wpmath0000006_10

Projection_(relational_algebra).html

  1. Π a 1 , , a n ( R ) \Pi_{a_{1},...,a_{n}}(R)
  2. a 1 , , a n a_{1},...,a_{n}
  3. R R
  4. { a 1 , , a n } \{a_{1},...,a_{n}\}
  5. Π a , b * 0.5 , c ( R ) \Pi_{a,\ b*0.5,\ c}(R)
  6. ( 3 , 7 ) (3,7)
  7. P e r s o n Person
  8. A g e Age
  9. W e i g h t Weight
  10. P e r s o n Person
  11. Π A g e , W e i g h t ( P e r s o n ) \Pi_{Age,Weight}(Person)
  12. Π a 1 , , a n ( R ) = { t [ a 1 , , a n ] : t R } \Pi_{a_{1},...,a_{n}}(R)=\{\ t[a_{1},...,a_{n}]:\ t\in R\ \}
  13. t [ a 1 , , a n ] t[a_{1},...,a_{n}]
  14. t t
  15. { a 1 , , a n } \{a_{1},...,a_{n}\}
  16. t [ a 1 , , a n ] = { ( a , v ) | ( a , v ) t , a a 1 , , a n } t[a_{1},...,a_{n}]=\{\ (a^{\prime},v)\ |\ (a^{\prime},v)\in t,\ a^{\prime}\in a% _{1},...,a_{n}\ \}
  17. Π a 1 , , a n ( R ) \Pi_{a_{1},...,a_{n}}(R)
  18. { a 1 , , a n } \{a_{1},...,a_{n}\}
  19. R R

Projective_object.html

  1. Hom ( P , - ) : 𝒞 𝐒𝐞𝐭 \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Set}
  2. 𝒞 \mathcal{C}
  3. P 𝒞 P\in\mathcal{C}
  4. Hom ( P , - ) : 𝒞 𝐀𝐛 \operatorname{Hom}(P,-)\colon\mathcal{C}\to\mathbf{Ab}
  5. 𝐀𝐛 \mathbf{Ab}
  6. Q Q
  7. 𝒞 \mathcal{C}
  8. Hom ( - , Q ) \operatorname{Hom}(-,Q)
  9. 𝒞 \mathcal{C}
  10. 𝐀𝐛 \mathbf{Ab}
  11. 𝒜 \mathcal{A}
  12. 𝒜 \mathcal{A}
  13. A A
  14. 𝒜 \mathcal{A}
  15. P P
  16. 𝒜 \mathcal{A}
  17. P A 0. P\longrightarrow A\longrightarrow 0.
  18. p : P A p\colon P\to A
  19. R R
  20. R R
  21. R . \mathcal{M}_{R}.
  22. R \mathcal{M}_{R}
  23. R \mathcal{M}_{R}
  24. R R
  25. R . \mathcal{M}_{R}.
  26. R \mathcal{M}_{R}
  27. R R
  28. R R
  29. M M
  30. F F
  31. R R
  32. X X
  33. M M
  34. X X
  35. M M
  36. π : F M \pi\colon F\to M

Projective_vector_field.html

  1. M M
  2. M M
  3. X X
  4. X a ; b = 1 2 h a b + F a b X_{a;b}=\frac{1}{2}h_{ab}+F_{ab}
  5. h a b = ( X g ) a b = X a ; b + X b ; a h_{ab}=(\mathcal{L}_{X}g)_{ab}=X_{a;b}+X_{b;a}
  6. F a b = 1 2 ( X a ; b - X b ; a ) F_{ab}=\frac{1}{2}(X_{a;b}-X_{b;a})
  7. X a X_{a}
  8. X X
  9. X X
  10. ψ \psi
  11. X a ; b c = R a b c d X d + 2 g a ( b ψ c ) X_{a;bc}\,=R_{abcd}X^{d}+2g_{a(b}\psi_{c)}
  12. h a b ; c = 2 g a b ψ c + g a c ψ b + g b c ψ a h_{ab;c}\,=2g_{ab}\psi_{c}+g_{ac}\psi_{b}+g_{bc}\psi_{a}
  13. P ( M ) P(M)
  14. dim P ( M ) n ( n + 2 ) \dim P(M)\leq n(n+2)
  15. X X
  16. X \nabla X
  17. X \nabla\nabla X
  18. X X
  19. h h
  20. F F
  21. ψ \psi
  22. M M
  23. X R a = b c d δ a ψ b ; c d - δ a ψ b ; d c \mathcal{L}_{X}R^{a}{}_{bcd}=\delta^{a}{}_{d}\psi_{b;c}-\delta^{a}{}_{c}\psi_{% b;d}
  24. X R a b = - 3 ψ a ; b \mathcal{L}_{X}R_{ab}=-3\psi_{a;b}
  25. P ( M ) P(M)
  26. h = 0 \nabla h=0
  27. ψ = 0 \psi=0
  28. M M
  29. P ( M ) P(M)
  30. A ( M ) A(M)
  31. dim A ( M ) n ( n + 1 ) \dim A(M)\leq n(n+1)
  32. X X
  33. h h
  34. F F
  35. M M
  36. X R a = b c d 0 \mathcal{L}_{X}R^{a}{}_{bcd}=0
  37. X R a b = 0 \mathcal{L}_{X}R_{ab}=0
  38. X C a = b c d 0 \mathcal{L}_{X}C^{a}{}_{bcd}=0
  39. h = X g = 2 c g h=\mathcal{L}_{X}g=2cg
  40. h = 0 \nabla h=0
  41. M M
  42. A ( M ) A(M)
  43. H ( M ) H(M)
  44. dim H ( M ) 1 2 n ( n + 1 ) + 1 \dim H(M)\leq\frac{1}{2}n(n+1)+1
  45. X X
  46. F F
  47. c c
  48. h = X g = 0 h=\mathcal{L}_{X}g=0
  49. c = 0 c=0
  50. M M
  51. H ( M ) H(M)
  52. K ( M ) K(M)
  53. dim K ( M ) 1 2 n ( n + 1 ) \dim K(M)\leq\frac{1}{2}n(n+1)
  54. X X
  55. F F
  56. M M
  57. 𝕄 {\mathbb{M}}
  58. dim P ( 𝕄 ) = 24 \dim P({\mathbb{M}})=24

Proof_complexity.html

  1. n + 1 n+1
  2. n n
  3. A ( x , y ) B ( y , z ) A(x,y)\rightarrow B(y,z)
  4. y y
  5. y y
  6. A A
  7. B B
  8. C ( y ) C(y)
  9. A ( x , y ) C ( y ) A(x,y)\rightarrow C(y)
  10. C ( y ) B ( y , z ) C(y)\rightarrow B(y,z)
  11. A ( x , y ) A(x,y)
  12. B ( y , z ) B(y,z)
  13. y y
  14. A ( x , y ) B ( y , z ) A(x,y)\rightarrow B(y,z)
  15. C C
  16. C ( y ) C(y)
  17. A ( x , y ) B ( y , z ) A(x,y)\rightarrow B(y,z)
  18. A ( x , y ) B ( y , z ) A(x,y)\rightarrow B(y,z)

Proofs_involving_covariant_derivatives.html

  1. R a b m n ; l + R a b l m ; n + R a b n l ; m = 0 R_{abmn;l}+R_{ablm;n}+R_{abnl;m}=0\,\!
  2. g b n g a m ( R a b m n ; l + R a b l m ; n + R a b n l ; m ) = 0 , g^{bn}g^{am}(R_{abmn;l}+R_{ablm;n}+R_{abnl;m})=0,\,\!
  3. g b n ( R m - b m n ; l R m + b m l ; n R m ) b n l ; m = 0 , g^{bn}(R^{m}{}_{bmn;l}-R^{m}{}_{bml;n}+R^{m}{}_{bnl;m})=0,\,\!
  4. g b n ( R b n ; l - R b l ; n - R b ) m n l ; m = 0 , g^{bn}(R_{bn;l}-R_{bl;n}-R_{b}{}^{m}{}_{nl;m})=0,\,\!
  5. R n - n ; l R n - l ; n R n m = n l ; m 0. R^{n}{}_{n;l}-R^{n}{}_{l;n}-R^{nm}{}_{nl;m}=0.\,\!
  6. R ; l - R n - l ; n R m = l ; m 0. R_{;l}-R^{n}{}_{l;n}-R^{m}{}_{l;m}=0.\,\!
  7. R ; l = 2 R m , l ; m R_{;l}=2R^{m}{}_{l;m},\,\!
  8. m R m = l 1 2 l R \nabla_{m}R^{m}{}_{l}={1\over 2}\nabla_{l}R\,\!
  9. l R l = m 1 2 m R \nabla_{l}R^{l}{}_{m}={1\over 2}\nabla_{m}R\,\!
  10. l R l - m 1 2 δ l l m R = 0 \nabla_{l}R^{l}{}_{m}-{1\over 2}\delta^{l}{}_{m}\nabla_{l}R=0\,\!
  11. δ l = m g l , m \delta^{l}{}_{m}=g^{l}{}_{m},\,\!
  12. l R l - m 1 2 l g l R m = 0. \nabla_{l}R^{l}{}_{m}-{1\over 2}\nabla_{l}g^{l}{}_{m}R=0.\,\!
  13. l ( R l - m 1 2 g l R m ) = 0 , \nabla_{l}\left(R^{l}{}_{m}-{1\over 2}g^{l}{}_{m}R\right)=0,\,\!
  14. l ( R l m - 1 2 g l m R ) = 0. \nabla_{l}\left(R^{lm}-{1\over 2}g^{lm}R\right)=0.\,\!
  15. l G l m = 0 , \nabla_{l}G^{lm}=0,\,\!

Proofs_of_quadratic_reciprocity.html

  1. ( q p ) = ( - 1 ) u q u / p , \left(\frac{q}{p}\right)=(-1)^{\sum_{u}\left\lfloor qu/p\right\rfloor},
  2. x \left\lfloor x\right\rfloor
  3. ( 7 11 ) = ( - 1 ) 14 / 11 + 28 / 11 + 42 / 11 + 56 / 11 + 70 / 11 = ( - 1 ) 1 + 2 + 3 + 5 + 6 = ( - 1 ) 17 = - 1. \left(\frac{7}{11}\right)=(-1)^{\left\lfloor 14/11\right\rfloor+\left\lfloor 2% 8/11\right\rfloor+\left\lfloor 42/11\right\rfloor+\left\lfloor 56/11\right% \rfloor+\left\lfloor 70/11\right\rfloor}=(-1)^{1+2+3+5+6}=(-1)^{17}=-1.
  4. Σ u q u / p \Sigma_{u}\left\lfloor qu/p\right\rfloor
  5. ( q p ) = ( - 1 ) μ , \left(\frac{q}{p}\right)=(-1)^{\mu},
  6. ( p q ) = ( - 1 ) ν , \left(\frac{p}{q}\right)=(-1)^{\nu},
  7. ( p - 1 2 ) ( q - 1 2 ) , \left(\frac{p-1}{2}\right)\left(\frac{q-1}{2}\right),
  8. ( q p ) ( p q ) = ( - 1 ) μ + ν = ( - 1 ) ( p - 1 ) ( q - 1 ) / 4 . \left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{\mu+\nu}=(-1)^{(p-1)(q-% 1)/4}.
  9. ( - 1 ) r ( 2 ) 2 q ( - 1 ) r ( 4 ) 4 q ( - 1 ) r ( p - 1 ) ( p - 1 ) q 2 4 ( p - 1 ) (mod p ) . (-1)^{r(2)}2q\cdot(-1)^{r(4)}4q\cdot\cdots\cdot(-1)^{r(p-1)}(p-1)q\equiv 2% \cdot 4\cdot\cdots\cdot(p-1)\,\text{ (mod }p).
  10. q ( p - 1 ) / 2 ( - 1 ) r ( 2 ) + r ( 4 ) + + r ( p - 1 ) (mod p ) . q^{(p-1)/2}\equiv(-1)^{r(2)+r(4)+\cdots+r(p-1)}\,\text{ (mod }p).
  11. q u p = q u p + r ( u ) p , \frac{qu}{p}=\left\lfloor\frac{qu}{p}\right\rfloor+\frac{r(u)}{p},
  12. q u / p \left\lfloor qu/p\right\rfloor
  13. q ( p - 1 ) / 2 ( - 1 ) u q u / p (mod p ) . q^{(p-1)/2}\equiv(-1)^{\sum_{u}\left\lfloor qu/p\right\rfloor}\,\text{ (mod }p).
  14. L = 𝐐 ( ζ p ) , L=\mathbf{Q}(\zeta_{p}),
  15. G = Gal ( L / 𝐐 ) ( \Z / p \Z ) × , G=\operatorname{Gal}(L/\mathbf{Q})\cong(\Z/p\Z)^{\times},
  16. σ a ( ζ p ) = ζ p a \sigma_{a}(\zeta_{p})=\zeta_{p}^{a}
  17. a ( \Z / p \Z ) × . a\in(\Z/p\Z)^{\times}.
  18. 𝐐 ( p * ) , \mathbf{Q}(\sqrt{p^{*}}),
  19. p * = { p if p = 1 (mod 4 ) , - p if p = 3 (mod 4 ) . p^{*}=\begin{cases}p&\mbox{if }~{}p=1\,\text{ (mod }4),\\ -p&\mbox{if }~{}p=3\,\text{ (mod }4).\end{cases}
  20. ( 𝐙 / p 𝐙 ) × (\mathbf{Z}/p\mathbf{Z})^{\times}
  21. ( q p ) = 1 σ q H σ q fixes 𝐐 ( p * ) . \left(\frac{q}{p}\right)=1\quad\iff\quad\sigma_{q}\in H\quad\iff\quad\sigma_{q% }\mbox{ fixes }~{}\mathbf{Q}(\sqrt{p^{*}}).
  22. ϕ Gal ( L / 𝐐 ) \phi\in\operatorname{Gal}(L/\mathbf{Q})
  23. ϕ \phi
  24. ϕ ( x ) x q (mod β ) \phi(x)\equiv x^{q}\,\text{ (mod }\beta)\,\!
  25. ϕ \phi
  26. ϕ fixes K q splits completely in K . \phi\mbox{ fixes }~{}K\quad\iff\quad q\mbox{ splits completely in }~{}K.
  27. ϕ ( x ) x q (mod δ ) \phi(x)\equiv x^{q}\,\text{ (mod }\delta)\,\!
  28. ϕ | K Gal ( K / 𝐐 ) \phi|_{K}\in\operatorname{Gal}(K/\mathbf{Q})
  29. ϕ \phi
  30. ord ( ϕ | K ) = [ O K / δ O K : 𝐙 / q 𝐙 ] . \operatorname{ord}(\phi|_{K})=[O_{K}/\delta O_{K}:\mathbf{Z}/q\mathbf{Z}].
  31. ϕ ( ζ p ) = ζ p q ; \phi(\zeta_{p})=\zeta_{p}^{q};
  32. ϕ \phi
  33. ( q p ) = 1 q splits completely in 𝐐 ( p * ) . \left(\frac{q}{p}\right)=1\quad\iff\quad q\mbox{ splits completely in }~{}% \mathbf{Q}(\sqrt{p^{*}}).
  34. q splits completely in 𝐐 ( p * ) ( p * q ) = 1. q\mbox{ splits completely in }~{}\mathbf{Q}(\sqrt{p^{*}})\quad\iff\quad\left(% \frac{p^{*}}{q}\right)=1.
  35. ( p * q ) = ( q p ) \left(\frac{p^{*}}{q}\right)=\left(\frac{q}{p}\right)
  36. ( p * q ) = ( - p q ) = ( - 1 q ) ( p q ) = { + ( p q ) if q = 1 (mod 4 ) , - ( p q ) if q = 3 (mod 4 ) \left(\frac{p^{*}}{q}\right)=\left(\frac{-p}{q}\right)=\left(\frac{-1}{q}% \right)\left(\frac{p}{q}\right)=\begin{cases}+\left(\frac{p}{q}\right)&\mbox{% if }~{}q=1\,\text{ (mod }4),\\ -\left(\frac{p}{q}\right)&\mbox{if }~{}q=3\,\text{ (mod }4)\end{cases}
  37. ( p * q ) = 1. \left(\frac{p^{*}}{q}\right)=1.
  38. x 2 p * (mod q ) , x^{2}\equiv p^{*}\,\text{ (mod }q),\,\!
  39. x 2 - p * = c q x^{2}-p^{*}=cq\,\!
  40. K = 𝐐 ( p * ) , K=\mathbf{Q}(\sqrt{p^{*}}),
  41. ( x - p * , q ) (x-\sqrt{p^{*}},q)
  42. x - p * x-\sqrt{p^{*}}
  43. ( x + p * ) = ( x + p * ) ( x - p * , q ) = ( c q , q ( x + p * ) ) (x+\sqrt{p^{*}})=(x+\sqrt{p^{*}})(x-\sqrt{p^{*}},q)=(cq,q(x+\sqrt{p^{*}}))
  44. ( q ) β , (q)\subsetneq\beta,
  45. a + b p * β ( q ) , a+b\sqrt{p^{*}}\in\beta\setminus(q),
  46. p * = 1 (mod 4 ) , p^{*}=1\,\text{ (mod }4),
  47. 𝐙 [ 1 + p * 2 ] , \mathbf{Z}\left[\frac{1+\sqrt{p^{*}}}{2}\right],
  48. a + b p * β ( q ) , a+b\sqrt{p^{*}}\in\beta\setminus(q),
  49. ( a + b p * ) ( a - b p * ) = a 2 - b 2 p * β 𝐙 = ( q ) , (a+b\sqrt{p^{*}})(a-b\sqrt{p^{*}})=a^{2}-b^{2}p^{*}\in\beta\cap\mathbf{Z}=(q),
  50. q a 2 - b 2 p * . q\mid a^{2}-b^{2}p^{*}.\,\!
  51. a + b p * . a+b\sqrt{p^{*}}.
  52. p * = ( a b - 1 ) 2 (mod q ) p^{*}=(ab^{-1})^{2}\,\text{ (mod }q)\,\!

Property_of_Baire.html

  1. A A
  2. X X
  3. U X U\subseteq X
  4. Δ A , U A\mathbin{\Delta}U

Proportional_control.html

  1. P out = K p e ( t ) + p 0 P_{\mathrm{out}}=K_{p}\,{e(t)+p0}
  2. p 0 p0
  3. P out P_{\mathrm{out}}
  4. K p K_{p}
  5. e ( t ) e(t)
  6. e ( t ) = S P - P V e(t)=SP-PV

Proportional_division.html

  1. n × ( n - 1 ) / 2 = O ( n 2 ) n\times(n-1)/2=O(n^{2})
  2. 1 / n 1/n

Proportional_navigation.html

  1. a n = N λ ˙ V a_{n}=N\dot{\lambda}V
  2. a n a_{n}
  3. N N
  4. λ ˙ \dot{\lambda}
  5. a = N V r × Ω \vec{a}=N\vec{V}_{r}\times\vec{\Omega}
  6. Ω \Omega
  7. Ω = R × V r R R \vec{\Omega}=\frac{\vec{R}\times\vec{V}_{r}}{\vec{R}\cdot\vec{R}}
  8. V r = V t - V m \vec{V}_{r}=\vec{V}_{t}-\vec{V}_{m}
  9. R = R t - R m \vec{R}=\vec{R}_{t}-\vec{R}_{m}
  10. a = - N | V r | R | R | × Ω \vec{a}=-N|\vec{V}_{r}|\frac{\vec{R}}{|\vec{R}|}\times\vec{\Omega}
  11. a = - N | V r | V m | V m | × Ω \vec{a}=-N|\vec{V}_{r}|\frac{\vec{V}_{m}}{|\vec{V}_{m}|}\times\vec{\Omega}

Proportionally_fair.html

  1. i i
  2. w i = 1 / c i w_{i}=1/c_{i}
  3. c i c_{i}
  4. P = T α R β P=\frac{T^{\alpha}}{R^{\beta}}
  5. T T
  6. R R
  7. α \alpha
  8. β \beta
  9. α \alpha
  10. β \beta
  11. α = 0 \alpha=0
  12. β = 1 \beta=1
  13. α = 1 \alpha=1
  14. β = 0 \beta=0
  15. T T
  16. α 1 \alpha\approx 1
  17. β 1 \beta\approx 1

Pseudo-Anosov_map.html

  1. ϕ i j ϕ j = ϕ i , \phi_{ij}\circ\phi_{j}=\phi_{i},
  2. ϕ ( x , y ) = ( f ( x , y ) , c ± y ) \phi(x,y)=(f(x,y),c\pm y)
  3. f : S S f:S\to S

Pseudo-polynomial_time.html

  1. { 2 , 3 , , n } \{2,3,\dots,\sqrt{n}\}
  2. n n
  3. log ( n ) \log(n)
  4. O ( m n ) O(mn)
  5. O ( ( log n ) 6 ) O((\log{n})^{6})

Pseudoholomorphic_curve.html

  1. X X
  2. J J
  3. C C
  4. j j
  5. X X
  6. f : C X f:C\to X
  7. ¯ j , J f := 1 2 ( d f + J d f j ) = 0. \bar{\partial}_{j,J}f:=\frac{1}{2}(df+J\circ df\circ j)=0.
  8. J 2 = - 1 J^{2}=-1
  9. J d f = d f j , J\circ df=df\circ j,
  10. d f df
  11. J J
  12. T x f ( C ) T x X T_{x}f(C)\subseteq T_{x}X
  13. ν \nu
  14. ¯ j , J f = ν . \bar{\partial}_{j,J}f=\nu.
  15. ( j , J , ν ) (j,J,\nu)
  16. ν \nu
  17. X X
  18. C C
  19. g g
  20. n n
  21. C C
  22. 2 - 2 g - n 2-2g-n
  23. C C
  24. C C
  25. X X
  26. C C
  27. j = J = [ 0 - 1 1 0 ] , j=J=\begin{bmatrix}0&-1\\ 1&0\end{bmatrix},
  28. d f = [ d u / d x d u / d y d v / d x d v / d y ] , df=\begin{bmatrix}du/dx&du/dy\\ dv/dx&dv/dy\end{bmatrix},
  29. f ( x , y ) = ( u ( x , y ) , v ( x , y ) ) f(x,y)=(u(x,y),v(x,y))
  30. J d f = d f j J\circ df=df\circ j
  31. { d u / d x = d v / d y d v / d x = - d u / d y . \begin{cases}du/dx=dv/dy\\ dv/dx=-du/dy.\end{cases}
  32. J J
  33. ω \omega
  34. J J
  35. ω \omega
  36. ω ( v , J v ) > 0 \omega(v,Jv)>0
  37. v v
  38. ( v , w ) = 1 2 ( ω ( v , J w ) + ω ( w , J v ) ) (v,w)=\frac{1}{2}\left(\omega(v,Jw)+\omega(w,Jv)\right)
  39. X X
  40. ω \omega
  41. ω \omega
  42. J J
  43. ω \omega
  44. ω \omega

Pseudopotential.html

  1. r c r_{c}
  2. r c r_{c}
  3. V ^ 𝑝𝑠 = V 𝑝𝑠 loc + i , j D i j | p i p j | \hat{V}_{\,\textit{ps}}=V^{\textrm{loc}}_{\,\textit{ps}}+\sum_{i,j}D_{ij}|p_{i% }\rangle\langle p_{j}|
  4. V 𝑝𝑠 loc V^{\textrm{loc}}_{\,\textit{ps}}
  5. p i p_{i}
  6. D i j D_{ij}
  7. 0 = ϕ 𝐑 , i | ϕ 𝐑 , j - ϕ ~ 𝐑 , i | ϕ ~ 𝐑 , j 0=\langle\phi_{\mathbf{R},i}|\phi_{\mathbf{R},j}\rangle-\langle\tilde{\phi}_{% \mathbf{R},i}|\tilde{\phi}_{\mathbf{R},j}\rangle
  8. ϕ 𝐑 , i \phi_{\mathbf{R},i}
  9. ϕ ~ 𝐑 , i \tilde{\phi}_{\mathbf{R},i}
  10. 𝐑 \mathbf{R}
  11. r c r_{c}
  12. q 𝐑 , i j = ϕ 𝐑 , i | ϕ 𝐑 , j - ϕ ~ 𝐑 , i | ϕ ~ 𝐑 , j q_{\mathbf{R},ij}=\langle\phi_{\mathbf{R},i}|\phi_{\mathbf{R},j}\rangle-% \langle\tilde{\phi}_{\mathbf{R},i}|\tilde{\phi}_{\mathbf{R},j}\rangle
  13. H ^ | Ψ i = ϵ i S ^ | Ψ i \hat{H}|\Psi_{i}\rangle=\epsilon_{i}\hat{S}|\Psi_{i}\rangle
  14. S ^ \hat{S}
  15. S ^ = 1 + 𝐑 , i , j | p 𝐑 , i q 𝐑 , i j p 𝐑 , j | \hat{S}=1+\sum_{\mathbf{R},i,j}|p_{\mathbf{R},i}\rangle q_{\mathbf{R},ij}% \langle p_{\mathbf{R},j}|
  16. p 𝐑 , i p_{\mathbf{R},i}
  17. p 𝐑 , i | ϕ ~ 𝐑 , j r < r c = δ i , j \langle p_{\mathbf{R},i}|\tilde{\phi}_{\mathbf{R},j}\rangle_{r<r_{c}}=\delta_{% i,j}
  18. V V
  19. r r
  20. V ( r ) = 2 π 2 m b δ ( r ) V(r)=\frac{2\pi\hbar^{2}}{m}b\,\delta(r)
  21. \hbar
  22. 2 π 2\pi
  23. m m
  24. δ ( r ) \delta(r)
  25. b b
  26. r = 0 r=0
  27. δ \delta

Pseudorandom_ensemble.html

  1. U = { U n } n U=\{U_{n}\}_{n\in\mathbb{N}}
  2. X = { X n } n X=\{X_{n}\}_{n\in\mathbb{N}}
  3. X X
  4. X X
  5. U U

Pseudorapidity.html

  1. η \eta
  2. η - ln [ tan ( θ 2 ) ] , \eta\equiv-\ln\left[\tan\left(\frac{\theta}{2}\right)\right],
  3. θ \theta
  4. 𝐩 \mathbf{p}
  5. θ = 2 arctan ( e - η ) . \theta=2\arctan\left(e^{-\eta}\right).
  6. 𝐩 \mathbf{p}
  7. η = 1 2 ln ( | 𝐩 | + p L | 𝐩 | - p L ) = artanh ( p L | 𝐩 | ) , \eta=\frac{1}{2}\ln\left(\frac{\left|\mathbf{p}\right|+p\text{L}}{\left|% \mathbf{p}\right|-p\text{L}}\right)=\operatorname{artanh}\left(\frac{p_{L}}{% \left|\mathbf{p}\right|}\right),
  8. p L p\text{L}
  9. p z p_{z}
  10. m p E p η y m\ll p\Rightarrow E\approx p\Rightarrow\eta\approx y
  11. y 1 2 ln ( E + p L E - p L ) y\equiv\frac{1}{2}\ln\left(\frac{E+p\text{L}}{E-p\text{L}}\right)
  12. | 𝐩 | \left|\mathbf{p}\right|
  13. p L p\text{L}
  14. θ \theta
  15. θ \theta
  16. Δ y \Delta y
  17. Δ η \Delta\eta
  18. | η | |\eta|
  19. y = ln ( m 2 + p T 2 cosh 2 η + p T sinh η m 2 + p T 2 ) , y=\ln\left(\frac{\sqrt{m^{2}+p_{T}^{2}\cosh^{2}\eta}+p_{T}\sinh\eta}{\sqrt{m^{% 2}+p_{T}^{2}}}\right),
  20. p T p x 2 + p y 2 p\text{T}\equiv\sqrt{p_{x}^{2}+p_{y}^{2}}
  21. ( Δ R ) 2 ( Δ η ) 2 + ( Δ ϕ ) 2 \left(\Delta R\right)^{2}\equiv\left(\Delta\eta\right)^{2}+\left(\Delta\phi% \right)^{2}
  22. Δ ϕ \Delta\phi
  23. θ \theta
  24. η \eta
  25. θ \theta
  26. η \eta
  27. θ = 90 \theta=90
  28. η ( θ ) = - η ( 180 - θ ) \eta(\theta)=-\eta(180^{\circ}-\theta)
  29. p T p\text{T}
  30. ϕ \phi
  31. η \eta
  32. ( p x , p y , p z ) (p_{x},p_{y},p_{z})
  33. z z
  34. p x = p T cos ϕ p_{x}=p\text{T}\cos\phi
  35. p y = p T sin ϕ p_{y}=p\text{T}\sin\phi
  36. p z = p T sinh η p_{z}=p\text{T}\sinh{\eta}
  37. | p | = p T cosh η |p|=p\text{T}\cosh{\eta}

Puiseux_series.html

  1. i = k a i T i / n , \sum_{i=k}^{\infty}a_{i}T^{i/n},
  2. k k
  3. n n
  4. P ( x , y ) = 0 P(x,y)=0
  5. y y
  6. x x
  7. x x
  8. f = k = k 0 + c k T k / n f=\sum_{k=k_{0}}^{+\infty}c_{k}T^{k/n}
  9. k 0 k_{0}
  10. k 0 k_{0}
  11. K ( ( T 1 / n ) ) K(\!(T^{1/n})\!)
  12. T 1 / n T^{1/n}
  13. T 1 / 2 T^{1/2}
  14. T 15 / 30 T^{15/30}
  15. K ( ( T ) ) K(\!(T)\!)
  16. K ( ( T n ) ) K(\!(T_{n})\!)
  17. K ( ( T m ) ) K ( ( T n ) ) K(\!(T_{m})\!)\to K(\!(T_{n})\!)
  18. T m ( T n ) n / m T_{m}\mapsto(T_{n})^{n/m}
  19. \mathbb{Q}
  20. v ( f ) v(f)
  21. f = k = k 0 + c k T k / n f=\sum_{k=k_{0}}^{+\infty}c_{k}T^{k/n}
  22. k / n k/n
  23. c k c_{k}
  24. k / n k/n
  25. c k c_{k}
  26. exp ( - v ( f ) ) \exp(-v(f))
  27. f = k = k 0 + c k T k / n f=\sum_{k=k_{0}}^{+\infty}c_{k}T^{k/n}
  28. w ^ ( f ) \hat{w}(f)
  29. f = k = k 0 + c k T k / n f=\sum_{k=k_{0}}^{+\infty}c_{k}T^{k/n}
  30. ω v + w ( c k ) , \omega\cdot v+w(c_{k}),
  31. v = k / n v=k/n
  32. c k c_{k}
  33. w ^ \hat{w}
  34. × Γ \mathbb{Q}\times\Gamma
  35. K = K=\mathbb{C}
  36. X 2 - X = T - 1 X^{2}-X=T^{-1}
  37. X = T - 1 / 2 + 1 2 + 1 8 T 1 / 2 - 1 128 T 3 / 2 + X=T^{-1/2}+\frac{1}{2}+\frac{1}{8}T^{1/2}-\frac{1}{128}T^{3/2}+\cdots
  38. X = - T - 1 / 2 + 1 2 - 1 8 T 1 / 2 + 1 128 T 3 / 2 + X=-T^{-1/2}+\frac{1}{2}-\frac{1}{8}T^{1/2}+\frac{1}{128}T^{3/2}+\cdots
  39. T - 1 T^{-1}
  40. X p - X = T - 1 X^{p}-X=T^{-1}
  41. - 1 p -\frac{1}{p}
  42. X = T - 1 / p + X 1 X=T^{-1/p}+X_{1}
  43. X p = T - 1 + X 1 p , so X 1 p - X 1 = T - 1 / p X^{p}=T^{-1}+{X_{1}}^{p},\,\text{ so }{X_{1}}^{p}-X_{1}=T^{-1/p}
  44. X 1 X_{1}
  45. - 1 p 2 -\frac{1}{p^{2}}
  46. T - 1 / p + T - 1 / p 2 + T - 1 / p 3 + ; T^{-1/p}+T^{-1/p^{2}}+T^{-1/p^{3}}+\cdots;\,
  47. K ( ( T ) ) K(\!(T)\!)
  48. F ( x , y ) = 0 F(x,y)=0
  49. F ( x , f ( x ) ) = 0 F(x,f(x))=0
  50. x = t n + x=t^{n}+\cdots
  51. y = c t k + y=ct^{k}+\cdots
  52. f = c T k / n + f=cT^{k/n}+\cdots
  53. T 1 / n T^{1/n}
  54. y ( t ) = f ( x ( t ) ) y(t)=f(x(t))
  55. x ( t ) 1 / n = t + x(t)^{1/n}=t+\cdots
  56. y 2 = x 3 + x 2 y^{2}=x^{3}+x^{2}
  57. t ( t 2 - 1 , t 3 - t ) t\mapsto(t^{2}-1,t^{3}-t)
  58. y = x + 1 2 x 2 - 1 8 x 3 + y=x+\frac{1}{2}x^{2}-\frac{1}{8}x^{3}+\cdots
  59. y = - x - 1 2 x 2 + 1 8 x 3 + y=-x-\frac{1}{2}x^{2}+\frac{1}{8}x^{3}+\cdots
  60. y = - ( x + 1 ) 1 / 2 + ( x + 1 ) 3 / 2 y=-(x+1)^{1/2}+(x+1)^{3/2}
  61. y 2 = x 3 y^{2}=x^{3}
  62. t ( t 2 , t 3 ) t\mapsto(t^{2},t^{3})
  63. y = x 3 / 2 y=x^{3/2}
  64. K = K=\mathbb{C}
  65. | x | |x|
  66. f = e c e T e f=\sum_{e}c_{e}T^{e}\,
  67. c e 0 c_{e}\neq 0
  68. k = 1 + T k + 1 k \sum_{k=1}^{+\infty}T^{k+\frac{1}{k}}
  69. \mathbb{Q}
  70. \mathbb{R}

