3})2\sin\theta\cos\theta\dot{\theta}+I_{3}\ddot{\psi}\cos% \theta-I_{3}\dot{\psi}\sin\theta\dot{\theta}=0\,,
  219. L t = d d t ( i = 1 n q ˙ i L q ˙ i + j = 1 s ζ ˙ j L ζ ˙ j - L ) . \frac{\partial L}{\partial t}=\frac{d}{dt}\left(\sum_{i=1}^{n}\dot{q}_{i}\frac% {\partial L}{\partial\dot{q}_{i}}+\sum_{j=1}^{s}\dot{\zeta}_{j}\frac{\partial L% }{\partial\dot{\zeta}_{j}}-L\right)\,.
  220. L / t = 0 ∂L/∂t=0
  221. E = i = 1 n q ˙ i L q ˙ i + j = 1 s ζ ˙ j L ζ ˙ j - L , E=\sum_{i=1}^{n}\dot{q}_{i}\frac{\partial L}{\partial\dot{q}_{i}}+\sum_{j=1}^{% s}\dot{\zeta}_{j}\frac{\partial L}{\partial\dot{\zeta}_{j}}-L\,,
  222. L L
  223. R R
  224. R R
  225. R R
  226. E = R - j = 1 s ζ ˙ j R ζ ˙ j . E=R-\sum_{j=1}^{s}\dot{\zeta}_{j}\frac{\partial R}{\partial\dot{\zeta}_{j}}\,.
  227. R R
  228. s = 0 s=0
  229. E = R E=R
  230. R R
  231. s = 0 s=0
  232. E = R = H E=R=H
  233. R t = d d t ( R - j = 1 s ζ ˙ j R ζ ˙ j ) , \frac{\partial R}{\partial t}=\dfrac{d}{dt}\left(R-\sum_{j=1}^{s}\dot{\zeta}_{% j}\frac{\partial R}{\partial\dot{\zeta}_{j}}\right)\,,
  234. R R
  235. L L
  236. u u
  237. v v
  238. d ( u v ) = u d v + v d u d(uv)=udv+vdu

Rowbottom_cardinal.html

  1. ω \aleph_{\omega}

Rule_30.html

  1. n n
  2. n n
  3. C C
  4. D D
  5. i i
  6. i + 1 i+1

Runge's_theorem.html

  1. ( r n ) n 𝒩 (r_{n})_{n\in\mathcal{N}}
  2. ( r n ) n 𝒩 (r_{n})_{n\in\mathcal{N}}
  3. ( r n ) n 𝒩 (r_{n})_{n\in\mathcal{N}}
  4. ( r n ) n 𝒩 (r_{n})_{n\in\mathcal{N}}
  5. ( p n ) (p_{n})
  6. f ( w ) = 1 2 π i B f ( z ) d z z - w f(w)={1\over 2\pi i}\int_{B}{f(z)\,dz\over z-w}

Rushbrooke_inequality.html

  1. f = - k T lim N 1 N log Z N f=-kT\lim_{N\rightarrow\infty}\frac{1}{N}\log Z_{N}
  2. M ( T , H ) = def lim N 1 N ( i σ i ) = - ( f H ) T M(T,H)\ \stackrel{\mathrm{def}}{=}\ \lim_{N\rightarrow\infty}\frac{1}{N}\left(% \sum_{i}\sigma_{i}\right)=-\left(\frac{\partial f}{\partial H}\right)_{T}
  3. σ i \sigma_{i}
  4. χ T ( T , H ) = ( M H ) T \chi_{T}(T,H)=\left(\frac{\partial M}{\partial H}\right)_{T}
  5. c H = - T ( 2 f T 2 ) H . c_{H}=-T\left(\frac{\partial^{2}f}{\partial T^{2}}\right)_{H}.
  6. α , α , β , γ , γ \alpha,\alpha^{\prime},\beta,\gamma,\gamma^{\prime}
  7. δ \delta
  8. M ( t , 0 ) ( - t ) β for t 0 M(t,0)\simeq(-t)^{\beta}\mbox{ for }~{}t\uparrow 0
  9. M ( 0 , H ) | H | 1 / δ sign ( H ) for H 0 M(0,H)\simeq|H|^{1/\delta}\operatorname{sign}(H)\mbox{ for }~{}H\rightarrow 0
  10. χ T ( t , 0 ) { ( t ) - γ , for t 0 ( - t ) - γ , for t 0 \chi_{T}(t,0)\simeq\begin{cases}(t)^{-\gamma},&\textrm{for}\ t\downarrow 0\\ (-t)^{-\gamma^{\prime}},&\textrm{for}\ t\uparrow 0\end{cases}
  11. c H ( t , 0 ) { ( t ) - α for t 0 ( - t ) - α for t 0 c_{H}(t,0)\simeq\begin{cases}(t)^{-\alpha}&\textrm{for}\ t\downarrow 0\\ (-t)^{-\alpha^{\prime}}&\textrm{for}\ t\uparrow 0\end{cases}
  12. t = def T - T c T c t\ \stackrel{\mathrm{def}}{=}\ \frac{T-T_{c}}{T_{c}}
  13. χ T ( c H - c M ) = T ( M T ) H 2 \chi_{T}(c_{H}-c_{M})=T\left(\frac{\partial M}{\partial T}\right)_{H}^{2}
  14. c h , c M and χ T 0 c_{h},c_{M}\mbox{ and }~{}\chi_{T}\geq 0
  15. c H T χ T ( M T ) H 2 c_{H}\geq\frac{T}{\chi_{T}}\left(\frac{\partial M}{\partial T}\right)_{H}^{2}
  16. H = 0 , t > 0 H=0,t>0
  17. ( - t ) - α constant ( - t ) γ ( - t ) 2 ( β - 1 ) (-t)^{-\alpha^{\prime}}\geq\mathrm{constant}\cdot(-t)^{\gamma^{\prime}}(-t)^{2% (\beta-1)}
  18. α + 2 β + γ 2. \alpha^{\prime}+2\beta+\gamma^{\prime}\geq 2.

Rusty_bolt_effect.html

  1. E out = n = 1 K n E in n E\text{out}=\sum_{n=1}^{\infty}{K_{n}E\text{in}^{n}}
  2. E out = K 1 E in + K 2 E in 2 + K 3 E in 3 + K 4 E in 4 + K 5 E in 5 + E\text{out}=K_{1}E\text{in}+K_{2}E\text{in}^{2}+K_{3}E\text{in}^{3}+K_{4}E% \text{in}^{4}+K_{5}E\text{in}^{5}+...
  3. E out \displaystyle E\text{out}
  4. E f 1 + f 2 = k E f 1 E f 2 E f 1 - f 2 = k E f 1 E f 2 \begin{aligned}\displaystyle E_{f_{1}+f_{2}}&\displaystyle=kE_{f_{1}}E_{f_{2}}% \\ \displaystyle E_{f_{1}-f_{2}}&\displaystyle=kE_{f_{1}}E_{f_{2}}\end{aligned}
  5. E f 1 + f 2 + f 3 = k E f 1 E f 2 E f 3 E f 1 - f 2 + f 3 = k E f 1 E f 2 E f 3 E f 1 + f 2 - f 3 = k E f 1 E f 2 E f 3 E f 1 - f 2 - f 3 = k E f 1 E f 2 E f 3 \begin{aligned}\displaystyle E_{f_{1}+f_{2}+f_{3}}&\displaystyle=kE_{f_{1}}E_{% f_{2}}E_{f_{3}}\\ \displaystyle E_{f_{1}-f_{2}+f_{3}}&\displaystyle=kE_{f_{1}}E_{f_{2}}E_{f_{3}}% \\ \displaystyle E_{f_{1}+f_{2}-f_{3}}&\displaystyle=kE_{f_{1}}E_{f_{2}}E_{f_{3}}% \\ \displaystyle E_{f_{1}-f_{2}-f_{3}}&\displaystyle=kE_{f_{1}}E_{f_{2}}E_{f_{3}}% \end{aligned}

Saccheri_quadrilateral.html

  1. - 1 -1
  2. s s
  3. l l
  4. b b
  5. cosh s = cosh b cosh 2 l - sinh 2 l . \cosh s=\cosh b\cdot\cosh^{2}l-\sinh^{2}l.

Saddle-node_bifurcation.html

  1. d x d t = r + x 2 . \frac{dx}{dt}=r+x^{2}.
  2. x x
  3. r r
  4. r < 0 r<0
  5. - - r -\sqrt{-r}
  6. + - r +\sqrt{-r}
  7. r = 0 r=0
  8. r > 0 r>0
  9. d x d t = f ( r , x ) \tfrac{dx}{dt}=f(r,x)
  10. x = 0 x=0
  11. r = 0 r=0
  12. f x ( 0 , 0 ) = 0 \tfrac{\partial f}{\partial x}(0,0)=0
  13. d x d t = r ± x 2 \frac{dx}{dt}=r\pm x^{2}
  14. 2 f x 2 ( 0 , 0 ) 0 \tfrac{\partial^{2}f}{\partial x^{2}}(0,0)\neq 0
  15. f r ( 0 , 0 ) 0 \tfrac{\partial f}{\partial r}(0,0)\neq 0
  16. d x d t = α - x 2 \frac{dx}{dt}=\alpha-x^{2}
  17. d y d t = - y . \frac{dy}{dt}=-y.
  18. α \alpha
  19. α \alpha
  20. α = 0 \alpha=0
  21. α \alpha
  22. p x px
  23. p p
  24. x x

SAIDI.html

  1. SAIDI = U i N i N T \mbox{SAIDI}~{}=\frac{\sum{U_{i}N_{i}}}{N_{T}}
  2. N i N_{i}
  3. U i U_{i}
  4. i i
  5. N T N_{T}
  6. SAIDI = sum of all customer interruption durations total number of customers served \mbox{SAIDI}~{}=\frac{\mbox{sum of all customer interruption durations}~{}}{% \mbox{total number of customers served}~{}}

SAIFI.html

  1. SAIFI = λ i N i N T \mbox{SAIFI}~{}=\frac{\sum{\lambda_{i}N_{i}}}{N_{T}}
  2. λ i \lambda_{i}
  3. N i N_{i}
  4. i i
  5. N T N_{T}
  6. SAIFI = total number of customer interruptions total number of customers served \mbox{SAIFI}~{}=\frac{\mbox{total number of customer interruptions}~{}}{\mbox{% total number of customers served}~{}}

Samarium-neodymium_dating.html

  1. × 10 1 1 \times 10^{1}1
  2. × 10 8 \times 10^{8}
  3. × 10 6 \times 10^{6}
  4. ε N d ( t ) = [ ( N 143 d N 144 d ) s a m p l e ( t ) ( N 143 d N 144 d ) C H U R ( t ) - 1 ] * 10000 \varepsilon_{Nd(t)}=\left[\frac{\left(\frac{{}^{143}Nd}{{}^{144}Nd}\right)_{% sample(t)}}{\left(\frac{{}^{143}Nd}{{}^{144}Nd}\right)_{CHUR(t)}}-1\right]*10000
  5. T C H U R = ( 1 λ ) l n [ 1 + ( N 143 d N 144 d ) s a m p l e - ( N 143 d N 144 d ) C H U R ( S 147 m N 144 d ) s a m p l e - ( S 147 m N 144 d ) C H U R ] T_{CHUR}=(\frac{1}{\lambda})ln\left[1+\frac{\left(\frac{{}^{143}Nd}{{}^{144}Nd% }\right)_{sample}-\left(\frac{{}^{143}Nd}{{}^{144}Nd}\right)_{CHUR}}{\left(% \frac{{}^{147}Sm}{{}^{144}Nd}\right)_{sample}-\left(\frac{{}^{147}Sm}{{}^{144}% Nd}\right)_{CHUR}}\right]

