wpmath0000002_19

Spheroid.html

  1. x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1
  2. x 2 + y 2 a 2 + z 2 c 2 = 1. \frac{x^{2}+y^{2}}{a^{2}}+\frac{z^{2}}{c^{2}}=1.
  3. S prolate = 2 π a 2 ( 1 + c a e sin - 1 e ) where e 2 = 1 - a 2 c 2 . S_{\rm prolate}=2\pi a^{2}\left(1+\frac{c}{ae}\sin^{-1}e\right)\qquad\mbox{% where}~{}\qquad e^{2}=1-\frac{a^{2}}{c^{2}}.
  4. S oblate S_{\rm oblate}
  5. ( 4 π / 3 ) a 2 c 4.19 a 2 c (4\pi/3)a^{2}c\approx 4.19\,a^{2}c
  6. ( π / 6 ) A 2 C 0.523 A 2 C (\pi/6)A^{2}C\approx 0.523\,A^{2}C
  7. σ ( β , λ ) = ( a cos β cos λ , a cos β sin λ , c sin β ) ; \vec{\sigma}(\beta,\lambda)=(a\cos\beta\cos\lambda,a\cos\beta\sin\lambda,c\sin% \beta);\,\!
  8. β \beta\,\!
  9. λ \lambda\,\!
  10. - π 2 < β < + π 2 -\frac{\pi}{2}<\beta<+\frac{\pi}{2}\,\!
  11. - π < λ < + π -\pi<\lambda<+\pi\,\!
  12. K ( β , λ ) = c 2 ( a 2 + ( c 2 - a 2 ) cos 2 β ) 2 ; K(\beta,\lambda)={c^{2}\over(a^{2}+(c^{2}-a^{2})\cos^{2}\beta)^{2}};\,\!
  13. H ( β , λ ) = c ( 2 a 2 + ( c 2 - a 2 ) cos 2 β ) 2 a ( a 2 + ( c 2 - a 2 ) cos 2 β ) 3 / 2 . H(\beta,\lambda)={c(2a^{2}+(c^{2}-a^{2})\cos^{2}\beta)\over 2a(a^{2}+(c^{2}-a^% {2})\cos^{2}\beta)^{3/2}}.\,\!
  14. f = a - b a = 1 - b : a . f=\frac{a-b}{a}=1-b:a.

Spin_glass.html

  1. d d
  2. H = - i j J i j S i S j , H=-\sum_{\langle ij\rangle}J_{ij}S_{i}S_{j},
  3. S i S_{i}
  4. i i
  5. J i j J_{ij}
  6. i i
  7. j j
  8. J i j J_{ij}
  9. f [ J i j ] = - 1 β ln 𝒵 [ J i j ] f[J_{ij}]=-\dfrac{1}{\beta}\ln\mathcal{Z}\left[J_{ij}\right]
  10. 𝒵 [ J i j ] = Tr S e ( - β H ) \mathcal{Z}\left[J_{ij}\right]=\operatorname{Tr}_{S}e^{\left(-\beta H\right)}
  11. J i j J_{ij}
  12. J i j J_{ij}
  13. J 0 J_{0}
  14. J 2 J^{2}
  15. P ( J i j ) = N 2 π J 2 exp { - N 2 J 2 ( J i j - J 0 N ) 2 } . P(J_{ij})=\sqrt{\dfrac{N}{2\pi J^{2}}}\exp\left\{-\dfrac{N}{2J^{2}}\left(J_{ij% }-\dfrac{J_{0}}{N}\right)^{2}\right\}.
  16. m = 0 m=0
  17. q = i = 1 N S i α S i β 0 q=\sum_{i=1}^{N}S^{\alpha}_{i}S^{\beta}_{i}\neq 0
  18. α , β \alpha,\beta
  19. q q
  20. q q
  21. m m
  22. q q
  23. β f = \displaystyle\beta f=
  24. H = - i < j J i j S i S j H=-\sum_{i<j}J_{ij}S_{i}S_{j}
  25. J i j , S i , S j J_{ij},S_{i},S_{j}
  26. r r
  27. r N r\leq N
  28. N N
  29. H = - i 1 < i 2 < < i r J i 1 i r S i 1 S i r H=-\sum_{i_{1}<i_{2}<\cdots<i_{r}}J_{i_{1}\dots i_{r}}S_{i_{1}}\cdots S_{i_{r}}
  30. J i 1 i r , S i 1 , , S i r J_{i_{1}\dots i_{r}},S_{i_{1}},\dots,S_{i_{r}}
  31. r r\to\infty
  32. J 0 N \dfrac{J_{0}}{N}
  33. J 2 N \dfrac{J^{2}}{N}
  34. P ( J i 1 i r ) = N r - 1 J 2 π r ! exp { - N r - 1 J 2 r ! ( J i 1 i r - J 0 r ! 2 N r - 1 ) } P(J_{i_{1}\cdots i_{r}})=\sqrt{\dfrac{N^{r-1}}{J^{2}\pi r!}}\exp\left\{-\dfrac% {N^{r-1}}{J^{2}r!}\left(J_{i_{1}\cdots i_{r}}-\dfrac{J_{0}r!}{2N^{r-1}}\right)\right\}
  35. m m
  36. q q
  37. m m
  38. q q
  39. β f \displaystyle\beta f
  40. T f T_{f}
  41. T c T_{c}
  42. T f T_{f}

Spiral_galaxy.html

  1. I ( r ) = I 0 e - r / R D I(r)=I_{0}e^{-r/R_{D}}
  2. R D R_{D}
  3. I 0 I_{0}
  4. R o p t = 3.2 R D R_{opt}=3.2R_{D}
  5. L t o t = 2 π I 0 R D 2 L_{tot}=2\pi I_{0}R^{2}_{D}
  6. r / R D r/R_{D}

Splitting_field.html

  1. p ( X ) = i = 1 deg ( p ) ( X - a i ) p(X)=\prod_{i=1}^{\deg(p)}(X-a_{i})
  2. i i
  3. ( X - a i ) L [ X ] (X-a_{i})\in L[X]
  4. F = K 0 , K 1 , K r - 1 , K r = K F=K_{0},K_{1},\ldots K_{r-1},K_{r}=K
  5. f 1 ( X ) f 2 ( X ) f k ( X ) f_{1}(X)f_{2}(X)\cdots f_{k}(X)
  6. π : K i [ X ] K i [ X ] / ( f ( X ) ) \pi:K_{i}[X]\to K_{i}[X]/(f(X))
  7. f ( π ( X ) ) = π ( f ( X ) ) = f ( X ) mod f ( X ) = 0 f(\pi(X))=\pi(f(X))=f(X)\ \bmod\ f(X)=0
  8. [ K i + 1 : K i ] [K_{i+1}:K_{i}]
  9. [ K r : K r - 1 ] [ K 2 : K 1 ] [ K 1 : F ] [K_{r}:K_{r-1}]\cdots[K_{2}:K_{1}][K_{1}:F]
  10. c n - 1 α n - 1 + c n - 2 α n - 2 + + c 1 α + c 0 c_{n-1}\alpha^{n-1}+c_{n-2}\alpha^{n-2}+\cdots+c_{1}\alpha+c_{0}
  11. f ( X ) = X n + b n - 1 X n - 1 + + b 1 X + b 0 . f(X)=X^{n}+b_{n-1}X^{n-1}+\cdots+b_{1}X+b_{0}.
  12. α n = - ( b n - 1 α n - 1 + + b 1 α + b 0 ) . \alpha^{n}=-(b_{n-1}\alpha^{n-1}+\cdots+b_{1}\alpha+b_{0}).
  13. α n α m - n = - ( b n - 1 α n - 1 + + b 1 α + b 0 ) α m - n = - ( b n - 1 α m - 1 + + b 1 α m - n + 1 + b 0 α m - n ) \alpha^{n}\alpha^{m-n}=-\left(b_{n-1}\alpha^{n-1}+\cdots+b_{1}\alpha+b_{0}% \right)\alpha^{m-n}=-\left(b_{n-1}\alpha^{m-1}+\cdots+b_{1}\alpha^{m-n+1}+b_{0% }\alpha^{m-n}\right)
  14. g ( α ) = α 5 + α 2 g(\alpha)=\alpha^{5}+\alpha^{2}
  15. g ( α ) h ( α ) = ( α 5 + α 2 ) ( α 3 + 1 ) = α 8 + 2 α 5 + α 2 = ( α 7 ) α + 2 α 5 + α 2 = 2 α 5 + α 2 + 2 α . g(\alpha)h(\alpha)=\left(\alpha^{5}+\alpha^{2}\right)\left(\alpha^{3}+1\right)% =\alpha^{8}+2\alpha^{5}+\alpha^{2}=\left(\alpha^{7}\right)\alpha+2\alpha^{5}+% \alpha^{2}=2\alpha^{5}+\alpha^{2}+2\alpha.
  16. ( a 1 + b 1 x ) + ( a 2 + b 2 x ) = ( a 1 + a 2 ) + ( b 1 + b 2 ) x , (a_{1}+b_{1}x)+(a_{2}+b_{2}x)=(a_{1}+a_{2})+(b_{1}+b_{2})x,
  17. ( a 1 + b 1 x ) ( a 2 + b 2 x ) = a 1 a 2 + ( a 1 b 2 + b 1 a 2 ) x + ( b 1 b 2 ) x 2 ( a 1 a 2 - b 1 b 2 ) + ( a 1 b 2 + b 1 a 2 ) x . (a_{1}+b_{1}x)(a_{2}+b_{2}x)=a_{1}a_{2}+(a_{1}b_{2}+b_{1}a_{2})x+(b_{1}b_{2})x% ^{2}\equiv(a_{1}a_{2}-b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})x\,.
  18. ( a 1 , b 1 ) + ( a 2 , b 2 ) = ( a 1 + a 2 , b 1 + b 2 ) , (a_{1},b_{1})+(a_{2},b_{2})=(a_{1}+a_{2},b_{1}+b_{2}),
  19. ( a 1 , b 1 ) ( a 2 , b 2 ) = ( a 1 a 2 - b 1 b 2 , a 1 b 2 + b 1 a 2 ) . (a_{1},b_{1})\cdot(a_{2},b_{2})=(a_{1}a_{2}-b_{1}b_{2},a_{1}b_{2}+b_{1}a_{2}).
  20. ( a 1 + i b 1 ) + ( a 2 + i b 2 ) = ( a 1 + a 2 ) + i ( b 1 + b 2 ) , (a_{1}+ib_{1})+(a_{2}+ib_{2})=(a_{1}+a_{2})+i(b_{1}+b_{2}),
  21. ( a 1 + i b 1 ) ( a 2 + i b 2 ) = ( a 1 a 2 - b 1 b 2 ) + i ( a 1 b 2 + a 2 b 1 ) . (a_{1}+ib_{1})\cdot(a_{2}+ib_{2})=(a_{1}a_{2}-b_{1}b_{2})+i(a_{1}b_{2}+a_{2}b_% {1}).
  22. ( a 1 , b 1 ) + ( a 2 , b 2 ) = ( a 1 + a 2 , b 1 + b 2 ) , (a_{1},b_{1})+(a_{2},b_{2})=(a_{1}+a_{2},b_{1}+b_{2}),
  23. ( a 1 , b 1 ) ( a 2 , b 2 ) = ( a 1 a 2 - b 1 b 2 , a 1 b 2 + b 1 a 2 ) . (a_{1},b_{1})\cdot(a_{2},b_{2})=(a_{1}a_{2}-b_{1}b_{2},a_{1}b_{2}+b_{1}a_{2})\,.
  24. 2 3 \sqrt[3]{2}
  25. ω 1 = 1 \omega_{1}=1\,
  26. ω 2 = - 1 2 + 3 2 i , \omega_{2}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i,
  27. ω 3 = - 1 2 - 3 2 i . \omega_{3}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i.
  28. ω 3 = 1 / ω 2 \omega_{3}=1/\omega_{2}
  29. L = 𝐐 ( 2 3 , ω 2 ) = { a + b ω 2 + c 2 3 + d 2 3 ω 2 + e 2 2 3 + f 2 2 3 ω 2 | a , b , c , d , e , f 𝐐 } {L=\mathbf{Q}(\sqrt[3]{2},\omega_{2})=\{a+b\omega_{2}+c\sqrt[3]{2}+d\sqrt[3]{2% }\omega_{2}+e\sqrt[3]{2^{2}}+f\sqrt[3]{2^{2}}\omega_{2}\,|\,a,b,c,d,e,f\in% \mathbf{Q}\}}

Splitting_lemma.html

  1. 0 A 𝑞 B 𝑟 C 0 0\rightarrow A\overset{q}{\longrightarrow}B\overset{r}{\longrightarrow}C\rightarrow 0
  2. 0 A q t B r u C 0. 0\rightarrow A{{q\atop\longrightarrow}\atop{\longleftarrow\atop t}}B{{r\atop% \longrightarrow}\atop{\longleftarrow\atop u}}C\rightarrow 0.
  3. C B / q ( A ) C\cong B/q(A)
  4. B = q ( A ) u ( C ) A C B=q(A)\oplus u(C)\cong A\oplus C
  5. V ker T im T V\approx\ker T\oplus\operatorname{im}\,T
  6. t × r : B A × C t\times r\colon B\to A\times C
  7. C A × C C\to A\times C
  8. C B C\to B
  9. B S 3 B\cong S_{3}
  10. C = B / A { ± 1 } C=B/A\cong\{\pm 1\}
  11. 0 A q B r C 0 0\rightarrow A\stackrel{q}{\longrightarrow}B\stackrel{r}{\longrightarrow}C% \rightarrow 0\,
  12. S 3 S_{3}
  13. S 3 S_{3}

Spoke-hub_distribution_paradigm.html

  1. n ( n - 1 ) 2 \frac{n(n-1)}{2}

Spoke.html

  1. l = d 2 + r 1 2 + r 2 2 - 2 r 1 r 2 cos ( a ) - r 3 l=\sqrt{d^{2}+{r_{1}}^{2}+{r_{2}}^{2}-2\,r_{1}r_{2}\cos(a)}-r_{3}
  2. l 2 = ( r 2 - r 1 ) 2 + d 2 l^{2}=(r_{2}-r_{1})^{2}+d^{2}
  3. l = ( d - r 3 ) 2 + ( r 2 - r 1 ) 2 . l=\sqrt{(d-r_{3})^{2}+(r_{2}-r_{1})^{2}}.

Spontaneous_process.html

  1. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,

Spontaneous_symmetry_breaking.html

  1. = μ ϕ μ ϕ - V ( ϕ ) . \mathcal{L}=\partial^{\mu}\phi\partial_{\mu}\phi-V(\phi).

Spring_(device).html

  1. F = - k x , F=-kx,
  2. F = m a - k x = m a . F=ma\quad\Rightarrow\quad-kx=ma.\,
  3. - k x = m d 2 x d t 2 . -kx=m\frac{d^{2}x}{dt^{2}}.\,
  4. x x
  5. d 2 x d t 2 + k m x = 0 , \frac{d^{2}x}{dt^{2}}+\frac{k}{m}x=0,\,
  6. x ( t ) = A sin ( t k m ) + B cos ( t k m ) . x(t)=A\sin\left(t\sqrt{\frac{k}{m}}\right)+B\cos\left(t\sqrt{\frac{k}{m}}% \right).\,
  7. A A
  8. B B
  9. B = 0 B=0
  10. F m a x = E d 4 ( L - n d ) 16 ( 1 + ν ) ( D - d ) 3 n F_{max}=\frac{Ed^{4}(L-nd)}{16(1+\nu)(D-d)^{3}n}
  11. ν \nu

Square_matrix.html

  1. [ a 11 0 0 0 a 22 0 0 0 a 33 ] \begin{bmatrix}a_{11}&0&0\\ 0&a_{22}&0\\ 0&0&a_{33}\end{bmatrix}
  2. [ a 11 0 0 a 21 a 22 0 a 31 a 32 a 33 ] \begin{bmatrix}a_{11}&0&0\\ a_{21}&a_{22}&0\\ a_{31}&a_{32}&a_{33}\end{bmatrix}
  3. [ a 11 a 12 a 13 0 a 22 a 23 0 0 a 33 ] \begin{bmatrix}a_{11}&a_{12}&a_{13}\\ 0&a_{22}&a_{23}\\ 0&0&a_{33}\end{bmatrix}
  4. I 1 = [ 1 ] , I 2 = [ 1 0 0 1 ] , , I n = [ 1 0 0 0 1 0 0 0 1 ] . I_{1}=\begin{bmatrix}1\end{bmatrix},\ I_{2}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\ \cdots,\ I_{n}=\begin{bmatrix}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{bmatrix}.
  5. [ 1 / 4 0 0 1 / 4 ] \begin{bmatrix}1/4&0\\ 0&1/4\\ \end{bmatrix}
  6. [ 1 / 4 0 0 - 1 / 4 ] \begin{bmatrix}1/4&0\\ 0&-1/4\end{bmatrix}
  7. A T = A - 1 , A^{\mathrm{T}}=A^{-1},\,
  8. A T A = A A T = I , A^{\mathrm{T}}A=AA^{\mathrm{T}}=I,\,
  9. tr ( 𝖠𝖡 ) = i = 1 m j = 1 n A i j B j i = tr ( 𝖡𝖠 ) . \scriptstyle\operatorname{tr}(\mathsf{AB})=\sum_{i=1}^{m}\sum_{j=1}^{n}A_{ij}B% _{ji}=\operatorname{tr}(\mathsf{BA}).
  10. det [ a b c d ] = a d - b c . \det\begin{bmatrix}a&b\\ c&d\end{bmatrix}=ad-bc.
  11. det ( 𝖠 - λ 𝖨 ) = 0. \det(\mathsf{A}-\lambda\mathsf{I})=0.

Square_number.html

  1. 3 × 3 3 × 3
  2. n n
  3. n × n n×n
  4. n n
  5. 9 \sqrt{9}
  6. n n
  7. n n
  8. m \lfloor\sqrt{m}\rfloor
  9. m m
  10. x \lfloor x\rfloor
  11. x x
  12. n 2 - ( n - 1 ) 2 = 2 n - 1 n^{2}-(n-1)^{2}=2n-1
  13. n 2 = ( n - 1 ) 2 + ( n - 1 ) + n n^{2}=(n-1)^{2}+(n-1)+n
  14. m m
  15. m m
  16. m = 1 < s u p > 2 = 1 m=1<sup>2=1
  17. 1 × 1 1 × 1
  18. n n
  19. n n
  20. n n
  21. n 2 = k = 1 n ( 2 k - 1 ) . n^{2}=\sum_{k=1}^{n}(2k-1).
  22. n n
  23. n 2 = ( n - 1 ) 2 + ( n - 1 ) + n = ( n - 1 ) 2 + ( 2 n - 1 ) n^{2}=(n-1)^{2}+(n-1)+n=(n-1)^{2}+(2n-1)
  24. n n
  25. ( n 1 ) (n− 1)
  26. ( n 2 ) (n− 2)
  27. 4 k + 3 4k+3
  28. p p
  29. m m
  30. p p
  31. m m
  32. p p
  33. m p m∕p
  34. m m
  35. m m
  36. m m
  37. k k
  38. n n
  39. k n k−n
  40. m m
  41. 100 3 100−3
  42. k n k−n
  43. k + n k+n
  44. k k
  45. k k≥
  46. m \sqrt{m}
  47. n = 0 N n 2 = 0 2 + 1 2 + 2 2 + 3 2 + 4 2 + + N 2 \sum_{n=0}^{N}n^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+\cdots+N^{2}
  48. N ( N + 1 ) ( 2 N + 1 ) 6 . \frac{N(N+1)(2N+1)}{6}.
  49. n = m × ( m + 1 ) n=m×(m+1)
  50. n = 6 × ( 6 + 1 ) = 42 n=6×(6+1)=42
  51. n = m < s u p > 2 n=m<sup>2

Square_wave.html

  1. f f
  2. t t
  3. x square ( t ) = 4 π k = 1 sin ( 2 π ( 2 k - 1 ) f t ) ( 2 k - 1 ) = 4 π ( sin ( 2 π f t ) + 1 3 sin ( 6 π f t ) + 1 5 sin ( 10 π f t ) + ) \begin{aligned}\displaystyle x_{\mathrm{square}}(t)&\displaystyle{}=\frac{4}{% \pi}\sum_{k=1}^{\infty}{\sin{\left(2\pi(2k-1)ft\right)}\over(2k-1)}\\ &\displaystyle{}=\frac{4}{\pi}\left(\sin(2\pi ft)+{1\over 3}\sin(6\pi ft)+{1% \over 5}\sin(10\pi ft)+\cdots\right)\end{aligned}
  4. 2 π ( 2 k - 1 ) f 2π(2k-1)f
  5. x ( t ) = sgn ( sin [ t ] ) \ x(t)=\operatorname{sgn}(\sin[t])
  6. v ( t ) = sgn ( cos [ t ] ) \ v(t)=\operatorname{sgn}(\cos[t])
  7. x ( t ) = n = - + ( t - n T ) = n = - + ( u [ t - n T + 1 2 ] - u [ t - n T - 1 2 ] ) \ x(t)=\sum_{n=-\infty}^{+\infty}\sqcap(t-nT)=\sum_{n=-\infty}^{+\infty}\left(% u\left[t-nT+{1\over 2}\right]-u\left[t-nT-{1\over 2}\right]\right)
  8. x ( t ) = { 1 , | t | < T 1 0 , T 1 < | t | 1 2 T \ x(t)=\begin{cases}1,&|t|<T_{1}\\ 0,&T_{1}<|t|\leq{1\over 2}T\end{cases}
  9. x ( t + T ) = x ( t ) \ x(t+T)=x(t)
  10. y ( x ) = a × csc ( 2 π p x ) | sin ( 2 π p x ) | y(x)=a\times\csc\left(\frac{2\pi}{p}x\right)\left|\sin\left(\frac{2\pi}{p}x% \right)\right|
  11. y ( x ) = m ( 2 ν x - 2 ν x + 1 ) y(x)=m\left(2\lfloor\nu x\rfloor-\lfloor 2\nu x\rfloor+1\right)
  12. y ( x ) = m ( - 1 ) ν x , y(x)=m\left(-1\right)^{\lfloor\nu x\rfloor},

Squaring_the_circle.html

  1. π \pi
  2. π \pi
  3. d d
  4. d d
  5. π \scriptstyle\sqrt{\pi}
  6. 355 113 = 3.1415929203539823008 \tfrac{355}{113}=3.1415929203539823008\dots
  7. ( 9 2 + 19 2 22 ) 1 / 4 = 2143 22 4 = 3.1415926525826461252 \left(9^{2}+\frac{19^{2}}{22}\right)^{1/4}=\sqrt[4]{\frac{2143}{22}}=3.1415926% 525826461252\dots
  8. 6 5 ( 1 + φ ) and 40 3 - 2 3 = 3.14153333870509461863 \frac{6}{5}(1+\varphi)\,\text{ and }\sqrt{{40\over 3}-2\sqrt{3}\ }=3.141533338% 70509461863\dots
  9. 2 25 36 + ( 1 6 + 348 + 5 143 36 ( 72 + 143 ) ) 2 = 1.77245384141934376 2\cdot\sqrt{\frac{25}{36}+\left(\frac{1}{6}+\frac{348+5\sqrt{143}}{36\left(72+% \sqrt{143}\right)}\right)^{2}}=1{.}77245384141934376\dots
  10. 1.77245384141934376 2 = 3.141592619962188 1{.}77245384141934376\dots^{2}=3.141592619962188\dots
  11. π \sqrt{\pi}
  12. π . \pi.

St._Stephen's_Cathedral,_Vienna.html

  1. 3 / 2 \sqrt{3}/2

Stall_(fluid_mechanics).html

  1. L = n W L=nW
  2. L L
  3. n n
  4. W W
  5. V s t = V s n V_{st}=V_{s}\sqrt{n}
  6. V s t V_{st}
  7. V s V_{s}
  8. n n
  9. L L
  10. W W
  11. n \sqrt{n}

Standard_ML.html

  1. f ( x ) = x 3 - x - 1 f(x)=x^{3}-x-1
  2. x = 3 x=3
  3. f ( x ) = 3 x 2 - 1 f^{\prime}(x)=3x^{2}-1
  4. f ( 3 ) = 27 - 1 = 26 f^{\prime}(3)=27-1=26

Standard_molar_entropy.html

  1. S = k = 1 N Δ S k = k = 1 N d q k T d T S^{\circ}=\sum_{k=1}^{N}\Delta S_{k}=\sum_{k=1}^{N}\int\frac{dq_{k}}{T}\,dT

Standard_score.html

  1. z = x - μ σ z={x-\mu\over\sigma}
  2. γ \gamma
  3. P ( L < X < U ) = γ , P(L<X<U)=\gamma,
  4. P ( L - μ σ < Z < U - μ σ ) = γ . P\left(\frac{L-\mu}{\sigma}<Z<\frac{U-\mu}{\sigma}\right)=\gamma.
  5. P ( - z < Z < z ) = γ P\left(-z<Z<z\right)=\gamma
  6. L = μ - z σ , U = μ + z σ L=\mu-z\sigma,\ U=\mu+z\sigma
  7. E [ X ] \operatorname{E}[X]
  8. σ ( X ) = Var ( X ) : \sigma(X)=\sqrt{\operatorname{Var}(X)}:
  9. Z = X - E [ X ] σ ( X ) Z={X-\operatorname{E}[X]\over\sigma(X)}
  10. X 1 , , X n \ X_{1},\dots,X_{n}
  11. X ¯ = 1 n i = 1 n X i \bar{X}={1\over n}\sum_{i=1}^{n}X_{i}
  12. Z = X ¯ - E [ X ] σ ( X ) / n . Z=\frac{\bar{X}-\operatorname{E}[X]}{\sigma(X)/\sqrt{n}}.

Standardized_moment.html

  1. μ k σ k \frac{\mu_{k}}{\sigma^{k}}\!
  2. μ k \mu_{k}
  3. x k x^{k}
  4. μ k ( λ X ) = λ k μ k ( X ) \mu_{k}(\lambda X)=\lambda^{k}\mu_{k}(X)
  5. σ μ \frac{\sigma}{\mu}
  6. μ \mu

Statistical_power.html

  1. power = ( reject H 0 | H 1 is true ) \mbox{power}~{}=\mathbb{P}\big(\mbox{reject }~{}H_{0}\big|H_{1}\mbox{ is true}% ~{}\big)
  2. A i A_{i}
  3. B i B_{i}
  4. D i = B i - A i D_{i}=B_{i}-A_{i}
  5. H 0 : μ D = 0 H_{0}:\mu_{D}=0
  6. H 1 : μ D > 0 H_{1}:\mu_{D}>0
  7. T n = D ¯ n - 0 σ ^ D / n . T_{n}=\frac{\bar{D}_{n}-0}{\hat{\sigma}_{D}/\sqrt{n}}.
  8. D ¯ n = 1 n i = 1 n D i \bar{D}_{n}=\frac{1}{n}\sum_{i=1}^{n}D_{i}
  9. σ ^ D / n \hat{\sigma}_{D}/\sqrt{n}
  10. α = 0.05 \alpha=0.05
  11. Φ \Phi
  12. T n > 1.64. T_{n}>1.64.
  13. μ D = θ \mu_{D}=\theta
  14. B ( θ ) = P ( T n > 1.64 | μ D = θ ) = P ( D ¯ n - 0 σ ^ D / n > 1.64 | μ D = θ ) = P ( D ¯ n - θ + θ σ ^ D / n > 1.64 | μ D = θ ) = P ( D ¯ n - θ σ ^ D / n > 1.64 - θ σ ^ D / n | μ D = θ ) = P ( D ¯ n - θ σ ^ D / n > 1.64 - θ σ ^ D / n | μ D = θ ) = 1 - P ( D ¯ n - θ σ ^ D / n < 1.64 - θ σ ^ D / n | μ D = θ ) \begin{array}[]{ccl}B(\theta)&=&P(T_{n}>1.64|\mu_{D}=\theta)\\ &=&P(\frac{\bar{D}_{n}-0}{\hat{\sigma}_{D}/\sqrt{n}}>1.64|\mu_{D}=\theta)\\ &=&P\left(\frac{\bar{D}_{n}-\theta+\theta}{\hat{\sigma}_{D}/\sqrt{n}}>1.64% \right|\mu_{D}=\theta)\\ &=&P\left(\frac{\bar{D}_{n}-\theta}{\hat{\sigma}_{D}/\sqrt{n}}>1.64-\frac{% \theta}{\hat{\sigma}_{D}/\sqrt{n}}\right|\mu_{D}=\theta)\\ &=&P\left(\frac{\bar{D}_{n}-\theta}{\hat{\sigma}_{D}/\sqrt{n}}>1.64-\frac{% \theta}{\hat{\sigma}_{D}/\sqrt{n}}\right|\mu_{D}=\theta)\\ &=&1-P\left(\frac{\bar{D}_{n}-\theta}{\hat{\sigma}_{D}/\sqrt{n}}<1.64-\frac{% \theta}{\hat{\sigma}_{D}/\sqrt{n}}\right|\mu_{D}=\theta)\\ \end{array}
  15. T n T_{n}
  16. B ( θ ) 1 - Φ ( 1.64 - θ σ ^ D / n ) . B(\theta)\approx 1-\Phi(1.64-\frac{\theta}{\hat{\sigma}_{D}/\sqrt{n}}).
  17. θ \theta
  18. θ \theta
  19. θ \theta
  20. θ \theta
  21. α \alpha
  22. θ = 0 \theta=0
  23. θ > 1 \theta>1
  24. B ( 1 ) 1 - Φ ( 1.64 - n / σ ^ D ) > 0.90 , B(1)\approx 1-\Phi(1.64-\sqrt{n}/\hat{\sigma}_{D})>0{.}90\ ,
  25. Φ ( 1.64 - n / σ ^ D ) < 0.10. \Phi(1.64-\sqrt{n}/\hat{\sigma}_{D})<0.10.
  26. n / σ ^ D > 1.64 - z 0.10 = 1.64 + 1.28 3 \sqrt{n}/\hat{\sigma}_{D}>1.64-z_{0.10}=1.64+1.28\approx 3
  27. n > 9 σ ^ D 2 , n>9\,\hat{\sigma}_{D}^{2},
  28. z 0.10 z_{0.10}
  29. Φ \Phi

Steam_locomotive.html

  1. t = c P d 2 s D t=\frac{cPd^{2}s}{D}

Steinhaus–Moser_notation.html

  1. M ( n , 1 , 3 ) = n n M(n,1,3)=n^{n}
  2. M ( n , 1 , p + 1 ) = M ( n , n , p ) M(n,1,p+1)=M(n,n,p)
  3. M ( n , m + 1 , p ) = M ( M ( n , 1 , p ) , m , p ) M(n,m+1,p)=M(M(n,1,p),m,p)
  4. M ( 2 , 1 , 5 ) M(2,1,5)
  5. M ( 10 , 1 , 5 ) M(10,1,5)
  6. M ( 2 , 1 , M ( 2 , 1 , 5 ) ) M(2,1,M(2,1,5))
  7. f ( x ) = x x f(x)=x^{x}
  8. f 256 ( 256 ) = f 258 ( 2 ) f^{256}(256)=f^{258}(2)
  9. ( 256 256 ) 256 256 = 256 256 257 (256^{\,\!256})^{256^{256}}=256^{256^{257}}
  10. ( 256 256 257 ) 256 256 257 = 256 256 257 × 256 256 257 = 256 256 257 + 256 257 (256^{\,\!256^{257}})^{256^{256^{257}}}=256^{256^{257}\times 256^{256^{257}}}=% 256^{256^{257+256^{257}}}
  11. 256 256 256 257 256^{\,\!256^{256^{257}}}
  12. 256 256 256 256 257 {\,\!256^{256^{256^{256^{257}}}}}
  13. 256 256 256 256 256 257 {\,\!256^{256^{256^{256^{256^{257}}}}}}
  14. M ( 256 , 256 , 3 ) ( 256 ) 256 257 M(256,256,3)\approx(256\uparrow)^{256}257
  15. ( 256 ) 256 (256\uparrow)^{256}
  16. f ( n ) = 256 n f(n)=256^{n}
  17. 256 257 256\uparrow\uparrow 257
  18. n n n^{n}
  19. 256 n 256^{n}
  20. 10 n 10^{n}
  21. M ( 256 , 1 , 3 ) 3.23 × 10 616 M(256,1,3)\approx 3.23\times 10^{616}
  22. M ( 256 , 2 , 3 ) 10 1.99 × 10 619 M(256,2,3)\approx 10^{\,\!1.99\times 10^{619}}
  23. log 10 616 \log_{10}616
  24. M ( 256 , 3 , 3 ) 10 10 1.99 × 10 619 M(256,3,3)\approx 10^{\,\!10^{1.99\times 10^{619}}}
  25. 619 619
  26. 1.99 × 10 619 1.99\times 10^{619}
  27. M ( 256 , 4 , 3 ) 10 10 10 1.99 × 10 619 M(256,4,3)\approx 10^{\,\!10^{10^{1.99\times 10^{619}}}}
  28. M ( 256 , 256 , 3 ) ( 10 ) 255 1.99 × 10 619 M(256,256,3)\approx(10\uparrow)^{255}1.99\times 10^{619}
  29. ( 10 ) 255 (10\uparrow)^{255}
  30. f ( n ) = 10 n f(n)=10^{n}
  31. 10 257 < mega < 10 258 10\uparrow\uparrow 257<\,\text{mega}<10\uparrow\uparrow 258
  32. moser < 3 3 4 2 , \mathrm{moser}<3\rightarrow 3\rightarrow 4\rightarrow 2,
  33. moser < f 3 ( 4 ) = f ( f ( f ( 4 ) ) ) , where f ( n ) = 3 n 3. \mathrm{moser}<f^{3}(4)=f(f(f(4))),\,\text{ where }f(n)=3\uparrow^{n}3.
  34. moser 3 3 64 2 < f 64 ( 4 ) = Graham’s number . \mathrm{moser}\ll 3\rightarrow 3\rightarrow 64\rightarrow 2<f^{64}(4)=\,\text{% Graham's number}.

Stellar_parallax.html

  1. d ( pc ) = 1 / p ( arcsec ) . d(\mathrm{pc})=1/p(\mathrm{arcsec}).

