wpmath0000003_10

Lowest_common_denominator.html

  1. 2 3 = 6 9 = 12 18 = 144 216 = 200 , 000 300 , 000 . \frac{2}{3}=\frac{6}{9}=\frac{12}{18}=\frac{144}{216}=\frac{200,000}{300,000}.
  2. 5 12 + 6 12 = 11 12 \frac{5}{12}+\frac{6}{12}=\frac{11}{12}
  3. 5 12 < 11 12 \frac{5}{12}<\frac{11}{12}
  4. 5 12 + 11 18 \frac{5}{12}+\frac{11}{18}
  5. 5 12 \frac{5}{12}
  6. 11 18 \frac{11}{18}
  7. 1 2 + 2 3 = 3 6 + 4 6 = 7 6 \frac{1}{2}+\frac{2}{3}\;=\;\frac{3}{6}+\frac{4}{6}\;=\;\frac{7}{6}
  8. 5 12 + 11 18 = 15 36 + 22 36 = 37 36 \frac{5}{12}+\frac{11}{18}\;=\;\frac{15}{36}+\frac{22}{36}\;=\;\frac{37}{36}
  9. 5 12 + 11 18 = 90 216 + 132 216 = 222 216 \frac{5}{12}+\frac{11}{18}=\frac{90}{216}+\frac{132}{216}=\frac{222}{216}
  10. a b c + c b 2 d = a b d b 2 c d + c 2 b 2 c d = a b d + c 2 b 2 c d \frac{a}{bc}+\frac{c}{b^{2}d}\;=\;\frac{abd}{b^{2}cd}+\frac{c^{2}}{b^{2}cd}\;=% \;\frac{abd+c^{2}}{b^{2}cd}

Löwenheim–Skolem_theorem.html

  1. ar : S func S rel 0 \operatorname{ar}:S_{\operatorname{func}}\cup S_{\operatorname{rel}}% \rightarrow\mathbb{N}_{0}
  2. σ \sigma\,
  3. φ ( y , x 1 , , x n ) , \varphi(y,x_{1},\ldots,x_{n})\,,
  4. f φ : M n M f_{\varphi}:M^{n}\to M
  5. a 1 , , a n M a_{1},\ldots,a_{n}\in M
  6. M φ ( f φ ( a 1 , , a n ) , a 1 , , a n ) M\models\varphi(f_{\varphi}(a_{1},\dots,a_{n}),a_{1},\dots,a_{n})
  7. M ¬ y φ ( y , a 1 , , a n ) . M\models\neg\exists y\varphi(y,a_{1},\dots,a_{n})\,.
  8. φ \varphi
  9. f φ . f_{\varphi}\,.
  10. f φ f_{\varphi}
  11. F F\,
  12. M M\,
  13. F ( A ) = { b M b = f φ ( a 1 , , a n ) ; φ σ ; a 1 , , a n A } F(A)=\{b\in M\mid b=f_{\varphi}(a_{1},\dots,a_{n});\,\varphi\in\sigma;\,a_{1},% \dots,a_{n}\in A\}
  14. A M . A\subseteq M\,.
  15. F F\,
  16. F ω . F^{\omega}\,.
  17. A M A\subseteq M
  18. | A | = κ \left|A\right|=\kappa
  19. N = F ω ( A ) , N=F^{\omega}(A)\,,
  20. | N | = κ . \left|N\right|=\kappa\,.
  21. N N\,
  22. M M\,
  23. f φ f_{\varphi}
  24. f φ f_{\varphi}
  25. f φ f_{\varphi}
  26. M y φ ( y , a 1 , , a n ) . M\models\exists y\varphi(y,a_{1},\dots,a_{n})\,.
  27. F F\,
  28. F ( A ) F(A)\,
  29. A A\,
  30. M M\,
  31. | F ( A ) | | A | + | σ | + 0 . \left|F(A)\right|\leq\left|A\right|+\left|\sigma\right|+\aleph_{0}\,.

Lucas_number.html

  1. L n := { 2 if n = 0 ; 1 if n = 1 ; L n - 1 + L n - 2 if n > 1. L_{n}:=\begin{cases}2&\,\text{if }n=0;\\ 1&\,\text{if }n=1;\\ L_{n-1}+L_{n-2}&\,\text{if }n>1.\\ \end{cases}
  2. 2 , 1 , 3 , 4 , 7 , 11 , 18 , 29 , 47 , 76 , 123 , 2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\;\ldots\;
  3. L n L_{n}
  4. - 5 n 5 -5\leq{}n\leq 5
  5. L - n = ( - 1 ) n L n . L_{-n}=(-1)^{n}L_{n}.\!
  6. L n = F n - 1 + F n + 1 = F n + 2 F n - 1 = F n + 2 - F n - 2 \,L_{n}=F_{n-1}+F_{n+1}=F_{n}+2F_{n-1}=F_{n+2}-F_{n-2}
  7. L m + n = L m + 1 F n + L m F n - 1 \,L_{m+n}=L_{m+1}F_{n}+L_{m}F_{n-1}
  8. L n 2 = 5 F n 2 + 4 ( - 1 ) n \,L_{n}^{2}=5F_{n}^{2}+4(-1)^{n}
  9. n n\,
  10. L n F n \frac{L_{n}}{F_{n}}
  11. 5 . \sqrt{5}.
  12. F 2 n = L n F n \,F_{2n}=L_{n}F_{n}
  13. F n + k + ( - 1 ) k F n - k = L k F n \,F_{n+k}+(-1)^{k}F_{n-k}=L_{k}F_{n}
  14. F n = L n - 1 + L n + 1 5 = L n - 3 + L n + 3 10 \,F_{n}={L_{n-1}+L_{n+1}\over 5}={L_{n-3}+L_{n+3}\over 10}
  15. L n = φ n + ( 1 - φ ) n = φ n + ( - φ ) - n = ( 1 + 5 2 ) n + ( 1 - 5 2 ) n , L_{n}=\varphi^{n}+(1-\varphi)^{n}=\varphi^{n}+(-\varphi)^{-n}=\left({1+\sqrt{5% }\over 2}\right)^{n}+\left({1-\sqrt{5}\over 2}\right)^{n}\,,
  16. φ \varphi
  17. n > 1 n>1
  18. ( - φ ) - n (-\varphi)^{-n}
  19. L n L_{n}
  20. φ n \varphi^{n}
  21. φ n + 1 / 2 \varphi^{n}+1/2
  22. φ n + 1 / 2 \lfloor\varphi^{n}+1/2\rfloor
  23. F n = φ n - ( 1 - φ ) n 5 , F_{n}=\frac{\varphi^{n}-(1-\varphi)^{n}}{\sqrt{5}}\,,
  24. φ n = L n + F n 5 2 . \varphi^{n}={{L_{n}+F_{n}\sqrt{5}}\over 2}\,.

Lucas_primality_test.html

  1. a ( n - 1 ) / q 1 ( mod n ) a^{({n-1})/q}\ \not\equiv\ 1\;\;(\mathop{{\rm mod}}n)\,
  2. a n - 1 1 ( mod n ) if and only if ord ( a ) | ( n - 1 ) . a^{n-1}\equiv 1\;\;(\mathop{{\rm mod}}n)\ \,\text{ if and only if }\,\text{ % ord}(a)|(n-1).
  3. 17 35 70 1 ( mod 71 ) 17^{35}\ \equiv\ 70\ \not\equiv\ 1\;\;(\mathop{{\rm mod}}71)
  4. 17 14 25 1 ( mod 71 ) 17^{14}\ \equiv\ 25\ \not\equiv\ 1\;\;(\mathop{{\rm mod}}71)
  5. 17 10 1 1 ( mod 71 ) . 17^{10}\ \equiv\ 1\ \equiv\ 1\;\;(\mathop{{\rm mod}}71).
  6. 11 70 1 ( mod 71 ) . 11^{70}\ \equiv\ 1\;\;(\mathop{{\rm mod}}71).
  7. 11 35 70 1 ( mod 71 ) 11^{35}\ \equiv\ 70\ \not\equiv\ 1\;\;(\mathop{{\rm mod}}71)
  8. 11 14 54 1 ( mod 71 ) 11^{14}\ \equiv\ 54\ \not\equiv\ 1\;\;(\mathop{{\rm mod}}71)
  9. 11 10 32 1 ( mod 71 ) . 11^{10}\ \equiv\ 32\ \not\equiv\ 1\;\;(\mathop{{\rm mod}}71).
  10. \not\equiv
  11. \not\equiv

Lucas–Lehmer_primality_test.html

  1. { s i } \{s_{i}\}
  2. s i = { 4 if i = 0 ; s i - 1 2 - 2 otherwise. s_{i}=\begin{cases}4&\,\text{if }i=0;\\ s_{i-1}^{2}-2&\,\text{otherwise.}\end{cases}
  3. s p - 2 0 ( mod M p ) . s_{p-2}\equiv 0\;\;(\mathop{{\rm mod}}M_{p}).
  4. k ( k mod 2 n ) + k / 2 n ( mod 2 n - 1 ) . k\equiv(k\,\bmod\,2^{n})+\lfloor k/2^{n}\rfloor\;\;(\mathop{{\rm mod}}2^{n}-1).
  5. p log p 2 O ( log * p ) p\log p\ 2^{O(\log^{*}p)}
  6. s i = { 4 if i = 0 ; s i - 1 2 - 2 otherwise. s_{i}=\begin{cases}4&\,\text{if }i=0;\\ s_{i-1}^{2}-2&\,\text{otherwise.}\end{cases}
  7. s p - 2 0 ( mod M p ) . s_{p-2}\equiv 0\;\;(\mathop{{\rm mod}}M_{p}).
  8. s i {\langle}s_{i}{\rangle}
  9. ω = 2 + 3 \omega=2+\sqrt{3}
  10. ω ¯ = 2 - 3 \bar{\omega}=2-\sqrt{3}
  11. s i = ω 2 i + ω ¯ 2 i s_{i}=\omega^{2^{i}}+\bar{\omega}^{2^{i}}
  12. s 0 = ω 2 0 + ω ¯ 2 0 = ( 2 + 3 ) + ( 2 - 3 ) = 4 s_{0}=\omega^{2^{0}}+\bar{\omega}^{2^{0}}=\left(2+\sqrt{3}\right)+\left(2-% \sqrt{3}\right)=4
  13. s n = s n - 1 2 - 2 = ( ω 2 n - 1 + ω ¯ 2 n - 1 ) 2 - 2 = ω 2 n + ω ¯ 2 n + 2 ( ω ω ¯ ) 2 n - 1 - 2 = ω 2 n + ω ¯ 2 n . \begin{aligned}\displaystyle s_{n}&\displaystyle=s_{n-1}^{2}-2\\ &\displaystyle=\left(\omega^{2^{n-1}}+\bar{\omega}^{2^{n-1}}\right)^{2}-2\\ &\displaystyle=\omega^{2^{n}}+\bar{\omega}^{2^{n}}+2(\omega\bar{\omega})^{2^{n% -1}}-2\\ &\displaystyle=\omega^{2^{n}}+\bar{\omega}^{2^{n}}.\end{aligned}
  14. ω ω ¯ = ( 2 + 3 ) ( 2 - 3 ) = 1. \omega\bar{\omega}=\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)=1.
  15. s p - 2 0 ( mod M p ) s_{p-2}\equiv 0\;\;(\mathop{{\rm mod}}M_{p})
  16. M p M_{p}
  17. s p - 2 0 ( mod M p ) . s_{p-2}\equiv 0\;\;(\mathop{{\rm mod}}M_{p}).
  18. ω 2 p - 2 + ω ¯ 2 p - 2 = k M p \omega^{2^{p-2}}+\bar{\omega}^{2^{p-2}}=kM_{p}
  19. ω 2 p - 2 = k M p - ω ¯ 2 p - 2 . \omega^{2^{p-2}}=kM_{p}-\bar{\omega}^{2^{p-2}}.
  20. ω 2 p - 2 \omega^{2^{p-2}}
  21. ( ω 2 p - 2 ) 2 = k M p ω 2 p - 2 - ( ω ω ¯ ) 2 p - 2 . \left(\omega^{2^{p-2}}\right)^{2}=kM_{p}\omega^{2^{p-2}}-(\omega\bar{\omega})^% {2^{p-2}}.
  22. ω 2 p - 1 = k M p ω 2 p - 2 - 1. ( 1 ) \omega^{2^{p-1}}=kM_{p}\omega^{2^{p-2}}-1.\qquad\qquad(1)
  23. q \mathbb{Z}_{q}
  24. X = { a + b 3 a , b q } . X=\left\{a+b\sqrt{3}\mid a,b\in\mathbb{Z}_{q}\right\}.
  25. X X
  26. ( a + 3 b ) ( c + 3 d ) = [ ( a c + 3 b d ) mod q ] + 3 [ ( a d + b c ) mod q ] . \left(a+\sqrt{3}b\right)\left(c+\sqrt{3}d\right)=[(ac+3bd)\,\bmod\,q]+\sqrt{3}% [(ad+bc)\,\bmod\,q].
  27. | X | . |X|.
  28. ω \omega
  29. ω ¯ \bar{\omega}
  30. | X * | . |X^{*}|.
  31. | X * | | X | - 1 = q 2 - 1. |X^{*}|\leq|X|-1=q^{2}-1.
  32. M p 0 ( mod q ) M_{p}\equiv 0\;\;(\mathop{{\rm mod}}q)
  33. ω X \omega\in X
  34. k M p ω 2 p - 2 = 0 kM_{p}\omega^{2^{p-2}}=0
  35. ω 2 p - 1 = - 1 \omega^{2^{p-1}}=-1
  36. ω 2 p = 1. \omega^{2^{p}}=1.
  37. ω \omega
  38. ω 2 p - 1 . \omega^{2^{p}-1}.
  39. ω \omega
  40. 2 p . 2^{p}.
  41. ω 2 p - 1 1 \omega^{2^{p-1}}\neq 1
  42. 2 p - 1 . 2^{p-1}.
  43. 2 p . 2^{p}.
  44. 2 p | X * | q 2 - 1 < q 2 . 2^{p}\leq|X^{*}|\leq q^{2}-1<q^{2}.
  45. M p M_{p}
  46. q 2 M p = 2 p - 1. q^{2}\leq M_{p}=2^{p}-1.
  47. 2 p < 2 p - 1 2^{p}<2^{p}-1
  48. M p M_{p}
  49. M p M_{p}
  50. s p - 2 0 ( mod M p ) s_{p-2}\equiv 0\;\;(\mathop{{\rm mod}}M_{p})
  51. 2 p - 1 7 ( mod 12 ) 2^{p}-1\equiv 7\;\;(\mathop{{\rm mod}}12)
  52. p > 1 p>1
  53. ( 3 | M p ) = - 1. (3|M_{p})=-1.
  54. M p . M_{p}.
  55. 3 M p - 1 2 - 1 ( mod M p ) . 3^{\frac{M_{p}-1}{2}}\equiv-1\;\;(\mathop{{\rm mod}}M_{p}).
  56. M p M_{p}
  57. 2 p 1 ( mod M p ) 2^{p}\equiv 1\;\;(\mathop{{\rm mod}}M_{p})
  58. 2 2 p + 1 = ( 2 p + 1 2 ) 2 ( mod M p ) . 2\equiv 2^{p+1}=\left(2^{\frac{p+1}{2}}\right)^{2}\;\;(\mathop{{\rm mod}}M_{p}).
  59. 2 M p - 1 2 1 ( mod M p ) . 2^{\frac{M_{p}-1}{2}}\equiv 1\;\;(\mathop{{\rm mod}}M_{p}).
  60. 24 M p - 1 2 ( 2 M p - 1 2 ) 3 ( 3 M p - 1 2 ) ( 1 ) 3 ( - 1 ) - 1 ( mod M p ) . 24^{\frac{M_{p}-1}{2}}\equiv\left(2^{\frac{M_{p}-1}{2}}\right)^{3}\left(3^{% \frac{M_{p}-1}{2}}\right)\equiv(1)^{3}(-1)\equiv-1\;\;(\mathop{{\rm mod}}M_{p}).
  61. σ = 2 3 \sigma=2\sqrt{3}
  62. X = { a + b 3 a , b M p } . X=\{a+b\sqrt{3}\mid a,b\in\mathbb{Z}_{M_{p}}\}.
  63. ( 6 + σ ) M p = 6 M p + ( 2 M p ) ( 3 M p ) = 6 + 2 ( 3 M p - 1 2 ) 3 = 6 + 2 ( - 1 ) 3 = 6 - σ , \begin{aligned}\displaystyle(6+\sigma)^{M_{p}}&\displaystyle=6^{M_{p}}+\left(2% ^{M_{p}}\right)\left(\sqrt{3}^{M_{p}}\right)\\ &\displaystyle=6+2\left(3^{\frac{M_{p}-1}{2}}\right)\sqrt{3}\\ &\displaystyle=6+2(-1)\sqrt{3}\\ &\displaystyle=6-\sigma,\end{aligned}
  64. ( x + y ) M p x M p + y M p ( mod M p ) (x+y)^{M_{p}}\equiv x^{M_{p}}+y^{M_{p}}\;\;(\mathop{{\rm mod}}M_{p})
  65. a M p a ( mod M p ) a^{M_{p}}\equiv a\;\;(\mathop{{\rm mod}}M_{p})
  66. σ \sigma
  67. ω = ( 6 + σ ) 2 24 . \omega=\frac{(6+\sigma)^{2}}{24}.
  68. ω M p + 1 2 \omega^{\frac{M_{p}+1}{2}}
  69. ω M p + 1 2 = ( 6 + σ ) M p + 1 24 M p + 1 2 = ( 6 + σ ) ( 6 + σ ) M p 24 24 M p - 1 2 = ( 6 + σ ) ( 6 - σ ) - 24 = - 1. \begin{aligned}\displaystyle\omega^{\frac{M_{p}+1}{2}}&\displaystyle=\frac{(6+% \sigma)^{M_{p}+1}}{24^{\frac{M_{p}+1}{2}}}\\ &\displaystyle=\frac{(6+\sigma)(6+\sigma)^{M_{p}}}{24\cdot 24^{\frac{M_{p}-1}{% 2}}}\\ &\displaystyle=\frac{(6+\sigma)(6-\sigma)}{-24}\\ &\displaystyle=-1.\end{aligned}
  70. ω ¯ M p + 1 4 \bar{\omega}^{\frac{M_{p}+1}{4}}
  71. ω ω ¯ = 1 \omega\bar{\omega}=1
  72. ω M p + 1 2 ω ¯ M p + 1 4 = - ω ¯ M p + 1 4 ω M p + 1 4 + ω ¯ M p + 1 4 = 0 ω 2 p - 1 + 1 4 + ω ¯ 2 p - 1 + 1 4 = 0 ω 2 p - 2 + ω ¯ 2 p - 2 = 0 s p - 2 = 0. \begin{aligned}\displaystyle\omega^{\frac{M_{p}+1}{2}}\bar{\omega}^{\frac{M_{p% }+1}{4}}&\displaystyle=-\bar{\omega}^{\frac{M_{p}+1}{4}}\\ \displaystyle\omega^{\frac{M_{p}+1}{4}}+\bar{\omega}^{\frac{M_{p}+1}{4}}&% \displaystyle=0\\ \displaystyle\omega^{\frac{2^{p}-1+1}{4}}+\bar{\omega}^{\frac{2^{p}-1+1}{4}}&% \displaystyle=0\\ \displaystyle\omega^{2^{p-2}}+\bar{\omega}^{2^{p-2}}&\displaystyle=0\\ \displaystyle s_{p-2}&\displaystyle=0.\end{aligned}
  73. s p - 2 s_{p-2}
  74. M p . M_{p}.
  75. q = / q \mathbb{Z}_{q}=\mathbb{Z}/q\mathbb{Z}
  76. X = q [ T ] / T 2 - 3 X=\mathbb{Z}_{q}[T]/\langle T^{2}-3\rangle
  77. ω + T 2 - 3 \omega+\langle T^{2}-3\rangle
  78. ω ¯ + T 2 - 3 \bar{\omega}+\langle T^{2}-3\rangle
  79. ω \omega
  80. ω ¯ \bar{\omega}
  81. [ 3 ] \mathbb{Z}[\sqrt{3}]
  82. 3 \sqrt{3}

Ludwig_Boltzmann.html

  1. S = k log W S=k\cdot\log W\,
  2. S = k B ln W S=k_{B}\ln W\,
  3. W = N ! i 1 N i ! W=N!\prod_{i}\frac{1}{N_{i}!}
  4. ! !
  5. f t + v f x + F m f v = f t | collision \frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}+\frac{F}{m}\frac{% \partial f}{\partial v}=\frac{\partial f}{\partial t}\left.{\!\!\frac{}{}}% \right|_{\mathrm{collision}}

Lux_(disambiguation).html

  1. 4 n + 2 4n+2
  2. n n

Lyapunov_function.html

  1. V : n V:\mathbb{R}^{n}\to\mathbb{R}
  2. V V
  3. V ( 0 ) = 0 V(0)=0\,
  4. V ( x ) > 0 x U { 0 } V(x)>0\quad\forall x\in U\setminus\{0\}
  5. U U
  6. x = 0. x=0.
  7. g : n n g:\mathbb{R}^{n}\to\mathbb{R}^{n}
  8. y ˙ = g ( y ) \dot{y}=g(y)\,
  9. y * y^{*}\,
  10. 0 = g ( y * ) . 0=g(y^{*}).\,
  11. x = y - y * x=y-y^{*}\,
  12. x ˙ = y ˙ = g ( y ) = g ( x + y * ) = f ( x ) \dot{x}=\dot{y}=g(y)=g(x+y^{*})=f(x)\,
  13. f ( 0 ) = 0. f(0)=0.\,
  14. f ( x ) f(x)
  15. x * = 0 x^{*}=0\,
  16. x ˙ = f ( x ) . \dot{x}=f(x).\,
  17. V ˙ ( x ) = d d t V ( x ( t ) ) = V x d x d t = V x ˙ = V f ( x ) \dot{V}(x)=\frac{d}{dt}V(x(t))=\frac{\partial V}{\partial x}\cdot\frac{dx}{dt}% =\nabla V\cdot\dot{x}=\nabla V\cdot f(x)
  18. V V
  19. V V
  20. V ˙ ( x ) 0 x { 0 } \dot{V}(x)\leq 0\quad\forall x\in\mathcal{B}\setminus\{0\}
  21. \mathcal{B}
  22. 0
  23. V V
  24. V ˙ ( x ) < 0 x { 0 } \dot{V}(x)<0\quad\forall x\in\mathcal{B}\setminus\{0\}
  25. \mathcal{B}
  26. 0
  27. V V
  28. V ˙ ( x ) < 0 x n { 0 } , \dot{V}(x)<0\quad\forall x\in\mathbb{R}^{n}\setminus\{0\},
  29. V ( x ) V(x)
  30. x V ( x ) . \|x\|\to\infty\Rightarrow V(x)\to\infty.
  31. \mathbb{R}
  32. x ˙ = - x . \dot{x}=-x.
  33. V ( x ) = | x | V(x)=|x|
  34. { 0 } \mathbb{R}\setminus\{0\}
  35. V ˙ ( x ) = V ( x ) f ( x ) = sgn ( x ) ( - x ) = - | x | < 0. \dot{V}(x)=V^{\prime}(x)f(x)=\mathrm{sgn}(x)\cdot(-x)=-|x|<0.

Lyapunov_stability.html

  1. x e x_{e}
  2. x e x_{e}
  3. x e x_{e}
  4. x e x_{e}
  5. x e x_{e}
  6. x e x_{e}
  7. x e x_{e}
  8. x ˙ = f ( x ( t ) ) , x ( 0 ) = x 0 \dot{x}=f(x(t)),\;\;\;\;x(0)=x_{0}
  9. x ( t ) 𝒟 n x(t)\in\mathcal{D}\subseteq\mathbb{R}^{n}
  10. 𝒟 \mathcal{D}
  11. f : 𝒟 n f:\mathcal{D}\rightarrow\mathbb{R}^{n}
  12. 𝒟 \mathcal{D}
  13. f f
  14. x e x_{e}
  15. f ( x e ) = 0 f(x_{e})=0
  16. ϵ > 0 \epsilon>0
  17. δ > 0 \delta>0
  18. x ( 0 ) - x e < δ \|x(0)-x_{e}\|<\delta
  19. t 0 t\geq 0
  20. x ( t ) - x e < ϵ \|x(t)-x_{e}\|<\epsilon
  21. δ > 0 \delta>0
  22. x ( 0 ) - x e < δ \|x(0)-x_{e}\|<\delta
  23. lim t x ( t ) - x e = 0 \lim_{t\rightarrow\infty}\|x(t)-x_{e}\|=0
  24. α , β , δ > 0 \alpha,\beta,\delta>0
  25. x ( 0 ) - x e < δ \|x(0)-x_{e}\|<\delta
  26. x ( t ) - x e α x ( 0 ) - x e e - β t \|x(t)-x_{e}\|\leq\alpha\|x(0)-x_{e}\|e^{-\beta t}
  27. t 0 t\geq 0
  28. δ \delta
  29. ϵ \epsilon
  30. ϵ \epsilon
  31. α x ( 0 ) - x e e - β t \alpha\|x(0)-x_{e}\|e^{-\beta t}
  32. y ( t ) - x ( t ) 0 \|y(t)-x(t)\|\rightarrow 0
  33. t t\rightarrow\infty
  34. V ( x ) : n V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}
  35. V ( x ) 0 V(x)\geq 0
  36. x = 0 x=0
  37. V ˙ ( x ) = d d t V ( x ) 0 \dot{V}(x)=\frac{d}{dt}V(x)\leq 0
  38. x = 0 x=0
  39. V ˙ ( x ) \dot{V}(x)
  40. V ( 0 ) = 0 V(0)=0
  41. V ( x ) = 1 / ( 1 + | x | ) V(x)=1/(1+|x|)
  42. x ˙ ( t ) = x \dot{x}(t)=x
  43. V ˙ ( x ) 0 \dot{V}(x)\leq 0
  44. x = 0 x=0
  45. ϵ > 0 δ > 0 y X [ d ( x , y ) < δ n 𝐍 d ( f n ( x ) , f n ( y ) ) < ϵ ] . \forall\epsilon>0\ \exists\delta>0\ \forall y\in X\ \left[d(x,y)<\delta% \Rightarrow\forall n\in\mathbf{N}\ d\left(f^{n}(x),f^{n}(y)\right)<\epsilon% \right].
  46. δ > 0 [ d ( x , y ) < δ lim n d ( f n ( x ) , f n ( y ) ) = 0 ] . \exists\delta>0\left[d(x,y)<\delta\Rightarrow\lim_{n\to\infty}d\left(f^{n}(x),% f^{n}(y)\right)=0\right].
  47. 𝐱 ˙ = A 𝐱 \dot{\,\textbf{x}}=A\,\textbf{x}
  48. A A
  49. A A
  50. A T M + M A A^{T}M+MA
  51. M = M T M=M^{T}
  52. V ( x ) = x T M x V(x)=x^{T}Mx
  53. 𝐱 t + 1 = A 𝐱 t {\,\textbf{x}_{t+1}}=A\,\textbf{x}_{t}
  54. A A
  55. { A 1 , , A m } \{A_{1},\dots,A_{m}\}
  56. 𝐱 t + 1 = A i t 𝐱 t , A i t { A 1 , , A m } {\,\textbf{x}_{t+1}}=A_{i_{t}}\,\textbf{x}_{t},\quad A_{i_{t}}\in\{A_{1},\dots% ,A_{m}\}
  57. { A 1 , , A m } \{A_{1},\dots,A_{m}\}
  58. 𝐱 ˙ = f(x,u) \dot{\,\textbf{x}}=\,\textbf{f(x,u)}
  59. y ¨ + y - ε ( y ˙ 3 3 - y ˙ ) = 0. \ddot{y}+y-\varepsilon\left(\frac{\dot{y}^{3}}{3}-\dot{y}\right)=0.
  60. y ¨ = y = 0. \ddot{y}=y=0.
  61. x 1 = y , x 2 = y ˙ x_{1}=y,x_{2}=\dot{y}
  62. x 2 ˙ = - x 1 + ε ( x 2 3 3 - x 2 ) . \dot{x_{2}}=-x_{1}+\varepsilon\left(\frac{{x_{2}}^{3}}{3}-{x_{2}}\right).
  63. V = 1 2 ( x 1 2 + x 2 2 ) V=\frac{1}{2}\left(x_{1}^{2}+x_{2}^{2}\right)
  64. V ˙ = x 1 x ˙ 1 + x 2 x ˙ 2 = x 1 x 2 - x 1 x 2 + ε x 2 4 3 - ε x 2 2 = ε x 2 4 3 - ε x 2 2 \dot{V}=x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}=x_{1}x_{2}-x_{1}x_{2}+\varepsilon% \frac{x_{2}^{4}}{3}-\varepsilon{x_{2}^{2}}=\varepsilon\frac{x_{2}^{4}}{3}-% \varepsilon{x_{2}^{2}}
  65. ε \varepsilon
  66. x 2 2 < 3. x_{2}^{2}<3.
  67. V ˙ \dot{V}
  68. x 1 x_{1}
  69. x 1 x_{1}
  70. f ˙ ( t ) 0 \dot{f}(t)\to 0
  71. f ( t ) f(t)
  72. t t\to\infty
  73. f ( t ) = sin ( ln ( t ) ) , t > 0 f(t)=\sin(\ln(t)),\;t>0
  74. f ( t ) f(t)
  75. t t\to\infty
  76. f ˙ ( t ) 0 \dot{f}(t)\to 0
  77. f ( t ) = sin ( t 2 ) / t , t > 0 f(t)=\sin(t^{2})/t,\;t>0
  78. f ( t ) f(t)
  79. f ˙ 0 \dot{f}\leq 0
  80. f ˙ 0 \dot{f}\to 0
  81. t t\to\infty
  82. f ( t ) f(t)
  83. t t\to\infty
  84. f ˙ \dot{f}
  85. f ¨ \ddot{f}
  86. f ˙ ( t ) 0 \dot{f}(t)\to 0
  87. t t\to\infty
  88. V ˙ \dot{V}
  89. V ( x , t ) V(x,t)
  90. V ( x , t ) V(x,t)
  91. V ˙ ( x , t ) \dot{V}(x,t)
  92. V ˙ ( x , t ) \dot{V}(x,t)
  93. V ¨ \ddot{V}
  94. V ˙ ( x , t ) 0 \dot{V}(x,t)\to 0
  95. t t\to\infty
  96. e ˙ = - e + g w ( t ) \dot{e}=-e+g\cdot w(t)
  97. g ˙ = - e w ( t ) . \dot{g}=-e\cdot w(t).
  98. w w
  99. w ( t ) w(t)
  100. V = e 2 + g 2 V=e^{2}+g^{2}
  101. V ˙ = - 2 e 2 0. \dot{V}=-2e^{2}\leq 0.
  102. V ( t ) V ( 0 ) V(t)<=V(0)
  103. e e
  104. g g
  105. e e
  106. V ¨ = - 4 e ( - e + g w ) \ddot{V}=-4e(-e+g\cdot w)
  107. e e
  108. g g
  109. w w
  110. V ˙ 0 \dot{V}\to 0
  111. t t\to\infty
  112. e 0 e\to 0