Pumping_lemma_for_regular_languages.html

  1. ( L Σ * ) ( regular ( L ) ( ( p 1 ) ( ( w L ) ( ( | w | p ) ( ( x , y , z Σ * ) ( w = x y z ( | y | 1 | x y | p ( i 0 ) ( x y i z L ) ) ) ) ) ) ) ) \begin{array}[]{l}(\forall L\subseteq\Sigma^{*})\\ \quad(\mbox{regular}~{}(L)\Rightarrow\\ \quad((\exists p\geq 1)((\forall w\in L)((|w|\geq p)\Rightarrow\\ \quad((\exists x,y,z\in\Sigma^{*})(w=xyz\land(|y|\geq 1\land|xy|\leq p\land(% \forall i\geq 0)(xy^{i}z\in L))))))))\end{array}
  2. L \displaystyle L

Punching.html

  1. F = 0.7 t L ( U T S ) F=0.7tL(UTS)

Pythagoras_tree_(fractal).html

  1. 1 2 2 n \tfrac{1}{2}\sqrt{2}^{n}

P′′.html

  1. 𝒫 ′′ \mathcal{P}^{\prime\prime}

Q-theta_function.html

  1. θ ( z ; q ) = n = 0 ( 1 - q n z ) ( 1 - q n + 1 / z ) \theta(z;q)=\prod_{n=0}^{\infty}(1-q^{n}z)\left(1-q^{n+1}/z\right)
  2. θ ( z ; q ) = ( z ; q ) ( q / z ; q ) \theta(z;q)=(z;q)_{\infty}(q/z;q)_{\infty}
  3. ( ) (\cdot\cdot)_{\infty}

Q_meter.html

  1. Q = 2 π × Peak Energy Stored Energy dissipated per cycle . Q=2\pi\times\frac{\mbox{Peak Energy Stored}~{}}{\mbox{Energy dissipated per % cycle}~{}}.\,
  2. Q = X L R = ω L R Q=\frac{X_{L}}{R}=\frac{\omega L}{R}
  3. X L X_{L}
  4. L L
  5. ω \omega
  6. R R
  7. R R
  8. Q = F B W Q=\frac{F}{BW}
  9. F F
  10. B W BW

Q_value_(nuclear_science).html

  1. Q = K ( Final ) - K ( Initial ) = ( m I n i t i a l - m F i n a l ) c 2 Q=K_{(\,\text{Final})}-K_{(\,\text{Initial})}=(m_{Initial}-m_{Final})c^{2}
  2. Q = ( m n - m p - m ν ¯ - m e ) c 2 = K p + K e + K ν ¯ = 0.782 M e V Q=(m\text{n}-m\text{p}-m_{\mathrm{\overline{\nu}}}-m\text{e})c^{2}=K_{p}+K_{e}% +K_{\overline{\nu}}=0.782MeV
  3. ν ¯ \overline{ν}

Quadratic_differential.html

  1. U U
  2. f ( z ) d z d z f(z)dz\otimes dz
  3. z z
  4. f f
  5. U U
  6. f f
  7. μ \mu
  8. R R
  9. q q
  10. R R
  11. ( μ - 1 ) * ( q ) (\mu^{-1})^{*}(q)
  12. ω \omega
  13. ω ω \omega\otimes\omega
  14. q q
  15. | q | |q|
  16. q q
  17. q = f ( z ) d z d z q=f(z)dz\otimes dz
  18. | f ( z ) | ( d x 2 + d y 2 ) |f(z)|(dx^{2}+dy^{2})
  19. z = x + i y z=x+iy
  20. f f
  21. z z
  22. f ( z ) = 0 f(z)=0

Quadratic_variation.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. [ X ] t = lim P 0 k = 1 n ( X t k - X t k - 1 ) 2 [X]_{t}=\lim_{\|P\|\rightarrow 0}\sum_{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}
  3. [ X , Y ] t = lim P 0 k = 1 n ( X t k - X t k - 1 ) ( Y t k - Y t k - 1 ) . [X,Y]_{t}=\lim_{\|P\|\to 0}\sum_{k=1}^{n}\left(X_{t_{k}}-X_{t_{k-1}}\right)% \left(Y_{t_{k}}-Y_{t_{k-1}}\right).
  4. [ X , Y ] t = 1 2 ( [ X + Y ] t - [ X ] t - [ Y ] t ) . [X,Y]_{t}=\tfrac{1}{2}([X+Y]_{t}-[X]_{t}-[Y]_{t}).
  5. [ X ] t = 0 < s t ( Δ X s ) 2 . [X]_{t}=\sum_{0<s\leq t}(\Delta X_{s})^{2}.
  6. k = 1 n ( X t k - X t k - 1 ) 2 max k n | X t k - X t k - 1 | k = 1 n | X t k - X t k - 1 | max | u - v | P | X u - X v | V t ( X ) . \begin{aligned}\displaystyle\sum_{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}&% \displaystyle\leq\max_{k\leq n}|X_{t_{k}}-X_{t_{k-1}}|\sum_{k=1}^{n}|X_{t_{k}}% -X_{t_{k-1}}|\\ &\displaystyle\leq\max_{|u-v|\leq\|P\|}|X_{u}-X_{v}|V_{t}(X).\end{aligned}
  7. P \|P\|
  8. X t = X 0 + 0 t σ s d B s + 0 t μ s d s , X_{t}=X_{0}+\int_{0}^{t}\sigma_{s}\,dB_{s}+\int_{0}^{t}\mu_{s}\,ds,
  9. [ X ] t = 0 t σ s 2 d s . [X]_{t}=\int_{0}^{t}\sigma_{s}^{2}\,ds.
  10. X t Y t = X 0 Y 0 + 0 t X s - d Y s + 0 t Y s - d X s + [ X , Y ] t , X_{t}Y_{t}=X_{0}Y_{0}+\int_{0}^{t}X_{s-}\,dY_{s}+\int_{0}^{t}Y_{s-}\,dX_{s}+[X% ,Y]_{t},
  11. d ( X t Y t ) = X t - d Y t + Y t - d X t + d X t d Y t , \,d(X_{t}Y_{t})=X_{t-}\,dY_{t}+Y_{t-}\,dX_{t}+\,dX_{t}\,dY_{t},
  12. d X t d Y t = d [ X , Y ] t . \,dX_{t}\,dY_{t}=\,d[X,Y]_{t}.
  13. 𝔼 ( ( 0 t H d M ) 2 ) = 𝔼 ( 0 t H 2 d [ M ] ) . \mathbb{E}\left(\left(\int_{0}^{t}H\,dM\right)^{2}\right)=\mathbb{E}\left(\int% _{0}^{t}H^{2}\,d[M]\right).
  14. c p 𝔼 ( [ M ] t p / 2 ) 𝔼 ( ( M t * ) p ) C p 𝔼 ( [ M ] t p / 2 ) . c_{p}\mathbb{E}([M]_{t}^{p/2})\leq\mathbb{E}((M^{*}_{t})^{p})\leq C_{p}\mathbb% {E}([M]_{t}^{p/2}).

Quadratrix.html

  1. y = x cot π x 2 a . y=x\cot\frac{\pi x}{2a}.
  2. π \pi
  3. π \pi
  4. π \pi

Quadrifolium.html

  1. r = cos ( 2 θ ) , r=\cos(2\theta),\,
  2. ( x 2 + y 2 ) 3 = ( x 2 - y 2 ) 2 . (x^{2}+y^{2})^{3}=(x^{2}-y^{2})^{2}.\,
  3. r = sin ( 2 θ ) r=\sin(2\theta)\,
  4. ( x 2 + y 2 ) 3 = 4 x 2 y 2 . (x^{2}+y^{2})^{3}=4x^{2}y^{2}.\,
  5. ( x 2 - y 2 ) 4 + 837 ( x 2 + y 2 ) 2 + 108 x 2 y 2 = 16 ( x 2 + 7 y 2 ) ( y 2 + 7 x 2 ) ( x 2 + y 2 ) + 729 ( x 2 + y 2 ) . (x^{2}-y^{2})^{4}+837(x^{2}+y^{2})^{2}+108x^{2}y^{2}=16(x^{2}+7y^{2})(y^{2}+7x% ^{2})(x^{2}+y^{2})+729(x^{2}+y^{2}).\,
  6. 1 2 π \tfrac{1}{2}\pi

Quadrupole_ion_trap.html

  1. d 2 u d ξ 2 + [ a u - 2 q u cos ( 2 ξ ) ] u = 0 ( 1 ) \frac{d^{2}u}{d\xi^{2}}+[a_{u}-2q_{u}\cos(2\xi)]u=0\qquad\qquad(1)\!
  2. u u
  3. ξ \xi
  4. ξ = Ω t / 2 \xi=\Omega t/2
  5. a u a_{u}\,
  6. q u q_{u}\,
  7. Ω \Omega\,
  8. d 2 u d t 2 = Ω 2 4 d 2 u d ξ 2 ( 2 ) \frac{d^{2}u}{dt^{2}}=\frac{\Omega^{2}}{4}\frac{d^{2}u}{d\xi^{2}}\qquad\qquad(% 2)\!
  9. 4 Ω 2 d 2 u d t 2 + [ a u - 2 q u cos ( Ω t ) ] u = 0 ( 3 ) \frac{4}{\Omega^{2}}\frac{d^{2}u}{dt^{2}}+[a_{u}-2q_{u}\cos(\Omega t)]u=0% \qquad\qquad(3)\!
  10. m d 2 u d t 2 + m Ω 2 4 [ a u - 2 q u cos ( Ω t ) ] u = 0 ( 3 ) m\frac{d^{2}u}{dt^{2}}+m\frac{\Omega^{2}}{4}[a_{u}-2q_{u}\cos(\Omega t)]u=0% \qquad\qquad(3)\!
  11. F x = m a = m d 2 x d t 2 = - e ϕ x ( 4 ) F_{x}=ma=m\frac{d^{2}x}{dt^{2}}=-e\frac{\partial\phi}{\partial x}\qquad\qquad(% 4)\!
  12. ϕ \phi\,
  13. ϕ = ϕ 0 r 0 2 ( λ x 2 + σ y 2 + γ z 2 ) ( 5 ) \phi=\frac{\phi_{0}}{r_{0}^{2}}\big(\lambda x^{2}+\sigma y^{2}+\gamma z^{2}% \big)\qquad\qquad(5)\!
  14. ϕ 0 \phi_{0}\,
  15. λ \lambda\,
  16. σ \sigma\,
  17. γ \gamma\,
  18. r 0 r_{0}\,
  19. 2 ϕ 0 = 0 \nabla^{2}\phi_{0}=0\,
  20. λ + σ + γ = 0 \lambda+\sigma+\gamma=0\,
  21. λ = σ = 1 \lambda=\sigma=1\,
  22. γ = - 2 \gamma=-2\,
  23. λ = - σ = 1 \lambda=-\sigma=1\,
  24. γ = 0 \gamma=0\,
  25. x = r cos θ {x}={r}\,\cos\theta
  26. y = r sin θ {y}={r}\,\sin\theta
  27. z = z {z}={z}\,
  28. sin 2 θ + cos 2 θ = 1 \sin^{2}\theta+\cos^{2}\theta=1\,
  29. ϕ r , z = ϕ 0 r 0 2 ( r 2 - 2 z 2 ) . ( 6 ) \phi_{r,z}=\frac{\phi_{0}}{r_{0}^{2}}\big(r^{2}-2z^{2}\big).\qquad\qquad(6)\!
  30. ϕ 0 = U + V cos Ω t . ( 7 ) \phi_{0}=U+V\cos\Omega t.\qquad\qquad(7)\!
  31. Ω = 2 π ν \Omega=2\pi\nu
  32. ν \nu
  33. λ = 1 \lambda=1
  34. ϕ x = 2 x r 0 2 ( U + V cos Ω t ) . ( 8 ) \frac{\partial\phi}{\partial x}=\frac{2x}{r_{0}^{2}}\big(U+V\cos\Omega t\big).% \qquad\qquad(8)\!
  35. m d 2 x d t 2 = - 2 e r 0 2 ( U + V cos Ω t ) x . ( 9 ) m\frac{d^{2}x}{dt^{2}}=-\frac{2e}{r_{0}^{2}}\big(U+V\cos\Omega t\big)x.\qquad% \qquad(9)\!
  36. a x = 8 e U m r 0 2 Ω 2 ( 10 ) a_{x}=\frac{8eU}{mr_{0}^{2}\Omega^{2}}\qquad\qquad(10)\!
  37. q x = - 4 e V m r 0 2 Ω 2 . ( 11 ) q_{x}=-\frac{4eV}{mr_{0}^{2}\Omega^{2}}.\qquad\qquad(11)\!
  38. q x = q y q_{x}=q_{y}\,
  39. a z = - 8 e U m r 0 2 Ω 2 ( 12 ) a_{z}=-\frac{8eU}{mr_{0}^{2}\Omega^{2}}\qquad\qquad(12)\!
  40. q z = 4 e V m r 0 2 Ω 2 . ( 13 ) q_{z}=\frac{4eV}{mr_{0}^{2}\Omega^{2}}.\qquad\qquad(13)\!
  41. q u q_{u}
  42. a u a_{u}

Quantifier_elimination.html

  1. α \alpha
  2. α Q F \alpha_{QF}
  3. α Q F \alpha_{QF}
  4. α \alpha
  5. x . i = 1 n L i \exists x.\bigwedge_{i=1}^{n}L_{i}
  6. L i L_{i}
  7. F F
  8. j = 1 m i = 1 n L i j , \bigvee_{j=1}^{m}\bigwedge_{i=1}^{n}L_{ij},
  9. x . j = 1 m i = 1 n L i j \exists x.\bigvee_{j=1}^{m}\bigwedge_{i=1}^{n}L_{ij}
  10. j = 1 m x . i = 1 n L i j . \bigvee_{j=1}^{m}\exists x.\bigwedge_{i=1}^{n}L_{ij}.
  11. x . F \forall x.F
  12. F F
  13. ¬ F \lnot F
  14. x . F \forall x.F
  15. ¬ x . ¬ F . \lnot\exists x.\lnot F.

Quantum_calculus.html

  1. q = e i h = e 2 π i q=e^{ih}=e^{2\pi i\hbar}\,
  2. = h 2 π \scriptstyle\hbar=\frac{h}{2\pi}\,
  3. d q ( f ( x ) ) = f ( q x ) - f ( x ) d_{q}(f(x))=f(qx)-f(x)\,
  4. d h ( f ( x ) ) = f ( x + h ) - f ( x ) d_{h}(f(x))=f(x+h)-f(x)\,
  5. D q ( f ( x ) ) = d q ( f ( x ) ) d q ( x ) = f ( q x ) - f ( x ) ( q - 1 ) x D_{q}(f(x))=\frac{d_{q}(f(x))}{d_{q}(x)}=\frac{f(qx)-f(x)}{(q-1)x}
  6. D h ( f ( x ) ) = d h ( f ( x ) ) d h ( x ) = f ( x + h ) - f ( x ) h D_{h}(f(x))=\frac{d_{h}(f(x))}{d_{h}(x)}=\frac{f(x+h)-f(x)}{h}
  7. f ( x ) d q x \int f(x)d_{q}x
  8. f ( x ) d q x = ( 1 - q ) j = 0 x q j f ( x q j ) \int f(x)d_{q}x=(1-q)\sum_{j=0}^{\infty}xq^{j}f(xq^{j})
  9. f ( x ) d h x \int f(x)d_{h}x
  10. a b f ( x ) d h x \int_{a}^{b}f(x)d_{h}x
  11. x n x^{n}
  12. n n
  13. n x n - 1 nx^{n-1}
  14. D q ( x n ) = q n - 1 q - 1 x n - 1 = [ n ] q x n - 1 D_{q}(x^{n})=\frac{q^{n}-1}{q-1}x^{n-1}=[n]_{q}\ x^{n-1}
  15. [ n ] q = q n - 1 q - 1 [n]_{q}=\frac{q^{n}-1}{q-1}
  16. D h ( x n ) = x n - 1 + h x n - 2 + + h n - 1 D_{h}(x^{n})=x^{n-1}+hx^{n-2}+\cdots+h^{n-1}
  17. [ n ] q x n - 1 [n]_{q}x^{n-1}
  18. x n x^{n}
  19. x n x^{n}
  20. ( x ) n := x ( x - 1 ) ( x - n + 1 ) . (x)_{n}:=x(x-1)\cdots(x-n+1).

Quantum_cascade_laser.html

  1. n i n_{i}
  2. i i
  3. τ i f \tau_{if}
  4. W i f W_{if}
  5. i i
  6. f f
  7. d n 3 d t = I in + n 1 τ 13 + n 2 τ 23 - n 3 τ 31 - n 3 τ 32 \frac{\mathrm{d}n_{3}}{\mathrm{d}t}=I_{\mathrm{in}}+\frac{n_{1}}{\tau_{13}}+% \frac{n_{2}}{\tau_{23}}-\frac{n_{3}}{\tau_{31}}-\frac{n_{3}}{\tau_{32}}
  8. d n 2 d t = n 3 τ 32 + n 1 τ 12 - n 2 τ 21 - n 2 τ 23 \frac{\mathrm{d}n_{2}}{\mathrm{d}t}=\frac{n_{3}}{\tau_{32}}+\frac{n_{1}}{\tau_% {12}}-\frac{n_{2}}{\tau_{21}}-\frac{n_{2}}{\tau_{23}}
  9. d n 1 d t = n 2 τ 21 + n 3 τ 31 - n 1 τ 13 - n 1 τ 12 - I out \frac{\mathrm{d}n_{1}}{\mathrm{d}t}=\frac{n_{2}}{\tau_{21}}+\frac{n_{3}}{\tau_% {31}}-\frac{n_{1}}{\tau_{13}}-\frac{n_{1}}{\tau_{12}}-I_{\mathrm{out}}
  10. I in = I out = I I_{\mathrm{in}}=I_{\mathrm{out}}=I
  11. d n i d t = j = 1 N n j τ j i - n i j = 1 N 1 τ i j + I ( δ i N - δ i 1 ) \frac{\mathrm{d}n_{i}}{\mathrm{d}t}=\sum\limits_{j=1}^{N}\frac{n_{j}}{\tau_{ji% }}-n_{i}\sum\limits_{j=1}^{N}\frac{1}{\tau_{ij}}+I(\delta_{iN}-\delta_{i1})
  12. n 1 τ 12 = n 2 τ 23 = 0 \frac{n_{1}}{\tau_{12}}=\frac{n_{2}}{\tau_{23}}=0
  13. n 3 τ 32 = n 2 τ 21 \frac{n_{3}}{\tau_{32}}=\frac{n_{2}}{\tau_{21}}
  14. τ 32 > τ 21 \tau_{32}>\tau_{21}
  15. W 21 > W 32 W_{21}>W_{32}
  16. n 3 > n 2 n_{3}>n_{2}
  17. n 3 n 2 = τ 32 τ 21 = W 21 W 32 \frac{n_{3}}{n_{2}}=\frac{\tau_{32}}{\tau_{21}}=\frac{W_{21}}{W_{32}}
  18. N 2 D N_{\mathrm{2D}}
  19. i = 1 N n i = N 2 D \sum\limits_{i=1}^{N}n_{i}=N_{\mathrm{2D}}
  20. N 2 D N_{\mathrm{2D}}
  21. W 32 W_{32}
  22. W 32 W_{32}
  23. W 21 W_{21}
  24. W 21 W_{21}

Quantum_defect.html

  1. ω p ~{}\omega_{\rm p}~{}
  2. ω s ~{}\omega_{\rm s}~{}
  3. q = ω p - ω s ~{}q=\hbar\omega_{\rm p}-\hbar\omega_{\rm s}~{}
  4. E B = - R h c n 2 E\text{B}=-\dfrac{Rhc}{n^{2}}
  5. E B = - R h c ( n - δ l ) 2 E\text{B}=-\dfrac{Rhc}{(n-\delta_{l})^{2}}

Quantum_network.html

  1. k k
  2. n n
  3. 2 n 2^{n}
  4. n n
  5. 2 k n 2^{kn}
  6. k 2 n k2^{n}
  7. | A |A\rangle
  8. | R a |R_{a}\rangle
  9. | R b |R_{b}\rangle
  10. | B |B\rangle
  11. | R a |R_{a}\rangle
  12. | R b |R_{b}\rangle
  13. | R a |R_{a}\rangle
  14. | B |B\rangle
  15. | A |A\rangle
  16. | B |B\rangle

Quantum_programming.html

  1. | ϕ |\phi\rangle
  2. | ϕ | ϕ |\phi\rangle\otimes|\phi\rangle

Quantum_register.html

  1. | Ψ = | ψ 1 | ψ 2 | ψ n |\Psi\rangle=|\psi\rangle_{1}\otimes|\psi\rangle_{2}\otimes\ldots\otimes|\psi% \rangle_{n}
  2. = 1 2 n \mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\ldots\otimes\mathcal{% H}_{n}

Quantum_tomography.html

  1. P ( E i | ρ ) = Trace ( E i ρ ) \mathrm{P}(E_{i}|\rho)=\mathrm{Trace}(E_{i}\rho)
  2. E i E_{i}
  3. ρ \rho
  4. p i p_{i}
  5. P ( E i | ρ ) \mathrm{P}(E_{i}|\rho)
  6. E i E_{i}
  7. S S
  8. T T
  9. S T = Tr [ S T ] = S T S\cdot T=\mathrm{Tr}[S^{\dagger}T]=\vec{S}^{\dagger}\vec{T}
  10. T \vec{T}
  11. T T
  12. S \vec{S}^{\dagger}
  13. S T \vec{S}^{\dagger}\vec{T}
  14. d \mathbb{C}^{d}
  15. A A
  16. A = ( E 1 E 2 E 3 ) A=\begin{pmatrix}\vec{E}_{1}^{\dagger}\\ \vec{E}_{2}^{\dagger}\\ \vec{E}_{3}^{\dagger}\\ \vdots\end{pmatrix}
  17. ρ \vec{\rho}
  18. A ρ = ( E 1 ρ E 2 ρ E 3 ρ ) = ( E 1 ρ E 2 ρ E 3 ρ ) = ( P ( E 1 | ρ ) P ( E 2 | ρ ) P ( E 3 | ρ ) ) ( p 1 p 2 p 3 ) = p A\vec{\rho}=\begin{pmatrix}E_{1}^{\dagger}\vec{\rho}\\ E_{2}^{\dagger}\vec{\rho}\\ E_{3}^{\dagger}\vec{\rho}\\ \vdots\end{pmatrix}=\begin{pmatrix}E_{1}\cdot\rho\\ E_{2}\cdot\rho\\ E_{3}\cdot\rho\\ \vdots\end{pmatrix}=\begin{pmatrix}\mathrm{P}(E_{1}|\rho)\\ \mathrm{P}(E_{2}|\rho)\\ \mathrm{P}(E_{3}|\rho)\\ \vdots\end{pmatrix}\approx\begin{pmatrix}p_{1}\\ p_{2}\\ p_{3}\\ \vdots\end{pmatrix}=\vec{p}
  19. p \vec{p}
  20. ρ \vec{\rho}
  21. ρ \displaystyle\rho
  22. E i E_{i}
  23. σ x , σ y , σ z \sigma_{x},\sigma_{y},\sigma_{z}
  24. A A
  25. A T A^{T}
  26. A T A ρ = A T p A^{T}A\vec{\rho}=A^{T}\vec{p}
  27. ρ \vec{\rho}
  28. ρ = ( A T A ) - 1 A T p \vec{\rho}=(A^{T}A)^{-1}A^{T}\vec{p}
  29. A T A A^{T}A
  30. θ \theta
  31. w ( q , θ ) w(q,\theta)
  32. θ \theta
  33. q q
  34. w ( q , θ ) w(q,\theta)
  35. W ( x , p ) \mathrm{W}(x,p)
  36. n ^ i = a ^ i a ^ i \hat{n}_{i}=\hat{a}_{i}^{\dagger}\hat{a}_{i}
  37. a ^ 1 = 2 - 1 / 2 ( α ^ - α L O ) \hat{a}_{1}=2^{-1/2}(\hat{\alpha}-\alpha_{LO})
  38. a ^ 2 = 2 - 1 / 2 \hat{a}_{2}=2^{-1/2}
  39. ( α ^ + α L O ) (\hat{\alpha}+\alpha_{LO})
  40. a ^ \hat{a}
  41. n ^ 21 = n ^ 2 - n ^ 1 = α L O * a ^ + α L O a ^ \hat{n}_{21}=\hat{n}_{2}-\hat{n}_{1}=\alpha^{*}_{LO}\hat{a}+\alpha_{LO}\hat{a}% ^{\dagger}
  42. q ^ = 2 - 1 / 2 ( a ^ + a ^ ) \hat{q}=2^{-1/2}(\hat{a}^{\dagger}+\hat{a})
  43. n ^ 21 = 2 1 / 2 | α ^ L O | q ^ θ \hat{n}_{21}=2^{1/2}|\hat{\alpha}_{LO}|\hat{q}_{\theta}
  44. q ^ θ \hat{q}_{\theta}
  45. q ^ θ \hat{q}_{\theta}
  46. { | y j y j | } \{|y_{j}\rangle\langle y_{j}|\}
  47. f j f_{j}
  48. ρ ^ \hat{\rho}
  49. L ( ρ ^ ) = j y j | ρ ^ | y j f j L(\hat{\rho})=\prod_{j}\langle y_{j}|\hat{\rho}|y_{j}\rangle^{f_{j}}
  50. y j | ρ ^ | y j \langle y_{j}|\hat{\rho}|y_{j}\rangle
  51. y j y_{j}
  52. ρ ^ \hat{\rho}
  53. ϵ \epsilon
  54. 1 N + 2 \scriptstyle\frac{1}{N+2}
  55. Π l \Pi_{l}
  56. Tr [ Π l ρ m ] = P ( l | ρ m ) \displaystyle\mathrm{Tr}[\Pi_{l}\rho_{m}]=\mathrm{P}(l|\rho_{m})
  57. Π l \Pi_{l}
  58. Π l \Pi_{l}
  59. Pr ( m | n ) = Tr { ρ ^ r e t r [ n ] Θ ^ m } , \mathrm{Pr}\left(m|n\right)=\mathrm{Tr}\{\hat{\rho}_{retr}^{[n]}\hat{\Theta}_{% m}\},
  60. ρ ^ r e t r [ n ] \hat{\rho}_{retr}^{[n]}
  61. Θ ^ m \hat{\Theta}_{m}
  62. Π ^ n \hat{\Pi}_{n}
  63. ρ ^ m \hat{\rho}_{m}
  64. m Θ ^ m = 1 ^ . \sum_{m}\,\hat{\Theta}_{m}=\hat{1}.
  65. ρ ^ r e t r [ n ] \hat{\rho}_{retr}^{[n]}
  66. Θ ^ m \hat{\Theta}_{m}
  67. ρ ^ r e t r [ n ] = Π ^ n Tr { Π n } , \hat{\rho}_{retr}^{[n]}=\frac{\hat{\Pi}_{n}}{\mathrm{Tr}\{\Pi_{n}\}},
  68. Θ ^ m = D 𝒫 m ρ ^ m , \hat{\Theta}_{m}=D\mathcal{P}_{m}\hat{\rho}_{m},
  69. D D
  70. 𝒫 m \mathcal{P}_{m}
  71. ρ ^ m \hat{\rho}_{m}
  72. ρ ^ [ ? ] = m 𝒫 m ρ ^ m = 1 ^ / D , \hat{\rho}^{[?]}=\sum_{m}\,\mathcal{P}_{m}\hat{\rho}_{m}=\hat{1}/D,
  73. Pr ( m | n ) \mathrm{Pr}\left(m|n\right)
  74. 𝒫 m \mathcal{P}_{m}
  75. Θ ^ m \hat{\Theta}_{m}
  76. Pr ( m | n ) \mathrm{Pr}\left(m|n\right)
  77. ( ρ ) \mathcal{E}(\rho)
  78. ( ρ ) = i A i ρ A i \mathcal{E}(\rho)=\sum_{i}A_{i}\rho A_{i}^{\dagger}
  79. ρ ( ) \rho\in\mathcal{B(H)}
  80. A i \displaystyle A_{i}
  81. i A i A i I \textstyle\sum_{i}A_{i}^{\dagger}A_{i}\leq I
  82. Tr [ ( ρ ) ] 1 \mathrm{Tr}[\mathcal{E}(\rho)]\leq 1
  83. { E i } \displaystyle\{E_{i}\}
  84. ( ) \mathcal{B(H)}
  85. A i \displaystyle A_{i}
  86. A i = m a i m E m \displaystyle A_{i}=\sum_{m}a_{im}E_{m}
  87. ( ρ ) = m , n χ m n E m ρ E n \mathcal{E}(\rho)=\sum_{m,n}\chi_{mn}E_{m}\rho E_{n}^{\dagger}
  88. χ m n = i a m i a n i * \chi_{mn}=\sum_{i}a_{mi}a_{ni}^{*}
  89. χ \displaystyle\chi
  90. \mathcal{E}
  91. { E i } \displaystyle\{E_{i}\}
  92. d 2 d^{2}
  93. ρ j \rho_{j}
  94. d d
  95. \mathcal{H}
  96. ρ j \rho_{j}
  97. ( ρ ) \mathcal{E}(\rho)
  98. ρ j \rho_{j}
  99. ( ρ j ) = k c j k ρ k \textstyle\mathcal{E}(\rho_{j})=\sum_{k}c_{jk}\rho_{k}
  100. ρ j \rho_{j}
  101. c j k c_{jk}
  102. E m ρ j E n = k B m , n , j , k ρ k E_{m}\rho_{j}E_{n}^{\dagger}=\sum_{k}B_{m,n,j,k}\rho_{k}
  103. B B
  104. k c j k ρ k = ( ρ j ) = m , n χ m , n E m ρ j E n = m , n k χ m , n B m , n , j , k ρ k \sum_{k}c_{jk}\rho_{k}=\mathcal{E}(\rho_{j})=\sum_{m,n}\chi_{m,n}E_{m}\rho_{j}% E_{n}^{\dagger}=\sum_{m,n}\sum_{k}\chi_{m,n}B_{m,n,j,k}\rho_{k}
  105. ρ k \rho_{k}
  106. c j k = m , n χ m , n B m , n , j , k \displaystyle c_{jk}=\sum_{m,n}\chi_{m,n}B_{m,n,j,k}
  107. B B
  108. χ \chi
  109. χ m , n = j , k B m , n , j , k - 1 c j k \chi_{m,n}=\sum_{j,k}B^{-1}_{m,n,j,k}c_{jk}

Quantum_yield.html

  1. Φ = # molecules decomposed # photons absorbed \Phi=\frac{\rm\#\ molecules\ decomposed}{\rm\#\ photons\ absorbed}
  2. Φ = # photons emitted # photons absorbed \Phi=\frac{\rm\#\ photons\ emitted}{\rm\#\ photons\ absorbed}
  3. Φ = μ mol CO 2 fixed μ mol photons absorbed \Phi=\frac{\rm\mu mol\ CO_{2}\ fixed}{\rm\mu mol\ photons\ absorbed}
  4. Φ = Φ R × 𝐼𝑛𝑡 𝐼𝑛𝑡 R 1 - 10 - A R 1 - 10 - A n 2 n R 2 \Phi=\Phi_{\mathrm{R}}\times\frac{\mathit{Int}}{\mathit{Int}_{\mathrm{R}}}% \frac{1-10^{-A_{\mathrm{R}}}}{1-10^{-A}}\frac{{n}^{2}}{{n_{\mathrm{R}}}^{2}}
  5. Φ \Phi

Quasi-algebraically_closed_field.html

  1. \infty

Quasi-finite_morphism.html

  1. 𝒪 X , x κ ( f ( x ) ) \mathcal{O}_{X,x}\otimes\kappa(f(x))
  2. κ ( f ( x ) ) \kappa(f(x))
  3. 𝒪 f - 1 ( f ( x ) ) , x \mathcal{O}_{f^{-1}(f(x)),x}
  4. X X Y X\hookrightarrow X^{\prime}\to Y

Quasi-invariant_measure.html

  1. μ = T * ( μ ) μ . \mu^{\prime}=T_{*}(\mu)\approx\mu.