Stellar_wind.html

  1. M ˙ > 10 - 3 \scriptstyle\dot{M}>10^{-3}
  2. M ˙ < 10 - 6 \scriptstyle\dot{M}<10^{-6}

Step_function.html

  1. f : f:\mathbb{R}\rightarrow\mathbb{R}
  2. f ( x ) = i = 0 n α i χ A i ( x ) f(x)=\sum\limits_{i=0}^{n}\alpha_{i}\chi_{A_{i}}(x)\,
  3. x x
  4. n 0 , n\geq 0,
  5. α i \alpha_{i}
  6. A i A_{i}
  7. χ A \chi_{A}\,
  8. 1 A 1_{A}
  9. A A
  10. χ A ( x ) = { 1 if x A , 0 if x A . \chi_{A}(x)=\begin{cases}1&\mbox{if }~{}x\in A,\\ 0&\mbox{if }~{}x\notin A.\\ \end{cases}
  11. A i A_{i}
  12. A i A j = A_{i}\cap A_{j}=\emptyset
  13. i j i\neq j
  14. i = 0 n A i = . \cup_{i=0}^{n}A_{i}=\mathbb{R}.
  15. f = 4 χ [ - 5 , 1 ) + 3 χ ( 0 , 6 ) f=4\chi_{[-5,1)}+3\chi_{(0,6)}\,
  16. f = 0 χ ( - , - 5 ) + 4 χ [ - 5 , 0 ] + 7 χ ( 0 , 1 ) + 3 χ [ 1 , 6 ) + 0 χ [ 6 , ) . f=0\chi_{(-\infty,-5)}+4\chi_{[-5,0]}+7\chi_{(0,1)}+3\chi_{[1,6)}+0\chi_{[6,% \infty)}.\,
  17. A 0 = . A_{0}=\mathbb{R}.
  18. A i , A_{i},
  19. i = 0 , 1 , , n , i=0,1,\dots,n,
  20. f ( x ) = α i f(x)=\alpha_{i}\,
  21. x A i . x\in A_{i}.
  22. f = i = 0 n α i χ A i \textstyle f=\sum\limits_{i=0}^{n}\alpha_{i}\chi_{A_{i}}\,
  23. f d x = i = 0 n α i ( A i ) , \textstyle\int\!f\,dx=\sum\limits_{i=0}^{n}\alpha_{i}\ell(A_{i}),\,
  24. ( A ) \ell(A)
  25. A , A,
  26. A i A_{i}

Stepper_motor.html

  1. f = 100 2 π 2 p M h J r f=\frac{100}{2\pi}\sqrt{\frac{2p\cdot M_{h}}{J_{r}}}

Stereographic_projection.html

  1. ( X , Y ) = ( x 1 - z , y 1 - z ) , (X,Y)=\left(\frac{x}{1-z},\frac{y}{1-z}\right),
  2. ( x , y , z ) = ( 2 X 1 + X 2 + Y 2 , 2 Y 1 + X 2 + Y 2 , - 1 + X 2 + Y 2 1 + X 2 + Y 2 ) . (x,y,z)=\left(\frac{2X}{1+X^{2}+Y^{2}},\frac{2Y}{1+X^{2}+Y^{2}},\frac{-1+X^{2}% +Y^{2}}{1+X^{2}+Y^{2}}\right).
  3. ( R , Θ ) = ( sin φ 1 - cos φ , θ ) = ( cot φ 2 , θ ) , (R,\Theta)=\left(\frac{\sin\varphi}{1-\cos\varphi},\theta\right)=\left(\cot% \frac{\varphi}{2},\theta\right),
  4. ( φ , θ ) = ( 2 arctan ( 1 R ) , Θ ) . (\varphi,\theta)=\left(2\arctan\left(\frac{1}{R}\right),\Theta\right).
  5. ( R , Θ ) = ( r 1 - z , θ ) , (R,\Theta)=\left(\frac{r}{1-z},\theta\right),
  6. ( r , θ , z ) = ( 2 R 1 + R 2 , Θ , R 2 - 1 R 2 + 1 ) . (r,\theta,z)=\left(\frac{2R}{1+R^{2}},\Theta,\frac{R^{2}-1}{R^{2}+1}\right).
  7. P 3 P 2 P\in\mathbb{Q}^{3}\iff P^{\prime}\in\mathbb{Q}^{2}
  8. d A = 4 ( 1 + X 2 + Y 2 ) 2 d X d Y . dA=\frac{4}{(1+X^{2}+Y^{2})^{2}}\;dX\;dY.
  9. 4 ( 1 + X 2 + Y 2 ) 2 ( d X 2 + d Y 2 ) , \frac{4}{(1+X^{2}+Y^{2})^{2}}\;(dX^{2}+dY^{2}),
  10. R = e Θ / a , R=e^{\Theta/a},\,
  11. N O P P ′′ O S O P : O N = O S : O P ′′ O P O P ′′ = r 2 \triangle NOP^{\prime}\sim\triangle P^{\prime\prime}OS\implies OP^{\prime}:ON=% OS:OP^{\prime\prime}\implies OP^{\prime}\cdot OP^{\prime\prime}=r^{2}
  12. 1 2 \frac{1}{2}
  13. z = 1 2 z=−\frac{1}{2}
  14. ( x , y , z ) ( ξ , η ) = ( x 1 2 - z , y 1 2 - z ) , (x,y,z)\rightarrow(\xi,\eta)=\left(\frac{x}{\frac{1}{2}-z},\frac{y}{\frac{1}{2% }-z}\right),
  15. ( ξ , η ) ( x , y , z ) = ( ξ 1 + ξ 2 + η 2 , η 1 + ξ 2 + η 2 , - 1 + ξ 2 + η 2 2 + 2 ξ 2 + 2 η 2 ) . (\xi,\eta)\rightarrow(x,y,z)=\left(\frac{\xi}{1+\xi^{2}+\eta^{2}},\frac{\eta}{% 1+\xi^{2}+\eta^{2}},\frac{-1+\xi^{2}+\eta^{2}}{2+2\xi^{2}+2\eta^{2}}\right).
  16. Q P ¯ \scriptstyle\overline{QP}
  17. x i x_{i}
  18. X i X_{i}
  19. X i = x i 1 - x 0 X_{i}=\frac{x_{i}}{1-x_{0}}
  20. S 2 = j = 1 n X j 2 S^{2}=\sum_{j=1}^{n}X_{j}^{2}
  21. x 0 = S 2 - 1 S 2 + 1 x_{0}=\frac{S^{2}-1}{S^{2}+1}
  22. x i = 2 X i S 2 + 1 x_{i}=\frac{2X_{i}}{S^{2}+1}
  23. Q P ¯ \scriptstyle\overline{QP}
  24. ζ = x + i y 1 - z , \zeta=\frac{x+iy}{1-z},
  25. ( x , y , z ) = ( 2 R e ( ζ ) 1 + ζ ¯ ζ , 2 I m ( ζ ) 1 + ζ ¯ ζ , - 1 + ζ ¯ ζ 1 + ζ ¯ ζ ) . (x,y,z)=\left(\frac{2\mathrm{Re}(\zeta)}{1+\bar{\zeta}\zeta},\frac{2\mathrm{Im% }(\zeta)}{1+\bar{\zeta}\zeta},\frac{-1+\bar{\zeta}\zeta}{1+\bar{\zeta}\zeta}% \right).
  26. ξ = x - i y 1 + z , \xi=\frac{x-iy}{1+z},
  27. ( x , y , z ) = ( 2 R e ( ξ ) 1 + ξ ¯ ξ , 2 I m ( ξ ) 1 + ξ ¯ ξ , 1 - ξ ¯ ξ 1 + ξ ¯ ξ ) . (x,y,z)=\left(\frac{2\mathrm{Re}(\xi)}{1+\bar{\xi}\xi},\frac{2\mathrm{Im}(\xi)% }{1+\bar{\xi}\xi},\frac{1-\bar{\xi}\xi}{1+\bar{\xi}\xi}\right).
  28. ( 2 m n n 2 + m 2 , n 2 - m 2 n 2 + m 2 ) \left(\frac{2mn}{n^{2}+m^{2}},\frac{n^{2}-m^{2}}{n^{2}+m^{2}}\right)
  29. cos x = t 2 - 1 t 2 + 1 , sin x = 2 t t 2 + 1 . \cos x=\frac{t^{2}-1}{t^{2}+1},\quad\sin x=\frac{2t}{t^{2}+1}.
  30. d x = 2 d t t 2 + 1 . dx=\frac{2\,dt}{t^{2}+1}.

Stirling's_approximation.html

  1. ln ( n ! ) = n ln ( n ) - n + O ( ln ( n ) ) \ln(n!)=n\ln(n)-n+O(\ln(n))
  2. n ! 2 π n ( n e ) n . n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.
  3. n ! 2 π n ( n e ) n \frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}
  4. ln ( n ! ) = ln ( 1 ) + ln ( 2 ) + + ln ( n ) . \ln(n!)=\ln(1)+\ln(2)+\cdots+\ln(n).
  5. 1 2 ( ln ( 1 ) + ln ( n ) ) = 1 2 ln ( n ) , \tfrac{1}{2}(\ln(1)+\ln(n))=\tfrac{1}{2}\ln(n),
  6. ln ( n ! ) - 1 2 ln ( n ) 1 n ln ( x ) d x = n ln ( n ) - n + 1 , \ln(n!)-\tfrac{1}{2}\ln(n)\approx\int_{1}^{n}\ln(x)\,{\rm d}x=n\ln(n)-n+1,
  7. ln ( n ! ) - 1 2 ln ( n ) \displaystyle\ln(n!)-\tfrac{1}{2}\ln(n)
  8. lim n ( ln ( n ! ) - n ln ( n ) + n - 1 2 ln ( n ) ) = 1 - k = 2 m ( - 1 ) k B k k ( k - 1 ) + lim n R m , n . \lim_{n\to\infty}\left(\ln(n!)-n\ln(n)+n-\tfrac{1}{2}\ln(n)\right)=1-\sum_{k=2% }^{m}\frac{(-1)^{k}B_{k}}{k(k-1)}+\lim_{n\to\infty}R_{m,n}.
  9. R m , n = lim n R m , n + O ( 1 n m ) , R_{m,n}=\lim_{n\to\infty}R_{m,n}+O\left(\frac{1}{n^{m}}\right),
  10. ln ( n ! ) = n ln ( n e ) + 1 2 ln ( n ) + y + k = 2 m ( - 1 ) k B k k ( k - 1 ) n k - 1 + O ( 1 n m ) . \ln(n!)=n\ln\left(\frac{n}{e}\right)+\tfrac{1}{2}\ln(n)+y+\sum_{k=2}^{m}\frac{% (-1)^{k}B_{k}}{k(k-1)n^{k-1}}+O\left(\frac{1}{n^{m}}\right).
  11. n ! = e y n ( n e ) n ( 1 + O ( 1 n ) ) . n!=e^{y}\sqrt{n}\left(\frac{n}{e}\right)^{n}\left(1+O\left(\frac{1}{n}\right)% \right).
  12. e y = 2 π e^{y}=\sqrt{2\pi}
  13. n ! = 2 π n ( n e ) n ( 1 + O ( 1 n ) ) . n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\left(1+O\left(\frac{1}{n}\right)% \right).
  14. 2 π n \sqrt{2\pi n}
  15. ln ( n ! ) = j = 1 n ln ( j ) \ln(n!)=\sum_{j=1}^{n}\ln(j)
  16. j = 1 n ln ( j ) 1 n ln ( x ) d x = n ln ( n ) - n + 1. \sum_{j=1}^{n}\ln(j)\approx\int_{1}^{n}\ln(x)\,{\rm d}x=n\ln(n)-n+1.
  17. n ! n!
  18. n ! = 0 x n e - x d x . n!=\int_{0}^{\infty}x^{n}e^{-x}dx.
  19. x = n y x=ny
  20. n ! = 0 e n ln x - x d x = e n ln n n 0 e n ( ln y - y ) d y . n!=\int_{0}^{\infty}e^{n\ln x-x}dx=e^{n\ln n}n\int_{0}^{\infty}e^{n(\ln y-y)}dy.
  21. 0 e n ( ln y - y ) d y 2 π n e - n \int_{0}^{\infty}e^{n(\ln y-y)}dy\sim\sqrt{\frac{2\pi}{n}}e^{-n}
  22. n ! e n ln n n 2 π n e - n = 2 π n ( n e ) n . n!\sim e^{n\ln n}n\sqrt{\frac{2\pi}{n}}e^{-n}=\sqrt{2\pi n}\left(\frac{n}{e}% \right)^{n}.
  23. 0 e n ( ln y - y ) d y 2 π n e - n ( 1 + 1 12 n ) \int_{0}^{\infty}e^{n(\ln y-y)}dy\sim\sqrt{\frac{2\pi}{n}}e^{-n}\left(1+\frac{% 1}{12n}\right)
  24. n ! e n ln n n 2 π n e - n ( 1 + 1 12 n ) = 2 π n ( n e ) n ( 1 + 1 12 n ) . n!\sim e^{n\ln n}n\sqrt{\frac{2\pi}{n}}e^{-n}\left(1+\frac{1}{12n}\right)=% \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\left(1+\frac{1}{12n}\right).
  25. n ! \displaystyle n!
  26. | ln ( S ( n , t ) n ! ) | , \left|\ln\left(\frac{S(n,t)}{n!}\right)\right|,
  27. ln n ! \displaystyle\ln n!
  28. 2 π n n + 1 / 2 e - n n ! e n n + 1 / 2 e - n \sqrt{2\pi}\ n^{n+1/2}e^{-n}\leq n!\leq e\ n^{n+1/2}e^{-n}
  29. n ! n n + 1 / 2 e - n \frac{n!}{n^{n+1/2}e^{-n}}
  30. 2 π = 2.5066 \sqrt{2\pi}=2.5066\ldots
  31. e = 2.71828 e=2.71828\ldots
  32. n ! = Γ ( n + 1 ) , n!=\Gamma(n+1),
  33. ln Γ ( z ) = z ln z - z + 1 2 ln 2 π z + 0 2 arctan ( t z ) e 2 π t - 1 d t . \ln\Gamma(z)=z\ln z-z+\frac{1}{2}\ln\frac{2\pi}{z}+\int_{0}^{\infty}\frac{2% \arctan(\frac{t}{z})}{e^{2\pi t}-1}\,{\rm d}t.
  34. ln Γ ( z ) z ln z - z + 1 2 ln 2 π z + n = 1 B 2 n 2 n ( 2 n - 1 ) z 2 n - 1 \ln\Gamma(z)\sim z\ln z-z+\frac{1}{2}\ln\frac{2\pi}{z}+\sum_{n=1}^{\infty}% \frac{B_{2n}}{2n(2n-1)z^{2n-1}}
  35. Γ ( z ) = 2 π z ( z e ) z ( 1 + O ( 1 z ) ) . \Gamma(z)=\sqrt{\frac{2\pi}{z}}~{}{\left(\frac{z}{e}\right)}^{z}\left(1+O\left% (\frac{1}{z}\right)\right).
  36. 0 2 arctan ( t x ) e 2 π t - 1 d t = ln Γ ( 1 + x ) - x ln x + x - 1 2 ln 2 π x . \int_{0}^{\infty}\frac{2\arctan(\frac{t}{x})}{e^{2\pi t}-1}dt=\ln\Gamma(1+x)-x% \ln x+x-\frac{1}{2}\ln 2\pi x.
  37. z n ¯ = z ( z + 1 ) ( z + n - 1 ) ; z^{\bar{n}}=z(z+1)\cdots(z+n-1);
  38. 0 2 arctan ( t x ) e 2 π t - 1 d t = n = 1 c n ( x + 1 ) n ¯ \int_{0}^{\infty}\frac{2\arctan(\frac{t}{x})}{e^{2\pi t}-1}dt=\sum_{n=1}^{% \infty}\frac{c_{n}}{(x+1)^{\bar{n}}}
  39. c n = 1 n 0 1 x n ¯ ( x - 1 2 ) d x = 1 2 n k = 1 n k | s ( n , k ) | ( k + 1 ) ( k + 2 ) c_{n}=\frac{1}{n}\int_{0}^{1}x^{\bar{n}}\left(x-\tfrac{1}{2}\right)\,{\rm d}x=% \frac{1}{2n}\sum_{k=1}^{n}\frac{k|s(n,k)|}{(k+1)(k+2)}
  40. ln Γ ( x ) = x ln x - x + 1 2 ln 2 π x + 1 12 ( x + 1 ) + 1 12 ( x + 1 ) ( x + 2 ) + + 59 360 ( x + 1 ) ( x + 2 ) ( x + 3 ) + 29 60 ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 4 ) + \begin{aligned}\displaystyle\ln\Gamma(x)=x\ln x-x+\frac{1}{2}\ln\frac{2\pi}{x}% +\frac{1}{12(x+1)}+\frac{1}{12(x+1)(x+2)}+\\ \displaystyle+\frac{59}{360(x+1)(x+2)(x+3)}+\frac{29}{60(x+1)(x+2)(x+3)(x+4)}+% \cdots\end{aligned}
  41. Γ ( z ) 2 π z ( z e z sinh 1 z + 1 810 z 6 ) z , \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{z}{e}\sqrt{z\sinh\frac{1}{z}+% \frac{1}{810z^{6}}}\right)^{z},
  42. 2 ln ( Γ ( z ) ) ln ( 2 π ) - ln ( z ) + z ( 2 ln ( z ) + ln ( z sinh 1 z + 1 810 z 6 ) - 2 ) , 2\ln(\Gamma(z))\approx\ln(2\pi)-\ln(z)+z\left(2\ln(z)+\ln\left(z\sinh\frac{1}{% z}+\frac{1}{810z^{6}}\right)-2\right),
  43. Γ ( z ) 2 π z ( 1 e ( z + 1 12 z - 1 10 z ) ) z , \Gamma(z)\approx\sqrt{\frac{2\pi}{z}}\left(\frac{1}{e}\left(z+\frac{1}{12z-% \frac{1}{10z}}\right)\right)^{z},
  44. ln ( Γ ( z ) ) 1 2 [ ln ( 2 π ) - ln ( z ) ] + z [ ln ( z + 1 12 z - 1 10 z ) - 1 ] . \ln(\Gamma(z))\approx\tfrac{1}{2}\left[\ln(2\pi)-\ln(z)\right]+z\left[\ln\left% (z+\frac{1}{12z-\frac{1}{10z}}\right)-1\right].
  45. ln ( n ! ) n ln ( n ) - n + 1 6 ln ( n ( 1 + 4 n ( 1 + 2 n ) ) ) + 1 2 ln ( π ) . \ln(n!)\approx n\ln(n)-n+\tfrac{1}{6}\ln(n(1+4n(1+2n)))+\tfrac{1}{2}\ln(\pi).
  46. n ! [ constant ] n n + 1 / 2 e - n . n!\sim[{\rm constant}]\cdot n^{n+1/2}e^{-n}.
  47. 2 π \sqrt{2\pi}

Stirling_number.html

  1. s ( n , k ) s(n,k)\,
  2. c ( n , k ) = [ n k ] = | s ( n , k ) | c(n,k)=\left[{n\atop k}\right]=|s(n,k)|\,
  3. S ( n , k ) = { n k } = S n ( k ) S(n,k)=\left\{\begin{matrix}n\\ k\end{matrix}\right\}=S_{n}^{(k)}\,
  4. ( x ) n = k = 0 n s ( n , k ) x k . (x)_{n}=\sum_{k=0}^{n}s(n,k)x^{k}.
  5. ( x ) n (x)_{n}
  6. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) . (x)_{n}=x(x-1)(x-2)\cdots(x-n+1).
  7. x n ¯ x^{\underline{n\!}}
  8. x n ¯ x^{\overline{n\!}}
  9. c ( n , k ) = [ n k ] = | s ( n , k ) | = ( - 1 ) n - k s ( n , k ) c(n,k)=\left[{n\atop k}\right]=|s(n,k)|=(-1)^{n-k}s(n,k)
  10. 1 - 1 1 2 - 3 1 - 6 11 - 6 1 24 - 50 35 - 10 1 - 120 274 - 225 85 - 15 1 \begin{array}[]{ccccccccccc}&&&&&~{}~{}1&&&&&\\ &&&&-1&&~{}~{}1&&&&\\ &&&2&&-3&&~{}~{}1&&&\\ &&-6&&11&&-6&&~{}~{}1&&\\ &24&&-50&&35&&-10&&~{}~{}1&\\ -120&&274&&-225&&85&&-15&&~{}~{}1\\ \end{array}
  11. s ( n , k ) = s ( n - 1 , k - 1 ) - ( n - 1 ) s ( n - 1 , k ) s(n,k)=s(n-1,k-1)-(n-1)s(n-1,k)
  12. S ( n , k ) S(n,k)
  13. { n k } \textstyle\{{n\atop k}\}
  14. k = 0 n S ( n , k ) = B n \sum_{k=0}^{n}S(n,k)=B_{n}
  15. k = 0 n S ( n , k ) ( x ) k = x n . \sum_{k=0}^{n}S(n,k)(x)_{k}=x^{n}.
  16. L ( n , k ) = ( n - 1 k - 1 ) n ! k ! L(n,k)={n-1\choose k-1}\frac{n!}{k!}
  17. j = 0 n s ( n , j ) S ( j , k ) = δ n k \sum_{j=0}^{n}s(n,j)S(j,k)=\delta_{nk}
  18. j = 0 n S ( n , j ) s ( j , k ) = δ n k , \sum_{j=0}^{n}S(n,j)s(j,k)=\delta_{nk},
  19. δ n k \delta_{nk}
  20. s n k = s ( n , k ) . s_{nk}=s(n,k).\,
  21. S n k = S ( n , k ) . S_{nk}=S(n,k).
  22. s - 1 = S s^{-1}=S\,
  23. L ( n , k ) : L(n,k):
  24. ( - 1 ) n L ( n , k ) = z ( - 1 ) z s ( n , z ) S ( z , k ) , (-1)^{n}L(n,k)=\sum_{z}(-1)^{z}s(n,z)S(z,k),
  25. L ( 0 , 0 ) = 1 L(0,0)=1
  26. L ( n , k ) = 0 L(n,k)=0
  27. k > n k>n
  28. s ( n , k ) = j = 0 n - k ( - 1 ) j ( n - 1 + j n - k + j ) ( 2 n - k n - k - j ) S ( n - k + j , j ) s(n,k)=\sum_{j=0}^{n-k}(-1)^{j}{n-1+j\choose n-k+j}{2n-k\choose n-k-j}S(n-k+j,j)
  29. S ( n , k ) = j = 0 n - k ( - 1 ) j ( n - 1 + j n - k + j ) ( 2 n - k n - k - j ) s ( n - k + j , j ) . S(n,k)=\sum_{j=0}^{n-k}(-1)^{j}{n-1+j\choose n-k+j}{2n-k\choose n-k-j}s(n-k+j,% j).

Stochastic_calculus.html

  1. H d X \int H\,dX
  2. X X
  3. 0 t X s - d Y s := 0 t X s - d Y s + 1 2 [ X , Y ] t c , \int_{0}^{t}X_{s-}\circ dY_{s}:=\int_{0}^{t}X_{s-}dY_{s}+\frac{1}{2}\left[X,Y% \right]_{t}^{c},
  4. 0 t X s Y s \int_{0}^{t}X_{s}\,\partial Y_{s}

Stochastic_context-free_grammar.html

  1. G G
  2. G = ( M , T , R , S , P ) G=(M,T,R,S,P)
  3. M M
  4. T T
  5. R R
  6. S S
  7. P P
  8. { a , b } \left\{a,b\right\}
  9. S S
  10. ϵ \epsilon
  11. S a S , S b S , S ϵ S\to aS,S\to bS,S\to\epsilon
  12. S a S | b S | ϵ S\to aS|bS|\epsilon
  13. S S
  14. a a
  15. b b
  16. ϵ \epsilon
  17. S a S a b S a b b S a b b S\Rightarrow aS\Rightarrow abS\Rightarrow abbS\Rightarrow abb
  18. S a S a | b S b | a a | b b S\to aSa|bSb|aa|bb
  19. a a b a a b a a aabaabaa
  20. a a
  21. S a S a a a S a a a a b S b a a a a b a a b a a S\Rightarrow aSa\Rightarrow aaSaa\Rightarrow aabSbaa\Rightarrow aabaabaa
  22. x x
  23. P ( x | θ ) P(x|\theta)
  24. θ \theta
  25. x P ( x | θ ) = 1 \sum_{\,\text{x}}P(x|\theta)=1
  26. S \mathbf{\mathit{S}}
  27. L \mathbf{\mathit{L}}
  28. s s
  29. F \mathbf{\mathit{F}}
  30. S 𝐿𝑆 | L \mathit{S\to LS|L}
  31. L s | 𝑑𝐹𝑑 \mathit{L\to s|dFd}
  32. F 𝑑𝐹𝑑 | 𝐿𝑆 \mathit{F\to dFd|LS}
  33. α ( i , j , v ) \alpha(i,j,v)
  34. i , j , v i,j,v
  35. W v W_{v}
  36. x i , , x j xi,...,xj
  37. β ( i , j , v ) \beta(i,j,v)
  38. x x
  39. x i , , x j xi,...,xj
  40. α \alpha
  41. β \beta
  42. α \alpha
  43. β \beta
  44. x x
  45. P ( x | θ ) P(x|\theta)
  46. α \alpha
  47. β \beta
  48. γ ( i , j , v ) \gamma(i,j,v)
  49. π ^ \hat{\pi}
  50. log P ( x , π ^ | θ ) \log P(x,\hat{\pi}|\theta)
  51. O ( L 2 M ) O(L^{2}M)
  52. O ( L 3 M 3 ) O(L^{3}M^{3})
  53. P a W b probabilities of pairwise interactions between 16 possible pairs P\to aWb~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{probabilities of pairwise % interactions between 16 possible pairs}
  54. L a W probabilities of generating 4 possible single bases on the left L\to aW~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{probabilities of generating 4 % possible single bases on the left}
  55. R W a probabilities of generating 4 possible single bases on the right R\to Wa~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{probabilities of % generating 4 possible single bases on the right}
  56. B S S bifurcation with a probability of 1 B\to SS~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{bifurcation with a probability of 1}
  57. S W start with a probability of 1 S\to W~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{start with a probability of 1}
  58. E ϵ end with a probability of 1 E\to\epsilon~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{end with a probability% of 1}
  59. l o g e ^ log\hat{e}
  60. P , L , R P,~{}L,~{}R
  61. l o g P ( x , π ^ | θ ) log\,\text{P}(x,\hat{\pi}|\theta)
  62. O ( M a D + M b D 2 ) O(M_{a}D+M_{b}D^{2})
  63. X X
  64. Y Y
  65. X Y XY
  66. Y X YX
  67. X X XX
  68. X , Y X,Y
  69. w h i l e X Y whileX\neq Y
  70. C XY + 1 first sequence pair C_{\,\text{XY}}+1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{first sequence% pair}
  71. C YX + 1 second sequence pair C_{\,\text{YX}}+1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{second % sequence pair}
  72. Calculate mutation rates. \,\text{Calculate mutation rates.}
  73. Let r XY = mutation of base X to base Y = K C XY / P x P s \,\text{Let}~{}r_{\,\text{XY}}=\,\text{mutation of base X to base Y}=K~{}C_{\,% \text{XY}}/P_{x}P_{s}
  74. Let r XX = the negative of the rate of X mutation to other bases = - r XY \,\text{Let}~{}r_{\,\text{XX}}=\,\text{the negative of the rate of X mutation % to other bases}=-\sum r_{\,\text{XY}}
  75. P s = the probability that the base is not paired. P_{s}=\,\text{the probability that the base is not paired.}
  76. P X r X Y = P Y r Y X PX^{r}XY=PY^{r}YX
  77. C C
  78. σ \sigma
  79. D D
  80. l l
  81. D = ( C 1 , C 2 , C l ) D=(C_{1},~{}C_{2},...C_{l})
  82. T T
  83. M M
  84. P ( σ | M ) P(\sigma|M)
  85. T T
  86. P ( D | T , M ) P(D|T,M)
  87. = P ( D , σ | T , M ) =\sum P(D,\sigma|T,M)
  88. = P ( D | σ , T , M ) P ( σ | T , M ) =\sum P(D|\sigma,T,M)P(\sigma|T,M)
  89. = P ( D | σ , T , M ) P ( σ | M ) =\sum P(D|\sigma,T,M)P(\sigma|M)
  90. S L S ( 80 % ) | L ( 20 % ) S\to LS(80\%)|L(20\%)
  91. L s ( 70 % ) | d F d ( 30 % ) L\to s(70\%)|dFd(30\%)
  92. F d F d ( 60.4 % ) | L S ( 39.6 % ) F\to dFd(60.4\%)|LS(39.6\%)
  93. P ( σ | D , T , M ) P(\sigma|D,T,M)
  94. σ M A P = a r g m 𝜎 a x P ( D | σ , T M L , M ) P ( σ | M ) \sigma^{MAP}=arg\underset{\sigma}{m}axP(D|\sigma,T^{M}L,M)P(\sigma|M)
  95. T T

Stochastic_matrix.html

  1. s y m b o l X t symbol{X}_{t}
  2. i i
  3. j j
  4. P r ( j | i ) = P i , j Pr(j|i)=P_{i,j}
  5. P i , j P_{i,j}
  6. i t h i^{th}
  7. j t h j^{th}
  8. P = ( p 1 , 1 p 1 , 2 p 1 , j p 2 , 1 p 2 , 2 p 2 , j p i , 1 p i , 2 p i , j ) . P=\left(\begin{matrix}p_{1,1}&p_{1,2}&\dots&p_{1,j}&\dots\\ p_{2,1}&p_{2,2}&\dots&p_{2,j}&\dots\\ \vdots&\vdots&\ddots&\vdots&\ddots\\ p_{i,1}&p_{i,2}&\dots&p_{i,j}&\dots\\ \vdots&\vdots&\ddots&\vdots&\ddots\end{matrix}\right).
  9. i i
  10. j P i , j = 1. \sum_{j}P_{i,j}=1.\,
  11. k k
  12. P k P^{k}
  13. P P
  14. i i
  15. j j
  16. ( i , j ) t h (i,j)^{th}
  17. P P
  18. ( P 2 ) i , j . \left(P^{2}\right)_{i,j}.
  19. P P
  20. P k P^{k}
  21. s y m b o l π symbol{\pi}
  22. { 1 , , n } \{1,...,n\}
  23. s y m b o l π P = s y m b o l π . symbol{\pi}P=symbol{\pi}.
  24. s y m b o l 1 symbol{1}
  25. { 1 , , n } \{1,...,n\}
  26. i i
  27. lim k ( P k ) i , j = s y m b o l π j , \lim_{k\rightarrow\infty}\left(P^{k}\right)_{i,j}=symbol{\pi}_{j},
  28. s y m b o l π j symbol{\pi}_{j}
  29. j t h j^{th}
  30. s y m b o l π symbol{\pi}
  31. j j
  32. i i
  33. P = [ 0 0 1 / 2 0 1 / 2 0 0 1 0 0 1 / 4 1 / 4 0 1 / 4 1 / 4 0 0 1 / 2 0 1 / 2 0 0 0 0 1 ] . P=\begin{bmatrix}0&0&1/2&0&1/2\\ 0&0&1&0&0\\ 1/4&1/4&0&1/4&1/4\\ 0&0&1/2&0&1/2\\ 0&0&0&0&1\end{bmatrix}.
  34. [ 0 , 1 , 0 , 0 , 0 ] [0,1,0,0,0]
  35. s y m b o l τ = [ 0 , 1 , 0 , 0 ] symbol{\tau}=[0,1,0,0]
  36. T = [ 0 0 1 / 2 0 0 0 1 0 1 / 4 1 / 4 0 1 / 4 0 0 1 / 2 0 ] , T=\begin{bmatrix}0&0&1/2&0\\ 0&0&1&0\\ 1/4&1/4&0&1/4\\ 0&0&1/2&0\end{bmatrix}\,,
  37. ( I - T ) - 1 s y m b o l 1 = [ 2.75 4.5 3.5 2.75 ] , (I-T)^{-1}symbol{1}=\begin{bmatrix}2.75\\ 4.5\\ 3.5\\ 2.75\end{bmatrix}\,,
  38. I I
  39. 𝟏 \mathbf{1}
  40. E [ K ] = s y m b o l τ ( I + T + T 2 + ) s y m b o l 1 = s y m b o l τ ( I - T ) - 1 s y m b o l 1 = 4.5. E[K]=symbol{\tau}(I+T+T^{2}+\cdots)symbol{1}=symbol{\tau}(I-T)^{-1}symbol{1}=4% .5.
  41. E [ K ( K - 1 ) ( K - n + 1 ) ] = n ! s y m b o l τ ( I - T ) - n T n - 1 1 . E[K(K-1)\dots(K-n+1)]=n!symbol{\tau}(I-{T})^{-n}{T}^{n-1}\mathbf{1}\,.

Stokes'_law.html

  1. F d = 6 π μ R V F_{d}=6\pi\,\mu\,R\,V\,
  2. F g = ( ρ p - ρ f ) g 4 3 π R 3 , F_{g}=\left(\rho_{p}-\rho_{f}\right)\,g\,\frac{4}{3}\pi\,R^{3},
  3. V = 2 9 ( ρ p - ρ f ) μ g R 2 V=\frac{2}{9}\frac{\left(\rho_{p}-\rho_{f}\right)}{\mu}g\,R^{2}
  4. p = η 2 𝐮 = - η × 𝐬𝐲𝐦𝐛𝐨𝐥 ω , 𝐮 = 0 , \begin{aligned}&\displaystyle\nabla p=\eta\,\nabla^{2}\mathbf{u}=-\eta\,\nabla% \times\mathbf{symbol{\omega}},\\ &\displaystyle\nabla\cdot\mathbf{u}=0,\end{aligned}
  5. s y m b o l ω = × 𝐮 . symbol{\omega}=\nabla\times\mathbf{u}.
  6. 2 s y m b o l ω = 0 \nabla^{2}symbol{\omega}=0
  7. 2 p = 0. \nabla^{2}p=0.
  8. u z = - 1 r ψ z , u r = 1 r ψ r , u_{z}=-\frac{1}{r}\frac{\partial\psi}{\partial z},\qquad u_{r}=\frac{1}{r}% \frac{\partial\psi}{\partial r},
  9. ω φ = u z z - u r r = - r ( 1 r ψ r ) - 1 r 2 ψ z 2 . \omega_{\varphi}=\frac{\partial u_{z}}{\partial z}-\frac{\partial u_{r}}{% \partial r}=-\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial\psi}{% \partial r}\right)-\frac{1}{r}\,\frac{\partial^{2}\psi}{\partial z^{2}}.
  10. 2 ω φ = 1 r r ( r ω φ r ) + 2 ω φ z 2 - ω φ r 2 = 0. \nabla^{2}\omega_{\varphi}=\frac{1}{r}\frac{\partial}{\partial r}\left(r\,% \frac{\partial\omega_{\varphi}}{\partial r}\right)+\frac{\partial^{2}\omega_{% \varphi}}{\partial z^{2}}-\frac{\omega_{\varphi}}{r^{2}}=0.
  11. ψ = - 1 2 u r 2 [ 1 - 3 2 R r 2 + z 2 + 1 2 ( R r 2 + z 2 ) 3 ] . \psi=-\frac{1}{2}\,u\,r^{2}\,\left[1-\frac{3}{2}\frac{R}{\sqrt{r^{2}+z^{2}}}+% \frac{1}{2}\left(\frac{R}{\sqrt{r^{2}+z^{2}}}\right)^{3}\;\right].
  12. s y m b o l σ = 3 η u 2 R 𝐞 z symbol{\sigma}=\frac{3\,\eta\,u}{2\,R}\,\mathbf{e}_{z}

Stoma.html

  1. E = ( e i - e a ) / P r E=(e_{i}-e_{a})/Pr
  2. E = ( e i - e a ) g / P E=(e_{i}-e_{a})g/P
  3. g = E P / ( e i - e a ) g=EP/(e_{i}-e_{a})
  4. A = ( C a - C i ) g / 1.6 P A=(C_{a}-C_{i})g/1.6P

Strähle_construction.html

  1. N m O A + B A + ( O A - 3 B A ) × m O A + B A - 2 B A × m N^{m}\doteq\frac{OA+BA+(OA-3BA)\times{m}}{OA+BA-2BA\times{m}}
  2. 24 + 10 m 24 - 7 m \frac{24+10m}{24-7m}
  3. O A - B A = 1 \scriptstyle OA-BA=1
  4. O A + B A = N \scriptstyle OA+BA=\sqrt{N}
  5. N m N m + N ( 1 - m ) m + N ( 1 - m ) N^{m}\doteq\frac{Nm+\sqrt{N}(1-m)}{m+\sqrt{N}(1-m)}
  6. f ( m ) = m ( 1 - m ) ( 1 - 2 m ) f(m)=m(1-m)(1-2m)
  7. f ( x ) = x ( 1 - x 2 a ) \scriptstyle f(x)=x(1-x^{2a})
  8. 41 29 \scriptstyle\frac{41}{29}
  9. 2 \scriptstyle\sqrt{2}
  10. 2 \scriptstyle\sqrt{2}
  11. y = ( a x + b ) ( c x + d ) \scriptstyle y=\frac{(ax+b)}{(cx+d)}
  12. 2 \scriptstyle\sqrt{2}
  13. 17 12 \scriptstyle\frac{17}{12}
  14. p + 2 q p + q \scriptstyle\frac{p+2q}{p+q}
  15. p q \scriptstyle\frac{p}{q}
  16. 17 / 9 \scriptstyle\sqrt{17/9}
  17. O A / O B = 8.27 \scriptstyle OA/OB=8.27

Stream_function.html

  1. ψ ( x , y , t ) \psi(x,y,t)
  2. P P
  3. ( x , y ) (x,y)
  4. t t
  5. ψ = A P ( u d y - v d x ) . \psi=\int_{A}^{P}\left(u\,\,\text{d}y-v\,\,\text{d}x\right).
  6. ψ \psi
  7. A P AP
  8. ( u , v ) (u,v)
  9. ( + d y , - d x ) (+\,\text{d}y,-\,\text{d}x)
  10. ( d x , d y ) . (\,\text{d}x,\,\text{d}y).
  11. A A
  12. A A
  13. ψ . \psi.
  14. δ P = ( δ x , δ y ) \delta P=(\delta x,\delta y)
  15. P P
  16. δ ψ = u δ y - v δ x , \delta\psi=u\,\delta y-v\,\delta x,
  17. u x + v y = 0. \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.
  18. δ ψ = ψ x δ x + ψ y δ y , \delta\psi=\frac{\partial\psi}{\partial x}\,\delta x+\frac{\partial\psi}{% \partial y}\,\delta y,
  19. u = + ψ y u=+\frac{\partial\psi}{\partial y}
  20. v = - ψ x v=-\frac{\partial\psi}{\partial x}
  21. ψ . \psi.
  22. ψ \psi
  23. s y m b o l ψ : symbol{\psi}:
  24. 𝐮 = × s y m b o l ψ \mathbf{u}=\nabla\times symbol{\psi}
  25. s y m b o l ψ = ( 0 , 0 , ψ ) symbol{\psi}=(0,0,\psi)
  26. 𝐮 = ( u , v , 0 ) \mathbf{u}=(u,v,0)
  27. u = ψ y , v = - ψ x u=\frac{\partial\psi}{\partial y},\qquad v=-\frac{\partial\psi}{\partial x}
  28. u u
  29. v v
  30. x x
  31. y y
  32. 𝐮 = 𝐳 × ψ ( - ψ y , ψ x , 0 ) \mathbf{u}=\mathbf{z}\times\nabla\psi^{\prime}\equiv(-\psi^{\prime}_{y},\psi^{% \prime}_{x},0)
  33. 𝐳 = ( 0 , 0 , 1 ) \mathbf{z}=(0,0,1)
  34. + z +z
  35. ψ = - ψ \psi^{\prime}=-\psi
  36. u = - ψ y , v = ψ x u=-\frac{\partial\psi^{\prime}}{\partial y},\qquad v=\frac{\partial\psi^{% \prime}}{\partial x}
  37. u x + v y = 0 \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
  38. × ( ψ 𝐳 ) = ψ × 𝐳 + ψ × 𝐳 = ψ × 𝐳 = 𝐳 × ψ . \nabla\times\left(\psi\mathbf{z}\right)=\psi\nabla\times\mathbf{z}+\nabla\psi% \times\mathbf{z}=\nabla\psi\times\mathbf{z}=\mathbf{z}\times\nabla\psi^{\prime}.
  39. s y m b o l ψ = ψ 𝐳 symbol{\psi}=\psi\mathbf{z}
  40. δ ψ = q δ n \delta\psi=q\delta n\,
  41. q = ψ n q=\frac{\partial\psi}{\partial n}\,
  42. δ ψ = u δ y \delta\psi=u\delta y\,
  43. δ ψ = - v δ x \delta\psi=-v\delta x\,
  44. u = ψ y u=\frac{\partial\psi}{\partial y}\,
  45. v = - ψ x v=-\frac{\partial\psi}{\partial x}\,
  46. δ ψ = v r ( r δ θ ) , \delta\psi=v_{r}(r\delta\theta),\,
  47. δ ψ = - v θ δ r , \delta\psi=-v_{\theta}\delta r,\,
  48. v r = 1 r ψ θ , v_{r}=\frac{1}{r}\frac{\partial\psi}{\partial\theta},\,
  49. v θ = - ψ r . v_{\theta}=-\frac{\partial\psi}{\partial r}.\,
  50. δ ψ i n = u δ y + v δ x . \delta\psi_{in}=u\delta y+v\delta x.\,
  51. δ ψ o u t = ( u + u x δ x ) δ y + ( v + v y δ y ) δ x . \delta\psi_{out}=\left(u+\frac{\partial u}{\partial x}\delta x\right)\delta y+% \left(v+\frac{\partial v}{\partial y}\delta y\right)\delta x.\,
  52. δ ψ i n = δ ψ o u t \delta\psi_{in}=\delta\psi_{out}\,
  53. u δ y + v δ x = ( u + u x δ x ) δ y + ( v + v y δ y ) δ x u\delta y+v\delta x\ =\left(u+\frac{\partial u}{\partial x}\delta x\right)% \delta y+\left(v+\frac{\partial v}{\partial y}\delta y\right)\delta x\,
  54. u x + v y = 0. \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.
  55. 2 ψ x y - 2 ψ y x = 0. \frac{\partial^{2}\psi}{\partial x\partial y}-\frac{\partial^{2}\psi}{\partial y% \partial x}=0.
  56. 2 ψ = - ω \nabla^{2}\psi=-\omega
  57. 2 ψ = + ω \nabla^{2}\psi^{\prime}=+\omega
  58. s y m b o l ω = × 𝐮 symbol{\omega}=\nabla\times\mathbf{u}
  59. 𝐮 \mathbf{u}
  60. s y m b o l ω = ( 0 , 0 , ω ) , symbol{\omega}=(0,0,\omega),
  61. z z
  62. ω \omega
  63. P = ( x , y ) P=(x,y)
  64. Q = ( x + d x , y + d y ) Q=(x+dx,y+dy)
  65. ψ ( x + d x , y + d y ) - ψ ( x , y ) = ψ x d x + ψ y d y \psi(x+dx,y+dy)-\psi(x,y)={\partial\psi\over\partial x}dx+{\partial\psi\over% \partial y}dy
  66. = ψ d s y m b o l r \qquad\qquad=\nabla\psi\cdot dsymbol{r}
  67. ψ \psi
  68. C C
  69. P P
  70. Q Q
  71. d s y m b o l r dsymbol{r}
  72. ψ = C \psi=C
  73. P P
  74. 0 = ψ ( x + d x , y + d y ) - ψ ( x , y ) = ψ d s y m b o l r 0=\psi(x+dx,y+dy)-\psi(x,y)=\nabla\psi\cdot dsymbol{r}
  75. ψ \nabla\psi
  76. ψ = C \psi=C
  77. s y m b o l u ψ = 0 symbol{u}\cdot\nabla\psi=0
  78. s y m b o l u symbol{u}
  79. ψ \psi
  80. s y m b o l u ψ = ψ y ψ x + ( - ψ x ) ψ y = 0. symbol{u}\cdot\nabla\psi={\partial\psi\over\partial y}{\partial\psi\over% \partial x}+\Big(-{\partial\psi\over\partial x}\Big){\partial\psi\over\partial y% }=0.