Lyman_series.html

  1. 1 λ = R H ( 1 - 1 n 2 ) ( R H 1.0968 × 10 7 m - 1 13.6 eV h c ) {1\over\lambda}=R\text{H}\left(1-\frac{1}{n^{2}}\right)\qquad\left(R\text{H}% \approx 1.0968{\times}10^{7}\,\,\text{m}^{-1}\approx\frac{13.6\,\,\text{eV}}{% hc}\right)
  2. n = n=\infty
  3. n = n=\infty
  4. n n
  5. E n = - m e 4 2 ( 4 π ε 0 ) 2 1 n 2 = - 13.6 eV n 2 . E_{n}=-\frac{me^{4}}{2(4\pi\varepsilon_{0}\hbar)^{2}}\,\frac{1}{n^{2}}=-\frac{% 13.6\,\,\text{eV}}{n^{2}}.
  6. E i E\text{i}
  7. E f E\text{f}
  8. λ = h c E i - E f . \lambda=\frac{hc}{E\text{i}-E\text{f}}.
  9. λ = 12398.4 Å eV E i - E f . \lambda=\frac{12398.4\,{\rm\AA}\,\,\text{eV}}{E\text{i}-E\text{f}}.
  10. 1 λ = E i - E f 12398.4 Å eV = ( 12398.4 13.6 Å ) - 1 ( 1 m 2 - 1 n 2 ) = R H ( 1 m 2 - 1 n 2 ) \frac{1}{\lambda}=\frac{E\text{i}-E\text{f}}{12398.4\,{\rm\AA}\,\,\text{eV}}=% \left(\frac{12398.4}{13.6}\,{\rm\AA}\right)^{-1}\left(\frac{1}{m^{2}}-\frac{1}% {n^{2}}\right)=R\text{H}\left(\frac{1}{m^{2}}-\frac{1}{n^{2}}\right)
  11. R H R\text{H}
  12. 1 λ = R H ( 1 - 1 n 2 ) \frac{1}{\lambda}=R\text{H}\left(1-\frac{1}{n^{2}}\right)

Mach_wave.html

  1. μ = arcsin ( 1 M ) \mu=\arcsin\left(\frac{1}{M}\right)

Machining.html

  1. R M R = v f d {R}_{MR}=vfd\,\!
  2. R M R {R}_{MR}\,\!
  3. v v\,\!
  4. f f\,\!
  5. d d\,\!

Magic_cube.html

  1. M 3 ( n ) = n ( n 3 + 1 ) 2 . M_{3}(n)=\frac{n(n^{3}+1)}{2}.

Magic_hypercube.html

  1. M k ( n ) = n ( n k + 1 ) 2 M_{k}(n)=\frac{n(n^{k}+1)}{2}
  2. 1 2 \frac{1}{2}

Magic_number_(physics).html

  1. 2 ( ( n 1 ) + ( n 2 ) + ( n 3 ) ) 2({\textstyle\left({{n}\atop{1}}\right)}+{\textstyle\left({{n}\atop{2}}\right)% }+{\textstyle\left({{n}\atop{3}}\right)})

Magic_number_(sports).html

  1. G R L , G R T GR_{L},GR_{T}
  2. E = G R L + G R T 2 - G B L + 1 E=\frac{GR_{L}+GR_{T}}{2}-GBL+1
  3. G R L GR_{L}
  4. G R T GR_{T}
  5. E = 8 + 7 2 - 3.5 + 1 E=\frac{8+7}{2}-3.5+1

Magnetic_moment.html

  1. s y m b o l τ = s y m b o l μ × 𝐁 symbol{\tau}=symbol{\mu}\times\mathbf{B}
  2. s y m b o l τ symbol{\tau}
  3. 𝐁 \mathbf{B}
  4. s y m b o l μ symbol{\mu}
  5. N m / T = A m 2 = J / T , \,\text{N}{\cdot}\,\text{m}/\,\text{T}=\,\text{A}{\cdot}\,\text{m}^{2}=\,\text% {J}/\,\text{T},
  6. 1 statA cm 2 = 3.33564095 10 - 14 A m 2 1~{}\,\text{statA}{\cdot}\,\text{cm}^{2}=3.33564095\cdot 10^{-14}~{}\,\text{A}% {\cdot}\,\text{m}^{2}
  7. 1 erg / G = 1 abA cm 2 = 10 - 3 A m 2 1~{}\,\text{erg}/\,\text{G}=1~{}\,\text{abA}{\cdot}\,\text{cm}^{2}=10^{-3}~{}% \,\text{A}{\cdot}\,\text{m}^{2}
  8. μ = p s y m b o l . \mathbf{\mu}=psymbol{\ell}.
  9. s y m b o l μ = 1 2 𝐫 × 𝐣 symbol{\mu}=\frac{1}{2}\mathbf{r}\times\mathbf{j}
  10. s y m b o l l = 𝐫 × ( ρ 𝐯 ) symbol{l}=\mathbf{r}\times(\rho\mathbf{v})
  11. s y m b o l \Mu = 1 2 V 𝐫 × 𝐣 d V , symbol{\Mu}=\frac{1}{2}\iiint_{V}\mathbf{r}\times\mathbf{j}\,{\rm d}V,
  12. s y m b o l \Mu = 1 2 q 𝐫 × 𝐯 symbol{\Mu}=\frac{1}{2}q\,\mathbf{r}\times\mathbf{v}
  13. s y m b o l L = 𝐫 × 𝐩 = m 𝐫 × 𝐯 symbol{L}=\mathbf{r}\times\mathbf{p}=m\mathbf{r}\times\mathbf{v}
  14. s y m b o l \Mu = 1 2 S 𝐫 × 𝐣 d S , symbol{\Mu}=\frac{1}{2}\iint_{S}\mathbf{r}\times\mathbf{j}\,{\rm d}S,
  15. s y m b o l \Mu = I 2 S 𝐫 × d 𝐫 . symbol{\Mu}=\frac{I}{2}\int_{\partial S}\mathbf{r}\times{\rm d}\mathbf{r}.
  16. s y m b o l \Mu = I 𝐒 . symbol{\Mu}=I\mathbf{S}.
  17. s y m b o l μ = 𝐫 × 𝐣 symbol{\mu}=\mathbf{r}\times\mathbf{j}
  18. s y m b o l \Mu = 2 I 𝐒 . symbol{\Mu}=2I\mathbf{S}.
  19. s y m b o l μ = N I 𝐒 . symbol{\mu}=NI\mathbf{S}.
  20. U = - s y m b o l μ 𝐁 U=-symbol{\mu}\cdot\mathbf{B}
  21. 𝐅 loop = ( s y m b o l μ 𝐁 ) \mathbf{F}\text{loop}=\nabla\left(symbol{\mu}\cdot\mathbf{B}\right)
  22. 𝐅 dipole = ( s y m b o l μ ) 𝐁 \mathbf{F}\text{dipole}=\left(symbol{\mu}\cdot\nabla\right)\mathbf{B}
  23. 𝐅 loop = 𝐅 dipole + s y m b o l μ × ( × 𝐁 ) \mathbf{F}\text{loop}=\mathbf{F}\text{dipole}+symbol{\mu}\times\left(\nabla% \times\mathbf{B}\right)
  24. × 𝐁 = 0 ∇×\mathbf{B}=0
  25. 𝐀 ( 𝐫 ) = μ 0 4 π 𝐦 × 𝐫 | 𝐫 | 3 = μ 0 4 π 𝐦 × 𝐫 ^ | 𝐫 | 2 \mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{|% \mathbf{r}|^{3}}=\frac{\mu_{0}}{4\pi}\frac{\mathbf{m}\times\hat{\mathbf{r}}}{|% \mathbf{r}|^{2}}
  26. 𝐁 ( 𝐫 ) = × 𝐀 = μ 0 4 π ( 3 𝐫 ( 𝐦 𝐫 ) | 𝐫 | 5 - 𝐦 | 𝐫 | 3 ) = μ 0 4 π | r | 3 ( 3 ( m r ^ ) r ^ - m ) . \mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left(% \frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{|\mathbf{r}|^{5}}-\frac{{\mathbf% {m}}}{|\mathbf{r}|^{3}}\right)=\frac{\mu_{0}}{4\pi|r|^{3}}\left(3\left(m\cdot% \hat{r}\right)\hat{r}-m\right).
  27. ψ ( 𝐫 ) = 𝐦 𝐫 4 π | 𝐫 | 3 = 𝐦 𝐫 ^ 4 π | 𝐫 | 2 , \psi(\mathbf{r})=\frac{\mathbf{m}\cdot\mathbf{r}}{4\pi|\mathbf{r}|^{3}}=\frac{% \mathbf{m}\cdot\hat{\mathbf{r}}}{4\pi|\mathbf{r}|^{2}},
  28. 𝐇 ( 𝐫 ) = - ψ = 1 4 π ( 3 𝐫 ( 𝐦 𝐫 ) | 𝐫 | 5 - 𝐦 | 𝐫 | 3 ) = 1 4 π | r | 3 ( 3 ( m r ^ ) r ^ - m ) . {\mathbf{H}}({\mathbf{r}})=-\nabla\psi=\frac{1}{4\pi}\left(\frac{3\mathbf{r}(% \mathbf{m}\cdot\mathbf{r})}{|\mathbf{r}|^{5}}-\frac{{\mathbf{m}}}{|\mathbf{r}|% ^{3}}\right)=\frac{1}{4\pi|r|^{3}}\left(3\left(m\cdot\hat{r}\right)\hat{r}-m% \right).
  29. 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].
  30. 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].
  31. 𝐁 = μ 0 ( 𝐇 + 𝐌 ) \mathbf{B}=\mu_{0}\left(\mathbf{H}+\mathbf{M}\right)
  32. 𝐌 ( 𝐱 ) = 𝐦 δ ( 𝐱 ) \mathbf{M}(\mathbf{x})=\mathbf{m}\delta(\mathbf{x})
  33. F = ( m 2 B 1 ) , F=\nabla\left(m_{2}\cdot B_{1}\right),
  34. F ( r , m 1 , m 2 ) = 3 μ 0 4 π | r | 4 ( m 2 ( m 1 r ^ ) + m 1 ( m 2 r ^ ) + r ^ ( m 1 m 2 ) - 5 r ^ ( m 1 r ^ ) ( m 2 r ^ ) ) , F(r,m_{1},m_{2})=\frac{3\mu_{0}}{4\pi|r|^{4}}\left(m_{2}(m_{1}\cdot\hat{r})+m_% {1}(m_{2}\cdot\hat{r})+\hat{r}(m_{1}\cdot m_{2})-5\hat{r}(m_{1}\cdot\hat{r})(m% _{2}\cdot\hat{r})\right),
  35. \mathbf{r̂}
  36. r r
  37. F = 3 μ 0 4 π | r | 4 ( ( r ^ × m 1 ) × m 2 + ( r ^ × m 2 ) × m 1 - 2 r ^ ( m 1 m 2 ) + 5 r ^ ( r ^ × m 1 ) ( r ^ × m 2 ) ) . F=\frac{3\mu_{0}}{4\pi|r|^{4}}\left((\hat{r}\times m_{1})\times m_{2}+(\hat{r}% \times m_{2})\times m_{1}-2\hat{r}(m_{1}\cdot m_{2})+5\hat{r}(\hat{r}\times m_% {1})\cdot(\hat{r}\times m_{2})\right).
  38. s y m b o l τ = m 2 × B 1 . symbol\tau=m_{2}\times B_{1}.
  39. m Atom = g J μ B J ( J + 1 ) m\text{Atom}=g_{J}\mu_{B}\sqrt{J(J+1)}
  40. m Atom ( z ) = - m g J μ B m\text{Atom}(z)=-mg_{J}\mu_{B}
  41. - J , - ( J - 1 ) 0 + ( J - 1 ) , + J -J,-(J-1)\cdots 0\cdots+(J-1),+J
  42. 1 γ d 𝐦 d t = 𝐦 × 𝐇 eff - λ γ m 𝐦 × d 𝐦 d t \frac{1}{\gamma}\frac{{\rm d}\mathbf{m}}{{\rm d}t}=\mathbf{m\times H\text{eff}% }-\frac{\lambda}{\gamma m}\mathbf{m}\times\frac{{\rm d}\mathbf{m}}{{\rm d}t}
  43. γ \scriptstyle\gamma
  44. 𝐦 S = - g S μ B 𝐒 , \mathbf{m}\text{S}=-\frac{g\text{S}\mu\text{B}\mathbf{S}}{\hbar},

Magnetic_quantum_number.html

  1. 𝐋 \mathbf{L}
  2. 𝐋 = ( + 1 ) \mathbf{L}=\hbar\sqrt{\ell(\ell+1)}
  3. = h / 2 π \hbar=h/2\pi
  4. 𝐋 𝐳 = m . \mathbf{L_{z}}=m\hbar.
  5. ( m l = - , - + 1 , , 0 , , - 1 , ) (m_{l}=-\ell,-\ell+1,...,0,...,\ell-1,\ell)
  6. = 0 , m = 0 \ell=0,\quad m=0
  7. = 1 , m = - 1 , 0 , + 1 \ell=1,\quad m=-1,0,+1
  8. = 2 , m = - 2 , - 1 , 0 , + 1 , + 2 \ell=2,\quad m=-2,-1,0,+1,+2
  9. = 3 , m = - 3 , - 2 , - 1 , 0 , + 1 , + 2 , + 3 \ell=3,\quad m=-3,-2,-1,0,+1,+2,+3
  10. = 4 , m = - 4 , - 3 , - 2 , - 1 , 0 , + 1 , + 2 , + 3 , + 4 \ell=4,\quad m=-4,-3,-2,-1,0,+1,+2,+3,+4

Magnetocrystalline_anisotropy.html

  1. E / V = K 1 ( α 2 + β 2 ) = K 1 ( 1 - γ 2 ) . E/V=K_{1}\left(\alpha^{2}+\beta^{2}\right)=K_{1}\left(1-\gamma^{2}\right).
  2. E / V = K 1 sin 2 θ . \displaystyle E/V=K_{1}\sin^{2}\theta.
  3. E θ = 0 and 2 E θ 2 > 0. \frac{\partial E}{\partial\theta}=0\qquad\,\text{and}\qquad\frac{\partial^{2}E% }{\partial\theta^{2}}>0.
  4. E / V = K 1 sin 2 θ + K 2 sin 4 θ + K 3 sin 6 θ cos 6 ϕ \displaystyle E/V=K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta+K_{3}\sin^{6}\theta% \cos 6\phi
  5. × 10 < s u p > 4 J / m 3 ×10<sup>4J/m^{3}
  6. K 1 K_{1}
  7. K 2 K_{2}
  8. 45 45
  9. 15 15
  10. 120 120
  11. 3 3
  12. 550 550
  13. 89 89
  14. 27 27
  15. E / V = K 1 sin 2 θ + K 2 sin 4 θ + K 3 sin 4 θ sin 2 ϕ \displaystyle E/V=K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta+K_{3}\sin^{4}\theta% \sin 2\phi
  16. E / V = K 1 sin 2 θ + K 2 sin 4 θ + K 3 cos θ sin 3 θ cos 3 ϕ \displaystyle E/V=K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta+K_{3}\cos\theta\sin^% {3}\theta\cos 3\phi
  17. E / V = K 1 ( α 2 β 2 + β 2 γ 2 + γ 2 α 2 ) + K 2 α 2 β 2 γ 2 . E/V=K_{1}\left(\alpha^{2}\beta^{2}+\beta^{2}\gamma^{2}+\gamma^{2}\alpha^{2}% \right)+K_{2}\alpha^{2}\beta^{2}\gamma^{2}.
  18. Align l t ; 100 > &lt;100>
  19. Align l t ; 111 Align g t ; &lt;111&gt;
  20. < v a r > K 1 > 0 <var>K_{1}>0
  21. K 2 = + K_{2}=+\infty
  22. - 9 K 1 / 4 -9K_{1}/4
  23. K 2 = - 9 K 1 / 4 K_{2}=-9K_{1}/4
  24. - 9 K 1 -9K_{1}
  25. K 2 = - 9 K 1 K_{2}=-9K_{1}
  26. - -\infty
  27. 100 \langle 100\rangle
  28. 100 \langle 100\rangle
  29. 111 \langle 111\rangle
  30. 110 \langle 110\rangle
  31. 111 \langle 111\rangle
  32. 100 \langle 100\rangle
  33. 111 \langle 111\rangle
  34. 110 \langle 110\rangle
  35. 110 \langle 110\rangle
  36. < v a r > K 1 Align l t ; 0 <var>K_{1}&lt;0
  37. K 2 = + K_{2}=+\infty
  38. - 9 K 1 / 4 -9K_{1}/4
  39. K 2 = - 9 K 1 / 4 K_{2}=-9K_{1}/4
  40. - 9 K 1 -9K_{1}
  41. K 2 = - 9 K 1 K_{2}=-9K_{1}
  42. - -\infty
  43. 111 \langle 111\rangle
  44. 110 \langle 110\rangle
  45. 110 \langle 110\rangle
  46. 110 \langle 110\rangle
  47. 111 \langle 111\rangle
  48. 100 \langle 100\rangle
  49. 100 \langle 100\rangle
  50. 100 \langle 100\rangle
  51. 111 \langle 111\rangle
  52. × 10 < s u p > 4 J / m 3 ×10<sup>4J/m^{3}
  53. K 1 K_{1}
  54. K 2 K_{2}
  55. 4.8 4.8
  56. ± 0.5 \pm 0.5
  57. - 0.5 -0.5
  58. - 0.2 -0.2
  59. - 1.1 -1.1
  60. - 0.3 -0.3
  61. - 0.62 -0.62
  62. - 0.25 -0.25
  63. - 0.25 -0.25
  64. < v a r > K 1 <var>K_{1}
  65. K 1 K_{1}
  66. K 1 K^{\prime}_{1}
  67. 4.7 4.7
  68. 4.7 4.7
  69. - 0.60 -0.60
  70. - 0.59 -0.59
  71. - 1.10 -1.10
  72. - 1.36 -1.36
  73. < v a r > α 2 + β 2 + γ 2 < v a r 1 <var>α^{2}+β^{2}+γ^{2}<var>=1

Magnetomotive_force.html

  1. = Φ , \mathcal{F}=\Phi\mathcal{R},

Magnetorheological_fluid.html

  1. τ = τ y ( H ) + η d v d z , τ > τ y \tau=\tau_{y}(H)+\eta\frac{dv}{dz},\tau>\tau_{y}
  2. τ \tau
  3. τ y \tau_{y}
  4. H H
  5. η \eta
  6. d v d z \frac{dv}{dz}

Magnification.html

  1. MA = tan ε tan ε 0 \mathrm{MA}=\frac{\tan\varepsilon}{\tan\varepsilon_{0}}
  2. ε 0 {\varepsilon_{0}}
  3. ε {\varepsilon}
  4. M = f f - d o M={f\over f-d_{o}}
  5. f f
  6. d o d_{o}
  7. M M
  8. M M
  9. d i d_{i}
  10. h i h_{i}
  11. h o h_{o}
  12. M = - d i d o = h i h o M=-{d_{i}\over d_{o}}={h_{i}\over h_{o}}
  13. M = d i d o = h i h o = f d o - f = d i - f f M={d_{i}\over d_{o}}={h_{i}\over h_{o}}={f\over d_{o}-f}={d_{i}-f\over f}
  14. M = f o f e M={f_{o}\over f_{e}}
  15. f o f_{o}
  16. f e f_{e}
  17. MA = 25 cm f \mathrm{MA}={25\ \mathrm{cm}\over f}\quad
  18. f f
  19. MA = 25 cm f + 1 \mathrm{MA}={25\ \mathrm{cm}\over f}+1\quad
  20. MA = M o × M e \mathrm{MA}=M_{o}\times M_{e}
  21. M o M_{o}
  22. M e M_{e}
  23. f o f_{o}
  24. d d
  25. M o = d f o M_{o}={d\over f_{o}}
  26. f e f_{e}
  27. MA = 1 / M = D Objective / D Ramsden \mathrm{MA}=1/M=D_{\mathrm{Objective}}/{D_{\mathrm{Ramsden}}}

Magnitude_(mathematics).html

  1. | r | = r , if r 0 \left|r\right|=r,\,\text{ if }r\,\text{ ≥ }0
  2. | r | = - r , if r < 0. \left|r\right|=-r,\,\text{ if }r<0.
  3. | z | = a 2 + b 2 \left|z\right|=\sqrt{a^{2}+b^{2}}
  4. ( - 3 ) 2 + 4 2 = 5 \sqrt{(-3)^{2}+4^{2}}=5
  5. | z | = z z * = ( a + b i ) ( a - b i ) = a 2 - a b i + a b i - b 2 i 2 = a 2 + b 2 \left|z\right|=\sqrt{zz^{*}}=\sqrt{(a+bi)(a-bi)}=\sqrt{a^{2}-abi+abi-b^{2}i^{2% }}=\sqrt{a^{2}+b^{2}}
  6. i 2 = - 1 i^{2}=-1
  7. 𝐱 := x 1 2 + x 2 2 + + x n 2 . \|\mathbf{x}\|:=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}.
  8. 𝐱 := 𝐱 𝐱 . \|\mathbf{x}\|:=\sqrt{\mathbf{x}\cdot\mathbf{x}}.
  9. 𝐱 , \left\|\mathbf{x}\right\|,
  10. | 𝐱 | . \left|\mathbf{x}\right|.

Mahler's_theorem.html

  1. ( Δ f ) ( x ) = f ( x + 1 ) - f ( x ) (\Delta f)(x)=f(x+1)-f(x)\,
  2. f ( x ) = k = 0 ( Δ k f ) ( 0 ) ( x k ) , f(x)=\sum_{k=0}^{\infty}(\Delta^{k}f)(0){x\choose k},
  3. ( x k ) = x ( x - 1 ) ( x - 2 ) ( x - k + 1 ) k ! {x\choose k}=\frac{x(x-1)(x-2)\cdots(x-k+1)}{k!}

Mann–Whitney_U_test.html

  1. U 1 = R 1 - n 1 ( n 1 + 1 ) 2 U_{1}=R_{1}-{n_{1}(n_{1}+1)\over 2}\,\!
  2. U 2 = R 2 - n 2 ( n 2 + 1 ) 2 U_{2}=R_{2}-{n_{2}(n_{2}+1)\over 2}\,\!
  3. U 1 + U 2 = R 1 - n 1 ( n 1 + 1 ) 2 + R 2 - n 2 ( n 2 + 1 ) 2 . U_{1}+U_{2}=R_{1}-{n_{1}(n_{1}+1)\over 2}+R_{2}-{n_{2}(n_{2}+1)\over 2}.\,\!
  4. U 1 + U 2 = n 1 n 2 . U_{1}+U_{2}=n_{1}n_{2}.\,\!
  5. m U = n 1 n 2 2 , m_{U}=\frac{n_{1}n_{2}}{2},\,
  6. σ U = n 1 n 2 ( n 1 + n 2 + 1 ) 12 . \sigma_{U}=\sqrt{n_{1}n_{2}(n_{1}+n_{2}+1)\over 12}.\,
  7. 3 / π 3/\pi
  8. A U C 1 = U 1 n 1 n 2 AUC_{1}={U_{1}\over n_{1}n_{2}}
  9. M = 1 c ( c - 1 ) A U C k , l M={1\over c(c-1)}\sum AUC_{k,l}
  10. R k , l R_{k,l}
  11. A U C k , l AUC_{k,l}
  12. A U C k , k AUC_{k,k}
  13. A U C k , l A U C l , k AUC_{k,l}\neq AUC_{l,k}
  14. M M
  15. A U C k , l AUC_{k,l}
  16. A U C l , k AUC_{l,k}
  17. F X ( t ) < F Y ( t ) F_{X}(t)<F_{Y}(t)

Many-one_reduction.html

  1. A = f - 1 ( B ) A=f^{-1}(B)
  2. A m B . A\leq_{m}B.
  3. A 1 B . A\leq_{1}B.
  4. A , B A,B\subseteq\mathbb{N}
  5. A A
  6. B B
  7. A m B A\leq_{m}B
  8. f f
  9. A = f - 1 ( B ) . A=f^{-1}(B).
  10. f f
  11. A A
  12. B B
  13. A 1 B . A\leq_{1}B.
  14. A m B and B m A A\leq_{m}B\,\mathrm{and}\,B\leq_{m}A
  15. A A
  16. B B
  17. A m B . A\equiv_{m}B.
  18. A 1 B and B 1 A A\leq_{1}B\,\mathrm{and}\,B\leq_{1}A
  19. A A
  20. B B
  21. A 1 B . A\equiv_{1}B.
  22. A m B A\leq_{m}B
  23. A m B . \mathbb{N}\setminus A\leq_{m}\mathbb{N}\setminus B.

Marginal_cost.html

  1. Marginal Cost ( M C ) = d C d Q \,\text{Marginal Cost }(MC)=\frac{\ dC}{\ dQ}
  2. M C = Δ C Δ Q MC=\frac{\Delta C}{\Delta Q}
  3. A T C = C 0 + Δ C Q ATC=\frac{C_{0}+\Delta C}{Q}
  4. M C = Δ V C Δ Q MC=\frac{\Delta VC}{\Delta Q}
  5. Δ V C = w Δ L \Delta VC={w\Delta L}
  6. M C = w Δ L Δ Q MC=\frac{w\Delta L}{\Delta Q}
  7. Δ L Δ Q \frac{\Delta L}{\Delta Q}
  8. 1 M P L \frac{1}{MPL}
  9. M C = w M P L MC=\frac{w}{MPL}

Marginal_distribution.html

  1. Pr ( X = x ) = y Pr ( X = x , Y = y ) = y Pr ( X = x | Y = y ) Pr ( Y = y ) , \Pr(X=x)=\sum_{y}\Pr(X=x,Y=y)=\sum_{y}\Pr(X=x|Y=y)\Pr(Y=y),
  2. p X ( x ) = y p X , Y ( x , y ) d y = y p X | Y ( x | y ) p Y ( y ) d y , p_{X}(x)=\int_{y}p_{X,Y}(x,y)\,\operatorname{d}\!y=\int_{y}p_{X|Y}(x|y)\,p_{Y}% (y)\,\operatorname{d}\!y,
  3. p X ( x ) = y p X | Y ( x | y ) p Y ( y ) d y = 𝔼 Y [ p X | Y ( x | Y ) ] p_{X}(x)=\int_{y}p_{X|Y}(x|y)\,p_{Y}(y)\,\operatorname{d}\!y=\mathbb{E}_{Y}[p_% {X|Y}(x|Y)]
  4. 𝔼 Y [ f ( Y ) ] = y f ( y ) p Y ( y ) d y \mathbb{E}_{Y}[f(Y)]=\int_{y}f(y)p_{Y}(y)\,\operatorname{d}\!y

Marginal_propensity_to_consume.html

  1. 𝑀𝑃𝐶 \mathit{MPC}
  2. C C
  3. Y Y
  4. C C
  5. Y Y
  6. 𝑀𝑃𝐶 = d C d Y \mathit{MPC}=\frac{dC}{dY}
  7. 𝑀𝑃𝐶 = Δ C Δ Y \mathit{MPC}=\frac{\Delta C}{\Delta Y}
  8. Δ C \Delta C
  9. Δ Y \Delta Y
  10. 𝑀𝑃𝐶 = Δ C / Δ Y \mathit{MPC}=\Delta C/\Delta Y
  11. Δ C = 50 \Delta C=50
  12. Δ Y = 60 \Delta Y=60
  13. 𝑀𝑃𝐶 = Δ C / Δ Y = 50 / 60 = 0.83 \mathit{MPC}=\Delta C/\Delta Y=50/60=0.83
  14. $ 400 / $ 500 \$400/\$500
  15. Y = C + I Y=C+I
  16. Δ Y = Δ C + Δ I \Delta Y=\Delta C+\Delta I
  17. Δ Y = c Δ Y + Δ I \Delta Y=c\Delta Y+\Delta I
  18. c c
  19. 𝑀𝑃𝐶 \mathit{MPC}
  20. Δ Y - c Δ Y = Δ I \Delta Y-c\Delta Y=\Delta I
  21. Δ Y ( 1 - c ) = Δ I \Delta Y(1-c)=\Delta I
  22. Δ Y = Δ I ( 1 - c ) \Delta Y=\frac{\Delta I}{(1-c)}
  23. Δ Y Δ I = 1 ( 1 - c ) \frac{\Delta Y}{\Delta I}=\frac{1}{(1-c)}
  24. K = 1 ( 1 - c ) K=\frac{1}{(1-c)}
  25. K K
  26. K = Δ Y Δ I ) K=\frac{\Delta Y}{\Delta I})
  27. c c
  28. K K
  29. 1 / ( 1 - M P C ) 1/(1-MPC)
  30. K = 1 / 𝑀𝑃𝑆 K=1/\mathit{MPS}
  31. Δ C / Δ Y \Delta C/\Delta Y
  32. Δ S / Δ Y \Delta S/\Delta Y
  33. 1 - 𝑀𝑃𝐶 1-\mathit{MPC}
  34. K K
  35. \infty
  36. 0 < 𝑀𝑃𝐶 < 1 0<\mathit{MPC}<1
  37. 1 < K < 1<K<\infty
  38. 𝑀𝑃𝐶 < 𝐴𝑃𝐶 \mathit{MPC}<\mathit{APC}
  39. 0 Δ C / Δ Y < 1 0\leq\Delta C/\Delta Y<1

Marginal_propensity_to_save.html

  1. M P S = Change in Savings Change in Income MPS=\frac{\,\text{Change in Savings}}{\,\text{Change in Income}}
  2. M P S = d S d Y MPS=\frac{dS}{dY}
  3. 1 + c + c 2 + = 1 1 - c = 1 1 - M P C = 1 M P S 1+c+c^{2}+\cdots=\frac{1}{1-c}=\frac{1}{1-MPC}=\frac{1}{MPS}
  4. Simple Multiplier = 1 M P S \,\text{Simple Multiplier}=\frac{1}{MPS}

