wpmath0000001_13

Large_Magellanic_Cloud.html

  1. ( m - M ) 0 = (m-M)_{0}=

LaTeX.html

  1. E E

Latin_square.html

  1. k = 1 n ( k ! ) n / k L ( n ) ( n ! ) 2 n n n 2 . \prod_{k=1}^{n}\left(k!\right)^{n/k}\geq L(n)\geq\frac{\left(n!\right)^{2n}}{n% ^{n^{2}}}.
  2. [ 1 ] [ 1 2 2 1 ] [ 1 2 3 2 3 1 3 1 2 ] \begin{bmatrix}1\end{bmatrix}\quad\begin{bmatrix}1&2\\ 2&1\end{bmatrix}\quad\begin{bmatrix}1&2&3\\ 2&3&1\\ 3&1&2\end{bmatrix}
  3. [ 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 ] [ 1 2 3 4 2 4 1 3 3 1 4 2 4 3 2 1 ] \begin{bmatrix}1&2&3&4\\ 2&1&4&3\\ 3&4&1&2\\ 4&3&2&1\end{bmatrix}\quad\begin{bmatrix}1&2&3&4\\ 2&4&1&3\\ 3&1&4&2\\ 4&3&2&1\end{bmatrix}
  4. [ 1 2 3 4 5 2 3 5 1 4 3 5 4 2 1 4 1 2 5 3 5 4 1 3 2 ] [ 1 2 3 4 5 2 4 1 5 3 3 5 4 2 1 4 1 5 3 2 5 3 2 1 4 ] \begin{bmatrix}1&2&3&4&5\\ 2&3&5&1&4\\ 3&5&4&2&1\\ 4&1&2&5&3\\ 5&4&1&3&2\end{bmatrix}\quad\begin{bmatrix}1&2&3&4&5\\ 2&4&1&5&3\\ 3&5&4&2&1\\ 4&1&5&3&2\\ 5&3&2&1&4\end{bmatrix}
  5. 2 \mathbb{Z}_{2}
  6. 3 \mathbb{Z}_{3}
  7. 2 × 2 \mathbb{Z}_{2}\times\mathbb{Z}_{2}
  8. 4 \mathbb{Z}_{4}
  9. 5 \mathbb{Z}_{5}
  10. A B C D [ 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 ] E F G H [ 1 3 4 2 2 4 3 1 3 1 2 4 4 2 1 3 ] I J K L [ 1 4 2 3 2 3 1 4 3 2 4 1 4 1 3 2 ] \begin{matrix}A\\ B\\ C\\ D\\ \end{matrix}\begin{bmatrix}1&2&3&4\\ 2&1&4&3\\ 3&4&1&2\\ 4&3&2&1\\ \end{bmatrix}\quad\begin{matrix}E\\ F\\ G\\ H\\ \end{matrix}\begin{bmatrix}1&3&4&2\\ 2&4&3&1\\ 3&1&2&4\\ 4&2&1&3\\ \end{bmatrix}\quad\begin{matrix}I\\ J\\ K\\ L\\ \end{matrix}\begin{bmatrix}1&4&2&3\\ 2&3&1&4\\ 3&2&4&1\\ 4&1&3&2\\ \end{bmatrix}
  11. 12 12 123 124 \begin{matrix}12&12&123&124\\ \end{matrix}
  12. 1 2 1234 4 \begin{matrix}1&2&1234&4\\ \end{matrix}

Latitude.html

  1. i i
  2. m ( ϕ ) = π 180 R ϕ degrees = R ϕ radians . m(\phi)=\frac{\pi}{180}R\phi_{\rm degrees}=R\phi_{\rm radians}.
  3. f = a - b a , e 2 = 2 f - f 2 , b = a ( 1 - f ) = a 1 - e 2 . \begin{aligned}\displaystyle f&\displaystyle=\frac{a-b}{a},\qquad e^{2}=2f-f^{% 2},\qquad b=a(1-f)=a\sqrt{1-e^{2}}.\end{aligned}
  4. m ( ϕ ) = 0 ϕ M ( ϕ ) d ϕ = a ( 1 - e 2 ) 0 ϕ ( 1 - e 2 sin 2 ϕ ) - 3 / 2 d ϕ m(\phi)=\int_{0}^{\phi}M(\phi)d\phi=a(1-e^{2})\int_{0}^{\phi}\left(1-e^{2}\sin% ^{2}\phi\right)^{-3/2}d\phi
  5. M ( ϕ ) M(\phi)
  6. m p = m ( π / 2 ) m_{p}=m(\pi/2)\,
  7. δ m ( ϕ ) \displaystyle\delta m(\phi)
  8. ϕ \phi
  9. Δ LAT 1 \Delta^{1}_{\rm LAT}
  10. Δ LONG 1 \Delta^{1}_{\rm LONG}
  11. π \pi
  12. Δ LAT 1 = π a ( 1 - e 2 ) 180 ( 1 - e 2 sin 2 ϕ ) 3 / 2 \Delta^{1}_{\rm LAT}=\frac{\pi a(1-e^{2})}{180(1-e^{2}\sin^{2}\phi)^{3/2}}\,
  13. ϕ - 0.5 \phi-0.5
  14. ϕ + 0.5 \phi+0.5
  15. Δ LAT 1 = 111132.954 - 559.822 cos 2 ϕ + 1.175 cos 4 ϕ \Delta^{1}_{\rm LAT}=111132.954-559.822\cos 2\phi+1.175\cos 4\phi
  16. Δ LONG 1 = π a cos ϕ 180 ( 1 - e 2 sin 2 ϕ ) 1 / 2 \Delta^{1}_{\rm LONG}=\frac{\pi a\cos\phi}{180(1-e^{2}\sin^{2}\phi)^{1/2}}\,
  17. ψ ( ϕ ) = tan - 1 [ ( 1 - e 2 ) tan ϕ ] . \psi(\phi)=\tan^{-1}\left[(1-e^{2})\tan\phi\right]\;\!.
  18. ϕ \scriptstyle\phi
  19. u ( ϕ ) u(\phi)
  20. β ( ϕ ) = tan - 1 [ 1 - e 2 tan ϕ ] \beta(\phi)=\tan^{-1}\left[\sqrt{1-e^{2}}\tan\phi\right]\,\!
  21. p 2 a 2 + z 2 b 2 = 1. \frac{p^{2}}{a^{2}}+\frac{z^{2}}{b^{2}}=1.
  22. p = a cos β , z = b sin β ; p=a\cos\beta,\qquad z=b\sin\beta;
  23. μ ( ϕ ) = π 2 m ( ϕ ) m p \mu(\phi)={\displaystyle\frac{\pi}{2}\frac{m(\phi)}{m_{p}}}\,\!
  24. m ( ϕ ) = a ( 1 - e 2 ) 0 ϕ ( 1 - e 2 sin 2 ϕ ) - 3 / 2 d ϕ , m(\phi)=a(1-e^{2})\int_{0}^{\phi}\left(1-e^{2}\sin^{2}\phi\right)^{-3/2}d\phi,
  25. m p = m ( π / 2 ) . m_{p}=m(\pi/2).\,
  26. R = 2 m p π R=\frac{2m_{p}}{\pi}
  27. ξ ( ϕ ) = sin - 1 ( q ( ϕ ) q p ) \xi(\phi)=\sin^{-1}\left(\frac{q(\phi)}{q_{p}}\right)\,\!
  28. q ( ϕ ) \displaystyle q(\phi)
  29. q p = q ( π / 2 ) = 1 - 1 - e 2 2 e ln ( 1 - e 1 + e ) = 1 + 1 - e 2 e tanh - 1 e , q_{p}=q(\pi/2)=1-\frac{1-e^{2}}{2e}\ln\left(\frac{1-e}{1+e}\right)=1+\frac{1-e% ^{2}}{e}\tanh^{-1}e,\,
  30. R q = a q p / 2 . R_{q}=a\sqrt{q_{p}/2}.\,
  31. χ ( ϕ ) \displaystyle\chi(\phi)
  32. ψ ( ϕ ) \displaystyle\psi(\phi)
  33. y ( ϕ ) = E 2 π ψ ( ϕ ) . y(\phi)=\frac{E}{2\pi}\psi(\phi).
  34. ψ ( ϕ ) \displaystyle\psi(\phi)
  35. ϕ \phi\,\!
  36. ϕ \phi\,\!
  37. ϕ - β \phi-\beta\,\!
  38. ϕ - ξ \phi-\xi\,\!
  39. ϕ - μ \phi-\mu\,\!
  40. ϕ - χ \phi-\chi\,\!
  41. ϕ - ψ \phi-\psi\,\!

Laurent_polynomial.html

  1. 𝔽 \mathbb{F}
  2. 𝔽 \mathbb{F}
  3. 𝔽 \mathbb{F}
  4. 𝔽 \mathbb{F}
  5. p = k p k X k , p k 𝔽 p=\sum_{k}p_{k}X^{k},\quad p_{k}\in\mathbb{F}
  6. ( i a i X i ) + ( i b i X i ) = i ( a i + b i ) X i \left(\sum_{i}a_{i}X^{i}\right)+\left(\sum_{i}b_{i}X^{i}\right)=\sum_{i}(a_{i}% +b_{i})X^{i}
  7. ( i a i X i ) ( j b j X j ) = k ( i , j : i + j = k a i b j ) X k . \left(\sum_{i}a_{i}X^{i}\right)\cdot\left(\sum_{j}b_{j}X^{j}\right)=\sum_{k}% \left(\sum_{i,j:i+j=k}a_{i}b_{j}\right)X^{k}.

Laurent_series.html

  1. f ( z ) = n = - a n ( z - c ) n f(z)=\sum_{n=-\infty}^{\infty}a_{n}(z-c)^{n}
  2. a n = 1 2 π i γ f ( z ) d z ( z - c ) n + 1 . a_{n}=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(z)\,\mathrm{d}z}{(z-c)^{n+1}}.\,
  3. γ \gamma
  4. f ( z ) f(z)
  5. f ( z ) f(z)
  6. γ \gamma
  7. γ \gamma
  8. | z - c | = ϱ |z-c|=\varrho
  9. r < ϱ < R r<\varrho<R
  10. f f
  11. γ \gamma
  12. γ \gamma
  13. a n a_{n}
  14. f ( z ) f(z)
  15. f ( z ) f(z)
  16. f ( z ) f(z)
  17. f ( x ) = e - 1 / x 2 f(x)=e^{-1/x^{2}}
  18. f ( 0 ) = 0 f(0)=0
  19. n = 0 N ( - 1 ) n x - 2 n n ! \sum_{n=0}^{N}(-1)^{n}\,{x^{-2n}\over n!}
  20. n = - a n ( z - c ) n \sum_{n=-\infty}^{\infty}a_{n}(z-c)^{n}
  21. r \displaystyle r
  22. f ( z ) = 1 + 2 i 5 k = 0 ( 1 ( 2 i ) k + 1 - 1 ) z k . f(z)=\frac{1+2i}{5}\sum_{k=0}^{\infty}\left(\frac{1}{(2i)^{k+1}}-1\right)z^{k}.
  23. 1 z \frac{1}{z}
  24. 2 i z \frac{2i}{z}
  25. f ( z ) = e z z + e 1 z . f(z)={e^{z}\over z}+e^{\frac{1}{z}}.
  26. f ( z ) = + ( 1 3 ! ) z - 3 + ( 1 2 ! ) z - 2 + 2 z - 1 + 2 + ( 1 2 ! ) z + ( 1 3 ! ) z 2 + ( 1 4 ! ) z 3 + f(z)=\cdots+\left({1\over 3!}\right)z^{-3}+\left({1\over 2!}\right)z^{-2}+2z^{% -1}+2+\left({1\over 2!}\right)z+\left({1\over 3!}\right)z^{2}+\left({1\over 4!% }\right)z^{3}+\cdots
  27. ( z - c ) - k - 1 \left(z-c\right)^{-k-1}
  28. γ n = - a n ( z - c ) n - k - 1 d z = γ n = - b n ( z - c ) n - k - 1 d z . \oint_{\gamma}\sum_{n=-\infty}^{\infty}a_{n}\left(z-c\right)^{n-k-1}\mathrm{d}% z=\oint_{\gamma}\sum_{n=-\infty}^{\infty}b_{n}\left(z-c\right)^{n-k-1}\mathrm{% d}z.
  29. r + ϵ | z - c | R - ϵ r+\epsilon\leq|z-c|\leq R-\epsilon
  30. γ ( z - c ) n - k - 1 d z = 2 π i δ n k \oint_{\gamma}(z-c)^{n-k-1}dz=2\pi i\delta_{nk}
  31. a k = b k a_{k}=b_{k}
  32. k = - - 1 a k ( z - c ) k . \sum_{k=-\infty}^{-1}a_{k}(z-c)^{k}.
  33. z k z^{k}
  34. z = e π i w z=e^{\pi iw}

Law_of_dilution.html

  1. α α
  2. K d = [ A + ] [ B - ] [ A B ] = α 2 1 - α c 0 K_{d}=\cfrac{[A^{+}][B^{-}]}{[AB]}=\frac{\alpha^{2}}{1-\alpha}\cdot c_{0}
  3. K c = Λ c 2 ( Λ 0 - Λ c ) Λ 0 c K_{c}=\cfrac{\Lambda_{c}^{2}}{(\Lambda_{0}-\Lambda_{c})\Lambda_{0}}\cdot c
  4. A B A + + B - AB\rightleftharpoons A^{+}+B^{-}
  5. α α
  6. ( 1 - α ) (1-α)
  7. K d = [ A + ] [ B - ] [ A B ] = ( α c 0 ) ( α c 0 ) ( 1 - α ) c 0 = α 2 1 - α c 0 K_{d}=\cfrac{[A^{+}][B^{-}]}{[AB]}=\cfrac{(\alpha c_{0})(\alpha c_{0})}{(1-% \alpha)c_{0}}=\cfrac{\alpha^{2}}{1-\alpha}\cdot c_{0}
  8. K d = α 2 1 - α c 0 α 2 c 0 K_{d}=\frac{\alpha^{2}}{1-\alpha}\cdot c_{0}\approx\alpha^{2}c_{0}
  9. α = K d c 0 \alpha=\sqrt{\cfrac{K_{d}}{c_{0}}}
  10. [ A + ] = [ B - ] = α c 0 = K d c 0 [A^{+}]=[B^{-}]=\alpha c_{0}=\sqrt{K_{d}c_{0}}
  11. Λ c \Lambda_{c}
  12. Λ 0 \Lambda_{0}

Law_of_excluded_middle.html

  1. * 𝟐 11. . p p \mathbf{*2\cdot 11}.\ \ \vdash.\ p\ \vee\thicksim p
  2. a a
  3. b b
  4. a b a^{b}
  5. 2 \sqrt{2}
  6. 2 2 \sqrt{2}^{\sqrt{2}}
  7. a = 2 a=\sqrt{2}
  8. b = 2 b=\sqrt{2}
  9. 2 2 \sqrt{2}^{\sqrt{2}}
  10. a = 2 2 a=\sqrt{2}^{\sqrt{2}}
  11. b = 2 b=\sqrt{2}
  12. a b = ( 2 2 ) 2 = 2 ( 2 2 ) = 2 2 = 2 a^{b}=\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=\sqrt{2}^{\left(\sqrt{2}% \cdot\sqrt{2}\right)}=\sqrt{2}^{2}=2
  13. \forall
  14. f = { 1 if Goldbach’s conjecture is true 0 else \qquad\displaystyle f=\begin{cases}1&\,\text{if Goldbach's conjecture is true}% \\ 0&\,\text{else}\end{cases}
  15. f ( n ) = { 1 0 n occurs in the decimal representation of π 0 else \qquad\displaystyle f(n)=\begin{cases}1&0^{n}\,\text{ occurs in the decimal % representation of }\pi\\ 0&\,\text{else}\end{cases}
  16. 0 n 0^{n}
  17. π \pi
  18. 0 N 0^{N}
  19. π \pi
  20. N N
  21. π \pi

Law_of_noncontradiction.html

  1. * 𝟑 24. . ( p . p ) \mathbf{*3\cdot 24}.\ \ \vdash.\thicksim(p.\thicksim p)

Least_common_multiple.html

  1. 2 21 + 1 6 = 4 42 + 7 42 = 11 42 {2\over 21}+{1\over 6}={4\over 42}+{7\over 42}={11\over 42}
  2. lcm ( a , b ) = | a b | gcd ( a , b ) . \operatorname{lcm}(a,b)=\frac{|a\cdot b|}{\operatorname{gcd}(a,b)}.
  3. lcm ( 21 , 6 ) = 21 6 gcd ( 21 , 6 ) = 21 6 gcd ( 3 , 6 ) = 21 6 3 = 126 3 = 42. \operatorname{lcm}(21,6)={21\cdot 6\over\operatorname{gcd}(21,6)}={21\cdot 6% \over\operatorname{gcd}(3,6)}={21\cdot 6\over 3}=\frac{126}{3}=42.
  4. lcm ( a , b ) = ( | a | gcd ( a , b ) ) | b | = ( | b | gcd ( a , b ) ) | a | . \operatorname{lcm}(a,b)=\left({|a|\over\operatorname{gcd}(a,b)}\right)\cdot|b|% =\left({|b|\over\operatorname{gcd}(a,b)}\right)\cdot|a|.
  5. lcm ( 21 , 6 ) = 21 gcd ( 21 , 6 ) 6 = 21 gcd ( 3 , 6 ) 6 = 21 3 6 = 7 6 = 42. \operatorname{lcm}(21,6)={21\over\operatorname{gcd}(21,6)}\cdot 6={21\over% \operatorname{gcd}(3,6)}\cdot 6={21\over 3}\cdot 6=7\cdot 6=42.
  6. 90 = 2 1 3 2 5 1 = 2 3 3 5. 90=2^{1}\cdot 3^{2}\cdot 5^{1}=2\cdot 3\cdot 3\cdot 5.\,\!
  7. 8 = 2 3 8\;\,\;\,=2^{3}\,\!
  8. 9 = 3 2 9\;\,\;\,=3^{2}\,\!
  9. 21 = 3 1 7 1 21\;\,=3^{1}\cdot 7^{1}\,\!
  10. lcm ( 8 , 9 , 21 ) = 2 3 3 2 7 1 = 8 9 7 = 504. \operatorname{lcm}(8,9,21)=2^{3}\cdot 3^{2}\cdot 7^{1}=8\cdot 9\cdot 7=504.\,\!
  11. n = 2 n 2 3 n 3 5 n 5 7 n 7 = p p n p , n=2^{n_{2}}3^{n_{3}}5^{n_{5}}7^{n_{7}}\cdots=\prod_{p}p^{n_{p}},\;
  12. a = p p a p \;a=\prod_{p}p^{a_{p}}\;\;
  13. b = p p b p \;b=\prod_{p}p^{b_{p}}\;\;
  14. gcd ( a , b ) = p p min ( a p , b p ) \gcd(a,b)=\prod_{p}p^{\min(a_{p},b_{p})}\;
  15. lcm ( a , b ) = p p max ( a p , b p ) . \operatorname{lcm}(a,b)=\prod_{p}p^{\max(a_{p},b_{p})}.\;
  16. min ( x , y ) + max ( x , y ) = x + y , \min(x,y)+\max(x,y)=x+y,\;\;
  17. gcd ( a , b ) lcm ( a , b ) = a b . \gcd(a,b)\,\operatorname{lcm}(a,b)=a\,b.\;
  18. 4 = 2 2 3 0 , 6 = 2 1 3 1 , gcd ( 4 , 6 ) = 2 1 3 0 , lcm ( 4 , 6 ) = 2 2 3 1 . 4=2^{2}3^{0},\;\;6=2^{1}3^{1},\;\;\gcd(4,6)=2^{1}3^{0},\;\;\operatorname{lcm}(% 4,6)=2^{2}3^{1}.\;
  19. 1 3 = 2 0 3 - 1 5 0 , 2 5 = 2 1 3 0 5 - 1 , gcd ( 1 3 , 2 5 ) = 2 0 3 - 1 5 - 1 , lcm ( 1 3 , 2 5 ) = 2 1 3 0 5 0 , \tfrac{1}{3}=2^{0}3^{-1}5^{0},\;\;\tfrac{2}{5}=2^{1}3^{0}5^{-1},\;\;\gcd(% \tfrac{1}{3},\tfrac{2}{5})=2^{0}3^{-1}5^{-1},\;\;\operatorname{lcm}(\tfrac{1}{% 3},\tfrac{2}{5})=2^{1}3^{0}5^{0},\;\;
  20. 1 6 = 2 - 1 3 - 1 , 3 4 = 2 - 2 3 1 , gcd ( 1 6 , 3 4 ) = 2 - 2 3 - 1 , lcm ( 1 6 , 3 4 ) = 2 - 1 3 1 . \tfrac{1}{6}=2^{-1}3^{-1},\;\;\tfrac{3}{4}=2^{-2}3^{1},\;\;\gcd(\tfrac{1}{6},% \tfrac{3}{4})=2^{-2}3^{-1},\;\;\operatorname{lcm}(\tfrac{1}{6},\tfrac{3}{4})=2% ^{-1}3^{1}.\;\;
  21. lcm ( a , b ) = lcm ( b , a ) , \operatorname{lcm}(a,b)=\operatorname{lcm}(b,a),\;
  22. gcd ( a , b ) = gcd ( b , a ) . \gcd(a,b)=\gcd(b,a).\;
  23. lcm ( a , lcm ( b , c ) ) = lcm ( lcm ( a , b ) , c ) , \operatorname{lcm}(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\operatorname% {lcm}(a,b),c),\;
  24. gcd ( a , gcd ( b , c ) ) = gcd ( gcd ( a , b ) , c ) . \gcd(a,\gcd(b,c))=\gcd(\gcd(a,b),c).\;
  25. lcm ( a , gcd ( a , b ) ) = a , \operatorname{lcm}(a,\gcd(a,b))=a,\;
  26. gcd ( a , lcm ( a , b ) ) = a . \gcd(a,\operatorname{lcm}(a,b))=a.\;
  27. lcm ( a , a ) = a , \operatorname{lcm}(a,a)=a,\;
  28. gcd ( a , a ) = a . \gcd(a,a)=a.\;
  29. a b a = lcm ( a , b ) , a\geq b\iff a=\operatorname{lcm}(a,b),\;
  30. a b a = gcd ( a , b ) . a\leq b\iff a=\gcd(a,b).\;
  31. lcm ( a , gcd ( b , c ) ) = gcd ( lcm ( a , b ) , lcm ( a , c ) ) , \operatorname{lcm}(a,\gcd(b,c))=\gcd(\operatorname{lcm}(a,b),\operatorname{lcm% }(a,c)),\;
  32. gcd ( a , lcm ( b , c ) ) = lcm ( gcd ( a , b ) , gcd ( a , c ) ) . \gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c)).\;
  33. gcd ( lcm ( a , b ) , lcm ( b , c ) , lcm ( a , c ) ) = lcm ( gcd ( a , b ) , gcd ( b , c ) , gcd ( a , c ) ) . \gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(b,c),\operatorname{lcm}(a,c))=% \operatorname{lcm}(\gcd(a,b),\gcd(b,c),\gcd(a,c)).\;
  34. | { ( x , y ) : lcm ( x , y ) = D } | = 3 ω ( D ) , |\{(x,y)\;:\;\operatorname{lcm}(x,y)=D\}|=3^{\omega(D)},\;

Lebesgue_measure.html

  1. E E\subseteq\mathbb{R}
  2. I = [ a , b ] I=[a,b]
  3. l ( I ) = b - a l(I)=b-a
  4. λ * ( E ) \lambda^{*}(E)
  5. λ * ( E ) = inf { k = 1 l ( I k ) : ( I k ) k is a sequence of disjoint open intervals with E k = 1 I k } \lambda^{*}(E)=\operatorname{inf}\left\{\sum_{k=1}^{\infty}l(I_{k}):{(I_{k})_{% k\in\mathbb{N}}}\,\text{ is a sequence of disjoint open intervals with }E% \subseteq\bigcup_{k=1}^{\infty}I_{k}\right\}
  6. λ ( E ) = λ * ( E ) \lambda(E)=\lambda^{*}(E)
  7. A A\subseteq\mathbb{R}
  8. λ * ( A ) = λ * ( A E ) + λ * ( A E c ) \lambda^{*}(A)=\lambda^{*}(A\cap E)+\lambda^{*}(A\cap E^{c})
  9. E E
  10. I I
  11. E E
  12. E E
  13. E E
  14. E E
  15. E E
  16. E E
  17. A A
  18. E E
  19. A A
  20. A A
  21. E E
  22. A A
  23. E E
  24. A A
  25. E E
  26. A A
  27. A A
  28. A A
  29. E E
  30. A A
  31. E E
  32. E E
  33. E E
  34. λ ( A ) = | I 1 | | I 2 | | I n | . \lambda(A)=|I_{1}|\cdot|I_{2}|\cdots|I_{n}|.
  35. { , { 1 , 2 , 3 , 4 } , { 1 , 2 } , { 3 , 4 } , { 1 , 3 } , { 2 , 4 } } \{\emptyset,\{1,2,3,4\},\{1,2\},\{3,4\},\{1,3\},\{2,4\}\}
  36. δ > 0 \delta>0
  37. A A
  38. δ \delta
  39. δ A = { δ x : x A } \delta A=\{\delta x:x\in A\}
  40. δ n λ ( A ) . \delta^{n}\lambda\,(A).
  41. | det ( T ) | λ ( A ) |\det(T)|\,\lambda\,(A)
  42. λ ( [ 0 , 1 ] × [ 0 , 1 ] × × [ 0 , 1 ] ) = 1. \lambda([0,1]\times[0,1]\times\cdots\times[0,1])=1.
  43. \cup
  44. B = i = 1 n [ a i , b i ] , B=\prod_{i=1}^{n}[a_{i},b_{i}]\,,
  45. vol ( B ) = i = 1 n ( b i - a i ) . \operatorname{vol}(B)=\prod_{i=1}^{n}(b_{i}-a_{i})\,.
  46. λ * ( A ) = inf { B 𝒞 vol ( B ) : 𝒞 is a countable collection of boxes whose union covers A } . \lambda^{*}(A)=\inf\Bigl\{\sum_{B\in\mathcal{C}}\operatorname{vol}(B):\mathcal% {C}\,\text{ is a countable collection of boxes whose union covers }A\Bigr\}.
  47. λ * ( S ) = λ * ( S A ) + λ * ( S A ) . \lambda^{*}(S)=\lambda^{*}(S\cap A)+\lambda^{*}(S\setminus A)\,.

Legendre_symbol.html

  1. ( a p ) = { 1 if a is a quadratic residue modulo p and a 0 ( mod p ) , - 1 if a is a quadratic non-residue modulo p , 0 if a 0 ( mod p ) . \left(\frac{a}{p}\right)=\begin{cases}1&\,\text{ if }a\,\text{ is a quadratic % residue modulo }p\,\text{ and }a\not\equiv 0\;\;(\mathop{{\rm mod}}p),\\ -1&\,\text{ if }a\,\text{ is a quadratic non-residue modulo }p,\\ 0&\,\text{ if }a\equiv 0\;\;(\mathop{{\rm mod}}p).\end{cases}
  2. ( a p ) a p - 1 2 ( mod p ) and ( a p ) { - 1 , 0 , 1 } . \left(\frac{a}{p}\right)\equiv a^{\frac{p-1}{2}}\;\;(\mathop{{\rm mod}}p)\quad% \,\text{ and }\quad\left(\frac{a}{p}\right)\in\{-1,0,1\}.
  3. a R p a\mathrm{R}p
  4. a N p a\mathrm{N}p
  5. ( a p ) = ( b p ) . \left(\frac{a}{p}\right)=\left(\frac{b}{p}\right).
  6. ( a b p ) = ( a p ) ( b p ) . \left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right).
  7. ( x 2 p ) = { 1 if p x 0 if p x . \left(\frac{x^{2}}{p}\right)=\begin{cases}1&\mbox{if }~{}p\nmid x\\ 0&\mbox{if }~{}p\mid x.\end{cases}
  8. ( a p ) \left(\frac{a}{p}\right)
  9. ( - 1 p ) = ( - 1 ) p - 1 2 = { 1 if p 1 ( mod 4 ) - 1 if p 3 ( mod 4 ) . \left(\frac{-1}{p}\right)=(-1)^{\frac{p-1}{2}}=\begin{cases}1&\mbox{ if }~{}p% \equiv 1\;\;(\mathop{{\rm mod}}4)\\ -1&\mbox{ if }~{}p\equiv 3\;\;(\mathop{{\rm mod}}4).\end{cases}
  10. ( 2 p ) = ( - 1 ) p 2 - 1 8 = { 1 if p 1 or 7 ( mod 8 ) - 1 if p 3 or 5 ( mod 8 ) . \left(\frac{2}{p}\right)=(-1)^{\tfrac{p^{2}-1}{8}}=\begin{cases}1&\mbox{ if }~% {}p\equiv 1\mbox{ or }~{}7\;\;(\mathop{{\rm mod}}8)\\ -1&\mbox{ if }~{}p\equiv 3\mbox{ or }~{}5\;\;(\mathop{{\rm mod}}8).\end{cases}
  11. ( a p ) \left(\frac{a}{p}\right)
  12. ( 3 p ) = ( - 1 ) p + 1 6 = { 1 if p 1 or 11 ( mod 12 ) - 1 if p 5 or 7 ( mod 12 ) . \left(\frac{3}{p}\right)=(-1)^{\big\lfloor\frac{p+1}{6}\big\rfloor}=\begin{% cases}1&\mbox{ if }~{}p\equiv 1\mbox{ or }~{}11\;\;(\mathop{{\rm mod}}12)\\ -1&\mbox{ if }~{}p\equiv 5\mbox{ or }~{}7\;\;(\mathop{{\rm mod}}12).\end{cases}
  13. ( 5 p ) = ( - 1 ) p + 2 5 = { 1 if p 1 or 4 ( mod 5 ) - 1 if p 2 or 3 ( mod 5 ) . \left(\frac{5}{p}\right)=(-1)^{\big\lfloor\frac{p+2}{5}\big\rfloor}=\begin{% cases}1&\mbox{ if }~{}p\equiv 1\mbox{ or }~{}4\;\;(\mathop{{\rm mod}}5)\\ -1&\mbox{ if }~{}p\equiv 2\mbox{ or }~{}3\;\;(\mathop{{\rm mod}}5).\end{cases}
  14. F p - ( p 5 ) 0 ( mod p ) , F p ( p 5 ) ( mod p ) . F_{p-\left(\frac{p}{5}\right)}\equiv 0\;\;(\mathop{{\rm mod}}p),\qquad F_{p}% \equiv\left(\frac{p}{5}\right)\;\;(\mathop{{\rm mod}}p).
  15. ( 2 5 ) = - 1 , F 3 = 2 , F 2 = 1 , ( 3 5 ) = - 1 , F 4 = 3 , F 3 = 2 , ( 5 5 ) = 0 , F 5 = 5 , ( 7 5 ) = - 1 , F 8 = 21 , F 7 = 13 , ( 11 5 ) = 1 , F 10 = 55 , F 11 = 89. \begin{aligned}\displaystyle(\tfrac{2}{5})&\displaystyle=-1,&&\displaystyle F_% {3}=2,F_{2}=1,\\ \displaystyle(\tfrac{3}{5})&\displaystyle=-1,&&\displaystyle F_{4}=3,F_{3}=2,% \\ \displaystyle(\tfrac{5}{5})&\displaystyle=0,&&\displaystyle F_{5}=5,\\ \displaystyle(\tfrac{7}{5})&\displaystyle=-1,&&\displaystyle F_{8}=21,F_{7}=13% ,\\ \displaystyle(\tfrac{11}{5})&\displaystyle=1,&&\displaystyle F_{10}=55,F_{11}=% 89.\end{aligned}
  16. ( q p ) ( p q ) = ( - 1 ) p - 1 2 q - 1 2 . \left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{\tfrac{p-1}{2}\tfrac{q-% 1}{2}}.
  17. ( a p ) a p - 1 2 ( mod p ) . \left(\frac{a}{p}\right)\equiv a^{\tfrac{p-1}{2}}\;\;(\mathop{{\rm mod}}p).
  18. k = 0 p - 1 ζ a k 2 = ( a p ) k = 0 p - 1 ζ k 2 , ζ = e 2 π i p \sum_{k=0}^{p-1}\zeta^{ak^{2}}=\left(\frac{a}{p}\right)\sum_{k=0}^{p-1}\zeta^{% k^{2}},\qquad\zeta=e^{\frac{2\pi i}{p}}
  19. ( p q ) = sgn ( i = 1 q - 1 2 k = 1 p - 1 2 ( k p - i q ) ) . \left(\frac{p}{q}\right)=\operatorname{sgn}\left(\prod_{i=1}^{\frac{q-1}{2}}% \prod_{k=1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right)\right).
  20. ( p q ) \left(\frac{p}{q}\right)
  21. ( q p ) . \left(\frac{q}{p}\right).
  22. ( q p ) = n = 1 p - 1 2 sin ( 2 π q n p ) sin ( 2 π n p ) . \left(\frac{q}{p}\right)=\prod_{n=1}^{\frac{p-1}{2}}\frac{\sin\left(\frac{2\pi qn% }{p}\right)}{\sin\left(\frac{2\pi n}{p}\right)}.
  23. ( a n ) \left(\frac{a}{n}\right)
  24. ( a p ) n \left(\tfrac{a}{p}\right)_{n}
  25. ( 12345 331 ) \displaystyle\left(\frac{12345}{331}\right)
  26. ( 12345 331 ) = ( 98 331 ) = ( 2 7 2 331 ) = ( 2 331 ) = ( - 1 ) 331 2 - 1 8 = - 1. \left(\frac{12345}{331}\right)=\left(\frac{98}{331}\right)=\left(\frac{2\cdot 7% ^{2}}{331}\right)=\left(\frac{2}{331}\right)=(-1)^{\tfrac{331^{2}-1}{8}}=-1.

