wpmath0000002_12

Logical_assertion.html

  1. ( p ) ( x ( mod 2 ) 0 ) (\vdash p)\rightarrow(x\;\;(\mathop{{\rm mod}}2)\equiv 0)
  2. ( ( p ) ( x ( mod 2 ) 0 ) ) \vdash\left((\vdash p)\rightarrow(x\;\;(\mathop{{\rm mod}}2)\equiv 0)\right)

Logical_biconditional.html

  1. A B ~{}A\leftrightarrow B
  2. ~{}\leftrightarrow~{}
  3. ~{}\leftrightarrow~{}
  4. x 1 x 2 x 3 x n ~{}x_{1}\leftrightarrow x_{2}\leftrightarrow x_{3}\leftrightarrow...% \leftrightarrow x_{n}
  5. ( ( ( x 1 x 2 ) x 3 ) ) x n ~{}(((x_{1}\leftrightarrow x_{2})\leftrightarrow x_{3})\leftrightarrow...)% \leftrightarrow x_{n}
  6. x i ~{}x_{i}~{}
  7. ( x 1 and and x n ) ( ¬ x 1 and and ¬ x n ) (~{}x_{1}\and...\and x_{n}~{})~{}~{}(\neg x_{1}\and...\and\neg x_{n})
  8. x 1 x n ~{}x_{1}\leftrightarrow...\leftrightarrow x_{n}
  9. ¬ ( ¬ x 1 ¬ x n ) \neg~{}(\neg x_{1}\oplus...\oplus\neg x_{n})
  10. A B ¬ ( A B ) ~{}A\leftrightarrow B~{}~{}\Leftrightarrow~{}~{}\neg(A\oplus B)
  11. ¬ \Leftrightarrow\neg
  12. A B C ~{}A\leftrightarrow B\leftrightarrow C~{}~{}\Leftrightarrow
  13. A B C ~{}A\oplus B\oplus C
  14. \leftrightarrow
  15. ~{}~{}\Leftrightarrow~{}~{}
  16. \oplus
  17. ~{}~{}\Leftrightarrow~{}~{}
  18. A B C ~{}A\leftrightarrow B\leftrightarrow C
  19. ( A B ) and ( B C ) (A\leftrightarrow B)\and(B\leftrightarrow C)
  20. and \and
  21. ~{}~{}\Leftrightarrow~{}~{}
  22. A B A\leftrightarrow B
  23. \Leftrightarrow
  24. B A B\leftrightarrow A
  25. \Leftrightarrow
  26. A ~{}A
  27. ~{}~{}~{}\leftrightarrow~{}~{}~{}
  28. ( B C ) (B\leftrightarrow C)
  29. \Leftrightarrow
  30. ( A B ) (A\leftrightarrow B)
  31. ~{}~{}~{}\leftrightarrow~{}~{}~{}
  32. C ~{}C
  33. ~{}~{}~{}\leftrightarrow~{}~{}~{}
  34. \Leftrightarrow
  35. \Leftrightarrow
  36. ~{}~{}~{}\leftrightarrow~{}~{}~{}
  37. A ~{}A~{}
  38. ~{}\leftrightarrow~{}
  39. A ~{}A~{}
  40. \Leftrightarrow
  41. 1 ~{}1~{}
  42. \nLeftrightarrow
  43. A ~{}A~{}
  44. ~{}\leftrightarrow~{}
  45. \Leftrightarrow
  46. \nLeftrightarrow
  47. A B A\rightarrow B
  48. \nRightarrow
  49. ( A C ) (A\leftrightarrow C)
  50. \rightarrow
  51. ( B C ) (B\leftrightarrow C)
  52. \nRightarrow
  53. \Leftrightarrow
  54. \rightarrow
  55. A and B A\and B
  56. \Rightarrow
  57. A B A\leftrightarrow B
  58. \Rightarrow
  59. A B A\leftrightarrow B
  60. \nRightarrow
  61. A B AB
  62. \nRightarrow

Logical_equivalence.html

  1. p q p\equiv q
  2. p q p\Leftrightarrow q
  3. f e f\rightarrow e
  4. ¬ e ¬ f \neg e\rightarrow\neg f

Logical_NOR.html

  1. p q ¯ \overline{p\lor q}
  2. $\or$
  3. ¬ ( p q ) \neg(p\lor q)
  4. p + q ¯ \overline{p+q}
  5. \downarrow
  6. P Q P\downarrow Q
  7. \Leftrightarrow
  8. ¬ ( P Q ) \neg(PQ)
  9. \Leftrightarrow
  10. ¬ \neg
  11. \downarrow
  12. ¬ P \neg P
  13. \Leftrightarrow
  14. P P P\downarrow P
  15. ¬ \neg
  16. \Leftrightarrow
  17. P Q P\rightarrow Q
  18. \Leftrightarrow
  19. ( ( P P ) Q ) \Big((P\downarrow P)\downarrow Q\Big)
  20. \downarrow
  21. ( ( P P ) Q ) \Big((P\downarrow P)\downarrow Q\Big)
  22. \Leftrightarrow
  23. \downarrow
  24. P and Q P\and Q
  25. \Leftrightarrow
  26. ( P P ) (P\downarrow P)
  27. \downarrow
  28. ( Q Q ) (Q\downarrow Q)
  29. \Leftrightarrow
  30. \downarrow
  31. P Q PQ
  32. \Leftrightarrow
  33. ( P Q ) (P\downarrow Q)
  34. \downarrow
  35. ( P Q ) (P\downarrow Q)
  36. \Leftrightarrow
  37. \downarrow

Logistic_function.html

  1. f ( x ) = L 1 + e - k ( x - x 0 ) f(x)=\frac{L}{1+\mathrm{e}^{-k(x-x_{0})}}
  2. d d x f ( x ) = f ( x ) ( 1 - f ( x ) ) . \frac{d}{dx}f(x)=f(x)\cdot(1-f(x)).\,
  3. 1 - f ( x ) = f ( - x ) . 1-f(x)=f(-x).\,
  4. x f ( x ) - 1 / 2 x\mapsto f(x)-1/2
  5. d d x f ( x ) = f ( x ) ( 1 - f ( x ) ) \frac{d}{dx}f(x)=f(x)(1-f(x))
  6. f ( x ) = e x e x + e x 0 f(x)=\frac{e^{x}}{e^{x}+e^{x_{0}}}
  7. f ( x ) = e x e x + 1 = 1 1 + e - x f(x)=\frac{e^{x}}{e^{x}+1}\!=\frac{1}{1+e^{-x}}\!
  8. 2 f ( x ) = 1 + tanh ( x 2 ) 2\,f(x)=1+\tanh\left(\frac{x}{2}\right)
  9. tanh ( x ) = 2 f ( 2 x ) - 1. \tanh(x)=2\,f(2\,x)-1.
  10. tanh ( x ) = e x - e - x e x + e - x = e x ( 1 - e - 2 x ) e x ( 1 + e - 2 x ) = f ( 2 x ) - e - 2 x 1 + e - 2 x = f ( 2 x ) - e - 2 x + 1 - 1 1 + e - 2 x = 2 f ( 2 x ) - 1. \tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=\frac{e^{x}\cdot\left(1-e^{-2x}% \right)}{e^{x}\cdot\left(1+e^{-2x}\right)}=f(2x)-\frac{e^{-2x}}{1+e^{-2x}}=f(2% x)-\frac{e^{-2x}+1-1}{1+e^{-2x}}=2\,f(2\,x)-1.
  11. d P d t = r P ( 1 - P K ) \frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)
  12. P 0 P_{0}
  13. P ( t ) = K P 0 e r t K + P 0 ( e r t - 1 ) P(t)=\frac{KP_{0}e^{rt}}{K+P_{0}\left(e^{rt}-1\right)}
  14. lim t P ( t ) = K . \lim_{t\to\infty}P(t)=K.\,
  15. d n d τ = n ( 1 - n ) \frac{dn}{d\tau}=n(1-n)
  16. d P d t = r P ( 1 - P K ( t ) ) \frac{dP}{dt}=rP\left(1-\frac{P}{K(t)}\right)
  17. K ( t + T ) = K ( t ) . K(t+T)=K(t).\,
  18. p = P ( a + b x ) p=P(a+bx)\,
  19. g ( h ) = 1 1 + e - 2 β h g(h)=\frac{1}{1+e^{-2\beta h}}\!
  20. X = r ( 1 - X K ) X X^{\prime}=r\left(1-\frac{X}{K}\right)X
  21. X = F ( X ) X , F ( X ) 0 X^{\prime}=F\left(X\right)X,F^{\prime}(X)\leq 0
  22. X = r ( 1 - X K ) X - c ( t ) X , X^{\prime}=r\left(1-\frac{X}{K}\right)X-c(t)X,
  23. 1 T 0 T c ( t ) d t > r lim t + x ( t ) = 0 \frac{1}{T}\int_{0}^{T}{c(t)\,dt}>r\rightarrow\lim_{t\rightarrow+\infty}x(t)=0

Logistic_regression.html

  1. p ( y x ) p(y\mid x)
  2. σ ( t ) \sigma(t)
  3. σ ( t ) [ 0 , 1 ] \sigma(t)\in[0,1]
  4. t t
  5. σ ( t ) \sigma(t)
  6. σ ( t ) = e t e t + 1 = 1 1 + e - t , \sigma(t)=\frac{e^{t}}{e^{t}+1}=\frac{1}{1+e^{-t}},
  7. t t
  8. x x
  9. t t
  10. t = β 0 + β 1 x t=\beta_{0}+\beta_{1}x
  11. F ( x ) = 1 1 + e - ( β 0 + β 1 x ) F(x)=\frac{1}{1+e^{-(\beta_{0}+\beta_{1}x)}}
  12. F ( x ) F(x)
  13. Y i Y_{i}
  14. P ( Y i = 1 X ) P(Y_{i}=1\mid X)
  15. X i X_{i}
  16. X X
  17. β \beta
  18. g g
  19. g ( F ( x ) ) = ln F ( x ) 1 - F ( x ) = β 0 + β 1 x , g(F(x))=\ln\frac{F(x)}{1-F(x)}=\beta_{0}+\beta_{1}x,
  20. F ( x ) 1 - F ( x ) = e β 0 + β 1 x . \frac{F(x)}{1-F(x)}=e^{\beta_{0}+\beta_{1}x}.
  21. g ( ) g(\cdot)
  22. g ( F ( x ) ) g(F(x))
  23. ln \ln
  24. F ( x ) F(x)
  25. x x
  26. F ( x ) F(x)
  27. F ( x ) F(x)
  28. β 0 \beta_{0}
  29. β 1 x \beta_{1}x
  30. e e
  31. x x
  32. x x
  33. odds = e β 0 + β 1 x . \,\text{odds}=e^{\beta_{0}+\beta_{1}x}.
  34. O R = odds ( x + 1 ) / odds ( x ) = F ( x + 1 ) 1 - F ( x + 1 ) F ( x ) 1 - F ( x ) = e β 0 + β 1 ( x + 1 ) / e β 0 + β 1 x = e β 1 OR=\operatorname{odds}(x+1)/\operatorname{odds}(x)=\frac{\frac{F(x+1)}{1-F(x+1% )}}{\frac{F(x)}{1-F(x)}}=e^{\beta_{0}+\beta_{1}(x+1)}/e^{\beta_{0}+\beta_{1}x}% =e^{\beta_{1}}
  35. β 1 \beta_{1}
  36. e β 1 e^{\beta_{1}}
  37. a d b c \frac{ad}{bc}
  38. β 0 + β 1 x \beta_{0}+\beta_{1}x
  39. β 0 + β 1 x 1 + β 2 x 2 + + β m x m . \beta_{0}+\beta_{1}x_{1}+\beta_{2}x_{2}+\cdots+\beta_{m}x_{m}.
  40. β j \beta_{j}
  41. D = - 2 ln likelihood of the fitted model likelihood of the saturated model . D=-2\ln\frac{\,\text{likelihood of the fitted model}}{\,\text{likelihood of % the saturated model}}.
  42. χ s - p 2 , \chi^{2}_{s-p},
  43. D null = - 2 ln likelihood of null model likelihood of the saturated model D_{\,\text{null}}=-2\ln\frac{\,\text{likelihood of null model}}{\,\text{% likelihood of the saturated model}}
  44. D fitted = - 2 ln likelihood of fitted model likelihood of the saturated model . D_{\,\text{fitted}}=-2\ln\frac{\,\text{likelihood of fitted model}}{\,\text{% likelihood of the saturated model}}.
  45. D null - D fitted = ( - 2 ln likelihood of null model likelihood of the saturated model ) - ( - 2 ln likelihood of fitted model likelihood of the saturated model ) = - 2 ( ln likelihood of null model likelihood of the saturated model - ln likelihood of fitted model likelihood of the saturated model ) = - 2 ln ( likelihood of null model likelihood of the saturated model ) ( likelihood of fitted model likelihood of the saturated model ) = - 2 ln likelihood of the null model likelihood of fitted model . \begin{aligned}\displaystyle D\text{null}-D\text{fitted}&\displaystyle=\left(-% 2\ln\frac{\,\text{likelihood of null model}}{\,\text{likelihood of the % saturated model}}\right)-\left(-2\ln\frac{\,\text{likelihood of fitted model}}% {\,\text{likelihood of the saturated model}}\right)\\ &\displaystyle=-2\left(\ln\frac{\,\text{likelihood of null model}}{\,\text{% likelihood of the saturated model}}-\ln\frac{\,\text{likelihood of fitted % model}}{\,\text{likelihood of the saturated model}}\right)\\ \displaystyle=&\displaystyle-2\ln\frac{\left(\frac{\,\text{likelihood of null % model}}{\,\text{likelihood of the saturated model}}\right)}{\left(\frac{\,% \text{likelihood of fitted model}}{\,\text{likelihood of the saturated model}}% \right)}\\ \displaystyle=&\displaystyle-2\ln\frac{\,\text{likelihood of the null model}}{% \,\text{likelihood of fitted model}}.\end{aligned}
  46. R 2 L = D null - D fitted D null . R^{2}\text{L}=\frac{D\text{null}-D\text{fitted}}{D\text{null}}.
  47. χ 2 \chi^{2}
  48. W j = B j 2 S E B j 2 W_{j}=\frac{B^{2}_{j}}{SE^{2}_{B_{j}}}
  49. β j \beta_{j}
  50. β 0 \beta_{0}
  51. β 0 \beta_{0}
  52. β 0 * ^ = β 0 ^ + log π 1 - π - log π ~ 1 - π ~ \hat{\beta_{0}^{*}}=\hat{\beta_{0}}+\log{{\pi}\over{1-\pi}}-\log{{\tilde{\pi}}% \over{1-\tilde{\pi}}}
  53. π \pi
  54. π ~ \tilde{\pi}
  55. Y i x 1 , i , , x m , i \displaystyle Y_{i}\mid x_{1,i},\ldots,x_{m,i}
  56. f ( i ) f(i)
  57. f ( i ) = β 0 + β 1 x 1 , i + + β m x m , i , f(i)=\beta_{0}+\beta_{1}x_{1,i}+\cdots+\beta_{m}x_{m,i},
  58. β 0 , , β m \beta_{0},\ldots,\beta_{m}
  59. f ( i ) = s y m b o l β 𝐗 i , f(i)=symbol\beta\cdot\mathbf{X}_{i},
  60. logit ( 𝔼 [ Y i x 1 , i , , x m , i ] ) = logit ( p i ) = ln ( p i 1 - p i ) = β 0 + β 1 x 1 , i + + β m x m , i \operatorname{logit}(\mathbb{E}[Y_{i}\mid x_{1,i},\ldots,x_{m,i}])=% \operatorname{logit}(p_{i})=\ln\left(\frac{p_{i}}{1-p_{i}}\right)=\beta_{0}+% \beta_{1}x_{1,i}+\cdots+\beta_{m}x_{m,i}
  61. logit ( 𝔼 [ Y i 𝐗 i ] ) = logit ( p i ) = ln ( p i 1 - p i ) = s y m b o l β 𝐗 i \operatorname{logit}(\mathbb{E}[Y_{i}\mid\mathbf{X}_{i}])=\operatorname{logit}% (p_{i})=\ln\left(\frac{p_{i}}{1-p_{i}}\right)=symbol\beta\cdot\mathbf{X}_{i}
  62. ( - , + ) (-\infty,+\infty)
  63. e β e^{\beta}
  64. 𝔼 [ Y i 𝐗 i ] = p i = logit - 1 ( s y m b o l β 𝐗 i ) = 1 1 + e - s y m b o l β 𝐗 i \mathbb{E}[Y_{i}\mid\mathbf{X}_{i}]=p_{i}=\operatorname{logit}^{-1}(symbol% \beta\cdot\mathbf{X}_{i})=\frac{1}{1+e^{-symbol\beta\cdot\mathbf{X}_{i}}}
  65. Pr ( Y i = y i 𝐗 i ) = p i y i ( 1 - p i ) 1 - y i = ( e s y m b o l β 𝐗 i 1 + e s y m b o l β 𝐗 i ) y i ( 1 - e s y m b o l β 𝐗 i 1 + e s y m b o l β 𝐗 i ) 1 - y i = e s y m b o l β 𝐗 i y i 1 + e s y m b o l β 𝐗 i \operatorname{Pr}(Y_{i}=y_{i}\mid\mathbf{X}_{i})={p_{i}}^{y_{i}}(1-p_{i})^{1-y% _{i}}=\left(\frac{e^{symbol\beta\cdot\mathbf{X}_{i}}}{1+e^{symbol\beta\cdot% \mathbf{X}_{i}}}\right)^{y_{i}}\left(1-\frac{e^{symbol\beta\cdot\mathbf{X}_{i}% }}{1+e^{symbol\beta\cdot\mathbf{X}_{i}}}\right)^{1-y_{i}}=\frac{e^{symbol\beta% \cdot\mathbf{X}_{i}\cdot y_{i}}}{1+e^{symbol\beta\cdot\mathbf{X}_{i}}}
  66. Y i = s y m b o l β 𝐗 i + ε Y_{i}^{\ast}=symbol\beta\cdot\mathbf{X}_{i}+\varepsilon\,
  67. ε Logistic ( 0 , 1 ) \varepsilon\sim\operatorname{Logistic}(0,1)\,
  68. Y i = { 1 if Y i > 0 i.e. - ε < s y m b o l β 𝐗 i , 0 otherwise. Y_{i}=\begin{cases}1&\,\text{if }Y_{i}^{\ast}>0\ \,\text{ i.e. }-\varepsilon<% symbol\beta\cdot\mathbf{X}_{i},\\ 0&\,\text{otherwise.}\end{cases}
  69. Pr ( ε < x ) = logit - 1 ( x ) \Pr(\varepsilon<x)=\operatorname{logit}^{-1}(x)
  70. Pr ( Y i = 1 𝐗 i ) \displaystyle\Pr(Y_{i}=1\mid\mathbf{X}_{i})
  71. Y i 0 \displaystyle Y_{i}^{0\ast}
  72. ε 0 \displaystyle\varepsilon_{0}
  73. Pr ( ε 0 = x ) = Pr ( ε 1 = x ) = e - x e - e - x \Pr(\varepsilon_{0}=x)=\Pr(\varepsilon_{1}=x)=e^{-x}e^{-e^{-x}}
  74. Y i = { 1 if Y i 1 > Y i 0 , 0 otherwise. Y_{i}=\begin{cases}1&\,\text{if }Y_{i}^{1\ast}>Y_{i}^{0\ast},\\ 0&\,\text{otherwise.}\end{cases}
  75. s y m b o l β = s y m b o l β 1 - s y m b o l β 0 symbol\beta=symbol\beta_{1}-symbol\beta_{0}
  76. ε = ε 1 - ε 0 \varepsilon=\varepsilon_{1}-\varepsilon_{0}
  77. ε = ε 1 - ε 0 Logistic ( 0 , 1 ) . \varepsilon=\varepsilon_{1}-\varepsilon_{0}\sim\operatorname{Logistic}(0,1).
  78. Pr ( Y i = 1 𝐗 i ) \displaystyle\Pr(Y_{i}=1\mid\mathbf{X}_{i})
  79. ln Pr ( Y i = 0 ) \displaystyle\ln\Pr(Y_{i}=0)
  80. - l n Z -lnZ
  81. Pr ( Y i = 0 ) \displaystyle\Pr(Y_{i}=0)
  82. Z = e s y m b o l β 0 𝐗 i + e s y m b o l β 1 𝐗 i Z=e^{symbol\beta_{0}\cdot\mathbf{X}_{i}}+e^{symbol\beta_{1}\cdot\mathbf{X}_{i}}
  83. Pr ( Y i = 0 ) = e s y m b o l β 0 𝐗 i e s y m b o l β 0 𝐗 i + e s y m b o l β 1 𝐗 i Pr ( Y i = 1 ) = e s y m b o l β 1 𝐗 i e s y m b o l β 0 𝐗 i + e s y m b o l β 1 𝐗 i \begin{aligned}\displaystyle\Pr(Y_{i}=0)&\displaystyle=\frac{e^{symbol\beta_{0% }\cdot\mathbf{X}_{i}}}{e^{symbol\beta_{0}\cdot\mathbf{X}_{i}}+e^{symbol\beta_{% 1}\cdot\mathbf{X}_{i}}}\\ \displaystyle\Pr(Y_{i}=1)&\displaystyle=\frac{e^{symbol\beta_{1}\cdot\mathbf{X% }_{i}}}{e^{symbol\beta_{0}\cdot\mathbf{X}_{i}}+e^{symbol\beta_{1}\cdot\mathbf{% X}_{i}}}\end{aligned}
  84. Pr ( Y i = c ) = e s y m b o l β c 𝐗 i h e s y m b o l β h 𝐗 i \Pr(Y_{i}=c)=\frac{e^{symbol\beta_{c}\cdot\mathbf{X}_{i}}}{\sum_{h}e^{symbol% \beta_{h}\cdot\mathbf{X}_{i}}}
  85. Pr ( Y i = c ) = softmax ( c , s y m b o l β 0 𝐗 i , s y m b o l β 1 𝐗 i , ) . \Pr(Y_{i}=c)=\operatorname{softmax}(c,symbol\beta_{0}\cdot\mathbf{X}_{i},% symbol\beta_{1}\cdot\mathbf{X}_{i},\dots).
  86. Pr ( Y i = 0 ) \Pr(Y_{i}=0)
  87. Pr ( Y i = 1 ) \Pr(Y_{i}=1)
  88. Pr ( Y i = 0 ) + Pr ( Y i = 1 ) = 1 \Pr(Y_{i}=0)+\Pr(Y_{i}=1)=1
  89. Pr ( Y i = 1 ) \displaystyle\Pr(Y_{i}=1)
  90. s y m b o l β 0 = 0. symbol\beta_{0}=\mathbf{0}.
  91. e s y m b o l β 0 𝐗 i = e 𝟎 𝐗 i = 1 e^{symbol\beta_{0}\cdot\mathbf{X}_{i}}=e^{\mathbf{0}\cdot\mathbf{X}_{i}}=1
  92. Pr ( Y i = 1 ) = e s y m b o l β 1 𝐗 i 1 + e s y m b o l β 1 𝐗 i = 1 1 + e - s y m b o l β 1 𝐗 i = p i \Pr(Y_{i}=1)=\frac{e^{symbol\beta_{1}\cdot\mathbf{X}_{i}}}{1+e^{symbol\beta_{1% }\cdot\mathbf{X}_{i}}}=\frac{1}{1+e^{-symbol\beta_{1}\cdot\mathbf{X}_{i}}}=p_{i}
  93. s y m b o l β = s y m b o l β 1 - s y m b o l β 0 symbol\beta=symbol\beta_{1}-symbol\beta_{0}
  94. p i = 1 1 + e - ( β 0 + β 1 x 1 , i + + β k x k , i ) . p_{i}=\frac{1}{1+e^{-(\beta_{0}+\beta_{1}x_{1,i}+\cdots+\beta_{k}x_{k,i})}}.\,
  95. y = 1 1 + e - f ( X ) y=\frac{1}{1+e^{-f(X)}}
  96. d y d X = y ( 1 - y ) d f d X . \frac{\mathrm{d}y}{\mathrm{d}X}=y(1-y)\frac{\mathrm{d}f}{\mathrm{d}X}.\,
  97. Y i Bin ( n i , p i ) , for i = 1 , , n Y_{i}\ \sim\operatorname{Bin}(n_{i},p_{i}),\,\text{ for }i=1,\dots,n
  98. p i = 𝔼 [ Y i n i | 𝐗 i ] , p_{i}=\mathbb{E}\left[\left.\frac{Y_{i}}{n_{i}}\,\right|\,\mathbf{X}_{i}\right],
  99. logit ( 𝔼 [ Y i n i | 𝐗 i ] ) = logit ( p i ) = ln ( p i 1 - p i ) = s y m b o l β 𝐗 i , \operatorname{logit}\left(\mathbb{E}\left[\left.\frac{Y_{i}}{n_{i}}\,\right|\,% \mathbf{X}_{i}\right]\right)=\operatorname{logit}(p_{i})=\ln\left(\frac{p_{i}}% {1-p_{i}}\right)=symbol\beta\cdot\mathbf{X}_{i},
  100. Pr ( Y i = y i 𝐗 i ) = ( n i y i ) p i y i ( 1 - p i ) n i - y i = ( n i y i ) ( 1 1 + e - s y m b o l β 𝐗 i ) y i ( 1 - 1 1 + e - s y m b o l β 𝐗 i ) n i - y i \operatorname{Pr}(Y_{i}=y_{i}\mid\mathbf{X}_{i})={n_{i}\choose y_{i}}p_{i}^{y_% {i}}(1-p_{i})^{n_{i}-y_{i}}={n_{i}\choose y_{i}}\left(\frac{1}{1+e^{-symbol% \beta\cdot\mathbf{X}_{i}}}\right)^{y_{i}}\left(1-\frac{1}{1+e^{-symbol\beta% \cdot\mathbf{X}_{i}}}\right)^{n_{i}-y_{i}}
  101. σ ( x ) \sigma(x)
  102. Φ ( π 8 x ) \Phi(\sqrt{\frac{\pi}{8}}x)
  103. Y i Y_{i}\ast
  104. s y m b o l β 𝐗 i symbol\beta\cdot\mathbf{X}_{i}
  105. μ \displaystyle\mu
  106. μ \displaystyle\mu
  107. 1 2 \frac{1}{2}
  108. 1 2 \frac{1}{2}
  109. Accuracy = T P + T N T P + F P + F N + T N \,\text{Accuracy}=\frac{TP+TN}{TP+FP+FN+TN}
  110. Precision = Positive predictive value = T P T P + F P \,\text{Precision}=\,\text{Positive predictive value}=\frac{TP}{TP+FP}\,
  111. Negative predictive value = T N T N + F N \,\text{Negative predictive value}=\frac{TN}{TN+FN}
  112. Recall = Sensitivity = T P T P + F N \,\text{Recall}=\,\text{Sensitivity}=\frac{TP}{TP+FN}\,
  113. Specificity = T N T N + F P \,\text{Specificity}=\frac{TN}{TN+FP}

Logit.html

  1. p p
  2. p / ( 1 p ) p/(1−p)
  3. logit ( p ) = log ( p 1 - p ) = log ( p ) - log ( 1 - p ) = - log ( 1 p - 1 ) . \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right)=\log(p)-\log(1-p)=-\log% \left(\frac{1}{p}-1\right).\!\,
  4. α \alpha
  5. logit - 1 ( α ) = 1 1 + exp ( - α ) = exp ( α ) exp ( α ) + 1 \operatorname{logit}^{-1}(\alpha)=\frac{1}{1+\operatorname{exp}(-\alpha)}=% \frac{\operatorname{exp}(\alpha)}{\operatorname{exp}(\alpha)+1}
  6. log ( R ) = log ( p 1 / ( 1 - p 1 ) p 2 / ( 1 - p 2 ) ) = log ( p 1 1 - p 1 ) - log ( p 2 1 - p 2 ) = logit ( p 1 ) - logit ( p 2 ) . \operatorname{log}(R)=\log\left(\frac{{p_{1}}/(1-p_{1})}{{p_{2}}/(1-p_{2})}% \right)=\log\left(\frac{p_{1}}{1-p_{1}}\right)-\log\left(\frac{p_{2}}{1-p_{2}}% \right)=\operatorname{logit}(p_{1})-\operatorname{logit}(p_{2}).\!\,
  7. logit ( x ) \operatorname{logit}(x)
  8. Φ - 1 ( x ) / π 8 \Phi^{-1}(x)/\sqrt{\frac{\pi}{8}}
  9. Φ - 1 ( x ) \Phi^{-1}(x)
  10. Φ ( x ) \Phi(x)
  11. Φ ( x ) = - x 1 2 π e - z 2 2 d z \Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-\frac{z^{2}}{2}}% \operatorname{d}\!z

Long_s.html

  1. a b f ( x ) d x \int\limits_{a}^{b}f(x)\;\mathrm{d}x

Longest_common_subsequence_problem.html

  1. N N
  2. n 1 , , n N n_{1},...,n_{N}
  3. 2 n 1 2^{n_{1}}
  4. O ( 2 n 1 i > 1 n i ) . O\left(2^{n_{1}}\sum_{i>1}n_{i}\right).
  5. O ( N i = 1 N n i ) . O\left(N\prod_{i=1}^{N}n_{i}\right).
  6. L C S ( X i , Y j ) = { if i = 0 or j = 0 L C S ( X i - 1 , Y j - 1 ) x i if x i = y j longest ( L C S ( X i , Y j - 1 ) , L C S ( X i - 1 , Y j ) ) if x i y j LCS\left(X_{i},Y_{j}\right)=\begin{cases}&\mbox{ if }~{}\ i=0\mbox{ or }~{}j=0% \\ \textrm{ }LCS\left(X_{i-1},Y_{j-1}\right)\frown x_{i}&\mbox{ if }~{}x_{i}=y_{j% }\\ \mbox{longest}~{}\left(LCS\left(X_{i},Y_{j-1}\right),LCS\left(X_{i-1},Y_{j}% \right)\right)&\mbox{ if }~{}x_{i}\neq y_{j}\\ \end{cases}
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  67. X 1 m X_{1\dots m}
  68. Y 1 n Y_{1\dots n}
  69. | S C S ( X , Y ) | = n + m - | L C S ( X , Y ) | . \left|SCS(X,Y)\right|=n+m-\left|LCS(X,Y)\right|.
  70. d ( X , Y ) = n + m - 2 | L C S ( X , Y ) | . d^{\prime}(X,Y)=n+m-2\cdot\left|LCS(X,Y)\right|.
  71. x i x_{i}
  72. y j y_{j}
  73. x i x_{i}
  74. y j y_{j}
  75. Y Y
  76. X X
  77. Y Y
  78. X X
  79. Y Y
  80. i i
  81. j j
  82. X 1.. i X_{1..i}
  83. Y 1.. j Y_{1..j}
  84. X X
  85. Y Y
  86. X 1.. i - 1 X_{1..i-1}
  87. Y 1.. j Y_{1..j}
  88. X 1.. i X_{1..i}
  89. Y 1.. j - 1 Y_{1..j-1}
  90. 2 × min ( n , m ) 2\times\min(n,m)
  91. min ( m , n ) + 1 \min(m,n)+1
  92. r r
  93. n > m n>m
  94. O ( ( n + r ) log ( n ) ) O((n+r)\log(n))

