wpmath0000002_14

Nomogram.html

  1. 1 1 / A + 1 / B \frac{1}{1/A+1/B}
  2. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Non-uniform_rational_B-spline.html

  1. d n C ( u ) / d u n d^{n}C(u)/du^{n}
  2. R n R n R^{n}\to R^{n}
  3. N i , n ( u ) N_{i,n}(u)
  4. i i
  5. i i
  6. n n
  7. N i , n N_{i,n}
  8. n n
  9. N i , 0 N_{i,0}
  10. N i , n N_{i,n}
  11. N i , n - 1 N_{i,n-1}
  12. N i + 1 , n - 1 N_{i+1,n-1}
  13. n n
  14. n - 1 n-1
  15. N i , n N_{i,n}
  16. N 1 , 1 N_{1,1}
  17. N 2 , 1 N_{2,1}
  18. f f
  19. g g
  20. N i , n = f i , n N i , n - 1 + g i + 1 , n N i + 1 , n - 1 N_{i,n}=f_{i,n}N_{i,n-1}+g_{i+1,n}N_{i+1,n-1}
  21. f i f_{i}
  22. N i , n - 1 N_{i,n-1}
  23. g i + 1 g_{i+1}
  24. N i + 1 , n - 1 N_{i+1,n-1}
  25. N i , 1 N_{i,1}
  26. u u
  27. k i k_{i}
  28. i i
  29. f f
  30. g g
  31. f i , n ( u ) = u - k i k i + n - k i f_{i,n}(u)={{u-k_{i}}\over{k_{i+n}-k_{i}}}
  32. g i , n ( u ) = k i + n - u k i + n - k i g_{i,n}(u)={{k_{i+n}-u}\over{k_{i+n}-k_{i}}}
  33. f f
  34. g g
  35. n n
  36. u u
  37. N i , n N_{i,n}
  38. C ( u ) = i = 1 k N i , n w i j = 1 k N j , n w j P i = i = 1 k N i , n w i P i i = 1 k N i , n w i C(u)=\sum_{i=1}^{k}{\frac{N_{i,n}w_{i}}{\sum_{j=1}^{k}N_{j,n}w_{j}}}{P}_{i}=% \frac{\sum_{i=1}^{k}{N_{i,n}w_{i}{P}_{i}}}{\sum_{i=1}^{k}{N_{i,n}w_{i}}}
  39. k k
  40. P i {P}_{i}
  41. w i w_{i}
  42. C ( u ) = i = 1 k R i , n ( u ) P i C(u)=\sum_{i=1}^{k}R_{i,n}(u){P}_{i}
  43. R i , n ( u ) = N i , n ( u ) w i j = 1 k N j , n ( u ) w j R_{i,n}(u)={N_{i,n}(u)w_{i}\over\sum_{j=1}^{k}N_{j,n}(u)w_{j}}
  44. u u
  45. v v
  46. i i
  47. j j
  48. S ( u , v ) = i = 1 k j = 1 l R i , j ( u , v ) P i , j S(u,v)=\sum_{i=1}^{k}\sum_{j=1}^{l}R_{i,j}(u,v){P}_{i,j}
  49. R i , j ( u , v ) = N i , n ( u ) N j , m ( v ) w i , j p = 1 k q = 1 l N p , n ( u ) N q , m ( v ) w p , q R_{i,j}(u,v)=\frac{N_{i,n}(u)N_{j,m}(v)w_{i,j}}{\sum_{p=1}^{k}\sum_{q=1}^{l}N_% {p,n}(u)N_{q,m}(v)w_{p,q}}
  50. n n
  51. n - 1 n-1
  52. n n
  53. κ \kappa
  54. κ = | r ( t ) × r ′′ ( t ) | | r ( t ) | 3 \kappa=\frac{|r^{\prime}(t)\times r^{\prime\prime}(t)|}{|r^{\prime}(t)|^{3}}
  55. κ = | r ′′ ( s o ) | \kappa=|r^{\prime\prime}(s_{o})|
  56. κ \kappa
  57. 2 2 \scriptstyle\frac{\sqrt{2}}{2}
  58. 2 2 \scriptstyle\frac{\sqrt{2}}{2}
  59. 2 2 \scriptstyle\frac{\sqrt{2}}{2}
  60. 2 2 \scriptstyle\frac{\sqrt{2}}{2}
  61. { 0 , 0 , 0 , π / 2 , π / 2 , π , π , 3 π / 2 , 3 π / 2 , 2 π , 2 π , 2 π } \{0,0,0,\pi/2,\pi/2,\pi,\pi,3\pi/2,3\pi/2,2\pi,2\pi,2\pi\}\,
  62. t t
  63. ( sin ( t ) , cos ( t ) ) (\sin(t),\cos(t))
  64. cos ( t ) \cos(t)
  65. t t
  66. 2 π 2\pi
  67. π / 2 \pi/2

Noncommutative_geometry.html

  1. x y xy
  2. y x yx

Nonlinear_dimensionality_reduction.html

  1. m × n m\times n
  2. 𝐗 \mathbf{X}
  3. C = 1 m i = 1 m 𝐱 i 𝐱 i 𝖳 . C=\frac{1}{m}\sum_{i=1}^{m}{\mathbf{x}_{i}\mathbf{x}_{i}^{\mathsf{T}}}.
  4. C = 1 m i = 1 m Φ ( 𝐱 i ) Φ ( 𝐱 i ) 𝖳 . C=\frac{1}{m}\sum_{i=1}^{m}{\Phi(\mathbf{x}_{i})\Phi(\mathbf{x}_{i})^{\mathsf{% T}}}.
  5. Φ ( 𝐱 ) \Phi(\mathbf{x})
  6. Φ \Phi
  7. E ( W ) = i | 𝐗 i - j 𝐖 i j 𝐗 j | 𝟤 E(W)=\sum_{i}|{\mathbf{X}_{i}-\sum_{j}{\mathbf{W}_{ij}\mathbf{X}_{j}}|}^{% \mathsf{2}}
  8. j 𝐖 i j = 1 \sum_{j}{\mathbf{W}_{ij}}=1
  9. C ( Y ) = i | 𝐘 i - j 𝐖 i j 𝐘 j | 𝟤 C(Y)=\sum_{i}|{\mathbf{Y}_{i}-\sum_{j}{\mathbf{W}_{ij}\mathbf{Y}_{j}}|}^{% \mathsf{2}}
  10. 𝐗 = [ x 1 , x 2 , , x n ] Ω 𝐑 𝐃 \mathbf{X}=[x_{1},x_{2},\ldots,x_{n}]\in\Omega\subset\mathbf{R^{D}}
  11. 𝐝 \mathbf{d}
  12. μ \mu
  13. k \mathit{k}
  14. k ( x , y ) = k ( y , x ) , k(x,y)=k(y,x),\,
  15. k ( x , y ) 0 x , y , k k(x,y)\geq 0\qquad\forall x,y,k
  16. K i j = { e - x i - x j 2 2 / σ 2 if x i x j 0 otherwise K_{ij}=\begin{cases}e^{-\|x_{i}-x_{j}\|^{2}_{2}/\sigma^{2}}&\,\text{if }x_{i}% \sim x_{j}\\ 0&\,\text{otherwise}\end{cases}
  17. x i x j x_{i}\sim x_{j}
  18. x i x_{i}
  19. x j x_{j}
  20. σ \sigma
  21. x i - x j 2 σ \|x_{i}-x_{j}\|_{2}\gg\sigma
  22. K i j = 0 K_{ij}=0
  23. x i - x j 2 σ \|x_{i}-x_{j}\|_{2}\ll\sigma
  24. K i j = 1 K_{ij}=1
  25. σ \sigma
  26. K K
  27. D D
  28. P = D - 1 K . P=D^{-1}K.\,
  29. P P
  30. P ( x i , x j ) P(x_{i},x_{j})
  31. x i x_{i}
  32. x j x_{j}
  33. x i x_{i}
  34. x j x_{j}
  35. P t ( x i , x j ) P^{t}(x_{i},x_{j})
  36. P t P^{t}
  37. P P
  38. P P
  39. K K
  40. D t D_{t}
  41. t N {}_{t\in N}
  42. D t 2 ( x , y ) = || p t ( x , ) - p t ( y , ) || 2 D_{t}^{2}(x,y)=||p_{t}(x,\cdot)-p_{t}(y,\cdot)||^{2}
  43. D t D_{t}
  44. D t ( x , y ) D_{t}(x,y)
  45. D t ( x , y ) D_{t}(x,y)
  46. D t D_{t}
  47. D t D_{t}

Nonlinear_distortion.html

  1. v = k = 1 a k u k v=\sum_{k=1}^{\infty}a_{k}u^{k}
  2. a 2 a_{2}
  3. a 3 a_{3}
  4. a 3 a_{3}
  5. ω \omega
  6. 3 ω 3\omega
  7. v = ( a 1 + 3 4 a 3 ) s i n ( ω t ) - 1 4 a 3 s i n ( 3 ω t ) v=(a_{1}+\frac{3}{4}a_{3})sin(\omega t)-\frac{1}{4}a_{3}sin(3\omega t)
  8. 3 ω 3\omega

Nonlinear_system.html

  1. f ( x ) f(x)
  2. f ( x + y ) = f ( x ) + f ( y ) ; \textstyle f(x+y)\ =f(x)\ +f(y);
  3. f ( α x ) = α f ( x ) . \textstyle f(\alpha x)\ =\alpha f(x).
  4. f ( α x + β y ) = α f ( x ) + β f ( y ) f(\alpha x+\beta y)=\alpha f(x)+\beta f(y)\,
  5. f ( x ) = C f(x)=C\,
  6. f ( x ) f(x)
  7. C = 0 C=0
  8. f ( x ) = C f(x)=C
  9. x x
  10. f ( x ) f(x)
  11. f ( x ) f(x)
  12. x x
  13. x 2 + x - 1 = 0 . x^{2}+x-1=0\,.
  14. d u d x = - u 2 \frac{\operatorname{d}u}{\operatorname{d}x}=-u^{2}\,
  15. u = 1 x + C u=\frac{1}{x+C}
  16. d u d x + u 2 = 0 \frac{\operatorname{d}u}{\operatorname{d}x}+u^{2}=0\,
  17. d 2 θ d t 2 + sin ( θ ) = 0 \frac{d^{2}\theta}{dt^{2}}+\sin(\theta)=0\,
  18. θ \theta
  19. d θ / d t d\theta/dt
  20. d θ C 0 + 2 cos ( θ ) = t + C 1 \int\frac{d\theta}{\sqrt{C_{0}+2\cos(\theta)}}=t+C_{1}\,
  21. C 0 = 0 C_{0}=0
  22. θ = 0 \theta=0
  23. d 2 θ d t 2 + θ = 0 \frac{d^{2}\theta}{dt^{2}}+\theta=0\,
  24. sin ( θ ) θ \sin(\theta)\approx\theta
  25. θ 0 \theta\approx 0
  26. θ = π \theta=\pi
  27. d 2 θ d t 2 + π - θ = 0 \frac{d^{2}\theta}{dt^{2}}+\pi-\theta=0\,
  28. sin ( θ ) π - θ \sin(\theta)\approx\pi-\theta
  29. θ π \theta\approx\pi
  30. | θ | |\theta|
  31. θ = π / 2 \theta=\pi/2
  32. sin ( θ ) 1 \sin(\theta)\approx 1
  33. d 2 θ d t 2 + 1 = 0. \frac{d^{2}\theta}{dt^{2}}+1=0.

Normal.html

  1. T 4 T_{4}

Normal_(geometry).html

  1. a x + b y + c z + d = 0 ax+by+cz+d=0
  2. ( a , b , c ) (a,b,c)
  3. 𝐫 ( α , β ) = 𝐚 + α 𝐛 + β 𝐜 \mathbf{r}(\alpha,\beta)=\mathbf{a}+\alpha\mathbf{b}+\beta\mathbf{c}
  4. 𝐛 × 𝐜 \mathbf{b}\times\mathbf{c}
  5. 𝐫 = 𝐚 0 + α 1 𝐚 1 + + α n 𝐚 n \mathbf{r}=\mathbf{a}_{0}+\alpha_{1}\mathbf{a}_{1}+\cdots+\alpha_{n}\mathbf{a}% _{n}
  6. A = [ 𝐚 1 𝐚 n ] A=[\mathbf{a}_{1}\dots\mathbf{a}_{n}]
  7. 𝐱 s × 𝐱 t . {\partial\mathbf{x}\over\partial s}\times{\partial\mathbf{x}\over\partial t}.
  8. ( x , y , z ) (x,y,z)
  9. F ( x , y , z ) = 0 F(x,y,z)=0
  10. ( x , y , z ) (x,y,z)
  11. F ( x , y , z ) . \nabla F(x,y,z).
  12. F ( x , y , z ) = 0 F(x,y,z)=0
  13. F F
  14. f ( x , y ) f(x,y)
  15. x , y x,y
  16. f ( x , y ) = a 00 + a 01 y + a 10 x + a 11 x y f(x,y)=a_{00}+a_{01}y+a_{10}x+a_{11}xy
  17. F ( x , y , z ) = z - f ( x , y ) = 0 F(x,y,z)=z-f(x,y)=0
  18. F ( x , y , z ) \nabla F(x,y,z)
  19. F ( x , y , z ) = f ( x , y ) - z F(x,y,z)=f(x,y)-z
  20. F / z \partial{F}/\partial{z}
  21. f ( x , y ) \nabla f(x,y)
  22. F ( x , y , z ) = 𝐤 ^ - f ( x , y ) \nabla F(x,y,z)=\hat{\mathbf{k}}-\nabla f(x,y)
  23. 𝐤 ^ \hat{\mathbf{k}}
  24. F ( x , y , z ) = 𝐤 ^ - f ( x , y ) x 𝐢 ^ - f ( x , y ) y 𝐣 ^ \nabla F(x,y,z)=\hat{\mathbf{k}}-\frac{\partial{f(x,y)}}{\partial{x}}\hat{% \mathbf{i}}-\frac{\partial{f(x,y)}}{\partial{y}}\hat{\mathbf{j}}
  25. 𝐢 ^ \hat{\mathbf{i}}
  26. 𝐣 ^ \hat{\mathbf{j}}
  27. ( W n ) ( M t ) = 0 \iff(Wn)\cdot(Mt)=0
  28. ( W n ) T ( M t ) = 0 \iff(Wn)^{T}(Mt)=0
  29. ( n T W T ) ( M t ) = 0 \iff(n^{T}W^{T})(Mt)=0
  30. n T ( W T M ) t = 0 \iff n^{T}(W^{T}M)t=0
  31. W T M = I W^{T}M=I
  32. W = M - 1 T W={M^{-1}}^{T}
  33. W n Wn
  34. M t Mt
  35. ( n - 1 ) (n-1)
  36. n n
  37. ( x 1 , x 2 , , x n ) \scriptstyle(x_{1},x_{2},\ldots,x_{n})
  38. F ( x 1 , x 2 , , x n ) = 0 \scriptstyle F(x_{1},x_{2},\ldots,x_{n})=0
  39. F F
  40. F F
  41. F ( x 1 , x 2 , , x n ) = ( F x 1 , F x 2 , , F x n ) . \nabla F(x_{1},x_{2},\ldots,x_{n})=\left(\tfrac{\partial F}{\partial x_{1}},% \tfrac{\partial F}{\partial x_{2}},\ldots,\tfrac{\partial F}{\partial x_{n}}% \right)\,.
  42. f 1 ( x 1 , , x n ) , , f k ( x 1 , , x n ) . f_{1}(x_{1},\ldots,x_{n}),\ldots,f_{k}(x_{1},\ldots,x_{n}).
  43. x y = 0 , z = 0 . x\,y=0,\quad z=0\,.

Normal_matrix.html

  1. A A
  2. A * A = A A * A^{*}A=AA^{*}
  3. A A
  4. A A
  5. A A
  6. A = ( 1 1 0 0 1 1 1 0 1 ) A=\begin{pmatrix}1&1&0\\ 0&1&1\\ 1&0&1\end{pmatrix}
  7. A A * = ( 2 1 1 1 2 1 1 1 2 ) = A * A . AA^{*}=\begin{pmatrix}2&1&1\\ 1&2&1\\ 1&1&2\end{pmatrix}=A^{*}A.
  8. A A
  9. A e 1 2 = A * e 1 2 . \left\|Ae_{1}\right\|^{2}=\left\|A^{*}e_{1}\right\|^{2}.
  10. n n
  11. n n
  12. A A
  13. A A
  14. Λ Λ
  15. U U
  16. Λ Λ
  17. A A
  18. U U
  19. A A
  20. Λ Λ
  21. U U
  22. A A
  23. B B
  24. A A
  25. B B
  26. B B
  27. 𝐑 \mathbf{R}
  28. A A
  29. B B
  30. A B = B A AB=BA
  31. A B AB
  32. A + B A+B
  33. U U
  34. A A
  35. B B
  36. A A
  37. B B
  38. A A
  39. n × n n×n
  40. A A
  41. A A
  42. A A
  43. x x
  44. A A
  45. A A
  46. tr ( A * A ) = j | λ j | 2 . \operatorname{tr}(A^{*}A)=\sum\nolimits_{j}|\lambda_{j}|^{2}.
  47. A A
  48. n 1 ≤n−1
  49. A A
  50. U U
  51. U U
  52. P P
  53. A = U P A=UP
  54. U U
  55. P P
  56. A A
  57. N N
  58. 1 i n 1≤i≤n
  59. A A
  60. A A
  61. A A
  62. sup x = 1 A x = sup x = 1 | A x , x | = max { | λ | : λ σ ( A ) } \sup_{\|x\|=1}\|Ax\|=\sup_{\|x\|=1}|\langle Ax,x\rangle|=\max\{|\lambda|:% \lambda\in\sigma(A)\}
  63. 2 × 2 2×2
  64. a + b i ( a b - b a ) , a+bi\mapsto\begin{pmatrix}a&b\\ -b&a\end{pmatrix},
  65. A A
  66. P P
  67. λ < s u b > j ¯ = P ( λ j ) \overline{λ<sub>j}=P(λ_{j})

Normal_number.html

  1. 2 \sqrt{2}
  2. lim n N S ( a , n ) n = 1 b \lim_{n\to\infty}\frac{N_{S}(a,n)}{n}=\frac{1}{b}
  3. lim n N S ( w , n ) n = 1 b | w | \lim_{n\to\infty}\frac{N_{S}(w,n)}{n}=\frac{1}{b^{|w|}}
  4. Ω \ \Omega
  5. 2 \sqrt{2}
  6. 123 , 456 , 789 9 , 999 , 999 , 999 = 0. 0123456789 ¯ \frac{123,\!456,\!789}{9,\!999,\!999,\!999}=0.\overline{0123456789}
  7. d j = j d j - 1 / ( j - 1 ) , d_{j}=j^{d_{j-1}/(j-1)}\ ,
  8. ξ = j = 2 ( 1 - 1 d j ) . \xi=\prod_{j=2}^{\infty}\left({1-\frac{1}{d_{j}}}\right)\ .
  9. x q x\cdot q
  10. A 𝒩 A\subseteq\mathcal{N}
  11. α < 1 \alpha<1
  12. | A { 1 , , n } | n α |A\cap\{1,\ldots,n\}|\geq n^{\alpha}
  13. a 1 , a 2 , a 3 , a_{1},a_{2},a_{3},\ldots
  14. 0. a 1 a 2 a 3 0.a_{1}a_{2}a_{3}\ldots
  15. k 1 k\geq 1
  16. m 1 < m 2 < m 3 < m_{1}<m_{2}<m_{3}<\cdots
  17. m { m 1 , m 2 , } . m\in\{m_{1},m_{2},\ldots\}.
  18. Σ \Sigma
  19. Σ \Sigma
  20. Σ = { 0 , 1 } \Sigma=\{0,1\}
  21. q 0 [ 0 , 1 ] q_{0}\in[0,1]
  22. q 1 = 1 - q 0 q_{1}=1-q_{0}
  23. | Σ | |\Sigma|
  24. lim sup n d ( S n ) = , \limsup_{n\to\infty}d(S\upharpoonright n)=\infty,
  25. d ( S n ) d(S\upharpoonright n)
  26. f : Σ * Σ * × Q f:\Sigma^{*}\to\Sigma^{*}\times Q
  27. lim inf n | C ( S n ) | n < 1 , \liminf_{n\to\infty}\frac{|C(S\upharpoonright n)|}{n}<1,
  28. | C ( S n ) | |C(S\upharpoonright n)|
  29. ( b k x ) k = 0 {\left(b^{k}x\right)}_{k=0}^{\infty}
  30. lim n 1 n k = 0 n - 1 e 2 π i m b k x = 0 for all integers m 1. \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi imb^{k}x}=0\quad\,% \text{ for all integers }m\geq 1.
  31. ( x β k ) k = 0 \left({x\beta^{k}}\right)_{k=0}^{\infty}

Normal_operator.html

  1. T x 2 = T * T x , x = T T * x , x = T * x 2 \|Tx\|^{2}=\langle T^{*}Tx,x\rangle=\langle TT^{*}x,x\rangle=\|T^{*}x\|^{2}
  2. T T 1 + i T 2 T\equiv T_{1}+iT_{2}
  3. T 1 := T + T * 2 T_{1}:=\frac{T+T^{*}}{2}
  4. i T 2 := T - T * 2 , i\,T_{2}:=\frac{T-T^{*}}{2},
  5. T 1 T 2 = T 2 T 1 T_{1}T_{2}=T_{2}T_{1}
  6. λ ¯ \overline{\lambda}
  7. N 1 N_{1}
  8. N 2 N_{2}
  9. N 1 A = A N 2 N_{1}A=AN_{2}
  10. N 1 * A = A N 2 * N_{1}^{*}A=AN_{2}^{*}
  11. X X * = P V T ( s y m b o l 1 H - P V ) 2 T * P V = P V T ( s y m b o l 1 H - P V ) T * P V = P V T T * P V - P V T P V T * P V XX^{*}=P_{V}T(symbol{1}_{H}-P_{V})^{2}T^{*}P_{V}=P_{V}T(symbol{1}_{H}-P_{V})T^% {*}P_{V}=P_{V}TT^{*}P_{V}-P_{V}TP_{V}T^{*}P_{V}
  12. tr ( X X * ) \displaystyle\operatorname{tr}(XX^{*})
  13. 2 \ell^{2}
  14. N * N = N N * . N^{*}N=NN^{*}.
  15. N x = N * x \|Nx\|=\|N^{*}x\|\qquad
  16. 𝒟 ( N ) = 𝒟 ( N * ) . \qquad\mathcal{D}(N)=\mathcal{D}(N^{*}).

Northern_pike.html

  1. W = c L b W=cL^{b}\!\,

Norton's_theorem.html

  1. I total = 15 V 2 k Ω + ( 1 k Ω ( 1 k Ω + 1 k Ω ) ) = 5.625 mA . I_{\mathrm{total}}={15\mathrm{V}\over 2\,\mathrm{k}\Omega+(1\,\mathrm{k}\Omega% \|(1\,\mathrm{k}\Omega+1\,\mathrm{k}\Omega))}=5.625\mathrm{mA}.
  2. I No = 1 k Ω + 1 k Ω ( 1 k Ω + 1 k Ω + 1 k Ω ) I total I_{\mathrm{No}}={1\,\mathrm{k}\Omega+1\,\mathrm{k}\Omega\over(1\,\mathrm{k}% \Omega+1\,\mathrm{k}\Omega+1\,\mathrm{k}\Omega)}\cdot I_{\mathrm{total}}
  3. = 2 / 3 5.625 mA = 3.75 mA . =2/3\cdot 5.625\mathrm{mA}=3.75\mathrm{mA}.
  4. R eq = 1 k Ω + ( 2 k Ω ( 1 k Ω + 1 k Ω ) ) = 2 k Ω . R_{\mathrm{eq}}=1\,\mathrm{k}\Omega+(2\,\mathrm{k}\Omega\|(1\,\mathrm{k}\Omega% +1\,\mathrm{k}\Omega))=2\,\mathrm{k}\Omega.
  5. R T h = R N o R_{Th}=R_{No}\!
  6. V T h = I N o R N o V_{Th}=I_{No}R_{No}\!
  7. V T h R T h = I N o \frac{V_{Th}}{R_{Th}}=I_{No}\!

Nth_root.html

  1. r n = x , r^{n}=x,
  2. \sqrt{\,\,}
  3. \surd{}
  4. x \sqrt{x}\!\,
  5. x \surd x
  6. x 3 \sqrt[3]{x}\!\,
  7. x 4 \sqrt[4]{x}
  8. x n \sqrt[n]{x}
  9. \sqrt{\,\,}
  10. x n = x 1 / n \sqrt[n]{x}\,=\,x^{1/n}
  11. r n = x . r^{n}=x.\!\,
  12. x n \sqrt[n]{x}
  13. - 2 5 = - 1.148698354 \sqrt[5]{-2}\,=-1.148698354\ldots
  14. 2 = 1.414213562 \sqrt{2}=1.414213562\ldots
  15. r 2 = x . r^{2}=x.\!\,
  16. 25 = 5. \sqrt{25}=5.\!\,
  17. r 3 = x . r^{3}=x.\!\,
  18. x 3 \sqrt[3]{x}
  19. 8 3 = 2 and - 8 3 = - 2. \sqrt[3]{8}\,=\,2\quad\,\text{and}\quad\sqrt[3]{-8}\,=-2.
  20. a b n = a n b n , \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}\,,
  21. a b n = a n b n . \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\,.
  22. x 1 / n x^{1/n}
  23. a m n = ( a m ) 1 n = a m n . \sqrt[n]{a^{m}}=\left(a^{m}\right)^{\frac{1}{n}}=a^{\frac{m}{n}}.
  24. - 1 × - 1 = - 1 \sqrt{-1}\times\sqrt{-1}=-1
  25. - 1 × - 1 = 1 \sqrt{-1\times-1}=1
  26. 32 5 \sqrt{\tfrac{32}{5}}
  27. 32 5 = 16 2 5 = 4 2 5 \sqrt{\tfrac{32}{5}}=\sqrt{\tfrac{16\cdot 2}{5}}=4\sqrt{\tfrac{2}{5}}
  28. 4 2 5 = 4 2 5 4\sqrt{\tfrac{2}{5}}=\frac{4\sqrt{2}}{\sqrt{5}}
  29. 4 2 5 = 4 2 5 5 5 = 4 10 5 = 4 5 10 \frac{4\sqrt{2}}{\sqrt{5}}=\frac{4\sqrt{2}}{\sqrt{5}}\cdot\frac{\sqrt{5}}{% \sqrt{5}}=\frac{4\sqrt{10}}{5}=\frac{4}{5}\sqrt{10}
  30. 1 a 3 + b 3 = a 2 3 - a b 3 + b 2 3 ( a 3 + b 3 ) ( a 2 3 - a b 3 + b 2 3 ) = a 2 3 - a b 3 + b 2 3 a + b . \frac{1}{\sqrt[3]{a}+\sqrt[3]{b}}=\frac{\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{% b^{2}}}{(\sqrt[3]{a}+\sqrt[3]{b})(\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}% )}=\frac{\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}}{a+b}\,.
  31. 3 + 2 2 = 1 + 2 \sqrt{3+2\sqrt{2}}=1+\sqrt{2}\,
  32. ( 1 + x ) s / t = n = 0 k = 0 n - 1 ( s - k t ) n ! t n x n (1+x)^{s/t}=\sum_{n=0}^{\infty}\frac{\prod_{k=0}^{n-1}(s-kt)}{n!t^{n}}x^{n}
  33. | x | < 1 |x|<1
  34. 34 5 = 2.024397458 , \sqrt[5]{34}=2.024397458\ldots,
  35. x k + 1 = 1 n ( ( n - 1 ) x k + A x k n - 1 ) x_{k+1}=\frac{1}{n}\left({(n-1)x_{k}+\frac{A}{x_{k}^{n-1}}}\right)
  36. x n + y n x + y n x n - 1 . \sqrt[n]{x^{n}+y}\approx x+\frac{y}{nx^{n-1}}.
  37. 34 5 = 32 + 2 5 2 + 2 5 16 = 2.025. \sqrt[5]{34}=\sqrt[5]{32+2}\approx 2+\frac{2}{5\cdot 16}=2.025.
  38. z n = x n + y n = x + y n x n - 1 + ( n - 1 ) y 2 x + ( n + 1 ) y 3 n x n - 1 + ( 2 n - 1 ) y 2 x + ( 2 n + 1 ) y 5 n x n - 1 + ( 3 n - 1 ) y 2 x + ; \sqrt[n]{z}=\sqrt[n]{x^{n}+y}=x+\cfrac{y}{nx^{n-1}+\cfrac{(n-1)y}{2x+\cfrac{(n% +1)y}{3nx^{n-1}+\cfrac{(2n-1)y}{2x+\cfrac{(2n+1)y}{5nx^{n-1}+\cfrac{(3n-1)y}{2% x+\ddots}}}}}};
  39. z n = x + 2 x y n ( 2 z - y ) - y - ( 1 2 n 2 - 1 ) y 2 3 n ( 2 z - y ) - ( 2 2 n 2 - 1 ) y 2 5 n ( 2 z - y ) - ( 3 2 n 2 - 1 ) y 2 7 n ( 2 z - y ) - . \sqrt[n]{z}=x+\cfrac{2x\cdot y}{n(2z-y)-y-\cfrac{(1^{2}n^{2}-1)y^{2}}{3n(2z-y)% -\cfrac{(2^{2}n^{2}-1)y^{2}}{5n(2z-y)-\cfrac{(3^{2}n^{2}-1)y^{2}}{7n(2z-y)-% \ddots}}}}.
  40. 34 5 = 2 + 1 40 + 4 4 + 6 120 + 9 4 + 11 200 + 14 4 + = 2 + 4 1 165 - 1 - 4 6 495 - 9 11 825 - 14 16 1155 - . \sqrt[5]{34}=2+\cfrac{1}{40+\cfrac{4}{4+\cfrac{6}{120+\cfrac{9}{4+\cfrac{11}{2% 00+\cfrac{14}{4+\ddots}}}}}}=2+\cfrac{4\cdot 1}{165-1-\cfrac{4\cdot 6}{495-% \cfrac{9\cdot 11}{825-\cfrac{14\cdot 16}{1155-\ddots}}}}.
  41. x ( 20 p + x ) c x(20p+x)\leq c
  42. x 2 + 20 x p c x^{2}+20xp\leq c
  43. P ( n , i ) P(n,i)
  44. i i
  45. n n
  46. P ( 4 , 1 ) = 4 P(4,1)=4
  47. i = 0 n - 1 10 i P ( n , i ) p i x n - i \sum_{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}
  48. y y
  49. 10 n 10^{n}
  50. p p
  51. p = 0 p=0
  52. x x
  53. y c y\leq c
  54. x x
  55. y y
  56. c c
  57. r n = x , r^{n}=x,
  58. n log 10 r = log 10 x hence log 10 r = log 10 x n . n\log_{10}r=\log_{10}x\quad\quad\,\text{hence}\quad\quad\log_{10}r=\frac{\log_% {10}x}{n}.
  59. r = 10 log 10 x n . r=10^{\frac{\log_{10}x}{n}}.
  60. | r | n = | x | , |r|^{n}=|x|,
  61. 4 −4
  62. 2 i 2i
  63. 2 i −2i
  64. i i
  65. 1 2 ( 1 + i ) and - 1 2 ( 1 + i ) . \tfrac{1}{\sqrt{2}}(1+i)\quad\,\text{and}\quad-\tfrac{1}{\sqrt{2}}(1+i).
  66. r e i θ = ± r e i θ / 2 . \sqrt{re^{i\theta}}\,=\,\pm\sqrt{r}\,e^{i\theta/2}.
  67. r e i θ = r e i θ / 2 \sqrt{re^{i\theta}}\,=\,\sqrt{r}\,e^{i\theta/2}
  68. z \scriptstyle\sqrt{z}
  69. z \scriptstyle z
  70. 1 , ω , ω 2 , , ω n - 1 , 1,\;\omega,\;\omega^{2},\;\ldots,\;\omega^{n-1},
  71. ω = e 2 π i / n = cos ( 2 π n ) + i sin ( 2 π n ) \omega\,=\,e^{2\pi i/n}\,=\,\cos\left(\frac{2\pi}{n}\right)+i\sin\left(\frac{2% \pi}{n}\right)
  72. 2 π / n 2\pi/n
  73. i i
  74. - i -i
  75. η , η ω , η ω 2 , , η ω n - 1 , \eta,\;\eta\omega,\;\eta\omega^{2},\;\ldots,\;\eta\omega^{n-1},
  76. 2 4 , i 2 4 , - 2 4 , and - i 2 4 . \sqrt[4]{2},\quad i\sqrt[4]{2},\quad-\sqrt[4]{2},\quad\,\text{and}\quad-i\sqrt% [4]{2}.
  77. r e i θ n = r n e i θ / n . \sqrt[n]{re^{i\theta}}\,=\,\sqrt[n]{r}\,e^{i\theta/n}.
  78. r = a 2 + b 2 r=\sqrt{a^{2}+b^{2}}
  79. θ \theta
  80. cos θ = a / r , \cos\theta=a/r,
  81. sin θ = b / r , \sin\theta=b/r,
  82. tan θ = b / a . \tan\theta=b/a.
  83. θ / n \theta/n
  84. θ \theta
  85. x 5 = x + 1 x^{5}=x+1\,

Nuclear_isomer.html

  1. \hbar

Nuclear_magneton.html

  1. μ N = e 2 m p \mu_{\mathrm{N}}={{e\hbar}\over{2m_{\mathrm{p}}}}
  2. μ N = e 2 m p c \mu_{\mathrm{N}}={{e\hbar}\over{2m_{\mathrm{p}}c}}

Nuclear_weapon_design.html

  1. U 235 + n Sr 95 + Xe 139 + 2 n + 180 MeV \ {}^{235}\mathrm{U}+n\longrightarrow{}^{95}\mathrm{Sr}+{}^{139}\mathrm{Xe}+2n% +180\ \mathrm{MeV}
  2. D 2 + 3 T 4 He + n + 17.6 MeV \ {}^{2}\mathrm{D}+^{3}\!\mathrm{T}\longrightarrow^{4}\!\!\mathrm{He}+n+17.6\ % \mathrm{MeV}
  3. Li 6 + n 4 He + 3 T + 3 5 MeV \ {}^{6}\mathrm{Li}+n\longrightarrow^{4}\!\!\mathrm{He}+^{3}\!\mathrm{T}+5\ % \mathrm{MeV}

Number_line.html

  1. \mathbb{R}

Numerical_digit.html

  1. f ( x ) f(x)\,
  2. x x\,
  3. A + B = C A+B=C\,
  4. f ( f ( A ) + f ( B ) ) = f ( C ) f(f(A)+f(B))=f(C)\,

Numerical_integration.html

  1. a b f ( x ) d x \int_{a}^{b}\!f(x)\,dx
  2. f ( x ) f(x)
  3. x = a b x=\sqrt{ab}
  4. a b f ( x ) d x ( b - a ) f ( a + b 2 ) . \int_{a}^{b}f(x)\,dx\approx(b-a)\,f\left(\frac{a+b}{2}\right).
  5. a b f ( x ) d x ( b - a ) f ( a ) + f ( b ) 2 . \int_{a}^{b}f(x)\,dx\approx(b-a)\,\frac{f(a)+f(b)}{2}.
  6. a b f ( x ) d x b - a n ( f ( a ) 2 + k = 1 n - 1 ( f ( a + k b - a n ) ) + f ( b ) 2 ) \int_{a}^{b}f(x)\,dx\approx\frac{b-a}{n}\left({f(a)\over 2}+\sum_{k=1}^{n-1}% \left(f\left(a+k\frac{b-a}{n}\right)\right)+{f(b)\over 2}\right)
  7. | a b f ( x ) d x - ( b - a ) f ( a ) | = | a b ( x - a ) f ( v x ) d x | \left|\int_{a}^{b}f(x)\,dx-(b-a)f(a)\right|=\left|\int_{a}^{b}(x-a)f^{\prime}(% v_{x})\,dx\right|
  8. | a b f ( x ) d x - ( b - a ) f ( a ) | ( b - a ) 2 2 sup a x b | f ( x ) | \left|\int_{a}^{b}f(x)\,dx-(b-a)f(a)\right|\leq{(b-a)^{2}\over 2}\sup_{a\leq x% \leq b}\left|f^{\prime}(x)\right|
  9. n - 1 2 sup 0 x 1 | f ( x ) | {n^{-1}\over 2}\sup_{0\leq x\leq 1}\left|f^{\prime}(x)\right|
  10. f ( x ) = x f(x)=x
  11. - + f ( x ) d x = - 1 + 1 f ( t 1 - t 2 ) 1 + t 2 ( 1 - t 2 ) 2 d t , \int_{-\infty}^{+\infty}f(x)\,dx=\int_{-1}^{+1}f\left(\frac{t}{1-t^{2}}\right)% \frac{1+t^{2}}{(1-t^{2})^{2}}\,dt,
  12. a + f ( x ) d x = 0 1 f ( a + t 1 - t ) d t ( 1 - t ) 2 \int_{a}^{+\infty}f(x)\,dx=\int_{0}^{1}f\left(a+\frac{t}{1-t}\right)\frac{dt}{% (1-t)^{2}}
  13. - a f ( x ) d x = 0 1 f ( a - 1 - t t ) d t t 2 \int_{-\infty}^{a}f(x)\,dx=\int_{0}^{1}f\left(a-\frac{1-t}{t}\right)\frac{dt}{% t^{2}}
  14. F ( x ) = a x f ( u ) d u F(x)=\int_{a}^{x}f(u)\,du
  15. d F ( x ) d x = f ( x ) , F ( a ) = 0. \frac{dF(x)}{dx}=f(x),\quad F(a)=0.

