wpmath0000001_10

Gauss–Legendre_algorithm.html

  1. a 0 = 1 b 0 = 1 2 t 0 = 1 4 p 0 = 1. a_{0}=1\qquad b_{0}=\frac{1}{\sqrt{2}}\qquad t_{0}=\frac{1}{4}\qquad p_{0}=1.\!
  2. a n a_{n}\!
  3. b n b_{n}\!
  4. a n + 1 = a n + b n 2 , < c o d e > b n + 1 = a n b n , < / c o d e > < b r / > < c o d e > t n + 1 = t n - p n ( a n - a n + 1 ) 2 , < / c o d e > < b r / > < c o d e > p n + 1 = 2 p n . < / c o d e > < b r / > < c o d e > < / c o d e > \begin{aligned}\displaystyle a_{n+1}&\displaystyle=\frac{a_{n}+b_{n}}{2},\\ \displaystyle\par <code>                     b_{n+1}&\displaystyle = \sqrt{a_{% n} b_{n}},\\ \displaystyle</code><br/><code>                     t_{n+1}&\displaystyle = t_% {n} - p_{n}(a_{n}-a_{n+1})^{2},\\ \displaystyle</code><br/><code>                     p_{n+1}&\displaystyle = 2p% _{n}.</code><br/><code>\end{aligned}</code>
  5. π ( a n + 1 + b n + 1 ) 2 4 t n + 1 . \pi\approx\frac{(a_{n+1}+b_{n+1})^{2}}{4t_{n+1}}.\!
  6. 3.140 3.140\dots\!
  7. 3.14159264 3.14159264\dots\!
  8. 3.1415926535897932382 3.1415926535897932382\dots\!
  9. a n + 1 = a n + b n 2 , b n + 1 = a n b n , \begin{aligned}\displaystyle a_{n+1}&\displaystyle=\frac{a_{n}+b_{n}}{2},\\ \displaystyle b_{n+1}&\displaystyle=\sqrt{a_{n}b_{n}},\end{aligned}
  10. a 0 = 1 a_{0}=1\!
  11. b 0 = cos φ b_{0}=\cos\varphi\!
  12. π 2 K ( sin φ ) {\pi\over 2K(\sin\varphi)}\!
  13. K ( k ) K(k)\!
  14. K ( k ) = 0 π / 2 d θ 1 - k 2 sin 2 θ . K(k)=\int_{0}^{\pi/2}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}.\!
  15. c 0 = sin φ c_{0}=\sin\varphi\!
  16. c i + 1 = a i - a i + 1 c_{i+1}=a_{i}-a_{i+1}\!
  17. i = 0 2 i - 1 c i 2 = 1 - E ( sin φ ) K ( sin φ ) \sum_{i=0}^{\infty}2^{i-1}c_{i}^{2}=1-{E(\sin\varphi)\over K(\sin\varphi)}\!
  18. E ( k ) E(k)\!
  19. E ( k ) = 0 π / 2 1 - k 2 sin 2 θ d θ . E(k)=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}\theta}\,d\theta.\!
  20. φ \varphi\!
  21. θ \theta\!
  22. φ + θ = 1 2 π \varphi+\theta={1\over 2}\pi\!
  23. K ( sin φ ) E ( sin θ ) + K ( sin θ ) E ( sin φ ) - K ( sin φ ) K ( sin θ ) = 1 2 π . K(\sin\varphi)E(\sin\theta)+K(\sin\theta)E(\sin\varphi)-K(\sin\varphi)K(\sin% \theta)={1\over 2}\pi.\!
  24. φ = θ = π 4 \varphi=\theta={\pi\over 4}\!
  25. a 0 = 1 a_{0}=1\!
  26. b 0 = sin π 4 = 1 2 b_{0}=\sin{\pi\over 4}=\frac{1}{\sqrt{2}}\!

Gematria.html

  1. f ( x ) = ( 10 f l o o r ( ( x - 1 ) ÷ 9 ) ) × ( ( ( x - 1 ) r e m 9 ) + 1 ) f(x)=\left(10^{floor\left(\left(x-1\right)\div 9\right)}\right)\times\left(% \left(\left(x-1\right)\ rem\ 9\right)+1\right)

General_relativity.html

  1. R μ ν R_{\mu\nu}
  2. G μ ν G_{\mu\nu}
  3. R = g μ ν R μ ν R=g^{\mu\nu}R_{\mu\nu}\,
  4. R μ ν = R α μ α ν . R_{\mu\nu}={R^{\alpha}}_{\mu\alpha\nu}.\,
  5. T μ ν T_{\mu\nu}
  6. R μ ν = 0. R_{\mu\nu}=0.\,
  7. 10 - 21 10^{-21}
  8. R μ ν - 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν R_{\mu\nu}-{\textstyle 1\over 2}R\,g_{\mu\nu}+\Lambda\ g_{\mu\nu}=\frac{8\pi G% }{c^{4}}\,T_{\mu\nu}
  9. g μ ν g_{\mu\nu}
  10. 10 - 33 10^{-33}

Generalized_mean.html

  1. x 1 , , x n x_{1},\dots,x_{n}
  2. M p ( x 1 , , x n ) = ( 1 n i = 1 n x i p ) 1 p . M_{p}(x_{1},\dots,x_{n})=\left(\frac{1}{n}\sum_{i=1}^{n}x_{i}^{p}\right)^{% \frac{1}{p}}.
  3. M 0 ( x 1 , , x n ) = i = 1 n x i n M_{0}(x_{1},\dots,x_{n})=\sqrt[n]{\prod_{i=1}^{n}x_{i}}
  4. w i = 1 \sum w_{i}=1
  5. M p ( x 1 , , x n ) \displaystyle M_{p}(x_{1},\dots,x_{n})
  6. M ( x 1 , , x n ) \displaystyle M_{\infty}(x_{1},\dots,x_{n})
  7. lim p 0 M p = M 0 \textstyle\lim_{p\to 0}M_{p}=M_{0}
  8. M p ( x 1 , , x n ) = exp ( ln [ ( i = 1 n w i x i p ) 1 / p ] ) = exp ( ln ( i = 1 n w i x i p ) p ) M_{p}(x_{1},\dots,x_{n})=\exp{\left(\ln{\left[\left(\sum_{i=1}^{n}w_{i}x_{i}^{% p}\right)^{1/p}\right]}\right)}=\exp{\left(\frac{\ln{\left(\sum_{i=1}^{n}w_{i}% x_{i}^{p}\right)}}{p}\right)}
  9. lim p 0 ln ( i = 1 n w i x i p ) p = lim p 0 i = 1 n w i x i p ln x i i = 1 n w i x i p 1 = lim p 0 i = 1 n w i x i p ln x i i = 1 n w i x i p = i = 1 n w i ln x i = ln ( i = 1 n x i w i ) \lim_{p\to 0}\frac{\ln{\left(\sum_{i=1}^{n}w_{i}x_{i}^{p}\right)}}{p}=\lim_{p% \to 0}\frac{\frac{\sum_{i=1}^{n}w_{i}x_{i}^{p}\ln{x_{i}}}{\sum_{i=1}^{n}w_{i}x% _{i}^{p}}}{1}=\lim_{p\to 0}\frac{\sum_{i=1}^{n}w_{i}x_{i}^{p}\ln{x_{i}}}{\sum_% {i=1}^{n}w_{i}x_{i}^{p}}=\sum_{i=1}^{n}w_{i}\ln{x_{i}}=\ln{\left(\prod_{i=1}^{% n}x_{i}^{w_{i}}\right)}
  10. lim p 0 M p ( x 1 , , x n ) = exp ( ln ( i = 1 n x i w i ) ) = i = 1 n x i w i = M 0 ( x 1 , , x n ) \lim_{p\to 0}M_{p}(x_{1},\dots,x_{n})=\exp{\left(\ln{\left(\prod_{i=1}^{n}x_{i% }^{w_{i}}\right)}\right)}=\prod_{i=1}^{n}x_{i}^{w_{i}}=M_{0}(x_{1},\dots,x_{n})
  11. lim p M p = M \textstyle\lim_{p\to\infty}M_{p}=M_{\infty}
  12. lim p - M p = M - \textstyle\lim_{p\to-\infty}M_{p}=M_{-\infty}
  13. x 1 x n x_{1}\geq\dots\geq x_{n}
  14. lim p M p ( x 1 , , x n ) = lim p ( i = 1 n w i x i p ) 1 / p = x 1 lim p ( i = 1 n w i ( x i x 1 ) p ) 1 / p = x 1 = M ( x 1 , , x n ) . \lim_{p\to\infty}M_{p}(x_{1},\dots,x_{n})=\lim_{p\to\infty}\left(\sum_{i=1}^{n% }w_{i}x_{i}^{p}\right)^{1/p}=x_{1}\lim_{p\to\infty}\left(\sum_{i=1}^{n}w_{i}% \left(\frac{x_{i}}{x_{1}}\right)^{p}\right)^{1/p}=x_{1}=M_{\infty}(x_{1},\dots% ,x_{n}).
  15. M - M_{-\infty}
  16. M - ( x 1 , , x n ) = 1 M ( 1 / x 1 , , 1 / x n ) . M_{-\infty}(x_{1},\dots,x_{n})=\frac{1}{M_{\infty}(1/x_{1},\dots,1/x_{n})}.
  17. b x 1 , , b x n b\cdot x_{1},\dots,b\cdot x_{n}
  18. M p ( x 1 , , x n k ) = M p ( M p ( x 1 , , x k ) , M p ( x k + 1 , , x 2 k ) , , M p ( x ( n - 1 ) k + 1 , , x n k ) ) M_{p}(x_{1},\dots,x_{n\cdot k})=M_{p}(M_{p}(x_{1},\dots,x_{k}),M_{p}(x_{k+1},% \dots,x_{2\cdot k}),\dots,M_{p}(x_{(n-1)\cdot k+1},\dots,x_{n\cdot k}))
  19. p M p ( x 1 , , x n ) 0 \frac{\partial}{\partial p}M_{p}(x_{1},\dots,x_{n})\geq 0
  20. M - ( x 1 , , x n ) = lim p - M p ( x 1 , , x n ) = min { x 1 , , x n } M_{-\infty}(x_{1},\dots,x_{n})=\lim_{p\to-\infty}M_{p}(x_{1},\dots,x_{n})=\min% \{x_{1},\dots,x_{n}\}
  21. M - 1 ( x 1 , , x n ) = n 1 x 1 + + 1 x n M_{-1}(x_{1},\dots,x_{n})=\frac{n}{\frac{1}{x_{1}}+\dots+\frac{1}{x_{n}}}
  22. M 0 ( x 1 , , x n ) = lim p 0 M p ( x 1 , , x n ) = x 1 x n n M_{0}(x_{1},\dots,x_{n})=\lim_{p\to 0}M_{p}(x_{1},\dots,x_{n})=\sqrt[n]{x_{1}% \cdot\dots\cdot x_{n}}
  23. M 1 ( x 1 , , x n ) = x 1 + + x n n M_{1}(x_{1},\dots,x_{n})=\frac{x_{1}+\dots+x_{n}}{n}
  24. M 2 ( x 1 , , x n ) = x 1 2 + + x n 2 n M_{2}(x_{1},\dots,x_{n})=\sqrt{\frac{x_{1}^{2}+\dots+x_{n}^{2}}{n}}
  25. M 3 ( x 1 , , x n ) = x 1 3 + + x n 3 n 3 M_{3}(x_{1},\dots,x_{n})=\sqrt[3]{\frac{x_{1}^{3}+\dots+x_{n}^{3}}{n}}
  26. M + ( x 1 , , x n ) = lim p M p ( x 1 , , x n ) = max { x 1 , , x n } M_{+\infty}(x_{1},\dots,x_{n})=\lim_{p\to\infty}M_{p}(x_{1},\dots,x_{n})=\max% \{x_{1},\dots,x_{n}\}
  27. w i [ 0 ; 1 ] \displaystyle w_{i}\in[0;1]
  28. i = 1 n w i x i p p i = 1 n w i x i q q \sqrt[p]{\sum_{i=1}^{n}w_{i}x_{i}^{p}}\geq\sqrt[q]{\sum_{i=1}^{n}w_{i}x_{i}^{q}}
  29. i = 1 n w i x i p p i = 1 n w i x i q q \sqrt[p]{\sum_{i=1}^{n}\frac{w_{i}}{x_{i}^{p}}}\geq\sqrt[q]{\sum_{i=1}^{n}% \frac{w_{i}}{x_{i}^{q}}}
  30. i = 1 n w i x i - p - p = 1 i = 1 n w i 1 x i p p 1 i = 1 n w i 1 x i q q = i = 1 n w i x i - q - q \sqrt[-p]{\sum_{i=1}^{n}w_{i}x_{i}^{-p}}=\sqrt[p]{\frac{1}{\sum_{i=1}^{n}w_{i}% \frac{1}{x_{i}^{p}}}}\leq\sqrt[q]{\frac{1}{\sum_{i=1}^{n}w_{i}\frac{1}{x_{i}^{% q}}}}=\sqrt[-q]{\sum_{i=1}^{n}w_{i}x_{i}^{-q}}
  31. i = 1 n w i x i - q - q i = 1 n x i w i i = 1 n w i x i q q \begin{aligned}\displaystyle\sqrt[-q]{\sum_{i=1}^{n}w_{i}x_{i}^{-q}}&% \displaystyle\leq\prod_{i=1}^{n}x_{i}^{w_{i}}&\displaystyle\leq\sqrt[q]{\sum_{% i=1}^{n}w_{i}x_{i}^{q}}\\ \end{aligned}
  32. log ( i = 1 n x i w i ) = i = 1 n w i log ( x i ) \displaystyle\log\left(\prod_{i=1}^{n}x_{i}^{w_{i}}\right)=\sum_{i=1}^{n}w_{i}% \log(x_{i})
  33. i = 1 n x i w i i = 1 n w i x i \prod_{i=1}^{n}x_{i}^{w_{i}}\leq\sum_{i=1}^{n}w_{i}x_{i}
  34. i = 1 n w i x i p p i = 1 n x i w i i = 1 n w i x i q q \sqrt[p]{\sum_{i=1}^{n}w_{i}x_{i}^{p}}\leq\prod_{i=1}^{n}x_{i}^{w_{i}}\leq% \sqrt[q]{\sum_{i=1}^{n}w_{i}x_{i}^{q}}
  35. f ( x ) = x q p f(x)=x^{\frac{q}{p}}
  36. f ′′ ( x ) = ( q p ) ( q p - 1 ) x q p - 2 f^{\prime\prime}(x)=\left(\frac{q}{p}\right)\left(\frac{q}{p}-1\right)x^{\frac% {q}{p}-2}
  37. f ( i = 1 n w i x i p ) \displaystyle f\left(\sum_{i=1}^{n}w_{i}x_{i}^{p}\right)
  38. i = 1 n w i x i p p i = 1 n w i x i q q \sqrt[p]{\sum_{i=1}^{n}w_{i}x_{i}^{p}}\leq\sqrt[q]{\sum_{i=1}^{n}w_{i}x_{i}^{q}}
  39. M f ( x 1 , , x n ) = f - 1 ( 1 n i = 1 n f ( x i ) ) M_{f}(x_{1},\dots,x_{n})=f^{-1}\left({\frac{1}{n}\cdot\sum_{i=1}^{n}{f(x_{i})}% }\right)

Genus–differentia_definition.html

  1. n > 1 n>1
  2. n n

Geodesy.html

  1. X , Y X,Y
  2. Z Z
  3. Z Z
  4. X X
  5. X X
  6. s s
  7. α \alpha
  8. x x
  9. y y
  10. x x
  11. y y
  12. x x
  13. x x
  14. y y
  15. α \alpha
  16. s s
  17. x = s cos α y = s sin α \begin{matrix}x&=&s\cos\alpha\\ y&=&s\sin\alpha\end{matrix}
  18. s = x 2 + y 2 α = arctan ( y / x ) . \begin{matrix}s&=&\sqrt{x^{2}+y^{2}}\\ \alpha&=&\arctan{(y/x)}.\end{matrix}
  19. z z
  20. x x
  21. y y

Geographic_coordinate_system.html

  1. ϕ \phi
  2. λ \lambda
  3. h h
  4. ϕ \phi
  5. λ \lambda
  6. - 11 {}^{-11}
  7. x x
  8. y y
  9. z z
  10. x x^{\prime}
  11. y y^{\prime}
  12. z z^{\prime}
  13. x x
  14. x x^{\prime}
  15. 111132.92 - 559.82 cos 2 φ + 1.175 cos 4 φ - 0.0023 cos 6 φ 111132.92-559.82\,\cos 2\varphi+1.175\,\cos 4\varphi-0.0023\,\cos 6\varphi
  16. 111412.84 cos φ - 93.5 cos 3 φ - 0.118 cos 5 φ 111412.84\,\cos\varphi-93.5\,\cos 3\varphi-0.118\,\cos 5\varphi
  17. φ \scriptstyle{\varphi}\,\!
  18. π 180 M r cos φ \frac{\pi}{180}M_{r}\cos\varphi\!
  19. M r \scriptstyle{M_{r}}\,\!
  20. φ \scriptstyle{\varphi}\,\!
  21. π 180 a cos β \frac{\pi}{180}a\cos\beta\,\!
  22. a a
  23. tan β = b a tan φ \scriptstyle{\tan\beta=\frac{b}{a}\tan\varphi}\,\!
  24. β \scriptstyle{\beta}\,\!

Geographic_information_system.html

  1. tan S = ( z x ) 2 + ( z y ) 2 \tan S=\sqrt{\left(\frac{\partial z}{\partial x}\right)^{2}+\left(\frac{% \partial z}{\partial y}\right)^{2}}
  2. tan A = ( ( - z y ) ( z x ) ) \tan A=\left({\frac{\left({\frac{-\partial z}{\partial y}}\right)}{\left({% \frac{\partial z}{\partial x}}\right)}}\right)
  3. A = 270 + arctan ( ( z x ) ( z y ) ) - 90 ( ( z y ) | z y | ) A=270^{\circ}+\arctan\left({\frac{\left({\frac{\partial z}{\partial x}}\right)% }{\left({\frac{\partial z}{\partial y}}\right)}}\right)-90^{\circ}\left({\frac% {\left({\frac{\partial z}{\partial y}}\right)}{\left|{\frac{\partial z}{% \partial y}}\right|}}\right)

Geometric_algebra.html

  1. A ( B C ) = ( A B ) C A(BC)=(AB)C
  2. A ( B + C ) = A B + A C A(B+C)=AB+AC
  3. ( B + C ) A = B A + C A (B+C)A=BA+CA
  4. a 2 = g ( a , a ) a^{2}=g(a,a)\in\mathbb{R}
  5. a b = 1 2 ( a b + b a ) + 1 2 ( a b - b a ) ab=\frac{1}{2}(ab+ba)+\frac{1}{2}(ab-ba)
  6. a b := 1 2 ( a b + b a ) = 1 2 ( ( a + b ) 2 - a 2 - b 2 ) = g ( a , b ) , a\cdot b:=\frac{1}{2}(ab+ba)=\frac{1}{2}((a+b)^{2}-a^{2}-b^{2})=g(a,b),
  7. a b := 1 2 ( a b - b a ) = - ( b a ) a\wedge b:=\frac{1}{2}(ab-ba)=-(b\wedge a)
  8. a 1 a 2 a r = 1 r ! σ 𝔖 r sgn ( σ ) a σ ( 1 ) a σ ( 2 ) a σ ( r ) , a_{1}\wedge a_{2}\wedge\dots\wedge a_{r}=\frac{1}{r!}\sum_{\sigma\in\mathfrak{% S}_{r}}\operatorname{sgn}(\sigma)a_{\sigma(1)}a_{\sigma(2)}\dots a_{\sigma(r)},
  9. sgn ( σ ) \operatorname{sgn}(\sigma)
  10. r n r\leq n
  11. { a 1 , , a r } \{a_{1},...,a_{r}\}
  12. [ 𝐀 ] i j = a i a j [\mathbf{A}]_{ij}=a_{i}\cdot a_{j}
  13. k , l [ 𝐎 ] i k [ 𝐀 ] k l [ 𝐎 T ] l j = k , l [ 𝐎 ] i k [ 𝐎 ] j l [ 𝐀 ] k l = [ 𝐃 ] i j \sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{A}]_{kl}[\mathbf{O}^{\mathrm{T}}]_{lj}=% \sum_{k,l}[\mathbf{O}]_{ik}[\mathbf{O}]_{jl}[\mathbf{A}]_{kl}=[\mathbf{D}]_{ij}
  14. { e 1 , , e r } \{e_{1},...,e_{r}\}
  15. e i = j [ 𝐎 ] i j a j e_{i}=\sum_{j}[\mathbf{O}]_{ij}a_{j}
  16. e i e j = [ 𝐃 ] i j e_{i}\cdot e_{j}=[\mathbf{D}]_{ij}
  17. { e 1 , , e r } \{e_{1},...,e_{r}\}
  18. e i e j e_{i}\neq e_{j}
  19. e 1 e 2 e r = e 1 e 2 e r = ( j [ 𝐎 ] 1 j a j ) ( j [ 𝐎 ] 2 j a j ) ( j [ 𝐎 ] r j a j ) = det [ 𝐎 ] a 1 a 2 a r \begin{array}[]{rl}e_{1}e_{2}\cdots e_{r}&=e_{1}\wedge e_{2}\wedge\cdots\wedge e% _{r}\\ &=\left(\sum_{j}[\mathbf{O}]_{1j}a_{j}\right)\wedge\left(\sum_{j}[\mathbf{O}]_% {2j}a_{j}\right)\wedge\cdots\wedge\left(\sum_{j}[\mathbf{O}]_{rj}a_{j}\right)% \\ &=\det[\mathbf{O}]a_{1}\wedge a_{2}\wedge\cdots\wedge a_{r}\end{array}
  20. e ^ i = 1 | e i e i | e i , \hat{e}_{i}=\frac{1}{\sqrt{|e_{i}\cdot e_{i}|}}e_{i},
  21. 𝒢 ( p , q ) \mathcal{G}(p,q)
  22. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  23. 𝒢 ( 1 , 3 ) \mathcal{G}(1,3)
  24. 𝒢 ( 4 , 1 ) \mathcal{G}(4,1)
  25. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  26. { 1 , e 1 , e 2 , e 3 , e 1 e 2 , e 1 e 3 , e 2 e 3 , e 1 e 2 e 3 } \{1,e_{1},e_{2},e_{3},e_{1}e_{2},e_{1}e_{3},e_{2}e_{3},e_{1}e_{2}e_{3}\}
  27. ( Σ i α i B i ) ( Σ j β j B j ) = Σ i , j α i β j B i B j (\Sigma_{i}\alpha_{i}B_{i})(\Sigma_{j}\beta_{j}B_{j})=\Sigma_{i,j}\alpha_{i}% \beta_{j}B_{i}B_{j}\,
  28. { e 1 , , e n } \{e_{1},\cdots,e_{n}\}
  29. { e i e j 1 i < j n } \{e_{i}e_{j}\mid 1\leq i<j\leq n\}
  30. A A
  31. A r \langle A\rangle_{r}
  32. A = r = 0 n A r A=\sum_{r=0}^{n}\langle A\rangle_{r}
  33. a b = a b + a b = a b 0 + a b 2 ab=a\cdot b+a\wedge b=\langle ab\rangle_{0}+\langle ab\rangle_{2}
  34. a b 0 = a b \langle ab\rangle_{0}=a\cdot b\,
  35. a b 2 = a b \langle ab\rangle_{2}=a\wedge b\,
  36. a b i = 0 \langle ab\rangle_{i}=0\,
  37. A A
  38. A + = A 0 + A 2 + A 4 + A^{+}=\langle A\rangle_{0}+\langle A\rangle_{2}+\langle A\rangle_{4}+\cdots
  39. A - = A 1 + A 3 + A 5 + A^{-}=\langle A\rangle_{1}+\langle A\rangle_{3}+\langle A\rangle_{5}+\cdots
  40. 𝒢 + ( 2 , 0 ) 𝒢 ( 0 , 1 ) \mathcal{G}^{+}(2,0)\cong\mathcal{G}(0,1)
  41. 𝒢 + ( 1 , 3 ) 𝒢 ( 3 , 0 ) \mathcal{G}^{+}(1,3)\cong\mathcal{G}(3,0)
  42. { b 1 , b 2 , b k } \{b_{1},b_{2},\cdots b_{k}\}
  43. I I
  44. I I′
  45. I = ± I I=±I′
  46. 𝒢 ( n , 0 ) \mathcal{G}(n,0)
  47. { α 0 + α 1 I α i } \{\alpha_{0}+\alpha_{1}I\mid\alpha_{i}\in\mathbb{R}\}
  48. { α 0 + α 1 I α i } \{\alpha_{0}+\alpha_{1}I\mid\alpha_{i}\in\mathbb{R}\}
  49. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  50. { e 3 e 2 , e 1 e 3 , e 2 e 1 } \{e_{3}e_{2},e_{1}e_{3},e_{2}e_{1}\}\,
  51. { α 0 + i α 1 + j α 2 + k α 3 α i } \{\alpha_{0}+i\alpha_{1}+j\alpha_{2}+k\alpha_{3}\mid\alpha_{i}\in\mathbb{R}\}
  52. { e i } \{e_{i}\}
  53. { e i } \{e_{i}\}
  54. { e i } \{e^{i}\}
  55. e i e j = δ i , j e^{i}\cdot e_{j}=\delta^{i}{}_{j},
  56. ϵ = e 1 e n \epsilon=e_{1}\wedge\cdots\wedge e_{n}
  57. { e i } \{e_{i}\}
  58. e i = ( - 1 ) i - 1 ( e 1 e ˇ i e n ) ϵ - 1 , e^{i}=(-1)^{i-1}(e_{1}\wedge\cdots\wedge\check{e}_{i}\wedge\cdots\wedge e_{n})% \epsilon^{-1},
  59. e ˇ i \check{e}_{i}
  60. C D := r , s C r D s r + s C\wedge D:=\sum_{r,s}\langle\langle C\rangle_{r}\langle D\rangle_{s}\rangle_{r% +s}
  61. C × D := 1 2 ( C D - D C ) C\times D:=\tfrac{1}{2}(CD-DC)
  62. C D := r , s C r D s r + s - n C\;\triangledown\;D:=\sum_{r,s}\langle\langle C\rangle_{r}\langle D\rangle_{s}% \rangle_{r+s-n}
  63. C D := r , s C r D s s - r \,C\;\big\lrcorner\;D:=\sum_{r,s}\langle\langle C\rangle_{r}\langle D\rangle_{% s}\rangle_{s-r}
  64. C D := r , s C r D s r - s \,C\;\big\llcorner\;D:=\sum_{r,s}\langle\langle C\rangle_{r}\langle D\rangle_{% s}\rangle_{r-s}
  65. C * D := r , s C r D s 0 \,C*D:=\sum_{r,s}\langle\langle C\rangle_{r}\langle D\rangle_{s}\rangle_{0}
  66. C D := r , s C r D s | s - r | \,C\bullet D:=\sum_{r,s}\langle\langle C\rangle_{r}\langle D\rangle_{s}\rangle% _{|s-r|}
  67. C H D := r 0 , s 0 C r D s | s - r | \,C\bullet_{H}D:=\sum_{r\neq 0,s\neq 0}\langle\langle C\rangle_{r}\langle D% \rangle_{s}\rangle_{|s-r|}
  68. a b = a b + a b ab=a\cdot b+a\wedge b
  69. a B = a B + a B aB=a\;\big\lrcorner\;B+a\wedge B
  70. 𝒫 b ( a ) = ( a b - 1 ) b \mathcal{P}_{b}(a)=(a\cdot b^{-1})b
  71. 𝒫 B ( A ) = ( A B - 1 ) B \mathcal{P}_{B}(A)=(A\;\big\lrcorner\;B^{-1})\;\big\lrcorner\;B
  72. \mathbb{N}
  73. a = a m m - 1 = ( a m + a m ) m - 1 = a m + a m \,a=amm^{-1}=(a\cdot m+a\wedge m)m^{-1}=a_{\|m}+a_{\perp m}
  74. a m = ( a m ) m - 1 \,a_{\|m}=(a\cdot m)m^{-1}
  75. a m = a - a m = ( a m ) m - 1 . \,a_{\perp m}=a-a_{\|m}=(a\wedge m)m^{-1}.
  76. 𝒫 B ( A ) = ( A B - 1 ) B \,\mathcal{P}_{B}(A)=(A\;\big\lrcorner\;B^{-1})\;\big\lrcorner\;B
  77. 𝒫 B ( A ) = A - 𝒫 B ( A ) . \,\mathcal{P}_{B}^{\perp}(A)=A-\mathcal{P}_{B}(A).
  78. c c = n c n - 1 . \,c\mapsto c^{\prime}=ncn^{-1}.
  79. A N A N - 1 . \,A\mapsto NAN^{-1}.
  80. c = - c m + c m = - ( c m ) m - 1 + ( c m ) m - 1 = ( - m c - m c ) m - 1 = - m c m - 1 \!c^{\prime}={-c_{\|m}+c_{\perp m}}={-(c\cdot m)m^{-1}+(c\wedge m)m^{-1}}={(-m% \cdot c-m\wedge c)m^{-1}}=-mcm^{-1}
  81. a a = - M a M - 1 \!a\mapsto a^{\prime}=-MaM^{-1}
  82. M = p q r \!M=pq\ldots r
  83. M - 1 = ( p q r ) - 1 = r - 1 q - 1 p - 1 . \!M^{-1}=(pq\ldots r)^{-1}=r^{-1}\ldots q^{-1}p^{-1}.
  84. ( a b c ) = a b c = ( - m a m - 1 ) ( - m b m - 1 ) ( - m c m - 1 ) = - m a ( m - 1 m ) b ( m - 1 m ) c m - 1 = - m a b c m - 1 (abc)^{\prime}=a^{\prime}b^{\prime}c^{\prime}=(-mam^{-1})(-mbm^{-1})(-mcm^{-1}% )=-ma(m^{-1}m)b(m^{-1}m)cm^{-1}=-mabcm^{-1}\,
  85. ( a b c d ) = a b c d = ( - m a m - 1 ) ( - m b m - 1 ) ( - m c m - 1 ) ( - m d m - 1 ) = m a b c d m - 1 . (abcd)^{\prime}=a^{\prime}b^{\prime}c^{\prime}d^{\prime}=(-mam^{-1})(-mbm^{-1}% )(-mcm^{-1})(-mdm^{-1})=mabcdm^{-1}.\,
  86. A M α ( A ) M - 1 , \,A\mapsto M\alpha(A)M^{-1},
  87. a a
  88. b b
  89. a b = ( ( a b ) b - 1 ) b = a b b a\wedge b=((a\wedge b)b^{-1})b=a_{\perp b}b
  90. a b a\wedge b
  91. i = 1 n a i \bigwedge_{i=1}^{n}a_{i}
  92. R = a 1 a 2 . a r R=a_{1}a_{2}....a_{r}
  93. R = ( a 1 a 2 . a r ) = a r . a 2 a 1 R^{\dagger}=(a_{1}a_{2}....a_{r})^{\dagger}=a_{r}....a_{2}a_{1}
  94. R = a b R=ab
  95. R R = a b b a = a b 2 a = a 2 b 2 = R R RR^{\dagger}=abba=ab^{2}a=a^{2}b^{2}=R^{\dagger}R
  96. R R
  97. ( R v R ) 2 = R v 2 R = v 2 R R = v 2 (RvR^{\dagger})^{2}=Rv^{2}R^{\dagger}=v^{2}RR^{\dagger}=v^{2}
  98. R v R RvR
  99. v v
  100. ( R v 1 R ) ( R v 2 R ) = v 1 v 2 (Rv_{1}R^{\dagger})\cdot(Rv_{2}R^{\dagger})=v_{1}\cdot v_{2}
  101. R R
  102. R = e - B θ 2 R=e^{-\frac{B\theta}{2}}
  103. θ \theta
  104. B B
  105. ( n r ) × ( n r ) {\left({{n}\atop{r}}\right)}\times{\left({{n}\atop{r}}\right)}
  106. 𝖿 ¯ ( a 1 a 2 a r ) = f ( a 1 ) f ( a 2 ) f ( a r ) . \underline{\mathsf{f}}(a_{1}\wedge a_{2}\wedge\cdots\wedge a_{r})=f(a_{1})% \wedge f(a_{2})\wedge\cdots\wedge f(a_{r}).
  107. n a n - 1 n A n - 1 , nan^{-1}\mapsto nAn^{-1},
  108. R a R R A R . RaR^{\dagger}\mapsto RAR^{\dagger}.
  109. p = t + α v p=t+\alpha\ v
  110. B ( p - q ) = 0 B\wedge(p-q)=0
  111. B ( t + α v - q ) = 0 B\wedge(t+\alpha v-q)=0
  112. α = B ( q - t ) B v \alpha=\frac{B\wedge(q-t)}{B\wedge v}
  113. p = t + ( B ( q - t ) B v ) v p=t+\left(\frac{B\wedge(q-t)}{B\wedge v}\right)v
  114. a × b = - I ( a b ) . a\times b=-I(a\wedge b)\,.
  115. u ^ \hat{u}
  116. v ^ \hat{v}
  117. 𝐫 = r ( u ^ cos θ + v ^ sin θ ) = r u ^ ( cos θ + u ^ v ^ sin θ ) \mathbf{r}=r(\hat{u}\cos\theta+\hat{v}\sin\theta)=r\hat{u}(\cos\theta+\hat{u}% \hat{v}\sin\theta)
  118. i = u ^ v ^ = u ^ v ^ {i}=\hat{u}\hat{v}=\hat{u}\wedge\hat{v}
  119. i 2 = - 1 {i}^{2}=-1
  120. 𝐫 = r u ^ e i θ \mathbf{r}=r\hat{u}e^{{i}\theta}
  121. d 𝐫 d θ = r u ^ i e i θ = 𝐫 i \frac{d\mathbf{r}}{d\theta}=r\hat{u}{i}e^{{i}\theta}=\mathbf{r}{i}
  122. τ = d W d θ = F d r d θ = F ( 𝐫 i ) \tau=\frac{dW}{d\theta}=F\cdot\frac{dr}{d\theta}=F\cdot(\mathbf{r}{i})
  123. τ = 𝐫 × F \tau=\mathbf{r}\times F
  124. u ^ {\hat{u}}
  125. v ^ {\hat{v}}
  126. F = ( E + i c B ) e 0 {F}=({E}+ic{B})e_{0}
  127. e 0 e_{0}
  128. J {J}
  129. D F = μ 0 J DF=\mu_{0}J
  130. D F = 0 D\wedge F=0
  131. D F = μ 0 J D\cdot F=\mu_{0}J
  132. F = D A F=D\wedge A
  133. D D A = μ 0 J D\cdot D\wedge A=\mu_{0}J
  134. F = D A F=DA
  135. D A = 0 D\cdot A=0
  136. D 2 A = μ 0 J D^{2}A=\mu_{0}J
  137. D F DF
  138. D F = D F + D F DF=D\cdot F+D\wedge F
  139. D D
  140. \bigtriangledown
  141. \bigtriangledown
  142. \nabla
  143. = 2 \Box=\bigtriangledown^{2}
  144. γ 0 \gamma_{0}
  145. γ 0 = 1 c t \gamma_{0}\cdot\bigtriangledown=\frac{1}{c}\frac{\partial}{\partial t}
  146. γ 0 = \gamma_{0}\wedge\bigtriangledown=\nabla
  147. e < m t p l > β e^{<}mtpl>{{\beta}}
  148. β {\beta}
  149. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  150. 𝒢 ( 2 , 0 ) \mathcal{G}(2,0)
  151. P P
  152. Z = e 1 P = e 1 ( x e 1 + y e 2 ) = x ( 1 ) + y ( e 1 e 2 ) Z={e_{1}}P={e_{1}}(x{e_{1}}+y{e_{2}})=x(1)+y({e_{1}}{e_{2}})\,
  153. ( e 1 e 2 ) 2 = e 1 e 2 e 1 e 2 = - e 1 e 1 e 2 e 2 = - 1 ({e_{1}}{e_{2}})^{2}={e_{1}}{e_{2}}{e_{1}}{e_{2}}=-{e_{1}}{e_{1}}{e_{2}}{e_{2}% }=-1\,
  154. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  155. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  156. 𝒢 ( 1 , 3 ) \mathcal{G}(1,3)
  157. A d A f = A d x f \int_{A}dA\nabla f=\oint_{\partial A}dxf
  158. f = f + f \nabla f=\nabla\cdot f+\nabla\wedge f
  159. 1 D 1D
  160. a a
  161. b b
  162. A d A f = A d x f \int_{A}dA\nabla f=\oint_{\partial A}dxf
  163. a b d x f = a b d x f = a b d f = f ( b ) - f ( a ) \int_{a}^{b}dx\nabla f=\int_{a}^{b}dx\cdot\nabla f=\int_{a}^{b}df=f(b)-f(a)
  164. 3 \mathcal{E}^{3}
  165. 𝒢 4 , 1 \mathcal{G}^{4,1}
  166. e + \,e_{+}
  167. e - \,e_{-}
  168. e + 2 = + 1 \,{e_{+}}^{2}=+1
  169. e - 2 = - 1 \,{e_{-}}^{2}=-1
  170. 𝒢 ( 3 , 0 ) \mathcal{G}(3,0)
  171. n = e - + e + n_{\infty}=e_{-}+e_{+}
  172. n o = 1 2 ( e - - e + ) n_{o}=\tfrac{1}{2}(e_{-}-e_{+})
  173. n n o = - 1 n_{\infty}\cdot n_{o}=-1
  174. i , j , k i,j,k