Longitudinal_wave.html

  1. y ( x , t ) = y 0 cos ( ω ( t - x c ) ) y(x,t)=y_{0}\cos\Bigg(\omega\left(t-\frac{x}{c}\right)\Bigg)
  2. f = ω 2 π . f=\frac{\omega}{2\pi}.
  3. y ( x , t ) = y 0 cos ( k x - ω t + φ ) y(x,t)\,=y_{0}\cos(kx-\omega t+\varphi)

Loop_quantum_gravity.html

  1. { q i , p j } = δ i j \{q_{i},p_{j}\}=\delta_{ij}
  2. { f , g } = i = 1 N ( f q i g p i - f p i g q i ) . \{f,g\}=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial q_{i}}\frac{\partial g}% {\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q% _{i}}\right).
  3. f ( q i , p j ) f(q_{i},p_{j})
  4. g ( q i , p j ) g(q_{i},p_{j})
  5. q ˙ i = { q i , H } \dot{q}_{i}=\{q_{i},H\}
  6. p ˙ i = { p i , H } \dot{p}_{i}=\{p_{i},H\}
  7. H H
  8. F ( q , p ) F(q,p)
  9. d d t F ( q i , p i ) = { F , H } . {d\over dt}F(q_{i},p_{i})=\{F,H\}.
  10. S U ( 2 ) SU(2)
  11. G j ( x ) = 0 G_{j}(x)=0
  12. x x
  13. SU ( 2 ) \mathrm{SU}(2)
  14. A a i ( x ) A_{a}^{i}(x)
  15. i i
  16. λ j ( x ) \lambda^{j}(x)
  17. G ( λ ) = d 3 x G j ( x ) λ j ( x ) G(\lambda)=\int d^{3}xG_{j}(x)\lambda^{j}(x)
  18. SU ( 2 ) \mathrm{SU}(2)
  19. C a ( x ) = 0 C_{a}(x)=0
  20. N ( x ) \vec{N}(x)
  21. C ( N ) = d 3 x C a ( x ) N a ( x ) C(\vec{N})=\int d^{3}xC_{a}(x)N^{a}(x)
  22. N a ( x ) N^{a}(x)
  23. H ( x ) = 0 H(x)=0
  24. N ( x ) N(x)
  25. H ( N ) = d 3 x H ( x ) N ( x ) H(N)=\int d^{3}xH(x)N(x)
  26. N ( x ) N(x)
  27. A a i ( x ) A_{a}^{i}(x)
  28. q q
  29. E ~ i a ( x ) \tilde{E}^{a}_{i}(x)
  30. V \mathcal{L}_{V}
  31. V V
  32. { G ( λ ) , G ( μ ) } = G ( [ λ , μ ] ) \{G(\lambda),G(\mu)\}=G([\lambda,\mu])
  33. [ λ , μ ] k = λ i μ j ϵ i j k [\lambda,\mu]^{k}=\lambda_{i}\mu_{j}\epsilon^{ijk}
  34. { C ( N ) , C ( M ) } = C ( N M ) \{C(\vec{N}),C(\vec{M})\}=C(\mathcal{L}_{\vec{N}}\vec{M})
  35. { C ( N ) , G ( λ ) } = G ( N λ ) \{C(\vec{N}),G(\lambda)\}=G(\mathcal{L}_{\vec{N}}\lambda)
  36. λ \lambda
  37. { C ( N ) , H ( M ) } = H ( N M ) \{C(\vec{N}),H(M)\}=H(\mathcal{L}_{\vec{N}}M)
  38. { H ( N ) , H ( M ) } = C ( K ) \{H(N),H(M)\}=C(K)
  39. K K
  40. C I C_{I}
  41. { C I , C J } = f I J K C K \{C_{I},C_{J}\}=f_{IJ}^{K}C_{K}
  42. f I J K f_{IJ}^{K}
  43. O O
  44. { G j , O } G j = C a = H = 0 = { C a , O } G j = C a = H = 0 = { H , O } G j = C a = H = 0 = 0 \{G_{j},O\}_{G_{j}=C_{a}=H=0}=\{C_{a},O\}_{G_{j}=C_{a}=H=0}=\{H,O\}_{G_{j}=C_{% a}=H=0}=0
  45. G j = 0 G_{j}=0
  46. H , C a H,C_{a}
  47. E ~ i a \tilde{E}_{i}^{a}
  48. E i a E_{i}^{a}
  49. i = 1 , 2 , 3 i=1,2,3
  50. E ~ i a = det ( q ) E i a \tilde{E}_{i}^{a}=\sqrt{\operatorname{det}(q)}E_{i}^{a}
  51. det ( q ) q a b = E ~ i a E ~ j b δ i j \operatorname{det}(q)q^{ab}=\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\delta^{ij}
  52. δ i j \delta^{ij}
  53. q a b q^{ab}
  54. E i a E_{i}^{a}
  55. i i
  56. K a i = K a b E ~ a i / det ( q ) K_{a}^{i}=K_{ab}\tilde{E}^{ai}/\sqrt{\operatorname{det}(q)}
  57. A a i = Γ a i - i K a i A_{a}^{i}=\Gamma_{a}^{i}-iK_{a}^{i}
  58. SU ( 2 ) \operatorname{SU}(2)
  59. Γ a i \Gamma_{a}^{i}
  60. Γ a i = Γ a j k ϵ j k i \Gamma_{a}^{i}=\Gamma_{ajk}\epsilon^{jki}
  61. A a i A_{a}^{i}
  62. 𝒟 a \mathcal{D}_{a}
  63. E ~ i a \tilde{E}^{a}_{i}
  64. A a i A_{a}^{i}
  65. G i = 𝒟 a E ~ i a = 0 G^{i}=\mathcal{D}_{a}\tilde{E}_{i}^{a}=0
  66. C a = E ~ i b F a b i - A a i ( 𝒟 b E ~ i b ) = V a - A a i G i = 0 C_{a}=\tilde{E}_{i}^{b}F^{i}_{ab}-A_{a}^{i}(\mathcal{D}_{b}\tilde{E}_{i}^{b})=% V_{a}-A_{a}^{i}G^{i}=0
  67. H ~ = ϵ i j k E ~ i a E ~ j b F a b i = 0 \tilde{H}=\epsilon_{ijk}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}F^{i}_{ab}=0
  68. F a b i F^{i}_{ab}
  69. A a i A_{a}^{i}
  70. V a V_{a}
  71. SU ( 2 ) \operatorname{SU}(2)
  72. A a i A^{i}_{a}
  73. Ψ ( A a i ) \Psi(A^{i}_{a})
  74. q q
  75. ψ ( q ) \psi(q)
  76. A ^ a i Ψ ( A ) = A a i Ψ ( A ) \hat{A}_{a}^{i}\Psi(A)=A_{a}^{i}\Psi(A)
  77. q ^ ψ ( q ) = q ψ ( q ) \hat{q}\psi(q)=q\psi(q)
  78. E i a ~ ^ Ψ ( A ) = - i δ Ψ ( A ) δ A a i \hat{\tilde{E_{i}^{a}}}\Psi(A)=-i{\delta\Psi(A)\over\delta A_{a}^{i}}
  79. p ^ ψ ( q ) = - i d ψ ( q ) / d q \hat{p}\psi(q)=-i\hbar d\psi(q)/dq
  80. SU ( 2 ) \operatorname{SU}(2)
  81. A A
  82. E ~ \tilde{E}
  83. E ~ \tilde{E}
  84. G ^ j | ψ = 0 \hat{G}_{j}|\psi\rangle=0
  85. C ^ a | ψ = 0 \hat{C}_{a}|\psi\rangle=0
  86. H ~ ^ | ψ = 0 \hat{\tilde{H}}|\psi\rangle=0
  87. H ~ = det ( q ) H \tilde{H}=\sqrt{\operatorname{det}(q)}H
  88. G ( λ ) = d 3 x λ j ( D a E a ) j G(\lambda)=\int d^{3}x\lambda^{j}(D_{a}E^{a})^{j}
  89. { G ( λ ) , A a i } = a λ i + g ϵ i j k A a j λ k = ( D a λ ) i . \{G(\lambda),A_{a}^{i}\}=\partial_{a}\lambda^{i}+g\epsilon^{ijk}A_{a}^{j}% \lambda^{k}=(D_{a}\lambda)^{i}.
  90. G ^ j Ψ ( A ) = - i D a δ λ Ψ [ A ] δ A a j = 0. \hat{G}_{j}\Psi(A)=-iD_{a}{\delta\lambda\Psi[A]\over\delta A_{a}^{j}}=0.
  91. Ψ \Psi
  92. λ \lambda
  93. [ 1 + d 3 x λ j ( x ) G ^ j ] Ψ ( A ) = Ψ [ A + D λ ] = Ψ [ A ] , \Big[1+\int d^{3}x\lambda^{j}(x)\hat{G}_{j}\Big]\Psi(A)=\Psi[A+D\lambda]=\Psi[% A],
  94. Ψ [ A ] \Psi[A]
  95. C I = 0 C_{I}=0
  96. Ψ \Psi
  97. C ^ I Ψ = 0 \hat{C}_{I}\Psi=0
  98. C ^ I \hat{C}_{I}
  99. h γ [ A ] λ p h i h_{\gamma}[A]\lambda^{p}hi
  100. ( h e ) α β = U α γ - 1 ( x ) ( h e ) γ σ U σ β ( y ) (h^{\prime}_{e})_{\alpha\beta}=U_{\alpha\gamma}^{-1}(x)(h_{e})_{\gamma\sigma}U% _{\sigma\beta}(y)
  101. x = y x=y
  102. α = β \alpha=\beta
  103. ( h e ) α α = U α γ - 1 ( x ) ( h e ) γ σ U σ α ( x ) = [ U σ α ( x ) U α γ - 1 ( x ) ] ( h e ) γ σ = δ σ γ ( h e ) γ σ = ( h e ) γ γ (h^{\prime}_{e})_{\alpha\alpha}=U_{\alpha\gamma}^{-1}(x)(h_{e})_{\gamma\sigma}% U_{\sigma\alpha}(x)=[U_{\sigma\alpha}(x)U_{\alpha\gamma}^{-1}(x)](h_{e})_{% \gamma\sigma}=\delta_{\sigma\gamma}(h_{e})_{\gamma\sigma}=(h_{e})_{\gamma\gamma}
  104. Tr h γ = Tr h γ . \operatorname{Tr}h^{\prime}_{\gamma}=\operatorname{Tr}h_{\gamma}.
  105. W γ [ A ] W_{\gamma}[A]
  106. h γ [ A ] = 𝒫 exp { - γ 0 γ 1 d s γ ˙ a A a i ( γ ( s ) ) T i } h_{\gamma}[A]=\mathcal{P}\exp\Big\{-\int_{\gamma_{0}}^{\gamma_{1}}ds\dot{% \gamma}^{a}A_{a}^{i}(\gamma(s))T_{i}\Big\}
  107. γ \gamma
  108. s s
  109. 𝒫 \mathcal{P}
  110. s s
  111. T i T_{i}
  112. SU ( 2 ) \operatorname{SU}(2)
  113. [ T i , T j ] = 2 i ϵ i j k T k [T^{i},T^{j}]=2i\epsilon^{ijk}T^{k}
  114. ( N + 1 ) × ( N + 1 ) (N+1)\times(N+1)
  115. N = 1 , 2 , 3 , N=1,2,3,\dots
  116. SU ( 2 ) \operatorname{SU}(2)
  117. N / 2 N/2
  118. Ψ [ A ] = γ Ψ [ γ ] W γ [ A ] \Psi[A]=\sum_{\gamma}\Psi[\gamma]W_{\gamma}[A]
  119. exp ( i k x ) \exp(ikx)
  120. k k
  121. ψ [ x ] = d k ψ ( k ) exp ( i k x ) \psi[x]=\int dk\psi(k)\exp(ikx)
  122. ψ ( k ) \psi(k)
  123. Ψ [ γ ] = [ d A ] Ψ [ A ] W γ [ A ] \Psi[\gamma]=\int[dA]\Psi[A]W_{\gamma}[A]
  124. O ^ \hat{O}
  125. Φ [ A ] = O ^ Ψ [ A ] E q 1 \Phi[A]=\hat{O}\Psi[A]\qquad Eq\;1
  126. O ^ \hat{O}^{\prime}
  127. Ψ [ γ ] \Psi[\gamma]
  128. Φ [ γ ] = O ^ Ψ [ γ ] E q 2 \Phi[\gamma]=\hat{O}^{\prime}\Psi[\gamma]\qquad Eq\;2
  129. Φ [ γ ] \Phi[\gamma]
  130. Φ [ γ ] = [ d A ] Φ [ A ] W γ [ A ] E q 3. \Phi[\gamma]=\int[dA]\Phi[A]W_{\gamma}[A]\qquad Eq\;3.
  131. O ^ \hat{O}^{\prime}
  132. Ψ [ γ ] \Psi[\gamma]
  133. O ^ \hat{O}
  134. Ψ [ A ] \Psi[A]
  135. E q 2 Eq\;2
  136. E q 3 Eq\;3
  137. E q 1 Eq\;1
  138. E q 3 Eq\;3
  139. O ^ Ψ [ γ ] = [ d A ] W γ [ A ] O ^ Ψ [ A ] \hat{O}^{\prime}\Psi[\gamma]=\int[dA]W_{\gamma}[A]\hat{O}\Psi[A]
  140. O ^ Ψ [ γ ] = [ d A ] ( O ^ W γ [ A ] ) Ψ [ A ] \hat{O}^{\prime}\Psi[\gamma]=\int[dA](\hat{O}^{\dagger}W_{\gamma}[A])\Psi[A]
  141. O ^ \hat{O}^{\dagger}
  142. O ^ \hat{O}
  143. Ψ [ A ] \Psi[A]
  144. O ^ \hat{O}^{\prime}
  145. O ^ \hat{O}^{\dagger}
  146. A A
  147. W γ [ A ] W_{\gamma}[A]
  148. γ \gamma
  149. O ^ \hat{O}^{\prime}
  150. γ \gamma
  151. Ψ [ γ ] \Psi[\gamma]
  152. Ψ [ γ ] \Psi[\gamma]
  153. γ \gamma
  154. E ~ i a \tilde{E}^{a}_{i}
  155. H ~ ^ W γ [ A ] = - ϵ i j k F ^ a b k δ δ A a i δ δ A b j W γ [ A ] \hat{\tilde{H}}^{\dagger}W_{\gamma}[A]=-\epsilon_{ijk}\hat{F}^{k}_{ab}{\delta% \over\delta A_{a}^{i}}\;{\delta\over\delta A_{b}^{j}}W_{\gamma}[A]
  156. γ ˙ a \dot{\gamma}^{a}
  157. γ \gamma
  158. F ^ a b i γ ˙ a γ ˙ b \hat{F}^{i}_{ab}\dot{\gamma}^{a}\dot{\gamma}^{b}
  159. F a b i F^{i}_{ab}
  160. a a
  161. b b
  162. γ \gamma
  163. Ψ [ γ ] \Psi[\gamma]
  164. Σ \Sigma
  165. x 3 = 0 x^{3}=0
  166. Σ \Sigma
  167. sin θ \sin\theta
  168. θ \theta
  169. u \vec{u}
  170. v \vec{v}
  171. A = u v sin θ = u 2 v 2 ( 1 - cos 2 θ ) A=\|\vec{u}\|\|\vec{v}\|\sin\theta=\sqrt{\|\vec{u}\|^{2}\|\vec{v}\|^{2}(1-\cos% ^{2}\theta)}
  172. = u 2 v 2 - ( u v ) 2 \quad=\sqrt{\|\vec{u}\|^{2}\|\vec{v}\|^{2}-(\vec{u}\cdot\vec{v})^{2}}
  173. Σ \Sigma
  174. A Σ = Σ d x 1 d x 2 det ( q ( 2 ) ) A_{\Sigma}=\int_{\Sigma}dx^{1}dx^{2}\sqrt{\operatorname{det}(q^{(2)})}
  175. det ( q ( 2 ) ) = q 11 q 22 - q 12 2 \operatorname{det}(q^{(2)})=q_{11}q_{22}-q_{12}^{2}
  176. Σ \Sigma
  177. det ( q ( 2 ) ) = ϵ 3 a b ϵ 3 c d q a c q b c 2 \operatorname{det}(q^{(2)})={\epsilon^{3ab}\epsilon^{3cd}q_{ac}q_{bc}\over 2}
  178. q a b = ϵ a c d ϵ b e f q c e q d f 3 ! det ( q ) q^{ab}={\epsilon^{acd}\epsilon^{bef}q_{ce}q_{df}\over 3!\operatorname{det}(q)}
  179. det ( q ( 2 ) ) \operatorname{det}(q^{(2)})
  180. E ~ i a E ~ b i = det ( q ) q a b \tilde{E}^{a}_{i}\tilde{E}^{bi}=\operatorname{det}(q)q^{ab}
  181. A Σ = Σ d x 1 d x 2 E ~ i 3 E ~ 3 i A_{\Sigma}=\int_{\Sigma}dx^{1}dx^{2}\sqrt{\tilde{E}^{3}_{i}\tilde{E}^{3i}}
  182. E ~ i 3 \tilde{E}^{3}_{i}
  183. E ~ ^ i 3 δ δ A 3 i \hat{\tilde{E}}^{3}_{i}\sim{\delta\over\delta A_{3}^{i}}
  184. A Σ A_{\Sigma}
  185. N = 2 J N=2J
  186. J J
  187. i T i T i = J ( J + 1 ) 1 \sum_{i}T^{i}T^{i}=J(J+1)1
  188. A ^ Σ W γ [ A ] = 8 π Planck 2 β I j I ( j I + 1 ) W γ [ A ] \hat{A}_{\Sigma}W_{\gamma}[A]=8\pi\ell_{\,\text{Planck}}^{2}\beta\sum_{I}\sqrt% {j_{I}(j_{I}+1)}W_{\gamma}[A]
  189. I I
  190. Σ \Sigma
  191. R R
  192. V = R d 3 x det ( q ) = 1 6 R d x 3 ϵ a b c ϵ i j k E ~ i a E ~ j b E ~ k c V=\int_{R}d^{3}x\sqrt{\operatorname{det}(q)}={1\over 6}\int_{R}dx^{3}\sqrt{% \epsilon_{abc}\epsilon^{ijk}\tilde{E}^{a}_{i}\tilde{E}^{b}_{j}\tilde{E}^{c}_{k}}
  193. γ ˙ a \dot{\gamma}^{a}
  194. SU ( 2 ) \operatorname{SU}(2)
  195. SU ( 2 ) \operatorname{SU}(2)
  196. 𝔸 \mathbb{A}
  197. 𝔹 \mathbb{B}
  198. Tr ( 𝔸 ) Tr ( 𝔹 ) = Tr ( 𝔸 𝔹 ) + Tr ( 𝔸 𝔹 - 1 ) \operatorname{Tr}(\mathbb{A})\operatorname{Tr}(\mathbb{B})=\operatorname{Tr}(% \mathbb{A}\mathbb{B})+\operatorname{Tr}(\mathbb{A}\mathbb{B}^{-1})
  199. γ \gamma
  200. η \eta
  201. W γ [ A ] W η [ A ] = W γ η [ A ] + W γ η - 1 [ A ] W_{\gamma}[A]W_{\eta}[A]=W_{\gamma\circ\eta}[A]+W_{\gamma\circ\eta^{-1}}[A]
  202. η - 1 \eta^{-1}
  203. η \eta
  204. γ η \gamma\circ\eta
  205. γ \gamma
  206. η \eta
  207. W γ [ A ] = W γ - 1 [ A ] W_{\gamma}[A]=W_{\gamma^{-1}}[A]
  208. T r ( 𝔸 𝔹 ) = T r ( 𝔹 𝔸 ) Tr(\mathbb{A}\mathbb{B})=Tr(\mathbb{B}\mathbb{A})
  209. W γ η [ A ] = W η γ [ A ] W_{\gamma\circ\eta}[A]=W_{\eta\circ\gamma}[A]
  210. γ \gamma
  211. SL ( 2 , ) \operatorname{SL}(2,\mathbb{C})
  212. SU ( 2 ) \operatorname{SU}(2)
  213. SL ( 2 , ) \operatorname{SL}(2,\mathbb{C})
  214. SU ( 2 ) \operatorname{SU}(2)
  215. a e b I = 0 \nabla_{a}e_{b}^{I}=0
  216. Γ a i \Gamma_{a}^{i}
  217. W W
  218. W W
  219. J = | x a x b | J=\Big|{\partial x^{a}\over\partial x^{{}^{\prime}b}}\Big|
  220. T b a = J W x a x c x d x b T d c {T^{\prime}}^{a\dots}_{b\dots}=J^{W}{\partial x^{{}^{\prime}a}\over\partial x^% {c}}\dots{\partial x^{d}\over\partial x^{{}^{\prime}b}}T^{c\dots}_{d\dots}
  221. det ( q ) H \sqrt{\operatorname{det}(q)}H
  222. A a i = Γ a i + β K a i A_{a}^{i}=\Gamma_{a}^{i}+\beta K_{a}^{i}
  223. β \beta
  224. H = ϵ i j k F a b k E ~ i a E ~ j b det ( q ) + 2 β 2 + 1 β 2 ( E ~ i a E ~ j b - E ~ j a E ~ i b ) det ( q ) ( A a i - Γ a i ) ( A b j - Γ b j ) = H E + H H={\epsilon_{ijk}F_{ab}^{k}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}\over\sqrt{% \operatorname{det}(q)}}+2{\beta^{2}+1\over\beta^{2}}{(\tilde{E}_{i}^{a}\tilde{% E}_{j}^{b}-\tilde{E}_{j}^{a}\tilde{E}_{i}^{b})\over\sqrt{\operatorname{det}(q)% }}(A_{a}^{i}-\Gamma_{a}^{i})(A_{b}^{j}-\Gamma_{b}^{j})=H_{E}+H^{\prime}
  225. Γ a i \Gamma_{a}^{i}
  226. β = ± i \beta=\pm i
  227. Γ a i \Gamma_{a}^{i}
  228. 1 / det ( q ) 1/\sqrt{\operatorname{det}(q)}
  229. β \beta
  230. 1 / det ( q ) 1/\sqrt{\operatorname{det}(q)}
  231. { A c k , V } = ϵ a b c ϵ i j k E ~ i a E ~ j b det ( q ) \{A_{c}^{k},V\}={\epsilon_{abc}\epsilon^{ijk}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b% }\over\sqrt{\operatorname{det}(q)}}
  232. V V
  233. A c k A_{c}^{k}
  234. V V
  235. K = d 3 x K a i E ~ i a K=\int d^{3}xK_{a}^{i}\tilde{E}_{i}^{a}
  236. K a i = { A a i , K } K_{a}^{i}=\{A_{a}^{i},K\}
  237. A a i - Γ a i = β K a i = β { A a i , K } A_{a}^{i}-\Gamma_{a}^{i}=\beta K_{a}^{i}=\beta\{A_{a}^{i},K\}
  238. K K
  239. K = - { V , d 3 x H E } K=-\{V,\int d^{3}xH_{E}\}
  240. K K
  241. Kin \mathcal{H}_{\,\text{Kin}}
  242. ϕ , ψ Kin \langle\phi,\psi\rangle_{\,\text{Kin}}
  243. C ^ I \hat{C}_{I}
  244. C ^ 1 \hat{C}_{1}
  245. 1 Kin \mathcal{H}_{1}\subset\mathcal{H}_{\,\text{Kin}}
  246. P 1 P_{1}
  247. Kin \mathcal{H}_{\,\text{Kin}}
  248. 1 \mathcal{H}_{1}
  249. ϕ , ψ 1 \langle\phi,\psi\rangle_{1}
  250. ϕ , ψ 1 = P ϕ , P ψ Kin \langle\phi,\psi\rangle_{1}=\langle P\phi,P\psi\rangle_{\,\text{Kin}}
  251. C ^ I \hat{C}_{I}
  252. Ψ Kin \Psi\in\mathcal{H}_{\,\text{Kin}}
  253. C ^ I Ψ = 0 \hat{C}_{I}\Psi=0
  254. I I
  255. C ^ = ( i d d x - k ) \hat{C}=\Big(i{d\over dx}-k\Big)
  256. L 2 ( , d x ) L_{2}(\mathbb{R},dx)
  257. \mathbb{R}
  258. exp ( - i k x ) \exp(-ikx)
  259. - d x ψ * ( x ) ψ ( x ) = - d x e i k x e - i k x = - d x = \int_{-\infty}^{\infty}dx\psi^{*}(x)\psi(x)=\int_{-\infty}^{\infty}dxe^{ikx}e^% {-ikx}=\int_{-\infty}^{\infty}dx=\infty
  260. Kin \mathcal{H}_{\,\text{Kin}}
  261. 𝒮 \mathcal{S}
  262. Kin \mathcal{H}_{\,\text{Kin}}
  263. 𝒮 \mathcal{S}
  264. Kin \mathcal{H}_{\,\text{Kin}}
  265. Kin \mathcal{H}_{\,\text{Kin}}
  266. 𝒮 \mathcal{S}^{\prime}
  267. 𝒮 \mathcal{S}
  268. 𝒮 Kin 𝒮 \mathcal{S}\subset\mathcal{H}_{\,\text{Kin}}\subset\mathcal{S}^{\prime}
  269. 𝒮 \mathcal{S}^{\prime}
  270. ϕ , ψ 1 = P ϕ , ψ Kin . \langle\phi,\psi\rangle_{1}=\langle P\phi,\psi\rangle_{\,\text{Kin}}.
  271. P P
  272. Kin \mathcal{H}_{\,\text{Kin}}
  273. Diff \mathcal{H}_{\,\text{Diff}}
  274. s s
  275. s s^{\prime}
  276. ψ s \psi_{s}
  277. ψ s \psi_{s^{\prime}}
  278. s s
  279. s s^{\prime}
  280. Diff \mathcal{H}_{\,\text{Diff}}
  281. [ C ^ ( N ) , H ^ ( M ) ] H ^ ( N M ) [\hat{C}(\vec{N}),\hat{H}(M)]\propto\hat{H}(\mathcal{L}_{\vec{N}}M)
  282. ψ s D i f f \psi_{s}\in\mathcal{H}_{Diff}
  283. ( C ( N ) H ^ ( M ) - H ^ ( M ) C ( N ) ) ψ s H ^ ( N M ) ψ s (\vec{C}(\vec{N})\hat{H}(M)-\hat{H}(M)\vec{C}(\vec{N}))\psi_{s}\propto\hat{H}(% \mathcal{L}_{\vec{N}}M)\psi_{s}
  284. C ( N ) ψ s = 0 \vec{C}(\vec{N})\psi_{s}=0
  285. C ( N ) [ H ^ ( M ) ψ s ] H ^ ( N M ) ψ s 0 \vec{C}(\vec{N})[\hat{H}(M)\psi_{s}]\propto\hat{H}(\mathcal{L}_{\vec{N}}M)\psi% _{s}\not=0
  286. H ^ ( M ) ψ s \hat{H}(M)\psi_{s}
  287. D i f f \mathcal{H}_{Diff}
  288. Diff \mathcal{H}_{\,\text{Diff}}
  289. δ ( x ) \delta(x)
  290. x = 0 x=0
  291. δ ( x ) = e i k x d k \delta(x)=\int e^{ikx}dk
  292. x Σ δ ( H ^ ( x ) ) \prod_{x\in\Sigma}\delta(\hat{H}(x))
  293. H ^ ( x ) = 0 \hat{H}(x)=0
  294. x x
  295. Σ \Sigma
  296. [ d N ] e i d 3 x N ( x ) H ^ ( x ) \int[dN]e^{i\int d^{3}xN(x)\hat{H}(x)}
  297. [ d N ] e i d 3 x N ( x ) H ^ ( x ) s int s fin Diff \biggl\langle\int[dN]e^{i\int d^{3}xN(x)\hat{H}(x)}s_{\,\text{int}}s_{\,\text{% fin}}\biggr\rangle_{\,\text{Diff}}
  298. s int s_{\,\text{int}}
  299. s fin s_{\,\text{fin}}
  300. [ d N ] ( 1 + i d 3 x N ( x ) H ^ ( x ) + i 2 2 ! [ d 3 x N ( x ) H ^ ( x ) ] [ d 3 x N ( x ) H ^ ( x ) ] + ) s int , s fin Diff \biggl\langle\int[dN](1+i\int d^{3}xN(x)\hat{H}(x)+{i^{2}\over 2!}[\int d^{3}% xN(x)\hat{H}(x)][\int d^{3}x^{\prime}N(x^{\prime})\hat{H}(x^{\prime})]+\dots)s% _{\,\text{int}},s_{\,\text{fin}}\biggr\rangle_{\,\text{Diff}}
  301. H ^ \hat{H}
  302. N ( x n ) N(x_{n})
  303. N N
  304. H n o p H_{nop}
  305. H ^ \hat{H}
  306. H ^ ( x ) \hat{H}(x)
  307. [ d N ] e i d 3 x N ( x ) H ^ ( x ) \int[dN]e^{i\int d^{3}xN(x)\hat{H}(x)}
  308. B a b I J B_{ab}^{IJ}
  309. B B
  310. B a b I J = 1 2 ( E a I E b J - E b I E a J ) B_{ab}^{IJ}={1\over 2}(E^{I}_{a}E^{J}_{b}-E^{I}_{b}E^{J}_{a})
  311. B B
  312. μ A ^ μ \partial_{\mu}\hat{A}^{\mu}
  313. D i f f \mathcal{H}_{Diff}
  314. P h y s \mathcal{H}_{Phys}
  315. n - n-
  316. W ( x 1 , , x n ) = 0 | ϕ ( x n ) ϕ ( x 1 ) | 0 W(x_{1},\dots,x_{n})=\langle 0|\phi(x_{n})\dots\phi(x_{1})|0\rangle
  317. n - n-
  318. n - n-
  319. H ( x ) = 0 H(x)=0
  320. x x
  321. M = d 3 x [ H ( x ) ] 2 det ( q ( x ) ) M=\int d^{3}x{[H(x)]^{2}\over\sqrt{\operatorname{det}(q(x))}}
  322. H ( x ) H(x)
  323. M M
  324. H ( x ) H(x)
  325. H ( x ) H(x)
  326. M M
  327. M M
  328. C ( N ) C(\vec{N})
  329. { M , C ( N ) } = 0 \{M,C(\vec{N})\}=0
  330. s u ( 2 ) su(2)
  331. { M , M } = 0 \{M,M\}=0
  332. { { M , O } , O } M = 0 = 0 \{\{M,O\},O\}_{M=0}=0
  333. O O
  334. M ^ := d 3 x ( H det ( q ( x ) ) 1 / 4 ) ^ ( x ) ( H det ( q ( x ) ) 1 / 4 ) ^ ( x ) \hat{M}:=\int d^{3}x\widehat{\left({H\over\det(q(x))^{1/4}}\right)}^{\dagger}(% x)\widehat{\left({H\over\det(q(x))^{1/4}}\right)}(x)
  335. ( H det ( q ( x ) ) 1 / 4 ) ^ ( x ) Ψ = 0 \widehat{\left({H\over\det(q(x))^{1/4}}\right)}(x)\Psi=0
  336. x x
  337. M ^ Ψ = 0 \hat{M}\Psi=0
  338. M ^ Ψ = 0 \hat{M}\Psi=0
  339. 0 = < Ψ , M ^ Ψ d 3 x ( H det ( q ( x ) ) 1 / 4 ) ^ ( x ) Ψ 2 E q 4 0=<\Psi,\hat{M}\Psi>=\int d^{3}x\left\|\widehat{\left({H\over\det(q(x))^{1/4}}% \right)}(x)\Psi\right\|^{2}\quad Eq\;4
  340. ( H det ( q ( x ) ) 1 / 4 ) ^ ( x ) Ψ = 0 \widehat{\left({H\over\det(q(x))^{1/4}}\right)}(x)\Psi=0
  341. M ^ \hat{M}
  342. Q M Q_{M}
  343. Q M Q_{M}
  344. H K i n H_{Kin}
  345. H D i f f H_{Diff}
  346. M ^ \hat{M}
  347. H D i f f H_{Diff}
  348. M ^ \hat{M}
  349. H D i f f H_{Diff}
  350. Q M Q_{M}
  351. M ^ \hat{M}
  352. Q M Q_{M}
  353. M ¯ ^ \hat{\overline{M}}
  354. M ^ \hat{M}
  355. M ¯ ^ \hat{\overline{M}}
  356. M ^ \hat{M}
  357. Ψ \Psi
  358. ( H det ( q ( x ) ) 1 / 4 ) ^ ( x ) Ψ 0 \widehat{\left({H\over\det(q(x))^{1/4}}\right)}(x)\Psi\not=0
  359. M ^ Ψ = 0 \hat{M}\Psi=0
  360. Q M E Q_{M_{E}}
  361. H K i n H_{Kin}
  362. Q M E Q_{M_{E}}
  363. M ^ \hat{M}
  364. M ^ := M ^ - m i n ( s p e c ( M ^ ) ) 1 ^ \hat{M}^{\prime}:=\hat{M}-min(spec(\hat{M}))\hat{1}
  365. lim 0 m i n ( s p e c ( M ^ ) ) = 0 \lim_{\hbar\rightarrow 0}min(spec(\hat{M}))=0
  366. M ^ \hat{M}^{\prime}
  367. M M
  368. M E M_{E}
  369. M E = Σ d 3 x H ( x ) 2 - q a b V a ( x ) V b ( x ) d e t ( q ) M_{E}=\int_{\Sigma}d^{3}x{H(x)^{2}-q^{ab}V_{a}(x)V_{b}(x)\over\sqrt{det(q)}}
  370. H ( x ) = 0 H(x)=0
  371. V a ( x ) = 0 V_{a}(x)=0
  372. x x
  373. Σ \Sigma
  374. ϕ , ψ Phys = lim T ϕ , - T T d t e i t M ^ E ψ \langle\phi,\psi\rangle_{\,\text{Phys}}=\lim_{T\rightarrow\infty}\biggl\langle% \phi,\int_{-T}^{T}dte^{it\hat{M}_{E}}\psi\biggr\rangle
  375. δ ( M E ^ ) = lim T - T T d t e i t M ^ E \delta(\hat{M_{E}})=\lim_{T\rightarrow\infty}\int_{-T}^{T}dte^{it\hat{M}_{E}}
  376. t t
  377. e i t M ^ E = lim n [ e i t M ^ E / n ] n = lim n [ 1 + i t M ^ E / n ] n . e^{it\hat{M}_{E}}=\lim_{n\rightarrow\infty}[e^{it\hat{M}_{E}/n}]^{n}=\lim_{n% \rightarrow\infty}[1+it\hat{M}_{E}/n]^{n}.
  378. [ 1 + i t M ^ E / n ] [1+it\hat{M}_{E}/n]
  379. n n
  380. S BH = k B A 4 P 2 , S_{\,\text{BH}}=\frac{k_{\,\text{B}}A}{4\ell_{\,\text{P}}^{2}},
  381. A A
  382. k B k_{\,\text{B}}
  383. P = G / c 3 \ell_{\,\text{P}}=\sqrt{G\hbar/c^{3}}
  384. A A
  385. 4 π 4\pi
  386. S = A / 4 S=A/4
  387. S S
  388. A A
  389. n n
  390. n n
  391. n n
  392. n n