Quasi-Lie_algebra.html

  1. [ x , x ] = 0 [x,x]=0
  2. [ x , y ] = - [ y , x ] [x,y]=-[y,x]
  3. 2 [ x , x ] = 0. 2[x,x]=0.

Quasiconformal_mapping.html

  1. f z ¯ = μ ( z ) f z , \frac{\partial f}{\partial\bar{z}}=\mu(z)\frac{\partial f}{\partial z},
  2. d s 2 = Ω ( z ) 2 | d z + μ ( z ) d z ¯ | 2 , ds^{2}=\Omega(z)^{2}\left|\,dz+\mu(z)\,d\bar{z}\right|^{2},
  3. | f z | 2 | d z + μ ( z ) d z ¯ | 2 \left|\frac{\partial f}{\partial z}\right|^{2}\left|\,dz+\mu(z)\,d\bar{z}% \right|^{2}
  4. d z d z ¯ dzd\bar{z}
  5. ( 1 + | μ | ) 2 | f z | 2 , ( 1 - | μ | ) 2 | f z | 2 . (1+|\mu|)^{2}\textstyle{\left|\frac{\partial f}{\partial z}\right|^{2}},\qquad% (1-|\mu|)^{2}\textstyle{\left|\frac{\partial f}{\partial z}\right|^{2}}.
  6. K ( z ) = 1 + | μ ( z ) | 1 - | μ ( z ) | . K(z)=\frac{1+|\mu(z)|}{1-|\mu(z)|}.
  7. K = sup z D | K ( z ) | = 1 + μ 1 - μ K=\sup_{z\in D}|K(z)|=\frac{1+\|\mu\|_{\infty}}{1-\|\mu\|_{\infty}}
  8. z z | z | s z\mapsto z\,|z|^{s}
  9. max ( 1 + s , 1 1 + s ) \max(1+s,\frac{1}{1+s})
  10. μ < 1 \|\mu\|_{\infty}<1

Quasinormal_subgroup.html

  1. H H
  2. K K
  3. G G
  4. h k hk
  5. h H h\in H
  6. k K k\in K
  7. k h k^{\prime}h^{\prime}
  8. k K k^{\prime}\in K
  9. h H h^{\prime}\in H

Quasiperiodic_function.html

  1. f f
  2. ω \omega
  3. f ( z + ω ) = g ( z , f ( z ) ) f(z+\omega)=g(z,f(z))
  4. g g
  5. f f
  6. f ( z + ω ) = f ( z ) + C f(z+\omega)=f(z)+C
  7. f ( z + ω ) = C f ( z ) f(z+\omega)=Cf(z)
  8. f ( z ) = sin ( A z ) + sin ( B z ) f(z)=\sin(Az)+\sin(Bz)
  9. ϑ ( z + τ ; τ ) = e - 2 π i z - π i τ ϑ ( z ; τ ) , \vartheta(z+\tau;\tau)=e^{-2\pi iz-\pi i\tau}\vartheta(z;\tau),
  10. f ( z + ω ) = f ( z ) + a z + b f(z+\omega)=f(z)+az+b
  11. ζ ( z + ω ) = ζ ( z ) + η \zeta(z+\omega)=\zeta(z)+\eta
  12. f ( z + ω ) = f ( z ) f(z+\omega)=f(z)

Quaternion-Kähler_manifold.html

  1. K = K 0 SU ( 2 ) . K=K_{0}\cdot\operatorname{SU}(2).
  2. P n . \mathbb{H}\operatorname{P}^{n}.

Quaternion_algebra.html

  1. A F K A\otimes_{F}K
  2. F = F=\mathbb{R}
  3. \mathbb{R}
  4. { 1 , i , j , k } \{1,i,j,k\}
  5. i 2 = a i^{2}=a
  6. j 2 = b j^{2}=b
  7. i j = k ij=k
  8. j i = - k ji=-k
  9. k 2 = i j i j = - i i j j = - a b k^{2}=ijij=-iijj=-ab
  10. F = F=\mathbb{R}
  11. N ( t + x i + y j + z k ) = t 2 - a x 2 - b y 2 + a b z 2 N(t+xi+yj+zk)=t^{2}-ax^{2}-by^{2}+abz^{2}
  12. a x 2 + b y 2 = z 2 ax^{2}+by^{2}=z^{2}
  13. N ( x y ) = N ( x ) N ( y ) N(xy)=N(x)N(y)
  14. \mathbb{Q}
  15. B B
  16. \mathbb{Q}
  17. ν \nu
  18. \mathbb{Q}
  19. ν \mathbb{Q}_{\nu}
  20. p \mathbb{Q}_{p}
  21. \mathbb{R}
  22. B ν := ν B B_{\nu}:=\mathbb{Q}_{\nu}\otimes_{\mathbb{Q}}B
  23. ν \mathbb{Q}_{\nu}
  24. B ν B_{\nu}
  25. ν \mathbb{Q}_{\nu}
  26. B B
  27. ν \nu
  28. B ν B_{\nu}
  29. ν \mathbb{Q}_{\nu}
  30. ν \nu
  31. B ν B_{\nu}
  32. ν \mathbb{Q}_{\nu}
  33. \infty
  34. \infty
  35. \infty
  36. \infty

Quaternionic_projective_space.html

  1. n \mathbb{HP}^{n}
  2. [ q 0 , q 1 , , q n ] [q_{0},q_{1},\ldots,q_{n}]
  3. q i q_{i}
  4. [ c q 0 , c q 1 , c q n ] [cq_{0},cq_{1}\ldots,cq_{n}]
  5. n \mathbb{HP}^{n}
  6. n + 1 { ( 0 , , 0 ) } \mathbb{H}^{n+1}\setminus\{(0,\ldots,0)\}
  7. × \mathbb{H}^{\times}
  8. n + 1 \mathbb{H}^{n+1}
  9. n \mathbb{HP}^{n}
  10. 𝕊 4 n + 3 \mathbb{S}^{4n+3}
  11. Sp ( 1 ) \,\text{Sp}(1)
  12. 𝕊 4 n + 3 \mathbb{S}^{4n+3}
  13. n \mathbb{HP}^{n}
  14. Sp ( 1 ) 𝕊 4 n + 3 n . \mathrm{Sp}(1)\to\mathbb{S}^{4n+3}\to\mathbb{HP}^{n}.
  15. n \mathbb{HP}^{n}
  16. 2 n \mathbb{H}^{2n}
  17. n \mathbb{HP}^{n}
  18. \mathbb{HP}^{\infty}
  19. \mathbb{HP}^{\infty}
  20. π i ( ) = π i ( B S 3 ) π i - 1 ( S 3 ) \pi_{i}(\mathbb{HP}^{\infty})=\pi_{i}(BS^{3})\cong\pi_{i-1}(S^{3})
  21. i i
  22. π i ( ) \pi_{i}(\mathbb{HP}^{\infty})\otimes\mathbb{Q}\cong\mathbb{Q}
  23. i = 4 i=4
  24. π i ( ) = 0 \pi_{i}(\mathbb{HP}^{\infty})\otimes\mathbb{Q}=0
  25. i 4 i\neq 4
  26. \mathbb{HP}^{\infty}
  27. K ( , 4 ) K(\mathbb{Q},4)
  28. K ( , 4 ) \mathbb{HP}^{\infty}_{\mathbb{Q}}\simeq K(\mathbb{Z},4)_{\mathbb{Q}}
  29. 2 \mathbb{HP}^{2}
  30. 2 / U ( 1 ) \mathbb{HP}^{2}/\mathrm{U}(1)

Quick_ratio.html

  1. Quick (Acid Test) Ratio = Cash and Cash Equivalent + Marketable Securities + Accounts Receivable Current Liabilities \mbox{Quick (Acid Test) Ratio}~{}={\mbox{Cash and Cash Equivalent}~{}+\mbox{% Marketable Securities}~{}+\mbox{Accounts Receivable}~{}\over\mbox{Current % Liabilities}~{}}
  2. Acid Test Ratio = (Current Assets - Inventory ) Current Liabilities \mbox{Acid Test Ratio}~{}={\mbox{(Current Assets}~{}-\mbox{Inventory}~{})\over% \mbox{Current Liabilities}~{}}

Quickselect.html

  1. n ( 2 + 2 log 2 + o ( 1 ) ) 3.4 n + o ( n ) n(2+2\log 2+o(1))\leq 3.4n+o(n)
  2. 1.5 n + O ( n 1 / 2 ) 1.5n+O(n^{1/2})

Quote_notation.html

  1. x x
  2. y y
  3. x y x^{\prime}y
  4. z z
  5. y y
  6. a a
  7. a a
  8. w w
  9. a a
  10. x x
  11. x y x^{\prime}y
  12. y - x z / w y-xz/w
  13. x = 12 x=12
  14. y = 345 y=345
  15. z = 1000 z=1000
  16. a = 9 a=9
  17. w = 99 w=99
  18. 345 - 12 * 1000 99 345-\dfrac{12*1000}{99}
  19. 7385 33 \Rightarrow\dfrac{7385}{33}

Rabinovich–Fabrikant_equations.html

  1. x ˙ = y ( z - 1 + x 2 ) + γ x \dot{x}=y(z-1+x^{2})+\gamma x\,
  2. y ˙ = x ( 3 z + 1 - x 2 ) + γ y \dot{y}=x(3z+1-x^{2})+\gamma y\,
  3. z ˙ = - 2 z ( α + x y ) , \dot{z}=-2z(\alpha+xy),\,
  4. 𝐱 ~ 0 = ( 0 , 0 , 0 ) \tilde{\mathbf{x}}_{0}=(0,0,0)
  5. 𝐱 ~ 1 , 2 = ( ± q - , - α q - , 1 - ( 1 - γ α ) q - 2 ) \tilde{\mathbf{x}}_{1,2}=\left(\pm q_{-},-\frac{\alpha}{q_{-}},1-\left(1-\frac% {\gamma}{\alpha}\right)q_{-}^{2}\right)
  6. 𝐱 ~ 3 , 4 = ( ± q + , - α q + , 1 - ( 1 - γ α ) q + 2 ) \tilde{\mathbf{x}}_{3,4}=\left(\pm q_{+},-\frac{\alpha}{q_{+}},1-\left(1-\frac% {\gamma}{\alpha}\right)q_{+}^{2}\right)
  7. q ± = 1 ± 1 - γ α ( 1 - 3 γ 4 α ) 2 ( 1 - 3 γ 4 α ) q_{\pm}=\sqrt{\frac{1\pm\sqrt{1-\gamma\alpha\left(1-\frac{3\gamma}{4\alpha}% \right)}}{2\left(1-\frac{3\gamma}{4\alpha}\right)}}

Radial_basis_function.html

  1. ϕ ( 𝐱 ) = ϕ ( 𝐱 ) \phi(\mathbf{x})=\phi(\|\mathbf{x}\|)
  2. ϕ ( 𝐱 , 𝐜 ) = ϕ ( 𝐱 - 𝐜 ) \phi(\mathbf{x},\mathbf{c})=\phi(\|\mathbf{x}-\mathbf{c}\|)
  3. ϕ \phi
  4. ϕ ( 𝐱 ) = ϕ ( 𝐱 ) \phi(\mathbf{x})=\phi(\|\mathbf{x}\|)
  5. 𝐱 \|\mathbf{x}\|
  6. r = 𝐱 - 𝐱 i r=\|\mathbf{x}-\mathbf{x}_{i}\|\;
  7. ϕ ( r ) = e - ( ε r ) 2 \phi(r)=e^{-(\varepsilon r)^{2}}\,
  8. ϕ ( r ) = 1 + ( ε r ) 2 \phi(r)=\sqrt{1+(\varepsilon r)^{2}}
  9. ϕ ( r ) = 1 1 + ( ε r ) 2 \phi(r)=\frac{1}{1+(\varepsilon r)^{2}}
  10. ϕ ( r ) = 1 1 + ( ε r ) 2 \phi(r)=\frac{1}{\sqrt{1+(\varepsilon r)^{2}}}
  11. ϕ ( r ) = r k , k = 1 , 3 , 5 , \phi(r)=r^{k},\;k=1,3,5,\dots
  12. ϕ ( r ) = r k ln ( r ) , k = 2 , 4 , 6 , \phi(r)=r^{k}\ln(r),\;k=2,4,6,\dots
  13. ϕ ( r ) = r 2 ln ( r ) \phi(r)=r^{2}\ln(r)\;
  14. y ( 𝐱 ) = i = 1 N w i ϕ ( 𝐱 - 𝐱 i ) , y(\mathbf{x})=\sum_{i=1}^{N}w_{i}\,\phi(\|\mathbf{x}-\mathbf{x}_{i}\|),
  15. y ( 𝐱 ) = i = 1 N w i ϕ ( 𝐱 - 𝐱 i ) , y(\mathbf{x})=\sum_{i=1}^{N}w_{i}\,\phi(\|\mathbf{x}-\mathbf{x}_{i}\|),

Radiant_exitance.html

  1. M e = Φ e A , M_{\mathrm{e}}=\frac{\partial\Phi_{\mathrm{e}}}{\partial A},
  2. M e = σ T 4 , M_{\mathrm{e}}^{\circ}=\sigma T^{4},
  3. M e = ε M e = ε σ T 4 , M_{\mathrm{e}}=\varepsilon M_{\mathrm{e}}^{\circ}=\varepsilon\sigma T^{4},
  4. M e , ν = M e ν , M_{\mathrm{e},\nu}=\frac{\partial M_{\mathrm{e}}}{\partial\nu},
  5. M e , λ = M e λ , M_{\mathrm{e},\lambda}=\frac{\partial M_{\mathrm{e}}}{\partial\lambda},
  6. M e , ν = π L e , Ω , ν = 2 π h ν 3 c 2 1 e h ν k T - 1 , M_{\mathrm{e},\nu}^{\circ}=\pi L_{\mathrm{e},\Omega,\nu}^{\circ}=\frac{2\pi% \mathrm{h}\nu^{3}}{c^{2}}\frac{1}{e^{\frac{\mathrm{h}\nu}{\mathrm{k}T}}-1},
  7. M e , λ = π L e , Ω , λ = 2 π h c 2 λ 5 1 e h c λ k T - 1 , M_{\mathrm{e},\lambda}^{\circ}=\pi L_{\mathrm{e},\Omega,\lambda}^{\circ}=\frac% {2\pi\mathrm{h}c^{2}}{\lambda^{5}}\frac{1}{e^{\frac{\mathrm{h}c}{\lambda% \mathrm{k}T}}-1},
  8. M e , ν = ε M e , ν = 2 π h ε ν 3 c 2 1 e h ν k T - 1 , M_{\mathrm{e},\nu}=\varepsilon M_{\mathrm{e},\nu}^{\circ}=\frac{2\pi\mathrm{h}% \varepsilon\nu^{3}}{c^{2}}\frac{1}{e^{\frac{\mathrm{h}\nu}{\mathrm{k}T}}-1},
  9. M e , λ = ε M e , λ = 2 π h ε c 2 λ 5 1 e h c λ k T - 1 . M_{\mathrm{e},\lambda}=\varepsilon M_{\mathrm{e},\lambda}^{\circ}=\frac{2\pi% \mathrm{h}\varepsilon c^{2}}{\lambda^{5}}\frac{1}{e^{\frac{\mathrm{h}c}{% \lambda\mathrm{k}T}}-1}.

Radiant_flux.html

  1. Φ e = Q e t , \Phi_{\mathrm{e}}=\frac{\partial Q_{\mathrm{e}}}{\partial t},
  2. Φ e , ν = Φ e ν , \Phi_{\mathrm{e},\nu}=\frac{\partial\Phi_{\mathrm{e}}}{\partial\nu},
  3. Φ e , λ = Φ e λ , \Phi_{\mathrm{e},\lambda}=\frac{\partial\Phi_{\mathrm{e}}}{\partial\lambda},
  4. Φ e = Σ 𝐒 𝐧 ^ d A = Σ | 𝐒 | cos α d A , \Phi_{\mathrm{e}}=\oint_{\Sigma}\mathbf{S}\cdot\mathbf{\hat{n}}\,\mathrm{d}A=% \oint_{\Sigma}|\mathbf{S}|\cos\alpha\,\mathrm{d}A,
  5. Φ e = Σ | 𝐒 | cos α d A , \Phi_{\mathrm{e}}=\oint_{\Sigma}\langle|\mathbf{S}|\rangle\cos\alpha\,\mathrm{% d}A,

Radiative_transfer.html

  1. I ν I_{\nu}
  2. d a da\,
  3. 𝐫 \mathbf{r}
  4. d t dt\,
  5. d Ω d\Omega
  6. 𝐧 ^ \hat{\mathbf{n}}
  7. ν \nu\,
  8. ν + d ν \nu+d\nu\,
  9. d E ν = I ν ( 𝐫 , 𝐧 ^ , t ) cos θ d ν d a d Ω d t dE_{\nu}=I_{\nu}(\mathbf{r},\hat{\mathbf{n}},t)\cos\theta\ d\nu\,da\,d\Omega\,dt
  10. θ \theta
  11. 𝐧 ^ \hat{\mathbf{n}}
  12. 1 c t I ν + Ω ^ I ν + ( k ν , s + k ν , a ) I ν = j ν + 1 4 π c k ν , s Ω I ν d Ω \frac{1}{c}\frac{\partial}{\partial t}I_{\nu}+\hat{\Omega}\cdot\nabla I_{\nu}+% (k_{\nu,s}+k_{\nu,a})I_{\nu}=j_{\nu}+\frac{1}{4\pi c}k_{\nu,s}\int_{\Omega}I_{% \nu}d\Omega
  13. j ν j_{\nu}
  14. k ν , s k_{\nu,s}
  15. k ν , a k_{\nu,a}
  16. I ν ( s ) = I ν ( s 0 ) e - τ ν ( s 0 , s ) + s 0 s j ν ( s ) e - τ ν ( s , s ) d s I_{\nu}(s)=I_{\nu}(s_{0})e^{-\tau_{\nu}(s_{0},s)}+\int_{s_{0}}^{s}j_{\nu}(s^{% \prime})e^{-\tau_{\nu}(s^{\prime},s)}\,ds^{\prime}
  17. τ ν ( s 1 , s 2 ) \tau_{\nu}(s_{1},s_{2})
  18. s 1 s_{1}
  19. s 2 s_{2}
  20. τ ν ( s 1 , s 2 ) = def s 1 s 2 α ν ( s ) d s \tau_{\nu}(s_{1},s_{2})\ \stackrel{\mathrm{def}}{=}\ \int_{s_{1}}^{s_{2}}% \alpha_{\nu}(s)\,ds
  21. j ν α ν = B ν ( T ) \frac{j_{\nu}}{\alpha_{\nu}}=B_{\nu}(T)
  22. B ν ( T ) B_{\nu}(T)
  23. I ν ( s ) = I ν ( s 0 ) e - τ ν ( s 0 , s ) + s 0 s B ν ( T ( s ) ) α ν ( s ) e - τ ν ( s , s ) d s I_{\nu}(s)=I_{\nu}(s_{0})e^{-\tau_{\nu}(s_{0},s)}+\int_{s_{0}}^{s}B_{\nu}(T(s^% {\prime}))\alpha_{\nu}(s^{\prime})e^{-\tau_{\nu}(s^{\prime},s)}\,ds^{\prime}
  24. μ = cos θ \mu=\cos\theta
  25. I ν ( μ , z ) = a ( z ) + μ b ( z ) I_{\nu}(\mu,z)=a(z)+\mu b(z)
  26. z z
  27. μ \mu
  28. d μ = - sin θ d θ d\mu=-\sin\theta d\theta
  29. μ \mu
  30. J ν = 1 2 - 1 1 I ν d μ = a J_{\nu}=\frac{1}{2}\int^{1}_{-1}I_{\nu}d\mu=a
  31. H ν = 1 2 - 1 1 μ I ν d μ = b 3 H_{\nu}=\frac{1}{2}\int^{1}_{-1}\mu I_{\nu}d\mu=\frac{b}{3}
  32. K ν = 1 2 - 1 1 μ 2 I ν d μ = a 3 K_{\nu}=\frac{1}{2}\int^{1}_{-1}\mu^{2}I_{\nu}d\mu=\frac{a}{3}
  33. K ν = 1 / 3 J ν K_{\nu}=1/3J_{\nu}
  34. J ν J_{\nu}
  35. H ν H_{\nu}
  36. z z
  37. μ d I ν d z = - α ν ( I ν - B ν ) + σ ν ( J ν - I ν ) \mu\frac{dI_{\nu}}{dz}=-\alpha_{\nu}(I_{\nu}-B_{\nu})+\sigma_{\nu}(J_{\nu}-I_{% \nu})
  38. d H ν d z = α ν ( B ν - J ν ) \frac{dH_{\nu}}{dz}=\alpha_{\nu}(B_{\nu}-J_{\nu})
  39. μ \mu
  40. d K ν d z = - ( α ν + σ ν ) H ν \frac{dK_{\nu}}{dz}=-(\alpha_{\nu}+\sigma_{\nu})H_{\nu}
  41. z z
  42. d 2 J ν d z 2 = 3 α ν ( α ν + σ ν ) ( J ν - B ν ) \frac{d^{2}J_{\nu}}{dz^{2}}=3\alpha_{\nu}(\alpha_{\nu}+\sigma_{\nu})(J_{\nu}-B% _{\nu})

Radical_axis.html

  1. R 2 = d 1 2 - r 1 2 = d 2 2 - r 2 2 R^{2}=d_{1}^{2}-r_{1}^{2}=d_{2}^{2}-r_{2}^{2}
  2. d 1 2 - r 1 2 = d 2 2 - r 2 2 d_{1}^{2}-r_{1}^{2}=d_{2}^{2}-r_{2}^{2}
  3. L 2 + x 1 2 - r 1 2 = L 2 + x 2 2 - r 2 2 L^{2}+x_{1}^{2}-r_{1}^{2}=L^{2}+x_{2}^{2}-r_{2}^{2}
  4. x 1 2 - x 2 2 = r 1 2 - r 2 2 x_{1}^{2}-x_{2}^{2}=r_{1}^{2}-r_{2}^{2}
  5. x 1 - x 2 = r 1 2 - r 2 2 D x_{1}-x_{2}=\frac{r_{1}^{2}-r_{2}^{2}}{D}
  6. 2 x 1 = D + r 1 2 - r 2 2 D 2x_{1}=D+\frac{r_{1}^{2}-r_{2}^{2}}{D}
  7. 2 x 2 = D - r 1 2 - r 2 2 D 2x_{2}=D-\frac{r_{1}^{2}-r_{2}^{2}}{D}
  8. det [ g k p e i m f j n ] : det [ g k p f j n d h l ] : det [ g k p d h l e i m ] . \det\begin{bmatrix}g&k&p\\ e&i&m\\ f&j&n\end{bmatrix}:\det\begin{bmatrix}g&k&p\\ f&j&n\\ d&h&l\end{bmatrix}:\det\begin{bmatrix}g&k&p\\ d&h&l\\ e&i&m\end{bmatrix}.

Rainbow_table.html

  1. 𝐚𝐚𝐚𝐚𝐚𝐚 𝐻 281 DAF40 𝑅 sgfnyd 𝐻 920 ECF10 𝑅 𝐤𝐢𝐞𝐛𝐠𝐭 \mathbf{aaaaaa}\,\xrightarrow[\;H\;]{}\,\mathrm{281DAF40}\,\xrightarrow[\;R\;]% {}\,\mathrm{sgfnyd}\,\xrightarrow[\;H\;]{}\,\mathrm{920ECF10}\,\xrightarrow[\;% R\;]{}\,\mathbf{kiebgt}
  2. 920 E C F 10 𝑅 𝐤𝐢𝐞𝐛𝐠𝐭 \mathrm{920ECF10}\,\xrightarrow[\;R\;]{}\,\mathbf{kiebgt}
  3. 𝐚𝐚𝐚𝐚𝐚𝐚 𝐻 281 DAF40 𝑅 sgfnyd 𝐻 920 ECF10 \mathbf{aaaaaa}\,\xrightarrow[\;H\;]{}\,\mathrm{281DAF40}\,\xrightarrow[\;R\;]% {}\,\mathrm{sgfnyd}\,\xrightarrow[\;H\;]{}\,\mathrm{920ECF10}
  4. FB107E70 𝑅 bvtdll 𝐻 0 EE80890 𝑅 𝐤𝐢𝐞𝐛𝐠𝐭 \mathrm{FB107E70}\,\xrightarrow[\;R\;]{}\,\mathrm{bvtdll}\,\xrightarrow[\;H\;]% {}\,\mathrm{0EE80890}\,\xrightarrow[\;R\;]{}\,\mathbf{kiebgt}
  5. T = | P | T=|P|
  6. t = | P | / ( 2 L ) t=|P|/(2L)

Raised-cosine_filter.html

  1. β = 1 \beta=1
  2. f f
  3. 1 2 T \frac{1}{2T}
  4. T T
  5. H ( f ) = { T , | f | 1 - β 2 T T 2 [ 1 + cos ( π T β [ | f | - 1 - β 2 T ] ) ] , 1 - β 2 T < | f | 1 + β 2 T 0 , otherwise H(f)=\begin{cases}T,&|f|\leq\frac{1-\beta}{2T}\\ \frac{T}{2}\left[1+\cos\left(\frac{\pi T}{\beta}\left[|f|-\frac{1-\beta}{2T}% \right]\right)\right],&\frac{1-\beta}{2T}<|f|\leq\frac{1+\beta}{2T}\\ 0,&\mbox{otherwise}\end{cases}
  6. 0 β 1 0\leq\beta\leq 1
  7. β \beta
  8. T T
  9. h ( t ) = sinc ( t T ) cos ( π β t T ) 1 - 4 β 2 t 2 T 2 h(t)=\mathrm{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}% \right)}{1-\frac{4\beta^{2}t^{2}}{T^{2}}}
  10. β \beta
  11. 1 2 T \frac{1}{2T}
  12. Δ f \Delta f
  13. β = Δ f ( 1 2 T ) = Δ f R S / 2 = 2 T Δ f \beta=\frac{\Delta f}{\left(\frac{1}{2T}\right)}=\frac{\Delta f}{R_{S}/2}=2T\Delta f
  14. R S = 1 T R_{S}=\frac{1}{T}
  15. β \beta
  16. β \beta
  17. β = 0 \beta=0
  18. β \beta
  19. lim β 0 H ( f ) = rect ( f T ) \lim_{\beta\rightarrow 0}H(f)=\mathrm{rect}(fT)
  20. rect ( . ) \mathrm{rect}(.)
  21. sinc ( t T ) \mathrm{sinc}\left(\frac{t}{T}\right)
  22. β = 1 \beta=1
  23. β = 1 \beta=1
  24. H ( f ) | β = 1 = { T 2 [ 1 + cos ( π f T ) ] , | f | 1 T 0 , otherwise H(f)|_{\beta=1}=\left\{\begin{matrix}\frac{T}{2}\left[1+\cos\left(\pi fT\right% )\right],&|f|\leq\frac{1}{T}\\ 0,&\mbox{otherwise}\end{matrix}\right.
  25. B W = 1 2 R S ( β + 1 ) BW=\frac{1}{2}R_{S}(\beta+1)
  26. n T nT
  27. n n
  28. n = 0 n=0
  29. H ( f ) H(f)
  30. H R ( f ) H T ( f ) = H ( f ) H_{R}(f)\cdot H_{T}(f)=H(f)
  31. | H R ( f ) | = | H T ( f ) | = | H ( f ) | |H_{R}(f)|=|H_{T}(f)|=\sqrt{|H(f)|}