Geometric_distribution.html

  1. 6 + p 2 1 - p 6+\frac{p^{2}}{1-p}\!
  2. - ( 1 - p ) log 2 ( 1 - p ) - p log 2 p p \tfrac{-(1-p)\log_{2}(1-p)-p\log_{2}p}{p}\!
  3. p e t 1 - ( 1 - p ) e t \frac{pe^{t}}{1-(1-p)e^{t}}\!
  4. t < - ln ( 1 - p ) t<-\ln(1-p)\!
  5. p e i t 1 - ( 1 - p ) e i t \frac{pe^{it}}{1-(1-p)\,e^{it}}\!
  6. 0 < p 1 0<p\leq 1
  7. k { 0 , 1 , 2 , 3 , } k\in\{0,1,2,3,\dots\}\!
  8. ( 1 - p ) k p (1-p)^{k}\,p\!
  9. 1 - ( 1 - p ) k + 1 1-(1-p)^{k+1}\!
  10. 1 - p p \frac{1-p}{p}\!
  11. - 1 log 2 ( 1 - p ) - 1 \left\lceil\frac{-1}{\log_{2}(1-p)}\right\rceil\!-1
  12. - 1 / log 2 ( 1 - p ) -1/\log_{2}(1-p)
  13. 0
  14. 1 - p p 2 \frac{1-p}{p^{2}}\!
  15. 2 - p 1 - p \frac{2-p}{\sqrt{1-p}}\!
  16. 6 + p 2 1 - p 6+\frac{p^{2}}{1-p}\!
  17. - ( 1 - p ) log 2 ( 1 - p ) - p log 2 p p \tfrac{-(1-p)\log_{2}(1-p)-p\log_{2}p}{p}\!
  18. p 1 - ( 1 - p ) e t \frac{p}{1-(1-p)e^{t}}\!
  19. p 1 - ( 1 - p ) e i t \frac{p}{1-(1-p)\,e^{it}}\!
  20. Pr ( X = k ) = ( 1 - p ) k - 1 p \Pr(X=k)=(1-p)^{k-1}\,p\,
  21. Pr ( Y = k ) = ( 1 - p ) k p \Pr(Y=k)=(1-p)^{k}\,p\,
  22. E ( X ) = 1 p , var ( X ) = 1 - p p 2 . \mathrm{E}(X)=\frac{1}{p},\qquad\mathrm{var}(X)=\frac{1-p}{p^{2}}.
  23. E ( Y ) = 1 - p p , var ( Y ) = 1 - p p 2 . \mathrm{E}(Y)=\frac{1-p}{p},\qquad\mathrm{var}(Y)=\frac{1-p}{p^{2}}.
  24. κ n \kappa_{n}
  25. κ n + 1 = μ ( μ + 1 ) d κ n d μ . \kappa_{n+1}=\mu(\mu+1)\frac{d\kappa_{n}}{d\mu}.
  26. E ( Y ) \displaystyle\mathrm{E}(Y)
  27. p ^ = ( 1 n i = 1 n k i ) - 1 = n i = 1 n k i . \widehat{p}=\left(\frac{1}{n}\sum_{i=1}^{n}k_{i}\right)^{-1}=\frac{n}{\sum_{i=% 1}^{n}k_{i}}.\!
  28. p Beta ( α + n , β + i = 1 n ( k i - 1 ) ) . p\sim\mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^{n}(k_{i}-1)\right).\!
  29. p ^ \widehat{p}
  30. p ^ = ( 1 + 1 n i = 1 n k i ) - 1 = n i = 1 n k i + n . \widehat{p}=\left(1+\frac{1}{n}\sum_{i=1}^{n}k_{i}\right)^{-1}=\frac{n}{\sum_{% i=1}^{n}k_{i}+n}.\!
  31. p Beta ( α + n , β + i = 1 n k i ) . p\sim\mathrm{Beta}\left(\alpha+n,\ \beta+\sum_{i=1}^{n}k_{i}\right).\!
  32. p ^ \widehat{p}
  33. G X ( s ) \displaystyle G_{X}(s)
  34. Pr ( D = d ) = q 100 d 1 + q 100 + q 200 + + q 900 , \Pr(D=d)={q^{100d}\over 1+q^{100}+q^{200}+\cdots+q^{900}},
  35. { ( p - 1 ) Pr ( k ) + Pr ( k + 1 ) = 0 , Pr ( 0 ) = p } \{(p-1)\Pr(k)+\Pr(k+1)=0,\Pr(0)=p\}
  36. Z = m = 1 r Y m Z=\sum_{m=1}^{r}Y_{m}
  37. W = min m 1 , , r Y m W=\min_{m\in 1,\dots,r}Y_{m}\,
  38. p = 1 - m ( 1 - p m ) . p=1-\prod_{m}(1-p_{m}).
  39. k = 1 k X k \sum_{k=1}^{\infty}k\,X_{k}
  40. Y = X , Y=\lfloor X\rfloor,
  41. \lfloor\quad\rfloor
  42. ln ( U ) / ln ( 1 - p ) \lfloor\ln(U)/\ln(1-p)\rfloor
  43. p p
  44. U U
  45. p = 1 n p=\frac{1}{n}
  46. n n\rightarrow\infty
  47. λ = 1 n \lambda=\frac{1}{n}
  48. P ( X > a ) = ( 1 - p ) a = ( 1 - 1 n ) n 1 n ( a ) = [ ( 1 - 1 n ) n ] 1 n ( a ) n [ e - 1 ] 1 n ( a ) = e - 1 n a P\left(X>a\right)={{\left(1-p\right)}^{a}}={{\left(1-\frac{1}{n}\right)}^{n% \frac{1}{n}\left(a\right)}}={{\left[{{\left(1-\frac{1}{n}\right)}^{n}}\right]}% ^{\frac{1}{n}\left(a\right)}}\xrightarrow[n\to\infty]{}{{\left[{{e}^{-1}}% \right]}^{\frac{1}{n}\left(a\right)}}={{e}^{-\frac{1}{n}a}}

Geometric_mean.html

  1. { x i } i = 1 N \{x_{i}\}_{i=1}^{N}
  2. ( i = 1 N x i ) 1 / N \left(\prod_{i=1}^{N}x_{i}\right)^{1/N}
  3. 2 8 = 4 \sqrt{2\cdot 8}=4
  4. 4 1 1 / 32 3 = 1 / 2 \sqrt[3]{4\cdot 1\cdot 1/32}=1/2
  5. a a
  6. b b
  7. a a
  8. b b
  9. a a
  10. b b
  11. c c
  12. { a 1 , a 2 , , a n } \{a_{1},a_{2},\ldots,a_{n}\}
  13. ( i = 1 n a i ) 1 / n = a 1 a 2 a n n . \left(\prod_{i=1}^{n}a_{i}\right)^{1/n}=\sqrt[n]{a_{1}a_{2}\cdots a_{n}}.
  14. a n a_{n}
  15. h n h_{n}
  16. a n + 1 = a n + h n 2 , a 0 = x a_{n+1}=\frac{a_{n}+h_{n}}{2},\quad a_{0}=x
  17. h n + 1 = 2 1 a n + 1 h n , h 0 = y h_{n+1}=\frac{2}{\frac{1}{a_{n}}+\frac{1}{h_{n}}},\quad h_{0}=y
  18. h n + 1 h_{n+1}
  19. a n a_{n}
  20. h n h_{n}
  21. x x
  22. y y
  23. a i h i = a i + h i a i + h i h i a i = a i + h i 1 a i + 1 h i = a i + 1 h i + 1 \sqrt{a_{i}h_{i}}=\sqrt{\frac{a_{i}+h_{i}}{\frac{a_{i}+h_{i}}{h_{i}a_{i}}}}=% \sqrt{\frac{a_{i}+h_{i}}{\frac{1}{a_{i}}+\frac{1}{h_{i}}}}=\sqrt{a_{i+1}h_{i+1}}
  24. ( i = 1 n a i ) 1 / n = exp [ 1 n i = 1 n ln a i ] \left(\prod_{i=1}^{n}a_{i}\right)^{1/n}=\exp\left[\frac{1}{n}\sum_{i=1}^{n}\ln a% _{i}\right]
  25. a i a_{i}
  26. f ( x ) = log x f(x)=\log x
  27. b ( log b ( 2 ) + log b ( 8 ) ) / 2 = 4 , b^{(\log_{b}(2)+\log_{b}(8))/2}=4,
  28. b b
  29. e e
  30. a 0 a_{0}
  31. a n a_{n}
  32. ( a n a 0 ) 1 n , \left(\frac{a_{n}}{a_{0}}\right)^{\frac{1}{n}},
  33. n n
  34. a 0 , , a n a_{0},\ldots,a_{n}
  35. a k a_{k}
  36. a k + 1 a_{k+1}
  37. a k + 1 / a k a_{k+1}/a_{k}
  38. ( a 1 a 0 a 2 a 1 a n a n - 1 ) 1 n = ( a n a 0 ) 1 n \left(\frac{a_{1}}{a_{0}}\frac{a_{2}}{a_{1}}\cdots\frac{a_{n}}{a_{n-1}}\right)% ^{\frac{1}{n}}=\left(\frac{a_{n}}{a_{0}}\right)^{\frac{1}{n}}
  39. 𝐺𝑀 ( X i Y i ) = 𝐺𝑀 ( X i ) 𝐺𝑀 ( Y i ) \mathit{GM}\left(\frac{X_{i}}{Y_{i}}\right)=\frac{\mathit{GM}(X_{i})}{\mathit{% GM}(Y_{i})}
  40. 1.80 × 1.166666 × 1.428571 3 = 1.442249 \sqrt[3]{1.80\times 1.166666\times 1.428571}=1.442249
  41. ( X - X min ) / ( X norm - X min ) (X-X_{\min})/(X_{\mathrm{norm}}-X_{\min})
  42. 2.35 × 4 3 1.7701 \sqrt{2.35\times\frac{4}{3}}\approx 1.7701
  43. 16 : 9 = 1.77 7 ¯ 16:9=1.77\overline{7}
  44. 16 : 9 16:9
  45. 1.77 7 ¯ : 1 1.77\overline{7}:1
  46. 1.55 5 ¯ 1.55\overline{5}
  47. 16 : 9 16:9
  48. 4 : 3 = 12 : 9 4:3=12:9
  49. 16 9 × 4 3 1.5396 13.8 : 9 , \sqrt{\frac{16}{9}\times\frac{4}{3}}\approx 1.5396\approx 13.8:9,
  50. n 1 = n 0 n 2 n_{1}=\sqrt{n_{0}n_{2}}

Geometric_series.html

  1. 1 2 + 1 4 + 1 8 + 1 16 + \frac{1}{2}\,+\,\frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,\cdots
  2. 1 2 + 1 4 + 1 8 + 1 16 + \frac{1}{2}\,+\,\frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,\cdots
  3. a + a r + a r 2 + a r 3 + a+ar+ar^{2}+ar^{3}+\cdots
  4. r = 1 2 r=\frac{1}{2}
  5. a = 1 2 a=\frac{1}{2}
  6. s = 1 + 2 3 + 4 9 + 8 27 + s\;=\;1\,+\,\frac{2}{3}\,+\,\frac{4}{9}\,+\,\frac{8}{27}\,+\,\cdots
  7. 2 3 s = 2 3 + 4 9 + 8 27 + 16 81 + \frac{2}{3}s\;=\;\frac{2}{3}\,+\,\frac{4}{9}\,+\,\frac{8}{27}\,+\,\frac{16}{81% }\,+\,\cdots
  8. s - 2 3 s = 1 , so s = 3. s\,-\,\frac{2}{3}s\;=\;1,\;\;\;\mbox{so }~{}s=3.
  9. r 1 r\neq 1
  10. a + a r + a r 2 + a r 3 + + a r n - 1 = k = 0 n - 1 a r k = a 1 - r n 1 - r , a+ar+ar^{2}+ar^{3}+\cdots+ar^{n-1}=\sum_{k=0}^{n-1}ar^{k}=a\,\frac{1-r^{n}}{1-% r},
  11. Let s = a + a r + a r 2 + a r 3 + + a r n - 1 . \displaystyle\,\text{Let }s=a+ar+ar^{2}+ar^{3}+\cdots+ar^{n-1}.
  12. a + a r + a r 2 + a r 3 + a r 4 + = k = 0 a r k = a 1 - r , for | r | < 1. a+ar+ar^{2}+ar^{3}+ar^{4}+\cdots=\sum_{k=0}^{\infty}ar^{k}=\frac{a}{1-r},\,% \text{ for }|r|<1.
  13. 1 + r + r 2 + r 3 + = 1 1 - r , 1\,+\,r\,+\,r^{2}\,+\,r^{3}\,+\,\cdots\;=\;\frac{1}{1-r},
  14. Let s = 1 + r + r 2 + r 3 + . \displaystyle\,\text{Let }s=1+r+r^{2}+r^{3}+\cdots.
  15. 1 + r + r 2 + r 3 + \displaystyle 1+r+r^{2}+r^{3}+\cdots
  16. g ( K ) = r K 1 - r \displaystyle g(K)=\frac{r^{K}}{1-r}
  17. 1 = g ( 0 ) - g ( 1 ) , r = g ( 1 ) - g ( 2 ) , r 2 = g ( 2 ) - g ( 3 ) , \displaystyle 1=g(0)-g(1),r=g(1)-g(2),r^{2}=g(2)-g(3),\cdots
  18. S = 1 + r + r 2 + r 3 + = ( g ( 0 ) - g ( 1 ) ) + ( g ( 1 ) - g ( 2 ) ) + ( g ( 2 ) - g ( 3 ) ) + \displaystyle S=1+r+r^{2}+r^{3}+...=(g(0)-g(1))+(g(1)-g(2))+(g(2)-g(3))+\cdots
  19. | r | < 1 \begin{aligned}\displaystyle\left|r\right|<1\end{aligned}
  20. g ( K ) 0 as K \begin{aligned}\displaystyle g(K)\longrightarrow 0\,\text{ as }K\to\infty\end{aligned}
  21. g ( 0 ) = 1 1 - r . \begin{aligned}\displaystyle g(0)=\frac{1}{1-r}.\end{aligned}
  22. r 1 r\neq 1
  23. k = a b r k = r a - r b + 1 1 - r , \sum_{k=a}^{b}r^{k}=\frac{r^{a}-r^{b+1}}{1-r},
  24. a , b a,b\in\mathbb{N}
  25. b = n - 1 n = b + 1 b=n-1\Rightarrow n=b+1
  26. k = a b r k = k = 0 n - 1 r k - k = 0 a - 1 r k = 1 - r n 1 - r - 1 - r a 1 - r = 1 - r n - 1 + r a 1 - r = r a - r b + 1 1 - r \begin{aligned}\displaystyle\sum_{k=a}^{b}r^{k}&\displaystyle=\sum_{k=0}^{n-1}% r^{k}-\sum_{k=0}^{a-1}r^{k}\\ &\displaystyle=\frac{1-r^{n}}{1-r}-\frac{1-r^{a}}{1-r}\\ &\displaystyle=\frac{1-r^{n}-1+r^{a}}{1-r}\\ &\displaystyle=\frac{r^{a}-r^{b+1}}{1-r}\end{aligned}
  27. 0.7777 = 7 10 + 7 100 + 7 1000 + 7 10000 + . 0.7777\ldots\;=\;\frac{7}{10}\,+\,\frac{7}{100}\,+\,\frac{7}{1000}\,+\,\frac{7% }{10000}\,+\,\cdots.
  28. 0.7777 = a 1 - r = 7 / 10 1 - 1 / 10 = 7 9 . 0.7777\ldots\;=\;\frac{a}{1-r}\;=\;\frac{7/10}{1-1/10}\;=\;\frac{7}{9}.
  29. 0.123412341234 = a 1 - r = 1234 / 10000 1 - 1 / 10000 = 1234 9999 . 0.123412341234\ldots\;=\;\frac{a}{1-r}\;=\;\frac{1234/10000}{1-1/10000}\;=\;% \frac{1234}{9999}.
  30. 0.09090909 = 09 99 = 1 11 . 0.09090909\ldots\;=\;\frac{09}{99}\;=\;\frac{1}{11}.
  31. 0.143814381438 = 1438 9999 . 0.143814381438\ldots\;=\;\frac{1438}{9999}.
  32. 0.9999 = 9 9 = 1. 0.9999\ldots\;=\;\frac{9}{9}\;=\;1.
  33. n n
  34. 1 + 2 ( 1 8 ) + 4 ( 1 8 ) 2 + 8 ( 1 8 ) 3 + . 1\,+\,2\left(\frac{1}{8}\right)\,+\,4\left(\frac{1}{8}\right)^{2}\,+\,8\left(% \frac{1}{8}\right)^{3}\,+\,\cdots.
  35. 1 + 1 4 + 1 16 + 1 64 + . 1\,+\,\frac{1}{4}\,+\,\frac{1}{16}\,+\,\frac{1}{64}\,+\,\cdots.
  36. n = 0 4 - n = 1 + 4 - 1 + 4 - 2 + 4 - 3 + = 4 3 . \sum_{n=0}^{\infty}4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots={4\over 3}.\;
  37. 1 1 - r = 1 1 - 1 4 = 4 3 . \frac{1}{1-r}\;=\;\frac{1}{1-\frac{1}{4}}\;=\;\frac{4}{3}.
  38. 1 + 3 ( 1 9 ) + 12 ( 1 9 ) 2 + 48 ( 1 9 ) 3 + . 1\,+\,3\left(\frac{1}{9}\right)\,+\,12\left(\frac{1}{9}\right)^{2}\,+\,48\left% (\frac{1}{9}\right)^{3}\,+\,\cdots.
  39. 1 + a 1 - r = 1 + 1 3 1 - 4 9 = 8 5 . 1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}.
  40. r = 1 / 2 r=1/2
  41. I I
  42. I I
  43. I I
  44. n = 1 $ 100 ( 1 + I ) n , \sum_{n=1}^{\infty}\frac{\$100}{(1+I)^{n}},
  45. $ 100 ( 1 + I ) + $ 100 ( 1 + I ) 2 + $ 100 ( 1 + I ) 3 + $ 100 ( 1 + I ) 4 + . \frac{\$100}{(1+I)}\,+\,\frac{\$100}{(1+I)^{2}}\,+\,\frac{\$100}{(1+I)^{3}}\,+% \,\frac{\$100}{(1+I)^{4}}\,+\,\cdots.
  46. I I
  47. $ 100 / ( 1 + I ) 1 - 1 / ( 1 + I ) = $ 100 I . \frac{\$100/(1+I)}{1-1/(1+I)}\;=\;\frac{\$100}{I}.
  48. I I
  49. 1 1 - x = 1 + x + x 2 + x 3 + x 4 + \frac{1}{1-x}=1+x+x^{2}+x^{3}+x^{4}+\cdots
  50. | x | < 1 |x|<1
  51. tan - 1 ( x ) = d x 1 + x 2 = d x 1 - ( - x 2 ) = ( 1 + ( - x 2 ) + ( - x 2 ) 2 + ( - x 2 ) 3 + ) d x = ( 1 - x 2 + x 4 - x 6 + ) d x = x - x 3 3 + x 5 5 - x 7 7 + = n = 0 ( - 1 ) n 2 n + 1 x 2 n + 1 \begin{aligned}\displaystyle\tan^{-1}(x)&\displaystyle=\int\frac{dx}{1+x^{2}}% \\ &\displaystyle=\int\frac{dx}{1-(-x^{2})}\\ &\displaystyle=\int\left(1+\left(-x^{2}\right)+\left(-x^{2}\right)^{2}+\left(-% x^{2}\right)^{3}+\cdots\right)dx\\ &\displaystyle=\int\left(1-x^{2}+x^{4}-x^{6}+\cdots\right)dx\\ &\displaystyle=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots\\ &\displaystyle=\sum^{\infty}_{n=0}\frac{(-1)^{n}}{2n+1}x^{2n+1}\end{aligned}
  52. n = 1 n x n - 1 = 1 ( 1 - x ) 2 for | x | < 1. \sum^{\infty}_{n=1}nx^{n-1}=\frac{1}{(1-x)^{2}}\quad\,\text{ for }|x|<1.
  53. n = 2 n ( n - 1 ) x n - 2 = 2 ( 1 - x ) 3 for | x | < 1 , \sum^{\infty}_{n=2}n(n-1)x^{n-2}=\frac{2}{(1-x)^{3}}\quad\,\text{ for }|x|<1,
  54. n = 3 n ( n - 1 ) ( n - 2 ) x n - 3 = 6 ( 1 - x ) 4 for | x | < 1. \sum^{\infty}_{n=3}n(n-1)(n-2)x^{n-3}=\frac{6}{(1-x)^{4}}\quad\,\text{ for }|x% |<1.

Georg_Cantor.html

  1. \aleph

George_Boole.html

  1. mes { x 1 π a k x - b k t } = a k π t \mathrm{mes}\left\{x\in\mathbb{R}\,\mid\,\Re\frac{1}{\pi}\sum\frac{a_{k}}{x-b_% {k}}\geq t\right\}=\frac{\sum a_{k}}{\pi t}

Geostationary_orbit.html

  1. Δ t = 2 c R 2 + r 2 - 2 R r cos φ 253 ms \Delta t=\frac{2}{c}\sqrt{R^{2}+r^{2}-2Rr\cos\varphi}\approx 253\,\mathrm{ms}
  2. 𝐅 c = 𝐅 g \mathbf{F}\text{c}=\mathbf{F}\text{g}
  3. m 𝐚 c = m 𝐠 m\mathbf{a}\text{c}=m\mathbf{g}
  4. | 𝐚 c | = ω 2 r |\mathbf{a}\text{c}|=\omega^{2}r
  5. | 𝐠 | = G M r 2 |\mathbf{g}|=\frac{GM}{r^{2}}
  6. r 3 = G M ω 2 r = G M ω 2 3 r^{3}=\frac{GM}{\omega^{2}}\to r=\sqrt[3]{\frac{GM}{\omega^{2}}}
  7. r = μ ω 2 3 r=\sqrt[3]{\frac{\mu}{\omega^{2}}}
  8. ω 2 π rad 86 164 s 7.2921 × 10 - 5 rad / s \omega\approx\frac{2\mathrm{\pi}~{}\mathrm{rad}}{86\,164~{}\mathrm{s}}\approx 7% .2921\times 10^{-5}~{}\mathrm{rad}/\mathrm{s}
  9. v = ω r 3.0746 km / s 11 068 km / h 6877.8 mph . v=\omega r\approx 3.0746~{}\mathrm{km}/\mathrm{s}\approx 11\,068~{}\mathrm{km}% /\mathrm{h}\approx 6877.8~{}\mathrm{mph}\,\text{.}
  10. × 10 - 11 \times 10^{-}11
  11. × 10 2 4 \times 10^{2}4

Geostatistics.html

  1. Z ( 𝐱 ) Z(\mathbf{x})
  2. 𝐱 \mathbf{x}
  3. 𝐱 \mathbf{x}
  4. Z ( 𝐱 ) Z(\mathbf{x})
  5. Z ( 𝐱 ) Z(\mathbf{x})
  6. F ( z , 𝐱 ) = Prob { Z ( 𝐱 ) z information } . F(\mathit{z},\mathbf{x})=\operatorname{Prob}\{Z(\mathbf{x})\leqslant\mathit{z}% \mid\,\text{information}\}.
  7. Z Z
  8. 𝐱 \mathbf{x}
  9. 𝐱 \mathbf{x}
  10. Z ( 𝐱 ) Z(\mathbf{x})
  11. Z ( 𝐱 ) Z(\mathbf{x})
  12. Z ( 𝐱 ) Z(\mathbf{x})
  13. Z Z
  14. Z ( 𝐱 ) Z(\mathbf{x})
  15. f ( z , 𝐱 ) f(z,\mathbf{x})
  16. f ( z , 𝐱 ) f(z,\mathbf{x})
  17. Z Z
  18. N N
  19. N N
  20. F ( 𝐳 , 𝐱 ) = Prob { Z ( 𝐱 1 ) z 1 , Z ( 𝐱 2 ) z 2 , , Z ( 𝐱 N ) z N } . F(\mathbf{z},\mathbf{x})=\operatorname{Prob}\{Z(\mathbf{x}_{1})\leqslant z_{1}% ,Z(\mathbf{x}_{2})\leqslant z_{2},...,Z(\mathbf{x}_{N})\leqslant z_{N}\}.

Geosynchronous_orbit.html

  1. a = μ ( P 2 π ) 2 3 a=\sqrt[3]{\mu\left(\frac{P}{2\pi}\right)^{2}}

Geotechnical_engineering.html

  1. c c
  2. σ \sigma\,\!
  3. tan ( ϕ ) \tan(\phi\,\!)
  4. σ \sigma\,\!
  5. ϕ \phi\,\!
  6. c c

Gerolamo_Cardano.html

  1. a x 3 + b x + c = 0 ax^{3}+bx+c=0

Gini_coefficient.html

  1. δ ( x - x 0 ) \delta(x-x_{0})
  2. { 1 b - a a x b 0 otherwise \begin{cases}\frac{1}{b-a}&a\leq x\leq b\\ 0&\mathrm{otherwise}\end{cases}
  3. b - a 3 ( b + a ) \frac{b-a}{3(b+a)}
  4. λ e - x λ , x > 0 \lambda e^{-x\lambda},\,\,x>0
  5. 1 / 2 1/2
  6. { α k α x α + 1 x k 0 x < k \begin{cases}\frac{\alpha k^{\alpha}}{x^{\alpha+1}}&x\geq k\\ 0&x<k\end{cases}
  7. { 1 0 < α < 1 1 2 α - 1 α 1 \begin{cases}1&0<\alpha<1\\ \frac{1}{2\alpha-1}&\alpha\geq 1\end{cases}
  8. 2 - k / 2 e - x / 2 x k / 2 - 1 Γ ( k / 2 ) \frac{2^{-k/2}e^{-x/2}x^{k/2-1}}{\Gamma(k/2)}
  9. 2 Γ ( 1 + k 2 ) k Γ ( k / 2 ) π \frac{2\Gamma\left(\frac{1+k}{2}\right)}{k\,\Gamma(k/2)\sqrt{\pi}}
  10. e - x / θ x k - 1 θ - k Γ ( k ) \frac{e^{-x/\theta}x^{k-1}\theta^{-k}}{\Gamma(k)}
  11. Γ ( 2 k + 1 2 ) k Γ ( k ) π \frac{\Gamma\left(\frac{2k+1}{2}\right)}{k\,\Gamma(k)\sqrt{\pi}}
  12. B = 0 1 L ( X ) d X . B=\int_{0}^{1}L(X)dX.
  13. G = 1 n ( n + 1 - 2 ( i = 1 n ( n + 1 - i ) y i i = 1 n y i ) ) G=\frac{1}{n}\left(n+1-2\left(\frac{\sum\limits_{i=1}^{n}\;(n+1-i)y_{i}}{\sum% \limits_{i=1}^{n}y_{i}}\right)\right)
  14. G = 2 Σ i = 1 n i y i n Σ i = 1 n y i - n + 1 n G=\frac{2\Sigma_{i=1}^{n}\;iy_{i}}{n\Sigma_{i=1}^{n}y_{i}}-\frac{n+1}{n}
  15. G = 1 - Σ i = 1 n f ( y i ) ( S i - 1 + S i ) S n G=1-\frac{\Sigma_{i=1}^{n}\;f(y_{i})(S_{i-1}+S_{i})}{S_{n}}
  16. S i = Σ j = 1 i f ( y j ) y j S_{i}=\Sigma_{j=1}^{i}\;f(y_{j})\,y_{j}\,
  17. S 0 = 0 S_{0}=0\,
  18. n n\rightarrow\infty
  19. G = 1 - 1 μ 0 ( 1 - F ( y ) ) 2 d y = 1 μ 0 F ( y ) ( 1 - F ( y ) ) d y G=1-\frac{1}{\mu}\int_{0}^{\infty}(1-F(y))^{2}dy=\frac{1}{\mu}\int_{0}^{\infty% }F(y)(1-F(y))dy
  20. G ( S ) = 1 n - 1 ( n + 1 - 2 ( Σ i = 1 n ( n + 1 - i ) y i Σ i = 1 n y i ) ) G(S)=\frac{1}{n-1}\left(n+1-2\left(\frac{\Sigma_{i=1}^{n}\;(n+1-i)y_{i}}{% \Sigma_{i=1}^{n}y_{i}}\right)\right)
  21. G ( S ) = 1 - 2 n - 1 ( n - Σ i = 1 n i y i Σ i = 1 n y i ) G(S)=1-\frac{2}{n-1}\left(n-\frac{\Sigma_{i=1}^{n}\;iy_{i}}{\Sigma_{i=1}^{n}y_% {i}}\right)
  22. σ \sigma
  23. G = erf ( σ 2 ) G=\operatorname{erf}\left(\frac{\sigma}{2}\right)
  24. erf \operatorname{erf}
  25. G = 2 ϕ ( σ 2 ) - 1 G=2\phi\left(\frac{\sigma}{\sqrt{2}}\right)-1
  26. ϕ \phi
  27. G 1 = 1 - k = 1 n ( X k - X k - 1 ) ( Y k + Y k - 1 ) G_{1}=1-\sum_{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})
  28. k = A + N ( 0 , s 2 / y k ) k=A+\ N(0,s^{2}/y_{k})
  29. G = N + 1 N - 1 - 2 N ( N - 1 ) μ ( Σ i = 1 n P i X i ) G=\frac{N+1}{N-1}-\frac{2}{N(N-1)\mu}(\Sigma_{i=1}^{n}\;P_{i}X_{i})
  30. μ \mu
  31. X i X_{i}
  32. r j = x j / x ¯ r_{j}=x_{j}/\overline{x}
  33. x j x_{j}
  34. x ¯ \overline{x}
  35. r j = 1 r_{j}=1
  36. r j = 1 r_{j}=1
  37. Inequality = Σ j p j f ( r j ) , \,\text{Inequality}=\Sigma_{j}\,p_{j}\,f(r_{j})\,,
  38. A U C = ( G + 1 ) / 2 AUC=(G+1)/2
  39. G 1 G_{1}

Global-warming_potential.html

  1. R F = n = 1 100 A b s i * F i / ( p a t h l e n g t h * d e n s i t y ) RF=\sum_{n=1}^{100}Abs_{i}*F_{i}/(pathlength*density)
  2. G W P ( x ) = 0 T H a x [ x ( t ) ] d t 0 T H a r [ r ( t ) ] d t GWP\left(x\right)=\frac{\int_{0}^{TH}a_{x}\cdot\left[x(t)\right]dt}{\int_{0}^{% TH}a_{r}\cdot\left[r(t)\right]dt}
  3. 4 {}_{4}

Global_illumination.html

  1. L = ( 1 - T ) - 1 L e L=(1-T)^{-1}L^{e}\,
  2. L = i = 0 T i L e L=\sum_{i=0}^{\infty}T^{i}L^{e}
  3. L n t l e + = L ( n - 1 ) L_{n}tl_{e}+=L^{(n-1)}

