wpmath0000014_4

Expectations_hypothesis.html

  1. ( 1 + i l t ) n = ( 1 + i s t year 1 ) ( 1 + i s t year 2 ) ( 1 + i s t year n ) , (1+i_{lt})^{n}=(1+i_{st}^{\,\text{year 1}})(1+i_{st}^{\,\text{year 2}})\cdots(% 1+i_{st}^{\,\text{year n}}),
  2. d t d_{t}
  3. t = 1 , 2 , t=1,2,...
  4. r r
  5. t t
  6. P t = n = t + 1 ( 1 1 + r ) n - t 𝔼 [ d n t ] P_{t}=\sum_{n=t+1}^{\infty}\left(\frac{1}{1+r}\right)^{n-t}\mathbb{E}[d_{n}% \mid\mathcal{F}_{t}]
  7. t \mathcal{F}_{t}
  8. t t
  9. m t m_{t}
  10. t t
  11. P t = n = t + 1 m n B ( t , n ) = m t + 1 1 + r ( t , t + 1 ) + 1 1 + r ( t , t + 1 ) 𝔼 [ P t + 1 t ] P_{t}=\sum_{n=t+1}^{\infty}m_{n}B(t,n)=\frac{m_{t+1}}{1+r(t,t+1)}+\frac{1}{1+r% (t,t+1)}\mathbb{E}[P_{t+1}\mid\mathcal{F}_{t}]
  12. r ( t , T ) r(t,T)
  13. t t
  14. T T
  15. B ( t , T ) B(t,T)
  16. t t
  17. T T
  18. B ( t , T ) = 𝔼 [ ( 1 + r ( t , t + 1 ) ) - 1 ( 1 + r ( T - 1 , T ) ) - 1 t ] = 1 1 + r ( t , t + 1 ) 𝔼 [ B ( t + 1 , T ) t ] B(t,T)=\mathbb{E}[(1+r(t,t+1))^{-1}\cdots(1+r(T-1,T))^{-1}\mid\mathcal{F}_{t}]% =\frac{1}{1+r(t,t+1)}\mathbb{E}[B(t+1,T)\mid\mathcal{F}_{t}]

Expected_value_of_including_uncertainty.html

  1. d D the decision being made, chosen from space D x X the uncertain quantity, with true value in space X U ( d , x ) the utility function f ( x ) your prior subjective probability distribution (density function) on x \begin{array}[]{ll}d\in D&\,\text{the decision being made, chosen from space }% D\\ x\in X&\,\text{the uncertain quantity, with true value in space }X\\ U(d,x)&\,\text{the utility function}\\ f(x)&\,\text{your prior subjective probability distribution (density function)% on }x\end{array}
  2. E [ x ] E[x]
  3. d i u = arg max d U ( d , E [ x ] ) d_{iu}={\arg\max_{d}}~{}U(d,E[x])
  4. d * = arg max d U ( d , x ) f ( x ) d x d^{*}={\arg\max_{d}}{\int U(d,x)f(x)\,dx}
  5. E V I U = X [ U ( d * , x ) - U ( d i u , x ) ] f ( x ) d x EVIU=\int_{X}\left[U(d^{*},x)-U(d_{iu},x)\right]f(x)\,dx

Experiments_of_Rayleigh_and_Brace.html

  1. 1 6000 \tfrac{1}{6000}

Exponential_random_graph_models.html

  1. Y Y
  2. n n
  3. m m
  4. { Y i j : i = 1 , , n ; j = 1 , , n } \{Y_{ij}:i=1,\dots,n;j=1,\dots,n\}
  5. Y i j = 1 Y_{ij}=1
  6. ( i , j ) (i,j)
  7. Y i j = 0 Y_{ij}=0
  8. y y
  9. s ( y ) s(y)
  10. P ( Y = y | θ ) = exp ( θ T s ( y ) ) c ( θ ) P(Y=y|\theta)=\frac{\exp(\theta^{T}s(y))}{c(\theta)}
  11. θ \theta
  12. s ( y ) s(y)
  13. c ( θ ) c(\theta)
  14. n n
  15. n n
  16. 2 n ( n - 1 ) / 2 2^{n(n-1)/2}

Exponentially_closed_field.html

  1. F F\,
  2. E E\,
  3. F F\,
  4. F F\,
  5. 1 + 1 / n < E ( 1 ) < n 1+1/n<E(1)<n\,
  6. n n\,
  7. E E\,
  8. F F\,
  9. E E\,
  10. a x a^{x}\,
  11. 1 < a F 1<a\in F
  12. F F\,
  13. F F\,
  14. n n\,
  15. n n\,
  16. F F\,
  17. E ( 1 n E - 1 ( a ) ) n = E ( E - 1 ( a ) ) = a E\left(\frac{1}{n}E^{-1}(a)\right)^{n}=E(E^{-1}(a))=a
  18. a > 0 a>0
  19. E E\,
  20. E ( x ) = a x E(x)=a^{x}\,
  21. 1 < a F 1<a\in F\,
  22. F F\,
  23. E ( 2 ) = a 2 E(\sqrt{2})=a^{\sqrt{2}}
  24. 1 < a 1<a\,
  25. F F\,
  26. E 2 : F F + E_{2}:F\rightarrow F^{+}
  27. E 2 ( x + y ) = E 2 ( x ) E 2 ( y ) E_{2}(x+y)=E_{2}(x)E_{2}(y)\,
  28. E 2 ( 1 ) = 2 E_{2}(1)=2
  29. E 2 E_{2}\,

Exponentially_modified_Gaussian_distribution.html

  1. f ( x ; μ , σ , λ ) = λ 2 e λ 2 ( 2 μ + λ σ 2 - 2 x ) erfc ( μ + λ σ 2 - x 2 σ ) f(x;\mu,\sigma,\lambda)=\frac{\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda% \sigma^{2}-2x)}\operatorname{erfc}\left(\frac{\mu+\lambda\sigma^{2}-x}{\sqrt{2% }\sigma}\right)
  2. erfc ( x ) = 1 - erf ( x ) = 2 π x e - t 2 d t . \begin{aligned}\displaystyle\operatorname{erfc}(x)&\displaystyle=1-% \operatorname{erf}(x)\\ &\displaystyle=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-t^{2}}\,dt.\end{aligned}
  3. { σ 2 f ′′ ( x ) + f ( x ) ( λ σ 2 - μ + x ) + λ f ( x ) ( x - μ ) = 0 , f ( 0 ) = 1 2 λ e 1 2 λ ( λ σ 2 + 2 μ ) erfc ( λ σ 2 + μ 2 σ ) , f ( 0 ) = λ e - μ 2 2 σ 2 ( 2 - π λ σ e ( λ σ 2 + μ ) 2 2 σ 2 erfc ( λ σ 2 + μ 2 σ ) ) 2 π σ } \left\{\begin{array}[]{l}\sigma^{2}f^{\prime\prime}(x)+f^{\prime}(x)\left(% \lambda\sigma^{2}-\mu+x\right)+\lambda f(x)(x-\mu)=0,\\ f(0)=\frac{1}{2}\lambda e^{\frac{1}{2}\lambda\left(\lambda\sigma^{2}+2\mu% \right)}\,\text{erfc}\left(\frac{\lambda\sigma^{2}+\mu}{\sqrt{2}\sigma}\right)% ,\\ f^{\prime}(0)=\frac{\lambda e^{-\frac{\mu^{2}}{2\sigma^{2}}}\left(\sqrt{2}-% \sqrt{\pi}\lambda\sigma e^{\frac{\left(\lambda\sigma^{2}+\mu\right)^{2}}{2% \sigma^{2}}}\,\text{erfc}\left(\frac{\lambda\sigma^{2}+\mu}{\sqrt{2}\sigma}% \right)\right)}{2\sqrt{\pi}\sigma}\end{array}\right\}
  4. f ( x ; h , μ , σ , τ ) = h σ τ π 2 exp ( 1 2 ( σ τ ) 2 - x - μ σ ) erfc ( 1 2 ( σ τ - x - μ σ ) ) , f(x;h,\mu,\sigma,\tau)=\frac{h\sigma}{\tau}\sqrt{\frac{\pi}{2}}\exp\left(\frac% {1}{2}\left(\frac{\sigma}{\tau}\right)^{2}-\frac{x-\mu}{\sigma}\right)% \operatorname{erfc}\left(\frac{1}{\sqrt{2}}\ \left(\frac{\sigma}{\tau}-\frac{x% -\mu}{\sigma}\right)\right),
  5. h h
  6. τ = 1 λ \tau=\frac{1}{\lambda}
  7. f ( x ; h , μ , σ , τ ) = h exp ( - 1 2 ( x - μ σ ) 2 ) σ τ π 2 erfcx ( 1 2 ( σ τ - x - μ σ ) ) , f(x;h,\mu,\sigma,\tau)=h\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right% )^{2}\right)\frac{\sigma}{\tau}\sqrt{\frac{\pi}{2}}\operatorname{erfcx}\left(% \frac{1}{\sqrt{2}}\ \left(\frac{\sigma}{\tau}-\frac{x-\mu}{\sigma}\right)% \right),
  8. erfcx ( t ) = exp ( t 2 ) erfc ( t ) \operatorname{erfcx}\left(t\right)=\exp{\left(t^{2}\right)}\operatorname{erfc}% \left(t\right)
  9. f ( x ; h , μ , σ , τ ) = h exp ( - 1 2 ( x - μ σ ) 2 ) ( 1 - ( x - μ ) τ σ 2 ) , f(x;h,\mu,\sigma,\tau)=\frac{h\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}% \right)^{2}\right)}{\left(1-\frac{\left(x-\mu\right)\tau}{\sigma^{2}}\right)},
  10. z = 1 2 ( σ τ - x - μ σ ) z=\frac{1}{\sqrt{2}}\left(\frac{\sigma}{\tau}-\frac{x-\mu}{\sigma}\right)
  11. m = μ + ν m=\mu+\nu
  12. s 2 = σ 2 + ν 2 s^{2}=\sigma^{2}+\nu^{2}
  13. γ 1 = 2 ν 3 ( σ 2 + ν 2 ) 3 / 2 \gamma_{1}=\frac{2\nu^{3}}{(\sigma^{2}+\nu^{2})^{3/2}}
  14. μ ¯ = m - s ( γ 1 2 ) 1 / 3 \overline{\mu}=m-s\left(\frac{\gamma_{1}}{2}\right)^{1/3}
  15. σ 2 ¯ = s 2 [ 1 - ( γ 1 2 ) 2 / 3 ] \overline{\sigma^{2}}=s^{2}\left[1-\left(\frac{\gamma_{1}}{2}\right)^{2/3}\right]
  16. ν ¯ = s ( γ 1 2 ) 1 / 3 . \overline{\nu}=s\left(\frac{\gamma_{1}}{2}\right)^{1/3}.
  17. mean - median standard deviation \frac{\,\text{mean}-\,\text{median}}{\,\text{standard deviation}}

Exportation_(logic).html

  1. ( ( P and Q ) R ) ( P ( Q R ) ) ((P\and Q)\to R)\Leftrightarrow(P\to(Q\to R))
  2. \Leftrightarrow
  3. ( ( P and Q ) R ) ( P ( Q R ) ) ((P\and Q)\to R)\vdash(P\to(Q\to R))
  4. \vdash
  5. ( P ( Q R ) ) (P\to(Q\to R))
  6. ( ( P and Q ) R ) ((P\and Q)\to R)
  7. ( P and Q ) R P ( Q R ) . \frac{(P\and Q)\to R}{P\to(Q\to R)}.
  8. ( P and Q ) R (P\and Q)\to R
  9. P ( Q R ) P\to(Q\to R)
  10. ( ( P and Q ) R ) ) ( P ( Q R ) ) ) ((P\and Q)\to R))\to(P\to(Q\to R)))
  11. P P
  12. Q Q
  13. R R

Exsphere_(polyhedra).html

  1. δ δ
  2. tan δ 2 = r e x r i n . \tan\frac{\delta}{2}=\frac{r_{ex}}{r_{in}}.
  3. δ δ
  4. a a
  5. 3 a 3a
  6. δ = π - a r c c o s ( 1 / 3 ) δ=π-arccos(1/3)
  7. δ δ
  8. ( 0 , - 1 , g ) , ( g , 0 , 1 ) , ( 0 , 1 , g ) , (0,-1,g),(g,0,1),(0,1,g),
  9. ( 1 , - g , 0 ) , ( g , 0 , 1 ) , ( 0 , - 1 , g ) , (1,-g,0),(g,0,1),(0,-1,g),
  10. g g
  11. ( g , 1 , 1 - g ) , ( - g , 1 , g - 1 ) (g,1,1-g),(-g,1,g-1)
  12. ( g - 1 , g , 1 ) , ( - g , - 1 , g - 1 ) (g-1,g,1),(-g,-1,g-1)
  13. ( 2 g - 2 , 0 , 2 g ) ( g - 1 , 0 , g ) (2g-2,0,2g)\sim(g-1,0,g)
  14. ( g 2 - g + 1 , - g - ( g - 1 ) 2 , 1 - g + g 2 ) = ( 2 , - 2 , 2 ) ( 1 , - 1 , 1 ) (g^{2}-g+1,-g-(g-1)^{2},1-g+g^{2})=(2,-2,2)\sim(1,-1,1)
  15. cos δ = ( g - 1 ) 1 + g 1 ( g - 1 ) 2 + g 2 3 = 2 g - 1 3 = 5 3 0.74535599. \cos\delta=\frac{(g-1)\cdot 1+g\cdot 1}{\sqrt{(g-1)^{2}+g^{2}}\sqrt{3}}=\frac{% 2g-1}{3}=\frac{\surd 5}{3}\approx 0.74535599.
  16. δ 0.72973 rad 41.81 \therefore\delta\approx 0.72973\,\mathrm{rad}\approx 41.81^{\circ}
  17. tan δ 2 = sin δ 1 + cos δ = 2 3 + 5 0.3819660 \therefore\tan\frac{\delta}{2}=\frac{\sin\delta}{1+\cos\delta}=\frac{2}{3+% \surd 5}\approx 0.3819660
  18. a a
  19. r e x = a ( 3 + 5 ) 3 0.1102641 a . r_{ex}=\frac{a}{(3+\sqrt{5})\sqrt{3}}\approx 0.1102641a.

Extension_neural_network.html

  1. R R
  2. N N
  3. C C
  4. V V
  5. R = [ Y u s u f H e i g h t , 178 c m W e i g h t 98 k g ] R=\begin{bmatrix}Yusuf&Height,&178cm\\ &Weight&98kg\end{bmatrix}
  6. H e i g h t Height
  7. W e i g h t Weight
  8. V V
  9. A = { ( x , y ) | x U , y = K ( x ) } A=\left\{(x,y)|x\in U,y=K(x)\right\}
  10. A A
  11. U U
  12. K K
  13. x x
  14. y y
  15. x x
  16. K ( x ) K(x)
  17. x x
  18. [ - , ] \left[-\infty,\infty\right]
  19. x x
  20. [ 0 , 1 ] \left[0,1\right]
  21. X i n X_{in}
  22. X o u t X_{out}
  23. x x
  24. X i n X_{in}
  25. X o u t X_{out}
  26. n n
  27. n c n_{c}
  28. i i
  29. x i j p x^{p}_{ij}
  30. p p
  31. o i k o_{ik}
  32. k k
  33. o i k o_{ik}
  34. k = 1 , 2 , . , n c k=1,2,....,n_{c}
  35. k * k^{*}
  36. i i
  37. j j
  38. N m N_{m}
  39. N p N_{p}
  40. E τ E_{\tau}
  41. i i
  42. p p
  43. X i p = { x i 1 p , x i 2 p , , x i n p } X_{i}^{p}=\{x_{i1}^{p},x_{i2}^{p},...,x_{in}^{p}\}
  44. 1 p n c 1\leq p\leq n_{c}
  45. X i p X_{i}^{p}
  46. X i p X_{i}^{p}
  47. k * = p k^{*}=p
  48. k * p k^{*}\neq p
  49. z p j n e w = z p j o l d + η ( x i j p - z p j o l d ) z_{pj}^{new}=z_{pj}^{old}+\eta(x_{ij}^{p}-z_{pj}^{old})
  50. z k * j n e w = z k * j o l d - η ( x i j p - z k * j o l d ) z_{k^{*}j}^{new}=z_{k^{*}j}^{old}-\eta(x_{ij}^{p}-z_{k^{*}j}^{old})
  51. w p j L ( n e w ) = w p j L ( o l d ) + η ( x i j p - z p j o l d ) w_{pj}^{L(new)}=w_{pj}^{L(old)}+\eta(x_{ij}^{p}-z_{pj}^{old})
  52. w p j U ( n e w ) = w p j U ( o l d ) + η ( x i j p - z p j o l d ) w_{pj}^{U(new)}=w_{pj}^{U(old)}+\eta(x_{ij}^{p}-z_{pj}^{old})
  53. w k * j L ( n e w ) = w k * j L ( o l d ) - η ( x i j p - z k * j o l d ) w_{k^{*}j}^{L(new)}=w_{k^{*}j}^{L(old)}-\eta(x_{ij}^{p}-z_{k^{*}j}^{old})
  54. w k * j U ( n e w ) = w k * j U ( o l d ) - η ( x i j p - z k * j o l d ) w_{k^{*}j}^{U(new)}=w_{k^{*}j}^{U(old)}-\eta(x_{ij}^{p}-z_{k^{*}j}^{old})
  55. i i
  56. p p
  57. j j
  58. i i
  59. k * k^{*}
  60. j j
  61. i i
  62. E D A > E D B ED_{A}>ED_{B}
  63. i i
  64. E D B > E D A ED_{B}>ED_{A}
  65. i i

Extensions_of_Fisher's_method.html

  1. X = - 2 i = 1 k log e ( p i ) χ 2 ( 2 k ) . X=-2\sum_{i=1}^{k}\log_{e}(p_{i})\sim\chi^{2}(2k).
  2. E [ c χ 2 ( k ) ] = c k , \operatorname{E}[c\chi^{2}(k^{\prime})]=ck^{\prime},
  3. Var [ c χ 2 ( k ) ] = 2 c 2 k . \operatorname{Var}[c\chi^{2}(k^{\prime})]=2c^{2}k^{\prime}.

Extranatural_transformation.html

  1. F : A × B op × B D F:A\times B^{\mathrm{op}}\times B\rightarrow D
  2. G : A × C op × C D G:A\times C^{\mathrm{op}}\times C\rightarrow D
  3. η ( a , b , c ) : F ( a , b , b ) G ( a , c , c ) \eta(a,b,c):F(a,b,b)\rightarrow G(a,c,c)
  4. η ( - , b , c ) \eta(-,b,c)
  5. ( g : b b ) Mor B \forall(g:b\rightarrow b^{\prime})\in\mathrm{Mor}\,B
  6. a A \forall a\in A
  7. c C \forall c\in C
  8. F ( a , b , b ) F ( 1 , 1 , g ) F ( a , b , b ) F ( 1 , g , 1 ) | η ( a , b , c ) | F ( a , b , b ) η ( a , b , c ) G ( a , c , c ) \begin{matrix}F(a,b^{\prime},b)&\xrightarrow{F(1,1,g)}&F(a,b^{\prime},b^{% \prime})\\ _{F(1,g,1)}|&&_{\eta(a,b^{\prime},c)}|\\ F(a,b,b)&\xrightarrow{\eta(a,b,c)}&G(a,c,c)\end{matrix}
  9. ( h : c c ) Mor C \forall(h:c\rightarrow c^{\prime})\in\mathrm{Mor}\,C
  10. a A \forall a\in A
  11. b B \forall b\in B
  12. F ( a , b , b ) η ( a , b , c ) G ( a , c , c ) η ( a , b , c ) | G ( 1 , h , 1 ) | G ( a , c , c ) G ( 1 , 1 , h ) G ( a , c , c ) \begin{matrix}F(a,b,b)&\xrightarrow{\eta(a,b,c^{\prime})}&G(a,c^{\prime},c^{% \prime})\\ _{\eta(a,b,c)}|&&_{G(1,h,1)}|\\ G(a,c,c)&\xrightarrow{G(1,1,h)}&G(a,c,c^{\prime})\end{matrix}

Faber_polynomials.html

  1. f ( z ) = z - 1 + a 0 + a 1 z + \displaystyle f(z)=z^{-1}+a_{0}+a_{1}z+\cdots
  2. P m ( f ) - z - m \displaystyle P_{m}(f)-z^{-m}

Fabius_function.html

  1. n = 1 2 - n ξ n , \sum_{n=1}^{\infty}2^{-n}\xi_{n},

Fabric_sound_evaluation_system.html

  1. Y = C 0 + i = 1 k C i X i - X i ¯ θ i Y=C_{0}+\sum_{i=1}^{k}C_{i}\frac{X_{i}-\bar{X_{i}}}{\theta^{i}}
  2. Y Y
  3. X i X_{i}
  4. i i
  5. X i ¯ \bar{X_{i}}
  6. i i
  7. θ i \theta^{i}
  8. i i
  9. C 0 C_{0}
  10. C i C_{i}

Factor_regression_model.html

  1. 𝐲 n = 𝐀𝐱 n + 𝐁𝐳 n + 𝐜 + 𝐞 n \mathbf{y}_{n}=\mathbf{A}\mathbf{x}_{n}+\mathbf{B}\mathbf{z}_{n}+\mathbf{c}+% \mathbf{e}_{n}
  2. 𝐲 n \mathbf{y}_{n}
  3. n n
  4. G × 1 G\times 1
  5. 𝐱 n \mathbf{x}_{n}
  6. n n
  7. L x L_{x}
  8. 𝐀 \mathbf{A}
  9. 𝐳 n \mathbf{z}_{n}
  10. n n
  11. L z L_{z}
  12. 𝐁 \mathbf{B}
  13. 𝐜 \mathbf{c}
  14. 𝐞 n \mathbf{e}_{n}
  15. 𝐲 n = 𝐀𝐱 n + 𝐜 + 𝐞 n \mathbf{y}_{n}=\mathbf{A}\mathbf{x}_{n}+\mathbf{c}+\mathbf{e}_{n}
  16. 𝐲 n = 𝐁𝐳 n + 𝐜 + 𝐞 n \mathbf{y}_{n}=\mathbf{B}\mathbf{z}_{n}+\mathbf{c}+\mathbf{e}_{n}
  17. 𝐲 n = 𝐀𝐱 n + 𝐁𝐳 n + 𝐜 + 𝐞 n = [ 𝐀 𝐁 ] [ 𝐱 n 𝐳 n ] + 𝐜 + 𝐞 n = 𝐃𝐟 n + 𝐜 + 𝐞 n \begin{aligned}&\displaystyle\mathbf{y}_{n}=\mathbf{A}\mathbf{x}_{n}+\mathbf{B% }\mathbf{z}_{n}+\mathbf{c}+\mathbf{e}_{n}\\ \displaystyle=&\displaystyle\begin{bmatrix}\mathbf{A}&\mathbf{B}\end{bmatrix}% \begin{bmatrix}\mathbf{x}_{n}\\ \mathbf{z}_{n}\end{bmatrix}+\mathbf{c}+\mathbf{e}_{n}\\ \displaystyle=&\displaystyle\mathbf{D}\mathbf{f}_{n}+\mathbf{c}+\mathbf{e}_{n}% \end{aligned}
  18. 𝐃 = [ 𝐀 𝐁 ] \mathbf{D}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\end{bmatrix}
  19. 𝐟 n = [ 𝐱 n 𝐳 n ] \mathbf{f}_{n}=\begin{bmatrix}\mathbf{x}_{n}\\ \mathbf{z}_{n}\end{bmatrix}

Factorization_of_polynomials_over_finite_fields.html

  1. - 1 = a 2 , -1=a^{2},
  2. P = ( x 2 + a ) ( x 2 - a ) . P=(x^{2}+a)(x^{2}-a).
  3. 2 = b 2 , 2=b^{2},
  4. P = ( x 2 + b x + 1 ) ( x 2 - b x + 1 ) . P=(x^{2}+bx+1)(x^{2}-bx+1).
  5. - 2 = c 2 , -2=c^{2},
  6. P = ( x 2 + c x - 1 ) ( x 2 - c x - 1 ) . P=(x^{2}+cx-1)(x^{2}-cx-1).
  7. f = x 11 + 2 x 9 + 2 x 8 + x 6 + x 5 + 2 x 3 + 2 x 2 + 1 𝐅 3 [ x ] , f=x^{11}+2x^{9}+2x^{8}+x^{6}+x^{5}+2x^{3}+2x^{2}+1\in\mathbf{F}_{3}[x],
  8. c = gcd ( f , f ) = x 9 + 2 x 6 + x 3 + 2. c=\gcd(f,f^{\prime})=x^{9}+2x^{6}+x^{3}+2.
  9. y = x + 2 y=x+2
  10. z = x + 1 z=x+1
  11. R = x + 1 R=x+1
  12. i = 2 i=2
  13. w = x + 2 w=x+2
  14. y = x + 2 y=x+2
  15. z = 1 z=1
  16. R = x + 1 R=x+1
  17. i = 3 i=3
  18. w = x + 2 w=x+2
  19. R R
  20. y = 1 y=1
  21. z = x + 2 z=x+2
  22. i = 5 i=5
  23. w = 1 w=1
  24. f = ( x + 1 ) ( x 2 + 1 ) 3 ( x + 2 ) 4 . f=(x+1)(x^{2}+1)^{3}(x+2)^{4}.
  25. i := 1 ; S := , f * := f ; i:=1;\qquad S:=\emptyset,\qquad f^{*}:=f;
  26. deg f * 2 i \deg f^{*}\geq 2i
  27. g = gcd ( f * , x q i - x ) g=\gcd(f^{*},x^{q^{i}}-x)
  28. S := S ( g , i ) S:=S\cup{(g,i)}
  29. S := S ( f * , deg f * ) S:=S\cup{(f^{*},\deg f^{*})}
  30. x q i - x 𝐅 q [ x ] x^{q^{i}}-x\in\mathbf{F}_{q}[x]
  31. g = gcd ( f * , x q i - x ) g=\gcd\left(f^{*},x^{q^{i}}-x\right)
  32. g = gcd ( f * , ( x q i - x mod f * ) ) . g=\gcd\left(f^{*},\left(x^{q^{i}}-x\mod f^{*}\right)\right).
  33. x q i - x mod f * , x^{q^{i}}-x\mod f^{*},
  34. x q i - 1 mod f * x^{q^{i-1}}\mod f^{*}
  35. O ( log ( q ) deg ( f ) 2 ) O\left(\log(q)\deg(f)^{2}\right)
  36. O ( log ( q ) deg ( f ) 3 ) O\left(\log(q)\deg(f)^{3}\right)
  37. O ( deg ( f ) 2 ( log ( q ) + deg ( f ) ) ) O\left(\deg(f)^{2}(\log(q)+\deg(f))\right)
  38. O ( deg ( f ) 2 ( log ( q ) + deg ( f ) ) ) . O\left(\deg(f)^{2}(\log(q)+\deg(f))\right).
  39. f 1 , , f r f_{1},\ldots,f_{r}
  40. { u } { ( gcd ( g , u ) , u / gcd ( g , u ) ) } \,\setminus\,\{u\}\cup\{(\gcd(g,u),u/\gcd(g,u))\}
  41. q d - 1 2 q d 1 2 . \frac{q^{d}-1}{2q^{d}}\sim\tfrac{1}{2}.
  42. 2.5 log 2 r 2.5\log_{2}r
  43. O ( d n 2 log ( r ) log ( q ) ) O(dn^{2}\log(r)\log(q))
  44. O ( n 2 ( log ( r ) log ( q ) + n ) ) O(n^{2}(\log(r)\log(q)+n))
  45. v v q - v ( mod f ) v\to v^{q}-v\;\;(\mathop{{\rm mod}}f)
  46. g := h q d - 1 2 - 1 ( mod f ) g:=h^{\frac{q^{d}-1}{2}}-1\;\;(\mathop{{\rm mod}}f)
  47. g := h q - 1 2 - 1 ( mod f ) . g:=h^{\frac{q-1}{2}}-1\;\;(\mathop{{\rm mod}}f).
  48. O ( n 2 log ( r ) log ( q ) ) O(n^{2}\log(r)\log(q))
  49. O ( n 2 ( log ( q ) + n ) ) O(n^{2}(\log(q)+n))
  50. p i ( x ) , p i ( x q ) , p i ( x q 2 ) , p i ( x q d - 1 ) . p_{i}(x),p_{i}(x^{q}),p_{i}\left(x^{q^{2}}\right),p_{i}\left(x^{q^{d-1}}\right).
  51. B = { α R : p 1 ( α ) , , p k ( α ) 𝐅 q } = { u R : u q = u } \begin{aligned}\displaystyle B&\displaystyle=\left\{\alpha\in R\ :\ p_{1}(% \alpha),\cdots,p_{k}(\alpha)\in\mathbf{F}_{q}\right\}\\ &\displaystyle=\{u\in R\ :\ u^{q}=u\}\end{aligned}
  52. s = ( Y - x ) ( Y - x q ) ( Y - x q d - 1 ) = s 0 + + s d - 1 Y d - 1 + Y d \begin{aligned}\displaystyle s&\displaystyle=(Y-x)\left(Y-x^{q}\right)\cdots% \left(Y-x^{q^{d-1}}\right)\\ &\displaystyle=s_{0}+\cdots+s_{d-1}Y^{d-1}+Y^{d}\end{aligned}
  53. { s 0 , , s d - 1 } \{s_{0},\dots,s_{d-1}\}
  54. p i ( s ) = g i p_{i}(s)=g_{i}
  55. p i ( s h ) p j ( s h ) . p_{i}(s_{h})\neq p_{j}(s_{h}).
  56. v v q - v ( mod f ) v\to v^{q}-v\;\;(\mathop{{\rm mod}}f)
  57. n / p i = n i n/p_{i}=n_{i}
  58. gcd ( f , x q n i - x ) = 1 \gcd\left(f,x^{q^{n_{i}}}-x\right)=1
  59. x q n - x x^{q^{n}}-x
  60. x q n - x x^{q^{n}}-x
  61. x q n i - x . x^{q^{n_{i}}}-x.
  62. n j = n / p j n_{j}=n/p_{j}
  63. h := x q n i - x mod f h:=x^{q^{n_{i}}}-x\bmod f
  64. g := x q n - x mod f g:=x^{q^{n}}-x\bmod f
  65. x q n i mod f x^{q^{n_{i}}}\bmod f
  66. n 1 , , n k n_{1},\ldots,n_{k}
  67. O ( n 2 ( n + log q ) ) O(n^{2}(n+\log q))
  68. x q n i - x ( mod f ) x^{q^{n_{i}}}-x\;\;(\mathop{{\rm mod}}f)
  69. O ( n 2 ( n + log q ) ) O(n^{2}(n+\log q))
  70. O ( n log n ) O(n\log n)
  71. O ( n ( log n ) 2 ) O(n(\log n)^{2})
  72. x q n i - x mod f x^{q^{n_{i}}}-x\bmod f
  73. O ( n 2 log n log q ) O(n^{2}\log n\log q)
  74. O ( k n ( log n ) 2 ) O(kn(\log n)^{2})
  75. O ( n 2 log n log q ) O(n^{2}\log n\log q)

Fannes–Audenaert_inequality.html

  1. ρ \rho
  2. σ \sigma
  3. d d
  4. | S ( ρ ) - S ( σ ) | T log ( d - 1 ) + H [ { T , 1 - T } ] |S(\rho)-S(\sigma)|\leq T\log(d-1)+H[\{T,1-T\}]
  5. H [ { p i } ] = - p i log p i H[\{p_{i}\}]=-\sum p_{i}\log p_{i}\,
  6. { p i } \{p_{i}\}
  7. S ( ρ ) = H [ { λ i } ] S(\rho)=H[\{\lambda_{i}\}]\,
  8. ρ \rho
  9. λ i \lambda_{i}
  10. T ( ρ , σ ) = 1 2 || ρ - σ || 1 = 1 2 Tr [ ( ρ - σ ) ( ρ - σ ) ] T(\rho,\sigma)=\frac{1}{2}||\rho-\sigma||_{1}=\frac{1}{2}\mathrm{Tr}\left[% \sqrt{(\rho-\sigma)^{\dagger}(\rho-\sigma)}\right]
  11. ρ = Diag ( 1 - T , T / ( d - 1 ) , , T / ( d - 1 ) ) \rho=\mathrm{Diag}(1-T,T/(d-1),\dots,T/(d-1))\,
  12. σ = Diag ( 1 , 0 , , 0 ) \sigma=\mathrm{Diag}(1,0,\dots,0)\,
  13. | S ( ρ ) - S ( σ ) | 2 T log ( d ) - 2 T log 2 T |S(\rho)-S(\sigma)|\leq 2T\log(d)-2T\log 2T
  14. T 1 / 2 e T\leq 1/2e
  15. | S ( ρ ) - S ( σ ) | 2 T log ( d ) + 1 / ( e log 2 ) |S(\rho)-S(\sigma)|\leq 2T\log(d)+1/(e\log 2)

Faraday's_ice_pail_experiment.html

  1. S 𝐄 d 𝐀 = 1 ϵ 0 ( Q + Q i n d u c e d ) = 0 \iint\limits_{S}\mathbf{E}\cdot d\mathbf{A}=\frac{1}{\epsilon_{0}}(Q+Q_{% induced})=0\,

Favard's_theorem.html

  1. y n + 1 = ( x - c n ) y n - d n y n - 1 y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}

Fay's_trisecant_identity.html

  1. E ( x , v ) E ( u , y ) θ ( z + u x ω ) θ ( z + v y ω ) - E ( x , u ) E ( v , y ) θ ( z + v x ω ) θ ( z + u y ω ) = E ( x , y ) E ( u , v ) θ ( z ) θ ( z + u + v x + y ω ) \begin{aligned}&\displaystyle E(x,v)E(u,y)\theta\left(z+\int_{u}^{x}\omega% \right)\theta\left(z+\int_{v}^{y}\omega\right)\\ \displaystyle-&\displaystyle E(x,u)E(v,y)\theta\left(z+\int_{v}^{x}\omega% \right)\theta\left(z+\int_{u}^{y}\omega\right)\\ \displaystyle=&\displaystyle E(x,y)E(u,v)\theta(z)\theta\left(z+\int_{u+v}^{x+% y}\omega\right)\end{aligned}
  2. u + v x + y ω = u x ω + v y ω = u y ω + v x ω \begin{aligned}&\displaystyle\int_{u+v}^{x+y}\omega=\int_{u}^{x}\omega+\int_{v% }^{y}\omega=\int_{u}^{y}\omega+\int_{v}^{x}\omega\end{aligned}

Fåhræus_effect.html

  1. r 0 r_{0}
  2. H t H_{t}
  3. H 0 H_{0}
  4. H t H 0 = 1 2 - ( 1 - ( δ r 0 ) ) 2 \frac{H_{t}}{H_{0}}=\frac{1}{2-(1-(\frac{\delta}{r_{0}}))^{2}}
  5. H t H_{t}
  6. H 0 H_{0}
  7. δ \delta
  8. r 0 r_{0}
  9. H t H_{t}
  10. H d H_{d}
  11. H t H d = H d + ( 1 - H d ) ( 1 + 1.7 e x p ( - 0.415 D ) - 0.6 e x p ( - 0.011 D ) ) \frac{H_{t}}{H_{d}}=H_{d}+(1-H_{d})(1+1.7exp(-0.415D)-0.6exp(-0.011D))
  12. H t H_{t}
  13. H d H_{d}
  14. D D

Feature_scaling.html

  1. x = x - min ( x ) max ( x ) - min ( x ) x^{\prime}=\frac{x-\,\text{min}(x)}{\,\text{max}(x)-\,\text{min}(x)}
  2. x x
  3. x x^{\prime}
  4. x = x - x ¯ σ x^{\prime}=\frac{x-\bar{x}}{\sigma}
  5. x x
  6. x ¯ \bar{x}
  7. σ \sigma
  8. x = x || x || x^{\prime}=\frac{x}{||x||}

Fedor_Baranov.html

  1. C = F F + M ( 1 - e - ( F + M ) T ) N 0 C=\frac{F}{F+M}(1-e^{-(F+M)T})N_{0}

Feedback-controlled_electromigration.html

  1. P = I / G P=I/G

Fei–Ranis_model_of_economic_growth.html

  1. Phase 1 : A L ( from figure ) = M P = 0 and A B ( from figure ) = A P \,\text{Phase 1}:AL(\,\text{from figure})=MP=0\,\text{ and }AB(\,\text{from % figure})=AP\,
  2. Phase2 : A P > M P \,\text{Phase2}:AP>MP\,
  3. M P = Real wages = A B = Constant institutional wages (CIW) MP=\,\text{Real wages}=AB=\,\text{Constant institutional wages (CIW)}\,
  4. Phase3 : MP > CIW \,\text{Phase3}:\,\text{MP}>\,\text{CIW}\,
  5. R = t s O t R=\frac{ts}{Ot}
  6. S = t E O t S=\frac{tE}{Ot}
  7. T = t s t e T=\frac{ts}{te}
  8. T = t s t e , T=\frac{ts}{te},
  9. T = t s / O t t e / O t = R S , or T = R S T=\frac{ts/Ot}{te/Ot}=\frac{R}{S},\,\text{ or }T=\frac{R}{S}\,
  10. K t = K o + S o + Π o K_{t}=K_{o}+S_{o}+\Pi_{o}\,

Fekete_problem.html

  1. 1 i < j N x i - x j - s \sum_{1\leq i<j\leq N}\|x_{i}-x_{j}\|^{-s}
  2. 1 i < j N log x i - x j - 1 \sum_{1\leq i<j\leq N}\log\|x_{i}-x_{j}\|^{-1}

Fekete–Szegő_inequality.html

  1. f ( z ) = z + a 2 z 2 + a 3 z 3 + f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots

Fenchel–Moreau_theorem.html

  1. f * * f f^{**}\leq f
  2. ( X , τ ) (X,\tau)
  3. f : X { ± } f:X\to\mathbb{R}\cup\{\pm\infty\}
  4. f = f * * f=f^{**}
  5. f f
  6. f + f\equiv+\infty
  7. f - f\equiv-\infty

Fermat's_right_triangle_theorem.html

  1. y 2 = x ( x - 1 ) ( x + 1 ) y^{2}=x(x-1)(x+1)
  2. x 4 - y 4 = z 2 x^{4}-y^{4}=z^{2}
  3. n = 4 n=4
  4. a 2 a^{2}
  5. b 2 b^{2}
  6. c 2 c^{2}
  7. d 2 d^{2}
  8. a 2 + d 2 = b 2 a^{2}+d^{2}=b^{2}
  9. b 2 + d 2 = c 2 b^{2}+d^{2}=c^{2}
  10. ( d , b ) (d,b)
  11. b 4 - d 4 = ( b 2 - d 2 ) ( b 2 + d 2 ) = a 2 c 2 b^{4}-d^{4}=(b^{2}-d^{2})(b^{2}+d^{2})=a^{2}c^{2}
  12. b 4 - d 4 = e 2 . b^{4}-d^{4}=e^{2}.
  13. n = 4 n=4
  14. ( x , y ) (x,y)
  15. y 2 = x ( x + 1 ) ( x - 1 ) . y^{2}=x(x+1)(x-1).
  16. x x
  17. y y
  18. z z
  19. x = 2 p q x=2pq
  20. y = p 2 - q 2 y=p^{2}-q^{2}
  21. z = p 2 + q 2 z=p^{2}+q^{2}
  22. p p
  23. q q
  24. p q ( p 2 - q 2 ) pq(p^{2}-q^{2})
  25. p p
  26. q q
  27. p + q p+q
  28. p - q p-q
  29. p + q = r 2 p+q=r^{2}
  30. p - q = s 2 p-q=s^{2}
  31. r r
  32. s s
  33. p p
  34. q q
  35. ( r - s ) (r-s)
  36. ( r + s ) (r+s)
  37. u = ( r - s ) / 2 u=(r-s)/2
  38. v = ( r + s ) / 2 v=(r+s)/2
  39. u 2 + v 2 = p u^{2}+v^{2}=p
  40. u u
  41. v v
  42. ( u v ) / 2 = q / 4 (uv)/2=q/4
  43. q q
  44. u v uv
  45. q / 4 q/4

