wpmath0000002_8

Green's_theorem.html

  1. C ( L d x + M d y ) = D ( M x - L y ) d x d y \oint_{C}(L\,dx+M\,dy)=\iint_{D}\left(\frac{\partial M}{\partial x}-\frac{% \partial L}{\partial y}\right)\,dx\,dy
  2. C L d x = D ( - L y ) d A ( 1 ) \oint_{C}L\,dx=\iint_{D}\left(-\frac{\partial L}{\partial y}\right)\,dA\qquad% \mathrm{(1)}
  3. C M d y = D ( M x ) d A ( 2 ) \oint_{C}M\,dy=\iint_{D}\left(\frac{\partial M}{\partial x}\right)\,dA\qquad% \mathrm{(2)}
  4. D = { ( x , y ) | a x b , g 1 ( x ) y g 2 ( x ) } D=\{(x,y)|a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}
  5. D L y d A \displaystyle\iint_{D}\frac{\partial L}{\partial y}\,dA
  6. C 1 L ( x , y ) d x = a b L ( x , g 1 ( x ) ) d x . \int_{C_{1}}L(x,y)\,dx=\int_{a}^{b}L(x,g_{1}(x))\,dx.
  7. C 3 L ( x , y ) d x = - - C 3 L ( x , y ) d x = - a b L ( x , g 2 ( x ) ) d x . \int_{C_{3}}L(x,y)\,dx=-\int_{-C_{3}}L(x,y)\,dx=-\int_{a}^{b}L(x,g_{2}(x))\,dx.
  8. C 4 L ( x , y ) d x = C 2 L ( x , y ) d x = 0. \int_{C_{4}}L(x,y)\,dx=\int_{C_{2}}L(x,y)\,dx=0.
  9. C L d x = C 1 L ( x , y ) d x + C 2 L ( x , y ) d x + C 3 L ( x , y ) d x + C 4 L ( x , y ) d x = - a b L ( x , g 2 ( x ) ) d x + a b L ( x , g 1 ( x ) ) d x . ( 4 ) \begin{aligned}\displaystyle\int_{C}L\,dx&\displaystyle=\int_{C_{1}}L(x,y)\,dx% +\int_{C_{2}}L(x,y)\,dx+\int_{C_{3}}L(x,y)\,dx+\int_{C_{4}}L(x,y)\,dx\\ &\displaystyle=-\int_{a}^{b}L(x,g_{2}(x))\,dx+\int_{a}^{b}L(x,g_{1}(x))\,dx.% \qquad\mathrm{(4)}\end{aligned}
  10. 𝐅 = ( L , M , 0 ) \mathbf{F}=(L,M,0)
  11. C ( L d x + M d y ) = C ( L , M , 0 ) ( d x , d y , d z ) = C 𝐅 d 𝐫 . \oint_{C}(L\,dx+M\,dy)=\oint_{C}(L,M,0)\cdot(dx,dy,dz)=\oint_{C}\mathbf{F}% \cdot d\mathbf{r}.
  12. C 𝐅 d 𝐫 = S × 𝐅 𝐧 ^ d S . \oint_{C}\mathbf{F}\cdot d\mathbf{r}=\iint_{S}\nabla\times\mathbf{F}\cdot% \mathbf{\hat{n}}\,dS.
  13. S S
  14. D D
  15. 𝐧 ^ \mathbf{\hat{n}}
  16. × 𝐅 𝐧 ^ = [ ( 0 y - M z ) 𝐢 + ( L z - 0 x ) 𝐣 + ( M x - L y ) 𝐤 ] 𝐤 = ( M x - L y ) . \nabla\times\mathbf{F}\cdot\mathbf{\hat{n}}=\left[\left(\frac{\partial 0}{% \partial y}-\frac{\partial M}{\partial z}\right)\mathbf{i}+\left(\frac{% \partial L}{\partial z}-\frac{\partial 0}{\partial x}\right)\mathbf{j}+\left(% \frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\mathbf{k}% \right]\cdot\mathbf{k}=\left(\frac{\partial M}{\partial x}-\frac{\partial L}{% \partial y}\right).
  17. S × 𝐅 𝐧 ^ d S = D ( M x - L y ) d A . \iint_{S}\nabla\times\mathbf{F}\cdot\mathbf{\hat{n}}\,dS=\iint_{D}\left(\frac{% \partial M}{\partial x}-\frac{\partial L}{\partial y}\right)\,dA.
  18. C L d x + M d y = D ω = D d ω = D L y d y d x + M x d x d y = D ( M x - L y ) d x d y . \oint_{C}L\,dx+M\,dy=\oint_{\partial D}\omega=\int_{D}\,d\omega=\int_{D}\frac{% \partial L}{\partial y}\,dy\wedge\,dx+\frac{\partial M}{\partial x}\,dx\wedge% \,dy=\iint_{D}\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y% }\right)\,dx\,dy.
  19. D ( 𝐅 ) d A = C 𝐅 𝐧 ^ d s , \iint_{D}\left(\nabla\cdot\mathbf{F}\right)dA=\oint_{C}\mathbf{F}\cdot\mathbf{% \hat{n}}\,ds,
  20. 𝐅 \nabla\cdot\mathbf{F}
  21. 𝐅 \mathbf{F}
  22. 𝐧 ^ \mathbf{\hat{n}}
  23. 𝐧 ^ \mathbf{\hat{n}}
  24. d 𝐫 = ( d x , d y ) d\mathbf{r}=(dx,dy)
  25. ( d y , - d x ) (dy,-dx)
  26. d x 2 + d y 2 = d s . \sqrt{dx^{2}+dy^{2}}=ds.
  27. ( d y , - d x ) = 𝐧 ^ d s . (dy,-dx)=\mathbf{\hat{n}}\,ds.
  28. C ( L d x + M d y ) = C ( M , - L ) ( d y , - d x ) = C ( M , - L ) 𝐧 ^ d s . \oint_{C}(L\,dx+M\,dy)=\oint_{C}(M,-L)\cdot(dy,-dx)=\oint_{C}(M,-L)\cdot% \mathbf{\hat{n}}\,ds.
  29. 𝐅 = ( M , - L ) \mathbf{F}=(M,-L)
  30. C ( M , - L ) 𝐧 ^ d s = D ( ( M , - L ) ) d A = D ( M x - L y ) d A . \oint_{C}(M,-L)\cdot\mathbf{\hat{n}}\,ds=\iint_{D}\left(\nabla\cdot(M,-L)% \right)dA=\iint_{D}\left(\frac{\partial M}{\partial x}-\frac{\partial L}{% \partial y}\right)\,dA.
  31. A = D d A . A=\iint_{D}dA.
  32. M x - L y = 1. \frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}=1.
  33. A = C ( L d x + M d y ) . A=\oint_{C}(L\,dx+M\,dy).
  34. A = C x d y = - C y d x = 1 2 C ( - y d x + x d y ) . A=\oint_{C}x\,dy=-\oint_{C}y\,dx=\tfrac{1}{2}\oint_{C}(-y\,dx+x\,dy).

Greisen–Zatsepin–Kuzmin_limit.html

  1. γ CMB \gamma_{\rm CMB}
  2. Δ \Delta
  3. γ CMB + p Δ + p + π 0 , \gamma_{\rm CMB}+p\rightarrow\Delta^{+}\rightarrow p+\pi^{0},
  4. γ CMB + p Δ + n + π + . \gamma_{\rm CMB}+p\rightarrow\Delta^{+}\rightarrow n+\pi^{+}.
  5. ν + ν ¯ Z hadrons \nu+\bar{\nu}\rightarrow Z\rightarrow\,\text{hadrons}
  6. E = m Z 2 2 m ν = 4.2 × 10 21 ( eV m ν ) eV E=\frac{m_{Z}^{2}}{2m_{\nu}}=4.2\times 10^{21}\left(\frac{\,\text{eV}}{m_{\nu}% }\right)\,\text{eV}
  7. m Z m_{Z}
  8. m ν m_{\nu}

Ground_state.html

  1. x = 0 x=0
  2. ψ ( 0 ) = 0 \psi(0)=0
  3. ψ | H | ψ = d x ( - 2 2 m ψ * d 2 ψ d x 2 + V ( x ) | ψ ( x ) | 2 ) \left\langle\psi|H|\psi\right\rangle=\int dx\;\left(-\frac{\hbar^{2}}{2m}\psi^% {*}\frac{d^{2}\psi}{dx^{2}}+V(x)|\psi(x)|^{2}\right)
  4. V ( x ) V(x)
  5. x = 0 x=0
  6. x [ - ϵ , ϵ ] x\in[-\epsilon,\epsilon]
  7. ψ ( x ) \psi^{\prime}(x)
  8. ψ ( x ) = ψ ( x ) , x < - ϵ \psi^{\prime}(x)=\psi(x),x<-\epsilon
  9. ψ ( x ) = - ψ ( x ) , x > ϵ \psi^{\prime}(x)=-\psi(x),x>\epsilon
  10. x [ - ϵ , ϵ ] x\in[-\epsilon,\epsilon]
  11. ψ ( x ) \psi^{\prime}(x)
  12. ψ ( x ) - c x \psi(x)\approx-cx
  13. x = 0 x=0
  14. ψ ( x ) = N { | ψ ( x ) | | x | > ϵ c ϵ | x | ϵ \psi^{\prime}(x)=N\left\{\begin{array}[]{ll}|\psi(x)|&|x|>\epsilon\\ c\epsilon&|x|\leq\epsilon\end{array}\right.
  15. N = 1 1 + | c | 2 ϵ 3 / 3 N=\frac{1}{\sqrt{1+|c|^{2}\epsilon^{3}/3}}
  16. | d ψ / d x | 2 < | d ψ / d x | 2 |d\psi^{\prime}/dx|^{2}<|d\psi/dx|^{2}
  17. V ( x ) 0 V(x)\geq 0
  18. x [ - ϵ , ϵ ] x\in[-\epsilon,\epsilon]
  19. ψ \psi^{\prime}
  20. | ψ | < | ψ | |\psi^{\prime}|<|\psi|
  21. x [ - ϵ , ϵ ] x\in[-\epsilon,\epsilon]
  22. V a v g ϵ = - ϵ ϵ d x V ( x ) | ψ | 2 = ϵ 3 | c | 2 1 + | c | 2 ϵ 3 / 3 - ϵ ϵ V ( x ) 2 ϵ 4 | c | 2 3 V ( 0 ) + . {V^{\epsilon}_{avg}}^{\prime}=\int_{-\epsilon}^{\epsilon}dx\;V(x)|\psi^{\prime% }|^{2}=\frac{\epsilon^{3}|c|^{2}}{1+|c|^{2}\epsilon^{3}/3}\int_{-\epsilon}^{% \epsilon}V(x)\approx\frac{2\epsilon^{4}|c|^{2}}{3}V(0)+\dots\;.
  23. ϵ \epsilon
  24. \dots
  25. ψ \psi
  26. V a v g ϵ = - ϵ ϵ d x V ( x ) | ψ | 2 = - ϵ ϵ d x | c | 2 | x | 2 V ( x ) 2 ϵ 4 | c | 2 3 V ( 0 ) + . V^{\epsilon}_{avg}=\int_{-\epsilon}^{\epsilon}dx\;V(x)|\psi|^{2}=\int_{-% \epsilon}^{\epsilon}dx\;|c|^{2}|x|^{2}V(x)\approx\frac{2\epsilon^{4}|c|^{2}}{3% }V(0)+\dots\;.
  27. ψ \psi^{\prime}
  28. ϵ \epsilon
  29. ψ \psi
  30. ψ \psi^{\prime}
  31. h 2 n 2 8 m L 2 \frac{h^{2}n^{2}}{8mL^{2}}

