wpmath0000002_16

Probability-generating_function.html

  1. G ( z ) = E ( z X ) = x = 0 p ( x ) z x , G(z)=\operatorname{E}(z^{X})=\sum_{x=0}^{\infty}p(x)z^{x},
  2. G ( z ) = G ( z 1 , , z d ) = E ( z 1 X 1 z d X d ) = x 1 , , x d = 0 p ( x 1 , , x d ) z 1 x 1 z d x d , G(z)=G(z_{1},\ldots,z_{d})=\operatorname{E}\bigl(z_{1}^{X_{1}}\cdots z_{d}^{X_% {d}}\bigr)=\sum_{x_{1},\ldots,x_{d}=0}^{\infty}p(x_{1},\ldots,x_{d})z_{1}^{x_{% 1}}\cdots z_{d}^{x_{d}},
  3. p ( k ) = Pr ( X = k ) = G ( k ) ( 0 ) k ! . p(k)=\operatorname{Pr}(X=k)=\frac{G^{(k)}(0)}{k!}.
  4. E ( 1 ) = G ( 1 - ) = i = 0 f ( i ) = 1. \operatorname{E}(1)=G(1^{-})=\sum_{i=0}^{\infty}f(i)=1.
  5. E ( X ) = G ( 1 - ) . \operatorname{E}\left(X\right)=G^{\prime}(1^{-}).
  6. E ( X ( X - 1 ) ( X - k + 1 ) ) \textrm{E}(X(X-1)\cdots(X-k+1))
  7. E ( X ! ( X - k ) ! ) = G ( k ) ( 1 - ) , k 0. \textrm{E}\left(\frac{X!}{(X-k)!}\right)=G^{(k)}(1^{-}),\quad k\geq 0.
  8. Var ( X ) = G ′′ ( 1 - ) + G ( 1 - ) - [ G ( 1 - ) ] 2 . \operatorname{Var}(X)=G^{\prime\prime}(1^{-})+G^{\prime}(1^{-})-\left[G^{% \prime}(1^{-})\right]^{2}.
  9. G X ( e t ) = M X ( t ) G_{X}(e^{t})=M_{X}(t)
  10. G X ( t ) G_{X}(t)
  11. M X ( t ) M_{X}(t)
  12. S n = i = 1 n a i X i , S_{n}=\sum_{i=1}^{n}a_{i}X_{i},
  13. G S n ( z ) = E ( z S n ) = E ( z i = 1 n a i X i , ) = G X 1 ( z a 1 ) G X 2 ( z a 2 ) G X n ( z a n ) . G_{S_{n}}(z)=\operatorname{E}(z^{S_{n}})=\operatorname{E}(z^{\sum_{i=1}^{n}a_{% i}X_{i},})=G_{X_{1}}(z^{a_{1}})G_{X_{2}}(z^{a_{2}})\cdots G_{X_{n}}(z^{a_{n}}).
  14. S n = i = 1 n X i , S_{n}=\sum_{i=1}^{n}X_{i},
  15. G S n ( z ) = G X 1 ( z ) G X 2 ( z ) G X n ( z ) . G_{S_{n}}(z)=G_{X_{1}}(z)G_{X_{2}}(z)\cdots G_{X_{n}}(z).
  16. G S ( z ) = G X 1 ( z ) G X 2 ( 1 / z ) . G_{S}(z)=G_{X_{1}}(z)G_{X_{2}}(1/z).
  17. G S N ( z ) = G N ( G X ( z ) ) . G_{S_{N}}(z)=G_{N}(G_{X}(z)).
  18. G S N ( z ) = E ( z S N ) = E ( z i = 1 N X i ) = E ( E ( z i = 1 N X i | N ) ) = E ( ( G X ( z ) ) N ) = G N ( G X ( z ) ) . G_{S_{N}}(z)=\operatorname{E}(z^{S_{N}})=\operatorname{E}(z^{\sum_{i=1}^{N}X_{% i}})=\operatorname{E}\big(\operatorname{E}(z^{\sum_{i=1}^{N}X_{i}}|N)\big)=% \operatorname{E}\big((G_{X}(z))^{N}\big)=G_{N}(G_{X}(z)).
  19. f i = Pr { N = i } f_{i}=\Pr\{N=i\}
  20. G X i G_{X_{i}}
  21. X i X_{i}
  22. G S N ( z ) = i 1 f i k = 1 i G X i ( z ) . G_{S_{N}}(z)=\sum_{i\geq 1}f_{i}\prod_{k=1}^{i}G_{X_{i}}(z).
  23. G ( z ) = ( z c ) . G(z)=\left(z^{c}\right).\,
  24. G ( z ) = [ ( 1 - p ) + p z ] n . G(z)=\left[(1-p)+pz\right]^{n}.\,
  25. G ( z ) = ( p z 1 - ( 1 - p ) z ) r . G(z)=\left(\frac{pz}{1-(1-p)z}\right)^{r}.
  26. | z | < 1 1 - p |z|<\frac{1}{1-p}
  27. G ( z ) = e λ ( z - 1 ) . G(z)=\textrm{e}^{\lambda(z-1)}.\;\,

Probability_mass_function.html

  1. \subseteq
  2. f X ( x ) = Pr ( X = x ) = Pr ( { s S : X ( s ) = x } ) . f_{X}(x)=\Pr(X=x)=\Pr(\{s\in S:X(s)=x\}).
  3. x A f X ( x ) = 1 \sum_{x\in A}f_{X}(x)=1
  4. \notin
  5. ( A , 𝒜 , P ) (A,\mathcal{A},P)
  6. ( B , ) (B,\mathcal{B})
  7. X : A B X\colon A\to B
  8. X * ( P ) X_{*}(P)
  9. f X : B f_{X}\colon B\to\mathbb{R}
  10. f X ( b ) = P ( X - 1 ( b ) ) = [ X * ( P ) ] ( { b } ) f_{X}(b)=P(X^{-1}(b))=[X_{*}(P)](\{b\})
  11. ( B , , μ ) (B,\mathcal{B},\mu)
  12. f = d X * P / d μ f=dX_{*}P/d\mu
  13. P ( X = b ) = P ( X - 1 ( { b } ) ) := X - 1 ( { b } ) d P = { b } f d μ = f ( b ) , P(X=b)=P(X^{-1}(\{b\})):=\int_{X^{-1}(\{b\})}dP=\int_{\{b\}}fd\mu=f(b),
  14. f X ( x ) = { 1 2 , x { 0 , 1 } , 0 , x { 0 , 1 } . f_{X}(x)=\begin{cases}\frac{1}{2},&x\in\{0,1\},\\ 0,&x\notin\{0,1\}.\end{cases}

Probability_vector.html

  1. x 0 = [ 0.5 0.25 0.25 ] , x 1 = [ 0 1 0 ] , x 2 = [ 0.65 0.35 ] , x 3 = [ 0.3 0.5 0.07 0.1 0.03 ] . x_{0}=\begin{bmatrix}0.5\\ 0.25\\ 0.25\end{bmatrix},\;x_{1}=\begin{bmatrix}0\\ 1\\ 0\end{bmatrix},\;x_{2}=\begin{bmatrix}0.65&0.35\end{bmatrix},\;x_{3}=\begin{% bmatrix}0.3&0.5&0.07&0.1&0.03\end{bmatrix}.
  2. p p
  3. p = [ p 1 p 2 p n ] or p = [ p 1 p 2 p n ] p=\begin{bmatrix}p_{1}\\ p_{2}\\ \vdots\\ p_{n}\end{bmatrix}\quad\,\text{or}\quad p=\begin{bmatrix}p_{1}&p_{2}&\cdots&p_% {n}\end{bmatrix}
  4. i = 1 n p i = 1 \sum_{i=1}^{n}p_{i}=1
  5. 0 p i 1 0\leq p_{i}\leq 1
  6. i i
  7. 1 / n 1/n
  8. 1 / n 1/n
  9. 1 / n 1/\sqrt{n}
  10. n σ 2 + 1 / n \sqrt{n\sigma^{2}+1/n}
  11. σ 2 \sigma^{2}

Probable_prime.html

  1. ( a p ) (\tfrac{a}{p})
  2. ( a p ) (\tfrac{a}{p})
  3. a d 1 ( mod n ) , a^{d}\equiv 1\;\;(\mathop{{\rm mod}}n),\;
  4. a d 2 r - 1 ( mod n ) for some 0 r s - 1. a^{d\cdot 2^{r}}\equiv-1\;\;(\mathop{{\rm mod}}n)\,\text{ for some }0\leq r% \leq s-1.\,
  5. d d
  6. s s
  7. 96 = d 2 s 96=d\cdot 2^{s}
  8. d d
  9. s = 0 s=0
  10. d d
  11. 96 96
  12. s s
  13. d = 3 d=3
  14. s = 5 s=5
  15. 96 = 3 2 5 96=3\cdot 2^{5}
  16. a a
  17. 97 97
  18. 2 2
  19. a d ( mod n ) a^{d}\;\;(\mathop{{\rm mod}}n)
  20. 2 3 ( mod 97 ) 2^{3}\;\;(\mathop{{\rm mod}}97)
  21. 1 ( mod 97 ) 1\;\;(\mathop{{\rm mod}}97)
  22. 2 3 2 r ( mod 97 ) 2^{3\cdot 2^{r}}\;\;(\mathop{{\rm mod}}97)
  23. 0 r < s 0\leq r<s
  24. 96 ( mod 97 ) 96\;\;(\mathop{{\rm mod}}97)
  25. 97 97
  26. 97 97
  27. r = 0 : 2 3 8 ( mod 97 ) r=0:2^{3}\equiv 8\;\;(\mathop{{\rm mod}}97)
  28. r = 1 : 2 6 64 ( mod 97 ) r=1:2^{6}\equiv 64\;\;(\mathop{{\rm mod}}97)
  29. r = 2 : 2 12 22 ( mod 97 ) r=2:2^{12}\equiv 22\;\;(\mathop{{\rm mod}}97)
  30. r = 3 : 2 24 96 ( mod 97 ) r=3:2^{24}\equiv 96\;\;(\mathop{{\rm mod}}97)
  31. 97 97

Process_gain.html

  1. G p = 1.5 B n Δ f 2 W 3 G_{p}=\cfrac{1.5B_{n}\Delta f^{2}}{W^{3}}

Product_(category_theory).html

  1. 𝒞 \mathcal{C}
  2. X 1 X_{1}
  3. X 2 X_{2}
  4. X X
  5. X 1 X_{1}
  6. X 2 X_{2}
  7. X 1 × X 2 X_{1}\times X_{2}
  8. π 1 : X X 1 , π 2 : X X 2 \pi_{1}:X\to X_{1},\pi_{2}:X\to X_{2}
  9. Y Y
  10. f 1 : Y X 1 , f 2 : Y X 2 f_{1}:Y\to X_{1},f_{2}:Y\to X_{2}
  11. f : Y X f:Y\to X
  12. f f
  13. f 1 f_{1}
  14. f 2 f_{2}
  15. f 1 , f 2 \langle f_{1},f_{2}\rangle
  16. π 1 \pi_{1}
  17. π 2 \pi_{2}
  18. I I
  19. X X
  20. ( X i ) i I (X_{i})_{i\in I}
  21. π i : X X i \pi_{i}:X\to X_{i}
  22. Y Y
  23. I I
  24. f i : Y X i f_{i}:Y\to X_{i}
  25. f : Y X f:Y\to X
  26. i I i\in I
  27. i I X i \prod_{i\in I}X_{i}
  28. I = { 1 , , n } I=\{1,\ldots,n\}
  29. X 1 × × X n X_{1}\times\cdots\times X_{n}
  30. f 1 , , f n \langle f_{1},\ldots,f_{n}\rangle
  31. f f
  32. - , - \langle-,-\rangle
  33. f 1 , f 2 , i { 1 , 2 } , π i f 1 , f 2 = f i \forall f_{1},\forall f_{2},\forall i\in\{1,2\},\ \pi_{i}\circ\langle f_{1},f_% {2}\rangle=f_{i}
  34. f f
  35. f , π 1 f , π 2 f = f \forall f,\ \langle\pi_{1}\circ f,\pi_{2}\circ f\rangle=f
  36. I I
  37. { f } i \{f\}_{i}
  38. J J
  39. 𝒞 𝒥 {\mathcal{C}}^{\mathcal{J}}
  40. 𝒞 × 𝒞 \mathcal{C}\times\mathcal{C}
  41. Δ : 𝒞 𝒞 × 𝒞 \Delta:\mathcal{C}\to\mathcal{C}\times\mathcal{C}
  42. X X
  43. ( X , X ) (X,X)
  44. f f
  45. ( f , f ) (f,f)
  46. X 1 × X 2 X_{1}\times X_{2}
  47. 𝒞 \mathcal{C}
  48. Δ \Delta
  49. ( X 1 , X 2 ) (X_{1},X_{2})
  50. 𝒞 × 𝒞 \mathcal{C}\times\mathcal{C}
  51. X X
  52. 𝒞 \mathcal{C}
  53. ( X , X ) ( X 1 , X 2 ) (X,X)\to(X_{1},X_{2})
  54. i I X i := { ( x i ) i I | x i X i i I } \prod_{i\in I}X_{i}:=\{(x_{i})_{i\in I}|x_{i}\in X_{i}\,\forall i\in I\}
  55. π j : i I X i X j , π j ( ( x i ) i I ) := x j \pi_{j}:\prod_{i\in I}X_{i}\to X_{j}\mathrm{,}\quad\pi_{j}((x_{i})_{i\in I}):=% x_{j}
  56. f i : Y X i f_{i}:Y\to X_{i}
  57. f : Y i I X i , f ( y ) := ( f i ( y ) ) i I f:Y\to\prod_{i\in I}X_{i}\mathrm{,}\quad f(y):=(f_{i}(y))_{i\in I}
  58. I I
  59. G G
  60. G \mathbb{Z}\to G
  61. G G
  62. I I
  63. I I
  64. 𝒞 I 𝒞 {\mathcal{C}}^{I}\to\mathcal{C}
  65. f 1 : X 1 Y 1 , f 2 : X 2 Y 2 f_{1}:X_{1}\to Y_{1},f_{2}:X_{2}\to Y_{2}
  66. X 1 × X 2 Y 1 × Y 2 X_{1}\times X_{2}\to Y_{1}\times Y_{2}
  67. f 1 π 1 , f 2 π 2 \langle f_{1}\circ\pi_{1},f_{2}\circ\pi_{2}\rangle
  68. { X } i , { Y } i , f i : X i Y i \{X\}_{i},\{Y\}_{i},f_{i}:X_{i}\to Y_{i}
  69. i I X i i I Y i \prod_{i\in I}X_{i}\to\prod_{i\in I}Y_{i}
  70. { f i π i } i \{f_{i}\circ\pi_{i}\}_{i}
  71. 𝒞 \mathcal{C}
  72. 1 1
  73. 𝒞 \mathcal{C}
  74. X × ( Y × Z ) ( X × Y ) × Z X × Y × Z X\times(Y\times Z)\simeq(X\times Y)\times Z\simeq X\times Y\times Z
  75. X × 1 1 × X X X\times 1\simeq 1\times X\simeq X
  76. X × Y Y × X X\times Y\simeq Y\times X
  77. X × ( Y + Z ) ( X × Y ) + ( X × Z ) . X\times(Y+Z)\simeq(X\times Y)+(X\times Z).

Product_rule.html

  1. ( f g ) = f g + f g (f\cdot g)^{\prime}=f^{\prime}\cdot g+f\cdot g^{\prime}\,\!
  2. d d x ( u v ) = u d v d x + v d u d x \dfrac{d}{dx}(u\cdot v)=u\cdot\dfrac{dv}{dx}+v\cdot\dfrac{du}{dx}
  3. d ( u v ) = u d v + v d u d(uv)=u\,dv+v\,du
  4. d d x ( u v w ) = d u d x v w + u d v d x w + u v d w d x \dfrac{d}{dx}(u\cdot v\cdot w)=\dfrac{du}{dx}\cdot v\cdot w+u\cdot\dfrac{dv}{% dx}\cdot w+u\cdot v\cdot\dfrac{dw}{dx}
  5. d ( u v ) \displaystyle d(u\cdot v)
  6. d ( u v ) = v d u + u d v d(u\cdot v)=v\cdot du+u\cdot dv\,\!
  7. d d x ( u v ) = v d u d x + u d v d x \frac{d}{dx}(u\cdot v)=v\cdot\frac{du}{dx}+u\cdot\frac{dv}{dx}\,\!
  8. ( u v ) = v u + u v . (u\cdot v)^{\prime}=v\cdot u^{\prime}+u\cdot v^{\prime}.\,\!
  9. h ( x ) = lim a 0 h ( x + a ) - h ( x ) a = lim a 0 f ( x + a ) g ( x + a ) - f ( x ) g ( x ) a h^{\prime}(x)=\lim_{a\to 0}\frac{h(x+a)-h(x)}{a}=\lim_{a\to 0}\frac{f(x+a)g(x+% a)-f(x)g(x)}{a}
  10. = lim a 0 f ( x + a ) g ( x + a ) - f ( x ) g ( x + a ) + f ( x ) g ( x + a ) - f ( x ) g ( x ) a =\lim_{a\to 0}\frac{f(x+a)g(x+a)-f(x)g(x+a)+f(x)g(x+a)-f(x)g(x)}{a}
  11. = lim a 0 [ f ( x + a ) - f ( x ) ] g ( x + a ) + f ( x ) [ g ( x + a ) - g ( x ) ] a =\lim_{a\to 0}\frac{[f(x+a)-f(x)]\cdot g(x+a)+f(x)\cdot[g(x+a)-g(x)]}{a}
  12. = lim a 0 f ( x + a ) - f ( x ) a lim a 0 g ( x + a ) + lim a 0 f ( x ) lim a 0 g ( x + a ) - g ( x ) a =\lim_{a\to 0}\frac{f(x+a)-f(x)}{a}\cdot\lim_{a\to 0}g(x+a)+\lim_{a\to 0}f(x)% \cdot\lim_{a\to 0}\frac{g(x+a)-g(x)}{a}
  13. = f ( x ) g ( x ) + f ( x ) g ( x ) =f^{\prime}(x)g(x)+f(x)g^{\prime}(x)
  14. lim Δ x 0 Δ h Δ x \lim_{\Delta x\to 0}{\Delta h\over\Delta x}
  15. Δ h = Δ f g ( x 0 ) + f ( x 0 ) Δ g + Δ f Δ g \Delta h=\Delta fg(x_{0})+f(x_{0})\Delta g+\Delta f\Delta g
  16. Δ h Δ x = Δ f g ( x 0 ) + f ( x 0 ) Δ g + Δ f Δ g Δ x = Δ f Δ x g ( x 0 ) + f ( x 0 ) Δ g Δ x + Δ f Δ g Δ x \frac{\Delta h}{\Delta x}=\frac{\Delta fg(x_{0})+f(x_{0})\Delta g+\Delta f% \Delta g}{\Delta x}=\frac{\Delta f}{\Delta x}g(x_{0})+f(x_{0})\frac{\Delta g}{% \Delta x}+\frac{\Delta f\Delta g}{\Delta x}
  17. lim Δ x 0 ( Δ f Δ x g ( x 0 ) ) = f ( x 0 ) g ( x 0 ) \lim_{\Delta x\to 0}\left(\frac{\Delta f}{\Delta x}g(x_{0})\right)=f^{\prime}(% x_{0})g(x_{0})
  18. lim Δ x 0 ( f ( x 0 ) Δ g Δ x ) = f ( x 0 ) g ( x 0 ) \lim_{\Delta x\to 0}\left(f(x_{0})\frac{\Delta g}{\Delta x}\right)=f(x_{0})g^{% \prime}(x_{0})
  19. lim Δ x 0 Δ f Δ g Δ x = lim Δ x 0 ( Δ f Δ x Δ g Δ x Δ x ) = lim Δ x 0 Δ f Δ x lim Δ x 0 Δ g Δ x lim Δ x 0 Δ x = f ( x 0 ) g ( x 0 ) 0 = 0 \lim_{\Delta x\to 0}\frac{\Delta f\Delta g}{\Delta x}=\lim_{\Delta x\to 0}% \left(\frac{\Delta f}{\Delta x}\frac{\Delta g}{\Delta x}\Delta x\right)=\lim_{% \Delta x\to 0}{\frac{\Delta f}{\Delta x}}\cdot\lim_{\Delta x\to 0}{\frac{% \Delta g}{\Delta x}}\cdot\lim_{\Delta x\to 0}{\Delta x}=f^{\prime}(x_{0})g^{% \prime}(x_{0})\cdot 0=0
  20. lim Δ x 0 Δ h Δ x \lim_{\Delta x\to 0}\frac{\Delta h}{\Delta x}
  21. h ( x 0 ) = lim Δ x 0 Δ h Δ x = lim Δ x 0 ( Δ f Δ x g ( x 0 ) ) + lim Δ x 0 ( f ( x 0 ) Δ g Δ x ) + lim Δ x 0 ( Δ f Δ g Δ x ) = f ( x 0 ) g ( x 0 ) + f ( x 0 ) g ( x 0 ) + 0 = f ( x 0 ) g ( x 0 ) + f ( x 0 ) g ( x 0 ) \begin{aligned}\displaystyle h^{\prime}(x_{0})&\displaystyle=\lim_{\Delta x\to 0% }\frac{\Delta h}{\Delta x}\\ &\displaystyle=\lim_{\Delta x\to 0}\left(\frac{\Delta f}{\Delta x}g(x_{0})% \right)+\lim_{\Delta x\to 0}\left(f(x_{0})\frac{\Delta g}{\Delta x}\right)+% \lim_{\Delta x\to 0}\left(\frac{\Delta f\Delta g}{\Delta x}\right)\\ &\displaystyle=f^{\prime}(x_{0})g(x_{0})+f(x_{0})g^{\prime}(x_{0})+0\\ &\displaystyle=f^{\prime}(x_{0})g(x_{0})+f(x_{0})g^{\prime}(x_{0})\\ \end{aligned}
  22. f , g : f,g:\mathbb{R}\rightarrow\mathbb{R}
  23. x x
  24. f ( x + h ) = f ( x ) + f ( x ) h + ψ 1 ( h ) g ( x + h ) = g ( x ) + g ( x ) h + ψ 2 ( h ) f(x+h)=f(x)+f^{\prime}(x)h+\psi_{1}(h)\qquad\qquad g(x+h)=g(x)+g^{\prime}(x)h+% \psi_{2}(h)
  25. lim h 0 ψ 1 ( h ) h = lim h 0 ψ 2 ( h ) h = 0 \lim_{h\to 0}\frac{\psi_{1}(h)}{h}=\lim_{h\to 0}\frac{\psi_{2}(h)}{h}=0
  26. ψ 1 , ψ 2 o ( h ) \psi_{1},\psi_{2}\sim o(h)
  27. f g ( x + h ) - f g ( x ) = ( f ( x ) + f ( x ) h + ψ 1 ( h ) ) ( g ( x ) + g ( x ) h + ψ 2 ( h ) ) - f g ( x ) = f ( x ) g ( x ) h + f ( x ) g ( x ) h + O ( h ) \displaystyle fg(x+h)-fg(x)=(f(x)+f^{\prime}(x)h+\psi_{1}(h))(g(x)+g^{\prime}(% x)h+\psi_{2}(h))-fg(x)=f^{\prime}(x)g(x)h+f(x)g^{\prime}(x)h+O(h)
  28. h h
  29. ln f = ln ( u v ) = ln u + ln v . \ln f=\ln(u\cdot v)=\ln u+\ln v.\,
  30. 1 f d f d x = 1 u d u d x + 1 v d v d x {1\over f}{df\over dx}={1\over u}{du\over dx}+{1\over v}{dv\over dx}\,
  31. d f d x = v d u d x + u d v d x . {df\over dx}=v{du\over dx}+u{dv\over dx}.\,
  32. q ( x ) = x 2 4 q(x)={x^{2}\over 4}
  33. f = q ( u + v ) - q ( u - v ) , f=q(u+v)-q(u-v),
  34. f = q ( u + v ) ( u + v ) - q ( u - v ) ( u - v ) f^{\prime}=q^{\prime}(u+v)(u^{\prime}+v^{\prime})-q^{\prime}(u-v)(u^{\prime}-v% ^{\prime})
  35. = ( 1 2 ( u + v ) ( u + v ) ) - ( 1 2 ( u - v ) ( u - v ) ) =\left({1\over 2}(u+v)(u^{\prime}+v^{\prime})\right)-\left({1\over 2}(u-v)(u^{% \prime}-v^{\prime})\right)
  36. = 1 2 ( u u + v u + u v + v v ) - 1 2 ( u u - v u - u v + v v ) ={1\over 2}(uu^{\prime}+vu^{\prime}+uv^{\prime}+vv^{\prime})-{1\over 2}(uu^{% \prime}-vu^{\prime}-uv^{\prime}+vv^{\prime})
  37. = v u + u v =vu^{\prime}+uv^{\prime}
  38. = u v + u v . =uv^{\prime}+u^{\prime}v.\,
  39. d ( a b ) d x = ( a b ) a d a d x + ( a b ) b d b d x = b d a d x + a d b d x . {d(ab)\over dx}=\frac{\partial(ab)}{\partial a}\frac{da}{dx}+\frac{\partial(ab% )}{\partial b}\frac{db}{dx}=b\frac{da}{dx}+a\frac{db}{dx}.\,
  40. d ( u v ) d x \frac{d(uv)}{dx}\,
  41. = st ( ( u + d u ) ( v + d v ) - u v d x ) =\operatorname{st}\left(\frac{(u+\mathrm{d}u)(v+\mathrm{d}v)-uv}{\mathrm{d}x}\right)
  42. = st ( u v + u d v + v d u + d v d u - u v d x ) =\operatorname{st}\left(\frac{uv+u\cdot\mathrm{d}v+v\cdot\mathrm{d}u+\mathrm{d% }v\cdot\mathrm{d}u-uv}{\mathrm{d}x}\right)
  43. = st ( u d v + ( v + d v ) d u d x ) =\operatorname{st}\left(\frac{u\cdot\mathrm{d}v+(v+\mathrm{d}v)\cdot\mathrm{d}% u}{\mathrm{d}x}\right)
  44. = u d v d x + v d u d x ={u}\frac{dv}{dx}+{v}\frac{du}{dx}
  45. d ( u v ) \displaystyle d(uv)
  46. d u d v = u v ( d x ) 2 = 0 du\cdot dv=u^{\prime}v^{\prime}(dx)^{2}=0\,\!
  47. d ( u v w ) d x = d u d x v w + u d v d x w + u v d w d x \frac{d(uvw)}{dx}=\frac{du}{dx}vw+u\frac{dv}{dx}w+uv\frac{dw}{dx}\,\!
  48. f 1 , , f k f_{1},\dots,f_{k}
  49. d d x [ i = 1 k f i ( x ) ] = i = 1 k ( d d x f i ( x ) j i f j ( x ) ) = ( i = 1 k f i ( x ) ) ( i = 1 k f i ( x ) f i ( x ) ) . \frac{d}{dx}\left[\prod_{i=1}^{k}f_{i}(x)\right]=\sum_{i=1}^{k}\left(\frac{d}{% dx}f_{i}(x)\prod_{j\neq i}f_{j}(x)\right)=\left(\prod_{i=1}^{k}f_{i}(x)\right)% \left(\sum_{i=1}^{k}\frac{f^{\prime}_{i}(x)}{f_{i}(x)}\right).
  50. ( u v ) ( n ) ( x ) = k = 0 n ( n k ) u ( n - k ) ( x ) v ( k ) ( x ) . (uv)^{(n)}(x)=\sum_{k=0}^{n}{n\choose k}\cdot u^{(n-k)}(x)\cdot v^{(k)}(x).
  51. ( i = 1 k f i ) ( n ) = j 1 + j 2 + + j k = n ( n j 1 , j 2 , , j k ) i = 1 k f i ( j i ) . \left(\prod_{i=1}^{k}f_{i}\right)^{(n)}=\sum_{j_{1}+j_{2}+...+j_{k}=n}{n% \choose j_{1},j_{2},...,j_{k}}\prod_{i=1}^{k}f_{i}^{(j_{i})}.
  52. n x 1 x n ( u v ) = S | S | u i S x i n - | S | v i S x i {\partial^{n}\over\partial x_{1}\,\cdots\,\partial x_{n}}(uv)=\sum_{S}{% \partial^{|S|}u\over\prod_{i\in S}\partial x_{i}}\cdot{\partial^{n-|S|}v\over% \prod_{i\not\in S}\partial x_{i}}
  53. 3 x 1 x 2 x 3 ( u v ) = u 3 v x 1 x 2 x 3 + u x 1 2 v x 2 x 3 + u x 2 2 v x 1 x 3 + u x 3 2 v x 1 x 2 + 2 u x 1 x 2 v x 3 + 2 u x 1 x 3 v x 2 + 2 u x 2 x 3 v x 1 + 3 u x 1 x 2 x 3 v . \begin{aligned}&\displaystyle{}\quad{\partial^{3}\over\partial x_{1}\,\partial x% _{2}\,\partial x_{3}}(uv)\\ \\ &\displaystyle{}=u\cdot{\partial^{3}v\over\partial x_{1}\,\partial x_{2}\,% \partial x_{3}}+{\partial u\over\partial x_{1}}\cdot{\partial^{2}v\over% \partial x_{2}\,\partial x_{3}}+{\partial u\over\partial x_{2}}\cdot{\partial^% {2}v\over\partial x_{1}\,\partial x_{3}}+{\partial u\over\partial x_{3}}\cdot{% \partial^{2}v\over\partial x_{1}\,\partial x_{2}}\\ \\ &\displaystyle{}\qquad+{\partial^{2}u\over\partial x_{1}\,\partial x_{2}}\cdot% {\partial v\over\partial x_{3}}+{\partial^{2}u\over\partial x_{1}\,\partial x_% {3}}\cdot{\partial v\over\partial x_{2}}+{\partial^{2}u\over\partial x_{2}\,% \partial x_{3}}\cdot{\partial v\over\partial x_{1}}+{\partial^{3}u\over% \partial x_{1}\,\partial x_{2}\,\partial x_{3}}\cdot v.\end{aligned}
  54. ( D ( x , y ) B ) ( u , v ) = B ( u , y ) + B ( x , v ) ( u , v ) X × Y . (D_{\left(}x,y\right)\,B)\left(u,v\right)=B\left(u,y\right)+B\left(x,v\right)% \qquad\forall(u,v)\in X\times Y.
  55. ( f g ) = f g + f g (f\cdot g)^{\prime}=f\;^{\prime}\cdot g+f\cdot g\;^{\prime}\,
  56. ( f g ) = f g + f g (f\cdot g)^{\prime}=f\;^{\prime}\cdot g+f\cdot g\;^{\prime}\,
  57. ( f × g ) = f × g + f × g (f\times g)^{\prime}=f\;^{\prime}\times g+f\times g\;^{\prime}\,
  58. ( f × x ) f × g + g × f (f\times x)^{\prime}\neq f^{\prime}\times g+g\times f^{\prime}
  59. ( f × x ) = f × g - g × f (f\times x)^{\prime}=f^{\prime}\times g-g\times f^{\prime}
  60. ( f g ) = f g + f g \nabla(f\cdot g)=\nabla f\cdot g+f\cdot\nabla g\,
  61. d d x x n = n x n - 1 {d\over dx}x^{n}=nx^{n-1}\,\!
  62. d d x x n + 1 = d d x ( x n x ) = x d d x x n + x n d d x x (the product rule is used here) = x ( n x n - 1 ) + x n 1 (the induction hypothesis is used here) = ( n + 1 ) x n . \begin{aligned}\displaystyle{d\over dx}x^{n+1}&\displaystyle{}={d\over dx}% \left(x^{n}\cdot x\right)\\ &\displaystyle{}=x{d\over dx}x^{n}+x^{n}{d\over dx}x\qquad\mbox{(the product % rule is used here)}\\ &\displaystyle{}=x\left(nx^{n-1}\right)+x^{n}\cdot 1\qquad\mbox{(the induction% hypothesis is used here)}\\ &\displaystyle{}=(n+1)x^{n}.\end{aligned}

Profit_maximization.html

  1. d 2 R d Q 2 < d 2 C d Q 2 \frac{\operatorname{d}^{2}R}{{\operatorname{d}Q}^{2}}<\frac{\operatorname{d}^{% 2}C}{{\operatorname{d}Q}^{2}}

Progressive_tax.html

  1. $ 3 , 000 + ( $ 25 , 000 - $ 20 , 000 ) × 30 % = $ 1 , 500 + $ 3 , 000 = $ 4 , 500. \$3,000+(\$25,000-\$20,000)\times 30\%=\$1,500+\$3,000=\$4,500.
  2. $ 25 , 000 × 30 % - $ 3 , 000 = $ 7 , 500 - $ 3 , 000 = $ 4 , 500. \$25,000\times 30\%-\$3,000=\$7,500-\$3,000=\$4,500.

