wpmath0000003_7

Homeomorphism_(graph_theory).html

  1. G G
  2. G G^{\prime}
  3. G G
  4. G G^{\prime}
  5. L ( g ) = { G i ( g ) } L(g)=\{G_{i}^{(g)}\}
  6. G i ( g ) G_{i}^{(g)}
  7. L ( 0 ) = { K 5 , K 3 , 3 } L(0)=\{K_{5},K_{3,3}\}

Homogeneous_function.html

  1. f ( α 𝐯 ) = α k f ( 𝐯 ) f(\alpha\mathbf{v})=\alpha^{k}f(\mathbf{v})
  2. f ( x , y ) = x f(x,y)=x
  3. x y > 0 xy>0
  4. f ( x , y ) = 0 f(x,y)=0
  5. x y 0 xy\leq 0
  6. f ( α x , α y ) = α f ( x , y ) f(\alpha x,\alpha y)=\alpha f(x,y)
  7. α , x , y \alpha,x,y
  8. y = 0 , x 0 y=0,x\neq 0
  9. f ( α 𝐯 ) = α f ( 𝐯 ) f(\alpha\mathbf{v})=\alpha f(\mathbf{v})
  10. f ( α 𝐯 1 , , α 𝐯 n ) = α n f ( 𝐯 1 , , 𝐯 n ) f(\alpha\mathbf{v}_{1},\ldots,\alpha\mathbf{v}_{n})=\alpha^{n}f(\mathbf{v}_{1}% ,\ldots,\mathbf{v}_{n})
  11. f ( x , y , z ) = x 5 y 2 z 3 f(x,y,z)=x^{5}y^{2}z^{3}\,
  12. f ( α x , α y , α z ) = ( α x ) 5 ( α y ) 2 ( α z ) 3 = α 10 x 5 y 2 z 3 = α 10 f ( x , y , z ) . f(\alpha x,\alpha y,\alpha z)=(\alpha x)^{5}(\alpha y)^{2}(\alpha z)^{3}=% \alpha^{10}x^{5}y^{2}z^{3}=\alpha^{10}f(x,y,z).\,
  13. x 5 + 2 x 3 y 2 + 9 x y 4 x^{5}+2x^{3}y^{2}+9xy^{4}\,
  14. f ( v ) = g ( v , v , , v ) . f(v)=g(v,v,\dots,v).
  15. g ( v 1 , v 2 , , v n ) = 1 n ! t 1 t 2 t n f ( t 1 v 1 + + t n v n ) . g(v_{1},v_{2},\dots,v_{n})=\frac{1}{n!}\frac{\partial}{\partial t_{1}}\frac{% \partial}{\partial t_{2}}\cdots\frac{\partial}{\partial t_{n}}f(t_{1}v_{1}+% \cdots+t_{n}v_{n}).
  16. f ( x ) = ln x f(x)=\ln x
  17. f ( 5 x ) = ln 5 x = ln 5 + f ( x ) f(5x)=\ln 5x=\ln 5+f(x)
  18. f ( 10 x ) = ln 10 + f ( x ) f(10x)=\ln 10+f(x)
  19. f ( 15 x ) = ln 15 + f ( x ) f(15x)=\ln 15+f(x)
  20. k k
  21. f ( α x ) = α k f ( x ) f(\alpha\cdot x)=\alpha^{k}\cdot f(x)
  22. f ( x ) = x + 5 f(x)=x+5
  23. f ( α x ) = α k f ( x ) f(\alpha x)=\alpha^{k}f(x)\,
  24. 𝐱 f ( 𝐱 ) = k f ( 𝐱 ) . \mathbf{x}\cdot\nabla f(\mathbf{x})=kf(\mathbf{x}).
  25. g ( α ) = f ( α 𝐱 ) \textstyle g(\alpha)=f(\alpha\mathbf{x})
  26. α 𝐱 f ( α 𝐱 ) = k f ( α 𝐱 ) \textstyle\alpha\mathbf{x}\cdot\nabla f(\alpha\mathbf{x})=kf(\alpha\mathbf{x})
  27. g ( α ) = 𝐱 f ( α 𝐱 ) = k α f ( α 𝐱 ) = k α g ( α ) . g^{\prime}(\alpha)=\mathbf{x}\cdot\nabla f(\alpha\mathbf{x})=\frac{k}{\alpha}f% (\alpha\mathbf{x})=\frac{k}{\alpha}g(\alpha).
  28. g ( α ) - k α g ( α ) = 0 \textstyle g^{\prime}(\alpha)-\frac{k}{\alpha}g(\alpha)=0
  29. g ( α ) = g ( 1 ) α k \textstyle g(\alpha)=g(1)\alpha^{k}
  30. f ( α 𝐱 ) = g ( α ) = α k g ( 1 ) = α k f ( 𝐱 ) \textstyle f(\alpha\mathbf{x})=g(\alpha)=\alpha^{k}g(1)=\alpha^{k}f(\mathbf{x})
  31. f / x i \partial f/\partial x_{i}
  32. 𝐱 \mathbf{x}\cdot\nabla
  33. n f ( t x ) φ ( x ) d x = t k n f ( x ) φ ( x ) d x \int_{\mathbb{R}^{n}}f(tx)\varphi(x)\,dx=t^{k}\int_{\mathbb{R}^{n}}f(x)\varphi% (x)\,dx
  34. φ \varphi
  35. t - n n f ( y ) φ ( y / t ) d y = t k n f ( y ) φ ( y ) d y t^{-n}\int_{\mathbb{R}^{n}}f(y)\varphi(y/t)\,dy=t^{k}\int_{\mathbb{R}^{n}}f(y)% \varphi(y)\,dy
  36. φ \varphi
  37. t - n S , φ μ t = t k S , φ t^{-n}\langle S,\varphi\circ\mu_{t}\rangle=t^{k}\langle S,\varphi\rangle
  38. φ \varphi
  39. I ( x , y ) d y d x + J ( x , y ) = 0 , I(x,y)\frac{\mathrm{d}y}{\mathrm{d}x}+J(x,y)=0,
  40. x d v d x = - J ( 1 , v ) I ( 1 , v ) - v . x\frac{\mathrm{d}v}{\mathrm{d}x}=-\frac{J(1,v)}{I(1,v)}-v.

Homogeneous_space.html

  1. ρ : G Aut 𝐂 ( X ) \rho:G\to\mathrm{Aut}_{\mathbf{C}}(X)
  2. S n - 1 O ( n ) / O ( n - 1 ) S^{n-1}\cong\mathrm{O}(n)/\mathrm{O}(n-1)
  3. S n - 1 SO ( n ) / SO ( n - 1 ) S^{n-1}\cong\mathrm{SO}(n)/\mathrm{SO}(n-1)
  4. P n - 1 PO ( n ) / PO ( n - 1 ) \mathrm{P}^{n-1}\cong\mathrm{PO}(n)/\mathrm{PO}(n-1)
  5. Gr ( r , n ) = O ( n ) / ( O ( r ) × O ( n - r ) ) \mathrm{Gr}(r,n)=\mathrm{O}(n)/(\mathrm{O}(r)\times\mathrm{O}(n-r))
  6. H o = g H o g - 1 ( 1 ) H_{o^{\prime}}=gH_{o}g^{-1}\qquad\qquad(1)
  7. 1 2 N ( N + 1 ) \tfrac{1}{2}N(N+1)
  8. ξ i ( a ) \xi^{(a)}_{i}
  9. ξ [ i ; k ] ( a ) = C b c a ξ i ( b ) ξ k ( c ) \xi^{(a)}_{[i;k]}=C^{a}_{\ bc}\xi^{(b)}_{i}\xi^{(c)}_{k}
  10. C b c a C^{a}_{\ bc}
  11. C b c a = 0 C^{a}_{\ bc}=0
  12. C b c a = ε b c a C^{a}_{\ bc}=\varepsilon^{a}_{\ bc}
  13. ε b c a \varepsilon^{a}_{\ bc}

Homology_sphere.html

  1. { b , ( o 1 , 0 ) ; ( a 1 , b 1 ) , , ( a r , b r ) } \{b,(o_{1},0);(a_{1},b_{1}),\dots,(a_{r},b_{r})\}\,
  2. b + b 1 / a 1 + + b r / a r = 1 / ( a 1 a r ) . b+b_{1}/a_{1}+\cdots+b_{r}/a_{r}=1/(a_{1}\cdots a_{r}).

Homothetic_transformation.html

  1. M S + λ S M , M\mapsto S+\lambda\overrightarrow{SM},
  2. O M λ O M . \overrightarrow{OM}\mapsto\lambda\overrightarrow{OM}.

Homotopy_lifting_property.html

  1. π : E B \pi\colon E\to B
  2. X X\,
  3. ( X , π ) (X,\pi)\,
  4. π \pi\,
  5. X X\,
  6. f : X × [ 0 , 1 ] B f\colon X\times[0,1]\to B\,
  7. f ~ 0 : X E \tilde{f}_{0}\colon X\to E
  8. f 0 = f | X × { 0 } f_{0}=f|_{X\times\{0\}}
  9. f 0 = π f ~ 0 f_{0}=\pi\circ\tilde{f}_{0}\,
  10. f ~ : X × [ 0 , 1 ] E \tilde{f}\colon X\times[0,1]\to E
  11. f f\,
  12. f = π f ~ f=\pi\circ\tilde{f}\,
  13. f ~ 0 = f ~ | X × { 0 } \tilde{f}_{0}=\tilde{f}|_{X\times\{0\}}\,
  14. f ~ \tilde{f}
  15. π \pi\,
  16. π \pi\,
  17. π \pi\,
  18. X X\,
  19. X Y X\supseteq Y
  20. T := ( X × { 0 } ) ( Y × [ 0 , 1 ] ) X × [ 0 , 1 ] T\colon=(X\times\{0\})\cup(Y\times[0,1])\ \subseteq\ X\times[0,1]
  21. π : E B \pi\colon E\to B\,
  22. ( X , Y , π ) (X,Y,\pi)\,
  23. f : X × [ 0 , 1 ] B f\colon X\times[0,1]\to B\,
  24. g ~ : T E \tilde{g}\colon T\to E
  25. g = f | T g=f|_{T}\,
  26. f ~ : X × [ 0 , 1 ] E \tilde{f}\colon X\times[0,1]\to E
  27. f f\,
  28. π f ~ = f \pi\tilde{f}=f\,
  29. g ~ \tilde{g}\,
  30. f ~ | T = g ~ \tilde{f}|_{T}=\tilde{g}\,
  31. ( X , π ) (X,\pi)\,
  32. Y = Y=\emptyset
  33. T T\,
  34. X × { 0 } X\times\{0\}
  35. ( X , Y ) (X,Y)\,
  36. π \pi\,
  37. π \pi\,
  38. π \pi\,

Homotopy_principle.html

  1. C 1 C^{1}
  2. ( u 1 , u 2 , , u m ) (u_{1},u_{2},\dots,u_{m})
  3. Ψ ( u 1 , u 2 , , u m , J f k ) = 0 \Psi(u_{1},u_{2},\dots,u_{m},J^{k}_{f})=0\!\,
  4. J f k J^{k}_{f}
  5. J f k J^{k}_{f}
  6. y 1 , y 2 , , y N . y_{1},y_{2},\dots,y_{N}.
  7. Ψ ( u 1 , u 2 , , u m , y 1 , y 2 , , y N ) = 0 \Psi(u_{1},u_{2},\dots,u_{m},y_{1},y_{2},\dots,y_{N})=0\!\,
  8. y j = k f u i 1 u i k . y_{j}={\partial^{k}f\over\partial u_{i_{1}}\ldots\partial u_{i_{k}}}.\!\,
  9. Ψ ( u 1 , u 2 , , u m , y 1 , y 2 , , y N ) = 0 \Psi(u_{1},u_{2},\dots,u_{m},y_{1},y_{2},\dots,y_{N})=0\!\,
  10. f ( x ) 0. f^{\prime}(x)\neq 0.
  11. f : [ 0 , 1 ] 𝐑 f\colon[0,1]\to\mathbf{R}
  12. f ( 0 ) = a f(0)=a
  13. f ( 1 ) = b , f(1)=b,
  14. a b , a\neq b,
  15. f ( t ) = a + ( b - a ) t f(t)=a+(b-a)t
  16. θ n θ \theta\mapsto n\theta
  17. z z n z\mapsto z^{n}
  18. x x
  19. y y
  20. α \alpha
  21. x ˙ sin α = y ˙ cos α . \dot{x}\sin\alpha=\dot{y}\cos\alpha.\,
  22. C 0 C^{0}
  23. C 0 C^{0}
  24. C C^{\infty}
  25. M m M^{m}
  26. 𝐑 m + 1 \mathbf{R}^{m+1}
  27. C 1 C^{1}
  28. f t f_{t}
  29. f 0 = f f_{0}=f
  30. f 1 = - f f_{1}=-f
  31. t t
  32. grad ( f t ) \operatorname{grad}(f_{t})
  33. C 1 C^{1}
  34. S 2 S^{2}
  35. C 1 C^{1}
  36. C 1 C^{1}

Hopf_algebra.html

  1. S ( c ( 1 ) ) c ( 2 ) = c ( 1 ) S ( c ( 2 ) ) = ϵ ( c ) 1 for all c H . S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\epsilon(c)1\qquad\mbox{ for all }~{}c\in H.
  2. a ( m n ) := Δ ( a ) ( m n ) = ( a 1 a 2 ) ( m n ) = ( a 1 m a 2 n ) a(m\otimes n):=\Delta(a)(m\otimes n)=(a_{1}\otimes a_{2})(m\otimes n)=(a_{1}m% \otimes a_{2}n)
  3. a ( m ) := ϵ ( a ) m a(m):=\epsilon(a)m
  4. ( a f ) ( m ) := f ( S ( a ) m ) (af)(m):=f(S(a)m)
  5. H * ( G ) H * ( G × G ) H * ( G ) H * ( G ) H^{*}(G)\rightarrow H^{*}(G\times G)\cong H^{*}(G)\otimes H^{*}(G)
  6. ( Δ ( 1 ) 1 ) ( 1 Δ ( 1 ) ) = ( 1 Δ ( 1 ) ) ( Δ ( 1 ) 1 ) = ( Δ Id ) Δ ( 1 ) (\Delta(1)\otimes 1)(1\otimes\Delta(1))=(1\otimes\Delta(1))(\Delta(1)\otimes 1% )=(\Delta\otimes\mbox{Id}~{})\Delta(1)
  7. ϵ ( a b c ) = ϵ ( a b ( 1 ) ) ϵ ( b ( 2 ) c ) = ϵ ( a b ( 2 ) ) ϵ ( b ( 1 ) c ) \epsilon(abc)=\sum\epsilon(ab_{(1)})\epsilon(b_{(2)}c)=\sum\epsilon(ab_{(2)})% \epsilon(b_{(1)}c)
  8. S ( a ( 1 ) ) a ( 2 ) = 1 ( 1 ) ϵ ( a 1 ( 2 ) ) S(a_{(1)})a_{(2)}=1_{(1)}\epsilon(a1_{(2)})
  9. a ( 1 ) S ( a ( 2 ) ) = ϵ ( 1 ( 1 ) a ) 1 ( 2 ) a_{(1)}S(a_{(2)})=\epsilon(1_{(1)}a)1_{(2)}
  10. S ( a ( 1 ) ) a ( 2 ) S ( a ( 3 ) ) = S ( a ) S(a_{(1)})a_{(2)}S(a_{(3)})=S(a)

Hopf_fibration.html

  1. S 1 S 3 𝑝 S 2 , S^{1}\hookrightarrow S^{3}\xrightarrow{\ p\,}S^{2},
  2. S 0 S 1 S 1 , S^{0}\hookrightarrow S^{1}\to S^{1},
  3. S 1 S 3 S 2 , S^{1}\hookrightarrow S^{3}\to S^{2},
  4. S 3 S 7 S 4 , S^{3}\hookrightarrow S^{7}\to S^{4},
  5. S 7 S 15 S 8 . S^{7}\hookrightarrow S^{15}\to S^{8}.
  6. p ( z 0 , z 1 ) = ( 2 z 0 z 1 , | z 0 | 2 - | z 1 | 2 ) . p(z_{0},z_{1})=(2z_{0}z_{1}^{\ast},\left|z_{0}\right|^{2}-\left|z_{1}\right|^{% 2}).
  7. 2 z 0 z 1 2 z 0 z 1 + ( | z 0 | 2 - | z 1 | 2 ) 2 = 4 | z 0 | 2 | z 1 | 2 + | z 0 | 4 - 2 | z 0 | 2 | z 1 | 2 + | z 1 | 4 = ( | z 0 | 2 + | z 1 | 2 ) 2 = 1 2z_{0}z_{1}^{\ast}\cdot 2z_{0}^{\ast}z_{1}+\left(\left|z_{0}\right|^{2}-\left|% z_{1}\right|^{2}\right)^{2}=4\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+% \left|z_{0}\right|^{4}-2\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{% 1}\right|^{4}=\left(\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}\right)^{2}=1
  8. z 0 = e i ( ξ 1 + ξ 2 ) sin η z_{0}=e^{i\,(\xi_{1}+\xi_{2})}\sin\eta
  9. z 1 = e i ( ξ 1 - ξ 2 ) cos η . z_{1}=e^{i\,(\xi_{1}-\xi_{2})}\cos\eta.
  10. x 1 = cos ( ξ 1 + ξ 2 ) sin η x_{1}=\cos(\xi_{1}+\xi_{2})\sin\eta
  11. x 2 = sin ( ξ 1 + ξ 2 ) sin η x_{2}=\sin(\xi_{1}+\xi_{2})\sin\eta
  12. x 3 = cos ( ξ 1 - ξ 2 ) cos η x_{3}=\cos(\xi_{1}-\xi_{2})\cos\eta
  13. x 4 = sin ( ξ 1 - ξ 2 ) cos η x_{4}=\sin(\xi_{1}-\xi_{2})\cos\eta
  14. z = cos ( 2 η ) z=\cos(2\eta)
  15. x = sin ( 2 η ) cos ξ 1 x=\sin(2\eta)\cos\xi_{1}
  16. y = sin ( 2 η ) sin ξ 1 y=\sin(2\eta)\sin\xi_{1}
  17. q = x 1 + i x 2 + j x 3 + k x 4 . q=x_{1}+{i}x_{2}+{j}x_{3}+{k}x_{4}.\,\!
  18. p = i y 1 + j y 2 + k y 3 . p={i}y_{1}+{j}y_{2}+{k}y_{3}.\,\!
  19. p q p q * p\mapsto qpq^{*}\,\!
  20. [ x 1 + i x 2 x 3 + i x 4 - x 3 + i x 4 x 1 - i x 2 ] . \begin{bmatrix}x_{1}+ix_{2}&x_{3}+ix_{4}\\ -x_{3}+ix_{4}&x_{1}-ix_{2}\end{bmatrix}.\,\!
  21. [ 1 - 2 ( y 2 + z 2 ) 2 ( x y - w z ) 2 ( x z + w y ) 2 ( x y + w z ) 1 - 2 ( x 2 + z 2 ) 2 ( y z - w x ) 2 ( x z - w y ) 2 ( y z + w x ) 1 - 2 ( x 2 + y 2 ) ] . \begin{bmatrix}1-2(y^{2}+z^{2})&2(xy-wz)&2(xz+wy)\\ 2(xy+wz)&1-2(x^{2}+z^{2})&2(yz-wx)\\ 2(xz-wy)&2(yz+wx)&1-2(x^{2}+y^{2})\end{bmatrix}.
  22. ( 2 ( x z + w y ) , 2 ( y z - w x ) , 1 - 2 ( x 2 + y 2 ) ) , \Big(2(xz+wy),2(yz-wx),1-2(x^{2}+y^{2})\Big),\,\!
  23. q θ = cos θ + k sin θ q_{\theta}=\cos\theta+{k}\sin\theta
  24. q ( a , b , c ) = 1 2 ( 1 + c ) ( 1 + c - i b + j a ) q_{(a,b,c)}=\frac{1}{\sqrt{2(1+c)}}(1+c-{i}b+{j}a)
  25. 1 2 ( 1 + c ) ( ( 1 + c ) cos ( θ ) , a sin ( θ ) - b cos ( θ ) , a cos ( θ ) + b sin ( θ ) , ( 1 + c ) sin ( θ ) ) . \frac{1}{\sqrt{2(1+c)}}\Big((1+c)\cos(\theta),a\sin(\theta)-b\cos(\theta),a% \cos(\theta)+b\sin(\theta),(1+c)\sin(\theta)\Big).\,\!
  26. ( 0 , cos ( θ ) , - sin ( θ ) , 0 ) , \Big(0,\cos(\theta),-\sin(\theta),0\Big),
  27. 𝐯 ( x , y , z ) = A ( a 2 + x 2 + y 2 + z 2 ) - 2 ( 2 ( - a y + x z ) , 2 ( a x + y z ) , a 2 - x 2 - y 2 + z 2 ) \mathbf{v}(x,y,z)=A\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-2}\left(2(-ay+xz),2(% ax+yz),a^{2}-x^{2}-y^{2}+z^{2}\right)
  28. p ( x , y , z ) = - A 2 B ( a 2 + x 2 + y 2 + z 2 ) - 3 , p(x,y,z)=-A^{2}B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-3},
  29. ρ ( x , y , z ) = 3 B ( a 2 + x 2 + y 2 + z 2 ) - 1 \rho(x,y,z)=3B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-1}
  30. S 3 S 7 S 4 . S^{3}\hookrightarrow S^{7}\to S^{4}.

Horseshoe_map.html

  1. f f
  2. S S
  3. S S
  4. f f
  5. 1 a \frac{1}{a}
  6. S S
  7. S S
  8. f f
  9. H n = f n ( S 0 ) S 0 . H_{n}=f^{n}(S_{0})\cap S_{0}.
  10. V n = f - n ( H n ) V_{n}=f^{-n}(H_{n})
  11. Λ Λ
  12. Λ Λ
  13. n n→∞
  14. Λ Λ
  15. Λ Λ
  16. A A
  17. B B
  18. A A
  19. B B
  20. A A
  21. B B
  22. Λ A A \displaystyle\Lambda_{A\bullet A}
  23. Λ < s u b > B A Λ<sub>B•A

Hough_transform.html

  1. r = x cos θ + y sin θ r=x\cos\theta+y\sin\theta
  2. r r
  3. θ \theta
  4. x x
  5. ( r , θ ) (r,\theta)
  6. ( r , θ ) (r,\theta)
  7. r = x cos θ + y sin θ r=x\cos\theta+y\sin\theta
  8. ( r , θ ) (r,\theta)
  9. n log ( n ) n\log(n)
  10. 10 5 10^{5}
  11. θ , ϕ , ρ \theta,\phi,\rho
  12. z = a x x + a y y + d z=a_{x}x+a_{y}y+d
  13. a x a_{x}
  14. a y a_{y}
  15. d d
  16. a x a_{x}
  17. a y a_{y}
  18. n n
  19. ρ \rho
  20. 𝒪 ( A m - 2 ) \mathcal{O}\left({A^{m-2}}\right)
  21. A A
  22. m m

Householder_transformation.html

  1. x - 2 x , v v = x - 2 v ( v H x ) , x-2\langle x,v\rangle v=x-2v(v\text{H}x),
  2. P = I - 2 v v H P=I-2vv\text{H}\,
  3. P = P H , P=P\text{H},
  4. P - 1 = P H , P^{-1}=P\text{H},
  5. P 2 = I P^{2}=I
  6. ± 1 \pm 1
  7. u u
  8. v v
  9. P u = u Pu=u
  10. n - 1 n-1
  11. n - 1 n-1
  12. v v
  13. P v = - v Pv=-v
  14. α \displaystyle\alpha
  15. α = - sgn ( a 21 ) j = 2 n a j 1 2 \displaystyle\alpha=-\operatorname{sgn}(a_{21})\sqrt{\sum_{j=2}^{n}a_{j1}^{2}}
  16. r = 1 2 ( α 2 - a 21 α ) r=\sqrt{\frac{1}{2}(\alpha^{2}-a_{21}\alpha)}
  17. α \displaystyle\alpha
  18. v ( 1 ) = [ v 1 v 2 v n ] , v^{(1)}=\begin{bmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n}\end{bmatrix},
  19. v 1 = 0 ; v_{1}=0;
  20. v 2 = a 21 - α 2 r v_{2}=\frac{a_{21}-\alpha}{2r}
  21. v k = a k 1 2 r v_{k}=\frac{a_{k1}}{2r}
  22. P 1 = I - 2 v ( 1 ) ( v ( 1 ) ) T \displaystyle P^{1}=I-2v^{(1)}(v^{(1)})\text{T}
  23. A ( 2 ) = P 1 A P 1 \displaystyle A^{(2)}=P^{1}AP^{1}
  24. P 1 \displaystyle P^{1}
  25. A ( 2 ) \displaystyle A^{(2)}
  26. α = - sgn ( a k + 1 , k k ) j = k + 1 n ( a j k k ) 2 \displaystyle\alpha=-\operatorname{sgn}(a^{k}_{k+1,k})\sqrt{\sum_{j=k+1}^{n}(a% ^{k}_{jk})^{2}}
  27. r = 1 2 ( α 2 - a k + 1 , k k α ) r=\sqrt{\frac{1}{2}(\alpha^{2}-a^{k}_{k+1,k}\alpha)}
  28. v 1 k = v 2 k = = v k k = 0 ; v^{k}_{1}=v^{k}_{2}=\cdots=v^{k}_{k}=0;
  29. v k + 1 k = a k + 1 , k k - α 2 r v^{k}_{k+1}=\frac{a^{k}_{k+1,k}-\alpha}{2r}
  30. v j k = a j k k 2 r v^{k}_{j}=\frac{a^{k}_{jk}}{2r}
  31. P k = I - 2 v ( k ) ( v ( k ) ) T \displaystyle P^{k}=I-2v^{(k)}(v^{(k)})\text{T}
  32. A ( k + 1 ) = P k A ( k ) P k \displaystyle A^{(k+1)}=P^{k}A^{(k)}P^{k}
  33. 𝐀 = [ 4 1 - 2 2 1 2 0 1 - 2 0 3 - 2 2 1 - 2 - 1 ] , \mathbf{A}=\begin{bmatrix}4&1&-2&2\\ 1&2&0&1\\ -2&0&3&-2\\ 2&1&-2&-1\end{bmatrix},
  34. = [ 1 0 0 0 0 - 1 / 3 2 / 3 - 2 / 3 0 2 / 3 2 / 3 1 / 3 0 - 2 / 3 1 / 3 2 / 3 ] , \mathbf{}=\begin{bmatrix}1&0&0&0\\ 0&-1/3&2/3&-2/3\\ 0&2/3&2/3&1/3\\ 0&-2/3&1/3&2/3\end{bmatrix},
  35. [ 4 - 3 0 0 - 3 10 / 3 1 4 / 3 0 1 5 / 3 - 4 / 3 0 4 / 3 - 4 / 3 - 1 ] , \mathbf{}\begin{bmatrix}4&-3&0&0\\ -3&10/3&1&4/3\\ 0&1&5/3&-4/3\\ 0&4/3&-4/3&-1\end{bmatrix},
  36. [ 1 0 0 0 0 1 0 0 0 0 - 3 / 5 - 4 / 5 0 0 - 4 / 5 3 / 5 ] , \mathbf{}\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&-3/5&-4/5\\ 0&0&-4/5&3/5\end{bmatrix},
  37. [ 4 - 3 0 0 - 3 10 / 3 - 5 / 3 0 0 - 5 / 3 - 33 / 25 68 / 75 0 0 68 / 75 149 / 75 ] , \mathbf{}\begin{bmatrix}4&-3&0&0\\ -3&10/3&-5/3&0\\ 0&-5/3&-33/25&68/75\\ 0&0&68/75&149/75\end{bmatrix},
  38. Trace ( U U H ) N = j = 2 N | λ j | 2 N = 1 , det ( U U H ) = j = 1 N | λ j | 2 = 1. \frac{\mbox{Trace}~{}(UU\text{H})}{N}=\frac{\sum_{j=2}^{N}|\lambda_{j}|^{2}}{N% }=1,\mbox{det}~{}(UU\text{H})=\prod_{j=1}^{N}|\lambda_{j}|^{2}=1.
  39. U U T = I . UU\text{T}=I.