Group_cohomology.html

  1. G n G^{n}
  2. H n ( G , M ) H^{n}(G,M)
  3. M G = { x M | g G : g x = x } . M^{G}=\{x\in M\ |\ \forall g\in G:\ gx=x\}.
  4. 0 L M N 0 0\to L\to M\to N\to 0
  5. 0 L G M G N G δ 0 H 1 ( G , L ) H 1 ( G , M ) H 1 ( G , N ) δ 1 H 2 ( G , L ) 0\to L^{G}\to M^{G}\to N^{G}\overset{\delta^{0}}{\to}H^{1}(G,L)\to H^{1}(G,M)% \to H^{1}(G,N)\overset{\delta^{1}}{\to}H^{2}(G,L)\to\cdots
  6. d n : C n ( G , M ) C n + 1 ( G , M ) d^{n}:C^{n}(G,M)\rightarrow C^{n+1}(G,M)
  7. ( d n φ ) ( g 1 , , g n + 1 ) = g 1 φ ( g 2 , , g n + 1 ) \left(d^{n}\varphi\right)(g_{1},\dots,g_{n+1})=g_{1}\cdot\varphi(g_{2},\dots,g% _{n+1})
  8. + i = 1 n ( - 1 ) i φ ( g 1 , , g i - 1 , g i g i + 1 , g i + 2 , , g n + 1 ) {}+\sum_{i=1}^{n}(-1)^{i}\varphi(g_{1},\dots,g_{i-1},g_{i}g_{i+1},g_{i+2},% \dots,g_{n+1})
  9. + ( - 1 ) n + 1 φ ( g 1 , , g n ) {}+(-1)^{n+1}\varphi(g_{1},\dots,g_{n})
  10. d n + 1 d n = 0 d^{n+1}\circ d^{n}=0
  11. Z n ( G , M ) = ker ( d n ) Z^{n}(G,M)=\operatorname{ker}(d^{n})
  12. { B 0 ( G , M ) = 0 B n ( G , M ) = im ( d n - 1 ) , n 1 \begin{cases}B^{0}(G,M)={0}\\ B^{n}(G,M)=\operatorname{im}(d^{n-1}),\ n\geq 1\end{cases}
  13. H n ( G , M ) = Z n ( G , M ) / B n ( G , M ) . H^{n}(G,M)=Z^{n}(G,M)/B^{n}(G,M).
  14. H n ( G , M ) = Ext 𝐙 [ G ] n ( 𝐙 , M ) , H^{n}(G,M)=\operatorname{Ext}^{n}_{\mathbf{Z}[G]}(\mathbf{Z},M),
  15. F n F n - 1 F 0 𝐙 . \dots\to F_{n}\to F_{n-1}\to\dots\to F_{0}\to\mathbf{Z}.
  16. f n : 𝐙 [ G n + 1 ] 𝐙 [ G n ] , ( g 0 , g 1 , , g n ) i = 0 n ( - 1 ) i ( g 0 , , g i ^ , , g n ) . f_{n}:\mathbf{Z}[G^{n+1}]\to\mathbf{Z}[G^{n}],\quad(g_{0},g_{1},\dots,g_{n})% \mapsto\sum_{i=0}^{n}(-1)^{i}(g_{0},\dots,\widehat{g_{i}},\dots,g_{n}).
  17. Hom G ( F n , M ) Hom G ( F n - 1 , M ) Hom G ( F 0 , M ) Hom G ( 𝐙 , M ) . \cdots\leftarrow\operatorname{Hom}_{G}(F_{n},M)\leftarrow\operatorname{Hom}_{G% }(F_{n-1},M)\leftarrow\dots\leftarrow\operatorname{Hom}_{G}(F_{0},M)\leftarrow% \operatorname{Hom}_{G}(\mathbf{Z},M).
  18. H n ( G , M ) = H n ( Hom G ( F , M ) ) H^{n}(G,M)=H^{n}({\rm Hom}_{G}(F,M))
  19. g ϕ n ( g 1 , g 2 , , g n ) = ϕ n ( g g 1 , g g 2 , , g g n ) . g\phi_{n}(g_{1},g_{2},\ldots,g_{n})=\phi_{n}(gg_{1},gg_{2},\ldots,gg_{n}).
  20. δ ϕ 2 ( g 1 , g 2 , g 3 ) = ϕ 2 ( g 2 , g 3 ) - ϕ 2 ( g 1 , g 3 ) + ϕ 2 ( g 1 , g 2 ) . \delta\phi_{2}(g_{1},g_{2},g_{3})=\phi_{2}(g_{2},g_{3})-\phi_{2}(g_{1},g_{3})+% \phi_{2}(g_{1},g_{2}).
  21. φ \varphi
  22. φ 2 ( g 1 , g 2 ) = ϕ 3 ( 1 , g 1 , g 1 g 2 ) φ 3 ( g 1 , g 2 , g 3 ) = ϕ 4 ( 1 , g 1 , g 1 g 2 , g 1 g 2 g 3 ) , \begin{aligned}\displaystyle\varphi_{2}(g_{1},g_{2})&\displaystyle=\phi_{3}(1,% g_{1},g_{1}g_{2})\\ \displaystyle\varphi_{3}(g_{1},g_{2},g_{3})&\displaystyle=\phi_{4}(1,g_{1},g_{% 1}g_{2},g_{1}g_{2}g_{3}),\end{aligned}
  23. d φ 2 ( g 1 , g 2 , g 3 ) = δ ϕ 3 ( 1 , g 1 , g 1 g 2 , g 1 g 2 g 3 ) = ϕ 3 ( g 1 , g 1 g 2 , g 1 g 2 g 3 ) - ϕ 3 ( 1 , g 1 g 2 , g 1 g 2 g 3 ) + ϕ 3 ( 1 , g 1 , g 1 g 2 g 3 ) - ϕ 3 ( 1 , g 1 , g 1 g 2 ) = g 1 ϕ 3 ( 1 , g 2 , g 2 g 3 ) - ϕ 3 ( 1 , g 1 g 2 , g 1 g 2 g 3 ) + ϕ 3 ( 1 , g 1 , g 1 g 2 g 3 ) - ϕ 3 ( 1 , g 1 , g 1 g 2 ) = g 1 φ 2 ( g 2 , g 3 ) - φ 2 ( g 1 g 2 , g 3 ) + φ 2 ( g 1 , g 2 g 3 ) - φ 2 ( g 1 , g 2 ) , \begin{aligned}\displaystyle d\varphi_{2}(g_{1},g_{2},g_{3})&\displaystyle=% \delta\phi_{3}(1,g_{1},g_{1}g_{2},g_{1}g_{2}g_{3})\\ &\displaystyle=\phi_{3}(g_{1},g_{1}g_{2},g_{1}g_{2}g_{3})-\phi_{3}(1,g_{1}g_{2% },g_{1}g_{2}g_{3})+\phi_{3}(1,g_{1},g_{1}g_{2}g_{3})-\phi_{3}(1,g_{1},g_{1}g_{% 2})\\ &\displaystyle=g_{1}\phi_{3}(1,g_{2},g_{2}g_{3})-\phi_{3}(1,g_{1}g_{2},g_{1}g_% {2}g_{3})+\phi_{3}(1,g_{1},g_{1}g_{2}g_{3})-\phi_{3}(1,g_{1},g_{1}g_{2})\\ &\displaystyle=g_{1}\varphi_{2}(g_{2},g_{3})-\varphi_{2}(g_{1}g_{2},g_{3})+% \varphi_{2}(g_{1},g_{2}g_{3})-\varphi_{2}(g_{1},g_{2}),\end{aligned}
  24. M G := M / D M , M_{G}:=M/DM,\,
  25. H n ( G , M ) H_{n}\left(G,M\right)
  26. H n ( G , M ) = Tor n 𝐙 [ G ] ( 𝐙 , M ) H_{n}(G,M)=\operatorname{Tor}_{n}^{\mathbf{Z}[G]}(\mathbf{Z},M)
  27. F n 𝐙 [ G ] M F n - 1 𝐙 [ G ] M F 0 𝐙 [ G ] M 𝐙 𝐙 [ G ] M . \dots\to F_{n}\otimes_{\mathbf{Z}[G]}M\to F_{n-1}\otimes_{\mathbf{Z}[G]}M\to% \dots\to F_{0}\otimes_{\mathbf{Z}[G]}M\to\mathbf{Z}\otimes_{\mathbf{Z}[G]}M.
  28. H n ( G , M ) = H n ( F 𝐙 [ G ] M ) H_{n}(G,M)=H_{n}(F\otimes_{\mathbf{Z}[G]}M)
  29. H 0 ( G , A ) = A G , H^{0}(G,A)=A^{G},\,
  30. φ φ \ \varphi\sim\varphi^{\prime}
  31. a φ ( g ) = φ ( g ) ( g a ) \ a\varphi^{\prime}(g)=\varphi(g)\cdot(ga)
  32. π 0 ( X ; x ) \ \pi_{0}(X;x)
  33. 1 A B C 1 1\to A\to B\to C\to 1\,
  34. 1 A G B G C G H 1 ( G , A ) H 1 ( G , B ) H 1 ( G , C ) . 1\to A^{G}\to B^{G}\to C^{G}\to H^{1}(G,A)\to H^{1}(G,B)\to H^{1}(G,C).\,
  35. H * ( G ; M ) = n H n ( G ; M ) H^{*}(G;M)=\bigoplus_{n}H^{n}(G;M)\,
  36. H * ( G 1 × G 2 ; k ) H * ( G 1 ; k ) H * ( G 2 ; k ) . H^{*}(G_{1}\times G_{2};k)\cong H^{*}(G_{1};k)\otimes H^{*}(G_{2};k).\,
  37. H * ( G ; k ) k [ x ] , H^{*}(G;k)\cong k[x],\,
  38. H * ( G ; k ) k [ x 1 , , x r ] H^{*}(G;k)\cong k[x_{1},\ldots,x_{r}]
  39. H 2 ( G , 𝐙 ) ( R [ F , F ] ) / [ F , R ] H_{2}(G,\mathbf{Z})\cong(R\cap[F,F])/[F,R]

Growth_accounting.html

  1. Y = F ( A , K , L ) Y=F(A,K,L)
  2. d Y / d K = M P K = r {dY}/{dK}=MPK=r
  3. d Y / d L = M P L = w {dY}/{dL}=MPL=w
  4. d Y = F A d A + F K d K + F L d L dY=F_{A}dA+F_{K}dK+F_{L}dL
  5. F i F_{i}
  6. d Y = F A d A + M P K d K + M P L d L = F A d A + r d K + w d L dY=F_{A}dA+MPKdK+MPLdL=F_{A}dA+rdK+wdL
  7. d Y / Y = ( F A A / Y ) ( d A / A ) + ( r K / Y ) * ( d K / K ) + ( w L / Y ) * ( d L / L ) {dY}/{Y}=({F_{A}}A/{Y})({dA}/{A})+(r{K}/{Y})*({dK}/{K})+(w{L}/{Y})*({dL}/{L})
  8. g i = d i / i g_{i}={di}/{i}
  9. g Y = ( F A A / Y ) * g A + ( r K / Y ) * g K + ( w L / Y ) * g L g_{Y}=({F_{A}}A/{Y})*g_{A}+({rK}/{Y})*g_{K}+({wL}/{Y})*g_{L}
  10. r K / Y {rK}/{Y}
  11. α \alpha
  12. w L / Y {wL}/{Y}
  13. 1 - α 1-\alpha
  14. g Y = F A A / Y * g A + α * g K + ( 1 - α ) * g L g_{Y}={F_{A}}A/{Y}*g_{A}+\alpha*g_{K}+(1-\alpha)*g_{L}
  15. α \alpha
  16. g Y g_{Y}
  17. g K g_{K}
  18. g L g_{L}
  19. F A A Y * g A \frac{F_{A}A}{Y}*g_{A}
  20. S o l o w R e s i d u a l = g Y - α * g K - ( 1 - α ) * g L SolowResidual=g_{Y}-\alpha*g_{K}-(1-\alpha)*g_{L}
  21. S o l o w R e s i d u a l = g ( Y / L ) - α * g ( K / L ) SolowResidual=g_{(Y/L)}-\alpha*g_{(K/L)}

Gudermannian_function.html

  1. gd x = 0 x d t cosh t - < x < . {\rm{gd}}\,x=\int_{0}^{x}\frac{\mathrm{d}t}{\cosh t}\qquad-\infty<x<\infty.
  2. gd x = arcsin ( tanh x ) = arctan ( sinh x ) = arccsc ( coth x ) = sgn ( x ) . arccos ( sech x ) = sgn ( x ) . arcsec ( cosh x ) = 2 arctan [ tanh ( 1 2 x ) ] = 2 arctan ( e x ) - 1 2 π . \begin{aligned}\displaystyle{\rm{gd}}\,x&\displaystyle=\arcsin\left(\tanh x% \right)=\mathrm{arctan}\left(\sinh x\right)=\mathrm{arccsc}\left(\coth x\right% )\\ &\displaystyle=\mbox{sgn}~{}(x).\mathrm{arccos}\left(\mathrm{sech}\,x\right)=% \mbox{sgn}~{}(x).\mathrm{arcsec}\left(\cosh x\right)\\ &\displaystyle=2\,\arctan\left[\tanh\left(\tfrac{1}{2}x\right)\right]\\ &\displaystyle=2\,\arctan(e^{x})-\tfrac{1}{2}\pi.\end{aligned}\,\!
  3. \arccot ( csch x ) \arccot(\mathrm{csch}\,x)
  4. sin gd x = tanh x ; csc gd x = coth x ; cos gd x = sech x ; sec gd x = cosh x ; tan gd x = sinh x ; cot gd x = csch x ; tan ( 1 2 gd x ) = tanh 1 2 x . \begin{aligned}\displaystyle\sin\,\mathrm{gd}\,x&\displaystyle=\tanh x;\quad% \csc\,\mathrm{gd}\,x=\coth x;\\ \displaystyle\cos\,\mathrm{gd}\,x&\displaystyle=\mathrm{sech}\,x;\quad\,\sec\,% \mathrm{gd}\,x=\cosh x;\\ \displaystyle\tan\,\mathrm{gd}\,x&\displaystyle=\sinh x;\quad\,\cot\,\mathrm{% gd}\,x=\mathrm{csch}\,x;\\ \displaystyle\tan(\tfrac{1}{2}\mathrm{gd}\,x)&\displaystyle=\tanh\tfrac{1}{2}x% .\end{aligned}
  5. gd - 1 x = 0 x d t cos t - π / 2 < x < π / 2 = ln | 1 + sin x cos x | = 1 2 ln | 1 + sin x 1 - sin x | = ln | tan x + sec x | = ln | tan ( 1 4 π + 1 2 x ) | = arctanh ( sin x ) = arcsinh ( tan x ) = arccoth ( csc x ) = arccsch ( cot x ) = sgn ( x ) . arccosh ( sec x ) = sgn ( x ) . arcsech ( cos x ) . \begin{aligned}\displaystyle\operatorname{gd}^{-1}\,x&\displaystyle=\int_{0}^{% x}\frac{\mathrm{d}t}{\cos t}\qquad-\pi/2<x<\pi/2\\ &\displaystyle=\ln\left|\frac{1+\sin x}{\cos x}\right|=\tfrac{1}{2}\ln\left|% \frac{1+\sin x}{1-\sin x}\right|\\ &\displaystyle=\ln\left|\tan x+\sec x\right|=\ln\left|\tan\left(\tfrac{1}{4}% \pi+\tfrac{1}{2}x\right)\right|\\ &\displaystyle=\mathrm{arctanh}\,(\sin x)=\mathrm{arcsinh}\,(\tan x)\\ &\displaystyle=\mathrm{arccoth}\,(\csc x)=\mathrm{arccsch}\,(\cot x)\\ &\displaystyle=\mbox{sgn}~{}(x).\mathrm{arccosh}\,(\sec x)=\mbox{sgn}~{}(x).% \mathrm{arcsech}\,(\cos x).\end{aligned}
  6. sinh gd - 1 x = tan x ; csch gd - 1 x = cot x ; cosh gd - 1 x = sec x ; sech gd - 1 x = cos x ; tanh gd - 1 x = sin x ; coth gd - 1 x = csc x . \begin{aligned}\displaystyle\sinh\,\operatorname{gd}^{-1}\,x&\displaystyle=% \tan x;\quad\mathrm{csch}\,\operatorname{gd}^{-1}\,x=\cot x;\\ \displaystyle\cosh\,\operatorname{gd}^{-1}\,x&\displaystyle=\mathrm{sec}\,x;% \quad\,\mathrm{sech}\,\operatorname{gd}^{-1}\,x=\cos x;\\ \displaystyle\tanh\,\operatorname{gd}^{-1}\,x&\displaystyle=\sin x;\quad\,% \coth\,\operatorname{gd}^{-1}\,x=\mathrm{csc}\,x.\end{aligned}
  7. d d x gd x = sech x ; d d x gd - 1 x = sec x . \frac{\mathrm{d}}{\mathrm{d}x}\;\mathrm{gd}\,x=\mathrm{sech}\,x;\quad\frac{% \mathrm{d}}{\mathrm{d}x}\;\operatorname{gd}^{-1}\,x=\sec x.
  8. 𝔖 𝔦 𝔫 \mathfrak{Sin}
  9. 𝔬 𝔰 \mathfrak{Cos}
  10. u = 0 ϕ sec t d t = ln tan ( 1 4 π + 1 2 ϕ ) u=\int_{0}^{\phi}\sec t\,\mathrm{d}t=\ln\,\tan\left(\tfrac{1}{4}\pi+\tfrac{1}{% 2}\phi\right)
  11. gd u = i - 1 ln tan ( 1 4 π + 1 2 u i ) \operatorname{gd}\,u=i^{-1}\ln\,\tan\left(\tfrac{1}{4}\pi+\tfrac{1}{2}ui\right)
  12. 1 2 π - gd x \tfrac{1}{2}\pi-\mathrm{gd}\,x

Gumbel_distribution.html

  1. μ + β γ \mu+\beta\,\gamma\!
  2. γ \gamma
  3. μ - β ln ( ln ( 2 ) ) \mu-\beta\,\ln(\ln(2))\!
  4. μ \mu\!
  5. π 2 6 β 2 \frac{\pi^{2}}{6}\,\beta^{2}\!
  6. 12 6 ζ ( 3 ) π 3 1.14 \frac{12\sqrt{6}\,\zeta(3)}{\pi^{3}}\approx 1.14\!
  7. 12 5 \frac{12}{5}
  8. ln ( β ) + γ + 1 \ln(\beta)+\gamma+1\!
  9. Γ ( 1 + β t ) e μ t \Gamma(1+\beta\,t)\,e^{\mu\,t}\!
  10. Γ ( 1 + i β t ) e i μ t \Gamma(1+i\,\beta\,t)\,e^{i\,\mu\,t}\!
  11. F ( x ; μ , β ) = e - e - ( x - μ ) / β . F(x;\mu,\beta)=e^{-e^{-(x-\mu)/\beta}}.\,
  12. μ - β ln ( ln 2 ) , \mu-\beta\ln\left(\ln 2\right),
  13. E ( X ) = μ + γ β , \operatorname{E}(X)=\mu+\gamma\beta,
  14. γ \gamma
  15. 0.5772. \approx 0.5772.
  16. β π / 6 . \beta\pi/\sqrt{6}.
  17. μ = 0 \mu=0
  18. β = 1 \beta=1
  19. F ( x ) = e - e ( - x ) F(x)=e^{-e^{(-x)}}\,
  20. f ( x ) = e - ( x + e - x ) . f(x)=e^{-(x+e^{-x})}.
  21. - ln ( ln ( 2 ) ) 0.3665 -\ln(\ln(2))\approx 0.3665
  22. γ \gamma
  23. π / 6 1.2825. \pi/\sqrt{6}\approx 1.2825.
  24. κ n = ( n - 1 ) ! ζ ( n ) . \kappa_{n}=(n-1)!\zeta(n).
  25. Q ( p ) Q(p)
  26. Q ( p ) = μ - β ln ( - ln ( p ) ) , Q(p)=\mu-\beta\ln(-\ln(p)),
  27. Q ( U ) Q(U)
  28. μ \mu
  29. β \beta
  30. U U
  31. ( 0 , 1 ) (0,1)
  32. G ( y ) = P ( Y y ) = P ( X - y | X 0 ) = ( F ( 0 ) - F ( - y ) ) / F ( 0 ) G(y)=P(Y\leq y)=P(X\geq-y|X\leq 0)=(F(0)-F(-y))/F(0)
  33. g ( y ) = f ( - y ) / F ( 0 ) g(y)=f(-y)/F(0)
  34. F F
  35. - ln [ - ln ( F ) ] = ( x - μ ) / β -\ln[-\ln(F)]=(x-\mu)/\beta
  36. F F
  37. x x
  38. / β /\beta