Projective_space.html

  1. ~{}
  2. U i = { [ x 0 : : x n ] , x i 0 } , i = 0 , , n . U_{i}=\{[x_{0}:\cdots:x_{n}],x_{i}\neq 0\},\quad i=0,\dots,n.
  3. [ x 0 : : x n ] ( x 0 x i , , x i x i ^ , , x n x i ) [x_{0}:\cdots:x_{n}]\mapsto\left(\frac{x_{0}}{x_{i}},\dots,\widehat{\frac{x_{i% }}{x_{i}}},\dots,\frac{x_{n}}{x_{i}}\right)
  4. [ y 0 : : y i - 1 : 1 : y i + 1 : : y n ] ( y 0 , , y i ^ , y n ) [y_{0}:\cdots:y_{i-1}:1:y_{i+1}:\cdots:y_{n}]\leftarrow\left(y_{0},\dots,% \widehat{y_{i}},\dots y_{n}\right)
  5. x 1 x , y 1 y . x\mapsto\frac{1}{x},\,y\mapsto\frac{1}{y}.
  6. \cdots
  7. = X - 1 X 0 X n = P . \varnothing=X_{-1}\subset X_{0}\subset\cdots X_{n}=P.
  8. X i X_{i}
  9. i i
  10. n n
  11. n - 1 n-1

Prony_equation.html

  1. h f = L D ( a V + b V 2 ) h_{f}=\frac{L}{D}(aV+bV^{2})

Proof_of_Bertrand's_postulate.html

  1. n 1 n\geq 1
  2. p p
  3. n < p 2 n n<p\leq 2n
  4. p r p^{r}
  5. ( 2 n n ) := ( 2 n ) ! ( n ! ) 2 {\textstyle\left({{2n}\atop{n}}\right)}:=\frac{(2n)!}{(n!)^{2}}
  6. 2 n 2n
  7. 2 n \sqrt{2n}
  8. r r
  9. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  10. ( 2 n 3 , n ) \left(\tfrac{2n}{3},n\right)
  11. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  12. n n
  13. O ( θ n ) O(\theta^{n})
  14. θ < 4 \theta<4
  15. 4 n / 2 n 4^{n}/2n
  16. n n
  17. n n
  18. 2 n 2n
  19. n > 468 n>468
  20. n n
  21. n > 0 n>0
  22. 4 n 2 n ( 2 n n ) . \frac{4^{n}}{2n}\leq{\left({{2n}\atop{n}}\right)}.
  23. 4 n = ( 1 + 1 ) 2 n = k = 0 2 n ( 2 n k ) = 2 + k = 1 2 n - 1 ( 2 n k ) 2 n ( 2 n n ) , 4^{n}=(1+1)^{2n}=\sum_{k=0}^{2n}{\left({{2n}\atop{k}}\right)}=2+\sum_{k=1}^{2n% -1}{\left({{2n}\atop{k}}\right)}\leq 2n{\left({{2n}\atop{n}}\right)},
  24. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  25. 2 n 2n
  26. p p
  27. R ( p , n ) R(p,n)
  28. r r
  29. p r p^{r}
  30. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  31. p p
  32. p R ( p , n ) 2 n p^{R(p,n)}\leq 2n
  33. p p
  34. n ! n!
  35. j = 1 n p j , \sum_{j=1}^{\infty}\left\lfloor\frac{n}{p^{j}}\right\rfloor,
  36. R ( p , n ) = j = 1 2 n p j - 2 j = 1 n p j = j = 1 ( 2 n p j - 2 n p j ) . R(p,n)=\sum_{j=1}^{\infty}\left\lfloor\frac{2n}{p^{j}}\right\rfloor-2\sum_{j=1% }^{\infty}\left\lfloor\frac{n}{p^{j}}\right\rfloor=\sum_{j=1}^{\infty}\left(% \left\lfloor\frac{2n}{p^{j}}\right\rfloor-2\left\lfloor\frac{n}{p^{j}}\right% \rfloor\right).
  37. n / p j mod 1 < 1 / 2 n/p^{j}\bmod 1<1/2
  38. n / p j mod 1 1 / 2 n/p^{j}\bmod 1\geq 1/2
  39. j > log p ( 2 n ) j>\log_{p}(2n)
  40. R ( p , n ) log p ( 2 n ) , R(p,n)\leq\log_{p}(2n),
  41. p R ( p , n ) p log p 2 n = 2 n . p^{R(p,n)}\leq p^{\log_{p}{2n}}=2n.
  42. p p
  43. 2 n 3 < p n \frac{2n}{3}<p\leq n
  44. R ( p , n ) = 0. R(p,n)=0.
  45. p p
  46. ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 {\textstyle\left({{2n}\atop{n}}\right)}=\frac{(2n)!}{(n!)^{2}}
  47. p p
  48. 2 p 2p
  49. 2 n ! 2n!
  50. p p
  51. p p
  52. n ! n!
  53. p p
  54. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  55. p p
  56. 3 p 3p
  57. p p
  58. 2 p 2p
  59. p p
  60. x # = p x p , x\#=\prod_{p\leq x}p,
  61. p p
  62. x x
  63. x 3 x\geq 3
  64. x # < 2 2 x - 3 x\#<2^{2x-3}
  65. x # = x # x\#=\lfloor x\rfloor\#
  66. x = n x=n
  67. ( 2 n n ) {\left({{2n}\atop{n}}\right)}
  68. n + 1 p 2 n - 1 n+1\leq p\leq 2n-1
  69. ( 2 n - 1 ) # / ( n ) # ( 2 n n ) < 2 2 n - 2 (2n-1)\#/(n)\#\leq{\left({{2n}\atop{n}}\right)}<2^{2n-2}
  70. n = 3 n=3
  71. n # = 6 < 8. n\#=6<8.
  72. n = 4 n=4
  73. n # = 6 < 32. n\#=6<32.
  74. n n
  75. n # = ( 2 m - 1 ) # < 2 2 ( 2 m - 1 ) - 3 n\#=(2m-1)\#<2^{2(2m-1)-3}
  76. n n
  77. n # = ( 2 m ) # < 2 2 ( 2 m ) - 3 n\#=(2m)\#<2^{2(2m)-3}
  78. n # < 2 2 n - 3 n\#<2^{2n-3}
  79. p > 2 n , p>\sqrt{2n},
  80. ( 2 n n ) \textstyle{2n\choose n}
  81. 2 n \sqrt{2n}
  82. ( 2 n ) 2 n (2n)^{\sqrt{2n}}
  83. 4 n 2 n ( 2 n n ) = ( p 2 n p R ( p , n ) ) ( 2 n < p 2 n 3 p R ( p , n ) ) < ( 2 n ) 2 n 1 < p 2 n 3 p = ( 2 n ) 2 n ( 2 n 3 ) # ( 2 n ) 2 n 4 2 n / 3 . \frac{4^{n}}{2n}\leq{\left({{2n}\atop{n}}\right)}=\left(\prod_{p\leq\sqrt{2n}}% p^{R(p,n)}\right)\left(\prod_{\sqrt{2n}<p\leq\frac{2n}{3}}p^{R(p,n)}\right)<(2% n)^{\sqrt{2n}}\prod_{1<p\leq\frac{2n}{3}}p=(2n)^{\sqrt{2n}}\Big(\frac{2n}{3}% \Big)\#\leq(2n)^{\sqrt{2n}}4^{2n/3}.
  84. log 4 3 n ( 2 n + 1 ) log 2 n . {\frac{\log 4}{3}}n\leq(\sqrt{2n}+1)\log 2n\;.
  85. n < 468. n<468.
  86. n 5 n\geq 5
  87. n < p 2 n n<p\leq 2n
  88. n < 64 n<64
  89. n 4 n\geq 4
  90. 4 n n < ( 2 n n ) . \frac{4^{n}}{n}<{\left({{2n}\atop{n}}\right)}.
  91. 4 4 4 = 64 < 70 = ( 8 4 ) , \frac{4^{4}}{4}=64<70={\left({{8}\atop{4}}\right)},
  92. n - 1 n-1
  93. ( 2 n n ) = 2 2 n - 1 n ( 2 ( n - 1 ) n - 1 ) > 2 2 n - 1 n 4 n - 1 n - 1 > 2 2 4 n - 1 n = 4 n n . {\left({{2n}\atop{n}}\right)}=2\,\frac{2n-1}{n}{\left({{2(n-1)}\atop{n-1}}% \right)}>2\,\frac{2n-1}{n}\frac{4^{n-1}}{n-1}>2\cdot 2\,\frac{4^{n-1}}{n}=% \frac{4^{n}}{n}.
  94. π ( x ) \pi(x)
  95. n n
  96. n n
  97. π ( n ) 1 3 n + 2. \pi(n)\leq\frac{1}{3}n+2.
  98. p = 2 , 3 p=2,3
  99. p 1 p\equiv 1
  100. p 5 ( mod 6 ) p\equiv 5\;\;(\mathop{{\rm mod}}6)
  101. π ( n ) \pi(n)
  102. k 1 k\equiv 1
  103. k 5 ( mod 6 ) k\equiv 5\;\;(\mathop{{\rm mod}}6)
  104. 1 1
  105. 2 , 3 2,3
  106. π ( n ) n + 5 6 + n + 1 6 + 1 n + 5 6 + n + 1 6 + 1 = 1 3 n + 2. \pi(n)\leq\left\lfloor\frac{n+5}{6}\right\rfloor+\left\lfloor\frac{n+1}{6}% \right\rfloor+1\leq\frac{n+5}{6}+\frac{n+1}{6}+1=\frac{1}{3}n+2.
  107. n 5 n\geq 5
  108. 2 n 3 \sqrt{2n}\geq 3
  109. 2 n # 3 # = 6 \sqrt{2n}\#\geq 3\#=6
  110. 4 n n \displaystyle\frac{4^{n}}{n}
  111. 2 3 n log 2 < 1 3 2 n log 2 n + 3 log n 2 \frac{2}{3}n\log 2<\frac{1}{3}\sqrt{2n}\log 2n+3\log\frac{n}{2}
  112. 2 3 n \frac{2}{3}n
  113. log 2 < 2 log n n + 9 4 log n 2 n 2 + log 2 2 n f ( n ) . \log 2<\sqrt{2}\cdot\frac{\log\sqrt{n}}{\sqrt{n}}+\frac{9}{4}\frac{\log\frac{n% }{2}}{\frac{n}{2}}+\frac{\log 2}{\sqrt{2n}}\equiv f(n)\;.
  114. g ( x ) = log x x g(x)=\frac{\log x}{x}
  115. x e x\geq e
  116. f ( n ) f(n)
  117. n e 2 > 2 e n\geq e^{2}>2e
  118. f ( 2 6 ) log 2 = 2 3 8 + 9 4 5 32 + 2 16 = 0.97 < 1 < f ( n ) log 2 , \frac{f(2^{6})}{\log 2}=\sqrt{2}\cdot\frac{3}{8}+\frac{9}{4}\cdot\frac{5}{32}+% \frac{\sqrt{2}}{16}=0.97\dots<1<\frac{f(n)}{\log 2},
  119. n < 2 6 = 64 n<2^{6}=64

Proof_theory.html

  1. Π 1 0 \Pi^{0}_{1}
  2. Π 1 0 \Pi^{0}_{1}

Proofs_of_Fermat's_little_theorem.html

  1. a p a ( mod p ) a^{p}\equiv a\;\;(\mathop{{\rm mod}}p)\,\!
  2. a p - 1 1 ( mod p ) ( X ) a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p)\quad\quad(X)
  3. a p a ( mod p ) a^{p}\equiv a\;\;(\mathop{{\rm mod}}p)\,\!
  4. T n ( x ) = { { n x } 0 x < 1 , 1 x = 1 , T_{n}(x)=\begin{cases}\{nx\}&0\leq x<1,\\ 1&x=1,\end{cases}
  5. T m ( T n ( x ) ) = T m n ( x ) . T_{m}(T_{n}(x))=T_{mn}(x).\,
  6. T m ( T n ( 1 ) ) = T m ( 1 ) = 1 = T m n ( 1 ) . T_{m}(T_{n}(1))=T_{m}(1)=1=T_{mn}(1).\,\!
  7. T m ( T n ( x ) ) = { m { n x } } . T_{m}(T_{n}(x))=\{m\{nx\}\}.\,\!
  8. { m { n x } } = { m n x } . \{m\{nx\}\}=\{mnx\}.\,\!
  9. { m { n x } } = { m n x - m k } = { m n x } \{m\{nx\}\}=\{mnx-mk\}=\{mnx\}\,\!
  10. T a p ( x ) = T a ( T a ( T a ( x ) ) ) , p times \begin{matrix}T_{a^{p}}(x)=&\underbrace{T_{a}(T_{a}(\cdots T_{a}(x)\cdots))},% \\ &p\,\textrm{ times}\\ \end{matrix}
  11. x 0 , T a ( x 0 ) , T a ( T a ( x 0 ) ) , T a ( T a ( T a ( x 0 ) ) ) , . x_{0},T_{a}(x_{0}),T_{a}(T_{a}(x_{0})),T_{a}(T_{a}(T_{a}(x_{0}))),\ldots.\,\!
  12. x 0 , T a ( x 0 ) , T a 2 ( x 0 ) , T a 3 ( x 0 ) , . x_{0},T_{a}(x_{0}),T_{a^{2}}(x_{0}),T_{a^{3}}(x_{0}),\ldots.
  13. ( x + y ) p x p + y p ( mod p ) . (x+y)^{p}\equiv x^{p}+y^{p}\;\;(\mathop{{\rm mod}}p).\,
  14. ( x + y ) p = x p + y p (x+y)^{p}=x^{p}+y^{p}\,\!
  15. ( k + 1 ) p k p + 1 p ( mod p ) . (k+1)^{p}\equiv k^{p}+1^{p}\;\;(\mathop{{\rm mod}}p).\,
  16. ( k + 1 ) p k + 1 ( mod p ) , (k+1)^{p}\equiv k+1\;\;(\mathop{{\rm mod}}p),\,
  17. ( x + y ) n = i = 0 n ( n i ) x n - i y i , (x+y)^{n}=\sum_{i=0}^{n}{n\choose i}x^{n-i}y^{i},
  18. ( n i ) = n ! i ! ( n - i ) ! , {n\choose i}=\frac{n!}{i!(n-i)!},
  19. ( 4 2 ) = 6 , {4\choose 2}=6,
  20. ( x 1 + x 2 + + x m ) n = k 1 , k 2 , , k m ( n k 1 , k 2 , , k m ) x 1 k 1 x 2 k 2 x m k m . (x_{1}+x_{2}+\cdots+x_{m})^{n}=\sum_{k_{1},k_{2},\ldots,k_{m}}{n\choose k_{1},% k_{2},\ldots,k_{m}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}.
  21. a p = k 1 , k 2 , , k a ( p k 1 , k 2 , , k a ) a^{p}=\sum_{k_{1},k_{2},\ldots,k_{a}}{p\choose k_{1},k_{2},\ldots,k_{a}}
  22. ( p k 1 , k 2 , , k a ) 0 ( mod p ) {p\choose k_{1},k_{2},\ldots,k_{a}}\equiv 0\;\;(\mathop{{\rm mod}}p)\,\!
  23. ( p k 1 , k 2 , , k a ) 1 ( mod p ) {p\choose k_{1},k_{2},\ldots,k_{a}}\equiv 1\;\;(\mathop{{\rm mod}}p)\,\!
  24. k j = p k_{j}=p
  25. a , 2 a , 3 a , , ( p - 1 ) a ( A ) a,2a,3a,\ldots,(p-1)a\quad\quad(A)
  26. 1 , 2 , 3 , , p - 1. ( B ) 1,2,3,\ldots,p-1.\quad\quad\quad(B)
  27. a × 2 a × 3 a × × ( p - 1 ) a 1 × 2 × 3 × × ( p - 1 ) ( mod p ) . a\times 2a\times 3a\times\cdots\times(p-1)a\equiv 1\times 2\times 3\times% \cdots\times(p-1)\;\;(\mathop{{\rm mod}}p).
  28. a p - 1 ( p - 1 ) ! ( p - 1 ) ! ( mod p ) . a^{p-1}(p-1)!\equiv(p-1)!\;\;(\mathop{{\rm mod}}p).
  29. a p - 1 1 ( mod p ) . a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p).\,\!
  30. 3 , 6 , 9 , 12 , 15 , 18 ; 3,6,9,12,15,18;\,\!
  31. 3 , 6 , 2 , 5 , 1 , 4 , 3,6,2,5,1,4,\,\!
  32. 1 , 2 , 3 , 4 , 5 , 6. 1,2,3,4,5,6.\,\!
  33. 3 × 6 × 9 × 12 × 15 × 18 3 × 6 × 2 × 5 × 1 × 4 1 × 2 × 3 × 4 × 5 × 6 ( mod 7 ) ; 3\times 6\times 9\times 12\times 15\times 18\equiv 3\times 6\times 2\times 5% \times 1\times 4\equiv 1\times 2\times 3\times 4\times 5\times 6\;\;(\mathop{{% \rm mod}}7);\,\!
  34. 3 6 ( 1 × 2 × 3 × 4 × 5 × 6 ) ( 1 × 2 × 3 × 4 × 5 × 6 ) ( mod 7 ) . 3^{6}(1\times 2\times 3\times 4\times 5\times 6)\equiv(1\times 2\times 3\times 4% \times 5\times 6)\;\;(\mathop{{\rm mod}}7).\,\!
  35. 3 6 1 ( mod 7 ) , 3^{6}\equiv 1\;\;(\mathop{{\rm mod}}7),\,\!
  36. u x u y ( mod p ) , ux\equiv uy\;\;(\mathop{{\rm mod}}p),\,\!
  37. x y ( mod p ) . x\equiv y\;\;(\mathop{{\rm mod}}p).\,\!
  38. u x u y ( mod p ) , ux\equiv uy\;\;(\mathop{{\rm mod}}p),\,\!
  39. x y ( mod p ) . x\equiv y\;\;(\mathop{{\rm mod}}p).\,\!
  40. a , 2 a , 3 a , , ( p - 1 ) a , a,2a,3a,\ldots,(p-1)a,\,\!
  41. 1 , 2 , 3 , , p - 1. 1,2,3,\ldots,p-1.\,\!
  42. k a m a ( mod p ) , ka\equiv ma\;\;(\mathop{{\rm mod}}p),\,\!
  43. k m ( mod p ) . k\equiv m\;\;(\mathop{{\rm mod}}p).\,\!
  44. a × 3 a × 7 a × 9 a 1 × 3 × 7 × 9 ( mod 10 ) . a\times 3a\times 7a\times 9a\equiv 1\times 3\times 7\times 9\;\;(\mathop{{\rm mod% }}10).\,\!
  45. a φ ( 10 ) 1 ( mod 10 ) . {a^{\varphi(10)}}\equiv 1\;\;(\mathop{{\rm mod}}10).\,\!
  46. a k 1 ( mod p ) . a^{k}\equiv 1\;\;(\mathop{{\rm mod}}p).\,\!
  47. a p - 1 a k m ( a k ) m 1 m 1 ( mod p ) . a^{p-1}\equiv a^{km}\equiv(a^{k})^{m}\equiv 1^{m}\equiv 1\;\;(\mathop{{\rm mod% }}p).\,\!
  48. b x + p y = 1. bx+py=1.\,\!
  49. b x 1 ( mod p ) . bx\equiv 1\;\;(\mathop{{\rm mod}}p).\,\!

Propellant_mass_fraction.html

  1. ζ = m p m 0 \zeta=\frac{m_{p}}{m_{0}}
  2. m 0 = m f + m p m_{0}=m_{f}+m_{p}
  3. ζ = m 0 - m f m 0 = m p m p + m f = 1 - m f m 0 \zeta=\frac{m_{0}-m_{f}}{m_{0}}=\frac{m_{p}}{m_{p}+m_{f}}=1-\frac{m_{f}}{m_{0}}
  4. ζ \zeta
  5. m p m_{p}
  6. m 0 m_{0}
  7. m f m_{f}
  8. 1 - ( 342 , 100 / 1 , 708 , 500 ) = 0.7998 \displaystyle 1-(342,100/1,708,500)=0.7998
  9. Δ v = - v e ln m f m 0 \displaystyle\Delta v=-v_{e}\ln\frac{m_{f}}{m_{0}}
  10. m f / m 0 \displaystyle m_{f}/m_{0}
  11. Δ v \displaystyle\Delta v
  12. v e \displaystyle v_{e}
  13. v e = g n I s p \displaystyle v_{e}=g_{n}I_{sp}

Proportionality_(mathematics).html

  1. y x \tfrac{y}{x}
  2. a c = b d \tfrac{a}{c}\ =\ \tfrac{b}{d}
  3. d c = b a = d + b c + a . \tfrac{d}{c}\ =\ \tfrac{b}{a}\ =\ \tfrac{d\,+\,b}{c\,+\,a}.
  4. - 1 k \tfrac{-1}{k}
  5. c a = k \tfrac{c}{a}\ =\ k
  6. a c = 1 k . \tfrac{a}{c}\ =\ \tfrac{1}{k}.
  7. a b = c d \tfrac{a}{b}\ =\ \tfrac{c}{d}
  8. y = k x . y=kx.\,
  9. y x y\propto x
  10. k = y x k=\frac{y}{x}\,
  11. y = k x y=kx\,
  12. x = ( 1 k ) y , x=\left(\frac{1}{k}\right)y,
  13. y = k x y={k\over x}
  14. y = k a x . y=ka^{x}.\,
  15. y = k log a ( x ) . y=k\log_{a}(x).\,

Prospect_theory.html

  1. U = i = 1 n w ( p i ) v ( x i ) U=\sum_{i=1}^{n}w(p_{i})v(x_{i})
  2. U U
  3. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  4. p 1 , p 2 , , p n p_{1},p_{2},\dots,p_{n}
  5. v \scriptstyle v
  6. w w
  7. v ( - 15 ) \scriptstyle v(-15)
  8. w ( 0.01 ) × v ( - 1000 ) + w ( 0.99 ) × v ( 0 ) = w ( 0.01 ) × v ( - 1000 ) \scriptstyle w(0.01)\times v(-1000)+w(0.99)\times v(0)=w(0.01)\times v(-1000)
  9. w ( 0.01 ) > 0.01 \scriptstyle w(0.01)>0.01
  10. v ( - 15 ) / v ( - 1000 ) > 0.015 \scriptstyle v(-15)/v(-1000)>0.015
  11. w ( 0.01 ) \scriptstyle w(0.01)
  12. v ( - 15 ) / v ( - 1000 ) \scriptstyle v(-15)/v(-1000)
  13. w ( 0.01 ) > v ( - 15 ) / v ( - 1000 ) \scriptstyle w(0.01)>v(-15)/v(-1000)
  14. w ( 0.01 ) × v ( - 1000 ) < v ( - 15 ) \scriptstyle w(0.01)\times v(-1000)<v(-15)
  15. v \scriptstyle v
  16. v ( 985 ) \scriptstyle v(985)
  17. w ( 0.99 ) × v ( 1000 ) \scriptstyle w(0.99)\times v(1000)

Protein_structure_prediction.html

  1. α \alpha
  2. β \beta
  3. ϕ \phi
  4. ψ \psi

Proxima_Centauri.html

  1. ρ \begin{smallmatrix}\rho_{\odot}\end{smallmatrix}

Pseudo-Riemannian_manifold.html

  1. p \scriptstyle p
  2. n \scriptstyle n
  3. M \scriptstyle M
  4. T p M \scriptstyle T_{p}M
  5. n \scriptstyle n
  6. p \scriptstyle p
  7. g \scriptstyle g
  8. g : T p M × T p M . g:T_{p}M\times T_{p}M\to\mathbb{R}.
  9. X , Y , Z T p M \scriptstyle X,Y,Z\in T_{p}M
  10. p \scriptstyle p
  11. M \scriptstyle M
  12. g ( X , Y ) = g ( Y , X ) \,g(X,Y)=g(Y,X)
  13. g ( a X + Y , Z ) = a g ( X , Z ) + g ( Y , Z ) \,g(aX+Y,Z)=ag(X,Z)+g(Y,Z)
  14. a \scriptstyle a\in\mathbb{R}
  15. g \scriptstyle g
  16. X T p M X\in T_{p}M
  17. g ( X , Y ) = 0 \,g(X,Y)=0
  18. Y T p M Y\in T_{p}M
  19. ( M , g ) (M,g)
  20. M M
  21. g g
  22. n \mathbb{R}^{n}
  23. n - 1 , 1 \mathbb{R}^{n-1,1}
  24. p , q \mathbb{R}^{p,q}
  25. g = d x 1 2 + + d x p 2 - d x p + 1 2 - - d x p + q 2 g=dx_{1}^{2}+\cdots+dx_{p}^{2}-dx_{p+1}^{2}-\cdots-dx_{p+q}^{2}

Pseudovector.html

  1. 𝐩 = 𝐚 × 𝐛 . \mathbf{p}=\mathbf{a}\times\mathbf{b}.\,
  2. 𝐯 = R 𝐯 \mathbf{v}^{\prime}=R\mathbf{v}
  3. 𝐯 = ( det R ) ( R 𝐯 ) \mathbf{v}^{\prime}=(\det R)(R\mathbf{v})
  4. 𝐯 𝟑 = 𝐯 𝟏 + 𝐯 𝟐 = ( det R ) ( R 𝐯 𝟏 ) + ( det R ) ( R 𝐯 𝟐 ) = ( det R ) ( R ( 𝐯 𝟏 + 𝐯 𝟐 ) ) = ( det R ) ( R 𝐯 𝟑 ) . \mathbf{v_{3}}^{\prime}=\mathbf{v_{1}}^{\prime}+\mathbf{v_{2}}^{\prime}=(\det R% )(R\mathbf{v_{1}})+(\det R)(R\mathbf{v_{2}})=(\det R)(R(\mathbf{v_{1}}+\mathbf% {v_{2}}))=(\det R)(R\mathbf{v_{3}}).
  5. 𝐯 𝟑 = 𝐯 𝟏 + 𝐯 𝟐 = ( R 𝐯 𝟏 ) + ( det R ) ( R 𝐯 𝟐 ) = R ( 𝐯 𝟏 + ( det R ) 𝐯 𝟐 ) . \mathbf{v_{3}}^{\prime}=\mathbf{v_{1}}^{\prime}+\mathbf{v_{2}}^{\prime}=(R% \mathbf{v_{1}})+(\det R)(R\mathbf{v_{2}})=R(\mathbf{v_{1}}+(\det R)\mathbf{v_{% 2}}).
  6. | 𝐯 𝟑 | = | 𝐯 𝟏 + 𝐯 𝟐 | , |\mathbf{v_{3}}|=|\mathbf{v_{1}}+\mathbf{v_{2}}|,
  7. | 𝐯 𝟑 | = | 𝐯 𝟏 - 𝐯 𝟐 | |\mathbf{v_{3}}^{\prime}|=|\mathbf{v_{1}}^{\prime}-\mathbf{v_{2}}^{\prime}|
  8. ( R 𝐯 𝟏 ) × ( R 𝐯 𝟐 ) = ( det R ) ( R ( 𝐯 𝟏 × 𝐯 𝟐 ) ) (R\mathbf{v_{1}})\times(R\mathbf{v_{2}})=(\det R)(R(\mathbf{v_{1}}\times% \mathbf{v_{2}}))
  9. 𝐯 𝟑 = 𝐯 𝟏 × 𝐯 𝟐 = ( R 𝐯 𝟏 ) × ( R 𝐯 𝟐 ) = ( det R ) ( R ( 𝐯 𝟏 × 𝐯 𝟐 ) ) = ( det R ) ( R 𝐯 𝟑 ) . \mathbf{v_{3}}^{\prime}=\mathbf{v_{1}}^{\prime}\times\mathbf{v_{2}}^{\prime}=(% R\mathbf{v_{1}})\times(R\mathbf{v_{2}})=(\det R)(R(\mathbf{v_{1}}\times\mathbf% {v_{2}}))=(\det R)(R\mathbf{v_{3}}).
  10. 𝐚𝐛 = 𝐚 𝐛 + 𝐚 𝐛 , \mathbf{ab}=\mathbf{a\cdot b}+\mathbf{a\wedge b}\ ,
  11. 𝐚 × 𝐛 = ( a 2 b 3 - a 3 b 2 ) 𝐞 1 + ( a 3 b 1 - a 1 b 3 ) 𝐞 2 + ( a 1 b 2 - a 2 b 1 ) 𝐞 3 , \mathbf{a}\times\mathbf{b}=(a^{2}b^{3}-a^{3}b^{2})\mathbf{e}_{1}+(a^{3}b^{1}-a% ^{1}b^{3})\mathbf{e}_{2}+(a^{1}b^{2}-a^{2}b^{1})\mathbf{e}_{3},
  12. 𝐚 𝐛 = ( a 2 b 3 - a 3 b 2 ) 𝐞 23 + ( a 3 b 1 - a 1 b 3 ) 𝐞 31 + ( a 1 b 2 - a 2 b 1 ) 𝐞 12 . \mathbf{a}\wedge\mathbf{b}=(a^{2}b^{3}-a^{3}b^{2})\mathbf{e}_{23}+(a^{3}b^{1}-% a^{1}b^{3})\mathbf{e}_{31}+(a^{1}b^{2}-a^{2}b^{1})\mathbf{e}_{12}\ .
  13. 𝐚 𝐛 = i 𝐚 × 𝐛 , \mathbf{a}\ \wedge\ \mathbf{b}=\mathit{i}\ \mathbf{a}\ \times\ \mathbf{b}\ ,
  14. i 2 = - 1 . \mathit{i}^{2}=-1\ .

Pulse-position_modulation.html

  1. 2 M 2^{M}
  2. M / T M/T

Pulse-width_modulation.html

  1. y m i n y_{min}
  2. y m a x y_{max}
  3. f ( t ) f(t)
  4. T T
  5. y m i n y_{min}
  6. y m a x y_{max}
  7. y ¯ = 1 T 0 T f ( t ) d t . \bar{y}=\frac{1}{T}\int^{T}_{0}f(t)\,dt.
  8. f ( t ) f(t)
  9. y m a x y_{max}
  10. 0 < t < D T 0<t<D\cdot T
  11. y m i n y_{min}
  12. D T < t < T D\cdot T<t<T
  13. y ¯ = 1 T ( 0 D T y m a x d t + D T T y m i n d t ) \displaystyle\bar{y}=\frac{1}{T}\left(\int_{0}^{DT}y_{max}\,dt+\int_{DT}^{T}y_% {min}\,dt\right)
  14. y m i n = 0 y_{min}=0
  15. y ¯ = D y m a x \bar{y}=D\cdot y_{max}
  16. y ¯ \bar{y}
  17. sin x / x \sin x/x

Purchasing_power.html

  1. C t = C ( 1 + i ) - t = C ( 1 + i ) t C_{t}=C(1+i)^{-t}\,=\frac{C}{(1+i)^{t}}\,

Put_option.html

  1. 𝒪 \mathcal{O}
  2. Π \Pi
  3. 0
  4. T + T\in\mathbb{R}^{+}
  5. K K\in\mathbb{R}
  6. S : [ 0 , T ] S:[0,T]\to\mathbb{R}
  7. Π \Pi
  8. 𝒪 \mathcal{O}
  9. 𝒪 \mathcal{O}
  10. - S T + K -S_{T}+K
  11. 0
  12. K - S T 0 K-S_{T}\geq 0
  13. K - S T < 0 K-S_{T}<0
  14. ( K - S T ) 0 (K-S_{T})\vee 0
  15. ( K - S T ) + (K-S_{T})^{+}

Put–call_parity.html

  1. C - P = D ( F - K ) C-P=D(F-K)\,
  2. D F = S D\cdot F=S
  3. C - P = S - D K . C-P=S-D\cdot K.\,
  4. C + D K = P + S . C+D\cdot K=P+S.\,
  5. C ( t ) - P ( t ) = S ( t ) - K B ( t , T ) C(t)-P(t)=S(t)-K\cdot B(t,T)\,
  6. C ( t ) C(t)
  7. t t
  8. P ( t ) P(t)
  9. S ( t ) S(t)
  10. K K
  11. B ( t , T ) B(t,T)
  12. T . T.
  13. r r
  14. B ( t , T ) = e - r ( T - t ) . B(t,T)=e^{-r(T-t)}.\,
  15. r r
  16. r r
  17. r = l n ( 1 + i ) r=ln(1+i)
  18. i i
  19. C ( t ) - P ( t ) + D ( t ) = S ( t ) - K B ( t , T ) C(t)-P(t)+D(t)=S(t)-K\cdot B(t,T)\,
  20. C ( t ) - P ( t ) = S ( t ) - K B ( t , T ) - D ( t ) , C(t)-P(t)=S(t)-K\cdot B(t,T)\ -D(t),
  21. t t
  22. C ( t ) - P ( t ) = S ( t ) - K B ( t , T ) C(t)-P(t)=S(t)-K\cdot B(t,T)\,
  23. d d
  24. d d
  25. 1 - d 1-d

Pyroelectricity.html

  1. p i = P S , i T p_{i}=\frac{\partial P_{S,i}}{\partial T}

Pyrometer.html

  1. j = ε σ T 4 j^{\star}=\varepsilon\sigma T^{4}

Pythagorean_comma.html

  1. apotome - limma 113.69 - 90.23 23.46 cents \hbox{apotome}-\hbox{limma}\approx 113.69-90.23\approx 23.46~{}\hbox{cents}\!
  2. apotome limma = 3 7 / 2 11 2 8 / 3 5 = 3 12 2 19 = 531441 524288 = 1.0136432647705078125 \frac{\hbox{apotome}}{\hbox{limma}}=\frac{3^{7}/2^{11}}{2^{8}/3^{5}}=\frac{3^{% 12}}{2^{19}}=\frac{531441}{524288}=1.0136432647705078125\!
  3. twelve fifths seven octaves = ( 3 2 ) 12 / 2 7 = 3 12 2 19 = 531441 524288 = 1.0136432647705078125 \frac{\hbox{twelve fifths}}{\hbox{seven octaves}}=\left(\tfrac{3}{2}\right)^{1% 2}\!\!\Big/\,2^{7}=\frac{3^{12}}{2^{19}}=\frac{531441}{524288}=1.0136432647705% 078125\!
  4. 262144 ( 9 8 ) 6 = 531441 262144\cdot\left(\textstyle{\frac{9}{8}}\right)^{6}=531441
  5. 262144 ( 2 1 ) 1 = 524288 262144\cdot\left(\textstyle{\frac{2}{1}}\right)^{1}=524288

Pythagorean_expectation.html

  1. Win = runs scored 2 runs scored 2 + runs allowed 2 = 1 1 + ( runs allowed / runs scored ) 2 \mathrm{Win}=\frac{\,\text{runs scored}^{2}}{\,\text{runs scored}^{2}+\,\text{% runs allowed}^{2}}=\frac{1}{1+(\,\text{runs allowed}/\,\text{runs scored})^{2}}
  2. Win = 897 2 897 2 + 697 2 = 0.623525865 \mathrm{Win}=\frac{\,\text{897}^{2}}{\,\text{897}^{2}+\,\text{697}^{2}}=0.6235% 25865
  3. Win = runs scored 1.83 runs scored 1.83 + runs allowed 1.83 = 1 1 + ( runs allowed / runs scored ) 1.83 \mathrm{Win}=\frac{\,\text{runs scored}^{1.83}}{\,\text{runs scored}^{1.83}+\,% \text{runs allowed}^{1.83}}=\frac{1}{1+(\,\text{runs allowed}/\,\text{runs % scored})^{1.83}}
  4. Exponent = 1.50 log ( R + R A G ) + 0.45 \mathrm{Exponent}=1.50\cdot\log\left(\frac{R+RA}{G}\right)+0.45
  5. Exponent = ( R + R A G ) .287 \mathrm{Exponent}=\left(\frac{R+RA}{G}\right)^{.287}
  6. Win = points for 13.91 points for 13.91 + points against 13.91 . \mathrm{Win}=\frac{\,\text{points for}^{13.91}}{\,\text{points for}^{13.91}+\,% \text{points against}^{13.91}}.
  7. Pythagorean wins = Points For 2.37 Points For 2.37 + Points Against 2.37 × 16. \,\text{Pythagorean wins}=\frac{\,\text{Points For}^{2.37}}{\,\text{Points For% }^{2.37}+\,\text{Points Against}^{2.37}}\times 16.

Q_factor.html

  1. Δ f \Delta f
  2. f c / Δ f f_{c}/\Delta f
  3. Q = def f r Δ f = ω r Δ ω , Q\ \stackrel{\mathrm{def}}{=}\ \frac{f_{r}}{\Delta f}=\frac{\omega_{r}}{\Delta% \omega},\,
  4. Q = def 2 π × Energy Stored Energy dissipated per cycle = 2 π f r × Energy Stored Power Loss . Q\ \stackrel{\mathrm{def}}{=}\ 2\pi\times\frac{\,\text{Energy Stored}}{\,\text% {Energy dissipated per cycle}}=2\pi f_{r}\times\frac{\,\text{Energy Stored}}{% \,\text{Power Loss}}.\,
  5. Q ( ω ) = ω × Maximum Energy Stored Power Loss , Q(\omega)=\omega\times\frac{\,\text{Maximum Energy Stored}}{\,\text{Power Loss% }},\,
  6. Q = 1 / 2 Q=1/2
  7. Q = 1 / 2 Q=1/\sqrt{2}
  8. Q = 1 / 3 Q=1/\sqrt{3}
  9. 2 π 2\pi
  10. e - 2 π e^{-2\pi}
  11. Δ f = f 0 Q \Delta f=\frac{f_{0}}{Q}\,
  12. f 0 f_{0}
  13. Δ f \Delta f
  14. f 0 f_{0}
  15. ω 0 = 2 π f 0 \omega_{0}=2\pi f_{0}
  16. Q = 1 2 ζ = ω 0 2 α = τ ω 0 2 , Q=\frac{1}{2\zeta}={\omega_{0}\over 2\alpha}={\tau\omega_{0}\over 2},
  17. ζ = 1 2 Q = α ω 0 = 1 τ ω 0 . \zeta=\frac{1}{2Q}={\alpha\over\omega_{0}}={1\over\tau\omega_{0}}.
  18. e - α t e^{-\alpha t}
  19. e - t / τ e^{-t/\tau}
  20. α = ω 0 2 Q = ζ ω 0 = 1 τ \alpha={\omega_{0}\over 2Q}=\zeta\omega_{0}={1\over\tau}
  21. τ = 2 Q ω 0 = 1 ζ ω 0 = 1 α \tau={2Q\over\omega_{0}}={1\over\zeta\omega_{0}}={1\over\alpha}
  22. e - 2 α t e^{-2\alpha t}
  23. e - 2 t / τ e^{-2t/\tau}
  24. H ( s ) = ω 0 2 s 2 + ω 0 Q 2 ζ ω 0 = 2 α s + ω 0 2 H(s)=\frac{\omega_{0}^{2}}{s^{2}+\underbrace{\frac{\omega_{0}}{Q}}_{2\zeta% \omega_{0}=2\alpha}s+\omega_{0}^{2}}\,
  25. Q > 0.5 Q>0.5
  26. - α -\alpha
  27. α \alpha
  28. H ( s ) = ω 0 2 s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{\omega_{0}^{2}}{s^{2}+\frac{\omega_{0}}{Q}s+\omega_{0}^{2}}
  29. H ( s ) = ω 0 Q s s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{\frac{\omega_{0}}{Q}s}{s^{2}+\frac{\omega_{0}}{Q}s+\omega_{0}^{2}}
  30. H ( s ) = s 2 + ω 0 2 s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{s^{2}+\omega_{0}^{2}}{s^{2}+\frac{\omega_{0}}{Q}s+\omega_{0}^{2}}
  31. H ( s ) = s 2 s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{s^{2}}{s^{2}+\frac{\omega_{0}}{Q}s+\omega_{0}^{2}}
  32. Q = 1 R L C = ω 0 L R Q=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{\omega_{0}L}{R}
  33. R R
  34. L L
  35. C C
  36. Q = R C L = R ω 0 L = ω 0 R C Q=R\sqrt{\frac{C}{L}}=\frac{R}{\omega_{0}L}=\omega_{0}RC
  37. Q L = X L R L = ω 0 L R L Q_{L}=\frac{X_{L}}{R_{L}}=\frac{\omega_{0}L}{R_{L}}
  38. ω 0 \omega_{0}
  39. L L
  40. X L X_{L}
  41. R L R_{L}
  42. Q C = X C R C = 1 ω 0 C R C Q_{C}=\frac{X_{C}}{R_{C}}=\frac{1}{\omega_{0}CR_{C}}
  43. ω 0 \omega_{0}
  44. C C
  45. X C X_{C}
  46. R C R_{C}
  47. Q = 1 ( 1 / Q L + 1 / Q C ) Q=\frac{1}{(1/Q_{L}+1/Q_{C})}
  48. Q = M k D , Q=\frac{\sqrt{Mk}}{D},\,
  49. F damping = - D v F_{\,\text{damping}}=-Dv
  50. v v
  51. Q = 2 π f o P , Q=\frac{2\pi f_{o}\,\mathcal{E}}{P},\,
  52. f o f_{o}
  53. \mathcal{E}
  54. P = - d d t P=-\frac{d\mathcal{E}}{dt}