Marginal_rate_of_substitution.html

  1. M R S x y = - m indif = - ( d y / d x ) MRS_{xy}=-m_{\mathrm{indif}}=-(dy/dx)\,
  2. M R S x y = M U x / M U y MRS_{xy}=MU_{x}/MU_{y}\,
  3. M R S x y 0 \ MRS_{xy}\geq 0
  4. U ( x , y ) U(x,y)
  5. M U x = U / x \ MU_{x}=\partial U/\partial x
  6. M U y = U / y \ MU_{y}=\partial U/\partial y
  7. M U x \ MU_{x}
  8. M U y \ MU_{y}
  9. d U = ( U / x ) d x + ( U / y ) d y \ dU=(\partial U/\partial x)dx+(\partial U/\partial y)dy
  10. d U = M U x d x + M U y d y \ dU=MU_{x}dx+MU_{y}dy
  11. d U d x = M U x d x d x + M U y d y d x \frac{dU}{dx}=MU_{x}\frac{dx}{dx}+MU_{y}\frac{dy}{dx}
  12. d U d x = M U x + M U y d y d x \frac{dU}{dx}=MU_{x}+MU_{y}\frac{dy}{dx}
  13. 0 = M U x + M U y d y d x 0=MU_{x}+MU_{y}\frac{dy}{dx}
  14. - d y d x = M U x M U y -\frac{dy}{dx}=\frac{MU_{x}}{MU_{y}}
  15. M R S x y = M U x / M U y . \ MRS_{xy}=MU_{x}/MU_{y}.\,
  16. m indif = m budget \ m_{\mathrm{indif}}=m_{\mathrm{budget}}
  17. - ( M R S x y ) = - ( P x / P y ) \ -(MRS_{xy})=-(P_{x}/P_{y})
  18. M R S x y = P x / P y \ MRS_{xy}=P_{x}/P_{y}
  19. M U x / M U y = P x / P y \ MU_{x}/MU_{y}=P_{x}/P_{y}
  20. M U x / P x = M U y / P y \ MU_{x}/P_{x}=MU_{y}/P_{y}

Market_impact.html

  1. λ = | Δ Price t | Volume t \lambda=\frac{|\Delta\mathrm{Price}_{t}|}{\mathrm{Volume}_{t}}
  2. Λ R 0 = s o + Λ L 1 \Lambda_{R_{0}}=s_{o}+\Lambda_{L_{1}}
  3. p 1 p 2 p_{1}\leq p_{2}
  4. Λ R 0 \Lambda_{R_{0}}
  5. Λ R 0 = 0 \Lambda_{R_{0}}=0
  6. s o s_{o}
  7. Λ L 1 \Lambda_{L_{1}}
  8. x a = f l - 1 ( p 2 ) x_{a}=f_{l}^{-1}(p_{2})
  9. p 2 p_{2}
  10. ϕ l \phi_{l}
  11. ϕ l ( x a ) \phi_{l}(x_{a})

Mass_in_special_relativity.html

  1. p μ = m v μ p^{\mu}=mv^{\mu}\,
  2. F μ = m A μ . F^{\mu}=mA^{\mu}.\!
  3. E 2 - ( p c ) 2 = ( m c 2 ) 2 E^{2}-(pc)^{2}=(mc^{2})^{2}\,\!
  4. E 2 - ( p c ) 2 = 0 E^{2}-(pc)^{2}=0\,\!
  5. E = p c E=pc\,\!
  6. E 0 = m c 2 E_{0}=mc^{2}\,\!
  7. E = ( m c 2 ) 2 + ( p c ) 2 E=\sqrt{(mc^{2})^{2}+(pc)^{2}}\,\!
  8. ( c , v ) (c,\vec{v})
  9. ( E , p c ) (E,\vec{p}c)
  10. p c = E v c pc=E{v\over c}
  11. E 2 = ( m c 2 ) 2 + E 2 v 2 c 2 , E^{2}=(mc^{2})^{2}+E^{2}{v^{2}\over c^{2}},
  12. E = m c 2 1 - v 2 c 2 E={mc^{2}\over\sqrt{1-\displaystyle{v^{2}\over c^{2}}}}
  13. p = m v 1 - v 2 c 2 . p={mv\over\sqrt{1-\displaystyle{v^{2}\over c^{2}}}}.
  14. E 0 = m c 2 E_{0}=mc^{2}\,
  15. E = γ m c 2 E=\gamma mc^{2}\,
  16. p = m v γ . p=mv\gamma\,.
  17. m 2 = E 2 - p 2 m^{2}=E^{2}-p^{2}\,\!
  18. E 2 - p 2 E^{2}-p^{2}
  19. p \vec{p}
  20. p \vec{p}
  21. m = E 2 - ( p c ) 2 c 2 m=\frac{\sqrt{E^{2}-(pc)^{2}}}{c^{2}}
  22. m 2 = ( E ) 2 - p 2 m^{2}=\left(\sum E\right)^{2}-\left\|\sum\vec{p}\ \right\|^{2}
  23. p \vec{p}
  24. m e m = ( 4 / 3 ) E e m / c 2 m_{em}=(4/3)E_{em}/c^{2}
  25. m L = γ 3 m m_{L}=\gamma^{3}m
  26. m T = γ m m_{T}=\gamma m
  27. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  28. m L m_{L}
  29. m T m_{T}
  30. m T m_{T}
  31. m m
  32. γ \gamma
  33. f x = m γ 3 a x = m L a x , f_{x}=m\gamma^{3}a_{x}=m_{L}a_{x},\,
  34. f y = m γ a y = m T a y , f_{y}=m\gamma a_{y}=m_{T}a_{y},\,
  35. f z = m γ a z = m T a z . f_{z}=m\gamma a_{z}=m_{T}a_{z}.\,
  36. m r e l = E c 2 m_{rel}=\frac{E}{c^{2}}\!
  37. m 0 = E 0 c 2 m_{0}=\frac{E_{0}}{c^{2}}\!
  38. m r e l m 0 = γ \frac{m_{rel}}{m_{0}}=\gamma\!
  39. m r e l = E / c 2 m_{rel}=E/c^{2}\!
  40. m r e l = γ m 0 m_{rel}=\gamma m_{0}\!
  41. γ \gamma
  42. γ \gamma
  43. 𝐩 = m r e l 𝐯 \mathbf{p}=m_{rel}\mathbf{v}
  44. 𝐟 = d ( m r e l 𝐯 ) d t , \mathbf{f}=\frac{d(m_{rel}\mathbf{v})}{dt},\!
  45. 𝐟 = m r e l 𝐚 \mathbf{f}=m_{rel}\mathbf{a}
  46. m r e l m_{rel}\,
  47. d ( m r e l 𝐯 ) {d(m_{rel}\mathbf{v})}\!
  48. E = m c 2 E=mc^{2}
  49. p = m 0 v 1 - v 2 c 2 p={m_{0}v\over{\sqrt{1-\frac{v^{2}}{c^{2}}}}}\!

Mass_spectrum.html

  1. m / e m/e
  2. m / z m/z

Mass–energy_equivalence.html

  1. E E
  2. m m
  3. c c
  4. E = m c 2 E=mc^{2}
  5. E [ u s s u b , u 0 ] E[u^{\prime}ssub^{\prime},u^{\prime}0^{\prime}]
  6. E E
  7. m m
  8. ε ε
  9. M M
  10. m m
  11. E r 2 - | p | 2 c 2 = m 0 2 c 4 E r 2 - ( p c ) 2 = ( m 0 c 2 ) 2 \begin{aligned}\displaystyle E_{r}^{2}-|\vec{p}\,|^{2}c^{2}&\displaystyle=m_{0% }^{2}c^{4}\\ \displaystyle E_{r}^{2}-(pc)^{2}&\displaystyle=(m_{0}c^{2})^{2}\end{aligned}
  12. E r = ( m 0 c 2 ) 2 + ( p c ) 2 E_{r}=\sqrt{(m_{0}c^{2})^{2}+(pc)^{2}}\,\!
  13. E E
  14. m m
  15. E = h f E=hf
  16. h h
  17. f f
  18. E = m E=m
  19. E / m E/m
  20. E / m = E/m=
  21. 89 , 875 , 517 , 873 , 681 , 764 J / k g 89,875,517,873,681,764J/kg
  22. E k = m 0 ( γ - 1 ) c 2 = m 0 c 2 1 - v 2 c 2 - m 0 c 2 , E_{k}=m_{0}(\gamma-1)c^{2}=\frac{m_{0}c^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-m_{% 0}c^{2},
  23. v v
  24. m 0 m_{0}
  25. γ \gamma
  26. E k = 1 2 m 0 v 2 + E_{k}=\frac{1}{2}m_{0}v^{2}+\cdots
  27. P = m 0 v 1 - v 2 c 2 . P=\frac{m_{0}v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.
  28. m m
  29. m = m 0 1 - v 2 c 2 m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  30. E k = m c 2 - m 0 c 2 . E_{k}=mc^{2}-m_{0}c^{2}.\,
  31. E = m c 2 E=mc^{2}\,
  32. m rel = m 0 1 - v 2 c 2 . m_{\mathrm{rel}}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.
  33. m 0 m_{0}
  34. m m
  35. E = m 0 c 2 [ 1 + 1 2 ( v c ) 2 + 3 8 ( v c ) 4 + 5 16 ( v c ) 6 + ] . E=m_{0}c^{2}\left[1+\frac{1}{2}\left(\frac{v}{c}\right)^{2}+\frac{3}{8}\left(% \frac{v}{c}\right)^{4}+\frac{5}{16}\left(\frac{v}{c}\right)^{6}+\ldots\right].
  36. v c \frac{v}{c}
  37. E m 0 c 2 + 1 2 m 0 v 2 . E\approx m_{0}c^{2}+\frac{1}{2}m_{0}v^{2}.
  38. m L = m 0 ( 1 - v 2 c 2 ) 3 , m T = m 0 1 - v 2 c 2 m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=% \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  39. m 0 = 4 3 E e m c 2 m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}
  40. m e m = E e m / c 2 . m_{em}=E_{em}/c^{2}\,.
  41. m 0 = 4 3 E e m c 2 m_{0}=\tfrac{4}{3}\tfrac{E_{em}}{c^{2}}
  42. L L
  43. Δ m Δm
  44. L L
  45. E k = m c 2 ( 1 1 - v 2 c 2 - 1 ) E_{k}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
  46. ( H 0 - E 0 ) - ( H 1 - E 1 ) = E ( 1 1 - v 2 c 2 - 1 ) \left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left(\frac{1}{\sqrt{1-% \frac{v^{2}}{c^{2}}}}-1\right)
  47. H E H−E
  48. K 0 - K 1 = E ( 1 1 - v 2 c 2 - 1 ) K_{0}-K_{1}=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)
  49. v c \frac{v}{c}
  50. K 0 - K 1 = E c 2 v 2 2 . K_{0}-K_{1}=\frac{E}{c^{2}}\frac{v^{2}}{2}.
  51. M M
  52. v v
  53. P = M v P=Mv
  54. E 2 \frac{E}{2}
  55. 1 v c 1−\frac{v}{c}
  56. c c
  57. v c \frac{v}{c}
  58. Δ P ΔP
  59. Δ P = v c E 2 c . \Delta P={v\over c}{E\over 2c}.
  60. Δ P ΔP
  61. Δ P ΔP
  62. 2 Δ P = v E c 2 . 2\Delta P=v{E\over c^{2}}.
  63. P = M v - 2 Δ P = ( M - E c 2 ) v . P^{\prime}=Mv-2\Delta P=\left(M-{E\over c^{2}}\right)v.
  64. c < s u p > 2 c<sup>2
  65. E = m c < s u p > 2 E=mc<sup>2
  66. K < s u b > 0 K 1 = L / V 2 v 2 / 2 K<sub>0−K_{1}=L/V^{2}v^{2}/2
  67. E = m c < s u p > 2 E=mc<sup>2
  68. v = d x ( 4 ) / d t \,v=dx^{(4)}/dt
  69. v ~ = d x ( 4 ) / d τ \tilde{v}=dx^{(4)}/d\tau
  70. d τ = d t 1 - ( v 2 / c 2 ) d\tau=dt\cdot\sqrt{1-(v^{2}/c^{2})}
  71. p ( 4 ) = m 0 d x ( 4 ) / d τ p^{(4)}=m_{0}\cdot dx^{(4)}/d\tau
  72. d τ
  73. d t dt
  74. C < s u b > p C<sub>p

Master_theorem.html

  1. f ( n ) f(n)
  2. T ( n ) = a T ( n b ) + f ( n ) T(n)=a\;T\left(\frac{n}{b}\right)+f(n)
  3. T ( n ) = a T ( n b ) + f ( n ) where a 1 , b > 1 T(n)=a\;T\!\left(\frac{n}{b}\right)+f(n)\;\;\;\;\mbox{where}~{}\;\;a\geq 1% \mbox{, }~{}b>1
  4. f ( n ) O ( n c ) f(n)\in O\left(n^{c}\right)
  5. c < log b a c<\log_{b}a
  6. T ( n ) Θ ( n log b a ) T(n)\in\Theta\left(n^{\log_{b}a}\right)
  7. T ( n ) = 8 T ( n 2 ) + 1000 n 2 T(n)=8T\left(\frac{n}{2}\right)+1000n^{2}
  8. a = 8 , b = 2 , f ( n ) = 1000 n 2 a=8,\,b=2,\,f(n)=1000n^{2}
  9. f ( n ) O ( n c ) f(n)\in O\left(n^{c}\right)
  10. c = 2 c=2
  11. log b a = log 2 8 = 3 > c \log_{b}a=\log_{2}8=3>c
  12. T ( n ) Θ ( n log b a ) = Θ ( n 3 ) T(n)\in\Theta\left(n^{\log_{b}a}\right)=\Theta\left(n^{3}\right)
  13. T ( n ) = 1001 n 3 - 1000 n 2 T(n)=1001n^{3}-1000n^{2}
  14. T ( 1 ) = 1 T(1)=1
  15. f ( n ) = Θ ( n c log k n ) f(n)=\Theta\left(n^{c}\log^{k}n\right)
  16. c = log b a c=\log_{b}a
  17. T ( n ) = Θ ( n c log k + 1 n ) T(n)=\Theta\left(n^{c}\log^{k+1}n\right)
  18. T ( n ) = 2 T ( n 2 ) + 10 n T(n)=2T\left(\frac{n}{2}\right)+10n
  19. a = 2 , b = 2 , c = 1 , f ( n ) = 10 n a=2,\,b=2,\,c=1,\,f(n)=10n
  20. f ( n ) = Θ ( n c log k n ) f(n)=\Theta\left(n^{c}\log^{k}n\right)
  21. c = 1 , k = 0 c=1,k=0
  22. log b a = log 2 2 = 1 \log_{b}a=\log_{2}2=1
  23. c = log b a c=\log_{b}a
  24. T ( n ) = Θ ( n log b a log k + 1 n ) = Θ ( n 1 log 1 n ) = Θ ( n log n ) T(n)=\Theta\left(n^{\log_{b}a}\log^{k+1}n\right)=\Theta\left(n^{1}\log^{1}n% \right)=\Theta\left(n\log n\right)
  25. T ( n ) = n + 10 n log 2 n T(n)=n+10n\log_{2}n
  26. T ( 1 ) = 1 T(1)=1
  27. f ( n ) = Ω ( n c ) f(n)=\Omega\left(n^{c}\right)
  28. c > log b a c>\log_{b}a
  29. a f ( n b ) k f ( n ) af\left(\frac{n}{b}\right)\leq kf(n)
  30. k < 1 k<1
  31. n n
  32. T ( n ) = Θ ( f ( n ) ) T\left(n\right)=\Theta\left(f(n)\right)
  33. T ( n ) = 2 T ( n 2 ) + n 2 T(n)=2T\left(\frac{n}{2}\right)+n^{2}
  34. a = 2 , b = 2 , f ( n ) = n 2 a=2,\,b=2,\,f(n)=n^{2}
  35. f ( n ) = Ω ( n c ) f(n)=\Omega\left(n^{c}\right)
  36. c = 2 c=2
  37. log b a = log 2 2 = 1 \log_{b}a=\log_{2}2=1
  38. c > log b a c>\log_{b}a
  39. 2 ( n 2 4 ) k n 2 2\left(\frac{n^{2}}{4}\right)\leq kn^{2}
  40. k = 1 / 2 k=1/2
  41. T ( n ) = Θ ( f ( n ) ) = Θ ( n 2 ) . T\left(n\right)=\Theta\left(f(n)\right)=\Theta\left(n^{2}\right).
  42. T ( n ) = 2 n 2 - n T(n)=2n^{2}-n
  43. T ( 1 ) = 1 T(1)=1
  44. T ( n ) = 2 n T ( n 2 ) + n n T(n)=2^{n}T\left(\frac{n}{2}\right)+n^{n}
  45. T ( n ) = 2 T ( n 2 ) + n log n T(n)=2T\left(\frac{n}{2}\right)+\frac{n}{\log n}
  46. n log b a n^{\log_{b}a}
  47. T ( n ) = 0.5 T ( n 2 ) + n T(n)=0.5T\left(\frac{n}{2}\right)+n
  48. T ( n ) = 64 T ( n 8 ) - n 2 log n T(n)=64T\left(\frac{n}{8}\right)-n^{2}\log n
  49. T ( n ) = T ( n 2 ) + n ( 2 - cos n ) T(n)=T\left(\frac{n}{2}\right)+n(2-\cos n)
  50. f ( n ) f(n)
  51. n log b a n^{\log_{b}a}
  52. f ( n ) n log b a = n log n n l o g 2 2 = n n log n = 1 log n \frac{f(n)}{n^{\log_{b}a}}=\frac{\frac{n}{\log n}}{n^{log_{2}2}}=\frac{n}{n% \log n}=\frac{1}{\log n}
  53. 1 log n < n ϵ \frac{1}{\log n}<n^{\epsilon}
  54. ϵ > 0 \epsilon>0
  55. T ( n ) = T ( n 2 ) + O ( 1 ) T(n)=T\left(\frac{n}{2}\right)+O(1)
  56. O ( log n ) O(\log n)
  57. c = log b a c=\log_{b}a
  58. a = 1 , b = 2 , c = 0 , k = 0 a=1,b=2,c=0,k=0
  59. T ( n ) = 2 T ( n 2 ) + O ( 1 ) T(n)=2T\left(\frac{n}{2}\right)+O(1)
  60. O ( n ) O(n)
  61. c < log b a c<\log_{b}a
  62. a = 2 , b = 2 , c = 0 a=2,b=2,c=0
  63. T ( n ) = 2 T ( n 2 ) + O ( log n ) T(n)=2T\left(\frac{n}{2}\right)+O(\log n)
  64. O ( n ) O(n)
  65. p = 1 p=1
  66. g ( u ) = log ( u ) g(u)=\log(u)
  67. Θ ( 2 n - log n ) \Theta(2n-\log n)
  68. T ( n ) = 2 T ( n 2 ) + O ( n ) T(n)=2T\left(\frac{n}{2}\right)+O(n)
  69. O ( n log n ) O(n\log n)
  70. c = log b a c=\log_{b}a
  71. a = 2 , b = 2 , c = 1 , k = 0 a=2,b=2,c=1,k=0

Matching_(graph_theory).html

  1. ν ( G ) \nu(G)
  2. G G
  3. | A B | 2 | B A | . |A\setminus B|\leq 2|B\setminus A|.
  4. | A | = | A B | + | A B | 2 | B A | + 2 | B A | = 2 | B | . |A|=|A\cap B|+|A\setminus B|\leq 2|B\cap A|+2|B\setminus A|=2|B|.
  5. k 0 m k x k . \sum_{k\geq 0}m_{k}x^{k}.
  6. k 0 ( - 1 ) k m k x n - 2 k , \sum_{k\geq 0}(-1)^{k}m_{k}x^{n-2k},
  7. G = ( V = ( X , Y ) , E ) G=(V=(X,Y),E)
  8. Y \ Y
  9. O ( E ) \ O(E)
  10. O ( V E ) \ O(VE)
  11. s s
  12. X \ X
  13. t \ t
  14. Y \ Y
  15. s \ s
  16. t \ t
  17. X \ X
  18. Y \ Y
  19. O ( V E ) O(\sqrt{V}E)
  20. O ( V 2.376 ) O(V^{2.376})
  21. O ~ ( E 10 / 7 ) \tilde{O}(E^{10/7})
  22. k k
  23. | X | < | Y | |X|<|Y|
  24. O ( min { | X | k , E } + k min { k 2 , E } ) O\left(\min\{|X|k,E\}+\sqrt{k}\min\{k^{2},E\}\right)
  25. λ \lambda
  26. O ( min { | X | k , | X | | Y | λ , E } + k 2 + k 2.5 λ ) O\left(\min\left\{|X|k,\frac{|X||Y|}{\lambda},E\right\}+k^{2}+\frac{k^{2.5}}{% \lambda}\right)
  27. O ( V 2 E ) O(V^{2}E)
  28. O ( V 2 log V + V E ) O(V^{2}\log{V}+VE)
  29. O ( V 2.376 ) O(V^{2.376})
  30. O ( V E ) O(VE)
  31. O ~ ( V 2.376 ) \tilde{O}(V^{2.376})
  32. O ( V 1 / 2 E ) O(V^{1/2}E)
  33. O ( ( V / log V ) 1 / 2 E ) O((V/\log V)^{1/2}E)
  34. E = Θ ( V 2 ) E=\Theta(V^{2})
  35. O ( V + E ) O(V+E)

Mathematics_of_oscillation.html

  1. a n a_{n}
  2. ω ( a n ) \omega(a_{n})
  3. a n a_{n}
  4. ω ( a n ) = lim sup a n - lim inf a n . \omega(a_{n})=\lim\sup a_{n}-\lim\inf a_{n}.
  5. lim sup a n \lim\sup a_{n}
  6. lim inf a n \lim\inf a_{n}
  7. f f
  8. f f
  9. I I
  10. f f
  11. ω f ( I ) = sup x I f ( x ) - inf x I f ( x ) . \omega_{f}(I)=\sup_{x\in I}f(x)-\inf_{x\in I}f(x).
  12. f : X f:X\to\mathbb{R}
  13. X X
  14. f f
  15. U U
  16. ω f ( U ) = sup x U f ( x ) - inf x U f ( x ) . \omega_{f}(U)=\sup_{x\in U}f(x)-\inf_{x\in U}f(x).
  17. f f
  18. x 0 x_{0}
  19. ϵ 0 \epsilon\to 0
  20. f f
  21. ϵ \epsilon
  22. x 0 x_{0}
  23. ω f ( x 0 ) = lim ϵ 0 ω f ( x 0 - ϵ , x 0 + ϵ ) . \omega_{f}(x_{0})=\lim_{\epsilon\to 0}\omega_{f}(x_{0}-\epsilon,x_{0}+\epsilon).
  24. x 0 x_{0}
  25. x 0 x_{0}
  26. f : X f:X\to\mathbb{R}
  27. ω f ( x 0 ) = lim ϵ 0 ω f ( B ϵ ( x 0 ) ) . \omega_{f}(x_{0})=\lim_{\epsilon\to 0}\omega_{f}(B_{\epsilon}(x_{0})).
  28. ω f ( x 0 ) = 0. \omega_{f}(x_{0})=0.
  29. ω ( x ) = inf { diam ( f ( U ) ) U is a neighborhood of x } \omega(x)=\inf\left\{\mathrm{diam}(f(U))\mid U\mathrm{\ is\ a\ neighborhood\ % of\ }x\right\}

