wpmath0000004_14

Sphenocorona.html

  1. V = ( 1 2 1 + 3 3 2 + 13 + 3 6 ) a 3 1.51535... a 3 V=\left(\frac{1}{2}\sqrt{1+3\sqrt{\frac{3}{2}+\sqrt{13+3\sqrt{6}}}}\right)a^{3% }\approx 1.51535...a^{3}
  2. A = ( 2 + 3 3 ) a 2 7.19615... a 2 A=\left(2+3\sqrt{3}\right)a^{2}\approx 7.19615...a^{2}

Sphere-world.html

  1. R 2 - r 2 R^{2}-r^{2}
  2. R 2 - r 2 R^{2}-r^{2}
  3. 2 π 2\pi

Sphere_of_influence_(astrodynamics).html

  1. r S O I r_{SOI}
  2. r S O I = a ( m M ) 2 / 5 r_{SOI}=a\left(\frac{m}{M}\right)^{2/5}
  3. a a
  4. m m
  5. M M

Spherical_cap.html

  1. a a
  2. h h
  3. V = π h 6 ( 3 a 2 + h 2 ) V=\frac{\pi h}{6}(3a^{2}+h^{2})
  4. A = 2 π r h A=2\pi rh
  5. h h
  6. r r
  7. h h
  8. 2 r 2r
  9. a a
  10. h h
  11. r r
  12. r 2 = ( r - h ) 2 + a 2 = r 2 + h 2 - 2 r h + a 2 r^{2}=(r-h)^{2}+a^{2}=r^{2}+h^{2}-2rh+a^{2}
  13. r = a 2 + h 2 2 h r=\frac{a^{2}+h^{2}}{2h}
  14. A = 2 π ( a 2 + h 2 ) 2 h h = π ( a 2 + h 2 ) A=2\pi\frac{(a^{2}+h^{2})}{2h}h=\pi(a^{2}+h^{2})
  15. h = r - r 2 - a 2 \scriptstyle h=r-\sqrt{r^{2}-a^{2}}
  16. h = r + r 2 - a 2 \scriptstyle h=r+\sqrt{r^{2}-a^{2}}
  17. a = h ( 2 r - h ) \scriptstyle a=\sqrt{h(2r-h)}
  18. V = π h 2 3 ( 3 r - h ) V=\frac{\pi h^{2}}{3}(3r-h)
  19. V = V ( 1 ) - V ( 2 ) V=V^{(1)}-V^{(2)}
  20. V ( 1 ) = 4 π 3 r 1 3 + 4 π 3 r 2 3 V^{(1)}=\frac{4\pi}{3}r_{1}^{3}+\frac{4\pi}{3}r_{2}^{3}
  21. V ( 2 ) = π h 1 2 3 ( 3 r 1 - h 1 ) + π h 2 2 3 ( 3 r 2 - h 2 ) V^{(2)}=\frac{\pi h_{1}^{2}}{3}(3r_{1}-h_{1})+\frac{\pi h_{2}^{2}}{3}(3r_{2}-h% _{2})
  22. V ( 2 ) = π 12 d ( r 1 + r 2 - d ) 2 [ d 2 + 2 d ( r 1 + r 2 ) - 3 ( r 1 - r 2 ) 2 ] V^{(2)}=\frac{\pi}{12d}(r_{1}+r_{2}-d)^{2}[d^{2}+2d(r_{1}+r_{2})-3(r_{1}-r_{2}% )^{2}]
  23. n n
  24. h h
  25. r r
  26. n n
  27. V = π n - 1 2 r n Γ ( n + 1 2 ) 0 arccos ( r - h r ) sin n ( t ) d t V=\frac{\pi^{\frac{n-1}{2}}\,r^{n}}{\,\Gamma\left(\frac{n+1}{2}\right)}\int% \limits_{0}^{\arccos\left(\frac{r-h}{r}\right)}\sin^{n}(t)\,\mathrm{d}t
  28. Γ \Gamma
  29. Γ ( z ) = 0 t z - 1 e - t d t \Gamma(z)=\int_{0}^{\infty}t^{z-1}\mathrm{e}^{-t}\,\mathrm{d}t
  30. V V
  31. C n = π n / 2 / Γ [ 1 + n 2 ] C_{n}={\scriptstyle\pi^{n/2}/\Gamma[1+\frac{n}{2}]}
  32. F 1 2 {}_{2}F_{1}
  33. I x ( a , b ) I_{x}(a,b)
  34. V = C n r n ( 1 2 - r - h r Γ [ 1 + n 2 ] π Γ [ n + 1 2 ] F 1 2 ( 1 2 , 1 - n 2 ; 3 2 ; ( r - h r ) 2 ) ) = 1 2 C n r n I ( 2 r h - h 2 ) / r 2 ( n + 1 2 , 1 2 ) V=C_{n}\,r^{n}\left(\frac{1}{2}\,-\,\frac{r-h}{r}\,\frac{\Gamma[1+\frac{n}{2}]% }{\sqrt{\pi}\,\Gamma[\frac{n+1}{2}]}{}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1-n}{% 2};\tfrac{3}{2};\left(\tfrac{r-h}{r}\right)^{2}\right)\right)=\frac{1}{2}C_{n}% \,r^{n}I_{(2rh-h^{2})/r^{2}}\left(\frac{n+1}{2},\frac{1}{2}\right)
  35. A A
  36. A n = 2 π n / 2 / Γ [ n 2 ] A_{n}={\scriptstyle 2\pi^{n/2}/\Gamma[\frac{n}{2}]}
  37. A = 1 2 A n r n - 1 I ( 2 r h - h 2 ) / r 2 ( n - 1 2 , 1 2 ) A=\frac{1}{2}A_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left(\frac{n-1}{2},\frac{1}{2% }\right)
  38. 0 h r \scriptstyle 0\leq h\leq r
  39. A = A n p n - 2 ( q ) , V = V n p n ( q ) A=A_{n}p_{n-2}(q),V=V_{n}p_{n}(q)
  40. q = 1 - h / r ( 0 q 1 ) , p n ( q ) = ( 1 - G n ( q ) / G n ( 1 ) ) / 2 q=1-h/r(0\leq q\leq 1),p_{n}(q)=(1-G_{n}(q)/G_{n}(1))/2
  41. G n ( q ) = 0 q ( 1 - t 2 ) ( n - 1 ) / 2 d t G_{n}(q)=\int\limits_{0}^{q}(1-t^{2})^{(n-1)/2}dt
  42. n = 2 k + 1 : n=2k+1:
  43. G n ( q ) = i = 0 k ( - 1 ) i ( k i ) q 2 i + 1 2 i + 1 G_{n}(q)=\sum_{i=0}^{k}(-1)^{i}{\left({{k}\atop{i}}\right)}\frac{q^{2i+1}}{2i+1}
  44. n n\to\infty
  45. q / n = c o n s t . q/n=const.
  46. p n ( q ) 1 - F ( q / n ) p_{n}(q)\to 1-F(\sqrt{q/n})
  47. F ( ) F()

Spherical_pendulum.html

  1. ( r , θ , ϕ ) (r,\theta,\phi)
  2. L = 1 2 m r 2 ( θ ˙ 2 + sin 2 θ ϕ ˙ 2 ) + m g r cos θ . L=\frac{1}{2}mr^{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\ \dot{\phi}^{2}\right% )+mgr\cos\theta.
  3. d d t ( m r 2 θ ˙ ) - m r 2 sin θ cos θ ϕ ˙ 2 + m g r sin θ = 0 \frac{d}{dt}\left(mr^{2}\dot{\theta}\right)-mr^{2}\sin\theta\cos\theta\dot{% \phi}^{2}+mgr\sin\theta=0
  4. d d t ( m r 2 sin 2 θ ϕ ˙ ) = 0 \frac{d}{dt}\left(mr^{2}\sin^{2}\theta\,\dot{\phi}\right)=0
  5. H = P θ θ ˙ + P ϕ ϕ ˙ - L H=P_{\theta}\dot{\theta}+P_{\phi}\dot{\phi}-L
  6. P θ = L θ ˙ = m r 2 θ ˙ P_{\theta}=\frac{\partial L}{\partial\dot{\theta}}=mr^{2}\dot{\theta}
  7. P ϕ = L ϕ ˙ = m r 2 ϕ ˙ sin 2 θ P_{\phi}=\frac{\partial L}{\partial\dot{\phi}}=mr^{2}\dot{\phi}\sin^{2}\theta

Spherical_trigonometry.html

  1. A \displaystyle A^{\prime}
  2. cos a = cos b cos c + sin b sin c cos A , \cos a=\cos b\cos c+\sin b\sin c\cos A,\!
  3. cos b = cos c cos a + sin c sin a cos B , \cos b=\cos c\cos a+\sin c\sin a\cos B,\!
  4. cos c = cos a cos b + sin a sin b cos C , \cos c=\cos a\cos b+\sin a\sin b\cos C,\!
  5. cos a 1 - a 2 / 2 \cos a\approx 1-a^{2}/2
  6. sin A sin a = sin B sin b = sin C sin c . \frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}.
  7. ( 0 , 0 , 1 ) (0,\,0,\,1)
  8. ( sin c , 0 , cos c ) (\sin c,\,0,\,\cos c)
  9. ( sin b cos A , sin b sin A , cos b ) (\sin b\cos A,\,\sin b\sin A,\,\cos b)
  10. sin c sin b cos A + cos c cos b \sin c\,\sin b\,\cos A+\cos c\,\cos b
  11. cos a = cos b cos c + sin b sin c cos A . \cos a=\cos b\,\cos c+\sin b\,\sin c\,\cos A.
  12. cos A = cos a - cos b cos c sin b sin c . \cos A=\frac{\cos a\,-\,\cos b\,\cos c}{\sin b\,\sin c}.
  13. sin 2 A = 1 - cos 2 A \sin^{2}A=1-\cos^{2}A
  14. cos A \cos A
  15. sin 2 A = 1 - ( cos a - cos b cos c sin b sin c ) 2 = ( 1 - cos 2 b ) ( 1 - cos 2 c ) - ( cos a - cos b cos c ) 2 sin 2 b sin 2 c sin A sin a = [ 1 - cos 2 a - cos 2 b - cos 2 c + 2 cos a cos b cos c ] 1 / 2 sin a sin b sin c . \begin{aligned}\displaystyle\sin^{2}\!A&\displaystyle=1-\left(\frac{\cos a-% \cos b\,\cos c}{\sin b\,\sin c}\right)^{2}\\ &\displaystyle=\frac{(1-\cos^{2}\!b)(1-\cos^{2}\!c)-(\cos a-\cos b\,\cos c)^{2% }}{\sin^{2}\!b\,\sin^{2}\!c}\\ \displaystyle\frac{\sin A}{\sin a}&\displaystyle=\frac{[1-\cos^{2}\!a-\cos^{2}% \!b-\cos^{2}\!c+2\cos a\cos b\cos c]^{1/2}}{\sin a\sin b\sin c}.\end{aligned}
  16. a , b , c a,\;b,\;c
  17. cos A = - cos B cos C + sin B sin C cos a , \cos A=-\cos B\,\cos C+\sin B\,\sin C\,\cos a,
  18. cos B = - cos C cos A + sin C sin A cos b , \cos B=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,
  19. cos C = - cos A cos B + sin A sin B cos c . \cos C=-\cos A\,\cos B+\sin A\,\sin B\,\cos c.
  20. cos ( inner side ) cos ( inner angle ) = cot ( outer side ) sin ( inner side ) - cot ( outer angle ) sin ( inner angle ) , \cos(\,\text{inner side})\cos(\,\text{inner angle})=\cot(\,\text{outer side})% \sin(\,\text{inner side})\ -\ \cot(\,\text{outer angle})\sin(\,\text{inner % angle}),
  21. (CT1) cos b cos C = cot a sin b - cot A sin C , ( a C b A ) (CT2) cos b cos A = cot c sin b - cot C sin A , ( C b A c ) (CT3) cos c cos A = cot b sin c - cot B sin A , ( b A c B ) (CT4) cos c cos B = cot a sin c - cot A sin B , ( A c B a ) (CT5) cos a cos B = cot c sin a - cot C sin B , ( c B a C ) (CT6) cos a cos C = cot b sin a - cot B sin C , ( B a C b ) . \begin{array}[]{lll}\,\text{(CT1)}&\cos b\,\cos C=\cot a\,\sin b-\cot A\,\sin C% ,&(aCbA)\\ \,\text{(CT2)}&\cos b\,\cos A=\cot c\,\sin b-\cot C\,\sin A,&(CbAc)\\ \,\text{(CT3)}&\cos c\,\cos A=\cot b\,\sin c-\cot B\,\sin A,&(bAcB)\\ \,\text{(CT4)}&\cos c\,\cos B=\cot a\,\sin c-\cot A\,\sin B,&(AcBa)\\ \,\text{(CT5)}&\cos a\,\cos B=\cot c\,\sin a-\cot C\,\sin B,&(cBaC)\\ \,\text{(CT6)}&\cos a\,\cos C=\cot b\,\sin a-\cot B\,\sin C,&(BaCb).\end{array}
  22. cos c \cos c
  23. cos a \displaystyle\cos a
  24. sin a sin b \sin a\sin b
  25. 2 s = ( a + b + c ) 2s=(a+b+c)
  26. 2 S = ( A + B + C ) 2S=(A+B+C)
  27. sin 1 2 A = [ sin ( s - b ) sin ( s - c ) sin b sin c ] 1 / 2 \displaystyle\sin{\textstyle\frac{1}{2}}A=\left[\frac{\sin(s{-}b)\sin(s{-}c)}{% \sin b\sin c}\right]^{1/2}
  28. sin 1 2 ( A + B ) cos 1 2 C = cos 1 2 ( a - b ) cos 1 2 c \displaystyle\frac{\sin{\textstyle\frac{1}{2}}(A{+}B)}{\cos{\textstyle\frac{1}% {2}}C}=\frac{\cos{\textstyle\frac{1}{2}}(a{-}b)}{\cos{\textstyle\frac{1}{2}}c}
  29. tan 1 2 ( A + B ) = cos 1 2 ( a - b ) cos 1 2 ( a + b ) cot 1 2 C \displaystyle{\tan{\textstyle\frac{1}{2}}(A{+}B)}=\frac{\cos{\textstyle\frac{1% }{2}}(a{-}b)}{\cos{\textstyle\frac{1}{2}}(a{+}b)}\cot{\textstyle\frac{1}{2}C}
  30. a a
  31. sin a = tan ( π / 2 - B ) tan b = cos ( π / 2 - c ) cos ( π / 2 - A ) = cot B tan b = sin c sin A . \sin a=\tan(\pi/2{-}B)\,\tan b=\cos(\pi/2{-}c)\,\cos(\pi/2{-}A)=\cot B\,\tan b% =\sin c\,\sin A.
  32. (R1) \displaystyle\,\text{(R1)}
  33. (Q1) \displaystyle\,\text{(Q1)}
  34. cos a = ( cos a cos c + sin a sin c cos B ) cos c + sin b sin c cos A \cos a=(\cos a\,\cos c+\sin a\,\sin c\,\cos B)\cos c+\sin b\,\sin c\,\cos A
  35. cos a sin 2 c = sin a cos c sin c cos B + sin b sin c cos A \cos a\,\sin^{2}c=\sin a\,\cos c\,\sin c\,\cos B+\sin b\,\sin c\,\cos A
  36. sin c \sin c
  37. cos a sin c = sin a cos c cos B + sin b cos A \cos a\sin c=\sin a\,\cos c\,\cos B+\sin b\,\cos A
  38. Area of polygon (on the unit sphere) E n = Σ - ( n - 2 ) π . \,\text{Area of polygon (on the unit sphere)}\equiv E_{n}=\Sigma-(n-2)\pi.
  39. Area of triangle (on the unit sphere) E = E 3 = A + B + C - π , \,\text{Area of triangle (on the unit sphere)}\equiv E=E_{3}=A+B+C-\pi,
  40. A + B + C = π + 4 π × Area of triangle Area of the sphere . \displaystyle A+B+C=\pi+\frac{4\pi\times\,\text{Area of triangle}}{\,\text{% Area of the sphere}}.
  41. tan 1 4 E = tan 1 2 s tan 1 2 ( s - a ) tan 1 2 ( s - b ) tan 1 2 ( s - c ) \tan\tfrac{1}{4}E=\sqrt{\tan\tfrac{1}{2}s\,\tan\tfrac{1}{2}(s{-}a)\,\tan\tfrac% {1}{2}(s{-}b)\,\tan\tfrac{1}{2}(s{-}c)}
  42. s = ( a + b + c ) / 2 s=(a+b+c)/2
  43. a = b 1 2 c a=b\approx\frac{1}{2}c
  44. tan E 2 = tan 1 2 a tan 1 2 b sin C 1 + tan 1 2 a tan 1 2 b cos C . \tan\frac{E}{2}=\frac{\tan\frac{1}{2}a\tan\frac{1}{2}b\sin C}{1+\tan\frac{1}{2% }a\tan\frac{1}{2}b\cos C}.
  45. tan E 4 2 = sin 1 2 ( ϕ 2 + ϕ 1 ) cos 1 2 ( ϕ 2 - ϕ 1 ) tan λ 2 - λ 1 2 . \tan\frac{E_{4}}{2}=\frac{\sin\frac{1}{2}(\phi_{2}+\phi_{1})}{\cos\frac{1}{2}(% \phi_{2}-\phi_{1})}\tan\frac{\lambda_{2}-\lambda_{1}}{2}.
  46. ϕ , λ \phi,\lambda
  47. ϕ 1 , ϕ 2 , λ 2 - λ 1 \phi_{1},\phi_{2},\lambda_{2}-\lambda_{1}
  48. E 4 1 2 ( ϕ 2 + ϕ 1 ) ( λ 2 - λ 1 ) E_{4}\approx\frac{1}{2}(\phi_{2}+\phi_{1})(\lambda_{2}-\lambda_{1})

Spin_foam.html

  1. Z := Γ w ( Γ ) [ j f , i e f A f ( j f ) e A e ( j f , i e ) v A v ( j f , i e ) ] Z:=\sum_{\Gamma}w(\Gamma)\left[\sum_{j_{f},i_{e}}\prod_{f}A_{f}(j_{f})\prod_{e% }A_{e}(j_{f},i_{e})\prod_{v}A_{v}(j_{f},i_{e})\right]
  2. Γ \Gamma
  3. f f
  4. e e
  5. v v
  6. Γ \Gamma
  7. w ( Γ ) w(\Gamma)
  8. j j
  9. i i
  10. A v ( j f , i e ) A_{v}(j_{f},i_{e})
  11. A e ( j f , i e ) A_{e}(j_{f},i_{e})
  12. A f ( j f ) A_{f}(j_{f})
  13. A f ( j f ) = dim ( j f ) A_{f}(j_{f})=\dim(j_{f})

Spin_isomers_of_hydrogen.html

  1. | , ( 1 / 2 ) ( | + | ) |\uparrow\uparrow\rangle,(1/\sqrt{2})(|\uparrow\downarrow\rangle+|\downarrow% \uparrow\rangle)
  2. | |\downarrow\downarrow\rangle
  3. ( 1 / 2 ) ( | - | ) (1/\sqrt{2})(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)
  4. E J = J ( J + 1 ) 2 2 I ; g J = 2 J + 1 E_{J}=\frac{J(J+1)\hbar^{2}}{2I};\,\text{ }g_{J}=2J+1
  5. Z rot = J = 0 g J e - E J / k B T Z_{\,\text{rot}}=\sum\limits_{J=0}^{\infty}{g_{J}e^{-E_{J}/{k_{B}T}\;}}
  6. Z para = even J ( 2 J + 1 ) e - J ( J + 1 ) 2 / 2 I k B T Z ortho = 3 odd J ( 2 J + 1 ) e - J ( J + 1 ) 2 / 2 I k B T \begin{aligned}\displaystyle Z_{\,\text{para}}&\displaystyle=\sum\limits_{\,% \text{even }J}{(2J+1)e^{{-J(J+1)\hbar^{2}}/{2Ik_{B}T}\;}}\\ \displaystyle Z_{\,\text{ortho}}&\displaystyle=3\sum\limits_{\,\text{odd }J}{(% 2J+1)e^{{-J(J+1)\hbar^{2}}/{2Ik_{B}T}\;}}\end{aligned}
  7. Z equil = J = 0 ( 2 - ( - 1 ) J ) ( 2 J + 1 ) e - J ( J + 1 ) 2 / 2 I k B T Z_{\,\text{equil}}=\sum\limits_{J=0}^{\infty}{\left(2-(-1)^{J}\right)(2J+1)e^{% {-J(J+1)\hbar^{2}}/{2Ik_{B}T}\;}}
  8. U rot = R T 2 ( ln Z rot T ) C v , rot = U rot T \begin{aligned}\displaystyle U_{\,\text{rot}}&\displaystyle=RT^{2}\left(\frac{% \partial\ln Z_{\,\text{rot}}}{\partial T}\right)\\ \displaystyle C_{v,\,\text{ rot}}&\displaystyle=\frac{\partial U_{\,\text{rot}% }}{\partial T}\end{aligned}
  9. E J = 1 - E J = 0 k B = 2 θ r o t = 2 k B I = 174.98 K \frac{E_{J=1}-E_{J=0}}{k_{B}}=2\theta_{rot}=\frac{\hbar^{2}}{k_{B}I}=174.98\,% \text{ K}

Spin_quantum_number.html

  1. s s
  2. 𝐬 = s ( s + 1 ) \|\mathbf{s}\|=\sqrt{s\,(s+1)}\,\hbar
  3. 𝐬 \mathbf{s}
  4. 𝐬 \|\mathbf{s}\|
  5. s s
  6. \hbar
  7. s z = m s s_{z}=m_{s}\,\hbar
  8. m m
  9. s s
  10. s s
  11. s s
  12. 2 s + 1 2s+1
  13. m m
  14. s s
  15. [ S i , S j ] = i ϵ i j k S k [S_{i},S_{j}]=i\hbar\epsilon_{ijk}S_{k}
  16. [ S i , S 2 ] = 0 \left[S_{i},S^{2}\right]=0
  17. S 2 S^{2}
  18. S z S_{z}
  19. S 2 | s , m s = 2 s ( s + 1 ) | s , m s S^{2}|s,m_{s}\rangle={\hbar}^{2}s(s+1)|s,m_{s}\rangle
  20. S z | s , m s = m s | s , m s S_{z}|s,m_{s}\rangle=\hbar m_{s}|s,m_{s}\rangle
  21. S ± | s , m s = s ( s + 1 ) - m s ( m s ± 1 ) | s , m s ± 1 S_{\pm}|s,m_{s}\rangle=\hbar\sqrt{s(s+1)-m_{s}(m_{s}\pm 1)}|s,m_{s}\pm 1\rangle
  22. S ± = S x ± i S y S_{\pm}=S_{x}\pm iS_{y}
  23. n n
  24. l l
  25. m m
  26. n n
  27. n n
  28. l l
  29. l l
  30. n n
  31. m m
  32. m m
  33. l l
  34. l l
  35. n n
  36. l l
  37. m m
  38. n n
  39. l l
  40. m m
  41. l l
  42. m m
  43. S = 1 2 ( 1 2 + 1 ) = 3 2 S=\hbar\sqrt{\frac{1}{2}\left(\frac{1}{2}+1\right)}=\frac{\sqrt{3}}{2}\hbar
  44. \hbar
  45. 𝐒 𝐳 = ± 1 2 \mathbf{S_{z}}=\pm\frac{1}{2}\hbar
  46. μ 𝐬 = - e 2 m g S \mathbf{\mu_{s}}=-\frac{e}{2m}gS
  47. e e
  48. g g
  49. μ 𝐳 = ± 1 2 g μ B \mathbf{\mu_{z}}=\pm\frac{1}{2}g{\mu_{B}}
  50. μ B \mu_{B}
  51. s s

Spirograph.html

  1. C o C_{o}
  2. R R
  3. C i C_{i}
  4. r < R r<R
  5. C o C_{o}
  6. C i C_{i}
  7. C o C_{o}
  8. A A
  9. C i C_{i}
  10. ρ < r \rho<r
  11. C i C_{i}
  12. A A
  13. A A
  14. X X
  15. A A
  16. T T
  17. C o C_{o}
  18. B B
  19. C i C_{i}
  20. T T
  21. B B
  22. C i C_{i}
  23. T T
  24. C i C_{i}
  25. C o C_{o}
  26. C i C_{i}
  27. B B
  28. C i C_{i}
  29. T T
  30. C o C_{o}
  31. ( X ^ , Y ^ ) (\hat{X},\hat{Y})
  32. C i C_{i}
  33. X X
  34. Y Y
  35. t t
  36. T T
  37. C o C_{o}
  38. t ^ \hat{t}
  39. C i C_{i}
  40. B B
  41. B B
  42. T T
  43. t R = ( t - t ^ ) r tR=(t-\hat{t})r
  44. t ^ = - R - r r t . \hat{t}=-\frac{R-r}{r}t.
  45. t ^ < 0 \hat{t}<0
  46. ( x c , y c ) (x_{c},y_{c})
  47. C i C_{i}
  48. R - r R-r
  49. C i C_{i}
  50. x c = ( R - r ) cos t , y c = ( R - r ) sin t . \begin{array}[]{rcl}x_{c}&=&(R-r)\cos t,\\ y_{c}&=&(R-r)\sin t.\end{array}
  51. t ^ \hat{t}
  52. A A
  53. ( x ^ , y ^ ) (\hat{x},\hat{y})
  54. x ^ = ρ cos t ^ , y ^ = ρ sin t ^ . \begin{array}[]{rcl}\hat{x}&=&\rho\cos\hat{t},\\ \hat{y}&=&\rho\sin\hat{t}.\end{array}
  55. A A
  56. x = x c + x ^ = ( R - r ) cos t + ρ cos t ^ , y = y c + y ^ = ( R - r ) sin t + ρ sin t ^ , \begin{array}[]{rcrcl}x&=&x_{c}+\hat{x}&=&(R-r)\cos t+\rho\cos\hat{t},\\ y&=&y_{c}+\hat{y}&=&(R-r)\sin t+\rho\sin\hat{t},\\ \end{array}
  57. ρ \rho
  58. t t
  59. t ^ \hat{t}
  60. A A
  61. t t
  62. x = x c + x ^ = ( R - r ) cos t + ρ cos R - r r t , y = y c + y ^ = ( R - r ) sin t - ρ sin R - r r t . \begin{array}[]{rcrcl}x&=&x_{c}+\hat{x}&=&(R-r)\cos t+\rho\cos\frac{R-r}{r}t,% \\ y&=&y_{c}+\hat{y}&=&(R-r)\sin t-\rho\sin\frac{R-r}{r}t.\\ \end{array}
  63. sin \sin
  64. R R
  65. C o C_{o}
  66. l = ρ r l=\frac{\rho}{r}
  67. k = r R . k=\frac{r}{R}.
  68. 0 l 1 0\leq l\leq 1
  69. A A
  70. C i C_{i}
  71. 0 k 1 0\leq k\leq 1
  72. C i C_{i}
  73. C o C_{o}
  74. ρ R = l k , \frac{\rho}{R}=lk,
  75. x ( t ) = R [ ( 1 - k ) cos t + l k cos 1 - k k t ] , y ( t ) = R [ ( 1 - k ) sin t - l k sin 1 - k k t ] . \begin{array}[]{rcl}x(t)&=&R\left[(1-k)\cos t+lk\cos\frac{1-k}{k}t\right],\\ y(t)&=&R\left[(1-k)\sin t-lk\sin\frac{1-k}{k}t\right].\\ \end{array}
  76. R R
  77. R R
  78. k = 0 k=0
  79. k = 1 k=1
  80. k = 0 k=0
  81. R R
  82. C i C_{i}
  83. k = 0 k=0
  84. sin \sin
  85. cos \cos
  86. k = 1 k=1
  87. C i C_{i}
  88. r r
  89. R R
  90. C o C_{o}
  91. r = R r=R
  92. C i C_{i}
  93. C o C_{o}
  94. l = 1 l=1
  95. A A
  96. C i C_{i}