Lens_(optics).html

  1. 1 f = ( n - 1 ) [ 1 R 1 - 1 R 2 + ( n - 1 ) d n R 1 R 2 ] , \frac{1}{f}=(n-1)\left[\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(n-1)d}{nR_{1}R_{% 2}}\right],
  2. f f
  3. n n
  4. R 1 R_{1}
  5. R 2 R_{2}
  6. d d
  7. 1 f ( n - 1 ) [ 1 R 1 - 1 R 2 ] . \frac{1}{f}\approx\left(n-1\right)\left[\frac{1}{R_{1}}-\frac{1}{R_{2}}\right].
  8. 1 S 1 + 1 S 2 = 1 f \frac{1}{S_{1}}+\frac{1}{S_{2}}=\frac{1}{f}
  9. x 1 x 2 = f 2 , x_{1}x_{2}=f^{2},\!
  10. x 1 = S 1 - f x_{1}=S_{1}-f
  11. x 2 = S 2 - f x_{2}=S_{2}-f
  12. BFL = f 2 ( d - f 1 ) d - ( f 1 + f 2 ) . \mbox{BFL}~{}=\frac{f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}.
  13. M = - f 2 f 1 , M=-\frac{f_{2}}{f_{1}},

Leon_M._Lederman.html

  1. Υ \Upsilon\,

Leonhard_Euler.html

  1. e e
  2. i i
  3. e x = n = 0 x n n ! = lim n ( 1 0 ! + x 1 ! + x 2 2 ! + + x n n ! ) . e^{x}=\sum_{n=0}^{\infty}{x^{n}\over n!}=\lim_{n\to\infty}\left(\frac{1}{0!}+% \frac{x}{1!}+\frac{x^{2}}{2!}+\cdots+\frac{x^{n}}{n!}\right).
  4. e e
  5. n = 1 1 n 2 = lim n ( 1 1 2 + 1 2 2 + 1 3 2 + + 1 n 2 ) = π 2 6 . \sum_{n=1}^{\infty}{1\over n^{2}}=\lim_{n\to\infty}\left(\frac{1}{1^{2}}+\frac% {1}{2^{2}}+\frac{1}{3^{2}}+\cdots+\frac{1}{n^{2}}\right)=\frac{\pi^{2}}{6}.
  6. φ φ
  7. e i φ = cos φ + i sin φ . e^{i\varphi}=\cos\varphi+i\sin\varphi.\,
  8. e i π + 1 = 0 e^{i\pi}+1=0\,
  9. e e
  10. i i
  11. π \pi
  12. V V
  13. E E
  14. F F
  15. e e
  16. π \pi
  17. γ = lim n ( 1 + 1 2 + 1 3 + 1 4 + + 1 n - ln ( n ) ) . \gamma=\lim_{n\rightarrow\infty}\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+% \cdots+\frac{1}{n}-\ln(n)\right).
  18. ρ t + ( ρ u ) = 0 \displaystyle{\partial\rho\over\partial t}+\nabla\cdot(\rho u)=0
  19. \otimes
  20. F = π 2 E I ( K L ) 2 F=\frac{\pi^{2}EI}{(KL)^{2}}
  21. F F
  22. E E
  23. I I
  24. L L
  25. K K
  26. K K
  27. K K
  28. K K
  29. K K
  30. K L KL
  31. a + b n n = x \frac{a+b^{n}}{n}=x

Lepton.html

  1. 1 / 2 {1}/{2}
  2. μ = g Q 4 m , \mu=g\frac{Q\hbar}{4m},
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. Γ ( μ - e - + ν e ¯ + ν μ ) = K 1 G F 2 m μ 5 , \Gamma\left(\mu^{-}\rightarrow e^{-}+\bar{\nu_{e}}+\nu_{\mu}\right)=K_{1}G_{F}% ^{2}m_{\mu}^{5},
  7. Γ ( τ - e - + ν e ¯ + ν τ ) = K 2 G F 2 m τ 5 , \Gamma\left(\tau^{-}\rightarrow e^{-}+\bar{\nu_{e}}+\nu_{\tau}\right)=K_{2}G_{% F}^{2}m_{\tau}^{5},
  8. Γ ( τ - e - + ν e ¯ + ν τ ) = Γ ( τ - μ - + ν μ ¯ + ν τ ) . \Gamma\left(\tau^{-}\rightarrow e^{-}+\bar{\nu_{e}}+\nu_{\tau}\right)=\Gamma% \left(\tau^{-}\rightarrow\mu^{-}+\bar{\nu_{\mu}}+\nu_{\tau}\right).
  9. τ l = B ( l - e - + ν e ¯ + ν l ) Γ ( l - e - + ν e ¯ + ν l ) , \tau_{l}=\frac{B\left(l^{-}\rightarrow e^{-}+\bar{\nu_{e}}+\nu_{l}\right)}{% \Gamma\left(l^{-}\rightarrow e^{-}+\bar{\nu_{e}}+\nu_{l}\right)},
  10. τ τ τ μ = B ( τ - e - + ν e ¯ + ν τ ) B ( μ - e - + ν e ¯ + ν μ ) ( m μ m τ ) 5 . \frac{\tau_{\tau}}{\tau_{\mu}}=\frac{B\left(\tau^{-}\rightarrow e^{-}+\bar{\nu% _{e}}+\nu_{\tau}\right)}{B\left(\mu^{-}\rightarrow e^{-}+\bar{\nu_{e}}+\nu_{% \mu}\right)}\left(\frac{m_{\mu}}{m_{\tau}}\right)^{5}.
  11. 1 / 2 {1}/{2}
  12. 1 / 2 {1}/{2}
  13. 1 / 2 {1}/{2}
  14. 1 / 2 {1}/{2}
  15. 1 / 2 {1}/{2}
  16. 1 / 2 {1}/{2}

Lever.html

  1. T 1 = M 1 a = M 2 b = T 2 , T_{1}=M_{1}a=M_{2}b=T_{2},\!
  2. M A = M 2 M 1 = a b . MA=\frac{M_{2}}{M_{1}}=\frac{a}{b}.\!
  3. M A = F B F A = a b . MA=\frac{F_{B}}{F_{A}}=\frac{a}{b}.
  4. a = | 𝐫 A - 𝐫 P | , b = | 𝐫 B - 𝐫 P | , a=|\mathbf{r}_{A}-\mathbf{r}_{P}|,\quad b=|\mathbf{r}_{B}-\mathbf{r}_{P}|,
  5. 𝐫 A - 𝐫 P = a 𝐞 A , 𝐫 B - 𝐫 P = b 𝐞 B . \mathbf{r}_{A}-\mathbf{r}_{P}=a\mathbf{e}_{A},\quad\mathbf{r}_{B}-\mathbf{r}_{% P}=b\mathbf{e}_{B}.
  6. 𝐯 A = θ ˙ a 𝐞 A , 𝐯 B = θ ˙ b 𝐞 B , \mathbf{v}_{A}=\dot{\theta}a\mathbf{e}_{A}^{\perp},\quad\mathbf{v}_{B}=\dot{% \theta}b\mathbf{e}_{B}^{\perp},
  7. F θ = 𝐅 A 𝐯 A θ ˙ - 𝐅 B 𝐯 B θ ˙ = a ( 𝐅 A 𝐞 A ) - b ( 𝐅 B 𝐞 B ) = a F A - b F B , F_{\theta}=\mathbf{F}_{A}\cdot\frac{\partial\mathbf{v}_{A}}{\partial\dot{% \theta}}-\mathbf{F}_{B}\cdot\frac{\partial\mathbf{v}_{B}}{\partial\dot{\theta}% }=a(\mathbf{F}_{A}\cdot\mathbf{e}_{A}^{\perp})-b(\mathbf{F}_{B}\cdot\mathbf{e}% _{B}^{\perp})=aF_{A}-bF_{B},
  8. F θ = a F A - b F B = 0. F_{\theta}=aF_{A}-bF_{B}=0.\,\!
  9. M A = F B F A = a b , MA=\frac{F_{B}}{F_{A}}=\frac{a}{b},

Lie_algebra.html

  1. [ x , y ] [x,y]
  2. 𝔤 \,\mathfrak{g}
  3. [ , ] : 𝔤 × 𝔤 𝔤 [\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}
  4. [ a x + b y , z ] = a [ x , z ] + b [ y , z ] , [ z , a x + b y ] = a [ z , x ] + b [ z , y ] [ax+by,z]=a[x,z]+b[y,z],\quad[z,ax+by]=a[z,x]+b[z,y]
  5. 𝔤 \mathfrak{g}
  6. 𝔤 \,\mathfrak{g}
  7. [ x , x ] = 0 [x,x]=0
  8. 𝔤 \mathfrak{g}
  9. [ x , [ y , z ] ] + [ z , [ x , y ] ] + [ y , [ z , x ] ] = 0 [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0
  10. 𝔤 \mathfrak{g}
  11. x , y x , y = y , x y , x x,yx,y=−y,xy,x
  12. 𝔤 \mathfrak{g}
  13. 𝔤 \mathfrak{g}
  14. 𝔰 𝔲 ( n ) \mathfrak{su}(n)
  15. 𝔤 \mathfrak{g}
  16. 𝔤 \mathfrak{g}
  17. 𝔤 \mathfrak{g}
  18. [ [ x , y ] , z ] [[x,y],z]
  19. [ x , [ y , z ] ] [x,[y,z]]
  20. 𝔥 𝔤 \mathfrak{h}\subseteq\mathfrak{g}
  21. I 𝔤 I\subseteq\mathfrak{g}
  22. [ 𝔤 , I ] I , [\mathfrak{g},I]\subseteq I,
  23. 𝔤 \mathfrak{g}
  24. f : 𝔤 𝔤 , f ( [ x , y ] ) = [ f ( x ) , f ( y ) ] , f:\mathfrak{g}\to\mathfrak{g^{\prime}},\quad f([x,y])=[f(x),f(y)],
  25. 𝔤 \mathfrak{g}
  26. 𝔤 \mathfrak{g}
  27. 𝔤 / I \mathfrak{g}/I
  28. 𝔤 \mathfrak{g}
  29. [ x , s ] = 0 [x,s]=0
  30. 𝔤 \mathfrak{g}
  31. 𝔤 \mathfrak{g}
  32. [ x , s ] [x,s]
  33. 𝔤 \mathfrak{g}
  34. 𝔤 \mathfrak{g^{\prime}}
  35. 𝔤 𝔤 \mathfrak{g}\oplus\mathfrak{g^{\prime}}
  36. ( x , x ) , x 𝔤 , x 𝔤 \mathfrak{}(x,x^{\prime}),\,x\in\mathfrak{g},x^{\prime}\in\mathfrak{g^{\prime}}
  37. [ ( x , x ) , ( y , y ) ] = ( [ x , y ] , [ x , y ] ) , x , y 𝔤 , x , y 𝔤 . [(x,x^{\prime}),(y,y^{\prime})]=([x,y],[x^{\prime},y^{\prime}]),\quad x,y\in% \mathfrak{g},\,x^{\prime},y^{\prime}\in\mathfrak{g^{\prime}}.
  38. 𝔤 \mathfrak{g}
  39. 𝔦 \mathfrak{i}
  40. 𝔤 𝔤 / 𝔦 \mathfrak{g}\to\mathfrak{g}/\mathfrak{i}
  41. 𝔤 \mathfrak{g}
  42. 𝔦 \mathfrak{i}
  43. 𝔤 / 𝔦 \mathfrak{g}/\mathfrak{i}
  44. * *
  45. [ a , b ] = a * b - b * a . [a,b]=a*b-b*a.
  46. 𝔤 𝔩 n ( F ) . \mathfrak{gl}_{n}(F).
  47. 𝔤 𝔩 ( V ) \mathfrak{gl}(V)
  48. 𝔤 \mathfrak{g}
  49. π : 𝔤 𝔤 𝔩 ( V ) . \pi:\mathfrak{g}\to\mathfrak{gl}(V).
  50. ad : 𝔤 𝔤 𝔩 ( 𝔤 ) \operatorname{ad}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})
  51. ad ( x ) ( y ) = [ x , y ] \operatorname{ad}(x)(y)=[x,y]
  52. 𝔤 \mathfrak{g}
  53. 𝔤 \mathfrak{g}
  54. 𝔤 \mathfrak{g}
  55. δ : 𝔤 𝔤 \delta:\mathfrak{g}\rightarrow\mathfrak{g}
  56. δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ] \delta([x,y])=[\delta(x),y]+[x,\delta(y)]
  57. ad ( x ) \operatorname{ad}(x)
  58. ad \operatorname{ad}
  59. 𝔤 𝔩 ( 𝔤 ) \mathfrak{gl}(\mathfrak{g})
  60. 𝔤 \mathfrak{g}
  61. ad \operatorname{ad}
  62. 𝔤 \mathfrak{g}
  63. 𝔤 \mathfrak{g}
  64. 𝔲 ( n ) \mathfrak{u}(n)
  65. 𝔤 𝔩 n ( F ) \mathfrak{gl}_{n}(F)
  66. 𝔰 𝔩 n ( F ) . \mathfrak{sl}_{n}(F).
  67. G G
  68. 𝔤 \mathfrak{g}
  69. G G
  70. 𝔤 \mathfrak{g}
  71. X X
  72. e x p ( t X ) G exp(tX)∈G
  73. t t
  74. 𝔤 \mathfrak{g}
  75. x x
  76. y y
  77. z z
  78. [ x , y ] = z , [ x , z ] = 0 , [ y , z ] = 0 [x,y]=z,\quad[x,z]=0,\quad[y,z]=0
  79. x = ( 0 1 0 0 0 0 0 0 0 ) , y = ( 0 0 0 0 0 1 0 0 0 ) , z = ( 0 0 1 0 0 0 0 0 0 ) . x=\left(\begin{array}[]{ccc}0&1&0\\ 0&0&0\\ 0&0&0\end{array}\right),\quad y=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&0&0\end{array}\right),\quad z=\left(\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 0&0&0\end{array}\right)~{}.\quad
  80. ( 1 a c 0 1 b 0 0 1 ) = e b y e c z e a x . \left(\begin{array}[]{ccc}1&a&c\\ 0&1&b\\ 0&0&1\end{array}\right)=e^{by}e^{cz}e^{ax}~{}.
  81. 𝔰 𝔲 ( 2 ) \mathfrak{su}(2)
  82. 𝔰 𝔬 ( 3 ) , \mathfrak{so}(3),
  83. [ L x , L y ] = i L z [L_{x},L_{y}]=i\hbar L_{z}
  84. [ L y , L z ] = i L x [L_{y},L_{z}]=i\hbar L_{x}
  85. [ L z , L x ] = i L y [L_{z},L_{x}]=i\hbar L_{y}
  86. i i
  87. L [ X , Y ] f = L X ( L Y f ) - L Y ( L X f ) . L_{[X,Y]}f=L_{X}(L_{Y}f)-L_{Y}(L_{X}f).\,
  88. 𝔤 \mathfrak{g}
  89. 𝔤 \mathfrak{g}
  90. K n K^{n}
  91. T n , T^{n},
  92. 𝔨 n , \mathfrak{k}^{n},
  93. 𝔤 \mathfrak{g}
  94. 𝔤 > [ 𝔤 , 𝔤 ] > [ [ 𝔤 , 𝔤 ] , 𝔤 ] > [ [ [ 𝔤 , 𝔤 ] , 𝔤 ] , 𝔤 ] > \mathfrak{g}>[\mathfrak{g},\mathfrak{g}]>[[\mathfrak{g},\mathfrak{g}],% \mathfrak{g}]>[[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}]>\cdots
  95. 𝔤 \mathfrak{g}
  96. ad ( u ) : 𝔤 𝔤 , ad ( u ) v = [ u , v ] \operatorname{ad}(u):\mathfrak{g}\to\mathfrak{g},\quad\operatorname{ad}(u)v=[u% ,v]
  97. 𝔤 \mathfrak{g}
  98. 𝔤 > [ 𝔤 , 𝔤 ] > [ [ 𝔤 , 𝔤 ] , [ 𝔤 , 𝔤 ] ] > [ [ [ 𝔤 , 𝔤 ] , [ 𝔤 , 𝔤 ] ] , [ [ 𝔤 , 𝔤 ] , [ 𝔤 , 𝔤 ] ] ] > \mathfrak{g}>[\mathfrak{g},\mathfrak{g}]>[[\mathfrak{g},\mathfrak{g}],[% \mathfrak{g},\mathfrak{g}]]>[[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},% \mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]]>\cdots
  99. 𝔤 \mathfrak{g}
  100. 𝔤 \mathfrak{g}
  101. 𝔤 \mathfrak{g}
  102. K ( u , v ) = tr ( ad ( u ) ad ( v ) ) , K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),
  103. 𝔤 \mathfrak{g}
  104. 𝔤 \mathfrak{g}
  105. K ( 𝔤 , [ 𝔤 , 𝔤 ] ) = 0. K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0.
  106. 𝔤 \mathfrak{g}
  107. X X
  108. e x p ( t X ) G exp(tX)∈G
  109. t t
  110. e x p exp
  111. 𝔤 𝔩 n ( ) \mathfrak{gl}_{n}(\mathbb{C})
  112. GL n ( ) \mathrm{GL}_{n}(\mathbb{C})
  113. n × n n×n
  114. 𝔰 𝔩 n ( ) \mathfrak{sl}_{n}(\mathbb{C})
  115. SL n ( ) \mathrm{SL}_{n}(\mathbb{C})
  116. n × n n×n
  117. 𝔬 ( n ) \mathfrak{o}(n)
  118. O ( n ) \mathrm{O}(n)
  119. 𝔰 𝔬 ( n ) \mathfrak{so}(n)
  120. SO ( n ) \mathrm{SO}(n)
  121. n × n n×n
  122. 𝔲 ( n ) \mathfrak{u}(n)
  123. U ( n ) \mathrm{U}(n)
  124. n × n n×n
  125. 𝔰 𝔲 ( n ) \mathfrak{su}(n)
  126. SU ( n ) \mathrm{SU}(n)
  127. n × n n×n
  128. [ X , Y ] [X,Y]
  129. X X
  130. Y Y
  131. [ X , Y ] = X Y - Y X [X,Y]=XY-YX
  132. X , Y X,Y
  133. 𝔤 \mathfrak{g}
  134. 𝔤 \mathfrak{g}
  135. Hom ( Γ ( 𝔤 ) , H ) Hom ( 𝔤 , L ( H ) ) . \mathrm{Hom}(\Gamma(\mathfrak{g}),H)\cong\mathrm{Hom}(\mathfrak{g},\mathrm{L}(% H)).
  136. 𝔤 L ( Γ ( 𝔤 ) ) \mathfrak{g}\rightarrow\mathrm{L}(\Gamma(\mathfrak{g}))
  137. Γ ( 𝔤 ) \Gamma(\mathfrak{g})
  138. Γ ( L ( H ) ) H \Gamma(\mathrm{L}(H))\rightarrow H
  139. H H
  140. H H
  141. G G
  142. G H G→H
  143. 𝐋 ( G ) 𝐋 ( H ) \mathbf{L}(G)→\mathbf{L}(H)
  144. [ , ] ( id + τ A , A ) = 0 [\cdot,\cdot]\circ(\mathrm{id}+\tau_{A,A})=0
  145. [ , ] ( [ , ] id ) ( id + σ + σ 2 ) = 0 [\cdot,\cdot]\circ([\cdot,\cdot]\otimes\mathrm{id})\circ(\mathrm{id}+\sigma+% \sigma^{2})=0

Lie_group.html

  1. μ : G × G G μ ( x , y ) = x y \mu:G\times G\to G\quad\mu(x,y)=xy
  2. ( x , y ) x - 1 y (x,y)\mapsto x^{-1}y
  3. GL ( 2 , 𝐑 ) = { A = ( a b c d ) : det A = a d - b c 0 } . \operatorname{GL}(2,\mathbf{R})=\left\{A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}:\det A=ad-bc\neq 0\right\}.
  4. φ \varphi
  5. SO ( 2 , 𝐑 ) = { ( cos φ - sin φ sin φ cos φ ) : φ 𝐑 / 2 π 𝐙 } . \operatorname{SO}(2,\mathbf{R})=\left\{\begin{pmatrix}\cos\varphi&-\sin\varphi% \\ \sin\varphi&\cos\varphi\end{pmatrix}:\varphi\in\mathbf{R}/2\pi\mathbf{Z}\right\}.
  6. 𝔤 . \mathfrak{g}.
  7. 𝔤 \mathfrak{g}
  8. 𝔤 \mathfrak{g}
  9. ϕ : G H \phi\colon G\to H
  10. ϕ * \phi_{*}
  11. ϕ * \phi_{*}
  12. ϕ * : 𝔤 𝔥 \phi_{*}\colon\mathfrak{g}\to\mathfrak{h}
  13. ϕ * \phi_{*}
  14. 𝔤 \mathfrak{g}
  15. 𝔤 \mathfrak{g}
  16. exp ( A ) = 1 + A + A 2 2 ! + A 3 3 ! + \exp(A)=1+A+\frac{A^{2}}{2!}+\frac{A^{3}}{3!}+\cdots
  17. 𝔤 \mathfrak{g}
  18. 𝔤 \mathfrak{g}
  19. c ( s + t ) = c ( s ) c ( t ) c(s+t)=c(s)c(t)
  20. exp ( v ) = c ( 1 ) . \exp(v)=c(1).
  21. 𝔤 \mathfrak{g}
  22. 𝔤 \mathfrak{g}
  23. 𝔤 \mathfrak{g}
  24. exp ( u ) exp ( v ) = exp ( u + v + 1 2 [ u , v ] + 1 12 [ [ u , v ] , v ] - 1 12 [ [ u , v ] , u ] - ) , \exp(u)\,\exp(v)=\exp\left(u+v+\tfrac{1}{2}[u,v]+\tfrac{1}{12}[\,[u,v],v]-% \tfrac{1}{12}[\,[u,v],u]-\cdots\right),
  25. ϕ : G H \phi:G\to H
  26. ϕ * : 𝔤 𝔥 \phi_{*}:\mathfrak{g}\to\mathfrak{h}
  27. x 𝔤 x\in\mathfrak{g}
  28. ϕ ( exp ( x ) ) = exp ( ϕ * ( x ) ) . \phi(\exp(x))=\exp(\phi_{*}(x)).\,
  29. S U ( ) SU(\infty)

Life_expectancy.html

  1. e x e_{x}
  2. x x
  3. x x
  4. x + n x+n
  5. p x n \,{}_{n}p_{x}\!
  6. x x
  7. x x
  8. x + 1 x+1
  9. q x q_{x}\!
  10. x x
  11. e x \,e_{x}\!
  12. x x
  13. K ( x ) K(x)
  14. x x
  15. e x = E [ K ( x ) ] = k = 0 k P r ( K ( x ) = k ) = k = 0 k k p x q x + k . e_{x}=E[K(x)]=\sum_{k=0}^{\infty}k\,Pr(K(x)=k)=\sum_{k=0}^{\infty}k\,\,_{k}p_{% x}\,\,q_{x+k}.
  16. p x k q x + k = p x k - p x k + 1 {}_{k}p_{x}\,q_{x+k}={}_{k}p_{x}-{}_{k+1}p_{x}
  17. e x = k = 1 p x k . e_{x}=\sum_{k=1}^{\infty}\,{}_{k}p_{x}.
  18. x x
  19. e x + 1 / 2 e_{x}+1/2

Lift_(force).html

  1. d p d R = ρ v 2 R \frac{\operatorname{d}p}{\operatorname{d}R}=\rho\frac{v^{2}}{R}
  2. L = 1 2 ρ v 2 A C L L=\tfrac{1}{2}\rho v^{2}AC_{L}
  3. C L C_{L}
  4. L = p 𝐧 𝐤 d A , L=\oint p\mathbf{n}\cdot\mathbf{k}\;\mathrm{d}A,
  5. D p = p 𝐧 𝐢 d A , Y = p 𝐧 𝐣 d A . \begin{aligned}\displaystyle D_{p}&\displaystyle=\oint p\mathbf{n}\cdot\mathbf% {i}\;\mathrm{d}A,\\ \displaystyle Y&\displaystyle=\oint p\mathbf{n}\cdot\mathbf{j}\;\mathrm{d}A.% \end{aligned}
  6. Γ \Gamma
  7. L L^{\prime}
  8. L = ρ v Γ L^{\prime}=\rho v\Gamma\,
  9. ρ \rho
  10. v v
  11. - L -L^{\prime}
  12. - L -L^{\prime}

Light.html

  1. n 1 sin θ 1 = n 2 sin θ 2 . n_{1}\sin\theta_{1}=n_{2}\sin\theta_{2}\ .
  2. θ 1 \theta_{1}
  3. θ 2 \theta_{2}

Lighthouse.html

  1. d = 1.17 H d=1.17\sqrt{H}

Likelihood-ratio_test.html

  1. D \displaystyle D
  2. f ( x | θ ) f(x|\theta)
  3. θ \theta
  4. H 0 \displaystyle H_{0}
  5. Λ \Lambda
  6. Λ ( x ) = L ( θ 0 | x ) L ( θ 1 | x ) = f ( i x i | θ 0 ) f ( i x i | θ 1 ) \Lambda(x)=\frac{L(\theta_{0}|x)}{L(\theta_{1}|x)}=\frac{f(\cup_{i}\,x_{i}|% \theta_{0})}{f(\cup_{i}\,x_{i}|\theta_{1})}
  7. Λ ( x ) = L ( θ 0 x ) sup { L ( θ x ) : θ { θ 0 , θ 1 } } , \Lambda(x)=\frac{L(\theta_{0}\mid x)}{\sup\{\,L(\theta\mid x):\theta\in\{% \theta_{0},\theta_{1}\}\}},
  8. L ( θ | x ) L(\theta|x)
  9. sup \sup
  10. Λ > c \Lambda>c
  11. H 0 H_{0}
  12. Λ < c \Lambda<c
  13. H 0 H_{0}
  14. q q
  15. Λ = c . \Lambda=c.
  16. c , q c,\;q
  17. α \alpha
  18. q P ( Λ = c | H 0 ) + P ( Λ < c | H 0 ) = α q\cdot P(\Lambda=c\;|\;H_{0})+P(\Lambda<c\;|\;H_{0})=\alpha
  19. α \alpha
  20. θ \theta
  21. Θ 0 \Theta_{0}
  22. Θ \Theta
  23. H 0 : θ Θ 0 H 1 : θ Θ 0 \begin{aligned}\displaystyle H_{0}&\displaystyle:&\displaystyle\theta\in\Theta% _{0}\\ \displaystyle H_{1}&\displaystyle:&\displaystyle\theta\in\Theta_{0}^{% \complement}\end{aligned}
  24. L ( θ | x ) = f ( x | θ ) L(\theta|x)=f(x|\theta)
  25. f ( x | θ ) f(x|\theta)
  26. θ \theta
  27. x x
  28. Λ ( x ) = sup { L ( θ x ) : θ Θ 0 } sup { L ( θ x ) : θ Θ } . \Lambda(x)=\frac{\sup\{\,L(\theta\mid x):\theta\in\Theta_{0}\,\}}{\sup\{\,L(% \theta\mid x):\theta\in\Theta\,\}}.
  29. sup \sup
  30. { x | Λ c } \{x|\Lambda\leq c\}
  31. c c
  32. 0 c 1 0\leq c\leq 1
  33. x x
  34. n n
  35. \infty
  36. - 2 log ( Λ ) -2\log(\Lambda)
  37. χ 2 \chi^{2}
  38. Θ \Theta
  39. Θ 0 \Theta_{0}
  40. Λ \Lambda
  41. - 2 log ( Λ ) -2\log(\Lambda)
  42. χ 2 \chi^{2}
  43. X X
  44. k 1 H k_{1H}
  45. k 1 T k_{1T}
  46. k 2 H k_{2H}
  47. k 2 T k_{2T}
  48. Θ \Theta
  49. p 1 H p_{1H}
  50. p 1 T p_{1T}
  51. p 2 H p_{2H}
  52. p 2 T p_{2T}
  53. i = 1 , 2 i=1,2
  54. j = H , T j=H,T
  55. H H
  56. 0 p i j 1 0\leq p_{ij}\leq 1
  57. p i H + p i T = 1 p_{iH}+p_{iT}=1
  58. H 0 H_{0}
  59. p 1 j = p 2 j p_{1j}=p_{2j}
  60. n i j n_{ij}
  61. p i j p_{ij}
  62. H H
  63. n i j = k i j k i H + k i T . n_{ij}=\frac{k_{ij}}{k_{iH}+k_{iT}}.
  64. p i j p_{ij}
  65. H 0 H_{0}
  66. m i j = k 1 j + k 2 j k 1 H + k 2 H + k 1 T + k 2 T , m_{ij}=\frac{k_{1j}+k_{2j}}{k_{1H}+k_{2H}+k_{1T}+k_{2T}},
  67. i i
  68. H H
  69. H 0 H_{0}
  70. χ 2 ( 1 ) \chi^{2}(1)
  71. χ 2 \chi^{2}
  72. - 2 log Λ = 2 i , j k i j log n i j m i j . -2\log\Lambda=2\sum_{i,j}k_{ij}\log\frac{n_{ij}}{m_{ij}}.

Likelihood_function.html

  1. ( θ | x ) = P ( x | θ ) \mathcal{L}(\theta|x)=P(x|\theta)
  2. ( θ | x ) = p θ ( x ) = P θ ( X = x ) , \mathcal{L}(\theta|x)=p_{\theta}(x)=P_{\theta}(X=x),\,
  3. P ( X = x | θ ) P(X=x|\theta)
  4. P ( X = x ; θ ) P(X=x;\theta)
  5. ( θ | x ) = f θ ( x ) , \mathcal{L}(\theta|x)=f_{\theta}(x),\,
  6. f ( x | θ ) f(x|\theta)
  7. ( α , β | x ) = β α Γ ( α ) x α - 1 e - β x \mathcal{L}(\alpha,\beta\,|\,x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha% -1}e^{-\beta x}
  8. log ( α , β | x ) = α log β - log Γ ( α ) + ( α - 1 ) log x - β x . \log\mathcal{L}(\alpha,\beta\,|\,x)=\alpha\log\beta-\log\Gamma(\alpha)+(\alpha% -1)\log x-\beta x.\,
  9. log ( α , β | x ) β = α β - x \frac{\partial\log\mathcal{L}(\alpha,\beta\,|\,x)}{\partial\beta}=\frac{\alpha% }{\beta}-x
  10. log ( α , β | x 1 , , x n ) β = log ( α , β | x 1 ) β + + log ( α , β | x n ) β = n α β - i = 1 n x i . \frac{\partial\log\mathcal{L}(\alpha,\beta\,|\,x_{1},\ldots,x_{n})}{\partial% \beta}=\frac{\partial\log\mathcal{L}(\alpha,\beta\,|\,x_{1})}{\partial\beta}+% \cdots+\frac{\partial\log\mathcal{L}(\alpha,\beta\,|\,x_{n})}{\partial\beta}=% \frac{n\alpha}{\beta}-\sum_{i=1}^{n}x_{i}.
  11. β ^ = α x ¯ . \hat{\beta}=\frac{\alpha}{\bar{x}}.
  12. β ^ \hat{\beta}
  13. x ¯ = 1 n i = 1 n x i \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  14. x f ( x θ ) , x\mapsto f(x\mid\theta),\!
  15. θ f ( x θ ) , \theta\mapsto f(x\mid\theta),\!
  16. ( θ x ) = f ( x θ ) , \mathcal{L}(\theta\mid x)=f(x\mid\theta),\!
  17. approx ( θ x in interval j ) = f ( x * θ ) Δ j , \mathcal{L}\text{approx}(\theta\mid x\,\text{ in interval }j)=f(x_{*}\mid% \theta)\Delta_{j},\!
  18. approx ( θ x in interval j ) = f ( x * θ ) , \mathcal{L}\text{approx}(\theta\mid x\,\text{ in interval }j)=f(x_{*}\mid% \theta),\!
  19. ( θ x ) = f ( x θ ) . \mathcal{L}(\theta\mid x)=f(x\mid\theta).\!
  20. approx ( θ x in interval j containing discrete mass k ) = p k ( θ ) + f ( x * θ ) Δ j , \mathcal{L}\text{approx}(\theta\mid x\,\text{ in interval }j\,\text{ % containing discrete mass }k)=p_{k}(\theta)+f(x_{*}\mid\theta)\Delta_{j},\!
  21. x * x_{*}
  22. ( θ x ) = p k ( θ ) , \mathcal{L}(\theta\mid x)=p_{k}(\theta),\!
  23. p H p\text{H}
  24. p H 2 p\text{H}^{2}
  25. p H = 0.5 p\text{H}=0.5
  26. P ( HH | p H = 0.5 ) = 0.25. P(\,\text{HH}|p\text{H}=0.5)=0.25.
  27. p H = 0.5 p\text{H}=0.5
  28. ( p H = 0.5 | HH ) = P ( HH | p H = 0.5 ) = 0.25. \mathcal{L}(p\text{H}=0.5|\,\text{HH})=P(\,\text{HH}|p\text{H}=0.5)=0.25.
  29. p H = 0.5 p\text{H}=0.5
  30. p H = 1 p\text{H}=1
  31. p H = 1 p\text{H}=1
  32. p H > 0 p\text{H}>0
  33. p H p\text{H}
  34. p H p\text{H}
  35. P ( n | N ) = [ n N ] N P(n|N)=\frac{[n\leq N]}{N}
  36. N = 1 P ( n | N ) = N [ N n ] N = N = n 1 N \sum_{N=1}^{\infty}P(n|N)=\sum_{N}\frac{[N\geq n]}{N}=\sum_{N=n}^{\infty}\frac% {1}{N}
  37. P ( { n 1 , n 2 } | N ) = [ n 2 N ] ( N 2 ) . P(\{n_{1},n_{2}\}|N)=\frac{[n_{2}\leq N]}{{\left({{N}\atop{2}}\right)}}.
  38. N = 1 P ( { n 1 , n 2 } | N ) = N [ N n 2 ] ( N 2 ) = 2 n 2 - 1 \sum_{N=1}^{\infty}P(\{n_{1},n_{2}\}|N)=\sum_{N}\frac{[N\geq n_{2}]}{{\left({{% N}\atop{2}}\right)}}=\frac{2}{n_{2}-1}
  39. θ ^ \hat{\theta}
  40. θ ^ \hat{\theta}
  41. ( θ | x ) / ( θ ^ | x ) . \mathcal{L}(\theta|x)/\mathcal{L}(\hat{\theta}|x).
  42. { θ : ( θ | x ) / ( θ ^ | x ) 0.10 } , \{\theta:\mathcal{L}(\theta|x)/\mathcal{L}(\hat{\theta}|x)\geq 0.10\},
  43. { θ : ( θ | x ) / ( θ ^ | x ) p / 100 } . \{\theta:\mathcal{L}(\theta|x)/\mathcal{L}(\hat{\theta}|x)\geq p/100\}.
  44. θ ^ \hat{\theta}

Likelihood_principle.html

  1. L ( θ X = 3 ) = ( 12 3 ) θ 3 ( 1 - θ ) 9 = 220 θ 3 ( 1 - θ ) 9 L\left(\theta\mid X=3\right)=\begin{pmatrix}12\\ 3\end{pmatrix}\;\theta^{3}(1-\theta)^{9}=220\;\theta^{3}(1-\theta)^{9}
  2. L ( θ Y = 12 ) = ( 11 2 ) θ 3 ( 1 - θ ) 9 = 55 θ 3 ( 1 - θ ) 9 . L\left(\theta\mid Y=12\right)=\begin{pmatrix}11\\ 2\end{pmatrix}\;\theta^{3}(1-\theta)^{9}=55\;\theta^{3}(1-\theta)^{9}.
  3. Λ = L ( a | X = x ) L ( b | X = x ) = P ( X = x | a ) P ( X = x | b ) \Lambda={L(a|X=x)\over L(b|X=x)}={P(X=x|a)\over P(X=x|b)}
  4. θ \theta
  5. θ \theta
  6. T ( X ) T(X)
  7. θ \theta
  8. x 1 x_{1}
  9. x 2 x_{2}
  10. T ( x 1 ) = T ( x 2 ) T(x_{1})=T(x_{2})
  11. θ \theta
  12. p 3 ( 1 - p ) 9 p^{3}\;(1-p)^{9}
  13. ( ( 12 9 ) + ( 12 10 ) + ( 12 11 ) + ( 12 12 ) ) ( 1 2 ) 12 \left({12\choose 9}+{12\choose 10}+{12\choose 11}+{12\choose 12}\right)\left({% 1\over 2}\right)^{12}
  14. 1 - ( ( 10 2 ) ( 1 2 ) 11 + ( 9 2 ) ( 1 2 ) 10 + + ( 2 2 ) ( 1 2 ) 3 ) 1-\left({10\choose 2}\left({1\over 2}\right)^{11}+{9\choose 2}\left({1\over 2}% \right)^{10}+\cdots+{2\choose 2}\left({1\over 2}\right)^{3}\right)

Limit_(category_theory).html

  1. ϕ \phi
  2. ϕ \phi
  3. F x , A = { G F V ( x ) A G } F_{x,A}=\{G\in F\mid V(x)\cup A\subset G\}
  4. I x , A : F x , A F I_{x,A}:F_{x,A}\to F
  5. I x , A I_{x,A}
  6. \bullet\rightrightarrows\bullet
  7. \bullet\rightarrow\bullet\leftarrow\bullet
  8. s , t : i Ob ( J ) F ( i ) f Hom ( J ) F ( cod ( f ) ) s,t:\prod_{i\in\mathrm{Ob}(J)}F(i)\rightrightarrows\prod_{f\in\mathrm{Hom}(J)}% F(\mathrm{cod}(f))
  9. s = ( F ( f ) π F ( dom ( f ) ) ) f Hom ( J ) t = ( π F ( cod ( f ) ) ) f Hom ( J ) . \begin{aligned}\displaystyle s&\displaystyle=\bigl(F(f)\circ\pi_{F(\mathrm{dom% }(f))}\bigr)_{f\in\mathrm{Hom}(J)}\\ \displaystyle t&\displaystyle=\bigl(\pi_{F(\mathrm{cod}(f))}\bigr)_{f\in% \mathrm{Hom}(J)}.\end{aligned}
  10. Δ : 𝒞 𝒞 𝒥 \Delta:\mathcal{C}\to\mathcal{C}^{\mathcal{J}}
  11. lim : 𝒞 𝒥 𝒞 \mathrm{lim}:\mathcal{C}^{\mathcal{J}}\to\mathcal{C}
  12. Hom ( N , lim F ) Cone ( N , F ) \mathrm{Hom}(N,\mathrm{lim}F)\cong\mathrm{Cone}(N,F)
  13. colim : 𝒞 𝒥 𝒞 \mathrm{colim}:\mathcal{C}^{\mathcal{J}}\to\mathcal{C}
  14. Hom ( colim F , N ) Cocone ( F , N ) . \mathrm{Hom}(\mathrm{colim}F,N)\cong\mathrm{Cocone}(F,N).
  15. Hom ( N , lim F ) lim Hom ( N , F - ) \mathrm{Hom}(N,\mathrm{lim}F)\cong\mathrm{lim}\,\mathrm{Hom}(N,F-)
  16. lim Hom ( N , F - ) = Cone ( N , F ) . \mathrm{lim}\,\mathrm{Hom}(N,F-)=\mathrm{Cone}(N,F).
  17. Hom ( colim F , N ) lim Hom ( F - , N ) \mathrm{Hom}(\mathrm{colim}F,N)\cong\mathrm{lim}\,\mathrm{Hom}(F-,N)
  18. colim J lim I F ( i , j ) lim I colim J F ( i , j ) . \mathrm{colim}_{J}\,\mathrm{lim}_{I}F(i,j)\rightarrow\mathrm{lim}_{I}\,\mathrm% {colim}_{J}F(i,j).
  19. Hom \operatorname{Hom}
  20. Hom \operatorname{Hom}