Global_Positioning_System.html

  1. ( x - x i ) 2 + ( y - y i ) 2 + ( z - z i ) 2 = ( [ t ~ i - b - s i ] c ) 2 , i = 1 , 2 , , n (x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}=\bigl([\tilde{t}_{i}-b-s_{i}]c\bigr)% ^{2},\;i=1,2,\dots,n
  2. p i = ( t ~ i - s i ) c p_{i}=\left(\tilde{t}_{i}-s_{i}\right)c
  3. ( x - x i ) 2 + ( y - y i ) 2 + ( z - z i ) 2 + b c = p i , i = 1 , 2 , , n \sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}}+bc=p_{i},\;i=1,2,...,n
  4. ( x ^ , y ^ , z ^ , b ^ ) = arg min ( x , y , z , b ) i ( ( x - x i ) 2 + ( y - y i ) 2 + ( z - z i ) 2 + b c - p i ) 2 \left(\hat{x},\hat{y},\hat{z},\hat{b}\right)=\underset{\left(x,y,z,b\right)}{% \arg\min}\sum_{i}\left(\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}}+bc-p_{% i}\right)^{2}
  5. 1 s 1575.42 × 10 6 = 0.63475 ns 1 ns \frac{1\,\mathrm{s}}{1575.42\times 10^{6}}=0.63475\,\mathrm{ns}\approx 1\,% \mathrm{ns}
  6. 1 s 1023 × 10 3 = 977.5 ns 1000 ns \frac{1\,\mathrm{s}}{1023\times 10^{3}}=977.5\,\mathrm{ns}\approx 1000\,% \mathrm{ns}
  7. ϕ ( r i , s j , t k ) \ \phi(r_{i},s_{j},t_{k})
  8. t k \ \ t_{k}
  9. ϕ \ \phi
  10. ϕ i , j , k = ϕ ( r i , s j , t k ) \ \phi_{i,j,k}=\phi(r_{i},s_{j},t_{k})
  11. Δ r , Δ s , Δ t \ \Delta^{r},\Delta^{s},\Delta^{t}
  12. α i , j , k \ \alpha_{i,j,k}
  13. Δ r , Δ s , Δ t \ \Delta^{r},\Delta^{s},\Delta^{t}
  14. Δ r ( α i , j , k ) = α i + 1 , j , k - α i , j , k \ \Delta^{r}(\alpha_{i,j,k})=\alpha_{i+1,j,k}-\alpha_{i,j,k}
  15. Δ s ( α i , j , k ) = α i , j + 1 , k - α i , j , k \ \Delta^{s}(\alpha_{i,j,k})=\alpha_{i,j+1,k}-\alpha_{i,j,k}
  16. Δ t ( α i , j , k ) = α i , j , k + 1 - α i , j , k \ \Delta^{t}(\alpha_{i,j,k})=\alpha_{i,j,k+1}-\alpha_{i,j,k}
  17. α i , j , k a n d β l , m , n \ \alpha_{i,j,k}\ and\ \beta_{l,m,n}
  18. ( a α i , j , k + b β l , m , n ) \ (a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})
  19. Δ r ( a α i , j , k + b β l , m , n ) = a Δ r ( α i , j , k ) + b Δ r ( β l , m , n ) \ \Delta^{r}(a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})=a\ \Delta^{r}(\alpha_{i,j,k})% +b\ \Delta^{r}(\beta_{l,m,n})
  20. Δ s ( a α i , j , k + b β l , m , n ) = a Δ s ( α i , j , k ) + b Δ s ( β l , m , n ) \ \Delta^{s}(a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})=a\ \Delta^{s}(\alpha_{i,j,k})% +b\ \Delta^{s}(\beta_{l,m,n})
  21. Δ t ( a α i , j , k + b β l , m , n ) = a Δ t ( α i , j , k ) + b Δ t ( β l , m , n ) \ \Delta^{t}(a\ \alpha_{i,j,k}+b\ \beta_{l,m,n})=a\ \Delta^{t}(\alpha_{i,j,k})% +b\ \Delta^{t}(\beta_{l,m,n})
  22. Δ s ( ϕ 1 , 1 , 1 ) = ϕ 1 , 2 , 1 - ϕ 1 , 1 , 1 \ \Delta^{s}(\phi_{1,1,1})=\phi_{1,2,1}-\phi_{1,1,1}
  23. Δ r ( Δ s ( ϕ 1 , 1 , 1 ) ) \displaystyle\Delta^{r}(\Delta^{s}(\phi_{1,1,1}))
  24. Δ t ( Δ r ( Δ s ( ϕ 1 , 1 , 1 ) ) ) \ \Delta^{t}(\Delta^{r}(\Delta^{s}(\phi_{1,1,1})))

Glossary_of_topology.html

  1. T 1 T_{1}
  2. τ \tau
  3. τ \tau
  4. B B
  5. τ \tau
  6. X X
  7. B B
  8. B B
  9. ( X , τ ) (X,\tau)
  10. σ \sigma
  11. σ \sigma
  12. X X
  13. τ \tau
  14. A A
  15. A \partial A
  16. b d bd
  17. A A
  18. X X
  19. \R n \R^{n}
  20. \varnothing
  21. f f
  22. X X / Z X\rightarrow X/Z
  23. Z X Z\subset X
  24. ( 0 , 1 ) (0,1)
  25. ( 1 , 2 ) (1,2)
  26. f f
  27. f : X Y f\colon X\rightarrow Y
  28. Y Y
  29. f f
  30. U Y U\subset Y
  31. U U
  32. Y Y
  33. f - 1 ( U ) f^{-1}(U)
  34. X X

Gluon.html

  1. ( r b ¯ + b r ¯ ) / 2 . (r\bar{b}+b\bar{r})/\sqrt{2}.
  2. ( r r ¯ + b b ¯ + g g ¯ ) / 3 . (r\bar{r}+b\bar{b}+g\bar{g})/\sqrt{3}.
  3. ( r b ¯ + b r ¯ ) / 2 (r\bar{b}+b\bar{r})/\sqrt{2}
  4. - i ( r b ¯ - b r ¯ ) / 2 -i(r\bar{b}-b\bar{r})/\sqrt{2}
  5. ( r g ¯ + g r ¯ ) / 2 (r\bar{g}+g\bar{r})/\sqrt{2}
  6. - i ( r g ¯ - g r ¯ ) / 2 -i(r\bar{g}-g\bar{r})/\sqrt{2}
  7. ( b g ¯ + g b ¯ ) / 2 (b\bar{g}+g\bar{b})/\sqrt{2}
  8. - i ( b g ¯ - g b ¯ ) / 2 -i(b\bar{g}-g\bar{b})/\sqrt{2}
  9. ( r r ¯ - b b ¯ ) / 2 (r\bar{r}-b\bar{b})/\sqrt{2}
  10. ( r r ¯ + b b ¯ - 2 g g ¯ ) / 6 . (r\bar{r}+b\bar{b}-2g\bar{g})/\sqrt{6}.
  11. r ¯ \overline{r}
  12. g ¯ \overline{g}
  13. b ¯ \overline{b}

Glycine.html

  1. K = [ H 3 N + CH 2 CO 2 - ] [ H 2 NCH 2 CO 2 H ] K=\mathrm{\frac{[H_{3}N^{+}CH_{2}CO_{2}^{-}]}{[H_{2}NCH_{2}CO_{2}H]}}

GM.html

  1. μ = G M \mu=GM

Goldbach's_conjecture.html

  1. 1 / ln m 1/\ln m\,\!
  2. 1 / [ ln m ln ( n - m ) ] 1\big/\big[\ln m\,\ln(n-m)\big]
  3. m = 3 n / 2 1 ln m 1 ln ( n - m ) n 2 ln 2 n . \sum_{m=3}^{n/2}\frac{1}{\ln m}{1\over\ln(n-m)}\approx\frac{n}{2\ln^{2}n}.
  4. n = p 1 + + p c n=p_{1}+\cdots+p_{c}
  5. p 1 p c p_{1}\leq\cdots\leq p_{c}
  6. ( p p γ c , p ( n ) ( p - 1 ) c ) 2 x 1 x c : x 1 + + x c = n d x 1 d x c - 1 ln x 1 ln x c \left(\prod_{p}\frac{p\gamma_{c,p}(n)}{(p-1)^{c}}\right)\int_{2\leq x_{1}\leq% \cdots\leq x_{c}:x_{1}+\cdots+x_{c}=n}\frac{dx_{1}\cdots dx_{c-1}}{\ln x_{1}% \cdots\ln x_{c}}
  7. γ c , p ( n ) \gamma_{c,p}(n)
  8. n = q 1 + + q c mod p n=q_{1}+\cdots+q_{c}\mod p
  9. q 1 , , q c 0 mod p q_{1},\ldots,q_{c}\neq 0\mod p
  10. c = 2 c=2
  11. 2 Π 2 ( p | n ; p 3 p - 1 p - 2 ) 2 n d x ln 2 x 2 Π 2 ( p | n ; p 3 p - 1 p - 2 ) n ln 2 n 2\Pi_{2}\left(\prod_{p|n;p\geq 3}\frac{p-1}{p-2}\right)\int_{2}^{n}\frac{dx}{% \ln^{2}x}\approx 2\Pi_{2}\left(\prod_{p|n;p\geq 3}\frac{p-1}{p-2}\right)\frac{% n}{\ln^{2}n}
  12. Π 2 \Pi_{2}
  13. Π 2 := p 3 ( 1 - 1 ( p - 1 ) 2 ) = 0.6601618158 . \Pi_{2}:=\prod_{p\geq 3}\left(1-\frac{1}{(p-1)^{2}}\right)=0.6601618158\ldots.
  14. C N 1 - c CN^{1-c}
  15. e 3100 2 × 10 1346 e^{3100}\approx 2\times 10^{1346}

Golden_ratio.html

  1. a + b a = a b φ \frac{a+b}{a}=\frac{a}{b}\equiv\varphi
  2. a + b a = a b = def φ , \frac{a+b}{a}=\frac{a}{b}\ \stackrel{\,\text{def}}{=}\ \varphi,
  3. φ \varphi
  4. ϕ \phi
  5. φ = 1 + 5 2 = 1.6180339887 . \varphi=\frac{1+\sqrt{5}}{2}=1.6180339887\ldots.
  6. 1 + 1 1 + 1 1 + 1 1 + 1 1 + 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}
  7. 1 + 5 2 \frac{1+\sqrt{5}}{2}
  8. 13 8 + n = 0 ( - 1 ) ( n + 1 ) ( 2 n + 1 ) ! ( n + 2 ) ! n ! 4 ( 2 n + 3 ) \frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{(2n+3)}}
  9. a + b a = a b = φ . \frac{a+b}{a}=\frac{a}{b}=\varphi.
  10. a + b a = 1 + b a = 1 + 1 φ . \frac{a+b}{a}=1+\frac{b}{a}=1+\frac{1}{\varphi}.
  11. 1 + 1 φ = φ . 1+\frac{1}{\varphi}=\varphi.
  12. φ + 1 = φ 2 \varphi+1=\varphi^{2}
  13. φ 2 - φ - 1 = 0. {\varphi}^{2}-\varphi-1=0.
  14. φ = 1 + 5 2 = 1.61803 39887 \varphi=\frac{1+\sqrt{5}}{2}=1.61803\,39887\dots
  15. φ = 1 - 5 2 = - 0.6180 339887 \varphi=\frac{1-\sqrt{5}}{2}=-0.6180\,339887\dots
  16. φ = 1 + 5 2 = 1.61803 39887 \varphi=\frac{1+\sqrt{5}}{2}=1.61803\,39887\dots
  17. n m = m n - m . ( * ) \frac{n}{m}=\frac{m}{n-m}.\qquad(*)
  18. 1 + 5 2 \textstyle\frac{1+\sqrt{5}}{2}
  19. 2 ( 1 + 5 2 ) - 1 = 5 \textstyle 2\left(\frac{1+\sqrt{5}}{2}\right)-1=\sqrt{5}
  20. - 1 φ = 1 - φ = 1 - 5 2 = - 0.61803 39887 . -\frac{1}{\varphi}=1-\varphi=\frac{1-\sqrt{5}}{2}=-0.61803\,39887\dots.
  21. Φ \Phi
  22. Φ = 1 φ = φ - 1 = 0.61803 39887 . \Phi={1\over\varphi}=\varphi^{-1}=0.61803\,39887\ldots.
  23. Φ \Phi
  24. Φ = φ - 1 = 1.61803 39887 - 1 = 0.61803 39887 . \Phi=\varphi-1=1.61803\,39887\ldots-1=0.61803\,39887\ldots.
  25. 1 φ = φ - 1 , {1\over\varphi}=\varphi-1,
  26. 1 Φ = Φ + 1. {1\over\Phi}=\Phi+1.
  27. φ = [ 1 ; 1 , 1 , 1 , ] = 1 + 1 1 + 1 1 + 1 1 + \varphi=[1;1,1,1,\dots]=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}
  28. φ - 1 = [ 0 ; 1 , 1 , 1 , ] = 0 + 1 1 + 1 1 + 1 1 + \varphi^{-1}=[0;1,1,1,\dots]=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}
  29. φ = 1 + 1 + 1 + 1 + . \varphi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}.
  30. φ = 13 8 + n = 0 ( - 1 ) ( n + 1 ) ( 2 n + 1 ) ! ( n + 2 ) ! n ! 4 ( 2 n + 3 ) . \varphi=\frac{13}{8}+\sum_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{(n+2)!n!4^{% (2n+3)}}.
  31. φ = 1 + 2 sin ( π / 10 ) = 1 + 2 sin 18 \varphi=1+2\sin(\pi/10)=1+2\sin 18^{\circ}
  32. φ = 1 2 csc ( π / 10 ) = 1 2 csc 18 \varphi={1\over 2}\csc(\pi/10)={1\over 2}\csc 18^{\circ}
  33. φ = 2 cos ( π / 5 ) = 2 cos 36 \varphi=2\cos(\pi/5)=2\cos 36^{\circ}
  34. φ = 2 sin ( 3 π / 10 ) = 2 sin 54 . \varphi=2\sin(3\pi/10)=2\sin 54^{\circ}.
  35. Φ \Phi
  36. | A B | | B C | = | A C | | A B | = ϕ \tfrac{|AB|}{|BC|}=\tfrac{|AC|}{|AB|}=\phi
  37. b a = 1 + 5 2 . {b\over a}={{1+\sqrt{5}}\over 2}.
  38. F ( n ) = φ n - ( 1 - φ ) n 5 = φ n - ( - φ ) - n 5 . F\left(n\right)={{\varphi^{n}-(1-\varphi)^{n}}\over{\sqrt{5}}}={{\varphi^{n}-(% -\varphi)^{-n}}\over{\sqrt{5}}}.
  39. lim n F ( n + 1 ) F ( n ) = φ . \lim_{n\to\infty}\frac{F(n+1)}{F(n)}=\varphi.
  40. n = 1 | F ( n ) φ - F ( n + 1 ) | = φ . \sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)|=\varphi.
  41. lim n F ( n + a ) F ( n ) = φ a , \lim_{n\to\infty}\frac{F(n+a)}{F(n)}={\varphi}^{a},
  42. a = 1 a=1
  43. φ n + 1 = φ n + φ n - 1 . \varphi^{n+1}=\varphi^{n}+\varphi^{n-1}.
  44. 3 φ 3 - 5 φ 2 + 4 \displaystyle 3\varphi^{3}-5\varphi^{2}+4
  45. φ k = F k φ + F k - 1 , \varphi^{k}=F_{k}\varphi+F_{k-1},
  46. F k F_{k}
  47. x 2 = a x + b x^{2}=ax+b
  48. \Q ( α ) \Q(\alpha)
  49. \Q \Q
  50. { 1 , α , , α n - 1 } \{1,\alpha,\dots,\alpha^{n-1}\}
  51. φ ± = ( 1 ± 5 ) / 2 \varphi_{\pm}=(1\pm\sqrt{5})/2
  52. x , 1 / ( 1 - x ) , ( x - 1 ) / x , x,1/(1-x),(x-1)/x,
  53. 1 / x , 1 - x , x / ( x - 1 ) 1/x,1-x,x/(x-1)
  54. 1 / 2 1/2
  55. PSL ( 2 , 𝐙 ) \operatorname{PSL}(2,\mathbf{Z})
  56. S 3 , S_{3},
  57. { 0 , 1 , } \{0,1,\infty\}
  58. S 3 S 2 S_{3}\to S_{2}
  59. C 3 < S 3 C_{3}<S_{3}
  60. ( ) ( 01 ) ( 0 1 ) ()(01\infty)(0\infty 1)
  61. φ 2 = φ + 1 = 2.618 \varphi^{2}=\varphi+1=2.618\dots
  62. 1 φ = φ - 1 = 0.618 . {1\over\varphi}=\varphi-1=0.618\dots.
  63. φ n = φ n - 1 + φ n - 2 = φ F n + F n - 1 . \varphi^{n}=\varphi^{n-1}+\varphi^{n-2}=\varphi\cdot\operatorname{F}_{n}+% \operatorname{F}_{n-1}.
  64. n / 2 - 1 = m \lfloor n/2-1\rfloor=m
  65. φ n = φ n - 1 + φ n - 3 + + φ n - 1 - 2 m + φ n - 2 - 2 m \!\ \varphi^{n}=\varphi^{n-1}+\varphi^{n-3}+\cdots+\varphi^{n-1-2m}+\varphi^{n% -2-2m}
  66. φ n - φ n - 1 = φ n - 2 . \!\ \varphi^{n}-\varphi^{n-1}=\varphi^{n-2}.
  67. ( 5 ) \mathbb{Q}(\sqrt{5})
  68. ( 5 ) \mathbb{Q}(\sqrt{5})
  69. φ n = L n + F n 5 2 \varphi^{n}={{L_{n}+F_{n}\sqrt{5}}\over 2}
  70. L n L_{n}
  71. n n
  72. 4 log ( φ ) 4\log(\varphi)
  73. φ = 1 + 5 2 \varphi={1+\sqrt{5}\over 2}
  74. x n + 1 = ( x n + 5 / x n ) 2 x_{n+1}=\frac{(x_{n}+5/x_{n})}{2}
  75. x n + 1 = x n 2 + 1 2 x n - 1 , x_{n+1}=\frac{x_{n}^{2}+1}{2x_{n}-1},
  76. x n + 1 = x n 2 + 2 x n x n 2 + 1 . x_{n+1}=\frac{x_{n}^{2}+2x_{n}}{x_{n}^{2}+1}.
  77. φ \sqrt{\varphi}
  78. φ \sqrt{\varphi}
  79. 1 : φ : φ 1:\sqrt{\varphi}:\varphi
  80. φ = φ 2 - 1 \sqrt{\varphi}=\sqrt{\varphi^{2}-1}
  81. φ = 1 + φ \varphi=\sqrt{1+\varphi}
  82. φ \sqrt{\varphi}
  83. φ 4 / π \sqrt{\varphi}\approx 4/\pi

Golden_ratio_base.html

  1. 5 \sqrt{5}
  2. 1 + 5 2 1+\frac{\sqrt{5}}{2}
  3. k = 0 φ - 2 k = 1 1 - φ - 2 = φ \sum_{k=0}^{\infty}\varphi^{-2k}=\frac{1}{1-\varphi^{-2}}=\varphi
  4. 5 \sqrt{5}
  5. 5 \sqrt{5}
  6. 2 \sqrt{2}
  7. 5 \sqrt{5}
  8. 5 \sqrt{5}
  9. 5 \sqrt{5}

Golomb_ruler.html

  1. A = { a 1 , a 2 , , a m } a 1 < a 2 < < a m A=\left\{a_{1},a_{2},...,a_{m}\right\}\quad a_{1}<a_{2}<...<a_{m}
  2. i , j , k , l { 1 , 2 , , m } , a i - a j = a k - a l i = k and j = l . \forall i,j,k,l\in\left\{1,2,...,m\right\},a_{i}-a_{j}=a_{k}-a_{l}\iff i=k\and j% =l.
  3. m m
  4. a m - a 1 a_{m}-a_{1}
  5. a 1 = 0 a_{1}=0
  6. m > 2 m>2
  7. a 2 - a 1 < a m - a m - 1 a_{2}-a_{1}<a_{m}-a_{m-1}
  8. f : { 1 , 2 , , m } { 0 , 1 , , n } f:\left\{1,2,...,m\right\}\to\left\{0,1,...,n\right\}
  9. f ( 1 ) = 0 f(1)=0
  10. f ( m ) = n f(m)=n
  11. i , j , k , l { 1 , 2 , , m } , f ( i ) - f ( j ) = f ( k ) - f ( l ) i = k and j = l . \forall i,j,k,l\in\left\{1,2,...,m\right\},f(i)-f(j)=f(k)-f(l)\iff i=k\and j=l.
  12. m m
  13. n n
  14. f ( 2 ) < f ( m ) - f ( m - 1 ) f(2)<f(m)-f(m-1)
  15. m > 2 m>2
  16. m m
  17. 2 p k + ( k 2 mod p ) , k [ 0 , p - 1 ] 2pk+(k^{2}\,\bmod\,p),k\in[0,p-1]

Gospel_of_John.html

  1. 𝔓 52 \mathfrak{P}^{52}
  2. 𝔓 90 \mathfrak{P}^{90}
  3. 𝔓 75 \mathfrak{P}^{75}
  4. 𝔓 66 \mathfrak{P}^{66}

Gödel's_completeness_theorem.html

  1. A \models A
  2. ¬ A \lnot A
  3. ¬ A \lnot A
  4. ¬ A \lnot A
  5. A \vdash A

Gödel's_ontological_proof.html

  1. Ax. 1. { P ( φ ) x [ φ ( x ) ψ ( x ) ] } P ( ψ ) Ax. 2. P ( ¬ φ ) ¬ P ( φ ) Th. 1. P ( φ ) x [ φ ( x ) ] Df. 1. G ( x ) φ [ P ( φ ) φ ( x ) ] Ax. 3. P ( G ) Th. 2. x G ( x ) Df. 2. φ ess x φ ( x ) ψ { ψ ( x ) y [ φ ( y ) ψ ( y ) ] } Ax. 4. P ( φ ) P ( φ ) Th. 3. G ( x ) G ess x Df. 3. E ( x ) φ [ φ ess x y φ ( y ) ] Ax. 5. P ( E ) Th. 4. x G ( x ) \begin{array}[]{rl}\,\text{Ax. 1.}&\left\{P(\varphi)\wedge\Box\;\forall x[% \varphi(x)\to\psi(x)]\right\}\to P(\psi)\\ \,\text{Ax. 2.}&P(\neg\varphi)\leftrightarrow\neg P(\varphi)\\ \,\text{Th. 1.}&P(\varphi)\to\Diamond\;\exists x[\varphi(x)]\\ \,\text{Df. 1.}&G(x)\iff\forall\varphi[P(\varphi)\to\varphi(x)]\\ \,\text{Ax. 3.}&P(G)\\ \,\text{Th. 2.}&\Diamond\;\exists x\;G(x)\\ \,\text{Df. 2.}&\varphi\,\text{ ess }x\iff\varphi(x)\wedge\forall\psi\left\{% \psi(x)\to\Box\;\forall y[\varphi(y)\to\psi(y)]\right\}\\ \,\text{Ax. 4.}&P(\varphi)\to\Box\;P(\varphi)\\ \,\text{Th. 3.}&G(x)\to G\,\text{ ess }x\\ \,\text{Df. 3.}&E(x)\iff\forall\varphi[\varphi\,\text{ ess }x\to\Box\;\exists y% \;\varphi(y)]\\ \,\text{Ax. 5.}&P(E)\\ \,\text{Th. 4.}&\Box\;\exists x\;G(x)\end{array}

Grade_of_service.html

  1. Grade of Service = number of successful calls number of offered calls ( 1 ) \mbox{Grade of Service}~{}=\frac{\mbox{number of successful calls}~{}}{\mbox{% number of offered calls}~{}}\qquad(1)
  2. t d t_{d}
  3. t o t_{o}
  4. Grade of Service = ( A N N ! ) ( k = 0 N A k k ! ) ( 2 ) \mbox{Grade of Service}~{}=\frac{\left(\frac{A^{N}}{N!}\right)}{\left(\sum_{k=% 0}^{N}\frac{A^{k}}{k!}\right)}\qquad(2)

Graded-index_fiber.html

  1. Pulse dispersion = k δ n n 1 l c , \mathrm{Pulse~{}dispersion}=\frac{k\delta n\ n_{1}\ l}{c}\,\!,
  2. δ n \delta n\,\!
  3. n 1 n_{1}\,\!
  4. l l\,\!
  5. c 3 × 10 8 m / s c\approx 3\times 10^{8}~{}\mathrm{m/s}\,\!
  6. k k\,\!

Gradient.html

  1. T T
  2. ( x , y , z ) (x,y,z)
  3. T ( x , y , z ) T(x,y,z)
  4. f \vec{\nabla}f
  5. ( f ( x ) ) 𝐯 = D 𝐯 f ( x ) . (\nabla f(x))\cdot\mathbf{v}=D_{\mathbf{v}}f(x).
  6. f = f x 1 𝐞 1 + + f x n 𝐞 n \nabla f=\frac{\partial f}{\partial x_{1}}\mathbf{e}_{1}+\cdots+\frac{\partial f% }{\partial x_{n}}\mathbf{e}_{n}
  7. f = f x 𝐢 + f y 𝐣 + f z 𝐤 \nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}% \mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}
  8. f ( x , y , z ) = 2 x + 3 y 2 - sin ( z ) f(x,y,z)=\ 2x+3y^{2}-\sin(z)
  9. f = f x 𝐢 + f y 𝐣 + f z 𝐤 = 2 𝐢 + 6 y 𝐣 - cos ( z ) 𝐤 . \nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}% \mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}=2\mathbf{i}+6y\mathbf{j}-% \cos(z)\mathbf{k}.
  10. f ( x ) f ( x 0 ) + ( f ) x 0 ( x - x 0 ) f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot(x-x_{0})
  11. ( f ) x 0 (\nabla f)_{x_{0}}
  12. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  13. ( f ) x v = d f x ( v ) (\nabla f)_{x}\cdot v=\mathrm{d}f_{x}(v)
  14. ( f x 1 , , f x n ) \left(\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{\partial x_{n}% }\right)
  15. ( f ) i = d f i 𝖳 (\nabla f)_{i}=\mathrm{d}f^{\mathsf{T}}_{i}
  16. lim h 0 f ( x + h ) - f ( x ) - f ( x ) h h = 0 \lim_{h\to 0}\frac{\|f(x+h)-f(x)-\nabla f(x)\cdot h\|}{\|h\|}=0
  17. ( α f + β g ) ( a ) = α f ( a ) + β g ( a ) . \nabla\left(\alpha f+\beta g\right)(a)=\alpha\nabla f(a)+\beta\nabla g(a).
  18. ( f g ) ( a ) = f ( a ) g ( a ) + g ( a ) f ( a ) . \nabla(fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).
  19. ( f g ) ( c ) = f ( a ) g ( c ) , (f\circ g)^{\prime}(c)=\nabla f(a)\cdot g^{\prime}(c),
  20. ( f g ) ( c ) = ( D g ( c ) ) 𝖳 ( f ( a ) ) \nabla(f\circ g)(c)=(Dg(c))^{\mathsf{T}}(\nabla f(a))
  21. ( h f ) ( a ) = h ( f ( a ) ) f ( a ) . \nabla(h\circ f)(a)=h^{\prime}(f(a))\nabla f(a).
  22. g ( f , X ) = X f , i.e., g x ( ( f ) x , X x ) = ( X f ) ( x ) g(\nabla f,X)=\partial_{X}f,\qquad\,\text{i.e.,}\quad g_{x}((\nabla f)_{x},X_{% x})=(\partial_{X}f)(x)
  23. j = 1 n X j ( φ ( x ) ) x j ( f φ - 1 ) | φ ( x ) , \sum_{j=1}^{n}X^{j}(\varphi(x))\frac{\partial}{\partial x_{j}}(f\circ\varphi^{% -1})\Big|_{\varphi(x)},
  24. f = g i k f x k x i . \nabla f=g^{ik}\frac{\partial f}{\partial x^{k}}\frac{\partial}{\partial x^{i}}.
  25. ( X f ) ( x ) = d f x ( X x ) . (\partial_{X}f)(x)=df_{x}(X_{x})\ .
  26. = g : T * M T M \sharp=\sharp^{g}\colon T^{*}M\to TM
  27. f ( ρ , ϕ , z ) = f ρ 𝐞 ρ + 1 ρ f ϕ 𝐞 ϕ + f z 𝐞 z \nabla f(\rho,\phi,z)=\frac{\partial f}{\partial\rho}\mathbf{e}_{\rho}+\frac{1% }{\rho}\frac{\partial f}{\partial\phi}\mathbf{e}_{\phi}+\frac{\partial f}{% \partial z}\mathbf{e}_{z}
  28. f ( r , θ , ϕ ) = f r 𝐞 r + 1 r f θ 𝐞 θ + 1 r sin θ f ϕ 𝐞 ϕ \nabla f(r,\theta,\phi)=\frac{\partial f}{\partial r}\mathbf{e}_{r}+\frac{1}{r% }\frac{\partial f}{\partial\theta}\mathbf{e}_{\theta}+\frac{1}{r\sin\theta}% \frac{\partial f}{\partial\phi}\mathbf{e}_{\phi}
  29. 𝐟 = g j k f i x j 𝐞 i 𝐞 k \nabla\mathbf{f}=g^{jk}\frac{\partial{{f}^{i}}}{\partial{{x}_{j}}}{{\mathbf{e}% }_{i}}{{\mathbf{e}}_{k}}
  30. f i x j = ( f 1 , f 2 , f 3 ) ( x 1 , x 2 , x 3 ) \frac{\partial{{f}_{i}}}{\partial{{x}_{j}}}=\frac{\partial({{f}_{1}},{{f}_{2}}% ,{{f}_{3}})}{\partial({{x}_{1}},{{x}_{2}},{{x}_{3}})}
  31. 𝐟 = g j k ( f i x j + Γ i j l f l ) 𝐞 i 𝐞 k \nabla\mathbf{f}=g^{jk}\left(\frac{\partial{{f}^{i}}}{\partial{{x}_{j}}}+{% \Gamma^{i}}_{jl}f^{l}\right){{\mathbf{e}}_{i}}{{\mathbf{e}}_{k}}
  32. a 𝐟 b = g a c c 𝐟 b \nabla^{a}\mathbf{f}^{b}=g^{ac}\nabla_{c}\mathbf{f}^{b}
  33. c \nabla_{c}

Grand_Unified_Theory.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  3. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  4. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  5. S U ( 5 ) SU(5)
  6. 𝟓 + 𝟏𝟎 \mathbf{5}+\mathbf{10}
  7. S U ( 5 ) SU(5)
  8. S U ( 5 ) S U ( 3 ) × S U ( 2 ) × U ( 1 ) SU(5)\supset SU(3)\times SU(2)\times U(1)
  9. S U ( 5 ) SU(5)
  10. S U ( 5 ) SU(5)
  11. 𝟓 \mathbf{5}
  12. 𝟏𝟎 \mathbf{10}
  13. 𝟓 \mathbf{5}
  14. 𝟏𝟎 \mathbf{10}
  15. S O ( 10 ) S U ( 5 ) S U ( 3 ) × S U ( 2 ) × U ( 1 ) SO(10)\supset SU(5)\supset SU(3)\times SU(2)\times U(1)
  16. 𝟏𝟔 \mathbf{16}
  17. 𝟓 ¯ \overline{\mathbf{5}}
  18. 𝟏𝟎 \mathbf{10}
  19. S U ( 5 ) SU(5)
  20. S U ( 5 ) SU(5)
  21. S O ( 10 ) SO(10)
  22. S O ( 10 ) SO(10)
  23. 15 × 15 15×15
  24. 𝟏𝟎 + 𝟓 \mathbf{10}+\mathbf{5}
  25. S U ( 5 ) SU(5)
  26. S O ( 10 ) SO(10)
  27. 𝟔𝟒 \mathbf{64}
  28. 𝟔𝟒 = 𝟖 + 𝟓𝟔 \mathbf{64}=\mathbf{8}+\mathbf{56}
  29. S U ( 8 ) SU(8)
  30. S U ( 5 ) SU(5)
  31. O ( 16 ) O(16)
  32. S p ( 8 ) Sp(8)
  33. S p ( 4 ) Sp(4)
  34. 4 × 4 4×4
  35. 𝟏𝟔 \mathbf{16}
  36. S p ( 8 ) Sp(8)
  37. S U ( 4 ) SU(4)
  38. S U ( 3 ) × U ( 1 ) SU(3)×U(1)
  39. [ e + i e ¯ + j v + k v ¯ u r + i u r ¯ + j d r + k d r ¯ u g + i u g ¯ + j d g + k d g ¯ u b + i u b ¯ + j d b + k d b ¯ ] L \begin{bmatrix}e+i\overline{e}+jv+k\overline{v}\\ u_{r}+i\overline{u_{r}}+jd_{r}+k\overline{d_{r}}\\ u_{g}+i\overline{u_{g}}+jd_{g}+k\overline{d_{g}}\\ u_{b}+i\overline{u_{b}}+jd_{b}+k\overline{d_{b}}\\ \end{bmatrix}_{L}
  40. 4 × 4 4×4
  41. 4 × 4 4×4
  42. S p ( 8 ) × S U ( 2 ) Sp(8)×SU(2)
  43. S U ( 4 , H ) L × H R = S p ( 8 ) × S U ( 2 ) S U ( 4 ) × S U ( 2 ) S U ( 3 ) × S U ( 2 ) × U ( 1 ) SU(4,H)_{L}\times H_{R}=Sp(8)\times SU(2)\supset SU(4)\times SU(2)\supset SU(3% )\times SU(2)\times U(1)
  44. ψ \psi
  45. A μ a b A^{ab}_{\mu}
  46. 4 × 4 4×4
  47. S p ( 8 ) Sp(8)
  48. B μ B_{\mu}
  49. ψ a ¯ γ μ ( A μ a b ψ b + ψ a B μ ) \overline{\psi^{a}}\gamma_{\mu}\left(A^{ab}_{\mu}\psi^{b}+\psi^{a}B_{\mu}\right)
  50. ψ = [ a e μ e ¯ b τ μ ¯ τ ¯ c ] \psi=\begin{bmatrix}a&e&\mu\\ \overline{e}&b&\tau\\ \overline{\mu}&\overline{\tau}&c\end{bmatrix}
  51. [ ψ A , ψ B ] J 3 ( O ) [\psi_{A},\psi_{B}]\subset J_{3}(O)
  52. O ( 10 ) O(10)
  53. S U ( N ) SU(N)
  54. S U ( 5 ) SU(5)
  55. S O ( 10 ) SO(10)
  56. Λ GUT 10 16 GeV \Lambda_{\,\text{GUT}}\approx 10^{16}\,\,\text{GeV}
  57. S O ( 10 ) SO(10)
  58. S O ( 10 ) SO(10)
  59. S U ( 5 ) SU(5)
  60. S U ( 5 ) SU(5)
  61. S U ( 5 ) × U ( 1 ) SU(5)×U(1)
  62. S U ( 4 ) × S U ( 2 ) × S U ( 2 ) SU(4)×SU(2)×SU(2)
  63. S O ( 10 ) SO(10)
  64. S O ( 10 ) × U ( 1 ) SO(10)×U(1)
  65. S U ( 3 ) × S U ( 3 ) × S U ( 3 ) SU(3)×SU(3)×SU(3)
  66. S U ( 6 ) SU(6)
  67. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  68. S O ( 10 ) SO(10)
  69. S O ( 10 ) SO(10)
  70. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  71. S U ( 5 ) SU(5)
  72. S O ( 10 ) SO(10)
  73. S U ( 5 ) SU(5)
  74. ( 3 , 2 ) - 5 6 (3,2)_{-\frac{5}{6}}
  75. S U ( 5 ) SU(5)
  76. ( 3 , 2 ) 1 6 (3,2)_{\frac{1}{6}}
  77. S U ( 5 ) SU(5)
  78. T ( 3 , 1 ) - 1 3 T(3,1)_{-\frac{1}{3}}
  79. T ¯ ( 3 ¯ , 1 ) 1 3 \bar{T}(\bar{3},1)_{\frac{1}{3}}
  80. S U ( 5 ) SU(5)
  81. S U ( 5 ) SU(5)
  82. S O ( 10 ) SO(10)
  83. S O ( 10 ) SO(10)
  84. S O ( 10 ) SO(10)