Numerical_methods_for_ordinary_differential_equations.html

  1. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , ( 1 ) y^{\prime}(t)=f(t,y(t)),\qquad y(t_{0})=y_{0},\qquad\qquad(1)
  2. y ( t ) y ( t + h ) - y ( t ) h , ( 2 ) y^{\prime}(t)\approx\frac{y(t+h)-y(t)}{h},\qquad\qquad(2)
  3. y ( t + h ) y ( t ) + h y ( t ) y(t+h)\approx y(t)+hy^{\prime}(t)\qquad\qquad
  4. y ( t + h ) y ( t ) + h f ( t , y ( t ) ) . ( 3 ) y(t+h)\approx y(t)+hf(t,y(t)).\qquad\qquad(3)
  5. y n + 1 = y n + h f ( t n , y n ) . ( 4 ) y_{n+1}=y_{n}+hf(t_{n},y_{n}).\qquad\qquad(4)
  6. y ( t ) y ( t ) - y ( t - h ) h , ( 5 ) y^{\prime}(t)\approx\frac{y(t)-y(t-h)}{h},\qquad\qquad(5)
  7. y n + 1 = y n + h f ( t n + 1 , y n + 1 ) . ( 6 ) y_{n+1}=y_{n}+hf(t_{n+1},y_{n+1}).\qquad\qquad(6)
  8. y ( t ) = - A y + 𝒩 ( y ) , ( 7 ) y^{\prime}(t)=-A\,y+\mathcal{N}(y),\qquad\qquad\qquad(7)
  9. - A y -Ay
  10. 𝒩 ( y ) \mathcal{N}(y)
  11. e A t e^{At}
  12. [ t n , t n + 1 = t n + h ] [t_{n},t_{n+1}=t_{n}+h]
  13. y n + 1 = e - A h y n + 0 h e - ( h - τ ) A 𝒩 ( y ( t n + τ ) ) d τ . y_{n+1}=e^{-Ah}y_{n}+\int_{0}^{h}e^{-(h-\tau)A}\mathcal{N}\left(y\left(t_{n}+% \tau\right)\right)\,d\tau.
  14. 𝒩 ( y ( t n + τ ) ) \mathcal{N}(y(t_{n}+\tau))
  15. y n + 1 = e - A h y n + A - 1 ( 1 - e - A h ) 𝒩 ( y ( t n ) ) . ( 8 ) y_{n+1}=e^{-Ah}y_{n}+A^{-1}(1-e^{-Ah})\mathcal{N}(y(t_{n}))\ .\qquad\qquad(8)
  16. α k y n + k + α k - 1 y n + k - 1 + + α 0 y n \alpha_{k}y_{n+k}+\alpha_{k-1}y_{n+k-1}+\cdots+\alpha_{0}y_{n}
  17. = h [ β k f ( t n + k , y n + k ) + β k - 1 f ( t n + k - 1 , y n + k - 1 ) + + β 0 f ( t n , y n ) ] . =h\left[\beta_{k}f(t_{n+k},y_{n+k})+\beta_{k-1}f(t_{n+k-1},y_{n+k-1})+\cdots+% \beta_{0}f(t_{n},y_{n})\right].
  18. lim h 0 + max n = 0 , 1 , , t * / h y n , h - y ( t n ) = 0. \lim_{h\to 0+}\max_{n=0,1,\dots,\lfloor t^{*}/h\rfloor}\|y_{n,h}-y(t_{n})\|=0.
  19. y n + k = Ψ ( t n + k ; y n , y n + 1 , , y n + k - 1 ; h ) . y_{n+k}=\Psi(t_{n+k};y_{n},y_{n+1},\dots,y_{n+k-1};h).\,
  20. δ n + k h = Ψ ( t n + k ; y ( t n ) , y ( t n + 1 ) , , y ( t n + k - 1 ) ; h ) - y ( t n + k ) . \delta^{h}_{n+k}=\Psi\left(t_{n+k};y(t_{n}),y(t_{n+1}),\dots,y(t_{n+k-1});h% \right)-y(t_{n+k}).
  21. lim h 0 δ n + k h h = 0. \lim_{h\to 0}\frac{\delta^{h}_{n+k}}{h}=0.
  22. p p
  23. δ n + k h = O ( h p + 1 ) as h 0. \delta^{h}_{n+k}=O(h^{p+1})\quad\mbox{as }~{}h\to 0.
  24. u i + 1 - u i - 1 2 h = u ( x i ) + 𝒪 ( h 2 ) , \frac{u_{i+1}-u_{i-1}}{2h}=u^{\prime}(x_{i})+\mathcal{O}(h^{2}),
  25. u i + 1 - 2 u i + u i - 1 h 2 = u ′′ ( x i ) + 𝒪 ( h 2 ) . \frac{u_{i+1}-2u_{i}+u_{i-1}}{h^{2}}=u^{\prime\prime}(x_{i})+\mathcal{O}(h^{2}).
  26. h = x i - x i - 1 h=x_{i}-x_{i-1}
  27. d 2 u d x 2 - u = 0 , \frac{d^{2}u}{dx^{2}}-u=0,
  28. u ( 0 ) = 0 , u(0)=0,
  29. u ( 1 ) = 1. u(1)=1.
  30. u i ′′ = u i + 1 - 2 u i + u i - 1 h 2 u^{\prime\prime}_{i}=\frac{u_{i+1}-2u_{i}+u_{i-1}}{h^{2}}
  31. u i + 1 - 2 u i + u i - 1 h 2 - u i = 0 , i = 1 , 2 , 3 , , n - 1. \frac{u_{i+1}-2u_{i}+u_{i-1}}{h^{2}}-u_{i}=0,\quad\forall i={1,2,3,...,n-1}.
  32. u ( 0 ) = u 0 u(0)=u_{0}
  33. u ( 1 ) = u n u(1)=u_{n}

Numerical_stability.html

  1. f f
  2. f f
  3. f f
  4. x x
  5. y y
  6. y y
  7. y y
  8. Δ y = y * y Δy=y*−y
  9. x x
  10. f ( x + Δ x ) = y * f(x+Δx)=y*
  11. | Δ x | | x | \frac{|\Delta x|}{|x|}
  12. x x
  13. x x
  14. x x
  15. x x
  16. f ( x + Δ x ) y * f(x+Δx)−y*

Numerically_controlled_oscillator.html

  1. Δ F \Delta F
  2. ϕ n \phi_{n}
  3. ϕ n \phi_{n}
  4. ϕ 0 \phi_{0}
  5. GRR = 2 N GCD ( Δ F , 2 N ) \mbox{GRR}~{}=\frac{2^{N}}{\mbox{GCD}~{}(\Delta F,2^{N})}
  6. Δ F \Delta F
  7. F o u t = Δ F 2 N F c l o c k F_{out}=\frac{\Delta F}{2^{N}}F_{clock}
  8. F r e s = F c l o c k 2 N F_{res}=\frac{F_{clock}}{2^{N}}
  9. Δ F / 2 N \Delta F/2^{N}
  10. n W = 2 W GCD ( Δ F , 2 W ) - 1 n_{W}=\frac{2^{W}}{\mbox{GCD}~{}(\Delta F,2^{W})}-1
  11. ζ m a x = 2 - M π GCD ( Δ F , 2 W ) sin ( π 2 - P GCD ( Δ F , 2 W ) ) \zeta_{max}=2^{-M}\frac{\pi\mbox{GCD}~{}(\Delta F,2^{W})}{\sin\left(\pi\cdot 2% ^{-P}\mbox{GCD}~{}(\Delta F,2^{W})\right)}
  12. ζ m a x - 6.02 P dBc . \zeta_{max}\approx-6.02\cdot P\;\mbox{dBc}~{}.

Observable.html

  1. | a |a\rangle
  2. 𝐀 \mathbf{A}
  3. a a
  4. 𝐀 \mathbf{A}
  5. | a |a\rangle
  6. a a
  7. | a . |a\rangle.
  8. 𝐀 \scriptstyle\mathbf{A}
  9. | a \scriptstyle|a\rangle
  10. a a
  11. | ϕ \scriptstyle|\phi\rangle\in\mathcal{H}
  12. a a
  13. | a | ϕ | 2 \scriptstyle|\langle a|\phi\rangle|^{2}
  14. 𝐀𝐁 - 𝐁𝐀 0. \mathbf{A}\mathbf{B}-\mathbf{B}\mathbf{A}\neq\mathbf{0}.
  15. 𝐀 \scriptstyle\mathbf{A}
  16. 𝐁 \scriptstyle\mathbf{B}
  17. L 2 ( \reals ) L^{2}(\reals)

Observable_universe.html

  1. a ( t ) = 1 1 + z \!a(t)=\frac{1}{1+z}
  2. 1 / 1092.64 {1}/{1092.64}
  3. ρ c \rho_{c}
  4. ρ c = 3 H 0 2 8 π G \rho_{c}=\frac{3H_{0}^{2}}{8\pi G}
  5. H 0 H_{0}
  6. 4 / 3 {4}/{3}
  7. 4 3 π ρ ( c H ) 3 \frac{4}{3}\pi\rho\left(\frac{c}{H}\right)^{3}
  8. c 3 2 G H \frac{c^{3}}{2GH}

Octagon.html

  1. A = 2 cot π 8 a 2 = 2 ( 1 + 2 ) a 2 4.828427125 a 2 . A=2\cot\frac{\pi}{8}a^{2}=2(1+\sqrt{2})a^{2}\simeq 4.828427125\,a^{2}.
  2. A = 4 sin π 4 R 2 = 2 2 R 2 2.828427 R 2 . A=4\sin\frac{\pi}{4}R^{2}=2\sqrt{2}R^{2}\simeq 2.828427\,R^{2}.
  3. A = 8 tan π 8 r 2 = 8 ( 2 - 1 ) r 2 3.3137085 r 2 . A=8\tan\frac{\pi}{8}r^{2}=8(\sqrt{2}-1)r^{2}\simeq 3.3137085\,r^{2}.
  4. A = S 2 - a 2 , \,\!A=S^{2}-a^{2},
  5. S = a 2 + a + a 2 = ( 1 + 2 ) a S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a
  6. S 2.414 a S\approx 2.414a
  7. A = ( ( 1 + 2 ) a ) 2 - a 2 = 2 ( 1 + 2 ) a 2 A=((1+\sqrt{2})a)^{2}-a^{2}=2(1+\sqrt{2})a^{2}
  8. A 4.828 a 2 A\approx 4.828a^{2}
  9. A = 2 ( 2 - 1 ) S 2 A=2(\sqrt{2}-1)S^{2}
  10. A 0.828 S 2 A\approx 0.828S^{2}
  11. A = 2 a S . \ A=2aS.
  12. a S / 2.414 a\approx S/2.414
  13. e = a / 2 , e=a/\sqrt{2},
  14. e = ( S - a ) / 2 \,\!e=(S-a)/2
  15. Area = 2 a 2 ( 2 + 1 ) \,\text{Area}=2a^{2}(\sqrt{2}+1)

Odds.html

  1. 2 : 5 = ( 2 / 7 ) : ( 5 / 7 ) . 2:5=(2/7):(5/7).
  2. o f o_{f}
  3. o a o_{a}
  4. o f \displaystyle o_{f}
  5. p \displaystyle p
  6. p : q p:q
  7. o f \displaystyle o_{f}
  8. o f , o_{f},
  9. o f : 1 , o_{f}:1,
  10. 1 : ( 1 / o f ) = 1 : o a , 1:(1/o_{f})=1:o_{a},
  11. p \displaystyle p
  12. 01 ¯ \overline{01}
  13. 0099 ¯ \overline{0099}
  14. o f o_{f}
  15. o a o_{a}
  16. p p
  17. q q
  18. 1 ¯ \overline{1}
  19. 90 ¯ \overline{90}
  20. 09 ¯ \overline{09}
  21. 01 ¯ \overline{01}
  22. 0099 ¯ \overline{0099}
  23. 90 ¯ \overline{90}
  24. p 1 - p \frac{p}{1-p}
  25. 1 - p p \frac{1-p}{p}

Oil_drop_experiment.html

  1. F d = 6 π r η v 1 F_{d}=6\pi r\eta v_{1}\,
  2. s y m b o l w = 4 π 3 r 3 ( ρ - ρ a i r ) s y m b o l g symbol{w}=\frac{4\pi}{3}r^{3}(\rho-\rho_{air})symbol{g}
  3. r 2 = 9 η v 1 2 g ( ρ - ρ a i r ) . r^{2}=\frac{9\eta v_{1}}{2g(\rho-\rho_{air})}.\,
  4. F E = q E F_{E}=qE\,
  5. E = V d E=\frac{V}{d}\,
  6. q s y m b o l E - s y m b o l w = 6 π \etasymbol ( r v 2 ) = | s y m b o l v 2 v 1 | s y m b o l w qsymbol{E}-symbol{w}=6\pi\etasymbol{(r\cdot v_{2})}=\left|symbol{\frac{v_{2}}{% v_{1}}}\right|symbol{w}

Oil_well.html

  1. E L o i l = W I × L O E N R I [ P o + ( P g × G O R ) / 1 , 000 ] × ( 1 - T ) {EL}_{oil}=\frac{{WI}\times{LOE}}{{NRI}[{P_{o}}+({P_{g}}\times{GOR})/1,000]% \times(1-{T})}
  2. E L g a s = W I × L O E N R I [ ( P o × Y ) + P g ] × ( 1 - T ) {EL}_{gas}=\frac{{WI}\times{LOE}}{{NRI}[({P_{o}}\times{Y})+{P_{g}}]\times(1-{T% })}
  3. E L o i l {EL}_{oil}
  4. E L g a s {EL}_{gas}
  5. P o , P g {P}_{o},{P}_{g}
  6. L O E {LOE}
  7. W I {WI}
  8. N R I {NRI}
  9. G O R {GOR}
  10. Y {Y}
  11. T {T}

One-form.html

  1. \mathbb{R}
  2. α : T M , α x = α | T x M : T x M \alpha:TM\rightarrow{\mathbb{R}},\quad\alpha_{x}=\alpha|_{T_{x}M}:T_{x}M% \rightarrow{\mathbb{R}}
  3. α x = f 1 ( x ) d x 1 + f 2 ( x ) d x 2 + + f n ( x ) d x n \alpha_{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots+f_{n}(x)\,dx_{n}
  4. mean ( v ) = [ 1 / n , 1 / n , , 1 / n ] v . \operatorname{mean}(v)=[1/n,1/n,\dots,1/n]\cdot v.
  5. NPV ( R ( t ) ) = w , R = t = 0 R ( t ) ( 1 + i ) t d t . \mathrm{NPV}(R(t))=\langle w,R\rangle=\int_{t=0}^{\infty}\frac{R(t)}{(1+i)^{t}% }\,dt.
  6. d θ . d\theta.
  7. θ ( x , y ) \theta(x,y)
  8. atan2 ( y , x ) = arctan ( y / x ) . \operatorname{atan2}(y,x)=\operatorname{arctan}(y/x).
  9. d θ \displaystyle d\theta
  10. U U\subseteq\mathbb{R}
  11. ( a , b ) (a,b)
  12. f : U f:U\to\mathbb{R}
  13. x 0 U x_{0}\in U
  14. d f ( x 0 , d x ) : d x f ( x 0 ) d x df(x_{0},dx):dx\mapsto f^{\prime}(x_{0})dx
  15. x d f ( x , d x ) x\mapsto df(x,dx)
  16. f d f f\mapsto df

Operational_semantics.html

  1. C 1 , C 2 C_{1},C_{2}
  2. s s
  3. E E
  4. V V
  5. L L
  6. E , s V L := E , s ( s ( L V ) ) \frac{\langle E,s\rangle\Rightarrow V}{\langle L:=E\,,\,s\rangle% \longrightarrow(s\uplus(L\mapsto V))}
  7. E E
  8. s s
  9. V V
  10. L := E L:=E
  11. s s
  12. L = V L=V
  13. C 1 , s s C 1 ; C 2 , s C 2 , s C 1 , s C 1 , s C 1 ; C 2 , s C 1 ; C 2 , s 𝐬𝐤𝐢𝐩 , s s \frac{\langle C_{1},s\rangle\longrightarrow s^{\prime}}{\langle C_{1};C_{2}\,,% s\rangle\longrightarrow\langle C_{2},s^{\prime}\rangle}\quad\quad\frac{\langle C% _{1},s\rangle\longrightarrow\langle C_{1}^{\prime},s^{\prime}\rangle}{\langle C% _{1};C_{2}\,,s\rangle\longrightarrow\langle C_{1}^{\prime};C_{2}\,,s^{\prime}% \rangle}\quad\quad\frac{}{\langle\mathbf{skip},s\rangle\longrightarrow s}
  14. C 1 C_{1}
  15. s s
  16. s s^{\prime}
  17. C 1 ; C 2 C_{1};C_{2}
  18. s s
  19. C 2 C_{2}
  20. s s^{\prime}
  21. C 1 C_{1}
  22. C 2 C_{2}
  23. C 1 C_{1}
  24. s s
  25. C 1 C_{1}^{\prime}
  26. s s^{\prime}
  27. C 1 ; C 2 C_{1};C_{2}
  28. s s
  29. C 1 ; C 2 C_{1}^{\prime};C_{2}
  30. s s^{\prime}
  31. C 1 C_{1}
  32. C 1 ; C 2 C_{1};C_{2}
  33. C 1 C_{1}
  34. C 2 C_{2}
  35. B B
  36. B , s 𝐭𝐫𝐮𝐞 𝐰𝐡𝐢𝐥𝐞 B 𝐝𝐨 C , s C ; 𝐰𝐡𝐢𝐥𝐞 B 𝐝𝐨 C , s B , s 𝐟𝐚𝐥𝐬𝐞 𝐰𝐡𝐢𝐥𝐞 B 𝐝𝐨 C , s s \frac{\langle B,s\rangle\Rightarrow\mathbf{true}}{\langle\mathbf{while}\ B\ % \mathbf{do}\ C,s\rangle\longrightarrow\langle C;\mathbf{while}\ B\ \mathbf{do}% \ C,s\rangle}\quad\frac{\langle B,s\rangle\Rightarrow\mathbf{false}}{\langle% \mathbf{while}\ B\ \mathbf{do}\ C,s\rangle\longrightarrow s}
  37. e = v | ( e e ) | x v = λ x . e C = [ ] | ( C e ) | ( v C ) e=v\;|\;(e\;e)\;|\;x\quad\quad v=\lambda x.e\quad\quad C=\left[\,\right]\;|\;(% C\;e)\;|\;(v\;C)
  38. C C
  39. [ ] \left[\,\right]
  40. ( λ x . e v ) e [ x / v ] ( β ) (\lambda x.e\;v)\longrightarrow e\,\left[x/v\right]\quad(\mathrm{\beta})
  41. ( ( λ x . x λ x . x ) λ x . ( x x ) ) ((\lambda x.x\;\lambda x.x)\lambda x.(x\;x))
  42. ( [ ] λ x . ( x x ) ) ([\,]\;\lambda x.(x\;x))

Operator_(physics).html

  1. L ( q , q ˙ , t ) L(q,\dot{q},t)
  2. H ( q , p , t ) H(q,p,t)
  3. q ˙ = d q / d t \dot{q}=\mathrm{d}q/\mathrm{d}t
  4. p = L q ˙ p=\frac{\partial L}{\partial\dot{q}}
  5. S G , H ( S ( q , p ) ) = H ( q , p ) S\in G,H(S(q,p))=H(q,p)
  6. X ( a ) X({a})
  7. r r + a {r}\rightarrow{r}+{a}
  8. p p {p}\rightarrow{p}
  9. U ( t 0 ) U(t_{0})
  10. r ( t ) r ( t + t 0 ) {r}(t)\rightarrow{r}(t+t_{0})
  11. p ( t ) p ( t + t 0 ) {p}(t)\rightarrow{p}(t+t_{0})
  12. R ( n ^ , θ ) R({\hat{n}},\theta)
  13. r R ( n ^ , θ ) r {r}\rightarrow R({\hat{n}},\theta){r}
  14. p R ( n ^ , θ ) p {p}\rightarrow R({\hat{n}},\theta){p}
  15. G ( v ) G({v})
  16. r r + v t {r}\rightarrow{r}+{v}t
  17. p p + m v {p}\rightarrow{p}+m{v}
  18. P P
  19. r - r {r}\rightarrow-{r}
  20. p - p {p}\rightarrow-{p}
  21. T T
  22. r r ( - t ) {r}\rightarrow{r}(-t)
  23. p - p ( - t ) {p}\rightarrow-{p}(-t)
  24. R ( s y m b o l n ^ , θ ) R(\hat{symbol{n}},\theta)
  25. s y m b o l n ^ \hat{symbol{n}}
  26. I + ϵ A I+\epsilon A
  27. I I
  28. ϵ \epsilon
  29. A A
  30. T a f ( x ) = f ( x - a ) T_{a}f(x)=f(x-a)
  31. a = ϵ a=\epsilon
  32. T ϵ f ( x ) = f ( x - ϵ ) f ( x ) - ϵ f ( x ) . T_{\epsilon}f(x)=f(x-\epsilon)\approx f(x)-\epsilon f^{\prime}(x).
  33. T ϵ f ( x ) = ( I - ϵ D ) f ( x ) T_{\epsilon}f(x)=(I-\epsilon D)f(x)
  34. D D
  35. a a
  36. T a f ( x ) = lim N T a / N T a / N f ( x ) T_{a}f(x)=\lim_{N\to\infty}T_{a/N}\cdots T_{a/N}f(x)
  37. \cdots
  38. N N
  39. N N
  40. T a f ( x ) = lim N ( I - ( a / N ) D ) N f ( x ) . T_{a}f(x)=\lim_{N\to\infty}(I-(a/N)D)^{N}f(x).
  41. T a f ( x ) = exp ( - a D ) f ( x ) . T_{a}f(x)=\exp(-aD)f(x).
  42. T a f ( x ) = ( I - a D + a 2 D 2 2 ! - a 3 D 3 3 ! + ) f ( x ) . T_{a}f(x)=\left(I-aD+{a^{2}D^{2}\over 2!}-{a^{3}D^{3}\over 3!}+\cdots\right)f(% x).
  43. f ( x ) - a f ( x ) + a 2 2 ! f ′′ ( x ) - a 3 3 ! f ′′′ ( x ) + f(x)-af^{\prime}(x)+{a^{2}\over 2!}f^{\prime\prime}(x)-{a^{3}\over 3!}f^{% \prime\prime\prime}(x)+\cdots
  44. f ( x - a ) f(x-a)
  45. T a f ( x ) T_{a}f(x)
  46. - - - | ψ ( r ) | 2 d 3 r = - - - ψ ( r ) * ψ ( r ) d 3 r < \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|\psi({r}% )|^{2}{\rm d}^{3}{r}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\psi({r})^{*}\psi({r}){\rm d}^{3}{r}<\infty
  47. - - - | ψ ( r ) | 2 d 3 r = 1 \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|\psi({r}% )|^{2}{\rm d}^{3}{r}=1
  48. | ψ i |\psi_{i}\rangle
  49. | ψ = i c i | ϕ i |\psi\rangle=\sum_{i}c_{i}|\phi_{i}\rangle
  50. | ϕ i |\phi_{i}\rangle
  51. | ψ |\psi\rangle
  52. | ψ = c ( ϕ ) d ϕ | ϕ i |\psi\rangle=\int c(\phi){\rm d}\phi|\phi_{i}\rangle
  53. | ϕ |\phi\rangle
  54. A ^ \hat{A}
  55. A ^ ψ = a ψ , \hat{A}\psi=a\psi,
  56. A ^ \hat{A}
  57. A ^ ψ = A ^ ψ ( 𝐫 ) = A ^ 𝐫 ψ = 𝐫 A ^ ψ a ψ = a ψ ( 𝐫 ) = a 𝐫 ψ = 𝐫 a ψ \begin{aligned}&\displaystyle\hat{A}\psi=\hat{A}\psi(\mathbf{r})=\hat{A}\left% \langle\mathbf{r}\mid\psi\right\rangle=\left\langle\mathbf{r}\mid\hat{A}\mid% \psi\right\rangle\\ &\displaystyle a\psi=a\psi(\mathbf{r})=a\left\langle\mathbf{r}\mid\psi\right% \rangle=\left\langle\mathbf{r}\mid a\mid\psi\right\rangle\\ \end{aligned}
  58. | ψ \left|\psi\right\rangle
  59. 𝐀 ^ = j = 1 n 𝐞 j A ^ j \mathbf{\hat{A}}=\sum_{j=1}^{n}\mathbf{e}_{j}\hat{A}_{j}
  60. 𝐀 ^ ψ = ( j = 1 n 𝐞 j A ^ j ) ψ = j = 1 n ( 𝐞 j A ^ j ψ ) = j = 1 n ( 𝐞 j a j ψ ) \mathbf{\hat{A}}\psi=\left(\sum_{j=1}^{n}\mathbf{e}_{j}\hat{A}_{j}\right)\psi=% \sum_{j=1}^{n}\left(\mathbf{e}_{j}\hat{A}_{j}\psi\right)=\sum_{j=1}^{n}\left(% \mathbf{e}_{j}a_{j}\psi\right)
  61. A ^ j ψ = a j ψ . \hat{A}_{j}\psi=a_{j}\psi.
  62. 𝐀 ^ ψ = 𝐀 ^ ψ ( 𝐫 ) = 𝐀 ^ 𝐫 ψ = 𝐫 𝐀 ^ ψ ( j = 1 n 𝐞 j A ^ j ) ψ = ( j = 1 n 𝐞 j A ^ j ) ψ ( 𝐫 ) = ( j = 1 n 𝐞 j A ^ j ) 𝐫 ψ = 𝐫 j = 1 n 𝐞 j A ^ j ψ \begin{aligned}&\displaystyle\mathbf{\hat{A}}\psi=\mathbf{\hat{A}}\psi(\mathbf% {r})=\mathbf{\hat{A}}\left\langle\mathbf{r}\mid\psi\right\rangle=\left\langle% \mathbf{r}\mid\mathbf{\hat{A}}\mid\psi\right\rangle\\ &\displaystyle\left(\sum_{j=1}^{n}\mathbf{e}_{j}\hat{A}_{j}\right)\psi=\left(% \sum_{j=1}^{n}\mathbf{e}_{j}\hat{A}_{j}\right)\psi(\mathbf{r})=\left(\sum_{j=1% }^{n}\mathbf{e}_{j}\hat{A}_{j}\right)\left\langle\mathbf{r}\mid\psi\right% \rangle=\left\langle\mathbf{r}\mid\sum_{j=1}^{n}\mathbf{e}_{j}\hat{A}_{j}\mid% \psi\right\rangle\\ \end{aligned}\,\!
  63. A ^ \hat{A}
  64. B ^ \hat{B}
  65. [ A ^ , B ^ ] = A ^ B ^ - B ^ A ^ \left[\hat{A},\hat{B}\right]=\hat{A}\hat{B}-\hat{B}\hat{A}
  66. [ A ^ , B ^ ] ψ = A ^ B ^ ψ - B ^ A ^ ψ . \left[\hat{A},\hat{B}\right]\psi=\hat{A}\hat{B}\psi-\hat{B}\hat{A}\psi.
  67. [ A ^ , B ^ ] ψ = 0 , \left[\hat{A},\hat{B}\right]\psi=0,
  68. Δ A = 0 \Delta A=0
  69. Δ B = 0 \Delta B=0
  70. [ A ^ , B ^ ] ψ \displaystyle\left[\hat{A},\hat{B}\right]\psi
  71. [ A ^ , B ^ ] ψ 0 , \left[\hat{A},\hat{B}\right]\psi\neq 0,
  72. Δ A Δ B 2 \Delta A\Delta B\geq\frac{\hbar}{2}
  73. A ^ \langle\hat{A}\rangle
  74. A ^ \hat{A}
  75. A ^ = R ψ * ( 𝐫 ) A ^ ψ ( 𝐫 ) d 3 𝐫 = ψ | A ^ | ψ . \langle\hat{A}\rangle=\int_{R}\psi^{*}\left(\mathbf{r}\right)\hat{A}\psi\left(% \mathbf{r}\right)\mathrm{d}^{3}\mathbf{r}=\langle\psi|\hat{A}|\psi\rangle.
  76. F ( A ^ ) = R ψ ( 𝐫 ) * [ F ( A ^ ) ψ ( 𝐫 ) ] d 3 𝐫 = ψ | F ( A ^ ) | ψ , \langle F(\hat{A})\rangle=\int_{R}\psi(\mathbf{r})^{*}\left[F(\hat{A})\psi(% \mathbf{r})\right]\mathrm{d}^{3}\mathbf{r}=\langle\psi|F(\hat{A})|\psi\rangle,
  77. F ( A ^ ) = A ^ 2 A ^ 2 = R ψ * ( 𝐫 ) A ^ 2 ψ ( 𝐫 ) d 3 𝐫 = ψ | A ^ 2 | ψ \begin{aligned}&\displaystyle F(\hat{A})=\hat{A}^{2}\\ &\displaystyle\Rightarrow\langle\hat{A}^{2}\rangle=\int_{R}\psi^{*}\left(% \mathbf{r}\right)\hat{A}^{2}\psi\left(\mathbf{r}\right)\mathrm{d}^{3}\mathbf{r% }=\langle\psi|\hat{A}^{2}|\psi\rangle\\ \end{aligned}\,\!
  78. A ^ = A ^ \hat{A}=\hat{A}^{\dagger}
  79. ϕ i | A ^ | ϕ j = ϕ j | A ^ | ϕ i * . \langle\phi_{i}|\hat{A}|\phi_{j}\rangle=\langle\phi_{j}|\hat{A}|\phi_{i}% \rangle^{*}.
  80. ϕ j \phi_{j}
  81. A i j = ϕ i | A ^ | ϕ j , A_{ij}=\langle\phi_{i}|\hat{A}|\phi_{j}\rangle,
  82. A ^ = ( A 11 A 12 A 1 n A 21 A 22 A 2 n A n 1 A n 2 A n n ) \hat{A}=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nn}\\ \end{pmatrix}
  83. det ( A ^ - a I ^ ) = 0 , \det\left(\hat{A}-a\hat{I}\right)=0,
  84. I ^ = i | ϕ i ϕ i | \hat{I}=\sum_{i}|\phi_{i}\rangle\langle\phi_{i}|
  85. I ^ = | ϕ ϕ | d ϕ \hat{I}=\int|\phi\rangle\langle\phi|d\phi
  86. A ^ \hat{A}
  87. A ^ - 1 \hat{A}^{-1}
  88. A ^ A ^ - 1 = A ^ - 1 A ^ = I ^ \hat{A}\hat{A}^{-1}=\hat{A}^{-1}\hat{A}=\hat{I}
  89. det ( A ^ ) 0 \det(\hat{A})\neq 0
  90. x ^ = x y ^ = y z ^ = z \begin{aligned}\displaystyle\hat{x}=x\\ \displaystyle\hat{y}=y\\ \displaystyle\hat{z}=z\end{aligned}
  91. 𝐫 ^ = 𝐫 \mathbf{\hat{r}}=\mathbf{r}\,\!
  92. p ^ x = - i x p ^ y = - i y p ^ z = - i z \begin{aligned}\displaystyle\hat{p}_{x}&\displaystyle=-i\hbar\frac{\partial}{% \partial x}\\ \displaystyle\hat{p}_{y}&\displaystyle=-i\hbar\frac{\partial}{\partial y}\\ \displaystyle\hat{p}_{z}&\displaystyle=-i\hbar\frac{\partial}{\partial z}\end{aligned}
  93. 𝐩 ^ = - i \mathbf{\hat{p}}=-i\hbar\nabla\,\!
  94. p ^ x = - i x - q A x p ^ y = - i y - q A y p ^ z = - i z - q A z \begin{aligned}\displaystyle\hat{p}_{x}=-i\hbar\frac{\partial}{\partial x}-qA_% {x}\\ \displaystyle\hat{p}_{y}=-i\hbar\frac{\partial}{\partial y}-qA_{y}\\ \displaystyle\hat{p}_{z}=-i\hbar\frac{\partial}{\partial z}-qA_{z}\end{aligned}
  95. 𝐩 ^ = P ^ - q A = - i - q A \begin{aligned}\displaystyle\mathbf{\hat{p}}&\displaystyle={\hat{P}}-q{A}\\ &\displaystyle=-i\hbar\nabla-q{A}\\ \end{aligned}\,\!
  96. T ^ x = - 2 2 m 2 x 2 T ^ y = - 2 2 m 2 y 2 T ^ z = - 2 2 m 2 z 2 \begin{aligned}\displaystyle\hat{T}_{x}&\displaystyle=-\frac{\hbar^{2}}{2m}% \frac{\partial^{2}}{\partial x^{2}}\\ \displaystyle\hat{T}_{y}&\displaystyle=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}% }{\partial y^{2}}\\ \displaystyle\hat{T}_{z}&\displaystyle=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}% }{\partial z^{2}}\\ \end{aligned}
  97. T ^ = 𝐩 ^ 𝐩 ^ 2 m = ( - i ) ( - i ) 2 m = - 2 2 m 2 \begin{aligned}\displaystyle\hat{T}&\displaystyle=\frac{\mathbf{\hat{p}}\cdot% \mathbf{\hat{p}}}{2m}\\ &\displaystyle=\frac{(-i\hbar\nabla)\cdot(-i\hbar\nabla)}{2m}\\ &\displaystyle=\frac{-\hbar^{2}}{2m}\nabla^{2}\end{aligned}\,\!
  98. T ^ x = 1 2 m ( - i x - q A x ) 2 T ^ y = 1 2 m ( - i y - q A y ) 2 T ^ z = 1 2 m ( - i z - q A z ) 2 \begin{aligned}\displaystyle\hat{T}_{x}&\displaystyle=\frac{1}{2m}\left(-i% \hbar\frac{\partial}{\partial x}-qA_{x}\right)^{2}\\ \displaystyle\hat{T}_{y}&\displaystyle=\frac{1}{2m}\left(-i\hbar\frac{\partial% }{\partial y}-qA_{y}\right)^{2}\\ \displaystyle\hat{T}_{z}&\displaystyle=\frac{1}{2m}\left(-i\hbar\frac{\partial% }{\partial z}-qA_{z}\right)^{2}\end{aligned}\,\!
  99. T ^ = 𝐩 ^ 𝐩 ^ 2 m = 1 2 m ( - i - q A ) ( - i - q A ) = 1 2 m ( - i - q A ) 2 \begin{aligned}\displaystyle\hat{T}&\displaystyle=\frac{\mathbf{\hat{p}}\cdot% \mathbf{\hat{p}}}{2m}\\ &\displaystyle=\frac{1}{2m}(-i\hbar\nabla-q{A})\cdot(-i\hbar\nabla-q{A})\\ &\displaystyle=\frac{1}{2m}(-i\hbar\nabla-q{A})^{2}\end{aligned}\,\!
  100. T ^ x x = J ^ x 2 2 I x x T ^ y y = J ^ y 2 2 I y y T ^ z z = J ^ z 2 2 I z z \begin{aligned}\displaystyle\hat{T}_{xx}&\displaystyle=\frac{\hat{J}_{x}^{2}}{% 2I_{xx}}\\ \displaystyle\hat{T}_{yy}&\displaystyle=\frac{\hat{J}_{y}^{2}}{2I_{yy}}\\ \displaystyle\hat{T}_{zz}&\displaystyle=\frac{\hat{J}_{z}^{2}}{2I_{zz}}\\ \end{aligned}\,\!
  101. T ^ = J ^ J ^ 2 I \hat{T}=\frac{{\hat{J}}\cdot{\hat{J}}}{2I}\,\!
  102. V ^ = V ( 𝐫 , t ) = V \hat{V}=V\left(\mathbf{r},t\right)=V\,\!
  103. E ^ = i t \hat{E}=i\hbar\frac{\partial}{\partial t}\,\!
  104. E ^ = E \hat{E}=E\,\!
  105. H ^ = T ^ + V ^ = p ^ p ^ 2 m + V = p ^ 2 2 m + V \begin{aligned}\displaystyle\hat{H}&\displaystyle=\hat{T}+\hat{V}\\ &\displaystyle=\frac{{\hat{p}}\cdot{\hat{p}}}{2m}+V\\ &\displaystyle=\frac{\hat{p}^{2}}{2m}+V\\ \end{aligned}\,\!
  106. L ^ x = - i ( y z - z y ) L ^ y = - i ( z x - x z ) L ^ z = - i ( x y - y x ) \begin{aligned}\displaystyle\hat{L}_{x}&\displaystyle=-i\hbar\left(y{\partial% \over\partial z}-z{\partial\over\partial y}\right)\\ \displaystyle\hat{L}_{y}&\displaystyle=-i\hbar\left(z{\partial\over\partial x}% -x{\partial\over\partial z}\right)\\ \displaystyle\hat{L}_{z}&\displaystyle=-i\hbar\left(x{\partial\over\partial y}% -y{\partial\over\partial x}\right)\end{aligned}
  107. 𝐋 ^ = 𝐫 × - i \mathbf{\hat{L}}=\mathbf{r}\times-i\hbar\nabla
  108. S ^ x = 2 σ x S ^ y = 2 σ y S ^ z = 2 σ z \begin{aligned}\displaystyle\hat{S}_{x}={\hbar\over 2}\sigma_{x}\\ \displaystyle\hat{S}_{y}={\hbar\over 2}\sigma_{y}\\ \displaystyle\hat{S}_{z}={\hbar\over 2}\sigma_{z}\end{aligned}
  109. σ x = ( 0 1 1 0 ) \sigma_{x}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}
  110. σ y = ( 0 - i i 0 ) \sigma_{y}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}
  111. σ z = ( 1 0 0 - 1 ) \sigma_{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  112. 𝐒 ^ = 2 s y m b o l σ \mathbf{\hat{S}}={\hbar\over 2}symbol{\sigma}\,\!
  113. J ^ x = L ^ x + S ^ x J ^ y = L ^ y + S ^ y J ^ z = L ^ z + S ^ z \begin{aligned}\displaystyle\hat{J}_{x}&\displaystyle=\hat{L}_{x}+\hat{S}_{x}% \\ \displaystyle\hat{J}_{y}&\displaystyle=\hat{L}_{y}+\hat{S}_{y}\\ \displaystyle\hat{J}_{z}&\displaystyle=\hat{L}_{z}+\hat{S}_{z}\end{aligned}
  114. 𝐉 ^ = 𝐋 ^ + 𝐒 ^ = - i r × + 2 s y m b o l σ \begin{aligned}\displaystyle\mathbf{\hat{J}}&\displaystyle=\mathbf{\hat{L}}+% \mathbf{\hat{S}}\\ &\displaystyle=-i\hbar{r}\times\nabla+\frac{\hbar}{2}symbol{\sigma}\end{aligned}
  115. d ^ x = q x ^ d ^ y = q y ^ d ^ z = q z ^ \begin{aligned}\displaystyle\hat{d}_{x}&\displaystyle=q\hat{x}\\ \displaystyle\hat{d}_{y}&\displaystyle=q\hat{y}\\ \displaystyle\hat{d}_{z}&\displaystyle=q\hat{z}\end{aligned}
  116. 𝐝 ^ = q 𝐫 ^ \mathbf{\hat{d}}=q\mathbf{\hat{r}}
  117. p ^ = - i x \hat{p}=-i\hbar\frac{\partial}{\partial x}
  118. p ^ ψ = - i x ψ , \hat{p}\psi=-i\hbar\frac{\partial}{\partial x}\psi,
  119. p ^ \hat{p}
  120. - i x ψ = p ψ . -i\hbar\frac{\partial}{\partial x}\psi=p\psi.
  121. 𝐩 ^ = - i . \mathbf{\hat{p}}=-i\hbar\nabla.
  122. 𝐞 x p ^ x + 𝐞 y p ^ y + 𝐞 z p ^ z = - i ( 𝐞 x x + 𝐞 y y + 𝐞 z z ) , \mathbf{e}_{\mathrm{x}}\hat{p}_{x}+\mathbf{e}_{\mathrm{y}}\hat{p}_{y}+\mathbf{% e}_{\mathrm{z}}\hat{p}_{z}=-i\hbar\left(\mathbf{e}_{\mathrm{x}}\frac{\partial}% {\partial x}+\mathbf{e}_{\mathrm{y}}\frac{\partial}{\partial y}+\mathbf{e}_{% \mathrm{z}}\frac{\partial}{\partial z}\right),
  123. p ^ x = - i x , p ^ y = - i y , p ^ z = - i z \hat{p}_{x}=-i\hbar\frac{\partial}{\partial x},\quad\hat{p}_{y}=-i\hbar\frac{% \partial}{\partial y},\quad\hat{p}_{z}=-i\hbar\frac{\partial}{\partial z}\,\!
  124. 𝐩 ^ \mathbf{\hat{p}}
  125. p ^ x ψ = - i x ψ = p x ψ p ^ y ψ = - i y ψ = p y ψ p ^ z ψ = - i z ψ = p z ψ \begin{aligned}\displaystyle\hat{p}_{x}\psi&\displaystyle=-i\hbar\frac{% \partial}{\partial x}\psi=p_{x}\psi\\ \displaystyle\hat{p}_{y}\psi&\displaystyle=-i\hbar\frac{\partial}{\partial y}% \psi=p_{y}\psi\\ \displaystyle\hat{p}_{z}\psi&\displaystyle=-i\hbar\frac{\partial}{\partial z}% \psi=p_{z}\psi\\ \end{aligned}\,\!