Limit_superior_and_limit_inferior.html

  1. lim inf n x n := lim n ( inf m n x m ) \liminf_{n\to\infty}x_{n}:=\lim_{n\to\infty}\Big(\inf_{m\geq n}x_{m}\Big)
  2. lim inf n x n := sup n 0 inf m n x m = sup { inf { x m : m n } : n 0 } . \liminf_{n\to\infty}x_{n}:=\sup_{n\geq 0}\,\inf_{m\geq n}x_{m}=\sup\{\,\inf\{% \,x_{m}:m\geq n\,\}:n\geq 0\,\}.
  3. lim sup n x n := lim n ( sup m n x m ) \limsup_{n\to\infty}x_{n}:=\lim_{n\to\infty}\Big(\sup_{m\geq n}x_{m}\Big)
  4. lim sup n x n := inf n 0 sup m n x m = inf { sup { x m : m n } : n 0 } . \limsup_{n\to\infty}x_{n}:=\inf_{n\geq 0}\,\sup_{m\geq n}x_{m}=\inf\{\,\sup\{% \,x_{m}:m\geq n\,\}:n\geq 0\,\}.
  5. lim ¯ n x n := lim inf n x n \underline{\lim}_{n\to\infty}x_{n}:=\liminf_{n\to\infty}x_{n}
  6. lim ¯ n x n := lim sup n x n \overline{\lim}_{n\to\infty}x_{n}:=\limsup_{n\to\infty}x_{n}
  7. lim inf n x n lim sup n x n . \liminf_{n\to\infty}x_{n}\leq\limsup_{n\to\infty}x_{n}.
  8. ( x n ) (x_{n})
  9. x n x_{n}
  10. b b
  11. ε \varepsilon
  12. N N
  13. x n < b + ε x_{n}<b+\varepsilon
  14. n > N n>N
  15. b + ε b+\varepsilon
  16. x n x_{n}
  17. b b
  18. ε \varepsilon
  19. N N
  20. x n > b - ε x_{n}>b-\varepsilon
  21. n > N n>N
  22. b - ε b-\varepsilon
  23. lim sup n ( - x n ) = - lim inf n x n \limsup_{n\to\infty}(-x_{n})=-\liminf_{n\to\infty}x_{n}
  24. \mathbb{R}
  25. lim inf n x n = lim sup n x n \liminf_{n\to\infty}x_{n}=\limsup_{n\to\infty}x_{n}
  26. lim n x n \lim_{n\to\infty}x_{n}
  27. \mathbb{R}
  28. lim inf n x n = lim n x n = \liminf_{n\to\infty}x_{n}=\infty\;\;\Rightarrow\;\;\lim_{n\to\infty}x_{n}=\infty
  29. lim sup n x n = - lim n x n = - . \limsup_{n\to\infty}x_{n}=-\infty\;\;\Rightarrow\;\;\lim_{n\to\infty}x_{n}=-\infty.
  30. I = lim inf n x n I=\liminf_{n\to\infty}x_{n}
  31. S = lim sup n x n S=\limsup_{n\to\infty}x_{n}
  32. x k n x_{k_{n}}
  33. x h n x_{h_{n}}
  34. x n x_{n}
  35. k n k_{n}
  36. h n h_{n}
  37. lim inf n x n + ϵ > x h n x k n > lim sup n x n - ϵ \liminf_{n\to\infty}x_{n}+\epsilon>x_{h_{n}}\;\;\;\;\;\;\;\;\;x_{k_{n}}>% \limsup_{n\to\infty}x_{n}-\epsilon
  38. n 0 n_{0}\in\mathbb{N}
  39. n n 0 n\geq n_{0}
  40. lim inf n x n - ϵ < x n < lim sup n x n + ϵ \liminf_{n\to\infty}x_{n}-\epsilon<x_{n}<\limsup_{n\to\infty}x_{n}+\epsilon
  41. Λ \Lambda
  42. x n x_{n}
  43. Λ \Lambda
  44. λ \lambda
  45. x n x_{n}
  46. λ \lambda
  47. inf n x n lim inf n x n lim sup n x n sup n x n \inf_{n}x_{n}\leq\liminf_{n\to\infty}x_{n}\leq\limsup_{n\to\infty}x_{n}\leq% \sup_{n}x_{n}
  48. { a n } , { b n } \{a_{n}\},\{b_{n}\}
  49. - \infty-\infty
  50. - + -\infty+\infty
  51. lim sup n ( a n + b n ) lim sup n ( a n ) + lim sup n ( b n ) . \limsup_{n\to\infty}(a_{n}+b_{n})\leq\limsup_{n\to\infty}(a_{n})+\limsup_{n\to% \infty}(b_{n}).
  52. lim inf n ( a n + b n ) lim inf n ( a n ) + lim inf n ( b n ) . \liminf_{n\to\infty}(a_{n}+b_{n})\geq\liminf_{n\to\infty}(a_{n})+\liminf_{n\to% \infty}(b_{n}).
  53. a n a a_{n}\to a
  54. lim sup n a n \limsup_{n\to\infty}a_{n}
  55. lim inf n a n \liminf_{n\to\infty}a_{n}
  56. a a
  57. lim inf n x n = - 1 \liminf_{n\to\infty}x_{n}=-1
  58. lim sup n x n = + 1. \limsup_{n\to\infty}x_{n}=+1.
  59. lim inf n ( p n + 1 - p n ) , \liminf_{n\to\infty}(p_{n+1}-p_{n}),
  60. + +\infty
  61. lim sup x a f ( x ) = lim ε 0 ( sup { f ( x ) : x E B ( a ; ε ) { a } } ) \limsup_{x\to a}f(x)=\lim_{\varepsilon\to 0}(\sup\{f(x):x\in E\cap B(a;% \varepsilon)\setminus\{a\}\})
  62. lim inf x a f ( x ) = lim ε 0 ( inf { f ( x ) : x E B ( a ; ε ) { a } } ) \liminf_{x\to a}f(x)=\lim_{\varepsilon\to 0}(\inf\{f(x):x\in E\cap B(a;% \varepsilon)\setminus\{a\}\})
  63. lim sup x a f ( x ) = inf ε > 0 ( sup { f ( x ) : x E B ( a ; ε ) { a } } ) \limsup_{x\to a}f(x)=\inf_{\varepsilon>0}(\sup\{f(x):x\in E\cap B(a;% \varepsilon)\setminus\{a\}\})
  64. lim inf x a f ( x ) = sup ε > 0 ( inf { f ( x ) : x E B ( a ; ε ) { a } } ) . \liminf_{x\to a}f(x)=\sup_{\varepsilon>0}(\inf\{f(x):x\in E\cap B(a;% \varepsilon)\setminus\{a\}\}).
  65. lim sup x a f ( x ) = inf { sup { f ( x ) : x E U { a } } : U open , a U , E U { a } } \limsup_{x\to a}f(x)=\inf\{\sup\{f(x):x\in E\cap U\setminus\{a\}\}:U\ \mathrm{% open},a\in U,E\cap U\setminus\{a\}\neq\emptyset\}
  66. lim inf x a f ( x ) = sup { inf { f ( x ) : x E U { a } } : U open , a U , E U { a } } \liminf_{x\to a}f(x)=\sup\{\inf\{f(x):x\in E\cap U\setminus\{a\}\}:U\ \mathrm{% open},a\in U,E\cap U\setminus\{a\}\neq\emptyset\}
  67. d ( x , y ) := { 0 if x = y , 1 if x y . d(x,y):=\begin{cases}0&\,\text{if }x=y,\\ 1&\,\text{if }x\neq y.\end{cases}
  68. I n := inf { X m : m { n , n + 1 , n + 2 , } } = m = n X m = X n X n + 1 X n + 2 . I_{n}:=\inf\{X_{m}:m\in\{n,n+1,n+2,\ldots\}\}=\bigcap_{m=n}^{\infty}X_{m}=X_{n% }\cap X_{n+1}\cap X_{n+2}\cap\cdots.
  69. lim inf n X n : = lim n inf { X m : m { n , n + 1 , } } = sup { inf { X m : m { n , n + 1 , } } : n { 1 , 2 , } } = n = 1 ( m = n X m ) . \begin{aligned}\displaystyle\liminf_{n\to\infty}X_{n}&\displaystyle:=\lim_{n% \to\infty}\inf\{X_{m}:m\in\{n,n+1,\ldots\}\}\\ &\displaystyle=\sup\{\inf\{X_{m}:m\in\{n,n+1,\ldots\}\}:n\in\{1,2,\dots\}\}\\ &\displaystyle={\bigcup_{n=1}^{\infty}}\left({\bigcap_{m=n}^{\infty}}X_{m}% \right).\end{aligned}
  70. J n := sup { X m : m { n , n + 1 , n + 2 , } } = m = n X m = X n X n + 1 X n + 2 . J_{n}:=\sup\{X_{m}:m\in\{n,n+1,n+2,\ldots\}\}=\bigcup_{m=n}^{\infty}X_{m}=X_{n% }\cup X_{n+1}\cup X_{n+2}\cup\cdots.
  71. lim sup n X n : = lim n sup { X m : m { n , n + 1 , } } = inf { sup { X m : m { n , n + 1 , } } : n { 1 , 2 , } } = n = 1 ( m = n X m ) . \begin{aligned}\displaystyle\limsup_{n\to\infty}X_{n}&\displaystyle:=\lim_{n% \to\infty}\sup\{X_{m}:m\in\{n,n+1,\ldots\}\}\\ &\displaystyle=\inf\{\sup\{X_{m}:m\in\{n,n+1,\ldots\}\}:n\in\{1,2,\dots\}\}\\ &\displaystyle={\bigcap_{n=1}^{\infty}}\left({\bigcup_{m=n}^{\infty}}X_{m}% \right).\end{aligned}
  72. { X n } = { { 0 } , { 1 } , { 0 } , { 1 } , { 0 } , { 1 } , } . \{X_{n}\}=\{\{0\},\{1\},\{0\},\{1\},\{0\},\{1\},\dots\}.
  73. { X n } = { { 50 } , { 20 } , { - 100 } , { - 25 } , { 0 } , { 1 } , { 0 } , { 1 } , { 0 } , { 1 } , } . \{X_{n}\}=\{\{50\},\{20\},\{-100\},\{-25\},\{0\},\{1\},\{0\},\{1\},\{0\},\{1\}% ,\dots\}.
  74. { X n } = { { 0 } , { 1 } , { 1 / 2 } , { 1 / 2 } , { 2 / 3 } , { 1 / 3 } , { 3 / 4 } , { 1 / 4 } , } . \{X_{n}\}=\{\{0\},\{1\},\{1/2\},\{1/2\},\{2/3\},\{1/3\},\{3/4\},\{1/4\},\dots\}.
  75. lim inf X := inf { x Y : x is a limit point of X } \liminf X:=\inf\{x\in Y:x\,\text{ is a limit point of }X\}\,
  76. lim sup X := sup { x Y : x is a limit point of X } \limsup X:=\sup\{x\in Y:x\,\text{ is a limit point of }X\}\,
  77. { B ¯ 0 : B 0 B } \bigcap\{\overline{B}_{0}:B_{0}\in B\}
  78. B ¯ 0 \overline{B}_{0}
  79. B 0 B_{0}
  80. lim sup B := sup { B ¯ 0 : B 0 B } \limsup B:=\sup\bigcap\{\overline{B}_{0}:B_{0}\in B\}
  81. lim sup B = inf { sup B 0 : B 0 B } \limsup B=\inf\{\sup B_{0}:B_{0}\in B\}
  82. lim inf B := inf { B ¯ 0 : B 0 B } \liminf B:=\inf\bigcap\{\overline{B}_{0}:B_{0}\in B\}
  83. lim inf B = sup { inf B 0 : B 0 B } \liminf B=\sup\{\inf B_{0}:B_{0}\in B\}
  84. X X
  85. ( x α ) α A (x_{\alpha})_{\alpha\in A}
  86. ( A , ) (A,{\leq})
  87. x α X x_{\alpha}\in X
  88. α A \alpha\in A
  89. B B
  90. B := { { x α : α 0 α } : α 0 A } . B:=\{\{x_{\alpha}:\alpha_{0}\leq\alpha\}:\alpha_{0}\in A\}.\,
  91. B B
  92. X X
  93. ( x n ) (x_{n})
  94. x n X x_{n}\in X
  95. n n\in\mathbb{N}
  96. \mathbb{N}
  97. C C
  98. C := { { x n : n 0 n } : n 0 } . C:=\{\{x_{n}:n_{0}\leq n\}:n_{0}\in\mathbb{N}\}.\,
  99. C C

Line-of-sight_propagation.html

  1. horizon miles 2 × height feet . \mathrm{horizon}_{\mathrm{miles}}\approx\sqrt{2\times\mathrm{height}_{\mathrm{% feet}}}.
  2. horizon km 3.57 height metres \mathrm{horizon}_{\mathrm{km}}\approx 3.57\cdot\sqrt{\mathrm{height}_{\mathrm{% metres}}}
  3. d 2 = ( R + h ) 2 - R 2 = 2 R h + h 2 d^{2}=(R+h)^{2}-R^{2}=2\cdot R\cdot h+h^{2}
  4. d 2 R h d\approx\sqrt{2\cdot R\cdot h}
  5. d 3.57 h d\approx 3.57\cdot\sqrt{h}
  6. d 1.23 h d\approx 1.23\cdot\sqrt{h}
  7. d 2 k R h d\approx\sqrt{2\cdot k\cdot R\cdot h}
  8. d 4.12 h d\approx 4.12\cdot\sqrt{h}
  9. d 1.41 h d\approx 1.41\cdot\sqrt{h}
  10. d 4.12 1500 = 160 km. d\approx 4.12\cdot\sqrt{1500}=160\mbox{ km.}

Linear_algebra.html

  1. T : V W T:V\to W
  2. T ( u + v ) = T ( u ) + T ( v ) , T ( a v ) = a T ( v ) T(u+v)=T(u)+T(v),\quad T(av)=aT(v)
  3. T ( a u + b v ) = T ( a u ) + T ( b v ) = a T ( u ) + b T ( v ) \quad T(au+bv)=T(au)+T(bv)=aT(u)+bT(v)
  4. a 1 v 1 + a 2 v 2 + + a k v k , a_{1}v_{1}+a_{2}v_{2}+\cdots+a_{k}v_{k},
  5. dim ( U 1 + U 2 ) = dim U 1 + dim U 2 - dim ( U 1 U 2 ) \dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2})
  6. a 1 v 1 + a 2 v 2 + + a n v n . a_{1}v_{1}+a_{2}v_{2}+\cdots+a_{n}v_{n}.\,
  7. T v - λ v = ( T - λ I ) v = 0 , Tv-\lambda v=(T-\lambda\,\,\text{I})v=0,
  8. T ( v ) = T ( a 1 v 1 ) + + T ( a n v n ) = a 1 T ( v 1 ) + + a n T ( v n ) = a 1 λ 1 v 1 + + a n λ n v n . T(v)=T(a_{1}v_{1})+\cdots+T(a_{n}v_{n})=a_{1}T(v_{1})+\cdots+a_{n}T(v_{n})=a_{% 1}\lambda_{1}v_{1}+\cdots+a_{n}\lambda_{n}v_{n}.
  9. , : V × V F \langle\cdot,\cdot\rangle:V\times V\rightarrow F
  10. u , v = v , u ¯ . \langle u,v\rangle=\overline{\langle v,u\rangle}.
  11. a u , v = a u , v . \langle au,v\rangle=a\langle u,v\rangle.
  12. u + v , w = u , w + v , w . \langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle.
  13. v , v 0 \langle v,v\rangle\geq 0
  14. v 2 = v , v , \|v\|^{2}=\langle v,v\rangle,
  15. | u , v | u v . |\langle u,v\rangle|\leq\|u\|\cdot\|v\|.
  16. | u , v | u v 1 , \frac{|\langle u,v\rangle|}{\|u\|\cdot\|v\|}\leq 1,
  17. u , v = 0 \langle u,v\rangle=0
  18. a i = v , v i a_{i}=\langle v,v_{i}\rangle
  19. T u , v = u , T * v . \langle Tu,v\rangle=\langle u,T^{*}v\rangle.
  20. 2 x \displaystyle 2x
  21. L 2 + 3 2 L 1 L 2 L_{2}+\tfrac{3}{2}L_{1}\rightarrow L_{2}
  22. L 3 + L 1 L 3 L_{3}+L_{1}\rightarrow L_{3}
  23. 2 x \displaystyle 2x
  24. L 3 + - 4 L 2 L 3 L_{3}+-4L_{2}\rightarrow L_{3}
  25. 2 x \displaystyle 2x
  26. z = - 1 ( L 3 ) z=-1\quad(L_{3})
  27. y = 3 ( L 2 ) y=3\quad(L_{2})
  28. x = 2 ( L 1 ) x=2\quad(L_{1})
  29. A x = b . Ax=b.
  30. x 0 + N = { x 0 + n : n N } x_{0}+N=\{x_{0}+n:n\in N\}
  31. f ( x ) = a 0 2 + n = 1 [ a n cos ( n x ) + b n sin ( n x ) ] . f(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\,[a_{n}\cos(nx)+b_{n}\sin(nx)].
  32. f , g = 1 π - π π f ( x ) g ( x ) d x . \langle f,g\rangle=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)g(x)\,dx.
  33. f , h k = a 0 2 h 0 , h k + n = 1 [ a n h n , h k + b n g n , h k ] , \langle f,h_{k}\rangle=\frac{a_{0}}{2}\langle h_{0},h_{k}\rangle+\sum_{n=1}^{% \infty}\,[a_{n}\langle h_{n},h_{k}\rangle+b_{n}\langle\ g_{n},h_{k}\rangle],
  34. f , h k = a k \langle f,h_{k}\rangle=a_{k}
  35. a k = 1 π - π π f ( x ) cos ( k x ) d x . a_{k}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(kx)\,dx.
  36. - - - | ϕ | 2 d x d y d z \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|\phi|^{2% }dxdydz
  37. H = - 2 2 m 2 + V ( x , y , z ) H=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(x,y,z)
  38. λ : a x + b y + c = 0 , \lambda:ax+by+c=0,
  39. λ : [ a b c ] { x y 1 } = 0 , \lambda:\begin{bmatrix}a&b&c\end{bmatrix}\begin{Bmatrix}x\\ y\\ 1\end{Bmatrix}=0,
  40. A 𝐱 = 0 , A\mathbf{x}=0,
  41. λ 1 : a 1 x + b 1 y + c 1 = 0 , λ 2 : a 2 x + b 2 y + c 2 = 0 , \lambda_{1}:a_{1}x+b_{1}y+c_{1}=0,\quad\lambda_{2}:a_{2}x+b_{2}y+c_{2}=0,
  42. λ 1 , 2 : [ a 1 b 1 c 1 a 2 b 2 c 2 ] { x y 1 } = { 0 0 } , \lambda_{1,2}:\begin{bmatrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\end{bmatrix}\begin{Bmatrix}x\\ y\\ 1\end{Bmatrix}=\begin{Bmatrix}0\\ 0\end{Bmatrix},
  43. B 𝐱 = 0. B\mathbf{x}=0.
  44. x 1 = | b 1 c 1 b 2 c 2 | , x 2 = - | a 1 c 1 a 2 c 2 | , x 3 = | a 1 b 1 a 2 b 2 | x_{1}=\begin{vmatrix}b_{1}&c_{1}\\ b_{2}&c_{2}\end{vmatrix},x_{2}=-\begin{vmatrix}a_{1}&c_{1}\\ a_{2}&c_{2}\end{vmatrix},x_{3}=\begin{vmatrix}a_{1}&b_{1}\\ a_{2}&b_{2}\end{vmatrix}
  45. λ 1 , 2 , 3 : [ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 ] { x y 1 } = { 0 0 0 } . \lambda_{1,2,3}:\begin{bmatrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{bmatrix}\begin{Bmatrix}x\\ y\\ 1\end{Bmatrix}=\begin{Bmatrix}0\\ 0\\ 0\end{Bmatrix}.
  46. C 𝐱 = 0. C\mathbf{x}=0.
  47. λ : [ a b ] { x y } = c , \lambda:\begin{bmatrix}a&b\end{bmatrix}\begin{Bmatrix}x\\ y\end{Bmatrix}=c,
  48. A 𝐱 = c . A\mathbf{x}=c.
  49. A ( α 𝐱 + β 𝐲 ) = α A 𝐱 + β A 𝐲 = α c + β d . A(\alpha\mathbf{x}+\beta\mathbf{y})=\alpha A\mathbf{x}+\beta A\mathbf{y}=% \alpha c+\beta d.
  50. A 𝐱 = A ( x 𝐢 + y 𝐣 ) = x A 𝐢 + y A 𝐣 = [ A 𝐢 A 𝐣 ] { x y } = [ a b ] { x y } = c . A\mathbf{x}=A(x\mathbf{i}+y\mathbf{j})=xA\mathbf{i}+yA\mathbf{j}=\begin{% bmatrix}A\mathbf{i}&A\mathbf{j}\end{bmatrix}\begin{Bmatrix}x\\ y\end{Bmatrix}=\begin{bmatrix}a&b\end{bmatrix}\begin{Bmatrix}x\\ y\end{Bmatrix}=c.
  51. λ : A 𝐱 = [ A 𝐯 A 𝐰 ] { α β } = [ d e ] { α β } = c , \lambda:A\mathbf{x}=\begin{bmatrix}A\mathbf{v}&A\mathbf{w}\end{bmatrix}\begin{% Bmatrix}\alpha\\ \beta\end{Bmatrix}=\begin{bmatrix}d&e\end{bmatrix}\begin{Bmatrix}\alpha\\ \beta\end{Bmatrix}=c,
  52. [ a b ] [ v 1 w 1 v 2 w 2 ] = [ d e ] . \begin{bmatrix}a&b\end{bmatrix}\begin{bmatrix}v_{1}&w_{1}\\ v_{2}&w_{2}\end{bmatrix}=\begin{bmatrix}d&e\end{bmatrix}.
  53. { x y } = [ v 1 w 1 v 2 w 2 ] { α β } . \begin{Bmatrix}x\\ y\end{Bmatrix}=\begin{bmatrix}v_{1}&w_{1}\\ v_{2}&w_{2}\end{bmatrix}\begin{Bmatrix}\alpha\\ \beta\end{Bmatrix}.
  54. σ = [ σ 1 σ 2 ] = 1 v 1 w 2 - v 2 w 1 [ w 2 - w 1 ] , τ = [ τ 1 τ 2 ] = 1 v 1 w 2 - v 2 w 1 [ - v 2 v 1 ] , \sigma=\begin{bmatrix}\sigma_{1}&\sigma_{2}\end{bmatrix}=\frac{1}{v_{1}w_{2}-v% _{2}w_{1}}\begin{bmatrix}w_{2}&-w_{1}\end{bmatrix},\tau=\begin{bmatrix}\tau_{1% }&\tau_{2}\end{bmatrix}=\frac{1}{v_{1}w_{2}-v_{2}w_{1}}\begin{bmatrix}-v_{2}&v% _{1}\end{bmatrix},
  55. σ 𝐱 = α , τ 𝐱 = β , \sigma\mathbf{x}=\alpha,\tau\mathbf{x}=\beta,
  56. [ σ 1 σ 2 τ 1 τ 2 ] { x y } = { α β } . \begin{bmatrix}\sigma_{1}&\sigma_{2}\\ \tau_{1}&\tau_{2}\end{bmatrix}\begin{Bmatrix}x\\ y\end{Bmatrix}=\begin{Bmatrix}\alpha\\ \beta\end{Bmatrix}.
  57. σ 𝐯 = 1 , σ 𝐰 = 0 , τ 𝐰 = 1 , τ 𝐯 = 0. \sigma\mathbf{v}=1,\sigma\mathbf{w}=0,\tau\mathbf{w}=1,\tau\mathbf{v}=0.
  58. [ σ 1 σ 2 τ 1 τ 2 ] [ v 1 w 1 v 2 w 2 ] = [ 1 0 0 1 ] . \begin{bmatrix}\sigma_{1}&\sigma_{2}\\ \tau_{1}&\tau_{2}\end{bmatrix}\begin{bmatrix}v_{1}&w_{1}\\ v_{2}&w_{2}\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}.
  59. A 𝐱 = [ a b ] { x y } = c . A\mathbf{x}=\begin{bmatrix}a&b\end{bmatrix}\begin{Bmatrix}x\\ y\end{Bmatrix}=c.
  60. b y = c - a x . by=c-ax.
  61. 𝐱 ( t ) = { 0 c / b } + t { 1 - a / b } = 𝐩 + t 𝐡 . \mathbf{x}(t)=\begin{Bmatrix}0\\ c/b\end{Bmatrix}+t\begin{Bmatrix}1\\ -a/b\end{Bmatrix}=\mathbf{p}+t\mathbf{h}.
  62. A 𝐡 = [ a b ] { 1 - a / b } = 0. A\mathbf{h}=\begin{bmatrix}a&b\end{bmatrix}\begin{Bmatrix}1\\ -a/b\end{Bmatrix}=0.

Linear_combination.html

  1. a 1 v 1 + a 2 v 2 + a 3 v 3 + + a n v n . a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+\cdots+a_{n}v_{n}.\,
  2. ( a 1 , a 2 , a 3 ) = ( a 1 , 0 , 0 ) + ( 0 , a 2 , 0 ) + ( 0 , 0 , a 3 ) (a_{1},a_{2},a_{3})=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\,
  3. = a 1 ( 1 , 0 , 0 ) + a 2 ( 0 , 1 , 0 ) + a 3 ( 0 , 0 , 1 ) =a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\,
  4. = a 1 e 1 + a 2 e 2 + a 3 e 3 . =a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}.\,
  5. cos t = 1 2 e i t + 1 2 e - i t \cos t=\begin{matrix}\frac{1}{2}\end{matrix}e^{it}+\begin{matrix}\frac{1}{2}% \end{matrix}e^{-it}\,
  6. 2 sin t = ( - i ) e i t + ( i ) e - i t . 2\sin t=(-i)e^{it}+(i)e^{-it}.\,
  7. a 1 ( 1 ) + a 2 ( x + 1 ) + a 3 ( x 2 + x + 1 ) = x 2 - 1. a_{1}(1)+a_{2}(x+1)+a_{3}(x^{2}+x+1)=x^{2}-1.\,
  8. ( a 1 ) + ( a 2 x + a 2 ) + ( a 3 x 2 + a 3 x + a 3 ) = x 2 - 1 (a_{1})+(a_{2}x+a_{2})+(a_{3}x^{2}+a_{3}x+a_{3})=x^{2}-1\,
  9. a 3 x 2 + ( a 2 + a 3 ) x + ( a 1 + a 2 + a 3 ) = 1 x 2 + 0 x + ( - 1 ) . a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})=1x^{2}+0x+(-1).\,
  10. a 3 = 1 , a 2 + a 3 = 0 , a 1 + a 2 + a 3 = - 1. a_{3}=1,\quad a_{2}+a_{3}=0,\quad a_{1}+a_{2}+a_{3}=-1.\,
  11. x 2 - 1 = - 1 - ( x + 1 ) + ( x 2 + x + 1 ) = - p 1 - p 2 + p 3 x^{2}-1=-1-(x+1)+(x^{2}+x+1)=-p_{1}-p_{2}+p_{3}\,
  12. 0 x 3 + a 3 x 2 + ( a 2 + a 3 ) x + ( a 1 + a 2 + a 3 ) 0x^{3}+a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})\,
  13. = 1 x 3 + 0 x 2 + 0 x + ( - 1 ) . =1x^{3}+0x^{2}+0x+(-1).\,
  14. 0 = 1 0=1\,
  15. Sp ( v 1 , , v n ) := { a 1 v 1 + + a n v n : a 1 , , a n K } . \mathrm{Sp}(v_{1},\ldots,v_{n}):=\{a_{1}v_{1}+\cdots+a_{n}v_{n}:a_{1},\ldots,a% _{n}\in K\}.\,
  16. v = a i v i = b i v i where ( a i ) ( b i ) . v=\sum a_{i}v_{i}=\sum b_{i}v_{i}\,\text{ where }(a_{i})\neq(b_{i}).
  17. c i := a i - b i c_{i}:=a_{i}-b_{i}
  18. 0 = c i v i . 0=\sum c_{i}v_{i}.
  19. 𝐑 n \mathbf{R}^{n}
  20. a i = 1 \sum a_{i}=1
  21. a i 0 a_{i}\geq 0
  22. a i 0 a_{i}\geq 0
  23. a i = 1 \sum a_{i}=1
  24. 𝐑 \mathbf{R}^{\infty}
  25. ( 2 , 3 , - 5 , 0 , ) (2,3,-5,0,\dots)
  26. 2 v 1 + 3 v 2 - 5 v 3 + 0 v 4 + 2v_{1}+3v_{2}-5v_{3}+0v_{4}+\cdots
  27. 𝐑 n \mathbf{R}^{n}
  28. a 1 v 1 b 1 + + a n v n b n a_{1}v_{1}b_{1}+\cdots+a_{n}v_{n}b_{n}\,

Linear_congruential_generator.html

  1. X n + 1 = ( a X n + c ) mod m X_{n+1}=\left(aX_{n}+c\right)~{}~{}\bmod~{}~{}m
  2. X X
  3. m , 0 < m m,\,0<m
  4. a , 0 < a < m a,\,0<a<m
  5. c , 0 c < m c,\,0\leq c<m
  6. X 0 , 0 X 0 < m X_{0},\,0\leq X_{0}<m
  7. c \,c
  8. m \,m
  9. a - 1 \,a-1
  10. m \,m
  11. a - 1 \,a-1
  12. m \,m

Linear_cryptanalysis.html

  1. P 1 P 3 C 1 = K 2 . P_{1}\oplus P_{3}\oplus C_{1}=K_{2}.
  2. P i 1 P i 2 C j 1 C j 2 = K k 1 K k 2 P_{i_{1}}\oplus P_{i_{2}}\oplus\cdots\oplus C_{j_{1}}\oplus C_{j_{2}}\oplus% \cdots=K_{k_{1}}\oplus K_{k_{2}}\oplus\cdots

Linear_equation.html

  1. a x = b . ax=b.
  2. x = b a . x=\frac{b}{a}.
  3. y = m x + b , y=mx+b,\,
  4. A x + B y = C , Ax+By=C,\,
  5. a x + b y + c = 0 , ax+by+c=0,\,
  6. y = m x + b , y=mx+b,
  7. y - y 1 = m ( x - x 1 ) , y-y_{1}=m(x-x_{1}),\,
  8. y - y 1 = y 2 - y 1 x 2 - x 1 ( x - x 1 ) , y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1}),\,
  9. ( x 2 - x 1 ) ( y - y 1 ) = ( y 2 - y 1 ) ( x - x 1 ) . (x_{2}-x_{1})(y-y_{1})=(y_{2}-y_{1})(x-x_{1}).\,
  10. x ( y 2 - y 1 ) - y ( x 2 - x 1 ) = x 1 y 2 - x 2 y 1 x\,(y_{2}-y_{1})-y\,(x_{2}-x_{1})=x_{1}y_{2}-x_{2}y_{1}
  11. | x y 1 x 1 y 1 1 x 2 y 2 1 | = 0 . \begin{vmatrix}x&y&1\\ x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\end{vmatrix}=0\,.
  12. x a + y b = 1 , \frac{x}{a}+\frac{y}{b}=1,\,
  13. A x + B y = C , Ax+By=C,\,
  14. ( A B ) ( x y ) = ( C ) . \begin{pmatrix}A&B\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}C\end{pmatrix}.
  15. A 1 x + B 1 y = C 1 , A_{1}x+B_{1}y=C_{1},\,
  16. A 2 x + B 2 y = C 2 , A_{2}x+B_{2}y=C_{2},\,
  17. ( A 1 B 1 A 2 B 2 ) ( x y ) = ( C 1 C 2 ) . \begin{pmatrix}A_{1}&B_{1}\\ A_{2}&B_{2}\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}C_{1}\\ C_{2}\end{pmatrix}.
  18. x = T t + U x=Tt+U\,
  19. y = V t + W . y=Vt+W.\,
  20. x = ( p - h ) t + h x=(p-h)t+h\,
  21. y = ( q - k ) t + k . y=(q-k)t+k.\,
  22. P 1 P_{1}
  23. P 2 P_{2}
  24. P P
  25. det ( P 1 P , P 1 P 2 ) = 0. \det(\overrightarrow{P_{1}P},\overrightarrow{P_{1}P_{2}})=0.
  26. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},\,y_{1})
  27. P 2 = ( x 2 , y 2 ) P_{2}=(x_{2},\,y_{2})
  28. P = ( x , y ) P=(x,\,y)
  29. P 1 P = ( x - x 1 , y - y 1 ) \overrightarrow{P_{1}P}=(x-x_{1},\,y-y_{1})
  30. P 1 P 2 = ( x 2 - x 1 , y 2 - y 1 ) \overrightarrow{P_{1}P_{2}}=(x_{2}-x_{1},\,y_{2}-y_{1})
  31. det ( x - x 1 y - y 1 x 2 - x 1 y 2 - y 1 ) = 0. \det\begin{pmatrix}x-x_{1}&y-y_{1}\\ x_{2}-x_{1}&y_{2}-y_{1}\end{pmatrix}=0.
  32. ( x - x 1 ) ( y 2 - y 1 ) - ( y - y 1 ) ( x 2 - x 1 ) = 0. (x-x_{1})(y_{2}-y_{1})-(y-y_{1})(x_{2}-x_{1})=0.
  33. ( x - x 1 ) ( y 2 - y 1 ) = ( y - y 1 ) ( x 2 - x 1 ) . (x-x_{1})(y_{2}-y_{1})=(y-y_{1})(x_{2}-x_{1}).
  34. ( x 2 - x 1 ) (x_{2}-x_{1})
  35. x 1 = x 2 x_{1}=x_{2}
  36. y = b y=b\,
  37. x = a x=a\,
  38. f ( x 1 + x 2 ) = f ( x 1 ) + f ( x 2 ) f(x_{1}+x_{2})=f(x_{1})+f(x_{2})
  39. f ( a x ) = a f ( x ) , f(ax)=af(x),\,
  40. a 1 x 1 + a 2 x 2 + + a n x n = b , a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}=b,
  41. x 1 = b - a 2 a 1 x 2 - - a n a 1 x n . x_{1}=b-\frac{a_{2}}{a_{1}}x_{2}-\cdots-\frac{a_{n}}{a_{1}}x_{n}.