Gustav_Fechner.html

  1. S = K ln I S=K\ln I

Gδ_set.html

  1. 𝚷 2 0 \mathbf{\Pi}^{0}_{2}
  2. D = { f C ( [ 0 , 1 ] ) : f is not differentiable at any point of [ 0 , 1 ] } D=\left\{f\in C([0,1]):f\,\text{ is not differentiable at any point of }[0,1]\right\}
  3. C ( [ 0 , 1 ] ) C([0,1])
  4. ( 𝒳 , ρ ) (\mathcal{X},\rho)
  5. A 𝒳 A\subset\mathcal{X}
  6. A A
  7. 𝒳 \mathcal{X}
  8. σ \sigma
  9. A A
  10. ρ | A \rho|A
  11. ( A , σ ) (A,\sigma)
  12. G δ G_{\delta}
  13. f f
  14. G δ G_{\delta}
  15. p p
  16. Π 2 0 \Pi^{0}_{2}
  17. n n
  18. U U
  19. p p
  20. d ( f ( x ) , f ( y ) ) < 1 / n d(f(x),f(y))<1/n
  21. x , y x,y
  22. U U
  23. n n
  24. p p
  25. U U
  26. n n
  27. ( 𝒳 , 𝒯 ) (\mathcal{X},\mathcal{T})
  28. G 𝒳 G\subset\mathcal{X}
  29. 𝒯 \mathcal{T}
  30. G G
  31. 𝒳 \mathcal{X}

H-infinity_methods_in_control_theory.html

  1. [ z v ] = 𝐏 ( s ) [ w u ] = [ P 11 ( s ) P 12 ( s ) P 21 ( s ) P 22 ( s ) ] [ w u ] \begin{bmatrix}z\\ v\end{bmatrix}=\mathbf{P}(s)\,\begin{bmatrix}w\\ u\end{bmatrix}=\begin{bmatrix}P_{11}(s)&P_{12}(s)\\ P_{21}(s)&P_{22}(s)\end{bmatrix}\,\begin{bmatrix}w\\ u\end{bmatrix}
  2. u = 𝐊 ( s ) v u=\mathbf{K}(s)\,v
  3. z = F ( 𝐏 , 𝐊 ) w z=F_{\ell}(\mathbf{P},\mathbf{K})\,w
  4. F F_{\ell}
  5. F ( 𝐏 , 𝐊 ) = P 11 + P 12 𝐊 ( I - P 22 𝐊 ) - 1 P 21 F_{\ell}(\mathbf{P},\mathbf{K})=P_{11}+P_{12}\,\mathbf{K}\,(I-P_{22}\,\mathbf{% K})^{-1}\,P_{21}
  6. \mathcal{H}_{\infty}
  7. 𝐊 \mathbf{K}
  8. F ( 𝐏 , 𝐊 ) F_{\ell}(\mathbf{P},\mathbf{K})
  9. \mathcal{H}_{\infty}
  10. 2 \mathcal{H}_{2}
  11. F ( 𝐏 , 𝐊 ) F_{\ell}(\mathbf{P},\mathbf{K})
  12. || F ( 𝐏 , 𝐊 ) || = sup ω σ ¯ ( F ( 𝐏 , 𝐊 ) ( j ω ) ) ||F_{\ell}(\mathbf{P},\mathbf{K})||_{\infty}=\sup_{\omega}\bar{\sigma}(F_{\ell% }(\mathbf{P},\mathbf{K})(j\omega))
  13. σ ¯ \bar{\sigma}
  14. F ( 𝐏 , 𝐊 ) ( j ω ) F_{\ell}(\mathbf{P},\mathbf{K})(j\omega)

HAL::S.html

  1. x = a 2 + b i 2 x=a^{2}+b_{i}^{2}
  2. i i

Hall's_marriage_theorem.html

  1. W S W\subseteq S
  2. | W | | A W A | . |W|\leq\Bigl|\bigcup_{A\in W}A\Bigr|.
  3. I I
  4. | i I A i | |\bigcup_{i\in I}A_{i}|
  5. | I | |I|
  6. I I
  7. N G ( W ) N_{G}(W)
  8. | W | | N G ( W ) | . |W|\leq|N_{G}(W)|.

Hamiltonian_mechanics.html

  1. s y m b o l r = ( s y m b o l q , s y m b o l p ) symbol{r}=(symbol{q},symbol{p})
  2. q i , p i q_{i},p_{i}
  3. = ( s y m b o l q , s y m b o l p , t ) \mathcal{H}=\mathcal{H}(symbol{q},symbol{p},t)
  4. = T + V , T = p 2 2 m , V = V ( q ) . \mathcal{H}=T+V,\quad T=\frac{p^{2}}{2m},\quad V=V(q).
  5. q i q_{i}
  6. q ˙ i \dot{q}_{i}
  7. p i ( q i , q ˙ i , t ) = q ˙ i . p_{i}(q_{i},\dot{q}_{i},t)=\frac{\partial\mathcal{L}}{\partial{\dot{q}_{i}}}\,.
  8. q ˙ i \dot{q}_{i}
  9. p i p_{i}
  10. \mathcal{H}
  11. \mathcal{L}
  12. = i q ˙ i q ˙ i - = i q ˙ i p i - . \mathcal{H}=\sum_{i}{\dot{q}_{i}}\frac{\partial\mathcal{L}}{\partial{\dot{q}_{% i}}}-\mathcal{L}=\sum_{i}{\dot{q}_{i}}p_{i}-\mathcal{L}\,.
  13. q i q_{i}\,
  14. q ˙ i : \dot{q}_{i}:
  15. d = i ( q i d q i + q ˙ i d q ˙ i ) + t d t . \mathrm{d}\mathcal{L}=\sum_{i}\left(\frac{\partial\mathcal{L}}{\partial q_{i}}% \mathrm{d}q_{i}+\frac{\partial\mathcal{L}}{\partial{\dot{q}_{i}}}\mathrm{d}{% \dot{q}_{i}}\right)+\frac{\partial\mathcal{L}}{\partial t}\mathrm{d}t\,.
  16. p i = q ˙ i . p_{i}=\frac{\partial\mathcal{L}}{\partial{\dot{q}_{i}}}\,.
  17. d = i ( q i d q i + p i d q ˙ i ) + t d t . \mathrm{d}\mathcal{L}=\sum_{i}\left(\frac{\partial\mathcal{L}}{\partial q_{i}}% \mathrm{d}q_{i}+p_{i}\mathrm{d}{\dot{q}_{i}}\right)+\frac{\partial\mathcal{L}}% {\partial t}\mathrm{d}t\,.
  18. d = i ( q i d q i + d ( p i q ˙ i ) - q ˙ i d p i ) + t d t \mathrm{d}\mathcal{L}=\sum_{i}\left(\frac{\partial\mathcal{L}}{\partial q_{i}}% \mathrm{d}q_{i}+\mathrm{d}\left(p_{i}{\dot{q}_{i}}\right)-{\dot{q}_{i}}\mathrm% {d}p_{i}\right)+\frac{\partial\mathcal{L}}{\partial t}\mathrm{d}t\,
  19. d ( i p i q ˙ i - ) = i ( - q i d q i + q ˙ i d p i ) - t d t . \mathrm{d}\left(\sum_{i}p_{i}{\dot{q}_{i}}-\mathcal{L}\right)=\sum_{i}\left(-% \frac{\partial\mathcal{L}}{\partial q_{i}}\mathrm{d}q_{i}+{\dot{q}_{i}}\mathrm% {d}p_{i}\right)-\frac{\partial\mathcal{L}}{\partial t}\mathrm{d}t\,.
  20. d = i ( - q i d q i + q ˙ i d p i ) - t d t . \mathrm{d}\mathcal{H}=\sum_{i}\left(-\frac{\partial\mathcal{L}}{\partial q_{i}% }\mathrm{d}q_{i}+{\dot{q}_{i}}\mathrm{d}p_{i}\right)-\frac{\partial\mathcal{L}% }{\partial t}\mathrm{d}t.
  21. \mathcal{H}
  22. \mathcal{L}
  23. d = i ( q i d q i + p i d p i ) + t d t . \mathrm{d}\mathcal{H}=\sum_{i}\left(\frac{\partial\mathcal{H}}{\partial q_{i}}% \mathrm{d}q_{i}+\frac{\partial\mathcal{H}}{\partial p_{i}}\mathrm{d}p_{i}% \right)+\frac{\partial\mathcal{H}}{\partial t}\mathrm{d}t.
  24. i ( - q i d q i + q ˙ i d p i ) - t d t = i ( q i d q i + p i d p i ) + t d t . \sum_{i}\left(-\frac{\partial\mathcal{L}}{\partial q_{i}}\mathrm{d}q_{i}+{\dot% {q}_{i}}\mathrm{d}p_{i}\right)-\frac{\partial\mathcal{L}}{\partial t}\mathrm{d% }t=\sum_{i}\left(\frac{\partial\mathcal{H}}{\partial q_{i}}\mathrm{d}q_{i}+% \frac{\partial\mathcal{H}}{\partial p_{i}}\mathrm{d}p_{i}\right)+\frac{% \partial\mathcal{H}}{\partial t}\mathrm{d}t.
  25. q i = - q i , p i = q ˙ i , t = - t . \frac{\partial\mathcal{H}}{\partial q_{i}}=-\frac{\partial\mathcal{L}}{% \partial q_{i}}\,,\quad\frac{\partial\mathcal{H}}{\partial p_{i}}=\dot{q}_{i}% \,,\quad\frac{\partial\mathcal{H}}{\partial t}=-{\partial\mathcal{L}\over% \partial t}\,.
  26. d d t q ˙ i - q i = 0 . \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial{\dot{q}_{i}}% }-\frac{\partial\mathcal{L}}{\partial q_{i}}=0\,.
  27. q i = p ˙ i . \frac{\partial\mathcal{L}}{\partial q_{i}}={\dot{p}}_{i}\,.
  28. q j = - p ˙ j , p j = q ˙ j , t = - t . \frac{\partial\mathcal{H}}{\partial q_{j}}=-\dot{p}_{j}\,,\quad\frac{\partial% \mathcal{H}}{\partial p_{j}}=\dot{q}_{j}\,,\quad\frac{\partial\mathcal{H}}{% \partial t}=-{\partial\mathcal{L}\over\partial t}\,.
  29. { q j | j = 1 , , N } \left\{q_{j}\ |\ j=1,\ldots,N\right\}
  30. { q ˙ j | j = 1 , , N } . \left\{\dot{q}_{j}\ |\ j=1,\ldots,N\right\}.
  31. ( q j , q ˙ j , t ) \mathcal{L}(q_{j},\dot{q}_{j},t)
  32. p j = q ˙ j . p_{j}={\partial\mathcal{L}\over\partial\dot{q}_{j}}.
  33. ( q j , p j , t ) = ( i q ˙ i p i ) - ( q j , q ˙ j , t ) . \mathcal{H}\left(q_{j},p_{j},t\right)=\left(\sum_{i}\dot{q}_{i}p_{i}\right)-% \mathcal{L}(q_{j},\dot{q}_{j},t).
  34. \mathcal{H}
  35. d \displaystyle\mathrm{d}\mathcal{H}
  36. q j = - p ˙ j , p j = q ˙ j , t = - t . \frac{\partial\mathcal{H}}{\partial q_{j}}=-\dot{p}_{j},\qquad\frac{\partial% \mathcal{H}}{\partial p_{j}}=\dot{q}_{j},\qquad\frac{\partial\mathcal{H}}{% \partial t}=-{\partial\mathcal{L}\over\partial t}.
  37. d f d t = { f , } + f t \frac{\mathrm{d}f}{\mathrm{d}t}=\{f,\mathcal{H}\}+\frac{\partial f}{\partial t}
  38. d d t f = t f + { f , } . \frac{\mathrm{d}}{\mathrm{d}t}f=\frac{\partial}{\partial t}f+\{\,f,\mathcal{H}% \,\}.
  39. p ˙ i , q ˙ i {\dot{p}_{i}},{\dot{q}_{i}}
  40. t ρ = - { ρ , } . \frac{\partial}{\partial t}\rho=-\{\,\rho,\mathcal{H}\,\}.
  41. G ˙ i = 0 , φ ˙ i = F ( G ) , \dot{G}_{i}=0,\qquad\dot{\varphi}_{i}=F(G),
  42. ( q , p ) = 1 2 p , p q \mathcal{H}(q,p)=\frac{1}{2}\langle p,p\rangle_{q}
  43. , q \langle\cdot,\cdot\rangle_{q}
  44. T q * Q T_{q}^{*}Q
  45. ( x , y , z , p x , p y , p z ) = 1 2 ( p x 2 + p y 2 ) . \mathcal{H}(x,y,z,p_{x},p_{y},p_{z})=\frac{1}{2}\left(p_{x}^{2}+p_{y}^{2}% \right).
  46. p z p_{z}
  47. q i = x i q_{i}=x_{i}
  48. = i 1 2 m x ˙ i 2 + i e x ˙ i A i - e ϕ , \mathcal{L}=\sum_{i}\tfrac{1}{2}m\dot{x}_{i}^{2}+\sum_{i}e\dot{x}_{i}A_{i}-e\phi,
  49. ϕ \phi
  50. A i A_{i}
  51. p i = x ˙ i = m x ˙ i + e A i . p_{i}=\frac{\partial\mathcal{L}}{\partial\dot{x}_{i}}=m\dot{x}_{i}+eA_{i}.
  52. x ˙ i = p i - e A i m . \dot{x}_{i}=\frac{p_{i}-eA_{i}}{m}.
  53. = { i x ˙ i p i } - = i ( p i - e A i ) 2 2 m + e ϕ . \mathcal{H}=\left\{\sum_{i}\dot{x}_{i}p_{i}\right\}-\mathcal{L}=\sum_{i}\frac{% (p_{i}-eA_{i})^{2}}{2m}+e\phi.
  54. ( t ) = - m c 2 1 - x ˙ ( t ) 2 c 2 - e ϕ ( x ( t ) , t ) + e x ˙ ( t ) A ( x ( t ) , t ) . \mathcal{L}(t)=-mc^{2}\sqrt{1-\frac{{\dot{\vec{x}}(t)}^{2}}{c^{2}}}-e\phi(\vec% {x}(t),t)+e\dot{\vec{x}}(t)\cdot\vec{A}(\vec{x}(t),t)\,.
  55. P ( t ) = ( t ) x ˙ ( t ) = m x ˙ ( t ) 1 - x ˙ ( t ) 2 c 2 + e A ( x ( t ) , t ) , \vec{P}\,(t)=\frac{\partial\mathcal{L}(t)}{\partial\dot{\vec{x}}(t)}=\frac{m% \dot{\vec{x}}(t)}{\sqrt{1-\frac{{\dot{\vec{x}}(t)}^{2}}{c^{2}}}}+e\vec{A}(\vec% {x}(t),t)\,,
  56. x ˙ ( t ) = P ( t ) - e A ( x ( t ) , t ) m 2 + 1 c 2 ( P ( t ) - e A ( x ( t ) , t ) ) 2 . \dot{\vec{x}}(t)=\frac{\vec{P}\,(t)-e\vec{A}(\vec{x}(t),t)}{\sqrt{m^{2}+\frac{% 1}{c^{2}}{\left(\vec{P}\,(t)-e\vec{A}(\vec{x}(t),t)\right)}^{2}}}\,.
  57. ( t ) = x ˙ ( t ) P ( t ) - ( t ) = c m 2 c 2 + ( P ( t ) - e A ( x ( t ) , t ) ) 2 + e ϕ ( x ( t ) , t ) . \mathcal{H}(t)=\dot{\vec{x}}(t)\cdot\vec{P}\,(t)-\mathcal{L}(t)=c\sqrt{m^{2}c^% {2}+{\left(\vec{P}\,(t)-e\vec{A}(\vec{x}(t),t)\right)}^{2}}+e\phi(\vec{x}(t),t% )\,.
  58. P ˙ = - x = e ( A ) x ˙ - e ϕ \dot{\vec{P}}=-\frac{\partial\mathcal{H}}{\partial\vec{x}}=e(\vec{\nabla}\vec{% A})\cdot\dot{\vec{x}}-e\vec{\nabla}\phi\,
  59. d d t ( m x ˙ 1 - x ˙ 2 c 2 ) = e E + e x ˙ × B . \frac{d}{dt}\left(\frac{m\dot{\vec{x}}}{\sqrt{1-\frac{\dot{\vec{x}}^{2}}{c^{2}% }}}\right)=e\vec{E}+e\dot{\vec{x}}\times\vec{B}\,.
  60. p = γ m x ˙ ( t ) , \vec{p}=\gamma m\dot{\vec{x}}(t)\,,
  61. ( t ) = x ˙ ( t ) p ( t ) + m c 2 γ + e ϕ ( x ( t ) , t ) = γ m c 2 + e ϕ ( x ( t ) , t ) = E + V . \mathcal{H}(t)=\dot{\vec{x}}(t)\cdot\vec{p}\,(t)+\frac{mc^{2}}{\gamma}+e\phi(% \vec{x}(t),t)=\gamma mc^{2}+e\phi(\vec{x}(t),t)=E+V\,.
  62. p \vec{p}
  63. P \vec{P}
  64. E = γ m c 2 , E=\gamma mc^{2}\,,
  65. V = e ϕ . V=e\phi\,.