Lorentz_group.html

  1. ( t , x , y , z ) t 2 - x 2 - y 2 - z 2 (t,x,y,z)\mapsto t^{2}-x^{2}-y^{2}-z^{2}
  2. G G
  3. V V
  4. S V S⊂V
  5. S S
  6. G G
  7. g s S g G , s S gs∈S∀g∈G,∀s∈S
  8. g G g∈G
  9. Q ( x ) = x 0 2 - x 1 2 - x 2 2 - x 3 2 . Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}.
  10. Q ( x ) = c o n s t . Q(x)=const.
  11. T T
  12. ( m , 0 , 0 , 0 ) (m,0,0,0)
  13. m 0 m≠0
  14. S O ( 3 ) SO(3)
  15. X = [ t + z x - i y x + i y t - z ] . X=\left[\begin{matrix}t+z&x-iy\\ x+iy&t-z\end{matrix}\right].
  16. det X = t 2 - x 2 - y 2 - z 2 . \det\,X=t^{2}-x^{2}-y^{2}-z^{2}.
  17. X P X P * X\mapsto PXP^{*}
  18. P * P^{*}
  19. P P
  20. ξ = u + i v ξ=u+iv
  21. [ u 2 + v 2 + 1 2 u - 2 v u 2 + v 2 - 1 ] \left[\begin{matrix}u^{2}+v^{2}+1\\ 2u\\ -2v\\ u^{2}+v^{2}-1\end{matrix}\right]
  22. N = 2 [ u 2 + v 2 u + i v u - i v 1 ] . N=2\left[\begin{matrix}u^{2}+v^{2}&u+iv\\ u-iv&1\end{matrix}\right].
  23. P 1 = [ exp ( i θ / 2 ) 0 0 exp ( - i θ / 2 ) ] P_{1}=\left[\begin{matrix}\exp(i\theta/2)&0\\ 0&\exp(-i\theta/2)\end{matrix}\right]
  24. ξ ξ
  25. Q 1 = [ 1 0 0 0 0 cos ( θ ) - sin ( θ ) 0 0 sin ( θ ) cos ( θ ) 0 0 0 0 1 ] = exp ( θ [ 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 0 0 ] ) . Q_{1}=\left[\begin{matrix}1&0&0&0\\ 0&\cos(\theta)&-\sin(\theta)&0\\ 0&\sin(\theta)&\cos(\theta)&0\\ 0&0&0&1\end{matrix}\right]=\exp\left(\theta\left[\begin{matrix}0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\end{matrix}\right]\right)~{}.
  26. z z
  27. θ θ
  28. z z
  29. θ θ
  30. P 2 = [ exp ( β / 2 ) 0 0 exp ( - β / 2 ) ] P_{2}=\left[\begin{matrix}\exp(\beta/2)&0\\ 0&\exp(-\beta/2)\end{matrix}\right]
  31. ξ ξ
  32. Q 2 = [ cosh ( β ) 0 0 sinh ( β ) 0 1 0 0 0 0 1 0 sinh ( β ) 0 0 cosh ( β ) ] = exp ( β [ 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ] ) . Q_{2}=\left[\begin{matrix}\cosh(\beta)&0&0&\sinh(\beta)\\ 0&1&0&0\\ 0&0&1&0\\ \sinh(\beta)&0&0&\cosh(\beta)\end{matrix}\right]=\exp\left(\beta\left[\begin{% matrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{matrix}\right]\right)~{}.
  33. z z
  34. β β
  35. β β
  36. P 3 = P 2 P 1 = P 1 P 2 = [ exp ( ( β + i θ ) / 2 ) 0 0 exp ( - ( β + i θ ) / 2 ) ] P_{3}=P_{2}P_{1}=P_{1}P_{2}=\left[\begin{matrix}\exp\left((\beta+i\theta)/2% \right)&0\\ 0&\exp\left(-(\beta+i\theta)/2\right)\end{matrix}\right]
  37. ξ ξ
  38. Q 3 = Q 2 Q 1 = Q 1 Q 2 . Q_{3}=Q_{2}Q_{1}=Q_{1}Q_{2}.
  39. z z
  40. z z
  41. P 4 = [ 1 α 0 1 ] P_{4}=\left[\begin{matrix}1&\alpha\\ 0&1\end{matrix}\right]
  42. ξ ξ
  43. Q 4 = [ 1 + | α | 2 / 2 R e ( α ) I m ( α ) - | α | 2 / 2 R e ( α ) 1 0 - R e ( α ) - I m ( α ) 0 1 I m ( α ) | α | 2 / 2 R e ( α ) I m ( α ) 1 - | α | 2 / 2 ] Q_{4}=\left[\begin{matrix}1+|\alpha|^{2}/2&Re(\alpha)&Im(\alpha)&-|\alpha|^{2}% /2\\ Re(\alpha)&1&0&-Re(\alpha)\\ -Im(\alpha)&0&1&Im(\alpha)\\ |\alpha|^{2}/2&Re(\alpha)&Im(\alpha)&1-|\alpha|^{2}/2\end{matrix}\right]
  44. = exp [ 0 R e ( α ) I m ( α ) 0 R e ( α ) 0 0 - R e ( α ) - I m ( α ) 0 0 I m ( α ) 0 R e ( α ) I m ( α ) 0 ] . ~{}=\exp\left[\begin{matrix}0&Re(\alpha)&Im(\alpha)&0\\ Re(\alpha)&0&0&-Re(\alpha)\\ -Im(\alpha)&0&0&Im(\alpha)\\ 0&Re(\alpha)&Im(\alpha)&0\end{matrix}\right]~{}.
  45. α α
  46. [ t x y z ] [ t x y z ] + R e ( α ) [ x t - z 0 x ] + I m ( α ) [ y 0 z - t y ] + | α | 2 2 [ t - z 0 0 t - z ] . \left[\begin{matrix}t\\ x\\ y\\ z\end{matrix}\right]\rightarrow\left[\begin{matrix}t\\ x\\ y\\ z\end{matrix}\right]+Re(\alpha)\;\left[\begin{matrix}x\\ t-z\\ 0\\ x\end{matrix}\right]+Im(\alpha)\;\left[\begin{matrix}y\\ 0\\ z-t\\ y\end{matrix}\right]+\frac{|\alpha|^{2}}{2}\;\left[\begin{matrix}t-z\\ 0\\ 0\\ t-z\end{matrix}\right].
  47. I m ( α ) Im(α)
  48. α α
  49. x ( t + z ) + ( t - z ) x . x\,\left(\partial_{t}+\partial_{z}\right)+(t-z)\,\partial_{x}.
  50. f ( t , x , y , z ) f(t,x,y,z)
  51. f ( t , x , y , z ) = F ( y , t - z , t 2 - x 2 - z 2 ) , f(t,x,y,z)=F(y,\,t-z,\,t^{2}-x^{2}-z^{2}),
  52. F F
  53. F F
  54. y = c 1 , t - z = c 2 , t 2 - x 2 - z 2 = c 3 . y=c_{1},~{}~{}~{}~{}t-z=c_{2},~{}~{}~{}~{}t^{2}-x^{2}-z^{2}=c_{3}.
  55. y y
  56. c c
  57. α α
  58. R e ( α ) Re(α)
  59. x x
  60. y y
  61. h h
  62. z z
  63. x x
  64. y y
  65. α α
  66. c c
  67. c c
  68. c c
  69. J J
  70. E E
  71. - y x + x y i J z , - z y + y z i J x , - x z + z x J y ; -y\partial_{x}+x\partial_{y}\equiv iJ_{z}~{},\qquad-z\partial_{y}+y\partial_{z% }\equiv iJ_{x}~{},\qquad-x\partial_{z}+z\partial_{x}\equiv J_{y}~{};
  72. x t + t x i K x , y t + t y i K y , z t + t z i K z . x\partial_{t}+t\partial_{x}\equiv iK_{x}~{},\qquad y\partial_{t}+t\partial_{y}% \equiv iK_{y}~{},\qquad z\partial_{t}+t\partial_{z}\equiv iK_{z}.
  73. - y x + x y . -y\partial_{x}+x\partial_{y}.
  74. x λ = - y , y λ = x , x ( 0 ) = x 0 , y ( 0 ) = y 0 . \frac{\partial x}{\partial\lambda}=-y,\;\frac{\partial y}{\partial\lambda}=x,% \;x(0)=x_{0},\;y(0)=y_{0}.
  75. x ( λ ) = x 0 cos ( λ ) - y 0 sin ( λ ) , y ( λ ) = x 0 sin ( λ ) + y 0 cos ( λ ) x(\lambda)=x_{0}\cos(\lambda)-y_{0}\sin(\lambda),\;y(\lambda)=x_{0}\sin(% \lambda)+y_{0}\cos(\lambda)
  76. [ t x y z ] = [ 1 0 0 0 0 cos ( λ ) - sin ( λ ) 0 0 sin ( λ ) cos ( λ ) 0 0 0 0 1 ] [ t 0 x 0 y 0 z 0 ] \left[\begin{matrix}t\\ x\\ y\\ z\end{matrix}\right]=\left[\begin{matrix}1&0&0&0\\ 0&\cos(\lambda)&-\sin(\lambda)&0\\ 0&\sin(\lambda)&\cos(\lambda)&0\\ 0&0&0&1\end{matrix}\right]\left[\begin{matrix}t_{0}\\ x_{0}\\ y_{0}\\ z_{0}\end{matrix}\right]
  77. λ λ
  78. i J z = [ 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 0 0 ] , iJ_{z}=\left[\begin{matrix}0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\end{matrix}\right]~{},
  79. σ 1 = [ 0 1 1 0 ] , σ 2 = [ 0 - i i 0 ] , σ 3 = [ 1 0 0 - 1 ] . \sigma_{1}=\left[\begin{matrix}0&1\\ 1&0\end{matrix}\right],\;\;\sigma_{2}=\left[\begin{matrix}0&-i\\ i&0\end{matrix}\right],\;\;\sigma_{3}=\left[\begin{matrix}1&0\\ 0&-1\end{matrix}\right].
  80. u \partial_{u}\,\!
  81. [ 1 α 0 1 ] \left[\begin{matrix}1&\alpha\\ 0&1\end{matrix}\right]
  82. [ 1 + α 2 / 2 α 0 - α 2 / 2 α 1 0 - α 0 0 1 0 α 2 / 2 α 0 1 - α 2 / 2 ] \left[\begin{matrix}1+\alpha^{2}/2&\alpha&0&-\alpha^{2}/2\\ \alpha&1&0&-\alpha\\ 0&0&1&0\\ \alpha^{2}/2&\alpha&0&1-\alpha^{2}/2\end{matrix}\right]
  83. X 1 = X_{1}=\,\!
  84. x ( t + z ) + ( t - z ) x x(\partial_{t}+\partial_{z})+(t-z)\partial_{x}\,\!
  85. v \partial_{v}\,\!
  86. [ 1 i α 0 1 ] \left[\begin{matrix}1&i\alpha\\ 0&1\end{matrix}\right]
  87. [ 1 + α 2 / 2 0 α - α 2 / 2 0 1 0 0 α 0 1 - α α 2 / 2 0 α 1 - α 2 / 2 ] \left[\begin{matrix}1+\alpha^{2}/2&0&\alpha&-\alpha^{2}/2\\ 0&1&0&0\\ \alpha&0&1&-\alpha\\ \alpha^{2}/2&0&\alpha&1-\alpha^{2}/2\end{matrix}\right]
  88. X 2 = X_{2}=\,\!
  89. y ( t + z ) + ( t - z ) y y(\partial_{t}+\partial_{z})+(t-z)\partial_{y}\,\!
  90. 1 2 ( u u + v v ) \frac{1}{2}\left(u\partial_{u}+v\partial_{v}\right)
  91. [ exp ( β 2 ) 0 0 exp ( - β 2 ) ] \left[\begin{matrix}\exp\left(\frac{\beta}{2}\right)&0\\ 0&\exp\left(-\frac{\beta}{2}\right)\end{matrix}\right]
  92. [ cosh ( β ) 0 0 sinh ( β ) 0 1 0 0 0 0 1 0 sinh ( β ) 0 0 cosh ( β ) ] \left[\begin{matrix}\cosh(\beta)&0&0&\sinh(\beta)\\ 0&1&0&0\\ 0&0&1&0\\ \sinh(\beta)&0&0&\cosh(\beta)\end{matrix}\right]
  93. X 3 = X_{3}=\,\!
  94. z t + t z z\partial_{t}+t\partial_{z}\,\!
  95. 1 2 ( - v u + u v ) \frac{1}{2}\left(-v\partial_{u}+u\partial_{v}\right)
  96. [ exp ( i θ 2 ) 0 0 exp ( - i θ 2 ) ] \left[\begin{matrix}\exp\left(\frac{i\theta}{2}\right)&0\\ 0&\exp\left(\frac{-i\theta}{2}\right)\end{matrix}\right]
  97. [ 1 0 0 0 0 cos ( θ ) - sin ( θ ) 0 0 sin ( θ ) cos ( θ ) 0 0 0 0 1 ] \left[\begin{matrix}1&0&0&0\\ 0&\cos(\theta)&-\sin(\theta)&0\\ 0&\sin(\theta)&\cos(\theta)&0\\ 0&0&0&1\end{matrix}\right]
  98. X 4 = X_{4}=\,\!
  99. - y x + x y -y\partial_{x}+x\partial_{y}\,\!
  100. v 2 - u 2 - 1 2 u - u v v \frac{v^{2}-u^{2}-1}{2}\partial_{u}-uv\,\partial_{v}
  101. [ cos ( θ 2 ) - sin ( θ 2 ) sin ( θ 2 ) cos ( θ 2 ) ] \left[\begin{matrix}\cos\left(\frac{\theta}{2}\right)&-\sin\left(\frac{\theta}% {2}\right)\\ \sin\left(\frac{\theta}{2}\right)&\cos\left(\frac{\theta}{2}\right)\end{matrix% }\right]
  102. [ 1 0 0 0 0 cos ( θ ) 0 sin ( θ ) 0 0 1 0 0 - sin ( θ ) 0 cos ( θ ) ] \left[\begin{matrix}1&0&0&0\\ 0&\cos(\theta)&0&\sin(\theta)\\ 0&0&1&0\\ 0&-\sin(\theta)&0&\cos(\theta)\end{matrix}\right]
  103. X 5 = X_{5}=\,\!
  104. - x z + z x -x\partial_{z}+z\partial_{x}\,\!
  105. u v u + 1 - u 2 + v 2 2 v uv\,\partial_{u}+\frac{1-u^{2}+v^{2}}{2}\partial_{v}
  106. [ cos ( θ 2 ) i sin ( θ 2 ) i sin ( θ 2 ) cos ( θ 2 ) ] \left[\begin{matrix}\cos\left(\frac{\theta}{2}\right)&i\sin\left(\frac{\theta}% {2}\right)\\ i\sin\left(\frac{\theta}{2}\right)&\cos\left(\frac{\theta}{2}\right)\end{% matrix}\right]
  107. [ 1 0 0 0 0 1 0 0 0 0 cos ( θ ) - sin ( θ ) 0 0 sin ( θ ) cos ( θ ) ] \left[\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos(\theta)&-\sin(\theta)\\ 0&0&\sin(\theta)&\cos(\theta)\end{matrix}\right]
  108. X 6 = X_{6}=\,\!
  109. - z y + y z -z\partial_{y}+y\partial_{z}\,\!
  110. σ 2 = [ 0 i - i 0 ] . \sigma_{2}=\left[\begin{matrix}0&i\\ -i&0\end{matrix}\right].
  111. exp ( i θ 2 σ 2 ) = [ cos ( θ / 2 ) - sin ( θ / 2 ) sin ( θ / 2 ) cos ( θ / 2 ) ] . \exp\left(\frac{i\theta}{2}\,\sigma_{2}\right)=\left[\begin{matrix}\cos(\theta% /2)&-\sin(\theta/2)\\ \sin(\theta/2)&\cos(\theta/2)\end{matrix}\right].
  112. ξ cos ( θ / 2 ) ξ - sin ( θ / 2 ) sin ( θ / 2 ) ξ + cos ( θ / 2 ) . \xi\mapsto\frac{\cos(\theta/2)\,\xi-\sin(\theta/2)}{\sin(\theta/2)\,\xi+\cos(% \theta/2)}.
  113. d ξ d θ | θ = 0 = - 1 + ξ 2 2 . \frac{d\xi}{d\theta}|_{\theta=0}=-\frac{1+\xi^{2}}{2}.
  114. - 1 + ξ 2 2 ξ . -\frac{1+\xi^{2}}{2}\,\partial_{\xi}.
  115. ξ = u + i v \xi=u+iv
  116. - 1 + u 2 - v 2 2 u - u v v . -\frac{1+u^{2}-v^{2}}{2}\,\partial_{u}-uv\,\partial_{v}.
  117. X P X P * X\mapsto PXP^{*}
  118. [ 1 0 0 0 0 cos ( θ ) 0 sin ( θ ) 0 0 1 0 0 - sin ( θ ) 0 cos ( θ ) ] . \left[\begin{matrix}1&0&0&0\\ 0&\cos(\theta)&0&\sin(\theta)\\ 0&0&1&0\\ 0&-\sin(\theta)&0&\cos(\theta)\end{matrix}\right].
  119. θ θ
  120. θ θ
  121. z x - x z . z\partial_{x}-x\partial_{z}.\,\!
  122. y y
  123. X 1 X_{1}
  124. X 3 X_{3}
  125. X 4 X_{4}
  126. X 3 + a X 4 X_{3}+aX_{4}
  127. a 0 a\neq 0
  128. a a
  129. X 1 , X 2 X_{1},X_{2}
  130. X 1 , X 3 X_{1},X_{3}
  131. X 3 , X 4 X_{3},X_{4}
  132. X 1 , X 2 , X 3 X_{1},X_{2},X_{3}
  133. X 1 , X 2 , X 4 X_{1},X_{2},X_{4}
  134. X 2 , X 2 , X 3 + a X 4 X_{2},X_{2},X_{3}+aX_{4}
  135. a 0 a\neq 0
  136. X 1 , X 3 , X 5 X_{1},X_{3},X_{5}
  137. X 4 , X 5 , X 6 X_{4},X_{5},X_{6}
  138. X 1 , X 2 , X 3 , X 4 X_{1},X_{2},X_{3},X_{4}
  139. ( x 1 , x 2 , , x n , x n + 1 ) x 1 2 + x 2 2 + + x n 2 - x n + 1 2 . (x_{1},x_{2},\ldots,x_{n},x_{n+1})\mapsto x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}% -x_{n+1}^{2}.

Lorenz_cipher.html

  1. ψ \psi
  2. ψ \psi
  3. ψ \psi
  4. ψ \psi
  5. ψ \psi
  6. μ \mu
  7. μ \mu
  8. χ \chi
  9. χ \chi
  10. χ \chi
  11. χ \chi
  12. χ \chi
  13. χ \chi
  14. ψ \psi
  15. μ \mu
  16. μ \mu
  17. μ \mu
  18. μ \mu
  19. μ \mu

Lorenzo_Mascheroni.html

  1. γ \gamma

Loss_of_significance.html

  1. q r p q\leq r\leq p
  2. 2 - p 1 - y x 2 - q 2^{-p}\leq 1-\frac{y}{x}\leq 2^{-q}
  3. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  4. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  5. c c
  6. b b
  7. a = 1 a=1
  8. b = 200 b=200
  9. c = - 0.000015 c=-0.000015
  10. x 2 + 200 x - 0.000015 = 0. x^{2}+200x-0.000015=0.
  11. b 2 - 4 a c = 200 2 + 4 × 1 × 0.000015 = 200.00000015 \sqrt{b^{2}-4ac}=\sqrt{200^{2}+4\times 1\times 0.000015}=200.00000015\ldots
  12. ( - 200 - 200.00000015 ) / 2 = - 200.000000075 , (-200-200.00000015)/2=-200.000000075,
  13. ( - 200 + 200.00000015 ) / 2 = 0.000000075. (-200+200.00000015)/2=0.000000075.
  14. ( - 200 - 200.0000001 ) / 2 = - 200.00000005 , (-200-200.0000001)/2=-200.00000005,
  15. ( - 200 + 200.0000001 ) / 2 = 0.00000005. (-200+200.0000001)/2=0.00000005.
  16. x 1 = - b - sgn ( b ) b 2 - 4 a c 2 a , x 2 = 2 c - b - sgn ( b ) b 2 - 4 a c = c a x 1 . \begin{aligned}\displaystyle x_{1}&\displaystyle=\frac{-b-\operatorname{sgn}(b% )\,\sqrt{b^{2}-4ac}}{2a},\\ \displaystyle x_{2}&\displaystyle=\frac{2c}{-b-\operatorname{sgn}(b)\,\sqrt{b^% {2}-4ac}}=\frac{c}{ax_{1}}.\end{aligned}
  17. sgn ( b ) \operatorname{sgn}(b)
  18. b b
  19. b b
  20. b b
  21. 1.786737589984535 1.786737589984535
  22. 1.149782767465722 × 10 - 8 1.149782767465722\times 10^{-8}
  23. x 2 - 1.786737601482363 x + 2.054360090947453 × 10 - 8 = 0 x^{2}-1.786737601482363x+2.054360090947453\times 10^{-8}=0
  24. Δ = 1.786737578486707 \sqrt{\Delta}=1.786737578486707
  25. x 1 = ( 1.786737601482363 + 1.786737578486707 ) / 2 = 1.786737589984535 x_{1}=(1.786737601482363+1.786737578486707)/2=1.786737589984535
  26. x 2 = ( 1.786737601482363 - 1.786737578486707 ) / 2 = 0.000000011497828 x_{2}=(1.786737601482363-1.786737578486707)/2=0.000000011497828
  27. x 2 x_{2}
  28. x 1 = ( 1.786737601482363 + 1.786737578486707 ) / 2 = 1.786737589984535 x_{1}=(1.786737601482363+1.786737578486707)/2=1.786737589984535
  29. x 2 = 2.054360090947453 × 10 - 8 / 1.786737589984535 = 1.149782767465722 × 10 - 8 x_{2}=2.054360090947453\times 10^{-8}/1.786737589984535=1.149782767465722% \times 10^{-8}
  30. x 2 x_{2}
  31. b b
  32. b 2 - 4 a c \sqrt{b^{2}-4ac}
  33. b 2 b^{2}
  34. - 4 a c -4ac
  35. b 2 - 4 a c b^{2}-4ac
  36. 94906265.625 x 2 - 189812534 x + 94906268.375 94906265.625x^{2}-189812534x+94906268.375
  37. Δ = 7.5625 \Delta=7.5625
  38. x 1 = 1.000000028975958 x_{1}=1.000000028975958
  39. x 2 = 1.000000000000000 x_{2}=1.000000000000000
  40. Δ \Delta
  41. x 1 = 1.000000014487979 x_{1}=1.000000014487979
  42. x 2 = 1.000000014487979 x_{2}=1.000000014487979

Lottery.html

  1. 13983816 = 49 ! 6 ! 43 ! 13983816=\frac{49!}{6!\,43!}
  2. 258890850 = 15 75 ! 5 ! 70 ! 258890850=15\,\frac{75!}{5!\,70!}
  3. 622614630 = 90 ! 6 ! 84 ! 622614630=\frac{90!}{6!\,84!}

Lower_limit_topology.html

  1. { [ x , + ) } { ( - , x - 1 n ) | n } . \bigl\{[x,+\infty)\bigr\}\cup\Bigl\{\bigl(-\infty,x-\tfrac{1}{n}\bigr)\,\Big|% \,n\in\mathbb{N}\Bigr\}.

Lowest_temperature_recorded_on_Earth.html

  1. Δ U = Δ Q - Δ W \Delta U=\Delta Q-\Delta W
  2. Δ Q = 0 \Delta Q=0
  3. Δ W > 0 \Delta W>0
  4. Δ U < 0 \Rightarrow\Delta U<0

Lozenge.html

  1. P \lozenge P
  2. P P

Luminosity_function.html

  1. y ¯ ( λ ) \overline{y}(\lambda)
  2. V ( λ ) V(\lambda)\,
  3. F = 683.002 lm / W 0 y ¯ ( λ ) J ( λ ) d λ F=683.002\ \mathrm{lm/W}\cdot\int^{\infty}_{0}\overline{y}(\lambda)J(\lambda)d\lambda
  4. F F\,
  5. J ( λ ) J(\lambda)\,
  6. y ¯ ( λ ) \overline{y}(\lambda)
  7. V ( λ ) V(\lambda)\,
  8. λ \lambda\,
  9. y ¯ ( λ ) \overline{y}(\lambda)
  10. V ( λ ) V(\lambda)
  11. V M ( λ ) V_{M}(\lambda)
  12. V ( λ ) V^{\prime}(\lambda)

Luminous_intensity.html

  1. y ¯ ( λ ) \textstyle\overline{y}(\lambda)
  2. I v = 683 y ¯ ( λ ) I e , I_{\mathrm{v}}=683\cdot\overline{y}(\lambda)\cdot I_{\mathrm{e}},
  3. y ¯ ( λ ) \textstyle\overline{y}(\lambda)
  4. I v = 683 0 y ¯ ( λ ) d I e ( λ ) d λ d λ . I_{\mathrm{v}}=683\int^{\infty}_{0}\overline{y}(\lambda)\cdot\frac{dI_{\mathrm% {e}}(\lambda)}{d\lambda}\,d\lambda.

