wpmath0000005_11

Rendering_equation.html

  1. L o ( 𝐱 , ω o , λ , t ) = L e ( 𝐱 , ω o , λ , t ) + Ω f r ( 𝐱 , ω i , ω o , λ , t ) L i ( 𝐱 , ω i , λ , t ) ( ω i 𝐧 ) d ω i L_{\,\text{o}}(\mathbf{x},\,\omega_{\,\text{o}},\,\lambda,\,t)\,=\,L_{e}(% \mathbf{x},\,\omega_{\,\text{o}},\,\lambda,\,t)\ +\,\int_{\Omega}f_{r}(\mathbf% {x},\,\omega_{\,\text{i}},\,\omega_{\,\text{o}},\,\lambda,\,t)\,L_{\,\text{i}}% (\mathbf{x},\,\omega_{\,\text{i}},\,\lambda,\,t)\,(\omega_{\,\text{i}}\,\cdot% \,\mathbf{n})\,\operatorname{d}\omega_{\,\text{i}}
  2. λ \lambda\,\!
  3. t t\,\!
  4. 𝐱 \mathbf{x}
  5. 𝐧 \mathbf{n}
  6. ω o \omega_{\,\text{o}}
  7. ω i \omega_{\,\text{i}}
  8. L o ( 𝐱 , ω o , λ , t ) L_{\,\text{o}}(\mathbf{x},\,\omega_{\,\text{o}},\,\lambda,\,t)
  9. λ \lambda\,\!
  10. ω o \omega_{\,\text{o}}
  11. t t\,\!
  12. 𝐱 \mathbf{x}\,\!
  13. L e ( 𝐱 , ω o , λ , t ) L_{e}(\mathbf{x},\,\omega_{\,\text{o}},\,\lambda,\,t)
  14. Ω \Omega
  15. 𝐧 \mathbf{n}
  16. ω i \omega_{\,\text{i}}
  17. Ω d ω i \int_{\Omega}\dots\,\operatorname{d}\omega_{\,\text{i}}
  18. Ω \Omega
  19. f r ( 𝐱 , ω i , ω o , λ , t ) f_{r}(\mathbf{x},\,\omega_{\,\text{i}},\,\omega_{\,\text{o}},\,\lambda,\,t)
  20. ω i \omega_{\,\text{i}}
  21. ω o \omega_{\,\text{o}}
  22. 𝐱 \mathbf{x}\,\!
  23. t t\,\!
  24. λ \lambda\,\!
  25. L i ( 𝐱 , ω i , λ , t ) L_{\,\text{i}}(\mathbf{x},\,\omega_{\,\text{i}},\,\lambda,\,t)
  26. λ \lambda\,\!
  27. 𝐱 \mathbf{x}\,\!
  28. ω i \omega_{\,\text{i}}
  29. t t\,\!
  30. ω i 𝐧 \omega_{\,\text{i}}\cdot\mathbf{n}
  31. cos θ i \cos\theta_{i}
  32. L o L_{\,\text{o}}\,\!
  33. t ; t\,\!;
  34. L o L_{\,\text{o}}\,\!
  35. L o L_{o}
  36. L i L_{i}
  37. L o L_{o}

Renewal_theory.html

  1. i i
  2. i + 1 i+1
  3. S 1 , S 2 , S 3 , S 4 , S 5 , S_{1},S_{2},S_{3},S_{4},S_{5},\ldots
  4. 0 < 𝔼 [ S i ] < . 0<\mathbb{E}[S_{i}]<\infty.
  5. S i S_{i}
  6. i i
  7. 𝔼 [ S i ] \mathbb{E}[S_{i}]
  8. S i S_{i}
  9. J n = i = 1 n S i , J_{n}=\sum_{i=1}^{n}S_{i},
  10. J n J_{n}
  11. n n
  12. [ J n , J n + 1 ] [J_{n},J_{n+1}]
  13. ( X t ) t 0 (X_{t})_{t\geq 0}
  14. X t = n = 1 𝕀 { J n t } = sup { n : J n t } X_{t}=\sum^{\infty}_{n=1}\mathbb{I}_{\{J_{n}\leq t\}}=\sup\left\{\,n:J_{n}\leq t% \,\right\}
  15. 𝕀 \mathbb{I}
  16. { S i : i 1 } \{S_{i}:i\geq 1\}
  17. W 1 , W 2 , W_{1},W_{2},\ldots
  18. 𝔼 | W i | < . \mathbb{E}|W_{i}|<\infty.\,
  19. Y t = i = 1 X t W i Y_{t}=\sum_{i=1}^{X_{t}}W_{i}
  20. S i S_{i}
  21. W i W_{i}
  22. Y t Y_{t}
  23. S 1 , S 2 , S_{1},S_{2},\ldots
  24. W 1 , W 2 , W_{1},W_{2},\ldots
  25. W i W_{i}
  26. S i S_{i}
  27. W 1 , W 2 , W_{1},W_{2},\ldots
  28. S i S_{i}
  29. W i W_{i}
  30. Y t Y_{t}
  31. t t
  32. m ( t ) = 𝔼 [ X t ] . m(t)=\mathbb{E}[X_{t}].\,
  33. lim t 1 t m ( t ) = 1 / 𝔼 [ S 1 ] . \lim_{t\to\infty}\frac{1}{t}m(t)=1/\mathbb{E}[S_{1}].
  34. lim t X t t = 1 𝔼 [ S 1 ] . \lim_{t\to\infty}\frac{X_{t}}{t}=\frac{1}{\mathbb{E}[S_{1}]}.
  35. { X t t ; t 0 } \left\{\frac{X_{t}}{t};t\geq 0\right\}
  36. S n ¯ = a 𝕀 { S n > a } \overline{S_{n}}=a\mathbb{I}\{S_{n}>a\}
  37. a a
  38. 0 < F ( a ) = p < 1 0<F(a)=p<1
  39. X t ¯ \overline{X_{t}}
  40. X t X_{t}
  41. { n a ; n } \{na;n\in\mathbb{N}\}
  42. p p
  43. X t ¯ i = 1 [ a t ] Geometric ( p ) 𝔼 [ X t ¯ 2 ] C 1 t + C 2 t 2 P ( X t t > x ) E [ X t 2 ] t 2 x 2 E [ X t ¯ 2 ] t 2 x 2 C x 2 . \begin{aligned}\displaystyle\overline{X_{t}}&\displaystyle\leq\sum_{i=1}^{[at]% }\mathrm{Geometric}(p)\\ \displaystyle\mathbb{E}\left[\,\overline{X_{t}}^{2}\,\right]&\displaystyle\leq C% _{1}t+C_{2}t^{2}\\ \displaystyle P\left(\frac{X_{t}}{t}>x\right)&\displaystyle\leq\frac{E\left[X_% {t}^{2}\right]}{t^{2}x^{2}}\leq\frac{E\left[\overline{X_{t}}^{2}\right]}{t^{2}% x^{2}}\leq\frac{C}{x^{2}}.\end{aligned}
  44. g ( t ) = 𝔼 [ Y t ] . g(t)=\mathbb{E}[Y_{t}].\,
  45. lim t 1 t g ( t ) = 𝔼 [ W 1 ] 𝔼 [ S 1 ] . \lim_{t\to\infty}\frac{1}{t}g(t)=\frac{\mathbb{E}[W_{1}]}{\mathbb{E}[S_{1}]}.
  46. m ( t ) = F S ( t ) + 0 t m ( t - s ) f S ( s ) d s m(t)=F_{S}(t)+\int_{0}^{t}m(t-s)f_{S}(s)\,ds
  47. F S F_{S}
  48. S 1 S_{1}
  49. f S f_{S}
  50. m ( t ) = 𝔼 [ X t ] = 𝔼 [ 𝔼 ( X t S 1 ) ] . m(t)=\mathbb{E}[X_{t}]=\mathbb{E}[\mathbb{E}(X_{t}\mid S_{1})].\,
  51. 𝔼 ( X t S 1 = s ) = 𝕀 { t s } ( 1 + 𝔼 [ X t - s ] ) . \mathbb{E}(X_{t}\mid S_{1}=s)=\mathbb{I}_{\{t\geq s\}}\left(1+\mathbb{E}[X_{t-% s}]\right).\,
  52. m ( t ) \displaystyle m(t)
  53. ( X t ) t 0 (X_{t})_{t\geq 0}
  54. ( Y t ) t 0 (Y_{t})_{t\geq 0}
  55. lim t 1 t X t = 1 𝔼 S 1 \lim_{t\to\infty}\frac{1}{t}X_{t}=\frac{1}{\mathbb{E}S_{1}}
  56. lim t 1 t Y t = 1 𝔼 S 1 𝔼 W 1 \lim_{t\to\infty}\frac{1}{t}Y_{t}=\frac{1}{\mathbb{E}S_{1}}\mathbb{E}W_{1}
  57. ( X t ) t 0 (X_{t})_{t\geq 0}
  58. J X t t J X t + 1 J_{X_{t}}\leq t\leq J_{X_{t}+1}
  59. t 0 t\geq 0
  60. J X t X t t X t J X t + 1 X t \frac{J_{X_{t}}}{X_{t}}\leq\frac{t}{X_{t}}\leq\frac{J_{X_{t}+1}}{X_{t}}
  61. 0 < 𝔼 S i < 0<\mathbb{E}S_{i}<\infty
  62. X t X_{t}\to\infty
  63. t t\to\infty
  64. J X t X t = J n n = 1 n i = 1 n S i 𝔼 S 1 \frac{J_{X_{t}}}{X_{t}}=\frac{J_{n}}{n}=\frac{1}{n}\sum_{i=1}^{n}S_{i}\to% \mathbb{E}S_{1}
  65. J X t + 1 X t = J X t + 1 X t + 1 X t + 1 X t = J n + 1 n + 1 n + 1 n 𝔼 S 1 1 \frac{J_{X_{t}+1}}{X_{t}}=\frac{J_{X_{t}+1}}{X_{t}+1}\frac{X_{t}+1}{X_{t}}=% \frac{J_{n+1}}{n+1}\frac{n+1}{n}\to\mathbb{E}S_{1}\cdot 1
  66. t / X t t/X_{t}
  67. 1 t X t 1 𝔼 S 1 \frac{1}{t}X_{t}\to\frac{1}{\mathbb{E}S_{1}}
  68. ( Y t ) t 0 (Y_{t})_{t\geq 0}
  69. 1 t Y t = X t t 1 X t Y t 1 𝔼 S 1 𝔼 W 1 \frac{1}{t}Y_{t}=\frac{X_{t}}{t}\frac{1}{X_{t}}Y_{t}\to\frac{1}{\mathbb{E}S_{1% }}\cdot\mathbb{E}W_{1}
  70. Y t Y_{t}
  71. ( S X t + 1 > x ) ( S 1 > x ) = 1 - F S ( x ) \mathbb{P}(S_{X_{t}+1}>x)\geq\mathbb{P}(S_{1}>x)=1-F_{S}(x)
  72. J X t J_{X_{t}}
  73. S X t + 1 S_{X_{t}+1}
  74. ( S X t + 1 > x ) = 0 ( S X t + 1 > x J X t = s ) f S ( s ) d s = 0 ( S X t + 1 > x | S X t + 1 > t - s ) f S ( s ) d s = 0 ( S X t + 1 > x , S X t + 1 > t - s ) ( S X t + 1 > t - s ) f S ( s ) d s = 0 1 - F ( max { x , t - s } ) 1 - F ( t - s ) f S ( s ) d s = 0 min { 1 - F ( x ) 1 - F ( t - s ) , 1 - F ( t - s ) 1 - F ( t - s ) } f S ( s ) d s = 0 min { 1 - F ( x ) 1 - F ( t - s ) , 1 } f S ( s ) d s 1 - F ( x ) = ( S 1 > x ) \begin{aligned}\displaystyle\mathbb{P}(S_{X_{t}+1}>x)&\displaystyle{}=\int_{0}% ^{\infty}\mathbb{P}(S_{X_{t}+1}>x\mid J_{X_{t}}=s)f_{S}(s)\,ds\\ &\displaystyle{}=\int_{0}^{\infty}\mathbb{P}(S_{X_{t}+1}>x|S_{X_{t}+1}>t-s)f_{% S}(s)\,ds\\ &\displaystyle{}=\int_{0}^{\infty}\frac{\mathbb{P}(S_{X_{t}+1}>x\,,\,S_{X_{t}+% 1}>t-s)}{\mathbb{P}(S_{X_{t}+1}>t-s)}f_{S}(s)\,ds\\ &\displaystyle{}=\int_{0}^{\infty}\frac{1-F(\max\{x,t-s\})}{1-F(t-s)}f_{S}(s)% \,ds\\ &\displaystyle{}=\int_{0}^{\infty}\min\left\{\frac{1-F(x)}{1-F(t-s)},\frac{1-F% (t-s)}{1-F(t-s)}\right\}f_{S}(s)\,ds\\ &\displaystyle{}=\int_{0}^{\infty}\min\left\{\frac{1-F(x)}{1-F(t-s)},1\right\}% f_{S}(s)\,ds\\ &\displaystyle{}\geq 1-F(x)\\ &\displaystyle{}=\mathbb{P}(S_{1}>x)\end{aligned}
  75. R ( t ) = 1 - k = 1 K α k l = 1 K α l ( 1 - R k ( t ) ) j = 1 , j k K α j t ( 1 - R j ( u ) ) d u R(t)=1-\sum_{k=1}^{K}\frac{\alpha_{k}}{\sum_{l=1}^{K}\alpha_{l}}(1-R_{k}(t))% \prod_{j=1,j\neq k}^{K}\alpha_{j}\int_{t}^{\infty}(1-R_{j}(u))\,\text{d}u
  76. ( Y t ) t 0 (Y_{t})_{t\geq 0}
  77. 𝔼 W \displaystyle\mathbb{E}W
  78. 1 t Y t 𝔼 W 𝔼 S = 4 ( 1200 t + 200 ) t 2 + 4 t - 2 t 2 \frac{1}{t}Y_{t}\simeq\frac{\mathbb{E}W}{\mathbb{E}S}=\frac{4(1200t+200)}{t^{2% }+4t-2t^{2}}
  79. t 4 ( 1200 t + 200 ) t 2 + 4 t - 2 t 2 = 4 ( 4 t - t 2 ) ( 1200 ) - ( 4 - 2 t ) ( 1200 t + 200 ) ( t 2 + 4 t - 2 t 2 ) 2 , \frac{\partial}{\partial t}\frac{4(1200t+200)}{t^{2}+4t-2t^{2}}=4\frac{(4t-t^{% 2})(1200)-(4-2t)(1200t+200)}{(t^{2}+4t-2t^{2})^{2}},
  80. 0 \displaystyle 0
  81. 0 = 3 t 2 + t - 2 = ( 3 t - 2 ) ( t + 1 ) . 0=3t^{2}+t-2=(3t-2)(t+1).

Representation_theory_of_SU(2).html

  1. λ 0 λ≥0
  2. 2 λ + 1 2λ+1
  3. 𝔰 𝔲 ( 2 ) \mathfrak{su}(2)
  4. e e
  5. f f
  6. h h
  7. [ h , e ] = e [h,e]=e
  8. [ h , f ] = - f [h,f]=-f
  9. [ e , f ] = h [e,f]=h
  10. i −i
  11. 𝔰 𝔲 ( 2 ) \mathfrak{su}(2)
  12. ρ ( h ) ρ(h)
  13. x x
  14. α α
  15. h [ x ] = α x h[x]=\alpha x
  16. h [ e [ x ] ] = ( α + 1 ) e [ x ] h[e[x]]=(\alpha+1)e[x]
  17. h [ f [ x ] ] = ( α - 1 ) f [ x ] h[f[x]]=(\alpha-1)f[x]
  18. e e
  19. f f
  20. e e
  21. f f
  22. h 2 + e f + f e h^{2}+ef+fe
  23. λ ( λ + 1 ) λ(λ+1)
  24. h 2 + e f + f e h^{2}+ef+fe
  25. I 2 I^{2}
  26. I 1 2 + I 2 2 + I 3 2 I_{1}^{2}+I_{2}^{2}+I_{3}^{2}
  27. x x
  28. h h
  29. α 1 \alpha_{1}
  30. e ( x ) = 0 e(x)=0
  31. ( h 2 + e f + f e ) x = ( α 1 2 + α 1 ) x = λ ( λ + 1 ) x (h^{2}+ef+fe)x=(\alpha_{1}^{2}+\alpha_{1})x=\lambda(\lambda+1)x
  32. x x
  33. α 1 \alpha_{1}
  34. λ λ
  35. λ 1 −λ−1
  36. x x
  37. f ( x ) = 0 f(x)=0
  38. ( α 2 2 - α 2 ) x = λ ( λ + 1 ) x (\alpha_{2}^{2}-\alpha_{2})x=\lambda(\lambda+1)x
  39. α 2 \alpha_{2}
  40. λ + 1 λ+1
  41. λ −λ
  42. λ < λ + 1 λ<λ+1

Representation_theory_of_the_Galilean_group.html

  1. t , x , y , z t,x,y,z
  2. n n
  3. [ E , P i ] = 0 [E,P_{i}]=0
  4. [ P i , P j ] = 0 [P_{i},P_{j}]=0
  5. [ L i j , E ] = 0 [L_{ij},E]=0
  6. [ C i , C j ] = 0 [C_{i},C_{j}]=0
  7. [ L i j , L k l ] = i [ δ i k L j l - δ i l L j k - δ j k L i l + δ j l L i k ] [L_{ij},L_{kl}]=i\hbar[\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+% \delta_{jl}L_{ik}]
  8. [ L i j , P k ] = i [ δ i k P j - δ j k P i ] [L_{ij},P_{k}]=i\hbar[\delta_{ik}P_{j}-\delta_{jk}P_{i}]
  9. [ L i j , C k ] = i [ δ i k C j - δ j k C i ] [L_{ij},C_{k}]=i\hbar[\delta_{ik}C_{j}-\delta_{jk}C_{i}]
  10. [ C i , E ] = i P i [C_{i},E]=i\hbar P_{i}
  11. [ C i , P j ] = i M δ i j . [C_{i},P_{j}]=i\hbar M\delta_{ij}~{}.
  12. M M
  13. M E - P 2 2 ME-{P^{2}\over 2}
  14. W W
  15. W M L + P × C , \vec{W}\equiv M\vec{L}+\vec{P}\times\vec{C}~{},
  16. n n
  17. W i j = M L i j + P i C j - P j C i W_{ij}=ML_{ij}+P_{i}C_{j}-P_{j}C_{i}
  18. W i j k = P i L j k + P j L k i + P k L i j , W_{ijk}=P_{i}L_{jk}+P_{j}L_{ki}+P_{k}L_{ij}~{},
  19. m m
  20. m E < s u b > 0 mE<sub>0
  21. w = m s w=ms
  22. s s
  23. L L
  24. C C
  25. W i j = M S i j W_{ij}=MS_{ij}
  26. L i j = S i j + X i P j - X j P i L_{ij}=S_{ij}+X_{i}P_{j}-X_{j}P_{i}
  27. C i = M X i - P i t . C_{i}=MX_{i}-P_{i}t~{}.
  28. E E
  29. m m
  30. E E
  31. m E = m E 0 + P 2 2 , mE=mE_{0}+{P^{2}\over 2}~{},
  32. E E
  33. P P
  34. m m
  35. E E
  36. E E
  37. M M
  38. L L
  39. s s
  40. m m
  41. m m
  42. E E
  43. s s
  44. E E
  45. m m
  46. E E
  47. m m
  48. m E - P 2 2 = - P 2 2 mE-{P^{2}\over 2}={-P^{2}\over 2}
  49. m m
  50. t = - P C P 2 , t=-{\vec{P}\cdot\vec{C}\over P^{2}}~{},
  51. C = W × P P 2 - P t , \vec{C}={\vec{W}\times\vec{P}\over P^{2}}-\vec{P}t~{},

Rescorla–Wagner_model.html

  1. Δ V X n + 1 = α X β ( λ - V t o t ) \Delta V^{n+1}_{X}=\alpha_{X}\beta(\lambda-V_{tot})
  2. V t o t = V X n + Δ V X n + 1 V_{tot}=V^{n}_{X}+\Delta V^{n+1}_{X}
  3. Δ V X \Delta V_{X}
  4. α \alpha
  5. β \beta
  6. λ \lambda
  7. V X V_{X}
  8. V t o t V_{tot}

Residence_time.html

  1. τ = System capacity to hold a substance Flow rate of the substance through the system \tau=\frac{\mbox{System capacity to hold a substance}~{}}{\mbox{Flow rate of % the substance through the system}~{}}
  2. τ = V q \tau=\frac{V}{q}
  3. τ \tau
  4. C = C o e - k τ C=C_{o}e^{-k\tau}
  5. τ \tau
  6. τ s \tau_{s}
  7. τ s = τ s , 0 e E s , a R T \tau_{s}=\tau_{s,0}\;e^{\frac{E_{s,a}}{RT}}
  8. τ s \tau_{s}
  9. τ s , 0 \tau_{s,0}
  10. E s , a E_{s,a}
  11. R R
  12. T T
  13. τ s \tau_{s}
  14. E s , a E_{s,a}
  15. τ s , 0 \tau_{s,0}
  16. E s , a E_{s,a}
  17. M R T = 1 N i = 1 m t i n i MRT=\frac{1}{N}\sum_{i=1}^{m}t_{i}n_{i}
  18. m m
  19. t i t_{i}
  20. n i n_{i}
  21. N N

Residual_entropy.html

  1. 2 N 2^{N}
  2. S = N k ln ( 2 ) S=Nk\ln(2)

Resistance_thermometer.html

  1. α = R 100 - R 0 100 R 0 \alpha=\frac{R_{100}-R_{0}}{100R_{0}}
  2. R 0 = R_{0}=
  3. R 100 = R_{100}=
  4. R T = R 0 [ 1 + A T + B T 2 + C T 3 ( T - 100 ) ] ( - 200 C < T < 0 C ) , R_{T}=R_{0}\left[1+AT+BT^{2}+CT^{3}(T-100)\right]\;(-200\;{}^{\circ}\mathrm{C}% <T<0\;{}^{\circ}\mathrm{C}),
  5. R T = R 0 [ 1 + A T + B T 2 ] ( 0 C T < 850 C ) . R_{T}=R_{0}\left[1+AT+BT^{2}\right]\;(0\;{}^{\circ}\mathrm{C}\leq T<850\;{}^{% \circ}\mathrm{C}).
  6. R T R_{T}
  7. R 0 R_{0}
  8. A = 3.9083 × 10 - 3 C - 1 A=3.9083\times 10^{-3}\;{}^{\circ}\mathrm{C}^{-1}
  9. B = - 5.775 × 10 - 7 C - 2 B=-5.775\times 10^{-7}\;{}^{\circ}\mathrm{C}^{-2}
  10. C = - 4.183 × 10 - 12 C - 4 . C=-4.183\times 10^{-12}\;{}^{\circ}\mathrm{C}^{-4}.
  11. T = - A + A 2 - 4 B ( 1 - R T 100 ) 2 B T=\frac{-A+\sqrt{A^{2}-4B(1-\frac{R_{T}}{100})}}{2B}
  12. V T V_{T}
  13. T = - A + A 2 - 40 B ( 0.1 - V T ) 2 B T=\frac{-A+\sqrt{A^{2}-40B(0.1-V_{T})}}{2B}

Resonance_Raman_spectroscopy.html

  1. E E
  2. E = h ν E=h\nu
  3. h h
  4. ν \nu
  5. Δ ν ¯ \Delta\bar{\nu}
  6. Δ ν ¯ \Delta\bar{\nu}
  7. Δ ν ¯ \Delta\bar{\nu}
  8. Δ ν ¯ = 0 \Delta\bar{\nu}=0
  9. Δ ν ¯ \Delta\bar{\nu}
  10. Δ ν ¯ \Delta\bar{\nu}
  11. I r I u \frac{I_{r}}{I_{u}}
  12. I r I_{r}
  13. I u I_{u}
  14. 3 4 \frac{3}{4}

Resting_metabolic_rate.html

  1. P = 500 + ( 22 L B M ) P=500+\left({22\cdot LBM}\right)

Restriction_(mathematics).html

  1. f A f{\restriction_{A}}
  2. f : E F f:E→F
  3. E E
  4. F F
  5. f f
  6. E E
  7. dom f E \mathrm{dom}\,f\subseteq E
  8. A A
  9. E E
  10. f f
  11. A A
  12. f | A : A F {f|}_{A}\colon A\to F
  13. f f
  14. A A
  15. f f
  16. A dom f A\cap\mathrm{dom}\,f
  17. f f
  18. ( x , f ( x ) ) (x,f(x))
  19. E × F E\times F
  20. f f
  21. A A
  22. G ( f | A ) = { ( x , f ( x ) ) G ( f ) x A } G({f|}_{A})=\{(x,f(x))\in G(f)\mid x\in A\}
  23. ( x , f ( x ) ) (x,f(x))
  24. G G
  25. f : ; x x 2 f:\mathbb{R}\to\mathbb{R};x\mapsto x^{2}
  26. + = [ 0 , ) \mathbb{R}_{+}=[0,\infty)
  27. f : + ; x x 2 f:\mathbb{R}_{+}\to\mathbb{R};x\mapsto x^{2}
  28. f : X Y f:X\rightarrow Y
  29. X X
  30. f | X = f f|_{X}=f
  31. A B dom f A\subseteq B\subseteq\mathrm{dom}f
  32. ( f | B ) | A = f | A (f|_{B})|_{A}=f|_{A}
  33. f f
  34. f f
  35. f ( x ) = x 2 f(x)=x^{2}
  36. x 0 x≥0
  37. f - 1 ( y ) = y . f^{-1}(y)=\sqrt{y}.
  38. x 0 x≤0
  39. y y
  40. σ a θ b ( R ) \sigma_{a\theta b}(R)
  41. σ a θ v ( R ) \sigma_{a\theta v}(R)
  42. a a
  43. b b
  44. θ \theta
  45. { < , , = , , , > } \{\;<,\leq,=,\neq,\geq,\;>\}
  46. v v
  47. R R
  48. σ a θ b ( R ) \sigma_{a\theta b}(R)
  49. R R
  50. θ \theta
  51. a a
  52. b b
  53. σ a θ v ( R ) \sigma_{a\theta v}(R)
  54. R R
  55. θ \theta
  56. a a
  57. v v
  58. X , Y X,Y
  59. A = X Y A=X\cup Y
  60. f : A B f:A\to B
  61. F ( U ) F(U)
  62. U U
  63. V U V\subseteq U

Retina_horizontal_cell.html

  1. \to
  2. \to
  3. \to
  4. P m P_{m}
  5. P o P_{o}
  6. P m P_{m}
  7. P m P_{m}
  8. P m P_{m}
  9. P o P_{o}
  10. P o P_{o}
  11. P m P_{m}
  12. P o P_{o}
  13. P o P_{o}
  14. P o P_{o}
  15. P o P_{o}
  16. P m P_{m}
  17. P m P_{m}
  18. P m P_{m}

Return_on_assets.html

  1. ROA = Net Income Average Total Assets \mathrm{ROA}=\frac{\mbox{Net Income}~{}}{\mbox{Average Total Assets}~{}}

Return_on_equity.html

  1. ROE = Net Income Shareholder Equity \mathrm{ROE}=\frac{\mbox{Net Income}~{}}{\mbox{Shareholder Equity}~{}}
  2. ROE = Net income Sales × Sales Total Assets × Total Assets Average Shareholder Equity \mathrm{ROE}=\frac{\mbox{Net income}~{}}{\mbox{Sales}~{}}\times\frac{\mbox{% Sales}~{}}{\mbox{Total Assets}~{}}\times\frac{\mbox{Total Assets}~{}}{\mbox{% Average Shareholder Equity}~{}}

Revealed_preference.html

  1. B B
  2. B B
  3. X = X 1 , X 2 X=X_{1},X_{2}
  4. p 1 , p 2 p_{1},p_{2}
  5. m m
  6. ( x 1 , x 2 ) X (x_{1},x_{2})\in X
  7. ( y 1 , y 2 ) X (y_{1},y_{2})\in X
  8. p 1 X 1 + p 2 X 2 m p_{1}X_{1}+p_{2}X_{2}\leq m
  9. p 1 x 1 + p 2 x 2 = m p_{1}x_{1}+p_{2}x_{2}=m
  10. p 1 y 1 + p 2 y 2 = m p_{1}y_{1}+p_{2}y_{2}=m
  11. ( x 1 , x 2 ) (x_{1},x_{2})
  12. ( y 1 , y 2 ) (y_{1},y_{2})
  13. ( x 1 , x 2 ) (x_{1},x_{2})
  14. ( y 1 , y 2 ) (y_{1},y_{2})
  15. ( x 1 , x 2 ) ( y 1 , y 2 ) (x_{1},x_{2})\succeq(y_{1},y_{2})
  16. 𝐚 𝐛 \mathbf{a}\succeq\mathbf{b}
  17. x , y B x C ( B , ) x , y B y C ( B , ) } x C ( B , ) \left.\begin{matrix}x,y\in B\\ x\in C(B,\succeq)\\ x,y\in B^{\prime}\\ y\in C(B^{\prime},\succeq)\end{matrix}\right\}~{}\Rightarrow~{}x\in C(B^{% \prime},\succeq)
  18. x x
  19. y y
  20. C ( B , ) B C(B,\succeq)\subset B
  21. B B
  22. \succeq
  23. B B
  24. B B^{\prime}
  25. B B^{\prime}
  26. p x ( p , m ) m x ( p , m ) x ( p , m ) p x ( p , m ) > m p\cdot x(p^{\prime},m^{\prime})\leq m~{}\wedge~{}x(p^{\prime},m^{\prime})\neq x% (p,m)~{}\Rightarrow~{}p^{\prime}\cdot x(p,m)>m^{\prime}~{}
  27. 𝐚 𝐛 \mathbf{a}\succeq\mathbf{b}
  28. 𝐚 𝐛 \mathbf{a}\succ\mathbf{b}~{}
  29. C ( A , B ) = A C(A,B)=A
  30. C ( B , C ) = B C(B,C)=B
  31. C ( C , A ) = C C(C,A)=C
  32. C C

Reversible_computing.html

  1. c ( a b ) c\oplus(a\cdot b)
  2. c = 0 c=0
  3. a b = 1 a\cdot b=1

Revolutions_per_minute.html

  1. 1 rad/s = 1 2 π Hz = 60 2 π rpm \begin{aligned}\displaystyle 1~{}\,\text{rad/s}&\displaystyle=\frac{1}{2\pi}~{% }\,\text{Hz}\\ &\displaystyle=\frac{60}{2\pi}~{}\,\text{rpm}\end{aligned}
  2. 1 rpm = 1 60 Hz = 2 π 60 rad/s \begin{aligned}\displaystyle 1~{}\,\text{rpm}&\displaystyle=\frac{1}{60}~{}\,% \text{Hz}\\ &\displaystyle=\frac{2\pi}{60}~{}\,\text{rad/s}\end{aligned}
  3. 1 Hz = 2 π rad/s = 60 rpm \begin{aligned}\displaystyle 1~{}\,\text{Hz}&\displaystyle=2\pi~{}\,\text{rad/% s}\\ &\displaystyle=60~{}\,\text{rpm}\end{aligned}
  4. ω = 2 π f , f = ω 2 π . \omega=2\pi f\,\,\!\,\text{, }\,\,f=\frac{\omega}{2\pi}\,\text{.}\,\!
  5. 162 / 3 16{2}/{3}
  6. 331 / 3 33{1}/{3}
  7. 5 / 18 {5}/{18}
  8. 5 / 9 {5}/{9}
  9. 3 / 4 {3}/{4}