Raised_cosine_distribution.html

  1. [ μ - s , μ + s ] [\mu-s,\mu+s]
  2. f ( x ; μ , s ) = 1 2 s [ 1 + cos ( x - μ s π ) ] f(x;\mu,s)=\frac{1}{2s}\left[1+\cos\left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]\,
  3. μ - s x μ + s \mu-s\leq x\leq\mu+s
  4. F ( x ; μ , s ) = 1 2 [ 1 + x - μ s + 1 π sin ( x - μ s π ) ] F(x;\mu,s)=\frac{1}{2}\left[1\!+\!\frac{x\!-\!\mu}{s}\!+\!\frac{1}{\pi}\sin% \left(\frac{x\!-\!\mu}{s}\,\pi\right)\right]
  5. μ - s x μ + s \mu-s\leq x\leq\mu+s
  6. x < μ - s x<\mu-s
  7. x > μ + s x>\mu+s
  8. μ = 0 \mu=0
  9. s = 1 s=1
  10. E ( x 2 n ) = 1 2 - 1 1 [ 1 + cos ( x π ) ] x 2 n d x E(x^{2n})=\frac{1}{2}\int_{-1}^{1}[1+\cos(x\pi)]x^{2n}\,dx
  11. = 1 n + 1 + 1 1 + 2 n 1 F 2 ( n + 1 2 ; 1 2 , n + 3 2 ; - π 2 4 ) =\frac{1}{n\!+\!1}+\frac{1}{1\!+\!2n}\,_{1}F_{2}\left(n\!+\!\frac{1}{2};\frac{% 1}{2},n\!+\!\frac{3}{2};\frac{-\pi^{2}}{4}\right)
  12. F 2 1 \,{}_{1}F_{2}
  13. { 2 s 3 f ′′ ( x ) - 2 π 2 s f ( x ) + π 2 = 0 , f ( 0 ) = 1 s cosh 2 ( π μ 2 s ) , f ( 0 ) = - π 2 s 2 sinh ( π μ s ) } \left\{\begin{array}[]{l}2s^{3}f^{\prime\prime}(x)-2\pi^{2}sf(x)+\pi^{2}=0,\\ f(0)=\frac{1}{s}\cosh^{2}\left(\frac{\pi\mu}{2s}\right),\\ f^{\prime}(0)=-\frac{\pi}{2s^{2}}\sinh\left(\frac{\pi\mu}{s}\right)\end{array}\right\}

Ramanujan_theta_function.html

  1. f ( a , b ) = n = - a n ( n + 1 ) / 2 b n ( n - 1 ) / 2 f(a,b)=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2}\;b^{n(n-1)/2}
  2. f ( a , b ) = ( - a ; a b ) ( - b ; a b ) ( a b ; a b ) . f(a,b)=(-a;ab)_{\infty}\;(-b;ab)_{\infty}\;(ab;ab)_{\infty}.
  3. ( a ; q ) n (a;q)_{n}
  4. f ( q , q ) = n = - q n 2 = ( - q ; q 2 ) 2 ( q 2 ; q 2 ) f(q,q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}={(-q;q^{2})_{\infty}^{2}(q^{2};q^{2}% )_{\infty}}
  5. f ( q , q 3 ) = n = 0 q n ( n + 1 ) / 2 = ( q 2 ; q 2 ) ( - q ; q ) f(q,q^{3})=\sum_{n=0}^{\infty}q^{n(n+1)/2}={(q^{2};q^{2})_{\infty}}{(-q;q)_{% \infty}}
  6. f ( - q , - q 2 ) = n = - ( - 1 ) n q n ( 3 n - 1 ) / 2 = ( q ; q ) f(-q,-q^{2})=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n(3n-1)/2}=(q;q)_{\infty}
  7. ϑ ( w , q ) = f ( q w 2 , q w - 2 ) \vartheta(w,q)=f(qw^{2},qw^{-2})

Ramanujan–Nagell_equation.html

  1. 2 n - 7 = x 2 2^{n}-7=x^{2}\,
  2. 2 b - 1 = y ( y + 1 ) 2 2^{b}-1=\frac{y(y+1)}{2}
  3. 8 ( 2 b - 1 ) = 4 y ( y + 1 ) \Leftrightarrow 8(2^{b}-1)=4y(y+1)
  4. 2 b + 3 - 8 = 4 y 2 + 4 y \Leftrightarrow 2^{b+3}-8=4y^{2}+4y
  5. 2 b + 3 - 7 = 4 y 2 + 4 y + 1 \Leftrightarrow 2^{b+3}-7=4y^{2}+4y+1
  6. 2 b + 3 - 7 = ( 2 y + 1 ) 2 \Leftrightarrow 2^{b+3}-7=(2y+1)^{2}
  7. y ( y + 1 ) 2 = ( x - 1 ) ( x + 1 ) 8 \frac{y(y+1)}{2}=\frac{(x-1)(x+1)}{8}
  8. x 2 + D = A B n x^{2}+D=AB^{n}
  9. D = 2 m - 1 D=2^{m}-1
  10. x 2 + D = A y n x^{2}+D=Ay^{n}

Ramond–Ramond_field.html

  1. C p C p + d Λ p - 1 + H Λ p - 3 C_{p}\rightarrow C_{p}+d\Lambda_{p-1}+H\wedge\Lambda_{p-3}
  2. Λ q \Lambda_{q}
  3. Λ q \Lambda_{q}
  4. G p + 1 G p + 1 + H d Λ p - 3 . G_{p+1}\rightarrow G_{p+1}+H\wedge d\Lambda_{p-3}.
  5. F p + 1 = G p + 1 + H C p - 2 F_{p+1}=G_{p+1}+H\wedge C_{p-2}
  6. d F p + 1 = H F p - 1 dF_{p+1}=H\wedge F_{p-1}
  7. 0 = d 2 C p 0=d^{2}C_{p}
  8. 0 = d 2 C p = d G p + 1 = d F p + 1 + H G p - 1 . 0=d^{2}C_{p}=dG_{p+1}=dF_{p+1}+H\wedge G_{p-1}.
  9. 𝒥 10 - p \mathcal{J}_{10-p}
  10. 𝒥 10 - p \mathcal{J}_{10-p}
  11. 𝒥 10 - p \mathcal{J}_{10-p}
  12. 𝒥 p + 2 \mathcal{J}_{p+2}
  13. 𝒥 9 - p = d 2 C 7 - p = d G 8 - p = d F 8 - p + H G 6 - p . \mathcal{J}_{9-p}=d^{2}C_{7-p}=dG_{8-p}=dF_{8-p}+H\wedge G_{6-p}.
  14. 𝒥 9 - p \mathcal{J}_{9-p}

Random_number_generation.html

  1. X n + 1 = ( a X n + b ) mod m X_{n+1}=(aX_{n}+b)\,\textrm{mod}\,m
  2. a a
  3. b b
  4. m m
  5. X n + 1 X_{n+1}
  6. X X

Rankine_vortex.html

  1. Γ \Gamma
  2. R R
  3. u θ ( r ) = { Γ r / ( 2 π R 2 ) r R , Γ / ( 2 π r ) r > R . u_{\theta}(r)=\begin{cases}\Gamma r/(2\pi R^{2})&r\leq R,\\ \Gamma/(2\pi r)&r>R.\end{cases}
  4. 𝐮 = u θ 𝐞 θ \mathbf{u}=u_{\theta}\ \mathbf{e_{\theta}}

Rasch_model.html

  1. X n i = x { 0 , 1 } X_{ni}=x\in\{0,1\}
  2. x = 1 x=1
  3. x = 0 x=0
  4. X n i = 1 X_{ni}=1
  5. Pr { X n i = 1 } = e β n - δ i 1 + e β n - δ i , \Pr\{X_{ni}=1\}=\frac{e^{{\beta_{n}}-{\delta_{i}}}}{1+e^{{\beta_{n}}-{\delta_{% i}}}},
  6. β n \beta_{n}
  7. n n
  8. δ i \delta_{i}
  9. i i
  10. Pr { X n i = 1 } \Pr\{X_{ni}=1\}
  11. β n - δ i \beta_{n}-\delta_{i}
  12. β 1 \beta_{1}
  13. β 2 \beta_{2}
  14. δ i \delta_{i}
  15. ( β 1 - δ i ) - ( β 2 - δ i ) (\beta_{1}-\delta_{i})-(\beta_{2}-\delta_{i})
  16. β 1 - β 2 \beta_{1}-\beta_{2}
  17. log - odds { X n 1 = 1 r n = 1 } = δ 2 - δ 1 , \operatorname{log-odds}\{X_{n1}=1\mid\ r_{n}=1\}=\delta_{2}-\delta_{1},\,
  18. r n r_{n}
  19. β n \beta_{n}
  20. r n = 1 r_{n}=1
  21. δ 2 - δ 1 \delta_{2}-\delta_{1}
  22. β n \beta_{n}
  23. X n i = 1 X_{ni}=1
  24. X n i = 1 X_{ni}=1

Rasch_model_estimation.html

  1. Pr { X n i = 1 } = exp ( β n - δ i ) 1 + exp ( β n - δ i ) , \Pr\{X_{ni}=1\}=\frac{\exp({\beta_{n}}-{\delta_{i}})}{1+\exp({\beta_{n}}-{% \delta_{i}})},
  2. β n \beta_{n}
  3. n n
  4. δ i \delta_{i}
  5. i i
  6. x n i x_{ni}
  7. Λ = n i exp ( x n i ( β n - δ i ) ) n i ( 1 + exp ( β n - δ i ) ) . \Lambda=\frac{\prod_{n}\prod_{i}\exp(x_{ni}(\beta_{n}-\delta_{i}))}{\prod_{n}% \prod_{i}(1+\exp(\beta_{n}-\delta_{i}))}.
  8. log Λ = n N β n r n - i I δ i s i - n N i I log ( 1 + exp ( β n - δ i ) ) \log\Lambda=\sum_{n}^{N}\beta_{n}r_{n}-\sum_{i}^{I}\delta_{i}s_{i}-\sum_{n}^{N% }\sum_{i}^{I}\log(1+\exp(\beta_{n}-\delta_{i}))
  9. r n = i I x n i r_{n}=\sum_{i}^{I}x_{ni}
  10. s i = n N x n i s_{i}=\sum_{n}^{N}x_{ni}
  11. δ i \delta_{i}
  12. β n \beta_{n}
  13. s i = n N p n i , i = 1 , , I s_{i}=\sum_{n}^{N}p_{ni},\quad i=1,\dots,I
  14. r n = i I p n i , n = 1 , , N r_{n}=\sum_{i}^{I}p_{ni},\quad n=1,\dots,N
  15. p n i = exp ( β n - δ i ) / ( 1 + exp ( β n - δ i ) ) p_{ni}=\exp(\beta_{n}-\delta_{i})/(1+\exp(\beta_{n}-\delta_{i}))
  16. δ i \delta_{i}
  17. ( I - 1 ) / I (I-1)/I
  18. Λ = n Pr { ( x n i ) r n } = exp ( i - s i δ i ) n γ r \Lambda=\prod_{n}\Pr\{(x_{ni})\mid r_{n}\}=\frac{\exp(\sum_{i}-s_{i}\delta_{i}% )}{\prod_{n}\gamma_{r}}
  19. γ r = ( x ) r exp ( - i x n i δ i ) \gamma_{r}=\sum_{(x)\mid r}\exp(-\sum_{i}x_{ni}\delta_{i})
  20. γ 2 = exp ( - δ 1 - δ 2 ) + exp ( - δ 1 - δ 3 ) + exp ( - δ 2 - δ 3 ) . \gamma_{2}=\exp(-\delta_{1}-\delta_{2})+\exp(-\delta_{1}-\delta_{3})+\exp(-% \delta_{2}-\delta_{3}).

Rational_mapping.html

  1. f : V W f\colon V\to W
  2. ( f U , U ) (f_{U},U)
  3. f U f_{U}
  4. U V U\subset V
  5. W W
  6. ( f U , U ) (f_{U},U)
  7. ( f U , U ) (f_{U}^{\prime},U^{\prime})
  8. f U f_{U}
  9. f U f_{U}^{\prime}
  10. U U U\cap U^{\prime}
  11. V V
  12. f f
  13. g : W V g\colon W\to V
  14. V V
  15. W W
  16. f : V W f\colon V\to W
  17. K ( W ) K ( V ) K(W)\to K(V)
  18. \mathbb{C}
  19. k 2 \mathbb{P}^{2}_{k}
  20. X X
  21. k 3 \mathbb{P}^{3}_{k}
  22. [ w : x : y : z ] [w:x:y:z]
  23. x y - w z = 0 xy-wz=0
  24. k 2 \mathbb{P}^{2}_{k}
  25. X X
  26. w = x = 0 w=x=0
  27. y = z = 0 y=z=0
  28. X X
  29. w 0 w\neq 0
  30. w = 1 w=1
  31. x y z xyz
  32. X X
  33. A ( X ) = k [ x , y , z ] / ( x y - z ) k [ x , y ] A(X)=k[x,y,z]/(xy-z)\cong k[x,y]
  34. p ( x , y , z ) p ( x , y , x y ) p(x,y,z)\mapsto p(x,y,xy)
  35. k ( x , y ) k(x,y)
  36. k 2 \mathbb{P}^{2}_{k}

Rational_normal_curve.html

  1. C C
  2. n n
  3. n = 2 n=2
  4. n = 3 n=3
  5. ν : 𝐏 1 𝐏 n \nu:\mathbf{P}^{1}\to\mathbf{P}^{n}
  6. S S : T SS:T
  7. ν : [ S : T ] [ S n : S n - 1 T : S n - 2 T 2 : : T n ] . \nu:[S:T]\mapsto\left[S^{n}:S^{n-1}T:S^{n-2}T^{2}:\cdots:T^{n}\right].
  8. ν : x ( x , x 2 , , x n ) . \nu:x\mapsto\left(x,x^{2},\ldots,x^{n}\right).
  9. ( x , x 2 , , x n ) . \left(x,x^{2},\ldots,x^{n}\right).
  10. F i , j ( X 0 , , X n ) = X i X j - X i + 1 X j - 1 F_{i,j}\left(X_{0},\ldots,X_{n}\right)=X_{i}X_{j}-X_{i+1}X_{j-1}
  11. [ X 0 : : X n ] [X_{0}:\cdots:X_{n}]
  12. n n
  13. [ a i : b i ] [a_{i}:b_{i}]
  14. n + 1 n+1
  15. G ( S , T ) = i = 0 n ( a i S - b i T ) G(S,T)=\prod_{i=0}^{n}\left(a_{i}S-b_{i}T\right)
  16. n + 1 n+1
  17. H i ( S , T ) = G ( S , T ) ( a i S - b i T ) H_{i}(S,T)=\frac{G(S,T)}{(a_{i}S-b_{i}T)}
  18. n n
  19. [ S : T ] [ H 0 ( S , T ) : H 1 ( S , T ) : : H n ( S , T ) ] [S:T]\mapsto\left[H_{0}(S,T):H_{1}(S,T):\cdots:H_{n}(S,T)\right]
  20. G ( S , T ) G(S,T)
  21. [ S : T ] [ 1 ( a 0 S - b 0 T ) : : 1 ( a n S - b n T ) ] [S:T]\mapsto\left[\frac{1}{(a_{0}S-b_{0}T)}:\cdots:\frac{1}{(a_{n}S-b_{n}T)}\right]
  22. S n , S n - 1 T , S n - 2 T 2 , , T n , S^{n},S^{n-1}T,S^{n-2}T^{2},\cdots,T^{n},
  23. n n
  24. K K
  25. G G
  26. G G
  27. n + 1 n+1
  28. n + 1 n+1
  29. C C
  30. n + 3 n+3
  31. n + 1 n+1
  32. n + 1 n+1
  33. S S : T = 0 : 11 SS:T=0:11
  34. S S : T = 1 : 00 SS:T=1:00
  35. ( n + 2 2 ) - 2 n - 1 {\left({{n+2}\atop{2}}\right)}-2n-1
  36. n > 2 n>2
  37. n 1 n−1
  38. C 𝐏 < s u p > n C⊂\mathbf{P}<sup>n

Rational_point.html

  1. ( 3 , 67 / 4 ) (3,−67/4)
  2. ( 1 , 5 , 0 ) (1,−5,0)
  3. n \mathbb{P}^{n}
  4. f 1 , , f m f_{1},\dots,f_{m}
  5. [ x 0 : : x n ] [x_{0}:\cdots:x_{n}]
  6. f j = 0 f_{j}=0
  7. ( 3 , 67 / 4 ) (3,−67/4)
  8. y + 67 / 4 = 2 ( x 3 ) y+67/4=2(x−3)
  9. ( a , b ) " + " ( r , s ) = ( a + r , b + s + 91 / 4 ) (a,b)"+"(r,s)=(a+r,b+s+91/4)
  10. ( 0 , 91 / 4 ) (0,−91/4)
  11. x 2 + y 2 + 1 = 0 x^{2}+y^{2}+1=0
  12. P = ( 2 , 3 ) P=(√2,3)
  13. 3 x < s u p > 2 2 y = 0 3x<sup>2−2y=0

Rational_sieve.html

  1. a k a_{k}
  2. z = p i P p i a i z=\prod_{p_{i}\in P}p_{i}^{a_{i}}
  3. b k b_{k}
  4. z + n = p i P p i b i z+n=\prod_{p_{i}\in P}p_{i}^{b_{i}}
  5. z z
  6. z + n z+n
  7. n n
  8. p i P p i a i p i P p i b i ( mod n ) \prod_{p_{i}\in P}p_{i}^{a_{i}}\equiv\prod_{p_{i}\in P}p_{i}^{b_{i}}\;% \operatorname{(mod}\;n\operatorname{)}
  9. n 1 / b b = n \lfloor n^{1/b}\rfloor^{b}=n

Rational_singularity.html

  1. X X
  2. f : Y X f\colon Y\rightarrow X
  3. Y Y
  4. f * f_{*}
  5. 𝒪 Y \mathcal{O}_{Y}
  6. R i f * 𝒪 Y = 0 R^{i}f_{*}\mathcal{O}_{Y}=0
  7. i > 0 i>0
  8. X X
  9. 𝒪 X R f * 𝒪 Y \mathcal{O}_{X}\rightarrow Rf_{*}\mathcal{O}_{Y}
  10. 𝒪 X f * 𝒪 Y \mathcal{O}_{X}\simeq f_{*}\mathcal{O}_{Y}
  11. X X
  12. x 2 + y 2 + z 2 = 0. x^{2}+y^{2}+z^{2}=0.\,

Rational_trigonometry.html

  1. ( x , y ) (x,y)
  2. a x + b y + c = 0 , ax+by+c=0,
  3. a , b a,b
  4. c c
  5. A 1 = ( x 1 , y 1 ) A_{1}=(x_{1},y_{1})
  6. A 2 = ( x 2 , y 2 ) A_{2}=(x_{2},y_{2})
  7. x x
  8. y y
  9. Q ( A 1 , A 2 ) = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 . Q(A_{1},A_{2})=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}.\,
  10. d 1 + d 2 - d 3 0 d_{1}+d_{2}-d_{3}\geq 0
  11. ( Q 1 + Q 2 - Q 3 ) 2 0 (Q_{1}+Q_{2}-Q_{3})^{2}\geq 0
  12. [ 0 , 1 ] [0,1]
  13. s ( 1 , 2 ) = Q ( B , C ) Q ( A , B ) = Q R . s(\ell_{1},\ell_{2})=\frac{Q(B,C)}{Q(A,B)}=\frac{Q}{R}.
  14. a 1 x + b 1 y = constant a_{1}x+b_{1}y=\mathrm{constant}
  15. a 2 x + b 2 y = constant a_{2}x+b_{2}y=\mathrm{constant}
  16. ( 0 , 0 ) (0,0)
  17. a 1 x + b 1 y = 0 a_{1}x+b_{1}y=0
  18. a 2 x + b 2 y = 0 a_{2}x+b_{2}y=0
  19. ( - b 1 , a 1 ) (-b_{1},a_{1})
  20. ( - b 2 , a 2 ) (-b_{2},a_{2})
  21. ( 0 , 0 ) , ( - b 1 , a 1 ) (0,0),(-b_{1},a_{1})
  22. ( - b 2 , a 2 ) (-b_{2},a_{2})
  23. Q 1 = ( b 1 2 + a 1 2 ) , Q_{1}=(b_{1}^{2}+a_{1}^{2}),
  24. Q 2 = ( b 2 2 + a 2 2 ) , Q_{2}=(b_{2}^{2}+a_{2}^{2}),
  25. Q 3 = ( b 1 - b 2 ) 2 + ( a 1 - a 2 ) 2 Q_{3}=(b_{1}-b_{2})^{2}+(a_{1}-a_{2})^{2}
  26. 1 - s = ( Q 1 + Q 2 - Q 3 ) 2 4 Q 1 Q 2 . 1-s=\frac{(Q_{1}+Q_{2}-Q_{3})^{2}}{4Q_{1}Q_{2}}.\,
  27. 1 - s = ( a 1 2 + a 2 2 + b 1 2 + b 2 2 - ( b 1 - b 2 ) 2 - ( a 1 - a 2 ) 2 ) 2 4 ( a 1 2 + b 1 2 ) ( a 2 2 + b 2 2 ) 1-s=\frac{(a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2}-(b_{1}-b_{2})^{2}-(a_{1}-a_% {2})^{2})^{2}}{4(a_{1}^{2}+b_{1}^{2})(a_{2}^{2}+b_{2}^{2})}\,
  28. ( 2 a 1 a 2 + 2 b 1 b 2 ) 2 , (2a_{1}a_{2}+2b_{1}b_{2})^{2},
  29. 1 - s = ( a 1 a 2 + b 1 b 2 ) 2 ( a 1 2 + b 1 2 ) ( a 2 2 + b 2 2 ) 1-s=\frac{(a_{1}a_{2}+b_{1}b_{2})^{2}}{(a_{1}^{2}+b_{1}^{2})(a_{2}^{2}+b_{2}^{% 2})}\,
  30. ( a 2 b 1 - a 1 b 2 ) 2 + ( a 1 a 2 + b 1 b 2 ) 2 = ( a 1 2 + b 1 2 ) ( a 2 2 + b 2 2 ) , (a_{2}b_{1}-a_{1}b_{2})^{2}+(a_{1}a_{2}+b_{1}b_{2})^{2}=(a_{1}^{2}+b_{1}^{2})(% a_{2}^{2}+b_{2}^{2}),
  31. s = ( a 1 b 2 - a 2 b 1 ) 2 ( a 1 2 + b 1 2 ) ( a 2 2 + b 2 2 ) . s=\frac{(a_{1}b_{2}-a_{2}b_{1})^{2}}{(a_{1}^{2}+b_{1}^{2})(a_{2}^{2}+b_{2}^{2}% )}.\,
  32. ( - b 1 , a 1 ) (-b_{1},a_{1})
  33. ( x 1 , y 1 ) , ( - b 2 , a 2 ) (x_{1},y_{1}),(-b_{2},a_{2})
  34. ( x 2 , y 2 ) (x_{2},y_{2})
  35. ( 0 , 0 ) (0,0)
  36. ( x 3 , y 3 ) (x_{3},y_{3})
  37. s = ( ( y 1 - y 3 ) ( x 2 - x 3 ) - ( y 2 - y 3 ) ( x 1 - x 3 ) ) 2 ( ( y 1 - y 3 ) 2 + ( x 1 - x 3 ) 2 ) ( ( y 2 - y 3 ) 2 + ( x 2 - x 3 ) 2 ) . s=\frac{((y_{1}-y_{3})(x_{2}-x_{3})-(y_{2}-y_{3})(x_{1}-x_{3}))^{2}}{((y_{1}-y% _{3})^{2}+(x_{1}-x_{3})^{2})((y_{2}-y_{3})^{2}+(x_{2}-x_{3})^{2})}.\,
  38. s , s s,s
  39. r r
  40. ( 2 s + r ) 2 = 2 ( 2 s 2 + r 2 ) + 4 s 2 r (2s+r)^{2}=2(2s^{2}+r^{2})+4s^{2}r
  41. 4 s 2 + 4 s r + r 2 = 4 s 2 + 2 r 2 + 4 s 2 r 4s^{2}+4sr+r^{2}=4s^{2}+2r^{2}+4s^{2}r
  42. s s
  43. r 2 + 4 s 2 r - 4 s r = 0 r^{2}+4s^{2}r-4sr=0
  44. r 2 - 4 s ( 1 - s ) r = 0 r^{2}-4s(1-s)r=0
  45. r = 0 r=0
  46. r = 4 s ( 1 - s ) r=4s(1-s)
  47. sin 2 2 θ = 4 sin 2 θ ( 1 - sin 2 θ ) \sin^{2}2\theta=4\sin^{2}\theta(1-\sin^{2}\theta)
  48. θ , θ \theta,\theta
  49. 2 θ 2\theta
  50. S 2 ( s ) = S 2 ( sin 2 θ ) = sin 2 ( 2 θ ) = r ( s ) S_{2}(s)=S_{2}(\sin^{2}\theta)=\sin^{2}(2\theta)=r(s)
  51. s s
  52. r r
  53. s s
  54. t t
  55. s s
  56. S 3 ( s ) = s ( 3 - 4 s ) 2 = t ( s ) S_{3}(s)=s(3-4s)^{2}=t(s)
  57. sin 2 ( 30 / 2 ) = ( 1 - cos 30 ) / 2 = ( 1 - 3 / 2 ) / 2 = ( 2 - 3 ) / 4 0.0667. \sin^{2}(30^{\circ}/2)=(1-\cos 30^{\circ})/2=(1-\sqrt{3}/2)/2=(2-\sqrt{3})/4% \approx 0.0667.
  58. s s
  59. S n ( s ) S_{n}(s)
  60. sin 2 ( n θ ) = S n ( sin 2 θ ) . \sin^{2}(n\theta)=S_{n}(\sin^{2}\theta).\,
  61. S n ( s ) = s k = 0 n - 1 n n - k ( 2 n - 1 - k k ) ( - 4 s ) n - 1 - k . S_{n}(s)=s\sum_{k=0}^{n-1}{n\over n-k}{2n-1-k\choose k}(-4s)^{n-1-k}.
  62. S n ( s ) = 1 2 - 1 4 ( 1 - 2 s + 2 s 2 - s ) n - 1 4 ( 1 - 2 s - 2 s 2 - s ) n . S_{n}(s)=\frac{1}{2}-\frac{1}{4}\left(1-2s+2\sqrt{s^{2}-s}\right)^{n}-\frac{1}% {4}\left(1-2s-2\sqrt{s^{2}-s}\right)^{n}.
  63. S n ( s ) = - 1 4 ( ( 1 - s + i s ) 2 n - 1 ) 2 ( 1 - s - i s ) 2 n . S_{n}(s)=-\frac{1}{4}\left(\left(\sqrt{1-s}+i\sqrt{s}\right)^{2n}-1\right)^{2}% \left(\sqrt{1-s}-i\sqrt{s}\right)^{2n}.
  64. S n ( s ) = sin 2 ( n arcsin ( s ) ) . S_{n}(s)=\sin^{2}\left(n\arcsin\left(\sqrt{s}\right)\right).
  65. S n + 1 ( s ) = 2 ( 1 - 2 s ) S n ( s ) - S n - 1 ( s ) + 2 s . S_{n+1}(s)=2(1-2s)S_{n}(s)-S_{n-1}(s)+2s.\,
  66. 1 - 2 S n ( s ) = T n ( 1 - 2 s ) . 1-2S_{n}(s)=T_{n}(1-2s).\,
  67. S n ( s ) = 1 - T n ( 1 - 2 s ) 2 = 1 - T n ( 1 - s ) 2 . S_{n}(s)={1-T_{n}(1-2s)\over 2}=1-T_{n}\left(\sqrt{1-s}\right)^{2}.
  68. 2 T n ( x ) 2 - 1 = T 2 n ( x ) 2T_{n}(x)^{2}-1=T_{2n}(x)\,
  69. S n ( S m ( s ) ) = S n m ( s ) . S_{n}(S_{m}(s))=S_{nm}(s).\,
  70. 0 1 ( S n ( s ) - 1 2 ) ( S m ( s ) - 1 2 ) d s s ( 1 - s ) = 0. \int_{0}^{1}\left(S_{n}(s)-{1\over 2}\right)\left(S_{m}(s)-{1\over 2}\right){% ds\over\sqrt{s(1-s)}}=0.
  71. n = 1 S n ( s ) x n = s x ( 1 + x ) ( 1 - x ) 3 + 4 s x ( 1 - x ) . \sum_{n=1}^{\infty}S_{n}(s)x^{n}={sx(1+x)\over(1-x)^{3}+4sx(1-x)}.
  72. n = 1 S n ( s ) n ! x n = 1 2 e x [ 1 - e - 2 s x cos ( 2 x s ( 1 - s ) ) ] . \sum_{n=1}^{\infty}{S_{n}(s)\over n!}x^{n}={1\over 2}e^{x}\left[1-e^{-2sx}\cos% \left(2x\sqrt{s(1-s)}\right)\right].
  73. s ( 1 - s ) y ′′ + ( 1 / 2 - s ) y + n 2 ( y - 1 / 2 ) = 0. s(1-s)y^{\prime\prime}+(1/2-s)y^{\prime}+n^{2}(y-1/2)=0.\,
  74. S 0 ( s ) \displaystyle S_{0}(s)
  75. ( Q 1 + Q 2 + Q 3 ) 2 = 2 ( Q 1 2 + Q 2 2 + Q 3 2 ) . (Q_{1}+Q_{2}+Q_{3})^{2}=2(Q_{1}^{2}+Q_{2}^{2}+Q_{3}^{2}).\,
  76. A B AB\,
  77. a x + b y + c = 0 ax+by+c=0\,
  78. a = A y - B y a=A_{y}-B_{y}\,
  79. b = B x - A x b=B_{x}-A_{x}\,
  80. c = A x B y - A y B x c=A_{x}B_{y}-A_{y}B_{x}\,
  81. ( A y - B y ) x + ( B x - A x ) y + ( A x B y - A y B x ) = 0. (A_{y}-B_{y})x+(B_{x}-A_{x})y+(A_{x}B_{y}-A_{y}B_{x})=0.\,
  82. x = ( B x - A x ) t + A x and y = ( B y - A y ) t + A y x=(B_{x}-A_{x})t+A_{x}\,\text{ and }y=(B_{y}-A_{y})t+A_{y}\,
  83. x = b t + A x x=bt+A_{x}\,
  84. y = - a t + A y . y=-at+A_{y}.\,
  85. C x = b λ + A x C_{x}=b\lambda\ +A_{x}
  86. C y = - a λ + A y . C_{y}=-a\lambda\ +A_{y}.\,
  87. Q ( A B ) \displaystyle Q(AB)
  88. Q ( B C ) ( C x - B x ) 2 + ( C y - B y ) 2 = ( ( b λ + A x ) - B x ) 2 + ( ( - a λ + A y ) - B y ) 2 = ( b λ + ( A x - B x ) ) 2 + ( - a λ + ( A y - B y ) ) 2 = ( b λ + ( - b ) ) 2 + ( - a λ + a ) 2 = b 2 ( λ - 1 ) 2 + a 2 ( - λ + 1 ) 2 = b 2 ( λ - 1 ) 2 + a 2 ( λ - 1 ) 2 = ( a 2 + b 2 ) ( λ - 1 ) 2 \begin{aligned}\displaystyle Q(BC)&\displaystyle\equiv(C_{x}-B_{x})^{2}+(C_{y}% -B_{y})^{2}\\ &\displaystyle=((b\lambda\ +A_{x})-B_{x})^{2}+((-a\lambda\ +A_{y})-B_{y})^{2}% \\ &\displaystyle=(b\lambda\ +(A_{x}-B_{x}))^{2}+(-a\lambda\ +(A_{y}-B_{y}))^{2}% \\ &\displaystyle=(b\lambda\ +(-b))^{2}+(-a\lambda\ +a)^{2}\\ &\displaystyle=b^{2}(\lambda\ -1)^{2}+a^{2}(-\lambda\ +1)^{2}\\ &\displaystyle=b^{2}(\lambda\ -1)^{2}+a^{2}(\lambda\ -1)^{2}\\ &\displaystyle=(a^{2}+b^{2})(\lambda\ -1)^{2}\end{aligned}
  89. Q ( A C ) ( C x - A x ) 2 + ( C y - A y ) 2 = ( ( b λ + A x ) - A x ) 2 + ( ( - a λ + A y ) - A y ) 2 = ( b λ + A x - A x ) 2 + ( - a λ + A y - A y ) 2 = ( b λ ) 2 + ( - a λ ) 2 = b 2 λ 2 + ( - a ) 2 λ 2 = b 2 λ 2 + a 2 λ 2 = ( a 2 + b 2 ) λ 2 \begin{aligned}\displaystyle Q(AC)&\displaystyle\equiv(C_{x}-A_{x})^{2}+(C_{y}% -A_{y})^{2}\\ &\displaystyle=((b\lambda\ +A_{x})-A_{x})^{2}+((-a\lambda\ +A_{y})-A_{y})^{2}% \\ &\displaystyle=(b\lambda\ +A_{x}-A_{x})^{2}+(-a\lambda\ +A_{y}-A_{y})^{2}\\ &\displaystyle=(b\lambda)^{2}+(-a\lambda)^{2}\\ &\displaystyle=b^{2}\lambda^{2}+(-a)^{2}\lambda^{2}\\ &\displaystyle=b^{2}\lambda^{2}+a^{2}\lambda^{2}\\ &\displaystyle=(a^{2}+b^{2})\lambda^{2}\end{aligned}
  90. ( - λ + 1 ) 2 = ( λ - 1 ) 2 (-\lambda\ +1)^{2}=(\lambda\ -1)^{2}
  91. ( Q ( A B ) + Q ( B C ) + Q ( A C ) ) 2 = 2 ( Q ( A B ) 2 + Q ( B C ) 2 + Q ( A C ) 2 ) (Q(AB)+Q(BC)+Q(AC))^{2}=2(Q(AB)^{2}+Q(BC)^{2}+Q(AC)^{2})\,
  92. ( ( a 2 + b 2 ) + ( a 2 + b 2 ) ( λ - 1 ) 2 + ( a 2 + b 2 ) λ 2 ) 2 = 2 ( ( a 2 + b 2 ) 2 + ( ( a 2 + b 2 ) ( λ - 1 ) 2 ) 2 + ( ( a 2 + b 2 ) λ 2 ) 2 ) ((a^{2}+b^{2})+(a^{2}+b^{2})(\lambda\ -1)^{2}+(a^{2}+b^{2})\lambda^{2})^{2}=2(% (a^{2}+b^{2})^{2}+((a^{2}+b^{2})(\lambda\ -1)^{2})^{2}+((a^{2}+b^{2})\lambda^{% 2})^{2})\,
  93. ( a 2 + b 2 ) 2 ( 1 + ( λ - 1 ) 2 + λ 2 ) 2 = 2 ( a 2 + b 2 ) 2 ( 1 + ( ( λ - 1 ) 2 ) 2 + ( λ 2 ) 2 ) (a^{2}+b^{2})^{2}(1+(\lambda\ -1)^{2}+\lambda^{2})^{2}=2(a^{2}+b^{2})^{2}(1+((% \lambda\ -1)^{2})^{2}+(\lambda^{2})^{2})\,
  94. A A\,
  95. B B\,
  96. ( a 2 + b 2 ) (a^{2}+b^{2})\,
  97. Q ( A B ) 2 = ( a 2 + b 2 ) 2 Q(AB)^{2}=(a^{2}+b^{2})^{2}\,
  98. ( 1 + λ 2 - 2 λ + 1 + λ 2 ) 2 = 2 ( 1 + ( λ 2 - 2 λ + 1 ) 2 + λ 4 ) (1+\lambda^{2}-2\lambda\ +1+\lambda^{2})^{2}=2(1+(\lambda^{2}-2\lambda\ +1)^{2% }+\lambda^{4})\,
  99. ( 2 λ 2 - 2 λ + 2 ) 2 = 2 ( 1 + λ 4 - 2 λ 3 + λ 2 - 2 λ 3 + 4 λ 2 - 2 λ + λ 2 - 2 λ + 1 + λ 4 ) (2\lambda^{2}-2\lambda\ +2)^{2}=2(1+\lambda^{4}-2\lambda^{3}+\lambda^{2}-2% \lambda^{3}+4\lambda^{2}-2\lambda+\lambda^{2}-2\lambda+1+\lambda^{4})\,
  100. 4 ( λ 2 - λ + 1 ) 2 = 2 ( 2 λ 4 - 4 λ 3 + 6 λ 2 - 4 λ + 2 ) 4(\lambda^{2}-\lambda\ +1)^{2}=2(2\lambda^{4}-4\lambda^{3}+6\lambda^{2}-4% \lambda+2)\,
  101. 4 ( λ 4 - λ 3 + λ 2 - λ 3 + λ 2 - λ + λ 2 - λ + 1 ) = 4 ( λ 4 - 2 λ 3 + 3 λ 2 - 2 λ + 1 ) 4(\lambda^{4}-\lambda^{3}+\lambda^{2}-\lambda^{3}+\lambda^{2}-\lambda+\lambda^% {2}-\lambda+1)=4(\lambda^{4}-2\lambda^{3}+3\lambda^{2}-2\lambda+1)\,
  102. λ 4 - 2 λ 3 + 3 λ 2 - 2 λ + 1 = λ 4 - 2 λ 3 + 3 λ 2 - 2 λ + 1 \lambda^{4}-2\lambda^{3}+3\lambda^{2}-2\lambda+1=\lambda^{4}-2\lambda^{3}+3% \lambda^{2}-2\lambda+1\,
  103. Q 1 + Q 2 = Q 3 . Q_{1}+Q_{2}=Q_{3}.\,
  104. Q ( A B ) + Q ( A C ) = Q ( B C ) Q(AB)+Q(AC)=Q(BC)\,
  105. s C = Q ( A B ) Q ( B C ) = Q ( B D ) Q ( A B ) = Q ( A D ) Q ( A C ) . s_{C}=\frac{Q(AB)}{Q(BC)}=\frac{Q(BD)}{Q(AB)}=\frac{Q(AD)}{Q(AC)}.
  106. s B = Q ( A C ) Q ( B C ) = Q ( D C ) Q ( A C ) = Q ( A D ) Q ( A B ) . s_{B}=\frac{Q(AC)}{Q(BC)}=\frac{Q(DC)}{Q(AC)}=\frac{Q(AD)}{Q(AB)}.
  107. ( x 1 , y 1 ) (x_{1},y_{1})
  108. ( x 2 , y 2 ) (x_{2},y_{2})
  109. s 1 = ( x 2 - x 2 ) 2 + ( y 2 - y 1 ) 2 ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 = ( y 2 - y 1 ) 2 ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 , s_{1}=\frac{(x_{2}-x_{2})^{2}+(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1% })^{2}}=\frac{(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}},
  110. s 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 2 ) 2 ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 = ( x 2 - x 1 ) 2 ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 . s_{2}=\frac{(x_{2}-x_{1})^{2}+(y_{2}-y_{2})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1% })^{2}}=\frac{(x_{2}-x_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}.
  111. s 1 + s 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 = 1. s_{1}+s_{2}=\frac{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}{(x_{2}-x_{1})^{2}+(y_{2% }-y_{1})^{2}}=1.\,
  112. s C + s B = 1. s_{C}+s_{B}=1.\,
  113. Q ( A B ) Q ( B C ) + Q ( A C ) Q ( B C ) = 1. \frac{Q(AB)}{Q(BC)}+\frac{Q(AC)}{Q(BC)}=1.\,
  114. Q ( B C ) Q(BC)
  115. Q ( A B ) + Q ( A C ) = Q ( B C ) . Q(AB)+Q(AC)=Q(BC).\,
  116. A 1 A 2 A 3 ¯ \overline{A_{1}A_{2}A_{3}}
  117. s 1 Q 1 = s 2 Q 2 = s 3 Q 3 . \frac{s_{1}}{Q_{1}}=\frac{s_{2}}{Q_{2}}=\frac{s_{3}}{Q_{3}}.\,
  118. A 1 A 2 A 3 ¯ \overline{A_{1}A_{2}A_{3}}
  119. ( Q 1 + Q 2 - Q 3 ) 2 = 4 Q 1 Q 2 ( 1 - s 3 ) . (Q_{1}+Q_{2}-Q_{3})^{2}=4Q_{1}Q_{2}(1-s_{3}).\,
  120. ( 1 - s 3 ) (1-s_{3})
  121. A 1 A 2 A 3 ¯ , \overline{A_{1}A_{2}A_{3}},
  122. ( s 1 + s 2 + s 3 ) 2 = 2 ( s 1 2 + s 2 2 + s 3 2 ) + 4 s 1 s 2 s 3 . (s_{1}+s_{2}+s_{3})^{2}=2(s_{1}^{2}+s_{2}^{2}+s_{3}^{2})+4s_{1}s_{2}s_{3}.\,
  123. sin ( a ) = sin ( b + c ) = sin ( b ) cos ( c ) + sin ( c ) cos ( b ) \sin(a)=\sin(b+c)=\sin(b)\cos(c)+\sin(c)\cos(b)
  124. F p F_{p}
  125. F p × F p F_{p}\times F_{p}
  126. p p
  127. \Q × \Q \Q\times\Q
  128. a x + b y = 0 , a 2 + b 2 = c 2 ( a , b , c \Q ) ax+by=0,a^{2}+b^{2}=c^{2}(a,b,c\in\Q)