Ferredoxin-thioredoxin_reductase.html

  1. \rightleftharpoons

Fictitious_domain_method.html

  1. D D
  2. D D
  3. Ω \Omega
  4. D D
  5. D n D\subset\mathbb{R}^{n}
  6. u ( x ) u(x)
  7. L u = - ϕ ( x ) , x = ( x 1 , x 2 , , x n ) D Lu=-\phi(x),x=(x_{1},x_{2},\dots,x_{n})\in D
  8. l u = g ( x ) , x D lu=g(x),x\in\partial D\,
  9. D D
  10. Ω \Omega
  11. D D
  12. D Ω D\subset\Omega
  13. Ω \Omega
  14. Ω \Omega
  15. u ϵ ( x ) u_{\epsilon}(x)
  16. L ϵ u ϵ = - ϕ ϵ ( x ) , x = ( x 1 , x 2 , , x n ) Ω L_{\epsilon}u_{\epsilon}=-\phi^{\epsilon}(x),x=(x_{1},x_{2},\dots,x_{n})\in\Omega
  17. l ϵ u ϵ = g ϵ ( x ) , x Ω l_{\epsilon}u_{\epsilon}=g^{\epsilon}(x),x\in\partial\Omega
  18. u ϵ ( x ) ϵ 0 u ( x ) , x D u_{\epsilon}(x)\xrightarrow[\epsilon\rightarrow 0]{}u(x),x\in D\,
  19. d 2 u d x 2 = - 2 , 0 < x < 1 ( 1 ) \frac{d^{2}u}{dx^{2}}=-2,\quad 0<x<1\quad(1)
  20. u ( 0 ) = 0 , u ( 1 ) = 0 u(0)=0,u(1)=0\,
  21. u ϵ ( x ) u_{\epsilon}(x)
  22. d d x k ϵ ( x ) d u ϵ d x = - ϕ ϵ ( x ) , 0 < x < 2 ( 2 ) \frac{d}{dx}k^{\epsilon}(x)\frac{du_{\epsilon}}{dx}=-\phi^{\epsilon}(x),0<x<2% \quad(2)
  23. k ϵ ( x ) k^{\epsilon}(x)
  24. k ϵ ( x ) = { 1 , 0 < x < 1 1 ϵ 2 , 1 < x < 2 k^{\epsilon}(x)=\begin{cases}1,&0<x<1\\ \frac{1}{\epsilon^{2}},&1<x<2\end{cases}
  25. ( 3 ) (3)
  26. ϕ ϵ ( x ) = { 2 , 0 < x < 1 2 c 0 , 1 < x < 2 \phi^{\epsilon}(x)=\begin{cases}2,&0<x<1\\ 2c_{0},&1<x<2\end{cases}
  27. u ϵ ( 0 ) = 0 , u ϵ ( 1 ) = 0 u_{\epsilon}(0)=0,u_{\epsilon}(1)=0
  28. x = 1 x=1
  29. [ u ϵ ( 0 ) ] = 0 , [ k ϵ ( x ) d u ϵ d x ] = 0 [u_{\epsilon}(0)]=0,\ \left[k^{\epsilon}(x)\frac{du_{\epsilon}}{dx}\right]=0
  30. [ ] [\cdot]
  31. [ p ( x ) ] = p ( x + 0 ) - p ( x - 0 ) [p(x)]=p(x+0)-p(x-0)\,
  32. u ( x ) - u ϵ ( x ) = O ( ϵ 2 ) , 0 < x < 1 u(x)-u_{\epsilon}(x)=O(\epsilon^{2}),\quad 0<x<1
  33. u ϵ ( x ) u_{\epsilon}(x)
  34. d 2 u ϵ d x 2 - c ϵ ( x ) u ϵ = - ϕ ϵ ( x ) , 0 < x < 2 ( 4 ) \frac{d^{2}u_{\epsilon}}{dx^{2}}-c^{\epsilon}(x)u_{\epsilon}=-\phi^{\epsilon}(% x),\quad 0<x<2\quad(4)
  35. ϕ ϵ ( x ) \phi^{\epsilon}(x)
  36. c ϵ ( x ) c^{\epsilon}(x)
  37. c ϵ ( x ) = { 1 , 0 < x < 1 1 ϵ 2 , 1 < x < 2 c^{\epsilon}(x)=\begin{cases}1,&0<x<1\\ \frac{1}{\epsilon^{2}},&1<x<2\end{cases}
  38. x = 1 x=1
  39. [ u ϵ ( 0 ) ] = 0 , [ d u ϵ d x ] = 0 [u_{\epsilon}(0)]=0,\ \left[\frac{du_{\epsilon}}{dx}\right]=0
  40. u ( x ) - u ϵ ( x ) = O ( ϵ ) , 0 < x < 1 u(x)-u_{\epsilon}(x)=O(\epsilon),\quad 0<x<1

Field_emission_microscopy.html

  1. M = L / R M=L/R
  2. R R
  3. L L

Field_of_view_in_video_games.html

  1. r = w h = tan ( H 2 ) tan ( V 2 ) r={w\over h}=\frac{\tan\left({H\over 2}\right)}{\tan\left({V\over 2}\right)}
  2. H = 2 arctan ( tan ( V 2 ) × w h ) H=2\arctan\left(\tan\left({V\over 2}\right)\times{w\over h}\right)
  3. V = 2 arctan ( tan ( H 2 ) × h w ) V=2\arctan\left(\tan\left({H\over 2}\right)\times{h\over w}\right)

Field_strength_meter.html

  1. E = 30 P d \mbox{E}~{}=\frac{\sqrt{30\cdot P}}{d}

File:CheiRank1.jpg.html

  1. x = log 10 P i x=\log_{10}P_{i}
  2. y = log 10 P i * y=\log_{10}{P}^{*}_{i}
  3. N = 285509 N=285509
  4. α = 0.85 \alpha=0.85

File:Double_Amici_prism_with_refraction_angles.svg.html

  1. α 1 \alpha_{1}
  2. α 2 \alpha_{2}
  3. θ i \theta_{i}
  4. θ i \theta^{\prime}_{i}
  5. δ \delta

File:L1infin.png.html

  1. f = | x - 1 | - 1 f=|x-1|^{-1}

File:Prism2.svg.html

  1. α 1 \alpha_{1}
  2. α 2 \alpha_{2}
  3. θ i \theta_{i}
  4. θ i \theta^{\prime}_{i}
  5. δ \delta

File:Sin_small_angle_approx_error.svg.html

  1. ϵ = 100 θ - sin θ sin θ \epsilon=100\tfrac{\theta-\sin\theta}{\sin\theta}

File:Triplet_prism_with_refraction_angles.svg.html

  1. α 1 \alpha_{1}
  2. α 2 \alpha_{2}
  3. α 3 \alpha_{3}
  4. θ i \theta_{i}
  5. θ i \theta^{\prime}_{i}
  6. δ \delta

File_dynamics.html

  1. MSD t 1 2 \mathrm{MSD}\approx t^{\frac{1}{2}}
  2. M S D t 1 + a 2 MSD\approx t^{\frac{1+a}{2}}
  3. M S D t 1 - γ 2 / ( 1 + a ) - γ MSD\approx t^{\frac{1-\gamma}{2/(1+a)-\gamma}}
  4. M S D l o g 2 ( t ) MSD\approx log^{2}(t)
  5. ξ 1 - α 3 \xi\approx\sqrt{1-\alpha^{3}}
  6. P ( 𝐱 , t 𝐱 𝟎 ) P(\mathbf{x},t\mid\mathbf{x_{0}})
  7. t P ( 𝐱 , t 𝐱 𝟎 ) = D Σ j = - M M x j 2 P ( 𝐱 , t 𝐱 𝟎 ) . \partial_{t}P(\mathbf{x},t\mid\mathbf{x_{0}})=D\Sigma_{j=-M}^{M}\partial^{2}_{% x_{j}}P(\mathbf{x},t\mid\mathbf{x_{0}}).
  8. P ( 𝐱 , t 𝐱 𝟎 ) P(\mathbf{x},t\mid\mathbf{x_{0}})
  9. 𝐱 = { x - M , x - M + 1 , , x M } \mathbf{x}=\{x_{-M},x_{-M+1},\ldots,x_{M}\}
  10. t t
  11. 𝐱 𝟎 \mathbf{x_{0}}
  12. t 0 t_{0}
  13. ( D x j P ( 𝐱 , t 𝐱 𝟎 ) ) x j = x j + 1 = ( D x j + 1 P ( 𝐱 , t 𝐱 𝟎 ) ) x j + 1 = x j ; j = - M , , M - 1 , \big(D\partial_{x_{j}}P(\mathbf{x},t\mid\mathbf{x_{0}})\big)_{x_{j}=x_{j+1}}=% \big(D\partial_{x_{j+1}}P(\mathbf{x},t\mid\mathbf{x_{0}})\big)_{x_{j+1}=x_{j}}% ;\qquad j=-M,\ldots,M-1,
  14. P ( 𝐱 , t x 0 ) = Π j = - M M δ ( x j - x 0 , j ) . P(\mathbf{x},t\rightarrow\infty\mid x_{0})=\Pi_{j=-M}^{M}\delta(x_{j}-x_{0,j}).
  15. x 0 , j = j Δ x_{0,j}=j\Delta
  16. Δ \Delta
  17. x - M x - M + 1 x M x_{-M}\leq x_{-M+1}\leq\cdots\leq x_{M}
  18. t P ( 𝐱 , t 𝐱 𝟎 ) = Σ j = - M M D j x j 2 P ( 𝐱 , t 𝐱 𝟎 ) . \partial_{t}P(\mathbf{x},t\mid\mathbf{x_{0}})=\Sigma_{j=-M}^{M}D_{j}\partial^{% 2}_{x_{j}}P(\mathbf{x},t\mid\mathbf{x_{0}}).
  19. ( D j x j P ( 𝐱 , t 𝐱 𝟎 ) ) x j = x j + 1 = ( D j + 1 x j + 1 P ( 𝐱 , t 𝐱 𝟎 ) ) x j + 1 = x j ; j = - M , , M - 1 , \big(D_{j}\partial_{x_{j}}P(\mathbf{x},t\mid\mathbf{x_{0}})\big)_{x_{j}=x_{j+1% }}=\big(D_{j+1}\partial_{x_{j+1}}P(\mathbf{x},t\mid\mathbf{x_{0}})\big)_{x_{j+% 1}=x_{j}};\qquad j=-M,\ldots,M-1,
  20. x 0 , j = s i g n ( j ) Δ j 1 / ( 1 - a ) ; 0 a 1. x0,j=sign(j)\Delta\mid j\mid^{1/(1-a)};\qquad 0\leq\mathit{a}\leq 1.
  21. W ( D ) = 1 - γ Λ ( D / Λ ) - γ , 0 γ 1 , W(D)=\frac{1-\gamma}{\Lambda}(D/\Lambda)^{-\gamma},\qquad 0\leq\gamma\leq 1,
  22. ψ α ( t ) k ( k t ) - 1 - α , 0 < α 1 \psi_{\alpha}(t)\sim k(kt)^{-1-\alpha},0<\alpha\leq 1
  23. k α ( t ) k_{\alpha}(t)
  24. t P ( 𝐱 , t 𝐱 𝟎 ) = Σ j = - M M D j x j 2 0 t k α ( t - u ) P ( 𝐱 , u 𝐱 𝟎 ) d u . \partial_{t}P(\mathbf{x},t\mid\mathbf{x_{0}})=\Sigma_{j=-M}^{M}D_{j}\partial_{% x_{j}}^{2}\int_{0}^{t}k_{\alpha}(t-u)P(\mathbf{x},u\mid\mathbf{x_{0}})\,du.
  25. k α ( t ) k_{\alpha}(t)
  26. ψ α ( t ) \psi_{\alpha}(t)
  27. k ¯ α ( s ) = s ψ ¯ α ( s ) 1 - ψ ¯ α ( s ) \bar{k}_{\alpha}(s)=\frac{s\bar{\psi}_{\alpha}(s)}{1-\bar{\psi}_{\alpha}(s)}
  28. f ( t ) f(t)
  29. f ¯ ( s ) = 0 f ( t ) e - s t d t \bar{f}(s)=\int_{0}^{\infty}f(t)e^{-st}\,dt
  30. k α ( t ) k_{\alpha}(t)
  31. ψ α ( t ) \psi_{\alpha}(t)
  32. ψ α ( t ) \psi_{\alpha}(t)
  33. t i t_{i}
  34. t i t_{i}
  35. t i P ( 𝐱 , 𝐭 𝐱 𝟎 ) = D i x i 2 0 t i k α ( t i - u i ) P ( 𝐱 , 𝐭 ( i ) , u i 𝐱 𝟎 ) d u i ; - M i M . \partial_{t_{i}}P(\mathbf{x},\mathbf{t}\mid\mathbf{x_{0}})=D_{i}\partial_{x_{i% }}^{2}\int_{0}^{t_{i}}k_{\alpha}(t_{i}-u_{i})P(\mathbf{x},\mathbf{t}^{{}^{% \prime}(i)},u_{i}\mid\mathbf{x_{0}})\,du_{i};\qquad-M\leq i\leq M.
  36. P ( 𝐱 , 𝐭 𝐱 𝟎 ) P(\mathbf{x},\mathbf{t}\mid\mathbf{x_{0}})
  37. 𝐭 = { t i } i = - M M \mathbf{t}=\{t_{i}\}_{i=-M}^{M}
  38. 𝐭 ( i ) = { t c } c = - M , c i M \mathbf{t}^{{}^{\prime}(i)}=\{t_{c}\}_{c=-M,c\neq i}^{M}
  39. P ( 𝐱 , 𝐱 𝟎 ) = 1 c N Σ { p } e - 1 / 4 D t Σ j = - M M ( x j - x 0 , j ( p ) ) 2 . P(\mathbf{x},\mid\mathbf{x_{0}})=\frac{1}{c_{N}}\Sigma_{\{p\}}e^{-1/4Dt\Sigma_% {j=-M}^{M}(x_{j}-x_{0,j}(p))^{2}}.
  40. p p
  41. N ! N!
  42. P ( r , t r 0 ) P(r,t\mid r_{0})
  43. P ( r , t r 0 ) 1 c N e - R d 2 / 2 τ , P(r,t\mid r_{0})\sim\frac{1}{c_{N}}e^{-R_{d}^{2}/\sqrt{2\tau}},
  44. R d = r d Δ R_{d}=r_{d}\Delta
  45. r d = r - r 0 r_{d}=r-r_{0}
  46. r 0 r_{0}
  47. τ = Δ - 2 D t \tau=\Delta^{-2}Dt
  48. R d 2 τ . \langle R_{d}^{2}\rangle\sim\sqrt{\tau}.
  49. P ( x , t x 0 ) 1 c N Σ { p } e - Σ j = - M M ( x j - x 0 , j ( p ) ) 2 / 4 t D j . P(x,t\mid x_{0})\approx\frac{1}{c_{N}}\Sigma_{\{p\}}e^{-\Sigma_{j=-M}^{M}(x_{j% }-x_{0,j}(p))^{2}/4tD_{j}}.
  50. P ( r , t r 0 ) 1 c N e - R d 2 / 4 τ τ ( 1 - a ) ) / ( 2 - γ ( 1 + a ) ) ; τ = Δ - 2 Λ t . P(r,t\mid r_{0})\sim\frac{1}{c_{N}}e^{-R_{d}^{2}/4\tau\tau^{(1-a))/(2-\gamma(1% +a))}};\qquad\tau=\Delta^{-2}\Lambda t.
  51. R d 2 = 2 τ ( 1 - γ ) / ( 2 c - γ ) , c = 1 / ( ( 1 + a ) ) . \langle R_{d}^{2}\rangle=2\tau^{(1-\gamma)/(2c-\gamma)},\qquad c=1/((1+a)).
  52. P ¯ ( x , s x 0 ) = 1 k ¯ α ( s ) P ¯ nrml ( x , s / k ¯ α ( s ) x 0 ) . \bar{P}(x,s\mid x_{0})=\frac{1}{\bar{k}_{\alpha}(s)}\bar{P}\text{nrml}(x,s/% \bar{k}_{\alpha}(s)\mid x_{0}).
  53. r ¯ 2 ( s ) = 1 k ¯ α ( s ) r ¯ 2 ( s / k ¯ α ( s ) ) nrml . \langle\bar{r}^{2}(s)\rangle=\frac{1}{\bar{k}_{\alpha}(s)}\langle\bar{r}^{2}(s% /\bar{k}_{\alpha}(s))\rangle\text{nrml}.
  54. α \alpha
  55. r 2 ( t ) r 2 ( t ) nrml α . \langle r^{2}(t)\rangle\sim\langle r^{2}(t)\rangle\text{nrml}^{\alpha}.
  56. r r free / n . \langle\mid r\mid\rangle\sim\langle\mid r\mid\rangle\text{free}/n.
  57. n n
  58. r \langle\mid r\mid\rangle
  59. r free \langle\mid r\mid\rangle\text{free}
  60. r free t α / 2 ) \langle\mid r\mid\rangle\text{free}\sim t^{\alpha/2})
  61. n n
  62. r \langle\mid r\mid\rangle
  63. r \langle\mid r\mid\rangle
  64. | r | r free n f ( n ) ; 0 < f ( n ) < 1. \langle|r|\rangle\sim\frac{\langle\mid r\mid\rangle\text{free}}{n}f(n);\qquad 0% <f(n)<1.
  65. r free / n ) \langle\mid r\mid\rangle\text{free}/n)
  66. 0 < f ( n ) < 1 0<f(n)<1
  67. α > 1 \alpha>1
  68. ( 1 / n ! ) f ( n ) (1/n!)\leq f(n)
  69. f ( n ) ( n m ) ( n - m ) ! 1 / n ! , f(n)\sim{\displaystyle\left({{n}\atop{m}}\right)}(n-m)!1/n!,
  70. ( n m ) ( n - m ) ! {\displaystyle\left({{n}\atop{m}}\right)}(n-m)!
  71. f ( n ) e - n / 2 f(n)\sim e^{-n/2}
  72. f ( n ) e - n / n 0 f(n)\sim e^{-n/n_{0}}
  73. n 0 n_{0}
  74. MSD ( α n 0 ) 2 ln 2 ( t ) . \mathrm{MSD}\sim\left(\frac{\alpha}{n_{0}}\right)^{2}\ln^{2}(t).
  75. M S D M A D 2 MSD\sim\mid MAD\mid^{2}
  76. α = 0.9 , 0.1 \alpha=0.9,0.1
  77. ξ ( α ) \xi(\alpha)
  78. ξ ( α ) 1 - α 3 \xi(\alpha)\approx\sqrt{1-\alpha^{3}}
  79. α = 0.9 \alpha=0.9
  80. α = 0.1 \alpha=0.1
  81. α \alpha

Film_capacitor.html

  1. C = ε A d C=\varepsilon\cdot{{A}\over{d}}
  2. X C = - 1 ω C X_{C}=-\frac{1}{\omega C}
  3. X L = ω L ESL X_{L}=\omega L_{\mathrm{ESL}}
  4. Z = | Z | e j θ \ Z=|Z|e^{j\theta}\quad
  5. tan δ \tan\delta
  6. R ESR R_{\mathrm{ESR}}
  7. X C = - 1 ω C X_{C}=-\frac{1}{\omega C}
  8. X L = ω L ESL X_{L}=\omega L_{\mathrm{ESL}}
  9. Z Z
  10. Z = R ESR 2 + ( X C + X L ) 2 Z=\sqrt{R_{\mathrm{ESR}}^{2}+(X_{\mathrm{C}}+X_{\mathrm{L}})^{2}}
  11. Z = R ESR 2 + ( X C + ( - X L ) ) 2 Z=\sqrt{R_{\mathrm{ESR}}^{2}+(X_{\mathrm{C}}+(-X_{\mathrm{L}}))^{2}}
  12. X C = - 1 ω C X_{C}=-\frac{1}{\omega C}
  13. X L = ω L ESL X_{L}=\omega L_{\mathrm{ESL}}
  14. X C = X L X_{C}=X_{L}
  15. R ESR R_{\mathrm{ESR}}
  16. tan δ = ESR ω C \tan\delta=\mbox{ESR}~{}\cdot\omega C
  17. R i s o l R_{isol}

Filter_(large_eddy_simulation).html

  1. ϕ ( s y m b o l x , t ) \phi(symbol{x},t)
  2. ϕ ( s y m b o l x , t ) ¯ = - - ϕ ( s y m b o l r , t ) G ( s y m b o l x - s y m b o l r , t - t ) d t d s y m b o l r , \overline{\phi(symbol{x},t)}=\displaystyle{\int_{-\infty}^{\infty}}\int_{-% \infty}^{\infty}\phi(symbol{r},t^{\prime})G(symbol{x}-symbol{r},t-t^{\prime})% dt^{\prime}dsymbol{r},
  3. G G
  4. ϕ ¯ = G ϕ . \overline{\phi}=G\star\phi.
  5. G G
  6. Δ \Delta
  7. τ c , \tau_{c},
  8. ϕ ¯ . \overline{\phi}.
  9. ϕ \phi
  10. ϕ = ϕ ¯ + ϕ . \phi=\bar{\phi}+\phi^{\prime}.
  11. ϕ = ( 1 - G ) ϕ . \phi^{\prime}=\left(1-G\right)\star\phi.
  12. ϕ ( s y m b o l x , t ) , \phi(symbol{x},t),
  13. ϕ \phi
  14. ϕ ^ ( s y m b o l k , ω ) , \hat{\phi}(symbol{k},\omega),
  15. s y m b o l k , symbol{k},
  16. ω , \omega,
  17. ϕ ^ \hat{\phi}
  18. G ^ ( s y m b o l k , ω ) : \hat{G}(symbol{k},\omega):
  19. ϕ ^ ¯ ( s y m b o l k , ω ) = ϕ ^ ( s y m b o l k , ω ) G ^ ( s y m b o l k , ω ) \overline{\hat{\phi}}(symbol{k},\omega)=\hat{\phi}(symbol{k},\omega)\hat{G}(% symbol{k},\omega)
  20. ϕ ^ ¯ = G ^ ϕ ^ . \overline{\hat{\phi}}=\hat{G}\hat{\phi}.
  21. Δ \Delta
  22. k c , k_{c},
  23. τ c \tau_{c}
  24. ω c . \omega_{c}.
  25. ϕ ^ \hat{\phi}
  26. ϕ ^ = ( 1 - G ^ ) ϕ ^ . \hat{\phi^{\prime}}=(1-\hat{G})\hat{\phi}.
  27. a ¯ = a , \overline{a}=a,
  28. - - G ( s y m b o l ξ , t ) d 3 s y m b o l ξ d t = 1. \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G(symbol{\xi},t^{\prime})d^{3}% symbol{\xi}dt^{\prime}=1.
  29. ϕ + ψ ¯ = ϕ ¯ + ψ ¯ . \overline{\phi+\psi}=\overline{\phi}+\overline{\psi}.
  30. ϕ s ¯ = ϕ ¯ s , s = s y m b o l x , t . \overline{\frac{\partial\phi}{\partial s}}=\frac{\partial\overline{\phi}}{% \partial s},\qquad s=symbol{x},t.
  31. [ f , g ] [f,g]
  32. f f
  33. g g
  34. [ f , g ] ϕ = f g ( ϕ ) - g f ( ϕ ) = f ( g ( ϕ ) ) - g ( f ( ϕ ) ) , [f,g]\phi=f\circ g(\phi)-g\circ f(\phi)=f(g(\phi))-g(f(\phi)),
  35. [ G , s ] = 0. \left[G\star,\frac{\partial}{\partial s}\right]=0.
  36. ϕ ¯ ¯ ϕ ¯ , G G ϕ = G 2 ϕ G ϕ , \begin{array}[]{rcl}\overline{\overline{\phi}}&\neq&\overline{\phi},\\ G\star G\star\phi=G^{2}\phi&\neq&G\star\phi,\end{array}
  37. ϕ ¯ = G ( 1 - G ) ϕ 0. \overline{\phi^{\prime}}=G\star(1-G)\star\phi\neq 0.
  38. Δ \Delta
  39. [ s y m b o l x , G ] ϕ = s y m b o l x ( G ϕ ) - G ϕ s y m b o l x = s y m b o l x Ω G ( s y m b o l x - s y m b o l r , Δ ( s y m b o l x , t ) ) ϕ ( s y m b o l r , t ) d s y m b o l r - G ϕ s y m b o l x = ( G Δ ϕ ) Δ x + d Ω G ( x - r , Δ ( x , t ) ) ϕ ( r , t ) s y m b o l n d S , \begin{array}[]{rcl}\left[\frac{\partial}{\partial symbol{x}},G\star\right]% \phi&=&\frac{\partial}{\partial symbol{x}}\left(G\star\phi\right)-G\star\frac{% \partial\phi}{\partial symbol{x}}\\ &=&\frac{\partial}{\partial symbol{x}}\int_{\Omega}G(symbol{x}-symbol{r},% \Delta(symbol{x},t))\phi(symbol{r},t)dsymbol{r}-G\star\frac{\partial\phi}{% \partial symbol{x}}\\ &=&\left(\frac{\partial G}{\partial\Delta}\star\phi\right)\frac{\partial\Delta% }{\partial x}+\int_{d\Omega}G(x-r,\Delta(x,t))\phi(r,t)symbol{n}dS\end{array},
  40. s y m b o l n symbol{n}
  41. Ω \Omega
  42. d Ω . d\Omega.
  43. Δ , \Delta,
  44. G G
  45. [ t , G ] = ( G Δ ϕ ) Δ t . \left[\frac{\partial}{\partial t},G\star\right]=\left(\frac{\partial G}{% \partial\Delta}\star\phi\right)\frac{\partial\Delta}{\partial t}.
  46. G ( s y m b o l x , t ) G(symbol{x},t)
  47. G ^ ( s y m b o l k , ω ) , \hat{G}(symbol{k},\omega),
  48. G ( s y m b o l x - s y m b o l r ) = { 1 Δ , if | s y m b o l x - s y m b o l r | Δ 2 , 0 , otherwise . G(symbol{x}-symbol{r})=\begin{cases}\frac{1}{\Delta},&\,\text{if}\left|symbol{% x}-symbol{r}\right|\leq\frac{\Delta}{2},\\ 0,&\,\text{otherwise}.\end{cases}
  49. G ^ ( s y m b o l k ) = sin ( 1 2 k Δ ) 1 2 k Δ . \hat{G}(symbol{k})=\frac{\sin{(\frac{1}{2}k\Delta)}}{\frac{1}{2}k\Delta}.
  50. G ( s y m b o l x - s y m b o l r ) = ( 6 π Δ 2 ) 1 2 exp ( - 6 ( s y m b o l x - r ) 2 Δ 2 ) . G(symbol{x}-symbol{r})=\left(\frac{6}{\pi\Delta^{2}}\right)^{\frac{1}{2}}\exp{% \left(-\frac{6(symbol{x-r})^{2}}{\Delta^{2}}\right)}.
  51. G ^ ( s y m b o l k ) = exp ( - s y m b o l k 2 Δ 2 24 ) . \hat{G}(symbol{k})=\exp{\left(-\frac{symbol{k}^{2}\Delta^{2}}{24}\right)}.
  52. G ( s y m b o l x - s y m b o l r ) = sin ( π ( s y m b o l x - r ) / Δ ) π ( s y m b o l x - r ) . G(symbol{x}-symbol{r})=\frac{\sin{(\pi(symbol{x-r})/\Delta)}}{\pi(symbol{x-r})}.
  53. G ^ ( s y m b o l k ) = H ( k c - | k | ) , k c = π Δ . \hat{G}(symbol{k})=H\left(k_{c}-\left|k\right|\right),\qquad k_{c}=\frac{\pi}{% \Delta}.

Financial_correlation.html

  1. Y = X 2 Y=X^{2}
  2. d S ( t ) S ( t ) \frac{dS(t)}{S(t)}
  3. σ ( t ) \ \sigma(t)
  4. d S ( t ) S ( t ) = μ d t + σ ( t ) d z 1 ( t ) \frac{dS(t)}{S(t)}=\mu\,dt+\sigma(t)\,dz_{1}(t)
  5. d σ 2 ( t ) = g [ σ L 2 - σ 2 ( t ) ] d t + ξ σ ( t ) d z 2 ( t ) d\sigma^{2}(t)=g[\sigma_{L}^{2}-\sigma^{2}(t)]\,dt+\xi\sigma(t)\,dz_{2}(t)
  6. μ \ \mu
  7. S S
  8. σ ( t ) \ \sigma(t)
  9. S S
  10. σ 2 ( t ) \ \sigma^{2}(t)
  11. σ L 2 \sigma_{L}^{2}
  12. ξ \ \xi
  13. d z ( t ) = ε t d t dz(t)=\varepsilon_{t}\sqrt{dt}
  14. ε t \ \varepsilon_{t}
  15. ε t \varepsilon_{t}
  16. S S
  17. d z 1 dz_{1}
  18. d z 2 dz_{2}
  19. ρ \ \rho
  20. Corr [ d z 1 ( t ) , d z 2 ( t ) ] = ρ d t \operatorname{Corr}[dz_{1}(t),dz_{2}(t)]=\rho\,dt
  21. d z 1 ( t ) = ρ d z 2 ( t ) + 1 - ρ d z 3 ( t ) dz_{1}(t)=\sqrt{\rho}dz_{2}(t)+\sqrt{1-\rho}\,dz_{3}(t)
  22. d z 2 ( t ) dz_{2}(t)
  23. d z 3 ( t ) dz_{3}(t)
  24. d z ( t ) dz(t)
  25. d z ( t ) dz(t^{\prime})
  26. 1 X = 1 { τ X T } 1_{X}=1_{\{\tau_{X}\leq T\}}
  27. 1 Y = 1 { τ Y T } 1_{Y}=1_{\{\tau_{Y}\leq T\}}
  28. τ X \tau_{X}
  29. X X
  30. τ Y \tau_{Y}
  31. Y Y
  32. X X
  33. T T
  34. 1 X 1_{X}
  35. Y Y
  36. P ( X ) P(X)
  37. P ( Y ) P(Y)
  38. X X
  39. Y Y
  40. P ( X Y ) P(XY)
  41. P ( X ) - ( P ( X ) ) 2 \sqrt{P(X)-(P(X))^{2}}
  42. 1 { τ X T } 1_{\{\tau_{X}\leq T\}}
  43. 1 { τ Y T } 1_{\{\tau_{Y}\leq T\}}
  44. ρ ( 1 { τ X T } , 1 { τ Y T } ) = P ( X Y ) - P ( X ) P ( Y ) P ( X ) - ( P ( X ) ) 2 P ( Y ) - ( P ( Y ) ) 2 \rho(1_{\{\tau_{X}\leq T\}},1_{\{\tau_{Y}\leq T\}})=\frac{P(XY)-P(X)P(Y)}{% \sqrt{P(X)-(P(X))^{2}}\sqrt{P(Y)-(P(Y))^{2}}}
  45. C : [ 0 , 1 ] n [ 0 , 1 ] C:[0,1]^{n}\rightarrow[0,1]
  46. u i u_{i}
  47. u i = u 1 , , u n , u i [ 0 , 1 ] u_{i}=u_{1},...,u_{n},u_{i}\in[0,1]
  48. i N i\in N
  49. C C
  50. C ( u 1 , , u n ) = F [ F 1 - 1 ( u 1 ) , , F n - 1 ( u n ) ] C(u_{1},\ldots,u_{n})=F[F_{1}^{-1}(u_{1}),\ldots,F_{n}^{-1}(u_{n})]
  51. F i \ F_{i}
  52. F i - 1 F_{i}^{-1}
  53. F i \ F_{i}
  54. F i - 1 ( u i ) F_{i}^{-1}(u_{i})
  55. Q i ( t ) Q_{i}(t)
  56. u i = Q i ( t ) \ u_{i}=Q_{i}(t)
  57. C G D ( u 1 , , u n ) = M n , R [ N 1 - 1 ( Q 1 ( t ) ) , , N n - 1 ( Q n ( t ) ) ] C_{GD}(u_{1},\ldots,u_{n})=M_{n,R}[N_{1}^{-1}(Q_{1}(t)),\ldots,N_{n}^{-1}(Q_{n% }(t))]
  58. N i - 1 N_{i}^{-1}
  59. Q i ( t ) Q_{i}(t)
  60. N i - 1 Q i ( t ) N_{i}^{-1}Q_{i}(t)
  61. M n , R M_{n,R}
  62. M n , R M_{n,R}
  63. x i = ρ M + 1 - ρ Z i x_{i}=\sqrt{\rho}M+\sqrt{1-\rho}Z_{i}
  64. M M
  65. Z i Z_{i}
  66. 0 ρ 1 0\leq\rho\leq 1
  67. x i x_{i}
  68. M M
  69. Z i Z_{i}
  70. x i x_{i}
  71. M M
  72. i = 1 , , n , x i = M i=1,...,n,\ x_{i}=M
  73. x i x_{i}
  74. i = 1 , , n , x i = Z i i=1,\ldots,n,\ x_{i}=Z_{i}
  75. x i x_{i}

Financial_ratio.html

  1. Gross Profit Net Sales \frac{\mbox{Gross Profit}~{}}{\mbox{Net Sales}~{}}
  2. Net Sales – COGS Net Sales \frac{\mbox{Net Sales -- COGS}~{}}{\mbox{Net Sales}~{}}
  3. Operating Income Net Sales \frac{\mbox{Operating Income}~{}}{\mbox{Net Sales}~{}}
  4. Net Profit Net Sales \frac{\mbox{Net Profit}~{}}{\mbox{Net Sales}~{}}
  5. Net Income Average Shareholders Equity \frac{\mbox{Net Income}~{}}{\mbox{Average Shareholders Equity}~{}}
  6. Net Income Average Total Assets \frac{\mbox{Net Income}~{}}{\mbox{Average Total Assets}~{}}
  7. Net Income Total Assets \frac{\mbox{Net Income}~{}}{\mbox{Total Assets}~{}}
  8. ( Net Income Net Sales ) ( Net Sales Total Assets ) \left(\frac{\mbox{Net Income}~{}}{\mbox{Net Sales}~{}}\right)\left(\frac{\mbox% {Net Sales}~{}}{\mbox{Total Assets}~{}}\right)
  9. ( Net Income Net Sales ) ( Net Sales Average Assets ) ( Average Assets Average Equity ) \left(\frac{\mbox{Net Income}~{}}{\mbox{Net Sales}~{}}\right)\left(\frac{\mbox% {Net Sales}~{}}{\mbox{Average Assets}~{}}\right)\left(\frac{\mbox{Average % Assets}~{}}{\mbox{Average Equity}~{}}\right)
  10. Net Income Fixed Assets + Working Capital \frac{\mbox{Net Income}~{}}{\mbox{Fixed Assets + Working Capital}~{}}
  11. EBIT(1 − Tax Rate) Invested Capital \frac{\mbox{EBIT(1 − Tax Rate)}~{}}{\mbox{Invested Capital}~{}}
  12. Expected Return Economic Capital \frac{\mbox{Expected Return}~{}}{\mbox{Economic Capital}~{}}
  13. Expected Return Value at Risk \frac{\mbox{Expected Return}~{}}{\mbox{Value at Risk}~{}}
  14. EBIT Capital Employed \frac{\mbox{EBIT}~{}}{\mbox{Capital Employed}~{}}
  15. Cash Flow Market Recapitalisation \frac{\mbox{Cash Flow}~{}}{\mbox{Market Recapitalisation}~{}}
  16. Non-Interest expense Revenue \frac{\mbox{Non-Interest expense}~{}}{\mbox{Revenue}~{}}
  17. Net debt Equity \frac{\mbox{Net debt}~{}}{\mbox{Equity}~{}}
  18. EBIT Total Assets \frac{\mbox{EBIT}~{}}{\mbox{Total Assets}~{}}
  19. Current Assets Current Liabilities \frac{\mbox{Current Assets}~{}}{\mbox{Current Liabilities}~{}}
  20. Current Assets – (Inventories + Prepayments) Current Liabilities \frac{\mbox{Current Assets -- (Inventories + Prepayments)}~{}}{\mbox{Current % Liabilities}~{}}
  21. Cash and Marketable Securities Current Liabilities \frac{\mbox{Cash and Marketable Securities}~{}}{\mbox{Current Liabilities}~{}}
  22. Operating Cash Flow Total Debts \frac{\mbox{Operating Cash Flow}~{}}{\mbox{Total Debts}~{}}
  23. Accounts Receivable Annual Credit Sales ÷ 365 Days \frac{\mbox{Accounts Receivable}~{}}{\mbox{Annual Credit Sales ÷ 365 Days}~{}}
  24. Percent Change in Net Operating Income Percent Change in Sales \frac{\mbox{Percent Change in Net Operating Income}~{}}{\mbox{Percent Change % in Sales}~{}}
  25. Accounts Receivable Total Annual Sales ÷ 365 Days \frac{\mbox{Accounts Receivable}~{}}{\mbox{Total Annual Sales ÷ 365 Days}~{}}
  26. Accounts Payable Annual Credit Purchases ÷ 365 Days \frac{\mbox{Accounts Payable}~{}}{\mbox{Annual Credit Purchases ÷ 365 Days}~{}}
  27. Net Sales Total Assets \frac{\mbox{Net Sales}~{}}{\mbox{Total Assets}~{}}
  28. Cost of Goods Sold Average Inventory \frac{\mbox{Cost of Goods Sold}~{}}{\mbox{Average Inventory}~{}}
  29. Net Credit Sales Average Net Receivables \frac{\mbox{Net Credit Sales}~{}}{\mbox{Average Net Receivables}~{}}
  30. 365 Days Inventory Turnover \frac{\mbox{365 Days}~{}}{\mbox{Inventory Turnover}~{}}
  31. ( Inventory Cost of Goods Sold ) 365 Days \left(\frac{\mbox{Inventory}~{}}{\mbox{Cost of Goods Sold}~{}}\right)\mbox{365% Days}~{}
  32. ( Receivables Net Sales ) 365 Days \left(\frac{\mbox{Receivables}~{}}{\mbox{Net Sales}~{}}\right)\mbox{365 Days}~{}
  33. ( Accounts Payables Purchases ) 365 Days \left(\frac{\mbox{Accounts Payables}~{}}{\mbox{Purchases}~{}}\right)\mbox{365 % Days}~{}
  34. Inventory Conversion Period + Receivables Conversion Period - Payables Conversion Period \mbox{Inventory Conversion Period + Receivables Conversion Period - Payables % Conversion Period}~{}
  35. Total Liabilities Total Assets \frac{\mbox{Total Liabilities}~{}}{\mbox{Total Assets}~{}}
  36. Long-term Debt + Value of Leases Average Shareholders Equity \frac{\mbox{Long-term Debt + Value of Leases}~{}}{\mbox{Average Shareholders % Equity}~{}}
  37. Long-term Debt Average Shareholders Equity \frac{\mbox{Long-term Debt}~{}}{\mbox{Average Shareholders Equity}~{}}
  38. EBIT Annual Interest Expense \frac{\mbox{EBIT}~{}}{\mbox{Annual Interest Expense}~{}}
  39. Net Income Annual Interest Expense \frac{\mbox{Net Income}~{}}{\mbox{Annual Interest Expense}~{}}
  40. Net Operating Income Total Debt Service \frac{\mbox{Net Operating Income}~{}}{\mbox{Total Debt Service}~{}}
  41. Net Earnings Number of Shares \frac{\mbox{Net Earnings}~{}}{\mbox{Number of Shares}~{}}
  42. Dividends Earnings \frac{\mbox{Dividends}~{}}{\mbox{Earnings}~{}}
  43. Dividends EPS \frac{\mbox{Dividends}~{}}{\mbox{EPS}~{}}
  44. Earnings per Share Dividend per Share \frac{\mbox{Earnings per Share}~{}}{\mbox{Dividend per Share}~{}}
  45. Market Price per Share Diluted EPS \frac{\mbox{Market Price per Share}~{}}{\mbox{Diluted EPS}~{}}
  46. Dividend Current Market Price \frac{\mbox{Dividend}~{}}{\mbox{Current Market Price}~{}}
  47. Market Price per Share Present Value of Cash Flow per Share \frac{\mbox{Market Price per Share}~{}}{\mbox{Present Value of Cash Flow per % Share}~{}}
  48. Market Price per Share Balance Sheet Price per Share \frac{\mbox{Market Price per Share}~{}}{\mbox{Balance Sheet Price per Share}~{}}
  49. Market Price per Share Gross Sales \frac{\mbox{Market Price per Share}~{}}{\mbox{Gross Sales}~{}}
  50. Price per Earnings Annual EPS Growth \frac{\mbox{Price per Earnings}~{}}{\mbox{Annual EPS Growth}~{}}
  51. Enterprise Value EBITDA \frac{\mbox{Enterprise Value}~{}}{\mbox{EBITDA}~{}}
  52. Enterprise Value Net Sales \frac{\mbox{Enterprise Value}~{}}{\mbox{Net Sales}~{}}