Reynolds_stress.html

  1. u i = u i ¯ + u i , u_{i}=\overline{u_{i}}+u_{i}^{\prime},\,
  2. 𝐮 ( 𝐱 , t ) \mathbf{u}(\mathbf{x},t)
  3. u i u_{i}
  4. x i x_{i}
  5. x i x_{i}
  6. 𝐱 \mathbf{x}
  7. u i ¯ \overline{u_{i}}
  8. u i u^{\prime}_{i}
  9. τ i j ρ u i u j ¯ , \tau^{\prime}_{ij}\equiv\rho\,\overline{u^{\prime}_{i}\,u^{\prime}_{j}},\,
  10. τ i j ′′ u i u j ¯ , \tau^{\prime\prime}_{ij}\equiv\overline{u^{\prime}_{i}\,u^{\prime}_{j}},\,
  11. u i u_{i}
  12. u i ¯ \overline{u_{i}}
  13. u i u^{\prime}_{i}
  14. u i = u i ¯ + u i u_{i}=\overline{u_{i}}+u^{\prime}_{i}
  15. a ¯ ¯ = a ¯ , a + b ¯ = a ¯ + b ¯ , a b ¯ ¯ = a ¯ b ¯ . \begin{aligned}\displaystyle\overline{\bar{a}}&\displaystyle=\bar{a},\\ \displaystyle\overline{a+b}&\displaystyle=\bar{a}+\bar{b},\\ \displaystyle\overline{a\bar{b}}&\displaystyle=\bar{a}\bar{b}.\end{aligned}
  16. ρ u i u j ¯ \rho\overline{u^{\prime}_{i}u^{\prime}_{j}}
  17. R i j R_{ij}
  18. R i j ρ u i u j ¯ R_{ij}\ \equiv\ \rho\overline{u^{\prime}_{i}u^{\prime}_{j}}
  19. u i x i = 0 , \frac{\partial u_{i}}{\partial x_{i}}=0,
  20. ρ D u i D t = - p x i + μ ( 2 u i x j x j ) , \rho\frac{Du_{i}}{Dt}=-\frac{\partial p}{\partial x_{i}}+\mu\left(\frac{% \partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}\right),
  21. D / D t D/Dt
  22. D D t = t + u j x j . \frac{D}{Dt}=\frac{\partial}{\partial t}+u_{j}\frac{\partial}{\partial x_{j}}.
  23. ( u i ¯ + u i ) x i = 0 , \frac{\partial\left(\overline{u_{i}}+u_{i}^{\prime}\right)}{\partial x_{i}}=0,
  24. ρ [ ( u i ¯ + u i ) t + ( u j ¯ + u j ) ( u i ¯ + u i ) x j ] = - ( p ¯ + p ) x i + μ [ 2 ( u i ¯ + u i ) x j x j ] . \rho\left[\frac{\partial\left(\overline{u_{i}}+u_{i}^{\prime}\right)}{\partial t% }+\left(\overline{u_{j}}+u_{j}^{\prime}\right)\frac{\partial\left(\overline{u_% {i}}+u_{i}^{\prime}\right)}{\partial x_{j}}\right]=-\frac{\partial\left(\bar{p% }+p^{\prime}\right)}{\partial x_{i}}+\mu\left[\frac{\partial^{2}\left(% \overline{u_{i}}+u_{i}^{\prime}\right)}{\partial x_{j}\partial x_{j}}\right].
  25. ( u j ¯ + u j ) ( u i ¯ + u i ) x j = ( u i ¯ + u i ) ( u j ¯ + u j ) x j - ( u i ¯ + u i ) ( u j ¯ + u j ) x j , \left(\overline{u_{j}}+u_{j}^{\prime}\right)\frac{\partial\left(\overline{u_{i% }}+u_{i}^{\prime}\right)}{\partial x_{j}}=\frac{\partial\left(\overline{u_{i}}% +u_{i}^{\prime}\right)\left(\overline{u_{j}}+u_{j}^{\prime}\right)}{\partial x% _{j}}-\left(\overline{u_{i}}+u_{i}^{\prime}\right)\frac{\partial\left(% \overline{u_{j}}+u_{j}^{\prime}\right)}{\partial x_{j}},
  26. ρ [ ( u i ¯ + u i ) t + ( u i ¯ + u i ) ( u j ¯ + u j ) x j ] = - ( p ¯ + p ) x i + μ [ 2 ( u i ¯ + u i ) x j x j ] . \rho\left[\frac{\partial\left(\overline{u_{i}}+u_{i}^{\prime}\right)}{\partial t% }+\frac{\partial\left(\overline{u_{i}}+u_{i}^{\prime}\right)\left(\overline{u_% {j}}+u_{j}^{\prime}\right)}{\partial x_{j}}\right]=-\frac{\partial\left(\bar{p% }+p^{\prime}\right)}{\partial x_{i}}+\mu\left[\frac{\partial^{2}\left(% \overline{u_{i}}+u_{i}^{\prime}\right)}{\partial x_{j}\partial x_{j}}\right].
  27. u i ¯ x i = 0 , \frac{\partial\overline{u_{i}}}{\partial x_{i}}=0,
  28. ρ [ u i ¯ t + u i ¯ u j ¯ x j + u i ¯ u j ¯ x j ] = - p ¯ x i + μ 2 u i ¯ x j x j . \rho\left[\frac{\partial\overline{u_{i}}}{\partial t}+\frac{\partial\overline{% u_{i}}\,\overline{u_{j}}}{\partial x_{j}}+\frac{\partial\overline{u_{i}^{% \prime}}\overline{u_{j}^{\prime}}}{\partial x_{j}}\right]=-\frac{\partial\bar{% p}}{\partial x_{i}}+\mu\frac{\partial^{2}\overline{u_{i}}}{\partial x_{j}% \partial x_{j}}.
  29. u i ¯ u j ¯ x j = u j ¯ u i ¯ x j + u i ¯ u j ¯ x j , \frac{\partial\overline{u_{i}}\,\overline{u_{j}}}{\partial x_{j}}=\overline{u_% {j}}\frac{\partial\overline{u_{i}}}{\partial x_{j}}+\overline{u_{i}}{\frac{% \partial\overline{u_{j}}}{\partial x_{j}}},
  30. ρ [ u i ¯ t + u j ¯ u i ¯ x j ] = - p ¯ x i + x j ( μ u i ¯ x j - ρ u i u j ¯ ) , \rho\left[\frac{\partial\overline{u_{i}}}{\partial t}+\overline{u_{j}}\frac{% \partial\overline{u_{i}}}{\partial x_{j}}\right]=-\frac{\partial\bar{p}}{% \partial x_{i}}+\frac{\partial}{\partial x_{j}}\left(\mu\frac{\partial\bar{u_{% i}}}{\partial x_{j}}-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right),
  31. ρ u i u j ¯ \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}
  32. μ u i ¯ x j \mu\frac{\partial\bar{u_{i}}}{\partial x_{j}}
  33. v i v j v k ¯ \overline{v^{\prime}_{i}v^{\prime}_{j}v^{\prime}_{k}}
  34. k k
  35. ϵ \epsilon

Rényi_entropy.html

  1. α \alpha
  2. α 0 \alpha\geq 0
  3. α 1 \alpha\neq 1
  4. H α ( X ) = 1 1 - α log ( i = 1 n p i α ) H_{\alpha}(X)=\frac{1}{1-\alpha}\log\Bigg(\sum_{i=1}^{n}p_{i}^{\alpha}\Bigg)
  5. X X
  6. 1 , 2 , , n 1,2,...,n
  7. p i Pr ( X = i ) p_{i}\doteq\Pr(X=i)
  8. i = 1 , , n i=1,\dots,n
  9. p i = 1 / n p_{i}=1/n
  10. i = 1 , , n i=1,\dots,n
  11. H α ( X ) = log n H_{\alpha}(X)=\log n
  12. X X
  13. H α ( X ) H_{\alpha}(X)
  14. α \alpha
  15. H α ( X ) = α 1 - α log ( P α ) H_{\alpha}(X)=\frac{\alpha}{1-\alpha}\log\left(\|P\|_{\alpha}\right)
  16. P = ( p 1 , , p n ) P=(p_{1},\dots,p_{n})
  17. \R n \R^{n}
  18. p i 0 p_{i}\geq 0
  19. i = 1 n p i = 1 \sum_{i=1}^{n}p_{i}=1
  20. α 0 \alpha\geq 0
  21. α \alpha
  22. α 0 \alpha\to 0
  23. X X
  24. α 1 \alpha\to 1
  25. α \alpha
  26. H 0 H_{0}
  27. H 0 ( X ) = log n = log | X | . H_{0}(X)=\log n=\log|X|.\,
  28. H α H_{\alpha}
  29. α 1 \alpha\rightarrow 1
  30. H 1 ( X ) = - i = 1 n p i log p i . H_{1}(X)=-\sum_{i=1}^{n}p_{i}\log p_{i}.
  31. α = 2 \alpha=2
  32. H 2 ( X ) = - log i = 1 n p i 2 = - log P ( X = Y ) H_{2}(X)=-\log\sum_{i=1}^{n}p_{i}^{2}=-\log P(X=Y)
  33. α \alpha\rightarrow\infty
  34. H α H_{\alpha}
  35. H H_{\infty}
  36. H ( X ) min i ( - log p i ) = - ( max i log p i ) = - log max i p i . H_{\infty}(X)\doteq\min_{i}(-\log p_{i})=-(\max_{i}\log p_{i})=-\log\max_{i}p_% {i}\,.
  37. H ( X ) H_{\infty}(X)
  38. b b
  39. 2 - b 2^{-b}
  40. H α H_{\alpha}
  41. α \alpha
  42. - d H α d α = 1 ( 1 - α ) 2 i = 1 n z i log ( z i / p i ) , -\frac{dH_{\alpha}}{d\alpha}=\frac{1}{(1-\alpha)^{2}}\sum_{i=1}^{n}z_{i}\log(z% _{i}/p_{i}),
  43. z i = p i α / j = 1 n p j α z_{i}=p_{i}^{\alpha}/\sum_{j=1}^{n}p_{j}^{\alpha}
  44. log n = H 0 H 1 H 2 H . \log n=H_{0}\geq H_{1}\geq H_{2}\geq H_{\infty}.
  45. α > 1 \alpha>1
  46. H 2 2 H . H_{2}\leq 2H_{\infty}.
  47. H 1 H_{1}
  48. X X
  49. D α ( P Q ) = 1 α - 1 log ( i = 1 n p i α q i α - 1 ) = 1 α - 1 log i = 1 n p i α q i 1 - α . D_{\alpha}(P\|Q)=\frac{1}{\alpha-1}\log\Bigg(\sum_{i=1}^{n}\frac{p_{i}^{\alpha% }}{q_{i}^{\alpha-1}}\Bigg)=\frac{1}{\alpha-1}\log\sum_{i=1}^{n}p_{i}^{\alpha}q% _{i}^{1-\alpha}.\,
  50. D 0 ( P Q ) = - log Q ( { i : p i > 0 } ) D_{0}(P\|Q)=-\log Q(\{i:p_{i}>0\})
  51. D 1 / 2 ( P Q ) = - 2 log i = 1 n p i q i D_{1/2}(P\|Q)=-2\log\sum_{i=1}^{n}\sqrt{p_{i}q_{i}}
  52. D 1 ( P Q ) = i = 1 n p i log p i q i D_{1}(P\|Q)=\sum_{i=1}^{n}p_{i}\log\frac{p_{i}}{q_{i}}
  53. D 2 ( P Q ) = log p i q i D_{2}(P\|Q)=\log\Big\langle\frac{p_{i}}{q_{i}}\Big\rangle
  54. D ( P Q ) = log sup i p i q i D_{\infty}(P\|Q)=\log\sup_{i}\frac{p_{i}}{q_{i}}
  55. H ( A , X ) = H ( A ) + 𝔼 a A [ H ( X | A = a ) ] H(A,X)=H(A)+\mathbb{E}_{a\sim A}\big[H(X|A=a)\big]
  56. D KL ( p ( x | a ) p ( a ) | | m ( x , a ) ) = D KL ( p ( a ) | | m ( a ) ) + 𝔼 p ( a ) { D KL ( p ( x | a ) | | m ( x | a ) ) } , D_{\mathrm{KL}}(p(x|a)p(a)||m(x,a))=D_{\mathrm{KL}}(p(a)||m(a))+\mathbb{E}_{p(% a)}\{D_{\mathrm{KL}}(p(x|a)||m(x|a))\},
  57. H α ( A , X ) = H α ( A ) + H α ( X ) H_{\alpha}(A,X)=H_{\alpha}(A)+H_{\alpha}(X)\;
  58. D α ( P ( A ) P ( X ) Q ( A ) Q ( X ) ) = D α ( P ( A ) Q ( A ) ) + D α ( P ( X ) Q ( X ) ) . D_{\alpha}(P(A)P(X)\|Q(A)Q(X))=D_{\alpha}(P(A)\|Q(A))+D_{\alpha}(P(X)\|Q(X)).
  59. H α ( p F ( x ; θ ) ) = 1 1 - α ( F ( α θ ) - α F ( θ ) + log E p [ e ( α - 1 ) k ( x ) ] ) H_{\alpha}(p_{F}(x;\theta))=\frac{1}{1-\alpha}\left(F(\alpha\theta)-\alpha F(% \theta)+\log E_{p}[e^{(\alpha-1)k(x)}]\right)
  60. D α ( p : q ) = J F , α ( θ : θ ) 1 - α D_{\alpha}(p:q)=\frac{J_{F,\alpha}(\theta:\theta^{\prime})}{1-\alpha}
  61. J F , α ( θ : θ ) = α F ( θ ) + ( 1 - α ) F ( θ ) - F ( α θ + ( 1 - α ) θ ) J_{F,\alpha}(\theta:\theta^{\prime})=\alpha F(\theta)+(1-\alpha)F(\theta^{% \prime})-F(\alpha\theta+(1-\alpha)\theta^{\prime})
  62. H 1 H 2 H_{1}\geq H_{2}
  63. i = 1 M p i log p i log i = 1 M p i 2 \sum\limits_{i=1}^{M}{p_{i}\log p_{i}}\leq\log\sum\limits_{i=1}^{M}{p_{i}^{2}}
  64. H H 2 H_{\infty}\leq H_{2}
  65. log i = 1 n p i 2 log sup i p i ( i = 1 n p i ) = log sup p i \log\sum\limits_{i=1}^{n}{p_{i}^{2}}\leq\log\sup_{i}p_{i}\left({\sum\limits_{i% =1}^{n}{p_{i}}}\right)=\log\sup p_{i}
  66. H 2 2 H H_{2}\leq 2H_{\infty}
  67. log i = 1 n p i 2 log sup i p i 2 = 2 log sup i p i \log\sum\limits_{i=1}^{n}{p_{i}^{2}}\geq\log\sup_{i}p_{i}^{2}=2\log\sup_{i}p_{i}

Rhombic_enneacontahedron.html

  1. η = 16 - 34 5 0.7947377530014315 \eta=16-\frac{34}{\sqrt{5}}\approx 0.7947377530014315

Ricco's_law.html

  1. C o n t r a s t = K A r e a . Contrast=\frac{K}{Area}.
  2. K = ( c 1 B - 1 / 4 + c 2 ) 2 K=(c_{1}B^{-1/4}+c_{2})^{2}
  3. C 1 / B . C\propto 1/\sqrt{B}.

Rice_distribution.html

  1. f ( x ν , σ ) = x σ 2 exp ( - ( x 2 + ν 2 ) 2 σ 2 ) I 0 ( x ν σ 2 ) , f(x\mid\nu,\sigma)=\frac{x}{\sigma^{2}}\exp\left(\frac{-(x^{2}+\nu^{2})}{2% \sigma^{2}}\right)I_{0}\left(\frac{x\nu}{\sigma^{2}}\right),
  2. χ X ( t ν , σ ) = exp ( - ν 2 2 σ 2 ) [ Ψ 2 ( 1 ; 1 , 1 2 ; ν 2 2 σ 2 , - 1 2 σ 2 t 2 ) + i 2 σ t Ψ 2 ( 3 2 ; 1 , 3 2 ; ν 2 2 σ 2 , - 1 2 σ 2 t 2 ) ] , \begin{aligned}&\displaystyle\chi_{X}(t\mid\nu,\sigma)\\ &\displaystyle\quad=\exp\left(-\frac{\nu^{2}}{2\sigma^{2}}\right)\left[\Psi_{2% }\left(1;1,\frac{1}{2};\frac{\nu^{2}}{2\sigma^{2}},-\frac{1}{2}\sigma^{2}t^{2}% \right)\right.\\ &\displaystyle\left.{}\qquad+i\sqrt{2}\sigma t\Psi_{2}\left(\frac{3}{2};1,% \frac{3}{2};\frac{\nu^{2}}{2\sigma^{2}},-\frac{1}{2}\sigma^{2}t^{2}\right)% \right],\end{aligned}
  3. Ψ 2 ( α ; γ , γ ; x , y ) \Psi_{2}\left(\alpha;\gamma,\gamma^{\prime};x,y\right)
  4. x x
  5. y y
  6. Ψ 2 ( α ; γ , γ ; x , y ) = n = 0 m = 0 ( α ) m + n ( γ ) m ( γ ) n x m y n m ! n ! , \Psi_{2}\left(\alpha;\gamma,\gamma^{\prime};x,y\right)=\sum_{n=0}^{\infty}\sum% _{m=0}^{\infty}\frac{(\alpha)_{m+n}}{(\gamma)_{m}(\gamma^{\prime})_{n}}\frac{x% ^{m}y^{n}}{m!n!},
  7. ( x ) n = x ( x + 1 ) ( x + n - 1 ) = Γ ( x + n ) Γ ( x ) (x)_{n}=x(x+1)\cdots(x+n-1)=\frac{\Gamma(x+n)}{\Gamma(x)}
  8. μ 1 = σ π / 2 L 1 / 2 ( - ν 2 / 2 σ 2 ) \mu_{1}^{{}^{\prime}}=\sigma\sqrt{\pi/2}\,\,L_{1/2}(-\nu^{2}/2\sigma^{2})
  9. μ 2 = 2 σ 2 + ν 2 \mu_{2}^{{}^{\prime}}=2\sigma^{2}+\nu^{2}\,
  10. μ 3 = 3 σ 3 π / 2 L 3 / 2 ( - ν 2 / 2 σ 2 ) \mu_{3}^{{}^{\prime}}=3\sigma^{3}\sqrt{\pi/2}\,\,L_{3/2}(-\nu^{2}/2\sigma^{2})
  11. μ 4 = 8 σ 4 + 8 σ 2 ν 2 + ν 4 \mu_{4}^{{}^{\prime}}=8\sigma^{4}+8\sigma^{2}\nu^{2}+\nu^{4}\,
  12. μ 5 = 15 σ 5 π / 2 L 5 / 2 ( - ν 2 / 2 σ 2 ) \mu_{5}^{{}^{\prime}}=15\sigma^{5}\sqrt{\pi/2}\,\,L_{5/2}(-\nu^{2}/2\sigma^{2})
  13. μ 6 = 48 σ 6 + 72 σ 4 ν 2 + 18 σ 2 ν 4 + ν 6 \mu_{6}^{{}^{\prime}}=48\sigma^{6}+72\sigma^{4}\nu^{2}+18\sigma^{2}\nu^{4}+\nu% ^{6}\,
  14. μ k = σ k 2 k / 2 Γ ( 1 + k / 2 ) L k / 2 ( - ν 2 / 2 σ 2 ) . \mu_{k}^{{}^{\prime}}=\sigma^{k}2^{k/2}\,\Gamma(1\!+\!k/2)\,L_{k/2}(-\nu^{2}/2% \sigma^{2}).\,
  15. L q ( x ) = L q ( 0 ) ( x ) = M ( - q , 1 , x ) = 1 F 1 ( - q ; 1 ; x ) L_{q}(x)=L_{q}^{(0)}(x)=M(-q,1,x)=\,_{1}F_{1}(-q;1;x)
  16. M ( a , b , z ) = 1 F 1 ( a ; b ; z ) M(a,b,z)=_{1}F_{1}(a;b;z)
  17. L 1 / 2 ( x ) \displaystyle L_{1/2}(x)
  18. μ 2 = 2 σ 2 + ν 2 - ( π σ 2 / 2 ) L 1 / 2 2 ( - ν 2 / 2 σ 2 ) . \mu_{2}=2\sigma^{2}+\nu^{2}-(\pi\sigma^{2}/2)\,L^{2}_{1/2}(-\nu^{2}/2\sigma^{2% }).
  19. L 1 / 2 2 ( ) L^{2}_{1/2}(\cdot)
  20. L 1 / 2 ( ) L_{1/2}(\cdot)
  21. L 1 / 2 ( 2 ) ( ) . L^{(2)}_{1/2}(\cdot).
  22. { σ 4 x 2 f ′′ ( x ) + ( 2 σ 2 x 3 - σ 4 x ) f ( x ) + f ( x ) ( σ 4 - v 2 x 2 + x 4 ) = 0 f ( 1 ) = exp ( - v 2 + 1 2 σ 2 ) I 0 ( v σ 2 ) σ 2 f ( 1 ) = exp ( - v 2 + 1 2 σ 2 ) ( ( σ 2 - 1 ) I 0 ( v σ 2 ) + v I 1 ( v σ 2 ) ) σ 4 } \left\{\begin{array}[]{l}\sigma^{4}x^{2}f^{\prime\prime}(x)+\left(2\sigma^{2}x% ^{3}-\sigma^{4}x\right)f^{\prime}(x)+f(x)\left(\sigma^{4}-v^{2}x^{2}+x^{4}% \right)=0\\ f(1)=\frac{\exp\left(-\frac{v^{2}+1}{2\sigma^{2}}\right)I_{0}\left(\frac{v}{% \sigma^{2}}\right)}{\sigma^{2}}\\ f^{\prime}(1)=\frac{\exp\left(-\frac{v^{2}+1}{2\sigma^{2}}\right)\left(\left(% \sigma^{2}-1\right)I_{0}\left(\frac{v}{\sigma^{2}}\right)+vI_{1}\left(\frac{v}% {\sigma^{2}}\right)\right)}{\sigma^{4}}\end{array}\right\}
  23. R Rice ( ν , σ ) R\sim\mathrm{Rice}\left(\nu,\sigma\right)
  24. R = X 2 + Y 2 R=\sqrt{X^{2}+Y^{2}}
  25. X N ( ν cos θ , σ 2 ) X\sim N\left(\nu\cos\theta,\sigma^{2}\right)
  26. Y N ( ν sin θ , σ 2 ) Y\sim N\left(\nu\sin\theta,\sigma^{2}\right)
  27. θ \theta
  28. R Rice ( ν , σ ) R\sim\mathrm{Rice}\left(\nu,\sigma\right)
  29. P P
  30. λ = ν 2 2 σ 2 . \lambda=\frac{\nu^{2}}{2\sigma^{2}}.
  31. X X
  32. R = σ X . R=\sigma\sqrt{X}.
  33. R Rice ( ν , 1 ) R\sim\,\text{Rice}\left(\nu,1\right)
  34. R 2 R^{2}
  35. ν 2 \nu^{2}
  36. R Rice ( ν , 1 ) R\sim\,\text{Rice}\left(\nu,1\right)
  37. R R
  38. ν \nu
  39. R Rice ( 0 , σ ) R\sim\,\text{Rice}\left(0,\sigma\right)
  40. R Rayleigh ( σ ) R\sim\,\text{Rayleigh}\left(\sigma\right)
  41. μ 2 = 4 - π 2 σ 2 \mu_{2}=\frac{4-\pi}{2}\sigma^{2}
  42. R Rice ( 0 , σ ) R\sim\,\text{Rice}\left(0,\sigma\right)
  43. R 2 R^{2}
  44. lim x - L ν ( x ) = | x | ν Γ ( 1 + ν ) . \lim_{x\rightarrow-\infty}L_{\nu}(x)=\frac{|x|^{\nu}}{\Gamma(1+\nu)}.
  45. r = μ 1 / μ 2 1 / 2 r=\mu^{{}^{\prime}}_{1}/\mu^{1/2}_{2}
  46. g ( θ ) = ξ ( θ ) [ 1 + r 2 ] - 2 , g(\theta)=\sqrt{\xi{(\theta)}\left[1+r^{2}\right]-2},
  47. θ \theta
  48. θ = ν σ \theta=\frac{\nu}{\sigma}
  49. ξ ( θ ) \xi{\left(\theta\right)}
  50. ξ ( θ ) = 2 + θ 2 - π 8 exp ( - θ 2 / 2 ) [ ( 2 + θ 2 ) I 0 ( θ 2 / 4 ) + θ 2 I 1 ( θ 2 / 4 ) ] 2 , \xi{\left(\theta\right)}=2+\theta^{2}-\frac{\pi}{8}\exp{(-\theta^{2}/2)}\left[% (2+\theta^{2})I_{0}(\theta^{2}/4)+\theta^{2}I_{1}(\theta^{2}/4)\right]^{2},
  51. I 0 I_{0}
  52. I 1 I_{1}
  53. ξ ( θ ) \xi{\left(\theta\right)}
  54. σ \sigma
  55. μ 2 \mu_{2}
  56. μ 2 = ξ ( θ ) σ 2 . \mu_{2}=\xi{\left(\theta\right)}\sigma^{2}.\,
  57. θ * \theta^{*}
  58. g g
  59. θ 0 {\theta}_{0}
  60. θ lowerbound = 0 {\theta}_{\mathrm{lowerbound}}=0
  61. r = π / ( 4 - π ) r=\sqrt{\pi/(4-\pi)}
  62. r = μ 1 / μ 2 1 / 2 r=\mu^{{}^{\prime}}_{1}/\mu^{1/2}_{2}
  63. | g i ( θ 0 ) - θ i - 1 | \left|g^{i}\left(\theta_{0}\right)-\theta_{i-1}\right|
  64. g i g^{i}
  65. g g
  66. i i
  67. θ n \theta_{n}
  68. n n
  69. θ * \theta^{*}
  70. θ * = g ( θ * ) \theta^{*}=g\left(\theta^{*}\right)
  71. ν \nu
  72. σ \sigma
  73. ξ ( θ ) \xi{\left(\theta\right)}
  74. σ = μ 2 1 / 2 ξ ( θ * ) , \sigma=\frac{\mu^{1/2}_{2}}{\sqrt{\xi\left(\theta^{*}\right)}},
  75. ν = ( μ 1 2 + ( ξ ( θ * ) - 2 ) σ 2 ) . \nu=\sqrt{\left(\mu^{{}^{\prime}~{}2}_{1}+\left(\xi\left(\theta^{*}\right)-2% \right)\sigma^{2}\right)}.

Richard_Schroeppel.html

  1. e 2 ln n ln ln n e^{\sqrt{2\ln{n}\ln{\ln{n}}}}
  2. e ln n ln ln n e^{\sqrt{\ln{n}\ln{\ln{n}}}}

Riemann_series_theorem.html

  1. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  2. \ell
  3. { S 1 , S 2 , S 3 , } , S n = k = 1 n a k , \left\{S_{1},\ S_{2},\ S_{3},\dots\right\},\quad S_{n}=\sum_{k=1}^{n}a_{k},
  4. \ell
  5. | S n - | ϵ . \left|S_{n}-\ell\right|\leq\ \epsilon.
  6. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  7. n = 1 | a n | \sum_{n=1}^{\infty}\left|a_{n}\right|
  8. σ \sigma
  9. b b
  10. a a
  11. σ ( a ) = b \sigma(a)=b
  12. x y x\neq y
  13. σ ( x ) σ ( y ) \sigma(x)\neq\sigma(y)
  14. { a 1 , a 2 , a 3 , } \left\{a_{1},\ a_{2},\ a_{3},\dots\right\}
  15. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  16. M M
  17. σ ( n ) \sigma(n)
  18. n = 1 a σ ( n ) = M . \sum_{n=1}^{\infty}a_{\sigma(n)}=M.
  19. σ ( n ) \sigma(n)
  20. n = 1 a σ ( n ) = . \sum_{n=1}^{\infty}a_{\sigma(n)}=\infty.
  21. - -\infty
  22. n = 1 ( - 1 ) n + 1 n \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}
  23. n = 1 | ( - 1 ) n + 1 n | \sum_{n=1}^{\infty}\bigg|\frac{(-1)^{n+1}}{n}\bigg|
  24. 1 - 1 2 + 1 3 - 1 4 + 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots
  25. 1 - 1 2 - 1 4 + 1 3 - 1 6 - 1 8 + 1 5 - 1 10 - 1 12 + 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-% \frac{1}{10}-\frac{1}{12}+\cdots
  26. 1 2 k - 1 - 1 2 ( 2 k - 1 ) - 1 4 k , k = 1 , 2 , . \frac{1}{2k-1}-\frac{1}{2(2k-1)}-\frac{1}{4k},\quad k=1,2,\dots.
  27. 1 2 k - 1 - 1 2 ( 2 k - 1 ) = 1 2 ( 2 k - 1 ) , \frac{1}{2k-1}-\frac{1}{2(2k-1)}=\frac{1}{2(2k-1)},
  28. 1 2 - 1 4 + 1 6 - 1 8 + 1 10 + + 1 2 ( 2 k - 1 ) - 1 2 ( 2 k ) + \frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}+\cdots+\frac{1}{2% (2k-1)}-\frac{1}{2(2k)}+\cdots
  29. = 1 2 ( 1 - 1 2 + 1 3 + ) = 1 2 ln ( 2 ) =\frac{1}{2}\left(1-\frac{1}{2}+\frac{1}{3}+\cdots\right)=\frac{1}{2}\ln(2)
  30. 1 + 1 2 + 1 3 + + 1 n = γ + ln n + o ( 1 ) , 1+{1\over 2}+{1\over 3}+\cdots+{1\over n}=\gamma+\ln n+o(1),
  31. 1 2 + 1 4 + 1 6 + + 1 2 q = 1 2 γ + 1 2 ln q + o ( 1 ) , {1\over 2}+{1\over 4}+{1\over 6}+\cdots+{1\over 2q}={1\over 2}\,\gamma+{1\over 2% }\ln q+o(1),
  32. 1 + 1 3 + 1 5 + + 1 2 p - 1 = 1 2 γ + 1 2 ln p + ln 2 + o ( 1 ) . {1}+{1\over 3}+{1\over 5}+\cdots+{1\over 2p-1}={1\over 2}\,\gamma+{1\over 2}% \ln p+\ln 2+o(1).
  33. 1 + 1 3 + + 1 2 a - 1 - 1 2 - 1 4 - - 1 2 b + 1 2 a + 1 + + 1 4 a - 1 - 1 2 b + 2 - {1}+{1\over 3}+\cdots+{1\over 2a-1}-{1\over 2}-{1\over 4}-\cdots-{1\over 2b}+{% 1\over 2a+1}+\cdots+{1\over 4a-1}-{1\over 2b+2}-\cdots
  34. S ( a + b ) n = 1 2 ln p + ln 2 - 1 2 ln q + o ( 1 ) = 1 2 ln ( a / b ) + ln 2 + o ( 1 ) . S_{(a+b)n}={1\over 2}\ln p+\ln 2-{1\over 2}\ln q+o(1)={1\over 2}\ln(a/b)+\ln 2% +o(1).
  35. 1 2 ln ( a / b ) + ln 2 = ln ( 2 a / b ) . {1\over 2}\ln(a/b)+\ln 2=\ln\bigl(2\sqrt{a/b}\bigr).
  36. ln ( 2 r ) , \ln\bigl(2\sqrt{r}\bigr),
  37. a n + a_{n}^{+}
  38. a n - a_{n}^{-}
  39. a n + = a n + | a n | 2 , a n - = a n - | a n | 2 . a_{n}^{+}=\frac{a_{n}+|a_{n}|}{2},\quad a_{n}^{-}=\frac{a_{n}-|a_{n}|}{2}.
  40. n = 1 a n + \sum_{n=1}^{\infty}a_{n}^{+}
  41. n = 1 a n - \sum_{n=1}^{\infty}a_{n}^{-}
  42. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  43. a n + a_{n}^{+}
  44. n = 1 p - 1 a n + M < n = 1 p a n + . \sum_{n=1}^{p-1}a_{n}^{+}\leq M<\sum_{n=1}^{p}a_{n}^{+}.
  45. a n + a_{n}^{+}
  46. + +\infty
  47. n = 1 p a n + = a σ ( 1 ) + + a σ ( m 1 ) , a σ ( j ) > 0 , σ ( 1 ) < < σ ( m 1 ) = p . \sum_{n=1}^{p}a_{n}^{+}=a_{\sigma(1)}+\cdots+a_{\sigma(m_{1})},\quad a_{\sigma% (j)}>0,\ \ \sigma(1)<\ldots<\sigma(m_{1})=p.
  48. a n - a_{n}^{-}
  49. a n - a_{n}^{-}
  50. - -\infty
  51. n = 1 p a n + + n = 1 q a n - < M n = 1 p a n + + n = 1 q - 1 a n - . \sum_{n=1}^{p}a_{n}^{+}+\sum_{n=1}^{q}a_{n}^{-}<M\leq\sum_{n=1}^{p}a_{n}^{+}+% \sum_{n=1}^{q-1}a_{n}^{-}.
  52. n = 1 p a n + + n = 1 q a n - = a σ ( 1 ) + + a σ ( m 1 ) + a σ ( m 1 + 1 ) + + a σ ( n 1 ) , \sum_{n=1}^{p}a_{n}^{+}+\sum_{n=1}^{q}a_{n}^{-}=a_{\sigma(1)}+\cdots+a_{\sigma% (m_{1})}+a_{\sigma(m_{1}+1)}+\cdots+a_{\sigma(n_{1})},
  53. σ ( m 1 + 1 ) < < σ ( n 1 ) = q . \sigma(m_{1}+1)<\ldots<\sigma(n_{1})=q.
  54. a p j + a_{p_{j}}^{+}
  55. | a q j - | |a_{q_{j}}^{-}|
  56. a p j + a_{p_{j}}^{+}
  57. a q j - a_{q_{j}}^{-}
  58. n = 1 a σ ( n ) = M . \sum_{n=1}^{\infty}a_{\sigma(n)}=M.
  59. A k + 1 = A k { σ ( k + 1 ) } ; S k + 1 = S k + a σ ( k + 1 ) . A_{k+1}=A_{k}\cup\{\sigma(k+1)\}\,;\quad S_{k+1}=S_{k}+a_{\sigma(k+1)}.
  60. 𝐑 { , - } \mathbf{R}\cup\{\infty,-\infty\}
  61. L = { a + t b : t 𝐑 } , a , b 𝐂 , b 0 , L=\{a+tb:t\in\mathbf{R}\},\quad a,b\in\mathbf{C},\ b\neq 0,

Riemann–Lebesgue_lemma.html

  1. f ^ ( z ) := d f ( x ) exp ( - i z x ) d x 0 as | z | . \hat{f}(z):=\int_{\mathbb{R}^{d}}f(x)\exp(-iz\cdot x)\,dx\rightarrow 0\,\text{% as }|z|\rightarrow\infty.
  2. 0 f ( t ) e - t z d t 0 \int_{0}^{\infty}f(t)e^{-tz}\,dt\to 0
  3. f ^ n 0. \hat{f}_{n}\ \to\ 0.
  4. | f ( x ) e - i z x d x | = | 1 i z f ( x ) e - i z x d x | 1 | z | | f ( x ) | d x 0 as z ± . \left|\int f(x)e^{-izx}\,dx\right|=\left|\int\frac{1}{iz}f^{\prime}(x)e^{-izx}% \,dx\right|\leq\frac{1}{|z|}\int|f^{\prime}(x)|dx\rightarrow 0\mbox{ as }~{}z% \rightarrow\pm\infty.
  5. ( 0 , ) (0,\infty)
  6. f f^{\prime}
  7. F ( t ) = G ( t ) / t F(t)=G(t)/t
  8. F ( z ) 0 F(z)\rightarrow 0
  9. | t | |t|\rightarrow\infty
  10. L 1 ( 0 , ) L^{1}(0,\infty)

Riemann–Siegel_theta_function.html

  1. θ ( t ) = arg ( Γ ( 2 i t + 1 4 ) ) - log π 2 t \theta(t)=\arg\left(\Gamma\left(\frac{2it+1}{4}\right)\right)-\frac{\log\pi}{2}t
  2. θ ( 0 ) = 0 \theta(0)=0
  3. θ ( t ) t 2 log t 2 π - t 2 - π 8 + 1 48 t + 7 5760 t 3 + \theta(t)\sim\frac{t}{2}\log\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\frac{1}{% 48t}+\frac{7}{5760t^{3}}+\cdots
  4. t 1 t\gg 1
  5. | t | < 1 / 2 |t|<1/2
  6. θ ( t ) = - t 2 log π + k = 0 ( - 1 ) k ψ ( 2 k ) ( 1 4 ) ( 2 k + 1 ) ! ( t 2 ) 2 k + 1 \theta(t)=-\frac{t}{2}\log\pi+\sum_{k=0}^{\infty}\frac{(-1)^{k}\psi^{(2k)}% \left(\frac{1}{4}\right)}{(2k+1)!}\left(\frac{t}{2}\right)^{2k+1}
  7. ψ ( 2 k ) \psi^{(2k)}
  8. 2 k 2k
  9. s = 1 / 2 + i t s=1/2+it
  10. ± 17.8455995405 \pm 17.8455995405\ldots
  11. ± 6.289835988 \pm 6.289835988\ldots
  12. 3.530972829 3.530972829\ldots
  13. θ ( 0 ) = - ln π + γ + π / 2 + 3 ln 2 2 = - 2.6860917 \theta^{\prime}(0)=-\frac{\ln\pi+\gamma+\pi/2+3\ln 2}{2}=-2.6860917\ldots
  14. log Γ ( z ) = - γ z - log z + n = 1 ( z n - log ( 1 + z n ) ) , \log\Gamma\left(z\right)=-\gamma z-\log z+\sum_{n=1}^{\infty}\left(\frac{z}{n}% -\log\left(1+\frac{z}{n}\right)\right),
  15. ( 2 i t + 1 ) / 4 (2it+1)/4
  16. θ ( t ) = - γ + log π 2 t - arctan 2 t + n = 1 ( t 2 n - arctan ( 2 t 4 n + 1 ) ) . \theta(t)=-\frac{\gamma+\log\pi}{2}t-\arctan 2t+\sum_{n=1}^{\infty}\left(\frac% {t}{2n}-\arctan\left(\frac{2t}{4n+1}\right)\right).
  17. arg z = log z - log z ¯ 2 i and Γ ( z ) ¯ = Γ ( z ¯ ) \arg z=\frac{\log z-\log\bar{z}}{2i}\quad\,\text{and}\quad\overline{\Gamma(z)}% =\Gamma(\bar{z})
  18. θ ( t ) = log Γ ( 2 i t + 1 4 ) - log Γ ( - 2 i t + 1 4 ) 2 i - log π 2 t , \theta(t)=\frac{\log\Gamma\left(\frac{2it+1}{4}\right)-\log\Gamma\left(\frac{-% 2it+1}{4}\right)}{2i}-\frac{\log\pi}{2}t,
  19. - 1 < ( t ) < 1 -1<\Re(t)<1
  20. - 5 < ( t ) < 5 -5<\Re(t)<5
  21. - 40 < ( t ) < 40 -40<\Re(t)<40
  22. ζ ( 1 2 + i t ) = e - i θ ( t ) Z ( t ) , \zeta\left(\frac{1}{2}+it\right)=e^{-i\theta(t)}Z(t),
  23. Z ( t ) = e i θ ( t ) ζ ( 1 2 + i t ) . Z(t)=e^{i\theta(t)}\zeta\left(\frac{1}{2}+it\right).
  24. t t
  25. Z ( t ) Z\left(t\right)
  26. sin ( θ ( t ) ) = 0 \sin\left(\,\theta(t)\,\right)=0
  27. t t
  28. θ ( t ) π \frac{\theta(t)}{\pi}
  29. g n g_{n}
  30. θ ( g n ) = n π . \theta\left(g_{n}\right)=n\pi.
  31. n n
  32. g n g_{n}
  33. θ ( g n ) \theta(g_{n})
  34. θ \theta
  35. Z ( t ) Z\left(t\right)
  36. g n g_{n}
  37. ζ ( 1 2 + i g n ) = cos ( θ ( g n ) ) Z ( g n ) = ( - 1 ) n Z ( g n ) , \zeta\left(\frac{1}{2}+ig_{n}\right)=\cos(\theta(g_{n}))Z(g_{n})=(-1)^{n}Z(g_{% n}),
  38. Z ( t ) Z\left(t\right)
  39. ( - 1 ) n Z ( g n ) > 0 (-1)^{n}\,Z\left(g_{n}\right)>0
  40. N ( T ) N\left(T\right)
  41. N ( T ) = θ ( T ) π + 1 + S ( T ) , N\left(T\right)=\frac{\theta(T)}{\pi}+1+S(T),
  42. S ( T ) S(T)
  43. log T \log T
  44. g n g_{n}
  45. N ( g n ) = n + 1. N\left(g_{n}\right)=n+1.