Rational_zeta_series.html

  1. x = n = 2 q n ζ ( n , m ) x=\sum_{n=2}^{\infty}q_{n}\zeta(n,m)
  2. x = n = 2 q n [ ζ ( n ) - k = 1 m - 1 k - n ] x=\sum_{n=2}^{\infty}q_{n}\left[\zeta(n)-\sum_{k=1}^{m-1}k^{-n}\right]
  3. 1 = n = 2 [ ζ ( n ) - 1 ] 1=\sum_{n=2}^{\infty}\left[\zeta(n)-1\right]
  4. 1 - γ = n = 2 1 n [ ζ ( n ) - 1 ] 1-\gamma=\sum_{n=2}^{\infty}\frac{1}{n}\left[\zeta(n)-1\right]
  5. log 2 = n = 1 1 n [ ζ ( 2 n ) - 1 ] \log 2=\sum_{n=1}^{\infty}\frac{1}{n}\left[\zeta(2n)-1\right]
  6. log π = n = 2 2 ( 3 / 2 ) n - 3 n [ ζ ( n ) - 1 ] \log\pi=\sum_{n=2}^{\infty}\frac{2(3/2)^{n}-3}{n}\left[\zeta(n)-1\right]
  7. 13 30 - π 8 = n = 1 1 4 2 n [ ζ ( 2 n ) - 1 ] \frac{13}{30}-\frac{\pi}{8}=\sum_{n=1}^{\infty}\frac{1}{4^{2n}}\left[\zeta(2n)% -1\right]
  8. n = 1 ( - 1 ) n t 2 n [ ζ ( 2 n ) - 1 ] = t 2 1 + t 2 + 1 - π t 2 - π t e 2 π t - 1 \sum_{n=1}^{\infty}(-1)^{n}t^{2n}\left[\zeta(2n)-1\right]=\frac{t^{2}}{1+t^{2}% }+\frac{1-\pi t}{2}-\frac{\pi t}{e^{2\pi t}-1}
  9. x e x - 1 = n = 0 B n t n n ! \frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}
  10. n = 1 t 2 n n ζ ( 2 n ) = log ( π t sin ( π t ) ) \sum_{n=1}^{\infty}\frac{t^{2n}}{n}\zeta(2n)=\log\left(\frac{\pi t}{\sin(\pi t% )}\right)
  11. ψ ( m ) ( z + 1 ) = k = 0 ( - 1 ) m + k + 1 ( m + k ) ! ζ ( m + k + 1 ) z k k ! \psi^{(m)}(z+1)=\sum_{k=0}^{\infty}(-1)^{m+k+1}(m+k)!\;\zeta(m+k+1)\;\frac{z^{% k}}{k!}
  12. n = 2 t n [ ζ ( n ) - 1 ] = - t [ γ + ψ ( 1 - t ) - t 1 - t ] \sum_{n=2}^{\infty}t^{n}\left[\zeta(n)-1\right]=-t\left[\gamma+\psi(1-t)-\frac% {t}{1-t}\right]
  13. k = 0 ( k + ν + 1 k ) [ ζ ( k + ν + 2 ) - 1 ] = ζ ( ν + 2 ) \sum_{k=0}^{\infty}{k+\nu+1\choose k}\left[\zeta(k+\nu+2)-1\right]=\zeta(\nu+2)
  14. ζ ( s , x + y ) = k = 0 ( s + k - 1 s - 1 ) ( - y ) k ζ ( s + k , x ) \zeta(s,x+y)=\sum_{k=0}^{\infty}{s+k-1\choose s-1}(-y)^{k}\zeta(s+k,x)
  15. k = 0 ( k + ν + 1 k + 1 ) [ ζ ( k + ν + 2 ) - 1 ] = 1 \sum_{k=0}^{\infty}{k+\nu+1\choose k+1}\left[\zeta(k+\nu+2)-1\right]=1
  16. k = 0 ( - 1 ) k ( k + ν + 1 k + 1 ) [ ζ ( k + ν + 2 ) - 1 ] = 2 - ( ν + 1 ) \sum_{k=0}^{\infty}(-1)^{k}{k+\nu+1\choose k+1}\left[\zeta(k+\nu+2)-1\right]=2% ^{-(\nu+1)}
  17. k = 0 ( - 1 ) k ( k + ν + 1 k + 2 ) [ ζ ( k + ν + 2 ) - 1 ] = ν [ ζ ( ν + 1 ) - 1 ] - 2 - ν \sum_{k=0}^{\infty}(-1)^{k}{k+\nu+1\choose k+2}\left[\zeta(k+\nu+2)-1\right]=% \nu\left[\zeta(\nu+1)-1\right]-2^{-\nu}
  18. k = 0 ( - 1 ) k ( k + ν + 1 k ) [ ζ ( k + ν + 2 ) - 1 ] = ζ ( ν + 2 ) - 1 - 2 - ( ν + 2 ) \sum_{k=0}^{\infty}(-1)^{k}{k+\nu+1\choose k}\left[\zeta(k+\nu+2)-1\right]=% \zeta(\nu+2)-1-2^{-(\nu+2)}
  19. S n = k = 0 ( k + n k ) [ ζ ( k + n + 2 ) - 1 ] S_{n}=\sum_{k=0}^{\infty}{k+n\choose k}\left[\zeta(k+n+2)-1\right]
  20. S n = ( - 1 ) n [ 1 + k = 1 n ζ ( k + 1 ) ] S_{n}=(-1)^{n}\left[1+\sum_{k=1}^{n}\zeta(k+1)\right]
  21. T n = k = 0 ( k + n - 1 k ) [ ζ ( k + n + 2 ) - 1 ] T_{n}=\sum_{k=0}^{\infty}{k+n-1\choose k}\left[\zeta(k+n+2)-1\right]
  22. T n = ( - 1 ) n + 1 [ n + 1 - ζ ( 2 ) + k = 1 n - 1 ( - 1 ) k ( n - k ) ζ ( k + 1 ) ] T_{n}=(-1)^{n+1}\left[n+1-\zeta(2)+\sum_{k=1}^{n-1}(-1)^{k}(n-k)\zeta(k+1)\right]
  23. k = 0 ( k + n - m k ) [ ζ ( k + n + 2 ) - 1 ] \sum_{k=0}^{\infty}{k+n-m\choose k}\left[\zeta(k+n+2)-1\right]
  24. k = 0 ζ ( k + n + 2 ) - 1 2 k ( n + k + 1 n + 1 ) = ( 2 n + 2 - 1 ) ζ ( n + 2 ) - 1 \sum_{k=0}^{\infty}\frac{\zeta(k+n+2)-1}{2^{k}}{{n+k+1}\choose{n+1}}=\left(2^{% n+2}-1\right)\zeta(n+2)-1
  25. n = 2 n m [ ζ ( n ) - 1 ] = 1 + k = 1 m k ! S ( m + 1 , k + 1 ) ζ ( k + 1 ) \sum_{n=2}^{\infty}n^{m}\left[\zeta(n)-1\right]=1\,+\sum_{k=1}^{m}k!\;S(m+1,k+% 1)\zeta(k+1)
  26. n = 2 ( - 1 ) n n m [ ζ ( n ) - 1 ] = - 1 + 1 - 2 m + 1 m + 1 B m + 1 - k = 1 m ( - 1 ) k k ! S ( m + 1 , k + 1 ) ζ ( k + 1 ) \sum_{n=2}^{\infty}(-1)^{n}n^{m}\left[\zeta(n)-1\right]=-1\,+\,\frac{1-2^{m+1}% }{m+1}B_{m+1}\,-\sum_{k=1}^{m}(-1)^{k}k!\;S(m+1,k+1)\zeta(k+1)
  27. B k B_{k}
  28. S ( m , k ) S(m,k)

Raychaudhuri_equation.html

  1. X \vec{X}
  2. θ ˙ = - θ 2 3 - 2 σ 2 + 2 ω 2 - E [ X ] a a + X ˙ a ; a \dot{\theta}=-\frac{\theta^{2}}{3}-2\sigma^{2}+2\omega^{2}-{E[\vec{X}]^{a}}_{a% }+{{\dot{X}^{a}}}_{;a}
  3. σ 2 = 1 2 σ m n σ m n , ω 2 = 1 2 ω m n ω m n \sigma^{2}=\frac{1}{2}\sigma_{mn}\,\sigma^{mn},\;\omega^{2}=\frac{1}{2}\omega_% {mn}\,\omega^{mn}
  4. σ a b = θ a b - 1 3 θ h a b \sigma_{ab}=\theta_{ab}-\frac{1}{3}\,\theta\,h_{ab}
  5. ω a b = h m a h n b X [ m ; n ] \omega_{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{[m;n]}
  6. θ a b = h m a h n b X ( m ; n ) \theta_{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{(m;n)}
  7. θ \theta
  8. h a b = g a b + X a X b h_{ab}=g_{ab}+X_{a}\,X_{b}
  9. X \vec{X}
  10. E [ X ] a b E[\vec{X}]_{ab}
  11. E [ X ] a a = R m n X m X n {E[\vec{X}]^{a}}_{a}=R_{mn}\,X^{m}\,X^{n}
  12. E [ X ] a b = 4 π ( μ + 3 p ) E[\vec{X}]_{ab}=4\pi(\mu+3p)
  13. 4 π μ 4\pi\mu
  14. X \vec{X}
  15. θ ˙ = - θ 2 3 - 2 σ 2 - E [ X ] a a \dot{\theta}=-\frac{\theta^{2}}{3}-2\sigma^{2}-{E[\vec{X}]^{a}}_{a}
  16. θ ˙ - θ 2 3 \dot{\theta}\leq-\frac{\theta^{2}}{3}
  17. τ \tau
  18. 1 θ 1 θ 0 + τ 3 \frac{1}{\theta}\geq\frac{1}{\theta_{0}}+\frac{\tau}{3}
  19. θ 0 \theta_{0}
  20. θ \theta
  21. - 3 / θ 0 -3/\theta_{0}
  22. θ 0 \theta_{0}
  23. θ ^ ˙ = - 1 2 θ ^ 2 - 2 σ ^ 2 + 2 ω ^ 2 - T μ ν U μ U ν \dot{\widehat{\theta}}=-\frac{1}{2}\widehat{\theta}^{2}-2\widehat{\sigma}^{2}+% 2\widehat{\omega}^{2}-T_{\mu\nu}U^{\mu}U^{\nu}
  24. 2 / θ ^ 0 2/\widehat{\theta}_{0}

Rayleigh_flow.html

  1. d M 2 M 2 = 1 + γ M 2 1 - M 2 ( 1 + γ - 1 2 M 2 ) d T 0 T 0 \ \frac{dM^{2}}{M^{2}}=\frac{1+\gamma M^{2}}{1-M^{2}}\left(1+\frac{\gamma-1}{2% }M^{2}\right)\frac{dT_{0}}{T_{0}}
  2. T 0 T 0 * = 2 ( γ + 1 ) M 2 ( 1 + γ M 2 ) 2 ( 1 + γ - 1 2 M 2 ) \ \frac{T_{0}}{T_{0}^{*}}=\frac{2\left(\gamma+1\right)M^{2}}{\left(1+\gamma M^% {2}\right)^{2}}\left(1+\frac{\gamma-1}{2}M^{2}\right)
  3. Δ S = Δ s c p = l n [ M 2 ( γ + 1 1 + γ M 2 ) γ + 1 γ ] \ \Delta S=\frac{\Delta s}{c_{p}}=ln\left[M^{2}\left(\frac{\gamma+1}{1+\gamma M% ^{2}}\right)^{\frac{\gamma+1}{\gamma}}\right]
  4. H = h h * = c p T c p T * = T T * T T * = [ ( γ + 1 ) M 1 + γ M 2 ] 2 \begin{aligned}\displaystyle H&\displaystyle=\frac{h}{h^{*}}=\frac{c_{p}T}{c_{% p}T^{*}}=\frac{T}{T^{*}}\\ \displaystyle\frac{T}{T^{*}}&\displaystyle=\left[\frac{\left(\gamma+1\right)M}% {1+\gamma M^{2}}\right]^{2}\end{aligned}
  5. A = A * = constant m ˙ = m ˙ * = constant \begin{aligned}\displaystyle A&\displaystyle=A^{*}=\mbox{constant}\\ \displaystyle\dot{m}&\displaystyle=\dot{m}^{*}=\mbox{constant}\\ \end{aligned}
  6. p p * = γ + 1 1 + γ M 2 ρ ρ * = 1 + γ M 2 ( γ + 1 ) M 2 T T * = ( γ + 1 ) 2 M 2 ( 1 + γ M 2 ) 2 v v * = ( γ + 1 ) M 2 1 + γ M 2 p 0 p 0 * = γ + 1 1 + γ M 2 [ ( 2 γ + 1 ) ( 1 + γ - 1 2 M 2 ) ] γ γ - 1 \begin{aligned}\displaystyle\frac{p}{p^{*}}&\displaystyle=\frac{\gamma+1}{1+% \gamma M^{2}}\\ \displaystyle\frac{\rho}{\rho^{*}}&\displaystyle=\frac{1+\gamma M^{2}}{\left(% \gamma+1\right)M^{2}}\\ \displaystyle\frac{T}{T^{*}}&\displaystyle=\frac{\left(\gamma+1\right)^{2}M^{2% }}{\left(1+\gamma M^{2}\right)^{2}}\\ \displaystyle\frac{v}{v^{*}}&\displaystyle=\frac{\left(\gamma+1\right)M^{2}}{1% +\gamma M^{2}}\\ \displaystyle\frac{p_{0}}{p_{0}^{*}}&\displaystyle=\frac{\gamma+1}{1+\gamma M^% {2}}\left[\left(\frac{2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2}M^{2}\right% )\right]^{\frac{\gamma}{\gamma-1}}\end{aligned}
  7. Δ S F = s - s i c p = l n [ ( M M i ) γ - 1 γ ( 1 + γ - 1 2 M i 2 1 + γ - 1 2 M 2 ) γ + 1 2 γ ] Δ S R = s - s i c p = l n [ ( M M i ) 2 ( 1 + γ M i 2 1 + γ M 2 ) γ + 1 γ ] \begin{aligned}\displaystyle\Delta S_{F}&\displaystyle=\frac{s-s_{i}}{c_{p}}=% ln\left[\left(\frac{M}{M_{i}}\right)^{\frac{\gamma-1}{\gamma}}\left(\frac{1+% \frac{\gamma-1}{2}M_{i}^{2}}{1+\frac{\gamma-1}{2}M^{2}}\right)^{\frac{\gamma+1% }{2\gamma}}\right]\\ \displaystyle\Delta S_{R}&\displaystyle=\frac{s-s_{i}}{c_{p}}=ln\left[\left(% \frac{M}{M_{i}}\right)^{2}\left(\frac{1+\gamma M_{i}^{2}}{1+\gamma M^{2}}% \right)^{\frac{\gamma+1}{\gamma}}\right]\end{aligned}
  8. ( 1 + γ - 1 2 M i 2 ) [ M i 2 ( 1 + γ M i 2 ) 2 ] = ( 1 + γ - 1 2 M 2 ) [ M 2 ( 1 + γ M 2 ) 2 ] \ \left(1+\frac{\gamma-1}{2}M_{i}^{2}\right)\left[\frac{M_{i}^{2}}{\left(1+% \gamma M_{i}^{2}\right)^{2}}\right]=\left(1+\frac{\gamma-1}{2}M^{2}\right)% \left[\frac{M^{2}}{\left(1+\gamma M^{2}\right)^{2}}\right]

Rayleigh–Ritz_method.html

  1. ( λ ~ i , 𝐱 ~ i ) (\tilde{\lambda}_{i},\tilde{\,\textbf{x}}_{i})
  2. A 𝐱 = λ 𝐱 A\,\textbf{x}=\lambda\,\textbf{x}
  3. A N × N A\in\mathbb{C}^{N\times N}
  4. V N × m V\in\mathbb{C}^{N\times m}
  5. R V * A V R\leftarrow V^{*}AV
  6. R 𝐯 i = λ ~ i 𝐯 i R\mathbf{v}_{i}=\tilde{\lambda}_{i}\mathbf{v}_{i}
  7. ( λ ~ i , 𝐱 ~ i ) = ( λ ~ i , V 𝐯 i ) (\tilde{\lambda}_{i},\tilde{\,\textbf{x}}_{i})=(\tilde{\lambda}_{i},V\,\textbf% {v}_{i})
  8. A 𝐱 ~ i - λ ~ i 𝐱 ~ i \|A\tilde{\,\textbf{x}}_{i}-\tilde{\lambda}_{i}\tilde{\,\textbf{x}}_{i}\|
  9. ω \omega
  10. 1 2 ω 2 Y 1 2 m 1 \frac{1}{2}\omega^{2}Y_{1}^{2}m_{1}
  11. 1 2 k 1 Y 1 2 \frac{1}{2}k_{1}Y_{1}^{2}
  12. i = 1 2 ( 1 2 ω 2 Y i 2 M i ) = i = 1 2 ( 1 2 K i Y i 2 ) \sum_{i=1}^{2}\left(\frac{1}{2}\omega^{2}Y_{i}^{2}M_{i}\right)=\sum_{i=1}^{2}% \left(\frac{1}{2}K_{i}Y_{i}^{2}\right)
  13. ω \omega
  14. d ω / d B = 0 d\omega/dB=0
  15. ω \omega
  16. ω \omega
  17. ω \omega

RE_(complexity).html

  1. R = RE co-RE . \mbox{R}~{}=\mbox{RE}~{}\cap\mbox{co-RE}~{}.

Reachability.html

  1. s s
  2. t t
  3. t t
  4. s s
  5. s s
  6. t t
  7. G = ( V , E ) G=(V,E)
  8. V V
  9. E E
  10. G G
  11. E E
  12. ( s , t ) (s,t)
  13. V V
  14. v 0 = s , v 1 , v 2 , , v k = t v_{0}=s,v_{1},v_{2},...,v_{k}=t
  15. ( v i - 1 , v i ) (v_{i-1},v_{i})
  16. E E
  17. 1 i k 1\leq i\leq k
  18. G G
  19. s s
  20. t t
  21. t t
  22. s s
  23. s s
  24. t t
  25. s s
  26. G G
  27. G G
  28. O ( 1 ) O(1)
  29. O ( | V | 3 ) O(|V|^{3})
  30. O ( | V | 2 ) O(|V|^{2})
  31. O ( 1 ) O(1)
  32. O ( n log n ) O(n\log{n})
  33. O ( n log n ) O(n\log{n})
  34. v v
  35. w w
  36. v v
  37. w w
  38. G G
  39. v 0 v_{0}
  40. v 0 v_{0}
  41. G G
  42. L i L_{i}
  43. L i + 1 L_{i+1}
  44. k k
  45. k k
  46. i = 0 k = V \bigcup_{i=0}^{k}=V
  47. G 0 , G 1 , , G k - 1 G_{0},G_{1},\ldots,G_{k-1}
  48. G i = r i L i L i + 1 G_{i}=r_{i}\cup L_{i}\cup L_{i+1}
  49. r i r_{i}
  50. L 0 L i - 1 L_{0}\ldots L_{i-1}
  51. G i G_{i}
  52. G G
  53. G i G_{i}
  54. G i G_{i}
  55. 1 / 2 1/2
  56. G i G_{i}
  57. S S
  58. Q S Q\in S
  59. Q Q
  60. v v
  61. G i G_{i}
  62. Q Q
  63. v v
  64. Q Q
  65. v v
  66. Q Q
  67. v v
  68. Q Q
  69. v v
  70. v v
  71. u u
  72. w w
  73. u u
  74. w w
  75. Q Q
  76. u u
  77. Q Q
  78. w w
  79. Q Q
  80. G 0 , G k G_{0}\ldots,G_{k}
  81. O ( log n ) O(\log{n})
  82. Q Q
  83. O ( 1 ) O(1)
  84. O ( 1 ) O(1)
  85. 1 / 2 1/2
  86. O ( n log n ) O(n\log n)
  87. O ( log n ) O(\log{n})
  88. s s
  89. t t
  90. G G
  91. O ( n ) O(n)
  92. O ( log n ) O(\log{n})
  93. O ( 1 ) O(1)
  94. s s
  95. t t
  96. G G
  97. i = n + 1 i=n+1
  98. s s
  99. i i
  100. i i
  101. t t
  102. n + 1 n+1
  103. s s
  104. 0
  105. s s
  106. t t
  107. 1 1
  108. n n
  109. u u
  110. v v
  111. L ( u ) = ( a 1 , a 2 ) L(u)=(a_{1},a_{2})
  112. L ( v ) = ( b 1 , b 2 ) L(v)=(b_{1},b_{2})
  113. L ( u ) < L ( v ) L(u)<L(v)
  114. a 1 b 1 a_{1}\leq b_{1}
  115. a 2 b 2 a_{2}\leq b_{2}
  116. a 1 a_{1}
  117. a 2 a_{2}
  118. b 1 b_{1}
  119. b 2 b_{2}
  120. v v
  121. u u
  122. L ( u ) < L ( v ) L(u)<L(v)
  123. O ( 1 ) O(1)
  124. k k
  125. u u
  126. v v
  127. s 1 , s 2 , , s k s_{1},s_{2},...,s_{k}

Real_interest_rate.html

  1. 5 % 5\%
  2. 2 % 2\%
  3. 3 % 3\%
  4. 1 + i = ( 1 + r ) ( 1 + π e ) 1+i=(1+r)(1+\pi_{e})
  5. i i
  6. r r
  7. π e \pi_{e}
  8. $ 1000 \$1000
  9. 10 % 10\%
  10. $ 1100 \$1100
  11. 10 % 10\%
  12. 25 % 25\%
  13. 15 % 15\%
  14. 0.042 % 0.042\%
  15. 15 % 15\%
  16. 5 % 5\%
  17. 10 % 10\%
  18. 0 % 0\%
  19. 10 % 10\%
  20. 4 % 4\%
  21. 6 % 6\%
  22. 1 + i 1 + π - 1 = r \frac{1+i}{1+\pi}-1=r
  23. 2 % 2\%
  24. 10 % 10\%
  25. 7.27 % 7.27\%
  26. 1 + 0.02 1 + 0.1 - 1 = - 0.0727 \frac{1+0.02}{1+0.1}-1=-0.0727
  27. 30 % 30\%
  28. 40 % 40\%
  29. 121 % 121\%
  30. 32 % 32\%
  31. 2 % 2\%
  32. 10 % 10\%
  33. r = i - π 1 + π × 100 r=\frac{i-\pi}{1+\pi}\times 100
  34. r r
  35. i i
  36. π π
  37. £ 200 , 000 £200,000
  38. i i
  39. £ 200 , 000 × 1.02 = £ 204 , 000. £200,000×1.02=£204,000.
  40. π π
  41. £ 200 , 000 × 1.1 = £ 220 , 000 £200,000×1.1=£220,000
  42. £ 200 , 000 £200,000
  43. i π = £ 204 , 000 £ 220 , 000 = £ 16 , 000 i−π=£204,000−£220,000=− £16,000
  44. £ 16 , 000 £16,000
  45. £ 204 , 000 £204,000
  46. £ 200 , 000 £200,000
  47. £ 200 , 000 £200,000
  48. £ 16 , 000 £16,000
  49. 2 % 2\%
  50. 100 100
  51. r = £ 16 , 000 £ 220 , 000 × 100 = 0.07272 × 100 = 7.27 % r=\frac{− £16,000}{£220,000}×100=− 0.07272×100=− 7.27\%
  52. 7.27 % 7.27\%