Operator_norm.html

  1. A v c v for all v V \|Av\|\leq c\|v\|\quad\mbox{ for all }~{}v\in V
  2. A o p = inf { c 0 : A v c v for all v V } \|A\|_{op}=\inf\{c\geq 0:\|Av\|\leq c\|v\|\mbox{ for all }~{}v\in V\}
  3. l 2 l^{2}
  4. l 2 = { ( a n ) n 1 : a n , n | a n | 2 < } . l^{2}=\{(a_{n})_{n\geq 1}:\;a_{n}\in\mathbb{C},\;\sum_{n}|a_{n}|^{2}<\infty\}.
  5. s = sup n | s n | . \|s\|_{\infty}=\sup_{n}|s_{n}|.
  6. ( a n ) T s ( s n a n ) . (a_{n})\stackrel{T_{s}}{\longrightarrow}(s_{n}\cdot a_{n}).
  7. T s o p = s . \|T_{s}\|_{op}=\|s\|_{\infty}.
  8. A o p \displaystyle\|A\|_{op}
  9. A o p 0 and A o p = 0 if and only if A = 0 , \|A\|_{op}\geq 0\mbox{ and }~{}\|A\|_{op}=0\mbox{ if and only if }~{}A=0,
  10. a A o p = | a | A o p for every scalar a , \|aA\|_{op}=|a|\|A\|_{op}\quad\mbox{ for every scalar }~{}a,
  11. A + B o p A o p + B o p . \|A+B\|_{op}\leq\|A\|_{op}+\|B\|_{op}.
  12. A v A o p v for every v V . \|Av\|\leq\|A\|_{op}\|v\|\quad\mbox{ for every }~{}v\in V.
  13. B A o p B o p A o p . \|BA\|_{op}\leq\|B\|_{op}\|A\|_{op}.
  14. 1 \ell_{1}
  15. 2 \ell_{2}
  16. \ell_{\infty}
  17. 1 \ell_{1}
  18. 1 \ell_{1}
  19. 2 \ell_{2}
  20. 2 \ell_{2}
  21. 2 \ell_{2}
  22. \ell_{\infty}
  23. 1 \ell_{1}
  24. A o p = A * o p \|A\|_{op}=\|A^{*}\|_{op}
  25. A * A o p = A o p 2 \|A^{*}A\|_{op}=\|A\|_{op}^{2}
  26. ρ ( A ) A o p . \rho(A)\leq\|A\|_{op}.
  27. ρ ( N ) = N o p . \rho(N)=\|N\|_{op}.
  28. P t ( f ) = f Ω t . P_{t}(f)=f\cdot\Omega_{t}.
  29. P t - P s o p = 1 , for all t s . \|P_{t}-P_{s}\|_{op}=1,\quad\mbox{for all}~{}\quad t\neq s.

Optical_microscope.html

  1. d = λ 2 N A d=\frac{\lambda}{2NA}

Optical_telescope.html

  1. R R
  2. D D
  3. D o b D_{ob}
  4. Φ \Phi
  5. D a D_{a}
  6. R R
  7. λ {\lambda}
  8. R = λ 10 6 = 550 10 6 = 0.00055 R=\frac{\lambda}{10^{6}}=\frac{550}{10^{6}}=0.00055
  9. Φ \Phi
  10. D a = 313 Π 10800 D_{a}=\frac{313\Pi}{10800}
  11. D a = 313 Π 10800 * 206265 = 1878 D_{a}=\frac{313\Pi}{10800}*206265=1878
  12. F = 2 R D * D o b * Φ D a = 2 * 0.00055 130 * 3474.2 * 206265 1878 3.22 F=\frac{\frac{2R}{D}*D_{ob}*\Phi}{D_{a}}=\frac{\frac{2*0.00055}{130}*3474.2*20% 6265}{1878}\approx 3.22
  13. α R \alpha_{R}
  14. sin ( α R ) = 1.22 λ D \sin(\alpha_{R})=1.22\frac{\lambda}{D}
  15. λ \lambda
  16. D D
  17. λ \lambda
  18. α R = 138 D \alpha_{R}=\frac{138}{D}
  19. α R \alpha_{R}
  20. D D
  21. α R \alpha_{R}
  22. α D = 116 D \alpha_{D}=\frac{116}{D}
  23. N N
  24. f f
  25. D D
  26. N = f D = 1200 254 4.7 N=\frac{f}{D}=\frac{1200}{254}\approx 4.7
  27. P P
  28. D D
  29. D p D_{p}
  30. P = ( D D p ) 2 = ( 254 7 ) 2 1316.7 P=\left(\frac{D}{D_{p}}\right)^{2}=\left(\frac{254}{7}\right)^{2}\approx 1316.7
  31. A A
  32. p = A 1 A 2 = π 5 2 π 1 2 = 25 p=\frac{A_{1}}{A_{2}}=\frac{\pi 5^{2}}{\pi 1^{2}}=25
  33. M M
  34. f f
  35. f e f_{e}
  36. M = f f e = 1200 3 = 400 M=\frac{f}{f_{e}}=\frac{1200}{3}=400
  37. M m M_{m}
  38. D D
  39. D e p D_{ep}
  40. D e p D_{ep}
  41. D D
  42. M M
  43. M m = D D e p = 254 7 36 M_{m}=\frac{D}{D_{ep}}=\frac{254}{7}\approx 36
  44. D e p = D M = 254 36 7 D_{ep}=\frac{D}{M}=\frac{254}{36}\approx 7
  45. v t v_{t}
  46. v a v_{a}
  47. M M
  48. v t = v a M = 52 81.25 = 0.64 v_{t}=\frac{v_{a}}{M}=\frac{52}{81.25}=0.64^{\circ}
  49. v m v_{m}
  50. B B
  51. f f
  52. v m = B * 180 π f 31.75 * 57.2958 1200 1.52 v_{m}=B*\frac{\frac{180}{\pi}}{f}\approx 31.75*\frac{57.2958}{1200}\approx 1.5% 2^{\circ}
  53. D D
  54. f f
  55. d d
  56. v a v_{a}
  57. M = f d = 650 8 = 81.25 M=\frac{f}{d}=\frac{650}{8}=81.25
  58. v t v_{t}
  59. v t = v a M = 52 81.25 = 0.64 v_{t}=\frac{v_{a}}{M}=\frac{52}{81.25}=0.64
  60. m m
  61. D D
  62. p p
  63. m = D d = 130 7 18.6 m=\frac{D}{d}=\frac{130}{7}\approx 18.6
  64. F m = 650 18.6 35 \frac{F}{m}=\frac{650}{18.6}\approx 35
  65. e e
  66. D D
  67. m m
  68. e = D m = 130 18.6 7 e=\frac{D}{m}=\frac{130}{18.6}\approx 7
  69. B B
  70. p p
  71. B = 2 * p 2 = 2 * 7 2 = 98 B=2*p^{2}=2*7^{2}=98

Optical_tweezers.html

  1. 𝐅 𝟏 = q ( 𝐄 𝟏 + d 𝐱 𝟏 d t × 𝐁 ) . \mathbf{F_{1}}=q\left(\mathbf{E_{1}}+\frac{d\mathbf{x_{1}}}{dt}\times\mathbf{B% }\right).
  2. 𝐩 = q 𝐝 , \mathbf{p}=q\mathbf{d},
  3. 𝐝 \mathbf{d}
  4. 𝐱 1 - 𝐱 2 . \mathbf{x}_{1}-\mathbf{x}_{2}.
  5. 𝐅 \displaystyle\mathbf{F}
  6. 𝐄 𝟏 \mathbf{E_{1}}
  7. q q
  8. 𝐱 \mathbf{x}
  9. 𝐩 \mathbf{p}
  10. 𝐅 \displaystyle\mathbf{F}
  11. 𝐩 = α 𝐄 \mathbf{p}=\alpha\mathbf{E}
  12. ( 𝐄 ) 𝐄 = ( 1 2 E 2 ) - 𝐄 × ( × 𝐄 ) \left(\mathbf{E}\cdot\nabla\right)\mathbf{E}=\nabla\left(\frac{1}{2}E^{2}% \right)-\mathbf{E}\times\left(\nabla\times\mathbf{E}\right)
  13. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  14. 𝐅 \displaystyle\mathbf{F}
  15. 𝐅 = 1 2 α E 2 = 2 π n 0 a 3 c ( m 2 - 1 m 2 + 2 ) I ( 𝐫 ) , \mathbf{F}=\frac{1}{2}\alpha\nabla E^{2}=\frac{2\pi n_{0}a^{3}}{c}\left(\frac{% m^{2}-1}{m^{2}+2}\right)\nabla I(\mathbf{r}),
  16. α = 4 π n 0 2 ϵ 0 a 3 ( m 2 - 1 ) / ( m 2 + 2 ) \alpha=4\pi n_{0}^{2}\epsilon_{0}a^{3}(m^{2}-1)/(m^{2}+2)
  17. a a
  18. n 0 n_{0}
  19. m = n 1 / n 0 m=n_{1}/n_{0}
  20. 𝐅 scat ( 𝐫 ) = k 4 α 2 6 π c n 0 3 ϵ 0 2 I ( 𝐫 ) z ^ = 8 π n 0 k 4 a 6 3 c ( m 2 - 1 m 2 + 2 ) 2 I ( 𝐫 ) z ^ . \mathbf{F}_{\,\text{scat}}(\mathbf{r})=\frac{k^{4}\alpha^{2}}{6\pi cn_{0}^{3}% \epsilon_{0}^{2}}I(\mathbf{r})\hat{z}=\frac{8\pi n_{0}k^{4}a^{6}}{3c}\left(% \frac{m^{2}-1}{m^{2}+2}\right)^{2}I(\mathbf{r})\hat{z}.
  21. 𝚫 𝐄 AC Stark = 3 π c 2 Γ μ 2 ω 0 3 δ 𝐈 ( 𝐫 , 𝐳 ) \mathbf{\Delta E}_{\,\text{AC Stark}}=\frac{3\pi c^{2}\Gamma\mu}{2\omega_{0}^{% 3}\delta}\mathbf{I(r,z)}
  22. Γ \Gamma
  23. μ \mu
  24. ω o \omega_{o}
  25. δ \delta
  26. ( λ ) (\lambda)
  27. ( w o ) (w_{o})
  28. ( P o ) (P_{o})
  29. I ( r , z ) = I 0 ( w 0 w ( z ) ) 2 e - 2 r 2 w 2 ( z ) I(r,z)=I_{0}\left(\frac{w_{0}}{w(z)}\right)^{2}e^{-\frac{2r^{2}}{w^{2}(z)}}
  30. w ( z ) = w 0 1 + ( z z R ) 2 w(z)=w_{0}\sqrt{1+\left(\frac{z}{z_{R}}\right)^{2}}
  31. z R = π w 0 2 λ z_{R}=\frac{\pi w_{0}^{2}}{\lambda}
  32. P 0 = 1 2 π I 0 w 0 2 P_{0}=\frac{1}{2}\pi I_{0}w_{0}^{2}
  33. z z
  34. r r
  35. r = 0 r=0
  36. z = 0 z=0
  37. 1 2 m ( ω z 2 z 2 + ω r 2 r 2 ) \frac{1}{2}m(\omega_{z}^{2}z^{2}+\omega_{r}^{2}r^{2})
  38. 2 I z 2 | r = 0 = 4 P 0 2 λ 2 π 3 w 0 6 z 2 = 1 2 m ω z 2 z 2 \frac{\partial^{2}I}{\partial z^{2}}\Biggr|_{r=0}=\frac{4P_{0}^{2}\lambda^{2}}% {\pi^{3}w_{0}^{6}}z^{2}=\frac{1}{2}m\omega_{z}^{2}z^{2}
  39. 2 I r 2 | z = 0 = 4 P 0 2 π w 0 4 r 2 = 1 2 m ω r 2 r 2 \frac{\partial^{2}I}{\partial r^{2}}\Biggr|_{z=0}=\frac{4P_{0}^{2}}{\pi w_{0}^% {4}}r^{2}=\frac{1}{2}m\omega_{r}^{2}r^{2}
  40. ω r = 8 P 0 π m w 0 4 \omega_{r}=\sqrt{\frac{8P_{0}}{\pi mw_{0}^{4}}}
  41. ω z = 8 P 0 λ 2 m π 3 w 0 6 \omega_{z}=\sqrt{\frac{8P_{0}\lambda^{2}}{m\pi^{3}w_{0}^{6}}}
  42. ω r ω z = π w 0 λ \frac{\omega_{r}}{\omega_{z}}=\frac{\pi w_{0}}{\lambda}

Option_style.html

  1. max { ( S - K ) , 0 } \max\{(S-K),0\}
  2. max { ( K - S ) , 0 } \max\{(K-S),0\}

Orbifold.html

  1. π \pi
  2. \subset
  3. \rightarrow
  4. \subset
  5. \subset
  6. \subset
  7. \rightarrow
  8. \subset
  9. \subset
  10. \subset
  11. \subset
  12. \subset
  13. \subset
  14. \rightarrow
  15. \cap
  16. \cap
  17. \subset
  18. \subset
  19. \supset
  20. \cap
  21. \supset
  22. \cap
  23. \cap
  24. \rightarrow
  25. \rightarrow
  26. \cap
  27. \rightarrow
  28. \cap
  29. \cap
  30. \rightarrow
  31. \rightarrow
  32. \rightarrow
  33. \rightarrow
  34. \rightarrow
  35. \rightarrow
  36. \rightarrow
  37. \rightarrow
  38. \rightarrow
  39. \rightarrow
  40. \rightarrow
  41. \rightarrow
  42. \rightarrow
  43. \rightarrow
  44. \rightarrow
  45. \rightarrow
  46. Γ A B Γ A B C Γ A C Γ A . \Gamma_{AB}\star_{\,\Gamma_{ABC}}\Gamma_{AC}\rightarrow\Gamma_{A}.
  47. - 7 \sqrt{-7}
  48. \subset
  49. \cap
  50. T 4 / 2 T^{4}/\mathbb{Z}_{2}\,
  51. T n / S n T^{n}/S_{n}
  52. T n T^{n}
  53. S n S_{n}
  54. 𝐑 = log 2 𝐑 + \mathbf{R}=\log_{2}\mathbf{R}^{+}
  55. S 1 = 𝐑 / 𝐙 S^{1}=\mathbf{R}/\mathbf{Z}
  56. T t := S 1 × × S 1 , T^{t}:=S^{1}\times\cdots\times S^{1},
  57. S t , S_{t},
  58. t = 1 + 1 + + 1 t=1+1+\cdots+1
  59. t = t t=t
  60. 3 = 2 + 1 3=2+1

Orbit_(dynamics).html

  1. Φ : U M \Phi:U\to M
  2. U T × M U\subset T\times M
  3. I ( x ) := { t T : ( t , x ) U } , I(x):=\{t\in T:(t,x)\in U\},
  4. γ x := { Φ ( t , x ) : t I ( x ) } M \gamma_{x}:=\{\Phi(t,x):t\in I(x)\}\subset M
  5. Φ ( t , x ) = x \Phi(t,x)=x\,
  6. I ( x ) = ( t x - , t x + ) I(x)=(t_{x}^{-},t_{x}^{+})
  7. γ x + := { Φ ( t , x ) : t ( 0 , t x + ) } \gamma_{x}^{+}:=\{\Phi(t,x):t\in(0,t_{x}^{+})\}
  8. γ x - := { Φ ( t , x ) : t ( t x - , 0 ) } \gamma_{x}^{-}:=\{\Phi(t,x):t\in(t_{x}^{-},0)\}
  9. γ x + = def { Φ ( t , x ) : t 0 } \gamma_{x}^{+}\ \overset{\underset{\mathrm{def}}{}}{=}\ \{\Phi(t,x):t\geq 0\}\,
  10. γ x - = def { Φ ( - t , x ) : t 0 } \gamma_{x}^{-}\ \overset{\underset{\mathrm{def}}{}}{=}\ \{\Phi(-t,x):t\geq 0\}\,
  11. γ x = def γ x - γ x + \gamma_{x}\ \overset{\underset{\mathrm{def}}{}}{=}\ \gamma_{x}^{-}\cup\gamma_{% x}^{+}\,
  12. Φ \Phi\,
  13. Φ : X X \Phi:X\to X\,
  14. X X\,
  15. t t\,
  16. t T t\in T\,
  17. x x\,
  18. x X x\in X\,
  19. Φ ( t , x ) \Phi(t,x)\,
  20. Φ t ( x ) \Phi^{t}(x)\,
  21. x t = Φ t ( x ) x_{t}=\Phi^{t}(x)\,
  22. x 0 x_{0}\,
  23. x x\,
  24. G G
  25. X X
  26. G . x X G.x\subset X
  27. S t a b G ( x ) Stab_{G}(x)
  28. G G
  29. G . x G.x
  30. X X
  31. S L 2 ( ) \ S L 2 ( ) SL_{2}(\mathbb{R})\backslash SL_{2}(\mathbb{Z})

Orbital_elements.html

  1. e e\,\!
  2. a a\,\!
  3. ω \omega\,\!
  4. M o M_{o}\,\!
  5. ν \nu\,\!
  6. ν \nu\,\!
  7. M M\,\!
  8. ν o \nu_{o}\,\!
  9. t t
  10. Ω , i , ω \Omega,i,\omega
  11. α , β , γ \alpha,\beta,\gamma
  12. x ^ , y ^ , z ^ \hat{x},\hat{y},\hat{z}
  13. I ^ , J ^ , K ^ \hat{I},\hat{J},\hat{K}
  14. I ^ , J ^ \hat{I},\hat{J}
  15. I ^ \hat{I}
  16. J ^ \hat{J}
  17. I ^ \hat{I}
  18. I ^ \hat{I}
  19. K ^ \hat{K}
  20. x ^ , y ^ \hat{x},\hat{y}
  21. x ^ \hat{x}
  22. z ^ \hat{z}
  23. y ^ \hat{y}
  24. x ^ \hat{x}
  25. z ^ \hat{z}
  26. I ^ , J ^ , K ^ \hat{I},\hat{J},\hat{K}
  27. x ^ , y ^ , z ^ \hat{x},\hat{y},\hat{z}
  28. Ω , i , ω \Omega,i,\omega
  29. x 1 = cos Ω cos ω - sin Ω cos i sin ω x_{1}=\cos\Omega\cdot\cos\omega-\sin\Omega\cdot\cos i\cdot\sin\omega
  30. x 2 = sin Ω cos ω + cos Ω cos i sin ω x_{2}=\sin\Omega\cdot\cos\omega+\cos\Omega\cdot\cos i\cdot\sin\omega
  31. x 3 = sin i sin ω x_{3}=\sin i\cdot\sin\omega
  32. y 1 = - cos Ω sin ω - sin Ω cos i cos ω y_{1}=-\cos\Omega\cdot\sin\omega-\sin\Omega\cdot\cos i\cdot\cos\omega
  33. y 2 = - sin Ω sin ω + cos Ω cos i cos ω y_{2}=-\sin\Omega\cdot\sin\omega+\cos\Omega\cdot\cos i\cdot\cos\omega
  34. y 3 = sin i cos ω y_{3}=\sin i\cdot\cos\omega
  35. z 1 = sin i sin Ω z_{1}=\sin i\cdot\sin\Omega
  36. z 2 = - sin i cos Ω z_{2}=-\sin i\cdot\cos\Omega
  37. z 3 = cos i z_{3}=\cos i\,
  38. x ^ = x 1 I ^ + x 2 J ^ + x 3 K ^ \hat{x}=x_{1}\hat{I}+x_{2}\hat{J}+x_{3}\hat{K}
  39. y ^ = y 1 I ^ + y 2 J ^ + y 3 K ^ \hat{y}=y_{1}\hat{I}+y_{2}\hat{J}+y_{3}\hat{K}
  40. z ^ = z 1 I ^ + z 2 J ^ + z 3 K ^ \hat{z}=z_{1}\hat{I}+z_{2}\hat{J}+z_{3}\hat{K}
  41. x ^ , y ^ , z ^ \hat{x},\hat{y},\hat{z}
  42. Ω , i , ω \Omega,i,\omega
  43. Ω = arg ( - z 2 , z 1 ) \Omega=\operatorname{arg}(-z_{2},z_{1})
  44. i = arg ( z 3 , z 1 2 + z 2 2 ) i=\operatorname{arg}(z_{3},\sqrt{{z_{1}}^{2}+{z_{2}}^{2}})
  45. ω = arg ( y 3 , x 3 ) \omega=\operatorname{arg}(y_{3},x_{3})
  46. arg ( x , y ) \operatorname{arg}(x,y)
  47. n = μ a 3 n=\sqrt{\frac{\mu}{a^{3}}}
  48. t 0 t_{0}
  49. [ e 0 , a 0 , i 0 , Ω 0 , ω 0 , M 0 ] [e_{0},a_{0},i_{0},\Omega_{0},\omega_{0},M_{0}]
  50. t 0 + δ t t_{0}+\delta t
  51. [ e 0 , a 0 , i 0 , Ω 0 , ω 0 , M 0 + n δ t ] [e_{0},a_{0},i_{0},\Omega_{0},\omega_{0},M_{0}+n\delta t]