Hubbert_peak_theory.html

  1. Q ( t ) = Q < m t p l > max 1 + a e - b t Q(t)={Q_{<}mtpl>{{\rm max}}\over{1+ae^{-bt}}}
  2. Q Q
  3. Q ( t ) Q(t)
  4. a a
  5. b b
  6. t < m t p l > max = 1 b ln ( a ) . t_{<}mtpl>{{\rm max}}={1\over b}\ln\left({a}\right).
  7. Q ( t ) Q(t)
  8. Q ( t ) = Q max / 2 Q(t)=Q\text{max}/2

Human_body_weight.html

  1. m = 1 2 a m + 4 m=\tfrac{1}{2}a_{m}+4
  2. m = 2 a y + 10 m=2a_{y}+10
  3. m = e 0.175571 a y + 2.197099 m=e^{0.175571a_{y}+2.197099}

Hurewicz_theorem.html

  1. h : π k ( X ) H k ( X ) h_{\ast}\colon\,\pi_{k}(X)\to H_{k}(X)\,\!
  2. h : π 1 ( X ) π 1 ( X ) / [ π 1 ( X ) , π 1 ( X ) ] . h_{\ast}\colon\,\pi_{1}(X)\to\pi_{1}(X)/[\pi_{1}(X),\pi_{1}(X)].\,\!
  3. H 1 ( X ) π 1 ( X ) / [ π 1 ( X ) , π 1 ( X ) ] . H_{1}(X)\cong\pi_{1}(X)/[\pi_{1}(X),\pi_{1}(X)].\,\!
  4. π n + 1 ( X ) H n + 1 ( X ) \pi_{n+1}(X)\to H_{n+1}(X)
  5. n 2 n\geq 2
  6. u n H n ( S n ) u_{n}\in H_{n}(S^{n})
  7. f π n ( X ) f\in\pi_{n}(X)
  8. f * ( u n ) H n ( X ) f_{*}(u_{n})\in H_{n}(X)
  9. h : π k ( X , A ) H k ( X , A ) h_{\ast}\colon\pi_{k}(X,A)\to H_{k}(X,A)\,\!
  10. π n ( X , A ) π n ( X C A ) . \pi_{n}(X,A)\to\pi_{n}(X\cup CA)\,\!.
  11. h : π k ( X ; A , B ) H k ( X ; A , B ) h_{\ast}\colon\pi_{k}(X;A,B)\to H_{k}(X;A,B)\,\!
  12. π i ( X ) = 0 \pi_{i}(X)\otimes\mathbb{Q}=0
  13. i r i\leq r
  14. h : π i ( X ) H i ( X ; ) h\otimes\mathbb{Q}:\pi_{i}(X)\otimes\mathbb{Q}\longrightarrow H_{i}(X;\mathbb{% Q})
  15. 1 i 2 r 1\leq i\leq 2r
  16. i = 2 r + 1 i=2r+1

Hurwitz_zeta_function.html

  1. ζ ( s , q ) = n = 0 1 ( q + n ) s . \zeta(s,q)=\sum_{n=0}^{\infty}\frac{1}{(q+n)^{s}}.
  2. R e ( s ) 1 Re(s)\leq 1
  3. ζ ( s , q ) = Γ ( 1 - s ) 1 2 π i C z s - 1 e q z 1 - e z d z \zeta(s,q)=\Gamma(1-s)\frac{1}{2\pi i}\int_{C}\frac{z^{s-1}e^{qz}}{1-e^{z}}dz
  4. C C
  5. ζ ( s , q ) \zeta(s,q)
  6. s s
  7. s 1 s\neq 1
  8. s = 1 s=1
  9. 1 1
  10. lim s 1 [ ζ ( s , q ) - 1 s - 1 ] = - Γ ( q ) Γ ( q ) = - ψ ( q ) \lim_{s\to 1}\left[\zeta(s,q)-\frac{1}{s-1}\right]=\frac{-\Gamma^{\prime}(q)}{% \Gamma(q)}=-\psi(q)
  11. Γ \Gamma
  12. ψ \psi
  13. ζ ( s , q ) = 1 s - 1 n = 0 1 n + 1 k = 0 n ( - 1 ) k ( n k ) ( q + k ) 1 - s . \zeta(s,q)=\frac{1}{s-1}\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{k=0}^{n}(-1)^{k}% {n\choose k}(q+k)^{1-s}.
  14. q 1 - s q^{1-s}
  15. Δ n q 1 - s = k = 0 n ( - 1 ) n - k ( n k ) ( q + k ) 1 - s \Delta^{n}q^{1-s}=\sum_{k=0}^{n}(-1)^{n-k}{n\choose k}(q+k)^{1-s}
  16. ζ ( s , q ) = 1 s - 1 n = 0 ( - 1 ) n n + 1 Δ n q 1 - s = 1 s - 1 log ( 1 + Δ ) Δ q 1 - s \begin{aligned}\displaystyle\zeta(s,q)&\displaystyle=\frac{1}{s-1}\sum_{n=0}^{% \infty}\frac{(-1)^{n}}{n+1}\Delta^{n}q^{1-s}\\ &\displaystyle=\frac{1}{s-1}{\log(1+\Delta)\over\Delta}q^{1-s}\end{aligned}
  17. ζ ( s , q ) = 1 Γ ( s ) 0 t s - 1 e - q t 1 - e - t d t \zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt
  18. s > 1 \Re s>1
  19. q > 0. \Re q>0.
  20. ζ ( 1 - s , x ) = 1 2 s [ e - i π s / 2 β ( x ; s ) + e i π s / 2 β ( 1 - x ; s ) ] \zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s)+e^{i\pi s/2}\beta(1-x;s% )\right]
  21. β ( x ; s ) = 2 Γ ( s + 1 ) n = 1 exp ( 2 π i n x ) ( 2 π n ) s = 2 Γ ( s + 1 ) ( 2 π ) s Li ( e 2 π i x ) s \beta(x;s)=2\Gamma(s+1)\sum_{n=1}^{\infty}\frac{\exp(2\pi inx)}{(2\pi n)^{s}}=% \frac{2\Gamma(s+1)}{(2\pi)^{s}}\mbox{Li}~{}_{s}(e^{2\pi ix})
  22. 0 x 1 0\leq x\leq 1
  23. Li s ( z ) \,\text{Li}_{s}(z)
  24. 1 m n 1\leq m\leq n
  25. ζ ( 1 - s , m n ) = 2 Γ ( s ) ( 2 π n ) s k = 1 n [ cos ( π s 2 - 2 π k m n ) ζ ( s , k n ) ] \zeta\left(1-s,\frac{m}{n}\right)=\frac{2\Gamma(s)}{(2\pi n)^{s}}\sum_{k=1}^{n% }\left[\cos\left(\frac{\pi s}{2}-\frac{2\pi km}{n}\right)\;\zeta\left(s,\frac{% k}{n}\right)\right]
  26. q ζ ( s , q ) = - s ζ ( s + 1 , q ) . \frac{\partial}{\partial q}\zeta(s,q)=-s\zeta(s+1,q).
  27. ζ ( s , x + y ) = k = 0 y k k ! k x k ζ ( s , x ) = k = 0 ( s + k - 1 s - 1 ) ( - y ) k ζ ( s + k , x ) . \zeta(s,x+y)=\sum_{k=0}^{\infty}\frac{y^{k}}{k!}\frac{\partial^{k}}{\partial x% ^{k}}\zeta(s,x)=\sum_{k=0}^{\infty}{s+k-1\choose s-1}(-y)^{k}\zeta(s+k,x).
  28. ζ ( s , q ) = 1 q s + n = 0 ( - q ) n ( s + n - 1 n ) ζ ( s + n ) , \zeta(s,q)=\frac{1}{q^{s}}+\sum_{n=0}^{\infty}(-q)^{n}{s+n-1\choose n}\zeta(s+% n),
  29. | q | < 1 |q|<1
  30. ζ ( s , N ) = k = 0 [ N + s - 1 k + 1 ] ( s + k - 1 s - 1 ) ( - 1 ) k ζ ( s + k , N ) \zeta(s,N)=\sum_{k=0}^{\infty}\left[N+\frac{s-1}{k+1}\right]{s+k-1\choose s-1}% (-1)^{k}\zeta(s+k,N)
  31. ζ ( s , q ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( q ) ( s - 1 ) n . \zeta(s,q)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}(q)\;% (s-1)^{n}.
  32. γ 0 ( q ) = - ψ ( q ) \gamma_{0}(q)=-\psi(q)
  33. γ 0 ( 1 ) = - ψ ( 1 ) = γ 0 = γ \gamma_{0}(1)=-\psi(1)=\gamma_{0}=\gamma
  34. β \beta
  35. B n ( x ) = - [ ( - i ) n β ( x ; n ) ] B_{n}(x)=-\Re\left[(-i)^{n}\beta(x;n)\right]
  36. z \Re z
  37. ζ ( - n , x ) = - B n + 1 ( x ) n + 1 . \zeta(-n,x)=-{B_{n+1}(x)\over n+1}.
  38. n = 0 n=0
  39. ζ ( 0 , x ) = 1 2 - x . \zeta(0,x)=\frac{1}{2}-x.
  40. ϑ ( z , τ ) \vartheta(z,\tau)
  41. 0 [ ϑ ( z , i t ) - 1 ] t s / 2 d t t = π - ( 1 - s ) / 2 Γ ( 1 - s 2 ) [ ζ ( 1 - s , z ) + ζ ( 1 - s , 1 - z ) ] \int_{0}^{\infty}\left[\vartheta(z,it)-1\right]t^{s/2}\frac{dt}{t}=\pi^{-(1-s)% /2}\Gamma\left(\frac{1-s}{2}\right)\left[\zeta(1-s,z)+\zeta(1-s,1-z)\right]
  42. s > 0 \Re s>0
  43. 0 [ ϑ ( n , i t ) - 1 ] t s / 2 d t t = 2 π - ( 1 - s ) / 2 Γ ( 1 - s 2 ) ζ ( 1 - s ) = 2 π - s / 2 Γ ( s 2 ) ζ ( s ) . \int_{0}^{\infty}\left[\vartheta(n,it)-1\right]t^{s/2}\frac{dt}{t}=2\ \pi^{-(1% -s)/2}\ \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)=2\ \pi^{-s/2}\ \Gamma\left(% \frac{s}{2}\right)\zeta(s).
  44. t 0 t\rightarrow 0
  45. ζ ( s , n / k ) = k s φ ( k ) χ χ ¯ ( n ) L ( s , χ ) , \zeta(s,n/k)=\frac{k^{s}}{\varphi(k)}\sum_{\chi}\overline{\chi}(n)L(s,\chi),
  46. L ( s , χ ) = 1 k s n = 1 k χ ( n ) ζ ( s , n k ) . L(s,\chi)=\frac{1}{k^{s}}\sum_{n=1}^{k}\chi(n)\;\zeta\left(s,\frac{n}{k}\right).
  47. k s ζ ( s ) = n = 1 k ζ ( s , n k ) , k^{s}\zeta(s)=\sum_{n=1}^{k}\zeta\left(s,\frac{n}{k}\right),
  48. p = 0 q - 1 ζ ( s , a + p / q ) = q s ζ ( s , q a ) . \sum_{p=0}^{q-1}\zeta(s,a+p/q)=q^{s}\,\zeta(s,qa).
  49. E n ( x ) E_{n}(x)
  50. E 2 n - 1 ( p q ) = ( - 1 ) n 4 ( 2 n - 1 ) ! ( 2 π q ) 2 n k = 1 q ζ ( 2 n , 2 k - 1 2 q ) cos ( 2 k - 1 ) π p q E_{2n-1}\left(\frac{p}{q}\right)=(-1)^{n}\frac{4(2n-1)!}{(2\pi q)^{2n}}\sum_{k% =1}^{q}\zeta\left(2n,\frac{2k-1}{2q}\right)\cos\frac{(2k-1)\pi p}{q}
  51. E 2 n ( p q ) = ( - 1 ) n 4 ( 2 n ) ! ( 2 π q ) 2 n + 1 k = 1 q ζ ( 2 n + 1 , 2 k - 1 2 q ) sin ( 2 k - 1 ) π p q E_{2n}\left(\frac{p}{q}\right)=(-1)^{n}\frac{4(2n)!}{(2\pi q)^{2n+1}}\sum_{k=1% }^{q}\zeta\left(2n+1,\frac{2k-1}{2q}\right)\sin\frac{(2k-1)\pi p}{q}
  52. ζ ( s , 2 p - 1 2 q ) = 2 ( 2 q ) s - 1 k = 1 q [ C s ( k q ) cos ( ( 2 p - 1 ) π k q ) + S s ( k q ) sin ( ( 2 p - 1 ) π k q ) ] \zeta\left(s,\frac{2p-1}{2q}\right)=2(2q)^{s-1}\sum_{k=1}^{q}\left[C_{s}\left(% \frac{k}{q}\right)\cos\left(\frac{(2p-1)\pi k}{q}\right)+S_{s}\left(\frac{k}{q% }\right)\sin\left(\frac{(2p-1)\pi k}{q}\right)\right]
  53. 1 p q 1\leq p\leq q
  54. C ν ( x ) C_{\nu}(x)
  55. S ν ( x ) S_{\nu}(x)
  56. χ ν \chi_{\nu}
  57. C ν ( x ) = Re χ ν ( e i x ) C_{\nu}(x)=\operatorname{Re}\,\chi_{\nu}(e^{ix})
  58. S ν ( x ) = Im χ ν ( e i x ) . S_{\nu}(x)=\operatorname{Im}\,\chi_{\nu}(e^{ix}).
  59. ψ ( m ) ( z ) = ( - 1 ) m + 1 m ! ζ ( m + 1 , z ) . \psi^{(m)}(z)=(-1)^{m+1}m!\zeta(m+1,z)\ .
  60. ζ ( - n , x ) = - B n + 1 ( x ) n + 1 . \zeta(-n,x)=-\frac{B_{n+1}(x)}{n+1}\ .
  61. Φ ( z , s , q ) = k = 0 z k ( k + q ) s \Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}
  62. ζ ( s , q ) = Φ ( 1 , s , q ) . \zeta(s,q)=\Phi(1,s,q).\,
  63. ζ ( s , a ) = a - s F s s + 1 ( 1 , a 1 , a 2 , a s ; a 1 + 1 , a 2 + 1 , a s + 1 ; 1 ) \zeta(s,a)=a^{-s}\cdot{}_{s+1}F_{s}(1,a_{1},a_{2},\ldots a_{s};a_{1}+1,a_{2}+1% ,\ldots a_{s}+1;1)
  64. a 1 = a 2 = = a s = a and a 𝒩 and s 𝒩 + . a_{1}=a_{2}=\ldots=a_{s}=a\,\text{ and }a\notin\mathcal{N}\,\text{ and }s\in% \mathcal{N}^{+}.
  65. ζ ( s , a ) = G s + 1 , s + 1 1 , s + 1 ( - 1 | 0 , 1 - a , , 1 - a 0 , - a , , - a ) s 𝒩 + . \zeta(s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1\;\left|\;\begin{matrix}0,1-a,% \ldots,1-a\\ 0,-a,\ldots,-a\end{matrix}\right)\right.\qquad\qquad s\in\mathcal{N}^{+}.

Hybrid_system.html

  1. x 1 x_{1}
  2. x 2 x_{2}
  3. x C = { x 1 > 0 } x\in C=\{x_{1}>0\}
  4. x 1 ˙ = x 2 , x 2 ˙ = - g \dot{x_{1}}=x_{2},\dot{x_{2}}=-g
  5. g g
  6. x D = { x 1 = 0 } x\in D=\{x_{1}=0\}
  7. x 1 + = x 1 , x 2 + = - γ x 2 x_{1}^{+}=x_{1},x_{2}^{+}=-\gamma x_{2}
  8. 0 < γ < 1 0<\gamma<1
  9. γ \gamma
  10. 0 λ x 1 0 0\leq\lambda\perp x_{1}\geq 0
  11. λ \lambda

Hydraulic_diameter.html

  1. D H = 4 A P D_{H}=\frac{4A}{P}
  2. D H = 4 ( π D 2 / 4 ) π D = D D_{H}=\frac{4(\pi D^{2}/4)}{\pi D}=D
  3. D H = 4 π ( D o 2 - D i 2 ) / 4 π ( D o + D i ) = D o - D i D_{H}=\frac{4\cdot\pi(D_{o}^{2}-D_{i}^{2})/4}{\pi(D_{o}+D_{i})}=D_{o}-D_{i}
  4. D H = 4 a 2 4 a = a D_{H}=\frac{4a^{2}}{4a}=a
  5. D H = 4 a b 2 ( a + b ) = 2 a b a + b D_{H}=\frac{4ab}{2\left(a+b\right)}=\frac{2ab}{a+b}
  6. D H = 4 a b 2 a + b D_{H}=\frac{4ab}{2a+b}

Hydraulic_engineering.html

  1. p = ρ g y p=\rho gy

Hydroelectricity.html

  1. P = ρ h r g k P=\rho hrgk
  2. P P
  3. ρ \rho
  4. h h
  5. r r
  6. g g
  7. k k

Hydrogen_line.html

  1. π × 1420.40575177 MHz = 4.46233627 GHz \pi\times 1420.40575177\,\text{ MHz}=4.46233627\,\text{ GHz}
  2. 2 π × 1420.40575177 MHz = 8.92467255 GHz 2\pi\times 1420.40575177\,\text{ MHz}=8.92467255\,\text{ GHz}

Hydrogeology.html

  1. h t = h ( t i ) h i - h i - 1 Δ t . \frac{\partial h}{\partial t}=h^{\prime}(t_{i})\approx\frac{h_{i}-h_{i-1}}{% \Delta t}.

Hydrostatics.html

  1. p ( z ) - p ( z 0 ) = 1 A z 0 z d z A d x d y ρ ( z ) g ( z ) = z 0 z d z ρ ( z ) g ( z ) p(z)-p(z_{0})=\frac{1}{A}\int_{z_{0}}^{z}dz^{\prime}\iint\limits_{A}dx^{\prime% }dy^{\prime}\,\rho(z^{\prime})g(z^{\prime})=\int_{z_{0}}^{z}dz^{\prime}\,\rho(% z^{\prime})g(z^{\prime})
  2. p - p 0 = ρ g h , \ p-p_{0}=\rho gh,
  3. p = ρ g H + p atm , \ p=\rho gH+p_{\mathrm{atm}},
  4. p ( h ) = p ( 0 ) e - M g h / k T \ p(h)=p(0)e^{-Mgh/kT}
  5. F = ρ g V F=\rho gV
  6. F H = p c A F_{H}=p_{c}A
  7. F V = ρ g V F_{V}=\rho gV

Hyperbolic_space.html

  1. x 0 2 - x 1 2 - - x n 2 = 1 , x 0 > 0. x_{0}^{2}-x_{1}^{2}-\cdots-x_{n}^{2}=1,\quad x_{0}>0.
  2. Q ( x ) = x 0 2 - x 1 2 - x 2 2 - - x n 2 , Q(x)=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2},
  3. B ( x , y ) = ( Q ( x + y ) - Q ( x ) - Q ( y ) ) / 2 = x 0 y 0 - x 1 y 1 - - x n y n . B(x,y)=(Q(x+y)-Q(x)-Q(y))/2=x_{0}y_{0}-x_{1}y_{1}-\cdots-x_{n}y_{n}.
  4. d ( x , y ) = arcosh B ( x , y ) . d(x,y)=\operatorname{arcosh}B(x,y).
  5. d ( x , y ) = arcosh ( B ( x , y ) Q ( x ) Q ( y ) ) . d(x,y)=\operatorname{arcosh}\left(\frac{B(x,y)}{\sqrt{Q(x)Q(y)}}\right).
  6. B n = { ( x 1 , , x n ) | x 1 2 + + x n 2 < 1 } B^{n}=\{(x_{1},\ldots,x_{n})|x_{1}^{2}+\cdots+x_{n}^{2}<1\}

Hypercharge.html

  1. Y = S + C + B + T + B Y=S+C+B^{\prime}+T+B
  2. Q = I 3 + 1 2 Y , Q=I_{3}+\frac{1}{2}Y,
  3. Y = 2 Q ¯ . Y=2\bar{Q}.
  4. Y = 1 3 ( n u + n d - 2 n s ) . Y={1\over 3}(n_{u}+n_{d}-2n_{s}).

Hypercholesterolemia.html

  1. \approx

Hyperelliptic_curve.html

  1. y 2 = f ( x ) y^{2}=f(x)
  2. C : y 2 = f ( x ) C:y^{2}=f(x)
  3. f ( x ) = x 5 - 2 x 4 - 7 x 3 + 8 x 2 + 12 x = x ( x + 1 ) ( x - 3 ) ( x + 2 ) ( x - 2 ) . f(x)=x^{5}-2x^{4}-7x^{3}+8x^{2}+12x=x(x+1)(x-3)(x+2)(x-2).
  4. y 2 = f ( x ) y^{2}=f(x)\,
  5. w 2 = v 2 g + 2 f ( 1 / v ) w^{2}=v^{2g+2}f(1/v)\,
  6. ( x , y ) ( 1 / x , y / x g + 1 ) (x,y)\mapsto(1/x,y/x^{g+1})
  7. ( v , w ) ( 1 / v , w / v g + 1 ) , (v,w)\mapsto(1/v,w/v^{g+1}),
  8. 2 - 2 g 1 = 2 ( 1 - g 0 ) - s X ( e s - 1 ) 2-2g_{1}=2(1-g_{0})-\sum_{s\in X}(e_{s}-1)

Hyperelliptic_curve_cryptography.html

  1. g g
  2. K K
  3. C : y 2 + h ( x ) y = f ( x ) K [ x , y ] C:y^{2}+h(x)y=f(x)\in K[x,y]
  4. h ( x ) K [ x ] h(x)\in K[x]
  5. g g
  6. f ( x ) K [ x ] f(x)\in K[x]
  7. 2 g + 1 2g+1
  8. K K
  9. C C
  10. J ( C ) J(C)
  11. K K
  12. G G
  13. g g
  14. G G
  15. G G
  16. a a
  17. G G
  18. g g
  19. g a g^{a}
  20. n n
  21. log ( n ) \log(n)
  22. G G
  23. n = p 1 r 1 p k r k n=p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}
  24. p 1 r 1 p k r k p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}
  25. n n
  26. G G
  27. p i p_{i}
  28. i = 1 , , k i=1,...,k
  29. p p
  30. n n
  31. G G
  32. p p
  33. G G
  34. p p
  35. # G = n \#G=n
  36. n n
  37. n p 4 \frac{n}{p}\leq 4
  38. g = 3 g=3
  39. C C
  40. 𝔽 q \mathbb{F}_{q}
  41. q q
  42. n n
  43. p p
  44. n n
  45. k k
  46. p | q k - 1 p|q^{k}-1
  47. J ( C ) J(C)
  48. p p
  49. 𝔽 q k * \mathbb{F}_{q^{k}}^{*}
  50. k k
  51. J ( C ) J(C)
  52. 𝔽 q k * \mathbb{F}_{q^{k}}^{*}
  53. k k
  54. q g q^{g}
  55. J ( C ) J(C)
  56. 𝔽 q k * \mathbb{F}_{q^{k}}^{*}
  57. k k
  58. 𝔽 q k * \mathbb{F}_{q^{k}}^{*}
  59. p p
  60. 𝔽 q . \mathbb{F}_{q}.
  61. 𝔽 q \mathbb{F}_{q}
  62. C C
  63. g g
  64. 𝔽 q \mathbb{F}_{q}
  65. q q
  66. C k C_{k}
  67. C C
  68. 𝔽 q k \mathbb{F}_{q^{k}}
  69. C k C_{k}
  70. [ ( q k - 1 ) 2 g , ( q k + 1 ) 2 g ] [(\sqrt{q}^{k}-1)^{2g},(\sqrt{q}^{k}+1)^{2g}]
  71. A k A_{k}
  72. C k C_{k}
  73. C = C 1 C=C_{1}
  74. Z C ( t ) = exp ( i = 1 A i t i i ) Z_{C}(t)=\exp(\sum_{i=1}^{\infty}{A_{i}\frac{t^{i}}{i}})
  75. Z C ( t ) = P ( t ) ( 1 - t ) ( 1 - q t ) Z_{C}(t)=\frac{P(t)}{(1-t)(1-qt)}
  76. P ( t ) P(t)
  77. 2 g 2g
  78. \mathbb{Z}
  79. P ( t ) P(t)
  80. P ( t ) = i = 1 g ( 1 - a i t ) ( 1 - a i ¯ t ) P(t)=\prod_{i=1}^{g}{(1-a_{i}t)(1-\bar{a_{i}}t)}
  81. a i a_{i}\in\mathbb{C}
  82. i = 1 , , g i=1,...,g
  83. a ¯ \bar{a}
  84. a a
  85. J ( C k ) J(C_{k})
  86. i = 1 g | 1 - a i k | 2 \prod_{i=1}^{g}{|1-a_{i}^{k}|^{2}}
  87. P ( t ) P(t)

Hyperfine_structure.html

  1. 𝐈 \mathbf{I}
  2. s y m b o l μ I = g I μ N 𝐈 , symbol{\mu}\text{I}=g\text{I}\mu\text{N}\mathbf{I},
  3. g I g\text{I}
  4. μ N \mu\text{N}
  5. H ^ D = - s y m b o l μ I 𝐁 . \hat{H}\text{D}=-symbol{\mu}\text{I}\cdot\mathbf{B}.
  6. 𝐁 𝐁 el = 𝐁 el l + 𝐁 el s . \mathbf{B}\equiv\mathbf{B}\text{el}=\mathbf{B}\text{el}^{l}+\mathbf{B}\text{el% }^{s}.
  7. 𝐁 el l = μ 0 4 π - e 𝐯 × - 𝐫 r 3 , \mathbf{B}\text{el}^{l}=\dfrac{\mu_{0}}{4\pi}\dfrac{-e\mathbf{v}\times-\mathbf% {r}}{r^{3}},
  8. 𝐁 el l = - 2 μ B μ 0 4 π 1 r 3 𝐫 × m e 𝐯 . \mathbf{B}\text{el}^{l}=-2\mu\text{B}\dfrac{\mu_{0}}{4\pi}\dfrac{1}{r^{3}}% \dfrac{\mathbf{r}\times m\text{e}\mathbf{v}}{\hbar}.
  9. 𝐁 el l = - 2 μ B μ 0 4 π 1 r 3 𝐥 . \mathbf{B}\text{el}^{l}=-2\mu\text{B}\dfrac{\mu_{0}}{4\pi}\dfrac{1}{r^{3}}% \mathbf{l}.
  10. 𝐋 \scriptstyle{\mathbf{L}}
  11. ϕ i l \scriptstyle{\phi^{l}_{i}}
  12. i 𝐥 i = i ϕ i l 𝐋 \scriptstyle{\sum_{i}\mathbf{l}_{i}=\sum_{i}\phi^{l}_{i}\mathbf{L}}
  13. ϕ i l = l ^ z i / L z \scriptstyle{\phi^{l}_{i}=\hat{l}_{z_{i}}/L_{z}}
  14. 𝐁 el l = - 2 μ B μ 0 4 π 1 L z i l ^ z i r i 3 𝐋 . \mathbf{B}\text{el}^{l}=-2\mu\text{B}\dfrac{\mu_{0}}{4\pi}\dfrac{1}{L_{z}}\sum% _{i}\dfrac{\hat{l}_{zi}}{r_{i}^{3}}\mathbf{L}.
  15. s y m b o l μ s = - g s μ B 𝐬 , symbol{\mu}\text{s}=-g_{s}\mu\text{B}\mathbf{s},
  16. 𝐁 el s = μ 0 4 π r 3 ( 3 ( s y m b o l μ s 𝐫 ^ ) 𝐫 ^ - s y m b o l μ s ) + 2 μ 0 3 s y m b o l μ s δ 3 ( 𝐫 ) . \mathbf{B}\text{el}^{s}=\dfrac{\mu_{0}}{4\pi r^{3}}\left(3(symbol{\mu}\text{s}% \cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-symbol{\mu}\text{s}\right)+\dfrac{2\mu_% {0}}{3}symbol{\mu}\text{s}\delta^{3}(\mathbf{r}).
  17. H ^ D = 2 g I μ N μ B μ 0 4 π 1 L z i l ^ z i r i 3 𝐈 𝐋 + g I μ N g s μ B μ 0 4 π 1 S z i s ^ z i r i 3 { 3 ( 𝐈 𝐫 ^ ) ( 𝐒 𝐫 ^ ) - 𝐈 𝐒 } + 2 3 g I μ N g s μ B μ 0 1 S z i s ^ z i δ 3 ( 𝐫 i ) 𝐈 𝐒 . \begin{array}[]{cl}\hat{H}_{D}=&2g\text{I}\mu\text{N}\mu\text{B}\dfrac{\mu_{0}% }{4\pi}\dfrac{1}{L_{z}}\sum_{i}\dfrac{\hat{l}_{zi}}{r_{i}^{3}}\mathbf{I}\cdot% \mathbf{L}\\ &+g\text{I}\mu\text{N}g\text{s}\mu\text{B}\dfrac{\mu_{0}}{4\pi}\dfrac{1}{S_{z}% }\sum_{i}\dfrac{\hat{s}_{zi}}{r_{i}^{3}}\left\{3(\mathbf{I}\cdot\hat{\mathbf{r% }})(\mathbf{S}\cdot\hat{\mathbf{r}})-\mathbf{I}\cdot\mathbf{S}\right\}\\ &+\frac{2}{3}g\text{I}\mu\text{N}g\text{s}\mu\text{B}\mu_{0}\dfrac{1}{S_{z}}% \sum_{i}\hat{s}_{zi}\delta^{3}(\mathbf{r}_{i})\mathbf{I}\cdot\mathbf{S}.\\ \end{array}
  18. H ^ D = 2 g I μ B μ N μ 0 4 π 𝐈 𝐍 r 3 , \hat{H}_{D}=2g_{I}\mu\text{B}\mu\text{N}\dfrac{\mu_{0}}{4\pi}\dfrac{\mathbf{I}% \cdot\mathbf{N}}{r^{3}},
  19. 𝐍 = 𝐥 - ( g s / 2 ) 𝐬 + 3 ( 𝐬 𝐫 ^ ) 𝐫 ^ . \mathbf{N}=\mathbf{l}-(g_{s}/2)\mathbf{s}+3(\mathbf{s}\cdot\hat{\mathbf{r}})% \hat{\mathbf{r}}.
  20. H ^ D \scriptstyle{\hat{H}\text{D}}
  21. H ^ D = 2 g I μ B μ N μ 0 4 π 𝐍 𝐉 𝐉 𝐉 𝐈 𝐉 r 3 . \hat{H}\text{D}=2g_{I}\mu\text{B}\mu\text{N}\dfrac{\mu_{0}}{4\pi}\dfrac{% \mathbf{N}\cdot\mathbf{J}}{\mathbf{J}\cdot\mathbf{J}}\dfrac{\mathbf{I}\cdot% \mathbf{J}}{r^{3}}.
  22. H ^ D = A ^ 𝐈 𝐉 , \hat{H}\text{D}=\hat{A}\mathbf{I}\cdot\mathbf{J},
  23. A ^ \scriptstyle{\langle\hat{A}\rangle}
  24. Δ E D = 1 2 A ^ [ F ( F + 1 ) - I ( I + 1 ) - J ( J + 1 ) ] . \Delta E\text{D}=\frac{1}{2}\langle\hat{A}\rangle[F(F+1)-I(I+1)-J(J+1)].
  25. I 1 \scriptstyle{I\geq 1}
  26. Q ¯ ¯ \scriptstyle{\underline{\underline{Q}}}
  27. Q i j = 1 e ( 3 x i x j - ( r ) 2 δ i j ) ρ ( 𝐫 ) d 3 r , Q_{ij}=\dfrac{1}{e}\int\left(3x_{i}^{\prime}x_{j}^{\prime}-(r^{\prime})^{2}% \delta_{ij}\right)\rho(\mathbf{r}^{\prime})d^{3}r^{\prime},
  28. T m 2 ( Q ) = 4 π 5 ρ ( 𝐫 ) ( r ) 2 Y m 2 ( θ , ϕ ) d 3 r . T^{2}_{m}(Q)=\sqrt{\frac{4\pi}{5}}\int\rho(\mathbf{r}^{\prime})(r^{\prime})^{2% }Y^{2}_{m}(\theta^{\prime},\phi^{\prime})d^{3}r^{\prime}.
  29. q ¯ ¯ \scriptstyle{\underline{\underline{q}}}
  30. q ¯ ¯ = 𝐄 , \underline{\underline{q}}=\nabla\otimes\mathbf{E},
  31. q i j = 2 V x i x j . q_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.
  32. T 2 ( q ) \scriptstyle{T^{2}(q)}
  33. T 0 2 ( q ) = 6 2 q z z T^{2}_{0}(q)=\frac{\sqrt{6}}{2}q_{zz}
  34. T + 1 2 ( q ) = - q x z - i q y z T^{2}_{+1}(q)=-q_{xz}-iq_{yz}
  35. T + 2 2 ( q ) = 1 2 ( q x x - q y y ) + i q x y , T^{2}_{+2}(q)=\frac{1}{2}(q_{xx}-q_{yy})+iq_{xy},
  36. T - m 2 ( q ) = ( - 1 ) m T + m 2 ( q ) * . T^{2}_{-m}(q)=(-1)^{m}T^{2}_{+m}(q)^{*}.
  37. H ^ Q = - e T 2 ( Q ) T 2 ( q ) = - e m ( - 1 ) m T m 2 ( Q ) T - m 2 ( q ) . \hat{H}_{Q}=-eT^{2}(Q)\cdot T^{2}(q)=-e\sum_{m}(-1)^{m}T^{2}_{m}(Q)T^{2}_{-m}(% q).
  38. I > 0 \scriptstyle{I>0}
  39. I 1 \scriptstyle{I\geq 1}
  40. I > 0 \scriptstyle{I>0}
  41. H ^ I I \scriptstyle{\hat{H}_{II}}
  42. H ^ I I = - α α s y m b o l μ α 𝐁 α , \hat{H}_{II}=-\sum_{\alpha\neq\alpha^{\prime}}symbol{\mu}_{\alpha}\cdot\mathbf% {B}_{\alpha^{\prime}},
  43. H ^ I I = μ 0 μ N 2 4 π α α g α g α R α α 3 { 𝐈 α 𝐈 α - 3 ( 𝐈 α 𝐑 ^ α α ) ( 𝐈 α 𝐑 ^ α α ) } . \hat{H}_{II}=\dfrac{\mu_{0}\mu\text{N}^{2}}{4\pi}\sum_{\alpha\neq\alpha^{% \prime}}\dfrac{g_{\alpha}g_{\alpha^{\prime}}}{R_{\alpha\alpha^{\prime}}^{3}}% \left\{\mathbf{I}_{\alpha}\cdot\mathbf{I}_{\alpha^{\prime}}-3(\mathbf{I}_{% \alpha}\cdot\hat{\mathbf{R}}_{\alpha\alpha^{\prime}})(\mathbf{I}_{\alpha^{% \prime}}\cdot\hat{\mathbf{R}}_{\alpha\alpha^{\prime}})\right\}.
  44. H ^ IR = e μ 0 μ N 4 π α α 1 R α α 3 { Z α g α M α 𝐈 α + Z α g α M α 𝐈 α } 𝐓 \hat{H}\text{IR}=\dfrac{e\mu_{0}\mu\text{N}\hbar}{4\pi}\sum_{\alpha\neq\alpha^% {\prime}}\dfrac{1}{R_{\alpha\alpha^{\prime}}^{3}}\left\{\dfrac{Z_{\alpha}g_{% \alpha^{\prime}}}{M_{\alpha}}\mathbf{I}_{\alpha^{\prime}}+\dfrac{Z_{\alpha^{% \prime}}g_{\alpha}}{M_{\alpha^{\prime}}}\mathbf{I}_{\alpha}\right\}\cdot% \mathbf{T}
  45. 1 299 , 792 , 458 \frac{1}{299,792,458}

Hypocycloid.html

  1. x ( θ ) = ( R - r ) cos θ + r cos ( R - r r θ ) x(\theta)=(R-r)\cos\theta+r\cos\left(\frac{R-r}{r}\theta\right)
  2. y ( θ ) = ( R - r ) sin θ - r sin ( R - r r θ ) , y(\theta)=(R-r)\sin\theta-r\sin\left(\frac{R-r}{r}\theta\right),
  3. x ( θ ) = r ( k - 1 ) cos θ + r cos ( ( k - 1 ) θ ) x(\theta)=r(k-1)\cos\theta+r\cos\left((k-1)\theta\right)\,
  4. y ( θ ) = r ( k - 1 ) sin θ - r sin ( ( k - 1 ) θ ) . y(\theta)=r(k-1)\sin\theta-r\sin\left((k-1)\theta\right).\,

Icosagon.html

  1. A = 5 t 2 ( 1 + 5 + 5 + 2 5 ) 31.56875757 t 2 . A={5}t^{2}(1+\sqrt{5}+\sqrt{5+2\sqrt{5}})\simeq 31.56875757t^{2}.

ID-based_encryption.html

  1. k \textstyle k
  2. 𝒫 \textstyle\mathcal{P}
  3. \textstyle\mathcal{M}
  4. 𝒞 \textstyle\mathcal{C}
  5. K m \textstyle K_{m}
  6. d \textstyle d
  7. 𝒫 \textstyle\mathcal{P}
  8. K m \textstyle K_{m}
  9. I D { 0 , 1 } * \textstyle ID\in\left\{0,1\right\}^{*}
  10. d \textstyle d
  11. I D \textstyle ID
  12. 𝒫 \textstyle\mathcal{P}
  13. m \textstyle m\in\mathcal{M}
  14. I D { 0 , 1 } * \textstyle ID\in\left\{0,1\right\}^{*}
  15. c 𝒞 \textstyle c\in\mathcal{C}
  16. d \textstyle d
  17. 𝒫 \textstyle\mathcal{P}
  18. c 𝒞 \textstyle c\in\mathcal{C}
  19. m \textstyle m\in\mathcal{M}
  20. m , I D { 0 , 1 } * : D e c r y p t ( E x t r a c t ( 𝒫 , K m , I D ) , 𝒫 , E n c r y p t ( 𝒫 , m , I D ) ) = m \forall m\in\mathcal{M},ID\in\left\{0,1\right\}^{*}:Decrypt\left(Extract\left(% \mathcal{P},K_{m},ID\right),\mathcal{P},Encrypt\left(\mathcal{P},m,ID\right)% \right)=m