Sasakian_manifold.html

  1. ( M , θ ) (M,\theta)
  2. g g
  3. ( M , g ) (M,g)
  4. ( M × > 0 ) (M\times{\mathbb{R}}^{>0})\,
  5. M M
  6. > 0 {\mathbb{R}}^{>0}
  7. t 2 g + d t 2 , t^{2}g+dt^{2},\,
  8. t t
  9. > 0 {\mathbb{R}}^{>0}
  10. M M
  11. θ \theta
  12. t 2 d θ + 2 t d t θ t^{2}\,d\theta+2t\,dt\cdot\theta\,
  13. t 2 d θ + 2 t d t θ . t^{2}\,d\theta+2t\,dt\cdot\theta.\,
  14. S 2 n - 1 2 n , * = n , * S^{2n-1}\hookrightarrow{\mathbb{R}}^{2n,*}={\mathbb{C}}^{n,*}
  15. S 2 n - 1 S^{2n-1}
  16. i n i\vec{n}
  17. n \vec{n}
  18. i i
  19. n {\mathbb{C}}^{n}
  20. 2 n + 1 {{\mathbb{R}}^{2n+1}}
  21. ( x , y , z ) (\vec{x},\vec{y},z)
  22. θ = 1 2 d z + i y i d x i \theta=\frac{1}{2}dz+\sum_{i}y_{i}\,dx_{i}
  23. g = i ( d x i ) 2 + ( d y i ) 2 + θ 2 . g=\sum_{i}(dx_{i})^{2}+(dy_{i})^{2}+\theta^{2}.
  24. 2 n - 1 n , * / 2 {\mathbb{P}}^{2n-1}{\mathbb{R}}\hookrightarrow{\mathbb{C}}^{n,*}/{\mathbb{Z}}_% {2}
  25. 2 {\mathbb{Z}}_{2}
  26. t / t . t\partial/\partial t.
  27. ξ = - J ( t / t ) . \xi=-J(t\partial/\partial t).
  28. M M
  29. M M
  30. M M
  31. L a , b , c L^{a,b,c}

Sauerbrey_equation.html

  1. Δ f = - 2 f 0 2 A ρ q μ q Δ m \Delta f=-\frac{2f_{0}^{2}}{A\sqrt{\rho_{q}\mu_{q}}}\Delta m
  2. f 0 f_{0}
  3. Δ f \Delta f
  4. Δ m \Delta m
  5. A A
  6. ρ q \rho_{q}
  7. ρ q \rho_{q}
  8. μ q \mu_{q}
  9. μ q \mu_{q}
  10. Δ f / f \Delta f/f
  11. Δ f / f \Delta f/f
  12. Δ m A = N q ρ q π Z f L tan - 1 [ Z tan ( π f U - f L f U ) ] \frac{\Delta m}{A}\ =\frac{N_{q}\rho_{q}}{\pi Zf_{L}}\tan^{-1}\left[Z\tan\left% (\pi\frac{f_{U}-f_{L}}{f_{U}}\right)\right]
  13. f L f_{L}
  14. f U f_{U}
  15. N q N_{q}
  16. Δ m \Delta m
  17. A A
  18. ρ q \rho_{q}
  19. ρ q \rho_{q}
  20. Z Z
  21. Z = ( ρ q μ q ρ f μ f ) Z=\sqrt{\left(\frac{\rho_{q}\mu_{q}}{\rho_{f}\mu_{f}}\ \right)}
  22. ρ f \rho_{f}
  23. μ q \mu_{q}
  24. μ q \mu_{q}
  25. μ f \mu_{f}
  26. Δ f = - f 0 3 / 2 ( η l ρ l / π ρ q μ q ) 1 / 2 \Delta f={-\ f_{0}^{3/2}(\eta_{l}\rho_{l}/\pi\rho_{q}\mu_{q})^{1/2}}
  27. ρ l \rho_{l}
  28. η l \eta_{l}

Scalar-vector-tensor_decomposition.html

  1. g μ ν = g μ ν + h μ ν g^{\prime}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}
  2. h μ ν h_{\mu\nu}
  3. h 00 = - 2 ψ h_{00}=-2\psi
  4. h 0 i = w i h_{0i}=w_{i}
  5. h i j = 2 ( ϕ g i j + S i j ) h_{ij}=2(\phi g_{ij}+S_{ij})
  6. S i j S_{ij}
  7. g i j g_{ij}
  8. g i j S i j = 0 g^{ij}S_{ij}=0
  9. w i w_{i}
  10. S i j S_{ij}
  11. w i = w | | + i w , i w_{i}=w^{||}{}_{i}+w^{\perp}{}_{i},
  12. × 𝐰 | | = 𝟎 \nabla\times\mathbf{w}^{||}=\mathbf{0}
  13. 𝐰 = 0 \nabla\cdot\mathbf{w}^{\perp}=0
  14. i \nabla_{i}
  15. g i j g_{ij}
  16. w | | = i i A w^{||}{}_{i}=\nabla_{i}A
  17. 𝐰 \mathbf{w}
  18. S i j S_{ij}
  19. S i j = S | | + i j S i j + S T , i j S_{ij}=S^{||}{}_{ij}+S^{\perp}_{ij}+S^{T}{}_{ij},
  20. S | | = i j ( i j - 1 3 g i j 2 ) B S^{||}{}_{ij}=(\nabla_{i}\nabla_{j}-\frac{1}{3}g_{ij}\nabla^{2})B
  21. B B
  22. S S
  23. S = i j i S + j j S i S^{\perp}{}_{ij}=\nabla_{i}S^{\perp}{}_{j}+\nabla_{j}S^{\perp}{}_{i}
  24. S i S^{\perp}{}_{i}
  25. S T i j S^{T}{}_{ij}
  26. S T i j S^{T}{}_{ij}

Scalar_(mathematics).html

  1. k ( v 1 , v 2 , , v n ) k(v_{1},v_{2},\dots,v_{n})
  2. ( k v 1 , k v 2 , , k v n ) (kv_{1},kv_{2},\dots,kv_{n})

Scalar_electrodynamics.html

  1. ϕ ( x ) \phi(x)
  2. A μ ( x ) A_{\mu}(x)
  3. = ( D μ ϕ ) * D μ ϕ - U ( ϕ * ϕ ) - 1 4 F μ ν F μ ν , \mathcal{L}=(D_{\mu}\phi)^{*}D^{\mu}\phi-U(\phi^{*}\phi)-\frac{1}{4}F_{\mu\nu}% F^{\mu\nu}\ ,
  4. F μ ν = ( μ A ν - ν A μ ) F_{\mu\nu}=(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})
  5. D μ ϕ = ( μ ϕ - i e A μ ϕ ) D_{\mu}\phi=(\partial_{\mu}\phi-ieA_{\mu}\phi)
  6. ϕ \phi
  7. e e
  8. U ( ϕ * ϕ ) U(\phi^{*}\phi)
  9. λ ( x ) \lambda(x)
  10. ϕ ( x ) = e i e λ ( x ) ϕ ( x ) and A μ ( x ) = A μ ( x ) + μ λ ( x ) . \phi^{\prime}(x)=e^{ie\lambda(x)}\phi(x)\quad\textrm{and}\quad A_{\mu}^{\prime% }(x)=A_{\mu}(x)+\partial_{\mu}\lambda(x)\ .
  11. | ϕ | |\phi|
  12. e e
  13. | ϕ | |\phi|
  14. 2 + 1 2+1
  15. 2 π e \tfrac{2\pi}{e}
  16. J t o p μ = ϵ μ ν ρ F ν ρ . J_{top}^{\mu}=\epsilon^{\mu\nu\rho}F_{\nu\rho}\ .