QR_decomposition.html

  1. A = Q R , A=QR,\,
  2. A = Q R = Q [ R 1 0 ] = [ Q 1 , Q 2 ] [ R 1 0 ] = Q 1 R 1 , A=QR=Q\begin{bmatrix}R_{1}\\ 0\end{bmatrix}=\begin{bmatrix}Q_{1},Q_{2}\end{bmatrix}\begin{bmatrix}R_{1}\\ 0\end{bmatrix}=Q_{1}R_{1},
  3. A = [ 𝐚 1 , , 𝐚 n ] A=[\mathbf{a}_{1},\cdots,\mathbf{a}_{n}]
  4. 𝐯 , 𝐰 = 𝐯 𝐰 \langle\mathbf{v},\mathbf{w}\rangle=\mathbf{v}^{\top}\mathbf{w}
  5. 𝐯 , 𝐰 = 𝐯 * 𝐰 \langle\mathbf{v},\mathbf{w}\rangle=\mathbf{v}^{*}\mathbf{w}
  6. proj 𝐞 𝐚 = 𝐞 , 𝐚 𝐞 , 𝐞 𝐞 \mathrm{proj}_{\mathbf{e}}\mathbf{a}=\frac{\left\langle\mathbf{e},\mathbf{a}% \right\rangle}{\left\langle\mathbf{e},\mathbf{e}\right\rangle}\mathbf{e}
  7. 𝐮 1 \displaystyle\mathbf{u}_{1}
  8. 𝐚 i \mathbf{a}_{i}
  9. 𝐞 i \mathbf{e}_{i}
  10. 𝐚 1 \displaystyle\mathbf{a}_{1}
  11. 𝐞 i , 𝐚 i = 𝐮 i \langle\mathbf{e}_{i},\mathbf{a}_{i}\rangle=\|\mathbf{u}_{i}\|
  12. A = Q R A=QR
  13. Q = [ 𝐞 1 , , 𝐞 n ] and R = ( 𝐞 1 , 𝐚 1 𝐞 1 , 𝐚 2 𝐞 1 , 𝐚 3 0 𝐞 2 , 𝐚 2 𝐞 2 , 𝐚 3 0 0 𝐞 3 , 𝐚 3 ) . Q=\left[\mathbf{e}_{1},\cdots,\mathbf{e}_{n}\right]\qquad\,\text{and}\qquad R=% \begin{pmatrix}\langle\mathbf{e}_{1},\mathbf{a}_{1}\rangle&\langle\mathbf{e}_{% 1},\mathbf{a}_{2}\rangle&\langle\mathbf{e}_{1},\mathbf{a}_{3}\rangle&\ldots\\ 0&\langle\mathbf{e}_{2},\mathbf{a}_{2}\rangle&\langle\mathbf{e}_{2},\mathbf{a}% _{3}\rangle&\ldots\\ 0&0&\langle\mathbf{e}_{3},\mathbf{a}_{3}\rangle&\ldots\\ \vdots&\vdots&\vdots&\ddots\end{pmatrix}.
  14. A = ( 12 - 51 4 6 167 - 68 - 4 24 - 41 ) . A=\begin{pmatrix}12&-51&4\\ 6&167&-68\\ -4&24&-41\end{pmatrix}.
  15. Q Q
  16. Q T Q = I . \begin{matrix}Q^{T}\,Q=I.\end{matrix}
  17. Q Q
  18. U = ( 𝐮 1 𝐮 2 𝐮 3 ) = ( 12 - 69 - 58 / 5 6 158 6 / 5 - 4 30 - 33 ) ; U=\begin{pmatrix}\mathbf{u}_{1}&\mathbf{u}_{2}&\mathbf{u}_{3}\end{pmatrix}=% \begin{pmatrix}12&-69&-58/5\\ 6&158&6/5\\ -4&30&-33\end{pmatrix};
  19. Q = ( 𝐮 1 𝐮 1 𝐮 2 𝐮 2 𝐮 3 𝐮 3 ) = ( 6 / 7 - 69 / 175 - 58 / 175 3 / 7 158 / 175 6 / 175 - 2 / 7 6 / 35 - 33 / 35 ) . Q=\begin{pmatrix}\frac{\mathbf{u}_{1}}{\|\mathbf{u}_{1}\|}&\frac{\mathbf{u}_{2% }}{\|\mathbf{u}_{2}\|}&\frac{\mathbf{u}_{3}}{\|\mathbf{u}_{3}\|}\end{pmatrix}=% \begin{pmatrix}6/7&-69/175&-58/175\\ 3/7&158/175&6/175\\ -2/7&6/35&-33/35\end{pmatrix}.
  20. Q T A = Q T Q R = R ; \begin{matrix}Q^{T}A=Q^{T}Q\,R=R;\end{matrix}
  21. R = Q T A = ( 14 21 - 14 0 175 - 70 0 0 35 ) . \begin{matrix}R=Q^{T}A=\end{matrix}\begin{pmatrix}14&21&-14\\ 0&175&-70\\ 0&0&35\end{pmatrix}.
  22. x x
  23. e 1 e_{1}
  24. x x
  25. e 1 e_{1}
  26. x x
  27. e 1 e_{1}
  28. A A
  29. 𝐱 \mathbf{x}
  30. A A
  31. 𝐱 = | α | \|\mathbf{x}\|=|\alpha|
  32. 𝐱 \mathbf{x}
  33. x k x_{k}
  34. α = - e i arg x k 𝐱 \alpha=-\mathrm{e}^{\mathrm{i}\arg x_{k}}\|\mathbf{x}\|
  35. 𝐞 1 \mathbf{e}_{1}
  36. I I
  37. 𝐮 = 𝐱 - α 𝐞 1 , \mathbf{u}=\mathbf{x}-\alpha\mathbf{e}_{1},
  38. 𝐯 = 𝐮 𝐮 , \mathbf{v}={\mathbf{u}\over\|\mathbf{u}\|},
  39. Q = I - 2 𝐯𝐯 T . Q=I-2\mathbf{v}\mathbf{v}^{T}.
  40. A A
  41. Q = I - ( 1 + w ) 𝐯𝐯 H Q=I-(1+w)\mathbf{v}\mathbf{v}^{H}
  42. w = 𝐱 H 𝐯 / 𝐯 H 𝐱 w=\mathbf{x}^{H}\mathbf{v}\mathbf{/}\mathbf{v}^{H}\mathbf{x}
  43. 𝐱 H \mathbf{x}^{H}
  44. 𝐱 \mathbf{x}
  45. Q Q
  46. Q 𝐱 = ( α , 0 , , 0 ) T . Q\mathbf{x}=(\alpha,0,\cdots,0)^{T}.\,
  47. Q 1 A = [ α 1 0 A 0 ] Q_{1}A=\begin{bmatrix}\alpha_{1}&\star&\dots&\star\\ 0&&&\\ \vdots&&A^{\prime}&\\ 0&&&\end{bmatrix}
  48. Q k = ( I k - 1 0 0 Q k ) . Q_{k}=\begin{pmatrix}I_{k-1}&0\\ 0&Q_{k}^{\prime}\end{pmatrix}.
  49. t t
  50. t = min ( m - 1 , n ) t=\min(m-1,n)
  51. R = Q t Q 2 Q 1 A R=Q_{t}\cdots Q_{2}Q_{1}A
  52. Q = Q 1 T Q 2 T Q t T , Q=Q_{1}^{T}Q_{2}^{T}\cdots Q_{t}^{T},
  53. A = Q R A=QR
  54. A A
  55. 2 ( n - k + 1 ) 2 2(n-k+1)^{2}
  56. ( n - k + 1 ) 2 + ( n - k + 1 ) ( n - k ) + 2 (n-k+1)^{2}+(n-k+1)(n-k)+2
  57. 1 1
  58. 1 1
  59. 2 3 n 3 + n 2 + 1 3 n - 2 = O ( n 3 ) . \frac{2}{3}n^{3}+n^{2}+\frac{1}{3}n-2=O(n^{3}).
  60. A = ( 12 - 51 4 6 167 - 68 - 4 24 - 41 ) . A=\begin{pmatrix}12&-51&4\\ 6&167&-68\\ -4&24&-41\end{pmatrix}.
  61. 𝐚 1 = ( 12 , 6 , - 4 ) T \mathbf{a}_{1}=(12,6,-4)^{T}
  62. 𝐚 1 e 1 = ( 14 , 0 , 0 ) T . \|\mathbf{a}_{1}\|\;\mathrm{e}_{1}=(14,0,0)^{T}.
  63. 𝐮 = 𝐱 + α 𝐞 1 , \mathbf{u}=\mathbf{x}+\alpha\mathbf{e}_{1},
  64. 𝐯 = 𝐮 𝐮 . \mathbf{v}={\mathbf{u}\over\|\mathbf{u}\|}.
  65. α = - 14 \alpha=-14
  66. 𝐱 = 𝐚 1 = ( 12 , 6 , - 4 ) T \mathbf{x}=\mathbf{a}_{1}=(12,6,-4)^{T}
  67. 𝐮 = ( - 2 , 6 , - 4 ) T = ( 2 ) ( - 1 , 3 , - 2 ) T \mathbf{u}=(-2,6,-4)^{T}=({2})(-1,3,-2)^{T}
  68. 𝐯 = 1 14 ( - 1 , 3 , - 2 ) T \mathbf{v}={1\over\sqrt{14}}(-1,3,-2)^{T}
  69. Q 1 = I - 2 14 14 ( - 1 3 - 2 ) ( - 1 3 - 2 ) Q_{1}=I-{2\over\sqrt{14}\sqrt{14}}\begin{pmatrix}-1\\ 3\\ -2\end{pmatrix}\begin{pmatrix}-1&3&-2\end{pmatrix}
  70. = I - 1 7 ( 1 - 3 2 - 3 9 - 6 2 - 6 4 ) =I-{1\over 7}\begin{pmatrix}1&-3&2\\ -3&9&-6\\ 2&-6&4\end{pmatrix}
  71. = ( 6 / 7 3 / 7 - 2 / 7 3 / 7 - 2 / 7 6 / 7 - 2 / 7 6 / 7 3 / 7 ) . =\begin{pmatrix}6/7&3/7&-2/7\\ 3/7&-2/7&6/7\\ -2/7&6/7&3/7\\ \end{pmatrix}.
  72. Q 1 A = ( 14 21 - 14 0 - 49 - 14 0 168 - 77 ) , Q_{1}A=\begin{pmatrix}14&21&-14\\ 0&-49&-14\\ 0&168&-77\end{pmatrix},
  73. A = M 11 = ( - 49 - 14 168 - 77 ) . A^{\prime}=M_{11}=\begin{pmatrix}-49&-14\\ 168&-77\end{pmatrix}.
  74. Q 2 = ( 1 0 0 0 - 7 / 25 24 / 25 0 24 / 25 7 / 25 ) Q_{2}=\begin{pmatrix}1&0&0\\ 0&-7/25&24/25\\ 0&24/25&7/25\end{pmatrix}
  75. Q = Q 1 T Q 2 T = ( 6 / 7 - 69 / 175 58 / 175 3 / 7 158 / 175 - 6 / 175 - 2 / 7 6 / 35 33 / 35 ) Q=Q_{1}^{T}Q_{2}^{T}=\begin{pmatrix}6/7&-69/175&58/175\\ 3/7&158/175&-6/175\\ -2/7&6/35&33/35\end{pmatrix}
  76. Q = Q 1 T Q 2 T = ( 0.8571 - 0.3943 0.3314 0.4286 0.9029 - 0.0343 - 0.2857 0.1714 0.9429 ) Q=Q_{1}^{T}Q_{2}^{T}=\begin{pmatrix}0.8571&-0.3943&0.3314\\ 0.4286&0.9029&-0.0343\\ -0.2857&0.1714&0.9429\end{pmatrix}
  77. R = Q 2 Q 1 A = Q T A = ( 14 21 - 14 0 175 - 70 0 0 - 35 ) . R=Q_{2}Q_{1}A=Q^{T}A=\begin{pmatrix}14&21&-14\\ 0&175&-70\\ 0&0&-35\end{pmatrix}.
  78. A = ( 12 - 51 4 6 167 - 68 - 4 24 - 41 ) . A=\begin{pmatrix}12&-51&4\\ 6&167&-68\\ -4&24&-41\end{pmatrix}.
  79. 𝐚 31 = - 4 \mathbf{a}_{31}=-4
  80. G 1 G_{1}
  81. ( 6 , - 4 ) (6,-4)
  82. θ = arctan ( - ( - 4 ) 6 ) \theta=\arctan\left({-(-4)\over 6}\right)
  83. G 1 G_{1}
  84. G 1 = ( 1 0 0 0 cos ( θ ) - sin ( θ ) 0 sin ( θ ) cos ( θ ) ) G_{1}=\begin{pmatrix}1&0&0\\ 0&\cos(\theta)&-\sin(\theta)\\ 0&\sin(\theta)&\cos(\theta)\end{pmatrix}
  85. ( 1 0 0 0 0.83205 - 0.55470 0 0.55470 0.83205 ) \approx\begin{pmatrix}1&0&0\\ 0&0.83205&-0.55470\\ 0&0.55470&0.83205\end{pmatrix}
  86. G 1 A G_{1}A
  87. 𝐚 31 \mathbf{a}_{31}
  88. G 1 A ( 12 - 51 4 7.21110 125.6396 - 33.83671 0 112.6041 - 71.83368 ) G_{1}A\approx\begin{pmatrix}12&-51&4\\ 7.21110&125.6396&-33.83671\\ 0&112.6041&-71.83368\end{pmatrix}
  89. G 2 G_{2}
  90. G 3 G_{3}
  91. a 21 a_{21}
  92. a 32 a_{32}
  93. R R
  94. Q T Q^{T}
  95. Q T = G 3 G 2 G 1 Q^{T}=G_{3}G_{2}G_{1}
  96. G 3 G 2 G 1 A = Q T A = R G_{3}G_{2}G_{1}A=Q^{T}A=R
  97. A = Q R A=QR
  98. A = Q R A=QR
  99. det ( A ) = det ( Q ) det ( R ) . \det(A)=\det(Q)\cdot\det(R).
  100. | det ( Q ) | = 1 |\det(Q)|=1
  101. | det ( A ) | = | det ( R ) | = | i r i i | , |\det(A)|=|\det(R)|=\Big|\prod_{i}r_{ii}\Big|,
  102. r i i r_{ii}
  103. | i r i i | = | i λ i | , \Big|\prod_{i}r_{ii}\Big|=\Big|\prod_{i}\lambda_{i}\Big|,
  104. λ i \lambda_{i}
  105. A A
  106. A A
  107. A = Q ( R O ) , Q * Q = I , A=Q\begin{pmatrix}R\\ O\end{pmatrix},\qquad Q^{*}Q=I,
  108. O O
  109. Q Q
  110. | i r i i | = i σ i , \Big|\prod_{i}r_{ii}\Big|=\prod_{i}\sigma_{i},
  111. σ i \sigma_{i}
  112. A A
  113. A A
  114. R R
  115. i σ i = | i λ i | . {\prod_{i}\sigma_{i}}=\Big|{\prod_{i}\lambda_{i}}\Big|.
  116. A P = Q R A = Q R P T AP=QR\quad\iff A=QRP^{T}
  117. | r 11 | | r 22 | | r n n | |r_{11}|\geq|r_{22}|\geq\ldots\geq|r_{nn}|
  118. m < n m<n
  119. A x = b Ax=b
  120. m × n m\times n
  121. m m
  122. A T = Q R A^{T}=QR
  123. Q T = Q - 1 Q^{T}=Q^{-1}
  124. R = [ R 1 0 ] R=\begin{bmatrix}R_{1}\\ 0\end{bmatrix}
  125. R 1 R_{1}
  126. m × m m\times m
  127. ( n - m ) × m (n-m)\times m
  128. x = Q [ ( R 1 T ) - 1 b 0 ] x=Q\begin{bmatrix}(R_{1}^{T})^{-1}b\\ 0\end{bmatrix}
  129. R 1 - 1 R_{1}^{-1}
  130. ( R 1 T ) - 1 b (R_{1}^{T})^{-1}b
  131. x ^ \hat{x}
  132. m n m\geq n
  133. A x = b Ax=b
  134. A x ^ - b \|A\hat{x}-b\|
  135. A = Q R A=QR
  136. x ^ = R 1 - 1 ( Q 1 T b ) \hat{x}=R_{1}^{-1}(Q_{1}^{T}b)
  137. Q 1 Q_{1}
  138. m × n m\times n
  139. n n
  140. Q Q
  141. R 1 R_{1}
  142. x ^ \hat{x}
  143. R 1 R_{1}
  144. Q 1 Q_{1}
  145. R 1 R_{1}

Quadratic_form.html

  1. 4 x 2 + 2 x y - 3 y 2 4x^{2}+2xy-3y^{2}
  2. q ( x ) = a x 2 (unary) q(x)=ax^{2}\quad\textrm{(unary)}
  3. q ( x , y ) = a x 2 + b x y + c y 2 (binary) q(x,y)=ax^{2}+bxy+cy^{2}\quad\textrm{(binary)}
  4. q ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z (ternary) q(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz\quad\textrm{(ternary)}
  5. q ( x , y , z ) = d ( ( x , y , z ) , ( 0 , 0 , 0 ) ) 2 = ( x , y , z ) 2 = x 2 + y 2 + z 2 . q(x,y,z)=d((x,y,z),(0,0,0))^{2}=\|(x,y,z)\|^{2}=x^{2}+y^{2}+z^{2}.
  6. q A ( x 1 , , x n ) = i = 1 n j = 1 n a i j x i x j . q_{A}(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_{i}}{x_{j}}.
  7. λ 1 x ~ 1 2 + λ 2 x ~ 2 2 + + λ n x ~ n 2 , \lambda_{1}\tilde{x}_{1}^{2}+\lambda_{2}\tilde{x}_{2}^{2}+\cdots+\lambda_{n}% \tilde{x}_{n}^{2},
  8. ( - 1 ) n - . (-1)^{n_{-}}.
  9. q ( v ) = x T A x , q(v)=x^{\mathrm{T}}Ax,
  10. A B = S A S T . A\to B=SAS^{\mathrm{T}}.
  11. B = ( λ 1 0 0 0 λ 2 0 0 0 0 λ n ) B=\begin{pmatrix}\lambda_{1}&0&\cdots&0\\ 0&\lambda_{2}&\cdots&0\\ \vdots&\vdots&\ddots&0\\ 0&0&\cdots&\lambda_{n}\end{pmatrix}
  12. q ( x 1 , , x n ) = i = 1 n j = 1 n a i j x i x j , a i j K . q(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}{x_{i}}{x_{j}},\quad a% _{ij}\in K.
  13. q ( x ) = x T A x . q(x)=x^{\mathrm{T}}Ax.
  14. ψ ( x ) = φ ( C x ) . \psi(x)=\varphi(Cx).
  15. B = C T A C . B=C^{\mathrm{T}}AC.
  16. b q ( x , y ) = 1 2 ( q ( x + y ) - q ( x ) - q ( y ) ) = x T A y = y T A x . b_{q}(x,y)=\tfrac{1}{2}(q(x+y)-q(x)-q(y))=x^{\mathrm{T}}Ay=y^{\mathrm{T}}Ax.
  17. q ( x ) = b ( x , x ) q(x)=b(x,x)
  18. Q ( v ) = q ( v ) , v = [ v 1 , , v n ] T K n . Q(v)=q(v),\quad v=[v_{1},\ldots,v_{n}]^{\mathrm{T}}\in K^{n}.
  19. Q ( a v ) = a 2 Q ( v ) . Q(av)=a^{2}Q(v).
  20. B ( v , w ) = 1 2 ( Q ( v + w ) - Q ( v ) - Q ( w ) ) . B(v,w)=\tfrac{1}{2}(Q(v+w)-Q(v)-Q(w)).
  21. Q ( v ) = Q ( T v ) for all v V . Q(v)=Q^{\prime}(Tv)\,\text{ for all }v\in V.
  22. q ( x ) = a 1 x 1 2 + a 2 x 2 2 + + a n x n 2 . q(x)=a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+\ldots+a_{n}x_{n}^{2}.
  23. a 1 , , a n . \langle a_{1},\dots,a_{n}\rangle.
  24. 𝐱 = ( x , y , z ) T \mathbf{x}=(x,y,z)\text{T}
  25. A A
  26. 𝐱 T A 𝐱 + 𝐛 T 𝐱 = 1 \mathbf{x}\text{T}A\mathbf{x}+\mathbf{b}\text{T}\mathbf{x}=1
  27. A A
  28. A A
  29. λ i = 0 \lambda_{i}=0
  30. b i b_{i}
  31. b i 0 b_{i}\neq 0
  32. b i = 0 b_{i}=0
  33. i i
  34. 𝕓 \mathbb{b}
  35. a x 2 + 2 b x y + c y 2 ax^{2}+2bxy+cy^{2}
  36. ( a b b c ) \begin{pmatrix}a&b\\ b&c\end{pmatrix}
  37. a x 2 + b x y + c y 2 ax^{2}+bxy+cy^{2}
  38. ( a b / 2 b / 2 c ) . \begin{pmatrix}a&b/2\\ b/2&c\end{pmatrix}.
  39. w 2 + x 2 + y 2 + z 2 w^{2}+x^{2}+y^{2}+z^{2}
  40. a w 2 + b x 2 + c y 2 + d z 2 aw^{2}+bx^{2}+cy^{2}+dz^{2}

Quadratic_function.html

  1. f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,
  2. x 2 - x - 2 x^{2}-x-2\!
  3. f ( x ) = a x 2 + b x + c , a 0 f(x)=ax^{2}+bx+c,\quad a\neq 0
  4. y y
  5. f ( x , y ) = a x 2 + b y 2 + c x y + d x + e y + f f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!
  6. x x
  7. 4 4
  8. a x 2 + b x + c , ax^{2}+bx+c,\,\!
  9. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  10. f ( x , y ) = a x 2 + b y 2 + c x y + d x + e y + f , f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f,\,\!
  11. f ( x ) = a x 2 + b x + c f(x)=ax^{2}+bx+c\,\!
  12. f ( x ) = a ( x - x 1 ) ( x - x 2 ) f(x)=a(x-x_{1})(x-x_{2})\,\!
  13. f ( x ) = a ( x - h ) 2 + k f(x)=a(x-h)^{2}+k\,\!
  14. h h
  15. k k
  16. x x
  17. y y
  18. a > 0 a>0
  19. a Align l t ; 0 a&lt;0
  20. a a
  21. a a
  22. b b
  23. a a
  24. x x
  25. x = - b 2 a x=-\frac{b}{2a}
  26. b b
  27. y y
  28. c c
  29. y y
  30. ( h , k ) (h,k)
  31. f ( x ) = a x 2 + b x + c f(x)=ax^{2}+bx+c\,\!
  32. f ( x ) = a ( x + b 2 a ) 2 - b 2 - 4 a c 4 a , f(x)=a\left(x+\frac{b}{2a}\right)^{2}-\frac{b^{2}-4ac}{4a},
  33. ( - b 2 a , - Δ 4 a ) . \left(-\frac{b}{2a},-\frac{\Delta}{4a}\right).
  34. f ( x ) = a ( x - r 1 ) ( x - r 2 ) f(x)=a(x-r_{1})(x-r_{2})\,\!
  35. r 1 + r 2 2 \frac{r_{1}+r_{2}}{2}\,\!
  36. x x
  37. ( r 1 + r 2 2 , f ( r 1 + r 2 2 ) ) . \left(\frac{r_{1}+r_{2}}{2},f\left(\frac{r_{1}+r_{2}}{2}\right)\right).\!
  38. a Align l t ; 0 a&lt;0
  39. a > 0 a>0
  40. x = h = - b 2 a x=h=-\frac{b}{2a}
  41. f ( x ) = a x 2 + b x + c f ( x ) = 2 a x + b , f(x)=ax^{2}+bx+c\quad\Rightarrow\quad f^{\prime}(x)=2ax+b\,\!,
  42. x = - b 2 a x=-\frac{b}{2a}
  43. f ( x ) = a ( - b 2 a ) 2 + b ( - b 2 a ) + c = - ( b 2 - 4 a c ) 4 a = - Δ 4 a , f(x)=a\left(-\frac{b}{2a}\right)^{2}+b\left(-\frac{b}{2a}\right)+c=-\frac{(b^{% 2}-4ac)}{4a}=-\frac{\Delta}{4a}\,\!,
  44. ( - b 2 a , - Δ 4 a ) . \left(-\frac{b}{2a},-\frac{\Delta}{4a}\right).
  45. f ( x ) = a x 2 + b x + c f(x)=ax^{2}+bx+c\,
  46. x x
  47. f ( x ) = 0 f(x)=0
  48. a a
  49. b b
  50. c c
  51. x = - b ± Δ 2 a , x=\frac{-b\pm\sqrt{\Delta}}{2a},
  52. Δ = b 2 - 4 a c . \Delta=b^{2}-4ac\,.
  53. a x 2 + b x + c ax^{2}+bx+c\,
  54. max ( | a | , | b | , | c | ) | a | × ϕ , \frac{\max(|a|,|b|,|c|)}{|a|}\times\phi,\,
  55. ϕ \phi
  56. 1 + 5 2 . \frac{1+\sqrt{5}}{2}.
  57. a > 0 a>0\,\!
  58. y = ± a x 2 + b x + c y=\pm\sqrt{ax^{2}+bx+c}
  59. y p = a x 2 + b x + c y_{p}=ax^{2}+bx+c\,\!
  60. a < 0 a<0\,\!
  61. y = ± a x 2 + b x + c y=\pm\sqrt{ax^{2}+bx+c}
  62. y p = a x 2 + b x + c y_{p}=ax^{2}+bx+c\,\!
  63. f ( x ) = a x 2 + b x + c f(x)=ax^{2}+bx+c
  64. f ( n ) ( x ) f^{(n)}(x)
  65. f ( x ) f(x)
  66. f ( x ) f(x)
  67. f ( x ) = a ( x - x 0 ) 2 + x 0 f(x)=a(x-x_{0})^{2}+x_{0}
  68. f ( x ) = a ( x - x 0 ) 2 + x 0 = h ( - 1 ) ( g ( h ( x ) ) ) , f(x)=a(x-x_{0})^{2}+x_{0}=h^{(-1)}(g(h(x))),\,\!
  69. g ( x ) = a x 2 g(x)=ax^{2}\,\!
  70. h ( x ) = x - x 0 . h(x)=x-x_{0}.\,\!
  71. f ( n ) ( x ) = h ( - 1 ) ( g ( n ) ( h ( x ) ) ) f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))\,\!
  72. g ( n ) ( x ) g^{(n)}(x)
  73. g ( n ) ( x ) = a 2 n - 1 x 2 n . g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.\,\!
  74. f ( n ) ( x ) = a 2 n - 1 ( x - x 0 ) 2 n + x 0 f^{(n)}(x)=a^{2^{n}-1}(x-x_{0})^{2^{n}}+x_{0}\,\!
  75. x n + 1 = r x n ( 1 - x n ) , 0 x 0 < 1 x_{n+1}=rx_{n}(1-x_{n}),\quad 0\leq x_{0}<1
  76. θ \theta
  77. θ = 1 π sin - 1 ( x 0 1 / 2 ) \theta=\tfrac{1}{\pi}\sin^{-1}(x_{0}^{1/2})
  78. θ \theta
  79. x n x_{n}
  80. θ \theta
  81. θ \theta
  82. x n x_{n}
  83. x n = 1 2 - 1 2 ( 1 - 2 x 0 ) 2 n x_{n}=\frac{1}{2}-\frac{1}{2}(1-2x_{0})^{2^{n}}
  84. x 0 [ 0 , 1 ) x_{0}\in[0,1)
  85. ( 1 - 2 x 0 ) ( - 1 , 1 ) (1-2x_{0})\in(-1,1)
  86. x 0 x_{0}
  87. ( 1 - 2 x 0 ) 2 n (1-2x_{0})^{2^{n}}
  88. x n x_{n}
  89. 1 2 . \tfrac{1}{2}.
  90. f ( x , y ) = A x 2 + B y 2 + C x + D y + E x y + F f(x,y)=Ax^{2}+By^{2}+Cx+Dy+Exy+F\,\!
  91. f ( x , y ) f(x,y)\,\!
  92. z = 0 z=0\,\!
  93. 4 A B - E 2 < 0 4AB-E^{2}<0\,
  94. 4 A B - E 2 > 0 4AB-E^{2}>0\,
  95. x m = - 2 B C - D E 4 A B - E 2 , x_{m}=-\frac{2BC-DE}{4AB-E^{2}},
  96. y m = - 2 A D - C E 4 A B - E 2 . y_{m}=-\frac{2AD-CE}{4AB-E^{2}}.
  97. 4 A B - E 2 = 0 4AB-E^{2}=0\,
  98. D E - 2 C B = 2 A D - C E 0 DE-2CB=2AD-CE\neq 0\,
  99. 4 A B - E 2 = 0 4AB-E^{2}=0\,
  100. D E - 2 C B = 2 A D - C E = 0 DE-2CB=2AD-CE=0\,

Quadratic_irrational.html

  1. a + b c d {a+b\sqrt{c}\over d}
  2. d a + b c = a d - b d c a 2 - b 2 c . {d\over a+b\sqrt{c}}={ad-bd\sqrt{c}\over a^{2}-b^{2}c}.\,
  3. 3 = 1.732 = [ 1 ; 1 , 2 , 1 , 2 , 1 , 2 , ] \sqrt{3}=1.732\ldots=[1;1,2,1,2,1,2,\ldots]
  4. - b ± b 2 - 4 a c 2 a . \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.

Quadrature_mirror_filter.html

  1. π / 2 \pi/2
  2. H 1 ( z ) H_{1}(z)
  3. H 0 ( z ) H_{0}(z)
  4. H 1 ( z ) = H 0 ( - z ) H_{1}(z)=H_{0}(-z)
  5. Ω = π / 2 \Omega=\pi/2
  6. | H 1 ( e j Ω ) | = | H 0 ( e j ( π - Ω ) ) | |H_{1}(e^{j\Omega})|=|H_{0}(e^{j(\pi-\Omega)})|
  7. | H 0 ( e j Ω ) | 2 + | H 1 ( e j Ω ) | 2 = 1 |H_{0}(e^{j\Omega})|^{2}+|H_{1}(e^{j\Omega})|^{2}=1
  8. Ω \Omega
  9. 2 π 2\pi
  10. N N
  11. x n x_{n}
  12. y n = i = 0 M - 1 b i x n - i y_{n}=\sum_{i=0}^{M-1}b_{i}x_{n-i}
  13. m = 4 m=4
  14. b 0 1 + b 1 1 + b 2 1 + b 3 1 = 0 b_{0}\centerdot 1+b_{1}\centerdot 1+b_{2}\centerdot 1+b_{3}\centerdot 1=0
  15. b 0 0 + b 1 1 + b 2 2 + b 3 3 = 0 b_{0}\centerdot 0+b_{1}\centerdot 1+b_{2}\centerdot 2+b_{3}\centerdot 3=0
  16. x = α n + β x=\alpha n+\beta
  17. z n = i = 0 M - 1 c i x n - i z_{n}=\sum_{i=0}^{M-1}c_{i}x_{n-i}
  18. i = 0 M - 1 c i b i = 0 \sum_{i=0}^{M-1}c_{i}b_{i}=0

Quadric.html

  1. i , j = 1 D + 1 x i Q i j x j + i = 1 D + 1 P i x i + R = 0 \sum_{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\sum_{i=1}^{D+1}P_{i}x_{i}+R=0
  2. x Q x T + P x T + R = 0 xQx^{\mathrm{T}}+Px^{\mathrm{T}}+R=0\,
  3. x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}+{z^{2}\over c^{2}}=1\,
  4. x 2 a 2 + y 2 a 2 + z 2 b 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over a^{2}}+{z^{2}\over b^{2}}=1\,
  5. x 2 a 2 + y 2 a 2 + z 2 a 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over a^{2}}+{z^{2}\over a^{2}}=1\,
  6. x 2 a 2 + y 2 b 2 - z = 0 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-z=0\,
  7. x 2 a 2 + y 2 a 2 - z = 0 {x^{2}\over a^{2}}+{y^{2}\over a^{2}}-z=0\,
  8. x 2 a 2 - y 2 b 2 - z = 0 {x^{2}\over a^{2}}-{y^{2}\over b^{2}}-z=0\,
  9. x 2 a 2 + y 2 b 2 - z 2 c 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-{z^{2}\over c^{2}}=1\,
  10. x 2 a 2 + y 2 b 2 - z 2 c 2 = - 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-{z^{2}\over c^{2}}=-1\,
  11. x 2 a 2 + y 2 b 2 - z 2 c 2 = 0 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}-{z^{2}\over c^{2}}=0\,
  12. x 2 a 2 + y 2 a 2 - z 2 b 2 = 0 {x^{2}\over a^{2}}+{y^{2}\over a^{2}}-{z^{2}\over b^{2}}=0\,
  13. x 2 a 2 + y 2 b 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over b^{2}}=1\,
  14. x 2 a 2 + y 2 a 2 = 1 {x^{2}\over a^{2}}+{y^{2}\over a^{2}}=1\,
  15. x 2 a 2 - y 2 b 2 = 1 {x^{2}\over a^{2}}-{y^{2}\over b^{2}}=1\,
  16. x 2 + 2 a y = 0 x^{2}+2ay=0\,
  17. ( x 1 , , x D + 1 ) (x_{1},\dots,x_{D+1})
  18. [ X 0 , , X D + 1 ] [X_{0},\dots,X_{D+1}]
  19. x i = X i / X 0 x_{i}=X_{i}/X_{0}
  20. Q ( X ) = i j a i j X i X j = 0 Q(X)=\sum_{ij}a_{ij}X_{i}X_{j}=0\,
  21. Q ( X ) = ± X 0 2 ± X 1 2 ± ± X D + 1 2 Q(X)=\pm X_{0}^{2}\pm X_{1}^{2}\pm\cdots\pm X_{D+1}^{2}
  22. Q ( X ) = { X 0 2 + X 1 2 + X 2 2 + X 3 2 X 0 2 + X 1 2 + X 2 2 - X 3 2 X 0 2 + X 1 2 - X 2 2 - X 3 2 Q(X)=\begin{cases}X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\\ X_{0}^{2}+X_{1}^{2}+X_{2}^{2}-X_{3}^{2}\\ X_{0}^{2}+X_{1}^{2}-X_{2}^{2}-X_{3}^{2}\end{cases}
  23. X 0 2 - X 1 2 - X 2 2 = 0. X_{0}^{2}-X_{1}^{2}-X_{2}^{2}=0.\,
  24. f ( x ) = k g ( ( x - μ ) Σ - 1 ( x - μ ) T ) f(x)=k\cdot g((x-\mu)\Sigma^{-1}(x-\mu)^{T})
  25. k k
  26. x x
  27. n n
  28. μ \mu
  29. Σ \Sigma
  30. g g
  31. g ( z ) = e - z / 2 g(z)=e^{-z/2}
  32. z z

Quantitative_genetics.html

  1. Phenotypic correlation = cov ( P 1 , P 2 ) V P 1 * V P 2 \mbox{Phenotypic correlation}~{}=\frac{\mathrm{cov}(P_{1},P_{2})}{\sqrt{{V_{P_% {1}}*V_{P_{2}}}}}
  2. Genotypic correlation = cov ( G 1 , G 2 ) V G 1 * V G 2 \mbox{Genotypic correlation}~{}=\frac{\mathrm{cov}(G_{1},G_{2})}{\sqrt{{V_{G_{% 1}}*V_{G_{2}}}}}
  3. Environmental correlation = cov ( E 1 , E 2 ) V E 1 * V E 2 \mbox{Environmental correlation}~{}=\frac{\mathrm{cov}(E_{1},E_{2})}{\sqrt{{V_% {E_{1}}*V_{E_{2}}}}}
  4. Genic correlation = cov ( A 1 , A 2 ) V A 1 * V A 2 \mbox{Genic correlation}~{}=\frac{\mathrm{cov}(A_{1},A_{2})}{\sqrt{{V_{A_{1}}*% V_{A_{2}}}}}

Quantum_chaos.html

  1. ϵ = - 3.0 \epsilon=-3.0
  2. H = H s + ε H n s , H=H_{s}+\varepsilon H_{ns},\,
  3. H s H_{s}
  4. H n s H_{ns}
  5. H s H_{s}
  6. ϵ \epsilon
  7. N 3 N^{3}
  8. N N
  9. P ( s ) = e - s . P(s)=e^{-s}.
  10. s s
  11. P ( s ) = π 2 s e - π s 2 / 4 . P(s)=\frac{\pi}{2}se^{-\pi s^{2}/4}.
  12. g c ( E ) = k T k n = 1 1 2 sinh ( χ n k / 2 ) e i ( n S k - α n k π / 2 ) . g_{c}(E)=\sum_{k}T_{k}\sum_{n=1}^{\infty}\frac{1}{2\sinh{(\chi_{nk}/2)}}\,e^{i% (nS_{k}-\alpha_{nk}\pi/2)}.
  13. k k
  14. T k T_{k}
  15. S k S_{k}
  16. n S k nS_{k}
  17. n n
  18. n n
  19. α n k \alpha_{nk}
  20. 1 / sinh ( χ n k / 2 ) 1/\sinh{(\chi_{nk}/2)}
  21. χ n k \chi_{nk}
  22. sinh ( χ n k / 2 ) \sinh{(\chi_{nk}/2)}
  23. sin ( χ n k / 2 ) \sin{(\chi_{nk}/2)}
  24. χ n k \chi_{nk}
  25. χ n k = 2 π m \chi_{nk}=2\pi m
  26. m m
  27. sin ( χ n k / 2 ) = 0 \sin{(\chi_{nk}/2)}=0
  28. 1 / r 1/r
  29. n = 6 n=6
  30. d ( E ) = - 1 π I m ( T r ( G ( x , x , E ) ) d(E)=-\frac{1}{\pi}Im(Tr(G(x,x^{\prime},E))
  31. ϵ \epsilon
  32. f ( w ) = k n = 1 D 𝑛𝑘 i sin ( 2 π n w S k ~ - ϕ 𝑛𝑘 ) . f(w)=\sum_{k}\sum_{n=1}^{\infty}D^{i}_{\it nk}\sin(2\pi nw\tilde{S_{k}}-\phi_{% \it nk}).
  33. ϕ 𝑛𝑘 \phi_{\it nk}
  34. D 𝑛𝑘 i D^{i}_{\it nk}
  35. i i
  36. ϵ \epsilon
  37. w w
  38. D 𝑛𝑘 i D^{i}_{\it nk}
  39. H - H^{-}
  40. y ( 0 ) = 0 y(0)=0
  41. d 1 / 2 d x 1 / 2 V - 1 ( x ) = 2 π d N ( x ) d x \frac{d^{1/2}}{dx^{1/2}}V^{-1}(x)=2\sqrt{\pi}\frac{dN(x)}{dx}
  42. d N ( x ) d x \frac{dN(x)}{dx}
  43. H ( x , p ; R ) H(x,p;R)
  44. R R
  45. H ( x , p ; R ( t ) ) H(x,p;R(t))