Graph_theory.html

  1. G = ( V , E ) G=(V,E)
  2. V V
  3. E E
  4. V V
  5. V V
  6. E E
  7. V V
  8. E E
  9. | V | |V|
  10. | E | |E|
  11. { u , v } \{u,v\}
  12. u v uv
  13. K 3 , 3 K_{3,3}
  14. K 5 K_{5}

Grashof_number.html

  1. Gr L = g β ( T s - T ) L 3 ν 2 \mathrm{Gr}_{L}=\frac{g\beta(T_{s}-T_{\infty})L^{3}}{\nu^{2}}\,
  2. Gr D = g β ( T s - T ) D 3 ν 2 \mathrm{Gr}_{D}=\frac{g\beta(T_{s}-T_{\infty})D^{3}}{\nu^{2}}\,
  3. Gr D = g β ( T s - T ) D 3 ν 2 \mathrm{Gr}_{D}=\frac{g\beta(T_{s}-T_{\infty})D^{3}}{\nu^{2}}\,
  4. 10 8 < Gr L < 10 9 10^{8}<\mathrm{Gr}_{L}<10^{9}
  5. Gr c = g β * ( C a , s - C a , a ) L 3 ν 2 \mathrm{Gr}_{c}=\frac{g\beta^{*}(C_{a,s}-C_{a,a})L^{3}}{\nu^{2}}
  6. β * = - 1 ρ ( ρ C a ) T , p \beta^{*}=-\frac{1}{\rho}\left(\frac{\partial\rho}{\partial C_{a}}\right)_{T,p}
  7. β \mathrm{\beta}
  8. β = 1 v ( v T ) p = - 1 ρ ( ρ T ) p \beta=\frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_{p}=\frac{-1}{\rho% }\left(\frac{\partial\rho}{\partial T}\right)_{p}
  9. v v
  10. v v
  11. β \mathrm{\beta}
  12. ρ \mathrm{\rho}
  13. ρ = ρ o ( 1 - β Δ T ) \rho=\rho_{o}(1-\beta\Delta T)
  14. ρ o \rho_{o}
  15. ρ \rho
  16. Δ T = ( T - T o ) \Delta T=(T-T_{o})
  17. s ( ρ u r o n ) + y ( ρ v r o n ) = 0 \frac{\partial}{\partial s}(\rho ur_{o}^{n})+{\frac{\partial}{\partial y}}(% \rho vr_{o}^{n})=0
  18. s s
  19. u u
  20. y y
  21. v v
  22. r o r_{o}
  23. n n
  24. n n
  25. g g
  26. ρ ( u u s + v u y ) = y ( μ u y ) - d p d s + ρ g \rho\left(u\frac{\partial u}{\partial s}+v\frac{\partial u}{\partial y}\right)% =\frac{\partial}{\partial y}\left(\mu\frac{\partial u}{\partial y}\right)-% \frac{dp}{ds}+\rho g
  27. d p d s = ρ o g \frac{dp}{ds}=\rho_{o}g
  28. ρ ( u u s + v u y ) = μ ( 2 u y 2 ) + ρ g β ( T - T o ) \rho\left(u\frac{\partial u}{\partial s}+v\frac{\partial u}{\partial y}\right)% =\mu\left(\frac{\partial^{2}u}{\partial y^{2}}\right)+\rho g\beta(T-T_{o})
  29. ρ o - ρ = β ρ ( T - T o ) \rho_{o}-\rho=\beta\rho(T-T_{o})
  30. ν = μ ρ \nu=\frac{\mu}{\rho}
  31. u ( u s ) + v ( v y ) = ν ( 2 u y 2 ) + g β ( T - T o ) u\left(\frac{\partial u}{\partial s}\right)+v\left(\frac{\partial v}{\partial y% }\right)=\nu\left(\frac{\partial^{2}u}{\partial y^{2}}\right)+g\beta(T-T_{o})
  32. L c L_{c}
  33. V V
  34. V = Re L ν L c V=\frac{\mathrm{Re}_{L}\nu}{L_{c}}
  35. ( T s - T o ) (T_{s}-T_{o})
  36. s * = s L c s^{*}=\frac{s}{L_{c}}
  37. y * = y L c y^{*}=\frac{y}{L_{c}}
  38. u * = u V u^{*}=\frac{u}{V}
  39. v * = v V v^{*}=\frac{v}{V}
  40. T * = ( T - T o ) ( T s - T o ) T^{*}=\frac{(T-T_{o})}{(T_{s}-T_{o})}
  41. u * u * s * + v * u * y * = [ g β ( T s - T o ) L c 3 ν 2 ] T * Re L 2 + 1 Re L 2 u * y * 2 u^{*}\frac{\partial u^{*}}{\partial s^{*}}+v^{*}\frac{\partial u^{*}}{\partial y% ^{*}}=\left[\frac{g\beta(T_{s}-T_{o})L_{c}^{3}}{\nu^{2}}\right]\frac{T^{*}}{% \mathrm{Re}_{L}^{2}}+\frac{1}{\mathrm{Re}_{L}}\frac{\partial^{2}u^{*}}{% \partial{y^{*}}^{2}}
  42. T s T_{s}
  43. T o T_{o}
  44. L c L_{c}
  45. Gr = g β ( T s - T o ) L c 3 ν 2 \mathrm{Gr}=\frac{g\beta(T_{s}-T_{o})L_{c}^{3}}{\nu^{2}}
  46. F b F_{b}
  47. F b = ( ρ - ρ o ) g F_{b}=(\rho-\rho_{o})g
  48. F b = β g ρ o Δ T F_{b}=\beta g\rho_{o}\Delta T
  49. L L
  50. L \mathrm{L}
  51. μ \mu
  52. M Lt \mathrm{\frac{M}{Lt}}
  53. c p c_{p}
  54. Q MT \mathrm{\frac{Q}{MT}}
  55. k k
  56. Q LtT \mathrm{\frac{Q}{LtT}}
  57. β \beta
  58. 1 T \mathrm{\frac{1}{T}}
  59. g g
  60. L t 2 \mathrm{\frac{L}{t^{2}}}
  61. Δ T \Delta T
  62. T \mathrm{T}
  63. h h
  64. Q L 2 tT \mathrm{\frac{Q}{L^{2}tT}}
  65. μ , \mu,
  66. β \beta
  67. π \pi
  68. π 1 = L a μ b k c β d g e c p \pi_{1}=L^{a}\mu^{b}k^{c}\beta^{d}g^{e}c_{p}
  69. π 2 = L f μ g k h β i g j ρ \pi_{2}=L^{f}\mu^{g}k^{h}\beta^{i}g^{j}\rho
  70. π 3 = L k μ l k m β n g o Δ T \pi_{3}=L^{k}\mu^{l}k^{m}\beta^{n}g^{o}\Delta T
  71. π 4 = L q μ r k s β t g u h \pi_{4}=L^{q}\mu^{r}k^{s}\beta^{t}g^{u}h
  72. π \pi
  73. π 1 = μ ( c p ) k = P r \pi_{1}=\frac{\mu(c_{p})}{k}=Pr
  74. π 2 = l 3 g ρ 2 μ 2 \pi_{2}=\frac{l^{3}g\rho^{2}}{\mu^{2}}
  75. π 3 = β Δ T \pi_{3}=\beta\Delta T
  76. π 4 = h L k = N u \pi_{4}=\frac{hL}{k}=Nu
  77. π 2 \pi_{2}
  78. π 3 , \pi_{3},
  79. π 2 π 3 = β g ρ 2 Δ T L 3 μ 2 = Gr \pi_{2}\pi_{3}=\frac{\beta g\rho^{2}\Delta TL^{3}}{\mu^{2}}=\mathrm{Gr}
  80. ν = μ ρ \nu=\frac{\mu}{\rho}
  81. Δ T = ( T s - T o ) \Delta T=(T_{s}-T_{o})
  82. Gr = β g Δ T L 3 ν 2 \mathrm{Gr}=\frac{\beta g\Delta TL^{3}}{\nu^{2}}

Gravitational_constant.html

  1. G G
  2. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  3. G = 6.673 84 ( 80 ) × 10 - 11 m 3 kg - 1 s - 2 = 6.673 84 ( 80 ) × 10 - 11 N m 2 kg - 2 , G=6.673\ 84(80)\times 10^{-11}{\rm\ m^{3}\ kg^{-1}\ s^{-2}}=6.673\ 84(80)% \times 10^{-11}{\rm\ N\ m^{2}\ kg^{-2}},
  4. G 6.674 × 10 - 11 N ( m / kg ) 2 . G\approx 6.674\times 10^{-11}{\rm\ N\ (m/kg)^{2}}.
  5. G 6.674 × 10 - 8 cm 3 g - 1 s - 2 . G\approx 6.674\times 10^{-8}{\rm\ cm^{3}\ g^{-1}\ s^{-2}}.
  6. G = 4 π 2 AU 3 yr - 2 M - 1 G=4\pi^{2}{\rm\ AU^{3}}{\rm\ yr^{-2}}\ M_{\odot}^{-1}\,
  7. G 4.302 × 10 - 3 pc M - 1 ( km / s ) 2 . G\approx 4.302\times 10^{-3}{\rm\ pc}\ M_{\odot}^{-1}{\rm\ (km/s)^{2}}.\,
  8. G 0.8650 cm 3 g - 1 hr - 2 . G\approx 0.8650{\rm\ cm^{3}\ g^{-1}\ hr^{-2}}.
  9. G M = 3 π V / P 2 GM=3\pi V/P^{2}
  10. P 2 = 3 π G V M 10.896 hr 2 g cm - 3 V M . P^{2}=\frac{3\pi}{G}\frac{V}{M}\approx 10.896\ {\rm hr^{2}\ g\ cm^{-3}}\frac{V% }{M}.
  11. μ \scriptstyle\mu
  12. M M_{\oplus}
  13. μ = G M = ( 398 600.4418 ± 0.0008 ) km 3 s - 2 . \mu=GM_{\oplus}=(398\ 600.4418\pm 0.0008){\rm\ km^{3}\ s^{-2}}.
  14. μ = ( 1.327 124 400 × 10 11 ) km 3 s - 2 . \mu=(1.327\ 124\ 400\times 10^{11}){\rm\ km^{3}\ s^{-2}}.
  15. k = 0.017 202 098 95 A 3 2 D - 1 S - 1 2 k=0.017\ 202\ 098\ 95\ A^{\frac{3}{2}}\ D^{-1}\ S^{-\frac{1}{2}}
  16. A A\!
  17. D D\!
  18. S S\!

Gravitational_lens.html

  1. θ = 4 G M r c 2 \theta=\frac{4GM}{rc^{2}}
  2. r s r_{\mathrm{s}}
  3. r s = 2 G m / c 2 r_{\mathrm{s}}={2Gm}/{c^{2}}
  4. θ = 2 r s r \theta=2\frac{r_{\mathrm{s}}}{r}

Gravitational_redshift.html

  1. z z\,
  2. z = λ o - λ e λ e z=\frac{\lambda_{o}-\lambda_{e}}{\lambda_{e}}
  3. λ o \lambda_{o}\,
  4. λ e \lambda_{e}\,
  5. lim r + z ( r ) = 1 1 - r s R * - 1 \lim_{r\to+\infty}z(r)=\frac{1}{\sqrt{1-\frac{r_{s}}{R^{*}}}}-1
  6. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}
  7. G G
  8. M M
  9. c c
  10. R * R^{*}
  11. R * R^{*}
  12. r s r_{s}
  13. R * R^{*}
  14. r s r_{s}
  15. lim r + z approx ( r ) = 1 2 r s R * = G M c 2 R * \lim_{r\to+\infty}z_{\mathrm{approx}}(r)=\frac{1}{2}\frac{r_{s}}{R^{*}}=\frac{% GM}{c^{2}R^{*}}
  16. d s 2 = - r 2 d t 2 + d r 2 ds^{2}=-r^{2}dt^{2}+dr^{2}\,
  17. R ( r + d r ) - R ( r ) R = d r r = g d r {R(r+dr)-R(r)\over R}={dr\over r}=gdr\,
  18. d R d x = g = - d V d x {dR\over dx}=g=-{dV\over dx}\,
  19. R ( x ) = 1 - V ( x ) c 2 R(x)=1-{V(x)\over c^{2}}\,
  20. d t 2 dt^{2}
  21. R 2 = 1 - 2 V R^{2}=1-2V\,
  22. d s 2 = - ( 1 - 2 V ( r ) c 2 ) c 2 d t 2 + d x 2 + d y 2 + d z 2 ds^{2}=-\left(1-{2V(r)\over c^{2}}\right)c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}
  23. M M\;
  24. lim r + z ( r ) = 1 1 - ( 2 G M R * c 2 ) - 1 \lim_{r\to+\infty}z(r)=\frac{1}{\sqrt{1-\left(\frac{2GM}{R^{*}c^{2}}\right)}}-1
  25. G G\,
  26. M M\,
  27. R * R^{*}\,
  28. r , r,
  29. c c\,

Gravitational_singularity.html

  1. R μ ν ρ σ R μ ν ρ σ R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}

Gravity.html

  1. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  2. 1 / 20 {1}/{20}
  3. 2 / 20 {2}/{20}
  4. 3 / 20 {3}/{20}
  5. h = v 2 2 g h=\tfrac{v^{2}}{2g}

Gray_code.html

  1. n n / 2 n\oplus\lfloor n/2\rfloor
  2. g i g_{i}
  3. i i
  4. g 0 g_{0}
  5. b i b_{i}
  6. i i
  7. b 0 b_{0}
  8. b 0 = g 0 b_{0}=g_{0}
  9. b i = g i b i - 1 b_{i}=g_{i}\oplus b_{i-1}
  10. c o d e 0 = 0 code_{0}=0
  11. i > 0 i>0
  12. i i
  13. code i - 1 \mathrm{code}_{i-1}
  14. c o d e i code_{i}
  15. ( δ k ) (\delta_{k})
  16. λ k = | { j R n : δ j = k } | , for k R \lambda_{k}=|\{j\in\mathbb{Z}_{R^{n}}:\delta_{j}=k\}|\,,\text{ for }k\in% \mathbb{Z}_{R}
  17. λ k = R n / n \lambda_{k}=R^{n}/n
  18. R = 2 R=2
  19. R n R^{n}
  20. R n / n \lfloor R^{n}/n\rfloor
  21. R n / n \lceil R^{n}/n\rceil
  22. G G^{\prime}
  23. G = g 0 , , g 2 n - 1 G=g_{0},\ldots,g_{2^{n}-1}
  24. { g 0 } , { g 1 , , g k 2 } , { g k 2 + 1 , , g k 3 } , , { g k L - 2 + 1 , , g - 2 } , { g - 1 } \{g_{0}\},\{g_{1},\ldots,g_{k_{2}}\},\{g_{k_{2}+1},\ldots,g_{k_{3}}\},\ldots,% \{g_{k_{L-2}+1},\ldots,g_{-2}\},\{g_{-1}\}
  25. k 1 = 0 , k L - 1 = - 2 k_{1}=0,k_{L-1}=-2
  26. k L = - 1 k_{L}=-1
  27. 2 n 2^{n}
  28. ( n + 2 ) (n+2)
  29. 00 g 0 , 00g_{0},
  30. 00 g 1 , , 00 g k 2 , 01 g k 2 , , 01 g 1 , 11 g 1 , , 11 g k 2 , 00g_{1},\ldots,00g_{k_{2}},01g_{k_{2}},\ldots,01g_{1},11g_{1},\ldots,11g_{k_{2% }},
  31. 11 g k 2 + 1 , , 11 g k 3 , 01 g k 3 , , 01 g k 2 + 1 , 00 g k 2 + 1 , , 00 g k 3 , , 11g_{k_{2}+1},\ldots,11g_{k_{3}},01g_{k_{3}},\ldots,01g_{k_{2}+1},00g_{k_{2}+1% },\ldots,00g_{k_{3}},\ldots,
  32. 00 g - 2 , 00 g - 1 , 10 g - 1 , 10 g - 2 , , 10 g 0 , 11 g 0 , 11 g - 1 , 01 g - 1 , 01 g 0 00g_{-2},00g_{-1},10g_{-1},10g_{-2},\ldots,10g_{0},11g_{0},11g_{-1},01g_{-1},0% 1g_{0}
  33. m i = | { j : δ k j = i , 1 j L } | m_{i}=|\{j:\delta_{k_{j}}=i,1\leq j\leq L\}|
  34. λ k \lambda^{\prime}_{k}
  35. λ k = { 4 λ k - 2 m k , if 0 k < n L , otherwise \lambda^{\prime}_{k}=\begin{cases}4\lambda_{k}-2m_{k},&\,\text{if }0\leq k<n\\ L,&\,\text{ otherwise }\end{cases}
  36. R 0 mod 4 R\equiv 0\mod 4
  37. R n 0 mod n R^{n}\equiv 0\mod n
  38. Q n = ( V n , E n ) Q_{n}=(V_{n},E_{n})
  39. V n ( i ) = { v V n : v has weight i } V_{n}(i)=\{v\in V_{n}:v\,\text{ has weight }i\}
  40. 0 i n 0\leq i\leq n
  41. | V n ( i ) | = ( n i ) |V_{n}(i)|={\left({{n}\atop{i}}\right)}
  42. Q n ( i ) Q_{n}(i)
  43. Q n Q_{n}
  44. V n ( i ) V n ( i + 1 ) V_{n}(i)\cup V_{n}(i+1)
  45. E n ( i ) E_{n}(i)
  46. Q n ( i ) Q_{n}(i)
  47. Q n Q_{n}
  48. δ 1 E n ( i ) \delta_{1}\in E_{n}(i)
  49. δ 2 E n ( j ) \delta_{2}\in E_{n}(j)
  50. i j i\leq j
  51. P n , j P_{n,j}
  52. 2 ( n j ) 2{\left({{n}\atop{j}}\right)}
  53. E n ( j ) E_{n}(j)
  54. P 1 , 0 = ( 0 , 1 ) P_{1,0}=(0,1)
  55. P n , j = P_{n,j}=\emptyset
  56. j < 0 j<0
  57. j n j\geq n
  58. P n + 1 , j = 1 P n , j - 1 π n , 0 P n , j P_{n+1,j}=1P^{\pi_{n}}_{n,j-1},0P_{n,j}
  59. π n \pi_{n}
  60. P π P^{\pi}
  61. π \pi
  62. G n ( 1 ) G_{n}^{(1)}
  63. G n ( 2 ) G_{n}^{(2)}
  64. G n ( 1 ) = P n , 0 P n , 1 R P n , 2 P n , 3 R and G n ( 2 ) = P n , 0 R P n , 1 P n , 2 R P n , 3 G_{n}^{(1)}=P_{n,0}P_{n,1}^{R}P_{n,2}P_{n,3}^{R}\cdots\,\text{ and }G_{n}^{(2)% }=P_{n,0}^{R}P_{n,1}P_{n,2}^{R}P_{n,3}\cdots
  65. π n \pi_{n}
  66. π n = E - 1 ( π n - 1 2 ) \pi_{n}=E^{-1}(\pi_{n-1}^{2})
  67. P n , j P_{n,j}
  68. P n , j P_{n,j}
  69. Q 2 n + 1 ( n ) Q_{2n+1}(n)
  70. n 15 n\leq 15
  71. 2 2 \mathbb{Z}_{2}^{2}
  72. 4 \mathbb{Z}_{4}
  73. 2 2 m \mathbb{Z}_{2}^{2m}
  74. 4 m \mathbb{Z}_{4}^{m}
  75. 2 2 \mathbb{Z}_{2}^{2}
  76. 4 \mathbb{Z}_{4}

Great_circle.html

  1. θ = θ ( t ) , ϕ = ϕ ( t ) , a t b \theta=\theta(t),\quad\phi=\phi(t),\quad a\leq t\leq b
  2. d s = r θ 2 + ϕ 2 sin 2 θ d t . ds=r\sqrt{\theta^{\prime 2}+\phi^{\prime 2}\sin^{2}\theta}\,dt.
  3. S [ γ ] = r a b θ 2 + ϕ 2 sin 2 θ d t . S[\gamma]=r\int_{a}^{b}\sqrt{\theta^{\prime 2}+\phi^{\prime 2}\sin^{2}\theta}% \,dt.
  4. S [ γ ] r a b | θ ( t ) | d t r | θ ( b ) - θ ( a ) | . S[\gamma]\geq r\int_{a}^{b}|\theta^{\prime}(t)|\,dt\geq r|\theta(b)-\theta(a)|.
  5. x sin ϕ 0 - y cos ϕ 0 = 0 x\sin\phi_{0}-y\cos\phi_{0}=0

Greatest_common_divisor.html

  1. 54 × 1 = 27 × 2 = 18 × 3 = 9 × 6. 54\times 1=27\times 2=18\times 3=9\times 6.\,
  2. 1 , 2 , 3 , 6 , 9 , 18 , 27 , 54. 1,2,3,6,9,18,27,54.\,
  3. 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24. 1,2,3,4,6,8,12,24.\,
  4. 1 , 2 , 3 , 6. 1,2,3,6.\,
  5. gcd ( 54 , 24 ) = 6. \gcd(54,24)=6.\,
  6. 42 56 = 3 14 4 14 = 3 4 . \frac{42}{56}=\frac{3\cdot 14}{4\cdot 14}=\frac{3}{4}.
  7. gcd ( a , 0 ) = a \gcd(a,0)=a
  8. gcd ( a , b ) = gcd ( b , a mod b ) \gcd(a,b)=\gcd(b,a\,\mathrm{mod}\,b)
  9. a mod b = a - b a b a\,\mathrm{mod}\,b=a-b\left\lfloor{a\over b}\right\rfloor
  10. gcd ( a , a ) = a \gcd(a,a)=a
  11. gcd ( a , b ) = gcd ( a - b , b ) , \gcd(a,b)=\gcd(a-b,b)\quad,
  12. gcd ( a , b ) = gcd ( a , b - a ) , \gcd(a,b)=\gcd(a,b-a)\quad,
  13. exp [ O ( n log n ) ] \exp\left[O\left(\sqrt{n\log n}\right)\right]
  14. O ( log a + log b ) O(\log a+\log b)
  15. O ( ( log a + log b ) 2 ) O((\log a+\log b)^{2})
  16. gcd ( a , b ) = a b lcm ( a , b ) \gcd(a,b)=\frac{a\cdot b}{\operatorname{lcm}(a,b)}
  17. gcd ( a , b ) = a f ( b a ) , \gcd(a,b)=af\left(\frac{b}{a}\right),
  18. gcd ( a , b ) = log 2 k = 0 a - 1 ( 1 + e - 2 i π k b / a ) \gcd(a,b)=\log_{2}\prod_{k=0}^{a-1}(1+e^{-2i\pi kb/a})
  19. gcd ( a , b ) = k = 1 a exp ( 2 π i k b / a ) d | a c d ( k ) d \gcd(a,b)=\sum\limits_{k=1}^{a}\exp(2\pi ikb/a)\cdot\sum\limits_{d\left|a% \right.}\frac{c_{d}(k)}{d}
  20. gcd ( 2 a - 1 , 2 b - 1 ) = 2 gcd ( a , b ) - 1 \gcd(2^{a}-1,2^{b}-1)=2^{\gcd(a,b)}-1
  21. gcd ( n a - 1 , n b - 1 ) = n gcd ( a , b ) - 1 \gcd(n^{a}-1,n^{b}-1)=n^{\gcd(a,b)}-1\,
  22. gcd ( a , b ) \gcd(a,b)
  23. gcd ( a , b ) = k | a and k | b φ ( k ) . \gcd(a,b)=\sum_{k|a\;\hbox{and}\;k|b}\varphi(k).
  24. E ( 2 ) = d = 1 d 6 π 2 d 2 = 6 π 2 d = 1 1 d . \mathrm{E}(\mathrm{2})=\sum_{d=1}^{\infty}d\frac{6}{\pi^{2}d^{2}}=\frac{6}{\pi% ^{2}}\sum_{d=1}^{\infty}\frac{1}{d}.
  25. E ( k ) = d = 1 d 1 - k ζ ( k ) - 1 = ζ ( k - 1 ) ζ ( k ) . \mathrm{E}(k)=\sum_{d=1}^{\infty}d^{1-k}\zeta(k)^{-1}=\frac{\zeta(k-1)}{\zeta(% k)}.
  26. R = [ - 3 ] , a = 4 = 2 2 = ( 1 + - 3 ) ( 1 - - 3 ) , b = ( 1 + - 3 ) 2. R=\mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a=4=2\cdot 2=\left(1+\sqrt{-3}\,% \,\right)\left(1-\sqrt{-3}\,\,\right),\quad b=\left(1+\sqrt{-3}\,\,\right)% \cdot 2.

Greibach_normal_form.html

  1. A a A 1 A 2 A n A\to aA_{1}A_{2}\cdots A_{n}
  2. S ε S\to\varepsilon
  3. A A
  4. a a
  5. A 1 A 2 A n A_{1}A_{2}\ldots A_{n}
  6. S S

Gross_domestic_product.html

  1. G D P = R + I + P + S A + W GDP=R+I+P+SA+W

Grothendieck_topology.html

  1. \cup
  2. {}^{\ast}
  3. {}^{\ast}
  4. \in
  5. {}^{\ast}
  6. {}^{\ast}
  7. {}^{\ast}
  8. U i U_{i}
  9. \in
  10. F ( X ) α A F ( X α ) α , β A F ( X α × X X β ) F(X)\rightarrow\prod_{\alpha\in A}F(X_{\alpha}){{{}\atop\longrightarrow}\atop{% \longrightarrow\atop{}}}\prod_{\alpha,\beta\in A}F(X_{\alpha}\times_{X}X_{% \beta})
  11. \sube \sube
  12. \cup
  13. \cup
  14. C ~ \tilde{C}
  15. D ~ \tilde{D}
  16. u s : D ~ C ~ u_{s}:\tilde{D}\to\tilde{C}
  17. v ^ * \hat{v}^{*}
  18. v ^ * \hat{v}_{*}
  19. v ^ * \hat{v}_{*}
  20. v * : C ~ D ~ v_{*}:\tilde{C}\to\tilde{D}
  21. v ^ * \hat{v}^{*}
  22. C ~ D ~ \tilde{C}\to\tilde{D}
  23. C ~ D ~ \tilde{C}\to\tilde{D}

Group_(mathematics).html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.
  3. a a n factors , \underbrace{a\cdots a}_{n\,\text{ factors}},
  4. f ( x ) d x = f ( x + c ) d x \int f(x)\,dx=\int f(x+c)\,dx

Group_action.html

  1. φ : G × X X : ( g , x ) φ ( g , x ) \varphi:G\times X\to X:(g,x)\mapsto\varphi(g,x)
  2. φ : V × A A : ( v , t ) v + t \varphi:V\times A\to A:(v,t)\mapsto v+t
  3. z = cos 1 2 α + sin 1 2 α 𝐯 ^ \scriptstyle z=\cos\frac{1}{2}\alpha+\sin\frac{1}{2}\alpha\hat{\mathbf{v}}
  4. G . x = { g . x g G } . G.x=\left\{g.x\mid g\in G\right\}.
  5. G x = { g G g . x = x } . G_{x}=\{g\in G\mid g.x=x\}.
  6. ( H ) (H)
  7. ( H ) (H)
  8. G x G_{x}
  9. ( H ) (H)
  10. | G . x | = [ G : G x ] = | G | / | G x | . |G.x|=[G\,:\,G_{x}]=|G|/|G_{x}|.
  11. | X / G | = 1 | G | g G | X g | \left|X/G\right|=\frac{1}{\left|G\right|}\sum_{g\in G}\left|X^{g}\right|
  12. G = G X \scriptstyle G^{\prime}\;=\;G\,\ltimes\,X
  13. p : G G \scriptstyle p:\;G^{\prime}\,\rightarrow\,G
  14. g ( U ) U \scriptstyle g(U)\,\cap\,U\;\neq\;\emptyset

Group_delay_and_phase_delay.html

  1. x ( t ) \displaystyle x(t)
  2. y ( t ) \displaystyle y(t)
  3. y ( t ) = ( h * x ) ( t ) = def - x ( u ) h ( t - u ) d u y(t)=(h*x)(t)\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}x(u)h(t-u)\,du
  4. Y ( s ) = H ( s ) X ( s ) Y(s)=H(s)X(s)\,
  5. X ( s ) = { x ( t ) } = def - x ( t ) e - s t d t X(s)=\mathcal{L}\left\{x(t)\right\}\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty% }^{\infty}x(t)e^{-st}\,dt
  6. Y ( s ) = { y ( t ) } = def - y ( t ) e - s t d t Y(s)=\mathcal{L}\left\{y(t)\right\}\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty% }^{\infty}y(t)e^{-st}\,dt
  7. H ( s ) = { h ( t ) } = def - h ( t ) e - s t d t H(s)=\mathcal{L}\left\{h(t)\right\}\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty% }^{\infty}h(t)e^{-st}\,dt
  8. h ( t ) \displaystyle h(t)
  9. X ( s ) \displaystyle X(s)
  10. Y ( s ) \displaystyle Y(s)
  11. H ( s ) \displaystyle H(s)
  12. x ( t ) \displaystyle x(t)
  13. y ( t ) \displaystyle y(t)
  14. h ( t ) \displaystyle h(t)
  15. H ( s ) \displaystyle H(s)
  16. h ( t ) \displaystyle h(t)
  17. a ( t ) \displaystyle a(t)
  18. ω \displaystyle\omega
  19. x ( t ) = a ( t ) cos ( ω t + θ ) x(t)=a(t)\cos(\omega t+\theta)
  20. | d log ( a ( t ) ) d t | ω . \left|\frac{d\log\left(a(t)\right)}{dt}\right|\ll\omega\;.
  21. y ( t ) = | H ( i ω ) | a ( t - τ g ) cos ( ω ( t - τ ϕ ) + θ ) . y(t)=|H(i\omega)|\ a(t-\tau_{g})\cos\left(\omega(t-\tau_{\phi})+\theta\right)\;.
  22. τ g \displaystyle\tau_{g}
  23. τ ϕ \displaystyle\tau_{\phi}
  24. ω \displaystyle\omega
  25. τ g \displaystyle\tau_{g}
  26. τ ϕ \displaystyle\tau_{\phi}
  27. ω \displaystyle\omega
  28. H ( s ) \displaystyle H(s)
  29. x ( t ) = e i ω t x(t)=e^{i\omega t}
  30. y ( t ) \displaystyle y(t)
  31. ϕ \displaystyle\phi
  32. ϕ ( ω ) = def arg { H ( i ω ) } . \phi(\omega)\ \stackrel{\mathrm{def}}{=}\ \arg\left\{H(i\omega)\right\}\;.
  33. τ g \displaystyle\tau_{g}
  34. τ ϕ \displaystyle\tau_{\phi}
  35. ϕ \displaystyle\phi
  36. τ g ( ω ) = - d ϕ ( ω ) d ω \tau_{g}(\omega)=-\frac{d\phi(\omega)}{d\omega}
  37. τ ϕ ( ω ) = - ϕ ( ω ) ω \tau_{\phi}(\omega)=-\frac{\phi(\omega)}{\omega}
  38. τ g = - d ϕ d ω \tau_{g}=-\frac{d\phi}{d\omega}
  39. ϕ \phi
  40. ω \omega
  41. 2 π f 2\pi f
  42. f f
  43. τ g ( ω ) = - d ϕ d ω \tau_{g}(\omega)=-\frac{d\phi}{d\omega}
  44. ϕ ( ω ) = ϕ ( 0 ) - τ g ω \phi(\omega)=\phi(0)-\tau_{g}\omega
  45. τ g \tau_{g}

Group_homomorphism.html

  1. h ( u * v ) = h ( u ) h ( v ) h(u*v)=h(u)\cdot h(v)
  2. h ( u - 1 ) = h ( u ) - 1 . h\left(u^{-1}\right)=h(u)^{-1}.\,
  3. ker ( h ) { u G : h ( u ) = e H } . \operatorname{ker}(h)\equiv\left\{u\in G\colon h(u)=e_{H}\right\}.
  4. im ( h ) h ( G ) { h ( u ) : u G } . \operatorname{im}(h)\equiv h(G)\equiv\left\{h(u)\colon u\in G\right\}.
  5. h ( g - 1 u g ) \displaystyle h\left(g^{-1}\circ u\circ g\right)
  6. h ( g 1 ) \displaystyle h(g_{1})

Group_isomorphism.html

  1. \odot
  2. \odot
  3. f : G H f:G\rightarrow H
  4. f ( u * v ) = f ( u ) f ( v ) f(u*v)=f(u)\odot f(v)
  5. \odot
  6. ( G , * ) ( H , ) (G,*)\cong(H,\odot)
  7. G H G\cong H
  8. f : G H f:G\rightarrow H
  9. \odot
  10. f ( u ) f ( v ) = f ( u * v ) f(u)\odot f(v)=f(u*v)
  11. \odot
  12. \mathbb{R}
  13. \mathbb{R}
  14. ( , + ) ( + , × ) (\mathbb{R},+)\cong(\mathbb{R}^{+},\times)
  15. f ( x ) = e x f(x)=e^{x}
  16. \mathbb{Z}
  17. \mathbb{R}
  18. / \mathbb{R}/\mathbb{Z}
  19. S 1 S^{1}
  20. / S 1 \mathbb{R}/\mathbb{Z}\cong S^{1}
  21. f ( x + ) = e 2 π x i f(x+\mathbb{Z})=e^{2\pi xi}
  22. \mathbb{R}
  23. 2 = / 2 \mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}
  24. 2 × 2 \mathbb{Z}_{2}\times\mathbb{Z}_{2}
  25. \mathbb{R}
  26. \mathbb{C}
  27. \mathbb{C}
  28. \odot
  29. \odot
  30. \odot
  31. \odot
  32. \odot
  33. ( n , + n ) (\mathbb{Z}_{n},+_{n})
  34. < x { e , x , , x n - 1 } <x>=\{e,x,...,x^{n-1}\}
  35. G ( n , + n ) G\cong(\mathbb{Z}_{n},+_{n})
  36. φ : G n = { 0 , 1 , , n - 1 } \varphi:G\rightarrow\mathbb{Z}_{n}=\{0,1,...,n-1\}
  37. φ ( x a ) = a \varphi(x^{a})=a
  38. φ \varphi
  39. φ ( x a x b ) = φ ( x a + b ) = a + b = φ ( x a ) + n φ ( x b ) \varphi(x^{a}\cdot x^{b})=\varphi(x^{a+b})=a+b=\varphi(x^{a})+_{n}\varphi(x^{b})
  40. G ( n , + n ) G\cong(\mathbb{Z}_{n},+_{n})
  41. f : G H f:G\rightarrow H
  42. f ( e G ) = e H f(e_{G})=e_{H}
  43. f ( u - 1 ) = [ f ( u ) ] - 1 f(u^{-1})=\left[f(u)\right]^{-1}
  44. f ( u n ) = [ f ( u ) ] n f(u^{n})=\left[f(u)\right]^{n}
  45. f - 1 : H G f^{-1}:H\rightarrow G
  46. f : G G f:G\rightarrow G
  47. f ( u ) * f ( v ) = f ( u * v ) f(u)*f(v)=f(u*v)

Group_object.html

  1. \oplus

Group_representation.html

  1. ρ : G GL ( V ) \rho\colon G\to\mathrm{GL}(V)
  2. ρ ( g 1 g 2 ) = ρ ( g 1 ) ρ ( g 2 ) , for all g 1 , g 2 G . \rho(g_{1}g_{2})=\rho(g_{1})\rho(g_{2}),\qquad\,\text{for all }g_{1},g_{2}\in G.
  3. ker ρ = { g G ρ ( g ) = id } . \ker\rho=\left\{g\in G\mid\rho(g)=\mathrm{id}\right\}.
  4. α ρ ( g ) α - 1 = π ( g ) . \alpha\circ\rho(g)\circ\alpha^{-1}=\pi(g).
  5. ρ ( 1 ) = [ 1 0 0 1 ] ρ ( u ) = [ 1 0 0 u ] ρ ( u 2 ) = [ 1 0 0 u 2 ] . \rho\left(1\right)=\begin{bmatrix}1&0\\ 0&1\\ \end{bmatrix}\qquad\rho\left(u\right)=\begin{bmatrix}1&0\\ 0&u\\ \end{bmatrix}\qquad\rho\left(u^{2}\right)=\begin{bmatrix}1&0\\ 0&u^{2}\\ \end{bmatrix}.
  6. ρ ( 1 ) = [ 1 0 0 1 ] ρ ( u ) = [ u 0 0 1 ] ρ ( u 2 ) = [ u 2 0 0 1 ] . \rho\left(1\right)=\begin{bmatrix}1&0\\ 0&1\\ \end{bmatrix}\qquad\rho\left(u\right)=\begin{bmatrix}u&0\\ 0&1\\ \end{bmatrix}\qquad\rho\left(u^{2}\right)=\begin{bmatrix}u^{2}&0\\ 0&1\\ \end{bmatrix}.
  7. ρ ( 1 ) = [ 1 0 0 1 ] ρ ( u ) = [ a - b b a ] ρ ( u 2 ) = [ a b - b a ] \rho\left(1\right)=\begin{bmatrix}1&0\\ 0&1\\ \end{bmatrix}\qquad\rho\left(u\right)=\begin{bmatrix}a&-b\\ b&a\\ \end{bmatrix}\qquad\rho\left(u^{2}\right)=\begin{bmatrix}a&b\\ -b&a\\ \end{bmatrix}
  8. a = Re ( u ) = - 1 2 , b = Im ( u ) = 3 2 . a=\,\text{Re}(u)=-\tfrac{1}{2},\qquad b=\,\text{Im}(u)=\tfrac{\sqrt{3}}{2}.
  9. ρ ( 1 ) [ x ] = x \rho(1)[x]=x
  10. ρ ( g 1 g 2 ) [ x ] = ρ ( g 1 ) [ ρ ( g 2 ) [ x ] ] . \rho(g_{1}g_{2})[x]=\rho(g_{1})[\rho(g_{2})[x]].