Scalar–tensor_theory.html

  1. S = 1 c d 4 x - g 1 2 μ × [ Φ R - ω ( Φ ) Φ ( σ Φ ) 2 - V ( Φ ) + 2 μ m ( g μ ν , Ψ ) ] , S=\frac{1}{c}\int{d^{4}x\sqrt{-g}\frac{1}{2\mu}}\times\left[\Phi R-\frac{% \omega(\Phi)}{\Phi}(\partial_{\sigma}\Phi)^{2}-V(\Phi)+2\mu~{}\mathcal{L}_{m}(% g_{\mu\nu},\Psi)\right],
  2. g g
  3. R R
  4. g μ ν g_{\mu\nu}
  5. μ \mu
  6. L - 1 M - 1 T 2 L^{-1}M^{-1}T^{2}
  7. V ( Φ ) V(\Phi)
  8. m \mathcal{L}_{m}
  9. Ψ \Psi
  10. ω \omega
  11. μ \mu
  12. 8 π G / c 4 8\pi G/c^{4}
  13. G G
  14. Φ \Phi
  15. R μ ν - 1 2 g μ ν R = μ Φ T μ ν + 1 Φ [ μ ν - g μ ν ] Φ + ω ( Φ ) Φ 2 ( μ Φ ν Φ - 1 2 g μ ν ( α Φ ) 2 ) - g μ ν V ( Φ ) 2 Φ , R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{\mu}{\Phi}T_{\mu\nu}+\frac{1}{\Phi}[% \nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box]\Phi+\frac{\omega(\Phi)}{\Phi^{2}}(% \partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}(\partial_{\alpha}% \Phi)^{2})-g_{\mu\nu}\frac{V(\Phi)}{2\Phi},
  16. 2 ω ( Φ ) + 3 Φ Φ = μ Φ T - ω ( Φ ) Φ ( σ Φ ) 2 + V ( Φ ) - 2 V ( Φ ) Φ . \frac{2\omega(\Phi)+3}{\Phi}\Box\Phi=\frac{\mu}{\Phi}T-\frac{\omega^{\prime}(% \Phi)}{\Phi}(\partial_{\sigma}\Phi)^{2}+V^{\prime}(\Phi)-2\frac{V(\Phi)}{\Phi}.
  17. σ T μ σ = 0 , \nabla_{\sigma}T^{\mu\sigma}=0,
  18. T μ σ T^{\mu\sigma}
  19. T μ ν = - 2 - g δ ( - g m ) δ g μ ν . T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta g% ^{\mu\nu}}.
  20. g 00 = - 1 + 2 U c 2 + 𝒪 ( c - 3 ) , g 0 i = 𝒪 ( c - 2 ) , g i j = δ i j + 𝒪 ( c - 1 ) , g_{00}=-1+2\frac{U}{c^{2}}+\mathcal{O}(c^{-3}),~{}g_{0i}=\mathcal{O}(c^{-2}),~% {}g_{ij}=\delta_{ij}+\mathcal{O}(c^{-1}),
  21. U U
  22. U = 8 π G eff ρ + 𝒪 ( c - 1 ) , \triangle U=8\pi G_{\mathrm{eff}}~{}\rho+\mathcal{O}(c^{-1}),
  23. ρ \rho
  24. G eff = 2 ω 0 + 4 2 ω 0 + 3 G Φ 0 G_{\mathrm{eff}}=\frac{2\omega_{0}+4}{2\omega_{0}+3}\frac{G}{\Phi_{0}}
  25. 0 {}_{0}
  26. Φ 0 \Phi_{0}
  27. Φ 0 \Phi_{0}
  28. g i j δ i j + 𝒪 ( c - 3 ) g_{ij}\propto\delta_{ij}+\mathcal{O}(c^{-3})
  29. g 00 = - 1 + 2 W c 2 - β 2 W 2 c 4 + 𝒪 ( c - 5 ) g_{00}=-1+\frac{2W}{c^{2}}-\beta\frac{2W^{2}}{c^{4}}+\mathcal{O}(c^{-5})
  30. g 0 i = - ( γ + 1 ) 2 W i c 3 + 𝒪 ( c - 4 ) g_{0i}=-(\gamma+1)\frac{2W_{i}}{c^{3}}+\mathcal{O}(c^{-4})
  31. g i j = δ i j ( 1 + γ 2 W c 2 ) + 𝒪 ( c - 3 ) g_{ij}=\delta_{ij}\left(1+\gamma\frac{2W}{c^{2}}\right)+\mathcal{O}(c^{-3})
  32. W + 1 + 2 β - 3 γ c 2 W W + 2 c 2 ( 1 + γ ) t J = - 4 π G eff Σ + 𝒪 ( c - 3 ) , \Box W+\frac{1+2\beta-3\gamma}{c^{2}}W\triangle W+\frac{2}{c^{2}}(1+\gamma)% \partial_{t}J=-4\pi G_{\mathrm{eff}}\Sigma+\mathcal{O}(c^{-3}),
  33. W i - x i J = - 4 π G eff Σ i + 𝒪 ( c - 1 ) , \triangle W_{i}-\partial x_{i}J=-4\pi G_{\mathrm{eff}}\Sigma^{i}+\mathcal{O}(c% ^{-1}),
  34. J J
  35. J = t W + k W k + 𝒪 ( c - 1 ) . J=\partial_{t}W+\partial_{k}W_{k}+\mathcal{O}(c^{-1}).
  36. Σ = 1 c 2 ( T 00 + γ T k k ) , Σ i = 1 c T 0 i , \Sigma=\frac{1}{c^{2}}(T^{00}+\gamma T^{kk}),~{}~{}~{}\Sigma^{i}=\frac{1}{c}T^% {0i},
  37. γ = ω 0 + 1 ω 0 + 2 , β = 1 + ω 0 ( 2 ω 0 + 3 ) ( 2 ω 0 + 4 ) 2 , \gamma=\frac{\omega_{0}+1}{\omega_{0}+2},~{}~{}\beta=1+\frac{\omega_{0}^{% \prime}}{(2\omega_{0}+3)(2\omega_{0}+4)^{2}},
  38. G eff G_{\mathrm{eff}}
  39. G eff = 2 ω 0 + 4 2 ω 0 + 3 G , G_{\mathrm{eff}}=\frac{2\omega_{0}+4}{2\omega_{0}+3}G,
  40. G G
  41. μ \mu
  42. γ - 1 = ( 2.1 ± 2.3 ) × 10 - 5 \gamma-1=(2.1\pm 2.3)\times 10^{-5}
  43. ω 0 > 40000 \omega_{0}>40000
  44. ω \omega
  45. ω \omega
  46. β \beta
  47. | β - 1 | < 3 × 10 - 3 |\beta-1|<3\times 10^{-3}

Scalar–tensor–vector_gravity.html

  1. [ + , - , - , - ] [+,-,-,-]
  2. c = 1 c=1
  3. R μ ν = α Γ μ ν α - ν Γ μ α α + Γ μ ν α Γ α β β - Γ μ β α Γ α ν β . R_{\mu\nu}=\partial_{\alpha}\Gamma^{\alpha}_{\mu\nu}-\partial_{\nu}\Gamma^{% \alpha}_{\mu\alpha}+\Gamma^{\alpha}_{\mu\nu}\Gamma^{\beta}_{\alpha\beta}-% \Gamma^{\alpha}_{\mu\beta}\Gamma^{\beta}_{\alpha\nu}.
  4. G = - 1 16 π G ( R + 2 Λ ) - g , {\mathcal{L}}_{G}=-\frac{1}{16\pi G}\left(R+2\Lambda\right)\sqrt{-g},
  5. R R
  6. G G
  7. g g
  8. g μ ν g_{\mu\nu}
  9. Λ \Lambda
  10. ϕ μ \phi_{\mu}
  11. ϕ = - 1 4 π ω [ 1 4 B μ ν B μ ν - 1 2 μ 2 ϕ μ ϕ μ + V ϕ ( ϕ ) ] - g , {\mathcal{L}}_{\phi}=-\frac{1}{4\pi}\omega\left[\frac{1}{4}B^{\mu\nu}B_{\mu\nu% }-\frac{1}{2}\mu^{2}\phi_{\mu}\phi^{\mu}+V_{\phi}(\phi)\right]\sqrt{-g},
  12. B μ ν = μ ϕ ν - ν ϕ μ B_{\mu\nu}=\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}
  13. μ \mu
  14. ω \omega
  15. V ϕ V_{\phi}
  16. G G
  17. μ \mu
  18. ω \omega
  19. S = - 1 G [ 1 2 g μ ν ( μ G ν G G 2 + μ μ ν μ μ 2 - μ ω ν ω ) + V G ( G ) G 2 + V μ ( μ ) μ 2 + V ω ( ω ) ] - g , {\mathcal{L}}_{S}=-\frac{1}{G}\left[\frac{1}{2}g^{\mu\nu}\left(\frac{\nabla_{% \mu}G\nabla_{\nu}G}{G^{2}}+\frac{\nabla_{\mu}\mu\nabla_{\nu}\mu}{\mu^{2}}-% \nabla_{\mu}\omega\nabla_{\nu}\omega\right)+\frac{V_{G}(G)}{G^{2}}+\frac{V_{% \mu}(\mu)}{\mu^{2}}+V_{\omega}(\omega)\right]\sqrt{-g},
  20. μ \nabla_{\mu}
  21. g μ ν g_{\mu\nu}
  22. V G V_{G}
  23. V μ V_{\mu}
  24. V ω V_{\omega}
  25. S = ( G + ϕ + S + M ) d 4 x , S=\int{({\mathcal{L}}_{G}+{\mathcal{L}}_{\phi}+{\mathcal{L}}_{S}+{\mathcal{L}}% _{M})}~{}d^{4}x,
  26. M {\mathcal{L}}_{M}
  27. TP = - m + α ω q 5 ϕ μ u μ , {\mathcal{L}}_{\mathrm{TP}}=-m+\alpha\omega q_{5}\phi_{\mu}u^{\mu},
  28. m m
  29. α \alpha
  30. q 5 q_{5}
  31. u μ = d x μ / d s u^{\mu}=dx^{\mu}/ds
  32. q 5 = κ m q_{5}=\kappa m
  33. κ = G N / ω \kappa=\sqrt{G_{N}/\omega}
  34. M M
  35. r ¨ = - G N M r 2 [ 1 + α - α ( 1 + μ r ) e - μ r ] , \ddot{r}=-\frac{G_{N}M}{r^{2}}\left[1+\alpha-\alpha(1+\mu r)e^{-\mu r}\right],
  36. G N G_{N}
  37. α \alpha
  38. μ \mu
  39. M M
  40. μ = D M , \mu=\frac{D}{\sqrt{M}},
  41. α = G - G N G N M ( M + E ) 2 , \alpha=\frac{G_{\infty}-G_{N}}{G_{N}}\frac{M}{(\sqrt{M}+E)^{2}},
  42. G 20 G N G_{\infty}\simeq 20G_{N}
  43. D D
  44. E E
  45. D 6250 M 1 / 2 kpc - 1 , D\simeq 6250M_{\odot}^{1/2}\mathrm{kpc}^{-1},
  46. E 25000 M 1 / 2 , E\simeq 25000M_{\odot}^{1/2},
  47. M M_{\odot}

Scale_analysis_(mathematics).html

  1. w t + u w x + v w y + w w z - u 2 + v 2 R = - 1 ϱ p z - g + 2 Ω u cos φ + ν ( 2 w x 2 + 2 w y 2 + 2 w z 2 ) , ( 1 ) {{\partial w}\over{\partial t}}+u{\frac{\partial w}{\partial x}}+v{\frac{% \partial w}{\partial y}}+w{\frac{\partial w}{\partial z}}-{\frac{u^{2}+v^{2}}{% R}}=-{{\frac{1}{\varrho}}{\frac{\partial p}{\partial z}}}-g+2{\Omega u\cos% \varphi}+\nu\left({\frac{\partial^{2}w}{\partial x^{2}}}+{\frac{\partial^{2}w}% {\partial y^{2}}}+{\frac{\partial^{2}w}{\partial z^{2}}}\right),\qquad(1)
  2. w t \displaystyle{{\partial w}\over{\partial t}}
  3. 10 - 2 10 5 + 10 10 - 2 10 6 + 10 10 - 2 10 6 + 10 - 2 10 - 2 10 4 - 10 2 + 10 2 10 6 {\frac{10^{-2}}{10^{5}}}+10{\frac{10^{-2}}{10^{6}}}+10{\frac{10^{-2}}{10^{6}}}% +10^{-2}{\frac{10^{-2}}{10^{4}}}-{\frac{10^{2}+10^{2}}{10^{6}}}
  4. = - 1 1 10 4 10 4 - 10 + 2 × 10 - 4 × 10 + 10 - 5 ( 10 - 2 10 12 + 10 - 2 10 12 + 10 - 2 10 8 ) . ( 2 ) =-{{\frac{1}{1}}{\frac{10^{4}}{10^{4}}}}-10+2\times 10^{-4}\times 10+10^{-5}% \left({\frac{10^{-2}}{10^{12}}}+{\frac{10^{-2}}{10^{12}}}+{\frac{10^{-2}}{10^{% 8}}}\right).\qquad(2)
  5. 1 ϱ p z = - g . ( 3 ) {{\frac{1}{\varrho}}{\frac{\partial p}{\partial z}}}=-g.\qquad(3)
  6. ρ c P T t = k 2 T x 2 . \rho c_{P}{{\partial T}\over{\partial t}}=k{\frac{\partial^{2}T}{\partial x^{2% }}}.
  7. c = a + b c=a+b
  8. O ( a ) > O ( b ) O(a)>O(b)
  9. O ( c ) = O ( a ) O(c)=O(a)
  10. c = a - b c=a-b
  11. c = a + b c=a+b
  12. O ( a ) = O ( b ) O(a)=O(b)
  13. O ( a ) O ( b ) O ( c ) O(a)\thicksim O(b)\thicksim O(c)
  14. p = a b p=ab
  15. O ( p ) = O ( a ) O ( b ) O(p)=O(a)O(b)
  16. r = a b r=\frac{a}{b}
  17. O ( r ) = O ( a ) O ( b ) O(r)=\frac{O(a)}{O(b)}
  18. u x + v y = 0 , ( 1 ) {{\partial u}\over{\partial x}}+{\frac{\partial v}{\partial y}}=0,\qquad(1)
  19. u u x + v u y = - 1 ϱ P x + ν ( 2 u x 2 + 2 u y 2 ) , ( 2 ) u{{\partial u}\over{\partial x}}+v{\frac{\partial u}{\partial y}}=-{{\frac{1}{% \varrho}}{\frac{\partial P}{\partial x}}}+\nu\left({\frac{\partial^{2}u}{% \partial x^{2}}}+{\frac{\partial^{2}u}{\partial y^{2}}}\right),\qquad(2)
  20. u v x + v v y = - 1 ϱ P y + ν ( 2 v x 2 + 2 v y 2 ) , ( 3 ) u{{\partial v}\over{\partial x}}+v{\frac{\partial v}{\partial y}}=-{{\frac{1}{% \varrho}}{\frac{\partial P}{\partial y}}}+\nu\left({\frac{\partial^{2}v}{% \partial x^{2}}}+{\frac{\partial^{2}v}{\partial y^{2}}}\right),\qquad(3)
  21. v U δ L ( 4 ) \begin{aligned}\displaystyle v&\displaystyle\sim{\frac{U_{\infty}\delta}{L}}% \qquad(4)\end{aligned}
  22. v = 0 , u x = 0 ( 5 ) v=0,{{\partial u}\over{\partial x}}=0\qquad(5)
  23. P y = 0 ( 6 ) {{\partial P}\over{\partial y}}=0\qquad(6)
  24. d P d x = μ d 2 u d y 2 = c o n s t a n t ( 7 ) {{dP}\over{dx}}=\mu{{d^{2}u}\over{dy^{2}}}=constant\qquad(7)
  25. u = 0 , y = ± D 2 ( 8 ) u=0,y=\pm\frac{D}{2}\qquad(8)
  26. u = 3 2 U [ 1 - ( y D / 2 ) 2 ] ( 9 ) u=\frac{3}{2}U[1-{(\frac{y}{D/2})}^{2}]\qquad(9)
  27. U = D 2 12 μ ( - d P d x ) ( 10 ) U=\frac{D^{2}}{12\mu}(-\frac{dP}{dx})\qquad(10)