Hardy_space.html

  1. f H p . \|f\|_{H^{p}}.
  2. f H = sup | z | < 1 | f ( z ) | . \|f\|_{H^{\infty}}=\sup_{|z|<1}\left|f(z)\right|.
  3. f ~ ( e i θ ) = lim r 1 f ( r e i θ ) \tilde{f}\left(e^{i\theta}\right)=\lim_{r\to 1}f\left(re^{i\theta}\right)
  4. f ~ \tilde{f}
  5. f ~ L p = f H p . \|\tilde{f}\|_{L^{p}}=\|f\|_{H^{p}}.
  6. f ~ \tilde{f}
  7. g H p ( 𝐓 ) if and only if g L p ( 𝐓 ) and g ^ ( n ) = 0 for all n < 0 , g\in H^{p}\left(\mathbf{T}\right)\,\text{ if and only if }g\in L^{p}\left(% \mathbf{T}\right)\,\text{ and }\hat{g}(n)=0\,\text{ for all }n<0,
  8. n 𝐙 , g ^ ( n ) = 1 2 π 0 2 π g ( e i ϕ ) e - i n ϕ d ϕ . \forall n\in\mathbf{Z},\ \ \ \hat{g}(n)=\frac{1}{2\pi}\int_{0}^{2\pi}g\left(e^% {i\phi}\right)e^{-in\phi}\,\mathrm{d}\phi.
  9. f ~ \tilde{f}
  10. f ( r e i θ ) = 1 2 π 0 2 π P r ( θ - ϕ ) f ~ ( e i ϕ ) d ϕ , r < 1 , f\left(re^{i\theta}\right)=\frac{1}{2\pi}\int_{0}^{2\pi}P_{r}(\theta-\phi)% \tilde{f}\left(e^{i\phi}\right)\,\mathrm{d}\phi,\quad r<1,
  11. f ~ \tilde{f}
  12. f ~ \tilde{f}
  13. f ~ \tilde{f}
  14. f ~ \tilde{f}
  15. f ( z ) = n = 0 a n z n , | z | < 1. f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},\ \ \ |z|<1.
  16. f ( e i θ ) := F ~ ( e i θ ) = i cot ( θ 2 ) . f(e^{i\theta}):=\tilde{F}(e^{i\theta})=i\,\cot(\tfrac{\theta}{2}).
  17. G ( z ) = c exp ( 1 2 π - π π e i θ + z e i θ - z log ( φ ( e i θ ) ) d θ ) G(z)=c\,\exp\left(\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i% \theta}-z}\log\!\left(\varphi\!\left(e^{i\theta}\right)\right)\,\mathrm{d}% \theta\right)
  18. φ \varphi
  19. log ( φ ) \log(\varphi)
  20. φ \varphi
  21. lim r 1 - | G ( r e i θ ) | = φ ( e i θ ) \lim_{r\to 1^{-}}\left|G\left(re^{i\theta}\right)\right|=\varphi\left(e^{i% \theta}\right)
  22. lim r 1 - h ( r e i θ ) \lim_{r\to 1^{-}}h(re^{i\theta})
  23. ( M f ) ( e i θ ) = sup 0 < r < 1 | ( f * P r ) ( e i θ ) | , (Mf)(e^{i\theta})=\sup_{0<r<1}\left|(f*P_{r})\left(e^{i\theta}\right)\right|,
  24. e i φ P r ( θ - φ ) . e^{i\varphi}\rightarrow P_{r}(\theta-\varphi).
  25. u ( e i θ ) = a 0 2 + k 1 a k cos ( k θ ) + b k sin ( k θ ) v ( e i θ ) = k 1 a k sin ( k θ ) - b k cos ( k θ ) . u(e^{i\theta})=\frac{a_{0}}{2}+\sum_{k\geq 1}a_{k}\cos(k\theta)+b_{k}\sin(k% \theta)\longrightarrow v(e^{i\theta})=\sum_{k\geq 1}a_{k}\sin(k\theta)-b_{k}% \cos(k\theta).
  26. F ( z ) = n = 0 + c n z n , | z | < 1 F(z)=\sum_{n=0}^{+\infty}c_{n}z^{n},\quad|z|<1
  27. n = 0 + c n e i n θ \sum_{n=0}^{+\infty}c_{n}e^{in\theta}
  28. f H p = sup y > 0 ( | f ( x + i y ) | p d x ) 1 p . \|f\|_{H^{p}}=\sup_{y>0}\left(\int|f(x+iy)|^{p}\,\mathrm{d}x\right)^{\frac{1}{% p}}.
  29. f H = sup z 𝐇 | f ( z ) | . \|f\|_{H^{\infty}}=\sup_{z\in\mathbf{H}}|f(z)|.
  30. m ( z ) = i 1 + z 1 - z m(z)=i\cdot\frac{1+z}{1-z}
  31. ( M f ) ( z ) = π 1 - z f ( m ( z ) ) . (Mf)(z)=\frac{\sqrt{\pi}}{1-z}f(m(z)).
  32. f k ( x ) = 𝟏 [ 0 , 1 ] ( x - k ) - 𝟏 [ 0 , 1 ] ( x + k ) , k > 0. f_{k}(x)=\mathbf{1}_{[0,1]}(x-k)-\mathbf{1}_{[0,1]}(x+k),\ \ \ k>0.
  33. 𝐑 n f ( x ) x 1 i 1 x n i n d x , \int_{\mathbf{R}^{n}}f(x)x_{1}^{i_{1}}\ldots x_{n}^{i_{n}}\,\mathrm{d}x,
  34. f = c j a j , | c j | p < f=\sum c_{j}a_{j},\ \ \ \sum|c_{j}|^{p}<\infty
  35. M * = sup n 0 | M n | . M^{*}=\sup_{n\geq 0}\,|M_{n}|.
  36. M n = E ( f | Σ n ) M_{n}=E\bigl(f|\Sigma_{n}\bigr)
  37. S ( f ) = ( | M 0 | 2 + n = 0 | M n + 1 - M n | 2 ) 1 2 . S(f)=\left(|M_{0}|^{2}+\sum_{n=0}^{\infty}|M_{n+1}-M_{n}|^{2}\right)^{\frac{1}% {2}}.
  38. M t = F ( B t τ ) M_{t}=F(B_{t\wedge\tau})
  39. f = c k h k , f=\sum c_{k}h_{k},
  40. 0 1 ( | c k h k ( x ) | 2 ) 1 2 d x . \int_{0}^{1}\Bigl(\sum|c_{k}h_{k}(x)|^{2}\Bigr)^{\frac{1}{2}}\,\mathrm{d}x.

Hardy–Weinberg_principle.html

  1. q = 1 p q=1−p
  2. f ( A ) = f ( A ) \textstyle f^{\prime}(\,\text{A})=f(\,\text{A})
  3. f ( a ) = f ( a ) \textstyle f^{\prime}(\,\text{a})=f(\,\text{a})
  4. f t ( A ) = f t ( AA ) + 1 2 f t ( Aa ) f_{t}(\,\text{A})=f_{t}(\,\text{AA})+\frac{1}{2}f_{t}(\,\text{Aa})
  5. f t ( a ) = f t ( aa ) + 1 2 f t ( Aa ) f_{t}(\,\text{a})=f_{t}(\,\text{aa})+\frac{1}{2}f_{t}(\,\text{Aa})
  6. f 1 ( AA ) = p 2 = f 0 ( A ) 2 f_{1}(\,\text{AA})=p^{2}=f_{0}(\,\text{A})^{2}
  7. f 1 ( Aa ) = p q + q p = 2 p q = 2 f 0 ( A ) f 0 ( a ) f_{1}(\,\text{Aa})=pq+qp=2pq=2f_{0}(\,\text{A})f_{0}(\,\text{a})
  8. f 1 ( aa ) = q 2 = f 0 ( a ) 2 f_{1}(\,\text{aa})=q^{2}=f_{0}(\,\text{a})^{2}
  9. t > 1 t>1
  10. f 1 ( A ) = f 1 ( AA ) + 1 2 f 1 ( Aa ) = p 2 + p q = p ( p + q ) = p = f 0 ( A ) f_{1}(\,\text{A})=f_{1}(\,\text{AA})+\frac{1}{2}f_{1}(\,\text{Aa})=p^{2}+pq=p% \left(p+q\right)=p=f_{0}(\,\text{A})
  11. f 1 ( a ) = f 1 ( aa ) + 1 2 f 1 ( Aa ) = q 2 + p q = q ( p + q ) = q = f 0 ( a ) f_{1}(\,\text{a})=f_{1}(\,\text{aa})+\frac{1}{2}f_{1}(\,\text{Aa})=q^{2}+pq=q% \left(p+q\right)=q=f_{0}(\,\text{a})
  12. [ ( AA , AA ) , ( AA , Aa ) , ( AA , aa ) , ( Aa , Aa ) , ( Aa , aa ) , ( aa , aa ) ] \left[(\,\text{AA},\,\text{AA}),(\,\text{AA},\,\text{Aa}),(\,\text{AA},\,\text% {aa}),(\,\text{Aa},\,\text{Aa}),(\,\text{Aa},\,\text{aa}),(\,\text{aa},\,\text% {aa})\right]
  13. k = 3 k=3
  14. ( A A , a a ) (AA,aa)
  15. A a Aa
  16. 0 , 1 , 00 0,1,00
  17. [ f t + 1 ( AA ) , f t + 1 ( Aa ) , f t + 1 ( aa ) ] \displaystyle\left[f_{t+1}(\,\text{AA}),f_{t+1}(\,\text{Aa}),f_{t+1}(\,\text{% aa})\right]
  18. t + 1 t+1
  19. t t
  20. t = 1 t=1
  21. ( p + q + r ) 2 = p 2 + q 2 + r 2 + 2 p q + 2 p r + 2 q r (p+q+r)^{2}=p^{2}+q^{2}+r^{2}+2pq+2pr+2qr\,
  22. ( p 1 + + p n ) 2 (p_{1}+\cdots+p_{n})^{2}\,
  23. f ( A i A i ) = p i 2 f(A_{i}A_{i})=p_{i}^{2}\,
  24. f ( A i A j ) = 2 p i p j f(A_{i}A_{j})=2p_{i}p_{j}\,
  25. ( p + q ) 2 (p+q)^{2}\,
  26. ( p + q ) c (p+q)^{c}\,
  27. p 4 p^{4}
  28. 4 p 3 q 4p^{3}q
  29. 6 p 2 q 2 6p^{2}q^{2}
  30. 4 p q 3 4pq^{3}
  31. q 4 q^{4}
  32. n n
  33. c c
  34. ( p 1 + + p n ) c (p_{1}+\cdots+p_{n})^{c}
  35. ( p 1 + + p n ) c = k 1 , , k n : k 1 + + k n = c ( c k 1 , , k n ) p 1 k 1 p n k n (p_{1}+\cdots+p_{n})^{c}=\sum_{k_{1},\ldots,k_{n}\ \in\mathbb{N}:k_{1}+\cdots+% k_{n}=c}{c\choose k_{1},\ldots,k_{n}}p_{1}^{k_{1}}\cdots p_{n}^{k_{n}}
  36. q q
  37. q = f ( aa ) q=\sqrt{f(\,\text{aa})}
  38. p p
  39. q q
  40. p 2 p^{2}
  41. 2 p q 2pq
  42. p \displaystyle p
  43. q \displaystyle q
  44. Exp ( AA ) \displaystyle\mathrm{Exp}(\,\text{AA})
  45. χ 2 \displaystyle\chi^{2}
  46. prob [ n 12 | n 1 ] = ( n n 11 , n 12 , n 22 ) ( 2 n n 1 , n 2 ) 2 n 12 , \operatorname{prob}[n_{12}|n_{1}]=\frac{{\left({{n}\atop{n_{11},n_{12},n_{22}}% }\right)}}{{\left({{2n}\atop{n_{1},n_{2}}}\right)}}2^{n_{12}},
  47. n 1 = 2 n 11 + n 12 n_{1}=2n_{11}+n_{12}
  48. E ( f ( Aa ) ) = 2 p q \operatorname{E}(f(\,\text{Aa}))=2pq
  49. p + 2 q + r = 1. p+2q+r=1.
  50. p 1 + 2 q 1 + r 1 = ( p + q ) 2 + 2 ( p + q ) ( q + r ) + ( q + r ) 2 = 1 p_{1}+2q_{1}+r_{1}=(p+q)^{2}+2(p+q)(q+r)+(q+r)^{2}=1
  51. E 1 \displaystyle E_{1}
  52. p : 2 q : r = p 1 : 2 q 1 : r 1 \textstyle p:2q:r=p_{1}:2q_{1}:r_{1}
  53. q 2 = p r \textstyle q^{2}=pr
  54. p : 2 q : r = p 1 : 2 q 1 : r 1 \textstyle p:2q:r=p_{1}:2q_{1}:r_{1}
  55. q 2 = p r \textstyle q^{2}=pr
  56. p = p ( p + 2 q + r ) = p 2 + 2 p q + p r = p 2 + 2 p q + q 2 = ( p + q ) 2 = p 1 \begin{aligned}\displaystyle p&\displaystyle=p\left(p+2q+r\right)\\ &\displaystyle=p^{2}+2pq+pr\\ &\displaystyle=p^{2}+2pq+q^{2}\\ &\displaystyle=\left(p+q\right)^{2}\\ &\displaystyle=p_{1}\end{aligned}
  57. r = r 1 r=r_{1}
  58. p = p 1 \textstyle p=p_{1}
  59. q = q 1 \textstyle q=q_{1}
  60. r = r 1 \textstyle r=r_{1}
  61. p : 2 q : r = p 1 : 2 q 1 : r 1 \textstyle p:2q:r=p_{1}:2q_{1}:r_{1}
  62. sum \displaystyle\,\text{sum}
  63. p \displaystyle p
  64. p + 2 q + r = 0.83943 + 0.15771 + 0.00286 = 1.00000 p+2q+r=0.83943+0.15771+0.00286=1.00000\,
  65. E 0 = q 2 - p r = 0.00382. E_{0}=q^{2}-pr=0.00382.\,
  66. q \displaystyle q
  67. p 1 + 2 q 1 + r 1 = 0.84325 + 0.15007 + 0.00668 = 1.00000 p_{1}+2q_{1}+r_{1}=0.84325+0.15007+0.00668=1.00000\,
  68. E 1 = q 1 2 - p 1 r 1 = 0.00000 E_{1}=q_{1}^{2}-p_{1}r_{1}=0.00000\,
  69. ( θ 2 , 2 θ ( 1 - θ ) , ( 1 - θ ) 2 ) (\theta^{2},2\theta(1-\theta),(1-\theta)^{2})