Ideal_(order_theory).html

  1. \vee
  2. \vee
  3. \wedge
  4. \downarrow
  5. \downarrow
  6. \wedge
  7. \wedge
  8. \vee
  9. \vee
  10. \vee
  11. \vee
  12. \vee
  13. \vee
  14. \vee
  15. \vee
  16. \vee
  17. \vee
  18. \wedge

Ideal_norm.html

  1. A \mathcal{I}_{A}
  2. B \mathcal{I}_{B}
  3. N B / A : B A N_{B/A}\colon\mathcal{I}_{B}\to\mathcal{I}_{A}
  4. N B / A ( 𝔮 ) = 𝔭 [ B / 𝔮 : A / 𝔭 ] N_{B/A}(\mathfrak{q})=\mathfrak{p}^{[B/\mathfrak{q}:A/\mathfrak{p}]}
  5. 𝔮 \mathfrak{q}
  6. 𝔭 = 𝔮 A \mathfrak{p}=\mathfrak{q}\cap A
  7. 𝔮 \mathfrak{q}
  8. 𝔟 B \mathfrak{b}\in\mathcal{I}_{B}
  9. N B / A ( 𝔟 ) N_{B/A}(\mathfrak{b})
  10. { N L / K ( x ) | x 𝔟 } \{N_{L/K}(x)|x\in\mathfrak{b}\}
  11. 𝔞 A \mathfrak{a}\in\mathcal{I}_{A}
  12. N B / A ( 𝔞 B ) = 𝔞 n N_{B/A}(\mathfrak{a}B)=\mathfrak{a}^{n}
  13. n = [ L : K ] n=[L:K]
  14. N B / A ( x B ) = N L / K ( x ) A . N_{B/A}(xB)=N_{L/K}(x)A.
  15. L / K L/K
  16. 𝒪 K 𝒪 L \mathcal{O}_{K}\subset\mathcal{O}_{L}
  17. A = 𝒪 K , B = 𝒪 L A=\mathcal{O}_{K},B=\mathcal{O}_{L}
  18. 𝔟 𝒪 L \mathfrak{b}\in\mathcal{I}_{\mathcal{O}_{L}}
  19. N 𝒪 L / 𝒪 K ( 𝔟 ) = 𝒪 K σ Gal ( L / K ) σ ( 𝔟 ) , N_{\mathcal{O}_{L}/\mathcal{O}_{K}}(\mathfrak{b})=\mathcal{O}_{K}\cap\prod_{% \sigma\in\operatorname{Gal}(L/K)}\sigma(\mathfrak{b}),
  20. 𝒪 K \mathcal{I}_{\mathcal{O}_{K}}
  21. N 𝒪 L / 𝒪 K N_{\mathcal{O}_{L}/\mathcal{O}_{K}}
  22. N L / K N_{L/K}
  23. N L / K N_{L/K}
  24. K = K=\mathbb{Q}
  25. N 𝒪 L / N_{\mathcal{O}_{L}/\mathbb{Z}}\,
  26. \mathbb{Z}
  27. { ± 1 } \{\pm 1\}
  28. \mathbb{Z}
  29. L L
  30. K = K=\mathbb{Q}
  31. L L
  32. 𝒪 L \mathcal{O}_{L}
  33. 𝔞 \mathfrak{a}
  34. 𝒪 L \mathcal{O}_{L}
  35. 𝔞 \mathfrak{a}
  36. N ( 𝔞 ) := [ 𝒪 L : 𝔞 ] = | 𝒪 L / 𝔞 | . N(\mathfrak{a}):=\left[\mathcal{O}_{L}:\mathfrak{a}\right]=|\mathcal{O}_{L}/% \mathfrak{a}|.\,
  37. 𝔞 = ( a ) \mathfrak{a}=(a)
  38. N ( 𝔞 ) = | N L / ( a ) | N(\mathfrak{a})=|N_{L/\mathbb{Q}}(a)|
  39. 𝔞 \mathfrak{a}
  40. 𝔟 \mathfrak{b}
  41. 𝒪 L \mathcal{O}_{L}
  42. N ( 𝔞 𝔟 ) = N ( 𝔞 ) N ( 𝔟 ) N(\mathfrak{a}\cdot\mathfrak{b})=N(\mathfrak{a})N(\mathfrak{b})
  43. N : 𝒪 L > 0 × , N\colon\mathcal{I}_{\mathcal{O}_{L}}\to\mathbb{Q}_{>0}^{\times},
  44. 𝒪 L \mathcal{O}_{L}
  45. 𝔞 \mathfrak{a}
  46. a 𝔞 a\in\mathfrak{a}
  47. | N L / ( a ) | ( 2 π ) s | Δ L | N ( 𝔞 ) , |N_{L/\mathbb{Q}}(a)|\leq\left(\frac{2}{\pi}\right)^{s}\sqrt{|\Delta_{L}|}N(% \mathfrak{a}),
  48. Δ L \Delta_{L}
  49. L L
  50. s s
  51. L L
  52. \mathbb{C}
  53. L L

Ideal_number.html

  1. ( y ) \mathbb{Q}(y)
  2. [ y ] \mathbb{Z}[y]
  3. ( y ) \mathbb{Q}(y)
  4. ( y , w ) \mathbb{Q}(y,w)
  5. ι = ( - 8 - 16 y - 18 w + 12 w 2 + 10 y w + y w 2 ) / 23 \iota=(-8-16y-18w+12w^{2}+10yw+yw^{2})/23
  6. ι 6 - 2 ι 5 + 13 ι 4 - 15 ι 3 + 16 ι 2 + 28 ι + 8 = 0 \iota^{6}-2\iota^{5}+13\iota^{4}-15\iota^{3}+16\iota^{2}+28\iota+8=0
  7. [ y ] \mathbb{Z}[y]
  8. α = ( - 7 + 9 y - 33 w - 24 w 2 + 3 y w - 2 y w 2 ) / 23 \alpha=(-7+9y-33w-24w^{2}+3yw-2yw^{2})/23
  9. β = ( - 27 - 8 y - 9 w + 6 w 2 - 18 y w - 11 y w 2 ) / 23. \beta=(-27-8y-9w+6w^{2}-18yw-11yw^{2})/23.
  10. α 6 + 7 α 5 + 8 α 4 - 15 α 3 + 26 α 2 - 8 α + 8 = 0 \alpha^{6}+7\alpha^{5}+8\alpha^{4}-15\alpha^{3}+26\alpha^{2}-8\alpha+8=0\,
  11. β 6 + 4 β 5 + 35 β 4 + 112 β 3 + 162 β 2 + 108 β + 27 = 0 \beta^{6}+4\beta^{5}+35\beta^{4}+112\beta^{3}+162\beta^{2}+108\beta+27=0\,
  12. p 1 ( mod λ ) p\equiv 1\;\;(\mathop{{\rm mod}}\lambda)
  13. λ \lambda

Idempotent_element.html

  1. r . Ann ( S ) \mathrm{r.Ann}(S)

Identity_of_indiscernibles.html

  1. x y [ x = y P ( P x P y ) ] \forall x\forall y[x=y\rightarrow\forall P(Px\leftrightarrow Py)]
  2. x y [ P ( P x P y ) x = y ] \forall x\forall y[\forall P(Px\leftrightarrow Py)\rightarrow x=y]

IEEE_floating_point.html

  1. 1 / [ u r a d i c a l , u x ] 1/[u^{\prime}radical^{\prime},u^{\prime}x^{\prime}]
  2. s i n ( x ) sin(x)
  3. c o s ( x ) cos(x)
  4. t a n ( x ) tan(x)
  5. a s i n ( x ) asin(x)
  6. a c o s ( x ) acos(x)
  7. a t a n ( x ) atan(x)
  8. a t a n 2 ( y , x ) atan2(y,x)
  9. s i n P i ( x ) = s i n ( π x ) sinPi(x)=sin(πx)
  10. c o s P i ( x ) = c o s ( π x ) cosPi(x)=cos(πx)
  11. a t a n P i ( x ) = a t a n ( x ) / π atanPi(x)=atan(x)/π
  12. a t a n 2 P i ( y , x ) atan2Pi(y,x)
  13. s i n h ( x ) sinh(x)
  14. c o s h ( x ) cosh(x)
  15. t a n h ( x ) tanh(x)
  16. a s i n h ( x ) asinh(x)
  17. a c o s h ( x ) acosh(x)
  18. a t a n h ( x ) atanh(x)
  19. a s i n P i asinPi
  20. a c o s P i acosPi
  21. t a n P i tanPi
  22. 1 + p log 10 ( 2 ) 1+\lceil p\log_{10}(2)\rceil

Illustration_of_the_central_limit_theorem.html

  1. 2 \sqrt{2}
  2. X = { 1 with probability 1 / 3 , 2 with probability 1 / 3 , 3 with probability 1 / 3. X=\left\{\begin{matrix}1&\mbox{with}~{}\ \mbox{probability}~{}\ 1/3,\\ 2&\mbox{with}~{}\ \mbox{probability}~{}\ 1/3,\\ 3&\mbox{with}~{}\ \mbox{probability}~{}\ 1/3.\end{matrix}\right.
  3. { 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 2 + 1 = 3 2 + 2 = 4 2 + 3 = 5 3 + 1 = 4 3 + 2 = 5 3 + 3 = 6 } = { 2 with probability 1 / 9 3 with probability 2 / 9 4 with probability 3 / 9 5 with probability 2 / 9 6 with probability 1 / 9 } \left\{\begin{matrix}1+1&=&2\\ 1+2&=&3\\ 1+3&=&4\\ 2+1&=&3\\ 2+2&=&4\\ 2+3&=&5\\ 3+1&=&4\\ 3+2&=&5\\ 3+3&=&6\end{matrix}\right\}=\left\{\begin{matrix}2&\mbox{with}~{}\ \mbox{% probability}~{}\ 1/9\\ 3&\mbox{with}~{}\ \mbox{probability}~{}\ 2/9\\ 4&\mbox{with}~{}\ \mbox{probability}~{}\ 3/9\\ 5&\mbox{with}~{}\ \mbox{probability}~{}\ 2/9\\ 6&\mbox{with}~{}\ \mbox{probability}~{}\ 1/9\end{matrix}\right\}
  4. { 1 + 1 + 1 = 3 1 + 1 + 2 = 4 1 + 1 + 3 = 5 1 + 2 + 1 = 4 1 + 2 + 2 = 5 1 + 2 + 3 = 6 1 + 3 + 1 = 5 1 + 3 + 2 = 6 1 + 3 + 3 = 7 2 + 1 + 1 = 4 2 + 1 + 2 = 5 2 + 1 + 3 = 6 2 + 2 + 1 = 5 2 + 2 + 2 = 6 2 + 2 + 3 = 7 2 + 3 + 1 = 6 2 + 3 + 2 = 7 2 + 3 + 3 = 8 3 + 1 + 1 = 5 3 + 1 + 2 = 6 3 + 1 + 3 = 7 3 + 2 + 1 = 6 3 + 2 + 2 = 7 3 + 2 + 3 = 8 3 + 3 + 1 = 7 3 + 3 + 2 = 8 3 + 3 + 3 = 9 } = { 3 with probability 1 / 27 4 with probability 3 / 27 5 with probability 6 / 27 6 with probability 7 / 27 7 with probability 6 / 27 8 with probability 3 / 27 9 with probability 1 / 27 } \left\{\begin{matrix}1+1+1&=&3\\ 1+1+2&=&4\\ 1+1+3&=&5\\ 1+2+1&=&4\\ 1+2+2&=&5\\ 1+2+3&=&6\\ 1+3+1&=&5\\ 1+3+2&=&6\\ 1+3+3&=&7\\ 2+1+1&=&4\\ 2+1+2&=&5\\ 2+1+3&=&6\\ 2+2+1&=&5\\ 2+2+2&=&6\\ 2+2+3&=&7\\ 2+3+1&=&6\\ 2+3+2&=&7\\ 2+3+3&=&8\\ 3+1+1&=&5\\ 3+1+2&=&6\\ 3+1+3&=&7\\ 3+2+1&=&6\\ 3+2+2&=&7\\ 3+2+3&=&8\\ 3+3+1&=&7\\ 3+3+2&=&8\\ 3+3+3&=&9\end{matrix}\right\}=\left\{\begin{matrix}3&\mbox{with}~{}\ \mbox{% probability}~{}\ 1/27\\ 4&\mbox{with}~{}\ \mbox{probability}~{}\ 3/27\\ 5&\mbox{with}~{}\ \mbox{probability}~{}\ 6/27\\ 6&\mbox{with}~{}\ \mbox{probability}~{}\ 7/27\\ 7&\mbox{with}~{}\ \mbox{probability}~{}\ 6/27\\ 8&\mbox{with}~{}\ \mbox{probability}~{}\ 3/27\\ 9&\mbox{with}~{}\ \mbox{probability}~{}\ 1/27\end{matrix}\right\}

Image_(category_theory).html

  1. f : X Y f\colon X\to Y
  2. h : I Y h\colon I\to Y
  3. g : X I g\colon X\to I
  4. f = h g f=hg
  5. k : X Z k\colon X\to Z
  6. l : Z Y l\colon Z\to Y
  7. f = l k f=lk
  8. m : I Z m\colon I\to Z
  9. h = l m h=lm
  10. f : X Y f\colon X\to Y
  11. { f ( x ) | x X } \{f(x)~{}|~{}x\in X\}
  12. Y Y
  13. f f

Image_(mathematics).html

  1. f [ A ] = { y Y | y = f ( x ) for some x A } f[A]=\{\,y\in Y\,|\,y=f(x)\,\text{ for some }x\in A\,\}
  2. f - 1 [ B ] = { x X | f ( x ) B } f^{-1}[B]=\{\,x\in X\,|\,f(x)\in B\}
  3. f : 𝒫 ( X ) 𝒫 ( Y ) f^{\rightarrow}:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)
  4. f ( A ) = { f ( a ) | a A } f^{\rightarrow}(A)=\{f(a)\;|\;a\in A\}
  5. f : 𝒫 ( Y ) 𝒫 ( X ) f^{\leftarrow}:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)
  6. f ( B ) = { a X | f ( a ) B } f^{\leftarrow}(B)=\{a\in X\;|\;f(a)\in B\}
  7. f : 𝒫 ( X ) 𝒫 ( Y ) f_{\star}:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)
  8. f f^{\rightarrow}
  9. f : 𝒫 ( Y ) 𝒫 ( X ) f^{\star}:\mathcal{P}(Y)\rightarrow\mathcal{P}(X)
  10. f f^{\leftarrow}
  11. f ( x ) = { a , if x = 1 a , if x = 2 c , if x = 3. f(x)=\left\{\begin{matrix}a,&\mbox{if }~{}x=1\\ a,&\mbox{if }~{}x=2\\ c,&\mbox{if }~{}x=3.\end{matrix}\right.
  12. f ( s S A s ) = s S f ( A s ) f\left(\bigcup_{s\in S}A_{s}\right)=\bigcup_{s\in S}f(A_{s})
  13. f ( s S A s ) s S f ( A s ) f\left(\bigcap_{s\in S}A_{s}\right)\subseteq\bigcap_{s\in S}f(A_{s})
  14. f - 1 ( s S B s ) = s S f - 1 ( B s ) f^{-1}\left(\bigcup_{s\in S}B_{s}\right)=\bigcup_{s\in S}f^{-1}(B_{s})
  15. f - 1 ( s S B s ) = s S f - 1 ( B s ) f^{-1}\left(\bigcap_{s\in S}B_{s}\right)=\bigcap_{s\in S}f^{-1}(B_{s})

Image_segmentation.html

  1. δ \delta
  2. A 1 A_{1}
  3. δ \delta
  4. T T
  5. A j A_{j}
  6. A i A_{i}
  7. A n + 1 A_{n+1}
  8. λ \lambda
  9. λ \lambda
  10. λ \lambda
  11. λ \lambda
  12. λ \lambda
  13. O ( n log n ) O(n\log n)
  14. f f
  15. argmin u γ u 0 + ( u - f ) 2 d x . \operatorname*{argmin}_{u}\gamma\|\nabla u\|_{0}+\int(u-f)^{2}dx.
  16. u * u^{*}
  17. f f
  18. u * u^{*}
  19. γ > 0 \gamma>0
  20. u u
  21. argmin u , K γ | K | + μ K C | u | 2 d x + ( u - f ) 2 d x . \operatorname*{argmin}_{u,K}\gamma|K|+\mu\int_{K^{C}}|\nabla u|^{2}dx+\int(u-f% )^{2}dx.
  22. K K
  23. u u
  24. f f
  25. μ > 0 \mu>0
  26. μ \mu\to\infty
  27. P ( f P(f
  28. ) )
  29. f f
  30. ϵ Σ \epsilon\Sigma
  31. i i
  32. μ \mu
  33. σ \sigma
  34. P ( f P(f
  35. | l |l
  36. ) )
  37. 1 σ ( l i ) 2 π e - ( f i - μ ( l i ) ) 2 2 σ ( l i ) 2 \frac{1}{\sigma(l_{i})\sqrt{2\pi}}e^{-\frac{(f_{i}-\mu(l_{i}))^{2}}{2\sigma(l_% {i})^{2}}}
  38. α ( 1 - δ ( l \alpha(1-\delta(l
  39. - l -l
  40. ) + β Σ )+\beta\Sigma
  41. ϵ \epsilon
  42. ( 1 - δ ( l (1-\delta(l
  43. , l ,l
  44. ) ) ))
  45. α \alpha
  46. β \beta
  47. N ( i ) N(i)
  48. δ \delta
  49. Δ U = U \Delta U=U
  50. - U -U
  51. l i = { l i n e w , if Δ U 0 , l i n e w , if Δ U > 0 and δ < e - Δ U / T , l i o l d l_{i}=\begin{cases}l^{new}_{i},&\,\text{if}\ \ \Delta U<=0,\\ l^{new}_{i},&\,\text{if}\ \ \Delta U>0\ \ \,\text{and}\ \ \delta<e^{-\Delta U/% T},\\ l^{old}_{i}\end{cases}
  52. P ( λ | f i ) = P ( f i | λ ) P ( λ ) Σ λ ϵ Λ P ( f i | λ ) P ( λ ) P(\lambda|f_{i})=\frac{P(f_{i}|\lambda)P(\lambda)}{\Sigma_{\lambda\epsilon% \Lambda}P(f_{i}|\lambda)P(\lambda)}
  53. λ ϵ Λ \lambda\epsilon\Lambda
  54. P ( λ ) = Σ λ ϵ Λ P ( λ | f i ) | Ω | P(\lambda)=\frac{\Sigma_{\lambda\epsilon\Lambda}P(\lambda|f_{i})}{|\Omega|}
  55. Ω \Omega

Impedance_matching.html

  1. Z S = Z L * Z_{\mathrm{S}}=Z_{\mathrm{L}}^{*}\,
  2. Z S = Z L Z_{\mathrm{S}}=Z_{\mathrm{L}}\,
  3. Z 𝗅𝗈𝖺𝖽 = Z 𝗌𝗈𝗎𝗋𝖼𝖾 * Z_{\mathsf{load}}=Z_{\mathsf{source}}^{*}\,
  4. turns ratio = source resistance load resistance \,\text{turns ratio}=\sqrt{\frac{\,\text{source resistance}}{\,\text{load % resistance}}}
  5. R 1 R_{1}
  6. R 2 R_{2}
  7. X 1 X_{1}
  8. j R 1 X 1 R 1 + j X 1 \frac{jR_{1}X_{1}}{R_{1}+jX_{1}}
  9. R 2 = R 1 X 1 2 R 1 2 + X 1 2 R_{2}=\frac{R_{1}X_{1}^{2}}{R_{1}^{2}+X_{1}^{2}}
  10. X 1 X_{1}
  11. X 1 = R 2 R 1 2 R 1 - R 2 X_{1}=\sqrt{\frac{R_{2}R_{1}^{2}}{R_{1}-R_{2}}}
  12. R 1 R 2 R_{1}\gg R_{2}
  13. X 1 R 1 R 2 X_{1}\approx\sqrt{R_{1}R_{2}}\,
  14. Z c Z_{c}
  15. Γ 12 = Z 2 - Z 1 Z 2 + Z 1 \Gamma_{12}={Z_{2}-Z_{1}\over Z_{2}+Z_{1}}
  16. Γ 21 = Z 1 - Z 2 Z 1 + Z 2 \Gamma_{21}={Z_{1}-Z_{2}\over Z_{1}+Z_{2}}
  17. Γ 21 = - Γ 12 \Gamma_{21}=-\Gamma_{12}\,
  18. V i V_{i}\,
  19. I i I_{i}\,
  20. V t V_{t}\,
  21. I t I_{t}\,
  22. V r V_{r}\,
  23. I r I_{r}\,
  24. V i = Z c I i V_{i}=Z_{c}I_{i}\,
  25. V r = - Z c I r V_{r}=-Z_{c}I_{r}\,
  26. V t = Z L I t V_{t}=Z_{L}I_{t}\,
  27. V i V_{i}\,
  28. V r V_{r}\,
  29. V t V_{t}\,
  30. I i I_{i}\,
  31. I r I_{r}\,
  32. I t I_{t}\,
  33. Z c Z_{c}\,
  34. V t = V i + V r V_{t}=V_{i}+V_{r}\,
  35. I t = I i + I r I_{t}=I_{i}+I_{r}\,
  36. V r = Γ T L V i V_{r}=\Gamma_{TL}V_{i}\,
  37. I r = - Γ T L I i I_{r}=-\Gamma_{TL}I_{i}\,
  38. V t = ( 1 + Γ T L ) V i V_{t}=(1+\Gamma_{TL})V_{i}\,
  39. I t = ( 1 - Γ T L ) I i I_{t}=(1-\Gamma_{TL})I_{i}\,
  40. Γ T L \Gamma_{TL}\,
  41. Γ T L = Z L - Z c Z L + Z c = Γ L \Gamma_{TL}={Z_{L}-Z_{c}\over Z_{L}+Z_{c}}=\Gamma_{L}\,
  42. Z L Z_{L}
  43. Z c Z_{c}
  44. Γ = 0 \Gamma=0
  45. - Γ S T = Γ T S = Z s - Z c Z s + Z c = Γ S -\Gamma_{ST}=\Gamma_{TS}={Z_{s}-Z_{c}\over Z_{s}+Z_{c}}=\Gamma_{S}\,
  46. Z i n = Z C ( 1 + T 2 Γ L ) ( 1 - T 2 Γ L ) Z_{in}=Z_{C}\frac{(1+T^{2}\Gamma_{L})}{(1-T^{2}\Gamma_{L})}\,
  47. T , T\ ,
  48. T T\,
  49. Γ L = 0 \Gamma_{L}=0\,
  50. Z i n = Z C Z_{in}=Z_{C}\,
  51. V L = V S T ( 1 - Γ S ) ( 1 + Γ L ) 2 ( 1 - T 2 Γ S Γ L ) V_{L}=V_{S}\frac{T(1-\Gamma_{S})(1+\Gamma_{L})}{2(1-T^{2}\Gamma_{S}\Gamma_{L})}\,
  52. V S V_{S}\,
  53. Γ L = 0 \Gamma_{L}=0\,
  54. Γ S = 0 \Gamma_{S}=0\,
  55. V L = V S T 2 V_{L}=V_{S}\frac{T}{2}\,
  56. m 1 m_{1}
  57. m 2 m_{2}
  58. E 2 = 2 P 2 m 2 ( m 1 + m 2 ) 2 E_{2}=\frac{2P^{2}m_{2}}{(m_{1}+m_{2})^{2}}

Implicit_function_theorem.html

  1. f ( x , y ) = x 2 + y 2 f(x,y)=x^{2}+y^{2}
  2. ± 1 - x 2 \pm\sqrt{1-x^{2}}
  3. g 1 ( x ) = 1 - x 2 g_{1}(x)=\sqrt{1-x^{2}}
  4. g 2 ( x ) = - 1 - x 2 g_{2}(x)=-\sqrt{1-x^{2}}
  5. y = g 2 ( x ) y=g_{2}(x)
  6. g 1 ( x ) g_{1}(x)
  7. g 2 ( x ) g_{2}(x)
  8. g 1 ( x ) g_{1}(x)
  9. g 2 ( x ) g_{2}(x)
  10. { ( 𝐱 , g ( 𝐱 ) ) 𝐱 U } = { ( 𝐱 , 𝐲 ) U × V f ( 𝐱 , 𝐲 ) = 0 } . \{(\mathbf{x},g(\mathbf{x}))\mid\mathbf{x}\in U\}=\{(\mathbf{x},\mathbf{y})\in U% \times V\mid f(\mathbf{x},\mathbf{y})=0\}.
  11. ( D f ) ( 𝐚 , 𝐛 ) = [ f 1 x 1 ( 𝐚 , 𝐛 ) f 1 x n ( 𝐚 , 𝐛 ) f m x 1 ( 𝐚 , 𝐛 ) f m x n ( 𝐚 , 𝐛 ) | f 1 y 1 ( 𝐚 , 𝐛 ) f 1 y m ( 𝐚 , 𝐛 ) f m y 1 ( 𝐚 , 𝐛 ) f m y m ( 𝐚 , 𝐛 ) ] = [ X | Y ] (Df)(\mathbf{a},\mathbf{b})=\left[\begin{matrix}\frac{\partial f_{1}}{\partial x% _{1}}(\mathbf{a},\mathbf{b})&\cdots&\frac{\partial f_{1}}{\partial x_{n}}(% \mathbf{a},\mathbf{b})\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{m}}{\partial x_{1}}(\mathbf{a},\mathbf{b})&\cdots&\frac{% \partial f_{m}}{\partial x_{n}}(\mathbf{a},\mathbf{b})\end{matrix}\right|\left% .\begin{matrix}\frac{\partial f_{1}}{\partial y_{1}}(\mathbf{a},\mathbf{b})&% \cdots&\frac{\partial f_{1}}{\partial y_{m}}(\mathbf{a},\mathbf{b})\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{m}}{\partial y_{1}}(\mathbf{a},\mathbf{b})&\cdots&\frac{% \partial f_{m}}{\partial y_{m}}(\mathbf{a},\mathbf{b})\\ \end{matrix}\right]=[X|Y]
  12. { ( 𝐱 , g ( 𝐱 ) ) | 𝐱 U } = { ( 𝐱 , 𝐲 ) U × V | f ( 𝐱 , 𝐲 ) = 𝐜 } . \{(\mathbf{x},g(\mathbf{x}))|\mathbf{x}\in U\}=\{(\mathbf{x},\mathbf{y})\in U% \times V|f(\mathbf{x},\mathbf{y})=\mathbf{c}\}.
  13. g x j ( x ) = - ( f y ( x , g ( x ) ) ) - 1 f x j ( x , g ( x ) ) \frac{\partial g}{\partial x_{j}}(x)=-\left(\frac{\partial f}{\partial y}(x,g(% x))\right)^{-1}\frac{\partial f}{\partial x_{j}}(x,g(x))
  14. f ( x , y ) = x 2 + y 2 - 1 f(x,y)=x^{2}+y^{2}-1
  15. ( D f ) ( a , b ) = [ f x ( a , b ) f y ( a , b ) ] = [ 2 a 2 b ] (Df)(a,b)=\left[\frac{\partial f}{\partial x}(a,b)\ \ \frac{\partial f}{% \partial y}(a,b)\right]=[2a\ \ 2b]
  16. x = h ( y ) x=h(y)
  17. ( h ( y ) , y ) \left(h(y),y\right)
  18. x 2 + y 2 - 1 x^{2}+y^{2}-1
  19. 2 x d x + 2 y d y = 0 , 2xdx+2ydy=0,
  20. d y / d x = - x / y dy/dx=-x/y
  21. d x / d y = - y / x . dx/dy=-y/x.
  22. ( x 1 , , x m ) (x_{1},\ldots,x_{m})
  23. ( x 1 , , x m ) (x^{\prime}_{1},\ldots,x^{\prime}_{m})
  24. h 1 h m h_{1}\ldots h_{m}
  25. ( x 1 , , x m ) (x^{\prime}_{1},\ldots,x^{\prime}_{m})
  26. ( x 1 , , x m ) (x_{1},\ldots,x_{m})
  27. x 1 = h 1 ( x 1 , , x m ) , , x m = h m ( x 1 , , x m ) x^{\prime}_{1}=h_{1}(x_{1},\ldots,x_{m}),\ldots,x^{\prime}_{m}=h_{m}(x_{1},% \ldots,x_{m})
  28. ( x 1 , , x m ) (x^{\prime}_{1},\ldots,x^{\prime}_{m})
  29. ( x 1 , , x m ) (x_{1},\ldots,x_{m})
  30. ( x 1 , , x m , x 1 , , x m ) (x^{\prime}_{1},\ldots,x^{\prime}_{m},x_{1},\ldots,x_{m})
  31. f ( x 1 , , x m , x 1 , x m ) = ( h 1 ( x 1 , x m ) - x 1 , , h m ( x 1 , , x m ) - x m ) . f(x^{\prime}_{1},\ldots,x^{\prime}_{m},x_{1},\ldots x_{m})=(h_{1}(x_{1},\ldots x% _{m})-x^{\prime}_{1},\ldots,h_{m}(x_{1},\ldots,x_{m})-x^{\prime}_{m}).
  32. a = ( x 1 , , x m ) , b = ( x 1 , , x m ) a=(x^{\prime}_{1},\ldots,x^{\prime}_{m}),b=(x_{1},\ldots,x_{m})
  33. ( D f ) ( a , b ) = [ - 1 0 0 - 1 | h 1 x 1 ( b ) h 1 x m ( b ) h m x 1 ( b ) h m x m ( b ) ] = [ - 1 m | J ] . (Df)(a,b)=\left[\begin{matrix}-1&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&-1\end{matrix}\left|\begin{matrix}\frac{\partial h_{1}}{\partial x_{1% }}(b)&\cdots&\frac{\partial h_{1}}{\partial x_{m}}(b)\\ \vdots&\ddots&\vdots\\ \frac{\partial h_{m}}{\partial x_{1}}(b)&\cdots&\frac{\partial h_{m}}{\partial x% _{m}}(b)\\ \end{matrix}\right.\right]=[-1_{m}|J].
  34. ( x 1 , , x m ) (x_{1},\ldots,x_{m})
  35. ( x 1 , , x m ) (x^{\prime}_{1},\ldots,x^{\prime}_{m})
  36. J = [ x ( R , θ ) R x ( R , θ ) θ y ( R , θ ) R y ( R , θ ) θ ] = [ cos θ - R sin θ sin θ R cos θ ] . J=\begin{bmatrix}\frac{\partial x(R,\theta)}{\partial R}&\frac{\partial x(R,% \theta)}{\partial\theta}\\ \frac{\partial y(R,\theta)}{\partial R}&\frac{\partial y(R,\theta)}{\partial% \theta}\\ \end{bmatrix}=\begin{bmatrix}\cos\theta&-R\sin\theta\\ \sin\theta&R\cos\theta\end{bmatrix}.
  37. ( x 0 , y 0 ) X × Y (x_{0},y_{0})\in X\times Y
  38. f ( x 0 , y 0 ) = 0 f(x_{0},y_{0})=0
  39. y D f ( x 0 , y 0 ) ( 0 , y ) y\mapsto Df(x_{0},y_{0})(0,y)
  40. ( x , y ) U × V (x,y)\in U\times V
  41. f : R n × R m R n f:R^{n}\times R^{m}\to R^{n}
  42. f ( x 0 , y 0 ) = 0 f(x_{0},y_{0})=0
  43. A R n A\subset R^{n}
  44. B R m B\subset R^{m}
  45. f ( , y ) : A R n f(\cdot,y):A\to R^{n}
  46. A 0 R n A_{0}\subset R^{n}
  47. B 0 R m B_{0}\subset R^{m}
  48. y B 0 y\in B_{0}
  49. x = g ( y ) A 0 x=g(y)\in A_{0}