Split-complex_number.html

  1. z = x + j y z=x+jy
  2. j 2 = + 1 j^{2}=+1
  3. j 2 = - 1 j^{2}=-1
  4. z = z z * = z * z = x 2 - y 2 . \lVert z\rVert=zz^{*}=z^{*}z=x^{2}-y^{2}.
  5. z w = z w . \lVert zw\rVert=\lVert z\rVert\lVert w\rVert.
  6. z = z , z . \lVert z\rVert=\langle z,z\rangle.
  7. z 0 \lVert z\rVert\neq 0
  8. z - 1 = z * / z . z^{-1}=z^{*}/\lVert z\rVert.
  9. e = e * = e * e = 0. \lVert e\rVert=\lVert e^{*}\rVert=e^{*}e=0.
  10. ( a , b ) = a b . \lVert(a,b)\rVert=ab.
  11. 2 \sqrt{2}
  12. R R R\oplus R
  13. { ( a , b ) R R : a b = 1 } . \{(a,b)\in R\oplus R:ab=1\}.
  14. { cosh a + j sinh a : a R } \{\cosh a+j\ \sinh a:a\in R\}
  15. R R . R\oplus R.
  16. { z : z = a 2 } \{z:\lVert z\rVert=a^{2}\}
  17. { z : z = - a 2 } \{z:\lVert z\rVert=-a^{2}\}
  18. { z : z = 0 } . \{z:\lVert z\rVert=0\}.
  19. exp ( j θ ) = cosh ( θ ) + j sinh ( θ ) . \exp(j\theta)=\cosh(\theta)+j\sinh(\theta).\,
  20. z ± z z\mapsto\pm z
  21. z ± z * . z\mapsto\pm z^{*}.
  22. exp : ( , + ) SO + ( 1 , 1 ) \exp\colon(\mathbb{R},+)\to\mathrm{SO}^{+}(1,1)
  23. e j ( θ + ϕ ) = e j θ e j ϕ . e^{j(\theta+\phi)}=e^{j\theta}e^{j\phi}.\,
  24. z w = z w \lVert zw\rVert=\lVert z\rVert\lVert w\rVert
  25. z ( x y y x ) . z\mapsto\begin{pmatrix}x&y\\ y&x\end{pmatrix}.
  26. C = ( 1 0 0 - 1 ) . C=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}.
  27. ( cosh a sinh a sinh a cosh a ) . \begin{pmatrix}\cosh a&\sinh a\\ \sinh a&\cosh a\end{pmatrix}.
  28. z = x + j y z=x+jy
  29. ( u , v ) = ( x , y ) ( 1 1 1 - 1 ) = ( x , y ) S . (u,v)=(x,y)\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}=(x,y)S.
  30. u v = ( x + y ) ( x - y ) = x 2 - y 2 . uv=(x+y)(x-y)=x^{2}-y^{2}.
  31. ( cosh a , sinh a ) ( 1 1 1 - 1 ) = ( e a , e - a ) (\cosh a,\sinh a)\begin{pmatrix}1&1\\ 1&-1\end{pmatrix}=(e^{a},e^{-a})
  32. e b j e^{bj}\!
  33. σ : ( u , v ) ( r u , v / r ) , r = e b . \sigma:(u,v)\mapsto(ru,v/r),\quad r=e^{b}.
  34. Z = ( x y y x ) . Z=\begin{pmatrix}x&y\\ y&x\end{pmatrix}.
  35. J z = ( x + y 0 0 x - y ) , J_{z}=\begin{pmatrix}x+y&0\\ 0&x-y\end{pmatrix},
  36. Z = S J z S - 1 , Z=SJ_{z}S^{-1}\ ,
  37. S = ( 1 - 1 1 1 ) . S=\begin{pmatrix}1&-1\\ 1&1\end{pmatrix}.
  38. | y | |y|
  39. e a j e b j = e ( a + b ) j e^{aj}\ e^{bj}=e^{(a+b)j}
  40. { z = σ j e a j : σ \isin R } \{z=\sigma je^{aj}:\sigma\isin R\}

Splitting_of_prime_ideals_in_Galois_extensions.html

  1. A B K L \begin{array}[]{ccc}A&\hookrightarrow&B\\ \downarrow&&\downarrow\\ K&\hookrightarrow&L\end{array}
  2. p B = j P j e ( j ) pB=\prod_{j}P_{j}^{e(j)}
  3. p B = ( P j ) e . pB=(\prod P_{j})^{e}\ .
  4. F = A / p F=A/p
  5. [ L : K ] / [ F : F ] [L:K]/[F^{\prime}:F]
  6. | G | / | D | |G|/|D|
  7. F j = B / P j F_{j}=B/P_{j}
  8. ( 2 ) = ( 1 + i ) 2 , (2)=(1+i)^{2}\ ,
  9. O L / ( 1 + i ) O L O_{L}/(1+i)O_{L}
  10. a + b i a - b i modulo ( 1 + i ) , a+bi\equiv a-bi\textrm{\ modulo\ }(1+i)\ ,
  11. ( 13 ) = ( 2 + 3 i ) ( 2 - 3 i ) . (13)=(2+3i)(2-3i)\ .
  12. O L / ( 2 ± 3 i ) O L , O_{L}/(2\pm 3i)O_{L}\ ,
  13. ( a + b i ) 13 a + b i modulo ( 2 ± 3 i ) (a+bi)^{13}\equiv a+bi\textrm{\ modulo\ }(2\pm 3i)
  14. O L / ( 7 ) O L , O_{L}/(7)O_{L}\ ,
  15. ( a + b i ) 7 a - b i modulo 7 , (a+bi)^{7}\equiv a-bi\textrm{\ modulo\ }7\ ,
  16. h ( X ) = h 1 ( X ) e 1 h n ( X ) e n , h(X)=h_{1}(X)^{e_{1}}\cdots h_{n}(X)^{e_{n}},
  17. P O L = Q 1 e 1 Q n e n , PO_{L}=Q_{1}^{e_{1}}\cdots Q_{n}^{e_{n}},
  18. Q j = P O L + h j ( θ ) O L , Q_{j}=PO_{L}+h_{j}(\theta)O_{L},
  19. { y O L : y O L O K [ θ ] } ; \{y\in O_{L}:yO_{L}\subseteq O_{K}[\theta]\};
  20. i i
  21. i i
  22. X 2 + 1 = ( X + 1 ) 2 ( mod 2 ) . X^{2}+1=(X+1)^{2}\;\;(\mathop{{\rm mod}}2).
  23. Q = ( 2 ) 𝐙 [ i ] + ( i + 1 ) 𝐙 [ i ] = ( 1 + i ) 𝐙 [ i ] . Q=(2)\mathbf{Z}[i]+(i+1)\mathbf{Z}[i]=(1+i)\mathbf{Z}[i].
  24. Q = ( 7 ) 𝐙 [ i ] + ( i 2 + 1 ) 𝐙 [ i ] = 7 𝐙 [ i ] . Q=(7)\mathbf{Z}[i]+(i^{2}+1)\mathbf{Z}[i]=7\mathbf{Z}[i].
  25. X 2 + 1 = ( X + 5 ) ( X - 5 ) ( mod 13 ) . X^{2}+1=(X+5)(X-5)\;\;(\mathop{{\rm mod}}13).
  26. Q 1 = ( 13 ) 𝐙 [ i ] + ( i + 5 ) 𝐙 [ i ] = = ( 2 + 3 i ) 𝐙 [ i ] Q_{1}=(13)\mathbf{Z}[i]+(i+5)\mathbf{Z}[i]=\cdots=(2+3i)\mathbf{Z}[i]
  27. Q 2 = ( 13 ) 𝐙 [ i ] + ( i - 5 ) 𝐙 [ i ] = = ( 2 - 3 i ) 𝐙 [ i ] . Q_{2}=(13)\mathbf{Z}[i]+(i-5)\mathbf{Z}[i]=\cdots=(2-3i)\mathbf{Z}[i].

Splitting_theorem.html

  1. Ric ( M ) 0 {\rm Ric}(M)\geq 0
  2. d ( γ ( u ) , γ ( v ) ) = | u - v | d(\gamma(u),\gamma(v))=|u-v|
  3. u , v , u,v\in\mathbb{R},
  4. × L , \mathbb{R}\times L,
  5. L L
  6. Ric ( L ) 0. {\rm Ric}(L)\geq 0.

Square.html

  1. \ell
  2. P = 4 P=4\ell
  3. A = 2 . A=\ell^{2}.
  4. A = d 2 2 . A=\frac{d^{2}}{2}.
  5. A = 2 R 2 A=2R^{2}
  6. A = 4 r 2 . A=4r^{2}.
  7. A = 1 2 ( a 2 + c 2 ) = 1 2 ( b 2 + d 2 ) . A=\frac{1}{2}(a^{2}+c^{2})=\frac{1}{2}(b^{2}+d^{2}).
  8. max ( x 2 , y 2 ) = 1 \max(x^{2},y^{2})=1
  9. 2 \scriptstyle\sqrt{2}
  10. x 2 + y 2 = 2. x^{2}+y^{2}=2.
  11. | x - a | + | y - b | = r . \left|x-a\right|+\left|y-b\right|=r.
  12. 2 \scriptstyle\sqrt{2}
  13. π / 2 \pi/2
  14. π / 4 \pi/4
  15. 2 ( P H 2 - P E 2 ) = P D 2 - P B 2 . 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.
  16. π \pi

Square_(algebra).html

  1. x + 1 x+1
  2. x x
  3. x x
  4. x −x
  5. [ 0 , + ) [0, +∞)
  6. ( , 0 ] (−∞,0]
  7. x x
  8. ( 0 , 1 ) (0,1)
  9. i i
  10. l l
  11. n n
  12. p p
  13. ( p 1 ) / 2 (p−1)/2
  14. ( p 1 ) / 2 (p−1)/2
  15. I I
  16. x 2 I x^{2}\in I
  17. x I x\in I
  18. n n
  19. k k
  20. n n
  21. q q
  22. q q
  23. z z
  24. q ( z ) q(z)
  25. q q
  26. x ¯ \overline{x}
  27. x i - x ¯ x_{i}-\overline{x}

Square_cupola.html

  1. V = ( 1 + 2 2 3 ) a 3 1.94281... a 3 V=(1+\frac{2\sqrt{2}}{3})a^{3}\approx 1.94281...a^{3}
  2. A = ( 7 + 2 2 + 3 ) a 2 11.5605... a 2 A=(7+2\sqrt{2}+\sqrt{3})a^{2}\approx 11.5605...a^{2}
  3. C = ( 1 2 5 + 2 2 ) a 1.39897... a C=(\frac{1}{2}\sqrt{5+2\sqrt{2}})a\approx 1.39897...a

Square_gyrobicupola.html

  1. V = ( 2 + 4 2 3 ) a 3 3.88562... a 3 V=(2+\frac{4\sqrt{2}}{3})a^{3}\approx 3.88562...a^{3}
  2. A = 2 ( 5 + 3 ) a 2 13.4641... a 2 A=2(5+\sqrt{3})a^{2}\approx 13.4641...a^{2}

Square_pyramid.html

  1. H = 1 2 a H=\frac{1}{\sqrt{2}}a
  2. A = ( 1 + 3 ) a 2 A=(1+\sqrt{3})a^{2}
  3. V = 2 6 a 3 . V=\frac{\sqrt{2}}{6}a^{3}.
  4. A = l 2 + l l 2 + ( 2 h ) 2 A=l^{2}+l\sqrt{l^{2}+(2h)^{2}}
  5. V = 1 3 l 2 h . V=\frac{1}{3}l^{2}h.

Squeeze_mapping.html

  1. a a
  2. ( x , y ) ( a x , y / a ) (x,y)\mapsto(ax,y/a)
  3. a a
  4. { ( u , v ) : u v = constant } \{(u,v)\,:\,uv=\mathrm{constant}\}
  5. u = a x u=ax
  6. v = y / a v=y/a
  7. u v = x y uv=xy
  8. ( x , y ) (x,y)
  9. x y = 1 ) xy=1)
  10. r r
  11. s s
  12. x y xy
  13. x = u + v , y = u - v , x=u+v,\quad y=u-v\,,
  14. S O ( 2 ) SO(2)
  15. u - u , v - v u\mapsto-u,\quad v\mapsto-v
  16. x x
  17. y y
  18. x y , y x x↦y,y↦x
  19. x x , y y ) x↦−x,y↦−y)
  20. + +
  21. O ( 1 , 1 ) O(1,1)
  22. 4 4
  23. O ( 2 ) O(2)
  24. 2 2
  25. S O ( 1 , 1 ) SO(1,1)
  26. 2 2
  27. S O ( 2 ) SO(2)
  28. S O S L SO⊂SL
  29. S O ( 1 , 1 ) S L ( 2 ) SO(1,1) ⊂SL(2)
  30. v 1 = G x 2 v 2 = K G x 1 v_{1}=Gx_{2}\quad v_{2}=KGx_{1}
  31. x 2 2 - K x 1 2 = constant x_{2}^{2}-Kx_{1}^{2}=\mathrm{constant}

Squeezed_coherent_state.html

  1. Δ x Δ p = 2 \Delta x\Delta p=\frac{\hbar}{2}
  2. | 0 |0\rangle
  3. | α |\alpha\rangle
  4. Δ x Δ p \Delta x\neq\Delta p
  5. = 1 \hbar=1
  6. ψ ( x ) = C exp ( - ( x - x 0 ) 2 2 w 0 2 + i p 0 x ) \psi(x)=C\,\exp\left(-\frac{(x-x_{0})^{2}}{2w_{0}^{2}}+ip_{0}x\right)
  7. C , x 0 , w 0 , p 0 C,x_{0},w_{0},p_{0}
  8. w 0 w_{0}
  9. x ^ + i p ^ w 0 2 \hat{x}+i\hat{p}w_{0}^{2}
  10. x 0 + i p 0 w 0 2 x_{0}+ip_{0}w_{0}^{2}
  11. | α , ζ = D ( α ) S ( ζ ) | 0 |\alpha,\zeta\rangle=D(\alpha)S(\zeta)|0\rangle
  12. | 0 |0\rangle
  13. D ( α ) D(\alpha)
  14. S ( ζ ) S(\zeta)
  15. D ( α ) = exp ( α a ^ - α * a ^ ) and S ( ζ ) = exp ( 1 2 ( ζ * a ^ 2 - ζ a ^ 2 ) ) D(\alpha)=\exp(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})\qquad\,\text{and}% \qquad S(\zeta)=\exp(\frac{1}{2}(\zeta^{*}\hat{a}^{2}-\zeta\hat{a}^{\dagger 2}))
  16. a ^ \hat{a}
  17. a ^ \hat{a}^{\dagger}
  18. ω \omega
  19. a ^ = m ω 2 ( x - i p m ω ) and a ^ = m ω 2 ( x + i p m ω ) \hat{a}^{\dagger}=\sqrt{\frac{m\omega}{2\hbar}}\left(x-\frac{\mathrm{i}p}{m% \omega}\right)\qquad\,\text{and}\qquad\hat{a}=\sqrt{\frac{m\omega}{2\hbar}}% \left(x+\frac{\mathrm{i}p}{m\omega}\right)
  20. ζ \zeta
  21. x x
  22. p p
  23. ( Δ x ) 2 = 2 m ω e - 2 r and ( Δ p ) 2 = m ω 2 e 2 r (\Delta x)^{2}=\frac{\hbar}{2m\omega}\mathrm{e}^{-2r}\qquad\,\text{and}\qquad(% \Delta p)^{2}=\frac{m\hbar\omega}{2}\mathrm{e}^{2r}
  24. Δ x Δ p = 2 \Delta x\Delta p=\frac{\hbar}{2}
  25. π / 2 \pi/2
  26. | S M S V = S ( ζ ) | 0 |SMSV\rangle=S(\zeta)|0\rangle
  27. | S M S V = 1 cosh r n = 0 ( - tanh r ) n ( 2 n ) ! 2 n n ! | 2 n |SMSV\rangle=\frac{1}{\sqrt{\cosh r}}\sum_{n=0}^{\infty}(-\tanh r)^{n}\frac{% \sqrt{(2n)!}}{2^{n}n!}|2n\rangle
  28. | T M S V = S 2 ( ζ ) | 0 = exp ( - ζ a ^ b ^ + ζ * a ^ b ^ ) | 0 |TMSV\rangle=S_{2}(\zeta)|0\rangle=\exp(-\zeta\hat{a}\hat{b}+\zeta^{*}\hat{a}^% {\dagger}\hat{b}^{\dagger})|0\rangle
  29. | T M S V = 1 cosh r n = 0 ( tanh r ) n | n n |TMSV\rangle=\frac{1}{\cosh r}\sum_{n=0}^{\infty}(\tanh r)^{n}|nn\rangle
  30. χ ( 3 ) \chi^{(3)}
  31. χ ( 2 ) \chi^{(2)}

SRGB.html

  1. [ R linear G linear B linear ] = [ 3.2406 - 1.5372 - 0.4986 - 0.9689 1.8758 0.0415 0.0557 - 0.2040 1.0570 ] [ X Y Z ] \begin{bmatrix}R_{\mathrm{linear}}\\ G_{\mathrm{linear}}\\ B_{\mathrm{linear}}\end{bmatrix}=\begin{bmatrix}3.2406&-1.5372&-0.4986\\ -0.9689&1.8758&0.0415\\ 0.0557&-0.2040&1.0570\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}
  2. X = Y x / y , X=Yx/y,\,
  3. Z = Y ( 1 - x - y ) / y Z=Y(1-x-y)/y\,
  4. R linear R_{\mathrm{linear}}
  5. G linear G_{\mathrm{linear}}
  6. B linear B_{\mathrm{linear}}
  7. C linear C_{\mathrm{linear}}
  8. R linear R_{\mathrm{linear}}
  9. G linear G_{\mathrm{linear}}
  10. B linear B_{\mathrm{linear}}
  11. C srgb C_{\mathrm{srgb}}
  12. R srgb R_{\mathrm{srgb}}
  13. G srgb G_{\mathrm{srgb}}
  14. B srgb B_{\mathrm{srgb}}
  15. C srgb = { 12.92 C linear , C linear 0.0031308 ( 1 + a ) C linear 1 / 2.4 - a , C linear > 0.0031308 C_{\mathrm{srgb}}=\begin{cases}12.92C_{\mathrm{linear}},&C_{\mathrm{linear}}% \leq 0.0031308\\ (1+a)C_{\mathrm{linear}}^{1/2.4}-a,&C_{\mathrm{linear}}>0.0031308\end{cases}
  16. a = 0.055 a=0.055
  17. R srgb R_{\mathrm{srgb}}
  18. G srgb G_{\mathrm{srgb}}
  19. B srgb B_{\mathrm{srgb}}
  20. C linear = { C srgb 12.92 , C srgb 0.04045 ( C srgb + a 1 + a ) 2.4 , C srgb > 0.04045 C_{\mathrm{linear}}=\begin{cases}\frac{C_{\mathrm{srgb}}}{12.92},&C_{\mathrm{% srgb}}\leq 0.04045\\ \left(\frac{C_{\mathrm{srgb}}+a}{1+a}\right)^{2.4},&C_{\mathrm{srgb}}>0.04045% \end{cases}
  21. C C
  22. R R
  23. G G
  24. B B
  25. [ X Y Z ] = [ 0.4124 0.3576 0.1805 0.2126 0.7152 0.0722 0.0193 0.1192 0.9505 ] [ R linear G linear B linear ] \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}=\begin{bmatrix}0.4124&0.3576&0.1805\\ 0.2126&0.7152&0.0722\\ 0.0193&0.1192&0.9505\end{bmatrix}\begin{bmatrix}R_{\mathrm{linear}}\\ G_{\mathrm{linear}}\\ B_{\mathrm{linear}}\end{bmatrix}
  26. C linear C_{\mathrm{linear}}
  27. C srgb C_{\mathrm{srgb}}
  28. ( K 0 + a 1 + a ) γ = K 0 ϕ . \left(\frac{K_{0}+a}{1+a}\right)^{\gamma}=\frac{K_{0}}{\phi}.
  29. γ = 2.4 \gamma=2.4
  30. ϕ = 12.92 \phi=12.92
  31. K 0 K_{0}
  32. 0.0381548 0.0381548
  33. K 0 K_{0}
  34. 0.0404482 0.0404482
  35. K 0 = 0.04045 K_{0}=0.04045
  36. γ ( K 0 + a 1 + a ) γ - 1 ( 1 1 + a ) = 1 ϕ . \gamma\left(\frac{K_{0}+a}{1+a}\right)^{\gamma-1}\left(\frac{1}{1+a}\right)=% \frac{1}{\phi}.
  37. K 0 K_{0}
  38. ϕ \phi
  39. K 0 = a γ - 1 , ϕ = ( 1 + a ) γ ( γ - 1 ) γ - 1 ( a γ - 1 ) ( γ γ ) . K_{0}=\frac{a}{\gamma-1},\ \ \ \phi=\frac{(1+a)^{\gamma}(\gamma-1)^{\gamma-1}}% {(a^{\gamma-1})(\gamma^{\gamma})}.
  40. a = 0.055 a=0.055
  41. γ = 2.4 \gamma=2.4
  42. K 0 K_{0}
  43. 0.0392857 0.0392857
  44. ϕ \phi
  45. 12.9232102 12.9232102
  46. K 0 / ϕ K_{0}/\phi
  47. 0.00303993 0.00303993
  48. K 0 = 0.03928 K_{0}=0.03928
  49. ϕ = 12.92321 \phi=12.92321
  50. K 0 / ϕ = 0.00304 K_{0}/\phi=0.00304
  51. K 0 = 0.03928 K_{0}=0.03928
  52. ϕ = 12.92 \phi=12.92
  53. ϕ = 12.92 \phi=12.92
  54. K 0 K_{0}

Stable_distribution.html

  1. exp [ i t μ - | c t | α ( 1 - i β sgn ( t ) Φ ) ] , \exp\!\Big[\;it\mu-|c\,t|^{\alpha}\,(1-i\beta\,\mbox{sgn}~{}(t)\Phi)\;\Big],
  2. Φ = { tan π α 2 if α 1 - 2 π log | t | if α = 1 \Phi=\begin{cases}\tan\tfrac{\pi\alpha}{2}&\,\text{if }\alpha\neq 1\\ -\tfrac{2}{\pi}\log|t|&\,\text{if }\alpha=1\end{cases}
  3. f ( x ) = 1 2 π - φ ( t ) e - i x t d t f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\varphi(t)e^{-ixt}\,dt
  4. φ ( t ; α , β , c , μ ) = exp [ i t μ - | c t | α ( 1 - i β sgn ( t ) Φ ) ] \varphi(t;\alpha,\beta,c,\mu)=\exp\left[~{}it\mu\!-\!|ct|^{\alpha}\,(1\!-\!i% \beta\,\textrm{sgn}(t)\Phi)~{}\right]
  5. Φ = tan ( π α / 2 ) \Phi=\tan(\pi\alpha/2)\,
  6. Φ = - 2 π log | t | . \Phi=-\frac{2}{\pi}\log|t|.\,
  7. Φ = - 1 π ln ( t | c | ) \Phi^{\prime}=-\frac{1}{\pi}\ln\left(\frac{t}{|c|}\right)
  8. μ = μ + β | c | ln ( | c | ) π \mu^{\prime}=\mu+\frac{\beta|c|\ln(|c|)}{\pi}
  9. τ = t | c | \tau=\frac{t}{|c|}
  10. y = x - μ | c | y=\frac{x-\mu^{\prime}}{|c|}
  11. f ( x ; α , β , c , μ ) d x = f ( y ; α , β , 1 , 0 ) d y f(x;\alpha,\beta,c,\mu^{\prime})dx=f(y;\alpha,\beta,1,0)dy\,
  12. f ( x ; α , β , c , μ ) f(x;\alpha,\beta,c,\mu)
  13. Y = i = 1 N k i ( X i - μ ) Y=\sum_{i=1}^{N}k_{i}(X_{i}-\mu)\,
  14. s - 1 f ( y / s ; α , β , c , 0 ) s^{-1}f(y/s;\alpha,\beta,c,0)
  15. s = ( i = 1 N | k i | α ) 1 / α . s=\left(\sum_{i=1}^{N}|k_{i}|^{\alpha}\right)^{1/\alpha}.\,
  16. f ( x ) c α ( 1 + sgn ( x ) β ) sin ( π α / 2 ) Γ ( α + 1 ) / π | x | 1 + α f(x)\sim\frac{c^{\alpha}(1+\mbox{sgn}~{}(x)\beta)\sin(\pi\alpha/2)\Gamma(% \alpha+1)/\pi}{|x|^{1+\alpha}}
  17. exp [ i t μ 1 + i t μ 2 - | c 1 t | α - | c 2 t | α + i β 1 | c 1 t | α sgn ( t ) Φ + i β 2 | c 2 t | α sgn ( t ) Φ ] \exp\left[it\mu_{1}+it\mu_{2}-|c_{1}t|^{\alpha}-|c_{2}t|^{\alpha}+i\beta_{1}|c% _{1}t|^{\alpha}\textrm{sgn}(t)\Phi+i\beta_{2}|c_{2}t|^{\alpha}\,\textrm{sgn}(t% )\Phi\right]
  18. μ = μ 1 + μ 2 \mu=\mu_{1}+\mu_{2}\,
  19. | c | = ( | c 1 | α + | c 2 | α ) 1 / α |c|=(|c_{1}|^{\alpha}+|c_{2}|^{\alpha})^{1/\alpha}\,
  20. β = β 1 | c 1 | α + β 2 | c 2 | α | c | α \beta=\frac{\beta_{1}|c_{1}|^{\alpha}+\beta_{2}|c_{2}|^{\alpha}}{|c|^{\alpha}}
  21. f ( x ; α , β , c , μ ) = 1 π [ 0 e i t ( x - μ ) e - ( c t ) α ( 1 - i β Φ ) d t ] . f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[\int_{0}^{\infty}e^{it(x-\mu)}e^% {-(ct)^{\alpha}(1-i\beta\Phi)}\,dt\right].
  22. f ( x ; α , β , c , μ ) = 1 π [ 0 e i t ( x - μ ) n = 0 ( - q t α ) n n ! d t ] f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[\int_{0}^{\infty}e^{it(x-\mu)}% \sum_{n=0}^{\infty}\frac{(-qt^{\alpha})^{n}}{n!}\,dt\right]
  23. q = c α ( 1 - i β Φ ) q=c^{\alpha}(1-i\beta\Phi)
  24. f ( x ; α , β , c , μ ) = 1 π [ n = 1 ( - q ) n n ! ( i x - μ ) α n + 1 Γ ( α n + 1 ) ] f(x;\alpha,\beta,c,\mu)=\frac{1}{\pi}\Re\left[\sum_{n=1}^{\infty}\frac{(-q)^{n% }}{n!}\left(\frac{i}{x-\mu}\right)^{\alpha n+1}\Gamma(\alpha n+1)\right]

Stable_marriage_problem.html

  1. O ( n 2 ) O(n^{2})
  2. n n

Standard_gravitational_parameter.html

  1. μ = G M \mu=GM
  2. μ = r v 2 = r 3 ω 2 = 4 π 2 r 3 / T 2 \mu=rv^{2}=r^{3}\omega^{2}=4\pi^{2}r^{3}/T^{2}
  3. μ = 4 π 2 a 3 / T 2 \mu=4\pi^{2}a^{3}/T^{2}

Stanley's_reciprocity_theorem.html

  1. M [ a 1 a d ] [ 0 0 ] M\left[\begin{matrix}a_{1}\\ \vdots\\ a_{d}\end{matrix}\right]\geq\left[\begin{matrix}0\\ \vdots\\ 0\end{matrix}\right]
  2. F ( x 1 , , x d ) = ( a 1 , , a d ) cone x 1 a 1 x d a d . F(x_{1},\dots,x_{d})=\sum_{(a_{1},\dots,a_{d})\in{\rm cone}}x_{1}^{a_{1}}% \cdots x_{d}^{a_{d}}.
  3. F ( 1 / x 1 , , 1 / x d ) = ( - 1 ) d F int ( x 1 , , x d ) . F(1/x_{1},\dots,1/x_{d})=(-1)^{d}F_{\rm int}(x_{1},\dots,x_{d}).