Lumped_element_model.html

  1. ϕ B t = 0 \frac{\partial\phi_{B}}{\partial t}=0
  2. q t = 0 \frac{\partial q}{\partial t}=0
  3. L c λ L_{c}\ll\lambda
  4. L c L_{c}
  5. λ \lambda
  6. Q ˙ = T 1 - T 2 ( L k A ) \dot{Q}=\frac{T_{1}-T_{2}}{\left(\frac{L}{kA}\right)}
  7. L k A \frac{L}{kA}
  8. Q ˙ = T s u r f - T e n v r ( 1 h c o n v A s u r f ) \dot{Q}=\frac{T_{surf}-T_{envr}}{\left(\frac{1}{h_{conv}A_{surf}}\right)}
  9. 1 h c o n v A s u r f \frac{1}{h_{conv}A_{surf}}
  10. Q ˙ = T s u r f - T s u r r ( 1 h r A s u r f ) \dot{Q}=\frac{T_{surf}-T_{surr}}{\left(\frac{1}{h_{r}A_{surf}}\right)}
  11. 1 h r A \frac{1}{h_{r}A}
  12. h r = ϵ σ ( T s u r f 2 + T s u r r 2 ) ( T s u r f + T s u r r ) h_{r}=\epsilon\sigma(T_{surf}^{2}+T_{surr}^{2})(T_{surf}+T_{surr})
  13. A A
  14. L 1 L_{1}
  15. k 1 k_{1}
  16. L 2 L_{2}
  17. k 2 k_{2}
  18. T i T_{i}
  19. h i h_{i}
  20. T o T_{o}
  21. h o h_{o}
  22. Q ˙ = T i - T o R i + R 1 + R 2 + R o = T i - T 1 R i = T i - T 2 R i + R 1 = T i - T 3 R i + R 1 + R 2 = T 1 - T 2 R 1 = T 3 - T o R 0 \dot{Q}=\frac{T_{i}-T_{o}}{R_{i}+R_{1}+R_{2}+R_{o}}=\frac{T_{i}-T_{1}}{R_{i}}=% \frac{T_{i}-T_{2}}{R_{i}+R_{1}}=\frac{T_{i}-T_{3}}{R_{i}+R_{1}+R_{2}}=\frac{T_% {1}-T_{2}}{R_{1}}=\frac{T_{3}-T_{o}}{R_{0}}
  23. R i = 1 h i A R_{i}=\frac{1}{h_{i}A}
  24. R o = 1 h o A R_{o}=\frac{1}{h_{o}A}
  25. R 1 = L 1 k 1 A R_{1}=\frac{L_{1}}{k_{1}A}
  26. R 2 = L 2 k 2 A R_{2}=\frac{L_{2}}{k_{2}A}
  27. Rate of cooling Δ T \text{Rate of cooling}\sim\!\,\Delta T
  28. d Q d t = - h A ( T ( t ) - T env ) = - h A Δ T ( t ) \frac{dQ}{dt}=-h\cdot A(T(t)-T_{\,\text{env}})=-h\cdot A\Delta T(t)\quad
  29. C C
  30. T T
  31. Q = C T Q=CT
  32. C C
  33. C = d Q / d T C=dQ/dT
  34. d Q / d t = C ( d T / d t ) dQ/dt=C(dT/dt)
  35. d Q / d t dQ/dt
  36. T ( t ) T(t)
  37. t t
  38. T e n v T_{env}
  39. d T ( t ) d t = - r ( T ( t ) - T env ) = - r Δ T ( t ) \frac{dT(t)}{dt}=-r(T(t)-T_{\mathrm{env}})=-r\Delta T(t)\quad
  40. r = h A / C r=hA/C
  41. s - 1 s^{-1}
  42. t 0 t_{0}
  43. r = 1 / t 0 = Δ T / ( d T ( t ) / d t ) r=1/t_{0}=\Delta T/(dT(t)/dt)
  44. t 0 = C / h A t_{0}=C/hA
  45. C C
  46. c p c_{p}
  47. m m
  48. t 0 t_{0}
  49. m c p / h A mc_{p}/hA
  50. T ( t ) = T env + ( T ( 0 ) - T env ) e - r t . T(t)=T_{\mathrm{env}}+(T(0)-T_{\mathrm{env}})\ e^{-rt}.\quad
  51. Δ T ( t ) \Delta T(t)\quad
  52. T ( t ) - T env , T(t)-T_{\mathrm{env}}\ ,\quad
  53. Δ T ( 0 ) \Delta T(0)\quad
  54. Δ T ( t ) = Δ T ( 0 ) e - r t = Δ T ( 0 ) e - t / t 0 . \Delta T(t)=\Delta T(0)\ e^{-rt}=\Delta T(0)\ e^{-t/t_{0}}.\quad
  55. Δ T ( t ) \Delta T(t)
  56. d T ( t ) d t = d Δ T ( t ) d t = - 1 t 0 Δ T ( t ) \frac{dT(t)}{dt}=\frac{d\Delta T(t)}{dt}=-\frac{1}{t_{0}}\Delta T(t)\quad

Lyapunov_exponent.html

  1. δ 𝐙 0 \delta\mathbf{Z}_{0}
  2. | δ 𝐙 ( t ) | e λ t | δ 𝐙 0 | |\delta\mathbf{Z}(t)|\approx e^{\lambda t}|\delta\mathbf{Z}_{0}|\,
  3. λ \lambda
  4. λ = lim t lim δ 𝐙 0 0 1 t ln | δ 𝐙 ( t ) | | δ 𝐙 0 | \lambda=\lim_{t\to\infty}\lim_{\delta\mathbf{Z}_{0}\to 0}\frac{1}{t}\ln\frac{|% \delta\mathbf{Z}(t)|}{|\delta\mathbf{Z}_{0}|}
  5. δ 𝐙 0 0 \delta\mathbf{Z}_{0}\to 0
  6. x n + 1 = f ( x n ) x_{n+1}=f(x_{n})
  7. x 0 x_{0}
  8. λ ( x 0 ) = lim n 1 n i = 0 n - 1 ln | f ( x i ) | \lambda(x_{0})=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln|f^{\prime}(x_{i% })|
  9. f t f^{t}
  10. { λ 1 , λ 2 , , λ n } , \{\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\}\,,
  11. x 0 x_{0}
  12. J t ( x 0 ) = d f t ( x ) d x | x 0 J^{t}(x_{0})=\left.\frac{df^{t}(x)}{dx}\right|_{x_{0}}
  13. J t J^{t}
  14. x 0 x_{0}
  15. f t ( x 0 ) f^{t}(x_{0})
  16. L ( x 0 ) = lim t ( J t Transpose ( J t ) ) 1 / 2 t L(x_{0})=\lim_{t\rightarrow\infty}(J^{t}\cdot\mathrm{Transpose}(J^{t}))^{1/2t}
  17. L ( x 0 ) L(x_{0})
  18. Λ i ( x 0 ) \Lambda_{i}(x_{0})
  19. L ( x 0 ) L(x_{0})
  20. λ i \lambda_{i}
  21. λ i ( x 0 ) = log Λ i ( x 0 ) \lambda_{i}(x_{0})=\log\Lambda_{i}(x_{0})
  22. X ( t ) X(t)
  23. x 0 x_{0}
  24. exp ( d f t ( x ) d x | x 0 t ) \exp\left(\left.\frac{df^{t}(x)}{dx}\right|_{x_{0}}t\right)
  25. { α j ( X ( t ) ) } 1 n \{\alpha_{j}\big(X(t)\big)\}_{1}^{n}
  26. X ( t ) X(t)
  27. X ( t ) * X ( t ) X(t)^{*}X(t)
  28. λ m a x \lambda_{max}
  29. λ m a x = max j lim sup t 1 t ln α j ( X ( t ) ) . \lambda_{max}=\max\limits_{j}\limsup_{t\rightarrow\infty}\frac{1}{t}\ln\alpha_% {j}\big(X(t)\big).
  30. L L
  31. D K Y D_{KY}
  32. D K Y = k + i = 1 k λ i | λ k + 1 | D_{KY}=k+\sum_{i=1}^{k}\frac{\lambda_{i}}{|\lambda_{k+1}|}
  33. k k
  34. k k
  35. D K Y D_{KY}
  36. L L
  37. L L

Lyapunov_fractal.html

  1. λ \lambda
  2. λ < 0 \lambda<0
  3. λ > 0 \lambda>0
  4. [ 0 , 4 ] [0,4]
  5. S S
  6. ( a , b ) [ 0 , 4 ] × [ 0 , 4 ] (a,b)\in[0,4]\times[0,4]
  7. r n = a r_{n}=a
  8. S n = A S_{n}=A
  9. r n = b r_{n}=b
  10. S n = B S_{n}=B
  11. x 0 = 0.5 x_{0}=0.5
  12. x n + 1 = r n x n ( 1 - x n ) x_{n+1}=r_{n}x_{n}(1-x_{n})
  13. λ = lim N 1 N n = 1 N log | d x n + 1 d x n | = lim N 1 N n = 1 N log | r n ( 1 - 2 x n ) | \lambda=\lim_{N\rightarrow\infty}{1\over N}\sum_{n=1}^{N}\log\left|{dx_{n+1}% \over dx_{n}}\right|=\lim_{N\rightarrow\infty}{1\over N}\sum_{n=1}^{N}\log|r_{% n}(1-2x_{n})|
  14. λ \lambda
  15. N N
  16. ( a , b ) (a,b)
  17. λ \lambda

Lyman-alpha_line.html

  1. α \alpha
  2. n = 2 n=2
  3. n = 1 n=1
  4. l l
  5. n = 2 n=2
  6. j = 1 / 2 j=1/2
  7. j = 3 / 2 j=3/2
  8. j = 3 / 2 j=3/2
  9. n = 1 n=1
  10. j = 3 / 2 j=3/2

LyX.html

  1. 𝐋 𝐗 𝐘 \mathbf{L}\!{}_{\mathbf{\displaystyle Y}}\!\mathbf{X}

LZ77_and_LZ78.html

  1. \Longrightarrow
  2. \Longrightarrow
  3. \Longrightarrow
  4. \Longrightarrow
  5. \Longrightarrow

Machine_learning.html

  1. x D r x\approx Dr

Mafia_(party_game).html

  1. m P {\textstyle m\propto\sqrt{P}}
  2. W ( m , P ) m P , W(m,P)\approx\frac{m}{\sqrt{P}},
  3. π / 2 \sqrt{\pi/2}

Magnetic_circular_dichroism.html

  1. Δ A = A - - A + A - + A + \Delta A=\frac{A_{-}-A_{+}}{A_{-}+A_{+}}
  2. I 0 = 1 2 ( I - + I + ) I_{0}=\frac{1}{2}(I_{-}+I_{+})
  3. I Δ = I 0 2 [ ( 1 - sin ( δ 0 sin ω t ) ) 10 - A - + ( 1 + sin ( δ 0 sin ω t ) ) 10 - A + ] I_{\Delta}=\frac{I_{0}}{2}\left[\left(1-\sin\left(\delta_{0}\sin\omega t\right% )\right)10^{-A_{-}}+\left(1+\sin\left(\delta_{0}\sin\omega t\right)\right)10^{% -A_{+}}\right]
  4. Δ A = V a c 1.1515 V d c δ 0 sin ω t \Delta A=\frac{V_{ac}}{1.1515V_{dc}\delta_{0}\sin\omega t}
  5. A = - log ( V d c V e x ) A=-\log(\frac{V_{dc}}{V_{ex}})
  6. [ k ± ( a j ) ] = 0 k ( a j ) d ω - π 2 ( N a - N j ) ( α 2 n ) | a | m ± | j | 2 [k_{\pm}(a\longrightarrow j)]=\int_{0}^{\infty}k_{\mp}(a\longrightarrow j)d% \omega-{\pi^{2}\over\hbar}(N_{a}-N_{j})({\alpha_{2}\over n})\Big|\big\langle a% \Big|m_{\pm}\Big|j\big\rangle\Big|^{2}
  7. Δ k ± ( a j ) = 0 ( a j ) d ω - π 2 ( N a - N j ) ( α 2 n ) [ | a | m - | j | 2 - | a | m + | j | 2 ] \Delta k_{\pm}(a\longrightarrow j)=\int_{0}^{\infty}(a\longrightarrow j)d% \omega-{\pi^{2}\over\hbar}(N_{a}-N_{j})({\alpha_{2}\over n})[\Big|\big\langle a% \Big|m_{-}\Big|j\big\rangle\Big|^{2}-\Big|\big\langle a\Big|m_{+}\Big|j\big% \rangle\Big|^{2}]
  8. A E = γ D 0 f ( E ) \frac{A}{E}=\gamma D_{0}f(E)
  9. D 0 = ( 1 2 A ) a , λ ( | A α | m - | J λ | 2 + | A α | m + | J λ | 2 ) D_{0}=\left(\frac{1}{2}_{A}\right)\sum_{a,\lambda}\left(\left|\left\langle A_{% \alpha}\left|m_{-}\right|J_{\lambda}\right\rangle\right|^{2}+\left|\left% \langle A_{\alpha}\left|m_{+}\right|J_{\lambda}\right\rangle\right|^{2}\right)
  10. A : B : C = 1 Δ Γ : 1 Δ E : 1 Δ k T A:B:C=\frac{1}{\Delta\Gamma}:\frac{1}{\Delta E}:\frac{1}{\Delta kT}

Magnetic_declination.html

  1. M = T + V M=T+V
  2. C = M + D C=M+D
  3. C = T + V + D C=T+V+D
  4. C C
  5. M M
  6. T T
  7. V V
  8. D D
  9. V > 0 , D > 0 V>0,D>0
  10. V < 0 , D < 0 V<0,D<0

Magnetic_dipole.html

  1. 𝐦 \mathbf{m}
  2. 𝐀 ( 𝐫 ) = μ 0 4 π r 2 𝐦 × 𝐫 r = μ 0 4 π 𝐦 × 𝐫 r 3 , {\mathbf{A}}({\mathbf{r}})=\frac{\mu_{0}}{4\pi r^{2}}\frac{{\mathbf{m}}\times{% \mathbf{r}}}{r}=\frac{\mu_{0}}{4\pi}\frac{{\mathbf{m}}\times{\mathbf{r}}}{r^{3% }},
  3. 𝐁 ( 𝐫 ) = × 𝐀 = μ 0 4 π ( 3 𝐫 ( 𝐦 𝐫 ) r 5 - 𝐦 r 3 ) . \mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left(% \frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{r^{5}}-\frac{{\mathbf{m}}}{r^{3}% }\right).
  4. ψ ( 𝐫 ) = 𝐦 𝐫 4 π r 3 , \psi({\mathbf{r}})=\frac{{\mathbf{m}}\cdot{\mathbf{r}}}{4\pi r^{3}},
  5. 𝐇 ( 𝐫 ) = - ψ = 1 4 π ( 3 𝐫 ( 𝐦 𝐫 ) r 5 - 𝐦 r 3 ) = 𝐁 / μ 0 . {\mathbf{H}}({\mathbf{r}})=-\nabla\psi=\frac{1}{4\pi}\left(\frac{3\mathbf{r}(% \mathbf{m}\cdot\mathbf{r})}{r^{5}}-\frac{{\mathbf{m}}}{r^{3}}\right)=\mathbf{B% }/\mu_{0}.
  6. 𝐁 ( 𝐱 ) = μ 0 4 π [ 3 𝐧 ( 𝐧 𝐦 ) - 𝐦 | 𝐱 | 3 + 8 π 3 𝐦 δ ( 𝐱 ) ] , \mathbf{B}(\mathbf{x})=\frac{\mu_{0}}{4\pi}\left[\frac{3\mathbf{n}(\mathbf{n}% \cdot\mathbf{m})-\mathbf{m}}{|\mathbf{x}|^{3}}+\frac{8\pi}{3}\mathbf{m}\delta(% \mathbf{x})\right],
  7. δ ( 𝐱 ) δ(\mathbf{x})
  8. 𝐇 ( 𝐱 ) = 1 4 π [ 3 𝐧 ( 𝐧 𝐦 ) - 𝐦 | 𝐱 | 3 - 4 π 3 𝐦 δ ( 𝐱 ) ] . \mathbf{H}(\mathbf{x})=\frac{1}{4\pi}\left[\frac{3\mathbf{n}(\mathbf{n}\cdot% \mathbf{m})-\mathbf{m}}{|\mathbf{x}|^{3}}-\frac{4\pi}{3}\mathbf{m}\delta(% \mathbf{x})\right].
  9. 𝐌 ( 𝐱 ) = 𝐦 δ ( 𝐱 ) \mathbf{M}(\mathbf{x})=\mathbf{m}\delta(\mathbf{x})
  10. 𝐅 \mathbf{F}
  11. 𝐫 \mathbf{r}
  12. F = ( m 2 B 1 ) , F=\nabla\left(m_{2}\cdot B_{1}\right),
  13. F ( r , m 1 , m 2 ) = 3 μ 0 4 π | r | 5 [ ( m 1 r ) m 2 + ( m 2 r ) m 1 + ( m 1 m 2 ) r - 5 ( m 1 r ) ( m 2 r ) | r | 2 r ] , F(r,m_{1},m_{2})=\dfrac{3\mu_{0}}{4\pi|r|^{5}}\left[(m_{1}\cdot r)m_{2}+(m_{2}% \cdot r)m_{1}+(m_{1}\cdot m_{2})r-\dfrac{5(m_{1}\cdot r)(m_{2}\cdot r)}{|r|^{2% }}r\right],
  14. 𝐫 \mathbf{r}
  15. s y m b o l τ = m 2 × B 1 . symbol\tau=m_{2}\times B_{1}.
  16. < v a r > ψ <var>ψ

Magnetic_refrigeration.html

  1. Δ T a d = - H 0 H 1 ( T C ( T , H ) ) H ( M ( T , H ) T ) H d H \Delta T_{ad}=-\int_{H_{0}}^{H_{1}}\Bigg(\frac{T}{C(T,H)}\Bigg)_{H}{\Bigg(% \frac{\partial M(T,H)}{\partial T}\Bigg)}_{H}dH

Magnetic_susceptibility.html

  1. χ \chi
  2. χ v \chi_{v}
  3. χ \chi
  4. χ m \chi_{m}
  5. 𝐌 = χ v 𝐇 . \mathbf{M}=\chi_{v}\mathbf{H}.
  6. χ v \chi_{v}
  7. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) = μ 0 ( 1 + χ v ) 𝐇 = μ 𝐇 \mathbf{B}\ =\ \mu_{0}(\mathbf{H}+\mathbf{M})\ =\ \mu_{0}(1+\chi_{v})\mathbf{H% }\ =\ \mu\mathbf{H}
  8. ( 1 + χ v ) (1+\chi_{v})
  9. χ v \chi_{v}
  10. μ \mu
  11. μ = μ 0 ( 1 + χ v ) \mu=\mu_{0}(1+\chi_{v})\,
  12. 𝐈 = μ 0 𝐌 \mathbf{I}=\mu_{0}\mathbf{M}\,
  13. 𝐁 cgs = 𝐇 cgs + 4 π 𝐌 cgs = ( 1 + 4 π χ v cgs ) 𝐇 cgs \mathbf{B}^{\,\text{cgs}}\ =\ \mathbf{H}^{\,\text{cgs}}+4\pi\mathbf{M}^{\,% \text{cgs}}\ =\ (1+4\pi\chi_{v}^{\,\text{cgs}})\mathbf{H}^{\,\text{cgs}}
  14. χ v SI = 4 π χ v cgs \chi_{v}^{\,\text{SI}}=4\pi\chi_{v}^{\,\text{cgs}}
  15. χ mass = χ v / ρ \chi\text{mass}=\chi_{v}/\rho
  16. χ mol = M χ mass = M χ v / ρ \chi\text{mol}=M\chi\text{mass}=M\chi_{v}/\rho
  17. M i = χ i j H j M_{i}=\chi_{ij}H_{j}
  18. χ i j d = M i H j \chi^{d}_{ij}=\frac{\partial M_{i}}{\partial H_{j}}
  19. χ i j d \chi^{d}_{ij}
  20. χ mol \chi_{\,\text{mol}}
  21. χ mass \chi_{\,\text{mass}}
  22. χ v \chi_{v}
  23. ρ \rho
  24. χ = ( 1 / 3 ) χ | | + ( 2 / 3 ) χ \chi=(1/3)\chi_{||}+(2/3)\chi_{\perp}
  25. χ \chi_{\perp}
  26. χ | | \chi_{||}
  27. χ | | \chi_{||}

Magnetometer.html

  1. nT / Hz \rm{nT}/\sqrt{\rm{Hz}}

Magnus_effect.html

  1. ρ \rho
  2. F / L = ρ V G F/L=\rho VG
  3. G = ( 2 π r ) 2 ω G=(2\pi r)^{2}\omega

Mahlo_cardinal.html

  1. \in

Main_diagonal.html

  1. A A
  2. A i , j A_{i,j}
  3. i = j i=j
  4. [ \color r e d 1 0 0 0 \color r e d 1 0 0 0 \color r e d 1 ] [ \color r e d 1 0 0 0 0 \color r e d 1 0 0 0 0 \color r e d 1 0 ] [ \color r e d 1 0 0 0 \color r e d 1 0 0 0 \color r e d 1 0 0 0 ] \begin{bmatrix}\color{red}{1}&0&0\\ 0&\color{red}{1}&0\\ 0&0&\color{red}{1}\end{bmatrix}\qquad\begin{bmatrix}\color{red}{1}&0&0&0\\ 0&\color{red}{1}&0&0\\ 0&0&\color{red}{1}&0\end{bmatrix}\qquad\begin{bmatrix}\color{red}{1}&0&0\\ 0&\color{red}{1}&0\\ 0&0&\color{red}{1}\\ 0&0&0\end{bmatrix}
  5. N N
  6. B B
  7. B i , j B_{i,j}
  8. i + j = N + 1 i+j=N+1
  9. [ 0 0 \color r e d 1 0 \color r e d 1 0 \color r e d 1 0 0 ] \begin{bmatrix}0&0&\color{red}{1}\\ 0&\color{red}{1}&0\\ \color{red}{1}&0&0\end{bmatrix}

Malleability_(cryptography).html

  1. m m
  2. f ( m ) f(m)
  3. f f
  4. m m
  5. m m
  6. f ( m ) f(m)
  7. f f
  8. m m
  9. k k
  10. E ( m ) = m S ( k ) E(m)=m\oplus S(k)
  11. m t m\oplus t
  12. t t
  13. E ( m ) t = m t S ( k ) = E ( m t ) E(m)\oplus t=m\oplus t\oplus S(k)=E(m\oplus t)
  14. m m
  15. E ( m ) = m e mod n E(m)=m^{e}\bmod n
  16. ( e , n ) (e,n)
  17. m t mt
  18. t t
  19. E ( m ) t e mod n = ( m t ) e mod n = E ( m t ) E(m)\cdot t^{e}\bmod n=(mt)^{e}\bmod n=E(mt)
  20. m m
  21. E ( m ) = ( g b , m A b ) E(m)=(g^{b},mA^{b})
  22. ( g , A ) (g,A)
  23. ( c 1 , c 2 ) (c_{1},c_{2})
  24. ( c 1 , t c 2 ) (c_{1},t\cdot c_{2})
  25. t m tm
  26. t t
  27. m 1 m_{1}
  28. m 2 m_{2}
  29. m 1 + m 2 m_{1}+m_{2}
  30. m 1 m_{1}
  31. m 2 m_{2}
  32. m 1 m 2 m_{1}m_{2}

Manganese_dioxide.html

  1. \overrightarrow{\leftarrow}

Mannose.html

  1. C 1 4 {}^{4}C_{1}

Maple_(software).html

  1. cos ( x a ) d x \int\cos\left(\frac{x}{a}\right)dx
  2. a sin ( x a ) a\sin\left(\frac{x}{a}\right)
  3. [ 1 2 3 a b c x y z ] \begin{bmatrix}1&2&3\\ a&b&c\\ x&y&z\end{bmatrix}
  4. b z - c y + 3 a y - 2 a z + 2 x c - 3 x b bz-cy+3ay-2az+2xc-3xb
  5. x - 1 3 x 3 + 2 15 x 5 - 17 315 x 7 x-\frac{1}{3}\,x^{3}+\frac{2}{15}\,x^{5}-\frac{17}{315}\,x^{7}
  6. + 62 2835 x 9 - 1382 155925 x 11 + 21844 6081075 x 13 + O ( x 15 ) +\frac{62}{2835}\,x^{9}-\frac{1382}{155925}\,x^{11}+\frac{21844}{6081075}\,x^{% 13}+O(x^{15})
  7. x sin ( x ) x\cdot\sin(x)
  8. x x
  9. x 2 + y 2 x^{2}+y^{2}
  10. x x
  11. y y
  12. f := 2 k 2 / cosh ( k ( x - 4 k 2 t ) ) 2 f:=2\cdot k^{2}/\cosh(k\cdot(x-4\cdot k^{2}\cdot t))^{2}
  13. ( 1 + A t + B t 2 ) e c t (1+A\cdot t+B\cdot t^{2})\cdot e^{c\cdot t}
  14. 1 s - c + A ( s - c ) 2 + 2 B ( s - c ) 3 \frac{1}{s-c}+\frac{A}{(s-c)^{2}}+\frac{2B}{(s-c)^{3}}
  15. e a x e^{ax}
  16. I π ( Dirac ( w + 1 ) - Dirac ( w - 1 ) ) \mathrm{I}\pi\,(\mathrm{Dirac}(w+1)-\mathrm{Dirac}(w-1))
  17. f f
  18. f ( x ) - 3 - 1 1 ( x y + x 2 y 2 ) f ( y ) d y = h ( x ) f(x)-3\int_{-1}^{1}(xy+x^{2}y^{2})f(y)dy=h(x)
  19. f ( x ) = - 1 1 ( - 15 x 2 y 2 - 3 x y ) h ( y ) d y + h ( x ) f\left(x\right)=\int_{-1}^{1}\!\left(-15\,{x}^{2}{y}^{2}-3\,xy\right)h\left(y% \right){dy}+h\left(x\right)

Margin_of_error.html

  1. Standard error p ( 1 - p ) n \,\text{Standard error}\approx\sqrt{\frac{p(1-p)}{n}}
  2. 1.29 / n \approx 1.29/\sqrt{n}\,
  3. 0.98 / n \approx 0.98/\sqrt{n}\,
  4. 0.82 / n \approx 0.82/\sqrt{n}\,
  5. = erf - 1 ( X ) 2 n =\frac{\,\text{erf}^{-1}(X)}{\sqrt{2n}}
  6. FPC = N - n N - 1 . \operatorname{FPC}=\sqrt{\frac{N-n}{N-1}}.
  7. Standard error of difference = p + q - ( p - q ) 2 n = p + q - p 2 + 2 p q - q 2 n . \,\text{Standard error of difference}=\sqrt{\frac{p+q-(p-q)^{2}}{n}}=\sqrt{% \frac{p+q-p^{2}+2pq-q^{2}}{n}}.

Marginalism.html

  1. S 1 S_{1}
  2. S 2 S_{2}
  3. Δ U = U ( S 2 ) - U ( S 1 ) \Delta U=U(S_{2})-U(S_{1})\,
  4. S 1 S_{1}
  5. S 2 S_{2}
  6. g g\,
  7. g g\,
  8. Δ U Δ g | c . p . \left.\frac{\Delta U}{\Delta g}\right|_{c.p.}
  9. g g\,
  10. lim Δ g 0 Δ U Δ g | c . p . \lim_{\Delta g\to 0}{\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}}
  11. U g Δ U Δ g | c . p . \frac{\partial U}{\partial g}\approx\left.\frac{\Delta U}{\Delta g}\right|_{c.% p.}
  12. 2 U g 2 < 0 \frac{\partial^{2}U}{\partial g^{2}}<0
  13. M R S A B 1 M R S B A MRS_{AB}\neq\frac{1}{MRS_{BA}}
  14. M R S A B = 1 M R S B A MRS_{AB}=\frac{1}{MRS_{BA}}
  15. - 1 -1
  16. M R S S G = 2 sheep goat MRS_{SG}=\frac{2\,\text{ sheep}}{\,}\text{goat}
  17. M R S G S = 2 goat sheep 1 goat 2 sheep = 1 ( 2 sheep goat ) = 1 M R S S G MRS_{GS}=\frac{2\,\text{ goat}}{\,}\text{sheep}\neq\frac{1\,\text{ goat}}{2\,% \text{ sheep}}=\frac{1}{\left(\frac{2\,\text{ sheep}}{\,}\text{goat}\right)}=% \frac{1}{MRS_{SG}}
  18. M R S I B = 1 oz ice cream 1 g banana = 1 ( 1 g banana 1 oz ice cream ) = 1 M R S B I MRS_{IB}=\frac{1\,\text{ oz ice cream}}{1\,\text{ g banana}}=\frac{1}{\left(% \frac{1\,\text{ g banana}}{1\,\text{ oz ice cream}}\right)}=\frac{1}{MRS_{BI}}
  19. A A
  20. B B
  21. A A
  22. B B

Marin_Mersenne.html

  1. f = 1 2 L F μ , f=\frac{1}{2L}\sqrt{\frac{F}{\mu}},
  2. 2 3 - 2 4 \sqrt[4]{\frac{2}{3-\sqrt{2}}}
  3. 2 12 \sqrt[12]{2}

Markov's_inequality.html

  1. X X
  2. a > 0 a> 0
  3. ( X a ) 𝔼 ( X ) a . \mathbb{P}(X\geq a)\leq\frac{\mathbb{E}(X)}{a}.
  4. ( X , Σ , μ ) (X, Σ,μ)
  5. f f
  6. ε > 0 ε>0
  7. μ ( { x X : | f ( x ) | ε } ) 1 ε X | f | d μ . \mu(\{x\in X:|f(x)|\geq\varepsilon\})\leq{1\over\varepsilon}\int_{X}|f|\,d\mu.
  8. φ φ
  9. X X
  10. a 0 a≥0
  11. φ ( a ) > 0 φ(a)>0
  12. ( | X | a ) 𝔼 ( φ ( | X | ) ) φ ( a ) \mathbb{P}(|X|\geq a)\leq\frac{\mathbb{E}(\varphi(|X|))}{\varphi(a)}
  13. a I ( X a ) X aI_{(X\geq a)}\leq X\,
  14. 𝔼 \mathbb{E}
  15. 𝔼 ( a I ( X a ) ) 𝔼 ( X ) . \mathbb{E}(aI_{(X\geq a)})\leq\mathbb{E}(X).\,
  16. a 𝔼 ( I ( X a ) ) = a ( 1 ( X a ) + 0 ( X < a ) ) = a ( X a ) . a\mathbb{E}(I_{(X\geq a)})=a(1\cdot\mathbb{P}(X\geq a)+0\cdot\mathbb{P}(X<a))=% a\mathbb{P}(X\geq a).\,
  17. a ( X a ) 𝔼 ( X ) a\mathbb{P}(X\geq a)\leq\mathbb{E}(X)\,
  18. f f
  19. s ( x ) = { ε , if f ( x ) ε 0 , if f ( x ) < ε s(x)=\begin{cases}\varepsilon,&\,\text{if }f(x)\geq\varepsilon\\ 0,&\,\text{if }f(x)<\varepsilon\end{cases}
  20. 0 s ( x ) f ( x ) 0\leq s(x)\leq f(x)
  21. X f ( x ) d μ X s ( x ) d μ = ε μ ( { x X : f ( x ) ε } ) \int_{X}f(x)\,d\mu\geq\int_{X}s(x)\,d\mu=\varepsilon\mu(\{x\in X:\,f(x)\geq% \varepsilon\})
  22. ε > 0 \varepsilon>0
  23. ε \varepsilon
  24. μ ( { x X : f ( x ) ε } ) 1 ε X f d μ . \mu(\{x\in X:\,f(x)\geq\varepsilon\})\leq{1\over\varepsilon}\int_{X}f\,d\mu.
  25. ( | X - 𝔼 ( X ) | a ) Var ( X ) a 2 , \mathbb{P}(|X-\mathbb{E}(X)|\geq a)\leq\frac{\mathrm{Var}(X)}{a^{2}},
  26. Var ( X ) = 𝔼 [ ( X - 𝔼 ( X ) ) 2 ] . \operatorname{Var}(X)=\mathbb{E}[(X-\mathbb{E}(X))^{2}].
  27. ( X - 𝔼 ( X ) ) 2 (X-\mathbb{E}(X))^{2}
  28. a 2 a^{2}
  29. ( ( X - 𝔼 ( X ) ) 2 a 2 ) Var ( X ) a 2 , \mathbb{P}((X-\mathbb{E}(X))^{2}\geq a^{2})\leq\frac{\operatorname{Var}(X)}{a^% {2}},
  30. ( | X - 𝔼 ( X ) | a ) = ( ( X - 𝔼 ( X ) ) 2 a 2 ) MI 𝔼 ( ( X - 𝔼 ( X ) ) 2 ) a 2 = Var ( X ) a 2 \mathbb{P}(|X-\mathbb{E}(X)|\geq a)=\mathbb{P}\left((X-\mathbb{E}(X))^{2}\geq a% ^{2}\right)\overset{\underset{\mathrm{MI}}{}}{\leq}\frac{\mathbb{E}\left({(X-% \mathbb{E}(X))}^{2}\right)}{a^{2}}=\frac{\operatorname{Var}(X)}{a^{2}}
  31. ( | X | a ) ( φ ( | X | ) φ ( a ) ) MI = 𝔼 ( φ ( | X | ) ) φ ( a ) \mathbb{P}(|X|\geq a)\leq\mathbb{P}(\varphi(|X|)\geq\varphi(a))\overset{% \underset{\mathrm{MI}}{}}{\leq}=\frac{\mathbb{E}(\varphi(|X|))}{\varphi(a)}
  32. X X
  33. X X
  34. Q X ( 1 - p ) 𝔼 ( X ) p , Q_{X}(1-p)\leq\frac{\mathbb{E}(X)}{p},
  35. p ( X Q X ( 1 - p ) ) MI 𝔼 ( X ) Q X ( 1 - p ) . p\leq\mathbb{P}(X\geq Q_{X}(1-p))\overset{\underset{\mathrm{MI}}{}}{\leq}\frac% {\mathbb{E}(X)}{Q_{X}(1-p)}.
  36. M 0 M\succeq 0
  37. a > 0 a>0
  38. ( M a I ) tr ( E ( M ) ) n a . \mathbb{P}(M\npreceq a\cdot I)\leq\frac{\mathrm{tr}\left(E(M)\right)}{na}.