Riesz_function.html

  1. Riesz ( x ) = - k = 1 ( - x ) k ( k - 1 ) ! ζ ( 2 k ) . {\rm Riesz}(x)=-\sum_{k=1}^{\infty}\frac{(-x)^{k}}{(k-1)!\zeta(2k)}.
  2. F ( x ) = 1 2 Riesz ( 4 π 2 x ) F(x)=\frac{1}{2}{\rm Riesz}(4\pi^{2}x)
  3. x 2 coth x 2 = n = 0 c n x n = 1 + 1 12 x 2 - 1 720 x 4 + \frac{x}{2}\coth\frac{x}{2}=\sum_{n=0}^{\infty}c_{n}x^{n}=1+\frac{1}{12}x^{2}-% \frac{1}{720}x^{4}+\cdots
  4. F ( x ) = k = 1 x k c 2 k ( k - 1 ) ! = 12 x - 720 x 2 + 15120 x 3 - F(x)=\sum_{k=1}^{\infty}\frac{x^{k}}{c_{2k}(k-1)!}=12x-720x^{2}+15120x^{3}-\cdots
  5. x exp ( - x ) x\ \exp(-x)
  6. F ( x ) = k = 1 k k + 1 ¯ x k B 2 k . F(x)=\sum_{k=1}^{\infty}\frac{k^{\overline{k+1}}x^{k}}{B_{2k}}.
  7. n k ¯ n^{\overline{k}}
  8. Riesz ( x ) = O ( x e ) ( as x ) \operatorname{Riesz}(x)=O(x^{e})\qquad(\,\text{as }x\to\infty)
  9. 𝐌 ( Riesz ( z ) ) = 0 Riesz ( z ) z s d z z {\mathbf{M}}({\rm Riesz}(z))=\int_{0}^{\infty}{\rm Riesz(z)}z^{s}\frac{dz}{z}
  10. ( s ) > - 1 \Re(s)>-1
  11. 0 1 Riesz ( z ) z s d z z \int_{0}^{1}{\rm Riesz}(z)z^{s}\frac{dz}{z}
  12. ( s ) < - 1 2 \Re(s)<-\frac{1}{2}
  13. 1 Riesz ( z ) z s d z z \int_{1}^{\infty}{\rm Riesz}(z)z^{s}\frac{dz}{z}
  14. - 1 < ( s ) < - 1 2 -1<\Re(s)<-\frac{1}{2}
  15. Γ ( s + 1 ) ζ ( - 2 s ) = 𝐌 ( Riesz ( z ) ) \frac{\Gamma(s+1)}{\zeta(-2s)}={\mathbf{M}}({\rm Riesz}(z))
  16. Riesz ( z ) = c - i c + i Γ ( s + 1 ) ζ ( - 2 s ) z - s d s {\rm Riesz}(z)=\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s+1)}{\zeta(-2s)}z^{-s% }ds
  17. f ( x ) f(x)
  18. e x p ( - x ) - 1 = 0 d t f ( t ) t ρ ( x / t ) exp(-x)-1=\int_{0}^{\infty}dt\frac{f(t)}{t}\rho(\sqrt{x/t})
  19. ρ ( x ) = x - x \rho(x)=x-\lfloor x\rfloor
  20. × 10 1 7 \times 10^{1}7
  21. F ( z ) F(z)
  22. | z | < 9 |z|<9
  23. Riesz ( x ) = k = 1 ( - 1 ) k + 1 x k ( k - 1 ) ! ζ ( 2 k ) . {\rm Riesz}(x)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}x^{k}}{(k-1)!\zeta(2k)}.
  24. k = 1 ( - 1 ) k + 1 x k ( k - 1 ) ! = x exp ( - x ) \sum_{k=1}^{\infty}\frac{(-1)^{k+1}x^{k}}{(k-1)!}=x\exp(-x)
  25. n = 1 Riesz ( x / n 2 ) = x exp ( - x ) . \ {\sum_{n=1}^{\infty}\rm Riesz(x/n^{2})=x\exp(-x)}.
  26. Riesz ( x ) = x exp ( - x ) - k = 1 ( ζ ( 2 k ) - 1 ) ( ( - 1 ) k + 1 ( k - 1 ) ! ζ ( 2 k ) ) x k {\rm Riesz}(x)=x\exp(-x)-\sum_{k=1}^{\infty}\left(\zeta(2k)-1\right)\left(% \frac{(-1)^{k+1}}{(k-1)!\zeta(2k)}\right)x^{k}
  27. Riesz ( x ) = k = 1 ( - 1 ) k + 1 x k ( k - 1 ) ! ζ ( 2 k ) = k = 1 ( - 1 ) k + 1 x k ( k - 1 ) ! ( n = 1 μ ( n ) n - 2 k ) {\rm Riesz}(x)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}x^{k}}{(k-1)!\zeta(2k)}=\sum% _{k=1}^{\infty}\frac{(-1)^{k+1}x^{k}}{(k-1)!}\left(\sum_{n=1}^{\infty}\mu(n)n^% {-2k}\right)
  28. k = 1 n = 1 ( - 1 ) k + 1 ( x / n 2 ) k ( k - 1 ) ! = x n = 1 μ ( n ) n 2 exp ( - x n 2 ) . \sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{(-1)^{k+1}\left(x/n^{2}\right)^{k}% }{(k-1)!}=x\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}}\exp\left(-\frac{x}{n^{2}}% \right).
  29. Riesz ( x ) = x ( 6 π 2 + n = 1 μ ( n ) n 2 ( exp ( - x n 2 ) - 1 ) ) {\rm Riesz}(x)=x\left(\frac{6}{\pi^{2}}+\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}% }\left(\exp\left(-\frac{x}{n^{2}}\right)-1\right)\right)
  30. Riesz ( x ) = < m t p l > Riesz ( x ) x - x ( n = 1 μ ( n ) n 4 exp ( - x n 2 ) ) {\rm Riesz}^{\prime}(x)=\frac{<}{m}tpl>{{\rm Riesz(x)}}{x}-x\left(\sum_{n=1}^{% \infty}\frac{\mu(n)}{n^{4}}\exp\left(-\frac{x}{n^{2}}\right)\right)
  31. Riesz ( x ) = < m t p l > Riesz ( x ) x + x ( - 90 π 4 + n = 1 μ ( n ) n 4 ( 1 - exp ( - x n 2 ) ) ) . {\rm Riesz}^{\prime}(x)=\frac{<}{m}tpl>{{\rm Riesz(x)}}{x}+x\left(-\frac{90}{% \pi^{4}}+\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{4}}\left(1-\exp\left(-\frac{x}{n^% {2}}\right)\right)\right).
  32. Riesz ( x ) = c o n s t × x 1 / 4 sin ( ϕ - 1 2 γ 1 log ( x ) ) {\rm Riesz}(x)=const\times x^{1/4}\sin\left(\phi-\frac{1}{2}\gamma_{1}\log(x)\right)
  33. γ 1 = 14.13472514... \gamma_{1}=14.13472514...
  34. c o n s t = 7.7750627... × 10 - 5 const=7.7750627...\times 10^{-5}
  35. ϕ = - 0.54916... = ( - 31 , 46447 ) \phi=-0.54916...=(-31,46447^{\circ})

Riesz–Fischer_theorem.html

  1. S N f ( x ) = n = - N N F n e i n x , S_{N}f(x)=\sum_{n=-N}^{N}F_{n}\,\mathrm{e}^{inx},
  2. F n = 1 2 π - π π f ( x ) e - i n x d x , F_{n}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,\mathrm{e}^{-inx}\,\mathrm{d}x,
  3. lim N S N f - f 2 = 0 , \lim_{N\to\infty}\left\|S_{N}f-f\right\|_{2}=0,
  4. 2 \left\|\cdot\right\|_{2}
  5. { a n } \left\{a_{n}\right\}\,
  6. n = - | a n | 2 < , \sum_{n=-\infty}^{\infty}\left|a_{n}\right|^{2}<\infty,
  7. a n a_{n}
  8. x , y = 0 \langle x,y\rangle=0
  9. y A | x , y | 2 x 2 . \sum_{y\in A}|\langle x,y\rangle|^{2}\leq\|x\|^{2}.
  10. y A x , y y \sum_{y\in A}\langle x,y\rangle\,y
  11. x - y B x , y y < ε \|x-\sum_{y\in B}\langle x,y\rangle y\|<\varepsilon
  12. x 2 = y A | x , y | 2 . \|x\|^{2}=\sum_{y\in A}|\langle x,y\rangle|^{2}.
  13. c n c_{n}
  14. f = lim n k = 0 n c k ϕ k f=\lim_{n\to\infty}\sum_{k=0}^{n}c_{k}\phi_{k}
  15. ( f , ϕ n ) (f,\phi_{n})
  16. a n 2 \sum a_{n}^{2}
  17. a b f ( x ) φ n ( x ) d x = a n \int_{a}^{b}f(x)\varphi_{n}(x)\,\mathrm{d}x=a_{n}
  18. G n ( x ) = a x g n ( t ) d t , G_{n}(x)=\int_{a}^{x}g_{n}(t)\,\mathrm{d}t,
  19. u n p < \sum\|u_{n}\|_{p}<\infty
  20. | f - k = 0 n u k | p d μ ( > n | u | ) p d μ 0 as n . \int\left|f-\sum_{k=0}^{n}u_{k}\right|^{p}\,\mathrm{d}\mu\leq\int\left(\sum_{% \ell>n}|u_{\ell}|\right)^{p}\,\mathrm{d}\mu\rightarrow 0\,\text{ as }n% \rightarrow\infty.
  21. | k = 0 n u k | p k = 0 n | u k | p when p < 1 \left|\sum_{k=0}^{n}u_{k}\right|^{p}\leq\sum_{k=0}^{n}|u_{k}|^{p}\,\text{ when% }p<1

Rigid_rotor.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. μ = m 1 m 2 m 1 + m 2 \mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}
  4. R R
  5. R R
  6. θ \theta\,
  7. φ \varphi\,
  8. R R
  9. T T
  10. 2 T = μ R 2 [ θ ˙ 2 + ( φ ˙ sin θ ) 2 ] = μ R 2 ( θ ˙ φ ˙ ) ( 1 0 0 sin 2 θ ) ( θ ˙ φ ˙ ) = μ ( θ ˙ φ ˙ ) ( h θ 2 0 0 h φ 2 ) ( θ ˙ φ ˙ ) , 2T=\mu R^{2}\big[\dot{\theta}^{2}+(\dot{\varphi}\,\sin\theta)^{2}\big]=\mu R^{% 2}\big(\dot{\theta}\;\;\dot{\varphi}\Big)\begin{pmatrix}1&0\\ 0&\sin^{2}\theta\\ \end{pmatrix}\begin{pmatrix}\dot{\theta}\\ \dot{\varphi}\end{pmatrix}=\mu\Big(\dot{\theta}\;\;\dot{\varphi}\Big)\begin{% pmatrix}h_{\theta}^{2}&0\\ 0&h_{\varphi}^{2}\\ \end{pmatrix}\begin{pmatrix}\dot{\theta}\\ \dot{\varphi}\end{pmatrix},
  11. h θ = R h_{\theta}=R\,
  12. h φ = R sin θ h_{\varphi}=R\sin\theta\,
  13. R R
  14. 2 = 1 h θ h φ [ θ h φ h θ θ + φ h θ h φ φ ] = 1 R 2 [ 1 sin θ θ sin θ θ + 1 sin 2 θ 2 φ 2 ] . \nabla^{2}=\frac{1}{h_{\theta}h_{\varphi}}\left[\frac{\partial}{\partial\theta% }\frac{h_{\varphi}}{h_{\theta}}\frac{\partial}{\partial\theta}+\frac{\partial}% {\partial\varphi}\frac{h_{\theta}}{h_{\varphi}}\frac{\partial}{\partial\varphi% }\right]=\frac{1}{R^{2}}\left[\frac{1}{\sin\theta}\frac{\partial}{\partial% \theta}\sin\theta\frac{\partial}{\partial\theta}+\frac{1}{\sin^{2}\theta}\frac% {\partial^{2}}{\partial\varphi^{2}}\right].
  15. H = 1 2 μ R 2 [ p θ 2 + p φ 2 sin 2 θ ] . H=\frac{1}{2\mu R^{2}}\left[p^{2}_{\theta}+\frac{p^{2}_{\varphi}}{\sin^{2}% \theta}\right].
  16. I I
  17. I = μ R 2 I=\mu R^{2}
  18. μ \mu
  19. R R
  20. H ^ Ψ = E Ψ \hat{H}\Psi=E\Psi
  21. Ψ \Psi
  22. H ^ \hat{H}
  23. H ^ = - 2 2 μ 2 \hat{H}=-\frac{\hbar^{2}}{2\mu}\nabla^{2}
  24. \hbar
  25. 2 \nabla^{2}
  26. H ^ = - 2 2 I [ 1 sin θ θ ( sin θ θ ) + 1 sin 2 θ 2 φ 2 ] \hat{H}=-\frac{\hbar^{2}}{2I}\left[{1\over\sin\theta}{\partial\over\partial% \theta}\left(\sin\theta{\partial\over\partial\theta}\right)+{1\over{\sin^{2}% \theta}}{\partial^{2}\over\partial\varphi^{2}}\right]
  27. H ^ Y m ( θ , φ ) = 2 2 I ( + 1 ) Y m ( θ , φ ) . \hat{H}Y_{\ell}^{m}(\theta,\varphi)=\frac{\hbar^{2}}{2I}\ell(\ell+1)Y_{\ell}^{% m}(\theta,\varphi).
  28. Y m ( θ , φ ) Y_{\ell}^{m}(\theta,\varphi)
  29. m m\,
  30. E = 2 2 I ( + 1 ) E_{\ell}={\hbar^{2}\over 2I}\ell\left(\ell+1\right)
  31. 2 + 1 2\ell+1
  32. \ell\,
  33. m = - , - + 1 , , m=-\ell,-\ell+1,\dots,\ell
  34. E = B ( + 1 ) with B 2 2 I . E_{\ell}=B\;\ell\left(\ell+1\right)\quad\textrm{with}\quad B\equiv\frac{\hbar^% {2}}{2I}.
  35. B ¯ B h c = h 8 π 2 c I , \bar{B}\equiv\frac{B}{hc}=\frac{h}{8\pi^{2}cI},
  36. B ¯ \bar{B}
  37. B ¯ ( R ) \bar{B}(R)
  38. R R
  39. B e = B ¯ ( R e ) B_{e}=\bar{B}(R_{e})
  40. R e R_{e}
  41. R R
  42. \ell
  43. 2 B ¯ 2\bar{B}
  44. Δ l = ± 1 \Delta l=\pm 1
  45. ψ 2 | μ z | ψ 1 = ( μ z ) 21 = ψ 2 * μ z ψ 1 d τ . \langle\psi_{2}|\mu_{z}|\psi_{1}\rangle=\left(\mu_{z}\right)_{21}=\int\psi_{2}% ^{*}\mu_{z}\psi_{1}\,\mathrm{d}\tau.
  46. ( μ z ) l , m ; l , m = μ 0 2 π d ϕ 0 π Y l m ( θ , ϕ ) * cos θ Y l m ( θ , ϕ ) d cos θ . \left(\mu_{z}\right)_{l,m;l^{\prime},m^{\prime}}=\mu\int_{0}^{2\pi}\mathrm{d}% \phi\int_{0}^{\pi}Y_{l^{\prime}}^{m^{\prime}}\left(\theta,\phi\right)^{*}\cos% \theta\,Y_{l}^{m}\,\left(\theta,\phi\right)\;\mathrm{d}\cos\theta.
  47. μ cos θ \mu\cos\theta\,
  48. μ \mu
  49. Y l m ( θ , ϕ ) Y_{l}^{m}\,\left(\theta,\phi\right)
  50. l l
  51. m m
  52. l l^{\prime}
  53. m m^{\prime}
  54. Δ m = 0 and Δ l = ± 1 \Delta m=0\quad\hbox{and}\quad\Delta l=\pm 1
  55. R R
  56. l l
  57. D ¯ \bar{D}
  58. E ¯ l = E l h c = B ¯ l ( l + 1 ) - D ¯ l 2 ( l + 1 ) 2 \bar{E}_{l}={E_{l}\over hc}=\bar{B}l\left(l+1\right)-\bar{D}l^{2}\left(l+1% \right)^{2}
  59. D ¯ = 4 B ¯ 3 s y m b o l ω ¯ 2 \bar{D}={4\bar{B}^{3}\over\bar{symbol\omega}^{2}}
  60. s y m b o l ω ¯ \bar{symbol\omega}
  61. s y m b o l ω ¯ = 1 2 π c k μ \bar{symbol\omega}={1\over 2\pi c}\sqrt{k\over\mu}
  62. α \alpha\,
  63. y y
  64. y y^{\prime}
  65. β \beta\,
  66. y y^{\prime}
  67. α \alpha\,
  68. φ \varphi\,
  69. β \beta\,
  70. θ \theta\,
  71. β \beta\,
  72. α \alpha\,
  73. γ \gamma\,
  74. z ′′ - y - z z^{\prime\prime}-y^{\prime}-z
  75. z - y - z z-y-z
  76. 𝐑 ( α , β , γ ) = ( cos α - sin α 0 sin α cos α 0 0 0 1 ) ( cos β 0 sin β 0 1 0 - sin β 0 cos β ) ( cos γ - sin γ 0 sin γ cos γ 0 0 0 1 ) \mathbf{R}(\alpha,\beta,\gamma)=\begin{pmatrix}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\\ \end{pmatrix}\begin{pmatrix}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{pmatrix}
  77. 𝐫 ( 0 ) \mathbf{r}(0)
  78. 𝒫 \mathcal{P}
  79. 𝐫 ( 0 ) \mathbf{r}(0)
  80. 𝒫 \mathcal{P}
  81. 𝐫 ( 0 ) \mathbf{r}(0)
  82. 𝒫 \mathcal{P}
  83. 𝒫 \mathcal{P}
  84. 𝒫 \mathcal{P}
  85. 𝐫 ( α , β , γ ) = 𝐑 ( α , β , γ ) 𝐫 ( 0 ) . \mathbf{r}(\alpha,\beta,\gamma)=\mathbf{R}(\alpha,\beta,\gamma)\mathbf{r}(0).
  86. 𝒫 \mathcal{P}
  87. 𝐑 ( α , β , γ ) ( 0 0 r ) = ( r cos α sin β r sin α sin β r cos β ) , \mathbf{R}(\alpha,\beta,\gamma)\begin{pmatrix}0\\ 0\\ r\\ \end{pmatrix}=\begin{pmatrix}r\cos\alpha\sin\beta\\ r\sin\alpha\sin\beta\\ r\cos\beta\\ \end{pmatrix},
  88. 𝐫 ( 0 ) \mathbf{r}(0)
  89. 𝐈 ( t ) \mathbf{I}(t)
  90. 𝐑 ( α , β , γ ) - 1 𝐈 ( t ) 𝐑 ( α , β , γ ) = 𝐈 ( 0 ) with 𝐈 ( 0 ) = ( I 1 0 0 0 I 2 0 0 0 I 3 ) , \mathbf{R}(\alpha,\beta,\gamma)^{-1}\;\mathbf{I}(t)\;\mathbf{R}(\alpha,\beta,% \gamma)=\mathbf{I}(0)\quad\hbox{with}\quad\mathbf{I}(0)=\begin{pmatrix}I_{1}&0% &0\\ 0&I_{2}&0\\ 0&0&I_{3}\\ \end{pmatrix},
  91. 𝐈 ( t ) \mathbf{I}(t)
  92. t = 0 t=0
  93. t = 0 t=0
  94. T = 1 2 [ I 1 ω x 2 + I 2 ω y 2 + I 3 ω z 2 ] T=\frac{1}{2}\left[I_{1}\omega_{x}^{2}+I_{2}\omega_{y}^{2}+I_{3}\omega_{z}^{2}\right]
  95. ( ω x ω y ω z ) = ( - sin β cos γ sin γ 0 sin β sin γ cos γ 0 cos β 0 1 ) ( α ˙ β ˙ γ ˙ ) . \begin{pmatrix}\omega_{x}\\ \omega_{y}\\ \omega_{z}\\ \end{pmatrix}=\begin{pmatrix}-\sin\beta\cos\gamma&\sin\gamma&0\\ \sin\beta\sin\gamma&\cos\gamma&0\\ \cos\beta&0&1\\ \end{pmatrix}\begin{pmatrix}\dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{pmatrix}.
  96. s y m b o l ω = ( ω x , ω y , ω z ) symbol{\omega}=(\omega_{x},\omega_{y},\omega_{z})
  97. s y m b o l ω symbol{\omega}
  98. s y m b o l ω symbol{\omega}
  99. 2 T = ( α ˙ β ˙ γ ˙ ) 𝐠 ( α ˙ β ˙ γ ˙ ) , 2T=\begin{pmatrix}\dot{\alpha}&\dot{\beta}&\dot{\gamma}\end{pmatrix}\;\mathbf{% g}\;\begin{pmatrix}\dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{pmatrix},
  100. 𝐠 \mathbf{g}
  101. 𝐠 = ( I 1 sin 2 β cos 2 γ + I 2 sin 2 β sin 2 γ + I 3 cos 2 β ( I 2 - I 1 ) sin β sin γ cos γ I 3 cos β ( I 2 - I 1 ) sin β sin γ cos γ I 1 sin 2 γ + I 2 cos 2 γ 0 I 3 cos β 0 I 3 ) . \mathbf{g}=\begin{pmatrix}I_{1}\sin^{2}\beta\cos^{2}\gamma+I_{2}\sin^{2}\beta% \sin^{2}\gamma+I_{3}\cos^{2}\beta&(I_{2}-I_{1})\sin\beta\sin\gamma\cos\gamma&I% _{3}\cos\beta\\ (I_{2}-I_{1})\sin\beta\sin\gamma\cos\gamma&I_{1}\sin^{2}\gamma+I_{2}\cos^{2}% \gamma&0\\ I_{3}\cos\beta&0&I_{3}\\ \end{pmatrix}.
  102. 𝐋 \mathbf{L}
  103. L i \quad L_{i}
  104. 𝐋 = 𝐈 ( 0 ) s y m b o l ω or L i = T ω i , i = x , y , z . \mathbf{L}=\mathbf{I}(0)\;symbol{\omega}\quad\hbox{or}\quad L_{i}=\frac{% \partial T}{\partial\omega_{i}},\;\;i=x,\,y,\,z.
  105. L i \quad L_{i}
  106. 𝐋 \mathbf{L}
  107. T = 1 2 [ L x 2 I 1 + L y 2 I 2 + L z 2 I 3 ] . T=\frac{1}{2}\left[\frac{L_{x}^{2}}{I_{1}}+\frac{L_{y}^{2}}{I_{2}}+\frac{L_{z}% ^{2}}{I_{3}}\right].
  108. ( p α p β p γ ) = def ( T / α ˙ T / β ˙ T / γ ˙ ) = 𝐠 ( α ˙ β ˙ γ ˙ ) , \begin{pmatrix}p_{\alpha}\\ p_{\beta}\\ p_{\gamma}\\ \end{pmatrix}\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\partial T/{\partial% \dot{\alpha}}\\ \partial T/{\partial\dot{\beta}}\\ \partial T/{\partial\dot{\gamma}}\\ \end{pmatrix}=\mathbf{g}\begin{pmatrix}\;\,\dot{\alpha}\\ \dot{\beta}\\ \dot{\gamma}\\ \end{pmatrix},
  109. 𝐠 \mathbf{g}
  110. 2 T = ( p α p β p γ ) 𝐠 - 1 ( p α p β p γ ) , 2T=\begin{pmatrix}p_{\alpha}&p_{\beta}&p_{\gamma}\end{pmatrix}\;\mathbf{g}^{-1% }\;\begin{pmatrix}p_{\alpha}\\ p_{\beta}\\ p_{\gamma}\\ \end{pmatrix},
  111. sin 2 β 𝐠 - 1 = {\scriptstyle\sin^{2}\beta}\;\;\mathbf{g}^{-1}=
  112. ( cos 2 γ I 1 + sin 2 γ I 2 ( 1 I 2 - 1 I 1 ) sin β sin γ cos γ - cos β cos 2 γ I 1 - cos β sin 2 γ I 2 ( 1 I 2 - 1 I 1 ) sin β sin γ cos γ sin 2 β sin 2 γ I 1 + sin 2 β cos 2 γ I 2 ( 1 I 1 - 1 I 2 ) sin β cos β sin γ cos γ - cos β cos 2 γ I 1 - cos β sin 2 γ I 2 ( 1 I 1 - 1 I 2 ) sin β cos β sin γ cos γ cos 2 β cos 2 γ I 1 + cos 2 β sin 2 γ I 2 + sin 2 β I 3 ) . \begin{pmatrix}\frac{\cos^{2}\gamma}{I_{1}}+\frac{\sin^{2}\gamma}{I_{2}}&\left% (\frac{1}{I_{2}}-\frac{1}{I_{1}}\right){\scriptstyle\sin\beta\sin\gamma\cos% \gamma}&-\frac{\cos\beta\cos^{2}\gamma}{I_{1}}-\frac{\cos\beta\sin^{2}\gamma}{% I_{2}}\\ \left(\frac{1}{I_{2}}-\frac{1}{I_{1}}\right){\scriptstyle\sin\beta\sin\gamma% \cos\gamma}&\frac{\sin^{2}\beta\sin^{2}\gamma}{I_{1}}+\frac{\sin^{2}\beta\cos^% {2}\gamma}{I_{2}}&\left(\frac{1}{I_{1}}-\frac{1}{I_{2}}\right){\scriptstyle% \sin\beta\cos\beta\sin\gamma\cos\gamma}\\ -\frac{\cos\beta\cos^{2}\gamma}{I_{1}}-\frac{\cos\beta\sin^{2}\gamma}{I_{2}}&% \left(\frac{1}{I_{1}}-\frac{1}{I_{2}}\right){\scriptstyle\sin\beta\cos\beta% \sin\gamma\cos\gamma}&\frac{\cos^{2}\beta\cos^{2}\gamma}{I_{1}}+\frac{\cos^{2}% \beta\sin^{2}\gamma}{I_{2}}+\frac{\sin^{2}\beta}{I_{3}}\\ \end{pmatrix}.
  113. - 2 -\hbar^{2}
  114. T = 1 2 I 1 sin 2 β ( ( p α - p γ cos β ) cos γ - p β sin β sin γ ) 2 + 1 2 I 2 sin 2 β ( ( p α - p γ cos β ) sin γ + p β sin β cos γ ) 2 + p γ 2 2 I 3 . \begin{array}[]{lcl}T&=&\frac{1}{2I_{1}\sin^{2}\beta}\left((p_{\alpha}-p_{% \gamma}\cos\beta)\cos\gamma-p_{\beta}\sin\beta\sin\gamma\right)^{2}\\ &&+\frac{1}{2I_{2}\sin^{2}\beta}\left((p_{\alpha}-p_{\gamma}\cos\beta)\sin% \gamma+p_{\beta}\sin\beta\cos\gamma\right)^{2}+\frac{p_{\gamma}^{2}}{2I_{3}}.% \\ \end{array}
  115. p α - i α p_{\alpha}\longrightarrow-i\hbar\frac{\partial}{\partial\alpha}
  116. p β p_{\beta}
  117. p γ p_{\gamma}
  118. p α p_{\alpha}
  119. \hbar
  120. 𝒫 ^ i \hat{\mathcal{P}}_{i}
  121. p β p_{\beta}
  122. cos β \cos\beta
  123. sin β \sin\beta
  124. - 1 2 2 -\tfrac{1}{2}\hbar^{2}
  125. q 1 , q 2 , q 3 α , β , γ q^{1},\,q^{2},\,q^{3}\equiv\alpha,\,\beta,\,\gamma
  126. H ^ = - 2 2 | g | - 1 / 2 q i | g | 1 / 2 g i j q j , \hat{H}=-\tfrac{\hbar^{2}}{2}\;|g|^{-1/2}\frac{\partial}{\partial q^{i}}|g|^{1% /2}g^{ij}\frac{\partial}{\partial q^{j}},
  127. | g | |g|
  128. | g | = I 1 I 2 I 3 sin 2 β and g i j = ( 𝐠 - 1 ) i j . |g|=I_{1}\,I_{2}\,I_{3}\,\sin^{2}\beta\quad\hbox{and}\quad g^{ij}=(\mathbf{g}^% {-1})_{ij}.
  129. H ^ \hat{H}
  130. H ^ = 1 2 [ 𝒫 x 2 I 1 + 𝒫 y 2 I 2 + 𝒫 z 2 I 3 ] . \hat{H}=\tfrac{1}{2}\left[\frac{\mathcal{P}_{x}^{2}}{I_{1}}+\frac{\mathcal{P}_% {y}^{2}}{I_{2}}+\frac{\mathcal{P}_{z}^{2}}{I_{3}}\right].
  131. 𝒫 ^ i \hat{\mathcal{P}}_{i}
  132. 𝒫 2 D m m j ( α , β , γ ) * = 2 j ( j + 1 ) D m m j ( α , β , γ ) * with 𝒫 2 = 𝒫 x 2 + 𝒫 y 2 + 𝒫 z 2 , \mathcal{P}^{2}\,D^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)^{*}=\hbar^{2}j(j+1)D% ^{j}_{m^{\prime}m}(\alpha,\beta,\gamma)^{*}\quad\hbox{with}\quad\mathcal{P}^{2% }=\mathcal{P}^{2}_{x}+\mathcal{P}_{y}^{2}+\mathcal{P}_{z}^{2},
  133. I = I 1 = I 2 = I 3 I=I_{1}=I_{2}=I_{3}
  134. ( 2 j + 1 ) 2 (2j+1)^{2}
  135. 2 j ( j + 1 ) 2 I \tfrac{\hbar^{2}j(j+1)}{2I}
  136. I 1 = I 2 I_{1}=I_{2}
  137. I 3 < I 1 = I 2 I_{3}<I_{1}=I_{2}
  138. H ^ = 1 2 [ 𝒫 2 I 1 + 𝒫 z 2 ( 1 I 3 - 1 I 1 ) ] , \hat{H}=\tfrac{1}{2}\left[\frac{\mathcal{P}^{2}}{I_{1}}+\mathcal{P}_{z}^{2}% \Big(\frac{1}{I_{3}}-\frac{1}{I_{1}}\Big)\right],
  139. 𝒫 z 2 D m k j ( α , β , γ ) * = 2 k 2 D m k j ( α , β , γ ) * . \mathcal{P}_{z}^{2}\,D^{j}_{mk}(\alpha,\beta,\gamma)^{*}=\hbar^{2}k^{2}\,D^{j}% _{mk}(\alpha,\beta,\gamma)^{*}.
  140. H ^ D m k j ( α , β , γ ) * = E j k D m k j ( α , β , γ ) * with E j k / 2 = j ( j + 1 ) 2 I 1 + k 2 ( 1 2 I 3 - 1 2 I 1 ) . \hat{H}\,D^{j}_{mk}(\alpha,\beta,\gamma)^{*}=E_{jk}D^{j}_{mk}(\alpha,\beta,% \gamma)^{*}\quad\hbox{with}\quad E_{jk}/\hbar^{2}=\frac{j(j+1)}{2I_{1}}+k^{2}% \left(\frac{1}{2I_{3}}-\frac{1}{2I_{1}}\right).
  141. E j 0 E_{j0}
  142. 2 j + 1 2j+1
  143. m = - j , - j + 1 , , j m=-j,-j+1,\dots,j
  144. 2 ( 2 j + 1 ) 2(2j+1)
  145. I 1 I 2 I 3 I_{1}\neq I_{2}\neq I_{3}