Star_(game_theory).html

  1. * *
  2. * 1 *1
  3. * 2 + * 3 *2+*3

Star_height.html

  1. h ( ) = 0 \scriptstyle h\left(\emptyset\right)\,=\,0
  2. h ( ε ) = 0 \scriptstyle h\left(\varepsilon\right)\,=\,0
  3. h ( a ) = 0 \scriptstyle h\left(a\right)\,=\,0
  4. h ( E F ) = h ( E F ) = max ( h ( E ) , h ( F ) ) \scriptstyle h\left(EF\right)\,=\,h\left(E\,\mid\,F\right)\,=\,\max\left(\,h(E% ),h(F)\,\right)
  5. h ( E * ) = h ( E ) + 1. \scriptstyle h\left(E^{*}\right)\,=\,h(E)+1.
  6. \scriptstyle\emptyset
  7. ( b a a * b ) * a a * \scriptstyle\left(b\,\mid\,aa^{*}b\right)^{*}aa^{*}
  8. ( a b ) * a \scriptstyle(a\,\mid\,b)^{*}a
  9. r ( G ) = 1 + min v V r ( G - v ) , r(G)=1+\min_{v\in V}r(G-v),\,
  10. h ( E c ) = h ( E ) \scriptstyle h\left(E^{c}\right)\,=\,h(E)
  11. ( a b ) * a , \scriptstyle(a\,\mid\,b)^{*}a,
  12. c a \scriptstyle\emptyset^{c}a

Star_refinement.html

  1. X X
  2. 𝒰 = ( U i ) i I \mathcal{U}=(U_{i})_{i\in I}
  3. X X
  4. X = i I U i X=\bigcup_{i\in I}U_{i}
  5. S S
  6. X X
  7. S S
  8. 𝒰 \mathcal{U}
  9. U i U_{i}
  10. S S
  11. st ( S , 𝒰 ) = { U i : i I , S U i } . \mathrm{st}(S,\mathcal{U})=\bigcup\big\{U_{i}:i\in I,\ S\cap U_{i}\neq% \emptyset\big\}.
  12. x X x\in X
  13. st ( x , 𝒰 ) \mathrm{st}(x,\mathcal{U})
  14. st ( { x } , 𝒰 ) \mathrm{st}(\{x\},\mathcal{U})
  15. 𝒰 = ( U i ) i I \mathcal{U}=(U_{i})_{i\in I}
  16. X X
  17. 𝒱 = ( V j ) j J \mathcal{V}=(V_{j})_{j\in J}
  18. X X
  19. U i U_{i}
  20. V j V_{j}
  21. 𝒰 \mathcal{U}
  22. 𝒱 \mathcal{V}
  23. x X x\in X
  24. st ( x , 𝒰 ) \mathrm{st}(x,\mathcal{U})
  25. V j V_{j}
  26. 𝒰 \mathcal{U}
  27. 𝒱 \mathcal{V}
  28. i I i\in I
  29. st ( U i , 𝒰 ) \mathrm{st}(U_{i},\mathcal{U})
  30. V j V_{j}

Starling_equation.html

  1. J v = K f ( [ P c - P i ] - σ [ π c - π i ] ) \ J_{v}=K_{\mathrm{f}}([P_{\mathrm{c}}-P_{\mathrm{i}}]-\sigma[\pi_{\mathrm{c}}% -\pi_{\mathrm{i}}])
  2. J v J_{v}
  3. [ P c - P i ] - σ [ π c - π i ] [P_{\mathrm{c}}-P_{\mathrm{i}}]-\sigma[\pi_{\mathrm{c}}-\pi_{\mathrm{i}}]
  4. [ P c - P i ] - σ [ π c - π i ] [P_{\mathrm{c}}-P_{\mathrm{i}}]-\sigma[\pi_{\mathrm{c}}-\pi_{\mathrm{i}}]

Statically_indeterminate.html

  1. F = 0 \sum\vec{F}=0
  2. M = 0 \sum\vec{M}=0
  3. H A = F h V C = F v a a + b + c V A = F v - V C \begin{aligned}\displaystyle H_{A}&\displaystyle=F_{h}\\ \displaystyle V_{C}&\displaystyle=\frac{F_{v}\cdot a}{a+b+c}\\ \displaystyle V_{A}&\displaystyle=F_{v}-V_{C}\end{aligned}
  4. F h = 0 F_{h}=0

Stationary_point.html

  1. x x
  2. x y xy
  3. x x 3 x\mapsto x^{3}
  4. C 1 C^{1}
  5. f : f\colon\mathbb{R}\to\mathbb{R}
  6. C 1 C^{1}

Statistical_learning_theory.html

  1. I = 1 R V I=\frac{1}{R}V
  2. X X
  3. Y Y
  4. Z = X Y Z=X\otimes Y
  5. p ( z ) = p ( x , y ) p(z)=p(\vec{x},y)
  6. n n
  7. S = { ( x 1 , y 1 ) , , ( x n , y n ) } = { z 1 , , z n } S=\{(\vec{x}_{1},y_{1}),\dots,(\vec{x}_{n},y_{n})\}=\{\vec{z}_{1},\dots,\vec{z% }_{n}\}
  8. x i \vec{x}_{i}
  9. y i y_{i}
  10. f : X Y f:X\mapsto Y
  11. f ( x ) y f(\vec{x})\sim y
  12. \mathcal{H}
  13. f : X Y f:X\mapsto Y
  14. V ( f ( x ) , y ) V(f(\vec{x}),y)
  15. f ( x ) f(\vec{x})
  16. y y
  17. I [ f ] = X Y V ( f ( x ) , y ) p ( x , y ) d x d y I[f]=\displaystyle\int_{X\otimes Y}V(f(\vec{x}),y)p(\vec{x},y)d\vec{x}dy
  18. f f
  19. f f
  20. inf f I [ f ] \inf_{f\in\mathcal{H}}I[f]
  21. p ( x , y ) p(\vec{x},y)
  22. I S [ f ] = 1 n i = 1 n V ( f ( x i ) , y i ) I_{S}[f]=\frac{1}{n}\displaystyle\sum_{i=1}^{n}V(f(\vec{x}_{i}),y_{i})
  23. f S f_{S}
  24. f S f_{S}
  25. V ( f ( x ) , y ) = ( y - f ( x ) ) 2 V(f(\vec{x}),y)=(y-f(\vec{x}))^{2}
  26. V ( f ( x ) , y ) = | y - f ( x ) | V(f(\vec{x}),y)=|y-f(\vec{x})|
  27. V ( f ( x , y ) ) = θ ( - y f ( x ) ) V(f(\vec{x},y))=\theta(-yf(\vec{x}))
  28. θ \theta
  29. V ( f ( x , y ) ) = ( - y f ( x ) ) + V(f(\vec{x},y))=(-yf(\vec{x}))_{+}
  30. \mathcal{H}
  31. \mathcal{H}
  32. \mathcal{H}
  33. p p
  34. 1 n i = 1 n V ( f ( x i , y i ) ) + γ f 2 \frac{1}{n}\displaystyle\sum_{i=1}^{n}V(f(\vec{x}_{i},y_{i}))+\gamma\|f\|_{% \mathcal{H}}^{2}
  35. γ \gamma
  36. \mathcal{H}

Steel_square.html

  1. c = 12 2 + 12 2 c=\sqrt{12^{2}+12^{2}}\,
  2. L = c 2 + P 2 L=\sqrt{c^{2}+P^{2}}\,
  3. t a n g e n t = c L tangent=\frac{c}{L}\,
  4. t a n g e n t = 16 z tangent=\frac{16}{z}\,
  5. t a n g e n t = P b . tangent=\frac{P}{b}.\,
  6. t a n g e n t = P a 2 + b 2 . tangent=\frac{P}{\sqrt{a^{2}+b^{2}}}.\,

Stefan_problem.html

  1. u u
  2. 0
  3. x x
  4. f ( t ) f(t)
  5. [ 0 , s ( t ) ] [0,s(t)]
  6. s ( t ) s(t)
  7. u u
  8. s s
  9. u t \displaystyle\frac{\partial u}{\partial t}
  10. s s
  11. u u
  12. f f

Stein's_lemma.html

  1. E ( g ( X ) ( X - μ ) ) = σ 2 E ( g ( X ) ) . E\bigl(g(X)(X-\mu)\bigr)=\sigma^{2}E\bigl(g^{\prime}(X)\bigr).
  2. Cov ( g ( X ) , Y ) = E ( g ( X ) ) Cov ( X , Y ) . \operatorname{Cov}(g(X),Y)=E(g^{\prime}(X))\operatorname{Cov}(X,Y).
  3. φ ( x ) = 1 2 π e - x 2 / 2 \varphi(x)={1\over\sqrt{2\pi}}e^{-x^{2}/2}
  4. 1 σ φ ( x - μ σ ) . {1\over\sigma}\varphi\left({x-\mu\over\sigma}\right).
  5. f η ( x ) = exp ( η T ( x ) - Ψ ( η ) ) h ( x ) . f_{\eta}(x)=\exp(\eta^{\prime}T(x)-\Psi(\eta))h(x).
  6. ( a , b ) (a,b)
  7. a , b a,b
  8. - , -\infty,\infty
  9. x a or b x\rightarrow a\,\text{ or }b
  10. exp ( η T ( x ) ) h ( x ) g ( x ) 0 \exp(\eta^{\prime}T(x))h(x)g(x)\rightarrow 0
  11. g g
  12. E | g ( X ) | < E|g^{\prime}(X)|<\infty
  13. exp ( η T ( x ) ) h ( x ) 0 \exp(\eta^{\prime}T(x))h(x)\rightarrow 0
  14. a , b a,b
  15. E ( ( h ( X ) / h ( X ) + η i T i ( X ) ) g ( X ) ) = - E g ( X ) . E((h^{\prime}(X)/h(X)+\sum\eta_{i}T_{i}^{\prime}(X))g(X))=-Eg^{\prime}(X).
  16. X X
  17. \mathbb{R}
  18. E | g ( X ) | < and E | g ( X ) | < E|g(X)|<\infty\,\text{ and }E|g^{\prime}(X)|<\infty
  19. lim x f η ( x ) g ( x ) 0 \lim_{x\rightarrow\infty}f_{\eta}(x)g(x)\not=0
  20. g ( x ) = 1 g(x)=1
  21. f η ( x ) f_{\eta}(x)
  22. f ( x ) = { 1 x [ n , n + 2 - n ) 0 otherwise f(x)=\begin{cases}1&x\in[n,n+2^{-n})\\ 0&\,\text{otherwise}\end{cases}
  23. f f

Stein_manifold.html

  1. X X
  2. n n
  3. X X
  4. K X K\subset X
  5. K ¯ = { z X : | f ( z ) | sup K | f | f 𝒪 ( X ) } , \bar{K}=\{z\in X:|f(z)|\leq\sup_{K}|f|\ \forall f\in\mathcal{O}(X)\},
  6. X X
  7. 𝒪 ( X ) \mathcal{O}(X)
  8. X X
  9. X X
  10. x y x\neq y
  11. X X
  12. f 𝒪 ( X ) f\in\mathcal{O}(X)
  13. f ( x ) f ( y ) . f(x)\neq f(y).
  14. H 1 ( X , 𝒪 X * ) = 0 H^{1}(X,\mathcal{O}_{X}^{*})=0
  15. H 1 ( X , 𝒪 X ) H 1 ( X , 𝒪 X * ) H 2 ( X , ) H 2 ( X , 𝒪 X ) H^{1}(X,\mathcal{O}_{X})\longrightarrow H^{1}(X,\mathcal{O}_{X}^{*})% \longrightarrow H^{2}(X,\mathbb{Z})\longrightarrow H^{2}(X,\mathcal{O}_{X})
  16. H 1 ( X , 𝒪 X ) = H 2 ( X , 𝒪 X ) = 0 H^{1}(X,\mathcal{O}_{X})=H^{2}(X,\mathcal{O}_{X})=0
  17. H 2 ( X , ) = 0 H^{2}(X,\mathbb{Z})=0
  18. n \mathbb{C}^{n}
  19. n \mathbb{C}^{n}
  20. X X
  21. n n
  22. 2 n + 1 \mathbb{C}^{2n+1}
  23. X X
  24. x X x\in X
  25. n n
  26. X X
  27. x x
  28. ψ \psi
  29. X X
  30. i ¯ ψ > 0 i\partial\bar{\partial}\psi>0
  31. { z X , ψ ( z ) c } \{z\in X,\psi(z)\leq c\}
  32. X X
  33. c c
  34. ψ \psi
  35. { z | - ψ ( z ) c } \{z|-\infty\leq\psi(z)\leq c\}

Stellar_structure.html

  1. ρ ( r ) \rho(r)
  2. T ( r ) T(r)
  3. P ( r ) P(r)
  4. l ( r ) l(r)
  5. ϵ ( r ) \epsilon(r)
  6. d r \mbox{d}~{}r
  7. r r
  8. λ \lambda
  9. λ T / | T | \lambda\ll T/|\nabla T|
  10. d P d r = - G m ρ r 2 {\mbox{d}~{}P\over\mbox{d}~{}r}=-{Gm\rho\over r^{2}}
  11. m ( r ) m(r)
  12. r r
  13. d m d r = 4 π r 2 ρ . {\mbox{d}m\over\mbox{d}~{}r}=4\pi r^{2}\rho.
  14. r = 0 r=0
  15. r = R r=R
  16. d l d r = 4 π r 2 ρ ( ϵ - ϵ ν ) {\mbox{d}~{}l\over\mbox{d}~{}r}=4\pi r^{2}\rho(\epsilon-\epsilon_{\nu})
  17. ϵ ν \epsilon_{\nu}
  18. d T d r = - 1 k l 4 π r 2 , {\mbox{d}~{}T\over\mbox{d}~{}r}=-{1\over k}{l\over 4\pi r^{2}},
  19. d T d r = - 3 κ ρ l 64 π r 2 σ T 3 , {\mbox{d}~{}T\over\mbox{d}~{}r}=-{3\kappa\rho l\over 64\pi r^{2}\sigma T^{3}},
  20. κ \kappa
  21. σ \sigma
  22. d T d r = ( 1 - 1 γ ) T P d P d r , {\mbox{d}~{}T\over\mbox{d}~{}r}=\left(1-{1\over\gamma}\right){T\over P}{\mbox{% d}~{}P\over\mbox{d}~{}r},
  23. γ = c p / c v \gamma=c_{p}/c_{v}
  24. γ = 5 / 3 \gamma=5/3
  25. r = R r=R
  26. r = 0 r=0
  27. P ( R ) = 0 P(R)=0
  28. m ( 0 ) = 0 m(0)=0
  29. m ( R ) = M m(R)=M
  30. T ( R ) = T e f f T(R)=T_{eff}

Stern–Gerlach_experiment.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. | ψ = c 1 | ψ j = + 2 + c 2 | ψ j = - 2 |\psi\rangle=c_{1}\left|\psi_{j=+\frac{\hbar}{2}}\right\rangle+c_{2}\left|\psi% _{j=-\frac{\hbar}{2}}\right\rangle
  4. | ψ |\psi\rangle
  5. 1 / 2 {1}/{2}
  6. I X IX

Stevens'_power_law.html

  1. a a
  2. ψ ( I ) = k I a , \psi(I)=kI^{a},\,\!
  3. I I

Stiefel–Whitney_class.html

  1. E E
  2. E E
  3. w ( E ) w(E)
  4. H ( X ; 𝐙 / 2 𝐙 ) = i 0 H i ( X ; 𝐙 / 2 𝐙 ) H^{\ast}(X;\mathbf{Z}/2\mathbf{Z})=\bigoplus_{i\geq 0}H^{i}(X;\mathbf{Z}/2% \mathbf{Z})
  5. X X
  6. E E
  7. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  8. w ( E ) w(E)
  9. i i
  10. E E
  11. w ( E ) w(E)
  12. E E
  13. F F
  14. X X
  15. E E
  16. F F
  17. E E
  18. w ( E ) w(E)
  19. w ( F ) w(F)
  20. E F E→F
  21. E E
  22. F F
  23. w ( E ) w(E)
  24. w ( F ) w(F)
  25. E E
  26. F F
  27. L L
  28. L L
  29. L L
  30. E E
  31. F F
  32. E E
  33. F F
  34. X X
  35. E E
  36. F F
  37. X X
  38. E E
  39. X X
  40. G G
  41. w ( γ 1 1 ) = 1 + a H * ( 𝐏 1 ( 𝐑 ) ; 𝐙 2 ) = 𝐙 2 [ a ] / ( a 2 ) w(\gamma^{1}_{1})=1+a\in H^{*}(\mathbf{P}^{1}(\mathbf{R});\mathbf{Z}_{2})=% \mathbf{Z}_{2}[a]/(a^{2})
  42. w i = 0 H i ( X ) w_{i}=0\in H^{i}(X)
  43. w ( E ) H rank E ( X ) w(E)\in H^{\leq\mathrm{rank}E}(X)
  44. w ( E F ) = w ( E ) w ( F ) w(E\oplus F)=w(E)\smallsmile w(F)
  45. f : X X f:X^{\prime}\to X
  46. G r n = G r n ( 𝐑 ) Gr_{n}=Gr_{n}(\mathbf{R}^{\infty})
  47. γ n G r n \gamma^{n}\to Gr_{n}
  48. W G r n ( V ) W\in Gr_{n}(V)
  49. f * γ n V e c t n ( X ) f^{*}\gamma^{n}\in Vect_{n}(X)
  50. [ X ; G r n ] [X;Gr_{n}]
  51. V e c t n ( X ) Vect_{n}(X)
  52. 𝐏 ( 𝐑 ) = 𝐑 / 𝐑 * \mathbf{P}^{\infty}(\mathbf{R})=\mathbf{R}^{\infty}/\mathbf{R}^{*}
  53. π 1 ( 𝐏 ( 𝐑 ) ) = 𝐙 / 2 𝐙 π i ( 𝐏 ( 𝐑 ) ) = π i ( S ) = 0 i > 1 \begin{aligned}\displaystyle\pi_{1}(\mathbf{P}^{\infty}(\mathbf{R}))&% \displaystyle=\mathbf{Z}/2\mathbf{Z}\\ \displaystyle\pi_{i}(\mathbf{P}^{\infty}(\mathbf{R}))&\displaystyle=\pi_{i}(S^% {\infty})=0&&\displaystyle i>1\end{aligned}
  54. [ X ; 𝐏 ( 𝐑 ) ] = H 1 ( X ; 𝐙 / 2 𝐙 ) [X;\mathbf{P}^{\infty}(\mathbf{R})]=H^{1}(X;\mathbf{Z}/2\mathbf{Z})
  55. H 1 ( 𝐏 ( 𝐑 ) ; 𝐙 / 2 𝐙 ) = 𝐙 / 2 𝐙 H^{1}(\mathbf{P}^{\infty}(\mathbf{R});\mathbf{Z}/2\mathbf{Z})=\mathbf{Z}/2% \mathbf{Z}
  56. s 1 , , s s_{1},\ldots,s_{\ell}
  57. \ell
  58. w k - + 1 = = w k = 0 w_{k-\ell+1}=\cdots=w_{k}=0
  59. γ 1 1 \gamma_{1}^{1}
  60. i * θ 1 ( γ 1 ) = θ 1 ( i * γ 1 ) = θ 1 ( γ 1 1 ) = w 1 ( γ 1 1 ) = w 1 ( i * γ 1 ) = i * w 1 ( γ 1 ) i^{*}\theta_{1}(\gamma^{1})=\theta_{1}(i^{*}\gamma^{1})=\theta_{1}(\gamma_{1}^% {1})=w_{1}(\gamma_{1}^{1})=w_{1}(i^{*}\gamma^{1})=i^{*}w_{1}(\gamma^{1})
  61. θ 1 ( γ 1 ) = w 1 ( γ 1 ) \theta_{1}(\gamma^{1})=w_{1}(\gamma^{1})
  62. f * : H * ( X ; 𝐙 / 2 𝐙 ) ) H * ( X ; 𝐙 / 2 𝐙 ) f^{*}:H^{*}(X;\mathbf{Z}/2\mathbf{Z}))\to H^{*}(X^{\prime};\mathbf{Z}/2\mathbf% {Z})
  63. f * E = λ 1 λ n f^{*}E=\lambda_{1}\oplus\cdots\oplus\lambda_{n}
  64. λ i X \lambda_{i}\to X^{\prime}
  65. V e c t 1 ( X ) Vect_{1}(X)
  66. f * θ ( E ) = θ ( f * E ) = θ ( λ 1 λ n ) = θ ( λ 1 ) θ ( λ n ) = w ( λ 1 ) w ( λ n ) = w ( f * E ) = f * w ( E ) . f^{*}\theta(E)=\theta(f^{*}E)=\theta(\lambda_{1}\oplus\cdots\oplus\lambda_{n})% =\theta(\lambda_{1})\cdots\theta(\lambda_{n})=w(\lambda_{1})\cdots w(\lambda_{% n})=w(f^{*}E)=f^{*}w(E).
  67. e ( T S n ) = χ ( T S n ) [ S n ] = 2 [ S n ] 0 e(TS^{n})=\chi(TS^{n})[S^{n}]=2[S^{n}]\not=0
  68. w 1 3 , w 1 w 2 , w 3 w_{1}^{3},w_{1}w_{2},w_{3}
  69. w 2 w 4 k - 1 . w_{2}w_{4k-1}.
  70. v k x = S q k ( x ) , v_{k}\cup x=Sq^{k}(x),
  71. v k x , μ = S q k ( x ) , μ \langle v_{k}\cup x,\mu\rangle=\langle Sq^{k}(x),\mu\rangle
  72. β w i H i + 1 ( X ; 𝐙 ) \beta w_{i}\in H^{i+1}(X;\mathbf{Z})
  73. β : H i ( X ; 𝐙 / 2 ) H i + 1 ( X ; 𝐙 ) . \beta\colon H^{i}(X;\mathbf{Z}/2)\to H^{i+1}(X;\mathbf{Z}).
  74. w 2 i w_{2^{i}}
  75. S q i ( w j ) = t = 0 i ( j + t - i - 1 t ) w i - t w j + t . Sq^{i}(w_{j})=\sum_{t=0}^{i}{j+t-i-1\choose t}w_{i-t}w_{j+t}.

Stochastic_gradient_descent.html

  1. Q ( w ) = i = 1 n Q i ( w ) , Q(w)=\sum_{i=1}^{n}Q_{i}(w),
  2. w * w^{*}
  3. Q ( w ) Q(w)
  4. Q i Q_{i}
  5. i i
  6. Q i ( w ) Q_{i}(w)
  7. i i
  8. Q ( w ) Q(w)
  9. w := w - η Q ( w ) = w - η i = 1 n Q i ( w ) , w:=w-\eta\nabla Q(w)=w-\eta\sum_{i=1}^{n}\nabla Q_{i}(w),
  10. η \eta
  11. Q ( w ) Q(w)
  12. w := w - η Q i ( w ) . w:=w-\eta\nabla Q_{i}(w).
  13. w w
  14. η \eta
  15. i = 1 , 2 , , n \!i=1,2,...,n
  16. w := w - η Q i ( w ) . \!w:=w-\eta\nabla Q_{i}(w).
  17. η \eta
  18. y = w 1 + w 2 x y=\!w_{1}+w_{2}x
  19. ( x 1 , y 1 ) , , ( x n , y n ) \!(x_{1},y_{1}),\ldots,(x_{n},y_{n})
  20. Q ( w ) = i = 1 n Q i ( w ) = i = 1 n ( w 1 + w 2 x i - y i ) 2 . Q(w)=\sum_{i=1}^{n}Q_{i}(w)=\sum_{i=1}^{n}\left(w_{1}+w_{2}x_{i}-y_{i}\right)^% {2}.
  21. [ w 1 w 2 ] := [ w 1 w 2 ] - η [ 2 ( w 1 + w 2 x i - y i ) 2 x i ( w 1 + w 2 x i - y i ) ] . \begin{bmatrix}w_{1}\\ w_{2}\end{bmatrix}:=\begin{bmatrix}w_{1}\\ w_{2}\end{bmatrix}-\eta\begin{bmatrix}2(w_{1}+w_{2}x_{i}-y_{i})\\ 2x_{i}(w_{1}+w_{2}x_{i}-y_{i})\end{bmatrix}.
  22. t t
  23. k k
  24. Δ w Δw
  25. Δ w := η Q i ( w ) + α Δ w \Delta w:=\eta\nabla Q_{i}(w)+\alpha\Delta w
  26. w := w - η Δ w w:=w-\eta\Delta w
  27. w ¯ = 1 t i = 0 t - 1 w i \bar{w}=\frac{1}{t}\sum_{i=0}^{t-1}w_{i}
  28. w w
  29. η η
  30. G = τ = 1 t g τ g τ 𝖳 G=\sum_{\tau=1}^{t}g_{\tau}g_{\tau}^{\mathsf{T}}
  31. g τ = Q i ( w ) g_{\tau}=\nabla Q_{i}(w)
  32. τ τ
  33. G j , j = τ = 1 t g τ , j 2 G_{j,j}=\sum_{\tau=1}^{t}g_{\tau,j}^{2}
  34. w := w - η diag ( G ) - 1 2 g w:=w-\eta\,\mathrm{diag}(G)^{-\frac{1}{2}}\circ g
  35. w j := w j - η G j , j g j . w_{j}:=w_{j}-\frac{\eta}{\sqrt{}}{G_{j,j}}g_{j}.
  36. G i = τ = 1 t g τ 2 \sqrt{G_{i}}=\sqrt{\sum_{\tau=1}^{t}g_{\tau}^{2}}

Stock_and_flow.html

  1. 3 1 2 3\tfrac{1}{2}
  2. t \,t\,
  3. Q ( t ) \,Q(t)\,
  4. d Q ( t ) d t \,\frac{dQ(t)}{dt}\,
  5. K ( t ) \,K(t)\,
  6. I g ( t ) \,I^{g}(t)\,
  7. D ( t ) \,D(t)\,
  8. d K ( t ) d t = I g ( t ) - D ( t ) = I n ( t ) \frac{dK(t)}{dt}=I^{g}(t)-D(t)=I^{n}(t)
  9. I n ( t ) \,I^{n}(t)\,
  10. Stock A = 0 t - Flow d t \ \,\text{Stock A}=\int_{0}^{t}-\,\text{Flow }\,dt
  11. Stock B = 0 t Flow d t \ \,\text{Stock B}=\int_{0}^{t}\,\text{Flow }\,dt
  12. 1 ) Flow = sin ( 5 t ) 1)\ \,\text{Flow}=\sin(5t)
  13. 2.1 ) Stock A - = Flow 2.1)\ \,\text{Stock A}\ -=\,\text{Flow}
  14. 2.2 ) Stock B + = Flow 2.2)\ \,\text{Stock B}\ +=\,\text{Flow}

Stock_valuation.html

  1. P P
  2. E E
  3. G G
  4. K K
  5. D D
  6. P = ( E * G K 2 ) + ( D K ) P=(\frac{E*G}{K^{2}})+(\frac{D}{K})
  7. K K
  8. P = D i = 1 ( 1 + g 1 + k ) i = D 1 + g k - g P=D\cdot\sum_{i=1}^{\infty}\left(\frac{1+g}{1+k}\right)^{i}=D\cdot\frac{1+g}{k% -g}
  9. P \ P
  10. D \ D
  11. k \ k
  12. g \ g
  13. k = g \ k=g