Harmonic_number.html

  1. H n , 1 H_{n,1}
  2. n = x n=\lfloor{x}\rfloor
  3. γ + ln ( x ) \gamma+\ln(x)
  4. H n = 1 + 1 2 + 1 3 + + 1 n = k = 1 n 1 k . H_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\sum_{k=1}^{n}\frac{1}{k}.
  5. H n H_{n}
  6. H n = H n - 1 + 1 n . H_{n}=H_{n-1}+\frac{1}{n}.
  7. k = 1 n H k = ( n + 1 ) H n + 1 - ( n + 1 ) . \sum_{k=1}^{n}H_{k}=(n+1)H_{n+1}-(n+1).
  8. H n = 1 n ! [ n + 1 2 ] . H_{n}=\frac{1}{n!}\left[{n+1\atop 2}\right].
  9. f n ( x ) = x n n ! ( log x - H n ) f_{n}(x)=\frac{x^{n}}{n!}(\log x-H_{n})
  10. f n ( x ) = f n - 1 ( x ) . f_{n}^{\prime}(x)=f_{n-1}(x).
  11. f 1 ( x ) = x ( log x - 1 ) f_{1}(x)=x(\log x-1)
  12. n = 1 H n n 2 n = 1 12 π 2 ; \sum_{n=1}^{\infty}\frac{H_{n}}{n\cdot 2^{n}}=\frac{1}{12}\pi^{2};
  13. n = 1 H n 2 ( n + 1 ) 2 = 11 360 π 4 ; \sum_{n=1}^{\infty}\frac{H_{n}^{2}}{(n+1)^{2}}=\frac{11}{360}\pi^{4};
  14. n = 1 H n 2 n 2 = 17 360 π 4 ; \sum_{n=1}^{\infty}\frac{H_{n}^{2}}{n^{2}}=\frac{17}{360}\pi^{4};
  15. n = 1 H n n 3 = 1 72 π 4 ; \sum_{n=1}^{\infty}\frac{H_{n}}{n^{3}}=\frac{1}{72}\pi^{4};
  16. H n = 0 1 1 - x n 1 - x d x . H_{n}=\int_{0}^{1}\frac{1-x^{n}}{1-x}\,dx.
  17. 1 - x n 1 - x = 1 + x + + x n - 1 . \frac{1-x^{n}}{1-x}=1+x+\cdots+x^{n-1}.
  18. H n = 0 1 1 - x n 1 - x d x = - 1 0 1 - ( 1 - u ) n u d u = 0 1 1 - ( 1 - u ) n u d u = 0 1 [ k = 1 n ( - 1 ) k - 1 ( n k ) u k - 1 ] d u = k = 1 n ( - 1 ) k - 1 ( n k ) 0 1 u k - 1 d u = k = 1 n ( - 1 ) k - 1 1 k ( n k ) . \begin{aligned}\displaystyle H_{n}&\displaystyle=\int_{0}^{1}\frac{1-x^{n}}{1-% x}\,dx\\ &\displaystyle=-\int_{1}^{0}\frac{1-(1-u)^{n}}{u}\,du\\ &\displaystyle=\int_{0}^{1}\frac{1-(1-u)^{n}}{u}\,du\\ &\displaystyle=\int_{0}^{1}\left[\sum_{k=1}^{n}(-1)^{k-1}{\left({{n}\atop{k}}% \right)}u^{k-1}\right]\,du\\ &\displaystyle=\sum_{k=1}^{n}(-1)^{k-1}{\left({{n}\atop{k}}\right)}\int_{0}^{1% }u^{k-1}\,du\\ &\displaystyle=\sum_{k=1}^{n}(-1)^{k-1}\frac{1}{k}{\left({{n}\atop{k}}\right)}% .\end{aligned}
  19. x 1 = 1 , , x n = n x_{1}=1,\ldots,x_{n}=n
  20. Π k ( 1 , , n ) = ( - 1 ) n - k ( k - 1 ) ! ( n - k ) ! \Pi_{k}(1,\ldots,n)=(-1)^{n-k}(k-1)!(n-k)!
  21. H n = H n , 1 = k = 1 n 1 k = ( - 1 ) n - 1 n ! k = 1 n 1 k 2 Π k ( 1 , , n ) = k = 1 n ( - 1 ) k - 1 1 k ( n k ) . H_{n}=H_{n,1}=\sum_{k=1}^{n}\frac{1}{k}=(-1)^{n-1}n!\sum_{k=1}^{n}\frac{1}{k^{% 2}\Pi_{k}(1,\ldots,n)}=\sum_{k=1}^{n}(-1)^{k-1}\frac{1}{k}{\left({{n}\atop{k}}% \right)}.
  22. 1 n 1 x d x \int_{1}^{n}{1\over x}\,dx
  23. lim n ( H n - ln n ) = γ , \lim_{n\to\infty}\left(H_{n}-\ln n\right)=\gamma,
  24. H n ln n + γ + 1 2 n - k = 1 B 2 k 2 k n 2 k = ln n + γ + 1 2 n - 1 12 n 2 + 1 120 n 4 - , H_{n}\sim\ln{n}+\gamma+\frac{1}{2n}-\sum_{k=1}^{\infty}\frac{B_{2k}}{2kn^{2k}}% =\ln{n}+\gamma+\frac{1}{2n}-\frac{1}{12n^{2}}+\frac{1}{120n^{4}}-\cdots,
  25. B k B_{k}
  26. H α = 0 1 1 - x α 1 - x d x . H_{\alpha}=\int_{0}^{1}\frac{1-x^{\alpha}}{1-x}\,dx\,.
  27. H α = H α - 1 + 1 α , H_{\alpha}=H_{\alpha-1}+\frac{1}{\alpha}\,,
  28. H 1 - α - H α = π cot ( π α ) - 1 α + 1 1 - α . H_{1-\alpha}-H_{\alpha}=\pi\cot{(\pi\alpha)}-\frac{1}{\alpha}+\frac{1}{1-% \alpha}\,.
  29. H 3 4 = 4 3 - 3 ln 2 + π 2 H_{\frac{3}{4}}=\tfrac{4}{3}-3\ln{2}+\tfrac{\pi}{2}
  30. H 2 3 = 3 2 ( 1 - ln 3 ) + 3 π 6 H_{\frac{2}{3}}=\tfrac{3}{2}(1-\ln{3})+\sqrt{3}\tfrac{\pi}{6}
  31. H 1 2 = 2 - 2 ln 2 H_{\frac{1}{2}}=2-2\ln{2}
  32. H 1 3 = 3 - π 2 3 - 3 2 ln 3 H_{\frac{1}{3}}=3-\tfrac{\pi}{2\sqrt{3}}-\tfrac{3}{2}\ln{3}
  33. H 1 4 = 4 - π 2 - 3 ln 2 H_{\frac{1}{4}}=4-\tfrac{\pi}{2}-3\ln{2}
  34. H 1 6 = 6 - π 2 3 - 2 ln 2 - 3 2 ln 3 H_{\frac{1}{6}}=6-\tfrac{\pi}{2}\sqrt{3}-2\ln{2}-\tfrac{3}{2}\ln{3}
  35. H 1 8 = 8 - π 2 - 4 ln 2 - 1 2 { π + ln ( 2 + 2 ) - ln ( 2 - 2 ) } H_{\frac{1}{8}}=8-\tfrac{\pi}{2}-4\ln{2}-\tfrac{1}{\sqrt{2}}\left\{\pi+\ln% \left(2+\sqrt{2}\right)-\ln\left(2-\sqrt{2}\right)\right\}
  36. H 1 12 = 12 - 3 ( ln 2 + ln 3 2 ) - π ( 1 + 3 2 ) + 2 3 ln ( 2 - 3 ) H_{\frac{1}{12}}=12-3\left(\ln{2}+\tfrac{\ln{3}}{2}\right)-\pi\left(1+\tfrac{% \sqrt{3}}{2}\right)+2\sqrt{3}\ln\left(\sqrt{2-\sqrt{3}}\right)
  37. H x = x k = 1 1 k ( x + k ) . H_{x}=x\sum_{k=1}^{\infty}\frac{1}{k(x+k)}\,.
  38. 0 1 H x d x = γ , \int_{0}^{1}H_{x}\,dx=\gamma\,,
  39. 0 n H x d x = n γ + ln ( n ! ) . \int_{0}^{n}H_{x}\,dx=n\gamma+\ln{(n!)}\,.
  40. n = 1 z n H n = - ln ( 1 - z ) 1 - z , \sum_{n=1}^{\infty}z^{n}H_{n}=\frac{-\ln(1-z)}{1-z},
  41. n = 1 z n n ! H n = - e z k = 1 1 k ( - z ) k k ! = e z Ein ( z ) \sum_{n=1}^{\infty}\frac{z^{n}}{n!}H_{n}=-e^{z}\sum_{k=1}^{\infty}\frac{1}{k}% \frac{(-z)^{k}}{k!}=e^{z}\mbox{Ein}(z)
  42. Ein ( z ) = E ( z ) 1 + γ + ln z = Γ ( 0 , z ) + γ + ln z \mbox{Ein}(z)=\mbox{E}~{}_{1}(z)+\gamma+\ln z=\Gamma(0,z)+\gamma+\ln z\,
  43. ψ ( n ) = H n - 1 - γ . \psi(n)=H_{n-1}-\gamma.
  44. γ = lim n ( H n - ln ( n + 1 2 ) ) \gamma=\lim_{n\to\infty}{\left(H_{n}-\ln\left(n+{1\over 2}\right)\right)}
  45. σ ( n ) H n + ln ( H n ) e H n , \sigma(n)\leq H_{n}+\ln(H_{n})e^{H_{n}},
  46. λ ϕ ( x ) = - 1 1 ϕ ( x ) - ϕ ( y ) | x - y | d y \lambda\phi(x)=\int_{-1}^{1}\frac{\phi(x)-\phi(y)}{|x-y|}dy
  47. λ = 2 H n \lambda=2H_{n}
  48. H 0 = 0. H_{0}=0.
  49. H n , m = k = 1 n 1 k m . H_{n,m}=\sum_{k=1}^{n}\frac{1}{k^{m}}.
  50. H n , m = H n ( m ) = H m ( n ) . H_{n,m}=H_{n}^{(m)}=H_{m}(n).
  51. H n , 0 = n H_{n,0}=n
  52. H n = k = 1 n 1 k . H_{n}=\sum_{k=1}^{n}\frac{1}{k}.
  53. lim n H n , m = ζ ( m ) . \lim_{n\rightarrow\infty}H_{n,m}=\zeta(m).
  54. k = 1 n k m \sum_{k=1}^{n}k^{m}
  55. 0 a H x , 2 d x = a π 2 6 - H a \int_{0}^{a}H_{x,2}\,dx=a\frac{\pi^{2}}{6}-H_{a}
  56. 0 a H x , 3 d x = a A - 1 2 H a , 2 , \int_{0}^{a}H_{x,3}\,dx=aA-\frac{1}{2}H_{a,2},
  57. k = 1 n H k , m = ( n + 1 ) H n , m - H n , m - 1 \sum_{k=1}^{n}H_{k,m}=(n+1)H_{n,m}-H_{n,m-1}
  58. m 0 m\geq 0
  59. H n , m = k = 1 n - 1 H k , m - 1 k ( k + 1 ) + H n , m - 1 n H_{n,m}=\sum_{k=1}^{n-1}\frac{H_{k,m-1}}{k(k+1)}+\frac{H_{n,m-1}}{n}
  60. H 4 , 3 = H 1 , 2 1 2 + H 2 , 2 2 3 + H 3 , 2 3 4 + H 4 , 2 4 H_{4,3}=\frac{H_{1,2}}{1\cdot 2}+\frac{H_{2,2}}{2\cdot 3}+\frac{H_{3,2}}{3% \cdot 4}+\frac{H_{4,2}}{4}
  61. n = 1 z n H n , m = Li m ( z ) 1 - z , \sum_{n=1}^{\infty}z^{n}H_{n,m}=\frac{\mathrm{Li}_{m}(z)}{1-z},
  62. Li m ( z ) \mathrm{Li}_{m}(z)
  63. m > 1 m>1
  64. H q / p , m = ζ ( m ) - p m k = 1 1 ( q + p k ) m H_{q/p,m}=\zeta(m)-p^{m}\sum_{k=1}^{\infty}\frac{1}{(q+pk)^{m}}
  65. ζ ( m ) \zeta(m)
  66. H a , m = H a - 1 , m + 1 a m H_{a,m}=H_{a-1,m}+\frac{1}{a^{m}}
  67. H 1 4 , 2 = 16 - 8 G - 5 6 π 2 H_{\frac{1}{4},2}=16-8G-\tfrac{5}{6}\pi^{2}
  68. H 1 2 , 2 = 4 - π 2 3 H_{\frac{1}{2},2}=4-\tfrac{\pi^{2}}{3}
  69. H 3 4 , 2 = 8 G + 16 9 - 5 6 π 2 H_{\frac{3}{4},2}=8G+\tfrac{16}{9}-\tfrac{5}{6}\pi^{2}
  70. H 1 4 , 3 = 64 - 27 ζ ( 3 ) - π 3 H_{\frac{1}{4},3}=64-27\zeta(3)-\pi^{3}
  71. H 1 2 , 3 = 8 - 6 ζ ( 3 ) H_{\frac{1}{2},3}=8-6\zeta(3)
  72. H 3 4 , 3 = ( 4 3 ) 3 - 27 ζ ( 3 ) + π 3 H_{\frac{3}{4},3}={(\tfrac{4}{3})}^{3}-27\zeta(3)+\pi^{3}
  73. H 2 x = 1 2 ( H x + H x - 1 2 ) + ln 2 H_{2x}=\frac{1}{2}\left(H_{x}+H_{x-\frac{1}{2}}\right)+\ln{2}
  74. H 3 x = 1 3 ( H x + H x - 1 3 + H x - 2 3 ) + ln 3 , H_{3x}=\frac{1}{3}\left(H_{x}+H_{x-\frac{1}{3}}+H_{x-\frac{2}{3}}\right)+\ln{3},
  75. H n x = 1 n ( H x + H x - 1 n + H x - 2 n + + H x - n - 1 n ) + ln n . H_{nx}=\frac{1}{n}\left(H_{x}+H_{x-\frac{1}{n}}+H_{x-\frac{2}{n}}+\cdots+H_{x-% \frac{n-1}{n}}\right)+\ln{n}.
  76. H 2 x , 2 = 1 2 ( ζ ( 2 ) + 1 2 ( H x , 2 + H x - 1 2 , 2 ) ) H_{2x,2}=\frac{1}{2}\left(\zeta(2)+\frac{1}{2}\left(H_{x,2}+H_{x-\frac{1}{2},2% }\right)\right)
  77. H 3 x , 2 = 1 9 ( 6 ζ ( 2 ) + H x , 2 + H x - 1 3 , 2 + H x - 2 3 , 2 ) , H_{3x,2}=\frac{1}{9}\left(6\zeta(2)+H_{x,2}+H_{x-\frac{1}{3},2}+H_{x-\frac{2}{% 3},2}\right),
  78. ζ ( n ) \zeta(n)
  79. a 1 1 - x s 1 - x d x = - k = 1 1 k ( s k ) ( a - 1 ) k , \int_{a}^{1}\frac{1-x^{s}}{1-x}\,dx=-\sum_{k=1}^{\infty}\frac{1}{k}{s\choose k% }(a-1)^{k},
  80. k = 0 ( s k ) ( - x ) k = ( 1 - x ) s , \sum_{k=0}^{\infty}{s\choose k}(-x)^{k}=(1-x)^{s},
  81. H s = ψ ( s + 1 ) + γ = 0 1 1 - x s 1 - x d x , H_{s}=\psi(s+1)+\gamma=\int_{0}^{1}\frac{1-x^{s}}{1-x}\,dx,
  82. ψ ( x ) \psi(x)
  83. H s , 2 = - k = 1 ( - 1 ) k k ( s k ) H k . H_{s,2}=-\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}{s\choose k}H_{k}.
  84. d n H x d x n = ( - 1 ) n + 1 n ! [ ζ ( n + 1 ) - H x , n + 1 ] \frac{d^{n}H_{x}}{dx^{n}}=(-1)^{n+1}n!\left[\zeta(n+1)-H_{x,n+1}\right]
  85. d n H x , 2 d x n = ( - 1 ) n + 1 ( n + 1 ) ! [ ζ ( n + 2 ) - H x , n + 2 ] \frac{d^{n}H_{x,2}}{dx^{n}}=(-1)^{n+1}(n+1)!\left[\zeta(n+2)-H_{x,n+2}\right]
  86. d n H x , 3 d x n = ( - 1 ) n + 1 1 2 ( n + 2 ) ! [ ζ ( n + 3 ) - H x , n + 3 ] . \frac{d^{n}H_{x,3}}{dx^{n}}=(-1)^{n+1}\frac{1}{2}(n+2)!\left[\zeta(n+3)-H_{x,n% +3}\right].
  87. H x , 2 = n = 1 ( - 1 ) n + 1 ( n + 1 ) x n ζ ( n + 2 ) H_{x,2}=\sum_{n=1}^{\infty}(-1)^{n+1}(n+1)x^{n}\zeta(n+2)
  88. H x , 3 = 1 2 n = 1 ( - 1 ) n + 1 ( n + 1 ) ( n + 2 ) x n ζ ( n + 3 ) . H_{x,3}=\frac{1}{2}\sum_{n=1}^{\infty}(-1)^{n+1}(n+1)(n+2)x^{n}\zeta(n+3).
  89. H 1 a = 1 a ( ζ ( 2 ) - 1 a ζ ( 3 ) + 1 a 2 ζ ( 4 ) - 1 a 3 ζ ( 5 ) + ) H_{\frac{1}{a}}=\frac{1}{a}\left(\zeta(2)-\frac{1}{a}\zeta(3)+\frac{1}{a^{2}}% \zeta(4)-\frac{1}{a^{3}}\zeta(5)+\cdots\right)
  90. H 1 a , 2 = 1 a ( 2 ζ ( 3 ) - 3 a ζ ( 4 ) + 4 a 2 ζ ( 5 ) - 5 a 3 ζ ( 6 ) + ) H_{\frac{1}{a},2}=\frac{1}{a}\left(2\zeta(3)-\frac{3}{a}\zeta(4)+\frac{4}{a^{2% }}\zeta(5)-\frac{5}{a^{3}}\zeta(6)+\cdots\right)
  91. H 1 a , 3 = 1 2 a ( 2 3 ζ ( 4 ) - 3 4 a ζ ( 5 ) + 4 5 a 2 ζ ( 6 ) - 5 6 a 3 ζ ( 7 ) + ) . H_{\frac{1}{a},3}=\frac{1}{2a}\left(2\cdot 3\zeta(4)-\frac{3\cdot 4}{a}\zeta(5% )+\frac{4\cdot 5}{a^{2}}\zeta(6)-\frac{5\cdot 6}{a^{3}}\zeta(7)+\cdots\right).
  92. H n ( 0 ) = 1 n . H_{n}^{(0)}=\frac{1}{n}.
  93. H n ( r ) = k = 1 n H k ( r - 1 ) . H_{n}^{(r)}=\sum_{k=1}^{n}H_{k}^{(r-1)}.
  94. H n = H n ( 1 ) H_{n}=H_{n}^{(1)}