Linear_feedback_shift_register.html

  1. x 16 + x 14 + x 13 + x 11 + 1. x^{16}+x^{14}+x^{13}+x^{11}+1.\,
  2. 2 n - 1 2^{n}-1
  3. x 2 + x + 1 x^{2}+x+1
  4. x 3 + x 2 + 1 x^{3}+x^{2}+1
  5. x 4 + x 3 + 1 x^{4}+x^{3}+1
  6. x 5 + x 3 + 1 x^{5}+x^{3}+1
  7. x 6 + x 5 + 1 x^{6}+x^{5}+1
  8. x 7 + x 6 + 1 x^{7}+x^{6}+1
  9. x 8 + x 6 + x 5 + x 4 + 1 x^{8}+x^{6}+x^{5}+x^{4}+1
  10. x 9 + x 5 + 1 x^{9}+x^{5}+1
  11. x 10 + x 7 + 1 x^{10}+x^{7}+1
  12. x 11 + x 9 + 1 x^{11}+x^{9}+1
  13. x 12 + x 11 + x 10 + x 4 + 1 x^{12}+x^{11}+x^{10}+x^{4}+1
  14. x 13 + x 12 + x 11 + x 8 + 1 x^{13}+x^{12}+x^{11}+x^{8}+1
  15. x 14 + x 13 + x 12 + x 2 + 1 x^{14}+x^{13}+x^{12}+x^{2}+1
  16. x 15 + x 14 + 1 x^{15}+x^{14}+1
  17. x 16 + x 14 + x 13 + x 11 + 1 x^{16}+x^{14}+x^{13}+x^{11}+1
  18. x 17 + x 14 + 1 x^{17}+x^{14}+1
  19. x 18 + x 11 + 1 x^{18}+x^{11}+1
  20. x 19 + x 18 + x 17 + x 14 + 1 x^{19}+x^{18}+x^{17}+x^{14}+1

Linear_filter.html

  1. H ( ω ) H(\omega)
  2. | H ( ω ) | |H(\omega)|
  3. y ( t ) = 0 T x ( t - τ ) h ( τ ) d τ y(t)=\int_{0}^{T}x(t-\tau)\,h(\tau)\,d\tau
  4. y k = i = 0 N x k - i h i y_{k}=\sum_{i=0}^{N}x_{k-i}\,h_{i}
  5. | H ( ω ) | |H(\omega)|
  6. Δ f \Delta f
  7. Δ f \Delta f
  8. N = 1 / ( Δ f T ) N=1/(\Delta f\,T)
  9. Δ f \Delta f
  10. H ( s ) = K ω 0 2 s 2 + ω 0 Q s + ω 0 2 . H(s)=\frac{K\omega^{2}_{0}}{s^{2}+\frac{\omega_{0}}{Q}s+\omega^{2}_{0}}.
  11. H ( s ) = K ω 0 Q s s 2 + ω 0 Q s + ω 0 2 . H(s)=\frac{K\frac{\omega_{0}}{Q}s}{s^{2}+\frac{\omega_{0}}{Q}s+\omega^{2}_{0}}.
  12. ω 0 \omega_{0}
  13. s = σ + j ω s=\sigma+j\omega

Linear_map.html

  1. V W V→W
  2. V = W V=W
  3. V V
  4. f ( 𝐱 + 𝐲 ) = f ( 𝐱 ) + f ( 𝐲 ) f(\mathbf{x}+\mathbf{y})=f(\mathbf{x})+f(\mathbf{y})\!
  5. f ( α 𝐱 ) = α f ( 𝐱 ) f(\alpha\mathbf{x})=\alpha f(\mathbf{x})\!
  6. f ( a 1 𝐱 1 + + a m 𝐱 m ) = a 1 f ( 𝐱 1 ) + + a m f ( 𝐱 m ) . f(a_{1}\mathbf{x}_{1}+\cdots+a_{m}\mathbf{x}_{m})=a_{1}f(\mathbf{x}_{1})+% \cdots+a_{m}f(\mathbf{x}_{m}).\!
  7. f ( 𝟎 V ) = f ( 0 𝟎 V ) = 0 f ( 𝟎 V ) = 𝟎 W . f(\mathbf{0}_{V})=f(0\cdot\mathbf{0}_{V})=0\cdot f(\mathbf{0}_{V})=\mathbf{0}_% {W}.
  8. v c v v\mapsto cv
  9. ( : I ( ) D ( ) / ) \textstyle\left(\ \int\!:\ I(\Re)\to\ D(\Re)/\Re\ \right)
  10. c 1 𝐯 1 + + c n 𝐯 n . c_{1}\mathbf{v}_{1}+\cdots+c_{n}\mathbf{v}_{n}.
  11. f ( c 1 𝐯 1 + + c n 𝐯 n ) = c 1 f ( 𝐯 1 ) + + c n f ( 𝐯 n ) , f(c_{1}\mathbf{v}_{1}+\cdots+c_{n}\mathbf{v}_{n})=c_{1}f(\mathbf{v}_{1})+% \cdots+c_{n}f(\mathbf{v}_{n}),
  12. f ( 𝐯 j ) = a 1 j 𝐰 1 + + a m j 𝐰 m . f(\mathbf{v}_{j})=a_{1j}\mathbf{w}_{1}+\cdots+a_{mj}\mathbf{w}_{m}.
  13. ( a 1 j , , a m j ) T (a_{1j},...,a_{mj})\text{T}
  14. 𝐌 = ( a 1 j . * . * . a m j ) \mathbf{M}=\begin{pmatrix}&&&a_{1j}&&\\ &&&.&&\\ &*&&.&&*\\ &&&.&&\\ &&&a_{mj}&&\end{pmatrix}
  15. T T
  16. B B
  17. T T
  18. B B^{\prime}
  19. B B^{\prime}
  20. B B
  21. B B
  22. B B^{\prime}
  23. [ v ] B [\vec{v}]_{B^{\prime}}
  24. [ T ( v ) ] B [T(\vec{v})]_{B^{\prime}}
  25. A [ v ] B = [ T ( v ) ] B A^{\prime}[\vec{v}]_{B^{\prime}}=[T(\vec{v})]_{B^{\prime}}
  26. [ v ] B [\vec{v}]_{B^{\prime}}
  27. P - 1 A P P^{-1}AP
  28. P - 1 A P [ v ] B = [ T ( v ) ] B P^{-1}AP[\vec{v}]_{B^{\prime}}=[T(\vec{v})]_{B^{\prime}}
  29. 𝐀 = ( 0 - 1 1 0 ) \mathbf{A}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  30. 𝐀 = ( cos θ - sin θ sin θ cos θ ) \mathbf{A}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}
  31. 𝐀 = ( 1 0 0 - 1 ) \mathbf{A}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  32. 𝐀 = ( - 1 0 0 1 ) \mathbf{A}=\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}
  33. 𝐀 = ( 2 0 0 2 ) \mathbf{A}=\begin{pmatrix}2&0\\ 0&2\end{pmatrix}
  34. 𝐀 = ( 1 m 0 1 ) \mathbf{A}=\begin{pmatrix}1&m\\ 0&1\end{pmatrix}
  35. 𝐀 = ( k 0 0 1 / k ) \mathbf{A}=\begin{pmatrix}k&0\\ 0&1/k\end{pmatrix}
  36. 𝐀 = ( 0 0 0 1 ) . \mathbf{A}=\begin{pmatrix}0&0\\ 0&1\end{pmatrix}.
  37. ker ( f ) = { x V : f ( x ) = 0 } \operatorname{\ker}(f)=\{\,x\in V:f(x)=0\,\}
  38. im ( f ) = { w W : w = f ( x ) , x V } \operatorname{im}(f)=\{\,w\in W:w=f(x),x\in V\,\}
  39. dim ( ker ( f ) ) + dim ( im ( f ) ) = dim ( V ) . \dim(\ker(f))+\dim(\operatorname{im}(f))=\dim(V).
  40. coker f := W / f ( V ) = W / im ( f ) . \mathrm{coker}\,f:=W/f(V)=W/\mathrm{im}(f).
  41. 0 ker f V W coker f 0. 0\to\ker f\to V\to W\to\mathrm{coker}\,f\to 0.
  42. { a n } { b n } \{a_{n}\}\mapsto\{b_{n}\}
  43. 0 + 0 = 0 + 1 \aleph_{0}+0=\aleph_{0}+1
  44. { a n } { c n } \{a_{n}\}\mapsto\{c_{n}\}
  45. ind f := dim ker f - dim coker f , \mathrm{ind}\,f:=\dim\ker f-\dim\mathrm{coker}\,f,
  46. B [ v ] = A B [ u ] B[v^{\prime}]=AB[u^{\prime}]
  47. [ v ] = B - 1 A B [ u ] = A [ u ] . [v^{\prime}]=B^{-1}AB[u^{\prime}]=A^{\prime}[u^{\prime}].

Linear_model.html

  1. ( Y i , X i 1 , , X i p ) , i = 1 , , n (Y_{i},X_{i1},\ldots,X_{ip}),\,i=1,\ldots,n
  2. Y i = β 0 + β 1 ϕ 1 ( X i 1 ) + + β p ϕ p ( X i p ) + ε i i = 1 , , n Y_{i}=\beta_{0}+\beta_{1}\phi_{1}(X_{i1})+\cdots+\beta_{p}\phi_{p}(X_{ip})+% \varepsilon_{i}\qquad i=1,\ldots,n
  3. ϕ 1 , , ϕ p \phi_{1},\ldots,\phi_{p}
  4. Y ^ i = β 0 + β 1 ϕ 1 ( X i 1 ) + + β p ϕ p ( X i p ) ( i = 1 , , n ) , \hat{Y}_{i}=\beta_{0}+\beta_{1}\phi_{1}(X_{i1})+\cdots+\beta_{p}\phi_{p}(X_{ip% })\qquad(i=1,\ldots,n),
  5. S = i = 1 n ( Y i - β 0 - β 1 ϕ 1 ( X i 1 ) - - β p ϕ p ( X i p ) ) 2 . S=\sum_{i=1}^{n}\left(Y_{i}-\beta_{0}-\beta_{1}\phi_{1}(X_{i1})-\cdots-\beta_{% p}\phi_{p}(X_{ip})\right)^{2}.
  6. X t = c + ε t + i = 1 p ϕ i X t - i + i = 1 q θ i ε t - i . X_{t}=c+\varepsilon_{t}+\sum_{i=1}^{p}\phi_{i}X_{t-i}+\sum_{i=1}^{q}\theta_{i}% \varepsilon_{t-i}.\,

Linear_motor.html

  1. ( F = I L × B ) (\vec{F}=I\vec{L}\times\vec{B})

Linear_polarization.html

  1. 𝐄 ( 𝐫 , t ) = 𝐄 Re { | ψ exp [ i ( k z - ω t ) ] } \mathbf{E}(\mathbf{r},t)=\mid\mathbf{E}\mid\mathrm{Re}\left\{|\psi\rangle\exp% \left[i\left(kz-\omega t\right)\right]\right\}
  2. 𝐁 ( 𝐫 , t ) = 𝐳 ^ × 𝐄 ( 𝐫 , t ) / c \mathbf{B}(\mathbf{r},t)=\hat{\mathbf{z}}\times\mathbf{E}(\mathbf{r},t)/c
  3. ω = c k \omega=ck
  4. c c
  5. 𝐄 \mid\mathbf{E}\mid
  6. | ψ = def ( ψ x ψ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) |\psi\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}\psi_{x}\\ \psi_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}
  7. α x , α y \alpha_{x},\alpha_{y}
  8. α x = α y = def α \alpha_{x}=\alpha_{y}\ \stackrel{\mathrm{def}}{=}\ \alpha
  9. θ \theta
  10. | ψ = ( cos θ sin θ ) exp ( i α ) |\psi\rangle=\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix}\exp\left(i\alpha\right)
  11. | x = def ( 1 0 ) |x\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}1\\ 0\end{pmatrix}
  12. | y = def ( 0 1 ) |y\rangle\ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix}0\\ 1\end{pmatrix}
  13. | ψ = cos θ exp ( i α ) | x + sin θ exp ( i α ) | y = ψ x | x + ψ y | y |\psi\rangle=\cos\theta\exp\left(i\alpha\right)|x\rangle+\sin\theta\exp\left(i% \alpha\right)|y\rangle=\psi_{x}|x\rangle+\psi_{y}|y\rangle

Linear_prediction.html

  1. x ^ ( n ) = i = 1 p a i x ( n - i ) \widehat{x}(n)=\sum_{i=1}^{p}a_{i}x(n-i)\,
  2. x ^ ( n ) \widehat{x}(n)
  3. x ( n - i ) x(n-i)
  4. a i a_{i}
  5. e ( n ) = x ( n ) - x ^ ( n ) e(n)=x(n)-\widehat{x}(n)\,
  6. x ( n ) x(n)
  7. a i a_{i}
  8. e ( n ) = x ( n ) - x ^ ( n ) e(n)=\|x(n)-\widehat{x}(n)\|\,
  9. \|\cdot\|
  10. x ^ ( n ) \widehat{x}(n)
  11. a i a_{i}
  12. i = 1 p a i R ( j - i ) = - R ( j ) , \sum_{i=1}^{p}a_{i}R(j-i)=-R(j),
  13. R ( i ) = E { x ( n ) x ( n - i ) } \ R(i)=E\{x(n)x(n-i)\}\,
  14. R a = - r , Ra=-r,\,
  15. e ( n ) = x ( n ) - x ^ ( n ) = x ( n ) - i = 1 p a i x ( n - i ) = - i = 0 p a i x ( n - i ) e(n)=x(n)-\widehat{x}(n)=x(n)-\sum_{i=1}^{p}a_{i}x(n-i)=-\sum_{i=0}^{p}a_{i}x(% n-i)
  16. a i a_{i}
  17. a 0 = - 1 a_{0}=-1
  18. R a = [ 1 , 0 , , 0 ] T \ Ra=[1,0,...,0]^{\mathrm{T}}

Linear_programming.html

  1. maximize \displaystyle\,\text{maximize}
  2. ( ) T (\cdot)^{\mathrm{T}}
  3. f ( x 1 , x 2 ) = c 1 x 1 + c 2 x 2 f(x_{1},x_{2})=c_{1}x_{1}+c_{2}x_{2}
  4. a 11 x 1 + a 12 x 2 b 1 a 21 x 1 + a 22 x 2 b 2 a 31 x 1 + a 32 x 2 b 3 \begin{matrix}a_{11}x_{1}+a_{12}x_{2}&\leq b_{1}\\ a_{21}x_{1}+a_{22}x_{2}&\leq b_{2}\\ a_{31}x_{1}+a_{32}x_{2}&\leq b_{3}\\ \end{matrix}
  5. x 1 0 x 2 0 \begin{matrix}x_{1}\geq 0\\ x_{2}\geq 0\end{matrix}
  6. max { c T x | A x b and x 0 } \max\{c^{\mathrm{T}}x\;|\;Ax\leq b\and x\geq 0\}
  7. S 1 x 1 + S 2 x 2 S_{1}\cdot x_{1}+S_{2}\cdot x_{2}
  8. x 1 + x 2 L x_{1}+x_{2}\leq L
  9. F 1 x 1 + F 2 x 2 F F_{1}\cdot x_{1}+F_{2}\cdot x_{2}\leq F
  10. P 1 x 1 + P 2 x 2 P P_{1}\cdot x_{1}+P_{2}\cdot x_{2}\leq P
  11. x 1 0 , x 2 0 x_{1}\geq 0,x_{2}\geq 0
  12. [ S 1 S 2 ] [ x 1 x 2 ] \begin{bmatrix}S_{1}&S_{2}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}
  13. [ 1 1 F 1 F 2 P 1 P 2 ] [ x 1 x 2 ] [ L F P ] , [ x 1 x 2 ] [ 0 0 ] . \begin{bmatrix}1&1\\ F_{1}&F_{2}\\ P_{1}&P_{2}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\leq\begin{bmatrix}L\\ F\\ P\end{bmatrix},\,\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}\geq\begin{bmatrix}0\\ 0\end{bmatrix}.
  14. [ 1 - 𝐜 T 0 0 𝐀 𝐈 ] [ Z 𝐱 𝐱 s ] = [ 0 𝐛 ] \begin{bmatrix}1&-\mathbf{c}^{T}&0\\ 0&\mathbf{A}&\mathbf{I}\end{bmatrix}\begin{bmatrix}Z\\ \mathbf{x}\\ \mathbf{x}_{s}\end{bmatrix}=\begin{bmatrix}0\\ \mathbf{b}\end{bmatrix}
  15. S 1 x 1 + S 2 x 2 S_{1}\cdot x_{1}+S_{2}\cdot x_{2}
  16. x 1 + x 2 + x 3 = L x_{1}+x_{2}+x_{3}=L
  17. F 1 x 1 + F 2 x 2 + x 4 = F F_{1}\cdot x_{1}+F_{2}\cdot x_{2}+x_{4}=F
  18. P 1 x 1 + P 2 x 2 + x 5 = P P_{1}\cdot x_{1}+P_{2}\cdot x_{2}+x_{5}=P
  19. x 1 , x 2 , x 3 , x 4 , x 5 0 x_{1},x_{2},x_{3},x_{4},x_{5}\geq 0
  20. x 3 , x 4 , x 5 x_{3},x_{4},x_{5}
  21. [ 1 - S 1 - S 2 0 0 0 0 1 1 1 0 0 0 F 1 F 2 0 1 0 0 P 1 P 2 0 0 1 ] [ Z x 1 x 2 x 3 x 4 x 5 ] = [ 0 L F P ] , [ x 1 x 2 x 3 x 4 x 5 ] 0. \begin{bmatrix}1&-S_{1}&-S_{2}&0&0&0\\ 0&1&1&1&0&0\\ 0&F_{1}&F_{2}&0&1&0\\ 0&P_{1}&P_{2}&0&0&1\\ \end{bmatrix}\begin{bmatrix}Z\\ x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\end{bmatrix}=\begin{bmatrix}0\\ L\\ F\\ P\end{bmatrix},\,\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\end{bmatrix}\geq 0.
  22. 2 x 1 - x 2 2x_{1}-x_{2}
  23. x 1 - x 2 1 x_{1}-x_{2}\leq 1
  24. - x 1 + x 2 - 2 -x_{1}+x_{2}\leq-2
  25. x 1 , x 2 0 x_{1},x_{2}\geq 0
  26. L y L + F y F + P y P L\cdot y_{L}+F\cdot y_{F}+P\cdot y_{P}
  27. y L + F 1 y F + P 1 y P S 1 y_{L}+F_{1}\cdot y_{F}+P_{1}\cdot y_{P}\geq S_{1}
  28. y L + F 2 y F + P 2 y P S 2 y_{L}+F_{2}\cdot y_{F}+P_{2}\cdot y_{P}\geq S_{2}
  29. y L , y F , y P 0 y_{L},y_{F},y_{P}\geq 0
  30. [ L F P ] [ y L y F y P ] \begin{bmatrix}L&F&P\end{bmatrix}\begin{bmatrix}y_{L}\\ y_{F}\\ y_{P}\end{bmatrix}
  31. [ 1 F 1 P 1 1 F 2 P 2 ] [ y L y F y P ] [ S 1 S 2 ] , [ y L y F y P ] 0. \begin{bmatrix}1&F_{1}&P_{1}\\ 1&F_{2}&P_{2}\end{bmatrix}\begin{bmatrix}y_{L}\\ y_{F}\\ y_{P}\end{bmatrix}\geq\begin{bmatrix}S_{1}\\ S_{2}\end{bmatrix},\,\begin{bmatrix}y_{L}\\ y_{F}\\ y_{P}\end{bmatrix}\geq 0.
  32. i = 1 m c i x i + j = 1 n d j t j \sum_{i=1}^{m}{c_{i}x_{i}}+\sum_{j=1}^{n}{d_{j}t_{j}}
  33. i = 1 m a i j x i + e j t j g j \sum_{i=1}^{m}{a_{ij}x_{i}}+e_{j}t_{j}\geq g_{j}
  34. 1 j n 1\leq j\leq n
  35. f i x i + j = 1 n b i j t j h i f_{i}x_{i}+\sum_{j=1}^{n}{b_{ij}t_{j}}\geq h_{i}
  36. 1 i m 1\leq i\leq m
  37. x i 0 , t j 0 x_{i}\geq 0,\,t_{j}\geq 0
  38. 1 i m , 1 j n 1\leq i\leq m,1\leq j\leq n
  39. i = 1 m c i x i + j = 1 n d j t j \sum_{i=1}^{m}{c_{i}x_{i}}+\sum_{j=1}^{n}{d_{j}t_{j}}
  40. i = 1 m a i j x i y j + e j t j y j g j y j \sum_{i=1}^{m}{a_{ij}x_{i}}\cdot y_{j}+e_{j}t_{j}\cdot y_{j}\geq g_{j}\cdot y_% {j}
  41. 1 j n 1\leq j\leq n
  42. f i x i s i + j = 1 n b i j t j s i h i s i f_{i}x_{i}\cdot s_{i}+\sum_{j=1}^{n}{b_{ij}t_{j}}\cdot s_{i}\geq h_{i}\cdot s_% {i}
  43. 1 i m 1\leq i\leq m
  44. x i 0 , t j 0 x_{i}\geq 0,\,t_{j}\geq 0
  45. 1 i m , 1 j n 1\leq i\leq m,1\leq j\leq n
  46. y j 0 , s i 0 y_{j}\geq 0,\,s_{i}\geq 0
  47. 1 j n , 1 i m 1\leq j\leq n,1\leq i\leq m
  48. j = 1 n g j y j + i = 1 m h i s i \sum_{j=1}^{n}{g_{j}y_{j}}+\sum_{i=1}^{m}{h_{i}s_{i}}
  49. j = 1 n a i j y j + f i s i c i \sum_{j=1}^{n}{a_{ij}y_{j}}+f_{i}s_{i}\leq c_{i}
  50. 1 i m 1\leq i\leq m
  51. e j y j + i = 1 m b i j s i d j e_{j}y_{j}+\sum_{i=1}^{m}{b_{ij}s_{i}}\leq d_{j}
  52. 1 j n 1\leq j\leq n
  53. y j 0 , s i 0 y_{j}\geq 0,\,s_{i}\geq 0
  54. 1 j n , 1 i m 1\leq j\leq n,1\leq i\leq m
  55. O ( n 3.5 L ) O(n^{3.5}L)
  56. P = { x A x 0 } P=\{x\mid Ax\geq 0\}
  57. { max c x x P } \{\max cx\mid x\in P\}
  58. x * x^{*}
  59. P P
  60. { max c x x P } \{\max cx\mid x\in P\}

Linear_regression.html

  1. { y i , x i 1 , , x i p } i = 1 n \{y_{i},\,x_{i1},\ldots,x_{ip}\}_{i=1}^{n}
  2. y i = β 1 x i 1 + + β p x i p + ε i = 𝐱 i T s y m b o l β + ε i , i = 1 , , n , y_{i}=\beta_{1}x_{i1}+\cdots+\beta_{p}x_{ip}+\varepsilon_{i}=\mathbf{x}^{\rm T% }_{i}symbol\beta+\varepsilon_{i},\qquad i=1,\ldots,n,
  3. 𝐲 = 𝐗 s y m b o l β + s y m b o l ε , \mathbf{y}=\mathbf{X}symbol\beta+symbol\varepsilon,\,
  4. 𝐲 = ( y 1 y 2 y n ) , \mathbf{y}=\begin{pmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{pmatrix},\quad
  5. 𝐗 = ( 𝐱 1 T 𝐱 2 T 𝐱 n T ) = ( x 11 x 1 p x 21 x 2 p x n 1 x n p ) , \mathbf{X}=\begin{pmatrix}\mathbf{x}^{\rm T}_{1}\\ \mathbf{x}^{\rm T}_{2}\\ \vdots\\ \mathbf{x}^{\rm T}_{n}\end{pmatrix}=\begin{pmatrix}x_{11}&\cdots&x_{1p}\\ x_{21}&\cdots&x_{2p}\\ \vdots&\ddots&\vdots\\ x_{n1}&\cdots&x_{np}\end{pmatrix},
  6. s y m b o l β = ( β 1 β 2 β p ) , s y m b o l ε = ( ε 1 ε 2 ε n ) . symbol\beta=\begin{pmatrix}\beta_{1}\\ \beta_{2}\\ \vdots\\ \beta_{p}\end{pmatrix},\quad symbol\varepsilon=\begin{pmatrix}\varepsilon_{1}% \\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{pmatrix}.
  7. y i y_{i}\,
  8. x i 1 , x i 2 , , x i p x_{i1},\,x_{i2},\,\ldots,\,x_{ip}\,
  9. 𝐗 \mathbf{X}
  10. s y m b o l β symbol\beta\,
  11. ε i \varepsilon_{i}\,
  12. h i = β 1 t i + β 2 t i 2 + ε i , h_{i}=\beta_{1}t_{i}+\beta_{2}t_{i}^{2}+\varepsilon_{i},
  13. h i = 𝐱 i T s y m b o l β + ε i . h_{i}=\mathbf{x}^{\rm T}_{i}symbol\beta+\varepsilon_{i}.
  14. ( - , ) (-\infty,\infty)
  15. ε i 𝐱 i \varepsilon_{i}\perp\mathbf{x}_{i}
  16. f f
  17. [ a , b ] [a,b]
  18. g W g\in W
  19. W W
  20. C [ a , b ] C[a,b]
  21. Area = a b | f ( x ) - g ( x ) | d x \,\text{Area }=\int_{a}^{b}\left|f(x)-g(x)\right|\,dx
  22. W W
  23. a b [ f ( x ) - g ( x ) ] 2 d x \int_{a}^{b}[f(x)-g(x)]^{2}\,dx
  24. g g
  25. f f
  26. W W
  27. f - g 2 \lVert f-g\rVert^{2}
  28. f - g \lVert f-g\rVert
  29. a b [ f ( x ) - g ( x ) ] 2 d x = f - g , f - g = f - g 2 \int_{a}^{b}[f(x)-g(x)]^{2}\,dx=\left\langle f-g,f-g\right\rangle=\lVert f-g% \rVert^{2}
  30. f f
  31. g subspace W g\in\,\text{ subspace }W
  32. f f
  33. f , g \left\langle f,g\right\rangle
  34. f f
  35. [ a , b ] [a,b]
  36. W W
  37. C [ a , b ] C[a,b]
  38. f f
  39. W W
  40. g = f , w 1 w 1 + f , w 2 w 2 + + f , w n w n g=\left\langle f,\vec{w}_{1}\right\rangle\vec{w}_{1}+\left\langle f,\vec{w}_{2% }\right\rangle\vec{w}_{2}+\dots+\left\langle f,\vec{w}_{n}\right\rangle\vec{w}% _{n}
  41. B = { w 1 , w 2 , , w n } B=\{\vec{w}_{1},\vec{w}_{2},\dots,\vec{w}_{n}\}
  42. W W

Linear_search.html

  1. { n if k = 0 n + 1 k + 1 if 1 k n . \begin{cases}n&\mbox{if }~{}k=0\\ \displaystyle\frac{n+1}{k+1}&\mbox{if }~{}1\leq k\leq n.\end{cases}
  2. n + 1 2 \frac{n+1}{2}
  3. ( n + 2 ) ( n - 1 ) 2 n \displaystyle\frac{(n+2)(n-1)}{2n}

Linear_span.html

  1. span ( S ) = { 0 + i = 1 k λ i v i | k , v i S , λ i 𝐊 } . \operatorname{span}(S)=\left\{{0+\sum_{i=1}^{k}\lambda_{i}v_{i}\Big|k\in% \mathbb{N},v_{i}\in S,\lambda_{i}\in\mathbf{K}}\right\}.
  2. Sp ¯ ( E ) \overline{\operatorname{Sp}}(E)
  3. Span ¯ ( E ) \overline{\operatorname{Span}}(E)
  4. Sp ¯ ( E ) = { u X | ϵ > 0 x Sp ( E ) : x - u < ϵ } . \overline{\operatorname{Sp}}(E)=\{u\in X|\forall\epsilon>0\,\exists x\in% \operatorname{Sp}(E):\|x-u\|<\epsilon\}.
  5. Sp ¯ ( E ) \overline{\operatorname{Sp}}(E)
  6. Sp ¯ ( E ) = Sp ( E ) ¯ \overline{\operatorname{Sp}}(E)=\overline{\operatorname{Sp}(E)}
  7. Sp ¯ ( E ) \overline{\operatorname{Sp}}(E)
  8. Sp ( E ) \operatorname{Sp}(E)
  9. E = ( Sp ( E ) ) = ( Sp ( E ) ¯ ) . E^{\perp}=(\operatorname{Sp}(E))^{\perp}=(\overline{\operatorname{Sp}(E)})^{% \perp}.

Linear_subspace.html

  1. x - x = 0 x-x={0}
  2. 0 x = 0 0x={0}
  3. \exist c K : 𝐯 = c 𝐯 (or 𝐯 = 1 c 𝐯 ) \exist c\in K:\mathbf{v}^{\prime}=c\mathbf{v}\,\text{ (or }\mathbf{v}=\frac{1}% {c}\mathbf{v}^{\prime}\,\text{)}
  4. \exist c K : 𝐅 = c 𝐅 (or 𝐅 = 1 c 𝐅 ) \exist c\in K:\mathbf{F}^{\prime}=c\mathbf{F}\,\text{ (or }\mathbf{F}=\frac{1}% {c}\mathbf{F}^{\prime}\,\text{)}
  5. { [ x 1 x 2 x n ] K n : a 11 x 1 + a 12 x 2 + + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + + a 2 n x n = 0 a m 1 x 1 + a m 2 x 2 + + a m n x n = 0 } . \left\{\left[\!\!\begin{array}[]{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array}\!\!\right]\in K^{n}:\begin{aligned}\displaystyle a_{11}x_{1}&% &\displaystyle\;+&&\displaystyle a_{12}x_{2}&&\displaystyle\;+\cdots+&&% \displaystyle a_{1n}x_{n}&&\displaystyle\;=0&\\ \displaystyle a_{21}x_{1}&&\displaystyle\;+&&\displaystyle a_{22}x_{2}&&% \displaystyle\;+\cdots+&&\displaystyle a_{2n}x_{n}&&\displaystyle\;=0&\\ \displaystyle\vdots&&&&\displaystyle\vdots&&&&\displaystyle\vdots&&% \displaystyle\vdots&\\ \displaystyle a_{m1}x_{1}&&\displaystyle\;+&&\displaystyle a_{m2}x_{2}&&% \displaystyle\;+\cdots+&&\displaystyle a_{mn}x_{n}&&\displaystyle\;=0&\end{% aligned}\right\}.
  6. x + 3 y + 2 z = 0 and 2 x - 4 y + 5 z = 0 x+3y+2z=0\;\;\;\;\,\text{and}\;\;\;\;2x-4y+5z=0
  7. A 𝐱 = 0. A\mathbf{x}=\mathbf{0}.
  8. A = [ 1 3 2 2 - 4 5 ] . A=\left[\begin{aligned}\displaystyle 1&&\displaystyle 3&&\displaystyle 2&\\ \displaystyle 2&&\displaystyle\;\;-4&&\displaystyle\;\;\;\;5&\end{aligned}\,% \right]\,\text{.}
  9. { [ x 1 x 2 x n ] K n : x 1 = a 11 t 1 + a 12 t 2 + + a 1 m t m x 2 = a 21 t 1 + a 22 t 2 + + a 2 m t m x n = a n 1 t 1 + a n 2 t 2 + + a n m t m for some t 1 , , t m K } . \left\{\left[\!\!\begin{array}[]{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array}\!\!\right]\in K^{n}:\begin{aligned}\displaystyle x_{1}&&% \displaystyle\;=&&\displaystyle a_{11}t_{1}&&\displaystyle\;+&&\displaystyle a% _{12}t_{2}&&\displaystyle\;+\cdots+&&\displaystyle a_{1m}t_{m}&\\ \displaystyle x_{2}&&\displaystyle\;=&&\displaystyle a_{21}t_{1}&&% \displaystyle\;+&&\displaystyle a_{22}t_{2}&&\displaystyle\;+\cdots+&&% \displaystyle a_{2m}t_{m}&\\ \displaystyle\vdots&&&&\displaystyle\vdots&&&&\displaystyle\vdots&&&&% \displaystyle\vdots&\\ \displaystyle x_{n}&&\displaystyle\;=&&\displaystyle a_{n1}t_{1}&&% \displaystyle\;+&&\displaystyle a_{n2}t_{2}&&\displaystyle\;+\cdots+&&% \displaystyle a_{nm}t_{m}&\\ \end{aligned}\,\text{ for some }t_{1},\ldots,t_{m}\in K\right\}.
  10. x = 2 t 1 + 3 t 2 , y = 5 t 1 - 4 t 2 , and z = - t 1 + 2 t 2 x=2t_{1}+3t_{2},\;\;\;\;y=5t_{1}-4t_{2},\;\;\;\;\,\text{and}\;\;\;\;z=-t_{1}+2% t_{2}
  11. [ x y z ] = t 1 [ 2 5 - 1 ] + t 2 [ 3 - 4 2 ] . \begin{bmatrix}x&\\ y&\\ z&\end{bmatrix}\;=\;t_{1}\!\begin{bmatrix}2&\\ 5&\\ -1&\end{bmatrix}+t_{2}\!\begin{bmatrix}3&\\ -4&\\ 2&\end{bmatrix}.
  12. t 1 𝐯 1 + + t k 𝐯 k . t_{1}\mathbf{v}_{1}+\cdots+t_{k}\mathbf{v}_{k}.
  13. Span { 𝐯 1 , , 𝐯 k } = { t 1 𝐯 1 + + t k 𝐯 k : t 1 , , t k K } . \,\text{Span}\{\mathbf{v}_{1},\ldots,\mathbf{v}_{k}\}=\left\{t_{1}\mathbf{v}_{% 1}+\cdots+t_{k}\mathbf{v}_{k}:t_{1},\ldots,t_{k}\in K\right\}.
  14. x = t 1 , y = 0 , z = t 2 . x=t_{1},\;\;\;y=0,\;\;\;z=t_{2}.
  15. ( t 1 , 0 , t 2 ) = t 1 ( 1 , 0 , 0 ) + t 2 ( 0 , 0 , 1 ) . (t_{1},0,t_{2})=t_{1}(1,0,0)+t_{2}(0,0,1)\,\text{.}\,
  16. 𝐱 = A 𝐭 where A = [ 2 3 5 - 4 - 1 2 ] . \mathbf{x}=A\mathbf{t}\;\;\;\;\,\text{where}\;\;\;\;A=\left[\begin{aligned}% \displaystyle 2&&\displaystyle 3&\\ \displaystyle 5&&\displaystyle\;\;-4&\\ \displaystyle-1&&\displaystyle 2&\end{aligned}\,\right]\,\text{.}
  17. t 1 𝐯 1 + + t k 𝐯 k u 1 𝐯 1 + + u k 𝐯 k t_{1}\mathbf{v}_{1}+\cdots+t_{k}\mathbf{v}_{k}\;\neq\;u_{1}\mathbf{v}_{1}+% \cdots+u_{k}\mathbf{v}_{k}
  18. x 1 = 2 x 2 and x 3 = 5 x 4 . x_{1}=2x_{2}\;\;\;\;\,\text{and}\;\;\;\;x_{3}=5x_{4}.
  19. ( 2 t 1 , t 1 , 5 t 2 , t 2 ) = t 1 ( 2 , 1 , 0 , 0 ) + t 2 ( 0 , 0 , 5 , 1 ) . (2t_{1},t_{1},5t_{2},t_{2})=t_{1}(2,1,0,0)+t_{2}(0,0,5,1).\,
  20. U + W = { 𝐮 + 𝐰 : 𝐮 U , 𝐰 W } . U+W=\left\{\mathbf{u}+\mathbf{w}\colon\mathbf{u}\in U,\mathbf{w}\in W\right\}.
  21. max ( dim U , dim W ) dim ( U + W ) dim ( U ) + dim ( W ) . \max(\dim U,\dim W)\leq\dim(U+W)\leq\dim(U)+\dim(W).
  22. dim ( U + W ) = dim ( U ) + dim ( W ) - dim ( U W ) . \dim(U+W)=\dim(U)+\dim(W)-\dim(U\cap W).
  23. U U
  24. W W
  25. V V
  26. U + W U+W
  27. U W U\cap W
  28. [ 1 0 - 3 0 2 0 0 1 5 0 - 1 4 0 0 0 1 7 - 9 0 0 0 0 0 0 ] \left[\begin{aligned}\displaystyle 1&&\displaystyle 0&&\displaystyle-3&&% \displaystyle 0&&\displaystyle 2&&\displaystyle 0\\ \displaystyle 0&&\displaystyle 1&&\displaystyle 5&&\displaystyle 0&&% \displaystyle-1&&\displaystyle 4\\ \displaystyle 0&&\displaystyle 0&&\displaystyle 0&&\displaystyle 1&&% \displaystyle 7&&\displaystyle-9\\ \displaystyle 0&&\displaystyle\;\;\;\;\;0&&\displaystyle\;\;\;\;\;0&&% \displaystyle\;\;\;\;\;0&&\displaystyle\;\;\;\;\;0&&\displaystyle\;\;\;\;\;0% \end{aligned}\,\right]
  29. 𝐜 3 = - 3 𝐜 1 + 5 𝐜 2 𝐜 5 = 2 𝐜 1 - 𝐜 2 + 7 𝐜 4 𝐜 6 = 4 𝐜 2 - 9 𝐜 4 \begin{aligned}\displaystyle\mathbf{c}_{3}&\displaystyle=-3\mathbf{c}_{1}+5% \mathbf{c}_{2}\\ \displaystyle\mathbf{c}_{5}&\displaystyle=2\mathbf{c}_{1}-\mathbf{c}_{2}+7% \mathbf{c}_{4}\\ \displaystyle\mathbf{c}_{6}&\displaystyle=4\mathbf{c}_{2}-9\mathbf{c}_{4}\end{aligned}
  30. x 3 = - 3 x 1 + 5 x 2 x 5 = 2 x 1 - x 2 + 7 x 4 x 6 = 4 x 2 - 9 x 4 . \begin{aligned}\displaystyle x_{3}&\displaystyle=-3x_{1}+5x_{2}\\ \displaystyle x_{5}&\displaystyle=2x_{1}-x_{2}+7x_{4}\\ \displaystyle x_{6}&\displaystyle=4x_{2}-9x_{4}.\end{aligned}