Quantum_decoherence.html

  1. ψ ( x 1 , x 2 , , x N ) \psi(x_{1},x_{2},...,x_{N})
  2. | ψ |\psi\rangle
  3. | ψ = i | i i | ψ |\psi\rangle=\sum_{i}|i\rangle\langle i|\psi\rangle
  4. | i |i\rangle
  5. | ϵ |\epsilon\rangle
  6. | 𝑏𝑒𝑓𝑜𝑟𝑒 = i | i | ϵ i | ψ . |\mathit{before}\rangle=\sum_{i}|i\rangle|\epsilon\rangle\langle i|\psi\rangle.
  7. | i | ϵ |i\rangle|\epsilon\rangle
  8. | i | ϵ |i\rangle\otimes|\epsilon\rangle
  9. | i | ϵ |i\rangle|\epsilon\rangle
  10. | ϵ i |\epsilon_{i}\rangle
  11. | 𝑏𝑒𝑓𝑜𝑟𝑒 |\mathit{before}\rangle
  12. | 𝑎𝑓𝑡𝑒𝑟 = i | ϵ i i | ψ |\mathit{after}\rangle=\sum_{i}|\epsilon_{i}\rangle\langle i|\psi\rangle
  13. i | j = δ i j \langle i|j\rangle=\delta_{ij}
  14. ϵ i | ϵ j = δ i j \langle\epsilon_{i}|\epsilon_{j}\rangle=\delta_{ij}
  15. | i | ϵ |i\rangle|\epsilon\rangle
  16. | i , ϵ i = | i | ϵ i |i,\epsilon_{i}\rangle=|i\rangle|\epsilon_{i}\rangle
  17. | 𝑏𝑒𝑓𝑜𝑟𝑒 |\mathit{before}\rangle
  18. | 𝑎𝑓𝑡𝑒𝑟 = i | i , ϵ i i | ψ |\mathit{after}\rangle=\sum_{i}|i,\epsilon_{i}\rangle\langle i|\psi\rangle
  19. i , ϵ i | j , ϵ j = i | j ϵ i | ϵ j = δ i j ϵ i | ϵ j = δ i j \langle i,\epsilon_{i}|j,\epsilon_{j}\rangle=\langle i|j\rangle\langle\epsilon% _{i}|\epsilon_{j}\rangle=\delta_{ij}\langle\epsilon_{i}|\epsilon_{j}\rangle=% \delta_{ij}
  20. ϵ i | ϵ j δ i j \langle\epsilon_{i}|\epsilon_{j}\rangle\approx\delta_{ij}
  21. | i |i\rangle
  22. i i
  23. | ϵ j = | ϵ |\epsilon_{j}\rangle=|\epsilon^{\prime}\rangle
  24. ψ \psi
  25. ϕ \phi
  26. ψ \psi
  27. 𝑝𝑟𝑜𝑏 𝑏𝑒𝑓𝑜𝑟𝑒 ( ψ ϕ ) = | ψ | ϕ | 2 = | i ψ i * ϕ i | 2 = i | ψ i * ϕ i | 2 + i j ; i j ψ i * ψ j ϕ j * ϕ i \mathit{prob}_{\mathit{before}}(\psi\rightarrow\phi)=|\langle\psi|\phi\rangle|% ^{2}=|\sum_{i}\psi^{*}_{i}\phi_{i}|^{2}=\sum_{i}|\psi_{i}^{*}\phi_{i}|^{2}+% \sum_{ij;i\neq j}\psi^{*}_{i}\psi_{j}\phi^{*}_{j}\phi_{i}
  28. ψ i = i | ψ , ψ i * = ψ | i \psi_{i}=\langle i|\psi\rangle,\psi_{i}^{*}=\langle\psi|i\rangle
  29. ϕ i = i | ϕ \phi_{i}=\langle i|\phi\rangle
  30. i j i\neq j
  31. ψ \psi
  32. ϕ \phi
  33. ψ \psi
  34. E i E_{i}
  35. 𝑝𝑟𝑜𝑏 𝑎𝑓𝑡𝑒𝑟 ( ψ ϕ ) = j | 𝑎𝑓𝑡𝑒𝑟 | ϕ , ϵ j | 2 = j | i ψ i * i , ϵ i | ϕ , ϵ j | 2 = j | i ψ i * i | ϕ ϵ i | ϵ j | 2 \mathit{prob}_{\mathit{after}}(\psi\rightarrow\phi)=\sum_{j}|\langle\mathit{% after}|\phi,\epsilon_{j}\rangle|^{2}=\sum_{j}|\sum_{i}\psi_{i}^{*}\langle i,% \epsilon_{i}|\phi,\epsilon_{j}\rangle|^{2}=\sum_{j}|\sum_{i}\psi_{i}^{*}% \langle i|\phi\rangle\langle\epsilon_{i}|\epsilon_{j}\rangle|^{2}
  36. ϵ i | ϵ j δ i j \langle\epsilon_{i}|\epsilon_{j}\rangle\approx\delta_{ij}
  37. 𝑝𝑟𝑜𝑏 𝑎𝑓𝑡𝑒𝑟 ( ψ ϕ ) j | ψ j * j | ϕ | 2 = i | ψ i * ϕ i | 2 \mathit{prob}_{\mathit{after}}(\psi\rightarrow\phi)\approx\sum_{j}|\psi_{j}^{*% }\langle j|\phi\rangle|^{2}=\sum_{i}|\psi^{*}_{i}\phi_{i}|^{2}
  38. Σ i \Sigma_{i}
  39. i j ; i j ψ i * ψ j ϕ j * ϕ i \sum_{ij;i\neq j}\psi^{*}_{i}\psi_{j}\phi^{*}_{j}\phi_{i}
  40. ρ = | 𝑏𝑒𝑓𝑜𝑟𝑒 𝑏𝑒𝑓𝑜𝑟𝑒 | = | ψ ψ | | ϵ ϵ | \rho=|\mathit{before}\rangle\langle\mathit{before}|=|\psi\rangle\langle\psi|% \otimes|\epsilon\rangle\langle\epsilon|
  41. | ϵ |\epsilon\rangle
  42. ρ 𝑠𝑦𝑠 = 𝑇𝑟 𝑒𝑛𝑣 ( ρ ) = | ψ ψ | ϵ | ϵ = | ψ ψ | \rho_{\mathit{sys}}=\mathit{Tr}_{\mathit{env}}(\rho)=|\psi\rangle\langle\psi|% \langle\epsilon|\epsilon\rangle=|\psi\rangle\langle\psi|
  43. 𝑝𝑟𝑜𝑏 𝑏𝑒𝑓𝑜𝑟𝑒 ( ψ ϕ ) = ϕ | ρ 𝑠𝑦𝑠 | ϕ = ϕ | ψ ψ | ϕ = | ψ | ϕ | 2 = i | ψ i * ϕ i | 2 + i j ; i j ψ i * ψ j ϕ j * ϕ i \mathit{prob}_{\mathit{before}}(\psi\rightarrow\phi)=\langle\phi|\rho_{\mathit% {sys}}|\phi\rangle=\langle\phi|\psi\rangle\langle\psi|\phi\rangle={|\langle% \psi|\phi\rangle|}^{2}=\sum_{i}|\psi_{i}^{*}\phi_{i}|^{2}+\sum_{ij;i\neq j}% \psi^{*}_{i}\psi_{j}\phi^{*}_{j}\phi_{i}
  44. ψ i = i | ψ , ψ i * = ψ | i \psi_{i}=\langle i|\psi\rangle,\psi_{i}^{*}=\langle\psi|i\rangle
  45. ϕ i = i | ϕ \phi_{i}=\langle i|\phi\rangle
  46. ρ = | 𝑎𝑓𝑡𝑒𝑟 𝑎𝑓𝑡𝑒𝑟 | = i , j ψ i ψ j * | i , ϵ i j , ϵ j | = i , j ψ i ψ j * | i j | | ϵ i ϵ j | \rho=|\mathit{after}\rangle\langle\mathit{after}|=\sum_{i,j}\psi_{i}\psi_{j}^{% *}|i,\epsilon_{i}\rangle\langle j,\epsilon_{j}|=\sum_{i,j}\psi_{i}\psi_{j}^{*}% |i\rangle\langle j|\otimes|\epsilon_{i}\rangle\langle\epsilon_{j}|
  47. ρ 𝑠𝑦𝑠 = 𝑇𝑟 𝑒𝑛𝑣 ( i , j ψ i ψ j * | i j | | ϵ i ϵ j | ) = i , j ψ i ψ j * | i j | ϵ i | ϵ j = i , j ψ i ψ j * | i j | δ i j = i | ψ i | 2 | i i | \rho_{\mathit{sys}}=\mathit{Tr}_{\mathit{env}}(\sum_{i,j}\psi_{i}\psi_{j}^{*}|% i\rangle\langle j|\otimes|\epsilon_{i}\rangle\langle\epsilon_{j}|)=\sum_{i,j}% \psi_{i}\psi_{j}^{*}|i\rangle\langle j|\langle\epsilon_{i}|\epsilon_{j}\rangle% =\sum_{i,j}\psi_{i}\psi_{j}^{*}|i\rangle\langle j|\delta_{ij}=\sum_{i}|\psi_{i% }|^{2}|i\rangle\langle i|
  48. j | ϕ j | 2 | j j | \sum_{j}|\phi_{j}|^{2}|j\rangle\langle j|
  49. 𝑝𝑟𝑜𝑏 𝑎𝑓𝑡𝑒𝑟 ( ψ ϕ ) = i , j | ψ i | 2 | ϕ j | 2 j | i i | j = i | ψ i * ϕ i | 2 \mathit{prob}_{\mathit{after}}(\psi\rightarrow\phi)=\sum_{i,j}|\psi_{i}|^{2}|% \phi_{j}|^{2}\langle j|i\rangle\langle i|j\rangle=\sum_{i}|\psi_{i}^{*}\phi_{i% }|^{2}
  50. i j ; i j ψ i * ψ j ϕ j * ϕ i \sum_{ij;i\neq j}\psi^{*}_{i}\psi_{j}\phi^{*}_{j}\phi_{i}
  51. 𝒮 \mathcal{H_{S}}
  52. \mathcal{H_{B}}
  53. H ^ = H ^ S I ^ B + I ^ S H ^ B + H ^ I \hat{H}=\hat{H}_{S}\otimes\hat{I}_{B}+\hat{I}_{S}\otimes\hat{H}_{B}+\hat{H}_{I}
  54. H ^ S , H ^ B \hat{H}_{S},\hat{H}_{B}
  55. H ^ I \hat{H}_{I}
  56. I ^ S , I ^ B \hat{I}_{S},\hat{I}_{B}
  57. ρ S B ( t ) = U ^ ( t ) ρ S B ( 0 ) U ^ ( t ) \rho_{SB}(t)=\hat{U}(t)\rho_{SB}(0)\hat{U^{\dagger}}(t)
  58. U ^ = e - i H ^ t \hat{U}=e^{\frac{-i\hat{H}t}{\hbar}}
  59. ρ S B = ρ S ρ B \rho_{SB}=\rho_{S}\otimes\rho_{B}
  60. ρ S B ( t ) = U ^ ( t ) [ ρ S ( 0 ) ρ B ( 0 ) ] U ^ ( t ) . \rho_{SB}(t)=\hat{U}(t)[\rho_{S}(0)\otimes\rho_{B}(0)]\hat{U^{\dagger}}(t).
  61. H ^ I = i S i ^ B ^ i , \hat{H}_{I}=\sum_{i}\hat{S_{i}}\otimes\hat{B}_{i},
  62. S i ^ B i ^ \hat{S_{i}}\otimes\hat{B_{i}}
  63. S ^ i , B ^ i \hat{S}_{i},\hat{B}_{i}
  64. ρ S ( t ) = T r B [ U ^ ( t ) [ ρ S ( 0 ) ρ B ( 0 ) ] U ^ ( t ) ] . \rho_{S}(t)=Tr_{B}[\hat{U}(t)[\rho_{S}(0)\otimes\rho_{B}(0)]\hat{U^{\dagger}}(% t)].
  65. ρ S ( t ) \mathbf{\mathit{\rho}}_{S}(t)
  66. ρ B ( 0 ) = j a j | j j | . \rho_{B}(0)=\sum_{j}a_{j}|{j}\rangle\langle{j}|.
  67. ρ S ( t ) = l A l ^ ρ S ( 0 ) A ^ l \rho_{S}(t)=\sum_{l}\hat{A_{l}}\rho_{S}(0)\hat{A}^{\dagger}_{l}
  68. A l ^ , A ^ l \hat{A_{l}},\hat{A}^{\dagger}_{l}
  69. A l ^ = a j k | U ^ | j . \hat{A_{l}}=\sqrt{a_{j}}\langle{k}|\hat{U}|{j}\rangle.
  70. 𝑇𝑟 ( ρ S ( t ) ) = 1 \mathbf{\mathit{Tr}}(\mathbf{\mathit{\rho}}_{S}(t))=1
  71. l A ^ l A l ^ = I ^ S . \sum_{l}\hat{A}^{\dagger}_{l}\hat{A_{l}}=\hat{I}_{S}.
  72. ρ 𝐒 ( t ) \mathbf{\rho_{S}}(t)
  73. ρ S ( t ) = - i [ 𝐇 ~ 𝐒 , ρ S ( t ) ] + L D [ ρ S ( t ) ] \rho_{S}^{\prime}(t)=\frac{-i}{\hbar}\big[\mathbf{\tilde{H}_{S}},\rho_{S}(t)% \big]+L_{D}\big[\rho_{S}(t)\big]
  74. 𝐇 ~ 𝐒 = 𝐇 𝐒 + Δ \mathbf{\tilde{H}_{S}}=\mathbf{H_{S}}+\Delta
  75. 𝐇 𝐒 \mathbf{H_{S}}
  76. Δ \Delta
  77. L D L_{D}
  78. L D [ ρ S ( t ) ] = 1 2 α , β = 1 M b α β ( [ 𝐅 α , ρ S ( t ) 𝐅 β ] + [ 𝐅 α ρ S ( t ) , 𝐅 β ] ) . L_{D}\big[\rho_{S}(t)\big]=\frac{1}{2}\sum_{\alpha,\beta=1}^{M}b_{\alpha\beta}% \big(\big[\mathbf{F}_{\alpha},\rho_{S}(t)\mathbf{F}^{\dagger}_{\beta}\big]+% \big[\mathbf{F}_{\alpha}\rho_{S}(t),\mathbf{F}^{\dagger}_{\beta}\big]\big).
  79. { 𝐅 α } α = 1 M \big\{\mathbf{F}_{\alpha}\big\}_{\alpha=1}^{M}
  80. S \mathcal{H}_{S}
  81. b α β \mathbf{\mathit{b}}_{\alpha\beta}
  82. L D \mathit{L}_{D}
  83. L D [ ρ S ( t ) ] = 0 L_{D}\big[\rho_{S}(t)\big]=0
  84. \mathcal{H}
  85. | | , | | |\uparrow\rangle\langle\uparrow|,|\downarrow\rangle\langle\downarrow|
  86. ( | 0 0 | , | 1 1 | ) \big(|0\rangle\langle 0|,|1\rangle\langle 1|\big)
  87. J z ^ \hat{J_{z}}
  88. ϕ \phi
  89. | 0 |0\rangle
  90. | 1 |1\rangle
  91. J z ^ \hat{J_{z}}
  92. | 0 |0\rangle
  93. | 1 |1\rangle
  94. | 0 | 0 , | 1 e i ϕ | 1 . |0\rangle\rightarrow|0\rangle,|1\rangle\rightarrow e^{i\phi}|1\rangle.
  95. R z ( ϕ ) = ( 1 0 0 e i ϕ ) . R_{z}(\phi)=\begin{pmatrix}1&0\\ 0&e^{i\phi}\end{pmatrix}.
  96. | ψ j = a | 0 j + b | 1 j |\psi_{j}\rangle=a|0_{j}\rangle+b|1_{j}\rangle
  97. e i ϕ \mathbf{\mathit{e}}^{i\phi}
  98. ϕ \phi
  99. ρ j = - R z ( ϕ ) | ψ j ψ j | R z ( ϕ ) p ( ϕ ) d ϕ \rho_{j}=\int\limits_{-\infty}^{\infty}R_{z}(\phi)|\psi_{j}\rangle\langle\psi_% {j}|R^{\dagger}_{z}(\phi)p(\phi)\,d{\phi}
  100. p ( ϕ ) \mathbf{\mathit{p}}(\phi)
  101. p ( ϕ ) \mathbf{\mathit{p}}(\phi)
  102. p ( ϕ ) = ( 4 π α ) - 1 2 e - ϕ 2 4 α p(\phi)=(4\pi\alpha)^{-\frac{1}{2}}e^{\frac{-\phi^{2}}{4\alpha}}
  103. ρ j = ( | a | 2 a b * e - α a * b e - α | b | 2 ) . \rho_{j}=\begin{pmatrix}|a|^{2}&ab^{*}e^{-\alpha}\\ a^{*}be^{-\alpha}&|b|^{2}\end{pmatrix}.
  104. α \alpha
  105. | 0 0 | , | 1 1 | |0\rangle\langle 0|,|1\rangle\langle 1|
  106. ϵ \epsilon
  107. ϵ \epsilon
  108. H ϵ H_{\epsilon}
  109. H = H A H ϵ H=H_{A}\otimes H_{\epsilon}
  110. ϵ \epsilon
  111. ϵ \epsilon
  112. | in |\mathrm{in}\rangle
  113. c 1 | ψ 1 + c 2 | ψ 2 c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle
  114. | ψ 1 |\psi_{1}\rangle
  115. | ψ 2 |\psi_{2}\rangle
  116. { | e i } i \{|e_{i}\rangle\}_{i}
  117. U ( | ψ 1 | in ) U(|\psi_{1}\rangle\otimes|\mathrm{in}\rangle)
  118. U ( | ψ 2 | in ) U(|\psi_{2}\rangle\otimes|\mathrm{in}\rangle)
  119. i | e i | f 1 i \sum_{i}|e_{i}\rangle\otimes|f_{1i}\rangle
  120. i | e i | f 2 i \sum_{i}|e_{i}\rangle\otimes|f_{2i}\rangle
  121. H ϵ H_{\epsilon}
  122. | f 1 i |f_{1i}\rangle
  123. | f 1 j |f_{1j}\rangle
  124. | f 2 i |f_{2i}\rangle
  125. | f 2 j |f_{2j}\rangle
  126. | f 1 i |f_{1i}\rangle
  127. | f 2 j |f_{2j}\rangle
  128. i ( f 1 i | f 1 i + f 2 i | f 2 i ) | e i e i | \sum_{i}(\langle f_{1i}|f_{1i}\rangle+\langle f_{2i}|f_{2i}\rangle)|e_{i}% \rangle\langle e_{i}|

Quantum_fluctuation.html

  1. Δ E Δ t h 4 π \Delta E\Delta t\approx{h\over 4\pi}
  2. m c 2 mc^{2}
  3. φ t ( x ) {\displaystyle\varphi_{t}(x)}
  4. t t
  5. φ ~ t ( k ) {\displaystyle\tilde{\varphi}_{t}(k)}
  6. ρ 0 [ φ t ] = exp [ - 1 d 3 k ( 2 π ) 3 φ ~ t * ( k ) | k | 2 + m 2 φ ~ t ( k ) ] . \rho_{0}[\varphi_{t}]=\exp{\left[-\frac{1}{\hbar}\int\frac{d^{3}k}{(2\pi)^{3}}% \tilde{\varphi}_{t}^{*}(k)\sqrt{|k|^{2}+m^{2}}\;\tilde{\varphi}_{t}(k)\right]}.
  7. φ t ( x ) {\displaystyle\varphi_{t}(x)}
  8. t t
  9. ρ E [ φ t ] = exp [ - H [ φ t ] / k B T ] = exp [ - 1 k B T d 3 k ( 2 π ) 3 φ ~ t * ( k ) 1 2 ( | k | 2 + m 2 ) φ ~ t ( k ) ] . \rho_{E}[\varphi_{t}]=\exp{[-H[\varphi_{t}]/k_{\mathrm{B}}T]}=\exp{\left[-% \frac{1}{k_{\mathrm{B}}T}\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{\varphi}_{t}^{*}(% k){\scriptstyle\frac{1}{2}}(|k|^{2}+m^{2})\;\tilde{\varphi}_{t}(k)\right]}.
  10. \hbar
  11. k B T k_{\mathrm{B}}T
  12. k B k_{\mathrm{B}}
  13. | k | 2 + m 2 \sqrt{|k|^{2}+m^{2}}
  14. 1 2 ( | k | 2 + m 2 ) {\scriptstyle\frac{1}{2}}(|k|^{2}+m^{2})

Quantum_indeterminacy.html

  1. Pr ( λ ) = E ( λ ) ψ ψ \operatorname{Pr}(\lambda)=\langle\operatorname{E}(\lambda)\psi\mid\psi\rangle
  2. σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 - i i 0 ) , σ 3 = ( 1 0 0 - 1 ) \sigma_{1}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  3. 1 2 ( 1 , 1 ) , 1 2 ( 1 , - 1 ) \frac{1}{\sqrt{2}}(1,1),\frac{1}{\sqrt{2}}(1,-1)
  4. ( 1 , 0 ) , ( 0 , 1 ) (1,0),(0,1)\quad
  5. ψ = 1 2 ( 1 , 1 ) , \psi=\frac{1}{\sqrt{2}}(1,1),
  6. E A ( U ) = U λ d E ( λ ) , \operatorname{E}_{A}(U)=\int_{U}\lambda d\operatorname{E}(\lambda),
  7. D A ( U ) = Tr ( E A ( U ) S ) . \operatorname{D}_{A}(U)=\operatorname{Tr}(\operatorname{E}_{A}(U)S).

Quantum_superposition.html

  1. | 0 |0\rangle
  2. | 1 |1\rangle
  3. | 0 |0\rangle
  4. | 1 |1\rangle
  5. c c
  6. c 1 + c 2 c_{1}\mid\uparrow\rangle+c_{2}\mid\downarrow\rangle
  7. p up = c 1 2 p\text{up}=\mid c_{1}\mid^{2}
  8. p down = c 2 2 p\text{down}=\mid c_{2}\mid^{2}
  9. p up or down = p up + p down = 1 p\text{up or down}=p\text{up}+p\text{down}=1
  10. 3 i / 5 3i/5
  11. 4 / 5 4/5
  12. | ψ = 3 5 i | + 4 5 | . |\psi\rangle={3\over 5}i|\uparrow\rangle+{4\over 5}|\downarrow\rangle.
  13. | 3 i 5 | 2 = 9 25 \left|\;\frac{3i}{5}\;\right|^{2}=\frac{9}{25}
  14. | 4 5 | 2 = 16 25 \left|\;\frac{4}{5}\;\right|^{2}=\frac{16}{25}
  15. 9 25 + 16 25 = 1 \frac{9}{25}+\frac{16}{25}=1
  16. α \alpha
  17. | ψ α | ψ |\psi\rangle\approx\alpha|\psi\rangle
  18. ψ \psi
  19. \mid\uparrow\rangle\to\mid\downarrow\rangle
  20. 3 i 5 + 4 5 \mid\downarrow\rangle\to\frac{3i}{5}\mid\uparrow\rangle+\frac{4}{5}\mid\downarrow\rangle
  21. c 1 + c 2 c 1 ( ) + c 2 ( 3 i 5 + 4 5 ) c_{1}\mid\uparrow\rangle+c_{2}\mid\downarrow\rangle\to c_{1}\left(\mid% \downarrow\rangle\right)+c_{2}\left(\frac{3i}{5}\mid\uparrow\rangle+\frac{4}{5% }\mid\downarrow\rangle\right)
  22. x x
  23. | x |x\rangle
  24. x ψ ( x ) | x \sum_{x}\psi(x)|x\rangle
  25. x x
  26. \reals \reals
  27. ψ ( x ) \psi(x)
  28. x ψ + ( x ) | x , + ψ - ( x ) | x , \sum_{x}\psi_{+}(x)|x,\uparrow\rangle+\psi_{-}(x)|x,\downarrow\rangle\,
  29. | x , y |x,y\rangle
  30. x y A ( x , y ) | x , y \sum_{xy}A(x,y)|x,y\rangle\,
  31. P ( x , y ) P(x,y)
  32. P ( x , y ) = P x ( x ) P y ( y ) P(x,y)=P_{x}(x)P_{y}(y)\,
  33. A ( x , y ) = ψ x ( x ) ψ y ( y ) A(x,y)=\psi_{x}(x)\psi_{y}(y)\,
  34. x ρ ( x ) | x \sum_{x}\rho(x)|x\rangle
  35. ρ \rho
  36. x | 1 + y | 2 + z | 3 x|1\rangle+y|2\rangle+z|3\rangle\,
  37. x , y , z x,y,z
  38. x + y + z = 1 x+y+z=1\,
  39. A | 1 + B | 2 + C | 3 = ( A r + i A i ) | 1 + ( B r + i B i ) | 2 + ( C r + i C i ) | 3 A|1\rangle+B|2\rangle+C|3\rangle=(A_{r}+iA_{i})|1\rangle+(B_{r}+iB_{i})|2% \rangle+(C_{r}+iC_{i})|3\rangle\,
  40. A r 2 + A i 2 + B r 2 + B i 2 + C r 2 + C i 2 = 1 A_{r}^{2}+A_{i}^{2}+B_{r}^{2}+B_{i}^{2}+C_{r}^{2}+C_{i}^{2}=1\,
  41. n ψ n | n \sum_{n}\psi_{n}|n\rangle\,
  42. ( ψ - 2 , ψ - 1 , ψ 0 , ψ 1 , ψ 2 ) \scriptstyle(...\psi_{-2},\psi_{-1},\psi_{0},\psi_{1},\psi_{2}...)
  43. ψ n * ψ n = 1 \sum\psi_{n}^{*}\psi_{n}=1
  44. ( P - 2 , P - 1 , P 0 , P 1 , P 2 , ) (...P_{-2},P_{-1},P_{0},P_{1},P_{2},...)
  45. K x y ( t ) \scriptstyle K_{x\rightarrow y}(t)
  46. P y ( t 0 + t ) = x P x ( t 0 ) K x y ( t ) P_{y}(t_{0}+t)=\sum_{x}P_{x}(t_{0})K_{x\rightarrow y}(t)\,
  47. y K x y = 1 \sum_{y}K_{x\rightarrow y}=1\,
  48. K x y ( 0 ) = δ x y \scriptstyle K{x\rightarrow y}(0)=\delta_{xy}
  49. P y ( t + d t ) = P y ( t ) + d t x P x R x y P_{y}(t+dt)=P_{y}(t)+dt\sum_{x}P_{x}R_{x\rightarrow y}\,
  50. R x y \scriptstyle R_{x\rightarrow y}
  51. R x y = K x y ( d t ) - δ x y d t R_{x\rightarrow y}={K_{x\rightarrow y}(dt)-\delta_{xy}\over dt}\,
  52. d P y d t = x P x R x y {dP_{y}\over dt}=\sum_{x}P_{x}R_{x\rightarrow y}\,
  53. y R x y = 0 \sum_{y}R_{x\rightarrow y}=0\,
  54. R x y \scriptstyle R_{x\rightarrow y}
  55. R x x R_{x\rightarrow x}
  56. d P x d t = c ( P x + 1 - 2 P x + P x - 1 ) {dP_{x}\over dt}=c(P_{x+1}-2P_{x}+P_{x-1})\,
  57. P ( x , t ) t = c 2 P x 2 {\partial P(x,t)\over\partial t}=c{\partial^{2}P\over\partial x^{2}}\,
  58. ψ n ( t ) = m U n m ( t ) ψ m \psi_{n}(t)=\sum_{m}U_{nm}(t)\psi_{m}\,
  59. U U
  60. n U n m * U n p = δ m p \sum_{n}U^{*}_{nm}U_{np}=\delta_{mp}\,
  61. U U = I U^{\dagger}U=I\,
  62. H m n = i d d t U m n H_{mn}=i{d\over dt}U_{mn}
  63. ( I + i H d t ) ( I - i H d t ) = I (I+iH^{\dagger}dt)(I-iHdt)=I\,
  64. H - H = 0 H^{\dagger}-H=0\,
  65. ψ \psi
  66. i d ψ n d t = c * ψ n + 1 + c ψ n - 1 i{d\psi_{n}\over dt}=c^{*}\psi_{n+1}+c\psi_{n-1}
  67. ψ ψ e i 2 c t \psi\rightarrow\psi e^{i2ct}
  68. i d ψ n d t = c ψ n + 1 - 2 c ψ n + c ψ n - 1 i{d\psi_{n}\over dt}=c\psi_{n+1}-2c\psi_{n}+c\psi_{n-1}
  69. i ψ t = - 2 ψ x 2 i{\partial\psi\over\partial t}=-{\partial^{2}\psi\over\partial x^{2}}
  70. i ψ t = - 2 ψ x 2 + V ( x ) ψ i{\partial\psi\over\partial t}=-{\partial^{2}\psi\over\partial x^{2}}+V(x)\psi
  71. K m n \scriptstyle K_{m\rightarrow n}
  72. K x y ( T ) = x ( t ) t K x ( t ) x ( t + 1 ) K_{x\rightarrow y}(T)=\sum_{x(t)}\prod_{t}K_{x(t)x(t+1)}\,
  73. x ( t ) x(t)
  74. x ( 0 ) = 0 x(0)=0
  75. x ( T ) = y x(T)=y
  76. ρ n \rho_{n}
  77. ρ n K n m = ρ m K m n \rho_{n}K_{n\rightarrow m}=\rho_{m}K_{m\rightarrow n}\,
  78. ρ m \rho_{m}
  79. ρ n R n m = ρ m R m n \rho_{n}R_{n\rightarrow m}=\rho_{m}R_{m\rightarrow n}\,
  80. p n = ρ n p n p^{\prime}_{n}=\sqrt{\rho_{n}}\;p_{n}\,
  81. ρ n R n m 1 ρ m = H n m \sqrt{\rho_{n}}R_{n\rightarrow m}{1\over\sqrt{\rho_{m}}}=H_{nm}\,
  82. H n m = H m n H_{nm}=H_{mn}\,
  83. i d d t ψ n = H n m ψ m i{d\over dt}\psi_{n}=\sum H_{nm}\psi_{m}\,
  84. U ( t ) = e - i H t U(t)=e^{-iHt}\,
  85. K ( t ) = e - H t K^{\prime}(t)=e^{-Ht}\,
  86. | ψ = 1 2 ( | 00...0 + | 11...1 ) |\psi\rangle=\frac{1}{\sqrt{2}}\bigg(|00...0\rangle+|11...1\rangle\bigg)
  87. ψ n q \psi^{q}_{n}
  88. A n m = q q ψ n * q ψ m q A_{nm}=\sum_{q}q\psi^{*q}_{n}\psi^{q}_{m}
  89. I I II