Harmonic_series_(mathematics).html

  1. n = 1 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + . \sum_{n=1}^{\infty}\,\frac{1}{n}\;\;=\;\;1\,+\,\frac{1}{2}\,+\,\frac{1}{3}\,+% \,\frac{1}{4}\,+\,\frac{1}{5}\,+\,\cdots.\!
  2. 1 100 k = 1 n 1 k \frac{1}{100}\sum_{k=1}^{n}\frac{1}{k}
  3. 10 2 k = 1 n 1 k . \frac{10}{2}\sum_{k=1}^{n}\frac{1}{k}.
  4. 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 + \displaystyle 1\;\;+\;\;\frac{1}{2}\;\;+\;\;\frac{1}{3}\,+\,\frac{1}{4}\;\;+\;% \;\frac{1}{5}\,+\,\frac{1}{6}\,+\,\frac{1}{7}\,+\,\frac{1}{8}\;\;+\;\;\frac{1}% {9}\,+\,\cdots
  5. 1 + ( 1 2 ) + ( 1 4 + 1 4 ) + ( 1 8 + 1 8 + 1 8 + 1 8 ) + ( 1 16 + + 1 16 ) + \displaystyle 1+\left(\frac{1}{2}\right)+\left(\frac{1}{4}+\frac{1}{4}\right)+% \left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\left(\frac{1}{16% }+\cdots+\frac{1}{16}\right)+\cdots
  6. n = 1 2 k 1 n 1 + k 2 \sum_{n=1}^{2^{k}}\,\frac{1}{n}\;\geq\;1+\frac{k}{2}
  7. area of rectangles = 1 + 1 2 + 1 3 + 1 4 + 1 5 + . \begin{array}[]{c}\,\text{area of}\\ \,\text{rectangles}\end{array}=1\,+\,\frac{1}{2}\,+\,\frac{1}{3}\,+\,\frac{1}{% 4}\,+\,\frac{1}{5}\,+\,\cdots.
  8. area under curve = 1 1 x d x = . \begin{array}[]{c}\,\text{area under}\\ \,\text{curve}\end{array}=\int_{1}^{\infty}\frac{1}{x}\,dx\;=\;\infty.
  9. n = 1 k 1 n > 1 k + 1 1 x d x = ln ( k + 1 ) . \sum_{n=1}^{k}\,\frac{1}{n}\;>\;\int_{1}^{k+1}\frac{1}{x}\,dx\;=\;\ln(k+1).
  10. n = 1 k 1 n = ln ( k ) + γ + ε k < ln ( k ) + 1 \sum_{n=1}^{k}\,\frac{1}{n}\;=\;\ln(k)+\gamma+\varepsilon_{k}<\ln(k)+1
  11. γ \gamma
  12. ε k 1 2 k \varepsilon_{k}\sim\frac{1}{2k}
  13. k k
  14. p prime 1 p = 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + 1 13 + 1 17 + = . \sum_{p\,\text{ prime }}\frac{1}{p}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{% 1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\cdots=\infty.
  15. H n = k = 1 n 1 k , H_{n}=\sum_{k=1}^{n}\frac{1}{k},\!
  16. n = 1 ( - 1 ) n + 1 n = 1 - 1 2 + 1 3 - 1 4 + 1 5 - \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\;=\;1\,-\,\frac{1}{2}\,+\,\frac{1}{3}% \,-\,\frac{1}{4}\,+\,\frac{1}{5}\,-\,\cdots
  17. 1 - 1 2 + 1 3 - 1 4 + 1 5 - = ln 2. 1\,-\,\frac{1}{2}\,+\,\frac{1}{3}\,-\,\frac{1}{4}\,+\,\frac{1}{5}\,-\,\cdots\;% =\;\ln 2.
  18. ln 2 \ln 2
  19. 1 1 ( 1 1 - 1 2 ) + 1 2 ( 2 3 - 2 4 ) + 1 4 ( 4 5 - 4 6 + 4 7 - 4 8 ) + = ln 2. \frac{1}{1}\left(\frac{1}{1}-\frac{1}{2}\right)\,+\,\frac{1}{2}\left(\frac{2}{% 3}-\frac{2}{4}\right)\,+\,\frac{1}{4}\left(\frac{4}{5}-\frac{4}{6}+\frac{4}{7}% -\frac{4}{8}\right)+\cdots\;=\;\ln 2.
  20. n = 0 ( - 1 ) n 2 n + 1 = 1 - 1 3 + 1 5 - 1 7 + = π 4 . \sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}\;\;=\;\;1\,-\,\frac{1}{3}\,+\,\frac{1% }{5}\,-\,\frac{1}{7}\,+\,\cdots\;\;=\;\;\frac{\pi}{4}.
  21. n = 0 1 a n + b , \sum_{n=0}^{\infty}\frac{1}{an+b},\!
  22. a 0 a\neq 0
  23. b b
  24. n = 1 1 n p , \sum_{n=1}^{\infty}\frac{1}{n^{p}},\!
  25. n = 2 1 n ln ( n ) p , \sum_{n=2}^{\infty}\frac{1}{n\ln(n)^{p}},\!
  26. lim sup u 0 + φ ( u 2 ) φ ( u ) < 1 2 \limsup_{u\to 0^{+}}\frac{\varphi(\frac{u}{2})}{\varphi(u)}<\frac{1}{2}
  27. n = 1 s n n , \sum_{n=1}^{\infty}\frac{s_{n}}{n},\!
  28. C 2 C_{2}

Hartree.html

  1. E h = 2 m e a 0 2 = m e ( e 2 4 π ϵ 0 ) 2 = m e c 2 α 2 = c α a 0 E_{\mathrm{h}}={\hbar^{2}\over{m_{\mathrm{e}}a^{2}_{0}}}=m_{\mathrm{e}}\left(% \frac{e^{2}}{4\pi\epsilon_{0}\hbar}\right)^{2}=m_{\mathrm{e}}c^{2}\alpha^{2}={% \hbar c\alpha\over{a_{0}}}
  2. × 10 18 \times 10^{−}18

Hawking_radiation.html

  1. d s 2 = - ( 1 - 2 M r ) d t 2 + 1 1 - 2 M / r d r 2 + r 2 d Ω 2 . ds^{2}=-\left(1-{2M\over r}\right)dt^{2}+{1\over 1-2M/r}dr^{2}+r^{2}d\Omega^{2}.
  2. r = 2 M + u 2 / 2 M r=2M+u^{2}/2M
  3. d s 2 = - u 2 4 M 2 d t 2 + 4 d u 2 + d X 2 = - ρ 2 d τ 2 + d ρ 2 + d X 2 , ds^{2}=-{u^{2}\over 4M^{2}}dt^{2}+4du^{2}+dX_{\perp}^{2}=-\rho^{2}d\tau^{2}+d% \rho^{2}+dX_{\perp}^{2},
  4. τ = t / 4 M \tau=t/4M
  5. ρ = 2 u \rho=2u
  6. u 0 u\rightarrow 0
  7. β ( u ) = 2 π ρ = ( 4 π ) u = 4 π 2 M ( r - 2 M ) ; \beta(u)=2\pi\rho=(4\pi)u=4\pi\sqrt{2M(r-2M)};
  8. β ( r ) = 4 π 2 M ( r - 2 M ) 1 - 2 M / r 1 - 2 M / r . \beta(r^{\prime})=4\pi\sqrt{2M(r-2M)}\sqrt{1-2M/r^{\prime}\over 1-2M/r}.
  9. β ( ) = ( 4 π ) 2 M r \beta(\infty)=(4\pi)\sqrt{2Mr}\;
  10. r r
  11. 2 M 2M
  12. β = 8 π M . \beta=8\pi M.
  13. T H = 1 8 π M . T_{H}={1\over 8\pi M}.
  14. G = c = = k B = 1 G=c=\hbar=k\text{B}=1
  15. T H = κ 2 π , T_{H}=\frac{\kappa}{2\pi},
  16. κ \kappa
  17. T = c 3 8 π G M k B ( 1.227 × 10 23 kg M K = 6.169 × 10 - 8 K × M M ) , T={\hbar\,c^{3}\over 8\pi GMk\text{B}}\;\quad\left(\approx{1.227\times 10^{23}% \;\,\text{kg}\over M}\;\,\text{K}=6.169\times 10^{-8}\;\,\text{K}\times{\,% \text{M}_{\odot}\over M}\right),
  18. \hbar
  19. d S = d Q T = 8 π M d Q . dS={dQ\over T}=8\pi MdQ.
  20. d S = 8 π M d M = d ( 4 π M 2 ) . dS=8\pi MdM=d(4\pi M^{2}).
  21. S = π R 2 = A 4 . S=\pi R^{2}={A\over 4}.
  22. M M
  23. σ = π 2 k B 4 60 3 c 2 \sigma=\frac{\pi^{2}k_{B}^{4}}{60\hbar^{3}c^{2}}\;
  24. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}\;
  25. g = G M r s 2 = c 4 4 G M g=\frac{GM}{r_{s}^{2}}=\frac{c^{4}}{4GM}\;
  26. E = k B T = g 2 π c = 2 π c ( c 4 4 G M ) = c 3 8 π G M E=k_{B}T=\frac{\hbar g}{2\pi c}=\frac{\hbar}{2\pi c}\left(\frac{c^{4}}{4GM}% \right)=\frac{\hbar c^{3}}{8\pi GM}\;
  27. T H = c 3 8 π G M k B T_{H}=\frac{\hbar c^{3}}{8\pi GMk_{B}}\;
  28. b = h c / ( 4.9651 k B ) = 2.8978 × 10 - 3 m K b=hc/(4.9651k_{B})=2.8978\times 10^{-3}\mathrm{m}\,\mathrm{K}
  29. λ m a x = b T H = 8 π 2 4.9651 r s = 15.902 r s \lambda_{max}=\frac{b}{T_{H}}=\frac{8\pi^{2}}{4.9651}\,r_{s}=15.902\,r_{s}\;
  30. r s r_{s}
  31. A s = 4 π r s 2 = 4 π ( 2 G M c 2 ) 2 = 16 π G 2 M 2 c 4 A_{s}=4\pi r_{s}^{2}=4\pi\left(\frac{2GM}{c^{2}}\right)^{2}=\frac{16\pi G^{2}M% ^{2}}{c^{4}}\;
  32. P = A s j = A s ϵ σ T 4 P=A_{s}j^{\star}=A_{s}\epsilon\sigma T^{4}\;
  33. ϵ = 1 \epsilon=1\;
  34. P = A s ϵ σ T H 4 = ( 16 π G 2 M 2 c 4 ) ( π 2 k B 4 60 3 c 2 ) ( c 3 8 π G M k B ) 4 = c 6 15360 π G 2 M 2 P=A_{s}\epsilon\sigma T_{H}^{4}=\left(\frac{16\pi G^{2}M^{2}}{c^{4}}\right)% \left(\frac{\pi^{2}k_{B}^{4}}{60\hbar^{3}c^{2}}\right)\left(\frac{\hbar c^{3}}% {8\pi GMk_{B}}\right)^{4}=\frac{\hbar c^{6}}{15360\pi G^{2}M^{2}}\;
  35. P = c 6 15360 π G 2 M 2 P=\frac{\hbar c^{6}}{15360\pi G^{2}M^{2}}\;
  36. P P
  37. \hbar
  38. c c
  39. G G
  40. M M_{\odot}
  41. P = c 6 15360 π G 2 M 2 = 9.004 × 10 - 29 W P=\frac{\hbar c^{6}}{15360\pi G^{2}M_{\odot}^{2}}=9.004\times 10^{-29}\;\,% \text{W}\;
  42. K ev = c 6 15360 π G 2 = 3.562 × 10 32 W kg 2 K_{\operatorname{ev}}=\frac{\hbar c^{6}}{15360\pi G^{2}}=3.562\times 10^{32}\;% \,\text{W}\cdot\,\text{kg}^{2}\;
  43. P = - d E d t = K ev M 2 P=-\frac{dE}{dt}=\frac{K_{\operatorname{ev}}}{M^{2}}\;
  44. E = M c 2 E=Mc^{2}\;
  45. P = - d E d t = - ( d d t ) M c 2 = - c 2 d M d t P=-\frac{dE}{dt}=-\left(\frac{d}{dt}\right)Mc^{2}=-c^{2}\frac{dM}{dt}\;
  46. - c 2 d M d t = K ev M 2 -c^{2}\frac{dM}{dt}=\frac{K_{\operatorname{ev}}}{M^{2}}\;
  47. M 2 d M = - K ev c 2 d t M^{2}dM=-\frac{K_{\operatorname{ev}}}{c^{2}}dt\;
  48. M 0 M_{0}
  49. t ev t_{\operatorname{ev}}\;
  50. M 0 0 M 2 d M = - K ev c 2 0 t ev d t \int_{M_{0}}^{0}M^{2}dM=-\frac{K_{\operatorname{ev}}}{c^{2}}\int_{0}^{t_{% \operatorname{ev}}}dt\;
  51. t ev = c 2 M 0 3 3 K ev = ( c 2 M 0 3 3 ) ( 15360 π G 2 c 6 ) = 5120 π G 2 M 0 3 c 4 = 8.410 × 10 - 17 [ M 0 kg ] 3 s t_{\operatorname{ev}}=\frac{c^{2}M_{0}^{3}}{3K_{\operatorname{ev}}}=\left(% \frac{c^{2}M_{0}^{3}}{3}\right)\left(\frac{15360\pi G^{2}}{\hbar c^{6}}\right)% =\frac{5120\pi G^{2}M_{0}^{3}}{\hbar c^{4}}=8.410\times 10^{-17}\left[\frac{M_% {0}}{\mathrm{kg}}\right]^{3}\mathrm{s}\;
  52. t ev = 5120 π G 2 M 0 3 c 4 t_{\operatorname{ev}}=\frac{5120\pi G^{2}M_{0}^{3}}{\hbar c^{4}}\;
  53. M 0 M_{0}
  54. m P m_{P}
  55. t ev = 5120 π G 2 m P 3 c 4 = 5120 π t P = 5120 π G c 5 = 8.671 × 10 - 40 s t_{\operatorname{ev}}=\frac{5120\pi G^{2}m_{P}^{3}}{\hbar c^{4}}=5120\pi t_{P}% =5120\pi\sqrt{\frac{\hbar G}{c^{5}}}=8.671\times 10^{-40}\;\,\text{s}\;
  56. t ev = 5120 π G c 5 t_{\operatorname{ev}}=5120\pi\sqrt{\frac{\hbar G}{c^{5}}}\;
  57. t P t_{P}
  58. M M_{\odot}
  59. t ev = 5120 π G 2 M 3 c 4 = 6.617 × 10 74 s t_{\operatorname{ev}}=\frac{5120\pi G^{2}M_{\odot}^{3}}{\hbar c^{4}}=6.617% \times 10^{74}\;\,\text{s}\;
  60. M M
  61. T u = 2.725 K T_{u}=2.725\;\,\text{K}\;
  62. M H c 3 8 π G k B T u 4.503 × 10 22 kg M_{H}\leq\frac{\hbar c^{3}}{8\pi Gk_{B}T_{u}}\leq 4.503\times 10^{22}\;\,\text% {kg}\;
  63. M H c 3 8 π G k B T u M_{H}\leq\frac{\hbar c^{3}}{8\pi Gk_{B}T_{u}}\;
  64. M H M = 7.539 × 10 - 3 = 0.754 % \frac{M_{H}}{M_{\oplus}}=7.539\times 10^{-3}=0.754\;\%\;
  65. M M_{\oplus}
  66. P = 3.563 45 × 10 32 [ kg M ] 2 W P=3.563\,45\times 10^{32}\left[\frac{\mathrm{kg}}{M}\right]^{2}\mathrm{W}\;
  67. t ev = 8.407 16 × 10 - 17 [ M 0 kg ] 3 s 2.66 × 10 - 24 [ M 0 kg ] 3 yr t_{\mathrm{ev}}=8.407\,16\times 10^{-17}\left[\frac{M_{0}}{\mathrm{kg}}\right]% ^{3}\mathrm{s}\ \ \approx\ 2.66\times 10^{-24}\left[\frac{M_{0}}{\mathrm{kg}}% \right]^{3}\mathrm{yr}\;
  68. M 0 = 2.282 71 × 10 5 [ t ev s ] 1 / 3 kg 7.2 × 10 7 [ t ev yr ] 1 / 3 kg M_{0}=2.282\,71\times 10^{5}\left[\frac{t_{\mathrm{ev}}}{\mathrm{s}}\right]^{1% /3}\mathrm{kg}\ \ \approx\ 7.2\times 10^{7}\left[\frac{t_{\mathrm{ev}}}{% \mathrm{yr}}\right]^{1/3}\mathrm{kg}\;
  69. τ 1 M * ( M B H M * ) ( n + 3 ) / ( n + 1 ) \tau\sim{1\over M_{*}}\Bigl({M_{BH}\over M_{*}}\Bigr)^{(n+3)/(n+1)}
  70. M * M_{*}