Improper_integral.html

  1. \infty
  2. - -\infty
  3. lim b a b f ( x ) d x , lim a - a b f ( x ) d x , \lim_{b\to\infty}\int_{a}^{b}f(x)\,\mathrm{d}x,\qquad\lim_{a\to-\infty}\int_{a% }^{b}f(x)\,\mathrm{d}x,
  4. lim c b - a c f ( x ) d x , lim c a + c b f ( x ) d x , \lim_{c\to b^{-}}\int_{a}^{c}f(x)\,\mathrm{d}x,\quad\lim_{c\to a^{+}}\int_{c}^% {b}f(x)\,\mathrm{d}x,
  5. 1 / x 2 1/{x^{2}}
  6. 1 1 x 2 d x = lim b 1 b 1 x 2 d x = lim b ( - 1 b + 1 1 ) = 1. \int_{1}^{\infty}\frac{1}{x^{2}}\,\mathrm{d}x=\lim_{b\to\infty}\int_{1}^{b}% \frac{1}{x^{2}}\,\mathrm{d}x=\lim_{b\to\infty}\left(-\frac{1}{b}+\frac{1}{1}% \right)=1.
  7. 1 / x 1/\sqrt{x}
  8. 0 1 1 x d x = lim a 0 + a 1 1 x d x = lim a 0 + ( 2 - 2 a ) = 2. \int_{0}^{1}\frac{1}{\sqrt{x}}\,\mathrm{d}x=\lim_{a\to 0^{+}}\int_{a}^{1}\frac% {1}{\sqrt{x}}\,\mathrm{d}x=\lim_{a\to 0^{+}}(2-2\sqrt{a})=2.
  9. lim t a t f ( x ) d x \lim_{t\to\infty}\int_{a}^{t}f(x)\,\mathrm{d}x
  10. lim b 1 b 1 x d x = . \lim_{b\to\infty}\int_{1}^{b}\frac{1}{x}\,\mathrm{d}x=\infty.
  11. lim b 1 b x sin x d x , \lim_{b\to\infty}\int_{1}^{b}x\sin x\,\mathrm{d}x,
  12. - f ( x ) d x \int_{-\infty}^{\infty}f(x)\,\mathrm{d}x
  13. - f ( x ) d x = lim a - lim b a b f ( x ) d x \int_{-\infty}^{\infty}f(x)\,\mathrm{d}x=\lim_{a\to-\infty}\lim_{b\to\infty}% \int_{a}^{b}f(x)\,\mathrm{d}x
  14. lim a - a c f ( x ) d x + lim b c b f ( x ) d x \lim_{a\to-\infty}\int_{a}^{c}f(x)\,\mathrm{d}x+\lim_{b\to\infty}\int_{c}^{b}f% (x)\,\mathrm{d}x
  15. - e - x 2 d x = π \int_{-\infty}^{\infty}e^{-x^{2}}\,\mathrm{d}x=\sqrt{\pi}
  16. - e x d x \int_{-\infty}^{\infty}e^{x}\,\mathrm{d}x
  17. - x d x \int_{-\infty}^{\infty}x\,\mathrm{d}x
  18. lim a - a c x d x + lim b c b x d x \lim_{a\to-\infty}\int_{a}^{c}x\,\mathrm{d}x+\lim_{b\to\infty}\int_{c}^{b}x\,% \mathrm{d}x
  19. - \infty-\infty
  20. p . v . - x d x = lim b - b b x d x = 0. \operatorname{p.v.}\int_{-\infty}^{\infty}x\,\mathrm{d}x=\lim_{b\to\infty}\int% _{-b}^{b}x\,\mathrm{d}x=0.
  21. 1 1 x 2 d x \int_{1}^{\infty}\frac{1}{x^{2}}\,\mathrm{d}x
  22. 0 sin x x d x \int_{0}^{\infty}\frac{\sin x}{x}\,\mathrm{d}x
  23. 0 sin x x d x = - \int_{0}^{\infty}\frac{\sin x}{x}\,\mathrm{d}x=\infty-\infty
  24. a c f ( x ) d x \int_{a}^{c}f(x)\,\mathrm{d}x\,
  25. lim b c - a b f ( x ) d x \lim_{b\to c^{-}}\int_{a}^{b}f(x)\,\mathrm{d}x\,
  26. a b | f ( x ) | d x \int_{a}^{b}|f(x)|\,\mathrm{d}x
  27. a f ( x ) d x , and a | f ( x ) | d x \int_{a}^{\infty}f(x)\,\mathrm{d}x,\quad\mbox{and}~{}\ \int_{a}^{\infty}|f(x)|% \,\mathrm{d}x
  28. 0 d x 1 + x 2 \int_{0}^{\infty}\frac{\mathrm{d}x}{1+x^{2}}
  29. lim b 0 b d x 1 + x 2 = lim b arctan b = π 2 , \lim_{b\to\infty}\int_{0}^{b}\frac{\mathrm{d}x}{1+x^{2}}=\lim_{b\to\infty}% \arctan{b}=\frac{\pi}{2},
  30. 0 sin ( x ) x d x \int_{0}^{\infty}\frac{\sin(x)}{x}\,\mathrm{d}x
  31. 0 | sin ( x ) x | d x = . \int_{0}^{\infty}\left|\frac{\sin(x)}{x}\right|\,\mathrm{d}x=\infty.
  32. f ( x ) = sin ( x ) / x f(x)=\sin(x)/x
  33. 0 sin ( x ) x d x = lim b 0 b sin ( x ) x d x = π 2 . \int_{0}^{\infty}\frac{\sin(x)}{x}\,\mathrm{d}x=\lim_{b\rightarrow\infty}\int_% {0}^{b}\frac{\sin(x)}{x}\,\mathrm{d}x=\frac{\pi}{2}.
  34. lim a 0 + ( - 1 - a d x x + a 1 d x x ) = 0 , \lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x}+\int_{a}^{1}% \frac{\mathrm{d}x}{x}\right)=0,
  35. lim a 0 + ( - 1 - a d x x + 2 a 1 d x x ) = - ln 2. \lim_{a\rightarrow 0+}\left(\int_{-1}^{-a}\frac{\mathrm{d}x}{x}+\int_{2a}^{1}% \frac{\mathrm{d}x}{x}\right)=-\ln 2.
  36. - 1 1 d x x ( which gives - + ) . \int_{-1}^{1}\frac{\mathrm{d}x}{x}{\ }\left(\mbox{which}~{}\ \mbox{gives}~{}\ % -\infty+\infty\right).
  37. lim a - a a 2 x d x x 2 + 1 = 0 , \lim_{a\rightarrow\infty}\int_{-a}^{a}\frac{2x\,\mathrm{d}x}{x^{2}+1}=0,
  38. lim a - 2 a a 2 x d x x 2 + 1 = - ln 4. \lim_{a\rightarrow\infty}\int_{-2a}^{a}\frac{2x\,\mathrm{d}x}{x^{2}+1}=-\ln 4.
  39. - 2 x d x x 2 + 1 ( which gives - + ) . \int_{-\infty}^{\infty}\frac{2x\,\mathrm{d}x}{x^{2}+1}{\ }\left(\mbox{which}~{% }\ \mbox{gives}~{}\ -\infty+\infty\right).
  40. 0 f ( x ) d x \int_{0}^{\infty}f(x)\,\mathrm{d}x
  41. lim λ 0 λ ( 1 - x λ ) α f ( x ) d x \lim_{\lambda\to\infty}\int_{0}^{\lambda}\left(1-\frac{x}{\lambda}\right)^{% \alpha}f(x)\,\mathrm{d}x
  42. 0 sin x d x \int_{0}^{\infty}\sin x\,\mathrm{d}x
  43. 2 \mathbb{R}^{2}
  44. f ( x , y ) = log ( x 2 + y 2 ) f(x,y)=\log(x^{2}+y^{2})
  45. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  46. [ - a , a ] n [-a,a]^{n}
  47. a > 0 a>0
  48. n \mathbb{R}^{n}
  49. lim a [ - a , a ] n f , \lim_{a\to\infty}\int_{[-a,a]^{n}}f,
  50. n \mathbb{R}^{n}
  51. f ~ \tilde{f}
  52. n \mathbb{R}^{n}
  53. f ~ ( x ) = { f ( x ) x A 0 x A \tilde{f}(x)=\begin{cases}f(x)&x\in A\\ 0&x\not\in A\end{cases}
  54. f ~ \tilde{f}
  55. [ - a , a ] n [-a,a]^{n}
  56. A f = [ - a , a ] n f ~ . \int_{A}f=\int_{[-a,a]^{n}}\tilde{f}.
  57. n \mathbb{R}^{n}
  58. A f = lim a A [ - a , a ] n f = lim a [ - a , a ] n f ~ . \int_{A}f=\lim_{a\to\infty}\int_{A\cap[-a,a]^{n}}f=\lim_{a\to\infty}\int_{[-a,% a]^{n}}\tilde{f}.
  59. M > 0 M>0
  60. f M = min { f , M } f_{M}=\min\{f,M\}
  61. A f = lim M A f M \int_{A}f=\lim_{M\to\infty}\int_{A}f_{M}
  62. f + = max { f , 0 } f_{+}=\max\{f,0\}
  63. f - = max { - f , 0 } f_{-}=\max\{-f,0\}
  64. f = f + - f - f=f_{+}-f_{-}
  65. f + f_{+}
  66. f - f_{-}
  67. f + f_{+}
  68. f - f_{-}
  69. A f = A f + - A f - . \int_{A}f=\int_{A}f_{+}-\int_{A}f_{-}.
  70. A | f | = A f + + A f - . \int_{A}|f|=\int_{A}f_{+}+\int_{A}f_{-}.

Incenter.html

  1. ( x a , y a ) (x_{a},y_{a})
  2. ( x b , y b ) (x_{b},y_{b})
  3. ( x c , y c ) (x_{c},y_{c})
  4. a a
  5. b b
  6. c c
  7. ( a x a + b x b + c x c a + b + c , a y a + b y b + c y c a + b + c ) = a ( x a , y a ) + b ( x b , y b ) + c ( x c , y c ) a + b + c . \bigg(\frac{ax_{a}+bx_{b}+cx_{c}}{a+b+c},\frac{ay_{a}+by_{b}+cy_{c}}{a+b+c}% \bigg)=\frac{a(x_{a},y_{a})+b(x_{b},y_{b})+c(x_{c},y_{c})}{a+b+c}.
  8. 1 : 1 : 1. \ 1:1:1.
  9. a : b : c \ a:b:c
  10. a a
  11. b b
  12. c c
  13. sin ( A ) : sin ( B ) : sin ( C ) \sin(A):\sin(B):\sin(C)
  14. A A
  15. B B
  16. C C
  17. I A I A C A A B + I B I B A B B C + I C I C B C C A = 1. \frac{IA\cdot IA}{CA\cdot AB}+\frac{IB\cdot IB}{AB\cdot BC}+\frac{IC\cdot IC}{% BC\cdot CA}=1.
  18. I A I B I C = 4 R r 2 , IA\cdot IB\cdot IC=4Rr^{2},
  19. O I 2 = R ( R - 2 r ) , OI^{2}=R(R-2r),
  20. I N = 1 2 ( R - 2 r ) < 1 2 R . IN=\frac{1}{2}(R-2r)<\frac{1}{2}R.
  21. I H 2 = 2 r 2 - 4 R 2 cos A cos B cos C . IH^{2}=2r^{2}-4R^{2}\cos A\cos B\cos C.
  22. I G < H G , I H < H G , I G < I O , 2 I N < I O . IG<HG,\quad IH<HG,\quad IG<IO,\quad 2IN<IO.
  23. d s < d u < d v < 1 3 ; \frac{d}{s}<\frac{d}{u}<\frac{d}{v}<\frac{1}{3};
  24. d < 1 3 e ; d<\frac{1}{3}e;
  25. d < 1 2 R . d<\frac{1}{2}R.
  26. B X C X \tfrac{BX}{CX}

Incidence_(geometry).html

  1. P P
  2. l l
  3. P I l PIl
  4. P I l PIl
  5. ( P , l ) (P,l)
  6. P P
  7. U U
  8. W W
  9. V V
  10. d i m U + d i m W d i m ( U + W ) dimU+dimW−dim(U+W)
  11. 𝐏 ( V ) \mathbf{P}(V)
  12. V V
  13. d i m V 1 dimV−1
  14. L L
  15. M M
  16. P P
  17. d i m L + d i m M d i m P dimL+dimM≥dimP
  18. P G ( 2 , F ) PG(2,F)
  19. F F
  20. P G ( 2 , F ) PG(2,F)
  21. V V
  22. F F
  23. 𝐏 ( V ) = P G ( 2 , F ) \mathbf{P}(V)=PG(2,F)
  24. V V
  25. V V
  26. V V
  27. V V
  28. 𝐏 ( V ) \mathbf{P}(V)
  29. a a , b , c aa,b,c
  30. ( x , y , z ) (x,y,z)
  31. ( x : y : z ) (x:y:z)
  32. ( a , b , c ) (a,b,c)
  33. a a : b : c aa:b:c
  34. P = ( x , y , z ) P=(x,y,z)
  35. l = a a , b , c l=aa,b,c
  36. P I l PIl
  37. a x + b y + c z = 0 ax+by+cz=0
  38. a x + b y + c z = [ a , b , c ] ( x , y , z ) = ( a , b , c ) L ( x , y , z ) P = ax+by+cz=[a,b,c]\cdot(x,y,z)=(a,b,c)_{L}\cdot(x,y,z)_{P}=
  39. = [ a : b : c ] ( x : y : z ) = ( a , b , c ) ( x y z ) = 0. =[a:b:c]\cdot(x:y:z)=(a,b,c)\left(\begin{matrix}x\\ y\\ z\end{matrix}\right)=0.
  40. l l
  41. a x + b y + c z = 0 ax+by+cz=0
  42. ( x y z x 1 y 1 z 1 x 2 y 2 z 2 ) ( a b c ) = ( 0 0 0 ) . \left(\begin{matrix}x&y&z\\ x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\end{matrix}\right)\left(\begin{matrix}a\\ b\\ c\end{matrix}\right)=\left(\begin{matrix}0\\ 0\\ 0\end{matrix}\right).
  43. | x y z x 1 y 1 z 1 x 2 y 2 z 2 | = 0. \left|\begin{matrix}x&y&z\\ x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\end{matrix}\right|=0.
  44. l l
  45. l = a a , b , c l=aa,b,c
  46. P = ( x , y , z ) P=(x,y,z)
  47. | x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 | = 0 , \left|\begin{matrix}x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\\ x_{3}&y_{3}&z_{3}\end{matrix}\right|=0,
  48. l = a a , b , c l=aa,b,c
  49. P P
  50. P P
  51. a x + b y + c z = 0 ax+by+cz=0
  52. P P
  53. ( a b c a 1 b 1 c 1 a 2 b 2 c 2 ) ( x y z ) = ( 0 0 0 ) . \left(\begin{matrix}a&b&c\\ a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\end{matrix}\right)\left(\begin{matrix}x\\ y\\ z\end{matrix}\right)=\left(\begin{matrix}0\\ 0\\ 0\end{matrix}\right).
  54. | a b c a 1 b 1 c 1 a 2 b 2 c 2 | = 0. \left|\begin{matrix}a&b&c\\ a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\end{matrix}\right|=0.
  55. a , b a,b
  56. c c
  57. P P
  58. P P
  59. | a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 | = 0. \left|\begin{matrix}a_{1}&b_{1}&c_{1}\\ a_{2}&b_{2}&c_{2}\\ a_{3}&b_{3}&c_{3}\end{matrix}\right|=0.
  60. d i m ( L + M ) = d i m L + d i m M d i m ( L M ) dim(L+M)=dimL+dimM−dim(L∩M)
  61. d i m L + d i m M > d i m P dimL+dimM>dimP
  62. d i m ( L M ) > 0 dim(L∩M)>0

Incidence_matrix.html

  1. ( b i j ) (b_{ij})
  2. b i j = 1 b_{ij}=1
  3. v i v_{i}
  4. x j x_{j}
  5. ( 1 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 ) \begin{pmatrix}1&1&1&0\\ 1&0&0&0\\ 0&1&0&1\\ 0&0&1&1\\ \end{pmatrix}
  6. [ b i j ] [b_{ij}]
  7. b i j = - 1 b_{ij}=-1
  8. x j x_{j}
  9. v i v_{i}
  10. 1 1
  11. v i v_{i}
  12. A ( L ( G ) ) = B ( G ) T B ( G ) - 2 I m A(L(G))=B(G)^{T}B(G)-2I_{m}
  13. A ( L ( G ) ) A(L(G))
  14. I m I_{m}
  15. B ( G ) B ( G ) T . B(G)B(G)^{T}.
  16. [ b i j ] [b_{ij}]
  17. b i j = 1 b_{ij}=1
  18. p i p_{i}
  19. L j L_{j}

Inclusion_map.html

  1. A A
  2. B B
  3. ι \iota
  4. x x
  5. A A
  6. x x
  7. B B
  8. ι : A B , ι ( x ) = x . \iota:A\rightarrow B,\qquad\iota(x)=x.
  9. \hookrightarrow
  10. ι : A X \iota:A\rightarrow X
  11. \star
  12. ι ( x y ) = ι ( x ) ι ( y ) \iota(x\star y)=\iota(x)\star\iota(y)
  13. \star

Inclusion–exclusion_principle.html

  1. | A B | = | A | + | B | - | A B | , |A\cup B|=|A|+|B|-|A\cap B|,
  2. | A B C | = | A | + | B | + | C | - | A B | - | A C | - | B C | + | A B C | . |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|.
  3. n n
  4. n n
  5. n n
  6. n n
  7. | i = 1 n A i | = i = 1 n | A i | - 1 i < j n | A i A j | + 1 i < j < k n | A i A j A k | - + ( - 1 ) n - 1 | A 1 A n | . \biggl|\bigcup_{i=1}^{n}A_{i}\biggr|=\sum_{i=1}^{n}\left|A_{i}\right|\;-\sum_{% 1\leq i<j\leq n}\left|A_{i}\cap A_{j}\right|\;+\sum_{1\leq i<j<k\leq n}\left|A% _{i}\cap A_{j}\cap A_{k}\right|\;-\ \ldots\ +\;\left(-1\right)^{n-1}\left|A_{1% }\cap\cdots\cap A_{n}\right|.
  8. | i = 1 n A i | = k = 1 n ( - 1 ) k + 1 ( 1 i 1 < < i k n | A i 1 A i k | ) \Biggl|\bigcup_{i=1}^{n}A_{i}\Biggr|=\sum_{k=1}^{n}(-1)^{k+1}\left(\sum_{1\leq i% _{1}<\cdots<i_{k}\leq n}\left|A_{i_{1}}\cap\cdots\cap A_{i_{k}}\right|\right)
  9. | i = 1 n A i | = J { 1 , 2 , , n } ( - 1 ) | J | - 1 | j J A j | . \Biggl|\bigcup_{i=1}^{n}A_{i}\Biggr|=\sum_{\emptyset\neq J\subseteq\{1,2,% \ldots,n\}}(-1)^{|J|-1}\Biggl|\bigcap_{j\in J}A_{j}\Biggr|.
  10. A i ¯ \bar{A_{i}}
  11. | i = 1 n A i ¯ | = | S - i = 1 n A i | = | S | - i = 1 n | A i | + 1 i < j n | A i A j | - + ( - 1 ) n | A 1 A n | . \biggl|\bigcap_{i=1}^{n}\bar{A_{i}}\biggr|=\biggl|S-\bigcup_{i=1}^{n}A_{i}% \biggr|=\left|S\right|\;-\sum_{i=1}^{n}\left|A_{i}\right|\;+\sum_{1\leq i<j% \leq n}\left|A_{i}\cap A_{j}\right|\;-\;\ldots\ +\;\left(-1\right)^{n}\left|A_% {1}\cap\cdots\cap A_{n}\right|.
  12. W = | m = 1 n A m | . W=\biggl|\bigcup_{m=1}^{n}A_{m}\biggr|.
  13. W = m 1 = 1 n | A m 1 | - 1 m 1 < m 2 n | A m 1 A m 2 | + 1 m 1 < m 2 < m 3 n | A m 1 A m 2 A m 3 | - + ( - 1 ) p - 1 1 m 1 < < m p n | A m 1 A m p | . W=\sum_{m_{1}=1}^{n}\left|A_{m_{1}}\right|{}-\sum_{1\leq m_{1}<m_{2}\leq n}% \left|A_{m_{1}}\cap A_{m_{2}}\right|{}+\sum_{1\leq m_{1}<m_{2}<m_{3}\leq n}% \left|A_{m_{1}}\cap A_{m_{2}}\cap A_{m_{3}}\right|{}-\cdots{}+(-1)^{p-1}\sum_{% 1\leq m_{1}<\cdots<m_{p}\leq n}\left|A_{m_{1}}\cap\cdots\cap A_{m_{p}}\right|\cdots.
  14. A m 1 A m p A_{m_{1}}\cap\cdots\cap A_{m_{p}}
  15. m m
  16. ( n p ) {n\choose p}
  17. W = ( n 1 ) | A 1 | - ( n 2 ) | A 1 A 2 | + ( n 3 ) | A 1 A 2 A 3 | - + ( - 1 ) p - 1 ( n p ) | A 1 A p | . W={n\choose 1}\left|A_{1}\right|-{n\choose 2}\left|A_{1}\cap A_{2}\right|+{n% \choose 3}\left|A_{1}\cap A_{2}\cap A_{3}\right|-\cdots+(-1)^{p-1}{n\choose p}% \left|A_{1}\cap\cdots\cap A_{p}\right|\cdots.
  18. | A 1 A p | \left|A_{1}\cap\cdots\cap A_{p}\right|
  19. W = ( n 1 ) ( n - 1 ) ! - ( n 2 ) ( n - 2 ) ! + ( n 3 ) ( n - 3 ) ! - + ( - 1 ) p - 1 ( n p ) ( n - p ) ! W = p = 1 n ( - 1 ) p - 1 ( n p ) ( n - p ) ! . W={n\choose 1}(n-1)!-{n\choose 2}(n-2)!+{n\choose 3}(n-3)!-\cdots+(-1)^{p-1}{n% \choose p}(n-p)!\cdots W=\sum_{p=1}^{n}(-1)^{p-1}{n\choose p}(n-p)!.
  20. ( n p ) = n ! p ! ( n - p ) ! {n\choose p}=\frac{n!}{p!(n-p)!}
  21. W = p = 1 n ( - 1 ) p - 1 n ! p ! . W=\sum_{p=1}^{n}(-1)^{p-1}\,\frac{n!}{p!}.
  22. Q = 1 - W n ! = p = 0 n ( - 1 ) p p ! , Q=1-\frac{W}{n!}=\sum_{p=0}^{n}\frac{(-1)^{p}}{p!},
  23. A J := j J A j A_{J}:=\bigcap_{j\in J}A_{j}
  24. | i = 1 n A i | = k = 1 n ( - 1 ) k - 1 ( n k ) α k . \biggl|\bigcup_{i=1}^{n}A_{i}\biggr|=\sum_{k=1}^{n}(-1)^{k-1}{\left({{n}\atop{% k}}\right)}\alpha_{k}.
  25. | S i = 1 n A i | = k = 0 n ( - 1 ) k ( n k ) α k . \biggl|S\setminus\bigcup_{i=1}^{n}A_{i}\biggr|=\sum_{k=0}^{n}(-1)^{k}{\left({{% n}\atop{k}}\right)}\alpha_{k}.
  26. A = S A_{\emptyset}=S
  27. J [ n ] ( - 1 ) | J | | A J | . \sum_{J\subseteq[n]}(-1)^{|J|}|A_{J}|.
  28. I J ( - 1 ) | J | - | I | | A J | . \sum_{I\subseteq J}(-1)^{|J|-|I|}|A_{J}|.
  29. B k = A I { k } B_{k}=A_{I\cup\{k\}}
  30. N I N\setminus I
  31. B = A I B_{\emptyset}=A_{I}
  32. K N I ( - 1 ) | K | | B K | . \sum_{K\subseteq N\setminus I}(-1)^{|K|}|B_{K}|.
  33. ( Ω , , ) \scriptstyle(\Omega,\mathcal{F},\mathbb{P})
  34. ( A 1 A 2 ) = ( A 1 ) + ( A 2 ) - ( A 1 A 2 ) , \mathbb{P}(A_{1}\cup A_{2})=\mathbb{P}(A_{1})+\mathbb{P}(A_{2})-\mathbb{P}(A_{% 1}\cap A_{2}),
  35. ( A 1 A 2 A 3 ) = ( A 1 ) + ( A 2 ) + ( A 3 ) - ( A 1 A 2 ) - ( A 1 A 3 ) - ( A 2 A 3 ) + ( A 1 A 2 A 3 ) \mathbb{P}(A_{1}\cup A_{2}\cup A_{3})=\mathbb{P}(A_{1})+\mathbb{P}(A_{2})+% \mathbb{P}(A_{3})\qquad{}-\mathbb{P}(A_{1}\cap A_{2})-\mathbb{P}(A_{1}\cap A_{% 3})-\mathbb{P}(A_{2}\cap A_{3})\qquad{}+\mathbb{P}(A_{1}\cap A_{2}\cap A_{3})
  36. ( i = 1 n A i ) = i = 1 n ( A i ) - i < j ( A i A j ) + i < j < k ( A i A j A k ) - + ( - 1 ) n - 1 ( i = 1 n A i ) , \mathbb{P}\biggl(\bigcup_{i=1}^{n}A_{i}\biggr){}=\sum_{i=1}^{n}\mathbb{P}(A_{i% })-\sum_{i<j}\mathbb{P}(A_{i}\cap A_{j})\qquad+\sum_{i<j<k}\mathbb{P}(A_{i}% \cap A_{j}\cap A_{k})-\ \cdots\ +(-1)^{n-1}\,\mathbb{P}\biggl(\bigcap_{i=1}^{n% }A_{i}\biggr),
  37. ( i = 1 n A i ) = k = 1 n ( ( - 1 ) k - 1 I { 1 , , n } | I | = k ( A I ) ) , \mathbb{P}\biggl(\bigcup_{i=1}^{n}A_{i}\biggr)=\sum_{k=1}^{n}\left((-1)^{k-1}% \sum_{\scriptstyle I\subset\{1,\ldots,n\}\atop\scriptstyle|I|=k}\mathbb{P}(A_{% I})\right),
  38. A I := i I A i A_{I}:=\bigcap_{i\in I}A_{i}
  39. \mathbb{P}
  40. a k = ( A I ) for every I { 1 , , n } with | I | = k , a_{k}=\mathbb{P}(A_{I})\quad\,\text{for every}\quad I\subset\{1,\ldots,n\}% \quad\,\text{with}\quad|I|=k,
  41. ( i = 1 n A i ) = k = 1 n ( - 1 ) k - 1 ( n k ) a k \mathbb{P}\biggl(\bigcup_{i=1}^{n}A_{i}\biggr)=\sum_{k=1}^{n}(-1)^{k-1}{\left(% {{n}\atop{k}}\right)}a_{k}
  42. ( n k ) \scriptstyle{\left({{n}\atop{k}}\right)}
  43. g ( A ) = S A f ( S ) g(A)=\sum_{S\subseteq A}f(S)
  44. f ( A ) = S A ( - 1 ) | A | - | S | g ( S ) ( * * ) f(A)=\sum_{S\subseteq A}(-1)^{\left|A\right|-\left|S\right|}g(S)\qquad(**)
  45. m ¯ = { 1 , 2 , , m } \underline{m}=\{1,2,\ldots,m\}
  46. f ( m ¯ ) = 0 f(\underline{m})=0
  47. f ( S ) = | i m ¯ \ S A i \ i S A i | and f ( S ) = ( i m ¯ \ S A i \ i S A i ) f(S)=\bigg|\bigcap_{i\in\underline{m}\backslash S}A_{i}\bigg\backslash\bigcup_% {i\in S}A_{i}\bigg|\qquad\,\text{and}\qquad f(S)=\mathbb{P}\bigg(\bigcap_{i\in% \underline{m}\backslash S}A_{i}\bigg\backslash\bigcup_{i\in S}A_{i}\bigg)
  48. S S
  49. S m ¯ S\subsetneq\underline{m}
  50. g ( A ) = | i m ¯ \ A A i | , g ( m ¯ ) = | i m ¯ A i | and g ( A ) = ( i m ¯ \ A A i ) , g ( m ¯ ) = ( i m ¯ A i ) g(A)=\bigg|\bigcap_{i\in\underline{m}\backslash A}A_{i}\bigg|,~{}~{}g(% \underline{m})=\bigg|\bigcup_{i\in\underline{m}}A_{i}\bigg|\qquad\,\text{and}% \qquad g(A)=\mathbb{P}\bigg(\bigcap_{i\in\underline{m}\backslash A}A_{i}\bigg)% ,~{}~{}g(\underline{m})=\mathbb{P}\bigg(\bigcup_{i\in\underline{m}}A_{i}\bigg)
  51. A A
  52. A m ¯ A\subsetneq\underline{m}
  53. a a
  54. i m ¯ \ A A i \cap_{i\in\underline{m}\backslash A}A_{i}
  55. A i A_{i}
  56. A i A_{i}
  57. i A i\in A
  58. \ - \cap\backslash\!\!\cup\!\,\text{-}
  59. { A i i m ¯ \ A } \{A_{i}\mid i\in\underline{m}\backslash A\}
  60. A i A_{i}
  61. a a
  62. a a
  63. S S
  64. A A
  65. g ( A ) g(A)
  66. f ( m ¯ ) = 0 f(\underline{m})=0
  67. A = m ¯ A=\underline{m}
  68. m ¯ T ( - 1 ) | T | - 1 g ( m ¯ \ T ) = S m ¯ ( - 1 ) m - | S | - 1 g ( S ) = g ( m ¯ ) \sum_{\underline{m}\supseteq T\supsetneq\varnothing}(-1)^{\left|T\right|-1}g(% \underline{m}\backslash T)=\sum_{\varnothing\subseteq S\subsetneq\underline{m}% }(-1)^{m-\left|S\right|-1}g(S)=g(\underline{m})
  69. n n
  70. f ( A ) = S A μ ( A - S ) g ( S ) ( * * * ) f(A)=\sum_{S\subseteq A}\mu(A-S)g(S)\qquad(***)
  71. A - S A-S
  72. ( A - S ) S = A (A-S)\uplus S=A
  73. μ \mu
  74. ( A - S ) (A-S)
  75. ( - 1 ) | A | - | S | (-1)^{\left|A\right|-\left|S\right|}
  76. A - S A-S
  77. g ( S ) = T S f ( T ) g(S)=\sum_{T\subseteq S}f(T)
  78. f ( A ) f(A)
  79. T T
  80. T A T\subsetneq A
  81. f ( T ) f(T)
  82. T T
  83. T A T\subsetneq A
  84. a A a\in A
  85. a T a\not\in T
  86. A - S A-S
  87. f ( T ) f(T)
  88. S S
  89. T S A T\subseteq S\subseteq A
  90. f ( T ) f(T)
  91. S S
  92. A - S A-S
  93. a a
  94. S S
  95. A - S A-S
  96. a a
  97. A ¯ k \scriptstyle\overline{A}_{k}
  98. A k A \scriptstyle A_{k}\,\subseteq\,A
  99. i = 1 n A i = i = 1 n A ¯ i ¯ \bigcap_{i=1}^{n}A_{i}=\overline{\bigcup_{i=1}^{n}\overline{A}_{i}}
  100. j = 0 n ( n j ) ( - 1 ) j ( n - j ) k . \sum_{j=0}^{n}{\left({{n}\atop{j}}\right)}(-1)^{j}(n-j)^{k}.
  101. S ( n , k ) = 1 k ! t = 0 k ( - 1 ) t ( k t ) ( k - t ) n . S(n,k)=\frac{1}{k!}\sum_{t=0}^{k}(-1)^{t}{\left({{k}\atop{t}}\right)}(k-t)^{n}.
  102. ( m ) n = m ( m - 1 ) ( m - 2 ) ( m - n + 1 ) . (m)_{n}=m(m-1)(m-2)\cdots(m-n+1).
  103. r n ( B ) = t = 0 n ( - 1 ) t ( m - t ) n - t r t ( B ) . r_{n}(B)=\sum_{t=0}^{n}(-1)^{t}(m-t)_{n-t}\;r_{t}(B^{\prime}).
  104. n = p 1 a 1 p 2 a 2 p r a r . n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{r}^{a_{r}}.
  105. ϕ ( n ) = n - i = 1 r n p i + 1 i < j r n p i p j - = n i = 1 r ( 1 - 1 p i ) . \phi(n)=n-\sum_{i=1}^{r}\frac{n}{p_{i}}+\sum_{1\leq i<j\leq r}\frac{n}{p_{i}p_% {j}}-\cdots=n\prod_{i=1}^{r}\left(1-\frac{1}{p_{i}}\right).
  106. 1 A 1 A n j = 1 k ( - 1 ) j - 1 1 i 1 < < i j n p i 1 p i j 1 A i 1 A i j . 1_{A_{1}\cup\cdots\cup A_{n}}\geq\sum_{j=1}^{k}(-1)^{j-1}\sum_{1\leq i_{1}<% \cdots<i_{j}\leq n}p_{i_{1}}\dots p_{i_{j}}\,1_{A_{i_{1}}\cap\cdots\cap A_{i_{% j}}}.
  107. A 1 , A 2 , , A t A_{1},A_{2},\dots,A_{t}
  108. A 1 , A 2 , , A t A_{1},A_{2},\dots,A_{t}
  109. | { A i 1 i t } | - | { A i A j 1 i < j t } | + + ( - 1 ) t + 1 | { A 1 A 2 A t } | = ( t 1 ) - ( t 2 ) + + ( - 1 ) t + 1 ( t t ) . |\{A_{i}\mid 1\leq i\leq t\}|-|\{A_{i}\cap A_{j}\mid 1\leq i<j\leq t\}|+\cdots% +(-1)^{t+1}|\{A_{1}\cap A_{2}\cap\cdots\cap A_{t}\}|={\left({{t}\atop{1}}% \right)}-{\left({{t}\atop{2}}\right)}+\cdots+(-1)^{t+1}{\left({{t}\atop{t}}% \right)}.
  110. 0 = ( 1 - 1 ) t = ( t 0 ) - ( t 1 ) + ( t 2 ) - + ( - 1 ) t ( t t ) 0=(1-1)^{t}={\left({{t}\atop{0}}\right)}-{\left({{t}\atop{1}}\right)}+{\left({% {t}\atop{2}}\right)}-\cdots+(-1)^{t}{\left({{t}\atop{t}}\right)}
  111. ( t 0 ) = 1 {\left({{t}\atop{0}}\right)}=1
  112. 1 = ( t 1 ) - ( t 2 ) + + ( - 1 ) t + 1 ( t t ) , 1={\left({{t}\atop{1}}\right)}-{\left({{t}\atop{2}}\right)}+\cdots+(-1)^{t+1}{% \left({{t}\atop{t}}\right)},
  113. 𝟏 S : X { 0 , 1 } \mathbf{1}_{S}\colon X\to\{0,1\}\,
  114. 𝟏 S ( x ) := { 1 if x S , 0 if x S . \mathbf{1}_{S}(x):=\begin{cases}1&\,\text{if }x\in S,\\ 0&\,\text{if }x\notin S.\end{cases}
  115. A A
  116. B B
  117. X X
  118. 𝟏 A 𝟏 B = 𝟏 A B . \mathbf{1}_{A}\cdot\mathbf{1}_{B}=\mathbf{1}_{A\cap B}.
  119. i = 1 n A i \cup_{i=1}^{n}A_{i}
  120. A I = i I A i . A_{I}=\bigcap_{i\in I}A_{i}.
  121. ( 1 A - 1 A 1 ) ( 1 A - 1 A 2 ) ( 1 A - 1 A n ) = 0 , (1_{A}-1_{A_{1}})(1_{A}-1_{A_{2}})\cdots(1_{A}-1_{A_{n}})\,=\,0,

Income_inequality_metrics.html

  1. 4.97 5 4.97\approx 5

Income–consumption_curve.html

  1. Δ X n 1 \Delta X_{n}^{1}
  2. m m^{\prime}
  3. m m
  4. p 1 p_{1}
  5. Δ X n 1 = X 1 ( p 1 , m ) - X 1 ( p 1 , m ) . \Delta X_{n}^{1}=X^{1}(p_{1},m)-X^{1}(p_{1},m^{\prime}).
  6. u ( X 1 , X 2 ) = X 1 a X 2 ( 1 - a ) u(X_{1},X_{2})=X_{1}^{a}X_{2}^{(1-a)}\,
  7. X 1 = a m / p 1 . X_{1}=am/p_{1}.
  8. X 2 = ( 1 - a ) m / p 2 X_{2}=(1-a)m/p_{2}\,
  9. X 1 = X 1 ( p 1 , p 2 , m ) X^{1}=X^{1}(p_{1},p_{2},m)\,

Incomplete_Fermi–Dirac_integral.html

  1. F j ( x , b ) = 1 Γ ( j + 1 ) b t j exp ( t - x ) + 1 d t . F_{j}(x,b)=\frac{1}{\Gamma(j+1)}\int_{b}^{\infty}\frac{t^{j}}{\exp(t-x)+1}\,dt.