Linked_list.html

  1. n n
  2. O ( n ) O(n)
  3. K + B * n K+B*n
  4. K K
  5. B B
  6. n n
  7. K K
  8. B B

Liouville_number.html

  1. 0 < | x - p q | < 1 q n . 0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{n}}.
  2. x = k = 1 a k b k ! x=\sum_{k=1}^{\infty}\frac{a_{k}}{b^{k!}}\;
  3. x = ( 0. a 1 a 2 000 a 3 00000000000000000 a 4 000... ) b . x=(0.a_{1}a_{2}000a_{3}00000000000000000a_{4}000...)_{b}\;.
  4. q n = b n ! ; p n = q n k = 1 n a k b k ! . q_{n}=b^{n!}\,;\quad p_{n}=q_{n}\sum_{k=1}^{n}\frac{a_{k}}{b^{k!}}\;.
  5. 0 < | x - p n q n | = k = n + 1 a k b k ! k = n + 1 b - 1 b k ! < k = ( n + 1 ) ! b - 1 b k = b - 1 b ( n + 1 ) ! k = 0 1 b k = b - 1 b ( n + 1 ) ! 1 b - 1 = b b ( n + 1 ) ! b n ! b ( n + 1 ) ! = 1 q n n , 0<\left|x-\frac{p_{n}}{q_{n}}\right|=\sum_{k=n+1}^{\infty}\frac{a_{k}}{b^{k!}}% \leq\sum_{k=n+1}^{\infty}\frac{b-1}{b^{k!}}<\sum_{k=(n+1)!}^{\infty}\frac{b-1}% {b^{k}}=\frac{b-1}{b^{(n+1)!}}\sum_{k=0}^{\infty}\frac{1}{b^{k}}=\frac{b-1}{b^% {(n+1)!}}\cdot\frac{1}{b-1}=\frac{b}{b^{(n+1)!}}\leq\frac{b^{n!}}{b^{(n+1)!}}=% \frac{1}{{q_{n}}^{n}}\,,
  6. n n ! = n n ! + n ! - n ! = ( n + 1 ) ! - n ! . n\cdot n!=n\cdot n!+n!-n!=(n+1)!-n!\;.
  7. 0 < | x - p q | < 1 q n . 0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{n}}\,.
  8. | x - p q | = | c d - p q | = | c q d p | d q \left|x-\frac{p}{q}\right|=\left|\frac{c}{d}-\frac{p}{q}\right|=\frac{|cq —dp|% }{dq}
  9. | x - p q | = | c q d p | d q = 0 , \left|x-\frac{p}{q}\right|=\frac{|cq —dp|}{dq}=0\,,
  10. | x - p q | = | c q d p | d q 1 d q \left|x-\frac{p}{q}\right|=\frac{|cq —dp|}{dq}\geq\frac{1}{dq}
  11. | x - p q | 1 d q > 1 2 n - 1 q 1 q n . \left|x-\frac{p}{q}\right|\geq\frac{1}{dq}>\frac{1}{2^{n-1}q}\geq\frac{1}{q^{n% }}\,.
  12. V n , q = p = - ( p q - 1 q n , p q + 1 q n ) V_{n,q}=\bigcup\limits_{p=-\infty}^{\infty}\left(\frac{p}{q}-\frac{1}{q^{n}},% \frac{p}{q}+\frac{1}{q^{n}}\right)
  13. L q = 2 V n , q . L\subseteq\bigcup\limits_{q=2}^{\infty}V_{n,q}.
  14. L ( - m , m ) q = 2 V n , q ( - m , m ) q = 2 p = - m q m q ( p q - 1 q n , p q + 1 q n ) . L\cap(-m,m)\subseteq\bigcup\limits_{q=2}^{\infty}V_{n,q}\cap(-m,m)\subseteq% \bigcup\limits_{q=2}^{\infty}\bigcup\limits_{p=-mq}^{mq}\left(\frac{p}{q}-% \frac{1}{q^{n}},\frac{p}{q}+\frac{1}{q^{n}}\right).
  15. | ( p q + 1 q n ) - ( p q - 1 q n ) | = 2 q n \left|\left(\frac{p}{q}+\frac{1}{q^{n}}\right)-\left(\frac{p}{q}-\frac{1}{q^{n% }}\right)\right|=\frac{2}{q^{n}}
  16. m ( L ( - m , m ) ) q = 2 p = - m q m q 2 q n = q = 2 2 ( 2 m q + 1 ) q n ( 4 m + 1 ) q = 2 1 q n - 1 ( 4 m + 1 ) 1 d q q n - 1 4 m + 1 n - 2 . m(L\cap(-m,\,m))\leq\sum\limits_{q=2}^{\infty}\sum_{p=-mq}^{mq}\frac{2}{q^{n}}% =\sum\limits_{q=2}^{\infty}\frac{2(2mq+1)}{q^{n}}\leq(4m+1)\sum\limits_{q=2}^{% \infty}\frac{1}{q^{n-1}}\leq(4m+1)\int^{\infty}_{1}\frac{dq}{q^{n-1}}\leq\frac% {4m+1}{n-2}.
  17. lim n 4 m + 1 n - 2 = 0 \lim_{n\to\infty}\frac{4m+1}{n-2}=0
  18. U n = q = 2 p = - { x 𝐑 : 0 < | x - p q | < 1 q n } = q = 2 p = - ( p q - 1 q n , p q + 1 q n ) { p q } U_{n}=\bigcup\limits_{q=2}^{\infty}\bigcup\limits_{p=-\infty}^{\infty}\left\{x% \in\mathbf{R}:0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{n}}\right\}=\bigcup% \limits_{q=2}^{\infty}\bigcup\limits_{p=-\infty}^{\infty}\left(\frac{p}{q}-% \frac{1}{q^{n}},\frac{p}{q}+\frac{1}{q^{n}}\right)\setminus\left\{\frac{p}{q}\right\}
  19. L = n = 1 U n . L=\bigcap\limits_{n=1}^{\infty}U_{n}.
  20. 0 < | x - p q | < 1 q μ 0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{\mu}}
  21. 1 10 n | x - p q | 1 q μ \frac{1}{10^{n}}\geq\left|x-\frac{p}{q}\right|\geq\frac{1}{q^{\mu}}
  22. | α - p q | > A q n \left|\alpha-\frac{p}{q}\right|>\frac{A}{q^{n}}
  23. A < min ( 1 , 1 M , | α - α 1 | , | α - α 2 | , , | α - α m | ) A<\min\left(1,\frac{1}{M},\left|\alpha-\alpha_{1}\right|,\left|\alpha-\alpha_{% 2}\right|,\ldots,\left|\alpha-\alpha_{m}\right|\right)
  24. | α - p q | A q n A < min ( 1 , 1 M , | α - α 1 | , | α - α 2 | , , | α - α m | ) \left|\alpha-\frac{p}{q}\right|\leq\frac{A}{q^{n}}\leq A<\min\left(1,\frac{1}{% M},\left|\alpha-\alpha_{1}\right|,\left|\alpha-\alpha_{2}\right|,\ldots,\left|% \alpha-\alpha_{m}\right|\right)
  25. f ( α ) - f ( p q ) = ( α - p q ) f ( x 0 ) f(\alpha)-f(\tfrac{p}{q})=(\alpha-\frac{p}{q})\cdot f^{\prime}(x_{0})
  26. | α - p q | = | f ( α ) - f ( p q ) | | f ( x 0 ) | = | f ( p q ) f ( x 0 ) | \left|\alpha-\frac{p}{q}\right|=\frac{\left|f(\alpha)-f(\tfrac{p}{q})\right|}{% |f^{\prime}(x_{0})|}=\left|\frac{f(\tfrac{p}{q})}{f^{\prime}(x_{0})}\right|
  27. i = 0 n \sum_{i=0}^{n}
  28. | f ( p q ) | = | i = 0 n c i p i q - i | = 1 q n | i = 0 n c i p i q n - i | 1 q n \left|f\left(\frac{p}{q}\right)\right|=\left|\sum_{i=0}^{n}c_{i}p^{i}q^{-i}% \right|=\frac{1}{q^{n}}\left|\sum_{i=0}^{n}c_{i}p^{i}q^{n-i}\right|\geq\frac{1% }{q^{n}}
  29. | α - p q | = | f ( p q ) f ( x 0 ) | 1 M q n > A q n | α - p q | \left|\alpha-\frac{p}{q}\right|=\left|\frac{f(\tfrac{p}{q})}{f^{\prime}(x_{0})% }\right|\geq\frac{1}{Mq^{n}}>\frac{A}{q^{n}}\geq\left|\alpha-\frac{p}{q}\right|
  30. | x - p q | > A q n \left|x-\frac{p}{q}\right|>\frac{A}{q^{n}}
  31. | x - a b | < 1 b m = 1 b r + n = 1 b r b n 1 2 r 1 b n A b n \left|x-\frac{a}{b}\right|<\frac{1}{b^{m}}=\frac{1}{b^{r+n}}=\frac{1}{b^{r}b^{% n}}\leq\frac{1}{2^{r}}\frac{1}{b^{n}}\leq\frac{A}{b^{n}}

Lipschitz_continuity.html

  1. d Y ( f ( x 1 ) , f ( x 2 ) ) K d X ( x 1 , x 2 ) . d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2}).
  2. d Y ( f ( x 1 ) , f ( x 2 ) ) d X ( x 1 , x 2 ) K . \frac{d_{Y}(f(x_{1}),f(x_{2}))}{d_{X}(x_{1},x_{2})}\leq K.
  3. d Y ( f ( x ) , f ( y ) ) M d X ( x , y ) α d_{Y}(f(x),f(y))\leq Md_{X}(x,y)^{\alpha}
  4. 1 K d X ( x 1 , x 2 ) d Y ( f ( x 1 ) , f ( x 2 ) ) K d X ( x 1 , x 2 ) \frac{1}{K}d_{X}(x_{1},x_{2})\leq d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_% {2})
  5. | x | |x|
  6. x \sqrt{x}
  7. D f ( x ) K \|Df(x)\|\leq K
  8. D f , U K \|Df\|_{\infty,U}\leq K
  9. sup α f α \sup_{\alpha}f_{\alpha}
  10. inf α f α \inf_{\alpha}f_{\alpha}
  11. f ~ ( x ) := inf u U { f ( u ) + k d ( x , u ) } , \tilde{f}(x):=\inf_{u\in U}\{f(u)+k\,d(x,u)\},
  12. ( x 1 - x 2 ) T ( F ( x 1 ) - F ( x 2 ) ) C x 1 - x 2 2 (x_{1}-x_{2})^{T}(F(x_{1})-F(x_{2}))\leq C\|x_{1}-x_{2}\|^{2}
  13. { F : 𝐑 2 𝐑 , F ( x , y ) = - 50 ( y - cos ( x ) ) \begin{cases}F:\mathbf{R}^{2}\to\mathbf{R},\\ F(x,y)=-50(y-\cos(x))\end{cases}
  14. F ( x ) = e - x F(x)=e^{-x}

Liquid_crystal.html

  1. S = P 2 ( cos θ ) = 3 cos 2 θ - 1 2 S=\langle P_{2}(\cos\theta)\rangle=\left\langle\frac{3\cos^{2}\theta-1}{2}\right\rangle
  2. θ \theta
  3. ρ ( z ) \rho(z)
  4. ρ ( 𝐫 ) = ρ ( z ) = ρ 0 + ρ 1 cos ( q s z - ϕ ) + \rho(\mathbf{r})=\rho(z)=\rho_{0}+\rho_{1}\cos\left(q_{s}z-\phi\right)+\cdots\,
  5. ψ ( 𝐫 ) = ρ 1 ( 𝐫 ) e i ϕ ( 𝐫 ) \psi(\mathbf{r})=\rho_{1}(\mathbf{r})e^{i\phi(\mathbf{r})}
  6. ρ 0 \rho_{0}
  7. ψ = 0 \psi=0
  8. ψ \psi
  9. σ = cos ( 2 π z i d ) ( 3 2 cos 2 θ i - 1 2 ) \sigma=\left\langle\cos\left(\frac{2\pi z_{i}}{d}\right)\left(\frac{3}{2}\cos^% {2}\theta_{i}-\frac{1}{2}\right)\right\rangle
  10. U i ( θ i , z i ) = - U 0 ( S + α σ cos ( 2 π z i d ) ) ( 3 2 cos 2 θ i - 1 2 ) U_{i}(\theta_{i},z_{i})=-U_{0}\left(S+\alpha\sigma\cos\left(\frac{2\pi z_{i}}{% d}\right)\right)\left(\frac{3}{2}\cos^{2}\theta_{i}-\frac{1}{2}\right)

Lisp_(programming_language).html

  1. O ( n ) O(n)

List_of_agnostics.html

  1. E = m c 2 E=mc^{2}

List_of_equations_in_classical_mechanics.html

  1. m = λ d m=\int\lambda\mathrm{d}\ell
  2. m = σ d S m=\iint\sigma\mathrm{d}S
  3. m = ρ d V m=\iiint\rho\mathrm{d}V\,\!
  4. 𝐦 = 𝐫 m \mathbf{m}=\mathbf{r}m\,\!
  5. x i x_{i}\,\!
  6. 𝐦 = i = 1 N 𝐫 i m i \mathbf{m}=\sum_{i=1}^{N}\mathbf{r}_{\mathrm{i}}m_{i}\,\!
  7. x i x_{i}\,\!
  8. 𝐦 = ρ ( 𝐫 ) x i d 𝐫 \mathbf{m}=\int\rho\left(\mathbf{r}\right)x_{i}\mathrm{d}\mathbf{r}\,\!
  9. 𝐦 i = 𝐫 i m i \mathbf{m}_{\mathrm{i}}=\mathbf{r}_{\mathrm{i}}m_{i}\,\!
  10. 𝐫 com = 1 M i 𝐫 i m i = 1 M i 𝐦 i \mathbf{r}_{\mathrm{com}}=\frac{1}{M}\sum_{i}\mathbf{r}_{\mathrm{i}}m_{i}=% \frac{1}{M}\sum_{i}\mathbf{m}_{\mathrm{i}}\,\!
  11. 𝐫 com = 1 M d 𝐦 = 1 M 𝐫 d m = 1 M 𝐫 ρ d V \mathbf{r}_{\mathrm{com}}=\frac{1}{M}\int\mathrm{d}\mathbf{m}=\frac{1}{M}\int% \mathbf{r}\mathrm{d}m=\frac{1}{M}\int\mathbf{r}\rho\mathrm{d}V\,\!
  12. μ = ( m 1 m 2 ) / ( m 1 + m 2 ) \mu=\left(m_{1}m_{2}\right)/\left(m_{1}+m_{2}\right)\,\!
  13. I = i 𝐦 i 𝐫 i = i | 𝐫 i | 2 m I=\sum_{i}\mathbf{m}_{\mathrm{i}}\cdot\mathbf{r}_{\mathrm{i}}=\sum_{i}\left|% \mathbf{r}_{\mathrm{i}}\right|^{2}m\,\!
  14. I = | 𝐫 | 2 d m = 𝐫 d 𝐦 = | 𝐫 | 2 ρ d V I=\int\left|\mathbf{r}\right|^{2}\mathrm{d}m=\int\mathbf{r}\cdot\mathrm{d}% \mathbf{m}=\int\left|\mathbf{r}\right|^{2}\rho\mathrm{d}V\,\!
  15. 𝐯 = d 𝐫 / d t \mathbf{v}=\mathrm{d}\mathbf{r}/\mathrm{d}t\,\!
  16. 𝐚 = d 𝐯 / d t = d 2 𝐫 / d t 2 \mathbf{a}=\mathrm{d}\mathbf{v}/\mathrm{d}t=\mathrm{d}^{2}\mathbf{r}/\mathrm{d% }t^{2}\,\!
  17. 𝐣 = d 𝐚 / d t = d 3 𝐫 / d t 3 \mathbf{j}=\mathrm{d}\mathbf{a}/\mathrm{d}t=\mathrm{d}^{3}\mathbf{r}/\mathrm{d% }t^{3}\,\!
  18. s y m b o l ω = 𝐧 ^ ( d θ / d t ) symbol{\omega}=\mathbf{\hat{n}}\left(\mathrm{d}\theta/\mathrm{d}t\right)\,\!
  19. s y m b o l α = d s y m b o l ω / d t = 𝐧 ^ ( d 2 θ / d t 2 ) symbol{\alpha}=\mathrm{d}symbol{\omega}/\mathrm{d}t=\mathbf{\hat{n}}\left(% \mathrm{d}^{2}\theta/\mathrm{d}t^{2}\right)\,\!
  20. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}\,\!
  21. 𝐅 = d 𝐩 / d t \mathbf{F}=\mathrm{d}\mathbf{p}/\mathrm{d}t\,\!
  22. 𝐉 = Δ 𝐩 = t 1 t 2 𝐅 d t \mathbf{J}=\Delta\mathbf{p}=\int_{t_{1}}^{t_{2}}\mathbf{F}\mathrm{d}t\,\!
  23. 𝐋 = ( 𝐫 - 𝐫 0 ) × 𝐩 \mathbf{L}=\left(\mathbf{r}-\mathbf{r}_{0}\right)\times\mathbf{p}\,\!
  24. s y m b o l τ = ( 𝐫 - 𝐫 0 ) × 𝐅 = d 𝐋 / d t symbol{\tau}=\left(\mathbf{r}-\mathbf{r}_{0}\right)\times\mathbf{F}=\mathrm{d}% \mathbf{L}/\mathrm{d}t\,\!
  25. Δ 𝐋 = t 1 t 2 s y m b o l τ d t \Delta\mathbf{L}=\int_{t_{1}}^{t_{2}}symbol{\tau}\mathrm{d}t\,\!
  26. W = C 𝐅 d 𝐫 W=\int_{C}\mathbf{F}\cdot\mathrm{d}\mathbf{r}\,\!
  27. Δ W ON = - Δ W BY \Delta W_{\mathrm{ON}}=-\Delta W_{\mathrm{BY}}\,\!
  28. Δ W = - Δ V \Delta W=-\Delta V\,\!
  29. P = d E / d t P=\mathrm{d}E/\mathrm{d}t\,\!
  30. q ˙ , Q ˙ \dot{q},\dot{Q}\,\!
  31. q ˙ d q / d t \dot{q}\equiv\mathrm{d}q/\mathrm{d}t\,\!
  32. p = L / q ˙ p=\partial L/\partial\dot{q}\,\!
  33. L ( 𝐪 , 𝐪 ˙ , t ) = T ( 𝐪 ˙ ) - V ( 𝐪 , 𝐪 ˙ , t ) L(\mathbf{q},\mathbf{\dot{q}},t)=T(\mathbf{\dot{q}})-V(\mathbf{q},\mathbf{\dot% {q}},t)\,\!
  34. 𝐪 = 𝐪 ( t ) \mathbf{q}=\mathbf{q}(t)\,\!
  35. H ( 𝐩 , 𝐪 , t ) = 𝐩 𝐪 ˙ - L ( 𝐪 , 𝐪 ˙ , t ) H(\mathbf{p},\mathbf{q},t)=\mathbf{p}\cdot\mathbf{\dot{q}}-L(\mathbf{q},% \mathbf{\dot{q}},t)\,\!
  36. 𝒮 \scriptstyle{\mathcal{S}}\,\!
  37. 𝒮 = t 1 t 2 L ( 𝐪 , 𝐪 ˙ , t ) d t \mathcal{S}=\int_{t_{1}}^{t_{2}}L(\mathbf{q},\mathbf{\dot{q}},t)\mathrm{d}t\,\!
  38. n ^ = e ^ r × e ^ θ {\hat{n}}={\hat{e}}_{r}\times{\hat{e}}_{\theta}\,\!
  39. e ^ r \scriptstyle{\hat{e}}_{r}\,\!
  40. e ^ θ \scriptstyle{\hat{e}}_{\theta}\,\!
  41. 𝐯 average = Δ 𝐫 Δ t \mathbf{v}_{\mathrm{average}}={\Delta\mathbf{r}\over\Delta t}
  42. 𝐯 = d 𝐫 d t \mathbf{v}={d\mathbf{r}\over dt}
  43. s y m b o l ω = n ^ d θ d t symbol{\omega}={\hat{n}}\frac{{\rm d}\theta}{{\rm d}t}\,\!
  44. 𝐯 = s y m b o l ω × 𝐫 \mathbf{v}=symbol{\omega}\times\mathbf{r}\,\!
  45. 𝐚 average = Δ 𝐯 Δ t \mathbf{a}_{\mathrm{average}}=\frac{\Delta\mathbf{v}}{\Delta t}
  46. 𝐚 = d 𝐯 d t = d 2 𝐫 d t 2 \mathbf{a}=\frac{d\mathbf{v}}{dt}=\frac{d^{2}\mathbf{r}}{dt^{2}}
  47. s y m b o l α = d s y m b o l ω d t = n ^ d 2 θ d t 2 symbol{\alpha}=\frac{{\rm d}symbol{\omega}}{{\rm d}t}={\hat{n}}\frac{{\rm d}^{% 2}\theta}{{\rm d}t^{2}}\,\!
  48. 𝐚 = s y m b o l α × 𝐫 + s y m b o l ω × 𝐯 \mathbf{a}=symbol{\alpha}\times\mathbf{r}+symbol{\omega}\times\mathbf{v}\,\!
  49. 𝐣 average = Δ 𝐚 Δ t \mathbf{j}_{\mathrm{average}}=\frac{\Delta\mathbf{a}}{\Delta t}
  50. 𝐣 = d 𝐚 d t = d 2 𝐯 d t 2 = d 3 𝐫 d t 3 \mathbf{j}=\frac{d\mathbf{a}}{dt}=\frac{d^{2}\mathbf{v}}{dt^{2}}=\frac{d^{3}% \mathbf{r}}{dt^{3}}
  51. s y m b o l ζ = d s y m b o l α d t = n ^ d 2 ω d t 2 = n ^ d 3 θ d t 3 symbol{\zeta}=\frac{{\rm d}symbol{\alpha}}{{\rm d}t}={\hat{n}}\frac{{\rm d}^{2% }\omega}{{\rm d}t^{2}}={\hat{n}}\frac{{\rm d}^{3}\theta}{{\rm d}t^{3}}\,\!
  52. 𝐣 = s y m b o l ζ × 𝐫 + s y m b o l α × 𝐚 \mathbf{j}=symbol{\zeta}\times\mathbf{r}+symbol{\alpha}\times\mathbf{a}\,\!
  53. 𝐩 = m 𝐯 \mathbf{p}=m\mathbf{v}
  54. 𝐩 = s y m b o l ω × 𝐦 \mathbf{p}=symbol{\omega}\times\mathbf{m}\,\!
  55. 𝐋 = 𝐫 × 𝐩 = 𝐈 s y m b o l ω \mathbf{L}=\mathbf{r}\times\mathbf{p}=\mathbf{I}\cdot symbol{\omega}
  56. 𝐅 = d 𝐩 d t = d ( m 𝐯 ) d t = m 𝐚 + 𝐯 d m d t \begin{aligned}\displaystyle\mathbf{F}&\displaystyle=\frac{d\mathbf{p}}{dt}=% \frac{d(m\mathbf{v})}{dt}\\ &\displaystyle=m\mathbf{a}+\mathbf{v}\frac{{\rm d}m}{{\rm d}t}\\ \end{aligned}\,\!
  57. d 𝐩 i d t = 𝐅 E + i j 𝐅 i j \frac{\mathrm{d}\mathbf{p}_{i}}{\mathrm{d}t}=\mathbf{F}_{E}+\sum_{i\neq j}% \mathbf{F}_{ij}\,\!
  58. s y m b o l τ = d 𝐋 d t = 𝐫 × 𝐅 = d ( 𝐈 s y m b o l ω ) d t symbol{\tau}=\frac{{\rm d}\mathbf{L}}{{\rm d}t}=\mathbf{r}\times\mathbf{F}=% \frac{{\rm d}(\mathbf{I}\cdot symbol{\omega})}{{\rm d}t}\,\!
  59. s y m b o l τ = d L d t = d ( I \cdotsymbol ω ) d t = d I d t \cdotsymbol ω + I \cdotsymbol α \begin{aligned}\displaystyle symbol{\tau}&\displaystyle=\frac{{\rm d}{L}}{{\rm d% }t}=\frac{{\rm d}({I}\cdotsymbol{\omega})}{{\rm d}t}\\ &\displaystyle=\frac{{\rm d}{I}}{{\rm d}t}\cdotsymbol{\omega}+{I}\cdotsymbol{% \alpha}\\ \end{aligned}\,\!
  60. d 𝐋 i d t = s y m b o l τ E + i j s y m b o l τ i j \frac{\mathrm{d}\mathbf{L}_{i}}{\mathrm{d}t}=symbol{\tau}_{E}+\sum_{i\neq j}% symbol{\tau}_{ij}\,\!
  61. 𝐘 = d 𝐅 d t = d 2 𝐩 d t 2 = d 2 ( m 𝐯 ) d t 2 = m 𝐣 + 𝟐 𝐚 d m d t + 𝐯 d 2 m d t 2 \begin{aligned}\displaystyle\mathbf{Y}&\displaystyle=\frac{d\mathbf{F}}{dt}=% \frac{d^{2}\mathbf{p}}{dt^{2}}=\frac{d^{2}(m\mathbf{v})}{dt^{2}}\\ &\displaystyle=m\mathbf{j}+\mathbf{2a}\frac{{\rm d}m}{{\rm d}t}+\mathbf{v}% \frac{{\rm d^{2}}m}{{\rm d}t^{2}}\\ \end{aligned}\,\!
  62. 𝐘 = m 𝐣 \mathbf{Y}=m\mathbf{j}
  63. s y m b o l \Rho = d τ d t = 𝐫 × 𝐘 = d ( 𝐈 s y m b o l α ) d t symbol{\Rho}=\frac{{\rm d}\mathbf{\tau}}{{\rm d}t}=\mathbf{r}\times\mathbf{Y}=% \frac{{\rm d}(\mathbf{I}\cdot symbol{\alpha})}{{\rm d}t}\,\!
  64. Δ 𝐩 = 𝐅 d t \Delta\mathbf{p}=\int\mathbf{F}dt
  65. Δ 𝐩 = 𝐅 Δ t \Delta\mathbf{p}=\mathbf{F}\Delta t
  66. Δ 𝐋 = s y m b o l τ d t \Delta\mathbf{L}=\int symbol{\tau}dt
  67. Δ 𝐋 = s y m b o l τ Δ t \Delta\mathbf{L}=symbol{\tau}\Delta t
  68. s y m b o l Ω = w r I s y m b o l ω symbol{\Omega}=\frac{wr}{Isymbol{\omega}}
  69. W = Δ T = C ( 𝐅 d 𝐫 + s y m b o l τ 𝐧 d θ ) W=\Delta T=\int_{C}\left(\mathbf{F}\cdot\mathrm{d}\mathbf{r}+symbol{\tau}\cdot% \mathbf{n}{\mathrm{d}\theta}\right)\,\!
  70. Δ E k = W = 1 2 m ( v 2 - v 0 2 ) \Delta E_{k}=W=\frac{1}{2}m(v^{2}-{v_{0}}^{2})
  71. Δ E p = 1 2 k ( r 2 - r 1 ) 2 \Delta E_{p}=\frac{1}{2}k(r_{2}-r_{1})^{2}\,\!
  72. 𝐈 s y m b o l α + s y m b o l ω × ( 𝐈 s y m b o l ω ) = s y m b o l τ \mathbf{I}\cdot symbol{\alpha}+symbol{\omega}\times\left(\mathbf{I}\cdot symbol% {\omega}\right)=symbol{\tau}\,\!
  73. 𝐫 = r ( t ) = r e ^ r \mathbf{r}={r}(t)=r{\hat{e}}_{r}\,\!
  74. 𝐫 = r ( r , θ , t ) = r e ^ r \mathbf{r}={r}\left(r,\theta,t\right)=r{\hat{e}}_{r}
  75. 𝐯 = e ^ r d r d t + r ω e ^ θ \mathbf{v}={\hat{e}}_{r}\frac{\mathrm{d}r}{\mathrm{d}t}+r\omega{\hat{e}}_{\theta}
  76. 𝐩 = m ( e ^ r d r d t + r ω e ^ θ ) \mathbf{p}=m\left({\hat{e}}_{r}\frac{\mathrm{d}r}{\mathrm{d}t}+r\omega{\hat{e}% }_{\theta}\right)
  77. 𝐋 = m r × ( e ^ r d r d t + r ω e ^ θ ) \mathbf{L}=m{r}\times\left({\hat{e}}_{r}\frac{\mathrm{d}r}{\mathrm{d}t}+r% \omega{\hat{e}}_{\theta}\right)
  78. 𝐚 = ( d 2 r d t 2 - r ω 2 ) e ^ r + ( r α + 2 ω d r d t ) e ^ θ \mathbf{a}=\left(\frac{\mathrm{d}^{2}r}{\mathrm{d}t^{2}}-r\omega^{2}\right){% \hat{e}}_{r}+\left(r\alpha+2\omega\frac{\mathrm{d}r}{{\rm d}t}\right){\hat{e}}% _{\theta}
  79. 𝐅 = - m ω 2 R e ^ r = - ω 2 𝐦 \mathbf{F}_{\bot}=-m\omega^{2}R{\hat{e}}_{r}=-\omega^{2}\mathbf{m}\,\!
  80. 𝐅 c = 2 ω m d r d t e ^ θ = 2 ω m v e ^ θ \mathbf{F}_{c}=2\omega m\frac{{\rm d}r}{{\rm d}t}{\hat{e}}_{\theta}=2\omega mv% {\hat{e}}_{\theta}\,\!
  81. 𝐅 c = m 𝐚 c = - 2 m s y m b o l ω × v \mathbf{F}_{c}=m\mathbf{a}_{c}=-2msymbol{\omega\times v}
  82. d 2 d θ 2 ( 1 𝐫 ) + 1 𝐫 = - μ 𝐫 2 𝐥 2 𝐅 ( 𝐫 ) \frac{d^{2}}{d\theta^{2}}\left(\frac{1}{\mathbf{r}}\right)+\frac{1}{\mathbf{r}% }=-\frac{\mu\mathbf{r}^{2}}{\mathbf{l}^{2}}\mathbf{F}(\mathbf{r})
  83. v = v 0 + a t v=v_{0}+at\,
  84. ω 1 = ω 0 + α t \omega_{1}=\omega_{0}+\alpha t\,
  85. s = 1 2 ( v 0 + v ) t s=\frac{1}{2}(v_{0}+v)t
  86. θ = 1 2 ( ω 0 + ω 1 ) t \theta=\frac{1}{2}(\omega_{0}+\omega_{1})t
  87. s = v 0 t + 1 2 a t 2 s=v_{0}t+\frac{1}{2}at^{2}
  88. θ = ω 0 t + 1 2 α t 2 \theta=\omega_{0}t+\frac{1}{2}\alpha t^{2}
  89. v 2 = v 0 2 + 2 a s v^{2}=v_{0}^{2}+2as\,
  90. ω 1 2 = ω 0 2 + 2 α θ \omega_{1}^{2}=\omega_{0}^{2}+2\alpha\theta
  91. s = v t - 1 2 a t 2 s=vt-\frac{1}{2}at^{2}
  92. θ = ω 1 t - 1 2 α t 2 \theta=\omega_{1}t-\frac{1}{2}\alpha t^{2}
  93. 𝐫 = 𝐫 + 𝐕 t \mathbf{r}^{\prime}=\mathbf{r}+\mathbf{V}t\,\!
  94. 𝐯 = 𝐯 + 𝐕 \mathbf{v}^{\prime}=\mathbf{v}+\mathbf{V}\,\!
  95. 𝐚 = 𝐚 \mathbf{a}^{\prime}=\mathbf{a}
  96. 𝐚 = 𝐚 + 𝐀 \mathbf{a}^{\prime}=\mathbf{a}+\mathbf{A}
  97. 𝐅 = 𝐅 - 𝐅 app \mathbf{F}^{\prime}=\mathbf{F}-\mathbf{F}_{\mathrm{app}}
  98. θ = θ + Ω t \theta^{\prime}=\theta+\Omega t\,\!
  99. s y m b o l ω = s y m b o l ω + s y m b o l Ω symbol{\omega}^{\prime}=symbol{\omega}+symbol{\Omega}\,\!
  100. s y m b o l α = s y m b o l α symbol{\alpha}^{\prime}=symbol{\alpha}
  101. s y m b o l α = s y m b o l α + s y m b o l Λ symbol{\alpha}^{\prime}=symbol{\alpha}+symbol{\Lambda}
  102. s y m b o l τ = s y m b o l τ - s y m b o l τ app symbol{\tau}^{\prime}=symbol{\tau}-symbol{\tau}_{\mathrm{app}}
  103. d 𝐓 d t = d 𝐓 d t - s y m b o l Ω × 𝐓 \frac{{\rm d}\mathbf{T}^{\prime}}{{\rm d}t}=\frac{{\rm d}\mathbf{T}}{{\rm d}t}% -symbol{\Omega}\times\mathbf{T}
  104. d 2 x d t 2 = - ω 2 x \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=-\omega^{2}x\,\!
  105. x = A sin ( ω t + ϕ ) x=A\sin\left(\omega t+\phi\right)\,\!
  106. d 2 θ d t 2 = - ω 2 θ \frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}=-\omega^{2}\theta\,\!
  107. θ = Θ sin ( ω t + ϕ ) \theta=\Theta\sin\left(\omega t+\phi\right)\,\!
  108. d 2 x d t 2 + b d x d t + ω 2 x = 0 \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+b\frac{\mathrm{d}x}{\mathrm{d}t}+% \omega^{2}x=0\,\!
  109. x = A e - b t / 2 m cos ( ω ) x=Ae^{-bt/2m}\cos\left(\omega^{\prime}\right)\,\!
  110. ω res = ω 2 - ( b 4 m ) 2 \omega_{\mathrm{res}}=\sqrt{\omega^{2}-\left(\frac{b}{4m}\right)^{2}}\,\!
  111. γ = b / m \gamma=b/m\,\!
  112. τ = 1 / γ \tau=1/\gamma\,\!
  113. d 2 θ d t 2 + b d θ d t + ω 2 θ = 0 \frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}+b\frac{\mathrm{d}\theta}{\mathrm{% d}t}+\omega^{2}\theta=0\,\!
  114. θ = Θ e - κ t / 2 m cos ( ω ) \theta=\Theta e^{-\kappa t/2m}\cos\left(\omega\right)\,\!
  115. ω res = ω 2 - ( κ 4 m ) 2 \omega_{\mathrm{res}}=\sqrt{\omega^{2}-\left(\frac{\kappa}{4m}\right)^{2}}\,\!
  116. γ = κ / m \gamma=\kappa/m\,\!
  117. τ = 1 / γ \tau=1/\gamma\,\!
  118. ω = k m \omega=\sqrt{\frac{k}{m}}\,\!
  119. ω = k m - ( b 2 m ) 2 \omega^{\prime}=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}\right)^{2}}\,\!
  120. ω = κ I \omega=\sqrt{\frac{\kappa}{I}}\,\!
  121. ω = g L \omega=\sqrt{\frac{g}{L}}\,\!
  122. ω = g L [ 1 + k = 1 n = 1 k ( 2 n - 1 ) n = 1 m ( 2 n ) sin 2 n Θ ] \omega=\sqrt{\frac{g}{L}}\left[1+\sum_{k=1}^{\infty}\frac{\prod_{n=1}^{k}\left% (2n-1\right)}{\prod_{n=1}^{m}\left(2n\right)}\sin^{2n}\Theta\right]\,\!
  123. U = m 2 ( x ) 2 = m ( ω A ) 2 2 cos 2 ( ω t + ϕ ) U=\frac{m}{2}\left(x\right)^{2}=\frac{m\left(\omega A\right)^{2}}{2}\cos^{2}(% \omega t+\phi)\,\!
  124. U max m 2 ( ω A ) 2 U_{\mathrm{max}}\frac{m}{2}\left(\omega A\right)^{2}\,\!
  125. T = ω 2 m 2 ( d x d t ) 2 = m ( ω A ) 2 2 sin 2 ( ω t + ϕ ) T=\frac{\omega^{2}m}{2}\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^{2}=\frac{% m\left(\omega A\right)^{2}}{2}\sin^{2}\left(\omega t+\phi\right)\,\!
  126. E = T + U E=T+U\,\!
  127. E = m ( ω A ) 2 2 e - b t / m E=\frac{m\left(\omega A\right)^{2}}{2}e^{-bt/m}\,\!