Hazard_ratio.html

  1. h ( t ) = lim Δ t 0 observed events in interval [ t , t + Δ t ] / N ( t ) Δ t h(t)=\lim_{\Delta t\rightarrow 0}\frac{\mathrm{observed\;events\;in\;interval}% [t,t+\Delta t]/N(t)}{\Delta t}
  2. t t
  3. t + Δ t t+\Delta t
  4. t t
  5. Δ t \Delta t
  6. Δ t \Delta t
  7. h 0 ( t ) h_{0}(t)
  8. log h ( t ) = f ( h 0 ( t ) , α + β 1 X 1 + + β k X k ) . \log h(t)=f(h_{0}(t),\alpha+\beta_{1}X_{1}+\cdots+\beta_{k}X_{k}).\,
  9. e β e^{\beta}
  10. β \beta
  11. β \beta
  12. β \beta
  13. S 1 ( t ) = S 0 ( t ) r S_{1}(t)=S_{0}(t)^{r}
  14. S 0 ( t ) = .2 S_{0}(t)=.2
  15. S 1 ( t ) = .2 2 = .04 S_{1}(t)=.2^{2}=.04
  16. e β e^{\beta}

Head-related_transfer_function.html

  1. 1 {}_{1}
  2. 1 {}_{1}
  3. 2 {}_{2}
  4. 2 {}_{2}
  5. 2 {}_{2}
  6. 2 {}_{2}
  7. 1 {}_{1}
  8. 1 {}_{1}
  9. 1 {}_{1}
  10. 2 {}_{2}
  11. 2 {}_{2}
  12. 1 {}_{1}
  13. 2 {}_{2}
  14. 2 {}_{2}
  15. 2 {}_{2}
  16. 1 {}_{1}
  17. 1 {}_{1}
  18. 2 {}_{2}
  19. 1 {}_{1}
  20. 2 {}_{2}
  21. N {}_{N}
  22. T {}^{T}
  23. $^{ }$
  24. T {}^{T}
  25. β {}_{β}
  26. a = 1 A \sum_{a=1}^{A}
  27. a {}_{a}
  28. n = 1 N \sum_{n=1}^{N}
  29. n {}_{n}
  30. n , a {}_{n,a}
  31. 2 {}^{2}
  32. n = 1 N \sum_{n=1}^{N}
  33. n {}_{n}
  34. $^{ }$
  35. $^{ }$
  36. n = 1 N \sum_{n=1}^{N}
  37. n {}_{n}
  38. n {}_{n}
  39. N {}^{N}
  40. n {}^{n}
  41. $^{ }$
  42. $^{ }$
  43. d , k {}_{d,k}
  44. n = 1 N \sum_{n=1}^{N}
  45. n {}_{n}
  46. n , d , k {}_{n,d,k}
  47. n , d , k {}_{n,d,k}
  48. N × D × K {}^{N×D×K}
  49. n , d , k {}_{n,d,k}
  50. $^{ }$
  51. d , k {}_{d,k}

Headphones.html

  1. d B ( S P L ) / V = d B ( S P L ) / m W - 10 * log 10 I m p e d a n c e 1000 dB(SPL)/V=dB(SPL)/mW-10*\log 10{\frac{Impedance}{1000}}

Health_economics.html

  1. M C H K = r + δ MC_{HK}=r+\delta\,
  2. δ \delta\,

Heart_rate.html

  1. Age in days = EHR ( 0.3 ) + 6 \mathrm{Age\ in\ days}=\ \mathrm{EHR}(0.3)+6

Heart_valve.html

  1. Δ p {\Delta}p
  2. a Q t + b Q 2 = Δ p a{{\partial}Q\over{\partial}t}+bQ^{2}={\Delta}p
  3. ρ ( u t + u u x ) + p x = 0 {\rho}\left({{\partial}u\over{\partial}t}+{u{\partial}u\over{\partial}x}\right% )+{{\partial}p\over{\partial}x}=0
  4. A t + x ( A u ) = 0 {{\partial}A\over{\partial}t}+{{\partial}\over{\partial}x}(Au)=0
  5. A ( x , t ) = A 0 ( 1 - [ 1 - Λ ( t ) ] x L ) 2 A(x,t)=A_{0}\left(1-[1-{\Lambda}(t)]{x\over{L}}\right)^{2}
  6. 0 L p ( x , t ) A x d x = [ A 0 - A ( L , t ) ] p ( L , t ) \int_{0}^{L}p(x,t){{\partial}A\over{\partial}x}\,dx=[A_{0}-A(L,t)]\,p(L,t)
  7. Λ 2 ( t ) = A ( L , t ) A 0 \Lambda^{2}(t)={A(L,t)\over A_{0}}

Heat_capacity.html

  1. J K \mathrm{\tfrac{J}{K}}
  2. C Q Δ T , C\equiv\frac{Q}{\Delta T},
  3. C ( T ) δ Q d T , C(T)\equiv\frac{\delta Q}{dT},
  4. Q Q
  5. Δ T \Delta T
  6. [ c ] = J kg K [c]=\mathrm{\tfrac{J}{kg\cdot K}}
  7. [ C mol ] = J mol K [C_{\mathrm{mol}}]=\mathrm{\tfrac{J}{mol\cdot K}}
  8. [ s ] = J m 3 K [s]=\mathrm{\tfrac{J}{m^{3}\cdot K}}
  9. Δ e s y s t e m = e i n - e o u t \Delta e_{system}=e_{in}-e_{out}
  10. d U = δ Q - δ W . {\ \mathrm{d}U=\delta Q-\delta W}.
  11. d U = δ Q - P d V . {\ \mathrm{d}U=\delta Q-P\mathrm{d}V}.
  12. ( U T ) V = ( Q T ) V = C V . \left(\frac{\partial U}{\partial T}\right)_{V}=\left(\frac{\partial Q}{% \partial T}\right)_{V}=C_{V}.
  13. H = U + P V . {\ H=U+PV}.
  14. d H = δ Q + V d P , {\ \mathrm{d}H=\delta Q+V\mathrm{d}P},
  15. ( H T ) P = ( Q T ) P = C P . \left(\frac{\partial H}{\partial T}\right)_{P}=\left(\frac{\partial Q}{% \partial T}\right)_{P}=C_{P}.
  16. ( U T ) V = ( Q T ) V = C V . \left(\frac{\partial U}{\partial T}\right)_{V}=\left(\frac{\partial Q}{% \partial T}\right)_{V}=C_{V}.
  17. ( H T ) P = ( Q T ) P = C P . \left(\frac{\partial H}{\partial T}\right)_{P}=\left(\frac{\partial Q}{% \partial T}\right)_{P}=C_{P}.
  18. C P - C V = T ( P T ) V , n ( V T ) P , n C_{P}-C_{V}=T\left(\frac{\partial P}{\partial T}\right)_{V,n}\left(\frac{% \partial V}{\partial T}\right)_{P,n}
  19. C P - C V = V T α 2 β T C_{P}-C_{V}=VT\frac{\alpha^{2}}{\beta_{T}}
  20. α \alpha
  21. β T \beta_{T}
  22. P V = n R T PV=nRT
  23. C P - C V = T ( P T ) V , n ( V T ) P , n C_{P}-C_{V}=T\left(\frac{\partial P}{\partial T}\right)_{V,n}\left(\frac{% \partial V}{\partial T}\right)_{P,n}
  24. P = n R T V ( P T ) V , n = n R V P=\frac{nRT}{V}\Rightarrow\left(\frac{\partial P}{\partial T}\right)_{V,n}=% \frac{nR}{V}
  25. V = n R T P ( V T ) P , n = n R P V=\frac{nRT}{P}\Rightarrow\left(\frac{\partial V}{\partial T}\right)_{P,n}=% \frac{nR}{P}
  26. T ( P T ) V , n ( V T ) P , n = T ( n R V ) ( n R P ) = ( n R T V ) ( n R P ) = ( P ) ( n R P ) = n R T\left(\frac{\partial P}{\partial T}\right)_{V,n}\left(\frac{\partial V}{% \partial T}\right)_{P,n}=T\left(\frac{nR}{V}\right)\left(\frac{nR}{P}\right)=% \left(\frac{nRT}{V}\right)\left(\frac{nR}{P}\right)=\left(P\right)\left(\frac{% nR}{P}\right)=nR
  27. C P - C V = n R C_{P}-C_{V}=nR
  28. c = C m , c={\partial C\over\partial m},
  29. c = E m = C m = C ρ V , c=E_{m}={C\over m}={C\over{\rho V}},
  30. C C
  31. m m
  32. V V
  33. ρ = m V \rho=\frac{m}{V}
  34. d P = 0 dP=0
  35. d V = 0 dV=0
  36. c P = ( C m ) P , c_{P}=\left(\frac{\partial C}{\partial m}\right)_{P},
  37. c V = ( C m ) V . c_{V}=\left(\frac{\partial C}{\partial m}\right)_{V}.
  38. c P - c V = α 2 T ρ β T . c_{P}-c_{V}=\frac{\alpha^{2}T}{\rho\beta_{T}}.
  39. c c
  40. C V - 1 CV^{-1}\,
  41. c V c_{V}\,
  42. m m
  43. c m c_{m}\,
  44. c m = C m = c v o l u m e t r i c ρ . c_{m}=\frac{C}{m}=\frac{c_{volumetric}}{\rho}.
  45. C P , m = ( C n ) P C_{P,m}=\left(\frac{\partial C}{\partial n}\right)_{P}
  46. C V , m = ( C n ) V C_{V,m}=\left(\frac{\partial C}{\partial n}\right)_{V}
  47. C i , m = ( C n ) C_{i,m}=\left(\frac{\partial C}{\partial n}\right)
  48. C * = C n R = C N k C^{*}={C\over nR}={C\over{Nk}}
  49. C * C^{*}\,
  50. c ^ \hat{c}
  51. S * = S / N k S^{*}=S/Nk
  52. C * = d S * d ln T C^{*}={dS^{*}\over d\ln T}
  53. T d S = δ Q T\,dS=\delta Q\,
  54. S ( T f ) = T = 0 T f δ Q T = 0 T f δ Q d T d T T = 0 T f C ( T ) d T T . S(T_{f})=\int_{T=0}^{T_{f}}\frac{\delta Q}{T}=\int_{0}^{T_{f}}\frac{\delta Q}{% dT}\frac{dT}{T}=\int_{0}^{T_{f}}C(T)\,\frac{dT}{T}.
  55. U Pot = - 2 U Kin , U\text{Pot}=-2U\text{Kin},\,
  56. U = - U Kin , U=-U\text{Kin},\,
  57. 3 / 2 {3}/{2}
  58. 3 / 2 {3}/{2}
  59. 1 / 2 {1}/{2}
  60. 3 / 2 {3}/{2}
  61. C V = ( U T ) V = 3 2 N k B = 3 2 n R C_{V}=\left(\frac{\partial U}{\partial T}\right)_{V}=\frac{3}{2}N\,k_{B}=\frac% {3}{2}n\,R
  62. C V , m = C V n = 3 2 R C_{V,m}=\frac{C_{V}}{n}=\frac{3}{2}R
  63. C V C_{V}
  64. C V , m C_{V,m}
  65. k B k_{B}
  66. C p , m = C V , m + R = 5 2 R C_{p,m}=C_{V,m}+R=\frac{5}{2}R
  67. f = 3 n a f=3n_{a}\,
  68. f vib = f - f trans - f rot = 6 - 3 - 2 = 1 f_{\mathrm{vib}}=f-f_{\mathrm{trans}}-f_{\mathrm{rot}}=6-3-2=1\,
  69. R R
  70. C V , m = 3 R 2 + R + R = 7 R 2 = 3.5 R C_{V,m}=\frac{3R}{2}+R+R=\frac{7R}{2}=3.5R
  71. C V , m = 3 R 2 + R = 5 R 2 = 2.5 R C_{V,m}=\frac{3R}{2}+R=\frac{5R}{2}=2.5R
  72. E = 1 2 m ( v x 2 + v y 2 + v z 2 ) E=\frac{1}{2}\,m\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right)
  73. [ v x , v y , v z ] [v_{x},v_{y},v_{z}]
  74. E = 1 2 ( I 1 ω 1 2 + I 2 ω 2 2 + I 3 ω 3 2 ) E=\frac{1}{2}\,\left(I_{1}\omega_{1}^{2}+I_{2}\omega_{2}^{2}+I_{3}\omega_{3}^{% 2}\right)
  75. [ ω 1 , ω 2 , ω 3 ] [\omega_{1},\omega_{2},\omega_{3}]
  76. v = 3 n - 3 - n r v=3n-3-n_{r}
  77. n r n_{r}
  78. C V , m = f 2 R C_{V,m}=\frac{f}{2}R