Orbital_mechanics.html

  1. - G M r -\frac{GM}{r}\,
  2. v 2 2 \frac{v^{2}}{2}\,
  3. v 2 2 - G M r \frac{v^{2}}{2}-\frac{GM}{r}\,
  4. r r
  5. r r
  6. v 2 G M r v\geq\sqrt{\frac{2GM}{r}}
  7. r = p 1 + e cos θ r=\frac{p}{1+e\cos\theta}
  8. μ = G ( m 1 + m 2 ) \mu=G(m_{1}+m_{2})\,
  9. p = h 2 / μ p=h^{2}/\mu\,
  10. v = G M r \ v=\sqrt{\frac{GM}{r}\ }
  11. G G
  12. v = 2 G M r = 2 G M r . \ v=\sqrt{2}\sqrt{\frac{GM}{r}\ }=\sqrt{\frac{2GM}{r}\ }.
  13. r p = p 1 + e r_{p}=\frac{p}{1+e}
  14. r a = p 1 - e r_{a}=\frac{p}{1-e}
  15. 2 a = r p + r a 2a=r_{p}+r_{a}
  16. a = p 1 - e 2 a=\frac{p}{1-e^{2}}
  17. r = a ( 1 - e 2 ) 1 + e cos θ r=\frac{a(1-e^{2})}{1+e\cos\theta}
  18. T T\,\!
  19. T = 2 π a 3 μ T=2\pi\sqrt{a^{3}\over{\mu}}
  20. μ \mu\,
  21. a a\,\!
  22. a a\,\!
  23. v v\,
  24. v = μ ( 2 r - 1 a ) v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}
  25. μ \mu\,
  26. r r\,
  27. a a\,\!
  28. 1 a {1\over{a}}
  29. ϵ \epsilon\,
  30. v 2 2 - μ r = - μ 2 a = ϵ < 0 {v^{2}\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0
  31. v v\,
  32. r r\,
  33. a a\,
  34. μ \mu\,
  35. r = h 2 μ 1 1 + cos θ r={{h^{2}}\over{\mu}}{{1}\over{1+\cos\theta}}
  36. r r\,
  37. h h\,
  38. θ \theta\,
  39. μ \mu\,
  40. ϵ = v 2 2 - μ r = 0 \epsilon={v^{2}\over 2}-{\mu\over{r}}=0
  41. v v\,
  42. v = 2 μ r v=\sqrt{2\mu\over{r}}
  43. r = h 2 μ 1 1 + e cos θ r={{h^{2}}\over{\mu}}{{1}\over{1+e\cos\theta}}
  44. ϵ \epsilon\,
  45. ϵ = v 2 2 - μ r = μ - 2 a \epsilon={v^{2}\over 2}-{\mu\over{r}}={\mu\over{-2a}}
  46. v v\,
  47. r r\,
  48. a a\,
  49. μ \mu\,
  50. v v_{\infty}\,\!
  51. v = μ - a v_{\infty}=\sqrt{\mu\over{-a}}\,\!
  52. μ \mu\,\!
  53. a a\,\!
  54. 2 ϵ = C 3 = v 2 2\epsilon=C_{3}=v_{\infty}^{2}\,\!
  55. M = E - ϵ sin E M=E-\epsilon\cdot\sin E
  56. ϵ \displaystyle\epsilon
  57. θ \theta
  58. E E
  59. θ \theta
  60. t t
  61. E E
  62. E E
  63. E E
  64. E E
  65. ϵ \textstyle\epsilon
  66. E = { n = 1 M n 3 n ! lim θ 0 ( d n - 1 d θ n - 1 ( θ θ - sin ( θ ) 3 n ) ) , ϵ = 1 n = 1 M n n ! lim θ 0 ( d n - 1 d θ n - 1 ( θ θ - ϵ sin ( θ ) n ) ) , ϵ 1 E=\begin{cases}\displaystyle\sum_{n=1}^{\infty}{\frac{M^{\frac{n}{3}}}{n!}}% \lim_{\theta\to 0}\left(\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}}% \left(\frac{\theta}{\sqrt[3]{\theta-\sin(\theta)}}^{n}\right)\right),&\epsilon% =1\\ \displaystyle\sum_{n=1}^{\infty}{\frac{M^{n}}{n!}}\lim_{\theta\to 0}\left(% \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}}\left(\frac{\theta}{\theta-% \epsilon\cdot\sin(\theta)}^{n}\right)\right),&\epsilon\neq 1\end{cases}
  67. E = { x + 1 60 x 3 + 1 1400 x 5 + 1 25200 x 7 + 43 17248000 x 9 + 1213 7207200000 x 11 + 151439 12713500800000 x 13 | x = ( 6 M ) 1 3 , ϵ = 1 1 1 - ϵ M - ϵ ( 1 - ϵ ) 4 M 3 3 ! + ( 9 ϵ 2 + ϵ ) ( 1 - ϵ ) 7 M 5 5 ! - ( 225 ϵ 3 + 54 ϵ 2 + ϵ ) ( 1 - ϵ ) 10 M 7 7 ! + ( 11025 ϵ 4 + 4131 ϵ 3 + 243 ϵ 2 + ϵ ) ( 1 - ϵ ) 13 M 9 9 ! , ϵ 1 E=\begin{cases}\displaystyle x+\frac{1}{60}x^{3}+\frac{1}{1400}x^{5}+\frac{1}{% 25200}x^{7}+\frac{43}{17248000}x^{9}+\frac{1213}{7207200000}x^{11}+\frac{15143% 9}{12713500800000}x^{13}\cdots\ |\ x=(6M)^{\frac{1}{3}},&\epsilon=1\\ \\ \displaystyle\frac{1}{1-\epsilon}M-\frac{\epsilon}{(1-\epsilon)^{4}}\frac{M^{3% }}{3!}+\frac{(9\epsilon^{2}+\epsilon)}{(1-\epsilon)^{7}}\frac{M^{5}}{5!}-\frac% {(225\epsilon^{3}+54\epsilon^{2}+\epsilon)}{(1-\epsilon)^{10}}\frac{M^{7}}{7!}% +\frac{(11025\epsilon^{4}+4131\epsilon^{3}+243\epsilon^{2}+\epsilon)}{(1-% \epsilon)^{13}}\frac{M^{9}}{9!}\cdots,&\epsilon\neq 1\end{cases}
  68. E E
  69. E E
  70. ϵ \epsilon
  71. e = 1 e=1
  72. E - sin E E-\sin E
  73. r S O I r_{SOI}
  74. r S O I = a p ( m p m s ) 2 / 5 r_{SOI}=a_{p}\left(\frac{m_{p}}{m_{s}}\right)^{2/5}
  75. a p a_{p}
  76. m p m_{p}
  77. m s m_{s}
  78. x 0 x_{0}
  79. v 0 v_{0}
  80. t = 0 t=0
  81. x 0 ( t ) x_{0}(t)
  82. v 0 ( t ) v_{0}(t)
  83. x 0 ( t ) x_{0}(t)
  84. v 0 ( t ) v_{0}(t)

Orbital_period.html

  1. T T\,
  2. T = 2 π a 3 / μ T=2\pi\sqrt{a^{3}/\mu}
  3. a a\,
  4. μ = G M \mu=GM\,
  5. k m 3 / s 2 km^{3}/s^{2}
  6. G G\,
  7. M M\,
  8. M = ρ V = ρ 4 3 π a 3 M=\rho V=\rho{\frac{4}{3}}\pi a^{3}
  9. T = 3 π G ρ T=\sqrt{\frac{3\pi}{G\rho}}
  10. T = 1.41 T=1.41
  11. T = 3.30 T=3.30
  12. T T\,
  13. T = 2 π a 3 G ( M 1 + M 2 ) T=2\pi\sqrt{\frac{a^{3}}{G\left(M_{1}+M_{2}\right)}}
  14. a a\,
  15. M 1 + M 2 M_{1}+M_{2}\,
  16. G G\,
  17. P 1 P_{1}
  18. P 2 P_{2}
  19. P 1 < P 2 P_{1}<P_{2}
  20. 1 P s y n = 1 P 1 - 1 P 2 \frac{1}{P_{syn}}=\frac{1}{P_{1}}-\frac{1}{P_{2}}

Orbital_speed.html

  1. v o 2 π a T v_{o}\approx{2\pi a\over T}
  2. v o μ a v_{o}\approx\sqrt{\mu\over a}
  3. v v
  4. a a
  5. T T
  6. μ = G M μ=GM
  7. v o G ( m 1 + m 2 ) r v_{o}\approx\sqrt{G(m_{1}+m_{2})\over r}
  8. r r
  9. G G
  10. v o G M r v_{o}\approx\sqrt{\frac{GM}{r}}
  11. r r
  12. v o v e 2 v_{o}\approx\frac{v_{e}}{\sqrt{2}}
  13. M M
  14. e e
  15. v o = 2 π a T [ 1 - 1 4 e 2 - 3 64 e 4 - 5 256 e 6 - 175 16384 e 8 - ] v_{o}=\frac{2\pi a}{T}\left[1-\frac{1}{4}e^{2}-\frac{3}{64}e^{4}-\frac{5}{256}% e^{6}-\frac{175}{16384}e^{8}-\dots\right]
  16. v = μ ( 2 r - 1 a ) v=\sqrt{\mu\left({2\over r}-{1\over a}\right)}
  17. μ μ
  18. r r
  19. a a
  20. v = 1.327 × 10 20 m 3 s - 2 ( 2 1.471 × 10 11 m - 1 1.496 × 10 11 m ) 30 , 300 m / s v=\sqrt{1.327\times 10^{20}~{}m^{3}s^{-2}\cdot\left({2\over 1.471\times 10^{11% }~{}m}-{1\over 1.496\times 10^{11}~{}m}\right)}\approx 30,300~{}m/s

Order_of_operations.html

  1. 1 + 3 + 5 = 4 + 5 = 2 + 5 = 7. \sqrt{1+3}+5=\sqrt{4}+5=2+5=7.\,
  2. 1 + 2 3 + 4 + 5 = 3 7 + 5. \frac{1+2}{3+4}+5=\frac{3}{7}+5.
  3. [ ( 1 + 2 ) - 3 ] - ( 4 - 5 ) = [ 3 - 3 ] - ( - 1 ) = 1. [(1+2)-3]-(4-5)=[3-3]-(-1)=1.\,
  4. 1 ÷ 2 × x = 1 × 1 2 × x = 1 2 x . 1\div 2\times x=1\times\tfrac{1}{2}\times x=\tfrac{1}{2}x.
  5. 10 - 3 + 2 10-3+2\,
  6. 10 + ( - 3 ) + 2 10+(-3)+2\,
  7. a b c = a ( b c ) a^{b^{c}}=a^{(b^{c})}
  8. ( a b ) c (a^{b})^{c}
  9. a p q a^{p^{q}}
  10. ( a p ) q (a^{p})^{q}
  11. a ( p q ) a^{(p^{q})}
  12. 1 + 2 × 3 = 9 , 1+2\times 3=9,\;
  13. 1 + 2 × 3 = 7. 1+2\times 3=7.\;
  14. a ( b c ) = a b c a^{(b^{c})}=a^{b^{c}}
  15. ( a b ) c . (a^{b})^{c}.
  16. N \sqrt{N}

Order_statistic.html

  1. x 1 = 6 , x 2 = 9 , x 3 = 3 , x 4 = 8 , x_{1}=6,\ \ x_{2}=9,\ \ x_{3}=3,\ \ x_{4}=8,\,
  2. i i
  3. x i x_{i}
  4. x ( 1 ) = 3 , x ( 2 ) = 6 , x ( 3 ) = 8 , x ( 4 ) = 9 , x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\,
  5. ( [ u i t a l i c s c o r r e c t i o n , u i ] ) ([u^{\prime}italicscorrection^{\prime},u^{\prime}i^{\prime}])
  6. [ u i t a l i c s c o r r e c t i o n , u i ] [u^{\prime}italicscorrection^{\prime},u^{\prime}i^{\prime}]
  7. X ( 1 ) = min { X 1 , , X n } X_{(1)}=\min\{\,X_{1},\ldots,X_{n}\,\}
  8. n n
  9. [ u i t a l i c s c o r r e c t i o n , u n ] [u^{\prime}italicscorrection^{\prime},u^{\prime}n^{\prime}]
  10. X ( n ) = max { X 1 , , X n } . X_{(n)}=\max\{\,X_{1},\ldots,X_{n}\,\}.
  11. Range { X 1 , , X n } = X ( n ) - X ( 1 ) . {\rm Range}\{\,X_{1},\ldots,X_{n}\,\}=X_{(n)}-X_{(1)}.
  12. n n
  13. n = 2 m + 1 n=2m+1
  14. m m
  15. X ( m + 1 ) X_{(m+1)}
  16. n n
  17. n = 2 m n=2m
  18. X ( m ) X_{(m)}
  19. X ( m + 1 ) X_{(m+1)}
  20. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  21. F X F_{X}
  22. U i = F X ( X i ) U_{i}=F_{X}(X_{i})
  23. U 1 , , U n U_{1},\ldots,U_{n}
  24. U ( i ) = F X ( X ( i ) ) U_{(i)}=F_{X}(X_{(i)})
  25. U ( k ) U_{(k)}
  26. [ u , u + d u ] [u,\ u+du]
  27. n ! ( k - 1 ) ! ( n - k ) ! u k - 1 ( 1 - u ) n - k d u + O ( d u 2 ) , {n!\over(k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\,du+O(du^{2}),
  28. U ( k ) B ( k , n + 1 - k ) . U_{(k)}\sim B(k,n+1-k).
  29. U ( k ) U_{(k)}
  30. O ( d u 2 ) O(du^{2})
  31. ( 0 , u ) (0,u)
  32. ( u , u + d u ) (u,u+du)
  33. ( u + d u , 1 ) (u+du,1)
  34. n ! ( k - 1 ) ! ( n - k ) ! u k - 1 d u ( 1 - u - d u ) n - k {n!\over(k-1)!(n-k)!}u^{k-1}\cdot du\cdot(1-u-du)^{n-k}
  35. f U ( i ) , U ( j ) ( u , v ) d u d v = n ! u i - 1 ( i - 1 ) ! ( v - u ) j - i - 1 ( j - i - 1 ) ! ( 1 - v ) n - j ( n - j ) ! d u d v f_{U_{(i)},U_{(j)}}(u,v)\,du\,dv=n!{u^{i-1}\over(i-1)!}{(v-u)^{j-i-1}\over(j-i% -1)!}{(1-v)^{n-j}\over(n-j)!}\,du\,dv
  36. O ( d u d v ) O(du\,dv)
  37. ( 0 , u ) (0,u)
  38. ( u , u + d u ) (u,u+du)
  39. ( u + d u , v ) (u+du,v)
  40. ( v , v + d v ) (v,v+dv)
  41. ( v + d v , 1 ) (v+dv,1)
  42. f U ( 1 ) , U ( 2 ) , , U ( n ) ( u 1 , u 2 , , u n ) d u 1 d u n = n ! d u 1 d u n . f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n})\,du_{1}\cdots du_% {n}=n!\,du_{1}\cdots du_{n}.
  43. 0 < u 1 < < u n < 1 0<u_{1}<\cdots<u_{n}<1
  44. d F X ( x ) = f X ( x ) d x dF_{X}(x)=f_{X}(x)\,dx
  45. u = F X ( x ) u=F_{X}(x)
  46. d u = f X ( x ) d x du=f_{X}(x)\,dx
  47. f X ( k ) ( x ) = n ! ( k - 1 ) ! ( n - k ) ! [ F X ( x ) ] k - 1 [ 1 - F X ( x ) ] n - k f X ( x ) f_{X_{(k)}}(x)=\frac{n!}{(k-1)!(n-k)!}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)
  48. f X ( j ) , X ( k ) ( x , y ) = n ! ( j - 1 ) ! ( k - j - 1 ) ! ( n - k ) ! [ F X ( x ) ] j - 1 [ F X ( y ) - F X ( x ) ] k - 1 - j [ 1 - F X ( y ) ] n - k f X ( x ) f X ( y ) f_{X_{(j)},X_{(k)}}(x,y)=\frac{n!}{(j-1)!(k-j-1)!(n-k)!}[F_{X}(x)]^{j-1}[F_{X}% (y)-F_{X}(x)]^{k-1-j}[1-F_{X}(y)]^{n-k}f_{X}(x)f_{X}(y)
  49. x y x\leq y
  50. f X ( 1 ) , , X ( n ) ( x 1 , , x n ) = n ! f X ( x 1 ) f X ( x n ) f_{X_{(1)},\ldots,X_{(n)}}(x_{1},\ldots,x_{n})=n!f_{X}(x_{1})\cdots f_{X}(x_{n})
  51. x 1 x 2 x n . x_{1}\leq x_{2}\leq\dots\leq x_{n}.
  52. ( 6 3 ) 2 - 6 = 5 16 31 % . {6\choose 3}2^{-6}={5\over 16}\approx 31\%.
  53. [ ( 6 2 ) + ( 6 3 ) + ( 6 4 ) ] 2 - 6 = 25 32 78 % . \left[{6\choose 2}+{6\choose 3}+{6\choose 4}\right]2^{-6}={25\over 32}\approx 7% 8\%.
  54. U ( n p ) A N ( p , p ( 1 - p ) n ) . U_{(\lceil np\rceil)}\sim AN\left(p,\frac{p(1-p)}{n}\right).
  55. X ( n p ) A N ( F - 1 ( p ) , p ( 1 - p ) n [ f ( F - 1 ( p ) ) ] 2 ) X_{(\lceil np\rceil)}\sim AN\left(F^{-1}(p),\frac{p(1-p)}{n[f(F^{-1}(p))]^{2}}\right)
  56. B ( k , n + 1 - k ) = d X X + Y , B(k,n+1-k)\ \stackrel{\mathrm{d}}{=}\ \frac{X}{X+Y},
  57. X = i = 1 k Z i , Y = i = k + 1 n + 1 Z i , X=\sum_{i=1}^{k}Z_{i},\quad Y=\sum_{i=k+1}^{n+1}Z_{i},
  58. X 1 , X 2 , , X n X_{1},X_{2},...,X_{n}
  59. F ( x ) F(x)
  60. f ( x ) f(x)
  61. k th k\text{th}
  62. p 1 = P ( X < x ) = F ( x ) - f ( x ) , p 2 = P ( X = x ) = f ( x ) , and p 3 = P ( X > x ) = 1 - F ( x ) . p_{1}=P(X<x)=F(x)-f(x),\ p_{2}=P(X=x)=f(x),\,\text{ and }p_{3}=P(X>x)=1-F(x).
  63. k th k\text{th}
  64. P ( X ( k ) x ) = P ( there are at most n - k observations greater than x ) , = j = 0 n - k ( n j ) p 3 j ( p 1 + p 2 ) n - j . \begin{aligned}\displaystyle P(X_{(k)}\leq x)&\displaystyle=P(\,\text{there % are at most }n-k\,\text{ observations greater than }x),\\ &\displaystyle=\sum_{j=0}^{n-k}{n\choose j}p_{3}^{j}(p_{1}+p_{2})^{n-j}.\end{aligned}
  65. P ( X ( k ) < x ) P(X_{(k)}<x)
  66. P ( X ( k ) < x ) = P ( there are at most n - k observations greater than or equal to x ) , = j = 0 n - k ( n j ) ( p 2 + p 3 ) j ( p 1 ) n - j . \begin{aligned}\displaystyle P(X_{(k)}<x)&\displaystyle=P(\,\text{there are at% most }n-k\,\text{ observations greater than or equal to }x),\\ &\displaystyle=\sum_{j=0}^{n-k}{n\choose j}(p_{2}+p_{3})^{j}(p_{1})^{n-j}.\end% {aligned}
  67. X k X_{k}
  68. P ( X ( k ) = x ) = P ( X ( k ) x ) - P ( X ( k ) < x ) , = j = 0 n - k ( n j ) ( p 3 j ( p 1 + p 2 ) n - j - ( p 2 + p 3 ) j ( p 1 ) n - j ) , = j = 0 n - k ( n j ) ( ( 1 - F ( x ) ) j ( F ( x ) ) n - j - ( 1 - F ( x ) + f ( x ) ) j ( F ( x ) - f ( x ) ) n - j ) . \begin{aligned}\displaystyle P(X_{(k)}=x)&\displaystyle=P(X_{(k)}\leq x)-P(X_{% (k)}<x),\\ &\displaystyle=\sum_{j=0}^{n-k}{n\choose j}\left(p_{3}^{j}(p_{1}+p_{2})^{n-j}-% (p_{2}+p_{3})^{j}(p_{1})^{n-j}\right),\\ &\displaystyle=\sum_{j=0}^{n-k}{n\choose j}\left((1-F(x))^{j}(F(x))^{n-j}-(1-F% (x)+f(x))^{j}(F(x)-f(x))^{n-j}\right).\end{aligned}

Order_topology.html

  1. ( a , ) = { x a < x } (a,\infty)=\{x\mid a<x\}
  2. ( - , b ) = { x x < b } (-\infty,b)=\{x\mid x<b\}
  3. ( a , b ) = { x a < x < b } (a,b)=\{x\mid a<x<b\}
  4. Z = { - 1 } ( 0 , 1 ) Z=\{-1\}\cup(0,1)
  5. ( - 1 , ) (-1,\infty)
  6. [ 0 , λ ) = { α α < λ } [0,\lambda)=\{\alpha\mid\alpha<\lambda\}\,
  7. [ 0 , λ ] = { α α λ } [0,\lambda]=\{\alpha\mid\alpha\leq\lambda\}\,
  8. ( ω 1 + 1 ) × ( ω + 1 ) (\omega_{1}+1)\times(\omega+1)
  9. ( ω 1 , ω ) (\omega_{1},\omega)
  10. ω 1 × ω \omega_{1}\times\omega

Ordered_exponential.html

  1. a : K A . a\mathrel{:}K\to A.\,
  2. OE [ a ] ( t ) 𝒯 { e 0 t a ( t ) d t } n = 0 1 n ! 0 t 0 t n 𝒯 { a ( t 1 ) a ( t n ) } d t 1 d t n \operatorname{OE}[a](t)\equiv\mathcal{T}\left\{e^{\int_{0}^{t}a(t^{\prime})\,% dt^{\prime}}\right\}\equiv\sum_{n=0}^{\infty}\frac{1}{n!}\int_{0}^{t}\cdots% \int_{0}^{t^{\prime}_{n}}\mathcal{T}\left\{a(t^{\prime}_{1})\cdots a(t^{\prime% }_{n})\right\}\,dt^{\prime}_{1}\cdots dt^{\prime}_{n}
  3. 𝒯 \mathcal{T}
  4. 𝒯 { a ( 1.2 ) a ( 9.5 ) a ( 4.1 ) } = a ( 9.5 ) a ( 4.1 ) a ( 1.2 ) . \mathcal{T}\left\{a(1.2)a(9.5)a(4.1)\right\}=a(9.5)a(4.1)a(1.2).
  5. OE : ( K A ) ( K A ) . \operatorname{OE}\mathrel{:}\left(K\to A\right)\to\left(K\to A\right).
  6. OE [ a ] ( t ) = 0 t e a ( t ) d t lim N ( e a ( t N ) Δ t e a ( t N - 1 ) Δ t e a ( t 1 ) Δ t e a ( t 0 ) Δ t ) \operatorname{OE}[a](t)=\prod_{0}^{t}e^{a(t^{\prime})\,dt^{\prime}}\equiv\lim_% {N\rightarrow\infty}\left(e^{a(t_{N})\Delta t}e^{a(t_{N-1})\Delta t}\cdots e^{% a(t_{1})\Delta t}e^{a(t_{0})\Delta t}\right)
  7. d d t OE [ a ] ( t ) \displaystyle\frac{d}{dt}\operatorname{OE}[a](t)
  8. OE [ a ] ( t ) = 1 + 0 t a ( t ) OE [ a ] ( t ) d t . \operatorname{OE}[a](t)=1+\int_{0}^{t}a(t^{\prime})\operatorname{OE}[a](t^{% \prime})\,dt^{\prime}.
  9. OE [ a ] ( t ) = 1 + 0 t a ( t 1 ) d t 1 + 0 t 0 t 1 a ( t 1 ) a ( t 2 ) d t 2 d t 1 + . \operatorname{OE}[a](t)=1+\int_{0}^{t}a(t_{1})\,dt_{1}+\int_{0}^{t}\int_{0}^{t% _{1}}a(t_{1})a(t_{2})\,dt_{2}\,dt_{1}+\cdots.
  10. M M
  11. e T M e\in TM
  12. g : e g e g:e\mapsto ge
  13. x M x\in M
  14. d e ( x ) + J ( x ) e ( x ) = 0 de(x)+\operatorname{J}(x)e(x)=0
  15. d d
  16. J ( x ) \operatorname{J}(x)
  17. e ( x ) e(x)
  18. J ( x ) \operatorname{J}(x)
  19. e ( y ) = P e x p ( - x y J ( γ ( t ) ) γ ( t ) d t ) e ( x ) e(y)=\operatorname{P}exp(-\int_{x}^{y}J(\gamma(t))\gamma^{\prime}(t)dt)e(x)
  20. P \operatorname{P}
  21. γ ( t ) M \gamma(t)\in M
  22. J ( x ) \operatorname{J}(x)
  23. γ \gamma
  24. | u | , | v | |u|,|v|
  25. x , x + u , x + u + v , x + v , x,x+u,x+u+v,x+v,
  26. OE [ - J ] e ( x ) = e x p ( - J ( x + v ) ( - v ) ) e x p ( - J ( x + u + v ) ( - u ) ) e x p ( - J ( x + u ) v ) e x p ( - J ( x ) u ) e ( x ) = ( 1 - J ( x + v ) ( - v ) ) ( 1 - J ( x + u + v ) ( - u ) ) ( 1 - J ( x + u ) v ) ( 1 - J ( x ) u ) e ( x ) \operatorname{OE}[-\operatorname{J}]e(x)=exp(-\operatorname{J}(x+v)(-v))exp(-% \operatorname{J}(x+u+v)(-u))exp(-\operatorname{J}(x+u)v)exp(-\operatorname{J}(% x)u)e(x)=(1-\operatorname{J}(x+v)(-v))(1-\operatorname{J}(x+u+v)(-u))(1-% \operatorname{J}(x+u)v)(1-\operatorname{J}(x)u)e(x)
  27. OE [ - J ] g OE [ J ] g - 1 \operatorname{OE}[-\operatorname{J}]\mapsto g\operatorname{OE}[\operatorname{J% }]g^{-1}
  28. - J ( x ) -\operatorname{J}(x)
  29. | u | , | v | |u|,|v|

Orders_of_magnitude_(area).html

  1. G c 3 \frac{G\hbar}{c^{3}}

Orders_of_magnitude_(length).html

  1. 10 10 10 122 10^{10^{10^{122}}}
  2. 10 10 10 122 10^{10^{10^{122}}}
  3. 10 10 115 10^{10^{115}}
  4. 10 10 115 10^{10^{115}}
  5. 10 10 115 10^{10^{115}}
  6. 10 10 115 10^{10^{115}}
  7. 10 10 115 10^{10^{115}}
  8. 10 10 10 122 10^{10^{10^{122}}}
  9. 10 10 10 122 10^{10^{10^{122}}}
  10. 10 10 10 122 10^{10^{10^{122}}}
  11. 10 10 10 122 10^{10^{10^{122}}}
  12. 10 10 10 122 10^{10^{10^{122}}}

Orders_of_magnitude_(numbers).html

  1. × 10 6176 \times 10^{−}6176
  2. × 10 4966 \times 10^{−}4966
  3. × 10 4951 \times 10^{−}4951
  4. × 10 398 \times 10^{−}398
  5. × 10 324 \times 10^{−}324
  6. × 10 101 \times 10^{−}101
  7. × 10 45 \times 10^{−}45
  8. × 10 9 \times 10^{9}
  9. 2 2 n + 1 2^{2^{n}}+1
  10. × 10 1 0 \times 10^{1}0
  11. × 10 1 0 \times 10^{1}0
  12. × 10 1 0 \times 10^{1}0
  13. × 10 1 1 \times 10^{1}1
  14. × 10 1 1 \times 10^{1}1
  15. × 10 1 1 \times 10^{1}1
  16. 19683 3 19683^{3}
  17. 27 9 27^{9}
  18. 3 27 3^{27}
  19. 3 3 3 3^{3^{3}}
  20. 3 3 3\uparrow\uparrow 3
  21. 3 2 3\uparrow\uparrow\uparrow 2
  22. × 10 1 3 \times 10^{1}3
  23. × 10 1 6 \times 10^{1}6
  24. × 10 1 8 \times 10^{1}8
  25. × 10 1 8 \times 10^{1}8
  26. × 10 1 8 \times 10^{1}8
  27. × 10 1 9 \times 10^{1}9
  28. × 10 1 9 \times 10^{1}9
  29. × 10 1 9 \times 10^{1}9
  30. × 10 1 9 \times 10^{1}9
  31. × 10 2 1 \times 10^{2}1
  32. × 10 2 1 \times 10^{2}1
  33. × 10 2 2 \times 10^{2}2
  34. × 10 2 3 \times 10^{2}3
  35. × 10 2 4 \times 10^{2}4
  36. × 10 2 7 \times 10^{2}7
  37. × 10 2 8 \times 10^{2}8
  38. × 10 3 4 \times 10^{3}4
  39. 2 2 7 - 1 - 1 2^{2^{7}-1}-1
  40. × 10 3 8 \times 10^{3}8
  41. × 10 3 8 \times 10^{3}8
  42. × 10 3 8 \times 10^{3}8
  43. × 10 4 0 \times 10^{4}0
  44. × 10 4 4 \times 10^{4}4
  45. × 10 4 5 \times 10^{4}5
  46. × 10 4 6 \times 10^{4}6
  47. × 10 5 3 \times 10^{5}3
  48. × 10 5 7 \times 10^{5}7
  49. × 10 6 0 \times 10^{6}0
  50. × 10 6 3 \times 10^{6}3
  51. × 10 6 7 \times 10^{6}7
  52. × 10 7 2 \times 10^{7}2
  53. × 10 7 4 \times 10^{7}4
  54. × 10 7 7 \times 10^{7}7
  55. × 10 9 6 \times 10^{9}6
  56. 10 10 100 10^{10^{100}}
  57. × 10 1 16 \times 10^{1}16
  58. × 10 1 20 \times 10^{1}20
  59. × 10 1 60 \times 10^{1}60
  60. × 10 1 81 \times 10^{1}81
  61. × 10 1 85 \times 10^{1}85
  62. × 10 2 45 \times 10^{2}45
  63. × 10 3 08 \times 10^{3}08
  64. × 10 3 84 \times 10^{3}84
  65. × 10 1 083 \times 10^{1}083
  66. × 10 4 932 \times 10^{4}932
  67. × 10 4 932 \times 10^{4}932
  68. × 10 6 144 \times 10^{6}144
  69. 25 1 , 312 , 000 1.956 × 10 1 , 834 , 097 25^{1,312,000}\approx 1.956\times 10^{1,834,097}
  70. 10 10 100 10^{10^{100}}
  71. 10 10 100 10^{10^{100}}
  72. 10 10 1 , 834 , 102 10^{10^{1,834,102}}
  73. 10 10 10 , 000 , 000 10^{10^{10,000,000}}
  74. 10 10 10 34 10^{\,\!10^{10^{34}}}
  75. 10 10 10 963 10^{\,\!10^{10^{963}}}

Ordinate.html

  1. ( x abscissa , y ordinate ) (\overbrace{x}\text{abscissa},\overbrace{y}\text{ordinate})

Orientability.html

  1. K 2 × S 1 K^{2}\times S^{1}
  2. K 2 K^{2}
  3. H n ( M , M { p } ; ) H_{n}(M,M\setminus\{p\};\mathbb{Z})
  4. H n ( M , M ; ) H_{n}(M,\partial M;\mathbb{Z})
  5. \mathbb{Z}
  6. G L + ( n ) GL^{+}(n)
  7. GL ( n , 𝐑 ) \operatorname{GL}(n,\mathbf{R})
  8. π 0 ( GL ( n , 𝐑 ) ) = 𝐙 / 2 \pi_{0}(\operatorname{GL}(n,\mathbf{R}))=\mathbf{Z}/2
  9. σ ± : O ( p , q ) { - 1 , + 1 } . \sigma_{\pm}:\operatorname{O}(p,q)\to\{-1,+1\}.
  10. O ( M ) × σ + { - 1 , + 1 } \operatorname{O}(M)\times_{\sigma_{+}}\{-1,+1\}
  11. O ( M ) × σ - { - 1 , + 1 } . \operatorname{O}(M)\times_{\sigma_{-}}\{-1,+1\}.

Orthocentric_system.html

  1. A B 2 + C H 2 = A C 2 + B H 2 = B C 2 + A H 2 = 4 R 2 AB^{2}+CH^{2}=AC^{2}+BH^{2}=BC^{2}+AH^{2}=4R^{2}
  2. B C sin A = A C sin B = A B sin C = H A | cos A | = H B | cos B | = H C | cos C | = 2 R . \frac{BC}{\sin{A}}=\frac{AC}{\sin{B}}=\frac{AB}{\sin{C}}=\frac{HA}{|\cos{A}|}=% \frac{HB}{|\cos{B}|}=\frac{HC}{|\cos{C}|}=2R.