Find_first_set.html

  1. log 2 x \lfloor\log_{2}x\rfloor
  2. log 2 x y log 2 x + log 2 y \lceil\log_{2}xy\rceil\leq\lceil\log_{2}x\rceil+\lceil\log_{2}y\rceil

Finite_extensions_of_local_fields.html

  1. L / K L/K
  2. l / k l/k
  3. G G
  4. L / K L/K
  5. 𝒪 L / 𝒪 L 𝔭 \mathcal{O}_{L}/\mathcal{O}_{L}\mathfrak{p}
  6. 𝔭 \mathfrak{p}
  7. 𝒪 K \mathcal{O}_{K}
  8. [ L : K ] = [ l : k ] [L:K]=[l:k]
  9. G G
  10. π \pi
  11. K K
  12. π \pi
  13. L L
  14. L / K L/K
  15. Gal ( l / k ) \operatorname{Gal}(l/k)
  16. L / K L/K
  17. l / k l/k
  18. G G
  19. L / K L/K
  20. G G
  21. L = K [ π ] L=K[\pi]
  22. π \pi
  23. N ( L / K ) N(L/K)
  24. K K

First-difference_estimator.html

  1. Δ y i t \Delta y_{it}
  2. Δ x i t \Delta x_{it}
  3. c i c_{i}
  4. y i t = x i t β + c i + u i t , t = 1 , T , y_{it}=x_{it}\beta+c_{i}+u_{it},t=1,...T,
  5. y i t - 1 = x i t - 1 β + c i + u i t - 1 , t = 2 , T . y_{it-1}=x_{it-1}\beta+c_{i}+u_{it-1},t=2,...T.
  6. Δ y i t = y i t - y i t - 1 = Δ x i t β + Δ u i t , t = 2 , T , \Delta y_{it}=y_{it}-y_{it-1}=\Delta x_{it}\beta+\Delta u_{it},t=2,...T,
  7. c i c_{i}
  8. β ^ F D \hat{\beta}_{FD}
  9. β ^ F D = ( Δ X Δ X ) - 1 Δ X Δ y \hat{\beta}_{FD}=(\Delta X^{\prime}\Delta X)^{-1}\Delta X^{\prime}\Delta y
  10. Δ X Δ X \Delta X^{\prime}\Delta X
  11. r a n k [ Δ X Δ X ] = k rank[\Delta X^{\prime}\Delta X]=k
  12. A v a ^ r ( β ^ F D ) = σ ^ u 2 ( Δ X Δ X ) - 1 , Av\hat{a}r(\hat{\beta}_{FD})=\hat{\sigma}^{2}_{u}(\Delta X^{\prime}\Delta X)^{% -1},
  13. σ ^ u 2 \hat{\sigma}^{2}_{u}
  14. σ ^ u 2 = [ n ( T - 1 ) - K ] - 1 u ^ u ^ . \hat{\sigma}^{2}_{u}=[n(T-1)-K]^{-1}\hat{u}^{\prime}\hat{u}.
  15. E [ u i t - u i t - 1 | x i t - x i t - 1 ] = 0 E[u_{it}-u_{it-1}|x_{it}-x_{it-1}]=0
  16. E [ β ^ F D ] = β E[\hat{\beta}_{FD}]=\beta
  17. p l i m β ^ = β plim\hat{\beta}=\beta
  18. u i t u_{it}
  19. T = 2 T=2
  20. u i t u_{it}
  21. u i t u_{it}
  22. Δ u i t \Delta u_{it}

Fiscal_sustainability.html

  1. B t = i = 1 ( 1 + r ) - i P B t + i B_{t}=\sum_{i=1}^{\infty}(1+r)^{-i}PB_{t+i}
  2. B t B_{t}
  3. r r
  4. P B t {PB}_{t}
  5. I T G A P = ( r - g ) ( b t - i = 1 ( 1 + g 1 + r ) i p b t + i ) 1 + g ITGAP=\frac{(r-g)(b_{t}-\sum_{i=1}^{\infty}(\frac{1+g}{1+r})^{i}pb_{t+i})}{1+g}
  6. b t b_{t}
  7. r r
  8. g g
  9. p b t pb_{t}

Fitting_ideal.html

  1. a j 1 m 1 + + a j n m n = 0 ( for j = 1 , 2 , ) a_{j1}m_{1}+\cdots+a_{jn}m_{n}=0\ (\,\text{for }j=1,2,\dots)\,

Five-limit_tuning.html

  1. f E = 5 1 3 0 2 - 2 f C = 5 4 256 Hz = 320 Hz f_{E}=5^{1}\cdot 3^{0}\cdot 2^{-2}\cdot f_{C}={5\over 4}\cdot 256\ \mathrm{Hz}% =320\ \mathrm{Hz}
  2. f E = 5 0 3 5 2 - 6 f C = 81 64 256 Hz = 324 Hz f_{E}=5^{0}\cdot 3^{5}\cdot 2^{-6}\cdot f_{C}={81\over 64}\cdot 256\ \mathrm{% Hz}=324\ \mathrm{Hz}
  3. 1 1 2 3 5 4 = 10 12 = 5 6 . {1\over 1}\cdot{2\over 3}\cdot{5\over 4}={10\over 12}={5\over 6}.
  4. 5 6 2 1 = 10 6 = 5 3 . {5\over 6}\cdot{2\over 1}={10\over 6}={5\over 3}.
  5. S 1 = 5 4 ÷ 6 5 = 25 24 70.672 cents S_{1}={{5\over 4}\div{6\over 5}}={25\over 24}\approx 70.672\ \hbox{cents}
  6. S 2 = 9 8 ÷ 16 15 = 135 128 92.179 cents S_{2}={{9\over 8}\div{16\over 15}}={135\over 128}\approx 92.179\ \hbox{cents}
  7. S 3 = 16 15 111.731 cents S_{3}={16\over 15}\approx 111.731\ \hbox{cents}
  8. S 4 = 9 5 ÷ 5 3 = 27 25 133.238 cents S_{4}={{9\over 5}\div{5\over 3}}={27\over 25}\approx 133.238\ \hbox{cents}
  9. S E = 2 12 = 100.000 cents . S_{E}=\sqrt[12]{2}=100.000\ \hbox{cents}.
  10. m 2 A 1 {m2\over A1}
  11. S 3 S 2 = 16 15 ÷ 135 128 {S_{3}\over S_{2}}={{16\over 15}\div{135\over 128}}
  12. 2048 2025 {2048\over 2025}
  13. 19.6 19.6~{}
  14. L D D S = G D L D {{LD}\over{DS}}={{GD}\over{LD}}
  15. S 2 S 1 = 135 128 ÷ 25 24 {S_{2}\over S_{1}}={{135\over 128}\div{25\over 24}}
  16. 81 80 {81\over 80}
  17. 21.5 21.5~{}
  18. S 4 S 3 = 27 25 ÷ 16 15 {S_{4}\over S_{3}}={{27\over 25}\div{16\over 15}}
  19. m 2 A 1 {m2\over A1}
  20. S 3 S 1 = 16 15 ÷ 25 24 {S_{3}\over S_{1}}={{16\over 15}\div{25\over 24}}
  21. 128 125 {128\over 125}
  22. 41.1 41.1~{}
  23. S 4 S 2 = 27 25 ÷ 135 128 {S_{4}\over S_{2}}={{27\over 25}\div{135\over 128}}
  24. m 2 A 1 {m2\over A1}
  25. S 4 S 1 = 27 25 ÷ 25 24 {S_{4}\over S_{1}}={{27\over 25}\div{25\over 24}}
  26. 648 625 {648\over 625}
  27. 62.6 62.6~{}
  28. 81 64 80 81 = 1 5 4 1 = 5 4 {81\over 64}\cdot{80\over 81}={{1\cdot 5}\over{4\cdot 1}}={5\over 4}
  29. 32 27 81 80 = 2 3 1 5 = 6 5 {32\over 27}\cdot{81\over 80}={{2\cdot 3}\over{1\cdot 5}}={6\over 5}

FKT_algorithm.html

  1. PerfMatch ( G ) = M P M ( | V | ) ( i , j ) M A i j . \operatorname{PerfMatch}(G)=\sum_{M\in PM(|V|)}\prod_{(i,j)\in M}A_{ij}.
  2. pf ( A ) = M P M ( n ) sgn ( M ) ( i , j ) M A i j , \operatorname{pf}(A)=\sum_{M\in PM(n)}\operatorname{sgn}(M)\prod_{(i,j)\in M}A% _{ij},
  3. pf ( A ) 2 = det ( A ) , \operatorname{pf}(A)^{2}=\det(A),

Flat_vector_bundle.html

  1. π : E X \pi:E\to X
  2. : Γ ( X , E ) Γ ( X , Ω X 1 E ) \nabla:\Gamma(X,E)\to\Gamma(X,\Omega_{X}^{1}\otimes E)
  3. Ω X * ( E ) = Ω X * E \Omega_{X}^{*}(E)=\Omega^{*}_{X}\otimes E
  4. 𝒪 X \mathcal{O}_{X}
  5. Ω X * ( E ) \Omega_{X}^{*}(E)
  6. d 2 = 0 d^{2}=0
  7. Ω X * ( E ) \Omega_{X}^{*}(E)
  8. \ { 0 } , \mathbb{C}\backslash\{0\},
  9. - d z z -\frac{dz}{z}
  10. Λ top M \Lambda^{\mathrm{top}}M

Flip-flop_(electronics).html

  1. Q Q
  2. Q ¯ \overline{Q}
  3. Q ¯ \overline{Q}
  4. Q ¯ \overline{Q}
  5. S R ¯ \overline{SR}
  6. S R ¯ \overline{SR}
  7. S ¯ \overline{S}
  8. R ¯ \overline{R}
  9. S R ¯ \overline{SR}
  10. S R ¯ \overline{SR}
  11. S ¯ \overline{S}
  12. R ¯ \overline{R}
  13. Q ¯ \overline{Q}
  14. S R ¯ \overline{SR}
  15. S R ¯ \overline{SR}
  16. Q ¯ \overline{Q}
  17. S R ¯ \overline{SR}
  18. S R ¯ \overline{SR}
  19. Q ¯ \overline{Q}
  20. Q ¯ \overline{Q}
  21. Q n e x t = T Q = T Q ¯ + T ¯ Q Q_{next}=T\oplus Q=T\overline{Q}+\overline{T}Q
  22. T T
  23. Q Q
  24. Q n e x t Q_{next}
  25. Q Q
  26. Q n e x t Q_{next}
  27. T T
  28. Q n e x t = J Q ¯ + K ¯ Q Q_{next}=J\overline{Q}+\overline{K}Q

Flood_Studies_Report.html

  1. P R = 0.829 P I M P + 25 S O I L + 0.078 U C W I - 20.7 PR=0.829PIMP+25SOIL+0.078UCWI-20.7

Flory–Fox_equation.html

  1. T g = T g , - K M n T_{g}=T_{g,\infty}-\frac{K}{M_{n}}
  2. T g = T g , - K ( M n M w ) 1 2 . T_{g}=T_{g,\infty}-\frac{K}{(M_{n}M_{w})^{\frac{1}{2}}}.
  3. 1 T g = 1 T g , + K T g , 2 1 M n \frac{1}{T_{g}}=\frac{1}{T_{g,\infty}}+\frac{K}{T_{g,\infty}^{2}}\frac{1}{M_{n}}
  4. 1 T g = w 1 T g , 1 + w 2 T g , 2 . \frac{1}{T_{g}}=\frac{w_{1}}{T_{g,1}}+\frac{w_{2}}{T_{g,2}}.

Flory–Rehner_equation.html

  1. - [ ln ( 1 - ν 2 ) + ν 2 + χ 1 ν 2 2 ] = V 1 n ( ν 2 1 3 - ν 2 2 ) -\left[\ln{\left(1-\nu_{2}\right)}+\nu_{2}+\chi_{1}\nu_{2}^{2}\right]=V_{1}n% \left(\nu_{2}^{\frac{1}{3}}-\frac{\nu_{2}}{2}\right)
  2. ν 2 \nu_{2}
  3. V 1 V_{1}
  4. n n
  5. χ 1 \chi_{1}
  6. - [ ln ( 1 - ν 2 ) + ν 2 + χ 1 ν 2 2 ] = V 1 ν ¯ M c ( 1 - 2 M c M ) ( ν 2 1 3 - ν 2 2 ) -\left[\ln{\left(1-\nu_{2}\right)}+\nu_{2}+\chi_{1}\nu_{2}^{2}\right]=\frac{V_% {1}}{\bar{\nu}M_{c}}\left(1-\frac{2M_{c}}{M}\right)\left(\nu_{2}^{\frac{1}{3}}% -\frac{\nu_{2}}{2}\right)
  7. ν ¯ \bar{\nu}
  8. M M
  9. M c M_{c}

Flow_conditioning.html

  1. U y U m a x = [ 1 - Y R ] 1 / n {\frac{U_{y}}{U_{max}}}=\left[1-\frac{Y}{R}\right]^{1/n}
  2. 10 5 10^{5}
  3. 10 6 10^{6}
  4. 10 6 10^{6}
  5. n = 1 f . n=\frac{1}{\sqrt{f}}.
  6. P i p e D i a m e t e r s = 4.4 D [ R e ] 1 / 6 PipeDiameters=4.4D\left[R_{e}\right]^{1/6}
  7. q m = ( C d ) ( E v ) ( Y ) [ π 4 ] ( d ) 2 2 ρ Δ P q_{m}=(C_{d})(E_{v})(Y)\left[\frac{\pi}{4}\right](d)^{2}\sqrt{2\rho\Delta P}
  8. q m q_{m}
  9. C d C_{d}
  10. E v E_{v}
  11. ρ \rho
  12. Δ P \Delta P
  13. β \beta
  14. β \beta
  15. Δ C d \Delta Cd
  16. β \beta
  17. ( β ) 3.5 (\beta)^{3.5}
  18. β \beta
  19. Δ C d - ( β ) 3.5 \Delta Cd-(\beta)^{3.5}
  20. V ¯ f l o w = [ 1 T a b - 1 T b a ] [ D i s t S o u n d p a t h 2 cos ϕ ] \bar{V}_{flow}=\left[\frac{1}{T_{ab}}-\frac{1}{T_{ba}}\right]\left[\frac{Dist_% {Soundpath}}{2\cos\phi}\right]

Fluid_queue.html

  1. d X ( t ) d t = { r i if X ( t ) > 0 max ( r i , 0 ) if X ( t ) = 0. \frac{\mathrm{d}X(t)}{\mathrm{d}t}=\begin{cases}r_{i}&\,\text{ if }X(t)>0\\ \max(r_{i},0)&\,\text{ if }X(t)=0.\end{cases}
  2. Q = ( - α α β - β ) Q=\begin{pmatrix}-\alpha&\alpha\\ \beta&-\beta\end{pmatrix}
  3. F ( x , 1 ) = β α + β ( 1 - e ( β μ - α λ - μ ) x ) F(x,1)=\frac{\beta}{\alpha+\beta}\left(1-e^{\left(\frac{\beta}{\mu}-\frac{% \alpha}{\lambda-\mu}\right)x}\right)
  4. F ( x , 2 ) = α α + β - β ( λ - μ ) α + β e ( β μ - α λ - μ ) x F(x,2)=\frac{\alpha}{\alpha+\beta}-\frac{\beta\left(\lambda-\mu\right)}{\alpha% +\beta}e^{\left(\frac{\beta}{\mu}-\frac{\alpha}{\lambda-\mu}\right)x}
  5. ( λ - μ ) β ( μ ( α + β ) - β λ ) ( α + β ) ( μ , λ - μ ) . \frac{(\lambda-\mu)\beta}{(\mu(\alpha+\beta)-\beta\lambda)(\alpha+\beta)}(\mu,% \lambda-\mu).
  6. Q = ( - α α β - β ) Q=\begin{pmatrix}-\alpha&\alpha\\ \beta&-\beta\end{pmatrix}
  7. W ( s ) = β λ + s λ - β μ + α μ - 4 β α μ ( μ - λ ) + ( s λ + β ( λ - μ ) + α μ ) 2 2 β ( λ - μ ) W^{\ast}(s)=\frac{\beta\lambda+s\lambda-\beta\mu+\alpha\mu-\sqrt{4\beta\alpha% \mu(\mu-\lambda)+(s\lambda+\beta(\lambda-\mu)+\alpha\mu)^{2}}}{2\beta(\lambda-% \mu)}
  8. 𝔼 ( W ) = λ α μ + β ( λ - μ ) . \mathbb{E}(W)=\frac{\lambda}{\alpha\mu+\beta(\lambda-\mu)}.

Focused_information_criterion.html

  1. μ \mu
  2. μ ^ j \hat{\mu}_{j}
  3. j j
  4. r j r_{j}
  5. μ ^ j \hat{\mu}_{j}
  6. r ^ j \hat{r}_{j}
  7. r j = b j 2 + τ j 2 r_{j}=b_{j}^{2}+\tau_{j}^{2}
  8. j j
  9. r ^ j \hat{r}_{j}
  10. y y
  11. x x
  12. b j b_{j}

Fold_number.html

  1. f = antilog 10 i = 1 n F i n = 10 i = 1 n F i n f=\,\text{antilog}_{10}\frac{\textstyle\sum_{i=1}^{n}F_{i}}{n}=10^{\frac{% \textstyle\sum_{i=1}^{n}F_{i}}{n}}

Folkman's_theorem.html

  1. x T = i T x { i } , x_{T}=\sum_{i\in T}x_{\{i\}},

Force_between_magnets.html

  1. 𝐦 = I 𝐀 \mathbf{m}=I\mathbf{A}
  2. m = q m d m=q_{m}d\,
  3. 𝐅 = ( 𝐦 𝐁 ) \mathbf{F}=\nabla\left(\mathbf{m}\cdot\mathbf{B}\right)
  4. F = μ q m 1 q m 2 4 π r 2 F={{\mu q_{m1}q_{m2}}\over{4\pi r^{2}}}
  5. F = μ 0 H 2 A 2 = B 2 A 2 μ 0 F=\frac{\mu_{0}H^{2}A}{2}=\frac{B^{2}A}{2\mu_{0}}
  6. F = [ B 0 2 A 2 ( L 2 + R 2 ) π μ 0 L 2 ] [ 1 x 2 + 1 ( x + 2 L ) 2 - 2 ( x + L ) 2 ] F=\left[\frac{B_{0}^{2}A^{2}\left(L^{2}+R^{2}\right)}{\pi\mu_{0}L^{2}}\right]% \left[{\frac{1}{x^{2}}}+{\frac{1}{(x+2L)^{2}}}-{\frac{2}{(x+L)^{2}}}\right]
  7. B 0 = μ 0 2 M B_{0}\,=\,\frac{\mu_{0}}{2}M
  8. R R
  9. h h
  10. h h
  11. F ( x ) = π μ 0 4 M 2 R 4 [ 1 x 2 + 1 ( x + 2 h ) 2 - 2 ( x + h ) 2 ] F(x)=\frac{\pi\mu_{0}}{4}M^{2}R^{4}\left[\frac{1}{x^{2}}+\frac{1}{(x+2h)^{2}}-% \frac{2}{(x+h)^{2}}\right]
  12. M M
  13. x x
  14. x x
  15. B 0 B_{0}
  16. M M
  17. B 0 = ( μ 0 / 2 ) * M B_{0}=(\mu_{0}/2)*M
  18. m = M V m=MV
  19. V V
  20. V = π R 2 h V=\pi R^{2}h
  21. h x h\ll x
  22. F ( x ) = 3 π μ 0 2 M 2 R 4 h 2 1 x 4 = 3 μ 0 2 π M 2 V 2 1 x 4 = 3 μ 0 2 π m 1 m 2 1 x 4 F(x)=\frac{3\pi\mu_{0}}{2}M^{2}R^{4}h^{2}\frac{1}{x^{4}}=\frac{3\mu_{0}}{2\pi}% M^{2}V^{2}\frac{1}{x^{4}}=\frac{3\mu_{0}}{2\pi}m_{1}m_{2}\frac{1}{x^{4}}
  23. 𝐁 ( 𝐦 , 𝐫 ) = μ 0 4 π r 3 ( 3 ( 𝐦 𝐫 ^ ) 𝐫 ^ - 𝐦 ) + 2 μ 0 3 𝐦 δ 3 ( 𝐫 ) \mathbf{B}(\mathbf{m},\mathbf{r})=\frac{\mu_{0}}{4\pi r^{3}}\left(3(\mathbf{m}% \cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}\right)+\frac{2\mu_{0}}{3}% \mathbf{m}\delta^{3}(\mathbf{r})
  24. 𝐫 ^ = 𝐫 / r \hat{\mathbf{r}}=\mathbf{r}/r
  25. B z ( 𝐫 ) = μ 0 4 π m 1 ( 3 cos 2 θ - 1 r 3 ) B_{z}(\mathbf{r})=\frac{\mu_{0}}{4\pi}m_{1}\left(\frac{3\cos^{2}\theta-1}{r^{3% }}\right)
  26. B x ( 𝐫 ) = μ 0 4 π m 1 ( 3 cos θ sin θ r 3 ) B_{x}(\mathbf{r})=\frac{\mu_{0}}{4\pi}m_{1}\left(\frac{3\cos\theta\sin\theta}{% r^{3}}\right)
  27. 𝐅 ( 𝐫 , 𝐦 1 , 𝐦 2 ) = 3 μ 0 4 π r 5 [ ( 𝐦 1 𝐫 ) 𝐦 2 + ( 𝐦 2 𝐫 ) 𝐦 1 + ( 𝐦 1 𝐦 2 ) 𝐫 - 5 ( 𝐦 1 𝐫 ) ( 𝐦 2 𝐫 ) r 2 𝐫 ] \mathbf{F}(\mathbf{r},\mathbf{m}_{1},\mathbf{m}_{2})=\frac{3\mu_{0}}{4\pi r^{5% }}\left[(\mathbf{m}_{1}\cdot\mathbf{r})\mathbf{m}_{2}+(\mathbf{m}_{2}\cdot% \mathbf{r})\mathbf{m}_{1}+(\mathbf{m}_{1}\cdot\mathbf{m}_{2})\mathbf{r}-\frac{% 5(\mathbf{m}_{1}\cdot\mathbf{r})(\mathbf{m}_{2}\cdot\mathbf{r})}{r^{2}}\mathbf% {r}\right]
  28. 𝐅 ( z , m 1 , m 2 ) = - 3 μ 0 m 1 m 2 2 π z 4 \mathbf{F}(z,m_{1},m_{2})=-\frac{3\mu_{0}m_{1}m_{2}}{2\pi z^{4}}
  29. F r ( 𝐫 , α , β ) = - 3 μ 0 4 π m 2 m 1 r 4 [ 2 cos ( ϕ - α ) cos ( ϕ - β ) - sin ( ϕ - α ) sin ( ϕ - β ) ] F_{r}(\mathbf{r},\alpha,\beta)=-\frac{3\mu_{0}}{4\pi}\frac{m_{2}m_{1}}{r^{4}}% \left[2\cos(\phi-\alpha)\cos(\phi-\beta)-\sin(\phi-\alpha)\sin(\phi-\beta)\right]
  30. F ϕ ( 𝐫 , α , β ) = - 3 μ 0 4 π m 2 m 1 r 4 sin ( 2 ϕ - α - β ) F_{\phi}(\mathbf{r},\alpha,\beta)=-\frac{3\mu_{0}}{4\pi}\frac{m_{2}m_{1}}{r^{4% }}\sin(2\phi-\alpha-\beta)

Forces_on_sails.html

  1. V 1 V_{1}
  2. V 2 V_{2}
  3. β T \beta_{T}
  4. β \beta
  5. θ \theta
  6. V 1 = V B + V T c o s ( β T ) V_{1}=V_{B}+V_{T}\cdot cos(\beta_{T})
  7. V 2 V T s i n ( β T ) c o s ( θ ) V_{2}\approx V_{T}\cdot sin(\beta_{T})\cdot cos(\theta)
  8. V A = V 1 2 + V 2 2 \ V_{A}=\sqrt{{V_{1}}^{2}+{V_{2}}^{2}}
  9. β = a c o s ( V 1 V A ) \beta=acos({V_{1}\over V_{A}})
  10. F T F_{T}
  11. F T = C E F_{T}=CE
  12. E E
  13. C C
  14. E = q m a x S = 1 2 ρ V A 2 S E=q_{max}S=\frac{1}{2}\rho{V_{A}}^{2}S
  15. F T = 1 2 ρ V A 2 S C F_{T}=\frac{1}{2}\rho{V_{A}}^{2}SC
  16. 𝐅 𝐓 \mathbf{F_{T}}
  17. F T F_{T}
  18. q m a x q_{max}
  19. ρ \rho
  20. ρ \rho
  21. S S
  22. C C
  23. V A {V_{A}}
  24. d S dS
  25. d 𝐅 d\mathbf{F}
  26. d 𝐅 d\mathbf{F}
  27. d P dP
  28. 𝐧 \mathbf{n}
  29. d 𝐅 = - d P d S 𝐧 d\mathbf{F}=-dP\,dS\,\mathbf{n}
  30. V A V_{A}
  31. ρ \rho
  32. g g
  33. z z
  34. P P
  35. V A 2 2 + g z + P ρ = constant \frac{{V_{A}}^{2}}{2}+gz+\frac{P}{\rho}=\mathrm{constant}
  36. V A 2 2 + P ρ = constant \frac{{V_{A}}^{2}}{2}+\frac{P}{\rho}=\mathrm{constant}
  37. V 0 V_{0}
  38. V 0 V_{0}
  39. P 0 ρ = V A 2 2 + P ρ \frac{P_{0}}{\rho}=\frac{{V_{A}}^{2}}{2}+\frac{P}{\rho}
  40. P 0 = ρ V A 2 2 + P P_{0}=\frac{\rho{V_{A}}^{2}}{2}+P
  41. P P
  42. P 0 P_{0}
  43. q q
  44. P 0 P_{0}
  45. d 𝐅 d\mathbf{F}
  46. P 0 P_{0}
  47. P 0 P_{0}
  48. P = q \ P=q
  49. q = 1 2 ρ V A 2 q=\frac{1}{2}\rho{V_{A}}^{2}
  50. q = 1 2 ρ V A 2 = P q=\frac{1}{2}\rho{V_{A}}^{2}=P
  51. d 𝐅 = - d E d S 𝐧 d\mathbf{F}=-dE\,dS\,\mathbf{n}
  52. d E dE
  53. d V dV
  54. V 0 V_{0}
  55. V 0 V_{0}
  56. q m a x = ρ V 0 2 2 q_{max}=\frac{\rho V_{0}^{2}}{2}
  57. d E = k q m a x dE=kq_{max}
  58. k \,k
  59. k \,k
  60. F T = C E F_{T}=CE
  61. E E
  62. = q m a x S = 1 2 ρ S V 0 2 =q_{max}S=\frac{1}{2}\rho SV_{0}^{2}
  63. C C
  64. S \ S
  65. S i \ S_{i}
  66. S e \ S_{e}
  67. S = S i + S e \ S=S_{i}+S_{e}
  68. F T = C q m a x S = C i q m a x S i + C e q m a x S e F_{T}=Cq_{max}S=C_{i}q_{max}S_{i}+C_{e}q_{max}S_{e}
  69. S \ S
  70. S c \ S_{c}
  71. α i \ \alpha_{i}
  72. α e \ \alpha_{e}
  73. S e = α e S c \ S_{e}=\alpha_{e}S_{c}
  74. S i = α i S c \ S_{i}=\alpha_{i}S_{c}
  75. F T = q m a x S c ( C i α i + c e α e ) = q m a x S c C c F_{T}=q_{max}S_{c}(C_{i}\alpha_{i}+c_{e}\alpha_{e})=q_{max}S_{c}C_{c}
  76. C c \ C_{c}
  77. S c \ S_{c}
  78. c c \ c_{c}
  79. c i \ c_{i}
  80. c e \ c_{e}
  81. α i \ \alpha_{i}
  82. α e \ \alpha_{e}
  83. α e α i 1 \ \alpha_{e}\approx\alpha_{i}\approx 1
  84. F T = C c E = q m a x S c C c F_{T}=C_{c}E=q_{max}S_{c}C_{c}
  85. C c \ C_{c}
  86. C \ C
  87. S c \ S_{c}
  88. S \ S
  89. 𝐅 𝐓 \mathbf{F_{T}}
  90. 𝐅 𝐑 \mathbf{F_{R}}
  91. 𝐅 𝐇 \mathbf{F_{H}}
  92. θ \ \theta
  93. 𝐅 𝐥𝐚𝐭 = 𝐅 𝐇 c o s ( θ ) \mathbf{F_{lat}}=\mathbf{F_{H}}cos(\theta)
  94. 𝐅 𝐯𝐞𝐫𝐭 = 𝐅 𝐇 s i n ( θ ) \mathbf{F_{vert}}=\mathbf{F_{H}}sin(\theta)
  95. α \alpha
  96. 𝐅 𝐓 = 𝐅 x + 𝐅 y + 𝐅 z \mathbf{F_{T}}=\mathbf{F}_{x}+\mathbf{F}_{y}+\mathbf{F}_{z}
  97. F a x i s = 1 2 ρ S C a x i s v 2 F_{axis}=\frac{1}{2}\rho SC_{axis}v^{2}
  98. x \ x
  99. x \ x
  100. 𝐃 \mathbf{D}
  101. C D \ C_{D}
  102. y y
  103. z z
  104. z z
  105. 𝐋 \mathbf{L}
  106. C z \ C_{z}
  107. C N \ C_{N}
  108. C L \ C_{L}
  109. α \alpha
  110. F R \ {F_{R}}
  111. F l a t \ F_{lat}
  112. β \beta
  113. L \ {L}
  114. D \ D
  115. F l a t = L c o s ( β ) + D s i n ( β ) \ F_{lat}=L\cdot cos(\beta)+D\cdot sin(\beta)
  116. F R = L s i n ( β ) - D c o s ( β ) \ F_{R}=L\cdot sin(\beta)-D\cdot cos(\beta)
  117. V T \ {V_{T}}
  118. V A e \ V_{Ae}
  119. β T \beta_{T}
  120. β A e \beta_{Ae}
  121. θ \ \theta
  122. V 1 V_{1}
  123. V 2 V_{2}
  124. V B V_{B}
  125. V 1 = V B + V T c o s ( β T ) V_{1}=V_{B}+V_{T}\cdot cos(\beta_{T})
  126. V 2 V T s i n ( β T ) c o s ( θ ) V_{2}\approx V_{T}\cdot sin(\beta_{T})\cdot cos(\theta)
  127. V A e = V 1 2 + V 2 2 \ V_{Ae}=\sqrt{V_{1}^{2}+V_{2}^{2}}
  128. β A e = a c o s ( V 1 V A e ) \beta_{Ae}=acos({V_{1}\over V_{Ae}})
  129. α \alpha
  130. β \beta
  131. 𝐅 𝐓 \mathbf{F_{T}}
  132. δ \delta
  133. α \alpha
  134. β = ϵ A + ϵ H \beta=\epsilon_{A}+\epsilon_{H}
  135. F T = 1 2 × 1.2 × 10 × 0.9 × 8.3 2 = 372 N F_{T}=\frac{1}{2}\times 1.2\times 10\times 0.9\times 8.3^{2}=372N
  136. α \alpha
  137. F T = ρ V A 2 S a i r cos ( π 2 - α ) F_{T}=\rho{V_{A}}^{2}S_{air}\cos(\frac{\pi}{2}-\alpha)
  138. S = S a i r S=S_{air}
  139. F = ρ V A 2 S F=\rho{V_{A}}^{2}S
  140. C D = 2 = C D m a x C_{D}=2=C_{D_{max}}
  141. C D m a x C_{D_{max}}
  142. Re = ρ V A l μ = V A l ν \mathrm{Re}={{\rho{V_{A}}l}\over{\mu}}={{{V_{A}}l}\over{\nu}}
  143. V A V_{A}
  144. l l
  145. ν \nu
  146. ν = μ / ρ \nu=\mu/\rho
  147. ρ \rho
  148. μ \mu
  149. η \eta
  150. Re = 10 6 \mathrm{Re}=10^{6}
  151. F T = 1 2 × 1.2 × 10 × 1.5 × 8.3 2 = 620 n e w t o n F_{T}=\frac{1}{2}\times 1.2\times 10\times 1.5\times 8.3^{2}=620\ newton
  152. V A V_{A}
  153. V T V_{T}
  154. D D
  155. α \alpha
  156. β \beta
  157. β \beta
  158. α \alpha
  159. λ \lambda
  160. L = 1 2 ρ V A 2 A C L L=\tfrac{1}{2}\rho{V_{A}}^{2}AC_{L}
  161. D = 1 2 ρ V A 2 A C D D=\tfrac{1}{2}\rho{V_{A}}^{2}AC_{D}
  162. α \alpha
  163. α \alpha
  164. α \alpha
  165. β \beta
  166. ϵ A \epsilon_{A}
  167. ϵ H \epsilon_{H}
  168. β = ϵ A + ϵ H \beta=\epsilon_{A}+\epsilon_{H}
  169. Λ = b 2 S \Lambda={b^{2}\over S}
  170. b b
  171. S S
  172. C i = C L 2 π × Λ × e Ci={{{C_{L}}^{2}}\over{\pi\times\Lambda\times e}}
  173. C L C_{L}
  174. π \pi
  175. p i 3.1416 pi\approx 3.1416
  176. Λ \Lambda
  177. A R AR
  178. e e
  179. L = 1 2 ρ S C L V A 2 L=\frac{1}{2}\rho SC_{L}{V_{A}}^{2}
  180. D i = 1 2 ρ S C i V A 2 D_{i}=\frac{1}{2}\rho SC_{i}{V_{A}}^{2}
  181. C i = C L 2 π Λ e C_{i}={{{C_{L}}^{2}}\over{\pi\Lambda e}}
  182. Λ = b 2 S \Lambda={b^{2}\over S}
  183. D i = 2 L 2 ρ V A 2 e b 2 D_{i}={2{L^{2}}\over\rho{V_{A}}^{2}eb^{2}}
  184. Λ \Lambda
  185. Λ \Lambda
  186. C L C_{L}
  187. Λ \Lambda
  188. V A V_{A}
  189. β \beta
  190. α \alpha
  191. β - α \beta-\alpha
  192. β \beta
  193. U = μ k l n ( z + z 0 z 0 ) U=\frac{\mu^{\prime}}{k}\ ln(\frac{z+z_{0}}{z_{0}})
  194. k k
  195. z z
  196. z 0 z_{0}
  197. μ \mu^{\prime}
  198. U U
  199. F = 1 2 ρ S C V 2 F=\frac{1}{2}\rho SCV^{2}
  200. C C
  201. V V
  202. Re = U l ν \mathrm{Re}={{{\mathrm{U}}l}\over{\nu}}
  203. U U
  204. l l
  205. ν {\nu}
  206. P e x t r a c t a b l e P_{extractable}
  207. P e x t r a c t a b l e m a x P_{extractable}^{max}
  208. P e x t r a c t a b l e m a x = 16 27 P a r r i v i n g o n s a i l P_{extractable}^{max}=\frac{16}{27}P_{arrivingonsail}
  209. P a r r i v i n g o n s a i l = P k i n e t i c = 1 2 ρ S V A 3 P_{arrivingonsail}=P_{kinetic}=\frac{1}{2}\rho S{V_{A}}^{3}\,
  210. ρ \rho
  211. S S
  212. V A V_{A}
  213. F = 1 2 ρ S C V A 2 F=\frac{1}{2}\rho SC{V_{A}}^{2}
  214. S \ S
  215. C \ C
  216. C S \ CS
  217. S \ S
  218. C \ C
  219. V A V_{A}
  220. V T V_{T}
  221. V B V_{B}
  222. V T 2 = V A 2 + V B 2 - 2 V A V B c o s ( β T - π ) {V_{T}}^{2}={V_{A}}^{2}+{V_{B}}^{2}-2V_{A}V_{B}cos(\beta_{T}-\pi)
  223. β T \beta_{T}
  224. α \alpha
  225. β \beta
  226. D D
  227. F R F_{R}
  228. F H F_{H}
  229. λ \lambda
  230. L L
  231. F R F_{R}
  232. F V E R T F_{VERT}
  233. Δ \Delta
  234. C i = C L 2 π Λ e C_{i}={{{C_{L}}^{2}}\over{\pi\Lambda e}}
  235. C L C_{L}
  236. π \pi
  237. Λ \Lambda
  238. Λ = b 2 S \Lambda={b^{2}\over S}
  239. e e
  240. C L C_{L}
  241. e e
  242. Λ \Lambda
  243. C i \ C_{i}
  244. C i \ C_{i}
  245. C i \ C_{i}
  246. δ \ \delta
  247. C i = C L 2 π Λ e = C L 2 π Λ ( 1 + δ ) C_{i}={{{C_{L}}^{2}}\over{\pi\Lambda e}}={{{C_{L}}^{2}}\over{\pi\Lambda}}(1+\delta)
  248. L / D = C L / C i = π Λ e C L L/D=C_{L}/C_{i}={{\pi\Lambda e}\over C_{L}}
  249. C L = C L ( Λ Λ + 2 ) C_{L}={C_{L}}^{\prime}({{\Lambda}\over{\Lambda+2}})
  250. Λ \Lambda
  251. Λ = b 2 S \Lambda={b^{2}\over S}
  252. C L C_{L}
  253. C L = 2 π ( α + α 0 ) \ {C_{L}}^{\prime}=2\pi(\alpha+\alpha 0)
  254. L / D = ( e α + α 0 ) ( Λ + 2 2 ) L/D=({{e}\over{\alpha+\alpha 0}})({{\Lambda+2}\over{2}})
  255. e e
  256. Λ \Lambda
  257. α \alpha
  258. α 0 \alpha 0
  259. F R = L s i n ( β ) - D t c o s ( β ) > 0 {F_{R}}=Lsin(\beta)-{D_{t}}cos(\beta)>0
  260. D t = ( L / D ) α - 1 L {D_{t}}={(L/D)_{\alpha}}^{-1}L
  261. ( D / L ) α < t a n ( β ) \ {(D/L)_{\alpha}}<tan(\beta)
  262. α \ \alpha
  263. β \ \beta
  264. ( L / D ) α \ {(L/D)_{\alpha}}
  265. F R \ {F_{R}}
  266. D t \ {D_{t}}
  267. L \ {L}
  268. P = 𝐅 𝐓 𝐕 𝐁 P=\mathbf{F_{T}}\cdot\mathbf{V_{B}}
  269. \cdot
  270. 𝐅 𝐓 \mathbf{F_{T}}
  271. 𝐕 𝐁 \mathbf{V_{B}}
  272. 𝐕 𝐁 𝐕 𝐁 = V B 2 \mathbf{V_{B}}\cdot\mathbf{V_{B}}={V_{B}}^{2}
  273. V B \ V_{B}
  274. P \ P
  275. 0 = F R + R I \ 0=F_{R}+R_{I}
  276. F R F_{R}
  277. R I R_{I}
  278. R I = 1 2 ρ w a t e r S b C V B 2 R_{I}=\frac{1}{2}\rho_{water}S_{b}C{V_{B}}^{2}
  279. ρ w a t e r \ \rho_{water}
  280. C C
  281. S b \ S_{b}
  282. V B \ V_{B}
  283. b = 1 2 ρ w a t e r S b C \ b=\frac{1}{2}\rho_{water}S_{b}C
  284. R I = b V B 2 = - F R R_{I}=b{V_{B}}^{2}=-F_{R}
  285. V B = ( F R b ) 1 2 V_{B}=({F_{R}\over b})^{\frac{1}{2}}
  286. P = ( F R ) 3 2 ( 1 b ) 1 2 P=(F_{R})^{\frac{3}{2}}(\frac{1}{b})^{\frac{1}{2}}
  287. L L
  288. D D
  289. F R = L s i n ( β ) - D c o s ( β ) F_{R}=Lsin(\beta)-Dcos(\beta)
  290. L = ( L / D ) α D L={(L/D)_{\alpha}}D
  291. α \alpha
  292. β \beta
  293. P = ( L ( s i n ( β ) - ( L / D ) α - 1 c o s ( β ) ) ) 3 2 × ( 1 b ) 1 2 P=(L(sin(\beta)-{(L/D)_{\alpha}}^{-1}cos(\beta)))^{\frac{3}{2}}\times(\frac{1}% {b})^{\frac{1}{2}}
  294. α \alpha
  295. β \beta
  296. L = 1 2 ρ a i r S C V A 2 L=\frac{1}{2}\rho_{air}SC{V_{A}}^{2}
  297. C C
  298. S S
  299. V A {V_{A}}
  300. 𝟎 = M s a i l / G + M h u l l / G \mathbf{0}=\mathbf{\mathrm{}}{M}_{sail/G}+\mathbf{\mathrm{}}{M}_{hull/G}
  301. M s a i l / G = 𝐅 𝐟𝐨𝐫𝐰𝐚𝐫𝐝 × d i s t a n c e G - E \ \mathbf{\mathrm{}}{M}_{sail/G}=\mathbf{F_{forward}}\times distance_{G-E}
  302. M h u l l / G \ \mathbf{\mathrm{}}{M}_{hull/G}
  303. M h e e l m a x \ M_{heel_{max}}
  304. M p i t c h m a x \ M_{pitch_{max}}
  305. M h e e l m a x M p i t c h m a x \ M_{heel_{max}}<<M_{pitch_{max}}
  306. 10 × M h e e l m a x = M p i t c h m a x \ 10\times M_{heel_{max}}=M_{pitch_{max}}
  307. D p i t c h \ D_{pitch}
  308. D h e e l \ D_{heel}
  309. M p i t c h = D p i t c h × ( l i f t × s i n ( β ) - d r a g × c o s ( β ) ) = D p i t c h × l i f t × ( s i n ( β ) - ( L / D ) α - 1 × c o s ( β ) ) \ M_{pitch}=D_{pitch}\times(lift\times sin(\beta)-drag\times cos(\beta))=D_{% pitch}\times lift\times(sin(\beta)-{(L/D)_{\alpha}}^{-1}\times cos(\beta))
  310. M h e e l = D h e e l × ( l i f t × c o s ( β ) + d r a g × s i n ( β ) ) = D h e e l × l i f t × ( c o s ( β ) + ( L / D ) α - 1 × s i n ( β ) ) \ M_{heel}=D_{heel}\times(lift\times cos(\beta)+drag\times sin(\beta))=D_{heel% }\times lift\times(cos(\beta)+{(L/D)_{\alpha}}^{-1}\times sin(\beta))
  311. β \ \beta
  312. M h e e l \ M_{heel}
  313. M p i t c h \ M_{pitch}
  314. M p i t c h o r h e e l = A × G M × s i n ( ϕ ) \ M_{pitchorheel}=A\times GM\times sin(\phi)
  315. ϕ \ \phi
  316. G M \ GM
  317. A \ A
  318. M h e e l \ M_{heel}
  319. M h e e l = D h e e l ( l i f t × c o s ( β ) + d r a g × s i n ( β ) ) \ M_{heel}=D_{heel}(lift\times cos(\beta)+drag\times sin(\beta))
  320. F = 1 2 × ρ × S × C × V 2 F=\frac{1}{2}\times\rho\times S\times C\times V^{2}
  321. V = a × c o s ( ϕ ) V=a\times cos(\phi)
  322. M h e e l = p r e s s u r e × S × A c o s ( ϕ ) n \ M_{heel}=pressure\times S\times A{cos(\phi)}^{n}
  323. p r e s s u r e \ pressure
  324. A c o s ( ϕ ) n \ A{cos(\phi)}^{n}
  325. ϕ \ \phi
  326. S \ S
  327. A \ A
  328. V 2 = b × ( 1 - a × ϕ ) V^{2}=b\times(1-a\times\phi)
  329. M p i t c h m a x \ M_{pitch_{max}}
  330. M h e e l m a x \ M_{heel_{max}}
  331. D h e e l D p i t c h \ D_{heel}\approx D_{pitch}
  332. D h e e l = D p i t c h \ D_{heel}=D_{pitch}
  333. β 180 deg \ \beta\approx 180\deg
  334. β = 180 \ \beta=180
  335. M p i t c h = D p i t c h × d r a g \ M_{pitch}=D_{pitch}\times drag
  336. M h e e l = - D h e e l × l i f t \ M_{heel}=-D_{heel}\times lift
  337. M p i t c h = D p i t c h × d r a g × 1 \ M_{pitch}=D_{pitch}\times drag\times 1
  338. β 120 deg \ \beta\approx 120\deg
  339. M p i t c h = - D p i t c h × d r a g × c o s ( β ) \ M_{pitch}=-D_{pitch}\times drag\times cos(\beta)
  340. M h e e l = D h e e l × d r a g × s i n ( β ) \ M_{heel}=D_{heel}\times drag\times sin(\beta)
  341. M h e e l m a x M p i t c h m a x \ M_{heel_{max}}<<M_{pitch_{max}}
  342. 10 = M p i t c h m a x M h e e l m a x \ 10=\frac{M_{pitch_{max}}}{M_{heel_{max}}}
  343. β = 174.3 deg \ \beta=174.3\deg
  344. 3.5 = M p i t c h m a x M h e e l m a x \ 3.5=\frac{M_{pitch_{max}}}{M_{heel_{max}}}
  345. M h e e l = D h e e l × d r a g × s i n ( β ) \ M_{heel}=D_{heel}\times drag\times sin(\beta)
  346. F p r o p u l s i v e = - d r a g × c o s ( β ) \ F_{propulsive}=-drag\times cos(\beta)
  347. β 90 deg \ \beta\approx 90\deg
  348. F p r o p u l s i v e = l i f t × s i n ( β ) - d r a g × c o s ( β ) \ F_{propulsive}=lift\times sin(\beta)-drag\times cos(\beta)
  349. β = 90 \ \beta=90
  350. l i f t d r a g \ lift>>drag
  351. M p i t c h = D p i t c h × l i f t \ M_{pitch}=D_{pitch}\times lift
  352. M h e e l = - D h e e l × d r a g \ M_{heel}=-D_{heel}\times drag
  353. l i f t d r a g \ lift>>drag
  354. M p i t c h m a x > D p i t c h × l i f t \ M_{pitch_{max}}>D_{pitch}\times lift
  355. β = 90 deg \ \beta=90\deg
  356. β = 70 \ \beta=70
  357. F p r o p u l s i v e = d r a g × c o s ( β ) \ F_{propulsive}=drag\times cos(\beta)
  358. c o s ( β ) \ cos(\beta)
  359. M p i t c h = D p i t c h × ( l i f t × s i n ( β ) - d r a g × c o s ( β ) ) = D p i t c h × l i f t × ( s i n ( β ) - ( L / D ) α - 1 × c o s ( β ) ) \ M_{pitch}=D_{pitch}\times(lift\times sin(\beta)-drag\times cos(\beta))=D_{% pitch}\times lift\times(sin(\beta)-{(L/D)_{\alpha}}^{-1}\times cos(\beta))
  360. M h e e l = D h e e l × ( l i f t × c o s ( β ) + d r a g × s i n ( β ) ) = D h e e l × l i f t × ( c o s ( β ) + ( L / D ) α - 1 × s i n ( β ) ) \ M_{heel}=D_{heel}\times(lift\times cos(\beta)+drag\times sin(\beta))=D_{heel% }\times lift\times(cos(\beta)+{(L/D)_{\alpha}}^{-1}\times sin(\beta))
  361. M h e e l m a x M p i t c h m a x \ M_{heel_{max}}<<M_{pitch_{max}}
  362. 10 = M p i t c h m a x M h e e l m a x \ 10=\frac{M_{pitch_{max}}}{M_{heel_{max}}}
  363. D p i t c h × l i f t × ( s i n ( β ) - ( L / D ) α - 1 × c o s ( β ) ) D h e e l × l i f t × ( c o s ( β ) + ( L / D ) α - 1 × s i n ( β ) ) = 10 \ \frac{D_{pitch}\times lift\times(sin(\beta)-{(L/D)_{\alpha}}^{-1}\times cos(% \beta))}{D_{heel}\times lift\times(cos(\beta)+{(L/D)_{\alpha}}^{-1}\times sin(% \beta))}=10
  364. β = π - t a n - 1 ( 1 10 ) - t a n - 1 ( L / D ) \ \beta=\pi-tan^{-1}(\frac{1}{10})-tan^{-1}(L/D)
  365. t a n - 1 \ tan^{-1}
  366. t a n ( t a n - 1 ( x ) ) = x \ tan(tan^{-1}(x))=x
  367. β \ \beta
  368. L / D 1 \ L/D>>1
  369. L / D > 10 \ L/D>10
  370. t a n - 1 ( L / D ) π 2 \ tan^{-1}(L/D)\approx\frac{\pi}{2}
  371. 3.5 = M p i t c h m a x M h e e l m a x \ 3.5=\frac{M_{pitch_{max}}}{M_{heel_{max}}}
  372. M h e e l = D h e e l × l i f t × ( c o s ( β ) + ( L / D ) α - 1 × s i n ( β ) ) \ M_{heel}=D_{heel}\times lift\times(cos(\beta)+{(L/D)_{\alpha}}^{-1}\times sin% (\beta))
  373. M p i t c h m a x > D p i t c h × d r a g \ M_{pitch_{max}}>D_{pitch}\times drag
  374. M p i t c h m a x > D p i t c h × l i f t \ M_{pitch_{max}}>D_{pitch}\times lift
  375. M h e e l m a x > - D h e e l × d r a g × s i n ( β ) \ M_{heel_{max}}>-D_{heel}\times drag\times sin(\beta)
  376. M h e e l m a x > D h e e l × l i f t × ( c o s ( β ) + ( L / D ) α - 1 × s i n ( β ) ) \ M_{heel_{max}}>D_{heel}\times lift\times(cos(\beta)+{(L/D)_{\alpha}}^{-1}% \times sin(\beta))
  377. F f o r w a r d = d r a g × c o s ( β ) \ F_{forward}=drag\times cos(\beta)
  378. F f o r w a r d = d r a g \ F_{forward}=drag
  379. F f o r w a r d = l i f t \ F_{forward}=lift
  380. F f o r w a r d = l i f t × ( s i n ( β ) - ( L / D ) α - 1 × c o s ( β ) ) \ F_{forward}=lift\times(sin(\beta)-{(L/D)_{\alpha}}^{-1}\times cos(\beta))
  381. M h e e l m a x > D h e e l × F f o r w a r d × s i n ( β ) c o s ( β ) \ M_{heel_{max}}>D_{heel}\times F_{forward}\times\frac{sin(\beta)}{cos(\beta)}
  382. M p i t c h m a x > D p i t c h × F f o r w a r d \ M_{pitch_{max}}>D_{pitch}\times F_{forward}
  383. M p i t c h m a x > D p i t c h × F f o r w a r d \ M_{pitch_{max}}>D_{pitch}\times F_{forward}
  384. M h e e l m a x > D h e e l × F f o r w a r d × s i n ( π - β - t a n - 1 ( L / D ) ) \ M_{heel_{max}}>D_{heel}\times F_{forward}\times sin(\pi-\beta-tan^{-1}(L/D))
  385. s i n ( π - β - t a n - 1 ( L / D ) ) s i n ( π 2 - β ) \ sin(\pi-\beta-tan^{-1}(L/D))\approx sin(\frac{\pi}{2}-\beta)
  386. P = ( F f o r w a r d ) 3 2 × ( 1 b ) 1 2 P=(F_{forward})^{\frac{3}{2}}\times(\frac{1}{b})^{\frac{1}{2}}
  387. V t r u e w i n d 2 = V a p p a r e n t w i n d 2 + V b o a t 2 - 2 × V a p p a r e n t w i n d × V b o a t × c o s ( β - π ) {V_{truewind}}^{2}={V_{apparentwind}}^{2}+{V_{boat}}^{2}-2\times V_{% apparentwind}\times V_{boat}\times cos(\beta-\pi)
  388. F = 1 2 × ρ × S × C × V w i n d 2 F=\frac{1}{2}\times\rho\times S\times C\times{V_{wind}}^{2}
  389. V w i n d = μ k l n ( z + z 0 z 0 ) V_{wind}=\frac{\mu^{\prime}}{k}\ ln(\frac{z+z0}{z0})
  390. C M ( 1 / 4 c ) = - π / 4 ( A 1 - A 2 ) \ C_{M}(1/4c)=-\pi/4(A_{1}-A_{2})