Stolper–Samuelson_theorem.html

  1. P ( C ) = a r + b w , P(C)=ar+bw,\,
  2. P ( W ) = c r + d w P(W)=cr+dw\,

Stone's_theorem_on_one-parameter_unitary_groups.html

  1. H H
  2. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  3. t 0 𝐑 , ξ H : lim t t 0 U t ξ = U t 0 ξ \forall t_{0}\in\mathbf{R},~{}\xi\in H:\qquad\lim_{t\to t_{0}}U_{t}\xi=U_{t_{0% }}\xi
  4. s , t 𝐑 : U t + s = U t U s . \forall s,t\in\mathbf{R}:\qquad U_{t+s}=U_{t}U_{s}.
  5. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  6. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  7. A A
  8. t 𝐑 : U t = e i t A . \forall t\in\mathbf{R}:\qquad U_{t}=e^{itA}.
  9. A A
  10. H H
  11. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  12. t 𝐑 : U t := e i t A \forall t\in\mathbf{R}:\qquad U_{t}:=e^{itA}
  13. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  14. i A iA
  15. A A
  16. 𝐑 \mathbf{R}
  17. 𝐑 \mathbf{R}
  18. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  19. 𝐑 \mathbf{R}
  20. H H
  21. ρ ρ
  22. H H
  23. f C c ( 𝐑 ) : ρ ( f ) := 𝐑 f ( t ) U t d t , \forall f\in C_{c}(\mathbf{R}):\qquad\rho(f):=\int_{\mathbf{R}}f(t)U_{t}\,dt,
  24. ρ ρ
  25. τ τ
  26. H H
  27. τ τ
  28. 𝐑 \mathbf{R}
  29. A A
  30. i A iA
  31. ( U t ) t 𝐑 (U_{t})_{t\in\mathbf{R}}
  32. 𝐑 \mathbf{R}
  33. ( L 1 ( 𝐑 ) , ) (L^{1}(\mathbf{R}),\star)
  34. ( L 1 ( 𝐑 ) , ) (L^{1}(\mathbf{R}),\star)
  35. ( T t ψ ) ( x ) = ψ ( x + t ) \left(T_{t}\psi\right)(x)=\psi(x+t)
  36. d d x = i 1 i d d x \frac{d}{dx}=i\frac{1}{i}\frac{d}{dx}
  37. 𝐑 \mathbf{R}
  38. T t = e t d d x . T_{t}=e^{t\,\frac{d}{dx}}.
  39. H H
  40. H H
  41. Q , P Q,P
  42. L < s u p > 2 ( 𝐑 ) L<sup>2(\mathbf{R})

Stone–von_Neumann_theorem.html

  1. 𝐑 \mathbf{R}
  2. x x
  3. p p
  4. [ x ψ ] ( x 0 ) = x 0 ψ ( x 0 ) [x\psi](x_{0})=x_{0}\psi(x_{0})
  5. [ p ψ ] ( x 0 ) = - i ψ x ( x 0 ) [p\psi](x_{0})=-i\hbar\frac{\partial\psi}{\partial x}(x_{0})
  6. V V
  7. 𝐑 \mathbf{R}
  8. 2 π
  9. x x
  10. p p
  11. [ x , p ] = x p - p x = i . [x,p]=xp-px=i\hbar.
  12. p p
  13. x x
  14. T r a c e ( A B ) = T r a c e ( B A ) Trace(AB)=Trace(BA)
  15. p p
  16. x x
  17. Q Q
  18. P P
  19. Q Q , P = i QQ,P=i
  20. s s
  21. t t
  22. e i t Q e i s P - e - i s t e i s P e i t Q = 0. e^{itQ}e^{isP}-e^{-ist}e^{isP}e^{itQ}=0.
  23. U ( t ) U(t)
  24. V ( s ) V(s)
  25. U ( t ) U(t)
  26. V ( s ) V(s)
  27. U ( t ) U(t)
  28. V ( s ) V(s)
  29. U ( t ) U(t)
  30. V ( s ) V(s)
  31. H H
  32. W * U ( t ) W = e i t x and W * V ( s ) W = e i s p , W^{*}U(t)W=e^{itx}\quad\mbox{and}~{}\quad W^{*}V(s)W=e^{isp},
  33. p p
  34. x x
  35. W W
  36. U U
  37. x x
  38. P P
  39. P P
  40. Q Q
  41. n n
  42. 2 n 2n
  43. 𝐑 \mathbf{R}
  44. 𝐑 \mathbf{R}
  45. 𝐙 \mathbf{Z}
  46. p p
  47. p p
  48. 𝐙 / p 𝐙 \mathbf{Z}/p\mathbf{Z}
  49. K H H KHH
  50. K R 𝐑 KR\mathbf{R}
  51. K K
  52. K R 𝐑 KR\mathbf{R}
  53. K R 𝐑 KR\mathbf{R}
  54. K R 𝐑 KR\mathbf{R}
  55. K H H A KHH→A
  56. G G
  57. G G
  58. f f ^ ( γ ) = G γ ( t ) ¯ f ( t ) d μ ( t ) f\mapsto{\hat{f}}(\gamma)=\int_{G}\overline{\gamma(t)}f(t)d\mu(t)
  59. C * ( G ) C*(G)
  60. G G
  61. C * ( G ) C*(G)
  62. G G
  63. 𝐑 \mathbf{R}
  64. G G
  65. C C
  66. ρ ρ
  67. s s
  68. G G
  69. f f
  70. ( s f ) ( t ) = f ( t + s ) . (s\cdot f)(t)=f(t+s).
  71. G G
  72. ( s f ) ^ ( γ ) = γ ( s ) f ^ ( γ ) . \widehat{(s\cdot f)}(\gamma)=\gamma(s)\hat{f}(\gamma).
  73. C C
  74. C * ( G ^ ) ρ ^ G C^{*}(\hat{G})\rtimes_{\hat{\rho}}G
  75. U ( s ) U(s)
  76. G G
  77. V ( γ ) V(γ)
  78. U ( s ) V ( γ ) U * ( s ) = γ ( s ) V ( γ ) . U(s)V(\gamma)U^{*}(s)=\gamma(s)V(\gamma).\;
  79. C 0 ( G ) ρ G C_{0}(G)\rtimes_{\rho}G
  80. 𝒦 ( L 2 ( G ) ) {\mathcal{K}}(L^{2}(G))
  81. G = 𝐑 G=\mathbf{R}
  82. P P
  83. Q Q
  84. n n
  85. ( n + 2 ) × ( n + 2 ) (n+ 2) × (n+ 2)
  86. M ( a , b , c ) = [ 1 a c 0 1 n b 0 0 1 ] . \mathrm{M}(a,b,c)=\begin{bmatrix}1&a&c\\ 0&1_{n}&b\\ 0&0&1\end{bmatrix}.
  87. M ( 0 , 0 , c ) M(0, 0,c)
  88. n = 1 n=1
  89. P = [ 0 1 0 0 0 0 0 0 0 ] , Q = [ 0 0 0 0 0 1 0 0 0 ] , z = [ 0 0 1 0 0 0 0 0 0 ] , P=\begin{bmatrix}0&1&0\\ 0&0&0\\ 0&0&0\end{bmatrix},\qquad Q=\begin{bmatrix}0&0&0\\ 0&0&1\\ 0&0&0\end{bmatrix},\qquad z=\begin{bmatrix}0&0&1\\ 0&0&0\\ 0&0&0\end{bmatrix},
  90. z = l o g M ( 0 , 0 , 1 ) = e x p ( z ) 1 z=logM(0,0,1)=exp(z)−1
  91. h h
  92. [ U h ( M ( a , b , c ) ) ] ψ ( x ) = e i ( b x + h c ) ψ ( x + h a ) . \left[U_{h}(\mathrm{M}(a,b,c))\right]\psi(x)=e^{i(b\cdot x+hc)}\psi(x+ha).
  93. h a ha
  94. π π
  95. π π′
  96. π ( z ) = π ( z ) π(z)=π′(z)
  97. z z
  98. h h
  99. α h : M ( a , b , c ) M ( - h - 1 b , h a , c - a b ) \alpha_{h}:\mathrm{M}(a,b,c)\to\mathrm{M}\left(-h^{-1}b,ha,c-ab\right)
  100. W W
  101. g g
  102. W U h ( g ) W * = U h α ( g ) . WU_{h}(g)W^{*}=U_{h}\alpha(g).
  103. W W
  104. W W
  105. W W
  106. W W
  107. 𝐑 n e - i x p e i ( b x + h c ) ψ ( x + h a ) d x = e i ( h a p + h ( c - b a ) ) 𝐑 n e - i y ( p - b ) ψ ( y ) d y . \int_{\mathbf{R}^{n}}e^{-ix\cdot p}e^{i(b\cdot x+hc)}\psi(x+ha)\ dx=e^{i(ha% \cdot p+h(c-b\cdot a))}\int_{\mathbf{R}^{n}}e^{-iy\cdot(p-b)}\psi(y)\ dy.
  108. ( α h ) 2 M ( a , b , c ) = M ( - a , - b , c ) . (\alpha_{h})^{2}\mathrm{M}(a,b,c)=\mathrm{M}(-a,-b,c).
  109. W 1 U h W 1 * = U h α 2 ( g ) W_{1}U_{h}W_{1}^{*}=U_{h}\alpha^{2}(g)
  110. [ W 1 ψ ] ( x ) = ψ ( - x ) . [W_{1}\psi](x)=\psi(-x).
  111. a j = z j , a j * = z j , a_{j}=\frac{\partial}{\partial z_{j}},\qquad a_{j}^{*}=z_{j},
  112. [ a j , a k * ] = δ j , k . \left[a_{j},a_{k}^{*}\right]=\delta_{j,k}.
  113. a [ u s u , u b = , u j , u p = 217 ] a[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}j^{\prime},u^{\prime}p% =\u{2}217^{\prime}]
  114. a [ u s u , u b = , u j , u p = 217 ] a[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}j^{\prime},u^{\prime}p% =\u{2}217^{\prime}]
  115. a [ u s u , u b = , u j , u p = 217 ] a[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}j^{\prime},u^{\prime}p% =\u{2}217^{\prime}]
  116. K K
  117. K = 𝐙 / p 𝐙 K=\mathbf{Z}/p\mathbf{Z}
  118. p p
  119. ω ω
  120. K K
  121. 𝐓 \mathbf{T}
  122. h h
  123. K K
  124. [ U h M ( a , b , c ) ψ ] ( x ) = ω ( b x + h c ) ψ ( x + h a ) . [U_{h}\mathrm{M}(a,b,c)\psi](x)=\omega(b\cdot x+hc)\psi(x+ha).
  125. h h
  126. χ χ
  127. χ ( M ( a , b , c ) ) = { | K | n ω ( h c ) if a = b = 0 0 otherwise \chi(\mathrm{M}(a,b,c))=\begin{cases}|K|^{n}\ \omega(hc)&\,\text{ if }a=b=0\\ 0&\,\text{otherwise}\end{cases}
  128. 1 | H n ( 𝐊 ) | g H n ( K ) | χ ( g ) | 2 = 1 | K | 2 n + 1 | K | 2 n | K | = 1. \frac{1}{|H_{n}(\mathbf{K})|}\sum_{g\in H_{n}(K)}|\chi(g)|^{2}=\frac{1}{|K|^{2% n+1}}|K|^{2n}|K|=1.
  129. H < s u b > n ( 𝐙 / p 𝐙 ) H<sub>n(\mathbf{Z}/p\mathbf{Z})
  130. x < s u p > n t ; n , p = i n x n 1 x<sup>nt;n,p=iℏnx^{n− 1}

Storage_tube.html

  1. V c r 1 V_{cr1}
  2. V c r 1 V_{cr1}
  3. V c r 1 V_{cr1}
  4. V c r 2 V_{cr2}
  5. V c r 1 V_{cr1}
  6. V c r 2 V_{cr2}
  7. V c r 1 V_{cr1}
  8. V c r 2 V_{cr2}
  9. V c r 1 V_{cr1}
  10. V c r 2 V_{cr2}
  11. V c r 1 V_{cr1}
  12. V c r 1 V_{cr1}
  13. V c r 2 V_{cr2}
  14. V c r 2 V_{cr2}

Strassen_algorithm.html

  1. 𝐂 = 𝐀𝐁 𝐀 , 𝐁 , 𝐂 R 2 n × 2 n \mathbf{C}=\mathbf{A}\mathbf{B}\qquad\mathbf{A},\mathbf{B},\mathbf{C}\in R^{2^% {n}\times 2^{n}}
  2. 𝐀 = [ 𝐀 1 , 1 𝐀 1 , 2 𝐀 2 , 1 𝐀 2 , 2 ] , 𝐁 = [ 𝐁 1 , 1 𝐁 1 , 2 𝐁 2 , 1 𝐁 2 , 2 ] , 𝐂 = [ 𝐂 1 , 1 𝐂 1 , 2 𝐂 2 , 1 𝐂 2 , 2 ] \mathbf{A}=\begin{bmatrix}\mathbf{A}_{1,1}&\mathbf{A}_{1,2}\\ \mathbf{A}_{2,1}&\mathbf{A}_{2,2}\end{bmatrix}\mbox{ , }\mathbf{B}=\begin{% bmatrix}\mathbf{B}_{1,1}&\mathbf{B}_{1,2}\\ \mathbf{B}_{2,1}&\mathbf{B}_{2,2}\end{bmatrix}\mbox{ , }\mathbf{C}=\begin{% bmatrix}\mathbf{C}_{1,1}&\mathbf{C}_{1,2}\\ \mathbf{C}_{2,1}&\mathbf{C}_{2,2}\end{bmatrix}
  3. 𝐀 i , j , 𝐁 i , j , 𝐂 i , j R 2 n - 1 × 2 n - 1 \mathbf{A}_{i,j},\mathbf{B}_{i,j},\mathbf{C}_{i,j}\in R^{2^{n-1}\times 2^{n-1}}
  4. 𝐂 1 , 1 = 𝐀 1 , 1 𝐁 1 , 1 + 𝐀 1 , 2 𝐁 2 , 1 \mathbf{C}_{1,1}=\mathbf{A}_{1,1}\mathbf{B}_{1,1}+\mathbf{A}_{1,2}\mathbf{B}_{% 2,1}
  5. 𝐂 1 , 2 = 𝐀 1 , 1 𝐁 1 , 2 + 𝐀 1 , 2 𝐁 2 , 2 \mathbf{C}_{1,2}=\mathbf{A}_{1,1}\mathbf{B}_{1,2}+\mathbf{A}_{1,2}\mathbf{B}_{% 2,2}
  6. 𝐂 2 , 1 = 𝐀 2 , 1 𝐁 1 , 1 + 𝐀 2 , 2 𝐁 2 , 1 \mathbf{C}_{2,1}=\mathbf{A}_{2,1}\mathbf{B}_{1,1}+\mathbf{A}_{2,2}\mathbf{B}_{% 2,1}
  7. 𝐂 2 , 2 = 𝐀 2 , 1 𝐁 1 , 2 + 𝐀 2 , 2 𝐁 2 , 2 \mathbf{C}_{2,2}=\mathbf{A}_{2,1}\mathbf{B}_{1,2}+\mathbf{A}_{2,2}\mathbf{B}_{% 2,2}
  8. 𝐌 1 := ( 𝐀 1 , 1 + 𝐀 2 , 2 ) ( 𝐁 1 , 1 + 𝐁 2 , 2 ) \mathbf{M}_{1}:=(\mathbf{A}_{1,1}+\mathbf{A}_{2,2})(\mathbf{B}_{1,1}+\mathbf{B% }_{2,2})
  9. 𝐌 2 := ( 𝐀 2 , 1 + 𝐀 2 , 2 ) 𝐁 1 , 1 \mathbf{M}_{2}:=(\mathbf{A}_{2,1}+\mathbf{A}_{2,2})\mathbf{B}_{1,1}
  10. 𝐌 3 := 𝐀 1 , 1 ( 𝐁 1 , 2 - 𝐁 2 , 2 ) \mathbf{M}_{3}:=\mathbf{A}_{1,1}(\mathbf{B}_{1,2}-\mathbf{B}_{2,2})
  11. 𝐌 4 := 𝐀 2 , 2 ( 𝐁 2 , 1 - 𝐁 1 , 1 ) \mathbf{M}_{4}:=\mathbf{A}_{2,2}(\mathbf{B}_{2,1}-\mathbf{B}_{1,1})
  12. 𝐌 5 := ( 𝐀 1 , 1 + 𝐀 1 , 2 ) 𝐁 2 , 2 \mathbf{M}_{5}:=(\mathbf{A}_{1,1}+\mathbf{A}_{1,2})\mathbf{B}_{2,2}
  13. 𝐌 6 := ( 𝐀 2 , 1 - 𝐀 1 , 1 ) ( 𝐁 1 , 1 + 𝐁 1 , 2 ) \mathbf{M}_{6}:=(\mathbf{A}_{2,1}-\mathbf{A}_{1,1})(\mathbf{B}_{1,1}+\mathbf{B% }_{1,2})
  14. 𝐌 7 := ( 𝐀 1 , 2 - 𝐀 2 , 2 ) ( 𝐁 2 , 1 + 𝐁 2 , 2 ) \mathbf{M}_{7}:=(\mathbf{A}_{1,2}-\mathbf{A}_{2,2})(\mathbf{B}_{2,1}+\mathbf{B% }_{2,2})
  15. 𝐂 1 , 1 = 𝐌 1 + 𝐌 4 - 𝐌 5 + 𝐌 7 \mathbf{C}_{1,1}=\mathbf{M}_{1}+\mathbf{M}_{4}-\mathbf{M}_{5}+\mathbf{M}_{7}
  16. 𝐂 1 , 2 = 𝐌 3 + 𝐌 5 \mathbf{C}_{1,2}=\mathbf{M}_{3}+\mathbf{M}_{5}
  17. 𝐂 2 , 1 = 𝐌 2 + 𝐌 4 \mathbf{C}_{2,1}=\mathbf{M}_{2}+\mathbf{M}_{4}
  18. 𝐂 2 , 2 = 𝐌 1 - 𝐌 2 + 𝐌 3 + 𝐌 6 \mathbf{C}_{2,2}=\mathbf{M}_{1}-\mathbf{M}_{2}+\mathbf{M}_{3}+\mathbf{M}_{6}
  19. f ( n ) f(n)
  20. l l
  21. O ( [ 7 + o ( 1 ) ] n ) = O ( N log 2 7 + o ( 1 ) ) O ( N 2.8074 ) O([7+o(1)]^{n})=O(N^{\log_{2}7+o(1)})\approx O(N^{2.8074})
  22. ϕ : 𝐀 × 𝐁 𝐂 \phi:\mathbf{A}\times\mathbf{B}\rightarrow\mathbf{C}
  23. R ( ϕ / 𝐅 ) = min { r | f i 𝐀 * , g i 𝐁 * , w i 𝐂 , 𝐚 𝐀 , 𝐛 𝐁 , ϕ ( 𝐚 , 𝐛 ) = i = 1 r f i ( 𝐚 ) g i ( 𝐛 ) w i } R(\phi/\mathbf{F})=\min\left\{r\left|\exists f_{i}\in\mathbf{A}^{*},g_{i}\in% \mathbf{B}^{*},w_{i}\in\mathbf{C},\forall\mathbf{a}\in\mathbf{A},\mathbf{b}\in% \mathbf{B},\phi(\mathbf{a},\mathbf{b})=\sum_{i=1}^{r}f_{i}(\mathbf{a})g_{i}(% \mathbf{b})w_{i}\right.\right\}
  24. [ 1 0 0 0 ] : 𝐚 \begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\mathbf{a}
  25. [ 1 0 0 0 ] : 𝐛 \begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\mathbf{b}
  26. [ 1 0 0 0 ] \begin{bmatrix}1&0\\ 0&0\end{bmatrix}
  27. [ 1 0 0 1 ] : 𝐚 \begin{bmatrix}1&0\\ 0&1\end{bmatrix}:\mathbf{a}
  28. [ 1 0 0 1 ] : 𝐛 \begin{bmatrix}1&0\\ 0&1\end{bmatrix}:\mathbf{b}
  29. [ 1 0 0 1 ] \begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  30. [ 0 1 0 0 ] : 𝐚 \begin{bmatrix}0&1\\ 0&0\end{bmatrix}:\mathbf{a}
  31. [ 0 0 1 0 ] : 𝐛 \begin{bmatrix}0&0\\ 1&0\end{bmatrix}:\mathbf{b}
  32. [ 1 0 0 0 ] \begin{bmatrix}1&0\\ 0&0\end{bmatrix}
  33. [ 0 0 1 1 ] : 𝐚 \begin{bmatrix}0&0\\ 1&1\end{bmatrix}:\mathbf{a}
  34. [ 1 0 0 0 ] : 𝐛 \begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\mathbf{b}
  35. [ 0 0 1 - 1 ] \begin{bmatrix}0&0\\ 1&-1\end{bmatrix}
  36. [ 1 0 0 0 ] : 𝐚 \begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\mathbf{a}
  37. [ 0 1 0 0 ] : 𝐛 \begin{bmatrix}0&1\\ 0&0\end{bmatrix}:\mathbf{b}
  38. [ 0 1 0 0 ] \begin{bmatrix}0&1\\ 0&0\end{bmatrix}
  39. [ 1 0 0 0 ] : 𝐚 \begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\mathbf{a}
  40. [ 0 1 0 - 1 ] : 𝐛 \begin{bmatrix}0&1\\ 0&-1\end{bmatrix}:\mathbf{b}
  41. [ 0 1 0 1 ] \begin{bmatrix}0&1\\ 0&1\end{bmatrix}
  42. [ 0 1 0 0 ] : 𝐚 \begin{bmatrix}0&1\\ 0&0\end{bmatrix}:\mathbf{a}
  43. [ 0 0 0 1 ] : 𝐛 \begin{bmatrix}0&0\\ 0&1\end{bmatrix}:\mathbf{b}
  44. [ 0 1 0 0 ] \begin{bmatrix}0&1\\ 0&0\end{bmatrix}
  45. [ 0 0 0 1 ] : 𝐚 \begin{bmatrix}0&0\\ 0&1\end{bmatrix}:\mathbf{a}
  46. [ - 1 0 1 0 ] : 𝐛 \begin{bmatrix}-1&0\\ 1&0\end{bmatrix}:\mathbf{b}
  47. [ 1 0 1 0 ] \begin{bmatrix}1&0\\ 1&0\end{bmatrix}
  48. [ 0 0 1 0 ] : 𝐚 \begin{bmatrix}0&0\\ 1&0\end{bmatrix}:\mathbf{a}
  49. [ 1 0 0 0 ] : 𝐛 \begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\mathbf{b}
  50. [ 0 0 1 0 ] \begin{bmatrix}0&0\\ 1&0\end{bmatrix}
  51. [ 1 1 0 0 ] : 𝐚 \begin{bmatrix}1&1\\ 0&0\end{bmatrix}:\mathbf{a}
  52. [ 0 0 0 1 ] : 𝐛 \begin{bmatrix}0&0\\ 0&1\end{bmatrix}:\mathbf{b}
  53. [ - 1 1 0 0 ] \begin{bmatrix}-1&1\\ 0&0\end{bmatrix}
  54. [ 0 0 0 1 ] : 𝐚 \begin{bmatrix}0&0\\ 0&1\end{bmatrix}:\mathbf{a}
  55. [ 0 0 1 0 ] : 𝐛 \begin{bmatrix}0&0\\ 1&0\end{bmatrix}:\mathbf{b}
  56. [ 0 0 1 0 ] \begin{bmatrix}0&0\\ 1&0\end{bmatrix}
  57. [ - 1 0 1 0 ] : 𝐚 \begin{bmatrix}-1&0\\ 1&0\end{bmatrix}:\mathbf{a}
  58. [ 1 1 0 0 ] : 𝐛 \begin{bmatrix}1&1\\ 0&0\end{bmatrix}:\mathbf{b}
  59. [ 0 0 0 1 ] \begin{bmatrix}0&0\\ 0&1\end{bmatrix}
  60. [ 0 0 1 0 ] : 𝐚 \begin{bmatrix}0&0\\ 1&0\end{bmatrix}:\mathbf{a}
  61. [ 0 1 0 0 ] : 𝐛 \begin{bmatrix}0&1\\ 0&0\end{bmatrix}:\mathbf{b}
  62. [ 0 0 0 1 ] \begin{bmatrix}0&0\\ 0&1\end{bmatrix}
  63. [ 0 1 0 - 1 ] : 𝐚 \begin{bmatrix}0&1\\ 0&-1\end{bmatrix}:\mathbf{a}
  64. [ 0 0 1 1 ] : 𝐛 \begin{bmatrix}0&0\\ 1&1\end{bmatrix}:\mathbf{b}
  65. [ 1 0 0 0 ] \begin{bmatrix}1&0\\ 0&0\end{bmatrix}
  66. [ 0 0 0 1 ] : 𝐚 \begin{bmatrix}0&0\\ 0&1\end{bmatrix}:\mathbf{a}
  67. [ 0 0 0 1 ] : 𝐛 \begin{bmatrix}0&0\\ 0&1\end{bmatrix}:\mathbf{b}
  68. [ 0 0 0 1 ] \begin{bmatrix}0&0\\ 0&1\end{bmatrix}
  69. 𝐚𝐛 = i = 1 8 f i ( 𝐚 ) g i ( 𝐛 ) w i \mathbf{a}\mathbf{b}=\sum_{i=1}^{8}f_{i}(\mathbf{a})g_{i}(\mathbf{b})w_{i}
  70. 𝐚𝐛 = i = 1 7 f i ( 𝐚 ) g i ( 𝐛 ) w i \mathbf{a}\mathbf{b}=\sum_{i=1}^{7}f_{i}(\mathbf{a})g_{i}(\mathbf{b})w_{i}
  71. L = Θ ( R ) L=\Theta(R)
  72. 1 2 R L R . \frac{1}{2}R\leq L\leq R.
  73. O ( n 2 ) O(n^{2})
  74. M 2 M_{2}
  75. ( A 2 , 1 + A 2 , 2 ) (A_{2,1}+A_{2,2})
  76. A 2 , 2 A_{2,2}
  77. A 2 , 1 A_{2,1}
  78. O ( n 2 ) O(n^{2})
  79. O ( n log 2 3 ) O(n^{\log_{2}3})
  80. O ( n 2 ) O(n^{2})

Stress_concentration.html

  1. 2 a 2a
  2. 2 b 2b
  3. σ \sigma
  4. σ m a x = σ ( 1 + 2 a b ) = σ ( 1 + 2 a ρ ) \sigma_{max}=\sigma\left(1+2\cfrac{a}{b}\right)=\sigma\left(1+2\sqrt{\cfrac{a}% {\rho}}\right)
  5. σ m a x \sigma_{max}
  6. σ \sigma

Strict_conditional.html

  1. ( p q ) \Box(p\rightarrow q)
  2. \Box
  3. \Box

Strong_CP_problem.html

  1. = - 1 4 F μ ν F μ ν - n f g 2 θ 32 π 2 F μ ν F ~ μ ν + ψ ¯ ( i γ μ D μ - m e i θ γ 5 ) ψ {\mathcal{L}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{n_{f}g^{2}\theta}{32\pi^{% 2}}F_{\mu\nu}\tilde{F}^{\mu\nu}+\bar{\psi}(i\gamma^{\mu}D_{\mu}-me^{i\theta^{% \prime}\gamma_{5}})\psi
  2. θ ~ \scriptstyle{\tilde{\theta}}