Mathieu_group.html

  1. q + 1 q+1
  2. x 4 x^{4}
  3. 7 x 4 7x^{4}

Matrix_mechanics.html

  1. 0 T P d X = n h \int_{0}^{T}P\;dX=nh
  2. X ( t ) = n = - e 2 π i n t / T X n X(t)=\sum_{n=-\infty}^{\infty}e^{2\pi int/T}X_{n}
  3. X n = X - n * X_{n}=X_{-n}^{*}
  4. E n - E m h ( n - m ) / T E_{n}-E_{m}\approx h(n-m)/T
  5. ( n m ) (n−m)
  6. X n m = X m n * X_{nm}=X_{mn}^{*}
  7. X n m ( t ) = e 2 π i ( E n - E m ) t / h X n m ( 0 ) X_{nm}(t)=e^{2\pi i(E_{n}-E_{m})t/h}X_{nm}(0)
  8. ( X P ) m n = k = 0 X m k P k n (XP)_{mn}=\sum_{k=0}^{\infty}X_{mk}P_{kn}
  9. k ( X n k P k m - P n k X k m ) = i h 2 π δ n m \sum_{k}(X_{nk}P_{km}-P_{nk}X_{km})={ih\over 2\pi}~{}\delta_{nm}
  10. H = 1 2 ( P 2 + X 2 ) H={1\over 2}(P^{2}+X^{2})
  11. H H
  12. E E
  13. X ( t ) = 2 E cos ( t ) , P ( t ) = 2 E sin ( t ) X(t)=\sqrt{2E}\cos(t),\qquad P(t)=\sqrt{2E}\sin(t)
  14. P d X PdX
  15. 2 ¯ E √\overline{2}{E}
  16. 2 π E 2πE
  17. E = n h 2 π E={nh\over 2\pi}
  18. ħ = 1 ħ=1
  19. X ( t ) X(t)
  20. P ( t ) P(t)
  21. A ( t ) = X ( t ) + i P ( t ) = 2 E e i t , A ( t ) = X ( t ) - i P ( t ) = 2 E e - i t A(t)=X(t)+iP(t)=\sqrt{2E}\,e^{it},\quad A^{\dagger}(t)=X(t)-iP(t)=\sqrt{2E}\,e% ^{-it}
  22. A A
  23. A ( t ) A(t)
  24. ( m n ) (m−n)
  25. A A
  26. A A
  27. 2 X ( 0 ) = h 2 π [ 0 1 0 0 0 1 0 2 0 0 0 2 0 3 0 0 0 3 0 4 ] , \sqrt{2}X(0)=\sqrt{\frac{h}{2\pi}}\;\begin{bmatrix}0&\sqrt{1}&0&0&0&\cdots\\ \sqrt{1}&0&\sqrt{2}&0&0&\cdots\\ 0&\sqrt{2}&0&\sqrt{3}&0&\cdots\\ 0&0&\sqrt{3}&0&\sqrt{4}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \end{bmatrix},
  28. 2 P ( 0 ) = h 2 π [ 0 i 1 0 0 0 - i 1 0 i 2 0 0 0 - i 2 0 i 3 0 0 0 - i 3 0 i 4 ] , \sqrt{2}P(0)=\sqrt{\frac{h}{2\pi}}\;\begin{bmatrix}0&i\sqrt{1}&0&0&0&\cdots\\ -i\sqrt{1}&0&i\sqrt{2}&0&0&\cdots\\ 0&-i\sqrt{2}&0&i\sqrt{3}&0&\cdots\\ 0&0&-i\sqrt{3}&0&i\sqrt{4}&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \end{bmatrix},
  29. X ( t ) X(t)
  30. P ( t ) P(t)
  31. X m n ( t ) = X m n ( 0 ) e i ( E m - E n ) t , P m n ( t ) = P m n ( 0 ) e i ( E m - E n ) t X_{mn}(t)=X_{mn}(0)e^{i(E_{m}-E_{n})t},\quad P_{mn}(t)=P_{mn}(0)e^{i(E_{m}-E_{% n})t}
  32. X X
  33. P P
  34. X P + P X XP+PX
  35. [ X , P ] = ( X P - P X ) [X,P]=(XP-PX)
  36. X P P X XP−PX
  37. i ħ
  38. H = 1 2 ( X 2 + P 2 ) H={1\over 2}(X^{2}+P^{2})
  39. H = 1 2 P 2 + 1 2 X 2 + ϵ X 3 . H={1\over 2}P^{2}+{1\over 2}X^{2}+\epsilon X^{3}~{}.
  40. X X
  41. P P
  42. d X d t = P d P d t = - X - 3 ϵ X 2 . {dX\over dt}=P\quad{dP\over dt}=-X-3\epsilon X^{2}~{}.
  43. H H
  44. X X
  45. P P
  46. d H d t = P * d P d t + ( X + 3 ϵ X 2 ) * d X d t = 0 , {dH\over dt}=P*{dP\over dt}+(X+3\epsilon X^{2})*{dX\over dt}=0~{},
  47. A B A∗B
  48. A * B = 1 2 ( A B + B A ) A*B={1\over 2}(AB+BA)~{}
  49. H H
  50. H H
  51. ħ ħ
  52. 0 T k P m k ( t ) d X k n d t d t ? J m n . \int_{0}^{T}\sum_{k}P_{mk}(t){dX_{kn}\over dt}dt\,\,\stackrel{\scriptstyle?}{% \approx}\,\,J_{mn}~{}.
  53. ħ ħ
  54. ħ ħ
  55. d d J 0 T P d X = 1 {d\over dJ}\int_{0}^{T}PdX=1
  56. = 0 T d t ( d P d J d X d t + P d d J d X d t ) = 0 T d t ( d P d J d X d t - d P d t d X d J ) =\int_{0}^{T}dt\left({dP\over dJ}{dX\over dt}+P{d\over dJ}{dX\over dt}\right)=% \int_{0}^{T}dt\left({dP\over dJ}{dX\over dt}-{dP\over dt}{dX\over dJ}\right)\,
  57. 2 π T 0 T d t ( d p d J d X d θ - d P d θ d X d J ) = 1 . {2\pi\over T}\int_{0}^{T}dt\left({dp\over dJ}{dX\over d\theta}-{dP\over d% \theta}{dX\over dJ}\right)=1\,.
  58. A ( m + r ) ( n + r ) - A m n r ( d A d J ) m n A_{(m+r)(n+r)}-A_{mn}\approx r\;\left({dA\over dJ}\right)_{mn}\,
  59. i k A m ( m + k ) ( T 2 π d A d t ) m ( m + k ) = ( d A d θ ) m ( m + k ) . ikA_{m(m+k)}\approx\left({T\over 2\pi}{dA\over dt}\right)_{m(m+k)}=\left({dA% \over d\theta}\right)_{m(m+k)}\,.
  60. ( d A / d J ) r r (dA/dJ)rr
  61. ( A B - B A ) [ 0 , k ] = r = - ( A [ 0 , r ] B [ r , k ] - A [ r , k ] B [ 0 , r ] ) (AB-BA)[0,k]=\sum_{r=-\infty}^{\infty}\left(A[0,r]B[r,k]-A[r,k]B[0,r]\right)
  62. = r ( A [ - r + k , k ] + ( r - k ) d A d J [ r ] ) ( B [ 0 , k - r ] + r d B d J [ r - k ] ) - r A [ r , k ] B [ 0 , r ] . =\sum_{r}\left(\;A[-r+k,k]+(r-k){dA\over dJ}[r]\;\right)\left(\;B[0,k-r]+r{dB% \over dJ}[r-k]\;\right)-\sum_{r}A[r,k]B[0,r]\,.
  63. r r
  64. r ( A [ r , k ] - r d A d J [ k - r ] ) ( B [ 0 , r ] + ( k - r ) d B d J [ r ] ) - r A [ r , k ] B [ 0 , r ] \sum_{r^{\prime}}(\;A[r^{\prime},k]-r^{\prime}{dA\over dJ}[k-r^{\prime}]\;)% \left(\;B[0,r^{\prime}]+(k-r^{\prime}){dB\over dJ}[r^{\prime}]\;\right)-\sum_{% r}A[r,k]B[0,r]\,
  65. r ( d B d J [ r ] ( k - r ) A [ r , k ] - d A d J [ k - r ] r B [ 0 , r ] ) \sum_{r^{\prime}}\left(\;{dB\over dJ}[r^{\prime}](k-r^{\prime})A[r^{\prime},k]% -{dA\over dJ}[k-r^{\prime}]r^{\prime}B[0,r^{\prime}]\right)
  66. ( A B - B A ) [ 0 , k ] = r ( d B d J [ r ] i d A d θ [ k - r ] - d A d J [ k - r ] i d B d θ [ r ] ) (AB-BA)[0,k]=\sum_{r^{\prime}}\left(\;{dB\over dJ}[r^{\prime}]i{dA\over d% \theta}[k-r^{\prime}]-{dA\over dJ}[k-r^{\prime}]i{dB\over d\theta}[r^{\prime}]% \right)\,
  67. i i
  68. k k
  69. i h 2 π { X , P } PB [ X , P ] X P - P X = i h 2 π \frac{ih}{2\pi}\{X,P\}_{\mathrm{PB}}\qquad\qquad\longmapsto\qquad\qquad[X,P]% \equiv XP-PX=\frac{ih}{2\pi}\,
  70. m n ψ m * A m n ψ n \sum_{mn}\psi_{m}^{*}A_{mn}\psi_{n}
  71. A m n ( t ) = e i ( E m - E n ) t A m n ( 0 ) A_{mn}(t)=e^{i(E_{m}-E_{n})t}A_{mn}(0)
  72. d A m n d t = i ( E m - E n ) A m n {dA_{mn}\over dt}=i(E_{m}-E_{n})A_{mn}
  73. d A d t = i ( H A - A H ) . {dA\over dt}=i(HA-AH).
  74. A ( t ) = e i H t A ( 0 ) e - i H t . \,A(t)=e^{iHt}A(0)e^{-iHt}.
  75. | ψ ( t ) = e - i H t | ψ ( 0 ) , d | ψ d t = - i H | ψ . |\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle,\;\;\;\;{d|\psi\rangle\over dt}=-iH|% \psi\rangle.
  76. A = ( X + t 2 P ) A=(X+t^{2}P)
  77. [ a , b c ] = a b c - b c a = a b c - b a c + b a c - b c a = [ a , b ] c + b [ a , c ] [a,bc]=abc-bca=abc-bac+bac-bca=[a,b]c+b[a,c]\,
  78. [ P , X n ] = - i n X n - 1 [P,X^{n}]=-in~{}X^{n-1}\,
  79. | ψ = x ψ ( x ) | x |\psi\rangle=\int_{x}\psi(x)|x\rangle\,
  80. X | ψ = x x ψ ( x ) | x . X|\psi\rangle=\int_{x}x\psi(x)|x\rangle~{}.
  81. ψ ψ
  82. D x ψ ( x ) | x = x ψ ( x ) | x D\int_{x}\psi(x)|x\rangle=\int_{x}\psi^{\prime}(x)|x\rangle\,
  83. ( D X - X D ) | ψ = x [ ( x ψ ( x ) ) - x ψ ( x ) ] | x = x ψ ( x ) | x = | ψ (DX-XD)|\psi\rangle=\int_{x}\left[\left(x\psi(x)\right)^{\prime}-x\psi^{\prime% }(x)\right]|x\rangle=\int_{x}\psi(x)|x\rangle=|\psi\rangle\,
  84. [ P + i D , X ] = 0 [P+iD,X]=0\,
  85. ( P + i D ) | x = f ( x ) | x (P+iD)|x\rangle=f(x)|x\rangle\,
  86. | x |x\rangle
  87. f ( x ) f(x)
  88. ψ ( x ) e - i f ( x ) ψ ( x ) \psi(x)\rightarrow e^{-if(x)}\psi(x)\,
  89. i D i D + f ( X ) iD\rightarrow iD+f(X)\,
  90. P x ψ ( x ) | x = x - i ψ ( x ) | x P\int_{x}\psi(x)|x\rangle=\int_{x}-i\psi^{\prime}(x)|x\rangle\,
  91. [ P 2 2 m + V ( X ) ] x ψ x | x = x [ - 1 2 m 2 x 2 + V ( x ) ] ψ x | x \left[{P^{2}\over 2m}+V(X)\right]\int_{x}\psi_{x}|x\rangle=\int_{x}\left[-{1% \over 2m}{\partial^{2}\over\partial x^{2}}+V(x)\right]\psi_{x}|x\rangle
  92. [ X i , X j ] = 0 [X_{i},X_{j}]=0\,
  93. [ P i , P j ] = 0 [P_{i},P_{j}]=0\,
  94. [ X i , P j ] = i δ i j . [X_{i},P_{j}]=i\delta_{ij}\,.
  95. A / t = 0 ∂A/∂t=0
  96. i d A d t = [ A , H ] = A H - H A i\hbar{dA\over dt}=[A,H]=AH-HA
  97. p 2 / 2 m + V ( x ) p^{2}/2m+V(x)
  98. d X d t = P m , d P d t = - V {dX\over dt}={P\over m},\quad{dP\over dt}=-\nabla V
  99. d d t X = d d t ψ | X | ψ = 1 m ψ | P | ψ = 1 m P \frac{d}{dt}\langle X\rangle=\frac{d}{dt}\langle\psi|X|\psi\rangle=\frac{1}{m}% \langle\psi|P|\psi\rangle=\frac{1}{m}\langle P\rangle
  100. d d t P = d d t ψ | P | ψ = ψ | ( - V ) | ψ = - V . \frac{d}{dt}\langle P\rangle=\frac{d}{dt}\langle\psi|P|\psi\rangle=\langle\psi% |(-\nabla V)|\psi\rangle=-\langle\nabla V\rangle\,.
  101. x , p p x,pp
  102. x , p x,p
  103. x x + d x = x + H p d t x\rightarrow x+dx=x+{\partial H\over\partial p}dt
  104. p p + d p = p - H x d t . p\rightarrow p+dp=p-{\partial H\over\partial x}dt~{}.
  105. G G
  106. G G
  107. d x = G p d s = { G , X } d s dx={\partial G\over\partial p}ds=\{G,X\}ds\,
  108. d p = - G x d s = { G , P } d s . dp=-{\partial G\over\partial x}ds=\{G,P\}ds\,.
  109. A ( x , p ) A(x,p)
  110. d A = A x d x + A p d p = { A , G } d s . dA={\partial A\over\partial x}dx+{\partial A\over\partial p}dp=\{A,G\}ds\,.
  111. G G
  112. G G
  113. d A = i [ G , A ] d s . dA=i[G,A]ds\,.
  114. A = U A U A^{\prime}=U^{\dagger}AU\,
  115. s s
  116. U U
  117. U = U - 1 . U^{\dagger}=U^{-1}\,.
  118. d X = i [ X , P ] d s = d s dX=i[X,P]ds=ds\,
  119. d P = i [ P , P ] d s = 0 . dP=i[P,P]ds=0\,.
  120. X X + s I . X\rightarrow X+sI~{}.
  121. d H d s = i [ L , H ] = 0 . {dH\over ds}=i[L,H]=0\,.
  122. d L d t = i [ H , L ] = 0 {dL\over dt}=i[H,L]=0\,
  123. U - 1 H U = H U^{-1}HU=H\,
  124. U H = H U UH=HU
  125. P ( ω ) = 2 3 ω 4 | d i | 2 . P(\omega)={2\over 3}{\omega^{4}}|d_{i}|^{2}\,.
  126. P i j = 2 3 ( E i - E j ) 4 | X i j | 2 . P_{ij}={2\over 3}(E_{i}-E_{j})^{4}|X_{ij}|^{2}\,.
  127. L i = ϵ i j k X j P k L_{i}=\epsilon_{ijk}X^{j}P^{k}\,
  128. [ L i , X j ] = i ϵ i j k X k [L_{i},X_{j}]=i\epsilon_{ijk}X_{k}\,
  129. [ L z , X ] = i Y [L_{z},X]=iY\,
  130. [ L z , Y ] = - i X [L_{z},Y]=-iX\,
  131. X + i Y , X i Y X+iY,X−iY
  132. [ L z , X + i Y ] = ( X + i Y ) [L_{z},X+iY]=(X+iY)\,
  133. [ L z , X - i Y ] = - ( X - i Y ) [L_{z},X-iY]=-(X-iY)\,
  134. L z ( ( X + i Y ) | m ) = ( X + i Y ) L z | m + ( X + i Y ) | m = ( m + 1 ) ( X + i Y ) | m L_{z}((X+iY)|m\rangle)=(X+iY)L_{z}|m\rangle+(X+iY)|m\rangle=(m+1)(X+iY)|m\rangle\,
  135. m m
  136. z z
  137. j P i j x j i - X i j p j i = i j 2 m ( E i - E j ) | X i j | 2 = i \sum_{j}P_{ij}x_{ji}-X_{ij}p_{ji}=i\sum_{j}2m(E_{i}-E_{j})|X_{ij}|^{2}=i\,
  138. j 2 m ( E i - E j ) | X i j | 2 = 1 \sum_{j}2m(E_{i}-E_{j})|X_{ij}|^{2}=1\,

Matrix_normal_distribution.html

  1. 𝐌 \mathbf{M}
  2. 𝐔 \mathbf{U}
  3. 𝐕 \mathbf{V}
  4. 𝒩 n , p ( 𝐌 , 𝐔 , 𝐕 ) \mathcal{MN}_{n,p}(\mathbf{M},\mathbf{U},\mathbf{V})
  5. p ( 𝐗 𝐌 , 𝐔 , 𝐕 ) = exp ( - 1 2 tr [ 𝐕 - 1 ( 𝐗 - 𝐌 ) T 𝐔 - 1 ( 𝐗 - 𝐌 ) ] ) ( 2 π ) n p / 2 | 𝐕 | n / 2 | 𝐔 | p / 2 p(\mathbf{X}\mid\mathbf{M},\mathbf{U},\mathbf{V})=\frac{\exp\left(-\frac{1}{2}% \,\mathrm{tr}\left[\mathbf{V}^{-1}(\mathbf{X}-\mathbf{M})^{T}\mathbf{U}^{-1}(% \mathbf{X}-\mathbf{M})\right]\right)}{(2\pi)^{np/2}|\mathbf{V}|^{n/2}|\mathbf{% U}|^{p/2}}
  6. tr \mathrm{tr}
  7. 𝐗 𝒩 n × p ( 𝐌 , 𝐔 , 𝐕 ) , \mathbf{X}\sim\mathcal{MN}_{n\times p}(\mathbf{M},\mathbf{U},\mathbf{V}),
  8. vec ( 𝐗 ) 𝒩 n p ( vec ( 𝐌 ) , 𝐕 𝐔 ) \mathrm{vec}(\mathbf{X})\sim\mathcal{N}_{np}(\mathrm{vec}(\mathbf{M}),\mathbf{% V}\otimes\mathbf{U})
  9. \otimes
  10. vec ( 𝐌 ) \mathrm{vec}(\mathbf{M})
  11. 𝐌 \mathbf{M}
  12. - 1 2 tr [ 𝐕 - 1 ( 𝐗 - 𝐌 ) T 𝐔 - 1 ( 𝐗 - 𝐌 ) ] \displaystyle\;\;\;\;-\frac{1}{2}\,\text{tr}\left[\mathbf{V}^{-1}(\mathbf{X}-% \mathbf{M})^{T}\mathbf{U}^{-1}(\mathbf{X}-\mathbf{M})\right]
  13. | 𝐕 𝐔 | = | 𝐕 | n | 𝐔 | p . |\mathbf{V}\otimes\mathbf{U}|=|\mathbf{V}|^{n}|\mathbf{U}|^{p}.
  14. 𝐗 𝒩 n × p ( 𝐌 , 𝐔 , 𝐕 ) \mathbf{X}\sim\mathcal{MN}_{n\times p}(\mathbf{M},\mathbf{U},\mathbf{V})
  15. E [ 𝐗 ] = 𝐌 E[\mathbf{X}]=\mathbf{M}
  16. E [ ( 𝐗 - 𝐌 ) ( 𝐗 - 𝐌 ) T ] = 𝐔 tr ( 𝐕 ) E[(\mathbf{X}-\mathbf{M})(\mathbf{X}-\mathbf{M})^{T}]=\mathbf{U}\operatorname{% tr}(\mathbf{V})
  17. E [ ( 𝐗 - 𝐌 ) T ( 𝐗 - 𝐌 ) ] = 𝐕 tr ( 𝐔 ) E[(\mathbf{X}-\mathbf{M})^{T}(\mathbf{X}-\mathbf{M})]=\mathbf{V}\operatorname{% tr}(\mathbf{U})
  18. tr \operatorname{tr}
  19. E [ 𝐗𝐀𝐗 T ] \displaystyle E[\mathbf{X}\mathbf{A}\mathbf{X}^{T}]
  20. 𝐗 T 𝒩 p × n ( 𝐌 T , 𝐕 , 𝐔 ) \mathbf{X}^{T}\sim\mathcal{MN}_{p\times n}(\mathbf{M}^{T},\mathbf{V},\mathbf{U})
  21. 𝐃𝐗𝐂 𝒩 n × p ( 𝐃𝐌𝐂 , 𝐃𝐔𝐃 T , 𝐂 T 𝐕𝐂 ) \mathbf{DXC}\sim\mathcal{MN}_{n\times p}(\mathbf{DMC},\mathbf{DUD}^{T},\mathbf% {C}^{T}\mathbf{VC})
  22. 𝐘 i 𝒩 p ( s y m b o l μ , s y m b o l Σ ) with i { 1 , , n } \mathbf{Y}_{i}\sim\mathcal{N}_{p}({symbol\mu},{symbol\Sigma})\,\text{ with }i% \in\{1,\ldots,n\}
  23. 𝐗 \mathbf{X}
  24. 𝐘 i \mathbf{Y}_{i}
  25. 𝐗 𝒩 n × p ( 𝐌 , 𝐔 , 𝐕 ) \mathbf{X}\sim\mathcal{MN}_{n\times p}(\mathbf{M},\mathbf{U},\mathbf{V})
  26. 𝐌 \mathbf{M}
  27. s y m b o l μ {symbol\mu}
  28. 𝐌 = 𝟏 n × s y m b o l μ T \mathbf{M}=\mathbf{1}_{n}\times{symbol\mu}^{T}
  29. 𝐔 \mathbf{U}
  30. 𝐕 = s y m b o l Σ \mathbf{V}={symbol\Sigma}
  31. 𝐗 1 , 𝐗 2 , , 𝐗 k \mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{k}
  32. i = 1 k 𝒩 n × p ( 𝐗 i 𝐌 , 𝐔 , 𝐕 ) . \prod_{i=1}^{k}\mathcal{MN}_{n\times p}(\mathbf{X}_{i}\mid\mathbf{M},\mathbf{U% },\mathbf{V}).
  33. 𝐌 = 1 k i = 1 k 𝐗 i \mathbf{M}=\frac{1}{k}\sum_{i=1}^{k}\mathbf{X}_{i}
  34. 𝐔 = 1 k p i = 1 k ( 𝐗 i - 𝐌 ) 𝐕 - 1 ( 𝐗 i - 𝐌 ) T \mathbf{U}=\frac{1}{kp}\sum_{i=1}^{k}(\mathbf{X}_{i}-\mathbf{M})\mathbf{V}^{-1% }(\mathbf{X}_{i}-\mathbf{M})^{T}
  35. 𝐕 = 1 k n i = 1 k ( 𝐗 i - 𝐌 ) T 𝐔 - 1 ( 𝐗 i - 𝐌 ) , \mathbf{V}=\frac{1}{kn}\sum_{i=1}^{k}(\mathbf{X}_{i}-\mathbf{M})^{T}\mathbf{U}% ^{-1}(\mathbf{X}_{i}-\mathbf{M}),
  36. 𝒩 n × p ( 𝐗 𝐌 , 𝐔 , 𝐕 ) = 𝒩 n × p ( 𝐗 𝐌 , s 𝐔 , 1 / s 𝐕 ) . \mathcal{MN}_{n\times p}(\mathbf{X}\mid\mathbf{M},\mathbf{U},\mathbf{V})=% \mathcal{MN}_{n\times p}(\mathbf{X}\mid\mathbf{M},s\mathbf{U},1/s\mathbf{V}).
  37. 𝐗 \mathbf{X}
  38. 𝐗 𝒩 n × p ( 𝟎 , 𝐈 , 𝐈 ) . \mathbf{X}\sim\mathcal{MN}_{n\times p}(\mathbf{0},\mathbf{I},\mathbf{I}).
  39. 𝐘 = 𝐌 + 𝐀𝐗𝐁 , \mathbf{Y}=\mathbf{M}+\mathbf{A}\mathbf{X}\mathbf{B},
  40. 𝐘 𝒩 n × p ( 𝐌 , 𝐀𝐀 T , 𝐁 T 𝐁 ) , \mathbf{Y}\sim\mathcal{MN}_{n\times p}(\mathbf{M},\mathbf{AA}^{T},\mathbf{B}^{% T}\mathbf{B}),

Matrix_ring.html

  1. M = i I R M=\bigoplus_{i\in I}R
  2. 𝔽 𝕄 I ( R ) \mathbb{CFM}_{I}(R)\,
  3. 𝔽 𝕄 I ( R ) \mathbb{RFM}_{I}(R)
  4. 𝔽 𝕄 I ( R ) \mathbb{RCFM}_{I}(R)
  5. 𝔽 𝕄 I ( D ) \mathbb{CFM}_{I}(D)
  6. 𝔽 𝕄 I ( D ) \mathbb{RFM}_{I}(D)
  7. [ 1 0 0 0 ] [ 1 1 0 0 ] = [ 1 1 0 0 ] \begin{bmatrix}1&0\\ 0&0\end{bmatrix}\begin{bmatrix}1&1\\ 0&0\end{bmatrix}=\begin{bmatrix}1&1\\ 0&0\end{bmatrix}\,
  8. [ 1 1 0 0 ] [ 1 0 0 0 ] = [ 1 0 0 0 ] \begin{bmatrix}1&1\\ 0&0\end{bmatrix}\begin{bmatrix}1&0\\ 0&0\end{bmatrix}=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}\,
  9. [ 0 1 0 0 ] [ 0 1 0 0 ] = [ 0 0 0 0 ] \begin{bmatrix}0&1\\ 0&0\end{bmatrix}\begin{bmatrix}0&1\\ 0&0\end{bmatrix}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\,

Matrix_similarity.html

  1. B = P - 1 A P B=P^{-1}AP
  2. A P - 1 A P A\mapsto P^{-1}AP
  3. X I < s u b > n A XI<sub>n−A

Maximal_torus.html

  1. T = { diag ( e i θ 1 , e i θ 2 , , e i θ n ) : j , θ j } . T=\left\{\mathrm{diag}(e^{i\theta_{1}},e^{i\theta_{2}},\dots,e^{i\theta_{n}}):% \forall j,\theta_{j}\in\mathbb{R}\right\}.
  2. 𝔤 \mathfrak{g}
  3. 𝔤 \mathfrak{g}
  4. W ( T , G ) := N G ( T ) / C G ( T ) . W(T,G):=N_{G}(T)/C_{G}(T).
  5. T = T 0 T=T_{0}
  6. G f ( g ) d g = | W | - 1 T f ( t ) | Δ ( t ) | 2 d t , \displaystyle{\int_{G}f(g)\,dg=|W|^{-1}\int_{T}f(t)|\Delta(t)|^{2}\,dt,}

Maximum_flow_problem.html

  1. N = ( V , E ) \scriptstyle N=(V,E)
  2. s , t V \scriptstyle s,t\in V
  3. N \scriptstyle N
  4. c : E + \scriptstyle c:E\to\mathbb{R}^{+}
  5. c u v \scriptstyle c_{uv}
  6. c ( u , v ) \scriptstyle c(u,v)
  7. f : E + \scriptstyle f:E\to\mathbb{R}^{+}
  8. f u v \scriptstyle f_{uv}
  9. f ( u , v ) \scriptstyle f(u,v)
  10. f u v c u v \scriptstyle f_{uv}\leq c_{uv}
  11. ( u , v ) E \scriptstyle(u,v)\in E
  12. u : ( u , v ) E f u v = u : ( v , u ) E f v u \scriptstyle\sum_{u:(u,v)\in E}f_{uv}=\sum_{u:(v,u)\in E}f_{vu}
  13. v V { s , t } \scriptstyle v\in V\setminus\{s,t\}
  14. | f | = v : ( s , v ) E f s v \scriptstyle|f|=\sum_{v:(s,v)\in E}f_{sv}
  15. s \scriptstyle s
  16. N \scriptstyle N
  17. | f | \scriptstyle|f|
  18. s \scriptstyle s
  19. t \scriptstyle t
  20. G \scriptstyle G
  21. f \scriptstyle f
  22. G \scriptstyle G
  23. G f \scriptstyle G_{f}
  24. G \scriptstyle G
  25. f \scriptstyle f
  26. G f \scriptstyle G_{f}
  27. G \scriptstyle G
  28. e = ( u , v ) \scriptstyle e=(u,v)
  29. G f \scriptstyle G_{f}
  30. c e - f ( e ) \scriptstyle c_{e}-f(e)
  31. e = ( v , u ) \scriptstyle e^{\prime}=(v,u)
  32. G f \scriptstyle G_{f}
  33. f ( e ) \scriptstyle f(e)
  34. O ( E V ) \scriptscriptstyle O(E\sqrt{V})
  35. O ( E V log E / V log V V ) \scriptscriptstyle O(EV\log_{E/V\log V}V)
  36. O ( E min ( V 2 / 3 , E ) log ( V 2 / E ) log U ) \scriptscriptstyle O(E\min(V^{2/3},\sqrt{E})\log(V^{2}/E)\log{U})
  37. O ( V E ) \scriptscriptstyle O(VE)
  38. E O ( V 16 15 - ϵ ) E\leq O(V^{{16\over 15}-\epsilon})
  39. E > V 1 + ϵ E>V^{1+\epsilon}
  40. N = ( V , E ) N=(V,E)
  41. c : V + c:V\mapsto\mathbb{R}^{+}
  42. c ( v ) c(v)
  43. f f
  44. i V f i v c ( v ) v V \ { s , t } \sum_{i\in V}f_{iv}\leq c(v)\qquad\forall v\in V\backslash\{s,t\}
  45. N N
  46. N N
  47. v V v\in V
  48. v in v_{\,\text{in}}
  49. v out v_{\,\text{out}}
  50. v in v_{\,\text{in}}
  51. v v
  52. v out v_{\,\text{out}}
  53. v v
  54. c ( v ) c(v)
  55. v in v_{\,\text{in}}
  56. v out v_{\,\text{out}}
  57. v in v_{\,\text{in}}
  58. v out v_{\,\text{out}}
  59. r ( S - { k } ) = i , j { S - { k } } , i < j r i j \scriptstyle r(S-\{k\})=\sum_{i,j\in\{S-\{k\}\},i<j}r_{ij}
  60. c c
  61. s s
  62. f i f_{i}
  63. p i p_{i}
  64. p i p_{i}
  65. f i f_{i}
  66. t t
  67. v i v_{i}
  68. t t
  69. d i d_{i}
  70. d i d_{i}
  71. v i v_{i}
  72. MaximumFlowValue ( G ) = i v d i \operatorname{MaximumFlowValue}\,(G)=\sum_{i\in v}d_{i}
  73. N N
  74. k k
  75. m m
  76. 1 / m 1/m
  77. i i
  78. D i D_{i}
  79. i i
  80. [ D i ] [D_{i}]
  81. k k
  82. k k
  83. k k

Maximum_sustainable_yield.html

  1. N t = K 1 + K - N 0 N 0 e - r t N_{t}=\frac{K}{1+\frac{K-N_{0}}{N_{0}}e^{-rt}}
  2. N t N_{t}
  3. K K
  4. N 0 N_{0}
  5. r r
  6. r r
  7. K K
  8. N 0 N_{0}
  9. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)
  10. d N d t \frac{dN}{dt}
  11. N K \frac{N}{K}
  12. r N rN
  13. N K \frac{N}{K}
  14. N = K 2 N=\frac{K}{2}
  15. d N d t = r N ( 1 - N K ) - H \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)-H
  16. d N d t = 0 \frac{dN}{dt}=0
  17. r N ( 1 - N K ) = H rN\left(1-\frac{N}{K}\right)=H
  18. N M S Y N_{MSY}
  19. H 1 H_{1}
  20. N a N_{a}
  21. N b N_{b}
  22. H 2 H_{2}
  23. N M S Y N_{MSY}
  24. N = K 2 N=\frac{K}{2}
  25. H H
  26. H 2 H_{2}
  27. H 3 H_{3}
  28. N N
  29. N b N_{b}
  30. H 1 H_{1}
  31. N b N_{b}
  32. H 1 H_{1}
  33. N M S Y N_{MSY}
  34. H 2 H_{2}
  35. H H
  36. K K
  37. r r
  38. H = K r 4 H=\frac{Kr}{4}

Maxwell_material.html

  1. σ Total \sigma_{\mathrm{Total}}
  2. ε Total \varepsilon_{\mathrm{Total}}
  3. σ Total = σ D = σ S \sigma_{\mathrm{Total}}=\sigma_{D}=\sigma_{S}
  4. ε Total = ε D + ε S \varepsilon_{\mathrm{Total}}=\varepsilon_{D}+\varepsilon_{S}
  5. d ε Total d t = d ε D d t + d ε S d t = σ η + 1 E d σ d t \frac{d\varepsilon_{\mathrm{Total}}}{dt}=\frac{d\varepsilon_{D}}{dt}+\frac{d% \varepsilon_{S}}{dt}=\frac{\sigma}{\eta}+\frac{1}{E}\frac{d\sigma}{dt}
  6. 1 E d σ d t + σ η = d ε d t \frac{1}{E}\frac{d\sigma}{dt}+\frac{\sigma}{\eta}=\frac{d\varepsilon}{dt}
  7. σ ˙ E + σ η = ε ˙ \frac{\dot{\sigma}}{E}+\frac{\sigma}{\eta}=\dot{\varepsilon}
  8. ε 0 \varepsilon_{0}
  9. η E \frac{\eta}{E}
  10. σ ( t ) E ε 0 \frac{\sigma(t)}{E\varepsilon_{0}}
  11. E η t \frac{E}{\eta}t
  12. t 1 t_{1}
  13. ε back = - σ ( t 1 ) E = ε 0 exp ( - E η t 1 ) . \varepsilon_{\mathrm{back}}=-\frac{\sigma(t_{1})}{E}=\varepsilon_{0}\exp\left(% -\frac{E}{\eta}t_{1}\right).
  14. ε irreversible = ε 0 ( 1 - exp ( - E η t 1 ) ) . \varepsilon_{\mathrm{irreversible}}=\varepsilon_{0}\left(1-\exp\left(-\frac{E}% {\eta}t_{1}\right)\right).
  15. σ 0 \sigma_{0}
  16. ε ( t ) = σ 0 E + t σ 0 η \varepsilon(t)=\frac{\sigma_{0}}{E}+t\frac{\sigma_{0}}{\eta}
  17. t 1 t_{1}
  18. ε reversible = σ 0 E , \varepsilon_{\mathrm{reversible}}=\frac{\sigma_{0}}{E},
  19. ε irreversible = t 1 σ 0 η . \varepsilon_{\mathrm{irreversible}}=t_{1}\frac{\sigma_{0}}{\eta}.
  20. E * ( ω ) = 1 1 / E - i / ( ω η ) = E η 2 ω 2 + i ω E 2 η η 2 ω 2 + E 2 E^{*}(\omega)=\frac{1}{1/E-i/(\omega\eta)}=\frac{E\eta^{2}\omega^{2}+i\omega E% ^{2}\eta}{\eta^{2}\omega^{2}+E^{2}}
  21. E 1 ( ω ) = E η 2 ω 2 η 2 ω 2 + E 2 = ( η / E ) 2 ω 2 ( η / E ) 2 ω 2 + 1 E = τ 2 ω 2 τ 2 ω 2 + 1 E E_{1}(\omega)=\frac{E\eta^{2}\omega^{2}}{\eta^{2}\omega^{2}+E^{2}}=\frac{(\eta% /E)^{2}\omega^{2}}{(\eta/E)^{2}\omega^{2}+1}E=\frac{\tau^{2}\omega^{2}}{\tau^{% 2}\omega^{2}+1}E
  22. E 2 ( ω ) = ω E 2 η η 2 ω 2 + E 2 = ( η / E ) ω ( η / E ) 2 ω 2 + 1 E = τ ω τ 2 ω 2 + 1 E E_{2}(\omega)=\frac{\omega E^{2}\eta}{\eta^{2}\omega^{2}+E^{2}}=\frac{(\eta/E)% \omega}{(\eta/E)^{2}\omega^{2}+1}E=\frac{\tau\omega}{\tau^{2}\omega^{2}+1}E
  23. τ η / E \tau\equiv\eta/E
  24. E 1 E \frac{E_{1}}{E}
  25. E 2 E \frac{E_{2}}{E}
  26. E 2 ω η \frac{E_{2}}{\omega\eta}
  27. ω τ \omega\tau