Markov_process.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. ( t , t T ) (\mathcal{F}_{t},\ t\in T)
  3. T T
  4. ( S , 𝒮 ) (S,\mathcal{S})
  5. X = ( X t , t T ) X=(X_{t},\ t\in T)
  6. { t } \{\mathcal{F}_{t}\}
  7. A 𝒮 A\in\mathcal{S}
  8. s , t T s,t\in T
  9. S S
  10. T = T=\mathbb{N}
  11. ( X n = x n | X n - 1 = x n - 1 , X n - 2 = x n - 2 , , X 0 = x 0 ) = ( X n = x n | X n - 1 = x n - 1 ) \mathbb{P}(X_{n}=x_{n}|X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots,X_{0}=x_{0})=% \mathbb{P}(X_{n}=x_{n}|X_{n-1}=x_{n-1})
  12. X n X_{n}
  13. X 0 = 10 X_{0}=10
  14. { X n : n } \{X_{n}:n\in\mathbb{N}\}
  15. X t X_{t}
  16. X t X_{t}
  17. X n X_{n}
  18. X 0 = 0 X_{0}=0
  19. { X n : n } \{X_{n}:n\in\mathbb{N}\}
  20. X 6 = $ 0.50 X_{6}=\$0.50
  21. X 6 X_{6}
  22. X 7 $ 0.60 X_{7}\geq\$0.60
  23. X 6 X_{6}
  24. X 7 X_{7}
  25. X 6 X_{6}
  26. Y ( t ) = { X ( s ) : s [ a ( t ) , b ( t ) ] } . Y(t)=\big\{X(s):s\in[a(t),b(t)]\,\big\}.

Markov_property.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. ( s , s I ) (\mathcal{F}_{s},\ s\in I)
  3. I I
  4. ( S , 𝒮 ) (S,\mathcal{S})
  5. ( S , 𝒮 ) (S,\mathcal{S})
  6. X = ( X t , t I ) X=(X_{t},\ t\in I)
  7. A 𝒮 A\in\mathcal{S}
  8. s , t I s,t\in I
  9. s < t s<t
  10. ( X t A | s ) = ( X t A | X s ) . \mathbb{P}(X_{t}\in A|\mathcal{F}_{s})=\mathbb{P}(X_{t}\in A|X_{s}).
  11. S S
  12. I = I=\mathbb{N}
  13. ( X n = x n | X n - 1 = x n - 1 , , X 0 = x 0 ) = ( X n = x n | X n - 1 = x n - 1 ) \mathbb{P}(X_{n}=x_{n}|X_{n-1}=x_{n-1},\dots,X_{0}=x_{0})=\mathbb{P}(X_{n}=x_{% n}|X_{n-1}=x_{n-1})
  14. 𝔼 [ f ( X t ) | s ] = 𝔼 [ f ( X t ) | σ ( X s ) ] \mathbb{E}[f(X_{t})|\mathcal{F}_{s}]=\mathbb{E}[f(X_{t})|\sigma(X_{s})]
  15. t s 0 t\geq s\geq 0
  16. f : S f:S\rightarrow\mathbb{R}
  17. X = ( X t : t 0 ) X=(X_{t}:t\geq 0)
  18. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  19. { t } t 0 \{\mathcal{F}_{t}\}_{t\geq 0}
  20. t 0 t\geq 0
  21. t + \mathcal{F}_{t+}
  22. s \mathcal{F}_{s}
  23. s > t s>t
  24. τ \tau
  25. Ω \Omega
  26. τ + = { A : { τ = t } A t + , t 0 } \mathcal{F}_{\tau^{+}}=\{A\in\mathcal{F}:\{\tau=t\}\cap A\in\mathcal{F}_{t+},% \,\forall t\geq 0\}
  27. X X
  28. τ \tau
  29. { τ < } \{\tau<\infty\}
  30. t 0 t\geq 0
  31. X τ + t X_{\tau+t}
  32. τ + \mathcal{F}_{\tau^{+}}
  33. X τ X_{\tau}
  34. τ = t \tau=t

Martingale_(betting_system).html

  1. 2 43 2^{43}
  2. q 42 q^{42}
  3. q = 1 / 2 q=1/2
  4. 2 42 2^{42}
  5. q > 1 / 2 q>1/2
  6. i = 1 n B 2 i - 1 = B ( 2 n - 1 ) \sum_{i=1}^{n}B\cdot 2^{i-1}=B(2^{n}-1)
  7. ( 1 - q n ) B - q n B ( 2 n - 1 ) = B ( 1 - ( 2 q ) n ) (1-q^{n})\cdot B-q^{n}\cdot B(2^{n}-1)=B(1-(2q)^{n})

Martingale_(probability_theory).html

  1. 𝐄 ( | X n | ) < \mathbf{E}(|X_{n}|)<\infty
  2. 𝐄 ( X n + 1 X 1 , , X n ) = X n . \mathbf{E}(X_{n+1}\mid X_{1},\ldots,X_{n})=X_{n}.
  3. 𝐄 ( X n + 1 - X n X 1 , , X n ) = 0 \mathbf{E}(X_{n+1}-X_{n}\mid X_{1},\ldots,X_{n})=0
  4. 𝐄 ( X n + 1 X 1 , , X n ) - X n = 0 \mathbf{E}(X_{n+1}\mid X_{1},\ldots,X_{n})-X_{n}=0
  5. n n
  6. n + 1 n+1
  7. 𝐄 ( | Y n | ) < \mathbf{E}(|Y_{n}|)<\infty
  8. 𝐄 ( Y n + 1 X 1 , , X n ) = Y n . \mathbf{E}(Y_{n+1}\mid X_{1},\ldots,X_{n})=Y_{n}.
  9. 𝐄 ( | Y t | ) < \mathbf{E}(|Y_{t}|)<\infty
  10. 𝐄 ( Y t { X τ , τ s } ) = Y s s t . \mathbf{E}(Y_{t}\mid\{X_{\tau},\tau\leq s\})=Y_{s}\quad\forall s\leq t.
  11. s s
  12. Y : T × Ω S Y:T\times\Omega\to S
  13. Σ * \Sigma_{*}
  14. 𝐄 𝐏 ( | Y t | ) < + ; \mathbf{E}_{\mathbf{P}}(|Y_{t}|)<+\infty;
  15. 𝐄 𝐏 ( [ Y t - Y s ] χ F ) = 0 , \mathbf{E}_{\mathbf{P}}\left([Y_{t}-Y_{s}]\chi_{F}\right)=0,
  16. Y s = 𝐄 𝐏 ( Y t | Σ s ) , Y_{s}=\mathbf{E}_{\mathbf{P}}(Y_{t}|\Sigma_{s}),
  17. X n + 1 = X n ± 1 X_{n+1}=X_{n}\pm 1
  18. Y n = ( q / p ) X n . Y_{n}=(q/p)^{X_{n}}.
  19. E [ Y n + 1 X 1 , , X n ] \displaystyle E[Y_{n+1}\mid X_{1},\dots,X_{n}]
  20. Y n = i = 1 n g ( X i ) f ( X i ) Y_{n}=\prod_{i=1}^{n}\frac{g(X_{i})}{f(X_{i})}
  21. { r X n : n = 1 , 2 , 3 , } \{\,r^{X_{n}}:n=1,2,3,\dots\,\}
  22. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},\ldots
  23. E [ X n + 1 | X 1 , , X n ] X n . {}E[X_{n+1}|X_{1},\ldots,X_{n}]\geq X_{n}.
  24. E [ X t | { X τ : τ s } ] X s s t . {}E[X_{t}|\{X_{\tau}:\tau\leq s\}]\geq X_{s}\quad\forall s\leq t.
  25. E [ X n + 1 | X 1 , , X n ] X n . {}E[X_{n+1}|X_{1},\ldots,X_{n}]\leq X_{n}.
  26. E [ X t | { X τ : τ s } ] X s s t . {}E[X_{t}|\{X_{\tau}:\tau\leq s\}]\leq X_{s}\quad\forall s\leq t.
  27. ( X t ) t > 0 (X_{t})_{t>0}
  28. τ \tau
  29. ( X t τ ) t > 0 (X_{t}^{\tau})_{t>0}
  30. X t τ := X min { τ , t } X_{t}^{\tau}:=X_{\min\{\tau,t\}}

Masnavi.html

  1. Problem / Theme Complication Resolution \mathrm{Problem/Theme\longrightarrow Complication\longrightarrow Resolution}

Mass_spectrometry.html

  1. 𝐅 = Q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=Q(\mathbf{E}+\mathbf{v}\times\mathbf{B})
  2. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  3. ( m / Q ) 𝐚 = 𝐄 + 𝐯 × 𝐁 . (m/Q)\mathbf{a}=\mathbf{E}+\mathbf{v}\times\mathbf{B}.

Material_implication_(rule_of_inference).html

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

Mathematical_constants_by_continued_fraction_representation.html

  1. 0
  2. \mathbb{Z}
  3. ϕ - 1 \phi^{-1}
  4. 𝔸 \mathbb{A}\setminus\mathbb{Q}
  5. C C
  6. \mathbb{R}
  7. C 2 C_{2}
  8. \mathbb{R}
  9. γ \gamma
  10. \mathbb{R}
  11. Ω \Omega
  12. 𝔸 \mathbb{R}\setminus\mathbb{A}
  13. β \beta^{\star}
  14. \mathbb{R}
  15. K K
  16. ( ? ) \mathbb{R}(\setminus\mathbb{Q}?)
  17. 1 M ( 1 , 2 ) \frac{1}{M(1,\sqrt{2})}
  18. 𝔸 \mathbb{R}\setminus\mathbb{A}
  19. B 4 B_{4}
  20. \mathbb{R}
  21. C 2 C_{2}
  22. 𝔸 \mathbb{R}\setminus\mathbb{A}
  23. C 2 = 0.110111001011101111000 2 C_{2}=0.110111001011101111000\ldots_{2}
  24. G G
  25. \mathbb{R}
  26. 1 2 \frac{1}{2}
  27. \mathbb{Q}
  28. β \beta
  29. \mathbb{R}
  30. M M
  31. \mathbb{R}
  32. C M R B C_{{}_{MRB}}
  33. \mathbb{R}
  34. C 10 C_{10}
  35. 𝔸 \mathbb{R}\setminus\mathbb{A}
  36. 4.57540 × 10 165 4.57540\times 10^{165}
  37. C 10 C_{10}
  38. 1 1
  39. \mathbb{N}
  40. ϕ \phi
  41. 𝔸 \mathbb{A}\setminus\mathbb{Q}
  42. E E
  43. \mathbb{R}\setminus\mathbb{Q}
  44. B 2 B_{2}
  45. \mathbb{R}
  46. 1.8304 < B 2 < 2.347 1.8304<B_{2}<2.347
  47. 2 \sqrt{2}
  48. 𝔸 \mathbb{A}\setminus\mathbb{Q}
  49. μ \mu
  50. \mathbb{R}
  51. B B
  52. \mathbb{R}
  53. ρ \rho
  54. 𝔸 \mathbb{A}\setminus\mathbb{Q}
  55. ζ ( 3 ) \zeta(3)
  56. \mathbb{R}\setminus\mathbb{Q}
  57. V V
  58. \mathbb{R}
  59. 2 2
  60. \mathbb{N}
  61. 2 2 2^{\sqrt{2}}
  62. 𝔸 \mathbb{R}\setminus\mathbb{A}
  63. α \alpha
  64. \mathbb{R}
  65. e e
  66. 𝔸 \mathbb{R}\setminus\mathbb{A}
  67. K 0 K_{0}
  68. \mathbb{R}
  69. F F
  70. \mathbb{R}
  71. P 2 P_{2}
  72. 𝔸 \mathbb{R}\setminus\mathbb{A}
  73. 3 3
  74. \mathbb{N}
  75. ψ \psi
  76. \mathbb{R}\setminus\mathbb{Q}
  77. π \pi
  78. 𝔸 \mathbb{R}\setminus\mathbb{A}
  79. 4 4
  80. \mathbb{N}
  81. δ \delta
  82. \mathbb{R}
  83. 5 5
  84. \mathbb{N}
  85. e π e^{\pi}
  86. 𝔸 \mathbb{R}\setminus\mathbb{A}
  87. ( - 1 ) - i (-1)^{-i}

Mathematical_fallacy.html

  1. 16 64 = 16 / 6 / 4 = 1 4 . \frac{16}{64}=\frac{16\!\!\!/}{6\!\!\!/4}=\frac{1}{4}.
  2. 16 64 = 1 4 \textstyle\frac{16}{64}=\frac{1}{4}
  3. 2 = 1 2=1
  4. a a
  5. b b
  6. a = b a=b
  7. a a
  8. a 2 = a b a^{2}=ab
  9. b 2 b^{2}
  10. a 2 - b 2 = a b - b 2 a^{2}-b^{2}=ab-b^{2}
  11. ( a - b ) ( a + b ) = b ( a - b ) (a-b)(a+b)=b(a-b)
  12. ( a - b ) (a-b)
  13. a + b = b a+b=b
  14. a = b a=b
  15. b + b = b b+b=b
  16. 2 b = b 2b=b
  17. b b
  18. 2 = 1 2=1
  19. u = 1 log x \textstyle u=\frac{1}{\log x}
  20. d v = d x x \textstyle dv=\frac{dx}{x}
  21. 1 x log x d x = 1 + 1 x log x d x \int\frac{1}{x\,\log x}\,dx=1+\int\frac{1}{x\,\log x}\,dx
  22. a b 1 x log x d x = 1 | a b + a b 1 x log x d x = 0 + a b 1 x log x d x = a b 1 x log x d x \int_{a}^{b}\frac{1}{x\,\log x}\,dx=1|_{a}^{b}+\int_{a}^{b}\frac{1}{x\,\log x}% \,dx=0+\int_{a}^{b}\frac{1}{x\,\log x}\,dx=\int_{a}^{b}\frac{1}{x\,\log x}\,dx
  23. 5 = 4 5=4
  24. - 20 = - 20 -20=-20
  25. 25 - 45 = 16 - 36 25-45=16-36
  26. 5 2 - 5 * 9 = 4 2 - 4 * 9 5^{2}-5*9=4^{2}-4*9
  27. 81 / 4 81/4
  28. 5 2 - 5 * 9 + 81 / 4 = 4 2 - 4 * 9 + 81 / 4 5^{2}-5*9+81/4=4^{2}-4*9+81/4
  29. ( 5 - 9 / 2 ) 2 = ( 4 - 9 / 2 ) 2 (5-9/2)^{2}=(4-9/2)^{2}
  30. 5 - 9 / 2 = 4 - 9 / 2 5-9/2=4-9/2
  31. 9 / 2 9/2
  32. 5 = 4 5=4
  33. 5 - 9 / 2 = - ( 4 - 9 / 2 ) . 5-9/2=-(4-9/2).
  34. 1 = 1 = ( - 1 ) ( - 1 ) = - 1 - 1 = i i = - 1. 1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1.
  35. x y = x y \textstyle\sqrt{xy}=\sqrt{x}\sqrt{y}
  36. cos 2 x = 1 - sin 2 x \cos^{2}x=1-\sin^{2}x
  37. cos x = ( 1 - sin 2 x ) 1 2 \cos x=(1-\sin^{2}x)^{\frac{1}{2}}
  38. 1 + cos x = 1 + ( 1 - sin 2 x ) 1 2 . 1+\cos x=1+(1-\sin^{2}x)^{\frac{1}{2}}.
  39. 1 - 1 = 1 + ( 1 - 0 ) 1 2 1-1=1+(1-0)^{\frac{1}{2}}
  40. 0 = 2 0=2
  41. x 2 = a 2 x^{2}=a^{2}
  42. x = ± a x=\pm a
  43. x 2 = 4 x^{2}=4\,
  44. x = 4 = 2 , x=\sqrt{4}=2,
  45. - 1 = ( - 1 ) 2 4 = ( ( - 1 ) 2 ) 1 4 = 1 1 4 = 1 \sqrt{-1}=(-1)^{\frac{2}{4}}=((-1)^{2})^{\frac{1}{4}}=1^{\frac{1}{4}}=1
  46. e 2 π i \displaystyle e^{2\pi i}
  47. i i
  48. 1 n \sqrt[n]{1}

Mathematical_morphology.html

  1. d \mathbb{R}^{d}
  2. d \mathbb{Z}^{d}
  3. E = 2 E=\mathbb{R}^{2}
  4. E = 2 E=\mathbb{Z}^{2}
  5. E = 2 E=\mathbb{Z}^{2}
  6. A B = { z E | B z A } A\ominus B=\{z\in E|B_{z}\subseteq A\}
  7. B z = { b + z | b B } B_{z}=\{b+z|b\in B\}
  8. z E \forall z\in E
  9. A B = b B A - b A\ominus B=\bigcap_{b\in B}A_{-b}
  10. A B = b B A b A\oplus B=\bigcup_{b\in B}A_{b}
  11. A B = B A = a A B a A\oplus B=B\oplus A=\bigcup_{a\in A}B_{a}
  12. A B = { z E | ( B s ) z A } A\oplus B=\{z\in E|(B^{s})_{z}\cap A\neq\varnothing\}
  13. B s = { x E | - x B } B^{s}=\{x\in E|-x\in B\}
  14. A B = ( A B ) B A\circ B=(A\ominus B)\oplus B
  15. A B = B x A B x A\circ B=\bigcup_{B_{x}\subseteq A}B_{x}
  16. A B = ( A B ) B A\bullet B=(A\oplus B)\ominus B
  17. A B = ( A c B s ) c A\bullet B=(A^{c}\circ B^{s})^{c}
  18. X c = { x E | x X } X^{c}=\{x\in E|x\not\in X\}
  19. A C A\subseteq C
  20. A B C B A\oplus B\subseteq C\oplus B
  21. A B C B A\ominus B\subseteq C\ominus B
  22. A B A B A A B A B A\ominus B\subseteq A\circ B\subseteq A\subseteq A\bullet B\subseteq A\oplus B
  23. ( A B ) C = A ( B C ) (A\oplus B)\oplus C=A\oplus(B\oplus C)
  24. ( A B ) C = A ( B C ) (A\ominus B)\ominus C=A\ominus(B\oplus C)
  25. A B = ( A c B s ) c A\oplus B=(A^{c}\ominus B^{s})^{c}
  26. A B = ( A c B s ) c A\bullet B=(A^{c}\circ B^{s})^{c}
  27. A ( C B ) A\subseteq(C\ominus B)
  28. ( A B ) C (A\oplus B)\subseteq C
  29. A B A A\circ B\subseteq A
  30. A A B A\subseteq A\bullet B
  31. { , - } \mathbb{R}\cup\{\infty,-\infty\}
  32. \mathbb{R}
  33. \infty
  34. - -\infty
  35. ( f b ) ( x ) = sup y E [ f ( y ) + b ( x - y ) ] (f\oplus b)(x)=\sup_{y\in E}[f(y)+b(x-y)]
  36. ( f b ) ( x ) = inf y E [ f ( y ) - b ( y - x ) ] (f\ominus b)(x)=\inf_{y\in E}[f(y)-b(y-x)]
  37. f b = ( f b ) b f\circ b=(f\ominus b)\oplus b
  38. f b = ( f b ) b f\bullet b=(f\oplus b)\ominus b
  39. b ( x ) = { 0 , x B , - , otherwise b(x)=\left\{\begin{array}[]{ll}0,&x\in B,\\ -\infty,&\mbox{otherwise}\end{array}\right.
  40. B E B\subseteq E
  41. ( f b ) ( x ) = sup z B s f ( x + z ) (f\oplus b)(x)=\sup_{z\in B^{s}}f(x+z)
  42. ( f b ) ( x ) = inf z B f ( x + z ) (f\ominus b)(x)=\inf_{z\in B}f(x+z)
  43. ( L , ) (L,\leq)
  44. \wedge
  45. \vee
  46. \emptyset
  47. { X i } \{X_{i}\}
  48. δ : L L \delta\colon L\rightarrow L
  49. i δ ( X i ) = δ ( i X i ) \bigvee_{i}\delta(X_{i})=\delta\left(\bigvee_{i}X_{i}\right)
  50. δ ( ) = \delta(\emptyset)=\emptyset
  51. ε : L L \varepsilon\colon L\rightarrow L
  52. i ε ( X i ) = ε ( i X i ) \bigwedge_{i}\varepsilon(X_{i})=\varepsilon\left(\bigwedge_{i}X_{i}\right)
  53. ε ( U ) = U \varepsilon(U)=U
  54. δ \delta
  55. ε \varepsilon
  56. X ε ( Y ) δ ( X ) Y X\leq\varepsilon(Y)\Leftrightarrow\delta(X)\leq Y
  57. X , Y L X,Y\in L
  58. δ \delta
  59. ε \varepsilon
  60. ( ε , δ ) (\varepsilon,\delta)
  61. γ : L L \gamma\colon L\rightarrow L
  62. ϕ : L L \phi\colon L\rightarrow L
  63. γ = δ ε \gamma=\delta\varepsilon
  64. ϕ = ε δ \phi=\varepsilon\delta
  65. \leq
  66. { , - } \mathbb{R}\cup\{\infty,-\infty\}
  67. \leq
  68. \vee
  69. \wedge
  70. f g f\leq g
  71. f ( x ) g ( x ) , x E f(x)\leq g(x),\forall x\in E
  72. f g f\wedge g
  73. ( f g ) ( x ) = f ( x ) g ( x ) (f\wedge g)(x)=f(x)\wedge g(x)
  74. f g f\vee g
  75. ( f g ) ( x ) = f ( x ) g ( x ) (f\vee g)(x)=f(x)\vee g(x)

Mathematical_proof.html

  1. P ( n ) P(n)
  2. n n
  3. 𝐍 \mathbf{N}
  4. P ( 1 ) P(1)
  5. P ( n ) P(n)
  6. n = 1 n=1
  7. P ( n + 1 ) P(n+1)
  8. P ( n ) P(n)
  9. P ( n ) P(n)
  10. P ( n + 1 ) P(n+1)
  11. P ( n ) P(n)
  12. n n
  13. 2 n + 1 2n+ 1
  14. n = 1 n=1
  15. 2 n + 1 = 2 ( 1 ) + 1 = 3 2n+ 1 = 2(1) + 1 = 3
  16. 3 3
  17. P ( 1 ) P(1)
  18. 2 n + 1 2n+ 1
  19. n n
  20. 2 ( n + 1 ) + 1 = ( 2 n + 1 ) + 2 2(n+1) + 1 =(2n+1) + 2
  21. 2 n + 1 2n+ 1
  22. ( 2 n + 1 ) + 2 (2n+1) + 2
  23. 2 2
  24. P ( n + 1 ) P(n+1)
  25. P ( n ) P(n)
  26. 2 n + 1 2n+ 1
  27. n n
  28. 2 \sqrt{2}
  29. 2 \sqrt{2}
  30. 2 = a b \sqrt{2}={a\over b}
  31. b 2 = a b\sqrt{2}=a
  32. 2 \sqrt{2}
  33. a b a^{b}
  34. 2 2 \sqrt{2}^{\sqrt{2}}
  35. a = b = 2 a=b=\sqrt{2}
  36. 2 2 \sqrt{2}^{\sqrt{2}}
  37. a = 2 2 a=\sqrt{2}^{\sqrt{2}}
  38. b = 2 b=\sqrt{2}
  39. ( 2 2 ) 2 = 2 2 = 2 \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=\sqrt{2}^{2}=2
  40. a b . a^{b}.

Mathematics_of_paper_folding.html

  1. B Q = 2 A P 1 + A P . BQ=\frac{2AP}{1+AP}.
  2. x x
  3. 2 x 1 + x \frac{2x}{1+x}
  4. 1 - x 1 + x \frac{1-x}{1+x}
  5. 1 - x 2 2 \frac{1-x^{2}}{2}
  6. 1 + x 2 1 + x \frac{1+x^{2}}{1+x}
  7. L = π t 6 ( 2 n + 4 ) ( 2 n - 1 ) L=\tfrac{\pi t}{6}(2^{n}+4)(2^{n}-1)

Matrix_addition.html

  1. A + B = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ] + [ b 11 b 12 b 1 n b 21 b 22 b 2 n b m 1 b m 2 b m n ] = [ a 11 + b 11 a 12 + b 12 a 1 n + b 1 n a 21 + b 21 a 22 + b 22 a 2 n + b 2 n a m 1 + b m 1 a m 2 + b m 2 a m n + b m n ] \begin{aligned}\displaystyle{A}+{B}&\displaystyle=\begin{bmatrix}a_{11}&a_{12}% &\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\\ \end{bmatrix}+\begin{bmatrix}b_{11}&b_{12}&\cdots&b_{1n}\\ b_{21}&b_{22}&\cdots&b_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ b_{m1}&b_{m2}&\cdots&b_{mn}\\ \end{bmatrix}\\ &\displaystyle=\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots&a_{1n}+b_{1n}% \\ a_{21}+b_{21}&a_{22}+b_{22}&\cdots&a_{2n}+b_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots&a_{mn}+b_{mn}\\ \end{bmatrix}\\ \end{aligned}\,\!
  2. [ 1 3 1 0 1 2 ] + [ 0 0 7 5 2 1 ] = [ 1 + 0 3 + 0 1 + 7 0 + 5 1 + 2 2 + 1 ] = [ 1 3 8 5 3 3 ] \begin{bmatrix}1&3\\ 1&0\\ 1&2\end{bmatrix}+\begin{bmatrix}0&0\\ 7&5\\ 2&1\end{bmatrix}=\begin{bmatrix}1+0&3+0\\ 1+7&0+5\\ 1+2&2+1\end{bmatrix}=\begin{bmatrix}1&3\\ 8&5\\ 3&3\end{bmatrix}
  3. [ 1 3 1 0 1 2 ] - [ 0 0 7 5 2 1 ] = [ 1 - 0 3 - 0 1 - 7 0 - 5 1 - 2 2 - 1 ] = [ 1 3 - 6 - 5 - 1 1 ] \begin{bmatrix}1&3\\ 1&0\\ 1&2\end{bmatrix}-\begin{bmatrix}0&0\\ 7&5\\ 2&1\end{bmatrix}=\begin{bmatrix}1-0&3-0\\ 1-7&0-5\\ 1-2&2-1\end{bmatrix}=\begin{bmatrix}1&3\\ -6&-5\\ -1&1\end{bmatrix}
  4. A B = [ A s y m b o l 0 s y m b o l 0 B ] = [ a 11 a 1 n 0 0 a m 1 a m n 0 0 0 0 b 11 b 1 q 0 0 b p 1 b p q ] {A}\oplus{B}=\begin{bmatrix}{A}&symbol{0}\\ symbol{0}&{B}\end{bmatrix}=\begin{bmatrix}a_{11}&\cdots&a_{1n}&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ a_{m1}&\cdots&a_{mn}&0&\cdots&0\\ 0&\cdots&0&b_{11}&\cdots&b_{1q}\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&b_{p1}&\cdots&b_{pq}\end{bmatrix}
  5. [ 1 3 2 2 3 1 ] [ 1 6 0 1 ] = [ 1 3 2 0 0 2 3 1 0 0 0 0 0 1 6 0 0 0 0 1 ] \begin{bmatrix}1&3&2\\ 2&3&1\end{bmatrix}\oplus\begin{bmatrix}1&6\\ 0&1\end{bmatrix}=\begin{bmatrix}1&3&2&0&0\\ 2&3&1&0&0\\ 0&0&0&1&6\\ 0&0&0&0&1\end{bmatrix}
  6. i = 1 n A i = diag ( A 1 , A 2 , A 3 A n ) = [ A 1 s y m b o l 0 s y m b o l 0 s y m b o l 0 A 2 s y m b o l 0 s y m b o l 0 s y m b o l 0 A n ] \bigoplus_{i=1}^{n}{A}_{i}={\rm diag}({A}_{1},{A}_{2},{A}_{3}\cdots{A}_{n})=% \begin{bmatrix}{A}_{1}&symbol{0}&\cdots&symbol{0}\\ symbol{0}&{A}_{2}&\cdots&symbol{0}\\ \vdots&\vdots&\ddots&\vdots\\ symbol{0}&symbol{0}&\cdots&{A}_{n}\\ \end{bmatrix}\,\!
  7. 𝐈 k \mathbf{I}_{k}
  8. 𝐀 𝐁 = 𝐀 𝐈 m + 𝐈 n 𝐁 . \mathbf{A}\oplus\mathbf{B}=\mathbf{A}\otimes\mathbf{I}_{m}+\mathbf{I}_{n}% \otimes\mathbf{B}.