Strong_generating_set.html

  1. G S n G\leq S_{n}
  2. { 1 , 2 , , n } . \{1,2,\ldots,n\}.
  3. B = ( β 1 , β 2 , , β r ) B=(\beta_{1},\beta_{2},\ldots,\beta_{r})
  4. β i { 1 , 2 , , n } , \beta_{i}\in\{1,2,\ldots,n\},
  5. B B
  6. B B
  7. G G
  8. B i = ( β 1 , β 2 , , β i ) , B_{i}=(\beta_{1},\beta_{2},\ldots,\beta_{i}),\,
  9. G ( i ) G^{(i)}
  10. B i B_{i}
  11. B B
  12. S G S\subseteq G
  13. S G ( i ) = G ( i ) \langle S\cap G^{(i)}\rangle=G^{(i)}
  14. i i
  15. 1 i r 1\leq i\leq r
  16. G ( i ) G ( j ) G^{(i)}\neq G^{(j)}
  17. i j i\neq j

Strong_pseudoprime.html

  1. a d 1 mod n a^{d}\equiv 1\mod n
  2. a d 2 r - 1 mod n for some 0 r < s . a^{d\cdot 2^{r}}\equiv-1\mod n\quad\mbox{ for some }~{}0\leq r<s.
  3. < v a r > a ± 1 m o d < v a r > n < / v a r > <var>a≡±1mod<var>n</var>

Strongly_interacting_massive_particle.html

  1. Ω \Omega

Strömgren_sphere.html

  1. n e = n p n_{e}=n_{p}
  2. N R N_{R}
  3. N R = n = 2 N n N_{R}=\sum_{n=2}^{\infty}N_{n}
  4. N n N_{n}
  5. N n N_{n}
  6. n e = n p n_{e}=n_{p}
  7. N n = n e n p β n ( T e ) = n e 2 β n ( T e ) N_{n}=n_{e}n_{p}\beta_{n}(T_{e})=n_{e}^{2}\beta_{n}(T_{e})
  8. β n ( T e ) \beta_{n}(T_{e})
  9. T e T_{e}
  10. N R = n e 2 β 2 ( T e ) N_{R}=n_{e}^{2}\beta_{2}(T_{e})
  11. β 2 ( T e ) \beta_{2}(T_{e})
  12. β 2 ( T e ) 2 × 10 - 16 T e - 3 / 4 [ m 3 s - 1 ] \beta_{2}(T_{e})\approx 2\times 10^{-16}T_{e}^{-3/4}\ \mathrm{[m^{3}s^{-1}]}
  13. n n
  14. 0 x 1 0\leq x\leq 1
  15. n e = x n n_{e}=xn
  16. n h = ( 1 - x ) n n_{h}=(1-x)n
  17. α 0 \alpha_{0}
  18. J J
  19. N I N_{I}
  20. N I = α 0 n h J N_{I}=\alpha_{0}n_{h}J
  21. J J
  22. S * S_{*}
  23. α 0 n h J ( r ) = 3 S * 4 π r 3 \alpha_{0}n_{h}J(r)=\frac{3S_{*}}{4\pi r^{3}}
  24. R S R_{S}
  25. 4 π 3 ( n x ) 2 β 2 R S 3 = S * \frac{4\pi}{3}(nx)^{2}\beta_{2}R_{S}^{3}=S_{*}
  26. R S = ( 3 4 π S * n 2 β 2 ) 1 3 R_{S}=\left(\frac{3}{4\pi}\frac{S_{*}}{n^{2}\beta_{2}}\right)^{\frac{1}{3}}

Structural_proof_theory.html

  1. A 1 , , A m B 1 , , B n A_{1},\dots,A_{m}\vdash B_{1},\dots,B_{n}

Structural_rule.html

  1. Γ Σ Γ , A Σ \frac{\Gamma\vdash\Sigma}{\Gamma,A\vdash\Sigma}
  2. Γ Σ Γ A , Σ \frac{\Gamma\vdash\Sigma}{\Gamma\vdash A,\Sigma}
  3. Γ , A , A Σ Γ , A Σ \frac{\Gamma,A,A\vdash\Sigma}{\Gamma,A\vdash\Sigma}
  4. Γ A , A , Σ Γ A , Σ \frac{\Gamma\vdash A,A,\Sigma}{\Gamma\vdash A,\Sigma}
  5. Γ 1 , A , Γ 2 , B , Γ 3 Σ Γ 1 , B , Γ 2 , A , Γ 3 Σ \frac{\Gamma_{1},A,\Gamma_{2},B,\Gamma_{3}\vdash\Sigma}{\Gamma_{1},B,\Gamma_{2% },A,\Gamma_{3}\vdash\Sigma}
  6. Γ Σ 1 , A , Σ 2 , B , Σ 3 Γ Σ 1 , B , Σ 2 , A , Σ 3 \frac{\Gamma\vdash\Sigma_{1},A,\Sigma_{2},B,\Sigma_{3}}{\Gamma\vdash\Sigma_{1}% ,B,\Sigma_{2},A,\Sigma_{3}}

Structuring_element.html

  1. 3 × 3 3\times 3
  2. 21 × 21 21\times 21

Sub-Riemannian_manifold.html

  1. M M
  2. M M
  3. H ( M ) T ( M ) H(M)\subset T(M)
  4. H ( M ) T ( M ) H(M)\subset T(M)
  5. γ \gamma
  6. M M
  7. γ ˙ ( t ) H γ ( t ) ( M ) \dot{\gamma}(t)\in H_{\gamma(t)}(M)
  8. t t
  9. H ( M ) H(M)
  10. x M x\in M
  11. A ( x ) , [ A , B ] ( x ) , [ A , [ B , C ] ] ( x ) , [ A , [ B , [ C , D ] ] ] ( x ) , T x ( M ) A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\ldots\in T_{x}(M)
  12. A , B , C , D , A,B,C,D,\dots
  13. ( M , H , g ) (M,H,g)
  14. M M
  15. H H
  16. g g
  17. H H
  18. d ( x , y ) = inf 0 1 g ( γ ˙ ( t ) , γ ˙ ( t ) ) d t , d(x,y)=\inf\int_{0}^{1}\sqrt{g(\dot{\gamma}(t),\dot{\gamma}(t))}\,dt,
  19. γ : [ 0 , 1 ] M \gamma:[0,1]\to M
  20. γ ( 0 ) = x \gamma(0)=x
  21. γ ( 1 ) = y \gamma(1)=y
  22. x x
  23. y y
  24. α \alpha
  25. 2 × S 1 . \mathbb{R}^{2}\times S^{1}.
  26. 2 × S 1 . \mathbb{R}^{2}\times S^{1}.
  27. α \alpha
  28. β \beta
  29. { α , β , [ α , β ] } \{\alpha,\beta,[\alpha,\beta]\}
  30. H H
  31. α \alpha
  32. β \beta
  33. H H

Subadditivity.html

  1. f : A B f\colon A\to B
  2. x , y A , f ( x + y ) f ( x ) + f ( y ) . \forall x,y\in A,f(x+y)\leq f(x)+f(y).
  3. x , y 0 \forall x,y\geq 0
  4. x + y x + y . \sqrt{x+y}\leq\sqrt{x}+\sqrt{y}.
  5. { a n } , n 1 \left\{a_{n}\right\},n\geq 1
  6. ( 1 ) a n + m a n + a m (1)\qquad a_{n+m}\leq a_{n}+a_{m}
  7. { a n } n = 1 {\left\{a_{n}\right\}}_{n=1}^{\infty}
  8. lim n a n n \displaystyle\lim_{n\to\infty}\frac{a_{n}}{n}
  9. inf a n n \inf\frac{a_{n}}{n}
  10. - -\infty
  11. a n + m a n + a m . a_{n+m}\geq a_{n}+a_{m}.
  12. a n = log n ! a_{n}=\log n!
  13. f ( x ) f ( x + y ) - f ( y ) f(x)\geq f(x+y)-f(y)
  14. f ( 0 ) f ( 0 + y ) - f ( y ) = 0 f(0)\geq f(0+y)-f(y)=0
  15. f : [ 0 , ) [ 0 , ) f:[0,\infty)\to[0,\infty)
  16. f ( 0 ) 0 f(0)\geq 0
  17. f ( x ) y x + y f ( 0 ) + x x + y f ( x + y ) f(x)\geq\textstyle{\frac{y}{x+y}}f(0)+\textstyle{\frac{x}{x+y}}f(x+y)
  18. f ( x ) f(x)
  19. f ( y ) f(y)

Subbayya_Sivasankaranarayana_Pillai.html

  1. k 6 k\geq 6
  2. ( 3 k + 1 ) / ( 2 k - 1 ) [ 1.5 k ] + 1 (3^{k}+1)/(2^{k}-1)\leq[1.5^{k}]+1
  3. k 7. k\geq 7.
  4. g ( k ) = 2 k + l - 2 g(k)=2^{k}+l-2
  5. l l
  6. ( 3 / 2 ) k \leq(3/2)^{k}
  7. g ( 6 ) = 73 g(6)=73

Subgroup_growth.html

  1. s n ( G ) s_{n}^{\triangleleft}(G)
  2. ζ G , p ( s ) = ν = 0 s p n ( G ) p - n s \zeta_{G,p}(s)=\sum_{\nu=0}^{\infty}s_{p^{n}}(G)p^{-ns}
  3. [ x , y ] = z , [ x , z ] = [ y , z ] = 1. [x,y]=z,[x,z]=[y,z]=1.
  4. ( a , b , c ) ( a , b , c ) = ( a + a , b + b , c + c + a b ) . (a,b,c)\circ(a^{\prime},b^{\prime},c^{\prime})=(a+a^{\prime},b+b^{\prime},c+c^% {\prime}+ab^{\prime}).
  5. G = x , y , z y , z z 1 G=\langle x,y,z\rangle\triangleright\langle y,z\rangle\triangleright\langle z% \rangle\triangleright 1
  6. g 2 y , z , g 3 z g_{2}\in\langle y,z\rangle,g_{3}\in\langle z\rangle
  7. ζ G , p ( s ) = 1 ( 1 - p - 1 ) 3 | a 11 | p s - 1 | a 22 | p s - 2 | a 33 | p s - 3 d μ , \zeta_{G,p}(s)=\frac{1}{(1-p^{-1})^{3}}\int_{\mathcal{M}}|a_{11}|_{p}^{s-1}|a_% {22}|_{p}^{s-2}|a_{33}|_{p}^{s-3}\;d\mu,
  8. | | p |\cdot|_{p}
  9. \mathcal{M}
  10. { a 11 , a 12 , a 13 , a 22 , a 23 , a 33 } \{a_{11},a_{12},a_{13},a_{22},a_{23},a_{33}\}
  11. { x a 11 y a 12 z a 13 , y a 22 z a 23 , z a 33 } \{x^{a_{11}}y^{a_{12}}z^{a_{13}},y^{a_{22}}z^{a_{23}},z^{a_{33}}\}
  12. a 33 | a 11 a 22 a_{33}|a_{11}\cdot a_{22}
  13. ζ G , p ( s ) = a 0 b 0 c = 0 a + b p - a s - b ( s - 1 ) - c ( s - 2 ) = 1 - p 3 - 3 s ( 1 - p - s ) ( 1 - p 1 - s ) ( 1 - p 2 - 2 s ) ( 1 - p 2 - 3 s ) \zeta_{G,p}(s)=\sum_{a\geq 0}\sum_{b\geq 0}\sum_{c=0}^{a+b}p^{-as-b(s-1)-c(s-2% )}=\frac{1-p^{3-3s}}{(1-p^{-s})(1-p^{1-s})(1-p^{2-2s})(1-p^{2-3s})}
  14. ζ G ( s ) = ζ ( s ) ζ ( s - 1 ) ζ ( 2 s - 2 ) ζ ( 2 s - 3 ) ζ ( 3 s - 3 ) . \zeta_{G}(s)=\frac{\zeta(s)\zeta(s-1)\zeta(2s-2)\zeta(2s-3)}{\zeta(3s-3)}.
  15. ζ G , p ( s ) \zeta_{G,p}(s)
  16. s = 1 \Re s=1
  17. s = α \Re s=\alpha
  18. n x s n ( G ) x α log k x \sum_{n\leq x}s_{n}(G)\sim x^{\alpha}\log^{k}x
  19. g ( h U ) = ( g h ) U . g(hU)=(gh)U.
  20. G / U G/U
  21. G / U G/U
  22. { 1 , , n } , \{1,\ldots,n\},
  23. { 2 , , n } \{2,\ldots,n\}
  24. ( n - 1 ) ! (n-1)!
  25. s n ( G ) s_{n}(G)
  26. ( n - 1 ) ! (n-1)!
  27. s n ( G ) = h n ( G ) ( n - 1 ) ! - ν = 1 n - 1 h n - ν ( G ) s ν ( G ) ( n - ν ) ! , s_{n}(G)=\frac{h_{n}(G)}{(n-1)!}-\sum_{\nu=1}^{n-1}\frac{h_{n-\nu}(G)s_{\nu}(G% )}{(n-\nu)!},
  28. h n ( G ) h_{n}(G)
  29. φ : G S n . \varphi:G\rightarrow S_{n}.
  30. h n ( G ) h_{n}(G)
  31. s n ( G ) s_{n}(G)
  32. h n ( G ) h_{n}(G)
  33. s n ( G ) s_{n}(G)
  34. F 2 F_{2}
  35. F 2 F_{2}
  36. F 2 S n , F_{2}\rightarrow S_{n},
  37. h n ( F 2 ) = ( n ! ) 2 . h_{n}(F_{2})=(n!)^{2}.
  38. s n ( F 2 ) n n ! . s_{n}(F_{2})\sim n\cdot n!.
  39. h n ( G ) h_{n}(G)

Subjective_expected_utility.html

  1. { x i } \{x_{i}\}
  2. u ( x i ) , u(x_{i}),
  3. P ( x i ) . P(x_{i}).
  4. i u ( x i ) P ( x i ) . \sum_{i}\;u(x_{i})\;P(x_{i}).
  5. { x i } \{x_{i}\}
  6. { y j } , \{y_{j}\},
  7. j u ( y j ) P ( y j ) . \sum_{j}\;u(y_{j})\;P(y_{j}).
  8. x ( = { x i } ) x(=\{x_{i}\})
  9. y ( = { y i } ) y(=\{y_{i}\})
  10. s ( = { s i } ) s(=\{s_{i}\})
  11. t ( = { t i } ) t(=\{t_{i}\})
  12. λ x + ( 1 - λ ) s \lambda x+(1-\lambda)s
  13. λ y + ( 1 - λ ) t \lambda y+(1-\lambda)t
  14. 0 < λ < 1 0<\lambda<1

Subspace_topology.html

  1. ( X , τ ) (X,\tau)
  2. S S
  3. X X
  4. S S
  5. τ S = { S U U τ } . \tau_{S}=\{S\cap U\mid U\in\tau\}.
  6. S S
  7. S S
  8. ( X , τ ) (X,\tau)
  9. S S
  10. ( X , τ ) (X,\tau)
  11. S S
  12. X X
  13. ι : S X \iota:S\hookrightarrow X
  14. ι \iota
  15. S S
  16. X X
  17. S S
  18. ι \iota
  19. ι - 1 ( U ) \iota^{-1}(U)
  20. U U
  21. X X
  22. S S
  23. X X
  24. ι \iota
  25. S S
  26. ι \iota
  27. S S
  28. X X
  29. ι \iota
  30. S S
  31. X X
  32. ( X , τ ) (X,\tau)
  33. S S
  34. X X
  35. S S
  36. X X
  37. X X
  38. ( S , τ S ) (S,\tau_{S})
  39. ( X , τ ) (X,\tau)
  40. S S
  41. X X
  42. ( S , τ S ) (S,\tau_{S})
  43. ( X , τ ) (X,\tau)
  44. S τ S\in\tau
  45. S S
  46. \mathbb{R}
  47. \mathbb{R}
  48. \mathbb{Q}
  49. \mathbb{R}
  50. \mathbb{Q}
  51. \mathbb{R}
  52. \mathbb{R}
  53. \mathbb{R}
  54. \mathbb{R}
  55. \mathbb{R}
  56. \mathbb{R}
  57. \mathbb{R}
  58. Y Y
  59. X X
  60. i : Y X i:Y\to X
  61. Z Z
  62. f : Z Y f:Z\to Y
  63. i f i\circ f
  64. Y Y
  65. S S
  66. X X
  67. f : X Y f:X\to Y
  68. S S
  69. f : X Y f:X\to Y
  70. f : X f ( X ) f:X\to f(X)
  71. S S
  72. S S
  73. X X
  74. A A
  75. S S
  76. A A
  77. X X
  78. A A
  79. S S
  80. X X
  81. S S
  82. X X
  83. S τ S\in\tau
  84. S S
  85. S S
  86. X X
  87. S S
  88. X X
  89. X S τ X\setminus S\in\tau
  90. S S
  91. S S
  92. X X
  93. B B
  94. X X
  95. B S = { U S : U B } B_{S}=\{U\cap S:U\in B\}
  96. S S

Suffix_tree.html

  1. S S
  2. S S
  3. S S
  4. O ( n log n ) O(n\log n)
  5. S S
  6. n n
  7. S S
  8. n n
  9. n n
  10. S S
  11. χ α \chi\alpha
  12. χ \chi
  13. α \alpha
  14. α \alpha
  15. S S
  16. n n
  17. Θ ( n ) \Theta(n)
  18. O ( n ) O(n)
  19. O ( n log n ) O(n\log n)
  20. S S
  21. n n
  22. D = { S 1 , S 2 , , S K } D=\{S_{1},S_{2},\dots,S_{K}\}
  23. n = | n 1 | + | n 2 | + + | n K | n=|n_{1}|+|n_{2}|+\cdots+|n_{K}|
  24. P P
  25. m m
  26. O ( m ) O(m)
  27. P 1 , , P q P_{1},\dots,P_{q}
  28. m m
  29. O ( m ) O(m)
  30. z z
  31. P 1 , , P q P_{1},\dots,P_{q}
  32. m m
  33. O ( m + z ) O(m+z)
  34. n n
  35. P P
  36. P [ i m ] P[i\dots m]
  37. D D
  38. Θ ( m ) \Theta(m)
  39. P P
  40. S i S_{i}
  41. S j S_{j}
  42. Θ ( n i + n j ) \Theta(n_{i}+n_{j})
  43. Θ ( n + z ) \Theta(n+z)
  44. Θ ( n ) \Theta(n)
  45. Θ ( n ) \Theta(n)
  46. Θ ( n ) \Theta(n)
  47. Σ \Sigma
  48. D D
  49. O ( n + z ) O(n+z)
  50. z z
  51. Θ ( n ) \Theta(n)
  52. i i
  53. S i S_{i}
  54. D D
  55. Θ ( n ) \Theta(n)
  56. Θ ( n ) \Theta(n)
  57. S i [ p . . n i ] S_{i}[p..n_{i}]
  58. S j [ q . . n j ] S_{j}[q..n_{j}]
  59. Θ ( 1 ) \Theta(1)
  60. O ( k n + z ) O(kn+z)
  61. z z
  62. Θ ( n ) \Theta(n)
  63. Θ ( g n ) \Theta(gn)
  64. g g
  65. Θ ( k n ) \Theta(kn)
  66. k k
  67. z z
  68. O ( n log n + z ) O(n\log n+z)
  69. O ( k n log ( n / k ) + z ) O(kn\log(n/k)+z)
  70. k k
  71. D D
  72. k = 2 , , K k=2,\dots,K
  73. Θ ( n ) \Theta(n)
  74. Θ ( 1 ) \Theta(1)
  75. Θ ( n ) \Theta(n)
  76. O ( n 2 ) O(n^{2})
  77. Θ ( n ) \Theta(n)
  78. 2 n 2n
  79. σ \sigma
  80. O ( σ ) O(\sigma)
  81. Θ ( 1 ) \Theta(1)
  82. Θ ( 1 ) \Theta(1)
  83. O ( log σ ) O(\log\sigma)
  84. Θ ( 1 ) \Theta(1)
  85. Θ ( 1 ) \Theta(1)
  86. Θ ( 1 ) \Theta(1)
  87. Θ ( 1 ) \Theta(1)
  88. O ( σ ) O(\sigma)
  89. O ( log σ ) O(\log\sigma)
  90. O ( log σ ) O(\log\sigma)
  91. O ( 1 ) O(1)
  92. O ( log σ ) O(\log\sigma)
  93. O ( σ ) O(\sigma)
  94. O ( 1 ) O(1)
  95. O ( 1 ) O(1)
  96. O ( 1 ) O(1)
  97. O ( 1 ) O(1)

Sumset.html

  1. A + B = { a + b : a A , b B } . A+B=\{a+b:a\in A,b\in B\}.
  2. n A = A + + A , nA=A+\cdots+A,
  3. 4 = , 4\Box=\mathbb{N},
  4. \Box

Sun-synchronous_orbit.html

  1. Δ Ω = - 2 π J 2 μ p 2 3 2 cos i \Delta\Omega=-2\pi\ \frac{J_{2}}{\mu\ p^{2}}\ \frac{3}{2}\ \cos i\,
  2. J 2 J_{2}\,
  3. μ \mu\,
  4. p p
  5. i i
  6. ρ \rho
  7. Δ Ω / P = ρ \Delta\Omega/P=\rho
  8. 2 π a a μ 2\pi\ a\sqrt{\frac{a}{\mu}}\,
  9. p a p\approx a\,
  10. ρ - 3 J 2 cos i 2 a 7 / 2 μ 1 / 2 = - ( 360 ° per year ) × ( a / 12352 km ) - 7 / 2 cos i = - ( 360 ° per year ) × ( P / 3.795 hrs ) - 7 / 3 cos i \rho\approx-\frac{3J_{2}\cos i}{2a^{7/2}\mu^{1/2}}=-(360\,\text{° per year})% \times(a/12352\,\text{ km})^{-7/2}\cos i=-(360\,\text{° per year})\times(P/3.7% 95\,\text{ hrs})^{-7/3}\cos i
  11. ρ \rho
  12. cos i - ρ μ 3 2 J 2 a 7 2 = - ( a / 12352 km ) 7 / 2 = - ( P / 3.795 hrs ) 7 / 3 , \cos i\ \approx\ -\frac{\rho\ \sqrt{\mu}}{\frac{3}{2}\ J_{2}}\ a^{\frac{7}{2}}% =-(a/12352\,\text{ km})^{7/2}=-(P/3.795\,\text{ hrs})^{7/3},
  13. 1 1 2 1\tfrac{1}{2}
  14. 1 3 5 1\tfrac{3}{5}
  15. 1 5 7 1\tfrac{5}{7}
  16. 1 11 13 1\tfrac{11}{13}
  17. 2 2
  18. 2 2 11 2\tfrac{2}{11}
  19. 2 2 5 2\tfrac{2}{5}
  20. 2 2 3 2\tfrac{2}{3}
  21. 3 3
  22. 3 3 7 3\tfrac{3}{7}

Sun_Zhiwei.html

  1. π \pi
  2. m m
  3. p k , , p n ( k < n ) p_{k},\ldots,p_{n}\ (k<n)
  4. 2 m + 2.2 m 2m+2.2\sqrt{m}
  5. m = p n - p n - 1 + + ( - 1 ) n - k p k m=p_{n}-p_{n-1}+...+(-1)^{n-k}p_{k}
  6. p j p_{j}
  7. j j
  8. \Z \Z
  9. π \pi

Super-Poincaré_algebra.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 P_{\mu}
  3. σ μ \sigma^{\mu}
  4. g μ ν g^{\mu\nu}
  5. { γ μ , γ ν } = 2 g μ ν \{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}
  6. σ μ ν = i 2 [ γ μ , γ ν ] \sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]
  7. [ M μ ν , Q α ] = 1 2 ( σ μ ν ) α β Q β [M^{\mu\nu},Q_{\alpha}]=\frac{1}{2}(\sigma^{\mu\nu})_{\alpha}^{\beta}Q_{\beta}
  8. [ Q α , P μ ] = 0 [Q_{\alpha},P^{\mu}]=0
  9. { 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}
  10. ( 1 2 , 0 ) V ( 0 , 1 2 ) V * \left(\frac{1}{2},0\right)\otimes V\oplus\left(0,\frac{1}{2}\right)\otimes V^{*}
  11. ( 1 / 2 , 0 ) (1/2,0)
  12. ( 0 , 1 / 2 ) (0,1/2)
  13. [ ( 1 2 , 0 ) V ] [ ( 0 , 1 2 ) V * ] [(\frac{1}{2},0)\otimes V]\otimes[(0,\frac{1}{2})\otimes V^{*}]
  14. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2},0)\otimes(0,\frac{1}{2})
  15. V V * V\otimes V^{*}
  16. [ ( 1 2 , 0 ) V ] [ ( 1 2 , 0 ) V ] [(\frac{1}{2},0)\otimes V]\otimes[(\frac{1}{2},0)\otimes V]
  17. N 2 N^{2}
  18. u ( 1 ) u(1)
  19. u ( 1 ) u(1)
  20. u ( 1 ) u(1)
  21. u ( 1 ) u(1)
  22. s u ( 2 ) u ( 1 ) su(2)\oplus u(1)
  23. s u ( 2 ) su(2)
  24. u ( 1 ) u(1)
  25. u ( 1 ) u(1)
  26. u ( 1 ) u(1)
  27. u ( 1 ) A u ( 1 ) B u ( 1 ) C u(1)_{A}\oplus u(1)_{B}\oplus u(1)_{C}
  28. u ( 1 ) B u(1)_{B}
  29. u ( 1 ) C u(1)_{C}
  30. u ( 1 ) A u(1)_{A}
  31. u ( 1 ) B u(1)_{B}
  32. u ( 1 ) C u(1)_{C}
  33. s u ( 2 ) u ( 1 ) A u ( 1 ) B su(2)\oplus u(1)_{A}\oplus u(1)_{B}
  34. s u ( 2 ) su(2)
  35. u ( 1 ) u(1)
  36. u ( 1 ) A u(1)_{A}
  37. u ( 1 ) B u(1)_{B}

Supercommutative_algebra.html

  1. y x = ( - 1 ) | x | | y | x y . yx=(-1)^{|x||y|}xy.\,
  2. [ x , y ] = x y - ( - 1 ) | x | | y | y x [x,y]=xy-(-1)^{|x||y|}yx\,
  3. x y + y x = 0 xy+yx=0\,
  4. x 2 = 0. x^{2}=0.\,

Supercritical_flow.html

  1. F r = def U g h Fr\ \stackrel{\mathrm{def}}{=}\ \frac{U}{\sqrt{gh}}
  2. F r < 1 Fr<1
  3. F r > 1 Fr>1
  4. F r 1 Fr\approx 1

Superhard_material.html

  1. σ c = σ 0 + k gb d \sigma_{c}=\sigma_{0}+\frac{k\text{gb}}{\sqrt{d}}
  2. σ c = k crack 2 E γ s π a 0 1 d \sigma_{c}=k\text{crack}\sqrt{\frac{2E\gamma_{s}}{\pi a_{0}}}\propto\frac{1}{% \sqrt{d}}

Supermanifold.html

  1. ( x , θ , θ ¯ ) (x,\theta,\bar{\theta})
  2. θ \theta\,
  3. θ ¯ \bar{\theta}
  4. C ( p ) Λ ( ξ 1 , ξ q ) . C^{\infty}(\mathbb{R}^{p})\otimes\Lambda^{\bullet}(\xi_{1},\dots\xi_{q}).
  5. ω = i d ξ i d x i , \omega=\sum_{i}d\xi_{i}\wedge dx_{i},
  6. x i x_{i}
  7. ξ i \xi_{i}
  8. i d p i d q i + j ε j 2 ( d ξ j ) 2 , \sum_{i}dp_{i}\wedge dq_{i}+\sum_{j}\frac{\varepsilon_{j}}{2}(d\xi_{j})^{2},
  9. p i , q i p_{i},q_{i}
  10. ξ i \xi_{i}
  11. ε j \varepsilon_{j}
  12. { F , G } = r F z i ω i j ( z ) l G z j . \{F,G\}=\frac{\partial_{r}F}{\partial z^{i}}\omega^{ij}(z)\frac{\partial_{l}G}% {\partial z^{j}}.
  13. r \partial_{r}
  14. l \partial_{l}
  15. n | n {\mathcal{R}}^{n|n}
  16. Δ H = 1 2 ρ r z a ( ρ ω i j ( z ) l H z j ) \Delta H=\frac{1}{2\rho}\frac{\partial_{r}}{\partial z^{a}}\left(\rho\omega^{% ij}(z)\frac{\partial_{l}H}{\partial z^{j}}\right)
  17. Δ = r x a l θ a \Delta=\frac{\partial_{r}}{\partial x^{a}}\frac{\partial_{l}}{\partial\theta_{% a}}
  18. ω = d x a d θ a \omega=dx^{a}\wedge d\theta_{a}
  19. Δ 2 = 0 \Delta^{2}=0
  20. P T M P\subset TM
  21. S 2 P T M / P S^{2}P\mapsto TM/P
  22. G L ( P ) × G L ( T M / P ) GL(P)\times GL(TM/P)