List_of_Fourier-related_transforms.html

  1. n = - f ( n T ) δ ( t - n T ) , \scriptstyle\sum_{n=-\infty}^{\infty}f(nT)\cdot\delta(t-nT),\,

List_of_logarithmic_identities.html

  1. log b ( 1 ) = 0 \log_{b}(1)=0\!\,
  2. b 0 = 1 b^{0}=1\!\,
  3. log b ( b ) = 1 \log_{b}(b)=1\!\,
  4. b 1 = b b^{1}=b\!\,
  5. b log b ( x ) = x because antilog b ( log b ( x ) ) = x b^{\log_{b}(x)}=x\,\text{ because }\operatorname{antilog}_{b}(\log_{b}(x))=x\,
  6. log b ( b x ) = x because log b ( antilog b ( x ) ) = x \log_{b}(b^{x})=x\,\text{ because }\log_{b}(\operatorname{antilog}_{b}(x))=x\,
  7. b c = x , log b ( x ) = c b^{c}=x\,\text{, }\log_{b}(x)=c\,
  8. log b ( x y ) = log b ( x ) + log b ( y ) \log_{b}(xy)=\log_{b}(x)+\log_{b}(y)\!\,
  9. b c b d = b c + d b^{c}\cdot b^{d}=b^{c+d}\!\,
  10. log b ( x y ) = log b ( x ) - log b ( y ) \log_{b}\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right)=\log_{b}(x)-\log_% {b}(y)
  11. b c - d = b c b d b^{c-d}=\tfrac{b^{c}}{b^{d}}
  12. log b ( x d ) = d log b ( x ) \log_{b}(x^{d})=d\log_{b}(x)\!\,
  13. ( b c ) d = b c d (b^{c})^{d}=b^{cd}\!\,
  14. log b ( x y ) = log b ( x ) y \log_{b}\!\left(\!\sqrt[y]{x}\right)=\begin{matrix}\frac{\log_{b}(x)}{y}\end{matrix}
  15. x y = x 1 / y \sqrt[y]{x}=x^{1/y}
  16. x log b ( y ) = y log b ( x ) x^{\log_{b}(y)}=y^{\log_{b}(x)}\!\,
  17. x log b ( y ) = b log b ( x ) log b ( y ) = b log b ( y ) log b ( x ) = y log b ( x ) x^{\log_{b}(y)}=b^{\log_{b}(x)\log_{b}(y)}=b^{\log_{b}(y)\log_{b}(x)}=y^{\log_% {b}(x)}\!\,
  18. c log b ( x ) + d log b ( y ) = log b ( x c y d ) c\log_{b}(x)+d\log_{b}(y)=\log_{b}(x^{c}y^{d})\!\,
  19. log b ( x c y d ) = log b ( x c ) + log b ( y d ) \log_{b}(x^{c}y^{d})=\log_{b}(x^{c})+\log_{b}(y^{d})\!\,
  20. b b
  21. x x
  22. y y
  23. b 1 b\neq 1
  24. c c
  25. d d
  26. x y = b log b ( x ) b log b ( y ) = b log b ( x ) + log b ( y ) log b ( x y ) = log b ( b log b ( x ) + log b ( y ) ) = log b ( x ) + log b ( y ) xy=b^{\log_{b}(x)}b^{\log_{b}(y)}=b^{\log_{b}(x)+\log_{b}(y)}\Rightarrow\log_{% b}(xy)=\log_{b}(b^{\log_{b}(x)+\log_{b}(y)})=\log_{b}(x)+\log_{b}(y)
  27. x y = ( b log b ( x ) ) y = b y log b ( x ) log b ( x y ) = y log b ( x ) x^{y}=(b^{\log_{b}(x)})^{y}=b^{y\log_{b}(x)}\Rightarrow\log_{b}(x^{y})=y\log_{% b}(x)
  28. log b ( x y ) = log b ( x y - 1 ) = log b ( x ) + log b ( y - 1 ) = log b ( x ) - log b ( y ) \log_{b}\bigg(\frac{x}{y}\bigg)=\log_{b}(xy^{-1})=\log_{b}(x)+\log_{b}(y^{-1})% =\log_{b}(x)-\log_{b}(y)
  29. log b ( x y ) = log b ( x 1 y ) = 1 y log b ( x ) \log_{b}(\sqrt[y]{x})=\log_{b}(x^{\frac{1}{y}})=\frac{1}{y}\log_{b}(x)
  30. log b a = log d a log d b \log_{b}a={\log_{d}a\over\log_{d}b}
  31. c = log b a c=\log_{b}a
  32. b c = a b^{c}=a
  33. log d \log_{d}
  34. log d b c = log d a \log_{d}b^{c}=\log_{d}a
  35. c c
  36. c log d b = log d a c\log_{d}b=\log_{d}a
  37. c = log d a log d b c=\frac{\log_{d}a}{\log_{d}b}
  38. c = log b a c=\log_{b}a
  39. log b a = log d a log d b \log_{b}a=\frac{\log_{d}a}{\log_{d}b}
  40. log b a = 1 log a b \log_{b}a=\frac{1}{\log_{a}b}
  41. log b n a = log b a n \log_{b^{n}}a={{\log_{b}a}\over n}
  42. b log a d = d log a b b^{\log_{a}d}=d^{\log_{a}b}
  43. - log b a = log b ( 1 a ) = log 1 b a -\log_{b}a=\log_{b}\left({1\over a}\right)=\log_{1\over b}a
  44. log b 1 a 1 log b n a n = log b π ( 1 ) a 1 log b π ( n ) a n , \log_{b_{1}}a_{1}\,\cdots\,\log_{b_{n}}a_{n}=\log_{b_{\pi(1)}}a_{1}\,\cdots\,% \log_{b_{\pi(n)}}a_{n},\,
  45. π \scriptstyle\pi\,
  46. log b w log a x log d c log d z = log d w log b x log a c log d z . \log_{b}w\cdot\log_{a}x\cdot\log_{d}c\cdot\log_{d}z=\log_{d}w\cdot\log_{b}x% \cdot\log_{a}c\cdot\log_{d}z.\,
  47. log b ( a + c ) = log b a + log b ( 1 + c a ) \log_{b}(a+c)=\log_{b}a+\log_{b}\left(1+\frac{c}{a}\right)
  48. log b ( a - c ) = log b a + log b ( 1 - c a ) \log_{b}(a-c)=\log_{b}a+\log_{b}\left(1-\frac{c}{a}\right)
  49. a a
  50. c c
  51. c > a c>a
  52. a = c a=c
  53. log b i = 0 N a i = log b a 0 + log b ( 1 + i = 1 N a i a 0 ) = log b a 0 + log b ( 1 + i = 1 N b ( log b a i - log b a 0 ) ) \log_{b}\sum\limits_{i=0}^{N}a_{i}=\log_{b}a_{0}+\log_{b}\left(1+\sum\limits_{% i=1}^{N}\frac{a_{i}}{a_{0}}\right)=\log_{b}a_{0}+\log_{b}\left(1+\sum\limits_{% i=1}^{N}b^{\left(\log_{b}a_{i}-\log_{b}a_{0}\right)}\right)
  54. a 0 > a 1 > > a N a_{0}>a_{1}>\ldots>a_{N}
  55. x log ( log ( x ) ) log ( x ) = log ( x ) x^{\frac{\log(\log(x))}{\log(x)}}=\log(x)
  56. lim x 0 + log a x = - if a > 1 \lim_{x\to 0^{+}}\log_{a}x=-\infty\quad\mbox{if }~{}a>1
  57. lim x 0 + log a x = + if a < 1 \lim_{x\to 0^{+}}\log_{a}x=+\infty\quad\mbox{if }~{}a<1
  58. lim x + log a x = + if a > 1 \lim_{x\to+\infty}\log_{a}x=+\infty\quad\mbox{if }~{}a>1
  59. lim x + log a x = - if a < 1 \lim_{x\to+\infty}\log_{a}x=-\infty\quad\mbox{if }~{}a<1
  60. lim x 0 + x b log a x = 0 if b > 0 \lim_{x\to 0^{+}}x^{b}\log_{a}x=0\quad\mbox{if }~{}b>0
  61. lim x + 1 x b log a x = 0 if b > 0 \lim_{x\to+\infty}{1\over x^{b}}\log_{a}x=0\quad\mbox{if }~{}b>0
  62. d d x ln x = 1 x , {d\over dx}\ln x={1\over x},
  63. d d x log b x = 1 x ln b , {d\over dx}\log_{b}x={1\over x\ln b},
  64. x > 0 x>0
  65. b > 0 b>0
  66. b 1 b\neq 1
  67. ln x = 1 x 1 t d t \ln x=\int_{1}^{x}\frac{1}{t}dt
  68. log a x d x = x ( log a x - log a e ) + C \int\log_{a}x\,dx=x(\log_{a}x-\log_{a}e)+C
  69. x [ n ] = x n ( log ( x ) - H n ) x^{\left[n\right]}=x^{n}(\log(x)-H_{n})
  70. H n H_{n}
  71. x [ 0 ] = log x x^{\left[0\right]}=\log x
  72. x [ 1 ] = x log ( x ) - x x^{\left[1\right]}=x\log(x)-x
  73. x [ 2 ] = x 2 log ( x ) - 3 2 x 2 x^{\left[2\right]}=x^{2}\log(x)-\begin{matrix}\frac{3}{2}\end{matrix}\,x^{2}
  74. x [ 3 ] = x 3 log ( x ) - 11 6 x 3 x^{\left[3\right]}=x^{3}\log(x)-\begin{matrix}\frac{11}{6}\end{matrix}\,x^{3}
  75. d d x x [ n ] = n x [ n - 1 ] \frac{d}{dx}\,x^{\left[n\right]}=n\,x^{\left[n-1\right]}
  76. x [ n ] d x = x [ n + 1 ] n + 1 + C \int x^{\left[n\right]}\,dx=\frac{x^{\left[n+1\right]}}{n+1}+C
  77. Log ( z ) = ln ( | z | ) + i Arg ( z ) \operatorname{Log}(z)=\ln(|z|)+i\operatorname{Arg}(z)
  78. e Log ( z ) = z e^{\operatorname{Log}(z)}=z
  79. log ( z ) = ln ( | z | ) + i arg ( z ) \log(z)=\ln(|z|)+i\arg(z)
  80. log ( z ) = Log ( z ) + 2 π i k \log(z)=\operatorname{Log}(z)+2\pi ik
  81. e log ( z ) = z e^{\log(z)}=z
  82. Log ( 1 ) = 0 \operatorname{Log}(1)=0
  83. Log ( e ) = 1 \operatorname{Log}(e)=1
  84. log ( 1 ) = 0 + 2 π i k \log(1)=0+2\pi ik
  85. log ( e ) = 1 + 2 π i k \log(e)=1+2\pi ik
  86. Log ( z 1 ) + Log ( z 2 ) = Log ( z 1 z 2 ) ( mod 2 π i ) \operatorname{Log}(z_{1})+\operatorname{Log}(z_{2})=\operatorname{Log}(z_{1}z_% {2})\;\;(\mathop{{\rm mod}}2\pi i)
  87. Log ( z 1 ) - Log ( z 2 ) = Log ( z 1 / z 2 ) ( mod 2 π i ) \operatorname{Log}(z_{1})-\operatorname{Log}(z_{2})=\operatorname{Log}(z_{1}/z% _{2})\;\;(\mathop{{\rm mod}}2\pi i)
  88. log ( z 1 ) + log ( z 2 ) = log ( z 1 z 2 ) \log(z_{1})+\log(z_{2})=\log(z_{1}z_{2})
  89. log ( z 1 ) - log ( z 2 ) = log ( z 1 / z 2 ) \log(z_{1})-\log(z_{2})=\log(z_{1}/z_{2})
  90. z 1 z 2 = e z 2 Log ( z 1 ) {z_{1}}^{z_{2}}=e^{z_{2}\operatorname{Log}(z_{1})}
  91. Log ( z 1 z 2 ) = z 2 Log ( z 1 ) ( mod 2 π i ) \operatorname{Log}{\left({z_{1}}^{z_{2}}\right)}=z_{2}\operatorname{Log}(z_{1}% )\;\;(\mathop{{\rm mod}}2\pi i)
  92. z 1 z 2 = e z 2 log ( z 1 ) {z_{1}}^{z_{2}}=e^{z_{2}\log(z_{1})}
  93. log ( z 1 z 2 ) = z 2 log ( z 1 ) + 2 π i k 2 \log{\left({z_{1}}^{z_{2}}\right)}=z_{2}\log(z_{1})+2\pi ik_{2}
  94. log ( z 1 z 2 ) = z 2 Log ( z 1 ) + z 2 2 π i k 1 + 2 π i k 2 \log{\left({z_{1}}^{z_{2}}\right)}=z_{2}\operatorname{Log}(z_{1})+z_{2}2\pi ik% _{1}+2\pi ik_{2}

List_of_mathematical_functions.html

  1. x 1 2 x^{\frac{1}{2}}\!
  2. x 1 3 x^{\frac{1}{3}}\!

List_of_thermodynamic_properties.html

  1. a a
  2. μ i \mu_{i}
  3. N i N_{i}
  4. β S \beta_{S}
  5. κ \kappa
  6. β T \beta_{T}
  7. κ \kappa
  8. K f K_{f}
  9. ρ \rho
  10. K b K_{b}
  11. H H
  12. h h
  13. S S
  14. T T
  15. s s
  16. f f
  17. R , R ¯ R,\bar{R}
  18. R S R_{S}
  19. G G
  20. g g
  21. Ξ \Xi
  22. Ω \Omega
  23. Q Q
  24. C p C_{p}
  25. c p c_{p}
  26. C v C_{v}
  27. c v c_{v}
  28. A A
  29. F F
  30. Φ \Phi
  31. U U
  32. u u
  33. π T \pi_{T}
  34. m m
  35. N i N_{i}
  36. μ i \mu_{i}
  37. p p
  38. V V
  39. T T
  40. S S
  41. k k
  42. α \alpha
  43. α L \alpha_{L}
  44. α A \alpha_{A}
  45. α V \alpha_{V}
  46. χ \chi
  47. V V
  48. P P
  49. v v
  50. W W

List_of_zeta_functions.html

  1. ζ ( s ) = n = 1 1 n s . \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.

Lithium_carbonate.html

  1. \overrightarrow{\leftarrow}

Loading_coil.html

  1. cosh ( γ d ) = cosh ( γ d ) + Z 2 Z 0 sinh ( γ d ) \cosh(\gamma^{\prime}d)=\cosh(\gamma d)+\frac{Z}{2Z_{0}}\sinh(\gamma d)
  2. γ \gamma\!\,
  3. γ \gamma^{\prime}\!\,
  4. d d\!\,
  5. Z Z\!\,
  6. Z 0 Z_{0}\!\,
  7. ω c = 1 L 1 2 C 1 2 \omega_{c}=\frac{1}{\sqrt{L_{\frac{1}{2}}C_{\frac{1}{2}}}}
  8. Z 0 = L 1 2 C 1 2 Z_{0}=\sqrt{\frac{L_{\frac{1}{2}}}{C_{\frac{1}{2}}}}
  9. L 1 2 L_{\frac{1}{2}}
  10. C 1 2 C_{\frac{1}{2}}
  11. L = Z 0 ω c L=\frac{Z_{0}}{\omega_{c}}
  12. d = 2 ω c Z 0 C d=\frac{2}{\omega_{c}Z_{0}C}
  13. λ c d = π v Z 0 C \frac{\lambda_{c}}{d}=\pi vZ_{0}C
  14. R G = L C \frac{R}{G}=\frac{L}{C}
  15. R R
  16. L L
  17. G G
  18. C C
  19. R G L C \frac{R}{G}\gg\frac{L}{C}

Local_exchange_carrier.html

  1. m + 0.5 p ( q + | q - n | ) m+0.5p(q+|q-n|)
  2. m m
  3. n n
  4. p p
  5. q q
  6. m + ( t - 1 ) p m+(t-1)p
  7. m m
  8. p p
  9. t t
  10. m m
  11. p p

Location_parameter.html

  1. x 0 x_{0}
  2. f x 0 ( x ) = f ( x - x 0 ) . f_{x_{0}}(x)=f(x-x_{0}).
  3. x 0 x_{0}
  4. x 0 x_{0}
  5. f x 0 , θ ( x ) = f θ ( x - x 0 ) f_{x_{0},\theta}(x)=f_{\theta}(x-x_{0})
  6. x 0 x_{0}
  7. f θ f_{\theta}
  8. x 0 x_{0}
  9. f W ( w ) , f_{W}(w),
  10. X = x 0 + W X=x_{0}+W
  11. f x 0 ( x ) = f W ( x - x 0 ) f_{x_{0}}(x)=f_{W}(x-x_{0})

Logarithm.html

  1. 1000 1000
  2. 10 10
  3. 3 3
  4. 10 10
  5. 3 3
  6. 1000 1000
  7. b b
  8. x x
  9. b b
  10. 1 1
  11. x x
  12. b b
  13. y y
  14. 10 10
  15. b = 10 b=10
  16. e e
  17. 2.718 ≈2.718
  18. 2 2
  19. b = 2 b=2
  20. log b ( x y ) = log b ( x ) + log b ( y ) , \log_{b}(xy)=\log_{b}(x)+\log_{b}(y),\,
  21. b b
  22. x x
  23. y y
  24. b 1 b≠1
  25. 2 3 = 2 × 2 × 2 = 8. 2^{3}=2\times 2\times 2=8.\,
  26. b n = b × b × × b n factors . b^{n}=\underbrace{b\times b\times\cdots\times b}_{n\,\text{ factors}}.
  27. b y = x . b^{y}=x.\,
  28. = =
  29. log 2 ( 1 2 ) = - 1 , \log_{2}\!\left(\frac{1}{2}\right)=-1,\,
  30. 2 - 1 = 1 2 1 = 1 2 . 2^{-1}=\frac{1}{2^{1}}=\frac{1}{2}.
  31. x = b log b ( x ) x=b^{\log_{b}(x)}
  32. y = b log b ( y ) y=b^{\log_{b}(y)}
  33. log b ( x y ) = log b ( x ) + log b ( y ) \log_{b}(xy)=\log_{b}(x)+\log_{b}(y)
  34. log 3 ( 243 ) = log 3 ( 9 27 ) = log 3 ( 9 ) + log 3 ( 27 ) = 2 + 3 = 5 \log_{3}(243)=\log_{3}(9\cdot 27)=\log_{3}(9)+\log_{3}(27)=2+3=5
  35. log b ( x y ) = log b ( x ) - log b ( y ) \log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)
  36. log 2 ( 16 ) = log 2 ( 64 4 ) = log 2 ( 64 ) - log 2 ( 4 ) = 6 - 2 = 4 \log_{2}(16)=\log_{2}\!\left(\frac{64}{4}\right)=\log_{2}(64)-\log_{2}(4)=6-2=4
  37. log b ( x p ) = p log b ( x ) \log_{b}(x^{p})=p\log_{b}(x)
  38. log 2 ( 64 ) = log 2 ( 2 6 ) = 6 log 2 ( 2 ) = 6 \log_{2}(64)=\log_{2}(2^{6})=6\log_{2}(2)=6
  39. log b x p = log b ( x ) p \log_{b}\sqrt[p]{x}=\frac{\log_{b}(x)}{p}
  40. log 10 1000 = 1 2 log 10 1000 = 3 2 = 1.5 \log_{10}\sqrt{1000}=\frac{1}{2}\log_{10}1000=\frac{3}{2}=1.5
  41. log b ( x ) = log k ( x ) log k ( b ) . \log_{b}(x)=\frac{\log_{k}(x)}{\log_{k}(b)}.\,
  42. log b ( x ) = log 10 ( x ) log 10 ( b ) = log e ( x ) log e ( b ) . \log_{b}(x)=\frac{\log_{10}(x)}{\log_{10}(b)}=\frac{\log_{e}(x)}{\log_{e}(b)}.\,
  43. b = x 1 log b ( x ) . b=x^{\frac{1}{\log_{b}(x)}}.
  44. log 10 ( 10 x ) = log 10 ( 10 ) + log 10 ( x ) = 1 + log 10 ( x ) . \log_{10}(10x)=\log_{10}(10)+\log_{10}(x)=1+\log_{10}(x).
  45. cos α cos β = 1 2 [ cos ( α + β ) + cos ( α - β ) ] \cos\,\alpha\,\cos\,\beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)]
  46. N = 10 7 ( 1 - 10 - 7 ) L . N=10^{7}{(1-10^{-7})}^{L}.\,
  47. L = log ( 1 - 10 - 7 ) ( N 10 7 ) 10 7 log 1 e ( N 10 7 ) = - 10 7 log e ( N 10 7 ) , L=\log_{(1-10^{-7})}\!\left(\frac{N}{10^{7}}\right)\approx 10^{7}\log_{\frac{1% }{e}}\!\left(\frac{N}{10^{7}}\right)=-10^{7}\log_{e}\!\left(\frac{N}{10^{7}}% \right),
  48. ( 1 - 10 - 7 ) 10 7 1 e . {(1-10^{-7})}^{10^{7}}\approx\frac{1}{e}.\,
  49. f ( t u ) = f ( t ) + f ( u ) . f(tu)=f(t)+f(u).\,
  50. e x = lim n ( 1 + x / n ) n , e^{x}=\lim_{n\rightarrow\infty}(1+x/n)^{n},
  51. ln ( x ) = lim n n ( x 1 / n - 1 ) . \ln(x)=\lim_{n\rightarrow\infty}n(x^{1/n}-1).
  52. c d = b log b ( c ) b log b ( d ) = b log b ( c ) + log b ( d ) cd=b^{\log_{b}(c)}\,b^{\log_{b}(d)}=b^{\log_{b}(c)+\log_{b}(d)}\,
  53. c d = c d - 1 = b log b ( c ) - log b ( d ) . \frac{c}{d}=cd^{-1}=b^{\log_{b}(c)-\log_{b}(d)}.\,
  54. c d = ( b log b ( c ) ) d = b d log b ( c ) c^{d}=(b^{\log_{b}(c)})^{d}=b^{d\log_{b}(c)}\,
  55. c d = c 1 d = b 1 d log b ( c ) . \sqrt[d]{c}=c^{\frac{1}{d}}=b^{\frac{1}{d}\log_{b}(c)}.\,
  56. log 10 ( 3542 ) = log 10 ( 10 354.2 ) = 1 + log 10 ( 354.2 ) 1 + log 10 ( 354 ) . \log_{10}(3542)=\log_{10}(10\cdot 354.2)=1+\log_{10}(354.2)\approx 1+\log_{10}% (354).\,
  57. f ( x ) = b x . f(x)=b^{x}.\,
  58. b x = y b^{x}=y\,
  59. log b ( x y ) = log b ( x ) + log b ( y ) . \log_{b}(xy)=\log_{b}(x)+\log_{b}(y).
  60. f ( x y ) = f ( x ) + f ( y ) . f(xy)=f(x)+f(y).
  61. log b ( b x ) = x log b ( b ) = x . \log_{b}\left(b^{x}\right)=x\log_{b}(b)=x.
  62. b log b ( y ) = y b^{\log_{b}(y)}=y
  63. d d x log b ( x ) = 1 x ln ( b ) . \frac{d}{dx}\log_{b}(x)=\frac{1}{x\ln(b)}.
  64. d d x ln ( f ( x ) ) = f ( x ) f ( x ) . \frac{d}{dx}\ln(f(x))=\frac{f^{\prime}(x)}{f(x)}.
  65. ln ( x ) d x = x ln ( x ) - x + C . \int\ln(x)\,dx=x\ln(x)-x+C.
  66. ln ( t ) = 1 t 1 x d x . \ln(t)=\int_{1}^{t}\frac{1}{x}\,dx.
  67. ln ( t u ) = 1 t u 1 x d x = ( 1 ) 1 t 1 x d x + t t u 1 x d x = ( 2 ) ln ( t ) + 1 u 1 w d w = ln ( t ) + ln ( u ) . \ln(tu)=\int_{1}^{tu}\frac{1}{x}\,dx\ \stackrel{(1)}{=}\int_{1}^{t}\frac{1}{x}% \,dx+\int_{t}^{tu}\frac{1}{x}\,dx\ \stackrel{(2)}{=}\ln(t)+\int_{1}^{u}\frac{1% }{w}\,dw=\ln(t)+\ln(u).
  68. ln ( t r ) = 1 t r 1 x d x = 1 t 1 w r ( r w r - 1 d w ) = r 1 t 1 w d w = r ln ( t ) . \ln(t^{r})=\int_{1}^{t^{r}}\frac{1}{x}dx=\int_{1}^{t}\frac{1}{w^{r}}\left(rw^{% r-1}\,dw\right)=r\int_{1}^{t}\frac{1}{w}\,dw=r\ln(t).
  69. 1 + 1 2 + 1 3 + + 1 n = k = 1 n 1 k , 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\sum_{k=1}^{n}\frac{1}{k},
  70. k = 1 n 1 k - ln ( n ) , \sum_{k=1}^{n}\frac{1}{k}-\ln(n),
  71. ln ( x ) = - lim ϵ 0 ϵ d t t ( e - x t - e - t ) \ln(x)=-\lim_{\epsilon\to 0}\int_{\epsilon}^{\infty}\frac{dt}{t}\left(e^{-xt}-% e^{-t}\right)
  72. π \pi
  73. 2 - 3 \sqrt{2-\sqrt{3}}
  74. log 2 ( x 2 ) = 2 log 2 ( x ) . \log_{2}(x^{2})=2\log_{2}(x).\,
  75. ln ( n + 1 ) = ln ( n ) + 2 k = 0 1 2 k + 1 ( 1 2 n + 1 ) 2 k + 1 . \ln(n+1)=\ln(n)+2\sum_{k=0}^{\infty}\frac{1}{2k+1}\left(\frac{1}{2n+1}\right)^% {2k+1}.
  76. ln ( x ) π 2 M ( 1 , 2 2 - m / x ) - m ln ( 2 ) . \ln(x)\approx\frac{\pi}{2M(1,2^{2-m}/x)}-m\ln(2).
  77. x 2 m > 2 p / 2 . x\,2^{m}>2^{p/2}.\,
  78. S = - k i p i ln ( p i ) . S=-k\sum_{i}p_{i}\ln(p_{i}).\,
  79. 466 440 493 466 1.059 2 12 . \frac{466}{440}\approx\frac{493}{466}\approx 1.059\approx\sqrt[12]{2}.
  80. 2 1 72 1.0097 2^{\frac{1}{72}}\approx 1.0097
  81. 2 1 12 1.0595 2^{\frac{1}{12}}\approx 1.0595
  82. 5 4 = 1.25 \tfrac{5}{4}=1.25
  83. 2 4 12 = 2 3 1.2599 \begin{aligned}\displaystyle 2^{\frac{4}{12}}&\displaystyle=\sqrt[3]{2}\\ &\displaystyle\approx 1.2599\end{aligned}
  84. 2 6 12 = 2 1.4142 \begin{aligned}\displaystyle 2^{\frac{6}{12}}&\displaystyle=\sqrt{2}\\ &\displaystyle\approx 1.4142\end{aligned}
  85. 2 12 12 = 2 2^{\frac{12}{12}}=2
  86. log 2 12 ( r ) = 12 log 2 ( r ) \log_{\sqrt[12]{2}}(r)=12\log_{2}(r)
  87. 1 6 \tfrac{1}{6}\,
  88. 1 1\,
  89. 3.8631 \approx 3.8631\,
  90. 4 4\,
  91. 6 6\,
  92. 12 12\,
  93. log 2 1200 ( r ) = 1200 log 2 ( r ) \log_{\sqrt[1200]{2}}(r)=1200\log_{2}(r)
  94. 16 2 3 16\tfrac{2}{3}\,
  95. 100 100\,
  96. 386.31 \approx 386.31\,
  97. 400 400\,
  98. 600 600\,
  99. 1200 1200\,
  100. x ln ( x ) , \frac{x}{\ln(x)},
  101. Li ( x ) = 2 x 1 ln ( t ) d t . \mathrm{Li}(x)=\int_{2}^{x}\frac{1}{\ln(t)}\,dt.
  102. ln ( n ! ) = ln ( 1 ) + ln ( 2 ) + + ln ( n ) . \ln(n!)=\ln(1)+\ln(2)+\cdots+\ln(n).\,
  103. e a = z . e^{a}=z.\,
  104. r = x 2 + y 2 . r=\sqrt{x^{2}+y^{2}}.\,
  105. z = r ( cos φ + i sin φ ) = r e i φ . \begin{array}[]{lll}z&=&r\left(\cos\varphi+i\sin\varphi\right)\\ &=&re^{i\varphi}.\end{array}\,
  106. a = ln ( r ) + i ( φ + 2 n π ) , a=\ln(r)+i(\varphi+2n\pi),\,
  107. b n = x , b^{n}=x,\,
  108. Li s ( z ) = k = 1 z k k s . \operatorname{Li}_{s}(z)=\sum_{k=1}^{\infty}{z^{k}\over k^{s}}.

Logarithmic_integral_function.html

  1. li ( x ) = 0 x d t ln t . {\rm li}(x)=\int_{0}^{x}\frac{dt}{\ln t}.\;
  2. 1 / ln ( t ) 1/\ln(t)
  3. li ( x ) = lim ε 0 + ( 0 1 - ε d t ln t + 1 + ε x d t ln t ) . {\rm li}(x)=\lim_{\varepsilon\to 0+}\left(\int_{0}^{1-\varepsilon}\frac{dt}{% \ln t}+\int_{1+\varepsilon}^{x}\frac{dt}{\ln t}\right).\;
  4. Li ( x ) = li ( x ) - li ( 2 ) {\rm Li}(x)={\rm li}(x)-{\rm li}(2)\,
  5. Li ( x ) = 2 x d t ln t {\rm Li}(x)=\int_{2}^{x}\frac{dt}{\ln t}\,
  6. li ( x ) = Ei ( ln x ) , \hbox{li}(x)=\hbox{Ei}(\ln x),\,\!
  7. li ( e u ) = Ei ( u ) = γ + ln | u | + n = 1 u n n n ! for u 0 , {\rm li}(e^{u})=\hbox{Ei}(u)=\gamma+\ln|u|+\sum_{n=1}^{\infty}{u^{n}\over n% \cdot n!}\quad\,\text{ for }u\neq 0\;,
  8. li ( x ) = γ + ln ln x + x n = 1 ( - 1 ) n - 1 ( ln x ) n n ! 2 n - 1 k = 0 ( n - 1 ) / 2 1 2 k + 1 . {\rm li}(x)=\gamma+\ln\ln x+\sqrt{x}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(\ln x)% ^{n}}{n!\,2^{n-1}}\sum_{k=0}^{\lfloor(n-1)/2\rfloor}\frac{1}{2k+1}.
  9. - ( Γ ( 0 , - ln 2 ) + i π ) -(\Gamma\left(0,-\ln 2\right)+i\,\pi)
  10. Γ ( a , x ) \Gamma\left(a,x\right)
  11. li ( x ) = O ( x ln x ) . {\rm li}(x)=O\left({x\over\ln x}\right)\;.
  12. O O
  13. li ( x ) x ln x k = 0 k ! ( ln x ) k {\rm li}(x)\sim\frac{x}{\ln x}\sum_{k=0}^{\infty}\frac{k!}{(\ln x)^{k}}
  14. li ( x ) x / ln x 1 + 1 ln x + 2 ( ln x ) 2 + 6 ( ln x ) 3 + . \frac{{\rm li}(x)}{x/\ln x}\sim 1+\frac{1}{\ln x}+\frac{2}{(\ln x)^{2}}+\frac{% 6}{(\ln x)^{3}}+\cdots.
  15. li ( x ) - x ln x = O ( x ln 2 x ) . {\rm li}(x)-{x\over\ln x}=O\left({x\over\ln^{2}x}\right)\;.
  16. π ( x ) Li ( x ) \pi(x)\sim\operatorname{Li}(x)
  17. π ( x ) \pi(x)
  18. x x

Logarithmic_spiral.html

  1. ( r , θ ) (r,\theta)
  2. r = a e b θ r=ae^{b\theta}\,
  3. θ = 1 b ln ( r / a ) , \theta=\frac{1}{b}\ln(r/a),
  4. e e
  5. a a
  6. b b
  7. x ( t ) = r ( t ) cos ( t ) = a e b t cos ( t ) x(t)=r(t)\cos(t)=ae^{bt}\cos(t)\,
  8. y ( t ) = r ( t ) sin ( t ) = a e b t sin ( t ) y(t)=r(t)\sin(t)=ae^{bt}\sin(t)\,
  9. a a
  10. b b
  11. ( r , θ ) (r,\theta)
  12. arccos 𝐫 ( θ ) , 𝐫 ( θ ) 𝐫 ( θ ) 𝐫 ( θ ) = arctan 1 b = ϕ . \arccos\frac{\langle\mathbf{r}(\theta),\mathbf{r}^{\prime}(\theta)\rangle}{\|% \mathbf{r}(\theta)\|\|\mathbf{r}^{\prime}(\theta)\|}=\arctan\frac{1}{b}=\phi.
  13. 𝐫 ( θ ) \mathbf{r}(\theta)
  14. b b
  15. b = 0 b=0
  16. ϕ = π 2 \textstyle\phi=\frac{\pi}{2}
  17. a a
  18. b b
  19. e 2 π b e^{2\pi b}
  20. P P
  21. θ \theta
  22. - -\infty
  23. r cos ( ϕ ) \textstyle\frac{r}{\cos(\phi)}
  24. r r
  25. P P
  26. 2 π i 2\pi i
  27. x x k x\mapsto x^{k}
  28. k k

Logic_gate.html

  1. A B A\cdot B
  2. A A
  3. B B
  4. A + B A+B
  5. A ¯ \overline{A}
  6. A A
  7. A B ¯ \overline{A\cdot B}
  8. A B A\uparrow B
  9. A + B ¯ \overline{A+B}
  10. A - B A-B
  11. A B A\oplus B
  12. A B ¯ \overline{A\oplus B}
  13. A B {A\odot B}

Logical_conjunction.html

  1. \land
  2. \cdot
  3. \cdot
  4. A and B ~{}A\and B
  5. A A
  6. B B
  7. A and B A\and B
  8. A , A,
  9. B B
  10. A and B \vdash A\and B
  11. A and B A\and B
  12. A \vdash A
  13. A and B A\and B
  14. B \vdash B
  15. A and B A\and B
  16. \Leftrightarrow
  17. B and A B\and A
  18. \Leftrightarrow
  19. A ~{}A
  20. and ~{}~{}~{}\and~{}~{}~{}
  21. ( B and C ) (B\and C)
  22. \Leftrightarrow
  23. ( A and B ) (A\and B)
  24. and ~{}~{}~{}\and~{}~{}~{}
  25. C ~{}C
  26. and ~{}~{}~{}\and~{}~{}~{}
  27. \Leftrightarrow
  28. \Leftrightarrow
  29. and ~{}~{}~{}\and~{}~{}~{}
  30. A ~{}A
  31. and \and
  32. ( B C ) (BC)
  33. \Leftrightarrow
  34. ( A and B ) (A\and B)
  35. $\or$
  36. ( A and C ) (A\and C)
  37. and \and
  38. \Leftrightarrow
  39. \Leftrightarrow
  40. $\or$
  41. A ~{}A
  42. and \and
  43. ( B C ) (B\oplus C)
  44. \Leftrightarrow
  45. ( A and B ) (A\and B)
  46. \oplus
  47. ( A and C ) (A\and C)
  48. and \and
  49. \Leftrightarrow
  50. \Leftrightarrow
  51. \oplus
  52. A ~{}A
  53. and \and
  54. ( B C ) (B\nrightarrow C)
  55. \Leftrightarrow
  56. ( A and B ) (A\and B)
  57. \nrightarrow
  58. ( A and C ) (A\and C)
  59. and \and
  60. \Leftrightarrow
  61. \Leftrightarrow
  62. \nrightarrow
  63. A ~{}A
  64. and \and
  65. ( B and C ) (B\and C)
  66. \Leftrightarrow
  67. ( A and B ) (A\and B)
  68. and \and
  69. ( A and C ) (A\and C)
  70. and \and
  71. \Leftrightarrow
  72. \Leftrightarrow
  73. and \and
  74. A ~{}A~{}
  75. and ~{}\and~{}
  76. A ~{}A~{}
  77. \Leftrightarrow
  78. A A~{}
  79. and ~{}\and~{}
  80. \Leftrightarrow
  81. A B A\rightarrow B
  82. \Rightarrow
  83. ( A and C ) (A\and C)
  84. \rightarrow
  85. ( B and C ) (B\and C)
  86. \Rightarrow
  87. \Leftrightarrow
  88. \rightarrow
  89. A and B A\and B
  90. \Rightarrow
  91. A and B A\and B
  92. \Rightarrow
  93. A and B A\and B
  94. \Rightarrow
  95. A B AB
  96. \Rightarrow