Mayer–Vietoris_sequence.html

  1. H n + 1 ( X ) * H n ( A B ) ( i * , j * ) H n ( A ) H n ( B ) k * - l * H n ( X ) * * H n - 1 ( A B ) H 0 ( A ) H 0 ( B ) k * - l * H 0 ( X ) 0. \begin{aligned}\displaystyle\cdots\rightarrow H_{n+1}(X)&\displaystyle% \xrightarrow{\partial_{*}}\,H_{n}(A\cap B)\,\xrightarrow{(i_{*},j_{*})}\,H_{n}% (A)\oplus H_{n}(B)\,\xrightarrow{k_{*}-l_{*}}\,H_{n}(X)\xrightarrow{\partial_{% *}}\\ &\displaystyle\quad\xrightarrow{\partial_{*}}\,H_{n-1}(A\cap B)\rightarrow% \cdots\rightarrow H_{0}(A)\oplus H_{0}(B)\,\xrightarrow{k_{*}-l_{*}}\,H_{0}(X)% \rightarrow\,0.\end{aligned}
  2. \oplus
  3. H ~ 0 ( A B ) ( i * , j * ) H ~ 0 ( A ) H ~ 0 ( B ) k * - l * H ~ 0 ( X ) 0. \cdots\rightarrow\tilde{H}_{0}(A\cap B)\,\xrightarrow{(i_{*},j_{*})}\,\tilde{H% }_{0}(A)\oplus\tilde{H}_{0}(B)\,\xrightarrow{k_{*}-l_{*}}\,\tilde{H}_{0}(X)% \rightarrow\,0.
  4. H 1 ( X ) ( H 1 ( A ) H 1 ( B ) ) / Ker ( k * - l * ) H_{1}(X)\cong(H_{1}(A)\oplus H_{1}(B))/\,\text{Ker}(k_{*}-l_{*})
  5. Ker ( k * - l * ) Im ( i * , j * ) . \,\text{Ker}(k_{*}-l_{*})\cong\,\text{Im}(i_{*},j_{*}).
  6. 0 H ~ n ( S k ) * H ~ n - 1 ( S k - 1 ) 0 \cdots\rightarrow 0\rightarrow\tilde{H}_{n}\left(S^{k}\right)\xrightarrow{% \partial_{*}}\,\tilde{H}_{n-1}\left(S^{k-1}\right)\rightarrow 0\rightarrow\cdots\!
  7. H ~ n ( S k ) δ k n = { if n = k 0 if n k \tilde{H}_{n}\left(S^{k}\right)\cong\delta_{kn}\,\mathbb{Z}=\left\{\begin{% matrix}\mathbb{Z}&\mbox{if }~{}n=k\\ 0&\mbox{if }~{}n\neq k\end{matrix}\right.
  8. 0 H 2 ( X ) 𝛼 H 1 ( X ) 0 0\rightarrow H_{2}(X)\rightarrow\,\mathbb{Z}\ \xrightarrow{\alpha}\ \mathbb{Z}% \oplus\mathbb{Z}\rightarrow\,H_{1}(X)\rightarrow 0\!
  9. H ~ n ( X ) δ 1 n ( 2 ) = { 2 if n = 1 0 if n 1 \tilde{H}_{n}\left(X\right)\cong\delta_{1n}\,(\mathbb{Z}\oplus\mathbb{Z}_{2})=% \left\{\begin{matrix}\mathbb{Z}\oplus\mathbb{Z}_{2}&\mbox{if }~{}n=1\\ 0&\mbox{if }~{}n\neq 1\end{matrix}\right.
  10. H ~ n ( K L ) H ~ n ( K ) H ~ n ( L ) \tilde{H}_{n}(K\vee L)\cong\tilde{H}_{n}(K)\oplus\tilde{H}_{n}(L)
  11. H ~ n ( S 2 S 2 ) δ 2 n ( ) = { if n = 2 0 if n 2 \tilde{H}_{n}\left(S^{2}\vee S^{2}\right)\cong\delta_{2n}\,(\mathbb{Z}\oplus% \mathbb{Z})=\left\{\begin{matrix}\mathbb{Z}\oplus\mathbb{Z}&\mbox{if }~{}n=2\\ 0&\mbox{if }~{}n\neq 2\end{matrix}\right.
  12. H ~ n ( S Y ) H ~ n - 1 ( Y ) \tilde{H}_{n}(SY)\cong\tilde{H}_{n-1}(Y)
  13. H n ( A B , C D ) ( i * , j * ) H n ( A , C ) H n ( B , D ) k * - l * H n ( X , Y ) * H n - 1 ( A B , C D ) \cdots\rightarrow H_{n}(A\cap B,C\cap D)\,\xrightarrow{(i_{*},j_{*})}\,H_{n}(A% ,C)\oplus H_{n}(B,D)\,\xrightarrow{k_{*}-l_{*}}\,H_{n}(X,Y)\,\xrightarrow{% \partial_{*}}\,H_{n-1}(A\cap B,C\cap D)\rightarrow\cdots
  14. ( g h ) * = g * h * (g\circ h)_{*}=g_{*}\circ h_{*}
  15. H n ( X ; G ) H n ( A ; G ) H n ( B ; G ) H n ( A B ; G ) H n + 1 ( X ; G ) \cdots\rightarrow H^{n}(X;G)\rightarrow H^{n}(A;G)\oplus H^{n}(B;G)\rightarrow H% ^{n}(A\cap B;G)\rightarrow H^{n+1}(X;G)\rightarrow\cdots
  16. H n ( X ) 𝜌 H n ( U ) H n ( V ) Δ H n ( U V ) d * H n + 1 ( X ) \cdots\rightarrow H^{n}(X)\,\xrightarrow{\rho}\,H^{n}(U)\oplus H^{n}(V)\,% \xrightarrow{\Delta}\,H^{n}(U\cap V)\,\xrightarrow{d^{*}}\,H^{n+1}(X)\rightarrow\cdots
  17. 0 C n ( A B ) 𝛼 C n ( A ) C n ( B ) 𝛽 C n ( A + B ) 0 0\rightarrow C_{n}(A\cap B)\,\xrightarrow{\alpha}\,C_{n}(A)\oplus C_{n}(B)\,% \xrightarrow{\beta}\,C_{n}(A+B)\rightarrow 0
  18. 0 Ω n ( X ) Ω n ( U ) Ω n ( V ) Ω n ( U V ) 0 0\rightarrow\Omega^{n}(X)\rightarrow\Omega^{n}(U)\oplus\Omega^{n}(V)% \rightarrow\Omega^{n}(U\cap V)\rightarrow 0

MD5CRK.html

  1. 2 N 2^{N}
  2. N N
  3. 2 N ! ( 2 N - K ) ! × 2 N K 2^{N}!\over{(2^{N}-K)!\times{2^{N}}^{K}}
  4. K K
  5. K K
  6. 1 1 - e K × ( 1 - K ) 2 N + 1 1\over{1-e^{K\times(1-K)\over 2^{N+1}}}
  7. 1.17741 × 2 N / 2 = 1.17741 × 2 64 {1.17741\times 2^{N/2}}={1.17741\times 2^{64}}
  8. 2.17 × 10 19 / 12.25 × 10 12 1 , 770 , 000 {2.17\times 10^{19}/12.25\times 10^{12}\approx 1,770,000}

Mealy_machine.html

  1. ( S , S 0 , Σ , Λ , T , G ) (S,S_{0},\Sigma,\Lambda,T,G)
  2. S S
  3. S 0 S_{0}
  4. S S
  5. Σ \Sigma
  6. Λ \Lambda
  7. T : S × Σ S T:S\times\Sigma\rightarrow S
  8. G : S × Σ Λ G:S\times\Sigma\rightarrow\Lambda
  9. T : S × Σ S × Λ T:S\times\Sigma\rightarrow S\times\Lambda
  10. Σ Σ
  11. ( S × Σ , ( x , i ) ( T ( x , i ) , G ( x , i ) ) ) (S×Σ,(x,i)→(T(x,i),G(x,i)))
  12. ( x , i ) (x,i)
  13. x x
  14. i i
  15. S < s u b > i S<sub>i

Mean_field_theory.html

  1. = 0 + Δ \mathcal{H}=\mathcal{H}_{0}+\Delta\mathcal{H}
  2. F F 0 = def 0 - T S 0 F\leq F_{0}\ \stackrel{\mathrm{def}}{=}\ \langle\mathcal{H}\rangle_{0}-TS_{0}
  3. S 0 S_{0}
  4. 0 \mathcal{H}_{0}
  5. 0 = i = 1 N h i ( ξ i ) \mathcal{H}_{0}=\sum_{i=1}^{N}h_{i}\left(\xi_{i}\right)
  6. ( ξ i ) \left(\xi_{i}\right)
  7. = ( i , j ) 𝒫 V i , j ( ξ i , ξ j ) \mathcal{H}=\sum_{(i,j)\in\mathcal{P}}V_{i,j}\left(\xi_{i},\xi_{j}\right)
  8. 𝒫 \mathcal{P}
  9. Tr i f ( ξ i ) {\rm Tr}_{i}f(\xi_{i})
  10. f f
  11. F 0 = F_{0}=\,\!
  12. Tr 1 , 2 , . . , N ( ξ 1 , ξ 2 , , ξ N ) P 0 ( N ) ( ξ 1 , ξ 2 , , ξ N ) {\rm Tr}_{1,2,..,N}\mathcal{H}(\xi_{1},\xi_{2},...,\xi_{N})P^{(N)}_{0}(\xi_{1}% ,\xi_{2},...,\xi_{N})
  13. + k T Tr 1 , 2 , . . , N P 0 ( N ) ( ξ 1 , ξ 2 , , ξ N ) log P 0 ( N ) ( ξ 1 , ξ 2 , , ξ N ) +kT\,{\rm Tr}_{1,2,..,N}P^{(N)}_{0}(\xi_{1},\xi_{2},...,\xi_{N})\log P^{(N)}_{% 0}(\xi_{1},\xi_{2},...,\xi_{N})
  14. P 0 ( N ) ( ξ 1 , ξ 2 , , ξ N ) P^{(N)}_{0}(\xi_{1},\xi_{2},...,\xi_{N})
  15. ( ξ 1 , ξ 2 , , ξ N ) (\xi_{1},\xi_{2},...,\xi_{N})
  16. P 0 ( N ) ( ξ 1 , ξ 2 , , ξ N ) = 1 Z 0 ( N ) e - β 0 ( ξ 1 , ξ 2 , , ξ N ) = i = 1 N 1 Z 0 e - β h i ( ξ i ) = def i = 1 N P 0 ( i ) ( ξ i ) P^{(N)}_{0}(\xi_{1},\xi_{2},...,\xi_{N})=\frac{1}{Z^{(N)}_{0}}e^{-\beta% \mathcal{H}_{0}(\xi_{1},\xi_{2},...,\xi_{N})}=\prod_{i=1}^{N}\frac{1}{Z_{0}}e^% {-\beta h_{i}\left(\xi_{i}\right)}\ \stackrel{\mathrm{def}}{=}\ \prod_{i=1}^{N% }P^{(i)}_{0}(\xi_{i})
  17. Z 0 Z_{0}
  18. F 0 = ( i , j ) 𝒫 Tr i , j V i , j ( ξ i , ξ j ) P 0 ( i ) ( ξ i ) P 0 ( j ) ( ξ j ) + k T i = 1 N Tr i P 0 ( i ) ( ξ i ) log P 0 ( i ) ( ξ i ) . F_{0}=\sum_{(i,j)\in\mathcal{P}}{\rm Tr}_{i,j}V_{i,j}\left(\xi_{i},\xi_{j}% \right)P^{(i)}_{0}(\xi_{i})P^{(j)}_{0}(\xi_{j})+kT\sum_{i=1}^{N}{\rm Tr}_{i}P^% {(i)}_{0}(\xi_{i})\log P^{(i)}_{0}(\xi_{i}).
  19. P 0 ( i ) P^{(i)}_{0}
  20. P 0 ( i ) ( ξ i ) = 1 Z 0 e - β h i M F ( ξ i ) i = 1 , 2 , . . , N P^{(i)}_{0}(\xi_{i})=\frac{1}{Z_{0}}e^{-\beta h_{i}^{MF}(\xi_{i})}\qquad i=1,2% ,..,N
  21. h i M F ( ξ i ) = { j | ( i , j ) 𝒫 } Tr j V i , j ( ξ i , ξ j ) P 0 ( j ) ( ξ j ) . h_{i}^{MF}(\xi_{i})=\sum_{\{j|(i,j)\in\mathcal{P}\}}{\rm Tr}_{j}V_{i,j}\left(% \xi_{i},\xi_{j}\right)P^{(j)}_{0}(\xi_{j}).
  22. d d
  23. H = - J i , j s i s j - h i s i H=-J\sum_{\langle i,j\rangle}s_{i}s_{j}-h\sum_{i}s_{i}
  24. i , j \sum_{\langle i,j\rangle}
  25. i , j \langle i,j\rangle
  26. s i = ± 1 s_{i}=\pm 1
  27. s j s_{j}
  28. m i s i m_{i}\equiv\langle s_{i}\rangle
  29. H = - J i , j ( m i + δ s i ) ( m j + δ s j ) - h i s i H=-J\sum_{\langle i,j\rangle}(m_{i}+\delta s_{i})(m_{j}+\delta s_{j})-h\sum_{i% }s_{i}
  30. δ s i s i - m i \delta s_{i}\equiv s_{i}-m_{i}
  31. H H M F - J i , j ( m i m j + m i δ s j + m j δ s i ) - h i s i H\approx H^{MF}\equiv-J\sum_{\langle i,j\rangle}(m_{i}m_{j}+m_{i}\delta s_{j}+% m_{j}\delta s_{i})-h\sum_{i}s_{i}
  32. H M F = - J i , j ( m 2 + 2 m ( s i - m ) ) - h i s i H^{MF}=-J\sum_{\langle i,j\rangle}\left(m^{2}+2m(s_{i}-m)\right)-h\sum_{i}s_{i}
  33. i , j = 1 2 i j n n ( i ) \sum_{\langle i,j\rangle}=\frac{1}{2}\sum_{i}\sum_{j\in nn(i)}
  34. n n ( i ) nn(i)
  35. i i
  36. 1 / 2 1/2
  37. H M F = J m 2 N z 2 - ( h + m J z ) h eff i s i H^{MF}=\frac{Jm^{2}Nz}{2}-\underbrace{(h+mJz)}_{h^{\mathrm{eff}}}\sum_{i}s_{i}
  38. z z
  39. h eff = h + J z m h^{\mathrm{eff}}=h+Jzm
  40. h h
  41. d d
  42. z = 2 d z=2d
  43. Z = e - β J m 2 N z / 2 [ 2 cosh ( h + m J z k B T ) ] N Z=e^{-\beta Jm^{2}Nz/2}\left[2\cosh\left(\frac{h+mJz}{k_{B}T}\right)\right]^{N}
  44. N N
  45. m m
  46. h eff h^{\mathrm{eff}}
  47. m m
  48. h eff h^{\mathrm{eff}}
  49. m m
  50. T c T_{c}
  51. m = 0 m=0
  52. T < T c T<T_{c}
  53. m = ± m 0 m=\pm m_{0}
  54. T c T_{c}
  55. T c = J z k B T_{c}=\frac{Jz}{k_{B}}
  56. Δ \Delta

Measure-preserving_dynamical_system.html

  1. ( X , , μ , T ) (X,\mathcal{B},\mu,T)
  2. X X
  3. \mathcal{B}
  4. X X
  5. μ : [ 0 , 1 ] \mu:\mathcal{B}\rightarrow[0,1]
  6. T : X X T:X\rightarrow X
  7. μ \mu
  8. A μ ( T - 1 ( A ) ) = μ ( A ) \forall A\in\mathcal{B}\;\;\mu(T^{-1}(A))=\mu(A)
  9. T 0 = i d X : X X T_{0}=id_{X}:X\rightarrow X
  10. T s T t = T t + s T_{s}\circ T_{t}=T_{t+s}
  11. T s - 1 = T - s T_{s}^{-1}=T_{-s}
  12. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  13. ( Y , , ν , S ) (Y,\mathcal{B},\nu,S)
  14. φ : X Y \varphi:X\to Y
  15. B B\in\mathcal{B}
  16. μ ( φ - 1 B ) = ν ( B ) \mu(\varphi^{-1}B)=\nu(B)
  17. ( Y , , ν , S ) (Y,\mathcal{B},\nu,S)
  18. ( X , 𝒜 , μ , T ) (X,\mathcal{A},\mu,T)
  19. ψ : Y X \psi:Y\to X
  20. x = ψ ( φ x ) x=\psi(\varphi x)
  21. y = φ ( ψ y ) y=\varphi(\psi y)
  22. ( X , , T , μ ) (X,\mathcal{B},T,\mu)
  23. T n x Q a n . T^{n}x\in Q_{a_{n}}.\,
  24. ( X , , T , μ ) (X,\mathcal{B},T,\mu)
  25. T - 1 Q = { T - 1 Q 1 , , T - 1 Q k } . T^{-1}Q=\{T^{-1}Q_{1},\ldots,T^{-1}Q_{k}\}.\,
  26. Q R = { Q i R j i = 1 , , k , j = 1 , , m , μ ( Q i R j ) > 0 } . Q\vee R=\{Q_{i}\cap R_{j}\mid i=1,\ldots,k,\ j=1,\ldots,m,\ \mu(Q_{i}\cap R_{j% })>0\}.\,
  27. n = 0 N T - n Q = { Q i 0 T - 1 Q i 1 T - N Q i N where i = 1 , , k , = 0 , , N , μ ( Q i 0 T - 1 Q i 1 T - N Q i N ) > 0 } \bigvee_{n=0}^{N}T^{-n}Q=\left\{Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap\cdots\cap T^% {-N}Q_{i_{N}}\,\text{ where }i_{\ell}=1,\ldots,k,\ \ell=0,\ldots,N,\ \mu\left(% Q_{i_{0}}\cap T^{-1}Q_{i_{1}}\cap\cdots\cap T^{-N}Q_{i_{N}}\right)>0\right\}
  28. H ( Q ) = - m = 1 k μ ( Q m ) log μ ( Q m ) . H(Q)=-\sum_{m=1}^{k}\mu(Q_{m})\log\mu(Q_{m}).
  29. ( X , , T , μ ) (X,\mathcal{B},T,\mu)
  30. h μ ( T , Q ) = lim N 1 N H ( n = 0 N T - n Q ) . h_{\mu}(T,Q)=\lim_{N\rightarrow\infty}\frac{1}{N}H\left(\bigvee_{n=0}^{N}T^{-n% }Q\right).\,
  31. ( X , , T , μ ) (X,\mathcal{B},T,\mu)
  32. h μ ( T ) = sup Q h μ ( T , Q ) . h_{\mu}(T)=\sup_{Q}h_{\mu}(T,Q).\,

Measurement_in_quantum_mechanics.html

  1. x ( t ) \vec{x}(t)
  2. p ( t ) \vec{p}(t)
  3. H ^ \hat{H}
  4. H ^ = T ^ + V ^ = p ^ 2 2 m + V ( x ^ ) {\hat{H}}=\hat{T}+\hat{V}={\hat{p}^{2}\over 2m}+V(\hat{x})
  5. p ^ {\hat{p}}
  6. p ^ = - i x {\hat{p}}=-i\hbar{\partial\over\partial x}
  7. p ^ = p {\hat{p}}=p
  8. x ^ {\hat{x}}
  9. x ^ = x {\hat{x}}=x
  10. x ^ = i p {\hat{x}}=i\hbar{\partial\over\partial p}
  11. O ^ \hat{O}
  12. O ^ \hat{O}
  13. | 1 , | 2 , | 3 , |1\rangle,|2\rangle,|3\rangle,...
  14. O 1 , O 2 , O 3 , O_{1},O_{2},O_{3},...
  15. | ψ |\psi\rangle
  16. O ^ \hat{O}
  17. | ψ |\psi\rangle
  18. | ψ = c 1 | 1 + c 2 | 2 + c 3 | 3 + = n c n | n |\psi\rangle=c_{1}|1\rangle+c_{2}|2\rangle+c_{3}|3\rangle+\cdots=\sum_{n}c_{n}% |n\rangle
  19. c 1 , c 2 , c_{1},c_{2},\ldots
  20. O 1 , O 2 , O 3 , O_{1},O_{2},O_{3},...
  21. Pr ( O n ) = | n | ψ | 2 | ψ | ψ | = | c n | 2 k | c k | 2 \Pr(O_{n})=\frac{|\langle n|\psi\rangle|^{2}}{|\langle\psi|\psi\rangle|}=\frac% {|c_{n}|^{2}}{\sum_{k}|c_{k}|^{2}}
  22. | ψ |\psi\rangle
  23. ψ | ψ = 1 \langle\psi|\psi\rangle=1
  24. Pr ( O n ) = | n | ψ | 2 = | c n | 2 \Pr(O_{n})=|\langle n|\psi\rangle|^{2}=|c_{n}|^{2}
  25. O n O_{n}
  26. | n |n\rangle
  27. | ψ = | n |\psi^{\prime}\rangle=|n\rangle
  28. O ^ {\hat{O}}
  29. O n O_{n}
  30. O ^ \hat{O}
  31. O ^ \hat{O}
  32. | x |x\rangle
  33. x x
  34. | ψ |\psi\rangle
  35. O ^ \hat{O}
  36. | ψ |\psi\rangle
  37. | ψ = a b c ( x ) | x d x |\psi\rangle=\int_{a}^{b}c(x)|x\rangle\,dx
  38. c ( x ) c(x)
  39. ( a , b ) (a,b)
  40. Pr ( d < x < e ) = d e | x | ψ | 2 d x a b ψ | ψ d x = d e | c ( x ) | 2 d x a b | c ( x ) | 2 d x \Pr(d<x<e)=\frac{\int_{d}^{e}|\langle x|\psi\rangle|^{2}\,dx}{\int_{a}^{b}% \langle\psi|\psi\rangle\,dx}=\frac{\int_{d}^{e}|c(x)|^{2}\,dx}{\int_{a}^{b}|c(% x)|^{2}\,dx}
  41. ( d , e ) ( a , b ) (d,e)\subseteq(a,b)
  42. | ψ |\psi\rangle
  43. a b ψ | ψ d x = 1 \int_{a}^{b}\langle\psi|\psi\rangle\,dx=1
  44. Pr ( d < x < e ) = d e | c ( x ) | 2 d x \Pr(d<x<e)=\int_{d}^{e}|c(x)|^{2}\,dx
  45. x x
  46. | x |x\rangle
  47. | ψ = | x . |\psi^{\prime}\rangle=|x\rangle.
  48. O ^ {\hat{O}}
  49. O 1 , O 2 , O 3 , O_{1},O_{2},O_{3},\ldots
  50. V 1 , V 2 , V_{1},V_{2},\ldots
  51. P n P_{n}
  52. V n V_{n}
  53. O ^ {\hat{O}}
  54. O 1 , O 2 , O 3 , O_{1},O_{2},O_{3},\ldots
  55. Pr ( O n ) = Tr ( P n ρ ) \Pr(O_{n})=\mathrm{Tr}(P_{n}\rho)
  56. Tr \mathrm{Tr}
  57. ρ = P n ρ P n Tr ( P n ρ ) \rho^{\prime}=\frac{P_{n}\rho P_{n}}{\mathrm{Tr}(P_{n}\rho)}
  58. ρ ′′ = n P n ρ P n \rho^{\prime\prime}=\sum_{n}P_{n}\rho P_{n}
  59. ρ ′′ \rho^{\prime\prime}
  60. ρ \rho^{\prime}
  61. n n
  62. O ^ \hat{O}
  63. | ψ |\psi\rangle
  64. ψ | O ^ | ψ \langle\psi|\hat{O}|\psi\rangle
  65. ψ | O ^ 2 | ψ - ( ψ | O ^ | ψ ) 2 \langle\psi|\hat{O}^{2}|\psi\rangle-(\langle\psi|\hat{O}|\psi\rangle)^{2}
  66. ψ | O ^ 2 | ψ - ( ψ | O ^ | ψ ) 2 \sqrt{\langle\psi|\hat{O}^{2}|\psi\rangle-(\langle\psi|\hat{O}|\psi\rangle)^{2}}
  67. | ψ 1 |\psi_{1}\rangle
  68. E 1 = π 2 2 2 m L 2 E_{1}=\frac{\pi^{2}\hbar^{2}}{2mL^{2}}
  69. x | ψ 1 = 2 L sin ( π x L ) \langle x|\psi_{1}\rangle=\sqrt{\frac{2}{L}}~{}{\rm sin}\left(\frac{\pi x}{L}\right)
  70. E 1 E_{1}
  71. Pr ( S < x < S + d S ) = 2 L sin 2 ( π S L ) d S . \Pr(S<x<S+dS)=\frac{2}{L}~{}{\rm sin}^{2}\left(\frac{\pi S}{L}\right)dS.
  72. | x = S |x=S\rangle
  73. | x = S |x=S\rangle
  74. | ψ n |\psi_{n}\rangle
  75. | x = S = n | ψ n ψ n | x = S = n | ψ n 2 L sin ( n π S L ) |x=S\rangle=\sum_{n}|\psi_{n}\rangle\left\langle\psi_{n}|x=S\right\rangle=\sum% _{n}|\psi_{n}\rangle\sqrt{\frac{2}{L}}~{}{\rm sin}\left(\frac{n\pi S}{L}\right)
  76. E n E_{n}
  77. Pr ( E n ) = | ψ n | S | 2 = 2 L sin 2 ( n π S L ) \Pr(E_{n})=|\langle\psi_{n}|S\rangle|^{2}=\frac{2}{L}~{}{\rm sin}^{2}\left(% \frac{n\pi S}{L}\right)
  78. E n E_{n}
  79. | ψ n |\psi_{n}\rangle
  80. | ψ = n c n | ψ n \scriptstyle|\psi\rangle=\sum_{n}c_{n}|\psi_{n}\rangle
  81. | ψ n \scriptstyle|\psi_{n}\rangle
  82. | ψ \scriptstyle|\psi\rangle
  83. | ϕ \scriptstyle|\phi\rangle
  84. | ψ | ϕ \scriptstyle|\psi\rangle|\phi\rangle
  85. | ψ | ϕ n c n | ψ n | ϕ n (measurement of the first kind), |\psi\rangle|\phi\rangle\rightarrow\sum_{n}c_{n}|\psi_{n}\rangle|\phi_{n}% \rangle\quad\,\text{(measurement of the first kind),}
  86. | ϕ n \scriptstyle|\phi_{n}\rangle
  87. n | c n | 2 | ψ n ψ n | \scriptstyle\sum_{n}|c_{n}|^{2}|\psi_{n}\rangle\langle\psi_{n}|
  88. | c n | 2 \scriptstyle|c_{n}|^{2}
  89. | ψ n . \scriptstyle|\psi_{n}\rangle.
  90. | ψ n | c n | 2 | ψ n ψ n | |\psi\rangle\rightarrow\sum_{n}|c_{n}|^{2}|\psi_{n}\rangle\langle\psi_{n}|
  91. | ψ n | c n | 2 | ψ n ψ n | | ψ n |\psi\rangle\rightarrow\sum_{n}|c_{n}|^{2}|\psi_{n}\rangle\langle\psi_{n}|% \rightarrow|\psi_{n}\rangle
  92. | ψ n | c n | 2 P n , P n = i | ψ n i ψ n i | , |\psi\rangle\rightarrow\sum_{n}|c_{n}|^{2}P_{n},\;P_{n}=\sum_{i}|\psi_{ni}% \rangle\langle\psi_{ni}|,
  93. | ψ n i \scriptstyle|\psi_{ni}\rangle
  94. ρ \scriptstyle\rho
  95. ρ n P n ρ P n . \rho\rightarrow\sum_{n}P_{n}\rho P_{n}.
  96. | ψ | ϕ n c n | χ n | ϕ n , |\psi\rangle|\phi\rangle\rightarrow\sum_{n}c_{n}|\chi_{n}\rangle|\phi_{n}\rangle,
  97. | χ n \scriptstyle|\chi_{n}\rangle
  98. | χ n \scriptstyle|\chi_{n}\rangle
  99. | c n | 2 . \scriptstyle|c_{n}|^{2}.
  100. | e \scriptstyle|e\rangle
  101. n c n | ψ n | ϕ n | e n , \sum_{n}c_{n}|\psi_{n}\rangle|\phi_{n}\rangle|e_{n}\rangle,
  102. | ψ | ψ n \scriptstyle|\psi\rangle\rightarrow|\psi_{n}\rangle
  103. | ϕ | ϕ n \scriptstyle|\phi\rangle\rightarrow|\phi_{n}\rangle
  104. | e | e n \scriptstyle|e\rangle\rightarrow|e_{n}\rangle
  105. { | ψ n } \{|\psi_{n}\rangle\}
  106. { | ϕ n } \{|\phi_{n}\rangle\}
  107. { | e n } \{|e_{n}\rangle\}

Mechanical_efficiency.html

  1. Efficiency = Measured performance Ideal performance \,\text{Efficiency}=\frac{\,\text{Measured performance}}{\,\text{Ideal % performance}}

Medial_magma.html

  1. ( x y ) ( u v ) = ( x u ) ( y v ) (x\cdot y)\cdot(u\cdot v)=(x\cdot u)\cdot(y\cdot v)
  2. x y u v = x u y v xy\cdot uv=xu\cdot yv
  3. m n m≠n
  4. x + y x+y
  5. x y = m x + n y x\cdot y=mx+ny
  6. M × M M×M
  7. ( x , y ) ( u , v ) = ( x u , y v ) (x,y)∙(u,v)=(x∙u,y∙v)
  8. M M
  9. M × M M×M
  10. ( x , y ) (x,y)
  11. x y x∙y
  12. ( u , v ) (u,v)
  13. u v u∙v
  14. ( x u , y v ) (x∙u,y∙v)
  15. ( x u ) ( y v ) (x∙u)∙(y∙v)
  16. M M
  17. M × M M×M
  18. M M
  19. f f
  20. g g
  21. f g f∙g
  22. ( f g ) ( x ) = f ( x ) g ( x ) (f\cdot g)(x)=f(x)\cdot g(x)
  23. A A
  24. A A
  25. A A
  26. x y = φ ( x ) + ψ ( y ) + c x∗y=φ(x)+ψ(y)+c
  27. c c
  28. A A
  29. A A
  30. f ( g ( x 11 , , x 1 n ) , , g ( x m 1 , , x m n ) ) = g ( f ( x 11 , , x m 1 ) , , f ( x 1 n , , x m n ) ) . f(g(x_{11},\ldots,x_{1n}),\ldots,g(x_{m1},\ldots,x_{mn}))=g(f(x_{11},\ldots,x_% {m1}),\ldots,f(x_{1n},\ldots,x_{mn})).