Incomplete_gamma_function.html

  1. Γ ( s , x ) = x t s - 1 e - t d t , \Gamma(s,x)=\int_{x}^{\infty}t^{s-1}\,e^{-t}\,{\rm d}t,\,\!
  2. γ ( s , x ) = 0 x t s - 1 e - t d t . \gamma(s,x)=\int_{0}^{x}t^{s-1}\,e^{-t}\,{\rm d}t.\,\!
  3. Γ ( s , x ) = ( s - 1 ) Γ ( s - 1 , x ) + x s - 1 e - x \Gamma(s,x)=(s-1)\Gamma(s-1,x)+x^{s-1}e^{-x}
  4. γ ( s , x ) = ( s - 1 ) γ ( s - 1 , x ) - x s - 1 e - x \gamma(s,x)=(s-1)\gamma(s-1,x)-x^{s-1}e^{-x}
  5. Γ ( s ) = 0 t s - 1 e - t d t \Gamma(s)=\int_{0}^{\infty}t^{s-1}\,e^{-t}\,{\rm d}t
  6. Γ ( s ) = Γ ( s , 0 ) \Gamma(s)=\Gamma(s,0)
  7. γ ( s , x ) + Γ ( s , x ) = Γ ( s ) . \gamma(s,x)+\Gamma(s,x)=\Gamma(s).
  8. γ ( s , x ) = k = 0 x s e - x x k s ( s + 1 ) ( s + k ) = x s Γ ( s ) e - x k = 0 x k Γ ( s + k + 1 ) \gamma(s,x)=\sum_{k=0}^{\infty}\frac{x^{s}e^{-x}x^{k}}{s(s+1)...(s+k)}=x^{s}\,% \Gamma(s)\,e^{-x}\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(s+k+1)}
  9. γ * \gamma^{*}
  10. γ * ( s , z ) := e - z k = 0 z k Γ ( s + k + 1 ) \gamma^{*}(s,z):=e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(s+k+1)}
  11. γ ( s , z ) = z s Γ ( s ) γ * ( s , z ) \gamma(s,z)=z^{s}\,\Gamma(s)\,\gamma^{*}(s,z)
  12. e s * 2 k π i e^{s*2k\pi i}
  13. γ ( s , z ) z s Γ ( s ) γ * ( s , 0 ) = z s Γ ( s ) / Γ ( s + 1 ) = z s / s \gamma(s,z)\asymp z^{s}\,\Gamma(s)\,\gamma^{*}(s,0)=z^{s}\,\Gamma(s)/\Gamma(s+% 1)=z^{s}/s
  14. u v t s - 1 e - t d t = γ ( s , v ) - γ ( s , u ) \int_{u}^{v}t^{s-1}\,e^{-t}\,{\rm d}t=\gamma(s,v)-\gamma(s,u)
  15. γ ( s , z ) = 0 z t s - 1 e - t d t , ( s ) > 0. \gamma(s,z)=\int_{0}^{z}t^{s-1}\,e^{-t}\,{\rm d}t,\,\Re(s)>0.
  16. Γ ( s ) = 0 t s - 1 e - t d t = lim x γ ( s , x ) \Gamma(s)=\int_{0}^{\infty}t^{s-1}\,e^{-t}\,{\rm d}t=\lim_{x\rightarrow\infty}% \gamma(s,x)
  17. 1 R e ( s ) 2 1≤Re(s)≤2
  18. γ * ( s , x ) = k = 0 ( - x ) k k ! Γ ( s ) ( s + k ) . \gamma^{*}(s,x)=\sum_{k=0}^{\infty}\frac{(-x)^{k}}{k!\,\Gamma(s)(s+k)}.
  19. γ * \gamma^{*}
  20. γ ( s , z ) - 1 s = - 1 s + z s k = 0 ( - z ) k k ! ( s + k ) = z s - 1 s + z s k = 1 ( - z ) k k ! ( s + k ) , ( s ) > - 1 , s 0 \gamma(s,z)-\frac{1}{s}=-\frac{1}{s}+z^{s}\,\sum_{k=0}^{\infty}\frac{(-z)^{k}}% {k!(s+k)}=\frac{z^{s}-1}{s}+z^{s}\,\sum_{k=1}^{\infty}\frac{(-z)^{k}}{k!(s+k)}% ,\quad\Re(s)>-1,\,s\neq 0
  21. z s - 1 s ln ( z ) , Γ ( s ) - 1 s = 1 s - γ + O ( s ) - 1 s - γ \frac{z^{s}-1}{s}\rightarrow\ln(z),\quad\Gamma(s)-\frac{1}{s}=\frac{1}{s}-% \gamma+O(s)-\frac{1}{s}\rightarrow-\gamma
  22. γ \gamma
  23. Γ ( 0 , z ) = lim s 0 ( Γ ( s ) - 1 s - ( γ ( s , z ) - 1 s ) ) = - γ - ln ( z ) - k = 1 ( - z ) k k ( k ! ) \Gamma(0,z)=\lim_{s\rightarrow 0}\left(\Gamma(s)-\tfrac{1}{s}-(\gamma(s,z)-% \tfrac{1}{s})\right)=-\gamma-\ln(z)-\sum_{k=1}^{\infty}\frac{(-z)^{k}}{k\,(k!)}
  24. E 1 ( z ) E_{1}(z)
  25. Γ ( - n , z ) \Gamma(-n,z)
  26. Γ ( s , z ) \Gamma(s,z)
  27. Γ ( s ) \Gamma(s)
  28. ( s i , z i ) ( s , 0 ) (s_{i},z_{i})\rightarrow(s,0)
  29. Γ ( s ) = ( s - 1 ) ! \Gamma(s)=(s-1)!
  30. Γ ( s , x ) = ( s - 1 ) ! e - x k = 0 s - 1 x k k ! \Gamma(s,x)=(s-1)!\,e^{-x}\sum_{k=0}^{s-1}\frac{x^{k}}{k!}
  31. Γ ( s , 0 ) = Γ ( s ) , ( s ) > 0 \Gamma(s,0)=\Gamma(s),\Re(s)>0
  32. Γ ( 1 , x ) = e - x , \Gamma(1,x)=e^{-x},
  33. γ ( 1 , x ) = 1 - e - x , \gamma(1,x)=1-e^{-x},
  34. Γ ( 0 , x ) = - Ei ( - x ) \Gamma(0,x)=-{\rm Ei}(-x)
  35. x > 0 , x>0,
  36. Γ ( s , x ) = x s E 1 - s ( x ) , \Gamma(s,x)=x^{s}\,{\rm E}_{1-s}(x),
  37. Γ ( 1 2 , x ) = π erfc ( x ) , \Gamma\left(\tfrac{1}{2},x\right)=\sqrt{\pi}\,{\rm erfc}\left(\sqrt{x}\right),
  38. γ ( 1 2 , x ) = π erf ( x ) . \gamma\left(\tfrac{1}{2},x\right)=\sqrt{\pi}\,{\rm erf}\left(\sqrt{x}\right).
  39. Ei \mathrm{Ei}
  40. E n \mathrm{E_{n}}
  41. erf \mathrm{erf}
  42. erfc \mathrm{erfc}
  43. erfc ( x ) = 1 - erf ( x ) \operatorname{erfc}(x)=1-\operatorname{erf}(x)
  44. γ ( s , x ) x s 1 s \frac{\gamma(s,x)}{x^{s}}\rightarrow\frac{1}{s}
  45. x 0 , x\rightarrow 0,
  46. Γ ( s , x ) x s - 1 s \frac{\Gamma(s,x)}{x^{s}}\rightarrow-\frac{1}{s}
  47. x 0 x\rightarrow 0
  48. ( s ) < 0 \Re(s)<0\,
  49. γ ( s , x ) Γ ( s ) \gamma(s,x)\rightarrow\Gamma(s)
  50. x , x\rightarrow\infty,
  51. Γ ( s , x ) x s - 1 e - x 1 \frac{\Gamma(s,x)}{x^{s-1}e^{-x}}\rightarrow 1
  52. x , x\rightarrow\infty,
  53. Γ ( s , z ) z s - 1 e - z k = 0 Γ ( s ) Γ ( s - k ) z - k \Gamma(s,z)\sim z^{s-1}e^{-z}\,\sum_{k=0}\frac{\Gamma(s)}{\Gamma(s-k)}z^{-k}
  54. | z | |z|\to\infty
  55. | arg z | < 3 2 π |\!\arg z|<\tfrac{3}{2}\pi
  56. γ ( s , z ) = k = 0 z s e - z z k s ( s + 1 ) ( s + k ) \gamma(s,z)=\sum_{k=0}^{\infty}\frac{z^{s}e^{-z}z^{k}}{s(s+1)...(s+k)}
  57. γ ( s , z ) = k = 0 ( - 1 ) k k ! z s + k s + k = z s s M ( s , s + 1 , - z ) , \gamma(s,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}\frac{z^{s+k}}{s+k}=\frac{z^% {s}}{s}M(s,s+1,-z),
  58. γ ( s , z ) = s - 1 z s e - z M ( 1 , s + 1 , z ) \gamma(s,z)=\frac{}{}s^{-1}z^{s}e^{-z}M(1,s+1,z)
  59. M ( 1 , s + 1 , z ) = 1 + z ( s + 1 ) + z 2 ( s + 1 ) ( s + 2 ) + z 3 ( s + 1 ) ( s + 2 ) ( s + 3 ) + M(1,s+1,z)=1+\frac{z}{(s+1)}+\frac{z^{2}}{(s+1)(s+2)}+\frac{z^{3}}{(s+1)(s+2)(% s+3)}+\cdots
  60. Γ ( s , z ) = e - z U ( 1 - s , 1 - s , z ) = z s e - z Γ ( 1 - s ) 0 e - u u s ( z + u ) d u = = e - z z s U ( 1 , 1 + s , z ) = e - z 0 e - u ( z + u ) s - 1 d u = e - z z s 0 e - z u ( 1 + u ) s - 1 d u . \begin{aligned}\displaystyle\Gamma(s,z)&\displaystyle=e^{-z}U(1-s,1-s,z)=\frac% {z^{s}e^{-z}}{\Gamma(1-s)}\int_{0}^{\infty}\frac{e^{-u}}{u^{s}(z+u)}{\rm d}u=% \\ &\displaystyle=e^{-z}z^{s}U(1,1+s,z)=e^{-z}\int_{0}^{\infty}e^{-u}(z+u)^{s-1}{% \rm d}u=e^{-z}z^{s}\int_{0}^{\infty}e^{-zu}(1+u)^{s-1}{\rm d}u.\end{aligned}
  61. γ ( s , z ) = z s e - z s - s z s + 1 + z s + 2 - ( s + 1 ) z s + 3 + 2 z s + 4 - ( s + 2 ) z s + 5 + 3 z s + 6 - . \gamma(s,z)=\cfrac{z^{s}e^{-z}}{s-\cfrac{sz}{s+1+\cfrac{z}{s+2-\cfrac{(s+1)z}{% s+3+\cfrac{2z}{s+4-\cfrac{(s+2)z}{s+5+\cfrac{3z}{s+6-\ddots}}}}}}}.
  62. Γ ( s , z ) = z s e - z z + 1 - s 1 + 1 z + 2 - s 1 + 2 z + 3 - s 1 + \Gamma(s,z)=\cfrac{z^{s}e^{-z}}{z+\cfrac{1-s}{1+\cfrac{1}{z+\cfrac{2-s}{1+% \cfrac{2}{z+\cfrac{3-s}{1+\ddots}}}}}}
  63. Γ ( s , z ) = z s e - z 1 + z - s + s - 1 3 + z - s + 2 ( s - 2 ) 5 + z - s + 3 ( s - 3 ) 7 + z - s + 4 ( s - 4 ) 9 + z - s + \Gamma(s,z)=\cfrac{z^{s}e^{-z}}{1+z-s+\cfrac{s-1}{3+z-s+\cfrac{2(s-2)}{5+z-s+% \cfrac{3(s-3)}{7+z-s+\cfrac{4(s-4)}{9+z-s+\ddots}}}}}
  64. Γ ( s , z ) \displaystyle\Gamma(s,z)
  65. γ ( s , x ) \gamma(s,x)
  66. Γ ( s , x ) \Gamma(s,x)
  67. P ( s , x ) = γ ( s , x ) Γ ( s ) , P(s,x)=\frac{\gamma(s,x)}{\Gamma(s)},
  68. Q ( s , x ) = Γ ( s , x ) Γ ( s ) = 1 - P ( s , x ) . Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}=1-P(s,x).
  69. P ( s , x ) P(s,x)
  70. s s
  71. s > 0 s>0
  72. Q ( s , λ ) Q(s,\lambda)
  73. X X
  74. Poi ( λ ) {\rm Poi}(\lambda)
  75. P r ( X < s ) = i < s e - λ λ i i ! = Γ ( s , λ ) Γ ( s ) = Q ( s , λ ) . Pr(X<s)=\sum_{i<s}e^{-\lambda}\frac{\lambda^{i}}{i!}=\frac{\Gamma(s,\lambda)}{% \Gamma(s)}=Q(s,\lambda).
  76. Γ ( s , x ) \Gamma(s,x)
  77. Γ ( s , x ) x = - x s - 1 e - x \frac{\partial\Gamma(s,x)}{\partial x}=-x^{s-1}e^{-x}
  78. s s
  79. Γ ( s , x ) s = ln x Γ ( s , x ) + x T ( 3 , s , x ) \frac{\partial\Gamma(s,x)}{\partial s}=\ln x\Gamma(s,x)+x\,T(3,s,x)
  80. 2 Γ ( s , x ) s 2 = ln 2 x Γ ( s , x ) + 2 x [ ln x T ( 3 , s , x ) + T ( 4 , s , x ) ] \frac{\partial^{2}\Gamma(s,x)}{\partial s^{2}}=\ln^{2}x\Gamma(s,x)+2x[\ln x\,T% (3,s,x)+T(4,s,x)]
  81. T ( m , s , x ) T(m,s,x)
  82. T ( m , s , x ) = G m - 1 , m m , 0 ( 0 , 0 , , 0 s - 1 , - 1 , , - 1 | x ) . T(m,s,x)=G_{m-1,\,m}^{\,m,\,0}\!\left(\left.\begin{matrix}0,0,\dots,0\\ s-1,-1,\dots,-1\end{matrix}\;\right|\,x\right).
  83. m Γ ( s , x ) s m = ln m x Γ ( s , x ) + m x n = 0 m - 1 P n m - 1 ln m - n - 1 x T ( 3 + n , s , x ) \frac{\partial^{m}\Gamma(s,x)}{\partial s^{m}}=\ln^{m}x\Gamma(s,x)+mx\,\sum_{n% =0}^{m-1}P_{n}^{m-1}\ln^{m-n-1}x\,T(3+n,s,x)
  84. P j n P_{j}^{n}
  85. P j n = ( n j ) j ! = n ! ( n - j ) ! . P_{j}^{n}=\left(\begin{array}[]{l}n\\ j\end{array}\right)j!=\frac{n!}{(n-j)!}.
  86. T ( m , s , x ) s = ln x T ( m , s , x ) + ( m - 1 ) T ( m + 1 , s , x ) \frac{\partial T(m,s,x)}{\partial s}=\ln x~{}T(m,s,x)+(m-1)T(m+1,s,x)
  87. T ( m , s , x ) x = - 1 x [ T ( m - 1 , s , x ) + T ( m , s , x ) ] \frac{\partial T(m,s,x)}{\partial x}=-\frac{1}{x}[T(m-1,s,x)+T(m,s,x)]
  88. T ( m , s , x ) T(m,s,x)
  89. | z | < 1 |z|<1
  90. T ( m , s , z ) = - ( - 1 ) m - 1 ( m - 2 ) ! d m - 2 d t m - 2 [ Γ ( s - t ) z t - 1 ] | t = 0 + n = 0 ( - 1 ) n z s - 1 + n n ! ( - s - n ) m - 1 T(m,s,z)=-\frac{(-1)^{m-1}}{(m-2)!}\frac{{\rm d}^{m-2}}{{\rm d}t^{m-2}}\left[% \Gamma(s-t)z^{t-1}\right]\Big|_{t=0}+\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{s-1+n% }}{n!(-s-n)^{m-1}}
  91. | z | 1 |z|\geq 1
  92. T ( 2 , s , x ) = Γ ( s , x ) / x T(2,s,x)=\Gamma(s,x)/x
  93. x T ( 3 , 1 , x ) = E 1 ( x ) x\,T(3,1,x)={\rm E}_{1}(x)
  94. E 1 ( x ) {\rm E}_{1}(x)
  95. T ( m , s , x ) T(m,s,x)
  96. x t s - 1 ln m t e t d t = m s m x t s - 1 e t d t = m s m Γ ( s , x ) \int_{x}^{\infty}\frac{t^{s-1}\ln^{m}t}{e^{t}}{\rm d}t=\frac{\partial^{m}}{% \partial s^{m}}\int_{x}^{\infty}\frac{t^{s-1}}{e^{t}}{\rm d}t=\frac{\partial^{% m}}{\partial s^{m}}\Gamma(s,x)
  97. x b - 1 γ ( s , x ) d x = 1 b ( x b γ ( s , x ) + Γ ( s + b , x ) ) . \int x^{b-1}\gamma(s,x)\mathrm{d}x=\frac{1}{b}\left(x^{b}\gamma(s,x)+\Gamma(s+% b,x)\right).
  98. x b - 1 Γ ( s , x ) d x = 1 b ( x b Γ ( s , x ) - Γ ( s + b , x ) ) , \int x^{b-1}\Gamma(s,x)\mathrm{d}x=\frac{1}{b}\left(x^{b}\Gamma(s,x)-\Gamma(s+% b,x)\right),
  99. - γ ( s 2 , z 2 π ) ( z 2 π ) s 2 e - 2 π i k z d z = Γ ( 1 - s 2 , k 2 π ) ( k 2 π ) 1 - s 2 . \int_{-\infty}^{\infty}\frac{\gamma\left(\frac{s}{2},z^{2}\pi\right)}{(z^{2}% \pi)^{\frac{s}{2}}}e^{-2\pi ikz}\mathrm{d}z=\frac{\Gamma\left(\frac{1-s}{2},k^% {2}\pi\right)}{(k^{2}\pi)^{\frac{1-s}{2}}}.
  100. P ( a , x ) P(a,x)
  101. Q ( a , x ) Q(a,x)
  102. γ ( a , x ) \gamma(a,x)
  103. Γ ( a , x ) \Gamma(a,x)

Incompressible_flow.html

  1. ρ \rho
  2. ρ \rho
  3. m = V ρ d V . {m}={\iiint\limits_{V}\!\rho\,\mathrm{d}V}.
  4. m t = - S ( 𝐉 d 𝐒 ) . {\partial m\over\partial t}={-\iint\limits_{S}\!\!\!\!\!\!\!\!\!\!\!\subset\!% \supset(\mathbf{J}\cdot\mathrm{d}\mathbf{S})}.
  5. V ρ t d V = - V ( 𝐉 ) d V , {\iiint\limits_{V}{\partial\rho\over\partial t}\,\mathrm{d}V}={-\iiint\limits_% {V}\left(\nabla\cdot\mathbf{J}\right)\,\mathrm{d}V},
  6. ρ t = - 𝐉 . {\partial\rho\over\partial t}=-\nabla\cdot\mathbf{J}.
  7. 𝐉 = ρ 𝐮 . {\mathbf{J}}={\rho\mathbf{u}}.
  8. ρ t + ( ρ 𝐮 ) = ρ t + ρ 𝐮 + ρ ( 𝐮 ) = 0. {\partial\rho\over\partial t}+{\nabla\cdot\left(\rho\mathbf{u}\right)}={% \partial\rho\over\partial t}+{\nabla\rho\cdot\mathbf{u}}+{\rho\left(\nabla% \cdot\mathbf{u}\right)}=0.
  9. d ρ d t = ρ t + ρ x d x d t + ρ y d y d t + ρ z d z d t . {\mathrm{d}\rho\over\mathrm{d}t}={\partial\rho\over\partial t}+{\partial\rho% \over\partial x}{\mathrm{d}x\over\mathrm{d}t}+{\partial\rho\over\partial y}{% \mathrm{d}y\over\mathrm{d}t}+{\partial\rho\over\partial z}{\mathrm{d}z\over% \mathrm{d}t}.
  10. D ρ D t = ρ t + ρ 𝐮 . {D\rho\over Dt}={\partial\rho\over\partial t}+{\nabla\rho\cdot\mathbf{u}}.
  11. D ρ D t = - ρ ( 𝐮 ) . {D\rho\over Dt}={-\rho\left(\nabla\cdot\mathbf{u}\right)}.
  12. 𝐮 = 0. {\nabla\cdot\mathbf{u}}=0.
  13. β = 1 ρ d ρ d p . \beta={\frac{1}{\rho}}{\frac{\mathrm{d}\rho}{\mathrm{d}p}}.
  14. 𝐮 = 0. \nabla\cdot\mathbf{u}=0.\,
  15. D ρ D t = ρ t + 𝐮 ρ = 0 \frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+\mathbf{u}\cdot\nabla\rho=0
  16. ρ t \tfrac{\partial\rho}{\partial t}
  17. 𝐮 ρ \mathbf{u}\cdot\nabla\rho
  18. ρ = constant \rho=\,\text{constant}
  19. ρ t = 0 \frac{\partial\rho}{\partial t}=0
  20. ρ = 0 \nabla\rho=0
  21. D ρ D t = ρ t + 𝐮 ρ = 0 𝐮 = 0 \frac{D\rho}{Dt}=\frac{\partial\rho}{\partial t}+\mathbf{u}\cdot\nabla\rho=0% \Rightarrow\nabla\cdot\mathbf{u}=0
  22. ν = μ ρ \nu=\tfrac{\mu}{\rho}
  23. 𝐮 = 0 {\nabla\cdot\mathbf{u}=0}
  24. ( ρ o 𝐮 ) = 0 {\nabla\cdot\left(\rho_{o}\mathbf{u}\right)=0}
  25. ( α 𝐮 ) = β \nabla\cdot\left(\alpha\mathbf{u}\right)=\beta
  26. ( α 𝐮 ) = β \nabla\cdot\left(\alpha\mathbf{u}\right)=\beta
  27. α \alpha
  28. β \beta