Logical_connective.html

  1. n n
  2. P P
  3. P P
  4. P P
  5. Q Q
  6. \nleftrightarrow
  7. P P
  8. Q Q
  9. \wedge
  10. $$\or$$
  11. \rightarrow
  12. \Rightarrow
  13. \supset
  14. \leftrightarrow
  15. \equiv
  16. = =
  17. \wedge
  18. $\or$
  19. \rightarrow
  20. \rightarrow
  21. \leftrightarrow
  22. P ¯ \overline{P}
  23. \Rightarrow
  24. \bigwedge
  25. \bigvee
  26. ¬ P Q ¬P∨Q
  27. P Q P→Q
  28. P P
  29. Q Q
  30. \vee
  31. \wedge
  32. \bot
  33. \bot
  34. \nleftrightarrow
  35. \nleftrightarrow
  36. \nrightarrow
  37. \nleftarrow
  38. \nrightarrow
  39. \nleftarrow
  40. \nrightarrow
  41. \nleftarrow
  42. \nrightarrow
  43. \top
  44. \nleftarrow
  45. \top
  46. \nrightarrow
  47. \leftrightarrow
  48. \nleftarrow
  49. \leftrightarrow
  50. \lor
  51. \leftrightarrow
  52. \bot
  53. \lor
  54. \leftrightarrow
  55. \nleftrightarrow
  56. \lor
  57. \nleftrightarrow
  58. \top
  59. \land
  60. \leftrightarrow
  61. \bot
  62. \land
  63. \leftrightarrow
  64. \nleftrightarrow
  65. \land
  66. \nleftrightarrow
  67. \top
  68. a · ( b + c ) = ( a · b ) + ( a · c ) a·(b+c)=(a·b)+(a·c)
  69. a a
  70. b b
  71. c c
  72. \land
  73. \lor
  74. a ( a b ) = a a\land(a\lor b)=a
  75. a a
  76. b b
  77. \vee
  78. \wedge
  79. \top
  80. \bot
  81. ¬ \neg
  82. \leftrightarrow
  83. \nleftrightarrow
  84. \top
  85. \bot
  86. ¬ \neg
  87. \vee
  88. \wedge
  89. \top
  90. \rightarrow
  91. \leftrightarrow
  92. \vee
  93. \wedge
  94. \nleftrightarrow
  95. \bot
  96. f ( f ( a ) ) = a f(f(a))=a
  97. ¬ \neg
  98. \wedge
  99. \wedge
  100. \vee
  101. \vee
  102. \rightarrow
  103. P Q ¬ R S P\vee Q\wedge{\neg R}\rightarrow S
  104. ( P ( Q ( ¬ R ) ) ) S (P\vee(Q\wedge(\neg R)))\rightarrow S
  105. ¬ \neg
  106. \wedge
  107. \vee
  108. \rightarrow
  109. P Q P∧Q
  110. P Q P∨Q
  111. P P
  112. Q Q

Logical_disjunction.html

  1. A B ~{}AB
  2. A A
  3. B B
  4. A B AB
  5. A B AB
  6. \Leftrightarrow
  7. B A BA
  8. \Leftrightarrow
  9. A ~{}A
  10. ~{}~{}~{}~{}~{}~{}
  11. ( B C ) (BC)
  12. \Leftrightarrow
  13. ( A B ) (AB)
  14. ~{}~{}~{}~{}~{}~{}
  15. C ~{}C
  16. ~{}~{}~{}~{}~{}~{}
  17. \Leftrightarrow
  18. \Leftrightarrow
  19. ~{}~{}~{}~{}~{}~{}
  20. A ~{}A
  21. $\or$
  22. ( B and C ) (B\and C)
  23. \Leftrightarrow
  24. ( A B ) (AB)
  25. and \and
  26. ( A C ) (AC)
  27. $\or$
  28. \Leftrightarrow
  29. \Leftrightarrow
  30. and \and
  31. A ~{}A
  32. $\or$
  33. ( B C ) (B\leftrightarrow C)
  34. \Leftrightarrow
  35. ( A B ) (AB)
  36. \leftrightarrow
  37. ( A C ) (AC)
  38. $\or$
  39. \Leftrightarrow
  40. \Leftrightarrow
  41. \leftrightarrow
  42. A ~{}A
  43. $\or$
  44. ( B C ) (B\rightarrow C)
  45. \Leftrightarrow
  46. ( A B ) (AB)
  47. \rightarrow
  48. ( A C ) (AC)
  49. $\or$
  50. \Leftrightarrow
  51. \Leftrightarrow
  52. \rightarrow
  53. A ~{}A
  54. $\or$
  55. ( B C ) (BC)
  56. \Leftrightarrow
  57. ( A B ) (AB)
  58. $\or$
  59. ( A C ) (AC)
  60. $\or$
  61. \Leftrightarrow
  62. \Leftrightarrow
  63. $\or$
  64. A ~{}A~{}
  65. ~{}~{}
  66. A ~{}A~{}
  67. \Leftrightarrow
  68. A A~{}
  69. ~{}~{}
  70. \Leftrightarrow
  71. A B A\rightarrow B
  72. \Rightarrow
  73. ( A C ) (AC)
  74. \rightarrow
  75. ( B C ) (BC)
  76. \Rightarrow
  77. \Leftrightarrow
  78. \rightarrow
  79. A and B A\and B
  80. \Rightarrow
  81. A B AB
  82. \Rightarrow
  83. A B AB
  84. \Rightarrow
  85. A B AB
  86. \Rightarrow
  87. 11 = 1 11=1
  88. 1 + 1 = 10 1+1=10
  89. $\or$
  90. $\or$
  91. A B AB
  92. ¬ A B \neg AB
  93. A ¬ B ¬ C D ¬ E . A\neg B\neg CD\neg E.

Logistic_map.html

  1. ( 1 ) x n + 1 = r x n ( 1 - x n ) (1)\qquad x_{n+1}=rx_{n}(1-x_{n})
  2. x n x_{n}
  3. r = 4 r=4
  4. μ = 2 \mu=2
  5. r - 1 r \frac{r-1}{r}
  6. r - 1 r \frac{r-1}{r}
  7. 1 + 6 1+\sqrt{6}
  8. \dots
  9. 1 + 8 1+\sqrt{8}
  10. c 2 k . c2^{k}.
  11. c 2 k * , c2^{k^{*}},
  12. c 2 k c2^{k}
  13. k < k * . k<k^{*}.
  14. π - 1 x - 1 / 2 ( 1 - x ) - 1 / 2 \pi^{-1}x^{-1/2}(1-x)^{-1/2}
  15. x n = sin 2 ( 2 n θ π ) x_{n}=\sin^{2}(2^{n}\theta\pi)
  16. θ \theta
  17. θ = 1 π sin - 1 ( x 0 1 / 2 ) \theta=\tfrac{1}{\pi}\sin^{-1}(x_{0}^{1/2})
  18. θ \theta
  19. x n x_{n}
  20. θ \theta
  21. θ \theta
  22. x n x_{n}
  23. x n x_{n}
  24. x n = - α 2 n - α - 2 n + 2 4 x_{n}=\frac{-\alpha^{2^{n}}-\alpha^{-2^{n}}+2}{4}
  25. α \alpha
  26. α = - 8 x 0 + 4 ± ( - 8 x 0 + 4 ) 2 - 16 4 \alpha=\frac{-8x_{0}+4\pm\sqrt{(-8x_{0}+4)^{2}-16}}{4}
  27. α \alpha
  28. x n = 1 2 - 1 2 ( 1 - 2 x 0 ) 2 n x_{n}=\frac{1}{2}-\frac{1}{2}(1-2x_{0})^{2^{n}}
  29. x 0 [ 0 , 1 ) x_{0}\in[0,1)
  30. ( 1 - 2 x 0 ) ( - 1 , 1 ) (1-2x_{0})\in(-1,1)
  31. x 0 x_{0}
  32. ( 1 - 2 x 0 ) 2 n (1-2x_{0})^{2^{n}}
  33. x n x_{n}
  34. 1 2 . \tfrac{1}{2}.
  35. x n + 1 = 4 x n ( 1 - x n ) x_{n+1}=4x_{n}(1-x_{n})\,
  36. y n + 1 = { 2 y n 0 y n < 0.5 2 y n - 1 0.5 y n < 1 , y_{n+1}=\begin{cases}2y_{n}&0\leq y_{n}<0.5\\ 2y_{n}-1&0.5\leq y_{n}<1,\end{cases}
  37. x n = sin 2 ( 2 π y n ) x_{n}=\sin^{2}(2\pi y_{n})
  38. μ = 2 \mu=2
  39. 2 ( 2 k - 1 - 1 ) / k 2(2^{k-1}-1)/k
  40. 2 ( 2 13 - 1 - 1 ) / 13 = 630 2(2^{13-1}-1)/13=630

Longitude.html

  1. Δ LONG 1 = π 180 a cos ϕ \Delta^{1}_{\rm LONG}=\frac{\pi}{180}a\cos\phi\,\!
  2. ϕ \phi
  3. Δ LAT 1 \Delta^{1}_{\rm LAT}
  4. Δ LONG 1 \Delta^{1}_{\rm LONG}
  5. Δ LONG 1 = π a cos ϕ 180 ( 1 - e 2 sin 2 ϕ ) \Delta^{1}_{\rm LONG}=\frac{\pi a\cos\phi}{180\cdot\sqrt{(1-e^{2}\sin^{2}\phi)% }}\,
  6. e 2 = a 2 - b 2 a 2 e^{2}=\frac{a^{2}-b^{2}}{a^{2}}
  7. Δ LONG 1 = π 180 a cos ψ \Delta^{1}_{\rm LONG}=\frac{\pi}{180}a\cos\psi\,\!
  8. tan ψ = b a tan ϕ \tan\psi=\frac{b}{a}\tan\phi

Lorentz_force.html

  1. 𝐅 = q [ 𝐄 + ( 𝐯 × 𝐁 ) ] \mathbf{F}=q\left[\mathbf{E}+(\mathbf{v}\times\mathbf{B})\right]
  2. 𝐅 ( 𝐫 , 𝐫 ˙ , t , q ) = q [ 𝐄 ( 𝐫 , t ) + 𝐫 ˙ × 𝐁 ( 𝐫 , t ) ] \mathbf{F}(\mathbf{r},\mathbf{\dot{r}},t,q)=q[\mathbf{E}(\mathbf{r},t)+\mathbf% {\dot{r}}\times\mathbf{B}(\mathbf{r},t)]
  3. d 𝐅 = d q ( 𝐄 + 𝐯 × 𝐁 ) \mathrm{d}\mathbf{F}=\mathrm{d}q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}% \right)\,\!
  4. 𝐟 = ρ ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{f}=\rho\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\,\!
  5. 𝐉 = ρ 𝐯 \mathbf{J}=\rho\mathbf{v}\,\!
  6. 𝐅 = ( ρ 𝐄 + 𝐉 × 𝐁 ) d V . \mathbf{F}=\iiint\!(\rho\mathbf{E}+\mathbf{J}\times\mathbf{B})\,\mathrm{d}V.\,\!
  7. 𝐟 = \cdotsymbol σ - 1 c 2 𝐒 t \mathbf{f}=\nabla\cdotsymbol{\sigma}-\dfrac{1}{c^{2}}\dfrac{\partial\mathbf{S}% }{\partial t}\,\!
  8. 𝐅 = q 2 𝐯 × 𝐁 . \mathbf{F}=\frac{q}{2}\mathbf{v}\times\mathbf{B}.
  9. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})
  10. 𝐅 = I s y m b o l × 𝐁 \mathbf{F}=Isymbol{\ell}\times\mathbf{B}\,\!
  11. 𝐅 = I d s y m b o l × 𝐁 \mathbf{F}=I\int\mathrm{d}symbol{\ell}\times\mathbf{B}
  12. = - d Φ B d t \mathcal{E}=-\frac{\mathrm{d}\Phi_{B}}{\mathrm{d}t}
  13. Φ B = Σ ( t ) d 𝐀 𝐁 ( 𝐫 , t ) \Phi_{B}=\iint_{\Sigma(t)}\mathrm{d}\mathbf{A}\cdot\mathbf{B}(\mathbf{r},t)
  14. = Σ ( t ) d s y m b o l 𝐅 / q \mathcal{E}=\oint_{\partial\Sigma(t)}\mathrm{d}symbol{\ell}\cdot\mathbf{F}/q
  15. 𝐄 = 𝐅 / q \mathbf{E}=\mathbf{F}/q
  16. × 𝐄 = - 𝐁 t . \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\ .
  17. Σ ( t ) d s y m b o l 𝐄 ( 𝐫 , t ) = - Σ ( t ) d 𝐀 d 𝐁 ( 𝐫 , t ) d t \oint_{\partial\Sigma(t)}\mathrm{d}symbol{\ell}\cdot\mathbf{E}(\mathbf{r},\ t)% =-\ \iint_{\Sigma(t)}\mathrm{d}\mathbf{A}\cdot{{\mathrm{d}\,\mathbf{B}(\mathbf% {r},\ t)}\over\mathrm{d}t}
  18. Σ ( t ) d s y m b o l 𝐅 / q ( 𝐫 , t ) = - d d t Σ ( t ) d 𝐀 𝐁 ( 𝐫 , t ) . \oint_{\partial\Sigma(t)}\mathrm{d}symbol{\ell}\cdot\mathbf{F}/q(\mathbf{r},\ % t)=-\frac{\mathrm{d}}{\mathrm{d}t}\iint_{\Sigma(t)}\mathrm{d}\mathbf{A}\cdot% \mathbf{B}(\mathbf{r},\ t).
  19. Σ ( t ) d s y m b o l 𝐅 / q ( 𝐫 , t ) = - Σ ( t ) d 𝐀 t 𝐁 ( 𝐫 , t ) + Σ ( t ) 𝐯 × 𝐁 d s y m b o l \oint_{\partial\Sigma(t)}\mathrm{d}symbol{\ell}\cdot\mathbf{F}/q(\mathbf{r},t)% =-\iint_{\Sigma(t)}\mathrm{d}\mathbf{A}\cdot\frac{\partial}{\partial t}\mathbf% {B}(\mathbf{r},t)+\oint_{\partial\Sigma(t)}\!\!\!\!\mathbf{v}\times\mathbf{B}% \,\mathrm{d}symbol{\ell}
  20. Σ ( t ) d s y m b o l 𝐅 / q ( 𝐫 , t ) = Σ ( t ) d s y m b o l 𝐄 ( 𝐫 , t ) + Σ ( t ) 𝐯 × 𝐁 ( 𝐫 , t ) d s y m b o l \oint_{\partial\Sigma(t)}\mathrm{d}symbol{\ell}\cdot\mathbf{F}/q(\mathbf{r},\ % t)=\oint_{\partial\Sigma(t)}\mathrm{d}symbol{\ell}\cdot\mathbf{E}(\mathbf{r},% \ t)+\oint_{\partial\Sigma(t)}\!\!\!\!\mathbf{v}\times\mathbf{B}(\mathbf{r},\ % t)\,\mathrm{d}symbol{\ell}
  21. 𝐅 = q 𝐄 ( 𝐫 , t ) + q 𝐯 × 𝐁 ( 𝐫 , t ) . \mathbf{F}=q\,\mathbf{E}(\mathbf{r},\ t)+q\,\mathbf{v}\times\mathbf{B}(\mathbf% {r},\ t).
  22. 𝐄 = - ϕ - 𝐀 t \mathbf{E}=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}
  23. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  24. 𝐅 = q [ - ϕ - 𝐀 t + 𝐯 × ( × 𝐀 ) ] \mathbf{F}=q\left[-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}+\mathbf{v}% \times(\nabla\times\mathbf{A})\right]
  25. d 𝐀 d t = 𝐀 t + ( 𝐯 ) 𝐀 \frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t}=\frac{\partial\mathbf{A}}{\partial t}% +(\mathbf{v}\cdot\nabla)\mathbf{A}
  26. 𝐅 = q [ - ( ϕ - 𝐯 𝐀 ) - d 𝐀 d t ] \mathbf{F}=q\left[-\nabla(\phi-\mathbf{v}\cdot\mathbf{A})-\frac{d\mathbf{A}}{% \mathrm{d}t}\right]
  27. L = m 2 𝐫 ˙ 𝐫 ˙ + q 𝐀 𝐫 ˙ - q ϕ L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf{% \dot{r}}-q\phi
  28. q 𝐀 ( 𝐫 , t ) q\mathbf{A}(\mathbf{r},t)
  29. q 𝐀 ( 𝐫 , t ) 𝐫 ˙ q\mathbf{A}(\mathbf{r},t)\cdot\mathbf{\dot{r}}
  30. q ϕ ( 𝐫 , t ) q\phi(\mathbf{r},t)
  31. V = q ϕ - q 𝐀 𝐫 ˙ V=q\phi-q\mathbf{A}\cdot\mathbf{\dot{r}}
  32. T = m 2 𝐫 ˙ 𝐫 ˙ T=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}
  33. L = T - V = m 2 𝐫 ˙ 𝐫 ˙ + q 𝐀 𝐫 ˙ - q ϕ L=T-V=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+q\mathbf{A}\cdot\mathbf% {\dot{r}}-q\phi
  34. L = m 2 ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) + q ( x ˙ A x + y ˙ A y + z ˙ A z ) - q ϕ L=\frac{m}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})+q(\dot{x}A_{x}+\dot{y}A_{y}% +\dot{z}A_{z})-q\phi
  35. d d t L x ˙ = L x \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial\dot{x}}=\frac{% \partial L}{\partial x}
  36. d d t L x ˙ \displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial\dot{x}}
  37. L x = - q ϕ x + q ( A x x x ˙ + A y x y ˙ + A z x z ˙ ) \frac{\partial L}{\partial x}=-q\frac{\partial\phi}{\partial x}+q\left(\frac{% \partial A_{x}}{\partial x}\dot{x}+\frac{\partial A_{y}}{\partial x}\dot{y}+% \frac{\partial A_{z}}{\partial x}\dot{z}\right)
  38. m x ¨ + q ( A x t + A x x x ˙ + A x y y ˙ + A x z z ˙ ) = - q ϕ x + q ( A x x x ˙ + A y x y ˙ + A z x z ˙ ) m\ddot{x}+q\left(\frac{\partial A_{x}}{\partial t}+\frac{\partial A_{x}}{% \partial x}\dot{x}+\frac{\partial A_{x}}{\partial y}\dot{y}+\frac{\partial A_{% x}}{\partial z}\dot{z}\right)=-q\frac{\partial\phi}{\partial x}+q\left(\frac{% \partial A_{x}}{\partial x}\dot{x}+\frac{\partial A_{y}}{\partial x}\dot{y}+% \frac{\partial A_{z}}{\partial x}\dot{z}\right)
  39. F x = - q ( ϕ x + A x t ) + q [ y ˙ ( A y x - A x y ) + z ˙ ( A z x - A x z ) ] = q E x + q [ y ˙ ( × 𝐀 ) z - z ˙ ( × 𝐀 ) y ] = q E x + q [ 𝐫 ˙ × ( × 𝐀 ) ] x = q E x + q ( 𝐫 ˙ × 𝐁 ) x \begin{aligned}\displaystyle F_{x}&\displaystyle=-q\left(\frac{\partial\phi}{% \partial x}+\frac{\partial A_{x}}{\partial t}\right)+q\left[\dot{y}\left(\frac% {\partial A_{y}}{\partial x}-\frac{\partial A_{x}}{\partial y}\right)+\dot{z}% \left(\frac{\partial A_{z}}{\partial x}-\frac{\partial A_{x}}{\partial z}% \right)\right]\\ &\displaystyle=qE_{x}+q[\dot{y}(\nabla\times\mathbf{A})_{z}-\dot{z}(\nabla% \times\mathbf{A})_{y}]\\ &\displaystyle=qE_{x}+q[\mathbf{\dot{r}}\times(\nabla\times\mathbf{A})]_{x}\\ &\displaystyle=qE_{x}+q(\mathbf{\dot{r}}\times\mathbf{B})_{x}\end{aligned}
  40. 𝐅 = q ( 𝐄 + 𝐫 ˙ × 𝐁 ) \mathbf{F}=q(\mathbf{E}+\mathbf{\dot{r}}\times\mathbf{B})
  41. L = - m c 2 1 - ( 𝐫 ˙ c ) 2 + e 𝐀 ( 𝐫 ) 𝐫 ˙ - e ϕ ( 𝐫 ) L=-mc^{2}\sqrt{1-\left(\frac{\dot{\mathbf{r}}}{c}\right)^{2}}+e\mathbf{A}(% \mathbf{r})\cdot\dot{\mathbf{r}}-e\phi(\mathbf{r})\,\!
  42. d 𝐏 d t = L 𝐫 = e 𝐀 𝐫 𝐫 ˙ - e ϕ 𝐫 \frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=\frac{\partial L}{\partial\mathbf{r}}% =e{\partial\mathbf{A}\over\partial\mathbf{r}}\cdot\dot{\mathbf{r}}-e{\partial% \phi\over\partial\mathbf{r}}\,\!
  43. 𝐏 - e 𝐀 = m 𝐫 ˙ 1 - ( 𝐫 ˙ c ) 2 \mathbf{P}-e\mathbf{A}=\frac{m\dot{\mathbf{r}}}{\sqrt{1-\left(\frac{\dot{% \mathbf{r}}}{c}\right)^{2}}}\,
  44. d 𝐫 d t = 𝐩 ( ( 𝐏 - e 𝐀 ) 2 + ( m c 2 ) 2 + e ϕ ) \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}=\frac{\partial}{\partial\mathbf{p}}% \left(\sqrt{(\mathbf{P}-e\mathbf{A})^{2}+(mc^{2})^{2}}+e\phi\right)\,\!
  45. d 𝐩 d t = - 𝐫 ( ( 𝐏 - e 𝐀 ) 2 + ( m c 2 ) 2 + e ϕ ) \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=-{\partial\over\partial\mathbf{r}}% \left(\sqrt{(\mathbf{P}-e\mathbf{A})^{2}+(mc^{2})^{2}}+e\phi\right)\,\!
  46. d d t ( m 𝐫 ˙ 1 - ( 𝐫 ˙ c ) 2 ) = e ( 𝐄 + 𝐫 ˙ × 𝐁 ) . \frac{\mathrm{d}}{\mathrm{d}t}\left({m\dot{\mathbf{r}}\over\sqrt{1-\left(\frac% {\dot{\mathbf{r}}}{c}\right)^{2}}}\right)=e\left(\mathbf{E}+\dot{\mathbf{r}}% \times\mathbf{B}\right).\,\!
  47. 𝐅 = q cgs ( 𝐄 cgs + 𝐯 c × 𝐁 cgs ) . \mathbf{F}=q_{\mathrm{cgs}}\left(\mathbf{E}_{\mathrm{cgs}}+\frac{\mathbf{v}}{c% }\times\mathbf{B}_{\mathrm{cgs}}\right).
  48. q cgs = q SI 4 π ϵ 0 , 𝐄 cgs = 4 π ϵ 0 𝐄 SI , 𝐁 cgs = 4 π / μ 0 𝐁 SI q_{\mathrm{cgs}}=\frac{q_{\mathrm{SI}}}{\sqrt{4\pi\epsilon_{0}}},\quad\mathbf{% E}_{\mathrm{cgs}}=\sqrt{4\pi\epsilon_{0}}\,\mathbf{E}_{\mathrm{SI}},\quad% \mathbf{B}_{\mathrm{cgs}}={\sqrt{4\pi/\mu_{0}}}\,{\mathbf{B}_{\mathrm{SI}}}
  49. p α = ( p 0 , p 1 , p 2 , p 3 ) = ( γ m c , p x , p y , p z ) , p^{\alpha}=\left(p_{0},p_{1},p_{2},p_{3}\right)=\left(\gamma mc,p_{x},p_{y},p_% {z}\right)\,,
  50. F α β = ( 0 E x / c E y / c E z / c - E x / c 0 B z - B y - E y / c - B z 0 B x - E z / c B y - B x 0 ) F^{\alpha\beta}=\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\ -E_{x}/c&0&B_{z}&-B_{y}\\ -E_{y}/c&-B_{z}&0&B_{x}\\ -E_{z}/c&B_{y}&-B_{x}&0\end{pmatrix}
  51. U β = ( U 0 , U 1 , U 2 , U 3 ) = γ ( - c , v x , v y , v z ) , U_{\beta}=\left(U_{0},U_{1},U_{2},U_{3}\right)=\gamma\left(-c,v_{x},v_{y},v_{z% }\right)\,,
  52. γ ( v ) = 1 1 - v 2 c 2 = 1 1 - v x 2 + v y 2 + v z 2 c 2 \gamma(v)=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v_{x}^% {2}+v_{y}^{2}+v_{z}^{2}}{c^{2}}}}
  53. F μ ν = Λ μ α Λ ν β F α β , F^{\prime\mu\nu}={\Lambda^{\mu}}_{\alpha}{\Lambda^{\nu}}_{\beta}F^{\alpha\beta% }\,,
  54. d p 1 d τ = q U β F 1 β = q ( U 0 F 10 + U 1 F 11 + U 2 F 12 + U 3 F 13 ) . \frac{\mathrm{d}p^{1}}{\mathrm{d}\tau}=qU_{\beta}F^{1\beta}=q\left(U_{0}F^{10}% +U_{1}F^{11}+U_{2}F^{12}+U_{3}F^{13}\right).\,
  55. d p 1 d τ = q [ U 0 ( - E x c ) + U 2 ( B z ) + U 3 ( - B y ) ] . \frac{\mathrm{d}p^{1}}{\mathrm{d}\tau}=q\left[U_{0}\left(\frac{-E_{x}}{c}% \right)+U_{2}(B_{z})+U_{3}(-B_{y})\right].\,
  56. d p 1 d τ = q γ [ - c ( - E x c ) + v y B z + v z ( - B y ) ] = q γ ( E x + v y B z - v z B y ) = q γ [ E x + ( 𝐯 × 𝐁 ) x ] . \begin{aligned}\displaystyle\frac{\mathrm{d}p^{1}}{\mathrm{d}\tau}&% \displaystyle=q\gamma\left[-c\left(\frac{-E_{x}}{c}\right)+v_{y}B_{z}+v_{z}(-B% _{y})\right]\\ &\displaystyle=q\gamma\left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right)\\ &\displaystyle=q\gamma\left[E_{x}+\left(\mathbf{v}\times\mathbf{B}\right)_{x}% \right]\,.\end{aligned}
  57. d 𝐩 d τ = q γ ( 𝐄 + 𝐯 × 𝐁 ) , \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}\tau}=q\gamma\left(\mathbf{E}+\mathbf{v}% \times\mathbf{B}\right)\,,
  58. d t = γ ( v ) d τ , dt=\gamma(v)d\tau\,,
  59. d 𝐩 d t = q ( 𝐄 + 𝐯 × 𝐁 ) . \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=q\left(\mathbf{E}+\mathbf{v}\times% \mathbf{B}\right)\,.
  60. 𝐩 = γ ( v ) m 0 𝐯 . \mathbf{p}=\gamma(v)m_{0}\mathbf{v}\,.
  61. \mathcal{F}
  62. γ 0 \gamma_{0}
  63. 𝐄 = ( γ 0 ) γ 0 \mathbf{E}=(\mathcal{F}\cdot\gamma_{0})\gamma_{0}
  64. i 𝐁 = ( γ 0 ) γ 0 i\mathbf{B}=(\mathcal{F}\wedge\gamma_{0})\gamma_{0}
  65. \mathcal{F}
  66. γ 0 \gamma_{0}
  67. v = x ˙ v=\dot{x}
  68. v 2 = 1 , v^{2}=1,
  69. 𝐯 = c v γ 0 / ( v γ 0 ) . \mathbf{v}=cv\wedge\gamma_{0}/(v\cdot\gamma_{0}).