Rijndael_S-box.html

  1. [ 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 ] [ x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 ] + [ 1 1 0 0 0 1 1 0 ] \begin{bmatrix}1&0&0&0&1&1&1&1\\ 1&1&0&0&0&1&1&1\\ 1&1&1&0&0&0&1&1\\ 1&1&1&1&0&0&0&1\\ 1&1&1&1&1&0&0&0\\ 0&1&1&1&1&1&0&0\\ 0&0&1&1&1&1&1&0\\ 0&0&0&1&1&1&1&1\end{bmatrix}\begin{bmatrix}x_{0}\\ x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\\ x_{6}\\ x_{7}\end{bmatrix}+\begin{bmatrix}1\\ 1\\ 0\\ 0\\ 0\\ 1\\ 1\\ 0\end{bmatrix}
  2. [ 0 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 ] [ x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 ] + [ 1 0 1 0 0 0 0 0 ] \begin{bmatrix}0&0&1&0&0&1&0&1\\ 1&0&0&1&0&0&1&0\\ 0&1&0&0&1&0&0&1\\ 1&0&1&0&0&1&0&0\\ 0&1&0&1&0&0&1&0\\ 0&0&1&0&1&0&0&1\\ 1&0&0&1&0&1&0&0\\ 0&1&0&0&1&0&1&0\end{bmatrix}\begin{bmatrix}x_{0}\\ x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\\ x_{6}\\ x_{7}\end{bmatrix}+\begin{bmatrix}1\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{bmatrix}
  3. b i = b i b ( i + 4 ) mod 8 b ( i + 5 ) mod 8 b ( i + 6 ) mod 8 b ( i + 7 ) mod 8 c i b^{\prime}_{i}=b_{i}\oplus b_{(i+4)\operatorname{mod}8}\oplus b_{(i+5)% \operatorname{mod}8}\oplus b_{(i+6)\operatorname{mod}8}\oplus b_{(i+7)% \operatorname{mod}8}\oplus c_{i}
  4. b o u t = ( b i n × 31 d ) mod 257 d 99 d b_{out}=(b_{in}\times 31_{d})\operatorname{mod}257_{d}\oplus 99d

Rindler_coordinates.html

  1. d s 2 = - d T 2 + d X 2 + d Y 2 + d Z 2 , T , X , Y , Z . ds^{2}=-dT^{2}+dX^{2}+dY^{2}+dZ^{2},\;\forall T,X,Y,Z\,.
  2. 0 < X < , - X < T < X \scriptstyle 0\,<\,X\,<\,\infty,\;-X\,<\,T\,<\,X
  3. t = 1 g arctanh ( T X ) , x = X 2 - T 2 , y = Y , z = Z . t=\frac{1}{g}\operatorname{arctanh}\left(\frac{T}{X}\right),\;x=\sqrt{X^{2}-T^% {2}},\;y=Y,\;z=Z\,.
  4. T = x sinh ( g t ) , X = x cosh ( g t ) , Y = y , Z = z . T=x\,\sinh(gt),\;X=x\,\cosh(gt),\;Y=y,\;Z=z\,.
  5. d s 2 = - g 2 x 2 d t 2 + d x 2 + d y 2 + d z 2 , x > 0 , t , y , z . ds^{2}=-g^{2}x^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2},\;\forall x>0,\forall t,y,z\,.
  6. t = 1 g arctanh ( T X ) , x = X 2 - T 2 , y = Y , z = Z t=\frac{1}{g}\operatorname{arctanh}\left(\frac{T}{X}\right),\;x=\sqrt{X^{2}-T^% {2}},\;y=Y,\;z=Z
  7. t = c g arctanh ( c T X ) X c T c 2 T g X X c 2 T g t T t c 2 g \begin{aligned}\displaystyle t&\displaystyle=\frac{c}{g}\operatorname{arctanh}% \left(\frac{cT}{X}\right)\;\overset{X\,\gg\,cT}{\approx}\;\frac{c^{2}T}{gX}\\ \displaystyle X&\displaystyle\approx\frac{c^{2}T}{gt}\;\overset{T\,\approx\,t}% {\approx}\;\frac{c^{2}}{g}\end{aligned}
  8. d σ 0 = - x d t , d σ 1 = d x , d σ 2 = d y , d σ 3 = d z d\sigma^{0}=-x\,dt,\;\;d\sigma^{1}=dx,\;\;d\sigma^{2}=dy,\;\;d\sigma^{3}=dz
  9. e 0 = 1 x t , e 1 = x , e 2 = y , e 3 = z \vec{e}_{0}=\frac{1}{x}\partial_{t},\;\;\vec{e}_{1}=\partial_{x},\;\;\vec{e}_{% 2}=\partial_{y},\;\;\vec{e}_{3}=\partial_{z}
  10. e 0 \scriptstyle\vec{e}_{0}
  11. x = x 0 , y = y 0 , z = z 0 \scriptstyle x\;=\;x_{0},\;y\;=\;y_{0},\;z\;=\;z_{0}
  12. e 0 e 0 = 1 x e 1 \nabla_{\vec{e}_{0}}\vec{e}_{0}=\frac{1}{x}\vec{e}_{1}
  13. x \scriptstyle\partial_{x}
  14. t = t 0 \scriptstyle t\;=\;t_{0}
  15. T = X = 0 \scriptstyle T\;=\;X\;=\;0
  16. d t = 0 \scriptstyle dt\;=\;0
  17. d σ 2 = d x 2 + d y 2 + d z 2 , x > 0 , y , z \scriptstyle d\sigma^{2}\;=\;dx^{2}\,+\,dy^{2}\,+\,dz^{2},\;\forall x\,>\,0,\;% \forall y,\,z
  18. f 0 = T , f 1 = X , f 2 = Y , f 3 = Z \vec{f}_{0}=\partial_{T},\;\vec{f}_{1}=\partial_{X},\;\vec{f}_{2}=\partial_{Y}% ,\;\vec{f}_{3}=\partial_{Z}
  19. f 0 = 1 x cosh ( t ) t - sinh ( t ) x f 1 = - 1 x sinh ( t ) t + cosh ( t ) x f 2 = y , f 3 = z \begin{aligned}\displaystyle\vec{f}_{0}&\displaystyle=\frac{1}{x}\cosh(t)\,% \partial_{t}-\sinh(t)\,\partial_{x}\\ \displaystyle\vec{f}_{1}&\displaystyle=-\frac{1}{x}\sinh(t)\,\partial_{t}+% \cosh(t)\,\partial_{x}\\ \displaystyle\vec{f}_{2}&\displaystyle=\partial_{y},\;\vec{f}_{3}=\partial_{z}% \end{aligned}
  20. f 0 \scriptstyle\vec{f}_{0}
  21. f 0 f 0 = 0 \scriptstyle\nabla_{\vec{f}_{0}}\vec{f}_{0}\;=\;0
  22. x = 0 \scriptstyle x\;=\;0
  23. t = t 0 , x = x 0 , y = y 0 , z = z 0 \scriptstyle t\;=\;t_{0},\;x\;=\;x_{0},\;y\;=\;y_{0},\;z\;=\;z_{0}
  24. t = arctanh ( s x 0 ) , - x 0 < s < x 0 x = x 0 2 - s 2 , - x 0 < s < x 0 y = y 0 z = z 0 \begin{aligned}\displaystyle t&\displaystyle=\operatorname{arctanh}\left(\frac% {s}{x_{0}}\right),\;-x_{0}<s<x_{0}\\ \displaystyle x&\displaystyle=\sqrt{x_{0}^{2}-s^{2}},\;-x_{0}<s<x_{0}\\ \displaystyle y&\displaystyle=y_{0}\\ \displaystyle z&\displaystyle=z_{0}\end{aligned}
  25. s \scriptstyle s
  26. x 0 = 1 \scriptstyle x_{0}\;=\;1
  27. s { - 1 2 , 0 , 1 2 } \scriptstyle s\,\in\,\left\{-\frac{1}{2},\;0,\;\frac{1}{2}\right\}
  28. t ¨ + 2 x x ˙ t ˙ = 0 , x ¨ + x t ˙ 2 = 0 , y ¨ = 0 , z ¨ = 0 \ddot{t}+\frac{2}{x}\,\dot{x}\,\dot{t}=0,\;\ddot{x}+x\,\dot{t}^{2}=0,\;\ddot{y% }=0,\;\ddot{z}=0
  29. t ˙ = E x 2 , y ˙ = P , z ˙ = Q \dot{t}=\frac{E}{x^{2}},\;\;\dot{y}=P,\;\;\dot{z}=Q
  30. ϵ = - x 2 t ˙ 2 + x ˙ 2 + y ˙ 2 + z ˙ 2 \scriptstyle\epsilon\;=\;-x^{2}\,\dot{t}^{2}\,+\,\dot{x}^{2}\,+\,\dot{y}^{2}\,% +\,\dot{z}^{2}
  31. ϵ { - 1 , 0 , 1 } \scriptstyle\epsilon\;\in\;\left\{-1,\,0,\,1\right\}
  32. x ˙ 2 = ( ϵ + E 2 x 2 ) - P 2 - Q 2 \dot{x}^{2}=\left(\epsilon+\frac{E^{2}}{x^{2}}\right)-P^{2}-Q^{2}
  33. E 2 x 2 - P 2 - Q 2 \scriptstyle\frac{E^{2}}{x^{2}}\,-\,P^{2}\,-\,Q^{2}
  34. E \scriptstyle E
  35. 0 < x < E P 2 + Q 2 \scriptstyle 0\,<\,x\,<\,\frac{E}{\sqrt{P^{2}\,+\,Q^{2}}}
  36. t - t 0 = arctanh ( 1 E [ s ( P 2 + Q 2 ) - E 2 - ( P 2 + Q 2 ) x 0 2 ] ) + arctanh ( 1 E E 2 - ( P 2 + Q 2 ) x 0 2 ) x = x 0 2 + 2 s E 2 - ( P 2 + Q 2 ) x 0 2 - s 2 ( P 2 + Q 2 ) y - y 0 = P s ; z - z 0 = Q s \begin{aligned}\displaystyle t-t_{0}&\displaystyle=\operatorname{arctanh}\left% (\frac{1}{E}\left[s\left(P^{2}+Q^{2}\right)-\sqrt{E^{2}-\left(P^{2}+Q^{2}% \right)x_{0}^{2}}\right]\right)+\\ &\displaystyle\quad\quad\operatorname{arctanh}\left(\frac{1}{E}\sqrt{E^{2}-(P^% {2}+Q^{2})x_{0}^{2}}\right)\\ \displaystyle x&\displaystyle=\sqrt{x_{0}^{2}+2s\sqrt{E^{2}-(P^{2}+Q^{2})x_{0}% ^{2}}-s^{2}(P^{2}+Q^{2})}\\ \displaystyle y-y_{0}&\displaystyle=Ps;\;\;z-z_{0}=Qs\end{aligned}
  37. t = 0 \scriptstyle t\,=\,0
  38. d s 2 = g 00 d t 2 + g j k d x j d x k , j , k { 1 , 2 , 3 } ds^{2}=g_{00}\,dt^{2}+g_{jk}\,dx^{j}\,dx^{k},\;\;j,\;k\in\{1,2,3\}
  39. t = 0 \scriptstyle t\;=\;0
  40. d ρ 2 = 1 - g 00 ( g j k d x j d x k ) d\rho^{2}=\frac{1}{-g_{00}}\left(g_{jk}\,dx^{j}\,dx^{k}\right)
  41. t = 0 \scriptstyle t\;=\;0
  42. t \scriptstyle\partial_{t}
  43. d ρ 2 = 1 x 2 ( d x 2 + d y 2 + d z 2 ) , x > 0 , y , z d\rho^{2}=\frac{1}{x^{2}}\left(dx^{2}+dy^{2}+dz^{2}\right),\;\;\forall x>0,\;% \;\forall y,z
  44. t , y , z , - z y + y z \partial_{t},\;\;\partial_{y},\;\;\partial_{z},\;\;-z\,\partial_{y}+y\,% \partial_{z}
  45. exp ( ± t ) ( y x t ± [ y x - x y ] ) \displaystyle\exp(\pm t)\,\left(\frac{y}{x}\,\partial_{t}\pm\left[y\,\partial_% {x}-x\,\partial_{y}\right]\right)
  46. T \scriptstyle\partial_{T}
  47. t = t 0 \scriptstyle t\;=\;t_{0}
  48. x = x 0 , y = 0 , z = 0 \scriptstyle x\;=\;x_{0},\;y\;=\;0,\;z\;=\;0
  49. x = x 0 + h , y = 0 , z = 0 \scriptstyle x\;=\;x_{0}\,+\,h,\;y\;=\;0,\;\;z\;=\;0
  50. d y = d z = 0 \scriptstyle dy\;=\;dz\;=\;0
  51. t - t 0 = log ( x / x 0 ) t-t_{0}=\log(x/x_{0})
  52. x 0 log ( 1 + h x 0 ) = h - h 2 2 x 0 + O ( h 3 ) x_{0}\,\log\left(1+\frac{h}{x_{0}}\right)=h-\frac{h^{2}}{2\,x_{0}}+O\left(h^{3% }\right)
  53. h + 1 x 0 + O ( h 3 ) \scriptstyle h\,+\,\frac{1}{x_{0}}\,+\,O\left(h^{3}\right)
  54. h > 0 \scriptstyle h\,>\,0

Ring-imaging_Cherenkov_detector.html

  1. n n
  2. θ c \theta_{c}
  3. v v
  4. cos θ c = c n v \cos\theta_{c}=\frac{c}{nv}
  5. c c
  6. v v
  7. n n
  8. θ c \theta_{c}
  9. θ c \theta_{c}
  10. θ c \theta_{c}
  11. θ c \theta_{c}
  12. θ c \theta_{c}
  13. n n
  14. c / n v > 1 c/nv>1
  15. N c N_{c}
  16. θ c \theta_{c}
  17. θ c \theta_{c}
  18. σ \sigma
  19. n n
  20. N c N_{c}
  21. σ \sigma
  22. N c N_{c}
  23. q q
  24. θ c \theta_{c}
  25. N = N c q 2 sin 2 ( θ c ) 1 - 1 n 2 N=\dfrac{N_{c}q^{2}\sin^{2}(\theta_{c})}{1-\dfrac{1}{n^{2}}}
  26. σ m = σ N \sigma_{m}=\frac{\sigma}{\sqrt{N}}
  27. σ m \sigma_{m}
  28. σ m \sigma_{m}
  29. n n
  30. N c N_{c}
  31. σ \sigma
  32. f f
  33. r = f θ c r=f\theta_{c}
  34. θ c 1 \theta_{c}\ll 1

Ring_oscillator.html

  1. f = 1 2 * t * n f={\frac{1}{2*t*n}}

Riser_(casting).html

  1. t riser = 1.25 t casting t_{\,\text{riser}}=1.25t_{\,\text{casting}}
  2. ( V A ) n riser = 1.25 ( V A ) n casting \left(\frac{V}{A}\right)^{n}\text{riser}=1.25\left(\frac{V}{A}\right)^{n}\text% {casting}

Ritz_method.html

  1. Ψ \Psi
  2. E 0 E_{0}
  3. E 0 Ψ | H ^ | Ψ Ψ | Ψ . E_{0}\leq\frac{\langle\Psi|\hat{H}|\Psi\rangle}{\langle\Psi|\Psi\rangle}.
  4. { Ψ i } \left\{\Psi_{i}\right\}
  5. Ψ = i = 1 N c i Ψ i . \Psi=\sum_{i=1}^{N}c_{i}\Psi_{i}.
  6. ε = i = 1 N c i Ψ i | H ^ | i = 1 N c i Ψ i i = 1 N c i Ψ i | i = 1 N c i Ψ i = i = 1 N j = 1 N c i * c j H i j i = 1 N j = 1 N c i * c j S i j A B . \varepsilon=\frac{\left\langle\displaystyle\sum_{i=1}^{N}c_{i}\Psi_{i}\right|% \hat{H}\left|\displaystyle\sum_{i=1}^{N}c_{i}\Psi_{i}\right\rangle}{\left% \langle\left.\displaystyle\sum_{i=1}^{N}c_{i}\Psi_{i}\right|\displaystyle\sum_% {i=1}^{N}c_{i}\Psi_{i}\right\rangle}=\frac{\displaystyle\sum_{i=1}^{N}% \displaystyle\sum_{j=1}^{N}c_{i}^{*}c_{j}H_{ij}}{\displaystyle\sum_{i=1}^{N}% \displaystyle\sum_{j=1}^{N}c_{i}^{*}c_{j}S_{ij}}\equiv\frac{A}{B}.
  7. { c i } \left\{c_{i}\right\}
  8. { c i * } \left\{c_{i}^{*}\right\}
  9. ε \varepsilon
  10. { c i * } \left\{c_{i}^{*}\right\}
  11. ε c k * = j = 1 N c j ( H k j - ε S k j ) B = 0 , \frac{\partial\varepsilon}{\partial c_{k}^{*}}=\frac{\displaystyle\sum_{j=1}^{% N}c_{j}(H_{kj}-\varepsilon S_{kj})}{B}=0,
  12. j = 1 N c j ( H k j - ε S k j ) = 0 for k = 1 , 2 , , N . \sum_{j=1}^{N}c_{j}\left(H_{kj}-\varepsilon S_{kj}\right)=0\quad\,\text{for}% \quad k=1,2,\dots,N.
  13. ε \varepsilon
  14. { c j } \left\{c_{j}\right\}
  15. det ( H - ε S ) = 0 , \det\left(H-\varepsilon S\right)=0,
  16. ε \varepsilon
  17. ε i \varepsilon_{i}
  18. ε i \varepsilon_{i}
  19. ε 0 \varepsilon_{0}
  20. { c j } \left\{c_{j}\right\}
  21. H k j H_{kj}
  22. S k j S_{kj}
  23. ϵ \epsilon
  24. c c

RKKY_interaction.html

  1. H ( 𝐑 i j ) = 𝐈 i 𝐈 j 4 | Δ k m k m | 2 m * ( 2 π ) 3 R i j 4 2 [ 2 k m R i j cos ( 2 k m R i j ) - sin ( 2 k m R i j ) ] H(\mathbf{R}_{ij})=\frac{\mathbf{I}_{i}\cdot\mathbf{I}_{j}}{4}\frac{\left|% \Delta k_{m}k_{m}\right|^{2}m^{*}}{(2\pi)^{3}R_{ij}^{4}\hbar^{2}}\left[2k_{m}R% _{ij}\cos(2k_{m}R_{ij})-\sin(2k_{m}R_{ij})\right]
  2. R i j R_{ij}
  3. 𝐈 i \mathbf{I}_{i}
  4. Δ k m k m \Delta k_{m}k_{m}
  5. m * m^{*}
  6. k m k_{m}

RL_circuit.html

  1. Z L = L s Z_{L}\ =\ Ls
  2. s = σ + j ω s\ =\ \sigma+j\omega
  3. j 2 = - 1 j^{2}=-1
  4. σ \scriptstyle\sigma
  5. ω \scriptstyle\omega
  6. 𝐕 ( t ) \displaystyle\mathbf{V}(t)
  7. v ( t ) = Re { V ( t ) } = A e σ t cos ( ω t + ϕ ) v(t)=\mathrm{Re}\left\{V(t)\right\}=Ae^{\sigma t}\cos(\omega t+\phi)
  8. σ = 0 \sigma=0
  9. s = j ω s=j\omega
  10. V L ( s ) = L s R + L s V i n ( s ) V_{L}(s)=\frac{Ls}{R+Ls}V_{in}(s)
  11. V R ( s ) = R R + L s V i n ( s ) V_{R}(s)=\frac{R}{R+Ls}V_{in}(s)
  12. I ( s ) = V i n ( s ) R + L s I(s)=\frac{V_{in}(s)}{R+Ls}
  13. H L ( s ) = V L ( s ) V i n ( s ) = L s R + L s = G L e j ϕ L H_{L}(s)={V_{L}(s)\over V_{in}(s)}={Ls\over R+Ls}=G_{L}e^{j\phi_{L}}
  14. H R ( s ) = V R ( s ) V i n ( s ) = R R + L s = G R e j ϕ R H_{R}(s)={V_{R}(s)\over V_{in}(s)}={R\over R+Ls}=G_{R}e^{j\phi_{R}}
  15. s = - R L s=-{R\over L}
  16. G L = | H L ( ω ) | = | V L ( ω ) V i n ( ω ) | = ω L R 2 + ( ω L ) 2 G_{L}=|H_{L}(\omega)|=\left|\frac{V_{L}(\omega)}{V_{in}(\omega)}\right|=\frac{% \omega L}{\sqrt{R^{2}+\left(\omega L\right)^{2}}}
  17. G R = | H R ( ω ) | = | V R ( ω ) V i n ( ω ) | = R R 2 + ( ω L ) 2 G_{R}=|H_{R}(\omega)|=\left|\frac{V_{R}(\omega)}{V_{in}(\omega)}\right|=\frac{% R}{\sqrt{R^{2}+\left(\omega L\right)^{2}}}
  18. ϕ L = H L ( s ) = tan - 1 ( R ω L ) \phi_{L}=\angle H_{L}(s)=\tan^{-1}\left(\frac{R}{\omega L}\right)
  19. ϕ R = H R ( s ) = tan - 1 ( - ω L R ) \phi_{R}=\angle H_{R}(s)=\tan^{-1}\left(-\frac{\omega L}{R}\right)
  20. V L \displaystyle V_{L}
  21. h L ( t ) = δ ( t ) - R L e - t R L u ( t ) = δ ( t ) - 1 τ e - 1 τ t u ( t ) h_{L}(t)=\delta(t)-{R\over L}e^{-t\frac{R}{L}}u(t)=\delta(t)-{1\over\tau}e^{-% \frac{1}{\tau}t}u(t)
  22. τ = L R \tau={L\over R}
  23. h R ( t ) = R L e - t R L u ( t ) = 1 τ e - 1 τ t u ( t ) h_{R}(t)={R\over L}e^{-t\frac{R}{L}}u(t)={1\over\tau}e^{-\frac{1}{\tau}t}u(t)
  24. i ( t ) = i ( 0 ) e - R L t = i ( 0 ) e - 1 τ t i(t)=i(0)e^{-\frac{R}{L}t}=i(0)e^{-\frac{1}{\tau}t}
  25. ω \scriptstyle\omega\;\to\;\infty
  26. G L \displaystyle G_{L}
  27. ω 0 \scriptstyle\omega\;\to\;0
  28. G L \displaystyle G_{L}
  29. G L = G R = 1 2 G_{L}=G_{R}=\frac{1}{\sqrt{2}}
  30. ω c = R L \omega_{c}=\frac{R}{L}
  31. f c = R 2 π L f_{c}=\frac{R}{2\pi L}
  32. ω 0 \scriptstyle\omega\;\to\;0
  33. ϕ L \displaystyle\phi_{L}
  34. ω \omega\to\infty
  35. ϕ L \displaystyle\phi_{L}
  36. V L \scriptstyle V_{L}
  37. V R \scriptstyle V_{R}
  38. j ω s \scriptstyle j\omega\;\to\;s
  39. V i n = 0 \scriptstyle V_{in}\;=\;0
  40. t = 0 \scriptstyle t\;=\;0
  41. V i n = V \scriptstyle V_{in}\;=\;V
  42. V i n ( s ) \displaystyle V_{in}(s)
  43. V L ( t ) \displaystyle V_{L}(t)
  44. τ = L R \scriptstyle\tau\;=\;\frac{L}{R}
  45. 1 e \scriptstyle\frac{1}{e}
  46. τ \scriptstyle\tau
  47. V L \scriptstyle V_{L}
  48. V ( 1 e ) \scriptstyle V\left(\frac{1}{e}\right)
  49. V R \scriptstyle V_{R}
  50. V ( 1 - 1 e ) \scriptstyle V\left(1\,-\,\frac{1}{e}\right)
  51. ( 1 - 1 e ) \scriptstyle\left(1\,-\,\frac{1}{e}\right)
  52. τ \scriptstyle\tau
  53. t = N τ \scriptstyle t\;=\;N\tau
  54. t = ( N + 1 ) τ \scriptstyle t\;=\;(N\,+\,1)\tau
  55. t = N τ \scriptstyle t\;=\;N\tau
  56. τ \scriptstyle\tau
  57. 5 τ \scriptstyle 5\tau
  58. V \scriptstyle V
  59. τ \tau
  60. 5 τ \scriptstyle 5\tau
  61. I \scriptstyle I
  62. V R \scriptstyle\frac{V}{R}
  63. V i n \displaystyle V_{in}
  64. V L \scriptstyle V_{L}
  65. V o u t V_{out}
  66. V i n \scriptstyle V_{in}
  67. I R \displaystyle I_{R}

Robinson–Schensted_correspondence.html

  1. λ 𝒫 n ( t λ ) 2 = n ! \sum_{\lambda\in\mathcal{P}_{n}}(t_{\lambda})^{2}=n!
  2. 𝒫 n \mathcal{P}_{n}
  3. n n
  4. n n
  5. λ λ
  6. σ σ
  7. σ = ( 1 2 3 n σ 1 σ 2 σ 3 σ n ) \sigma=\begin{pmatrix}1&2&3&\cdots&n\\ \sigma_{1}&\sigma_{2}&\sigma_{3}&\cdots&\sigma_{n}\end{pmatrix}
  8. ( P 0 , Q 0 ) , ( P 1 , Q 1 ) , , ( P n , Q n ) , (P_{0},Q_{0}),(P_{1},Q_{1}),\cdots,(P_{n},Q_{n}),
  9. P < s u b > 0 = Q 0 P<sub>0=Q_{0}

Rodrigues'_rotation_formula.html

  1. S O ( 3 ) SO(3)
  2. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  3. S O ( 3 ) SO(3)
  4. S O ( 3 ) SO(3)
  5. 𝐯 \mathbf{v}
  6. 𝐤 \mathbf{k}
  7. 𝐯 \mathbf{v}
  8. θ θ
  9. 𝐯 \mathbf{v}
  10. θ θ
  11. k k
  12. k k
  13. 𝐤 \mathbf{k}
  14. 𝐯 \mathbf{v}
  15. 𝐤 \mathbf{k}
  16. θ θ
  17. 𝐯 = ( 𝐤 𝐯 ) 𝐤 \mathbf{v}_{\parallel}=(\mathbf{k}\cdot\mathbf{v})\mathbf{k}
  18. 𝐯 \mathbf{v}
  19. 𝐤 \mathbf{k}
  20. 𝐯 \mathbf{v}
  21. 𝐤 \mathbf{k}
  22. 𝐯 = 𝐯 - 𝐯 = 𝐯 - ( 𝐤 𝐯 ) 𝐤 \mathbf{v}_{\perp}=\mathbf{v}-\mathbf{v}_{\parallel}=\mathbf{v}-(\mathbf{k}% \cdot\mathbf{v})\mathbf{k}
  23. 𝐯 \mathbf{v}
  24. 𝐤 \mathbf{k}
  25. 𝐯 \mathbf{v}
  26. 𝐤 \mathbf{k}
  27. 𝐰 = 𝐤 × 𝐯 . \mathbf{w}=\mathbf{k}\times\mathbf{v}.
  28. 𝐰 \mathbf{w}
  29. 𝐰 \mathbf{w}
  30. 𝐤 \mathbf{k}
  31. 𝐰 = 𝐤 × 𝐯 = 𝐤 × ( 𝐯 + 𝐯 ) = 𝐤 × 𝐯 + 𝐤 × 𝐯 = 𝐤 × 𝐯 , \mathbf{w}=\mathbf{k}\times\mathbf{v}=\mathbf{k}\times(\mathbf{v}_{\parallel}+% \mathbf{v}_{\perp})=\mathbf{k}\times\mathbf{v}_{\parallel}+\mathbf{k}\times% \mathbf{v}_{\perp}=\mathbf{k}\times\mathbf{v}_{\perp},
  32. 𝐤 \mathbf{k}
  33. 𝐰 \mathbf{w}
  34. 90 ° 90°
  35. 𝐤 \mathbf{k}
  36. θ θ
  37. 𝐤 \mathbf{k}
  38. 𝐯 rot \displaystyle\mathbf{v}_{\perp\ \mathrm{rot}}
  39. 𝐤 \mathbf{k}
  40. 𝐯 \mathbf{v}
  41. 𝐤 \mathbf{k}
  42. θ θ
  43. 𝐤 \mathbf{k}
  44. 𝐤 \mathbf{k}
  45. 𝐯 rot \displaystyle\mathbf{v}_{\mathrm{rot}}
  46. 𝐯 \mathbf{v}
  47. 𝐤 \mathbf{k}
  48. 𝐊 \mathbf{K}
  49. 𝐤 \mathbf{k}
  50. 𝐊 = [ 0 - k 3 k 2 k 3 0 - k 1 - k 2 k 1 0 ] . \mathbf{K}=\left[\begin{array}[]{ccc}0&-k_{3}&k_{2}\\ k_{3}&0&-k_{1}\\ -k_{2}&k_{1}&0\end{array}\right].
  51. 𝐊𝐯 = 𝐤 × 𝐯 \mathbf{K}\mathbf{v}=\mathbf{k}\times\mathbf{v}
  52. 𝐯 \mathbf{v}
  53. 𝐊 \mathbf{K}
  54. 𝐯 rot = 𝐯 cos θ + ( 𝐤 × 𝐯 ) sin θ + 𝐤 ( 𝐤 𝐯 ) ( 1 - cos θ ) = 𝐯 cos θ + ( 𝐊𝐯 ) sin θ + 𝐤 ( 𝐤 𝐯 ) ( 1 - cos θ ) \begin{aligned}\displaystyle\mathbf{v}_{\mathrm{rot}}&\displaystyle=\mathbf{v}% \cos\theta+(\mathbf{k}\times\mathbf{v})\sin\theta+\mathbf{k}(\mathbf{k}\cdot% \mathbf{v})(1-\cos\theta)\\ &\displaystyle=\mathbf{v}\cos\theta+(\mathbf{K}\mathbf{v})\sin\theta+\mathbf{k% }(\mathbf{k}\cdot\mathbf{v})(1-\cos\theta)\\ \end{aligned}
  55. 𝐯 \mathbf{v}
  56. θ θ
  57. 𝐯 rot \displaystyle\mathbf{v}_{\mathrm{rot}}
  58. 𝐚 × ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐚 𝐜 ) - 𝐜 ( 𝐚 𝐛 ) \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf% {c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b})
  59. 𝐚 = 𝐛 = 𝐤 \mathbf{a}=\mathbf{b}=\mathbf{k}
  60. 𝐜 = 𝐯 \mathbf{c}=\mathbf{v}
  61. ( 𝐤 ( 𝐤 𝐯 ) - 𝐯 ) = 𝐤 × ( 𝐤 × 𝐯 ) (\mathbf{k}(\mathbf{k}\cdot\mathbf{v})-\mathbf{v})=\mathbf{k}\times(\mathbf{k}% \times\mathbf{v})
  62. 𝐤 ( 𝐤 𝐯 ) - 𝐯 = 𝐊 2 𝐯 . \mathbf{k}(\mathbf{k}\cdot\mathbf{v})-\mathbf{v}=\mathbf{K}^{2}\mathbf{v}~{}.
  63. 𝐯 rot = 𝐯 + ( sin θ ) 𝐊𝐯 + ( 1 - cos θ ) 𝐊 2 𝐯 , \mathbf{v}_{\mathrm{rot}}=\mathbf{v}+(\sin\theta)\mathbf{K}\mathbf{v}+(1-\cos% \theta)\mathbf{K}^{2}\mathbf{v}~{},
  64. 𝐯 rot = 𝐑𝐯 \begin{aligned}\displaystyle\mathbf{v}_{\mathrm{rot}}&\displaystyle=\mathbf{R}% \mathbf{v}\end{aligned}
  65. 𝐑 \mathbf{R}
  66. 𝐊 \mathbf{K}
  67. 𝐤 \mathbf{k}
  68. θ θ
  69. 𝐑 \mathbf{R}
  70. 𝐤 \mathbf{k}
  71. θ θ
  72. 𝐑 \mathbf{R}
  73. S O ( 3 ) SO(3)
  74. 𝐊 \mathbf{K}
  75. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  76. 𝐊 \mathbf{K}
  77. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  78. 𝐑 = exp ( θ 𝐊 ) . \mathbf{R}=\exp(\theta\mathbf{K})~{}.
  79. 𝐑 ( θ ) 𝐑 ( ϕ ) = 𝐑 ( θ + ϕ ) , 𝐑 ( 0 ) = 𝐈 , \mathbf{R}(\theta)\mathbf{R}(\phi)=\mathbf{R}(\theta+\phi),\quad\mathbf{R}(0)=% \mathbf{I}~{},
  80. θ θ