Streamlines,_streaklines,_and_pathlines.html

  1. d x S d s × u ( x S ) = 0 , {d\vec{x}_{S}\over ds}\times\vec{u}(\vec{x}_{S})=0,
  2. × \times
  3. x S ( s ) \vec{x}_{S}(s)
  4. u = ( u , v , w ) , \vec{u}=(u,v,w),
  5. x S = ( x S , y S , z S ) , \vec{x}_{S}=(x_{S},y_{S},z_{S}),
  6. d x S u = d y S v = d z S w , {dx_{S}\over u}={dy_{S}\over v}={dz_{S}\over w},
  7. s s
  8. s x S ( s ) . s\mapsto\vec{x}_{S}(s).
  9. { d x P d t = u P ( x P , t ) x P ( t 0 ) = x P 0 \begin{cases}\displaystyle\frac{d\vec{x}_{P}}{dt}=\vec{u}_{P}(\vec{x}_{P},t)\\ \vec{x}_{P}(t_{0})=\vec{x}_{P0}\end{cases}
  10. P P
  11. x P \vec{x}_{P}
  12. u \vec{u}
  13. x P \vec{x}_{P}
  14. t t
  15. { d x P d t = u P ( x P , t ) x P ( t = τ P ) = x P 0 \begin{cases}\displaystyle\frac{d\vec{x}_{P}}{dt}=\vec{u}_{P}(\vec{x}_{P},t)\\ \vec{x}_{P}(t=\tau_{P})=\vec{x}_{P0}\end{cases}
  16. u P \vec{u}_{P}
  17. P P
  18. x P \vec{x}_{P}
  19. t t
  20. τ P \tau_{P}
  21. x P ( t , τ P ) \vec{x}_{P}(t,\tau_{P})
  22. 0 τ P t 0 0\leq\tau_{P}\leq t_{0}
  23. t 0 t_{0}
  24. a 0 a_{0}
  25. x \vec{x}
  26. a 0 a_{0}
  27. x \vec{x}
  28. a 0 a_{0}

Strength_of_materials.html

  1. σ = F A \sigma=\frac{F}{A}
  2. Δ σ = σ max - σ min \Delta\sigma=\sigma_{\mathrm{max}}-\sigma_{\mathrm{min}}
  3. F S = U T S / R FS=UTS/R
  4. R = U T S / F S R=UTS/FS
  5. R R

Stress_(mechanics).html

  1. F F
  2. σ \sigma
  3. σ \sigma
  4. σ \sigma
  5. τ \tau
  6. d d
  7. d d
  8. T T
  9. n n
  10. T = s y m b o l σ ( n ) T=symbol{\sigma}(n)
  11. s y m b o l σ symbol{\sigma}
  12. s y m b o l σ ( α u + β v ) = \alphasymbol σ ( u ) + \betasymbol σ ( v ) symbol{\sigma}(\alpha u+\beta v)=\alphasymbol{\sigma}(u)+\betasymbol{\sigma}(v)
  13. u , v u,v
  14. α , β \alpha,\beta
  15. s y m b o l σ symbol{\sigma}
  16. s y m b o l σ symbol{\sigma}
  17. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  18. x , y , z x,y,z
  19. [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] \begin{bmatrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{bmatrix}\quad\quad\quad
  20. [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] \quad\quad\quad\begin{bmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\ \end{bmatrix}
  21. T = s y m b o l σ ( n ) T=symbol{\sigma}(n)
  22. n n
  23. n 1 , n 2 , n 3 n_{1},n_{2},n_{3}
  24. T = n \cdotsymbol σ = s y m b o l σ T n T T=n\cdotsymbol{\sigma}=symbol{\sigma}^{T}\cdot n^{T}
  25. [ T 1 T 2 T 3 ] = [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] [ n 1 n 2 n 3 ] \begin{bmatrix}T_{1}\\ T_{2}\\ T_{3}\end{bmatrix}=\begin{bmatrix}\sigma_{11}&\sigma_{21}&\sigma_{31}\\ \sigma_{12}&\sigma_{22}&\sigma_{32}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{bmatrix}\begin{bmatrix}n_{1}\\ n_{2}\\ n_{3}\end{bmatrix}
  26. T T
  27. n n
  28. σ 12 = σ 21 \sigma_{12}=\sigma_{21}
  29. σ 13 = σ 31 \sigma_{13}=\sigma_{31}
  30. σ 23 = σ 32 \sigma_{23}=\sigma_{32}
  31. [ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] \begin{bmatrix}\sigma_{x}&\tau_{xy}&\tau_{xz}\\ \tau_{xy}&\sigma_{y}&\tau_{yz}\\ \tau_{xz}&\tau_{yz}&\sigma_{z}\end{bmatrix}
  32. σ x , σ y , σ z \sigma_{x},\sigma_{y},\sigma_{z}
  33. τ x y , τ x z , τ y z \tau_{xy},\tau_{xz},\tau_{yz}
  34. s y m b o l σ symbol{\sigma}
  35. e 1 , e 2 , e 3 e_{1},e_{2},e_{3}
  36. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  37. s y m b o l σ e i = λ i e i symbol{\sigma}e_{i}=\lambda_{i}e_{i}
  38. e 1 , e 2 , e 3 e_{1},e_{2},e_{3}
  39. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  40. s y m b o l σ symbol{\sigma}
  41. s y m b o l P symbol{P}
  42. s y m b o l P symbol{P}
  43. s y m b o l P = J s y m b o l σ s y m b o l F - T symbol{P}=J~{}symbol{\sigma}~{}symbol{F}^{-T}~{}
  44. s y m b o l F symbol{F}
  45. J = \detsymbol F J=\detsymbol{F}
  46. P i L = J σ i k F L k - 1 = J σ i k X L x k P_{iL}=J~{}\sigma_{ik}~{}F^{-1}_{Lk}=J~{}\sigma_{ik}~{}\cfrac{\partial X_{L}}{% \partial x_{k}}~{}\,\!
  47. s y m b o l S symbol{S}
  48. s y m b o l S = J s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T . symbol{S}=J~{}symbol{F}^{-1}\cdotsymbol{\sigma}\cdotsymbol{F}^{-T}~{}.
  49. S I L = J F I k - 1 F L m - 1 σ k m = J X I x k X L x m σ k m S_{IL}=J~{}F^{-1}_{Ik}~{}F^{-1}_{Lm}~{}\sigma_{km}=J~{}\cfrac{\partial X_{I}}{% \partial x_{k}}~{}\cfrac{\partial X_{L}}{\partial x_{m}}~{}\sigma_{km}\!\,\!

Stress–energy_tensor.html

  1. T α β = T β α . T^{\alpha\beta}=T^{\beta\alpha}.
  2. T μ ν = ( T 00 T 01 T 02 T 03 T 10 T 11 T 12 T 13 T 20 T 21 T 22 T 23 T 30 T 31 T 32 T 33 ) . T^{\mu\nu}=\begin{pmatrix}T^{00}&T^{01}&T^{02}&T^{03}\\ T^{10}&T^{11}&T^{12}&T^{13}\\ T^{20}&T^{21}&T^{22}&T^{23}\\ T^{30}&T^{31}&T^{32}&T^{33}\end{pmatrix}.
  3. T 00 = ρ , T^{00}=\rho,
  4. T 00 = ϵ 0 2 ( E 2 c 2 + B 2 ) , T^{00}={\epsilon_{0}\over 2}\left({E^{2}\over c^{2}}+B^{2}\right),
  5. T 0 i = T i 0 . T^{0i}=T^{i0}.
  6. T i k T^{ik}
  7. T i i T^{ii}
  8. T i k i k T^{ik}\quad i\neq k
  9. T μ ν = T α β g α μ g β ν , T_{\mu\nu}=T^{\alpha\beta}g_{\alpha\mu}g_{\beta\nu},
  10. T μ = ν T μ α g α ν , T^{\mu}{}_{\nu}=T^{\mu\alpha}g_{\alpha\nu},
  11. 𝔗 μ = ν T μ - g ν . \mathfrak{T}^{\mu}{}_{\nu}=T^{\mu}{}_{\nu}\sqrt{-g}\,.
  12. 0 = T μ ν = ; ν ν T μ ν . 0=T^{\mu\nu}{}_{;\nu}=\nabla_{\nu}T^{\mu\nu}{}.\!
  13. 0 = T μ ν = , ν ν T μ ν . 0=T^{\mu\nu}{}_{,\nu}=\partial_{\nu}T^{\mu\nu}.\!
  14. 0 = N T μ ν d 3 s ν 0=\int_{\partial N}T^{\mu\nu}\mathrm{d}^{3}s_{\nu}\!
  15. N \partial N
  16. d 3 s ν \mathrm{d}^{3}s_{\nu}
  17. 0 = ( x α T μ ν - x μ T α ν ) , ν . 0=(x^{\alpha}T^{\mu\nu}-x^{\mu}T^{\alpha\nu})_{,\nu}.\!
  18. 0 = div T = T μ ν = ; ν ν T μ ν = T μ ν + , ν Γ μ T σ ν σ ν + Γ ν T μ σ σ ν 0=\operatorname{div}T=T^{\mu\nu}{}_{;\nu}=\nabla_{\nu}T^{\mu\nu}=T^{\mu\nu}{}_% {,\nu}+\Gamma^{\mu}{}_{\sigma\nu}T^{\sigma\nu}+\Gamma^{\nu}{}_{\sigma\nu}T^{% \mu\sigma}
  19. Γ μ σ ν \Gamma^{\mu}{}_{\sigma\nu}
  20. ξ μ \xi^{\mu}
  21. 0 = ν ( ξ μ T μ ν ) = 1 - g ν ( - g ξ μ T μ ν ) 0=\nabla_{\nu}(\xi^{\mu}T_{\mu}^{\nu})=\frac{1}{\sqrt{-g}}\partial_{\nu}(\sqrt% {-g}\ \xi^{\mu}T_{\mu}^{\nu})
  22. 0 = N - g ξ μ T μ ν d 3 s ν = N ξ μ 𝔗 μ ν d 3 s ν 0=\int_{\partial N}\sqrt{-g}\ \xi^{\mu}T_{\mu}^{\nu}\ \mathrm{d}^{3}s_{\nu}=% \int_{\partial N}\xi^{\mu}\mathfrak{T}_{\mu}^{\nu}\ \mathrm{d}^{3}s_{\nu}
  23. R μ ν - 1 2 R g μ ν = 8 π G c 4 T μ ν , R_{\mu\nu}-{1\over 2}R\,g_{\mu\nu}={8\pi G\over c^{4}}T_{\mu\nu},
  24. R μ ν R_{\mu\nu}
  25. R R
  26. g μ ν g_{\mu\nu}\,
  27. G G
  28. 𝐱 p ( t ) \mathbf{x}\text{p}(t)
  29. T α β ( 𝐱 , t ) = m v α ( t ) v β ( t ) 1 - ( v / c ) 2 δ ( 𝐱 - 𝐱 p ( t ) ) = E v α ( t ) v β ( t ) c 2 δ ( 𝐱 - 𝐱 p ( t ) ) T^{\alpha\beta}(\mathbf{x},t)=\frac{m\,v^{\alpha}(t)v^{\beta}(t)}{\sqrt{1-(v/c% )^{2}}}\;\,\delta(\mathbf{x}-\mathbf{x}\text{p}(t))=E\frac{v^{\alpha}(t)v^{% \beta}(t)}{c^{2}}\;\,\delta(\mathbf{x}-\mathbf{x}\text{p}(t))
  30. v α v^{\alpha}\!
  31. v α = ( 1 , d 𝐱 p d t ( t ) ) , v^{\alpha}=\left(1,\frac{d\mathbf{x}\text{p}}{dt}(t)\right)\,,
  32. E = p 2 c 2 + m 2 c 4 E=\sqrt{p^{2}c^{2}+m^{2}c^{4}}
  33. T α β = ( ρ + p c 2 ) u α u β + p g α β T^{\alpha\beta}\,=\left(\rho+{p\over c^{2}}\right)u^{\alpha}u^{\beta}+pg^{% \alpha\beta}
  34. ρ \rho
  35. p p
  36. u α u^{\alpha}
  37. g α β g^{\alpha\beta}
  38. u α u β g α β = - c 2 . u^{\alpha}u^{\beta}g_{\alpha\beta}=-c^{2}\,.
  39. u α = ( 1 , 0 , 0 , 0 ) , u^{\alpha}=(1,0,0,0)\,,
  40. g α β = ( - c - 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) g^{\alpha\beta}\,=\left(\begin{matrix}-c^{-2}&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}\right)\,
  41. T α β = ( ρ 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p ) . T^{\alpha\beta}=\left(\begin{matrix}\rho&0&0&0\\ 0&p&0&0\\ 0&0&p&0\\ 0&0&0&p\end{matrix}\right).
  42. T μ ν = 1 μ 0 ( F μ α g α β F ν β - 1 4 g μ ν F δ γ F δ γ ) T^{\mu\nu}=\frac{1}{\mu_{0}}\left(F^{\mu\alpha}g_{\alpha\beta}F^{\nu\beta}-% \frac{1}{4}g^{\mu\nu}F_{\delta\gamma}F^{\delta\gamma}\right)
  43. F μ ν F_{\mu\nu}
  44. ϕ \phi
  45. T μ ν = 2 m ( g μ α g ν β + g μ β g ν α - g μ ν g α β ) α ϕ ¯ β ϕ - g μ ν m c 2 ϕ ¯ ϕ . T^{\mu\nu}=\frac{\hbar^{2}}{m}(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu% \alpha}-g^{\mu\nu}g^{\alpha\beta})\partial_{\alpha}\bar{\phi}\partial_{\beta}% \phi-g^{\mu\nu}mc^{2}\bar{\phi}\phi.
  46. T μ ν = 2 - g δ ( matter - g ) δ g μ ν = 2 δ matter δ g μ ν + g μ ν matter . T^{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L}_{\mathrm{matter}}\sqrt{% -g})}{\delta g_{\mu\nu}}=2\frac{\delta\mathcal{L}_{\mathrm{matter}}}{\delta g_% {\mu\nu}}+g^{\mu\nu}\mathcal{L}_{\mathrm{matter}}.
  47. matter \mathcal{L}_{\mathrm{matter}}

Stress–strain_curve.html

  1. σ \sigma
  2. ϵ \epsilon
  3. stress = F A \mathrm{stress}=\tfrac{F}{A}

Strike_price.html

  1. max [ ( S - K ) ; 0 ] \max\left[(S-K);0\right]
  2. ( S - K ) + (S-K)^{+}
  3. ( x ) + = { 0 x < 0 x x 0 (x)^{+}=\{^{x\ \ x\geq 0}_{0\ \ x<0}
  4. max [ ( K - S ) ; 0 ] \max\left[(K-S);0\right]
  5. ( K - S ) + (K-S)^{+}
  6. 1 S K 1_{S\geq K}
  7. 1 { } 1_{\{\}}

String_field_theory.html

  1. n n
  2. | Ψ = d 26 p ( T ( p ) c 1 e i p X | 0 + A μ ( p ) X μ c 1 e i p X | 0 + χ ( p ) c 0 e i p X | 0 + ) , |\Psi\rangle=\int d^{26}p\left(T(p)c_{1}e^{ip\cdot X}|0\rangle+A_{\mu}(p)% \partial X^{\mu}c_{1}e^{ip\cdot X}|0\rangle+\chi(p)c_{0}e^{ip\cdot X}|0\rangle% +\ldots\right),
  3. | 0 |0\rangle
  4. T ( p ) T(p)
  5. A μ ( p ) A_{\mu}(p)
  6. χ ( p ) \chi(p)
  7. T T
  8. A μ A_{\mu}
  9. χ \chi
  10. Q B | Ψ = 0 Q_{B}|\Psi\rangle=0
  11. | Ψ | Ψ + Q B | Λ |\Psi\rangle\sim|\Psi\rangle+Q_{B}|\Lambda\rangle
  12. | Ψ |\Psi\rangle
  13. | Ψ |\Psi\rangle
  14. S free open ( Ψ ) = 1 2 Ψ | Q B | Ψ , S_{\,\text{free open}}(\Psi)=\tfrac{1}{2}\langle\Psi|Q_{B}|\Psi\rangle\ ,
  15. Ψ | \langle\Psi|
  16. | Ψ |\Psi\rangle
  17. ( L 0 - L ~ 0 ) | Ψ = 0 (L_{0}-\tilde{L}_{0})|\Psi\rangle=0
  18. ( b 0 - b ~ 0 ) | Ψ = 0 (b_{0}-\tilde{b}_{0})|\Psi\rangle=0
  19. S free closed = 1 2 Ψ | ( c 0 - c ~ 0 ) Q B | Ψ . S_{\,\text{free closed}}=\tfrac{1}{2}\langle\Psi|(c_{0}-\tilde{c}_{0})Q_{B}|% \Psi\rangle\ .
  20. S ( Ψ ) = 1 2 Ψ | Q B | Ψ + 1 3 Ψ , Ψ , Ψ S(\Psi)=\tfrac{1}{2}\langle\Psi|Q_{B}|\Psi\rangle+\tfrac{1}{3}\langle\Psi,\Psi% ,\Psi\rangle
  21. Ψ \Psi
  22. Ψ 1 , Ψ 2 , Ψ 3 \langle\Psi_{1},\Psi_{2},\Psi_{3}\rangle
  23. * *
  24. Σ | Ψ 1 * Ψ 2 = Σ , Ψ 1 , Ψ 2 . \langle\Sigma|\Psi_{1}*\Psi_{2}\rangle=\langle\Sigma,\Psi_{1},\Psi_{2}\rangle\ .
  25. * *
  26. Ψ i \Psi_{i}
  27. g n ( Ψ ) gn(\Psi)
  28. Ψ \Psi
  29. S ( Ψ ) S(\Psi)
  30. Ψ Ψ + Q B Λ + Ψ * Λ - Λ * Ψ , \Psi\to\Psi+Q_{B}\Lambda+\Psi*\Lambda-\Lambda*\Psi\ ,
  31. Λ \Lambda
  32. Ψ e - Λ ( Ψ + Q B ) e Λ \Psi\to e^{-\Lambda}(\Psi+Q_{B})e^{\Lambda}
  33. e Λ = 1 + Λ + 1 2 Λ * Λ + 1 3 ! Λ * Λ * Λ + e^{\Lambda}=1+\Lambda+\tfrac{1}{2}\Lambda*\Lambda+\tfrac{1}{3!}\Lambda*\Lambda% *\Lambda+\ldots
  34. Q B Ψ + Ψ * Ψ = 0 . Q_{B}\Psi+\Psi*\Psi=0\left.\right.\ .
  35. Ψ \Psi
  36. S ( Ψ ) S(\Psi)
  37. b 0 Ψ = 0 . b_{0}\Psi=0\left.\right.\ .
  38. S gauge-fixed = 1 2 Ψ | c 0 L 0 | Ψ + 1 3 Ψ , Ψ , Ψ , S_{\,\text{gauge-fixed}}=\tfrac{1}{2}\langle\Psi|c_{0}L_{0}|\Psi\rangle+\tfrac% {1}{3}\langle\Psi,\Psi,\Psi\rangle\ ,
  39. Ψ \Psi
  40. π \pi
  41. T T
  42. b b
  43. T T
  44. \infty
  45. η 0 | Ψ = 0 \eta_{0}|\Psi\rangle=0
  46. S ( Ψ ) = 1 2 Ψ | Y ( i ) Y ( - i ) Q B | Ψ + 1 3 Ψ | Y ( i ) Y ( - i ) | Ψ * Ψ , S(\Psi)=\tfrac{1}{2}\langle\Psi|Y(i)Y(-i)Q_{B}|\Psi\rangle+\tfrac{1}{3}\langle% \Psi|Y(i)Y(-i)|\Psi*\Psi\rangle\ ,
  47. Y ( z ) = - ξ e - 2 ϕ c ( z ) Y(z)=-\partial\xi e^{-2\phi}c(z)
  48. - 1 2 -\tfrac{1}{2}
  49. Y ( i ) Y ( - i ) Q B Ψ = 0 . Y(i)Y(-i)Q_{B}\Psi=0\left.\right.\ .
  50. Y ( i ) Y ( - i ) Y(i)Y(-i)
  51. Q B Q_{B}
  52. S = 1 2 e - Φ Q B e Φ | e - Φ η 0 e Φ - 1 2 0 1 d t e - Φ ^ t e Φ ^ | { e - Φ ^ Q B e Φ ^ , e - Φ ^ η 0 e Φ ^ } S=\tfrac{1}{2}\langle e^{-\Phi}Q_{B}e^{\Phi}|e^{-\Phi}\eta_{0}e^{\Phi}\rangle-% \tfrac{1}{2}\int_{0}^{1}dt\langle e^{-\hat{\Phi}}\partial_{t}e^{\hat{\Phi}}|\{% e^{-\hat{\Phi}}Q_{B}e^{\hat{\Phi}},e^{-\hat{\Phi}}\eta_{0}e^{\hat{\Phi}}\}\rangle
  53. * *
  54. { , } \{,\}
  55. Φ ^ ( t ) \hat{\Phi}(t)
  56. Φ ^ ( 0 ) = 0 \hat{\Phi}(0)=0
  57. Φ ^ ( 1 ) = Φ \hat{\Phi}(1)=\Phi
  58. Φ \Phi
  59. ξ \xi
  60. η 0 ( e - Φ Q B e Φ ) = 0. \eta_{0}\left(e^{-\Phi}Q_{B}e^{\Phi}\right)=0.
  61. e Φ e Q B Λ e Φ e η 0 Λ . e^{\Phi}\to e^{Q_{B}\Lambda}e^{\Phi}e^{\eta_{0}\Lambda^{\prime}}.
  62. \hbar
  63. S ( Ψ ) = g 0 ( g c ) g - 1 n 0 1 n ! { Ψ n } g S(\Psi)=\hbar\sum_{g\geq 0}(\hbar g_{c})^{g-1}\sum_{n\geq 0}\frac{1}{n!}\{\Psi% ^{n}\}_{g}
  64. { Ψ n } g \{\Psi^{n}\}_{g}
  65. n n
  66. g g
  67. g c g_{c}

Strong_cardinal.html

  1. V λ M V_{\lambda}\subseteq M

Structural_analysis.html

  1. M A = 0 = - 10 * 1 + 2 * R B R B = 5 \sum M_{A}=0=-10*1+2*R_{B}\Rightarrow R_{B}=5
  2. F y = 0 = R A y + R B - 10 R A y = 5 \sum F_{y}=0=R_{Ay}+R_{B}-10\Rightarrow R_{Ay}=5
  3. F x = 0 = R A x \sum F_{x}=0=R_{Ax}
  4. F y = 0 = R A y + F A D sin ( 60 ) = 5 + F A D 3 2 F A D = - 10 3 \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)=5+F_{AD}\frac{\sqrt{3}}{2}\Rightarrow F_{AD% }=-\frac{10}{\sqrt{3}}
  5. F x = 0 = R A x + F A D cos ( 60 ) + F A B = 0 - 10 3 1 2 + F A B F A B = 5 3 \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-\frac{10}{\sqrt{3}}\frac{1}{2}+F_{% AB}\Rightarrow F_{AB}=\frac{5}{\sqrt{3}}
  6. F y = 0 = - 10 - F A D sin ( 60 ) - F B D sin ( 60 ) = - 10 - ( - 10 3 ) 3 2 - F B D 3 2 F B D = - 10 3 \sum F_{y}=0=-10-F_{AD}\sin(60)-F_{BD}\sin(60)=-10-\left(-\frac{10}{\sqrt{3}}% \right)\frac{\sqrt{3}}{2}-F_{BD}\frac{\sqrt{3}}{2}\Rightarrow F_{BD}=-\frac{10% }{\sqrt{3}}
  7. F x = 0 = - F A D cos ( 60 ) + F B D cos ( 60 ) + F C D = - 10 3 1 2 + 10 3 1 2 + F C D F C D = 0 \sum F_{x}=0=-F_{AD}\cos(60)+F_{BD}\cos(60)+F_{CD}=-\frac{10}{\sqrt{3}}\frac{1% }{2}+\frac{10}{\sqrt{3}}\frac{1}{2}+F_{CD}\Rightarrow F_{CD}=0
  8. F y = 0 = - F B C F B C = 0 \sum F_{y}=0=-F_{BC}\Rightarrow F_{BC}=0
  9. F x = - F C D = - 0 = 0 v e r i f i e d \sum F_{x}=-F_{CD}=-0=0\Rightarrow verified
  10. F y = R B + F B D sin ( 60 ) + F B C = 5 + ( - 10 3 ) 3 2 + 0 = 0 v e r i f i e d \sum F_{y}=R_{B}+F_{BD}\sin(60)+F_{BC}=5+\left(-\frac{10}{\sqrt{3}}\right)% \frac{\sqrt{3}}{2}+0=0\Rightarrow verified
  11. F x = - F A B - F B D cos ( 60 ) = 5 3 - 10 3 1 2 = 0 v e r i f i e d \sum F_{x}=-F_{AB}-F_{BD}\cos(60)=\frac{5}{\sqrt{3}}-\frac{10}{\sqrt{3}}\frac{% 1}{2}=0\Rightarrow verified
  12. M D = 0 = - 5 * 1 + 3 * F A B F A B = 5 3 \sum M_{D}=0=-5*1+\sqrt{3}*F_{AB}\Rightarrow F_{AB}=\frac{5}{\sqrt{3}}
  13. F y = 0 = R A y - F B D sin ( 60 ) - 10 = 5 - F B D 3 2 - 10 F B D = - 10 3 \sum F_{y}=0=R_{Ay}-F_{BD}\sin(60)-10=5-F_{BD}\frac{\sqrt{3}}{2}-10\Rightarrow F% _{BD}=-\frac{10}{\sqrt{3}}
  14. F x = 0 = F A B + F B D cos ( 60 ) + F C D = 5 3 - 10 3 1 2 + F C D F C D = 0 \sum F_{x}=0=F_{AB}+F_{BD}\cos(60)+F_{CD}=\frac{5}{\sqrt{3}}-\frac{10}{\sqrt{3% }}\frac{1}{2}+F_{CD}\Rightarrow F_{CD}=0
  15. M B = 0 = 3 * F C D F C D = 0 \sum M_{B}=0=\sqrt{3}*F_{CD}\Rightarrow F_{CD}=0
  16. F y = 0 = F B D sin ( 60 ) + R B = F B D 3 2 + 5 F B D = - 10 3 \sum F_{y}=0=F_{BD}\sin(60)+R_{B}=F_{BD}\frac{\sqrt{3}}{2}+5\Rightarrow F_{BD}% =-\frac{10}{\sqrt{3}}
  17. F x = 0 = - F A B - F B D cos ( 60 ) - F C D = - F A B - ( - 10 3 ) 1 2 - 0 F A B = 5 3 \sum F_{x}=0=-F_{AB}-F_{BD}\cos(60)-F_{CD}=-F_{AB}-\left(-\frac{10}{\sqrt{3}}% \right)\frac{1}{2}-0\Rightarrow F_{AB}=\frac{5}{\sqrt{3}}