Orthogonal_group.html

  1. n n
  2. O ( n ) O(n)
  3. n n
  4. n × n n×n
  5. 1 1
  6. 1 −1
  7. O ( n ) O(n)
  8. 1 1
  9. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  10. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  11. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  12. n n
  13. F F
  14. O ( n , F ) O(n,F)
  15. n × n n×n
  16. F F
  17. G L ( n , F ) GL(n,F)
  18. O ( n , F ) = { Q GL ( n , F ) Q 𝖳 Q = Q Q 𝖳 = I } \mathrm{O}(n,F)=\{Q\in\mathrm{GL}(n,F)\mid Q^{\mathsf{T}}Q=QQ^{\mathsf{T}}=I\}
  19. Q Q
  20. Q Q
  21. I I
  22. 1 1
  23. 1 −1
  24. 1 1
  25. 1 −1
  26. O ( p , q ) O(p,q)
  27. p p
  28. q q
  29. Ω ( n , F ) Ω(n,F)
  30. O ( n , F ) O(n,F)
  31. F F
  32. Ω ( n , F ) Ω(n,F)
  33. O ( n , F ) O(n,F)
  34. S O ( n , F ) SO(n,F)
  35. 1 1
  36. 1 −1
  37. n n
  38. n n
  39. 1 1
  40. O ( n , F ) O(n,F)
  41. S O ( n , F ) SO(n,F)
  42. S O ( n , F ) SO(n,F)
  43. G O GO
  44. O ( p , q ) O(p,q)
  45. 𝐑 \mathbf{R}
  46. I −I
  47. O ( 2 k ) O(2k)
  48. O ( 2 k + 1 ) O(2k+1)
  49. n n
  50. n = 2 k n=2k
  51. n = 2 k + 1 n=2k+1
  52. p p
  53. r r
  54. 𝔰 𝔬 ( 2 r + 1 ) \mathfrak{so}(2r+1)
  55. 𝔰 𝔬 ( 2 r ) \mathfrak{so}(2r)
  56. 𝐑 \mathbf{R}
  57. O ( n , 𝐑 ) O(n,\mathbf{R})
  58. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  59. O ( n ) O(n)
  60. S O ( n ) SO(n)
  61. n ( n 1 ) / 2 n(n−1)/2
  62. O ( n , 𝐑 ) O(n,\mathbf{R})
  63. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  64. O ( n , 𝐑 ) O(n,\mathbf{R})
  65. E ( n ) E(n)
  66. O ( n , 𝐑 ) = E ( n ) G L ( n , 𝐑 ) O(n,\mathbf{R})=E(n)∩GL(n,\mathbf{R})
  67. n = 3 n=3
  68. ( n 1 ) (n− 1)
  69. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  70. O ( n , 𝐑 ) O(n,\mathbf{R})
  71. n n
  72. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  73. n n
  74. O ( n , 𝐑 ) O(n,\mathbf{R})
  75. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  76. k k
  77. k k
  78. O ( 2 , 𝐑 ) O(2,\mathbf{R})
  79. S O ( 2 , 𝐑 ) SO(2,\mathbf{R})
  80. [ R 1 R k 0 0 ± 1 ± 1 ] \begin{bmatrix}\begin{matrix}R_{1}&&\\ &\ddots&\\ &&R_{k}\end{matrix}&0\\ 0&\begin{matrix}\pm 1&&\\ &\ddots&\\ &&\pm 1\end{matrix}\\ \end{bmatrix}
  81. S O ( 3 , 𝐑 ) SO(3,\mathbf{R})
  82. n n
  83. n n
  84. v v v↦−v
  85. O ( 2 , 𝐑 ) O(2,\mathbf{R})
  86. S O ( 2 , 𝐑 ) SO(2,\mathbf{R})
  87. U ( 1 ) U(1)
  88. e x p ( φ i ) = c o s φ + i s i n φ exp(φi)=cos φ+isin φ
  89. 1 1
  90. [ cos ( ϕ ) - sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) ] . \begin{bmatrix}\cos(\phi)&-\sin(\phi)\\ \sin(\phi)&\cos(\phi)\end{bmatrix}.
  91. S O ( 3 , 𝐑 ) SO(3,\mathbf{R})
  92. S O ( 3 ) SO(3)
  93. T T
  94. S O ( 2 n ) SO(2n)
  95. [ R 1 0 0 R n ] , \begin{bmatrix}R_{1}&&0\\ &\ddots&\\ 0&&R_{n}\end{bmatrix},
  96. SO ( 2 n ) SO ( 2 n ) × { 1 } < SO ( 2 n + 1 ) \mathrm{SO}(2n)\cong\mathrm{SO}(2n)\times\{1\}<\mathrm{SO}(2n+1)
  97. S O ( 2 n + 1 ) SO(2n+1)
  98. S O ( 2 n + 1 ) SO(2n+1)
  99. { ± 1 } n S n \{\pm 1\}^{n}\rtimes S_{n}
  100. [ 1 0 0 1 ] or [ 0 1 1 0 ] , \begin{bmatrix}1&0\\ 0&1\end{bmatrix}\quad\,\text{or}\quad\begin{bmatrix}0&1\\ 1&0\end{bmatrix},
  101. ± 1 ±1
  102. S O ( 2 n ) SO(2n)
  103. H n - 1 S n < { ± 1 } n S n H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}
  104. S O ( 2 n + 1 ) SO(2n+1)
  105. ( ϵ 1 , , ϵ n ) ϵ 1 ϵ n (\epsilon_{1},\ldots,\epsilon_{n})\mapsto\epsilon_{1}\cdots\epsilon_{n}
  106. S O ( 2 n ) SO(2n)
  107. S O ( 2 n ) SO(2n)
  108. S O ( 2 n ) S O ( 2 n + 1 ) SO(2n)→SO(2n+1)
  109. S O ( 2 n + 1 ) SO(2n+1)
  110. [ 0 1 1 0 ] \begin{bmatrix}0&1\\ 1&0\end{bmatrix}
  111. 1 −1
  112. S O ( 2 n ) SO(2n)
  113. O ( 1 ) = S < s u p > 0 O(1)=S<sup>0
  114. S O ( 2 ) SO(2)
  115. S < s u p > 1 S<sup>1
  116. S O ( 3 ) SO(3)
  117. 𝐑 P < s u p > 3 \mathbf{R}P<sup>3
  118. S O ( 4 ) SO(4)
  119. S U ( 2 ) × S U ( 2 ) = S < s u p > 3 × S 3 SU(2)×SU(2)=S<sup>3×S^{3}
  120. n > 2 n>2
  121. S O ( n , 𝐑 ) SO(n,\mathbf{R})
  122. S p i n ( n ) Spin(n)
  123. n = 2 n=2
  124. S p i n ( 2 ) Spin(2)
  125. O ( 0 ) O ( 1 ) O ( 2 ) O = k = 0 O ( k ) \mathrm{O}(0)\subset\mathrm{O}(1)\subset\mathrm{O}(2)\subset\cdots\subset O=% \bigcup_{k=0}^{\infty}\mathrm{O}(k)
  126. O ( n + 1 ) O(n+ 1)
  127. O ( n ) O ( n + 1 ) S n , \mathrm{O}(n)\to\mathrm{O}(n+1)\to S^{n},
  128. O ( n + 1 ) O(n+ 1)
  129. O ( n ) O ( n + 1 ) O(n)→O(n+ 1)
  130. ( n 1 ) (n− 1)
  131. n > k + 1 n>k+1
  132. O O
  133. π 0 ( O ) \displaystyle\pi_{0}(O)
  134. O O
  135. π 0 ( K O ) \displaystyle\pi_{0}(KO)
  136. O ( 2 ) O(2)
  137. O O
  138. O O
  139. O / U O/U
  140. K O = B O × 𝐙 KO=BO×\mathbf{Z}
  141. K S p = B S p × 𝐙 KSp=BSp×\mathbf{Z}
  142. R R
  143. 𝐑 \mathbf{R}
  144. 𝐂 \mathbf{C}
  145. 𝐇 \mathbf{H}
  146. 𝐎 \mathbf{O}
  147. 𝐂 \mathbf{C}
  148. O ( n , 𝐂 ) O(n,\mathbf{C})
  149. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  150. n ( n 1 ) / 2 n(n−1)/2
  151. 𝐂 \mathbf{C}
  152. 𝐑 \mathbf{R}
  153. O ( n , 𝐂 ) O(n,\mathbf{C})
  154. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  155. n 2 n≥2
  156. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  157. n > 2 n>2
  158. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  159. S O ( 2 , 𝐂 ) SO(2,\mathbf{C})
  160. q q
  161. p p
  162. O ( 2 n + 1 , q ) O(2n+ 1,q)
  163. V V
  164. G G
  165. V = L 1 L 2 L m W , V=L_{1}\oplus L_{2}\oplus\cdots\oplus L_{m}\oplus W,
  166. W W
  167. W W
  168. G G
  169. W W
  170. G G
  171. W W
  172. G G
  173. n = 1 n=1
  174. 2 ( q ϵ ) 2(q−ϵ)
  175. O ( n , q ) O(n,q)
  176. | O ( 2 n + 1 , q ) | = 2 q n i = 0 n - 1 ( q 2 n - q 2 i ) . |\mathrm{O}(2n+1,q)|=2q^{n}\prod_{i=0}^{n-1}(q^{2n}-q^{2i}).
  177. 1 −1
  178. | O ( 2 n , q ) | = 2 ( q n - 1 ) i = 1 n - 1 ( q 2 n - q 2 i ) . |\mathrm{O}(2n,q)|=2(q^{n}-1)\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).
  179. 1 −1
  180. | O ( 2 n , q ) | = 2 ( q n + ( - 1 ) n + 1 ) i = 1 n - 1 ( q 2 n - q 2 i ) . |\mathrm{O}(2n,q)|=2(q^{n}+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).
  181. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  182. 0
  183. D ( f ) = r a n k ( I f ) m o d u l o 2 D(f)=rank(I−f)modulo2
  184. I I
  185. O ( n , F ) O(n,F)
  186. F F
  187. 0
  188. 1 1
  189. S O ( n , F ) SO(n,F)
  190. O ( n , F ) O(n,F)
  191. 1 1
  192. O ( n , F ) O(n,F)
  193. ± 1 ±1
  194. 1 1
  195. 𝐮 \mathbf{u}
  196. 𝐯 \mathbf{v}
  197. 𝐯 + B ( 𝐯 , 𝐮 ) / Q ( 𝐮 ) · 𝐮 \mathbf{v}+B(\mathbf{v},\mathbf{u})/Q(\mathbf{u}) ·\mathbf{u}
  198. B B
  199. Q Q
  200. 𝐯 \mathbf{v}
  201. 𝐯 2 · B ( 𝐯 , 𝐮 ) / Q ( 𝐮 ) · 𝐮 \mathbf{v}−2·B(\mathbf{v},\mathbf{u})/Q(\mathbf{u}) ·\mathbf{u}
  202. I = I I=−I
  203. 2 n + 1 2n+1
  204. 2 n 2n
  205. 2 n 2n
  206. F F
  207. F F
  208. n n
  209. n n
  210. 1 μ 2 Pin V O V 1 1\rightarrow\mu_{2}\rightarrow\mathrm{Pin}_{V}\rightarrow\mathrm{O_{V}}\rightarrow 1
  211. F F
  212. O ( n , F ) O(n,F)
  213. S O ( n , F ) SO(n,F)
  214. n × n n×n
  215. , ,
  216. 𝔬 ( n , F ) \mathfrak{o}(n,F)
  217. 𝔰 𝔬 ( n , F ) \mathfrak{so}(n,F)
  218. n n
  219. n = 2 k + 1 n=2k+1
  220. n = 2 r n=2r
  221. 𝐯 𝐰 \mathbf{v}∧\mathbf{w}
  222. 𝐯 𝐰 𝐯 * 𝐰 - 𝐰 * 𝐯 , \mathbf{v}\wedge\mathbf{w}\mapsto\mathbf{v}^{*}\otimes\mathbf{w}-\mathbf{w}^{*% }\otimes\mathbf{v},
  223. 𝐯 \mathbf{v}
  224. 𝔰 𝔬 ( p , q ) . \mathfrak{so}(p,q).
  225. O ( n ) U ( n ) S p ( n ) = U S p ( 2 n ) O(n)⊂U(n)⊂Sp(n)=USp(2n)
  226. U S p ( n ) U ( n ) O ( 2 n ) USp(n)⊂U(n)⊂O(2n)
  227. U ( n ) / O ( n ) U(n)/O(n)
  228. O ( n ) O ( n - 1 ) \mathrm{O}(n)\supset\mathrm{O}(n-1)
  229. O ( 2 n ) U ( n ) SU ( n ) \mathrm{O}(2n)\supset\mathrm{U}(n)\supset\mathrm{SU}(n)
  230. U ( n ) U(n)
  231. S U ( n ) SU(n)
  232. O ( 2 n ) USp ( n ) \mathrm{O}(2n)\supset\mathrm{USp}(n)
  233. O ( 7 ) G 2 \mathrm{O}(7)\supset\mathrm{G}_{2}
  234. O ( n ) O(n)
  235. U ( n ) SU ( n ) O ( n ) \mathrm{U}(n)\supset\mathrm{SU}(n)\supset\mathrm{O}(n)
  236. USp ( 2 n ) O ( n ) \mathrm{USp}(2n)\supset\mathrm{O}(n)
  237. G 2 O ( 3 ) \mathrm{G}_{2}\supset\mathrm{O}(3)
  238. F 4 O ( 9 ) \mathrm{F}_{4}\supset\mathrm{O}(9)
  239. E 6 O ( 10 ) \mathrm{E}_{6}\supset\mathrm{O}(10)
  240. E 7 O ( 12 ) \mathrm{E}_{7}\supset\mathrm{O}(12)
  241. E 8 O ( 16 ) \mathrm{E}_{8}\supset\mathrm{O}(16)
  242. C O ( n ) CO(n)
  243. n n
  244. n n
  245. ± 1 ±1
  246. C S O ( n ) CSO(n)
  247. O ( n ) P O ( n ) O(n)→PO(n)
  248. S p i n ( n ) S O ( n ) Spin(n)→SO(n)
  249. S O ( n ) P S O ( n ) SO(n)→PSO(n)
  250. P S O ( 2 k ) PSO(2k)
  251. P S O ( 2 k + 1 ) PSO(2k+ 1)
  252. S O ( 2 k + 1 ) SO(2k+ 1)
  253. S p i n ( n ) Spin(n)
  254. S O ( n ) SO(n)
  255. P S O ( n ) PSO(n)
  256. 𝔰 𝔬 ( n , ) \mathfrak{so}(n,{\mathbb{R}})
  257. O ( n ) O(n)
  258. V < s u b > n ( 𝐑 n ) V<sub>n(\mathbf{R}^{n})
  259. O ( n ) G L ( n , 𝐙 ) O(n)∩GL(n,\mathbf{Z})
  260. ± 1 ±1
  261. S O ( 2 k + 1 ) P S O ( 2 k + 1 ) SO(2k+ 1)≅PSO(2k+ 1)
  262. S O ( 2 k ) SO(2k)

Orthogonal_matrix.html

  1. Q T Q = Q Q T = I , Q^{\mathrm{T}}Q=QQ^{\mathrm{T}}=I,
  2. Q T = Q - 1 , Q^{\mathrm{T}}=Q^{-1},\,
  3. u v = ( Q u ) ( Q v ) {u}\cdot{v}=\left(Q{u}\right)\cdot\left(Q{v}\right)\,
  4. v T v = ( Q v ) T ( Q v ) = v T Q T Q v . {v}^{\mathrm{T}}{v}=(Q{v})^{\mathrm{T}}(Q{v})={v}^{\mathrm{T}}Q^{\mathrm{T}}Q{% v}.
  5. [ 1 0 0 1 ] ( identity transformation ) \begin{bmatrix}1&0\\ 0&1\\ \end{bmatrix}\qquad(\,\text{identity transformation})
  6. R ( 16.26 ) = [ cos θ - sin θ sin θ cos θ ] = [ 0.96 - 0.28 0.28 0.96 ] ( rotation by 16.26 ) R(16.26^{\circ})=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}=\begin{bmatrix}0.96&-0.28\\ 0.28&\;\;\,0.96\\ \end{bmatrix}\qquad(\,\text{rotation by }16.26^{\circ})
  7. [ 1 0 0 - 1 ] ( reflection across x -axis ) \begin{bmatrix}1&0\\ 0&-1\\ \end{bmatrix}\qquad(\,\text{reflection across }x\,\text{-axis})
  8. [ 0 - 0.80 - 0.60 0.80 - 0.36 0.48 0.60 0.48 - 0.64 ] ( rotoinversion: axis ( 0 , - 3 / 5 , 4 / 5 ) , angle 90 ) \begin{bmatrix}0&-0.80&-0.60\\ 0.80&-0.36&\;\;\,0.48\\ 0.60&\;\;\,0.48&-0.64\end{bmatrix}\qquad\left(\begin{aligned}&\displaystyle\,% \text{rotoinversion:}\\ &\displaystyle\,\text{axis }(0,-3/5,4/5),\,\text{ angle }90^{\circ}\end{% aligned}\right)
  9. [ 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 ] ( permutation of coordinate axes ) \begin{bmatrix}0&0&0&1\\ 0&0&1&0\\ 1&0&0&0\\ 0&1&0&0\end{bmatrix}\qquad(\,\text{permutation of coordinate axes})
  10. [ p t q u ] , \begin{bmatrix}p&t\\ q&u\end{bmatrix},
  11. 1 \displaystyle 1
  12. [ cos θ - sin θ sin θ cos θ ] (rotation), [ cos θ sin θ sin θ - cos θ ] (reflection) \begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}\,\text{ (rotation), }\qquad\begin{bmatrix}\cos\theta&\sin\theta% \\ \sin\theta&-\cos\theta\\ \end{bmatrix}\,\text{ (reflection)}
  13. [ 0 1 1 0 ] . \begin{bmatrix}0&1\\ 1&0\end{bmatrix}.
  14. [ - 1 0 0 0 - 1 0 0 0 - 1 ] and [ 0 - 1 0 1 0 0 0 0 - 1 ] \begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}\,\text{ and }\begin{bmatrix}0&-1&0\\ 1&0&0\\ 0&0&-1\end{bmatrix}
  15. [ cos ( α ) cos ( γ ) - sin ( α ) sin ( β ) sin ( γ ) - sin ( α ) cos ( β ) - cos ( α ) sin ( γ ) - sin ( α ) sin ( β ) cos ( γ ) cos ( α ) sin ( β ) sin ( γ ) + sin ( α ) cos ( γ ) cos ( α ) cos ( β ) cos ( α ) sin ( β ) cos ( γ ) - sin ( α ) sin ( γ ) cos ( β ) sin ( γ ) - sin ( β ) cos ( β ) cos ( γ ) ] \begin{bmatrix}\cos(\alpha)\cos(\gamma)-\sin(\alpha)\sin(\beta)\sin(\gamma)&-% \sin(\alpha)\cos(\beta)&-\cos(\alpha)\sin(\gamma)-\sin(\alpha)\sin(\beta)\cos(% \gamma)\\ \cos(\alpha)\sin(\beta)\sin(\gamma)+\sin(\alpha)\cos(\gamma)&\cos(\alpha)\cos(% \beta)&\cos(\alpha)\sin(\beta)\cos(\gamma)-\sin(\alpha)\sin(\gamma)\\ \cos(\beta)\sin(\gamma)&-\sin(\beta)&\cos(\beta)\cos(\gamma)\end{bmatrix}
  16. Q = I - 2 v v T v T v . Q=I-2{{v}{v}^{\mathrm{T}}\over{v}^{\mathrm{T}}{v}}.
  17. 1 = det ( I ) = det ( Q T Q ) = det ( Q T ) det ( Q ) = ( det ( Q ) ) 2 . 1=\det(I)=\det(Q^{\mathrm{T}}Q)=\det(Q^{\mathrm{T}})\det(Q)=(\det(Q))^{2}.
  18. [ 2 0 0 1 2 ] \begin{bmatrix}2&0\\ 0&\frac{1}{2}\end{bmatrix}
  19. [ 0 O ( n ) 0 0 0 1 ] \begin{bmatrix}&&&0\\ &O(n)&&\vdots\\ &&&0\\ 0&\cdots&0&1\end{bmatrix}
  20. ( n - 1 ) + ( n - 2 ) + + 1 = n ( n - 1 ) / 2 (n-1)+(n-2)+\cdots+1=n(n-1)/2
  21. P T Q P = [ R 1 R k ] ( n even ) , P T Q P = [ R 1 R k 1 ] ( n odd ) . P^{\mathrm{T}}QP=\begin{bmatrix}R_{1}&&\\ &\ddots&\\ &&R_{k}\end{bmatrix}\ (n\,\text{ even}),\ P^{\mathrm{T}}QP=\begin{bmatrix}R_{1% }&&&\\ &\ddots&&\\ &&R_{k}&\\ &&&1\end{bmatrix}\ (n\,\text{ odd}).
  22. P T Q P = [ R 1 R k 0 0 ± 1 ± 1 ] , P^{\mathrm{T}}QP=\begin{bmatrix}\begin{matrix}R_{1}&&\\ &\ddots&\\ &&R_{k}\end{matrix}&0\\ 0&\begin{matrix}\pm 1&&\\ &\ddots&\\ &&\pm 1\end{matrix}\\ \end{bmatrix},
  23. Q T Q = I Q^{\mathrm{T}}Q=I
  24. Q ˙ T Q + Q T Q ˙ = 0 \dot{Q}^{\mathrm{T}}Q+Q^{\mathrm{T}}\dot{Q}=0
  25. Q ˙ T = - Q ˙ . \dot{Q}^{\mathrm{T}}=-\dot{Q}.
  26. 𝔰 𝔬 ( 3 ) \mathfrak{so}(3)
  27. Ω = [ 0 - z θ y θ z θ 0 - x θ - y θ x θ 0 ] . \Omega=\begin{bmatrix}0&-z\theta&y\theta\\ z\theta&0&-x\theta\\ -y\theta&x\theta&0\end{bmatrix}.
  28. exp ( Ω ) = [ 1 - 2 s 2 + 2 x 2 s 2 2 x y s 2 - 2 z s c 2 x z s 2 + 2 y s c 2 x y s 2 + 2 z s c 1 - 2 s 2 + 2 y 2 s 2 2 y z s 2 - 2 x s c 2 x z s 2 - 2 y s c 2 y z s 2 + 2 x s c 1 - 2 s 2 + 2 z 2 s 2 ] . \exp(\Omega)=\begin{bmatrix}1-2s^{2}+2x^{2}s^{2}&2xys^{2}-2zsc&2xzs^{2}+2ysc\\ 2xys^{2}+2zsc&1-2s^{2}+2y^{2}s^{2}&2yzs^{2}-2xsc\\ 2xzs^{2}-2ysc&2yzs^{2}+2xsc&1-2s^{2}+2z^{2}s^{2}\end{bmatrix}.
  29. R = [ 0 0 0 0 0 0 0 0 0 ] . R=\begin{bmatrix}\star&\star&\star\\ 0&\star&\star\\ 0&0&\star\\ 0&0&0\\ 0&0&0\end{bmatrix}.
  30. [ 3 1 7 5 ] [ 1.8125 0.0625 3.4375 2.6875 ] [ 0.8 - 0.6 0.6 0.8 ] \begin{bmatrix}3&1\\ 7&5\end{bmatrix}\rightarrow\begin{bmatrix}1.8125&0.0625\\ 3.4375&2.6875\end{bmatrix}\rightarrow\cdots\rightarrow\begin{bmatrix}0.8&-0.6% \\ 0.6&0.8\end{bmatrix}
  31. [ 3 1 7 5 ] [ 1.41421 - 1.06066 1.06066 1.41421 ] [ 0.8 - 0.6 0.6 0.8 ] \begin{bmatrix}3&1\\ 7&5\end{bmatrix}\rightarrow\begin{bmatrix}1.41421&-1.06066\\ 1.06066&1.41421\end{bmatrix}\rightarrow\begin{bmatrix}0.8&-0.6\\ 0.6&0.8\end{bmatrix}
  32. [ 3 1 7 5 ] [ 0.393919 - 0.919145 0.919145 0.393919 ] \begin{bmatrix}3&1\\ 7&5\end{bmatrix}\rightarrow\begin{bmatrix}0.393919&-0.919145\\ 0.919145&0.393919\end{bmatrix}
  33. Q Q
  34. M M
  35. M M
  36. R R
  37. Q = M ( M T M ) - 1 2 Q=M(M^{\mathrm{T}}M)^{-\frac{1}{2}}
  38. Q n + 1 = 2 M ( Q n - 1 M + M T Q n ) - 1 Q_{n+1}=2M(Q_{n}^{-1}M+M^{\mathrm{T}}Q_{n})^{-1}
  39. Q 0 = M Q_{0}=M
  40. M M

Orthogonality.html

  1. x , y \langle x,y\rangle
  2. x y x\,\bot\,y
  3. f , g w = a b f ( x ) g ( x ) w ( x ) d x . \langle f,g\rangle_{w}=\int_{a}^{b}f(x)g(x)w(x)\,dx.
  4. a b f ( x ) g ( x ) w ( x ) d x = 0. \int_{a}^{b}f(x)g(x)w(x)\,dx=0.
  5. f w = f , f w \|f\|_{w}=\sqrt{\langle f,f\rangle_{w}}
  6. f i , f j = a b f i ( x ) f j ( x ) w ( x ) d x = f i 2 δ i , j = f j 2 δ i , j \langle f_{i},f_{j}\rangle=\int_{a}^{b}f_{i}(x)f_{j}(x)w(x)\,dx=\|f_{i}\|^{2}% \delta_{i,j}=\|f_{j}\|^{2}\delta_{i,j}
  7. f i , f j = a b f i ( x ) f j ( x ) w ( x ) d x = δ i , j \langle f_{i},f_{j}\rangle=\int_{a}^{b}f_{i}(x)f_{j}(x)w(x)\,dx=\delta_{i,j}
  8. δ i , j = { 1 if i = j 0 if i j \delta_{i,j}=\left\{\begin{matrix}1&\mathrm{if}\ i=j\\ 0&\mathrm{if}\ i\neq j\end{matrix}\right.
  9. 𝐯 k = i = 0 a i + k < n n / a 𝐞 i \mathbf{v}_{k}=\sum_{i=0\atop ai+k<n}^{n/a}\mathbf{e}_{i}
  10. - 1 1 ( 2 t + 3 ) ( 45 t 2 + 9 t - 17 ) d t = 0 \int_{-1}^{1}\left(2t+3\right)\left(45t^{2}+9t-17\right)\,dt=0
  11. 1 / 1 - x 2 . 1/\sqrt{1-x^{2}}.
  12. ψ m \psi_{m}
  13. ψ n \psi_{n}
  14. ψ m | ψ n = 0 \langle\psi_{m}|\psi_{n}\rangle=0
  15. ψ m \psi_{m}
  16. ψ n \psi_{n}

Orthographic_projection.html

  1. P = [ 1 0 0 0 1 0 0 0 0 ] P=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&0\\ \end{bmatrix}
  2. P v = [ 1 0 0 0 1 0 0 0 0 ] [ v x v y v z ] = [ v x v y 0 ] Pv=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&0\\ \end{bmatrix}\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\end{bmatrix}=\begin{bmatrix}v_{x}\\ v_{y}\\ 0\end{bmatrix}
  3. P = [ 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] P=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\end{bmatrix}
  4. P v = [ 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] [ v x v y v z 1 ] = [ v x v y 0 1 ] Pv=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\\ 1\end{bmatrix}=\begin{bmatrix}v_{x}\\ v_{y}\\ 0\\ 1\end{bmatrix}
  5. P = [ 2 r i g h t - l e f t 0 0 - r i g h t + l e f t r i g h t - l e f t 0 2 t o p - b o t t o m 0 - t o p + b o t t o m t o p - b o t t o m 0 0 - 2 f a r - n e a r f a r + n e a r f a r - n e a r 0 0 0 1 ] P=\begin{bmatrix}\frac{2}{right-left}&0&0&-\frac{right+left}{right-left}\\ 0&\frac{2}{top-bottom}&0&-\frac{top+bottom}{top-bottom}\\ 0&0&\frac{-2}{far-near}&\frac{far+near}{far-near}\\ 0&0&0&1\end{bmatrix}
  6. P = S T = [ 2 r i g h t - l e f t 0 0 0 0 2 t o p - b o t t o m 0 0 0 0 2 f a r - n e a r 0 0 0 0 1 ] [ 1 0 0 - l e f t + r i g h t 2 0 1 0 - t o p + b o t t o m 2 0 0 - 1 f a r + n e a r 2 0 0 0 1 ] P=ST=\begin{bmatrix}\frac{2}{right-left}&0&0&0\\ 0&\frac{2}{top-bottom}&0&0\\ 0&0&\frac{2}{far-near}&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}1&0&0&-\frac{left+right}{2}\\ 0&1&0&-\frac{top+bottom}{2}\\ 0&0&-1&\frac{far+near}{2}\\ 0&0&0&1\end{bmatrix}
  7. P - 1 = [ r i g h t - l e f t 2 0 0 l e f t + r i g h t 2 0 t o p - b o t t o m 2 0 t o p + b o t t o m 2 0 0 f a r - n e a r - 2 f a r + n e a r 2 0 0 0 1 ] P^{-1}=\begin{bmatrix}\frac{right-left}{2}&0&0&\frac{left+right}{2}\\ 0&\frac{top-bottom}{2}&0&\frac{top+bottom}{2}\\ 0&0&\frac{far-near}{-2}&\frac{far+near}{2}\\ 0&0&0&1\end{bmatrix}

Orthonormal_basis.html

  1. ( x , y , z ) = x e 1 + y e 2 + z e 3 , (x,y,z)=xe_{1}+ye_{2}+ze_{3},\,
  2. x = b B x , b b 2 b . x=\sum_{b\in B}{\langle x,b\rangle\over\lVert b\rVert^{2}}b.
  3. x = b B x , b b x=\sum_{b\in B}\langle x,b\rangle b
  4. x 2 = b B | x , b | 2 . \|x\|^{2}=\sum_{b\in B}|\langle x,b\rangle|^{2}.
  5. Φ ( x ) , Φ ( y ) = x , y \langle\Phi(x),\Phi(y)\rangle=\langle x,y\rangle
  6. V n ( 𝐑 n ) V_{n}(\mathbf{R}^{n})
  7. V k ( 𝐑 n ) V_{k}(\mathbf{R}^{n})
  8. k < n k<n

Orthonormality.html

  1. 𝐱 = 𝐱 𝐱 \|\mathbf{x}\|=\sqrt{\mathbf{x}\cdot\mathbf{x}}
  2. x 1 x 2 + y 1 y 2 = 0 x_{1}x_{2}+y_{1}y_{2}=0\quad
  3. x 1 2 + y 1 2 = 1 \sqrt{{x_{1}}^{2}+{y_{1}}^{2}}=1
  4. x 2 2 + y 2 2 = 1 \sqrt{{x_{2}}^{2}+{y_{2}}^{2}}=1
  5. ( 2 ) (2)
  6. ( 3 ) (3)
  7. ( 1 ) (1)
  8. cos θ 1 cos θ 2 + sin θ 1 sin θ 2 = 0 \cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}=0
  9. tan θ 1 = - cot θ 2 \tan\theta_{1}=-\cot\theta_{2}
  10. tan ( θ 1 ) = tan ( θ 2 + π 2 ) \tan(\theta_{1})=\tan\left(\theta_{2}+\tfrac{\pi}{2}\right)
  11. θ 1 = θ 2 + π 2 \Rightarrow\theta_{1}=\theta_{2}+\tfrac{\pi}{2}
  12. 𝒱 \mathcal{V}
  13. { u 1 , u 2 , , u n , } 𝒱 \left\{u_{1},u_{2},\ldots,u_{n},\ldots\right\}\in\mathcal{V}
  14. i , j : u i , u j = δ i j \forall i,j:\langle u_{i},u_{j}\rangle=\delta_{ij}
  15. δ i j \delta_{ij}\,
  16. , \langle\cdot,\cdot\rangle
  17. 𝒱 \mathcal{V}
  18. || a 1 e 1 + a 2 e 2 + + a n e n || 2 = | a 1 | 2 + | a 2 | 2 + + | a n | 2 ||a_{1}e_{1}+a_{2}e_{2}+\cdots+a_{n}e_{n}||^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots% +|a_{n}|^{2}
  19. 𝒱 \mathcal{V}
  20. 𝒱 \mathcal{V}
  21. \vdots
  22. ϕ ( x ) \phi(x)
  23. ψ ( x ) \psi(x)
  24. [ a , b ] [a,b]
  25. ( 1 ) ϕ ( x ) , ψ ( x ) = a b ϕ ( x ) ψ ( x ) d x = 0 , and (1)\quad\langle\phi(x),\psi(x)\rangle=\int_{a}^{b}\phi(x)\psi(x)dx=0,\quad{\rm and}
  26. ( 2 ) || ϕ ( x ) || 2 = || ψ ( x ) || 2 = [ a b | ϕ ( x ) | 2 d x ] 1 2 = [ a b | ψ ( x ) | 2 d x ] 1 2 = 1. (2)\quad||\phi(x)||_{2}=||\psi(x)||_{2}=\left[\int_{a}^{b}|\phi(x)|^{2}dx% \right]^{\frac{1}{2}}=\left[\int_{a}^{b}|\psi(x)|^{2}dx\right]^{\frac{1}{2}}=1.
  27. f , g = - π π f ( x ) g ( x ) d x \langle f,g\rangle=\int_{-\pi}^{\pi}f(x)g(x)dx
  28. { 1 2 π , sin ( x ) π , sin ( 2 x ) π , , sin ( n x ) π , cos ( x ) π , cos ( 2 x ) π , , cos ( n x ) π } , n \left\{\frac{1}{\sqrt{2\pi}},\frac{\sin(x)}{\sqrt{\pi}},\frac{\sin(2x)}{\sqrt{% \pi}},\ldots,\frac{\sin(nx)}{\sqrt{\pi}},\frac{\cos(x)}{\sqrt{\pi}},\frac{\cos% (2x)}{\sqrt{\pi}},\ldots,\frac{\cos(nx)}{\sqrt{\pi}}\right\},\quad n\in\mathbb% {N}