Rolling_resistance.html

  1. F = C r r N \ F=C_{rr}N
  2. F F
  3. C r r C_{rr}
  4. N N
  5. C r r C_{rr}
  6. C r r = 0.01 \ C_{rr}=0.01
  7. C r r C_{rr}
  8. C r r C_{rr}
  9. C r r C_{rr}
  10. 2000 C r r 2000C_{rr}
  11. C r r C_{rr}
  12. 1000 g C r r 1000gC_{rr}
  13. C r r C_{rr}
  14. C r r C_{rr}
  15. C r r C_{rr}
  16. c o s ( θ ) = 1 cos(\theta)=1
  17. C r r = z / d \ C_{rr}=\sqrt{z/d}
  18. z z
  19. d d
  20. C r r = 0.0048 ( 18 / D ) 1 2 ( 100 / W ) 1 4 \ C_{rr}=0.0048(18/D)^{\frac{1}{2}}(100/W)^{\frac{1}{4}}
  21. D D
  22. W W
  23. C r r \ C_{rr}
  24. b \ b
  25. F = N b r \ F=\frac{Nb}{r}
  26. F F
  27. r r
  28. b b
  29. N N
  30. T T
  31. R r R_{r}
  32. T = V s Ω R r T=\frac{V_{s}}{\Omega}R_{r}
  33. V s V_{s}
  34. Ω \Omega
  35. V s / Ω V_{s}/\Omega
  36. F = k N 0.5 F=kN^{0.5}
  37. F F
  38. N N
  39. k k
  40. F F
  41. N N
  42. d N N = 2 d F F {\operatorname{d}N\over N}=2{\operatorname{d}F\over F}

Root-mean-square_speed.html

  1. v rms = 3 R T M m v_{\mathrm{rms}}=\sqrt{{3RT}\over{M_{m}}}
  2. v rms = 3 k T m v_{\mathrm{rms}}=\sqrt{{3kT}\over{m}}
  3. E k = 3 2 n R T = 3 2 N k T E_{\mathrm{k}}={{3}\over{2}}nRT=\frac{3}{2}NkT
  4. E k , molecule = 1 2 m v 2 E_{\mathrm{k,molecule}}={{1}\over{2}}mv^{2}
  5. 1 2 n M v 2 = E k {{1}\over{2}}nMv^{2}=E_{\mathrm{k}}
  6. v rms = 2 E k m v_{\mathrm{rms}}=\sqrt{{2E_{\mathrm{k}}}\over{m}}
  7. v rms = 0 v 2 p ( v ) d v v_{\mathrm{rms}}=\sqrt{\int_{0}^{\infty}v^{2}\ p(v)dv}\,\!
  8. = 0 4 π ( m 2 π k T ) 3 2 v 4 e - v 2 m 2 k T d v =\sqrt{\int_{0}^{\infty}4\pi\left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}}v^{4}% \ e^{-\frac{v^{2}m}{2kT}}dv}\,\!
  9. = 4 π ( m 2 π k T ) 3 2 3 8 π 1 2 ( 2 k T m ) 5 2 =\sqrt{4\pi\left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}}\frac{3}{8}\pi^{\frac{1% }{2}}\left(\frac{2kT}{m}\right)^{\frac{5}{2}}}\,\!
  10. = 3 k T m =\sqrt{\frac{3kT}{m}}

Root_test.html

  1. lim sup n | a n | n , \limsup_{n\rightarrow\infty}\sqrt[n]{|a_{n}|},
  2. a n a_{n}
  3. n = 1 a n . \sum_{n=1}^{\infty}a_{n}.
  4. C = lim sup n | a n | n , C=\limsup_{n\rightarrow\infty}\sqrt[n]{|a_{n}|},
  5. lim n | a n | n , \lim_{n\rightarrow\infty}\sqrt[n]{|a_{n}|},
  6. 1 / n 2 \textstyle\sum 1/{n^{2}}
  7. 1 / n \textstyle\sum 1/n
  8. f ( z ) = n = 0 c n ( z - p ) n f(z)=\sum_{n=0}^{\infty}c_{n}(z-p)^{n}
  9. 1 / lim sup n | c n | n , 1/\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_{n}|}},
  10. a n n < k < 1 , \sqrt[n]{a_{n}}<k<1,
  11. a n < k n < 1. a_{n}<k^{n}<1.
  12. n = N k n \sum_{n=N}^{\infty}k^{n}
  13. n = N a n \sum_{n=N}^{\infty}a_{n}
  14. | a n | n . \sqrt[n]{|a_{n}|}.
  15. | a n | n > 1 \sqrt[n]{|a_{n}|}>1
  16. | a n | n = | c n ( z - p ) n | n < 1 , \sqrt[n]{|a_{n}|}=\sqrt[n]{|c_{n}(z-p)^{n}|}<1,
  17. | c n | n | z - p | < 1 \sqrt[n]{|c_{n}|}\cdot|z-p|<1
  18. | z - p | < 1 / | c n | n |z-p|<1/\sqrt[n]{|c_{n}|}
  19. | z - p | < 1 / lim sup n | c n | n , |z-p|<1/\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_{n}|}},
  20. R 1 / lim sup n | c n | n . R\leq 1/\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_{n}|}}.
  21. | a n | n = | c n ( z - p ) n | n = 1 , \sqrt[n]{|a_{n}|}=\sqrt[n]{|c_{n}(z-p)^{n}|}=1,
  22. R = 1 / lim sup n | c n | n . R=1/\limsup_{n\rightarrow\infty}{\sqrt[n]{|c_{n}|}}.

Rooted_product_of_graphs.html

  1. v i v_{i}
  2. v i v_{i}
  3. h 1 h_{1}
  4. G H := ( V , E ) G\circ H:=(V,E)
  5. V = { ( g i , h j ) : 1 i n , 1 j m } V=\left\{(g_{i},h_{j}):1\leq i\leq n,1\leq j\leq m\right\}
  6. E = { ( ( g i , h 1 ) , ( g k , h 1 ) ) : ( g i , g k ) E ( G ) } i = 1 n { ( ( g i , h j ) , ( g i , h k ) ) : ( h j , h k ) E ( H ) } E=\left\{((g_{i},h_{1}),(g_{k},h_{1})):(g_{i},g_{k})\in E(G)\right\}\cup% \bigcup_{i=1}^{n}\left\{((g_{i},h_{j}),(g_{i},h_{k})):(h_{j},h_{k})\in E(H)\right\}

Rotary_disc_shutter.html

  1. Shutter Angle 360 = Exposure Time Frame Interval \frac{\,\text{Shutter Angle}}{360^{\circ}}=\frac{\,\text{Exposure Time}}{\,% \text{Frame Interval}}

Rotating_reference_frame.html

  1. ( x , y , z ) \left(x^{\prime},y^{\prime},z^{\prime}\right)
  2. ( x , y , z ) \left(x,y,z\right)
  3. z z
  4. Ω \Omega
  5. t = 0 t=0
  6. x = x cos ( θ ( t ) ) - y sin ( θ ( t ) ) x=x^{\prime}\cos\left(\theta(t)\right)-y^{\prime}\sin\left(\theta(t)\right)
  7. y = x sin ( θ ( t ) ) + y cos ( θ ( t ) ) y=x^{\prime}\sin\left(\theta(t)\right)+y^{\prime}\cos\left(\theta(t)\right)
  8. x = x cos ( - θ ( t ) ) - y sin ( - θ ( t ) ) x^{\prime}=x\cos\left(-\theta(t)\right)-y\sin\left(-\theta(t)\right)
  9. y = x sin ( - θ ( t ) ) + y cos ( - θ ( t ) ) y^{\prime}=x\sin\left(-\theta(t)\right)+y\cos\left(-\theta(t)\right)
  10. s y m b o l ı ^ , s y m b o l ȷ ^ , s y m b o l k ^ \hat{symbol{\imath}},\ \hat{symbol{\jmath}},\ \hat{symbol{k}}
  11. s y m b o l ı ^ ( t ) = ( cos θ ( t ) , sin θ ( t ) ) \hat{symbol{\imath}}(t)=(\cos\theta(t),\ \sin\theta(t))
  12. s y m b o l ȷ ^ ( t ) = ( - sin θ ( t ) , cos θ ( t ) ) . \hat{symbol{\jmath}}(t)=(-\sin\theta(t),\ \cos\theta(t))\ .
  13. d d t s y m b o l ı ^ ( t ) = Ω ( - sin θ ( t ) , cos θ ( t ) ) = Ω s y m b o l ȷ ^ ; \frac{d}{dt}\hat{symbol{\imath}}(t)=\Omega(-\sin\theta(t),\ \cos\theta(t))=% \Omega\hat{symbol{\jmath}}\ ;
  14. d d t s y m b o l ȷ ^ ( t ) = Ω ( - cos θ ( t ) , - sin θ ( t ) ) = - Ω s y m b o l ı ^ , \frac{d}{dt}\hat{symbol{\jmath}}(t)=\Omega(-\cos\theta(t),\ -\sin\theta(t))=-% \Omega\hat{symbol{\imath}}\ ,
  15. Ω d d t θ ( t ) \Omega\equiv\frac{d}{dt}\theta(t)
  16. s y m b o l Ω symbol{\Omega}
  17. s y m b o l Ω = ( 0 , 0 , Ω ) symbol{\Omega}=(0,\ 0,\ \Omega)
  18. d d t s y m b o l u ^ = s y m b o l Ω × s y m b o l u ^ , \frac{d}{dt}\hat{symbol{u}}=symbol{\Omega\times}\hat{symbol{u}}\ ,
  19. s y m b o l u ^ \hat{symbol{u}}
  20. s y m b o l ı ^ \hat{symbol{\imath}}
  21. s y m b o l ȷ ^ \hat{symbol{\jmath}}
  22. s y m b o l ı ^ , s y m b o l ȷ ^ , s y m b o l k ^ \hat{symbol{\imath}},\ \hat{symbol{\jmath}},\ \hat{symbol{k}}
  23. Ω \Omega
  24. s y m b o l Ω symbol{\Omega}
  25. s y m b o l u ^ \hat{symbol{u}}
  26. d d t s y m b o l u ^ = s y m b o l Ω × u ^ . \frac{d}{dt}\hat{symbol{u}}=symbol{\Omega\times\hat{u}}\ .
  27. s y m b o l f symbol{f}
  28. s y m b o l f ( t ) = f x ( t ) s y m b o l ı ^ + f y ( t ) s y m b o l ȷ ^ + f z ( t ) s y m b o l k ^ , symbol{f}(t)=f_{x}(t)\hat{symbol{\imath}}+f_{y}(t)\hat{symbol{\jmath}}+f_{z}(t% )\hat{symbol{k}}\ ,
  29. d d t s y m b o l f = d f x d t s y m b o l ı ^ + d s y m b o l ı ^ d t f x + d f y d t s y m b o l ȷ ^ + d s y m b o l ȷ ^ d t f y + d f z d t s y m b o l k ^ + d s y m b o l k ^ d t f z \frac{d}{dt}symbol{f}=\frac{df_{x}}{dt}\hat{symbol{\imath}}+\frac{d\hat{symbol% {\imath}}}{dt}f_{x}+\frac{df_{y}}{dt}\hat{symbol{\jmath}}+\frac{d\hat{symbol{% \jmath}}}{dt}f_{y}+\frac{df_{z}}{dt}\hat{symbol{k}}+\frac{d\hat{symbol{k}}}{dt% }f_{z}
  30. = d f x d t s y m b o l ı ^ + d f y d t s y m b o l ȷ ^ + d f z d t s y m b o l k ^ + [ s y m b o l Ω × ( f x s y m b o l ı ^ + f y s y m b o l ȷ ^ + f z s y m b o l k ^ ) ] =\frac{df_{x}}{dt}\hat{symbol{\imath}}+\frac{df_{y}}{dt}\hat{symbol{\jmath}}+% \frac{df_{z}}{dt}\hat{symbol{k}}+[symbol{\Omega\times}(f_{x}\hat{symbol{\imath% }}+f_{y}\hat{symbol{\jmath}}+f_{z}\hat{symbol{k}})]
  31. = ( d s y m b o l f d t ) r + s y m b o l Ω × f ( t ) , =\left(\frac{dsymbol{f}}{dt}\right)_{r}+symbol{\Omega\times f}(t)\ ,
  32. ( d s y m b o l f d t ) r \left(\frac{dsymbol{f}}{dt}\right)_{r}
  33. s y m b o l f symbol{f}
  34. d d t s y m b o l f = [ ( d d t ) r + s y m b o l Ω × ] s y m b o l f . \frac{d}{dt}symbol{f}=\left[\left(\frac{d}{dt}\right)_{r}+symbol{\Omega\times}% \right]symbol{f}\ .
  35. 𝐯 = def d 𝐫 d t \mathbf{v}\ \stackrel{\mathrm{def}}{=}\ \frac{d\mathbf{r}}{dt}
  36. s y m b o l r ( t ) symbol{r}(t)
  37. s y m b o l r ( t ) symbol{r}(t)
  38. 𝐯 𝐢 = def d 𝐫 d t = ( d 𝐫 d t ) r + s y m b o l Ω × 𝐫 = 𝐯 r + s y m b o l Ω × 𝐫 , \mathbf{v_{i}}\ \stackrel{\mathrm{def}}{=}\ \frac{d\mathbf{r}}{dt}=\left(\frac% {d\mathbf{r}}{dt}\right)_{\mathrm{r}}+symbol\Omega\times\mathbf{r}=\mathbf{v}_% {\mathrm{r}}+symbol\Omega\times\mathbf{r}\ ,
  39. 𝐚 i = def ( d 2 𝐫 d t 2 ) i = ( d 𝐯 d t ) i = [ ( d d t ) r + s y m b o l Ω × ] [ ( d 𝐫 d t ) r + s y m b o l Ω × 𝐫 ] , \mathbf{a}_{\mathrm{i}}\ \stackrel{\mathrm{def}}{=}\ \left(\frac{d^{2}\mathbf{% r}}{dt^{2}}\right)_{\mathrm{i}}=\left(\frac{d\mathbf{v}}{dt}\right)_{\mathrm{i% }}=\left[\left(\frac{d}{dt}\right)_{\mathrm{r}}+symbol\Omega\times\right]\left% [\left(\frac{d\mathbf{r}}{dt}\right)_{\mathrm{r}}+symbol\Omega\times\mathbf{r}% \right]\ ,
  40. 𝐚 r = 𝐚 i - 2 s y m b o l Ω × 𝐯 r - s y m b o l Ω × ( s y m b o l Ω × 𝐫 ) - d s y m b o l Ω d t × 𝐫 \mathbf{a}_{\mathrm{r}}=\mathbf{a}_{\mathrm{i}}-2symbol\Omega\times\mathbf{v}_% {\mathrm{r}}-symbol\Omega\times(symbol\Omega\times\mathbf{r})-\frac{dsymbol% \Omega}{dt}\times\mathbf{r}
  41. 𝐚 r = def ( d 2 𝐫 d t 2 ) r \mathbf{a}_{\mathrm{r}}\ \stackrel{\mathrm{def}}{=}\ \left(\frac{d^{2}\mathbf{% r}}{dt^{2}}\right)_{\mathrm{r}}
  42. - s y m b o l Ω × ( s y m b o l Ω × 𝐫 ) -symbol\Omega\times(symbol\Omega\times\mathbf{r})
  43. - 2 s y m b o l Ω × 𝐯 r -2symbol\Omega\times\mathbf{v}_{\mathrm{r}}
  44. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  45. 𝐅 Coriolis = - 2 m s y m b o l Ω × 𝐯 r \mathbf{F}_{\mathrm{Coriolis}}=-2msymbol\Omega\times\mathbf{v}_{\mathrm{r}}
  46. 𝐅 centrifugal = - m s y m b o l Ω × ( s y m b o l Ω × 𝐫 ) \mathbf{F}_{\mathrm{centrifugal}}=-msymbol\Omega\times(symbol\Omega\times% \mathbf{r})
  47. 𝐅 Euler = - m d s y m b o l Ω d t × 𝐫 \mathbf{F}_{\mathrm{Euler}}=-m\frac{dsymbol\Omega}{dt}\times\mathbf{r}
  48. m m
  49. s y m b o l Ω = 0 . symbol{\Omega}=0\ .
  50. 𝐚 i \mathbf{a}_{\mathrm{i}}
  51. 𝐅 imp \mathbf{F}_{\mathrm{imp}}
  52. 𝐅 imp = m 𝐚 i \mathbf{F}_{\mathrm{imp}}=m\mathbf{a}_{\mathrm{i}}
  53. 𝐅 𝐫 = 𝐅 imp + 𝐅 centrifugal + 𝐅 Coriolis + 𝐅 Euler = m 𝐚 𝐫 . \mathbf{F_{r}}=\mathbf{F}_{\mathrm{imp}}+\mathbf{F}_{\mathrm{centrifugal}}+% \mathbf{F}_{\mathrm{Coriolis}}+\mathbf{F}_{\mathrm{Euler}}=m\mathbf{a_{r}}\ .

Rough_set.html

  1. I = ( 𝕌 , 𝔸 ) I=(\mathbb{U},\mathbb{A})
  2. 𝕌 \mathbb{U}
  3. 𝔸 \mathbb{A}
  4. a : 𝕌 V a a:\mathbb{U}\rightarrow V_{a}
  5. a 𝔸 a\in\mathbb{A}
  6. V a V_{a}
  7. a a
  8. a ( x ) a(x)
  9. V a V_{a}
  10. a a
  11. x x
  12. 𝕌 \mathbb{U}
  13. P 𝔸 P\subseteq\mathbb{A}
  14. IND ( P ) \mathrm{IND}(P)
  15. IND ( P ) = { ( x , y ) 𝕌 2 a P , a ( x ) = a ( y ) } \mathrm{IND}(P)=\left\{(x,y)\in\mathbb{U}^{2}\mid\forall a\in P,a(x)=a(y)\right\}
  16. IND ( P ) \mathrm{IND}(P)
  17. P P
  18. 𝕌 \mathbb{U}
  19. IND ( P ) \mathrm{IND}(P)
  20. 𝕌 / IND ( P ) \mathbb{U}/\mathrm{IND}(P)
  21. 𝕌 / P \mathbb{U}/P
  22. ( x , y ) IND ( P ) (x,y)\in\mathrm{IND}(P)
  23. x x
  24. y y
  25. P P
  26. P 1 P_{1}
  27. P 2 P_{2}
  28. P 3 P_{3}
  29. P 4 P_{4}
  30. P 5 P_{5}
  31. O 1 O_{1}
  32. O 2 O_{2}
  33. O 3 O_{3}
  34. O 4 O_{4}
  35. O 5 O_{5}
  36. O 6 O_{6}
  37. O 7 O_{7}
  38. O 8 O_{8}
  39. O 9 O_{9}
  40. O 10 O_{10}
  41. P = { P 1 , P 2 , P 3 , P 4 , P 5 } P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}
  42. { { O 1 , O 2 } { O 3 , O 7 , O 10 } { O 4 } { O 5 } { O 6 } { O 8 } { O 9 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{7},O_{10}\}\\ \{O_{4}\}\\ \{O_{5}\}\\ \{O_{6}\}\\ \{O_{8}\}\\ \{O_{9}\}\end{cases}
  43. { O 1 , O 2 } \{O_{1},O_{2}\}
  44. { O 3 , O 7 , O 10 } \{O_{3},O_{7},O_{10}\}
  45. P P
  46. [ x ] P [x]_{P}
  47. P = { P 1 } P=\{P_{1}\}
  48. { { O 1 , O 2 } { O 3 , O 5 , O 7 , O 9 , O 10 } { O 4 , O 6 , O 8 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{5},O_{7},O_{9},O_{10}\}\\ \{O_{4},O_{6},O_{8}\}\end{cases}
  49. X 𝕌 X\subseteq\mathbb{U}
  50. P P
  51. X X
  52. P P
  53. X X
  54. P P
  55. X = { O 1 , O 2 , O 3 , O 4 } X=\{O_{1},O_{2},O_{3},O_{4}\}
  56. P = { P 1 , P 2 , P 3 , P 4 , P 5 } P=\{P_{1},P_{2},P_{3},P_{4},P_{5}\}
  57. X X
  58. [ x ] P , [x]_{P},
  59. { O 3 , O 7 , O 10 } \{O_{3},O_{7},O_{10}\}
  60. X X
  61. O 3 O_{3}
  62. O 7 O_{7}
  63. O 10 O_{10}
  64. X X
  65. P P
  66. P P
  67. P P
  68. X X
  69. P ¯ X = { x [ x ] P X } {\underline{P}}X=\{x\mid[x]_{P}\subseteq X\}
  70. P ¯ X = { x [ x ] P X } {\overline{P}}X=\{x\mid[x]_{P}\cap X\neq\emptyset\}
  71. P P
  72. [ x ] P [x]_{P}
  73. P ¯ X = { O 1 , O 2 } { O 4 } {\underline{P}}X=\{O_{1},O_{2}\}\cup\{O_{4}\}
  74. [ x ] P [x]_{P}
  75. 𝕌 / P \mathbb{U}/P
  76. X X
  77. P P
  78. [ x ] P [x]_{P}
  79. P ¯ X = { O 1 , O 2 } { O 4 } { O 3 , O 7 , O 10 } {\overline{P}}X=\{O_{1},O_{2}\}\cup\{O_{4}\}\cup\{O_{3},O_{7},O_{10}\}
  80. [ x ] P [x]_{P}
  81. 𝕌 / P \mathbb{U}/P
  82. X ¯ \overline{X}
  83. X X
  84. X X
  85. 𝕌 - P ¯ X \mathbb{U}-{\overline{P}}X
  86. P ¯ X - P ¯ X {\overline{P}}X-{\underline{P}}X
  87. X X
  88. 𝕌 / P \mathbb{U}/P
  89. P ¯ X , P ¯ X \langle{\underline{P}}X,{\overline{P}}X\rangle
  90. X X
  91. X X
  92. X X
  93. α P ( X ) = | P ¯ X | | P ¯ X | \alpha_{P}(X)=\frac{\left|{\underline{P}}X\right|}{\left|{\overline{P}}X\right|}
  94. X X
  95. α P ( X ) \alpha_{P}(X)
  96. 0 α P ( X ) 1 0\leq\alpha_{P}(X)\leq 1
  97. X X
  98. X X
  99. α P ( X ) = 1 \alpha_{P}(X)=1
  100. X X
  101. P P
  102. P ¯ X = P ¯ X {\overline{P}}X={\underline{P}}X
  103. X X
  104. P P
  105. X X
  106. P ¯ X {\underline{P}}X\neq\emptyset
  107. P ¯ X = 𝕌 {\overline{P}}X=\mathbb{U}
  108. P P
  109. X X
  110. X X
  111. X X
  112. P ¯ X = {\underline{P}}X=\emptyset
  113. P ¯ X 𝕌 {\overline{P}}X\neq\mathbb{U}
  114. P P
  115. X X
  116. X X
  117. X X
  118. P ¯ X = {\underline{P}}X=\emptyset
  119. P ¯ X = 𝕌 {\overline{P}}X=\mathbb{U}
  120. P P
  121. X X
  122. X X
  123. P P
  124. X X
  125. RED P \mathrm{RED}\subseteq P
  126. [ x ] RED [x]_{\mathrm{RED}}
  127. [ x ] P [x]_{P}
  128. RED \mathrm{RED}
  129. P P
  130. RED \mathrm{RED}
  131. [ x ] ( RED - { a } ) [ x ] P [x]_{(\mathrm{RED}-\{a\})}\neq[x]_{P}
  132. a RED a\in\mathrm{RED}
  133. RED \mathrm{RED}
  134. [ x ] P [x]_{P}
  135. { P 3 , P 4 , P 5 } \{P_{3},P_{4},P_{5}\}
  136. { { O 1 , O 2 } { O 3 , O 7 , O 10 } { O 4 } { O 5 } { O 6 } { O 8 } { O 9 } \begin{cases}\{O_{1},O_{2}\}\\ \{O_{3},O_{7},O_{10}\}\\ \{O_{4}\}\\ \{O_{5}\}\\ \{O_{6}\}\\ \{O_{8}\}\\ \{O_{9}\}\end{cases}
  137. { P 3 , P 4 , P 5 } \{P_{3},P_{4},P_{5}\}
  138. [ x ] RED [ x ] P [x]_{\mathrm{RED}}\neq[x]_{P}
  139. { P 1 , P 2 , P 5 } \{P_{1},P_{2},P_{5}\}
  140. [ x ] P [x]_{P}
  141. { P 5 } \{P_{5}\}
  142. { P 5 } \{P_{5}\}
  143. { P 5 } \{P_{5}\}
  144. P P
  145. Q Q
  146. P P
  147. [ x ] P [x]_{P}
  148. Q Q
  149. [ x ] Q [x]_{Q}
  150. [ x ] Q = { Q 1 , Q 2 , Q 3 , , Q N } [x]_{Q}=\{Q_{1},Q_{2},Q_{3},\dots,Q_{N}\}
  151. Q i Q_{i}
  152. Q Q
  153. Q Q
  154. P P
  155. γ P ( Q ) \gamma_{P}(Q)
  156. γ P ( Q ) = i = 1 N | P ¯ Q i | | 𝕌 | 1 \gamma_{P}(Q)=\frac{\sum_{i=1}^{N}\left|{\underline{P}}Q_{i}\right|}{\left|% \mathbb{U}\right|}\leq 1
  157. Q i Q_{i}
  158. [ x ] Q [x]_{Q}
  159. P P
  160. P ¯ Q i {\underline{P}}Q_{i}
  161. X X
  162. P P
  163. Q i Q_{i}
  164. [ x ] Q [x]_{Q}
  165. P P
  166. Q Q
  167. γ P ( Q ) \gamma_{P}(Q)
  168. P P
  169. Q Q
  170. Q Q
  171. P P
  172. 𝒫 = { P 1 , P 2 , P 3 , , P n } \mathcal{P}=\{P_{1},P_{2},P_{3},\dots,P_{n}\}
  173. Q , Q 𝒫 Q,Q\notin\mathcal{P}
  174. P i a P j b P k c Q d P_{i}^{a}P_{j}^{b}\dots P_{k}^{c}\to Q^{d}
  175. ( P i = a ) and ( P j = b ) and and ( P k = c ) ( Q = d ) (P_{i}=a)\and(P_{j}=b)\and\dots\and(P_{k}=c)\to(Q=d)
  176. { a , b , c , } \{a,b,c,\dots\}
  177. 𝕌 \mathbb{U}
  178. d d
  179. Q Q
  180. d d
  181. Q Q
  182. Q = d Q=d
  183. Q d Q\neq d
  184. P 4 P_{4}
  185. { P 1 , P 2 , P 3 } \{P_{1},P_{2},P_{3}\}
  186. P 4 P_{4}
  187. { 1 , 2 } \{1,2\}
  188. P 4 = 1 P_{4}=1
  189. 𝕌 \mathbb{U}
  190. P 4 = 1 P_{4}=1
  191. P 4 1 P_{4}\neq 1
  192. P 4 1 P_{4}\neq 1
  193. P 4 = 2 P_{4}=2
  194. P 4 1 P_{4}\neq 1
  195. P 4 P_{4}
  196. P 4 = 1 P_{4}=1
  197. P 4 = 2 , 3 , 4 , e t c . P_{4}=2,3,4,etc.
  198. P 4 = 1 P_{4}=1
  199. { O 1 , O 2 , O 3 , O 7 , O 10 } \{O_{1},O_{2},O_{3},O_{7},O_{10}\}
  200. P 4 1 P_{4}\neq 1
  201. { O 4 , O 5 , O 6 , O 8 , O 9 } \{O_{4},O_{5},O_{6},O_{8},O_{9}\}
  202. P 4 = 1 P_{4}=1
  203. P 4 = 1 P_{4}=1
  204. P 4 1 P_{4}\neq 1
  205. { O 1 , O 2 , O 3 , O 7 , O 10 } \{O_{1},O_{2},O_{3},O_{7},O_{10}\}
  206. { O 4 , O 5 , O 6 , O 8 , O 9 } \{O_{4},O_{5},O_{6},O_{8},O_{9}\}
  207. P 4 = 1 P_{4}=1
  208. P 4 1 P_{4}\neq 1
  209. P 4 = 1 P_{4}=1
  210. O 4 O_{4}
  211. O 5 O_{5}
  212. O 6 O_{6}
  213. O 8 O_{8}
  214. O 9 O_{9}
  215. O 1 O_{1}
  216. P 1 1 , P 2 2 , P 3 0 P_{1}^{1},P_{2}^{2},P_{3}^{0}
  217. P 1 1 , P 2 2 P_{1}^{1},P_{2}^{2}
  218. P 1 1 , P 2 2 , P 3 0 P_{1}^{1},P_{2}^{2},P_{3}^{0}
  219. P 1 1 , P 2 2 , P 3 0 P_{1}^{1},P_{2}^{2},P_{3}^{0}
  220. P 1 1 , P 2 2 P_{1}^{1},P_{2}^{2}
  221. O 2 O_{2}
  222. P 1 1 , P 2 2 , P 3 0 P_{1}^{1},P_{2}^{2},P_{3}^{0}
  223. P 1 1 , P 2 2 P_{1}^{1},P_{2}^{2}
  224. P 1 1 , P 2 2 , P 3 0 P_{1}^{1},P_{2}^{2},P_{3}^{0}
  225. P 1 1 , P 2 2 , P 3 0 P_{1}^{1},P_{2}^{2},P_{3}^{0}
  226. P 1 1 , P 2 2 P_{1}^{1},P_{2}^{2}
  227. O 3 O_{3}
  228. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  229. P 2 0 P_{2}^{0}
  230. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  231. P 1 2 , P 2 0 , P 3 0 P_{1}^{2},P_{2}^{0},P_{3}^{0}
  232. P 2 0 P_{2}^{0}
  233. O 7 O_{7}
  234. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  235. P 2 0 P_{2}^{0}
  236. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  237. P 1 2 , P 2 0 , P 3 0 P_{1}^{2},P_{2}^{0},P_{3}^{0}
  238. P 2 0 P_{2}^{0}
  239. O 10 O_{10}
  240. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  241. P 2 0 P_{2}^{0}
  242. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  243. P 1 2 , P 2 0 , P 3 0 P_{1}^{2},P_{2}^{0},P_{3}^{0}
  244. P 2 0 P_{2}^{0}
  245. O 3 O_{3}
  246. O 6 O_{6}
  247. P 1 2 , P 3 0 P_{1}^{2},P_{3}^{0}
  248. P 4 = 1 P_{4}=1
  249. O 3 O_{3}
  250. O 6 O_{6}
  251. P 1 P_{1}
  252. P 3 P_{3}
  253. O 3 O_{3}
  254. P 1 = 2 P_{1}=2
  255. P 3 = 0 P_{3}=0
  256. O 3 O_{3}
  257. P 4 = 1 P_{4}=1
  258. P 1 P_{1}
  259. P 3 P_{3}
  260. { ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 ) and ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 ) ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 ) and ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 ) ( P 1 2 P 3 0 ) and ( P 2 0 ) and ( P 1 2 P 3 0 ) and ( P 1 2 P 2 0 P 3 0 ) and ( P 2 0 ) ( P 1 2 P 3 0 ) and ( P 2 0 ) and ( P 1 2 P 3 0 ) and ( P 1 2 P 2 0 P 3 0 ) and ( P 2 0 ) ( P 1 2 P 3 0 ) and ( P 2 0 ) and ( P 1 2 P 3 0 ) and ( P 1 2 P 2 0 P 3 0 ) and ( P 2 0 ) \begin{cases}(P_{1}^{1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2})\and(P_{1}^{% 1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2})% \\ (P_{1}^{1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2})\and(P_{1}^{1}P_{2}^{2}P_% {3}^{0})\and(P_{1}^{1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2})\\ (P_{1}^{2}P_{3}^{0})\and(P_{2}^{0})\and(P_{1}^{2}P_{3}^{0})\and(P_{1}^{2}P_{2}% ^{0}P_{3}^{0})\and(P_{2}^{0})\\ (P_{1}^{2}P_{3}^{0})\and(P_{2}^{0})\and(P_{1}^{2}P_{3}^{0})\and(P_{1}^{2}P_{2}% ^{0}P_{3}^{0})\and(P_{2}^{0})\\ (P_{1}^{2}P_{3}^{0})\and(P_{2}^{0})\and(P_{1}^{2}P_{3}^{0})\and(P_{1}^{2}P_{2}% ^{0}P_{3}^{0})\and(P_{2}^{0})\end{cases}
  261. P 4 = 1 P_{4}=1
  262. O 10 O_{10}
  263. P 1 P_{1}
  264. P 3 P_{3}
  265. P 2 P_{2}
  266. P 1 P_{1}
  267. P 3 P_{3}
  268. P 1 P_{1}
  269. P 2 P_{2}
  270. P 3 P_{3}
  271. P 2 P_{2}
  272. ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 ) and ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 P 3 0 ) and ( P 1 1 P 2 2 ) (P_{1}^{1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2})\and(P_{1}^{1}P_{2}^{2}P_% {3}^{0})\and(P_{1}^{1}P_{2}^{2}P_{3}^{0})\and(P_{1}^{1}P_{2}^{2})
  273. { O 1 , O 2 } \{O_{1},O_{2}\}
  274. P 1 1 P 2 2 P_{1}^{1}P_{2}^{2}
  275. ( P 1 = 1 ) ( P 2 = 2 ) ( P 4 = 1 ) (P_{1}=1)(P_{2}=2)\to(P_{4}=1)
  276. ( P 1 2 P 3 0 ) and ( P 2 0 ) and ( P 1 2 P 3 0 ) and ( P 1 2 P 2 0 P 3 0 ) and ( P 2 0 ) (P_{1}^{2}P_{3}^{0})\and(P_{2}^{0})\and(P_{1}^{2}P_{3}^{0})\and(P_{1}^{2}P_{2}% ^{0}P_{3}^{0})\and(P_{2}^{0})
  277. { O 3 , O 7 , O 10 } \{O_{3},O_{7},O_{10}\}
  278. P 1 2 P 2 0 P 3 0 P 2 0 P_{1}^{2}P_{2}^{0}P_{3}^{0}P_{2}^{0}
  279. ( P 1 = 2 and P 2 = 0 ) ( P 3 = 0 and P 2 = 0 ) ( P 4 = 1 ) (P_{1}=2\and P_{2}=0)(P_{3}=0\and P_{2}=0)\to(P_{4}=1)
  280. { ( P 1 = 1 ) ( P 4 = 1 ) ( P 2 = 2 ) ( P 4 = 1 ) ( P 1 = 2 ) and ( P 2 = 0 ) ( P 4 = 1 ) ( P 3 = 0 ) and ( P 2 = 0 ) ( P 4 = 1 ) \begin{cases}(P_{1}=1)\to(P_{4}=1)\\ (P_{2}=2)\to(P_{4}=1)\\ (P_{1}=2)\and(P_{2}=0)\to(P_{4}=1)\\ (P_{3}=0)\and(P_{2}=0)\to(P_{4}=1)\end{cases}
  281. P 4 = 2 P_{4}=2
  282. P 4 = 2 P_{4}=2
  283. X X
  284. ( d , w ) (d,w)
  285. X X
  286. T T
  287. t = ( a , v ) t=(a,v)
  288. [ T ] = t T [ t ] X . \emptyset\neq[T]=\bigcap_{t\in T}[t]\subseteq X.
  289. T T
  290. X X
  291. X X
  292. T T
  293. S S
  294. T T
  295. X X
  296. S S
  297. 𝕋 \mathbb{T}
  298. 𝕋 \mathbb{T}
  299. X X
  300. T T
  301. 𝕋 \mathbb{T}
  302. X X
  303. t 𝕋 [ T ] = X , \bigcup_{t\in\mathbb{T}}[T]=X,
  304. 𝕋 \mathbb{T}
  305. 𝕋 \mathbb{T}
  306. { ( P 1 , 1 ) ( P 4 , 1 ) ( P 5 , 0 ) ( P 4 , 1 ) ( P 1 , 0 ) ( P 4 , 2 ) ( P 2 , 1 ) ( P 4 , 2 ) \begin{cases}(P_{1},1)\to(P_{4},1)\\ (P_{5},0)\to(P_{4},1)\\ (P_{1},0)\to(P_{4},2)\\ (P_{2},1)\to(P_{4},2)\end{cases}
  307. α \textstyle\alpha
  308. β \textstyle\beta
  309. x x
  310. X X
  311. \R \textstyle\R
  312. x x
  313. X X
  314. x x
  315. \R \textstyle\R