Independent_component_analysis.html

  1. x = ( x 1 , , x m ) T x=(x_{1},\ldots,x_{m})^{T}
  2. s = ( s 1 , , s n ) T . s=(s_{1},\ldots,s_{n})^{T}.
  3. x , x,
  4. s = W x s=Wx\,
  5. s s
  6. F ( s 1 , , s n ) F(s_{1},\ldots,s_{n})
  7. x i x_{i}
  8. x = ( x 1 , , x m ) T x=(x_{1},\ldots,x_{m})^{T}
  9. s k s_{k}
  10. k = 1 , , n k=1,\ldots,n
  11. x i = a i , 1 s 1 + + a i , k s k + + a i , n s n x_{i}=a_{i,1}s_{1}+\cdots+a_{i,k}s_{k}+\cdots+a_{i,n}s_{n}
  12. a i , k a_{i,k}
  13. x = k = 1 n s k a k x=\sum_{k=1}^{n}s_{k}a_{k}
  14. x x
  15. a k = ( a 1 , k , , a m , k ) T a_{k}=(a_{1,k},\ldots,a_{m,k})^{T}
  16. a k a_{k}
  17. A = ( a 1 , , a n ) A=(a_{1},\ldots,a_{n})
  18. x = A s x=As
  19. s = ( s 1 , , s n ) T s=(s_{1},\ldots,s_{n})^{T}
  20. x 1 , , x N x_{1},\ldots,x_{N}
  21. x x
  22. A A
  23. s s
  24. w w
  25. s k = ( w T * x ) s_{k}=(w^{T}*x)
  26. s s
  27. x x
  28. W = A - 1 W=A^{-1}
  29. n = m n=m
  30. n > m n>m
  31. n N ( 0 , diag ( Σ ) ) n\sim N(0,\operatorname{diag}(\Sigma))
  32. x = A s + n x=As+n
  33. f ( | θ ) f(\cdot|\theta)
  34. θ \theta
  35. x = f ( s | θ ) + n x=f(s|\theta)+n
  36. s k s_{k}
  37. m m
  38. n n
  39. m n m\geq n
  40. A A
  41. x 1 , x 2 , , x m {x_{1},x_{2},\ldots,x_{m}}
  42. m m
  43. y 1 , y 2 , , y n {y_{1},y_{2},\ldots,y_{n}}
  44. n n
  45. G G
  46. g i j = 1 g_{ij}=1
  47. i i
  48. y i = 1 y_{i}=1
  49. j j
  50. g i j = 1 g_{ij}=1
  51. j j
  52. x j = 1 x_{j}=1
  53. x i = j = 1 n ( g i j y j ) , i = 1 , 2 , , m , x_{i}=\bigvee_{j=1}^{n}(g_{ij}\wedge y_{j}),i=1,2,\ldots,m,
  54. \wedge
  55. \vee
  56. G G
  57. G G
  58. X X
  59. X 0 X^{0}
  60. X X
  61. x i j = 0 , j x_{ij}=0,\forall j
  62. i i
  63. K = E [ ( 𝐲 - 𝐲 ¯ ) 4 ] ( E [ ( 𝐲 - 𝐲 ¯ ) 2 ] ) 2 - 3 K=\frac{\operatorname{E}[(\mathbf{y}-\mathbf{\overline{y}})^{4}]}{(% \operatorname{E}[(\mathbf{y}-\mathbf{\overline{y}})^{2}])^{2}}-3
  64. 𝐲 ¯ \mathbf{\overline{y}}
  65. 𝐲 \mathbf{y}
  66. 𝐲 \mathbf{y}
  67. 𝐲 = 𝐰 T 𝐱 \mathbf{y}=\mathbf{w}^{T}\mathbf{x}
  68. 𝐱 = ( x 1 , x 2 , , x M ) T \mathbf{x}=(x_{1},x_{2},\ldots,x_{M})^{T}
  69. 𝐰 \mathbf{w}
  70. 𝐬 \mathbf{s}
  71. 𝐲 \mathbf{y}
  72. 𝐲 = 𝐬 \mathbf{y}=\mathbf{s}
  73. 𝐲 \mathbf{y}
  74. 𝐰 \mathbf{w}
  75. S 1 S_{1}
  76. S 2 S_{2}
  77. S 1 S_{1}
  78. S 2 S_{2}
  79. 𝐰 \mathbf{w}
  80. 𝐱 \mathbf{x}
  81. 𝐳 \mathbf{z}
  82. 𝐳 = ( z 1 , z 2 , , z M ) T \mathbf{z}=(z_{1},z_{2},\ldots,z_{M})^{T}
  83. 𝐱 \mathbf{x}
  84. 𝐱 = 𝐔𝐃𝐕 T \mathbf{x}=\mathbf{U}\mathbf{D}\mathbf{V}^{T}
  85. U i = U i / E ( U i 2 ) U_{i}=U_{i}/\operatorname{E}(U_{i}^{2})
  86. 𝐳 = 𝐔 \mathbf{z}=\mathbf{U}
  87. 𝐰 \mathbf{w}
  88. 𝐲 = 𝐰 T 𝐳 \mathbf{y}=\mathbf{w}^{T}\mathbf{z}
  89. E [ ( 𝐰 T 𝐳 ) 2 ] = 1 \operatorname{E}[(\mathbf{w}^{T}\mathbf{z})^{2}]=1
  90. K = E [ 𝐲 4 ] ( E [ 𝐲 2 ] ) 2 - 3 = E [ ( 𝐰 T 𝐳 ) 4 ] - 3. K=\frac{\operatorname{E}[\mathbf{y}^{4}]}{(\operatorname{E}[\mathbf{y}^{2}])^{% 2}}-3=\operatorname{E}[(\mathbf{w}^{T}\mathbf{z})^{4}]-3.
  91. 𝐰 \mathbf{w}
  92. 𝐰 n e w = 𝐰 o l d - η E [ 𝐳 ( 𝐰 o l d T 𝐳 ) 3 ] . \mathbf{w}_{new}=\mathbf{w}_{old}-\eta\operatorname{E}[\mathbf{z}(\mathbf{w}_{% old}^{T}\mathbf{z})^{3}].
  93. η \eta
  94. 𝐰 \mathbf{w}
  95. 𝐰 n e w = 𝐰 n e w | 𝐰 n e w | \mathbf{w}_{new}=\frac{\mathbf{w}_{new}}{|\mathbf{w}_{new}|}
  96. 𝐰 o l d = 𝐰 n e w \mathbf{w}_{old}=\mathbf{w}_{new}
  97. 𝐰 \mathbf{w}
  98. J ( x ) = S ( y ) - S ( x ) J(x)=S(y)-S(x)\,
  99. S ( x ) = - p x ( u ) log p x ( u ) d u S(x)=-\int p_{x}(u)\log p_{x}(u)du
  100. J ( x ) = 1 12 ( E ( x 3 ) ) 2 + 1 48 ( k u r t ( x ) ) 2 J(x)=\frac{1}{12}(E(x^{3}))^{2}+\frac{1}{48}(kurt(x))^{2}
  101. J ( y ) = k 1 ( E ( G 1 ( y ) ) ) 2 + k 2 ( E ( G 2 ( y ) ) - E ( G 2 ( v ) ) 2 J(y)=k_{1}(E(G_{1}(y)))^{2}+k_{2}(E(G_{2}(y))-E(G_{2}(v))^{2}
  102. G 1 G_{1}
  103. G 2 G_{2}
  104. G 1 = 1 a 1 log ( cosh ( a 1 u ) ) G_{1}=\frac{1}{a_{1}}\log(\cosh(a_{1}u))
  105. G 2 = - exp ( - u 2 2 ) G_{2}=-\exp(-\frac{u^{2}}{2})
  106. 𝐱 \mathbf{x}
  107. g g
  108. 𝐖 \mathbf{W}
  109. 𝐘 = g ( 𝐲 ) \mathbf{Y}=g(\mathbf{y})
  110. 𝐲 = 𝐖𝐱 \mathbf{y}=\mathbf{Wx}
  111. 𝐖 \mathbf{W}
  112. 𝐖 \mathbf{W}
  113. 𝐘 \mathbf{Y}
  114. 𝐲 = g - 1 ( 𝐘 ) \mathbf{y}=g^{-1}(\mathbf{Y})
  115. g g
  116. p s p_{s}
  117. p 𝐲 p_{\mathbf{y}}
  118. Y Y
  119. 𝐱 \mathbf{x}
  120. 𝐘 \mathbf{Y}
  121. 𝐘 = g ( 𝐲 ) \mathbf{Y}=g(\mathbf{y})
  122. 𝐲 = 𝐖𝐱 \mathbf{y}=\mathbf{Wx}
  123. 𝐖 \mathbf{W}
  124. p 𝐲 p_{\mathbf{y}}
  125. 𝐘 \mathbf{Y}
  126. H ( 𝐘 ) = - 1 N t = 1 N ln p 𝐘 ( 𝐘 t ) H(\mathbf{Y})=-\frac{1}{N}\sum_{t=1}^{N}\ln p_{\mathbf{Y}}(\mathbf{Y}^{t})
  127. p 𝐘 p_{\mathbf{Y}}
  128. p 𝐲 p_{\mathbf{y}}
  129. p 𝐘 ( Y ) = p 𝐲 ( 𝐲 ) | 𝐘 𝐲 | p_{\mathbf{Y}}(Y)=\frac{p_{\mathbf{y}}(\mathbf{y})}{|\frac{\partial\mathbf{Y}}% {\partial\mathbf{y}}|}
  130. 𝐉 = 𝐘 𝐲 \mathbf{J}=\frac{\partial\mathbf{Y}}{\partial\mathbf{y}}
  131. | 𝐉 | = g ( 𝐲 ) |\mathbf{J}|=g^{\prime}(\mathbf{y})
  132. g g^{\prime}
  133. g = p s g^{\prime}=p_{s}
  134. p 𝐘 ( Y ) = p 𝐲 ( 𝐲 ) | 𝐘 𝐲 | = p 𝐲 ( 𝐲 ) p 𝐬 ( 𝐲 ) p_{\mathbf{Y}}(Y)=\frac{p_{\mathbf{y}}(\mathbf{y})}{|\frac{\partial\mathbf{Y}}% {\partial\mathbf{y}}|}=\frac{p_{\mathbf{y}}(\mathbf{y})}{p_{\mathbf{s}}(% \mathbf{y})}
  135. H ( 𝐘 ) = - 1 N t = 1 N ln p 𝐲 ( 𝐲 ) p 𝐬 ( 𝐲 ) H(\mathbf{Y})=-\frac{1}{N}\sum_{t=1}^{N}\ln\frac{p_{\mathbf{y}}(\mathbf{y})}{p% _{\mathbf{s}}(\mathbf{y})}
  136. p 𝐲 = p s p_{\mathbf{y}}=p_{s}
  137. p 𝐘 p_{\mathbf{Y}}
  138. H ( 𝐘 ) H({\mathbf{Y}})
  139. p 𝐲 ( 𝐲 ) = p 𝐱 ( 𝐱 ) | 𝐲 𝐱 | = p 𝐱 ( 𝐱 ) | 𝐖 | p_{\mathbf{y}}(\mathbf{y})=\frac{p_{\mathbf{x}}(\mathbf{x})}{|\frac{\partial% \mathbf{y}}{\partial\mathbf{x}}|}=\frac{p_{\mathbf{x}}(\mathbf{x})}{|\mathbf{W% }|}
  140. | 𝐖 | |\mathbf{W}|
  141. 𝐖 \mathbf{W}
  142. H ( 𝐘 ) = - 1 N t = 1 N ln p 𝐱 ( 𝐱 t ) | 𝐖 | p 𝐬 ( 𝐲 t ) H(\mathbf{Y})=-\frac{1}{N}\sum_{t=1}^{N}\ln\frac{p_{\mathbf{x}}(\mathbf{x}^{t}% )}{|\mathbf{W}|p_{\mathbf{s}}(\mathbf{y}^{t})}
  143. H ( 𝐘 ) = 1 N t = 1 N ln p 𝐬 ( 𝐲 t ) + ln | 𝐖 | + H ( 𝐱 ) H(\mathbf{Y})=\frac{1}{N}\sum_{t=1}^{N}\ln p_{\mathbf{s}}(\mathbf{y}^{t})+\ln|% \mathbf{W}|+H(\mathbf{x})
  144. H ( 𝐱 ) = - 1 N t = 1 N ln p 𝐱 ( 𝐱 t ) H(\mathbf{x})=-\frac{1}{N}\sum_{t=1}^{N}\ln p_{\mathbf{x}}(\mathbf{x}^{t})
  145. 𝐖 \mathbf{W}
  146. H 𝐱 H_{\mathbf{x}}
  147. h ( 𝐘 ) = 1 N t = 1 N ln p 𝐬 ( 𝐲 t ) + ln | 𝐖 | h(\mathbf{Y})=\frac{1}{N}\sum_{t=1}^{N}\ln p_{\mathbf{s}}(\mathbf{y}^{t})+\ln|% \mathbf{W}|
  148. p 𝐬 p_{\mathbf{s}}
  149. p 𝐬 = ( 1 - tanh ( 𝐬 ) 2 ) p_{\mathbf{s}}=(1-\tanh(\mathbf{s})^{2})
  150. h ( 𝐘 ) = 1 N i = 1 M t = 1 N ln ( 1 - tanh ( 𝐰 𝐢 𝐓 𝐱 𝐭 ) 2 ) + ln | 𝐖 | h(\mathbf{Y})=\frac{1}{N}\sum_{i=1}^{M}\sum_{t=1}^{N}\ln(1-\tanh(\mathbf{w_{i}% ^{T}x^{t}})^{2})+\ln|\mathbf{W}|
  151. 𝐱 \mathbf{x}
  152. 𝐲 \mathbf{y}
  153. p 𝐬 = g p_{\mathbf{s}}=g^{\prime}
  154. 𝐖 \mathbf{W}
  155. 𝐖 \mathbf{W}
  156. y y
  157. p s p_{s}
  158. p s p_{s}
  159. s s
  160. y = 𝐖 x y=\mathbf{W}x
  161. p s p_{s}
  162. s s
  163. p s p_{s}
  164. 𝐀 \mathbf{A}
  165. x x
  166. 𝐀 \mathbf{A}
  167. p s p_{s}
  168. 𝐀 \mathbf{A}
  169. 𝐋 ( 𝐖 ) \mathbf{L(W)}
  170. 𝐖 \mathbf{W}
  171. 𝐋 ( 𝐖 ) = p s ( 𝐖 x ) | 𝐖 | . \mathbf{L(W)}=p_{s}(\mathbf{W}x)|\mathbf{W}|.
  172. 𝐖 \mathbf{W}
  173. x x
  174. s s
  175. p s p_{s}
  176. 𝐖 \mathbf{W}
  177. 𝐋 ( 𝐖 ) \mathbf{L(W)}
  178. 𝐖 \mathbf{W}
  179. 𝐋 ( 𝐖 ) \mathbf{L(W)}
  180. ln 𝐋 ( 𝐖 ) \ln\mathbf{L(W)}
  181. ln 𝐋 ( 𝐖 ) = i t ln p s ( w i T x t ) + N ln | 𝐖 | \ln\mathbf{L(W)}=\sum_{i}\sum_{t}\ln p_{s}(w^{T}_{i}x_{t})+N\ln|\mathbf{W}|
  182. p s = ( 1 - tanh ( s ) 2 ) p_{s}=(1-\tanh(s)^{2})
  183. ln 𝐋 ( 𝐖 ) = 1 N i M t N ln ( 1 - tanh ( w i T x t ) 2 ) + ln | 𝐖 | \ln\mathbf{L(W)}={1\over N}\sum_{i}^{M}\sum_{t}^{N}\ln(1-\tanh(w^{T}_{i}x_{t})% ^{2})+\ln|\mathbf{W}|
  184. 𝐖 \mathbf{W}

Indeterminate_form.html

  1. lim x c f ( x ) g ( x ) . \lim_{x\to c}\frac{f(x)}{g(x)}.\!
  2. lim x 0 x x = 1 , ( 1 ) \lim_{x\to 0}\frac{x}{x}=1,\!~{}~{}(1)
  3. lim x 0 x 2 x = 0 , ( 2 ) \lim_{x\to 0}\frac{x^{2}}{x}=0,\!~{}~{}(2)
  4. lim x 0 sin ( x ) x = 1 , ( 3 ) \lim_{x\to 0}\frac{\sin(x)}{x}=1,\!~{}~{}(3)
  5. lim x 49 x - 49 x - 7 = 14 , ( 4 ) \lim_{x\to 49}\frac{x-49}{\sqrt{x}\,-7}=14,~{}~{}(4)
  6. lim x 0 a x x = a . ( 5 ) \lim_{x\to 0}\frac{ax}{x}=a.\!~{}~{}(5)
  7. lim x 0 x x 3 = . ( 6 ) \lim_{x\to 0}\frac{x}{x^{3}}=\infty.\!~{}~{}(6)
  8. lim x 0 x 0 = 1 , ( 7 ) \lim_{x\to 0}x^{0}=1,\!~{}~{}(7)
  9. lim x 0 + 0 x = 0. ( 8 ) \lim_{x\to 0^{+}}0^{x}=0.\!~{}~{}(8)
  10. lim x c f ( x ) = 0 \scriptstyle\lim_{x\to c}f(x)\;=\;0\!
  11. lim x c g ( x ) = 0 \scriptstyle\lim_{x\to c}g(x)\;=\;0
  12. lim x c f ( x ) g ( x ) . \lim_{x\to c}f(x)^{g(x)}.
  13. α \alpha
  14. β \beta
  15. lim β α = 1 \lim\frac{\beta}{\alpha}=1
  16. x sin x , x\sim\sin x,
  17. x arcsin x , x\sim\arcsin x,
  18. x tan x , x\sim\tan x,
  19. x arctan x , x\sim\arctan x,
  20. x ln ( 1 + x ) , x\sim\ln(1+x),
  21. 1 - cos x 1 2 x 2 , 1-\cos x\sim\frac{1}{2}x^{2},
  22. a x - 1 x ln a , a^{x}-1\sim x\ln a,
  23. ( 1 + x ) a - 1 a x . (1+x)^{a}-1\sim ax.
  24. lim x 0 1 x 3 [ ( 2 + cos x 3 ) x - 1 ] = lim x 0 e x ln 2 + cos x 3 - 1 x 3 = lim x 0 1 x 2 ln 2 + cos x 3 = lim x 0 1 x 2 ln ( cos x - 1 3 + 1 ) = lim x 0 cos x - 1 3 x 2 = - 1 6 \lim_{x\to 0}\frac{1}{x^{3}}[(\frac{2+\cos x}{3})^{x}-1]=\lim_{x\to 0}\frac{e^% {x\ln{\frac{2+\cos x}{3}}}-1}{x^{3}}=\lim_{x\to 0}\frac{1}{x^{2}}\ln\frac{2+% \cos x}{3}=\lim_{x\to 0}\frac{1}{x^{2}}\ln(\frac{\cos x-1}{3}+1)=\lim_{x\to 0}% \frac{\cos x-1}{3x^{2}}=-\frac{1}{6}
  25. α α \alpha\sim\alpha^{\prime}
  26. β β \beta\sim\beta^{\prime}
  27. lim β α = lim β β α β α α = lim β β lim α α lim β α = lim β α \lim\frac{\beta}{\alpha}=\lim\frac{\beta\beta^{\prime}\alpha^{\prime}}{\beta^{% \prime}\alpha^{\prime}\alpha}=\lim\frac{\beta}{\beta^{\prime}}\lim\frac{\alpha% ^{\prime}}{\alpha}\lim\frac{\beta^{\prime}}{\alpha^{\prime}}=\lim\frac{\beta^{% \prime}}{\alpha^{\prime}}
  28. lim x c f ( x ) g ( x ) = lim x c f ( x ) g ( x ) , \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)% },\!
  29. ln lim x c f ( x ) g ( x ) = lim x c ln f ( x ) 1 / g ( x ) . \ln\lim_{x\to c}f(x)^{g(x)}=\lim_{x\to c}\frac{\ln f(x)}{1/g(x)}.\!
  30. lim x c f ( x ) = 0 , lim x c g ( x ) = 0 \lim_{x\to c}f(x)=0,\ \lim_{x\to c}g(x)=0\!
  31. lim x c f ( x ) g ( x ) = lim x c 1 / g ( x ) 1 / f ( x ) \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{1/g(x)}{1/f(x)}\!
  32. lim x c f ( x ) = , lim x c g ( x ) = \lim_{x\to c}f(x)=\infty,\ \lim_{x\to c}g(x)=\infty\!
  33. lim x c f ( x ) g ( x ) = lim x c 1 / g ( x ) 1 / f ( x ) \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{1/g(x)}{1/f(x)}\!
  34. lim x c f ( x ) = 0 , lim x c g ( x ) = \lim_{x\to c}f(x)=0,\ \lim_{x\to c}g(x)=\infty\!
  35. lim x c f ( x ) g ( x ) = lim x c f ( x ) 1 / g ( x ) \lim_{x\to c}f(x)g(x)=\lim_{x\to c}\frac{f(x)}{1/g(x)}\!
  36. lim x c f ( x ) g ( x ) = lim x c g ( x ) 1 / f ( x ) \lim_{x\to c}f(x)g(x)=\lim_{x\to c}\frac{g(x)}{1/f(x)}\!
  37. lim x c f ( x ) = , lim x c g ( x ) = \lim_{x\to c}f(x)=\infty,\ \lim_{x\to c}g(x)=\infty\!
  38. lim x c ( f ( x ) - g ( x ) ) = lim x c 1 / g ( x ) - 1 / f ( x ) 1 / ( f ( x ) g ( x ) ) \lim_{x\to c}(f(x)-g(x))=\lim_{x\to c}\frac{1/g(x)-1/f(x)}{1/(f(x)g(x))}\!
  39. lim x c ( f ( x ) - g ( x ) ) = ln lim x c e f ( x ) e g ( x ) \lim_{x\to c}(f(x)-g(x))=\ln\lim_{x\to c}\frac{e^{f(x)}}{e^{g(x)}}\!
  40. lim x c f ( x ) = 0 + , lim x c g ( x ) = 0 \lim_{x\to c}f(x)=0^{+},\lim_{x\to c}g(x)=0\!
  41. lim x c f ( x ) g ( x ) = exp lim x c g ( x ) 1 / ln f ( x ) \lim_{x\to c}f(x)^{g(x)}=\exp\lim_{x\to c}\frac{g(x)}{1/\ln f(x)}\!
  42. lim x c f ( x ) g ( x ) = exp lim x c ln f ( x ) 1 / g ( x ) \lim_{x\to c}f(x)^{g(x)}=\exp\lim_{x\to c}\frac{\ln f(x)}{1/g(x)}\!
  43. lim x c f ( x ) = 1 , lim x c g ( x ) = \lim_{x\to c}f(x)=1,\ \lim_{x\to c}g(x)=\infty\!
  44. lim x c f ( x ) g ( x ) = exp lim x c ln f ( x ) 1 / g ( x ) \lim_{x\to c}f(x)^{g(x)}=\exp\lim_{x\to c}\frac{\ln f(x)}{1/g(x)}\!
  45. lim x c f ( x ) g ( x ) = exp lim x c g ( x ) 1 / ln f ( x ) \lim_{x\to c}f(x)^{g(x)}=\exp\lim_{x\to c}\frac{g(x)}{1/\ln f(x)}\!
  46. lim x c f ( x ) = , lim x c g ( x ) = 0 \lim_{x\to c}f(x)=\infty,\ \lim_{x\to c}g(x)=0\!
  47. lim x c f ( x ) g ( x ) = exp lim x c g ( x ) 1 / ln f ( x ) \lim_{x\to c}f(x)^{g(x)}=\exp\lim_{x\to c}\frac{g(x)}{1/\ln f(x)}\!
  48. lim x c f ( x ) g ( x ) = exp lim x c ln f ( x ) 1 / g ( x ) \lim_{x\to c}f(x)^{g(x)}=\exp\lim_{x\to c}\frac{\ln f(x)}{1/g(x)}\!

Index_notation.html

  1. 𝐚 = ( a 1 a 2 a n ) , 𝐚 = ( a 1 a 2 a n ) \mathbf{a}=\begin{pmatrix}a_{1}\\ a_{2}\\ \vdots\\ a_{n}\end{pmatrix},\quad\mathbf{a}=\begin{pmatrix}a_{1}&a_{2}&\cdots&a_{n}\\ \end{pmatrix}
  2. 𝐚 = ( 10 8 9 6 3 5 ) \mathbf{a}=\begin{pmatrix}10&8&9&6&3&5\\ \end{pmatrix}
  3. a 1 = 10 , a 2 = 8 a 6 = 5 a_{1}=10,\,a_{2}=8\cdots a_{6}=5
  4. 𝐚 + 𝐛 = 𝐜 \mathbf{a}+\mathbf{b}=\mathbf{c}
  5. a i + b i = c i a_{i}+b_{i}=c_{i}
  6. a 1 + b 1 = c 1 a_{1}+b_{1}=c_{1}
  7. a 2 + b 2 = c 2 a_{2}+b_{2}=c_{2}
  8. \vdots
  9. a n + b n = c n a_{n}+b_{n}=c_{n}
  10. 𝐀 = ( a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ) \mathbf{A}=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\\ \end{pmatrix}
  11. 𝐀 = ( 9 8 6 1 2 7 4 9 2 6 0 5 ) \mathbf{A}=\begin{pmatrix}9&8&6\\ 1&2&7\\ 4&9&2\\ 6&0&5\end{pmatrix}
  12. a 11 = 9 , a 12 = 8 , a 21 = 1 , a 23 = 7 a_{11}=9,\,a_{12}=8,a_{21}=1,\,\cdots\,\,a_{23}=7\,\cdots
  13. 𝐀 + 𝐁 = 𝐂 \mathbf{A}+\mathbf{B}=\mathbf{C}
  14. A i j + B i j = C i j A_{ij}+B_{ij}=C_{ij}
  15. A i 1 i 2 + B i 1 i 2 = C i 1 i 2 A_{i_{1}i_{2}\cdots}+B_{i_{1}i_{2}\cdots}=C_{i_{1}i_{2}\cdots}

Index_set.html

  1. S S
  2. J \sub J\sub\mathbb{N}
  3. f : J S f:J→S
  4. S S
  5. \mathbb{N}
  6. r r\in\mathbb{R}
  7. r r
  8. 𝟏 r : { 0 , 1 } \mathbf{1}_{r}\colon\mathbb{R}\rightarrow\{0,1\}
  9. 𝟏 r ( x ) := { 0 , if x r 1 , if x = r . \mathbf{1}_{r}(x):=\begin{cases}0,&\mbox{if }~{}x\neq r\\ 1,&\mbox{if }~{}x=r.\end{cases}
  10. 𝟏 r \mathbf{1}_{r}
  11. \mathbb{R}

Induced_representation.html

  1. H H
  2. G G
  3. G G
  4. G / H G/H
  5. H H
  6. G G
  7. G G
  8. K G G KGG
  9. K G G KGG
  10. G G
  11. K K
  12. K H H KHH
  13. ρ ρ
  14. G G
  15. σ σ
  16. H H
  17. G G
  18. ρ ρ
  19. I n d ( σ ) Ind(σ)
  20. H H
  21. R e s ( ρ ) Res(ρ)
  22. σ σ
  23. K = 𝐂 K=\mathbf{C}
  24. H H
  25. χ χ
  26. σ σ
  27. χ ( h ) = T r σ ( h ) χ(h)=Trσ(h)
  28. ψ ψ
  29. ψ ( g ) = x G / H χ ^ ( x - 1 g x ) , \psi(g)=\sum_{x\in G/H}\widehat{\chi}\left(x^{-1}gx\right),
  30. χ ^ ( g ) = { χ ( g ) g H 0 otherwise \widehat{\chi}(g)=\begin{cases}\chi(g)&g\in H\\ 0&\,\text{otherwise}\end{cases}
  31. I n d ( σ ) Ind(σ)
  32. G G
  33. H H
  34. σ σ
  35. σ σ
  36. V V
  37. V V
  38. G G
  39. H H
  40. σ σ
  41. K G G KGG
  42. K H H KHH
  43. G G
  44. g g
  45. H H
  46. G G
  47. H H
  48. G G
  49. ( π , V ) (π,V)
  50. H H
  51. n = G G : H n=GG:H
  52. H H
  53. G G
  54. G G
  55. G / H G/H
  56. I n d [ u s u , u b = , u H , u p = , u G ] π Ind[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}H^{\prime},u^{\prime% }p=^{\prime},u^{\prime}G^{\prime}]π
  57. W = i = 1 n x i V . W=\bigoplus_{i=1}^{n}x_{i}V.
  58. G G
  59. H H
  60. G G
  61. W W
  62. g i = 1 n x i v i = i = 1 n x j ( i ) π ( h i ) v i g\cdot\sum_{i=1}^{n}x_{i}v_{i}=\sum_{i=1}^{n}x_{j(i)}\pi(h_{i})v_{i}
  63. v i V v_{i}\in V
  64. I n d [ u s u , u b = , u H , u p = , u G ] π Ind[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}H^{\prime},u^{\prime% }p=^{\prime},u^{\prime}G^{\prime}]π
  65. G G
  66. H H
  67. G G
  68. H H
  69. ( π , V ) (π,V)
  70. H H
  71. Ind H G π = { f : G V : f ( h g ) = π ( h ) f ( g ) and f L 2 ( G ) } . \operatorname{Ind}_{H}^{G}\pi=\left\{f:G\to V\ :\ f(hg)=\pi(h)f(g)\,\text{ and% }f\in L^{2}(G)\right\}.
  72. G G
  73. Ind H G π = { f : G V : f ( h g ) = Δ G - 1 2 ( h ) Δ H 1 2 ( h ) π ( h ) f ( g ) and f L 2 ( G ) } . \operatorname{Ind}_{H}^{G}\pi=\left\{f:G\to V\ :\ f(hg)=\Delta_{G}^{-\frac{1}{% 2}}(h)\Delta_{H}^{\frac{1}{2}}(h)\pi(h)f(g)\,\text{ and }f\in L^{2}(G)\right\}.
  74. G G
  75. H H
  76. ind H G π = { f : G V : f ( h g ) = π ( h ) f ( g ) and f has compact support mod H } . \operatorname{ind}_{H}^{G}\pi=\left\{f:G\to V\ :\ f(hg)=\pi(h)f(g)\,\text{ and% }f\,\text{ has compact support mod }H\right\}.
  77. G / H G/H
  78. G G
  79. H H
  80. G G
  81. σ σ
  82. H H
  83. V V
  84. V × G V×G
  85. G G
  86. g [ ( x , g ) ] = ( x , g g - 1 ) g^{\prime}[(x,g)]=\left(x,gg^{\prime-1}\right)
  87. g g
  88. g g′
  89. G G
  90. x x
  91. V V
  92. ( x , g ) ( h [ x ] , h g ) . (x,g)\sim(h[x],hg).
  93. G G
  94. V × G / V×G/~{}
  95. G G
  96. g - 1 h g [ ( x , g ) ] = ( x , h - 1 g ) ( h [ x ] , g ) . g^{-1}hg[(x,g)]=(x,h^{-1}g)\sim(h[x],g).
  97. V × G / V×G/~{}
  98. G / H G/H
  99. H H
  100. V V
  101. σ σ
  102. V V
  103. G / H G/H

Inequality_of_arithmetic_and_geometric_means.html

  1. x x
  2. y y
  3. x + y 2 x y \frac{x+y}{2}\geq\sqrt{xy}
  4. x = y x=y
  5. 0 \displaystyle 0
  6. ( x + y ) < s u p > 2 4 x y (x+y)<sup>2≥4xy
  7. ln x + ln y 2 ln ( x + y 2 ) \frac{\ln x+\ln y}{2}\leq\ln\left(\frac{x+y}{2}\right)
  8. n n
  9. n n
  10. x 1 + x 2 + + x n n . \frac{x_{1}+x_{2}+\cdots+x_{n}}{n}.
  11. x 1 x 2 x n n . \sqrt[n]{x_{1}\cdot x_{2}\cdots x_{n}}.
  12. exp ( ln x 1 + ln x 2 + + ln x n n ) . \exp\left(\frac{\ln{x_{1}}+\ln{x_{2}}+\cdots+\ln{x_{n}}}{n}\right).
  13. n n
  14. x 1 + x 2 + + x n n x 1 x 2 x n n , \frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\geq\sqrt[n]{x_{1}\cdot x_{2}\cdots x_{n}}\,,
  15. n = 2 n=2
  16. n n
  17. n n
  18. n n
  19. 2 2
  20. n n
  21. 2 n - 1 n x 1 x 2 x n n 2^{n-1}n\sqrt[n]{x_{1}x_{2}\cdots x_{n}}
  22. n n
  23. x 1 + x 2 + + x n n x 1 x 2 x n n , {x_{1}+x_{2}+\cdots+x_{n}\over n}\geq\sqrt[n]{x_{1}x_{2}\cdots x_{n}},
  24. 2 n - 1 ( x 1 + x 2 + + x n ) 2 n - 1 n x 1 x 2 x n n 2^{n-1}(x_{1}+x_{2}+\cdots+x_{n})\geq 2^{n-1}n\sqrt[n]{x_{1}x_{2}\cdots x_{n}}\,
  25. n n
  26. n n
  27. f ( x , y , z ) = x y + y z + z x 3 f(x,y,z)=\frac{x}{y}+\sqrt{\frac{y}{z}}+\sqrt[3]{\frac{z}{x}}
  28. x x
  29. y y
  30. z z
  31. f ( x , y , z ) \displaystyle f(x,y,z)
  32. x 1 = x y , x 2 = x 3 = 1 2 y z , x 4 = x 5 = x 6 = 1 3 z x 3 . x_{1}=\frac{x}{y},\qquad x_{2}=x_{3}=\frac{1}{2}\sqrt{\frac{y}{z}},\qquad x_{4% }=x_{5}=x_{6}=\frac{1}{3}\sqrt[3]{\frac{z}{x}}.
  33. n = 6 n=6
  34. f ( x , y , z ) 6 x y 1 2 y z 1 2 y z 1 3 z x 3 1 3 z x 3 1 3 z x 3 6 = 6 1 2 2 3 3 3 x y y z z x 6 = 2 2 / 3 3 1 / 2 . \begin{aligned}\displaystyle f(x,y,z)&\displaystyle\geq 6\cdot\sqrt[6]{\frac{x% }{y}\cdot\frac{1}{2}\sqrt{\frac{y}{z}}\cdot\frac{1}{2}\sqrt{\frac{y}{z}}\cdot% \frac{1}{3}\sqrt[3]{\frac{z}{x}}\cdot\frac{1}{3}\sqrt[3]{\frac{z}{x}}\cdot% \frac{1}{3}\sqrt[3]{\frac{z}{x}}}\\ &\displaystyle=6\cdot\sqrt[6]{\frac{1}{2\cdot 2\cdot 3\cdot 3\cdot 3}\frac{x}{% y}\frac{y}{z}\frac{z}{x}}\\ &\displaystyle=2^{2/3}\cdot 3^{1/2}.\end{aligned}
  35. f ( x , y , z ) = 2 2 / 3 3 1 / 2 when x y = 1 2 y z = 1 3 z x 3 . f(x,y,z)=2^{2/3}\cdot 3^{1/2}\quad\mbox{when}~{}\quad\frac{x}{y}=\frac{1}{2}% \sqrt{\frac{y}{z}}=\frac{1}{3}\sqrt[3]{\frac{z}{x}}.
  36. ( x , y , z ) (x,y,z)
  37. ( x , y , z ) = ( x , 2 3 3 x , 3 3 2 x ) with x > 0. (x,y,z)=\biggr(x,\sqrt[3]{2}\sqrt{3}\,x,\frac{3\sqrt{3}}{2}\,x\biggr)\quad% \mbox{with}~{}\quad x>0.
  38. x x
  39. x 1 + x 2 + + x n n x 1 x 2 x n n \frac{x_{1}+x_{2}+\cdots+x_{n}}{n}\geq\sqrt[n]{x_{1}x_{2}\cdots x_{n}}
  40. ( x i + x j 2 ) 2 - x i x j = ( x i - x j 2 ) 2 > 0. \Bigl(\frac{x_{i}+x_{j}}{2}\Bigr)^{2}-x_{i}x_{j}=\Bigl(\frac{x_{i}-x_{j}}{2}% \Bigr)^{2}>0.
  41. α = x 1 + x 2 + + x n n , \alpha=\frac{x_{1}+x_{2}+\ldots+x_{n}}{n},
  42. x 1 + x 2 + + x n n = α = α α α n x 1 x 2 x n n . \frac{x_{1}+x_{2}+\ldots+x_{n}}{n}=\alpha=\sqrt[n]{\alpha\alpha\ldots\alpha}% \geq\sqrt[n]{x_{1}x_{2}\ldots x_{n}}.
  43. n = 2 n=2
  44. n n
  45. n 3 n≥3
  46. n 3 n≥3
  47. α = x 1 + + x n n \alpha=\frac{\ x_{1}+\cdots+x_{n}}{n}
  48. α n x 1 x 2 x n \alpha^{n}\geq x_{1}x_{2}\cdots x_{n}\,
  49. n = 1 n=1
  50. n n
  51. n + 1 n+1
  52. α α
  53. ( n + 1 ) α = x 1 + + x n + x n + 1 . (n+1)\alpha=\ x_{1}+\cdots+x_{n}+x_{n+1}.\,
  54. α α
  55. α α
  56. α α
  57. k k
  58. k = 1 k=1
  59. n = 2 n=2
  60. n = 2 n=2
  61. k > 1 k>1
  62. 2 2
  63. x 1 + x 2 + + x 2 k 2 k \displaystyle\frac{x_{1}+x_{2}+\cdots+x_{2^{k}}}{2^{k}}
  64. x 1 = x 2 = = x 2 k - 1 x_{1}=x_{2}=\cdots=x_{2^{k-1}}
  65. x 2 k - 1 + 1 = x 2 k - 1 + 2 = = x 2 k x_{2^{k-1}+1}=x_{2^{k-1}+2}=\cdots=x_{2^{k}}
  66. x 1 + x 2 + + x 2 k 2 k > x 1 x 2 x 2 k 2 k \frac{x_{1}+x_{2}+\cdots+x_{2^{k}}}{2^{k}}>\sqrt[2^{k}]{x_{1}x_{2}\cdots x_{2^% {k}}}
  67. n n
  68. 2 2
  69. m m
  70. 2 2
  71. n n
  72. n n
  73. α α
  74. x n + 1 = x n + 2 = = x m = α . x_{n+1}=x_{n+2}=\cdots=x_{m}=\alpha.
  75. α \displaystyle\alpha
  76. α m > x 1 x 2 x n α m - n \alpha^{m}>x_{1}x_{2}\cdots x_{n}\alpha^{m-n}
  77. α > x 1 x 2 x n n \alpha>\sqrt[n]{x_{1}x_{2}\cdots x_{n}}
  78. n = 1 n=1
  79. n n
  80. n + 1 n+1
  81. x 1 + + x n + x n + 1 n + 1 - ( x 1 x n x n + 1 ) 1 n + 1 0 \frac{x_{1}+\cdots+x_{n}+x_{n+1}}{n+1}-({x_{1}\cdots x_{n}x_{n+1}})^{\frac{1}{% n+1}}\geq 0
  82. n + 1 n+1
  83. n + 1 n+1
  84. f ( t ) = x 1 + + x n + t n + 1 - ( x 1 x n t ) 1 n + 1 , t > 0. f(t)=\frac{x_{1}+\cdots+x_{n}+t}{n+1}-({x_{1}\cdots x_{n}t})^{\frac{1}{n+1}},% \qquad t>0.
  85. f ( t ) 0 f(t)≥0
  86. t > 0 t>0
  87. f ( t ) = 0 f(t)=0
  88. t t
  89. f f
  90. f f
  91. f ( t ) = 1 n + 1 - 1 n + 1 ( x 1 x n ) 1 n + 1 t - n n + 1 , t > 0. f^{\prime}(t)=\frac{1}{n+1}-\frac{1}{n+1}({x_{1}\cdots x_{n}})^{\frac{1}{n+1}}% t^{-\frac{n}{n+1}},\qquad t>0.
  92. ( x 1 x n ) 1 n + 1 t 0 - n n + 1 = 1. ({x_{1}\cdots x_{n}})^{\frac{1}{n+1}}t_{0}^{-\frac{n}{n+1}}=1.
  93. t 0 n n + 1 = ( x 1 x n ) 1 n + 1 , t_{0}^{\frac{n}{n+1}}=({x_{1}\cdots x_{n}})^{\frac{1}{n+1}},
  94. t 0 = ( x 1 x n ) 1 n , t_{0}=({x_{1}\cdots x_{n}})^{\frac{1}{n}},
  95. f f
  96. f ( t ) > 0 f′′(t)>0
  97. t > 0 t>0
  98. f f
  99. f ( t 0 ) = x 1 + + x n + ( x 1 x n ) 1 / n n + 1 - ( x 1 x n ) 1 n + 1 ( x 1 x n ) 1 n ( n + 1 ) = x 1 + + x n n + 1 + 1 n + 1 ( x 1 x n ) 1 n - ( x 1 x n ) 1 n = x 1 + + x n n + 1 - n n + 1 ( x 1 x n ) 1 n = n n + 1 ( x 1 + + x n n - ( x 1 x n ) 1 n ) 0 , . \begin{aligned}\displaystyle f(t_{0})&\displaystyle=\frac{x_{1}+\cdots+x_{n}+(% {x_{1}\cdots x_{n}})^{1/n}}{n+1}-({x_{1}\cdots x_{n}})^{\frac{1}{n+1}}({x_{1}% \cdots x_{n}})^{\frac{1}{n(n+1)}}\\ &\displaystyle=\frac{x_{1}+\cdots+x_{n}}{n+1}+\frac{1}{n+1}({x_{1}\cdots x_{n}% })^{\frac{1}{n}}-({x_{1}\cdots x_{n}})^{\frac{1}{n}}\\ &\displaystyle=\frac{x_{1}+\cdots+x_{n}}{n+1}-\frac{n}{n+1}({x_{1}\cdots x_{n}% })^{\frac{1}{n}}\\ &\displaystyle=\frac{n}{n+1}\Bigl(\frac{x_{1}+\cdots+x_{n}}{n}-({x_{1}\cdots x% _{n}})^{\frac{1}{n}}\Bigr)\geq 0,\end{aligned}.
  100. x x
  101. f ( 1 ) = 0 f(1)=0
  102. f ( 1 ) = 0 f′(1)=0
  103. f ( x ) > 0 f′′(x)>0
  104. x x
  105. f f
  106. x = 1 x=1
  107. x x
  108. x = 1 x=1
  109. α > 0 α>0
  110. n n
  111. x 1 α x 2 α x n α e x 1 α - 1 e x 2 α - 1 e x n α - 1 = exp ( x 1 α - 1 + x 2 α - 1 + + x n α - 1 ) , ( * ) \begin{aligned}\displaystyle{\frac{x_{1}}{\alpha}\frac{x_{2}}{\alpha}\cdots% \frac{x_{n}}{\alpha}}&\displaystyle\leq{e^{\frac{x_{1}}{\alpha}-1}e^{\frac{x_{% 2}}{\alpha}-1}\cdots e^{\frac{x_{n}}{\alpha}-1}}\\ &\displaystyle=\exp\Bigl(\frac{x_{1}}{\alpha}-1+\frac{x_{2}}{\alpha}-1+\cdots+% \frac{x_{n}}{\alpha}-1\Bigr),\qquad(*)\end{aligned}
  112. x 1 α - 1 + x 2 α - 1 + + x n α - 1 \displaystyle\frac{x_{1}}{\alpha}-1+\frac{x_{2}}{\alpha}-1+\cdots+\frac{x_{n}}% {\alpha}-1
  113. ( * ) (*)
  114. x 1 x 2 x n α n e 0 = 1 , \frac{x_{1}x_{2}\cdots x_{n}}{\alpha^{n}}\leq e^{0}=1,
  115. x 1 x 2 x n n α . \sqrt[n]{x_{1}x_{2}\cdots x_{n}}\leq\alpha.
  116. w > 0 w>0
  117. w 1 x 1 + w 2 x 2 + + w n x n w x 1 w 1 x 2 w 2 x n w n w \frac{w_{1}x_{1}+w_{2}x_{2}+\cdots+w_{n}x_{n}}{w}\geq\sqrt[w]{x_{1}^{w_{1}}x_{% 2}^{w_{2}}\cdots x_{n}^{w_{n}}}
  118. ln ( w 1 x 1 + + w n x n w ) > w 1 w ln x 1 + + w n w ln x n = ln x 1 w 1 x 2 w 2 x n w n w . \begin{aligned}\displaystyle\ln\Bigl(\frac{w_{1}x_{1}+\cdots+w_{n}x_{n}}{w}% \Bigr)&\displaystyle>\frac{w_{1}}{w}\ln x_{1}+\cdots+\frac{w_{n}}{w}\ln x_{n}% \\ &\displaystyle=\ln\sqrt[w]{x_{1}^{w_{1}}x_{2}^{w_{2}}\cdots x_{n}^{w_{n}}}.% \end{aligned}
  119. w 1 x 1 + + w n x n w > x 1 w 1 x 2 w 2 x n w n w . \frac{w_{1}x_{1}+\cdots+w_{n}x_{n}}{w}>\sqrt[w]{x_{1}^{w_{1}}x_{2}^{w_{2}}% \cdots x_{n}^{w_{n}}}.