Student's_t-distribution.html

  1. 1 2 + x Γ ( ν + 1 2 ) × F 1 2 ( 1 2 , ν + 1 2 ; 3 2 ; - x 2 ν ) π ν Γ ( ν 2 ) \begin{matrix}\frac{1}{2}+x\Gamma\left(\frac{\nu+1}{2}\right)\times\\ \frac{\,{}_{2}F_{1}\left(\frac{1}{2},\frac{\nu+1}{2};\frac{3}{2};-\frac{x^{2}}% {\nu}\right)}{\sqrt{\pi\nu}\,\Gamma\left(\frac{\nu}{2}\right)}\end{matrix}
  2. ν ν - 2 \textstyle\frac{\nu}{\nu-2}
  3. 6 ν - 4 \textstyle\frac{6}{\nu-4}
  4. K ν / 2 ( ν | t | ) ( ν | t | ) ν / 2 Γ ( ν / 2 ) 2 ν / 2 - 1 \textstyle\frac{K_{\nu/2}\left(\sqrt{\nu}|t|\right)\cdot\left(\sqrt{\nu}|t|% \right)^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}}
  5. n \sqrt{n}
  6. f ( t ) = Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) ( 1 + t 2 ν ) - ν + 1 2 , f(t)=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})}\left% (1+\frac{t^{2}}{\nu}\right)^{-\frac{\nu+1}{2}},\!
  7. ν \nu
  8. Γ \Gamma
  9. f ( t ) = 1 ν B ( 1 2 , ν 2 ) ( 1 + t 2 ν ) - ν + 1 2 , f(t)=\frac{1}{\sqrt{\nu}\,B\left(\frac{1}{2},\frac{\nu}{2}\right)}\left(1+% \frac{t^{2}}{\nu}\right)^{-\frac{\nu+1}{2}}\!,
  10. ν \nu
  11. Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) = ( ν - 1 ) ( ν - 3 ) 5 3 2 ν ( ν - 2 ) ( ν - 4 ) 4 2 . \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})}=\frac{(% \nu-1)(\nu-3)\cdots 5\cdot 3}{2\sqrt{\nu}(\nu-2)(\nu-4)\cdots 4\cdot 2\,}.
  12. ν \nu
  13. Γ ( ν + 1 2 ) ν π Γ ( ν 2 ) = ( ν - 1 ) ( ν - 3 ) 4 2 π ν ( ν - 2 ) ( ν - 4 ) 5 3 . \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})}=\frac{(% \nu-1)(\nu-3)\cdots 4\cdot 2}{\pi\sqrt{\nu}(\nu-2)(\nu-4)\cdots 5\cdot 3\,}.\!
  14. ν \nu
  15. ν \nu
  16. F ( t ) = - t f ( u ) d u = 1 - 1 2 I x ( t ) ( ν 2 , 1 2 ) , F(t)=\int_{-\infty}^{t}f(u)\,du=1-\tfrac{1}{2}I_{x(t)}\left(\tfrac{\nu}{2},% \tfrac{1}{2}\right),
  17. x ( t ) = ν < m t p l > t 2 + ν . x(t)=\frac{\nu}{<}mtpl>{{t^{2}+\nu}}.
  18. - t f ( u ) d u = 1 2 + t Γ ( 1 2 ( ν + 1 ) ) π ν Γ ( ν 2 ) F 1 2 ( 1 2 , 1 2 ( ν + 1 ) ; 3 2 ; - t 2 ν ) \int_{-\infty}^{t}f(u)\,du=\tfrac{1}{2}+t\frac{\Gamma\left(\tfrac{1}{2}(\nu+1)% \right)}{\sqrt{\pi\nu}\,\Gamma\left(\tfrac{\nu}{2}\right)}{}_{2}F_{1}\left(% \tfrac{1}{2},\tfrac{1}{2}(\nu+1);\tfrac{3}{2};-\tfrac{t^{2}}{\nu}\right)
  19. F ( x ) = 1 2 + 1 π arctan ( x ) . F(x)=\tfrac{1}{2}+\tfrac{1}{\pi}\arctan(x).
  20. f ( x ) = 1 π ( 1 + x 2 ) . f(x)=\frac{1}{\pi(1+x^{2})}.
  21. F ( x ) = 1 2 + x 2 2 + x 2 . F(x)=\tfrac{1}{2}+\frac{x}{2\sqrt{2+x^{2}}}.
  22. f ( x ) = 1 ( 2 + x 2 ) 3 2 . f(x)=\frac{1}{\left(2+x^{2}\right)^{\frac{3}{2}}}.
  23. f ( x ) = 6 3 π ( 3 + x 2 ) 2 . f(x)=\frac{6\sqrt{3}}{\pi\left(3+x^{2}\right)^{2}}.
  24. f ( x ) = 1 2 π e - x 2 2 . f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}.
  25. x ¯ \displaystyle\bar{x}
  26. t = x ¯ - μ s / n . t=\frac{\bar{x}-\mu}{s/\sqrt{n}}.
  27. p ( μ D , I ) = p ( μ , σ 2 D , I ) d σ 2 = p ( μ D , σ 2 , I ) p ( σ 2 D , I ) d σ 2 \begin{aligned}\displaystyle p(\mu\mid D,I)=&\displaystyle\int p(\mu,\sigma^{2% }\mid D,I)\;d\sigma^{2}\\ \displaystyle=&\displaystyle\int p(\mu\mid D,\sigma^{2},I)\;p(\sigma^{2}\mid D% ,I)\;d\sigma^{2}\end{aligned}
  28. p ( μ σ 2 , I ) = const \scriptstyle{p(\mu\mid\sigma^{2},I)=\mbox{const}~{}}
  29. p ( σ 2 I ) 1 / σ 2 \scriptstyle{p(\sigma^{2}\mid I)\;\propto\;1/\sigma^{2}}
  30. p ( μ D , σ 2 , I ) N ( x ¯ , σ 2 / n ) p ( σ 2 D , I ) Scale - inv - χ 2 ( ν , s 2 ) \begin{aligned}\displaystyle p(\mu\mid D,\sigma^{2},I)\sim&\displaystyle N(% \bar{x},\sigma^{2}/n)\\ \displaystyle p(\sigma^{2}\mid D,I)\sim&\displaystyle\operatorname{Scale-inv-}% \chi^{2}(\nu,s^{2})\end{aligned}
  31. s 2 = ( x i - x ¯ ) 2 n - 1 s^{2}=\sum\frac{(x_{i}-\bar{x})^{2}}{n-1}
  32. p ( μ | D , I ) 0 1 σ 2 exp ( - 1 2 σ 2 n ( μ - x ¯ ) 2 ) σ - ν - 2 exp ( - ν s 2 / 2 σ 2 ) d σ 2 0 σ - ν - 3 exp ( - 1 2 σ 2 ( n ( μ - x ¯ ) 2 + ν s 2 ) ) d σ 2 \begin{aligned}\displaystyle p(\mu|D,I)&\displaystyle\propto\int_{0}^{\infty}% \frac{1}{\sqrt{\sigma^{2}}}\exp\left(-\frac{1}{2\sigma^{2}}n(\mu-\bar{x})^{2}% \right)\;\cdot\;\sigma^{-\nu-2}\exp(-\nu s^{2}/2\sigma^{2})\;d\sigma^{2}\\ &\displaystyle\propto\int_{0}^{\infty}\sigma^{-\nu-3}\exp\left(-\frac{1}{2% \sigma^{2}}\left(n(\mu-\bar{x})^{2}+\nu s^{2}\right)\right)\;d\sigma^{2}\end{aligned}
  33. z = A / 2 σ 2 \scriptstyle{z=A/2\sigma^{2}}
  34. A = n ( μ - x ¯ ) 2 + ν s 2 \scriptstyle{A=n(\mu-\bar{x})^{2}+\nu s^{2}}
  35. d z = - A 2 σ 4 d σ 2 , dz=-\frac{A}{2\sigma^{4}}d\sigma^{2},
  36. p ( μ | D , I ) A - ν + 1 2 0 z ( ν - 1 ) / 2 exp ( - z ) d z p(\mu|D,I)\propto\;A^{-\frac{\nu+1}{2}}\int_{0}^{\infty}z^{(\nu-1)/2}\exp(-z)% \,dz
  37. p ( μ D , I ) A - ν + 1 2 ( 1 + n ( μ - x ¯ ) 2 ν s 2 ) - ν + 1 2 \begin{aligned}\displaystyle p(\mu\mid D,I)\propto&\displaystyle\;A^{-\frac{% \nu+1}{2}}\\ \displaystyle\propto&\displaystyle\left(1+\frac{n(\mu-\bar{x})^{2}}{\nu s^{2}}% \right)^{-\frac{\nu+1}{2}}\end{aligned}
  38. t = μ - x ¯ s / n t=\frac{\mu-\bar{x}}{s/\sqrt{n}}
  39. T = Z V / ν = Z ν V , T=\frac{Z}{\sqrt{V/\nu}}=Z\sqrt{\frac{\nu}{V}},
  40. ( Z + μ ) ν V . (Z+\mu)\sqrt{\frac{\nu}{V}}.
  41. X ¯ n = 1 n ( X 1 + + X n ) \overline{X}_{n}=\frac{1}{n}(X_{1}+\cdots+X_{n})
  42. S n 2 = 1 n - 1 i = 1 n ( X i - X ¯ n ) 2 S_{n}^{\;2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}_{n}\right)^{2}
  43. V = ( n - 1 ) S n 2 σ 2 V=(n-1)\frac{S_{n}^{2}}{\sigma^{2}}
  44. Z = ( X ¯ n - μ ) n σ Z=\left(\overline{X}_{n}-\mu\right)\frac{\sqrt{n}}{\sigma}
  45. X ¯ n \overline{X}_{n}
  46. T Z V / v = ( X ¯ n - μ ) n S n , T\equiv\frac{Z}{\sqrt{V/v}}=\left(\overline{X}_{n}-\mu\right)\frac{\sqrt{n}}{S% _{n}},
  47. E ( ln ( ν + X 2 ) ) E(\ln(\nu+X^{2}))
  48. E ( T k ) = { 0 k odd , 0 < k < ν 1 π Γ ( ν 2 ) [ Γ ( k + 1 2 ) Γ ( ν - k 2 ) ν k 2 ] k even , 0 < k < ν . E(T^{k})=\begin{cases}0&k\,\text{ odd},\quad 0<k<\nu\\ \frac{1}{\sqrt{\pi}\Gamma\left(\frac{\nu}{2}\right)}\left[\Gamma\left(\frac{k+% 1}{2}\right)\Gamma\left(\frac{\nu-k}{2}\right)\nu^{\frac{k}{2}}\right]&k\,% \text{ even},\quad 0<k<\nu.\\ \end{cases}
  49. Y F ( ν 1 = 1 , ν 2 = ν ) Y\sim\mathrm{F}(\nu_{1}=1,\nu_{2}=\nu)
  50. A ( t | ν ) = F ν ( t ) - F ν ( - t ) = 1 - I ν ν + t 2 ( ν 2 , 1 2 ) , A(t|\nu)=F_{\nu}(t)-F_{\nu}(-t)=1-I_{\frac{\nu}{\nu+t^{2}}}\left(\frac{\nu}{2}% ,\frac{1}{2}\right),
  51. { ( ν + x 2 ) f ( x ) + ( ν + 1 ) x f ( x ) = 0 , f ( 1 ) = ν ν / 2 ( ν + 1 ) - ν 2 - 1 2 B ( ν 2 , 1 2 ) } \left\{\begin{array}[]{l}\left(\nu+x^{2}\right)f^{\prime}(x)+(\nu+1)xf(x)=0,\\ f(1)=\frac{\nu^{\nu/2}(\nu+1)^{-\frac{\nu}{2}-\frac{1}{2}}}{B\left(\frac{\nu}{% 2},\frac{1}{2}\right)}\end{array}\right\}
  52. μ \mu
  53. σ \sigma
  54. X = μ + σ T X=\mu+\sigma T
  55. T = X - μ σ T=\frac{X-\mu}{\sigma}
  56. x - μ σ \frac{x-\mu}{\sigma}
  57. ν \nu
  58. p ( x ν , μ , σ ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) π ν σ ( 1 + 1 ν ( x - μ σ ) 2 ) - ν + 1 2 p(x\mid\nu,\mu,\sigma)=\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})% \sqrt{\pi\nu}\sigma}\left(1+\frac{1}{\nu}\left(\frac{x-\mu}{\sigma}\right)^{2}% \right)^{-\frac{\nu+1}{2}}
  59. σ \sigma
  60. σ \sigma
  61. μ \mu
  62. σ \sigma
  63. s / n \scriptstyle{s/\sqrt{n}}
  64. s 2 = ( x i - x ¯ ) 2 n - 1 . s^{2}=\sum\frac{(x_{i}-\bar{x})^{2}}{n-1}.
  65. σ 2 \sigma^{2}
  66. p ( x ν , μ , σ 2 ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) π ν σ 2 ( 1 + 1 ν ( x - μ ) 2 σ 2 ) - ν + 1 2 p(x\mid\nu,\mu,\sigma^{2})=\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2}% )\sqrt{\pi\nu\sigma^{2}}}\left(1+\frac{1}{\nu}\frac{(x-\mu)^{2}}{\sigma^{2}}% \right)^{-\frac{\nu+1}{2}}
  67. E ( X ) = μ for ν > 1 , var ( X ) = σ 2 ν ν - 2 for ν > 2 , mode ( X ) = μ . \begin{aligned}\displaystyle\operatorname{E}(X)&\displaystyle=\mu\quad\quad% \quad\,\text{for }\,\nu>1,\\ \displaystyle\,\text{var}(X)&\displaystyle=\sigma^{2}\frac{\nu}{\nu-2}\,\quad% \,\text{for }\,\nu>2,\\ \displaystyle\,\text{mode}(X)&\displaystyle=\mu.\end{aligned}
  68. μ \mu
  69. a = ν / 2 a=\nu/2
  70. b = ν σ 2 / 2 b=\nu\sigma^{2}/2
  71. ν \nu
  72. σ 2 \sigma^{2}
  73. ν = 2 a , σ 2 = b / a \nu=2a,\sigma^{2}=b/a
  74. λ \lambda
  75. λ = 1 σ 2 \lambda=\frac{1}{\sigma^{2}}
  76. p ( x | ν , μ , λ ) = Γ ( ν + 1 2 ) Γ ( ν 2 ) ( λ π ν ) 1 2 ( 1 + λ ( x - μ ) 2 ν ) - ν + 1 2 . p(x|\nu,\mu,\lambda)=\frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu}{2})}% \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}}\left(1+\frac{\lambda(x-\mu)^% {2}}{\nu}\right)^{-\frac{\nu+1}{2}}.
  77. E ( X ) = μ for ν > 1 , var ( X ) = 1 λ ν ν - 2 for ν > 2 , mode ( X ) = μ . \begin{aligned}\displaystyle\operatorname{E}(X)&\displaystyle=\mu\quad\quad% \quad\,\text{for }\,\nu>1,\\ \displaystyle\,\text{var}(X)&\displaystyle=\frac{1}{\lambda}\frac{\nu}{\nu-2}% \,\quad\,\text{for }\,\nu>2,\\ \displaystyle\,\text{mode}(X)&\displaystyle=\mu.\end{aligned}
  78. μ \mu
  79. a = ν / 2 a=\nu/2
  80. b = ν / ( 2 λ ) b=\nu/(2\lambda)
  81. j = 1 k 1 ( r + j + a ) 2 + b 2 r = , - 1 , 0 , 1 , . \prod_{j=1}^{k}\frac{1}{(r+j+a)^{2}+b^{2}}\quad\quad r=\ldots,-1,0,1,\ldots.
  82. Pr ( - A < T < A ) = 0.9 , \Pr(-A<T<A)=0.9,
  83. Pr ( T < A ) = 0.95 , \Pr(T<A)=0.95,
  84. A = t ( 0.05 , n - 1 ) A=t_{(0.05,n-1)}
  85. Pr ( - A < X ¯ n - μ S n n < A ) = 0.9 , \Pr\left(-A<\frac{\overline{X}_{n}-\mu}{\frac{S_{n}}{\sqrt{n}}}<A\right)=0.9,
  86. Pr ( X ¯ n - A S n n < μ < X ¯ n + A S n n ) = 0.9. \Pr\left(\overline{X}_{n}-A\frac{S_{n}}{\sqrt{n}}<\mu<\overline{X}_{n}+A\frac{% S_{n}}{\sqrt{n}}\right)=0.9.
  87. X ¯ n ± A S n n \overline{X}_{n}\pm A\frac{S_{n}}{\sqrt{n}}
  88. UCL 1 - a = X ¯ n + t a , n - 1 S n n . \mathrm{UCL}_{1-a}=\overline{X}_{n}+t_{a,n-1}\frac{S_{n}}{\sqrt{n}}.
  89. X ¯ n \overline{X}_{n}
  90. \infty
  91. X ¯ n ± A S n n . \overline{X}_{n}\pm A\frac{S_{n}}{\sqrt{n}}.
  92. 10 + 1.37218 2 11 = 10.58510. 10+1.37218\frac{\sqrt{2}}{\sqrt{11}}=10.58510.
  93. 10 - 1.37218 2 11 = 9.41490. 10-1.37218\frac{\sqrt{2}}{\sqrt{11}}=9.41490.
  94. ( 10 - 1.37218 2 11 , 10 + 1.37218 2 11 ) = ( 9.41490 , 10.58510 ) . \left(10-1.37218\frac{\sqrt{2}}{\sqrt{11}},10+1.37218\frac{\sqrt{2}}{\sqrt{11}% }\right)=\left(9.41490,10.58510\right).

Subatomic_particle.html

  1. E = m c 2 E=mc^{2}\!
  2. 1 / 1836 {1}/{1836}

Subcategory.html

  1. Hom 𝒮 ( X , Y ) = Hom 𝒞 ( X , Y ) . \mathrm{Hom}_{\mathcal{S}}(X,Y)=\mathrm{Hom}_{\mathcal{C}}(X,Y).
  2. 0 M M M ′′ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0
  3. M M^{\prime}
  4. M ′′ M^{\prime\prime}

Subsequence.html

  1. A , B , D \langle A,B,D\rangle
  2. A , B , C , D , E , F \langle A,B,C,D,E,F\rangle
  3. A , B , C , D \langle A,B,C,D\rangle
  4. X = A , C , B , D , E , G , C , E , D , B , G X=\langle A,C,B,D,E,G,C,E,D,B,G\rangle
  5. Y = B , E , G , C , F , E , U , B , K Y=\langle B,E,G,C,F,E,U,B,K\rangle
  6. G = B , E , E . G=\langle B,E,E\rangle.
  7. B , E , E , B \langle B,E,E,B\rangle
  8. B , E , G , C , E , B \langle B,E,G,C,E,B\rangle

Subtraction.html

  1. 2 - 1 = 1 2-1=1
  2. 4 - 2 = 2 4-2=2
  3. 6 - 3 = 3 6-3=3
  4. 4 - 6 = - 2 4-6=-2
  5. - 15. -15.

Sufficient_statistic.html

  1. 𝐗 \mathbf{X}
  2. θ \theta
  3. T ( 𝐗 ) T(\mathbf{X})
  4. T ( 𝐗 ) T(\mathbf{X})
  5. p ( 𝐗 ) = h ( 𝐗 ) g ( θ , T ( 𝐗 ) ) p(\mathbf{X})=h(\mathbf{X})\,g(\theta,T(\mathbf{X}))\,
  6. θ \theta
  7. 𝐗 \mathbf{X}
  8. T ( 𝐗 ) T(\mathbf{X})
  9. Pr ( x | t , θ ) = Pr ( x | t ) . \Pr(x|t,\theta)=\Pr(x|t).\,
  10. Pr ( θ | t , x ) = Pr ( θ | t ) , \Pr(\theta|t,x)=\Pr(\theta|t),\,
  11. Pr ( θ , x | t ) = Pr ( θ | t ) Pr ( x | t ) , \Pr(\theta,x|t)=\Pr(\theta|t)\Pr(x|t),\,
  12. f θ ( x ) = h ( x ) g θ ( T ( x ) ) , f_{\theta}(x)=h(x)\,g_{\theta}(T(x)),\,\!
  13. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  14. i = 1 n f ( x i ; θ ) = g 1 [ u 1 ( x 1 , x 2 , , x n ) ; θ ] H ( x 1 , x 2 , , x n ) . \prod_{i=1}^{n}f(x_{i};\theta)=g_{1}\left[u_{1}(x_{1},x_{2},\dots,x_{n});% \theta\right]H(x_{1},x_{2},\dots,x_{n}).\,
  15. i = 1 n f ( x i ; θ ) = g 1 [ u 1 ( x 1 , x 2 , , x n ) ; θ ] H ( x 1 , x 2 , , x n ) . \prod_{i=1}^{n}f(x_{i};\theta)=g_{1}\left[u_{1}(x_{1},x_{2},\dots,x_{n});% \theta\right]H(x_{1},x_{2},\dots,x_{n}).\,
  16. J = [ w i / y j ] J=\left[w_{i}/y_{j}\right]
  17. i = 1 n f [ w i ( y 1 , y 2 , , y n ) ; θ ] = | J | g 1 ( y 1 ; θ ) H [ w 1 ( y 1 , y 2 , , y n ) , , w n ( y 1 , y 2 , , y n ) ] . \prod_{i=1}^{n}f\left[w_{i}(y_{1},y_{2},\dots,y_{n});\theta\right]=|J|g_{1}(y_% {1};\theta)H\left[w_{1}(y_{1},y_{2},\dots,y_{n}),\dots,w_{n}(y_{1},y_{2},\dots% ,y_{n})\right].
  18. g 1 ( y 1 ; θ ) g_{1}(y_{1};\theta)
  19. Y 1 Y_{1}
  20. H [ w 1 , , w n ] | J | H[w_{1},\dots,w_{n}]|J|
  21. g ( y 1 , , y n ; θ ) g(y_{1},\dots,y_{n};\theta)
  22. g 1 ( y 1 ; θ ) g_{1}(y_{1};\theta)
  23. h ( y 2 , , y n | y 1 ; θ ) h(y_{2},\dots,y_{n}|y_{1};\theta)
  24. Y 2 , , Y n Y_{2},\dots,Y_{n}
  25. Y 1 = y 1 Y_{1}=y_{1}
  26. H ( x 1 , x 2 , , x n ) H(x_{1},x_{2},\dots,x_{n})
  27. H [ w 1 ( y 1 , , y n ) , , w n ( y 1 , , y n ) ) ] H\left[w_{1}(y_{1},\dots,y_{n}),\dots,w_{n}(y_{1},\dots,y_{n}))\right]
  28. θ \theta
  29. θ \theta
  30. J J
  31. h ( y 2 , , y n | y 1 ; θ ) h(y_{2},\dots,y_{n}|y_{1};\theta)
  32. θ \theta
  33. Y 1 Y_{1}
  34. θ \theta
  35. g ( y 1 , , y n ; θ ) = g 1 ( y 1 ; θ ) h ( y 2 , , y n | y 1 ) , g(y_{1},\dots,y_{n};\theta)=g_{1}(y_{1};\theta)h(y_{2},\dots,y_{n}|y_{1}),\,
  36. h ( y 2 , , y n | y 1 ) h(y_{2},\dots,y_{n}|y_{1})
  37. θ \theta
  38. Y 2 Y n Y_{2}...Y_{n}
  39. X 1 X n X_{1}...X_{n}
  40. Θ \Theta
  41. Y 1 Y_{1}
  42. J J
  43. y 1 , , y n y_{1},\dots,y_{n}
  44. u 1 ( x 1 , , x n ) , , u n ( x 1 , , x n ) u_{1}(x_{1},\dots,x_{n}),\dots,u_{n}(x_{1},\dots,x_{n})
  45. x 1 , , x n x_{1},\dots,x_{n}
  46. g [ u 1 ( x 1 , , x n ) , , u n ( x 1 , , x n ) ; θ ] | J * | = g 1 [ u 1 ( x 1 , , x n ) ; θ ] h ( u 2 , , u n | u 1 ) | J * | \frac{g\left[u_{1}(x_{1},\dots,x_{n}),\dots,u_{n}(x_{1},\dots,x_{n});\theta% \right]}{|J*|}=g_{1}\left[u_{1}(x_{1},\dots,x_{n});\theta\right]\frac{h(u_{2},% \dots,u_{n}|u_{1})}{|J*|}
  47. J * J*
  48. y 1 , , y n y_{1},\dots,y_{n}
  49. x 1 , , x n x_{1},\dots,x_{n}
  50. f ( x 1 ; θ ) f ( x n ; θ ) f(x_{1};\theta)\cdots f(x_{n};\theta)
  51. X 1 , , X n X_{1},\dots,X_{n}
  52. h ( y 2 , , y n | y 1 ) h(y_{2},\dots,y_{n}|y_{1})
  53. h ( u 2 , , u n | u 1 ) h(u_{2},\dots,u_{n}|u_{1})
  54. θ \theta
  55. H ( x 1 , , x 2 ) = h ( u 2 , , u n | u 1 ) | J * | H(x_{1},\dots,x_{2})=\frac{h(u_{2},\dots,u_{n}|u_{1})}{|J*|}
  56. θ \theta
  57. ( X , T ( X ) ) (X,T(X))
  58. f θ ( x , t ) f_{\theta}(x,t)
  59. T T
  60. X X
  61. f θ ( x , t ) = f θ ( x ) f_{\theta}(x,t)=f_{\theta}(x)
  62. t = T ( x ) t=T(x)
  63. f θ ( x ) = f θ ( x , t ) = f θ | t ( x ) f θ ( t ) f_{\theta}(x)=f_{\theta}(x,t)=f_{\theta|t}(x)f_{\theta}(t)
  64. f θ ( x ) = a ( x ) b θ ( t ) f_{\theta}(x)=a(x)b_{\theta}(t)
  65. a ( x ) = f θ | t ( x ) a(x)=f_{\theta|t}(x)
  66. b θ ( t ) = f θ ( t ) b_{\theta}(t)=f_{\theta}(t)
  67. f θ ( x ) = a ( x ) b θ ( t ) f_{\theta}(x)=a(x)b_{\theta}(t)
  68. f θ ( t ) = x : T ( x ) = t f θ ( x , t ) = x : T ( x ) = t f θ ( x ) = x : T ( x ) = t a ( x ) b θ ( t ) = ( x : T ( x ) = t a ( x ) ) b θ ( t ) . \begin{aligned}\displaystyle f_{\theta}(t)&\displaystyle=\sum_{x:T(x)=t}f_{% \theta}(x,t)\\ &\displaystyle=\sum_{x:T(x)=t}f_{\theta}(x)\\ &\displaystyle=\sum_{x:T(x)=t}a(x)b_{\theta}(t)\\ &\displaystyle=\left(\sum_{x:T(x)=t}a(x)\right)b_{\theta}(t).\end{aligned}
  69. t t
  70. f θ | t ( x ) \displaystyle f_{\theta|t}(x)
  71. θ \theta
  72. T T
  73. f θ ( x ) f θ ( y ) \frac{f_{\theta}(x)}{f_{\theta}(y)}
  74. \Longleftrightarrow
  75. P θ P_{\theta}
  76. { L ( θ 1 | X ) L ( θ 2 | X ) } \left\{\frac{L(\theta_{1}|X)}{L(\theta_{2}|X)}\right\}
  77. P ( X | θ ) P(X|\theta)
  78. Pr { X = x } = Pr { X 1 = x 1 , X 2 = x 2 , , X n = x n } . \Pr\{X=x\}=\Pr\{X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n}\}.
  79. p x 1 ( 1 - p ) 1 - x 1 p x 2 ( 1 - p ) 1 - x 2 p x n ( 1 - p ) 1 - x n p^{x_{1}}(1-p)^{1-x_{1}}p^{x_{2}}(1-p)^{1-x_{2}}\cdots p^{x_{n}}(1-p)^{1-x_{n}% }\,\!
  80. p x i ( 1 - p ) n - x i = p T ( x ) ( 1 - p ) n - T ( x ) p^{\sum x_{i}}(1-p)^{n-\sum x_{i}}=p^{T(x)}(1-p)^{n-T(x)}\,\!
  81. f X ( x 1 , , x n ) = 1 θ 𝟏 { 0 x 1 θ } 1 θ 𝟏 { 0 x n θ } = 1 θ n 𝟏 { 0 min { x i } } 𝟏 { max { x i } θ } \begin{aligned}\displaystyle f_{X}(x_{1},\ldots,x_{n})&\displaystyle=\frac{1}{% \theta}\mathbf{1}_{\{0\leq x_{1}\leq\theta\}}\cdots\frac{1}{\theta}\mathbf{1}_% {\{0\leq x_{n}\leq\theta\}}\\ &\displaystyle=\frac{1}{\theta^{n}}\mathbf{1}_{\{0\leq\min\{x_{i}\}\}}\mathbf{% 1}_{\{\max\{x_{i}\}\leq\theta\}}\end{aligned}
  82. n + 1 n T ( X ) . \frac{n+1}{n}T(X).
  83. X 1 , , X n X_{1},...,X_{n}\,
  84. [ α , β ] [\alpha,\beta]\,
  85. α \alpha\,
  86. β \beta\,
  87. T ( X 1 n ) = ( min 1 i n X i , max 1 i n X i ) T(X_{1}^{n})=\left(\min_{1\leq i\leq n}X_{i},\max_{1\leq i\leq n}X_{i}\right)\,
  88. ( α , β ) (\alpha\,,\,\beta)
  89. X 1 n = ( X 1 , , X n ) X_{1}^{n}=(X_{1},\ldots,X_{n})
  90. f X 1 n ( x 1 n ) = i = 1 n ( 1 β - α ) 𝟏 { α x i β } = ( 1 β - α ) n 𝟏 { α x i β , i = 1 , , n } = ( 1 β - α ) n 𝟏 { α min 1 i n X i } 𝟏 { max 1 i n X i β } . \begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=\prod_{i=1% }^{n}\left({1\over\beta-\alpha}\right)\mathbf{1}_{\{\alpha\leq x_{i}\leq\beta% \}}=\left({1\over\beta-\alpha}\right)^{n}\mathbf{1}_{\{\alpha\leq x_{i}\leq% \beta,\,\forall\,i=1,\ldots,n\}}\\ &\displaystyle=\left({1\over\beta-\alpha}\right)^{n}\mathbf{1}_{\{\alpha\,\leq% \,\min_{1\leq i\leq n}X_{i}\}}\mathbf{1}_{\{\max_{1\leq i\leq n}X_{i}\,\leq\,% \beta\}}.\end{aligned}
  91. h ( x 1 n ) = 1 , g ( α , β ) ( x 1 n ) = ( 1 β - α ) n 𝟏 { α min 1 i n X i } 𝟏 { max 1 i n X i β } . \begin{aligned}\displaystyle h(x_{1}^{n})=1,\quad g_{(\alpha,\beta)}(x_{1}^{n}% )=\left({1\over\beta-\alpha}\right)^{n}\mathbf{1}_{\{\alpha\,\leq\,\min_{1\leq i% \leq n}X_{i}\}}\mathbf{1}_{\{\max_{1\leq i\leq n}X_{i}\,\leq\,\beta\}}.\end{aligned}
  92. h ( x 1 n ) h(x_{1}^{n})
  93. ( α , β ) (\alpha,\beta)
  94. g ( α , β ) ( x 1 n ) g_{(\alpha\,,\,\beta)}(x_{1}^{n})
  95. x 1 n x_{1}^{n}
  96. T ( X 1 n ) = ( min 1 i n X i , max 1 i n X i ) , T(X_{1}^{n})=\left(\min_{1\leq i\leq n}X_{i},\max_{1\leq i\leq n}X_{i}\right),\,
  97. T ( X 1 n ) = ( min 1 i n X i , max 1 i n X i ) T(X_{1}^{n})=\left(\min_{1\leq i\leq n}X_{i},\max_{1\leq i\leq n}X_{i}\right)\,
  98. ( α , β ) (\alpha\,,\,\beta)
  99. Pr ( X = x ) = P ( X 1 = x 1 , X 2 = x 2 , , X n = x n ) . \Pr(X=x)=P(X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n}).\,
  100. e - λ λ x 1 x 1 ! e - λ λ x 2 x 2 ! e - λ λ x n x n ! {e^{-\lambda}\lambda^{x_{1}}\over x_{1}!}\cdot{e^{-\lambda}\lambda^{x_{2}}% \over x_{2}!}\cdots{e^{-\lambda}\lambda^{x_{n}}\over x_{n}!}\,
  101. e - n λ λ ( x 1 + x 2 + + x n ) 1 x 1 ! x 2 ! x n ! e^{-n\lambda}\lambda^{(x_{1}+x_{2}+\cdots+x_{n})}\cdot{1\over x_{1}!x_{2}!% \cdots x_{n}!}\,
  102. X 1 , , X n X_{1},\dots,X_{n}
  103. σ 2 \sigma^{2}
  104. T ( X 1 n ) = X ¯ = 1 n i = 1 n X i T(X_{1}^{n})=\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}
  105. X 1 n = ( X 1 , , X n ) X_{1}^{n}=(X_{1},\dots,X_{n})
  106. f X 1 n ( x 1 n ) = i = 1 n 1 2 π σ 2 e - ( x i - θ ) 2 / ( 2 σ 2 ) = ( 2 π σ 2 ) - n / 2 e - i = 1 n ( x i - θ ) 2 / ( 2 σ 2 ) = ( 2 π σ 2 ) - n / 2 e - i = 1 n ( ( x i - x ¯ ) - ( θ - x ¯ ) ) 2 / ( 2 σ 2 ) = ( 2 π σ 2 ) - n / 2 exp ( - 1 2 σ 2 ( i = 1 n ( x i - x ¯ ) 2 + i = 1 n ( θ - x ¯ ) 2 - 2 i = 1 n ( x i - x ¯ ) ( θ - x ¯ ) ) ) . \begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=\prod_{i=1% }^{n}\tfrac{1}{\sqrt{2\pi\sigma^{2}}}\,e^{-(x_{i}-\theta)^{2}/(2\sigma^{2})}=(% 2\pi\sigma^{2})^{-n/2}\,e^{-\sum_{i=1}^{n}(x_{i}-\theta)^{2}/(2\sigma^{2})}\\ &\displaystyle=(2\pi\sigma^{2})^{-n/2}\,e^{-\sum_{i=1}^{n}((x_{i}-\overline{x}% )-(\theta-\overline{x}))^{2}/(2\sigma^{2})}\\ &\displaystyle=(2\pi\sigma^{2})^{-n/2}\,\exp\left({-1\over 2\sigma^{2}}\left(% \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}+\sum_{i=1}^{n}(\theta-\overline{x})^{2}% -2\sum_{i=1}^{n}(x_{i}-\overline{x})(\theta-\overline{x})\right)\right).\end{aligned}
  107. i = 1 n ( x i - x ¯ ) ( θ - x ¯ ) = 0 \sum_{i=1}^{n}(x_{i}-\overline{x})(\theta-\overline{x})=0
  108. f X 1 n ( x 1 n ) = ( 2 π σ 2 ) - n 2 e - 1 2 σ 2 ( i = 1 n ( x i - x ¯ ) 2 + n ( θ - x ¯ ) 2 ) = ( 2 π σ 2 ) - n 2 e - 1 2 σ 2 i = 1 n ( x i - x ¯ ) 2 e - n 2 σ 2 ( θ - x ¯ ) 2 . \begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=(2\pi% \sigma^{2})^{-n\over 2}\,e^{{-1\over 2\sigma^{2}}(\sum_{i=1}^{n}(x_{i}-% \overline{x})^{2}+n(\theta-\overline{x})^{2})}&\displaystyle=(2\pi\sigma^{2})^% {-n\over 2}\,e^{{-1\over 2\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\,% e^{{-n\over 2\sigma^{2}}(\theta-\overline{x})^{2}}.\end{aligned}
  109. h ( x 1 n ) = ( 2 π σ 2 ) - n 2 e - 1 2 σ 2 i = 1 n ( x i - x ¯ ) 2 , g θ ( x 1 n ) = e - n 2 σ 2 ( θ - x ¯ ) 2 . \begin{aligned}\displaystyle h(x_{1}^{n})=(2\pi\sigma^{2})^{-n\over 2}\,e^{{-1% \over 2\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}},\,\,\,g_{\theta}(x_{% 1}^{n})=e^{{-n\over 2\sigma^{2}}(\theta-\overline{x})^{2}}.\end{aligned}
  110. h ( x 1 n ) h(x_{1}^{n})
  111. θ \theta
  112. g θ ( x 1 n ) g_{\theta}(x_{1}^{n})
  113. x 1 n x_{1}^{n}
  114. T ( X 1 n ) = X ¯ = 1 n i = 1 n X i , T(X_{1}^{n})=\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i},
  115. T ( X 1 n ) = X ¯ = 1 n i = 1 n X i T(X_{1}^{n})=\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}
  116. θ \theta
  117. X 1 , , X n X_{1},\dots,X_{n}
  118. T ( X 1 n ) = i = 1 n X i T(X_{1}^{n})=\sum_{i=1}^{n}X_{i}
  119. X 1 n = ( X 1 , , X n ) X_{1}^{n}=(X_{1},\dots,X_{n})
  120. f X 1 n ( x 1 n ) = i = 1 n 1 θ e - 1 θ x i = 1 θ n e - 1 θ i = 1 n x i . \begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=\prod_{i=1% }^{n}{1\over\theta}\,e^{{-1\over\theta}x_{i}}={1\over\theta^{n}}\,e^{{-1\over% \theta}\sum_{i=1}^{n}x_{i}}.\end{aligned}
  121. h ( x 1 n ) = 1 , g θ ( x 1 n ) = 1 θ n e - 1 θ i = 1 n x i . \begin{aligned}\displaystyle h(x_{1}^{n})=1,\,\,\,g_{\theta}(x_{1}^{n})={1% \over\theta^{n}}\,e^{{-1\over\theta}\sum_{i=1}^{n}x_{i}}.\end{aligned}
  122. h ( x 1 n ) h(x_{1}^{n})
  123. θ \theta
  124. g θ ( x 1 n ) g_{\theta}(x_{1}^{n})
  125. x 1 n x_{1}^{n}
  126. T ( X 1 n ) = i = 1 n X i T(X_{1}^{n})=\sum_{i=1}^{n}X_{i}
  127. T ( X 1 n ) = i = 1 n X i T(X_{1}^{n})=\sum_{i=1}^{n}X_{i}
  128. θ \theta
  129. X 1 , , X n X_{1},\dots,X_{n}\,
  130. Γ ( α , β ) \Gamma(\alpha\,,\,\beta)\,\,
  131. α \alpha\,
  132. β \beta\,
  133. T ( X 1 n ) = ( i = 1 n x i , i = 1 n x i ) T(X_{1}^{n})=\left(\prod_{i=1}^{n}{x_{i}},\sum_{i=1}^{n}x_{i}\right)\,
  134. ( α , β ) (\alpha,\beta)
  135. X 1 n = ( X 1 , , X n ) X_{1}^{n}=(X_{1},\dots,X_{n})
  136. f X 1 n ( x 1 n ) = i = 1 n ( 1 Γ ( α ) β α ) x i α - 1 e - 1 β x i = ( 1 Γ ( α ) β α ) n ( i = 1 n x i ) α - 1 e - 1 β i = 1 n x i . \begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=\prod_{i=1% }^{n}\left({1\over\Gamma(\alpha)\beta^{\alpha}}\right)x_{i}^{\alpha-1}e^{{-1% \over\beta}x_{i}}&\displaystyle=\left({1\over\Gamma(\alpha)\beta^{\alpha}}% \right)^{n}\left(\prod_{i=1}^{n}x_{i}\right)^{\alpha-1}e^{{-1\over\beta}\sum_{% i=1}^{n}{x_{i}}}.\end{aligned}
  137. h ( x 1 n ) = 1 , g ( α , β ) ( x 1 n ) = ( 1 Γ ( α ) β α ) n ( i = 1 n x i ) α - 1 e - 1 β i = 1 n x i . \begin{aligned}\displaystyle h(x_{1}^{n})=1,\,\,\,g_{(\alpha\,,\,\beta)}(x_{1}% ^{n})=\left({1\over\Gamma(\alpha)\beta^{\alpha}}\right)^{n}\left(\prod_{i=1}^{% n}x_{i}\right)^{\alpha-1}e^{{-1\over\beta}\sum_{i=1}^{n}{x_{i}}}.\end{aligned}
  138. h ( x 1 n ) h(x_{1}^{n})
  139. ( α , β ) (\alpha\,,\,\beta)
  140. g ( α , β ) ( x 1 n ) g_{(\alpha\,,\,\beta)}(x_{1}^{n})
  141. x 1 n x_{1}^{n}
  142. T ( X 1 n ) = ( i = 1 n x i , i = 1 n x i ) , T(X_{1}^{n})=\left(\prod_{i=1}^{n}{x_{i}},\sum_{i=1}^{n}{x_{i}}\right),
  143. T ( X 1 n ) = ( i = 1 n x i , i = 1 n x i ) T(X_{1}^{n})=\left(\prod_{i=1}^{n}{x_{i}},\sum_{i=1}^{n}{x_{i}}\right)
  144. ( α , β ) . (\alpha\,,\,\beta).
  145. X n , n = 1 , 2 , 3 , X_{n},n=1,2,3,\dots
  146. T ( X 1 , , X n ) T(X_{1},\dots,X_{n})
  147. Pr ( θ | X = x ) = Pr ( θ | T ( X ) = t ( x ) ) . \Pr(\theta|X=x)=\Pr(\theta|T(X)=t(x)).\,
  148. E ^ [ Y | X ] \hat{E}[Y|X]
  149. E ^ [ θ | X ] = E ^ [ θ | T ( X ) ] . \hat{E}[\theta|X]=\hat{E}[\theta|T(X)].