Otto_cycle.html

  1. ( V 1 / V 2 ) ({V}_{1}/{V}_{2})
  2. ( P 3 / P 2 ) ({P}_{3}/{P}_{2})
  3. V 4 / V 3 {V}_{4}/{V}_{3}
  4. V 1 / V 2 {V}_{1}/{V}_{2}
  5. Δ E = E i n + E o u t = 0 \Delta{\mathit{E}}=\mathit{E}_{in}+\mathit{E}_{out}=\mathit{0}
  6. E i n \mathit{E}_{in}
  7. E o u t \mathit{E}_{out}
  8. W 1 - 2 + Q 2 - 3 + W 3 - 4 + Q 4 - 1 = 0 \mathit{W}_{1-2}+\mathit{Q}_{2-3}+\mathit{W}_{3-4}+\mathit{Q}_{4-1}=\mathit{0}
  9. ( W 1 - 2 < m t p l > m ) = U 1 - U 2 \left(\frac{\mathit{W}_{1-2}}{<}mtpl>{{m}}\right)=\mathit{U}_{1}-\mathit{U}_{2}
  10. ( Q 2 - 3 < m t p l > m ) = U 2 - U 3 \left(\frac{\mathit{Q}_{2-3}}{<}mtpl>{{m}}\right)=\mathit{U}_{2}-\mathit{U}_{3}
  11. ( W 3 - 4 < m t p l > m ) = U 3 - U 4 \left(\frac{\mathit{W}_{3-4}}{<}mtpl>{{m}}\right)=\mathit{U}_{3}-\mathit{U}_{4}
  12. ( Q 4 - 1 < m t p l > m ) = U 4 - U 1 \left(\frac{\mathit{Q}_{4-1}}{<}mtpl>{{m}}\right)=\mathit{U}_{4}-\mathit{U}_{1}
  13. W 1 - 2 + Q 2 - 3 + W 3 - 4 + Q 4 - 1 = ( U 1 - U 2 ) + ( U 2 - U 3 ) + ( U 3 - U 4 ) + ( U 4 - U 1 ) = 0 \mathit{W}_{1-2}+\mathit{Q}_{2-3}+\mathit{W}_{3-4}+\mathit{Q}_{4-1}=\left(% \mathit{U}_{1}-\mathit{U}_{2}\right)+\left(\mathit{U}_{2}-\mathit{U}_{3}\right% )+\left(\mathit{U}_{3}-\mathit{U}_{4}\right)+\left(\mathit{U}_{4}-\mathit{U}_{% 1}\right)=0
  14. ( U 1 - U 2 ) = ( 1 - 5 ) = - 4 \left(\mathit{U}_{1}-\mathit{U}_{2}\right)=\left(1-5\right)=-4
  15. ( U 2 - U 3 ) = ( 5 - 9 ) = - 4 \left({\mathit{U}_{2}-\mathit{U}_{3}}\right)=\left(5-9\right)=-4
  16. ( U 3 - U 4 ) = ( 9 - 4 ) = + 5 \left(\mathit{U}_{3}-\mathit{U}_{4}\right)=\left(9-4\right)=+5
  17. ( U 4 - U 1 ) = ( 4 - 1 ) = + 3 \left(\mathit{U}_{4}-\mathit{U}_{1}\right)=\left(4-1\right)=+3
  18. - 4 - 4 + 5 + 3 = 0 -4-4+5+3=0
  19. Σ W o r k = W 1 - 2 + W 3 - 4 = ( U 1 - U 2 ) + ( U 3 - U 4 ) = - 4 + 5 = + 1 \Sigma Work=\mathit{W}_{1-2}+\mathit{W}_{3-4}=\left(\mathit{U}_{1}-\mathit{U}_% {2}\right)+\left(\mathit{U}_{3}-\mathit{U}_{4}\right)=-4+5=+1
  20. Σ H e a t = Q 2 - 3 + Q 4 - 1 = ( U 2 - U 3 ) + ( U 4 - U 1 ) = - 4 + 3 = - 1 \Sigma Heat=\mathit{Q}_{2-3}+\mathit{Q}_{4-1}=\left(\mathit{U}_{2}-\mathit{U}_% {3}\right)+\left(\mathit{U}_{4}-\mathit{U}_{1}\right)=-4+3=-1
  21. η = W 1 - 2 + W 3 - 4 - Q 2 - 3 = ( U 1 - U 2 ) + ( U 3 - U 4 ) - ( U 2 - U 3 ) \eta=\frac{\mathit{W}_{1-2}+\mathit{W}_{3-4}}{-\mathit{Q}_{2-3}}=\frac{\left(% \mathit{U}_{1}-\mathit{U}_{2}\right)+\left(\mathit{U}_{3}-\mathit{U}_{4}\right% )}{-\left(\mathit{U}_{2}-\mathit{U}_{3}\right)}
  22. η = 1 - U 1 - U 4 ( U 2 - U 3 ) = 1 - ( 1 - 4 ) ( 5 - 9 ) = 0.25 \eta=1-\frac{\mathit{U}_{1}-\mathit{U}_{4}}{\left(\mathit{U}_{2}-\mathit{U}_{3% }\right)}=1-\frac{(1-4)}{(5-9)}=0.25
  23. η = Q 2 - 3 + Q 4 - 1 Q 2 - 3 = 1 + ( U 4 - U 1 ) ( U 2 - U 3 ) = 1 - ( U 1 - U 4 ) ( U 2 - U 3 ) \eta=\frac{\mathit{Q}_{2-3}+\mathit{Q}_{4-1}}{\mathit{Q}_{2-3}}=1+\frac{\left(% \mathit{U}_{4}-\mathit{U}_{1}\right)}{\left(\mathit{U}_{2}-\mathit{U}_{3}% \right)}=1-\frac{\left(\mathit{U}_{1}-\mathit{U}_{4}\right)}{\left(\mathit{U}_% {2}-\mathit{U}_{3}\right)}
  24. η = 1 - 1 - 4 5 - 9 = 1 - - 3 - 4 = 0.25 \eta=1-\frac{1-4}{5-9}=1-\frac{-3}{-4}=0.25
  25. c v = ( δ u δ T ) v {\mathit{c}_{v}}=\left(\frac{\delta{\mathit{u}}}{\delta{T}}\right)_{v}
  26. δ u = ( c v ) ( δ T ) \delta\mathit{u}=({\mathit{c}_{v}})({\delta{T}})
  27. η = 1 - ( c v ( T 4 - T 1 ) c v ( T 3 - T 2 ) ) \eta=1-\left(\frac{\mathit{c}_{v}(\mathit{T}_{4}-\mathit{T}_{1})}{\mathit{c}_{% v}(\mathit{T}_{3}-\mathit{T}_{2})}\right)
  28. η = 1 - ( T 1 T 2 ) ( T 4 / T 1 - 1 T 3 / T 2 - 1 ) \eta=1-\left(\frac{\mathit{T}_{1}}{\mathit{T}_{2}}\right)\left(\frac{\mathit{T% }_{4}/\mathit{T}_{1}-1}{\mathit{T}_{3}/\mathit{T}_{2}-1}\right)
  29. T 4 / T 1 = T 3 / T 2 {T}_{4}/{T}_{1}={T}_{3}/{T}_{2}
  30. η = 1 - ( T 1 T 2 ) \eta=1-\left(\frac{\mathit{T}_{1}}{\mathit{T}_{2}}\right)
  31. ( T 2 T 1 ) = ( p 2 p 1 ) ( γ - 1 ) / γ \left(\frac{{T}_{2}}{{T}_{1}}\right)=\left(\frac{{p}_{2}}{{p}_{1}}\right)^{(% \gamma-1)/{\gamma}}
  32. ( T 2 T 1 ) = ( V 1 V 2 ) ( γ - 1 ) \left(\frac{{T}_{2}}{{T}_{1}}\right)=\left(\frac{{V}_{1}}{{V}_{2}}\right)^{(% \gamma-1)}
  33. γ = ( c p c v ) {\gamma}=\left(\frac{\mathit{c}_{p}}{{c}_{v}}\right)
  34. γ {\gamma}
  35. R R
  36. c p 𝑙𝑛 ( V 1 V 2 ) - 𝑅𝑙𝑛 ( p 2 p 1 ) = 0 \mathit{{c}_{p}}\mathit{ln}\left(\frac{{V}_{1}}{{V}_{2}}\right)-\mathit{R}% \mathit{ln}\left(\frac{{p}_{2}}{{p}_{1}}\right)=0
  37. c v 𝑙𝑛 ( T 2 T 1 ) - 𝑅𝑙𝑛 ( V 2 V 1 ) = 0 \mathit{{c}_{v}}\mathit{ln}\left(\frac{{T}_{2}}{{T}_{1}}\right)-\mathit{R}% \mathit{ln}\left(\frac{{V}_{2}}{{V}_{1}}\right)=0
  38. c p = ( γ R γ - 1 ) \mathit{c}_{p}=\left(\frac{\gamma\mathit{R}}{\gamma-1}\right)
  39. c v = ( R γ - 1 ) \mathit{c}_{v}=\left(\frac{\mathit{R}}{\gamma-1}\right)
  40. r \mathit{r}
  41. ( V 1 / V 2 ) ({V}_{1}/{V}_{2})
  42. ( T 2 T 1 ) = ( V 1 V 2 ) ( γ - 1 ) = r ( γ - 1 ) \left(\frac{{T}_{2}}{{T}_{1}}\right)=\left(\frac{{V}_{1}}{{V}_{2}}\right)^{(% \gamma-1)}={r}^{(\gamma-1)}
  43. η = 1 - ( < m t p l > 1 r ( γ - 1 ) ) \eta=1-\left(\frac{<}{m}tpl>{{1}}{{r}^{(\gamma-1)}}\right)
  44. r \mathit{r}
  45. γ \gamma
  46. r \mathit{r}
  47. η \eta
  48. γ \gamma
  49. Σ W o r k = W 1 - 2 + W 3 - 4 = ( U 1 - U 2 ) + ( U 3 - U 4 ) = - 4 + 5 = + 1 \Sigma Work=\mathit{W}_{1-2}+\mathit{W}_{3-4}=\left(\mathit{U}_{1}-\mathit{U}_% {2}\right)+\left(\mathit{U}_{3}-\mathit{U}_{4}\right)=-4+5=+1
  50. M = P V R T M=\frac{PV}{RT}
  51. Σ W o r k = 1 J / ( k g * s t r o k e ) * 0.00121 k g = 0.00121 J / s t r o k e \Sigma Work=1J/(kg*stroke)*0.00121kg=0.00121\;J/stroke
  52. P = 16.7 * 0.00121 = 0.0202 J / s e c o r w a t t s P=16.7*0.00121=0.0202\;J/sec\;or\;watts

Outer_product.html

  1. 𝐮 𝐯 = 𝐮𝐯 T = [ u 1 u 2 u 3 u 4 ] [ v 1 v 2 v 3 ] = [ u 1 v 1 u 1 v 2 u 1 v 3 u 2 v 1 u 2 v 2 u 2 v 3 u 3 v 1 u 3 v 2 u 3 v 3 u 4 v 1 u 4 v 2 u 4 v 3 ] . \mathbf{u}\otimes\mathbf{v}=\mathbf{u}\mathbf{v}^{\mathrm{T}}=\begin{bmatrix}u% _{1}\\ u_{2}\\ u_{3}\\ u_{4}\end{bmatrix}\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}=\begin{bmatrix% }u_{1}v_{1}&u_{1}v_{2}&u_{1}v_{3}\\ u_{2}v_{1}&u_{2}v_{2}&u_{2}v_{3}\\ u_{3}v_{1}&u_{3}v_{2}&u_{3}v_{3}\\ u_{4}v_{1}&u_{4}v_{2}&u_{4}v_{3}\end{bmatrix}.
  2. ( 𝐮𝐯 T ) i j = u i v j (\mathbf{u}\mathbf{v}^{\mathrm{T}})_{ij}=u_{i}v_{j}
  3. 𝐮 𝐯 = 𝐮𝐯 H . \mathbf{u}\otimes\mathbf{v}=\mathbf{u}\mathbf{v}^{\mathrm{H}}.
  4. 𝐮 , 𝐯 = 𝐮 T 𝐯 \left\langle\mathbf{u},\mathbf{v}\right\rangle=\mathbf{u}^{\mathrm{T}}\mathbf{v}
  5. ( 𝐮𝐯 T ) 𝐱 = 𝐮 ( 𝐯 T 𝐱 ) (\mathbf{u}\mathbf{v}^{\mathrm{T}})\mathbf{x}=\mathbf{u}(\mathbf{v}^{\mathrm{T% }}\mathbf{x})
  6. 𝐮 = ( u 1 , u 2 , , u m ) \mathbf{u}=(u_{1},u_{2},\dots,u_{m})
  7. 𝐯 = ( v 1 , v 2 , , v n ) \mathbf{v}=(v_{1},v_{2},\dots,v_{n})
  8. 𝐮 𝐯 = 𝐀 = [ u 1 v 1 u 1 v 2 u 1 v n u 2 v 1 u 2 v 2 u 2 v n u m v 1 u m v 2 u m v n ] . \mathbf{u}\otimes\mathbf{v}=\mathbf{A}=\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&% \dots&u_{1}v_{n}\\ u_{2}v_{1}&u_{2}v_{2}&\dots&u_{2}v_{n}\\ \vdots&\vdots&\ddots&\vdots\\ u_{m}v_{1}&u_{m}v_{2}&\dots&u_{m}v_{n}\end{bmatrix}.
  9. 𝐜 = 𝐚 𝐛 , c i j = a i b j \mathbf{c}=\mathbf{a}\otimes\mathbf{b},\quad c_{ij}=a_{i}b_{j}
  10. 𝐓 = 𝐚 𝐛 𝐜 , T i j k = a i b j c k \mathbf{T}=\mathbf{a}\otimes\mathbf{b}\otimes\mathbf{c},\quad T_{ijk}=a_{i}b_{% j}c_{k}
  11. j = 1 e V i j U j k \sum_{j=1}^{e}V_{ij}U_{jk}
  12. C s t = V i j U h k C_{st}=V_{ij}U_{hk}
  13. w y * ( w ) x . w\mapsto y^{*}(w)x.

Outlier.html

  1. m m
  2. n n
  3. Q 1 Q_{1}
  4. Q 3 Q_{3}
  5. [ Q 1 - k ( Q 3 - Q 1 ) , Q 3 + k ( Q 3 - Q 1 ) ] \big[Q_{1}-k(Q_{3}-Q_{1}),Q_{3}+k(Q_{3}-Q_{1})\big]
  6. k k
  7. R e j e c t i o n R e g i o n < m t p l t α / 2 ( n - 1 ) n n - 2 + t α / 2 2 RejectionRegion<mtpl>{{=}}\frac{{t_{\alpha/2}}{\left(n-1\right)}}{\sqrt{n}% \sqrt{n-2+{t_{\alpha/2}^{2}}}}
  8. t α / 2 {t_{\alpha/2}}
  9. 1 - p ( y | x ) 1-p(y|x)
  10. y y
  11. x x
  12. t t
  13. H H
  14. I H ( x , y ) \displaystyle IH(\langle x,y\rangle)
  15. H H
  16. p ( h | t ) p(h|t)
  17. L H L\subset H
  18. I H L ( x , y ) = 1 - 1 | L | j = 1 | L | p ( y | x , g j ( t , α ) IH_{L}(\langle x,y\rangle)=1-\frac{1}{|L|}\sum_{j=1}^{|L|}p(y|x,g_{j}(t,\alpha)
  19. g j ( t , α ) g_{j}(t,\alpha)
  20. g j g_{j}
  21. t t
  22. α \alpha

Overburden_pressure.html

  1. p ( z ) = p 0 + g 0 z ρ ( z ) d z p(z)=p_{0}+g\int_{0}^{z}\rho(z)\,dz

Overpressure.html

  1. Δ p = 2410 ( m V ) 0.72 \Delta p=2410\left({m\over V}\right)^{0.72}
  2. m m\,
  3. V V\,

Õ.html

  1. f ( n ) = O ~ ( g ( n ) ) f(n)=\tilde{O}(g(n))
  2. f ( n ) = O ( g ( n ) log k g ( n ) ) f(n)=O(g(n)\log^{k}g(n))

P-adic_analysis.html

  1. p p
  2. ( Δ f ) ( x ) = f ( x + 1 ) - f ( x ) (\Delta f)(x)=f(x+1)-f(x)\,
  3. f ( x ) = k = 0 ( Δ k f ) ( 0 ) ( x k ) , f(x)=\sum_{k=0}^{\infty}(\Delta^{k}f)(0){x\choose k},
  4. ( x k ) = x ( x - 1 ) ( x - 2 ) ( x - k + 1 ) k ! {x\choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}
  5. p p
  6. p p
  7. p p
  8. f ( x ) f(x)
  9. f ( r ) 0 ( mod p k ) f(r)\equiv 0\;\;(\mathop{{\rm mod}}p^{k})
  10. f ( r ) 0 ( mod p ) f^{\prime}(r)\not\equiv 0\;\;(\mathop{{\rm mod}}p)
  11. f ( s ) 0 ( mod p k + m ) f(s)\equiv 0\;\;(\mathop{{\rm mod}}p^{k+m})
  12. r s ( mod p k ) . r\equiv s\;\;(\mathop{{\rm mod}}p^{k}).
  13. s = r + t p k s=r+tp^{k}
  14. t = - f ( r ) p k ( f ( r ) - 1 ) . t=-\frac{f(r)}{p^{k}}\cdot(f^{\prime}(r)^{-1}).

Packing_problems.html

  1. k k
  2. n n
  3. k n + 1 \scriptstyle k\leq n+1
  4. k k
  5. a 1 , . . , a k a_{1},..,a_{k}
  6. ( k - 1 ) \scriptstyle(k-1)
  7. 2 ( 1 - 1 k ) \scriptstyle\sqrt{2\big(1-\frac{1}{k}\big)}
  8. k \scriptstyle k
  9. k \scriptstyle k
  10. a 1 , . . , a k \scriptstyle a_{1},..,a_{k}
  11. r k := 1 + 2 ( 1 - 1 k ) \scriptstyle r_{k}:=1+\sqrt{2\big(1-\frac{1}{k}\big)}
  12. x 1 , , x k \scriptstyle x_{1},...,x_{k}
  13. k \scriptstyle k
  14. r \scriptstyle r
  15. x 0 \scriptstyle x_{0}
  16. { x 1 , . . x k } \scriptstyle\{x_{1},..x_{k}\}
  17. { a 1 , . . a k } \scriptstyle\{a_{1},..a_{k}\}
  18. x j \scriptstyle x_{j}
  19. a j \scriptstyle a_{j}
  20. 1 j k \scriptstyle 1\leq j\leq k
  21. 1 i < j k \scriptstyle 1\leq i<j\leq k
  22. a i - a j = 2 x i - x j \scriptstyle\|a_{i}-a_{j}\|=2\leq\|x_{i}-x_{j}\|
  23. a 0 \scriptstyle a_{0}
  24. 1 j k \scriptstyle 1\leq j\leq k
  25. a 0 - a j x 0 - x j \scriptstyle\|a_{0}-a_{j}\|\leq\|x_{0}-x_{j}\|
  26. r k 1 + a 0 - a j 1 + x 0 - x j r \scriptstyle r_{k}\leq 1+\|a_{0}-a_{j}\|\leq 1+\|x_{0}-x_{j}\|\leq r
  27. k \scriptstyle k
  28. r \scriptstyle r
  29. r r k \scriptstyle r\geq r_{k}
  30. r \scriptstyle r
  31. r 1 + 2 \scriptstyle r\geq 1+\sqrt{2}
  32. 2 e j \scriptstyle\sqrt{2}e_{j}
  33. { e j } j \scriptstyle\{e_{j}\}_{j}
  34. 1 + 2 \scriptstyle 1+\sqrt{2}
  35. r < 1 + 2 \scriptstyle r<1+\sqrt{2}
  36. 2 2 - ( r - 1 ) 2 \scriptstyle\big\lfloor\frac{2}{2-(r-1)^{2}}\big\rfloor

Pafnuty_Chebyshev.html

  1. X X
  2. X X
  3. a σ a\sigma
  4. 1 / a 2 1/a^{2}
  5. Pr ( | X - 𝐄 ( X ) | a σ ) 1 a 2 . \Pr(|X-{\mathbf{E}}(X)|\geq a\,\sigma)\leq\frac{1}{a^{2}}.
  6. n > 1 n>1
  7. p p
  8. n < p < 2 n n<p<2n
  9. π ( n ) \pi(n)
  10. n n
  11. π ( n ) \pi(n)
  12. n / log ( n ) n/\log(n)
  13. n n

Pair_production.html

  1. E E
  2. m m
  3. c c
  4. p γ = p e - + p e + + p R p_{\gamma}=p_{e-}+p_{e+}+p_{R}
  5. p R p_{R}
  6. A = ( A 0 , A ) A=(A^{0},A)
  7. A 2 = A μ A μ = - ( A 0 ) 2 + A A A^{2}=A^{\mu}A_{\mu}=-(A^{0})^{2}+A\cdot A
  8. ( p γ ) 2 = 0 (p_{\gamma})^{2}=0
  9. ( p e - ) 2 = - m e 2 c 2 (p_{e-})^{2}=-m_{e}^{2}c^{2}
  10. ( p γ ) 2 = ( p e - + p e + + p R ) 2 (p_{\gamma})^{2}=(p_{e-}+p_{e+}+p_{R})^{2}
  11. p R 0 p_{R}\approx 0
  12. ( p γ ) 2 = ( p e - ) 2 + 2 p e - p e + + ( p e + ) 2 (p_{\gamma})^{2}=(p_{e-})^{2}+2p_{e-}p_{e+}+(p_{e+})^{2}
  13. - 2 m e 2 c 2 + 2 ( E 2 c 2 + p e - p e + ) = 0 -2m_{e}^{2}c^{2}+2(\frac{E^{2}}{c^{2}}+p_{e-}\cdot p_{e+})=0
  14. 2 ( γ 2 - 1 ) m e 2 c 2 ( 1 + cos θ e ) = 0 2(\gamma^{2}-1)m_{e}^{2}c^{2}(1+\cos\theta_{e})=0
  15. θ e = π \theta_{e}=\pi
  16. ( E k p p ) t r = h ν - 2 m e c 2 (E_{k}^{pp})_{tr}=h\nu-2m_{e}c^{2}
  17. h h
  18. ν \nu
  19. 2 m e c 2 2m_{e}c^{2}
  20. ( E ¯ k p p ) t r = 1 2 ( h ν - 2 m e c 2 ) (\bar{E}_{k}^{pp})_{tr}=\frac{1}{2}(h\nu-2m_{e}c^{2})
  21. σ = α r e 2 Z 2 P ( E , Z ) \sigma=\alpha r_{e}^{2}Z^{2}P(E,Z)
  22. α \alpha
  23. r e r_{e}
  24. Z Z
  25. P ( E , Z ) P(E,Z)

Palindromic_number.html

  1. n = i = 0 k a i b i n=\sum_{i=0}^{k}a_{i}b^{i}
  2. 1221 4 = 151 8 = 77 14 = 55 20 = 33 34 = 11 104 1221_{4}=151_{8}=77_{14}=55_{20}=33_{34}=11_{104}
  3. 1991 10 = 7 C 7 16 1991_{10}=7C7_{16}

Paper_size.html

  1. 2 {}^{2}
  2. 2 \sqrt{2}
  3. 2 \sqrt{2}
  4. a a
  5. b b
  6. a / b = 2 a/b=\sqrt{2}
  7. c c
  8. c = a / 2 c=a/2
  9. b c = b a 2 = 2 a b = 2 2 = 2 \frac{b}{c}=\frac{b}{\frac{a}{2}}=\frac{2}{\frac{a}{b}}=\frac{2}{\sqrt{2}}=% \sqrt{2}
  10. 2 {}^{2}
  11. 2 {}^{2}
  12. 2 \sqrt{2}
  13. 2 {}^{2}
  14. 1 / 2 \sqrt{1/2}
  15. 2 {}^{2}
  16. 1 / 2 {1}/{2}
  17. 1 / 4 {1}/{4}
  18. 1 / 8 {1}/{8}
  19. 33 ¯ \overline{33}
  20. 77 ¯ \overline{77}
  21. 54 ¯ \overline{54}
  22. 55 ¯ \overline{55}
  23. 2 ¯ \overline{2}
  24. 54 ¯ \overline{54}
  25. 3 ¯ \overline{3}
  26. 27 ¯ \overline{27}
  27. 7 ¯ \overline{7}
  28. 3 ¯ \overline{3}
  29. 5 ¯ \overline{5}
  30. 6 ¯ \overline{6}
  31. 1 / 8 {1}/{8}
  32. 6 ¯ \overline{6}
  33. 3 ¯ \overline{3}
  34. 3 ¯ \overline{3}
  35. 3 ¯ \overline{3}
  36. 3 ¯ \overline{3}
  37. 27 ¯ \overline{27}
  38. 61 / 4 6{1}/{4}

Parabolic_antenna.html

  1. G = 4 π A λ 2 e A = ( π d λ ) 2 e A G=\frac{4\pi A}{\lambda^{2}}e_{A}=(\frac{\pi d}{\lambda})^{2}e_{A}
  2. A A
  3. d d
  4. λ \lambda
  5. e A e_{A}
  6. A = r 2 π A=r^{2}\pi
  7. r = d / 2 r=d/2
  8. θ = k λ / d \theta=k\lambda/d\,
  9. G = ( π k θ ) 2 e A G=\left(\frac{\pi k}{\theta}\right)^{2}\ e_{A}

Parabolic_reflector.html

  1. 4 f y = x 2 \scriptstyle 4fy=x^{2}
  2. f \scriptstyle f
  3. 4 F D = R 2 , \scriptstyle 4FD=R^{2},
  4. F \scriptstyle F
  5. D \scriptstyle D
  6. R \scriptstyle R
  7. P = 2 F \scriptstyle P=2F
  8. P = R 2 2 D ) \scriptstyle P=\frac{R^{2}}{2D})
  9. Q = P 2 + R 2 , \scriptstyle Q=\sqrt{P^{2}+R^{2}},
  10. F , \scriptstyle F,
  11. D , \scriptstyle D,
  12. R \scriptstyle R
  13. R Q P + P ln ( R + Q P ) , \scriptstyle\frac{RQ}{P}+P\ln\left(\frac{R+Q}{P}\right),
  14. ln ( x ) \scriptstyle\ln(x)
  15. x \scriptstyle x
  16. 1 2 π R 2 D , \scriptstyle\frac{1}{2}\pi R^{2}D,
  17. ( π R 2 D ) , \scriptstyle(\pi R^{2}D),
  18. ( 2 3 π R 2 D , \scriptstyle(\frac{2}{3}\pi R^{2}D,
  19. D = R ) , \scriptstyle D=R),
  20. ( 1 3 π R 2 D ) . \scriptstyle(\frac{1}{3}\pi R^{2}D).
  21. π R 2 \scriptstyle\pi R^{2}
  22. A = π R 6 D 2 ( ( R 2 + 4 D 2 ) 3 / 2 - R 3 ) \scriptstyle A=\frac{\pi R}{6D^{2}}\left((R^{2}+4D^{2})^{3/2}-R^{3}\right)
  23. D 0 \scriptstyle D\neq 0
  24. 1 20 \frac{1}{20}

Paraboloid.html

  1. x x
  2. y y
  3. z z
  4. z c = x 2 a 2 + y 2 b 2 . \frac{z}{c}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}.
  5. a a
  6. b b
  7. x x
  8. z z
  9. y y
  10. z z
  11. z c = y 2 b 2 - x 2 a 2 . \frac{z}{c}=\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}.
  12. σ ( u , v ) = ( u , v , u 2 a 2 + v 2 b 2 ) \vec{\sigma}(u,v)=\left(u,v,{u^{2}\over a^{2}}+{v^{2}\over b^{2}}\right)
  13. K ( u , v ) = 4 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 2 K(u,v)={4\over a^{2}b^{2}\left(1+{4u^{2}\over a^{4}}+{4v^{2}\over b^{4}}\right% )^{2}}
  14. H ( u , v ) = a 2 + b 2 + 4 u 2 a 2 + 4 v 2 b 2 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 3 / 2 H(u,v)={a^{2}+b^{2}+{4u^{2}\over a^{2}}+{4v^{2}\over b^{2}}\over a^{2}b^{2}% \left(1+{4u^{2}\over a^{4}}+{4v^{2}\over b^{4}}\right)^{3/2}}
  15. σ ( u , v ) = ( u , v , u 2 a 2 - v 2 b 2 ) \vec{\sigma}(u,v)=\left(u,v,{u^{2}\over a^{2}}-{v^{2}\over b^{2}}\right)
  16. K ( u , v ) = - 4 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 2 K(u,v)={-4\over a^{2}b^{2}\left(1+{4u^{2}\over a^{4}}+{4v^{2}\over b^{4}}% \right)^{2}}
  17. H ( u , v ) = - a 2 + b 2 - 4 u 2 a 2 + 4 v 2 b 2 a 2 b 2 ( 1 + 4 u 2 a 4 + 4 v 2 b 4 ) 3 / 2 . H(u,v)={-a^{2}+b^{2}-{4u^{2}\over a^{2}}+{4v^{2}\over b^{2}}\over a^{2}b^{2}% \left(1+{4u^{2}\over a^{4}}+{4v^{2}\over b^{4}}\right)^{3/2}}.
  18. z = x 2 a 2 - y 2 b 2 z={x^{2}\over a^{2}}-{y^{2}\over b^{2}}
  19. z = 1 2 ( x 2 + y 2 ) ( 1 a 2 - 1 b 2 ) + x y ( 1 a 2 + 1 b 2 ) z={1\over 2}(x^{2}+y^{2})\left({1\over a^{2}}-{1\over b^{2}}\right)+xy\left({1% \over a^{2}}+{1\over b^{2}}\right)
  20. a = b \ a=b
  21. z = 2 a 2 x y z={2\over a^{2}}xy
  22. a = 2 a=\sqrt{2}
  23. z = x 2 - y 2 2 . z={x^{2}-y^{2}\over 2}.
  24. z = x y \ z=xy
  25. 2 \mathbb{R}^{2}\rightarrow\mathbb{R}
  26. z 1 ( x , y ) = x 2 - y 2 2 z_{1}(x,y)={x^{2}-y^{2}\over 2}
  27. z 2 ( x , y ) = x y \ z_{2}(x,y)=xy
  28. f ( z ) = 1 2 z 2 = f ( x + i y ) = z 1 ( x , y ) + i z 2 ( x , y ) f(z)={1\over 2}z^{2}=f(x+iy)=z_{1}(x,y)+iz_{2}(x,y)
  29. \mathbb{R}\rightarrow\mathbb{R}
  30. f ( x ) = 1 2 x 2 . \ f(x)={1\over 2}x^{2}.
  31. 4 F D = R 2 , \scriptstyle 4FD=R^{2},
  32. F \scriptstyle F
  33. D \scriptstyle D
  34. R \scriptstyle R
  35. P = 2 F \scriptstyle P=2F
  36. P = R 2 2 D ) \scriptstyle P=\frac{R^{2}}{2D})
  37. Q = P 2 + R 2 , \scriptstyle Q=\sqrt{P^{2}+R^{2}},
  38. F , \scriptstyle F,
  39. D , \scriptstyle D,
  40. R \scriptstyle R
  41. R Q P + P ln ( R + Q P ) , \scriptstyle\frac{RQ}{P}+P\ln\left(\frac{R+Q}{P}\right),
  42. ln ( x ) \scriptstyle\ln(x)
  43. x \scriptstyle x
  44. 1 2 π R 2 D , \scriptstyle\frac{1}{2}\pi R^{2}D,
  45. ( π R 2 D ) , \scriptstyle(\pi R^{2}D),
  46. ( 2 3 π R 2 D , \scriptstyle(\frac{2}{3}\pi R^{2}D,
  47. D = R ) , \scriptstyle D=R),
  48. ( 1 3 π R 2 D ) . \scriptstyle(\frac{1}{3}\pi R^{2}D).
  49. π R 2 \scriptstyle\pi R^{2}
  50. A = π R 6 D 2 ( ( R 2 + 4 D 2 ) 3 / 2 - R 3 ) \scriptstyle A=\frac{\pi R}{6D^{2}}\left((R^{2}+4D^{2})^{3/2}-R^{3}\right)

Parallel_computing.html

  1. α \alpha
  2. lim P 1 1 - α P + α = 1 α \lim_{P\to\infty}\frac{1}{\frac{1-\alpha}{P}+\alpha}=\frac{1}{\alpha}
  3. α = 0.1 \alpha=0.1
  4. P P
  5. S ( P ) = P - α ( P - 1 ) = α + P ( 1 - α ) . S(P)=P-\alpha(P-1)\qquad=\alpha+P(1-\alpha).\,
  6. I j O i = , I_{j}\cap O_{i}=\varnothing,\,
  7. I i O j = , I_{i}\cap O_{j}=\varnothing,\,
  8. O i O j = . O_{i}\cap O_{j}=\varnothing.\,

Parallel_transport.html

  1. α \alpha
  2. X X
  3. E E
  4. γ ˙ ( t ) X = 0 for t I . \nabla_{\dot{\gamma}(t)}X=0\,\text{ for }t\in I.\,
  5. γ ˙ X = 0 \nabla_{\dot{\gamma}}X=0
  6. X γ ( 0 ) = e 0 . X_{\gamma(0)}=e_{0}.
  7. Γ ( γ ) s t : E γ ( s ) E γ ( t ) \Gamma(\gamma)_{s}^{t}:E_{\gamma(s)}\rightarrow E_{\gamma(t)}
  8. γ ˙ = 0 \scriptstyle{\nabla_{\dot{\gamma}}=0}
  9. Γ ( γ ) s t : E γ ( s ) E γ ( t ) \Gamma(\gamma)_{s}^{t}:E_{\gamma(s)}\rightarrow E_{\gamma(t)}
  10. Γ ( γ ) s s = I d \Gamma(\gamma)_{s}^{s}=Id
  11. Γ ( γ ) u t Γ ( γ ) s u = Γ ( γ ) s t . \Gamma(\gamma)_{u}^{t}\circ\Gamma(\gamma)_{s}^{u}=\Gamma(\gamma)_{s}^{t}.
  12. X V = lim h 0 Γ ( γ ) h 0 V γ ( h ) - V γ ( 0 ) h = d d t Γ ( γ ) t 0 V γ ( t ) | t = 0 . \nabla_{X}V=\lim_{h\to 0}\frac{\Gamma(\gamma)_{h}^{0}V_{\gamma(h)}-V_{\gamma(0% )}}{h}=\left.\frac{d}{dt}\Gamma(\gamma)_{t}^{0}V_{\gamma(t)}\right|_{t=0}.
  13. γ ˙ \dot{\gamma}
  14. γ \gamma
  15. Γ ( γ ) s t γ ˙ ( s ) = γ ˙ ( t ) . \Gamma(\gamma)_{s}^{t}\dot{\gamma}(s)=\dot{\gamma}(t).\,
  16. γ ˙ ( t ) γ ˙ = 0. \nabla_{\dot{\gamma}(t)}\dot{\gamma}=0.\,
  17. Γ ( γ ) s t X , Γ ( γ ) s t Y γ ( t ) = X , Y γ ( s ) . \langle\Gamma(\gamma)_{s}^{t}X,\Gamma(\gamma)_{s}^{t}Y\rangle_{\gamma(t)}=% \langle X,Y\rangle_{\gamma(s)}.
  18. Z X , Y = Z X , Y + X , Z Y . Z\langle X,Y\rangle=\langle\nabla_{Z}X,Y\rangle+\langle X,\nabla_{Z}Y\rangle.
  19. γ ˙ \dot{\gamma}
  20. d d t γ ˙ ( t ) , γ ˙ ( t ) = 2 γ ˙ ( t ) γ ˙ ( t ) , γ ˙ ( t ) = 0. \frac{d}{dt}\langle\dot{\gamma}(t),\dot{\gamma}(t)\rangle=2\langle\nabla_{\dot% {\gamma}(t)}\dot{\gamma}(t),\dot{\gamma}(t)\rangle=0.
  21. γ ˙ ( t ) \dot{\gamma}(t)
  22. dist ( γ ( t 1 ) , γ ( t 2 ) ) = A | t 1 - t 2 | . \mbox{dist}~{}\big(\gamma(t_{1}),\gamma(t_{2})\big)=A|t_{1}-t_{2}|.
  23. Γ ( γ ) s t : P γ ( s ) P γ ( t ) \Gamma(\gamma)_{s}^{t}:P_{\gamma(s)}\rightarrow P_{\gamma(t)}
  24. Γ γ ( s ) g u = g Γ γ ( s ) \Gamma_{\gamma(s)}gu=g\Gamma_{\gamma(s)}

Parallelogram.html

  1. K = b h . K=bh.
  2. K rect = ( B + A ) × H K\text{rect}=(B+A)\times H\,
  3. K tri = 1 2 A × H . K\text{tri}=\frac{1}{2}A\times H.\,
  4. K = K rect - 2 × K tri = ( ( B + A ) × H ) - ( A × H ) = B × H . K=K\text{rect}-2\times K\text{tri}=((B+A)\times H)-(A\times H)=B\times H.
  5. K = B C sin θ . K=B\cdot C\cdot\sin\theta.\,
  6. γ \gamma
  7. K = | tan γ | 2 | B 2 - C 2 | . K=\frac{|\tan\gamma|}{2}\cdot\left|B^{2}-C^{2}\right|.
  8. K = 2 S ( S - B ) ( S - C ) ( S - D 1 ) K=2\sqrt{S(S-B)(S-C)(S-D_{1})}
  9. S = ( B + C + D 1 ) / 2 S=(B+C+D_{1})/2
  10. 𝐚 , 𝐛 \R 2 \mathbf{a},\mathbf{b}\in\R^{2}
  11. V = [ a 1 a 2 b 1 b 2 ] \R 2 × 2 V=\begin{bmatrix}a_{1}&a_{2}\\ b_{1}&b_{2}\end{bmatrix}\in\R^{2\times 2}
  12. | det ( V ) | = | a 1 b 2 - a 2 b 1 | |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,
  13. 𝐚 , 𝐛 \R n \mathbf{a},\mathbf{b}\in\R^{n}
  14. V = [ a 1 a 2 a n b 1 b 2 b n ] \R 2 × n V=\begin{bmatrix}a_{1}&a_{2}&\dots&a_{n}\\ b_{1}&b_{2}&\dots&b_{n}\end{bmatrix}\in\R^{2\times n}
  15. det ( V V T ) \sqrt{\det(VV^{\mathrm{T}})}
  16. a , b , c \R 2 a,b,c\in\R^{2}
  17. K = | det [ a 1 a 2 1 b 1 b 2 1 c 1 c 2 1 ] | . K=\left|\det\begin{bmatrix}a_{1}&a_{2}&1\\ b_{1}&b_{2}&1\\ c_{1}&c_{2}&1\end{bmatrix}\right|.
  18. A B E C D E \angle ABE\cong\angle CDE
  19. B A E D C E \angle BAE\cong\angle DCE
  20. A E = C E AE=CE
  21. B E = D E . BE=DE.