Inertial_electrostatic_confinement.html

  1. N = B 2 μ 0 m c 2 N=\frac{B}{2\mu_{0}mc^{2}}
  2. μ 0 \mu_{0}

Infinitary_logic.html

  1. \cdots
  2. γ < δ A γ \lor_{\gamma<\delta}{A_{\gamma}}
  3. δ \delta
  4. γ < δ V γ : \forall_{\gamma<\delta}{V_{\gamma}:}
  5. V γ V_{\gamma}
  6. γ < δ \gamma<\delta
  7. \cdots
  8. V = { V γ | γ < δ < β } V=\{V_{\gamma}|\gamma<\delta<\beta\}
  9. A 0 A_{0}
  10. V 0 : V 1 ( A 0 ) \forall V_{0}:\forall V_{1}\cdots(A_{0})
  11. V 0 : V 1 ( A 0 ) \exists V_{0}:\exists V_{1}\cdots(A_{0})
  12. δ \delta
  13. A = { A γ | γ < δ < α } A=\{A_{\gamma}|\gamma<\delta<\alpha\}
  14. ( A 0 A 1 ) (A_{0}\lor A_{1}\lor\cdots)
  15. ( A 0 and A 1 and ) (A_{0}\and A_{1}\and\cdots)
  16. δ \delta
  17. L α , β L_{\alpha,\beta}
  18. γ \gamma
  19. A = { A γ | γ < δ < α } A=\{A_{\gamma}|\gamma<\delta<\alpha\}
  20. and γ < δ A γ \and_{\gamma<\delta}{A_{\gamma}}
  21. δ \delta
  22. γ \gamma
  23. 0 < δ < α 0<\delta<\alpha
  24. ( ( and ϵ < δ ( A δ A ϵ ) ) ( A δ and ϵ < δ A ϵ ) ) ((\and_{\epsilon<\delta}{(A_{\delta}\implies A_{\epsilon})})\implies(A_{\delta% }\implies\and_{\epsilon<\delta}{A_{\epsilon}}))
  25. γ < δ \gamma<\delta
  26. ( ( and ϵ < δ A ϵ ) A γ ) ((\and_{\epsilon<\delta}{A_{\epsilon}})\implies A_{\gamma})
  27. γ \gamma
  28. ( μ < γ ( and δ < γ A μ , δ ) ) (\lor_{\mu<\gamma}{(\and_{\delta<\gamma}{A_{\mu,\delta}})})
  29. μ δ ϵ < γ : A μ , δ = A ϵ \forall\mu\forall\delta\exists\epsilon<\gamma:A_{\mu,\delta}=A_{\epsilon}
  30. A μ , δ = ¬ A ϵ A_{\mu,\delta}=\neg A_{\epsilon}
  31. g γ γ ϵ < γ : { A ϵ , ¬ A ϵ } { A μ , g ( μ ) : μ < γ } \forall g\in\gamma^{\gamma}\exists\epsilon<\gamma:\{A_{\epsilon},\neg A_{% \epsilon}\}\subseteq\{A_{\mu,g(\mu)}:\mu<\gamma\}
  32. γ < α \gamma<\alpha
  33. ( ( and μ < γ ( δ < γ A μ , δ ) ) ( ϵ < γ γ ( and μ < γ A μ , γ ϵ ( μ ) ) ) ) ((\and_{\mu<\gamma}{(\lor_{\delta<\gamma}{A_{\mu,\delta}})})\implies(\lor_{% \epsilon<\gamma^{\gamma}}{(\and_{\mu<\gamma}{A_{\mu,\gamma_{\epsilon}(\mu)})}}))
  34. { γ ϵ : ϵ < γ γ } \{\gamma_{\epsilon}:\epsilon<\gamma^{\gamma}\}
  35. γ γ \gamma^{\gamma}
  36. L α , β L_{\alpha,\beta}
  37. κ ω \kappa\neq\omega
  38. L κ , κ L_{\kappa,\kappa}
  39. κ \kappa
  40. \subseteq
  41. κ \kappa
  42. κ ω \kappa\neq\omega
  43. L κ , κ L_{\kappa,\kappa}
  44. \subseteq
  45. κ \kappa
  46. γ < ω V γ : ¬ and γ < ω V γ + V γ . \forall_{\gamma<\omega}{V_{\gamma}:}\neg\and_{\gamma<\omega}{V_{\gamma+}\in V_% {\gamma}}.\,
  47. L ω , ω L_{\omega,\omega}
  48. L ω 1 , ω L_{\omega_{1},\omega}
  49. L ω , ω L_{\omega,\omega}
  50. L ω 1 , ω L_{\omega_{1},\omega}
  51. L α , α L_{\alpha,\alpha}
  52. α \alpha
  53. α \alpha

Infinite_impulse_response.html

  1. a i a_{i}
  2. y [ n ] \displaystyle y\left[n\right]
  3. P \ P
  4. b i \ b_{i}
  5. Q \ Q
  6. a i \ a_{i}
  7. x [ n ] \ x[n]
  8. y [ n ] \ y[n]
  9. y [ n ] = 1 a 0 ( i = 0 P b i x [ n - i ] - j = 1 Q a j y [ n - j ] ) \ y[n]=\frac{1}{a_{0}}\left(\sum_{i=0}^{P}b_{i}x[n-i]-\sum_{j=1}^{Q}a_{j}y[n-j% ]\right)
  10. j = 0 Q a j y [ n - j ] = i = 0 P b i x [ n - i ] \ \sum_{j=0}^{Q}a_{j}y[n-j]=\sum_{i=0}^{P}b_{i}x[n-i]
  11. j = 0 Q a j z - j Y ( z ) = i = 0 P b i z - i X ( z ) \ \sum_{j=0}^{Q}a_{j}z^{-j}Y(z)=\sum_{i=0}^{P}b_{i}z^{-i}X(z)
  12. H ( z ) \displaystyle H(z)
  13. a 0 \ a_{0}
  14. H ( z ) \displaystyle H(z)
  15. z - 1 z^{-1}
  16. z z
  17. z z
  18. H ( z ) H(z)
  19. 0 = j = 0 Q a j z - j \ 0=\sum_{j=0}^{Q}a_{j}z^{-j}
  20. a j 0 a_{j}\neq 0
  21. z z
  22. H ( z ) H(z)
  23. H ( z ) = B ( z ) A ( z ) = 1 1 - a z - 1 H(z)=\frac{B(z)}{A(z)}=\frac{1}{1-az^{-1}}
  24. a a
  25. 0 < | a | < 1 0<|a|<1
  26. H ( z ) H(z)
  27. a a
  28. h ( n ) = a n u ( n ) h(n)=a^{n}u(n)
  29. u ( n ) u(n)
  30. h ( n ) h(n)
  31. n 0 n\geq 0

Information_bottleneck_method.html

  1. T T\,
  2. min p ( t | x ) I ( X ; T ) - β I ( T ; Y ) \min_{p(t|x)}\,\,I(X;T)-\beta I(T;Y)
  3. I ( X ; T ) I ( T ; Y ) I(X;T)\,\,I(T;Y)
  4. X ; T X;T\,
  5. T ; Y T;Y\,
  6. β \beta
  7. X , Y X,Y\,
  8. Σ X X , Σ Y Y \Sigma_{XX},\,\,\Sigma_{YY}
  9. T T\,
  10. X X\,
  11. Y Y\,
  12. T T\,
  13. X , T = A X X,\,\,T=AX\,
  14. A A\,
  15. A A\,
  16. M M\,
  17. Ω = Σ X | Y Σ X X - 1 = I - Σ X Y Σ Y Y - 1 Σ X Y T Σ X X - 1 . \Omega=\Sigma_{X|Y}\Sigma_{XX}^{-1}=I-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{XY}^{% T}\Sigma_{XX}^{-1}.\,
  18. Ω = U Λ V T with Λ = Diag ( λ 1 λ 2 λ N ) \Omega=U\Lambda V^{T}\,\text{ with }\Lambda=\operatorname{Diag}\big(\lambda_{1% }\leq\lambda_{2}\cdots\lambda_{N}\big)\,
  19. β i C = λ i < 1 ( 1 - λ i ) - 1 . \beta_{i}^{C}\underset{\lambda_{i}<1}{=}(1-\lambda_{i})^{-1}.\,
  20. M M\,
  21. β M - 1 C < β β M C \beta_{M-1}^{C}<\beta\leq\beta_{M}^{C}
  22. A = [ w 1 U 1 , , w M U M ] T A=[w_{1}U_{1},\dots,w_{M}U_{M}]^{T}
  23. w i = ( β ( 1 - λ i ) / λ i r i w_{i}=\sqrt{(\beta(1-\lambda_{i})/\lambda_{i}r_{i}}
  24. r i = U i T Σ X X U i . r_{i}=U_{i}^{T}\Sigma_{XX}U_{i}.\,
  25. X = x i X={x_{i}}\,
  26. f f\,
  27. d i , j = f ( | x i - x j | ) d_{i,j}=f\Big(\Big|x_{i}-x_{j}\Big|\Big)
  28. P i , j = exp ( - λ d i , j ) P_{i,j}=\exp(-\lambda d_{i,j})\,
  29. λ > 0 \lambda>0\,
  30. P P\,
  31. t t\,
  32. p ( 0 ) p(0)\,
  33. p ( t ) = P t p ( 0 ) p(t)=P^{t}p(0)\,
  34. p ( ) p(\infty)\,
  35. P P\,
  36. p ( 0 ) p(0)\,
  37. d d\,
  38. Y Y\,
  39. p ( X , Y ) p(X,Y)\,
  40. c k c_{k}\,
  41. x i : p ( c k | x i ) x_{i}:\,\,\,p(c_{k}|x_{i})
  42. { p ( c | x ) = K p ( c ) exp ( - β D K L [ p ( y | x ) | | p ( y | c ) ] ) p ( y | c ) = x p ( y | x ) p ( c | x ) p ( x ) / p ( c ) p ( c ) = x p ( c | x ) p ( x ) \begin{cases}p(c|x)=Kp(c)\exp\Big(-\beta\,D^{KL}\Big[p(y|x)\,||\,p(y|c)\Big]% \Big)\\ p(y|c)=\textstyle\sum_{x}p(y|x)p(c|x)p(x)\big/p(c)\\ p(c)=\textstyle\sum_{x}p(c|x)p(x)\\ \end{cases}
  43. A i , j = p ( c i | x j ) = K p ( c i ) exp ( - β D K L [ p ( y | x j ) | | p ( y | c i ) ] ) A_{i,j}=p(c_{i}|x_{j})=Kp(c_{i})\exp\Big(-\beta\,D^{KL}\Big[p(y|x_{j})\,||\,p(% y|c_{i})\Big]\Big)
  44. D K L D^{KL}\,
  45. Y Y\,
  46. x x\,
  47. c c\,
  48. Y Y\,
  49. D K L ( a | | b ) D^{KL}(a||b)\,
  50. a , b a,b\,
  51. D K L ( a | | b ) = i p ( a i ) log ( p ( a i ) p ( b i ) ) D^{KL}(a||b)=\sum_{i}p(a_{i})\log\Big(\frac{p(a_{i})}{p(b_{i})}\Big)
  52. K K\,
  53. p ( y i | c k ) \displaystyle p(y_{i}|c_{k})
  54. p ( a , b ) = p ( a | b ) p ( b ) = p ( b | a ) p ( a ) p(a,b)=p(a|b)p(b)=p(b|a)p(a)\,
  55. c c\,
  56. p ( c i ) \displaystyle p(c_{i})
  57. p ( x ) p(x)\,
  58. P P\,
  59. D i , j K L = D K L [ p ( y | x j ) | | p ( y | c i ) ] ) D_{i,j}^{KL}=D^{KL}\Big[p(y|x_{j})\,||\,p(y|c_{i})\Big]\Big)
  60. p ( y i | c j ) p(y_{i}|c_{j})\,
  61. p ( c i | x j ) p(c_{i}|x_{j})\,
  62. x x^{\prime}\,
  63. X X\,
  64. x x^{\prime}\,
  65. X : X:\,\,
  66. p ~ ( x i ) = p ( x i | x ) = K exp ( - λ f ( | x i - x | ) ) \tilde{p}(x_{i})=p(x_{i}|x^{\prime})=K\exp\Big(-\lambda f\big(\Big|x_{i}-x^{% \prime}\Big|\big)\Big)
  67. K K\,
  68. p ~ ( c i ) = p ( c i | x ) = j p ( c i | x j ) p ( x j | x ) = j p ( c i | x j ) p ~ ( x j ) \displaystyle\tilde{p}(c_{i})=p(c_{i}|x^{\prime})=\sum_{j}p(c_{i}|x_{j})p(x_{j% }|x^{\prime})=\sum_{j}p(c_{i}|x_{j})\tilde{p}(x_{j})
  69. p ( y i | x ) = j p ( y i | c j ) p ( c j | x ) ) = j p ( y i | c j ) p ~ ( c j ) p(y_{i}|x^{\prime})=\sum_{j}p(y_{i}|c_{j})p(c_{j}|x^{\prime}))=\sum_{j}p(y_{i}% |c_{j})\tilde{p}(c_{j})\,
  70. β \beta\,
  71. u , v u,v\,
  72. ± 1 \pm 1\,
  73. y = sign ( u v ) y=\operatorname{sign}(uv)\,
  74. [ - 1 , 1 ] 2 [-1,1]^{2}\,
  75. λ = 3 , β = 2.5 \lambda=3,\,\beta=2.5
  76. d i , j = | x i - x j | 2 d_{i,j}=\Big|x_{i}-x_{j}\Big|^{2}
  77. x i = ( u i , v i ) T x_{i}=(u_{i},v_{i})^{T}\,
  78. p ( y | x ) p(y|x)\,
  79. P r ( y i = 1 ) = 1 if sign ( u i v i ) = 1 \displaystyle Pr(y_{i}=1)=1\,\text{ if }\operatorname{sign}(u_{i}v_{i})=1
  80. L = Pr ( 1 ) Pr ( - 1 ) = 1 L=\frac{\Pr(1)}{\Pr(-1)}=1
  81. x x^{\prime}\,
  82. u = 0 u=0\,
  83. v = 0 v=0\,
  84. c j c_{j}\,
  85. p ( c j | x i ) p(c_{j}|x_{i})\,
  86. p ( y k | c j ) p(y_{k}|c_{j})\,
  87. f ( . ) f(.)\,
  88. β \beta\,
  89. λ \lambda\,
  90. f f\,
  91. p ( c i | x j ) p(c_{i}|x_{j})\,