Route_assignment.html

  1. S a ( v a ) = t a ( 1 + 0.15 ( < m t p l > v a c a ) 4 ) S_{a}\left({v_{a}}\right)=t_{a}\left({1+0.15\left({\frac{<}{m}tpl>{{v_{a}}}{{c% _{a}}}}\right)^{4}}\right)
  2. min a 0 v a S a ( x ) d x \min\sum_{a}{\int_{0}^{v_{a}}{S_{a}\left(x\right)}}dx
  3. v a = i j r α i j a r x i j r v_{a}=\sum_{i}{\sum_{j}{\sum_{r}{\alpha_{ij}^{ar}x_{ij}^{r}}}}
  4. r x i j r = T i j \sum_{r}{x_{ij}^{r}=T_{ij}}
  5. v a 0 , x i j r 0 v_{a}\geq 0,\;x_{ij}^{r}\geq 0
  6. x i j r x_{ij}^{r}
  7. α i j a r \alpha_{ij}^{ar}
  8. S a = 15 ( 1 + 0.15 ( < m t p l > v a 1000 ) 4 ) S_{a}=15\left({1+0.15\left({\frac{<}{m}tpl>{{v_{a}}}{{1000}}}\right)^{4}}\right)
  9. S b = 20 ( 1 + 0.15 ( < m t p l > v b 3000 ) 4 ) S_{b}=20\left({1+0.15\left({\frac{<}{m}tpl>{{v_{b}}}{{3000}}}\right)^{4}}\right)
  10. v a + v b = 8000 v_{a}+v_{b}=8000
  11. t i j c i j = C t_{ij}c_{ij}=C
  12. c i j c_{ij}
  13. t i j t_{ij}

Routh–Hurwitz_stability_criterion.html

  1. p - q = w ( + ) - w ( - ) p-q=w(+\infty)-w(-\infty)
  2. P 0 ( y ) P_{0}(y)
  3. P 1 ( y ) P_{1}(y)
  4. f ( i y ) = P 0 ( y ) + i P 1 ( y ) f(iy)=P_{0}(y)+iP_{1}(y)
  5. P 0 ( y ) P_{0}(y)
  6. P 1 ( y ) P_{1}(y)
  7. f ( i y ) = P 0 ( y ) + i P 1 ( y ) f(iy)=P_{0}(y)+iP_{1}(y)
  8. P 0 ( y ) P_{0}(y)
  9. P 1 ( y ) P_{1}(y)
  10. f ( z ) = a z 2 + b z + c f(z)=az^{2}+bz+c
  11. c 0 c\neq 0
  12. P 0 ( y ) P_{0}(y)
  13. P 1 ( y ) P_{1}(y)
  14. f ( i y ) = - a y 2 + i b y + c = P 0 ( y ) + i P 1 ( y ) = - a y 2 + c + i ( b y ) . f(iy)=-ay^{2}+iby+c=P_{0}(y)+iP_{1}(y)=-ay^{2}+c+i(by).
  15. P 0 ( y ) = ( ( - a / b ) y ) P 1 ( y ) + c , P_{0}(y)=((-a/b)y)P_{1}(y)+c,
  16. P 2 ( y ) = - c , P_{2}(y)=-c,
  17. P 1 ( y ) = ( ( - b / c ) y ) P 2 ( y ) , P_{1}(y)=((-b/c)y)P_{2}(y),
  18. P 3 ( y ) = 0 P_{3}(y)=0
  19. ( P 0 ( y ) , P 1 ( y ) , P 2 ( y ) ) = ( c - a y 2 , b y , - c ) (P_{0}(y),P_{1}(y),P_{2}(y))=(c-ay^{2},by,-c)
  20. y = + y=+\infty
  21. c - a y 2 c-ay^{2}
  22. y = - y=-\infty
  23. w ( + ) - w ( - ) = 2 w(+\infty)-w(-\infty)=2
  24. w ( + ) = 2 w(+\infty)=2
  25. w ( - ) = 0 w(-\infty)=0
  26. a 2 a_{2}
  27. - 1 -1
  28. P ( s ) = a 2 s 2 + a 1 s + a 0 = 0 P(s)=a_{2}s^{2}+a_{1}s+a_{0}=0
  29. P ( s ) P(s)
  30. a n > 0 a_{n}>0
  31. P ( s ) = a 3 s 3 + a 2 s 2 + a 1 s + a 0 = 0 P(s)=a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}=0
  32. a n > 0 a_{n}>0
  33. a 2 a 1 > a 3 a 0 a_{2}a_{1}>a_{3}a_{0}
  34. P ( s ) = a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = 0 P(s)=a_{4}s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}=0
  35. a n > 0 a_{n}>0
  36. a 3 a 2 > a 4 a 1 a_{3}a_{2}>a_{4}a_{1}
  37. a 3 a 2 a 1 > a 4 a 1 2 + a 3 2 a 0 a_{3}a_{2}a_{1}>a_{4}a_{1}^{2}+a_{3}^{2}a_{0}
  38. D ( s ) = a n s n + a n - 1 s n - 1 + + a 1 s + a 0 D(s)=a_{n}s^{n}+a_{n-1}s^{n-1}+\cdots+a_{1}s+a_{0}
  39. a n a_{n}
  40. a n - 1 a_{n-1}
  41. b 1 b_{1}
  42. c 1 c_{1}
  43. \vdots
  44. b i b_{i}
  45. c i c_{i}
  46. b i = a n - 1 × a n - 2 i - a n × a n - 2 i - 1 a n - 1 . b_{i}=\frac{a_{n-1}\times{a_{n-2i}}-a_{n}\times{a_{n-2i-1}}}{a_{n-1}}.
  47. c i = b 1 × a n - 2 i - 1 - a n - 1 × b i + 1 b 1 . c_{i}=\frac{b_{1}\times{a_{n-2i-1}}-a_{n-1}\times{b_{i+1}}}{b_{1}}.
  48. s 6 + 2 s 5 + 8 s 4 + 12 s 3 + 20 s 2 + 16 s + 16 = 0. s^{6}+2s^{5}+8s^{4}+12s^{3}+20s^{2}+16s+16=0.\,
  49. A ( s ) = 2 s 4 + 12 s 2 + 16. A(s)=2s^{4}+12s^{2}+16.\,
  50. B ( s ) = 8 s 3 + 24 s 1 . B(s)=8s^{3}+24s^{1}.\,

Row-major_order.html

  1. [ 11 12 13 21 22 23 ] \begin{bmatrix}11&12&13\\ 21&22&23\end{bmatrix}
  2. N 1 × N 2 × × N d N_{1}\times N_{2}\times\cdots\times N_{d}
  3. ( n 1 , n 2 , , n d ) (n_{1},n_{2},\ldots,n_{d})
  4. n k [ 0 , N k - 1 ] n_{k}\in[0,N_{k}-1]
  5. n d + N d ( n d - 1 + N d - 1 ( n d - 2 + N d - 2 ( + N 2 n 1 ) ) ) ) = k = 1 d ( = k + 1 d N ) n k n_{d}+N_{d}\cdot(n_{d-1}+N_{d-1}\cdot(n_{d-2}+N_{d-2}\cdot(\cdots+N_{2}n_{1})% \cdots)))=\sum_{k=1}^{d}\left(\prod_{\ell=k+1}^{d}N_{\ell}\right)n_{k}
  6. n 1 + N 1 ( n 2 + N 2 ( n 3 + N 3 ( + N d - 1 n d ) ) ) ) = k = 1 d ( = 1 k - 1 N ) n k n_{1}+N_{1}\cdot(n_{2}+N_{2}\cdot(n_{3}+N_{3}\cdot(\cdots+N_{d-1}n_{d})\cdots)% ))=\sum_{k=1}^{d}\left(\prod_{\ell=1}^{k-1}N_{\ell}\right)n_{k}

Row_space.html

  1. c 1 𝐫 1 + c 2 𝐫 2 + + c m 𝐫 m , c_{1}\mathbf{r}_{1}+c_{2}\mathbf{r}_{2}+\cdots+c_{m}\mathbf{r}_{m},
  2. A = [ 1 0 2 0 1 0 ] , A=\begin{bmatrix}1&0&2\\ 0&1&0\end{bmatrix},
  3. c 1 ( 1 , 0 , 2 ) + c 2 ( 0 , 1 , 0 ) = ( c 1 , c 2 , 2 c 1 ) . c_{1}(1,0,2)+c_{2}(0,1,0)=(c_{1},c_{2},2c_{1}).\,
  4. A = [ 1 3 2 2 7 4 1 5 2 ] . A=\begin{bmatrix}1&3&2\\ 2&7&4\\ 1&5&2\end{bmatrix}.
  5. [ 1 3 2 2 7 4 1 5 2 ] r 2 - 2 r 1 [ 1 3 2 0 1 0 1 5 2 ] r 3 - r 1 [ 1 3 2 0 1 0 0 2 0 ] r 3 - 2 r 2 [ 1 3 2 0 1 0 0 0 0 ] r 1 - 3 r 2 [ 1 0 2 0 1 0 0 0 0 ] . \begin{bmatrix}1&3&2\\ 2&7&4\\ 1&5&2\end{bmatrix}\underbrace{\sim}_{r_{2}-2r_{1}}\begin{bmatrix}1&3&2\\ 0&1&0\\ 1&5&2\end{bmatrix}\underbrace{\sim}_{r_{3}-r_{1}}\begin{bmatrix}1&3&2\\ 0&1&0\\ 0&2&0\end{bmatrix}\underbrace{\sim}_{r_{3}-2r_{2}}\begin{bmatrix}1&3&2\\ 0&1&0\\ 0&0&0\end{bmatrix}\underbrace{\sim}_{r_{1}-3r_{2}}\begin{bmatrix}1&0&2\\ 0&1&0\\ 0&0&0\end{bmatrix}.
  6. rank ( A ) + nullity ( A ) = n , \operatorname{rank}(A)+\operatorname{nullity}(A)=n,
  7. A 𝐱 = [ 𝐫 1 𝐱 𝐫 2 𝐱 𝐫 m 𝐱 ] , A\mathbf{x}=\begin{bmatrix}\mathbf{r}_{1}\cdot\mathbf{x}\\ \mathbf{r}_{2}\cdot\mathbf{x}\\ \vdots\\ \mathbf{r}_{m}\cdot\mathbf{x}\end{bmatrix},

Rössler_attractor.html

  1. x , y x,y
  2. z z
  3. { d x d t = - y - z d y d t = x + a y d z d t = b + z ( x - c ) \begin{cases}\frac{dx}{dt}=-y-z\\ \frac{dy}{dt}=x+ay\\ \frac{dz}{dt}=b+z(x-c)\end{cases}
  4. a = 0.2 a=0.2
  5. b = 0.2 b=0.2
  6. c = 5.7 c=5.7
  7. a = 0.1 a=0.1
  8. b = 0.1 b=0.1
  9. c = 14 c=14
  10. b = 2 b=2
  11. c = 4 c=4
  12. a a
  13. x , y x,y
  14. a = 0.2 a=0.2
  15. b = 0.2 b=0.2
  16. c = 5.7 c=5.7
  17. z = 0 z=0
  18. x , y x,y
  19. { d x d t = - y d y d t = x + a y \begin{cases}\frac{dx}{dt}=-y\\ \frac{dy}{dt}=x+ay\end{cases}
  20. x , y x,y
  21. ( 0 - 1 1 a ) \begin{pmatrix}0&-1\\ 1&a\\ \end{pmatrix}
  22. ( a ± a 2 - 4 ) / 2 (a\pm\sqrt{a^{2}-4})/2
  23. 0 < a < 2 0<a<2
  24. x , y x,y
  25. z z
  26. a a
  27. x x
  28. c c
  29. c c
  30. x , y x,y
  31. x x
  32. c c
  33. z z
  34. z z
  35. - z -z
  36. d x / d t dx/dt
  37. x x
  38. x x
  39. y y
  40. z z
  41. { x = c ± c 2 - 4 a b 2 y = - ( c ± c 2 - 4 a b 2 a ) z = c ± c 2 - 4 a b 2 a \begin{cases}x=\frac{c\pm\sqrt{c^{2}-4ab}}{2}\\ y=-\left(\frac{c\pm\sqrt{c^{2}-4ab}}{2a}\right)\\ z=\frac{c\pm\sqrt{c^{2}-4ab}}{2a}\end{cases}
  42. ( c + c 2 - 4 a b 2 , - c - c 2 - 4 a b 2 a , c + c 2 - 4 a b 2 a ) \left(\frac{c+\sqrt{c^{2}-4ab}}{2},\frac{-c-\sqrt{c^{2}-4ab}}{2a},\frac{c+% \sqrt{c^{2}-4ab}}{2a}\right)
  43. ( c - c 2 - 4 a b 2 , - c + c 2 - 4 a b 2 a , c - c 2 - 4 a b 2 a ) \left(\frac{c-\sqrt{c^{2}-4ab}}{2},\frac{-c+\sqrt{c^{2}-4ab}}{2a},\frac{c-% \sqrt{c^{2}-4ab}}{2a}\right)
  44. ( 0 - 1 - 1 1 a 0 z 0 x - c ) \begin{pmatrix}0&-1&-1\\ 1&a&0\\ z&0&x-c\\ \end{pmatrix}
  45. - λ 3 + λ 2 ( a + x - c ) + λ ( a c - a x - 1 - z ) + x - c + a z = 0 -\lambda^{3}+\lambda^{2}(a+x-c)+\lambda(ac-ax-1-z)+x-c+az=0\,
  46. λ 1 = 0.0971028 + 0.995786 i \lambda_{1}=0.0971028+0.995786i\,
  47. λ 2 = 0.0971028 - 0.995786 i \lambda_{2}=0.0971028-0.995786i\,
  48. λ 3 = - 5.68718 \lambda_{3}=-5.68718\,
  49. v 1 = ( 0.7073 - 0.07278 - 0.7032 i 0.0042 - 0.0007 i ) v_{1}=\begin{pmatrix}0.7073\\ -0.07278-0.7032i\\ 0.0042-0.0007i\\ \end{pmatrix}
  50. v 2 = ( 0.7073 0.07278 + 0.7032 i 0.0042 + 0.0007 i ) v_{2}=\begin{pmatrix}0.7073\\ 0.07278+0.7032i\\ 0.0042+0.0007i\\ \end{pmatrix}
  51. v 3 = ( 0.1682 - 0.0286 0.9853 ) v_{3}=\begin{pmatrix}0.1682\\ -0.0286\\ 0.9853\\ \end{pmatrix}
  52. a = 0.2 a=0.2
  53. b = 0.2 b=0.2
  54. c = 5.7 c=5.7
  55. v 1 v_{1}
  56. v 2 v_{2}
  57. a = 0.2 a=0.2
  58. b = 0.2 b=0.2
  59. c = 5.7 c=5.7
  60. F P 1 FP_{1}
  61. v 1 v_{1}
  62. v 2 v_{2}
  63. v 3 v_{3}
  64. F P 1 FP_{1}
  65. v 3 v_{3}
  66. v 1 v_{1}
  67. v 2 v_{2}
  68. F P 1 FP_{1}
  69. v 3 v_{3}
  70. a = 0.2 a=0.2
  71. b = 0.2 b=0.2
  72. c = 5.7 c=5.7
  73. λ 1 = - 0.0000046 + 5.4280259 i \lambda_{1}=-0.0000046+5.4280259i
  74. λ 2 = - 0.0000046 - 5.4280259 i \lambda_{2}=-0.0000046-5.4280259i
  75. λ 3 = 0.1929830 \lambda_{3}=0.1929830
  76. v 1 = ( 0.0002422 + 0.1872055 i 0.0344403 - 0.0013136 i 0.9817159 ) v_{1}=\begin{pmatrix}0.0002422+0.1872055i\\ 0.0344403-0.0013136i\\ 0.9817159\\ \end{pmatrix}
  77. v 2 = ( 0.0002422 - 0.1872055 i 0.0344403 + 0.0013136 i 0.9817159 ) v_{2}=\begin{pmatrix}0.0002422-0.1872055i\\ 0.0344403+0.0013136i\\ 0.9817159\\ \end{pmatrix}
  78. v 3 = ( 0.0049651 - 0.7075770 0.7066188 ) v_{3}=\begin{pmatrix}0.0049651\\ -0.7075770\\ 0.7066188\\ \end{pmatrix}
  79. λ 1 \lambda_{1}
  80. λ 2 \lambda_{2}
  81. a = 0.1 a=0.1
  82. b = 0.1 b=0.1
  83. c = 14 c=14
  84. y , z y,z
  85. x = 0 x=0
  86. x x
  87. x = 0 x=0
  88. x = 0 x=0
  89. z = 0 z=0
  90. x = 0.1 x=0.1
  91. a = 0.1 a=0.1
  92. b = 0.1 b=0.1
  93. c = 14 c=14
  94. z z
  95. x x
  96. c c
  97. c c
  98. c c
  99. Z n Z_{n}
  100. Z n + 1 Z_{n+1}
  101. z z
  102. z n z_{n}
  103. z z
  104. z n + 1 z_{n+1}
  105. a = 0.2 a=0.2
  106. b = 0.2 b=0.2
  107. c = 5.7 c=5.7
  108. a a
  109. b b
  110. c c
  111. b b
  112. c c
  113. a a
  114. a a
  115. a 0 a\leq 0
  116. a = 0.1 a=0.1
  117. a = 0.2 a=0.2
  118. a = 0.3 a=0.3
  119. a = 0.35 a=0.35
  120. a = 0.38 a=0.38
  121. a a
  122. c c
  123. b b
  124. b b
  125. b b
  126. b b
  127. a a
  128. c c
  129. b b
  130. c c
  131. a = b = 0.1 a=b=0.1
  132. c c
  133. c c
  134. c c
  135. c c
  136. c c
  137. c c
  138. c = 12 c=12
  139. c c
  140. c c
  141. a = b = .1 a=b=.1
  142. c = 4 c=4
  143. c = 6 c=6
  144. c = 8.5 c=8.5
  145. c = 8.7 c=8.7
  146. c = 9 c=9
  147. c = 12 c=12
  148. c = 12.6 c=12.6
  149. c = 13 c=13
  150. c = 18 c=18

Rule_of_product.html

  1. { A , B , C } { X , Y } To choose one of these AND one of these \begin{matrix}&\underbrace{\left\{A,B,C\right\}}&&\underbrace{\left\{X,Y\right% \}}\\ \mathrm{To}\ \mathrm{choose}\ \mathrm{one}\ \mathrm{of}&\mathrm{these}&\mathrm% {AND}\ \mathrm{one}\ \mathrm{of}&\mathrm{these}\end{matrix}
  2. is to choose one of these . { A X , A Y , B X , B Y , C X , C Y } \begin{matrix}\mathrm{is}\ \mathrm{to}\ \mathrm{choose}\ \mathrm{one}\ \mathrm% {of}&\mathrm{these}.\\ &\overbrace{\left\{AX,AY,BX,BY,CX,CY\right\}}\end{matrix}
  3. | S 1 | | S 2 | | S n | = | S 1 × S 2 × × S n | |S_{1}|\cdot|S_{2}|\cdots|S_{n}|=|S_{1}\times S_{2}\times\cdots\times S_{n}|
  4. × \times

Rule_of_sum.html

  1. S 1 , S 2 , , S n S_{1},S_{2},...,S_{n}
  2. | S 1 | + | S 2 | + + | S n | = | S 1 S 2 S n | |S_{1}|+|S_{2}|+\cdots+|S_{n}|=|S_{1}\cup S_{2}\cup\cdots\cup S_{n}|
  3. | i = 1 n A i | = i = 1 n | A i | - i , j : 1 i < j n | A i A j | + i , j , k : 1 i < j < k n | A i A j A k | - + ( - 1 ) n - 1 | A 1 A n | . \begin{aligned}\displaystyle\biggl|\bigcup_{i=1}^{n}A_{i}\biggr|&\displaystyle% {}=\sum_{i=1}^{n}\left|A_{i}\right|-\sum_{i,j\,:\,1\leq i<j\leq n}\left|A_{i}% \cap A_{j}\right|\\ &\displaystyle{}\qquad+\sum_{i,j,k\,:\,1\leq i<j<k\leq n}\left|A_{i}\cap A_{j}% \cap A_{k}\right|-\ \cdots\ +\left(-1\right)^{n-1}\left|A_{1}\cap\cdots\cap A_% {n}\right|.\end{aligned}

Running_angle.html

  1. ϕ ( t ) = arctan ( Δ Y t Δ X t ) . \phi(t)=\arctan\left(\frac{\Delta Yt}{\Delta Xt}\right).

Rydberg_atom.html

  1. 𝐅 = m 𝐚 k e 2 r 2 = m v 2 r \mathbf{F}=m\mathbf{a}\Rightarrow{ke^{2}\over r^{2}}={mv^{2}\over r}
  2. m v r = n mvr=n\hbar
  3. r = n 2 2 k e 2 m . r={n^{2}\hbar^{2}\over ke^{2}m}.
  4. e - + A A * + e - e^{-}+A\rightarrow A^{*}+e^{-}
  5. A + + B A * + B + A^{+}+B\rightarrow A^{*}+B^{+}
  6. A + γ A * A+\gamma\rightarrow A^{*}
  7. U C = - e 2 4 π ε 0 r U\text{C}=-\dfrac{e^{2}}{4\pi\varepsilon_{0}r}
  8. U e e = e 2 4 π ε 0 i < j 1 | 𝐫 i - 𝐫 j | U_{ee}=\dfrac{e^{2}}{4\pi\varepsilon_{0}}\sum_{i<j}\dfrac{1}{|\mathbf{r}_{i}-% \mathbf{r}_{j}|}
  9. U pol = - e 2 α d ( 4 π ε 0 ) 2 r 4 U\text{pol}=-\dfrac{e^{2}\alpha\text{d}}{(4\pi\varepsilon_{0})^{2}r^{4}}
  10. E B = - R y n 2 E\text{B}=-\dfrac{Ry}{n^{2}}
  11. E B = - R y ( n - δ l ) 2 E\text{B}=-\dfrac{Ry}{(n-\delta_{l})^{2}}
  12. E S = - 𝐝 𝐅 . E\text{S}=-\mathbf{d}\cdot\mathbf{F}.
  13. F IT = e 12 π ε 0 a 0 2 n 5 . F\text{IT}=\dfrac{e}{12\pi\varepsilon_{0}a_{0}^{2}n^{5}}.
  14. χ \chi
  15. | τ | = | 𝐫 × 𝐅 | = | 𝐫 | | 𝐅 | sin θ |\mathbf{\tau}|=|\mathbf{r}\times\mathbf{F}|=|\mathbf{r}||\mathbf{F}|\sin\theta
  16. θ = π τ = 0 \theta=\pi\Rightarrow\mathbf{\tau}=0

S-wave.html

  1. 𝐮 = 0 \nabla\cdot\mathbf{u}=0
  2. τ i j = λ δ i j e k k + 2 μ e i j \tau_{ij}=\lambda\delta_{ij}e_{kk}+2\mu e_{ij}
  3. τ \tau
  4. λ \lambda
  5. μ \mu
  6. μ \mu
  7. δ i j \delta_{ij}
  8. e i j = 1 2 ( i u j + j u i ) e_{ij}=\frac{1}{2}\left(\partial_{i}u_{j}+\partial_{j}u_{i}\right)
  9. τ i j = λ δ i j k u k + μ ( i u j + j u i ) \tau_{ij}=\lambda\delta_{ij}\partial_{k}u_{k}+\mu\left(\partial_{i}u_{j}+% \partial_{j}u_{i}\right)
  10. ρ 2 u i t 2 = j τ i j \rho\frac{\partial^{2}u_{i}}{\partial t^{2}}=\partial_{j}\tau_{ij}
  11. ρ \rho
  12. ρ 2 u i t 2 \displaystyle\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}
  13. ρ s y m b o l u ¨ = ( λ + 2 μ ) ( \cdotsymbol u ) - μ × ( × s y m b o l u ) \rho\ddot{symbol{u}}=\left(\lambda+2\mu\right)\nabla(\nabla\cdotsymbol{u})-\mu% \nabla\times(\nabla\times symbol{u})
  14. 2 ( \timessymbol u ) - 1 β 2 2 t 2 ( \timessymbol u ) = 0 \nabla^{2}(\nabla\timessymbol{u})-\frac{1}{\beta^{2}}\frac{\partial^{2}}{% \partial t^{2}}\left(\nabla\timessymbol{u}\right)=0
  15. β \beta
  16. β 2 = μ ρ \beta^{2}=\frac{\mu}{\rho}
  17. ( 2 + k 2 ) s y m b o l u = 0 (\nabla^{2}+k^{2})symbol{u}=0

S3_Texture_Compression.html

  1. c 0 c_{0}
  2. c 1 c_{1}
  3. c 0 > c 1 c_{0}>c_{1}
  4. c 2 = 2 3 c 0 + 1 3 c 1 c_{2}={2\over 3}c_{0}+{1\over 3}c_{1}
  5. c 3 = 1 3 c 0 + 2 3 c 1 c_{3}={1\over 3}c_{0}+{2\over 3}c_{1}
  6. c 0 c 1 c_{0}\leq c_{1}
  7. c 2 = 1 2 c 0 + 1 2 c 1 c_{2}={1\over 2}c_{0}+{1\over 2}c_{1}
  8. c 3 c_{3}
  9. c 0 c_{0}
  10. c 3 c_{3}
  11. c 0 c_{0}
  12. c 1 c_{1}
  13. α 0 > α 1 \alpha_{0}>\alpha_{1}
  14. α 2 = 6 α 0 + 1 α 1 7 \alpha_{2}={{6\alpha_{0}+1\alpha_{1}}\over 7}
  15. α 3 = 5 α 0 + 2 α 1 7 \alpha_{3}={{5\alpha_{0}+2\alpha_{1}}\over 7}
  16. α 4 = 4 α 0 + 3 α 1 7 \alpha_{4}={{4\alpha_{0}+3\alpha_{1}}\over 7}
  17. α 5 = 3 α 0 + 4 α 1 7 \alpha_{5}={{3\alpha_{0}+4\alpha_{1}}\over 7}
  18. α 6 = 2 α 0 + 5 α 1 7 \alpha_{6}={{2\alpha_{0}+5\alpha_{1}}\over 7}
  19. α 7 = 1 α 0 + 6 α 1 7 \alpha_{7}={{1\alpha_{0}+6\alpha_{1}}\over 7}
  20. α 0 α 1 \alpha_{0}\leq\alpha_{1}
  21. α 2 = 4 α 0 + 1 α 1 5 \alpha_{2}={{4\alpha_{0}+1\alpha_{1}}\over 5}
  22. α 3 = 3 α 0 + 2 α 1 5 \alpha_{3}={{3\alpha_{0}+2\alpha_{1}}\over 5}
  23. α 4 = 2 α 0 + 3 α 1 5 \alpha_{4}={{2\alpha_{0}+3\alpha_{1}}\over 5}
  24. α 5 = 1 α 0 + 4 α 1 5 \alpha_{5}={{1\alpha_{0}+4\alpha_{1}}\over 5}
  25. α 6 = 0 \alpha_{6}=0
  26. α 7 = 255 \alpha_{7}=255
  27. α 0 \alpha_{0}
  28. α 7 \alpha_{7}

Sackur–Tetrode_equation.html

  1. S = k N ( ln [ V N ( 4 π m 3 h 2 U N ) 3 / 2 ] + 5 2 ) S=kN\left(\ln\left[\frac{V}{N}\left(\frac{4\pi m}{3h^{2}}\frac{U}{N}\right)^{3% /2}\right]+{\frac{5}{2}}\right)
  2. Λ \Lambda
  3. S k N = ln [ V N Λ 3 ] + 5 2 . \frac{S}{kN}=\ln\left[\frac{V}{N\Lambda^{3}}\right]+\frac{5}{2}.
  4. V N Λ 3 1 \frac{V}{N\Lambda^{3}}\gg 1

Sagnac_effect.html

  1. ω \omega
  2. t 1 t_{1}
  3. t 1 = 2 π R + Δ L c t_{1}=\frac{2\pi R+\Delta L}{c}
  4. Δ L \Delta L
  5. Δ L = R ω t 1 . \Delta L=R\omega t_{1}.\,
  6. Δ L \Delta L
  7. t 1 = 2 π R c - R ω . t_{1}=\frac{2\pi R}{c-R\omega}.
  8. t 2 = 2 π R c + R ω . t_{2}=\frac{2\pi R}{c+R\omega}.
  9. Δ t = t 1 - t 2 = 4 π R 2 ω c 2 - R 2 ω 2 . \Delta t=t_{1}-t_{2}=\frac{4\pi R^{2}\omega}{c^{2}-R^{2}\omega^{2}}.
  10. R ω = v c R\omega=v\ll c
  11. Δ t 4 π R 2 ω c 2 = 4 A ω c 2 , \Delta t\approx\frac{4\pi R^{2}\omega}{c^{2}}=\frac{4A\omega}{c^{2}},
  12. Δ ϕ = 2 π c Δ t λ \Delta\phi=\frac{2\pi c\Delta t}{\lambda}
  13. A A
  14. ω \omega
  15. x = r cos ( θ + ω t ) x=r\cos\left(\theta+\omega t\right)
  16. y = r sin ( θ + ω t ) y=r\sin\left(\theta+\omega t\right)
  17. d s 2 = ( c 2 - r 2 ω 2 ) d t 2 - d r 2 - r 2 d θ 2 - d z 2 - 2 r 2 ω d t d θ ds^{2}=(c^{2}-r^{2}\omega^{2})\,dt^{2}-dr^{2}-r^{2}\,d\theta^{2}-dz^{2}-2r^{2}% \omega\,dt\,d\theta
  18. t t
  19. r r
  20. θ \theta
  21. z z
  22. ω \omega
  23. c ± r ω c\pm r\omega
  24. ω = 0 \omega=0
  25. ω 0 \omega\neq 0
  26. r = 0 r=0
  27. c c

Saha_ionization_equation.html

  1. n i + 1 n e n i = 2 Λ 3 g i + 1 g i exp [ - ( ϵ i + 1 - ϵ i ) k B T ] \frac{n_{i+1}n_{e}}{n_{i}}=\frac{2}{\Lambda^{3}}\frac{g_{i+1}}{g_{i}}\exp\left% [-\frac{(\epsilon_{i+1}-\epsilon_{i})}{k_{B}T}\right]
  2. n i n_{i}\,
  3. g i g_{i}\,
  4. ϵ i \epsilon_{i}\,
  5. n e n_{e}\,
  6. Λ \Lambda\,
  7. Λ = def h 2 2 π m e k B T \Lambda\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{h^{2}}{2\pi m_{e}k_{B}T}}
  8. m e m_{e}\,
  9. T T\,
  10. k B k_{B}\,
  11. h h\,
  12. ( ϵ i + 1 - ϵ i ) (\epsilon_{i+1}-\epsilon_{i})
  13. ( i + 1 ) t h (i+1)^{th}
  14. n 1 = n e n_{1}=n_{e}
  15. n = n 0 + n 1 n=n_{0}+n_{1}
  16. n e 2 n - n e = 2 Λ 3 g 1 g 0 exp [ - ϵ k B T ] \frac{n_{e}^{2}}{n-n_{e}}=\frac{2}{\Lambda^{3}}\frac{g_{1}}{g_{0}}\exp\left[% \frac{-\epsilon}{k_{B}T}\right]
  17. ϵ \epsilon
  18. Z i N i = Z i + 1 Z e N i + 1 N e \frac{Z_{i}}{N_{i}}=\frac{Z_{i+1}Z_{e}}{N_{i+1}N_{e}}
  19. μ i = μ i + 1 + μ e \mu_{i}=\mu_{i+1}+\mu_{e}\,

Saltwater_intrusion.html

  1. z = ρ f ( ρ s - ρ f ) h z=\frac{\rho_{f}}{(\rho_{s}-\rho_{f})}h
  2. h h
  3. z z
  4. h h
  5. z z
  6. ρ f \rho_{f}
  7. ρ s \rho_{s}
  8. ρ f \rho_{f}
  9. ρ s \rho_{s}
  10. z = 40 h z\ =40h

Sample_size_determination.html

  1. p ^ = X / n \hat{p}=X/n
  2. p ^ \hat{p}
  3. ( p ^ - 2 0.25 / n , p ^ + 2 0.25 / n ) (\hat{p}-2\sqrt{0.25/n},\hat{p}+2\sqrt{0.25/n})
  4. 4 0.25 / n = W 4\sqrt{0.25/n}=W
  5. σ / n . \sigma/\sqrt{n}.
  6. ( x ¯ - 2 σ / n , x ¯ + 2 σ / n ) . (\bar{x}-2\sigma/\sqrt{n},\bar{x}+2\sigma/\sqrt{n}).
  7. 4 σ / n = W 4\sigma/\sqrt{n}=W
  8. E = N - B - T , E=N-B-T,
  9. H 0 : μ = 0 H_{0}:\mu=0
  10. H a : μ = μ * H_{a}:\mu=\mu^{*}
  11. Pr ( x ¯ > z α σ / n | H 0 true ) = α \Pr(\bar{x}>z_{\alpha}\sigma/\sqrt{n}|H_{0}\,\text{ true})=\alpha
  12. x ¯ \bar{x}
  13. z α σ / n z_{\alpha}\sigma/\sqrt{n}
  14. Pr ( x ¯ > z α σ / n | H a true ) 1 - β \Pr(\bar{x}>z_{\alpha}\sigma/\sqrt{n}|H_{a}\,\text{ true})\geq 1-\beta
  15. n ( z α + Φ - 1 ( 1 - β ) μ * / σ ) 2 n\geq\left(\frac{z_{\alpha}+\Phi^{-1}(1-\beta)}{\mu^{*}/\sigma}\right)^{2}
  16. Φ \Phi
  17. x ¯ w = h = 1 H W h x ¯ h , \bar{x}_{w}=\sum_{h=1}^{H}W_{h}\bar{x}_{h},
  18. Var ( x ¯ w ) = h = 1 H W h 2 Var ( x ¯ h ) . \operatorname{Var}(\bar{x}_{w})=\sum_{h=1}^{H}W_{h}^{2}\,\operatorname{Var}(% \bar{x}_{h}).
  19. W h W_{h}
  20. W h = N h / N W_{h}=N_{h}/N
  21. N = N h N=\sum{N_{h}}
  22. Var ( x ¯ w ) = h = 1 H W h 2 V a r h ( 1 n h - 1 N h ) , \operatorname{Var}(\bar{x}_{w})=\sum_{h=1}^{H}W_{h}^{2}\,Var_{h}\left(\frac{1}% {n_{h}}-\frac{1}{N_{h}}\right),
  23. n h / N h = k S h n_{h}/N_{h}=kS_{h}
  24. S h = V a r h S_{h}=\sqrt{Var_{h}}
  25. k k
  26. n h = n \sum{n_{h}}=n
  27. C h C_{h}
  28. n h N h = K S h C h , \frac{n_{h}}{N_{h}}=\frac{KS_{h}}{\sqrt{C_{h}}},
  29. K K
  30. n h = n \sum{n_{h}}=n
  31. n h = K W h S h C h . n_{h}=\frac{K^{\prime}W_{h}S_{h}}{\sqrt{C_{h}}}.