Scale_length_(string_instruments).html

  1. 2 12 \scriptstyle\sqrt[12]{2}
  2. 2 12 2 12 - 1 \scriptstyle\frac{\sqrt[12]{2}}{\sqrt[12]{2}-1}

Schaefer's_dichotomy_theorem.html

  1. R ( x i 1 , , x i n ) R(x_{i_{1}},\ldots,x_{i_{n}})
  2. x i j x_{i_{j}}
  3. f : D m D f:D^{m}\to D
  4. R D k R\subseteq D^{k}
  5. ( t 11 , , t 1 k ) , , ( t m 1 , , t m k ) (t_{11},\ldots,t_{1k}),\ldots,(t_{m1},\ldots,t_{mk})
  6. ( f ( t 11 , , t m 1 ) , , f ( t 1 k , , t m k ) ) (f(t_{11},\ldots,t_{m1}),\ldots,f(t_{1k},\ldots,t_{mk}))
  7. Majority ( x , y , z ) = ( x y ) ( x z ) ( y z ) ; \operatorname{Majority}(x,y,z)=(x\wedge y)\vee(x\wedge z)\vee(y\wedge z);
  8. Minority ( x , y , z ) = x y z . \operatorname{Minority}(x,y,z)=x\oplus y\oplus z.
  9. m ( x , y , y ) = m ( y , y , x ) = x . m(x,y,y)=m(y,y,x)=x.
  10. x - y + z x-y+z
  11. x y - 1 z . xy^{-1}z.

Schauder_basis.html

  1. v = n = 0 α n b n , v=\sum_{n=0}^{\infty}\alpha_{n}b_{n},
  2. lim n v - k = 0 n α k b k V = 0. \lim_{n\to\infty}\left\|v-\sum_{k=0}^{n}\alpha_{k}b_{k}\right\|_{V}=0.
  3. c k = 0 N α k b k V k = 0 N α k c k W C k = 0 N α k b k V . c\left\|\sum_{k=0}^{N}\alpha_{k}b_{k}\right\|_{V}\leq\left\|\sum_{k=0}^{N}% \alpha_{k}c_{k}\right\|_{W}\leq C\left\|\sum_{k=0}^{N}\alpha_{k}b_{k}\right\|_% {V}.
  4. v = k = 0 α k b k P n P n ( v ) = k = 0 n α k b k v=\sum_{k=0}^{\infty}\alpha_{k}b_{k}\ \ \overset{\textstyle P_{n}}{% \longrightarrow}\ \ P_{n}(v)=\sum_{k=0}^{n}\alpha_{k}b_{k}
  5. | b n * ( v ) | b n V = | α n | b n V = α n b n V = P n ( v ) - P n - 1 ( v ) V 2 C v V . |b^{*}_{n}(v)|\;\|b_{n}\|_{V}=|\alpha_{n}|\;\|b_{n}\|_{V}=\|\alpha_{n}b_{n}\|_% {V}=\|P_{n}(v)-P_{n-1}(v)\|_{V}\leq 2C\|v\|_{V}.
  6. b n = { b n , j } j = 0 V , b n , j = δ n , j , b_{n}=\{b_{n,j}\}_{j=0}^{\infty}\in V,\ \ b_{n,j}=\delta_{n,j},
  7. { 1 , cos ( x ) , sin ( x ) , cos ( 2 x ) , sin ( 2 x ) , cos ( 3 x ) , sin ( 3 x ) , } \{1,\cos(x),\sin(x),\cos(2x),\sin(2x),\cos(3x),\sin(3x),\ldots\}
  8. { 1 , e i x , e - i x , e 2 i x , e - 2 i x , e 3 i x , e - 3 i x , } . \left\{1,e^{ix},e^{-ix},e^{2ix},e^{-2ix},e^{3ix},e^{-3ix},\ldots\right\}.
  9. { f : x k = - + c k e i k x } P N { P N f : x k = - N N c k e i k x } \left\{f:x\to\sum_{k=-\infty}^{+\infty}c_{k}e^{ikx}\right\}\ \overset{P_{N}}{% \longrightarrow}\ \left\{P_{N}f:x\to\sum_{k=-N}^{N}c_{k}e^{ikx}\right\}
  10. e 1 e 1 , e 1 e 2 , e 2 e 2 , e 2 e 1 , , e 1 e n , e 2 e n , , e n e n , e n e n - 1 , , e n e 1 , \begin{aligned}&\displaystyle e_{1}\otimes e_{1},\ \ e_{1}\otimes e_{2},\;e_{2% }\otimes e_{2},\;e_{2}\otimes e_{1},\ldots,\\ &\displaystyle e_{1}\otimes e_{n},e_{2}\otimes e_{n},\ldots,e_{n}\otimes e_{n}% ,e_{n}\otimes e_{n-1},\ldots,e_{n}\otimes e_{1},\ldots\end{aligned}
  11. X ^ ε X 𝒦 ( X ) . X^{\prime}\widehat{\otimes}_{\varepsilon}X\simeq\mathcal{K}(X).
  12. α n b n \sum\alpha_{n}b_{n}
  13. k = 0 n ε k α k b k V C k = 0 n α k b k V \Bigl\|\sum_{k=0}^{n}\varepsilon_{k}\alpha_{k}b_{k}\Bigr\|_{V}\leq C\Bigl\|% \sum_{k=0}^{n}\alpha_{k}b_{k}\Bigr\|_{V}
  14. k = 0 n ε k α k b π ( k ) V C k = 0 n α k b k V . \Bigl\|\sum_{k=0}^{n}\varepsilon_{k}\alpha_{k}b_{\pi(k)}\Bigr\|_{V}\leq C\Bigl% \|\sum_{k=0}^{n}\alpha_{k}b_{k}\Bigr\|_{V}.
  15. V n = k = 0 n a k e k V_{n}=\sum_{k=0}^{n}a_{k}e_{k}
  16. V n c 0 = max 0 k n | a k | = 1 \|V_{n}\|_{c_{0}}=\max_{0\leq k\leq n}|a_{k}|=1
  17. φ n = sup { | f ( x ) | : x F n , x 1 } \varphi_{n}=\sup\{|f(x)|:x\in F_{n},\;\|x\|\leq 1\}
  18. f : x = { x n } 1 n = 0 x n , f:x=\{x_{n}\}\in\ell^{1}\ \rightarrow\ \sum_{n=0}^{\infty}x_{n},
  19. v = b B α b b v=\sum_{b\in B}\alpha_{b}b
  20. { b B α b 0 } \{b\in B\mid\alpha_{b}\neq 0\}

Scheduling_(production_processes).html

  1. C T m i n = m a x j = 1 , M { τ j } CT_{min}=\begin{matrix}max\\ j=1,M\end{matrix}\{\tau_{j}\}
  2. C T m i n = m a x j = 1 , M { τ j / N j } CT_{min}=\begin{matrix}max\\ j=1,M\end{matrix}\{\tau_{j}/N_{j}\}

Schlenk_equilibrium.html

  1. \overrightarrow{\leftarrow}

Schlieren.html

  1. 1 f = 1 d o + 1 d i \frac{1}{f}=\frac{1}{d_{o}}+\frac{1}{d_{i}}
  2. f f
  3. d o d_{o}
  4. d i d_{i}