Meet-in-the-middle_attack.html

  1. C \displaystyle C
  2. S u b C i p h e r 1 = E N C f 1 ( k f 1 , P ) , k f 1 K SubCipher_{1}=ENC_{f_{1}}(k_{f_{1}},P),\;\forall k_{f_{1}}\in K
  3. S u b C i p h e r 1 SubCipher_{1}
  4. k f 1 k_{f_{1}}
  5. S u b C i p h e r 1 = D E C b 1 ( k b 1 , C ) , k b 1 K SubCipher_{1}=DEC_{b_{1}}(k_{b_{1}},C),\;\forall k_{b_{1}}\in K
  6. S u b C i p h e r 1 SubCipher_{1}
  7. C = E N C k n ( E N C k n - 1 ( ( E N C k 1 ( P ) ) ) ) C=ENC_{k_{n}}(ENC_{k_{n-1}}(...(ENC_{k_{1}}(P))...))
  8. P = D E C k 1 ( D E C k 2 ( ( D E C k n ( C ) ) ) ) P=DEC_{k_{1}}(DEC_{k_{2}}(...(DEC_{k_{n}}(C))...))
  9. S u b C i p h e r 1 = E N C f 1 ( k f 1 , P ) SubCipher_{1}=ENC_{f_{1}}(k_{f_{1}},P)
  10. k f 1 k_{f_{1}}
  11. K K
  12. S u b C i p h e r 1 SubCipher_{1}
  13. k f 1 k_{f_{1}}
  14. H 1 H_{1}
  15. S u b C i p h e r n + 1 = D E C b n + 1 ( k b n + 1 , C ) SubCipher_{n+1}=DEC_{b_{n+1}}(k_{b_{n+1}},C)
  16. k b n + 1 k_{b_{n+1}}
  17. K K
  18. S u b C i p h e r n + 1 SubCipher_{n+1}
  19. k b n + 1 k_{b_{n+1}}
  20. H n + 1 H_{n+1}
  21. s 1 s_{1}
  22. S u b C i p h e r 1 = D E C b 1 ( k b 1 , s 1 ) SubCipher_{1}=DEC_{b_{1}}(k_{b_{1}},s_{1})
  23. k b 1 k_{b_{1}}
  24. K K
  25. S u b C i p h e r 1 SubCipher_{1}
  26. H 1 H_{1}
  27. k b 1 k_{b_{1}}
  28. k f 1 k_{f_{1}}
  29. T 1 T_{1}
  30. S u b C i p h e r 2 = E N C f 2 ( k f 2 , s 1 ) SubCipher_{2}=ENC_{f_{2}}(k_{f_{2}},s_{1})
  31. k f 2 k_{f_{2}}
  32. K K
  33. S u b C i p h e r 2 SubCipher_{2}
  34. k f 2 k_{f_{2}}
  35. H 2 H_{2}
  36. s 2 s_{2}
  37. S u b C i p h e r 2 = D E C b 2 ( k b 2 , s 2 ) SubCipher_{2}=DEC_{b_{2}}(k_{b_{2}},s_{2})
  38. k b 2 k_{b_{2}}
  39. K K
  40. S u b C i p h e r 2 SubCipher_{2}
  41. H 2 H_{2}
  42. T 1 T_{1}
  43. T 2 T_{2}
  44. s n s_{n}
  45. S u b C i p h e r n = D E C b n ( k b n , s n ) SubCipher_{n}=DEC_{b_{n}}(k_{b_{n}},s_{n})
  46. k b n k_{b_{n}}
  47. K K
  48. S u b C i p h e r n SubCipher_{n}
  49. H n H_{n}
  50. T n - 1 T_{n-1}
  51. k b n k_{b_{n}}
  52. k f n k_{f_{n}}
  53. T n T_{n}
  54. S u b C i p h e r n + 1 = E N C f n + 1 ( k f n + 1 , s n ) SubCipher_{n+1}=ENC_{f_{n}+1}(k_{f_{n}+1},s_{n})
  55. k f n + 1 k_{f_{n+1}}
  56. K K
  57. S u b C i p h e r n + 1 SubCipher_{n+1}
  58. H n + 1 H_{n+1}
  59. T n T_{n}
  60. ( k f 1 , k b 1 , k f 2 , k b 2 , , k f n + 1 , k b n + 1 ) (k_{f_{1}},k_{b_{1}},k_{f_{2}},k_{b_{2}},...,k_{f_{n+1}},k_{b_{n+1}})
  61. 2 | k f 1 | + 2 | k b n + 1 | + 2 | s 1 | 2^{|k_{f_{1}}|}+2^{|k_{b_{n+1}}|}+2^{|s_{1}|}
  62. ( 2 | k b 1 | + 2 | k f 2 | + 2 | s 2 | (2^{|k_{b_{1}}|}+2^{|k_{f_{2}}|}+2^{|s_{2}|}
  63. ( 2 | k b 2 | + 2 | k f 3 | + ) ) (2^{|k_{b_{2}}|}+2^{|k_{f_{3}}|}+...))
  64. T 2 , T 3 , , T n T_{2},T_{3},...,T_{n}
  65. T 1 T_{1}
  66. T i T_{i}
  67. T n T_{n}
  68. 2 | k f 1 | + 2 | k b n + 1 | + 2 | k | - | s n | 2^{|k_{f_{1}}|}+2^{|k_{b_{n+1}}|}+2^{|k|-|s_{n}|}...
  69. k k
  70. 1 / 2 | l | 1/2^{|l|}
  71. l l
  72. 2 | k | / 2 | l | 2^{|k|}/2^{|l|}
  73. 2 | k | - b 2^{|k|-b}
  74. 2 - b = 2 | k | - 2 b 2^{-b}=2^{|k|-2b}
  75. 1 / 2 | b | 1/2^{|b|}
  76. 2 | k | - b + 2 | k | - 2 b + 2 | k | - 3 b + 2 | k | - 4 b 2^{|k|-b}+2^{|k|-2b}+2^{|k|-3b}+2^{|k|-4b}
  77. | k | / n |k|/n
  78. S u b C i p h e r 1 = E N C f 1 ( k f 1 , P ) SubCipher_{1}=ENC_{f_{1}}(k_{f_{1}},P)
  79. k f 1 k_{f_{1}}
  80. K K
  81. S u b C i p h e r 1 SubCipher_{1}
  82. k f 1 k_{f_{1}}
  83. S u b C i p h e r 2 = D E C b 2 ( k b 2 , C ) SubCipher_{2}=DEC_{b_{2}}(k_{b_{2}},C)
  84. k b 2 k_{b_{2}}
  85. K K
  86. S u b C i p h e r 2 SubCipher_{2}
  87. k b 2 k_{b_{2}}
  88. S u b C i p h e r 1 SubCipher_{1}
  89. S u b C i p h e r 2 SubCipher_{2}
  90. S u b C i p h e r 1 = D E C b 1 ( k b 1 , s ) SubCipher_{1}=DEC_{b_{1}}(k_{b_{1}},s)
  91. k b 1 k_{b_{1}}
  92. K K
  93. S u b C i p h e r 1 SubCipher_{1}
  94. k b 1 k_{b_{1}}
  95. k f 1 k_{f_{1}}
  96. S u b C i p h e r 2 = E N C f 2 ( k f 2 , s ) SubCipher_{2}=ENC_{f_{2}}(k_{f_{2}},s)
  97. k f 2 k_{f_{2}}
  98. K K
  99. S u b C i p h e r 2 SubCipher_{2}
  100. ( k f 1 , k b 1 , k f 2 , k b 2 ) (k_{f_{1}},k_{b_{1}},k_{f_{2}},k_{b_{2}})
  101. 2 | k f 1 | + 2 | k b 2 | + 2 | s | ( 2 | k b 1 | + 2 | k f 2 | ) 2^{|k_{f_{1}}|}+2^{|k_{b_{2}}|}+2^{|s|}\cdot\left(2^{|k_{b_{1}}|}+2^{|k_{f_{2}% }|}\right)

Mellin_transform.html

  1. { f } ( s ) = φ ( s ) = 0 x s - 1 f ( x ) d x . \left\{\mathcal{M}f\right\}(s)=\varphi(s)=\int_{0}^{\infty}x^{s-1}f(x)dx.
  2. { - 1 φ } ( x ) = f ( x ) = 1 2 π i c - i c + i x - s φ ( s ) d s . \left\{\mathcal{M}^{-1}\varphi\right\}(x)=f(x)=\frac{1}{2\pi i}\int_{c-i\infty% }^{c+i\infty}x^{-s}\varphi(s)\,ds.
  3. { f } ( s ) = { f ( - ln x ) } ( s ) \left\{\mathcal{B}f\right\}(s)=\left\{\mathcal{M}f(-\ln x)\right\}(s)
  4. { f } ( s ) = { f ( e - x ) } ( s ) . \left\{\mathcal{M}f\right\}(s)=\left\{\mathcal{B}f(e^{-x})\right\}(s).
  5. d x x \frac{dx}{x}
  6. x a x x\mapsto ax
  7. d ( a x ) a x = d x x ; \frac{d(ax)}{ax}=\frac{dx}{x};
  8. d x dx
  9. d ( x + a ) = d x d(x+a)=dx
  10. { f } ( - s ) = { f } ( - i s ) = { f ( - ln x ) } ( - i s ) . \left\{\mathcal{F}f\right\}(-s)=\left\{\mathcal{B}f\right\}(-is)=\left\{% \mathcal{M}f(-\ln x)\right\}(-is).
  11. { f } ( s ) = { f ( e - x ) } ( s ) = { f ( e - x ) } ( - i s ) . \left\{\mathcal{M}f\right\}(s)=\left\{\mathcal{B}f(e^{-x})\right\}(s)=\left\{% \mathcal{F}f(e^{-x})\right\}(-is).
  12. c > 0 c>0
  13. ( y ) > 0 \Re(y)>0
  14. y - s y^{-s}
  15. e - y = 1 2 π i c - i c + i Γ ( s ) y - s d s e^{-y}=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(s)y^{-s}\;ds
  16. Γ ( s ) \Gamma(s)
  17. f ( x ) = { 0 x < 1 , x a x > 1 , , f(x)=\begin{cases}0&x<1,\\ x^{a}&x>1,\end{cases},
  18. f ( s ) = - 1 s + a , \mathcal{M}f(s)=-\frac{1}{s+a},
  19. ( s + a ) < 0. \Re(s+a)<0.
  20. L 2 ( 0 , ) L^{2}(0,\infty)
  21. 1 2 + i \tfrac{1}{2}+i\mathbb{R}
  22. ~ \tilde{\mathcal{M}}
  23. ~ : L 2 ( 0 , ) L 2 ( - , ) , { ~ f } ( s ) := 1 2 π 0 x - 1 2 + i s f ( x ) d x . \tilde{\mathcal{M}}\colon L^{2}(0,\infty)\to L^{2}(-\infty,\infty),\{\tilde{% \mathcal{M}}f\}(s):=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}x^{-\frac{1}{2}+is}f% (x)\,dx.
  24. { ~ f } ( s ) := 1 2 π { f } ( 1 2 + i s ) . \{\tilde{\mathcal{M}}f\}(s):=\tfrac{1}{\sqrt{2\pi}}\{\mathcal{M}f\}(\tfrac{1}{% 2}+is).
  25. \mathcal{M}
  26. ~ \tilde{\mathcal{M}}
  27. ~ \tilde{\mathcal{M}}
  28. ~ - 1 : L 2 ( - , ) L 2 ( 0 , ) , { ~ - 1 φ } ( x ) = 1 2 π - x - 1 2 - i s φ ( s ) d s . \tilde{\mathcal{M}}^{-1}\colon L^{2}(-\infty,\infty)\to L^{2}(0,\infty),\{% \tilde{\mathcal{M}}^{-1}\varphi\}(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{% \infty}x^{-\frac{1}{2}-is}\varphi(s)\,ds.
  29. ~ f L 2 ( - , ) = f L 2 ( 0 , ) \|\tilde{\mathcal{M}}f\|_{L^{2}(-\infty,\infty)}=\|f\|_{L^{2}(0,\infty)}
  30. f L 2 ( 0 , ) f\in L^{2}(0,\infty)
  31. 1 / 2 π 1/\sqrt{2\pi}
  32. X ( s ) = 0 x s d F X + ( x ) + γ 0 x s d F X - ( x ) , \mathcal{M}_{X}(s)=\int_{0}^{\infty}x^{s}dF_{X^{+}}(x)+\gamma\int_{0}^{\infty}% x^{s}dF_{X^{-}}(x),
  33. X ( i t ) \scriptstyle\mathcal{M}_{X}(it)
  34. X Y ( s ) = X ( s ) Y ( s ) \mathcal{M}_{XY}(s)=\mathcal{M}_{X}(s)\mathcal{M}_{Y}(s)
  35. 1 r r ( r f r ) = f r r + f r r {1\over r}{\partial\over\partial r}\left(r{\partial f\over\partial r}\right)=f% _{rr}+{f_{r}\over r}
  36. 2 f = 1 r r ( r f r ) + 1 r 2 2 f θ 2 \nabla^{2}f={1\over r}{\partial\over\partial r}\left(r{\partial f\over\partial r% }\right)+{1\over r^{2}}{\partial^{2}f\over\partial\theta^{2}}
  37. 2 f = 1 r r ( r f r ) + 1 r 2 2 f φ 2 + 2 f z 2 . \nabla^{2}f={1\over r}{\partial\over\partial r}\left(r{\partial f\over\partial r% }\right)+{1\over r^{2}}{\partial^{2}f\over\partial\varphi^{2}}+{\partial^{2}f% \over\partial z^{2}}.
  38. ( r 2 f r r + r f r , r s ) = s 2 ( f , r s ) = s 2 F \mathcal{M}\left(r^{2}f_{rr}+rf_{r},r\rightarrow s\right)=s^{2}\mathcal{M}% \left(f,r\rightarrow s\right)=s^{2}F
  39. r 2 f r r + r f r + f θ θ = 0 r^{2}f_{rr}+rf_{r}+f_{\theta\theta}=0
  40. 1 r r ( r f r ) + 1 r 2 2 f θ 2 = 0 {1\over r}{\partial\over\partial r}\left(r{\partial f\over\partial r}\right)+{% 1\over r^{2}}{\partial^{2}f\over\partial\theta^{2}}=0
  41. F θ θ + s 2 F = 0 F_{\theta\theta}+s^{2}F=0
  42. F ( s , θ ) = C 1 ( s ) cos ( s θ ) + C 2 ( s ) sin ( s θ ) F(s,\theta)=C_{1}(s)\cos(s\theta)+C_{2}(s)\sin(s\theta)
  43. f ( r , - θ 0 ) = a ( r ) , f ( r , θ 0 ) = b ( r ) f(r,-\theta_{0})=a(r),\quad f(r,\theta_{0})=b(r)
  44. F ( s , - θ 0 ) = A ( s ) , F ( s , θ 0 ) = B ( s ) F(s,-\theta_{0})=A(s),\quad F(s,\theta_{0})=B(s)
  45. F ( s , θ ) = A ( s ) sin ( s ( θ 0 - θ ) ) sin ( 2 θ 0 s ) + B ( s ) sin ( s ( θ 0 + θ ) ) sin ( 2 θ 0 s ) F(s,\theta)=A(s)\frac{\sin(s(\theta_{0}-\theta))}{\sin(2\theta_{0}s)}+B(s)% \frac{\sin(s(\theta_{0}+\theta))}{\sin(2\theta_{0}s)}
  46. f ( r , θ ) = r m 2 θ 0 cos m θ ( 0 x m - 1 a ( x ) x 2 m + 2 r m x m sin ( m θ ) + r 2 m d x + 0 x m - 1 b ( x ) x 2 m - 2 r m x m sin ( m θ ) + r 2 m d x ) f(r,\theta)=\frac{r^{m}}{2\theta_{0}}\cos m\theta\left(\int_{0}^{\infty}\frac{% x^{m-1}a(x)}{x^{2m}+2r^{m}x^{m}\sin(m\theta)+r^{2m}}dx+\int_{0}^{\infty}\frac{% x^{m-1}b(x)}{x^{2m}-2r^{m}x^{m}\sin(m\theta)+r^{2m}}dx\right)
  47. - 1 ( sin ( s ϕ ) sin ( 2 θ 0 s ) ; s r ) = 1 2 θ 0 r m sin ( m ϕ ) 1 + 2 r m cos ( m ϕ ) + r 2 m \mathcal{M}^{-1}\left(\frac{\sin(s\phi)}{\sin(2\theta_{0}s)};s\rightarrow r% \right)=\frac{1}{2\theta_{0}}\frac{r^{m}\sin(m\phi)}{1+2r^{m}\cos(m\phi)+r^{2m}}
  48. m = π 2 θ 0 m=\frac{\pi}{2\theta_{0}}

Membrane_potential.html

  1. 𝐄 \mathbf{E}
  2. E e q , K + = R T z F ln [ K + ] o [ K + ] i , E_{eq,K^{+}}=\frac{RT}{zF}\ln\frac{[K^{+}]_{o}}{[K^{+}]_{i}},
  3. τ = R C τ=RC
  4. C C
  5. E m = R T F ln ( P K [ K + ] out + P Na [ Na + ] out + P Cl [ Cl - ] in P K [ K + ] in + P Na [ Na + ] in + P Cl [ Cl - ] out ) E_{m}=\frac{RT}{F}\ln{\left(\frac{P_{\mathrm{K}}[\mathrm{K}^{+}]_{\mathrm{out}% }+P_{\mathrm{Na}}[\mathrm{Na}^{+}]_{\mathrm{out}}+P_{\mathrm{Cl}}[\mathrm{Cl}^% {-}]_{\mathrm{in}}}{P_{\mathrm{K}}[\mathrm{K}^{+}]_{\mathrm{in}}+P_{\mathrm{Na% }}[\mathrm{Na}^{+}]_{\mathrm{in}}+P_{\mathrm{Cl}}[\mathrm{Cl}^{-}]_{\mathrm{% out}}}\right)}

Membrane_transport.html

  1. Δ G = R T log C 2 C 1 \Delta G=RT\log\frac{C_{2}}{C_{1}}
  2. Δ G = R T log C i n s i d e C o u t s i d e + Z F Δ P \Delta G=RT\log\frac{C_{inside}}{C_{outside}}+ZF\Delta P
  3. Δ G = R T log C i n s i d e C o u t s i d e + Δ G b \Delta G=RT\log\frac{C_{inside}}{C_{outside}}+\Delta G^{b}

Mental_calculation.html

  1. ( 10 a + b ) ( 10 c + d ) (10a+b)\cdot(10c+d)
  2. = 100 ( a c ) + 10 ( b c ) + 10 ( a d ) + b d =100(a\cdot c)+10(b\cdot c)+10(a\cdot d)+b\cdot d
  3. 23 47 = 100 ( 2 4 ) + 10 ( 3 4 ) + 10 ( 2 7 ) + 3 7 23\cdot 47=100(2\cdot 4)+10(3\cdot 4)+10(2\cdot 7)+3\cdot 7
  4. ( a d + b c ) 10 (a\cdot d+b\cdot c)\cdot 10
  5. + b d {}+b\cdot d
  6. + a c 100 {}+a\cdot c\cdot 100
  7. 75 23 75\cdot 23
  8. a b c d ab\cdot cd
  9. a c 100 + ( a d + b c ) 10 + b d a\cdot c\cdot 100+(a\cdot d+b\cdot c)\cdot 10+b\cdot d
  10. a c a\cdot c
  11. ( a d + b c ) (a\cdot d+b\cdot c)
  12. b d b\cdot d
  13. E x : 35 * 75 Ex:35*75
  14. 35 - 5 = 30 = X 35-5=30=X
  15. 75 + 5 = 80 = Y 75+5=80=Y
  16. ( X * Y ) + 50 ( t 1 - t 2 - 1 ) + 75 (X*Y)+50(t_{1}-t_{2}-1)+75
  17. = 30 * 80 + 50 ( 7 - 3 - 1 ) + 75 = 2625 =30*80+50(7-3-1)+75=2625
  18. 33 * 67 33*67
  19. 33 - 3 = 30 = X 33-3=30=X
  20. 67 + 3 = 70 = Y 67+3=70=Y
  21. ( X * Y ) + u 1 * u 2 + u 2 ( T 1 - T 2 ) (X*Y)+u_{1}*u_{2}+u_{2}(T_{1}-T_{2})
  22. ( 30 * 70 ) + 7 * 3 + 3 ( 60 - 30 ) = 2100 + 21 + 90 = 2211 (30*70)+7*3+3(60-30)=2100+21+90=2211
  23. t R = 10 - y \,t_{R}=10-y\,
  24. b L = x - 5 \,b_{L}=x-5\,
  25. b R = y - 5 \,b_{R}=y-5\,
  26. 10 ( b L + b R ) + t L t R \,10(b_{L}+b_{R})+t_{L}t_{R}
  27. = 10 [ ( x - 5 ) + ( y - 5 ) ] + ( 10 - x ) ( 10 - y ) \,=10[(x-5)+(y-5)]+(10-x)(10-y)
  28. = 10 ( x + y - 10 ) + ( 100 - 10 x - 10 y + x y ) \,=10(x+y-10)+(100-10x-10y+xy)
  29. = [ 10 ( x + y ) - 100 ] + [ 100 - 10 ( x + y ) + x y ] \,=[10(x+y)-100]+[100-10(x+y)+xy]
  30. = [ 10 ( x + y ) - 10 ( x + y ) ] + [ 100 - 100 ] + x y \,=[10(x+y)-10(x+y)]+[100-100]+xy
  31. = x y \,=xy
  32. n 2 = n 2 n^{2}=n^{2}
  33. n 2 = ( n 2 - a 2 ) + a 2 n^{2}=(n^{2}-a^{2})+a^{2}
  34. n 2 = ( n 2 - a n + a n - a 2 ) + a 2 n^{2}=(n^{2}-an+an-a^{2})+a^{2}
  35. n 2 = ( n - a ) ( n + a ) + a 2 n^{2}=(n-a)(n+a)+a^{2}
  36. root known square root - known square - unknown square 2 × known square root \,\text{root }\simeq\,\text{ known square root}-\frac{\,\text{known square}-\,% \text{unknown square}}{2\times\,\text{known square root}}\,
  37. root 4 - 16 - 15 2 × 4 4 - 0.125 3.875 \begin{aligned}\displaystyle\,\text{root}&\displaystyle\simeq 4-\frac{16-15}{2% \times 4}\\ &\displaystyle\simeq 4-0.125\\ &\displaystyle\simeq 3.875\\ \end{aligned}\,\!
  38. root 2 = x \mathrm{root}^{2}=x\,\!
  39. root = a - b \mathrm{root}=a-b\,\!
  40. ( a - b ) 2 = x (a-b)^{2}=x\,\!
  41. a 2 - 2 a b + b 2 = x a^{2}-2ab+b^{2}=x\,\!
  42. + b 2 {}+b^{2}\,
  43. + b 2 {}+b^{2}\,
  44. b a 2 - x 2 a b\simeq\frac{a^{2}-x}{2a}\,\!
  45. root a - a 2 - x 2 a \mathrm{root}\simeq a-\frac{a^{2}-x}{2a}\,\!
  46. root a 2 + x 2 a \mathrm{root}\simeq\frac{a^{2}+x}{2a}\,\!
  47. 6 ¯ \overline{6}
  48. 3 ¯ \overline{3}
  49. 6 ¯ \overline{6}
  50. 3 ¯ \overline{3}

Mereology.html

  1. P P x y ( P x y and ¬ P y x ) . PPxy\leftrightarrow(Pxy\and\lnot Pyx).
  2. O x y z [ P z x and P z y ] . Oxy\leftrightarrow\exists z[Pzx\and Pzy].
  3. U x y z [ P x z and P y z ] . Uxy\leftrightarrow\exists z[Pxz\and Pyz].
  4. P x y z [ O z x O z y ] . Pxy\leftrightarrow\forall z[Ozx\rightarrow Ozy].
  5. P x x . \ Pxx.
  6. ( P x y and P y x ) x = y . (Pxy\and Pyx)\rightarrow x=y.
  7. ( P x y and P y z ) P x z . (Pxy\and Pyz)\rightarrow Pxz.
  8. P P x y z [ P z y and ¬ O z x ] . PPxy\rightarrow\exists z[Pzy\and\lnot Ozx].
  9. ¬ P y x z [ P z y and ¬ O z x ] . \lnot Pyx\rightarrow\exists z[Pzy\and\lnot Ozx].
  10. ¬ P x y z [ P z x and ¬ O z y and ¬ v [ P P v z ] ] . \lnot Pxy\rightarrow\exists z[Pzx\and\lnot Ozy\and\lnot\exists v[PPvz]].
  11. W x [ P x W ] . \exists W\forall x[PxW].
  12. N x [ P N x ] . \exists N\forall x[PNx].
  13. U x y z v [ O v z ( O v x O v y ) ] . Uxy\rightarrow\exists z\forall v[Ovz\leftrightarrow(OvxOvy)].
  14. O x y z v [ P v z ( P v x and P v y ) ] . Oxy\rightarrow\exists z\forall v[Pvz\leftrightarrow(Pvx\and Pvy)].
  15. x [ ϕ ( x ) ] z y [ O y z x [ ϕ ( x ) and O y x ] ] . \exists x[\phi(x)]\to\exists z\forall y[Oyz\leftrightarrow\exists x[\phi(x)% \and Oyx]].
  16. y [ P y x and z [ ¬ P P z y ] ] . \exists y[Pyx\and\forall z[\lnot PPzy]].

Mertens_conjecture.html

  1. M ( n ) = 1 k n μ ( k ) M(n)=\sum_{1\leq k\leq n}\mu(k)
  2. | M ( n ) | < n . \left|M(n)\right|<\sqrt{n}.\,
  3. m ( n ) = M ( n ) / n m(n)=M(n)/\sqrt{n}
  4. m ( n ) m(n)
  5. - 1 < m ( n ) < 1 -1<m(n)<1
  6. lim inf m ( n ) < - 1.009 \liminf m(n)<-1.009
  7. lim sup m ( n ) > 1.06 \limsup m(n)>1.06
  8. × 10 6 4 \times 10^{6}4
  9. × 10 4 0 \times 10^{4}0
  10. ( log log log n ) 5 / 4 (\log{\log{\log{n}}})^{5/4}
  11. m ( n ) 0.570591 m(n)\approx 0.570591
  12. 1 ζ ( s ) = n = 1 μ ( n ) n s , \frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}},
  13. ( s ) > 1 \Re(s)>1
  14. 1 ζ ( s ) = 0 x - s d M ( x ) \frac{1}{\zeta(s)}=\int_{0}^{\infty}x^{-s}\,dM(x)
  15. 1 s ζ ( s ) = { M } ( - s ) = 0 x - s M ( x ) d x x . \frac{1}{s\zeta(s)}=\left\{\mathcal{M}M\right\}(-s)=\int_{0}^{\infty}x^{-s}M(x% )\,\frac{dx}{x}.
  16. M ( x ) = 1 2 π i σ - i σ + i x s s ζ ( s ) d s M(x)=\frac{1}{2\pi i}\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{x^{s}}{s\zeta% (s)}\,ds
  17. M ( x ) = O ( x 1 2 + ϵ ) M(x)=O(x^{\frac{1}{2}+\epsilon})
  18. M ( x ) = O ( x 1 2 ) M(x)=O(x^{\frac{1}{2}})

Mertens_function.html

  1. M ( n ) = k = 1 n μ ( k ) M(n)=\sum_{k=1}^{n}\mu(k)
  2. M ( x ) = 1 k x μ ( k ) . M(x)=\sum_{1\leq k\leq x}\mu(k).
  3. 1 ζ ( s ) = p ( 1 - p - s ) = n = 1 μ ( n ) n s \frac{1}{\zeta(s)}=\prod_{p}(1-p^{-s})=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}
  4. ζ ( s ) \zeta(s)
  5. 1 2 π i c - i c + i x s s ζ ( s ) d s = M ( x ) \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{x^{s}}{s\zeta(s)}\,ds=M(x)
  6. 1 ζ ( s ) = s 1 M ( x ) x s + 1 d x \frac{1}{\zeta(s)}=s\int_{1}^{\infty}\frac{M(x)}{x^{s+1}}\,dx
  7. Re ( s ) > 1 \mathrm{Re}(s)>1
  8. ψ ( x ) = M ( x 2 ) log ( 2 ) + M ( x 3 ) log ( 3 ) + M ( x 4 ) log ( 4 ) + . \psi(x)=M\left(\frac{x}{2}\right)\log(2)+M\left(\frac{x}{3}\right)\log(3)+M% \left(\frac{x}{4}\right)\log(4)+\cdots.
  9. C F ( s ) e s t d s M ( e t ) . \oint_{C}F(s)e^{st}\,ds\sim M(e^{t}).
  10. ζ ( ρ ) \zeta(\rho)
  11. 1 2 π i C x s s ζ ( s ) d s = ρ x ρ ρ ζ ( ρ ) - 2 + n = 1 ( - 1 ) n - 1 ( 2 π ) 2 n ( 2 n ) ! n ζ ( 2 n + 1 ) x 2 n . \frac{1}{2\pi i}\oint_{C}\frac{x^{s}}{s\zeta(s)}\,ds=\sum_{\rho}\frac{x^{\rho}% }{\rho\zeta^{\prime}(\rho)}-2+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(2\pi)^{2n}}{% (2n)!n\zeta(2n+1)x^{2n}}.
  12. y ( x ) 2 - r = 1 N B 2 r ( 2 r ) ! D t 2 r - 1 y ( x t + 1 ) + x 0 x y ( u ) u 2 d u = x - 1 H ( log x ) \frac{y(x)}{2}-\sum_{r=1}^{N}\frac{B_{2r}}{(2r)!}D_{t}^{2r-1}y\left(\frac{x}{t% +1}\right)+x\int_{0}^{x}\frac{y(u)}{u^{2}}\,du=x^{-1}H(\log x)
  13. n = 1 μ ( n ) n g log n = t h ( t ) ζ ( 1 / 2 + i t ) + 2 n = 1 ( - 1 ) n ( 2 π ) 2 n ( 2 n ) ! ζ ( 2 n + 1 ) - g ( x ) e - x ( 2 n + 1 / 2 ) d x , \sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}}g\log n=\sum_{t}\frac{h(t)}{\zeta^{% \prime}(1/2+it)}+2\sum_{n=1}^{\infty}\frac{(-1)^{n}(2\pi)^{2n}}{(2n)!\zeta(2n+% 1)}\int_{-\infty}^{\infty}g(x)e^{-x(2n+1/2)}\,dx,
  14. 2 π g ( x ) = - h ( u ) e i u x d u . 2\pi g(x)=\int_{-\infty}^{\infty}h(u)e^{iux}\,du.
  15. M ( n ) = a n e 2 π i a M(n)=\sum_{a\in\mathcal{F}_{n}}e^{2\pi ia}
  16. n \mathcal{F}_{n}
  17. × 10 5 \times 10^{5}
  18. × 10 5 \times 10^{5}
  19. × 10 6 \times 10^{6}
  20. × 10 9 \times 10^{9}

Message_authentication_code.html

  1. k k
  2. k k
  3. k e y = ( a , b ) key=(a,b)
  4. m m
  5. t a g := ( a * m + b ) mod p tag:=(a*m+b)\mod p
  6. p p

Messier_4.html

  1. [ F e H ] = - 1.07 ± 0.01 \left[\frac{Fe}{H}\right]\ =\ -1.07\pm 0.01

Metacentric_height.html

  1. K M = K B + B M KM=KB+BM
  2. B M = I V BM=\frac{I}{V}
  3. R M = G Z Δ RM=GZ\cdot\Delta
  4. G Z = G M s i n ϕ GZ=GM\cdot sin\phi
  5. T = 2 π k g G M ¯ T=\frac{2\pi\,k}{\sqrt{g\overline{GM}}}
  6. G M ¯ \overline{GM}
  7. G M T ¯ \overline{GM_{T}}
  8. G M L ¯ \overline{GM_{L}}