Supermodular_function.html

  1. f : R k R f\colon R^{k}\to R
  2. f ( x y ) + f ( x y ) f ( x ) + f ( y ) f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)
  3. \isin \isin
  4. \uparrow
  5. \downarrow
  6. 2 f z i z j 0 for all i j . \frac{\partial^{2}f}{\partial z_{i}\,\partial z_{j}}\geq 0\mbox{ for all }~{}i% \neq j.
  7. f \,f\,
  8. z i \,z_{i}\,
  9. i 1 , 2 , , N i\in{1,2,\dots,N}
  10. z i [ a , b ] z_{i}\in[a,b]
  11. f \,f\,
  12. i \,i\,
  13. z i \,z_{i}\,
  14. d f / d z j df/dz_{j}
  15. z j \,z_{j}\,
  16. j \,j\,
  17. i \,i\,
  18. z i \,z_{i}\,
  19. j \,j\,
  20. z j \,z_{j}\,
  21. f \,f\,
  22. z i \,z_{i}\,
  23. z j \,z_{j}\,
  24. i \,i\,
  25. z i \,z_{i}\,
  26. z j \,z_{j}\,
  27. f : 2 S R f\colon 2^{S}\to R
  28. A B S A\subset B\subset S
  29. x S B x\in S\setminus B
  30. f ( A { x } ) - f ( A ) f ( B { x } ) - f ( B ) f(A\cup\{x\})-f(A)\leq f(B\cup\{x\})-f(B)
  31. f ( A ) + f ( B ) f ( A B ) + f ( A B ) f(A)+f(B)\leq f(A\cap B)+f(A\cup B)

Superparticular_ratio.html

  1. n + 1 n = 1 + 1 n . \frac{n+1}{n}=1+\frac{1}{n}.
  2. n = 1 ( 2 n 2 n - 1 2 n 2 n + 1 ) = 2 1 2 3 4 3 4 5 6 5 6 7 = 4 3 16 15 36 35 = 2 8 9 24 25 48 49 = π 2 \prod_{n=1}^{\infty}\left(\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\right)=\frac{2}{% 1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6% }{7}\cdots=\frac{4}{3}\cdot\frac{16}{15}\cdot\frac{36}{35}\cdots=2\cdot\frac{8% }{9}\cdot\frac{24}{25}\cdot\frac{48}{49}\cdots=\frac{\pi}{2}
  3. π \pi
  4. π / 4 = 3 4 5 4 7 8 11 12 13 12 17 16 \pi/4=\frac{3}{4}\cdot\frac{5}{4}\cdot\frac{7}{8}\cdot\frac{11}{12}\cdot\frac{% 13}{12}\cdot\frac{17}{16}\cdots

Superpotential.html

  1. = 1 \hbar=1
  2. Q 1 = 1 2 [ ( p - i W ) b + ( p + i W ) b ] Q_{1}=\frac{1}{2}\left[(p-iW)b+(p+iW)b^{\dagger}\right]
  3. Q 2 = i 2 [ ( p - i W ) b - ( p + i W ) b ] Q_{2}=\frac{i}{2}\left[(p-iW)b-(p+iW)b^{\dagger}\right]
  4. H = { Q 1 , Q 1 } = { Q 2 , Q 2 } = p 2 2 + W 2 2 + W 2 ( b b - b b ) H=\{Q_{1},Q_{1}\}=\{Q_{2},Q_{2}\}=\frac{p^{2}}{2}+\frac{W^{2}}{2}+\frac{W^{% \prime}}{2}(bb^{\dagger}-b^{\dagger}b)
  5. H = p 2 2 + W 2 2 ± W 2 H=\frac{p^{2}}{2}+\frac{W^{2}}{2}\pm\frac{W^{\prime}}{2}
  6. x 0 , 1 , 2 , 3 x_{0,1,2,3}
  7. θ , θ ¯ \theta,\bar{\theta}
  8. x 0 , 1 , 2 , 3 x_{0,1,2,3}
  9. θ \theta
  10. θ ¯ \bar{\theta}

Superreal_number.html

  1. \mathbb{R}
  2. \mathbb{R}

Superselection.html

  1. ψ 1 \psi_{1}
  2. ψ 2 \psi_{2}
  3. ψ 1 | H | ψ 2 = 0 \langle\psi_{1}|H|\psi_{2}\rangle=0
  4. H H
  5. ψ 1 | A | ψ 2 = 0 \langle\psi_{1}|A|\psi_{2}\rangle=0
  6. A A
  7. g ( a ψ ) = ( g a ) ( g ψ ) g(a\cdot\psi)=(ga)\cdot(g\psi)
  8. ψ ( x + L ) = e i θ ψ ( x ) . \psi(x+L)=e^{i\theta}\psi(x).
  9. ϕ 1 , ϕ 2 , ϕ 3 \phi_{1},\phi_{2},\phi_{3}
  10. | ϕ i | 2 + t ϕ 2 + λ ( ϕ i 2 ) 2 |\nabla\phi_{i}|^{2}+t\phi^{2}+\lambda(\phi_{i}^{2})^{2}\,
  11. ϕ \phi
  12. ϕ \phi
  13. ϕ \phi
  14. π 2 ( S O ( 3 ) / S O ( 2 ) ) = \scriptstyle\pi_{2}(SO(3)/SO(2))=\mathbb{Z}
  15. ϕ \phi

Supersymmetric_quantum_mechanics.html

  1. H H O ψ n ( x ) = ( - 2 2 m d 2 d x 2 + m ω 2 2 x 2 ) ψ n ( x ) = E n H O ψ n ( x ) , H^{HO}\psi_{n}(x)=\bigg(\frac{-\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{m% \omega^{2}}{2}x^{2}\bigg)\psi_{n}(x)=E_{n}^{HO}\psi_{n}(x),
  2. ψ n ( x ) \psi_{n}(x)
  3. n n
  4. H H O H^{HO}
  5. E n H O E_{n}^{HO}
  6. E n H O E_{n}^{HO}
  7. n n
  8. A = 2 m d d x + W ( x ) A=\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+W(x)
  9. A = - 2 m d d x + W ( x ) , A^{\dagger}=-\frac{\hbar}{\sqrt{2m}}\frac{d}{dx}+W(x),
  10. W ( x ) W(x)
  11. H H O H^{HO}
  12. H ( 1 ) H^{(1)}
  13. H ( 2 ) H^{(2)}
  14. H ( 1 ) = A A = - 2 2 m d 2 d x 2 - 2 m W ( x ) + W 2 ( x ) H^{(1)}=A^{\dagger}A=\frac{-\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-\frac{\hbar}{% \sqrt{2m}}W^{\prime}(x)+W^{2}(x)
  15. H ( 2 ) = A A = - 2 2 m d 2 d x 2 + 2 m W ( x ) + W 2 ( x ) . H^{(2)}=AA^{\dagger}=\frac{-\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{\hbar}{% \sqrt{2m}}W^{\prime}(x)+W^{2}(x).
  16. ψ 0 ( 1 ) ( x ) \psi_{0}^{(1)}(x)
  17. H ( 1 ) H^{(1)}
  18. H ( 1 ) ψ 0 ( 1 ) ( x ) = A A ψ 0 ( 1 ) ( x ) = A ( 2 m d d x + W ( x ) ) ψ 0 ( 1 ) ( x ) = 0. H^{(1)}\psi_{0}^{(1)}(x)=A^{\dagger}A\psi_{0}^{(1)}(x)=A^{\dagger}\bigg(\frac{% \hbar}{\sqrt{2m}}\frac{d}{dx}+W(x)\bigg)\psi_{0}^{(1)}(x)=0.
  19. ψ 0 ( x ) \psi_{0}(x)
  20. W ( x ) W(x)
  21. W ( x ) = - 2 m ( ψ 0 ( x ) ψ 0 ( x ) ) = = x m ω 2 / 2 W(x)=\frac{-\hbar}{\sqrt{2m}}\bigg(\frac{\psi_{0}^{\prime}(x)}{\psi_{0}(x)}% \bigg)==x\sqrt{m\omega^{2}/2}
  22. H ( 1 ) = - 2 2 m d 2 d x 2 - m ω 2 2 x 2 - ω 2 H^{(1)}=\frac{-\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-\frac{m\omega^{2}}{2}x^{2}-% \frac{\hbar\omega}{2}
  23. H ( 2 ) = - 2 2 m d 2 d x 2 - m ω 2 2 x 2 + ω 2 . H^{(2)}=\frac{-\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}-\frac{m\omega^{2}}{2}x^{2}+% \frac{\hbar\omega}{2}.
  24. H ( 1 ) = H ( 2 ) - ω = H H O - ω 2 . H^{(1)}=H^{(2)}-\hbar\omega=H^{HO}-\frac{\hbar\omega}{2}.
  25. H ( 1 ) H^{(1)}
  26. E 0 = 0 E_{0}=0
  27. ω . \hbar\omega.
  28. H ( 2 ) H^{(2)}
  29. H H O H^{HO}
  30. ω \hbar\omega
  31. ω / 2 \hbar\omega/2
  32. H H O H^{HO}
  33. E n H O = ω ( n + 1 / 2 ) E_{n}^{HO}=\hbar\omega(n+1/2)
  34. { A , B } = A B + B A . \{A,B\}=AB+BA.
  35. \mathcal{H}
  36. i , j = 1 , , N i,j=1,\ldots,N
  37. { Q i , Q j } = δ i j . \{Q_{i},Q^{\dagger}_{j}\}=\mathcal{H}\delta_{ij}.
  38. Q i Q_{i}
  39. b b
  40. b b^{\dagger}
  41. { b , b } = 1 \{b,b^{\dagger}\}=1
  42. b 2 = 0 b^{2}=0
  43. p p
  44. x x
  45. [ x , p ] = i [x,p]=i
  46. W W
  47. x x
  48. Q 1 = 1 2 [ ( p - i W ) b + ( p + i W ) b ] Q_{1}=\frac{1}{2}\left[(p-iW)b+(p+iW^{\dagger})b^{\dagger}\right]
  49. Q 2 = i 2 [ ( p - i W ) b - ( p + i W ) b ] Q_{2}=\frac{i}{2}\left[(p-iW)b-(p+iW^{\dagger})b^{\dagger}\right]
  50. H = { Q 1 , Q 1 } = { Q 2 , Q 2 } = ( p + { W } ) 2 2 + { W } 2 2 + { W } 2 ( b b - b b ) H=\{Q_{1},Q_{1}\}=\{Q_{2},Q_{2}\}=\frac{(p+\Im\{W\})^{2}}{2}+\frac{{\Re\{W\}}^% {2}}{2}+\frac{\Re\{W\}^{\prime}}{2}(bb^{\dagger}-b^{\dagger}b)
  51. { W } \Im\{W\}
  52. Q = ( p - i W ) b Q=(p-iW)b
  53. Q = ( p + i W ) b Q^{\dagger}=(p+iW^{\dagger})b^{\dagger}
  54. { Q , Q } = { Q , Q } = 0 \{Q,Q\}=\{Q^{\dagger},Q^{\dagger}\}=0
  55. { Q , Q } = 2 H \{Q^{\dagger},Q\}=2H
  56. b b^{\dagger}
  57. Q Q^{\dagger}
  58. b b^{\dagger}
  59. [ Q , x } = - i b [Q,x\}=-ib
  60. [ Q , b } = 0 [Q,b\}=0
  61. [ Q , b } = d x d t - i { W } [Q,b^{\dagger}\}=\frac{dx}{dt}-i\Re\{W\}
  62. [ Q , x } = i b [Q^{\dagger},x\}=ib^{\dagger}
  63. [ Q , b } = d x d t + i { W } [Q^{\dagger},b\}=\frac{dx}{dt}+i\Re\{W\}
  64. [ Q , b } = 0 [Q^{\dagger},b^{\dagger}\}=0
  65. b ( t ) b^{\dagger}(t)
  66. { W } \Re\{W\}
  67. F = { W } F=\Re\{W\}
  68. [ Q , x } = - i b [Q,x\}=-ib
  69. [ Q , b } = 0 [Q,b\}=0
  70. [ Q , b } = d x d t - i F [Q,b^{\dagger}\}=\frac{dx}{dt}-iF
  71. [ Q , F } = - d b d t [Q,F\}=-\frac{db}{dt}
  72. [ Q , x } = i b [Q^{\dagger},x\}=ib^{\dagger}
  73. [ Q , b } = d x d t + i F [Q^{\dagger},b\}=\frac{dx}{dt}+iF
  74. [ Q , b } = 0 [Q^{\dagger},b^{\dagger}\}=0
  75. [ Q , F } = d b d t [Q^{\dagger},F\}=\frac{db^{\dagger}}{dt}
  76. θ \theta
  77. θ ¯ \bar{\theta}
  78. { θ , θ } = { θ ¯ , θ ¯ } = { θ ¯ , θ } = 0 \{\theta,\theta\}=\{\bar{\theta},\bar{\theta}\}=\{\bar{\theta},\theta\}=0
  79. f ( t , θ ¯ , θ ) = x ( t ) - i θ b ( t ) - i θ ¯ b ( t ) + θ ¯ θ F ( t ) f(t,\bar{\theta},\theta)=x(t)-i\theta b(t)-i\bar{\theta}b^{\dagger}(t)+\bar{% \theta}\theta F(t)
  80. [ Q , f } = θ f - i θ ¯ t f , [Q,f\}=\frac{\partial}{\partial\theta}f-i\bar{\theta}\frac{\partial}{\partial t% }f,
  81. [ Q , f } = θ ¯ f - i θ t f . [Q^{\dagger},f\}=\frac{\partial}{\partial\bar{\theta}}f-i\theta\frac{\partial}% {\partial t}f.
  82. b b^{\dagger}
  83. W W
  84. x x
  85. H = ( p ) 2 2 + W 2 2 + W 2 ( b b - b b ) H=\frac{(p)^{2}}{2}+\frac{{W}^{2}}{2}+\frac{W^{\prime}}{2}(bb^{\dagger}-b^{% \dagger}b)
  86. V + ( x , a 1 ) = V - ( x , a 2 ) + R ( a 1 ) V_{+}(x,a_{1})=V_{-}(x,a_{2})+R(a_{1})
  87. a a
  88. l l
  89. - e 2 4 π ϵ 0 1 r + h 2 l ( l + 1 ) 2 m 1 r 2 - E 0 \frac{-e^{2}}{4\pi\epsilon_{0}}\frac{1}{r}+\frac{h^{2}l(l+1)}{2m}\frac{1}{r^{2% }}-E_{0}
  90. V - V_{-}
  91. W = 2 m h e 2 24 π ϵ 0 ( l + 1 ) - h ( l + 1 ) r 2 m W=\frac{\sqrt{2m}}{h}\frac{e^{2}}{24\pi\epsilon_{0}(l+1)}-\frac{h(l+1)}{r\sqrt% {2m}}
  92. V + = - e 2 4 π ϵ 0 1 r + h 2 ( l + 1 ) ( l + 2 ) 2 m 1 r 2 + e 4 m 32 π 2 h 2 ϵ 0 2 ( l + 1 ) 2 V_{+}=\frac{-e^{2}}{4\pi\epsilon_{0}}\frac{1}{r}+\frac{h^{2}(l+1)(l+2)}{2m}% \frac{1}{r^{2}}+\frac{e^{4}m}{32\pi^{2}h^{2}\epsilon_{0}^{2}(l+1)^{2}}
  93. l + 1 l+1
  94. l = 0 l=0
  95. V - V_{-}
  96. V + V_{+}
  97. E n = i = 1 n R ( a i ) E_{n}=\sum\limits_{i=1}^{n}R(a_{i})
  98. a i a_{i}

Susceptible_individual.html

  1. S = A L {S}=\frac{A}{L}

Sylvester's_law_of_inertia.html

  1. n 0 + n + + n - = n . n_{0}+n_{+}+n_{-}=n.
  2. Δ 0 = 1 , Δ 1 , , Δ n = det A . \Delta_{0}=1,\Delta_{1},\ldots,\Delta_{n}=\det A.
  3. Q ( x 1 , x 2 , , x n ) = i = 1 n a i x i 2 Q(x_{1},x_{2},\ldots,x_{n})=\sum_{i=1}^{n}a_{i}x_{i}^{2}

Sylvester–Gallai_theorem.html

  1. k 2 ( k - 3 ) t k - 3. \displaystyle\sum_{k\geq 2}(k-3)t_{k}\leq-3.\,\!
  2. t 2 3 + k 4 ( k - 3 ) t k . \displaystyle t_{2}\geqslant 3+\sum_{k\geq 4}(k-3)t_{k}.\,\!
  3. t 2 n t_{2}\geq\sqrt{n}
  4. t 2 n / 2 t_{2}\geq\lfloor n/2\rfloor
  5. 3 n / 4 3\lfloor n/4\rfloor
  6. t 2 ( n ) 6 n / 13 t_{2}(n)\geq\lceil 6n/13\rceil

Symbolic_Cholesky_decomposition.html

  1. L L
  2. A = ( a i j ) 𝕂 n × n A=(a_{ij})\in\mathbb{K}^{n\times n}
  3. 𝕂 \mathbb{K}
  4. A = L L T A=LL^{T}\,
  5. 𝒜 i \mathcal{A}_{i}
  6. j \mathcal{L}_{j}
  7. A A
  8. L L
  9. min j \min\mathcal{L}_{j}
  10. j \mathcal{L}_{j}
  11. π ( i ) \pi(i)\,\!
  12. A A\,
  13. π ( i ) := 0 for all i \displaystyle\pi(i):=0~{}\mbox{for all}~{}~{}i

Symmetric_algebra.html

  1. v w - w v . v\otimes w-w\otimes v.
  2. v w - w v . v\otimes w-w\otimes v.
  3. v 1 v k σ S k v σ ( 1 ) v σ ( k ) . v_{1}\cdots v_{k}\mapsto\sum_{\sigma\in S_{k}}v_{\sigma(1)}\otimes\cdots% \otimes v_{\sigma(k)}.
  4. v 1 v k 1 k ! σ S k v σ ( 1 ) v σ ( k ) . v_{1}\cdots v_{k}\mapsto\frac{1}{k!}\sum_{\sigma\in S_{k}}v_{\sigma(1)}\otimes% \cdots\otimes v_{\sigma(k)}.
  5. v w 1 2 ( v w + w v ) vw\mapsto\frac{1}{2}(v\otimes w+w\otimes v)
  6. S ( V * ) × V K S(V^{*})\times V\to K
  7. x ( 2 , 3 ) = 2 , and y ( 2 , 3 ) = 3. x(2,3)=2,\,\text{ and }y(2,3)=3.
  8. dim ( S k ( V ) ) = ( n + k - 1 k ) \operatorname{dim}(S^{k}(V))={\left({{n+k-1}\atop{k}}\right)}

Syntactic_monoid.html

  1. S M S\subset M
  2. m M m\in M
  3. S / m = { u M | u m S } . S\ /\ m=\{u\in M\;|\;um\in S\}.
  4. m S = { u M | m u S } . m\setminus S=\{u\in M\;|\;mu\in S\}.
  5. S = { ( s , t ) M × M | S / s = S / t } . \sim_{S}\;=\{(s,t)\in M\times M\,|\;S\ /\ s=S\ /\ t\}.
  6. S = { ( s , t ) M × M | s S = t S } . \,{}_{S}\sim\;=\{(s,t)\in M\times M\,|\;s\setminus S=t\setminus S\}.
  7. u S v x , y M ( x u y S x v y S ) . u\equiv_{S}v\Leftrightarrow\forall x,y\in M(xuy\in S\Leftrightarrow xvy\in S).
  8. ( M / s ) / t = M / ( t s ) (M\ /\ s)\ /\ t=M\ /\ (ts)
  9. s , t M s,t\in M
  10. M ( S ) = M / S . M(S)=M\ /\ \sim_{S}.
  11. M ( S ) M(S)
  12. { m L | m M } \{m\setminus L\,|\;m\in M\}
  13. x L = y L x\setminus L\,=y\setminus L
  14. { m L | m M } \{m\setminus L\,|\;m\in M\}
  15. { m L | m L } \{m\setminus L\,|\;m\in L\}
  16. { m L | m M } \{m\setminus L\,|\;m\in M\}
  17. Q = { m L | m M } Q=\{m\setminus L\,|\;m\in M\}
  18. F = { m L | m L } F=\{m\setminus L\,|\;m\in L\}
  19. y ( x L ) = ( x y ) L y\setminus(x\setminus L)=(xy)\setminus L
  20. { m L | m M } \{m\setminus L\,|\;m\in M\}

Synthetic_aperture_radar.html

  1. N 2 + N 2 \frac{N^{2}+N}{2}
  2. π \pi
  3. π \pi

System_F.html

  1. Λ α . λ x α . x : α . α α \vdash\Lambda\alpha.\lambda x^{\alpha}.x:\forall\alpha.\alpha\to\alpha
  2. α \alpha
  3. Λ \Lambda
  4. λ \lambda
  5. α \alpha
  6. α \alpha
  7. 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 \scriptstyle\mathsf{Boolean}
  8. α . α α α \scriptstyle\forall\alpha.\alpha\to\alpha\to\alpha
  9. α \scriptstyle\alpha
  10. 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 \scriptstyle\mathsf{Boolean}
  11. \to
  12. 𝐓 \scriptstyle\mathbf{T}
  13. 𝐅 \scriptstyle\mathbf{F}
  14. 𝐓 = Λ α . λ x α λ y α . x \mathbf{T}=\Lambda\alpha{.}\lambda x^{\alpha}\lambda y^{\alpha}{.}x
  15. 𝐅 = Λ α . λ x α λ y α . y \mathbf{F}=\Lambda\alpha{.}\lambda x^{\alpha}\lambda y^{\alpha}{.}y
  16. α . α α α \scriptstyle\forall\alpha.\alpha\to\alpha\to\alpha
  17. 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 \scriptstyle\mathsf{Boolean}
  18. α . α α α \scriptstyle\forall\alpha.\alpha\to\alpha\to\alpha
  19. 𝐓 \scriptstyle\mathbf{T}
  20. 𝐅 \scriptstyle\mathbf{F}
  21. λ \scriptstyle\lambda
  22. 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 \scriptstyle\mathsf{Boolean}\rightarrow\mathsf{Boolean}\rightarrow\mathsf{Boolean}
  23. AND = λ x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 λ y 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 . x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 y 𝐅 \mathrm{AND}=\lambda x^{\mathsf{Boolean}}\lambda y^{\mathsf{Boolean}}{.}x\,% \mathsf{Boolean}\,y\,\mathbf{F}
  24. OR = λ x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 λ y 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 . x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 𝐓 y \mathrm{OR}=\lambda x^{\mathsf{Boolean}}\lambda y^{\mathsf{Boolean}}{.}x\,% \mathsf{Boolean}\,\mathbf{T}\,y
  25. NOT = λ x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 . x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 𝐅 𝐓 \mathrm{NOT}=\lambda x^{\mathsf{Boolean}}{.}x\,\mathsf{Boolean}\,\mathbf{F}\,% \mathbf{T}
  26. 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 \scriptstyle\mathsf{Boolean}
  27. IFTHENELSE = Λ α . λ x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 λ y α λ z α . x α y z \mathrm{IFTHENELSE}=\Lambda\alpha.\lambda x^{\mathsf{Boolean}}\lambda y^{% \alpha}\lambda z^{\alpha}.x\alpha yz
  28. 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 \scriptstyle\mathsf{Boolean}
  29. 𝐓 \scriptstyle\mathbf{T}
  30. ISZERO = λ n α . ( α α ) α α . n 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 ( λ x 𝖡𝗈𝗈𝗅𝖾𝖺𝗇 . 𝐅 ) 𝐓 \mathrm{ISZERO}=\lambda n^{\forall\alpha.(\alpha\rightarrow\alpha)\rightarrow% \alpha\rightarrow\alpha}{.}n\,\mathsf{Boolean}\,(\lambda x^{\mathsf{Boolean}}{% .}\mathbf{F})\,\mathbf{T}
  31. K 1 K 2 S K_{1}\rightarrow K_{2}\rightarrow\dots\rightarrow S
  32. S S
  33. K i K_{i}
  34. m m
  35. S S
  36. α . ( K 1 1 [ α / S ] α ) ( K 1 m [ α / S ] α ) α \forall\alpha.(K_{1}^{1}[\alpha/S]\rightarrow\dots\rightarrow\alpha)\dots% \rightarrow(K_{1}^{m}[\alpha/S]\rightarrow\dots\rightarrow\alpha)\rightarrow\alpha
  37. N N
  38. 𝑧𝑒𝑟𝑜 : N \mathit{zero}:\mathrm{N}
  39. 𝑠𝑢𝑐𝑐 : N N \mathit{succ}:\mathrm{N}\rightarrow\mathrm{N}
  40. α . α ( α α ) α \forall\alpha.\alpha\to(\alpha\to\alpha)\to\alpha
  41. Λ α . λ x α . λ f α α . x \Lambda\alpha.\lambda x^{\alpha}.\lambda f^{\alpha\to\alpha}.x
  42. Λ α . λ x α . λ f α α . f x \Lambda\alpha.\lambda x^{\alpha}.\lambda f^{\alpha\to\alpha}.fx
  43. Λ α . λ x α . λ f α α . f ( f x ) \Lambda\alpha.\lambda x^{\alpha}.\lambda f^{\alpha\to\alpha}.f(fx)
  44. Λ α . λ x α . λ f α α . f ( f ( f x ) ) \Lambda\alpha.\lambda x^{\alpha}.\lambda f^{\alpha\to\alpha}.f(f(fx))
  45. α . ( α α ) α α \forall\alpha.(\alpha\rightarrow\alpha)\rightarrow\alpha\rightarrow\alpha
  46. n n
  47. n n
  48. F n F_{n}
  49. \star
  50. J K J\Rightarrow K
  51. J F n - 1 J\in F_{n-1}
  52. K F n K\in F_{n}
  53. F ω F_{\omega}
  54. F ω = 1 i F i F_{\omega}=\underset{1\leq i}{\bigcup}F_{i}
  55. λ \lambda
  56. Λ \Lambda
  57. \forall
  58. λ \lambda

Systematic_risk.html

  1. π 1 * u i ( x 1 i ) + π 2 * u i ( x 2 i ) \pi_{1}*u_{i}(x_{1i})+\pi_{2}*u_{i}(x_{2i})
  2. π 1 \pi_{1}
  3. π 2 \pi_{2}
  4. ω 1 = ( 1 , 0 ) \omega_{1}=(1,0)
  5. ω 2 = ( 0 , 1 ) \omega_{2}=(0,1)
  6. p 1 / p 2 = π 1 / π 2 p_{1}/p_{2}=\pi_{1}/\pi_{2}
  7. ω 1 = ( 2 , 0 ) \omega_{1}=(2,0)
  8. ω 2 = ( 0 , 1 ) \omega_{2}=(0,1)
  9. p 1 / p 2 < π 1 / π 2 p_{1}/p_{2}<\pi_{1}/\pi_{2}
  10. p 1 / π 1 < p 2 / π 2 p_{1}/\pi_{1}<p_{2}/\pi_{2}
  11. p 1 < p 2 p_{1}<p_{2}

Systemic_risk.html

  1. i = 1 , 2 i=1,2
  2. a i 0 a_{i}\geq 0
  3. T 0 T\geq 0
  4. d i 0 d_{i}\geq 0
  5. T T
  6. a i a_{i}
  7. s i 0 s_{i}\geq 0
  8. r i 0 r_{i}\geq 0
  9. r i = min { d i , a i } r_{i}=\min\{d_{i},a_{i}\}
  10. s i = ( a i - d i ) + . s_{i}=(a_{i}-d_{i})^{+}.
  11. s i s_{i}
  12. r i r_{i}
  13. a i a_{i}
  14. a i a_{i}
  15. r 1 = min { d 1 , a 1 + 0.05 s 2 + 0.2 r 2 } r_{1}=\min\{d_{1},a_{1}+0.05s_{2}+0.2r_{2}\}
  16. r 2 = min { d 2 , a 2 + 0.03 s 1 + 0.1 r 1 } r_{2}=\min\{d_{2},a_{2}+0.03s_{1}+0.1r_{1}\}
  17. s 1 = ( a 1 + 0.05 s 2 + 0.2 r 2 - d 1 ) + s_{1}=(a_{1}+0.05s_{2}+0.2r_{2}-d_{1})^{+}
  18. s 2 = ( a 2 + 0.03 s 1 + 0.1 r 1 - d 2 ) + . s_{2}=(a_{2}+0.03s_{1}+0.1r_{1}-d_{2})^{+}.