Formally_smooth_map.html

  1. f : A B f:A\to B
  2. N C N\subseteq C
  3. B C / N B\to C/N
  4. B C B\to C

Formation_(group_theory).html

  1. 1 A B C 1 1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1

Forney_algorithm.html

  1. Λ ( x ) = i = 1 ν ( 1 - x X i ) = 1 + i = 1 ν λ i x i \Lambda(x)=\prod_{i=1}^{\nu}(1-x\,X_{i})=1+\sum_{i=1}^{\nu}\lambda_{i}\,x^{i}
  2. X j = α i j X_{j}=\alpha^{i_{j}}
  3. s 0 = e 1 α ( c + 0 ) i 1 + e 2 α ( c + 0 ) i 2 + s_{0}=e_{1}\alpha^{(c+0)\,i_{1}}+e_{2}\alpha^{(c+0)\,i_{2}}+\cdots\,
  4. s 1 = e 1 α ( c + 1 ) i 1 + e 2 α ( c + 1 ) i 2 + s_{1}=e_{1}\alpha^{(c+1)\,i_{1}}+e_{2}\alpha^{(c+1)\,i_{2}}+\cdots\,
  5. \cdots\,
  6. Ω ( x ) = S ( x ) Λ ( x ) ( mod x 2 t ) \Omega(x)=S(x)\,\Lambda(x)\;\;(\mathop{{\rm mod}}x^{2t})\,
  7. S ( x ) S(x)
  8. S ( x ) = s 0 x 0 + s 1 x 1 + s 2 x 2 + + s 2 t - 1 x 2 t - 1 . S(x)=s_{0}x^{0}+s_{1}x^{1}+s_{2}x^{2}+\cdots+s_{2t-1}x^{2t-1}.
  9. e j = - X j 1 - c Ω ( X j - 1 ) Λ ( X j - 1 ) e_{j}=-\frac{X_{j}^{1-c}\,\Omega(X_{j}^{-1})}{\Lambda^{\prime}(X_{j}^{-1})}\,
  10. c c
  11. c = 1 c=1
  12. e j = - Ω ( X j - 1 ) Λ ( X j - 1 ) e_{j}=-\frac{\Omega(X_{j}^{-1})}{\Lambda^{\prime}(X_{j}^{-1})}
  13. Λ ( x ) = i = 1 ν i λ i x i - 1 \Lambda^{\prime}(x)=\sum_{i=1}^{\nu}i\,\cdot\,\lambda_{i}\,x^{i-1}
  14. Γ ( x ) = ( 1 - x α j i ) \Gamma(x)=\prod(1-x\,\alpha^{j_{i}})
  15. Ψ ( x ) = Λ ( x ) Γ ( x ) \Psi(x)=\Lambda(x)\,\Gamma(x)

Fourier_amplitude_sensitivity_testing.html

  1. Y = f ( 𝐗 ) = f ( X 1 , X 2 , , X n ) Y=f\left(\mathbf{X}\right)=f\left(X_{1},X_{2},\dots,X_{n}\right)
  2. 0 X j 1 , j = 1 , , n 0\leq X_{j}\leq 1,j=1,\dots,n
  3. f ( X 1 , X 2 , , X n ) = f 0 + j = 1 n f j ( X j ) + j = 1 n - 1 k = j + 1 n f j k ( X j , X k ) + + f 12 n f\left(X_{1},X_{2},\ldots,X_{n}\right)=f_{0}+\sum_{j=1}^{n}f_{j}\left(X_{j}% \right)+\sum_{j=1}^{n-1}\sum_{k=j+1}^{n}f_{jk}\left(X_{j},X_{k}\right)+\cdots+% f_{12\dots n}
  4. f 0 f_{0}
  5. 0 1 f j 1 j 2 j r ( X j 1 , X j 2 , , X j r ) d X j k = 0 , 1 k r . \int_{0}^{1}f_{j_{1}j_{2}\dots j_{r}}\left(X_{j_{1}},X_{j_{2}},\dots,X_{j_{r}}% \right)dX_{j_{k}}=0,\,\text{ }1\leq k\leq r.
  6. f ( 𝐗 ) f\left(\mathbf{X}\right)
  7. V j 1 j 2 j r = 0 1 0 1 f j 1 j 2 j r 2 ( X j 1 , X j 2 , , X j r ) d X j 1 d X j 2 d X j r . V_{j_{1}j_{2}\dots j_{r}}=\int_{0}^{1}\cdots\int_{0}^{1}f_{j_{1}j_{2}\dots j_{% r}}^{2}\left(X_{j_{1}},X_{j_{2}},\dots,X_{j_{r}}\right)dX_{j_{1}}dX_{j_{2}}% \dots dX_{j_{r}}.
  8. V = j = 1 n V j + j = 1 n - 1 k = j + 1 n V j k + + V 12 n . V=\sum_{j=1}^{n}V_{j}+\sum_{j=1}^{n-1}\sum_{k=j+1}^{n}V_{jk}+\cdots+V_{12\dots n}.
  9. S j 1 j 2 j r = V j 1 j 2 j r V S_{j_{1}j_{2}\dots j_{r}}=\frac{V_{j_{1}j_{2}\dots j_{r}}}{V}
  10. S j = V j V S_{j}=\frac{V_{j}}{V}
  11. X j X_{j}
  12. f ( 𝐗 ) f\left(\mathbf{X}\right)
  13. f ( X 1 , X 2 , , X n ) = m 1 = - m 2 = - m n = - C m 1 m 2 m n exp [ 2 π i ( m 1 X 1 + m 2 X 2 + + m n X n ) ] , for integers m 1 , m 2 , , m n f\left(X_{1},X_{2},\dots,X_{n}\right)=\sum_{m_{1}=-\infty}^{\infty}\sum_{m_{2}% =-\infty}^{\infty}\cdots\sum_{m_{n}=-\infty}^{\infty}C_{m_{1}m_{2}...m_{n}}% \exp\bigl[2\pi i\left(m_{1}X_{1}+m_{2}X_{2}+\cdots+m_{n}X_{n}\right)\bigr],\,% \text{ for integers }m_{1},m_{2},\dots,m_{n}
  14. C m 1 m 2 m n = 0 1 0 1 f ( X 1 , X 2 , , X n ) exp [ - 2 π i ( m 1 X 1 + m 2 X 2 + + m n X n ) ] . C_{m_{1}m_{2}...m_{n}}=\int_{0}^{1}\cdots\int_{0}^{1}f\left(X_{1},X_{2},\dots,% X_{n}\right)\exp\bigl[-2\pi i\left(m_{1}X_{1}+m_{2}X_{2}+\dots+m_{n}X_{n}% \right)\bigr].
  15. f 0 = C 00 0 f j = m j 0 C 0 m j 0 exp [ 2 π i m j X j ] f j k = m j 0 m k 0 C 0 m j m k 0 exp [ 2 π i ( m j X j + m k X k ) ] f 12 n = m 1 0 m 2 0 m n 0 C m 1 m 2 m n exp [ 2 π i ( m 1 X 1 + m 2 X 2 + + m n X n ) ] . \begin{aligned}\displaystyle f_{0}&\displaystyle=C_{00\dots 0}\\ \displaystyle f_{j}&\displaystyle=\sum_{m_{j}\neq 0}C_{0\dots m_{j}\dots 0}% \exp\bigl[2\pi im_{j}X_{j}\bigr]\\ \displaystyle f_{jk}&\displaystyle=\sum_{m_{j}\neq 0}\sum_{m_{k}\neq 0}C_{0% \dots m_{j}\dots m_{k}\dots 0}\exp\bigl[2\pi i\left(m_{j}X_{j}+m_{k}X_{k}% \right)\bigr]\\ \displaystyle f_{12\dots n}&\displaystyle=\sum_{m_{1}\neq 0}\sum_{m_{2}\neq 0}% \cdots\sum_{m_{n}\neq 0}C_{m_{1}m_{2}\dots m_{n}}\exp\bigl[2\pi i\left(m_{1}X_% {1}+m_{2}X_{2}+\cdots+m_{n}X_{n}\right)\bigr].\end{aligned}
  16. V j = 0 1 f j 2 ( X j ) d X j = m j 0 | C 0 m j 0 | 2 = 2 m j = 1 ( A m j 2 + B m j 2 ) \begin{aligned}\displaystyle V_{j}&\displaystyle=\int_{0}^{1}f_{j}^{2}\left(X_% {j}\right)dX_{j}\\ &\displaystyle=\sum_{m_{j}\neq 0}\left|C_{0\dots m_{j}\dots 0}\right|^{2}\\ &\displaystyle=2\sum_{m_{j}=1}^{\infty}\left(A_{m_{j}}^{2}+B_{m_{j}}^{2}\right% )\end{aligned}
  17. A m j A_{m_{j}}
  18. B m j B_{m_{j}}
  19. C 0 m j 0 C_{0\dots m_{j}\dots 0}
  20. A m j = 0 1 0 1 f ( X 1 , X 2 , , X n ) cos ( 2 π m j X j ) d X 1 d X 2 d X n A_{m_{j}}=\int_{0}^{1}\cdots\int_{0}^{1}f\left(X_{1},X_{2},\dots,X_{n}\right)% \cos\left(2\pi m_{j}X_{j}\right)dX_{1}dX_{2}\dots dX_{n}
  21. B m j = 0 1 0 1 f ( X 1 , X 2 , , X n ) sin ( 2 π m j X j ) d X 1 d X 2 d X n B_{m_{j}}=\int_{0}^{1}\cdots\int_{0}^{1}f\left(X_{1},X_{2},\dots,X_{n}\right)% \sin\left(2\pi m_{j}X_{j}\right)dX_{1}dX_{2}\dots dX_{n}
  22. s s
  23. X j ( s ) = 1 2 π ( ω j s mod 2 π ) , j = 1 , 2 , , n X_{j}\left(s\right)=\frac{1}{2\pi}\left(\omega_{j}s\,\text{ mod }2\pi\right),j% =1,2,\dots,n
  24. { ω j } \left\{\omega_{j}\right\}
  25. j = 1 n γ j ω j = 0 \sum_{j=1}^{n}\gamma_{j}\omega_{j}=0
  26. { γ j } \left\{\gamma_{j}\right\}
  27. γ j = 0 \gamma_{j}=0
  28. j j
  29. A m j = lim T 1 2 T - T T f ( X 1 ( s ) , X 2 ( s ) , , X n ( s ) ) cos ( 2 π m j X j ( s ) ) d s A_{m_{j}}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f\bigl(X_{1}\left(s\right)% ,X_{2}\left(s\right),\dots,X_{n}\left(s\right)\bigr)\cos\bigl(2\pi m_{j}X_{j}% \left(s\right)\bigr)ds
  30. B m j = lim T 1 2 T - T T f ( X 1 ( s ) , X 2 ( s ) , , X n ( s ) ) sin ( 2 π m j X j ( s ) ) d s B_{m_{j}}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f\bigl(X_{1}\left(s\right)% ,X_{2}\left(s\right),\dots,X_{n}\left(s\right)\bigr)\sin\bigl(2\pi m_{j}X_{j}% \left(s\right)\bigr)ds
  31. { ω j } \left\{\omega_{j}\right\}
  32. { ω j } \left\{\omega_{j}\right\}
  33. M M
  34. j = 1 n γ j ω j 0 \sum_{j=1}^{n}\gamma_{j}\omega_{j}\neq 0
  35. j = 1 n | γ j | M + 1 \sum_{j=1}^{n}\left|\gamma_{j}\right|\leq M+1
  36. M M
  37. M M\to\infty
  38. 2 π 2\pi
  39. A m j 1 2 π - π π f ( X 1 ( s ) , X 2 ( s ) , , X n ( s ) ) cos ( m j ω j s ) d s := A ^ m j B m j 1 2 π - π π f ( X 1 ( s ) , X 2 ( s ) , , X n ( s ) ) sin ( m j ω j s ) d s := B ^ m j \begin{aligned}\displaystyle A_{m_{j}}&\displaystyle\approx\frac{1}{2\pi}\int_% {-\pi}^{\pi}f\bigl(X_{1}\left(s\right),X_{2}\left(s\right),\dots,X_{n}\left(s% \right)\bigr)\cos\left(m_{j}\omega_{j}s\right)ds:=\hat{A}_{m_{j}}\\ \displaystyle B_{m_{j}}&\displaystyle\approx\frac{1}{2\pi}\int_{-\pi}^{\pi}f% \bigl(X_{1}\left(s\right),X_{2}\left(s\right),\dots,X_{n}\left(s\right)\bigr)% \sin\left(m_{j}\omega_{j}s\right)ds:=\hat{B}_{m_{j}}\end{aligned}
  40. M M
  41. A m j A_{m_{j}}
  42. B m j B_{m_{j}}
  43. A ^ m j \hat{A}_{m_{j}}
  44. B ^ m j \hat{B}_{m_{j}}
  45. M M
  46. M M
  47. X j ( s ) = 1 2 π ( ω j s mod 2 π ) X_{j}\left(s\right)=\frac{1}{2\pi}\left(\omega_{j}s\,\text{ mod }2\pi\right)
  48. ω j , j = 1 , , n \omega_{j},j=1,\dots,n
  49. s s
  50. \infty
  51. 2 π 2\pi
  52. A ^ m j = { 0 m j odd 1 π - π / 2 π / 2 f ( 𝐗 ( s ) ) cos ( m j ω j s ) d s m j even \hat{A}_{m_{j}}=\begin{cases}0&m_{j}\,\text{ odd}\\ \frac{1}{\pi}\int_{-\pi/2}^{\pi/2}f\bigl(\mathbf{X}(s)\bigr)\cos\left(m_{j}% \omega_{j}s\right)ds&m_{j}\,\text{ even}\end{cases}
  53. B ^ m j = { 1 π - π / 2 π / 2 f ( 𝐗 ( s ) ) sin ( m j ω j s ) d s m j odd 0 m j even \hat{B}_{m_{j}}=\begin{cases}\frac{1}{\pi}\int_{-\pi/2}^{\pi/2}f\bigl(\mathbf{% X}(s)\bigr)\sin\left(m_{j}\omega_{j}s\right)ds&m_{j}\,\text{ odd}\\ 0&m_{j}\,\text{ even}\end{cases}
  54. A ^ m j = 1 2 q + 1 k = - q q f ( 𝐗 ( s k ) ) cos ( m j ω j s k ) , m j even B ^ m j = 1 2 q + 1 k = - q q f ( 𝐗 ( s k ) ) sin ( m j ω j s k ) , m j odd \begin{aligned}\displaystyle\hat{A}_{m_{j}}&\displaystyle=\frac{1}{2q+1}\sum_{% k=-q}^{q}f\bigl(\mathbf{X}(s_{k})\bigr)\cos\left(m_{j}\omega_{j}s_{k}\right),m% _{j}\,\text{ even}\\ \displaystyle\hat{B}_{m_{j}}&\displaystyle=\frac{1}{2q+1}\sum_{k=-q}^{q}f\bigl% (\mathbf{X}(s_{k})\bigr)\sin\left(m_{j}\omega_{j}s_{k}\right),m_{j}\,\text{ % odd }\end{aligned}
  55. [ - π / 2 , π / 2 ] \left[-\pi/2,\pi/2\right]
  56. s k = π k 2 q + 1 , k = - q , , - 1 , 0 , 1 , , q . s_{k}=\frac{\pi k}{2q+1},k=-q,\dots,-1,0,1,\dots,q.
  57. 2 q + 1 2q+1
  58. 2 q + 1 N ω m a x + 1 2q+1\geq N\omega_{max}+1
  59. ω m a x \omega_{max}
  60. { ω k } \left\{\omega_{k}\right\}
  61. N N
  62. V j = 2 m j = 1 ( A m j 2 + B m j 2 ) 2 m j = 1 ( A ^ m j 2 + B ^ m j 2 ) 2 m j = 1 2 ( A ^ m j 2 + B ^ m j 2 ) = 2 ( A ^ m j = 2 2 + B ^ m j = 1 2 ) \begin{aligned}\displaystyle V_{j}&\displaystyle=2\sum_{m_{j}=1}^{\infty}\left% (A_{m_{j}}^{2}+B_{m_{j}}^{2}\right)\\ &\displaystyle\approx 2\sum_{m_{j}=1}^{\infty}\left(\hat{A}_{m_{j}}^{2}+\hat{B% }_{m_{j}}^{2}\right)\\ &\displaystyle\approx 2\sum_{m_{j}=1}^{2}\left(\hat{A}_{m_{j}}^{2}+\hat{B}_{m_% {j}}^{2}\right)\\ &\displaystyle=2\left(\hat{A}_{m_{j}=2}^{2}+\hat{B}_{m_{j}=1}^{2}\right)\end{aligned}
  63. N = 2 N=2
  64. m j m_{j}
  65. m k m_{k}
  66. m j ω j = m k ω k . m_{j}\omega_{j}=m_{k}\omega_{k}.
  67. f ( 𝐗 ) f\left(\mathbf{X}\right)
  68. V A ^ 0 [ f 2 ] - A ^ 0 [ f ] 2 V\approx\hat{A}_{0}\left[f^{2}\right]-\hat{A}_{0}\left[f\right]^{2}
  69. A ^ 0 [ f 2 ] \hat{A}_{0}\left[f^{2}\right]
  70. f 2 f^{2}
  71. A ^ 0 [ f ] 2 \hat{A}_{0}\left[f\right]^{2}
  72. f f

Fowler–Yang_equations.html

  1. ϕ t = z [ ( 1 - ϕ ) u ] , \frac{\partial\phi}{\partial t}=\frac{\partial}{\partial z}[(1-\phi)u],
  2. u = λ ϕ m [ 2 u z 2 - ( 1 - ϕ ) ] , u=\lambda\phi^{m}\left[\frac{\partial^{2}u}{\partial z^{2}}-(1-\phi)\right],
  3. ϕ \phi

Fowlkes–Mallows_index.html

  1. F M = T P T P + F P T P T P + F N FM=\sqrt{\frac{TP}{TP+FP}\cdot\frac{TP}{TP+FN}}
  2. T P TP
  3. F P FP
  4. F N FN
  5. n n
  6. A 1 A_{1}
  7. A 2 A_{2}
  8. A 1 A_{1}
  9. A 2 A_{2}
  10. k = 2 , , n - 1 k=2,\ldots,n-1
  11. k k
  12. M = [ m i , j ] ( i = 1 , , k and j = 1 , , k ) M=[m_{i,j}]\qquad(i=1,\ldots,k\,\text{ and }j=1,\ldots,k)
  13. m i , j m_{i,j}
  14. i i
  15. A 1 A_{1}
  16. j j
  17. A 2 A_{2}
  18. k k
  19. B k = T k P k Q k B_{k}=\frac{T_{k}}{\sqrt{P_{k}Q_{k}}}
  20. T k = i = 1 k j = 1 k m i , j 2 - n T_{k}=\sum_{i=1}^{k}\sum_{j=1}^{k}m_{i,j}^{2}-n
  21. P k = i = 1 k ( j = 1 k m i , j ) 2 - n P_{k}=\sum_{i=1}^{k}(\sum_{j=1}^{k}m_{i,j})^{2}-n
  22. Q k = j = 1 k ( i = 1 k m i , j ) 2 - n Q_{k}=\sum_{j=1}^{k}(\sum_{i=1}^{k}m_{i,j})^{2}-n
  23. B k B_{k}
  24. k k
  25. B k B_{k}
  26. k k
  27. k k
  28. 0 B k 1 0\leq B_{k}\leq 1
  29. T P TP
  30. A 1 A_{1}
  31. A 2 A_{2}
  32. F P FP
  33. A 1 A_{1}
  34. A 2 A_{2}
  35. F N FN
  36. A 2 A_{2}
  37. A 1 A_{1}
  38. T N TN
  39. A 1 A_{1}
  40. A 2 A_{2}
  41. T P + F P + F N + T N = n ( n - 1 ) / 2 TP+FP+FN+TN=n(n-1)/2
  42. F M = T P T P + F P T P T P + F N FM=\sqrt{\frac{TP}{TP+FP}\cdot\frac{TP}{TP+FN}}
  43. T P TP
  44. F P FP
  45. F N FN
  46. 1 1

Fox–Wright_function.html

  1. Ψ q p [ ( a 1 , A 1 ) ( a 2 , A 2 ) ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) ( b q , B q ) ; z ] = n = 0 Γ ( a 1 + A 1 n ) Γ ( a p + A p n ) Γ ( b 1 + B 1 n ) Γ ( b q + B q n ) z n n ! . {}_{p}\Psi_{q}\left[\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots&(a_{p},A_% {p})\\ (b_{1},B_{1})&(b_{2},B_{2})&\ldots&(b_{q},B_{q})\end{matrix};z\right]=\sum_{n=% 0}^{\infty}\frac{\Gamma(a_{1}+A_{1}n)\cdots\Gamma(a_{p}+A_{p}n)}{\Gamma(b_{1}+% B_{1}n)\cdots\Gamma(b_{q}+B_{q}n)}\,\frac{z^{n}}{n!}.
  2. Ψ q * p [ ( a 1 , A 1 ) ( a 2 , A 2 ) ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) ( b q , B q ) ; z ] = Γ ( b 1 ) Γ ( b q ) Γ ( a 1 ) Γ ( a p ) n = 0 Γ ( a 1 + A 1 n ) Γ ( a p + A p n ) Γ ( b 1 + B 1 n ) Γ ( b q + B q n ) z n n ! {}_{p}\Psi^{*}_{q}\left[\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots&(a_{p% },A_{p})\\ (b_{1},B_{1})&(b_{2},B_{2})&\ldots&(b_{q},B_{q})\end{matrix};z\right]=\frac{% \Gamma(b_{1})\cdots\Gamma(b_{q})}{\Gamma(a_{1})\cdots\Gamma(a_{p})}\sum_{n=0}^% {\infty}\frac{\Gamma(a_{1}+A_{1}n)\cdots\Gamma(a_{p}+A_{p}n)}{\Gamma(b_{1}+B_{% 1}n)\cdots\Gamma(b_{q}+B_{q}n)}\,\frac{z^{n}}{n!}
  3. Ψ q p [ ( a 1 , A 1 ) ( a 2 , A 2 ) ( a p , A p ) ( b 1 , B 1 ) ( b 2 , B 2 ) ( b q , B q ) ; z ] = H p , q + 1 1 , p [ - z | ( 1 - a 1 , A 1 ) ( 1 - a 2 , A 2 ) ( 1 - a p , A p ) ( 0 , 1 ) ( 1 - b 1 , B 1 ) ( 1 - b 2 , B 2 ) ( 1 - b q , B q ) ] . {}_{p}\Psi_{q}\left[\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots&(a_{p},A_% {p})\\ (b_{1},B_{1})&(b_{2},B_{2})&\ldots&(b_{q},B_{q})\end{matrix};z\right]=H^{1,p}_% {p,q+1}\left[-z\left|\begin{matrix}(1-a_{1},A_{1})&(1-a_{2},A_{2})&\ldots&(1-a% _{p},A_{p})\\ (0,1)&(1-b_{1},B_{1})&(1-b_{2},B_{2})&\ldots&(1-b_{q},B_{q})\end{matrix}\right% .\right].