Metalanguage.html

  1. \mathcal{L}
  2. \mathcal{L}
  3. \mathcal{L}

Metalogic.html

  1. 𝒟 \mathcal{D}
  2. 𝒟 \mathcal{D}
  3. 𝒟 \mathcal{D}
  4. \mathcal{I}
  5. \mathcal{I}
  6. 𝒟 \mathcal{D}
  7. \mathcal{I}

Method_of_complements.html

  1. b n - y b^{n}-y
  2. ( b n - 1 ) - y (b^{n}-1)-y
  3. ( b n - 1 ) (b^{n}-1)
  4. b - 1 b-1
  5. b n - 1 = b n - 1 n = ( b - 1 ) ( b n - 1 + b n - 2 + + b + 1 ) = ( b - 1 ) b n - 1 + + ( b - 1 ) b^{n}-1=b^{n}-1^{n}=(b-1)(b^{n-1}+b^{n-2}+...+b+1)=(b-1)b^{n-1}+...+(b-1)
  6. b - 1 b-1
  7. b - 1 b-1
  8. b n - 1 - x + y b^{n}-1-x+y
  9. b n - 1 - ( x - y ) b^{n}-1-(x-y)
  10. x - y x-y
  11. b - 1 b-1
  12. x - y x-y
  13. x + b n - y x+b^{n}-y
  14. x - y + b n x-y+b^{n}
  15. b n b^{n}
  16. b n b^{n}
  17. x - y + b n - b n x-y+b^{n}-b^{n}
  18. x - y x-y
  19. b n b^{n}
  20. b n / 2 b^{n}/2

Metric_map.html

  1. d Y ( f ( x ) , f ( y ) ) d X ( x , y ) . d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!
  2. T : X 𝒩 ( X ) T:X\to\mathcal{N}(X)
  3. L 0 L\geq 0
  4. H ( T x , T y ) L d ( x , y ) , H(Tx,Ty)\leq Ld(x,y),
  5. x , y X x,y\in X
  6. L = 1 L=1
  7. L < 1 L<1

Metric_modulation.html

  1. new tempo old tempo = number of pivot note values in new measure number of pivot note values in old measure \frac{\,\text{new tempo}}{\,\text{old tempo}}=\frac{\,\text{number of pivot % note values in new measure}}{\,\text{number of pivot note values in old % measure}}
  2. x 84 \displaystyle\qquad\frac{x}{84}
  3. = =

Metric_signature.html

  1. ( 1 0 0 - 1 ) , ( 0 1 1 0 ) . \begin{pmatrix}1&0\\ 0&-1\end{pmatrix},\quad\begin{pmatrix}0&1\\ 1&0\end{pmatrix}.
  2. n \mathbb{R}^{n}
  3. \R 4 \R^{4}
  4. ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  5. ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) \begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}
  6. d s 2 = c 2 d t 2 - d x 2 - d y 2 - d z 2 ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}

Micelle.html

  1. v o a e * l o \frac{v_{o}}{a_{e}*l_{o}}
  2. v o v_{o}
  3. l o l_{o}
  4. a e a_{e}

Microburst.html

  1. d w d t = - 1 ρ p z - g {dw\over dt}=-{1\over\rho}{\partial p\over\partial z}-g
  2. p = ρ R T v p=\rho RT_{v}
  3. B - ρ ρ ¯ g = g T v - T ¯ v T ¯ v B\equiv-{\rho^{\prime}\over\bar{\rho}}g=g{T^{\prime}_{v}-\bar{T}_{v}\over\bar{% T}_{v}}
  4. \ell
  5. d w d t = 1 ρ ¯ p z + B - g {dw^{\prime}\over dt}={1\over\bar{\rho}}{\partial p^{\prime}\over\partial z}+B% -g\ell
  6. - w max = 2 × NAPE -w_{\rm max}=\sqrt{2\times\hbox{NAPE}}
  7. NAPE = - SFC LFS B d z \hbox{NAPE}=-\int_{\rm SFC}^{\rm LFS}B\,dz

Microfiltration.html

  1. v = P F + P C 2 - P P v=\frac{P_{F}+P_{C}}{2}-P_{P}
  2. J v = 1 A M * d V d t = Δ P μ * ( R u + R c ) J_{v}=\frac{1}{A_{M}}*\frac{dV}{dt}=\frac{\Delta P}{\mu*(R_{u}+R_{c})}
  3. R c = r * V S A m R_{c}=r*\frac{V_{S}}{A_{m}}
  4. r = 180 * ( 1 - ϵ ) ϵ 3 * d s 2 r=\frac{180*(1-\epsilon)}{\epsilon^{3}*d_{s}^{2}}

Midpoint.html

  1. A = ( a 1 , a 2 , , a n ) A=(a_{1},a_{2},\dots,a_{n})
  2. B = ( b 1 , b 2 , , b n ) B=(b_{1},b_{2},\dots,b_{n})
  3. A + B 2 . \frac{A+B}{2}.
  4. a i + b i 2 . \frac{a_{i}+b_{i}}{2}.
  5. P P
  6. P P

Milankovitch_cycles.html

  1. Q ¯ day {\overline{Q}}^{\mathrm{day}}

Minimal_Supersymmetric_Standard_Model.html

  1. α - 1 ( M Z 0 ) \alpha^{-1}(M_{Z^{0}})
  2. b 0 MSSM b_{0}^{\mathrm{MSSM}}
  3. - 3 -3
  4. + 1 +1
  5. + 6 3 5 +6\frac{3}{5}
  6. α 1 - 1 \alpha^{-1}_{1}
  7. 3 5 \frac{3}{5}
  8. α 3 - 1 - α 2 - 1 α 2 - 1 - α 1 - 1 = b 0 3 - b 0 2 b 0 2 - b 0 1 \frac{\alpha^{-1}_{3}-\alpha^{-1}_{2}}{\alpha^{-1}_{2}-\alpha^{-1}_{1}}=\frac{% b_{0\,3}-b_{0\,2}}{b_{0\,2}-b_{0\,1}}
  9. α - 1 ( M Z 0 ) \alpha^{-1}(M_{Z^{0}})
  10. χ ~ 1 0 , , χ ~ 4 0 \tilde{\chi}_{1}^{0},\ldots,\tilde{\chi}_{4}^{0}
  11. χ ~ 1 ± \tilde{\chi}_{1}^{\pm}
  12. χ ~ 2 ± \tilde{\chi}_{2}^{\pm}
  13. t ~ \tilde{t}
  14. b ~ \tilde{b}
  15. t ~ 1 = e + i ϕ cos ( θ ) t L ~ + sin ( θ ) t R ~ \tilde{t}_{1}=e^{+i\phi}\cos(\theta)\tilde{t_{L}}+\sin(\theta)\tilde{t_{R}}
  16. t ~ 2 = e - i ϕ cos ( θ ) t R ~ - sin ( θ ) t L ~ \tilde{t}_{2}=e^{-i\phi}\cos(\theta)\tilde{t_{R}}-\sin(\theta)\tilde{t_{L}}
  17. b ~ \tilde{b}
  18. ϕ \phi
  19. θ \theta
  20. q ~ q ¯ ~ q N ~ 1 0 q ¯ N ~ 1 0 \tilde{q}\tilde{\bar{q}}\rightarrow q\tilde{N}^{0}_{1}\bar{q}\tilde{N}^{0}_{1}\rightarrow
  21. q ~ q ¯ ~ q N ~ 2 0 q ¯ N ~ 1 0 q N ~ 1 0 ¯ q ¯ N ~ 1 0 \tilde{q}\tilde{\bar{q}}\rightarrow q\tilde{N}^{0}_{2}\bar{q}\tilde{N}^{0}_{1}% \rightarrow q\tilde{N}^{0}_{1}\ell\bar{\ell}\bar{q}\tilde{N}^{0}_{1}\rightarrow
  22. g ~ g ~ ( q q ¯ ~ ) ( q ¯ q ~ ) ( q q ¯ N ~ 1 0 ) ( q ¯ q N ~ 1 0 ) \tilde{g}\tilde{g}\rightarrow(q\tilde{\bar{q}})(\bar{q}\tilde{q})\rightarrow(q% \bar{q}\tilde{N}^{0}_{1})(\bar{q}q\tilde{N}^{0}_{1})\rightarrow
  23. g ~ g ~ ( q ¯ q ~ ) ( q ¯ q ~ ) ( q q ¯ C ~ 1 + ) ( q q ¯ C ~ 1 + ) ( q q ¯ W + ) ( q q ¯ W + ) \tilde{g}\tilde{g}\rightarrow(\bar{q}\tilde{q})(\bar{q}\tilde{q})\rightarrow(q% \bar{q}\tilde{C}^{+}_{1})(q\bar{q}\tilde{C}^{+}_{1})\rightarrow(q\bar{q}W^{+})% (q\bar{q}W^{+})\rightarrow
  24. + + \ell^{+}\ell^{+}
  25. C ~ + ~ + ν \tilde{C}^{+}\rightarrow\tilde{\ell}^{+}\nu
  26. N ~ 0 ~ + - \tilde{N}^{0}\rightarrow\tilde{\ell}^{+}\ell^{-}
  27. q q
  28. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  29. q ~ \tilde{q}
  30. \ell
  31. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  32. ~ \tilde{\ell}
  33. W W
  34. W ~ \tilde{W}
  35. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  36. B B
  37. B ~ \tilde{B}
  38. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  39. g g
  40. g ~ \tilde{g}
  41. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  42. h u , h d h_{u},h_{d}
  43. h ~ u , h ~ d \tilde{h}_{u},\tilde{h}_{d}
  44. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}
  45. ( 3 , 2 ) 1 6 (3,2)_{\frac{1}{6}}
  46. ( 3 ¯ , 1 ) - 2 3 (\bar{3},1)_{-\frac{2}{3}}
  47. ( 3 ¯ , 1 ) 1 3 (\bar{3},1)_{\frac{1}{3}}
  48. ( 1 , 2 ) - 1 2 (1,2)_{-\frac{1}{2}}
  49. ( 1 , 1 ) 1 (1,1)_{1\frac{}{}}
  50. ( 1 , 2 ) 1 2 (1,2)_{\frac{1}{2}}
  51. ( 1 , 2 ) - 1 2 (1,2)_{-\frac{1}{2}}
  52. m h 0 2 m Z 0 2 cos 2 2 β m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos^{2}2\beta
  53. m h 0 2 m Z 0 2 cos 2 2 β + 3 π 2 m t 4 sin 4 β v 2 log m t ~ m t m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos^{2}2\beta+\frac{3}{\pi^{2}}\frac{m_{t}^{4}% \sin^{4}\beta}{v^{2}}\log\frac{m_{\tilde{t}}}{m_{t}}
  54. m t m_{t}
  55. m t ~ m_{\tilde{t}}
  56. = y t m t ~ a h u q ~ 3 u ~ 3 c \mathcal{L}=y_{t}\,m_{\tilde{t}}\,a\;h_{u}\tilde{q}_{3}\tilde{u}^{c}_{3}
  57. a a
  58. m h 0 2 m Z 0 2 cos 2 2 β + 3 π 2 m t 4 sin 4 β v 2 ( log m t ~ m t + a 2 ( 1 - a 2 / 12 ) ) m_{h^{0}}^{2}\leq m_{Z^{0}}^{2}\cos^{2}2\beta+\frac{3}{\pi^{2}}\frac{m_{t}^{4}% \sin^{4}\beta}{v^{2}}\left(\log\frac{m_{\tilde{t}}}{m_{t}}+a^{2}(1-a^{2}/12)\right)
  59. a 6 a\rightarrow\sqrt{6}
  60. W = μ H u H d + y u H u Q U c + y d H d Q D c + y l H d L E c W=\mu H_{u}H_{d}+y_{u}H_{u}QU^{c}+y_{d}H_{d}QD^{c}+y_{l}H_{d}LE^{c}
  61. m 1 2 λ ~ λ ~ + h.c. \mathcal{L}\supset m_{\frac{1}{2}}\tilde{\lambda}\tilde{\lambda}+\,\text{h.c.}
  62. λ ~ \tilde{\lambda}
  63. m 1 2 m_{\frac{1}{2}}
  64. m 0 2 ϕ ϕ \mathcal{L}\supset m^{2}_{0}\phi^{\dagger}\phi
  65. ϕ \phi
  66. m 0 m_{0}
  67. 3 × 3 3\times 3
  68. A A
  69. B B
  70. B μ h u h d + A h u q ~ u c ~ + A h d q ~ d c ~ + A h d l ~ e c ~ + h.c. \mathcal{L}\supset B_{\mu}h_{u}h_{d}+Ah_{u}\tilde{q}\tilde{u^{c}}+Ah_{d}\tilde% {q}\tilde{d^{c}}+Ah_{d}\tilde{l}\tilde{e^{c}}+\,\text{h.c.}
  71. A A
  72. 3 × 3 3\times 3
  73. m 3 / 2 Ψ μ α ( σ μ ν ) α β Ψ β + m 3 / 2 G α G α + h.c. \mathcal{L}\supset m_{3/2}\Psi_{\mu}^{\alpha}(\sigma^{\mu\nu})_{\alpha}^{\beta% }\Psi_{\beta}+m_{3/2}G^{\alpha}G_{\alpha}+\,\text{h.c.}
  74. m 0 m_{0}
  75. m 1 / 2 m_{1/2}
  76. A 0 A_{0}
  77. tan β \tan\beta
  78. sign ( μ ) \mathrm{sign}(\mu)
  79. tan β \tan\beta
  80. M A M_{A}
  81. μ \mu
  82. M 1 M_{1}
  83. M 2 M_{2}
  84. M 3 M_{3}
  85. m q ~ , m u ~ R , m d ~ R m_{\tilde{q}},m_{\tilde{u}_{R}},m_{\tilde{d}_{R}}
  86. m l ~ , m e ~ R m_{\tilde{l}},m_{\tilde{e}_{R}}
  87. m Q ~ , m t ~ R , m b ~ R m_{\tilde{Q}},m_{\tilde{t}_{R}},m_{\tilde{b}_{R}}
  88. m L ~ , m τ ~ R m_{\tilde{L}},m_{\tilde{\tau}_{R}}
  89. A t , A b , A τ A_{t},A_{b},A_{\tau}

Minimum_description_length.html

  1. P P
  2. C C
  3. C ( x ) C(x)
  4. - log 2 P ( x ) -\log_{2}P(x)
  5. C C
  6. P P

Minimum_phase.html

  1. H ( z ) H(z)
  2. \mathbb{H}
  3. i n v \mathbb{H}_{inv}
  4. \mathbb{H}
  5. i n v \mathbb{H}_{inv}
  6. 𝕀 \mathbb{I}
  7. i n v = 𝕀 \mathbb{H}_{inv}\,\mathbb{H}=\mathbb{I}
  8. x ~ \tilde{x}
  9. \mathbb{H}
  10. y ~ \tilde{y}
  11. x ~ = y ~ \mathbb{H}\,\tilde{x}=\tilde{y}
  12. i n v \mathbb{H}_{inv}
  13. y ~ \tilde{y}
  14. i n v y ~ = i n v x ~ = 𝕀 x ~ = x ~ \mathbb{H}_{inv}\,\tilde{y}=\mathbb{H}_{inv}\,\mathbb{H}\,\tilde{x}=\mathbb{I}% \,\tilde{x}=\tilde{x}
  15. i n v \mathbb{H}_{inv}
  16. x ~ \tilde{x}
  17. y ~ \tilde{y}
  18. \mathbb{H}
  19. h ( n ) h(n)
  20. i n v \mathbb{H}_{inv}
  21. h i n v ( n ) h_{inv}(n)
  22. ( h * h i n v ) ( n ) = k = - h ( k ) h i n v ( n - k ) = δ ( n ) (h*h_{inv})(n)=\sum_{k=-\infty}^{\infty}h(k)\,h_{inv}(n-k)=\delta(n)
  23. δ ( n ) \delta(n)
  24. i n v \mathbb{H}_{inv}
  25. \mathbb{H}
  26. i n v \mathbb{H}_{inv}
  27. h ( n ) = 0 n < 0 h(n)=0\,\,\forall\,n<0
  28. h i n v ( n ) = 0 n < 0 h_{inv}(n)=0\,\,\forall\,n<0
  29. n = - | h ( n ) | = h 1 < \sum_{n=-\infty}^{\infty}{\left|h(n)\right|}=\|h\|_{1}<\infty
  30. n = - | h i n v ( n ) | = h i n v 1 < \sum_{n=-\infty}^{\infty}{\left|h_{inv}(n)\right|}=\|h_{inv}\|_{1}<\infty
  31. ( h * h i n v ) ( n ) = δ ( n ) (h*h_{inv})(n)=\,\!\delta(n)
  32. H ( z ) H i n v ( z ) = 1 H(z)\,H_{inv}(z)=1
  33. H i n v ( z ) = 1 H ( z ) H_{inv}(z)=\frac{1}{H(z)}
  34. H ( z ) = A ( z ) D ( z ) H(z)=\frac{A(z)}{D(z)}
  35. H i n v ( z ) = D ( z ) A ( z ) H_{inv}(z)=\frac{D(z)}{A(z)}
  36. H i n v ( z ) H_{inv}(z)
  37. ( h * h i n v ) ( t ) = δ ( t ) (h*h_{inv})(t)=\,\!\delta(t)
  38. δ ( t ) \delta(t)
  39. δ ( t ) * x ( t ) = - δ ( t - τ ) x ( τ ) d τ = x ( t ) \delta(t)*x(t)=\int_{-\infty}^{\infty}\delta(t-\tau)x(\tau)d\tau=x(t)
  40. H ( s ) H i n v ( s ) = 1 H(s)\,H_{inv}(s)=1
  41. H i n v ( s ) = 1 H ( s ) H_{inv}(s)=\frac{1}{H(s)}
  42. H ( s ) = A ( s ) D ( s ) H(s)=\frac{A(s)}{D(s)}
  43. H i n v ( s ) = D ( s ) A ( s ) H_{inv}(s)=\frac{D(s)}{A(s)}
  44. H i n v ( s ) H_{inv}(s)
  45. H ( j ω ) = def H ( s ) | s = j ω H(j\omega)\ \stackrel{\mathrm{def}}{=}\ H(s)\Big|_{s=j\omega}
  46. arg [ H ( j ω ) ] = - { log ( | H ( j ω ) | ) } \arg\left[H(j\omega)\right]=-\mathcal{H}\{\log\left(|H(j\omega)|\right)\}
  47. log ( | H ( j ω ) | ) = log ( | H ( j ) | ) + { arg [ H ( j ω ) ] } \log\left(|H(j\omega)|\right)=\log\left(|H(j\infty)|\right)+\mathcal{H}\{\arg% \left[H(j\omega)\right]\}
  48. H ( j ω ) = | H ( j ω ) | e j arg [ H ( j ω ) ] = def e α ( ω ) e j ϕ ( ω ) = e α ( ω ) + j ϕ ( ω ) H(j\omega)=|H(j\omega)|e^{j\arg\left[H(j\omega)\right]}\ \stackrel{\mathrm{def% }}{=}\ e^{\alpha(\omega)}e^{j\phi(\omega)}=e^{\alpha(\omega)+j\phi(\omega)}
  49. α ( ω ) \alpha(\omega)
  50. ϕ ( ω ) \phi(\omega)
  51. ϕ ( ω ) = - { α ( ω ) } \phi(\omega)=-\mathcal{H}\{\alpha(\omega)\}
  52. α ( ω ) = α ( ) + { ϕ ( ω ) } \alpha(\omega)=\alpha(\infty)+\mathcal{H}\{\phi(\omega)\}
  53. { x ( t ) } = def x ^ ( t ) = 1 π - x ( τ ) t - τ d τ \mathcal{H}\{x(t)\}\ \stackrel{\mathrm{def}}{=}\ \widehat{x}(t)=\frac{1}{\pi}% \int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}\,d\tau
  54. n = m | h ( n ) | 2 m + \sum_{n=m}^{\infty}\left|h(n)\right|^{2}\,\,\,\,\,\,\,\forall\,m\in\mathbb{Z}^% {+}
  55. a a
  56. H ( z ) H(z)
  57. a a
  58. | a | < 1 \left|a\right|<1
  59. a = | a | e i θ a where θ a = Arg ( a ) a=\left|a\right|e^{i\theta_{a}}\,\mbox{ where }~{}\,\theta_{a}=\mbox{Arg}~{}(a)
  60. a a
  61. 1 - a z - 1 1-az^{-1}
  62. ϕ a ( ω ) = Arg ( 1 - a e - i ω ) \phi_{a}\left(\omega\right)=\mbox{Arg}~{}\left(1-ae^{-i\omega}\right)
  63. = Arg ( 1 - | a | e i θ a e - i ω ) =\mbox{Arg}~{}\left(1-\left|a\right|e^{i\theta_{a}}e^{-i\omega}\right)
  64. = Arg ( 1 - | a | e - i ( ω - θ a ) ) =\mbox{Arg}~{}\left(1-\left|a\right|e^{-i(\omega-\theta_{a})}\right)
  65. = Arg ( { 1 - | a | c o s ( ω - θ a ) } + i { | a | s i n ( ω - θ a ) } ) =\mbox{Arg}~{}\left(\left\{1-\left|a\right|cos(\omega-\theta_{a})\right\}+i% \left\{\left|a\right|sin(\omega-\theta_{a})\right\}\right)
  66. = Arg ( { | a | - 1 - cos ( ω - θ a ) } + i { sin ( ω - θ a ) } ) =\mbox{Arg}~{}\left(\left\{\left|a\right|^{-1}-\cos(\omega-\theta_{a})\right\}% +i\left\{\sin(\omega-\theta_{a})\right\}\right)
  67. ϕ a ( ω ) \phi_{a}(\omega)
  68. - d ϕ a ( ω ) d ω = sin 2 ( ω - θ a ) + cos 2 ( ω - θ a ) - | a | - 1 cos ( ω - θ a ) sin 2 ( ω - θ a ) + cos 2 ( ω - θ a ) + | a | - 2 - 2 | a | - 1 cos ( ω - θ a ) -\frac{d\phi_{a}(\omega)}{d\omega}=\frac{\sin^{2}(\omega-\theta_{a})+\cos^{2}(% \omega-\theta_{a})-\left|a\right|^{-1}\cos(\omega-\theta_{a})}{\sin^{2}(\omega% -\theta_{a})+\cos^{2}(\omega-\theta_{a})+\left|a\right|^{-2}-2\left|a\right|^{% -1}\cos(\omega-\theta_{a})}
  69. - d ϕ a ( ω ) d ω = | a | - cos ( ω - θ a ) | a | + | a | - 1 - 2 cos ( ω - θ a ) -\frac{d\phi_{a}(\omega)}{d\omega}=\frac{\left|a\right|-\cos(\omega-\theta_{a}% )}{\left|a\right|+\left|a\right|^{-1}-2\cos(\omega-\theta_{a})}
  70. θ a \theta_{a}
  71. a a
  72. a a
  73. ( a - 1 ) * (a^{-1})^{*}
  74. a a
  75. | a | \left|a\right|
  76. a a
  77. 1 - a z - 1 1-az^{-1}
  78. 1 - a i z - 1 1-a_{i}z^{-1}
  79. N N
  80. Arg ( i = 1 N ( 1 - a i z - 1 ) ) = i = 1 N Arg ( 1 - a i z - 1 ) \mbox{Arg}~{}\left(\prod_{i=1}^{N}\left(1-a_{i}z^{-1}\right)\right)=\sum_{i=1}% ^{N}\mbox{Arg}~{}\left(1-a_{i}z^{-1}\right)
  81. s + 10 s + 5 and s - 10 s + 5 \frac{s+10}{s+5}\qquad\,\text{and}\qquad\frac{s-10}{s+5}
  82. ( s + 1 ) ( s - 5 ) ( s + 10 ) ( s + 2 ) ( s + 4 ) ( s + 6 ) \frac{(s+1)(s-5)(s+10)}{(s+2)(s+4)(s+6)}

Minkowski_addition.html

  1. A + B = { 𝐚 + 𝐛 | 𝐚 A , 𝐛 B } . A+B=\{\mathbf{a}+\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}.
  2. A - B = { 𝐚 - 𝐛 | 𝐚 A , 𝐛 B } . A-B=\{\mathbf{a}-\mathbf{b}\,|\,\mathbf{a}\in A,\ \mathbf{b}\in B\}.
  3. 2 \mathbb{R}^{2}
  4. ( 1 , 0 ) (1, 0)
  5. [ 0 , 1 ] [0,1]
  6. [ 1 , 2 ] [1,2]
  7. [ 1 , 2 ] [1,2]
  8. [ 1 , 3 ] [1,3]
  9. [ 1 , 3 ] [1,3]
  10. [ 0 , 1 ] [0,1]
  11. [ 1 , 2 ] [1,2]
  12. [ 1 , 3 ] [1,3]
  13. S S
  14. μ S + λ S \mu S+\lambda S
  15. μ S + λ S = ( μ + λ ) S \mu S+\lambda S=(\mu+\lambda)S
  16. μ , λ 0 \mu,\lambda\geq 0
  17. μ , λ \mu,\lambda
  18. A + A 2 A A+A⊋2A
  19. B + B 2 B B+B⊋2B
  20. A + B = { z n | A ( { z } - B ) } . A+B=\{z\in\mathbb{R}^{n}\,|\,A\cap(\{z\}-B)\neq\emptyset\}.
  21. A + e B = { z n | μ [ A ( { z } - B ) ] > 0 } , A+_{\mathrm{e}}B=\{z\in\mathbb{R}^{n}\,|\,\mu\left[A\cap(\{z\}-B)\right]>0\},
  22. 1 A + B ( z ) = sup x n 1 A ( x ) 1 B ( z - x ) , 1_{A\,+\,B}(z)=\sup_{x\,\in\,\mathbb{R}^{n}}1_{A}(x)1_{B}(z-x),
  23. 1 A + e B ( z ) = ess sup x n 1 A ( x ) 1 B ( z - x ) , 1_{A\,+_{\mathrm{e}}\,B}(z)=\mathop{\mathrm{ess\,sup}}_{x\,\in\,\mathbb{R}^{n}% }1_{A}(x)1_{B}(z-x),

Minkowski–Bouligand_dimension.html

  1. dim box ( S ) := lim ε 0 log N ( ε ) log ( 1 / ε ) . \dim_{\rm box}(S):=\lim_{\varepsilon\to 0}\frac{\log N(\varepsilon)}{\log(1/% \varepsilon)}.
  2. N covering ( ε ) N_{\rm covering}(\varepsilon)
  3. N covering ( ε ) N^{\prime}_{\rm covering}(\varepsilon)
  4. N packing ( ε ) N_{\rm packing}(\varepsilon)
  5. N packing ( ε ) N covering ( ε ) N covering ( ε / 2 ) . N\text{packing}(\varepsilon)\leq N^{\prime}\text{covering}(\varepsilon)\leq N% \text{covering}(\varepsilon/2).\,
  6. dim box ( S ) = n - lim r 0 log vol ( S r ) log r , \dim\text{box}(S)=n-\lim_{r\to 0}\frac{\log\,\text{vol}(S_{r})}{\log r},
  7. S r S_{r}
  8. R n R^{n}
  9. S r S_{r}
  10. dim ( A 1 A n ) = max { dim A 1 , , dim A n } . \dim(A_{1}\cup\cdots\cup A_{n})=\max\{\dim A_{1},\dots,\dim A_{n}\}.\,
  11. dim upper box ( A + B ) dim upper box ( A ) + dim upper box ( B ) . \dim\text{upper box}(A+B)\leq\dim\text{upper box}(A)+\dim\text{upper box}(B).
  12. dim Haus dim lowerbox dim upperbox . \dim_{\operatorname{Haus}}\leq\dim_{\operatorname{lowerbox}}\leq\dim_{% \operatorname{upperbox}}.
  13. ε = 10 - 2 n \varepsilon=10^{-2^{n}}
  14. \mathbb{Q}
  15. dim Haus = 0 \dim_{\operatorname{Haus}}=0
  16. dim box = 1 \dim_{\operatorname{box}}=1
  17. \mathbb{R}
  18. dim box { 0 , 1 , 1 2 , 1 3 , 1 4 , } = 1 2 . \dim_{\operatorname{box}}\left\{0,1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots% \right\}=\frac{1}{2}.