Szemerédi–Trotter_theorem.html

  1. n n
  2. m m
  3. O ( n 2 3 m 2 3 + n + m ) , O\left(n^{\frac{2}{3}}m^{\frac{2}{3}}+n+m\right),
  4. n n
  5. k > 2 k>2
  6. k k
  7. O ( n 2 k 3 + n k ) . O\left(\frac{n^{2}}{k^{3}}+\frac{n}{k}\right).
  8. 2 m 2m
  9. k k
  10. k 1 k−1
  11. n n
  12. k 3 k≥3
  13. m m
  14. e e
  15. e = O ( n 2 3 m 2 3 + n + m ) . e=O\left(n^{\frac{2}{3}}m^{\frac{2}{3}}+n+m\right).
  16. n n
  17. m m
  18. e 7.5 n e≤7.5n
  19. e = O ( n 2 3 m 2 3 + n + m ) . e=O\left(n^{\frac{2}{3}}m^{\frac{2}{3}}+n+m\right).
  20. n ( n 1 ) / 2 n(n−1)/2
  21. k k
  22. k 2 k≥2
  23. k k
  24. k C k≤C
  25. C C
  26. k k
  27. k C k≥C
  28. k k
  29. m k mk
  30. m k = O ( n 2 3 m 2 3 + n + m ) , mk=O\left(n^{\frac{2}{3}}m^{\frac{2}{3}}+n+m\right),
  31. m k = O ( n 2 / 3 m 2 / 3 ) , m k = O ( n ) mk=O(n^{2/3}m^{2/3}),mk=O(n)
  32. m k = O ( m ) mk=O(m)
  33. k k
  34. m = O ( n 2 / k 3 + n / k ) m=O(n^{2}/k^{3}+n/k)
  35. P = { ( a , b ) 𝐙 2 : 1 a N ; 1 b 2 N 2 } , P=\left\{(a,b)\in\mathbf{Z}^{2}\ :\ 1\leq a\leq N;1\leq b\leq 2N^{2}\right\},
  36. L = { ( x , m x + b ) : m , b 𝐙 ; 1 m N ; 1 b N 2 } . L=\left\{(x,mx+b)\ :\ m,b\in\mathbf{Z};1\leq m\leq N;1\leq b\leq N^{2}\right\}.
  37. | P | = 2 N 3 |P|=2N^{3}
  38. | L | = N 3 |L|=N^{3}
  39. N N
  40. x { 1 , , N } x\in\{1,\cdots,N\}
  41. N 4 N^{4}
  42. n n
  43. S S
  44. m m
  45. H H
  46. S S
  47. S S
  48. H H
  49. O ( m 2 3 n d 3 + n d - 1 ) . O\left(m^{\frac{2}{3}}n^{\frac{d}{3}}+n^{d-1}\right).
  50. H H
  51. k k
  52. O ( n d k 3 + n d - 1 k ) . O\left(\frac{n^{d}}{k^{3}}+\frac{n^{d-1}}{k}\right).

Tachymeter_(watch).html

  1. T = 3600 t T=\frac{3600}{t}

Takens'_theorem.html

  1. f : M M . f:M\to M.
  2. k > 2 d A . k>2d_{A}.
  3. ϕ T ( x ) = ( α ( x ) , α ( f ( x ) ) , , α ( f k - 1 ( x ) ) ) \phi_{T}(x)=\left(\alpha(x),\alpha\left(f(x)\right),\dots,\alpha\left(f^{k-1}(% x)\right)\right)
  4. τ : y t - τ , y t - 2 τ \tau:y_{t-\tau},y_{t-2\tau}
  5. k k\to\infty

Tandem_mass_spectrometry.html

  1. A B + + M A + B + + M AB^{+}+M\to A+B^{+}+M
  2. [ M + n H ] n + + e - [ [ M + ( n - 1 ) H ] ( n - 1 ) + ] * f r a g m e n t s [M+nH]^{n+}+e^{-}\to\bigg[[M+(n-1)H]^{(n-1)+}\bigg]^{*}\to fragments
  3. [ M + n H ] n + + A - [ [ M + ( n - 1 ) H ] ( n - 1 ) + ] * + A f r a g m e n t s [M+nH]^{n+}+A^{-}\to\bigg[[M+(n-1)H]^{(n-1)+}\bigg]^{*}+A\to fragments
  4. [ M - n H ] n - + A + [ [ M - n H ] ( n + 1 ) - ] * + A f r a g m e n t s [M-nH]^{n-}+A^{+}\to\bigg[[M-nH]^{(n+1)-}\bigg]^{*}+A\to fragments
  5. [ M + H ] 1 + + H e + [ [ M + H ] 2 + ] * + H e 0 f r a g m e n t s [M+H]^{1+}+He^{+}\to\bigg[[M+H]^{2+}\bigg]^{*}+He^{0}\to fragments
  6. A B + + h ν A + B + AB^{+}+h\nu\to A+B^{+}
  7. h ν h\nu

Tangent_half-angle_formula.html

  1. sin α = 2 tan α 2 1 + tan 2 α 2 \sin\alpha=\frac{{2\tan\frac{\alpha}{2}}}{{1+\tan^{2}\frac{\alpha}{2}}}
  2. cos α = 1 - tan 2 α 2 1 + tan 2 α 2 \cos\alpha=\frac{{1-\tan^{2}\frac{\alpha}{2}}}{{1+\tan^{2}\frac{\alpha}{2}}}
  3. tan α = 2 tan α 2 1 - tan 2 α 2 \tan\alpha=\frac{{2\tan\frac{\alpha}{2}}}{{1-\tan^{2}\frac{\alpha}{2}}}
  4. tan ( η 2 ± θ 2 ) \displaystyle\tan\left(\frac{\eta}{2}\pm\frac{\theta}{2}\right)
  5. sin α = 2 sin α 2 cos α 2 = 2 sin α 2 cos α 2 cos 2 α 2 + sin 2 α 2 = 2 tan α 2 1 + tan 2 α 2 \sin\alpha=2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}=\frac{2\sin\frac{\alpha}{% 2}\cos\frac{\alpha}{2}}{\cos^{2}\frac{\alpha}{2}+\sin^{2}\frac{\alpha}{2}}=% \frac{2\tan\frac{\alpha}{2}}{1+\tan^{2}\frac{\alpha}{2}}
  6. tan α = 2 tan α 2 1 - tan 2 α 2 \tan\alpha=\frac{{2\tan\frac{\alpha}{2}}}{{1-\tan^{2}\frac{\alpha}{2}}}
  7. cos α = sin α tan α = 2 tan α 2 1 + tan 2 α 2 2 tan α 2 1 - tan 2 α 2 = 1 - tan 2 α 2 1 + tan 2 α 2 \cos\alpha=\frac{\sin\alpha}{\tan\alpha}=\frac{\frac{{2\tan\frac{\alpha}{2}}}{% {1+\tan^{2}\frac{\alpha}{2}}}}{\frac{{2\tan\frac{\alpha}{2}}}{{1-\tan^{2}\frac% {\alpha}{2}}}}=\frac{{1-\tan^{2}\frac{\alpha}{2}}}{{1+\tan^{2}\frac{\alpha}{2}}}
  8. t sin ϕ = 1 1 + cos ϕ \frac{t}{\sin\phi}=\frac{1}{1+\cos\phi}
  9. t = sin ϕ 1 + cos ϕ = sin ϕ ( 1 - cos ϕ ) ( 1 + cos ϕ ) ( 1 - cos ϕ ) = 1 - cos ϕ sin ϕ t=\frac{\sin\phi}{1+\cos\phi}=\frac{\sin\phi(1-\cos\phi)}{(1+\cos\phi)(1-\cos% \phi)}=\frac{1-\cos\phi}{\sin\phi}
  10. tan a + b 2 = sin a + b 2 cos a + b 2 = sin a + sin b cos a + cos b \tan\frac{a+b}{2}=\frac{\sin\frac{a+b}{2}}{\cos\frac{a+b}{2}}=\frac{\sin a+% \sin b}{\cos a+\cos b}
  11. cos φ = 1 - t 2 1 + t 2 , \cos\varphi=\frac{1-t^{2}}{1+t^{2}},
  12. sin φ = 2 t 1 + t 2 , \sin\varphi=\frac{2t}{1+t^{2}},
  13. tan φ = 2 t 1 - t 2 , \tan\varphi=\frac{2t}{1-t^{2}},
  14. cot φ = 1 - t 2 2 t , \cot\varphi=\frac{1-t^{2}}{2t},
  15. sec φ = 1 + t 2 1 - t 2 , \sec\varphi=\frac{1+t^{2}}{1-t^{2}},
  16. csc φ = 1 + t 2 2 t , \csc\varphi=\frac{1+t^{2}}{2t},
  17. e i φ = 1 + i t 1 - i t , e^{i\varphi}=\frac{1+it}{1-it},
  18. e - i φ = 1 - i t 1 + i t . e^{-i\varphi}=\frac{1-it}{1+it}.
  19. arctan t = 1 2 i ln 1 + i t 1 - i t . \arctan t=\frac{1}{2i}\ln\frac{1+it}{1-it}.
  20. t = tan 1 2 φ . t=\tan\tfrac{1}{2}\varphi.
  21. φ = 2 arctan t , \varphi=2\arctan t,\,
  22. d φ = 2 d t 1 + t 2 . d\varphi={{2\,dt}\over{1+t^{2}}}.
  23. t = tanh 1 2 θ = sinh θ cosh θ + 1 = cosh θ - 1 sinh θ t=\tanh\tfrac{1}{2}\theta=\frac{\sinh\theta}{\cosh\theta+1}=\frac{\cosh\theta-% 1}{\sinh\theta}
  24. cosh θ = 1 + t 2 1 - t 2 , \cosh\theta=\frac{1+t^{2}}{1-t^{2}},
  25. sinh θ = 2 t 1 - t 2 , \sinh\theta=\frac{2t}{1-t^{2}},
  26. tanh θ = 2 t 1 + t 2 , \tanh\theta=\frac{2t}{1+t^{2}},
  27. coth θ = 1 + t 2 2 t , \coth\theta=\frac{1+t^{2}}{2t},
  28. sech θ = 1 - t 2 1 + t 2 , \mathrm{sech}\,\theta=\frac{1-t^{2}}{1+t^{2}},
  29. csch θ = 1 - t 2 2 t , \mathrm{csch}\,\theta=\frac{1-t^{2}}{2t},
  30. e θ = 1 + t 1 - t , e^{\theta}=\frac{1+t}{1-t},
  31. e - θ = 1 - t 1 + t . e^{-\theta}=\frac{1-t}{1+t}.
  32. artanh t = 1 2 ln 1 + t 1 - t . \operatorname{artanh}t=\frac{1}{2}\ln\frac{1+t}{1-t}.
  33. t = tan 1 2 φ = tanh 1 2 θ t=\tan\tfrac{1}{2}\varphi=\tanh\tfrac{1}{2}\theta
  34. φ = 2 tan - 1 tanh 1 2 θ gd θ . \varphi=2\tan^{-1}\tanh\tfrac{1}{2}\theta\equiv\mathrm{gd}\,\theta.

Tarski_monster_group.html

  1. p p
  2. G G
  3. p p
  4. p p
  5. G G
  6. G G
  7. N G N\trianglelefteq G
  8. U G U\leq G
  9. N N
  10. N U NU
  11. p 2 p^{2}
  12. p > 10 75 p>10^{75}

Taut_submanifold.html

  1. q M q\in M\,
  2. L q : N 𝐑 , L q ( x ) = dist ( x , q ) 2 L_{q}:N\to\mathbf{R},\qquad L_{q}(x)=\operatorname{dist}(x,q)^{2}\,
  3. L q L_{q}

Tax_bracket.html

  1. $ 3 , 000 + ( $ 25 , 000 - $ 20 , 000 ) × 30 % = $ 3 , 000 + $ 1 , 500 = $ 4 , 500 , \$3,000+(\$25,000-\$20,000)\times 30\%=\$3,000+\$1,500=\$4,500,
  2. $ 25 , 000 × 30 % - $ 3 , 000 = $ 7 , 500 - $ 3 , 000 = $ 4 , 500. \$25,000\times 30\%-\$3,000=\$7,500-\$3,000=\$4,500.

Tax_rate.html

  1. t t
  2. i i
  3. = t i . =\frac{t}{i}.
  4. Δ t Δ i \frac{\Delta t}{\Delta i}
  5. t t
  6. t = 0.20 t=0.20
  7. a a
  8. p p
  9. t × p t\times p
  10. p - t × p p-t\times p
  11. a = t × p p - t × p = t 1 - t a=\frac{t\times p}{p-t\times p}=\frac{t}{1-t}

Taylor–Proudman_theorem.html

  1. Ω \Omega
  2. Ω \Omega
  3. ρ ( 𝐮 ) 𝐮 = 𝐅 - p , \rho({\mathbf{u}}\cdot\nabla){\mathbf{u}}={\mathbf{F}}-\nabla p,
  4. 𝐮 {\mathbf{u}}
  5. ρ \rho
  6. p p
  7. F = Φ F=\nabla\Phi
  8. 2 ρ 𝛀 × 𝐮 = Φ - p , 2\rho\mathbf{\Omega}\times{\mathbf{u}}=\nabla\Phi-\nabla p,
  9. Ω \Omega
  10. ( 𝛀 ) 𝐮 = 0. ({\mathbf{\Omega}}\cdot\nabla){\mathbf{u}}={\mathbf{0}}.
  11. × ( A × B ) = A ( B ) - ( A ) B + ( B ) A - B ( A ) \nabla\times(A\times B)=A(\nabla\cdot B)-(A\cdot\nabla)B+(B\cdot\nabla)A-B(% \nabla\cdot A)
  12. × ( p ) = 0 \nabla\times(\nabla p)=0
  13. × ( Φ ) = 0 \nabla\times(\nabla\Phi)=0
  14. 𝛀 = 0 \nabla\cdot{\mathbf{\Omega}}=0
  15. Ω x 𝐮 x + Ω y 𝐮 y + Ω z 𝐮 z = 0. \Omega_{x}\frac{\partial{\mathbf{u}}}{\partial x}+\Omega_{y}\frac{\partial{% \mathbf{u}}}{\partial y}+\Omega_{z}\frac{\partial{\mathbf{u}}}{\partial z}=0.
  16. Ω x = Ω y = 0 \Omega_{x}=\Omega_{y}=0
  17. 𝐮 z = 0 , \frac{\partial{\mathbf{u}}}{\partial z}=0,
  18. Ω z 0 \Omega_{z}\neq 0
  19. Ω \vec{\Omega}
  20. z ^ \hat{z}

Teleparallelism.html

  1. { X 1 , , X 4 } \{\mathrm{X}_{1},\dots,\mathrm{X}_{4}\}
  2. M M\,
  3. p M p\in M\,
  4. { X 1 ( p ) , , X 4 ( p ) } \{\mathrm{X}_{1}(p),\dots,\mathrm{X}_{4}(p)\}
  5. T p M T_{p}M\,
  6. T p M T_{p}M\,
  7. p p\,
  8. T M TM\,
  9. M M\,
  10. { X i } \{\mathrm{X}_{i}\}
  11. \nabla\,
  12. M M\,
  13. v ( f i X i ) = ( v f i ) X i ( p ) \nabla_{v}(f^{i}\mathrm{X}_{i})=(vf^{i})\mathrm{X}_{i}(p)\,
  14. v T p M v\in T_{p}M\,
  15. f i f^{i}\,
  16. M M\,
  17. f i X i f^{i}X_{i}\,
  18. M M\,
  19. \nabla\,
  20. { X i } \{X_{i}\}
  21. X i X j = 0 , \nabla_{\mathrm{X}_{i}}\mathrm{X}_{j}=0\,,
  22. W k = i j ω k ( X i X j ) 0 , W^{k}{}_{ij}=\omega^{k}(\nabla_{\mathrm{X}_{i}}\mathrm{X}_{j})\equiv 0\,,
  23. ω k \omega^{k}\,
  24. ω i ( X j ) = δ j i \omega^{i}(\mathrm{X}_{j})=\delta^{i}_{j}\,
  25. \nabla
  26. ( U , x μ ) (U,x^{\mu})
  27. μ \partial_{\mu}
  28. Γ β = μ ν h i β ν h μ i \Gamma^{\beta}{}_{\mu\nu}=h^{\beta}_{i}\partial_{\nu}h^{i}_{\mu}\,
  29. X i = h i μ μ i , μ = 1 , 2 , n \mathrm{X}_{i}=h^{\mu}_{i}\partial_{\mu}\quad{i,\mu}=1,2,\dots n
  30. { X i } \{X_{i}\}
  31. g g\,
  32. g ( X i , X j ) = η i j g(X_{i},X_{j})=\eta_{ij}\,
  33. η i j = diag ( - 1 , - 1 , - 1 , 1 ) \eta_{ij}={\mathrm{diag}}(-1,-1,-1,1)\,
  34. B a μ B^{a}{}_{\mu}
  35. π : M \pi\colon{\mathcal{M}}\to M
  36. p M p\in M
  37. p {\mathcal{M}}_{p}
  38. ( V , ψ ) (V,\psi)\,
  39. ψ = ( x μ , x a ) \psi=(x^{\mu},x^{a})\,
  40. x μ x^{\mu}\,
  41. p {\mathcal{M}}_{p}\,
  42. p {\mathcal{M}}_{p}
  43. T M TM
  44. x μ ( x μ , x a = ξ a ( p ) ) . x^{\mu}\to(x^{\mu},x^{a}=\xi^{a}(p)).
  45. D μ ξ a ( d ξ a ) μ + B a = μ μ ξ a + B a μ D_{\mu}\xi^{a}\equiv(d\xi^{a})_{\mu}+B^{a}{}_{\mu}=\partial_{\mu}\xi^{a}+B^{a}% {}_{\mu}
  46. x a x a + α a x^{a}\rightarrow x^{a}+\alpha^{a}
  47. B a μ B a - μ μ α a B^{a}{}_{\mu}\rightarrow B^{a}{}_{\mu}-\partial_{\mu}\alpha^{a}
  48. x a = ξ a ( p ) x^{a}=\xi^{a}(p)
  49. h a = μ μ ξ a + B a μ h^{a}{}_{\mu}=\partial_{\mu}\xi^{a}+B^{a}{}_{\mu}
  50. x a = ξ a ( p ) x^{a}=\xi^{a}(p)
  51. M {\mathcal{M}}\to M
  52. B a μ B^{a}{}_{\mu}
  53. h a = h a d μ x μ = ( μ ξ a + B a ) μ d x μ h^{a}=h^{a}{}_{\mu}dx^{\mu}=(\partial_{\mu}\xi^{a}+B^{a}{}_{\mu})dx^{\mu}
  54. ξ a \xi^{a}
  55. p {\mathcal{M}}_{p}
  56. {\mathcal{M}}
  57. T a μ ν ( D B a ) μ ν = D μ B a - ν D ν B a , μ T^{a}{}_{\mu\nu}\equiv(DB^{a})_{\mu\nu}=D_{\mu}B^{a}{}_{\nu}-D_{\nu}B^{a}{}_{% \mu},
  58. p {\mathcal{M}}_{p}

Temperature_coefficient.html

  1. d R R = α d T . \frac{dR}{R}=\alpha\,dT.
  2. R ( T ) = R ( T 0 ) ( 1 + α Δ T ) , R(T)=R(T_{0})(1+\alpha\Delta T),
  3. R = A e B T R=A\cdot e^{\frac{B}{T}}
  4. R = r e B T = R 0 e - B T 0 e B T R=r^{\infty}e^{\frac{B}{T}}=R_{0}e^{-\frac{B}{T_{0}}}e^{\frac{B}{T}}
  5. R 0 R_{0}
  6. T 0 T_{0}
  7. R 0 R_{0}
  8. R T C = Δ B r B r Δ T × 100 RTC=\frac{\Delta Br}{Br\Delta T}\times 100
  9. 𝜌 ( T ) = ρ 0 [ 1 + α 0 ( T - T 0 ) ] \operatorname{\rho}(T)=\rho_{0}[1+\alpha_{0}(T-T_{0})]
  10. α 0 = 1 ρ 0 [ δ ρ δ T ] T = T 0 \alpha_{0}=\frac{1}{\rho_{0}}\left[\frac{\delta\rho}{\delta T}\right]_{T=T_{0}}
  11. ρ 0 \rho_{0}
  12. 𝜌 ( T ) = S α B T \operatorname{\rho}(T)=S\alpha^{\frac{B}{T}}
  13. S S
  14. α \alpha
  15. b b
  16. α \alpha
  17. α T = ρ T \alpha_{T}=\frac{\partial\rho}{\partial T}
  18. ρ \rho
  19. α T \alpha_{T}
  20. α T \alpha_{T}
  21. α T \alpha_{T}

Template:SI_dimensionless_units.html

  1. 2 π 2\pi
  2. 4 π 4\pi

Template_matching.html

  1. S A D ( x , y ) = i = 0 T rows j = 0 T cols Diff ( x + i , y + j , i , j ) SAD(x,y)=\sum_{i=0}^{T_{\,\text{rows}}}\sum_{j=0}^{T_{\,\text{cols}}}{\,\text{% Diff}(x+i,y+j,i,j)}
  2. x = 0 S rows y = 0 S cols S A D ( x , y ) \sum_{x=0}^{S_{\,\text{rows}}}\sum_{y=0}^{S_{\,\text{cols}}}{SAD(x,y)}

Tensile_structure.html

  1. w = < m t p l > t 1 R 1 + t 2 R 2 w=\frac{<}{m}tpl>{{t_{1}}}{{R_{1}}}+\frac{{t_{2}}}{{R_{2}}}
  2. < m t p l > t 1 R 1 = - t 2 R 2 \frac{<}{m}tpl>{{t_{1}}}{{R_{1}}}=-\frac{{t_{2}}}{{R_{2}}}
  3. H = < m t p l > w S 2 8 d H=\frac{<}{m}tpl>{{wS^{2}}}{{8d}}
  4. V = < m t p l > w S 2 V=\frac{<}{m}tpl>{{wS}}{{2}}
  5. L = 2 R arcsin < m t p l > S 2 R L=2R\arcsin\frac{<}{m}tpl>{{S}}{{2R}}
  6. T = H 2 + V 2 T=\sqrt{H^{2}+V^{2}}
  7. T = ( w S 2 8 d ) 2 + ( w S 2 ) 2 T=\sqrt{\left(\frac{wS^{2}}{8d}\right)^{2}+\left(\frac{wS}{2}\right)^{2}}
  8. T = w R T=wR
  9. k = < m t p l > E A L k=\frac{<}{m}tpl>{{EA}}{{L}}
  10. e = < m t p l > T L E A e=\frac{<}{m}tpl>{{TL}}{{EA}}
  11. T 0 T_{0}
  12. e = L - L 0 = < m t p l > L 0 ( T - T 0 ) E A e=L-L_{0}=\frac{<}{m}tpl>{{L_{0}(T-T_{0})}}{{EA}}
  13. < m t p l > L 0 ( T - T 0 ) E A + L 0 = 2 T arcsin ( w S 2 T ) w {\frac{<}{m}tpl>{{L_{0}(T-T_{0})}}{{EA}}}+L_{0}=\frac{{2T\arcsin(\frac{{wS}}{{% 2T}})}}{{w}}
  14. T 0 T_{0}
  15. W = < m t p l > 4 T d L W=\frac{<}{m}tpl>{{4Td}}{{L}}
  16. d = < m t p l > W L 4 T d=\frac{<}{m}tpl>{{WL}}{{4T}}
  17. L = S 2 + 4 d 2 = S 2 + 4 ( < m t p l > W L 4 T ) 2 L=\sqrt{S^{2}+4d^{2}}=\sqrt{S^{2}+4\left(\frac{<}{m}tpl>{{WL}}{{4T}}\right)^{2}}
  18. L 0 + < m t p l > L 0 ( T - T 0 ) E A = S 2 + 4 ( W ( L 0 + L 0 ( T - T 0 ) E A ) 4 T ) 2 L_{0}+\frac{<}{m}tpl>{{L_{0}(T-T_{0})}}{{EA}}=\sqrt{S^{2}+4\left(\frac{{W(L_{0% }+\frac{{L_{0}(T-T_{0})}}{{EA}})}}{{4T}}\right)^{2}}
  19. T 0 T_{0}
  20. f 1 = < m t p l > ( T m ) 2 L f_{1}=\sqrt{\frac{<}{m}tpl>{{(\frac{{T}}{{m}})}}{{2L}}}

Tensiometer_(surface_tension).html

  1. γ \scriptstyle\gamma

Tesla_(unit).html

  1. T = V s m 2 = N A m = J A m 2 = H A m 2 = Wb m 2 = kg C s = N s C m = kg A s 2 \mathrm{T}=\dfrac{\mathrm{V}\cdot{\mathrm{s}}}{\mathrm{m}^{2}}=\dfrac{\mathrm{% N}}{\mathrm{A}{\cdot}\mathrm{m}}=\dfrac{\mathrm{J}}{\mathrm{A}{\cdot}\mathrm{m% }^{2}}=\dfrac{\mathrm{H}{\cdot}\mathrm{A}}{\mathrm{m}^{2}}=\dfrac{\mathrm{Wb}}% {\mathrm{m}^{2}}=\dfrac{\mathrm{kg}}{\mathrm{C}{\cdot}\mathrm{s}}=\dfrac{% \mathrm{N}{\cdot}\mathrm{s}}{\mathrm{C}{\cdot}\mathrm{m}}=\dfrac{\mathrm{kg}}{% \mathrm{A}{\cdot}\mathrm{s}^{2}}

Tetrakis_hexahedron.html

  1. a a
  2. a / 4 a/4
  3. a r c t a n ( 1 / 2 ) arctan(1/2)
  4. a a
  5. 3 a / 4 3a/4
  6. 5 a / 4 √5a/4
  7. 5 a / 8 √5a/8
  8. a r c c o s ( 2 / 3 ) arccos(2/3)
  9. 180 - 2 a r c c o s ( 2 / 3 ) 180-2arccos(2/3)
  10. n 7. n\geq 7.