Fraction_of_inspired_oxygen.html

  1. P A O 2 = P E O 2 - P i O 2 V D V t 1 - V D V t P_{A}O_{2}=\frac{P_{E}O_{2}-P_{i}O_{2}\frac{V_{D}}{V_{t}}}{1-\frac{V_{D}}{V_{t% }}}

Fractional_programming.html

  1. f , g , h j , j = 1 , , m f,g,h_{j},j=1,\ldots,m
  2. 𝐒 0 n \mathbf{S}_{0}\subset\mathbb{R}^{n}
  3. 𝐒 = { s y m b o l x 𝐒 0 : h j ( s y m b o l x ) 0 , j = 1 , , m } \mathbf{S}=\{symbol{x}\in\mathbf{S}_{0}:h_{j}(symbol{x})\leq 0,j=1,\ldots,m\}
  4. maximize s y m b o l x 𝐒 f ( s y m b o l x ) g ( s y m b o l x ) , \underset{symbol{x}\in\mathbf{S}}{\,\text{maximize}}\quad\frac{f(symbol{x})}{g% (symbol{x})},
  5. g ( s y m b o l x ) > 0 g(symbol{x})>0
  6. 𝐒 \mathbf{S}
  7. f , g , h j , j = 1 , , m f,g,h_{j},j=1,\ldots,m
  8. q ( s y m b o l x ) = f ( s y m b o l x ) / g ( s y m b o l x ) q(symbol{x})=f(symbol{x})/g(symbol{x})
  9. s y m b o l y = s y m b o l x g ( s y m b o l x ) ; t = 1 g ( s y m b o l x ) symbol{y}=\frac{symbol{x}}{g(symbol{x})};t=\frac{1}{g(symbol{x})}
  10. maximize s y m b o l y t 𝐒 0 t f ( s y m b o l y t ) subject to t g ( s y m b o l y t ) 1 , t 0. \begin{aligned}\displaystyle\underset{\frac{symbol{y}}{t}\in\mathbf{S}_{0}}{\,% \text{maximize}}&\displaystyle tf(\frac{symbol{y}}{t})\\ \displaystyle\,\text{subject to}&\displaystyle tg(\frac{symbol{y}}{t})\leq 1,% \\ &\displaystyle t\geq 0.\end{aligned}
  11. t g ( s y m b o l y t ) = 1 tg(\frac{symbol{y}}{t})=1
  12. minimize s y m b o l u sup s y m b o l x 𝐒 0 f ( s y m b o l x ) - s y m b o l u T s y m b o l h ( s y m b o l x ) g ( s y m b o l x ) subject to u i 0 , i = 1 , , m . \begin{aligned}\displaystyle\underset{symbol{u}}{\,\text{minimize}}&% \displaystyle\underset{symbol{x}\in\mathbf{S}_{0}}{\operatorname{sup}}\frac{f(% symbol{x})-symbol{u}^{T}symbol{h}(symbol{x})}{g(symbol{x})}\\ \displaystyle\,\text{subject to}&\displaystyle u_{i}\geq 0,\quad i=1,\dots,m.% \end{aligned}

Fractionalization.html

  1. P ( | Ψ 1 > | Ψ 2 > ) = e i θ | Ψ 2 > | Ψ 1 Align g t ; P(|\Psi_{1}>|\Psi_{2}>)=e^{i\theta}|\Psi_{2}>|\Psi_{1}&gt;

Fragility.html

  1. m := ( log 10 η ( T g / T ) ) T = T g = 1 ln 10 ( ln η ( T g / T ) ) T = T g = T g ln 10 ( - ln η T ) T = T g {m}:=\left({\partial{\log_{10}\eta}\over\partial\left(T_{g}/T\right)}\right)_{% T=Tg}=\frac{1}{\ln 10}\left({\partial{\ln\eta}\over\partial\left(T_{g}/T\right% )}\right)_{T=Tg}=\frac{T_{g}}{\ln 10}\left({-\partial{\ln\eta}\over\partial T}% \right)_{T=Tg}
  2. η \eta
  3. T g T_{g}

Frame_(video).html

  1. r = k 2 1 d 2 r=k^{2}\cdot\frac{1}{d^{2}}
  2. n = k 1 d n=k\cdot\frac{1}{d}

Fraunhofer_diffraction_(mathematics).html

  1. $\mathbf{ }$
  2. k k
  3. A ( x , , y ) ) A(x,,y))
  4. x , y , z x,y,z
  5. l , m l,m
  6. x , y x,y
  7. U ( x , y ) U(x,y)
  8. U ( x , y , z ) Aperture A ( x , y ) e - i 2 π λ ( l x + m y ) d x d y U(x,y,z)\propto\iint\text{Aperture}\,A(x^{\prime},y^{\prime})e^{-i\frac{2\pi}{% \lambda}(lx^{\prime}+my^{\prime})}dx^{\prime}\,dy^{\prime}
  9. U ( x , y , z ) Aperture A ( x , y ) e - i k ( l x + m y ) d x d y U(x,y,z)\propto\iint\text{Aperture}\,A(x^{\prime},y^{\prime})e^{-ik(lx^{\prime% }+my^{\prime})}dx^{\prime}\,dy^{\prime}
  10. U ( x , y , z ) Aperture A ( x , y ) e - i 2 π λ z ( x x + y y ) d x d y U(x,y,z)\propto\iint\text{Aperture}\,A(x^{\prime},y^{\prime})e^{-i\frac{2\pi}{% \lambda z}(x^{\prime}x+y^{\prime}y)}\,dx^{\prime}\,dy^{\prime}
  11. U ( x , y , z ) Aperture A ( x , y ) e - i k ( x x + y y ) z d x d y U(x,y,z)\propto\iint\text{Aperture}\,A(x^{\prime},y^{\prime})e^{-i\frac{k(x^{% \prime}x+y^{\prime}y)}{z}}\,dx^{\prime}\,dy^{\prime}
  12. f x = x / ( λ z ) = l / λ f_{x}=x/(\lambda z)=l/\lambda
  13. f y = y / ( λ z ) = m / λ f_{y}=y/(\lambda z)=m/\lambda
  14. U ( x , y , z ) f ^ [ A ( x , y ) ] f x f y U(x,y,z)\propto\hat{f}[A(x^{\prime},y^{\prime})]_{f_{x}f_{y}}
  15. Â Â
  16. A A
  17. U ( 𝐫 ) Aperture A ( 𝐫 ) e - i 𝐤 ( 𝐫 - 𝐫 ) d r = Aperture a 0 ( 𝐫 ) e i ( 𝐤 𝟎 - 𝐤 ) ( 𝐫 - 𝐫 ) d r U(\mathbf{r})\propto{\int\text{Aperture}A(\mathbf{r^{\prime}})e^{-i\mathbf{k}% \cdot(\mathbf{r^{\prime}}-\mathbf{r})}dr^{\prime}}={\int\text{Aperture}a_{0}(% \mathbf{r^{\prime}})e^{i\mathbf{(k_{0}-k)}\cdot(\mathbf{r^{\prime}}-\mathbf{r}% )}dr^{\prime}}
  18. 𝐫 a n d 𝐫 \mathbf{r}and\mathbf{r}
  19. 𝐤 \mathbf{k}
  20. ρ , ω ρ,ω
  21. x = ρ cos ω ; y = ρ sin ω ~{}x^{\prime}=\rho^{\prime}\cos\omega^{\prime};y^{\prime}=\rho^{\prime}\sin% \omega^{\prime}
  22. x = ρ cos ω ; y = ρ sin ω ~{}x=\rho\cos\omega;y=\rho\sin\omega
  23. ρ ρ
  24. A ( ρ ) A(ρ)
  25. d x d y dxdy
  26. U ( ρ , ω , z ) 0 0 2 π A ( ρ ) e - i 2 π λ z ( ρ ρ cos ω cos ω + ρ ρ sin ω sin ω ) ρ d ρ d ω 0 2 π 0 A ( ρ ) e - i 2 π λ z ρ ρ cos ( ω - ω ) d ω ρ d ρ \begin{aligned}\displaystyle U(\rho,\omega,z)&\displaystyle\propto\int_{0}^{% \infty}\int_{0}^{2\pi}A(\rho^{\prime})e^{-i\frac{2\pi}{\lambda z}(\rho\rho^{% \prime}\cos\omega\cos\omega^{\prime}+\rho\rho^{\prime}\sin\omega\sin\omega^{% \prime})}\rho^{\prime}d\rho^{\prime}d\omega^{\prime}\\ &\displaystyle\propto\int_{0}^{2\pi}\int_{0}^{\infty}A(\rho^{\prime})e^{-i% \frac{2\pi}{\lambda z}\rho\rho^{\prime}\cos(\omega-\omega^{\prime})}d\omega^{% \prime}\rho^{\prime}d\rho^{\prime}\end{aligned}
  27. J 0 ( p ) = 1 2 π 0 2 π e i p cos α d α J_{0}(p)=\frac{1}{2\pi}\int_{0}^{2\pi}e^{ip\cos\alpha}d\alpha
  28. U ( ρ , z ) 2 π 0 A ( ρ ) J 0 ( 2 π ρ ρ λ z ) ρ d ρ \begin{aligned}\displaystyle U(\rho,z)&\displaystyle\propto 2\pi\int_{0}^{% \infty}A(\rho^{\prime})J_{0}\left(\frac{2\pi\rho^{\prime}\rho}{\lambda z}% \right)\rho^{\prime}d\rho^{\prime}\end{aligned}
  29. ω ω
  30. 2 π
  31. ω ω
  32. U ( ρ , z ) U(ρ,z)
  33. A ( ρ ) A(ρ)
  34. A ( x , y ) = a e i 2 π c t / λ = a e i k c t ~{}A(x^{\prime},y^{\prime})=ae^{i2\pi ct/\lambda}=ae^{ikct}
  35. a a
  36. λ λ
  37. c c
  38. t t
  39. k k
  40. 2 π / λ 2π/λ
  41. t t
  42. 𝐫 \mathbf{r}
  43. I ( 𝐫 ) U ( 𝐫 ) U ¯ ( 𝐫 ) I(\mathbf{r})\propto U(\mathbf{r})\overline{U}(\mathbf{r})
  44. W W
  45. y y
  46. x 0 x0
  47. y y
  48. U ( x , z ) = a - W / 2 W / 2 e - 2 π i x x / ( λ z ) d x = - a λ z 2 π i x | e - 2 π i x x / ( λ z ) | - W / 2 W / 2 \begin{aligned}\displaystyle U(x,z)&\displaystyle=a\int_{-W/2}^{W/2}e^{{-2\pi ixx% ^{\prime}}/(\lambda z)}dx^{\prime}\\ &\displaystyle=-\frac{a\lambda z}{2\pi ix}|e^{{-2\pi ixx^{\prime}}/(\lambda z)% }|_{-W/2}^{W/2}\end{aligned}
  49. U ( x , z ) \displaystyle U(x,z)
  50. ( p ) = s i n ( p ) / p (p)=sin(p)/p
  51. s i n ( π p ) / π p sin(πp)/πp
  52. U ( θ ) = a W sinc [ π W sin θ λ ] U(\theta)=aW~{}\mathrm{sinc}\left[\frac{\pi W\sin\theta}{\lambda}\right]
  53. θ θ
  54. s i n θ x / z sinθ≈x/z
  55. s i n c sinc
  56. I ( θ ) sinc 2 [ π W sin θ λ ] sinc 2 [ k W sin θ 2 ] \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\operatorname{sinc}% ^{2}\left[\frac{\pi W\sin\theta}{\lambda}\right]\\ &\displaystyle\propto\operatorname{sinc}^{2}\left[\frac{kW\sin\theta}{2}\right% ]\end{aligned}
  57. U ( θ , ϕ ) sinc ( π W sin θ λ ) sinc ( π H sin ϕ λ ) sinc ( k W sin θ 2 ) sinc ( k H sin ϕ 2 ) \begin{aligned}\displaystyle U(\theta,\phi)&\displaystyle\propto\operatorname{% sinc}\left(\frac{\pi W\sin\theta}{\lambda}\right)\operatorname{sinc}\left(% \frac{\pi H\sin\phi}{\lambda}\right)\\ &\displaystyle\propto\operatorname{sinc}\left(\frac{kW\sin\theta}{2}\right)% \operatorname{sinc}\left(\frac{kH\sin\phi}{2}\right)\end{aligned}
  58. I ( θ , ϕ ) sinc 2 ( π W sin θ λ ) sinc 2 ( π H sin ϕ λ ) sinc 2 ( k W sin θ 2 ) sinc 2 ( k H sin ϕ 2 ) \begin{aligned}\displaystyle I(\theta,\phi)&\displaystyle\propto\operatorname{% sinc}^{2}\left(\frac{\pi W\sin\theta}{\lambda}\right)\operatorname{sinc}^{2}% \left(\frac{\pi H\sin\phi}{\lambda}\right)\\ &\displaystyle\propto\operatorname{sinc}^{2}\left(\frac{kW\sin\theta}{2}\right% )\operatorname{sinc}^{2}\left(\frac{kH\sin\phi}{2}\right)\end{aligned}
  59. θ θ
  60. φ φ
  61. x x
  62. z z
  63. y y
  64. z z
  65. W W
  66. U ( ρ , z ) = 2 π a 0 W / 2 J 0 ( 2 π ρ ρ / λ z ) ρ d ρ \begin{aligned}\displaystyle U(\rho,z)&\displaystyle=2\pi a\int_{0}^{W/2}J_{0}% (2\pi\rho^{\prime}\rho/\lambda z)\rho^{\prime}d\rho^{\prime}\end{aligned}
  67. d d x [ x n + 1 J n + 1 ( x ) ] = x n + 1 J n ( x ) \frac{d}{dx}\left[x^{n+1}J_{n+1}(x)\right]=x^{n+1}J_{n}(x)
  68. 0 x x J 0 ( x ) d x = x J 1 ( x ) \int_{0}^{x}x^{\prime}J_{0}(x^{\prime})dx^{\prime}=xJ_{1}(x)
  69. x = 2 π ρ λ z ρ x^{\prime}=\frac{2\pi\rho}{\lambda z}\rho^{\prime}
  70. π ρ W / λ z πρW/λz
  71. U ( ρ , z ) J 1 ( π W ρ / λ z ) π W ρ / λ z U(\rho,z)\propto\frac{J_{1}(\pi W\rho/\lambda z)}{\pi W\rho/\lambda z}
  72. ρ / z ρ/z
  73. θ θ
  74. U ( θ ) J 1 ( π W sin θ / λ ) π W sin θ / λ U(\theta)\propto\frac{J_{1}(\pi W\sin\theta/\lambda)}{\pi W\sin\theta/\lambda}
  75. Π ( W / 2 ) ~{}\Pi(W/2)
  76. F [ Π ( r / a ) ] = 2 π J 1 ( q a ) q ~{}F[\Pi(r/a)]=\frac{2\pi J_{1}(qa)}{q}
  77. q / 2 π q/2π
  78. ρ / λ z ρ/λz
  79. a a
  80. W / 2 W/2
  81. U ( ρ ) = 2 π J 1 ( π W ρ / λ z ) 2 π W ρ / λ z = 2 π J 1 ( π W sin θ / λ ) W sin θ / λ = 2 π J 1 ( k W sin θ / 2 ) k W sin θ / 2 \begin{aligned}\displaystyle U(\rho)&\displaystyle=\frac{2\pi J_{1}(\pi W\rho/% \lambda z)}{2\pi W\rho/\lambda z}\\ &\displaystyle=\frac{2\pi J_{1}(\pi W\sin\theta/\lambda)}{W\sin\theta/\lambda}% \\ &\displaystyle=\frac{2\pi J_{1}(kW\sin\theta/2)}{kW\sin\theta/2}\end{aligned}
  82. I ( θ ) [ J 1 ( π W sin θ / λ ) π W sin θ / λ ) ] 2 [ J 1 ( k W sin θ / 2 ) ( k W sin θ / 2 ) ] 2 \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\left[\frac{J_{1}(% \pi W\sin\theta/\lambda)}{\pi W\sin\theta/\lambda)}\right]^{2}\\ &\displaystyle\propto\left[\frac{J_{1}(kW\sin\theta/2)}{(kW\sin\theta/2)}% \right]^{2}\end{aligned}
  83. A ( ρ ) = exp ( - [ ρ σ ] 2 ) A(\rho^{\prime})=\exp{\left(-\left[\frac{\rho^{\prime}}{\sigma}\right]^{2}% \right)}
  84. U ( ρ , z ) = 2 π a 0 exp ( - [ ρ σ ] 2 ) J 0 ( 2 π ρ ρ / λ z ) ρ d ρ U(\rho,z)=2\pi a\int_{0}^{\infty}\exp{\left(-\left[\frac{\rho^{\prime}}{\sigma% }\right]^{2}\right)}J_{0}(2\pi\rho^{\prime}\rho/\lambda z)\rho^{\prime}\,d\rho% ^{\prime}
  85. F ν ( k ) = 0 f ( r ) J ν ( k r ) r d r F_{\nu}(k)=\int_{0}^{\infty}f(r)J_{\nu}(kr)\,r\,dr
  86. F ν [ e ( a r ) 2 / 2 ] = e - k 2 / 2 a 2 a 2 F_{\nu}[e^{(ar)^{2}/2}]=\frac{e^{-k^{2}/2a^{2}}}{a^{2}}
  87. U ( ρ , z ) e - [ π ρ σ λ z ] 2 \begin{aligned}\displaystyle U(\rho,z)&\displaystyle\propto e^{-[\frac{\pi\rho% \sigma}{\lambda z}]^{2}}\end{aligned}
  88. U ( θ ) e - [ π σ sin θ λ ] 2 U(\theta)\propto e^{-[\frac{\pi\sigma\sin\theta}{\lambda}]^{2}}
  89. I ( θ ) e - [ 2 π σ sin θ λ ] I(\theta)\propto e^{-[\frac{2\pi\sigma\sin\theta}{\lambda}]}
  90. λ λ
  91. z = 0 z=0
  92. y y
  93. S S
  94. θ θ
  95. ( 2 π / λ ) ( S / 2 ) s i n θ (2π/λ)(S/2)sinθ
  96. - ( 2 π / λ ) ( S / 2 ) s i n θ -(2π/λ)(S/2)sinθ
  97. U ( θ ) = a e i π S sin θ λ + a e - i π S sin θ λ = a ( cos π S sin θ λ + i sin π S sin θ λ ) + a ( cos π S sin θ λ - i sin π S sin θ λ ) = 2 a cos π S sin θ λ \begin{aligned}\displaystyle U(\theta)&\displaystyle=ae^{\frac{i\pi S\sin% \theta}{\lambda}}+ae^{-\frac{i\pi S\sin\theta}{\lambda}}\\ &\displaystyle=a(\cos{\frac{\pi S\sin\theta}{\lambda}}+i\sin{\frac{\pi S\sin% \theta}{\lambda}})+a(\cos{\frac{\pi S\sin\theta}{\lambda}}-i\sin{\frac{\pi S% \sin\theta}{\lambda}})\\ &\displaystyle=2a\cos{\frac{\pi S\sin\theta}{\lambda}}\end{aligned}
  98. a [ δ ( x - S / 2 ) + δ ( x + S / 2 ) ] ~{}a[\delta{(x-S/2)}+\delta{(x+S/2)}]
  99. δ δ
  100. f ^ [ δ ( x ) ] = 1 \hat{f}[\delta(x)]=1
  101. f ^ [ g ( x - a ) ] = e - 2 π i a f x f ^ [ g ( x ) ] \hat{f}[g(x-a)]=e^{-2\pi iaf_{x}}\hat{f}[g(x)]
  102. U ( x , z ) = f ^ [ δ ( x - W / 2 ) + δ ( x + W / 2 ) ] = e - i π S x / λ z + e i π S x / λ z = 2 cos π S x λ z \begin{aligned}\displaystyle U(x,z)&\displaystyle=\hat{f}[\delta{(x-W/2)}+% \delta{(x+W/2)}]\\ &\displaystyle=e^{-i\pi Sx/\lambda z}+e^{i\pi Sx/\lambda z}\\ &\displaystyle=2\cos\frac{\pi Sx}{\lambda z}\end{aligned}
  103. U ( θ ) = 2 cos π S sin θ λ U(\theta)=2\cos\frac{\pi S\sin\theta}{\lambda}
  104. I ( θ ) cos 2 [ π S sin θ λ ] cos 2 [ k S sin θ 2 ] \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\cos^{2}\left[\frac% {\pi S\sin\theta}{\lambda}\right]\\ &\displaystyle\propto\cos^{2}{[\frac{kS\sin\theta}{2}]}\end{aligned}
  105. W W
  106. U ( θ ) = a [ e i π S sin θ λ + e - i π S sin θ λ ] - W / 2 W / 2 e - 2 π i x sin θ / ( λ ) d x = 2 a cos π S sin θ λ W sinc π W sin θ λ \begin{aligned}\displaystyle U(\theta)&\displaystyle=a\left[e^{\frac{i\pi S% \sin\theta}{\lambda}}+e^{-\frac{i\pi S\sin\theta}{\lambda}}\right]\int_{-W/2}^% {W/2}e^{{-2\pi ix^{\prime}\sin\theta}/(\lambda)}dx^{\prime}\\ &\displaystyle=2a\cos{\frac{\pi S\sin\theta}{\lambda}}W~{}\mathrm{sinc}\frac{% \pi W\sin\theta}{\lambda}\end{aligned}
  107. a [ rect ( x - S / 2 W ) + rect ( x + S / 2 W ) ] a\left[\mathrm{rect}\left(\frac{x-S/2}{W}\right)+\mathrm{rect}\left(\frac{x+S/% 2}{W}\right)\right]
  108. f ^ ( rect ( a x ) ) = 1 | a | sinc ( ξ a ) \hat{f}(\mathrm{rect}(ax))=\displaystyle\frac{1}{|a|}\cdot\operatorname{sinc}% \left(\frac{\xi}{a}\right)
  109. ξ ξ
  110. s i n c sinc
  111. f ^ [ g ( x - a ) ] = e - 2 π i a f x f ^ [ g ( x ) ] \hat{f}[g(x-a)]=e^{-2\pi iaf_{x}}\hat{f}[g(x)]
  112. U ( x , z ) = f ^ [ a [ rect ( x - S / 2 W ) + rect ( x + S / 2 W ) ] ] = 2 W [ e - i π S x / λ z + e i π S x / λ z ] sin π W x λ z π W x λ z = 2 a cos π S x λ z W sinc π W x λ z \begin{aligned}\displaystyle U(x,z)&\displaystyle=\hat{f}\left[a\left[\mathrm{% rect}\left(\frac{x-S/2}{W}\right)+\mathrm{rect}\left(\frac{x+S/2}{W}\right)% \right]\right]\\ &\displaystyle=2W\left[e^{-i\pi Sx/\lambda z}+e^{i\pi Sx/\lambda z}\right]% \frac{\sin{\frac{\pi Wx}{\lambda z}}}{\frac{\pi Wx}{\lambda z}}\\ &\displaystyle=2a\cos{\frac{\pi Sx}{\lambda z}}W~{}\mathrm{sinc}\frac{\pi Wx}{% \lambda z}\end{aligned}
  113. U ( θ ) = 2 a cos π S sin θ λ W sinc π W sin θ λ U(\theta)=2a\cos{\frac{\pi S\sin\theta}{\lambda}}W~{}\mathrm{sinc}\frac{\pi W% \sin\theta}{\lambda}
  114. I ( θ ) cos 2 [ π S sin θ λ ] sinc 2 [ π W sin θ λ ] cos 2 [ k S sin θ 2 ] sinc 2 [ k W sin θ 2 ] \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\cos^{2}\left[{% \frac{\pi S\sin\theta}{\lambda}}\right]~{}\mathrm{sinc}^{2}\left[\frac{\pi W% \sin\theta}{\lambda}\right]\\ &\displaystyle\propto\cos^{2}\left[\frac{kS\sin\theta}{2}\right]\mathrm{sinc}^% {2}\left[\frac{kW\sin\theta}{2}\right]\end{aligned}
  115. S S
  116. θ θ
  117. U ( θ ) = a n = 1 N e - i 2 π n S sin θ λ = 1 - e - i 2 π N S sin θ / λ 1 - e - i 2 π S sin θ / λ \begin{aligned}\displaystyle U(\theta)&\displaystyle=a\sum_{n=1}^{N}e^{\frac{-% i2\pi nS\sin\theta}{\lambda}}\\ &\displaystyle=\frac{1-e^{-i2\pi NS\sin\theta/\lambda}}{1-e^{-i2\pi S\sin% \theta/\lambda}}\end{aligned}
  118. n = 0 N δ ( x - n S ) \sum_{n=0}^{N}\delta(x-nS)
  119. f ^ [ n = 0 N δ ( x - n S ) ] = n = 0 N e - i f x n S = 1 - e - i 2 π N S sin θ / λ 1 - e - i 2 π S sin θ / λ \begin{aligned}\displaystyle\hat{f}\left[\sum_{n=0}^{N}\delta(x-nS)\right]&% \displaystyle=\sum_{n=0}^{N}e^{-if_{x}nS}\\ &\displaystyle=\frac{1-e^{-i2\pi NS\sin\theta/\lambda}}{1-e^{-i2\pi S\sin% \theta/\lambda}}\end{aligned}
  120. I ( θ ) 1 - cos ( 2 π N S sin θ / λ ) 1 - cos ( 2 π S sin θ / λ ) sin 2 ( π N S sin θ / λ ) sin 2 ( π S sin θ / λ ) \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\frac{1-\cos(2\pi NS% \sin\theta/\lambda)}{1-\cos(2\pi S\sin\theta/\lambda)}\\ &\displaystyle\propto\frac{\sin^{2}(\pi NS\sin\theta/\lambda)}{\sin^{2}(\pi S% \sin\theta/\lambda)}\end{aligned}
  121. π S sin n θ / λ = n π , n = 0 , ± 1 , ± 2 , . . \pi S\sin_{n}\theta/\lambda=n\pi,n=0,\pm 1,\pm 2,.....
  122. sin θ n = n λ S , n = 0 , ± 1 ± 2 , . \sin\theta_{n}=\frac{n\lambda}{S},n=0,\pm 1\pm 2,....
  123. N N
  124. N N
  125. N N
  126. W W
  127. S S
  128. U ( θ , ϕ ) a n = 1 N e - i 2 π n S sin θ λ - W / 2 W / 2 e - 2 π i x x / ( λ z ) d x a sinc ( W sin θ λ ) 1 - e - i 2 π N S sin θ / λ 1 - e - i 2 π D sin θ / λ \begin{aligned}\displaystyle U(\theta,\phi)&\displaystyle\propto a\sum_{n=1}^{% N}e^{\frac{-i2\pi nS\sin\theta}{\lambda}}\int_{-W/2}^{W/2}e^{{-2\pi ixx^{% \prime}}/(\lambda z)}dx^{\prime}\\ &\displaystyle\propto a\mathrm{sinc}\left(\frac{W\sin\theta}{\lambda}\right)% \frac{1-e^{-i2\pi NS\sin\theta/\lambda}}{1-e^{-i2\pi D\sin\theta/\lambda}}\end% {aligned}
  129. n = 1 N rect [ x - n S W ] \sum_{n=1}^{N}\mathrm{rect}\left[\frac{x^{\prime}-nS}{W}\right]
  130. f ( x ) f(x)
  131. g ( x ) g(x)
  132. h ( x ) = ( f * g ) ( x ) = - f ( y ) g ( x - y ) d y , h(x)=(f*g)(x)=\int_{-\infty}^{\infty}f(y)g(x-y)\,dy,
  133. h ^ ( ξ ) = f ^ ( ξ ) g ^ ( ξ ) . \hat{h}(\xi)=\hat{f}(\xi)\cdot\hat{g}(\xi).
  134. rect ( x / W ) * n = 0 N δ ( x - n S ) \mathrm{rect}(x^{\prime}/W)*\sum_{n=0}^{N}\delta(x^{\prime}-nS)
  135. U ( x , z ) \displaystyle U(x,z)
  136. I ( θ ) sinc 2 ( W sin θ λ ) sin 2 ( π N S sin θ / λ ) sin 2 ( π S sin θ / λ ) \begin{aligned}\displaystyle I(\theta)&\displaystyle\propto\mathrm{sinc}^{2}% \left(\frac{W\sin\theta}{\lambda}\right)\frac{\sin^{2}(\pi NS\sin\theta/% \lambda)}{\sin^{2}(\pi S\sin\theta/\lambda)}\end{aligned}
  137. U ( x , y , z ) Aperture A ( x , y ) e - i 2 π λ [ ( l - l 0 ) x + ( m - m 0 ) y ] d x d y Aperture A ( x , y ) e - i k [ ( l - l 0 ) x + ( m - m 0 ) y ] d x d y \begin{aligned}\displaystyle U(x,y,z)&\displaystyle\propto\iint\text{Aperture}% \,A(x^{\prime},y^{\prime})e^{-i\frac{2\pi}{\lambda}[(l-l_{0})x^{\prime}+(m-m_{% 0})y^{\prime}]}dx^{\prime}\,dy^{\prime}\\ &\displaystyle\propto\iint\text{Aperture}\,A(x^{\prime},y^{\prime})e^{-ik[(l-l% _{0})x^{\prime}+(m-m_{0})y^{\prime}]}dx^{\prime}\,dy^{\prime}\end{aligned}
  138. x x
  139. y y
  140. sin θ n = n λ S + sin θ 0 , n = 0 , ± 1 , ± 2.... \sin\theta_{n}=\frac{n\lambda}{S}+\sin\theta_{0},n=0,\pm 1,\pm 2....

Free-fall_atomic_model.html

  1. 𝐋 = 1 2 m 𝐯 2 + Z e 2 r + Z e c [ 𝐯 ( μ × 𝐫 r 3 ) ] \mathbf{L}=\frac{1}{2}m\mathbf{v}^{2}+\frac{Ze^{2}}{r}+\frac{Ze}{c}\left[% \mathbf{v}\cdot\left(\frac{\mu\times\mathbf{r}}{r^{3}}\right)\right]

Free_boundary_problem.html

  1. T t = ( α 1 T ) + Q \frac{\partial T}{\partial t}=\nabla\cdot(\alpha_{1}\nabla T)+Q
  2. L V = α 1 ν T 1 - α 2 ν T 2 , LV=\alpha_{1}\partial_{\nu}T_{1}-\alpha_{2}\partial_{\nu}T_{2},
  3. - 2 u = f , u | Ω = g -\nabla^{2}u=f,\qquad u|_{\partial\Omega}=g
  4. E [ u ] = 1 2 Ω | u | 2 d x - Ω f u d x E[u]=\frac{1}{2}\int_{\Omega}|\nabla u|^{2}\,\mathrm{d}x-\int_{\Omega}fu\,% \mathrm{d}x
  5. u φ u\leq\varphi\,
  6. - 2 u = f in N , u = g -\nabla^{2}u=f\,\text{ in }N,\quad u=g
  7. u φ in | Ω , u = φ on Γ . u\leq\varphi\,\text{ in }|\Omega,\quad\nabla u=\nabla\varphi\,\text{ on }% \Gamma.\,
  8. F ( x ) ( y - x ) 0 for all y C . \nabla F(x)\cdot(y-x)\geq 0\,\text{ for all }y\in C.\,
  9. Ω ( 2 u + f ) ( v - u ) d x 0 for all v φ . \int_{\Omega}(\nabla^{2}u+f)(v-u)\,\mathrm{d}x\geq 0\,\text{ for all }v\leq\varphi.
  10. Ω u ( v - u ) d x Ω f ( v - u ) d x for all v φ . \int_{\Omega}\nabla u\cdot\nabla(v-u)\mathrm{d}x\leq\int_{\Omega}f(v-u)\,% \mathrm{d}x\,\text{ for all }v\leq\varphi.

Frege_system.html

  1. r = B 1 , , B n B , r=\frac{B_{1},\dots,B_{n}}{B},
  2. A B A\lor B
  3. A A
  4. A A B \frac{A}{A\lor B}
  5. ϕ \phi
  6. ϕ \phi

Frequency_ambiguity_resolution.html

  1. R a d i a l V e l o c i t y > 0.5 ( P R F × C T r a n s m i t F r e q u e n c y ) Radial\ Velocity>0.5\left(\frac{PRF\times C}{Transmit\ Frequency}\right)
  2. A p p a r e n t V e l o c i t y = ( T r u e V e l o c i t y ) M O D ( P R F × C 2 × T r a n s m i t F r e q u e n c y ) Apparent\ Velocity=(TrueVelocity)MOD\left(\frac{PRF\times C}{2\times Transmit% \ Frequency}\right)
  3. A m b i g u o u s V e l o c i t y = - 0.5 ( D o p p l e r F r e q u e n c y × C T r a n s m i t F r e q u e n c y ) Ambiguous\ Velocity=-0.5\left(\frac{Doppler\ Frequency\times C}{Transmit\ % Frequency}\right)
  4. T r u e V e l o c i t y = A m b i g u o u s V e l o c i t y + 0.5 × N ( P R F × C T r a n s m i t F r e q u e n c y ) True\ Velocity=Ambiguous\ Velocity+0.5\times N\left(\frac{PRF\times C}{% Transmit\ Frequency}\right)
  5. N = I n t e g e r B e t w e e n ± ( 0.5 × B a n d w i d t h P R F ) N=Integer\ Between\pm\left(\frac{0.5\times Bandwidth}{PRF}\right)

Frequency_domain_decomposition.html

  1. G ^ y y ( j ω ) \hat{G}_{yy}(j\omega)
  2. ω = ω i \omega=\omega_{i}
  3. G ^ y y ( j ω i ) = U i S i U i H \hat{G}_{yy}(j\omega_{i})=U_{i}S_{i}U_{i}^{H}
  4. U i = [ u i 1 , u i 2 , , u i m ] U_{i}=[u_{i1},u_{i2},...,u_{im}]
  5. u i j u_{ij}
  6. S i S_{i}
  7. s i j s_{ij}
  8. n n
  9. n n

Fresnel_zone_antenna.html

  1. x 2 b 2 + ( y - c ) 2 a 2 = 1 \frac{x^{2}}{b^{2}}+\frac{(y-c)^{2}}{a^{2}}=1
  2. r n = ( n + α ) λ f + ( n + α ) 2 λ 2 4 r_{n}=\sqrt{(n+\alpha)\lambda f+\frac{(n+\alpha)^{2}\lambda^{2}}{4}}

Freudenthal_spectral_theorem.html

  1. p ( e - p ) = 0 p\wedge(e-p)=0
  2. p 1 , p 2 , , p n p_{1},p_{2},\ldots,p_{n}
  3. p 1 , p 2 , , p n p_{1},p_{2},\ldots,p_{n}
  4. { s n } \{s_{n}\}
  5. { t n } \{t_{n}\}
  6. { s n } \{s_{n}\}
  7. { t n } \{t_{n}\}
  8. ( X , Σ ) (X,\Sigma)
  9. M σ M_{\sigma}
  10. σ \sigma
  11. ( X , Σ ) (X,\Sigma)
  12. M σ M_{\sigma}
  13. μ \mu
  14. μ \mu
  15. μ \mu
  16. ( X , Σ ) (X,\Sigma)
  17. ν \nu
  18. μ \mu
  19. μ \mu
  20. ( X , Σ ) (X,\Sigma)
  21. ν \nu
  22. L 1 ( X , Σ , μ ) L^{1}(X,\Sigma,\mu)
  23. μ \mu
  24. L 1 ( X , Σ , μ ) L^{1}(X,\Sigma,\mu)

Freydoon_Shahidi.html

  1. L L

Fréchet-Kolmogorov_theorem.html

  1. B B
  2. L p ( n ) L^{p}(\mathbb{R}^{n})
  3. p [ 1 , ) p\in[1,\infty)
  4. lim r | x | > r | f | p = 0 \lim_{r\to\infty}\int_{|x|>r}\left|f\right|^{p}=0
  5. lim a 0 τ a f - f L p ( n ) = 0 \lim_{a\to 0}\|\tau_{a}f-f\|_{L^{p}(\mathbb{R}^{n})}=0
  6. τ a f \tau_{a}f
  7. f f
  8. a a
  9. τ a f ( x ) = f ( x - a ) . \tau_{a}f(x)=f(x-a).
  10. ε > 0 δ > 0 \forall\varepsilon>0\,\,\exists\delta>0
  11. τ a f - f L p ( n ) < ε f B , a \|\tau_{a}f-f\|_{L^{p}(\mathbb{R}^{n})}<\varepsilon\,\,\forall f\in B,\forall a
  12. | a | < δ . |a|<\delta.

Frictional_contact_mechanics.html

  1. ξ \xi
  2. F w F_{w}
  3. e n e_{n}
  4. e n > 0 e_{n}>0
  5. p n p_{n}
  6. p n > 0 p_{n}>0
  7. e n 0 , p n 0 , e n p n = 0 e_{n}\geq 0,p_{n}\geq 0,e_{n}\cdot p_{n}=0\,\!
  8. e n , p n e_{n},p_{n}
  9. p = ( p x , p y ) T \vec{p}=(p_{x},p_{y})^{T}\,\!
  10. z z
  11. g g
  12. p < g \|\vec{p}\|<g\,\!
  13. s = ( s x , s y ) T = 0 \vec{s}=(s_{x},s_{y})^{T}=\vec{0}\,\!
  14. p = - g s / s \vec{p}=-g\vec{s}/\|\vec{s}\|\,\!
  15. p g = μ p n \|\vec{p}\|\leq g=\mu p_{n}\,\!
  16. μ \mu
  17. μ \mu
  18. T T
  19. s \|\vec{s}\|
  20. F h o l d = 400 N F_{hold}=400N
  21. T T
  22. T = 400 N T=400N
  23. 180 d e g 180deg
  24. F l o a d = 600 N F_{load}=600N
  25. T ( ϕ ) = T h o l d , ϕ [ ϕ h o l d , ϕ i n t f ] T ( ϕ ) = T l o a d exp ( - μ ϕ ) , ϕ [ ϕ i n t f , ϕ l o a d ] ϕ i n t f = log ( T l o a d / T h o l d ) / μ \begin{matrix}T(\phi)=T_{hold},&\phi\in[\phi_{hold},\phi_{intf}]\\ T(\phi)=T_{load}\exp(-\mu\phi),&\phi\in[\phi_{intf},\phi_{load}]\\ \phi_{intf}=\log(T_{load}/T_{hold})/\mu&\end{matrix}
  26. T h o l d T_{hold}
  27. ϕ = ϕ h o l d \phi=\phi_{hold}
  28. T l o a d T_{load}
  29. ϕ = ϕ l o a d \phi=\phi_{load}
  30. ϕ = ϕ i n t f \phi=\phi_{intf}
  31. ϕ i n t f \phi_{intf}
  32. T T
  33. ϕ \phi
  34. T T
  35. ϕ [ ϕ i n t f , ϕ l o a d ] \phi\in[\phi_{intf},\phi_{load}]
  36. 600 N 600N
  37. 400 N 400N
  38. 400 400
  39. 600 N 600N
  40. δ n \delta_{n}
  41. R R
  42. E , ν E,\nu
  43. p n ( x , y ) = p 0 1 - r 2 / a 2 r = x 2 + y 2 a a = R δ n p 0 = 2 π E * ( δ n / R ) 1 / 2 F n = 4 3 E * R 1 / 2 δ n 3 / 2 E * = E / 2 ( 1 - ν 2 ) \begin{array}[]{lll}p_{n}(x,y)=p_{0}\sqrt{1-r^{2}/a^{2}}&r=\sqrt{x^{2}+y^{2}}% \leq a&a=\sqrt{R\delta_{n}}\\ p_{0}=\frac{2}{\pi}E^{*}\left(\delta_{n}/R\right)^{1/2}&F_{n}=\frac{4}{3}E^{*}% R^{1/2}\delta_{n}^{3/2}&E^{*}=E/2(1-\nu^{2})\end{array}
  44. F x F_{x}
  45. μ F n \mu F_{n}
  46. δ x \delta_{x}
  47. p x ( x , y ) = μ p 0 ( 1 - r 2 / a 2 - c a 1 - r 2 / c 2 ) 0 r c p x ( x , y ) = μ p n ( x , y ) c r a p y ( x , y ) = 0 a r \begin{array}[]{ll}p_{x}(x,y)=\mu p_{0}\left(\sqrt{1-r^{2}/a^{2}}-\frac{c}{a}% \sqrt{1-r^{2}/c^{2}}\right)&0\leq r\leq c\\ p_{x}(x,y)=\mu p_{n}(x,y)&c\leq r\leq a\\ p_{y}(x,y)=0&a\leq r\end{array}
  48. 0 r c 0\leq r\leq c
  49. u x = δ x / 2 u_{x}=\delta_{x}/2
  50. u x = - δ x / 2 u_{x}=-\delta_{x}/2
  51. δ x \delta_{x}
  52. a r c a\leq r\leq c
  53. s x ( x , y ) = δ x + u x s p h e r e ( x , y ) - u x p l a n e ( x , y ) s_{x}(x,y)=\delta_{x}+u_{x}^{sphere}(x,y)-u_{x}^{plane}(x,y)
  54. s x ( x , y ) s_{x}(x,y)
  55. ξ \xi
  56. x [ - a , a ] x\in[-a,a]
  57. p n ( x ) = p 0 a a 2 - x 2 | x | a a 2 = 4 F n R / π E * p 0 = 2 F n / π a E * = E / 2 ( 1 - ν 2 ) \begin{array}[]{lll}p_{n}(x)=\frac{p_{0}}{a}\sqrt{a^{2}-x^{2}}&|x|\leq a&a^{2}% =4F_{n}R/\pi E^{*}\\ p_{0}=2F_{n}/\pi a&E^{*}=E/2(1-\nu^{2})&\end{array}
  58. 2 a 2a^{\prime}
  59. x = x + a - a x^{\prime}=x+a-a^{\prime}
  60. F x > 0 F_{x}>0
  61. ξ < 0 \xi<0
  62. p x ( x ) = 0 | x | a p x ( x ) = μ p 0 a ( a 2 - x 2 - a 2 - x 2 ) a - 2 a x a p x ( x ) = μ p n ( x ) x a - 2 a \begin{array}[]{ll}p_{x}(x)=0&|x|\geq a\\ p_{x}(x)=\frac{\mu p_{0}}{a}\left(\sqrt{a^{2}-x^{2}}-\sqrt{a^{\prime 2}-x^{% \prime 2}}\right)&a-2a^{\prime}\leq x\leq a\\ p_{x}(x)=\mu p_{n}(x)&x\leq a-2a^{\prime}\end{array}
  63. a = a 1 - | F x | / μ F n , for | F x | μ F n ξ = - s i g n ( F x ) μ ( a - a ) / R , i.e. | ξ | μ a / R F x = - s i g n ( ξ ) μ F n ( 1 - ( 1 + R | ξ | / μ a ) 2 ) \begin{array}[]{ll}a^{\prime}=a\sqrt{1-|F_{x}|/\mu F_{n}},&\mbox{for }~{}|F_{x% }|\leq\mu F_{n}\\ \xi=-sign(F_{x})\,\mu(a-a^{\prime})/R,&\mbox{i.e. }~{}|\xi|\leq\mu a/R\\ F_{x}=-sign(\xi)\,\mu F_{n}\left(1-\left(1+R|\xi|/\mu a\right)^{2}\right)\end{array}
  64. a = 0 a^{\prime}=0
  65. 1 / d i s t a n c e 2 1/distance^{2}
  66. 1 / d i s t a n c e 1/distance
  67. ( x , y , z ) T \R 3 (x,y,z)^{T}\in\R^{3}\,\!
  68. z > 0 z>0\,\!
  69. p y p_{y}

Fried_parameter.html

  1. r 0 r_{0}
  2. r 0 r_{0}
  3. r 0 r_{0}
  4. λ \lambda
  5. C n 2 C_{n}^{2}
  6. z z^{\prime}
  7. r 0 = [ 0.423 k 2 Path C n 2 ( z ) d z ] - 3 / 5 r_{0}=\left[0.423\,k^{2}\,\int_{\mathrm{Path}}C_{n}^{2}(z^{\prime})\,dz^{% \prime}\right]^{-3/5}
  8. k = 2 π / λ k=2\pi/\lambda
  9. ζ \zeta
  10. sec ζ \sec\zeta
  11. r 0 r_{0}
  12. r 0 r_{0}
  13. r 0 = [ 0.423 k 2 sec ζ Vertical C n 2 ( z ) d z ] - 3 / 5 = ( cos ζ ) 3 / 5 r 0 ( v e r t i c a l ) . r_{0}=\left[0.423\,k^{2}\,\sec\zeta\int_{\mathrm{Vertical}}C_{n}^{2}(z)\,dz% \right]^{-3/5}=(\cos\zeta)^{3/5}\ r_{0}^{(vertical)}.
  14. r 0 r_{0}
  15. λ / r 0 \lambda/r_{0}
  16. D D
  17. 1.22 λ / D 1.22\lambda/D
  18. D r 0 D\gg r_{0}
  19. r 0 r_{0}
  20. λ 6 / 5 \lambda^{6/5}
  21. λ = 0.5 μ m \lambda=0.5\mu m