Schoof's_algorithm.html

  1. E E
  2. 𝔽 q \mathbb{F}_{q}
  3. q = p n q=p^{n}
  4. p p
  5. n n
  6. 1 \geq 1
  7. 2 , 3 \neq 2,3
  8. y 2 = x 3 + A x + B y^{2}=x^{3}+Ax+B\,
  9. A , B 𝔽 q A,B\in\mathbb{F}_{q}
  10. 𝔽 q \mathbb{F}_{q}
  11. ( a , b ) 𝔽 q 2 (a,b)\in\mathbb{F}_{q}^{2}
  12. O O
  13. E ( 𝔽 q ) E(\mathbb{F}_{q})
  14. O O
  15. E ( 𝔽 q ) E(\mathbb{F}_{q})
  16. E ( 𝔽 q ) \sharp E(\mathbb{F}_{q})
  17. E / 𝔽 q E/\mathbb{F}_{q}
  18. 𝔽 q \mathbb{F}_{q}
  19. E ( 𝔽 q ) \sharp E(\mathbb{F}_{q})
  20. q + 1 - E ( 𝔽 q ) 2 q . \mid q+1-\sharp E(\mathbb{F}_{q})\mid\leq 2\sqrt{q}.
  21. E ( 𝔽 q ) \sharp E(\mathbb{F}_{q})
  22. t t
  23. q + 1 - E ( 𝔽 q ) q+1-\sharp E(\mathbb{F}_{q})
  24. t t
  25. N N
  26. N > 4 q N>4\sqrt{q}
  27. t t
  28. E ( 𝔽 q ) \sharp E(\mathbb{F}_{q})
  29. t ( mod N ) t\;\;(\mathop{{\rm mod}}N)
  30. N N
  31. t ( mod l ) t\;\;(\mathop{{\rm mod}}l)
  32. l l
  33. S = { l 1 , l 2 , , l r } S=\{l_{1},l_{2},...,l_{r}\}
  34. l i = N > 4 q \prod l_{i}=N>4\sqrt{q}
  35. t ( mod l i ) t\;\;(\mathop{{\rm mod}}l_{i})
  36. l i S l_{i}\in S
  37. t ( mod N ) t\;\;(\mathop{{\rm mod}}N)
  38. t ( mod l ) t\;\;(\mathop{{\rm mod}}l)
  39. l p l\neq p
  40. ϕ \phi
  41. l p l\neq p
  42. q = p q=p
  43. p p
  44. E E
  45. 𝔽 q \mathbb{F}_{q}
  46. E E
  47. 𝔽 q ¯ \overline{\mathbb{F}_{q}}
  48. 𝔽 q \mathbb{F}_{q}
  49. 𝔽 ¯ q \bar{\mathbb{F}}_{q}
  50. 𝔽 ¯ q \bar{\mathbb{F}}_{q}
  51. 𝔽 q \mathbb{F}_{q}
  52. ϕ : ( x , y ) ( x q , y q ) \phi:(x,y)\mapsto(x^{q},y^{q})
  53. E ( 𝔽 q ) E(\mathbb{F}_{q})
  54. O O
  55. E ( 𝔽 q ¯ ) E(\bar{\mathbb{F}_{q}})
  56. E ( 𝔽 q ) E(\mathbb{F}_{q})
  57. ϕ \phi
  58. ϕ 2 - t ϕ + q = 0 , \phi^{2}-t\phi+q=0,
  59. t = q + 1 - E ( 𝔽 q ) t=q+1-\sharp E(\mathbb{F}_{q})
  60. P = ( x , y ) E P=(x,y)\in E
  61. ( x q 2 , y q 2 ) + q ( x , y ) = t ( x q , y q ) (x^{q^{2}},y^{q^{2}})+q(x,y)=t(x^{q},y^{q})
  62. q ( x , y ) q(x,y)
  63. t ( x q , y q ) t(x^{q},y^{q})
  64. ( x , y ) (x,y)
  65. q q
  66. ( x q , y q ) (x^{q},y^{q})
  67. t t
  68. ( x q 2 , y q 2 ) (x^{q^{2}},y^{q^{2}})
  69. ( x q , y q ) (x^{q},y^{q})
  70. q ( x , y ) q(x,y)
  71. 𝔽 q [ x , y ] / ( y 2 - x 3 - A x - B ) \mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B)
  72. E E
  73. t t
  74. l l
  75. l l
  76. l l
  77. t l t_{l}
  78. t ( mod l ) t\;\;(\mathop{{\rm mod}}l)
  79. l 2 , p l\neq 2,p
  80. ( x , y ) (x,y)
  81. l l
  82. E [ l ] = { P E ( 𝔽 q ¯ ) l P = O } E[l]=\{P\in E(\bar{\mathbb{F}_{q}})\mid lP=O\}
  83. q P = q ¯ P qP=\bar{q}P
  84. q ¯ \bar{q}
  85. q q ¯ ( mod l ) q\equiv\bar{q}\;\;(\mathop{{\rm mod}}l)
  86. q ¯ < l / 2 \mid\bar{q}\mid<l/2
  87. ϕ ( O ) = O \phi(O)=O
  88. r r
  89. r ϕ ( P ) = ϕ ( r P ) r\phi(P)=\phi(rP)
  90. ϕ ( P ) \phi(P)
  91. P P
  92. ( x , y ) (x,y)
  93. E [ l ] E[l]
  94. t ( x q , y q ) = t ¯ ( x q , y q ) t(x^{q},y^{q})=\bar{t}(x^{q},y^{q})
  95. t t ¯ ( mod l ) t\equiv\bar{t}\;\;(\mathop{{\rm mod}}l)
  96. ( x q 2 , y q 2 ) + q ¯ ( x , y ) t ¯ ( x q , y q ) , (x^{q^{2}},y^{q^{2}})+\bar{q}(x,y)\equiv\bar{t}(x^{q},y^{q}),
  97. t ¯ \bar{t}
  98. q ¯ \bar{q}
  99. [ - ( l - 1 ) / 2 , ( l - 1 ) / 2 ] [-(l-1)/2,(l-1)/2]
  100. l l
  101. x x
  102. l l
  103. ( x q 2 , y q 2 ) + q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})+\bar{q}(x,y)
  104. l l
  105. E E
  106. l l
  107. 𝔽 q [ x , y ] / ( y 2 - x 3 - A x - B , ψ l ) \mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B,\psi_{l})
  108. X X
  109. Y Y
  110. ( X ( x , y ) , Y ( x , y ) ) := ( x q 2 , y q 2 ) + q ¯ ( x , y ) (X(x,y),Y(x,y)):=(x^{q^{2}},y^{q^{2}})+\bar{q}(x,y)
  111. y y
  112. ( l 2 - 3 ) / 2 (l^{2}-3)/2
  113. x x
  114. q ¯ ( x , y ) \bar{q}(x,y)
  115. q ¯ \bar{q}
  116. q ¯ ( x , y ) = ( x q ¯ , y q ¯ ) = ( x - ψ q ¯ - 1 ψ q ¯ + 1 ψ q ¯ 2 , ψ 2 q ¯ 2 ψ q ¯ 4 ) \bar{q}(x,y)=(x_{\bar{q}},y_{\bar{q}})=\left(x-\frac{\psi_{\bar{q}-1}\psi_{% \bar{q}+1}}{\psi^{2}_{\bar{q}}},\frac{\psi_{2\bar{q}}}{2\psi^{4}_{\bar{q}}}\right)
  117. ψ n \psi_{n}
  118. n n
  119. y q ¯ / y y_{\bar{q}}/y
  120. x x
  121. θ ( x ) \theta(x)
  122. ( x q 2 , y q 2 ) ± q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})\neq\pm\bar{q}(x,y)
  123. ( x q 2 , y q 2 ) = ± q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})=\pm\bar{q}(x,y)
  124. ψ l \psi_{l}
  125. ( x q 2 , y q 2 ) ± q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})\neq\pm\bar{q}(x,y)
  126. E ( 𝔽 q ) E(\mathbb{F}_{q})
  127. X ( x , y ) = ( y q 2 - y q ¯ x q 2 - x q ¯ ) 2 - x q 2 - x q ¯ . X(x,y)=\left(\frac{y^{q^{2}}-y_{\bar{q}}}{x^{q^{2}}-x_{\bar{q}}}\right)^{2}-x^% {q^{2}}-x_{\bar{q}}.
  128. x x
  129. t ¯ \bar{t}
  130. y y
  131. X X
  132. x x
  133. ( y q 2 - y q ¯ ) 2 = y 2 ( y q 2 - 1 - y q ¯ / y ) 2 (y^{q^{2}}-y_{\bar{q}})^{2}=y^{2}(y^{q^{2}-1}-y_{\bar{q}}/y)^{2}
  134. q 2 - 1 q^{2}-1
  135. y 2 y^{2}
  136. x 3 + A x + B x^{3}+Ax+B
  137. ( x 3 + A x + B ) ( ( x 3 + A x + B ) q 2 - 1 2 - θ ( x ) ) (x^{3}+Ax+B)((x^{3}+Ax+B)^{\frac{q^{2}-1}{2}}-\theta(x))
  138. X ( x ) ( x 3 + A x + B ) ( ( x 3 + A x + B ) q 2 - 1 2 - θ ( x ) ) mod ψ l ( x ) . X(x)\equiv(x^{3}+Ax+B)((x^{3}+Ax+B)^{\frac{q^{2}-1}{2}}-\theta(x))\bmod\psi_{l% }(x).
  139. X x t ¯ q mod ψ l ( x ) X\equiv x^{q}_{\bar{t}}\bmod\psi_{l}(x)
  140. t ¯ [ 0 , ( l - 1 ) / 2 ] \bar{t}\in[0,(l-1)/2]
  141. t ¯ \bar{t}
  142. ϕ 2 ( P ) t ¯ ϕ ( P ) + q ¯ P = O \phi^{2}(P)\mp\bar{t}\phi(P)+\bar{q}P=O
  143. l l
  144. P P
  145. Y Y
  146. y t ¯ q y_{\bar{t}}^{q}
  147. t ¯ \bar{t}
  148. t ¯ \bar{t}
  149. - t ¯ -\bar{t}
  150. t t ¯ ( mod l ) t\equiv\bar{t}\;\;(\mathop{{\rm mod}}l)
  151. t ¯ ( mod l ) \bar{t}\;\;(\mathop{{\rm mod}}l)
  152. t l t_{l}
  153. l l
  154. ( x q 2 , y q 2 ) = ± q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})=\pm\bar{q}(x,y)
  155. ( x q 2 , y q 2 ) = q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})=\bar{q}(x,y)
  156. l l
  157. q ¯ ( x , y ) = - q ¯ ( x , y ) \bar{q}(x,y)=-\bar{q}(x,y)
  158. t ¯ 0 \bar{t}\neq 0
  159. t ¯ ϕ ( P ) = 2 q ¯ P \bar{t}\phi(P)=2\bar{q}P
  160. t ¯ 2 q ¯ ( 2 q ) 2 ( mod l ) \bar{t}^{2}\bar{q}\equiv(2q)^{2}\;\;(\mathop{{\rm mod}}l)
  161. q q
  162. l l
  163. q w 2 ( mod l ) q\equiv w^{2}\;\;(\mathop{{\rm mod}}l)
  164. w ϕ ( x , y ) w\phi(x,y)
  165. 𝔽 q [ x , y ] / ( y 2 - x 3 - A x - B , ψ l ) \mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B,\psi_{l})
  166. q ¯ ( x , y ) = w ϕ ( x , y ) \bar{q}(x,y)=w\phi(x,y)
  167. t l t_{l}
  168. ± 2 w ( mod l ) \pm 2w\;\;(\mathop{{\rm mod}}l)
  169. q q
  170. l l
  171. w w
  172. - w -w
  173. ( x q 2 , y q 2 ) = + q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})=+\bar{q}(x,y)
  174. ( x q 2 , y q 2 ) = - q ¯ ( x , y ) (x^{q^{2}},y^{q^{2}})=-\bar{q}(x,y)
  175. t l = 0 t_{l}=0
  176. l = 2 l=2
  177. l = 2 l=2
  178. q q
  179. q + 1 - t t ( mod 2 ) q+1-t\equiv t\;\;(\mathop{{\rm mod}}2)
  180. t 2 0 ( mod 2 ) t_{2}\equiv 0\;\;(\mathop{{\rm mod}}2)
  181. E ( 𝔽 q ) E(\mathbb{F}_{q})
  182. ( x 0 , 0 ) (x_{0},0)
  183. t 2 0 ( mod 2 ) t_{2}\equiv 0\;\;(\mathop{{\rm mod}}2)
  184. x 3 + A x + B x^{3}+Ax+B
  185. 𝔽 q \mathbb{F}_{q}
  186. gcd ( x q - x , x 3 + A x + B ) 1 \gcd(x^{q}-x,x^{3}+Ax+B)\neq 1
  187. S S
  188. p p
  189. N = l S l > 4 q . N=\prod_{l\in S}l>4\sqrt{q}.
  190. t 2 = 0 t_{2}=0
  191. gcd ( x q - x , x 3 + A x + B ) 1 \gcd(x^{q}-x,x^{3}+Ax+B)\neq 1
  192. t 2 = 1 t_{2}=1
  193. ψ l \psi_{l}
  194. 𝔽 q [ x , y ] / ( y 2 - x 3 - A x - B , ψ l ) . \mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B,\psi_{l}).
  195. l S l\in S
  196. q ¯ \bar{q}
  197. q q ¯ ( mod l ) q\equiv\bar{q}\;\;(\mathop{{\rm mod}}l)
  198. q ¯ < l / 2 \mid\bar{q}\mid<l/2
  199. ( x q , y q ) (x^{q},y^{q})
  200. ( x q 2 , y q 2 ) (x^{q^{2}},y^{q^{2}})
  201. ( x q ¯ , y q ¯ ) (x_{\bar{q}},y_{\bar{q}})
  202. x q 2 x q ¯ x^{q^{2}}\neq x_{\bar{q}}
  203. ( X , Y ) (X,Y)
  204. 1 t ¯ ( l - 1 ) / 2 1\leq\bar{t}\leq(l-1)/2
  205. X = x t ¯ q X=x^{q}_{\bar{t}}
  206. Y = y t ¯ q Y=y^{q}_{\bar{t}}
  207. t l = t ¯ t_{l}=\bar{t}
  208. t l = - t ¯ t_{l}=-\bar{t}
  209. q q
  210. l l
  211. w w
  212. q w 2 ( mod l ) q\equiv w^{2}\;\;(\mathop{{\rm mod}}l)
  213. w ( x q , y q ) w(x^{q},y^{q})
  214. w ( x q , y q ) = ( x q 2 , y q 2 ) w(x^{q},y^{q})=(x^{q^{2}},y^{q^{2}})
  215. t l = 2 w t_{l}=2w
  216. w ( x q , y q ) = ( x q 2 , - y q 2 ) w(x^{q},y^{q})=(x^{q^{2}},-y^{q^{2}})
  217. t l = - 2 w t_{l}=-2w
  218. t l = 0 t_{l}=0
  219. t l = 0 t_{l}=0
  220. t t
  221. N N
  222. S S
  223. N > 4 q N>4\sqrt{q}
  224. t t
  225. E ( 𝔽 q ) = q + 1 - t \sharp E(\mathbb{F}_{q})=q+1-t
  226. ϕ ( P ) \phi(P)
  227. ϕ 2 ( P ) \phi^{2}(P)
  228. l l
  229. x q x^{q}
  230. y q y^{q}
  231. x q 2 x^{q^{2}}
  232. y q 2 y^{q^{2}}
  233. l l
  234. R = 𝔽 q [ x , y ] / ( y 2 - x 3 - A x - B , ψ l ) R=\mathbb{F}_{q}[x,y]/(y^{2}-x^{3}-Ax-B,\psi_{l})
  235. O ( log q ) O(\log q)
  236. ψ l \psi_{l}
  237. l 2 - 1 2 \frac{l^{2}-1}{2}
  238. O ( l 2 ) O(l^{2})
  239. O ( log q ) O(\log q)
  240. O ( log q ) O(\log q)
  241. l l
  242. O ( log q ) O(\log q)
  243. O ( l 2 ) = O ( log 2 q ) O(l^{2})=O(\log^{2}q)
  244. R R
  245. O ( log 4 q ) O(\log^{4}q)
  246. 𝔽 q \mathbb{F}_{q}
  247. O ( log 2 q ) O(\log^{2}q)
  248. l l
  249. O ( log 7 q ) O(\log^{7}q)
  250. O ( log q ) O(\log q)
  251. O ( log 8 q ) O(\log^{8}q)
  252. O ~ ( log 5 q ) \tilde{O}(\log^{5}q)
  253. S = { l 1 , , l s } S=\{l_{1},\ldots,l_{s}\}
  254. l l
  255. ϕ 2 - t ϕ + q = 0 \phi^{2}-t\phi+q=0
  256. 𝔽 l \mathbb{F}_{l}
  257. O ( l ) O(l)
  258. O ( l 2 ) O(l^{2})
  259. O ( log q ) O(\log q)
  260. O ( log 6 q ) O(\log^{6}q)
  261. O ~ ( log 4 q ) \tilde{O}(\log^{4}q)
  262. E ( 𝔽 p ) E(\mathbb{F}_{p})
  263. p p
  264. E ( 𝔽 2 m ) E(\mathbb{F}_{2^{m}})
  265. E ( 𝔽 q ) E(\mathbb{F}_{q})