Group_theory.html

  1. G = S | R . G=\langle S|R\rangle.
  2. m : G × G G , ( g , h ) g h , i : G G , g g - 1 , m:G\times G\to G,(g,h)\mapsto gh,\quad i:G\to G,g\mapsto g^{-1},
  3. x , y | x y x y x = e 〈x,y|xyxyx=e〉
  4. x 2 - 3 = 0 x^{2}-3=0
  5. + 3 +\sqrt{3}
  6. - 3 -\sqrt{3}
  7. n 1 1 n s = p prime 1 1 - p - s \begin{aligned}\displaystyle\sum_{n\geq 1}\frac{1}{n^{s}}&\displaystyle=\prod_% {p\,\text{ prime}}\frac{1}{1-p^{-s}}\\ \end{aligned}\!
  8. z = x y z=xy
  9. G = z , y | z < s u p > 3 = y = z . G=〈z,y|z<sup>3=y〉=〈z〉.

Group_velocity.html

  1. v g ω k v_{g}\ \equiv\ \frac{\partial\omega}{\partial k}\,
  2. α ( x , 0 ) = - d k A ( k ) e i k x , \alpha(x,0)=\int_{-\infty}^{\infty}dk\,A(k)e^{ikx},
  3. α ( x , t ) = - d k A ( k ) e i ( k x - ω t ) , \alpha(x,t)=\int_{-\infty}^{\infty}dk\,A(k)e^{i(kx-\omega t)},
  4. x = - d x | α ( x , t ) | 2 , \langle x\rangle=\int_{-\infty}^{\infty}dx\,|\alpha(x,t)|^{2},
  5. v m e a n v_{mean}
  6. v m e a n d x d t = - d k [ | A ( k ) | 2 ( d ω d k ) ] . v_{mean}\equiv\frac{d\langle x\rangle}{dt}=\int_{-\infty}^{\infty}dk\,\bigg[|A% (k)|^{2}\;\bigg(\frac{d\omega}{dk}\bigg)\bigg].
  7. d ω d k \frac{d\omega}{dk}
  8. A ( k ) A(k)
  9. d ω d k \frac{d\omega}{dk}
  10. ω ( k ) ω 0 + ( k - k 0 ) ω 0 \omega(k)\approx\omega_{0}+(k-k_{0})\omega^{\prime}_{0}
  11. ω 0 = ω ( k 0 ) \omega_{0}=\omega(k_{0})
  12. ω 0 = ω ( k ) k | k = k 0 \omega^{\prime}_{0}=\frac{\partial\omega(k)}{\partial k}|_{k=k_{0}}
  13. α ( x , t ) = e i ( k 0 x - ω 0 t ) - d k A ( k ) e i ( k - k 0 ) ( x - ω 0 t ) . \alpha(x,t)=e^{i(k_{0}x-\omega_{0}t)}\int_{-\infty}^{\infty}dk\,A(k)e^{i(k-k_{% 0})(x-\omega^{\prime}_{0}t)}.
  14. e i ( k 0 x - ω 0 t ) e^{i(k_{0}x-\omega_{0}t)}
  15. k 0 k_{0}
  16. ω 0 / k 0 \omega_{0}/k_{0}
  17. - d k A ( k ) e i ( k - k 0 ) ( x - ω 0 t ) \int_{-\infty}^{\infty}dk\,A(k)e^{i(k-k_{0})(x-\omega^{\prime}_{0}t)}
  18. ( x - ω 0 t ) (x-\omega^{\prime}_{0}t)
  19. ω 0 = ( d ω / d k ) k = k 0 \omega^{\prime}_{0}=(d\omega/dk)_{k=k_{0}}
  20. ω ( k ) ω 0 + ( k - k 0 ) ω 0 \omega(k)\approx\omega_{0}+(k-k_{0})\omega^{\prime}_{0}
  21. ω ( k ) \omega(k)
  22. ω ( k ) \omega(k)
  23. λ 0 = 2 π c ω , λ = 2 π k = 2 π v p ω , n = c v p = λ 0 λ , \lambda_{0}=\frac{2\pi c}{\omega},\;\;\lambda=\frac{2\pi}{k}=\frac{2\pi v_{p}}% {\omega},\;\;n=\frac{c}{v_{p}}=\frac{\lambda_{0}}{\lambda},
  24. v g = c n + ω n ω = c n - λ 0 n λ 0 = v p ( 1 + λ n n λ ) = v p - λ v p λ = v p + k v p k . v_{g}=\frac{c}{n+\omega\frac{\partial n}{\partial\omega}}=\frac{c}{n-\lambda_{% 0}\frac{\partial n}{\partial\lambda_{0}}}=v_{p}\left(1+\frac{\lambda}{n}\frac{% \partial n}{\partial\lambda}\right)=v_{p}-\lambda\frac{\partial v_{p}}{% \partial\lambda}=v_{p}+k\frac{\partial v_{p}}{\partial k}.
  25. v p = ω / k , v g = ω k , v_{p}=\omega/k,\quad v_{g}=\frac{\partial\omega}{\partial k},\,
  26. 𝐯 p = 𝐤 ^ ω | 𝐤 | , 𝐯 g = 𝐤 ω \mathbf{v}_{p}=\hat{\mathbf{k}}\frac{\omega}{|\mathbf{k}|},\quad\mathbf{v}_{g}% =\vec{\nabla}_{\mathbf{k}}\,\omega\,
  27. 𝐤 ω \vec{\nabla}_{\mathbf{k}}\,\omega
  28. ω \omega
  29. 𝐤 \mathbf{k}
  30. 𝐤 ^ \hat{\mathbf{k}}
  31. v g = ( ( Re k ) ω ) - 1 . v_{g}=\left(\frac{\partial(\operatorname{Re}k)}{\partial\omega}\right)^{-1}.
  32. n ¯ = n + i κ . \underline{n}=n+i\kappa.
  33. c v g = n + ω n ω . \frac{c}{v_{g}}=n+\omega\frac{\partial n}{\partial\omega}.
  34. k k
  35. ω \omega
  36. v g v_{g}
  37. v g v_{g}
  38. Re k \operatorname{Re}k
  39. v g v_{g}

Groupoid.html

  1. \ast
  2. \ast
  3. : - 1 G G , {}^{-1}:G\to G,
  4. * : G × G G . *:G\times G\rightharpoonup G.
  5. \ast
  6. id x \mathrm{id}_{x}
  7. comp x , y , z : G ( y , z ) × G ( x , y ) G ( x , z ) : ( g , f ) g f \mathrm{comp}_{x,y,z}:G(y,z)\times G(x,y)\rightarrow G(x,z):(g,f)\mapsto gf
  8. inv : G ( x , y ) G ( y , x ) : f f - 1 \mathrm{inv}:G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}
  9. f id x = f f\mathrm{id}_{x}=f
  10. id y f = f \mathrm{id}_{y}f=f
  11. ( h g ) f = h ( g f ) (hg)f=h(gf)
  12. f f - 1 = id y ff^{-1}=\mathrm{id}_{y}
  13. f - 1 f = id x f^{-1}f=\mathrm{id}_{x}
  14. comp \mathrm{comp}
  15. inv \mathrm{inv}
  16. inv \mathrm{inv}
  17. comp \mathrm{comp}
  18. inv \mathrm{inv}
  19. id \mathrm{id}
  20. G ( x * x - 1 , y * y - 1 ) G(x*x^{-1},y*y^{-1})
  21. y * y - 1 * f * x * x - 1 y*y^{-1}*f*x*x^{-1}
  22. f G ( x * x - 1 , y * y - 1 ) and g G ( y * y - 1 , z * z - 1 ) f\in G(x*x^{-1},y*y^{-1})\quad\mathrm{and}\quad g\in G(y*y^{-1},z*z^{-1})
  23. g * f G ( x * x - 1 , z * z - 1 ) g*f\in G(x*x^{-1},z*z^{-1})
  24. z * z - 1 * g * y * y - 1 and y * y - 1 * f * x * x - 1 z*z^{-1}*g*y*y^{-1}\quad\mathrm{and}\quad y*y^{-1}*f*x*x^{-1}
  25. z * z - 1 * g * y * y - 1 y * y - 1 * f * x * x - 1 = z * z - 1 * g * f * x * x - 1 z*z^{-1}*g*y*y^{-1}y*y^{-1}*f*x*x^{-1}=z*z^{-1}*g*f*x*x^{-1}
  26. H , K H,K
  27. G P D ( H , K ) GPD(H,K)
  28. H K H\to K
  29. H , K H,K
  30. G , H , K G,H,K
  31. G r p d ( G × H , K ) G r p d ( G , G P D ( H , K ) ) . Grpd(G\times H,K)\cong Grpd(G,GPD(H,K)).
  32. G , H , K G,H,K
  33. p : E B p:E\to B
  34. x x
  35. E E
  36. b b
  37. B B
  38. p ( x ) p(x)
  39. e e
  40. E E
  41. x x
  42. p ( e ) = b p(e)=b
  43. e e
  44. B B
  45. B B
  46. π 1 \pi_{1}
  47. π 1 ( X , x ) \pi_{1}(X,x)
  48. π 1 \pi_{1}
  49. π 1 \pi_{1}
  50. π 1 \pi_{1}
  51. \sim
  52. G × X G\times X
  53. G X G\ltimes X
  54. X G X\rtimes G
  55. ( h , y ) ( g , x ) = ( h g , x ) (h,y)(g,x)=(hg,x)
  56. [ Gr , Set ] [\mathrm{Gr},\mathrm{Set}]
  57. Gr \mathrm{Gr}
  58. ( Gr ) (\mathrm{Gr})
  59. Gr \mathrm{Gr}
  60. Gr \mathrm{Gr}
  61. Set \mathrm{Set}
  62. Hom ( Gr , - ) \mathrm{Hom}(\mathrm{Gr},-)
  63. ob ( Gr ) \mathrm{ob}(\mathrm{Gr})
  64. Hom ( Gr , Gr ) \mathrm{Hom}(\mathrm{Gr},\mathrm{Gr})
  65. Gr \mathrm{Gr}
  66. \sim

Grover's_algorithm.html

  1. N N
  2. O ( N ) O(N)
  3. N N
  4. N N
  5. U ω | ω = - | ω U_{\omega}|\omega\rangle=-|\omega\rangle
  6. U ω | x = | x for all x ω U_{\omega}|x\rangle=|x\rangle\qquad\mbox{for all}~{}\ x\neq\omega
  7. | ω |\omega\rangle
  8. | s |s\rangle
  9. | s = 1 N x = 1 N | x |s\rangle=\frac{1}{\sqrt{N}}\sum_{x=1}^{N}|x\rangle
  10. U s = 2 | s s | - I U_{s}=2\left|s\right\rangle\left\langle s\right|-I
  11. | s = 1 N x = 1 N | x |s\rangle=\frac{1}{\sqrt{N}}\sum_{x=1}^{N}|x\rangle
  12. U ω U_{\omega}
  13. U s U_{s}
  14. U s = 2 | s s | - I U_{s}=2\left|s\right\rangle\left\langle s\right|-I
  15. U ω = I - 2 | ω ω | U_{\omega}=I-2\left|\omega\right\rangle\left\langle\omega\right|
  16. ( I - 2 | ω ω | ) | ω = | ω - 2 | ω ω | ω = - | ω = U ω | ω (I-2|\omega\rangle\langle\omega|)|\omega\rangle=|\omega\rangle-2|\omega\rangle% \langle\omega|\omega\rangle=-|\omega\rangle=U_{\omega}|\omega\rangle
  17. ( I - 2 | ω ω | ) | x = | x - 2 | ω ω | x = | x = U ω | x (I-2|\omega\rangle\langle\omega|)|x\rangle=|x\rangle-2|\omega\rangle\langle% \omega|x\rangle=|x\rangle=U_{\omega}|x\rangle
  18. x ω x\neq\omega
  19. ω | s = s | ω = 1 N \langle\omega|s\rangle=\langle s|\omega\rangle=\frac{1}{\sqrt{N}}
  20. s | s = N 1 N 1 N = 1 \langle s|s\rangle=N\frac{1}{\sqrt{N}}\cdot\frac{1}{\sqrt{N}}=1
  21. U ω | s = ( I - 2 | ω ω | ) | s = | s - 2 | ω ω | s = | s - 2 N | ω U_{\omega}|s\rangle=(I-2|\omega\rangle\langle\omega|)|s\rangle=|s\rangle-2|% \omega\rangle\langle\omega|s\rangle=|s\rangle-\frac{2}{\sqrt{N}}|\omega\rangle
  22. U s ( | s - 2 N | ω ) = ( 2 | s s | - I ) ( | s - 2 N | ω ) = 2 | s s | s - | s - 4 N | s s | ω + 2 N | ω = U_{s}(|s\rangle-\frac{2}{\sqrt{N}}|\omega\rangle)=(2|s\rangle\langle s|-I)(|s% \rangle-\frac{2}{\sqrt{N}}|\omega\rangle)=2|s\rangle\langle s|s\rangle-|s% \rangle-\frac{4}{\sqrt{N}}|s\rangle\langle s|\omega\rangle+\frac{2}{\sqrt{N}}|% \omega\rangle=
  23. = 2 | s - | s - 4 N 1 N | s + 2 N | ω = | s - 4 N | s + 2 N | ω = N - 4 N | s + 2 N | ω =2|s\rangle-|s\rangle-\frac{4}{\sqrt{N}}\cdot\frac{1}{\sqrt{N}}|s\rangle+\frac% {2}{\sqrt{N}}|\omega\rangle=|s\rangle-\frac{4}{N}|s\rangle+\frac{2}{\sqrt{N}}|% \omega\rangle=\frac{N-4}{N}|s\rangle+\frac{2}{\sqrt{N}}|\omega\rangle
  24. U ω U_{\omega}
  25. U s U_{s}
  26. | ω | s | 2 = 1 / N \left|\langle\omega|s\rangle\right|^{2}=1/N
  27. | ω | U s U ω s | 2 9 / N \left|\langle\omega|U_{s}U_{\omega}s\rangle\right|^{2}\approx 9/N
  28. U ω U_{\omega}
  29. f f
  30. f ( x ) = 1 f(x)=1
  31. | x |x\rangle
  32. f ( x ) = 0 f(x)=0
  33. | x |x\rangle
  34. | q |q\rangle
  35. | x | q U ω | x | q f ( x ) |x\rangle|q\rangle\overset{U_{\omega}}{\longrightarrow}|x\rangle|q\oplus f(x)\rangle
  36. \oplus
  37. f ( x ) = 1 f(x)=1
  38. | x |x\rangle
  39. ( | 0 - | 1 ) / 2 (|0\rangle-|1\rangle)/\sqrt{2}
  40. ( | 1 - | 0 ) / 2 (|1\rangle-|0\rangle)/\sqrt{2}
  41. | x |x\rangle
  42. | x ( | 0 - | 1 ) / 2 U ω ( - 1 ) f ( x ) | x ( | 0 - | 1 ) / 2 |x\rangle\left(|0\rangle-|1\rangle\right)/\sqrt{2}\overset{U_{\omega}}{% \longrightarrow}(-1)^{f(x)}|x\rangle\left(|0\rangle-|1\rangle\right)/\sqrt{2}
  43. | x |x\rangle
  44. U ω U_{\omega}
  45. | x U ω ( - 1 ) f ( x ) | x |x\rangle\overset{U_{\omega}}{\longrightarrow}(-1)^{f(x)}|x\rangle
  46. | s |s\rangle
  47. | ω |\omega\rangle
  48. | ω |\omega\rangle
  49. | s = 1 N - 1 x ω | x |s^{\prime}\rangle=\frac{1}{\sqrt{N-1}}\sum_{x\neq\omega}|x\rangle
  50. | s |s\rangle
  51. | ω |\omega\rangle
  52. | s |s\rangle
  53. s | s = N - 1 N \langle s^{\prime}|s\rangle=\sqrt{\frac{N-1}{N}}
  54. θ / 2 \theta/2
  55. | s |s\rangle
  56. | s |s^{\prime}\rangle
  57. sin θ / 2 = 1 N \sin\theta/2=\frac{1}{\sqrt{N}}
  58. U ω U_{\omega}
  59. | ω |\omega\rangle
  60. | s |s^{\prime}\rangle
  61. | ω |\omega\rangle
  62. | s |s^{\prime}\rangle
  63. U s U_{s}
  64. | s |s\rangle
  65. | s |s^{\prime}\rangle
  66. | ω |\omega\rangle
  67. U s U_{s}
  68. U ω U_{\omega}
  69. U s U ω U_{s}U_{\omega}
  70. θ = 2 arcsin 1 N \theta=2\arcsin\frac{1}{\sqrt{N}}
  71. | ω |\omega\rangle
  72. | ω |\omega\rangle
  73. sin 2 ( ( r + 1 2 ) θ ) \sin^{2}\left(\left(r+\frac{1}{2}\right)\theta\right)
  74. r π N / 4 r\approx\pi\sqrt{N}/4
  75. U s U ω U_{s}U_{\omega}
  76. s s
  77. ω \omega
  78. U s U_{s}
  79. U ω U_{\omega}
  80. { | s , | ω } \{|s\rangle,|\omega\rangle\}
  81. U s : a | ω + b | s ( | ω | s ) ( - 1 0 2 / N 1 ) ( a b ) . U_{s}:a|\omega\rangle+b|s\rangle\mapsto(|\omega\rangle\,|s\rangle)\begin{% pmatrix}-1&0\\ 2/\sqrt{N}&1\end{pmatrix}\begin{pmatrix}a\\ b\end{pmatrix}.
  82. U ω : a | ω + b | s ( | ω | s ) ( - 1 - 2 / N 0 1 ) ( a b ) . U_{\omega}:a|\omega\rangle+b|s\rangle\mapsto(|\omega\rangle\,|s\rangle)\begin{% pmatrix}-1&-2/\sqrt{N}\\ 0&1\end{pmatrix}\begin{pmatrix}a\\ b\end{pmatrix}.
  83. { | ω , | s } \{|\omega\rangle,|s\rangle\}
  84. U s U ω U_{s}U_{\omega}
  85. U ω U_{\omega}
  86. U s U_{s}
  87. U s U ω = ( - 1 0 2 / N 1 ) ( - 1 - 2 / N 0 1 ) = ( 1 2 / N - 2 / N 1 - 4 / N ) . U_{s}U_{\omega}=\begin{pmatrix}-1&0\\ 2/\sqrt{N}&1\end{pmatrix}\begin{pmatrix}-1&-2/\sqrt{N}\\ 0&1\end{pmatrix}=\begin{pmatrix}1&2/\sqrt{N}\\ -2/\sqrt{N}&1-4/N\end{pmatrix}.
  88. t = arcsin ( 1 / N ) t=\arcsin(1/\sqrt{N})
  89. U s U ω = M ( exp ( 2 i t ) 0 0 exp ( - 2 i t ) ) M - 1 U_{s}U_{\omega}=M\begin{pmatrix}\exp(2it)&0\\ 0&\exp(-2it)\end{pmatrix}M^{-1}
  90. M = ( - i i exp ( i t ) exp ( - i t ) ) . M=\begin{pmatrix}-i&i\\ \exp(it)&\exp(-it)\end{pmatrix}.
  91. ( U s U ω ) r = M ( exp ( 2 r i t ) 0 0 exp ( - 2 r i t ) ) M - 1 (U_{s}U_{\omega})^{r}=M\begin{pmatrix}\exp(2rit)&0\\ 0&\exp(-2rit)\end{pmatrix}M^{-1}
  92. | ( ω | ω ω | s ) ( U s U ω ) r ( 0 1 ) | 2 = sin 2 ( ( 2 r + 1 ) t ) \left|\begin{pmatrix}\langle\omega|\omega\rangle&\langle\omega|s\rangle\end{% pmatrix}(U_{s}U_{\omega})^{r}\begin{pmatrix}0\\ 1\end{pmatrix}\right|^{2}=\sin^{2}\left((2r+1)t\right)
  93. 2 r t π / 2 2rt\approx\pi/2
  94. r = π / 4 t = π / 4 arcsin ( 1 / N ) π N / 4 r=\pi/4t=\pi/4\arcsin(1/\sqrt{N})\approx\pi\sqrt{N}/4
  95. ( | ω | s ) ( U s U ω ) r ( 0 1 ) ( | ω | s ) M ( i 0 0 - i ) M - 1 ( 0 1 ) = | w 1 cos ( t ) - | s sin ( t ) cos ( t ) . (|\omega\rangle\,|s\rangle)(U_{s}U_{\omega})^{r}\begin{pmatrix}0\\ 1\end{pmatrix}\approx(|\omega\rangle\,|s\rangle)M\begin{pmatrix}i&0\\ 0&-i\end{pmatrix}M^{-1}\begin{pmatrix}0\\ 1\end{pmatrix}=|w\rangle\frac{1}{\cos(t)}-|s\rangle\frac{\sin(t)}{\cos(t)}.
  96. π N 1 / 2 4 , π ( N / 2 ) 1 / 2 4 , π ( N / 4 ) 1 / 2 4 , \pi\frac{N^{1/2}}{4},\pi\frac{(N/2)^{1/2}}{4},\pi\frac{(N/4)^{1/2}}{4},\ldots
  97. π N 1 / 2 4 ( 1 + 1 2 + 1 2 + ) = π N 1 / 2 4 ( 2 + 2 ) \pi\frac{N^{1/2}}{4}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{2}+\cdots\right)=\pi% \frac{N^{1/2}}{4}\left(2+\sqrt{2}\right)
  98. N / k \sqrt{N/k}
  99. K K
  100. b = N / K b=N/K
  101. N / 2 N/2
  102. ( N - b ) / 2 (N-b)/2
  103. π / 4 N \pi/4\sqrt{N}
  104. K K
  105. n 1 n_{1}
  106. n 2 n_{2}
  107. G 1 G_{1}
  108. G 2 G_{2}
  109. j 1 j_{1}
  110. j 2 j_{2}
  111. j 1 j_{1}
  112. j 2 j_{2}

GTPase.html

  1. GTPase * GTP GTPase * GDP = k diss.GDP k cat.GTP \frac{\mbox{GTPase}~{}*\mbox{GTP}~{}}{\mbox{GTPase}~{}*\mbox{GDP}~{}}=\frac{k_% {\mbox{diss.GDP}}~{}}{k_{\mbox{cat.GTP}}~{}}

Guided_ray.html

  1. sin θ n o 2 - n c 2 \sin\theta\leq\sqrt{n_{o}^{2}-n_{c}^{2}}

Guitar.html

  1. 2 12 \sqrt[12]{2}
  2. 2 12 \sqrt[12]{2}

Gyrocompass.html

  1. ( X 1 , Y 1 , Z 1 ) \textstyle(X_{1},Y_{1},Z_{1})
  2. R R
  3. Ω \textstyle\Omega
  4. ( X 2 Y 2 Z 2 ) = ( cos Ω t sin Ω t 0 - sin Ω t cos Ω t 0 0 0 1 ) ( X 1 Y 1 Z 1 ) , \left(\begin{array}[]{c}X_{2}\\ Y_{2}\\ Z_{2}\end{array}\right)=\left(\begin{array}[]{ccc}\cos\Omega t&\sin\Omega t&0% \\ -\sin\Omega t&\cos\Omega t&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{c}X_{1}\\ Y_{1}\\ Z_{1}\end{array}\right),
  5. X ^ 1 \hat{X}_{1}
  6. ( X 1 = 1 , Y 1 = 0 , Z 1 = 0 ) T (X_{1}=1,Y_{1}=0,Z_{1}=0)^{T}
  7. ( X 2 = cos Ω t , Y 2 = - sin Ω t , Z 2 = 0 ) T (X_{2}=\cos\Omega t,Y_{2}=-\sin\Omega t,Z_{2}=0)^{T}
  8. Ω = ( 0 , 0 , Ω ) T \vec{\Omega}=(0,0,\Omega)^{T}
  9. X 2 X_{2}
  10. Z 2 \textstyle Z_{2}
  11. X 3 \textstyle X_{3}
  12. ( X 3 Y 3 Z 3 ) = ( cos Φ sin Φ 0 - sin Φ cos Φ 0 0 0 1 ) ( X 2 Y 2 Z 2 ) . \left(\begin{array}[]{c}X_{3}\\ Y_{3}\\ Z_{3}\end{array}\right)=\left(\begin{array}[]{ccc}\cos\Phi&\sin\Phi&0\\ -\sin\Phi&\cos\Phi&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{c}X_{2}\\ Y_{2}\\ Z_{2}\end{array}\right).
  13. Y 3 \textstyle Y_{3}
  14. δ \textstyle\delta
  15. Z 3 \textstyle Z_{3}
  16. Z 4 \textstyle Z_{4}
  17. ( X 4 Y 4 Z 4 ) = ( cos δ 0 - sin δ 0 1 0 sin δ 0 cos δ ) ( X 3 Y 3 Z 3 ) , \left(\begin{array}[]{c}X_{4}\\ Y_{4}\\ Z_{4}\end{array}\right)=\left(\begin{array}[]{ccc}\cos\delta&0&-\sin\delta\\ 0&1&0\\ \sin\delta&0&\cos\delta\end{array}\right)\left(\begin{array}[]{c}X_{3}\\ Y_{3}\\ Z_{3}\end{array}\right),
  18. Z ^ 3 \textstyle\hat{Z}_{3}
  19. ( X 3 = 0 , Y 3 = 0 , Z 3 = 1 ) T \textstyle(X_{3}=0,Y_{3}=0,Z_{3}=1)^{T}\,
  20. ( X 4 = - sin δ , Y 4 = 0 , Z 4 = cos δ ) T \textstyle(X_{4}=-\sin\delta,Y_{4}=0,Z_{4}=\cos\delta)^{T}\,
  21. ( X 5 Y 5 Z 5 ) = ( X 4 Y 4 Z 4 ) - ( 0 0 R ) , \left(\begin{array}[]{c}X_{5}\\ Y_{5}\\ Z_{5}\end{array}\right)=\left(\begin{array}[]{c}X_{4}\\ Y_{4}\\ Z_{4}\end{array}\right)-\left(\begin{array}[]{c}0\\ 0\\ R\end{array}\right)\,,
  22. ( X 5 = 0 , Y 5 = 0 , Z 5 = 0 ) T (X_{5}=0,Y_{5}=0,Z_{5}=0)^{T}
  23. ( X 4 = 0 , Y 4 = 0 , Z 4 = R ) T (X_{4}=0,Y_{4}=0,Z_{4}=R)^{T}
  24. R R
  25. X 5 X_{5}
  26. Z 5 Z_{5}
  27. ( X 6 Y 6 Z 6 ) = ( cos α sin α 0 - sin α cos α 0 0 0 1 ) ( X 5 Y 5 Z 5 ) . \left(\begin{array}[]{c}X_{6}\\ Y_{6}\\ Z_{6}\end{array}\right)=\left(\begin{array}[]{ccc}\cos\alpha&\sin\alpha&0\\ -\sin\alpha&\cos\alpha&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{c}X_{5}\\ Y_{5}\\ Z_{5}\end{array}\right).
  28. X 6 X_{6}
  29. ( X 7 Y 7 Z 7 ) = ( 1 0 0 0 cos ψ sin ψ 0 - sin ψ cos ψ ) ( X 6 Y 6 Z 6 ) . \left(\begin{array}[]{c}X_{7}\\ Y_{7}\\ Z_{7}\end{array}\right)=\left(\begin{array}[]{ccc}1&0&0\\ 0&\cos\psi&\sin\psi\\ 0&-\sin\psi&\cos\psi\end{array}\right)\left(\begin{array}[]{c}X_{6}\\ Y_{6}\\ Z_{6}\end{array}\right).
  30. \mathcal{L}
  31. K K
  32. = K = 1 2 ω T I ω + 1 2 M v CM 2 , \mathcal{L}=K=\frac{1}{2}\,\vec{\omega}^{T}I\vec{\omega}+\frac{1}{2}\,M\vec{v}% _{\rm CM}^{\,2}\,,
  33. M M
  34. v CM 2 = Ω 2 R 2 sin 2 δ = constant \vec{v}_{\rm CM}^{\,2}=\Omega^{2}R^{2}\sin^{2}\delta={\rm constant}
  35. I = ( I 1 0 0 0 I 2 0 0 0 I 2 ) , I=\left(\begin{array}[]{ccc}I_{1}&0&0\\ 0&I_{2}&0\\ 0&0&I_{2}\end{array}\right)\,,
  36. ω = ( 1 0 0 0 cos ψ sin ψ 0 - sin ψ cos ψ ) ( ψ ˙ 0 0 ) + ( 1 0 0 0 cos ψ sin ψ 0 - sin ψ cos ψ ) ( cos α sin α 0 - sin α cos α 0 0 0 1 ) ( 0 0 α ˙ ) + ( 1 0 0 0 cos ψ sin ψ 0 - sin ψ cos ψ ) ( cos α sin α 0 - sin α cos α 0 0 0 1 ) ( cos δ 0 - sin δ 0 1 0 sin δ 0 cos δ ) ( cos Φ sin Φ 0 - sin Φ cos Φ 0 0 0 1 ) × ( cos Ω t sin Ω t 0 - sin Ω t cos Ω t 0 0 0 1 ) ( 0 0 Ω ) = ( ψ ˙ 0 0 ) + ( 0 α ˙ sin ψ α ˙ cos ψ ) + ( - Ω sin δ cos α Ω ( sin δ sin α cos ψ + cos δ sin ψ ) Ω ( - sin δ sin α sin ψ + cos δ cos ψ ) ) . \begin{aligned}\displaystyle\vec{\omega}&\displaystyle=\left(\begin{array}[]{% ccc}1&0&0\\ 0&\cos\psi&\sin\psi\\ 0&-\sin\psi&\cos\psi\end{array}\right)\left(\begin{array}[]{c}\dot{\psi}\\ 0\\ 0\end{array}\right)+\left(\begin{array}[]{ccc}1&0&0\\ 0&\cos\psi&\sin\psi\\ 0&-\sin\psi&\cos\psi\end{array}\right)\left(\begin{array}[]{ccc}\cos\alpha&% \sin\alpha&0\\ -\sin\alpha&\cos\alpha&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{c}0\\ 0\\ \dot{\alpha}\end{array}\right)\\ &\displaystyle{}+\left(\begin{array}[]{ccc}1&0&0\\ 0&\cos\psi&\sin\psi\\ 0&-\sin\psi&\cos\psi\end{array}\right)\left(\begin{array}[]{ccc}\cos\alpha&% \sin\alpha&0\\ -\sin\alpha&\cos\alpha&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{ccc}\cos\delta&0&-\sin\delta\\ 0&1&0\\ \sin\delta&0&\cos\delta\end{array}\right)\left(\begin{array}[]{ccc}\cos\Phi&% \sin\Phi&0\\ -\sin\Phi&\cos\Phi&0\\ 0&0&1\end{array}\right)\\ &\displaystyle{}\times\left(\begin{array}[]{ccc}\cos\Omega t&\sin\Omega t&0\\ -\sin\Omega t&\cos\Omega t&0\\ 0&0&1\end{array}\right)\left(\begin{array}[]{c}0\\ 0\\ \Omega\end{array}\right)\\ &\displaystyle=\left(\begin{array}[]{c}\dot{\psi}\\ 0\\ 0\\ \end{array}\right)+\left(\begin{array}[]{c}0\\ \dot{\alpha}\sin\psi\\ \dot{\alpha}\cos\psi\end{array}\right)+\left(\begin{array}[]{c}-\Omega\sin% \delta\cos\alpha\\ \Omega(\sin\delta\sin\alpha\cos\psi+\cos\delta\sin\psi)\\ \Omega(-\sin\delta\sin\alpha\sin\psi+\cos\delta\cos\psi)\end{array}\right).% \end{aligned}
  37. = 1 2 [ I 1 ω 1 2 + I 2 ( ω 2 2 + ω 3 2 ) ] = 1 2 I 1 ( ψ ˙ - Ω sin δ cos α ) 2 + 1 2 I 2 { [ α ˙ sin ψ + Ω ( sin δ sin α cos ψ + cos δ sin ψ ) ] 2 + [ α ˙ cos ψ + Ω ( - sin δ sin α sin ψ + cos δ cos ψ ) ] 2 } = 1 2 I 1 ( ψ ˙ - Ω sin δ cos α ) 2 + 1 2 I 2 { α ˙ 2 + Ω 2 ( cos 2 δ + sin 2 α sin 2 δ ) + 2 α ˙ Ω cos δ } . \begin{aligned}\displaystyle\mathcal{L}&\displaystyle=\frac{1}{2}\,[I_{1}% \omega_{1}^{2}+I_{2}(\omega_{2}^{2}+\omega_{3}^{2})]\\ &\displaystyle=\frac{1}{2}\,I_{1}(\dot{\psi}-\Omega\sin\delta\cos\alpha)^{2}{}% +\frac{1}{2}\,I_{2}\{[\dot{\alpha}\sin\psi+\Omega(\sin\delta\sin\alpha\cos\psi% +\cos\delta\sin\psi)]^{2}{}+[\dot{\alpha}\cos\psi+\Omega(-\sin\delta\sin\alpha% \sin\psi+\cos\delta\cos\psi)]^{2}\}\\ &\displaystyle=\frac{1}{2}\,I_{1}(\dot{\psi}-\Omega\sin\delta\cos\alpha)^{2}+% \frac{1}{2}\,I_{2}\{\dot{\alpha}^{2}+\Omega^{2}(\cos^{2}\delta+\sin^{2}\alpha% \sin^{2}\delta){}+2\dot{\alpha}\Omega\cos\delta\}.\end{aligned}
  38. = 1 + 1 2 I 2 Ω 2 cos 2 δ + d d t ( I 2 α Ω cos δ ) , \mathcal{L}=\mathcal{L}_{1}+\frac{1}{2}\,I_{2}\Omega^{2}\cos^{2}\delta+\frac{d% }{dt}(I_{2}\alpha\Omega\cos\delta)\,,
  39. 1 = 1 2 I 1 ( ψ ˙ - Ω sin δ cos α ) 2 + 1 2 I 2 ( α ˙ 2 + Ω 2 sin 2 α sin 2 δ ) , \mathcal{L}_{1}=\frac{1}{2}\,I_{1}(\dot{\psi}-\Omega\sin\delta\cos\alpha)^{2}+% \frac{1}{2}\,I_{2}\,(\dot{\alpha}^{2}+\Omega^{2}\sin^{2}\alpha\sin^{2}\delta)\,,
  40. 1 / ψ = 0 \partial\mathcal{L}_{1}/\partial\psi=0
  41. L x 1 ψ ˙ = I 1 ( ψ ˙ - Ω sin δ cos α ) = constant . L_{x}\equiv\frac{\partial\mathcal{L}_{1}}{\partial\dot{\psi}}=I_{1}(\dot{\psi}% -\Omega\sin\delta\cos\alpha)=\mathrm{constant}.
  42. L \vec{L}
  43. L = I ω \vec{L}=I\vec{\omega}
  44. L x L_{x}
  45. α \alpha
  46. d d t ( 1 α ˙ ) = 1 α , \frac{d}{dt}\!\left(\frac{\partial\mathcal{L}_{1}}{\partial\dot{\alpha}}\right% )=\frac{\partial\mathcal{L}_{1}}{\partial\alpha}\,,
  47. I 2 α ¨ = I 1 Ω ( ψ ˙ - Ω sin δ cos α ) sin δ sin α + 1 2 I 2 Ω 2 sin 2 δ sin 2 α = L x Ω sin δ sin α + 1 2 I 2 Ω 2 sin 2 δ sin 2 α . \begin{aligned}\displaystyle I_{2}\ddot{\alpha}&\displaystyle=I_{1}\Omega(\dot% {\psi}-\Omega\sin\delta\cos\alpha)\sin\delta\sin\alpha+\frac{1}{2}\,I_{2}\,% \Omega^{2}\sin^{2}\delta\sin 2\alpha\\ &\displaystyle=L_{x}\Omega\sin\delta\sin\alpha+\frac{1}{2}\,I_{2}\,\Omega^{2}% \sin^{2}\delta\sin 2\alpha\,.\end{aligned}
  48. sin δ = 0 \sin\delta=0
  49. L x = I 1 ψ ˙ = constant I 2 α ¨ = 0 . \begin{aligned}\displaystyle L_{x}&\displaystyle=I_{1}\dot{\psi}=\mathrm{% constant}\\ \displaystyle I_{2}\ddot{\alpha}&\displaystyle=0\,.\end{aligned}
  50. sin δ 0 \sin\delta\neq 0
  51. α 0 \alpha\approx 0
  52. L x I 1 ( ψ ˙ - Ω sin δ ) , I 2 α ¨ ( L x Ω sin δ + I 2 Ω 2 sin 2 δ ) α . \begin{aligned}\displaystyle L_{x}&\displaystyle\approx I_{1}(\dot{\psi}-% \Omega\sin\delta)\,,\\ \displaystyle I_{2}\ddot{\alpha}&\displaystyle\approx(L_{x}\Omega\sin\delta+I_% {2}\,\Omega^{2}\sin^{2}\delta)\,\alpha\,.\end{aligned}
  53. L x < 0 , L_{x}<0\,,
  54. | ψ ˙ | Ω . |\dot{\psi}|\gg\Omega\,.
  55. L x < 0 L_{x}<0
  56. ψ ˙ < 0 \dot{\psi}<0
  57. L x - I 1 | ψ ˙ | constant , I 2 α ¨ - I 1 | ψ ˙ | Ω sin δ α . \begin{aligned}\displaystyle L_{x}&\displaystyle\approx-I_{1}|\dot{\psi}|% \approx\mathrm{constant}\,,\\ \displaystyle I_{2}\ddot{\alpha}&\displaystyle\approx-I_{1}|\dot{\psi}|\Omega% \sin\delta\,\alpha\,.\end{aligned}
  58. α A sin ( ω ~ t + B ) \alpha\approx A\sin(\tilde{\omega}t+B)
  59. ω ~ = I 1 sin δ I 2 | ψ ˙ | Ω , \tilde{\omega}=\sqrt{\frac{I_{1}\sin\delta}{I_{2}}}\,\sqrt{|\dot{\psi}|\Omega}\,,
  60. T = 2 π | ψ ˙ | Ω I 2 I 1 sin δ . T=\frac{2\pi}{\sqrt{|\dot{\psi}|\Omega}}\,\sqrt{\frac{I_{2}}{I_{1}\sin\delta}}\,.
  61. ω ~ \tilde{\omega}
  62. ψ ˙ < 0 \dot{\psi}<0
  63. X 7 X_{7}
  64. T T
  65. ψ ˙ \dot{\psi}
  66. δ \delta