Frobenius_determinant_theorem.html

  1. G G
  2. g 1 , g 2 , , g n g_{1},g_{2},\dots,g_{n}
  3. x g i x_{g_{i}}
  4. G G
  5. X G X_{G}
  6. a i j = x g i g j a_{ij}=x_{g_{i}g_{j}}
  7. det X G = j = 1 r P j ( x g 1 , x g 2 , , x g n ) deg P j \det X_{G}=\prod_{j=1}^{r}P_{j}(x_{g_{1}},x_{g_{2}},\dots,x_{g_{n}})^{\deg P_{% j}}

Frobenius_manifold.html

  1. X * Y = A ( X , Y ) . X*Y=A(X,Y).\,
  2. g ( X * Y , Z ) = g ( X , Y * Z ) . g(X*Y,Z)=g(X,Y*Z).\,
  3. g ( A ( X , Y ) , Z ) = X [ Y [ Z [ Φ ] ] ] g(A(X,Y),Z)=X[Y[Z[\Phi]]]\,
  4. Φ , a b e g e f Φ , c d f = Φ , a d e g e f Φ , b c f \Phi_{,abe}g^{ef}\Phi_{,cdf}=\Phi_{,ade}g^{ef}\Phi_{,bcf}\,

Frozen_orbit.html

  1. Δ z ^ \Delta\hat{z}\,
  2. J 2 J_{2}\,
  3. Δ z ^ = - 2 π J 2 μ p 2 3 2 cos i sin i g ^ \Delta\hat{z}\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos i\ \sin i% \quad\hat{g}
  4. Δ Ω = - 2 π J 2 μ p 2 3 2 cos i \Delta\Omega\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos i
  5. J 3 J_{3}\,
  6. Δ z ^ = 2 π J 3 μ p 3 3 2 cos i ( e h ( 1 - 15 4 sin 2 i ) g ^ - e g ( 1 - 5 4 sin 2 i ) h ^ ) \Delta\hat{z}\ =\ 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \cos i\ \left(% \ e_{h}\ (1-\frac{15}{4}\ \sin^{2}i)\ \hat{g}\ -\ e_{g}\ (1-\frac{5}{4}\ \sin^% {2}i)\ \hat{h}\right)
  7. Δ i = - 2 π J 3 μ p 3 3 2 cos i e g ( 1 - 5 4 sin 2 i ) \Delta i\ =\ -2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \cos i\ e_{g}\ (1-% \frac{5}{4}\ \sin^{2}i)
  8. Δ Ω = 2 π J 3 μ p 3 3 2 cos i sin i e h ( 1 - 15 4 sin 2 i ) \Delta\Omega\ =\ 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \frac{\cos i}{% \sin i}\ \ e_{h}\ (1-\frac{15}{4}\ \sin^{2}i)
  9. J 2 J_{2}\,
  10. ( Δ e g , Δ e h ) = - 2 π J 2 μ p 2 3 2 ( 3 2 sin 2 i - 1 ) ( - e h , e g ) + 2 π J 2 μ p 2 3 2 cos 2 i ( - e h , e g ) = - 2 π J 2 μ p 2 3 ( 5 4 sin 2 i - 1 ) ( - e h , e g ) \begin{aligned}\displaystyle(\Delta e_{g},\Delta e_{h})\ =&\displaystyle-2\pi% \ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\left(\frac{3}{2}\ \sin^{2}i\ -\ 1% \right)\ (-e_{h},e_{g})\ +\ 2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos^% {2}i(-e_{h},e_{g})\ =\\ &\displaystyle-2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ 3\left(\frac{5}{4}\ \sin^{2}i\ % -\ 1\right)\ (-e_{h},e_{g})\end{aligned}
  11. J 3 J_{3}\,
  12. 2 π J 3 μ p 3 3 2 sin i ( 5 4 sin 2 i - 1 ) ( ( 1 - e g 2 + 4 e h 2 ) g ^ - 5 e g e h h ^ ) 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \sin i\ \left(\frac{5}{4}\ \sin^{% 2}i\ -\ 1\right)\left((1-{e_{g}}^{2}+4\ {e_{h}}^{2})\ \hat{g}\ -\ 5\ e_{g}\ e_% {h}\ \hat{h}\right)
  13. Δ Ω \Delta\Omega\,
  14. 2 π J 3 μ p 3 3 2 sin i ( 5 4 sin 2 i - 1 ) g ^ 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \sin i\ \left(\frac{5}{4}\ \sin^{% 2}i\ -\ 1\right)\ \hat{g}
  15. J 3 J_{3}\,
  16. 2 π J 3 μ p 3 3 2 sin i ( 5 4 sin 2 i - 1 ) ( 1 , 0 ) 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \sin i\ \left(\frac{5}{4}\ \sin^{% 2}i\ -\ 1\right)\ (1\ ,\ 0)
  17. ( Δ e g , Δ e h ) = - 2 π J 2 μ p 2 3 ( 5 4 sin 2 i - 1 ) ( - ( e h + J 3 sin i J 2 2 p ) , e g ) (\Delta e_{g},\Delta e_{h})\ =\ -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ 3\left(\frac{% 5}{4}\ \sin^{2}i\ -\ 1\right)\ \left(-\left(e_{h}+\frac{J_{3}\ \sin i}{J_{2}\ % 2\ p}\right)\ ,\ e_{g}\right)
  18. ( 0 , - J 3 sin i J 2 2 p ) (\ 0\ ,\ -\frac{J_{3}\ \sin i}{J_{2}\ 2\ p}\ )\,
  19. - 2 π J 2 μ p 2 3 ( 5 4 sin 2 i - 1 ) -2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ 3\left(\frac{5}{4}\ \sin^{2}i\ -\ 1\right)\,
  20. μ = 398600.440 km 3 / s 2 \mu=398600.440\,\text{ km}^{3}/s^{2}\,
  21. J 2 = 1.7555 10 10 km 5 / s 2 J_{2}\ =\ 1.7555\ 10^{10}\,\text{ km}^{5}/s^{2}\,
  22. J 3 = - 2.619 10 11 km 6 / s 2 J_{3}\ =\ -2.619\ 10^{11}\,\text{ km}^{6}/s^{2}\,
  23. i = 90 deg i=90\,\text{ deg}\,
  24. p = 7200 km p=7200\,\text{ km}\,
  25. ( 0 , 0.001036 ) (\ 0\ ,\ 0.001036\ )\,
  26. ( 0 , 0.001036 ) (0\ ,\ 0.001036)\,
  27. J 2 J_{2}\,
  28. J 3 J_{3}\,
  29. e = - J 3 sin i J 2 2 p e=-\frac{J_{3}\ \sin i}{J_{2}\ 2\ p}\,
  30. ω = 90 deg \omega=\ 90\ \,\text{deg}\,
  31. J 2 J_{2}\,
  32. J 3 J_{3}\,
  33. i = 90 deg i=90\,\text{ deg}\,
  34. p = 7200 km p=7200\,\text{ km}\,
  35. J 2 J_{2}\,
  36. J 3 J_{3}\,
  37. ( 0 , 0.001036 ) (0\ ,\ 0.001036)\,
  38. ( 0 , 0.001285 ) (0\ ,\ 0.001285)\,
  39. ( 0.000069 , 0.001285 ) (0.000069\ ,\ 0.001285)\,
  40. ω = 87 deg \omega=\ 87\ \,\text{deg}\,
  41. ω = 90 deg \omega=\ 90\ \,\text{deg}\,
  42. J 3 J_{3}\,
  43. J 3 J_{3}\,
  44. F r r ^ F_{r}\ \hat{r}\,
  45. F λ λ ^ F_{\lambda}\ \hat{\lambda}\,
  46. λ ^ \hat{\lambda}\,
  47. r ^ \hat{r}\,
  48. λ ^ \hat{\lambda}\,
  49. ϕ ^ , λ ^ , r ^ \hat{\phi}\ ,\ \hat{\lambda}\ ,\ \hat{r}
  50. J 3 J_{3}\,
  51. F r = J 3 1 r 5 2 sin λ ( 5 sin 2 λ - 3 ) F λ = - J 3 1 r 5 3 2 cos λ ( 5 sin 2 λ - 1 ) \begin{aligned}&\displaystyle F_{r}=J_{3}\ \frac{1}{r^{5}}\ 2\ \sin\lambda\ % \left(5\sin^{2}\lambda\ -\ 3\right)\\ &\displaystyle F_{\lambda}=-J_{3}\ \frac{1}{r^{5}}\ \frac{3}{2}\ \cos\lambda\ % \left(5\ \sin^{2}\lambda\ -1\right)\end{aligned}
  52. F λ λ ^ F_{\lambda}\ \hat{\lambda}\,
  53. F t t ^ F_{t}\ \hat{t}
  54. F z z ^ F_{z}\ \hat{z}
  55. a ^ , b ^ , n ^ \hat{a}\ ,\ \hat{b}\ ,\ \hat{n}\,
  56. n ^ \hat{n}\,
  57. a ^ , b ^ \hat{a}\ ,\ \hat{b}\,
  58. a ^ \hat{a}\,
  59. r ^ , t ^ , z ^ \hat{r}\ ,\ \hat{t}\ ,\ \hat{z}\,
  60. t ^ , z ^ , \hat{t}\ ,\ \hat{z},
  61. a ^ , b ^ , n ^ \hat{a}\ ,\ \hat{b}\ ,\ \hat{n}\,
  62. r a = cos u r_{a}=\cos u\,
  63. r b = cos i sin u r_{b}=\cos i\ \sin u\,
  64. r n = sin i sin u r_{n}=\sin i\ \sin u\,
  65. t a = - sin u t_{a}=-\sin u\,
  66. t b = cos i cos u t_{b}=\cos i\ \cos u\,
  67. t n = sin i cos u t_{n}=\sin i\ \cos u\,
  68. z a = 0 z_{a}=0\,
  69. z b = - sin i z_{b}=-\sin i\,
  70. z n = cos i z_{n}=\cos i\,
  71. u u\,
  72. r ^ \hat{r}\,
  73. g ^ = a ^ \hat{g}=\hat{a}\,
  74. h ^ = cos i b ^ + sin i n ^ \hat{h}=\cos i\ \hat{b}\ +\ \sin i\ \hat{n}\,
  75. sin λ = r n = sin i sin u \sin\lambda=\ r_{n}\ =\ \sin i\ \sin u\,
  76. λ \lambda\,
  77. r ^ \hat{r}\,
  78. F r = J 3 1 r 5 2 sin i sin u ( 5 sin 2 i sin 2 u - 3 ) F_{r}=J_{3}\ \frac{1}{r^{5}}\ 2\ \sin i\ \sin u\,\ \left(5\sin^{2}i\ \sin^{2}u% \ -\ 3\right)
  79. n ^ \hat{n}\,
  80. t ^ , z ^ , \hat{t}\ ,\ \hat{z},
  81. sin i cos u t ^ + cos i z ^ \sin i\ \cos u\ \hat{t}\ +\ \cos i\ \hat{z}\,
  82. cos λ λ ^ \cos\lambda\ \hat{\lambda}\,
  83. λ ^ \hat{\lambda}\,
  84. λ ^ \hat{\lambda}
  85. F λ λ ^ = - J 3 1 r 5 3 2 ( 5 sin 2 λ - 1 ) cos λ λ ^ = - J 3 1 r 5 3 2 ( 5 sin 2 λ - 1 ) ( sin i cos u t ^ + cos i z ^ ) F_{\lambda}\ \hat{\lambda}\ =-J_{3}\ \frac{1}{r^{5}}\ \frac{3}{2}\ \left(5\ % \sin^{2}\lambda\ -1\right)\ \cos\lambda\ \hat{\lambda}\ =\ -J_{3}\ \frac{1}{r^% {5}}\ \frac{3}{2}\ \left(5\ \sin^{2}\lambda\ -1\right)\ (\sin i\ \cos u\ \hat{% t}\ +\ \cos i\ \hat{z})\,
  86. F t = - J 3 1 r 5 3 2 ( 5 sin 2 i sin 2 u - 1 ) sin i cos u F_{t}=\ -J_{3}\ \frac{1}{r^{5}}\ \frac{3}{2}\ \left(5\ \sin^{2}i\ \sin^{2}u\ -% 1\right)\ \sin i\ \cos u
  87. F z = - J 3 1 r 5 3 2 ( 5 sin 2 i sin 2 u - 1 ) cos i F_{z}=\ -J_{3}\ \frac{1}{r^{5}}\ \frac{3}{2}\ \left(5\ \sin^{2}i\ \sin^{2}u\ -% 1\right)\ \cos i
  88. z ^ \hat{z}\,
  89. Δ z ^ = 1 μ p [ g ^ 0 2 π F z r 3 cos u d u + h ^ 0 2 π F z r 3 sin u d u ] × z ^ \Delta\hat{z}\ =\ \frac{1}{\mu p}\left[\hat{g}\int\limits_{0}^{2\pi}F_{z}r^{3}% \cos u\ du+\ \hat{h}\int\limits_{0}^{2\pi}F_{z}r^{3}\sin u\ du\right]\quad% \times\ \hat{z}
  90. F z F_{z}\,
  91. Δ z ^ = - J 3 μ p 3 3 2 cos i [ g ^ 0 2 π ( p r ) 2 ( 5 sin 2 i sin 2 u - 1 ) cos u d u + h ^ 0 2 π ( p r ) 2 ( 5 sin 2 i sin 2 u - 1 ) sin u d u ] × z ^ \begin{aligned}&\displaystyle\Delta\hat{z}\ =-\frac{J_{3}}{\mu\ p^{3}}\ \frac{% 3}{2}\ \cos i\ \cdot\\ &\displaystyle\left[\hat{g}\int\limits_{0}^{2\pi}{\left(\frac{p}{r}\right)}^{2% }\left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)\cos u\ du\ +\hat{h}\int\limits_{0}^{% 2\pi}{\left(\frac{p}{r}\right)}^{2}\left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)% \sin u\ du\right]\quad\times\ \hat{z}\end{aligned}
  92. p r \frac{p}{r}\,
  93. p r = 1 + e cos θ = 1 + e g cos u + e h sin u \frac{p}{r}\ =\ 1+e\cdot\cos\theta\ =\ 1+e_{g}\cdot\cos u+e_{h}\cdot\sin u
  94. e g = e cos ω e_{g}=\ e\ \cos\omega
  95. e h = e sin ω e_{h}=\ e\ \sin\omega
  96. g ^ , h ^ \hat{g}\ ,\ \hat{h}\,
  97. 0 2 π cos m u sin n u d u \int\limits_{0}^{2\pi}\cos^{m}u\ \sin^{n}u\ du\,
  98. n n\,
  99. m m\,
  100. 0 2 π ( p r ) 2 ( 5 sin 2 i sin 2 u - 1 ) cos u d u = 2 e g ( 5 sin 2 i 0 2 π sin 2 u cos 2 u d u - 0 2 π cos 2 u d u ) = 2 π e g ( 5 4 sin 2 i - 1 ) \begin{aligned}\displaystyle\int\limits_{0}^{2\pi}{\left(\frac{p}{r}\right)}^{% 2}\left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)\cos u\ du&\displaystyle=\ 2\ e_{g}% \ \left(5\ \sin^{2}i\ \int\limits_{0}^{2\pi}\sin^{2}u\cos^{2}u\ du\ -\int% \limits_{0}^{2\pi}\cos^{2}u\ du\right)\\ &\displaystyle=\ 2\pi\ e_{g}\ (\frac{5}{4}\sin^{2}i-1)\end{aligned}
  101. 0 2 π ( p r ) 2 ( 5 sin 2 i sin 2 u - 1 ) sin u d u = 2 e h ( 5 sin 2 i 0 2 π sin 4 u d u - 0 2 π sin 2 u d u ) = 2 π e h ( 15 4 sin 2 i - 1 ) \begin{aligned}\displaystyle\int\limits_{0}^{2\pi}{\left(\frac{p}{r}\right)}^{% 2}\left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)\sin u\ du&\displaystyle=\ 2\ e_{h}% \ \left(5\ \sin^{2}i\ \int\limits_{0}^{2\pi}\sin^{4}u\ du\ -\int\limits_{0}^{2% \pi}\sin^{2}u\ du\right)\\ &\displaystyle=\ 2\pi\ e_{h}\ (\frac{15}{4}\sin^{2}i-1)\end{aligned}
  102. Δ z ^ = 2 π J 3 μ p 3 3 2 cos i [ e g ( 1 - 5 4 sin 2 i ) g ^ + e h ( 1 - 15 4 sin 2 i ) h ^ ] × z ^ = 2 π J 3 μ p 3 3 2 cos i [ e h ( 1 - 15 4 sin 2 i ) g ^ - e g ( 1 - 5 4 sin 2 i ) h ^ ] \begin{aligned}\displaystyle\Delta\hat{z}&\displaystyle=\ 2\pi\ \frac{J_{3}}{% \mu\ p^{3}}\ \frac{3}{2}\ \cos i\ \left[e_{g}\ (1-\frac{5}{4}\sin^{2}i)\ \hat{% g}+\ e_{h}\ (1-\frac{15}{4}\sin^{2}i)\ \hat{h}\right]\quad\times\ \hat{z}\\ &\displaystyle=\ 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \frac{3}{2}\ \cos i\ \left[\ % e_{h}\ (1-\frac{15}{4}\sin^{2}i)\ \hat{g}\ -e_{g}\ (1-\frac{5}{4}\sin^{2}i)\ % \hat{h}\right]\end{aligned}
  103. g ^ \hat{g}\,
  104. h ^ \hat{h}\,
  105. g ^ \hat{g}\,
  106. u u\,
  107. f z f_{z}\,
  108. z ^ \hat{z}\,
  109. Δ e ¯ = 1 μ 0 2 π ( - t ^ f r + ( 2 r ^ - V r V t t ^ ) f t ) r 2 d u \Delta\bar{e}\ =\frac{1}{\mu}\ \int\limits_{0}^{2\pi}\left(-\hat{t}\ f_{r}\ +% \ \left(2\ \hat{r}-\frac{V_{r}}{V_{t}}\ \hat{t}\right)\ f_{t}\right)\ r^{2}\ du
  110. r ^ , t ^ \hat{r}\ ,\hat{t}\,
  111. r ^ \hat{r}\,
  112. V r = μ p e sin θ V_{r}=\sqrt{\frac{\mu}{p}}\cdot e\cdot\sin\theta
  113. r ^ \hat{r}\,
  114. V t = μ p ( 1 + e cos θ ) V_{t}=\sqrt{\frac{\mu}{p}}\cdot(1+e\cdot\cos\theta)
  115. t ^ \hat{t}\,
  116. F r , F t F_{r}\ ,\ F_{t}\,
  117. Δ e ¯ = J 3 μ p 3 sin i 0 2 π ( - t ^ ( p r ) 3 2 sin u ( 5 sin 2 i sin 2 u - 3 ) - ( 2 r ^ - V r V t t ^ ) ( p r ) 3 3 2 ( 5 sin 2 i sin 2 u - 1 ) cos u ) d u \begin{aligned}&\displaystyle\Delta\bar{e}\ =\frac{J_{3}}{\mu\ p^{3}}\ \sin i% \ \cdot\\ &\displaystyle\int\limits_{0}^{2\pi}\left(-\hat{t}\ {\left(\frac{p}{r}\right)}% ^{3}\ 2\ \sin u\,\ \left(5\sin^{2}i\ \sin^{2}u\ -\ 3\right)\ -\ \left(2\ \hat{% r}-\frac{V_{r}}{V_{t}}\ \hat{t}\right)\ {\left(\frac{p}{r}\right)}^{3}\ \frac{% 3}{2}\ \left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)\ \cos u\right)du\end{aligned}
  118. V r V t = e g sin u - e h cos u p r \frac{V_{r}}{V_{t}}=\frac{e_{g}\cdot\sin u\ -\ e_{h}\cdot\cos u}{\frac{p}{r}}
  119. 0 2 π ( - t ^ ( p r ) 3 2 sin u ( 5 sin 2 i sin 2 u - 3 ) - ( 2 r ^ - V r V t t ^ ) ( p r ) 3 3 2 ( 5 sin 2 i sin 2 u - 1 ) cos u ) d u = - 10 sin 2 i 0 2 π t ^ ( p r ) 3 sin 3 u d u + 6 0 2 π t ^ ( p r ) 3 sin u d u - 15 sin 2 i 0 2 π r ^ ( p r ) 3 sin 2 u cos u d u + 3 0 2 π r ^ ( p r ) 3 cos u d u + 15 2 sin 2 i e g 0 2 π t ^ ( p r ) 2 sin 3 u cos u d u - 15 2 sin 2 i e h 0 2 π t ^ ( p r ) 2 sin 2 u cos 2 u d u - 3 2 e g 0 2 π t ^ ( p r ) 2 sin u cos u d u + 3 2 e h 0 2 π t ^ ( p r ) 2 cos 2 u d u \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\left(-\hat{t}\ {\left(% \frac{p}{r}\right)}^{3}\ 2\ \sin u\,\ \left(5\sin^{2}i\ \sin^{2}u\ -\ 3\right)% \ -\ \left(2\ \hat{r}-\frac{V_{r}}{V_{t}}\ \hat{t}\right)\ {\left(\frac{p}{r}% \right)}^{3}\ \frac{3}{2}\ \left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)\ \cos u% \right)\ du\ =\\ &\displaystyle-10\sin^{2}i\ \int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{3}\ \sin^{3}u\ du\\ &\displaystyle+6\ \int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}\right)}^{3% }\ \sin u\ du\\ &\displaystyle-15\ \sin^{2}i\int\limits_{0}^{2\pi}\hat{r}\ {\left(\frac{p}{r}% \right)}^{3}\ \sin^{2}u\ \cos u\ du\\ &\displaystyle+3\ \int\limits_{0}^{2\pi}\hat{r}\ {\left(\frac{p}{r}\right)}^{3% }\ \cos u\ du\\ &\displaystyle+\frac{15}{2}\sin^{2}i\ e_{g}\int\limits_{0}^{2\pi}\hat{t}\ {% \left(\frac{p}{r}\right)}^{2}\ \ \ \sin^{3}u\ \cos u\ du\\ &\displaystyle-\frac{15}{2}\sin^{2}i\ e_{h}\ \int\limits_{0}^{2\pi}\hat{t}\ {% \left(\frac{p}{r}\right)}^{2}\ \ \ \sin^{2}u\ \cos^{2}u\ du\\ &\displaystyle-\frac{3}{2}\ e_{g}\ \int\limits_{0}^{2\pi}\ \hat{t}\ {\left(% \frac{p}{r}\right)}^{2}\ \ \sin u\ \cos u\ du\\ &\displaystyle+\frac{3}{2}\ e_{h}\ \int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac% {p}{r}\right)}^{2}\ \ \cos^{2}u\ du\end{aligned}
  120. r ^ = cos u g ^ + sin u h ^ \hat{r}=\cos u\ \hat{g}\ +\ \sin u\ \hat{h}
  121. t ^ = - sin u g ^ + cos u h ^ \hat{t}=-\sin u\ \hat{g}\ +\ \cos u\ \hat{h}
  122. p r = 1 + e cos θ = 1 + e g cos u + e h sin u \frac{p}{r}\ =\ 1+e\cdot\cos\theta\ =\ 1+e_{g}\cdot\cos u+e_{h}\cdot\sin u
  123. 0 2 π cos m u sin n u d u \int\limits_{0}^{2\pi}\cos^{m}u\ \sin^{n}u\ du\,
  124. n n\,
  125. m m\,
  126. 0 2 π t ^ ( p r ) 3 sin 3 u d u = - g ^ 0 2 π ( p r ) 3 sin 4 u d u + h ^ 0 2 π ( p r ) 3 sin 3 u cos u d u = - g ^ ( 0 2 π sin 4 u d u + 3 e g 2 0 2 π cos 2 u sin 4 u d u + 3 e h 2 0 2 π sin 6 u d u ) + h ^ 6 e g e h 0 2 π cos 2 u sin 4 u d u = - g ^ ( 2 π ( 3 8 + 3 16 e g 2 + 15 16 e h 2 ) ) + h ^ ( 2 π ( 3 8 e g e h ) ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{3}\ \sin^{3}u\ du\ =-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p% }{r}\right)}^{3}\ \sin^{4}u\ du\ +\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{% p}{r}\right)}^{3}\ \sin^{3}u\ \cos u\ du\ =\\ &\displaystyle-\hat{g}\ \left(\int\limits_{0}^{2\pi}\ \sin^{4}u\ du\ +\ 3\ {e_% {g}}^{2}\ \int\limits_{0}^{2\pi}\ \cos^{2}u\ \sin^{4}u\ du\ \ +\ 3\ {e_{h}}^{2% }\ \int\limits_{0}^{2\pi}\ \sin^{6}u\ du\ \right)\\ &\displaystyle+\hat{h}\ 6\ e_{g}\ e_{h}\ \int\limits_{0}^{2\pi}\ \cos^{2}u\ % \sin^{4}u\ du=\\ &\displaystyle-\hat{g}\ \left(2\pi\left(\frac{3}{8}\ +\ \frac{3}{16}\ {e_{g}}^% {2}\ +\ \frac{15}{16}\ {e_{h}}^{2}\right)\right)+\hat{h}\ \left(2\pi\left(% \frac{3}{8}\ e_{g}\ e_{h}\right)\right)\end{aligned}
  127. 0 2 π t ^ ( p r ) 3 sin u d u = - g ^ 0 2 π ( p r ) 3 sin 2 u d u + h ^ 0 2 π ( p r ) 3 sin u cos u d u = - g ^ ( 0 2 π sin 2 u d u + 3 e g 2 0 2 π cos 2 u sin 2 u d u + 3 e h 2 0 2 π sin 6 u d u ) + h ^ 6 e g e h 0 2 π cos 2 u sin 2 u d u = - g ^ ( 2 π ( 1 2 + 3 8 e g 2 + 9 8 e h 2 ) ) + h ^ ( 2 π ( 3 4 e g e h ) ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{3}\ \sin u\ du\ =-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p}{r% }\right)}^{3}\ \sin^{2}u\ du\ +\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{p}{% r}\right)}^{3}\ \sin u\ \cos u\ du\ =\\ &\displaystyle-\hat{g}\ \left(\int\limits_{0}^{2\pi}\ \sin^{2}u\ du\ +\ 3\ {e_% {g}}^{2}\ \int\limits_{0}^{2\pi}\ \cos^{2}u\ \sin^{2}u\ du\ \ +\ 3\ {e_{h}}^{2% }\ \int\limits_{0}^{2\pi}\ \sin^{6}u\ du\ \right)\\ &\displaystyle+\hat{h}\ 6\ e_{g}\ e_{h}\ \int\limits_{0}^{2\pi}\ \cos^{2}u\ % \sin^{2}u\ du=\\ &\displaystyle-\hat{g}\ \left(2\pi\left(\frac{1}{2}\ +\ \frac{3}{8}\ {e_{g}}^{% 2}\ +\ \frac{9}{8}\ {e_{h}}^{2}\right)\right)+\hat{h}\ \left(2\pi\left(\frac{3% }{4}\ e_{g}\ e_{h}\right)\right)\end{aligned}
  128. 0 2 π r ^ ( p r ) 3 sin 2 u cos u d u = g ^ 0 2 π ( p r ) 3 sin 2 u cos 2 u d u + h ^ 0 2 π ( p r ) 3 sin 3 u cos u d u = g ^ ( 0 2 π sin 2 u cos 2 u d u + 3 e g 2 0 2 π cos 4 u sin 2 u d u + 3 e h 2 0 2 π sin 4 u cos 2 u d u ) + h ^ 6 e g e h 0 2 π cos 2 u sin 4 u d u = g ^ ( 2 π ( 1 8 + 3 16 e g 2 + 3 16 e h 2 ) ) + h ^ ( 2 π ( 3 8 e g e h ) ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{r}\ {\left(\frac{p}{r}% \right)}^{3}\ \sin^{2}u\ \cos u\ du\ =\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(% \frac{p}{r}\right)}^{3}\ \sin^{2}u\cos^{2}u\ du\ +\hat{h}\int\limits_{0}^{2\pi% }\ {\left(\frac{p}{r}\right)}^{3}\ \sin^{3}u\ \cos u\ du\ =\\ &\displaystyle\hat{g}\ \left(\int\limits_{0}^{2\pi}\ \sin^{2}u\cos^{2}u\ du\ +% \ 3\ {e_{g}}^{2}\ \int\limits_{0}^{2\pi}\ \cos^{4}u\ \sin^{2}u\ du\ \ +\ 3\ {e% _{h}}^{2}\ \int\limits_{0}^{2\pi}\ \sin^{4}u\ \cos^{2}u\ du\ \right)\\ &\displaystyle+\hat{h}\ 6\ e_{g}\ e_{h}\ \int\limits_{0}^{2\pi}\ \cos^{2}u\ % \sin^{4}u\ du=\\ &\displaystyle\hat{g}\ \left(2\pi\left(\frac{1}{8}\ +\ \frac{3}{16}\ {e_{g}}^{% 2}\ +\ \frac{3}{16}\ {e_{h}}^{2}\right)\right)+\hat{h}\ \left(2\pi\left(\frac{% 3}{8}\ e_{g}\ e_{h}\right)\right)\end{aligned}
  129. 0 2 π r ^ ( p r ) 3 cos u d u = g ^ 0 2 π ( p r ) 3 cos 2 u d u + h ^ 0 2 π ( p r ) 3 sin u cos u d u = g ^ ( 0 2 π cos 2 u d u + 3 e g 2 0 2 π cos 4 u d u + 3 e h 2 0 2 π sin 2 u cos 2 u d u ) + h ^ 6 e g e h 0 2 π cos 2 u sin 2 u d u = g ^ ( 2 π ( 1 2 + 9 8 e g 2 + 3 8 e h 2 ) ) + h ^ ( 2 π ( 3 4 e g e h ) ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{r}\ {\left(\frac{p}{r}% \right)}^{3}\ \cos u\ du\ =\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}% \right)}^{3}\ \cos^{2}u\ du\ +\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{p}{r% }\right)}^{3}\ \sin u\ \cos u\ du\ =\\ &\displaystyle\hat{g}\ \left(\int\limits_{0}^{2\pi}\cos^{2}u\ du\ +\ 3\ {e_{g}% }^{2}\ \int\limits_{0}^{2\pi}\ \cos^{4}u\ du\ \ +\ 3\ {e_{h}}^{2}\ \int\limits% _{0}^{2\pi}\ \sin^{2}u\ \cos^{2}u\ du\ \right)\\ &\displaystyle+\hat{h}\ 6\ e_{g}\ e_{h}\ \int\limits_{0}^{2\pi}\ \cos^{2}u\ % \sin^{2}u\ du=\\ &\displaystyle\hat{g}\ \left(2\pi\left(\frac{1}{2}\ +\ \frac{9}{8}\ {e_{g}}^{2% }\ +\ \frac{3}{8}\ {e_{h}}^{2}\right)\right)+\hat{h}\ \left(2\pi\left(\frac{3}% {4}\ e_{g}\ e_{h}\right)\right)\end{aligned}
  130. 0 2 π t ^ ( p r ) 2 sin 3 u cos u d u = - g ^ 0 2 π ( p r ) 2 sin 4 u cos u d u + h ^ 0 2 π ( p r ) 2 sin 3 u cos 2 u d u = - g ^ 2 e g 0 2 π sin 4 u cos 2 u d u + h ^ 2 e h 0 2 π sin 4 u cos 2 u d u = - g ^ ( 2 π 1 8 e g ) + h ^ ( 2 π 1 8 e h ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{2}\ \sin^{3}u\ \cos u\ du\ =-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left% (\frac{p}{r}\right)}^{2}\ \sin^{4}u\ \cos u\ du\ +\hat{h}\int\limits_{0}^{2\pi% }\ {\left(\frac{p}{r}\right)}^{2}\ \sin^{3}u\ \cos^{2}u\ du\ =\\ &\displaystyle-\hat{g}\ 2\ e_{g}\ \int\limits_{0}^{2\pi}\ \sin^{4}u\ \cos^{2}u% \ du+\hat{h}\ 2\ e_{h}\ \int\limits_{0}^{2\pi}\ \sin^{4}u\ \cos^{2}u\ du=-\hat% {g}\ \left(2\pi\frac{1}{8}\ e_{g}\right)+\hat{h}\ \left(2\pi\frac{1}{8}\ e_{h}% \right)\end{aligned}
  131. 0 2 π t ^ ( p r ) 2 sin 2 u cos 2 u d u = - g ^ 0 2 π ( p r ) 2 sin 3 u cos 2 u d u + h ^ 0 2 π ( p r ) 2 sin 2 u cos 3 u d u = - g ^ 2 e h 0 2 π sin 4 u cos 2 u d u + h ^ 2 e g 0 2 π sin 2 u cos 4 u d u = - g ^ ( 2 π 1 8 e h ) + h ^ ( 2 π 1 8 e g ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{2}\ \sin^{2}u\ \cos^{2}u\ du\ =-\hat{g}\ \int\limits_{0}^{2\pi}\ {% \left(\frac{p}{r}\right)}^{2}\ \sin^{3}u\ \cos^{2}u\ du\ +\hat{h}\int\limits_{% 0}^{2\pi}\ {\left(\frac{p}{r}\right)}^{2}\ \sin^{2}u\ \cos^{3}u\ du\ =\\ &\displaystyle-\hat{g}\ 2\ e_{h}\ \int\limits_{0}^{2\pi}\ \sin^{4}u\ \cos^{2}u% \ du+\hat{h}\ 2\ e_{g}\ \int\limits_{0}^{2\pi}\ \sin^{2}u\ \cos^{4}u\ du=-\hat% {g}\ \left(2\pi\frac{1}{8}\ e_{h}\right)+\hat{h}\ \left(2\pi\frac{1}{8}\ e_{g}% \right)\end{aligned}
  132. 0 2 π t ^ ( p r ) 2 sin u cos u d u = - g ^ 0 2 π ( p r ) 2 sin 2 u cos u d u + h ^ 0 2 π ( p r ) 2 sin u cos 2 u d u = - g ^ 2 e g 0 2 π sin 2 u cos 2 u d u + h ^ 2 e h 0 2 π sin 2 u cos 2 u d u = - g ^ ( 2 π 1 4 e g ) + h ^ ( 2 π 1 4 e h ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{2}\ \sin u\ \cos u\ du\ =-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(% \frac{p}{r}\right)}^{2}\ \sin^{2}u\ \cos u\ du\ +\hat{h}\int\limits_{0}^{2\pi}% \ {\left(\frac{p}{r}\right)}^{2}\ \sin u\ \cos^{2}u\ du\ =\\ &\displaystyle-\hat{g}\ 2\ e_{g}\ \int\limits_{0}^{2\pi}\ \sin^{2}u\ \cos^{2}u% \ du+\hat{h}\ 2\ e_{h}\ \int\limits_{0}^{2\pi}\ \sin^{2}u\ \cos^{2}u\ du=-\hat% {g}\ \left(2\pi\frac{1}{4}\ e_{g}\right)+\hat{h}\ \left(2\pi\frac{1}{4}\ e_{h}% \right)\end{aligned}
  133. 0 2 π t ^ ( p r ) 2 cos 2 u d u = - g ^ 0 2 π ( p r ) 2 sin u cos 2 u d u + h ^ 0 2 π ( p r ) 2 cos 3 u d u = - g ^ 2 e h 0 2 π sin 2 u cos 2 u d u + h ^ 2 e g 0 2 π cos 4 u d u = - g ^ ( 2 π 1 4 e h ) + h ^ ( 2 π 3 4 e g ) \begin{aligned}&\displaystyle\int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}% \right)}^{2}\ \cos^{2}u\ du\ =-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p% }{r}\right)}^{2}\ \sin u\ \cos^{2}u\ du\ +\hat{h}\int\limits_{0}^{2\pi}\ {% \left(\frac{p}{r}\right)}^{2}\ \cos^{3}u\ du\ =\\ &\displaystyle-\hat{g}\ 2\ e_{h}\ \int\limits_{0}^{2\pi}\ \sin^{2}u\ \cos^{2}u% \ du+\hat{h}\ 2\ e_{g}\ \int\limits_{0}^{2\pi}\ \cos^{4}u\ du=-\hat{g}\ \left(% 2\pi\frac{1}{4}\ e_{h}\right)+\hat{h}\ \left(2\pi\frac{3}{4}\ e_{g}\right)\end% {aligned}
  134. - 10 sin 2 i ( - g ^ ( 3 8 + 3 16 e g 2 + 15 16 e h 2 ) + h ^ ( 3 8 e g e h ) ) + 6 ( - g ^ ( 1 2 + 3 8 e g 2 + 9 8 e h 2 ) + h ^ ( 3 4 e g e h ) ) - 15 sin 2 i ( g ^ ( 1 8 + 3 16 e g 2 + 3 16 e h 2 ) + h ^ ( 3 8 e g e h ) ) + 3 ( g ^ ( 1 2 + 9 8 e g 2 + 3 8 e h 2 ) + h ^ ( 3 4 e g e h ) ) + 15 2 sin 2 i e g ( - g ^ ( 1 8 e g ) + h ^ ( 1 8 e h ) ) - 15 2 sin 2 i e h ( - g ^ ( 1 8 e h ) + h ^ ( 1 8 e g ) ) - 3 2 e g ( - g ^ ( 1 4 e g ) + h ^ ( 1 4 e h ) ) + 3 2 e h ( - g ^ ( 1 4 e h ) + h ^ ( 3 4 e g ) ) = 3 2 ( 5 4 sin 2 i - 1 ) ( ( 1 - e g 2 + 4 e h 2 ) g ^ - 5 e g e h h ^ ) \begin{aligned}&\displaystyle-10\sin^{2}i\ \left(-\hat{g}\ \left(\frac{3}{8}\ % +\ \frac{3}{16}\ {e_{g}}^{2}\ +\ \frac{15}{16}\ {e_{h}}^{2}\right)+\hat{h}\ % \left(\frac{3}{8}\ e_{g}\ e_{h}\right)\right)\\ &\displaystyle+6\ \left(-\hat{g}\ \left(\frac{1}{2}\ +\ \frac{3}{8}\ {e_{g}}^{% 2}\ +\ \frac{9}{8}\ {e_{h}}^{2}\right)+\hat{h}\ \left(\frac{3}{4}\ e_{g}\ e_{h% }\right)\right)\\ &\displaystyle-15\sin^{2}i\ \left(\hat{g}\ \left(\frac{1}{8}\ +\ \frac{3}{16}% \ {e_{g}}^{2}\ +\ \frac{3}{16}\ {e_{h}}^{2}\right)+\hat{h}\ \left(\frac{3}{8}% \ e_{g}\ e_{h}\right)\right)\\ &\displaystyle+3\left(\hat{g}\ \left(\frac{1}{2}\ +\ \frac{9}{8}\ {e_{g}}^{2}% \ +\ \frac{3}{8}\ {e_{h}}^{2}\right)+\hat{h}\ \left(\frac{3}{4}\ e_{g}\ e_{h}% \right)\right)\\ &\displaystyle+\frac{15}{2}\sin^{2}i\ e_{g}\ \left(-\hat{g}\ \left(\frac{1}{8}% \ e_{g}\right)+\hat{h}\ \left(\frac{1}{8}\ e_{h}\right)\right)\\ &\displaystyle-\frac{15}{2}\sin^{2}i\ e_{h}\ \left(-\hat{g}\ \left(\frac{1}{8}% \ e_{h}\right)+\hat{h}\ \left(\frac{1}{8}\ e_{g}\right)\right)\\ &\displaystyle-\frac{3}{2}\ e_{g}\ \left(-\hat{g}\ \left(\frac{1}{4}\ e_{g}% \right)+\hat{h}\ \left(\frac{1}{4}\ e_{h}\right)\right)\\ &\displaystyle+\frac{3}{2}\ e_{h}\ \left(-\hat{g}\ \left(\frac{1}{4}\ e_{h}% \right)+\hat{h}\ \left(\frac{3}{4}\ e_{g}\right)\right)=\\ &\displaystyle\frac{3}{2}\ \left(\frac{5}{4}\ \sin^{2}i\ -\ 1\right)\left((1-{% e_{g}}^{2}\ +\ 4\ {e_{h}}^{2})\hat{g}\ -\ 5\ e_{g}\ e_{h}\ \hat{h}\right)\end{aligned}
  135. Δ e ¯ = J 3 μ p 3 sin i 0 2 π ( - t ^ ( p r ) 3 2 sin u ( 5 sin 2 i sin 2 u - 3 ) - ( 2 r ^ - V r V t t ^ ) ( p r ) 3 3 2 ( 5 sin 2 i sin 2 u - 1 ) cos u ) d u = 2 π J 3 μ p 3 sin i 3 2 ( 5 4 sin 2 i - 1 ) ( ( 1 - e g 2 + 4 e h 2 ) g ^ - 5 e g e h h ^ ) \begin{aligned}&\displaystyle\Delta\bar{e}\ =\frac{J_{3}}{\mu\ p^{3}}\ \sin i% \ \cdot\\ &\displaystyle\int\limits_{0}^{2\pi}\left(-\hat{t}\ {\left(\frac{p}{r}\right)}% ^{3}\ 2\ \sin u\,\ \left(5\sin^{2}i\ \sin^{2}u\ -\ 3\right)\ -\ \left(2\ \hat{% r}-\frac{V_{r}}{V_{t}}\ \hat{t}\right)\ {\left(\frac{p}{r}\right)}^{3}\ \frac{% 3}{2}\ \left(5\ \sin^{2}i\ \sin^{2}u\ -1\right)\ \cos u\right)du=\\ &\displaystyle 2\pi\ \frac{J_{3}}{\mu\ p^{3}}\ \sin i\ \frac{3}{2}\ \left(% \frac{5}{4}\ \sin^{2}i\ -\ 1\right)\left((1-{e_{g}}^{2}\ +\ 4\ {e_{h}}^{2})% \hat{g}\ -\ 5\ e_{g}\ e_{h}\ \hat{h}\right)\end{aligned}