Schoof–Elkies–Atkin_algorithm.html

  1. S = { l 1 , , l s } S=\{l_{1},\ldots,l_{s}\}
  2. l l
  3. ϕ 2 - t ϕ + q = 0 \phi^{2}-t\phi+q=0
  4. 𝔽 l \mathbb{F}_{l}
  5. Φ l ( X , Y ) \Phi_{l}(X,Y)
  6. l l
  7. Φ l ( X , j ( E ) ) \Phi_{l}(X,j(E))
  8. j ( E ) j(E^{\prime})
  9. 𝔽 q \mathbb{F}_{q}
  10. l l
  11. f l ( X ) f_{l}(X)
  12. l l
  13. E E
  14. E E^{\prime}
  15. f l f_{l}
  16. O ( l ) O(l)
  17. O ( l 2 ) O(l^{2})
  18. E E
  19. l l
  20. Φ l ( X , j ( E ) ) \Phi_{l}(X,j(E))
  21. 𝔽 l [ X ] \mathbb{F}_{l}[X]
  22. l l
  23. l l
  24. O ~ ( log 4 q ) \tilde{O}(\log^{4}q)

Schottky_defect.html

  1. V T i ′′′′ + 2 V O \Leftrightarrow V_{Ti}^{\prime\prime\prime\prime}+2V_{O}^{\bullet\bullet}
  2. V B a ′′ + V T i ′′′′ + 3 V O \Leftrightarrow V_{Ba}^{\prime\prime}+V_{Ti}^{\prime\prime\prime\prime}+3V_{O}% ^{\bullet\bullet}

Schreier_vector.html

  1. X = { x 1 , x 2 , , x r } X=\{x_{1},x_{2},...,x_{r}\}
  2. Ω = { 1 , 2 , , n } \Omega=\{1,2,...,n\}
  3. ω Ω \omega\in\Omega
  4. ω \omega
  5. g G g\in G
  6. ω g = α \omega^{g}=\alpha
  7. α ω G \alpha\in\omega^{G}
  8. ω Ω \omega\in\Omega
  9. 𝐯 = ( v [ 1 ] , v [ 2 ] , , v [ n ] ) \mathbf{v}=(v[1],v[2],...,v[n])
  10. v [ ω ] = - 1 v[\omega]=-1
  11. α ω G { ω } , v [ α ] { 1 , , r } \alpha\in\omega^{G}\setminus\{{\omega}\},v[\alpha]\in\{1,...,r\}
  12. v [ α ] v[\alpha]
  13. v [ α ] = 0 v[\alpha]=0
  14. α ω G \alpha\notin\omega^{G}
  15. X = { x 1 , x 2 , , x r } X=\{x_{1},x_{2},...,x_{r}\}
  16. α x i \alpha^{x_{i}}
  17. α x i \alpha^{x_{i}}
  18. v [ α x i ] = i v[\alpha^{x_{i}}]=i
  19. g = x k g , α = α x k - 1 , k = v [ α ] g={x_{k}}g,\alpha=\alpha^{x_{k}^{-1}},k=v[\alpha]

Schur-convex_function.html

  1. f : d f:\mathbb{R}^{d}\rightarrow\mathbb{R}
  2. x , y d x,y\in\mathbb{R}^{d}
  3. x x
  4. y y
  5. f ( x ) f ( y ) f(x)\leq f(y)
  6. ( x i - x j ) ( f x i - f x j ) 0 (x_{i}-x_{j})(\frac{\partial f}{\partial x_{i}}-\frac{\partial f}{\partial x_{% j}})\geq 0
  7. x d x\in\mathbb{R}^{d}
  8. f ( x ) = min ( x ) f(x)=\min(x)
  9. f ( x ) = max ( x ) f(x)=\max(x)
  10. i = 1 d P i log 2 1 P i \sum_{i=1}^{d}{P_{i}\cdot\log_{2}{\frac{1}{P_{i}}}}
  11. i = 1 d x i k , k 1 \sum_{i=1}^{d}{x_{i}^{k}},k\geq 1
  12. f ( x ) = i = 1 n x i f(x)=\prod_{i=1}^{n}x_{i}
  13. x i > 0 x_{i}>0
  14. x i > 0 x_{i}>0
  15. x y x\succ y
  16. x x
  17. y y
  18. g g
  19. i = 1 n g ( x i ) \sum_{i=1}^{n}g(x_{i})
  20. X 1 , , X n X_{1},\dots,X_{n}
  21. E j = 1 n X j a j \,\text{E}\prod_{j=1}^{n}X_{j}^{a_{j}}
  22. a = ( a 1 , , a n ) a=(a_{1},\dots,a_{n})

Schwartz_space.html

  1. S ( 𝐑 n ) = { f C ( 𝐑 n ) : f α , β < α , β } , S\left(\mathbf{R}^{n}\right)=\left\{f\in C^{\infty}(\mathbf{R}^{n}):\|f\|_{% \alpha,\beta}<\infty\quad\forall\alpha,\beta\right\},
  2. f α , β = sup x 𝐑 n | x α D β f ( x ) | . \|f\|_{\alpha,\beta}=\sup_{x\in\mathbf{R}^{n}}\left|x^{\alpha}D^{\beta}f(x)% \right|.
  3. x i e - a | x | 2 S ( 𝐑 n ) . x^{i}e^{-a|x|^{2}}\in S(\mathbf{R}^{n}).