Quartic_function.html

  1. f ( x ) = a x 4 + b x 3 + c x 2 + d x + e , f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,
  2. f ( x ) = a x 4 + c x 2 + e . f(x)=ax^{4}+cx^{2}+e.
  3. a x 4 + b x 3 + c x 2 + d x + e = 0 , ax^{4}+bx^{3}+cx^{2}+dx+e=0,
  4. F G G H = 1 + 5 2 = the golden ratio . \frac{FG}{GH}=\frac{1+\sqrt{5}}{2}=\,\text{the golden ratio}.
  5. a x 4 + b x 3 + c x 2 + d x + e = 0 ax^{4}+bx^{3}+cx^{2}+dx+e=0
  6. a 0 , a\neq 0,
  7. Δ = 256 a 3 e 3 - 192 a 2 b d e 2 - 128 a 2 c 2 e 2 + 144 a 2 c d 2 e - 27 a 2 d 4 + 144 a b 2 c e 2 - 6 a b 2 d 2 e - 80 a b c 2 d e + 18 a b c d 3 + 16 a c 4 e - 4 a c 3 d 2 - 27 b 4 e 2 + 18 b 3 c d e - 4 b 3 d 3 - 4 b 2 c 3 e + b 2 c 2 d 2 \begin{aligned}\displaystyle\Delta\ =&\displaystyle 256a^{3}e^{3}-192a^{2}bde^% {2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}\\ &\displaystyle+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e% \\ &\displaystyle-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b% ^{2}c^{2}d^{2}\end{aligned}
  8. P = 8 a c - 3 b 2 P=8ac-3b^{2}
  9. P 8 a 2 \frac{P}{8a^{2}}
  10. Q = b 3 + 8 d a 2 - 4 a b c , Q=b^{3}+8da^{2}-4abc,
  11. Q 8 a 3 \frac{Q}{8a^{3}}
  12. Δ 0 = c 2 - 3 b d + 12 a e , \Delta_{0}=c^{2}-3bd+12ae,
  13. D = 64 a 3 e - 16 a 2 c 2 + 16 a b 2 c - 16 a 2 b d - 3 b 4 D=64a^{3}e-16a^{2}c^{2}+16ab^{2}c-16a^{2}bd-3b^{4}
  14. Δ < 0 \Delta<0
  15. Δ > 0 \Delta>0
  16. P P
  17. P P
  18. D D
  19. Δ = 0 \Delta=0
  20. P P
  21. D D
  22. P P
  23. D D
  24. Q Q
  25. Δ 0 = 0 \Delta_{0}=0
  26. D D
  27. D D
  28. P P
  29. P P
  30. Q Q
  31. Δ 0 = 0 \Delta_{0}=0
  32. - b 4 a -\frac{b}{4a}
  33. Δ \Delta
  34. P P
  35. D D
  36. Δ \Delta
  37. P P
  38. D D
  39. x 1 , x 2 , x 3 , x 4 x_{1},x_{2},x_{3},x_{4}
  40. a x 4 + b x 3 + c x 2 + d x + e = 0 ax^{4}+bx^{3}+cx^{2}+dx+e=0\,
  41. a a
  42. x 1 , 2 = - b 4 a - S ± 1 2 - 4 S 2 - 2 p + q S x 3 , 4 = - b 4 a + S ± 1 2 - 4 S 2 - 2 p - q S \begin{aligned}\displaystyle x_{1,2}&\displaystyle=-\frac{b}{4a}-S\pm\frac{1}{% 2}\sqrt{-4S^{2}-2p+\frac{q}{S}}\\ \displaystyle x_{3,4}&\displaystyle=-\frac{b}{4a}+S\pm\frac{1}{2}\sqrt{-4S^{2}% -2p-\frac{q}{S}}\end{aligned}
  43. p p
  44. q q
  45. p = 8 a c - 3 b 2 8 a 2 q = b 3 - 4 a b c + 8 a 2 d 8 a 3 \begin{aligned}\displaystyle p&\displaystyle=\frac{8ac-3b^{2}}{8a^{2}}\\ \displaystyle q&\displaystyle=\frac{b^{3}-4abc+8a^{2}d}{8a^{3}}\end{aligned}
  46. S = 1 2 - 2 3 p + 1 3 a ( Q + Δ 0 Q ) (if S = 0 , see Special cases of the formula, below) Q = Δ 1 + Δ 1 2 - 4 Δ 0 3 2 3 (if Q = 0 , see Special cases of the formula, below) \begin{aligned}\displaystyle S&\displaystyle=\frac{1}{2}\sqrt{-\frac{2}{3}\ p+% \frac{1}{3a}\left(Q+\frac{\Delta_{0}}{Q}\right)}&\displaystyle\,\text{(if }S=0% \,\text{, see Special cases of the formula, below)}\\ \displaystyle Q&\displaystyle=\ \sqrt[3]{\frac{\Delta_{1}+\sqrt{\Delta_{1}^{2}% -4\Delta_{0}^{3}}}{2}}&\displaystyle\,\text{(if }Q=0\,\text{, see Special % cases of the formula, below)}\end{aligned}
  47. Δ 0 = c 2 - 3 b d + 12 a e Δ 1 = 2 c 3 - 9 b c d + 27 b 2 e + 27 a d 2 - 72 a c e \begin{aligned}\displaystyle\Delta_{0}&\displaystyle=c^{2}-3bd+12ae\\ \displaystyle\Delta_{1}&\displaystyle=2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace\end{aligned}
  48. Δ 1 2 - 4 Δ 0 3 = - 27 Δ , \Delta_{1}^{2}-4\Delta_{0}^{3}=-27\Delta\ ,
  49. Δ \Delta
  50. Δ > 0 , \Delta>0,
  51. Q Q
  52. S S
  53. S = 1 2 - 2 3 p + 2 3 a Δ 0 cos ϕ 3 S=\frac{1}{2}\sqrt{-\frac{2}{3}\ p+\frac{2}{3a}\sqrt{\Delta_{0}}\cos\frac{\phi% }{3}}
  54. ϕ = arccos ( Δ 1 2 Δ 0 3 ) . \phi=\arccos\left(\frac{\Delta_{1}}{2\sqrt{\Delta_{0}^{3}}}\right).
  55. Δ 0 \Delta\neq 0
  56. Δ 0 = 0 , \Delta_{0}=0,
  57. Δ 1 2 - 4 Δ 0 3 = Δ 1 2 \sqrt{\Delta_{1}^{2}-4\Delta_{0}^{3}}=\sqrt{\Delta_{1}^{2}}
  58. Q 0 , Q\neq 0,
  59. Δ 1 2 \sqrt{\Delta_{1}^{2}}
  60. Δ 1 , \Delta_{1},
  61. Δ 1 . \Delta_{1}.
  62. S = 0 , S=0,
  63. Q Q
  64. S 0. S\neq 0.
  65. ( x + b 4 a ) 4 . \left(x+\tfrac{b}{4a}\right)^{4}.
  66. q q
  67. Δ = 0 \Delta=0
  68. Δ 0 = 0 , \Delta_{0}=0,
  69. Δ 1 = 0 , \Delta_{1}=0,
  70. Δ = 0 \Delta=0
  71. Δ 0 0 , \Delta_{0}\neq 0,
  72. Q ( x ) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 . Q(x)=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.
  73. Q ( x ) = ( x - x 1 ) ( b 3 x 3 + b 2 x 2 + b 1 x + b 0 ) Q(x)=(x-x_{1})(b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0})
  74. Q ( x ) = ( c 2 x 2 + c 1 x + c 0 ) ( d 2 x 2 + d 1 x + d 0 ) . Q(x)=(c_{2}x^{2}+c_{1}x+c_{0})(d_{2}x^{2}+d_{1}x+d_{0}).
  75. Δ \Delta
  76. a 4 + a 3 + a 2 + a 1 + a 0 = 0 , a_{4}+a_{3}+a_{2}+a_{1}+a_{0}=0,
  77. Q ( 1 ) = 0 Q(1)=0
  78. a 4 - a 3 + a 2 - a 1 + a 0 = 0 , a_{4}-a_{3}+a_{2}-a_{1}+a_{0}=0,
  79. Q ( - 1 ) = 0 Q(-1)=0
  80. a 4 x 3 + ( a 4 x 1 + a 3 ) x 2 + ( a 4 x 1 2 + a 3 x 1 + a 2 ) x + a 4 x 1 3 + a 3 x 1 2 + a 2 x 1 + a 1 . a_{4}x^{3}+(a_{4}x_{1}+a_{3})x^{2}+(a_{4}x_{1}^{2}+a_{3}x_{1}+a_{2})x+a_{4}x_{% 1}^{3}+a_{3}x_{1}^{2}+a_{2}x_{1}+a_{1}.
  81. a 0 , a 1 , a 2 , a 3 , a 4 a_{0},a_{1},a_{2},a_{3},a_{4}
  82. x 1 = p q , x_{1}=\frac{p}{q}\ ,
  83. Q ( p q ) Q\left(\frac{p}{q}\right)
  84. a 3 = a 1 = 0 , a_{3}=a_{1}=0,\,
  85. Q ( x ) = a 4 x 4 + a 2 x 2 + a 0 Q(x)=a_{4}x^{4}+a_{2}x^{2}+a_{0}\,\!
  86. z = x 2 . z=x^{2}.\,
  87. z , z,\,
  88. q ( z ) = a 4 z 2 + a 2 z + a 0 . q(z)=a_{4}z^{2}+a_{2}z+a_{0}.\,\!
  89. z + z_{+}\,
  90. z - z_{-}\,
  91. x 1 = + z + , x 2 = - z + , x 3 = + z - , x 4 = - z - . \begin{aligned}\displaystyle x_{1}&\displaystyle=+\sqrt{z_{+}},\\ \displaystyle x_{2}&\displaystyle=-\sqrt{z_{+}},\\ \displaystyle x_{3}&\displaystyle=+\sqrt{z_{-}},\\ \displaystyle x_{4}&\displaystyle=-\sqrt{z_{-}}.\end{aligned}
  92. P ( x ) = a 0 x 4 + a 1 x 3 + a 2 x 2 + a 1 m x + a 0 m 2 P(x)=a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}mx+a_{0}m^{2}
  93. P ( m x ) = x 4 m 2 P ( m x ) P(mx)=\frac{x^{4}}{m^{2}}P(\frac{m}{x})
  94. m = 1 m=1
  95. z = x + m x z=x+\frac{m}{x}
  96. P ( x ) x 2 = 0 \frac{P(x)}{x^{2}}=0
  97. a 0 z 2 + a 1 z + ( a 2 - 2 m a 0 ) = 0. a_{0}z^{2}+a_{1}z+(a_{2}-2ma_{0})=0.
  98. P ( x ) = 0 P(x)=0
  99. a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 ( 1 ) a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\qquad\qquad(1^{\prime})
  100. x 4 + a x 3 + b x 2 + c x + d = 0 , x^{4}+ax^{3}+bx^{2}+cx+d=0,
  101. a = a 3 a 4 , b = a 2 a 4 , c = a 1 a 4 , d = a 0 a 4 . a=\frac{a_{3}}{a_{4}},\quad b=\frac{a_{2}}{a_{4}},\quad c=\frac{a_{1}}{a_{4}},% \quad d=\frac{a_{0}}{a_{4}}.
  102. y - a 3 4 a 4 y-\frac{a_{3}}{4a_{4}}
  103. y 4 + p y 2 + q y + r = 0 , y^{4}+py^{2}+qy+r=0,
  104. p = 8 b - 3 a 2 8 = 8 a 2 a 4 - 3 a 3 2 8 a 4 2 q = a 3 - 4 a b + 8 c 8 = a 3 3 - 4 a 2 a 3 a 4 + 8 a 1 a 4 2 8 a 4 3 r = - 3 a 4 + 256 d - 64 c a + 16 a 2 b 256 = - 3 a 3 4 + 256 a 0 a 4 3 - 64 a 1 a 3 a 4 2 + 16 a 2 a 3 2 a 4 256 a 4 4 \begin{aligned}\displaystyle p=&\displaystyle\frac{8b-3a^{2}}{8}&\displaystyle% =&\displaystyle\frac{8a_{2}a_{4}-3a_{3}^{2}}{8a_{4}^{2}}\\ \displaystyle q=&\displaystyle\frac{a^{3}-4ab+8c}{8}&\displaystyle=&% \displaystyle\frac{a_{3}^{3}-4a_{2}a_{3}a_{4}+8a_{1}a_{4}^{2}}{8a_{4}^{3}}\\ \displaystyle r=&\displaystyle\frac{-3a^{4}+256d-64ca+16a^{2}b}{256}&% \displaystyle=&\displaystyle\frac{-3a_{3}^{4}+256a_{0}a_{4}^{3}-64a_{1}a_{3}a_% {4}^{2}+16a_{2}a_{3}^{2}a_{4}}{256a_{4}^{4}}\end{aligned}
  105. y 1 - a 4 = y 1 - a 3 4 a 4 , y 2 - a 4 = y 2 - a 3 4 a 4 , y 3 - a 4 = y 3 - a 3 4 a 4 , y 4 - a 4 = y 4 - a 3 4 a 4 . y_{1}-\frac{a}{4}=y_{1}-\frac{a_{3}}{4a_{4}},\;y_{2}-\frac{a}{4}=y_{2}-\frac{a% _{3}}{4a_{4}},\;y_{3}-\frac{a}{4}=y_{3}-\frac{a_{3}}{4a_{4}},\;y_{4}-\frac{a}{% 4}=y_{4}-\frac{a_{3}}{4a_{4}}.
  106. 2 u 2 y + 2 α y + y 2 2u^{2}y+2\alpha y+y^{2}
  107. ( - β ) 2 - 4 ( 2 y + α ) ( y 2 + 2 y α + α 2 - γ ) = 0. (-\beta)^{2}-4(2y+\alpha)(y^{2}+2y\alpha+\alpha^{2}-\gamma)=0.\,
  108. y 3 + 5 2 α y 2 + ( 2 α 2 - γ ) y + ( α 3 2 - α γ 2 - β 2 8 ) = 0. y^{3}+{5\over 2}\alpha y^{2}+(2\alpha^{2}-\gamma)y+\left({\alpha^{3}\over 2}-{% \alpha\gamma\over 2}-{\beta^{2}\over 8}\right)=0.
  109. ( u α + 2 y - β 2 α + 2 y ) 2 . \left(u\sqrt{\alpha+2y}-\frac{\beta}{2\sqrt{\alpha+2y}}\right)^{2}.
  110. α + 2 y = 0. \alpha+2y=0.
  111. β = 0 , \beta=0,
  112. α + 2 y 0. \alpha+2y\not=0.
  113. α + 2 y 0 , \alpha+2y\not=0,
  114. ( u 2 + α + y + u α + 2 y - β 2 α + 2 y ) ( u 2 + α + y - u α + 2 y + β 2 α + 2 y ) = 0 , \left(u^{2}+\alpha+y+u\sqrt{\alpha+2y}-\frac{\beta}{2\sqrt{\alpha+2y}}\right)% \left(u^{2}+\alpha+y-u\sqrt{\alpha+2y}+\frac{\beta}{2\sqrt{\alpha+2y}}\right)=0,
  115. u = ± 1 α + 2 y ± 2 - ( 3 α + 2 y ± 1 2 β α + 2 y ) 2 , u={\pm_{1}\sqrt{\alpha+2y}\pm_{2}\sqrt{-\left(3\alpha+2y\pm_{1}{2\beta\over% \sqrt{\alpha+2y}}\right)}\over 2},
  116. ± 1 \pm_{1}
  117. ± 2 \pm_{2}
  118. ± 1 \pm_{1}
  119. x = - a 3 4 a 4 + ± 1 α + 2 y ± 2 - ( 3 α + 2 y ± 1 2 β α + 2 y ) 2 . x=-{a_{3}\over 4a_{4}}+{\pm_{1}\sqrt{\alpha+2y}\pm_{2}\sqrt{-\left(3\alpha+2y% \pm_{1}{2\beta\over\sqrt{\alpha+2y}}\right)}\over 2}.
  120. 0 = x 4 + a x 3 + b x 2 + c x + d = ( x 2 + p x + q ) ( x 2 + r x + s ) = x 4 + ( p + r ) x 3 + ( q + s + p r ) x 2 + ( p s + q r ) x + q s \begin{array}[]{lcl}0=x^{4}+ax^{3}+bx^{2}+cx+d&=&(x^{2}+px+q)(x^{2}+rx+s)\\ &=&x^{4}+(p+r)x^{3}+(q+s+pr)x^{2}+(ps+qr)x+qs\end{array}
  121. a = p + r b = q + s + p r c = p s + q r d = q s \begin{array}[]{lcl}a&=&p+r\\ b&=&q+s+pr\\ c&=&ps+qr\\ d&=&qs\end{array}
  122. a = 0 a=0
  123. ( x - a / 4 ) (x-a/4)
  124. x x
  125. r = - p r=-p
  126. b + p 2 = s + q c = ( s - q ) p d = s q \begin{array}[]{lcl}b+p^{2}&=&s+q\\ c&=&(s-q)p\\ d&=&sq\end{array}
  127. s s
  128. q q
  129. p 2 ( b + p 2 ) 2 - c 2 = p 2 ( s + q ) 2 - p 2 ( s - q ) 2 = 4 p 2 s q = 4 p 2 d \begin{array}[]{lcl}p^{2}(b+p^{2})^{2}-c^{2}&=&p^{2}(s+q)^{2}-p^{2}(s-q)^{2}\\ &=&4p^{2}sq\\ &=&4p^{2}d\end{array}
  130. P = p 2 P=p^{2}
  131. P 3 + 2 b P 2 + ( b 2 - 4 d ) P - c 2 = 0 P^{3}+2bP^{2}+(b^{2}-4d)P-c^{2}=0\,
  132. r = - p 2 s = b + p 2 + c / p 2 q = b + p 2 - c / p \begin{array}[]{lcl}r&=&-p\\ 2s&=&b+p^{2}+c/p\\ 2q&=&b+p^{2}-c/p\end{array}
  133. p p
  134. P P
  135. P 3 + 2 b P 2 + ( b 2 - 4 d ) P - c 2 P^{3}+2bP^{2}+(b^{2}-4d)P-c^{2}\,
  136. b 2 - 4 d b^{2}-4d
  137. x 4 + a x 3 + b x 2 + c x + d = 0 x^{4}+ax^{3}+bx^{2}+cx+d=0
  138. s 0 = 1 2 ( x 0 + x 1 + x 2 + x 3 ) , s 1 = 1 2 ( x 0 - x 1 + x 2 - x 3 ) , s 2 = 1 2 ( x 0 + x 1 - x 2 - x 3 ) , s 3 = 1 2 ( x 0 - x 1 - x 2 + x 3 ) , \begin{aligned}\displaystyle s_{0}&\displaystyle=\tfrac{1}{2}(x_{0}+x_{1}+x_{2% }+x_{3}),\\ \displaystyle s_{1}&\displaystyle=\tfrac{1}{2}(x_{0}-x_{1}+x_{2}-x_{3}),\\ \displaystyle s_{2}&\displaystyle=\tfrac{1}{2}(x_{0}+x_{1}-x_{2}-x_{3}),\\ \displaystyle s_{3}&\displaystyle=\tfrac{1}{2}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}
  139. ( s 2 - s 1 2 ) ( s 2 - s 2 2 ) ( s 2 - s 3 2 ) . (s^{2}-s_{1}^{2})(s^{2}-s_{2}^{2})(s^{2}-s_{3}^{2}).
  140. s 6 + 2 b s 4 + ( b 2 - 4 d ) s 2 - c 2 {s}^{6}+2b\,{s}^{4}+({b}^{2}-4d)\,{s}^{2}-{c}^{2}
  141. F 1 = x 2 + s x + b 2 + s 2 2 - c 2 s F_{1}={x}^{2}+sx+\frac{b}{2}+\frac{s^{2}}{2}-\frac{c}{2s}
  142. F 2 = x 2 - s x + b 2 + s 2 2 + c 2 s F_{2}={x}^{2}-sx+\frac{b}{2}+\frac{s^{2}}{2}+\frac{c}{2s}
  143. F 1 F 2 = x 4 + b x 2 + c x + d F_{1}F_{2}=x^{4}+bx^{2}+cx+d
  144. x 4 + p x 2 + q x + r = 0 x^{4}+px^{2}+qx+r=0
  145. y 2 + p y + q x + r = 0 , y^{2}+py+qx+r=0,
  146. y - x 2 = 0 ; y-x^{2}=0;
  147. y = x 2 ; y=x^{2};
  148. P i := ( x i , x i 2 ) P_{i}:=(x_{i},x_{i}^{2})
  149. x i x_{i}
  150. y = x 2 , y=x^{2},
  151. F 1 ( X , Y , Z ) \displaystyle F_{1}(X,Y,Z)
  152. λ F 1 + μ F 2 \lambda F_{1}+\mu F_{2}
  153. [ λ , μ ] [\lambda,\mu]
  154. λ \lambda
  155. μ \mu
  156. ( 4 2 ) = 6 \textstyle{{\left({{4}\atop{2}}\right)}=6}
  157. Q 1 = L 12 + L 34 , Q_{1}=L_{12}+L_{34},
  158. Q 2 = L 13 + L 24 , Q_{2}=L_{13}+L_{24},
  159. Q 3 = L 14 + L 23 . Q_{3}=L_{14}+L_{23}.
  160. λ F 1 + μ F 2 \lambda F_{1}+\mu F_{2}
  161. λ \lambda
  162. μ , \mu,

Quater-imaginary_base.html

  1. d 3 d 2 d 1 d 0 . d - 1 d - 2 d - 3 \ldots d_{3}d_{2}d_{1}d_{0}.d_{-1}d_{-2}d_{-3}\ldots
  2. + d 3 b 3 + d 2 b 2 + d 1 b + d 0 + d - 1 b - 1 + d - 2 b - 2 + d - 3 b - 3 \ldots+d_{3}\cdot b^{3}+d_{2}\cdot b^{2}+d_{1}\cdot b+d_{0}+d_{-1}\cdot b^{-1}% +d_{-2}\cdot b^{-2}+d_{-3}\cdot b^{-3}\ldots
  3. b = 2 i b=2i
  4. ( 2 i ) 2 = - 4 (2i)^{2}=-4
  5. + d 3 ( 2 i ) 3 + d 2 ( 2 i ) 2 + d 1 ( 2 i ) + d 0 + d - 1 ( 2 i ) - 1 + d - 2 ( 2 i ) - 2 + d - 3 ( 2 i ) - 3 \ldots+d_{3}\cdot(2i)^{3}+d_{2}\cdot(2i)^{2}+d_{1}\cdot(2i)+d_{0}+d_{-1}\cdot(% 2i)^{-1}+d_{-2}\cdot(2i)^{-2}+d_{-3}\cdot(2i)^{-3}\ldots
  6. = [ d 4 ( - 4 ) 2 + d 2 ( - 4 ) 1 + d 0 + d - 2 ( - 4 ) - 1 + ] + 2 i [ + d 5 ( - 4 ) 2 + d 3 ( - 4 ) 1 + d 1 + d - 1 ( - 4 ) - 1 + d - 3 ( - 4 ) - 2 + ] =[...d_{4}\cdot(-4)^{2}+d_{2}\cdot(-4)^{1}+d_{0}+d_{-2}\cdot(-4)^{-1}+\ldots]+% 2i\cdot[...+d_{5}\cdot(-4)^{2}+d_{3}\cdot(-4)^{1}+d_{1}+d_{-1}\cdot(-4)^{-1}+d% _{-3}\cdot(-4)^{-2}+\ldots]
  7. d 4 d 2 d 0 . d - 2 \ldots d_{4}d_{2}d_{0}.d_{-2}\ldots
  8. d 5 d 3 d 1 . d - 1 d - 3 \ldots d_{5}d_{3}d_{1}.d_{-1}d_{-3}\ldots
  9. d 3 d 2 d 1 d 0 \ldots d_{3}d_{2}d_{1}d_{0}
  10. + d 3 b 3 + d 2 b 2 + d 1 b + d 0 \cdots+d_{3}\cdot b^{3}+d_{2}\cdot b^{2}+d_{1}\cdot b+d_{0}
  11. b = 2 i b=2i
  12. d d
  13. d w - 1 , d w - 2 , d 0 d_{w-1},d_{w-2},...d_{0}
  14. w w
  15. b b
  16. Q 2 D w d k = 0 w - 1 d k b k Q2D_{w}\vec{d}\equiv\sum_{k=0}^{w-1}d_{k}\cdot b^{k}
  17. 1101 2 i 1101_{2i}
  18. 1 ( 2 i ) 3 + 1 ( 2 i ) 2 + 0 ( 2 i ) 1 + 1 ( 2 i ) 0 = - 8 i - 4 + 0 + 1 = - 3 - 8 i 1\cdot(2i)^{3}+1\cdot(2i)^{2}+0\cdot(2i)^{1}+1\cdot(2i)^{0}=-8i-4+0+1=-3-8i
  19. 1030003 2 i 1030003_{2i}
  20. 1 ( 2 i ) 6 + 3 ( 2 i ) 4 + 3 ( 2 i ) 0 = - 64 + 3 16 + 3 = - 13 1\cdot(2i)^{6}+3\cdot(2i)^{4}+3\cdot(2i)^{0}=-64+3\cdot 16+3=-13
  21. 7 10 \displaystyle 7_{10}
  22. 7 10 = 010303 2 i = 10303 2 i . \begin{aligned}\displaystyle 7_{10}=010303_{2i}=10303_{2i}.\end{aligned}
  23. i 𝐙 ∈i\mathbf{Z}
  24. 6 i 10 = 30 2 i \displaystyle 6i_{10}=30_{2i}
  25. d 5 d 4 d 3 d 2 d 1 d 0 . d - 1 d - 2 d - 3 ...d_{5}d_{4}d_{3}d_{2}d_{1}d_{0}.d_{-1}d_{-2}d_{-3}...
  26. d 5 b 5 + d 4 b 4 + d 3 b 3 + d 2 b 2 + d 1 b + d 0 + d - 1 b - 1 + d - 2 b - 2 + d - 3 b - 3 d_{5}b^{5}+d_{4}b^{4}+d_{3}b^{3}+d_{2}b^{2}+d_{1}b+d_{0}+d_{-1}b^{-1}+d_{-2}b^% {-2}+d_{-3}b^{-3}
  27. 32 i d 5 + 16 d 4 - 8 i d 3 - 4 d 2 + 2 i d 1 + d 0 + 1 2 i d - 1 + 1 - 4 d - 2 + 1 - 8 i d - 3 32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}+\frac{1}{2i}d_{-1}+\frac{1}{-4}d% _{-2}+\frac{1}{-8i}d_{-3}
  28. = 32 i d 5 + 16 d 4 - 8 i d 3 - 4 d 2 + 2 i d 1 + d 0 - i 2 d - 1 - 1 4 d - 2 + i 8 d - 3 =32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}-\frac{i}{2}d_{-1}-\frac{1}{4}d_% {-2}+\frac{i}{8}d_{-3}
  29. i \displaystyle i
  30. i = 10.2 2 i \displaystyle i=10.2_{2i}
  31. 12320 2 i 12320_{2i}
  32. 1011 2 i 1011_{2i}
  33. 1102 2 i 1102_{2i}
  34. 1131 2 i 1131_{2i}
  35. ( 9 - 8 i ) ( 29 + 4 i ) = 293 - 196 i (9-8i)\cdot(29+4i)=293-196i
  36. 5 = 16 + ( 3 - 4 ) + 1 = 10301 2 i 5=16+(3\cdot-4)+1=10301_{2i}
  37. i = 2 i + 2 ( - 1 2 i ) = 10.2 2 i i=2i+2\left(-\frac{1}{2}i\right)=10.2_{2i}
  38. 7 3 4 - 7 1 2 i = 1 ( 16 ) + 1 ( - 8 i ) + 2 ( - 4 ) + 1 ( 2 i ) + 3 ( - 1 2 i ) + 1 ( - 1 4 ) = 11210.31 2 i 7\frac{3}{4}-7\frac{1}{2}i=1(16)+1(-8i)+2(-4)+1(2i)+3\left(-\frac{1}{2}i\right% )+1\left(-\frac{1}{4}\right)=11210.31_{2i}

Quaternion_group.html

  1. Q = - 1 , i , j , k ( - 1 ) 2 = 1 , i 2 = j 2 = k 2 = i j k = - 1 , \mathrm{Q}=\langle-1,i,j,k\mid(-1)^{2}=1,\;i^{2}=j^{2}=k^{2}=ijk=-1\rangle,\,\!
  2. i j = k , j i = - k , j k = i , k j = - i , k i = j , i k = - j . \begin{aligned}\displaystyle ij&\displaystyle=k,&\displaystyle ji&% \displaystyle=-k,\\ \displaystyle jk&\displaystyle=i,&\displaystyle kj&\displaystyle=-i,\\ \displaystyle ki&\displaystyle=j,&\displaystyle ik&\displaystyle=-j.\end{aligned}
  3. x , y x 4 = 1 , x 2 = y 2 , y - 1 x y = x - 1 . \langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\rangle.\,\!
  4. Q = { ± 1 , ± i , ± j , ± k } GL 2 ( 𝐂 ) \mathrm{Q}=\{\pm 1,\pm i,\pm j,\pm k\}\to\mathrm{GL}_{2}(\mathbf{C})
  5. 1 ( 1 0 0 1 ) 1\mapsto\begin{pmatrix}1&0\\ 0&1\end{pmatrix}
  6. i ( i 0 0 - i ) i\mapsto\begin{pmatrix}i&0\\ 0&-i\end{pmatrix}
  7. j ( 0 1 - 1 0 ) j\mapsto\begin{pmatrix}0&1\\ -1&0\end{pmatrix}
  8. k ( 0 i i 0 ) k\mapsto\begin{pmatrix}0&i\\ i&0\end{pmatrix}
  9. Q = { ± 1 , ± i , ± j , ± k } GL ( 2 , 3 ) \mathrm{Q}=\{\pm 1,\pm i,\pm j,\pm k\}\to\mathrm{GL}(2,3)
  10. 1 ( 1 0 0 1 ) 1\mapsto\begin{pmatrix}1&0\\ 0&1\end{pmatrix}
  11. i ( 1 1 1 - 1 ) i\mapsto\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}
  12. j ( - 1 1 1 1 ) j\mapsto\begin{pmatrix}-1&1\\ 1&1\end{pmatrix}
  13. k ( 0 - 1 1 0 ) k\mapsto\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  14. x 8 - 72 x 6 + 180 x 4 - 144 x 2 + 36 x^{8}-72x^{6}+180x^{4}-144x^{2}+36
  15. x , y x 2 n = y 4 = 1 , x n = y 2 , y - 1 x y = x - 1 . \langle x,y\mid x^{2n}=y^{4}=1,x^{n}=y^{2},y^{-1}xy=x^{-1}\rangle.\,\!
  16. ( ω n 0 0 ω ¯ n ) and ( 0 - 1 1 0 ) \left(\begin{array}[]{cc}\omega_{n}&0\\ 0&\overline{\omega}_{n}\end{array}\right)\mbox{ and }~{}\left(\begin{array}[]{% cc}0&-1\\ 1&0\end{array}\right)

Quaternions_and_spatial_rotation.html

  1. θ θ
  2. u \vec{u}
  3. u \vec{u}
  4. θ θ
  5. ( 2 , 3 , 4 ) (2, 3, 4)
  6. 2 𝐢 + 3 𝐣 + 4 𝐤 2\mathbf{i}+3\mathbf{j}+4\mathbf{k}
  7. 𝐢 \mathbf{i}
  8. 𝐣 \mathbf{j}
  9. 𝐤 \mathbf{k}
  10. θ θ
  11. u = ( u x , u y , u z ) = u x 𝐢 + u y 𝐣 + u z 𝐤 \vec{u}=(u_{x},u_{y},u_{z})=u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}\mathbf{k}
  12. 𝐪 = e θ 2 ( u x 𝐢 + u y 𝐣 + u z 𝐤 ) = cos θ 2 + ( u x 𝐢 + u y 𝐣 + u z 𝐤 ) sin θ 2 \mathbf{q}=e^{\frac{\theta}{2}{(u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}\mathbf{k% })}}=\cos\frac{\theta}{2}+(u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}\mathbf{k})% \sin\frac{\theta}{2}
  13. 𝐩 = ( p x , p y , p z ) = p x 𝐢 + p y 𝐣 + p z 𝐤 \mathbf{p}=(p_{x},p_{y},p_{z})=p_{x}\mathbf{i}+p_{y}\mathbf{j}+p_{z}\mathbf{k}
  14. 𝐩 \mathbf{p}
  15. 𝐪 \mathbf{q}
  16. 𝐩 = 𝐪𝐩𝐪 - 1 \mathbf{p^{\prime}}=\mathbf{q}\mathbf{p}\mathbf{q}^{-1}
  17. u \vec{u}
  18. 𝐪 \mathbf{q}
  19. 𝐪 - 1 = e - θ 2 ( u x 𝐢 + u y 𝐣 + u z 𝐤 ) = cos θ 2 - ( u x 𝐢 + u y 𝐣 + u z 𝐤 ) sin θ 2 . \mathbf{q}^{-1}=e^{-\frac{\theta}{2}{(u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}% \mathbf{k})}}=\cos\frac{\theta}{2}-(u_{x}\mathbf{i}+u_{y}\mathbf{j}+u_{z}% \mathbf{k})\sin\frac{\theta}{2}.
  20. 𝐩 \mathbf{p}
  21. 𝐪 \mathbf{q}
  22. 𝐩𝐪 \mathbf{pq}
  23. 𝐩𝐪 v ( 𝐩𝐪 ) - 1 = 𝐩𝐪 v 𝐪 - 1 𝐩 - 1 = 𝐩 ( 𝐪 v 𝐪 - 1 ) 𝐩 - 1 \mathbf{pq}\vec{v}(\mathbf{pq})^{-1}=\mathbf{pq}\vec{v}\mathbf{q}^{-1}\mathbf{% p}^{-1}=\mathbf{p}(\mathbf{q}\vec{v}\mathbf{q}^{-1})\mathbf{p}^{-1}
  24. 𝐪 \mathbf{q}
  25. 𝐩 \mathbf{p}
  26. 𝐪 - 1 ( 𝐪 v 𝐪 - 1 ) 𝐪 = v \mathbf{q}^{-1}(\mathbf{q}\vec{v}\mathbf{q}^{-1})\mathbf{q}=\vec{v}
  27. n n
  28. 𝐪 \mathbf{q}
  29. n n
  30. 𝐪 = 𝐪 2 𝐪 1 \mathbf{q}^{\prime}=\mathbf{q}_{2}\mathbf{q}_{1}
  31. 𝐪 \mathbf{q′}
  32. 𝐢 \mathbf{i}
  33. 𝐣 \mathbf{j}
  34. 𝐤 \mathbf{k}
  35. 𝐩 \mathbf{p}
  36. 𝐪 \mathbf{q}
  37. f f
  38. v = 𝐢 + 𝐣 + 𝐤 \vec{v}=\mathbf{i}+\mathbf{j}+\mathbf{k}
  39. 2 π 3 \frac{2π}{3}
  40. α = 2 π 3 \alpha=\frac{2\pi}{3}
  41. v \vec{v}
  42. 3 \sqrt{3}
  43. π 3 \frac{π}{3}
  44. 1 2 \frac{1}{2}
  45. c o s 60 ° = 0.5 cos 60°=0.5
  46. 3 2 \frac{\sqrt{3}}{2}
  47. s i n 60 ° 0.866 sin 60°≈0.866
  48. u = cos α 2 + sin α 2 1 v v = cos π 3 + sin π 3 1 3 v = 1 2 + 3 2 1 3 v = 1 2 + 3 2 𝐢 + 𝐣 + 𝐤 3 = 1 + 𝐢 + 𝐣 + 𝐤 2 \begin{array}[]{lll}u&=&\cos\frac{\alpha}{2}+\sin\frac{\alpha}{2}\cdot\frac{1}% {\|\vec{v}\|}\vec{v}\\ &=&\cos\frac{\pi}{3}+\sin\frac{\pi}{3}\cdot\frac{1}{\sqrt{3}}\vec{v}\\ &=&\frac{1}{2}+\frac{\sqrt{3}}{2}\cdot\frac{1}{\sqrt{3}}\vec{v}\\ &=&\frac{1}{2}+\frac{\sqrt{3}}{2}\cdot\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{% \sqrt{3}}\\ &=&\frac{1+\mathbf{i}+\mathbf{j}+\mathbf{k}}{2}\end{array}
  49. f f
  50. f ( a 𝐢 + b 𝐣 + c 𝐤 ) = u ( a 𝐢 + b 𝐣 + c 𝐤 ) u - 1 f(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})=u(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})u% ^{-1}
  51. u - 1 = 1 - 𝐢 - 𝐣 - 𝐤 2 u^{-1}=\frac{1-\mathbf{i}-\mathbf{j}-\mathbf{k}}{2}
  52. f ( a 𝐢 + b 𝐣 + c 𝐤 ) = 1 + 𝐢 + 𝐣 + 𝐤 2 ( a 𝐢 + b 𝐣 + c 𝐤 ) 1 - 𝐢 - 𝐣 - 𝐤 2 f(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})=\frac{1+\mathbf{i}+\mathbf{j}+\mathbf{k% }}{2}(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})\frac{1-\mathbf{i}-\mathbf{j}-% \mathbf{k}}{2}
  53. f ( a 𝐢 + b 𝐣 + c 𝐤 ) = c 𝐢 + a 𝐣 + b 𝐤 f(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})=c\mathbf{i}+a\mathbf{j}+b\mathbf{k}
  54. f f
  55. 𝐢𝐣 = 𝐤 , 𝐣𝐢 = - 𝐤 , 𝐣𝐤 = 𝐢 , 𝐤𝐣 = - 𝐢 , 𝐤𝐢 = 𝐣 , 𝐢𝐤 = - 𝐣 , 𝐢 2 = 𝐣 2 = 𝐤 2 = - 1 \begin{aligned}\displaystyle\mathbf{ij}&\displaystyle=\mathbf{k},&% \displaystyle\mathbf{ji}&\displaystyle=\mathbf{-k},\\ \displaystyle\mathbf{jk}&\displaystyle=\mathbf{i},&\displaystyle\mathbf{kj}&% \displaystyle=\mathbf{-i},\\ \displaystyle\mathbf{ki}&\displaystyle=\mathbf{j},&\displaystyle\mathbf{ik}&% \displaystyle=\mathbf{-j},\\ \displaystyle\mathbf{i}^{2}&\displaystyle=\mathbf{j}^{2}&\displaystyle=\mathbf% {k}^{2}&\displaystyle=-1\end{aligned}
  56. f ( a 𝐢 + b 𝐣 + c 𝐤 ) = \displaystyle f(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})=
  57. 𝐪 𝐩 𝐪 * \mathbf{q p q}*
  58. [ c + a x 2 ( 1 - c ) a x a y ( 1 - c ) - a z s a x a z ( 1 - c ) + a y s a y a x ( 1 - c ) + a z s c + a y 2 ( 1 - c ) a y a z ( 1 - c ) - a x s a z a x ( 1 - c ) - a y s a z a y ( 1 - c ) + a x s c + a z 2 ( 1 - c ) ] \begin{bmatrix}c+a_{x}^{2}(1-c)&a_{x}a_{y}(1-c)-a_{z}s&a_{x}a_{z}(1-c)+a_{y}s% \\ a_{y}a_{x}(1-c)+a_{z}s&c+a_{y}^{2}(1-c)&a_{y}a_{z}(1-c)-a_{x}s\\ a_{z}a_{x}(1-c)-a_{y}s&a_{z}a_{y}(1-c)+a_{x}s&c+a_{z}^{2}(1-c)\end{bmatrix}
  59. s s
  60. c c
  61. s i n θ sinθ
  62. c o s θ cosθ
  63. 𝟏 \mathbf{1}
  64. 𝐪 \displaystyle\mathbf{q}
  65. θ θ
  66. [ 1 - 2 q j 2 - 2 q k 2 2 ( q i q j - q k q r ) 2 ( q i q k + q j q r ) 2 ( q i q j + q k q r ) 1 - 2 q i 2 - 2 q k 2 2 ( q j q k - q i q r ) 2 ( q i q k - q j q r ) 2 ( q j q k + q i q r ) 1 - 2 q i 2 - 2 q j 2 ] \begin{bmatrix}1-2q_{j}^{2}-2q_{k}^{2}&2(q_{i}q_{j}-q_{k}q_{r})&2(q_{i}q_{k}+q% _{j}q_{r})\\ 2(q_{i}q_{j}+q_{k}q_{r})&1-2q_{i}^{2}-2q_{k}^{2}&2(q_{j}q_{k}-q_{i}q_{r})\\ 2(q_{i}q_{k}-q_{j}q_{r})&2(q_{j}q_{k}+q_{i}q_{r})&1-2q_{i}^{2}-2q_{j}^{2}\end{bmatrix}
  67. 𝐢 \mathbf{i}
  68. ( a + b 𝐢 ) ( c + d 𝐢 ) = a c + a d 𝐢 + b 𝐢 c + b 𝐢 d 𝐢 = a c + a d 𝐢 + b c 𝐢 + b d 𝐢 2 = ( a c - b d ) + ( b c + a d ) 𝐢 . (a+b\mathbf{i})(c+d\mathbf{i})=ac+ad\mathbf{i}+b\mathbf{i}c+b\mathbf{i}d% \mathbf{i}=ac+ad\mathbf{i}+bc\mathbf{i}+bd\mathbf{i}^{2}=(ac-bd)+(bc+ad)% \mathbf{i}.
  69. 𝐢 \mathbf{i}
  70. 𝐣 \mathbf{j}
  71. 𝐤 \mathbf{k}
  72. ( a + b 𝐢 + c 𝐣 + d 𝐤 ) ( e + f 𝐢 + g 𝐣 + h 𝐤 ) = (a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k})(e+f\mathbf{i}+g\mathbf{j}+h\mathbf{k})=
  73. ( a e - b f - c g - d h ) + ( a f + b e + c h - d g ) 𝐢 + ( a g - b h + c e + d f ) 𝐣 + ( a h + b g - c f + d e ) 𝐤 . (ae-bf-cg-dh)+(af+be+ch-dg)\mathbf{i}+(ag-bh+ce+df)\mathbf{j}+(ah+bg-cf+de)% \mathbf{k}.
  74. b 𝐢 + c 𝐣 + d 𝐤 b\mathbf{i}+c\mathbf{j}+d\mathbf{k}
  75. v = ( b , c , d ) \vec{v}=(b,c,d)
  76. a a
  77. 𝐑 \mathbf{R}
  78. a + b 𝐢 + c 𝐣 + d 𝐤 = a + v . a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}=a+\vec{v}.
  79. a + v = ( a , 0 ) + ( 0 , v ) . a+\vec{v}=(a,\vec{0})+(0,\vec{v}).
  80. v w = v × w - v w , \vec{v}\vec{w}=\vec{v}\times\vec{w}-\vec{v}\cdot\vec{w},
  81. v w \vec{v}\vec{w}
  82. v × w \vec{v}\times\vec{w}
  83. v w \vec{v}\cdot\vec{w}
  84. ( s + v ) ( t + w ) = ( s t - v w ) + ( s w + t v + v × w ) . (s+\vec{v})(t+\vec{w})=(st-\vec{v}\cdot\vec{w})+(s\vec{w}+t\vec{v}+\vec{v}% \times\vec{w}).
  85. ( s + v ) - 1 = ( s + v ) * s + v 2 = s - v s 2 + v 2 , (s+\vec{v})^{-1}=\frac{(s+\vec{v})^{*}}{\lVert s+\vec{v}\rVert^{2}}=\frac{s-% \vec{v}}{s^{2}+\lVert\vec{v}\rVert^{2}},
  86. u \vec{u}
  87. q = cos α 2 + u sin α 2 q=\cos\frac{\alpha}{2}+\vec{u}\sin\frac{\alpha}{2}
  88. v = q v q - 1 = ( cos α 2 + u sin α 2 ) v ( cos α 2 - u sin α 2 ) \vec{v^{\prime}}=q\vec{v}q^{-1}=\left(\cos\frac{\alpha}{2}+\vec{u}\sin\frac{% \alpha}{2}\right)\,\vec{v}\,\left(\cos\frac{\alpha}{2}-\vec{u}\sin\frac{\alpha% }{2}\right)
  89. v \vec{v}
  90. α \alpha
  91. u \vec{u}
  92. v = v cos 2 α 2 + ( u v - v u ) sin α 2 cos α 2 - u v u sin 2 α 2 = v cos 2 α 2 + 2 ( u × v ) sin α 2 cos α 2 - ( v ( u u ) - 2 u ( u v ) ) sin 2 α 2 = v ( cos 2 α 2 - sin 2 α 2 ) + ( u × v ) ( 2 sin α 2 cos α 2 ) + u ( u v ) ( 2 sin 2 α 2 ) = v cos α + ( u × v ) sin α + u ( u v ) ( 1 - cos α ) = ( v - u ( u v ) ) cos α + ( u × v ) sin α + u ( u v ) = v cos α + ( u × v ) sin α + v \begin{aligned}\displaystyle\vec{v^{\prime}}&\displaystyle=\vec{v}\cos^{2}% \frac{\alpha}{2}+(\vec{u}\vec{v}-\vec{v}\vec{u})\sin\frac{\alpha}{2}\cos\frac{% \alpha}{2}-\vec{u}\vec{v}\vec{u}\sin^{2}\frac{\alpha}{2}\\ &\displaystyle=\vec{v}\cos^{2}\frac{\alpha}{2}+2(\vec{u}\times\vec{v})\sin% \frac{\alpha}{2}\cos\frac{\alpha}{2}-(\vec{v}(\vec{u}\cdot\vec{u})-2\vec{u}(% \vec{u}\cdot\vec{v}))\sin^{2}\frac{\alpha}{2}\\ &\displaystyle=\vec{v}(\cos^{2}\frac{\alpha}{2}-\sin^{2}\frac{\alpha}{2})+(% \vec{u}\times\vec{v})(2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})+\vec{u}(\vec{% u}\cdot\vec{v})(2\sin^{2}\frac{\alpha}{2})\\ &\displaystyle=\vec{v}\cos\alpha+(\vec{u}\times\vec{v})\sin\alpha+\vec{u}(\vec% {u}\cdot\vec{v})(1-\cos\alpha)\\ &\displaystyle=(\vec{v}-\vec{u}(\vec{u}\cdot\vec{v}))\cos\alpha+(\vec{u}\times% \vec{v})\sin\alpha+\vec{u}(\vec{u}\cdot\vec{v})\\ &\displaystyle=\vec{v}_{\bot}\cos\alpha+(\vec{u}\times\vec{v})\sin\alpha+\vec{% v}_{\|}\end{aligned}
  93. v \vec{v}_{\bot}
  94. v \vec{v}_{\|}
  95. α α
  96. ( w , x , y ) (w,x,y)
  97. ( w , x , y ) = ( 1 , 0 , 0 ) (w,x,y)=(1, 0, 0)
  98. ( w , x , y ) = ( 1 , 0 , 0 ) (w,x,y)=(−1, 0, 0)
  99. w = 0 w=0
  100. ( w , x , y ) (w,x,y)
  101. ( x , y , 0 ) (x,y, 0)
  102. α = 2 cos - 1 w = 2 sin - 1 x 2 + y 2 \alpha=2\cos^{-1}w=2\sin^{-1}\sqrt{x^{2}+y^{2}}
  103. w , x , y , z w,x,y,z
  104. ( w , x , y , z ) (w,x,y,z)
  105. ( x , y , z ) (x,y,z)
  106. α = 2 cos - 1 w = 2 sin - 1 x 2 + y 2 + z 2 . \alpha=2\cos^{-1}w=2\sin^{-1}\sqrt{x^{2}+y^{2}+z^{2}}.
  107. 𝐚 × 𝐛 = 𝐜 \mathbf{a}×\mathbf{b}=\mathbf{c}
  108. u \vec{u}
  109. 𝐢𝐣 = 𝐤 \mathbf{ij}=\mathbf{k}
  110. 𝐣𝐢 = 𝐤 \mathbf{ji}=−\mathbf{k}
  111. 𝐪 \mathbf{q}
  112. 𝐳 = a + b 𝐢 + c 𝐣 + d 𝐤 \mathbf{z}=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}
  113. | 𝐳 | = 1 |\mathbf{z}|=1
  114. R = ( a 2 + b 2 - c 2 - d 2 2 b c - 2 a d 2 b d + 2 a c 2 b c + 2 a d a 2 - b 2 + c 2 - d 2 2 c d - 2 a b 2 b d - 2 a c 2 c d + 2 a b a 2 - b 2 - c 2 + d 2 ) . R=\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2bc-2ad&2bd+2ac\\ 2bc+2ad&a^{2}-b^{2}+c^{2}-d^{2}&2cd-2ab\\ 2bd-2ac&2cd+2ab&a^{2}-b^{2}-c^{2}+d^{2}\\ \end{pmatrix}.
  115. 𝐳 * = a - b 𝐢 - c 𝐣 - d 𝐤 \mathbf{z^{*}}=a-b\mathbf{i}-c\mathbf{j}-d\mathbf{k}
  116. 𝐪 \mathbf{q}
  117. Q Q
  118. Q Q
  119. 𝐪 \mathbf{q}
  120. Q Q
  121. K = 1 3 [ Q x x - Q y y - Q z z Q y x + Q x y Q z x + Q x z Q y z - Q z y Q y x + Q x y Q y y - Q x x - Q z z Q z y + Q y z Q z x - Q x z Q z x + Q x z Q z y + Q y z Q z z - Q x x - Q y y Q x y - Q y x Q y z - Q z y Q z x - Q x z Q x y - Q y x Q x x + Q y y + Q z z ] , K=\frac{1}{3}\begin{bmatrix}Q_{xx}-Q_{yy}-Q_{zz}&Q_{yx}+Q_{xy}&Q_{zx}+Q_{xz}&Q% _{yz}-Q_{zy}\\ Q_{yx}+Q_{xy}&Q_{yy}-Q_{xx}-Q_{zz}&Q_{zy}+Q_{yz}&Q_{zx}-Q_{xz}\\ Q_{zx}+Q_{xz}&Q_{zy}+Q_{yz}&Q_{zz}-Q_{xx}-Q_{yy}&Q_{xy}-Q_{yx}\\ Q_{yz}-Q_{zy}&Q_{zx}-Q_{xz}&Q_{xy}-Q_{yx}&Q_{xx}+Q_{yy}+Q_{zz}\end{bmatrix},
  122. ( x , y , z , w ) (x,y,z,w)
  123. Q Q
  124. Q Q
  125. R R
  126. 𝐪 = ( w , [ u v e c , u r ] ) \mathbf{q}=(w,[u^{\prime}vec^{\prime},u^{\prime}r^{\prime}])
  127. w w
  128. v new = v + 2 r × ( r × v + w v ) \vec{v}\text{new}=\vec{v}+2\vec{r}\times(\vec{r}\times\vec{v}+w\vec{v})
  129. v new = q v q - 1 \vec{v}\text{new}=q\vec{v}q^{-1}
  130. R R
  131. R R
  132. R R
  133. R R
  134. R R
  135. f ( v ) = 𝐳 l v 𝐳 r = ( a l - b l - c l - d l b l a l - d l c l c l d l a l - b l d l - c l b l a l ) ( a r - b r - c r - d r b r a r d r - c r c r - d r a r b r d r c r - b r a r ) ( w x y z ) . f(\vec{v})=\mathbf{z}_{\rm{l}}\vec{v}\mathbf{z}_{\rm{r}}=\begin{pmatrix}a_{\rm% {l}}&-b_{\rm{l}}&-c_{\rm{l}}&-d_{\rm{l}}\\ b_{\rm{l}}&a_{\rm{l}}&-d_{\rm{l}}&c_{\rm{l}}\\ c_{\rm{l}}&d_{\rm{l}}&a_{\rm{l}}&-b_{\rm{l}}\\ d_{\rm{l}}&-c_{\rm{l}}&b_{\rm{l}}&a_{\rm{l}}\end{pmatrix}\begin{pmatrix}a_{\rm% {r}}&-b_{\rm{r}}&-c_{\rm{r}}&-d_{\rm{r}}\\ b_{\rm{r}}&a_{\rm{r}}&d_{\rm{r}}&-c_{\rm{r}}\\ c_{\rm{r}}&-d_{\rm{r}}&a_{\rm{r}}&b_{\rm{r}}\\ d_{\rm{r}}&c_{\rm{r}}&-b_{\rm{r}}&a_{\rm{r}}\end{pmatrix}\begin{pmatrix}w\\ x\\ y\\ z\end{pmatrix}.
  136. ( 𝐳 l v ) 𝐳 r = 𝐳 l ( v 𝐳 r ) (\mathbf{z}_{\rm{l}}\vec{v})\mathbf{z}_{\rm{r}}=\mathbf{z}_{\rm{l}}(\vec{v}% \mathbf{z}_{\rm{r}})
  137. 𝐳 l v 𝐳 r = ( 1 - d t a b - d t a c - d t a d d t a b 1 - d t b c - d t b d d t a c d t b c 1 - d t c d d t a d d t b d d t c d 1 ) ( w x y z ) \mathbf{z}_{\rm{l}}\vec{v}\mathbf{z}_{\rm{r}}=\begin{pmatrix}1&-dt_{ab}&-dt_{% ac}&-dt_{ad}\\ dt_{ab}&1&-dt_{bc}&-dt_{bd}\\ dt_{ac}&dt_{bc}&1&-dt_{cd}\\ dt_{ad}&dt_{bd}&dt_{cd}&1\end{pmatrix}\begin{pmatrix}w\\ x\\ y\\ z\end{pmatrix}
  138. 𝐳 l = 1 + d t a b + d t c d 2 i + d t a c - d t b d 2 j + d t a d + d t b c 2 k \mathbf{z}_{\rm{l}}=1+{dt_{ab}+dt_{cd}\over 2}i+{dt_{ac}-dt_{bd}\over 2}j+{dt_% {ad}+dt_{bc}\over 2}k
  139. 𝐳 r = 1 + d t a b - d t c d 2 i + d t a c + d t b d 2 j + d t a d - d t b c 2 k \mathbf{z}_{\rm{r}}=1+{dt_{ab}-dt_{cd}\over 2}i+{dt_{ac}+dt_{bd}\over 2}j+{dt_% {ad}-dt_{bc}\over 2}k