FRW::CFT_duality.html

  1. 10 - 123 10^{-123}

Fubini's_theorem_on_differentiation.html

  1. I I\subseteq\mathbb{R}
  2. f k : I f_{k}:I\to\mathbb{R}
  3. s ( x ) := k = 1 f k ( x ) s(x):=\sum_{k=1}^{\infty}f_{k}(x)
  4. x I , x\in I,
  5. s ( x ) = k = 1 f k ( x ) s^{\prime}(x)=\sum_{k=1}^{\infty}f_{k}^{\prime}(x)
  6. k = 1 n f k ( x ) \sum_{k=1}^{n}f_{k}^{\prime}(x)

Fukaya_category.html

  1. ( M , ω ) (M,\omega)
  2. ( M ) \mathcal{F}(M)
  3. M M
  4. Hom ( L 0 , L 1 ) = F C ( L 0 , L 1 ) \mathrm{Hom}(L_{0},L_{1})=FC(L_{0},L_{1})
  5. A A_{\infty}

Fundamental_group_scheme.html

  1. k k
  2. X Spec ( k ) X\to\,\text{Spec}(k)
  3. X X
  4. x : Spec ( k ) X x:\,\text{Spec}(k)\to X
  5. π 1 ( X , x ) \pi_{1}(X,x)
  6. X X
  7. x x
  8. k k
  9. X X
  10. S S
  11. X X
  12. X S X\to S
  13. x : S X x:S\to X
  14. X X
  15. x x
  16. S S
  17. x x
  18. S S
  19. π 1 ( X , x ) \pi_{1}(X,x)
  20. S S

Fundamental_normality_test.html

  1. \mathcal{F}
  2. Ω \Omega
  3. \mathcal{F}
  4. \mathcal{F}
  5. Ω \Omega

G-parity.html

  1. 𝒢 ( π + π 0 π - ) = η G ( π + π 0 π - ) \mathcal{G}\begin{pmatrix}\pi^{+}\\ \pi^{0}\\ \pi^{-}\end{pmatrix}=\eta_{G}\begin{pmatrix}\pi^{+}\\ \pi^{0}\\ \pi^{-}\end{pmatrix}
  2. 𝒢 = 𝒞 e ( i π I 2 ) \mathcal{G}=\mathcal{C}\,e^{(i\pi I_{2})}
  3. 𝒞 \mathcal{C}
  4. Q ¯ = B ¯ = Y ¯ = 0 \bar{Q}=\bar{B}=\bar{Y}=0
  5. η G = η C ( - 1 ) I \eta_{G}=\eta_{C}\,(-1)^{I}
  6. η G = ( - 1 ) S + L + I \eta_{G}=(-1)^{S+L+I}\,
  7. η G = ( - 1 ) L + I \eta_{G}=(-1)^{L+I}\,

Gard_model.html

  1. v = n 1 n N G v=n_{1}\cdots n_{N_{G}}
  2. n 1 n N G n_{1}\cdots n_{N_{G}}
  3. d n i d t = ( k f ρ i N - k b n i ) ( 1 + j = 1 N G β i j n j N ) \frac{dn_{i}}{dt}=(k_{f}\rho_{i}N-k_{b}n_{i})\left(1+\sum_{j=1}^{N_{G}}\beta_{% ij}\frac{n_{j}}{N}\right)
  4. k f k_{f}
  5. k b k_{b}
  6. N = i = 1 N G n i N=\sum_{i=1}^{N_{G}}n_{i}

Garnir_relations.html

  1. λ = λ 1 + + λ r \lambda=\lambda_{1}+\cdots+\lambda_{r}
  2. S λ 1 × × S λ r S n S_{\lambda_{1}}\times\cdots\times S_{\lambda_{r}}\leqslant S_{n}
  3. S μ 1 × × S μ s S_{\mu_{1}}\times\cdots\times S_{\mu_{s}}
  4. μ = μ 1 + + μ s \mu=\mu_{1}+\cdots+\mu_{s}
  5. e T = σ S μ 1 × × S μ s sgn ( σ ) T σ . e_{T}=\sum_{\sigma\in S_{\mu_{1}}\times\cdots\times S_{\mu_{s}}}\operatorname{% sgn}(\sigma)T\sigma.
  6. S ( 1 + i π i ) = 0 S(1+\sum_{i}\pi_{i})=0
  7. g A , B g_{A,B}
  8. i sgn ( π i ) π i \sum_{i}\operatorname{sgn}(\pi_{i})\pi_{i}
  9. g A , B = 1 - ( 45 ) + ( 245 ) + ( 465 ) - ( 2465 ) + ( 25 ) ( 46 ) g_{A,B}=1-(45)+(245)+(465)-(2465)+(25)(46)

Gauss–Bonnet_gravity.html

  1. G = R 2 - 4 R μ ν R μ ν + R μ ν ρ σ R μ ν ρ σ G=R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}
  2. d D x - g G \int d^{D}x\sqrt{-g}\,G
  3. 1 8 π 2 d 4 x - g G = χ ( M ) \frac{1}{8\pi^{2}}\int d^{4}x\sqrt{-g}\,G=\chi(M)
  4. d D x - g f ( G ) \int d^{D}x\sqrt{-g}\,f\left(G\right)

Gebhart_factor.html

  1. B i j = Energy absorbed at A j originating as emission at A i Total radiation emitted from A i B_{ij}=\frac{\mbox{Energy absorbed at }~{}A_{j}\mbox{ originating as emission % at }~{}A_{i}}{\mbox{Total radiation emitted from }~{}A_{i}}
  2. B i j = Q i j ϵ i A i σ T i 4 B_{ij}=\frac{Q_{ij}}{\epsilon_{i}\cdot A_{i}\cdot\sigma\cdot T_{i}^{4}}
  3. B i j B_{ij}
  4. Q i j Q_{ij}
  5. ϵ \epsilon
  6. A A
  7. T T
  8. B i j B_{ij}
  9. q i = A i ϵ i σ T i 4 - j = 1 N s A j ϵ j σ B j i T j 4 q_{i}=A_{i}\cdot\epsilon_{i}\cdot\sigma\cdot T_{i}^{4}-\sum_{j=1}^{N_{s}}A_{j}% \cdot\epsilon_{j}\cdot\sigma\cdot B_{ji}\cdot T_{j}^{4}
  10. q i q_{i}
  11. ϵ i A i B i j = ϵ j A j B j i \epsilon_{i}\cdot A_{i}\cdot B_{ij}=\epsilon_{j}\cdot A_{j}\cdot B_{ji}
  12. q 1 - 2 = A 1 ϵ 1 B 12 σ ( T 1 4 - T 2 4 ) q_{1-2}=A_{1}\cdot\epsilon_{1}\cdot B_{12}\cdot\sigma\cdot(T_{1}^{4}-T_{2}^{4})
  13. B i j = F i j ϵ j + k = 1 N s ( ( 1 - ϵ k ) F i k B k j ) B_{ij}=F_{ij}\cdot\epsilon_{j}+\sum_{k=1}^{N_{s}}((1-\epsilon_{k})\cdot F_{ik}% \cdot B_{kj})
  14. F i j F_{ij}
  15. j = 1 N s ( B i j ) = 1 \sum_{j=1}^{N_{s}}(B_{ij})=1

Gell-Mann_and_Low_theorem.html

  1. | Ψ 0 |\Psi_{0}\rangle
  2. H 0 H_{0}
  3. E 0 E_{0}
  4. H = H 0 + g V H=H_{0}+gV
  5. g g
  6. V V
  7. H ϵ = H 0 + e - ϵ | t | g V H_{\epsilon}=H_{0}+e^{-\epsilon|t|}gV
  8. H H
  9. H 0 H_{0}
  10. ϵ 0 + \epsilon\rightarrow 0^{+}
  11. | t | |t|\rightarrow\infty
  12. U ϵ I U_{\epsilon I}
  13. ϵ 0 + \epsilon\rightarrow 0^{+}
  14. | Ψ ϵ ( ± ) = U ϵ I ( 0 , ± ) | Ψ 0 Ψ 0 | U ϵ I ( 0 , ± ) | Ψ 0 |\Psi^{(\pm)}_{\epsilon}\rangle=\frac{U_{\epsilon I}(0,\pm\infty)|\Psi_{0}% \rangle}{\langle\Psi_{0}|U_{\epsilon I}(0,\pm\infty)|\Psi_{0}\rangle}
  15. | Ψ ϵ ( ± ) |\Psi^{(\pm)}_{\epsilon}\rangle
  16. H H
  17. H ϵ H_{\epsilon}
  18. g = e ϵ θ g=e^{\epsilon\theta}
  19. i t 1 U ϵ ( t 1 , t 2 ) = H ϵ ( t 1 ) U ϵ ( t 1 , t 2 ) i\hbar\partial_{t_{1}}U_{\epsilon}(t_{1},t_{2})=H_{\epsilon}(t_{1})U_{\epsilon% }(t_{1},t_{2})
  20. U ϵ ( t , t ) = 1 U_{\epsilon}(t,t)=1
  21. U ϵ ( t 1 , t 2 ) = 1 + 1 i t 2 t 1 d t ( H 0 + e ϵ ( θ - | t | ) V ) U ϵ ( t , t 2 ) . U_{\epsilon}(t_{1},t_{2})=1+\frac{1}{i\hbar}\int_{t_{2}}^{t_{1}}dt^{\prime}(H_% {0}+e^{\epsilon(\theta-|t^{\prime}|)}V)U_{\epsilon}(t^{\prime},t_{2}).
  22. 0 t 2 t 1 0\geq t_{2}\geq t_{1}
  23. U ϵ ( t 1 , t 2 ) = 1 + 1 i θ + t 2 θ + t 1 d t ( H 0 + e ϵ t V ) U ϵ ( t - θ , t 2 ) . U_{\epsilon}(t_{1},t_{2})=1+\frac{1}{i\hbar}\int_{\theta+t_{2}}^{\theta+t_{1}}% dt^{\prime}(H_{0}+e^{\epsilon t^{\prime}}V)U_{\epsilon}(t^{\prime}-\theta,t_{2% }).
  24. θ U ϵ ( t 1 , t 2 ) = ϵ g g U ϵ ( t 1 , t 2 ) = t 1 U ϵ ( t 1 , t 2 ) + t 2 U ϵ ( t 1 , t 2 ) . \partial_{\theta}U_{\epsilon}(t_{1},t_{2})=\epsilon g\partial_{g}U_{\epsilon}(% t_{1},t_{2})=\partial_{t_{1}}U_{\epsilon}(t_{1},t_{2})+\partial_{t_{2}}U_{% \epsilon}(t_{1},t_{2}).
  25. - i t 1 U ϵ ( t 2 , t 1 ) = U ϵ ( t 2 , t 1 ) H ϵ ( t 1 ) -i\hbar\partial_{t_{1}}U_{\epsilon}(t_{2},t_{1})=U_{\epsilon}(t_{2},t_{1})H_{% \epsilon}(t_{1})
  26. i ϵ g g U ϵ ( t 1 , t 2 ) = H ϵ ( t 1 ) U ϵ ( t 1 , t 2 ) - U ϵ ( t 1 , t 2 ) H ϵ ( t 2 ) . i\hbar\epsilon g\partial_{g}U_{\epsilon}(t_{1},t_{2})=H_{\epsilon}(t_{1})U_{% \epsilon}(t_{1},t_{2})-U_{\epsilon}(t_{1},t_{2})H_{\epsilon}(t_{2}).
  27. H ϵ I , U ϵ I H_{\epsilon I},U_{\epsilon I}
  28. e i H 0 t 1 / e^{iH_{0}t_{1}/\hbar}
  29. e i H 0 t 2 / e^{iH_{0}t_{2}/\hbar}
  30. U ϵ I ( t 1 , t 2 ) = e i H 0 t 1 / U ϵ ( t 1 , t 2 ) e - i H 0 t 2 / . U_{\epsilon I}(t_{1},t_{2})=e^{iH_{0}t_{1}/\hbar}U_{\epsilon}(t_{1},t_{2})e^{-% iH_{0}t_{2}/\hbar}.
  31. t 2 t 1 0 t_{2}\geq t_{1}\geq 0
  32. t 1 , 2 t_{1,2}
  33. ( H ϵ , t = 0 - E 0 ± i ϵ g g ) U ϵ I ( 0 , ± ) | Ψ 0 = 0. \left(H_{\epsilon,t=0}-E_{0}\pm i\hbar\epsilon g\partial_{g}\right)U_{\epsilon I% }(0,\pm\infty)|\Psi_{0}\rangle=0.
  34. i ϵ g g ( U | Ψ 0 ) = ( H ϵ - E 0 ) U | Ψ 0 . i\hbar\epsilon g\partial_{g}\left(U|\Psi_{0}\rangle\right)=(H_{\epsilon}-E_{0}% )U|\Psi_{0}\rangle.
  35. Ψ ϵ \Psi_{\epsilon}
  36. g ( U | Ψ 0 ) \partial_{g}(U|\Psi_{0}\rangle)
  37. i ϵ g g | Ψ ϵ \displaystyle i\hbar\epsilon g\partial_{g}|\Psi_{\epsilon}\rangle
  38. E = E 0 + Ψ 0 | H ϵ - H 0 | Ψ ϵ E=E_{0}+\langle\Psi_{0}|H_{\epsilon}-H_{0}|\Psi_{\epsilon}\rangle
  39. ϵ 0 + \epsilon\rightarrow 0^{+}
  40. g g | Ψ ϵ g\partial_{g}|\Psi_{\epsilon}\rangle
  41. | Ψ ϵ |\Psi_{\epsilon}\rangle
  42. H H

Generality_of_algebra.html

  1. π - x 2 = sin x + 1 2 sin 2 x + 1 3 sin 3 x + \frac{\pi-x}{2}=\sin x+\frac{1}{2}\sin 2x+\frac{1}{3}\sin 3x+\cdots
  2. 0 < x < π 0<x<\pi
  3. 1 - r cos x 1 - 2 r cos x + r 2 = 1 + r cos x + r 2 cos 2 x + r 3 cos 3 x + \frac{1-r\cos x}{1-2r\cos x+r^{2}}=1+r\cos x+r^{2}\cos 2x+r^{3}\cos 3x+\cdots
  4. r = 1 r=1
  5. 0 = 1 2 + cos x + cos 2 x + cos 3 x + . 0=\frac{1}{2}+\cos x+\cos 2x+\cos 3x+\cdots.
  6. x x

Generalized_Clifford_algebra.html

  1. n n
  2. F F
  3. e j e k = ω j k e k e j e_{j}e_{k}=\omega_{jk}e_{k}e_{j}\,
  4. ω j k e l = e l ω j k \omega_{jk}e_{l}=e_{l}\omega_{jk}\,
  5. ω j k ω l m = ω l m ω j k \omega_{jk}\omega_{lm}=\omega_{lm}\omega_{jk}\,
  6. e j N j = 1 = ω j k N j = ω j k N k e_{j}^{N_{j}}=1=\omega_{jk}^{N_{j}}=\omega_{jk}^{N_{k}}\,
  7. j , k , l , m = 1 , , n ∀j,k,l,m=1,...,n
  8. ω j k = ω k j - 1 = e 2 π i ν k j / N k j \omega_{jk}=\omega_{kj}^{-1}=e^{2\pi i\nu_{kj}/N_{kj}}
  9. j , k = 1 , , n ∀j,k=1,...,n
  10. N k j = N_{kj}=
  11. ( N j , N k ) (N_{j},N_{k})
  12. F F
  13. n n
  14. p p
  15. N k = p N_{k}=p
  16. ν k j = 1 \nu_{kj}=1
  17. e j e k = ω e k e j e_{j}e_{k}=\omega\,e_{k}e_{j}\,
  18. ω e l = e l ω \omega e_{l}=e_{l}\omega\,
  19. e j p = 1 = ω p e_{j}^{p}=1=\omega^{p}\,
  20. ω = ω - 1 = e 2 π i / p \omega=\omega^{-1}=e^{2\pi i/p}
  21. p p
  22. ω = 1 , a n d p = 2 ω=−1,andp=2
  23. n × n n×n
  24. V = ( 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 ) V=\begin{pmatrix}0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ 0&0&\cdots&1&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&0&0&\cdots&0\end{pmatrix}
  25. U = ( 1 0 0 0 0 ω 0 0 0 0 ω 2 0 0 0 0 ω ( n - 1 ) ) U=\begin{pmatrix}1&0&0&\cdots&0\\ 0&\omega&0&\cdots&0\\ 0&0&\omega^{2}&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots&\omega^{(n-1)}\end{pmatrix}
  26. W = ( 1 1 1 1 1 ω ω 2 ω n - 1 1 ω 2 ( ω 2 ) 2 ω 2 ( n - 1 ) 1 ω n - 1 ω 2 ( n - 1 ) ω ( n - 1 ) 2 ) W=\begin{pmatrix}1&1&1&\cdots&1\\ 1&\omega&\omega^{2}&\cdots&\omega^{n-1}\\ 1&\omega^{2}&(\omega^{2})^{2}&\cdots&\omega^{2(n-1)}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&\omega^{n-1}&\omega^{2(n-1)}&\cdots&\omega^{(n-1)^{2}}\end{pmatrix}
  27. V U = ω U V VU=ωUV
  28. ω ω
  29. V V
  30. U U
  31. V V
  32. n = p = 2 n=p=2
  33. ω ω
  34. V = ( 0 1 1 0 ) V=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  35. U = ( 1 0 0 - 1 ) U=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  36. W = ( 1 1 1 - 1 ) W=\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}
  37. e 1 = ( 0 1 1 0 ) e_{1}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  38. e 2 = ( 0 - 1 1 0 ) e_{2}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  39. e 3 = ( 1 0 0 - 1 ) e_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  40. n = p = 4 n=p=4
  41. ω ω
  42. i i
  43. V = ( 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ) V=\begin{pmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\end{pmatrix}
  44. U = ( 1 0 0 0 0 i 0 0 0 0 - 1 0 0 0 0 - i ) U=\begin{pmatrix}1&0&0&0\\ 0&i&0&0\\ 0&0&-1&0\\ 0&0&0&-i\end{pmatrix}
  45. W = ( 1 1 1 1 1 i - 1 - i 1 - 1 1 - 1 1 - i - 1 i ) W=\begin{pmatrix}1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i\end{pmatrix}
  46. e < s u b > 1 , e 2 , e 3 e<sub>1,e_{2},e_{3}

Generalized_entropy_index.html

  1. α \alpha
  2. G E ( α ) = { 1 N α ( α - 1 ) i = 1 N [ ( y i y ¯ ) α - 1 ] , α 0 , 1 , 1 N i = 1 N y i y ¯ ln y i y ¯ , α = 1 , - 1 N i = 1 N ln y i y ¯ , α = 0. GE(\alpha)=\begin{cases}\frac{1}{N\alpha(\alpha-1)}\sum_{i=1}^{N}\left[\left(% \frac{y_{i}}{\overline{y}}\right)^{\alpha}-1\right],&\alpha\neq 0,1,\\ \frac{1}{N}\sum_{i=1}^{N}\frac{y_{i}}{\overline{y}}\ln\frac{y_{i}}{\overline{y% }},&\alpha=1,\\ -\frac{1}{N}\sum_{i=1}^{N}\ln\frac{y_{i}}{\overline{y}},&\alpha=0.\end{cases}
  3. y i y_{i}
  4. α \alpha
  5. ϵ = 1 - α \epsilon=1-\alpha
  6. A = 1 - e - G E A=1-e^{-GE}

Generalized_integer_gamma_distribution.html

  1. X X\!
  2. r r
  3. λ \lambda
  4. f X ( x ) = λ r Γ ( r ) e - λ x x r - 1 ( x > 0 ; λ , r > 0 ) f_{X}(x)=\frac{\lambda^{r}}{\Gamma(r)}\,e^{-\lambda x}x^{r-1}~{}~{}~{}~{}~{}~{% }(x>0;\,\lambda,r>0)
  5. X Γ ( r , λ ) . X\sim\Gamma(r,\lambda)\!.
  6. X j Γ ( r j , λ j ) X_{j}\sim\Gamma(r_{j},\lambda_{j})\!
  7. ( j = 1 , , p ) , (j=1,\dots,p),
  8. p p
  9. r j r_{j}
  10. λ j \lambda_{j}\!
  11. λ j \lambda_{j}
  12. Y = j = 1 p X j Y=\sum^{p}_{j=1}X_{j}
  13. p p
  14. r j r_{j}\!
  15. λ j \lambda_{j}\!
  16. ( j = 1 , , p ) (j=1,\dots,p)
  17. Y G I G ( r j , λ j ; p ) . Y\sim GIG(r_{j},\lambda_{j};p)\!.
  18. f Y GIG ( y | r 1 , , r p ; λ 1 , , λ p ) = K j = 1 p P j ( y ) e - λ j y , ( y > 0 ) f_{Y}^{\,\text{GIG}}(y|r_{1},\dots,r_{p};\lambda_{1},\dots,\lambda_{p})\,=\,K% \sum^{p}_{j=1}P_{j}(y)\,e^{-\lambda_{j}\,y}\,,~{}~{}~{}~{}(y>0)
  19. F Y GIG ( y | r 1 , , r j ; λ 1 , , λ p ) = 1 - K j = 1 p P j * ( y ) e - λ j y , ( y > 0 ) F_{Y}^{\,\text{GIG}}(y|r_{1},\dots,r_{j};\lambda_{1},\dots,\lambda_{p})\,=\,1-% K\sum^{p}_{j=1}P^{*}_{j}(y)\,e^{-\lambda_{j}\,y}\,,~{}~{}~{}~{}(y>0)
  20. K = j = 1 p λ j r j , P j ( y ) = k = 1 r j c j , k y k - 1 K=\prod^{p}_{j=1}\lambda_{j}^{r_{j}}~{},~{}~{}~{}~{}~{}P_{j}(y)=\sum^{r_{j}}_{% k=1}c_{j,k}\,y^{k-1}
  21. P j * ( y ) = k = 1 r j c j , k ( k - 1 ) ! i = 0 k - 1 y i i ! λ j k - i P^{*}_{j}(y)=\sum^{r_{j}}_{k=1}c_{j,k}\,(k-1)!\sum^{k-1}_{i=0}\frac{y^{i}}{i!% \,\lambda_{j}^{k-i}}
  22. c j , r j = 1 ( r j - 1 ) ! i = 1 p i j ( λ i - λ j ) - r i , j = 1 , , p , c_{j,r_{j}}=\frac{1}{(r_{j}-1)!}\,\mathop{\prod^{p}_{i=1}}_{i\neq j}(\lambda_{% i}-\lambda_{j})^{-r_{i}}~{},~{}~{}~{}~{}~{}~{}j=1,\ldots,p\,,
  23. c j , r j - k = 1 k i = 1 k ( r j - k + i - 1 ) ! ( r j - k - 1 ) ! R ( i , j , p ) c j , r j - ( k - i ) , ( k = 1 , , r j - 1 ; j = 1 , , p ) c_{j,r_{j}-k}=\frac{1}{k}\sum^{k}_{i=1}\frac{(r_{j}-k+i-1)!}{(r_{j}-k-1)!}\,R(% i,j,p)\,c_{j,r_{j}-(k-i)}\,,~{}~{}~{}~{}~{}~{}(k=1,\ldots,r_{j}-1;\,j=1,\ldots% ,p)
  24. R ( i , j , p ) = k = 1 p k j r k ( λ j - λ k ) - i ( i = 1 , , r j - 1 ) . R(i,j,p)=\mathop{\sum^{p}_{k=1}}_{k\neq j}r_{k}\left(\lambda_{j}-\lambda_{k}% \right)^{-i}~{}~{}~{}(i=1,\ldots,r_{j}-1)\,.
  25. p + 1 p+1
  26. Z = Y 1 + Y 2 , Z=Y_{1}+Y_{2}\!,
  27. Y 1 G I G ( r j , λ j ; p ) Y_{1}\sim GIG(r_{j},\lambda_{j};p)\!
  28. Y 2 Γ ( r , λ ) Y_{2}\sim\Gamma(r,\lambda)\!
  29. r r
  30. λ λ j \lambda\neq\lambda_{j}
  31. ( j = 1 , , p ) (j=1,\dots,p)
  32. Z Z\!
  33. f Z GNIG ( z | r 1 , , r p , r ; λ 1 , , λ p , λ ) = K λ r j = 1 p e - λ j z k = 1 r j { c j , k < m t p l > Γ ( k ) Γ ( k + r ) z k + r - 1 F 1 1 ( r , k + r , - ( λ - λ j ) z ) } , ( z > 0 ) \begin{array}[]{l}\displaystyle f_{Z}^{\,\text{GNIG}}(z|r_{1},\dots,r_{p},r;\,% \lambda_{1},\dots,\lambda_{p},\lambda)=\\ \displaystyle\quad\quad\quad K\lambda^{r}\sum\limits_{j=1}^{p}{e^{-\lambda_{j}% z}}\sum\limits_{k=1}^{r_{j}}{\left\{{c_{j,k}\frac{<}{m}tpl>{{\Gamma(k)}}{{% \Gamma(k+r)}}z^{k+r-1}{}_{1}F_{1}(r,k+r,-(\lambda-\lambda_{j})z)}\right\}}{\rm% ,}~{}~{}~{}~{}(z>0)\end{array}
  34. F Z GNIG ( z | r 1 , , r p , r ; λ 1 , , λ p , λ ) = λ r z r < m t p l > Γ ( r + 1 ) F 1 1 ( r , r + 1 , - λ z ) - K λ r j = 1 p e - λ j z k = 1 r j c j , k * i = 0 k - 1 z r + i λ j i Γ ( r + 1 + i ) F 1 1 ( r , r + 1 + i , - ( λ - λ j ) z ) ( z > 0 ) \begin{array}[]{l}\displaystyle F_{Z}^{\,\text{GNIG}}(z|r_{1},\ldots,r_{p},r;% \,\lambda_{1},\ldots,\lambda_{p},\lambda)=\frac{\lambda^{r}\,{z^{r}}}{<}mtpl>{% {\Gamma(r+1)}}{}_{1}F_{1}(r,r+1,-\lambda z)\\ \quad\quad\displaystyle-K\lambda^{r}\sum\limits_{j=1}^{p}{e^{-\lambda_{j}z}}% \sum\limits_{k=1}^{r_{j}}{c_{j,k}^{*}}\sum\limits_{i=0}^{k-1}{\frac{{z^{r+i}% \lambda_{j}^{i}}}{{\Gamma(r+1+i)}}}{}_{1}F_{1}(r,r+1+i,-(\lambda-\lambda_{j})z% )~{}~{}~{}~{}(z>0)\end{array}
  35. c j , k * = c j , k < m t p l > λ j k Γ ( k ) c_{j,k}^{*}=\frac{{c_{j,k}}}{<}mtpl>{{\lambda_{j}^{k}}}\Gamma(k)
  36. c j , k c_{j,k}
  37. F 1 1 ( a , b ; z ) {}_{1}F_{1}(a,b;z)

Generalized_inversive_congruential_pseudorandom_numbers.html

  1. m = p 1 , p r m=p_{1},\dots p_{r}
  2. p 1 , , p r 5 p_{1},\dots,p_{r}\geq 5
  3. m = { 0 , 1 , , m - 1 } \mathbb{Z}_{m}=\{0,1,...,m-1\}
  4. a , b m a,b\in\mathbb{Z}_{m}
  5. ( y n ) n 0 (y_{n})_{n\geqslant 0}
  6. m \mathbb{Z}_{m}
  7. y 0 = seed y_{0}={\rm seed}
  8. y n + 1 a y n φ ( m ) - 1 + b ( mod m ) , n 0 y_{n+1}\equiv ay_{n}^{\varphi(m)-1}+b\;\;(\mathop{{\rm mod}}m)\,\text{, }n\geqslant 0
  9. φ ( m ) = ( p 1 - 1 ) ( p r - 1 ) \varphi(m)=(p_{1}-1)\dots(p_{r}-1)
  10. 3 × 5 a = 2 , b = 3 3\times 5\,a=2,b=3
  11. y 0 = 1 y_{0}=1
  12. φ ( m ) = 2 × 4 = 8 \varphi(m)=2\times 4=8\,
  13. ( y n ) n 0 = ( 1 , 5 , 13 , 2 , 4 , 7 , 1 , ) (y_{n})_{n\geqslant 0}=(1,5,13,2,4,7,1,\dots)
  14. 1 i r 1\leq i\leq r
  15. p i = { 0 , 1 , , p i - 1 } , m i = m / p i \mathbb{Z}_{p_{i}}=\{0,1,\dots,p_{i}-1\},m_{i}=m/p_{i}
  16. a i , b i p i a_{i},b_{i}\in\mathbb{Z}_{p_{i}}
  17. a m i 2 a i ( mod p i ) and b m i b i ( mod p i ) . a\equiv m_{i}^{2}a_{i}\;\;(\mathop{{\rm mod}}p_{i})\;\,\text{and }\;b\equiv m_% {i}b_{i}\;\;(\mathop{{\rm mod}}p_{i})\,\text{. }
  18. ( y n ) n 0 (y_{n})_{n\geqslant 0}
  19. p i \mathbb{Z}_{p_{i}}
  20. y n + 1 ( i ) a i ( y n ( i ) ) p i - 2 + b i ( mod p i ) , n 0 where y 0 m i ( y 0 ( i ) ) ( mod p i ) is assumed. y_{n+1}^{(i)}\equiv a_{i}(y_{n}^{(i)})^{p_{i}-2}+b_{i}\;\;(\mathop{{\rm mod}}p% _{i})\;\,\text{, }n\geqslant 0\;\text{where}\;y_{0}\equiv m_{i}(y_{0}^{(i)})\;% \;(\mathop{{\rm mod}}p_{i})\;\,\text{is assumed. }
  21. ( y n ( i ) ) n 0 (y_{n}^{(i)})_{n\geqslant 0}
  22. 1 i r 1\leq i\leq r
  23. y n m 1 y n ( 1 ) + m 2 y n ( 2 ) + + m r y n ( r ) ( mod m ) . y_{n}\equiv m_{1}y_{n}^{(1)}+m_{2}y_{n}^{(2)}+\dots+m_{r}y_{n}^{(r)}\;\;(% \mathop{{\rm mod}}m).
  24. p 1 , , p r \mathbb{Z}_{p_{1}},\dots,\mathbb{Z}_{p_{r}}
  25. m . \mathbb{Z}_{m}.
  26. m i 0 ( mod p j ) , for i j , m_{i}\equiv 0\;\;(\mathop{{\rm mod}}p_{j}),\;\text{for}\;i\neq j,
  27. y n m 1 y n ( 1 ) + m 2 y n ( 2 ) + + m r y n ( r ) ( mod m ) y_{n}\equiv m_{1}y_{n}^{(1)}+m_{2}y_{n}^{(2)}+\dots+m_{r}y_{n}^{(r)}\;\;(% \mathop{{\rm mod}}m)
  28. y n m i ( y n ( i ) ) ( mod p i ) y_{n}\equiv m_{i}(y_{n}^{(i)})\;\;(\mathop{{\rm mod}}p_{i})
  29. 1 i r 1\leq i\leq r
  30. n 0 n\geqslant 0
  31. y 0 m i ( y 0 ( i ) ) ( mod p i ) y_{0}\equiv m_{i}(y_{0}^{(i)})\;\;(\mathop{{\rm mod}}p_{i})
  32. 1 i r 1\leq i\leq r
  33. 1 i r 1\leq i\leq r
  34. y n m i ( y n ( i ) ) ( mod p i ) y_{n}\equiv m_{i}(y_{n}^{(i)})\;\;(\mathop{{\rm mod}}p_{i})
  35. n 0 n\geqslant 0
  36. y n + 1 a y n φ ( m ) - 1 + b m i ( a i m i φ ( m ) ( y n ( i ) ) φ ( m ) - 1 + b i ) m i ( a i ( y n ( i ) ) p i - 2 + b i ) m i ( y n + 1 ( i ) ) ( mod p i ) y_{n+1}\equiv ay_{n}^{\varphi(m)-1}+b\equiv m_{i}(a_{i}m_{i}^{\varphi(m)}(y_{n% }^{(i)})^{\varphi(m)-1}+b_{i})\equiv m_{i}(a_{i}(y_{n}^{(i)})^{p_{i}-2}+b_{i})% \equiv m_{i}(y_{n+1}^{(i)})\;\;(\mathop{{\rm mod}}p_{i})
  37. D m s = D m ( x 0 , , x m - 1 ) D_{m}^{s}=D_{m}(x_{0},\dots,x_{m}-1)
  38. x n = ( x n , x n + 1 , , x n + s - 1 ) x_{n}=(x_{n},x_{n}+1,\dots,x_{n}+s-1)
  39. [ 0 , 1 ) s [0,1)^{s}
  40. s 2 s\geq 2
  41. s 2 s\geq 2
  42. D m s D_{m}^{s}
  43. D m D_{m}
  44. s {}^{s}
  45. × ×
  46. ( 2 π (\frac{2}{\pi}
  47. × ×
  48. log m + 7 5 ) s \log m+\frac{7}{5})^{s}
  49. × ×
  50. i = 1 r ( 2 s - 2 + s ( p i ) - 1 / 2 ) + s m - 1 \textstyle\prod_{i=1}^{r}(2s-2+s(p_{i})^{-1/2})+s_{m}^{-1}
  51. D m D_{m}
  52. s {}^{s}
  53. 1 2 ( π + 2 ) \frac{1}{2(\pi+2)}
  54. × ×
  55. m - 1 / 2 m^{-1/2}
  56. × ×
  57. i = 1 r ( p i - 3 p i - 1 ) 1 / 2 \textstyle\prod_{i=1}^{r}(\frac{p_{i}-3}{p_{i}-1})^{1/2}
  58. : :≥
  59. D m ( s ) = O ( m - 1 / 2 ( log m ) s ) D_{m}^{(s)}=O(m^{-1/2}(\log m)^{s})
  60. D m ( s ) D_{m}^{(s)}
  61. m - 1 / 2 m^{-1/2}
  62. s 2 s\geq 2
  63. ( log m ) / log log m (\log m)/\log\log m
  64. i = 1 r ( 2 s - 2 + s ( p i ) - 1 / 2 ) = O ( m ϵ ) \textstyle\prod_{i=1}^{r}(2s-2+s(p_{i})^{-1/2})=O{(m^{\epsilon})}
  65. ϵ > 0 \epsilon>0
  66. D m s = O ( m - 1 / 2 + ϵ ) D_{m}^{s}=O(m^{-1/2+\epsilon})
  67. ϵ > 0 \epsilon>0
  68. i = 1 r ( ( p i - 3 ) / ( p i - 1 ) ) 1 / 2 2 - r / 2 \textstyle\prod_{i=1}^{r}((p_{i}-3)/(p_{i}-1))^{1/2}\geqslant 2^{-r/2}
  69. m - 1 / 2 - ϵ m^{-1/2-\epsilon}
  70. ϵ > 0 \epsilon>0
  71. m - 1 / 2 ( log log m ) 1 / 2 m^{-1/2}(\log\log m)^{1/2}