Reciprocity_(electromagnetism).html

  1. 𝐉 1 \mathbf{J}_{1}
  2. 𝐄 1 \mathbf{E}_{1}
  3. 𝐇 1 \mathbf{H}_{1}
  4. exp ( - i ω t ) \exp(-i\omega t)
  5. 𝐉 2 \mathbf{J}_{2}
  6. 𝐄 2 \mathbf{E}_{2}
  7. 𝐇 2 \mathbf{H}_{2}
  8. V [ 𝐉 1 𝐄 2 - 𝐄 1 𝐉 2 ] d V = S [ 𝐄 1 × 𝐇 2 - 𝐄 2 × 𝐇 1 ] 𝐝𝐒 . \int_{V}\left[\mathbf{J}_{1}\cdot\mathbf{E}_{2}-\mathbf{E}_{1}\cdot\mathbf{J}_% {2}\right]dV=\oint_{S}\left[\mathbf{E}_{1}\times\mathbf{H}_{2}-\mathbf{E}_{2}% \times\mathbf{H}_{1}\right]\cdot\mathbf{dS}.
  9. 𝐉 1 𝐄 2 - 𝐄 1 𝐉 2 = [ 𝐄 1 × 𝐇 2 - 𝐄 2 × 𝐇 1 ] . \mathbf{J}_{1}\cdot\mathbf{E}_{2}-\mathbf{E}_{1}\cdot\mathbf{J}_{2}=\nabla% \cdot\left[\mathbf{E}_{1}\times\mathbf{H}_{2}-\mathbf{E}_{2}\times\mathbf{H}_{% 1}\right].
  10. 𝐉 1 \mathbf{J}_{1}
  11. 𝐉 2 \mathbf{J}_{2}
  12. 𝐉 1 𝐄 2 d V = 𝐄 1 𝐉 2 d V . \int\mathbf{J}_{1}\cdot\mathbf{E}_{2}\,dV=\int\mathbf{E}_{1}\cdot\mathbf{J}_{2% }\,dV.
  13. S ( 𝐄 1 × 𝐇 2 ) 𝐝𝐒 = S ( 𝐄 2 × 𝐇 1 ) 𝐝𝐒 . \oint_{S}(\mathbf{E}_{1}\times\mathbf{H}_{2})\cdot\mathbf{dS}=\oint_{S}(% \mathbf{E}_{2}\times\mathbf{H}_{1})\cdot\mathbf{dS}.
  14. 𝐉 \mathbf{J}
  15. 𝐉 ( e ) \mathbf{J}^{(e)}
  16. 𝐄 ( e ) \mathbf{E}^{(e)}
  17. 𝐉 ( e ) = σ 𝐄 ( e ) . \mathbf{J}^{(e)}=\sigma\mathbf{E}^{(e)}.
  18. 𝐄 \mathbf{E}
  19. 𝐄 ( e ) \mathbf{E}^{(e)}
  20. 𝐄 ( r ) \mathbf{E}^{(r)}
  21. 𝐄 = 𝐄 ( e ) + 𝐄 ( r ) \mathbf{E}=\mathbf{E}^{(e)}+\mathbf{E}^{(r)}
  22. 𝐉 ( e ) \mathbf{J}^{(e)}
  23. 𝐄 ( r ) \mathbf{E}^{(r)}
  24. σ 𝐄 1 ( e ) 𝐄 2 ( e ) \sigma\mathbf{E}_{1}^{(e)}\mathbf{E}_{2}^{(e)}
  25. 𝐉 = σ 𝐄 \mathbf{J}=\sigma\mathbf{E}
  26. V [ 𝐉 1 ( e ) 𝐄 2 ( r ) - 𝐄 1 ( r ) 𝐉 2 ( e ) ] d V = V [ σ 𝐄 1 ( e ) ( 𝐄 2 ( r ) + 𝐄 2 ( e ) ) - ( 𝐄 1 ( r ) + 𝐄 1 ( e ) ) σ 𝐄 2 ( e ) ] d V \int_{V}\left[\mathbf{J}_{1}^{(e)}\cdot\mathbf{E}_{2}^{(r)}-\mathbf{E}_{1}^{(r% )}\cdot\mathbf{J}_{2}^{(e)}\right]dV=\int_{V}\left[\sigma\mathbf{E}_{1}^{(e)}% \cdot(\mathbf{E}_{2}^{(r)}+\mathbf{E}_{2}^{(e)})-(\mathbf{E}_{1}^{(r)}+\mathbf% {E}_{1}^{(e)})\cdot\sigma\mathbf{E}_{2}^{(e)}\right]dV
  27. = V [ 𝐄 1 ( e ) 𝐉 2 - 𝐉 1 𝐄 2 ( e ) ] d V . =\int_{V}\left[\mathbf{E}_{1}^{(e)}\cdot\mathbf{J}_{2}-\mathbf{J}_{1}\cdot% \mathbf{E}_{2}^{(e)}\right]dV.
  28. n V 1 ( n ) I 2 ( n ) = n V 2 ( n ) I 1 ( n ) \sum_{n}V_{1}^{(n)}I_{2}^{(n)}=\sum_{n}V_{2}^{(n)}I_{1}^{(n)}\!
  29. V 1 V_{1}
  30. V 2 V_{2}
  31. V 1 ( 1 ) = V V_{1}^{(1)}=V
  32. V 2 ( 2 ) = V V_{2}^{(2)}=V
  33. I 1 ( 2 ) = I 2 ( 1 ) I_{1}^{(2)}=I_{2}^{(1)}
  34. O ^ \hat{O}
  35. 𝐉 \mathbf{J}
  36. 𝐄 \mathbf{E}
  37. 𝐉 = 1 i ω [ ( × 1 μ × ) - ω 2 ε ] 𝐄 O ^ 𝐄 \mathbf{J}=\frac{1}{i\omega}\left[\left(\nabla\times\frac{1}{\mu}\nabla\times% \right)-\;\omega^{2}\varepsilon\right]\mathbf{E}\equiv\hat{O}\mathbf{E}
  38. ( 𝐅 , 𝐆 ) = 𝐅 𝐆 d V (\mathbf{F},\mathbf{G})=\int\mathbf{F}\cdot\mathbf{G}\,dV
  39. 𝐅 \mathbf{F}
  40. 𝐆 \mathbf{G}
  41. ε ε + i σ / ω \varepsilon\rightarrow\varepsilon+i\sigma/\omega
  42. O ^ \hat{O}
  43. ( f , g ) (f,g)\!
  44. ( f , O ^ g ) = ( O ^ f , g ) (f,\hat{O}g)=(\hat{O}f,g)
  45. 𝐉 = O ^ 𝐄 \mathbf{J}=\hat{O}\mathbf{E}
  46. ( 𝐄 1 , O ^ 𝐄 2 ) = ( O ^ 𝐄 1 , 𝐄 2 ) (\mathbf{E}_{1},\hat{O}\mathbf{E}_{2})=(\hat{O}\mathbf{E}_{1},\mathbf{E}_{2})
  47. 𝐅 \mathbf{F}
  48. 𝐆 \mathbf{G}
  49. V 𝐅 ( × 𝐆 ) d V = V ( × 𝐅 ) 𝐆 d V - S ( 𝐅 × 𝐆 ) 𝐝𝐀 . \int_{V}\mathbf{F}\cdot(\nabla\times\mathbf{G})\,dV=\int_{V}(\nabla\times% \mathbf{F})\cdot\mathbf{G}\,dV-\oint_{S}(\mathbf{F}\times\mathbf{G})\cdot% \mathbf{dA}.
  50. ( 𝐄 1 , O ^ 𝐄 2 ) (\mathbf{E}_{1},\hat{O}\mathbf{E}_{2})
  51. ( O ^ 𝐄 1 , 𝐄 2 ) (\hat{O}\mathbf{E}_{1},\mathbf{E}_{2})
  52. 𝐫 ^ \hat{\mathbf{r}}
  53. 𝐫 ^ 𝐄 = 0 \hat{\mathbf{r}}\cdot\mathbf{E}=0
  54. 𝐇 = 𝐫 ^ × 𝐄 / Z \mathbf{H}=\hat{\mathbf{r}}\times\mathbf{E}/Z
  55. μ / ϵ \sqrt{\mu/\epsilon}
  56. 𝐄 1 × 𝐇 2 = 𝐄 1 × 𝐫 ^ × 𝐄 2 / Z \mathbf{E}_{1}\times\mathbf{H}_{2}=\mathbf{E}_{1}\times\hat{\mathbf{r}}\times% \mathbf{E}_{2}/Z
  57. 𝐫 ^ ( 𝐄 1 𝐄 2 ) / Z \hat{\mathbf{r}}(\mathbf{E}_{1}\cdot\mathbf{E}_{2})/Z
  58. 𝐄 2 × 𝐇 1 = 𝐫 ^ ( 𝐄 2 𝐄 1 ) / Z \mathbf{E}_{2}\times\mathbf{H}_{1}=\hat{\mathbf{r}}(\mathbf{E}_{2}\cdot\mathbf% {E}_{1})/Z
  59. O ^ \hat{O}
  60. 𝐄 = O ^ - 1 𝐉 \mathbf{E}=\hat{O}^{-1}\mathbf{J}
  61. O ^ \hat{O}
  62. G n m ( 𝐱 , 𝐱 ) G_{nm}(\mathbf{x}^{\prime},\mathbf{x})
  63. 𝐄 \mathbf{E}
  64. 𝐱 \mathbf{x}^{\prime}
  65. 𝐱 \mathbf{x}
  66. G G
  67. O ^ - 1 \hat{O}^{-1}
  68. G n m ( 𝐱 , 𝐱 ) = G m n ( 𝐱 , 𝐱 ) G_{nm}(\mathbf{x}^{\prime},\mathbf{x})=G_{mn}(\mathbf{x},\mathbf{x}^{\prime})
  69. O ^ \hat{O}
  70. × 1 μ × - ( ω 2 / c 2 ) ε \nabla\times\frac{1}{\mu}\nabla\times-(\omega^{2}/c^{2})\varepsilon
  71. ( 𝐅 , 𝐆 ) = 𝐅 * 𝐆 d V (\mathbf{F},\mathbf{G})=\int\mathbf{F}^{*}\cdot\mathbf{G}\,dV
  72. - V [ 𝐉 1 * 𝐄 2 + 𝐄 1 * 𝐉 2 ] d V = S [ 𝐄 1 * × 𝐇 2 + 𝐄 2 × 𝐇 1 * ] 𝐝𝐀 -\int_{V}\left[\mathbf{J}_{1}^{*}\cdot\mathbf{E}_{2}+\mathbf{E}_{1}^{*}\cdot% \mathbf{J}_{2}\right]dV=\oint_{S}\left[\mathbf{E}_{1}^{*}\times\mathbf{H}_{2}+% \mathbf{E}_{2}\times\mathbf{H}_{1}^{*}\right]\cdot\mathbf{dA}
  73. 1 / i ω 1/i\omega
  74. O ^ \hat{O}
  75. 𝐉 1 = 𝐉 2 \mathbf{J}_{1}=\mathbf{J}_{2}
  76. - 𝐉 * 𝐄 -\mathbf{J}^{*}\cdot\mathbf{E}
  77. ( 𝐉 1 , 𝐄 1 ) (\mathbf{J}_{1},\mathbf{E}_{1})
  78. ( 𝐉 2 , 𝐄 2 ) (\mathbf{J}_{2},\mathbf{E}_{2})
  79. ( 𝐉 1 , 𝐄 1 ) (\mathbf{J}_{1},\mathbf{E}_{1})
  80. ( ε 1 , μ 1 ) (\varepsilon_{1},\mu_{1})
  81. ( 𝐉 2 , 𝐄 2 ) (\mathbf{J}_{2},\mathbf{E}_{2})
  82. ( ε 1 T , μ 1 T ) (\varepsilon_{1}^{T},\mu_{1}^{T})
  83. 𝐉 1 𝐇 2 d V = 𝐇 1 𝐉 2 d V . \int\mathbf{J}_{1}\cdot\mathbf{H}_{2}\,dV=\int\mathbf{H}_{1}\cdot\mathbf{J}_{2% }\,dV.
  84. 𝐄 \mathbf{E}
  85. i ω 𝐉 i\omega\mathbf{J}
  86. 𝐇 \mathbf{H}
  87. × ( 𝐉 / ε ) \nabla\times(\mathbf{J}/\varepsilon)
  88. A A
  89. B B
  90. A A
  91. B B
  92. J J
  93. A A
  94. K K
  95. B B
  96. J J
  97. B B
  98. K K
  99. A A
  100. ϕ 1 \phi_{1}
  101. ρ 1 \rho_{1}
  102. - 2 ϕ 1 = ρ 1 / ε 0 -\nabla^{2}\phi_{1}=\rho_{1}/\varepsilon_{0}
  103. ε 0 \varepsilon_{0}
  104. ϕ 2 \phi_{2}
  105. ρ 2 \rho_{2}
  106. - 2 ϕ 2 = ρ 2 / ε 0 -\nabla^{2}\phi_{2}=\rho_{2}/\varepsilon_{0}
  107. ρ 1 ϕ 2 d V = ρ 2 ϕ 1 d V . \int\rho_{1}\phi_{2}dV=\int\rho_{2}\phi_{1}dV.
  108. ϕ 2 ( 2 ϕ 1 ) d V = ϕ 1 ( 2 ϕ 2 ) d V \int\phi_{2}(\nabla^{2}\phi_{1})dV=\int\phi_{1}(\nabla^{2}\phi_{2})dV
  109. 2 \nabla^{2}

Rectifiable_set.html

  1. E E
  2. n \mathbb{R}^{n}
  3. m m
  4. { f i } \{f_{i}\}
  5. f i : m n f_{i}:\mathbb{R}^{m}\to\mathbb{R}^{n}
  6. m m
  7. m \mathcal{H}^{m}
  8. E \ i = 0 f i ( m ) E\backslash\bigcup_{i=0}^{\infty}f_{i}\left(\mathbb{R}^{m}\right)
  9. f i f_{i}
  10. E E
  11. m m
  12. f : m n f:\mathbb{R}^{m}\to\mathbb{R}^{n}
  13. m ( E f ( m ) ) = 0. \mathcal{H}^{m}\left(E\cap f\left(\mathbb{R}^{m}\right)\right)=0.
  14. m m
  15. f : K E f:K\to E
  16. K K
  17. m \mathbb{R}^{m}
  18. m m
  19. m m
  20. ( ϕ , m ) (\phi,m)
  21. ϕ \phi
  22. m m
  23. ϕ ( E F ) = 0 \phi(E\setminus F)=0
  24. ( ϕ , m ) (\phi,m)
  25. ( ϕ , m ) (\phi,m)
  26. ϕ ( E ) < \phi(E)<\infty
  27. ( ϕ , m ) (\phi,m)
  28. ϕ \phi
  29. m m
  30. ϕ ( F ) > 0 \phi(F)>0
  31. ϕ = m \phi=\mathcal{H}^{m}
  32. X = n X=\mathbb{R}^{n}

Rectification_(geometry).html

  1. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  2. { p q , r } \begin{Bmatrix}p\\ q,r\end{Bmatrix}
  3. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  4. { q , p r } \begin{Bmatrix}q,p\\ r\end{Bmatrix}
  5. { q r } \begin{Bmatrix}q\\ r\end{Bmatrix}
  6. { p q , r , s } \begin{Bmatrix}p\\ q,r,s\end{Bmatrix}
  7. { p q , r } \begin{Bmatrix}p\\ q,r\end{Bmatrix}
  8. { q , p r , s } \begin{Bmatrix}q,p\\ r,s\end{Bmatrix}
  9. { q , p r } \begin{Bmatrix}q,p\\ r\end{Bmatrix}
  10. { q r , s } \begin{Bmatrix}q\\ r,s\end{Bmatrix}
  11. { r , q , p s } \begin{Bmatrix}r,q,p\\ s\end{Bmatrix}
  12. { r , q s } \begin{Bmatrix}r,q\\ s\end{Bmatrix}

Rectified_5-cell.html

  1. ( 2 5 , 2 6 , 2 3 , 0 ) \left(\sqrt{\frac{2}{5}},\ \frac{2}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0\right)
  2. ( 2 5 , 2 6 , - 1 3 , ± 1 ) \left(\sqrt{\frac{2}{5}},\ \frac{2}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm 1\right)
  3. ( 2 5 , - 2 6 , 1 3 , ± 1 ) \left(\sqrt{\frac{2}{5}},\ \frac{-2}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  4. ( 2 5 , - 2 6 , - 2 3 , 0 ) \left(\sqrt{\frac{2}{5}},\ \frac{-2}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  5. ( - 3 10 , 1 6 , 1 3 , ± 1 ) \left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  6. ( - 3 10 , 1 6 , - 2 3 , 0 ) \left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  7. ( - 3 10 , - 3 2 , 0 , 0 ) \left(\frac{-3}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0\right)

Recurrence_quantification_analysis.html

  1. x ( i ) = ( u ( i ) , u ( i + τ ) , , u ( i + τ ( m - 1 ) ) , \vec{x}(i)=(u(i),u(i+\tau),\ldots,u(i+\tau(m-1)),
  2. u ( i ) u(i)
  3. m m
  4. τ \tau
  5. RR = 1 N 2 i , j = 1 N 𝐑 ( i , j ) . \,\text{RR}=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\mathbf{R}(i,j).
  6. min \ell_{\min}
  7. DET = = min N P ( ) i , j = 1 N R ( i , j ) , \,\text{DET}=\frac{\sum_{\ell=\ell_{\min}}^{N}\ell\,P(\ell)}{\sum_{i,j=1}^{N}R% (i,j)},
  8. P ( ) P(\ell)
  9. \ell
  10. LAM = v = v min N v P ( v ) v = 1 N v P ( v ) , \,\text{LAM}=\frac{\sum_{v=v_{\min}}^{N}vP(v)}{\sum_{v=1}^{N}vP(v)},
  11. P ( v ) P(v)
  12. v v
  13. v min v_{\min}
  14. L = = min N P ( ) = min N P ( ) \,\text{L}=\frac{\sum_{\ell=\ell_{\min}}^{N}\ell\,P(\ell)}{\sum_{\ell=\ell_{% \min}}^{N}P(\ell)}
  15. T T = v = v min N v P ( v ) v = v min N P ( v ) TT=\frac{\sum_{v=v_{\min}}^{N}vP(v)}{\sum_{v=v_{\min}}^{N}P(v)}
  16. L max L_{\max}
  17. D I V = 1 L max DIV=\frac{1}{L_{\max}}
  18. p ( ) p(\ell)
  19. \ell
  20. P ( ) P(\ell)
  21. p ( ) = P ( ) = l min N P ( ) p(\ell)=\frac{P(\ell)}{\sum_{\ell=l_{\min}}^{N}P(\ell)}
  22. ENTR = - = min N p ( ) ln p ( ) , \,\text{ENTR}=-\sum_{\ell=\ell_{\min}}^{N}p(\ell)\ln p(\ell),
  23. RR k = 1 N - k j - i = k N - k 𝐑 ( i , j ) , \,\text{RR}_{k}=\frac{1}{N-k}\sum_{j-i=k}^{N-k}\mathbf{R}(i,j),
  24. TREND = i = 1 N ~ ( i - N ~ / 2 ) ( R R i - R R i ) i = 1 N ~ ( i - N ~ / 2 ) 2 , \,\text{TREND}=\frac{\sum_{i=1}^{\tilde{N}}(i-\tilde{N}/2)(RR_{i}-\langle RR_{% i}\rangle)}{\sum_{i=1}^{\tilde{N}}(i-\tilde{N}/2)^{2}},
  25. \langle\cdot\rangle
  26. N ~ < N \tilde{N}<N

Recursive_Bayesian_estimation.html

  1. x x
  2. z z
  3. p ( 𝐱 k | 𝐱 k - 1 , 𝐱 k - 2 , , 𝐱 0 ) = p ( 𝐱 k | 𝐱 k - 1 ) p(\,\textbf{x}_{k}|\,\textbf{x}_{k-1},\,\textbf{x}_{k-2},\dots,\,\textbf{x}_{0% })=p(\,\textbf{x}_{k}|\,\textbf{x}_{k-1})
  4. p ( 𝐳 k | 𝐱 k , 𝐱 k - 1 , , 𝐱 0 ) = p ( 𝐳 k | 𝐱 k ) p(\,\textbf{z}_{k}|\,\textbf{x}_{k},\,\textbf{x}_{k-1},\dots,\,\textbf{x}_{0})% =p(\,\textbf{z}_{k}|\,\textbf{x}_{k})
  5. p ( 𝐱 0 , , 𝐱 k , 𝐳 1 , , 𝐳 k ) = p ( 𝐱 0 ) i = 1 k p ( 𝐳 i | 𝐱 i ) p ( 𝐱 i | 𝐱 i - 1 ) . p(\,\textbf{x}_{0},\dots,\,\textbf{x}_{k},\,\textbf{z}_{1},\dots,\,\textbf{z}_% {k})=p(\,\textbf{x}_{0})\prod_{i=1}^{k}p(\,\textbf{z}_{i}|\,\textbf{x}_{i})p(% \,\textbf{x}_{i}|\,\textbf{x}_{i-1}).
  6. x k - 1 x_{k_{-}1}
  7. p ( 𝐱 k | 𝐳 1 : k - 1 ) = p ( 𝐱 k | 𝐱 k - 1 ) p ( 𝐱 k - 1 | 𝐳 1 : k - 1 ) d 𝐱 k - 1 p(\,\textbf{x}_{k}|\,\textbf{z}_{1:k-1})=\int p(\,\textbf{x}_{k}|\,\textbf{x}_% {k-1})p(\,\textbf{x}_{k-1}|\,\textbf{z}_{1:k-1})\,d\,\textbf{x}_{k-1}
  8. p ( 𝐱 k | 𝐳 1 : k ) = p ( 𝐳 k | 𝐱 k ) p ( 𝐱 k | 𝐳 1 : k - 1 ) p ( 𝐳 k | 𝐳 1 : k - 1 ) = α p ( 𝐳 k | 𝐱 k ) p ( 𝐱 k | 𝐳 1 : k - 1 ) p(\,\textbf{x}_{k}|\,\textbf{z}_{1:k})=\frac{p(\,\textbf{z}_{k}|\,\textbf{x}_{% k})p(\,\textbf{x}_{k}|\,\textbf{z}_{1:k-1})}{p(\,\textbf{z}_{k}|\,\textbf{z}_{% 1:k-1})}=\alpha\,p(\,\textbf{z}_{k}|\,\textbf{x}_{k})p(\,\textbf{x}_{k}|\,% \textbf{z}_{1:k-1})
  9. p ( 𝐳 k | 𝐳 1 : k - 1 ) = p ( 𝐳 k | 𝐱 k ) p ( 𝐱 k | 𝐳 1 : k - 1 ) d 𝐱 k p(\,\textbf{z}_{k}|\,\textbf{z}_{1:k-1})=\int p(\,\textbf{z}_{k}|\,\textbf{x}_% {k})p(\,\textbf{x}_{k}|\,\textbf{z}_{1:k-1})d\,\textbf{x}_{k}
  10. x x
  11. α \alpha

Recursive_data_type.html

  1. n a t = μ α .1 + α nat=\mu\alpha.1+\alpha
  2. μ α .1 + α \mu\alpha.1+\alpha
  3. μ α . T \mu\alpha.T
  4. T [ μ α . T / α ] T[\mu\alpha.T/\alpha]
  5. X [ Y / Z ] X[Y/Z]
  6. r o l l : T [ μ α . T / α ] μ α . T roll:T[\mu\alpha.T/\alpha]\to\mu\alpha.T
  7. u n r o l l : μ α . T T [ μ α . T / α ] unroll:\mu\alpha.T\to T[\mu\alpha.T/\alpha]
  8. μ α . T \mu\alpha.T
  9. T [ μ α . T / α ] T[\mu\alpha.T/\alpha]

Reduced_homology.html

  1. n + 1 C n n C n - 1 n - 1 2 C 1 1 C 0 0 0 \cdots\overset{\partial_{n+1}}{\longrightarrow\,}C_{n}\overset{\partial_{n}}{% \longrightarrow\,}C_{n-1}\overset{\partial_{n-1}}{\longrightarrow\,}\cdots% \overset{\partial_{2}}{\longrightarrow\,}C_{1}\overset{\partial_{1}}{% \longrightarrow\,}C_{0}\overset{\partial_{0}}{\longrightarrow\,}0
  2. H n ( X ) = ker n / im ( n + 1 ) H_{n}(X)=\ker\partial_{n}/\mathrm{im}(\partial_{n+1})
  3. n + 1 C n n C n - 1 n - 1 2 C 1 1 C 0 ϵ 0 \cdots\overset{\partial_{n+1}}{\longrightarrow\,}C_{n}\overset{\partial_{n}}{% \longrightarrow\,}C_{n-1}\overset{\partial_{n-1}}{\longrightarrow\,}\cdots% \overset{\partial_{2}}{\longrightarrow\,}C_{1}\overset{\partial_{1}}{% \longrightarrow\,}C_{0}\overset{\epsilon}{\longrightarrow\,}\mathbb{Z}\to 0
  4. ϵ ( i n i σ i ) = i n i \epsilon\left(\sum_{i}n_{i}\sigma_{i}\right)=\sum_{i}n_{i}
  5. H n ~ ( X ) = ker ( n ) / im ( n + 1 ) \tilde{H_{n}}(X)=\ker(\partial_{n})/\mathrm{im}(\partial_{n+1})
  6. H ~ 0 ( X ) = ker ( ϵ ) / im ( 1 ) \tilde{H}_{0}(X)=\ker(\epsilon)/\mathrm{im}(\partial_{1})
  7. H 0 ( X ) = H ~ 0 ( X ) H_{0}(X)=\tilde{H}_{0}(X)\oplus\mathbb{Z}
  8. H n ( X ) = H ~ n ( X ) H_{n}(X)=\tilde{H}_{n}(X)

Reduced_product.html

  1. i I S i \prod_{i\in I}S_{i}
  2. { i I : a i = b i } U \left\{i\in I:a_{i}=b_{i}\right\}\in U\,
  3. R ( ( a i 1 ) / , , ( a i n ) / ) { i I R S i ( a i 1 , , a i n ) } U . R((a^{1}_{i})/{\sim},\dots,(a^{n}_{i})/{\sim})\iff\{i\in I\mid R^{S_{i}}(a^{1}% _{i},\dots,a^{n}_{i})\}\in U.\,

Reduction_potential.html

  1. E h E_{h}
  2. E 0 E_{0}
  3. E 0 r E_{0}^{r}
  4. E 0 o E_{0}^{o}
  5. E o b s | r e f 2 = E o b s | r e f 1 - E r e f 2 | r e f 1 E_{obs|ref2}=E_{obs|ref1}-E_{ref2|ref1}
  6. o b s obs
  7. r e f 2 ref2
  8. r e f 1 ref1
  9. E 0 E_{0}
  10. 300 m V = E o b s | r e f 1 - 192 m V 300mV=E_{obs|ref1}-192mV
  11. E o b s | r e f 1 = 300 m V + 192 m V = 492 m V E_{obs|ref1}=300mV+192mV=492mV
  12. E o b s | S H E = E 0 = 492 m V - 0 m V = 492 m V E_{obs|SHE}=E_{0}=492mV-0mV=492mV
  13. 300 m V = E o b s | r e f 1 - 192 m V 300mV=E_{obs|ref1}-192mV
  14. E o b s | r e f 1 = 300 m V + 192 m V = 492 m V E_{obs|ref1}=300mV+192mV=492mV
  15. E o b s | S C E = 492 m V - 241 m V = 251 m V E_{obs|SCE}=492mV-241mV=251mV
  16. E 0 E_{0}
  17. E h E_{h}
  18. E h E_{h}
  19. E h E_{h}
  20. E h E_{h}
  21. a A + b B + n [ e - ] + h [ H + ] = c C + d D aA+bB+n[e^{-}]+h[H^{+}]=cC+dD
  22. E 0 E_{0}
  23. E 0 ( volts ) = - Δ G n F E_{0}(\textrm{volts})=-\frac{\Delta G^{\ominus}}{nF}
  24. Δ G \Delta G^{\ominus}
  25. n n
  26. F F
  27. E h E_{h}
  28. E h = E 0 + 0.05916 n log ( { A } a { B } b { C } c { D } d ) - 0.05916 h n pH E_{h}=E_{0}+\frac{0.05916}{n}\log\left(\frac{\{A\}^{a}\{B\}^{b}}{\{C\}^{c}\{D% \}^{d}}\right)-\frac{0.05916h}{n}\,\text{pH}
  29. E h E_{h}
  30. - 0.05916 h / n -0.05916h/n
  31. E h E_{h}
  32. E h E_{h}
  33. E h E_{h}
  34. h / n h/n
  35. E h E_{h}
  36. E h E_{h}
  37. E h E_{h}

Reed_reaction.html

  1. C l 2 h ν 2 C l Cl_{2}\xrightarrow{h\nu}2Cl\cdot
  2. R - H + C l R + H C l R-H+\cdot Cl\longrightarrow R\cdot+HCl
  3. R + : S O 2 R - S ˙ O 2 R\cdot+:\!SO_{2}\longrightarrow R-\dot{S}O_{2}
  4. R - S ˙ O 2 + C l 2 R - S O 2 - C l + C l R-\dot{S}O_{2}+Cl_{2}\longrightarrow R-SO_{2}-Cl+Cl\cdot
  5. R - H + S O 2 C l 2 R - C l + S O 2 + H C l R-H+SO_{2}Cl_{2}\rightarrow\ R-Cl+SO_{2}+HCl

Reflection_group.html

  1. 2 π / n 2\pi/n
  2. I 2 ( n ) . I_{2}(n).
  3. * *\infty\infty
  4. * 22 *22\infty
  5. * * **
  6. * 2222 *2222
  7. * 333 *333
  8. * 442 *442
  9. * 632 *632
  10. ( r i r j ) c i j = 1 (r_{i}r_{j})^{c_{ij}}=1
  11. π / c i j \pi/c_{ij}
  12. 2 π / c i j 2\pi/c_{ij}
  13. - 1 = 1 -1=1

Regular_singular_point.html

  1. \mathbb{C}
  2. \mathbb{C}
  3. i = 0 n p i ( z ) f ( i ) ( z ) = 0 \sum_{i=0}^{n}p_{i}(z)f^{(i)}(z)=0
  4. p n ( z ) = 1. p_{n}(z)=1.\,
  5. p n - i ( z ) p_{n-i}(z)\,
  6. f ′′ ( x ) + p 1 ( x ) f ( x ) + p 0 ( x ) f ( x ) = 0. f^{\prime\prime}(x)+p_{1}(x)f^{\prime}(x)+p_{0}(x)f(x)=0.\,
  7. w = 1 / x w=1/x
  8. d f d x = - w 2 d f d w \frac{df}{dx}=-w^{2}\frac{df}{dw}
  9. d 2 f d x 2 = w 4 d 2 f d w 2 + 2 w 3 d f d w \frac{d^{2}f}{dx^{2}}=w^{4}\frac{d^{2}f}{dw^{2}}+2w^{3}\frac{df}{dw}
  10. p 1 ( x ) p_{1}(x)
  11. p 2 ( x ) p_{2}(x)
  12. p 1 ( x ) p_{1}(x)
  13. p 2 ( x ) p_{2}(x)
  14. x 2 d 2 f d x 2 + x d f d x + ( x 2 - α 2 ) f = 0 x^{2}\frac{d^{2}f}{dx^{2}}+x\frac{df}{dx}+(x^{2}-\alpha^{2})f=0
  15. d 2 f d x 2 + 1 x d f d x + ( 1 - α 2 x 2 ) f = 0 \frac{d^{2}f}{dx^{2}}+\frac{1}{x}\frac{df}{dx}+\left(1-\frac{\alpha^{2}}{x^{2}% }\right)f=0
  16. x = 1 / w x=1/w
  17. d 2 f d w 2 + 1 w d f d w + [ 1 w 4 - α 2 w 2 ] f = 0 \frac{d^{2}f}{dw^{2}}+\frac{1}{w}\frac{df}{dw}+\left[\frac{1}{w^{4}}-\frac{% \alpha^{2}}{w^{2}}\right]f=0
  18. p 1 ( w ) = 1 / w p_{1}(w)=1/w
  19. d d x [ ( 1 - x 2 ) d d x f ] + n ( n + 1 ) f = 0. {d\over dx}\left[(1-x^{2}){d\over dx}f\right]+n(n+1)f=0.
  20. ( 1 - x 2 ) d 2 f d x 2 - 2 x d f d x + n ( n + 1 ) f = 0. (1-x^{2}){d^{2}f\over dx^{2}}-2x{df\over dx}+n(n+1)f=0.
  21. d 2 f d x 2 - 2 x ( 1 - x 2 ) d f d x + n ( n + 1 ) ( 1 - x 2 ) f = 0. {d^{2}f\over dx^{2}}-{2x\over(1-x^{2})}{df\over dx}+{n(n+1)\over(1-x^{2})}f=0.
  22. E ψ = - 2 2 m d 2 ψ d 2 x + V ( x ) ψ E\psi=-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{d^{2}x}+V(x)\psi
  23. V ( x ) = 1 2 m ω 2 x 2 . \displaystyle V(x)=\frac{1}{2}m\omega^{2}x^{2}.
  24. d 2 f d x 2 - 2 x d f d x + λ f = 0. \frac{d^{2}f}{dx^{2}}-2x\frac{df}{dx}+\lambda f=0.
  25. z ( 1 - z ) d 2 f d z 2 + [ c - ( a + b + 1 ) z ] d f d z - a b f = 0. z(1-z)\frac{d^{2}f}{dz^{2}}+\left[c-(a+b+1)z\right]\frac{df}{dz}-abf=0.
  26. d 2 f d z 2 + c - ( a + b + 1 ) z z ( 1 - z ) d f d z - a b z ( 1 - z ) f = 0. \frac{d^{2}f}{dz^{2}}+\frac{c-(a+b+1)z}{z(1-z)}\frac{df}{dz}-\frac{ab}{z(1-z)}% f=0.