Sum_rule_in_differentiation.html

  1. d d x ( u + v ) = d u d x + d v d x \frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx}
  2. d d x ( u + v + w + ) = d u d x + d v d x + d w d x + \frac{d}{dx}(u+v+w+\dots)=\frac{du}{dx}+\frac{dv}{dx}+\frac{dw}{dx}+\cdots
  3. h ( x ) = lim a 0 h ( x + a ) - h ( x ) a h^{\prime}(x)=\lim_{a\to 0}\frac{h(x+a)-h(x)}{a}
  4. = lim a 0 [ f ( x + a ) + g ( x + a ) ] - [ f ( x ) + g ( x ) ] a =\lim_{a\to 0}\frac{[f(x+a)+g(x+a)]-[f(x)+g(x)]}{a}
  5. = lim a 0 f ( x + a ) - f ( x ) + g ( x + a ) - g ( x ) a =\lim_{a\to 0}\frac{f(x+a)-f(x)+g(x+a)-g(x)}{a}
  6. = lim a 0 f ( x + a ) - f ( x ) a + lim a 0 g ( x + a ) - g ( x ) a =\lim_{a\to 0}\frac{f(x+a)-f(x)}{a}+\lim_{a\to 0}\frac{g(x+a)-g(x)}{a}
  7. = f ( x ) + g ( x ) =f^{\prime}(x)+g^{\prime}(x)
  8. y = u + v y=u+v\,
  9. y + Δ y = ( u + Δ u ) + ( v + Δ v ) = u + v + Δ u + Δ v = y + Δ u + Δ v . y+\Delta{y}=(u+\Delta{u})+(v+\Delta{v})=u+v+\Delta{u}+\Delta{v}=y+\Delta{u}+% \Delta{v}.\,
  10. Δ y = Δ u + Δ v . \Delta{y}=\Delta{u}+\Delta{v}.\,
  11. Δ y Δ x = Δ u Δ x + Δ v Δ x . \frac{\Delta{y}}{\Delta{x}}=\frac{\Delta{u}}{\Delta{x}}+\frac{\Delta{v}}{% \Delta{x}}.
  12. d y d x = d u d x + d v d x . \frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}.
  13. d d x ( u + v ) = d u d x + d v d x . \frac{d}{dx}\left(u+v\right)=\frac{du}{dx}+\frac{dv}{dx}.
  14. d d x ( u - v ) = d d x ( u + ( - v ) ) = d u d x + d d x ( - v ) . \frac{d}{dx}\left(u-v\right)=\frac{d}{dx}\left(u+(-v)\right)=\frac{du}{dx}+% \frac{d}{dx}\left(-v\right).
  15. d d x ( u - v ) = d u d x + ( - d v d x ) = d u d x - d v d x . \frac{d}{dx}\left(u-v\right)=\frac{du}{dx}+\left(-\frac{dv}{dx}\right)=\frac{% du}{dx}-\frac{dv}{dx}.
  16. d d x ( u ± v ) = d u d x ± d v d x . \frac{d}{dx}\left(u\pm v\right)=\frac{du}{dx}\pm\frac{dv}{dx}.
  17. d d x ( 1 i n f i ( x ) ) = d d x ( f 1 ( x ) + f 2 ( x ) + + f n ( x ) ) = d d x f 1 ( x ) + d d x f 2 ( x ) + + d d x f n ( x ) \frac{d}{dx}\left(\sum_{1\leq i\leq n}f_{i}(x)\right)=\frac{d}{dx}\left(f_{1}(% x)+f_{2}(x)+\cdots+f_{n}(x)\right)=\frac{d}{dx}f_{1}(x)+\frac{d}{dx}f_{2}(x)+% \cdots+\frac{d}{dx}f_{n}(x)
  18. d d x ( 1 i n f i ( x ) ) = 1 i n ( d d x f i ( x ) ) . \frac{d}{dx}\left(\sum_{1\leq i\leq n}f_{i}(x)\right)=\sum_{1\leq i\leq n}% \left(\frac{d}{dx}f_{i}(x)\right).
  19. i = 1 k f i ( x ) = g ( x ) + f k ( x ) \sum_{i=1}^{k}f_{i}(x)=g(x)+f_{k}(x)
  20. d d x ( i = 1 k f i ( x ) ) = d d x g ( x ) + d d x f k ( x ) . \frac{d}{dx}\left(\sum_{i=1}^{k}f_{i}(x)\right)=\frac{d}{dx}g(x)+\frac{d}{dx}f% _{k}(x).
  21. d d x g ( x ) = d d x ( i = 1 k - 1 f i ( x ) ) = i = 1 k - 1 d d x f i ( x ) \frac{d}{dx}g(x)=\frac{d}{dx}\left(\sum_{i=1}^{k-1}f_{i}(x)\right)=\sum_{i=1}^% {k-1}\frac{d}{dx}f_{i}(x)
  22. d d x ( i = 1 k f i ( x ) ) = i = 1 k - 1 d d x f i ( x ) + d d x f k ( x ) = i = 1 k d d x f i ( x ) \frac{d}{dx}\left(\sum_{i=1}^{k}f_{i}(x)\right)=\sum_{i=1}^{k-1}\frac{d}{dx}f_% {i}(x)+\frac{d}{dx}f_{k}(x)=\sum_{i=1}^{k}\frac{d}{dx}f_{i}(x)

Sum_rule_in_integration.html

  1. ( f + g ) d x = f d x + g d x \int\left(f+g\right)\,dx=\int f\,dx+\int g\,dx
  2. ( e x + cos x ) d x = e x d x + cos x d x = e x + sin x + C \int\left(e^{x}+\cos{x}\right)\,dx=\int e^{x}\,dx+\int\cos{x}\ \,dx=e^{x}+\sin% {x}+C
  3. ( u - v ) d x \int\left(u-v\right)dx
  4. = u + ( - v ) d x =\int u+\left(-v\right)\,dx
  5. = u d x + ( - v ) d x =\int u\,dx+\int\left(-v\right)\,dx
  6. = u d x + ( - v d x ) =\int u\,dx+\left(-\int v\,dx\right)
  7. = u d x - v d x =\int u\,dx-\int v\,dx
  8. ( u ± v ) d x = u d x ± v d x \int(u\pm v)\,dx=\int u\,dx\pm\int v\,dx
  9. r = a b f ( r , x ) d x = r = a b f ( r , x ) d x \int\sum^{b}_{r=a}f\left(r,x\right)\,dx=\sum^{b}_{r=a}\int f\left(r,x\right)\,dx
  10. r = a b f ( r , x ) d x \int\sum^{b}_{r=a}f(r,x)\,dx
  11. = f ( a , x ) + f ( ( a + 1 ) , x ) + f ( ( a + 2 ) , x ) + =\int f\left(a,x\right)+f((a+1),x)+f((a+2),x)+\dots
  12. + f ( ( b - 1 ) , x ) + f ( b , x ) d x +f((b-1),x)+f(b,x)\,dx
  13. = f ( a , x ) d x + f ( ( a + 1 ) , x ) d x + f ( ( a + 2 ) , x ) d x + =\int f(a,x)\,dx+\int f((a+1),x)\,dx+\int f((a+2),x)\,dx+\dots
  14. + f ( ( b - 1 ) , x ) d x + f ( b , x ) d x +\int f((b-1),x)\,dx+\int f(b,x)\,dx
  15. = r = a b f ( r , x ) d x =\sum^{b}_{r=a}\int f(r,x)\,dx
  16. u = d u d x d x u=\int\frac{du}{dx}\,dx
  17. v = d v d x d x v=\int\frac{dv}{dx}\,dx
  18. u + v = d u d x d x + d v d x d x (1) u+v=\int\frac{du}{dx}\,dx+\int\frac{dv}{dx}\,dx\quad\mbox{(1)}~{}
  19. d d x ( u + v ) = d u d x + d v d x \frac{d}{dx}\left(u+v\right)=\frac{du}{dx}+\frac{dv}{dx}
  20. u + v = ( d u d x + d v d x ) d x (2) u+v=\int\left(\frac{du}{dx}+\frac{dv}{dx}\right)\,dx\quad\mbox{(2)}~{}
  21. u + v = d u d x d x + d v d x d x u+v=\int\frac{du}{dx}\,dx+\int\frac{dv}{dx}\,dx
  22. u + v = ( d u d x + d v d x ) d x u+v=\int\left(\frac{du}{dx}+\frac{dv}{dx}\right)\,dx
  23. ( d u d x + d v d x ) d x = d u d x d x + d v d x d x \int\left(\frac{du}{dx}+\frac{dv}{dx}\right)\,dx=\int\frac{du}{dx}\,dx+\int% \frac{dv}{dx}\,dx
  24. f = d u d x f=\frac{du}{dx}
  25. g = d v d x g=\frac{dv}{dx}

Summation.html

  1. i = 1 100 i . \sum_{i\mathop{=}1}^{100}i.
  2. i = 1 n i = n ( n + 1 ) 2 \sum_{i\mathop{=}1}^{n}i=\frac{n(n+1)}{2}
  3. i = m n a i = a m + a m + 1 + a m + 2 + + a n - 1 + a n . \sum_{i\mathop{=}m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots+a_{n-1}+a_{n}.
  4. i = 3 6 i 2 = 3 2 + 4 2 + 5 2 + 6 2 = 86. \sum_{i\mathop{=}3}^{6}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}=86.
  5. a i 2 = i = 1 n a i 2 . \sum a_{i}^{2}=\sum_{i\mathop{=}1}^{n}a_{i}^{2}.
  6. 0 k < 100 f ( k ) \sum_{0\leq k<100}f(k)
  7. f ( k ) f(k)
  8. k k
  9. x S f ( x ) \sum_{x\mathop{\in}S}f(x)
  10. f ( x ) f(x)
  11. x x
  12. S S
  13. d | n μ ( d ) \sum_{d|n}\;\mu(d)
  14. μ ( d ) \mu(d)
  15. d d
  16. n n
  17. , \sum_{\ell,\ell^{\prime}}
  18. . \sum_{\ell}\sum_{\ell^{\prime}}.
  19. \prod
  20. \sum
  21. x x
  22. x x
  23. n = m n=m
  24. n = m - 1 n=m-1
  25. i = a a g ( i ) = g ( a ) \sum_{i=a}^{a}g(i)=g(a)
  26. \,
  27. i = a b g ( i ) = g ( b ) + i = a b - 1 g ( i ) \sum_{i=a}^{b}g(i)=g(b)+\sum_{i=a}^{b-1}g(i)
  28. k = a b f ( k ) = [ a , b ] f d μ \sum_{k\mathop{=}a}^{b}f(k)=\int_{[a,b]}f\,d\mu
  29. [ a , b ] [a,b]
  30. a a
  31. b b
  32. μ \mu
  33. k = a b f ( k ) = Δ - 1 f ( b + 1 ) - Δ - 1 f ( a ) \sum_{k=a}^{b}f(k)=\Delta^{-1}f(b+1)-\Delta^{-1}f(a)
  34. s = a - 1 b f ( s ) d s i = a b f ( i ) s = a b + 1 f ( s ) d s . \int_{s=a-1}^{b}f(s)\ ds\leq\sum_{i=a}^{b}f(i)\leq\int_{s=a}^{b+1}f(s)\ ds.
  35. s = a b + 1 f ( s ) d s i = a b f ( i ) s = a - 1 b f ( s ) d s . \int_{s=a}^{b+1}f(s)\ ds\leq\sum_{i=a}^{b}f(i)\leq\int_{s=a-1}^{b}f(s)\ ds.
  36. b - a n i = 0 n - 1 f ( a + i b - a n ) a b f ( x ) d x , \frac{b-a}{n}\sum_{i=0}^{n-1}f\left(a+i\frac{b-a}{n}\right)\approx\int_{a}^{b}% f(x)\ dx,
  37. n n\to\infty
  38. n = s t C f ( n ) = C n = s t f ( n ) \sum_{n=s}^{t}C\cdot f(n)=C\cdot\sum_{n=s}^{t}f(n)
  39. n = s t f ( n ) + n = s t g ( n ) = n = s t [ f ( n ) + g ( n ) ] \sum_{n=s}^{t}f(n)+\sum_{n=s}^{t}g(n)=\sum_{n=s}^{t}\left[f(n)+g(n)\right]
  40. \;
  41. n = s t f ( n ) - n = s t g ( n ) = n = s t [ f ( n ) - g ( n ) ] \sum_{n=s}^{t}f(n)-\sum_{n=s}^{t}g(n)=\sum_{n=s}^{t}\left[f(n)-g(n)\right]
  42. \;
  43. n = s t f ( n ) = n = s + p t + p f ( n - p ) \sum_{n=s}^{t}f(n)=\sum_{n=s+p}^{t+p}f(n-p)
  44. \;
  45. n B f ( n ) = m A f ( σ ( m ) ) \sum_{n\in B}f(n)=\sum_{m\in A}f(\sigma(m))
  46. n = s j f ( n ) + n = j + 1 t f ( n ) = n = s t f ( n ) \sum_{n=s}^{j}f(n)+\sum_{n=j+1}^{t}f(n)=\sum_{n=s}^{t}f(n)
  47. \;
  48. i = k 0 k 1 j = l 0 l 1 a i , j = j = l 0 l 1 i = k 0 k 1 a i , j \sum_{i=k_{0}}^{k_{1}}\sum_{j=l_{0}}^{l_{1}}a_{i,j}=\sum_{j=l_{0}}^{l_{1}}\sum% _{i=k_{0}}^{k_{1}}a_{i,j}
  49. \;
  50. k j i n a i , j = i = k n j = k i a i , j = j = k n i = j n a i , j \sum_{k\leq j\leq i\leq n}a_{i,j}=\sum_{i=k}^{n}\sum_{j=k}^{i}a_{i,j}=\sum_{j=% k}^{n}\sum_{i=j}^{n}a_{i,j}
  51. \;
  52. n = 0 t f ( 2 n ) + n = 0 t f ( 2 n + 1 ) = n = 0 2 t + 1 f ( n ) \sum_{n=0}^{t}f(2n)+\sum_{n=0}^{t}f(2n+1)=\sum_{n=0}^{2t+1}f(n)
  53. \;
  54. n = 0 t i = 0 z - 1 f ( z n + i ) = n = 0 z t + z - 1 f ( n ) \sum_{n=0}^{t}\sum_{i=0}^{z-1}f(z\cdot n+i)=\sum_{n=0}^{z\cdot t+z-1}f(n)
  55. \;
  56. i = s m j = t n a i c j = i = s m a i j = t n c j \sum_{i=s}^{m}\sum_{j=t}^{n}{a_{i}}{c_{j}}=\sum_{i=s}^{m}a_{i}\cdot\sum_{j=t}^% {n}c_{j}
  57. \;
  58. n = s t ln f ( n ) = ln n = s t f ( n ) \sum_{n=s}^{t}\ln f(n)=\ln\prod_{n=s}^{t}f(n)
  59. \;
  60. c [ n = s t f ( n ) ] = n = s t c f ( n ) c^{\left[\sum_{n=s}^{t}f(n)\right]}=\prod_{n=s}^{t}c^{f(n)}
  61. \;
  62. i = m n 1 = n + 1 - m \sum_{i=m}^{n}1=n+1-m
  63. \,
  64. i = 1 n 1 i = H n \sum_{i=1}^{n}\frac{1}{i}=H_{n}
  65. i = 1 n 1 i k = H n k \sum_{i=1}^{n}\frac{1}{i^{k}}=H^{k}_{n}
  66. i = m n i = n ( n + 1 ) 2 - m ( m - 1 ) 2 = ( n + 1 - m ) ( n + m ) 2 \sum_{i=m}^{n}i=\frac{n(n+1)}{2}-\frac{m(m-1)}{2}=\frac{(n+1-m)(n+m)}{2}
  67. i = 0 n i = i = 1 n i = n ( n + 1 ) 2 \sum_{i=0}^{n}i=\sum_{i=1}^{n}i=\frac{n(n+1)}{2}
  68. i = 0 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 = n 3 3 + n 2 2 + n 6 \sum_{i=0}^{n}i^{2}=\frac{n(n+1)(2n+1)}{6}=\frac{n^{3}}{3}+\frac{n^{2}}{2}+% \frac{n}{6}
  69. i = 0 n i 3 = ( n ( n + 1 ) 2 ) 2 = n 4 4 + n 3 2 + n 2 4 = [ i = 1 n i ] 2 \sum_{i=0}^{n}i^{3}=\left(\frac{n(n+1)}{2}\right)^{2}=\frac{n^{4}}{4}+\frac{n^% {3}}{2}+\frac{n^{2}}{4}=\left[\sum_{i=1}^{n}i\right]^{2}
  70. \,
  71. i = 0 n i 4 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 2 + 3 n - 1 ) 30 = n 5 5 + n 4 2 + n 3 3 - n 30 \sum_{i=0}^{n}i^{4}=\frac{n(n+1)(2n+1)(3n^{2}+3n-1)}{30}=\frac{n^{5}}{5}+\frac% {n^{4}}{2}+\frac{n^{3}}{3}-\frac{n}{30}
  72. \,
  73. i = 0 n i p = ( n + 1 ) p + 1 p + 1 + k = 1 p B k p - k + 1 ( p k ) ( n + 1 ) p - k + 1 , \sum_{i=0}^{n}i^{p}=\frac{(n+1)^{p+1}}{p+1}+\sum_{k=1}^{p}\frac{B_{k}}{p-k+1}{% p\choose k}(n+1)^{p-k+1},
  74. B k B_{k}
  75. i = 0 n i 3 = ( i = 0 n i ) 2 \sum_{i=0}^{n}i^{3}=\left(\sum_{i=0}^{n}i\right)^{2}
  76. m m\in\mathbb{N}
  77. ( i = m n i ) 2 = i = m n ( i 3 - i m ( m - 1 ) ) \left(\sum_{i=m}^{n}i\right)^{2}=\sum_{i=m}^{n}(i^{3}-im(m-1))
  78. \,
  79. i = m n i 3 = ( i = m n i ) 2 + m ( m - 1 ) i = m n i \sum_{i=m}^{n}i^{3}=\left(\sum_{i=m}^{n}i\right)^{2}+m(m-1)\sum_{i=m}^{n}i
  80. \,
  81. i = m n - 1 a i = a m - a n 1 - a \sum_{i=m}^{n-1}a^{i}=\frac{a^{m}-a^{n}}{1-a}
  82. i = 0 n - 1 a i = 1 - a n 1 - a \sum_{i=0}^{n-1}a^{i}=\frac{1-a^{n}}{1-a}
  83. i = 0 i=0
  84. i = 0 n - 1 i a i = a - n a n + ( n - 1 ) a n + 1 ( 1 - a ) 2 \sum_{i=0}^{n-1}ia^{i}=\frac{a-na^{n}+(n-1)a^{n+1}}{(1-a)^{2}}
  85. \,
  86. i = 0 n - 1 i 2 i = 2 + ( n - 2 ) 2 n \sum_{i=0}^{n-1}i2^{i}=2+(n-2)2^{n}
  87. i = 0 n - 1 i 2 i = 2 - n + 1 2 n - 1 \sum_{i=0}^{n-1}\frac{i}{2^{i}}=2-\frac{n+1}{2^{n-1}}
  88. i = 0 n ( n i ) = 2 n \sum_{i=0}^{n}{n\choose i}=2^{n}
  89. \,
  90. i = 1 n i ( n i ) = n 2 n - 1 \sum_{i=1}^{n}i{n\choose i}=n2^{n-1}
  91. \,
  92. i = 0 n i ! ( n i ) = i = 0 n P i n = n ! e \sum_{i=0}^{n}i!\cdot{n\choose i}=\sum_{i=0}^{n}{}_{n}P_{i}=\lfloor n!\cdot e\rfloor
  93. \,
  94. i = 0 n - 1 ( i k ) = ( n k + 1 ) \sum_{i=0}^{n-1}{i\choose k}={n\choose k+1}
  95. \,
  96. i = 0 n ( n i ) a ( n - i ) b i = ( a + b ) n \sum_{i=0}^{n}{n\choose i}a^{(n-i)}b^{i}=(a+b)^{n}
  97. i = 0 n i i ! = ( n + 1 ) ! - 1 \sum_{i=0}^{n}i\cdot i!=(n+1)!-1
  98. \,
  99. i = 1 n P k + 1 i + k = i = 1 n j = 0 k ( i + j ) = ( n + k + 1 ) ! ( n - 1 ) ! ( k + 2 ) \sum_{i=1}^{n}{}_{i+k}P_{k+1}=\sum_{i=1}^{n}\prod_{j=0}^{k}(i+j)=\frac{(n+k+1)% !}{(n-1)!(k+2)}
  100. \,
  101. i = 0 n ( m + i - 1 i ) = ( m + n n ) \sum_{i=0}^{n}{m+i-1\choose i}={m+n\choose n}
  102. \,
  103. i = 1 n i c Θ ( n c + 1 ) \sum_{i=1}^{n}i^{c}\in\Theta(n^{c+1})
  104. i = 1 n 1 i Θ ( log n ) \sum_{i=1}^{n}\frac{1}{i}\in\Theta(\log n)
  105. i = 1 n c i Θ ( c n ) \sum_{i=1}^{n}c^{i}\in\Theta(c^{n})
  106. i = 1 n log ( i ) c Θ ( n log ( n ) c ) \sum_{i=1}^{n}\log(i)^{c}\in\Theta(n\cdot\log(n)^{c})
  107. i = 1 n log ( i ) c i d Θ ( n d + 1 log ( n ) c ) \sum_{i=1}^{n}\log(i)^{c}\cdot i^{d}\in\Theta(n^{d+1}\cdot\log(n)^{c})
  108. i = 1 n log ( i ) c i d b i Θ ( n d log ( n ) c b n ) \sum_{i=1}^{n}\log(i)^{c}\cdot i^{d}\cdot b^{i}\in\Theta(n^{d}\cdot\log(n)^{c}% \cdot b^{n})
  109. i i
  110. q q
  111. x x
  112. k k
  113. k k

Summation_by_parts.html

  1. { f k } \{f_{k}\}
  2. { g k } \{g_{k}\}
  3. k = m n f k ( g k + 1 - g k ) = [ f n + 1 g n + 1 - f m g m ] - k = m n g k + 1 ( f k + 1 - f k ) . \sum_{k=m}^{n}f_{k}(g_{k+1}-g_{k})=\left[f_{n+1}g_{n+1}-f_{m}g_{m}\right]-\sum% _{k=m}^{n}g_{k+1}(f_{k+1}-f_{k}).
  4. Δ \Delta
  5. k = m n f k Δ g k = [ f n + 1 g n + 1 - f m g m ] - k = m n g k + 1 Δ f k , \sum_{k=m}^{n}f_{k}\Delta g_{k}=\left[f_{n+1}g_{n+1}-f_{m}g_{m}\right]-\sum_{k% =m}^{n}g_{k+1}\Delta f_{k},
  6. f d g = f g - g d f . \int f\,dg=fg-\int g\,df.
  7. k = 0 n f k g k = f 0 k = 0 n g k + j = 0 n - 1 ( f j + 1 - f j ) k = j + 1 n g k = f n k = 0 n g k - j = 0 n - 1 ( f j + 1 - f j ) k = 0 j g k , \begin{aligned}\displaystyle\sum_{k=0}^{n}f_{k}g_{k}&\displaystyle=f_{0}\sum_{% k=0}^{n}g_{k}+\sum_{j=0}^{n-1}(f_{j+1}-f_{j})\sum_{k=j+1}^{n}g_{k}\\ &\displaystyle=f_{n}\sum_{k=0}^{n}g_{k}-\sum_{j=0}^{n-1}\left(f_{j+1}-f_{j}% \right)\sum_{k=0}^{j}g_{k},\end{aligned}
  8. M = 1 M=1
  9. k = 0 n f k g k = i = 0 M - 1 f 0 ( i ) G i ( i + 1 ) + j = 0 n - M f j ( M ) G j + M ( M ) = = i = 0 M - 1 ( - 1 ) i f n - i ( i ) G ~ n - i ( i + 1 ) + ( - 1 ) M j = 0 n - M f j ( M ) G ~ j ( M ) ; \begin{aligned}\displaystyle\sum_{k=0}^{n}f_{k}g_{k}&\displaystyle=\sum_{i=0}^% {M-1}f_{0}^{(i)}G_{i}^{(i+1)}+\sum_{j=0}^{n-M}f^{(M)}_{j}G_{j+M}^{(M)}=\\ &\displaystyle=\sum_{i=0}^{M-1}\left(-1\right)^{i}f_{n-i}^{(i)}\tilde{G}_{n-i}% ^{(i+1)}+\left(-1\right)^{M}\sum_{j=0}^{n-M}f_{j}^{(M)}\tilde{G}_{j}^{(M)};% \end{aligned}
  10. f j ( M ) := k = 0 M ( - 1 ) M - k ( M k ) f j + k f_{j}^{(M)}:=\sum_{k=0}^{M}\left(-1\right)^{M-k}{M\choose k}f_{j+k}
  11. G j ( M ) := k = j n ( k - j + M - 1 M - 1 ) g k , G_{j}^{(M)}:=\sum_{k=j}^{n}{k-j+M-1\choose M-1}g_{k},
  12. G ~ j ( M ) := k = 0 j ( j - k + M - 1 M - 1 ) g k . \tilde{G}_{j}^{(M)}:=\sum_{k=0}^{j}{j-k+M-1\choose M-1}g_{k}.
  13. M = n + 1 M=n+1
  14. k = 0 n f k g k = i = 0 n f 0 ( i ) G i ( i + 1 ) = i = 0 n ( - 1 ) i f n - i ( i ) G ~ n - i ( i + 1 ) . \sum_{k=0}^{n}f_{k}g_{k}=\sum_{i=0}^{n}f_{0}^{(i)}G_{i}^{(i+1)}=\sum_{i=0}^{n}% (-1)^{i}f_{n-i}^{(i)}\tilde{G}_{n-i}^{(i+1)}.
  15. ( n k ) {n\choose k}
  16. ( a n ) (a_{n})\,
  17. ( b n ) (b_{n})\,
  18. n 𝒩 n\in\mathcal{N}
  19. S N = n = 0 N a n b n S_{N}=\sum_{n=0}^{N}a_{n}b_{n}
  20. B n = k = 0 n b k , B_{n}=\sum_{k=0}^{n}b_{k},
  21. n > 0 , n>0,\,
  22. b n = B n - B n - 1 b_{n}=B_{n}-B_{n-1}\,
  23. S N = a 0 b 0 + n = 1 N a n ( B n - B n - 1 ) , S_{N}=a_{0}b_{0}+\sum_{n=1}^{N}a_{n}(B_{n}-B_{n-1}),
  24. S N = a 0 b 0 - a 0 B 0 + a N B N + n = 0 N - 1 B n ( a n - a n + 1 ) . S_{N}=a_{0}b_{0}-a_{0}B_{0}+a_{N}B_{N}+\sum_{n=0}^{N-1}B_{n}(a_{n}-a_{n+1}).
  25. S N = a N B N - n = 0 N - 1 B n ( a n + 1 - a n ) . S_{N}=a_{N}B_{N}-\sum_{n=0}^{N-1}B_{n}(a_{n+1}-a_{n}).
  26. S N S_{N}\,
  27. a b f ( x ) g ( x ) d x = [ f ( x ) g ( x ) ] a b - a b f ( x ) g ( x ) d x \int_{a}^{b}f(x)g^{\prime}(x)\,dx=\left[f(x)g(x)\right]_{a}^{b}-\int_{a}^{b}f^% {\prime}(x)g(x)\,dx
  28. g g^{\prime}\,
  29. g g\,
  30. f f\,
  31. f f^{\prime}\,
  32. b n b_{n}\,
  33. B n B_{n}\,
  34. a n a_{n}\,
  35. a n + 1 - a n a_{n+1}-a_{n}\,
  36. b n \sum b_{n}
  37. a n a_{n}
  38. S N = n = 0 N a n b n S_{N}=\sum_{n=0}^{N}a_{n}b_{n}
  39. S M - S N = a M B M - a N B N + n = N M - 1 B n ( a n + 1 - a n ) = ( a M - a ) B M - ( a N - a ) B N + a ( B M - B N ) + n = N M - 1 B n ( a n + 1 - a n ) , \begin{aligned}\displaystyle S_{M}-S_{N}&\displaystyle=a_{M}B_{M}-a_{N}B_{N}+% \sum_{n=N}^{M-1}B_{n}(a_{n+1}-a_{n})\\ &\displaystyle=(a_{M}-a)B_{M}-(a_{N}-a)B_{N}+a(B_{M}-B_{N})+\sum_{n=N}^{M-1}B_% {n}(a_{n+1}-a_{n}),\end{aligned}
  40. a n a_{n}
  41. b n \sum b_{n}
  42. B N B_{N}
  43. N N
  44. B B
  45. a n - a a_{n}-a
  46. b n \sum b_{n}
  47. n = N M - 1 | B n | | a n + 1 - a n | B n = N M - 1 | a n + 1 - a n | = B | a N - a M | \sum_{n=N}^{M-1}|B_{n}||a_{n+1}-a_{n}|\leq B\sum_{n=N}^{M-1}|a_{n+1}-a_{n}|=B|% a_{N}-a_{M}|
  48. a n a_{n}
  49. N N\to\infty
  50. B N B_{N}
  51. N N
  52. n = 0 | a n + 1 - a n | < \sum_{n=0}^{\infty}|a_{n+1}-a_{n}|<\infty
  53. n = N M - 1 | a n + 1 - a n | \sum_{n=N}^{M-1}|a_{n+1}-a_{n}|
  54. N N
  55. lim a n = 0 \lim a_{n}=0
  56. S N = n = 0 N a n b n S_{N}=\sum_{n=0}^{N}a_{n}b_{n}
  57. | S | = | n = 0 a n b n | B n = 0 | a n + 1 - a n | |S|=\left|\sum_{n=0}^{\infty}a_{n}b_{n}\right|\leq B\sum_{n=0}^{\infty}|a_{n+1% }-a_{n}|

Sundial.html

  1. H E = 15 × t ( h o u r s ) H_{E}=15^{\circ}\times t(hours)
  2. C o r r e c t i o n = E o T ( m i n u t e s ) + 60 × Δ D S T ( h o u r s ) 15 Correction^{\circ}=\frac{EoT(minutes)+60\times\Delta DST(hours)}{15}
  3. tan H H = sin L tan ( 15 × t ) \tan H_{H}=\sin L\tan(15^{\circ}\times t)
  4. H H = tan - 1 [ sin L × tan ( 15 × t ) ] \ H_{H}=\tan^{-1}[\sin L\times\tan(15^{\circ}\times t)]
  5. H H H_{H}
  6. H H H_{H}
  7. H H H_{H}
  8. tan H V = cos L tan ( 15 × t ) \tan H_{V}=\cos L\tan(15^{\circ}\times t)
  9. H H H_{H}
  10. H H H_{H}
  11. X = H tan ( 15 × t ) X=H\tan(15^{\circ}\times t)
  12. H VD H\text{VD}
  13. H VD H\text{VD}
  14. tan H VD = cos L cos D cot ( 15 × t ) - s o sin L sin D \tan H\text{VD}=\frac{\cos L}{\cos D\cot(15^{\circ}\times t)-s_{o}\sin L\sin D}
  15. L L
  16. D D
  17. s o s_{o}
  18. s o s_{o}
  19. D = 0 D=0^{\circ}
  20. tan H V = cos L tan ( 15 × t ) \tan H\text{V}=\cos L\tan(15^{\circ}\times t)
  21. B B
  22. tan B = sin D cot L \tan B=\sin D\cot L
  23. D = 0 D=0^{\circ}
  24. D = 180 D=180^{\circ}
  25. B = 0 B=0^{\circ}
  26. G G
  27. sin G = cos D cos L \sin G=\cos D\cos L
  28. tan H R V = cos ( L + R ) tan ( 15 × t ) \tan H_{RV}=\cos(L+R)\tan(15^{\circ}\times t)
  29. R R
  30. H R V H_{RV}
  31. H R V H_{RV}
  32. tan H R V = sin ( L + I ) tan ( 15 × t ) \tan H_{RV}=\sin(L+I)\tan(15^{\circ}\times t)
  33. B = 90 - ( L + R ) B=90^{\circ}-(L+R)
  34. B = 180 - ( L + I ) B=180^{\circ}-(L+I)
  35. H RD H\text{RD}
  36. H RD H\text{RD}
  37. tan H RD = cos R cos L - sin R sin L cos D - s o sin R sin D cot ( 15 × t ) cos D cot ( 15 × t ) - s o sin D sin L \tan H\text{RD}=\frac{\cos R\cos L-\sin R\sin L\cos D-s_{o}\sin R\sin D\cot(15% ^{\circ}\times t)}{\cos D\cot(15^{\circ}\times t)-s_{o}\sin D\sin L}
  38. D < D c D<D_{c}
  39. - 90 < R < ( 90 - L ) -90^{\circ}<R<(90^{\circ}-L)
  40. I I
  41. R R
  42. I = ( 90 + R ) I=(90^{\circ}+R)
  43. tan H RD = sin I cos L + cos I sin L cos D + s o cos I sin D cot ( 15 × t ) cos D cot ( 15 × t ) - s o sin D sin L \tan H\text{RD}=\frac{\sin I\cos L+\cos I\sin L\cos D+s_{o}\cos I\sin D\cot(15% ^{\circ}\times t)}{\cos D\cot(15^{\circ}\times t)-s_{o}\sin D\sin L}
  44. D < D c D<D_{c}
  45. 0 < I < ( 180 - L ) 0^{\circ}<I<(180^{\circ}-L)
  46. L L
  47. s o s_{o}
  48. R R
  49. D D
  50. R R
  51. D D
  52. s o s_{o}
  53. s o s_{o}
  54. D c D_{c}
  55. cos D c = tan R tan L = - tan L cot I \cos D_{c}=\tan R\tan L=-\tan L\cot I
  56. B B
  57. tan B = sin D sin R cos D + cos R tan L = sin D cos I cos D - sin I tan L \tan B=\frac{\sin D}{\sin R\cos D+\cos R\tan L}=\frac{\sin D}{\cos I\cos D-% \sin I\tan L}
  58. G G
  59. sin G = cos L cos D cos R - sin L sin R = - cos L cos D sin I + sin L cos I \sin G=\cos L\cos D\cos R-\sin L\sin R=-\cos L\cos D\sin I+\sin L\cos I
  60. G = 0 G=0^{\circ}
  61. cos D = tan L tan R = - tan L cot I \cos D=\tan L\tan R=-\tan L\cot I
  62. D = D c D=D_{c}
  63. Y = R tan α tan δ Y=R\tan\alpha\tan\delta\,

Sunscreen.html

  1. SPF = A ( λ ) E ( λ ) d λ A ( λ ) E ( λ ) / MPF ( λ ) d λ , \mathrm{SPF}=\frac{\int A(\lambda)E(\lambda)d\lambda}{\int A(\lambda)E(\lambda% )/\mathrm{MPF}(\lambda)\,d\lambda},
  2. E ( λ ) E(\lambda)
  3. A ( λ ) A(\lambda)
  4. MPF ( λ ) \mathrm{MPF}(\lambda)
  5. λ \lambda

Super-Poulet_number.html

  1. Φ n ( 2 ) g c d ( n , Φ n ( 2 ) ) \frac{\Phi_{n}(2)}{gcd(n,\Phi_{n}(2))}

Supercompact_cardinal.html

  1. M λ M . {}^{\lambda}M\subseteq M\,.
  2. [ A ] < κ := { x A | | x | < κ } . [A]^{<\kappa}:=\{x\subseteq A||x|<\kappa\}\,.
  3. { x [ A ] < κ | a x } U \{x\in[A]^{<\kappa}|a\in x\}\in U
  4. a A a\in A
  5. f : [ A ] < κ A f:[A]^{<\kappa}\to A
  6. { x [ A ] < κ | f ( x ) x } U \{x\in[A]^{<\kappa}|f(x)\in x\}\in U
  7. U U
  8. a A a\in A
  9. { x [ A ] < κ | f ( x ) = a } U \{x\in[A]^{<\kappa}|f(x)=a\}\in U

Supercooling.html

  1. T L x | x = 0 > T x \left.\frac{\partial T_{L}}{\partial x}\right|_{x=0}>\frac{\partial T}{% \partial x}
  2. m C L x | x = 0 > T x m\left.\frac{\partial C_{L}}{\partial x}\right|_{x=0}>\frac{\partial T}{% \partial x}
  3. m = T L / C L m=\partial T_{L}/\partial C_{L}
  4. C L S C^{LS}
  5. C S L C^{SL}
  6. C L x | x = 0 = - C L S - C S L D / v \left.\frac{\partial C_{L}}{\partial x}\right|_{x=0}=-\frac{C^{LS}-C^{SL}}{D/v}
  7. C S L = C 0 C^{SL}=C_{0}
  8. k = C S L C L S k=\frac{C^{SL}}{C^{LS}}
  9. T x < m C 0 ( 1 - k ) v k D \frac{\partial T}{\partial x}<\frac{mC_{0}(1-k)v}{kD}

Supergroup_(physics).html

  1. μ : G × G G \mu:G\times G\rightarrow G
  2. i : G G i:G\rightarrow G
  3. e : 1 G e:1\rightarrow G

Supermassive_black_hole.html

  1. σ \sigma
  2. × 10 9 \times 10^{9}

Superplasticity.html

  1. σ d 2 \sigma\propto d^{2}\,
  2. σ \sigma\,

Superstrong_cardinal.html

  1. V j ( κ ) V_{j(\kappa)}
  2. V j n ( κ ) V_{j^{n}(\kappa)}

Supersymmetry.html

  1. { Q α , Q β ˙ ¯ } = 2 ( σ ) μ α β ˙ P μ \{Q_{\alpha},\bar{Q_{\dot{\beta}}}\}=2(\sigma{}^{\mu})_{\alpha\dot{\beta}}P_{\mu}
  2. P μ = - i μ P_{\mu}=-i\partial{}_{\mu}
  3. σ μ \sigma{}^{\mu}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 3 / 2 {3}/{2}
  7. 3 / 2 {3}/{2}
  8. 1 / 2 {1}/{2}
  9. 1 / 2 {1}/{2}
  10. 1 / 2 {1}/{2}
  11. 3 / 2 {3}/{2}
  12. 1 / 2 {1}/{2}
  13. 1 / 2 {1}/{2}
  14. 1 / 2 {1}/{2}
  15. 3 / 2 {3}/{2}
  16. 3 / 2 {3}/{2}
  17. 1 / 2 {1}/{2}
  18. 1 / 2 {1}/{2}
  19. 3 / 2 {3}/{2}

Surface_tension.html

  1. γ = 1 dyn cm = 1 erg cm 2 = 1 mN m = 0.001 N m = 0.001 J m 2 \gamma=1~{}\mathrm{\frac{dyn}{cm}}=1~{}\mathrm{\frac{erg}{cm^{2}}}=1~{}\mathrm% {\frac{mN}{m}}=0.001~{}\mathrm{\frac{N}{m}}=0.001~{}\mathrm{\frac{J}{m^{2}}}
  2. γ \gamma
  3. F F
  4. L L
  5. F / L F/L
  6. F / L F/L
  7. L L
  8. F F
  9. γ = 1 2 F L \gamma=\frac{1}{2}\frac{F}{L}
  10. 1 / 2 1/2
  11. γ L = F / 2 \gamma L=F/2
  12. F F
  13. γ \gamma
  14. F F
  15. F F
  16. Δ x \Delta x
  17. W = F Δ x W=F\Delta x
  18. Δ A = 2 L Δ x \Delta A=2L\Delta x
  19. γ = ( 1 / 2 ) F / L \gamma=(1/2)F/L
  20. Δ x \Delta x
  21. γ = F 2 L = F Δ x 2 L Δ x = W Δ A \gamma=\frac{F}{2L}=\frac{F\Delta x}{2L\Delta x}=\frac{W}{\Delta A}
  22. W W
  23. Δ p = γ ( 1 R x + 1 R y ) \Delta p\ =\ \gamma\left(\frac{1}{R_{x}}+\frac{1}{R_{y}}\right)
  24. γ \scriptstyle\gamma
  25. F w = 2 F s cos θ ρ A s L g = 2 γ L z cos θ F_{w}=2F_{s}\cos\theta\Leftrightarrow\rho A_{s}Lg=2\gamma\,L\,z\,\cos\theta
  26. θ \scriptstyle\theta
  27. γ ls - γ sa \scriptstyle\gamma_{\mathrm{ls}}-\gamma_{\mathrm{sa}}
  28. γ la \scriptstyle\gamma_{\mathrm{la}}
  29. γ la > γ ls - γ sa > 0 \gamma_{\mathrm{la}}\ >\ \gamma_{\mathrm{ls}}-\gamma_{\mathrm{sa}}\ >\ 0
  30. f la \scriptstyle f_{\mathrm{la}}
  31. f A \scriptstyle f_{\mathrm{A}}
  32. f A = f la sin θ f_{\mathrm{A}}\ =\ f_{\mathrm{la}}\sin\theta
  33. f la \scriptstyle f_{\mathrm{la}}
  34. f ls \scriptstyle f_{\mathrm{ls}}
  35. f ls - f sa = - f la cos θ f_{\mathrm{ls}}-f_{\mathrm{sa}}\ =\ -f_{\mathrm{la}}\cos\theta
  36. γ ls - γ sa = - γ la cos θ \gamma_{\mathrm{ls}}-\gamma_{\mathrm{sa}}\ =\ -\gamma_{\mathrm{la}}\cos\theta
  37. γ ls \scriptstyle\gamma_{\mathrm{ls}}
  38. γ la \scriptstyle\gamma_{\mathrm{la}}
  39. γ sa \scriptstyle\gamma_{\mathrm{sa}}
  40. θ \scriptstyle\theta
  41. γ ls - γ sa \scriptstyle\gamma_{\mathrm{ls}}-\gamma_{\mathrm{sa}}
  42. γ la \scriptstyle\gamma_{\mathrm{la}}
  43. θ \scriptstyle\theta
  44. γ la > 0 > γ ls - γ sa \gamma_{\mathrm{la}}\ >\ 0\ >\ \gamma_{\mathrm{ls}}-\gamma_{\mathrm{sa}}
  45. γ la = γ ls - γ sa > 0 θ = 180 \gamma_{\mathrm{la}}\ =\ \gamma_{\mathrm{ls}}-\gamma_{\mathrm{sa}}\ >\ 0\qquad% \theta\ =\ 180^{\circ}
  46. h = 2 γ la cos θ ρ g r h\ =\ \frac{2\gamma_{\mathrm{la}}\cos\theta}{\rho gr}
  47. h \scriptstyle h
  48. γ la \scriptstyle\gamma_{\mathrm{la}}
  49. ρ \scriptstyle\rho
  50. r \scriptstyle r
  51. g \scriptstyle g
  52. θ \scriptstyle\theta
  53. θ \scriptstyle\theta
  54. h = 2 γ g ρ h\ =\ 2\sqrt{\frac{\gamma}{g\rho}}
  55. h \scriptstyle h
  56. γ \scriptstyle\gamma
  57. g \scriptstyle g
  58. ρ \scriptstyle\rho
  59. h = 2 γ la ( 1 - cos θ ) g ρ . h\ =\ \sqrt{\frac{2\gamma_{\mathrm{la}}\left(1-\cos\theta\right)}{g\rho}}.
  60. R ( z ) = R 0 + A k cos ( k z ) \scriptstyle R\left(z\right)=R_{0}+A_{k}\cos\left(kz\right)
  61. R 0 \scriptstyle R_{0}
  62. A k \scriptstyle A_{k}
  63. z \scriptstyle z
  64. k \scriptstyle k
  65. d W = γ d A \scriptstyle dW\ =\ \gamma dA
  66. γ = ( G A ) T , P , n \gamma=\left(\frac{\partial G}{\partial A}\right)_{T,P,n}
  67. G \scriptstyle G
  68. A \scriptstyle A
  69. Δ G < 0 \scriptstyle\Delta G\ <\ 0
  70. G = H - T S \scriptstyle G\ =\ H\ -\ TS
  71. H \scriptstyle H
  72. S \scriptstyle S
  73. ( γ T ) A , P = - S A \left(\frac{\partial\gamma}{\partial T}\right)_{A,P}=-S^{A}
  74. H A = γ - T ( γ T ) P H^{A}\ =\ \gamma-T\left(\frac{\partial\gamma}{\partial T}\right)_{P}
  75. d T = d N = 0 dT=dN=0
  76. d F = - P d V + γ d A dF\ =-PdV\ +\gamma dA
  77. P P
  78. γ \gamma
  79. d F = 0 dF=0
  80. P d V = γ d A PdV\ =\gamma dA
  81. V = 4 3 π R 3 d V 4 π R 2 d R V=\frac{4}{3}\pi R^{3}\rightarrow dV\approx 4\pi R^{2}dR
  82. A = 4 π R 2 d A 8 π R d R A=4\pi R^{2}\rightarrow dA\approx 8\pi RdR
  83. P = 2 R γ P=\frac{2}{R}\gamma
  84. γ V 2 / 3 = k ( T C - T ) \gamma V^{2/3}=k(T_{C}-T)\,\!
  85. γ V 2 / 3 = k ( T C - T - 6 ) \gamma V^{2/3}=k\left(T_{C}-T-6\right)
  86. γ = γ o ( 1 - T T C ) n \gamma=\gamma^{o}\left(1-\frac{T}{T_{C}}\right)^{n}
  87. γ o \scriptstyle\gamma^{o}
  88. γ 0 \scriptstyle\gamma^{0}
  89. K 2 T c 1 3 P c 2 3 \scriptstyle K_{2}T^{\frac{1}{3}}_{c}P^{\frac{2}{3}}_{c}
  90. K 2 \scriptstyle K_{2}
  91. P c \scriptstyle P_{c}
  92. K 2 \scriptstyle K_{2}
  93. Γ = - 1 R T ( γ ln C ) T , P \Gamma\ =\ -\frac{1}{RT}\left(\frac{\partial\gamma}{\partial\ln C}\right)_{T,P}
  94. Γ \scriptstyle\Gamma
  95. C \scriptstyle C
  96. R \scriptstyle R
  97. T \scriptstyle T
  98. P v fog = P v o e V 2 γ R T r k P_{v}^{\mathrm{fog}}=P_{v}^{o}e^{\frac{V2\gamma}{RTr_{k}}}
  99. P v o \scriptstyle P_{v}^{o}
  100. V \scriptstyle V
  101. R \scriptstyle R
  102. r k r_{k}

Susceptance.html

  1. Y = G + j B Y=G+jB\,
  2. Y = 1 Z = 1 R + j X = ( R R 2 + X 2 ) + j ( - X R 2 + X 2 ) Y=\frac{1}{Z}=\frac{1}{R+jX}=\left(\frac{R}{R^{2}+X^{2}}\right)+j\left(\frac{-% X}{R^{2}+X^{2}}\right)\,
  3. B = I m ( Y ) = ( - X R 2 + X 2 ) = - X | Z | 2 B=Im(Y)=\left(\frac{-X}{R^{2}+X^{2}}\right)=\frac{-X}{|Z|^{2}}
  4. Z = R + j X Z=R+jX\,
  5. | Y | = G 2 + B 2 \left|Y\right|=\sqrt{G^{2}+B^{2}}\,

Suslin's_problem.html

  1. 1 . \aleph_{1}.