Pareto_interpolation.html

  1. κ ^ = ( P b - P a ( 1 / a θ ^ ) - ( 1 / b θ ^ ) ) 1 / θ ^ \widehat{\kappa}=\left(\frac{P_{b}-P_{a}}{\left(1/a^{\widehat{\theta}}\right)-% \left(1/b^{\widehat{\theta}}\right)}\right)^{1/\widehat{\theta}}
  2. θ ^ = log ( 1 - P a ) - log ( 1 - P b ) log ( b ) - log ( a ) . \widehat{\theta}\;=\;\frac{\log(1-P_{a})-\log(1-P_{b})}{\log(b)-\log(a)}.
  3. estimated median = κ ^ 2 1 / θ ^ , \mbox{estimated median}~{}=\widehat{\kappa}\cdot 2^{1/\widehat{\theta}},\,
  4. median = κ 2 1 / θ . \mbox{median}~{}=\kappa\,2^{1/\theta}.\,

Parity_(mathematics).html

  1. = { 2 k : k } =\{2k:k\in\mathbb{Z}\}
  2. = { 2 k + 1 : k } =\{2k+1:k\in\mathbb{Z}\}
  3. 0 + I 0+I
  4. 1 + I 1+I

Parse_tree.html

  1. [ S [ 𝑁𝑃 J o h n ] [ 𝑉𝑃 [ V h i t ] [ 𝑁𝑃 t h e [ N b a l l ] ] ] ] [_{S}\ [_{\mathit{NP}}\ John]\ [_{\mathit{VP}}\ [_{V}\ hit]\ [_{\mathit{NP}}\ % the\ [_{N}\ ball]]]]

Partial_fraction_decomposition.html

  1. f ( x ) g ( x ) \frac{f(x)}{g(x)}
  2. j f j ( x ) g j ( x ) \sum_{j}\frac{f_{j}(x)}{g_{j}(x)}
  3. R ( x ) = f ( x ) g ( x ) R(x)=\frac{f(x)}{g(x)}
  4. g ( x ) = P ( x ) Q ( x ) g(x)=P(x)\cdot Q(x)\,
  5. C P + D Q = 1 CP+DQ=1
  6. 1 g ( x ) = C P + D Q P Q = C Q + D P \frac{1}{g(x)}=\frac{CP+DQ}{PQ}=\frac{C}{Q}+\frac{D}{P}
  7. R = f ( x ) g ( x ) = D f ( x ) P + C f ( x ) Q , R=\frac{f(x)}{g(x)}=\frac{Df(x)}{P}+\frac{Cf(x)}{Q},
  8. G ( x ) F ( x ) n \frac{G(x)}{F(x)^{n}}
  9. X X^{\prime}
  10. X . X.
  11. c i 1 p i = α j : p i ( α j ) = 0 c i 1 ( α j ) p i ( α j ) 1 x - α j . \frac{c_{i1}}{p_{i}}=\sum_{\alpha_{j}:p_{i}(\alpha_{j})=0}\frac{c_{i1}(\alpha_% {j})}{p^{\prime}_{i}(\alpha_{j})}\frac{1}{x-\alpha_{j}}.
  12. P ( x ) P(x)
  13. Q ( x ) = ( x - α 1 ) ( x - α 2 ) ( x - α n ) Q(x)=(x-\alpha_{1})(x-\alpha_{2})\cdots(x-\alpha_{n})
  14. P ( x ) Q ( x ) = i = 1 n P ( α i ) Q ( α i ) 1 ( x - α i ) \frac{P(x)}{Q(x)}=\sum_{i=1}^{n}\frac{P(\alpha_{i})}{Q^{\prime}(\alpha_{i})}% \frac{1}{(x-\alpha_{i})}
  15. Q Q^{\prime}
  16. Q Q
  17. \geq
  18. P ( x ) Q ( x ) = E ( x ) + R ( x ) Q ( x ) , \frac{P(x)}{Q(x)}=E(x)+\frac{R(x)}{Q(x)},
  19. x 2 + 1 ( x + 2 ) ( x - 1 ) \color B l u e ( x 2 + x + 1 ) = a x + 2 + b x - 1 + \color O l i v e G r e e n c x + d \color B l u e x 2 + x + 1 . \frac{x^{2}+1}{(x+2)(x-1)\color{Blue}(x^{2}+x+1)}=\frac{a}{x+2}+\frac{b}{x-1}+% \frac{\color{OliveGreen}cx+d}{\color{Blue}x^{2}+x+1}.
  20. P ( x ) Q ( x ) = P ( x ) ( x - α ) r = c 1 x - α + c 2 ( x - α ) 2 + + c r ( x - α ) r . \frac{P(x)}{Q(x)}=\frac{P(x)}{(x-\alpha)^{r}}=\frac{c_{1}}{x-\alpha}+\frac{c_{% 2}}{(x-\alpha)^{2}}+\cdots+\frac{c_{r}}{(x-\alpha)^{r}}.
  21. 3 x + 5 ( 1 - 2 x ) 2 = A ( 1 - 2 x ) 2 + B ( 1 - 2 x ) . \frac{3x+5}{(1-2x)^{2}}=\frac{A}{(1-2x)^{2}}+\frac{B}{(1-2x)}.
  22. 3 x + 5 ( 1 - 2 x ) 2 = 13 / 2 ( 1 - 2 x ) 2 + - 3 / 2 ( 1 - 2 x ) . \frac{3x+5}{(1-2x)^{2}}=\frac{13/2}{(1-2x)^{2}}+\frac{-3/2}{(1-2x)}.
  23. f ( x ) = i ( a i 1 x - x i + a i 2 ( x - x i ) 2 + + a i k i ( x - x i ) k i ) . f(x)=\sum_{i}\left(\frac{a_{i1}}{x-x_{i}}+\frac{a_{i2}}{(x-x_{i})^{2}}+\cdots+% \frac{a_{ik_{i}}}{(x-x_{i})^{k_{i}}}\right).
  24. g i j ( x ) = ( x - x i ) j - 1 f ( x ) , g_{ij}(x)=(x-x_{i})^{j-1}f(x),
  25. a i j = Res ( g i j , x i ) . a_{ij}=\operatorname{Res}(g_{ij},x_{i}).
  26. a i j = 1 ( k i - j ) ! lim x x i d k i - j d x k i - j ( ( x - x i ) k i f ( x ) ) , a_{ij}=\frac{1}{(k_{i}-j)!}\lim_{x\to x_{i}}\frac{d^{k_{i}-j}}{dx^{k_{i}-j}}% \left((x-x_{i})^{k_{i}}f(x)\right),
  27. a i 1 = P ( x i ) Q ( x i ) , a_{i1}=\frac{P(x_{i})}{Q^{\prime}(x_{i})},
  28. f ( x ) = P ( x ) Q ( x ) . f(x)=\frac{P(x)}{Q(x)}.
  29. f ( x ) = p ( x ) q ( x ) f(x)=\frac{p(x)}{q(x)}
  30. q ( x ) = ( x - a 1 ) j 1 ( x - a m ) j m ( x 2 + b 1 x + c 1 ) k 1 ( x 2 + b n x + c n ) k n q(x)=(x-a_{1})^{j_{1}}\cdots(x-a_{m})^{j_{m}}(x^{2}+b_{1}x+c_{1})^{k_{1}}% \cdots(x^{2}+b_{n}x+c_{n})^{k_{n}}
  31. f ( x ) = p ( x ) q ( x ) = P ( x ) + i = 1 m r = 1 j i A i r ( x - a i ) r + i = 1 n r = 1 k i B i r x + C i r ( x 2 + b i x + c i ) r f(x)=\frac{p(x)}{q(x)}=P(x)+\sum_{i=1}^{m}\sum_{r=1}^{j_{i}}\frac{A_{ir}}{(x-a% _{i})^{r}}+\sum_{i=1}^{n}\sum_{r=1}^{k_{i}}\frac{B_{ir}x+C_{ir}}{(x^{2}+b_{i}x% +c_{i})^{r}}
  32. f ( x ) = 1 x 2 + 2 x - 3 f(x)=\frac{1}{x^{2}+2x-3}
  33. q ( x ) = x 2 + 2 x - 3 = ( x + 3 ) ( x - 1 ) q(x)=x^{2}+2x-3=(x+3)(x-1)
  34. f ( x ) = 1 x 2 + 2 x - 3 = A x + 3 + B x - 1 f(x)=\frac{1}{x^{2}+2x-3}=\frac{A}{x+3}+\frac{B}{x-1}
  35. 1 = A ( x - 1 ) + B ( x + 3 ) 1=A(x-1)+B(x+3)
  36. f ( x ) = 1 x 2 + 2 x - 3 = 1 4 ( - 1 x + 3 + 1 x - 1 ) f(x)=\frac{1}{x^{2}+2x-3}=\frac{1}{4}\left(\frac{-1}{x+3}+\frac{1}{x-1}\right)
  37. f ( x ) = x 3 + 16 x 3 - 4 x 2 + 8 x f(x)=\frac{x^{3}+16}{x^{3}-4x^{2}+8x}
  38. f ( x ) = 1 + 4 x 2 - 8 x + 16 x 3 - 4 x 2 + 8 x = 1 + 4 x 2 - 8 x + 16 x ( x 2 - 4 x + 8 ) f(x)=1+\frac{4x^{2}-8x+16}{x^{3}-4x^{2}+8x}=1+\frac{4x^{2}-8x+16}{x(x^{2}-4x+8)}
  39. 4 x 2 - 8 x + 16 x ( x 2 - 4 x + 8 ) = A x + B x + C x 2 - 4 x + 8 \frac{4x^{2}-8x+16}{x(x^{2}-4x+8)}=\frac{A}{x}+\frac{Bx+C}{x^{2}-4x+8}
  40. 4 x 2 - 8 x + 16 = A ( x 2 - 4 x + 8 ) + ( B x + C ) x 4x^{2}-8x+16=A(x^{2}-4x+8)+(Bx+C)x
  41. f ( x ) = 1 + 2 ( 1 x + x x 2 - 4 x + 8 ) f(x)=1+2\left(\frac{1}{x}+\frac{x}{x^{2}-4x+8}\right)
  42. f ( x ) = x 9 - 2 x 6 + 2 x 5 - 7 x 4 + 13 x 3 - 11 x 2 + 12 x - 4 x 7 - 3 x 6 + 5 x 5 - 7 x 4 + 7 x 3 - 5 x 2 + 3 x - 1 f(x)=\frac{x^{9}-2x^{6}+2x^{5}-7x^{4}+13x^{3}-11x^{2}+12x-4}{x^{7}-3x^{6}+5x^{% 5}-7x^{4}+7x^{3}-5x^{2}+3x-1}
  43. f ( x ) = x 2 + 3 x + 4 + 2 x 6 - 4 x 5 + 5 x 4 - 3 x 3 + x 2 + 3 x ( x - 1 ) 3 ( x 2 + 1 ) 2 f(x)=x^{2}+3x+4+\frac{2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)% ^{2}}
  44. 2 x 6 - 4 x 5 + 5 x 4 - 3 x 3 + x 2 + 3 x ( x - 1 ) 3 ( x 2 + 1 ) 2 = A x - 1 + B ( x - 1 ) 2 + C ( x - 1 ) 3 + D x + E x 2 + 1 + F x + G ( x 2 + 1 ) 2 \frac{2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)^{2}}=\frac{A}{x% -1}+\frac{B}{(x-1)^{2}}+\frac{C}{(x-1)^{3}}+\frac{Dx+E}{x^{2}+1}+\frac{Fx+G}{(% x^{2}+1)^{2}}
  45. 2 x 6 - 4 x 5 + 5 x 4 - 3 x 3 + x 2 + 3 x \displaystyle{}\quad 2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x
  46. 2 x 6 - 4 x 5 + 5 x 4 - 3 x 3 + x 2 + 3 x \displaystyle{}2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x
  47. 2 x 6 - 4 x 5 + 5 x 4 - 3 x 3 + x 2 + 3 x \displaystyle{}2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x
  48. A + D \displaystyle A+D
  49. f ( x ) = x 2 + 3 x + 4 + 1 ( x - 1 ) + 1 ( x - 1 ) 3 + x + 1 x 2 + 1 + 1 ( x 2 + 1 ) 2 . f(x)=x^{2}+3x+4+\frac{1}{(x-1)}+\frac{1}{(x-1)^{3}}+\frac{x+1}{x^{2}+1}+\frac{% 1}{(x^{2}+1)^{2}}.
  50. 2 6 - 4 5 + 5 4 - 3 3 + 2 + 3 = A ( 0 + 0 ) + B ( 4 + 0 ) + 8 + D 0 2\cdot 6-4\cdot 5+5\cdot 4-3\cdot 3+2+3=A\cdot(0+0)+B\cdot(4+0)+8+D\cdot 0
  51. f ( z ) = z 2 - 5 ( z 2 - 1 ) ( z 2 + 1 ) = z 2 - 5 ( z + 1 ) ( z - 1 ) ( z + i ) ( z - i ) f(z)=\frac{z^{2}-5}{(z^{2}-1)(z^{2}+1)}=\frac{z^{2}-5}{(z+1)(z-1)(z+i)(z-i)}
  52. P ( z i ) Q ( z i ) = z i 2 - 5 4 z i 3 \frac{P(z_{i})}{Q^{\prime}(z_{i})}=\frac{z_{i}^{2}-5}{4z_{i}^{3}}
  53. 1 , - 1 , 3 i 2 , - 3 i 2 1,-1,\tfrac{3i}{2},-\tfrac{3i}{2}
  54. f ( z ) = 1 z + 1 - 1 z - 1 + 3 i 2 1 z + i - 3 i 2 1 z - i f(z)=\frac{1}{z+1}-\frac{1}{z-1}+\frac{3i}{2}\frac{1}{z+i}-\frac{3i}{2}\frac{1% }{z-i}
  55. f ( x ) = 1 x 3 - 1 f(x)=\frac{1}{x^{3}-1}
  56. f ( x ) = 1 ( x - 1 ) ( x 2 + x + 1 ) f(x)=\frac{1}{(x-1)(x^{2}+x+1)}
  57. 1 ( x - 1 ) ( x 2 + x + 1 ) = A x - 1 + B x + C x 2 + x + 1 \frac{1}{(x-1)(x^{2}+x+1)}=\frac{A}{x-1}+\frac{Bx+C}{x^{2}+x+1}
  58. x 1 x\to 1
  59. A x - 1 \frac{A}{x-1}
  60. 1 ( x - 1 ) ( x 2 + x + 1 ) = A x - 1 \frac{1}{(x-1)(x^{2}+x+1)}=\frac{A}{x-1}
  61. A = lim x 1 1 x 2 + x + 1 = 1 3 A=\lim_{x\to 1}{\frac{1}{x^{2}+x+1}}=\frac{1}{3}
  62. x x\to\infty
  63. lim x A x - 1 + B x + C x 2 + x + 1 = A x + B x x 2 = A + B x . \lim_{x\to\infty}{\frac{A}{x-1}+\frac{Bx+C}{x^{2}+x+1}}=\frac{A}{x}+\frac{Bx}{% x^{2}}=\frac{A+B}{x}.
  64. A + B x = lim x 1 x 3 - 1 = 0 \frac{A+B}{x}=\lim_{x\to\infty}{\frac{1}{x^{3}-1}}=0
  65. B = - 1 3 B=-\frac{1}{3}
  66. x = 0 x=0
  67. - 1 = - A + C -1=-A+C
  68. C = - 2 3 C=-\frac{2}{3}
  69. 1 3 x - 1 + - 1 3 x - 2 3 x 2 + x + 1 \frac{\frac{1}{3}}{x-1}+\frac{-\frac{1}{3}x-\frac{2}{3}}{x^{2}+x+1}
  70. P ( x ) , Q ( x ) , A 1 ( x ) , , A r ( x ) P(x),Q(x),A_{1}(x),\dots,A_{r}(x)
  71. Q = j = 1 r ( x - λ j ) ν j , \textstyle Q=\prod_{j=1}^{r}(x-\lambda_{j})^{\nu_{j}},
  72. deg ( P ) < deg ( Q ) = j = 1 r ν j , \textstyle\deg(P)<\deg(Q)=\sum_{j=1}^{r}\nu_{j},
  73. deg A j < ν j for j = 1 , , r . \textstyle\deg A_{j}<\nu_{j}\,\text{ for }j=1,\dots,r.
  74. Q i = j i ( x - λ j ) ν j = Q ( x - λ i ) ν i for i = 1 , , r . \textstyle Q_{i}=\prod_{j\neq i}(x-\lambda_{j})^{\nu_{j}}=\frac{Q}{(x-\lambda_% {i})^{\nu_{i}}}\,\text{ for }i=1,\dots,r.
  75. P Q = j = 1 r A j ( x - λ j ) ν j \frac{P}{Q}=\sum_{j=1}^{r}\frac{A_{j}}{(x-\lambda_{j})^{\nu_{j}}}
  76. i \textstyle i
  77. A i ( x ) \textstyle A_{i}(x)
  78. P Q i \textstyle\frac{P}{Q_{i}}
  79. ν i - 1 \textstyle\nu_{i}-1
  80. λ i \textstyle\lambda_{i}
  81. A i ( x ) := k = 0 ν i - 1 1 k ! ( P Q i ) ( k ) ( λ i ) ( x - λ i ) k . A_{i}(x):=\sum_{k=0}^{\nu_{i}-1}\frac{1}{k!}\left(\frac{P}{Q_{i}}\right)^{(k)}% (\lambda_{i})\ (x-\lambda_{i})^{k}.
  82. P Q i = A i + O ( ( x - λ i ) ν i ) \frac{P}{Q_{i}}=A_{i}+O((x-\lambda_{i})^{\nu_{i}})\qquad
  83. x λ i ; x\to\lambda_{i};
  84. A i \textstyle A_{i}
  85. P Q i \textstyle\frac{P}{Q_{i}}
  86. ν i - 1 \textstyle\nu_{i}-1
  87. deg A i < ν i \textstyle\deg A_{i}<\nu_{i}
  88. A i \textstyle A_{i}
  89. λ i \textstyle\lambda_{i}
  90. P - Q i A i = O ( ( x - λ i ) ν i ) P-Q_{i}A_{i}=O((x-\lambda_{i})^{\nu_{i}})\qquad
  91. x λ i , x\to\lambda_{i},
  92. P - Q i A i \textstyle P-Q_{i}A_{i}
  93. ( x - λ i ) ν i . \textstyle(x-\lambda_{i})^{\nu_{i}}.
  94. j i \textstyle j\neq i
  95. Q j A j \textstyle Q_{j}A_{j}
  96. ( x - λ i ) ν i \textstyle(x-\lambda_{i})^{\nu_{i}}
  97. P - j = 1 r Q j A j \textstyle P-\sum_{j=1}^{r}Q_{j}A_{j}
  98. Q \textstyle Q
  99. deg ( P - j = 1 r Q j A j ) < deg ( Q ) \textstyle\deg\left(P-\sum_{j=1}^{r}Q_{j}A_{j}\right)<\deg(Q)
  100. P - j = 1 r Q j A j = 0 \textstyle P-\sum_{j=1}^{r}Q_{j}A_{j}=0
  101. Q \textstyle Q
  102. 1 18 = 1 2 - 1 3 - 1 3 2 . \frac{1}{18}=\frac{1}{2}-\frac{1}{3}-\frac{1}{3^{2}}.

Partition_(number_theory).html

  1. × 10 3 1 \times 10^{3}1
  2. × 10 1 06 \times 10^{1}06
  3. n = 0 p ( n ) x n = k = 1 ( 1 1 - x k ) . \sum_{n=0}^{\infty}p(n)x^{n}=\prod_{k=1}^{\infty}\left(\frac{1}{1-x^{k}}\right).
  4. ( 1 - x ) ( 1 - x 2 ) ( 1 - x 3 ) = 1 - x - x 2 + x 5 + x 7 - x 12 - x 15 + x 22 + x 26 - . (1-x)(1-x^{2})(1-x^{3})\dots=1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}% -\dots.
  5. ( - 1 ) m (-1)^{m}
  6. p ( 5 k + 4 ) 0 ( mod 5 ) p(5k+4)\equiv 0\;\;(\mathop{{\rm mod}}5)\,
  7. k = 0 p ( 5 k + 4 ) x k = 5 ( x 5 ) 5 ( x ) 6 \sum_{k=0}^{\infty}p(5k+4)x^{k}=5~{}\frac{(x^{5})^{5}_{\infty}}{(x)^{6}_{% \infty}}
  8. ( x ) (x)_{\infty}
  9. ( x ) = m = 1 ( 1 - x m ) . (x)_{\infty}=\prod_{m=1}^{\infty}(1-x^{m}).
  10. p ( 7 k + 5 ) 0 ( mod 7 ) p ( 11 k + 6 ) 0 ( mod 11 ) . \begin{aligned}\displaystyle p(7k+5)&\displaystyle\equiv 0\;\;(\mathop{{\rm mod% }}7)\\ \displaystyle p(11k+6)&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}}11).\end{aligned}
  11. k = 0 p ( 7 k + 5 ) x k = 7 ( x 7 ) 3 ( x ) 4 + 49 x ( x 7 ) 7 ( x ) 8 \sum_{k=0}^{\infty}p(7k+5)x^{k}=7~{}\frac{(x^{7})^{3}_{\infty}}{(x)^{4}_{% \infty}}+49x~{}\frac{(x^{7})^{7}_{\infty}}{(x)^{8}_{\infty}}
  12. p ( 13 k + a ) 0 ( mod 13 ) \scriptstyle p(13k\,+\,a)\;\equiv\;0\;\;(\mathop{{\rm mod}}13)
  13. p ( b k + a ) 0 ( mod b ) \scriptstyle p(bk\,+\,a)\;\equiv\;0\;\;(\mathop{{\rm mod}}b)
  14. p ( 11 3 13 k + 237 ) 0 ( mod 13 ) . p(11^{3}\cdot 13\cdot k+237)\equiv 0\;\;(\mathop{{\rm mod}}13).
  15. p ( n ) = p ( n - 1 ) + p ( n - 2 ) - p ( n - 5 ) - p ( n - 7 ) + p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots
  16. g k = k ( 3 k - 1 ) 2 g_{k}=\frac{k(3k-1)}{2}
  17. p ( n ) = k 0 ( - 1 ) k - 1 p ( n - k ( 3 k - 1 ) / 2 ) p(n)=\sum_{k\neq 0}(-1)^{k-1}p\left(n-k(3k-1)/2\right)
  18. p ( n ) = 1 2 2 k = 1 v k A k ( n ) d d n exp ( π k 2 3 ( n - 1 24 ) ) , p(n)=\frac{1}{2\sqrt{2}}\sum_{k=1}^{v}\sqrt{k}\,A_{k}(n)\,\frac{d}{dn}\exp% \left({\frac{\pi}{k}\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}}\,\right)\,,
  19. A k ( n ) = 0 m < k ; ( m , k ) = 1 e π i [ s ( m , k ) - 1 k 2 n m ] . A_{k}(n)=\sum_{0\,\leq\,m\,<\,k;\;(m,\,k)\,=\,1}e^{\pi i\left[s(m,\,k)\;-\;% \frac{1}{k}2nm\right]}.
  20. n \sqrt{n}
  21. p ( n ) = 1 π 2 k = 1 k A k ( n ) d d n ( 1 n - 1 24 sinh [ π k 2 3 ( n - 1 24 ) ] ) . p(n)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}\,A_{k}(n)\,\frac{d}{dn}% \left({\frac{1}{\sqrt{n-\frac{1}{24}}}\sinh\left[{\frac{\pi}{k}\sqrt{\frac{2}{% 3}\left(n-\frac{1}{24}\right)}}\right]}\right).
  22. exp ( π k 2 n 3 ) , \exp\left(\frac{\pi}{k}\sqrt{\frac{2n}{3}}\right),
  23. n = 0 q ( n ) x n = k = 1 ( 1 + x k ) = k = 1 1 1 - x 2 k - 1 . \sum_{n=0}^{\infty}q(n)x^{n}=\prod_{k=1}^{\infty}(1+x^{k})=\prod_{k=1}^{\infty% }\frac{1}{1-x^{2k-1}}.
  24. n 0 p k ( n ) x n = x k i = 1 k 1 1 - x i . \sum_{n\geq 0}p_{k}(n)x^{n}=x^{k}\cdot\prod_{i=1}^{k}\frac{1}{1-x^{i}}.
  25. t T ( 1 - x t ) - 1 . \prod_{t\in T}(1-x^{t})^{-1}.
  26. n 2 + 1 , \left\lfloor\frac{n}{2}+1\right\rfloor,
  27. log p ( n ) C n as n \log p(n)\sim C\sqrt{n}\mbox{ as }n\rightarrow\infty
  28. C = π 2 3 C=\pi\sqrt{\frac{2}{3}}
  29. log p A ( n ) C α n \log p_{A}(n)\sim C\sqrt{\alpha n}
  30. p A ( n ) = ( a A a - 1 ) n k - 1 ( k - 1 ) ! + O ( n k - 2 ) . p_{A}(n)=\left(\prod_{a\in A}a^{-1}\right)\cdot\frac{n^{k-1}}{(k-1)!}+O(n^{k-2% }).
  31. p ( N , M ; n ) = p ( N , M - 1 ; n ) + p ( N - 1 , M ; n - M ) p(N,M;n)=p(N,M-1;n)+p(N-1,M;n-M)
  32. p ( N , M ; n ) - p ( N , M - 1 ; n ) p(N,M;n)-p(N,M-1;n)
  33. ( k + ) q = ( k + k ) q = j = 1 k + ( 1 - q j ) j = 1 k ( 1 - q j ) j = 1 ( 1 - q j ) . {k+\ell\choose\ell}_{q}={k+\ell\choose k}_{q}=\frac{\prod^{k+\ell}_{j=1}(1-q^{% j})}{\prod^{k}_{j=1}(1-q^{j})\prod^{\ell}_{j=1}(1-q^{j})}.
  34. n = 0 M N p ( M , N ; n ) q n = ( M + N M ) q . \sum^{MN}_{n=0}p(M,N;n)q^{n}={M+N\choose M}_{q}.

Partition_function_(quantum_field_theory).html

  1. Z [ J ] = 𝒟 ϕ e i ( S [ ϕ ] + d d x J ( x ) ϕ ( x ) ) Z[J]=\int\mathcal{D}\phi e^{i(S[\phi]+\int d^{d}xJ(x)\phi(x))}
  2. ϕ \phi
  3. G n G_{n}
  4. G n ( x 1 , , x n ) Ω | T { ϕ ( x 1 ) ϕ ( x n ) } | Ω = 𝒟 ϕ ϕ ( x 1 ) ϕ ( x n ) exp ( i S [ ϕ ] / ) 𝒟 ϕ exp ( i S [ ϕ ] / ) G_{n}(x_{1},...,x_{n})\equiv\langle\Omega|T\{\phi(x_{1})\cdots\phi(x_{n})\}|% \Omega\rangle=\frac{\int\mathcal{D}\phi\,\phi(x_{1})\cdots\phi(x_{n})\exp(iS[% \phi]/\hbar)}{\int\mathcal{D}\phi\,\exp(iS[\phi]/\hbar)}
  5. 𝒟 ϕ \mathcal{D}\phi
  6. ϕ ( x ) \phi(x)
  7. S [ ϕ ] S[\phi]
  8. Z [ J ] Z[J]
  9. J J
  10. Z [ J ] = 𝒟 ϕ exp { i [ S [ ϕ ] + d 4 x J ( x ) ϕ ( x ) ) ] } Z[J]=\int\mathcal{D}\phi\exp\left\{\frac{i}{\hbar}\left[S[\phi]+\int d^{4}xJ(x% )\phi(x))\right]\right\}
  11. G n ( x 1 , , , x n ) G_{n}(x_{1},...,,x_{n})
  12. G n ( x 1 , , x n ) = ( - i ) n 1 Z [ 0 ] n Z J ( x 1 ) J ( x n ) | J = 0 G_{n}(x_{1},...,x_{n})=(-i\hbar)^{n}\frac{1}{Z[0]}\left.\frac{\partial^{n}Z}{% \partial J(x_{1})\cdots\partial J(x_{n})}\right|_{J=0}
  13. Z [ J ] Z[J]