The_Residents_Radio_Special.html

  1. \cong

Thermal_design_power.html

  1. P = C V 2 f P=CV^{2}f
  2. C C
  3. f f
  4. V V

Thermoacoustics.html

  1. c 2 2 v x 2 - 2 v t 2 = 0 c^{2}\frac{\partial^{2}v}{\partial x^{2}}-\frac{\partial^{2}v}{\partial t^{2}}=0
  2. v = v A r cos ( ω t - k x ) + v A l cos ( ω t + k x ) . v=v_{Ar}\cos(\omega t-kx)+v_{Al}\cos(\omega t+kx).
  3. δ p = c ρ 0 [ v A r cos ( ω t - k x ) - v A l cos ( ω t + k x ) ] . \delta p=c\rho_{0}[v_{Ar}\cos(\omega t-kx)-v_{Al}\cos(\omega t+kx)].
  4. δ x = v A r ω sin ( ω t - k x ) + v A l ω sin ( ω t + k x ) \delta x=\frac{v_{Ar}}{\omega}\sin(\omega t-kx)+\frac{v_{Al}}{\omega}\sin(% \omega t+kx)
  5. δ T = c M C p [ v A r cos ( ω t - k x ) - v A l cos ( ω t + k x ) ] . \delta T=\frac{cM}{C_{p}}[v_{Ar}\cos(\omega t-kx)-v_{Al}\cos(\omega t+kx)].
  6. v A r = v A l v_{Ar}=v_{Al}
  7. P = γ p 0 2 c A ( v A r 2 - v A l 2 ) P=\frac{\gamma p_{0}}{2c}A(v_{Ar}^{2}-v_{Al}^{2})
  8. v A r = v A l v_{Ar}=v_{Al}
  9. v A r = 0 v_{Ar}=0
  10. v A l = 0 v_{Al}=0
  11. δ κ 2 = 2 κ V m ω C p . \delta_{\kappa}^{2}=\frac{2\kappa V_{m}}{\omega C_{p}}.
  12. δ ν 2 = 2 η ω ρ \delta_{\nu}^{2}=\frac{2\eta}{\omega\rho}
  13. P r = η C p M κ . P_{r}=\frac{\eta C_{p}}{M\kappa}.
  14. δ ν 2 = P r δ κ 2 . \delta_{\nu}^{2}=P_{r}\delta_{\kappa}^{2}.

Three-phase.html

  1. V L 1 - N = sin ( θ ) * V P V_{L1-N}=\sin\left(\theta\right)*V_{P}\,\!
  2. V L 2 - N = sin ( θ - 2 3 π ) * V P = sin ( θ + 4 3 π ) * V P V_{L2-N}=\sin\left(\theta-\frac{2}{3}\pi\right)*V_{P}=\sin\left(\theta+\frac{4% }{3}\pi\right)*V_{P}
  3. V L 3 - N = sin ( θ - 4 3 π ) * V P = sin ( θ + 2 3 π ) * V P V_{L3-N}=\sin\left(\theta-\frac{4}{3}\pi\right)*V_{P}=\sin\left(\theta+\frac{2% }{3}\pi\right)*V_{P}
  4. V P V_{P}
  5. θ = 2 π f t \theta=2\pi ft\,\!
  6. t t
  7. f f
  8. P = V I = 1 R V 2 \scriptstyle P\,=\,VI\,=\,\frac{1}{R}V^{2}
  9. P L i = V L i 2 R P_{Li}=\frac{V_{Li}^{2}}{R}
  10. P T O T = i P L i P_{TOT}=\sum_{i}P_{Li}
  11. p = 1 V P 2 P T O T R \scriptstyle p\,=\,\frac{1}{V_{P}^{2}}P_{TOT}R
  12. p = sin 2 θ + sin 2 ( θ - 2 3 π ) + sin 2 ( θ - 4 3 π ) = 3 2 p=\sin^{2}\theta+\sin^{2}\left(\theta-\frac{2}{3}\pi\right)+\sin^{2}\left(% \theta-\frac{4}{3}\pi\right)=\frac{3}{2}
  13. P T O T = 3 V P 2 2 R P_{TOT}=\frac{3V_{P}^{2}}{2R}
  14. θ \theta
  15. Z = | Z | e j φ Z=|Z|e^{j\varphi}
  16. I P = V P | Z | I_{P}=\frac{V_{P}}{|Z|}
  17. I L 1 = I P sin ( θ - φ ) I_{L1}=I_{P}\sin\left(\theta-\varphi\right)
  18. I L 2 = I P sin ( θ - 2 3 π - φ ) I_{L2}=I_{P}\sin\left(\theta-\frac{2}{3}\pi-\varphi\right)
  19. I L 3 = I P sin ( θ - 4 3 π - φ ) I_{L3}=I_{P}\sin\left(\theta-\frac{4}{3}\pi-\varphi\right)
  20. P L 1 = V L 1 I L 1 = V P I P sin ( θ ) sin ( θ - φ ) P_{L1}=V_{L1}I_{L1}=V_{P}I_{P}\sin\left(\theta\right)\sin\left(\theta-\varphi\right)
  21. P L 2 = V L 2 I L 2 = V P I P sin ( θ - 2 3 π ) sin ( θ - 2 3 π - φ ) P_{L2}=V_{L2}I_{L2}=V_{P}I_{P}\sin\left(\theta-\frac{2}{3}\pi\right)\sin\left(% \theta-\frac{2}{3}\pi-\varphi\right)
  22. P L 3 = V L 3 I L 3 = V P I P sin ( θ - 4 3 π ) sin ( θ - 4 3 π - φ ) P_{L3}=V_{L3}I_{L3}=V_{P}I_{P}\sin\left(\theta-\frac{4}{3}\pi\right)\sin\left(% \theta-\frac{4}{3}\pi-\varphi\right)
  23. P L 1 = V P I P 2 [ cos φ - cos ( 2 θ - φ ) ] P_{L1}=\frac{V_{P}I_{P}}{2}\left[\cos\varphi-\cos\left(2\theta-\varphi\right)\right]
  24. P L 2 = V P I P 2 [ cos φ - cos ( 2 θ - 4 3 π - φ ) ] P_{L2}=\frac{V_{P}I_{P}}{2}\left[\cos\varphi-\cos\left(2\theta-\frac{4}{3}\pi-% \varphi\right)\right]
  25. P L 3 = V P I P 2 [ cos φ - cos ( 2 θ - 8 3 π - φ ) ] P_{L3}=\frac{V_{P}I_{P}}{2}\left[\cos\varphi-\cos\left(2\theta-\frac{8}{3}\pi-% \varphi\right)\right]
  26. P T O T = V P I P 2 { 3 cos φ - [ cos ( 2 θ - φ ) + cos ( 2 θ - 4 3 π - φ ) + cos ( 2 θ - 8 3 π - φ ) ] } P_{TOT}=\frac{V_{P}I_{P}}{2}\left\{3\cos\varphi-\left[\cos\left(2\theta-% \varphi\right)+\cos\left(2\theta-\frac{4}{3}\pi-\varphi\right)+\cos\left(2% \theta-\frac{8}{3}\pi-\varphi\right)\right]\right\}
  27. P T O T = 3 V P I P 2 cos φ P_{TOT}=\frac{3V_{P}I_{P}}{2}\cos\varphi
  28. P T O T = 3 V P 2 2 | Z | cos φ P_{TOT}=\frac{3V_{P}^{2}}{2|Z|}\cos\varphi
  29. I L 1 = V L 1 - N R , I L 2 = V L 2 - N R , I L 3 = V L 3 - N R - I N = I L 1 + I L 2 + I L 3 \begin{aligned}\displaystyle I_{L1}&\displaystyle=\frac{V_{L1-N}}{R},\;I_{L2}=% \frac{V_{L2-N}}{R},\;I_{L3}=\frac{V_{L3-N}}{R}\\ \displaystyle-I_{N}&\displaystyle=I_{L1}+I_{L2}+I_{L3}\end{aligned}
  30. i = I N R V P i=\frac{I_{N}R}{V_{P}}
  31. i \displaystyle i
  32. I L 1 I_{L1}
  33. I L 2 I_{L2}
  34. I L 3 I_{L3}
  35. I L 1 + I L 2 * cos 2 3 π + j * I L 2 * sin 2 3 π + I L 3 * cos 4 3 π + j * I L 3 * sin 4 3 π I_{L1}+I_{L2}*\cos\frac{2}{3}\pi+j*I_{L2}*\sin\frac{2}{3}\pi+I_{L3}*\cos\frac{% 4}{3}\pi+j*I_{L3}*\sin\frac{4}{3}\pi
  36. I L 1 - I L 2 * 0.5 - I L 3 * 0.5 + j * 3 2 * ( I L 2 - I L 3 ) I_{L1}-I_{L2}*0.5-I_{L3}*0.5+j*\frac{\sqrt{3}}{2}*\left(I_{L2}-I_{L3}\right)
  37. I L 1 2 + I L 2 2 + I L 3 2 - I L 1 * I L 2 - I L 1 * I L 3 - I L 2 * I L 3 \sqrt{I_{L1}^{2}+I_{L2}^{2}+I_{L3}^{2}-I_{L1}*I_{L2}-I_{L1}*I_{L3}-I_{L2}*I_{L% 3}}

Throughput_accounting.html

  1. Throughput = Sales revenue – Total Variable Costs \,\text{Throughput}=\,\text{Sales revenue – Total Variable Costs}
  2. Throughput accounting Ratio = Return per factory hour / Cost per factory hour \,\text{Throughput accounting Ratio}=\,\text{Return per factory hour}/\,\text{% Cost per factory hour}

Thrust-to-weight_ratio.html

  1. ( T W ) c r u i s e = 1 ( L D ) c r u i s e \left(\frac{T}{W}\right)_{cruise}=\frac{1}{(\frac{L}{D})_{cruise}}
  2. T W = ( η p V ) ( P W ) \frac{T}{W}=\left(\frac{\eta_{p}}{V}\right)\left(\frac{P}{W}\right)
  3. η p \eta_{p}\;
  4. V V\;
  5. P P\;
  6. T W = 3 , 820 kN ( 5 , 307 kg ) ( 9.807 m / s 2 ) = 0.07340 kN N = 73.40 N N = 73.40 \frac{T}{W}=\frac{3,820\ \mathrm{kN}}{(5,307\ \mathrm{kg})(9.807\ \mathrm{m/s^% {2}})}=0.07340\ \frac{\mathrm{kN}}{\mathrm{N}}=73.40\ \frac{\mathrm{N}}{% \mathrm{N}}=73.40

Thue_equation.html

  1. ( C 1 r ) C 2 (C_{1}r)^{C_{2}}

Thue–Siegel–Roth_theorem.html

  1. α \alpha
  2. α \alpha
  3. ϵ > 0 \epsilon>0
  4. | α - p q | < 1 q 2 + ϵ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{2+\epsilon}}
  5. p p
  6. q q
  7. | α - p q | > C ( α , ϵ ) q 2 + ϵ \left|\alpha-\frac{p}{q}\right|>\frac{C(\alpha,\epsilon)}{q^{2+\epsilon}}
  8. C ( α , ϵ ) C(\alpha,\epsilon)
  9. ϵ > 0 \epsilon>0
  10. α \alpha
  11. d / 2 + 1 + ϵ d/2+1+\epsilon
  12. | α - p q | < 1 q 2 log ( q ) 1 + ϵ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{2}\log(q)^{1+\epsilon}}
  13. | α - ξ | < 1 H ( ξ ) κ |\alpha-\xi|<\frac{1}{H(\xi)^{\kappa}}

Tikhonov_regularization.html

  1. x x
  2. A 𝐱 = 𝐛 , A\mathbf{x}=\mathbf{b},
  3. A A
  4. 𝐱 \mathbf{x}
  5. 𝐛 \mathbf{b}
  6. 𝐱 \mathbf{x}
  7. A A
  8. 𝐱 \mathbf{x}
  9. A 𝐱 - 𝐛 2 \|A\mathbf{x}-\mathbf{b}\|^{2}
  10. \left\|\cdot\right\|
  11. A 𝐱 - 𝐛 2 + Γ 𝐱 2 \|A\mathbf{x}-\mathbf{b}\|^{2}+\|\Gamma\mathbf{x}\|^{2}
  12. Γ \Gamma
  13. Γ = α I \Gamma=\alpha I
  14. x ^ \hat{x}
  15. x ^ = ( A T A + Γ T Γ ) - 1 A T 𝐛 \hat{x}=(A^{T}A+\Gamma^{T}\Gamma)^{-1}A^{T}\mathbf{b}
  16. Γ \Gamma
  17. Γ = 0 \Gamma=0
  18. x x
  19. x x
  20. A x - b P 2 + x - x 0 Q 2 \|Ax-b\|_{P}^{2}+\|x-x_{0}\|_{Q}^{2}\,
  21. x Q 2 \left\|x\right\|_{Q}^{2}
  22. x T Q x x^{T}Qx
  23. P P
  24. b b
  25. x 0 x_{0}
  26. x x
  27. Q Q
  28. x x
  29. Q = Γ T Γ Q=\Gamma^{T}\Gamma
  30. x * x^{*}
  31. x * = ( A T P A + Q ) - 1 ( A T P b + Q x 0 ) . x^{*}=(A^{T}PA+Q)^{-1}(A^{T}Pb+Qx_{0}).\,
  32. x * = x 0 + ( A T P A + Q ) - 1 ( A T P ( b - A x 0 ) ) . x^{*}=x_{0}+(A^{T}PA+Q)^{-1}(A^{T}P(b-Ax_{0})).\,
  33. A A
  34. x x
  35. b b
  36. A A
  37. A * A + Γ T Γ A^{*}A+\Gamma^{T}\Gamma
  38. Γ = α I \Gamma=\alpha I
  39. A = U Σ V T A=U\Sigma V^{T}\,
  40. σ i \sigma_{i}
  41. x ^ = V D U T b \hat{x}=VDU^{T}b
  42. D D
  43. D i i = σ i σ i 2 + α 2 D_{ii}=\frac{\sigma_{i}}{\sigma_{i}^{2}+\alpha^{2}}
  44. x ^ = i = 1 q f i u i T b σ i v i \hat{x}=\sum_{i=1}^{q}f_{i}\frac{u_{i}^{T}b}{\sigma_{i}}v_{i}
  45. f i = σ i 2 σ i 2 + α 2 f_{i}=\frac{\sigma_{i}^{2}}{\sigma_{i}^{2}+\alpha^{2}}
  46. q q
  47. A A
  48. α \alpha
  49. G = RSS τ 2 = X β ^ - y 2 [ Tr ( I - X ( X T X + α 2 I ) - 1 X T ) ] 2 G=\frac{\operatorname{RSS}}{\tau^{2}}=\frac{\left\|X\hat{\beta}-y\right\|^{2}}% {\left[\operatorname{Tr}\left(I-X(X^{T}X+\alpha^{2}I)^{-1}X^{T}\right)\right]^% {2}}
  50. RSS \operatorname{RSS}
  51. τ \tau
  52. RSS = y - i = 1 q ( u i b ) u i 2 + i = 1 q α 2 σ i 2 + α 2 ( u i b ) u i 2 \operatorname{RSS}=\left\|y-\sum_{i=1}^{q}(u_{i}^{\prime}b)u_{i}\right\|^{2}+% \left\|\sum_{i=1}^{q}\frac{\alpha^{2}}{\sigma_{i}^{2}+\alpha^{2}}(u_{i}^{% \prime}b)u_{i}\right\|^{2}
  53. RSS = RSS 0 + i = 1 q α 2 σ i 2 + α 2 ( u i b ) u i 2 \operatorname{RSS}=\operatorname{RSS}_{0}+\left\|\sum_{i=1}^{q}\frac{\alpha^{2% }}{\sigma_{i}^{2}+\alpha^{2}}(u_{i}^{\prime}b)u_{i}\right\|^{2}
  54. τ = m - i = 1 q σ i 2 σ i 2 + α 2 = m - q + i = 1 q α 2 σ i 2 + α 2 \tau=m-\sum_{i=1}^{q}\frac{\sigma_{i}^{2}}{\sigma_{i}^{2}+\alpha^{2}}=m-q+\sum% _{i=1}^{q}\frac{\alpha^{2}}{\sigma_{i}^{2}+\alpha^{2}}
  55. C M C_{M}
  56. C D C_{D}
  57. C M = σ M 2 I C_{M}=\sigma_{M}^{2}I
  58. C D = σ D 2 I C_{D}=\sigma_{D}^{2}I
  59. α = σ D / σ M \alpha={\sigma_{D}}/{\sigma_{M}}
  60. Γ \Gamma
  61. x x
  62. σ x \sigma_{x}
  63. b b
  64. σ b \sigma_{b}
  65. x x

Time-to-digital_converter.html

  1. f 0 f_{0}
  2. T T
  3. T = n T 0 T=n\cdot T_{0}
  4. n n
  5. T 0 = 1 / f 0 T_{0}=1/f_{0}
  6. n 1 n_{1}
  7. n 2 n_{2}
  8. p ( n 1 ) = 1 - c p(n_{1})=1-c
  9. q ( n 2 ) = c q(n_{2})=c
  10. c = F r c ( T / T 0 ) c=Frc(T/T_{0})
  11. T / T 0 T/T_{0}
  12. T = ( p n 1 + q n 2 ) T 0 T=(p\cdot n_{1}+q\cdot n_{2})\cdot T_{0}
  13. p p
  14. q q
  15. f 1 f_{1}
  16. f 2 f_{2}
  17. n 1 n_{1}
  18. n 2 n_{2}
  19. T T
  20. T = n 1 - 1 f 1 - n 2 - 1 f 2 T=\frac{n_{1}-1}{f_{1}}-\frac{n_{2}-1}{f_{2}}
  21. τ \tau
  22. τ L \tau_{L}
  23. τ L \tau_{L}
  24. τ B < τ L \tau_{B}<\tau_{L}
  25. T T
  26. T = n ( τ 1 - τ 2 ) T=n\cdot(\tau_{1}-\tau_{2})
  27. τ \tau
  28. τ \tau
  29. T = n T 0 + T start - T stop T=nT_{0}+T_{\mathrm{start}}-T_{\mathrm{stop}}

Time_evolution.html

  1. F t , s : X X t , s \operatorname{F}_{t,s}:X\rightarrow X\quad\forall t,s\in\mathbb{R}
  2. F u , t ( F t , s ( x ) ) = F u , s ( x ) . \operatorname{F}_{u,t}(\operatorname{F}_{t,s}(x))=\operatorname{F}_{u,s}(x).
  3. F u , t = F u - t , 0 u , t . \operatorname{F}_{u,t}=\operatorname{F}_{u-t,0}\quad\forall u,t\in\mathbb{R}.
  4. G t + s = G t G s . \operatorname{G}_{t+s}=\operatorname{G}_{t}\operatorname{G}_{s}.
  5. F u , t ( F t , s ( x ) ) = F u , s ( x ) . u t s . \operatorname{F}_{u,t}(\operatorname{F}_{t,s}(x))=\operatorname{F}_{u,s}(x).% \quad u\geq t\geq s.

Time–frequency_analysis.html

  1. x ( t ) = { cos ( π t ) ; t < 10 cos ( 3 π t ) ; 10 t < 20 cos ( 2 π t ) ; t > 20 x(t)=\begin{cases}\cos(\pi t);&t<10\\ \cos(3\pi t);&10\leq t<20\\ \cos(2\pi t);&t>20\end{cases}
  2. x 1 ( t ) = { cos ( π t ) ; t < 10 cos ( 3 π t ) ; 10 t < 20 cos ( 2 π t ) ; t > 20 x_{1}(t)=\begin{cases}\cos(\pi t);&t<10\\ \cos(3\pi t);&10\leq t<20\\ \cos(2\pi t);&t>20\end{cases}
  3. x 2 ( t ) = { cos ( π t ) ; t < 10 cos ( 2 π t ) ; 10 t < 20 cos ( 3 π t ) ; t > 20 x_{2}(t)=\begin{cases}\cos(\pi t);&t<10\\ \cos(2\pi t);&10\leq t<20\\ \cos(3\pi t);&t>20\end{cases}
  4. 1 2 π d d t ϕ ( t ) , \frac{1}{2\pi}\frac{d}{dt}\phi(t),
  5. ϕ ( t ) \phi(t)
  6. [ x y ] , \begin{bmatrix}x\\ y\end{bmatrix},
  7. [ x y ] \begin{bmatrix}x\\ y\end{bmatrix}
  8. [ a b c d ] = [ 1 λ z 0 1 ] , \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&\lambda z\\ 0&1\end{bmatrix},
  9. λ \lambda
  10. [ a b c d ] = [ 1 0 - 1 λ f 1 ] \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&0\\ -\frac{1}{\lambda f}&1\end{bmatrix}
  11. [ a b c d ] = [ 1 0 1 λ R 1 ] \begin{bmatrix}a&b\\ c&d\end{bmatrix}=\begin{bmatrix}1&0\\ \frac{1}{\lambda R}&1\end{bmatrix}
  12. [ a b c d ] [ x y ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}.

Time–frequency_representation.html

  1. E ( t , f ) E(t,f)
  2. 2 \mathbb{R}^{2}
  3. E 0 = Sup | E ( t , f ) | , ( t , f ) 2 E_{0}=\mbox{Sup}~{}|E(t,f)|,\,(t,f)\in\mathbb{R}^{2}
  4. 𝒞 = { ( t , f ) 2 : | E ( t , f ) | > T , T [ 0 , E 0 ] } \mathcal{C}=\{(t,f)\in\mathbb{R}^{2}:|E(t,f)|>T,\,\forall T\in[0,E_{0}]\}
  5. E ( t , f ) E(t,f)
  6. 𝒞 \mathcal{C}

Tired_light.html

  1. E ( x ) = E 0 exp ( - x R 0 ) E(x)=E_{0}\exp{(-\frac{x}{R_{0}})}
  2. E ( x ) E(x)
  3. x x
  4. E 0 E_{0}
  5. R 0 R_{0}
  6. R 0 R_{0}

Titanic_prime.html

  1. p = 10 999 + n , p=10^{999}+n,

Tobin's_q.html

  1. (Equity Market Value + Liabilities Market Value) (Equity Book Value + Liabilities Book Value) \frac{\,\text{(Equity Market Value + Liabilities Market Value)}}{\,\text{(% Equity Book Value + Liabilities Book Value)}}
  2. (Equity Market Value + Liabilities Book Value) (Equity Book Value + Liabilities Book Value) \frac{\,\text{(Equity Market Value + Liabilities Book Value)}}{\,\text{(Equity% Book Value + Liabilities Book Value)}}
  3. Equity Market Value Equity Book Value \frac{\,\text{Equity Market Value}}{\,\text{Equity Book Value}}
  4. ( n u m b e r o f s h a r e s ) × ( s h a r e p r i c e ) (numberofshares)\times(shareprice)
  5. q = value of stock market corporate net worth q=\frac{\,\text{value of stock market}}{\,\text{corporate net worth}}
  6. q = Market value of installed capital Replacement cost of capital q=\frac{\,\text{Market value of installed capital}}{\,\text{Replacement cost % of capital}}

Todd–Coxeter_algorithm.html

  1. G = X R G=\langle X\mid R\rangle
  2. X X
  3. R R
  4. X X^{\prime}
  5. X X
  6. H = h 1 , h 2 , , h s H=\langle h_{1},h_{2},\ldots,h_{s}\rangle
  7. h i h_{i}
  8. X X^{\prime}
  9. R R
  10. h i h_{i}
  11. H H
  12. H H
  13. X X^{\prime}
  14. C i C_{i}
  15. g j X g_{j}\in X^{\prime}
  16. C k = C i g j C_{k}=C_{i}g_{j}
  17. R R
  18. 1 = g n 1 g n 2 g n t 1=g_{n_{1}}g_{n_{2}}\cdots g_{n_{t}}
  19. R R
  20. g n i X g_{n_{i}}\in X^{\prime}
  21. H H
  22. C k = C i g n 1 g n 2 g n j C_{k}=C_{i}g_{n_{1}}g_{n_{2}}\cdots g_{n_{j}}
  23. ( i , t ) (i,t)
  24. g n 1 g n 2 g n t = 1 g_{n_{1}}g_{n_{2}}\cdots g_{n_{t}}=1
  25. H H
  26. h n = g n 1 g n 2 g n t h_{n}=g_{n_{1}}g_{n_{2}}\cdots g_{n_{t}}
  27. H H
  28. g n i X g_{n_{i}}\in X^{\prime}
  29. H H
  30. C k = H g n 1 g n 2 g n j C_{k}=Hg_{n_{1}}g_{n_{2}}\cdots g_{n_{j}}
  31. C i = C j g C_{i}=C_{j}g
  32. g X g\in X^{\prime}
  33. C i = C j g C_{i}=C_{j}g
  34. C j = C i g - 1 C_{j}=C_{i}g^{-1}
  35. C i = C j C_{i}=C_{j}
  36. i < j i<j
  37. g X g\in X^{\prime}
  38. | G : H | |G:H|
  39. G G
  40. H H

Token_bucket.html

  1. 1 / r 1/r
  2. 1 / r 1/r
  3. ( r * S ) / 1000 (r*S)/1000
  4. r r
  5. T max = { b / ( M - r ) if r < M otherwise T\text{max}=\begin{cases}b/(M-r)&\,\text{ if }r<M\\ \infty&\,\text{ otherwise }\end{cases}
  6. L max = T max * M L\text{max}=T\text{max}*M

Tolerance_interval.html

  1. μ \mu
  2. σ \sigma
  3. μ ± 1.96 σ \mu\pm 1.96\sigma
  4. μ ^ \hat{\mu}
  5. σ ^ \hat{\sigma}
  6. μ ^ ± 1.96 σ ^ \hat{\mu}\pm 1.96\hat{\sigma}
  7. γ \gamma
  8. γ \gamma
  9. y 1 , y 2 , , y n y_{1},y_{2},...,y_{n}
  10. μ 35 \mu\geq 35
  11. y n + 1 35 y_{n+1}\geq 35
  12. μ \mu
  13. p = .99 p=.99
  14. n = 15 n=15
  15. μ \mu
  16. σ 2 \sigma^{2}
  17. X X
  18. X 𝒩 ( μ , σ 2 ) X\sim\mathcal{N}(\mu,\sigma^{2})
  19. X ¯ \bar{X}
  20. X ¯ ± t n - 1 , 0.975 S / ( n ) \bar{X}\pm t_{n-1,0.975}S/\sqrt{(}n)
  21. t m , 1 - α t_{m,1-\alpha}
  22. X ¯ + t n - 1 , 0.95 S / n \bar{X}+t_{n-1,0.95}S/\sqrt{n}
  23. exp ( X ¯ + t n - 1 , 0.95 S / n ) \exp{\left(\bar{X}+t_{n-1,0.95}S/\sqrt{n}\right)}
  24. X ¯ + t n - 1 , 0.95 S ( 1 + 1 / n ) \bar{X}+t_{n-1,0.95}S\sqrt{\left(1+1/n\right)}
  25. X ¯ ± t n - 1 , 0.975 S / n \bar{X}\pm t_{n-1,0.975}S/\sqrt{n}
  26. μ \mu
  27. μ \mu

Tomographic_reconstruction.html

  1. θ \theta
  2. μ ( x , y ) \mu(x,y)
  3. r r
  4. θ \theta
  5. I = I 0 exp ( - μ ( x , y ) d s ) I=I_{0}\exp\left({-\int\mu(x,y)\,ds}\right)
  6. μ ( x ) \mu(x)
  7. x x
  8. p p
  9. r r
  10. θ \theta
  11. p ( r , θ ) = ln ( I / I 0 ) = - μ ( x , y ) d s p(r,\theta)=\ln(I/I_{0})=-\int\mu(x,y)\,ds
  12. r r
  13. ( x , y ) (x,y)
  14. θ \theta
  15. x cos θ + y sin θ = r x\cos\theta+y\sin\theta=r
  16. p ( r , θ ) = - - f ( x , y ) δ ( x cos θ + y sin θ - r ) d x d y p(r,\theta)=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(x,y)\delta(x\cos% \theta+y\sin\theta-r)\,dx\,dy
  17. f ( x , y ) f(x,y)
  18. μ ( x , y ) \mu(x,y)
  19. f ( x , y ) f(x,y)
  20. f ( x , y ) f(x,y)