Saturation_arithmetic.html

  1. x 2 - y 2 \sqrt{x^{2}-y^{2}}

Saturation_current.html

  1. I S = e A ( D p τ p n i 2 N D + D n τ n n i 2 N A ) , I_{\mathrm{S}}=eA\left(\sqrt{\frac{D_{\mathrm{p}}}{\tau_{\mathrm{p}}}}\frac{n_% {\mathrm{i}}^{2}}{N_{\mathrm{D}}}+\sqrt{\frac{D_{\mathrm{n}}}{\tau_{\mathrm{n}% }}}\frac{n_{\mathrm{i}}^{2}}{N_{\mathrm{A}}}\right),\,
  2. τ p , n \tau_{\mathrm{p,n}}

Savitzky–Golay_filter.html

  1. Y j = i = - ( m - 1 ) / 2 i = ( m - 1 ) / 2 C i y j + i m + 1 2 j n - m - 1 2 Y_{j}=\sum_{i=-(m-1)/2}^{i=(m-1)/2}C_{i}\,y_{j+i}\qquad\frac{m+1}{2}\leq j\leq n% -\frac{m-1}{2}
  2. Y j = 1 35 ( - 3 × y j - 2 + 12 × y j - 1 + 17 × y j + 12 × y j + 1 - 3 × y j + 2 ) Y_{j}=\frac{1}{35}(-3\times y_{j-2}+12\times y_{j-1}+17\times y_{j}+12\times y% _{j+1}-3\times y_{j+2})
  3. z = x - x ¯ h z={{x-\bar{x}}\over h}
  4. x ¯ {\bar{x}}
  5. 1 - m 2 , , 0 , , m - 1 2 {1-m\over 2},\ldots,0,\ldots,{m-1\over 2}
  6. Y = a 0 + a 1 z + a 2 z 2 + a k z k . Y=a_{0}+a_{1}z+a_{2}z^{2}\cdots+a_{k}z^{k}.
  7. 𝐚 = ( 𝐉 𝐓 𝐉 ) - 𝟏 𝐉 𝐓 𝐲 {\mathbf{a}}=\left({{\mathbf{J}}^{\mathbf{T}}{\mathbf{J}}}\right)^{-{\mathbf{1% }}}{\mathbf{J}}^{\mathbf{T}}{\mathbf{y}}
  8. 𝐉 = < m t p l > Y a \mathbf{J}=\mathbf{<}mtpl>{{\partial Y\over\partial a}}
  9. 𝐉 = ( 1 - 2 4 - 8 1 - 1 1 - 1 1 0 0 0 1 1 1 1 1 2 4 8 ) \mathbf{J}=\begin{pmatrix}1&-2&4&-8\\ 1&-1&1&-1\\ 1&0&0&0\\ 1&1&1&1\\ 1&2&4&8\end{pmatrix}
  10. 𝐉 𝐓 𝐉 = ( m z z 2 z 3 z z 2 z 3 z 4 z 2 z 3 z 4 z 5 z 3 z 4 z 5 z 6 ) = ( m 0 z 2 0 0 z 2 0 z 4 z 2 0 z 4 0 0 z 4 0 z 6 ) = ( 5 0 10 0 0 10 0 34 10 0 34 0 0 34 0 130 ) \mathbf{J^{T}J}=\begin{pmatrix}m&\sum z&\sum z^{2}&\sum z^{3}\\ \sum z&\sum z^{2}&\sum z^{3}&\sum z^{4}\\ \sum z^{2}&\sum z^{3}&\sum z^{4}&\sum z^{5}\\ \sum z^{3}&\sum z^{4}&\sum z^{5}&\sum z^{6}\\ \end{pmatrix}=\begin{pmatrix}m&0&\sum z^{2}&0\\ 0&\sum z^{2}&0&\sum z^{4}\\ \sum z^{2}&0&\sum z^{4}&0\\ 0&\sum z^{4}&0&\sum z^{6}\\ \end{pmatrix}=\begin{pmatrix}5&0&10&0\\ 0&10&0&34\\ 10&0&34&0\\ 0&34&0&130\\ \end{pmatrix}
  11. 𝐉 𝐓 𝐉 e v e n = ( 5 10 10 34 ) and 𝐉 𝐓 𝐉 o d d = ( 10 34 34 130 ) \mathbf{J^{T}J}_{even}=\begin{pmatrix}5&10\\ 10&34\\ \end{pmatrix}\quad\mathrm{and}\quad\mathbf{J^{T}J}_{odd}=\begin{pmatrix}10&34% \\ 34&130\\ \end{pmatrix}
  12. ( 𝐉 𝐓 𝐉 ) - 1 even = 1 70 ( 34 - 10 - 10 5 ) and ( 𝐉 𝐓 𝐉 ) - 1 odd = 1 144 ( 130 - 34 - 34 10 ) (\mathbf{J^{T}J})^{-1}\text{even}={1\over 70}\begin{pmatrix}34&-10\\ -10&5\\ \end{pmatrix}\quad\mathrm{and}\quad(\mathbf{J^{T}J})^{-1}\text{odd}={1\over 14% 4}\begin{pmatrix}130&-34\\ -34&10\\ \end{pmatrix}
  13. ( a 0 a 2 ) j = 1 70 ( 34 - 10 - 10 5 ) ( 1 1 1 1 1 4 1 0 1 4 ) ( y j - 2 y j - 1 y j y j + 1 y j + 2 ) \begin{pmatrix}{a_{0}}\\ {a_{2}}\\ \end{pmatrix}_{j}={1\over 70}\begin{pmatrix}34&-10\\ -10&5\end{pmatrix}\begin{pmatrix}1&1&1&1&1\\ 4&1&0&1&4\\ \end{pmatrix}\begin{pmatrix}y_{j-2}\\ y_{j-1}\\ y_{j}\\ y_{j+1}\\ y_{j+2}\end{pmatrix}
  14. ( a 1 a 3 ) j = 1 144 ( 130 - 34 - 34 10 ) ( - 2 - 1 0 1 2 - 8 - 1 0 1 8 ) ( y j - 2 y j - 1 y j y j + 1 y j + 2 ) \begin{pmatrix}a_{1}\\ a_{3}\\ \end{pmatrix}_{j}={1\over 144}\begin{pmatrix}130&-34\\ -34&10\\ \end{pmatrix}\begin{pmatrix}-2&-1&0&1&2\\ -8&-1&0&1&8\\ \end{pmatrix}\begin{pmatrix}y_{j-2}\\ y_{j-1}\\ y_{j}\\ y_{j+1}\\ y_{j+2}\\ \end{pmatrix}
  15. a 0 , j = 1 35 ( - 3 y j - 2 + 12 y j - 1 + 17 y j + 12 y j + 1 - 3 y j + 2 ) a_{0,j}={1\over 35}(-3y_{j-2}+12y_{j-1}+17y_{j}+12y_{j+1}-3y_{j+2})
  16. a 1 , j = 1 12 ( y j - 2 - 8 y j - 1 + 8 y j + 1 - y j + 2 ) a_{1,j}={1\over 12}(y_{j-2}-8y_{j-1}+8y_{j+1}-y_{j+2})
  17. a 2 , j = 1 14 ( 2 y j - 2 - y j - 1 - 2 y j - y j + 1 + 2 y j + 2 ) a_{2,j}={1\over 14}(2y_{j-2}-y_{j-1}-2y_{j}-y_{j+1}+2y_{j+2})
  18. a 3 , j = 1 12 ( - y j - 2 + 2 y j - 1 - 2 y j + 1 + y j + 2 ) a_{3,j}={1\over 12}(-y_{j-2}+2y_{j-1}-2y_{j+1}+y_{j+2})
  19. 𝐂 = ( 𝐉 𝐓 𝐉 ) - 𝟏 𝐉 𝐓 \mathbf{C=(J^{T}J)^{-1}J^{T}}
  20. ( C y ) j = Y j = i = - ( m - 1 ) / 2 i = ( m - 1 ) / 2 C i y j + i m + 1 2 j n - m - 1 2 (C\otimes y)_{j}\ =Y_{j}=\sum_{i=-(m-1)/2}^{i=(m-1)/2}C_{i}\,y_{j+i}\qquad% \frac{m+1}{2}\leq j\leq n-\frac{m-1}{2}
  21. ( Y 3 Y 4 Y 5 ) = 1 35 ( - 3 12 17 12 - 3 0 0 0 - 3 12 17 12 - 3 0 0 0 - 3 12 17 12 - 3 ) ( y 1 y 2 y 3 y 4 y 5 y 6 y 7 ) \begin{pmatrix}Y_{3}\\ Y_{4}\\ Y_{5}\\ \dots\end{pmatrix}={1\over 35}\begin{pmatrix}-3&12&17&12&-3&0&0&\dots\\ 0&-3&12&17&12&-3&0&\dots\\ 0&0&-3&12&17&12&-3&\dots\\ \dots\end{pmatrix}\begin{pmatrix}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\\ y_{5}\\ y_{6}\\ y_{7}\\ \dots\end{pmatrix}
  22. Y = a 0 + a 1 z + a 2 z 2 + a 3 z 3 = a 0 at z = 0 , x = x ¯ Y=a_{0}+a_{1}z+a_{2}z^{2}+a_{3}z^{3}=a_{0}\,\text{ at }z=0,x=\bar{x}
  23. < m t p l > d Y d x = 1 h ( a 1 + 2 a 2 z + 3 a 3 z 2 ) = 1 h a 1 at z = 0 , x = x ¯ \frac{<}{m}tpl>{{dY}}{{dx}}=\frac{1}{h}\left({a_{1}+2a_{2}z+3a_{3}z^{2}}\right% )=\frac{1}{h}a_{1}\,\text{ at }z=0,x=\bar{x}
  24. < m t p l > d 2 Y d x 2 = 1 h 2 ( 2 a 2 + 6 a 3 z ) = 2 h 2 a 2 at z = 0 , x = x ¯ \frac{<}{m}tpl>{{d^{2}Y}}{{dx^{2}}}=\frac{1}{{h^{2}}}\left({2a_{2}+6a_{3}z}% \right)=\frac{2}{h^{2}}a_{2}{\,\text{ at }}z=0,x=\bar{x}
  25. < m t p l > d 3 Y d x 3 = 6 h 3 a 3 \frac{<}{m}tpl>{{d^{3}Y}}{{dx^{3}}}=\frac{6}{{h^{3}}}a_{3}
  26. - ( m - 1 ) / 2 ( m - 1 ) / 2 z 2 = m ( m 2 - 1 ) 12 \sum_{-(m-1)/2}^{(m-1)/2}z^{2}={m(m^{2}-1)\over 12}
  27. z 4 = m ( m 2 - 1 ) ( 3 m 2 - 7 ) 240 \sum z^{4}={m(m^{2}-1)(3m^{2}-7)\over 240}
  28. z 6 = m ( m 2 - 1 ) ( 3 m 4 - 18 m 2 + 31 ) 1344 \sum z^{6}={m(m^{2}-1)(3m^{4}-18m^{2}+31)\over 1344}
  29. C 0 i = ( 3 m 2 - 7 - 20 i 2 ) / 4 m ( m 2 - 4 ) / 3 ; 1 - m 2 i m - 1 2 C_{0i}=\frac{{\left({3m^{2}-7-20i^{2}}\right)/4}}{{m\left({m^{2}-4}\right)/3}}% ;\quad\frac{1-m}{2}\leq i\leq\frac{m-1}{2}
  30. C 1 i = 5 ( 3 m 4 - 18 m 2 + 31 ) i - 28 ( 3 m 2 - 7 ) i 3 m ( m 2 - 1 ) ( 3 m 4 - 39 m 2 + 108 ) / 15 C_{1i}=\frac{{5\left({3m^{4}-18m^{2}+31}\right)i-28\left({3m^{2}-7}\right)i^{3% }}}{{m\left({m^{2}-1}\right)\left({3m^{4}-39m^{2}+108}\right)/15}}
  31. C 2 i = 12 m i 2 - m ( m 2 - 1 ) m 2 ( m 2 - 1 ) ( m 2 - 4 ) / 15 C_{2i}=\frac{{12mi^{2}-m\left({m^{2}-1}\right)}}{{m^{2}\left({m^{2}-1}\right)% \left({m^{2}-4}\right)/15}}
  32. C 3 i = - ( 3 m 2 - 7 ) i + 20 i 3 m ( m 2 - 1 ) ( 3 m 4 - 39 m 2 + 108 ) / 420 C_{3i}=\frac{{-\left({3m^{2}-7}\right)i+20i^{3}}}{{m\left({m^{2}-1}\right)% \left({3m^{4}-39m^{2}+108}\right)/420}}
  33. C 0 i = 1 m C_{0i}=\frac{1}{m}
  34. C 1 i = i m ( m 2 - 1 ) / 12 C_{1i}=\frac{i}{m(m^{2}-1)/12}
  35. Y = b 0 P 0 ( z ) + b 1 P 1 ( z ) + b k P k ( z ) . Y=b_{0}P^{0}(z)+b_{1}P^{1}(z)\cdots+b_{k}P^{k}(z).
  36. Y = a 0 + a 1 z + a 2 z 2 + a 3 z 3 Y=a_{0}+a_{1}z+a_{2}z^{2}+a_{3}z^{3}
  37. < m t p l > d Y d x = 1 h ( a 1 + 2 a 2 z + 3 a 3 z 2 ) \frac{<}{m}tpl>{{dY}}{{dx}}=\frac{1}{h}\left({a_{1}+2a_{2}z+3a_{3}z^{2}}\right)
  38. < m t p l > d 2 Y d x 2 = 1 h 2 ( 2 a 2 + 6 a 3 z ) \frac{<}{m}tpl>{{d^{2}Y}}{{dx^{2}}}=\frac{1}{{h^{2}}}\left({2a_{2}+6a_{3}z}\right)
  39. < m t p l > d 3 Y d x 3 = 6 h 3 a 3 \frac{<}{m}tpl>{{d^{3}Y}}{{dx^{3}}}=\frac{6}{{h^{3}}}a_{3}{\,\text{ }}
  40. U = i w i ( Y i - y i ) 2 U=\sum_{i}w_{i}(Y_{i}-y_{i})^{2}
  41. 𝐚 = ( 𝐉 𝐓 𝐖𝐉 ) - 1 𝐉 𝐓 𝐖𝐲 W i , i 1 \mathbf{a}=\left(\mathbf{J^{T}W}\mathbf{J}\right)^{-1}\mathbf{J^{T}W}\mathbf{y% }\qquad W_{i,i}\neq 1
  42. 𝐉 𝐓 𝐖𝐉 = ( m w i w i z i w i z i 2 w i z i w i z i 2 w i z i 3 w i z i 2 w i z i 3 w i z i 4 ) \mathbf{J^{T}WJ}=\begin{pmatrix}m\sum w_{i}&\sum w_{i}z_{i}&\sum w_{i}z_{i}^{2% }\\ \sum w_{i}z_{i}&\sum w_{i}z_{i}^{2}&\sum w_{i}z_{i}^{3}\\ \sum w_{i}z_{i}^{2}&\sum w_{i}z_{i}^{3}&\sum w_{i}z_{i}^{4}\end{pmatrix}
  43. 𝐂 = ( 𝐉 𝐓 𝐖𝐉 ) - 1 𝐉 𝐓 𝐖 \mathbf{C}=\left(\mathbf{J^{T}W}\mathbf{J}\right)^{-1}\mathbf{J^{T}W}
  44. v = x - x ¯ h ( x ) ; w = y - y ¯ h ( y ) v=\frac{x-\bar{x}}{h(x)};w=\frac{y-\bar{y}}{h(y)}
  45. Y = a 00 + a 10 v + a 01 w + a 20 v 2 + a 11 v w + a 02 w 2 + a 30 v 3 + a 21 v 2 w + a 12 v w 2 + a 03 w 3 Y=a_{00}+a_{10}v+a_{01}w+a_{20}v^{2}+a_{11}vw+a_{02}w^{2}+a_{30}v^{3}+a_{21}v^% {2}w+a_{12}vw^{2}+a_{03}w^{3}
  46. J row = 1 v w v 2 v w w 2 v 3 v 2 w v w 2 w 3 J\text{row}=1\ v\ w\ v^{2}\ vw\ w^{2}\ v^{3}\ v^{2}w\ vw^{2}\ w^{3}
  47. 𝐂 = ( 𝐉 T 𝐉 ) - 1 𝐉 T \mathbf{C}=\left(\mathbf{J}^{T}\mathbf{J}\right)^{-1}\mathbf{J}^{T}
  48. < m t p l > 1 m σ \sqrt{<}mtpl>{{1\over m}}\sigma
  49. 3 ( 3 m 2 - 7 ) 4 m ( m 2 - 4 ) σ \sqrt{\frac{3(3m^{2}-7)}{4m(m^{2}-4)}}\sigma
  50. F T ( θ ) = j = ( 1 - m ) / 2 j = ( m - 1 ) / 2 C j cos ( j θ ) FT(\theta)=\sum_{j=(1-m)/2}^{j=(m-1)/2}C_{j}\cos(j\theta)
  51. Y j = i = - ( m - 1 ) / 2 i = ( m - 1 ) / 2 C i y j + i Y_{j}\ =\sum_{i=-(m-1)/2}^{i=(m-1)/2}C_{i}\,y_{j+i}
  52. 𝐁 = 𝐂𝐀𝐂 T \mathbf{B}=\mathbf{C}\mathbf{A}\mathbf{C}^{T}
  53. 𝐁 = σ 2 9 ( 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 ) ( 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ) ( 1 0 0 1 1 0 1 1 1 0 1 1 0 0 1 ) = σ 2 9 ( 3 2 1 2 3 2 1 2 3 ) \mathbf{B}={\sigma^{2}\over 9}\begin{pmatrix}1&1&1&0&0\\ 0&1&1&1&0\\ 0&0&1&1&1\\ \end{pmatrix}\begin{pmatrix}1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ \end{pmatrix}\begin{pmatrix}1&0&0\\ 1&1&0\\ 1&1&1\\ 0&1&1\\ 0&0&1\\ \end{pmatrix}={\sigma^{2}\over 9}\begin{pmatrix}3&2&1\\ 2&3&2\\ 1&2&3\\ \end{pmatrix}
  54. ρ i j = B i j B i i B j j ( i j ) \rho_{ij}=\frac{B_{ij}}{\sqrt{B_{ii}B_{jj}}}(i\neq j)
  55. ρ i , i + 1 = 2 3 = 0.66 \rho_{i,i+1}={2\over 3}=0.66
  56. ρ i , i + 2 = 1 3 = 0.33 \rho_{i,i+2}={1\over 3}=0.33
  57. 𝐂𝐁𝐂 𝐓 = σ 2 81 ( 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 ) ( 3 2 1 0 0 2 3 2 0 0 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 ) ( 1 0 0 1 1 0 1 1 1 0 1 1 0 0 1 ) = σ 2 81 ( 19 16 10 4 1 16 19 16 10 4 10 16 19 16 10 4 10 16 19 16 1 4 10 16 19 ) \mathbf{CBC^{T}}={\sigma^{2}\over 81}\begin{pmatrix}1&1&1&0&0\\ 0&1&1&1&0\\ 0&0&1&1&1\\ \end{pmatrix}\begin{pmatrix}3&2&1&0&0\\ 2&3&2&0&0\\ 1&2&3&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{pmatrix}\begin{pmatrix}1&0&0\\ 1&1&0\\ 1&1&1\\ 0&1&1\\ 0&0&1\\ \end{pmatrix}={\sigma^{2}\over 81}\begin{pmatrix}19&16&10&4&1\\ 16&19&16&10&4\\ 10&16&19&16&10\\ 4&10&16&19&16\\ 1&4&10&16&19\\ \end{pmatrix}
  58. 19 / 81 σ \sqrt{19/81}\sigma
  59. ρ i , i + 1 = 16 19 = 0.84 , ρ i , i + 2 = 10 19 = 0.53 , ρ i , i + 3 = 4 19 = 0.21 , ρ i , i + 4 = 1 19 = 0.05 \rho_{i,i+1}={16\over 19}=0.84,\rho_{i,i+2}={10\over 19}=0.53,\rho_{i,i+3}={4% \over 19}=0.21,\rho_{i,i+4}={1\over 19}=0.05
  60. Y j = 1 35 ( - 3 × y j - 2 + 12 × y j - 1 + 17 × y j + 12 × y j + 1 - 3 × y j + 2 ) Y_{j}=\frac{1}{35}(-3\times y_{j-2}+12\times y_{j-1}+17\times y_{j}+12\times y% _{j+1}-3\times y_{j+2})
  61. Y j = 1 12 h ( 1 × y j - 2 - 8 × y j - 1 + 0 × y j + 8 × y j + 1 - 1 × y j + 2 ) Y^{\prime}_{j}=\frac{1}{12h}(1\times y_{j-2}-8\times y_{j-1}+0\times y_{j}+8% \times y_{j+1}-1\times y_{j+2})
  62. Y j ′′ = 1 7 h 2 ( 2 × y j - 2 - 1 × y j - 1 - 2 × y j - 1 × y j + 1 + 2 × y j + 2 ) Y^{\prime\prime}_{j}=\frac{1}{7h^{2}}(2\times y_{j-2}-1\times y_{j-1}-2\times y% _{j}-1\times y_{j+1}+2\times y_{j+2})

Savonius_wind_turbine.html

  1. P max = 0.36 kg m - 3 h r v 3 P_{\mathrm{max}}=0.36\,\mathrm{kg\,m^{-3}}\cdot h\cdot r\cdot v^{3}
  2. h h
  3. r r
  4. v v
  5. ω = λ v r \omega=\frac{\lambda\cdot v}{r}
  6. λ \lambda

Scalar_projection.html

  1. 𝐚 \mathbf{a}
  2. 𝐛 \mathbf{b}
  3. 𝐚 \mathbf{a}
  4. 𝐛 \mathbf{b}
  5. s = | 𝐚 | cos θ = 𝐚 𝐛 ^ , s=|\mathbf{a}|\cos\theta=\mathbf{a}\cdot\mathbf{\hat{b}},
  6. \cdot
  7. 𝐛 ^ \hat{\mathbf{b}}
  8. 𝐛 \mathbf{b}
  9. | 𝐚 | |\mathbf{a}|
  10. 𝐚 \mathbf{a}
  11. θ \theta
  12. 𝐚 \mathbf{a}
  13. 𝐛 \mathbf{b}
  14. 𝐚 \mathbf{a}
  15. 𝐛 \mathbf{b}
  16. 𝐛 \mathbf{b}
  17. 𝐚 \mathbf{a}
  18. 𝐛 \mathbf{b}
  19. 𝐛 ^ \mathbf{\hat{b}}
  20. 𝐚 \mathbf{a}
  21. 𝐛 \mathbf{b}
  22. θ \theta
  23. 𝐚 \mathbf{a}
  24. 𝐛 \mathbf{b}
  25. 𝐚 \mathbf{a}
  26. 𝐛 \mathbf{b}
  27. s = | 𝐚 | cos θ . s=|\mathbf{a}|\cos\theta.
  28. θ \theta
  29. θ \theta
  30. 𝐚 \mathbf{a}
  31. 𝐛 \mathbf{b}
  32. 𝐚 𝐛 \mathbf{a}\cdot\mathbf{b}
  33. 𝐚 𝐛 | 𝐚 | | 𝐛 | = cos θ \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}=\cos\theta\,
  34. s s\,
  35. s = | 𝐚 | cos θ = | 𝐚 | 𝐚 𝐛 | 𝐚 | | 𝐛 | = 𝐚 𝐛 | 𝐛 | s=|\mathbf{a}|\cos\theta=|\mathbf{a}|\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf% {a}|\,|\mathbf{b}|}=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}\,
  36. 90 < θ 180 90<\theta\leq 180
  37. 𝐚 1 \mathbf{a}_{1}
  38. | 𝐚 1 | |\mathbf{a}_{1}|
  39. s = | 𝐚 1 | s=|\mathbf{a}_{1}|
  40. 0 < θ 90 0<\theta\leq 90
  41. s = - | 𝐚 1 | s=-|\mathbf{a}_{1}|
  42. 90 < θ 180 90<\theta\leq 180

Scale-invariant_feature_transform.html

  1. [ u v ] = [ m 1 m 2 m 3 m 4 ] [ x y ] + [ t x t y ] \begin{bmatrix}u\\ v\end{bmatrix}=\begin{bmatrix}m1&m2\\ m3&m4\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}+\begin{bmatrix}tx\\ ty\end{bmatrix}
  2. [ x y 0 0 1 0 0 0 x y 0 1 . . ] [ m 1 m 2 m 3 m 4 t x t y ] = [ u v . . ] \begin{bmatrix}x&y&0&0&1&0\\ 0&0&x&y&0&1\\ ....\\ ....\end{bmatrix}\begin{bmatrix}m1\\ m2\\ m3\\ m4\\ tx\\ ty\end{bmatrix}=\begin{bmatrix}u\\ v\\ .\\ .\end{bmatrix}
  3. A 𝐱 ^ 𝐛 , A\hat{\mathbf{x}}\approx\mathbf{b},
  4. 𝐱 ^ \hat{\mathbf{x}}
  5. A T A 𝐱 ^ = A T 𝐛 . A^{T}\!A\hat{\mathbf{x}}=A^{T}\mathbf{b}.
  6. ( A T A ) - 1 A T (A^{T}A)^{-1}A^{T}
  7. 𝐱 ^ = ( A T A ) - 1 A T 𝐛 . \hat{\mathbf{x}}=(A^{T}\!A)^{-1}A^{T}\mathbf{b}.
  8. D ( x , y , σ ) D\left(x,y,\sigma\right)
  9. D ( x , y , σ ) = L ( x , y , k i σ ) - L ( x , y , k j σ ) D\left(x,y,\sigma\right)=L\left(x,y,k_{i}\sigma\right)-L\left(x,y,k_{j}\sigma\right)
  10. L ( x , y , k σ ) L\left(x,y,k\sigma\right)
  11. I ( x , y ) I\left(x,y\right)
  12. G ( x , y , k σ ) G\left(x,y,k\sigma\right)
  13. k σ k\sigma
  14. L ( x , y , k σ ) = G ( x , y , k σ ) * I ( x , y ) L\left(x,y,k\sigma\right)=G\left(x,y,k\sigma\right)*I\left(x,y\right)
  15. k i σ k_{i}\sigma
  16. k j σ k_{j}\sigma
  17. k i σ k_{i}\sigma
  18. k j σ k_{j}\sigma
  19. σ \sigma
  20. k i k_{i}
  21. D ( x , y , σ ) D\left(x,y,\sigma\right)
  22. D ( 𝐱 ) = D + D T 𝐱 𝐱 + 1 2 𝐱 T 2 D 𝐱 2 𝐱 D(\,\textbf{x})=D+\frac{\partial D^{T}}{\partial\,\textbf{x}}\,\textbf{x}+% \frac{1}{2}\,\textbf{x}^{T}\frac{\partial^{2}D}{\partial\,\textbf{x}^{2}}\,% \textbf{x}
  23. 𝐱 = ( x , y , σ ) \,\textbf{x}=\left(x,y,\sigma\right)
  24. 𝐱 ^ \hat{\,\textbf{x}}
  25. 𝐱 \,\textbf{x}
  26. 𝐱 ^ \hat{\,\textbf{x}}
  27. 0.5 0.5
  28. D ( 𝐱 ) D(\,\textbf{x})
  29. 𝐱 ^ \hat{\,\textbf{x}}
  30. 0.03 0.03
  31. 𝐲 + 𝐱 ^ \,\textbf{y}+\hat{\,\textbf{x}}
  32. 𝐲 \,\textbf{y}
  33. 𝐇 = [ D x x D x y D x y D y y ] \,\textbf{H}=\begin{bmatrix}D_{xx}&D_{xy}\\ D_{xy}&D_{yy}\end{bmatrix}
  34. α \alpha
  35. β \beta
  36. r = α / β r=\alpha/\beta
  37. D x x + D y y D_{xx}+D_{yy}
  38. D x x D y y - D x y 2 D_{xx}D_{yy}-D_{xy}^{2}
  39. R = Tr ( 𝐇 ) 2 / Det ( 𝐇 ) \,\text{R}=\operatorname{Tr}(\,\textbf{H})^{2}/\operatorname{Det}(\,\textbf{H})
  40. ( r + 1 ) 2 / r (r+1)^{2}/r
  41. r th r_{\,\text{th}}
  42. ( r th + 1 ) 2 / r th (r_{\,\text{th}}+1)^{2}/r_{\,\text{th}}
  43. r th = 10 r_{\,\text{th}}=10
  44. L ( x , y , σ ) L\left(x,y,\sigma\right)
  45. σ \sigma
  46. L ( x , y ) L\left(x,y\right)
  47. σ \sigma
  48. m ( x , y ) m\left(x,y\right)
  49. θ ( x , y ) \theta\left(x,y\right)
  50. m ( x , y ) = ( L ( x + 1 , y ) - L ( x - 1 , y ) ) 2 + ( L ( x , y + 1 ) - L ( x , y - 1 ) ) 2 m\left(x,y\right)=\sqrt{\left(L\left(x+1,y\right)-L\left(x-1,y\right)\right)^{% 2}+\left(L\left(x,y+1\right)-L\left(x,y-1\right)\right)^{2}}
  51. θ ( x , y ) = atan2 ( L ( x , y + 1 ) - L ( x , y - 1 ) , L ( x + 1 , y ) - L ( x - 1 , y ) ) \theta\left(x,y\right)=\mathrm{atan2}\left(L\left(x,y+1\right)-L\left(x,y-1% \right),L\left(x+1,y\right)-L\left(x-1,y\right)\right)
  52. σ \sigma
  53. σ \sigma