Schwarz_reflection_principle.html

  1. F ( z ¯ ) ¯ \overline{F(\bar{z})}
  2. F ( z ¯ ) = F ( z ) ¯ . F(\bar{z})=\overline{F(z)}.
  3. { z | Im ( z ) 0 } \left\{z\in\mathbb{C}\ |\ \mathrm{Im}(z)\geq 0\right\}
  4. { F ( z ) | Im ( z ) > 0 } \left\{F(z)\in\mathbb{C}\ |\ \mathrm{Im}(z)>0\right\}

Scipione_del_Ferro.html

  1. p p
  2. q q
  3. x x
  4. x 3 + p x = q x^{3}+px=q\,
  5. x 3 = p x + q x^{3}=px+q\,
  6. x 2 x^{2}
  7. x = x + a x=x^{\prime}+a
  8. a a
  9. x = a + b + a - b x=\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}
  10. x 2 = ( 2 a 2 - b ) x 0 + 2 a x^{2}=(2\sqrt{a^{2}-b})x^{0}+2a
  11. x = a + b 3 + a - b 3 x=\sqrt[3]{a+\sqrt{b}}+\sqrt[3]{a-\sqrt{b}}
  12. x 3 = ( 3 a 2 - b 3 ) x + 2 a x^{3}=(3\sqrt[3]{a^{2}-b})x+2a
  13. q 2 + q 2 4 + p 3 27 3 + q 2 - q 2 4 + p 3 27 3 \sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{\frac{q% }{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}

Scoring_rule.html

  1. S ( 𝐫 , i ) S(\mathbf{r},i)
  2. L ( 𝐫 , i ) = ln ( r i ) L(\mathbf{r},i)=\ln(r_{i})
  3. L ( 𝐫 , i ) = log b ( r i ) L(\mathbf{r},i)=\log_{b}(r_{i})
  4. b > 0 b>0
  5. Q ( 𝐫 , i ) = 2 r i - 𝐫 𝐫 = 2 r i - j = 1 C r j 2 Q(\mathbf{r},i)=2r_{i}-\mathbf{r}\cdot\mathbf{r}=2r_{i}-\sum_{j=1}^{C}r_{j}^{2}
  6. r i r_{i}
  7. B ( 𝐫 , i ) = j = 1 C ( y j - r j ) 2 B(\mathbf{r},i)=\sum_{j=1}^{C}(y_{j}-r_{j})^{2}
  8. y j = 1 y_{j}=1
  9. y j = 0 y_{j}=0
  10. S ( 𝐫 , i ) = r i 𝐫 = r i r 1 2 + + r c 2 S(\mathbf{r},i)=\frac{r_{i}}{\lVert\mathbf{r}\rVert}=\frac{r_{i}}{\sqrt{r_{1}^% {2}+\cdots+r_{c}^{2}}}
  11. S ( 𝐫 , i ) S(\mathbf{r},i)
  12. a + b S ( 𝐫 , i ) a+bS(\mathbf{r},i)
  13. b > 0 b>0
  14. r i r_{i}
  15. 1 - r i 1-r_{i}
  16. S S
  17. E ( S ) = U N C + R E L - R E S . E(S)=UNC+REL-RES.
  18. U N C = x ¯ ( 1 - x ¯ ) UNC=\bar{x}(1-\bar{x})
  19. R E L = E ( p - π ( p ) ) 2 REL=E(p-\pi(p))^{2}
  20. R E S = E ( π ( p ) - x ¯ ) 2 RES=E(\pi(p)-\bar{x})^{2}
  21. x ¯ \bar{x}
  22. x x
  23. π ( p ) \pi(p)
  24. p p
  25. π ( p ) = P ( x = 1 p ) \pi(p)=P(x=1\mid p)

Screw_axis.html

  1. D ( 𝐱 ) = A ( 𝐱 ) + 𝐝 . D(\mathbf{x})=A(\mathbf{x})+\mathbf{d}.
  2. 𝐝 = 𝐝 L + 𝐝 , 𝐝 L = ( 𝐝 𝐒 ) 𝐒 , 𝐝 = 𝐝 - 𝐝 L . \mathbf{d}=\mathbf{d}_{L}+\mathbf{d}_{\perp},\quad\mathbf{d}_{L}=(\mathbf{d}% \cdot\mathbf{S})\mathbf{S},\quad\mathbf{d}_{\perp}=\mathbf{d}-\mathbf{d}_{L}.
  3. D ( 𝐱 ) = ( A ( 𝐱 ) + 𝐝 ) + 𝐝 L . D(\mathbf{x})=(A(\mathbf{x})+\mathbf{d}_{\perp})+\mathbf{d}_{L}.
  4. D * ( 𝐂 ) = A ( 𝐂 ) + 𝐝 = 𝐂 . D^{*}(\mathbf{C})=A(\mathbf{C})+\mathbf{d}_{\perp}=\mathbf{C}.
  5. 𝐂 = [ I - A ] - 1 𝐝 , \mathbf{C}=[I-A]^{-1}\mathbf{d}_{\perp},
  6. D ( 𝐱 ) = D * ( 𝐱 ) + 𝐝 L , D(\mathbf{x})=D^{*}(\mathbf{x})+\mathbf{d}_{L},
  7. D * ( 𝐂 ) = A ( 𝐂 ) + 𝐝 = 𝐂 . D^{*}(\mathbf{C})=A(\mathbf{C})+\mathbf{d}_{\perp}=\mathbf{C}.
  8. [ A ] = [ I - B ] - 1 [ I + B ] , [A]=[I-B]^{-1}[I+B],
  9. 𝐛 = tan ϕ 2 𝐒 , \mathbf{b}=\tan\frac{\phi}{2}\mathbf{S},
  10. [ B ] 𝐲 = 𝐛 × 𝐲 . [B]\mathbf{y}=\mathbf{b}\times\mathbf{y}.
  11. 𝐂 = [ I - B ] - 1 [ I + B ] 𝐂 + 𝐝 , [ I - B ] 𝐂 = [ I + B ] 𝐂 + [ I - B ] 𝐝 , \mathbf{C}=[I-B]^{-1}[I+B]\mathbf{C}+\mathbf{d}_{\perp},\quad[I-B]\mathbf{C}=[% I+B]\mathbf{C}+[I-B]\mathbf{d}_{\perp},
  12. - 2 [ B ] 𝐂 = [ I - B ] 𝐝 . -2[B]\mathbf{C}=[I-B]\mathbf{d}_{\perp}.
  13. 𝐂 = 𝐛 × 𝐝 - 𝐛 × ( 𝐛 × 𝐝 ) 2 𝐛 𝐛 . \mathbf{C}=\frac{\mathbf{b}\times\mathbf{d}-\mathbf{b}\times(\mathbf{b}\times% \mathbf{d})}{2\mathbf{b}\cdot\mathbf{b}}.
  14. S ^ = cos ϕ ^ 2 + sin ϕ ^ 2 𝖲 . \hat{S}=\cos\frac{\hat{\phi}}{2}+\sin\frac{\hat{\phi}}{2}\mathsf{S}.
  15. 𝐪 S 𝐪 S - 1 + 𝐝 \mathbf{q}\mapsto S\mathbf{q}S^{-1}+\mathbf{d}
  16. S = cos θ + 𝐒 sin θ , 𝐒 2 = - 1 , S=\cos\theta+\mathbf{S}\sin\theta,\ \ \mathbf{S}^{2}=-1,

Second-order_arithmetic.html

  1. n ( n X S n X ) \forall n(n\in X\rightarrow Sn\in X)
  2. X n ( n X n < S S S S S S 0 S S S S S S S 0 ) \exists X\forall n(n\in X\leftrightarrow n<SSSSSS0\cdot SSSSSSS0)
  3. 0 \ 0
  4. S , \ S,
  5. \cdot
  6. m n [ S m = S n m = n ] . \forall m\forall n[Sm=Sn\rightarrow m=n].
  7. n [ 0 = n m [ S m = n ] ] . \forall n[0=n\lor\exists m[Sm=n]].
  8. m [ m + 0 = m ] . \forall m[m+0=m].
  9. m n [ m + S n = S ( m + n ) ] . \forall m\forall n[m+Sn=S(m+n)].
  10. m [ m 0 = 0 ] . \forall m[m\cdot 0=0].
  11. m n [ m S n = ( m n ) + m ] . \forall m\forall n[m\cdot Sn=(m\cdot n)+m].
  12. m [ m < S n ( m < n m = n ) ] . \forall m[m<Sn\leftrightarrow(m<n\lor m=n)].
  13. n [ 0 = n 0 < n ] . \forall n[0=n\lor 0<n].
  14. m n [ ( S m < n S m = n ) m < n ] . \forall m\forall n[(Sm<n\lor Sm=n)\leftrightarrow m<n].
  15. m X ( ( φ ( 0 ) n ( φ ( n ) φ ( S n ) ) n φ ( n ) ) \forall m_{\bullet}\forall X_{\bullet}((\varphi(0)\land\forall n(\varphi(n)% \rightarrow\varphi(Sn))\rightarrow\forall n\varphi(n))
  16. n X n\in X
  17. X ( ( 0 X n ( n X S n X ) ) n ( n X ) ) \forall X((0\in X\land\forall n(n\in X\rightarrow Sn\in X))\rightarrow\forall n% (n\in X))
  18. m X Z n ( n Z φ ( n ) ) \forall m_{\bullet}\forall X_{\bullet}\exists Z\forall n(n\in Z\leftrightarrow% \varphi(n))
  19. Z = { n | φ ( n ) } Z=\{n|\varphi(n)\}
  20. n Z n\not\in Z
  21. Z n ( n Z n Z ) \exists Z\forall n(n\in Z\leftrightarrow n\not\in Z)
  22. \mathcal{M}
  23. \mathcal{M}
  24. ω \omega
  25. ACA 0 \mathrm{ACA}_{0}
  26. ACA 0 \mathrm{ACA}_{0}
  27. ACA 0 \mathrm{ACA}_{0}
  28. ACA 0 \mathrm{ACA}_{0}
  29. ACA 0 \mathrm{ACA}_{0}
  30. ACA \mathrm{ACA}
  31. ACA 0 \mathrm{ACA}_{0}
  32. n ( n < t ) \forall n(n<t\rightarrow\cdots)
  33. n < t ( ) \exists n<t(\cdots)
  34. n ( n < t ) \exists n(n<t\land\cdots)
  35. RCA 0 \mathrm{RCA}_{0}
  36. ACA 0 \mathrm{ACA}_{0}
  37. m X ( ( n ( φ ( n ) ψ ( n ) ) ) Z n ( n Z φ ( n ) ) ) \forall m\forall X((\forall n(\varphi(n)\leftrightarrow\psi(n)))\rightarrow% \exists Z\forall n(n\in Z\leftrightarrow\varphi(n)))
  38. RCA 0 \mathrm{RCA}_{0}
  39. Π 2 0 \Pi^{0}_{2}
  40. RCA 0 \mathrm{RCA}_{0}
  41. RCA 0 \mathrm{RCA}_{0}
  42. RCA 0 \mathrm{RCA}_{0}
  43. RCA 0 \mathrm{RCA}_{0}
  44. RCA 0 \mathrm{RCA}_{0}
  45. Σ 2 1 \Sigma^{1}_{2}
  46. Π 3 1 \Pi^{1}_{3}