Relative_luminance.html

  1. y ¯ ( λ ) \overline{y}(\lambda)

Relative_risk.html

  1. R R = p event when exposed p event when non-exposed RR=\frac{p\text{event when exposed}}{p\text{event when non-exposed}}
  2. a a
  3. b b
  4. c c
  5. d d
  6. R R = a / ( a + b ) c / ( c + d ) = 20 / 100 1 / 100 = 20. RR=\frac{a/(a+b)}{c/(c+d)}=\frac{20/100}{1/100}=20.
  7. C I = log ( R R ) ± SE × z α CI=\log(RR)\pm\mathrm{SE}\times z_{\alpha}
  8. z α z_{\alpha}
  9. \scriptstyle\approx
  10. \scriptstyle\approx
  11. O R = a d b c OR=\frac{ad}{bc}
  12. confidence = signal noise × sample size . \,\text{confidence}=\frac{\,\text{signal}}{\,\text{noise}}\times\sqrt{\,\text{% sample size}}.
  13. X 𝒩 ( log ( R R ) , σ 2 ) . X\ \sim\ \mathcal{N}(\log(RR),\,\sigma^{2}).\,
  14. S E ( l o g ( R R ) ) = [ 1 / a + 1 / c ] - [ 1 / ( a + b ) + 1 / ( c + d ) ] SE(log(RR))=\sqrt{[1/a+1/c]-[1/(a+b)+1/(c+d)]}
  15. l o g e p 1 p 2 = l o g e a / ( a + b ) c / ( c + d ) log_{e}{\frac{p_{1}}{p_{2}}}=log_{e}{\frac{a/(a+b)}{c/(c+d)}}
  16. m = l o g e p 1 p 2 m=log_{e}{\frac{p_{1}}{p_{2}}}
  17. s 2 = b a ( a + b ) + d c ( c + d ) s^{2}=\frac{b}{a(a+b)}+\frac{d}{c(c+d)}
  18. C I = e m ± 1.96 s CI=e^{m\pm 1.96s}
  19. R R = a m / ( a m + b n ) c m / ( c m + d n ) = a ( d + b q ) b ( c + a q ) = a d { 1 + ( b / d ) q } b c { 1 + ( a / c ) q } . RR=\frac{am/(am+bn)}{cm/(cm+dn)}=\frac{a(d+bq)}{b(c+aq)}=\frac{ad\left\{1+(b/d% )q\right\}}{bc\left\{1+(a/c)q\right\}}.

Relativistic_aberration.html

  1. v v\,
  2. θ s \theta_{s}\,
  3. θ o \theta_{o}\,
  4. cos θ o = cos θ s - v c 1 - v c cos θ s \cos\theta_{o}=\frac{\cos\theta_{s}-\frac{v}{c}}{1-\frac{v}{c}\cos\theta_{s}}\,

Relativistic_electromagnetism.html

  1. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  2. 1 1 - v 2 / c 2 1\over\sqrt{1-v^{2}/c^{2}}
  3. | E | = σ ϵ 0 |E^{\prime}|={\sigma^{\prime}\over\epsilon_{0}}
  4. σ \sigma
  5. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  6. σ = σ 1 - v 2 / c 2 \sigma^{\prime}\ ={\sigma\over\sqrt{1-v^{2}/c^{2}}}
  7. E = E 1 - v 2 / c 2 E^{\prime}={E\over\sqrt{1-v^{2}/c^{2}}}
  8. E = E E^{\prime}=E
  9. E y = E y 1 - v 2 / c 2 E^{\prime}_{y}={E_{y}\over\sqrt{1-v^{2}/c^{2}}}
  10. E x = E x E^{\prime}_{x}=E_{x}
  11. E y E x = y x {E_{y}\over E_{x}}={y\over x}
  12. x = x 1 - v 2 / c 2 x^{\prime}=x\sqrt{1-v^{2}/c^{2}}
  13. E y = E y 1 - v 2 / c 2 E^{\prime}_{y}={E_{y}\over\sqrt{1-v^{2}/c^{2}}}
  14. E y E x = E y E x 1 - v 2 / c 2 = y x {E^{\prime}_{y}\over E^{\prime}_{x}}={E_{y}\over E_{x}\sqrt{1-v^{2}/c^{2}}}={y% ^{\prime}\over x^{\prime}}
  15. l - = l 1 - v 2 / c 2 l_{-}={l\sqrt{1-v^{2}/c^{2}}}
  16. l + = l / 1 - v 2 / c 2 l_{+}=l/\sqrt{1-v^{2}/c^{2}}
  17. v c v\ll c
  18. F m = Q v I 2 π ϵ 0 c 2 R F_{m}={QvI\over 2\pi\epsilon_{0}c^{2}R}
  19. λ = q l + - q l - = q l ( 1 - v 2 / c 2 - 1 / 1 - v 2 / c 2 ) q l ( 1 - 1 2 v 2 c 2 - 1 - 1 2 v 2 c 2 ) = - q l v 2 c 2 \lambda={q\over l}_{+}-{q\over l}_{-}={q\over l}\bigl(\sqrt{1-v^{2}/c^{2}}-1/% \sqrt{1-v^{2}/c^{2}}\bigr)\approx{q\over l}\Bigl(1-\frac{1}{2}{v^{2}\over c^{2% }}\,-\,1-\frac{1}{2}{v^{2}\over c^{2}}\Bigr)=-{q\over l}{v^{2}\over c^{2}}
  20. I = q t = q v l I={q\over t}=q{v\over l}
  21. F e = Q E = Q λ 2 π ϵ 0 R = Q q v 2 2 π ϵ 0 c 2 R l = Q v I 2 π ϵ 0 c 2 R F_{e}=QE=Q{\lambda\over 2\pi\epsilon_{0}R}={Qqv^{2}\over 2\pi\epsilon_{0}c^{2}% Rl}={QvI\over 2\pi\epsilon_{0}c^{2}R}
  22. F e = F m F_{e}=F_{m}
  23. l ( - ) = l 1 - v 2 / c 2 l_{(-)}={l\sqrt{1-v^{2}/c^{2}}}
  24. 1 - ( 2 v / c ) 2 {\sqrt{1-(2v/c)^{2}}}
  25. l ( + ) = l 1 - v 2 / c 2 l_{(+)}={l\over\sqrt{1-v^{2}/c^{2}}}
  26. l ( + ) = l 1 - v 2 / c 2 1 - ( 2 v / c ) 2 l_{(+)}={l\over\sqrt{1-v^{2}/c^{2}}}{\sqrt{1-(2v/c)^{2}}}
  27. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  28. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}

René-François_de_Sluse.html

  1. - y f y f x . {-y{\partial f\over\partial y}\over{\partial f\over\partial x}}.

Renninger_negative-result_experiment.html

  1. 1 - 2 - 40 = 1 - 10 - 12 1-2^{-40}=1-10^{-12}

Rent's_rule.html

  1. T = t g p T=tg^{p}
  2. p p
  3. p < 1 p<1
  4. t t
  5. T = t T=t
  6. g = 1 g=1
  7. p = 1 p=1
  8. T = t g T=tg
  9. p * p*
  10. p = 0 p=0
  11. p = 1 p=1
  12. p * p*
  13. p = p * p^{\prime}=p*
  14. p * p p p^{*}\leq p^{\prime}\leq p

Repeated_game.html

  1. δ \delta
  2. lim T t = 1 T 1 T ( v i t ) > 0 \lim_{T\to\infty}\sum_{t=1}^{T}\frac{1}{T}(v_{i}^{t})>0
  3. v i t v_{i}^{t}
  4. w i t w_{i}^{t}