Quintic_function.html

  1. g ( x ) = a x 5 + b x 4 + c x 3 + d x 2 + e x + f , g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,
  2. a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0. ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,
  3. x 5 - x + 1 = 0. x^{5}-x+1=0.
  4. x 5 - x 4 - x + 1 = 0 = ( x 2 + 1 ) ( x + 1 ) ( x - 1 ) 2 . x^{5}-x^{4}-x+1=0=(x^{2}+1)(x+1)(x-1)^{2}.
  5. a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0 , ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0,
  6. x = y - b 5 a x=y-\frac{b}{5a}
  7. y 5 + p y 3 + q y 2 + r y + s = 0 y^{5}+py^{3}+qy^{2}+ry+s=0
  8. p = 5 a c - 2 b 2 5 a 2 p=\frac{5ac-2b^{2}}{5a^{2}}
  9. q = 25 a 2 d - 15 a b c + 4 b 3 25 a 3 q=\frac{25a^{2}d-15abc+4b^{3}}{25a^{3}}
  10. r = 125 a 3 e - 50 a 2 b d + 15 a b 2 c - 3 b 4 125 a 4 r=\frac{125a^{3}e-50a^{2}bd+15ab^{2}c-3b^{4}}{125a^{4}}
  11. s = 3125 a 4 f - 625 a 3 b e + 125 a 2 b 2 d - 25 a b 3 c + 4 b 5 3125 a 5 s=\frac{3125a^{4}f-625a^{3}be+125a^{2}b^{2}d-25ab^{3}c+4b^{5}}{3125a^{5}}
  12. P 2 - 1024 z Δ P^{2}-1024z\Delta
  13. P = z 3 - z 2 ( 20 r + 3 p 2 ) - z ( 8 p 2 r - 16 p q 2 - 240 r 2 + 400 s q - 3 p 4 ) P=z^{3}-z^{2}(20r+3p^{2})-z(8p^{2}r-16pq^{2}-240r^{2}+400sq-3p^{4})
  14. - p 6 + 28 p 4 r - 16 p 3 q 2 - 176 p 2 r 2 - 80 p 2 s q + 224 p r q 2 - 64 q 4 {}-p^{6}+28p^{4}r-16p^{3}q^{2}-176p^{2}r^{2}-80p^{2}sq+224prq^{2}-64q^{4}
  15. + 4000 p s 2 + 320 r 3 - 1600 r s q {}+4000ps^{2}+320r^{3}-1600rsq
  16. Δ = - 128 p 2 r 4 + 3125 s 4 - 72 p 4 q r s + 560 p 2 q r 2 s + 16 p 4 r 3 + 256 r 5 + 108 p 5 s 2 \Delta=-128p^{2}r^{4}+3125s^{4}-72p^{4}qrs+560p^{2}qr^{2}s+16p^{4}r^{3}+256r^{% 5}+108p^{5}s^{2}
  17. - 1600 q r 3 s + 144 p q 2 r 3 - 900 p 3 r s 2 + 2000 p r 2 s 2 - 3750 p q s 3 + 825 p 2 q 2 s 2 {}-1600qr^{3}s+144pq^{2}r^{3}-900p^{3}rs^{2}+2000pr^{2}s^{2}-3750pqs^{3}+825p^% {2}q^{2}s^{2}
  18. + 2250 q 2 r s 2 + 108 q 5 s - 27 q 4 r 2 - 630 p q 3 r s + 16 p 3 q 3 s - 4 p 3 q 2 r 2 . {}+2250q^{2}rs^{2}+108q^{5}s-27q^{4}r^{2}-630pq^{3}rs+16p^{3}q^{3}s-4p^{3}q^{2% }r^{2}.
  19. x 5 + a x + b = 0 , x^{5}+ax+b=0,
  20. x 5 + 5 μ 4 ( 4 ν + 3 ) ν 2 + 1 x + 4 μ 5 ( 2 ν + 1 ) ( 4 ν + 3 ) ν 2 + 1 = 0 x^{5}+\frac{5\mu^{4}(4\nu+3)}{\nu^{2}+1}x+\frac{4\mu^{5}(2\nu+1)(4\nu+3)}{\nu^% {2}+1}=0
  21. μ \mu
  22. ν \nu
  23. x 5 + 5 e 4 ( 4 c + 3 ) c 2 + 1 x + - 4 e 5 ( 2 c - 11 ) c 2 + 1 = 0. x^{5}+\frac{5e^{4}(4c+3)}{c^{2}+1}x+\frac{-4e^{5}(2c-11)}{c^{2}+1}=0.
  24. b = 4 5 ( a + 20 ± 2 ( 20 - a ) ( 5 + a ) ) b=\frac{4}{5}\left(a+20\pm 2\sqrt{(20-a)(5+a)}\right)
  25. a = 5 ( 4 ν + 3 ) ν 2 + 1 . a=\frac{5(4\nu+3)}{\nu^{2}+1}.
  26. c = - m / l 5 c=-m/l^{5}
  27. e = 1 / l e=1/l
  28. x 5 + a x + b = 0 x^{5}+ax+b=0
  29. a = 5 l ( 3 l 5 - 4 m ) m 2 + l 10 b = 4 ( 11 l 5 + 2 m ) m 2 + l 10 a=\frac{5l(3l^{5}-4m)}{m^{2}+l^{10}}\qquad b=\frac{4(11l^{5}+2m)}{m^{2}+l^{10}}
  30. - 10 - 2 5 + 5 - 1 4 . \frac{\sqrt{-10-2\sqrt{5}}+\sqrt{5}-1}{4}.
  31. x 5 - 5 x 4 + 30 x 3 - 50 x 2 + 55 x - 21 = 0 , x^{5}-5x^{4}+30x^{3}-50x^{2}+55x-21=0,
  32. x = 1 + 2 5 - ( 2 5 ) 2 + ( 2 5 ) 3 - ( 2 5 ) 4 . x=1+\sqrt[5]{2}-\left(\sqrt[5]{2}\right)^{2}+\left(\sqrt[5]{2}\right)^{3}-% \left(\sqrt[5]{2}\right)^{4}.
  33. x 5 - 5 x + 12 = 0. x^{5}-5x+12=0.
  34. a = 2 φ - 1 a=\sqrt{2\varphi^{-1}}
  35. b = 2 φ b=\sqrt{2\varphi}
  36. c = 5 4 , c=\sqrt[4]{5}\,,
  37. φ = 1 + 5 2 \varphi=\frac{1+\sqrt{5}}{2}
  38. x = - 1.84208 x=-1.84208\dots
  39. - c x = ( a + c ) 2 ( b - c ) 5 + ( - a + c ) ( b - c ) 2 5 + ( a + c ) ( b + c ) 2 5 - ( - a + c ) 2 ( b + c ) 5 , -cx=\sqrt[5]{(a+c)^{2}(b-c)}+\sqrt[5]{(-a+c)(b-c)^{2}}+\sqrt[5]{(a+c)(b+c)^{2}% }-\sqrt[5]{(-a+c)^{2}(b+c)}\,,
  40. x = y 1 5 + y 2 5 + y 3 5 + y 4 5 , x=\sqrt[5]{y_{1}}+\sqrt[5]{y_{2}}+\sqrt[5]{y_{3}}+\sqrt[5]{y_{4}}\,,
  41. y 4 + 4 y 3 + 4 5 y 2 - 8 5 3 y - 1 5 5 = 0 . y^{4}+4y^{3}+\frac{4}{5}y^{2}-\frac{8}{5^{3}}y-\frac{1}{5^{5}}=0\,.
  42. P ( x ) = 0 P(x)=0
  43. p p
  44. Q ( y ) = 0 Q(y)=0
  45. p 1 p–1
  46. P P
  47. p p
  48. Q Q
  49. p p
  50. Q Q
  51. P ( x ) = 0 P(x)=0
  52. p p
  53. p p
  54. p p
  55. Q Q
  56. x 5 + 5 a x 3 + 5 a 2 x + b = 0 , x^{5}+5ax^{3}+5a^{2}x+b=0\,,
  57. y 2 + b y - a 5 = 0 , y^{2}+by-a^{5}=0\,,
  58. x k = ω k y i 5 - a ω k y i 5 , x_{k}=\omega^{k}\sqrt[5]{y_{i}}-\frac{a}{\omega^{k}\sqrt[5]{y_{i}}},
  59. x 5 + a x 2 + b x^{5}+ax^{2}+b
  60. x 5 - 2 s 3 x 2 - s 5 5 x^{5}-2s^{3}x^{2}-\frac{s^{5}}{5}
  61. x 5 - 100 s 3 x 2 - 1000 s 5 x^{5}-100s^{3}x^{2}-1000s^{5}
  62. x 5 - 5 s 3 x 2 - 3 s 5 x^{5}-5s^{3}x^{2}-3s^{5}
  63. x 5 - 5 s 3 x 2 + 15 s 5 x^{5}-5s^{3}x^{2}+15s^{5}
  64. x 5 - 25 s 3 x 2 - 300 s 5 x^{5}-25s^{3}x^{2}-300s^{5}
  65. x 5 - 10 x 3 - 20 x 2 - 1505 x - 7412 x^{5}-10x^{3}-20x^{2}-1505x-7412
  66. x 5 + 625 4 x + 3750 x^{5}+\frac{625}{4}x+3750
  67. x 5 - 22 5 x 3 - 11 25 x 2 + 462 125 x + 979 3125 x^{5}-\frac{22}{5}x^{3}-\frac{11}{25}x^{2}+\frac{462}{125}x+\frac{979}{3125}
  68. x 5 + 20 x 3 + 20 x 2 + 30 x + 10 x^{5}+20x^{3}+20x^{2}+30x+10
  69. ~{}\qquad~{}
  70. 2 5 - 2 5 2 + 2 5 3 - 2 5 4 \sqrt[5]{2}-\sqrt[5]{2}^{2}+\sqrt[5]{2}^{3}-\sqrt[5]{2}^{4}
  71. x 5 + 320 x 2 - 1000 x + 4288 x^{5}+320x^{2}-1000x+4288
  72. ~{}\qquad~{}
  73. x 5 + 40 x 2 - 69 x + 108 x^{5}+40x^{2}-69x+108
  74. ~{}\qquad~{}
  75. x 5 - 20 x 3 + 250 x - 400 x^{5}-20x^{3}+250x-400
  76. x 5 - 5 x 3 + 85 8 x - 13 / 2 x^{5}-5x^{3}+\frac{85}{8}x-13/2
  77. x 5 + 20 17 x + 21 17 x^{5}+\frac{20}{17}x+\frac{21}{17}
  78. x 5 - 4 13 x + 29 65 x^{5}-\frac{4}{13}x+\frac{29}{65}
  79. x 5 + 10 13 x + 3 13 x^{5}+\frac{10}{13}x+\frac{3}{13}
  80. x 5 + 110 ( 5 x 3 + 60 x 2 + 800 x + 8320 ) x^{5}+110(5x^{3}+60x^{2}+800x+8320)
  81. x 5 - 20 x 3 - 80 x 2 - 150 x - 656 x^{5}-20x^{3}-80x^{2}-150x-656
  82. x 5 - 40 x 3 + 160 x 2 + 1000 x - 5888 x^{5}-40x^{3}+160x^{2}+1000x-5888
  83. x 5 - 50 x 3 - 600 x 2 - 2000 x - 11200 x^{5}-50x^{3}-600x^{2}-2000x-11200
  84. x 5 + 110 ( 5 x 3 + 20 x 2 - 360 x + 800 ) x^{5}+110(5x^{3}+20x^{2}-360x+800)
  85. x 5 - 20 x 3 + 320 x 2 + 540 x + 6368 x^{5}-20x^{3}+320x^{2}+540x+6368
  86. ~{}\qquad~{}
  87. x 5 - 20 x 3 - 160 x 2 - 420 x - 8928 x^{5}-20x^{3}-160x^{2}-420x-8928
  88. ~{}\qquad~{}
  89. x 5 - 20 x 3 + 170 x + 208 x^{5}-20x^{3}+170x+208
  90. x 5 + x 4 - 4 x 3 - 3 x 2 + 3 x + 1 x^{5}+x^{4}-4x^{3}-3x^{2}+3x+1
  91. 2 cos ( 2 k π 11 ) 2\cos(\frac{2k\pi}{11})
  92. x 5 + x 4 - 12 x 3 - 21 x 2 + x + 5 x^{5}+x^{4}-12x^{3}-21x^{2}+x+5
  93. k = 0 5 e 2 i π 6 k 31 \sum_{k=0}^{5}e^{\frac{2i\pi 6^{k}}{31}}
  94. y 5 + y 4 - 16 y 3 + 5 y 2 + 21 y - 9 y^{5}+y^{4}-16y^{3}+5y^{2}+21y-9
  95. k = 0 7 e 2 i π 3 k 41 \sum_{k=0}^{7}e^{\frac{2i\pi 3^{k}}{41}}
  96. y 5 + y 4 - 24 y 3 - 17 y 2 + 41 y - 13 y^{5}+y^{4}-24y^{3}-17y^{2}+41y-13
  97. ~{}\qquad~{}
  98. y 5 + y 4 - 28 y 3 + 37 y 2 + 25 y + 1 y^{5}+y^{4}-28y^{3}+37y^{2}+25y+1
  99. x 5 + ( a - 3 ) x 4 + ( - a + b + 3 ) x 3 + ( a 2 - a - 1 - 2 b ) x 2 + b x + a = 0 x^{5}+(a-3)x^{4}+(-a+b+3)x^{3}+(a^{2}-a-1-2b)x^{2}+bx+a=0\,
  100. a , l , m a,l,m
  101. x 5 - 5 p ( 2 x 3 + a x 2 + b x ) - p c = 0 x^{5}-5p(2x^{3}+ax^{2}+bx)-pc=0\,
  102. p = l 2 ( 4 m 2 + a 2 ) - m 2 4 , b = l ( 4 m 2 + a 2 ) - 5 p - 2 m 2 , c = b ( a + 4 m ) - p ( a - 4 m ) - a 2 m 2 p=\frac{l^{2}(4m^{2}+a^{2})-m^{2}}{4},\qquad b=l(4m^{2}+a^{2})-5p-2m^{2},% \qquad c=\frac{b(a+4m)-p(a-4m)-a^{2}m}{2}
  103. t 5 + t - a = 0 t^{5}+t-a=0\,
  104. a a\,
  105. x 5 + a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\,
  106. x 5 - x + t = 0 . x^{5}-x+t=0\,.
  107. x 5 + a x + b x^{5}+ax+b
  108. x 5 + c x + d x^{5}+cx+d

Quotient_algebra.html

  1. 𝒜 \mathcal{A}
  2. a 1 , a 2 , , a n , b 1 , b 2 , , b n A a_{1},a_{2},\ldots,a_{n},b_{1},b_{2},\ldots,b_{n}\in A
  3. ( a 1 , b 1 ) E , ( a 2 , b 2 ) E , , ( a n , b n ) E (a_{1},b_{1})\in E,(a_{2},b_{2})\in E,\ldots,(a_{n},b_{n})\in E
  4. ( f ( a 1 , a 2 , , a n ) , f ( b 1 , b 2 , , b n ) ) E (f(a_{1},a_{2},\ldots,a_{n}),f(b_{1},b_{2},\ldots,b_{n}))\in E
  5. 𝒜 \mathcal{A}
  6. f i 𝒜 f^{\mathcal{A}}_{i}
  7. n i n_{i}
  8. 𝒜 \mathcal{A}
  9. 𝒜 \mathcal{A}
  10. f i 𝒜 / E : ( A / E ) n i A / E f^{\mathcal{A}/E}_{i}:(A/E)^{n_{i}}\to A/E
  11. f i 𝒜 / E ( [ a 1 ] E , , [ a n i ] E ) = [ f i 𝒜 ( a 1 , , a n i ) ] E f^{\mathcal{A}/E}_{i}([a_{1}]_{E},\ldots,[a_{n_{i}}]_{E})=[f^{\mathcal{A}}_{i}% (a_{1},\ldots,a_{n_{i}})]_{E}
  12. [ a ] E [a]_{E}
  13. 𝒜 = ( A , ( f i 𝒜 ) i I ) \mathcal{A}=(A,(f^{\mathcal{A}}_{i})_{i\in I})
  14. 𝒜 \mathcal{A}
  15. 𝒜 / E = ( A / E , ( f i 𝒜 / E ) i I ) \mathcal{A}/E=(A/E,(f^{\mathcal{A}/E}_{i})_{i\in I})
  16. 𝒜 \mathcal{A}
  17. 𝒜 \mathcal{A}
  18. 𝒜 / E \mathcal{A}/E
  19. ker h = { ( a , a ) A × A | h ( a ) = h ( a ) } \mathop{\mathrm{ker}}\,h=\{(a,a^{\prime})\in A\times A|h(a)=h(a^{\prime})\}
  20. 𝒜 \mathcal{A}
  21. 𝒜 \mathcal{A}
  22. 𝒜 \mathcal{A}
  23. 𝒜 / ker h \mathcal{A}/\mathop{\mathrm{ker}}\,h
  24. h : 𝒜 h:\mathcal{A}\to\mathcal{B}
  25. 𝒜 / ker h \mathcal{A}/\mathop{\mathrm{ker}}\,h
  26. \mathcal{B}
  27. ker h \mathop{\mathrm{ker}}\,h
  28. 𝒜 \mathcal{A}
  29. A × A A\times A
  30. Con ( 𝒜 ) \mathrm{Con}(\mathcal{A})
  31. 𝒜 \mathcal{A}
  32. : Con ( 𝒜 ) × Con ( 𝒜 ) Con ( 𝒜 ) \wedge:\mathrm{Con}(\mathcal{A})\times\mathrm{Con}(\mathcal{A})\to\mathrm{Con}% (\mathcal{A})
  33. E 1 E 2 = E 1 E 2 E_{1}\wedge E_{2}=E_{1}\cap E_{2}
  34. 𝒜 \mathcal{A}
  35. E 𝒜 = { F Con ( 𝒜 ) | E F } \langle E\rangle_{\mathcal{A}}=\bigcap\{F\in\mathrm{Con}(\mathcal{A})|E% \subseteq F\}
  36. 𝒜 \mathcal{A}
  37. : Con ( 𝒜 ) × Con ( 𝒜 ) Con ( 𝒜 ) \vee:\mathrm{Con}(\mathcal{A})\times\mathrm{Con}(\mathcal{A})\to\mathrm{Con}(% \mathcal{A})
  38. E 1 E 2 = E 1 E 2 𝒜 E_{1}\vee E_{2}=\langle E_{1}\cup E_{2}\rangle_{\mathcal{A}}
  39. 𝒜 \mathcal{A}
  40. ( 𝒜 , , ) (\mathcal{A},\wedge,\vee)
  41. 𝒜 \mathcal{A}
  42. α β = β α \alpha\circ\beta=\beta\circ\alpha
  43. α β = α β \alpha\circ\beta=\alpha\vee\beta

Quotient_ring.html

  1. \oplus
  2. R R R\oplus R
  3. R X , Y . R\langle X,Y\rangle.
  4. R X , Y / ( X 2 + 1 , Y 2 + 1 , X Y + Y X ) . R\langle X,Y\rangle/(X^{2}+1,Y^{2}+1,XY+YX).

Quotient_rule.html

  1. f ( x ) f(x)
  2. f ( x ) = g ( x ) h ( x ) f(x)=\frac{g(x)}{h(x)}
  3. h ( x ) 0 h(x)\not=0
  4. g ( x ) / h ( x ) g(x)/h(x)
  5. f ( x ) = g ( x ) h ( x ) - g ( x ) h ( x ) [ h ( x ) ] 2 . f^{\prime}(x)=\frac{g^{\prime}(x)h(x)-g(x)h^{\prime}(x)}{[h(x)]^{2}}.
  6. h ( a ) 0 h(a)\not=0
  7. g ( a ) g^{\prime}(a)
  8. h ( a ) h^{\prime}(a)
  9. f ( a ) f^{\prime}(a)
  10. f ( a ) = g ( a ) h ( a ) - g ( a ) h ( a ) [ h ( a ) ] 2 . f^{\prime}(a)=\frac{g^{\prime}(a)h(a)-g(a)h^{\prime}(a)}{[h(a)]^{2}}.
  11. f ( x ) = g ( x ) ( h ( x ) ) - 1 f(x)=g(x)(h(x))^{-1}
  12. f ′′ ( x ) = g ′′ ( x ) [ h ( x ) ] 2 - 2 g ( x ) h ( x ) h ( x ) + g ( x ) [ 2 [ h ( x ) ] 2 - h ( x ) h ′′ ( x ) ] [ h ( x ) ] 3 . f^{\prime\prime}(x)=\frac{g^{\prime\prime}(x)[h(x)]^{2}-2g^{\prime}(x)h(x)h^{% \prime}(x)+g(x)[2[h^{\prime}(x)]^{2}-h(x)h^{\prime\prime}(x)]}{[h(x)]^{3}}.
  13. ( 4 x - 2 ) / ( x 2 + 1 ) (4x-2)/(x^{2}+1)
  14. d d x [ ( 4 x - 2 ) x 2 + 1 ] \displaystyle\frac{d}{dx}\left[\frac{(4x-2)}{x^{2}+1}\right]
  15. g ( x ) = 4 x - 2 g(x)=4x-2
  16. h ( x ) = x 2 + 1 h(x)=x^{2}+1
  17. cos ( x ) x 2 - sin ( x ) 2 x x 4 \frac{\cos(x)x^{2}-\sin(x)2x}{x^{4}}
  18. f ( x ) = g ( x ) h ( x ) f(x)=\frac{g(x)}{h(x)}
  19. g ( x ) = f ( x ) h ( x ) g(x)=f(x)h(x)\mbox{ }~{}\,
  20. g ( x ) = f ( x ) h ( x ) + f ( x ) h ( x ) g^{\prime}(x)=f^{\prime}(x)h(x)+f(x)h^{\prime}(x)\mbox{ }~{}\,
  21. f ( x ) = g ( x ) - f ( x ) h ( x ) h ( x ) = g ( x ) - g ( x ) h ( x ) h ( x ) h ( x ) f^{\prime}(x)=\frac{g^{\prime}(x)-f(x)h^{\prime}(x)}{h(x)}=\frac{g^{\prime}(x)% -\frac{g(x)}{h(x)}\cdot h^{\prime}(x)}{h(x)}
  22. f ( x ) = g ( x ) h ( x ) - g ( x ) h ( x ) ( h ( x ) ) 2 f^{\prime}(x)=\frac{g^{\prime}(x)h(x)-g(x)h^{\prime}(x)}{\left(h(x)\right)^{2}}
  23. f = u v f=\frac{u}{v}
  24. ln f = ln u - ln v \ln f=\ln u-\ln v
  25. f f = u u - v v \frac{f^{\prime}}{f}=\frac{u^{\prime}}{u}-\frac{v^{\prime}}{v}
  26. f u / v = u v - u v u v \frac{f^{\prime}}{u/v}=\frac{u^{\prime}v-uv^{\prime}}{uv}
  27. f = u v - u v u v u v f^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{uv}\cdot\frac{u}{v}
  28. f = u v - u v v 2 f^{\prime}=\frac{u^{\prime}v-uv^{\prime}}{v^{2}}

Quotient_space_(topology).html

  1. ~{}
  2. X X
  3. Y = X / Y=X/~{}
  4. X X
  5. Y = { [ x ] : x X } = { { v X : v x } : x X } , Y=\{[x]:x\in X\}=\{\{v\in X:v\sim x\}:x\in X\},
  6. τ Y = { U Y : U = ( [ a ] U [ a ] ) τ X } . \tau_{Y}=\left\{U\subseteq Y:\bigcup U=\left(\bigcup_{[a]\in U}[a]\right)\in% \tau_{X}\right\}.
  7. q : X X / q:X→X/~{}
  8. X X
  9. τ Y = { U Y : q - 1 ( U ) τ X } . \tau_{Y}=\left\{U\subseteq Y:q^{-1}(U)\in\tau_{X}\right\}.
  10. q q
  11. f : X Y f:X\to Y
  12. f - 1 ( U ) f^{-1}(U)
  13. f f
  14. Y Y
  15. f f
  16. \sim
  17. X X
  18. q : X X / q:X\to X/{\sim}
  19. x , y X x,y\in X
  20. D 2 / D 2 D^{2}/\partial{D^{2}}
  21. X = X=\mathbb{R}

R-value_(insulation).html

  1. Q ˙ A \dot{Q}_{A}
  2. R = Δ T / Q ˙ A R=\Delta T/\dot{Q}_{A}
  3. U = 1 R = Q ˙ A Δ T = k L U=\frac{1}{R}=\frac{\dot{Q}_{A}}{\Delta T}=\frac{k}{L}

Rader's_FFT_algorithm.html

  1. p m p^{m}
  2. X k = n = 0 N - 1 x n e - 2 π i N n k k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}nk}\qquad k=0,\dots,N-1.
  3. X 0 = n = 0 N - 1 x n , X_{0}=\sum_{n=0}^{N-1}x_{n},
  4. X g - p = x 0 + q = 0 N - 2 x g q e - 2 π i N g - ( p - q ) p = 0 , , N - 2. X_{g^{-p}}=x_{0}+\sum_{q=0}^{N-2}x_{g^{q}}e^{-\frac{2\pi i}{N}g^{-(p-q)}}% \qquad p=0,\dots,N-2.
  5. a q = x g q a_{q}=x_{g^{q}}
  6. b q = e - 2 π i N g - q . b_{q}=e^{-\frac{2\pi i}{N}g^{-q}}.

Radical_of_an_ideal.html

  1. I \sqrt{I}
  2. I = { r R r n I for some positive integer n } . \sqrt{I}=\{r\in R\mid r^{n}\in I\ \hbox{for some positive integer}\ n\}.
  3. R / I R/I
  4. ( a + b ) n + m - 1 = i = 0 n + m - 1 ( n + m - 1 i ) a i b n + m - 1 - i . (a+b)^{n+m-1}=\sum_{i=0}^{n+m-1}{n+m-1\choose i}a^{i}b^{n+m-1-i}.
  5. I \sqrt{I}
  6. I \sqrt{I}
  7. I n = I \sqrt{I^{n}}=\sqrt{I}
  8. P = P \sqrt{P}=P
  9. I , J \sqrt{I},\sqrt{J}
  10. I , J I,J
  11. ann R ( M ) = 𝔭 supp M 𝔭 = 𝔭 ass M 𝔭 \sqrt{\operatorname{ann}_{R}(M)}=\bigcap_{\mathfrak{p}\in\operatorname{supp}M}% \mathfrak{p}=\bigcap_{\mathfrak{p}\in\operatorname{ass}M}\mathfrak{p}
  12. supp M \operatorname{supp}M
  13. ass M \operatorname{ass}M
  14. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  15. I ( V ( J ) ) = Rad ( J ) \operatorname{I}(\operatorname{V}(J))=\operatorname{Rad}(J)\,
  16. V ( J ) = { x k n | f ( x ) = 0 for all f J } \operatorname{V}(J)=\{x\in k^{n}\ |\ f(x)=0\mbox{ for all }~{}f\in J\}
  17. I ( S ) = { f k [ x 1 , x 2 , x n ] | f ( x ) = 0 for all x S } . \operatorname{I}(S)=\{f\in k[x_{1},x_{2},\ldots x_{n}]\ |\ f(x)=0\mbox{ for % all }~{}x\in S\}.
  18. I ( V ( - ) ) = Rad ( - ) \operatorname{I}(\operatorname{V}(-))=\operatorname{Rad}(-)\,
  19. R = I + J = I + J R=\sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I+J}
  20. I + J = R I+J=R

Radio_propagation.html

  1. ρ P 1 r 2 . \rho_{P}\propto\frac{1}{r^{2}}.