Haar_measure.html

  1. g S = { g . s : s S } . gS=\{g.s\,:\,s\in S\}.
  2. S g = { s . g : s S } . Sg=\{s.g\,:\,s\in S\}.
  3. μ ( g S ) = μ ( S ) . \mu(gS)=\mu(S).\quad
  4. μ ( E ) = inf { μ ( U ) : E U , U open } . \mu(E)=\inf\{\mu(U):E\subseteq U,U\,\text{ open}\}.
  5. μ ( E ) = sup { μ ( K ) : K E , K compact } . \mu(E)=\sup\{\mu(K):K\subseteq E,K\,\text{ compact}\}.
  6. μ A ( T ) = lim U [ T : U ] [ A : U ] \mu_{A}(T)=\lim_{U}\frac{[T:U]}{[A:U]}
  7. μ A ( T ) = lim U [ T : U ] [ A : U ] \mu_{A}(T)=\lim_{U}\frac{[T:U]}{[A:U]}
  8. S - 1 S^{-1}
  9. μ - 1 ( S ) = μ ( S - 1 ) \mu_{-1}(S)=\mu(S^{-1})\quad
  10. μ - 1 ( S g ) = μ ( ( S g ) - 1 ) = μ ( g - 1 S - 1 ) = μ ( S - 1 ) = μ - 1 ( S ) . \mu_{-1}(Sg)=\mu((Sg)^{-1})=\mu(g^{-1}S^{-1})=\mu(S^{-1})=\mu_{-1}(S).\quad
  11. μ ( S - 1 ) = k ν ( S ) \mu(S^{-1})=k\nu(S)\,
  12. S ν ( g - 1 S ) S\mapsto\nu(g^{-1}S)\quad
  13. ν ( g - 1 S ) = Δ ( g ) ν ( S ) . \nu(g^{-1}S)=\Delta(g)\nu(S).\quad
  14. { x a x + b : a \R { 0 } , b \R } = { [ a b 0 1 ] } \big\{x\mapsto ax+b:a\in\R\setminus\{0\},b\in\R\big\}=\big\{\begin{bmatrix}a&b% \\ \\ 0&1\end{bmatrix}\big\}
  15. G f ( s x ) d μ ( x ) = G f ( x ) d μ ( x ) \int_{G}f(sx)\ d\mu(x)=\int_{G}f(x)\ d\mu(x)
  16. μ ( S ) = S 1 | t | d t \mu(S)=\int_{S}\frac{1}{|t|}\,dt
  17. S S
  18. a , b a,b
  19. μ ( S ) = log ( b / a ) \mu(S)=\log(b/a)
  20. g g
  21. g S gS
  22. g a , g b g\cdot a,g\cdot b
  23. μ ( g S ) = log ( ( g b ) / ( g a ) ) = log ( b / a ) = μ ( S ) \mu(gS)=\log((g\cdot b)/(g\cdot a))=\log(b/a)=\mu(S)
  24. μ ( S ) = S 1 | det ( X ) | n d X \mu(S)=\int_{S}{1\over|\det(X)|^{n}}\,dX
  25. n 2 \mathbb{R}^{n^{2}}
  26. n × n n\times n
  27. μ ( S ) = 1 2 π m ( f - 1 ( S ) ) , \mu(S)=\frac{1}{2\pi}m\left(f^{-1}(S)\right),
  28. μ ( S ) = S 1 ( x 2 + y 2 + z 2 + w 2 ) 2 d x d y d z d w \mu(S)=\int_{S}\frac{1}{(x^{2}+y^{2}+z^{2}+w^{2})^{2}}\,dx\,dy\,dz\,dw

Haar_wavelet.html

  1. ψ ( t ) \psi(t)
  2. ψ ( t ) = { 1 0 t < 1 2 , - 1 1 2 t < 1 , 0 otherwise. \psi(t)=\begin{cases}1&0\leq t<\frac{1}{2},\\ -1&\frac{1}{2}\leq t<1,\\ 0&\mbox{otherwise.}\end{cases}
  3. ϕ ( t ) \phi(t)
  4. ϕ ( t ) = { 1 0 t < 1 , 0 otherwise. \phi(t)=\begin{cases}1&0\leq t<1,\\ 0&\mbox{otherwise.}\end{cases}
  5. ψ n , k ( t ) = 2 n / 2 ψ ( 2 n t - k ) , t 𝐑 . \psi_{n,k}(t)=2^{n/2}\psi(2^{n}t-k),\quad t\in\mathbf{R}.
  6. 𝐑 ψ n , k ( t ) d t = 0 , ψ n , k L 2 ( 𝐑 ) 2 = 𝐑 ψ n , k ( t ) 2 d t = 1. \int_{\mathbf{R}}\psi_{n,k}(t)\,dt=0,\quad\|\psi_{n,k}\|^{2}_{L^{2}(\mathbf{R}% )}=\int_{\mathbf{R}}\psi_{n,k}(t)^{2}\,dt=1.
  7. 𝐑 ψ n 1 , k 1 ( t ) ψ n 2 , k 2 ( t ) d t = δ n 1 , n 2 δ k 1 , k 2 , \int_{\mathbf{R}}\psi_{n_{1},k_{1}}(t)\psi_{n_{2},k_{2}}(t)\,dt=\delta_{n_{1},% n_{2}}\delta_{k_{1},k_{2}},
  8. I n 1 , k 1 I_{n_{1},k_{1}}
  9. I n 2 , k 2 I_{n_{2},k_{2}}
  10. I n 1 , k 1 I_{n_{1},k_{1}}
  11. ψ n 2 , k 2 \psi_{n_{2},k_{2}}
  12. { ψ n , k ( t ) ; n 𝐙 , k 𝐙 } . \{\psi_{n,k}(t)\;;\;n\in\mathbf{Z},\;k\in\mathbf{Z}\}.
  13. ϕ ( t ) , ϕ ( 2 t ) , ϕ ( 4 t ) , , ϕ ( 2 n t ) , \phi(t),\phi(2t),\phi(4t),\dots,\phi(2^{n}t),\dots
  14. ψ ( t ) , ψ ( 2 t ) , ψ ( 4 t ) , , ψ ( 2 n t ) , \psi(t),\psi(2t),\psi(4t),\dots,\psi(2^{n}t),\dots
  15. - 2 ( n + n 1 ) / 2 ψ ( 2 n t - k ) ψ ( 2 n 1 t - k 1 ) d t = δ n , n 1 δ k , k 1 . \int_{-\infty}^{\infty}2^{(n+n_{1})/2}\psi(2^{n}t-k)\psi(2^{n_{1}}t-k_{1})\,dt% =\delta_{n,n_{1}}\delta_{k,k_{1}}.
  16. ϕ ( t ) = ϕ ( 2 t ) + ϕ ( 2 t - 1 ) ψ ( t ) = ϕ ( 2 t ) - ϕ ( 2 t - 1 ) , \begin{aligned}\displaystyle\phi(t)&\displaystyle=\phi(2t)+\phi(2t-1)\\ \displaystyle\psi(t)&\displaystyle=\phi(2t)-\phi(2t-1),\end{aligned}
  17. χ w ( k , n ) = 2 n / 2 - x ( t ) ϕ ( 2 n t - k ) d t \chi_{w}(k,n)=2^{n/2}\int_{-\infty}^{\infty}x(t)\phi(2^{n}t-k)\,dt
  18. \Chi w ( k , n ) = 2 n / 2 - x ( t ) ψ ( 2 n t - k ) d t \Chi_{w}(k,n)=2^{n/2}\int_{-\infty}^{\infty}x(t)\psi(2^{n}t-k)\,dt
  19. χ w ( k , n ) = 2 - 1 / 2 ( χ w ( 2 k , n + 1 ) + χ w ( 2 k + 1 , n + 1 ) ) \chi_{w}(k,n)=2^{-1/2}\bigl(\chi_{w}(2k,n+1)+\chi_{w}(2k+1,n+1)\bigr)
  20. \Chi w ( k , n ) = 2 - 1 / 2 ( χ w ( 2 k , n + 1 ) - χ w ( 2 k + 1 , n + 1 ) ) . \Chi_{w}(k,n)=2^{-1/2}\bigl(\chi_{w}(2k,n+1)-\chi_{w}(2k+1,n+1)\bigr).
  21. { t [ 0 , 1 ] ψ n , k ( t ) ; n 𝒩 { 0 } , 0 k < 2 n } , \{t\in[0,1]\mapsto\psi_{n,k}(t)\;;\;n\in\mathcal{N}\cup\{0\},\;0\leq k<2^{n}\},
  22. r n ( t ) = 2 - n / 2 k = 0 2 n - 1 ψ n , k ( t ) , t [ 0 , 1 ] , n 0. r_{n}(t)=2^{-n/2}\sum_{k=0}^{2^{n}-1}\psi_{n,k}(t),\quad t\in[0,1],\ n\geq 0.
  23. s n , k ( t ) = 2 1 + n / 2 0 t ψ n , k ( u ) d u , t [ 0 , 1 ] , 0 k < 2 n . s_{n,k}(t)=2^{1+n/2}\int_{0}^{t}\psi_{n,k}(u)\,du,\quad t\in[0,1],\ 0\leq k<2^% {n}.
  24. f n + 1 = a 0 s 0 + a 1 s 1 + m = 0 n - 1 ( k = 0 2 m - 1 a m , k s m , k ) C ( [ 0 , 1 ] ) f_{n+1}=a_{0}s_{0}+a_{1}s_{1}+\sum_{m=0}^{n-1}\Bigl(\sum_{k=0}^{2^{m}-1}a_{m,k% }s_{m,k}\Bigr)\in C([0,1])
  25. f n + 2 - f n + 1 = k = 0 2 n - 1 ( f ( x n , k ) - f n + 1 ( x n , k ) ) s n , k = k = 0 2 n - 1 a n , k s n , k f_{n+2}-f_{n+1}=\sum_{k=0}^{2^{n}-1}\bigl(f(x_{n,k})-f_{n+1}(x_{n,k})\bigr)s_{% n,k}=\sum_{k=0}^{2^{n}-1}a_{n,k}s_{n,k}
  26. { f : x [ 0 , π ] n = 0 a n cos ( n x ) } { T ( f ) : z n = 0 a n z n , | z | 1 } . \left\{f:x\in[0,\pi]\rightarrow\sum_{n=0}^{\infty}a_{n}\cos(nx)\right\}% \longrightarrow\left\{T(f):z\rightarrow\sum_{n=0}^{\infty}a_{n}z^{n},\quad|z|% \leq 1\right\}.
  27. H 2 = [ 1 1 1 - 1 ] . H_{2}=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}.
  28. ( a 0 , a 1 , , a 2 n , a 2 n + 1 ) (a_{0},a_{1},\dots,a_{2n},a_{2n+1})
  29. ( ( a 0 , a 1 ) , , ( a 2 n , a 2 n + 1 ) ) \left(\left(a_{0},a_{1}\right),\dots,\left(a_{2n},a_{2n+1}\right)\right)
  30. H 2 H_{2}
  31. ( ( s 0 , d 0 ) , , ( s n , d n ) ) \left(\left(s_{0},d_{0}\right),\dots,\left(s_{n},d_{n}\right)\right)
  32. H 4 = [ 1 1 1 1 1 1 - 1 - 1 1 - 1 0 0 0 0 1 - 1 ] , H_{4}=\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&0&0\\ 0&0&1&-1\end{bmatrix},
  33. H 2 N = [ H N [ 1 , 1 ] I N [ 1 , - 1 ] ] H_{2N}=\begin{bmatrix}H_{N}\otimes[1,1]\\ I_{N}\otimes[1,-1]\end{bmatrix}
  34. I N = [ 1 0 0 0 1 0 0 0 1 ] I_{N}=\begin{bmatrix}1&0&\dots&0\\ 0&1&\dots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\dots&1\end{bmatrix}
  35. \otimes
  36. A B A\otimes B
  37. A A
  38. B B
  39. A B = [ a 11 B a 1 n B a m 1 B a m n B ] . A\otimes B=\begin{bmatrix}a_{11}B&\dots&a_{1n}B\\ \vdots&\ddots&\vdots\\ a_{m1}B&\dots&a_{mn}B\end{bmatrix}.
  40. H 8 H_{8}
  41. H 8 = [ 1 1 1 1 1 1 1 1 1 1 1 1 - 1 - 1 - 1 - 1 1 1 - 1 - 1 0 0 0 0 0 0 0 0 1 1 - 1 - 1 1 - 1 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 1 - 1 ] . H_{8}=\begin{bmatrix}1&1&1&1&1&1&1&1\\ 1&1&1&1&-1&-1&-1&-1\\ 1&1&-1&-1&0&0&0&0&\\ 0&0&0&0&1&1&-1&-1\\ 1&-1&0&0&0&0&0&0&\\ 0&0&1&-1&0&0&0&0\\ 0&0&0&0&1&-1&0&0&\\ 0&0&0&0&0&0&1&-1\end{bmatrix}.
  42. H H
  43. H H
  44. H 8 H_{8}
  45. H 8 H_{8}
  46. H 8 H_{8}
  47. H 4 = 1 2 [ 1 1 1 1 1 1 - 1 - 1 2 - 2 0 0 0 0 2 - 2 ] H_{4}=\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ \sqrt{2}&-\sqrt{2}&0&0\\ 0&0&\sqrt{2}&-\sqrt{2}\end{bmatrix}
  48. N = 2 k , k N=2^{k},k\in\mathbb{N}
  49. y n = H n x n y_{n}=H_{n}x_{n}
  50. H = H * , H - 1 = H T , i.e. H H T = I H=H^{*},H^{-1}=H^{T},\,\text{i.e. }HH^{T}=I
  51. I I
  52. H 4 T H 4 = 1 2 [ 1 1 2 0 1 1 - 2 0 1 - 1 0 2 1 - 1 0 - 2 ] 1 2 [ 1 1 1 1 1 1 - 1 - 1 2 - 2 0 0 0 0 2 - 2 ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] H_{4}^{T}H_{4}=\frac{1}{2}\begin{bmatrix}1&1&\sqrt{2}&0\\ 1&1&-\sqrt{2}&0\\ 1&-1&0&\sqrt{2}\\ 1&-1&0&-\sqrt{2}\end{bmatrix}\cdot\;\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ \sqrt{2}&-\sqrt{2}&0&0\\ 0&0&\sqrt{2}&-\sqrt{2}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}
  53. x n = H T y n x_{n}=H^{T}y_{n}
  54. x 4 = [ 1 , 2 , 3 , 4 ] T x_{4}=[1,2,3,4]^{T}
  55. y 4 = H 4 x 4 = 1 2 [ 1 1 1 1 1 1 - 1 - 1 2 - 2 0 0 0 0 2 - 2 ] [ 1 2 3 4 ] = [ 5 - 2 - 1 / 2 - 1 / 2 ] y_{4}=H_{4}x_{4}=\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ \sqrt{2}&-\sqrt{2}&0&0\\ 0&0&\sqrt{2}&-\sqrt{2}\end{bmatrix}\begin{bmatrix}1\\ 2\\ 3\\ 4\end{bmatrix}=\begin{bmatrix}5\\ -2\\ -1/\sqrt{2}\\ -1/\sqrt{2}\end{bmatrix}
  56. x 4 ^ = H 4 T y 4 = 1 2 [ 1 1 2 0 1 1 - 2 0 1 - 1 0 2 1 - 1 0 - 2 ] [ 5 - 2 - 1 / 2 - 1 / 2 ] = [ 1 2 3 4 ] \hat{x_{4}}=H_{4}^{T}y_{4}=\frac{1}{2}\begin{bmatrix}1&1&\sqrt{2}&0\\ 1&1&-\sqrt{2}&0\\ 1&-1&0&\sqrt{2}\\ 1&-1&0&-\sqrt{2}\end{bmatrix}\begin{bmatrix}5\\ -2\\ -1/\sqrt{2}\\ -1/\sqrt{2}\end{bmatrix}=\begin{bmatrix}1\\ 2\\ 3\\ 4\end{bmatrix}
  57. B n = H n A n H n T B_{n}=H_{n}A_{n}H_{n}^{T}
  58. A n A_{n}
  59. H n H_{n}
  60. A n = H n T B n H n A_{n}=H_{n}^{T}B_{n}H_{n}

Hahn–Banach_theorem.html

  1. V V
  2. f : V 𝐑 f:V→\mathbf{R}
  3. f ( γ x ) = γ f ( x ) f(γx)=γf(x)
  4. f ( x + y ) f ( x ) + f ( y ) f(x+y)≤f(x)+f(y)
  5. x , y V x,y∈V
  6. V V
  7. V V
  8. p : V 𝐑 p:V→\mathbf{R}
  9. φ : U 𝐑 φ:U→\mathbf{R}
  10. U V U⊆V
  11. p p
  12. U U
  13. φ ( x ) p ( x ) x U \varphi(x)\leq p(x)\qquad\forall x\in U
  14. ψ : V 𝐑 ψ:V→\mathbf{R}
  15. φ φ
  16. V V
  17. ψ ψ
  18. ψ ( x ) = φ ( x ) x U , \psi(x)=\varphi(x)\qquad\forall x\in U,
  19. ψ ( x ) p ( x ) x V . \psi(x)\leq p(x)\qquad\forall x\in V.
  20. 𝐊 = 𝐑 \mathbf{K}=\mathbf{R}
  21. 𝐂 \mathbf{C}
  22. V V
  23. 𝐊 \mathbf{K}
  24. p : V 𝐑 p:V→\mathbf{R}
  25. φ : U 𝐊 φ:U→\mathbf{K}
  26. 𝐊 \mathbf{K}
  27. 𝐊 \mathbf{K}
  28. U U
  29. V V
  30. p p
  31. U U
  32. | φ ( x ) | p ( x ) x U |\varphi(x)|\leq p(x)\qquad\forall x\in U
  33. ψ : V 𝐊 ψ:V→\mathbf{K}
  34. φ φ
  35. V V
  36. 𝐊 \mathbf{K}
  37. ψ ψ
  38. ψ ( x ) = φ ( x ) x U , \psi(x)=\varphi(x)\qquad\forall x\in U,
  39. | ψ ( x ) | p ( x ) x V . |\psi(x)|\leq p(x)\qquad\forall x\in V.
  40. 𝐂 \mathbf{C}
  41. x U x∈U
  42. i x U ix∈U
  43. φ ( i x ) = i φ ( x ) φ(ix)=iφ(x)
  44. ψ ψ
  45. φ φ
  46. ψ ψ
  47. V V
  48. p p
  49. p ( a x + b y ) | a | p ( x ) + | b | p ( y ) , x , y V , | a | + | b | 1. p(ax+by)\leq|a|\,p(x)+|b|\,p(y),\qquad x,y\in V,\quad|a|+|b|\leq 1.
  50. V V
  51. U U
  52. φ : U 𝐊 φ:U→\mathbf{K}
  53. ψ : V 𝐊 ψ:V→\mathbf{K}
  54. φ φ
  55. φ φ
  56. 𝐊 \mathbf{K}
  57. V V
  58. U U
  59. z z
  60. V V
  61. U U
  62. ψ : V 𝐊 ψ:V→\mathbf{K}
  63. ψ ( x ) = 0 ψ(x)=0
  64. x x
  65. U U
  66. ψ ( z ) = 1 ψ(z)=1
  67. V V
  68. z z
  69. V V
  70. ψ : V 𝐊 ψ:V→\mathbf{K}
  71. ψ ( z ) = [ u ! ! ] z [ u ! ! ] ψ(z)=[u^{\prime}!!^{\prime}]z[u^{\prime}!!^{\prime}]
  72. [ u ! ! ] ψ [ u ! ! ] 1 [u^{\prime}!!^{\prime}]ψ[u^{\prime}!!^{\prime}]≤1
  73. J J
  74. V V
  75. V V′′
  76. 𝐊 = 𝐑 \mathbf{K}=\mathbf{R}
  77. 𝐂 \mathbf{C}
  78. V V
  79. 𝐊 \mathbf{K}
  80. A , B A,B
  81. V V
  82. A A
  83. λ : V 𝐊 λ:V→\mathbf{K}
  84. t 𝐑 t∈\mathbf{R}
  85. a A , b B a∈A,b∈B
  86. V V
  87. A A
  88. B B
  89. λ : V 𝐊 λ:V→\mathbf{K}
  90. s , t 𝐑 s,t∈\mathbf{R}
  91. v V v∈V
  92. M M
  93. X X
  94. K K
  95. X X
  96. K M = K∩M=∅
  97. N N
  98. X X
  99. M M
  100. K N = K∩N=∅
  101. B B
  102. B B
  103. 0 , 11 0,11
  104. X X
  105. X X
  106. U U
  107. X , X * X,X*
  108. X X
  109. X * , X X*,X
  110. X X
  111. X * X*
  112. X X
  113. x X x∈X
  114. X * X*
  115. x x
  116. X X
  117. C a a , b * * Caa,b**
  118. F ( u ) = a b u ( x ) d ρ ( x ) , F(u)=\int^{b}_{a}u(x)d\rho(x),
  119. u C a a , b u∈Caa,b
  120. | F | = V ( ρ ) |F|=V(ρ)
  121. V ( ρ ) V(ρ)
  122. ρ ρ
  123. C a a , b Caa,b

Hail.html

  1. Z d r Z_{dr}
  2. Z h Z_{h}

Half-life.html

  1. N ( t ) \displaystyle N(t)
  2. τ \tau
  3. λ \lambda
  4. t 1 2 t_{\frac{1}{2}}
  5. τ \tau
  6. λ \lambda
  7. t 1 2 = ln ( 2 ) λ = τ ln ( 2 ) t_{\frac{1}{2}}=\frac{\ln(2)}{\lambda}=\tau\ln(2)
  8. N ( t ) = N 0 ( 1 2 ) t / t 1 2 N ( t ) = N 0 e - t τ N ( t ) = N 0 e - λ t \begin{aligned}\displaystyle N(t)&\displaystyle=N_{0}\left(\frac{1}{2}\right)^% {t/t_{\frac{1}{2}}}\\ \displaystyle N(t)&\displaystyle=N_{0}e^{-\frac{t}{\tau}}\\ \displaystyle N(t)&\displaystyle=N_{0}e^{-\lambda t}\end{aligned}
  9. t 1 2 t_{\frac{1}{2}}
  10. τ \tau
  11. ( 1 2 ) t / t 1 2 = e - t τ = e - λ t \left(\frac{1}{2}\right)^{t/t_{\frac{1}{2}}}=e^{-\frac{t}{\tau}}=e^{-\lambda t}
  12. ln ( ( 1 2 ) t / t 1 2 ) = ln ( e - t τ ) = ln ( e - λ t ) \ln\left(\left(\frac{1}{2}\right)^{t/t_{\frac{1}{2}}}\right)=\ln\left(e^{-% \frac{t}{\tau}}\right)=\ln\left(e^{-\lambda t}\right)
  13. t t 1 2 ln ( 1 2 ) = ( - t τ ) ln ( e ) = ( - λ t ) ln ( e ) \frac{t}{t_{\frac{1}{2}}}\ln\left(\frac{1}{2}\right)=\left(-\frac{t}{\tau}% \right)\ln(e)=(-\lambda t)\ln(e)
  14. t t 1 2 ln ( 1 2 ) = - t τ = - λ t \frac{t}{t_{\frac{1}{2}}}\ln\left(\frac{1}{2}\right)=-\frac{t}{\tau}=-\lambda t
  15. ln ( 1 2 ) = - ln ( 2 ) \ln\left(\frac{1}{2}\right)=-\ln(2)
  16. t 1 2 = τ ln 2 = ln 2 λ . t_{\frac{1}{2}}=\tau\ln 2=\frac{\ln 2}{\lambda}.
  17. N ( t ) \displaystyle N(t)
  18. N ( 0 ) = N 0 N(0)=N_{0}
  19. N ( t 1 2 ) = 1 2 N 0 N\left(t_{\frac{1}{2}}\right)=\frac{1}{2}N_{0}
  20. lim t N ( t ) = 0 \lim_{t\to\infty}N(t)=0
  21. 1 T 1 2 = 1 t 1 + 1 t 2 \frac{1}{T_{\frac{1}{2}}}=\frac{1}{t_{1}}+\frac{1}{t_{2}}
  22. 1 T 1 2 = 1 t 1 + 1 t 2 + 1 t 3 + \frac{1}{T_{\frac{1}{2}}}=\frac{1}{t_{1}}+\frac{1}{t_{2}}+\frac{1}{t_{3}}+\cdots
  23. R C ln ( 2 ) RC\ln(2)
  24. ln ( 2 ) L / R \ln(2)L/R
  25. ln ( 2 ) / λ \ln(2)/\lambda

Hall_effect.html

  1. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) = 0 \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)=0
  2. - B × E = E × B = B × ( v × B ) = v B 2 - B ( v B ) -B\times E=E\times B=B\times(v\times B)=vB^{2}-B(v\cdot B)
  3. v = E × B / B 2 v=E\times B/B^{2}
  4. v = E / B v=E/B
  5. V H = - I B n t e V_{H}=-\frac{IB}{nte}
  6. e \,e
  7. R H = E y j x B R_{H}=\frac{E_{y}}{j_{x}B}
  8. E y E_{y}
  9. R H = E y j x B = V H t I B = - 1 n e . R_{H}=\frac{E_{y}}{j_{x}B}=\frac{V_{H}t}{IB}=-\frac{1}{ne}.
  10. R H = p μ h 2 - n μ e 2 e ( p μ h + n μ e ) 2 R_{H}=\frac{p\mu_{h}^{2}-n\mu_{e}^{2}}{e(p\mu_{h}+n\mu_{e})^{2}}
  11. R H = ( p - n b 2 ) e ( p + n b ) 2 R_{H}=\frac{(p-nb^{2})}{e(p+nb)^{2}}
  12. b = μ e μ h b=\frac{\mu_{e}}{\mu_{h}}
  13. n \,n
  14. p \,p
  15. μ e \,\mu_{e}
  16. μ h \,\mu_{h}
  17. e \,e
  18. β = Ω e ν = e B m e ν \beta=\frac{\Omega_{e}}{\nu}=\frac{eB}{m_{e}\nu}
  19. β = tan ( θ ) . \beta=\tan(\theta).