Lorentz_transformation.html

  1. c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 - ( z 2 - z 1 ) 2 = 0 c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=0
  2. c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 - ( z 2 - z 1 ) 2 = c 2 ( t 2 - t 1 ) 2 - ( x 2 - x 1 ) 2 - ( y 2 - y 1 ) 2 - ( z 2 - z 1 ) 2 . c^{2}(t_{2}-t_{1})^{2}-(x_{2}-x_{1})^{2}-(y_{2}-y_{1})^{2}-(z_{2}-z_{1})^{2}=c% ^{2}(t_{2}^{\prime}-t_{1}^{\prime})^{2}-(x_{2}^{\prime}-x_{1}^{\prime})^{2}-(y% _{2}^{\prime}-y_{1}^{\prime})^{2}-(z_{2}^{\prime}-z_{1}^{\prime})^{2}.
  3. c 2 t 2 - x 2 - y 2 - z 2 = c 2 t 2 - x 2 - y 2 - z 2 c^{2}t^{2}-x^{2}-y^{2}-z^{2}=c^{2}t^{\prime 2}-x^{\prime 2}-y^{\prime 2}-z^{% \prime 2}
  4. O ( 1 , 3 ) O(1,3)
  5. t = γ ( t - v x c 2 ) x = γ ( x - v t ) y = y z = z \begin{aligned}\displaystyle t^{\prime}&\displaystyle=\gamma\left(t-\frac{vx}{% c^{2}}\right)\\ \displaystyle x^{\prime}&\displaystyle=\gamma\left(x-vt\right)\\ \displaystyle y^{\prime}&\displaystyle=y\\ \displaystyle z^{\prime}&\displaystyle=z\end{aligned}
  6. γ = 1 1 - β 2 \ \gamma=\frac{1}{\sqrt{1-{\beta^{2}}}}
  7. β = v c \ \beta=\frac{v}{c}
  8. [ c t x y z ] = [ γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ] [ c t x y z ] , \begin{bmatrix}ct^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}\begin{bmatrix}c\,t\\ x\\ y\\ z\end{bmatrix},
  9. t \displaystyle t
  10. t \displaystyle t^{\prime}
  11. [ c t x y z ] = [ γ 0 - β γ 0 0 1 0 0 - β γ 0 γ 0 0 0 0 1 ] [ c t x y z ] , \begin{bmatrix}ct^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&0&-\beta\gamma&0\\ 0&1&0&0\\ -\beta\gamma&0&\gamma&0\\ 0&0&0&1\\ \end{bmatrix}\begin{bmatrix}c\,t\\ x\\ y\\ z\end{bmatrix},
  12. [ c t x y z ] = [ γ 0 0 - β γ 0 1 0 0 0 0 1 0 - β γ 0 0 γ ] [ c t x y z ] . \begin{bmatrix}ct^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&0&0&-\beta\gamma\\ 0&1&0&0\\ 0&0&1&0\\ -\beta\gamma&0&0&\gamma\\ \end{bmatrix}\begin{bmatrix}c\,t\\ x\\ y\\ z\end{bmatrix}.
  13. 𝐗 = [ c t x y z ] , 𝐗 = [ c t x y z ] , \mathbf{X}=\begin{bmatrix}c\,t\\ x\\ y\\ z\end{bmatrix}\ ,\quad\mathbf{X}^{\prime}=\begin{bmatrix}c\,t^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix},
  14. 𝐗 = s y m b o l Λ ( v ) 𝐗 . \mathbf{X}^{\prime}=symbol{\Lambda}(v)\mathbf{X}.
  15. 𝐯 \mathbf{v}
  16. O O
  17. O O′
  18. 𝐯 \mathbf{v}
  19. F F
  20. O O′
  21. O O
  22. 𝐯 −\mathbf{v}
  23. F F′
  24. 𝐫 \mathbf{r}
  25. 𝐯 \mathbf{v}
  26. 𝐫 = 𝐫 + 𝐫 \mathbf{r}=\mathbf{r}_{\perp}+\mathbf{r}_{\|}
  27. 𝐫 𝐯 = 𝐫 𝐯 + 𝐫 𝐯 = r v \mathbf{r}\cdot\mathbf{v}=\mathbf{r}_{\bot}\cdot\mathbf{v}+\mathbf{r}_{% \parallel}\cdot\mathbf{v}=r_{\parallel}v
  28. \mathbf{•}
  29. 𝐯 \mathbf{v}
  30. t \displaystyle t^{\prime}
  31. γ ( 𝐯 ) = 1 1 - v 2 c 2 \gamma(\mathbf{v})=\frac{1}{\sqrt{1-\tfrac{v^{2}}{c^{2}}}}
  32. 𝐫 \mathbf{r′}
  33. 𝐫 = 𝐫 + ( γ - 1 ) 𝐫 - γ 𝐯 t . \mathbf{r}^{\prime}=\mathbf{r}+\left(\gamma-1\right)\mathbf{r}_{\parallel}-% \gamma\mathbf{v}t\,.
  34. 𝐯 \mathbf{v}
  35. 𝐫 = r 𝐯 v = ( 𝐫 𝐯 v ) 𝐯 v \mathbf{r}_{\parallel}=r_{\parallel}\dfrac{\mathbf{v}}{v}=\left(\dfrac{\mathbf% {r}\cdot\mathbf{v}}{v}\right)\frac{\mathbf{v}}{v}
  36. 𝐯 / v \mathbf{v}/v
  37. = =
  38. 𝐫 \mathbf{r}
  39. 𝐯 \mathbf{v}
  40. 𝐯 \mathbf{v}
  41. 𝐫 = 𝐫 + ( γ - 1 v 2 𝐫 𝐯 - γ t ) 𝐯 . \mathbf{r}^{\prime}=\mathbf{r}+\left(\frac{\gamma-1}{v^{2}}\mathbf{r}\cdot% \mathbf{v}-\gamma t\right)\mathbf{v}\,.
  42. [ c t 𝐫 ] = [ γ - γ s y m b o l β T - \gammasymbol β 𝐈 + ( γ - 1 ) s y m b o l β s y m b o l β T / β 2 ] [ c t 𝐫 ] \begin{bmatrix}ct^{\prime}\\ \mathbf{r^{\prime}}\end{bmatrix}=\begin{bmatrix}\gamma&-\gamma symbol{\beta}^{% \mathrm{T}}\\ -\gammasymbol{\beta}&\mathbf{I}+(\gamma-1)symbol{\beta}symbol{\beta}^{\mathrm{% T}}/\beta^{2}\\ \end{bmatrix}\begin{bmatrix}ct\\ \mathbf{r}\end{bmatrix}\,
  43. s y m b o l β = v c [ β x β y β z ] = 1 c [ v x v y v z ] [ β 1 β 2 β 3 ] = 1 c [ v 1 v 2 v 3 ] symbol{\beta}=\frac{{v}}{c}\equiv\begin{bmatrix}\beta_{x}\\ \beta_{y}\\ \beta_{z}\end{bmatrix}=\frac{1}{c}\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\end{bmatrix}\equiv\begin{bmatrix}\beta_{1}\\ \beta_{2}\\ \beta_{3}\end{bmatrix}=\frac{1}{c}\begin{bmatrix}v_{1}\\ v_{2}\\ v_{3}\end{bmatrix}
  44. s y m b o l β T = v T c [ β x β y β z ] = 1 c [ v x v y v z ] [ β 1 β 2 β 3 ] = 1 c [ v 1 v 2 v 3 ] symbol{\beta}^{\mathrm{T}}=\frac{{v}^{\mathrm{T}}}{c}\equiv\begin{bmatrix}% \beta_{x}&\beta_{y}&\beta_{z}\end{bmatrix}=\frac{1}{c}\begin{bmatrix}v_{x}&v_{% y}&v_{z}\end{bmatrix}\equiv\begin{bmatrix}\beta_{1}&\beta_{2}&\beta_{3}\end{% bmatrix}=\frac{1}{c}\begin{bmatrix}v_{1}&v_{2}&v_{3}\\ \end{bmatrix}
  45. β = | s y m b o l β | = β x 2 + β y 2 + β z 2 . \beta=|symbol{\beta}|=\sqrt{\beta_{x}^{2}+\beta_{y}^{2}+\beta_{z}^{2}}\,.
  46. [ c t x y z ] = [ γ - γ β x - γ β y - γ β z - γ β x 1 + ( γ - 1 ) β x 2 β 2 ( γ - 1 ) β x β y β 2 ( γ - 1 ) β x β z β 2 - γ β y ( γ - 1 ) β y β x β 2 1 + ( γ - 1 ) β y 2 β 2 ( γ - 1 ) β y β z β 2 - γ β z ( γ - 1 ) β z β x β 2 ( γ - 1 ) β z β y β 2 1 + ( γ - 1 ) β z 2 β 2 ] [ c t x y z ] . \begin{bmatrix}c\,t^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&-\gamma\,\beta_{x}&-\gamma\,% \beta_{y}&-\gamma\,\beta_{z}\\ -\gamma\,\beta_{x}&1+(\gamma-1)\dfrac{\beta_{x}^{2}}{\beta^{2}}&(\gamma-1)% \dfrac{\beta_{x}\beta_{y}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{x}\beta_{z}}{% \beta^{2}}\\ -\gamma\,\beta_{y}&(\gamma-1)\dfrac{\beta_{y}\beta_{x}}{\beta^{2}}&1+(\gamma-1% )\dfrac{\beta_{y}^{2}}{\beta^{2}}&(\gamma-1)\dfrac{\beta_{y}\beta_{z}}{\beta^{% 2}}\\ -\gamma\,\beta_{z}&(\gamma-1)\dfrac{\beta_{z}\beta_{x}}{\beta^{2}}&(\gamma-1)% \dfrac{\beta_{z}\beta_{y}}{\beta^{2}}&1+(\gamma-1)\dfrac{\beta_{z}^{2}}{\beta^% {2}}\\ \end{bmatrix}\begin{bmatrix}c\,t\\ x\\ y\\ z\end{bmatrix}\,.
  47. 𝐗 = s y m b o l Λ ( 𝐯 ) 𝐗 . \mathbf{X}^{\prime}=symbol{\Lambda}(\mathbf{v})\mathbf{X}.
  48. [ c t x y z ] = [ Λ 00 Λ 01 Λ 02 Λ 03 Λ 10 Λ 11 Λ 12 Λ 13 Λ 20 Λ 21 Λ 22 Λ 23 Λ 30 Λ 31 Λ 32 Λ 33 ] [ c t x y z ] . \begin{bmatrix}c\,t^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix}=\begin{bmatrix}\Lambda_{00}&\Lambda_{01}&\Lambda_{02}&% \Lambda_{03}\\ \Lambda_{10}&\Lambda_{11}&\Lambda_{12}&\Lambda_{13}\\ \Lambda_{20}&\Lambda_{21}&\Lambda_{22}&\Lambda_{23}\\ \Lambda_{30}&\Lambda_{31}&\Lambda_{32}&\Lambda_{33}\\ \end{bmatrix}\begin{bmatrix}c\,t\\ x\\ y\\ z\end{bmatrix}.
  49. Λ 00 = γ , Λ 0 i = Λ i 0 = - γ β i , Λ i j = Λ j i = ( γ - 1 ) β i β j β 2 + δ i j = ( γ - 1 ) v i v j v 2 + δ i j , \begin{aligned}\displaystyle\Lambda_{00}&\displaystyle=\gamma,\\ \displaystyle\Lambda_{0i}&\displaystyle=\Lambda_{i0}=-\gamma\beta_{i},\\ \displaystyle\Lambda_{ij}&\displaystyle=\Lambda_{ji}=(\gamma-1)\dfrac{\beta_{i% }\beta_{j}}{\beta^{2}}+\delta_{ij}=(\gamma-1)\dfrac{v_{i}v_{j}}{v^{2}}+\delta_% {ij},\\ \end{aligned}\,\!
  50. B ( 𝐮 ) B ( 𝐯 ) = B ( 𝐮 𝐯 ) Gyr [ 𝐮 , 𝐯 ] = Gyr [ 𝐮 , 𝐯 ] B ( 𝐯 𝐮 ) B(\mathbf{u})B(\mathbf{v})=B\left(\mathbf{u}\oplus\mathbf{v}\right)\mathrm{Gyr% }\left[\mathbf{u},\mathbf{v}\right]=\mathrm{Gyr}\left[\mathbf{u},\mathbf{v}% \right]B\left(\mathbf{v}\oplus\mathbf{u}\right)
  51. 𝐮 𝐯 \mathbf{u}\oplus\mathbf{v}
  52. Gyr [ 𝐮 , 𝐯 ] = ( 1 0 0 gyr [ 𝐮 , 𝐯 ] ) \mathrm{Gyr}[\mathbf{u},\mathbf{v}]=\begin{pmatrix}1&0\\ 0&\mathrm{gyr}[\mathbf{u},\mathbf{v}]\end{pmatrix}\,
  53. gyr [ 𝐮 , 𝐯 ] 𝐰 = ( 𝐮 𝐯 ) ( 𝐮 ( 𝐯 𝐰 ) ) \,\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))
  54. L ( 𝐮 , U ) L ( 𝐯 , V ) = L ( 𝐮 U 𝐯 , gyr [ 𝐮 , U 𝐯 ] U V ) L(\mathbf{u},U)L(\mathbf{v},V)=L(\mathbf{u}\oplus U\mathbf{v},\mathrm{gyr}[% \mathbf{u},U\mathbf{v}]UV)
  55. e ϕ = γ ( 1 + β ) = γ ( 1 + v c ) = 1 + v c 1 - v c , e^{\phi}=\gamma(1+\beta)=\gamma\left(1+\frac{v}{c}\right)=\sqrt{\frac{1+\tfrac% {v}{c}}{1-\tfrac{v}{c}}},
  56. e - ϕ = γ ( 1 - β ) = γ ( 1 - v c ) = 1 - v c 1 + v c . e^{-\phi}=\gamma(1-\beta)=\gamma\left(1-\frac{v}{c}\right)=\sqrt{\frac{1-% \tfrac{v}{c}}{1+\tfrac{v}{c}}}.
  57. ϕ = ln [ γ ( 1 + β ) ] = - ln [ γ ( 1 - β ) ] \phi=\ln\left[\gamma(1+\beta)\right]=-\ln\left[\gamma(1-\beta)\right]\,
  58. c t - x = e - ϕ ( c t - x ) \displaystyle ct-x=e^{-\phi}(ct^{\prime}-x^{\prime})
  59. γ = cosh ϕ = e ϕ + e - ϕ 2 , \gamma=\cosh\phi={e^{\phi}+e^{-\phi}\over 2},
  60. β γ = sinh ϕ = e ϕ - e - ϕ 2 , \beta\gamma=\sinh\phi={e^{\phi}-e^{-\phi}\over 2},
  61. β = tanh ϕ = e ϕ - e - ϕ e ϕ + e - ϕ . \beta=\tanh\phi={e^{\phi}-e^{-\phi}\over e^{\phi}+e^{-\phi}}.
  62. [ c t x y z ] = [ cosh ϕ - sinh ϕ 0 0 - sinh ϕ cosh ϕ 0 0 0 0 1 0 0 0 0 1 ] [ c t x y z ] . \begin{bmatrix}ct^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{bmatrix}=\begin{bmatrix}\cosh\phi&-\sinh\phi&0&0\\ -\sinh\phi&\cosh\phi&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}\begin{bmatrix}ct\\ x\\ y\\ z\end{bmatrix}\ .
  63. ϕ ϕ
  64. [ cosh ϕ - sinh ϕ 0 0 - sinh ϕ cosh ϕ 0 0 0 0 1 0 0 0 0 1 ] = exp ( - ϕ [ 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ] ) exp ( - ϕ K x ) , \begin{bmatrix}\cosh\phi&-\sinh\phi&0&0\\ -\sinh\phi&\cosh\phi&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}=\exp\left(-\phi\begin{bmatrix}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{bmatrix}\right)\equiv\exp(-\phi K_{x}),
  65. 𝐙 = s y m b o l Λ ( 𝐯 ) 𝐙 . \mathbf{Z}^{\prime}=symbol{\Lambda}(\mathbf{v})\mathbf{Z}.
  66. Z α = Λ α Z α α . Z^{\alpha^{\prime}}=\Lambda^{\alpha^{\prime}}{}_{\alpha}Z^{\alpha}\,.
  67. T θ ι κ α β ζ = Λ α Λ β μ ν Λ ζ Λ θ ρ Λ ι σ υ Λ κ T σ υ ϕ μ ν ρ ϕ T^{\alpha^{\prime}\beta^{\prime}\cdots\zeta^{\prime}}_{\theta^{\prime}\iota^{% \prime}\cdots\kappa^{\prime}}=\Lambda^{\alpha^{\prime}}{}_{\mu}\Lambda^{\beta^% {\prime}}{}_{\nu}\cdots\Lambda^{\zeta^{\prime}}{}_{\rho}\Lambda_{\theta^{% \prime}}{}^{\sigma}\Lambda_{\iota^{\prime}}{}^{\upsilon}\cdots\Lambda_{\kappa^% {\prime}}{}^{\phi}T^{\mu\nu\cdots\rho}_{\sigma\upsilon\cdots\phi}
  68. Λ χ ψ \Lambda_{\chi^{\prime}}{}^{\psi}\,
  69. Λ χ . ψ \Lambda^{\chi^{\prime}}{}_{\psi}\,.
  70. ( Δ t , Δ x , Δ y , Δ z ) = ( t B - t A , x B - x A , y B - y A , z B - z A ) , (\Delta t,\Delta x,\Delta y,\Delta z)=(t_{B}-t_{A},x_{B}-x_{A},y_{B}-y_{A},z_{% B}-z_{A})\ ,
  71. s 2 = - c 2 ( Δ t ) 2 + ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 . s^{2}=-c^{2}(\Delta t)^{2}+(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\ .
  72. η μ ν = [ - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] . \eta_{\mu\nu}=\begin{bmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\ .
  73. s 2 = [ c Δ t Δ x Δ y Δ z ] [ - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] [ c Δ t Δ x Δ y Δ z ] s^{2}=\begin{bmatrix}c\Delta t&\Delta x&\Delta y&\Delta z\end{bmatrix}\begin{% bmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}c\Delta t\\ \Delta x\\ \Delta y\\ \Delta z\end{bmatrix}
  74. s 2 = η μ ν x μ x ν . s^{2}=\eta_{\mu\nu}x^{\mu}x^{\nu}\ .
  75. s 2 = [ c Δ t Δ x Δ y Δ z ] [ - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] [ c Δ t Δ x Δ y Δ z ] s^{\prime 2}=\begin{bmatrix}c\Delta t^{\prime}&\Delta x^{\prime}&\Delta y^{% \prime}&\Delta z^{\prime}\end{bmatrix}\begin{bmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}c\Delta t^{\prime}\\ \Delta x^{\prime}\\ \Delta y^{\prime}\\ \Delta z^{\prime}\end{bmatrix}
  76. s 2 = η μ ν x μ x ν . s^{\prime 2}=\eta_{\mu\nu}x^{\prime\mu}x^{\prime\nu}\ .
  77. x μ = x ν Λ ν μ + C μ , x^{\prime\mu}=x^{\nu}\Lambda^{\mu}_{\nu}+C^{\mu}\ ,
  78. η μ ν Λ α μ Λ β ν = η α β . \eta_{\mu\nu}\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}=\eta_{\alpha\beta}\ .
  79. η μ ν Λ μ α Λ ν β = η α β \eta_{\mu\nu}{\Lambda^{\mu}}_{\alpha}{\Lambda^{\nu}}_{\beta}=\eta_{\alpha\beta}
  80. det ( Λ b a ) = ± 1 . \det(\Lambda^{a}_{b})=\pm 1\ .
  81. P 0 0 = 1 , P 1 1 = P 2 2 = P 3 3 = - 1 P^{0}_{0}=1,P^{1}_{1}=P^{2}_{2}=P^{3}_{3}=-1
  82. T 0 0 = - 1 , T 1 1 = T 2 2 = T 3 3 = 1 T^{0}_{0}=-1,T^{1}_{1}=T^{2}_{2}=T^{3}_{3}=1

Lorenz_curve.html

  1. F i = i / n F_{i}=i/n\,
  2. S i = Σ j = 1 i y j S_{i}=\Sigma_{j=1}^{i}\;y_{j}\,
  3. L i = S i / S n L_{i}=S_{i}/S_{n}\,
  4. F i = Σ j = 1 i f ( y j ) F_{i}=\Sigma_{j=1}^{i}\;f(y_{j})\,
  5. S i = Σ j = 1 i f ( y j ) y j S_{i}=\Sigma_{j=1}^{i}\;f(y_{j})\,y_{j}\,
  6. L i = S i / S n L_{i}=S_{i}/S_{n}\,
  7. L ( F ( x ) ) = - x t f ( t ) d t - t f ( t ) d t = - x t f ( t ) d t μ L(F(x))=\frac{\int_{-\infty}^{x}t\,f(t)\,dt}{\int_{-\infty}^{\infty}t\,f(t)\,% dt}=\frac{\int_{-\infty}^{x}t\,f(t)\,dt}{\mu}
  8. μ \mu
  9. L ( F ) = 0 F x ( F 1 ) d F 1 0 1 x ( F 1 ) d F 1 L(F)=\frac{\int_{0}^{F}x(F_{1})\,dF_{1}}{\int_{0}^{1}x(F_{1})\,dF_{1}}
  10. F - L X + c ( F ) = μ X μ X + c ( F - L X ( F ) ) F-L_{X+c}(F)=\frac{\mu_{X}}{\mu_{X}+c}(F-L_{X}(F))\,
  11. d L ( F ) d F = x ( F ) μ \frac{dL(F)}{dF}=\frac{x(F)}{\mu}
  12. d 2 L ( F ) d F 2 = 1 μ f ( x ( F ) ) \frac{d^{2}L(F)}{dF^{2}}=\frac{1}{\mu\,f(x(F))}\,
  13. F ( μ ) - L ( F ( μ ) ) = mean absolute deviation 2 μ F(\mu)-L(F(\mu))=\frac{\,\text{mean absolute deviation}}{2\,\mu}

Loudspeaker.html

  1. p ( θ ) = p 0 J 1 ( k a sin θ ) k a sin θ p(\theta)=\frac{p_{0}J_{1}(k_{a}\sin\theta)}{k_{a}\sin\theta}
  2. k a = 2 π a λ k_{a}=\frac{2\pi a}{\lambda}
  3. p 0 p_{0}
  4. a a
  5. λ \lambda
  6. λ = c f = speed of sound frequency \lambda=\frac{c}{f}=\frac{\,\text{speed of sound}}{\,\text{frequency}}
  7. θ \theta
  8. J 1 J_{1}

Louis_de_Broglie.html

  1. λ = h p = h m v 1 - v 2 c 2 \lambda=\frac{h}{p}=\frac{h}{{m}{v}}\sqrt{1-\frac{v^{2}}{c^{2}}}
  2. λ \lambda
  3. h h
  4. p p
  5. m m
  6. v v
  7. c c
  8. m c 2 = h ν mc^{2}=h\nu
  9. action h = - entropy k {\,\text{action}\over h}=-{\,\text{entropy}\over k}

Low-density_lipoprotein.html

  1. L C - H - k T L\approx C-H-kT
  2. L = C - H - 0.16 T L=C-H-0.16T

Low-pass_filter.html

  1. n \scriptstyle n
  2. 6 n \scriptstyle 6n
  3. 20 n \scriptstyle 20n
  4. Output Input = K 1 τ s + 1 \frac{\,\text{Output}}{\,\text{Input}}=K\frac{1}{\tau s+1}
  5. τ = R C \scriptstyle\tau\;=\;RC
  6. f c = 1 2 π τ = 1 2 π R C f_{\mathrm{c}}={1\over 2\pi\tau}={1\over 2\pi RC}
  7. ω c = 1 τ = 1 R C \omega_{\mathrm{c}}={1\over\tau}={1\over RC}
  8. V out \scriptstyle V_{\mathrm{out}}
  9. f c = 1 2 π R 2 C f_{\,\text{c}}=\frac{1}{2\pi R_{2}C}
  10. ω c = 1 R 2 C \omega_{\,\text{c}}=\frac{1}{R_{2}C}
  11. v in ( t ) - v out ( t ) = R i ( t ) v_{\,\text{in}}(t)-v_{\,\text{out}}(t)=R\;i(t)
  12. Q c ( t ) = C v out ( t ) Q_{c}(t)=C\,v_{\,\text{out}}(t)
  13. i ( t ) = d Q c d t i(t)=\frac{\operatorname{d}Q_{c}}{\operatorname{d}t}
  14. Q c ( t ) Q_{c}(t)
  15. t \scriptstyle t
  16. i ( t ) = C d v out d t \scriptstyle i(t)\;=\;C\frac{\operatorname{d}v_{\,\text{out}}}{\operatorname{d% }t}
  17. v in ( t ) - v out ( t ) = R C d v out d t v_{\,\text{in}}(t)-v_{\,\text{out}}(t)=RC\frac{\operatorname{d}v_{\,\text{out}% }}{\operatorname{d}t}
  18. Δ T \scriptstyle\Delta_{T}
  19. v in \scriptstyle v_{\,\text{in}}
  20. ( x 1 , x 2 , , x n ) \scriptstyle(x_{1},\,x_{2},\,\ldots,\,x_{n})
  21. v out \scriptstyle v_{\,\text{out}}
  22. ( y 1 , y 2 , , y n ) \scriptstyle(y_{1},\,y_{2},\,\ldots,\,y_{n})
  23. x i - y i = R C y i - y i - 1 Δ T x_{i}-y_{i}=RC\,\frac{y_{i}-y_{i-1}}{\Delta_{T}}
  24. y i = x i ( Δ T R C + Δ T ) Input contribution + y i - 1 ( R C R C + Δ T ) Inertia from previous output . y_{i}=\overbrace{x_{i}\left(\frac{\Delta_{T}}{RC+\Delta_{T}}\right)}^{\,\text{% Input contribution}}+\overbrace{y_{i-1}\left(\frac{RC}{RC+\Delta_{T}}\right)}^% {\,\text{Inertia from previous output}}.
  25. y i = α x i + ( 1 - α ) y i - 1 where α Δ T R C + Δ T y_{i}=\alpha x_{i}+(1-\alpha)y_{i-1}\qquad\,\text{where}\qquad\alpha\triangleq% \frac{\Delta_{T}}{RC+\Delta_{T}}
  26. 0 α 1 \scriptstyle 0\;\leq\;\alpha\;\leq\;1
  27. α \scriptstyle\alpha
  28. R C \scriptstyle RC
  29. Δ T \scriptstyle\Delta_{T}
  30. α \scriptstyle\alpha
  31. R C = Δ T ( 1 - α α ) RC=\Delta_{T}\left(\frac{1-\alpha}{\alpha}\right)
  32. f c = 1 2 π R C f_{c}=\frac{1}{2\pi RC}
  33. R C = 1 2 π f c RC=\frac{1}{2\pi f_{c}}
  34. α \alpha
  35. f c f_{c}
  36. α = 2 π Δ T f c 2 π Δ T f c + 1 \alpha=\frac{2\pi\Delta_{T}f_{c}}{2\pi\Delta_{T}f_{c}+1}
  37. f c = α ( 1 - α ) 2 π Δ T f_{c}=\frac{\alpha}{(1-\alpha)2\pi\Delta_{T}}
  38. α = 0.5 \scriptstyle\alpha\;=\;0.5
  39. R C \scriptstyle RC
  40. α 0.5 \scriptstyle\alpha\;\ll\;0.5
  41. R C \scriptstyle RC
  42. Δ T α R C \scriptstyle\Delta_{T}\;\approx\;\alpha RC
  43. R C \scriptstyle RC
  44. α \scriptstyle\alpha
  45. ( y 1 , y 2 , , y n ) \scriptstyle(y_{1},\,y_{2},\,\ldots,\,y_{n})
  46. ( x 1 , x 2 , , x n ) \scriptstyle(x_{1},\,x_{2},\,\ldots,\,x_{n})

Lp_space.html

  1. p p
  2. n n
  3. x 2 = ( x 1 2 + x 2 2 + + x n 2 ) 1 2 \ \|x\|_{2}=\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)^{\frac{1}{2}}
  4. x x
  5. y y
  6. p p
  7. p 1 p≥1
  8. p p
  9. x x
  10. x p = ( | x 1 | p + | x 2 | p + + | x n | p ) 1 p \ \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\cdots+|x_{n}|^{p}\right)^{\frac{1}{% p}}
  11. p p→∞
  12. x = max { | x 1 | , | x 2 | , , | x n | } \ \|x\|_{\infty}=\max\left\{|x_{1}|,|x_{2}|,\ldots,|x_{n}|\right\}
  13. p 1 p≥1
  14. p p
  15. p p
  16. p p
  17. x 2 x 1 \|x\|_{2}\leq\|x\|_{1}
  18. p p
  19. p p
  20. x x
  21. p p
  22. x x
  23. p 1 p≥1
  24. a 0 a≥0
  25. x 1 n x 2 \|x\|_{1}\leq\sqrt{n}\|x\|_{2}
  26. n n
  27. n > 1 n>1
  28. x p = ( | x 1 | p + | x 2 | p + + | x n | p ) 1 p \|x\|_{p}=\left(\left|x_{1}\right|^{p}+|x_{2}|^{p}+\cdots+|x_{n}|^{p}\right)^{% \frac{1}{p}}
  29. ( 1 , 1 , 1 , ) (1,1,1,...)
  30. p p
  31. p p
  32. p p
  33. ( 1 , 1 2 , , 1 n , 1 n + 1 , ) \left(1,\frac{1}{2},\ldots,\frac{1}{n},\frac{1}{n+1},\ldots\right)
  34. p > 1 p>1
  35. 1 p + 1 2 p + + 1 n p + 1 ( n + 1 ) p + , 1^{p}+\frac{1}{2^{p}}+\cdots+\frac{1}{n^{p}}+\frac{1}{(n+1)^{p}}+\cdots,
  36. p = 1 p=1
  37. p > 1 p>1
  38. x = sup ( | x 1 | , | x 2 | , , | x n | , | x n + 1 | , ) \ \|x\|_{\infty}=\sup(|x_{1}|,|x_{2}|,\ldots,|x_{n}|,|x_{n+1}|,\ldots)
  39. x = lim p x p \ \|x\|_{\infty}=\lim_{p\to\infty}\|x\|_{p}
  40. 1 p 1≤p≤∞
  41. p p
  42. p p
  43. ( S , Σ , μ ) (S,Σ,μ)
  44. S S
  45. 𝐂 \mathbf{C}
  46. 𝐑 \mathbf{R}
  47. p p
  48. f p ( S | f | p d μ ) 1 p < \|f\|_{p}\equiv\left({\int_{S}|f|^{p}\;\mathrm{d}\mu}\right)^{\frac{1}{p}}<\infty
  49. ( f + g ) ( x ) \displaystyle(f+g)(x)
  50. λ λ
  51. p p
  52. p p
  53. f + g p p 2 p - 1 ( f p p + g p p ) . \left\|f+g\right\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right).
  54. t t p t\mapsto t^{p}
  55. p 1 p\geq 1
  56. p p
  57. p ( S , μ ) \scriptstyle\mathcal{L}^{p}(S,\,\mu)
  58. f f
  59. f = 0 f=0
  60. p p
  61. 𝒩 ker ( p ) = { f : f = 0 μ -almost everywhere } \mathcal{N}\equiv\mathrm{ker}(\|\cdot\|_{p})=\{f:f=0\ \mu\,\text{-almost % everywhere}\}
  62. f f
  63. g g
  64. f = g f=g
  65. L p ( S , μ ) p ( S , μ ) / 𝒩 L^{p}(S,\mu)\equiv\mathcal{L}^{p}(S,\mu)/\mathcal{N}
  66. p = p=∞
  67. S S
  68. 𝐂 \mathbf{C}
  69. 𝐑 \mathbf{R}
  70. f f
  71. f inf { C 0 : | f ( x ) | C for almost every x } . \|f\|_{\infty}\equiv\inf\{C\geq 0:|f(x)|\leq C\mbox{ for almost every }~{}x\}.
  72. f = lim p f p . \|f\|_{\infty}=\lim_{p\to\infty}\|f\|_{p}.
  73. S S
  74. f , g = S f ( x ) g ( x ) ¯ d μ ( x ) \langle f,g\rangle=\int_{S}f(x)\overline{g(x)}\,\mathrm{d}\mu(x)
  75. 1 p 1≤p≤∞
  76. S = 𝐍 S=\mathbf{N}
  77. μ μ
  78. 𝐍 \mathbf{N}
  79. S S
  80. p p
  81. n n
  82. n n
  83. p p
  84. I I
  85. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  86. f κ p ( g ) ( f ) = f g d μ f\mapsto\kappa_{p}(g)(f)=\int fg\,\mathrm{d}\mu\
  87. f L p ( μ ) f\in L^{p}(\mu)
  88. I q , p = μ ( S ) 1 p - 1 q \|I\|_{q,p}=\mu(S)^{\frac{1}{p}-\frac{1}{q}}
  89. f = 1 f=1
  90. μ μ
  91. ( S , Σ , μ ) (S,Σ,μ)
  92. f f
  93. S S
  94. f = j = 1 n a j 𝟏 A j f=\sum_{j=1}^{n}a_{j}\mathbf{1}_{A_{j}}
  95. 𝟏 A j {\mathbf{1}}_{A_{j}}
  96. A j A_{j}
  97. j = 1 , , n j=1,...,n
  98. S S
  99. Σ Σ
  100. σ σ
  101. σ σ
  102. S S
  103. V S V⊂S
  104. A Σ A∈Σ
  105. V V
  106. ε > 0 ε>0
  107. F F
  108. U U
  109. F A U V and μ ( U ) - μ ( F ) = μ ( U F ) < ε F\subset A\subset U\subset V\quad\,\text{and}\quad\mu(U)-\mu(F)=\mu(U\setminus F% )<\varepsilon
  110. φ φ
  111. S S
  112. 0 φ 𝟏 V and S | 𝟏 A - φ | d μ < ε 0\leq\varphi\leq\mathbf{1}_{V}\quad\,\text{and}\quad\int_{S}|\mathbf{1}_{A}-% \varphi|\mathrm{d}\mu<\varepsilon
  113. S S
  114. p p
  115. μ μ
  116. d = 1 d=1
  117. d = 2 d=2
  118. f L p ( 𝐑 d ) : τ t f - f p 0 , as 𝐑 d t 0 , \forall f\in L^{p}(\mathbf{R}^{d}):\qquad\left\|\tau_{t}f-f\right\|_{p}\to 0,% \quad\,\text{ as }\mathbf{R}^{d}\ni t\to 0,
  119. ( τ t f ) ( x ) = f ( x - t ) . \left(\tau_{t}f\right)(x)=f(x-t).
  120. a , b 0 a,b≥0
  121. N p ( f + g ) N p ( f ) + N p ( g ) N_{p}(f+g)\leq N_{p}(f)+N_{p}(g)
  122. d p ( f , g ) = N p ( f - g ) = f - g p p d_{p}(f,g)=N_{p}(f-g)=\|f-g\|_{p}^{p}
  123. p 1 p≥1
  124. u , v u,v
  125. | u | + | v | p u p + v p \left\||u|+|v|\right\|_{p}\geq\|u\|_{p}+\|v\|_{p}
  126. 0
  127. p p
  128. 0
  129. S S
  130. ( S , Σ , μ ) (S,Σ,μ)
  131. μ μ
  132. μ ( S ) = 1 μ(S)=1
  133. μ μ
  134. μ μ
  135. ( S , Σ ) (S,Σ)
  136. 0
  137. V ε = { f : μ ( { x : | f ( x ) | > ε } ) < ε } , ε > 0 V_{\varepsilon}=\Bigl\{f:\mu\bigl(\{x:|f(x)|>\varepsilon\}\bigr)<\varepsilon% \Bigr\},\qquad\varepsilon>0
  138. d d
  139. d ( f , g ) = S φ ( | f ( x ) - g ( x ) | ) d μ ( x ) d(f,g)=\int_{S}\varphi\bigl(|f(x)-g(x)|\bigr)\,\mathrm{d}\mu(x)
  140. φ φ
  141. 0 , ) ) 0,∞))
  142. φ ( 0 ) = 0 φ(0)=0
  143. φ ( t ) > 0 φ(t)>0
  144. t > 0 t>0
  145. φ ( t ) = m i n ( t , 1 ) ) φ(t)=min(t,1))
  146. λ λ
  147. W ε = { f : λ ( { x : | f ( x ) | > ε and | x | < 1 ε } ) < ε } W_{\varepsilon}=\left\{f:\lambda\left(\left\{x:|f(x)|>\varepsilon\,\text{ and % }|x|<\frac{1}{\varepsilon}\right\}\right)<\varepsilon\right\}
  148. λ λ
  149. g g
  150. λ f ( t ) = μ { x S : | f ( x ) | > t } \lambda_{f}(t)=\mu\left\{x\in S:|f(x)|>t\right\}
  151. λ f ( t ) C p t p \lambda_{f}(t)\leq\frac{C^{p}}{t^{p}}
  152. f p , w = sup t > 0 t λ f 1 p ( t ) \|f\|_{p,w}=\sup_{t>0}~{}t\lambda_{f}^{\frac{1}{p}}(t)
  153. f p , w f p \|f\|_{p,w}\leq\|f\|_{p}
  154. || | f | || L p , = sup 0 < μ ( E ) < μ ( E ) - 1 r + 1 p ( E | f | r d μ ) 1 r |||f|||_{L^{p,\infty}}=\sup_{0<\mu(E)<\infty}\mu(E)^{-\frac{1}{r}+\frac{1}{p}}% \left(\int_{E}|f|^{r}\,d\mu\right)^{\frac{1}{r}}
  155. ( S , Σ , μ ) (S,Σ,μ)
  156. w : S 0 , ) ) w:S→0,∞))
  157. w w
  158. w d μ wdμ
  159. ν ν
  160. ν ( A ) A w ( x ) d μ ( x ) , A Σ , \nu(A)\equiv\int_{A}w(x)\,\mathrm{d}\mu(x),\qquad A\in\Sigma,
  161. w = d ν d w=d\frac{ν}{d}
  162. u L p ( S , w d μ ) ( S w ( x ) | u ( x ) | p d μ ( x ) ) 1 p \|u\|_{L^{p}(S,w\,\mathrm{d}\mu)}\equiv\left(\int_{S}w(x)|u(x)|^{p}\,\mathrm{d% }\mu(x)\right)^{\frac{1}{p}}
  163. λ λ
  164. w w
  165. ( L loc 1 ) \left(\scriptstyle L^{1}_{\,\text{loc}}\right)
  166. L p ( G ) \scriptstyle L^{p}(G)
  167. G G

Lua_(programming_language).html

  1. n n
  2. n n

Ludwig_von_Bertalanffy.html

  1. L ( t ) = r B ( L - L ( t ) ) L^{\prime}(t)=r_{B}\left(L_{\infty}-L(t)\right)
  2. r B r_{B}
  3. L L_{\infty}

Ludwig_Wittgenstein.html

  1. [ p ¯ , ξ ¯ , N ( ξ ¯ ) ] [\bar{p},\bar{\xi},N(\bar{\xi})]
  2. [ p ¯ , ξ ¯ , N ( ξ ¯ ) ] [\bar{p},\bar{\xi},N(\bar{\xi})]

Luminance.html

  1. L v = d 2 Φ v d A d Ω cos θ L_{\mathrm{v}}=\frac{\mathrm{d}^{2}\Phi_{\mathrm{v}}}{\mathrm{d}A\,\mathrm{d}{% \Omega}\cos\theta}
  2. L v L_{\mathrm{v}}
  3. Φ v \Phi_{\mathrm{v}}
  4. θ \theta\,
  5. A A
  6. Ω \Omega\,

Luminiferous_aether.html

  1. c n + ( 1 - 1 n 2 ) v . \frac{c}{n}+\left(1-\frac{1}{n^{2}}\right)v.
  2. v / c v/c
  3. v / c v/c

Luminosity.html

  1. L ν = S obs 4 π D L 2 ( 1 + z ) 1 + α L_{\nu}=\frac{S_{\mathrm{obs}}4\pi{D_{L}}^{2}}{(1+z)^{1+\alpha}}
  2. I ν α I\propto{\nu}^{\alpha}
  3. L = σ A T 4 L=\sigma AT^{4}
  4. L L
  5. F = L A F=\frac{L}{A}
  6. A A
  7. F F
  8. A = 4 π r 2 A=4\pi r^{2}
  9. F = L 4 π r 2 F=\frac{L}{4\pi r^{2}}\,
  10. r r
  11. L L
  12. T T
  13. R R
  14. L = 4 π R 2 σ T 4 L=4\pi R^{2}\sigma T^{4}\,
  15. × 10 8 \times 10^{−}8
  16. L L_{\odot}
  17. L L = ( R R ) 2 ( T T ) 4 \frac{L}{L_{\odot}}={\left(\frac{R}{R_{\odot}}\right)}^{2}{\left(\frac{T}{T_{% \odot}}\right)}^{4}
  18. R R_{\odot}
  19. T T_{\odot}
  20. L L ( M M ) 3.9 \frac{L}{L_{\odot}}\approx{\left(\frac{M}{M_{\odot}}\right)}^{3.9}
  21. m star = m - 2.5 log 10 [ L star L ( d d star ) 2 ] m_{\rm star}=m_{\odot}-2.5\log_{10}\left[\frac{L_{\rm star}}{L_{\odot}}\left(% \frac{d_{\odot}}{d_{\rm star}}\right)^{2}\right]
  22. m star m\text{star}
  23. m m_{\odot}
  24. L star L\text{star}
  25. L L_{\odot}
  26. d star d\text{star}
  27. d d_{\odot}
  28. m = - 26.73 m_{\odot}=-26.73
  29. d = 1.58 × 10 - 5 lyr d_{\odot}=1.58{\times}10^{-5}\,\,\text{lyr}
  30. m star = - 2.72 - 2.5 log ( L star / d star 2 ) m\text{star}=-2.72-2.5\,\log(L\text{star}/d\text{star}^{2})
  31. L star L\text{star}
  32. L L_{\odot}
  33. M bol,star - M bol , = - 2.5 log 10 L star L M\text{bol,star}-M_{\,\text{bol},\odot}=-2.5\log_{10}\frac{L\text{star}}{L_{% \odot}}
  34. L star L = 10 ( M bol , - M bol,star ) / 2.5 \frac{L\text{star}}{L_{\odot}}=10^{(M_{\,\text{bol},\odot}-M\text{bol,star})/2% .5}
  35. L L_{\odot}
  36. L star L\text{star}
  37. M bol , M_{\,\text{bol},\odot}
  38. M bol,star M\text{bol,star}
  39. R R_{\odot}
  40. R R_{\odot}
  41. L L_{\odot}
  42. I f : δ = d B D B T h e n : d B = δ D B A n d : R B = ( δ D B 2 ) If:{\delta}=\frac{d_{B}}{D_{B}}\qquad Then:d_{B}=\delta\cdot D_{B}\quad And:R_% {B}={\left({\frac{\delta\cdot D_{B}}{2}}\right)}
  43. δ {\delta}
  44. D B {D_{B}}
  45. d B {d_{B}}
  46. R B {R_{B}}
  47. S c e n a r i o I : R B = ( 0.05660 197.0 2 ) = 5.582 A U = 5.6 A U Scenario\quad I:\qquad R_{B}={\left({\frac{{0.05660}\cdot 197.0}{2}}\right)}=5% .582AU=5.6AU
  48. S c e n a r i o I I : R B = ( 0.04333 152.0 2 ) = 3.308 A U = 3.3 A U Scenario\ II:\qquad R_{B}={\left({\frac{{0.04333}\cdot 152.0}{2}}\right)}=3.30% 8AU=3.3AU
  49. Scenario I : d B = ( 149 , 597 , 871 km 696 , 000 km ) ( 5.6 AU ) = 1 , 204 R ( rounded ) \,\text{Scenario I}:\qquad d_{B}={\left({\frac{149,597,871\,\textrm{km}}{696,0% 00\,\textrm{km}}}\right)}{\left(5.6\,\textrm{AU}\right)}=1,204R_{\odot}\quad(% \,\text{rounded})
  50. Scenario II : d B = ( 149 , 597 , 871 km 696 , 000 km ) ( 3.3 AU ) = 710 R ( rounded ) \,\text{Scenario II}:\qquad d_{B}={\left({\frac{149,597,871\,\textrm{km}}{696,% 000\,\textrm{km}}}\right)}{\left(3.3\,\textrm{AU}\right)}=\quad 710R_{\odot}% \quad(\,\text{rounded})
  51. S c e n a r i o I : L B L = ( 1 , 204 1 ) 2 ( 3 , 300 5 , 778 ) 4 = 154 , 000 L ( r o u n d e d ) Scenario\quad I:\qquad\frac{L_{\rm B}}{L_{\odot}}={\left({\frac{1,204}{1}}% \right)}^{2}{\left({\frac{3,300}{5,778}}\right)}^{4}=154,000L_{\odot}(rounded)
  52. S c e n a r i o I I : L B L = ( 730 1 ) 2 ( 3 , 641 5 , 778 ) 4 = 84 , 000 L ( r o u n d e d ) Scenario\ II:\qquad\frac{L_{\rm B}}{L_{\odot}}=\quad{\left({\frac{730}{1}}% \right)}^{2}{\left({\frac{3,641}{5,778}}\right)}^{4}=\ 84,000L_{\odot}(rounded)