Switched-mode_power_supply.html

  1. 2 ¯ \overline{2}
  2. 2 ¯ \overline{2}
  3. V 2 = D V 1 \scriptstyle V_{2}=DV_{1}
  4. V 2 = 1 1 - D V 1 \scriptstyle V_{2}=\frac{1}{1-D}V_{1}
  5. V 2 = - D 1 - D V 1 \scriptstyle V_{2}=-\frac{D}{1-D}V_{1}
  6. V 2 = - D 1 - D V 1 \scriptstyle V_{2}=-\frac{D}{1-D}V_{1}
  7. V 2 = D 1 - D V 1 \scriptstyle V_{2}=\frac{D}{1-D}V_{1}
  8. V 2 = D 1 - D V 1 \scriptstyle V_{2}=\frac{D}{1-D}V_{1}

Symmetric_difference.html

  1. A B ~{}A\triangle B
  2. ~{}\setminus~{}
  3. = ~{}=~{}
  4. A B A\,\triangle\,B\,
  5. A B . A\ominus B.
  6. A B . A\oplus B.
  7. { 1 , 2 , 3 } \{1,2,3\}
  8. { 3 , 4 } \{3,4\}
  9. { 1 , 2 , 4 } \{1,2,4\}
  10. A B C ~{}A\triangle B\triangle C
  11. ~{}\triangle~{}
  12. = ~{}=~{}
  13. A B = ( A B ) ( B A ) , A\,\triangle\,B=(A\smallsetminus B)\cup(B\smallsetminus A),\,
  14. A B = { x : ( x A ) ( x B ) } . A\,\triangle\,B=\{x:(x\in A)\oplus(x\in B)\}.
  15. χ \chi
  16. χ ( A B ) = χ A χ B \chi_{(A\,\triangle\,B)}=\chi_{A}\oplus\chi_{B}
  17. [ x A B ] = [ x A ] [ x B ] [x\in A\,\triangle\,B]=[x\in A]\oplus[x\in B]
  18. A B = ( A B ) ( A B ) , A\,\triangle\,B=(A\cup B)\smallsetminus(A\cap B),
  19. A B A B A\triangle B\subseteq A\cup B
  20. A A
  21. B B
  22. D = A B D=A\triangle B
  23. I = A B I=A\cap B
  24. D D
  25. I I
  26. D D
  27. I I
  28. A B A\cup B
  29. A B = ( A B ) ( A B ) A\,\cup\,B=(A\,\triangle\,B)\,\triangle\,(A\cap B)
  30. A B = B A , A\,\triangle\,B=B\,\triangle\,A,\,
  31. ( A B ) C = A ( B C ) . (A\,\triangle\,B)\,\triangle\,C=A\,\triangle\,(B\,\triangle\,C).\,
  32. A = A , A\,\triangle\,\varnothing=A,\,
  33. A A = . A\,\triangle\,A=\varnothing.\,
  34. ( A B ) ( B C ) = A C . (A\,\triangle\,B)\,\triangle\,(B\,\triangle\,C)=A\,\triangle\,C.\,
  35. A ( B C ) = ( A B ) ( A C ) , A\cap(B\,\triangle\,C)=(A\cap B)\,\triangle\,(A\cap C),
  36. A B = A c B c A\triangle B=A^{c}\triangle B^{c}
  37. A c A^{c}
  38. B c B^{c}
  39. A A
  40. B B
  41. ( α A α ) ( α B α ) α ( A α B α ) \left(\bigcup_{\alpha\in\mathcal{I}}A_{\alpha}\right)\triangle\left(\bigcup_{% \alpha\in\mathcal{I}}B_{\alpha}\right)\subseteq\bigcup_{\alpha\in\mathcal{I}}% \left(A_{\alpha}\triangle B_{\alpha}\right)
  42. \mathcal{I}
  43. f : S T f:S\rightarrow T
  44. A , B T A,B\subseteq T
  45. f f
  46. f - 1 ( A Δ B ) = f - 1 ( A ) Δ f - 1 ( B ) f^{-1}\left(A\Delta B\right)=f^{-1}\left(A\right)\Delta f^{-1}\left(B\right)
  47. x y = ( x y ) ¬ ( x y ) = ( x ¬ y ) ( y ¬ x ) = x y . x\,\triangle\,y=(x\lor y)\land\lnot(x\land y)=(x\land\lnot y)\lor(y\land\lnot x% )=x\oplus y.
  48. M = { a M : | { A M : a A } | is odd } \triangle M=\left\{a\in\bigcup M:|\{A\in M:a\in A\}|\mbox{ is odd}~{}\right\}
  49. M \bigcup M
  50. M M
  51. M = { M 1 , M 2 , , M n } M=\{M_{1},M_{2},\ldots,M_{n}\}
  52. n 2 n\geq 2
  53. | M | |\triangle M|
  54. M \triangle M
  55. M M
  56. | M | = l = 1 n ( - 2 ) l - 1 i 1 i 2 i l | M i 1 M i 2 M i l | |\triangle M|=\sum_{l=1}^{n}(-2)^{l-1}\sum_{i_{1}\neq i_{2}\neq\ldots\neq i_{l% }}|M_{i_{1}}\cap M_{i_{2}}\cap\ldots\cap M_{i_{l}}|
  57. i 1 i 2 i l i_{1}\neq i_{2}\neq\ldots\neq i_{l}
  58. { i 1 , i 2 , , i l } \{i_{1},i_{2},\ldots,i_{l}\}
  59. { 1 , 2 , , n } \{1,2,\ldots,n\}
  60. ( n l ) {\left({{n}\atop{l}}\right)}
  61. d μ ( X , Y ) = μ ( X Y ) d_{\mu}(X,Y)=\mu(X\,\triangle\,Y)
  62. μ ( X Y ) = 0 \mu(X\,\triangle\,Y)=0
  63. μ ( X ) , μ ( Y ) < \mu(X),\mu(Y)<\infty
  64. | μ ( X ) - μ ( Y ) | μ ( X Y ) |\mu(X)-\mu(Y)|\leq\mu(X\,\triangle\,Y)
  65. | μ ( X ) - μ ( Y ) | = | ( μ ( X Y ) + μ ( X Y ) ) - ( μ ( X Y ) + μ ( Y X ) ) | = | μ ( X Y ) - μ ( Y X ) | | μ ( X Y ) | + | μ ( Y X ) | = μ ( X Y ) + μ ( Y X ) = μ ( ( X Y ) ( Y X ) ) = μ ( X Δ Y ) \begin{aligned}\displaystyle|\mu(X)-\mu(Y)|&\displaystyle=|(\mu(X\setminus Y)+% \mu(X\cap Y))-(\mu(X\cap Y)+\mu(Y\setminus X))|\\ &\displaystyle=|\mu(X\setminus Y)-\mu(Y\setminus X)|\\ &\displaystyle\leq|\mu(X\setminus Y)|+|\mu(Y\setminus X)|\\ &\displaystyle=\mu(X\setminus Y)+\mu(Y\setminus X)\\ &\displaystyle=\mu((X\setminus Y)\cup(Y\setminus X))\\ &\displaystyle=\mu(X\Delta Y)\end{aligned}
  66. S = ( Ω , 𝒜 , μ ) S=\left(\Omega,\mathcal{A},\mu\right)
  67. F , G 𝒜 F,G\in\mathcal{A}
  68. 𝒟 , 𝒜 \mathcal{D},\mathcal{E}\subseteq\mathcal{A}
  69. F G 𝒜 F\triangle G\in\mathcal{A}
  70. F = G [ 𝒜 , μ ] F=G\left[\mathcal{A},\mu\right]
  71. μ ( F G ) = 0 \mu\left(F\triangle G\right)=0
  72. = [ 𝒜 , μ ] =\left[\mathcal{A},\mu\right]
  73. 𝒜 \mathcal{A}
  74. 𝒟 [ 𝒜 , μ ] \mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A},\mu\right]
  75. D 𝒟 D\in\mathcal{D}
  76. E E\in\mathcal{E}
  77. D = E [ 𝒜 , μ ] D=E\left[\mathcal{A},\mu\right]
  78. [ 𝒜 , μ ] \subseteq\left[\mathcal{A},\mu\right]
  79. 𝒜 \mathcal{A}
  80. 𝒟 = [ 𝒜 , μ ] \mathcal{D}=\mathcal{E}\left[\mathcal{A},\mu\right]
  81. 𝒟 [ 𝒜 , μ ] \mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A},\mu\right]
  82. 𝒟 [ 𝒜 , μ ] \mathcal{E}\subseteq\mathcal{D}\left[\mathcal{A},\mu\right]
  83. = [ 𝒜 , μ ] =\left[\mathcal{A},\mu\right]
  84. 𝒜 \mathcal{A}
  85. 𝒟 \mathcal{D}
  86. 𝒜 \mathcal{A}
  87. = [ 𝒜 , μ ] =\left[\mathcal{A},\mu\right]
  88. D 𝒟 D\in\mathcal{D}
  89. 𝒟 \mathcal{D}
  90. 𝒟 \mathcal{D}
  91. 𝒟 \mathcal{D}
  92. σ \sigma
  93. 𝒜 \mathcal{A}
  94. 𝒟 \mathcal{D}
  95. F = G [ 𝒜 , μ ] F=G\left[\mathcal{A},\mu\right]
  96. | 𝟏 F - 𝟏 G | = 0 \left|\mathbf{1}_{F}-\mathbf{1}_{G}\right|=0
  97. [ 𝒜 , μ ] \left[\mathcal{A},\mu\right]

Symmetric_matrix.html

  1. A = A . A=A^{\top}.
  2. [ 1 7 3 7 4 - 5 3 - 5 6 ] . \begin{bmatrix}1&7&3\\ 7&4&-5\\ 3&-5&6\end{bmatrix}.
  3. Mat = n Sym n Skew , n \mbox{Mat}~{}_{n}=\mbox{Sym}~{}_{n}\oplus\mbox{Skew}~{}_{n},
  4. X = 1 2 ( X + X ) + 1 2 ( X - X ) . X=\frac{1}{2}(X+X^{\top})+\frac{1}{2}(X-X^{\top}).
  5. , \langle\cdot,\cdot\rangle
  6. A x , y = x , A y x , y n . \langle Ax,y\rangle=\langle x,Ay\rangle\quad\forall x,y\in\mathbb{R}^{n}.
  7. n \mathbb{R}^{n}
  8. V B V V^{\dagger}BV
  9. A = L L T A=LL^{T}
  10. P A P T = L T L T PAP^{T}=LTL^{T}
  11. P P
  12. L L
  13. T T
  14. D D
  15. A = Q Λ Q A=Q\Lambda Q^{\top}
  16. x x
  17. y y
  18. λ 1 \lambda_{1}
  19. λ 2 \lambda_{2}
  20. λ 1 x , y = A x , y = x , A y = λ 2 x , y \lambda_{1}\langle x,y\rangle=\langle Ax,y\rangle=\langle x,Ay\rangle=\lambda_% {2}\langle x,y\rangle
  21. x , y 0 \langle x,y\rangle\neq 0
  22. λ 1 = λ 2 \lambda_{1}=\lambda_{2}
  23. x , y = 0 \langle x,y\rangle=0
  24. q ( x 1 , , x n ) = i = 1 n λ i x i 2 q(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}\lambda_{i}x_{i}^{2}
  25. a i j = 0 a_{ij}=0
  26. a j i = 0 a_{ji}=0
  27. 1 i j n . 1\leq i\leq j\leq n.
  28. a i 1 i 2 a i 2 i 3 a i k i 1 = a i 2 i 1 a i 3 i 2 a i 1 i k a_{i_{1}i_{2}}a_{i_{2}i_{3}}\dots a_{i_{k}i_{1}}=a_{i_{2}i_{1}}a_{i_{3}i_{2}}% \dots a_{i_{1}i_{k}}
  29. ( i 1 , i 2 , , i k ) . (i_{1},i_{2},\dots,i_{k}).

Symmetric_relation.html

  1. a , b X , a R b b R a . \forall a,b\in X,\ aRb\Rightarrow\;bRa.

Symplectic_geometry.html

  1. ω = d p d q \omega=dp\wedge dq
  2. A = S ω . A=\int_{S}\omega.
  3. ( ( x 1 , x 2 ) , ( x 3 , x 4 ) , ( x 2 n - 1 , x 2 n ) ) ((x_{1},x_{2}),(x_{3},x_{4}),\ldots(x_{2n-1},x_{2n}))
  4. ω = d x 1 d x 2 + d x 3 d x 4 + + d x 2 n - 1 d x 2 n . \omega=dx_{1}\wedge dx_{2}+dx_{3}\wedge dx_{4}+\cdots+dx_{2n-1}\wedge dx_{2n}.
  5. A = V ω = V d x 1 d x 2 + V d x 3 d x 4 + + V d x 2 n - 1 d x 2 n . A=\int_{V}\omega=\int_{V}dx_{1}\wedge dx_{2}+\int_{V}dx_{3}\wedge dx_{4}+% \cdots+\int_{V}dx_{2n-1}\wedge dx_{2n}.

Symplectic_group.html

  1. S p ( 2 n , F ) Sp(2n,F)
  2. S p ( n ) Sp(n)
  3. 2 2
  4. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  5. S p ( n ) Sp(n)
  6. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  7. n n
  8. S p ( 2 n , F ) Sp(2n,F)
  9. 2 n 2n
  10. F F
  11. S p ( 2 n , F ) Sp(2n,F)
  12. 2 n × 2 n 2n×2n
  13. F F
  14. 1 1
  15. S L ( 2 n , F ) SL(2n,F)
  16. 2 n 2n
  17. F F
  18. V V
  19. S p ( V ) Sp(V)
  20. F F
  21. 𝐑 \mathbf{R}
  22. 𝐂 \mathbf{C}
  23. S p ( 2 n , F ) Sp(2n,F)
  24. n ( 2 n + 1 ) n(2n+1)
  25. S p ( 2 n , F ) Sp(2n,F)
  26. 2 2
  27. 2 n × 2 n 2n×2n
  28. S p ( 2 n , F ) Sp(2n,F)
  29. S p ( 2 n , F ) Sp(2n,F)
  30. n n
  31. S Sp ( 2 n , F ) iff S T Ω S = Ω S\in\operatorname{Sp}(2n,F)\quad\,\text{iff}\quad S\text{T}\Omega S=\Omega
  32. Ω = ( 0 I n - I n 0 ) . \Omega=\begin{pmatrix}0&I_{n}\\ -I_{n}&0\\ \end{pmatrix}.
  33. S p ( 2 n , F ) Sp(2n,F)
  34. 2 n × 2 n 2n×2n
  35. Ω A + A T Ω = 0. \Omega A+A^{\mathrm{T}}\Omega=0.
  36. n = 1 n=1
  37. S p ( 2 , F ) = S L ( 2 , F ) Sp(2,F)=SL(2,F)
  38. n > 1 n>1
  39. S p ( 2 n , F ) Sp(2n,F)
  40. S L ( 2 n , F ) SL(2n,F)
  41. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  42. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  43. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  44. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  45. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  46. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  47. 𝐬𝐩 ( 2 n , 𝐑 ) \mathbf{sp}(2n,\mathbf{R})
  48. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  49. S Sp ( 2 n , ) X , Y 𝔰 𝔭 ( 2 n , ) such that S = e X e Y . \forall\;S\in\operatorname{Sp}(2n,\mathbb{R})\;\exists\;X,Y\in\mathfrak{sp}(2n% ,\mathbb{R})\,\text{ such that }S=e^{X}e^{Y}.
  50. S S
  51. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  52. S = O Z O such that O , O Sp ( 2 n , ) SO ( 2 n ) U ( n ) and Z = ( D 0 0 D - 1 ) . S=OZO^{\prime}\quad\,\text{such that}\quad O,O^{\prime}\in\operatorname{Sp}(2n% ,\mathbb{R})\cap\operatorname{SO}(2n)\cong U(n)\quad\,\text{and}\quad Z=\begin% {pmatrix}D&0\\ 0&D^{-1}\end{pmatrix}.
  53. D D
  54. Z Z
  55. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  56. U ( n ) U(n)
  57. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  58. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  59. U ( n ) U(n)
  60. n ( n + 1 ) n(n+1)
  61. 𝐬𝐩 ( 2 n , 𝐅 ) \mathbf{sp}(2n,\mathbf{F})
  62. Q Q
  63. Q = ( A B C - A T ) Q=\begin{pmatrix}A&B\\ C&-A^{\mathrm{T}}\end{pmatrix}
  64. B B
  65. C C
  66. S p ( 2 , R ) Sp(2,R)
  67. 2 × 2 2×2
  68. 1 1
  69. ( 0 , 1 ) (0,1)
  70. ( 1 0 0 1 ) , ( 1 0 1 1 ) and ( 1 1 0 1 ) . \begin{pmatrix}1&0\\ 0&1\end{pmatrix},\quad\begin{pmatrix}1&0\\ 1&1\end{pmatrix}\quad\,\text{and}\quad\begin{pmatrix}1&1\\ 0&1\end{pmatrix}.
  71. S p ( 2 n , F ) Sp(2n,F)
  72. S p ( n ) Sp(n)
  73. S p ( n ) Sp(n)
  74. U S p ( 2 n ) USp(2n)
  75. S p ( n ) U ( 2 n ) S p ( 2 n , 𝐂 ) Sp(n)≅U(2n)∩Sp(2n,\mathbf{C})
  76. S p ( n ) Sp(n)
  77. S p ( n ) Sp(n)
  78. S p ( n ) Sp(n)
  79. G L ( n , 𝐇 ) GL(n,\mathbf{H})
  80. x , y = x ¯ 1 y 1 + + x ¯ n y n \langle x,y\rangle=\bar{x}_{1}y_{1}+\cdots+\bar{x}_{n}y_{n}
  81. S p ( n ) Sp(n)
  82. U ( n , 𝐇 ) U(n,\mathbf{H})
  83. 1 1
  84. S U ( 2 ) SU(2)
  85. 3 3
  86. S p ( n ) Sp(n)
  87. 𝐇 \mathbf{H}
  88. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  89. S p ( n ) Sp(n)
  90. 𝐬𝐩 ( 2 n , 𝐂 ) \mathbf{sp}(2n,\mathbf{C})
  91. S p ( n ) Sp(n)
  92. n ( 2 n + 1 ) n(2n+1)
  93. S p ( n ) Sp(n)
  94. n - b y - n n-by-n
  95. A + A = 0 A+A^{\dagger}=0
  96. A A
  97. S p ( n ) Sp(n)
  98. Sp ( n ) Sp ( n - 1 ) \operatorname{Sp}(n)\supset\operatorname{Sp}(n-1)
  99. Sp ( n ) U ( n ) \operatorname{Sp}(n)\supset\operatorname{U}(n)
  100. Sp ( 2 ) O ( 4 ) \operatorname{Sp}(2)\supset\operatorname{O}(4)
  101. SU ( 2 n ) Sp ( n ) \operatorname{SU}(2n)\supset\operatorname{Sp}(n)
  102. F 4 Sp ( 4 ) \operatorname{F}_{4}\supset\operatorname{Sp}(4)
  103. G 2 Sp ( 1 ) \operatorname{G}_{2}\supset\operatorname{Sp}(1)
  104. 𝐬𝐩 ( 2 ) = 𝐬𝐨 ( 5 ) \mathbf{sp}(2)=\mathbf{so}(5)
  105. 𝐬𝐩 ( 1 ) = 𝐬𝐨 ( 3 ) = 𝐬𝐮 ( 2 ) \mathbf{sp}(1)=\mathbf{so}(3)=\mathbf{su}(2)
  106. S p ( 2 n , 𝐂 ) Sp(2n,\mathbf{C})
  107. 𝐬𝐩 ( 2 n , 𝐂 ) \mathbf{sp}(2n,\mathbf{C})
  108. 𝐬𝐩 ( 2 n , 𝐑 ) \mathbf{sp}(2n,\mathbf{R})
  109. 𝐬𝐩 ( n ) \mathbf{sp}(n)
  110. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  111. S p ( n ) Sp(n)
  112. 𝐬𝐩 ( p , n p ) \mathbf{sp}(p,n−p)
  113. S p ( p , n p ) Sp(p,n−p)
  114. n n
  115. 𝐳 = ( q 1 , , q n , p 1 , , p n ) T . \mathbf{z}=(q_{1},\ldots,q_{n},p_{1},\ldots,p_{n})^{T}.
  116. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})
  117. 𝐙 = 𝐙 ( 𝐳 , t ) = ( Q 1 , , Q n , P 1 , , P n ) T \mathbf{Z}=\mathbf{Z}(\mathbf{z},t)=(Q_{1},\ldots,Q_{n},P_{1},\ldots,P_{n})^{T}
  118. 𝐙 ˙ = M ( 𝐳 , t ) 𝐳 ˙ , \dot{\mathbf{Z}}=M(\mathbf{z},t)\dot{\mathbf{z}},
  119. M ( 𝐳 , t ) SL ( 2 n , ) M(\mathbf{z},t)\in\mathrm{SL}(2n,\mathbb{R})
  120. t t
  121. 𝐳 \mathbf{z}
  122. n n
  123. 𝐳 ^ = ( q ^ 1 , , q ^ n , p ^ 1 , , p ^ n ) T . \mathbf{\hat{z}}=(\hat{q}_{1},\ldots,\hat{q}_{n},\hat{p}_{1},\ldots,\hat{p}_{n% })^{T}.
  124. [ 𝐳 ^ , 𝐳 ^ T ] = i Ω [\mathbf{\hat{z}},\mathbf{\hat{z}}^{T}]=i\hbar\Omega
  125. Ω = ( 𝟎 I n - I n 𝟎 ) \Omega=\begin{pmatrix}\mathbf{0}&I_{n}\\ -I_{n}&\mathbf{0}\end{pmatrix}
  126. n × n n×n
  127. H ^ = 1 2 𝐳 ^ T K 𝐳 ^ \hat{H}=\frac{1}{2}\mathbf{\hat{z}}^{T}K\mathbf{\hat{z}}
  128. K K
  129. 2 n × 2 n 2n×2n
  130. d 𝐳 ^ d t = Ω K 𝐳 ^ \frac{d\mathbf{\hat{z}}}{dt}=\Omega K\mathbf{\hat{z}}
  131. S p ( 2 n , 𝐑 ) Sp(2n,\mathbf{R})

Symplectic_matrix.html

  1. M T Ω M = Ω . M\text{T}\Omega M=\Omega\,.
  2. Ω = [ 0 I n - I n 0 ] \Omega=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\\ \end{bmatrix}
  3. M - 1 = Ω - 1 M T Ω . M^{-1}=\Omega^{-1}M\text{T}\Omega.
  4. Pf ( M T Ω M ) = det ( M ) Pf ( Ω ) . \mbox{Pf}~{}(M\text{T}\Omega M)=\det(M)\mbox{Pf}~{}(\Omega).
  5. M T Ω M = Ω M\text{T}\Omega M=\Omega
  6. Pf ( Ω ) 0 \mbox{Pf}~{}(\Omega)\neq 0
  7. M = ( A B C D ) M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}
  8. A T D - C T B = I A\text{T}D-C\text{T}B=I
  9. A T C = C T A A\text{T}C=C\text{T}A
  10. D T B = B T D . D\text{T}B=B\text{T}D.
  11. M - 1 = Ω - 1 M T Ω = ( D T - B T - C T A T ) . M^{-1}=\Omega^{-1}M\text{T}\Omega=\begin{pmatrix}D\text{T}&-B\text{T}\\ -C\text{T}&A\text{T}\end{pmatrix}.
  12. ω ( L u , L v ) = ω ( u , v ) . \omega(Lu,Lv)=\omega(u,v).
  13. M T Ω M = Ω . M\text{T}\Omega M=\Omega.
  14. Ω A T Ω A \Omega\mapsto A\text{T}\Omega A
  15. M A - 1 M A . M\mapsto A^{-1}MA.
  16. Ω = [ 0 1 - 1 0 0 0 0 1 - 1 0 ] . \Omega=\begin{bmatrix}\begin{matrix}0&1\\ -1&0\end{matrix}&&0\\ &\ddots&\\ 0&&\begin{matrix}0&1\\ -1&0\end{matrix}\end{bmatrix}.
  17. Ω a b = - g a c J c b \Omega_{ab}=-g_{ac}{J^{c}}_{b}
  18. g a c g_{ac}
  19. S S
  20. U U
  21. U ( 2 n , 𝐑 ) U(2n,\mathbf{R})
  22. S = U T D U for D = diag ( λ 1 , , λ n , λ 1 - 1 , , λ n - 1 ) , S=U\text{T}DU\quad\,\text{for}\quad D=\operatorname{diag}(\lambda_{1},\ldots,% \lambda_{n},\lambda_{1}^{-1},\ldots,\lambda_{n}^{-1}),
  23. D D
  24. S S
  25. S S
  26. S = U R for U U ( 2 n , ) and R Sp ( 2 n , ) Sym + ( 2 n , ) . S=UR\quad\,\text{for}\quad U\in\operatorname{U}(2n,\mathbb{R})\,\text{ and }R% \in\operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_{+}(2n,\mathbb{R}).
  27. S = O ( D 0 0 D - 1 ) O , S=O\begin{pmatrix}D&0\\ 0&D^{-1}\end{pmatrix}O^{\prime},
  28. O O
  29. O O
  30. D D

Symplectic_vector_space.html

  1. ω = [ 0 I n - I n 0 ] \omega=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\end{bmatrix}
  2. ω ( x i , y j ) = - ω ( y j , x i ) = δ i j \omega(x_{i},y_{j})=-\omega(y_{j},x_{i})=\delta_{ij}\,
  3. ω ( x i , x j ) = ω ( y i , y j ) = 0. \omega(x_{i},x_{j})=\omega(y_{i},y_{j})=0.\,
  4. ω ( x η , y ξ ) = ξ ( x ) - η ( y ) . \omega(x\oplus\eta,y\oplus\xi)=\xi(x)-\eta(y).
  5. ( v 1 * , , v n * ) . (v^{*}_{1},\ldots,v^{*}_{n}).
  6. ( x 1 , , x n , y 1 , , y n ) . (x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}).
  7. ω n = ( - 1 ) n / 2 x 1 * x n * y 1 * y n * . \omega^{n}=(-1)^{n/2}x^{*}_{1}\wedge\ldots\wedge x^{*}_{n}\wedge y^{*}_{1}% \wedge\ldots\wedge y^{*}_{n}.
  8. ω n = x 1 * y 1 * x n * y n * . \omega^{n}=x^{*}_{1}\wedge y^{*}_{1}\wedge\ldots\wedge x^{*}_{n}\wedge y^{*}_{% n}.
  9. W = { v V ω ( v , w ) = 0 for all w W } . W^{\perp}=\{v\in V\mid\omega(v,w)=0\mbox{ for all }~{}w\in W\}.
  10. ( W ) = W (W^{\perp})^{\perp}=W
  11. dim W + dim W = dim V . \dim W+\dim W^{\perp}=\dim V.

Symplectomorphism.html

  1. f : ( M , ω ) ( N , ω ) f:(M,\omega)\rightarrow(N,\omega^{\prime})
  2. f * ω = ω , f^{*}\omega^{\prime}=\omega,
  3. f * f^{*}
  4. f f
  5. M M
  6. M M
  7. X Γ ( T M ) X\in\Gamma^{\infty}(TM)
  8. X ω = 0. \mathcal{L}_{X}\omega=0.
  9. X X
  10. ϕ t : M M \phi_{t}:M\rightarrow M
  11. X X
  12. t t
  13. Γ ( T M ) \Gamma^{\infty}(TM)
  14. ( M , ω ) (M,\omega)
  15. Ham ( M , ω ) \mathop{\rm Ham}(M,\omega)