Radioactive_decay.html

  1. λ λ
  2. τ τ
  3. 2 \sqrt{2}
  4. A A
  5. N N
  6. t 1 / 2 = ln ( 2 ) λ = τ ln ( 2 ) t_{1/2}=\frac{\ln(2)}{\lambda}=\tau\ln(2)
  7. A = - d N d t = λ N A=-\frac{\mathrm{d}N}{\mathrm{d}t}=\lambda N
  8. S A a 0 = - d N d t | t = 0 = λ N 0 S_{A}a_{0}=-\frac{\mathrm{d}N}{\mathrm{d}t}\bigg|_{t=0}=\lambda N_{0}
  9. A A
  10. B B
  11. A B A→B
  12. d N −dN
  13. d t dt
  14. N N
  15. - d N d t N . -\frac{\mathrm{d}N}{\mathrm{d}t}\propto N.
  16. λ λ
  17. d N / N −dN/N
  18. d t dt
  19. N N
  20. N ( t ) = N 0 e - λ t = N 0 e - t / τ , N(t)=N_{0}\,e^{-{\lambda}t}=N_{0}\,e^{-t/\tau},\,\!
  21. N N
  22. t t
  23. t t
  24. N A + N B = N total = N A 0 , N_{A}+N_{B}=N_{\mathrm{total}}=N_{A0},
  25. N t o t a l N_{total}
  26. A A
  27. A A
  28. N A = N A 0 e - λ t N_{A}=N_{A0}e^{-{\lambda}t}\,\!
  29. B B
  30. A A
  31. N B = N A 0 - N A = N A 0 - N A 0 e - λ t = N A 0 ( 1 - e - λ t ) . N_{B}=N_{A0}-N_{A}=N_{A0}-N_{A0}e^{-{\lambda}t}=N_{A0}\left(1-e^{-{\lambda}t}% \right).
  32. N N
  33. P ( N ) = N N exp ( - N ) N ! . P(N)=\frac{\langle N\rangle^{N}\exp(-\langle N\rangle)}{N!}.
  34. A A
  35. B B
  36. B B
  37. C C
  38. A B C A→B→C
  39. A A
  40. B B
  41. B B
  42. C C
  43. A A
  44. B B
  45. B B
  46. B B
  47. A A
  48. C C
  49. d N B d t = - λ B N B + λ A N A . \frac{\mathrm{d}N_{B}}{\mathrm{d}t}=-\lambda_{B}N_{B}+\lambda_{A}N_{A}.
  50. d N B / d t dN_{B}/dt
  51. A A
  52. B B
  53. B B
  54. A A
  55. B B
  56. C C
  57. N A N_{A}
  58. A A
  59. N A 0 N_{A0}
  60. A A
  61. λ A λ_{A}
  62. A A
  63. B B
  64. N B = N A 0 λ A λ B - λ A ( e - λ A t - e - λ B t ) . N_{B}=\frac{N_{A0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}\left(e^{-\lambda_{A}t}% -e^{-\lambda_{B}t}\right).
  65. B B
  66. λ B λ_{B}
  67. lim λ B 0 [ N A 0 λ A λ B - λ A ( e - λ A t - e - λ B t ) ] = N A 0 λ A 0 - λ A ( e - λ A t - 1 ) = N A 0 ( 1 - e - λ A t ) , \lim_{\lambda_{B}\rightarrow 0}\left[\frac{N_{A0}\lambda_{A}}{\lambda_{B}-% \lambda_{A}}\left(e^{-\lambda_{A}t}-e^{-\lambda_{B}t}\right)\right]=\frac{N_{A% 0}\lambda_{A}}{0-\lambda_{A}}\left(e^{-\lambda_{A}t}-1\right)=N_{A0}\left(1-e^% {-\lambda_{A}t}\right),
  68. e λ B t e^{λ_{B}t}
  69. D D
  70. i i
  71. i = 1 , 2 , 3 , D i=_{1,2,3,...D}
  72. d N j d t = - λ j N j + λ j - 1 N ( j - 1 ) 0 e - λ j - 1 t . \frac{\mathrm{d}N_{j}}{\mathrm{d}t}=-\lambda_{j}N_{j}+\lambda_{j-1}N_{(j-1)0}e% ^{-\lambda_{j-1}t}.
  73. A B A→B
  74. A C A→C
  75. t t
  76. N = N A + N B + N C N=N_{A}+N_{B}+N_{C}
  77. d N A d t \displaystyle\frac{\mathrm{d}N_{A}}{\mathrm{d}t}
  78. λ λ
  79. λ = λ B + λ C . \lambda=\lambda_{B}+\lambda_{C}.
  80. d N A d t < 0 , d N B d t > 0 , d N C d t > 0. \frac{\mathrm{d}N_{A}}{\mathrm{d}t}<0,\frac{\mathrm{d}N_{B}}{\mathrm{d}t}>0,% \frac{\mathrm{d}N_{C}}{\mathrm{d}t}>0.
  81. N A N_{A}
  82. N A = N A 0 e - λ t . N_{A}=N_{A0}e^{-\lambda t}.
  83. λ λ
  84. λ B λ_{B}
  85. λ C λ_{C}
  86. B B
  87. C C
  88. N B = λ B λ N A 0 ( 1 - e - λ t ) , N_{B}=\frac{\lambda_{B}}{\lambda}N_{A0}\left(1-e^{-\lambda t}\right),
  89. N C = λ C λ N A 0 ( 1 - e - λ t ) . N_{C}=\frac{\lambda_{C}}{\lambda}N_{A0}\left(1-e^{-\lambda t}\right).
  90. λ B / λ λ_{B}/λ
  91. B B
  92. λ C / λ λ_{C}/λ
  93. C C
  94. N N
  95. A = λ N . A=λN.
  96. n = N / L n=N/L
  97. M = A r n = A r N / L M=A_{r}n=A_{r}N/L
  98. A r A_{r}
  99. A B A→B
  100. N = N 0 e - λ t = N 0 e - t / τ , N=N_{0}\,e^{-{\lambda}t}=N_{0}\,e^{-t/\tau},\,\!
  101. λ λ
  102. τ τ
  103. τ τ
  104. τ τ
  105. τ = 1 λ . \tau=\frac{1}{\lambda}.
  106. A B + C A→B+C
  107. λ = λ B + λ C \lambda=\lambda_{B}+\lambda_{C}\,
  108. 1 τ = λ = λ B + λ C = 1 τ B + 1 τ C \frac{1}{\tau}=\lambda=\lambda_{B}+\lambda_{C}=\frac{1}{\tau_{B}}+\frac{1}{% \tau_{C}}\,
  109. N = N 0 e - λ t = N 0 e - t / τ , N=N_{0}\,e^{-{\lambda}t}=N_{0}\,e^{-t/\tau},\,\!
  110. N = N 0 / 2 N=N_{0}/2
  111. t t
  112. t 1 / 2 = ln 2 λ = τ ln 2. t_{1/2}=\frac{\ln 2}{\lambda}=\tau\ln 2.
  113. l n ( 2 ) ln(2)
  114. e e
  115. τ τ
  116. τ τ
  117. t 1 / 2 t_{1/2}
  118. N ( t ) = N 0 e - t / τ = N 0 2 - t / t 1 / 2 . N(t)=N_{0}\,e^{-t/\tau}=N_{0}\,2^{-t/t_{1/2}}.\,\!
  119. τ τ
  120. t 1 / 2 t_{1/2}
  121. n t h n^{th}
  122. N = N 0 / n N=N_{0}/n
  123. t = T 1 / n t=T_{1/n}
  124. t 1 / n = ln n λ = τ ln n . t_{1/n}=\frac{\ln n}{\lambda}=\tau\ln n.
  125. N = N 0 e - t / τ , N=N_{0}\,e^{-t/\tau},
  126. N N 0 = 4 / 14 0.286 , \frac{N}{N_{0}}=4/14\approx 0.286,
  127. τ = T 1 / 2 ln 2 8267 \tau=\frac{T_{1/2}}{\ln 2}\approx 8267
  128. t = - τ ln N N 0 10360 t=-\tau\,\ln\frac{N}{N_{0}}\approx 10360

Radioisotope_thermoelectric_generator.html

  1. η t h = Desired Output Required Input = W o u t Q i n \eta_{th}=\frac{\,\text{Desired Output}}{\,\text{Required Input}}=\frac{W^{% \prime}_{out}}{Q^{\prime}_{in}}
  2. Δ E sys = Q i n + W i n - Q o u t - W o u t \Delta E^{\prime\mathrm{sys}}=Q^{\prime}_{in}+W^{\prime}_{in}-Q^{\prime}_{out}% -W^{\prime}_{out}\,
  3. W i n = 0 W^{\prime}_{in}=0\,
  4. W o u t = Q i n - Q o u t W^{\prime}_{out}=Q^{\prime}_{in}-Q^{\prime}_{out}\,
  5. η I I = η t h η t h , r e v \eta_{II}=\frac{\eta_{th}}{\eta_{th,rev}}
  6. η t h = 1 - T h e a t s i n k T h e a t s o u r c e \eta_{th}=1-\frac{T_{heatsink}}{T_{heatsource}}

Radix_point.html

  1. 1101.101 2 \displaystyle 1101.101_{2}

Railgun.html

  1. μ 0 \mu_{0}
  2. r r
  3. d d
  4. I I
  5. s s
  6. 𝐁 ( s ) = μ 0 I 4 π s ϕ ^ \mathbf{B}(s)=\frac{\mu_{0}I}{4\pi s}\hat{\phi}
  7. x = - x=-\infty
  8. d d
  9. d d
  10. B ( s ) = μ 0 I 4 π ( 1 s + 1 d - s ) z ^ B(s)=\frac{\mu_{0}I}{4\pi}\left(\frac{1}{s}+\frac{1}{d-s}\right)\hat{z}
  11. s s
  12. ϕ ^ \hat{\phi}
  13. z ^ \hat{z}
  14. x = - x=-\infty
  15. 𝐅 = I d s y m b o l × 𝐁 \mathbf{F}=I\int\mathrm{d}symbol{\ell}\times\mathbf{B}
  16. r r
  17. d - r d-r
  18. d s y m b o l \mathrm{d}symbol{\ell}
  19. y ^ \hat{y}
  20. 𝐅 = I r d - r d s y m b o l × μ 0 I 4 π ( 1 s + 1 d - s ) z ^ = μ 0 I 2 2 π ln ( d - r r ) x ^ \mathbf{F}=I\int_{r}^{d-r}\mathrm{d}symbol{\ell}\times\frac{\mu_{0}I}{4\pi}% \left(\frac{1}{s}+\frac{1}{d-s}\right)\hat{z}=\frac{\mu_{0}I^{2}}{2\pi}\ln{% \left(\frac{d-r}{r}\right)}\hat{x}
  21. l l
  22. F F
  23. d d
  24. l > 3 d l>3d
  25. V = I . R + d ( L . I ) d t V=I.R+\frac{\,\text{d}(L.I)}{\,\text{d}t}
  26. R = R . x R=R^{\prime}.x
  27. L = L . x L=L^{\prime}.x
  28. V = I . ( R x + L v ) + L x d I d t V=I.(R^{\prime}x+L^{\prime}v)+L^{\prime}x\frac{\,\text{d}I}{\,\text{d}t}
  29. I 2 L v I^{2}L^{\prime}v
  30. F = L I 2 2 F=\frac{L^{\prime}I^{2}}{2}
  31. L L^{\prime}
  32. μ \mu
  33. L L^{\prime}
  34. d v d t = L I 2 2 m \frac{\,\text{d}v}{\,\text{d}t}=\frac{L^{\prime}I^{2}}{2m}
  35. d x d t = v \frac{\,\text{d}x}{\,\text{d}t}=v
  36. L = μ 0 π ln ( d - r r ) L^{\prime}=\frac{\mu_{0}}{\pi}\ln{\left(\frac{d-r}{r}\right)}
  37. F = L I 2 2 = μ 0 I 2 2 π ln ( d - r r ) F=\frac{L^{\prime}I^{2}}{2}=\frac{\mu_{0}I^{2}}{2\pi}\ln{\left(\frac{d-r}{r}% \right)}

Ramsey's_theorem.html

  1. R ( r , s ) R(r,s)
  2. R ( r , s ) R(r,s)
  3. r r
  4. s s
  5. R ( r , s ) R(r,s)
  6. L L
  7. R ( r , s ) R(r,s)
  8. 1 2 ( n ) ( n 1 ) \frac{1}{2}(n)(n−1)
  9. c c
  10. c c
  11. n n
  12. R ( 3 , 3 ) = 6 R(3,3)=6
  13. R ( 4 , 2 ) = 4 R(4,2)=4
  14. R ( s , 2 ) = s R(s,2)=s
  15. s s
  16. s 1 s−1
  17. R ( s , 2 ) s R(s,2)≥s
  18. s s
  19. s s
  20. 2 2
  21. R ( 4 , 3 ) R ( 4 , 2 ) + R ( 3 , 3 ) 1 = 9 R(4,3)≤R(4,2)+R(3,3)−1=9
  22. R ( 4 , 4 ) R ( 4 , 3 ) + R ( 3 , 4 ) 18 R(4,4)≤R(4,3)+R(3,4)≤18
  23. ( 4 , 4 , 16 ) (4,4,16)
  24. 2 2
  25. 16 16
  26. 4 4
  27. 2 2
  28. 16 16
  29. ( 4 , 4 , 17 ) (4,4,17)
  30. 17 17
  31. R ( 4 , 4 ) = 18 R(4,4)=18
  32. R ( 4 , 5 ) = 25 R(4,5)=25
  33. R ( 5 , 5 ) R(5,5)
  34. 43 43
  35. 49 49
  36. R ( 5 , 5 ) = 43 R(5,5)=43
  37. ( 5 , 5 , 42 ) (5,5,42)
  38. ( 5 , 5 , 43 ) (5,5,43)
  39. R ( r , s ) R(r,s)
  40. r , s > 5 r,s>5
  41. R ( 6 , 6 ) R(6,6)
  42. R ( 8 , 8 ) R(8,8)
  43. R ( r , s ) R(r,s)
  44. r , s 10 r,s≤10
  45. R ( r , s ) R(r,s)
  46. R ( 1 , s ) = 1 R(1,s)=1
  47. R ( 2 , s ) = s R(2,s)=s
  48. s s
  49. R ( r , s ) = R ( s , r ) R(r,s)=R(s,r)
  50. R ( r , s ) R ( r 1 , s ) + R ( r , s 1 ) R(r,s)≤R(r−1,s)+R(r,s−1)
  51. R ( r , s ) ( r + s - 2 r - 1 ) . R(r,s)\leq{\left({{r+s-2}\atop{r-1}}\right)}.
  52. r = s r=s
  53. R ( s , s ) ( 1 + o ( 1 ) ) 4 s - 1 π s . R(s,s)\leq(1+o(1))\frac{4^{s-1}}{\sqrt{\pi s}}.
  54. R ( s , s ) ( 1 + o ( 1 ) ) s 2 e 2 s 2 , R(s,s)\geq(1+o(1))\frac{s}{\sqrt{2}e}2^{\frac{s}{2}},
  55. s = 10 s=10
  56. 101 R ( 10 , 10 ) 48620 101≤R(10,10)≤48620
  57. 4 4
  58. 2 \sqrt{2}
  59. ( 1 + o ( 1 ) ) 2 s e 2 s 2 R ( s , s ) s - c log s log log s 4 s , (1+o(1))\frac{\sqrt{2}s}{e}2^{\frac{s}{2}}\leq R(s,s)\leq s^{-\frac{c\log s}{% \log\log s}}4^{s},
  60. R ( 3 , t ) R(3,t)
  61. t 2 log t \tfrac{t^{2}}{\log t}
  62. n n
  63. Θ ( n log n ) . \Theta\left(\sqrt{n\log n}\right).
  64. R ( 3 , t ) R(3,t)
  65. R ( s , t ) R(s,t)
  66. s s
  67. t t
  68. c s t s + 1 2 ( log t ) s + 1 2 - 1 s - 2 R ( s , t ) c s t s - 1 ( log t ) s - 2 , c^{\prime}_{s}\frac{t^{\frac{s+1}{2}}}{(\log t)^{\frac{s+1}{2}-\frac{1}{s-2}}}% \leq R(s,t)\leq c_{s}\frac{t^{s-1}}{(\log t)^{s-2}},
  69. [ k ] ( n ) [k]^{(n)}
  70. [ k ] ( n ) [k]^{(n)}
  71. [ k ] ( n ) [k]^{(n)}
  72. C k 1 C^{1}_{k}
  73. C k 1 C^{1}_{k}
  74. C k + 1 1 C^{1}_{k+1}
  75. C k 1 C^{1}_{k}
  76. C k 2 C^{2}_{k}
  77. C k m C^{m}_{k}
  78. C k C k 1 C k 2 C_{k}\supseteq C^{1}_{k}\supseteq C^{2}_{k}\supseteq\dots
  79. | C k | c k ! n ! ( k - n ) ! |C_{k}|\leq c^{\frac{k!}{n!(k-n)!}}
  80. D k = C k C k 1 C k 2 D_{k}=C_{k}\cap C^{1}_{k}\cap C^{2}_{k}\cap\dots
  81. ( n ) \mathbb{N}^{(n)}

Random_walk.html

  1. X 0 , X 1 , X 2 , X_{0},X_{1},X_{2},\dots
  2. X t X_{t}
  3. t 0 t\geq 0
  4. d \mathbb{Z}^{d}
  5. \mathbb{Z}
  6. Z 1 , Z 2 , Z_{1},Z_{2},\dots
  7. S 0 = 0 S_{0}=0\,\!
  8. S n = j = 1 n Z j . S_{n}=\sum_{j=1}^{n}Z_{j}.
  9. { S n } \{S_{n}\}\,\!
  10. \mathbb{Z}
  11. E ( S n ) E(S_{n})\,\!
  12. S n S_{n}\,\!
  13. E ( S n ) = j = 1 n E ( Z j ) = 0. E(S_{n})=\sum_{j=1}^{n}E(Z_{j})=0.
  14. E ( Z n 2 ) = 1 E(Z_{n}^{2})=1
  15. E ( S n 2 ) = i = 1 n j = 1 n E ( Z j Z i ) = n . E(S_{n}^{2})=\sum_{i=1}^{n}\sum_{j=1}^{n}E(Z_{j}Z_{i})=n.
  16. E ( | S n | ) E(|S_{n}|)\,\!
  17. n \sqrt{n}
  18. lim n E ( | S n | ) n = 2 π . \lim_{n\to\infty}\frac{E(|S_{n}|)}{\sqrt{n}}=\sqrt{\frac{2}{\pi}}.
  19. N N
  20. \mathbb{Z}
  21. a / ( a + b ) a/(a+b)
  22. S n = k S_{n}=k
  23. ( n ( n + k ) / 2 ) n\choose(n+k)/2
  24. S n = k S_{n}=k
  25. 2 - n ( n ( n + k ) / 2 ) 2^{-n}{n\choose(n+k)/2}
  26. n n
  27. \mathbb{Z}
  28. P [ S 0 = k ] P[S_{0}=k]
  29. 2 P [ S 1 = k ] 2P[S_{1}=k]
  30. 2 2 P [ S 2 = k ] 2^{2}P[S_{2}=k]
  31. 2 3 P [ S 3 = k ] 2^{3}P[S_{3}=k]
  32. 2 4 P [ S 4 = k ] 2^{4}P[S_{4}=k]
  33. 2 5 P [ S 5 = k ] 2^{5}P[S_{5}=k]
  34. \mathbb{Z}
  35. i = 0 , ± 1 , ± 2 , . i=0,\pm 1,\pm 2,\dots.
  36. 0 < p < 1 \,0<p<1
  37. P i , i + 1 = p = 1 - P i , i - 1 . \,P_{i,i+1}=p=1-P_{i,i-1}.
  38. σ 2 = t δ t ε 2 , \sigma^{2}=\frac{t}{\delta t}\,\varepsilon^{2},
  39. ε \varepsilon
  40. δ t \delta t
  41. σ 2 = 6 D t \sigma^{2}=6\,D\,t
  42. D = ε 2 6 δ t D=\frac{\varepsilon^{2}}{6\delta t}
  43. R \vec{R}
  44. R x R_{x}
  45. R y R_{y}
  46. R z R_{z}
  47. D = ε 2 4 δ t D=\frac{\varepsilon^{2}}{4\delta t}
  48. D = ε 2 2 δ t D=\frac{\varepsilon^{2}}{2\delta t}
  49. Φ - 1 ( z , μ , σ ) \Phi^{-1}(z,\mu,\sigma)
  50. i = 0 n X i \sum_{i=0}^{n}{X_{i}}
  51. E | S n 2 | = σ n . \sqrt{E|S_{n}^{2}|}=\sigma\sqrt{n}.
  52. n \sqrt{n}
  53. n \sqrt{n}
  54. σ 2 \sigma^{2}
  55. t t
  56. t 2 / d w t^{2/d_{w}}
  57. d w d_{w}
  58. S ( t ) S(t)
  59. N N
  60. S N ( t ) S_{N}(t)
  61. X 1 , X 2 , , X k X_{1},X_{2},\dots,X_{k}
  62. X 1 = 0 X_{1}=0
  63. X i + 1 {X_{i+1}}
  64. X i X_{i}
  65. p v , w , k ( G ) p_{v,w,k}(G)
  66. p 0 , 0 , 2 k p_{0,0,2k}
  67. 2 k 2k
  68. p ( x , y ) p(x,y)
  69. ω \omega
  70. ω \omega
  71. ω \omega

Range_(mathematics).html

  1. f f
  2. f f
  3. f ( x ) = x 2 f(x)=x^{2}
  4. \mathbb{R}
  5. + \mathbb{R}^{+}
  6. x 2 x^{2}
  7. x x
  8. \mathbb{R}
  9. + \mathbb{R}^{+}
  10. f ( x ) = 2 x f(x)=2x

Rank-into-rank.html

  1. j n ( κ ) j^{n}(\kappa)

Rank_of_an_abelian_group.html

  1. α n α a α = 0 , n α , \sum_{\alpha}n_{\alpha}a_{\alpha}=0,\quad n_{\alpha}\in\mathbb{Z},
  2. 0 A B C 0 0\to A\to B\to C\to 0\;
  3. rank ( j J A j ) = j J rank ( A j ) , \operatorname{rank}\left(\bigoplus_{j\in J}A_{j}\right)=\sum_{j\in J}% \operatorname{rank}(A_{j}),
  4. rank ( M ) = dim R 0 M R R 0 \,\text{rank}(M)=\dim_{R_{0}}M\otimes_{R}R_{0}
  5. x 𝐙 q = 0 x\otimes_{\mathbf{Z}}q=0

Rate-monotonic_scheduling.html

  1. n n
  2. U = i = 1 n C i T i n ( 2 1 / n - 1 ) U=\sum_{i=1}^{n}\frac{C_{i}}{T_{i}}\leq n({2}^{1/n}-1)
  3. n n
  4. U 0.8284 U ≤ 0.8284
  5. lim n n ( 2 n - 1 ) = ln 2 0.693147 \lim_{n\rightarrow\infty}n(\sqrt[n]{2}-1)=\ln 2\approx 0.693147\ldots
  6. 1 8 + 2 5 + 2 10 = 0.725 \frac{1}{8}+\frac{2}{5}+\frac{2}{10}=0.725
  7. 3 3\,
  8. U = 3 ( 2 1 3 - 1 ) = 0.77976 U=3(2^{\frac{1}{3}}-1)=0.77976\ldots
  9. 0.725 < 0.77976 0.725<0.77976\ldots

Rate–distortion_theory.html

  1. inf Q Y | X ( y | x ) I Q ( Y ; X ) subject to D Q D * . \inf_{Q_{Y|X}(y|x)}I_{Q}(Y;X)\ \mbox{subject to}~{}\ D_{Q}\leq D^{*}.
  2. I ( Y ; X ) = H ( Y ) - H ( Y | X ) I(Y;X)=H(Y)-H(Y|X)\,
  3. H ( Y ) = - - P Y ( y ) log 2 ( P Y ( y ) ) d y H(Y)=-\int_{-\infty}^{\infty}P_{Y}(y)\log_{2}(P_{Y}(y))\,dy
  4. H ( Y | X ) = - - - Q Y | X ( y | x ) P X ( x ) log 2 ( Q Y | X ( y | x ) ) d x d y . H(Y|X)=-\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}Q_{Y|X}(y|x)P_{X}(x)\log% _{2}(Q_{Y|X}(y|x))\,dx\,dy.
  5. inf Q Y | X ( y | x ) E [ D Q [ X , Y ] ] subject to I Q ( Y ; X ) R . \inf_{Q_{Y|X}(y|x)}E[D_{Q}[X,Y]]\ \mbox{subject to}~{}\ I_{Q}(Y;X)\leq R.
  6. D Q = - - P X , Y ( x , y ) ( x - y ) 2 d x d y = - - Q Y | X ( y | x ) P X ( x ) ( x - y ) 2 d x d y . D_{Q}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}P_{X,Y}(x,y)(x-y)^{2}\,dx% \,dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}Q_{Y|X}(y|x)P_{X}(x)(x-y)^{% 2}\,dx\,dy.
  7. R ( D ) h ( X ) - h ( D ) R(D)\geq h(X)-h(D)\,
  8. R ( D ) = lim n R n ( D ) R(D)=\lim_{n\rightarrow\infty}R_{n}(D)
  9. R n ( D ) = 1 n inf Q Y n | X n 𝒬 I ( Y n , X n ) R_{n}(D)=\frac{1}{n}\inf_{Q_{Y^{n}|X^{n}}\in\mathcal{Q}}I(Y^{n},X^{n})
  10. 𝒬 = { Q Y n | X n ( Y n | X n , X 0 ) : E [ d ( X n , Y n ) ] D } \mathcal{Q}=\{Q_{Y^{n}|X^{n}}(Y^{n}|X^{n},X_{0}):E[d(X^{n},Y^{n})]\leq D\}
  11. R ( D ) = { 1 2 log 2 ( σ x 2 / D ) , if 0 D σ x 2 0 , if D > σ x 2 . R(D)=\left\{\begin{matrix}\frac{1}{2}\log_{2}(\sigma_{x}^{2}/D),&\mbox{if }~{}% 0\leq D\leq\sigma_{x}^{2}\\ \\ 0,&\mbox{if }~{}D>\sigma_{x}^{2}.\end{matrix}\right.

Ratio.html

  1. A B \tfrac{A}{B}
  2. 2 3 \tfrac{2}{3}
  3. 3 7 \tfrac{3}{7}
  4. 2 5 \tfrac{2}{5}
  5. 3 5 \tfrac{3}{5}
  6. ( a / b ) = 1 + 1 ( a / b ) (a/b)=1+\frac{1}{(a/b)}
  7. a b = 1 + 5 2 , \tfrac{a}{b}=\tfrac{1+\sqrt{5}}{2},
  8. a b = 1 + 2 , \tfrac{a}{b}=1+\sqrt{2},
  9. α : β : γ \alpha:\beta:\gamma
  10. α : β , \alpha:\beta,
  11. β : γ , \beta:\gamma,
  12. α : γ . \alpha:\gamma.
  13. α , β , γ , \alpha,\beta,\gamma,

Ray_transfer_matrix_analysis.html

  1. ( x 2 θ 2 ) = ( A B C D ) ( x 1 θ 1 ) , {x_{2}\choose\theta_{2}}=\begin{pmatrix}A&B\\ C&D\end{pmatrix}{x_{1}\choose\theta_{1}},
  2. A = x 2 x 1 | θ 1 = 0 B = x 2 θ 1 | x 1 = 0 , A={x_{2}\over x_{1}}\bigg|_{\theta_{1}=0}\qquad B={x_{2}\over\theta_{1}}\bigg|% _{x_{1}=0},
  3. C = θ 2 x 1 | θ 1 = 0 D = θ 2 θ 1 | x 1 = 0 . C={\theta_{2}\over x_{1}}\bigg|_{\theta_{1}=0}\qquad D={\theta_{2}\over\theta_% {1}}\bigg|_{x_{1}=0}.
  4. det ( 𝐌 ) = A D - B C = n 1 n 2 . \det(\mathbf{M})=AD-BC={n_{1}\over n_{2}}.
  5. 𝐒 = ( 1 d 0 1 ) \mathbf{S}=\begin{pmatrix}1&d\\ 0&1\end{pmatrix}
  6. ( x 2 θ 2 ) = 𝐒 ( x 1 θ 1 ) {x_{2}\choose\theta_{2}}=\mathbf{S}{x_{1}\choose\theta_{1}}
  7. x 2 = x 1 + d θ 1 θ 2 = θ 1 \begin{matrix}x_{2}&=&x_{1}+d\theta_{1}\\ \theta_{2}&=&\theta_{1}\end{matrix}
  8. 𝐋 = ( 1 0 - 1 f 1 ) \mathbf{L}=\begin{pmatrix}1&0\\ \frac{-1}{f}&1\end{pmatrix}
  9. 𝐋𝐒 = ( 1 0 - 1 f 1 ) ( 1 d 0 1 ) = ( 1 d - 1 f 1 - d f ) \mathbf{L}\mathbf{S}=\begin{pmatrix}1&0\\ \frac{-1}{f}&1\end{pmatrix}\begin{pmatrix}1&d\\ 0&1\end{pmatrix}=\begin{pmatrix}1&d\\ \frac{-1}{f}&1-\frac{d}{f}\end{pmatrix}
  10. 𝐒𝐋 = ( 1 d 0 1 ) ( 1 0 - 1 f 1 ) = ( 1 - d f d - 1 f 1 ) \mathbf{SL}=\begin{pmatrix}1&d\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ \frac{-1}{f}&1\end{pmatrix}=\begin{pmatrix}1-\frac{d}{f}&d\\ \frac{-1}{f}&1\end{pmatrix}
  11. ( 1 d 0 1 ) \begin{pmatrix}1&d\\ 0&1\end{pmatrix}
  12. ( 1 0 0 n 1 n 2 ) \begin{pmatrix}1&0\\ 0&\frac{n_{1}}{n_{2}}\end{pmatrix}
  13. ( 1 0 n 1 - n 2 R n 2 n 1 n 2 ) \begin{pmatrix}1&0\\ \frac{n_{1}-n_{2}}{R\cdot n_{2}}&\frac{n_{1}}{n_{2}}\end{pmatrix}
  14. ( 1 0 0 - 1 ) \begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  15. ( 1 0 - 2 R 1 ) \begin{pmatrix}1&0\\ -\frac{2}{R}&1\end{pmatrix}
  16. ( 1 0 - 1 f 1 ) \begin{pmatrix}1&0\\ -\frac{1}{f}&1\end{pmatrix}
  17. ( 1 0 n 2 - n 1 R 2 n 1 n 2 n 1 ) ( 1 t 0 1 ) ( 1 0 n 1 - n 2 R 1 n 2 n 1 n 2 ) \begin{pmatrix}1&0\\ \frac{n_{2}-n_{1}}{R_{2}n_{1}}&\frac{n_{2}}{n_{1}}\end{pmatrix}\begin{pmatrix}% 1&t\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ \frac{n_{1}-n_{2}}{R_{1}n_{2}}&\frac{n_{1}}{n_{2}}\end{pmatrix}
  18. ( k d n k 0 1 k ) \begin{pmatrix}k&\frac{d}{nk}\\ 0&\frac{1}{k}\end{pmatrix}
  19. ψ \psi
  20. ϕ \phi
  21. ϕ \phi
  22. ψ \psi
  23. 𝐌 = 𝐋𝐒 = ( 1 d - 1 f 1 - d f ) \mathbf{M}=\mathbf{L}\mathbf{S}=\begin{pmatrix}1&d\\ \frac{-1}{f}&1-\frac{d}{f}\end{pmatrix}
  24. 𝐌 ( x 1 θ 1 ) = ( x 2 θ 2 ) = λ ( x 1 θ 1 ) \mathbf{M}{x_{1}\choose\theta_{1}}={x_{2}\choose\theta_{2}}=\lambda{x_{1}% \choose\theta_{1}}
  25. [ 𝐌 - λ 𝐈 ] ( x 1 θ 1 ) = 0 \left[\mathbf{M}-\lambda\mathbf{I}\right]{x_{1}\choose\theta_{1}}=0
  26. det [ 𝐌 - λ 𝐈 ] = 0 \operatorname{det}\left[\mathbf{M}-\lambda\mathbf{I}\right]=0
  27. λ 2 - tr ( 𝐌 ) λ + det ( 𝐌 ) = 0 \lambda^{2}-\operatorname{tr}(\mathbf{M})\lambda+\operatorname{det}(\mathbf{M}% )=0
  28. tr ( 𝐌 ) = A + D = 2 - d f \operatorname{tr}(\mathbf{M})=A+D=2-{d\over f}
  29. det ( 𝐌 ) = A D - B C = 1 \operatorname{det}(\mathbf{M})=AD-BC=1
  30. λ 2 - 2 g λ + 1 = 0 \lambda^{2}-2g\lambda+1=0
  31. g = def tr ( 𝐌 ) 2 = 1 - d 2 f g\ \stackrel{\mathrm{def}}{=}\ {\operatorname{tr}(\mathbf{M})\over 2}=1-{d% \over 2f}
  32. λ ± = g ± g 2 - 1 \lambda_{\pm}=g\pm\sqrt{g^{2}-1}\,
  33. ( x N θ N ) = λ N ( x 1 θ 1 ) {x_{N}\choose\theta_{N}}=\lambda^{N}{x_{1}\choose\theta_{1}}
  34. g 2 > 1 g^{2}>1
  35. λ + λ - = 1 \lambda_{+}\lambda_{-}=1
  36. g 2 g^{2}
  37. λ ± = g ± i 1 - g 2 = cos ( ϕ ) ± i sin ( ϕ ) = e ± i ϕ \lambda_{\pm}=g\pm i\sqrt{1-g^{2}}=\cos(\phi)\pm i\sin(\phi)=e^{\pm i\phi}
  38. g 2 < 1 g^{2}<1
  39. r + r_{+}
  40. r - r_{-}
  41. λ + \lambda_{+}
  42. λ - \lambda_{-}
  43. λ + \lambda_{+}
  44. λ - \lambda_{-}
  45. c + r + + c - r - c_{+}r_{+}+c_{-}r_{-}
  46. c + c_{+}
  47. c - c_{-}
  48. 𝐌 N ( c + r + + c - r - ) = λ + N c + r + + λ - N c - r - = e i N ϕ c + r + + e - i N ϕ c - r - \mathbf{M}^{N}(c_{+}r_{+}+c_{-}r_{-})=\lambda_{+}^{N}c_{+}r_{+}+\lambda_{-}^{N% }c_{-}r_{-}=e^{iN\phi}c_{+}r_{+}+e^{-iN\phi}c_{-}r_{-}
  49. λ 0 \lambda_{0}
  50. 1 q = 1 R - i λ 0 π n w 2 \frac{1}{q}=\frac{1}{R}-\frac{i\lambda_{0}}{\pi nw^{2}}
  51. ( q 2 1 ) = k ( A B C D ) ( q 1 1 ) {q_{2}\choose 1}=k\begin{pmatrix}A&B\\ C&D\end{pmatrix}{q_{1}\choose 1}
  52. q 2 = k ( A q 1 + B ) q_{2}=k(Aq_{1}+B)\,
  53. 1 = k ( C q 1 + D ) 1=k(Cq_{1}+D)\,
  54. q 2 = A q 1 + B C q 1 + D q_{2}=\frac{Aq_{1}+B}{Cq_{1}+D}
  55. 1 q 2 = C + D / q 1 A + B / q 1 . {1\over q_{2}}={C+D/q_{1}\over A+B/q_{1}}.
  56. [ A B C D ] = [ 1 d 0 1 ] \begin{bmatrix}A&B\\ C&D\end{bmatrix}=\begin{bmatrix}1&d\\ 0&1\end{bmatrix}
  57. q 2 = A q 1 + B C q 1 + D = q 1 + d 1 = q 1 + d q_{2}=\frac{Aq_{1}+B}{Cq_{1}+D}=\frac{q_{1}+d}{1}=q_{1}+d
  58. [ A B C D ] = [ 1 0 - 1 / f 1 ] \begin{bmatrix}A&B\\ C&D\end{bmatrix}=\begin{bmatrix}1&0\\ -1/f&1\end{bmatrix}
  59. q 2 = A q 1 + B C q 1 + D = q 1 - q 1 f + 1 q_{2}=\frac{Aq_{1}+B}{Cq_{1}+D}=\frac{q_{1}}{-\frac{q_{1}}{f}+1}
  60. 1 q 2 = - q 1 f + 1 q 1 = 1 q 1 - 1 f \frac{1}{q_{2}}=\frac{-\frac{q_{1}}{f}+1}{q_{1}}=\frac{1}{q_{1}}-\frac{1}{f}