Matrix_decomposition.html

  1. A x = b Ax=b
  2. L ( U x ) = b L(Ux)=b
  3. U x = L - 1 b Ux=L^{-1}b
  4. A x = b Ax=b
  5. A = L U A=LU
  6. A = L D U A=LDU
  7. A = L U P A=LUP
  8. A x = b Ax=b
  9. A = C F A=CF
  10. A x = b Ax=b
  11. A = U T U A=U^{T}U
  12. A = Q R A=QR
  13. A x = b Ax=b
  14. Q T Q = I Q^{T}Q=I
  15. A x = b Ax=b
  16. R x = Q T b Rx=Q^{T}b
  17. A = V D V - 1 A=VDV^{-1}
  18. A V = V D AV=VD
  19. A = V D V - 1 A=VDV^{-1}
  20. A A
  21. V V
  22. V V T = I VV^{T}=I
  23. A = V D V T A=VDV^{T}
  24. x t + 1 = A x t x_{t+1}=Ax_{t}
  25. x 0 = c x_{0}=c
  26. x t = A t c x_{t}=A^{t}c
  27. x t = V D t V - 1 c x_{t}=VD^{t}V^{-1}c
  28. D t D^{t}
  29. A = U T U H A=UTU^{H}
  30. U H U^{H}
  31. A = V S V T A=VSV^{T}
  32. V T V^{T}
  33. V T V^{T}
  34. A = Q S Z H A=QSZ^{H}
  35. B = Q T Z H B=QTZ^{H}
  36. λ i = S i i / T i i \lambda_{i}=S_{ii}/T_{ii}
  37. A v = λ B v Av=\lambda Bv
  38. λ \lambda
  39. A = Q S Z T A=QSZ^{T}
  40. B = Q T Z T B=QTZ^{T}
  41. A = V D V T A=VDV^{T}
  42. V T V^{T}
  43. A A H AA^{H}
  44. A = U D V H A=UDV^{H}
  45. V H V^{H}
  46. A = U P A=UP
  47. A = Q S A=QS
  48. A A T AA^{T}
  49. A T A A^{T}A
  50. A = D 1 S D 2 A=D_{1}SD_{2}
  51. S α = { r e i θ r > 0 , | θ | α } S_{\alpha}=\{re^{i\theta}\in\mathbb{C}\mid r>0,|\theta|\leq\alpha\}
  52. A = C Z C H A=CZC^{H}
  53. Z = d i a g ( e i θ 1 , , e i θ n ) Z=diag(e^{i\theta_{1}},\ldots,e^{i\theta_{n}})
  54. | θ j | α |\theta_{j}|\leq\alpha

Matrix_multiplication.html

  1. 𝐀 \mathbf{A}
  2. n × m n×m
  3. 𝐁 \mathbf{B}
  4. m × p m×p
  5. 𝐀𝐁 \mathbf{AB}
  6. n × p n×p
  7. m m
  8. 𝐀 \mathbf{A}
  9. m m
  10. 𝐁 \mathbf{B}
  11. 𝐂 = 𝐀𝐁 \mathbf{C}=\mathbf{AB}
  12. 𝐀 \mathbf{A}
  13. 𝐚 \mathbf{a}
  14. A A
  15. a a
  16. i , j i,j
  17. 𝐀 \mathbf{A}
  18. 𝐀 \mathbf{A}
  19. λ λ
  20. λ 𝐀 λ\mathbf{A}
  21. 𝐀 \mathbf{A}
  22. λ 𝐀 λ\mathbf{A}
  23. ( λ 𝐀 ) i j = λ ( 𝐀 ) i j , (\lambda\mathbf{A})_{ij}=\lambda\left(\mathbf{A}\right)_{ij}\,,
  24. λ 𝐀 = λ ( A 11 A 12 A 1 m A 21 A 22 A 2 m A n 1 A n 2 A n m ) = ( λ A 11 λ A 12 λ A 1 m λ A 21 λ A 22 λ A 2 m λ A n 1 λ A n 2 λ A n m ) . \lambda\mathbf{A}=\lambda\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1m}\\ A_{21}&A_{22}&\cdots&A_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nm}\\ \end{pmatrix}=\begin{pmatrix}\lambda A_{11}&\lambda A_{12}&\cdots&\lambda A_{1% m}\\ \lambda A_{21}&\lambda A_{22}&\cdots&\lambda A_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ \lambda A_{n1}&\lambda A_{n2}&\cdots&\lambda A_{nm}\\ \end{pmatrix}\,.
  25. 𝐀 \mathbf{A}
  26. λ λ
  27. ( 𝐀 λ ) i j = ( 𝐀 ) i j λ , (\mathbf{A}\lambda)_{ij}=\left(\mathbf{A}\right)_{ij}\lambda\,,
  28. 𝐀 λ = ( A 11 A 12 A 1 m A 21 A 22 A 2 m A n 1 A n 2 A n m ) λ = ( A 11 λ A 12 λ A 1 m λ A 21 λ A 22 λ A 2 m λ A n 1 λ A n 2 λ A n m λ ) . \mathbf{A}\lambda=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1m}\\ A_{21}&A_{22}&\cdots&A_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nm}\\ \end{pmatrix}\lambda=\begin{pmatrix}A_{11}\lambda&A_{12}\lambda&\cdots&A_{1m}% \lambda\\ A_{21}\lambda&A_{22}\lambda&\cdots&A_{2m}\lambda\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}\lambda&A_{n2}\lambda&\cdots&A_{nm}\lambda\\ \end{pmatrix}\,.
  29. λ = 2 , 𝐀 = ( a b c d ) \lambda=2,\quad\mathbf{A}=\begin{pmatrix}a&b\\ c&d\\ \end{pmatrix}
  30. 2 𝐀 = 2 ( a b c d ) = ( 2 a 2 b 2 c 2 d ) = ( a 2 b 2 c 2 d 2 ) = ( a b c d ) 2 = 𝐀 2. 2\mathbf{A}=2\begin{pmatrix}a&b\\ c&d\\ \end{pmatrix}=\begin{pmatrix}2\!\cdot\!a&2\!\cdot\!b\\ 2\!\cdot\!c&2\!\cdot\!d\\ \end{pmatrix}=\begin{pmatrix}a\!\cdot\!2&b\!\cdot\!2\\ c\!\cdot\!2&d\!\cdot\!2\\ \end{pmatrix}=\begin{pmatrix}a&b\\ c&d\\ \end{pmatrix}2=\mathbf{A}2.
  31. λ = i , 𝐀 = ( i 0 0 j ) \lambda=i,\quad\mathbf{A}=\begin{pmatrix}i&0\\ 0&j\\ \end{pmatrix}
  32. i ( i 0 0 j ) = ( i 2 0 0 i j ) = ( - 1 0 0 k ) ( - 1 0 0 - k ) = ( i 2 0 0 j i ) = ( i 0 0 j ) i , i\begin{pmatrix}i&0\\ 0&j\\ \end{pmatrix}=\begin{pmatrix}i^{2}&0\\ 0&ij\\ \end{pmatrix}=\begin{pmatrix}-1&0\\ 0&k\\ \end{pmatrix}\neq\begin{pmatrix}-1&0\\ 0&-k\\ \end{pmatrix}=\begin{pmatrix}i^{2}&0\\ 0&ji\\ \end{pmatrix}=\begin{pmatrix}i&0\\ 0&j\\ \end{pmatrix}i\,,
  33. i , j , k i,j,k
  34. i j = + k ij=+k
  35. j i = k ji=−k
  36. i i
  37. 𝐀 \mathbf{A}
  38. j j
  39. 𝐁 \mathbf{B}
  40. i j ij
  41. 𝐀 \mathbf{A}
  42. n × m n×m
  43. 𝐁 \mathbf{B}
  44. m × p m×p
  45. 𝐀 = ( A 11 A 12 A 1 m A 21 A 22 A 2 m A n 1 A n 2 A n m ) , 𝐁 = ( B 11 B 12 B 1 p B 21 B 22 B 2 p B m 1 B m 2 B m p ) \mathbf{A}=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1m}\\ A_{21}&A_{22}&\cdots&A_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nm}\\ \end{pmatrix},\quad\mathbf{B}=\begin{pmatrix}B_{11}&B_{12}&\cdots&B_{1p}\\ B_{21}&B_{22}&\cdots&B_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ B_{m1}&B_{m2}&\cdots&B_{mp}\\ \end{pmatrix}
  46. 𝐀𝐁 \mathbf{AB}
  47. n × p n×p
  48. 𝐀𝐁 = ( ( 𝐀𝐁 ) 11 ( 𝐀𝐁 ) 12 ( 𝐀𝐁 ) 1 p ( 𝐀𝐁 ) 21 ( 𝐀𝐁 ) 22 ( 𝐀𝐁 ) 2 p ( 𝐀𝐁 ) n 1 ( 𝐀𝐁 ) n 2 ( 𝐀𝐁 ) n p ) \mathbf{A}\mathbf{B}=\begin{pmatrix}\left(\mathbf{AB}\right)_{11}&\left(% \mathbf{AB}\right)_{12}&\cdots&\left(\mathbf{AB}\right)_{1p}\\ \left(\mathbf{AB}\right)_{21}&\left(\mathbf{AB}\right)_{22}&\cdots&\left(% \mathbf{AB}\right)_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ \left(\mathbf{AB}\right)_{n1}&\left(\mathbf{AB}\right)_{n2}&\cdots&\left(% \mathbf{AB}\right)_{np}\\ \end{pmatrix}
  49. i , j i,j
  50. i i
  51. 𝐀 \mathbf{A}
  52. j j
  53. 𝐁 \mathbf{B}
  54. k = 1 , 2 , , m k=1,2,...,m
  55. k k
  56. ( 𝐀𝐁 ) i j = k = 1 m A i k B k j . (\mathbf{A}\mathbf{B})_{ij}=\sum_{k=1}^{m}A_{ik}B_{kj}\,.
  57. 𝐀𝐁 \mathbf{AB}
  58. 𝐀 \mathbf{A}
  59. 𝐁 \mathbf{B}
  60. m m
  61. k k
  62. 𝐀 \mathbf{A}
  63. 𝐁 \mathbf{B}
  64. 𝐀 \mathbf{A}
  65. 𝐁 \mathbf{B}
  66. [ \color B r o w n a 11 \color B r o w n a 12 \color O r a n g e a 31 \color O r a n g e a 32 ] 4 × 2 matrix [ \color P l u m b 12 \color V i o l e t b 13 \color P l u m b 22 \color V i o l e t b 23 ] 2 × 3 matrix = [ x 12 x 13 x 32 x 33 ] 4 × 3 matrix \overset{4\times 2\,\text{ matrix}}{\begin{bmatrix}{\color{Brown}{a_{11}}}&{% \color{Brown}{a_{12}}}\\ \cdot&\cdot\\ {\color{Orange}{a_{31}}}&{\color{Orange}{a_{32}}}\\ \cdot&\cdot\\ \end{bmatrix}}\overset{2\times 3\,\text{ matrix}}{\begin{bmatrix}\cdot&{\color% {Plum}{b_{12}}}&{\color{Violet}{b_{13}}}\\ \cdot&{\color{Plum}{b_{22}}}&{\color{Violet}{b_{23}}}\\ \end{bmatrix}}=\overset{4\times 3\,\text{ matrix}}{\begin{bmatrix}\cdot&x_{12}% &x_{13}\\ \cdot&\cdot&\cdot\\ \cdot&x_{32}&x_{33}\\ \cdot&\cdot&\cdot\\ \end{bmatrix}}
  67. x 12 \displaystyle x_{12}
  68. 𝐀 = ( a b c ) , 𝐁 = ( x y z ) , \mathbf{A}=\begin{pmatrix}a&b&c\end{pmatrix}\,,\quad\mathbf{B}=\begin{pmatrix}% x\\ y\\ z\end{pmatrix}\,,
  69. 𝐀𝐁 = ( a b c ) ( x y z ) = a x + b y + c z , \mathbf{AB}=\begin{pmatrix}a&b&c\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=ax+by+cz\,,
  70. 𝐁𝐀 = ( x y z ) ( a b c ) = ( x a x b x c y a y b y c z a z b z c ) . \mathbf{BA}=\begin{pmatrix}x\\ y\\ z\end{pmatrix}\begin{pmatrix}a&b&c\end{pmatrix}=\begin{pmatrix}xa&xb&xc\\ ya&yb&yc\\ za&zb&zc\end{pmatrix}\,.
  71. 𝐀𝐁 \mathbf{AB}
  72. 𝐁𝐀 \mathbf{BA}
  73. 1 × 1 1×1
  74. 3 × 3 3×3
  75. 𝐀𝐁 \mathbf{AB}
  76. 𝐁𝐀 \mathbf{BA}
  77. 𝐀 = ( a b c p q r u v w ) , 𝐁 = ( x y z ) , \mathbf{A}=\begin{pmatrix}a&b&c\\ p&q&r\\ u&v&w\end{pmatrix},\quad\mathbf{B}=\begin{pmatrix}x\\ y\\ z\end{pmatrix}\,,
  78. 𝐀𝐁 = ( a b c p q r u v w ) ( x y z ) = ( a x + b y + c z p x + q y + r z u x + v y + w z ) , \mathbf{AB}=\begin{pmatrix}a&b&c\\ p&q&r\\ u&v&w\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}ax+by+cz\\ px+qy+rz\\ ux+vy+wz\end{pmatrix}\,,
  79. 𝐁𝐀 \mathbf{BA}
  80. a , b , c , p , q , r , u , v , w a,b,c,p,q,r,u,v,w
  81. 𝐀 \mathbf{A}
  82. 𝐀 \mathbf{A}
  83. 𝐀 = ( a b c p q r u v w ) , 𝐁 = ( α β γ λ μ ν ρ σ τ ) , \mathbf{A}=\begin{pmatrix}a&b&c\\ p&q&r\\ u&v&w\end{pmatrix},\quad\mathbf{B}=\begin{pmatrix}\alpha&\beta&\gamma\\ \lambda&\mu&\nu\\ \rho&\sigma&\tau\\ \end{pmatrix}\,,
  84. 𝐀𝐁 = ( a b c p q r u v w ) ( α β γ λ μ ν ρ σ τ ) = ( a α + b λ + c ρ a β + b μ + c σ a γ + b ν + c τ p α + q λ + r ρ p β + q μ + r σ p γ + q ν + r τ u α + v λ + w ρ u β + v μ + w σ u γ + v ν + w τ ) , \mathbf{AB}=\begin{pmatrix}a&b&c\\ p&q&r\\ u&v&w\end{pmatrix}\begin{pmatrix}\alpha&\beta&\gamma\\ \lambda&\mu&\nu\\ \rho&\sigma&\tau\\ \end{pmatrix}=\begin{pmatrix}a\alpha+b\lambda+c\rho&a\beta+b\mu+c\sigma&a% \gamma+b\nu+c\tau\\ p\alpha+q\lambda+r\rho&p\beta+q\mu+r\sigma&p\gamma+q\nu+r\tau\\ u\alpha+v\lambda+w\rho&u\beta+v\mu+w\sigma&u\gamma+v\nu+w\tau\end{pmatrix}\,,
  85. 𝐁𝐀 = ( α β γ λ μ ν ρ σ τ ) ( a b c p q r u v w ) = ( α a + β p + γ u α b + β q + γ v α c + β r + γ w λ a + μ p + ν u λ b + μ q + ν v λ c + μ r + ν w ρ a + σ p + τ u ρ b + σ q + τ v ρ c + σ r + τ w ) . \mathbf{BA}=\begin{pmatrix}\alpha&\beta&\gamma\\ \lambda&\mu&\nu\\ \rho&\sigma&\tau\\ \end{pmatrix}\begin{pmatrix}a&b&c\\ p&q&r\\ u&v&w\end{pmatrix}=\begin{pmatrix}\alpha a+\beta p+\gamma u&\alpha b+\beta q+% \gamma v&\alpha c+\beta r+\gamma w\\ \lambda a+\mu p+\nu u&\lambda b+\mu q+\nu v&\lambda c+\mu r+\nu w\\ \rho a+\sigma p+\tau u&\rho b+\sigma q+\tau v&\rho c+\sigma r+\tau w\end{% pmatrix}\,.
  86. 𝐀𝐁 \mathbf{AB}
  87. 𝐁𝐀 \mathbf{BA}
  88. 𝐀𝐁 \mathbf{AB}
  89. 𝐁𝐀 \mathbf{BA}
  90. 𝐀 = ( a b c ) , 𝐁 = ( α β γ λ μ ν ρ σ τ ) , 𝐂 = ( x y z ) , \mathbf{A}=\begin{pmatrix}a&b&c\end{pmatrix}\,,\quad\mathbf{B}=\begin{pmatrix}% \alpha&\beta&\gamma\\ \lambda&\mu&\nu\\ \rho&\sigma&\tau\\ \end{pmatrix}\,,\quad\mathbf{C}=\begin{pmatrix}x\\ y\\ z\end{pmatrix}\,,
  91. 𝐀𝐁𝐂 \displaystyle\mathbf{ABC}
  92. 𝐂𝐁𝐀 \mathbf{CBA}
  93. 𝐀 ( 𝐁𝐂 ) = ( 𝐀𝐁 ) 𝐂 \mathbf{A}(\mathbf{BC})=(\mathbf{AB})\mathbf{C}
  94. 𝐀𝐁𝐂 \mathbf{ABC}
  95. 𝐀 = ( a b c x y z ) , 𝐁 = ( α ρ β σ γ τ ) , \mathbf{A}=\begin{pmatrix}a&b&c\\ x&y&z\end{pmatrix}\,,\quad\mathbf{B}=\begin{pmatrix}\alpha&\rho\\ \beta&\sigma\\ \gamma&\tau\\ \end{pmatrix}\,,
  96. 𝐀𝐁 = ( a b c x y z ) ( α ρ β σ γ τ ) = ( a α + b β + c γ a ρ + b σ + c τ x α + y β + z γ x ρ + y σ + z τ ) , \mathbf{A}\mathbf{B}=\begin{pmatrix}a&b&c\\ x&y&z\end{pmatrix}\begin{pmatrix}\alpha&\rho\\ \beta&\sigma\\ \gamma&\tau\\ \end{pmatrix}=\begin{pmatrix}a\alpha+b\beta+c\gamma&a\rho+b\sigma+c\tau\\ x\alpha+y\beta+z\gamma&x\rho+y\sigma+z\tau\\ \end{pmatrix}\,,
  97. 𝐁𝐀 = ( α ρ β σ γ τ ) ( a b c x y z ) = ( α a + ρ x α b + ρ y α c + ρ z β a + σ x β b + σ y β c + σ z γ a + τ x γ b + τ y γ c + τ z ) . \mathbf{B}\mathbf{A}=\begin{pmatrix}\alpha&\rho\\ \beta&\sigma\\ \gamma&\tau\\ \end{pmatrix}\begin{pmatrix}a&b&c\\ x&y&z\end{pmatrix}=\begin{pmatrix}\alpha a+\rho x&\alpha b+\rho y&\alpha c+% \rho z\\ \beta a+\sigma x&\beta b+\sigma y&\beta c+\sigma z\\ \gamma a+\tau x&\gamma b+\tau y&\gamma c+\tau z\end{pmatrix}\,.
  98. n n
  99. i = 1 n 𝐀 i = 𝐀 1 𝐀 2 𝐀 n . \prod_{i=1}^{n}\mathbf{A}_{i}=\mathbf{A}_{1}\mathbf{A}_{2}\cdots\mathbf{A}_{n}\,.
  100. ( 𝐀 1 𝐀 2 𝐀 n ) i 0 i n = i 1 = 1 s 1 i 2 = 1 s 2 i n - 1 = 1 s n - 1 ( 𝐀 1 ) i 0 i 1 ( 𝐀 2 ) i 1 i 2 ( 𝐀 3 ) i 2 i 3 ( 𝐀 n - 1 ) i n - 2 i n - 1 ( 𝐀 n ) i n - 1 i n \left(\mathbf{A}_{1}\mathbf{A}_{2}\cdots\mathbf{A}_{n}\right)_{i_{0}i_{n}}=% \sum_{i_{1}=1}^{s_{1}}\sum_{i_{2}=1}^{s_{2}}\cdots\sum_{i_{n-1}=1}^{s_{n-1}}% \left(\mathbf{A}_{1}\right)_{i_{0}i_{1}}\left(\mathbf{A}_{2}\right)_{i_{1}i_{2% }}\left(\mathbf{A}_{3}\right)_{i_{2}i_{3}}\cdots\left(\mathbf{A}_{n-1}\right)_% {i_{n-2}i_{n-1}}\left(\mathbf{A}_{n}\right)_{i_{n-1}i_{n}}
  101. 𝐁 = 𝐏 - 1 𝐀𝐏 \mathbf{B}=\mathbf{P}^{-1}\mathbf{A}\mathbf{P}
  102. 𝐏 \mathbf{P}
  103. 𝐀 \mathbf{A}
  104. 𝐁 \mathbf{B}
  105. n × n n×n
  106. 𝐀 \mathbf{A}
  107. k k
  108. 𝐀 k = 𝐀𝐀 𝐀 k times \mathbf{A}^{k}=\underset{k\mathrm{\,times}}{\mathbf{A}\mathbf{A}\cdots\mathbf{% A}}
  109. λ λ
  110. k k
  111. 𝐀 \mathbf{A}
  112. 𝐀 \mathbf{A}
  113. k k
  114. 𝐀 \mathbf{A}
  115. 𝐀 k = ( A 11 0 0 0 A 22 0 0 0 A n n ) k = ( A 11 k 0 0 0 A 22 k 0 0 0 A n n k ) \mathbf{A}^{k}=\begin{pmatrix}A_{11}&0&\cdots&0\\ 0&A_{22}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&A_{nn}\end{pmatrix}^{k}=\begin{pmatrix}A_{11}^{k}&0&\cdots&0\\ 0&A_{22}^{k}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&A_{nn}^{k}\end{pmatrix}
  116. U , V U,V
  117. W W
  118. S : V W S:V→W
  119. T : U V T:U→V
  120. S T : U W ST:U→W
  121. 𝐀 , 𝐁 \mathbf{A},\mathbf{B}
  122. 𝐂 \mathbf{C}
  123. S , T S,T
  124. S T ST
  125. 𝐀𝐁 = 𝐂 \mathbf{AB}=\mathbf{C}
  126. 𝐚 \mathbf{a}
  127. 𝐛 \mathbf{b}
  128. 𝐚 𝐛 \displaystyle\mathbf{a}\cdot\mathbf{b}
  129. n × m n×m
  130. m × p m×p
  131. 𝐁 \mathbf{B}
  132. 𝐀 = ( A 11 A 12 A 1 m A 21 A 22 A 2 m A n 1 A n 2 A n m ) = ( 𝐚 1 𝐚 2 𝐚 n ) , \mathbf{A}=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1m}\\ A_{21}&A_{22}&\cdots&A_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nm}\end{pmatrix}=\begin{pmatrix}\mathbf{a}_{1}\\ \mathbf{a}_{2}\\ \vdots\\ \mathbf{a}_{n}\end{pmatrix},
  133. 𝐁 = ( B 11 B 12 B 1 p B 21 B 22 B 2 p B m 1 B m 2 B m p ) = ( 𝐛 1 𝐛 2 𝐛 p ) \mathbf{B}=\begin{pmatrix}B_{11}&B_{12}&\cdots&B_{1p}\\ B_{21}&B_{22}&\cdots&B_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ B_{m1}&B_{m2}&\cdots&B_{mp}\end{pmatrix}=\begin{pmatrix}\mathbf{b}_{1}&\mathbf% {b}_{2}&\cdots&\mathbf{b}_{p}\end{pmatrix}
  134. 𝐚 i = ( A i 1 A i 2 A i m ) , 𝐛 i = ( B 1 i B 2 i B m i ) \mathbf{a}_{i}=\begin{pmatrix}A_{i1}&A_{i2}&\cdots&A_{im}\end{pmatrix}\,,\quad% \mathbf{b}_{i}=\begin{pmatrix}B_{1i}\\ B_{2i}\\ \vdots\\ B_{mi}\end{pmatrix}
  135. 𝐀𝐁 = ( 𝐚 1 𝐚 2 𝐚 n ) ( 𝐛 1 𝐛 2 𝐛 p ) = ( ( 𝐚 1 𝐛 1 ) ( 𝐚 1 𝐛 2 ) ( 𝐚 1 𝐛 p ) ( 𝐚 2 𝐛 1 ) ( 𝐚 2 𝐛 2 ) ( 𝐚 2 𝐛 p ) ( 𝐚 n 𝐛 1 ) ( 𝐚 n 𝐛 2 ) ( 𝐚 n 𝐛 p ) ) \mathbf{AB}=\begin{pmatrix}\mathbf{a}_{1}\\ \mathbf{a}_{2}\\ \vdots\\ \mathbf{a}_{n}\end{pmatrix}\begin{pmatrix}\mathbf{b}_{1}&\mathbf{b}_{2}&\dots&% \mathbf{b}_{p}\end{pmatrix}=\begin{pmatrix}(\mathbf{a}_{1}\cdot\mathbf{b}_{1})% &(\mathbf{a}_{1}\cdot\mathbf{b}_{2})&\dots&(\mathbf{a}_{1}\cdot\mathbf{b}_{p})% \\ (\mathbf{a}_{2}\cdot\mathbf{b}_{1})&(\mathbf{a}_{2}\cdot\mathbf{b}_{2})&\dots&% (\mathbf{a}_{2}\cdot\mathbf{b}_{p})\\ \vdots&\vdots&\ddots&\vdots\\ (\mathbf{a}_{n}\cdot\mathbf{b}_{1})&(\mathbf{a}_{n}\cdot\mathbf{b}_{2})&\dots&% (\mathbf{a}_{n}\cdot\mathbf{b}_{p})\end{pmatrix}
  136. 𝐀𝐁 = ( 𝐀𝐛 1 𝐀𝐛 2 𝐀𝐛 p ) = ( 𝐚 1 𝐁 𝐚 2 𝐁 𝐚 n 𝐁 ) \mathbf{AB}=\begin{pmatrix}\mathbf{A}\mathbf{b}_{1}&\mathbf{A}\mathbf{b}_{2}&% \dots&\mathbf{A}\mathbf{b}_{p}\end{pmatrix}=\begin{pmatrix}\mathbf{a}_{1}% \mathbf{B}\\ \mathbf{a}_{2}\mathbf{B}\\ \vdots\\ \mathbf{a}_{n}\mathbf{B}\end{pmatrix}
  137. 𝐚 𝐛 \displaystyle\mathbf{a}\otimes\mathbf{b}
  138. 𝐀 \mathbf{A}
  139. 𝐁 \mathbf{B}
  140. 𝐀𝐁 \displaystyle\mathbf{AB}
  141. 𝐚 ¯ i = ( A 1 i A 2 i A n i ) , 𝐛 ¯ i = ( B i 1 B i 2 B i p ) . \mathbf{\bar{a}}_{i}=\begin{pmatrix}A_{1i}\\ A_{2i}\\ \vdots\\ A_{ni}\end{pmatrix}\,,\quad\mathbf{\bar{b}}_{i}=\begin{pmatrix}B_{i1}&B_{i2}&% \cdots&B_{ip}\end{pmatrix}\,.
  142. ( \color B r o w n 1 \color O r a n g e 2 \color V i o l e t 3 \color B r o w n 4 \color O r a n g e 5 \color V i o l e t 6 \color B r o w n 7 \color O r a n g e 8 \color V i o l e t 9 ) ( \color B r o w n a \color B r o w n d \color O r a n g e b \color O r a n g e e \color V i o l e t c \color V i o l e t f ) = ( \color B r o w n 1 \color B r o w n 4 \color B r o w n 7 ) ( \color B r o w n a \color B r o w n d ) + ( \color O r a n g e 2 \color O r a n g e 5 \color O r a n g e 8 ) ( \color O r a n g e b \color O r a n g e e ) + ( \color V i o l e t 3 \color V i o l e t 6 \color V i o l e t 9 ) ( \color V i o l e t c \color V i o l e t f ) = ( \color B r o w n 1 a \color B r o w n 1 d \color B r o w n 4 a \color B r o w n 4 d \color B r o w n 7 a \color B r o w n 7 d ) + ( \color O r a n g e 2 b \color O r a n g e 2 e \color O r a n g e 5 b \color O r a n g e 5 e \color O r a n g e 8 b \color O r a n g e 8 e ) + ( \color V i o l e t 3 c \color V i o l e t 3 f \color V i o l e t 6 c \color V i o l e t 6 f \color V i o l e t 9 c \color V i o l e t 9 f ) = ( \color B r o w n 1 a + \color O r a n g e 2 b + \color V i o l e t 3 c \color B r o w n 1 d + \color O r a n g e 2 e + \color V i o l e t 3 f \color B r o w n 4 a + \color O r a n g e 5 b + \color V i o l e t 6 c \color B r o w n 4 d + \color O r a n g e 5 e + \color V i o l e t 6 f \color B r o w n 7 a + \color O r a n g e 8 b + \color V i o l e t 9 c \color B r o w n 7 d + \color O r a n g e 8 e + \color V i o l e t 9 f ) . \begin{aligned}\displaystyle\begin{pmatrix}{\color{Brown}1}&{\color{Orange}2}&% {\color{Violet}3}\\ {\color{Brown}4}&{\color{Orange}5}&{\color{Violet}6}\\ {\color{Brown}7}&{\color{Orange}8}&{\color{Violet}9}\\ \end{pmatrix}\begin{pmatrix}{\color{Brown}a}&{\color{Brown}d}\\ {\color{Orange}b}&{\color{Orange}e}\\ {\color{Violet}c}&{\color{Violet}f}\\ \end{pmatrix}&\displaystyle=\begin{pmatrix}{\color{Brown}1}\\ {\color{Brown}4}\\ {\color{Brown}7}\\ \end{pmatrix}\begin{pmatrix}{\color{Brown}{a}}&{\color{Brown}{d}}\\ \end{pmatrix}+\begin{pmatrix}{\color{Orange}2}\\ {\color{Orange}5}\\ {\color{Orange}8}\\ \end{pmatrix}\begin{pmatrix}{\color{Orange}{b}}&{\color{Orange}{e}}\\ \end{pmatrix}+\begin{pmatrix}{\color{Violet}3}\\ {\color{Violet}6}\\ {\color{Violet}9}\\ \end{pmatrix}\begin{pmatrix}{\color{Violet}c}&{\color{Violet}f}\\ \end{pmatrix}\\ &\displaystyle=\begin{pmatrix}{\color{Brown}{1a}}&{\color{Brown}{1d}}\\ {\color{Brown}{4a}}&{\color{Brown}{4d}}\\ {\color{Brown}{7a}}&{\color{Brown}{7d}}\\ \end{pmatrix}+\begin{pmatrix}{\color{Orange}{2b}}&{\color{Orange}{2e}}\\ {\color{Orange}{5b}}&{\color{Orange}{5e}}\\ {\color{Orange}{8b}}&{\color{Orange}{8e}}\\ \end{pmatrix}+\begin{pmatrix}{\color{Violet}{3c}}&{\color{Violet}{3f}}\\ {\color{Violet}{6c}}&{\color{Violet}{6f}}\\ {\color{Violet}{9c}}&{\color{Violet}{9f}}\\ \end{pmatrix}\\ &\displaystyle=\begin{pmatrix}{\color{Brown}{1a}}+{\color{Orange}{2b}}+{\color% {Violet}{3c}}&{\color{Brown}{1d}}+{\color{Orange}{2e}}+{\color{Violet}{3f}}\\ {\color{Brown}{4a}}+{\color{Orange}{5b}}+{\color{Violet}{6c}}&{\color{Brown}{4% d}}+{\color{Orange}{5e}}+{\color{Violet}{6f}}\\ {\color{Brown}{7a}}+{\color{Orange}{8b}}+{\color{Violet}{9c}}&{\color{Brown}{7% d}}+{\color{Orange}{8e}}+{\color{Violet}{9f}}\\ \end{pmatrix}.\end{aligned}
  143. m × p m×p
  144. p × n p×n
  145. O ( m n p ) O(mnp)
  146. 2 × 2 2×2
  147. O ( n log 2 7 ) O ( n 2.807 ) O(n^{\log_{2}7})\approx O(n^{2.807})
  148. k k
  149. k × k k×k
  150. n × n n×n
  151. 𝐀 , 𝐁 , 𝐂 \mathbf{A},\mathbf{B},\mathbf{C}
  152. 𝐀𝐁 = 𝐂 \mathbf{AB}=\mathbf{C}
  153. 𝐂 \mathbf{C}
  154. 𝐀 \mathbf{A}
  155. 𝐁 \mathbf{B}
  156. 𝐂 = ( 𝐂 11 𝐂 12 𝐂 21 𝐂 22 ) = ( 𝐀 11 𝐀 12 𝐀 21 𝐀 22 ) ( 𝐁 11 𝐁 12 𝐁 21 𝐁 22 ) = 𝐀𝐁 \mathbf{C}=\begin{pmatrix}\mathbf{C}_{11}&\mathbf{C}_{12}\\ \mathbf{C}_{21}&\mathbf{C}_{22}\\ \end{pmatrix}=\begin{pmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\\ \end{pmatrix}\begin{pmatrix}\mathbf{B}_{11}&\mathbf{B}_{12}\\ \mathbf{B}_{21}&\mathbf{B}_{22}\\ \end{pmatrix}=\mathbf{A}\mathbf{B}
  157. 𝐀 \mathbf{A}
  158. 𝐁 \mathbf{B}
  159. 𝐂 \mathbf{C}
  160. n n
  161. n n
  162. n / 2 n/2
  163. n / 2 n/2
  164. ( 𝐀 11 𝐀 12 𝐀 21 𝐀 22 ) ( 𝐁 11 𝐁 12 𝐁 21 𝐁 22 ) = ( 𝐀 11 𝐁 11 + 𝐀 12 𝐁 21 𝐀 11 𝐁 12 + 𝐀 12 𝐁 22 𝐀 21 𝐁 11 + 𝐀 22 𝐁 21 𝐀 21 𝐁 12 + 𝐀 22 𝐁 22 ) \begin{pmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\\ \end{pmatrix}\begin{pmatrix}\mathbf{B}_{11}&\mathbf{B}_{12}\\ \mathbf{B}_{21}&\mathbf{B}_{22}\\ \end{pmatrix}=\begin{pmatrix}\mathbf{A}_{11}\mathbf{B}_{11}+\mathbf{A}_{12}% \mathbf{B}_{21}&\mathbf{A}_{11}\mathbf{B}_{12}+\mathbf{A}_{12}\mathbf{B}_{22}% \\ \mathbf{A}_{21}\mathbf{B}_{11}+\mathbf{A}_{22}\mathbf{B}_{21}&\mathbf{A}_{21}% \mathbf{B}_{12}+\mathbf{A}_{22}\mathbf{B}_{22}\\ \end{pmatrix}
  165. M / 3 \sqrt{M}{/3}
  166. M / 3 \sqrt{M}{/3}
  167. M M
  168. p p
  169. p \sqrt{p}
  170. p \sqrt{p}
  171. 𝐀 \mathbf{A}
  172. 𝐁 \mathbf{B}
  173. 𝐀 𝐁 \mathbf{A}○\mathbf{B}
  174. i , j i,j
  175. 𝐀 \mathbf{A}
  176. i , j i,j
  177. 𝐁 \mathbf{B}
  178. ( 𝐀 𝐁 ) i j = A i j B i j , \left(\mathbf{A}\circ\mathbf{B}\right)_{ij}=A_{ij}B_{ij}\,,
  179. 𝐀 𝐁 = ( A 11 A 12 A 1 m A 21 A 22 A 2 m A n 1 A n 2 A n m ) ( B 11 B 12 B 1 m B 21 B 22 B 2 m B n 1 B n 2 B n m ) = ( A 11 B 11 A 12 B 12 A 1 m B 1 m A 21 B 21 A 22 B 22 A 2 m B 2 m A n 1 B n 1 A n 2 B n 2 A n m B n m ) \mathbf{A}\circ\mathbf{B}=\begin{pmatrix}A_{11}&A_{12}&\cdots&A_{1m}\\ A_{21}&A_{22}&\cdots&A_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}&A_{n2}&\cdots&A_{nm}\\ \end{pmatrix}\circ\begin{pmatrix}B_{11}&B_{12}&\cdots&B_{1m}\\ B_{21}&B_{22}&\cdots&B_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ B_{n1}&B_{n2}&\cdots&B_{nm}\\ \end{pmatrix}=\begin{pmatrix}A_{11}B_{11}&A_{12}B_{12}&\cdots&A_{1m}B_{1m}\\ A_{21}B_{21}&A_{22}B_{22}&\cdots&A_{2m}B_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ A_{n1}B_{n1}&A_{n2}B_{n2}&\cdots&A_{nm}B_{nm}\\ \end{pmatrix}
  180. m n mn
  181. 𝐀 : 𝐁 \mathbf{A}:\mathbf{B}
  182. 𝐀 : 𝐁 = i , j A i j B i j = vec ( 𝐀 ) 𝖳 vec ( 𝐁 ) = tr ( 𝐀 𝖳 𝐁 ) = tr ( 𝐀𝐁 𝖳 ) , \mathbf{A}:\mathbf{B}=\sum_{i,j}A_{ij}B_{ij}=\mathrm{vec}(\mathbf{A})^{\mathsf% {T}}\mathrm{vec}(\mathbf{B})=\mathrm{tr}(\mathbf{A}^{\mathsf{T}}\mathbf{B})=% \mathrm{tr}(\mathbf{A}\mathbf{B}^{\mathsf{T}}),
  183. 𝐀 \mathbf{A}
  184. 𝐁 \mathbf{B}
  185. m × n m×n
  186. p × q p×q
  187. 𝐀 𝐁 = ( A 11 𝐁 A 12 𝐁 A 1 n 𝐁 A 21 𝐁 A 22 𝐁 A 2 n 𝐁 A m 1 𝐁 A m 2 𝐁 A m n 𝐁 ) . \mathbf{A}\otimes\mathbf{B}=\begin{pmatrix}A_{11}\mathbf{B}&A_{12}\mathbf{B}&% \cdots&A_{1n}\mathbf{B}\\ A_{21}\mathbf{B}&A_{22}\mathbf{B}&\cdots&A_{2n}\mathbf{B}\\ \vdots&\vdots&\ddots&\vdots\\ A_{m1}\mathbf{B}&A_{m2}\mathbf{B}&\cdots&A_{mn}\mathbf{B}\\ \end{pmatrix}.
  188. m p × n q mp×nq
  189. O ( n < s u p > 2.376 ) O(n<sup>2.376)