Generalized_minimum-distance_decoding.html

  1. u , v n u,v\in\sum^{n}
  2. Δ ( u , v ) \Delta(u,v)
  3. C n C\subseteq\sum^{n}
  4. d = min Δ ( c 1 , c 2 ) d=\min{\Delta(c_{1},c_{2})}
  5. c 1 c 2 C c_{1}\neq c_{2}\in C
  6. m = ( m 1 , , m K ) [ Q ] K m=(m_{1},\ldots,m_{K})\in[Q]^{K}
  7. C out = [ Q ] K [ Q ] N , C in : [ q ] k [ q ] n C\text{out}=[Q]^{K}\rightarrow[Q]^{N},C\text{in}:[q]^{k}\rightarrow[q]^{n}
  8. D D
  9. d d
  10. C out C in ( m ) = ( C in ( C out ( m ) 1 ) , , C in ( C out ( m ) N ) ) C\text{out}\circ C\text{in}(m)=(C\text{in}(C\text{out}(m)_{1}),\ldots,C\text{% in}(C\text{out}(m)_{N}))
  11. C out ( m ) = ( ( C out ( m ) 1 , , ( m ) N ) ) C\text{out}(m)=((C\text{out}(m)_{1},\ldots,(m)_{N}))
  12. C out C\text{out}
  13. K = O ( log N ) K=O(\log{N})
  14. N N
  15. D M L D : n C D_{MLD}:\sum^{n}\rightarrow C
  16. y n y\in\sum_{n}
  17. D M L D ( y ) = arg min c C Δ ( c , y ) D_{MLD}(y)=\arg\min_{c\in C}\Delta(c,y)
  18. Pr [ ] \Pr[\bullet]
  19. S S
  20. S S
  21. Pr [ A ] 0 \Pr[A]\geq 0
  22. A A
  23. Pr [ S ] = 1 \Pr[S]=1
  24. Pr [ A B ] = Pr [ A ] + Pr [ B ] \Pr[A\cup B]=\Pr[A]+\Pr[B]
  25. A A
  26. B B
  27. X X
  28. 𝔼 = x Pr [ X = x ] \mathbb{E}=\sum_{x}\Pr[X=x]
  29. 𝐲 = ( y 1 , , y N ) [ q n ] N \mathbf{y}=(y_{1},\ldots,y_{N})\in[q^{n}]^{N}
  30. C out C\text{out}
  31. 𝐲 = ( y 1 , , y N ) [ q n ] N \mathbf{y}=(y_{1},\dots,y_{N})\in[q^{n}]^{N}
  32. 1 i N 1\leq i\leq N
  33. y i = M L D C in ( y i ) y_{i}^{\prime}=MLD_{C\text{in}}(y_{i})
  34. ω i = min ( Δ ( C in ( y i ) , y i ) , d 2 ) \omega_{i}=\min(\Delta(C\text{in}(y_{i}^{\prime}),y_{i}),{d\over 2})
  35. 1 i N 1\leq i\leq N
  36. 2 ω i d 2\omega_{i}\over d
  37. y i ′′ y_{i}^{\prime\prime}\leftarrow
  38. y i ′′ = y i y_{i}^{\prime\prime}=y_{i}^{\prime}
  39. C out C\text{out}
  40. 𝐲 ′′ = ( y 1 ′′ , , y N ′′ ) \mathbf{y}^{\prime\prime}=(y_{1}^{\prime\prime},\ldots,y_{N}^{\prime\prime})
  41. 𝐜 = ( c 1 , , c N ) C out C in [ q n ] N \mathbf{c}=(c_{1},\ldots,c_{N})\in C\text{out}\circ{C\text{in}}\subseteq[q^{n}% ]^{N}
  42. Δ ( 𝐜 , 𝐲 ) < D d 2 \Delta(\mathbf{c},\mathbf{y})<\frac{Dd}{2}
  43. 𝐜 \mathbf{c}
  44. D d 4 Dd\over 4
  45. 𝐲 ′′ \mathbf{y^{\prime\prime}}
  46. e e^{\prime}
  47. s s^{\prime}
  48. 𝐜 \mathbf{c}
  49. 𝔼 [ 2 e + s ] < D \mathbb{E}[2e^{\prime}+s^{\prime}]<D
  50. 2 e + s < D 2e^{\prime}+s^{\prime}<D
  51. 𝐜 \mathbf{c}
  52. 1 i N 1\leq i\leq N
  53. e i = Δ ( y i , c i ) e_{i}=\Delta(y_{i},c_{i})
  54. i = 1 N e i < D d 2 ( 1 ) \sum_{i=1}^{N}e_{i}<\frac{Dd}{2}\qquad\qquad(1)
  55. 1 i N 1\leq i\leq N
  56. X = i ? 1 X{{}_{i}^{?}}=1
  57. y i ′′ = ? , y_{i}^{\prime\prime}=?,
  58. X = i e 1 X{{}_{i}^{e}}=1
  59. C in ( y i ′′ ) c i C\text{in}(y_{i}^{\prime\prime})\neq c_{i}
  60. y i ′′ ? . y_{i}^{\prime\prime}\neq?.
  61. 1 i N 1\leq i\leq N
  62. 𝔼 [ 2 X + i e X ] i ? 2 e i d ( 2 ) \mathbb{E}[2X{{}_{i}^{e}+X{{}_{i}^{?}}}]\leq{{2e_{i}}\over d}\qquad\qquad(2)
  63. e = i X i e e^{\prime}=\sum_{i}X{{}_{i}^{e}}
  64. s = i X i ? s^{\prime}=\sum_{i}X{{}_{i}^{?}}
  65. 𝔼 [ 2 e + s ] 2 d i e i < D \mathbb{E}[2e^{\prime}+s^{\prime}]\leq{2\over d}\sum_{i}e_{i}<D
  66. i t h i^{\prime}th
  67. i t h i^{\prime}th
  68. ( c i = C in ( y i ) ) (c_{i}=C\text{in}(y_{i}^{\prime}))
  69. y i ′′ = ? y_{i}^{\prime\prime}=?
  70. X i e = 0 X_{i}^{e}=0
  71. Pr [ y i ′′ = ? ] = 2 ω i d \Pr[y_{i}^{\prime\prime}=?]={2\omega_{i}\over d}
  72. 𝔼 [ X i ? ] = Pr [ X i ? = 1 ] = 2 ω i d \mathbb{E}[X_{i}^{?}]=\Pr[X_{i}^{?}=1]={2\omega_{i}\over d}
  73. 𝔼 [ X i e ] = Pr [ X i e = 1 ] = 0 \mathbb{E}[X_{i}^{e}]=\Pr[X_{i}^{e}=1]=0
  74. ω i = min ( Δ ( C in ( y i ) , y i ) , d 2 ) Δ ( C in ( y i ) , y i ) = Δ ( c i , y i ) = e i \omega_{i}=\min(\Delta(C\text{in}(y_{i}^{\prime}),y_{i}),{d\over 2})\leq\Delta% (C\text{in}(y_{i}^{\prime}),y_{i})=\Delta(c_{i},y_{i})=e_{i}
  75. ( c i C in ( y i ) ) (c_{i}\neq C\text{in}(y_{i}^{\prime}))
  76. 𝔼 [ X i ? ] = 2 ω i d \mathbb{E}[X_{i}^{?}]={2\omega_{i}\over d}
  77. 𝔼 [ X i e ] = Pr [ X i e = 1 ] = 1 - 2 ω i d . \mathbb{E}[X_{i}^{e}]=\Pr[X_{i}^{e}=1]=1-{2\omega_{i}\over d}.
  78. c i C in ( y i ) c_{i}\neq C\text{in}(y_{i}^{\prime})
  79. e i + ω i d e_{i}+\omega_{i}\geq d
  80. ( ω i = Δ ( C in ( y i ) , y i ) < d 2 ) (\omega_{i}=\Delta(C\text{in}(y_{i}^{\prime}),y_{i})<{d\over 2})
  81. 𝔼 [ 2 X i e + X i ? ] = 2 - 2 ω i d 2 e i d . \mathbb{E}[2X_{i}^{e}+X_{i}^{?}]=2-{2\omega_{i}\over d}\leq{2e_{i}\over d}.
  82. C out C in C\text{out}\circ C\text{in}
  83. i i
  84. i i
  85. 𝐲 = ( y 1 , , y N ) [ q n ] N \mathbf{y}=(y_{1},\ldots,y_{N})\in[q^{n}]^{N}
  86. θ [ 0 , 1 ] \theta\in[0,1]
  87. 1 i N 1\leq i\leq N
  88. y i = M L D C in ( y i ) y_{i}^{\prime}=MLD_{C\text{in}}(y_{i})
  89. ω i = min ( Δ ( C in ( y i ) , y i ) , d 2 ) \omega_{i}=\min(\Delta(C\text{in}(y_{i}^{\prime}),y_{i}),{d\over 2})
  90. θ \theta
  91. y i ′′ y_{i}^{\prime\prime}\leftarrow
  92. y i ′′ = y i y_{i}^{\prime\prime}=y_{i}^{\prime}
  93. C out C\text{out}
  94. 𝐲 ′′ = ( y 1 ′′ , , y N ′′ ) \mathbf{y}^{\prime\prime}=(y_{1}^{\prime\prime},\ldots,y_{N}^{\prime\prime})
  95. Pr [ y i ′′ = ? ] = 2 ω i d . \Pr[y_{i}^{\prime\prime}=?]={2\omega_{i}\over d}.
  96. Pr [ y i ′′ = ? ] = Pr [ θ [ 0 , 2 ω i d ] ] = 2 ω i d . \Pr[y_{i}^{\prime\prime}=?]=\Pr[\theta\in[0,{2\omega_{i}\over d}]]={2\omega_{i% }\over d}.
  97. θ \theta
  98. 𝔼 [ 2 e + s ] \mathbb{E}[2e^{\prime}+s^{\prime}]
  99. [ 0 , 1 ] [0,1]
  100. Q = { 0 , 1 } { 2 ω 1 d , , 2 ω N d } Q=\{0,1\}\cup\{{2\omega_{1}\over d},\ldots,{2\omega_{N}\over d}\}
  101. i , ω i = min ( Δ ( 𝐲 𝐢 , 𝐲 𝐢 ) , d 2 ) i,\omega_{i}=\min(\Delta(\mathbf{y_{i}^{\prime}},\mathbf{y_{i}}),{d\over 2})
  102. Q = { 0 , 1 } { q 1 , , q m } Q=\{0,1\}\cup\{q_{1},\ldots,q_{m}\}
  103. q 1 < q 2 < < q m q_{1}<q_{2}<\cdots<q_{m}
  104. m d 2 m\leq\left\lfloor\frac{d}{2}\right\rfloor
  105. θ [ q i , q i + 1 ] \theta\in[q_{i},q_{i+1}]
  106. 𝐲 ′′ \mathbf{y^{\prime\prime}}
  107. θ Q \theta\in Q
  108. 𝐲 = ( y 1 , , y N ) [ q n ] N \mathbf{y}=(y_{1},\ldots,y_{N})\in[q^{n}]^{N}
  109. θ Q \theta\in Q
  110. y i = M L D C in ( y i ) y_{i}^{\prime}=MLD_{C\text{in}}(y_{i})
  111. 1 i N 1\leq i\leq N
  112. ω i = min ( Δ ( C in ( y i ) , y i ) , d 2 ) \omega_{i}=\min(\Delta(C\text{in}(y_{i}^{\prime}),y_{i}),{d\over 2})
  113. 1 i N 1\leq i\leq N
  114. θ \theta
  115. y i ′′ y_{i}^{\prime\prime}\leftarrow
  116. y i ′′ = y i y_{i}^{\prime\prime}=y_{i}^{\prime}
  117. C out C\text{out}
  118. 𝐲 ′′ = ( y 1 ′′ , , y N ′′ ) \mathbf{y^{\prime\prime}}=(y_{1}^{\prime\prime},\ldots,y_{N}^{\prime\prime})
  119. c θ c_{\theta}
  120. C out C in C\text{out}\circ C\text{in}
  121. c θ c_{\theta}
  122. 𝐲 \mathbf{y}
  123. < d D / 2 <dD/2
  124. O ( d ) O(d)
  125. O ( N Q n O ( 1 ) + N T out ) O(NQn^{O(1)}+NT\text{out})
  126. T out T\text{out}

Generic_Stream_Encapsulation.html

  1. y = x 32 + x 26 + x 23 + x 22 + x 16 + x 12 + x 11 + x 10 + x 8 + x 7 + x 5 + x 4 + x 2 + x 1 + x 0 y=x^{32}+x^{26}+x^{23}+x^{22}+x^{16}+x^{12}+x^{11}+x^{10}+x^{8}+x^{7}+x^{5}+x^% {4}+x^{2}+x^{1}+x^{0}

Genetic_algebra.html

  1. ( x 2 ) 2 = w ( x ) 2 x 2 (x^{2})^{2}=w(x)^{2}x^{2}
  2. e = a 2 e=a^{2}
  3. w ( a ) = 1 w(a)=1
  4. B = K e U e Z e B=Ke\oplus U_{e}\oplus Z_{e}
  5. U e = { a ker w : e a = a / 2 } U_{e}=\{a\in\ker w:ea=a/2\}
  6. Z e = { a ker w : e a = 0 } Z_{e}=\{a\in\ker w:ea=0\}
  7. U e 2 = 0 U_{e}^{2}=0
  8. c 1 , , c n c_{1},\ldots,c_{n}
  9. 1 + c 1 + + c n = 0 1+c_{1}+\cdots+c_{n}=0
  10. x n + c 1 w ( x ) x n - 1 + + c n w ( x ) n x^{n}+c_{1}w(x)x^{n-1}+\cdots+c_{n}w(x)^{n}
  11. a n + c 1 w ( a ) a n - 1 + + c n w ( a ) n = 0 a^{n}+c_{1}w(a)a^{n-1}+\cdots+c_{n}w(a)^{n}=0
  12. a B a\in B
  13. a k a^{k}
  14. ( a k - 1 ) a (a^{k-1})a

Genic_capture.html

  1. a a
  2. b b
  3. T = a + b C T=a+bC
  4. C C
  5. C C
  6. a a
  7. b b
  8. T T
  9. G T G a + b ¯ 2 G C + C ¯ 2 G b G_{T}\approx G_{a}+\bar{b}^{2}G_{C}+\bar{C}^{2}G_{b}
  10. G T G_{T}
  11. T T
  12. a a
  13. b b
  14. C C
  15. T T
  16. b ¯ 2 G C \bar{b}^{2}G_{C}
  17. G T G_{T}

Geometric_feature_learning.html

  1. x i = x i - 1 + σ i - 1 d i [ cos ( θ i - 1 + ϕ i ) sin ( θ i - 1 + ϕ i ) ] \textstyle\ x_{i}=x_{i-1}+\sigma_{i-1}d_{i}\begin{bmatrix}\cos(\theta_{i-1}+% \phi_{i})\\ \sin(\theta_{i-1}+\phi_{i})\end{bmatrix}
  2. θ i = θ i - 1 + Δ θ i \textstyle\ \theta_{i}=\theta_{i-1}+\Delta\theta_{i}
  3. σ i = σ i - 1 Δ σ i \textstyle\ \sigma_{i}=\sigma_{i-1}\Delta\sigma_{i}
  4. θ \textstyle\theta
  5. σ \textstyle\sigma
  6. f m a x \textstyle\ f_{max}
  7. I m a x = m a x 𝑓 m a x 𝐶 I ( C , F f ) \textstyle\ I_{max}=\underset{f}{max}\underset{C}{max}I(C,F_{f})
  8. I ( C , F f ) = - 𝐶 F f B E L ( F f , C ) log B E L ( C , F f ) B E L ( F f ) B E L ( C ) \textstyle\ I(C,F_{f})=-\underset{C}{\sum}\underset{F_{f}}{\sum}BEL(F_{f},C)% \log\frac{BEL(C,F_{f})}{BEL(F_{f})BEL(C)}
  9. f m a x \textstyle\ f_{max}
  10. f f m a x \textstyle\ f_{f_{max}}
  11. f f ( p ) ( I ) = m a x x I f f ( p ) ( x ) \textstyle\ f_{f_{(p)}}(I)=\underset{x\in I}{max}f_{f_{(p)}}(x)
  12. f f ( p ) ( x ) \textstyle f_{f_{(p)}}(x)
  13. f f ( p ) ( I ) = m a x { 0 , f ( p ) T ) f ( x ) f ( p ) f ( x ) } \textstyle f_{f_{(p)}}(I)=max\left\{0,\frac{f(p)^{T})f(x)}{\left\|f(p)\right\|% \left\|f(x)\right\|}\right\}
  14. K S D a , b ( X ) = m a x 𝛼 | c d f ( α | a ) - c d f ( α | b ) | \textstyle KSD_{a,b}(X)=\underset{\alpha}{max}\left|cdf(\alpha|a)-cdf(\alpha|b% )\right|
  15. f T ( X ) \textstyle f_{T}(X)
  16. F t = ϕ \textstyle F_{t}=\phi
  17. t T \textstyle t\in T
  18. t 1 \textstyle t_{1}
  19. t k \textstyle t_{k}
  20. i e w i t \textstyle\underset{i\in e}{\sum}w_{i}^{t}
  21. θ t \textstyle\theta_{t}
  22. w i t \textstyle w_{i}^{t}
  23. i e w i t > θ t \textstyle\underset{i\in e}{\sum}w_{i}^{t}>\theta_{t}
  24. i e w i t θ t \textstyle\underset{i\in e}{\sum}w_{i}^{t}\leq\theta_{t}
  25. ( x i , y i ) \textstyle(x_{i},y_{i})
  26. x i \textstyle x_{i}
  27. x R N \textstyle x\in R^{N}
  28. y i \textstyle y_{i}
  29. x i \textstyle x_{i}
  30. f ( x ) = s g n ( i = 1 l y i α i k ( x , x i ) + b ) = { 1 , p o s i t i v e i n p u t s - 1 , n e g a t i v e i n p u t s \textstyle f(x)=sgn\left(\sum_{i=1}^{l}y_{i}\alpha_{i}\cdot k(x,x_{i})+b\right% )=\left\{\begin{matrix}1,positive\;inputs\\ -1,negative\;inputs\end{matrix}\right.
  31. k ( x , x i ) = ϕ ( x ) ϕ ( x i ) \textstyle k(x,x_{i})=\phi(x)\cdot\phi(x_{i})

Geometric_stable_distribution.html

  1. φ ( t ; α , β , λ , μ ) = [ 1 + λ α | t | α ω - i μ t ] - 1 \varphi(t;\alpha,\beta,\lambda,\mu)=[1+\lambda^{\alpha}|t|^{\alpha}\omega-i\mu t% ]^{-1}
  2. ω = { 1 - i tan π α 2 β sign ( t ) if α 1 1 + i 2 π β log | t | sign ( t ) if α = 1 \omega=\begin{cases}1-i\tan\tfrac{\pi\alpha}{2}\beta\,\operatorname{sign}(t)&% \,\text{if }\alpha\neq 1\\ 1+i\tfrac{2}{\pi}\beta\log|t|\operatorname{sign}(t)&\,\text{if }\alpha=1\end{cases}
  3. α \alpha
  4. α \alpha
  5. β \beta
  6. β \beta
  7. β \beta
  8. β \beta
  9. φ ( t ; α , 0 , λ , μ ) = [ 1 + λ α | t | α - i μ t ] - 1 \varphi(t;\alpha,0,\lambda,\mu)=[1+\lambda^{\alpha}|t|^{\alpha}-i\mu t]^{-1}
  10. μ = 0 \mu=0
  11. β = 1 \beta=1
  12. α < 1 \alpha<1
  13. 0 < μ < 1 0<\mu<1
  14. β \beta
  15. λ > 0 \lambda>0
  16. μ \mu
  17. α \alpha
  18. β \beta
  19. μ \mu
  20. α \alpha
  21. f ( x | 0 , λ ) = 1 2 λ exp ( - | x | λ ) f(x|0,\lambda)=\frac{1}{2\lambda}\exp\left(-\frac{|x|}{\lambda}\right)\,\!
  22. 2 λ 2 2\lambda^{2}
  23. α < 2 \alpha<2
  24. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  25. Y = a n ( X 1 + X 2 + + X n ) + b n Y=a_{n}(X_{1}+X_{2}+\cdots+X_{n})+b_{n}
  26. X i X_{i}
  27. a n a_{n}
  28. b n b_{n}
  29. X 1 , X 2 , X_{1},X_{2},\dots
  30. Y = a N p ( X 1 + X 2 + + X N p ) + b N p Y=a_{N_{p}}(X_{1}+X_{2}+\cdots+X_{N_{p}})+b_{N_{p}}
  31. X i X_{i}
  32. a N p a_{N_{p}}
  33. b N p b_{N_{p}}
  34. N p N_{p}
  35. X i X_{i}
  36. Pr ( N p = n ) = ( 1 - p ) n - 1 p . \Pr(N_{p}=n)=(1-p)^{n-1}\,p\,.
  37. Y = a ( X 1 + X 2 + + X N p ) Y=a(X_{1}+X_{2}+\cdots+X_{N_{p}})
  38. X i X_{i}
  39. Φ ( t ; α , β , λ , μ ) = exp [ i t μ - | λ t | α ( 1 - i β sign ( t ) Ω ) ] , \Phi(t;\alpha,\beta,\lambda,\mu)=\exp\left[~{}it\mu\!-\!|\lambda t|^{\alpha}\,% (1\!-\!i\beta\operatorname{sign}(t)\Omega)~{}\right],
  40. Ω = { tan π α 2 if α 1 , - 2 π log | t | if α = 1. \Omega=\begin{cases}\tan\tfrac{\pi\alpha}{2}&\,\text{if }\alpha\neq 1,\\ -\tfrac{2}{\pi}\log|t|&\,\text{if }\alpha=1.\end{cases}
  41. φ ( t ; α , β , λ , μ ) = [ 1 - log ( Φ ( t ; α , β , λ , μ ) ) ] - 1 . \varphi(t;\alpha,\beta,\lambda,\mu)=[1-\log(\Phi(t;\alpha,\beta,\lambda,\mu))]% ^{-1}.

Geometric_tomography.html

  1. E n E^{n}

Geophysical_fluid_dynamics.html

  1. u u
  2. τ = μ d u d x , \tau=\mu\frac{du}{dx},
  3. ρ ( 𝐯 t Eulerian acceleration + 𝐯 𝐯 Advection ) Inertia (per volume) = - p Pressure gradient + μ 2 𝐯 Viscosity Divergence of stress + 𝐟 Other body forces . \overbrace{\rho\Big(\underbrace{\frac{\partial\mathbf{v}}{\partial t}}_{\begin% {smallmatrix}\,\text{Eulerian}\\ \,\text{acceleration}\end{smallmatrix}}+\underbrace{\mathbf{v}\cdot\nabla% \mathbf{v}}_{\begin{smallmatrix}\,\text{Advection}\end{smallmatrix}}\Big)}^{\,% \text{Inertia (per volume)}}=\overbrace{\underbrace{-\nabla p}_{\begin{% smallmatrix}\,\text{Pressure}\\ \,\text{gradient}\end{smallmatrix}}+\underbrace{\mu\nabla^{2}\mathbf{v}}_{\,% \text{Viscosity}}}^{\,\text{Divergence of stress}}+\underbrace{\mathbf{f}}_{% \begin{smallmatrix}\,\text{Other}\\ \,\text{body}\\ \,\text{forces}\end{smallmatrix}}.
  4. g g

Geopotential_model.html

  1. 𝐅 = - G m 1 m 2 r 2 𝐫 ^ \mathbf{F}=-G\frac{m_{1}m_{2}}{r^{2}}\mathbf{\hat{r}}
  2. 𝐅 ¯ = - G V ρ r 2 𝐫 ^ d x d y d z \mathbf{\bar{F}}=-G\int\limits_{V}\frac{\rho}{r^{2}}\mathbf{\hat{r}}\,dx\,dy\,dz
  3. u = - V ρ G r d x d y d z u\ =\ -\int\limits_{V}\rho\frac{G}{r}\,dx\,dy\,dz
  4. s = x 2 + y 2 + z 2 . s=\sqrt{x^{2}+y^{2}+z^{2}}\,.
  5. F ¯ = - G M R 2 r ^ \bar{F}=-\frac{GM}{R^{2}}\ \hat{r}
  6. u = - G M r u=-\frac{GM}{r}
  7. F x x + F y y + F z z = 0 \frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}+\frac{% \partial F_{z}}{\partial z}=0
  8. 2 u x 2 + 2 u y 2 + 2 u z 2 = 0 \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+% \frac{\partial^{2}u}{\partial z^{2}}=0
  9. ϕ = R ( r ) Θ ( θ ) Φ ( φ ) \phi\ =\ R(r)\ \Theta(\theta)\ \Phi(\varphi)
  10. g ( x , y , z ) \displaystyle g(x,y,z)
  11. x = r cos θ cos φ \displaystyle x=r\cos\theta\cos\varphi
  12. 1 r \frac{1}{r}
  13. 1 r 2 P 1 0 ( sin θ ) = 1 r 2 sin θ \frac{1}{r^{2}}P^{0}_{1}(\sin\theta)=\frac{1}{r^{2}}\sin\theta
  14. 1 r 2 P 1 1 ( sin θ ) cos φ = 1 r 2 cos θ cos φ \frac{1}{r^{2}}P^{1}_{1}(\sin\theta)\cos\varphi=\frac{1}{r^{2}}\cos\theta\cos\varphi
  15. 1 r 2 P 1 1 ( sin θ ) sin φ = 1 r 2 cos θ sin φ \frac{1}{r^{2}}P^{1}_{1}(\sin\theta)\sin\varphi=\frac{1}{r^{2}}\cos\theta\sin\varphi
  16. 1 r 3 P 2 0 ( sin θ ) = 1 r 3 1 2 ( 3 sin 2 θ - 1 ) \frac{1}{r^{3}}P^{0}_{2}(\sin\theta)=\frac{1}{r^{3}}\frac{1}{2}(3\sin^{2}% \theta-1)
  17. 1 r 3 P 2 1 ( sin θ ) cos φ = 1 r 3 3 sin θ cos θ cos φ \frac{1}{r^{3}}P^{1}_{2}(\sin\theta)\cos\varphi=\frac{1}{r^{3}}3\sin\theta\cos% \theta\ \cos\varphi
  18. 1 r 3 P 2 1 ( sin θ ) sin φ = 1 r 3 3 sin θ cos θ sin φ \frac{1}{r^{3}}P^{1}_{2}(\sin\theta)\sin\varphi=\frac{1}{r^{3}}3\sin\theta\cos% \theta\sin\varphi
  19. 1 r 3 P 2 2 ( sin θ ) cos 2 φ = 1 r 3 3 cos 2 θ cos 2 φ \frac{1}{r^{3}}P^{2}_{2}(\sin\theta)\cos 2\varphi=\frac{1}{r^{3}}3\cos^{2}% \theta\ \cos 2\varphi
  20. 1 r 3 P 2 2 ( sin θ ) sin 2 φ = 1 r 3 3 cos 2 θ sin 2 φ \frac{1}{r^{3}}P^{2}_{2}(\sin\theta)\sin 2\varphi=\frac{1}{r^{3}}3\cos^{2}% \theta\sin 2\varphi
  21. 1 r 4 P 3 0 ( sin θ ) = 1 r 4 1 2 sin θ ( 5 sin 2 θ - 3 ) \frac{1}{r^{4}}P^{0}_{3}(\sin\theta)=\frac{1}{r^{4}}\frac{1}{2}\sin\theta\ (5% \sin^{2}\theta-3)
  22. 1 r 4 P 3 1 ( sin θ ) cos φ = 1 r 4 3 2 ( 5 sin 2 θ - 1 ) cos θ cos φ \frac{1}{r^{4}}P^{1}_{3}(\sin\theta)\cos\varphi=\frac{1}{r^{4}}\frac{3}{2}\ (5% \ \sin^{2}\theta-1)\cos\theta\cos\varphi
  23. 1 r 4 P 3 1 ( sin θ ) sin φ = 1 r 4 3 2 ( 5 sin 2 θ - 1 ) cos θ sin φ \frac{1}{r^{4}}P^{1}_{3}(\sin\theta)\sin\varphi=\frac{1}{r^{4}}\frac{3}{2}\ (5% \ \sin^{2}\theta-1)\cos\theta\sin\varphi
  24. 1 r 4 P 3 2 ( sin θ ) cos 2 φ = 1 r 4 15 sin θ cos 2 θ cos 2 φ \frac{1}{r^{4}}P^{2}_{3}(\sin\theta)\cos 2\varphi=\frac{1}{r^{4}}15\sin\theta% \cos^{2}\theta\cos 2\varphi
  25. 1 r 4 P 3 2 ( sin θ ) sin 2 φ = 1 r 4 15 sin θ cos 2 θ sin 2 φ \frac{1}{r^{4}}P^{2}_{3}(\sin\theta)\sin 2\varphi=\frac{1}{r^{4}}15\sin\theta% \cos^{2}\theta\sin 2\varphi
  26. 1 r 4 P 3 3 ( sin θ ) cos 3 φ = 1 r 4 15 cos 3 θ cos 3 φ \frac{1}{r^{4}}P^{3}_{3}(\sin\theta)\cos 3\varphi=\frac{1}{r^{4}}15\cos^{3}% \theta\cos 3\varphi
  27. 1 r 4 P 3 3 ( sin θ ) sin 3 φ = 1 r 4 15 cos 3 θ sin 3 φ \frac{1}{r^{4}}P^{3}_{3}(\sin\theta)\sin 3\varphi=\frac{1}{r^{4}}15\cos^{3}% \theta\sin 3\varphi
  28. u = - μ r + n = 2 N z J n P n 0 ( sin θ ) r n + 1 + n = 2 N t m = 1 n P n m ( sin θ ) ( C n m cos m φ + S n m sin m φ ) r n + 1 u=-\frac{\mu}{r}+\sum_{n=2}^{N_{z}}\frac{J_{n}P^{0}_{n}(\sin\theta)}{r^{n+1}}+% \sum_{n=2}^{N_{t}}\sum_{m=1}^{n}\frac{P^{m}_{n}(\sin\theta)(C_{n}^{m}\cos m% \varphi+S_{n}^{m}\sin m\varphi)}{r^{n+1}}
  29. μ = G M \mu=GM
  30. P n 0 ( sin θ ) r n + 1 n = 0 , 1 , 2 , \frac{P^{0}_{n}(\sin\theta)}{r^{n+1}}\quad n=0,1,2,\dots
  31. P n m ( sin θ ) cos m φ r n + 1 , 1 m n n = 1 , 2 , \frac{P^{m}_{n}(\sin\theta)\cos m\varphi}{r^{n+1}}\,,\quad 1\leq m\leq n\quad n% =1,2,\dots
  32. P n m ( sin θ ) sin m φ r n + 1 \frac{P^{m}_{n}(\sin\theta)\sin m\varphi}{r^{n+1}}
  33. J n ~ = - J n μ R n \tilde{J_{n}}=-\frac{J_{n}}{\mu\ R^{n}}
  34. C n m ~ = - C n m μ R n \tilde{C_{n}^{m}}=-\frac{C_{n}^{m}}{\mu\ R^{n}}
  35. S n m ~ = - S n m μ R n \tilde{S_{n}^{m}}=-\frac{S_{n}^{m}}{\mu\ R^{n}}
  36. u = - μ r ( 1 + n = 2 N z J n ~ P n 0 ( sin θ ) ( r R ) n + n = 2 N t m = 1 n P n m ( sin θ ) ( C n m ~ cos m φ + S n m ~ sin m φ ) ( r R ) n ) u=-\frac{\mu}{r}\left(1+\sum_{n=2}^{N_{z}}\frac{\tilde{J_{n}}P^{0}_{n}(\sin% \theta)}{{(\frac{r}{R})}^{n}}+\sum_{n=2}^{N_{t}}\sum_{m=1}^{n}\frac{P^{m}_{n}(% \sin\theta)(\tilde{C_{n}^{m}}\cos m\varphi+\tilde{S_{n}^{m}}\sin m\varphi)}{{(% \frac{r}{R})}^{n}}\right)
  37. u = J 2 P 2 0 ( sin θ ) r 3 = J 2 1 r 3 1 2 ( 3 sin 2 θ - 1 ) = J 2 1 r 5 1 2 ( 3 z 2 - r 2 ) u=\frac{J_{2}\ P^{0}_{2}(\sin\theta)}{r^{3}}=J_{2}\frac{1}{r^{3}}\frac{1}{2}(3% \sin^{2}\theta-1)=J_{2}\frac{1}{r^{5}}\frac{1}{2}(3z^{2}-r^{2})
  38. φ ^ = - sin φ x ^ + cos φ y ^ θ ^ = - sin θ ( cos φ x ^ + sin φ y ^ ) + cos θ z ^ r ^ = cos θ ( cos φ x ^ + sin φ y ^ ) + sin θ z ^ \begin{aligned}&\displaystyle\hat{\varphi}=-\sin\varphi\hat{x}+\cos\varphi\hat% {y}\\ &\displaystyle\hat{\theta}=-\sin\theta\ (\cos\varphi\ \hat{x}+\sin\varphi\hat{% y})+\cos\theta\hat{z}\\ &\displaystyle\hat{r}=\cos\theta\ (\cos\varphi\ \hat{x}\ +\ \sin\varphi\ \hat{% y})+\ \sin\theta\ \hat{z}\end{aligned}
  39. φ ^ , θ ^ , r ^ \hat{\varphi}\ ,\ \hat{\theta}\ ,\ \hat{r}
  40. F θ = - 1 r u θ = - J 2 1 r 4 3 cos θ sin θ F r = - u r = J 2 1 r 4 3 2 ( 3 sin 2 θ - 1 ) \begin{aligned}&\displaystyle F_{\theta}=-\frac{1}{r}\ \frac{\partial u}{% \partial\theta}=-J_{2}\ \frac{1}{r^{4}}3\ \cos\theta\ \sin\theta\\ &\displaystyle F_{r}=-\frac{\partial u}{\partial r}=J_{2}\ \frac{1}{r^{4}}% \frac{3}{2}\ \left(3\sin^{2}\theta\ -\ 1\right)\end{aligned}
  41. F x = - u x = J 2 x r 7 ( 6 z 2 - 3 2 ( x 2 + y 2 ) ) \displaystyle F_{x}=-\frac{\partial u}{\partial x}=J_{2}\ \frac{x}{r^{7}}\left% (6z^{2}-\frac{3}{2}(x^{2}+y^{2})\right)
  42. u = J 3 P 3 0 ( sin θ ) r 4 = J 3 1 r 4 1 2 sin θ ( 5 sin 2 θ - 3 ) = J 3 1 r 7 1 2 z ( 5 z 2 - 3 r 2 ) u=\frac{J_{3}P^{0}_{3}(\sin\theta)}{r^{4}}=J_{3}\frac{1}{r^{4}}\frac{1}{2}\sin% \theta(5\sin^{2}\theta-3)=J_{3}\frac{1}{r^{7}}\frac{1}{2}z(5z^{2}-3r^{2})
  43. F θ = - 1 r u θ = - J 3 1 r 5 3 2 cos θ ( 5 sin 2 θ - 1 ) F r = - u r = J 3 1 r 5 2 sin θ ( 5 sin 2 θ - 3 ) \begin{aligned}&\displaystyle F_{\theta}=-\frac{1}{r}\frac{\partial u}{% \partial\theta}=-J_{3}\frac{1}{r^{5}}\frac{3}{2}\cos\theta\left(5\sin^{2}% \theta-1\right)\\ &\displaystyle F_{r}=-\frac{\partial u}{\partial r}=J_{3}\frac{1}{r^{5}}2\sin% \theta\left(5\sin^{2}\theta-3\right)\end{aligned}
  44. F x = - u x = J 3 x z r 9 ( 10 z 2 - 15 2 ( x 2 + y 2 ) ) F y = - u y = J 3 y z r 9 ( 10 z 2 - 15 2 ( x 2 + y 2 ) ) F z = - u z = J 3 1 r 9 ( 4 z 2 ( z 2 - 3 ( x 2 + y 2 ) ) + 3 2 ( x 2 + y 2 ) 2 ) \begin{aligned}&\displaystyle F_{x}=-\frac{\partial u}{\partial x}=J_{3}\frac{% xz}{r^{9}}\left(10z^{2}-\frac{15}{2}(x^{2}+y^{2})\right)\\ &\displaystyle F_{y}=-\frac{\partial u}{\partial y}=J_{3}\frac{yz}{r^{9}}\left% (10z^{2}-\frac{15}{2}(x^{2}+y^{2})\right)\\ &\displaystyle F_{z}=-\frac{\partial u}{\partial z}=J_{3}\frac{1}{r^{9}}\left(% 4z^{2}\ \left(z^{2}-3(x^{2}+y^{2})\right)+\frac{3}{2}(x^{2}+y^{2})^{2}\right)% \end{aligned}
  45. J 2 ~ = 1.0826 × 10 - 3 \tilde{J_{2}}=1.0826\times 10^{-3}
  46. J 3 ~ = - 2.532 × 10 - 6 \tilde{J_{3}}=-2.532\times 10^{-6}
  47. N z N_{z}
  48. N t N_{t}
  49. 2 f x 2 + 2 f y 2 + 2 f z 2 = 1 r 2 r ( r 2 f r ) + 1 r 2 cos θ θ ( cos θ f θ ) + 1 r 2 cos 2 θ 2 f φ 2 \frac{\partial^{2}f}{\partial x^{2}}\ +\ \frac{\partial^{2}f}{\partial y^{2}}% \ +\ \frac{\partial^{2}f}{\partial z^{2}}\ =\ {1\over r^{2}}{\partial\over% \partial r}\left(r^{2}{\partial f\over\partial r}\right)+{1\over r^{2}\cos% \theta}{\partial\over\partial\theta}\left(\cos\theta{\partial f\over\partial% \theta}\right)+{1\over r^{2}\cos^{2}\theta}{\partial^{2}f\over\partial\varphi^% {2}}
  50. r 2 ϕ ( 2 ϕ x 2 + 2 ϕ y 2 + 2 ϕ z 2 ) = 1 R d d r ( r 2 d R d r ) + 1 Θ cos θ d d θ ( cos θ d Θ d θ ) + 1 Φ cos 2 θ d 2 Φ d φ 2 \frac{r^{2}}{\phi}\left(\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial% ^{2}\phi}{\partial y^{2}}+\frac{\partial^{2}\phi}{\partial z^{2}}\right)\ =% \frac{1}{R}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)+\frac{1}{\Theta\cos% \theta}\frac{d}{d\theta}\left(\cos\theta\frac{d\Theta}{d\theta}\right)+\frac{1% }{\Phi\cos^{2}\theta}\frac{d^{2}\Phi}{d\varphi^{2}}
  51. 1 R d d r ( r 2 d R d r ) \frac{1}{R}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)
  52. r r
  53. 1 Θ cos θ d d θ ( cos θ d Θ d θ ) + 1 Φ cos 2 θ d 2 Φ d φ 2 \frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta\frac{d\Theta}{d% \theta}\right)+\frac{1}{\Phi\cos^{2}\theta}\frac{d^{2}\Phi}{d\varphi^{2}}
  54. 1 R d d r ( r 2 d R d r ) = λ \frac{1}{R}\frac{d}{dr}\left(r^{2}\frac{dR}{dr}\right)\ =\ \lambda
  55. 1 Θ cos θ d d θ ( cos θ d Θ d θ ) + 1 Φ cos 2 θ d 2 Φ d φ 2 = - λ \frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta\frac{d\Theta}{d% \theta}\right)+\frac{1}{\Phi\cos^{2}\theta}\frac{d^{2}\Phi}{d\varphi^{2}}\ =\ -\lambda
  56. λ \lambda
  57. 1 Θ cos θ d d θ ( cos θ d Θ d θ ) + λ cos 2 θ + 1 Φ d 2 Φ d φ 2 = 0 \frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta\frac{d\Theta}{% d\theta}\right)\ +\lambda\ \cos^{2}\theta\ +\ \frac{1}{\Phi}\frac{d^{2}\Phi}{d% \varphi^{2}}\ =\ 0
  58. θ \theta
  59. φ \varphi
  60. 1 Φ d 2 Φ d φ 2 = - m 2 \frac{1}{\Phi}\frac{d^{2}\Phi}{d\varphi^{2}}\ =\ -m^{2}
  61. 1 Θ cos θ d d θ ( cos θ d Θ d θ ) + λ cos 2 θ = m 2 \frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta\frac{d\Theta}{% d\theta}\right)\ +\lambda\ \cos^{2}\theta=\ m^{2}
  62. x = sin θ x=\sin\theta
  63. d d x ( ( 1 - x 2 ) d Θ d x ) + ( λ - m 2 1 - x 2 ) Θ = 0 \frac{d}{dx}\left((1-x^{2})\frac{d\Theta}{dx}\right)+\left(\lambda-\frac{m^{2}% }{1-x^{2}}\right)\Theta=0
  64. ϕ \phi
  65. R ( r ) = 1 r n + 1 R(r)=\frac{1}{r^{n+1}}
  66. λ = n ( n + 1 ) \lambda=n(n+1)
  67. d d x ( ( 1 - x 2 ) d P n d x ) + n ( n + 1 ) P n = 0 \frac{d}{dx}\left((1-x^{2})\ \frac{dP_{n}}{dx}\right)\ +\ n(n+1)\ P_{n}\ =\ 0
  68. ϕ = 1 r n + 1 P n ( sin θ ) \phi=\frac{1}{r^{n+1}}\ P_{n}(\sin\theta)
  69. P n m ( x ) P_{n}^{m}(x)
  70. d d x ( ( 1 - x 2 ) d P n m d x ) + ( n ( n + 1 ) - m 2 1 - x 2 ) P n m = 0 \frac{d}{dx}\left((1-x^{2})\ \frac{dP_{n}^{m}}{dx}\right)\ +\ \left(n(n+1)-% \frac{m^{2}}{1-x^{2}}\right)\ P_{n}^{m}\ =\ 0
  71. ϕ = 1 r n + 1 P n m ( sin θ ) ( a cos m φ + b sin m φ ) \phi=\frac{1}{r^{n+1}}\ P_{n}^{m}(\sin\theta)\ (a\ \cos m\varphi\ +\ b\ \sin m\varphi)
  72. P 0 ( x ) = 1 P n ( x ) = 1 2 n n ! d n ( x 2 - 1 ) n d x n n 1 \begin{aligned}&\displaystyle P_{0}(x)=1\\ &\displaystyle P_{n}(x)=\frac{1}{2^{n}n!}\ \frac{d^{n}(x^{2}-1)^{n}}{dx^{n}}% \quad n\geq 1\\ \end{aligned}
  73. P 0 ( x ) = 1 \displaystyle P_{0}(x)=1
  74. P n m ( x ) = ( 1 - x 2 ) m 2 d n P n d x n 1 m n P_{n}^{m}(x)\ =(1-x^{2})^{\frac{m}{2}}\ \frac{d^{n}P_{n}}{dx^{n}}\quad 1\leq m\leq n
  75. P n m ( sin θ ) = cos m θ d n P n d x n ( sin θ ) P_{n}^{m}(\sin\theta)=\cos^{m}\theta\ \frac{d^{n}P_{n}}{dx^{n}}(\sin\theta)