Rayleigh_distribution.html

  1. σ 2 ln ( 2 ) \sigma\sqrt{2\ln(2)}\,
  2. σ \sigma\,
  3. 4 - π 2 σ 2 \frac{4-\pi}{2}\sigma^{2}
  4. 2 π ( π - 3 ) ( 4 - π ) 3 / 2 \frac{2\sqrt{\pi}(\pi-3)}{(4-\pi)^{3/2}}
  5. - 6 π 2 - 24 π + 16 ( 4 - π ) 2 -\frac{6\pi^{2}-24\pi+16}{(4-\pi)^{2}}
  6. 1 + ln ( σ 2 ) + γ 2 1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}
  7. 1 + σ t e σ 2 t 2 / 2 π 2 ( erf ( σ t 2 ) + 1 ) 1+\sigma t\,e^{\sigma^{2}t^{2}/2}\sqrt{\frac{\pi}{2}}\left(\textrm{erf}\left(% \frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right)
  8. 1 - σ t e - σ 2 t 2 / 2 π 2 ( erfi ( σ t 2 ) - i ) 1\!-\!\sigma te^{-\sigma^{2}t^{2}/2}\sqrt{\frac{\pi}{2}}\!\left(\textrm{erfi}% \!\left(\frac{\sigma t}{\sqrt{2}}\right)\!-\!i\right)
  9. f ( x ; σ ) = x σ 2 e - x 2 / ( 2 σ 2 ) , x 0 , f(x;\sigma)=\frac{x}{\sigma^{2}}e^{-x^{2}/(2\sigma^{2})},\quad x\geq 0,
  10. σ \sigma
  11. F ( x ) = 1 - e - x 2 / ( 2 σ 2 ) F(x)=1-e^{-x^{2}/(2\sigma^{2})}
  12. x [ 0 , ) . x\in[0,\infty).
  13. Y = ( U , V ) Y=(U,V)
  14. f U ( u ; σ ) = e - u 2 / 2 σ 2 2 π σ 2 f_{U}(u;\sigma)=\frac{e^{-u^{2}/2\sigma^{2}}}{\sqrt{2\pi\sigma^{2}}}
  15. f V ( v ; σ ) f_{V}(v;\sigma)
  16. x x
  17. Y Y
  18. f ( x ; σ ) = 1 2 π σ 2 - d u - d v e - u 2 / 2 σ 2 e - v 2 / 2 σ 2 δ ( x - u 2 + v 2 ) . f(x;\sigma)=\frac{1}{2\pi\sigma^{2}}\int_{-\infty}^{\infty}du\,\int_{-\infty}^% {\infty}dv\,e^{-u^{2}/2\sigma^{2}}e^{-v^{2}/2\sigma^{2}}\delta(x-\sqrt{u^{2}+v% ^{2}}).
  19. f ( x ; σ ) = 1 2 π σ 2 0 2 π d ϕ 0 d r δ ( r - x ) r e - r 2 / 2 σ 2 = x σ 2 e - x 2 / 2 σ 2 , f(x;\sigma)=\frac{1}{2\pi\sigma^{2}}\int_{0}^{2\pi}\,d\phi\int_{0}^{\infty}dr% \,\delta(r-x)re^{-r^{2}/2\sigma^{2}}=\frac{x}{\sigma^{2}}e^{-x^{2}/2\sigma^{2}},
  20. μ k = σ k 2 k 2 Γ ( 1 + k 2 ) \mu_{k}=\sigma^{k}2^{\frac{k}{2}}\,\Gamma\left(1+\frac{k}{2}\right)
  21. Γ ( z ) \Gamma(z)
  22. μ ( X ) = σ π 2 1.253 σ \mu(X)=\sigma\sqrt{\frac{\pi}{2}}\ \approx 1.253\sigma
  23. var ( X ) = 4 - π 2 σ 2 0.429 σ 2 \textrm{var}(X)=\frac{4-\pi}{2}\sigma^{2}\approx 0.429\sigma^{2}
  24. σ \sigma
  25. f max = f ( σ ; σ ) = 1 σ e - 1 2 1 σ 0.606 f\text{max}=f(\sigma;\sigma)=\frac{1}{\sigma}e^{-\frac{1}{2}}\approx\frac{1}{% \sigma}0.606
  26. γ 1 = 2 π ( π - 3 ) ( 4 - π ) 3 2 0.631 \gamma_{1}=\frac{2\sqrt{\pi}(\pi-3)}{(4-\pi)^{\frac{3}{2}}}\approx 0.631
  27. γ 2 = - 6 π 2 - 24 π + 16 ( 4 - π ) 2 0.245 \gamma_{2}=-\frac{6\pi^{2}-24\pi+16}{(4-\pi)^{2}}\approx 0.245
  28. φ ( t ) = 1 - σ t e - 1 2 σ 2 t 2 π 2 [ erfi ( σ t 2 ) - i ] \varphi(t)=1-\sigma te^{-\frac{1}{2}\sigma^{2}t^{2}}\sqrt{\frac{\pi}{2}}\left[% \textrm{erfi}\left(\frac{\sigma t}{\sqrt{2}}\right)-i\right]
  29. erfi ( z ) \operatorname{erfi}(z)
  30. M ( t ) = 1 + σ t e 1 2 σ 2 t 2 π 2 [ erf ( σ t 2 ) + 1 ] M(t)=1+\sigma t\,e^{\frac{1}{2}\sigma^{2}t^{2}}\sqrt{\frac{\pi}{2}}\left[% \textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)+1\right]
  31. erf ( z ) \operatorname{erf}(z)
  32. H = 1 + ln ( σ 2 ) + γ 2 H=1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}
  33. γ \gamma
  34. { σ 2 x f ( x ) + f ( x ) ( x 2 - σ 2 ) = 0 f ( 1 ) = exp ( - 1 2 σ 2 ) σ 2 } \left\{\begin{array}[]{l}\sigma^{2}xf^{\prime}(x)+f(x)\left(x^{2}-\sigma^{2}% \right)=0\\ f(1)=\frac{\exp\left(-\frac{1}{2\sigma^{2}}\right)}{\sigma^{2}}\end{array}\right\}
  35. x i x_{i}
  36. σ \sigma
  37. σ 2 ^ 1 2 N i = 1 N x i 2 \widehat{\sigma^{2}}\approx\!\,\frac{1}{2N}\sum_{i=1}^{N}x_{i}^{2}
  38. σ ^ 1 2 N i = 1 N x i 2 \hat{\sigma}\approx\!\,\sqrt{\frac{1}{2N}\sum_{i=1}^{N}x_{i}^{2}}
  39. σ = σ ^ Γ ( N ) N Γ ( N + 1 2 ) = σ ^ 4 N N ! ( N - 1 ) ! N ( 2 N ) ! π \sigma=\hat{\sigma}\frac{\Gamma(N)\sqrt{N}}{\Gamma(N+\frac{1}{2})}=\hat{\sigma% }\frac{4^{N}N!(N-1)!\sqrt{N}}{(2N)!\sqrt{\pi}}
  40. χ 1 2 , χ 2 2 \chi_{1}^{2},\ \chi_{2}^{2}
  41. P r ( χ 2 ( 2 N ) χ 1 2 ) = α / 2 , P r ( χ 2 ( 2 N ) χ 2 2 ) = 1 - α / 2 Pr(\chi^{2}(2N)\leq\chi_{1}^{2})=\alpha/2,\quad Pr(\chi^{2}(2N)\leq\chi_{2}^{2% })=1-\alpha/2
  42. N x 2 ¯ χ 2 2 σ ^ 2 N x 2 ¯ χ 1 2 \frac{N\overline{x^{2}}}{\chi_{2}^{2}}\leq\widehat{\sigma}^{2}\leq\frac{N% \overline{x^{2}}}{\chi_{1}^{2}}
  43. X = σ - 2 ln ( U ) X=\sigma\sqrt{-2\ln(U)}\,
  44. σ \sigma
  45. R Rayleigh ( σ ) R\sim\mathrm{Rayleigh}(\sigma)
  46. R = X 2 + Y 2 R=\sqrt{X^{2}+Y^{2}}
  47. X N ( 0 , σ 2 ) X\sim N(0,\sigma^{2})
  48. Y N ( 0 , σ 2 ) Y\sim N(0,\sigma^{2})
  49. R Rayleigh ( 1 ) R\sim\mathrm{Rayleigh}(1)
  50. R 2 R^{2}
  51. N N
  52. [ Q = R 2 ] χ 2 ( N ) . [Q=R^{2}]\sim\chi^{2}(N)\ .
  53. R Rayleigh ( σ ) R\sim\mathrm{Rayleigh}(\sigma)
  54. i = 1 N R i 2 \sum_{i=1}^{N}R_{i}^{2}
  55. N N
  56. 2 σ 2 2\sigma^{2}
  57. [ Y = i = 1 N R i 2 ] Γ ( N , 2 σ 2 ) . \left[Y=\sum_{i=1}^{N}R_{i}^{2}\right]\sim\Gamma(N,2\sigma^{2}).
  58. σ \sigma
  59. λ \lambda
  60. λ = σ 2 . \lambda=\sigma\sqrt{2}.
  61. X X
  62. X Exponential ( λ ) X\sim\mathrm{Exponential}(\lambda)
  63. Y = 2 X σ 2 λ Rayleigh ( σ ) . Y=\sqrt{2X\sigma^{2}\lambda}\sim\mathrm{Rayleigh}(\sigma).
  64. f ( x ; σ ) = 1 2 π σ 1 σ 2 - d u - d v e - u 2 / 2 σ 1 2 e - v 2 / 2 σ 2 2 δ ( x - u 2 + v 2 ) . f(x;\sigma)=\frac{1}{2\pi\sigma_{1}\sigma_{2}}\int_{-\infty}^{\infty}du\,\int_% {-\infty}^{\infty}dv\,e^{-u^{2}/2\sigma_{1}^{2}}e^{-v^{2}/2\sigma_{2}^{2}}% \delta(x-\sqrt{u^{2}+v^{2}}).
  65. σ 1 \sigma_{1}
  66. σ 2 \sigma_{2}
  67. a = u σ 2 / σ 1 a=u\sigma_{2}/\sigma_{1}
  68. a / σ 2 = u / σ 1 a/\sigma_{2}=u/\sigma_{1}
  69. σ 1 σ 2 d a = d u \frac{\sigma_{1}}{\sigma_{2}}\mathrm{d}a=\mathrm{d}u
  70. f ( x ; σ ) = σ 1 σ 2 1 2 π σ 1 σ 2 - d a - d v e - a 2 / 2 σ 2 2 e - v 2 / 2 σ 2 2 δ ( x - ( σ 1 σ 2 a ) 2 + v 2 ) f(x;\sigma)=\frac{\sigma_{1}}{\sigma_{2}}\frac{1}{2\pi\sigma_{1}\sigma_{2}}% \int_{-\infty}^{\infty}da\,\int_{-\infty}^{\infty}dv\,e^{-a^{2}/2\sigma_{2}^{2% }}e^{-v^{2}/2\sigma_{2}^{2}}\delta\left(x-\sqrt{\left(\frac{\sigma_{1}}{\sigma% _{2}}a\right)^{2}+v^{2}}\right)
  71. { a = r cos ( ϕ ) v = r sin ( ϕ ) d a d v = r d r d ϕ \left\{\begin{array}[]{l}a=r\textrm{cos}(\phi)\\ v=r\textrm{sin}(\phi)\\ \textrm{d}a\textrm{d}v=r\textrm{d}r\textrm{d}\phi\end{array}\right.
  72. f ( x ; σ ) = σ 1 σ 2 1 2 π σ 1 σ 2 0 2 π d ϕ 0 r d r e - r 2 / 2 σ 2 2 δ ( x - ( σ 1 σ 2 a ) 2 + v 2 ) . f(x;\sigma)=\frac{\sigma_{1}}{\sigma_{2}}\frac{1}{2\pi\sigma_{1}\sigma_{2}}% \int_{0}^{2\pi}\textrm{d}\phi\,\int_{0}^{\infty}r\textrm{d}r\,e^{-r^{2}/2% \sigma_{2}^{2}}\delta\left(x-\sqrt{\left(\frac{\sigma_{1}}{\sigma_{2}}a\right)% ^{2}+v^{2}}\right).
  73. f ( x ; σ ) = 1 2 π σ 2 2 0 2 π d ϕ 0 r d r e - r 2 / 2 σ 2 2 δ ( x - ( σ 1 σ 2 a ) 2 + v 2 ) . f(x;\sigma)=\frac{1}{2\pi\sigma_{2}^{2}}\int_{0}^{2\pi}\textrm{d}\phi\,\int_{0% }^{\infty}r\textrm{d}r\,e^{-r^{2}/2\sigma_{2}^{2}}\delta\left(x-\sqrt{\left(% \frac{\sigma_{1}}{\sigma_{2}}a\right)^{2}+v^{2}}\right).

Rayleigh_fading.html

  1. R R
  2. p R ( r ) = 2 r Ω e - r 2 / Ω , r 0 p_{R}(r)=\frac{2r}{\Omega}e^{-r^{2}/\Omega},\ r\geq{}0
  3. Ω = E ( R 2 ) \Omega=E(R^{2})
  4. R ( τ ) = J 0 ( 2 π f d τ ) \,\!R(\tau)=J_{0}(2\pi f_{d}\tau)
  5. τ \,\!\tau
  6. f d f_{d}
  7. LCR = 2 π f d ρ e - ρ 2 \mathrm{LCR}=\sqrt{2\pi}f_{d}\rho e^{-\rho^{2}}
  8. f d f_{d}
  9. ρ \,\!\rho
  10. ρ = R thresh R rms . \rho=\frac{R_{\mathrm{thresh}}}{R_{\mathrm{rms}}}.
  11. ρ \,\!\rho
  12. AFD = e ρ 2 - 1 ρ f d 2 π . \mathrm{AFD}=\frac{e^{\rho^{2}}-1}{\rho f_{d}\sqrt{2\pi}}.
  13. ρ \rho
  14. AFD × LCR = 1 - e - ρ 2 . \mathrm{AFD}\times\mathrm{LCR}=1-e^{-\rho^{2}}.
  15. S ( ν ) = 1 π f d 1 - ( ν f d ) 2 , S(\nu)=\frac{1}{\pi f_{d}\sqrt{1-\left(\frac{\nu}{f_{d}}\right)^{2}}},
  16. ν \,\!\nu
  17. ν \,\!\nu
  18. ± f d \pm f_{d}
  19. α n \alpha_{n}
  20. k k
  21. n n
  22. f n = f d cos α n \,\!f_{n}=f_{d}\cos{\alpha_{n}}
  23. M M
  24. k t h k^{th}
  25. t t
  26. R ( t , k ) = 2 2 [ n = 1 M ( cos β n + j sin β n ) cos ( 2 π f n t + θ n , k ) + 1 2 ( cos α + j sin α ) cos 2 π f d t ] . R(t,k)=2\sqrt{2}\left[\sum_{n=1}^{M}\left(\cos{\beta_{n}}+j\sin{\beta_{n}}% \right)\cos{\left(2\pi f_{n}t+\theta_{n,k}\right)}+\frac{1}{\sqrt{2}}\left(% \cos{\alpha}+j\sin{\alpha}\right)\cos{2\pi f_{d}t}\right].
  27. α \,\!\alpha
  28. β n \,\!\beta_{n}
  29. θ n , k \,\!\theta_{n,k}
  30. α \,\!\alpha
  31. β n \,\!\beta_{n}
  32. R ( t ) R(t)
  33. β n = π n M + 1 \,\!\beta_{n}=\frac{\pi n}{M+1}
  34. θ n , k \,\!\theta_{n,k}
  35. θ n \,\!\theta_{n}
  36. θ n , k = β n + 2 π ( k - 1 ) M + 1 . \theta_{n,k}=\beta_{n}+\frac{2\pi(k-1)}{M+1}.
  37. α n = π ( n - 0.5 ) 2 M \alpha_{n}=\frac{\pi(n-0.5)}{2M}
  38. β n = π n M , \beta_{n}=\frac{\pi n}{M},
  39. R ( t , k ) = 2 M n = 1 M A k ( n ) ( cos β n + j sin β n ) cos ( 2 π f d t cos α n + θ n ) . R(t,k)=\sqrt{\frac{2}{M}}\sum_{n=1}^{M}A_{k}(n)\left(\cos{\beta_{n}}+j\sin{% \beta_{n}}\right)\cos{\left(2\pi f_{d}t\cos{\alpha_{n}}+\theta_{n}\right)}.
  40. A k ( n ) A_{k}(n)
  41. k k
  42. n n
  43. θ n \,\!\theta_{n}

Rayleigh_law.html

  1. H H
  2. M M
  3. M = χ 0 H + α R μ 0 H 2 . M=\chi_{0}H+\alpha_{R}\mu_{0}H^{2}.
  4. χ 0 \chi_{0}
  5. α R \alpha_{R}

Rayleigh_quotient.html

  1. R ( M , x ) R(M,x)
  2. R ( M , x ) := x * M x x * x . R(M,x):={x^{*}Mx\over x^{*}x}.
  3. x * x^{*}
  4. x x^{\prime}
  5. R ( M , c x ) = R ( M , x ) R(M,cx)=R(M,x)
  6. λ min \lambda_{\min}
  7. v min v_{\min}
  8. R ( M , x ) λ max R(M,x)\leq\lambda_{\max}
  9. R ( M , v max ) = λ max R(M,v_{\max})=\lambda_{\max}
  10. λ max \lambda_{\max}
  11. M M
  12. R ( M , x ) [ λ min , λ max ] R(M,x)\in\left[\lambda_{\min},\lambda_{\max}\right]
  13. R ( M , x ) = x * M x x * x = i = 1 n λ i y i 2 i = 1 n y i 2 R(M,x)={x^{*}Mx\over x^{*}x}=\frac{\sum_{i=1}^{n}\lambda_{i}y_{i}^{2}}{\sum_{i% =1}^{n}y_{i}^{2}}
  14. ( λ i , v i ) (\lambda_{i},v_{i})
  15. i i
  16. y i = v i * x y_{i}=v_{i}^{*}x
  17. i i
  18. v min , v max v_{\min},v_{\max}
  19. λ m a x = λ 1 λ 2 λ n = λ m i n \lambda_{max}=\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{n}=\lambda_{min}
  20. x x
  21. v 1 v_{1}
  22. y 1 = v 1 * x = 0 y_{1}=v_{1}^{*}x=0
  23. R ( M , x ) R(M,x)
  24. λ 2 \lambda_{2}
  25. x = v 2 x=v_{2}
  26. λ i \lambda_{i}
  27. M v i = A A v i = λ i v i Mv_{i}=A^{\prime}Av_{i}=\lambda_{i}v_{i}
  28. v i A A v i = v i λ i v i \Rightarrow v_{i}^{\prime}A^{\prime}Av_{i}=v_{i}^{\prime}\lambda_{i}v_{i}
  29. A v i 2 = λ i v i 2 \Rightarrow\left\|Av_{i}\right\|^{2}=\lambda_{i}\left\|v_{i}\right\|^{2}
  30. λ i = A v i 2 v i 2 0. \Rightarrow\lambda_{i}=\frac{\left\|Av_{i}\right\|^{2}}{\left\|v_{i}\right\|^{% 2}}\geq 0.
  31. M v i = λ i v i \displaystyle\qquad\qquad Mv_{i}=\lambda_{i}v_{i}
  32. x = i = 1 n α i v i , x=\sum_{i=1}^{n}\alpha_{i}v_{i},
  33. α i = x v i v i v i = x , v i v i 2 \alpha_{i}=\frac{x^{\prime}v_{i}}{v_{i}^{\prime}v_{i}}=\frac{\langle x,v_{i}% \rangle}{\left\|v_{i}\right\|^{2}}
  34. R ( M , x ) = x A A x x x = ( j = 1 n α j v j ) ( A A ) ( i = 1 n α i v i ) ( j = 1 n α j v j ) ( i = 1 n α i v i ) R(M,x)=\frac{x^{\prime}A^{\prime}Ax}{x^{\prime}x}=\frac{\left(\sum_{j=1}^{n}% \alpha_{j}v_{j}\right)^{\prime}\left(A^{\prime}A\right)\left(\sum_{i=1}^{n}% \alpha_{i}v_{i}\right)}{\left(\sum_{j=1}^{n}\alpha_{j}v_{j}\right)^{\prime}% \left(\sum_{i=1}^{n}\alpha_{i}v_{i}\right)}
  35. R ( M , x ) = i = 1 n α i 2 λ i i = 1 n α i 2 = i = 1 n λ i ( x v i ) 2 ( x x ) ( v i v i ) R(M,x)=\frac{\sum_{i=1}^{n}\alpha_{i}^{2}\lambda_{i}}{\sum_{i=1}^{n}\alpha_{i}% ^{2}}=\sum_{i=1}^{n}\lambda_{i}\frac{(x^{\prime}v_{i})^{2}}{(x^{\prime}x)(v_{i% }^{\prime}v_{i})}
  36. R ( M , x ) R(M,x)
  37. i = 1 n α i 2 λ i \sum_{i=1}^{n}\alpha_{i}^{2}\lambda_{i}
  38. i = 1 n α i 2 = 1 \sum_{i=1}^{n}\alpha_{i}^{2}=1
  39. α 1 = ± 1 \alpha_{1}=\pm 1
  40. α i = 0 \alpha_{i}=0
  41. R ( M , x ) = x T M x R(M,x)=x^{T}Mx
  42. x 2 = x T x = 1. \|x\|^{2}=x^{T}x=1.
  43. ( x ) = x T M x - λ ( x T x - 1 ) , \mathcal{L}(x)=x^{T}Mx-\lambda\left(x^{T}x-1\right),
  44. ( x ) \mathcal{L}(x)
  45. d ( x ) d x = 0 \frac{d\mathcal{L}(x)}{dx}=0
  46. 2 x T M T - 2 λ x T = 0 \therefore 2x^{T}M^{T}-2\lambda x^{T}=0
  47. M x = λ x \therefore Mx=\lambda x
  48. R ( M , x ) = x T M x x T x = λ x T x x T x = λ . R(M,x)=\frac{x^{T}Mx}{x^{T}x}=\lambda\frac{x^{T}x}{x^{T}x}=\lambda.
  49. x 1 , , x n x_{1},\cdots,x_{n}
  50. λ 1 , , λ n \lambda_{1},\cdots,\lambda_{n}
  51. L ( y ) = 1 w ( x ) ( - d d x [ p ( x ) d y d x ] + q ( x ) y ) L(y)=\frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y\right)
  52. y 1 , y 2 = a b w ( x ) y 1 ( x ) y 2 ( x ) d x \langle{y_{1},y_{2}}\rangle=\int_{a}^{b}w(x)y_{1}(x)y_{2}(x)\,dx
  53. y , L y y , y = a b y ( x ) ( - d d x [ p ( x ) d y d x ] + q ( x ) y ( x ) ) d x a b w ( x ) y ( x ) 2 d x . \frac{\langle{y,Ly}\rangle}{\langle{y,y}\rangle}=\frac{\int_{a}^{b}y(x)\left(-% \frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y(x)\right)dx}{\int_{a}^{b}{w(x% )y(x)^{2}}dx}.
  54. y , L y y , y \displaystyle\frac{\langle{y,Ly}\rangle}{\langle{y,y}\rangle}
  55. R ( A , B ; x ) := x * A x x * B x . R(A,B;x):=\frac{x^{*}Ax}{x^{*}Bx}.
  56. R ( D , C * x ) R(D,C^{*}x)
  57. D = C - 1 A C * - 1 D=C^{-1}A{C^{*}}^{-1}
  58. C C * CC^{*}
  59. R ( H ; x , y ) := y * H x y * y x * x R(H;x,y):=\frac{y^{*}Hx}{\sqrt{}}{y^{*}y\cdot x^{*}x}

Rayleigh–Jeans_law.html

  1. B λ ( T ) = 2 c k B T λ 4 , B_{\lambda}(T)=\frac{2ck_{B}T}{\lambda^{4}},
  2. B ν ( T ) = 2 ν 2 k B T c 2 . B_{\nu}(T)=\frac{2\nu^{2}k_{B}T}{c^{2}}.
  3. B λ ( T ) = 2 h c 2 λ 5 1 e h c λ k B T - 1 , B_{\lambda}(T)=\frac{2hc^{2}}{\lambda^{5}}~{}\frac{1}{e^{\frac{hc}{\lambda k_{% B}T}}-1},
  4. e x = 1 + x + x 2 2 ! + x 3 3 ! + . e^{x}=1+x+{x^{2}\over 2!}+{x^{3}\over 3!}+\cdots.
  5. e h c λ k B T 1 + h c λ k B T . e^{\frac{hc}{\lambda k_{B}T}}\approx 1+\frac{hc}{\lambda k_{B}T}.
  6. 1 e h c λ k B T - 1 1 h c λ k B T = λ k B T h c . \frac{1}{e^{\frac{hc}{\lambda k_{B}T}}-1}\approx\frac{1}{\frac{hc}{\lambda k_{% B}T}}=\frac{\lambda k_{B}T}{hc}.
  7. B λ ( T ) = 2 c k B T λ 4 , B_{\lambda}(T)=\frac{2ck_{B}T}{\lambda^{4}},
  8. h ν k B T h\nu\ll k_{B}T
  9. B ν ( T ) = 2 h ν 3 / c 2 e h ν k B T - 1 2 h ν 3 c 2 k B T h ν = 2 ν 2 k B T c 2 . B_{\nu}(T)=\frac{2h\nu^{3}/c^{2}}{e^{\frac{h\nu}{k_{B}T}}-1}\approx\frac{2h\nu% ^{3}}{c^{2}}\cdot\frac{k_{B}T}{h\nu}=\frac{2\nu^{2}k_{B}T}{c^{2}}.
  10. d P / d λ = B λ ( T ) dP/d{\lambda}=B_{\lambda}(T)
  11. d P / d ν = B ν ( T ) dP/d{\nu}=B_{\nu}(T)
  12. B λ ( T ) B ν ( T ) B_{\lambda}(T)\neq B_{\nu}(T)
  13. λ = c / ν \lambda=c/\nu
  14. B λ ( T ) B_{\lambda}(T)
  15. B ν ( T ) B_{\nu}(T)
  16. B λ d λ = d P = B ν d ν B_{\lambda}\,d\lambda=dP=B_{\nu}\,d\nu
  17. B λ ( T ) = B ν ( T ) × d ν d λ B_{\lambda}(T)=B_{\nu}(T)\times\frac{d\nu}{d\lambda}
  18. d ν d λ = d d λ ( c λ ) = - c λ 2 \frac{d\nu}{d\lambda}=\frac{d}{d\lambda}\left(\frac{c}{\lambda}\right)=-\frac{% c}{\lambda^{2}}
  19. B λ ( T ) = 2 k B T ( c λ ) 2 c 2 × c λ 2 = 2 c k B T λ 4 B_{\lambda}(T)=\frac{2k_{B}T\left(\frac{c}{\lambda}\right)^{2}}{c^{2}}\times% \frac{c}{\lambda^{2}}=\frac{2ck_{B}T}{\lambda^{4}}
  20. B λ ( T ) = 2 c 2 λ 5 h e h c λ k B T - 1 2 c k B T λ 4 B_{\lambda}(T)=\frac{2c^{2}}{\lambda^{5}}~{}\frac{h}{e^{\frac{hc}{\lambda k_{B% }T}}-1}\approx\frac{2ck_{B}T}{\lambda^{4}}
  21. B ν ( T ) = 2 h ν 3 / c 2 e h ν k B T - 1 2 k B T ν 2 c 2 B_{\nu}(T)=\frac{2h\nu^{3}/c^{2}}{e^{\frac{h\nu}{k_{B}T}}-1}\approx\frac{2k_{B% }T\nu^{2}}{c^{2}}
  22. I ( ν , T ) = π B ν ( T ) I(\nu,T)=\pi B_{\nu}(T)
  23. I ( λ , T ) = 2 π c 2 λ 5 h e h c λ k B T - 1 2 π c k B T λ 4 I(\lambda,T)=\frac{2\pi c^{2}}{\lambda^{5}}~{}\frac{h}{e^{\frac{hc}{\lambda k_% {B}T}}-1}\approx\frac{2\pi ck_{B}T}{\lambda^{4}}
  24. I ( ν , T ) = 2 π h ν 3 / c 2 e h ν k B T - 1 2 π k B T ν 2 c 2 I(\nu,T)=\frac{2\pi h\nu^{3}/c^{2}}{e^{\frac{h\nu}{k_{B}T}}-1}\approx\frac{2% \pi k_{B}T\nu^{2}}{c^{2}}
  25. u ( ν , T ) = 4 π c B ν ( T ) u(\nu,T)=\frac{4\pi}{c}B_{\nu}(T)
  26. u ( λ , T ) = 8 π c λ 5 h e h c λ k B T - 1 8 π k B T λ 4 u(\lambda,T)=\frac{8\pi c}{\lambda^{5}}~{}\frac{h}{e^{\frac{hc}{\lambda k_{B}T% }}-1}\approx\frac{8\pi k_{B}T}{\lambda^{4}}
  27. u ( ν , T ) = 8 π h ν 3 / c 3 e h ν k B T - 1 8 π k B T ν 2 c 3 u(\nu,T)=\frac{8\pi h\nu^{3}/c^{3}}{e^{\frac{h\nu}{k_{B}T}}-1}\approx\frac{8% \pi k_{B}T\nu^{2}}{c^{3}}

RC5.html

  1. O d d ( ( e - 2 ) * 2 w ) Odd((e-2)*2^{w})
  2. O d d ( ( ϕ - 2 ) * 2 w ) Odd((\phi-2)*2^{w})
  3. ϕ \phi

RC_circuit.html

  1. C d V d t + V R = 0 C\frac{dV}{dt}+\frac{V}{R}=0
  2. V ( t ) = V 0 e - t R C , V(t)=V_{0}e^{-\frac{t}{RC}}\ ,
  3. V 0 e \frac{V_{0}}{e}
  4. τ = R C . \tau=RC\ .
  5. Z C = 1 s C Z_{C}=\frac{1}{sC}
  6. s = σ + j ω s\ =\ \sigma+j\omega
  7. j j
  8. j 2 = - 1 j^{2}=-1
  9. σ \sigma
  10. ω \omega
  11. σ = 0 \sigma\ =\ 0
  12. s = j ω s\ =\ j\omega
  13. V C ( s ) = 1 / C s R + 1 / C s V i n ( s ) = 1 1 + R C s V i n ( s ) V_{C}(s)=\frac{1/Cs}{R+1/Cs}V_{in}(s)=\frac{1}{1+RCs}V_{in}(s)
  14. V R ( s ) = R R + 1 / C s V i n ( s ) = R C s 1 + R C s V i n ( s ) V_{R}(s)=\frac{R}{R+1/Cs}V_{in}(s)=\frac{RCs}{1+RCs}V_{in}(s)
  15. H C ( s ) = V C ( s ) V i n ( s ) = 1 1 + R C s H_{C}(s)={V_{C}(s)\over V_{in}(s)}={1\over 1+RCs}
  16. H R ( s ) = V R ( s ) V i n ( s ) = R C s 1 + R C s H_{R}(s)={V_{R}(s)\over V_{in}(s)}={RCs\over 1+RCs}
  17. s = - 1 R C s=-{1\over RC}
  18. G C = | H C ( j ω ) | = | V C ( j ω ) V i n ( j ω ) | = 1 1 + ( ω R C ) 2 G_{C}=|H_{C}(j\omega)|=\left|\frac{V_{C}(j\omega)}{V_{in}(j\omega)}\right|=% \frac{1}{\sqrt{1+\left(\omega RC\right)^{2}}}
  19. G R = | H R ( j ω ) | = | V R ( j ω ) V i n ( j ω ) | = ω R C 1 + ( ω R C ) 2 G_{R}=|H_{R}(j\omega)|=\left|\frac{V_{R}(j\omega)}{V_{in}(j\omega)}\right|=% \frac{\omega RC}{\sqrt{1+\left(\omega RC\right)^{2}}}
  20. ϕ C = H C ( j ω ) = tan - 1 ( - ω R C ) \phi_{C}=\angle H_{C}(j\omega)=\tan^{-1}\left(-\omega RC\right)
  21. ϕ R = H R ( j ω ) = tan - 1 ( 1 ω R C ) \phi_{R}=\angle H_{R}(j\omega)=\tan^{-1}\left(\frac{1}{\omega RC}\right)
  22. V C = G C V i n e j ϕ C V_{C}\ =\ G_{C}V_{in}e^{j\phi_{C}}
  23. V R = G R V i n e j ϕ R V_{R}\ =\ G_{R}V_{in}e^{j\phi_{R}}
  24. I ( s ) = V i n ( s ) R + 1 C s = C s 1 + R C s V i n ( s ) I(s)=\frac{V_{in}(s)}{R+\frac{1}{Cs}}={Cs\over 1+RCs}V_{in}(s)
  25. h C ( t ) = 1 R C e - t / R C u ( t ) = 1 τ e - t / τ u ( t ) h_{C}(t)={1\over RC}e^{-t/RC}u(t)={1\over\tau}e^{-t/\tau}u(t)
  26. τ = R C \tau\ =\ RC
  27. h R ( t ) = δ ( t ) - 1 R C e - t / R C u ( t ) = δ ( t ) - 1 τ e - t / τ u ( t ) h_{R}(t)=\delta(t)-{1\over RC}e^{-t/RC}u(t)=\delta(t)-{1\over\tau}e^{-t/\tau}u% (t)
  28. ω \omega\to\infty
  29. G C 0 G_{C}\to 0
  30. G R 1 G_{R}\to 1
  31. ω 0 \omega\to 0
  32. G C 1 G_{C}\to 1
  33. G R 0 G_{R}\to 0
  34. G C = G R = 1 2 G_{C}=G_{R}=\frac{1}{\sqrt{2}}
  35. ω c = 1 R C \omega_{c}=\frac{1}{RC}
  36. f c = 1 2 π R C f_{c}=\frac{1}{2\pi RC}
  37. ω 0 \omega\to 0
  38. ϕ C 0 \phi_{C}\to 0
  39. ϕ R 90 = π / 2 c \phi_{R}\to 90^{\circ}=\pi/2^{c}
  40. ω \omega\to\infty
  41. ϕ C - 90 = - π / 2 c \phi_{C}\to-90^{\circ}=-\pi/2^{c}
  42. ϕ R 0 \phi_{R}\to 0
  43. V C V_{C}
  44. V R V_{R}
  45. j ω s j\omega\to s
  46. V i n = 0 V_{in}=0
  47. t = 0 t=0
  48. V i n = V V_{in}=V
  49. V i n ( s ) = V 1 s V_{in}(s)=V\frac{1}{s}
  50. V C ( s ) = V 1 1 + s R C 1 s V_{C}(s)=V\frac{1}{1+sRC}\frac{1}{s}
  51. V R ( s ) = V s R C 1 + s R C 1 s V_{R}(s)=V\frac{sRC}{1+sRC}\frac{1}{s}
  52. V C ( t ) = V ( 1 - e - t / R C ) \,\!V_{C}(t)=V\left(1-e^{-t/RC}\right)
  53. V R ( t ) = V e - t / R C \,\!V_{R}(t)=Ve^{-t/RC}
  54. τ = R C \tau=RC
  55. 1 / e 1/e
  56. τ \tau
  57. V C V_{C}
  58. V ( 1 - 1 / e ) V(1-1/e)
  59. V R V_{R}
  60. V ( 1 / e ) V(1/e)
  61. ( 1 - 1 e ) \left(1-\frac{1}{e}\right)
  62. τ \tau
  63. t = N τ t=N\tau
  64. t = ( N + 1 ) τ t=(N+1)\tau
  65. t = N τ t=N\tau
  66. τ \tau
  67. 5 τ 5\tau
  68. V V
  69. τ \tau
  70. 5 τ 5\tau
  71. I I
  72. V i n - V C R = C d V C d t \frac{V_{in}-V_{C}}{R}=C\frac{dV_{C}}{dt}
  73. V R = V i n - V C \,\!V_{R}=V_{in}-V_{C}
  74. ω 1 R C \omega\gg\frac{1}{RC}
  75. I I
  76. I = V i n R + 1 / j ω C I=\frac{V_{in}}{R+1/j\omega C}
  77. ω C 1 R \omega C\gg\frac{1}{R}
  78. I V i n R I\approx\frac{V_{in}}{R}
  79. V C = 1 C 0 t I d t V_{C}=\frac{1}{C}\int_{0}^{t}Idt
  80. V C 1 R C 0 t V i n d t V_{C}\approx\frac{1}{RC}\int_{0}^{t}V_{in}dt
  81. ω 1 R C \omega\ll\frac{1}{RC}
  82. I I
  83. R 1 ω C R\ll\frac{1}{\omega C}
  84. I V i n 1 / j ω C I\approx\frac{V_{in}}{1/j\omega C}
  85. V i n I j ω C = V C V_{in}\approx\frac{I}{j\omega C}=V_{C}
  86. V R = I R = C d V C d t R V_{R}=IR=C\frac{dV_{C}}{dt}R
  87. V R R C d V i n d t V_{R}\approx RC\frac{dV_{in}}{dt}
  88. V o u t V_{out}
  89. V i n V_{in}
  90. I R = V i n R I_{R}=\frac{V_{in}}{R}\,
  91. I C = j ω C V i n I_{C}=j\omega CV_{in}\,
  92. I R = V i n R I_{R}=\frac{V_{in}}{R}
  93. I C = C d V i n d t I_{C}=C\frac{dV_{in}}{dt}
  94. V o u t I i n = R 1 + s R C \frac{V_{out}}{I_{in}}=\frac{R}{1+sRC}