Matrix_representation_of_conic_sections.html

  1. Q = def A x 2 + B x y + C y 2 + D x + E y + F = 0. Q\ \stackrel{\mathrm{def}}{=}\ Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.\,
  2. 𝐱 T A Q 𝐱 = 0 \mathbf{x}^{T}A_{Q}\mathbf{x}=0
  3. 𝐱 \mathbf{x}
  4. ( x y 1 ) \begin{pmatrix}x\\ y\\ 1\end{pmatrix}
  5. A Q A_{Q}
  6. A Q = ( A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ) . A_{Q}=\begin{pmatrix}A&B/2&D/2\\ B/2&C&E/2\\ D/2&E/2&F\end{pmatrix}.
  7. det A Q = 0 \det A_{Q}=0
  8. det A 33 \det A_{33}
  9. A 33 = [ A B / 2 B / 2 C ] . A_{33}=\begin{bmatrix}A&B/2\\ B/2&C\end{bmatrix}.
  10. det A 33 < 0 \det A_{33}<0
  11. det A 33 = 0 \det A_{33}=0
  12. det A 33 > 0 \det A_{33}>0
  13. A = C A=C
  14. B = 0 B=0
  15. det A 33 > 0 \det A_{33}>0
  16. det A Q 0 \det A_{Q}\neq 0
  17. ( A + C ) det A Q < 0 (A+C)\det A_{Q}<0
  18. ( A + C ) det A Q > 0 (A+C)\det A_{Q}>0
  19. x 2 + y 2 + 10 = 0 x^{2}+y^{2}+10=0
  20. det A Q = 0 \det A_{Q}=0
  21. det A 33 \det A_{33}
  22. det A 33 < 0 \det A_{33}<0
  23. det A 33 = 0 \det A_{33}=0
  24. D 2 + E 2 > 4 ( A + C ) F D^{2}+E^{2}>4(A+C)F
  25. D 2 + E 2 = 4 ( A + C ) F D^{2}+E^{2}=4(A+C)F
  26. D 2 + E 2 < 4 ( A + C ) F D^{2}+E^{2}<4(A+C)F
  27. det A 33 > 0 \det A_{33}>0
  28. Q Q
  29. Q = [ Q x , Q y ] = [ 0 , 0 ] . \nabla Q=[\frac{\partial Q}{\partial x},\frac{\partial Q}{\partial y}]=[0,0].
  30. A Q A_{Q}
  31. S = def { a 11 + a 12 x + a 13 y = 0 a 21 + a 22 x + a 23 y = 0 = def { D / 2 + A x + ( B / 2 ) y = 0 E / 2 + ( B / 2 ) x + C y = 0 S\ \stackrel{\mathrm{def}}{=}\ \left\{\begin{matrix}a_{11}+a_{12}x+a_{13}y&=&0% \\ a_{21}+a_{22}x+a_{23}y&=&0\end{matrix}\right.\ \stackrel{\mathrm{def}}{=}\ % \left\{\begin{matrix}D/2+Ax+(B/2)y&=&0\\ E/2+(B/2)x+Cy&=&0\end{matrix}\right.
  32. ( x c y c ) = ( A B / 2 B / 2 C ) - 1 ( - D / 2 - E / 2 ) = ( ( B E - 2 C D ) / ( 4 A C - B 2 ) ( D B - 2 A E ) / ( 4 A C - B 2 ) ) \begin{pmatrix}x_{c}\\ y_{c}\end{pmatrix}=\begin{pmatrix}A&B/2\\ B/2&C\end{pmatrix}^{-1}\begin{pmatrix}-D/2\\ -E/2\end{pmatrix}=\begin{pmatrix}(BE-2CD)/(4AC-B^{2})\\ (DB-2AE)/(4AC-B^{2})\end{pmatrix}
  33. a 1 , 2 = def { S ( x 0 , y 0 ) (center of the conic) u ( u x , u y ) (eigenvector of A 33 ) a_{1,2}\ \stackrel{\mathrm{def}}{=}\ \left\{\begin{matrix}S(x_{0},y_{0})&% \qquad\mbox{(center of the conic)}\\ \vec{u}(u_{x},u_{y})&\qquad\mbox{(eigenvector of }~{}A_{33})\end{matrix}\right.
  34. a 1 , 2 = def x - x 0 u x = y - y 0 u y a_{1,2}\ \stackrel{\mathrm{def}}{=}\ \frac{x-x_{0}}{u_{x}}=\frac{y-y_{0}}{u_{y}}
  35. V = def { e (axis) Q (the general equation of the conic) V\ \stackrel{\mathrm{def}}{=}\ \left\{\begin{matrix}&e&\qquad\mbox{(axis)}\\ &Q&\qquad\mbox{(the general equation of the conic)}\end{matrix}\right.
  36. 𝐩 T A Q 𝐱 = 0 \mathbf{p}^{T}A_{Q}\mathbf{x}=0
  37. λ 1 \lambda_{1}
  38. λ 2 \lambda_{2}
  39. λ 1 x 2 + λ 2 y 2 + det A Q det A 33 = 0 \lambda_{1}x^{\prime 2}+\lambda_{2}y^{\prime 2}+\frac{\det A_{Q}}{\det A_{33}}=0
  40. - det A Q det A 33 -\frac{\det A_{Q}}{\det A_{33}}
  41. x 2 a 2 + y 2 b 2 = 1. \frac{{x^{\prime}}^{2}}{a^{2}}+\frac{{y^{\prime}}^{2}}{b^{2}}=1.
  42. T : R S ( O , X , Y ) ( O = S , X , Y ) = def { t = O O = S α = arccos a 1 ( 1 0 ) | a 1 | T:RS(O,X,Y)\mapsto(O^{\prime}=S,X^{\prime},Y^{\prime})\ \stackrel{\mathrm{def}% }{=}\ \left\{\begin{aligned}\displaystyle\vec{t}&\displaystyle=\overrightarrow% {OO^{\prime}}=S\\ \displaystyle\alpha&\displaystyle=\operatorname{arccos}\frac{\vec{a}_{1}\cdot{% 1\choose 0}}{|\vec{a}_{1}|}\end{aligned}\right.

Matroid.html

  1. M M
  2. ( E , ) (E,\mathcal{I})
  3. E E
  4. \mathcal{I}
  5. E E
  6. \emptyset\in\mathcal{I}
  7. E E
  8. \mathcal{I}\neq\emptyset
  9. A A E A^{\prime}\subset A\subset E
  10. A A\in\mathcal{I}
  11. A A^{\prime}\in\mathcal{I}
  12. A A
  13. B B
  14. \mathcal{I}
  15. A A
  16. B B
  17. A A
  18. B B
  19. B B
  20. E E
  21. E E
  22. M M
  23. E E
  24. M M
  25. ( E , ) (E,\mathcal{B})
  26. E E
  27. \mathcal{B}
  28. E E
  29. \mathcal{B}
  30. A A
  31. B B
  32. \mathcal{B}
  33. a A B a\in A\setminus B
  34. b B A b\in B\setminus A
  35. A { a } { b } A\setminus\{a\}\cup\{b\}\in\mathcal{B}
  36. \mathcal{B}
  37. M M
  38. M M
  39. M M
  40. E E
  41. A A
  42. E E
  43. A A
  44. A A
  45. M M
  46. E E
  47. A A
  48. E E
  49. r ( A ) | A | r(A)\leq|A|
  50. A A
  51. B B
  52. E E
  53. r ( A B ) + r ( A B ) r ( A ) + r ( B ) r(A\cup B)+r(A\cap B)\leq r(A)+r(B)
  54. A A
  55. x x
  56. r ( A ) r ( A { x } ) r ( A ) + 1 r(A)\leq r(A\cup\{x\})\leq r(A)+1
  57. A B E A\subset B\subset E
  58. r ( A ) r ( B ) r ( E ) r(A)\leq r(B)\leq r(E)
  59. ( E , r ) (E,r)
  60. E E
  61. A A
  62. E E
  63. r ( A ) = | A | r(A)=|A|
  64. | A | - r ( A ) |A|-r(A)
  65. A A
  66. A A
  67. E E
  68. M M
  69. M M
  70. M M
  71. E E
  72. r r
  73. cl ( A ) \operatorname{cl}(A)
  74. A A
  75. E E
  76. cl ( A ) = { x E r ( A ) = r ( A { x } ) } \operatorname{cl}(A)=\Bigl\{x\in E\mid r(A)=r\bigl(A\cup\{x\}\bigr)\Bigr\}
  77. cl : 𝒫 ( E ) 𝒫 ( E ) \operatorname{cl}:\mathcal{P}(E)\to\mathcal{P}(E)
  78. 𝒫 \mathcal{P}
  79. X X
  80. E E
  81. X cl ( X ) X\subseteq\operatorname{cl}(X)
  82. X X
  83. E E
  84. cl ( X ) = cl ( cl ( X ) ) \operatorname{cl}(X)=\operatorname{cl}(\operatorname{cl}(X))
  85. X X
  86. Y Y
  87. E E
  88. X Y X\subseteq Y
  89. cl ( X ) cl ( Y ) \operatorname{cl}(X)\subseteq\operatorname{cl}(Y)
  90. a a
  91. b b
  92. E E
  93. Y Y
  94. E E
  95. a cl ( Y { b } ) cl ( Y ) a\in\operatorname{cl}(Y\cup\{b\})\setminus\operatorname{cl}(Y)
  96. b cl ( Y { a } ) cl ( Y ) b\in\operatorname{cl}(Y\cup\{a\})\setminus\operatorname{cl}(Y)
  97. cl : 𝒫 ( E ) 𝒫 ( E ) \operatorname{cl}:\mathcal{P}(E)\to\mathcal{P}(E)
  98. E E
  99. S S
  100. T T
  101. S T S\cap T
  102. S S
  103. T T
  104. S S
  105. T T
  106. S S
  107. U U
  108. S S
  109. T T
  110. E S E\setminus S
  111. ( M ) \mathcal{L}(M)
  112. L L
  113. E E
  114. S S
  115. S \bigvee S
  116. cl ( S ) = { x E x S } \operatorname{cl}(S)=\{x\in E\mid x\leq\bigvee S\}
  117. y y
  118. { x E x y } \{x\in E\mid x\leq y\}
  119. L L
  120. r r
  121. r - 1 r-1
  122. E E
  123. \mathcal{H}
  124. X X
  125. Y Y
  126. \mathcal{H}
  127. X Y X\subset Y
  128. x E x\in E
  129. Y , Z Y,Z\in\mathcal{H}
  130. x Y Z x\notin Y\cup Z
  131. X X\in\mathcal{H}
  132. ( Y Z ) { x } X (Y\cap Z)\cup\{x\}\subseteq X
  133. U k , n U_{k,n}
  134. U 0 , n U_{0,n}
  135. U n , n U_{n,n}
  136. M * ( G ) = ( E , I ) M^{*}(G)=(E,I)
  137. r * ( S ) = | S | - r ( M ) + r ( E S ) r^{*}(S)=|S|-r(M)+r\left(E\setminus S\right)
  138. r ( A ) = r ( A T ) - r ( T ) . r^{\prime}(A)=r(A\cup T)-r(T).
  139. r ( S ) + r ( E - S ) = r ( M ) r(S)+r(E-S)=r(M)
  140. P M P_{M}
  141. M M
  142. p M ( λ ) := S E ( - 1 ) | S | λ r ( M ) - r ( S ) , p_{M}(\lambda):=\sum_{S\subseteq E}(-1)^{|S|}\lambda^{r(M)-r(S)},
  143. p M ( λ ) := A μ ( , A ) λ r ( M ) - r ( A ) , p_{M}(\lambda):=\sum_{A}\mu(\emptyset,A)\lambda^{r(M)-r(A)}\ ,
  144. β ( M ) = ( - 1 ) r ( M ) - 1 p M ( 1 ) \beta(M)=(-1)^{r(M)-1}p_{M}^{\prime}(1)
  145. β ( M ) = ( - 1 ) r ( M ) X E ( - 1 ) | X | r ( X ) . \beta(M)=(-1)^{r(M)}\sum_{X\subseteq E}(-1)^{|X|}r(X)\ .
  146. T M * ( x , y ) = T M ( y , x ) , T_{M^{*}}(x,y)=T_{M}(y,x),
  147. T M ( x , y ) = S E ( x - 1 ) r ( M ) - r ( S ) ( y - 1 ) | S | - r ( S ) . T_{M}(x,y)=\sum_{S\subseteq E}(x-1)^{r(M)-r(S)}(y-1)^{|S|-r(S)}.
  148. R M ( u , v ) = S E u r ( M ) - r ( S ) v | S | - r ( S ) . R_{M}(u,v)=\sum_{S\subseteq E}u^{r(M)-r(S)}v^{|S|-r(S)}.
  149. p M ( λ ) = ( - 1 ) r ( M ) T M ( 1 - λ , 0 ) . p_{M}(\lambda)=(-1)^{r(M)}T_{M}(1-\lambda,0).
  150. F ( M ) = F ( M - e ) + F ( M / e ) F(M)=F(M-e)+F(M/e)
  151. e e
  152. F ( M M ) = F ( M ) F ( M ) F(M\oplus M^{\prime})=F(M)F(M^{\prime})
  153. x c l ( Y ) ( Y Y ) Y is finite and x c l ( Y ) . x\in cl(Y)\Leftrightarrow(\exists Y^{\prime}\subseteq Y)Y^{\prime}\,\text{ is % finite and }x\in cl(Y^{\prime}).
  154. x J I x\in J\setminus I
  155. I { x } I\cup\{x\}

Matter_wave.html

  1. λ = h p . \lambda=\frac{h}{p}.
  2. E = h ν E=h\nu
  3. p = E c = h λ p=\frac{E}{c}=\frac{h}{\lambda}
  4. ν \scriptstyle\nu
  5. λ \scriptstyle\lambda
  6. λ λ
  7. p p
  8. h h
  9. p = h λ . p=\frac{h}{\lambda}.
  10. λ λ
  11. p p
  12. f f
  13. E E
  14. \hbar
  15. 𝐤 \mathbf{k}
  16. ω \omega
  17. p = γ m 0 v p=\gamma m_{0}v
  18. E = m c 2 = γ m 0 c 2 E=mc^{2}=\gamma m_{0}c^{2}
  19. λ = h γ m 0 v = h m 0 v 1 - v 2 c 2 f = γ m 0 c 2 h = m 0 c 2 h / 1 - v 2 c 2 \begin{aligned}&\displaystyle\lambda=\,\,\frac{h}{\gamma m_{0}v}\,=\,\frac{h}{% m_{0}v}\,\,\,\,\sqrt{1-\frac{v^{2}}{c^{2}}}\\ &\displaystyle f=\frac{\gamma\,m_{0}c^{2}}{h}=\frac{m_{0}c^{2}}{h}\bigg/\sqrt{% 1-\frac{v^{2}}{c^{2}}}\end{aligned}
  20. m 0 m_{0}
  21. v v
  22. γ \gamma
  23. c c
  24. v g = ω k = ( E / ) ( p / ) = E p v_{g}=\frac{\partial\omega}{\partial k}=\frac{\partial(E/\hbar)}{\partial(p/% \hbar)}=\frac{\partial E}{\partial p}
  25. v g = E p = p ( 1 2 p 2 m ) = p m = v \begin{aligned}\displaystyle v_{g}&\displaystyle=\frac{\partial E}{\partial p}% =\frac{\partial}{\partial p}\left(\frac{1}{2}\frac{p^{2}}{m}\right)\\ &\displaystyle=\frac{p}{m}\\ &\displaystyle=v\end{aligned}
  26. v g = E p = p ( p 2 c 2 + m 0 2 c 4 ) = p c 2 p 2 c 2 + m 0 2 c 4 = p c 2 E \begin{aligned}\displaystyle v_{g}&\displaystyle=\frac{\partial E}{\partial p}% =\frac{\partial}{\partial p}\left(\sqrt{p^{2}c^{2}+m_{0}^{2}c^{4}}\right)\\ &\displaystyle=\frac{pc^{2}}{\sqrt{p^{2}c^{2}+m_{0}^{2}c^{4}}}\\ &\displaystyle=\frac{pc^{2}}{E}\end{aligned}
  27. m 0 m_{0}
  28. c c
  29. v p = E / p = c 2 / v v_{p}=E/p=c^{2}/v
  30. v g = p c 2 E = c 2 v p = v \begin{aligned}\displaystyle v_{g}&\displaystyle=\frac{pc^{2}}{E}\\ &\displaystyle=\frac{c^{2}}{v_{p}}\\ &\displaystyle=v\end{aligned}
  31. v p = ω k = E / p / = E p . v_{\mathrm{p}}=\frac{\omega}{k}=\frac{E/\hbar}{p/\hbar}=\frac{E}{p}.
  32. v p = E p = γ m 0 c 2 γ m 0 v = c 2 v = c β v_{\mathrm{p}}=\frac{E}{p}=\frac{\gamma m_{0}c^{2}}{\gamma m_{0}v}=\frac{c^{2}% }{v}=\frac{c}{\beta}
  33. γ \gamma
  34. v < c v<c
  35. v p > c , v_{\mathrm{p}}>c,\,
  36. v v
  37. f = ν 0 1 - v 2 c 2 . f=\nu_{0}\sqrt{1-\frac{v^{2}}{c^{2}}}\,.
  38. λ f = E / p = v p . \lambda f=E/p=v_{\mathrm{p}}\,.
  39. v p v_{\mathrm{p}}
  40. λ \lambda
  41. f f
  42. v p > c v_{\mathrm{p}}>c

Max-flow_min-cut_theorem.html

  1. N = ( V , E ) N=(V,E)
  2. s s
  3. t t
  4. N N
  5. c ( u , v ) c(u,v)
  6. f ( u , v ) f(u,v)
  7. ( u , v ) E : f u v c u v \forall(u,v)\in E:\qquad f_{uv}\leq c_{uv}
  8. v V { s , t } : { u : ( u , v ) E } f u v = { u : ( v , u ) E } f v u . \forall v\in V\setminus\{s,t\}:\qquad\sum\nolimits_{\{u:(u,v)\in E\}}f_{uv}=% \sum\nolimits_{\{u:(v,u)\in E\}}f_{vu}.
  9. | f | = v V f s v , |f|=\sum\nolimits_{v\in V}f_{sv},
  10. s s
  11. N N
  12. | f | |f|
  13. s s
  14. t t
  15. C = ( S , T ) C=(S,T)
  16. V V
  17. s S s∈S
  18. t T t∈T
  19. C C
  20. { ( u , v ) E : u S , v T } . \{(u,v)\in E\ :\ u\in S,v\in T\}.
  21. C C
  22. | f | = 0 |f|=0
  23. c ( S , T ) = ( u , v ) S × T c u v = ( i , j ) E c i j d i j , c(S,T)=\sum\nolimits_{(u,v)\in S\times T}c_{uv}=\sum\nolimits_{(i,j)\in E}c_{% ij}d_{ij},
  24. d i j = 1 d_{ij}=1
  25. i S i\in S
  26. j T j\in T
  27. c ( S , T ) c(S,T)
  28. S S
  29. T T
  30. | f | = s |f|=\nabla_{s}
  31. ( i , j ) E c i j d i j \sum_{(i,j)\in E}c_{ij}d_{ij}
  32. f i j c i j ( i , j ) E j : ( j , i ) E f j i - j : ( i , j ) E f i j 0 i V , i s , t s + j : ( j , s ) E f j s - j : ( s , j ) E f s j 0 - s + j : ( j , t ) E f j t - j : ( t , j ) E f t j 0 f i j 0 ( i , j ) E \begin{array}[]{rclr}f_{ij}&\leq&c_{ij}&(i,j)\in E\\ \sum_{j:(j,i)\in E}f_{ji}-\sum_{j:(i,j)\in E}f_{ij}&\leq&0&i\in V,i\neq s,t\\ \nabla_{s}+\sum_{j:(j,s)\in E}f_{js}-\sum_{j:(s,j)\in E}f_{sj}&\leq&0&\\ -\nabla_{s}+\sum_{j:(j,t)\in E}f_{jt}-\sum_{j:(t,j)\in E}f_{tj}&\leq&0&\\ f_{ij}&\geq&0&(i,j)\in E\\ \end{array}
  33. d i j - p i + p j 0 ( i , j ) E p s - p t 1 p i 0 i V d i j 0 ( i , j ) E \begin{array}[]{rclr}d_{ij}-p_{i}+p_{j}&\geq&0&(i,j)\in E\\ p_{s}-p_{t}&\geq&1&\\ p_{i}&\geq&0&i\in V\\ d_{ij}&\geq&0&(i,j)\in E\end{array}
  34. C = ( S , T ) C=(S,T)
  35. i S i\in S
  36. p i = 1 p_{i}=1
  37. p s p_{s}
  38. p t p_{t}
  39. S S
  40. T T
  41. c ( v ) c(v)
  42. f f
  43. v V { s , t } : i V f i v c ( v ) . \forall v\in V\setminus\{s,t\}:\qquad\sum\nolimits_{i\in V}f_{iv}\leq c(v).
  44. G = ( V , E ) G=(V,E)
  45. s s
  46. t t
  47. G G
  48. n n
  49. m m
  50. P P
  51. Q Q
  52. max { g } = i r ( p i ) - p i P r ( p i ) - q j Q c ( q j ) . \max\{g\}=\sum_{i}r(p_{i})-\sum_{p_{i}\in P}r(p_{i})-\sum_{q_{j}\in Q}c(q_{j}).
  53. P P
  54. Q Q
  55. min { g } = p i P r ( p i ) + q j Q c ( q j ) . \min\{g^{\prime}\}=\sum_{p_{i}\in P}r(p_{i})+\sum_{q_{j}\in Q}c(q_{j}).
  56. P P
  57. Q Q
  58. r ( p < s u b > i ) r(p<sub>i)
  59. n n
  60. i i
  61. i , j i,j
  62. P P
  63. Q Q
  64. max { g } = i P f i + i Q b i - i P , j Q j P , i Q p i j . \max\{g\}=\sum_{i\in P}f_{i}+\sum_{i\in Q}b_{i}-\sum_{i\in P,j\in Qj\in P,i\in Q% }p_{ij}.
  65. min { g } = i P , j Q j P , i Q p i j . \min\{g^{\prime}\}=\sum_{i\in P,j\in Qj\in P,i\in Q}p_{ij}.
  66. i , j i,j
  67. j , i j,i
  68. P P
  69. Q Q
  70. G = ( V , E ) G=(V,E)
  71. s s
  72. t t
  73. G G
  74. f f
  75. G G
  76. G G
  77. A A
  78. s s
  79. V A V−A
  80. c ( S , T ) = ( u , v ) S × T c u v c(S,T)=\sum\nolimits_{(u,v)\in S\times T}c_{uv}
  81. v a l u e ( f ) = f o u t ( A ) - f i n ( A c ) value(f)=f_{out}(A)-f_{in}(A^{c})
  82. A A
  83. G G
  84. ( x , y ) , x A , y A c (x,y),x\in A,y\in A^{c}
  85. y y
  86. s s
  87. y y
  88. ( x , y ) (x,y)
  89. G G
  90. ( y , x ) , x A , y A c (y,x),x\in A,y\in A^{c}
  91. f ( x , y ) > 0 f(x,y)>0
  92. x x
  93. y y
  94. G < s u b > f G<sub>f

Maxima_and_minima.html

  1. x x \sqrt[x]{x}
  2. x x \sqrt[x]{x}
  3. f ( x , y ) = x 2 + y 2 ( 1 - x ) 3 , x , y , f(x,y)=x^{2}+y^{2}(1-x)^{3},\qquad x,y\in\mathbb{R},

Maximal_element.html

  1. ( P , ) (P,\leq)
  2. S P S\subset P
  3. m S m\in S
  4. S S
  5. s S s\in S
  6. m s m\leq s
  7. m = s . m=s.
  8. 2 \sqrt{2}
  9. m m
  10. s S s\in S
  11. max S s \max S\leq s
  12. s max S s\leq\max S
  13. s = max S s=\max S
  14. \leq
  15. S S
  16. m m
  17. s S s\in S
  18. s m s\leq m
  19. m s m\leq s
  20. \leq
  21. S S
  22. m S m\in S
  23. s S s\in S
  24. s m s\leq m
  25. m s m\leq s
  26. m = s m=s
  27. s m s\leq m
  28. m m
  29. S S
  30. X X
  31. x X x\in X
  32. \preceq
  33. x , y X x,y\in X
  34. x y x\preceq y
  35. x x
  36. y y
  37. x y x\preceq y
  38. y x y\preceq x
  39. x x
  40. y y
  41. x = y x=y
  42. B X B\subset X
  43. x B x\in B
  44. y B y\in B
  45. y x y\preceq x
  46. x y x\prec y
  47. x y x\preceq y
  48. y x y\preceq x
  49. \preceq
  50. x x
  51. x B x\in B
  52. y x y\preceq x
  53. x y x\preceq y
  54. y x y\preceq x
  55. x y x\preceq y
  56. x = y x=y
  57. x y x\sim y
  58. x B x\in B
  59. y B y\in B
  60. y x . y\prec x.
  61. P P
  62. X X
  63. p P p\in P
  64. x X x\in X
  65. p ( x ) + p(x)\in\mathbb{R}_{+}
  66. Γ : P × + X \Gamma\colon P\times\mathbb{R}_{+}\rightarrow X
  67. Γ ( p , m ) = { x X p ( x ) m } . \Gamma(p,m)=\{x\in X\mid p(x)\leq m\}.
  68. p p
  69. m m
  70. \preceq
  71. Γ ( p , m ) \Gamma(p,m)
  72. D ( p , m ) = { x X x D(p,m)=\big\{x\in X\mid x
  73. Γ ( p , m ) } \Gamma(p,m)\big\}
  74. p p
  75. m m
  76. x * x^{*}
  77. x * D ( p , m ) x^{*}\in D(p,m)
  78. Q Q
  79. P P
  80. x P x\in P
  81. y Q y\in Q
  82. x y x\leq y
  83. L L
  84. P P
  85. P P
  86. y L y\in L
  87. x y x\leq y
  88. x L x\in L
  89. L L
  90. P P
  91. L L