Synchrotron_radiation.html

  1. 𝐚 𝐯 \mathbf{a}\perp\mathbf{v}
  2. γ \gamma
  3. γ \gamma
  4. 𝐁 ( 𝐫 , t ) = - μ 0 q 4 π [ c 𝐧 ^ × β γ 2 R 2 ( 1 - β 𝐧 ^ ) 3 + 𝐧 ^ × [ β ˙ + 𝐧 ^ × ( β × β ˙ ) ] R ( 1 - β 𝐧 ^ ) 3 ] retarded ( 1 ) \mathbf{B}(\mathbf{r},t)=-\frac{\mu_{0}q}{4\pi}\left[\frac{c\,\hat{\mathbf{n}}% \times\vec{\beta}}{\gamma^{2}R^{2}(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}}% )^{3}}+\frac{\hat{\mathbf{n}}\times[\,\dot{\vec{\beta}}+\hat{\mathbf{n}}\times% (\vec{\beta}\times\dot{\vec{\beta}})]}{R\,(1-\vec{\beta}\mathbf{\cdot}\hat{% \mathbf{n}})^{3}}\right]_{\mathrm{retarded}}\qquad(1)
  5. 𝐄 ( 𝐫 , t ) = q 4 π ε 0 [ 𝐧 ^ - β γ 2 R 2 ( 1 - β 𝐧 ^ ) 3 + 𝐧 ^ × [ ( 𝐧 ^ - β ) × β ˙ ] c R ( 1 - β 𝐧 ^ ) 3 ] retarded ( 2 ) \mathbf{E}(\mathbf{r},t)=\frac{q}{4\pi\varepsilon_{0}}\left[\frac{\hat{\mathbf% {n}}-\vec{\beta}}{\gamma^{2}R^{2}(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})% ^{3}}+\frac{\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\vec{\beta})\times\dot{% \vec{\beta}}\,]}{c\,R\,(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})^{3}}% \right]_{\mathrm{retarded}}\qquad\qquad(2)
  6. 𝐑 ( t ) = 𝐫 - 𝐫 0 ( t ) , \mathbf{R}(t^{\prime})=\mathbf{r}-\mathbf{r}_{0}(t^{\prime}),
  7. R ( t ) = | 𝐑 | , R(t^{\prime})=|\mathbf{R}|,
  8. 𝐧 ^ ( t ) = 𝐑 / R , \hat{\mathbf{n}}(t^{\prime})=\mathbf{R}/R,
  9. t t^{\prime}
  10. [ 𝐒 𝐧 ^ ] = q 2 16 π 2 ε 0 c { 1 R 2 | 𝐧 ^ × [ ( 𝐧 ^ - β ) × β ˙ ] ( 1 - β 𝐧 ^ ) 3 | 2 } retarded ( 3 ) [\mathbf{S\cdot}\hat{\mathbf{n}}]=\frac{q^{2}}{16\pi^{2}\varepsilon_{0}c}\left% \{\frac{1}{R^{2}}\left|\frac{\hat{\mathbf{n}}\times[(\hat{\mathbf{n}}-\vec{% \beta})\times\dot{\vec{\beta}}]}{(1-\vec{\beta}\mathbf{\cdot}\hat{\mathbf{n}})% ^{3}}\right|^{2}\right\}_{\,\text{retarded}}\qquad\qquad(3)
  11. β \vec{\beta}
  12. β ˙ \dot{\vec{\beta}}
  13. ( 1 - β 𝐧 ) (1-\vec{\beta}\mathbf{\cdot}\vec{\mathbf{n}})
  14. t = T 1 t^{\prime}=T_{1}
  15. t = T 2 t^{\prime}=T_{2}
  16. d P d Ω = R ( t ) 2 [ 𝐒 ( t ) 𝐧 ^ ( t ) ] d t d t = R ( t ) 2 𝐒 ( t ) 𝐧 ^ ( t ) [ 1 - β ( t ) 𝐧 ^ ( t ) ] \frac{\mathrm{d}P}{\mathrm{d}\mathit{\Omega}}=R(t^{\prime})^{2}\,[\mathbf{S}(t% ^{\prime})\mathbf{\cdot}\hat{\mathbf{n}}(t^{\prime})]\,\frac{\mathrm{d}t}{% \mathrm{d}t^{\prime}}=R(t^{\prime})^{2}\,\mathbf{S}(t^{\prime})\mathbf{\cdot}% \hat{\mathbf{n}}(t^{\prime})\,[1-\vec{\beta}(t^{\prime})\mathbf{\cdot}\hat{% \mathbf{n}}(t^{\prime})]
  17. = q 2 16 π 2 ε 0 c | 𝐧 ^ ( t ) × { [ 𝐧 ^ ( t ) - β ( t ) ] × β ˙ ( t ) } | 2 [ 1 - β ( t ) 𝐧 ( t ) ] 5 ( 4 ) =\frac{q^{2}}{16\pi^{2}\varepsilon_{0}c}\,\frac{|\hat{\mathbf{n}}(t^{\prime})% \times\{[\hat{\mathbf{n}}(t^{\prime})-\vec{\beta}(t^{\prime})]\times\dot{\vec{% \beta}}(t^{\prime})\}|^{2}}{[1-\vec{\beta}(t^{\prime})\mathbf{\cdot}\vec{% \mathbf{n}}(t^{\prime})]^{5}}\qquad\qquad(4)
  18. P = e 2 6 π ε 0 c γ 6 [ | β ˙ | 2 - | β × β ˙ | 2 ] ( 5 ) P=\frac{e^{2}}{6\pi\varepsilon_{0}c}\gamma^{6}\left[\left|\dot{\vec{\beta}}% \right|^{2}-\left|\vec{\beta}\times\dot{\vec{\beta}}\right|^{2}\right]\qquad(5)
  19. β ˙ \dot{\vec{\beta}}
  20. β \vec{\beta}
  21. β \vec{\beta}
  22. β ˙ \dot{\vec{\beta}}
  23. θ \theta
  24. ϕ \phi
  25. d P d Ω = q 2 16 π 2 ϵ 0 c | β ˙ | 2 ( 1 - β cos θ ) 3 [ 1 - sin 2 θ cos 2 ϕ γ 2 ( 1 - β cos θ ) 2 ] . ( 6 ) \frac{\mathrm{d}P}{\mathrm{d}\mathit{\Omega}}=\frac{q^{2}}{16\pi^{2}\epsilon_{% 0}c}\frac{|\dot{\vec{\beta}}|^{2}}{(1-\beta\cos\theta)^{3}}\left[1-\frac{\sin^% {2}\theta\cos^{2}\phi}{\gamma^{2}(1-\beta\cos\theta)^{2}}\right].\qquad(6)
  26. ( γ 1 ) (\gamma\gg 1)
  27. d P d Ω 2 π e 2 c 3 γ 6 | 𝐯 ˙ | 2 ( 1 + γ 2 θ 2 ) 3 [ 1 - 4 γ 2 θ 2 cos 2 ϕ ( 1 + γ 2 θ 2 ) 2 ] . ( 7 ) \frac{\mathrm{d}P}{\mathrm{d}\mathit{\Omega}}\simeq\frac{2}{\pi}\frac{e^{2}}{c% ^{3}}\gamma^{6}\frac{|\dot{\mathbf{v}}|^{2}}{(1+\gamma^{2}\theta^{2})^{3}}% \left[1-\frac{4\gamma^{2}\theta^{2}\cos^{2}\phi}{(1+\gamma^{2}\theta^{2})^{2}}% \right].\qquad\qquad(7)
  28. ( 1 - β cos θ ) (1-\beta\cos\theta)
  29. P = e 2 6 π ϵ 0 c | β ˙ | 2 γ 4 = e 2 c 6 π ϵ 0 β 4 γ 4 ρ 2 = e 4 6 π ϵ 0 m 4 c 5 E 2 B 2 , ( 8 ) P=\frac{e^{2}}{6\pi\epsilon_{0}c}\left|\dot{\vec{\beta}}\right|^{2}\gamma^{4}=% \frac{e^{2}c}{6\pi\epsilon_{0}}\frac{\beta^{4}\gamma^{4}}{\rho^{2}}=\frac{e^{4% }}{6\pi\epsilon_{0}m^{4}c^{5}}E^{2}B^{2},\qquad(8)
  30. 1 / m 4 1/m^{4}
  31. 1 / ρ 2 1/\rho^{2}
  32. B 2 B^{2}
  33. d 2 W d Ω = - d 2 P d Ω d t = c ε 0 - | R E ( t ) | 2 d t \frac{d^{2}W}{d\Omega}=\int_{-\infty}^{\infty}\frac{d^{2}P}{d\Omega}dt=c% \varepsilon_{0}\int_{-\infty}^{\infty}\left|R\vec{E}(t)\right|^{2}dt
  34. d 2 W d Ω = 2 c ε 0 0 | R E ( ω ) | 2 d ω \frac{d^{2}W}{d\Omega}=2c\varepsilon_{0}\int_{0}^{\infty}\left|R\vec{E}(\omega% )\right|^{2}d\omega
  35. d 3 W d Ω d ω = 2 c ε 0 R 2 | E ( ω ) | 2 = e 2 4 π ε 0 4 π 2 c | - n ^ × [ ( n ^ - β ) × β ˙ ] ( 1 - n ^ β ) 2 e i ω ( t - n ^ r ( t ) / c ) d t | 2 ( 9 ) \frac{d^{3}W}{d\Omega d\omega}=2c\varepsilon_{0}R^{2}\left|\vec{E}(\omega)% \right|^{2}=\frac{e^{2}}{4\pi\varepsilon_{0}4\pi^{2}c}\left|\int_{-\infty}^{% \infty}\frac{\hat{n}\times\left[\left(\hat{n}-\vec{\beta}\right)\times\dot{% \vec{\beta}}\right]}{\left(1-\hat{n}\cdot\vec{\beta}\right)^{2}}e^{i\omega(t-% \hat{n}\cdot\vec{r}(t)/c)}dt\right|^{2}\qquad(9)
  36. r ( t ) = ( ρ sin β c ρ t , ρ ( 1 - cos β c ρ t ) , 0 ) \vec{r}(t)=\left(\rho\sin\frac{\beta c}{\rho}t,\rho\left(1-\cos\frac{\beta c}{% \rho}t\right),0\right)
  37. n ^ × ( n ^ × β ) = β [ - ε sin ( β c t ρ ) + ε cos ( β c t ρ ) sin θ ] \hat{n}\times\left(\hat{n}\times\vec{\beta}\right)=\beta\left[-\vec{% \varepsilon}_{\parallel}\sin\left(\frac{\beta ct}{\rho}\right)+\vec{% \varepsilon}_{\perp}\cos\left(\frac{\beta ct}{\rho}\right)\sin\theta\right]
  38. ω ( t - n ^ r ( t ) c ) = ω [ t - ρ c sin ( β c t ρ ) cos θ ] \omega\left(t-\frac{\hat{n}\cdot\vec{r}(t)}{c}\right)=\omega\left[t-\frac{\rho% }{c}\sin\left(\frac{\beta ct}{\rho}\right)\cos\theta\right]
  39. ξ = ρ ω 3 c γ 3 ( 1 + γ 2 θ 2 ) 3 / 2 \xi=\frac{\rho\omega}{3c\gamma^{3}}\left(1+\gamma^{2}\theta^{2}\right)^{3/2}
  40. d 3 W d Ω d ω = e 2 16 π 3 ε 0 c ( 2 ω ρ 3 c γ 2 ) 2 ( 1 + γ 2 θ 2 ) 2 [ K 2 / 3 2 ( ξ ) + γ 2 θ 2 1 + γ 2 θ 2 K 1 / 3 2 ( ξ ) ] ( 10 ) \frac{d^{3}W}{d\Omega d\omega}=\frac{e^{2}}{16\pi^{3}\varepsilon_{0}c}\left(% \frac{2\omega\rho}{3c\gamma^{2}}\right)^{2}\left(1+\gamma^{2}\theta^{2}\right)% ^{2}\left[K_{2/3}^{2}(\xi)+\frac{\gamma^{2}\theta^{2}}{1+\gamma^{2}\theta^{2}}% K_{1/3}^{2}(\xi)\right]\qquad(10)
  41. K K
  42. ξ 1 \xi\gg 1
  43. ξ = 1 2 \xi=\frac{1}{2}
  44. θ = 0 \theta=0
  45. ω c = 3 2 c ρ γ 3 \omega\text{c}=\frac{3}{2}\frac{c}{\rho}\gamma^{3}
  46. ξ ( θ c ) ξ ( 0 ) + 1 \xi(\theta\text{c})\simeq\xi(0)+1
  47. θ c 1 γ ( 2 ω c ω ) 1 / 3 \theta\text{c}\simeq\frac{1}{\gamma}\left(\frac{2\omega\text{c}}{\omega}\right% )^{1/3}
  48. d W d ω = d 3 W d ω d Ω d Ω = 3 e 2 4 π ε 0 c γ ω ω c ω / ω c K 5 / 3 ( x ) d x \frac{dW}{d\omega}=\oint\frac{d^{3}W}{d\omega d\Omega}d\Omega=\frac{\sqrt{3}e^% {2}}{4\pi\varepsilon_{0}c}\gamma\frac{\omega}{\omega\text{c}}\int_{\omega/% \omega\text{c}}^{\infty}K_{5/3}(x)dx
  49. S ( y ) 9 3 8 π y y K 5 / 3 ( x ) d x S(y)\equiv\frac{9\sqrt{3}}{8\pi}y\int_{y}^{\infty}K_{5/3}(x)dx
  50. 0 S ( y ) d y = 1 \int_{0}^{\infty}S(y)dy=1
  51. y = ω ω c y=\frac{\omega}{\omega\text{c}}
  52. d W d ω = 2 e 2 γ 9 ε 0 c S ( y ) ( 11 ) \frac{dW}{d\omega}=\frac{2e^{2}\gamma}{9\varepsilon_{0}c}S(y)\qquad(11)
  53. d W d ω e 2 4 π ε 0 c ( ω ρ c ) 1 / 3 \frac{dW}{d\omega}\sim\frac{e^{2}}{4\pi\varepsilon_{0}c}\left(\frac{\omega\rho% }{c}\right)^{1/3}
  54. ω ω c \omega\ll\omega\text{c}
  55. d W d ω 3 π 2 e 2 4 π ε 0 c γ ( ω ω c ) 0.5 e - ω / ω c \frac{dW}{d\omega}\approx\sqrt{\frac{3\pi}{2}}\frac{e^{2}}{4\pi\varepsilon_{0}% c}\gamma\left(\frac{\omega}{\omega\text{c}}\right)^{0.5}e^{-\omega/\omega\text% {c}}
  56. ω ω c \omega\gg\omega\text{c}
  57. ξ K 5 / 3 ( x ) d x = 1 3 0 9 + 36 x 2 + 16 x 4 ( 3 + 4 x 2 ) 1 + x 2 / 3 exp [ - ξ ( 1 + 4 x 2 3 ) 1 + x 2 3 ] d x \int_{\xi}^{\infty}K_{5/3}(x)dx=\frac{1}{\sqrt{3}}\,\int_{0}^{\infty}\,\frac{9% +36x^{2}+16x^{4}}{(3+4x^{2})\sqrt{1+x^{2}/3}}\exp\left[-\xi\left(1+\frac{4x^{2% }}{3}\right)\sqrt{1+\frac{x^{2}}{3}}\right]\ dx
  58. ε c = ω c = 3 2 c ρ γ 3 \varepsilon_{c}=\hbar\omega\text{c}=\frac{3}{2}\frac{\hbar c}{\rho}\gamma^{3}
  59. K 2 / 3 2 ( ξ ) K_{2/3}^{2}(\xi)
  60. γ 2 θ 2 1 + γ 2 θ 2 K 1 / 3 2 ( ξ ) \frac{\gamma^{2}\theta^{2}}{1+\gamma^{2}\theta^{2}}K_{1/3}^{2}(\xi)
  61. θ = 0 \theta=0
  62. d 2 W d Ω = 0 d 3 W d ω d Ω d ω = 7 e 2 γ 5 64 π ε 0 ρ 1 ( 1 + γ 2 θ 2 ) 5 / 2 [ 1 + 5 7 γ 2 θ 2 1 + γ 2 θ 2 ] ( 12 ) \frac{d^{2}W}{d\Omega}=\int_{0}^{\infty}\frac{d^{3}W}{d\omega d\Omega}d\omega=% \frac{7e^{2}\gamma^{5}}{64\pi\varepsilon_{0}\rho}\frac{1}{(1+\gamma^{2}\theta^% {2})^{5/2}}\left[1+\frac{5}{7}\frac{\gamma^{2}\theta^{2}}{1+\gamma^{2}\theta^{% 2}}\right]\qquad(12)
  63. B = ( 0 , B 0 sin ( k u z ) , 0 ) \vec{B}=\left(0,B_{0}\sin(k\text{u}z),0\right)
  64. r ( t ) = λ u K 2 π γ sin ω u t x ^ + ( β z ¯ c t + λ u K 2 16 π γ 2 cos ( 2 ω u t ) ) z ^ \vec{r}(t)=\frac{\lambda\text{u}K}{2\pi\gamma}\sin\omega\text{u}t\cdot\hat{x}+% \left(\bar{\beta_{z}}ct+\frac{\lambda\text{u}K^{2}}{16\pi\gamma^{2}}\cos(2% \omega\text{u}t)\right)\cdot\hat{z}
  65. K = e B 0 λ u 2 π m c K=\frac{eB_{0}\lambda\text{u}}{2\pi mc}
  66. β z ¯ = 1 - 1 2 γ 2 ( 1 + K 2 2 ) \bar{\beta_{z}}=1-\frac{1}{2\gamma^{2}}\left(1+\frac{K^{2}}{2}\right)
  67. K K
  68. d = λ u β ¯ - λ u cos θ = n λ d=\frac{\lambda\text{u}}{\bar{\beta}}-\lambda\text{u}\cos\theta=n\lambda
  69. cos θ 1 - θ 2 2 \cos\theta\approx 1-\frac{\theta^{2}}{2}
  70. O ( θ 2 ) O(\theta^{2})
  71. λ n = λ u 2 γ 2 n ( 1 + K 2 2 + γ 2 θ 2 β ¯ ) \lambda_{n}=\frac{\lambda\text{u}}{2\gamma^{2}n}\left(\frac{1+\frac{K^{2}}{2}+% \gamma^{2}\theta^{2}}{\bar{\beta}}\right)
  72. β ¯ 1 \bar{\beta}\rightarrow 1
  73. λ n = λ u 2 γ 2 n ( 1 + K 2 2 + γ 2 θ 2 ) ( 13 ) \lambda_{n}=\frac{\lambda\text{u}}{2\gamma^{2}n}\left(1+\frac{K^{2}}{2}+\gamma% ^{2}\theta^{2}\right)\qquad(13)
  74. d 3 W d Ω d ω = e 2 4 π ε 0 4 π 2 c | - n ^ × [ ( n ^ - β ) × β ˙ ] ( 1 - n ^ β ) 2 e i ω ( t - n ^ r ( t ) / c ) d t | 2 \frac{d^{3}W}{d\Omega d\omega}=\frac{e^{2}}{4\pi\varepsilon_{0}4\pi^{2}c}\left% |\int_{-\infty}^{\infty}\frac{\hat{n}\times\left[\left(\hat{n}-\vec{\beta}% \right)\times\dot{\vec{\beta}}\right]}{\left(1-\hat{n}\cdot\vec{\beta}\right)^% {2}}e^{i\omega(t-\hat{n}\cdot\vec{r}(t)/c)}dt\right|^{2}
  75. N u N_{u}
  76. d 3 W d Ω d ω = e 2 ω 2 4 π ε 0 4 π 2 c | - λ u / 2 β ¯ c λ u / 2 β ¯ c n ^ × ( n ^ × β ) e i ω ( t - n ^ r ( t ) / c ) d t | 2 | 1 + e i δ + e 2 i δ + + e i ( N u - 1 ) δ | 2 ( 14 ) \frac{d^{3}W}{d\Omega d\omega}=\frac{e^{2}\omega^{2}}{4\pi\varepsilon_{0}4\pi^% {2}c}\left|\int_{-\lambda_{u}/2\bar{\beta}c}^{\lambda_{u}/2\bar{\beta}c}\hat{n% }\times\left(\hat{n}\times\vec{\beta}\right)e^{i\omega(t-\hat{n}\cdot\vec{r}(t% )/c)}dt\right|^{2}\left|1+e^{i\delta}+e^{2i\delta}+\cdots+e^{i(N_{u}-1)\delta}% \right|^{2}\qquad(14)
  77. β ¯ = β ( 1 - K 2 4 γ 2 ) \bar{\beta}=\beta\left(1-\frac{K^{2}}{4\gamma^{2}}\right)
  78. δ = 2 π ω ω res ( θ ) \delta=\frac{2\pi\omega}{\omega\text{res}(\theta)}
  79. ω res ( θ ) = 2 π c λ res ( θ ) \omega\text{res}(\theta)=\frac{2\pi c}{\lambda\text{res}(\theta)}
  80. λ res ( θ ) = λ u 2 γ 2 ( 1 + K 2 2 + γ 2 θ 2 ) \lambda\text{res}(\theta)=\frac{\lambda_{u}}{2\gamma^{2}}\left(1+\frac{K^{2}}{% 2}+\gamma^{2}\theta^{2}\right)
  81. d 3 W d Ω d ω = e 2 γ 2 N 2 4 π ε 0 c L ( N Δ ω n ω res ( θ ) ) F n ( K , θ , ϕ ) ( 15 ) \frac{d^{3}W}{d\Omega d\omega}=\frac{e^{2}\gamma^{2}N^{2}}{4\pi\varepsilon_{0}% c}L\left(N\frac{\Delta\omega_{n}}{\omega\text{res}(\theta)}\right)F_{n}(K,% \theta,\phi)\qquad(15)
  82. δ \delta
  83. L ( N Δ ω k ω res ( θ ) ) = sin 2 ( N π Δ ω k / ω res ( θ ) ) N 2 sin 2 ( π Δ ω k / ω res ( θ ) ) L\left(N\frac{\Delta\omega_{k}}{\omega\text{res}(\theta)}\right)=\frac{\sin^{2% }\left(N\pi\Delta\omega_{k}/\omega\text{res}(\theta)\right)}{N^{2}\sin^{2}% \left(\pi\Delta\omega_{k}/\omega\text{res}(\theta)\right)}
  84. F n F_{n}
  85. K K
  86. F n ( K , θ , ϕ ) | - λ u / 2 β ¯ c λ u / 2 β ¯ c n ^ × ( n ^ × β ) e i ω ( t - n ^ r ( t ) / c ) d t | 2 F_{n}(K,\theta,\phi)\propto\left|\int_{-\lambda_{u}/2\bar{\beta}c}^{\lambda_{u% }/2\bar{\beta}c}\hat{n}\times\left(\hat{n}\times\vec{\beta}\right)e^{i\omega(t% -\hat{n}\cdot\vec{r}(t)/c)}dt\right|^{2}
  87. θ = 0 \theta=0
  88. ϕ = 0 \phi=0
  89. d 3 W d Ω d ω = e 2 γ 2 N 2 4 π ε 0 c L ( N Δ ω n ω res ( 0 ) ) F n ( K , 0 , 0 ) \frac{d^{3}W}{d\Omega d\omega}=\frac{e^{2}\gamma^{2}N^{2}}{4\pi\varepsilon_{0}% c}L\left(N\frac{\Delta\omega_{n}}{\omega\text{res}(0)}\right)F_{n}(K,0,0)
  90. F n ( K , 0 , 0 ) = n 2 K 2 1 + K 2 / 2 [ J n + 1 2 ( Z ) - J n - 1 2 ( Z ) ] 2 F_{n}(K,0,0)=\frac{n^{2}K^{2}}{1+K^{2}/2}\left[J_{\frac{n+1}{2}}(Z)-J_{\frac{n% -1}{2}}(Z)\right]^{2}
  91. Z = n K 2 4 ( 1 + K 2 / 2 ) Z=\frac{nK^{2}}{4(1+K^{2}/2)}

Synthetic_division.html

  1. x - a , x-a,
  2. x - a x-a
  3. x 3 - 12 x 2 - 42 x - 3 \frac{x^{3}-12x^{2}-42}{x-3}
  4. 1 - 12 0 - 42 \begin{array}[]{cc}\begin{array}[]{r}\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&&\\ \hline\end{array}\end{array}
  5. - 1 x + 3 \begin{array}[]{rr}-1x&+3\end{array}
  6. 3 1 - 12 0 - 42 \begin{array}[]{cc}\begin{array}[]{r}\\ 3\\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&&\\ \hline\end{array}\end{array}
  7. 3 1 - 12 0 - 42 1 \begin{array}[]{cc}\begin{array}[]{r}\\ 3\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&&\\ \hline 1&&&\\ \end{array}\end{array}
  8. 3 1 - 12 0 - 42 3 1 \begin{array}[]{cc}\begin{array}[]{r}\\ 3\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &3&&\\ \hline 1&&&\\ \end{array}\end{array}
  9. 3 1 - 12 0 - 42 3 1 - 9 \begin{array}[]{cc}\begin{array}[]{c}\\ 3\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &3&&\\ \hline 1&-9&&\\ \end{array}\end{array}
  10. 3 1 - 12 0 - 42 3 - 27 - 81 1 - 9 - 27 - 123 \begin{array}[]{cc}\begin{array}[]{c}\\ 3\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &3&-27&-81\\ \hline 1&-9&-27&-123\end{array}\end{array}
  11. 1 - 9 - 27 - 123 \begin{array}[]{rrr|r}1&-9&-27&-123\end{array}
  12. 1 x 2 - 9 x - 27 - 123 \begin{array}[]{rrr|r}1x^{2}&-9x&-27&-123\end{array}
  13. x 3 - 12 x 2 - 42 x - 3 = x 2 - 9 x - 27 - 123 x - 3 \frac{x^{3}-12x^{2}-42}{x-3}=x^{2}-9x-27-\frac{123}{x-3}
  14. p ( x ) p(x)
  15. a a
  16. p ( x ) ( x - a ) \frac{p(x)}{(x-a)}
  17. x 3 - 12 x 2 - 42 x 2 + x - 3 \frac{x^{3}-12x^{2}-42}{x^{2}+x-3}
  18. 1 - 12 0 - 42 \begin{array}[]{|rrrr}1&-12&0&-42\end{array}
  19. - 1 x 2 - 1 x + 3 \begin{array}[]{rrr}-1x^{2}&-1x&+3\end{array}
  20. 3 - 1 1 - 12 0 - 42 \begin{array}[]{cc}\begin{array}[]{rr}\\ &3\\ -1&\\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&&\\ &&&\\ \hline\end{array}\end{array}
  21. 3 - 1 1 - 12 0 - 42 1 \begin{array}[]{cc}\begin{array}[]{rr}\\ &3\\ -1&\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&&\\ &&&\\ \hline 1&&&\\ \end{array}\end{array}
  22. 3 - 1 1 - 12 0 - 42 3 - 1 1 \begin{array}[]{cc}\begin{array}[]{rr}\\ &3\\ -1&\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&3&\\ &-1&&\\ \hline 1&&&\\ \end{array}\end{array}
  23. 3 - 1 1 - 12 0 - 42 3 - 1 1 - 13 \begin{array}[]{cc}\begin{array}[]{rr}\\ &3\\ -1&\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&3&\\ &-1&&\\ \hline 1&-13&&\\ \end{array}\end{array}
  24. 3 - 1 1 - 12 0 - 42 3 - 39 - 1 13 1 - 13 16 \begin{array}[]{cc}\begin{array}[]{rr}\\ &3\\ -1&\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&3&-39\\ &-1&13&\\ \hline 1&-13&16&\\ \end{array}\end{array}
  25. 3 - 1 1 - 12 0 - 42 3 - 39 - 1 13 1 - 13 16 - 81 \begin{array}[]{cc}\begin{array}[]{rr}\\ &3\\ -1&\\ \\ \end{array}&\begin{array}[]{|rrrr}1&-12&0&-42\\ &&3&-39\\ &-1&13&\\ \hline 1&-13&16&-81\\ \end{array}\end{array}
  26. 1 - 13 16 - 81 \begin{array}[]{rr|rr}1&-13&16&-81\end{array}
  27. 1 x - 13 16 x - 81 \begin{array}[]{rr|rr}1x&-13&16x&-81\end{array}
  28. x 3 - 12 x 2 - 42 x 2 + x - 3 = x - 13 + 16 x - 81 x 2 + x - 3 \frac{x^{3}-12x^{2}-42}{x^{2}+x-3}=x-13+\frac{16x-81}{x^{2}+x-3}
  29. g ( x ) g(x)
  30. h ( x ) = g ( x ) a h(x)=\frac{g(x)}{a}
  31. h ( x ) h(x)
  32. g ( x ) g(x)
  33. f ( x ) f(x)
  34. g ( x ) g(x)
  35. 6 x 3 + 5 x 2 - 7 3 x 2 - 2 x - 1 \frac{6x^{3}+5x^{2}-7}{3x^{2}-2x-1}
  36. 1 2 / 3 6 5 0 - 7 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&&\\ &&&\\ \hline&&&\\ &&&\\ \end{array}\end{array}
  37. g ( x ) g(x)
  38. g ( x ) g(x)
  39. f ( x ) f(x)
  40. 1 2 / 3 6 5 0 - 7 6 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&&\\ &&&\\ \hline 6&&&\\ &&&\\ \end{array}\end{array}
  41. 1 2 / 3 6 5 0 - 7 6 2 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&&\\ &&&\\ \hline 6&&&\\ 2&&&\\ \end{array}\end{array}
  42. 1 2 / 3 6 5 0 - 7 2 4 6 2 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&2&\\ &4&&\\ \hline 6&&&\\ 2&&&\\ \end{array}\end{array}
  43. 1 2 / 3 6 5 0 - 7 2 4 6 9 2 3 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&2&\\ &4&&\\ \hline 6&9&&\\ 2&3&&\\ \end{array}\end{array}
  44. 1 2 / 3 6 5 0 - 7 2 3 4 6 6 9 2 3 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&2&3\\ &4&6&\\ \hline 6&9&&\\ 2&3&&\\ \end{array}\end{array}
  45. 1 2 / 3 6 5 0 - 7 2 3 4 6 6 9 8 - 4 2 3 \begin{array}[]{cc}\begin{array}[]{rrr}\\ &1&\\ 2&&\\ \\ &&/3\\ \end{array}\begin{array}[]{|rrrr}6&5&0&-7\\ &&2&3\\ &4&6&\\ \hline 6&9&8&-4\\ 2&3&&\\ \end{array}\end{array}
  46. 2 x + 3 8 x - 4 \begin{array}[]{rr|rr}2x&+3&8x&-4\end{array}
  47. 6 x 3 + 5 x 2 - 7 3 x 2 - 2 x - 1 = 2 x + 3 + 8 x - 4 3 x 2 - 2 x - 1 \frac{6x^{3}+5x^{2}-7}{3x^{2}-2x-1}=2x+3+\frac{8x-4}{3x^{2}-2x-1}
  48. a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h i x 4 - j x 3 - k x 2 - l x - m = n x 3 + o x 2 + p x + q + r x 3 + s x 2 + t x + u i x 4 - j x 3 - k x 2 - l x - m \dfrac{ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h}{ix^{4}-jx^{3}-kx^{2}-lx% -m}=nx^{3}+ox^{2}+px+q+\dfrac{rx^{3}+sx^{2}+tx+u}{ix^{4}-jx^{3}-kx^{2}-lx-m}
  49. j k l m q j p j p k q k o j o k o l p l q l n j n k n l n m o m p m q m a b c d e f g h a o 0 p 0 q 0 r s t u n o p q \begin{array}[]{cc}\begin{array}[]{rrrr}\\ \\ \\ \\ j&k&l&m\\ \end{array}&\begin{array}[]{|rrrr|rrrr}&&&&qj&&&\\ &&&pj&pk&qk&&\\ &&oj&ok&ol&pl&ql&\\ &nj&nk&nl&nm&om&pm&qm\\ a&b&c&d&e&f&g&h\\ \hline a&o_{0}&p_{0}&q_{0}&r&s&t&u\\ n&o&p&q&&&&\\ \end{array}\end{array}

Syntonic_comma.html

  1. 81 64 80 81 = 1 5 4 1 = 5 4 {81\over 64}\cdot{80\over 81}={{1\cdot 5}\over{4\cdot 1}}={5\over 4}
  2. 32 27 81 80 = 2 3 1 5 = 6 5 {32\over 27}\cdot{81\over 80}={{2\cdot 3}\over{1\cdot 5}}={6\over 5}
  3. 3 2 3 4 3 2 3 4 4 5 = 81 80 {3\over 2}\cdot{3\over 4}\cdot{3\over 2}\cdot{3\over 4}\cdot{4\over 5}={81% \over 80}
  4. 3 2 1 2 3 2 3 2 1 2 3 2 1 2 8 5 = 81 80 {3\over 2}\cdot{1\over 2}\cdot{3\over 2}\cdot{3\over 2}\cdot{1\over 2}\cdot{3% \over 2}\cdot{1\over 2}\cdot{8\over 5}={81\over 80}
  5. 3 2 3 2 3 2 3 2 8 5 1 2 1 2 1 2 = 81 80 {3\over 2}\cdot{3\over 2}\cdot{3\over 2}\cdot{3\over 2}\cdot{8\over 5}\cdot{1% \over 2}\cdot{1\over 2}\cdot{1\over 2}={81\over 80}

System_dynamics.html

  1. Potential adopters = 0 t -New adopters d t \ \mbox{Potential adopters}~{}=\int_{0}^{t}\mbox{-New adopters }~{}\,dt
  2. Adopters = 0 t New adopters d t \ \mbox{Adopters}~{}=\int_{0}^{t}\mbox{New adopters }~{}\,dt
  3. 1 ) Probability that contact has not yet adopted = Potential adopters / ( Potential adopters + Adopters ) 1)\ \mbox{Probability that contact has not yet adopted}~{}=\mbox{Potential % adopters}~{}/(\mbox{Potential adopters }~{}+\mbox{ Adopters}~{})
  4. 2 ) Imitators = q Adopters Probability that contact has not yet adopted 2)\ \mbox{Imitators}~{}=q\cdot\mbox{Adopters}~{}\cdot\mbox{Probability that % contact has not yet adopted}~{}
  5. 3 ) Innovators = p Potential adopters 3)\ \mbox{Innovators}~{}=p\cdot\mbox{Potential adopters}~{}
  6. 4 ) New adopters = Innovators + Imitators 4)\ \mbox{New adopters}~{}=\mbox{Innovators}~{}+\mbox{Imitators}~{}
  7. 4.1 ) Potential adopters - = New adopters 4.1)\ \mbox{Potential adopters}~{}\ -=\mbox{New adopters }~{}
  8. 4.2 ) Adopters + = New adopters 4.2)\ \mbox{Adopters}~{}\ +=\mbox{New adopters }~{}
  9. p = 0.03 \ p=0.03
  10. q = 0.4 \ q=0.4
  11. 10 ) Valve New adopters = New adopters T i m e S t e p 10)\ \mbox{Valve New adopters}~{}\ =\mbox{New adopters}~{}\cdot TimeStep
  12. 10.1 ) Potential adopters - = Valve New adopters 10.1)\ \mbox{Potential adopters}~{}\ -=\mbox{Valve New adopters}~{}
  13. 10.2 ) Adopters + = Valve New adopters 10.2)\ \mbox{Adopters}~{}\ +=\mbox{Valve New adopters }~{}
  14. T i m e S t e p = 1 / 4 \ TimeStep=1/4
  15. Valve New adopters = New adopters T i m e S t e p \ \mbox{Valve New adopters}~{}\ =\mbox{New adopters }~{}\cdot TimeStep

System_of_linear_equations.html

  1. 3 x + 2 y - z = 1 2 x - 2 y + 4 z = - 2 - x + 1 2 y - z = 0 \begin{aligned}\displaystyle 3x&&\displaystyle\;+&&\displaystyle 2y&&% \displaystyle\;-&&\displaystyle z&&\displaystyle\;=&&\displaystyle 1&\\ \displaystyle 2x&&\displaystyle\;-&&\displaystyle 2y&&\displaystyle\;+&&% \displaystyle 4z&&\displaystyle\;=&&\displaystyle-2&\\ \displaystyle-x&&\displaystyle\;+&&\displaystyle\tfrac{1}{2}y&&\displaystyle\;% -&&\displaystyle z&&\displaystyle\;=&&\displaystyle 0&\end{aligned}
  2. x , y , z x,y,z
  3. x = 1 y = - 2 z = - 2 \begin{aligned}\displaystyle x&\displaystyle\,=&\displaystyle 1\\ \displaystyle y&\displaystyle\,=&\displaystyle-2\\ \displaystyle z&\displaystyle\,=&\displaystyle-2\end{aligned}
  4. 2 x \displaystyle 2x
  5. x x
  6. y y
  7. x = 3 - 3 2 y . x=3-\frac{3}{2}y.
  8. 4 ( 3 - 3 2 y ) + 9 y = 15. 4\left(3-\frac{3}{2}y\right)+9y=15.
  9. y y
  10. y = 1 y=1
  11. x x
  12. x = 3 / 2 x=3/2
  13. a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 a m 1 x 1 + a m 2 x 2 + + a m n x n = b m . \begin{aligned}\displaystyle a_{11}x_{1}&&\displaystyle\;+&&\displaystyle a_{1% 2}x_{2}&&\displaystyle\;+\cdots+&&\displaystyle a_{1n}x_{n}&&\displaystyle\;=&% &&\displaystyle b_{1}\\ \displaystyle a_{21}x_{1}&&\displaystyle\;+&&\displaystyle a_{22}x_{2}&&% \displaystyle\;+\cdots+&&\displaystyle a_{2n}x_{n}&&\displaystyle\;=&&&% \displaystyle b_{2}\\ \displaystyle\vdots&&&&\displaystyle\vdots&&&&\displaystyle\vdots&&&&&% \displaystyle\;\vdots\\ \displaystyle a_{m1}x_{1}&&\displaystyle\;+&&\displaystyle a_{m2}x_{2}&&% \displaystyle\;+\cdots+&&\displaystyle a_{mn}x_{n}&&\displaystyle\;=&&&% \displaystyle b_{m}.\\ \end{aligned}
  14. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  15. a 11 , a 12 , , a m n a_{11},a_{12},\ldots,a_{mn}
  16. b 1 , b 2 , , b m b_{1},b_{2},\ldots,b_{m}
  17. x 1 [ a 11 a 21 a m 1 ] + x 2 [ a 12 a 22 a m 2 ] + + x n [ a 1 n a 2 n a m n ] = [ b 1 b 2 b m ] x_{1}\begin{bmatrix}a_{11}\\ a_{21}\\ \vdots\\ a_{m1}\end{bmatrix}+x_{2}\begin{bmatrix}a_{12}\\ a_{22}\\ \vdots\\ a_{m2}\end{bmatrix}+\cdots+x_{n}\begin{bmatrix}a_{1n}\\ a_{2n}\\ \vdots\\ a_{mn}\end{bmatrix}=\begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{m}\end{bmatrix}
  18. A x = b A{x}={b}
  19. A = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ] , x = [ x 1 x 2 x n ] , b = [ b 1 b 2 b m ] A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix},\quad{x}=\begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{bmatrix},\quad{b}=\begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{m}\end{bmatrix}
  20. 3 x + 2 y = 6 and 6 x + 4 y = 12 3x+2y=6\;\;\;\;\,\text{and}\;\;\;\;6x+4y=12
  21. x - 2 y = - 1 3 x + 5 y = 8 4 x + 3 y = 7 \begin{aligned}\displaystyle x&&\displaystyle\;-&&\displaystyle 2y&&% \displaystyle\;=&&\displaystyle-1&\\ \displaystyle 3x&&\displaystyle\;+&&\displaystyle 5y&&\displaystyle\;=&&% \displaystyle 8&\\ \displaystyle 4x&&\displaystyle\;+&&\displaystyle 3y&&\displaystyle\;=&&% \displaystyle 7&\end{aligned}
  22. 3 x + 2 y = 6 and 3 x + 2 y = 12 3x+2y=6\;\;\;\;\,\text{and}\;\;\;\;3x+2y=12
  23. x + y = 1 2 x + y = 1 3 x + 2 y = 3 \begin{aligned}\displaystyle x&&\displaystyle\;+&&\displaystyle y&&% \displaystyle\;=&&\displaystyle 1&\\ \displaystyle 2x&&\displaystyle\;+&&\displaystyle y&&\displaystyle\;=&&% \displaystyle 1&\\ \displaystyle 3x&&\displaystyle\;+&&\displaystyle 2y&&\displaystyle\;=&&% \displaystyle 3&\end{aligned}
  24. ( x = 3 , y = - 2 , z = 6 ) (x=3,\;y=-2,\;z=6)
  25. ( 3 , - 2 , 6 ) (3,\,-2,\,6)
  26. x \displaystyle x
  27. x = - 7 z - 1 and y = 3 z + 2 . x=-7z-1\;\;\;\;\,\text{and}\;\;\;\;y=3z+2\,\text{.}
  28. y = - 3 7 x + 11 7 and z = - 1 7 x - 1 7 . y=-\frac{3}{7}x+\frac{11}{7}\;\;\;\;\,\text{and}\;\;\;\;z=-\frac{1}{7}x-\frac{% 1}{7}\,\text{.}
  29. x \displaystyle x
  30. - 4 y + 12 z = - 8 - 2 y + 7 z = - 2 \begin{aligned}\displaystyle-4y&&\displaystyle\;+&&\displaystyle 12z&&% \displaystyle\;=&&\displaystyle-8&\\ \displaystyle-2y&&\displaystyle\;+&&\displaystyle 7z&&\displaystyle\;=&&% \displaystyle-2&\end{aligned}
  31. x \displaystyle x
  32. [ 1 3 - 2 5 3 5 6 7 2 4 3 8 ] . \left[\begin{array}[]{rrr|r}1&3&-2&5\\ 3&5&6&7\\ 2&4&3&8\end{array}\right]\,\text{.}
  33. [ 1 3 - 2 5 3 5 6 7 2 4 3 8 ] \displaystyle\left[\begin{array}[]{rrr|r}1&3&-2&5\\ 3&5&6&7\\ 2&4&3&8\end{array}\right]
  34. x + 3 y - 2 z = 5 3 x + 5 y + 6 z = 7 2 x + 4 y + 3 z = 8 \begin{aligned}\displaystyle x&\displaystyle\;+&\displaystyle\;3y&% \displaystyle\;-&\displaystyle\;2z&\displaystyle\;=&\displaystyle\;5\\ \displaystyle 3x&\displaystyle\;+&\displaystyle\;5y&\displaystyle\;+&% \displaystyle\;6z&\displaystyle\;=&\displaystyle\;7\\ \displaystyle 2x&\displaystyle\;+&\displaystyle\;4y&\displaystyle\;+&% \displaystyle\;3z&\displaystyle\;=&\displaystyle\;8\end{aligned}
  35. x = | 5 3 - 2 7 5 6 8 4 3 | | 1 3 - 2 3 5 6 2 4 3 | , y = | 1 5 - 2 3 7 6 2 8 3 | | 1 3 - 2 3 5 6 2 4 3 | , z = | 1 3 5 3 5 7 2 4 8 | | 1 3 - 2 3 5 6 2 4 3 | . x=\frac{\,\left|\begin{matrix}5&3&-2\\ 7&5&6\\ 8&4&3\end{matrix}\right|\,}{\,\left|\begin{matrix}1&3&-2\\ 3&5&6\\ 2&4&3\end{matrix}\right|\,},\;\;\;\;y=\frac{\,\left|\begin{matrix}1&5&-2\\ 3&7&6\\ 2&8&3\end{matrix}\right|\,}{\,\left|\begin{matrix}1&3&-2\\ 3&5&6\\ 2&4&3\end{matrix}\right|\,},\;\;\;\;z=\frac{\,\left|\begin{matrix}1&3&5\\ 3&5&7\\ 2&4&8\end{matrix}\right|\,}{\,\left|\begin{matrix}1&3&-2\\ 3&5&6\\ 2&4&3\end{matrix}\right|\,}.
  36. A x = b A{x}={b}
  37. x = A - 1 b {x}=A^{-1}{b}
  38. A - 1 A^{-1}
  39. A g A^{g}
  40. x = A g b + ( I - A g A ) w {x}=A^{g}{b}+(I-A^{g}A){w}
  41. w {w}
  42. w = 0 {w}={0}
  43. A x = b A{x}={b}
  44. A A g b = b . AA^{g}{b}={b}.
  45. A g A^{g}
  46. A - 1 A^{-1}
  47. x = A - 1 b + ( I - A - 1 A ) w = A - 1 b + ( I - I ) w = A - 1 b {x}=A^{-1}{b}+(I-A^{-1}A){w}=A^{-1}{b}+(I-I){w}=A^{-1}{b}
  48. w {w}
  49. w {w}
  50. w {w}
  51. a 11 x 1 \displaystyle a_{11}x_{1}
  52. A 𝐱 = 𝟎 A\,\textbf{x}=\,\textbf{0}
  53. A 𝐱 = 𝐛 and A 𝐱 = 𝟎 . A\,\textbf{x}=\,\textbf{b}\qquad\,\text{and}\qquad A\,\textbf{x}=\,\textbf{0}% \,\text{.}
  54. { 𝐩 + 𝐯 : 𝐯 is any solution to A 𝐱 = 𝟎 } . \left\{\,\textbf{p}+\,\textbf{v}:\,\textbf{v}\,\text{ is any solution to }A\,% \textbf{x}=\,\textbf{0}\right\}.