Information_extraction.html

  1. M e r g e r B e t w e e n ( c o m p a n y 1 , c o m p a n y 2 , d a t e ) MergerBetween(company_{1},company_{2},date)

Information_geometry.html

  1. | V | = n |V|=n
  2. log b n \log_{b}n
  3. I ( v ) = log n I(v)=\log n
  4. I ( v ) = log 2 n I(v)=\log_{2}n
  5. C C
  6. V V
  7. v V v\in V
  8. C C
  9. c C c\in C
  10. I ( c ) = log 2 | C | I(c)=\log_{2}|C|
  11. c c
  12. v v
  13. I ( c v ) = log 2 | C v | I(c_{v})=\log_{2}|C_{v}|
  14. | C | | C v | \frac{|C|}{|C_{v}|}
  15. | C v | |C_{v}|
  16. | C | |C|
  17. I ( v ) = log 2 | C | | C v | I(v)=\log_{2}\frac{|C|}{|C_{v}|}
  18. v v
  19. I ( v ) = - log 2 p ( v ) I(v)=-\log_{2}p(v)
  20. p ( v ) I ( v ) p(v)I(v)
  21. v v
  22. H ( V ) = - p ( v ) log p ( v ) H(V)=-\sum p(v)\log p(v)
  23. H ( V ) H(V)
  24. V V
  25. p p
  26. V V
  27. ξ = [ ξ i ] n \xi=[\xi^{i}]\in\mathbb{R}^{n}
  28. V V
  29. supp ( V ) \,\text{supp}(V)
  30. ξ \xi
  31. ξ \xi
  32. V V
  33. V V
  34. V V
  35. V V
  36. V V
  37. p ( v ) = ξ i p i ( v ) = ξ i p i p(v)=\sum\xi^{i}p_{i}(v)=\xi^{i}p_{i}
  38. p i ( v j ) = 1 \sum p_{i}(v_{j})=1
  39. ξ i = 1 \sum\xi^{i}=1
  40. p i ( v ) p_{i}(v)
  41. m m
  42. ρ = A ξ + B \rho=A\xi+B
  43. I ( v ) = log p ( v ) = E ( v ) + ξ i F i ( v ) I(v)=\log p(v)=E(v)+\sum\xi^{i}F_{i}(v)
  44. e e
  45. p ( v ) p(v)
  46. e e
  47. p p
  48. log p \log p
  49. e e
  50. log p ( v ) = E ( v ) + ξ i F i ( v ) - ψ ( ξ ) \log p(v)=E(v)+\sum\xi^{i}F_{i}(v)-\psi(\xi)
  51. ψ ( ξ ) \psi(\xi)
  52. [ F i ] [ F i , 1 ] [F_{i}]\rightarrow[F_{i},1]
  53. V V
  54. ξ n \xi\in\mathbb{R}^{n}
  55. p ( v ; ξ ) p(v;\xi)
  56. ξ \xi
  57. [ ξ i ] [\xi^{i}]
  58. p ( v ; ξ ) p(v;\xi)
  59. M M
  60. i f = f ξ i := f ¯ ξ i \partial_{i}f=\frac{\partial f}{\partial\xi^{i}}:=\frac{\partial\bar{f}}{% \partial\xi^{i}}
  61. f ¯ = f ξ - 1 \bar{f}=f\circ\xi^{-1}
  62. f ( M ) f\in\mathcal{F}(M)
  63. M M
  64. f f
  65. M M
  66. ϕ = ( f ξ - 1 ) - 1 = ξ f - 1 \phi=(f\circ\xi^{-1})^{-1}=\xi\circ f^{-1}
  67. f f
  68. f = log f=\log
  69. f - 1 = exp f^{-1}=\exp
  70. log p \log p
  71. e e
  72. p p
  73. m m
  74. e e
  75. m m
  76. M M
  77. q q
  78. T q M = { X i i | X n , i = ξ i } T_{q}M=\left\{X^{i}\partial_{i}\Big|X\in\mathbb{R}^{n},\partial_{i}=\frac{% \partial}{\partial\xi^{i}}\right\}
  79. i \partial_{i}
  80. p ( v ; ξ ) p(v;\xi)
  81. X i i p X^{i}\partial_{i}p
  82. m m
  83. X i i log p X^{i}\partial_{i}\log p
  84. e e
  85. f f
  86. p p
  87. α \alpha
  88. 1 1
  89. 0
  90. - 1 -1
  91. m m
  92. α = - 1 \alpha=-1
  93. ( - 1 ) = 2 1 - α p 1 - α 2 = p \ell^{(-1)}=\frac{2}{1-\alpha}p^{\frac{1-\alpha}{2}}=p
  94. e e
  95. α = 1 \alpha=1
  96. = ( 1 ) = log p ( X ( e ) = 1 p X ( m ) \ell=\ell^{(1)}=\log p(X^{(e)}=\frac{1}{p}X^{(m)}
  97. 0
  98. α = 0 \alpha=0
  99. ( 0 ) = 2 1 - α p 1 - α 2 = 2 p \ell^{(0)}=\frac{2}{1-\alpha}p^{\frac{1-\alpha}{2}}=2\sqrt{p}
  100. X ( 0 ) = 1 p X ( m ) X^{(0)}=\frac{1}{\sqrt{p}}X^{(m)}
  101. α ( v ; ξ ) = E ( v ) + ξ i F i ( v ) \ell^{\alpha}(v;\xi)=E(v)+\xi^{i}F_{i}(v)
  102. α \alpha
  103. m m
  104. e e
  105. 0
  106. α \alpha
  107. α \alpha
  108. X ( α ) = X i i α X^{(\alpha)}=X^{i}\partial_{i}\ell^{\alpha}
  109. M M
  110. q q
  111. , q : T q × T q \langle\;,\;\rangle_{q}:T_{q}\times T_{q}\to\mathbb{R}
  112. || X || = X , X ||X||=\sqrt{\langle X,X\rangle}
  113. d s ds
  114. d V dV
  115. T q * M = { T q } T_{q}^{*}M=\{T_{q}\rightarrow\mathbb{R}\}
  116. T q T_{q}
  117. g i j = p i j = E ( i j ) g_{ij}=\sum{p\partial_{i}\ell\partial_{j}\ell}=E(\partial_{i}\ell\partial_{j}\ell)
  118. i \partial_{i}\ell
  119. i i
  120. e e
  121. g i j = - E ( i j ) g_{ij}=-E(\partial_{i}\partial_{j}\ell)
  122. j p i = ( j p i + p i j ) = j i p = 0 \partial_{j}\sum{p\partial_{i}\ell}=\sum(\partial_{j}p\partial_{i}\ell+p% \partial_{i}\partial_{j}\ell)=\partial_{j}\partial_{i}\sum p=0
  123. g i j = 1 p i p j p g_{ij}=\sum{\frac{1}{p}\partial_{i}p\partial_{j}p}
  124. g i j α = i ( α ) j ( - α ) g_{ij}^{\alpha}=\sum{\partial_{i}\ell^{(\alpha)}\partial_{j}\ell^{(-\alpha)}}
  125. ± 1 \pm 1
  126. 0
  127. g i j = D [ i j | | ] = D [ | | i j ] = - D [ i | | j ] g_{ij}=D[\partial_{i}\partial_{j}||]=D[||\partial_{i}\partial_{j}]=-D[\partial% _{i}||\partial_{j}]
  128. D ( p | | q ) 0 D(p||q)\geq 0
  129. 0
  130. p = q p=q
  131. D [ i | | ] = i D ( p | | p ) = 0 D[\partial_{i}||]=\partial_{i}D(p||p)=0
  132. D [ | | j ] = j D ( p | | p ) = 0 D[||\partial_{j}]=\partial^{\prime}_{j}D(p||p)=0
  133. i \partial_{i}
  134. i \partial^{\prime}_{i}
  135. D ( p | | q ) = D ( - 1 ) ( p | | q ) = D ( 1 ) ( q | | p ) = p log p q D(p||q)=D^{(-1)}(p||q)=D^{(1)}(q||p)=\sum{p\log\frac{p}{q}}
  136. ± 1 \pm 1
  137. g i j = - D [ i | | j ] = - j i p ( log p - log ; q ) = i p j q q = [ p = q ] = p i j g_{ij}=-D[\partial_{i}||\partial_{j}]=-\partial^{\prime}_{j}\partial_{i}\sum{p% (\log p-\log;q)}=\sum\frac{\partial_{i}p\partial_{j}q}{q}=[p=q]=\sum{p\partial% _{i}\ell\partial_{j}\ell}
  138. α ± 1 \alpha\neq\pm 1
  139. D ( α ) = 4 1 - α 2 ( 1 - p 1 - α 2 p 1 + α 2 ) D^{(\alpha)}=\frac{4}{1-\alpha^{2}}(1-\sum p^{\frac{1-\alpha}{2}}p^{\frac{1+% \alpha}{2}})
  140. D ( 0 ) ( p | | q ) = 2 ( ( p ) - ( q ) ) 2 = 4 ( 1 - p q ) D^{(0)}(p||q)=2\sum{(\sqrt{(}p)-\sqrt{(}q))^{2}}=4(1-\sum{\sqrt{pq}})
  141. 0
  142. - D ( 0 ) [ i | | j ] -D^{(0)}[\partial_{i}||\partial_{j}]
  143. D ( p | | q ) D(p||q)
  144. G Y + G ( Y | X ) = G X G_{Y}+G_{(Y|X)}=G_{X}
  145. p ( y ) = p ( y | x ) p ( x ) p(y)=p(y|x)p(x)
  146. G = g i j G=g_{ij}
  147. E [ X ( e ) ] = E [ X i i log p ] = 0 E[X^{(e)}]=E[X^{i}\partial_{i}\log p]=0
  148. B | V | B\in\mathbb{R}^{|V|}
  149. E [ B ] = 0 E[B]=0
  150. T p ( e ) T_{p}^{(e)}
  151. A | V | A\in\mathbb{R}^{|V|}
  152. E [ A - E [ A ] ] = 0 E[A-E[A]]=0
  153. A - E [ A ] T p ( e ) A-E[A]\in T_{p}^{(e)}
  154. X ( E [ A ] ) = X ( m ) A = X i i A = X i p i log p A = E [ X ( e ) A ] = E [ X ( e ) A ] - 0 = E [ X ( e ) A ] - E [ X ( e ) E [ A ] ] = E [ X ( e ) ( A - E [ A ] ) ] = E [ X ( e ) Y ( e ) ] = X , Y X(E[A])=\sum X^{(m)}A=\sum X^{i}\partial_{i}A=\sum X^{i}p\partial_{i}\log pA=E% [X^{(e)}A]=E[X^{(e)}A]-0=E[X^{(e)}A]-E[X^{(e)}E[A]]=E[X^{(e)}(A-E[A])]=E[X^{(e% )}Y^{(e)}]=\langle X,Y\rangle
  155. Y ( e ) = A - E [ A ] = grad E [ A ] Y^{(e)}=A-E[A]=\,\text{grad}E[A]
  156. || d E [ A ] || 2 = Y ( e ) , Y ( e ) = E [ ( A - E [ A ] ) 2 ] = V [ A ] ||dE[A]||^{2}=\langle Y^{(e)},Y^{(e)}\rangle=E[(A-E[A])^{2}]=V[A]
  157. || d E [ A ] || 2 = G - 1 ||dE[A]||^{2}=G^{-1}
  158. V [ A ] G - 1 V[A]\geq G^{-1}
  159. : T M × T M T M \nabla:TM\times TM\rightarrow TM
  160. X X
  161. Y Y
  162. T M TM
  163. X Y \nabla_{X}Y
  164. Y Y
  165. X X
  166. Y Y
  167. X X
  168. Y Y
  169. X X
  170. k \partial_{k}
  171. ( X Y ) k = X i ( i Y ) k = X i ( i Y k + Y j Γ i j k ) \left(\nabla_{X}Y\right)^{k}=X^{i}\left(\nabla_{i}Y\right)^{k}=X^{i}(\partial_% {i}Y^{k}+Y^{j}\Gamma_{ij}^{k})
  172. Γ i j k \Gamma_{ij}^{k}
  173. g i j g_{ij}
  174. Π q , q ( i ) \Pi_{q,{q^{\prime}}}(\partial_{i})
  175. q q
  176. q q^{\prime}
  177. T q M T_{q^{\prime}}M
  178. Π q , q ( j ) = ( j ) q - d ξ i Γ i j k ( k ) q \Pi_{q,q^{\prime}}(\partial_{j})=(\partial_{j})_{q^{\prime}}-d\xi^{i}\Gamma_{% ij}^{k}(\partial_{k})_{q^{\prime}}
  179. j \partial_{j}
  180. T q M T_{q^{\prime}}M
  181. j \partial_{j}
  182. * \nabla^{*}
  183. i g j k = i j , k + j , i * k = Γ i j , k + Γ i k , j * = 0 \partial_{i}g_{jk}=\langle\nabla_{\partial_{i}}\partial_{j},\partial_{k}% \rangle+\langle\partial_{j},\nabla^{*}_{\partial_{i}}\partial_{k}\rangle=% \Gamma_{ij,k}+\Gamma_{ik,j}*=0
  184. \nabla
  185. * \nabla^{*}
  186. α \alpha
  187. α \alpha
  188. Γ i j , k ( α ) = E [ ( i j + 1 - α 2 i j ) k ] \Gamma_{ij,k}^{(\alpha)}=E[(\partial_{i}\partial_{j}\ell+\frac{1-\alpha}{2}% \partial_{i}\ell\partial_{j}\ell)\partial_{k}\ell]
  189. Γ i j , k ( α ) = i j ( α ) k ( - α ) \Gamma_{ij,k}^{(\alpha)}=\sum\partial_{i}\partial_{j}\ell^{(\alpha)}\partial_{% k}\ell^{(-\alpha)}
  190. Γ i j , k ( α ) = - D ( α ) [ i j | | k ] ( D ( - α ) [ p | | q ] = D ( α ) [ q | | p ] ) \Gamma_{ij,k}^{(\alpha)}=-D^{(\alpha)}[\partial_{i}\partial_{j}||\partial_{k}]% \;(D^{(-\alpha)}[p||q]=D^{(\alpha)}[q||p])
  191. α = ± 1 , 0 \alpha=\pm 1,0
  192. Γ i j , k ( 0 ) \Gamma_{ij,k}^{(0)}
  193. Γ i j , k ( α ) = Γ i j , k ( 0 ) + α T i j k \Gamma_{ij,k}^{(\alpha)}=\Gamma_{ij,k}^{(0)}+\alpha T_{ijk}
  194. T i j k = 1 2 E [ i l j l k l ] T_{ijk}=\frac{1}{2}E[\partial_{i}l\partial_{j}l\partial_{k}l]
  195. Γ i j , k ( α ) + Γ i k , j ( - α ) = Γ i j , k ( 0 ) + α T i j k + Γ i k , j ( 0 ) - α T i j k = i g j k \Gamma_{ij,k}^{(\alpha)}+\Gamma_{ik,j}^{(-\alpha)}=\Gamma_{ij,k}^{(0)}+\alpha T% _{ijk}+\Gamma_{ik,j}^{(0)}-\alpha T_{ijk}=\partial_{i}g_{jk}
  196. Γ i j , k ( - α ) \Gamma_{ij,k}^{(-\alpha)}
  197. Γ i j , k ( α ) \Gamma_{ij,k}^{(\alpha)}
  198. Γ i j , k ( α ) = 0 \Gamma_{ij,k}^{(\alpha)}=0
  199. α \alpha
  200. - α -\alpha
  201. Γ i j , k ( - 1 ) = i j ( ξ i p i ) k ( 1 ) = 0 \Gamma_{ij,k}^{(-1)}=\sum\partial_{i}\partial_{j}(\xi^{i}p_{i})\partial_{k}% \ell^{(1)}=0
  202. Γ i j , k ( 1 ) = 0 \Gamma_{ij,k}^{(1)}=0
  203. D ( p | | q ) 0 D(p||q)\geq 0
  204. 0
  205. p = q p=q
  206. D [ i | | ] = i D ( p | | p ) = 0 D[\partial_{i}||]=\partial_{i}D(p||p)=0
  207. D [ | | j ] = j D ( p | | p ) = 0 D[||\partial_{j}]=\partial^{\prime}_{j}D(p||p)=0
  208. i \partial_{i}
  209. i \partial^{\prime}_{i}
  210. i \partial_{i}
  211. D [ i | | ] = - D [ | | i ] D[\partial_{i}||]=-D[||\partial_{i}]
  212. D [ i j | | ] = D [ | | i j ] = - D [ i | | j ] D[\partial_{i}\partial_{j}||]=D[||\partial_{i}\partial_{j}]=-D[\partial_{i}||% \partial_{j}]
  213. g i j = - D [ i | | j ] g_{ij}=-D[\partial_{i}||\partial_{j}]
  214. g i j g_{ij}
  215. i \partial_{i}
  216. i g j k = - D [ i j | | k ] - D [ j | | i k ] = Γ i j , k + Γ i k , j * \partial_{i}g_{jk}=-D[\partial_{i}\partial_{j}||\partial_{k}]-D[\partial_{j}||% \partial_{i}\partial_{k}]=\Gamma_{ij,k}+\Gamma_{ik,j}^{*}
  217. D ( p | | q ) D(p||q)
  218. D [ p | | q ] = 1 2 g i j ( q ) Δ ξ i Δ ξ j + 1 6 h i j k Δ ξ i Δ ξ j Δ ξ k + o ( | | Δ ξ | | 3 ) \displaystyle D[p||q]=\frac{1}{2}g_{ij}(q)\Delta\xi^{i}\Delta\xi^{j}+\frac{1}{% 6}h_{ijk}\Delta\xi^{i}\Delta\xi^{j}\Delta\xi^{k}+o(||\Delta\xi||^{3})
  219. D ( p | | q ) D(p||q)
  220. D ( p | | q ) = p f ( q p ) D(p||q)=\sum pf(\frac{q}{p})
  221. f f
  222. u > 0 u>0
  223. D ( p | | q ) ) f p q p = f ( 1 ) D(p||q))\geq f\sum p\frac{q}{p}=f(1)
  224. f ( u ) = u log u f(u)=u\log u
  225. D ( p | | q ) = D ( - 1 ) ( p | | q ) = D ( 1 ) ( q | | p ) = p log p q , D(p||q)=D^{(-1)}(p||q)=D^{(1)}(q||p)=\sum{p\log\frac{p}{q}},
  226. ± 1 \pm 1
  227. g i j = - D [ i | | j ] = - j i p ( log ; p - log ; q ) = i p j q q = i p j log q = [ p = q ] = p i j g_{ij}=-D[\partial_{i}||\partial_{j}]=-\partial^{\prime}_{j}\partial_{i}\sum{p% (\log;p-\log;q)}=\sum\frac{\partial_{i}p\partial_{j}q}{q}=\sum\partial_{i}p% \partial_{j}\log q=[p=q]=\sum{p\partial_{i}\ell\partial_{j}\ell}
  228. α ± 1 \alpha\neq\pm 1
  229. f f
  230. D ( α ) = 4 1 - α 2 ( 1 - p 1 - α 2 p 1 + α 2 ) . D^{(\alpha)}=\frac{4}{1-\alpha^{2}}(1-\sum p^{\frac{1-\alpha}{2}}p^{\frac{1+% \alpha}{2}}).
  231. 0
  232. D ( 0 ) ( p | | q ) = 2 ( ( p ) - ( q ) ) 2 = 4 ( 1 - p q ) . D^{(0)}(p||q)=2\sum{(\sqrt{(}p)-\sqrt{(}q))^{2}}=4(1-\sum{\sqrt{pq}}).
  233. - D ( 0 ) [ i | | j ] -D^{(0)}[\partial_{i}||\partial_{j}]
  234. S S
  235. S * S^{*}
  236. [ θ i ] [\theta^{i}]
  237. [ η j ] [\eta_{j}]
  238. i = θ i \partial_{i}=\frac{\partial}{\partial\theta^{i}}
  239. i = η i \partial^{i}=\frac{\partial}{\partial\eta_{i}}
  240. , : T S × T S * \langle,\rangle:TS\times TS^{*}\rightarrow\mathbb{R}
  241. 0 \geq 0
  242. \nabla
  243. S S
  244. * \nabla^{*}
  245. S * S^{*}
  246. X , X * \langle X,X^{*}\rangle
  247. X T S X\in TS
  248. X * T S * X^{*}\in TS^{*}
  249. \nabla
  250. * \nabla^{*}
  251. S S
  252. i \partial_{i}
  253. S S
  254. i , j \langle\partial_{i},\partial^{j}\rangle
  255. j \partial^{j}
  256. S * S^{*}
  257. i , j = δ i j \langle\partial_{i},\partial^{j}\rangle=\delta_{i}^{j}
  258. S * S^{*}
  259. f f
  260. S S
  261. η i = θ i f - 1 \eta_{i}=\theta^{i}\circ f^{-1}
  262. S S
  263. \nabla
  264. S S
  265. * \nabla^{*}
  266. S * S^{*}
  267. ( S , g , , * ) (S,g,\nabla,\nabla^{*})
  268. j = ( j θ i ) i = g i j i \partial^{j}=(\partial^{j}\theta^{i})\partial_{i}=g^{ij}\partial_{i}
  269. i = ( i η j ) j = g i j j \partial_{i}=(\partial_{i}\eta_{j})\partial^{j}=g_{ij}\partial^{j}
  270. j θ i = i θ j \partial^{j}\theta^{i}=\partial^{i}\theta^{j}
  271. η \eta
  272. ψ ( θ ) \psi(\theta)
  273. ϕ ( η ) \phi(\eta)
  274. i ψ = η i \partial_{i}\psi=\eta_{i}
  275. i ϕ = θ i \partial^{i}\phi=\theta^{i}
  276. ψ ( θ ) = max η { θ i η i - ϕ ( η ) } \psi(\theta)={\max}_{\eta}\{\theta^{i}\eta_{i}-\phi(\eta)\}
  277. ϕ ( η ) = max θ { θ i η i - ψ ( θ ) } \phi(\eta)={\max}_{\theta}\{\theta^{i}\eta_{i}-\psi(\theta)\}
  278. g i j = i , j = i η j = i j ψ g_{ij}=\langle\partial_{i},\partial_{j}\rangle=\partial_{i}\eta_{j}=\partial_{% i}\partial_{j}\psi
  279. g i j = i , j = i θ j = i j ϕ g^{ij}=\langle\partial^{i},\partial^{j}\rangle=\partial^{i}\theta^{j}=\partial% ^{i}\partial^{j}\phi
  280. i g j k = ( Γ i j , k = 0 ) + Γ i k , j * = i j k ψ \partial_{i}g_{jk}=(\Gamma_{ij,k}=0)+\Gamma_{ik,j}^{*}=\partial_{i}\partial_{j% }\partial_{k}\psi
  281. i g j k = Γ i j , k + ( Γ ( * ) i k , j = 0 ) = i j k ϕ \partial^{i}g^{jk}=\Gamma^{ij,k}+(\Gamma^{(*)ik,j}=0)=\partial^{i}\partial^{j}% \partial^{k}\phi
  282. D ( p | | q ) = ψ ( p ) + ϕ ( q ) - θ i ( p ) η i ( q ) D(p||q)=\psi(p)+\phi(q)-\theta^{i}(p)\eta_{i}(q)
  283. i , j = δ i j \langle\partial_{i},\partial^{j}\rangle=\delta_{i}^{j}
  284. i , j = δ i j \langle\partial_{i},\partial^{j}\rangle=\delta_{i}^{j}
  285. g i j = - D [ i | | j ] g_{ij}=-D[\partial_{i}||\partial_{j}]
  286. α = ± 1 \alpha=\pm 1
  287. ψ = ϕ = 1 2 ( θ i ) 2 \psi=\phi=\frac{1}{2}\sum{(\theta^{i})^{2}}
  288. D ( p | | q ) = 1 2 ( θ i ( p ) - θ i ( q ) ) 2 = 1 2 d ( p , q ) 2 D(p||q)=\frac{1}{2}\sum(\theta^{i}(p)-\theta^{i}(q))^{2}=\frac{1}{2}d(p,q)^{2}
  289. D ( p | | q ) + D ( q | | r ) - D ( p | | r ) = ( θ i ( p ) - θ i ( q ) ) ( η i ( r ) - η i ( q ) ) D(p||q)+D(q||r)-D(p||r)=(\theta^{i}(p)-\theta^{i}(q))(\eta_{i}(r)-\eta_{i}(q))
  290. ϕ ( η ) = θ i η i - ψ ( θ ) \phi(\eta)=\theta^{i}\eta_{i}-\psi(\theta)
  291. ( g , , * ) (g,\nabla,\nabla^{*})
  292. D ( p | | q ) + D ( q | | r ) - D ( p | | r ) = - 1 ( p ) , - 1 ( r ) + o ( max { | | ξ ( p ) - ξ ( q ) | | , | | ξ ( p ) - ξ ( r ) | | } 3 ) D(p||q)+D(q||r)-D(p||r)=\langle\mathcal{E}^{-1}(p),\mathcal{E}^{-1}(r)\rangle+% o(\max\{||\xi(p)-\xi(q)||,||\xi(p)-\xi(r)||\}^{3})
  293. \mathcal{E}
  294. p p
  295. r r
  296. q q
  297. ( θ i ( p ) - θ i ( q ) ) ( η i ( r ) - η i ( q ) ) = 0 (\theta^{i}(p)-\theta^{i}(q))(\eta_{i}(r)-\eta_{i}(q))=0
  298. D ( p | | r ) = D ( p | | q ) + D ( q | | r ) D(p||r)=D(p||q)+D(q||r)
  299. q S q\in S
  300. p , r M p,r\in M
  301. M M
  302. * \nabla^{*}
  303. D ( p | | q ) = min D ( p | | r ) D(p||q)=\min D(p||r)
  304. \nabla
  305. p p
  306. q q
  307. M M
  308. ( g , , * ) (g,\nabla,\nabla*)
  309. ( g γ , γ , γ * ) (g_{\gamma},\nabla_{\gamma},\nabla^{*}_{\gamma})
  310. γ : [ a , b ] S \gamma:[a,b]\rightarrow S
  311. D γ ( γ ( b ) | | γ ( a ) ) = g γ ( s ) μ ( t ) μ ( s ) d s d t D_{\gamma}(\gamma(b)||\gamma(a))=\int\int g_{\gamma}(s)\frac{\mu(t)}{\mu(s)}dsdt
  312. g γ = g i j γ ˙ i γ ˙ j g_{\gamma}=g_{ij}\dot{\gamma}^{i}\dot{\gamma}^{j}
  313. μ ( t ) = e a t Γ γ ( s ) d s \mu(t)=e^{\int_{a}^{t}{\Gamma_{\gamma}(s)}ds}
  314. Γ γ ( s ) = { γ ˙ i γ ˙ j Γ i j , k + γ ¨ j g i j } γ ˙ k / g γ \Gamma_{\gamma}(s)=\{\dot{\gamma}^{i}\dot{\gamma}^{j}\Gamma_{ij,k}+\ddot{% \gamma}^{j}g_{ij}\}\dot{\gamma}^{k}/g_{\gamma}
  315. Γ γ ( s ) = 0 \Gamma_{\gamma}(s)=0
  316. D γ ( γ ( b ) | | γ ( a ) ) = a b ( b - s ) g γ ( s ) d s D_{\gamma}(\gamma(b)||\gamma(a))=\int_{a}^{b}(b-s)g_{\gamma}(s)ds
  317. a b = Γ a b c c \nabla_{\partial_{a}}\partial_{b}=\Gamma_{ab}^{c}\partial_{c}
  318. p ( v ; θ ) = exp [ C ( v ) + θ i F i ( v ) - ψ ( θ ) ] p(v;\theta)=\exp[C(v)+\theta^{i}F_{i}(v)-\psi(\theta)]
  319. exp [ ψ ( θ ) ] = ( C ( v ) + θ i F i ) \exp[\psi(\theta)]=\sum(C(v)+\theta^{i}F_{i})
  320. i \partial_{i}
  321. η i ( θ ) = i ψ ( θ ) = F i ( v ) p ( v ; θ ) = E [ F i ] \eta_{i}(\theta)=\partial_{i}\psi(\theta)=\sum F_{i}(v)p(v;\theta)=E[F_{i}]
  322. ϕ ( θ ) = θ i η i ( θ ) - ψ ( θ ) = θ i p F i - ψ = E [ log ; p - C ] = - H ( p ) - E [ C ] \phi(\theta)=\theta^{i}\eta_{i}(\theta)-\psi(\theta)=\theta^{i}\sum pF_{i}-% \psi=E[\log;p-C]=-H(p)-E[C]
  323. H H
  324. θ i F i = log p - C ( v ) + ψ ( θ ) \theta^{i}F_{i}=\log p-C(v)+\psi(\theta)
  325. E [ ψ ( θ ) ] = ψ ( θ ) E[\psi(\theta)]=\psi(\theta)
  326. g i j = E [ i j ] = E [ ( F i - η i ) ( F j - η j ) ] = V [ η ] g_{ij}=E[\partial_{i}\ell\partial_{j}\ell]=E[(F_{i}-\eta_{i})(F_{j}-\eta_{j})]% =V[\eta]
  327. η \eta
  328. D ( p | | q ) = p ( log p - log q ) D(p||q)=\sum p(\log p-\log q)
  329. D ( p | | q ) + D ( q | | r ) - D ( p | | r ) = ( p - q ) ( log ; p - log ; q ) D(p||q)+D(q||r)-D(p||r)=\sum(p-q)(\log;p-\log;q)
  330. η i \eta_{i}
  331. H ( p ) + E [ C ] H(p)+E[C]
  332. C = 0 C=0
  333. α \alpha
  334. ( α ) = C ( v ) + θ i F i \ell^{(\alpha)}=C(v)+\theta^{i}F_{i}
  335. η i = F i ( - α ) \displaystyle\eta_{i}=\sum F_{i}\ell^{(-\alpha)}
  336. α = ± 1 \alpha=\pm 1
  337. q q
  338. D ( α ) ( p | | r ) = D ( α ) ( p | | q ) + D ( α ) ( q | | r ) - 1 - α 2 4 D ( α ) ( p | | q ) D ( α ) ( q | | r ) D^{(\alpha)}(p||r)=D^{(\alpha)}(p||q)+D^{(\alpha)}(q||r)-\frac{1-\alpha^{2}}{4% }D^{(\alpha)}(p||q)D^{(\alpha)}(q||r)

Initial_value_problem.html

  1. y ( t ) = f ( t , y ( t ) ) y^{\prime}(t)=f(t,y(t))
  2. f : Ω × n n f\colon\Omega\subset\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}^{n}
  3. Ω \Omega
  4. × n \mathbb{R}\times\mathbb{R}^{n}
  5. f f
  6. ( t 0 , y 0 ) Ω (t_{0},y_{0})\in\Omega
  7. y y
  8. y ( t 0 ) = y 0 y(t_{0})=y_{0}
  9. y i ( t ) = f i ( t , y 1 ( t ) , y 2 ( t ) , ) y_{i}^{\prime}(t)=f_{i}(t,y_{1}(t),y_{2}(t),\ldots)
  10. y ( t ) y(t)
  11. ( y 1 ( t ) , , y n ( t ) ) (y_{1}(t),\ldots,y_{n}(t))
  12. y y
  13. y ′′ ( t ) = f ( t , y ( t ) , y ( t ) ) y^{\prime\prime}(t)=f(t,y(t),y^{\prime}(t))
  14. y = 0.85 y y^{\prime}=0.85y
  15. y ( 0 ) = 19 y(0)=19
  16. y ( t ) y(t)
  17. y = d y d t y^{\prime}=\frac{dy}{dt}
  18. d y d t = 0.85 y \frac{dy}{dt}=0.85y
  19. y y
  20. t t
  21. d y y = 0.85 d t \frac{dy}{y}=0.85dt
  22. B B
  23. ln | y | = 0.85 t + B \ln|y|=0.85t+B
  24. ln \ln
  25. | y | = e B e 0.85 t |y|=e^{B}e^{0.85t}
  26. C C
  27. C = ± e B C=\pm e^{B}
  28. y = C e 0.85 t y=Ce^{0.85t}
  29. C C
  30. y ( 0 ) = 19 y(0)=19
  31. t t
  32. y y
  33. 19 = C e 0.85 * 0 19=Ce^{0.85*0}
  34. C = 19 C=19
  35. y ( t ) = 19 e 0.85 t y(t)=19e^{0.85t}
  36. y + 3 y = 6 t + 5 , y ( 0 ) = 3 y^{\prime}+3y=6t+5,\qquad y(0)=3
  37. y ( t ) = 2 e - 3 t + 2 t + 1. y(t)=2e^{-3t}+2t+1.\,
  38. y + 3 y \displaystyle y^{\prime}+3y

Injective_module.html

  1. R 𝔭 / R R_{\mathfrak{p}}/R
  2. 𝔭 \mathfrak{p}
  3. R ^ P \hat{R}_{P}

Injective_object.html

  1. \mathfrak{C}
  2. \mathcal{H}
  3. \mathfrak{C}
  4. Q Q
  5. \mathfrak{C}
  6. \mathcal{H}
  7. f : A Q f:A\to Q
  8. h : A B h:A\to B
  9. \mathcal{H}
  10. g : B Q g:B\to Q
  11. f f
  12. g h = f gh=f
  13. Q Q
  14. \mathcal{H}
  15. Q Q
  16. Q Q
  17. g g
  18. h h
  19. f f
  20. H o m ( - , Q ) Hom_{\mathfrak{C}}(-,Q)
  21. \mathcal{H}
  22. \mathcal{H}
  23. \mathfrak{C}
  24. \mathfrak{C}
  25. 0 A B C 0 0\to A\to B\to C\to 0
  26. \mathfrak{C}
  27. \mathfrak{C}
  28. \mathfrak{C}
  29. \mathfrak{C}
  30. \mathfrak{C}
  31. \mathfrak{C}

Inscribed_angle.html

  1. 2 ψ + 180 - θ = 180 . 2\psi+180^{\circ}-\theta=180^{\circ}.
  2. 2 ψ = θ , 2\psi=\theta,\,
  3. D V C = D V E + E V C . \angle DVC=\angle DVE+\angle EVC.\,
  4. ψ 0 = D V C , \psi_{0}=\angle DVC,
  5. ψ 1 = D V E , \psi_{1}=\angle DVE,
  6. ψ 2 = E V C , \psi_{2}=\angle EVC,
  7. ψ 0 = ψ 1 + ψ 2 . ( 1 ) \psi_{0}=\psi_{1}+\psi_{2}.\qquad\qquad(1)
  8. D O C = D O E + E O C . \angle DOC=\angle DOE+\angle EOC.
  9. θ 0 = D O C , \theta_{0}=\angle DOC,
  10. θ 1 = D O E , \theta_{1}=\angle DOE,
  11. θ 2 = E O C , \theta_{2}=\angle EOC,
  12. θ 0 = θ 1 + θ 2 . ( 2 ) \theta_{0}=\theta_{1}+\theta_{2}.\qquad\qquad(2)
  13. θ 1 = 2 ψ 1 \theta_{1}=2\psi_{1}
  14. θ 2 = 2 ψ 2 \theta_{2}=2\psi_{2}
  15. θ 0 = 2 ψ 1 + 2 ψ 2 \theta_{0}=2\psi_{1}+2\psi_{2}\,
  16. θ 0 = 2 ψ 0 . \theta_{0}=2\psi_{0}.\,
  17. D V C = E V C - D V E \angle DVC=\angle EVC-\angle DVE
  18. ψ 0 = D V C , \psi_{0}=\angle DVC,
  19. ψ 1 = D V E , \psi_{1}=\angle DVE,
  20. ψ 2 = E V C , \psi_{2}=\angle EVC,
  21. ψ 0 = ψ 2 - ψ 1 . ( 3 ) \psi_{0}=\psi_{2}-\psi_{1}.\qquad\qquad(3)
  22. D O C = E O C - D O E . \angle DOC=\angle EOC-\angle DOE.
  23. θ 0 = D O C , \theta_{0}=\angle DOC,
  24. θ 1 = D O E , \theta_{1}=\angle DOE,
  25. θ 2 = E O C , \theta_{2}=\angle EOC,
  26. θ 0 = θ 2 - θ 1 . ( 4 ) \theta_{0}=\theta_{2}-\theta_{1}.\qquad\qquad(4)
  27. θ 1 = 2 ψ 1 \theta_{1}=2\psi_{1}
  28. θ 2 = 2 ψ 2 \theta_{2}=2\psi_{2}
  29. θ 0 = 2 ψ 2 - 2 ψ 1 \theta_{0}=2\psi_{2}-2\psi_{1}
  30. θ 0 = 2 ψ 0 . \theta_{0}=2\psi_{0}.

Insertion_device.html

  1. K = q B λ u 2 π β m c K=\frac{qB\lambda_{u}}{2\pi\beta mc}
  2. λ u \lambda_{u}
  3. β = v / c \beta=v/c
  4. N N

Instantaneously_trained_neural_networks.html

  1. w i j = { - 1 , for x i = 0 + 1 , for x i = 1 r + s - 1 , for i = n + 1 w_{ij}=\begin{cases}-1,&\mbox{for }~{}x_{i}=0\\ +1,&\mbox{for }~{}x_{i}=1\\ r+s-1,&\mbox{for }~{}i=n+1\end{cases}
  2. r r
  3. s s
  4. y = { 1 if x i 0 0 if x i < 0 y=\left\{\begin{matrix}1&\mbox{if }~{}\sum x_{i}\geq 0\\ 0&\mbox{if }~{}\sum x_{i}<0\end{matrix}\right.

Instrumentation_amplifier.html

  1. V out V 2 - V 1 = ( 1 + 2 R 1 R gain ) R 3 R 2 \frac{V_{\mathrm{out}}}{V_{2}-V_{1}}=\left(1+{2R_{1}\over R_{\mathrm{gain}}}% \right){R_{3}\over R_{2}}
  2. R 2 R_{\,\text{2}}
  3. R 3 R_{\,\text{3}}
  4. R 3 / R 2 R_{\,\text{3}}/R_{\,\text{2}}
  5. R 2 R_{\,\text{2}}
  6. R gain R_{\,\text{gain}}
  7. R 3 / R 2 R_{\,\text{3}}/R_{\,\text{2}}
  8. R gain R_{\,\text{gain}}
  9. R 1 R_{\,\text{1}}
  10. R gain R_{\,\text{gain}}

Integer-valued_polynomial.html

  1. 1 2 t 2 + 1 2 t = 1 2 ( t ) ( t + 1 ) \frac{1}{2}t^{2}+\frac{1}{2}t=\frac{1}{2}(t)(t+1)
  2. ( t + k k ) {\left({{t+k}\atop{k}}\right)}

Integer_programming.html

  1. maximize 𝐜 T 𝐱 subject to A 𝐱 𝐛 , 𝐱 𝟎 , and 𝐱 n , \begin{aligned}&\displaystyle\,\text{maximize}&&\displaystyle\mathbf{c}^{% \mathrm{T}}\mathbf{x}\\ &\displaystyle\,\text{subject to}&&\displaystyle A\mathbf{x}\leq\mathbf{b},\\ &&&\displaystyle\mathbf{x}\geq\mathbf{0},\\ &\displaystyle\,\text{and}&&\displaystyle\mathbf{x}\in\mathbb{Z}^{n},\end{aligned}
  2. maximize 𝐜 T 𝐱 subject to A 𝐱 + 𝐬 = 𝐛 , 𝐬 𝟎 , and 𝐱 n , \begin{aligned}&\displaystyle\,\text{maximize}&&\displaystyle\mathbf{c}^{% \mathrm{T}}\mathbf{x}\\ &\displaystyle\,\text{subject to}&&\displaystyle A\mathbf{x}+\mathbf{s}=% \mathbf{b},\\ &&&\displaystyle\mathbf{s}\geq\mathbf{0},\\ &\displaystyle\,\text{and}&&\displaystyle\mathbf{x}\in\mathbb{Z}^{n},\end{aligned}
  3. 𝐜 , 𝐛 \mathbf{c},\mathbf{b}
  4. A A
  5. 𝐬 \mathbf{s}
  6. max y - x + y 1 3 x + 2 y 12 2 x + 3 y 12 x , y 0 x , y \begin{aligned}\displaystyle\max&\displaystyle\,\text{ }y\\ \displaystyle-x+y&\displaystyle\leq 1\\ \displaystyle 3x+2y&\displaystyle\leq 12\\ \displaystyle 2x+3y&\displaystyle\leq 12\\ \displaystyle x,y&\displaystyle\geq 0\\ \displaystyle x,y&\displaystyle\in\mathbb{Z}\end{aligned}
  7. ( 1 , 2 ) (1,2)
  8. ( 2 , 2 ) (2,2)
  9. ( 1.8 , 2.8 ) (1.8,2.8)
  10. G = ( V , E ) G=(V,E)
  11. min v V y v \displaystyle\min\sum_{v\in V}y_{v}
  12. y v y_{v}
  13. y v y_{v}
  14. v C v\in C
  15. v C v\not\in C
  16. y v y_{v}
  17. x i x_{i}
  18. 0 x U 0\leq x\leq U
  19. log 2 U + 1 \lfloor\log_{2}U\rfloor+1
  20. x = x 1 + 2 x 2 + 4 x 3 + + 2 log 2 U x log 2 U + 1 . x=x_{1}+2x_{2}+4x_{3}+\ldots+2^{\lfloor\log_{2}U\rfloor}x_{\lfloor\log_{2}U% \rfloor+1}.
  21. max 𝐜 T 𝐱 \max\mathbf{c}^{\mathrm{T}}\mathbf{x}
  22. A 𝐱 = 𝐛 A\mathbf{x}=\mathbf{b}
  23. A , 𝐛 , A,\mathbf{b},
  24. 𝐜 \mathbf{c}
  25. A A
  26. 𝐱 \mathbf{x}
  27. 𝐱 \mathbf{x}
  28. A 𝐱 = 𝐛 A\mathbf{x}=\mathbf{b}
  29. 𝐱 0 = [ x n 1 , x n 2 , , x n j ] \mathbf{x}_{0}=[x_{n_{1}},x_{n_{2}},\cdots,x_{n_{j}}]
  30. 𝐱 \mathbf{x}
  31. B B
  32. A A
  33. B 𝐱 0 = 𝐛 B\mathbf{x}_{0}=\mathbf{b}
  34. B B
  35. B B
  36. B B
  37. B B
  38. det ( B ) = ± 1 \det(B)=\pm 1
  39. B B
  40. 𝐱 0 = B - 1 𝐛 \mathbf{x}_{0}=B^{-1}\mathbf{b}
  41. B - 1 = B a d j det ( B ) = ± B a d j B^{-1}=\frac{B^{adj}}{\det(B)}=\pm B^{adj}
  42. B a d j B^{adj}
  43. B B
  44. B B
  45. B - 1 = ± B a d j is integral. x 0 = B - 1 b is integral. Every basic feasible solution is integral. \begin{aligned}&\displaystyle\Rightarrow B^{-1}=\pm B^{adj}\,\text{ is % integral.}\\ &\displaystyle\Rightarrow x_{0}=B^{-1}b\,\text{ is integral.}\\ &\displaystyle\Rightarrow\,\text{Every basic feasible solution is integral.}% \end{aligned}
  46. A A
  47. A A

Integral_equation.html

  1. f ( x ) = a b K ( x , t ) φ ( t ) d t . f(x)=\int_{a}^{b}K(x,t)\,\varphi(t)\,dt.
  2. φ φ
  3. f f
  4. φ ( x ) = f ( x ) + λ a b K ( x , t ) φ ( t ) d t . \varphi(x)=f(x)+\lambda\int_{a}^{b}K(x,t)\,\varphi(t)\,dt.
  5. λ λ
  6. f ( x ) = a x K ( x , t ) φ ( t ) d t f(x)=\int_{a}^{x}K(x,t)\,\varphi(t)\,dt
  7. φ ( x ) = f ( x ) + λ a x K ( x , t ) φ ( t ) d t . \varphi(x)=f(x)+\lambda\int_{a}^{x}K(x,t)\,\varphi(t)\,dt.
  8. f f
  9. f f
  10. j = 1 n w j K ( s i , t j ) u ( t j ) = f ( s i ) , i = 0 , 1 , , n . \sum_{j=1}^{n}w_{j}K\left(s_{i},t_{j}\right)u(t_{j})=f(s_{i}),\qquad i=0,1,% \cdots,n.
  11. n n
  12. n n
  13. n n
  14. u ( t 0 ) , u ( t 1 ) , , u ( t n ) . u(t_{0}),u(t_{1}),\cdots,u(t_{n}).
  15. f f
  16. φ ( x ) φ(x)
  17. φ ( x ) = f ( x ) + λ a x K ( x , t ) F ( x , t , φ ( t ) ) d t , \varphi(x)=f(x)+\lambda\int_{a}^{x}K(x,t)\,F(x,t,\varphi(t))\,dt,
  18. F F
  19. y ( t ) = λ x ( t ) + 0 k ( t - s ) x ( s ) d s , 0 t < . y(t)=\lambda x(t)+\int^{\infty}_{0}k(t-s)x(s)ds,\qquad 0\leq t<\infty.
  20. K ( x t ) K(xt)
  21. K ( t ) K(t)
  22. g ( s ) = s 0 d t K ( s t ) f ( t ) g(s)=s\int_{0}^{\infty}dtK(st)f(t)
  23. f ( t ) = n = 0 a n M ( n + 1 ) x n f(t)=\sum_{n=0}^{\infty}\frac{a_{n}}{M(n+1)}x^{n}
  24. g ( s ) = n = 0 a n s - n , M ( n + 1 ) = 0 d t K ( t ) t n g(s)=\sum_{n=0}^{\infty}a_{n}s^{-n},\qquad M(n+1)=\int_{0}^{\infty}dtK(t)t^{n}
  25. g ( s ) g(s)
  26. M ( n + 1 ) M(n+1)
  27. j M i , j v j = λ v i \sum_{j}M_{i,j}v_{j}=\lambda v_{i}
  28. 𝐯 \mathbf{v}
  29. λ λ
  30. i i
  31. j j
  32. x x
  33. y y
  34. K ( x , y ) φ ( y ) d y = λ φ ( x ) , \int K(x,y)\varphi(y)\mathrm{d}y=\lambda\varphi(x),
  35. j j
  36. y y
  37. 𝐌 \mathbf{M}
  38. 𝐯 \mathbf{v}
  39. K ( x , y ) K(x,y)
  40. φ ( y ) φ(y)
  41. j j
  42. K ( x , y ) K(x,y)
  43. K K
  44. x = y x=y