Halogen_lamp.html

  1. V 3 V^{3}
  2. V 1.3 V^{1.3}
  3. V - 14 V^{-14}

Hamiltonian_(quantum_mechanics).html

  1. H ^ = T ^ + V ^ \hat{H}=\hat{T}+\hat{V}
  2. V ^ = V = V ( r , t ) \hat{V}=V=V({r},t)
  3. T ^ = p ^ p ^ 2 m = p ^ 2 2 m = - 2 2 m 2 \hat{T}=\frac{{\hat{p}}\cdot{\hat{p}}}{2m}=\frac{\hat{p}^{2}}{2m}=-\frac{\hbar% ^{2}}{2m}\nabla^{2}
  4. p ^ = - i \hat{p}=-i\hbar\nabla
  5. 2 = 2 x 2 + 2 y 2 + 2 z 2 \nabla^{2}=\frac{\partial^{2}}{{\partial x}^{2}}+\frac{\partial^{2}}{{\partial y% }^{2}}+\frac{\partial^{2}}{{\partial z}^{2}}
  6. H ^ \displaystyle\hat{H}
  7. H ^ = n = 1 N T ^ n + V \hat{H}=\sum_{n=1}^{N}\hat{T}_{n}+V
  8. V = V ( r 1 , r 2 r N , t ) V=V({r}_{1},{r}_{2}\cdots{r}_{N},t)
  9. T ^ n = p n p n 2 m n \hat{T}_{n}=\frac{{p}_{n}\cdot{p}_{n}}{2m_{n}}
  10. n 2 = 2 x n 2 + 2 y n 2 + 2 z n 2 \nabla_{n}^{2}=\frac{\partial^{2}}{\partial x_{n}^{2}}+\frac{\partial^{2}}{% \partial y_{n}^{2}}+\frac{\partial^{2}}{\partial z_{n}^{2}}
  11. H ^ \displaystyle\hat{H}
  12. - 2 2 M i j -\frac{\hbar^{2}}{2M}\nabla_{i}\cdot\nabla_{j}
  13. V = i = 1 N V ( r i , t ) = V ( r 1 , t ) + V ( r 2 , t ) + + V ( r N , t ) V=\sum_{i=1}^{N}V({r}_{i},t)=V({r}_{1},t)+V({r}_{2},t)+\cdots+V({r}_{N},t)
  14. H ^ \displaystyle\hat{H}
  15. | ψ ( t ) \left|\psi(t)\right\rangle
  16. H | ψ ( t ) = i t | ψ ( t ) . H\left|\psi(t)\right\rangle=i\hbar{\partial\over\partial t}\left|\psi(t)\right\rangle.
  17. | ψ ( t ) = e - i H t / | ψ ( 0 ) . \left|\psi(t)\right\rangle=e^{-iHt/\hbar}\left|\psi(0)\right\rangle.
  18. U = e - i H t / U=e^{-iHt/\hbar}
  19. | a \left|a\right\rangle
  20. H | a = E a | a . H\left|a\right\rangle=E_{a}\left|a\right\rangle.
  21. H ^ = - 2 2 m 2 x 2 \hat{H}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}
  22. H ^ = - 2 2 m 2 \hat{H}=-\frac{\hbar^{2}}{2m}\nabla^{2}
  23. H ^ = - 2 2 m 2 x 2 + V 0 \hat{H}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+V_{0}
  24. H ^ = - 2 2 m 2 + V 0 \hat{H}=-\frac{\hbar^{2}}{2m}\nabla^{2}+V_{0}
  25. V = k 2 x 2 = m ω 2 2 x 2 V=\frac{k}{2}x^{2}=\frac{m\omega^{2}}{2}x^{2}
  26. ω 2 = k m \omega^{2}=\frac{k}{m}
  27. H ^ = - 2 2 m 2 x 2 + m ω 2 2 x 2 \hat{H}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial x^{2}}+\frac{m\omega% ^{2}}{2}x^{2}
  28. H ^ = - 2 2 m 2 + m ω 2 2 r 2 \hat{H}=-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{m\omega^{2}}{2}r^{2}
  29. r 2 = r r = | r | 2 = x 2 + y 2 + z 2 r^{2}={r}\cdot{r}=|{r}|^{2}=x^{2}+y^{2}+z^{2}
  30. H ^ \displaystyle\hat{H}
  31. H ^ = - 2 2 I x x J ^ x 2 - 2 2 I y y J ^ y 2 - 2 2 I z z J ^ z 2 \hat{H}=-\frac{\hbar^{2}}{2I_{xx}}\hat{J}_{x}^{2}-\frac{\hbar^{2}}{2I_{yy}}% \hat{J}_{y}^{2}-\frac{\hbar^{2}}{2I_{zz}}\hat{J}_{z}^{2}
  32. J ^ x \hat{J}_{x}\,\!
  33. J ^ y \hat{J}_{y}\,\!
  34. J ^ z \hat{J}_{z}\,\!
  35. V = q 1 q 2 4 π ϵ 0 | r | V=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}|{r}|}
  36. V j = 1 2 i j q i ϕ ( 𝐫 i ) = 1 8 π ε 0 i j q i q j | 𝐫 i - 𝐫 j | V_{j}=\frac{1}{2}\sum_{i\neq j}q_{i}\phi(\mathbf{r}_{i})=\frac{1}{8\pi% \varepsilon_{0}}\sum_{i\neq j}\frac{q_{i}q_{j}}{|\mathbf{r}_{i}-\mathbf{r}_{j}|}
  37. V = 1 8 π ε 0 j = 1 N i j q i q j | 𝐫 i - 𝐫 j | V=\frac{1}{8\pi\varepsilon_{0}}\sum_{j=1}^{N}\sum_{i\neq j}\frac{q_{i}q_{j}}{|% \mathbf{r}_{i}-\mathbf{r}_{j}|}
  38. H ^ \displaystyle\hat{H}
  39. V = - d ^ E V=-{\hat{d}}\cdot{E}
  40. V = - d ^ E V=-{\hat{d}}\cdot{E}
  41. H ^ = - d ^ E = - q E r ^ \hat{H}=-{\hat{d}}\cdot{E}=-q{E}\cdot{\hat{r}}
  42. V = - s y m b o l μ B V=-symbol{\mu}\cdot{B}
  43. H ^ = - s y m b o l μ B \hat{H}=-symbol{\mu}\cdot{B}
  44. s y m b o l μ S = g s e 2 m S symbol{\mu}_{S}=\frac{g_{s}e}{2m}{S}
  45. H ^ = g s e 2 m S B \hat{H}=\frac{g_{s}e}{2m}{S}\cdot{B}
  46. Π ^ = P ^ - q A {\hat{\Pi}}={\hat{P}}-q{A}
  47. P ^ {\hat{P}}
  48. P ^ = - i {\hat{P}}=-i\hbar\nabla
  49. T ^ = Π ^ Π ^ 2 m = 1 2 m ( P ^ - q A ) 2 \hat{T}=\frac{{\hat{\Pi}}\cdot{\hat{\Pi}}}{2m}=\frac{1}{2m}\left({\hat{P}}-q{A% }\right)^{2}
  50. V = q ϕ V=q\phi
  51. H ^ = 1 2 m ( - i - q A ) 2 + q ϕ \hat{H}=\frac{1}{2m}\left(-i\hbar\nabla-q{A}\right)^{2}+q\phi
  52. | a |a\rangle
  53. U | a U|a\rangle
  54. U H | a = U E a | a = E a ( U | a ) = H ( U | a ) . UH|a\rangle=UE_{a}|a\rangle=E_{a}(U|a\rangle)=H\;(U|a\rangle).
  55. | a |a\rangle
  56. U | a U|a\rangle
  57. U = I - i ϵ G + O ( ϵ 2 ) U=I-i\epsilon G+O(\epsilon^{2})\,
  58. [ H , G ] = 0 [H,G]=0\,
  59. t ψ ( t ) | G | ψ ( t ) = 1 i ψ ( t ) | [ G , H ] | ψ ( t ) = 0. \frac{\partial}{\partial t}\langle\psi(t)|G|\psi(t)\rangle=\frac{1}{i\hbar}% \langle\psi(t)|[G,H]|\psi(t)\rangle=0.
  60. ψ ( t ) | H = - i t ψ ( t ) | . \langle\psi(t)|H=-i\hbar{\partial\over\partial t}\langle\psi(t)|.
  61. { | n } \left\{\left|n\right\rangle\right\}
  62. n | n = δ n n . \langle n^{\prime}|n\rangle=\delta_{nn^{\prime}}.
  63. | ψ ( t ) \left|\psi\left(t\right)\right\rangle
  64. | ψ ( t ) = n a n ( t ) | n |\psi(t)\rangle=\sum_{n}a_{n}(t)|n\rangle
  65. a n ( t ) = n | ψ ( t ) . a_{n}(t)=\langle n|\psi(t)\rangle.
  66. H ( t ) = def ψ ( t ) | H | ψ ( t ) = n n a n * a n n | H | n \langle H(t)\rangle\ \stackrel{\mathrm{def}}{=}\ \langle\psi(t)|H|\psi(t)% \rangle=\sum_{nn^{\prime}}a_{n^{\prime}}^{*}a_{n}\langle n^{\prime}|H|n\rangle
  67. | ψ ( t ) \left|\psi\left(t\right)\right\rangle
  68. H a n * = n a n n | H | n = n | H | ψ \frac{\partial\langle H\rangle}{\partial a_{n^{\prime}}^{*}}=\sum_{n}a_{n}% \langle n^{\prime}|H|n\rangle=\langle n^{\prime}|H|\psi\rangle
  69. H a n * = i a n t \frac{\partial\langle H\rangle}{\partial a_{n^{\prime}}^{*}}=i\hbar\frac{% \partial a_{n^{\prime}}}{\partial t}
  70. H a n = - i a n * t \frac{\partial\langle H\rangle}{\partial a_{n}}=-i\hbar\frac{\partial a_{n}^{*% }}{\partial t}
  71. π n ( t ) = i a n * ( t ) \pi_{n}(t)=i\hbar a_{n}^{*}(t)
  72. H π n = a n t , H a n = - π n t \frac{\partial\langle H\rangle}{\partial\pi_{n}}=\frac{\partial a_{n}}{% \partial t},\quad\frac{\partial\langle H\rangle}{\partial a_{n}}=-\frac{% \partial\pi_{n}}{\partial t}
  73. a n a_{n}
  74. π n \pi_{n}
  75. H \langle H\rangle

Hammer.html

  1. E = m v 2 2 E={mv^{2}\over 2}

Hamming_code.html

  1. m m
  2. 2 m - 1 2^{m}-1
  3. 2 m - m - 1 2^{m}-m-1
  4. m m
  5. m m
  6. 2 m - 1 2^{m}-1
  7. 2 m - m - 1 2^{m}-m-1
  8. ( 2 m - 1 , 2 m - m - 1 ) (2^{m}-1,2^{m}-m-1)
  9. ( 2 m - m - 1 ) / ( 2 m - 1 ) (2^{m}-m-1)/(2^{m}-1)
  10. 𝐆 := ( I k - A T ) \mathbf{G}:=\begin{pmatrix}\begin{array}[]{c|c}I_{k}&-A\text{T}\\ \end{array}\end{pmatrix}
  11. 𝐇 := ( A I n - k ) \mathbf{H}:=\begin{pmatrix}\begin{array}[]{c|c}A&I_{n-k}\\ \end{array}\end{pmatrix}
  12. 𝐇 𝐆 T = 𝟎 \mathbf{H}\,\mathbf{G}\text{T}=\mathbf{0}
  13. 𝐆 \mathbf{G}
  14. 𝐇 \mathbf{H}
  15. 𝐆 := ( 1 0 0 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 1 1 ) 4 , 7 \mathbf{G}:=\begin{pmatrix}1&0&0&0&1&1&0\\ 0&1&0&0&1&0&1\\ 0&0&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{pmatrix}_{4,7}
  16. 𝐇 := ( 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 1 ) 3 , 7 . \mathbf{H}:=\begin{pmatrix}1&1&0&1&1&0&0\\ 1&0&1&1&0&1&0\\ 0&1&1&1&0&0&1\\ \end{pmatrix}_{3,7}.
  17. x \vec{x}
  18. x = a G \vec{x}=\vec{a}G
  19. a = a 1 a 2 a 3 a 4 \vec{a}=a_{1}a_{2}a_{3}a_{4}
  20. a i a_{i}
  21. F 2 F_{2}
  22. 𝐆 := ( 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 ) 4 , 8 \mathbf{G}:=\begin{pmatrix}1&1&1&0&0&0&0&1\\ 1&0&0&1&1&0&0&1\\ 0&1&0&1&0&1&0&1\\ 1&1&0&1&0&0&1&0\end{pmatrix}_{4,8}
  23. 𝐇 := ( 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 ) 4 , 8 . \mathbf{H}:=\begin{pmatrix}1&0&1&0&1&0&1&0\\ 0&1&1&0&0&1&1&0\\ 0&0&0&1&1&1&1&0\\ 1&1&1&1&1&1&1&1\end{pmatrix}_{4,8}.
  24. 𝐇 = ( 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 ) 4 , 8 . \mathbf{H}=\left(\begin{array}[]{cccc|cccc}0&1&1&1&1&0&0&0\\ 1&0&1&1&0&1&0&0\\ 1&1&0&1&0&0&1&0\\ 1&1&1&0&0&0&0&1\end{array}\right)_{4,8}.
  25. 𝐆 = ( 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 ) 4 , 8 . \mathbf{G}=\left(\begin{array}[]{cccc|cccc}1&0&0&0&0&1&1&1\\ 0&1&0&0&1&0&1&1\\ 0&0&1&0&1&1&0&1\\ 0&0&0&1&1&1&1&0\end{array}\right)_{4,8}.

Hamming_distance.html

  1. R n R^{n}

Handshaking.html

  1. x x
  2. y y
  3. x + 1 x+1
  4. y + 1 y+1
  5. x x
  6. y y
  7. x + 1 x+1
  8. y + 1 y+1

Harmonic_function.html

  1. 2 f x 1 2 + 2 f x 2 2 + + 2 f x n 2 = 0 \frac{\partial^{2}f}{\partial x_{1}^{2}}+\frac{\partial^{2}f}{\partial x_{2}^{% 2}}+\cdots+\frac{\partial^{2}f}{\partial x_{n}^{2}}=0
  2. 2 f = 0 \nabla^{2}f=0
  3. Δ f = 0 \textstyle\Delta f=0
  4. f ( x , y ) = e x sin y \,\!f(x,y)=e^{x}\sin y
  5. f ( x , y ) = Im ( e x + i y ) f(x,y)=\operatorname{Im}(e^{x+iy})
  6. e x + i y e^{x+iy}
  7. f ( x 1 , x 2 ) = ln ( x 1 2 + x 2 2 ) \,\!f(x_{1},x_{2})=\ln(x_{1}^{2}+x_{2}^{2})
  8. 2 { 0 } \mathbb{R}^{2}\setminus\{0\}
  9. r 2 = x 2 + y 2 + z 2 r^{2}=x^{2}+y^{2}+z^{2}
  10. 1 r \frac{1}{r}
  11. x r 3 \frac{x}{r^{3}}
  12. - ln ( r 2 - z 2 ) -\ln(r^{2}-z^{2})\,
  13. - ln ( r + z ) -\ln(r+z)\,
  14. x r 2 - z 2 \frac{x}{r^{2}-z^{2}}\,
  15. x r ( r + z ) \frac{x}{r(r+z)}\,
  16. f ( x 1 , , x n ) = ( x 1 2 + + x n 2 ) 1 - n / 2 \,\!f(x_{1},\dots,x_{n})=({x_{1}}^{2}+\cdots+{x_{n}}^{2})^{1-n/2}
  17. n { 0 } \mathbb{R}^{n}\setminus\{0\}
  18. f n ( x , y ) = 1 n exp ( n x ) cos ( n y ) \scriptstyle f_{n}(x,y)=\frac{1}{n}\exp(nx)\cos(ny)
  19. f = g \scriptstyle f\,^{\prime}=g
  20. u ( x ) = 1 ω n r n - 1 B ( x , r ) u d σ = n ω n r n B ( x , r ) u d V u(x)=\frac{1}{\omega_{n}r^{n-1}}\int_{\partial B(x,r)}u\,d\sigma=\frac{n}{% \omega_{n}r^{n}}\int_{B(x,r)}u\,dV
  21. χ r := 1 | B ( 0 , r ) | χ B ( 0 , r ) = 1 ω n r n χ B ( 0 , r ) \chi_{r}:=\frac{1}{|B(0,r)|}\chi_{B(0,r)}=\frac{1}{\omega_{n}r^{n}}\chi_{B(0,r)}
  22. 𝐑 n χ r d x = 1 \scriptstyle\int_{\mathbf{R}^{n}}\chi_{r}\,dx=1
  23. u ( x ) = u * χ r ( x ) u(x)=u*\chi_{r}(x)\;
  24. 0 = Δ u * w r , s = u * Δ w r , s = u * χ r - u * χ s 0=\Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi_{r}-u*\chi_{s}\;
  25. Ω \Omega
  26. dist ( x , Ω ) > r \mathrm{dist}(x,\partial\Omega)>r
  27. L loc 1 L^{1}_{\mathrm{loc}}\;
  28. u * χ r = u * χ s u*\chi_{r}=u*\chi_{s}\;
  29. u = u * χ r = u * χ r * * χ r , x Ω m r , u=u*\chi_{r}=u*\chi_{r}*\cdots*\chi_{r}\,,\qquad x\in\Omega_{mr},
  30. C m - 1 ( Ω m r ) C^{m-1}(\Omega_{mr})\;
  31. C m - 1 C^{m-1}\;
  32. C ( Ω ) C^{\infty}(\Omega)\;
  33. Δ u * w r , s = u * Δ w r , s = u * χ r - u * χ s = 0 \Delta u*w_{r,s}=u*\Delta w_{r,s}=u*\chi_{r}-u*\chi_{s}=0\;
  34. V V ¯ Ω , V\subset\overline{V}\subset\Omega,
  35. sup V u C inf V u \sup_{V}u\leq C\inf_{V}u
  36. Ω { x 0 } \scriptstyle\Omega\,\setminus\,\{x_{0}\}
  37. f ( x ) = o ( | x - x 0 | 2 - n ) , as x x 0 , f(x)=o\left(|x-x_{0}|^{2-n}\right),\qquad\,\text{as }x\to x_{0},
  38. Δ f = 0 \Delta f=0\,
  39. J ( u ) := Ω | u | 2 d x J(u):=\int_{\Omega}|\nabla u|^{2}\,dx
  40. u H 1 ( Ω ) u\in H^{1}(\Omega)
  41. v C c ( Ω ) , v\in C^{\infty}_{c}(\Omega),
  42. v H 0 1 ( Ω ) . v\in H^{1}_{0}(\Omega).
  43. Δ f = 0. \ \Delta f=0.
  44. D [ u ] = 1 2 M d u 2 d Vol D[u]=\frac{1}{2}\int_{M}\|du\|^{2}\,d\operatorname{Vol}

Harmonic_mean.html

  1. x 1 , x 2 , , x n x_{1},x_{2},...,x_{n}
  2. H = n 1 x 1 + 1 x 2 + + 1 x n = n i = 1 n 1 x i = n j = 1 n x j i = 1 n j = 1 n x j x i . H=\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}}=\frac{n}{% \sum_{i=1}^{n}\frac{1}{x_{i}}}=\frac{n\cdot\prod_{j=1}^{n}x_{j}}{\sum_{i=1}^{n% }\frac{\prod_{j=1}^{n}x_{j}}{x_{i}}}.
  3. 3 1 1 + 1 2 + 1 4 = 1 1 3 ( 1 1 + 1 2 + 1 4 ) = 12 7 . \frac{3}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}}=\frac{1}{\frac{1}{3}(\frac{1}{1}% +\frac{1}{2}+\frac{1}{4})}=\frac{12}{7}\,.
  4. min ( x 1 x n ) H ( x 1 x n ) n min ( x 1 x n ) \min(x_{1}\ldots x_{n})\leq H(x_{1}\ldots x_{n})\leq n\min(x_{1}\ldots x_{n})
  5. 1 / H ( 1 / x 1 1 / x n ) = A ( x 1 x n ) 1/H(1/x_{1}\ldots 1/x_{n})=A(x_{1}\ldots x_{n})
  6. H ( x 1 , x 2 , , x n ) = M - 1 ( x 1 , x 2 , , x n ) = n x 1 - 1 + x 2 - 1 + + x n - 1 H(x_{1},x_{2},...,x_{n})=M_{-1}(x_{1},x_{2},...,x_{n})=\frac{n}{x_{1}^{-1}+x_{% 2}^{-1}+...+x_{n}^{-1}}
  7. A 3 G 3 + G 3 H 3 + 1 3 4 ( 1 + A H ) 2 . \frac{A^{3}}{G^{3}}+\frac{G^{3}}{H^{3}}+1\leq\frac{3}{4}\left(1+\frac{A}{H}% \right)^{2}.
  8. w 1 w_{1}
  9. w n w_{n}
  10. x 1 x_{1}
  11. x n x_{n}
  12. i = 1 n w i i = 1 n w i x i . \frac{\sum_{i=1}^{n}w_{i}}{\sum_{i=1}^{n}\frac{w_{i}}{x_{i}}}.
  13. i = 1 h a r m o n i c A i \sum_{i=1}^{harmonic}A_{i}
  14. tan 2 A = 2 a b a + b * 1 b - a \tan 2A=\frac{2ab}{a+b}*\frac{1}{b-a}
  15. tan A = a b \tan A=\frac{a}{b}
  16. a a
  17. b b
  18. tan A = 3 7 , \tan A=\frac{3}{7},
  19. tan 2 A = 2 * 3 7 1 - ( 3 7 ) 2 = 21 20 ; \tan 2A=\frac{2*\frac{3}{7}}{1-(\frac{3}{7})^{2}}=\frac{21}{20};
  20. 2 * 3 * 7 3 + 7 * 1 7 - 3 = 21 20 . \frac{2*3*7}{3+7}*\frac{1}{7-3}=\frac{21}{20}.
  21. x 1 x_{1}
  22. x 2 x_{2}
  23. H = 2 x 1 x 2 x 1 + x 2 . H=\frac{2x_{1}x_{2}}{x_{1}+x_{2}}.
  24. A = x 1 + x 2 2 A=\frac{x_{1}+x_{2}}{2}
  25. G = x 1 x 2 , G=\sqrt{x_{1}x_{2}},
  26. H = G 2 A . H=\frac{G^{2}}{A}.
  27. G = A H G=\sqrt{AH}
  28. H ( x 1 , , x n ) = ( G ( x 1 , , x n ) ) n A ( x 2 x 3 x n , x 1 x 3 x n , , x 1 x 2 x n - 1 ) = ( G ( x 1 , , x n ) ) n A ( i = 1 n x i x 1 , i = 1 n x i x 2 , , i = 1 n x i x n ) . H(x_{1},\ldots,x_{n})=\frac{(G(x_{1},\ldots,x_{n}))^{n}}{A(x_{2}x_{3}\cdots x_% {n},x_{1}x_{3}\cdots x_{n},\ldots,x_{1}x_{2}\cdots x_{n-1})}=\frac{(G(x_{1},% \ldots,x_{n}))^{n}}{A\left(\frac{\prod_{i=1}^{n}x_{i}}{x_{1}},\frac{\prod_{i=1% }^{n}x_{i}}{x_{2}},\ldots,\frac{\prod_{i=1}^{n}x_{i}}{x_{n}}\right)}.
  29. n = 2 n=2
  30. H ( x 1 , x 2 ) = ( G ( x 1 , x 2 ) ) 2 A ( x 2 , x 1 ) = ( G ( x 2 , x 1 ) ) 2 A ( x 1 , x 2 ) H(x_{1},x_{2})=\frac{(G(x_{1},x_{2}))^{2}}{A(x_{2},x_{1})}=\frac{(G(x_{2},x_{1% }))^{2}}{A(x_{1},x_{2})}
  31. H = G 2 A H=\frac{G^{2}}{A}
  32. ( x 1 , x 2 ) . (x_{1},x_{2}).
  33. a - 2 + b - 2 = d - 2 a^{-2}+b^{-2}=d^{-2}

Harmonic_oscillator.html

  1. F = - k x \vec{F}=-k\vec{x}\,
  2. F = m a = m d 2 x d t 2 = m x ¨ = - k x . F=ma=m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=m\ddot{x}=-kx.
  3. x ( t ) = A cos ( ω t + ϕ ) , x(t)=A\cos\left(\omega t+\phi\right),
  4. ω = k m = 2 π T . \omega=\sqrt{\frac{k}{m}}=\frac{2\pi}{T}.
  5. 1 / T {1}/{T}
  6. U = 1 2 k x 2 . U=\frac{1}{2}kx^{2}.
  7. F = F e x t - k x - c d x d t = m d 2 x d t 2 . F=F_{ext}-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^{2}x}{\mathrm{% d}t^{2}}.
  8. F e x t = 0 F_{ext}=0
  9. d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = 0 , \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+2\zeta\omega_{0}\frac{\mathrm{d}x}{% \mathrm{d}t}+\omega_{0}^{\,2}x=0,
  10. ω 0 = k m \omega_{0}=\sqrt{\frac{k}{m}}
  11. ζ = c 2 m k \zeta=\frac{c}{2\sqrt{mk}}
  12. ω 1 = ω 0 1 - ζ 2 . \omega_{1}=\omega_{0}\sqrt{1-\zeta^{2}}.
  13. Q = 2 π × Energy stored Energy lost per cycle . Q=2\pi\times\frac{\,\text{Energy stored}}{\,\text{Energy lost per cycle}}.
  14. Q = 1 2 ζ . Q=\frac{1}{2\zeta}.
  15. F ( t ) - k x - c d x d t = m d 2 x d t 2 . F(t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{% 2}}.
  16. d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = F ( t ) m . \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+2\zeta\omega_{0}\frac{\mathrm{d}x}{% \mathrm{d}t}+\omega_{0}^{2}x=\frac{F(t)}{m}.
  17. d 2 z d t 2 + 2 ζ ω 0 d z d t + ω 0 2 z = 0 , \frac{\mathrm{d}^{2}z}{\mathrm{d}t^{2}}+2\zeta\omega_{0}\frac{\mathrm{d}z}{% \mathrm{d}t}+\omega_{0}^{2}z=0,
  18. z ( t ) = A e - ζ ω 0 t sin ( 1 - ζ 2 ω 0 t + ϕ ) , z(t)=A\mathrm{e}^{-\zeta\omega_{0}t}\ \sin\left(\sqrt{1-\zeta^{2}}\ \omega_{0}% t+\phi\right),
  19. x ( t ) = 1 - e - ζ ω 0 t sin ( 1 - ζ 2 ω 0 t + φ ) sin ( φ ) , x(t)=1-\mathrm{e}^{-\zeta\omega_{0}t}\frac{\sin\left(\sqrt{1-\zeta^{2}}\ % \omega_{0}t+\varphi\right)}{\sin(\varphi)},
  20. cos φ = ζ . \cos\varphi=\zeta.\,
  21. ω / ω 0 \omega/\omega_{0}
  22. ζ \zeta
  23. d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x = 1 m F 0 sin ( ω t ) , \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}+2\zeta\omega_{0}\frac{\mathrm{d}x}{% \mathrm{d}t}+\omega_{0}^{2}x=\frac{1}{m}F_{0}\sin(\omega t),
  24. F 0 \,\!F_{0}
  25. ω \,\!\omega
  26. F 0 \,\!F_{0}
  27. ω \,\!\omega
  28. ω 0 \,\!\omega_{0}
  29. ζ \,\!\zeta
  30. ϕ \,\!\phi
  31. x ( t ) = F 0 m Z m ω sin ( ω t + ϕ ) x(t)=\frac{F_{0}}{mZ_{m}\omega}\sin(\omega t+\phi)
  32. Z m = ( 2 ω 0 ζ ) 2 + 1 ω 2 ( ω 0 2 - ω 2 ) 2 Z_{m}=\sqrt{\left(2\omega_{0}\zeta\right)^{2}+\frac{1}{\omega^{2}}\left(\omega% _{0}^{2}-\omega^{2}\right)^{2}}
  33. ϕ = arctan ( 2 ω ω 0 ζ ω 2 - ω 0 2 ) \phi=\arctan\left(\frac{2\omega\omega_{0}\zeta}{\omega^{2}-\omega_{0}^{2}}\right)
  34. ω r = ω 0 1 - 2 ζ 2 \,\!\omega_{r}=\omega_{0}\sqrt{1-2\zeta^{2}}
  35. F 0 \,\!F_{0}
  36. ζ < 1 / 2 \,\zeta<1/\sqrt{2}
  37. F 0 = 0 \,\!F_{0}=0
  38. ω \omega
  39. β \beta
  40. ω s , ω i \omega_{s},\omega_{i}
  41. d 2 q d τ 2 + 2 ζ d q d τ + q = 0 \frac{\mathrm{d}^{2}q}{\mathrm{d}\tau^{2}}+2\zeta\frac{\mathrm{d}q}{\mathrm{d}% \tau}+q=0
  42. d 2 q d τ 2 + 2 ζ d q d τ + q = cos ( ω τ ) . \frac{\mathrm{d}^{2}q}{\mathrm{d}\tau^{2}}+2\zeta\frac{\mathrm{d}q}{\mathrm{d}% \tau}+q=\cos(\omega\tau).
  43. q t ( τ ) = { e - ζ τ ( c 1 e τ ζ 2 - 1 + c 2 e - τ ζ 2 - 1 ) ζ > 1 (overdamping) e - ζ τ ( c 1 + c 2 τ ) = e - τ ( c 1 + c 2 τ ) ζ = 1 (critical damping) e - ζ τ [ c 1 cos ( 1 - ζ 2 τ ) + c 2 sin ( 1 - ζ 2 τ ) ] ζ < 1 (underdamping) q_{t}(\tau)=\begin{cases}\mathrm{e}^{-\zeta\tau}\left(c_{1}\mathrm{e}^{\tau% \sqrt{\zeta^{2}-1}}+c_{2}\mathrm{e}^{-\tau\sqrt{\zeta^{2}-1}}\right)&\zeta>1\,% \text{ (overdamping)}\\ \mathrm{e}^{-\zeta\tau}(c_{1}+c_{2}\tau)=\mathrm{e}^{-\tau}(c_{1}+c_{2}\tau)&% \zeta=1\,\text{ (critical damping)}\\ \mathrm{e}^{-\zeta\tau}\left[c_{1}\cos\left(\sqrt{1-\zeta^{2}}\tau\right)+c_{2% }\sin\left(\sqrt{1-\zeta^{2}}\tau\right)\right]&\zeta<1\,\text{(underdamping)}% \end{cases}
  44. d 2 q d τ 2 + 2 ζ d q d τ + q = cos ( ω τ ) + i sin ( ω τ ) = e i ω τ . \frac{\mathrm{d}^{2}q}{\mathrm{d}\tau^{2}}+2\zeta\frac{\mathrm{d}q}{\mathrm{d}% \tau}+q=\cos(\omega\tau)+\mathrm{i}\sin(\omega\tau)=\mathrm{e}^{\mathrm{i}% \omega\tau}.
  45. q s ( τ ) = A e i ( ω τ + ϕ ) . \,\!q_{s}(\tau)=A\mathrm{e}^{\mathrm{i}(\omega\tau+\phi)}.
  46. q s = A e i ( ω τ + ϕ ) , d q s d τ = i ω A e i ( ω τ + ϕ ) , d 2 q s d τ 2 = - ω 2 A e i ( ω τ + ϕ ) . q_{s}=A\mathrm{e}^{\mathrm{i}(\omega\tau+\phi)},\ \frac{\mathrm{d}q_{s}}{% \mathrm{d}\tau}=\mathrm{i}\omega A\mathrm{e}^{\mathrm{i}(\omega\tau+\phi)},\ % \frac{\mathrm{d}^{2}q_{s}}{\mathrm{d}\tau^{2}}=-\omega^{2}A\mathrm{e}^{\mathrm% {i}(\omega\tau+\phi)}.
  47. - ω 2 A e i ( ω τ + ϕ ) + 2 ζ i ω A e i ( ω τ + ϕ ) + A e i ( ω τ + ϕ ) = ( - ω 2 A + 2 ζ i ω A + A ) e i ( ω τ + ϕ ) = e i ω τ . \,\!-\omega^{2}A\mathrm{e}^{\mathrm{i}(\omega\tau+\phi)}+2\zeta\mathrm{i}% \omega A\mathrm{e}^{\mathrm{i}(\omega\tau+\phi)}+A\mathrm{e}^{\mathrm{i}(% \omega\tau+\phi)}=(-\omega^{2}A\,+\,2\zeta\mathrm{i}\omega A\,+\,A)\mathrm{e}^% {\mathrm{i}(\omega\tau+\phi)}=\mathrm{e}^{\mathrm{i}\omega\tau}.
  48. - ω 2 A + 2 ζ i ω A + A = e - i ϕ = cos ϕ - i sin ϕ . \,\!-\omega^{2}A+2\zeta\mathrm{i}\omega A+A=\mathrm{e}^{-\mathrm{i}\phi}=\cos% \phi-\mathrm{i}\sin\phi.
  49. A ( 1 - ω 2 ) = cos ϕ 2 ζ ω A = - sin ϕ . A(1-\omega^{2})=\cos\phi\qquad 2\zeta\omega A=-\sin\phi.
  50. A 2 ( 1 - ω 2 ) 2 = cos 2 ϕ ( 2 ζ ω A ) 2 = sin 2 ϕ } A 2 [ ( 1 - ω 2 ) 2 + ( 2 ζ ω ) 2 ] = 1. \left.\begin{array}[]{rcl}A^{2}(1-\omega^{2})^{2}&=&\cos^{2}\phi\\ (2\zeta\omega A)^{2}&=&\sin^{2}\phi\end{array}\right\}\Rightarrow A^{2}[(1-% \omega^{2})^{2}+(2\zeta\omega)^{2}]=1.
  51. A = A ( ζ , ω ) = sign ( - sin ϕ 2 ζ ω ) 1 ( 1 - ω 2 ) 2 + ( 2 ζ ω ) 2 . A=A(\zeta,\omega)=\,\text{sign}\left(\frac{-\sin\phi}{2\zeta\omega}\right)% \frac{1}{\sqrt{(1-\omega^{2})^{2}+(2\zeta\omega)^{2}}}.
  52. tan ϕ = - 2 ζ ω 1 - ω 2 = 2 ζ ω ω 2 - 1 ϕ ϕ ( ζ , ω ) = arctan ( 2 ζ ω ω 2 - 1 ) . \tan\phi=-\frac{2\zeta\omega}{1-\omega^{2}}=\frac{2\zeta\omega}{\omega^{2}-1}% \Rightarrow\phi\equiv\phi(\zeta,\omega)=\arctan\left(\frac{2\zeta\omega}{% \omega^{2}-1}\right).
  53. q s ( τ ) = A ( ζ , ω ) cos ( ω τ + ϕ ( ζ , ω ) ) = A cos ( ω τ + ϕ ) . \,\!q_{s}(\tau)=A(\zeta,\omega)\cos(\omega\tau+\phi(\zeta,\omega))=A\cos(% \omega\tau+\phi).
  54. q ( τ ) = q t ( τ ) + q s ( τ ) . \,\!q(\tau)=q_{t}(\tau)+q_{s}(\tau).
  55. x x\,
  56. θ \theta\,\!
  57. q q\,
  58. ϕ \phi\,
  59. d x d t \frac{\mathrm{d}x}{\mathrm{d}t}\,
  60. d θ d t \frac{\mathrm{d}\theta}{\mathrm{d}t}\,
  61. d q d t \frac{\mathrm{d}q}{\mathrm{d}t}\,
  62. d ϕ d t \frac{\mathrm{d}\phi}{\mathrm{d}t}\,
  63. M M\,
  64. I I\,
  65. L L\,
  66. C C\,
  67. K K\,
  68. μ \mu\,
  69. 1 / C 1/C\,
  70. 1 / L 1/L\,
  71. γ \gamma\,
  72. Γ \Gamma\,
  73. R R\,
  74. G = 1 / R G=1/R\,
  75. F ( t ) F(t)\,
  76. τ ( t ) \tau(t)\,
  77. e e\,
  78. i i\,
  79. f n f_{n}\,
  80. 1 2 π K M \frac{1}{2\pi}\sqrt{\frac{K}{M}}\,
  81. 1 2 π μ I \frac{1}{2\pi}\sqrt{\frac{\mu}{I}}\,
  82. 1 2 π 1 L C \frac{1}{2\pi}\sqrt{\frac{1}{LC}}\,
  83. 1 2 π 1 L C \frac{1}{2\pi}\sqrt{\frac{1}{LC}}\,
  84. M x ¨ + γ x ˙ + K x = F M\ddot{x}+\gamma\dot{x}+Kx=F\,
  85. I θ ¨ + Γ θ ˙ + μ θ = τ I\ddot{\theta}+\Gamma\dot{\theta}+\mu\theta=\tau\,
  86. L q ¨ + R q ˙ + q / C = e L\ddot{q}+R\dot{q}+q/C=e\,
  87. C ϕ ¨ + G ϕ ˙ + ϕ / L = i C\ddot{\phi}+G\dot{\phi}+\phi/L=i\,
  88. V ( x ) = 1 2 k x 2 V(x)=\frac{1}{2}kx^{2}
  89. V ( x ) V(x)
  90. x x
  91. x = x 0 x=x_{0}
  92. V ( x ) = V ( x 0 ) + ( x - x 0 ) V ( x 0 ) + 1 2 ( x - x 0 ) 2 V ( 2 ) ( x 0 ) + O ( x - x 0 ) 3 V(x)=V(x_{0})+(x-x_{0})V^{\prime}(x_{0})+\frac{1}{2}(x-x_{0})^{2}V^{(2)}(x_{0}% )+O(x-x_{0})^{3}
  93. V ( x 0 ) V(x_{0})
  94. x 0 x_{0}
  95. V ( x ) = V ( x 0 ) + 1 2 ( x - x 0 ) 2 V ( 2 ) ( x 0 ) + O ( x - x 0 ) 3 V(x)=V(x_{0})+\frac{1}{2}(x-x_{0})^{2}V^{(2)}(x_{0})+O(x-x_{0})^{3}
  96. V ( x ) 1 2 x 2 V ( 2 ) ( 0 ) = 1 2 k x 2 V(x)\approx\frac{1}{2}x^{2}V^{(2)}(0)=\frac{1}{2}kx^{2}
  97. V ( x ) V(x)
  98. d 2 θ d t 2 + g θ = 0. {\mathrm{d}^{2}\theta\over\mathrm{d}t^{2}}+{g\over\ell}\theta=0.
  99. θ ( t ) = θ 0 cos ( g t ) | θ 0 | 1 \theta(t)=\theta_{0}\cos\left(\sqrt{g\over\ell}t\right)\quad\quad\quad\quad|% \theta_{0}|\ll 1
  100. θ 0 \theta_{0}
  101. 2 π 2\pi
  102. g \sqrt{g\over\ell}
  103. T 0 = 2 π g | θ 0 | 1. T_{0}=2\pi\sqrt{\ell\over g}\quad\quad\quad\quad|\theta_{0}|\ll 1.
  104. F ( t ) = - k x ( t ) F\left(t\right)=-kx\left(t\right)
  105. F ( t ) = - k x ( t ) = m d 2 d t 2 x ( t ) = m a . F(t)=-kx(t)=m\frac{\mathrm{d}^{2}}{\mathrm{d}{t}^{2}}x\left(t\right)=ma.
  106. x ( t ) = A cos ( k m t ) . x\left(t\right)=A\cos\left(\sqrt{k\over m}t\right).
  107. m m
  108. m m
  109. U = k x 2 / 2. U=k{x}^{2}/2.