Partition_function_(statistical_mechanics).html

  1. Z = s e - β E s Z=\sum_{s}\mathrm{e}^{-\beta E_{s}}
  2. s s
  3. β \beta
  4. 1 k B T \tfrac{1}{k_{B}T}
  5. E s E_{s}
  6. Z = 1 h 3 e - β H ( q , p ) d 3 q d 3 p Z=\frac{1}{h^{3}}\int\mathrm{e}^{-\beta H(q,p)}~{}d^{3}q~{}d^{3}p
  7. h h
  8. β \beta
  9. 1 k B T \tfrac{1}{k_{B}T}
  10. H ( q , p ) H(q,p)
  11. q q
  12. p p
  13. Z = tr ( e - β H ^ ) Z=\operatorname{tr}(\mathrm{e}^{-\beta\hat{H}})
  14. β \beta
  15. 1 k B T \tfrac{1}{k_{B}T}
  16. H ^ \hat{H}
  17. Z = 1 h q , p | e - β H ^ | q , p d q d p Z=\frac{1}{h}\int\langle q,p|\mathrm{e}^{-\beta\hat{H}}|q,p\rangle~{}dq~{}dp
  18. h h
  19. β \beta
  20. 1 k B T \tfrac{1}{k_{B}T}
  21. H ^ \hat{H}
  22. q q
  23. p p
  24. Z = j g j e - β E j Z=\sum_{j}g_{j}\cdot\mathrm{e}^{-\beta E_{j}}
  25. Z = 1 N ! h 3 N exp [ - β H ( p 1 p N , x 1 x N ) ] d 3 p 1 d 3 p N d 3 x 1 d 3 x N Z=\frac{1}{N!h^{3N}}\int\,\exp[-\beta H(p_{1}\cdots p_{N},x_{1}\cdots x_{N})]% \;d^{3}p_{1}\cdots d^{3}p_{N}\,d^{3}x_{1}\cdots d^{3}x_{N}
  26. Z = tr ( e - β H ^ ) Z=\operatorname{tr}(\mathrm{e}^{-\beta\hat{H}})
  27. s y m b o l 1 = | x , p x , p | d x d p h symbol{1}=\int|x,p\rangle\,\langle x,p|~{}\frac{dx\,dp}{h}
  28. | |
  29. \rangle
  30. Z = tr ( e - β H ^ | x , p x , p | ) d x d p h = x , p | e - β H ^ | x , p d x d p h Z=\int\operatorname{tr}\left(\mathrm{e}^{-\beta\hat{H}}|x,p\rangle\,\langle x,% p|\right)\frac{dx\,dp}{h}=\int\langle x,p|\mathrm{e}^{-\beta\hat{H}}|x,p% \rangle~{}\frac{dx\,dp}{h}
  31. x ^ \hat{x}
  32. p ^ \hat{p}
  33. p i = Ω ( E - E i ) \displaystyle p_{i}=\Omega\left(E-E_{i}\right)
  34. S / E = 1 / T \partial S/\partial E=1/T
  35. k ln p i = k ln Ω ( E - E i ) \displaystyle k\ln p_{i}=k\ln\Omega\left(E-E_{i}\right)
  36. Z \displaystyle Z
  37. P s = 1 Z e - β E s . P_{s}=\frac{1}{Z}\mathrm{e}^{-\beta E_{s}}.
  38. s P s = 1 Z s e - β E s = 1 Z Z = 1. \sum_{s}P_{s}=\frac{1}{Z}\sum_{s}\mathrm{e}^{-\beta E_{s}}=\frac{1}{Z}Z=1.
  39. E = s E s P s = 1 Z s E s e - β E s = - 1 Z β Z ( β , E 1 , E 2 , ) = - ln Z β \langle E\rangle=\sum_{s}E_{s}P_{s}=\frac{1}{Z}\sum_{s}E_{s}e^{-\beta E_{s}}=-% \frac{1}{Z}\frac{\partial}{\partial\beta}Z(\beta,E_{1},E_{2},\cdots)=-\frac{% \partial\ln Z}{\partial\beta}
  40. E = k B T 2 ln Z T . \langle E\rangle=k_{B}T^{2}\frac{\partial\ln Z}{\partial T}.
  41. E s = E s ( 0 ) + λ A s for all s E_{s}=E_{s}^{(0)}+\lambda A_{s}\qquad\mbox{for all}~{}\;s
  42. A = s A s P s = - 1 β λ ln Z ( β , λ ) . \langle A\rangle=\sum_{s}A_{s}P_{s}=-\frac{1}{\beta}\frac{\partial}{\partial% \lambda}\ln Z(\beta,\lambda).
  43. p i p_{i}
  44. S = - k B i p i ln ( p i ) S=-k_{B}\sum_{i}p_{i}\ln(p_{i})
  45. i p i = 1 \sum_{i}p_{i}=1
  46. E = i p i E i = U \langle E\rangle=\sum_{i}p_{i}E_{i}=U
  47. S S
  48. S = - k B i p i ln ( p i ) + λ 1 ( i p i - 1 ) + λ 2 ( i p i E i - U ) S=-k_{B}\sum_{i}p_{i}\ln(p_{i})+\lambda_{1}(\sum_{i}p_{i}-1)+\lambda_{2}(\sum_% {i}p_{i}E_{i}-U)
  49. λ 2 \lambda_{2}
  50. S S
  51. U U
  52. S U = - λ 2 = 1 T \frac{\partial S}{\partial U}=-\lambda_{2}=\frac{1}{T}
  53. S S
  54. p i p_{i}
  55. 0 \displaystyle 0
  56. p i p_{i}
  57. p i \displaystyle p_{i}
  58. β := 1 k B T \beta:=\frac{1}{k_{B}T}
  59. λ 1 \lambda_{1}
  60. 1 \displaystyle 1
  61. e 1 k B ( - 1 + λ 1 ) = 1 i e - β E i \mathrm{e}^{\frac{1}{k_{B}}(-1+\lambda_{1})}=\frac{1}{\sum_{i}\mathrm{e}^{-% \beta E_{i}}}
  62. Z = i e - β E i Z=\sum_{i}\mathrm{e}^{-\beta E_{i}}
  63. p i p_{i}
  64. Z Z
  65. p i \displaystyle p_{i}
  66. U U
  67. Z Z
  68. U \displaystyle U
  69. S S
  70. Z Z
  71. S \displaystyle S
  72. E = - ln Z β . \langle E\rangle=-\frac{\partial\ln Z}{\partial\beta}.
  73. ( Δ E ) 2 ( E - E ) 2 = 2 ln Z β 2 . \langle(\Delta E)^{2}\rangle\equiv\langle(E-\langle E\rangle)^{2}\rangle=\frac% {\partial^{2}\ln Z}{\partial\beta^{2}}.
  74. C v = E T = 1 k B T 2 ( Δ E ) 2 . C_{v}=\frac{\partial\langle E\rangle}{\partial T}=\frac{1}{k_{B}T^{2}}\langle(% \Delta E)^{2}\rangle.
  75. S - k B s P s ln P s = k B ( ln Z + β E ) = T ( k B T ln Z ) = - A T S\equiv-k_{B}\sum_{s}P_{s}\ln P_{s}=k_{B}(\ln Z+\beta\langle E\rangle)=\frac{% \partial}{\partial T}(k_{B}T\ln Z)=-\frac{\partial A}{\partial T}
  76. \langle
  77. \rangle
  78. A = E - T S = - k B T ln Z . A=\langle E\rangle-TS=-k_{B}T\ln Z.
  79. Z = j = 1 N ζ j . Z=\prod_{j=1}^{N}\zeta_{j}.
  80. Z = ζ N . Z=\zeta^{N}.
  81. Z = ζ N N ! . Z=\frac{\zeta^{N}}{N!}.
  82. 𝒵 \mathcal{Z}
  83. 𝒵 ( μ , V , T ) = i exp ( ( N i μ - E i ) / k B T ) . \mathcal{Z}(\mu,V,T)=\sum_{i}\exp((N_{i}\mu-E_{i})/k_{B}T).
  84. i i
  85. N i N_{i}
  86. E i E_{i}
  87. Φ G \Phi_{\rm G}
  88. - k B T ln 𝒵 = Φ G = E - T S - μ N . -k_{B}T\ln\mathcal{Z}=\Phi_{\rm G}=\langle E\rangle-TS-\mu\langle N\rangle.
  89. i i
  90. p i = 1 𝒵 exp ( ( N i μ - E i ) / k B T ) . p_{i}=\frac{1}{\mathcal{Z}}\exp((N_{i}\mu-E_{i})/k_{B}T).
  91. 𝒵 ( z , V , T ) = N i z N i Z ( N i , V , T ) , \mathcal{Z}(z,V,T)=\sum_{N_{i}}z^{N_{i}}Z(N_{i},V,T),
  92. z exp ( μ / k T ) z\equiv\exp(\mu/kT)
  93. Z ( N i , V , T ) Z(N_{i},V,T)

Pati–Salam_model.html

  1. S U ( 5 ) SU(5)
  2. S O ( 10 ) SO(10)
  3. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  4. ( 𝟒 , 𝟐 , 𝟏 ) (\mathbf{4},\mathbf{2},\mathbf{1})
  5. ( 𝟒 ¯ , 𝟏 , 𝟐 ) (\overline{\mathbf{4}},\mathbf{1},\mathbf{2})
  6. ( 𝟒 , 𝟏 , 𝟐 ) (\mathbf{4},\mathbf{1},\mathbf{2})
  7. ( 𝟒 ¯ , 𝟏 , 𝟐 ) (\overline{\mathbf{4}},\mathbf{1},\mathbf{2})
  8. ( 𝟒 ¯ , 𝟏 , 𝟐 ) (\overline{\mathbf{4}},\mathbf{1},\mathbf{2})
  9. ( 𝟔 , 𝟏 , 𝟏 ) (\mathbf{6},\mathbf{1},\mathbf{1})
  10. ( 1 3 0 0 0 0 1 3 0 0 0 0 1 3 0 0 0 0 - 1 ) SU ( 4 ) , ( 1 0 0 - 1 ) SU ( 2 ) R \begin{pmatrix}\frac{1}{3}&0&0&0\\ 0&\frac{1}{3}&0&0\\ 0&0&\frac{1}{3}&0\\ 0&0&0&-1\end{pmatrix}\in\,\text{SU}(4),\qquad\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\in\,\text{SU}(2)_{\,\text{R}}
  11. ( [ S U ( 4 ) × S U ( 2 ) L × S U ( 2 ) R ] / 𝐙 2 ) 𝐙 2 \left([SU(4)\times SU(2)_{L}\times SU(2)_{R}]/\mathbf{Z}_{2}\right)\rtimes% \mathbf{Z}_{2}
  12. S U ( 4 ) SU(4)
  13. S U ( 2 ) SU(2)
  14. ( 𝟒 , 𝟐 , 𝟏 ) ( 𝟒 ¯ , 𝟏 , 𝟐 ) (\mathbf{4},\mathbf{2},\mathbf{1})⊕(\overline{\mathbf{4}},\mathbf{1},\mathbf{2})
  15. ( 𝟒 , 𝟏 , 𝟐 ) ( 𝟒 ¯ , 𝟐 , 𝟏 ) (\mathbf{4},\mathbf{1},\mathbf{2})⊕(\overline{\mathbf{4}},\mathbf{2},\mathbf{1})
  16. π 2 ( S U ( 4 ) × S U ( 2 ) [ S U ( 3 ) × U ( 1 ) ] / 𝐙 3 ) = 𝐙 , \pi_{2}\left(\frac{SU(4)\times SU(2)}{[SU(3)\times U(1)]/\mathbf{Z}_{3}}\right% )=\mathbf{Z},
  17. S U ( 5 ) SU(5)
  18. S O ( 10 ) SO(10)
  19. S O ( 10 ) SO(10)
  20. S U ( 5 ) SU(5)
  21. N = 1 N=1
  22. 3 + 1 3+1
  23. 3 + 1 3+1
  24. S U ( 4 ) × S U ( 2 ) < s u b > L × S U ( 2 ) R SU(4)×SU(2)<sub>L×SU(2)_{R}
  25. S S ( 4 , 1 , 2 ) H ( 4 ¯ , 1 , 2 ) H S ( 1 , 2 , 2 ) H ( 1 , 2 , 2 ) H ( 6 , 1 , 1 ) H ( 4 , 1 , 2 ) H ( 4 , 1 , 2 ) H ( 6 , 1 , 1 ) H ( 4 ¯ , 1 , 2 ) H ( 4 ¯ , 1 , 2 ) H ( 1 , 2 , 2 ) H ( 4 , 2 , 1 ) i ( 4 ¯ , 1 , 2 ) j ( 4 , 1 , 2 ) H ( 4 ¯ , 1 , 2 ) i ϕ j \begin{matrix}S\\ S(4,1,2)_{H}(\bar{4},1,2)_{H}\\ S(1,2,2)_{H}(1,2,2)_{H}\\ (6,1,1)_{H}(4,1,2)_{H}(4,1,2)_{H}\\ (6,1,1)_{H}(\bar{4},1,2)_{H}(\bar{4},1,2)_{H}\\ (1,2,2)_{H}(4,2,1)_{i}(\bar{4},1,2)_{j}\\ (4,1,2)_{H}(\bar{4},1,2)_{i}\phi_{j}\\ \end{matrix}
  26. i i
  27. j j
  28. ( 𝟒 , 𝟐 , 𝟏 ) < s u b > H (\mathbf{4},\mathbf{2},\mathbf{1})<sub>H

Pattern_recognition.html

  1. n n
  2. 2 n - 1 2^{n}-1
  3. n n
  4. g : 𝒳 𝒴 g:\mathcal{X}\rightarrow\mathcal{Y}
  5. s y m b o l x 𝒳 symbol{x}\in\mathcal{X}
  6. y 𝒴 y\in\mathcal{Y}
  7. 𝐃 = { ( s y m b o l x 1 , y 1 ) , , ( s y m b o l x n , y n ) } \mathbf{D}=\{(symbol{x}_{1},y_{1}),\dots,(symbol{x}_{n},y_{n})\}
  8. h : 𝒳 𝒴 h:\mathcal{X}\rightarrow\mathcal{Y}
  9. g g
  10. s y m b o l x i symbol{x}_{i}
  11. y y
  12. 𝒳 \mathcal{X}
  13. 𝒳 \mathcal{X}
  14. g : 𝒳 𝒴 g:\mathcal{X}\rightarrow\mathcal{Y}
  15. 𝒳 \mathcal{X}
  16. 𝒴 \mathcal{Y}
  17. h : 𝒳 𝒴 h:\mathcal{X}\rightarrow\mathcal{Y}
  18. p ( label | s y m b o l x , s y m b o l θ ) = f ( s y m b o l x ; s y m b o l θ ) p({\rm label}|symbol{x},symbol\theta)=f\left(symbol{x};symbol{\theta}\right)
  19. s y m b o l x symbol{x}
  20. s y m b o l θ symbol{\theta}
  21. p ( s y m b o l x | label ) p({symbol{x}|\rm label})
  22. p ( label | s y m b o l θ ) p({\rm label}|symbol\theta)
  23. p ( label | s y m b o l x , s y m b o l θ ) = p ( s y m b o l x | label , symbol θ ) p ( label | symbol θ ) L all labels p ( s y m b o l x | L ) p ( L | s y m b o l θ ) . p({\rm label}|symbol{x},symbol\theta)=\frac{p({symbol{x}|\rm label,symbol% \theta})p({\rm label|symbol\theta})}{\sum_{L\in\,\text{all labels}}p(symbol{x}% |L)p(L|symbol\theta)}.
  24. p ( label | s y m b o l x , s y m b o l θ ) = p ( s y m b o l x | label , symbol θ ) p ( label | symbol θ ) L all labels p ( s y m b o l x | L ) p ( L | s y m b o l θ ) d L . p({\rm label}|symbol{x},symbol\theta)=\frac{p({symbol{x}|\rm label,symbol% \theta})p({\rm label|symbol\theta})}{\int_{L\in\,\text{all labels}}p(symbol{x}% |L)p(L|symbol\theta)\operatorname{d}L}.
  25. s y m b o l θ symbol\theta
  26. p ( s y m b o l θ ) p(symbol\theta)
  27. s y m b o l θ symbol\theta
  28. s y m b o l θ * = arg max s y m b o l θ p ( s y m b o l θ | 𝐃 ) symbol\theta^{*}=\arg\max_{symbol\theta}p(symbol\theta|\mathbf{D})
  29. s y m b o l θ * symbol\theta^{*}
  30. s y m b o l θ symbol\theta
  31. p ( s y m b o l θ | 𝐃 ) p(symbol\theta|\mathbf{D})
  32. s y m b o l θ symbol\theta
  33. p ( s y m b o l θ | 𝐃 ) = [ i = 1 n p ( y i | s y m b o l x i , s y m b o l θ ) ] p ( s y m b o l θ ) . p(symbol\theta|\mathbf{D})=\left[\prod_{i=1}^{n}p(y_{i}|symbol{x}_{i},symbol% \theta)\right]p(symbol\theta).
  34. s y m b o l θ * symbol{\theta}^{*}
  35. s y m b o l x symbol{x}
  36. s y m b o l θ symbol\theta
  37. p ( label | s y m b o l x ) = p ( label | s y m b o l x , s y m b o l θ ) p ( s y m b o l θ | 𝐃 ) d s y m b o l θ . p({\rm label}|symbol{x})=\int p({\rm label}|symbol{x},symbol\theta)p(symbol{% \theta}|\mathbf{D})\operatorname{d}symbol{\theta}.
  38. p ( label | s y m b o l θ ) p({\rm label}|symbol\theta)
  39. p ( label | s y m b o l θ ) p({\rm label}|symbol\theta)
  40. s y m b o l θ symbol\theta
  41. s y m b o l μ 1 symbol\mu_{1}
  42. s y m b o l μ 2 symbol\mu_{2}
  43. s y m b o l Σ symbol\Sigma

Pearson's_chi-squared_test.html

  1. 2 {}^{2}
  2. χ 2 \chi^{2}
  3. χ 2 \chi^{2}
  4. χ 2 \chi^{2}
  5. χ 2 \chi^{2}
  6. χ 2 \chi^{2}
  7. N N
  8. n n
  9. E i = N n , E_{i}=\frac{N}{n}\,,
  10. p = 1 p=1
  11. O i O_{i}
  12. N N
  13. p = s + 1 p=s+1
  14. s s
  15. p = 4 p=4
  16. p = 3 p=3
  17. n - p n-p
  18. n n
  19. χ 2 = i = 1 n ( O i - E i ) 2 E i = N i = 1 n p i ( O i / N - p i p i ) 2 \chi^{2}=\sum_{i=1}^{n}\frac{(O_{i}-E_{i})^{2}}{E_{i}}=N\sum_{i=1}^{n}p_{i}% \left(\frac{O_{i}/N-p_{i}}{p_{i}}\right)^{2}
  20. χ 2 \chi^{2}
  21. χ 2 \chi^{2}
  22. O i O_{i}
  23. N N
  24. E i = N p i E_{i}=Np_{i}
  25. p i p_{i}
  26. n n
  27. n n
  28. p p
  29. n - 1 - p n-1-p
  30. n - 1 n-1
  31. E i , j = N p i p j , E_{i,j}=Np_{i\cdot}p_{\cdot j},
  32. N N
  33. p i = O i N = j = 1 c O i , j N , p_{i\cdot}=\frac{O_{i\cdot}}{N}=\sum_{j=1}^{c}\frac{O_{i,j}}{N},
  34. p j = O j N = i = 1 r O i , j N p_{\cdot j}=\frac{O_{\cdot j}}{N}=\frac{\sum_{i=1}^{r}O_{i,j}}{N}
  35. χ 2 = i = 1 r j = 1 c ( O i , j - E i , j ) 2 E i , j \chi^{2}=\sum_{i=1}^{r}\sum_{j=1}^{c}{(O_{i,j}-E_{i,j})^{2}\over E_{i,j}}
  36. = N i , j p i p j ( ( O i , j / N ) - p i p j p i p j ) 2 \ \ \ \ =N\sum_{i,j}p_{i\cdot}p_{\cdot j}\left(\frac{(O_{i,j}/N)-p_{i\cdot}p_{% \cdot j}}{p_{i\cdot}p_{\cdot j}}\right)^{2}
  37. 60 n \frac{60}{n}
  38. ( O < s u b > i (\frac{O<sub>i}{−}
  39. N N
  40. χ 2 = ( 44 - 50 ) 2 50 + ( 56 - 50 ) 2 50 = 1.44. \chi^{2}={(44-50)^{2}\over 50}+{(56-50)^{2}\over 50}=1.44.

Pearson_product-moment_correlation_coefficient.html

  1. ρ X , Y = cov ( X , Y ) σ X σ Y \rho_{X,Y}=\frac{\operatorname{cov}(X,Y)}{\sigma_{X}\sigma_{Y}}
  2. cov \operatorname{cov}
  3. σ X \sigma_{X}
  4. X X
  5. cov ( X , Y ) = E [ ( X - μ X ) ( Y - μ Y ) ] \operatorname{cov}(X,Y)=E[(X-\mu_{X})(Y-\mu_{Y})]
  6. ρ X , Y = E [ ( X - μ X ) ( Y - μ Y ) ] σ X σ Y \rho_{X,Y}=\frac{E[(X-\mu_{X})(Y-\mu_{Y})]}{\sigma_{X}\sigma_{Y}}
  7. cov \operatorname{cov}
  8. σ X \sigma_{X}
  9. μ X \mu_{X}
  10. X X
  11. E E
  12. μ X = E ( X ) \mu_{X}=E(X)
  13. μ Y = E ( Y ) \mu_{Y}=E(Y)
  14. σ X 2 = E [ ( X - E ( X ) ) 2 ] = E ( X 2 ) - E ( X ) 2 \sigma_{X}^{2}=E[(X-E(X))^{2}]=E(X^{2})-E(X)^{2}
  15. σ Y 2 = E [ ( Y - E ( Y ) ) 2 ] = E ( Y 2 ) - E ( Y ) 2 \sigma_{Y}^{2}=E[(Y-E(Y))^{2}]=E(Y^{2})-E(Y)^{2}
  16. E [ ( X - μ X ) ( Y - μ Y ) ] = E [ ( X - E ( X ) ) ( Y - E ( Y ) ) ] = E ( X Y ) - E ( X ) E ( Y ) , E[(X-\mu_{X})(Y-\mu_{Y})]=E[(X-E(X))(Y-E(Y))]=E(XY)-E(X)E(Y),\,
  17. ρ X , Y = E ( X Y ) - E ( X ) E ( Y ) E ( X 2 ) - E ( X ) 2 E ( Y 2 ) - E ( Y ) 2 . \rho_{X,Y}=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^{2})-E(X)^{2}}~{}\sqrt{E(Y^{2})-E(Y% )^{2}}}.
  18. r = r x y = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 i = 1 n ( y i - y ¯ ) 2 r=r_{xy}=\frac{\sum^{n}_{i=1}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum^{n}_{i% =1}(x_{i}-\bar{x})^{2}}\sqrt{\sum^{n}_{i=1}(y_{i}-\bar{y})^{2}}}
  19. n , x i , y i n,x_{i},y_{i}
  20. x ¯ = 1 n i = 1 n x i \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  21. y ¯ \bar{y}
  22. r = r x y = n x i y i - x i y i n x i 2 - ( x i ) 2 n y i 2 - ( y i ) 2 . r=r_{xy}=\frac{n\sum x_{i}y_{i}-\sum x_{i}\sum y_{i}}{\sqrt{n\sum x_{i}^{2}-(% \sum x_{i})^{2}}~{}\sqrt{n\sum y_{i}^{2}-(\sum y_{i})^{2}}}.
  23. n , x i , y i n,x_{i},y_{i}
  24. r = r x y = x i y i - n x ¯ y ¯ ( x i 2 - n x ¯ 2 ) ( y i 2 - n y ¯ 2 ) . r=r_{xy}=\frac{\sum x_{i}y_{i}-n\bar{x}\bar{y}}{\sqrt{(\sum x_{i}^{2}-n\bar{x}% ^{2})}~{}\sqrt{(\sum y_{i}^{2}-n\bar{y}^{2})}}.
  25. n , x i , y i , x ¯ , y ¯ n,x_{i},y_{i},\bar{x},\bar{y}
  26. r = r x y = 1 n - 1 i = 1 n ( x i - x ¯ s x ) ( y i - y ¯ s y ) r=r_{xy}=\frac{1}{n-1}\sum^{n}_{i=1}\left(\frac{x_{i}-\bar{x}}{s_{x}}\right)% \left(\frac{y_{i}-\bar{y}}{s_{y}}\right)
  27. n , x i , y i , x ¯ , y ¯ n,x_{i},y_{i},\bar{x},\bar{y}
  28. s x , s y s_{x},s_{y}
  29. ( x i - x ¯ s x ) \left(\frac{x_{i}-\bar{x}}{s_{x}}\right)
  30. r = r x y = x i y i - n x ¯ y ¯ ( n - 1 ) s x s y r=r_{xy}=\frac{\sum x_{i}y_{i}-n\bar{x}\bar{y}}{(n-1)s_{x}s_{y}}
  31. n , x i , y i , x ¯ , y ¯ n,x_{i},y_{i},\bar{x},\bar{y}
  32. s x = 1 n - 1 i = 1 n ( x i - x ¯ ) 2 s_{x}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}
  33. φ \varphi
  34. φ \ \varphi
  35. φ \ \varphi
  36. θ \ \theta
  37. θ \ \theta
  38. cos θ = x y x y = 2.93 103 0.0983 = 0.920814711. \cos\theta=\frac{{x}\cdot{y}}{\left\|{x}\right\|\left\|{y}\right\|}=\frac{2.93% }{\sqrt{103}\sqrt{0.0983}}=0.920814711.
  39. cos θ = x y x y = 0.308 30.8 0.00308 = 1 = ρ x y , \cos\theta=\frac{{x}\cdot{y}}{\left\|{x}\right\|\left\|{y}\right\|}=\frac{0.30% 8}{\sqrt{30.8}\sqrt{0.00308}}=1=\rho_{xy},
  40. 1 - 1 - ρ 2 1-\sqrt{1-\rho^{2}}
  41. t = r n - 2 1 - r 2 t=r\sqrt{\frac{n-2}{1-r^{2}}}
  42. r = t n - 2 + t 2 . r=\frac{t}{\sqrt{n-2+t^{2}}}.
  43. f ( r ) = ( n - 2 ) 𝚪 ( n - 1 ) ( 1 - ρ 2 ) n - 1 2 ( 1 - r 2 ) n - 4 2 2 π 𝚪 ( n - 1 2 ) ( 1 - ρ r ) n - 3 2 𝐅 𝟏 𝟐 ( 1 2 , 1 2 ; 2 n - 1 2 ; ρ r + 1 2 ) f(r)=\frac{(n-2)\,\mathbf{\Gamma}(n-1)(1-\rho^{2})^{\frac{n-1}{2}}(1-r^{2})^{% \frac{n-4}{2}}}{\sqrt{2\pi}\,\mathbf{\Gamma}\left(n-\frac{1}{2}\right)(1-\rho r% )^{n-\frac{3}{2}}}\,\mathbf{{}_{2}F_{1}}\left(\frac{1}{2},\frac{1}{2};\frac{2n% -1}{2};\frac{\rho r+1}{2}\right)
  44. 𝚪 \mathbf{\Gamma}
  45. 𝐅 𝟏 𝟐 ( a , b ; c ; z ) \,\mathbf{{}_{2}F_{1}}(a,b;c;z)
  46. ρ = 0 \,\rho=0
  47. f ( r ) = ( 1 - r 2 ) n - 4 2 𝐁 ( 1 2 , n - 2 2 ) , f(r)=\frac{(1-r^{2})^{\frac{n-4}{2}}}{\mathbf{B}\left(\frac{1}{2},\frac{n-2}{2% }\right)},
  48. 𝐁 \mathbf{B}
  49. F ( r ) = 1 2 ln 1 + r 1 - r = arctanh ( r ) . F(r)={1\over 2}\ln{1+r\over 1-r}=\operatorname{arctanh}(r).
  50. mean = F ( ρ ) = arctanh ( ρ ) \,\text{mean}=F(\rho)=\operatorname{arctanh}(\rho)
  51. SE = 1 n - 3 . \,\text{SE}=\frac{1}{\sqrt{n-3}}.
  52. z = x - mean SE = [ F ( r ) - F ( ρ 0 ) ] n - 3 z=\frac{x-\,\text{mean}}{\,\text{SE}}=[F(r)-F(\rho_{0})]\sqrt{n-3}
  53. ρ = ρ 0 \rho=\rho_{0}
  54. ρ = 0 \rho=0
  55. ρ \rho
  56. 100 ( 1 - α ) % CI : arctanh ( ρ ) [ arctanh ( r ) ± z α / 2 S E ] 100(1-\alpha)\%\,\text{CI}:\operatorname{arctanh}(\rho)\in[\operatorname{% arctanh}(r)\pm z_{\alpha/2}SE]
  57. 100 ( 1 - α ) % CI : ρ [ tanh ( arctanh ( r ) - z α / 2 S E ) , tanh ( arctanh ( r ) + z α / 2 S E ) ] 100(1-\alpha)\%\,\text{CI}:\rho\in[\operatorname{tanh}(\operatorname{arctanh}(% r)-z_{\alpha/2}SE),\operatorname{tanh}(\operatorname{arctanh}(r)+z_{\alpha/2}% SE)]
  58. i ( Y i - Y ¯ ) 2 = i ( Y i - Y ^ i ) 2 + i ( Y ^ i - Y ¯ ) 2 , \sum_{i}(Y_{i}-\bar{Y})^{2}=\sum_{i}(Y_{i}-\hat{Y}_{i})^{2}+\sum_{i}(\hat{Y}_{% i}-\bar{Y})^{2},
  59. Y ^ i \hat{Y}_{i}
  60. 1 = i ( Y i - Y ^ i ) 2 i ( Y i - Y ¯ ) 2 + i ( Y ^ i - Y ¯ ) 2 i ( Y i - Y ¯ ) 2 . 1=\frac{\sum_{i}(Y_{i}-\hat{Y}_{i})^{2}}{\sum_{i}(Y_{i}-\bar{Y})^{2}}+\frac{% \sum_{i}(\hat{Y}_{i}-\bar{Y})^{2}}{\sum_{i}(Y_{i}-\bar{Y})^{2}}.
  61. Y ^ i \hat{Y}_{i}
  62. Y i - Y ^ i Y_{i}-\hat{Y}_{i}
  63. r ( Y , Y ^ ) = i ( Y i - Y ¯ ) ( Y ^ i - Y ¯ ) i ( Y i - Y ¯ ) 2 i ( Y ^ i - Y ¯ ) 2 = i ( Y i - Y ^ i + Y ^ i - Y ¯ ) ( Y ^ i - Y ¯ ) i ( Y i - Y ¯ ) 2 i ( Y ^ i - Y ¯ ) 2 = i [ ( Y i - Y ^ i ) ( Y ^ i - Y ¯ ) + ( Y ^ i - Y ¯ ) 2 ] i ( Y i - Y ¯ ) 2 i ( Y ^ i - Y ¯ ) 2 = i ( Y ^ i - Y ¯ ) 2 i ( Y i - Y ¯ ) 2 i ( Y ^ i - Y ¯ ) 2 = i ( Y ^ i - Y ¯ ) 2 i ( Y i - Y ¯ ) 2 . \begin{aligned}\displaystyle r(Y,\hat{Y})&\displaystyle=\frac{\sum_{i}(Y_{i}-% \bar{Y})(\hat{Y}_{i}-\bar{Y})}{\sqrt{\sum_{i}(Y_{i}-\bar{Y})^{2}\cdot\sum_{i}(% \hat{Y}_{i}-\bar{Y})^{2}}}\\ &\displaystyle=\frac{\sum_{i}(Y_{i}-\hat{Y}_{i}+\hat{Y}_{i}-\bar{Y})(\hat{Y}_{% i}-\bar{Y})}{\sqrt{\sum_{i}(Y_{i}-\bar{Y})^{2}\cdot\sum_{i}(\hat{Y}_{i}-\bar{Y% })^{2}}}\\ &\displaystyle=\frac{\sum_{i}[(Y_{i}-\hat{Y}_{i})(\hat{Y}_{i}-\bar{Y})+(\hat{Y% }_{i}-\bar{Y})^{2}]}{\sqrt{\sum_{i}(Y_{i}-\bar{Y})^{2}\cdot\sum_{i}(\hat{Y}_{i% }-\bar{Y})^{2}}}\\ &\displaystyle=\frac{\sum_{i}(\hat{Y}_{i}-\bar{Y})^{2}}{\sqrt{\sum_{i}(Y_{i}-% \bar{Y})^{2}\cdot\sum_{i}(\hat{Y}_{i}-\bar{Y})^{2}}}\\ &\displaystyle=\sqrt{\frac{\sum_{i}(\hat{Y}_{i}-\bar{Y})^{2}}{\sum_{i}(Y_{i}-% \bar{Y})^{2}}}.\end{aligned}
  64. r ( Y , Y ^ ) 2 = i ( Y ^ i - Y ¯ ) 2 i ( Y i - Y ¯ ) 2 r(Y,\hat{Y})^{2}=\frac{\sum_{i}(\hat{Y}_{i}-\bar{Y})^{2}}{\sum_{i}(Y_{i}-\bar{% Y})^{2}}
  65. r ( Y , Y ^ ) 2 r(Y,\hat{Y})^{2}
  66. r ( Y , Y ^ ) 2 = S S reg S S tot r(Y,\hat{Y})^{2}=\frac{SS\text{reg}}{SS\text{tot}}
  67. S S reg SS\text{reg}
  68. S S tot SS\text{tot}
  69. S S reg = i ( Y ^ i - Y ¯ ) 2 SS\text{reg}=\sum_{i}(\hat{Y}_{i}-\bar{Y})^{2}
  70. S S tot = i ( Y i - Y ¯ ) 2 SS\text{tot}=\sum_{i}(Y_{i}-\bar{Y})^{2}
  71. E ( r ) = ρ - ρ ( 1 - ρ 2 ) 2 n + , E\left(r\right)=\rho-\frac{\rho\left(1-\rho^{2}\right)}{2n}+\cdots,\quad
  72. ρ . \,\rho.
  73. ( 1 ) r a d j = r 𝐅 𝟏 𝟐 ( 1 2 , 1 2 ; n - 1 2 ; 1 - r 2 ) , (1)\qquad r_{adj}=r\mathbf{{}_{2}F_{1}}\left(\frac{1}{2},\frac{1}{2};\frac{n-1% }{2};1-r^{2}\right),
  74. r , n r,n
  75. 𝐅 𝟏 𝟐 ( a , b ; c ; z ) \,\mathbf{{}_{2}F_{1}}(a,b;c;z)
  76. ( 2 ) r = E ( r ) = r a d j - r a d j ( 1 - r a d j 2 ) 2 n . (2)\qquad r=E\left(r\right)=r_{adj}-\frac{r_{adj}\left(1-r_{adj}^{2}\right)}{2% n}.
  77. ( 3 ) r a d j = r [ 1 + 1 - r 2 2 n ] , (3)\qquad r_{adj}=r\left[1+\frac{1-r^{2}}{2n}\right],
  78. r , n r,n
  79. r a d j = 1 - ( 1 - r 2 ) ( n - 1 ) ( n - 2 ) . r_{adj}=\sqrt{1-\frac{(1-r^{2})(n-1)}{(n-2)}}.
  80. m ( x ; w ) = i w i x i i w i . \operatorname{m}(x;w)={\sum_{i}w_{i}x_{i}\over\sum_{i}w_{i}}.
  81. cov ( x , y ; w ) = i w i ( x i - m ( x ; w ) ) ( y i - m ( y ; w ) ) i w i . \operatorname{cov}(x,y;w)={\sum_{i}w_{i}(x_{i}-\operatorname{m}(x;w))(y_{i}-% \operatorname{m}(y;w))\over\sum_{i}w_{i}}.
  82. corr ( x , y ; w ) = cov ( x , y ; w ) cov ( x , x ; w ) cov ( y , y ; w ) . \operatorname{corr}(x,y;w)={\operatorname{cov}(x,y;w)\over\sqrt{\operatorname{% cov}(x,x;w)\operatorname{cov}(y,y;w)}}.
  83. Corr r ( X , Y ) = E [ X Y ] E X 2 E Y 2 . \,\text{Corr}_{r}(X,Y)=\frac{E[XY]}{\sqrt{EX^{2}\cdot EY^{2}}}.
  84. Corr r ( X , Y ) = Corr r ( Y , X ) = Corr r ( X , b Y ) Corr r ( X , a + b Y ) , a 0 , b > 0. \,\text{Corr}_{r}(X,Y)=\,\text{Corr}_{r}(Y,X)=\,\text{Corr}_{r}(X,bY)\neq\,% \text{Corr}_{r}(X,a+bY),\quad a\neq 0,b>0.
  85. r r x y = x i y i ( x i 2 ) ( y i 2 ) . rr_{xy}=\frac{\sum x_{i}y_{i}}{\sqrt{(\sum x_{i}^{2})(\sum y_{i}^{2})}}.
  86. r r x y , w = w i x i y i ( w i x i 2 ) ( w i y i 2 ) . rr_{xy,w}=\frac{\sum w_{i}x_{i}y_{i}}{\sqrt{(\sum w_{i}x_{i}^{2})(\sum w_{i}y_% {i}^{2})}}.
  87. K K
  88. T T
  89. s s
  90. K = round ( T s ) . K=\operatorname{round}\left(\frac{T}{s}\right).
  91. r ¯ s \bar{r}_{s}
  92. r ¯ s = 1 K k = 1 K r k , \bar{r}_{s}=\frac{1}{K}\sum\limits_{k=1}^{K}r_{k},
  93. r k r_{k}
  94. k k
  95. s s
  96. d X , Y = 1 - ρ X , Y . d_{X,Y}=1-\rho_{X,Y}.
  97. X i , j X_{i,j}
  98. Z m , m Z_{m,m}
  99. D = X - 1 m Z m , m X D=X-\frac{1}{m}Z_{m,m}X
  100. T = D ( D T D ) - 1 2 , T=D(D^{T}D)^{-\frac{1}{2}},
  101. d = x - 1 m Z 1 , m X , d=x-\frac{1}{m}Z_{1,m}X,
  102. t = d ( D T D ) - 1 2 . t=d(D^{T}D)^{-\frac{1}{2}}.