Representation_theory_of_the_Lorentz_group.html

  1. 1 2 \frac{1}{2}
  2. 1 2 \frac{1}{2}
  3. 1 2 \frac{1}{2}
  4. 1 2 \frac{1}{2}
  5. 1 2 \frac{1}{2}
  6. 1 2 \frac{1}{2}
  7. 1 2 \frac{1}{2}
  8. S O ( 3 ) SO(3)
  9. 𝐀 = 𝐉 + i 𝐊 2 , 𝐁 = 𝐉 - i 𝐊 2 . \mathbf{A}=\frac{\mathbf{J}+i\mathbf{K}}{2}\,,\quad\mathbf{B}=\frac{\mathbf{J}% -i\mathbf{K}}{2}\,.
  10. [ A i , A j ] = i ε i j k A k , [ B i , B j ] = i ε i j k B k , [ A i , B j ] = 0 , \left[A_{i},A_{j}\right]=i\varepsilon_{ijk}A_{k}\,,\quad\left[B_{i},B_{j}% \right]=i\varepsilon_{ijk}B_{k}\,,\quad\left[A_{i},B_{j}\right]=0\,,
  11. i , j , k i,j,k
  12. 1 , 2 , 3 1,2,3
  13. 𝐀 \mathbf{A}
  14. 𝐁 \mathbf{B}
  15. 𝐀 \mathbf{A}
  16. 𝐁 \mathbf{B}
  17. 𝐂 \mathbf{C}
  18. 𝐑 \mathbf{R}
  19. 𝐂 \mathbf{C}
  20. 𝐑 \mathbf{R}
  21. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  22. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  23. 𝐂 \mathbf{C}
  24. 𝐬𝐥 ( 2 , 𝐂 ) 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})⊕\mathbf{sl}(2,\mathbf{C})
  25. S U ( 2 ) SU(2)
  26. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  27. V V
  28. S L ( 2 , 𝐑 ) SL(2,\mathbf{R})
  29. V V
  30. S U ( 2 ) SU(2)
  31. V V
  32. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  33. V V
  34. 𝐬𝐥 ( 2 , 𝐑 ) \mathbf{sl}(2,\mathbf{R})
  35. V V
  36. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  37. V V
  38. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  39. V V
  40. 𝐬𝐥 ( 2 , 𝐂 ) 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})⊕\mathbf{sl}(2,\mathbf{C})
  41. S L ( 2 , 𝐂 ) × S L ( 2 , 𝐂 ) SL(2,\mathbf{C})×SL(2,\mathbf{C})
  42. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  43. S U ( 2 ) × S U ( 2 ) S L ( 2 , 𝐂 ) × S L ( 2 , 𝐂 ) SU(2) × SU(2)⊂SL(2,\mathbf{C}) × SL(2,\mathbf{C})
  44. S U ( 2 ) × S U ( 2 ) SU(2) × SU(2)
  45. S U ( 2 ) SU(2)
  46. S L ( 2 , 𝐂 ) × S L ( 2 , 𝐂 ) SL(2,\mathbf{C}) × SL(2,\mathbf{C})
  47. 𝐬𝐥 ( 2 , 𝐂 ) 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})⊕\mathbf{sl}(2,\mathbf{C})
  48. S L ( 2 , 𝐂 ) × S L ( 2 , 𝐂 ) SL(2,\mathbf{C}) × SL(2,\mathbf{C})
  49. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  50. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  51. μ μ
  52. μ = 0 , 1 , μ=0, 1, …
  53. 𝐬𝐥 ( 2 , 𝐂 ) 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})⊕\mathbf{sl}(2,\mathbf{C})
  54. 𝐠 \mathbf{g}
  55. I d Id
  56. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  57. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  58. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  59. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  60. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  61. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  62. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  63. ( μ , ν ) (μ,ν)
  64. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  65. ( μ , 0 ) (μ,0)
  66. ( 0 , ν ) (0,ν)
  67. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  68. ( ν , ν ) (ν,ν)
  69. ( μ , ν ) ( ν , μ ) (μ,ν)⊕(ν,μ)
  70. ( μ , ν ) (μ,ν)
  71. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  72. ϕ μ , ν ( X ) = ϕ μ ϕ ¯ ν ( X ) = ϕ μ ( X ) Id ν + 1 + Id μ + 1 ϕ ν ( X ) ¯ , X 𝔰 𝔩 ( 2 , ) \phi_{\mu,\nu}(X)=\phi_{\mu}\otimes\overline{\phi}_{\nu}(X)=\phi_{\mu}(X)% \otimes\mathrm{Id}_{\nu+1}+\mathrm{Id}_{\mu+1}\otimes\overline{\phi_{\nu}(X)},% \quad X\in\mathfrak{sl}(2,\mathbb{C})
  73. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  74. s l ( 2 , 𝐂 ) s l ( 2 , 𝐂 ) sl(2,\mathbf{C})⊕sl(2,\mathbf{C})
  75. 𝐉 \mathbf{J}
  76. 𝐊 \mathbf{K}
  77. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  78. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  79. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  80. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  81. m = μ / 2 m=μ/2
  82. n = ν / 2 n=ν/2
  83. ( m , n ) D ( m , n ) π m , n . (m,n)\equiv D^{(m,n)}\equiv\pi_{m,n}.
  84. V V
  85. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  86. ( 2 n + 1 ) (2n+ 1)
  87. n n
  88. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  89. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  90. n n
  91. m m
  92. 1 2 \frac{1}{2}
  93. n n
  94. 1 2 \frac{1}{2}
  95. ( m , n ) ( n , m ) (m,n)⊕(n,m)
  96. m n m≠n
  97. ( m , n ) (m,n)
  98. ( n , m ) (n,m)
  99. 1 2 \frac{1}{2}
  100. 1 2 \frac{1}{2}
  101. 1 2 \frac{1}{2}
  102. 1 2 \frac{1}{2}
  103. 1 2 \frac{1}{2}
  104. 1 2 \frac{1}{2}
  105. 1 2 \frac{1}{2}
  106. 1 2 \frac{1}{2}
  107. 3 2 \frac{3}{2}
  108. 3 2 \frac{3}{2}
  109. 1 2 \frac{1}{2}
  110. 1 2 \frac{1}{2}
  111. 𝐠 \mathbf{g}
  112. Γ ( 𝐠 ) Γ(\mathbf{g})
  113. e x p ( 𝐠 ) exp(\mathbf{g})
  114. L ( G ) L(G)
  115. G G
  116. X X
  117. t 𝐑 t∈\mathbf{R}
  118. G 𝐠 G↔\mathbf{g}
  119. 𝐠 = L ( G ) \mathbf{g}=L(G)
  120. G = Γ ( 𝐠 ) G=Γ(\mathbf{g})
  121. Γ ( L ( G ) ) = G Γ(L(G))=G
  122. L ( Γ ( 𝐠 ) ) = 𝐠 L(Γ(\mathbf{g}))=\mathbf{g}
  123. G G
  124. 𝐠 \mathbf{g}
  125. H H
  126. 𝐡 \mathbf{h}
  127. G G
  128. 𝐡 𝐠 \mathbf{h}⊂\mathbf{g}
  129. G , H G,H
  130. 𝐠 , 𝐡 \mathbf{g},\mathbf{h}
  131. Π : G H Π:G→H
  132. π : 𝐠 𝐡 π:\mathbf{g}→\mathbf{h}
  133. X 𝐠 X∈\mathbf{g}
  134. G G
  135. 𝐠 \mathbf{g}
  136. Π : G G L ( V ) Π:G→GL(V)
  137. V V
  138. π : 𝐠 E n d V π:\mathbf{g}→EndV
  139. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  140. X X
  141. Λ = e i X n = 0 ( i X ) n n ! \Lambda=e^{iX}\equiv\sum_{n=0}^{\infty}\frac{(iX)^{n}}{n!}
  142. π : 𝐬𝐨 ( 3 ; 1 ) 𝐠𝐥 ( V ) π:\mathbf{so}(3;1)→\mathbf{gl}(V)
  143. V V
  144. Π Π
  145. Π U ( e i X ) = e i π ( X ) , X 𝔰 𝔬 ( 3 ; 1 ) . \Pi_{U}(e^{iX})=e^{i\pi(X)},\quad X\in\mathfrak{so}(3;1).
  146. U U
  147. X 𝐬𝐨 ( 3 ; 1 ) X∈\mathbf{so}(3;1)
  148. Π ( g ) Π(g)
  149. Π ( g ) Π(g)
  150. X X
  151. X X
  152. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  153. U U
  154. Π Π
  155. g U g∉U
  156. g g
  157. Π Π
  158. h , k h,k
  159. i i
  160. Π ( g ) = Π U ( e i X n ) Π U ( e i X n - 1 ) Π U ( e i X 2 ) Π U ( e i X 1 ) = e i π ( X n ) e i π ( X n - 1 ) e i π ( X 2 ) e i π ( X 1 ) . \Pi(g)=\Pi_{U}(e^{iX_{n}})\Pi_{U}(e^{iX_{n-1}})\cdots\Pi_{U}(e^{iX_{2}})\Pi_{U% }(e^{iX_{1}})=e^{i\pi(X_{n})}e^{i\pi(X_{n-1})}\cdots e^{i\pi(X_{2})}e^{i\pi(X_% {1})}.
  161. n n
  162. Π ( g ) Π(g)
  163. g g
  164. U U
  165. h h
  166. g = ( g i + 1 h - 1 ) ( h g i - 1 ) , Π U ( g i + 1 h - 1 ) Π U ( h g i - 1 ) . g=\cdots(g_{i+1}h^{-1})(hg_{i}^{-1})\cdots,\qquad\cdots\Pi_{U}(g_{i+1}h^{-1})% \Pi_{U}(hg_{i}^{-1})\cdots.
  167. Π U ( g i + 1 h - 1 ) Π U ( h g i - 1 ) = Π U ( g i + 1 h - 1 h g i - 1 ) = Π U ( g i + 1 g i - 1 ) \cdots\Pi_{U}(g_{i+1}h^{-1})\Pi_{U}(hg_{i}^{-1})\cdots=\cdots\Pi_{U}(g_{i+1}h^% {-1}hg_{i}^{-1})\cdots=\cdots\Pi_{U}(g_{i+1}g_{i}^{-1})\cdots
  168. Π ( g ) Π(g)
  169. Π ( g ) Π(g)
  170. Π ( g ) Π(g)
  171. H H
  172. s = 0 s=0
  173. s = 1 s=1
  174. s s
  175. Π ( g ) Π(g)
  176. s = 0 s=0
  177. s = 1 s=1
  178. Π ( g ) Π(g)
  179. Π Π
  180. 0 t 1 2 0≤t≤\frac{1}{2}
  181. 1 2 t 1 \frac{1}{2}≤t≤1
  182. g h gh
  183. 1 2 \frac{1}{2}
  184. Π ( g h ) = Π ( g ) Π ( h ) Π(gh)=Π(g)Π(h)
  185. X 𝐬𝐨 ( 3 , 1 ) X∈\mathbf{so}(3,1)
  186. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  187. G L ( n , 𝐂 ) GL(n,\mathbf{C})
  188. p : G L ( n , 𝐂 ) P G L ( 2 , 𝐂 ) p:GL(n,\mathbf{C})→PGL(2,\mathbf{C})
  189. e x p : 𝐠𝐥 ( n , 𝐂 ) G L ( n , 𝐂 ) exp:\mathbf{gl}(n,\mathbf{C})→GL(n,\mathbf{C})
  190. π π
  191. p p
  192. X 𝐠𝐥 ( n , 𝐂 ) X∈\mathbf{gl}(n,\mathbf{C})
  193. e x p exp
  194. p p
  195. e x p : 𝐩𝐠𝐥 ( 2 , 𝐂 ) P G L ( 2 , 𝐂 ) exp:\mathbf{pgl}(2,\mathbf{C})→PGL(2,\mathbf{C})
  196. P G L ( 2 , 𝐂 ) PGL(2,\mathbf{C})
  197. e x p exp
  198. Π ( g ) Π(g)
  199. U = G U=G
  200. Π Π
  201. Π Π
  202. π π
  203. Π Π
  204. Π ( e i X ) = e i π ( X ) , X 𝔰 𝔬 ( 3 ; 1 ) . \Pi(e^{iX})=e^{i\pi(X)},\quad X\in\mathfrak{so}(3;1).
  205. X X
  206. g g
  207. Π Π
  208. Π Π
  209. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  210. [ σ i , σ j ] = 2 i ϵ i j k σ k [\sigma_{i},\sigma_{j}]=2i\epsilon_{ijk}\sigma_{k}
  211. [ j i , j j ] = i ϵ i j k j k , [ j i , k j ] = i ϵ i j k k k , [ k i , k j ] = - i ϵ i j k j k , [j_{i},j_{j}]=i\epsilon_{ijk}j_{k},\quad[j_{i},k_{j}]=i\epsilon_{ijk}k_{k},% \quad[k_{i},k_{j}]=-i\epsilon_{ijk}j_{k},
  212. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  213. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  214. G = { ( g , [ p g ] ) : g SO + ( 3 , 1 ) , [ p g ] π g } G=\{(g,[p_{g}]):g\in\mathrm{SO}^{+}(3,1),[p_{g}]\in\pi_{g}\}
  215. ( g 1 , [ p 1 ] ) ( g 2 , [ p 2 ] ) = ( g 1 g 2 , [ p 12 ] ) , g 1 , g 2 SO ( 3 ; 1 ) + , [ p 1 ] π g 1 , [ p 2 ] π g 2 , [ p 12 ] π g 12 , p 12 ( t ) = p 1 ( t ) p 2 ( t ) . (g_{1},[p_{1}])(g_{2},[p_{2}])=(g_{1}g_{2},[p_{12}]),\quad g_{1},g_{2}\in% \mathrm{SO}(3;1)^{+},\quad[p_{1}]\in\pi_{g_{1}},[p_{2}]\in\pi_{g_{2}},[p_{12}]% \in\pi_{g_{12}},\quad p_{12}(t)=p_{1}(t)\cdot p_{2}(t).
  216. G G
  217. G S L ( 2 , 𝐂 ) G≈SL(2,\mathbf{C})
  218. G S L ( 2 , 𝐂 ) G≈SL(2,\mathbf{C})
  219. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  220. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  221. 𝐠 \mathbf{g}
  222. G G
  223. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  224. 𝐡 \mathbf{h}
  225. 𝐏 ( A ) : 𝐡 𝐡 ; X A X A , X 𝐡 , A SL ( 2 , C ) . \mathbf{P}(A):\mathbf{h}\rightarrow\mathbf{h};\quad X\rightarrow A^{\dagger}XA% ,\quad X\in\mathbf{h},A\in\mathrm{SL}(2,C).
  226. X 𝐡 X∈\mathbf{h}
  227. 𝐡 \mathbf{h}
  228. X = ( ξ 4 + ξ 3 ξ 1 + i ξ 2 ξ 1 - i ξ 2 ξ 4 - ξ 3 ) X=\bigl(\begin{smallmatrix}\xi_{4}+\xi_{3}&\xi_{1}+i\xi_{2}\\ \xi_{1}-i\xi_{2}&\xi_{4}-\xi_{3}\\ \end{smallmatrix}\bigr)
  229. 𝐡 \mathbf{h}
  230. 𝐏 \mathbf{P}
  231. G L ( 𝐡 ) E n d 𝐡 GL(\mathbf{h})⊂End\mathbf{h}
  232. 𝐏 : S L ( 2 , 𝐂 ) G L ( 𝐡 ) \mathbf{P}:SL(2,\mathbf{C})→GL (\mathbf{h})
  233. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  234. A X = X A AX=XA
  235. A A
  236. A A
  237. ± I ±I
  238. d e t A = 1 detA=1
  239. 𝐡 \mathbf{h}
  240. X = ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) ( ξ 1 , ξ 2 , ξ 3 , ξ 4 ) = ( x , y , z , t ) = X . X=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})\leftrightarrow\overrightarrow{(\xi_{1},\xi% _{2},\xi_{3},\xi_{4})}=(x,y,z,t)=\overrightarrow{X}.
  241. 𝐏 ( A ) \mathbf{P}(A)
  242. 𝐡 \mathbf{h}
  243. 𝐩 \mathbf{p}
  244. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  245. 𝐩 ( A ) \mathbf{p}(A)
  246. O ( 3 ; 1 ) O(3;1)
  247. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  248. 𝐩 \mathbf{p}
  249. O ( 3 ; 1 ) O(3;1)
  250. π : 𝐬𝐥 ( 2 , 𝐂 ) 𝐬𝐨 ( 3 ; 1 ) π:\mathbf{sl}(2,\mathbf{C})→\mathbf{so}(3;1)
  251. 𝐏 \mathbf{P}
  252. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  253. G G
  254. S L ( 2 , C ) SL(2, C)
  255. s l ( 2 , C ) sl(2, C)
  256. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  257. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  258. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  259. μ μ
  260. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  261. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  262. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  263. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  264. ( μ , ν ) (μ,ν)
  265. ( μ , 0 ) (μ,0)
  266. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  267. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  268. S U ( 2 ) SU(2)
  269. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  270. S L ( 2 , 𝐑 ) SL(2,\mathbf{R})
  271. 𝐬𝐥 ( 2 , 𝐑 ) \mathbf{sl}(2,\mathbf{R})
  272. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  273. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  274. i i
  275. i i
  276. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  277. ( n + 1 ) (n+1)
  278. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  279. n 0 n≥0
  280. n n
  281. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  282. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  283. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  284. n n
  285. S U ( 2 ) SU(2)
  286. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  287. S L ( 2 , 𝐑 SL(2,\mathbf{R}
  288. 𝐬𝐥 ( 2 , 𝐑 ) \mathbf{sl}(2,\mathbf{R})
  289. ( μ , ν ) (μ,ν)
  290. μ μ
  291. ν ν
  292. ϕ μ , ν ( X ) P = - P z 1 ( X 11 z 1 + X 12 z 2 ) - P z 2 ( X 21 z 1 + X 22 z 2 ) - P z 1 ¯ ( X 11 ¯ z 1 ¯ + X 12 ¯ z 2 ¯ ) - P z 2 ¯ ( X 21 ¯ z 1 ¯ + X 22 ¯ z 2 ¯ ) , X 𝔰 𝔩 ( 2 , ) , \phi_{\mu,\nu}(X)P=-\frac{\partial P}{\partial z_{1}}(X_{11}z_{1}+X_{12}z_{2})% -\frac{\partial P}{\partial z_{2}}(X_{21}z_{1}+X_{22}z_{2})-\frac{\partial P}{% \partial\overline{z_{1}}}(\overline{X_{11}}\overline{z_{1}}+\overline{X_{12}}% \overline{z_{2}})-\frac{\partial P}{\partial\overline{z_{2}}}(\overline{X_{21}% }\overline{z_{1}}+\overline{X_{22}}\overline{z_{2}}),\quad X\in\mathfrak{sl}(2% ,\mathbb{C}),
  293. ϕ μ , ν ( H ) = - z 1 z 1 + z 2 z 2 - z 1 ¯ z 1 ¯ + z 2 ¯ z 2 ¯ , ϕ μ , ν ( X ) = - z 2 z 1 - z 2 ¯ z 1 ¯ , ϕ μ , ν ( Y ) = - z 1 z 2 - z 1 ¯ z 2 ¯ \phi_{\mu,\nu}(H)=-z_{1}\frac{\partial}{\partial z_{1}}+z_{2}\frac{\partial}{% \partial z_{2}}-\overline{z_{1}}\frac{\partial}{\partial\overline{z_{1}}}+% \overline{z_{2}}\frac{\partial}{\partial\overline{z_{2}}},\quad\phi_{\mu,\nu}(% X)=-z_{2}\frac{\partial}{\partial z_{1}}-\overline{z_{2}}\frac{\partial}{% \partial\overline{z_{1}}},\quad\phi_{\mu,\nu}(Y)=-z_{1}\frac{\partial}{% \partial z_{2}}-\overline{z_{1}}\frac{\partial}{\partial\overline{z_{2}}}
  294. e x p : 𝐬𝐥 ( 2 , 𝐂 ) S L ( 2 , 𝐂 ) exp:\mathbf{sl}(2,\mathbf{C})→SL(2,\mathbf{C})
  295. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  296. Q Q
  297. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  298. q = e x p ( Q ) q=exp(Q)
  299. g g
  300. G G
  301. 𝐠 \mathbf{g}
  302. g g
  303. e x p exp
  304. q q
  305. e - X e i π H = e ( 0 - 1 0 0 ) e i π ( 1 0 0 - 1 ) = ( 1 - 1 0 1 ) ( - 1 0 0 - 1 ) = ( - 1 1 0 - 1 ) = q . e^{-X}e^{i\pi H}=e^{\bigl(\begin{smallmatrix}0&-1\\ 0&0\\ \end{smallmatrix}\bigr)}e^{i\pi\bigl(\begin{smallmatrix}1&0\\ 0&-1\\ \end{smallmatrix}\bigr)}=\bigl(\begin{smallmatrix}1&-1\\ 0&1\\ \end{smallmatrix}\bigr)\bigl(\begin{smallmatrix}-1&0\\ 0&-1\\ \end{smallmatrix}\bigr)=\bigl(\begin{smallmatrix}-1&1\\ 0&-1\\ \end{smallmatrix}\bigr)=q.
  306. 𝐩 e x p : 𝐬𝐥 ( 2 , 𝐂 ) S O ( 3 , 1 ) \mathbf{p}∘exp:\mathbf{sl}(2,\mathbf{C})→SO(3,1)
  307. a a
  308. e x p exp
  309. b b
  310. 𝐩 \mathbf{p}
  311. 𝐩 ( b ) = 𝐩 ( a ) \mathbf{p}(b)=\mathbf{p}(a)
  312. e x p exp
  313. q q
  314. p p
  315. p = ( 1 - 1 0 1 ) = e - X , p - 1 q = ( 1 1 0 1 ) ( - 1 1 0 - 1 ) = ( - 1 0 0 - 1 ) = - I . p=\bigl(\begin{smallmatrix}1&-1\\ 0&1\\ \end{smallmatrix}\bigr)=e^{-X},\quad p^{-1}q=\bigl(\begin{smallmatrix}1&1\\ 0&1\\ \end{smallmatrix}\bigr)\bigl(\begin{smallmatrix}-1&1\\ 0&-1\\ \end{smallmatrix}\bigr)=\bigl(\begin{smallmatrix}-1&0\\ 0&-1\\ \end{smallmatrix}\bigr)=-I.
  316. 𝐩 \mathbf{p}
  317. e x p exp
  318. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  319. σ σ
  320. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  321. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  322. 𝐩 e x p : 𝐬𝐥 ( 2 , 𝐂 ) S O ( 3 , 1 ) = e x p σ \mathbf{p}∘exp:\mathbf{sl}(2,\mathbf{C})→SO(3,1)=exp∘σ
  323. X 𝐬𝐥 ( 2 , 𝐂 ) X∈\mathbf{sl}(2,\mathbf{C})
  324. 𝐩 e x p \mathbf{p}∘exp
  325. e x p σ exp∘σ
  326. S L ( 2 , C ) SL(2,C)
  327. ( Φ , V ) (Φ,V)
  328. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  329. ( Π , V ) (Π,V)
  330. k e r 𝐩 k e r Φ ker\mathbf{p}⊂ker Φ
  331. Φ Φ
  332. Π Π
  333. I I
  334. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  335. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  336. S L ( 2 , C ) SL(2,C)
  337. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  338. 𝐩 \mathbf{p}
  339. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  340. σ : 𝐬𝐨 ( 3 , 1 ) 𝐬𝐥 ( 3 , 𝐂 ) σ:\mathbf{so}(3,1)→\mathbf{sl}(3,\mathbf{C})
  341. e x p : 𝐬𝐥 ( 2 , 𝐂 ) S L ( 2 , 𝐂 ) exp:\mathbf{sl}(2,\mathbf{C})→SL(2,\mathbf{C})
  342. ( m , n ) (m,n)
  343. ( 2 m + 1 ) ( 2 n + 1 ) (2m+ 1)(2n+ 1)
  344. Π Π
  345. S U ( 2 ) × S U ( 2 ) SU(2)×SU(2)
  346. S U ( 2 ) SU(2)
  347. S U ( 2 ) SU(2)
  348. 𝐠 \mathbf{g}
  349. dim π μ = Π α R + α , μ + δ Π α R + α , δ , \operatorname{dim}\pi_{\mu}=\frac{\Pi_{\alpha\in R^{+}}\langle\alpha,\mu+% \delta\rangle}{\Pi_{\alpha\in R^{+}}\langle\alpha,\delta\rangle},
  350. δ δ
  351. 𝐠 \mathbf{g}
  352. 𝐡 𝐠 \mathbf{h}⊂\mathbf{g}
  353. 𝐡 * ) \mathbf{h}*)
  354. 𝐡 \mathbf{h}
  355. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  356. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  357. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  358. Π Π
  359. G G
  360. N = k e r Π N=kerΠ
  361. N N
  362. N N
  363. N N
  364. N Z N⊂Z
  365. Z Z
  366. G G
  367. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  368. V , W V,W
  369. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  370. ( m , n ) (m,n)
  371. U : G G L ( V ) U:G→GL(V)
  372. V V
  373. G G
  374. U ( G ) U ( V ) G L ( V ) U(G)⊂U(V)⊂GL(V)
  375. U ( V ) U(V)
  376. G L ( V ) GL(V)
  377. v ) v)
  378. k e r U kerU
  379. U U
  380. G G
  381. G G
  382. k e r U kerU
  383. G G
  384. U U
  385. k e r U kerU
  386. U U
  387. U U
  388. U ( G ) G . U(G)≈G.
  389. U ( G ) U(G)
  390. U ( G ) U(G)
  391. U ( V ) U(V)
  392. U ( G ) U ( V ) U(G)⊂U(V)
  393. U ( G ) U(G)
  394. G G
  395. 𝐀 \mathbf{A}
  396. 𝐁 \mathbf{B}
  397. 𝐉 \mathbf{J}
  398. 𝐊 \mathbf{K}
  399. ( m , n ) (m,n)
  400. S O ( 3 ) SO(3)
  401. ( m , n ) (m,n)
  402. S O ( 3 ) SO(3)
  403. m + n , m + n 1 , , | m n | m+n,m+n−1,…,|m−n|
  404. j j
  405. ( 2 j + 1 ) (2j+1)
  406. 1 2 \frac{1}{2}
  407. 1 2 \frac{1}{2}
  408. 𝐉 = 𝐀 + 𝐁 \mathbf{J}=\mathbf{A}+\mathbf{B}
  409. ( m + n ) (m+n)ℏ
  410. S O ( 3 ) SO(3)
  411. ( m , n ) (m,n)
  412. m + n m+n
  413. ( 1 2 , 0 ) (\frac{1}{2},0)
  414. ( 0 , 1 2 ) (0,\frac{1}{2})
  415. 2 2
  416. ( 0 , 3 2 ) (0,\frac{3}{2})
  417. ( 1 , 1 2 ) (1,\frac{1}{2})
  418. 2 3 2 + 1 = 4 2\frac{3}{2}+1=4
  419. ( 2 + 1 ) ( 2 1 2 + 1 ) = 6 (2+1)(2\frac{1}{2}+1)=6
  420. 3 2 \frac{3}{2}
  421. 1 2 \frac{1}{2}
  422. S O ( 3 ) SO(3)
  423. n 2 \frac{n}{2}
  424. n n
  425. S O ( 3 ) SO(3)
  426. μ μ
  427. I −I
  428. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  429. 𝐬𝐥 ( 2 , 𝐂 ) 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})⊕\mathbf{sl}(2,\mathbf{C})
  430. γ γ
  431. γ γ
  432. I W −I∈W
  433. π , σ π,σ
  434. π σ π≈σ
  435. Π Σ Π≈Σ
  436. π m , n * π m , n , Π m , n * Π m , n , 2 m , 2 n . \pi_{m,n}^{*}\cong\pi_{m,n},\quad\Pi_{m,n}^{*}\cong\Pi_{m,n},\quad 2m,2n\in% \mathbb{N}.
  437. π π
  438. π ¯ \overline{π}
  439. π π
  440. 𝐬𝐥 ( n , 𝐂 ) \mathbf{sl}(n,\mathbf{C})
  441. π ± ( X ) = 1 2 ( π ( X ) ± i π ( i - 1 X ) ) , \pi^{\pm}(X)=\frac{1}{2}(\pi(X)\pm i\pi(i^{-1}X)),
  442. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  443. π m , n ¯ = π m , n + + π m , n - ¯ = π m 2 n + 1 ¯ + π n ¯ 2 m + 1 ¯ = π n 2 m + 1 + π m ¯ 2 n + 1 = π n , m + + π n , m - = π n , m , Π m , n ¯ = Π n , m , 2 m , 2 n , \overline{\pi_{m,n}}=\overline{\pi_{m,n}^{+}+\pi_{m,n}^{-}}=\overline{\pi_{m}^% {\oplus_{2n+1}}}+\overline{\overline{\pi_{n}}^{\oplus_{2m+1}}}=\pi_{n}^{\oplus% _{2m+1}}+\overline{\pi_{m}}^{\oplus_{2n+1}}=\pi_{n,m}^{+}+\pi_{n,m}^{-}=\pi_{n% ,m},\quad\overline{\Pi_{m,n}}=\Pi_{n,m},\quad 2m,2n\in\mathbb{N},
  444. e x p ( X ¯ ) exp(\overline{X})
  445. e x p ( ¯ X ) \overline{exp(}{X}{)}
  446. ( m , n ) (m,n)
  447. m = n m=n
  448. ( m , n ) ( n , m ) (m,n)⊕(n,m)
  449. ( π , V ) (π,V)
  450. E n d V EndV
  451. π π
  452. π ( X ) ( A ) = [ π ( X ) , A ] , A End V , X 𝔤 . \pi(X)(A)=[\pi(X),A],\quad A\in\operatorname{End}V,\ X\in\mathfrak{g}.
  453. ( Π , V ) (Π,V)
  454. G G
  455. Π Π
  456. E n d V EndV
  457. G G
  458. Π Π
  459. ( Π , V ) (Π,V)
  460. E n d V EndV
  461. ( π , H ) (π,H)
  462. ( Π , H ) (Π,H)
  463. H H
  464. H H
  465. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2}, 0)⊕(0,\frac{1}{2})
  466. E n d ( H ) End(H)
  467. 1 2 \frac{1}{2}
  468. 1 2 \frac{1}{2}
  469. E n d H EndH
  470. H H
  471. E n d H EndH
  472. E n d H EndH
  473. E n d H EndH
  474. ( 0 , 0 ) (0,0)
  475. ( 0 , 0 ) (0,0)
  476. ( 1 2 , , 1 2 ) (\frac{1}{2},,\frac{1}{2})
  477. ( 1 2 , 1 2 ) (\frac{1}{2},\frac{1}{2})
  478. ( 1 , 0 ) ( 0 , 1 ) (1,0)⊕(0,1)
  479. 1 + 1 + 4 + 4 + 6 = 16 1+1+4+4+6=16
  480. C l 3 , 1 ( ) = ( 0 , 0 ) ( 1 2 , 1 2 ) [ ( 1 , 0 ) ( 0 , 1 ) ] ( 1 2 , 1 2 ) p ( 0 , 0 ) p , Cl_{3,1}(\mathbb{R})=(0,0)\oplus(\frac{1}{2},\frac{1}{2})\oplus[(1,0)\oplus(0,% 1)]\oplus(\frac{1}{2},\frac{1}{2})_{p}\oplus(0,0)_{p},
  481. ( 1 , 0 ) ( 0 , 1 ) (1,0)⊕(0,1)
  482. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2},0)⊕(0,\frac{1}{2})
  483. E n d H EndH
  484. ( m , n ) (m,n)
  485. m = n m=n
  486. O ( 3 , 1 ) O(3,1)
  487. P O ( 3 , 1 ) P∈O(3,1)
  488. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  489. P P
  490. P = d i a g ( 1 , 1 , 1 , 1 ) P=diag(1, −1, −1, −1)
  491. Ad P ( J i ) = P J i P - 1 = J i , Ad P ( K i ) = P K i P - 1 = - K i . \mathrm{Ad}_{P}(J_{i})=PJ_{i}P^{-1}=J_{i},\qquad\mathrm{Ad}_{P}(K_{i})=PK_{i}P% ^{-1}=-K_{i}.
  492. 𝐊 \mathbf{K}
  493. 𝐉 \mathbf{J}
  494. P P
  495. 𝐊 \mathbf{K}
  496. 𝐉 \mathbf{J}
  497. π π
  498. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  499. Π Π
  500. π π
  501. X 𝐬𝐨 ( 3 ; 1 ) X∈\mathbf{so}(3;1)
  502. P P
  503. Π Π
  504. Π ( P ) π ( B i ) Π ( P ) - 1 = π ( A i ) \Pi(P)\pi(B_{i})\Pi(P)^{-1}=\pi(A_{i})
  505. 𝐀 \mathbf{A}
  506. 𝐁 \mathbf{B}
  507. m = n m=n
  508. m n m≠n
  509. ( m , n ) ( n , m ) (m,n)⊕(n,m)
  510. Π ( P ) Π(P)
  511. ( m , n ) (m,n)
  512. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2}, 0)⊕(0,\frac{1}{2})
  513. ( 0 , 0 ) (0,0)
  514. T = d i a g ( 1 , 1 , 1 , 1 ) T=diag(−1, 1, 1, 1)
  515. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  516. T T
  517. P P
  518. O ( 3 , 1 ) O(3,1)
  519. H H
  520. T T
  521. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  522. i i
  523. Ψ Ψ
  524. E E
  525. E −E
  526. Π ( T ) Π(T)
  527. i i
  528. + H +H
  529. Π Π
  530. P P
  531. T T
  532. C C
  533. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  534. 𝐔 \mathbf{U}
  535. 𝐁 \mathbf{B}
  536. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  537. 𝐑 \mathbf{R}
  538. V V
  539. n n
  540. f V f∈V
  541. ( Π ( g ) f ) ( x ) = f ( Π x ( g ) - 1 x ) , x n , f V (\Pi(g)f)(x)=f(\Pi_{x}(g)^{-1}x),\qquad x\in\mathbb{R}^{n},f\in V
  542. Π f V Πf∈V
  543. n n
  544. Π Π
  545. V V
  546. G G
  547. n n
  548. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  549. S U ( 2 ) SU(2)
  550. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  551. S O ( 3 ) SO(3)
  552. f , g = 𝕊 2 f ¯ g d Ω = 0 2 π 0 π f ¯ g sin θ d θ d φ . \langle f,g\rangle=\int_{\mathbb{S}^{2}}\overline{f}gd\Omega=\int_{0}^{2\pi}% \int_{0}^{\pi}\overline{f}g\sin\theta d\theta d\varphi.
  553. f f
  554. | f = l = 1 m = - l m = l | Y m l Y m l | f , f ( θ , φ ) = l = 1 m = - l m = l f l m Y m l ( θ , φ ) , |f\rangle=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}|Y_{m}^{l}\rangle\langle Y_{m}^{% l}|f\rangle,\quad f(\theta,\varphi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}f_{lm}% Y^{l}_{m}(\theta,\varphi),
  555. f l m = Y m l , f = 𝕊 2 < ¯ m t p l > Y m l f_{lm}=\langle Y_{m}^{l},f\rangle=\int_{\mathbb{S}^{2}}\overline{<}mtpl>{{Y^{l% }_{m}}}
  556. S O ( 3 ) SO(3)
  557. ( Π ( R ) f ) ( θ ( x ) , φ ( x ) ) = l = 1 m = - l m = l m = - l m = l D m m ( l ) ( R ) f l m Y m l ( θ ( R - 1 x ) , φ ( R - 1 x ) ) , R SO ( 3 ) , x 𝕊 2 . (\Pi(R)f)(\theta(x),\varphi(x))=\sum_{l=1}^{\infty}\sum_{m=-l}^{m=l}\sum_{m^{% \prime}=-l}^{m^{\prime}=l}D^{(l)}_{mm^{\prime}}(R)f_{lm^{\prime}}Y^{l}_{m}(% \theta(R^{-1}x),\varphi(R^{-1}x)),\qquad R\in\mathrm{SO}(3),\quad x\in\mathbb{% S}^{2}.
  558. Π ( R ) f , Π ( R ) g = f , g f , g 𝕊 2 , R SO ( 3 ) . \langle\Pi(R)f,\Pi(R)g\rangle=\langle f,g\rangle\qquad\forall f,g\in\mathbb{S}% ^{2},\quad\forall R\in\mathrm{SO}(3).
  559. 𝐬𝐮 ( 2 ) \mathbf{su}(2)
  560. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  561. S O ( 3 ) SO(3)
  562. Π Π
  563. ( Π , V ) (Π,V)
  564. f , g U SO ( 3 ) Π ( R ) f , Π ( R ) g d g = 1 8 π 2 0 2 π 0 π 0 2 π Π ( R ) f , Π ( R ) g sin θ d φ d θ d ψ , f , g V , \langle f,g\rangle_{U}\equiv\int_{\mathrm{SO}(3)}\langle\Pi(R)f,\Pi(R)g\rangle dg% =\frac{1}{8\pi^{2}}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\langle\Pi(R)f,% \Pi(R)g\rangle\sin\theta d\varphi d\theta d\psi,\quad f,g\in V,
  565. S O ( 3 ) SO(3)
  566. 1 1
  567. V V
  568. M M
  569. a , b , c , d a,b,c,d
  570. Π f = ( a b c d ) , det Π f = 1. \Pi_{f}=\begin{pmatrix}a&b\\ c&d\end{pmatrix},\qquad\operatorname{det}\Pi_{f}=1.
  571. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  572. a , b , c , α , β , γ , α , β , γ a,b,c,α,β,γ,α′,β′,γ′
  573. w ( z ) = ( z - a z - b ) α ( z - c z - b ) 2 γ F 1 ( α + β + γ , α + β + γ ; 1 + α - α ; ( z - a ) ( c - b ) ( z - b ) ( c - a ) ) . w(z)=\left(\frac{z-a}{z-b}\right)^{\alpha}\left(\frac{z-c}{z-b}\right)^{\gamma% }\;_{2}F_{1}\left(\alpha+\beta+\gamma,\alpha+\beta^{\prime}+\gamma;1+\alpha-% \alpha^{\prime};\frac{(z-a)(c-b)}{(z-b)(c-a)}\right).
  574. u = A z + B C z + D and η = A a + B C a + D u=\frac{Az+B}{Cz+D}\quad\,\text{ and }\quad\eta=\frac{Aa+B}{Ca+D}
  575. A , B , C , D A,B,C,D
  576. π f Λ - 1 = ( A B C D ) , \pi_{f_{\Lambda}}^{-1}=\begin{pmatrix}A&B\\ C&D\end{pmatrix},
  577. Λ Λ
  578. B B
  579. G = S L ( 2 , 𝐂 ) G=SL(2,\mathbf{C})
  580. B B
  581. z z
  582. χ ν , k ( z 0 c z - 1 ) = r i ν e i k θ , \chi_{\nu,k}\begin{pmatrix}z&0\\ c&z^{-1}\end{pmatrix}=r^{i\nu}e^{ik\theta},
  583. k k
  584. ν ν
  585. k k
  586. k −k
  587. k = 0 k=0
  588. K = S U ( 2 ) K=SU(2)
  589. G G
  590. K K
  591. G / B = K / T G/B=K/T
  592. T = B K T=B∩K
  593. K K
  594. | z | = 1 |z|=1
  595. ν ν
  596. K K
  597. K K
  598. | k | + 2 m + 1 |k|+2m+1
  599. m m
  600. 𝐂 \mathbf{C}
  601. π ν , k ( a b c d ) - 1 f ( z ) = | c z + d | - 2 - i ν ( c z + d | c z + d | ) - k f ( a z + b c z + d ) . \pi_{\nu,k}\begin{pmatrix}a&b\\ c&d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-i\nu}\left({cz+d\over|cz+d|}\right)^{-k}f% \left({az+b\over cz+d}\right).
  602. B B
  603. G = B B s B G=B∪BsB
  604. s s
  605. ( 0 - 1 1 0 ) \begin{pmatrix}0&-1\\ 1&0\end{pmatrix}
  606. 𝔤 \mathfrak{g}
  607. G G
  608. K K
  609. 𝔤 \mathfrak{g}
  610. 𝐂 \mathbf{C}
  611. ( f , g ) = f ( z ) g ( w ) ¯ d z d w | z - w | 2 - t . (f,g)=\int\int{f(z)\overline{g(w)}\,dz\,dw\over|z-w|^{2-t}}.
  612. π t ( a b c d ) - 1 f ( z ) = | c z + d | - 2 - t f ( a z + b c z + d ) . \pi_{t}\begin{pmatrix}a&b\\ c&d\end{pmatrix}^{-1}f(z)=|cz+d|^{-2-t}f\left({az+b\over cz+d}\right).
  613. K K
  614. K = S U ( 2 ) K=SU(2)
  615. 𝔤 \mathfrak{g}
  616. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  617. I −I
  618. k k
  619. G G
  620. k = 0 k=0
  621. λ λ
  622. ρ ρ
  623. λ ( g ) f ( x ) = f ( g - 1 x ) , ρ ( g ) f ( x ) = f ( x g ) . \lambda(g)f(x)=f(g^{-1}x),\,\,\rho(g)f(x)=f(xg).
  624. f f
  625. π ν , k ( f ) = G f ( g ) π ( g ) d g \pi_{\nu,k}(f)=\int_{G}f(g)\pi(g)\,dg
  626. H H
  627. H = k 0 H S ( L 2 ( C ) ) L 2 ( R , c k ( ν 2 + k 2 ) 1 / 2 d ν ) , H=\bigoplus_{k\geq 0}HS(L^{2}(C))\otimes L^{2}(R,c_{k}(\nu^{2}+k^{2})^{1/2}d% \nu),
  628. c 0 = 1 / 4 π 3 / 2 , c k = 1 / ( 2 π ) 3 / 2 ( k 0 ) c_{0}=1/4\pi^{3/2},\,\,c_{k}=1/(2\pi)^{3/2}\,\,(k\neq 0)
  629. U U
  630. U ( f ) ( ν , k ) = π ν , k ( f ) U(f)(\nu,k)=\pi_{\nu,k}(f)
  631. H H
  632. U U
  633. U ( λ ( x ) ρ ( y ) f ) ( ν , k ) = π ν , k ( x ) - 1 π ν , k ( f ) π ν , k ( y ) . U(\lambda(x)\rho(y)f)(\nu,k)=\pi_{\nu,k}(x)^{-1}\pi_{\nu,k}(f)\pi_{\nu,k}(y).
  634. ( f 1 , f 2 ) = k 0 c k 2 - Tr ( π ν , k ( f 1 ) π ν , k ( f 2 ) * ) ( ν 2 + k 2 ) d ν . (f_{1},f_{2})=\sum_{k\geq 0}c_{k}^{2}\int_{-\infty}^{\infty}{\rm Tr}(\pi_{\nu,% k}(f_{1})\pi_{\nu,k}(f_{2})^{*})(\nu^{2}+k^{2})\,d\nu.
  635. f 2 * ( g ) = f 2 ( g - 1 ) ¯ f_{2}^{*}(g)=\overline{f_{2}(g^{-1})}
  636. f ( 1 ) = k 0 c k 2 - Tr ( π ν , k ( f ) ) ( ν 2 + k 2 ) d ν . f(1)=\sum_{k\geq 0}c_{k}^{2}\int_{-\infty}^{\infty}{\rm Tr}(\pi_{\nu,k}(f))(% \nu^{2}+k^{2})\,d\nu.
  637. f f
  638. C c ( G ) C^{\infty}_{c}(G)
  639. f f
  640. η η
  641. d i a g ( 1 , 1 , 1 , 1 ) diag(−1, 1, 1, 1)
  642. J 1 = J 23 = - J 32 = i ( 0 0 0 0 0 0 0 0 0 0 0 - 1 0 0 1 0 ) , J 2 = J 31 = - J 13 = i ( 0 0 0 0 0 0 0 1 0 0 0 0 0 - 1 0 0 ) , J 3 = J 12 = - J 21 = i ( 0 0 0 0 0 0 - 1 0 0 1 0 0 0 0 0 0 ) , K 1 = J 01 = J 10 = i ( 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ) , K 2 = J 02 = J 20 = i ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) , K 3 = J 03 = J 30 = i ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ) . \begin{aligned}\displaystyle J_{1}&\displaystyle=J^{23}=-J^{32}=i\biggl(\begin% {smallmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&-1\\ 0&0&1&0\\ \end{smallmatrix}\biggr),\\ \displaystyle J_{2}&\displaystyle=J^{31}=-J^{13}=i\biggl(\begin{smallmatrix}0&% 0&0&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&-1&0&0\\ \end{smallmatrix}\biggr),\\ \displaystyle J_{3}&\displaystyle=J^{12}=-J^{21}=i\biggl(\begin{smallmatrix}0&% 0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ \displaystyle K_{1}&\displaystyle=J^{01}=J^{10}=i\biggl(\begin{smallmatrix}0&1% &0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ \displaystyle K_{2}&\displaystyle=J^{02}=J^{20}=i\biggl(\begin{smallmatrix}0&0% &1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ \end{smallmatrix}\biggr),\\ \displaystyle K_{3}&\displaystyle=J^{03}=J^{30}=i\biggl(\begin{smallmatrix}0&0% &0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{smallmatrix}\biggr).\end{aligned}
  643. [ J μ ν , J ρ σ ] = i ( η σ μ J ρ ν + η ν σ J μ ρ - η ρ μ J σ ν - η ν ρ J μ σ ) . [J^{\mu\nu},J^{\rho\sigma}]=i(\eta^{\sigma\mu}J^{\rho\nu}+\eta^{\nu\sigma}J^{% \mu\rho}-\eta^{\rho\mu}J^{\sigma\nu}-\eta^{\nu\rho}J^{\mu\sigma}).
  644. [ J i , J j ] = i ϵ i j k J k , [ J i , K j ] = i ϵ i j k K k , [ K i , K j ] = - i ϵ i j k J k . [J_{i},J_{j}]=i\epsilon_{ijk}J_{k},\quad[J_{i},K_{j}]=i\epsilon_{ijk}K_{k},% \quad[K_{i},K_{j}]=-i\epsilon_{ijk}J_{k}.
  645. J J
  646. V V
  647. 𝐬𝐨 ( 3 ; 1 ) \mathbf{so}(3;1)
  648. ( m , n ) (m,n)
  649. m a , a m −m≤a,a′≤m
  650. n b , b n −n≤b,b′≤n
  651. ( π m , n ( J i ) ) a b , a b = δ b b ( J i ( m ) ) a a + δ a a ( J i ( n ) ) b b , ( π m , n ( K i ) ) a b , a b = i ( δ a a ( J i ( n ) ) b b - δ b b ( J i ( m ) ) a a ) , \begin{aligned}\displaystyle(\pi_{m,n}(J_{i}))_{a^{\prime}b^{\prime},ab}&% \displaystyle=\delta_{b^{\prime}b}(J_{i}^{(m)})_{a^{\prime}a}+\delta_{a^{% \prime}a}(J_{i}^{(n)})_{b^{\prime}b},\\ \displaystyle(\pi_{m,n}(K_{i}))_{a^{\prime}b^{\prime},ab}&\displaystyle=i(% \delta_{a^{\prime}a}(J_{i}^{(n)})_{b^{\prime}b}-\delta_{b^{\prime}b}(J_{i}^{(m% )})_{a^{\prime}a}),\end{aligned}
  652. δ δ
  653. ( 2 n + 1 ) (2n+1)
  654. 𝐬𝐨 ( 3 ) \mathbf{so}(3)
  655. ( J 3 ( j ) ) a a = a δ a a , ( J 1 ( j ) ± i J 2 ( j ) ) a a = ( j a ) ( j ± a + 1 ) δ a , a ± 1 . \begin{aligned}\displaystyle(J_{3}^{(j)})_{a^{\prime}a}&\displaystyle=a\delta_% {a^{\prime}a},\\ \displaystyle(J_{1}^{(j)}\pm iJ_{2}^{(j)})_{a^{\prime}a}&\displaystyle=\sqrt{(% j\mp a)(j\pm a+1)}\delta_{a^{\prime},a\pm 1}.\end{aligned}
  656. m = 1 2 m=\frac{1}{2}
  657. n = 0 n=0
  658. m = 0 m=0
  659. n = 1 2 n=\frac{1}{2}
  660. J i ( 1 2 ) = 1 2 σ i J_{i}^{(\frac{1}{2})}=\frac{1}{2}\sigma_{i}
  661. π ( 1 2 , 0 ) ( J i ) = 1 2 ( σ i 1 ( 1 ) + 1 ( 2 ) J i ( 0 ) ) = 1 2 σ i π ( 1 2 , 0 ) ( K i ) = i 2 ( 1 ( 2 ) J i ( 0 ) - σ i 1 ( 1 ) ) = - i 2 σ i , π ( 0 , 1 2 ) ( J i ) = 1 2 ( J i ( 0 ) 1 ( 2 ) + 1 ( 1 ) σ i ) = 1 2 σ i π ( 0 , 1 2 ) ( K i ) = i 2 ( 1 ( 1 ) σ i - J i ( 0 ) 1 ( 2 ) ) = + i 2 σ i . \begin{aligned}\displaystyle\pi_{(\frac{1}{2},0)}(J_{i})&\displaystyle=\frac{1% }{2}(\sigma_{i}\otimes 1_{(1)}+1_{(2)}\otimes J^{(0)}_{i})=\frac{1}{2}\sigma_{% i}\quad\pi_{(\frac{1}{2},0)}(K_{i})=\frac{i}{2}(1_{(2)}\otimes J^{(0)}_{i}-% \sigma_{i}\otimes 1_{(1)})=-\frac{i}{2}\sigma_{i},\\ \displaystyle\pi_{(0,\frac{1}{2})}(J_{i})&\displaystyle=\frac{1}{2}(J^{(0)}_{i% }\otimes 1_{(2)}+1_{(1)}\otimes\sigma_{i})=\frac{1}{2}\sigma_{i}\quad\pi_{(0,% \frac{1}{2})}(K_{i})=\frac{i}{2}(1_{(1)}\otimes\sigma_{i}-J^{(0)}_{i}\otimes 1% _{(2)})=+\frac{i}{2}\sigma_{i}.\end{aligned}
  662. ( 1 2 , 0 ) ( 0 , 1 2 ) (\frac{1}{2},0)⊕(0,\frac{1}{2})
  663. 𝐬𝐨 ( 3 , 1 ) \mathbf{so}(3,1)
  664. ( Ψ < s u b > L , Ψ R ) (Ψ<sub>L, Ψ_{R})
  665. S < s u b > α β g α β = 0 S<sub>αβg^{αβ}=0
  666. 3 2 \frac{3}{2}
  667. 3 2 \frac{3}{2}
  668. A A
  669. S U ( 2 ) SU(2)
  670. 𝐩 \mathbf{p}
  671. 2 : 1 2:1
  672. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  673. S O ( 3 , 1 ) < s u p > + SO(3,1)<sup>+
  674. S L ( 2 , 𝐂 ) SL(2,\mathbf{C})
  675. e x p σ l o g : S L ( 2 , 𝐂 ) S O ( 3 , 1 ) < s u p > + exp∘σ∘log:SL(2,\mathbf{C})→SO(3,1)<sup>+
  676. q = e x p ( Q ) q=exp(Q)
  677. Q Q
  678. λ , λ λ,−λ
  679. λ = i π + 2 π i k λ=iπ+2πik
  680. k k
  681. 𝐬𝐥 ( 2 , 𝐂 ) \mathbf{sl}(2,\mathbf{C})
  682. Q Q
  683. q q
  684. G G
  685. Z Z
  686. G G
  687. f : X Y f:X→Y
  688. X X
  689. f ( X ) f(X)
  690. A < s u b > 1 A<sub>1
  691. 1 2 \frac{1}{2}
  692. L < s u p > 2 ( 𝐒 2 ) L<sup>2(\mathbf{S}^{2})
  693. D D
  694. D < s u p > ( [ u e l l ] ) D<sup>([u^{\prime}ell^{\prime}])
  695. H H
  696. H S ( H ) HS(H)
  697. H H