Hash-based_message_authentication_code.html

  1. 𝐻𝑀𝐴𝐶 ( K , m ) = H ( ( K o p a d ) | | H ( ( K i p a d ) | | m ) ) \,\textit{HMAC}(K,m)=H\Bigl((K\oplus opad)\;||\;H\bigl((K\oplus ipad)\;||\;m% \bigr)\Bigr)

Hash_table.html

  1. = n k =\frac{n}{k}

Hausdorff_dimension.html

  1. C H d ( S ) := inf { i r i d : there is a cover of S by balls with radii r i > 0 } . C_{H}^{d}(S):=\inf\Bigl\{\sum_{i}r_{i}^{d}:\,\text{ there is a cover of }S\,% \text{ by balls with radii }r_{i}>0\Bigr\}.
  2. C H d ( S ) C_{H}^{d}(S)
  3. { B ( x i , r i ) : i I } \{B(x_{i},r_{i}):i\in I\}
  4. i I r i d < δ \sum_{i\in I}r_{i}^{d}<\delta
  5. dim H ( X ) := inf { d 0 : C H d ( X ) = 0 } . \dim_{\operatorname{H}}(X):=\inf\{d\geq 0:C_{H}^{d}(X)=0\}.
  6. dim Haus ( X ) dim ind ( X ) . \dim_{\mathrm{Haus}}(X)\geq\dim_{\operatorname{ind}}(X).
  7. inf Y dim Haus ( Y ) = dim ind ( X ) , \inf_{Y}\dim_{\operatorname{Haus}}(Y)=\dim_{\operatorname{ind}}(X),
  8. X = i I X i X=\bigcup_{i\in I}X_{i}
  9. dim Haus ( X ) = sup i I dim Haus ( X i ) . \dim_{\operatorname{Haus}}(X)=\sup_{i\in I}\dim_{\operatorname{Haus}}(X_{i}).
  10. dim Haus ( X × Y ) dim Haus ( X ) + dim Haus ( Y ) . \dim_{\operatorname{Haus}}(X\times Y)\geq\dim_{\operatorname{Haus}}(X)+\dim_{% \operatorname{Haus}}(Y).
  11. ψ i : 𝐑 n 𝐑 n , i = 1 , , m \psi_{i}:\mathbf{R}^{n}\rightarrow\mathbf{R}^{n},\quad i=1,\ldots,m
  12. i = 1 m ψ i ( V ) V , \bigcup_{i=1}^{m}\psi_{i}(V)\subseteq V,
  13. i = 1 m r i s = 1. \sum_{i=1}^{m}r_{i}^{s}=1.
  14. ( 1 2 ) s + ( 1 2 ) s + ( 1 2 ) s = 3 ( 1 2 ) s = 1. \left(\frac{1}{2}\right)^{s}+\left(\frac{1}{2}\right)^{s}+\left(\frac{1}{2}% \right)^{s}=3\left(\frac{1}{2}\right)^{s}=1.
  15. A ψ ( A ) = i = 1 m ψ i ( A ) A\mapsto\psi(A)=\bigcup_{i=1}^{m}\psi_{i}(A)
  16. H s ( ψ i ( E ) ψ j ( E ) ) = 0 , H^{s}\left(\psi_{i}(E)\cap\psi_{j}(E)\right)=0,

Hausdorff_maximal_principle.html

  1. \varnothing
  2. \varnothing
  3. { S T S A and S totally ordered } \{S\mid T\subseteq S\subseteq A\mbox{ and S totally ordered}~{}\}
  4. \subseteq
  5. M = P M=\bigcup P

Hausdorff_space.html

  1. { ( x , f ( x ) ) x X } \{(x,f(x))\mid x\in X\}
  2. ker ( f ) { ( x , x ) f ( x ) = f ( x ) } \operatorname{ker}(f)\triangleq\{(x,x^{\prime})\mid f(x)=f(x^{\prime})\}
  3. eq ( f , g ) = { x f ( x ) = g ( x ) } \mbox{eq}~{}(f,g)=\{x\mid f(x)=g(x)\}

Heapsort.html

  1. O ( n ) O(n)
  2. O ( n l o g n ) O(nlogn)
  3. O ( n l o g n ) O(nlogn)
  4. a 00 a00
  5. a e n d e n d aendend
  6. 1.5 n l o g n + O ( n ) 1.5nlogn+O(n)
  7. n l o g n + O ( 1 ) nlogn+O(1)

Hearts.html

  1. C 13 14 C 13 52 = 1 45 , 358 , 111 , 400 \frac{{}_{14}C_{13}}{{}_{52}C_{13}}=\frac{1}{45,358,111,400}

Heat_engine.html

  1. d W = d Q c - ( - d Q h ) dW\ =\ dQ_{c}\ -\ (-dQ_{h})
  2. d W = - P d V dW=-PdV
  3. d Q h = T h d S h dQ_{h}=T_{h}dS_{h}
  4. ( - d Q h ) (-dQ_{h})
  5. d Q c = T c d S c dQ_{c}=T_{c}dS_{c}
  6. η = - d W - d Q h = - d Q h - d Q c - d Q h = 1 - d Q c - d Q h \eta=\frac{-dW}{-dQ_{h}}=\frac{-dQ_{h}-dQ_{c}}{-dQ_{h}}=1-\frac{dQ_{c}}{-dQ_{h}}
  7. d S c = - d S h dS_{c}=-dS_{h}
  8. η max = 1 - T c d S c - T h d S h = 1 - T c T h \eta\text{max}=1-\frac{T_{c}dS_{c}}{-T_{h}dS_{h}}=1-\frac{T_{c}}{T_{h}}
  9. T h T_{h}
  10. T c T_{c}
  11. d S c dS_{c}
  12. d S h dS_{h}
  13. η = 1 - T c T h \eta=1-\sqrt{\frac{T_{c}}{T_{h}}}
  14. T c T_{c}
  15. T h T_{h}
  16. η \eta
  17. η \eta
  18. η \eta
  19. Δ Q = T Δ S \Delta Q=T\Delta S

Heat_pump.html

  1. C O P heating = Δ Q hot Δ A T hot T hot - T cool , COP\text{heating}=\frac{\Delta Q\text{hot}}{\Delta A}\leq\frac{T\text{hot}}{T% \text{hot}-T\text{cool}},
  2. C O P cooling = Δ Q cool Δ A T cool T hot - T cool , COP\text{cooling}=\frac{\Delta Q\text{cool}}{\Delta A}\leq\frac{T\text{cool}}{% T\text{hot}-T\text{cool}},
  3. Δ Q cool \Delta Q\text{cool}
  4. T cool T\text{cool}
  5. Δ Q hot \Delta Q\text{hot}
  6. T hot T\text{hot}
  7. Δ A \Delta A

Hebrew_alphabet.html

  1. 0 \aleph_{0}
  2. \mathbb{Z}
  3. n \aleph_{n}
  4. n \beth_{n}
  5. 0 \aleph_{0}
  6. 1 \beth_{1}

Hebrew_calendar.html

  1. 1 / 3 {1}/{3}
  2. 1 / 3 {1}/{3}
  3. 765433 25920 \tfrac{765433}{25920}
  4. 1 / 2 {1}/{2}
  5. 1 / 19 {1}/{19}
  6. 15 18 \tfrac{15}{18}
  7. 9 18 \tfrac{9}{18}
  8. 4 18 \tfrac{4}{18}
  9. 16 18 \tfrac{16}{18}
  10. 10 18 \tfrac{10}{18}
  11. 5 18 \tfrac{5}{18}
  12. 17 18 \tfrac{17}{18}
  13. 11 18 \tfrac{11}{18}
  14. 6 18 \tfrac{6}{18}
  15. 13 18 \tfrac{13}{18}
  16. 7 18 \tfrac{7}{18}
  17. 1 18 \tfrac{1}{18}
  18. 14 18 \tfrac{14}{18}
  19. 8 18 \tfrac{8}{18}
  20. 2 18 \tfrac{2}{18}
  21. 15 18 \tfrac{15}{18}
  22. 9 18 \tfrac{9}{18}
  23. 3 18 \tfrac{3}{18}

Heine–Borel_theorem.html

  1. C T = C K { U } C_{T}=C_{K}\cup\{U\}
  2. C T , C_{T}^{\prime},
  3. C K = C T { U } , C_{K}^{\prime}=C_{T}^{\prime}\setminus\{U\},
  4. T 0 = [ - a , a ] n T_{0}=[-a,a]^{n}
  5. T 0 T 1 T 2 T k T_{0}\supset T_{1}\supset T_{2}\supset\ldots\supset T_{k}\supset\ldots
  6. C ( K ) C^{\infty}(K)
  7. K n K\subset\mathbb{R}^{n}
  8. \Rightarrow
  9. \Leftarrow
  10. I 0 I_{0}
  11. U I 0 \mbox{ }~{}_{U_{I_{0}}}
  12. U I 0 \mbox{ }~{}_{U_{I_{0}}}
  13. U I 0 \mbox{ }~{}_{U_{I_{0}}}
  14. \Rightarrow
  15. \Leftarrow
  16. \Leftarrow

Heisuke_Hironaka.html

  1. ν \nu
  2. τ \tau

Helianthus_annuus.html

  1. r = c n , r=c\sqrt{n},
  2. θ = n × 137.5 , \theta=n\times 137.5^{\circ},

Henri_Poincaré.html

  1. t = t - v x / c 2 t^{\prime}=t-vx/c^{2}\,
  2. ( ε ) \ell\left(\varepsilon\right)
  3. ε \varepsilon
  4. = 1 \ell=1
  5. x 2 + y 2 + z 2 - c 2 t 2 x^{2}+y^{2}+z^{2}-c^{2}t^{2}
  6. c t - 1 ct\sqrt{-1}
  7. γ m \gamma m

Henry_John_Stephen_Smith.html

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

Herbivore.html

  1. R = E f / ( T s + T h ) R=Ef/(Ts+Th)

Hertz.html

  1. ω = 2 π f \omega=2\pi f\,
  2. f = ω 2 π f=\frac{\omega}{2\pi}\,

Heterodyne.html

  1. sin θ sin φ = 1 2 cos ( θ - φ ) - 1 2 cos ( θ + φ ) \sin\theta\sin\varphi=\frac{1}{2}\cos(\theta-\varphi)-\frac{1}{2}\cos(\theta+\varphi)
  2. sin ( 2 π f 1 t ) \sin(2\pi f_{1}t)\,
  3. sin ( 2 π f 2 t ) \sin(2\pi f_{2}t)\,
  4. sin ( 2 π f 1 t ) sin ( 2 π f 2 t ) = 1 2 cos [ 2 π ( f 1 - f 2 ) t ] - 1 2 cos [ 2 π ( f 1 + f 2 ) t ] \sin(2\pi f_{1}t)\sin(2\pi f_{2}t)=\frac{1}{2}\cos[2\pi(f_{1}-f_{2})t]-\frac{1% }{2}\cos[2\pi(f_{1}+f_{2})t]\,
  5. F ( v 1 + v 2 ) = F ( v 1 ) + F ( v 2 ) F(v_{1}+v_{2})=F(v_{1})+F(v_{2})\,
  6. F ( v ) = α 1 v + α 2 v 2 + α 3 v 3 + F(v)=\alpha_{1}v+\alpha_{2}v^{2}+\alpha_{3}v^{3}+\ldots\,
  7. v out = F ( A 1 sin ω 1 t + A 2 sin ω 2 t ) v\text{out}=F(A_{1}\sin\omega_{1}t+A_{2}\sin\omega_{2}t)\,
  8. v out = α 1 ( A 1 sin ω 1 t + A 2 sin ω 2 t ) + α 2 ( A 1 sin ω 1 t + A 2 sin ω 2 t ) 2 + v\text{out}=\alpha_{1}(A_{1}\sin\omega_{1}t+A_{2}\sin\omega_{2}t)+\alpha_{2}(A% _{1}\sin\omega_{1}t+A_{2}\sin\omega_{2}t)^{2}+\ldots\,
  9. v out = α 1 ( A 1 sin ω 1 t + A 2 sin ω 2 t ) + α 2 ( A 1 2 sin 2 ω 1 t + 2 A 1 A 2 sin ω 1 t sin ω 2 t + A 2 2 sin 2 ω 2 t ) + v\text{out}=\alpha_{1}(A_{1}\sin\omega_{1}t+A_{2}\sin\omega_{2}t)+\alpha_{2}(A% _{1}^{2}\sin^{2}\omega_{1}t+2A_{1}A_{2}\sin\omega_{1}t\sin\omega_{2}t+A_{2}^{2% }\sin^{2}\omega_{2}t)+\ldots\,
  10. v out = α 1 ( A 1 sin ω 1 t + A 2 sin ω 2 t ) + α 2 ( A 1 2 2 [ 1 - cos 2 ω 1 t ] + A 1 A 2 [ cos ( ω 1 t - ω 2 t ) - cos ( ω 1 t + ω 2 t ) ] + A 2 2 2 [ 1 - cos 2 ω 2 t ] ) + v\text{out}=\alpha_{1}(A_{1}\sin\omega_{1}t+A_{2}\sin\omega_{2}t)+\alpha_{2}% \left(\frac{A_{1}^{2}}{2}[1-\cos 2\omega_{1}t]+A_{1}A_{2}[\cos(\omega_{1}t-% \omega_{2}t)-\cos(\omega_{1}t+\omega_{2}t)]+\frac{A_{2}^{2}}{2}[1-\cos 2\omega% _{2}t]\right)+\ldots\,
  11. v out = α 2 A 1 A 2 cos ( ω 1 - ω 2 ) t - α 2 A 1 A 2 cos ( ω 1 + ω 2 ) t + v\text{out}=\alpha_{2}A_{1}A_{2}\cos(\omega_{1}-\omega_{2})t-\alpha_{2}A_{1}A_% {2}\cos(\omega_{1}+\omega_{2})t+\ldots\,

Hexagon.html

  1. 2 3 3 \tfrac{2\sqrt{3}}{3}
  2. A = 3 3 2 t 2 2.598076211 t 2 . A=\frac{3\sqrt{3}}{2}t^{2}\simeq 2.598076211t^{2}.
  3. A = 3 2 d t A=\frac{3}{2}d\cdot t
  4. A = 3 2 d 2 0.866025404 d 2 . A=\frac{\sqrt{3}}{2}d^{2}\simeq 0.866025404d^{2}.
  5. A = a p / 2 A=ap/2
  6. A = 2 a 2 3 3.464102 a 2 , A\ =\ {2}a^{2}\sqrt{3}\ \simeq\ 3.464102a^{2},
  7. d = t 3 \scriptstyle d\ =\ t\sqrt{3}
  8. a + c + e = b + d + f . a+c+e=b+d+f.
  9. d 1 a 2 \frac{d_{1}}{a}\leq 2
  10. d 2 a > 3 . \frac{d_{2}}{a}>\sqrt{3}.

Hierarchy.html

  1. square quadrilateral polygon shape \,\text{square}\subset\,\text{quadrilateral}\subset\,\text{polygon}\subset\,% \text{shape}\,
  2. H. sapiens Homo Primates Mammalia Animalia \,\text{H. sapiens}\subset\,\text{Homo}\subset\,\text{Primates}\subset\,\text{% Mammalia}\subset\,\text{Animalia}
  3. square quadrilateral polygon shape \,\text{square}\subsetneq\,\text{quadrilateral}\subsetneq\,\text{polygon}% \subsetneq\,\text{shape}\,
  4. x y x\subsetneq y\,

High-pass_filter.html

  1. f c = 1 2 π τ = 1 2 π R C , f_{c}=\frac{1}{2\pi\tau}=\frac{1}{2\pi RC},\,
  2. f c = 1 2 π τ = 1 2 π R 1 C , f_{c}=\frac{1}{2\pi\tau}=\frac{1}{2\pi R_{1}C},\,
  3. { V out ( t ) = I ( t ) R (V) Q c ( t ) = C ( V in ( t ) - V out ( t ) ) (Q) I ( t ) = d Q c d t (I) \begin{cases}V_{\,\text{out}}(t)=I(t)\,R&\,\text{(V)}\\ Q_{c}(t)=C\,\left(V_{\,\text{in}}(t)-V_{\,\text{out}}(t)\right)&\,\text{(Q)}\\ I(t)=\frac{\operatorname{d}Q_{c}}{\operatorname{d}t}&\,\text{(I)}\end{cases}
  4. Q c ( t ) Q_{c}(t)
  5. t t
  6. V out ( t ) = C ( d V in d t - d V out d t ) I ( t ) R = R C ( d V in d t - d V out d t ) V_{\,\text{out}}(t)=\overbrace{C\,\left(\frac{\operatorname{d}V_{\,\text{in}}}% {\operatorname{d}t}-\frac{\operatorname{d}V_{\,\text{out}}}{\operatorname{d}t}% \right)}^{I(t)}\,R=RC\,\left(\frac{\operatorname{d}V_{\,\text{in}}}{% \operatorname{d}t}-\frac{\operatorname{d}V_{\,\text{out}}}{\operatorname{d}t}\right)
  7. Δ T \Delta_{T}
  8. V in V_{\,\text{in}}
  9. ( x 1 , x 2 , , x n ) (x_{1},x_{2},\ldots,x_{n})
  10. V out V_{\,\text{out}}
  11. ( y 1 , y 2 , , y n ) (y_{1},y_{2},\ldots,y_{n})
  12. y i = R C ( x i - x i - 1 Δ T - y i - y i - 1 Δ T ) y_{i}=RC\,\left(\frac{x_{i}-x_{i-1}}{\Delta_{T}}-\frac{y_{i}-y_{i-1}}{\Delta_{% T}}\right)
  13. y i = R C R C + Δ T y i - 1 Decaying contribution from prior inputs + R C R C + Δ T ( x i - x i - 1 ) Contribution from change in input y_{i}=\overbrace{\frac{RC}{RC+\Delta_{T}}y_{i-1}}^{\,\text{Decaying % contribution from prior inputs}}+\overbrace{\frac{RC}{RC+\Delta_{T}}\left(x_{i% }-x_{i-1}\right)}^{\,\text{Contribution from change in input}}
  14. y i = α y i - 1 + α ( x i - x i - 1 ) where α R C R C + Δ T y_{i}=\alpha y_{i-1}+\alpha(x_{i}-x_{i-1})\qquad\,\text{where}\qquad\alpha% \triangleq\frac{RC}{RC+\Delta_{T}}
  15. 0 α 1 0\leq\alpha\leq 1
  16. α \alpha
  17. R C RC
  18. Δ T \Delta_{T}
  19. α \alpha
  20. R C = Δ T ( α 1 - α ) RC=\Delta_{T}\left(\frac{\alpha}{1-\alpha}\right)
  21. f c = 1 2 π R C f_{c}=\frac{1}{2\pi RC}
  22. R C = 1 2 π f c RC=\frac{1}{2\pi f_{c}}
  23. α \alpha
  24. f c f_{c}
  25. α = 1 2 π Δ T f c + 1 \alpha=\frac{1}{2\pi\Delta_{T}f_{c}+1}
  26. f c = 1 - α 2 π α Δ T f_{c}=\frac{1-\alpha}{2\pi\alpha\Delta_{T}}
  27. α = 0.5 \alpha=0.5
  28. R C RC
  29. α 0.5 \alpha\ll 0.5
  30. R C RC
  31. R C α Δ T RC\approx\alpha\Delta_{T}
  32. n n
  33. R C RC
  34. R C RC
  35. R C RC
  36. R C RC
  37. R C RC
  38. R C RC
  39. C = 1 2 π f R = 1 6.28 × 5000 × 10 = 3.18 × 10 - 6 C=\frac{1}{2\pi fR}=\frac{1}{6.28\times 5000\times 10}=3.18\times 10^{-6}

High-speed_rail.html

  1. s = 1 0.031 × 1.016 t + 1 s={1\over 0.031\times 1.016^{t}+1}

Hilbert's_basis_theorem.html

  1. R R
  2. R [ X ] R[X]
  3. X X
  4. R R
  5. R R
  6. R R
  7. R [ X ] R[X]
  8. R R
  9. R [ X ] R[X]
  10. R R
  11. R [ X 1 , , X n ] R[X_{1},\ldots,X_{n}]
  12. R R
  13. R [ X ] R[X]
  14. 𝔞 R [ X ] \mathfrak{a}\subseteq R[X]
  15. { f 0 , f 1 , } \{f_{0},f_{1},\ldots\}
  16. 𝔟 n \mathfrak{b}_{n}
  17. f 0 , , f n - 1 f_{0},\ldots,f_{n-1}
  18. f n f_{n}
  19. 𝔞 𝔟 n \mathfrak{a}\setminus\mathfrak{b}_{n}
  20. { deg ( f 0 ) , deg ( f 1 ) , } \{\deg(f_{0}),\deg(f_{1}),\ldots\}
  21. a n a_{n}
  22. f n f_{n}
  23. 𝔟 \mathfrak{b}
  24. R R
  25. a 0 , a 1 , a_{0},a_{1},\ldots
  26. R R
  27. ( a 0 ) ( a 0 , a 1 ) ( a 0 , a 1 , a 2 ) (a_{0})\subset(a_{0},a_{1})\subset(a_{0},a_{1},a_{2})\ldots
  28. 𝔟 = ( a 0 , , a N - 1 ) \mathfrak{b}=(a_{0},\ldots,a_{N-1})
  29. N N
  30. a N = i < N u i a i , u i R . a_{N}=\sum_{i<N}u_{i}a_{i},\qquad u_{i}\in R.
  31. g = i < N u i X deg ( f N ) - deg ( f i ) f i , g=\sum_{i<N}u_{i}X^{\deg(f_{N})-\deg(f_{i})}f_{i},
  32. f N f_{N}
  33. g 𝔟 N g\in\mathfrak{b}_{N}
  34. f N 𝔟 N f_{N}\notin\mathfrak{b}_{N}
  35. f N - g 𝔞 𝔟 N f_{N}-g\in\mathfrak{a}\setminus\mathfrak{b}_{N}
  36. f N f_{N}
  37. 𝔞 R [ X ] \mathfrak{a}\subseteq R[X]
  38. 𝔟 \mathfrak{b}
  39. 𝔞 \mathfrak{a}
  40. R R
  41. 𝔞 \mathfrak{a}
  42. f 0 , , f N - 1 f_{0},\ldots,f_{N-1}
  43. d d
  44. { deg ( f 0 ) , , deg ( f N - 1 ) } \{\deg(f_{0}),\ldots,\deg(f_{N-1})\}
  45. 𝔟 k \mathfrak{b}_{k}
  46. 𝔞 \mathfrak{a}
  47. k {}\leq k
  48. 𝔟 k \mathfrak{b}_{k}
  49. R R
  50. 𝔞 \mathfrak{a}
  51. f 0 ( k ) , , f N ( k ) - 1 ( k ) , f^{(k)}_{0},\cdots,f^{(k)}_{N^{(k)}-1},
  52. k {}\leq k
  53. 𝔞 * R [ X ] \mathfrak{a}^{*}\subseteq R[X]
  54. { f i , f j ( k ) : i < N , j < N ( k ) , k < d } . \left\{f_{i},f^{(k)}_{j}\ :\ i<N,j<N^{(k)},k<d\right\}.
  55. 𝔞 * 𝔞 \mathfrak{a}^{*}\subseteq\mathfrak{a}
  56. 𝔞 𝔞 * \mathfrak{a}\subseteq\mathfrak{a}^{*}
  57. h 𝔞 𝔞 * h\in\mathfrak{a}\setminus\mathfrak{a}^{*}
  58. a a
  59. deg ( h ) d \deg(h)\geq d
  60. a 𝔟 a\in\mathfrak{b}
  61. a = j u j a j a=\sum_{j}u_{j}a_{j}
  62. f j f_{j}
  63. h 0 j u j X deg ( h ) - deg ( f j ) f j , h_{0}\triangleq\sum_{j}u_{j}X^{\deg(h)-\deg(f_{j})}f_{j},
  64. h h
  65. h 0 𝔞 * h_{0}\in\mathfrak{a}^{*}
  66. h 𝔞 * h\notin\mathfrak{a}^{*}
  67. h - h 0 𝔞 𝔞 * h-h_{0}\in\mathfrak{a}\setminus\mathfrak{a}^{*}
  68. deg ( h - h 0 ) < deg ( h ) \deg(h-h^{\prime}_{0})<\deg(h)
  69. deg ( h ) = k < d \deg(h)=k<d
  70. a 𝔟 k a\in\mathfrak{b}_{k}
  71. a = j u j a j ( k ) a=\sum_{j}u_{j}a^{(k)}_{j}
  72. f j ( k ) f^{(k)}_{j}
  73. h 0 j u j X deg ( h ) - deg ( f j ( k ) ) f j ( k ) , h_{0}\triangleq\sum_{j}u_{j}X^{\deg(h)-\deg(f^{(k)}_{j})}f^{(k)}_{j},
  74. 𝔞 = 𝔞 * \mathfrak{a}=\mathfrak{a}^{*}
  75. X X
  76. R R
  77. R [ X 0 , , X n - 1 ] R[X_{0},\ldots,X_{n-1}]
  78. R n R^{n}
  79. 𝔞 R [ X 0 , , X n - 1 ] \mathfrak{a}\subset R[X_{0},\ldots,X_{n-1}]
  80. A A
  81. R R
  82. A R [ X 0 , , X n - 1 ] / 𝔞 A\simeq R[X_{0},\ldots,X_{n-1}]/\mathfrak{a}
  83. 𝔞 \mathfrak{a}
  84. 𝔞 \mathfrak{a}
  85. 𝔞 = ( p 0 , , p N - 1 ) \mathfrak{a}=(p_{0},\ldots,p_{N-1})
  86. A A

Hilbert's_paradox_of_the_Grand_Hotel.html

  1. n n
  2. c c
  3. n n
  4. c c
  5. i i
  6. 2 i 2i
  7. 3 n 3^{n}
  8. 5 n 5^{n}
  9. c c
  10. p n p^{n}
  11. p p
  12. c c
  13. n n
  14. c c
  15. s s
  16. c c
  17. 2 s 3 c 2^{s}3^{c}
  18. 2 5 3 4 2^{5}3^{4}
  19. 2 s 3 c 5 n 7 e 2^{s}3^{c}5^{n}7^{e}
  20. n n
  21. c c
  22. ( n 2 + n ) / 2 (n^{2}+n)/2
  23. n n
  24. ( ( c + n - 1 ) 2 + c + n - 1 ) / 2 ((c+n-1)^{2}+c+n-1)/2
  25. ( c + n - 1 ) (c+n-1)
  26. n n
  27. 2 s 3 c 5 a 2^{s}3^{c}5^{a}
  28. a a
  29. 0 \aleph_{0}

Histogram.html

  1. n = i = 1 k m i . n=\sum_{i=1}^{k}{m_{i}}.
  2. M i = j = 1 i m j . M_{i}=\sum_{j=1}^{i}{m_{j}}.
  3. k = max x - min x h . k=\left\lceil\frac{\max x-\min x}{h}\right\rceil.
  4. k = n , k=\sqrt{n},\,
  5. k = log 2 n + 1 , k=\lceil\log_{2}n+1\rceil,\,
  6. k = 1 + log 2 ( n ) + log 2 ( 1 + | g 1 | σ g 1 ) k=1+\log_{2}(n)+\log_{2}\left(1+\frac{|g_{1}|}{\sigma_{g_{1}}}\right)
  7. g 1 g_{1}
  8. σ g 1 = 6 ( n - 2 ) ( n + 1 ) ( n + 3 ) \sigma_{g_{1}}=\sqrt{\frac{6(n-2)}{(n+1)(n+3)}}
  9. h = 3.5 σ ^ n 1 / 3 , h=\frac{3.5\hat{\sigma}}{n^{1/3}},
  10. σ ^ \hat{\sigma}
  11. arg min J ^ ( h ) = arg min 2 ( n - 1 ) h - n + 1 n 2 ( n - 1 ) h k N k 2 \underset{h}{\operatorname{arg\,min}}\hat{J}(h)=\underset{h}{\operatorname{arg% \,min}}\frac{2}{(n-1)h}-\frac{n+1}{n^{2}(n-1)h}\sum_{k}N_{k}^{2}
  12. N k N_{k}
  13. h = 2 IQR ( x ) n 1 / 3 , h=2\frac{\operatorname{IQR}(x)}{n^{1/3}},
  14. arg min 2 m ¯ - v h 2 \underset{h}{\operatorname{arg\,min}}\frac{2\bar{m}-v}{h^{2}}
  15. m ¯ \textstyle\bar{m}
  16. v \textstyle v
  17. h \textstyle h
  18. m ¯ = 1 k i = 1 k m i \textstyle\bar{m}=\frac{1}{k}\sum_{i=1}^{k}m_{i}
  19. v = 1 k i = 1 k ( m i - m ¯ ) 2 \textstyle v=\frac{1}{k}\sum_{i=1}^{k}(m_{i}-\bar{m})^{2}
  20. n 1 / 3 n^{1/3}
  21. n n
  22. n n
  23. s s
  24. n h / s nh/s
  25. s / ( n h ) \sqrt{s/(nh)}
  26. h / s h/s
  27. h h
  28. s / n 1 / 3 s/n^{1/3}
  29. k k
  30. n 1 / 3 n^{1/3}

History_of_geometry.html

  1. V = 1 3 h ( x 1 2 + x 1 x 2 + x 2 2 ) . V=\frac{1}{3}h(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}).
  2. ( 3 , 4 , 5 ) (3,4,5)
  3. ( 5 , 12 , 13 ) (5,12,13)
  4. ( 8 , 15 , 17 ) (8,15,17)
  5. ( 7 , 24 , 25 ) (7,24,25)
  6. ( 12 , 35 , 37 ) (12,35,37)
  7. A = ( s - a ) ( s - b ) ( s - c ) ( s - d ) A=\sqrt{(s-a)(s-b)(s-c)(s-d)}
  8. s = a + b + c + d 2 . s=\frac{a+b+c+d}{2}.
  9. a , b , c a,b,c
  10. a = u 2 v + v , b = u 2 w + w , c = u 2 v + u 2 w - ( v + w ) a=\frac{u^{2}}{v}+v,\ \ b=\frac{u^{2}}{w}+w,\ \ c=\frac{u^{2}}{v}+\frac{u^{2}}% {w}-(v+w)
  11. u , v , u,v,
  12. w w
  13. ( a , b , c ) (a,b,c)
  14. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  15. 3 2 + 4 2 = 5 2 3^{2}+4^{2}=5^{2}
  16. 8 2 + 15 2 = 17 2 8^{2}+15^{2}=17^{2}
  17. 12 2 + 35 2 = 37 2 12^{2}+35^{2}=37^{2}
  18. ( a , b , c ) (a,b,c)
  19. c 2 = a 2 + b 2 c^{2}=a^{2}+b^{2}
  20. a , b , c a,b,c
  21. 13500 2 + 12709 2 = 18541 2 13500^{2}+12709^{2}=18541^{2}

History_of_logic.html

  1. M M
  2. N N
  3. O O
  4. A A
  5. B B
  6. C C
  7. D D
  8. i i
  9. j j
  10. i i
  11. j j
  12. A A
  13. B B
  14. C C
  15. D D
  16. M M
  17. N N
  18. O O
  19. M M
  20. N N
  21. O O
  22. A A
  23. B B
  24. C C
  25. D D
  26. A A
  27. B B
  28. C C
  29. D D
  30. M M
  31. N N
  32. O O