Systematic_sampling.html

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

T-symmetry.html

  1. T : t - t . T:t\mapsto-t.
  2. x \vec{x}\!
  3. a \vec{a}\!
  4. F \vec{F}\!
  5. E E\!
  6. ϕ \phi\!
  7. E \vec{E}\!
  8. D \vec{D}\!
  9. ρ \rho\!
  10. P \vec{P}\!
  11. t t\!
  12. v \vec{v}\!
  13. p \vec{p}\!
  14. l \vec{l}\!
  15. A \vec{A}\!
  16. B \vec{B}\!
  17. H \vec{H}\!
  18. j \vec{j}\!
  19. M \vec{M}\!
  20. S \vec{S}\!
  21. T = e - i π J y / K , T=e^{-i\pi J_{y}/\hbar}K,

T1_space.html

  1. U A := x A G x . U_{A}:=\bigcap_{x\in A}G_{x}.

Tacoma_Narrows_Bridge_(1940).html

  1. m x ¨ ( t ) + c x ˙ ( t ) + k x ( t ) = F c o s ( ω t ) m\ddot{x}(t)+c\dot{x}(t)+kx(t)=Fcos(\omega t)
  2. m m
  3. c c
  4. k k
  5. F F
  6. ω \omega
  7. t t
  8. ω \omega
  9. ω r = k / m \omega_{r}=\sqrt{k/m}
  10. ω r \omega_{r}
  11. f s f_{s}
  12. f s D U = S \frac{f_{s}D}{U}=S
  13. U U
  14. D D
  15. S S
  16. D D
  17. S S
  18. 2 π f s = ω 2\pi f_{s}=\omega

Tangent_bundle.html

  1. M M
  2. T M = x M T x M = x M { x } × T x M = x M { ( x , y ) | y T x M } . TM=\bigsqcup_{x\in M}T_{x}M=\bigcup_{x\in M}\left\{x\right\}\times T_{x}M=% \bigcup_{x\in M}\left\{(x,y)|\;y\in T_{x}M\right\}.
  3. T x M T_{x}M
  4. M M
  5. x x
  6. T M TM
  7. ( x , v ) (x,v)
  8. x x
  9. M M
  10. v v
  11. M M
  12. x x
  13. π : T M M \pi:TM\twoheadrightarrow M
  14. π ( x , v ) = x \pi(x,v)=x
  15. T x M T_{x}M
  16. x x
  17. T M TM
  18. M M
  19. T M TM
  20. M M
  21. M M
  22. ϕ α : U α 𝐑 n \phi_{\alpha}\colon U_{\alpha}\to\mathbf{R}^{n}
  23. ϕ ~ α : π - 1 ( U α ) 𝐑 2 n \widetilde{\phi}_{\alpha}\colon\pi^{-1}(U_{\alpha})\to\mathbf{R}^{2n}
  24. ϕ ~ α ( x , v i i ) = ( ϕ α ( x ) , v 1 , , v n ) \widetilde{\phi}_{\alpha}(x,v^{i}\partial_{i})=(\phi_{\alpha}(x),v^{1},\cdots,% v^{n})
  25. ϕ ~ α ( A π - 1 ( U α ) ) \widetilde{\phi}_{\alpha}(A\cap\pi^{-1}(U_{\alpha}))
  26. π - 1 ( U α U β ) \pi^{-1}(U_{\alpha}\cap U_{\beta})
  27. V : M T M V\colon M\to TM
  28. ( V + W ) x = V x + W x (V+W)_{x}=V_{x}+W_{x}\,
  29. ( f V ) x = f ( x ) V x (fV)_{x}=f(x)V_{x}\,
  30. T 2 M = T ( T M ) . T^{2}M=T(TM).\,
  31. T k M T^{k}M
  32. T ( T k - 1 M ) T(T^{k-1}M)
  33. D k f : T k M T k N D^{k}f:T^{k}M\to T^{k}N
  34. T W W × W , TW\cong W\times W,
  35. W T W W\to TW
  36. w ( w , w ) w\mapsto(w,w)
  37. T m M 𝐑 n T_{m}M\approx\mathbf{R}^{n}
  38. 𝐑 n . \mathbf{R}^{n}.
  39. \approx
  40. \cong
  41. T ( T M ) T ( M × 𝐑 n ) T M × T ( 𝐑 n ) T M × ( 𝐑 n × 𝐑 n ) T(TM)\approx T(M\times\mathbf{R}^{n})\cong TM\times T(\mathbf{R}^{n})\cong TM% \times(\mathbf{R}^{n}\times\mathbf{R}^{n})
  42. T T M T M TTM\to TM
  43. ( T M M ) × ( 𝐑 n × 𝐑 n 𝐑 n ) . (TM\to M)\times(\mathbf{R}^{n}\times\mathbf{R}^{n}\to\mathbf{R}^{n}).
  44. V = i v i v i | ( x , v ) . V=\sum_{i}\left.v^{i}\frac{\partial}{\partial v^{i}}\right|_{(x,v)}.
  45. ( x , v ) ( x , v , 0 , v ) (x,v)\mapsto(x,v,0,v)
  46. { 𝐑 × T M T M ( t , v ) t v \begin{cases}\mathbf{R}\times TM\to TM\\ (t,v)\longmapsto tv\end{cases}
  47. f v = f π f^{v}=f\circ\pi
  48. M M

Tangent_vector.html

  1. x x
  2. x x
  3. 𝐫 ( t ) \mathbf{r}(t)
  4. 𝐫 ( t ) \mathbf{r}^{\prime}(t)
  5. 𝐓 ( t ) = 𝐫 ( t ) | 𝐫 ( t ) | . \mathbf{T}(t)=\frac{\mathbf{r}^{\prime}(t)}{|\mathbf{r}^{\prime}(t)|}\,.
  6. 𝐫 ( t ) = { ( 1 + t 2 , e t , cos t ) | t } \mathbf{r}(t)=\{(1+t^{2},e^{t},\cos{t})|\ t\in\mathbb{R}\}
  7. 3 \mathbb{R}^{3}
  8. t = 0 t=0
  9. 𝐓 ( 0 ) = 𝐫 ( 0 ) | 𝐫 ( 0 ) | = ( 1 + t 2 , e t , cos t ) 4 t 2 + e 2 t + sin 2 t | t = 0 = ( 0 , 1 , 0 ) . \mathbf{T}(0)=\frac{\mathbf{r}^{\prime}(0)}{|\mathbf{r}^{\prime}(0)|}=\left.% \frac{(1+t^{2},e^{t},\cos{t})}{\sqrt{4t^{2}+e^{2t}+\sin^{2}{t}}}\right|_{t=0}=% (0,1,0)\,.
  10. 𝐫 ( t ) \mathbf{r}(t)
  11. 𝐫 ( t ) = ( x 1 ( t ) , x 2 ( t ) , , x n ( t ) ) \mathbf{r}(t)=(x^{1}(t),x^{2}(t),\ldots,x^{n}(t))
  12. 𝐫 = x i = x i ( t ) , a t b , \mathbf{r}=x^{i}=x^{i}(t),\quad a\leq t\leq b\,,
  13. 𝐓 = T i \mathbf{T}=T^{i}
  14. T i = d x i d t . T^{i}=\frac{dx^{i}}{dt}\,.
  15. u i = u i ( x 1 , x 2 , , x n ) , 1 i n u^{i}=u^{i}(x^{1},x^{2},\ldots,x^{n}),\quad 1\leq i\leq n
  16. 𝐓 ¯ = T ¯ i \bar{\mathbf{T}}=\bar{T}^{i}
  17. T ¯ i = d u i d t = u i x s d x s d t = T s u i x s \bar{T}^{i}=\frac{du^{i}}{dt}=\frac{\partial u^{i}}{\partial x^{s}}\frac{dx^{s% }}{dt}=T^{s}\frac{\partial u^{i}}{\partial x^{s}}
  18. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  19. 𝐯 \mathbf{v}
  20. n \mathbb{R}^{n}
  21. 𝐯 \mathbf{v}
  22. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  23. D 𝐯 f ( 𝐱 ) = d d t f ( 𝐱 + t 𝐯 ) | t = 0 = i = 1 n v i f x i ( 𝐱 ) . D_{\mathbf{v}}f(\mathbf{x})=\left.\frac{d}{dt}f(\mathbf{x}+t\mathbf{v})\right|% _{t=0}=\sum_{i=1}^{n}v_{i}\frac{\partial f}{\partial x_{i}}(\mathbf{x})\,.
  24. 𝐱 \mathbf{x}
  25. 𝐯 ( f ( 𝐱 ) ) D 𝐯 ( f ( 𝐱 ) ) . \mathbf{v}(f(\mathbf{x}))\equiv D_{\mathbf{v}}(f(\mathbf{x}))\,.
  26. f , g : n f,g:\mathbb{R}^{n}\rightarrow\mathbb{R}
  27. 𝐯 , 𝐰 \mathbf{v},\mathbf{w}
  28. n \mathbb{R}^{n}
  29. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  30. a , b a,b\in\mathbb{R}
  31. ( a 𝐯 + b 𝐰 ) ( f ) = a 𝐯 ( f ) + b 𝐰 ( f ) (a\mathbf{v}+b\mathbf{w})(f)=a\mathbf{v}(f)+b\mathbf{w}(f)
  32. 𝐯 ( a f + b g ) = a 𝐯 ( f ) + b 𝐯 ( g ) \mathbf{v}(af+bg)=a\mathbf{v}(f)+b\mathbf{v}(g)
  33. 𝐯 ( f g ) = f ( 𝐱 ) 𝐯 ( g ) + g ( 𝐱 ) 𝐯 ( f ) . \mathbf{v}(fg)=f(\mathbf{x})\mathbf{v}(g)+g(\mathbf{x})\mathbf{v}(f)\,.
  34. M M
  35. A ( M ) A(M)
  36. M M
  37. M M
  38. x x
  39. D v : A ( M ) D_{v}:A(M)\rightarrow\mathbb{R}
  40. f , g A ( M ) f,g\in A(M)
  41. a , b a,b\in\mathbb{R}
  42. D v ( a f + b g ) = a D v ( f ) + b D v ( g ) . D_{v}(af+bg)=aD_{v}(f)+bD_{v}(g)\,.
  43. D v ( f g ) = D v ( f ) g ( x ) + f ( x ) D v ( g ) . D_{v}(f\cdot g)=D_{v}(f)\cdot g(x)+f(x)\cdot D_{v}(g)\,.

Tap_and_die.html

  1. T D = M D - 1 N TD=MD-\frac{1}{N}
  2. T D TD
  3. M D MD
  4. N N
  5. 5 / 16 {5}/{16}
  6. T D = M D - p i t c h TD=MD-pitch
  7. T D TD
  8. M D MD

Tautochrone_curve.html

  1. π r / g \pi\sqrt{r/g}
  2. s ˙ 2 \scriptstyle\dot{s}^{2}
  3. y ( s ) = s 2 y(s)=s^{2}\,
  4. d y = 2 s d s dy=2s\,ds
  5. d y 2 = 4 s 2 d s 2 = 4 y ( d x 2 + d y 2 ) dy^{2}=4s^{2}\,ds^{2}=4y\,(dx^{2}+dy^{2})\,
  6. d x d y = 1 - 4 y 2 y {dx\over dy}={\sqrt{1-4y}\over 2\sqrt{y}}\,
  7. x = 1 - 4 u 2 d u x=\int\sqrt{1-4u^{2}}\,du
  8. u = y \scriptstyle u=\sqrt{y}
  9. x = 1 2 u 1 - 4 u 2 + 1 4 sin - 1 ( 2 u ) x={1\over 2}u\sqrt{1-4u^{2}}+{1\over 4}\sin^{-1}(2u)\,
  10. y = u 2 y=u^{2}\,
  11. θ = sin - 1 ( 2 u ) \theta=\sin^{-1}(2u)
  12. 8 x = 2 sin ( θ ) cos ( θ ) + 2 θ = sin ( 2 θ ) + 2 θ 8x=2\sin(\theta)\cos(\theta)+2\theta=\sin(2\theta)+2\theta\,
  13. 8 y = 2 sin ( θ ) 2 = 1 - cos ( 2 θ ) 8y=2\sin(\theta)^{2}=1-\cos(2\theta)\,
  14. d 2 s d t 2 = - k 2 s \frac{d^{2}s}{{dt}^{2}}=-k^{2}s
  15. s = A cos k t s=A\cos kt\,
  16. g sin θ = - k 2 s g\sin\theta=-k^{2}s\,
  17. g cos θ d θ = - k 2 d s g\cos\theta\,d\theta=-k^{2}\,ds\,
  18. d s = - g k 2 cos θ d θ ds=-\frac{g}{k^{2}}\cos\theta\,d\theta\,
  19. d s 2 \displaystyle ds^{2}
  20. d s \displaystyle ds
  21. d y d x \displaystyle\frac{dy}{dx}
  22. ϕ = - 2 θ \phi=-2\theta\,
  23. r = g 4 k 2 r=\frac{g}{4k^{2}}\,
  24. x x
  25. y y
  26. x \displaystyle x
  27. T = π 2 k T=\frac{\pi}{2k}
  28. r \displaystyle r
  29. 1 2 m v 2 \frac{1}{2}mv^{2}
  30. d s d t \frac{ds}{dt}
  31. s s
  32. y 0 y_{0}\,
  33. y y\,
  34. m g ( y 0 - y ) mg(y_{0}-y)\,
  35. 1 2 m ( d s d t ) 2 \displaystyle\frac{1}{2}m\left(\frac{ds}{dt}\right)^{2}
  36. d s = d s d y d y ds=\frac{ds}{dy}dy
  37. y = y 0 y=y_{0}
  38. y = 0 y=0
  39. T ( y 0 ) = y = y 0 y = 0 d t = 1 2 g 0 y 0 1 y 0 - y d s d y d y T(y_{0})=\int_{y=y_{0}}^{y=0}\,dt=\frac{1}{\sqrt{2g}}\int_{0}^{y_{0}}\frac{1}{% \sqrt{y_{0}-y}}\frac{ds}{dy}\,dy
  40. d s d y \frac{ds}{dy}
  41. T ( y 0 ) T(y_{0})\,
  42. d s d y \frac{ds}{dy}
  43. d s d y \frac{ds}{dy}
  44. 1 y \frac{1}{\sqrt{y}}
  45. [ T ( y 0 ) ] = 1 2 g [ 1 y ] [ d s d y ] \mathcal{L}[T(y_{0})]=\frac{1}{\sqrt{2g}}\mathcal{L}\left[\frac{1}{\sqrt{y}}% \right]\mathcal{L}\left[\frac{ds}{dy}\right]
  46. [ 1 y ] = π z - 1 2 \mathcal{L}\left[\frac{1}{\sqrt{y}}\right]=\sqrt{\pi}z^{-\frac{1}{2}}
  47. d s d y \frac{ds}{dy}
  48. T ( y 0 ) T(y_{0})\,
  49. [ d s d y ] = 2 g π z 1 2 [ T ( y 0 ) ] \mathcal{L}\left[\frac{ds}{dy}\right]=\sqrt{\frac{2g}{\pi}}z^{\frac{1}{2}}% \mathcal{L}[T(y_{0})]
  50. T ( y 0 ) T(y_{0})\,
  51. T ( y 0 ) T(y_{0})\,
  52. d s d y \frac{ds}{dy}
  53. d s d y \frac{ds}{dy}
  54. T ( y 0 ) = T 0 T(y_{0})=T_{0}\,
  55. 1 z \frac{1}{z}
  56. [ d s d y ] \displaystyle\mathcal{L}\left[\frac{ds}{dy}\right]
  57. d s d y = T 0 2 g π 1 y \frac{ds}{dy}=T_{0}\frac{\sqrt{2g}}{\pi}\frac{1}{\sqrt{y}}

Tax_horsepower.html

  1. RAC h.p. = D 2 × n 2.5 \,\text{RAC h.p.}=\frac{D^{2}\times n}{2.5}
  2. CV/PS = 0.4 × i × d 2 × S \,\text{CV/PS}=0.4\times i\times d^{2}\times S

Technicolor_(physics).html

  1. 𝒪 ( M bare 2 M physical 2 ) \mathcal{O}\left(\frac{M^{2}_{\mathrm{bare}}}{M^{2}_{\mathrm{physical}}}\right)
  2. ( 1 ) M W ± = 1 2 g F E W and M Z = 1 2 g 2 + g 2 F E W M W cos θ W . (1)\qquad M_{W^{\pm}}={\frac{1}{2}}gF_{EW}\quad{\rm and}\quad M_{Z}={\frac{1}{% 2}}\sqrt{g^{2}+g^{\prime\,2}}F_{EW}\equiv\frac{M_{W}}{\cos\theta_{W}}\,.
  3. T ¯ T T C 4 π F E W 3 \langle\bar{T}T\rangle_{TC}\cong 4\pi F_{EW}^{3}
  4. T ¯ T \bar{T}T
  5. q L ( or L ) T L T R q R ( or R ) q_{L}(\mathrm{or}\,\,\ell_{L})\rightarrow T_{L}\rightarrow T_{R}\rightarrow q_% {R}\,(\mathrm{or}\,\,\ell_{R})
  6. ( 2 ) m q , ( M E T C ) g E T C 2 T ¯ T E T C M E T C 2 4 π F E W 3 Λ E T C 2 . (2)\qquad m_{q,\ell}(M_{ETC})\cong\frac{g_{ETC}^{2}\langle\bar{T}T\rangle_{ETC% }}{M_{ETC}^{2}}\cong\frac{4\pi F_{EW}^{3}}{\Lambda_{ETC}^{2}}\,.
  7. T ¯ T E T C \langle\bar{T}T\rangle_{ETC}
  8. ( 3 ) T ¯ T E T C = exp ( Λ T C M E T C d μ μ γ m ( μ ) ) T ¯ T T C , (3)\qquad\langle\bar{T}T\rangle_{ETC}=\exp{\left(\int_{\Lambda_{TC}}^{M_{ETC}}% \frac{d\mu}{\mu}\gamma_{m}(\mu)\right)}\,\langle\bar{T}T\rangle_{TC}\,,
  9. T ¯ T \bar{T}T
  10. T ¯ T \bar{T}T
  11. g E T C 2 1 g^{2}_{ETC}\gtrsim 1
  12. ( 4 ) F E W 2 M π T 2 g E T C 2 T ¯ T T ¯ T E T C M E T C 2 16 π 2 F E W 6 Λ E T C 2 . (4)\qquad F_{EW}^{2}M_{\pi T}^{2}\cong\frac{g_{ETC}^{2}\langle\bar{T}T\bar{T}T% \rangle_{ETC}}{M_{ETC}^{2}}\cong\frac{16\pi^{2}F_{EW}^{6}}{\Lambda_{ETC}^{2}}\,.
  13. T ¯ T T ¯ T E T C T ¯ T E T C 2 \langle\bar{T}T\bar{T}T\rangle_{ETC}\cong\langle\bar{T}T\rangle^{2}_{ETC}
  14. m q , m_{q,\ell}
  15. q ¯ q \bar{q}q
  16. ¯ \bar{\ell}\ell
  17. K 0 K ¯ 0 K^{0}\leftrightarrow\bar{K}^{0}
  18. B 0 B ¯ 0 B^{0}\leftrightarrow\bar{B}^{0}
  19. m q , m_{q,\ell}
  20. q ¯ q \bar{q}q^{\prime}
  21. ¯ \bar{\ell}\ell^{\prime}
  22. K K ¯ K\hbox{--}\bar{K}
  23. \ell
  24. T ¯ T E T C T ¯ T T C 4 π F E W 3 \langle\bar{T}T\rangle_{ETC}\cong\langle\bar{T}T\rangle_{TC}\cong 4\pi F_{EW}^% {3}
  25. T ¯ T \bar{T}T
  26. K K ¯ K\hbox{--}\bar{K}
  27. T ¯ T E T C \langle\bar{T}T\rangle_{ETC}
  28. N ^ f \hat{N}_{f}
  29. N ^ f - N f \hat{N}_{f}-N_{f}
  30. N ^ f \hat{N}_{f}
  31. 0 < ( α I R - α χ S B ) / α I R 1 0<(\alpha_{IR}-\alpha_{\chi SB})/\alpha_{IR}\ll 1
  32. M E T C 2 M_{ETC}^{2}
  33. K K ¯ K\hbox{--}\bar{K}
  34. 𝒪 ( 10 3 TeV ) \mathcal{O}(10^{3}\hbox{ TeV})
  35. T ¯ T \bar{T}T
  36. ( 5 ) S \displaystyle(5)\qquad S
  37. ( 7 ) S = - 0.04 ± 0.09 ( - 0.07 ) , T = 0.02 ± 0.09 ( + 0.09 ) , (7)\qquad\begin{aligned}\displaystyle S&\displaystyle=-0.04\pm 0.09\,(-0.07),% \\ \displaystyle T&\displaystyle=0.02\pm 0.09\,(+0.09),\end{aligned}
  38. σ V , A 3 \sigma_{V,A}^{3}
  39. Z 0 b ¯ b Z^{0}\rightarrow\bar{b}b
  40. q ¯ q \bar{q}q
  41. ρ T ± , 0 \rho_{T}^{\pm,0}
  42. W L ± Z L 0 W_{L}^{\pm}Z_{L}^{0}
  43. W L + W L - W_{L}^{+}W_{L}^{-}
  44. F F E W F\ll F_{EW}
  45. Λ T C F E W \Lambda_{TC}\ll F_{EW}
  46. F F E W F\ll F_{EW}
  47. M π T 2 T ¯ T T ¯ T M E T C M_{\pi_{T}}^{2}\propto\langle\bar{T}T\bar{T}T\rangle_{M_{ETC}}
  48. W L ± , 0 π T W^{\pm,0}_{L}\pi_{T}
  49. ( F / F E W ) 2 1 (F/F_{EW})^{2}\ll 1
  50. M ρ T M_{\rho_{T}}
  51. t ¯ t \bar{t}t
  52. ρ T ± W L ± Z L 0 \rho_{T}^{\pm}\rightarrow W_{L}^{\pm}Z_{L}^{0}
  53. a T ± γ W L ± a_{T}^{\pm}\rightarrow\gamma W_{L}^{\pm}
  54. ω T γ Z L 0 \omega_{T}\rightarrow\gamma Z_{L}^{0}
  55. F F E W F\ll F_{EW}
  56. 10 - 42 cm 2 \lesssim 10^{-42}\,\mathrm{cm}^{2}

Telecine.html

  1. 23.976 29.97 = 4 5 \frac{23.976}{29.97}=\frac{4}{5}

Tensegrity.html

  1. s 2 = ( d - l ) 2 + d 2 + l 2 = 2 ( d - 1 2 l ) 2 + 3 2 l 2 s^{2}=(d-l)^{2}+d^{2}+l^{2}=2(d-\frac{1}{2}\,l)^{2}+\frac{3}{2}\,l^{2}
  2. s > 3 / 2 l s>\sqrt{3/2}\,l
  3. s = 2 l s=\sqrt{2}\,l
  4. s = 1 2 ( 5 - 1 ) l s=\frac{1}{2}(\sqrt{5}-1)l
  5. s = 3 / 2 l s=\sqrt{3/2}\,l
  6. d = 1 2 l d=\frac{1}{2}\,l

Tensor_(intrinsic_definition).html

  1. V V V * V * V\otimes\cdots\otimes V\otimes V^{*}\otimes\cdots\otimes V^{*}
  2. T n m ( V ) = V V m V * V * n . T^{m}_{n}(V)=\underbrace{V\otimes\dots\otimes V}_{m}\otimes\underbrace{V^{*}% \otimes\dots\otimes V^{*}}_{n}.
  3. V V * V\otimes V^{*}
  4. V * V * V^{*}\otimes V^{*}
  5. A = v w T . A=vw^{\mathrm{T}}.
  6. A = v 1 w 1 T + + v k w k T . A=v_{1}w_{1}^{\mathrm{T}}+\cdots+v_{k}w_{k}^{\mathrm{T}}.
  7. T = a b d T=a\otimes b\otimes\cdots\otimes d
  8. T i j k = a i b j c k d . T_{ij\dots}^{k\ell\dots}=a_{i}b_{j}\cdots c^{k}d^{\ell}\cdots.
  9. z k = i j T i j k x i y j z_{k}=\sum_{ij}T_{ijk}x_{i}y_{j}\,
  10. T n m ( V ) T^{m}_{n}(V)
  11. f : V 1 × V 2 × × V N 𝐑 f:V_{1}\times V_{2}\times\cdots\times V_{N}\to\mathbf{R}
  12. f L m + n ( V , V , , V m , V * , V * , , V * n ; W ) f\in L^{m+n}(\underbrace{V,V,\dots,V}_{m},\underbrace{V^{*},V^{*},\dots,V^{*}}% _{n};W)
  13. T f L ( V V V * V * ; W ) T_{f}\in L(V\otimes\cdots\otimes V\otimes V^{*}\otimes\cdots\otimes V^{*};W)
  14. f ( v 1 , , v m , α 1 , , α n ) = T f ( v 1 v m α 1 α n ) f(v_{1},\dots,v_{m},\alpha_{1},\dots,\alpha_{n})=T_{f}(v_{1}\otimes\cdots% \otimes v_{m}\otimes\alpha_{1}\otimes\cdots\otimes\alpha_{n})
  15. T n m ( V ) L ( V * V * V V ; ) L m + n ( V * , , V * , V , , V ; ) . T^{m}_{n}(V)\cong L(V^{*}\otimes\dots\otimes V^{*}\otimes V\otimes\dots\otimes V% ;\mathbb{R})\cong L^{m+n}(V^{*},\dots,V^{*},V,\dots,V;\mathbb{R}).
  16. T 0 1 ( V ) L ( V * ; ) V T^{1}_{0}(V)\cong L(V^{*};\mathbb{R})\cong V
  17. T 1 0 ( V ) L ( V ; ) = V * T^{0}_{1}(V)\cong L(V;\mathbb{R})=V^{*}
  18. T 1 1 ( V ) L ( V ; V ) . T^{1}_{1}(V)\cong L(V;V).

Tensor_contraction.html

  1. C : V * V k C:V^{*}\otimes V\rightarrow k
  2. f , v = f ( v ) \langle f,v\rangle=f(v)
  3. V * V V^{*}\otimes V
  4. V * V V^{*}\otimes V
  5. V * V L ( V , V ) V^{*}\otimes V\rightarrow L(V,V)
  6. f v g f\otimes v\mapsto g
  7. f i v j f^{i}\otimes v_{j}
  8. V V V * V * V\otimes\cdots\otimes V\otimes V^{*}\otimes\cdots\otimes V^{*}
  9. f ~ ( v ) = f γ v γ \tilde{f}(\vec{v})=f_{\gamma}v^{\gamma}
  10. f γ v γ = f 1 v 1 + f 2 v 2 + + f n v n f_{\gamma}v^{\gamma}=f_{1}v^{1}+f_{2}v^{2}+\cdots+f_{n}v^{n}
  11. f v f\otimes v
  12. 𝐓 = T i 𝐞 i j 𝐞 j \mathbf{T}=T^{i}{}_{j}\mathbf{e}_{i}\mathbf{e}^{j}
  13. T i 𝐞 i j 𝐞 j = T i δ i j = j T j = j T 1 + 1 + T n n T^{i}{}_{j}\mathbf{e}_{i}\cdot\mathbf{e}^{j}=T^{i}{}_{j}\delta_{i}{}^{j}=T^{j}% {}_{j}=T^{1}{}_{1}+\cdots+T^{n}{}_{n}
  14. T a b = b c b T a b = b c T a 1 + 1 c T a 2 + 2 c + T a n = n c U a . c T^{ab}{}_{bc}=\sum_{b}{T^{ab}{}_{bc}}=T^{a1}{}_{1c}+T^{a2}{}_{2c}+\cdots+T^{an% }{}_{nc}=U^{a}{}_{c}.
  15. 𝐓 = 𝐞 i 𝐞 j \mathbf{T}=\mathbf{e}^{i}\mathbf{e}^{j}
  16. g i j = 𝐞 i 𝐞 j g^{ij}=\mathbf{e}^{i}\cdot\mathbf{e}^{j}
  17. U ( x ) = i T i i ( x ) U(x)=\sum_{i}T^{i}_{i}(x)
  18. V α ; β V^{\alpha}{}_{;\beta}
  19. V α = ; β V α x β . V^{\alpha}{}_{;\beta}={\partial V^{\alpha}\over\partial x^{\beta}}.
  20. V α = ; α V 0 + ; 0 + V n ; n V^{\alpha}{}_{;\alpha}=V^{0}{}_{;0}+\cdots+V^{n}{}_{;n}
  21. div V = V α = ; α 0 \mathrm{div}V=V^{\alpha}{}_{;\alpha}=0
  22. T U T\otimes U
  23. Λ α β \Lambda^{\alpha}{}_{\beta}
  24. \Mu β γ \Mu^{\beta}{}_{\gamma}
  25. Λ α \Mu β β = γ \Nu α γ \Lambda^{\alpha}{}_{\beta}\Mu^{\beta}{}_{\gamma}=\Nu^{\alpha}{}_{\gamma}

Tensor_field.html

  1. V V V * V * V\otimes\cdots\otimes V\otimes V^{*}\otimes\cdots\otimes V^{*}
  2. E * F \scriptstyle E^{*}\otimes F
  3. T 0 1 ( M ) = T ( M ) = T M T_{0}^{1}(M)=T(M)=TM
  4. T 0 1 ( V ) T_{0}^{1}(V)
  5. 𝒯 n m ( M ) \mathcal{T}^{m}_{n}(M)
  6. 𝒯 ( M ) \mathcal{T}(M)
  7. 𝒯 * ( M ) \mathcal{T}^{*}(M)
  8. 𝒯 ( M ) \mathcal{T}(M)
  9. 𝒯 * ( M ) \mathcal{T}^{*}(M)
  10. 𝒯 * ( M ) \mathcal{T}^{*}(M)
  11. 𝒯 ( M ) \mathcal{T}(M)
  12. ( X , Y ) X Y (X,Y)\mapsto\nabla_{X}Y
  13. X ( f Y ) = ( X f ) Y + f X Y \nabla_{X}(fY)=(Xf)Y+f\nabla_{X}Y

Terminal_velocity.html

  1. V t = 2 m g ρ A C d V_{t}=\sqrt{\frac{2mg}{\rho AC_{d}}}
  2. V t V_{t}
  3. m m
  4. g g
  5. C d C_{d}
  6. ρ \rho
  7. A A
  8. m m
  9. ρ 𝒱 \rho\mathcal{V}
  10. 𝒱 \mathcal{V}
  11. m m
  12. m r = m - ρ 𝒱 m_{r}=m-\rho\mathcal{V}
  13. F n e t = m a = m g - 1 2 ρ v 2 A C d F_{net}=ma=mg-{1\over 2}\rho v^{2}AC_{\mathrm{d}}
  14. m g - 1 2 ρ v 2 A C d = 0 mg-{1\over 2}\rho v^{2}AC_{\mathrm{d}}=0
  15. v = 2 m g ρ A C d v=\sqrt{\frac{2mg}{\rho AC_{\mathrm{d}}}}
  16. m a = m d v d t = m g - 1 2 ρ v 2 A C d ma=m\frac{\mathrm{d}v}{\mathrm{d}t}=mg-\frac{1}{2}\rho v^{2}AC_{\mathrm{d}}
  17. 1 / 2 {1}/{2}
  18. d v d t = g - k v 2 m \frac{\mathrm{d}v}{\mathrm{d}t}=g-\frac{kv^{2}}{m}
  19. d t = d v g - k v 2 m dt=\frac{\mathrm{d}v}{g-\frac{kv^{2}}{m}}
  20. 0 t d t = 0 v d v g - k v 2 m = 1 g 0 v d v 1 - α 2 v 2 \int_{0}^{t}{\mathrm{d}t^{\prime}}=\int_{0}^{v}\frac{\mathrm{d}v^{\prime}}{g-% \frac{kv^{\prime 2}}{m}}={1\over g}\int_{0}^{v}\frac{\mathrm{d}v^{\prime}}{1-% \alpha^{2}v^{\prime 2}}
  21. k / m g {k}/{mg}
  22. 1 / 2 {1}/{2}
  23. t - 0 = 1 g [ ln ( 1 + α v ) 2 α - ln ( 1 - α v ) 2 α + C ] v = 0 v = v = 1 g [ ln 1 + α v 1 - α v 2 α + C ] v = 0 v = v t-0={1\over g}\left[{\ln(1+\alpha v^{\prime})\over 2\alpha}-\frac{\ln(1-\alpha v% ^{\prime})}{2\alpha}+C\right]_{v^{\prime}=0}^{v^{\prime}=v}={1\over g}\left[{% \ln\frac{1+\alpha v^{\prime}}{1-\alpha v^{\prime}}\over 2\alpha}+C\right]_{v^{% \prime}=0}^{v^{\prime}=v}
  24. t = 1 2 α g ln 1 + α v 1 - α v t={1\over 2\alpha g}\ln\frac{1+\alpha v}{1-\alpha v}
  25. 1 2 ln 1 + α v 1 - α v = arctanh ( α v ) \frac{1}{2}\ln\frac{1+\alpha v}{1-\alpha v}=\mathrm{arctanh}(\alpha v)
  26. t = arctanh ( α v ) α g t=\frac{\mathrm{arctanh}(\alpha v)}{\alpha g}
  27. 1 α tanh ( α g t ) = v \frac{1}{\alpha}\tanh(\alpha gt)=v
  28. v = m g k tanh ( k m g g t ) v=\sqrt{\frac{mg}{k}}\tanh\left(\sqrt{\frac{k}{mg}}gt\right)
  29. 1 / 2 {1}/{2}
  30. v = 2 m g ρ A C d tanh ( t g ρ A C d 2 m ) v=\sqrt{\frac{2mg}{\rho AC_{d}}}\tanh\left(t\sqrt{\frac{g\rho AC_{d}}{2m}}\right)
  31. v = 2 m g ρ A C d v=\sqrt{\frac{2mg}{\rho AC_{d}}}
  32. R e 1 Re\ll 1
  33. p = μ 2 𝐯 \nabla p=\mu\nabla^{2}{\mathbf{v}}
  34. 𝐯 {\mathbf{v}}
  35. p p
  36. μ \mu
  37. ( 6 ) D = 3 π μ d V or C d = 24 R e \quad(6)\qquad D=3\pi\mu dV\qquad\qquad\,\text{or}\qquad\qquad C_{d}=\frac{24}% {Re}
  38. R e = 1 μ ρ d V Re=\frac{1}{\mu}\rho dV
  39. C d C_{d}
  40. V t = g d 2 18 μ ( ρ s - ρ ) V_{t}=\frac{gd^{2}}{18\mu}\left(\rho_{s}-\rho\right)
  41. C d Re 2 = m g D 2 A ρ ν 2 C_{d}\mathrm{Re}^{2}=\frac{mgD^{2}}{A\rho\nu^{2}}
  42. C d Re 2 = 4 m g π ρ ν 2 C_{d}\mathrm{Re}^{2}=\frac{4mg}{\pi\rho\nu^{2}}
  43. × 10 1 0 \times 10^{1}0
  44. × 10 1 0 \times 10^{1}0
  45. ( 1 ) W = F b + D \quad(1)\qquad W=F_{b}+D
  46. W W
  47. F b F_{b}
  48. D D
  49. ( 2 ) \displaystyle(2)
  50. d d
  51. g g
  52. ρ \rho
  53. ρ s \rho_{s}
  54. A = 1 4 π d 2 A=\frac{1}{4}\pi d^{2}
  55. C d C_{d}
  56. V V
  57. V t V_{t}
  58. V t V_{t}
  59. ( 5 ) V t = 4 g d 3 C d ( ρ s - ρ ρ ) \quad(5)\qquad V_{t}=\sqrt{\frac{4gd}{3C_{d}}\left(\frac{\rho_{s}-\rho}{\rho}% \right)}