Mipmap.html

  1. 2 n 2^{n}
  2. 2 n 2^{n}
  3. 2 i 2^{i}
  4. 2 j 2^{j}

Mirror_matter.html

  1. π + μ + e + \pi^{+}\rightarrow\mu^{+}\rightarrow e^{+}

Missing_square_puzzle.html

  1. S = 13 × 5 2 = 32.5 \textstyle{S=\frac{13\times 5}{2}=32.5}

Mitchell's_embedding_theorem.html

  1. Fun ( 𝒜 , A b ) \mathcal{L}\subset\operatorname{Fun}(\mathcal{A},Ab)
  2. 𝒜 \mathcal{A}
  3. A b Ab
  4. H : 𝒜 H:\mathcal{A}\to\mathcal{L}
  5. H ( A ) = h A H(A)=h_{A}
  6. A 𝒜 A\in\mathcal{A}
  7. h A h_{A}
  8. h A ( X ) = Hom 𝒜 ( A , X ) h_{A}(X)=\operatorname{Hom}_{\mathcal{A}}(A,X)
  9. H H
  10. H H
  11. h A h_{A}
  12. H H
  13. \mathcal{L}
  14. \mathcal{L}
  15. I = A 𝒜 h A . I=\prod_{A\in\mathcal{A}}h_{A}.
  16. R := Hom ( I , I ) R:=\operatorname{Hom}_{\mathcal{L}}(I,I)
  17. G ( B ) = Hom ( B , I ) G(B)=\operatorname{Hom}_{\mathcal{L}}(B,I)
  18. G : R - Mod . G:\mathcal{L}\to R\operatorname{-Mod}.
  19. G H : 𝒜 R - Mod GH:\mathcal{A}\to R\operatorname{-Mod}

Mixing_(physics).html

  1. lim k μ ( T - k A B ) = μ ( A ) μ ( B ) \lim_{k\rightarrow\infty}\,\mu(T^{-k}A\cap B)=\mu(A)\cdot\mu(B)

Modal_logic.html

  1. P ¬ ¬ P ; \Diamond P\leftrightarrow\lnot\Box\lnot P;
  2. P ¬ ¬ P . \Box P\leftrightarrow\lnot\Diamond\lnot P.
  3. G , R \langle G,R\rangle
  4. w * w*
  5. v ( w , P ) v(w,P)
  6. G , R , v \langle G,R,v\rangle
  7. v ( w , P ) v(w,P)
  8. w P w\models P
  9. w ¬ P w\models\neg P
  10. w ⊧̸ P w\not\models P
  11. w ( P Q ) w\models(P\wedge Q)
  12. w P w\models P
  13. w Q w\models Q
  14. w P w\models\Box P
  15. u P u\models P
  16. w P w\models\Diamond P
  17. u P u\models P
  18. P \models P
  19. w * P w*\models P
  20. w P w\models\Diamond P
  21. u P u\models P
  22. w P w\models\Diamond P
  23. u P u⊧P
  24. P P , P P P→□◇P,□P→□□P
  25. P P □P→P
  26. ¬ ¬ p ¬◇¬p
  27. ¬ ¬ p ¬□¬p
  28. ( p q ) ( p q ) . □(p→q)→(□p→□q).
  29. p p □p→p
  30. p p \Box p\to\Box\Box p
  31. p p p\to\Box\Diamond p
  32. p p \Box p\to\Diamond p
  33. p p \Diamond p\to\Box\Diamond p
  34. p p \Box p\to\Diamond p
  35. p p p\to\Box\Diamond p
  36. ( p q ) ( p q ) \Box(p\to q)\to(\Box p\to\Box q)
  37. P \Diamond P
  38. P \Box P
  39. 1 P \Diamond_{1}P
  40. 1 P \Box_{1}P
  41. 2 P \Diamond_{2}P
  42. 2 P \Box_{2}P
  43. ϕ ϕ \Box\phi\to\phi
  44. ϕ ϕ \Box\phi\to\Diamond\phi
  45. ( K Q ) (K\to\Box Q)
  46. ( K Q ) \Box(K\to Q)
  47. ( K ( K and ¬ Q ) ) \Box(K\to(K\and\lnot Q))
  48. ¬ K \Box\lnot K
  49. ( K ( K and ¬ Q ) ) \Box(K\to(K\and\lnot Q))
  50. ( ¬ K ) ( K K and ¬ K ) \Box(\lnot K)\to\Box(K\to K\and\lnot K)
  51. ( K and ¬ K ( K and ¬ Q ) ) \Box(K\and\lnot K\to(K\and\lnot Q))
  52. Q K Q\to K

Model_checking.html

  1. M , s p M,s\models p
  2. k k
  3. k k
  4. k k

Modular_group.html

  1. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  2. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  3. ( a b c d ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}
  4. a p + b q c p + d q \frac{ap+bq}{cp+dq}
  5. ( a b c d ) SL ( 2 , 𝐙 ) \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{SL}(2,\mathbf{Z})
  6. r = a p + b q and s = c p + d q . r=ap+bq\quad\mbox{ and }~{}\quad s=cp+dq.
  7. ω 1 \omega_{1}
  8. ω 2 \omega_{2}
  9. Λ ( ω 1 , ω 2 ) = { m ω 1 + n ω 2 : m , n 𝐙 } \Lambda(\omega_{1},\omega_{2})=\{m\omega_{1}+n\omega_{2}:m,n\in\mathbf{Z}\}
  10. α 1 \alpha_{1}
  11. α 2 \alpha_{2}
  12. ( α 1 α 2 ) = ( a b c d ) ( ω 1 ω 2 ) \begin{pmatrix}\alpha_{1}\\ \alpha_{2}\end{pmatrix}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}\omega_{1}\\ \omega_{2}\end{pmatrix}
  13. p n - 1 / q n - 1 p_{n-1}/q_{n-1}
  14. p n / q n p_{n}/q_{n}
  15. ( p n - 1 p n q n - 1 q n ) \begin{pmatrix}p_{n-1}&p_{n}\\ q_{n-1}&q_{n}\end{pmatrix}
  16. T : z z + 1 T:z\mapsto z+1
  17. Γ S , T S 2 = I , ( S T ) 3 = I \Gamma\cong\langle S,T\mid S^{2}=I,(ST)^{3}=I\rangle
  18. Γ C 2 * C 3 \Gamma\cong C_{2}*C_{3}
  19. SL 2 ( 𝐑 ) ¯ PSL 2 ( 𝐑 ) \overline{\mathrm{SL}_{2}(\mathbf{R})}\to\mathrm{PSL}_{2}(\mathbf{R})
  20. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  21. ( a b c d ) z = a z + b c z + d \begin{pmatrix}a&b\\ c&d\end{pmatrix}\cdot z\,=\,\frac{az+b}{cz+d}
  22. R = { z 𝐇 : | z | > 1 , | Re ( z ) | < 1 2 } R=\left\{z\in\mathbf{H}:\left|z\right|>1,\,\left|\,\mbox{Re}~{}(z)\,\right|<% \frac{1}{2}\right\}
  23. 1 Γ ( N ) Γ PSL ( 2 , 𝐙 / N 𝐙 ) 1 1\to\Gamma(N)\to\Gamma\to\mbox{PSL}~{}(2,\mathbf{Z}/N\mathbf{Z})\to 1
  24. z a z + b c z + d z\mapsto\frac{az+b}{cz+d}
  25. z - 1 / z z\mapsto-1/z
  26. z z + λ q , z\mapsto z+\lambda_{q},
  27. λ q = 2 cos ( π / q ) . \lambda_{q}=2\cos(\pi/q).\,
  28. Γ C 2 * C 3 , \Gamma\cong C_{2}*C_{3},
  29. H q C 2 * C q , H_{q}\cong C_{2}*C_{q},
  30. λ 3 = 1 , \lambda_{3}=1,
  31. λ 4 = 2 , \lambda_{4}=\sqrt{2},
  32. λ 5 = 1 2 ( 1 + 5 ) , \lambda_{5}=\tfrac{1}{2}(1+\sqrt{5}),
  33. λ 6 = 3 . \lambda_{6}=\sqrt{3}.

Module_homomorphism.html

  1. f : M N f:M\to N
  2. f ( x + y ) = f ( x ) + f ( y ) , f(x+y)=f(x)+f(y),
  3. f ( r x ) = r f ( x ) . f(rx)=rf(x).
  4. f ( x r ) = f ( x ) r . f(xr)=f(x)r.
  5. Hom ( / n , / m ) = / gcd ( n , m ) \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n,\mathbb{Z}/m)=\mathbb{Z}/% \operatorname{gcd}(n,m)
  6. End R ( R ) = R \operatorname{End}_{R}(R)=R
  7. Hom R ( R , M ) = M \operatorname{Hom}_{R}(R,M)=M
  8. f f ( 1 ) f\mapsto f(1)
  9. Hom R ( M , R ) \operatorname{Hom}_{R}(M,R)
  10. M * M^{*}
  11. f : M N f:M\to N
  12. g : M N g:M^{\prime}\to N^{\prime}
  13. f g : M M N N , ( x , y ) ( f ( x ) , f ( y ) ) f\oplus g:M\oplus M^{\prime}\to N\oplus N^{\prime},\,(x,y)\mapsto(f(x),f(y))
  14. f g : M M N N , x y f ( x ) f ( y ) . f\otimes g:M\otimes M^{\prime}\to N\otimes N^{\prime},\,x\otimes y\mapsto f(x)% \otimes f(y).
  15. f : M N f:M\to N
  16. f * : N * M * , f * ( α ) = α f . f^{*}:N^{*}\to M^{*},\,f^{*}(\alpha)=\alpha\circ f.
  17. 0 A 𝑓 B 𝑔 C 0 0\to A\overset{f}{\to}B\overset{g}{\to}C\to 0
  18. 0 A 𝔪 𝑓 B 𝔪 𝑔 C 𝔪 0 0\to A_{\mathfrak{m}}\overset{f}{\to}B_{\mathfrak{m}}\overset{g}{\to}C_{% \mathfrak{m}}\to 0
  19. 𝔪 {\mathfrak{m}}
  20. 𝔪 {\mathfrak{m}}
  21. 0 K M 𝑓 N C 0 0\to K\to M\overset{f}{\to}N\to C\to 0
  22. f : M B , g : N B f:M\to B,g:N\to B
  23. 0 M × B N M × N ϕ B 0 0\to M\times_{B}N\to M\times N\overset{\phi}{\to}B\to 0
  24. ϕ ( x , y ) = f ( x ) - g ( x ) \phi(x,y)=f(x)-g(x)
  25. B A B\subset A
  26. A A / I , B / I A / I A\to A/I,B/I\to A/I
  27. B = A × A / I B / I . B=A\times_{A/I}B/I.
  28. ϕ : M M \phi:M\to M
  29. ϕ \phi
  30. ϕ \phi
  31. M N M\to N
  32. M N . M\oplus N.
  33. f - 1 f^{-1}
  34. { ( y , x ) | ( x , y ) f } \{(y,x)|(x,y)\in f\}
  35. D ( f ) N / { y | ( 0 , y ) f } D(f)\to N/\{y|(0,y)\in f\}
  36. D ( f ) D(f)

Moduli_space.html

  1. g \mathcal{M}_{g}
  2. ¯ g \overline{\mathcal{M}}_{g}
  3. 0 \mathcal{M}_{0}
  4. 1 \mathcal{M}_{1}
  5. g , n \mathcal{M}_{g,n}
  6. ¯ g , n \overline{\mathcal{M}}_{g,n}
  7. ¯ 1 , 1 \overline{\mathcal{M}}_{1,1}

Modus_ponendo_tollens.html

  1. ¬ ( A and B ) \neg(A\and B)
  2. A A\,\!
  3. ¬ B \therefore\neg B
  4. A | B A\,|\,B
  5. ¬ A \neg A\,\!
  6. B \therefore B

Molecular_mechanics.html

  1. E = E covalent + E noncovalent \ E=E\text{covalent}+E\text{noncovalent}\,
  2. E covalent = E bond + E angle + E dihedral \ E\text{covalent}=E\text{bond}+E\text{angle}+E\text{dihedral}
  3. E noncovalent = E electrostatic + E van der Waals \ E\text{noncovalent}=E\text{electrostatic}+E\text{van der Waals}

Molecular_orbital_theory.html

  1. ψ j = i = 1 n c i j χ i . \psi_{j}=\sum_{i=1}^{n}c_{ij}\chi_{i}.

Molniya_orbit.html

  1. ( 5 4 sin 2 i - 1 ) \left(\frac{5}{4}\ \sin^{2}i\ -\ 1\right)
  2. Ω \Omega
  3. ω \omega
  4. Δ Ω = - 2 π J 2 μ p 2 3 2 cos i \Delta\Omega=-2\pi\frac{J_{2}}{\mu p^{2}}\ \frac{3}{2}\cos i
  5. Δ ω = - 2 π J 2 μ p 2 3 ( 5 4 sin 2 i - 1 ) \Delta\omega=-2\pi\frac{J_{2}}{\mu p^{2}}\ 3\left(\frac{5}{4}\ \sin^{2}i\ -\ 1\right)
  6. Δ ω \Delta\omega
  7. Δ Ω \Delta\Omega

Moment_(mathematics).html

  1. 0
  2. n n
  3. μ n = - ( x - c ) n f ( x ) d x . \mu_{n}=\int_{-\infty}^{\infty}(x-c)^{n}\,f(x)\,dx.
  4. n n
  5. E [ X - n ] \operatorname{E}\left[X^{-n}\right]
  6. n n
  7. E [ ln n ( X ) ] . \operatorname{E}\left[\ln^{n}(X)\right].
  8. n n
  9. μ μ
  10. n n
  11. n n
  12. μ n = E [ X n ] = - x n d F ( x ) \mu^{\prime}_{n}=\operatorname{E}\left[X^{n}\right]=\int_{-\infty}^{\infty}x^{% n}\,dF(x)\,
  13. E E
  14. E [ | X n | ] = - | x n | d F ( x ) = , \operatorname{E}\left[\left|X^{n}\right|\right]=\int_{-\infty}^{\infty}|x^{n}|% \,dF(x)=\infty,
  15. n n
  16. ( n 1 ) (n−1)
  17. n n
  18. n n
  19. n n
  20. x = E [ ( x - μ ) n ] σ n . x=\frac{\operatorname{E}\left[(x-\mu)^{n}\right]}{\sigma^{n}}.
  21. γ γ
  22. μ γ σ / 6 μ−γσ/6
  23. μ γ σ / 2 μ−γσ/2
  24. κ κ
  25. κ κ
  26. E [ ( T 2 - a T - 1 ) 2 ] \operatorname{E}\left[(T^{2}-aT-1)^{2}\right]
  27. T = ( X μ ) / σ T=(X−μ)/σ
  28. μ 1 ( X + Y ) = μ 1 ( X ) + μ 1 ( Y ) Var ( X + Y ) = Var ( X ) + Var ( Y ) μ 3 ( X + Y ) = μ 3 ( X ) + μ 3 ( Y ) \begin{aligned}\displaystyle\mu_{1}(X+Y)&\displaystyle=\mu_{1}(X)+\mu_{1}(Y)\\ \displaystyle\operatorname{Var}(X+Y)&\displaystyle=\operatorname{Var}(X)+% \operatorname{Var}(Y)\\ \displaystyle\mu_{3}(X+Y)&\displaystyle=\mu_{3}(X)+\mu_{3}(Y)\end{aligned}
  29. k k
  30. k k
  31. 1 n i = 1 n X i k \frac{1}{n}\sum_{i=1}^{n}X^{k}_{i}
  32. k k
  33. n n
  34. 1 n - 1 i = 1 n ( X i - X ¯ ) 2 \frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}
  35. n n
  36. n 1 n−1
  37. X ¯ \bar{X}
  38. n n - 1 , \tfrac{n}{n-1},
  39. n n
  40. μ n - ( r ) = - r ( r - x ) n f ( x ) d x , \mu_{n}^{-}(r)=\int_{-\infty}^{r}(r-x)^{n}\,f(x)\,dx,
  41. μ n + ( r ) = r ( x - r ) n f ( x ) d x . \mu_{n}^{+}(r)=\int_{r}^{\infty}(x-r)^{n}\,f(x)\,dx.
  42. ( M , d ) (M,d)
  43. σ σ
  44. σ σ
  45. 1 p 1≤p≤∞
  46. μ μ
  47. M d ( x , x 0 ) p d μ ( x ) . \int_{M}d(x,x_{0})^{p}\,\mathrm{d}\mu(x).
  48. p p
  49. p p
  50. μ μ
  51. ( Ω , Σ , 𝐏 ) (Ω,Σ,\mathbf{P})
  52. X : Ω M X:Ω→M
  53. p p
  54. M d ( x , x 0 ) p d ( X * ( 𝐏 ) ) ( x ) Ω d ( X ( ω ) , x 0 ) p d 𝐏 ( ω ) , \int_{M}d(x,x_{0})^{p}\,\mathrm{d}\left(X_{*}(\mathbf{P})\right)(x)\equiv\int_% {\Omega}d(X(\omega),x_{0})^{p}\,\mathrm{d}\mathbf{P}(\omega),
  55. p p
  56. p p
  57. x < s u b > 0 M x<sub>0∈M

Moment_magnitude_scale.html

  1. M L M_{\mathrm{L}}
  2. M w M_{\mathrm{w}}
  3. w \mathrm{w}
  4. M w M_{\mathrm{w}}
  5. M w = 2 3 log 10 ( M 0 ) - 6.07 , M_{\mathrm{w}}={\frac{2}{3}}\log_{10}(M_{0})-6.07,
  6. M 0 M_{0}
  7. M 0 M_{0}
  8. f Δ E f_{\Delta E}
  9. m 1 m_{1}
  10. m 2 m_{2}
  11. f Δ E = 10 ( 3 2 ( m 1 + 6 ) ) 10 ( 3 2 ( m 2 + 6 ) ) = 10 3 2 ( m 1 - m 2 ) . f_{\Delta E}=\frac{10^{(\frac{3}{2}(m_{1}+6))}}{10^{(\frac{3}{2}(m_{2}+6))}}=1% 0^{\frac{3}{2}(m_{1}-m_{2})}.
  12. E s E_{s}
  13. M 0 M_{0}
  14. M 0 M_{0}
  15. E s E_{\mathrm{s}}
  16. E s = M 0 10 - 4.8 = M 0 1.6 × 10 - 5 , E_{\mathrm{s}}=M_{0}\cdot 10^{-4.8}=M_{0}\cdot 1.6\times 10^{-5},
  17. M e = 2 3 log 10 E s - 2.9 M_{\mathrm{e}}=\textstyle{\frac{2}{3}}\log_{10}E_{\mathrm{s}}-2.9
  18. E s E_{\mathrm{s}}
  19. M n = 2 3 log 10 m TNT Mt + 6 , M_{n}=\textstyle\frac{2}{3}\displaystyle\log_{10}\frac{m_{\mathrm{TNT}}}{\mbox% {Mt}~{}}+6,
  20. m TNT m_{\mathrm{TNT}}
  21. E s E_{s}
  22. M w M_{\mathrm{w}}
  23. M 0 M_{0}
  24. M w M_{\mathrm{w}}
  25. M w M_{\mathrm{w}}
  26. M 0 × 10 25 M_{0}\times 10^{25}
  27. M L M_{\mathrm{L}}
  28. M w M_{\mathrm{w}}

Monad_(category_theory).html

  1. F F
  2. G G
  3. F F
  4. G G
  5. G F G\circ F
  6. F F
  7. G G
  8. F G F\circ G
  9. G G
  10. F F
  11. T = G F T=G\circ F
  12. X X
  13. Free ( X ) \mathrm{Free}(X)
  14. X T ( X ) X\rightarrow T(X)
  15. X X
  16. Free ( X ) \mathrm{Free}(X)
  17. T ( T ( X ) ) T ( X ) T(T(X))\rightarrow T(X)
  18. I T I\rightarrow T
  19. T T T T\circ T\rightarrow T
  20. ( P , ) (P,\leq)
  21. x x
  22. y y
  23. x y x\leq y
  24. C C
  25. C C
  26. T : C C T\colon C\to C
  27. η : 1 C T \eta\colon 1_{C}\to T
  28. 1 C 1_{C}
  29. C C
  30. μ : T 2 T \mu\colon T^{2}\to T
  31. T 2 T^{2}
  32. T T T\circ T
  33. C C
  34. C C
  35. μ T μ = μ μ T \mu\circ T\mu=\mu\circ\mu T
  36. T 3 T T^{3}\to T
  37. μ T η = μ η T = 1 T \mu\circ T\eta=\mu\circ\eta T=1_{T}
  38. T T T\to T
  39. 1 T 1_{T}
  40. T T
  41. T T
  42. T μ T\mu
  43. μ T \mu T
  44. C C
  45. 𝐄𝐧𝐝 C \mathbf{End}_{C}
  46. C C
  47. C C
  48. C op C^{\mathrm{op}}
  49. U U
  50. C C
  51. 𝐒𝐞𝐭 \mathbf{Set}
  52. A A
  53. T ( A ) T(A)
  54. A A
  55. f : A B f\colon A\to B
  56. T ( f ) T(f)
  57. f f
  58. A A
  59. η A : A T ( A ) \eta_{A}\colon A\to T(A)
  60. a A a\in A
  61. { a } \{a\}
  62. μ A : T ( T ( A ) ) T ( A ) \mu_{A}\colon T(T(A))\to T(A)
  63. ( T , η , μ ) (T,\eta,\mu)
  64. C C
  65. T T
  66. ( x , h ) (x,h)
  67. x x
  68. C C
  69. h : T x x h\colon Tx\to x
  70. C C
  71. f : ( x , h ) ( x , h ) f\colon(x,h)\to(x^{\prime},h^{\prime})
  72. T T
  73. f : x x f\colon x\to x^{\prime}
  74. C C
  75. C T C^{T}
  76. T T
  77. T T
  78. C T C^{T}
  79. C C
  80. C C
  81. C T C^{T}
  82. x x
  83. ( T x , μ x ) (Tx,\mu_{x})
  84. T T
  85. C T C_{T}
  86. T T
  87. T T
  88. C T C^{T}
  89. ( T x , μ x ) (Tx,\mu_{x})
  90. x x
  91. C C
  92. ( F , G , η , ε ) (F,G,\eta,\varepsilon)
  93. C C
  94. D D
  95. F : C D F\colon C\to D
  96. G : D C G\colon D\to C
  97. η \eta
  98. ε \varepsilon
  99. ( G F , η , G ε F ) (GF,\eta,G\varepsilon F)
  100. ( T , η , μ ) (T,\eta,\mu)
  101. 𝐀𝐝𝐣 ( C , T ) \mathbf{Adj}(C,T)
  102. ( F , G , e , ε ) (F,G,e,\varepsilon)
  103. ( G F , e , G ε F ) = ( T , η , μ ) (GF,e,G\varepsilon F)=(T,\eta,\mu)
  104. C C
  105. ( F T , G T , η , μ T ) : C C T (F_{T},G_{T},\eta,\mu_{T}):C\to C_{T}
  106. C T C_{T}
  107. ( F T , G T , η , μ T ) : C C T (F^{T},G^{T},\eta,\mu^{T}):C\to C^{T}
  108. C T C^{T}
  109. ( F , G , η , ε ) (F,G,\eta,\varepsilon)
  110. C C
  111. D D
  112. D D
  113. C T C^{T}
  114. T = G F T=GF
  115. G : D C G\colon D\to C
  116. F F
  117. C C
  118. C = 𝐂𝐚𝐭 C=\mathbf{Cat}

Monad_(functional_programming).html

  1. liftM2 : M : monad , ( A 1 A 2 R ) M A 1 M A 2 M R = \,\text{liftM2}\colon\forall M\colon\,\text{monad},\;(A_{1}\to A_{2}\to R)\to M% \,A_{1}\to M\,A_{2}\to M\,R=
  2. o p m 1 m 2 bind m 1 ( a 1 bind m 2 ( a 2 return ( o p a 1 a 2 ) ) ) op\mapsto m_{1}\mapsto m_{2}\mapsto\,\text{bind}\;m_{1}\;(a_{1}\mapsto\,\text{% bind}\;m_{2}\;(a_{2}\mapsto\,\text{return}\;(op\,a_{1}\,a_{2})))
  3. S T × S S\rightarrow T\times S
  4. return : T S T × S = t s ( t , s ) \,\text{return}\colon T\rightarrow S\rightarrow T\times S=t\mapsto s\mapsto(t,s)
  5. bind : ( S T × S ) ( T S T × S ) S T × S \,\text{bind}\colon(S\rightarrow T\times S)\rightarrow(T\rightarrow S% \rightarrow T^{\prime}\times S)\rightarrow S\rightarrow T^{\prime}\times S
  6. = m k s ( k t s ) where ( t , s ) = m s \ =m\mapsto k\mapsto s\mapsto(k\ t\ s^{\prime})\quad\,\text{where}\;(t,s^{% \prime})=m\,s
  7. return : T E T = t e t \,\text{return}\colon T\rightarrow E\rightarrow T=t\mapsto e\mapsto t
  8. bind : ( E T ) ( T E T ) E T = r f e f ( r e ) e \,\text{bind}\colon(E\rightarrow T)\rightarrow(T\rightarrow E\rightarrow T^{% \prime})\rightarrow E\rightarrow T^{\prime}=r\mapsto f\mapsto e\mapsto f\,(r\,% e)\,e
  9. ask : E E = id E \,\text{ask}\colon E\rightarrow E=\,\text{id}_{E}
  10. local : ( E E ) ( E T ) E T = f c e c ( f e ) \,\text{local}\colon(E\rightarrow E)\rightarrow(E\rightarrow T)\rightarrow E% \rightarrow T=f\mapsto c\mapsto e\mapsto c\,(f\,e)
  11. return : T W × T = t ( ϵ , t ) \,\text{return}\colon T\rightarrow W\times T=t\mapsto(\epsilon,t)
  12. bind : ( W × T ) ( T W × T ) W × T = ( w , t ) f ( w * w , t ) where ( w , t ) = f t \,\text{bind}\colon(W\times T)\rightarrow(T\rightarrow W\times T^{\prime})% \rightarrow W\times T^{\prime}=(w,t)\mapsto f\mapsto(w*w^{\prime},\,t^{\prime}% )\quad\,\text{where}\;(w^{\prime},t^{\prime})=f\,t
  13. R R
  14. T T
  15. ( T R ) R \left(T\rightarrow R\right)\rightarrow R
  16. return : T ( T R ) R = t f f t \,\text{return}\colon T\rightarrow\left(T\rightarrow R\right)\rightarrow R=t% \mapsto f\mapsto f\,t
  17. bind : ( ( T R ) R ) ( T ( T R ) R ) ( T R ) R \,\text{bind}\colon\left(\left(T\rightarrow R\right)\rightarrow R\right)% \rightarrow\left(T\rightarrow\left(T^{\prime}\rightarrow R\right)\rightarrow R% \right)\rightarrow\left(T^{\prime}\rightarrow R\right)\rightarrow R
  18. = c f k c ( t f t k ) =c\mapsto f\mapsto k\mapsto c\,\left(t\mapsto f\,t\,k\right)
  19. call/cc : ( ( T ( T R ) R ) ( T R ) R ) ( T R ) R \,\text{call/cc}\colon\left(\left(T\rightarrow\left(T^{\prime}\rightarrow R% \right)\rightarrow R\right)\rightarrow\left(T\rightarrow R\right)\rightarrow R% \right)\rightarrow\left(T\rightarrow R\right)\rightarrow R
  20. = f k ( f ( t x k t ) k ) =f\mapsto k\mapsto\left(f\left(t\mapsto x\mapsto k\,t\right)\,k\right)
  21. extend extract = id \,\text{extend}\,\,\,\text{extract}=\,\text{id}
  22. extract ( extend f ) = f \,\text{extract}\circ(\,\text{extend}\,f)=f
  23. ( extend f ) ( extend g ) = extend ( f ( extend g ) ) (\,\text{extend}\,f)\circ(\,\text{extend}\,g)=\,\text{extend}\,(f\circ(\,\text% {extend}\,g))
  24. extract duplicate = id \,\text{extract}\circ\,\text{duplicate}=\,\text{id}
  25. fmap extract duplicate = id \,\text{fmap}\,\,\text{extract}\circ\,\text{duplicate}=\,\text{id}
  26. duplicate duplicate = fmap duplicate duplicate \,\text{duplicate}\circ\,\text{duplicate}=\,\text{fmap}\,\,\text{duplicate}% \circ\,\text{duplicate}
  27. fmap : ( A B ) W A W B = f extend ( f extract ) \,\text{fmap}:(A\rightarrow B)\rightarrow\mathrm{W}\,A\rightarrow\mathrm{W}\,B% =f\mapsto\,\text{extend}\,(f\circ\,\text{extract})
  28. duplicate : W A W W A = extend id \,\text{duplicate}:\mathrm{W}\,A\rightarrow\mathrm{W}\,\mathrm{W}\,A=\,\text{% extend}\,\,\text{id}
  29. extend : ( W A B ) W A W B = f ( fmap f ) duplicate \,\text{extend}:(\mathrm{W}\,A\rightarrow B)\rightarrow\mathrm{W}\,A% \rightarrow\mathrm{W}\,B=f\mapsto(\,\text{fmap}\,f)\circ\,\text{duplicate}
  30. T T
  31. C × T C\times T
  32. C C
  33. extract : ( C × T ) T = ( c , t ) t \,\text{extract}:(C\times T)\rightarrow T=(c,t)\mapsto t
  34. extend : ( ( C × A ) B ) C × A C × B = f ( c , a ) ( c , f ( c , a ) ) \,\text{extend}:((C\times A)\rightarrow B)\rightarrow C\times A\rightarrow C% \times B=f\mapsto(c,a)\mapsto(c,f\,(c,a))
  35. fmap : ( A B ) ( C × A ) ( C × B ) = f ( c , a ) ( c , f a ) \,\text{fmap}:(A\rightarrow B)\rightarrow(C\times A)\rightarrow(C\times B)=f% \mapsto(c,a)\mapsto(c,f\,a)
  36. duplicate : ( C × A ) ( C × ( C × A ) ) = ( c , a ) ( c , ( c , a ) ) \,\text{duplicate}:(C\times A)\rightarrow(C\times(C\times A))=(c,a)\mapsto(c,(% c,a))
  37. T T
  38. M T M\rightarrow T
  39. M M
  40. extract : ( M T ) T = f f ε \,\text{extract}:(M\rightarrow T)\rightarrow T=f\mapsto f\,\varepsilon
  41. extend : ( ( M A ) B ) ( M A ) M B = f g m f ( m g ( m * m ) ) \,\text{extend}:((M\rightarrow A)\rightarrow B)\rightarrow(M\rightarrow A)% \rightarrow M\rightarrow B=f\mapsto g\mapsto m\mapsto f\,(m^{\prime}\mapsto g% \,(m*m^{\prime}))
  42. fmap : ( A B ) ( M A ) M B = f g ( f g ) \,\text{fmap}:(A\rightarrow B)\rightarrow(M\rightarrow A)\rightarrow M% \rightarrow B=f\mapsto g\mapsto(f\circ g)
  43. duplicate : ( M A ) M ( M A ) = f m m f ( m * m ) \,\text{duplicate}:(M\rightarrow A)\rightarrow M\rightarrow(M\rightarrow A)=f% \mapsto m\mapsto m^{\prime}\mapsto f\,(m*m^{\prime})
  44. M M
  45. T T
  46. ( S T ) × S (S\rightarrow T)\times S
  47. extract : ( ( S T ) × S ) T = ( f , s ) f s \,\text{extract}:((S\rightarrow T)\times S)\rightarrow T=(f,s)\mapsto f\,s
  48. extend : ( ( ( S A ) × S ) B ) ( ( S A ) × S ) ( S B ) × S \,\text{extend}:(((S\rightarrow A)\times S)\rightarrow B)\rightarrow((S% \rightarrow A)\times S)\rightarrow(S\rightarrow B)\times S\,
  49. = f ( g , s ) ( ( s f ( g , s ) ) , s ) =f\mapsto(g,s)\mapsto((s^{\prime}\mapsto f\,(g,s^{\prime})),s)
  50. fmap : ( A B ) ( ( S A ) × S ) ( S B ) × S = f ( f , s ) ( f f , s ) \,\text{fmap}:(A\rightarrow B)\rightarrow((S\rightarrow A)\times S)\rightarrow% (S\rightarrow B)\times S=f\mapsto(f^{\prime},s)\mapsto(f\circ f^{\prime},s)
  51. duplicate : ( ( S A ) × S ) ( S ( ( S A ) × S ) ) × S = ( f , s ) ( ( s ( f , s ) ) , s ) \,\text{duplicate}:((S\rightarrow A)\times S)\rightarrow(S\rightarrow((S% \rightarrow A)\times S))\times S=(f,s)\mapsto((s^{\prime}\mapsto(f,s^{\prime})% ),s)