Second_derivative.html

  1. f f
  2. f f
  3. 𝐚 = d 𝐯 d t = d 2 s y m b o l x d t 2 \mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^{2}symbol{x}}{dt^{2}}
  4. d 2 d x 2 [ x n ] = n ( n - 1 ) x ( n - 2 ) = ( n 2 - n ) x ( n - 2 ) . \frac{d^{2}}{dx^{2}}[x^{n}]=n(n-1)x^{(n-2)}=(n^{2}-n)x^{(n-2)}.
  5. f ( x ) f(x)\!
  6. f ′′ ( x ) f^{\prime\prime}(x)\!
  7. f ′′ = ( f ) f^{\prime\prime}=(f^{\prime})^{\prime}\!
  8. d 2 y d x 2 . \frac{d^{2}y}{dx^{2}}.
  9. d 2 y d x 2 = d d x ( d y d x ) . \frac{d^{2}y}{dx^{2}}\,=\,\frac{d}{dx}\left(\frac{dy}{dx}\right).
  10. f ( x ) = x 3 , f(x)=x^{3},\!
  11. f ( x ) = 3 x 2 . f^{\prime}(x)=3x^{2}.\!
  12. f ′′ ( x ) = 6 x . f^{\prime\prime}(x)=6x.\!
  13. f ( x ) = sin ( 2 x ) f(x)=\sin(2x)
  14. - π / 4 -\pi/4
  15. 5 π / 4 5\pi/4
  16. π \pi
  17. π \pi
  18. f ( x ) = 0 f^{\prime}(x)=0\!
  19. f ′′ ( x ) < 0 \ f^{\prime\prime}(x)<0
  20. f \ f
  21. x \ x
  22. f ′′ ( x ) > 0 \ f^{\prime\prime}(x)>0
  23. f \ f
  24. x \ x
  25. f ′′ ( x ) = 0 \ f^{\prime\prime}(x)=0
  26. x \ x
  27. f ′′ ( x ) = lim h 0 f ( x + h ) - 2 f ( x ) + f ( x - h ) h 2 . f^{\prime\prime}(x)=\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^{2}}.
  28. f ( x + h ) - 2 f ( x ) + f ( x - h ) h 2 = f ( x + h ) - f ( x ) h - f ( x ) - f ( x - h ) h h . \frac{f(x+h)-2f(x)+f(x-h)}{h^{2}}=\frac{\frac{f(x+h)-f(x)}{h}-\frac{f(x)-f(x-h% )}{h}}{h}.
  29. f f
  30. sgn ( x ) \operatorname{sgn}(x)
  31. sgn ( x ) = { - 1 if x < 0 , 0 if x = 0 , 1 if x > 0. \operatorname{sgn}(x)=\begin{cases}-1&\,\text{if }x<0,\\ 0&\,\text{if }x=0,\\ 1&\,\text{if }x>0.\end{cases}
  32. x = 0 x=0
  33. x = 0 x=0
  34. lim h 0 sgn ( 0 + h ) - 2 sgn ( 0 ) + sgn ( 0 - h ) h 2 \displaystyle\lim_{h\to 0}\frac{\operatorname{sgn}(0+h)-2\operatorname{sgn}(0)% +\operatorname{sgn}(0-h)}{h^{2}}
  35. f ( x ) f ( a ) + f ( a ) ( x - a ) + 1 2 f ′′ ( a ) ( x - a ) 2 . f(x)\approx f(a)+f^{\prime}(a)(x-a)+\tfrac{1}{2}f^{\prime\prime}(a)(x-a)^{2}.
  36. x [ 0 , L ] x\in[0,L]
  37. v ( 0 ) = v ( L ) = 0 v(0)=v(L)=0
  38. λ j = - j 2 π 2 L 2 \lambda_{j}=-\frac{j^{2}\pi^{2}}{L^{2}}
  39. v j ( x ) = 2 L sin ( j π x L ) v_{j}(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{j\pi x}{L}\right)
  40. v j ′′ ( x ) = λ j v j ( x ) , j = 1 , , . v^{\prime\prime}_{j}(x)=\lambda_{j}v_{j}(x),\,j=1,\ldots,\infty.
  41. 2 f x 2 , 2 f y 2 , and 2 f z 2 \frac{\partial^{2}f}{\partial x^{2}},\;\frac{\partial^{2}f}{\partial y^{2}},\,% \text{ and }\frac{\partial^{2}f}{\partial z^{2}}
  42. 2 f x y , 2 f x z , and 2 f y z . \frac{\partial^{2}f}{\partial x\,\partial y},\;\frac{\partial^{2}f}{\partial x% \,\partial z},\,\text{ and }\frac{\partial^{2}f}{\partial y\,\partial z}.
  43. 2 \nabla^{2}
  44. 2 f = 2 f x 2 + 2 f y 2 + 2 f z 2 . \nabla^{2}f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y% ^{2}}+\frac{\partial^{2}f}{\partial z^{2}}.

Second_partial_derivative_test.html

  1. H ( x , y ) = ( f x x ( x , y ) f x y ( x , y ) f y x ( x , y ) f y y ( x , y ) ) H(x,y)=\begin{pmatrix}f_{xx}(x,y)&f_{xy}(x,y)\\ f_{yx}(x,y)&f_{yy}(x,y)\end{pmatrix}
  2. D ( x , y ) = det ( H ( x , y ) ) = f x x ( x , y ) f y y ( x , y ) - ( f x y ( x , y ) ) 2 D(x,y)=\det(H(x,y))=f_{xx}(x,y)f_{yy}(x,y)-\left(f_{xy}(x,y)\right)^{2}
  3. D ( a , b ) > 0 D(a,b)>0
  4. f x x ( a , b ) > 0 f_{xx}(a,b)>0
  5. ( a , b ) (a,b)
  6. D ( a , b ) > 0 D(a,b)>0
  7. f x x ( a , b ) < 0 f_{xx}(a,b)<0
  8. ( a , b ) (a,b)
  9. D ( a , b ) < 0 D(a,b)<0
  10. ( a , b ) (a,b)
  11. D ( a , b ) = 0 D(a,b)=0
  12. D ( a , b ) D(a,b)
  13. f x x ( a , b ) f_{xx}(a,b)
  14. D < 0 D<0
  15. f x x f y y < f x y 2 f_{xx}f_{yy}<f_{xy}^{2}
  16. f x x f_{xx}
  17. f y y f_{yy}
  18. D > 0 D>0
  19. f x x f y y > f x y 2 f_{xx}f_{yy}>f_{xy}^{2}
  20. f x x f_{xx}
  21. f y y f_{yy}
  22. f x x f_{xx}
  23. f y y f_{yy}
  24. f x y f_{xy}
  25. f ( x , y ) = ( x + y ) ( x y + x y 2 ) f(x,y)=(x+y)(xy+xy^{2})
  26. z = f ( x , y ) = ( x + y ) ( x y + x y 2 ) z=f(x,y)=(x+y)(xy+xy^{2})
  27. z x = y ( 2 x + y ) ( y + 1 ) \frac{\partial z}{\partial x}=y(2x+y)(y+1)
  28. z y = x ( 3 y 2 + 2 y ( x + 1 ) + x ) \frac{\partial z}{\partial y}=x\left(3y^{2}+2y(x+1)+x\right)
  29. ( 0 , 0 ) , ( 0 , - 1 ) , ( 1 , - 1 ) (0,0),(0,-1),(1,-1)
  30. ( 3 8 , - 3 4 ) \left(\frac{3}{8},-\frac{3}{4}\right)
  31. D ( a , b ) = f x x ( a , b ) f y y ( a , b ) - ( f x y ( a , b ) ) 2 = 2 b ( b + 1 ) 2 a ( a + 3 b + 1 ) - ( 2 a + 2 b + 4 a b + 3 b 2 ) 2 . \begin{aligned}\displaystyle D(a,b)&\displaystyle=f_{xx}(a,b)f_{yy}(a,b)-\left% (f_{xy}(a,b)\right)^{2}\\ &\displaystyle=2b(b+1)\cdot 2a(a+3b+1)-(2a+2b+4ab+3b^{2})^{2}.\end{aligned}
  32. D ( 0 , 0 ) = 0 ; D ( 0 , - 1 ) = - 1 ; D ( 1 , - 1 ) = - 1 ; D ( 3 8 , - 3 4 ) = 27 128 . D(0,0)=0;~{}~{}D(0,-1)=-1;~{}~{}D(1,-1)=-1;~{}~{}D\left(\frac{3}{8},-\frac{3}{% 4}\right)=\frac{27}{128}.
  33. ( 3 8 , - 3 4 ) \left(\frac{3}{8},-\frac{3}{4}\right)
  34. f x x = - 3 8 < 0 f_{xx}=-\frac{3}{8}<0

Sector_(instrument).html

  1. f ( n ) = L n / 250 f(n)=Ln/250
  2. f ( n ) = L ( n / 50 ) 1 / 2 f(n)=L(n/50)^{1/2}
  3. f ( n ) = L ( n / 148 ) 1 / 3 f(n)=L(n/148)^{1/3}
  4. f ( n ) = L ( 31 / 2 tan ( 180 / n ) / n ) 1 / 2 f(n)=L(31/2\tan(180/n)/n)^{1/2}

Secular_equilibrium.html

  1. d N B d t = λ A N A - λ B N B \frac{dN_{B}}{dt}=\lambda_{A}N_{A}-\lambda_{B}N_{B}
  2. λ = l n ( 2 ) / t 1 / 2 \lambda=ln(2)/t_{1/2}
  3. d N B / d t = 0 dN_{B}/dt=0
  4. N B = λ A λ B N A N_{B}=\frac{\lambda_{A}}{\lambda_{B}}N_{A}
  5. N A ( t ) = N A ( 0 ) e - λ A t N_{A}(t)=N_{A}(0)e^{-\lambda_{A}t}
  6. λ A t 1 \lambda_{A}t\ll 1

Sedimentation_coefficient.html

  1. s = v t a s=\frac{v_{t}}{a}
  2. v t v_{t}
  3. ω 2 r \omega^{2}r
  4. ω \omega
  5. v t = m r ω 2 6 π η r 0 {v_{t}}=\frac{mr\omega^{2}}{6\pi\eta r_{0}}
  6. s = v t r ω 2 = m 6 π η r 0 s=\frac{v_{t}}{r\omega^{2}}=\frac{m}{6\pi\eta r_{0}}

SEER-SEM.html

  1. S e S_{e}
  2. S e S_{e}
  3. S e S_{e}
  4. S e = N e w S i z e + E x i s t i n g S i z e * ( 0.4 * R e d e s i g n + 0.25 * R e i m p l + 0.35 * R e t e s t ) S_{e}=NewSize+ExistingSize*(0.4*Redesign+0.25*Reimpl+0.35*Retest)
  5. S e S_{e}
  6. S e S_{e}
  7. S e S_{e}
  8. S e S_{e}
  9. S e S_{e}
  10. S e = L x * ( A d j F a c t o r * U F P ) E n t r o p y 1.2 S_{e}=Lx*(AdjFactor*UFP)^{\frac{Entropy}{1.2}}
  11. L x Lx
  12. A d j F a c t o r AdjFactor
  13. K = D 0.4 * ( S e C t e ) E K=D^{0.4}*(\frac{S_{e}}{C_{te}})^{E}
  14. S e S_{e}
  15. C t e C_{te}
  16. D D
  17. E E
  18. t d = D - 0.2 * ( S e C t e ) 0.4 t_{d}=D^{-0.2}*(\frac{S_{e}}{C_{te}})^{0.4}
  19. 0.4 0.4