Maximum_likelihood.html

  1. θ ^ \scriptstyle\hat{\theta}
  2. f ( x 1 , x 2 , , x n θ ) = f ( x 1 | θ ) × f ( x 2 | θ ) × × f ( x n | θ ) . f(x_{1},x_{2},\ldots,x_{n}\mid\theta)=f(x_{1}|\theta)\times f(x_{2}|\theta)% \times\cdots\times f(x_{n}|\theta).
  3. ( θ ; x 1 , , x n ) = f ( x 1 , x 2 , , x n θ ) = i = 1 n f ( x i θ ) . \mathcal{L}(\theta\,;\,x_{1},\ldots,x_{n})=f(x_{1},x_{2},\ldots,x_{n}\mid% \theta)=\prod_{i=1}^{n}f(x_{i}\mid\theta).
  4. ; ;
  5. θ \theta
  6. x 1 , , x n x_{1},\ldots,x_{n}
  7. ln ( θ ; x 1 , , x n ) = i = 1 n ln f ( x i θ ) , \ln\mathcal{L}(\theta\,;\,x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\ln f(x_{i}\mid% \theta),
  8. ^ = 1 n ln . \hat{\ell}=\frac{1}{n}\ln\mathcal{L}.
  9. ^ \scriptstyle\hat{\ell}
  10. ^ ( θ ; x ) \scriptstyle\hat{\ell}(\theta;x)
  11. { θ ^ mle } { arg max θ Θ ^ ( θ ; x 1 , , x n ) } . \{\hat{\theta}_{\mathrm{mle}}\}\subseteq\{\underset{\theta\in\Theta}{% \operatorname{arg\,max}}\ \hat{\ell}(\theta\,;\,x_{1},\ldots,x_{n})\}.
  12. P ( θ x 1 , x 2 , , x n ) = f ( x 1 , x 2 , , x n θ ) P ( θ ) P ( x 1 , x 2 , , x n ) P(\theta\mid x_{1},x_{2},\ldots,x_{n})=\frac{f(x_{1},x_{2},\ldots,x_{n}\mid% \theta)P(\theta)}{P(x_{1},x_{2},\ldots,x_{n})}
  13. P ( θ ) P(\theta)
  14. P ( x 1 , x 2 , , x n ) P(x_{1},x_{2},\ldots,x_{n})
  15. f ( x 1 , x 2 , , x n θ ) P ( θ ) f(x_{1},x_{2},\ldots,x_{n}\mid\theta)P(\theta)
  16. P ( θ ) P(\theta)
  17. f ( x 1 , x 2 , , x n | θ ) f(x_{1},x_{2},\ldots,x_{n}|\theta)
  18. P ( θ ) P(\theta)
  19. ^ ( θ x ) = 1 n i = 1 n ln f ( x i θ ) , \hat{\ell}(\theta\mid x)=\frac{1}{n}\sum_{i=1}^{n}\ln f(x_{i}\mid\theta),
  20. ( θ ) = E [ ln f ( x i θ ) ] \ell(\theta)=\operatorname{E}[\,\ln f(x_{i}\mid\theta)\,]
  21. f ( θ 0 ) f(\cdot\mid\theta_{0})
  22. θ \theta
  23. θ ^ \scriptstyle\hat{\theta}
  24. θ ^ mle 𝑝 θ 0 . \hat{\theta}_{\mathrm{mle}}\ \xrightarrow{p}\ \theta_{0}.
  25. θ ^ mle a.s. θ 0 . \hat{\theta}_{\mathrm{mle}}\ \xrightarrow{\,\text{a.s.}}\ \theta_{0}.
  26. ^ ( θ x ) \scriptstyle\hat{\ell}(\theta\mid x)
  27. θ ^ \scriptstyle\hat{\theta}
  28. sup θ Θ ^ ( x θ ) - ( θ ) a.s. 0. \sup_{\theta\in\Theta}\big\|\;\hat{\ell}(x\mid\theta)-\ell(\theta)\;\big\|\ % \xrightarrow{\,\text{a.s.}}\ 0.
  29. n ( θ ^ mle - θ 0 ) 𝑑 𝒩 ( 0 , I - 1 ) . \sqrt{n}\big(\hat{\theta}_{\mathrm{mle}}-\theta_{0}\big)\ \xrightarrow{d}\ % \mathcal{N}(0,\,I^{-1}).
  30. θ ^ ( θ ^ x ) = 1 n i = 1 n θ ln f ( x i θ ^ ) = 0. \nabla_{\!\theta}\,\hat{\ell}(\hat{\theta}\mid x)=\frac{1}{n}\sum_{i=1}^{n}% \nabla_{\!\theta}\ln f(x_{i}\mid\hat{\theta})=0.
  31. 0 = 1 n i = 1 n θ ln f ( x i θ 0 ) + [ 1 n i = 1 n θ θ ln f ( x i θ ~ ) ] ( θ ^ - θ 0 ) , 0=\frac{1}{n}\sum_{i=1}^{n}\nabla_{\!\theta}\ln f(x_{i}\mid\theta_{0})+\Bigg[% \,\frac{1}{n}\sum_{i=1}^{n}\nabla_{\!\theta\theta}\ln f(x_{i}\mid\tilde{\theta% })\,\Bigg](\hat{\theta}-\theta_{0}),
  32. θ ~ \tilde{\theta}
  33. θ ^ \hat{\theta}
  34. n ( θ ^ - θ 0 ) = [ - 1 n i = 1 n θ θ ln f ( x i θ ~ ) ] - 1 1 n i = 1 n θ ln f ( x i θ 0 ) \sqrt{n}(\hat{\theta}-\theta_{0})=\Bigg[\,{-\frac{1}{n}\sum_{i=1}^{n}\nabla_{% \!\theta\theta}\ln f(x_{i}\mid\tilde{\theta})}\,\Bigg]^{-1}\frac{1}{\sqrt{n}}% \sum_{i=1}^{n}\nabla_{\!\theta}\ln f(x_{i}\mid\theta_{0})
  35. n ( θ ^ - θ 0 ) 𝑑 𝒩 ( 0 , H - 1 I H - 1 ) . \sqrt{n}(\hat{\theta}-\theta_{0})\ \ \xrightarrow{d}\ \ \mathcal{N}\big(0,\ H^% {-1}IH^{-1}\big).
  36. θ ^ \widehat{\theta}
  37. α ^ = g ( θ ^ ) . \widehat{\alpha}=g(\,\widehat{\theta}\,).\,
  38. L ¯ ( α ) = sup θ : α = g ( θ ) L ( θ ) . \bar{L}(\alpha)=\sup_{\theta:\alpha=g(\theta)}L(\theta).\,
  39. f Y ( y ) = f X ( x ) / | g ( x ) | f_{Y}(y)=f_{X}(x)/|g^{\prime}(x)|\,
  40. n ( θ ^ mle - θ 0 ) 𝑑 𝒩 ( 0 , I - 1 ) , \sqrt{n}(\hat{\theta}\text{mle}-\theta_{0})\ \ \xrightarrow{d}\ \ \mathcal{N}(% 0,\ I^{-1}),
  41. I j k = E X [ - 2 ln f θ 0 ( X t ) θ j θ k ] . I_{jk}=\operatorname{E}_{X}\bigg[\;{-\frac{\partial^{2}\ln f_{\theta_{0}}(X_{t% })}{\partial\theta_{j}\,\partial\theta_{k}}}\;\bigg].
  42. b s E [ ( θ ^ mle - θ 0 ) s ] = 1 n I s i I j k ( 1 2 K i j k + J j , i k ) b_{s}\equiv\operatorname{E}[(\hat{\theta}_{\mathrm{mle}}-\theta_{0})_{s}]=% \frac{1}{n}\cdot I^{si}I^{jk}\big(\tfrac{1}{2}K_{ijk}+J_{j,ik}\big)
  43. 1 2 K i j k + J j , i k = E [ 1 2 3 ln f θ 0 ( x t ) θ i θ j θ k + ln f θ 0 ( x t ) θ j 2 ln f θ 0 ( x t ) θ i θ k ] . \tfrac{1}{2}K_{ijk}+J_{j,ik}=\operatorname{E}\bigg[\;\frac{1}{2}\frac{\partial% ^{3}\ln f_{\theta_{0}}(x_{t})}{\partial\theta_{i}\,\partial\theta_{j}\,% \partial\theta_{k}}+\frac{\partial\ln f_{\theta_{0}}(x_{t})}{\partial\theta_{j% }}\frac{\partial^{2}\ln f_{\theta_{0}}(x_{t})}{\partial\theta_{i}\,\partial% \theta_{k}}\;\bigg].
  44. θ ^ mle * = θ ^ mle - b ^ . \hat{\theta}^{*}_{\mathrm{mle}}=\hat{\theta}_{\mathrm{mle}}-\hat{b}.
  45. n ^ \hat{n}
  46. Pr ( H = 49 p = 1 / 3 ) \displaystyle\Pr(\mathrm{H}=49\mid p=1/3)
  47. L ( p ) = f D ( H = 49 p ) = ( 80 49 ) p 49 ( 1 - p ) 31 , L(p)=f_{D}(\mathrm{H}=49\mid p)={\left({{80}\atop{49}}\right)}p^{49}(1-p)^{31},
  48. 0 \displaystyle{0}
  49. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  50. f ( x μ , σ 2 ) = 1 2 π σ exp ( - ( x - μ ) 2 2 σ 2 ) , f(x\mid\mu,\sigma^{2})=\frac{1}{\sqrt{2\pi}\ \sigma\ }\exp{\left(-\frac{(x-\mu% )^{2}}{2\sigma^{2}}\right)},
  51. f ( x 1 , , x n μ , σ 2 ) = i = 1 n f ( x i μ , σ 2 ) = ( 1 2 π σ 2 ) n / 2 exp ( - i = 1 n ( x i - μ ) 2 2 σ 2 ) , f(x_{1},\ldots,x_{n}\mid\mu,\sigma^{2})=\prod_{i=1}^{n}f(x_{i}\mid\mu,\sigma^{% 2})=\left(\frac{1}{2\pi\sigma^{2}}\right)^{n/2}\exp\left(-\frac{\sum_{i=1}^{n}% (x_{i}-\mu)^{2}}{2\sigma^{2}}\right),
  52. f ( x 1 , , x n μ , σ 2 ) = ( 1 2 π σ 2 ) n / 2 exp ( - i = 1 n ( x i - x ¯ ) 2 + n ( x ¯ - μ ) 2 2 σ 2 ) , f(x_{1},\ldots,x_{n}\mid\mu,\sigma^{2})=\left(\frac{1}{2\pi\sigma^{2}}\right)^% {n/2}\exp\left(-\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}+n(\bar{x}-\mu)^{2}}{2% \sigma^{2}}\right),
  53. x ¯ \bar{x}
  54. ( μ , σ ) = f ( x 1 , , x n μ , σ ) \mathcal{L}(\mu,\sigma)=f(x_{1},\ldots,x_{n}\mid\mu,\sigma)
  55. log ( ( μ , σ ) ) = ( - n / 2 ) log ( 2 π σ 2 ) - 1 2 σ 2 i = 1 n ( x i - μ ) 2 \log(\mathcal{L}(\mu,\sigma))=(-n/2)\log(2\pi\sigma^{2})-\frac{1}{2\sigma^{2}}% \sum_{i=1}^{n}(x_{i}-\mu)^{2}
  56. 0 = μ log ( ( μ , σ ) ) = 0 - - 2 n ( x ¯ - μ ) 2 σ 2 . \begin{aligned}\displaystyle 0&\displaystyle=\frac{\partial}{\partial\mu}\log(% \mathcal{L}(\mu,\sigma))=0-\frac{-2n(\bar{x}-\mu)}{2\sigma^{2}}.\end{aligned}
  57. μ ^ = x ¯ = i = 1 n x i n . \hat{\mu}=\bar{x}=\sum^{n}_{i=1}\frac{x_{i}}{n}.
  58. E [ μ ^ ] = μ , E\left[\widehat{\mu}\right]=\mu,\,
  59. μ ^ \widehat{\mu}
  60. 0 \displaystyle 0
  61. σ ^ 2 = 1 n i = 1 n ( x i - μ ) 2 . \widehat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\mu)^{2}.
  62. μ = μ ^ \mu=\widehat{\mu}
  63. σ ^ 2 = 1 n i = 1 n ( x i - x ¯ ) 2 = 1 n i = 1 n x i 2 - 1 n 2 i = 1 n j = 1 n x i x j . \widehat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}=\frac{1}{n}% \sum_{i=1}^{n}x_{i}^{2}-\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}.
  64. δ i μ - x i \delta_{i}\equiv\mu-x_{i}
  65. σ ^ 2 = 1 n i = 1 n ( μ - δ i ) 2 - 1 n 2 i = 1 n j = 1 n ( μ - δ i ) ( μ - δ j ) . \widehat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}(\mu-\delta_{i})^{2}-\frac{1}{n^% {2}}\sum_{i=1}^{n}\sum_{j=1}^{n}(\mu-\delta_{i})(\mu-\delta_{j}).
  66. E [ δ i ] = 0 E\left[\delta_{i}\right]=0
  67. E [ δ i 2 ] = σ 2 E[\delta_{i}^{2}]=\sigma^{2}
  68. E [ σ ^ 2 ] = n - 1 n σ 2 . E\left[\widehat{\sigma}^{2}\right]=\frac{n-1}{n}\sigma^{2}.
  69. σ ^ \widehat{\sigma}
  70. σ ^ \widehat{\sigma}
  71. θ = ( μ , σ 2 ) \theta=(\mu,\sigma^{2})
  72. θ ^ = ( μ ^ , σ ^ 2 ) . \widehat{\theta}=\left(\widehat{\mu},\widehat{\sigma}^{2}\right).
  73. log ( ( μ ^ , σ ^ ) ) = - n 2 ( log ( 2 π σ ^ 2 ) + 1 ) \log(\mathcal{L}(\hat{\mu},\hat{\sigma}))=\frac{-n}{2}(\log(2\pi\hat{\sigma}^{% 2})+1)
  74. f ( x , y ) = f ( x ) f ( y ) f(x,y)=f(x)f(y)\,
  75. ( x 1 , , x n ) (x_{1},\ldots,x_{n})\,
  76. ( μ 1 , , μ n ) (\mu_{1},\ldots,\mu_{n})\,
  77. Σ \Sigma
  78. f ( x 1 , , x n ) = 1 ( 2 π ) n / 2 det ( Σ ) exp ( - 1 2 [ x 1 - μ 1 , , x n - μ n ] Σ - 1 [ x 1 - μ 1 , , x n - μ n ] T ) f(x_{1},\ldots,x_{n})=\frac{1}{(2\pi)^{n/2}\sqrt{\,\text{det}(\Sigma)}}\exp% \left(-\frac{1}{2}\left[x_{1}-\mu_{1},\ldots,x_{n}-\mu_{n}\right]\Sigma^{-1}% \left[x_{1}-\mu_{1},\ldots,x_{n}-\mu_{n}\right]^{T}\right)
  79. f ( x , y ) = 1 2 π σ x σ y 1 - ρ 2 exp [ - 1 2 ( 1 - ρ 2 ) ( ( x - μ x ) 2 σ x 2 - 2 ρ ( x - μ x ) ( y - μ y ) σ x σ y + ( y - μ y ) 2 σ y 2 ) ] f(x,y)=\frac{1}{2\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^{2}}}\exp\left[-\frac{1}{% 2(1-\rho^{2})}\left(\frac{(x-\mu_{x})^{2}}{\sigma_{x}^{2}}-\frac{2\rho(x-\mu_{% x})(y-\mu_{y})}{\sigma_{x}\sigma_{y}}+\frac{(y-\mu_{y})^{2}}{\sigma_{y}^{2}}% \right)\right]
  80. x i x_{i}
  81. σ 2 \sigma^{2}
  82. x ^ i \hat{x}_{i}
  83. x ¯ \bar{x}
  84. σ ^ 2 \widehat{\sigma}^{2}
  85. σ ^ 2 = 1 n i = 1 n ( x ^ i - x ¯ ) 2 . \widehat{\sigma}^{2}=\frac{1}{n}\sum_{i=1}^{n}(\hat{x}_{i}-\bar{x})^{2}.

Maximum_power_transfer_theorem.html

  1. η η
  2. η = R load R load + R source = 1 1 + R source / R load . \eta=\frac{R_{\mathrm{load}}}{R_{\mathrm{load}}+R_{\mathrm{source}}}=\frac{1}{% 1+R_{\mathrm{source}}/R_{\mathrm{load}}}.
  3. R load = R source R_{\mathrm{load}}=R_{\mathrm{source}}
  4. η = 0.5 , \eta=0.5,
  5. R load = R_{\mathrm{load}}=\infty
  6. R source = 0 , R_{\mathrm{source}}=0,
  7. η = 1 , \eta=1,
  8. R load = 0 R_{\mathrm{load}}=0
  9. η = 0. \eta=0.
  10. V V
  11. I I
  12. I I
  13. I = V R S + R L . I=\frac{V}{R_{\mathrm{S}}+R_{\mathrm{L}}}.
  14. P L = I 2 R L = ( V R S + R L ) 2 R L = V 2 R S 2 / R L + 2 R S + R L . P_{\mathrm{L}}=I^{2}R_{\mathrm{L}}=\left(\frac{V}{R_{\mathrm{S}}+R_{\mathrm{L}% }}\right)^{2}R_{\mathrm{L}}=\frac{V^{2}}{R_{\mathrm{S}}^{2}/R_{\mathrm{L}}+2R_% {\mathrm{S}}+R_{\mathrm{L}}}.
  15. R S 2 / R L + 2 R S + R L R_{\mathrm{S}}^{2}/R_{\mathrm{L}}+2R_{\mathrm{S}}+R_{\mathrm{L}}
  16. d d R L ( R S 2 / R L + 2 R S + R L ) = - R S 2 / R L 2 + 1. \frac{d}{dR_{\mathrm{L}}}\left(R_{\mathrm{S}}^{2}/R_{\mathrm{L}}+2R_{\mathrm{S% }}+R_{\mathrm{L}}\right)=-R_{\mathrm{S}}^{2}/R_{\mathrm{L}}^{2}+1.
  17. R S 2 / R L 2 = 1 R_{\mathrm{S}}^{2}/R_{\mathrm{L}}^{2}=1
  18. R L = ± R S . R_{\mathrm{L}}=\pm R_{\mathrm{S}}.
  19. d 2 d R L 2 ( R S 2 / R L + 2 R S + R L ) = 2 R S 2 / R L 3 . {{d^{2}}\over{dR_{\mathrm{L}}^{2}}}\left({R_{\mathrm{S}}^{2}/R_{\mathrm{L}}+2R% _{\mathrm{S}}+R_{\mathrm{L}}}\right)={2R_{\mathrm{S}}^{2}}/{R_{\mathrm{L}}^{3}% }.\,\!
  20. R S R_{\mathrm{S}}\,\!
  21. R L R_{\mathrm{L}}\,\!
  22. R S = R L . R_{\mathrm{S}}=R_{\mathrm{L}}.\,\!
  23. | V S | |V_{\mathrm{S}}|
  24. Z S Z_{\mathrm{S}}
  25. Z L Z_{\mathrm{L}}
  26. | I | |I|
  27. | I | |I|
  28. | I | = | V S | | Z S + Z L | . |I|={|V_{\mathrm{S}}|\over|Z_{\mathrm{S}}+Z_{\mathrm{L}}|}.
  29. P L P_{\mathrm{L}}
  30. R L R_{\mathrm{L}}
  31. P L \displaystyle P_{\mathrm{L}}
  32. R S R_{\mathrm{S}}
  33. X S X_{\mathrm{S}}
  34. Z S Z_{\mathrm{S}}
  35. X L X_{\mathrm{L}}
  36. Z L Z_{\mathrm{L}}
  37. R L R_{\mathrm{L}}
  38. X L X_{\mathrm{L}}
  39. V S V_{\mathrm{S}}
  40. R S R_{\mathrm{S}}
  41. X S X_{\mathrm{S}}
  42. R L R_{\mathrm{L}}
  43. X L X_{\mathrm{L}}
  44. ( R S + R L ) 2 + ( X S + X L ) 2 (R_{\mathrm{S}}+R_{\mathrm{L}})^{2}+(X_{\mathrm{S}}+X_{\mathrm{L}})^{2}\,
  45. X L = - X S . X_{\mathrm{L}}=-X_{\mathrm{S}}.\,
  46. P L = 1 2 | V S | 2 R L ( R S + R L ) 2 P_{\mathrm{L}}={1\over 2}{{|V_{\mathrm{S}}|^{2}R_{\mathrm{L}}}\over{(R_{% \mathrm{S}}+R_{\mathrm{L}})^{2}}}\,\!
  47. R L R_{\mathrm{L}}
  48. R L = R S R_{\mathrm{L}}=R_{\mathrm{S}}
  49. R L = R S R_{\mathrm{L}}=R_{\mathrm{S}}\,\!
  50. X L = - X S X_{\mathrm{L}}=-X_{\mathrm{S}}\,\!
  51. Z L = Z S * . Z_{\mathrm{L}}=Z_{\mathrm{S}}^{*}.

Maxwell–Boltzmann_statistics.html

  1. ε i \varepsilon_{i}
  2. N i \langle N_{i}\rangle
  3. N i = g i e ( ε i - μ ) / k T = N Z g i e - ε i / k T \langle N_{i}\rangle=\frac{g_{i}}{e^{(\varepsilon_{i}-\mu)/kT}}=\frac{N}{Z}\,g% _{i}e^{-\varepsilon_{i}/kT}
  4. ε i \varepsilon_{i}
  5. N i \langle N_{i}\rangle
  6. ε i \varepsilon_{i}
  7. g i g_{i}
  8. ε i \varepsilon_{i}
  9. N = i N i N=\sum_{i}N_{i}\,
  10. Z = i g i e - ε i / k T Z=\sum_{i}g_{i}e^{-\varepsilon_{i}/kT}
  11. N i = 1 e ( ε i - μ ) / k T = N Z e - ε i / k T \langle N_{i}\rangle=\frac{1}{e^{(\varepsilon_{i}-\mu)/kT}}=\frac{N}{Z}\,e^{-% \varepsilon_{i}/kT}
  12. ε i \varepsilon_{i}
  13. Z = i e - ε i / k T Z=\sum_{i}e^{-\varepsilon_{i}/kT}
  14. N i = g i e ( ε i - μ ) / k T ± 1 . \langle N_{i}\rangle=\frac{g_{i}}{e^{(\varepsilon_{i}-\mu)/kT}\pm 1}.
  15. e ( ε min - μ ) / k T 1 , e^{(\varepsilon_{\rm min}-\mu)/kT}\gg 1,
  16. ε min \varepsilon_{\rm min}
  17. ε i \varepsilon_{i}
  18. ε \varepsilon
  19. ε 1 \varepsilon_{1}
  20. N 1 N_{1}
  21. ε 2 \varepsilon_{2}
  22. N 2 N_{2}
  23. ε i \varepsilon_{i}
  24. N i N_{i}
  25. i . i.
  26. N i N_{i}
  27. N i N_{i}
  28. N i N_{i}
  29. i i
  30. N N
  31. W = N ! N a ! ( N - N a ) ! × ( N - N a ) ! N b ! ( N - N a - N b ) ! × ( N - N a - N b ) ! N c ! ( N - N a - N b - N c ) ! × × ( N - - N l ) ! N k ! ( N - - N l - N k ) ! = = N ! N a ! N b ! N c ! N k ! ( N - - N l - N k ) ! \begin{aligned}\displaystyle W&\displaystyle=\frac{N!}{N_{a}!(N-N_{a})!}\times% \frac{(N-N_{a})!}{N_{b}!(N-N_{a}-N_{b})!}~{}\times\frac{(N-N_{a}-N_{b})!}{N_{c% }!(N-N_{a}-N_{b}-N_{c})!}\times\ldots\times\frac{(N-\ldots-N_{l})!}{N_{k}!(N-% \ldots-N_{l}-N_{k})!}=\\ \\ &\displaystyle=\frac{N!}{N_{a}!N_{b}!N_{c}!\ldots N_{k}!(N-\ldots-N_{l}-N_{k})% !}\end{aligned}
  32. W = N ! i = a , b , c , k 1 N i ! \begin{aligned}\displaystyle W&\displaystyle=N!\prod_{i=a,b,c,...}^{k}\frac{1}% {N_{i}!}\end{aligned}
  33. g i g_{i}
  34. g i g_{i}
  35. N i N_{i}
  36. g i g_{i}
  37. N i N_{i}
  38. g i g_{i}
  39. g i N i g_{i}^{N_{i}}
  40. g i g_{i}
  41. g i g_{i}
  42. W W
  43. N N
  44. i i
  45. g i g_{i}
  46. N i N_{i}
  47. W = N ! g i N i N i ! W=N!\prod\frac{g_{i}^{N_{i}}}{N_{i}!}
  48. S = k ln W S=k\,\ln W
  49. W = i ( N i + g i - 1 ) ! N i ! ( g i - 1 ) ! W=\prod_{i}\frac{(N_{i}+g_{i}-1)!}{N_{i}!(g_{i}-1)!}
  50. g i 1 g_{i}\gg 1
  51. g i N i g_{i}\gg N_{i}
  52. N ! N N e - N , N!\approx N^{N}e^{-N},
  53. W i ( N i + g i ) N i + g i N i N i g i g i i g i N i ( 1 + N i / g i ) g i N i N i W\approx\prod_{i}\frac{(N_{i}+g_{i})^{N_{i}+g_{i}}}{N_{i}^{N_{i}}g_{i}^{g_{i}}% }\approx\prod_{i}\frac{g_{i}^{N_{i}}(1+N_{i}/g_{i})^{g_{i}}}{N_{i}^{N_{i}}}
  54. ( 1 + N i / g i ) g i e N i (1+N_{i}/g_{i})^{g_{i}}\approx e^{N_{i}}
  55. g i N i g_{i}\gg N_{i}
  56. W i g i N i N i ! W\approx\prod_{i}\frac{g_{i}^{N_{i}}}{N_{i}!}
  57. N i N_{i}
  58. W W
  59. ( N = N i ) \left(N=\textstyle\sum N_{i}\right)
  60. ( E = N i ε i ) \left(E=\textstyle\sum N_{i}\varepsilon_{i}\right)
  61. W W
  62. ln ( W ) \ln(W)
  63. N i N_{i}
  64. f ( N 1 , N 2 , , N n ) = ln ( W ) + α ( N - N i ) + β ( E - N i ε i ) f(N_{1},N_{2},\ldots,N_{n})=\ln(W)+\alpha(N-\sum N_{i})+\beta(E-\sum N_{i}% \varepsilon_{i})
  65. ln W = ln [ i = 1 n g i N i N i ! ] i = 1 n ( N i ln g i - N i ln N i + N i ) \ln W=\ln\left[\prod\limits_{i=1}^{n}\frac{g_{i}^{N_{i}}}{N_{i}!}\right]% \approx\sum\limits_{i=1}^{n}\left(N_{i}\ln g_{i}-N_{i}\ln N_{i}+N_{i}\right)
  66. f ( N 1 , N 2 , , N n ) = α N + β E + i = 1 n ( N i ln g i - N i ln N i + N i - ( α + β ε i ) N i ) f(N_{1},N_{2},\ldots,N_{n})=\alpha N+\beta E+\sum\limits_{i=1}^{n}\left(N_{i}% \ln g_{i}-N_{i}\ln N_{i}+N_{i}-(\alpha+\beta\varepsilon_{i})N_{i}\right)
  67. f N i = ln g i - ln N i - ( α + β ε i ) = 0 \frac{\partial f}{\partial N_{i}}=\ln g_{i}-\ln N_{i}-(\alpha+\beta\varepsilon% _{i})=0
  68. i = 1 n i=1\ldots n
  69. N i N_{i}
  70. N i = g i e α + β ε i N_{i}=\frac{g_{i}}{e^{\alpha+\beta\varepsilon_{i}}}
  71. N i N_{i}
  72. ln W \ln W
  73. N 1 N\gg 1
  74. ln W = ( α + 1 ) N + β E \ln W=(\alpha+1)N+\beta E\,
  75. d E = 1 β d ln W - α β d N dE=\frac{1}{\beta}d\ln W-\frac{\alpha}{\beta}dN
  76. d E = T d S + μ d N dE=T\,dS+\mu\,dN
  77. S = k ln W S=k\,\ln W
  78. ln W \ln W
  79. β = 1 / k T \beta=1/kT
  80. α = - μ / k T \alpha=-\mu/kT
  81. N i = g i e ( ε i - μ ) / k T N_{i}=\frac{g_{i}}{e^{(\varepsilon_{i}-\mu)/kT}}
  82. N i = g i e ε i / k T / z N_{i}=\frac{g_{i}}{e^{\varepsilon_{i}/kT}/z}
  83. z = exp ( μ / k T ) z=\exp(\mu/kT)
  84. i N i = N \sum_{i}N_{i}=N\,
  85. N i = N g i e - ε i / k T Z N_{i}=N\frac{g_{i}e^{-\varepsilon_{i}/kT}}{Z}
  86. Z = i g i e - ε i / k T Z=\sum_{i}g_{i}e^{-\varepsilon_{i}/kT}
  87. ε i \varepsilon_{i}
  88. g i g_{i}
  89. ε i \varepsilon_{i}
  90. s 1 \;s_{1}
  91. Ω R ( s 1 ) \;\Omega_{R}(s_{1})
  92. Ω R ( s 1 ) = 2 Ω R ( s 2 ) \;\Omega_{R}(s_{1})=2\;\Omega_{R}(s_{2})
  93. s 1 \;s_{1}
  94. s 2 \;s_{2}
  95. P ( s i ) \;P(s_{i})
  96. s i \;s_{i}
  97. P ( s 1 ) P ( s 2 ) = Ω R ( s 1 ) Ω R ( s 2 ) . \frac{P(s_{1})}{P(s_{2})}=\frac{\Omega_{R}(s_{1})}{\Omega_{R}(s_{2})}.
  98. S R = k ln Ω R \;S_{R}=k\ln\Omega_{R}
  99. P ( s 1 ) P ( s 2 ) = e S R ( s 1 ) / k e S R ( s 2 ) / k = e ( S R ( s 1 ) - S R ( s 2 ) ) / k . \frac{P(s_{1})}{P(s_{2})}=\frac{e^{S_{R}(s_{1})/k}}{e^{S_{R}(s_{2})/k}}=e^{(S_% {R}(s_{1})-S_{R}(s_{2}))/k}.
  100. d S R = 1 T ( d U R + P d V R - μ d N R ) . dS_{R}=\frac{1}{T}(dU_{R}+P\,dV_{R}-\mu\,dN_{R}).
  101. d N R dN_{R}
  102. d V R = 0. dV_{R}=0.
  103. S R ( s 1 ) - S R ( s 2 ) = 1 T ( U R ( s 1 ) - U R ( s 2 ) ) = - 1 T ( E ( s 1 ) - E ( s 2 ) ) , S_{R}(s_{1})-S_{R}(s_{2})=\frac{1}{T}(U_{R}(s_{1})-U_{R}(s_{2}))=-\frac{1}{T}(% E(s_{1})-E(s_{2})),
  104. U R ( s i ) \;U_{R}(s_{i})
  105. E ( s i ) \;E(s_{i})
  106. s i s_{i}
  107. P ( s 1 ) , P ( s 2 ) P(s_{1}),\;P(s_{2})
  108. P ( s 1 ) P ( s 2 ) = e - E ( s 1 ) / k T e - E ( s 2 ) / k T , \frac{P(s_{1})}{P(s_{2})}=\frac{e^{-E(s_{1})/kT}}{e^{-E(s_{2})/kT}},
  109. P ( s ) = 1 Z e - E ( s ) / k T , P(s)=\frac{1}{Z}e^{-E(s)/kT},
  110. Z = s e - E ( s ) / k T , \;Z=\sum_{s}e^{-E(s)/kT},
  111. ε i \varepsilon_{i}
  112. P ( ε i ) = 1 Z g i e - ε i / k T P(\varepsilon_{i})=\frac{1}{Z}g_{i}e^{-\varepsilon_{i}/kT}
  113. Z = j g j e - ε j / k T , Z=\sum_{j}g_{j}e^{-\varepsilon_{j}/kT},
  114. ε i \varepsilon_{i}