Scale_(map).html

  1. ϕ \phi
  2. λ \lambda
  3. α \alpha
  4. β \beta
  5. α β \alpha\neq\beta
  6. μ ( λ , ϕ , α ) = lim Q P P Q P Q , \mu(\lambda,\,\phi,\,\alpha)=\lim_{Q\to P}\frac{P^{\prime}Q^{\prime}}{PQ},
  7. ( α = 0 ) (\alpha=0)
  8. h ( λ , ϕ ) h(\lambda,\,\phi)
  9. ( α = π / 2 ) (\alpha=\pi/2)
  10. k ( λ , ϕ ) k(\lambda,\,\phi)
  11. k ( λ , ϕ ) k(\lambda,\phi)
  12. x = a λ x=a\lambda
  13. y = a ϕ y=a\phi
  14. x = λ x=\lambda
  15. y = ϕ y=\phi
  16. x = ( R F ) a λ x=(RF)a\lambda
  17. y = ( R F ) a ϕ y=(RF)a\phi
  18. ϕ \phi
  19. λ \lambda
  20. ϕ \phi
  21. λ \lambda
  22. ϕ + δ ϕ \phi+\delta\phi
  23. λ + δ λ \lambda+\delta\lambda
  24. a δ ϕ a\delta\phi
  25. a a
  26. ϕ \phi
  27. ( a cos ϕ ) δ λ (a\cos\phi)\delta\lambda
  28. λ \lambda
  29. x = a λ x=a\lambda
  30. y y
  31. δ x = a δ λ \delta x=a\delta\lambda
  32. δ y \delta y
  33. k = δ x a cos ϕ δ λ = sec ϕ \quad k\;=\;\dfrac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad% \qquad{}
  34. h = δ y a δ ϕ = y ( ϕ ) a \quad h\;=\;\dfrac{\delta y}{a\delta\phi\,}=\dfrac{y^{\prime}(\phi)}{a}
  35. k = sec ϕ k=\sec\phi
  36. y ( ϕ ) y(\phi)
  37. k = sec 30 = 2 / 3 = 1.15 k=\sec 30^{\circ}=2/\sqrt{3}=1.15
  38. k = sec 45 = 2 = 1.414 k=\sec 45^{\circ}=\sqrt{2}=1.414
  39. k = sec 60 = 2 k=\sec 60^{\circ}=2
  40. k = sec 80 = 5.76 k=\sec 80^{\circ}=5.76
  41. k = sec 85 = 11.5 k=\sec 85^{\circ}=11.5
  42. x = a λ , x=a\lambda,
  43. y = a ϕ , y=a\phi,
  44. a a
  45. λ \lambda
  46. λ = 0 \lambda=0
  47. ϕ \phi
  48. λ \lambda
  49. ϕ \phi
  50. π \pi
  51. λ \lambda
  52. [ - π , π ] [-\pi,\pi]
  53. ϕ \phi
  54. [ - π / 2 , π / 2 ] [-\pi/2,\pi/2]
  55. y ( ϕ ) = 1 y^{\prime}(\phi)=1
  56. k = δ x a cos ϕ δ λ = sec ϕ \quad k\;=\;\dfrac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad% \qquad{}
  57. h = δ y a δ ϕ = 1 \quad h\;=\;\dfrac{\delta y}{a\delta\phi\,}=\,1
  58. sec ϕ \sec\phi
  59. sec ϕ \sec\phi
  60. β = 45 \beta=45^{\circ}
  61. y y
  62. x = a λ x=a\lambda\,
  63. y = a ln [ tan ( π 4 + ϕ 2 ) ] y=a\ln\left[\tan\left(\frac{\pi}{4}+\frac{\phi}{2}\right)\right]
  64. λ \lambda\,
  65. ϕ \phi\,
  66. y ( ϕ ) = a sec ϕ y^{\prime}(\phi)=a\sec\phi
  67. k = δ x a cos ϕ δ λ = sec ϕ . k\;=\;\dfrac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi.
  68. h = δ y a δ ϕ = sec ϕ . h\;=\;\dfrac{\delta y}{a\delta\phi\,}=\,\sec\phi.
  69. sec ϕ \sec\phi
  70. sec ϕ \sec\phi
  71. x = a λ y = a sin ϕ x=a\lambda\qquad\qquad y=a\sin\phi
  72. λ \lambda
  73. ϕ \phi
  74. y ( ϕ ) = cos ϕ y^{\prime}(\phi)=\cos\phi
  75. k = δ x a cos ϕ δ λ = sec ϕ \quad k\;=\;\dfrac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad% \qquad{}
  76. h = δ y a δ ϕ = cos ϕ \quad h\;=\;\dfrac{\delta y}{a\delta\phi\,}=\,\cos\phi
  77. k = sec ϕ k=\sec\phi
  78. sec ϕ \sec\phi
  79. sec ϕ \sec\phi
  80. k = 1.0004 k=1.0004
  81. sec ϕ \sec\phi
  82. ϕ = 1.62 \phi=1.62
  83. ϕ 1 \phi_{1}
  84. x = 0.9996 a λ y = 0.9996 a ln ( tan ( π 4 + ϕ 2 ) ) . x=0.9996a\lambda\qquad\qquad y=0.9996a\ln\left(\tan\left(\frac{\pi}{4}+\frac{% \phi}{2}\right)\right).
  85. k = 0.9996 sec ϕ . \quad k\;=0.9996\sec\phi.
  86. ϕ 1 \phi_{1}
  87. sec ϕ 1 = 1 / 0.9996 = 1.00004 \sec\phi_{1}=1/0.9996=1.00004
  88. ϕ 1 = 1.62 \phi_{1}=1.62
  89. ϕ 2 \phi_{2}
  90. sec ϕ 2 = 1.0004 / 0.9996 = 1.0008 \sec\phi_{2}=1.0004/0.9996=1.0008
  91. ϕ 2 = 2.29 \phi_{2}=2.29
  92. 1 < k < 1.0004 1<k<1.0004
  93. k 0 = 0.9996 k_{0}=0.9996
  94. k = 1 k=1
  95. ϕ 1 \phi_{1}
  96. | k - 1 | < 0.0004 |k-1|<0.0004
  97. (a) tan α = a cos ϕ δ λ a δ ϕ , \,\text{(a)}\quad\tan\alpha=\frac{a\cos\phi\,\delta\lambda}{a\,\delta\phi},
  98. (b) tan β = δ x δ y = a δ λ δ y . \,\text{(b)}\quad\tan\beta=\frac{\delta x}{\delta y}=\frac{a\delta\lambda}{% \delta y}.
  99. β \beta
  100. α \alpha
  101. (c) tan β = a sec ϕ y ( ϕ ) tan α . \,\text{(c)}\quad\tan\beta=\frac{a\sec\phi}{y^{\prime}(\phi)}\tan\alpha.\,
  102. y ( ϕ ) = a sec ϕ y^{\prime}(\phi)=a\sec\phi
  103. α = β \alpha=\beta
  104. y ( ϕ ) = a y^{\prime}(\phi)=a
  105. y ( ϕ ) = a cos ϕ y^{\prime}(\phi)=a\cos\phi
  106. α \alpha
  107. β \beta
  108. ϕ \phi
  109. α \alpha\,
  110. μ α . \mu_{\alpha}.
  111. μ α = lim Q P P Q P Q = lim Q P δ x 2 + δ y 2 a 2 δ ϕ 2 + a 2 cos 2 ϕ δ λ 2 . \mu_{\alpha}=\lim_{Q\to P}\frac{P^{\prime}Q^{\prime}}{PQ}=\lim_{Q\to P}\frac{% \sqrt{\delta x^{2}+\delta y^{2}}}{\sqrt{a^{2}\,\delta\phi^{2}+a^{2}\cos^{2}\!% \phi\,\delta\lambda^{2}}}.
  112. δ x = a δ λ \delta x=a\delta\lambda
  113. δ ϕ \delta\phi
  114. δ y \delta y
  115. μ α ( ϕ ) = sec ϕ [ sin α sin β ] . \mu_{\alpha}(\phi)=\sec\phi\left[\frac{\sin\alpha}{\sin\beta}\right].
  116. β \beta
  117. α \alpha
  118. ϕ \phi
  119. μ α \mu_{\alpha}
  120. y = a y^{\prime}=a
  121. tan β = sec ϕ tan α . \tan\beta=\sec\phi\tan\alpha.\,
  122. β \beta
  123. α \alpha
  124. ϕ \phi

Scale_height.html

  1. H = k T M g H=\frac{kT}{Mg}
  2. H = R T g H=\frac{RT}{g}
  3. d P d z = - g ρ \frac{dP}{dz}=-g\rho
  4. ρ = M P k T \rho=\frac{MP}{kT}
  5. d P P = - d z k T M g \frac{dP}{P}=\frac{-dz}{\frac{kT}{Mg}}
  6. d P P = - d z H \frac{dP}{P}=-\frac{dz}{H}
  7. P = P 0 exp ( - z H ) P=P_{0}\exp\left(-\frac{z}{H}\right)

Scale_space.html

  1. t t
  2. t \sqrt{t}
  3. t t
  4. f ( x , y ) f(x,y)
  5. L ( x , y ; t ) L(x,y;t)
  6. f ( x , y ) f(x,y)
  7. g ( x , y ; t ) = 1 2 π t e - ( x 2 + y 2 ) / 2 t g(x,y;t)=\frac{1}{2\pi t}e^{-(x^{2}+y^{2})/2t}\,
  8. L ( , ; t ) = g ( , ; t ) * f ( , ) , L(\cdot,\cdot;t)\ =g(\cdot,\cdot;t)*f(\cdot,\cdot),
  9. L L
  10. x , y x,y
  11. t t
  12. L L
  13. t 0 t\geq 0
  14. t = σ 2 t=\sigma^{2}
  15. t = 0 t=0
  16. g g
  17. L ( x , y ; 0 ) = f ( x , y ) , L(x,y;0)=f(x,y),
  18. t = 0 t=0
  19. f f
  20. t t
  21. L L
  22. f f
  23. σ = t \sigma=\sqrt{t}
  24. t t
  25. L ( x , y ; t ) L(x,y;t)
  26. t = 0 t=0
  27. f f
  28. L ( x , y ; t ) L(x,y;t)
  29. t = 1 t=1
  30. L ( x , y ; t ) L(x,y;t)
  31. t = 4 t=4
  32. L ( x , y ; t ) L(x,y;t)
  33. t = 16 t=16
  34. L ( x , y ; t ) L(x,y;t)
  35. t = 64 t=64
  36. L ( x , y ; t ) L(x,y;t)
  37. t = 256 t=256
  38. t L = 1 2 2 L , \partial_{t}L=\frac{1}{2}\nabla^{2}L,
  39. L ( x , y ; 0 ) = f ( x , y ) L(x,y;0)=f(x,y)
  40. L x m y n ( x , y ; t ) = ( x m y n L ) ( x , y ; t ) . L_{x^{m}y^{n}}(x,y;t)=\left(\partial_{x^{m}y^{n}}L\right)(x,y;t).
  41. L x m y n ( , ; t ) = x m y n g ( , ; t ) * f ( , ) . L_{x^{m}y^{n}}(\cdot,\cdot;t)=\partial_{x^{m}y^{n}}g(\cdot,\cdot;\,t)*f(\cdot,% \cdot).
  42. L v = L x 2 + L y 2 L_{v}=\sqrt{L_{x}^{2}+L_{y}^{2}}
  43. L = ( L x , L y ) T . \nabla L=(L_{x},L_{y})^{T}.
  44. L ~ v 2 = L x 2 L x x + 2 L x L y L x y + L y 2 L y y = 0 {\tilde{L}}_{v}^{2}=L_{x}^{2}\,L_{xx}+2\,L_{x}\,L_{y}\,L_{xy}+L_{y}^{2}\,L_{yy% }=0
  45. L ~ v 3 = L x 3 L x x x + 3 L x 2 L y L x x y + 3 L x L y 2 L x y y + L y 3 L y y y < 0. {\tilde{L}}_{v}^{3}=L_{x}^{3}\,L_{xxx}+3\,L_{x}^{2}\,L_{y}\,L_{xxy}+3\,L_{x}\,% L_{y}^{2}\,L_{xyy}+L_{y}^{3}\,L_{yyy}<0.
  46. 2 L = L x x + L y y \nabla^{2}L=L_{xx}+L_{yy}\,
  47. det H L ( x , y ; t ) = ( L x x L y y - L x y 2 ) . \operatorname{det}HL(x,y;t)=(L_{xx}L_{yy}-L_{xy}^{2}).
  48. L ξ m η n ( x , y ; t ) = t ( m + n ) γ / 2 L x m y n ( x , y ; t ) L_{\xi^{m}\eta^{n}}(x,y;t)=t^{(m+n)\gamma/2}L_{x^{m}y^{n}}(x,y;t)
  49. γ [ 0 , 1 ] \gamma\in[0,1]
  50. γ \gamma
  51. ξ = t γ / 2 x \partial_{\xi}=t^{\gamma/2}\partial_{x}\quad
  52. η = t γ / 2 y . \quad\partial_{\eta}=t^{\gamma/2}\partial_{y}.
  53. t 0 t_{0}
  54. s s
  55. s 2 t 0 s^{2}t_{0}

Scaled_inverse_chi-squared_distribution.html

  1. Γ ( ν 2 , τ 2 ν 2 x ) / Γ ( ν 2 ) \Gamma\left(\frac{\nu}{2},\frac{\tau^{2}\nu}{2x}\right)\left/\Gamma\left(\frac% {\nu}{2}\right)\right.
  2. ν τ 2 ν - 2 \frac{\nu\tau^{2}}{\nu-2}
  3. ν > 2 \nu>2\,
  4. ν τ 2 ν + 2 \frac{\nu\tau^{2}}{\nu+2}
  5. 2 ν 2 τ 4 ( ν - 2 ) 2 ( ν - 4 ) \frac{2\nu^{2}\tau^{4}}{(\nu-2)^{2}(\nu-4)}
  6. ν > 4 \nu>4\,
  7. 4 ν - 6 2 ( ν - 4 ) \frac{4}{\nu-6}\sqrt{2(\nu-4)}
  8. ν > 6 \nu>6\,
  9. 12 ( 5 ν - 22 ) ( ν - 6 ) ( ν - 8 ) \frac{12(5\nu-22)}{(\nu-6)(\nu-8)}
  10. ν > 8 \nu>8\,
  11. ν 2 + ln ( τ 2 ν 2 Γ ( ν 2 ) ) \frac{\nu}{2}\!+\!\ln\left(\frac{\tau^{2}\nu}{2}\Gamma\left(\frac{\nu}{2}% \right)\right)
  12. - ( 1 + ν 2 ) ψ ( ν 2 ) \!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\left(\frac{\nu}{2}\right)
  13. 2 Γ ( ν 2 ) ( - τ 2 ν t 2 ) ν 4 K ν 2 ( - 2 τ 2 ν t ) \frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-\tau^{2}\nu t}{2}\right)^{\!\!% \frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2\tau^{2}\nu t}\right)
  14. 2 Γ ( ν 2 ) ( - i τ 2 ν t 2 ) ν 4 K ν 2 ( - 2 i τ 2 ν t ) \frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\tau^{2}\nu t}{2}\right)^{\!\!% \frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\tau^{2}\nu t}\right)
  15. X Scale-inv- χ 2 ( ν , τ 2 ) X\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,\tau^{2})
  16. X τ 2 ν inv- χ 2 ( ν ) \frac{X}{\tau^{2}\nu}\sim\mbox{inv-}~{}\chi^{2}(\nu)
  17. X Scale-inv- χ 2 ( ν , τ 2 ) X\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,\tau^{2})
  18. X Inv-Gamma ( ν 2 , ν τ 2 2 ) X\sim\textrm{Inv-Gamma}\left(\frac{\nu}{2},\frac{\nu\tau^{2}}{2}\right)
  19. ( E ( 1 / X ) ) (E(1/X))
  20. ( E ( ln ( X ) ) (E(\ln(X))
  21. x > 0 x>0
  22. f ( x ; ν , τ 2 ) = ( τ 2 ν / 2 ) ν / 2 Γ ( ν / 2 ) exp [ - ν τ 2 2 x ] x 1 + ν / 2 f(x;\nu,\tau^{2})=\frac{(\tau^{2}\nu/2)^{\nu/2}}{\Gamma(\nu/2)}~{}\frac{\exp% \left[\frac{-\nu\tau^{2}}{2x}\right]}{x^{1+\nu/2}}
  23. ν \nu
  24. τ 2 \tau^{2}
  25. F ( x ; ν , τ 2 ) = Γ ( ν 2 , τ 2 ν 2 x ) / Γ ( ν 2 ) F(x;\nu,\tau^{2})=\Gamma\left(\frac{\nu}{2},\frac{\tau^{2}\nu}{2x}\right)\left% /\Gamma\left(\frac{\nu}{2}\right)\right.
  26. = Q ( ν 2 , τ 2 ν 2 x ) =Q\left(\frac{\nu}{2},\frac{\tau^{2}\nu}{2x}\right)
  27. Γ ( a , x ) \Gamma(a,x)
  28. Γ ( x ) \Gamma(x)
  29. Q ( a , x ) Q(a,x)
  30. φ ( t ; ν , τ 2 ) = \varphi(t;\nu,\tau^{2})=
  31. 2 Γ ( ν 2 ) ( - i τ 2 ν t 2 ) ν 4 K ν 2 ( - 2 i τ 2 ν t ) , \frac{2}{\Gamma(\frac{\nu}{2})}\left(\frac{-i\tau^{2}\nu t}{2}\right)^{\!\!% \frac{\nu}{4}}\!\!K_{\frac{\nu}{2}}\left(\sqrt{-2i\tau^{2}\nu t}\right),
  32. K ν 2 ( z ) K_{\frac{\nu}{2}}(z)
  33. { 2 x 2 f ( x ) + f ( x ) ( - ν τ 2 + ν x + 2 x ) = 0 , f ( 1 ) = 2 - ν / 2 e - ν τ 2 2 ( ν τ 2 ) ν / 2 Γ ( ν 2 ) } \left\{2x^{2}f^{\prime}(x)+f(x)\left(-\nu\tau^{2}+\nu x+2x\right)=0,f(1)=\frac% {2^{-\nu/2}e^{-\frac{\nu\tau^{2}}{2}}\left(\nu\tau^{2}\right)^{\nu/2}}{\Gamma% \left(\frac{\nu}{2}\right)}\right\}
  34. τ 2 \tau^{2}
  35. τ 2 = n / i = 1 n 1 x i . \tau^{2}=n/\sum_{i=1}^{n}\frac{1}{x_{i}}.
  36. ν 2 \frac{\nu}{2}
  37. ln ( ν 2 ) + ψ ( ν 2 ) = i = 1 n ln ( x i ) - n ln ( τ 2 ) , \ln(\frac{\nu}{2})+\psi(\frac{\nu}{2})=\sum_{i=1}^{n}\ln(x_{i})-n\ln(\tau^{2}),
  38. ψ ( x ) \psi(x)
  39. ν . \nu.
  40. x ¯ = 1 n i = 1 n x i \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  41. ν \nu
  42. ν 2 = x ¯ x ¯ - τ 2 . \frac{\nu}{2}=\frac{\bar{x}}{\bar{x}-\tau^{2}}.
  43. p ( σ 2 | D , I ) p ( σ 2 | I ) p ( D | σ 2 ) p(\sigma^{2}|D,I)\propto p(\sigma^{2}|I)\;p(D|\sigma^{2})
  44. ( σ 2 | D , μ ) = 1 ( 2 π σ ) n exp [ - i n ( x i - μ ) 2 2 σ 2 ] \mathcal{L}(\sigma^{2}|D,\mu)=\frac{1}{\left(\sqrt{2\pi}\sigma\right)^{n}}\;% \exp\left[-\frac{\sum_{i}^{n}(x_{i}-\mu)^{2}}{2\sigma^{2}}\right]
  45. p ( σ 2 | D , I , μ ) 1 σ n + 2 exp [ - i n ( x i - μ ) 2 2 σ 2 ] p(\sigma^{2}|D,I,\mu)\propto\frac{1}{\sigma^{n+2}}\;\exp\left[-\frac{\sum_{i}^% {n}(x_{i}-\mu)^{2}}{2\sigma^{2}}\right]
  46. p ( σ 2 s 2 | s 2 ) = p ( σ 2 s 2 | σ 2 ) p(\tfrac{\sigma^{2}}{s^{2}}|s^{2})=p(\tfrac{\sigma^{2}}{s^{2}}|\sigma^{2})
  47. p ( σ 2 | I , μ ) 1 σ n 0 + 2 exp [ - n 0 s 0 2 2 σ 2 ] p(\sigma^{2}|I^{\prime},\mu)\propto\frac{1}{\sigma^{n_{0}+2}}\;\exp\left[-% \frac{n_{0}s_{0}^{2}}{2\sigma^{2}}\right]
  48. p ( σ 2 | D , I , μ ) 1 σ n + n 0 + 2 exp [ - n s 2 + n 0 s 0 2 2 σ 2 ] p(\sigma^{2}|D,I^{\prime},\mu)\propto\frac{1}{\sigma^{n+n_{0}+2}}\;\exp\left[-% \frac{\sum{ns^{2}+n_{0}s_{0}^{2}}}{2\sigma^{2}}\right]
  49. p ( μ , σ 2 D , I ) 1 σ n + 2 exp [ - i n ( x i - μ ) 2 2 σ 2 ] = 1 σ n + 2 exp [ - i n ( x i - x ¯ ) 2 2 σ 2 ] exp [ - i n ( μ - x ¯ ) 2 2 σ 2 ] \begin{aligned}\displaystyle p(\mu,\sigma^{2}\mid D,I)&\displaystyle\propto% \frac{1}{\sigma^{n+2}}\exp\left[-\frac{\sum_{i}^{n}(x_{i}-\mu)^{2}}{2\sigma^{2% }}\right]\\ &\displaystyle=\frac{1}{\sigma^{n+2}}\exp\left[-\frac{\sum_{i}^{n}(x_{i}-\bar{% x})^{2}}{2\sigma^{2}}\right]\exp\left[-\frac{\sum_{i}^{n}(\mu-\bar{x})^{2}}{2% \sigma^{2}}\right]\end{aligned}
  50. p ( σ 2 | D , I ) 1 σ n + 2 exp [ - i n ( x i - x ¯ ) 2 2 σ 2 ] - exp [ - i n ( μ - x ¯ ) 2 2 σ 2 ] d μ = 1 σ n + 2 exp [ - i n ( x i - x ¯ ) 2 2 σ 2 ] 2 π σ 2 / n ( σ 2 ) - ( n + 1 ) / 2 exp [ - ( n - 1 ) s 2 2 σ 2 ] \begin{aligned}\displaystyle p(\sigma^{2}|D,I)\;\propto&\displaystyle\frac{1}{% \sigma^{n+2}}\;\exp\left[-\frac{\sum_{i}^{n}(x_{i}-\bar{x})^{2}}{2\sigma^{2}}% \right]\;\int_{-\infty}^{\infty}\exp\left[-\frac{\sum_{i}^{n}(\mu-\bar{x})^{2}% }{2\sigma^{2}}\right]d\mu\\ \displaystyle=&\displaystyle\frac{1}{\sigma^{n+2}}\;\exp\left[-\frac{\sum_{i}^% {n}(x_{i}-\bar{x})^{2}}{2\sigma^{2}}\right]\;\sqrt{2\pi\sigma^{2}/n}\\ \displaystyle\propto&\displaystyle(\sigma^{2})^{-(n+1)/2}\;\exp\left[-\frac{(n% -1)s^{2}}{2\sigma^{2}}\right]\end{aligned}
  51. n - 1 \scriptstyle{n-1}\;
  52. s 2 = ( x i - x ¯ ) 2 / ( n - 1 ) \scriptstyle{s^{2}=\sum(x_{i}-\bar{x})^{2}/(n-1)}
  53. X Scale-inv- χ 2 ( ν , τ 2 ) X\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,\tau^{2})
  54. k X Scale-inv- χ 2 ( ν , k τ 2 ) kX\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,k\tau^{2})\,
  55. X inv- χ 2 ( ν ) X\sim\mbox{inv-}~{}\chi^{2}(\nu)\,
  56. X Scale-inv- χ 2 ( ν , 1 / ν ) X\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,1/\nu)\,
  57. X Scale-inv- χ 2 ( ν , τ 2 ) X\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,\tau^{2})
  58. X τ 2 ν inv- χ 2 ( ν ) \frac{X}{\tau^{2}\nu}\sim\mbox{inv-}~{}\chi^{2}(\nu)\,
  59. X Scale-inv- χ 2 ( ν , τ 2 ) X\sim\mbox{Scale-inv-}~{}\chi^{2}(\nu,\tau^{2})
  60. X Inv-Gamma ( ν 2 , ν τ 2 2 ) X\sim\textrm{Inv-Gamma}\left(\frac{\nu}{2},\frac{\nu\tau^{2}}{2}\right)

Scanning_tunneling_spectroscopy.html

  1. d I / d V dI/dV
  2. I = 4 π e - [ f ( E F - e V + ϵ ) - f ( E F + ϵ ) ] ρ S ( E F - e V + ϵ ) ρ T ( E F + ϵ ) | M μ ν | 2 d ϵ , ( 1 ) I=\frac{4\pi e}{\hbar}\int_{-\infty}^{\infty}\left[f\left(E_{F}-eV+\epsilon% \right)-f\left(E_{F}+\epsilon\right)\right]\rho_{S}\left(E_{F}-eV+\epsilon% \right)\rho_{T}\left(E_{F}+\epsilon\right)\left|M_{\mu\nu}\right|^{2}\,d% \epsilon\ ,\qquad\qquad(1)
  3. f ( E ) f\left(E\right)
  4. ρ s \rho_{s}
  5. ρ T \rho_{T}
  6. M μ ν M_{\mu\nu}
  7. M μ ν = - 2 2 m Σ ( χ ν * ψ μ - ψ μ χ ν * ) d 𝐒 , ( 2 ) M_{\mu\nu}=-\frac{\hbar^{2}}{2m}\int_{\Sigma}\left(\chi_{\nu}^{*}\nabla\psi_{% \mu}-\psi_{\mu}\nabla\chi_{\nu}^{*}\right)\cdot\,d{\mathbf{S}}\ ,\qquad\qquad(2)
  8. ψ \psi
  9. χ \chi
  10. I 0 e V ρ S ( E F - e V + ϵ ) ρ T ( E F + ϵ ) d ϵ , ( 3 ) I\propto\int_{0}^{eV}\rho_{S}\left(E_{F}-eV+\epsilon\right)\rho_{T}\left(E_{F}% +\epsilon\right)\,d\epsilon\ ,\qquad\qquad(3)
  11. d I d V ρ S ( E F - e V ) , ( 4 ) \frac{dI}{dV}\propto\rho_{S}\left(E_{F}-eV\right)\ ,\qquad\qquad(4)
  12. I = 0 e V ρ S ( r , E ) ρ T ( r , E - e V ) T ( E , e V , r ) d E , ( 5 ) I=\int_{0}^{eV}\rho_{S}\left(r,E\right)\rho_{T}\left(r,E-eV\right)T\left(E,eV,% r\right)\,dE\ ,\qquad\qquad(5)
  13. ρ s \rho_{s}
  14. ρ t \rho_{t}
  15. T = exp ( - 2 Z 2 m ϕ s + ϕ t 2 + e V 2 - E ) , ( 6 ) T=\exp\left(-\frac{2Z\sqrt{2m}}{\hbar}\sqrt{\frac{\phi_{s}+\phi_{t}}{2}+\frac{% eV}{2}-E}\right)\ ,\qquad\qquad(6)
  16. ϕ s \phi_{s}
  17. ϕ t \phi_{t}
  18. Z Z
  19. I I
  20. I I
  21. d I / d V dI/dV
  22. d I / d V dI/dV
  23. d I / d V dI/dV
  24. d I / d V dI/dV
  25. d I / d V dI/dV
  26. d I / d V dI/dV
  27. d I d V = ρ s ( r , e V ) ρ t ( r , 0 ) T ( e V , e V , r ) + 0 e V ρ s ( r , E ) ρ t ( r , E - e V ) d T ( E , e V , r ) d V d E , ( 7 ) \frac{dI}{dV}=\rho_{s}\left(r,eV\right)\rho_{t}\left(r,0\right)T\left(eV,eV,r% \right)+\int_{0}^{eV}\rho_{s}\left(r,E\right)\rho_{t}\left(r,E-eV\right)\frac{% dT\left(E,eV,r\right)}{dV}\,dE\ ,\qquad\qquad(7)
  28. ρ s \rho_{s}
  29. ρ t \rho_{t}
  30. d I / d V dI/dV
  31. d I / d V dI/dV
  32. d I / d V dI/dV
  33. d I / d V dI/dV
  34. d I / d V = I / V dI/dV=I/V
  35. d I / d V dI/dV
  36. d I / d V dI/dV
  37. I / V I/V
  38. d I / d V I / V = ρ s ( r , e V ) ρ t ( r , 0 ) + 0 e V ρ s ( r , E ) ρ t ( r , E - e V ) T ( e V , e V , r ) d T ( E , e V , r ) d V d E 1 e V 0 e V ρ s ( r , E ) ρ t ( r , E - e V ) T ( E , e V , r ) T ( e V , e V , r ) d E . ( 8 ) \frac{dI/dV}{I/V}=\frac{\rho_{s}\left(r,eV\right)\rho_{t}\left(r,0\right)+\int% _{0}^{eV}\frac{\rho_{s}\left(r,E\right)\rho_{t}\left(r,E-eV\right)}{T\left(eV,% eV,r\right)}\frac{dT\left(E,eV,r\right)}{dV}\,dE}{\frac{1}{eV}\int_{0}^{eV}% \rho_{s}\left(r,E\right)\rho_{t}\left(r,E-eV\right)\frac{T\left(E,eV,r\right)}% {T\left(eV,eV,r\right)}\,dE}\ .\qquad\qquad(8)
  39. T ( E , e V , r ) T\left(E,eV,r\right)
  40. T ( e V , e V , r ) T\left(eV,eV,r\right)
  41. d I / d V I / V = d ( l o g I ) d ( l o g V ) = ρ s ( r , e V ) ρ t ( r , 0 ) + A ( V ) B ( V ) , ( 9 ) \frac{dI/dV}{I/V}=\frac{d\left(logI\right)}{d\left(logV\right)}=\frac{\rho_{s}% \left(r,eV\right)\rho_{t}\left(r,0\right)+A\left(V\right)}{B\left(V\right)}\ ,% \qquad\qquad(9)
  42. B ( V ) B\left(V\right)
  43. A ( V ) A\left(V\right)
  44. A ( V ) A\left(V\right)
  45. B ( V ) B\left(V\right)
  46. ( d I / d V ) / ( I / V ) \left(dI/dV\right)/\left(I/V\right)
  47. ρ s \rho_{s}
  48. ± ϕ / e \pm\phi/e
  49. ϕ \phi
  50. T = 300 K T=300\,K
  51. k B T 0.026 e V k_{B}T\approx 0.026\,eV
  52. 2 k B T 0.052 e V 2k_{B}T\approx 0.052\,eV
  53. Δ E 0.1 e V \Delta E\approx 0.1\,eV
  54. Δ x Δ k 1 / 2 \Delta x\Delta k\geq 1/2
  55. Δ E 2 k F 2 M * Δ x = 0.47 E F - E 0 r k F , ( 10 ) \Delta E\geq\frac{\hbar^{2}k_{F}}{2M^{*}\Delta x}=0.47\frac{E_{F}-E_{0}}{rk_{F% }}\ ,\qquad\qquad(10)
  56. E F E_{F}
  57. E 0 E_{0}
  58. k F k_{F}
  59. r r

Scapegoat_tree.html

  1. I ( v ) = max ( | left ( v ) - right ( v ) | - 1 , 0 ) I(v)=\operatorname{max}(|\operatorname{left}(v)-\operatorname{right}(v)|-1,0)
  2. I ( v ) = Ω ( | v | ) I(v)=\Omega(|v|)
  3. Ω \Omega
  4. v 0 v_{0}
  5. h ( v 0 ) = log ( | v 0 | + 1 ) h(v_{0})=\log(|v_{0}|+1)
  6. Ω ( | v 0 | ) \Omega(|v_{0}|)
  7. I ( v ) = Ω ( | v 0 | ) I(v)=\Omega(|v_{0}|)
  8. h ( v ) = h ( v 0 ) + Ω ( | v 0 | ) h(v)=h(v_{0})+\Omega(|v_{0}|)
  9. log ( | v | ) log ( | v 0 | + 1 ) + 1 \log(|v|)\leq\log(|v_{0}|+1)+1
  10. I ( v ) = Ω ( | v | ) I(v)=\Omega(|v|)
  11. Ω ( | v | ) \Omega(|v|)
  12. v v
  13. O ( log n ) O(\log n)
  14. O ( | v | ) O(|v|)
  15. O ( log n ) O(\log n)
  16. Ω ( | v | ) O ( log n ) + O ( | v | ) Ω ( | v | ) = O ( log n ) {\Omega(|v|)O(\log{n})+O(|v|)\over\Omega(|v|)}=O(\log{n})
  17. n / 2 - 1 n/2-1
  18. O ( log n ) O(\log{n})
  19. n / 2 n/2
  20. O ( log n ) + O ( n ) O(\log{n})+O(n)
  21. O ( n ) O(n)
  22. O ( log n ) O(\log{n})
  23. 1 < m t p l > n 2 O ( log n ) + O ( n ) n 2 = n 2 O ( log n ) + O ( n ) n 2 = O ( log n ) {\sum_{1}^{<}mtpl>{{n\over 2}}O(\log{n})+O(n)\over{n\over 2}}={{n\over 2}O(% \log{n})+O(n)\over{n\over 2}}=O(\log{n})