Heat_equation.html

  1. u t - α ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) = 0 \frac{\partial u}{\partial t}-\alpha\left(\frac{\partial^{2}u}{\partial x^{2}}% +\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}% \right)=0
  2. 𝐪 = - k u \mathbf{q}=-k\,\nabla u
  3. 𝐪 = - k u x \mathbf{q}=-k\frac{\partial u}{\partial x}\,
  4. Δ Q = c p ρ Δ u \Delta Q=c_{p}\rho\,\Delta u\,
  5. Q = c p ρ u . Q=c_{p}\rho u.\,
  6. x - Δ x ξ x + Δ x x-\Delta x\leq\xi\leq x+\Delta x
  7. t - Δ t τ t + Δ t t-\Delta t\leq\tau\leq t+\Delta t
  8. c p ρ x - Δ x x + Δ x [ u ( ξ , t + Δ t ) - u ( ξ , t - Δ t ) ] d ξ = c p ρ t - Δ t t + Δ t x - Δ x x + Δ x u τ d ξ d τ c_{p}\rho\int_{x-\Delta x}^{x+\Delta x}[u(\xi,t+\Delta t)-u(\xi,t-\Delta t)]\,% d\xi=c_{p}\rho\int_{t-\Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x}% \frac{\partial u}{\partial\tau}\,d\xi\,d\tau
  9. k t - Δ t t + Δ t [ u x ( x + Δ x , τ ) - u x ( x - Δ x , τ ) ] d τ = k t - Δ t t + Δ t x - Δ x x + Δ x 2 u ξ 2 d ξ d τ k\int_{t-\Delta t}^{t+\Delta t}\left[\frac{\partial u}{\partial x}(x+\Delta x,% \tau)-\frac{\partial u}{\partial x}(x-\Delta x,\tau)\right]\,d\tau=k\int_{t-% \Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x}\frac{\partial^{2}u}{% \partial\xi^{2}}\,d\xi\,d\tau
  10. t - Δ t t + Δ t x - Δ x x + Δ x [ c p ρ u τ - k u ξ ξ ] d ξ d τ = 0. \int_{t-\Delta t}^{t+\Delta t}\int_{x-\Delta x}^{x+\Delta x}[c_{p}\rho u_{\tau% }-ku_{\xi\xi}]\,d\xi\,d\tau=0.
  11. c p ρ u t - k u x x = 0. c_{p}\rho u_{t}-ku_{xx}=0.\,\!
  12. u t = k c p ρ u x x , u_{t}=\frac{k}{c_{p}\rho}u_{xx},
  13. u t = k c p ρ ( 2 u x 2 ) \frac{\partial u}{\partial t}=\frac{k}{c_{p}\rho}\left(\frac{\partial^{2}u}{% \partial x^{2}}\right)
  14. α = k c p ρ \alpha=\frac{k}{c_{p}\rho}\,\!
  15. u t = k c p ρ ( 2 u x 2 ) - μ u . \frac{\partial u}{\partial t}=\frac{k}{c_{p}\rho}\left(\frac{\partial^{2}u}{% \partial x^{2}}\right)-\mu u.
  16. T 4 - T s 4 T^{4}-T_{s}^{4}
  17. T 4 - T s 4 u 4 T^{4}-T_{s}^{4}\approx u^{4}
  18. u t = α ( 2 u x 2 ) - m u 4 \frac{\partial u}{\partial t}=\alpha\left(\frac{\partial^{2}u}{\partial x^{2}}% \right)-\ mu^{4}
  19. m = ϵ σ p / ρ A c p \ m=\epsilon\sigma p/\rho Ac_{p}
  20. u t = α ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) {\partial u\over\partial t}=\alpha\left({\partial^{2}u\over\partial x^{2}}+{% \partial^{2}u\over\partial y^{2}}+{\partial^{2}u\over\partial z^{2}}\right)
  21. = α ( u x x + u y y + u z z ) =\alpha(u_{xx}+u_{yy}+u_{zz})\quad
  22. u t \frac{\partial u}{\partial t}
  23. α = k c p ρ \alpha=\frac{k}{c_{p}\rho}
  24. u t = α 2 u = α Δ u , u_{t}=\alpha\nabla^{2}u=\alpha\Delta u,\quad\,\!
  25. u t = α ( 2 u x 2 + 2 u y 2 + 2 u z 2 ) + 1 c p ρ q . \frac{\partial u}{\partial t}=\alpha\left({\partial^{2}u\over\partial x^{2}}+{% \partial^{2}u\over\partial y^{2}}+{\partial^{2}u\over\partial z^{2}}\right)+% \frac{1}{c_{p}\rho}q.
  26. T ( t ) α T ( t ) = X ′′ ( x ) X ( x ) . \frac{T^{\prime}(t)}{\alpha T(t)}=\frac{X^{\prime\prime}(x)}{X(x)}.
  27. u ( x , t ) = n = 1 D n sin ( n π x L ) e - n 2 π 2 α t L 2 u(x,t)=\sum_{n=1}^{\infty}D_{n}\sin\left(\frac{n\pi x}{L}\right)e^{-\frac{n^{2% }\pi^{2}\alpha t}{L^{2}}}
  28. D n = 2 L 0 L f ( x ) sin ( n π x L ) d x . D_{n}=\frac{2}{L}\int_{0}^{L}f(x)\sin\left(\frac{n\pi x}{L}\right)\,dx.
  29. e n ( x ) = 2 L sin ( n π x L ) e_{n}(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)
  30. Δ e n = - n 2 π 2 L 2 e n . \Delta e_{n}=-\frac{n^{2}\pi^{2}}{L^{2}}e_{n}.
  31. e n , e m = 0 L e n ( x ) e m * ( x ) d x = δ m n \langle e_{n},e_{m}\rangle=\int_{0}^{L}e_{n}(x)e^{*}_{m}(x)dx=\delta_{mn}
  32. Q Q
  33. q t ( V ) = V Q ( x , t ) d x q_{t}(V)=\int_{V}Q(x,t)\,dx\quad
  34. 𝐇 ( x ) 𝐧 ( x ) d S \mathbf{H}(x)\cdot\mathbf{n}(x)\,dS
  35. q t ( V ) = - V 𝐇 ( x ) 𝐧 ( x ) d S q_{t}(V)=-\int_{\partial V}\mathbf{H}(x)\cdot\mathbf{n}(x)\,dS
  36. 𝐇 ( x ) = - 𝐀 ( x ) u ( x ) \mathbf{H}(x)=-\mathbf{A}(x)\cdot\nabla u(x)
  37. q t ( V ) \displaystyle q_{t}(V)
  38. t u ( x , t ) = κ ( x ) Q ( x , t ) \partial_{t}u(x,t)=\kappa(x)Q(x,t)\,
  39. t u ( x , t ) = κ ( x ) i , j x i ( a i j ( x ) x j u ( x , t ) ) \partial_{t}u(x,t)=\kappa(x)\sum_{i,j}\partial_{x_{i}}\bigl(a_{ij}(x)\partial_% {x_{j}}u(x,t)\bigr)
  40. A u ( x ) := i , j x i a i j ( x ) x j u ( x ) Au(x):=\sum_{i,j}\partial_{x_{i}}a_{ij}(x)\partial_{x_{j}}u(x)
  41. { u t ( x , t ) - k u x x ( x , t ) = 0 ( x , t ) 𝐑 × ( 0 , ) u ( x , 0 ) = δ ( x ) \begin{cases}u_{t}(x,t)-ku_{xx}(x,t)=0&(x,t)\in\mathbf{R}\times(0,\infty)\\ u(x,0)=\delta(x)&\end{cases}
  42. Φ ( x , t ) = 1 4 π k t exp ( - x 2 4 k t ) . \Phi(x,t)=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^{2}}{4kt}\right).
  43. { u t ( 𝐱 , t ) - k i = 1 n u x i x i ( 𝐱 , t ) = 0 ( 𝐱 , t ) 𝐑 n × ( 0 , ) u ( 𝐱 , 0 ) = δ ( 𝐱 ) \begin{cases}u_{t}(\mathbf{x},t)-k\sum_{i=1}^{n}u_{x_{i}x_{i}}(\mathbf{x},t)=0% &(\mathbf{x},t)\in\mathbf{R}^{n}\times(0,\infty)\\ u(\mathbf{x},0)=\delta(\mathbf{x})\end{cases}
  44. Φ ( 𝐱 , t ) = Φ ( x 1 , t ) Φ ( x 2 , t ) Φ ( x n , t ) = 1 ( 4 π k t ) n exp ( - 𝐱 𝐱 4 k t ) . \Phi(\mathbf{x},t)=\Phi(x_{1},t)\Phi(x_{2},t)\dots\Phi(x_{n},t)=\frac{1}{\sqrt% {(4\pi kt)^{n}}}\exp\left(-\frac{\mathbf{x}\cdot\mathbf{x}}{4kt}\right).
  45. u ( 𝐱 , t ) = 𝐑 n Φ ( 𝐱 - 𝐲 , t ) g ( 𝐲 ) d 𝐲 . u(\mathbf{x},t)=\int_{\mathbf{R}^{n}}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y}% )d\mathbf{y}.
  46. { u t ( 𝐱 , t ) - k i = 1 n u x i x i ( 𝐱 , t ) = 0 ( 𝐱 , t ) Ω × ( 0 , ) u ( 𝐱 , 0 ) = g ( 𝐱 ) 𝐱 Ω \begin{cases}u_{t}(\mathbf{x},t)-k\sum_{i=1}^{n}u_{x_{i}x_{i}}(\mathbf{x},t)=0% &(\mathbf{x},t)\in\Omega\times(0,\infty)\\ u(\mathbf{x},0)=g(\mathbf{x})&\mathbf{x}\in\Omega\end{cases}
  47. u t = k u x x + f . u_{t}=ku_{xx}+f.
  48. { u t = k u x x ( x , t ) 𝐑 × ( 0 , ) u ( x , 0 ) = g ( x ) I C \begin{cases}u_{t}=ku_{xx}&(x,t)\in\mathbf{R}\times(0,\infty)\\ u(x,0)=g(x)&IC\end{cases}
  49. u ( x , t ) = 1 4 π k t - exp ( - ( x - y ) 2 4 k t ) g ( y ) d y u(x,t)=\frac{1}{\sqrt{4\pi kt}}\int_{-\infty}^{\infty}\exp\left(-\frac{(x-y)^{% 2}}{4kt}\right)g(y)\,dy
  50. Φ ( x , t ) := 1 4 π k t exp ( - x 2 4 k t ) , \Phi(x,t):=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^{2}}{4kt}\right),
  51. ( t - k x 2 ) ( Φ * g ) = [ ( t - k x 2 ) Φ ] * g = 0. \left(\partial_{t}-k\partial_{x}^{2}\right)(\Phi*g)=\left[\left(\partial_{t}-k% \partial_{x}^{2}\right)\Phi\right]*g=0.
  52. Φ ( x , t ) = 1 t Φ ( x t ) \Phi(x,t)=\frac{1}{\sqrt{t}}\,\Phi\left(\frac{x}{\sqrt{t}}\right)
  53. - Φ ( x , t ) d x = 1 , \int_{-\infty}^{\infty}\Phi(x,t)\,dx=1,
  54. { u t = k u x x ( x , t ) [ 0 , ) × ( 0 , ) u ( x , 0 ) = g ( x ) I C u ( 0 , t ) = 0 B C \begin{cases}u_{t}=ku_{xx}&(x,t)\in[0,\infty)\times(0,\infty)\\ u(x,0)=g(x)&IC\\ u(0,t)=0&BC\end{cases}
  55. u ( x , t ) = 1 4 π k t 0 [ exp ( - ( x - y ) 2 4 k t ) - exp ( - ( x + y ) 2 4 k t ) ] g ( y ) d y u(x,t)=\frac{1}{\sqrt{4\pi kt}}\int_{0}^{\infty}\left[\exp\left(-\frac{(x-y)^{% 2}}{4kt}\right)-\exp\left(-\frac{(x+y)^{2}}{4kt}\right)\right]g(y)\,dy
  56. { u t = k u x x ( x , t ) [ 0 , ) × ( 0 , ) u ( x , 0 ) = g ( x ) I C u x ( 0 , t ) = 0 B C \begin{cases}u_{t}=ku_{xx}&(x,t)\in[0,\infty)\times(0,\infty)\\ u(x,0)=g(x)&IC\\ u_{x}(0,t)=0&BC\end{cases}
  57. u ( x , t ) = 1 4 π k t 0 [ exp ( - ( x - y ) 2 4 k t ) + exp ( - ( x + y ) 2 4 k t ) ] g ( y ) d y u(x,t)=\frac{1}{\sqrt{4\pi kt}}\int_{0}^{\infty}\left[\exp\left(-\frac{(x-y)^{% 2}}{4kt}\right)+\exp\left(-\frac{(x+y)^{2}}{4kt}\right)\right]g(y)\,dy
  58. { u t = k u x x ( x , t ) [ 0 , ) × ( 0 , ) u ( x , 0 ) = 0 I C u ( 0 , t ) = h ( t ) B C \begin{cases}u_{t}=ku_{xx}&(x,t)\in[0,\infty)\times(0,\infty)\\ u(x,0)=0&IC\\ u(0,t)=h(t)&BC\end{cases}
  59. u ( x , t ) = 0 t x 4 π k ( t - s ) 3 exp ( - x 2 4 k ( t - s ) ) h ( s ) d s , x > 0 u(x,t)=\int_{0}^{t}\frac{x}{\sqrt{4\pi k(t-s)^{3}}}\exp\left(-\frac{x^{2}}{4k(% t-s)}\right)h(s)\,ds,\qquad\forall x>0
  60. ψ ( x , t ) := - 2 k x Φ ( x , t ) = x 4 π k t 3 exp ( - x 2 4 k t ) \psi(x,t):=-2k\partial_{x}\Phi(x,t)=\frac{x}{\sqrt{4\pi kt^{3}}}\exp\left(-% \frac{x^{2}}{4kt}\right)
  61. t - k x 2 , \partial_{t}-k\partial^{2}_{x},
  62. ψ ( x , t ) = 1 x 2 ψ ( 1 , t x 2 ) \psi(x,t)=\frac{1}{x^{2}}\,\psi\left(1,\frac{t}{x^{2}}\right)
  63. 0 ψ ( x , t ) d t = 1 , \int_{0}^{\infty}\psi(x,t)\,dt=1,
  64. { u t = k u x x + f ( x , t ) ( x , t ) 𝐑 × ( 0 , ) u ( x , 0 ) = 0 I C \begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in\mathbf{R}\times(0,\infty)\\ u(x,0)=0&IC\end{cases}
  65. u ( x , t ) = 0 t - 1 4 π k ( t - s ) exp ( - ( x - y ) 2 4 k ( t - s ) ) f ( y , s ) d y d s u(x,t)=\int_{0}^{t}\int_{-\infty}^{\infty}\frac{1}{\sqrt{4\pi k(t-s)}}\exp% \left(-\frac{(x-y)^{2}}{4k(t-s)}\right)f(y,s)\,dy\,ds
  66. Φ ( x , t ) := 1 4 π k t exp ( - x 2 4 k t ) \Phi(x,t):=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^{2}}{4kt}\right)
  67. ( t - k x 2 ) ( Φ * f ) = f , \left(\partial_{t}-k\partial_{x}^{2}\right)(\Phi*f)=f,
  68. ( t - k x 2 ) Φ = δ , \left(\partial_{t}-k\partial_{x}^{2}\right)\Phi=\delta,
  69. { u t = k u x x + f ( x , t ) ( x , t ) [ 0 , ) × ( 0 , ) u ( x , 0 ) = 0 I C u ( 0 , t ) = 0 B C \begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in[0,\infty)\times(0,\infty)\\ u(x,0)=0&IC\\ u(0,t)=0&BC\end{cases}
  70. u ( x , t ) = 0 t 0 1 4 π k ( t - s ) ( exp ( - ( x - y ) 2 4 k ( t - s ) ) - exp ( - ( x + y ) 2 4 k ( t - s ) ) ) f ( y , s ) d y d s u(x,t)=\int_{0}^{t}\int_{0}^{\infty}\frac{1}{\sqrt{4\pi k(t-s)}}\left(\exp% \left(-\frac{(x-y)^{2}}{4k(t-s)}\right)-\exp\left(-\frac{(x+y)^{2}}{4k(t-s)}% \right)\right)f(y,s)\,dy\,ds
  71. { u t = k u x x + f ( x , t ) ( x , t ) [ 0 , ) × ( 0 , ) u ( x , 0 ) = 0 I C u x ( 0 , t ) = 0 B C \begin{cases}u_{t}=ku_{xx}+f(x,t)&(x,t)\in[0,\infty)\times(0,\infty)\\ u(x,0)=0&IC\\ u_{x}(0,t)=0&BC\end{cases}
  72. u ( x , t ) = 0 t 0 1 4 π k ( t - s ) ( exp ( - ( x - y ) 2 4 k ( t - s ) ) + exp ( - ( x + y ) 2 4 k ( t - s ) ) ) f ( y , s ) d y d s u(x,t)=\int_{0}^{t}\int_{0}^{\infty}\frac{1}{\sqrt{4\pi k(t-s)}}\left(\exp% \left(-\frac{(x-y)^{2}}{4k(t-s)}\right)+\exp\left(-\frac{(x+y)^{2}}{4k(t-s)}% \right)\right)f(y,s)\,dy\,ds
  73. { u t = k u x x + f ( x , t ) 𝐑 × ( 0 , ) u ( x , 0 ) = g ( x ) I C \begin{cases}u_{t}=ku_{xx}+f&(x,t)\in\mathbf{R}\times(0,\infty)\\ u(x,0)=g(x)&IC\end{cases}
  74. { v t = k v x x + f , w t = k w x x ( x , t ) 𝐑 × ( 0 , ) v ( x , 0 ) = 0 , w ( x , 0 ) = g ( x ) I C \begin{cases}v_{t}=kv_{xx}+f,\,w_{t}=kw_{xx}&(x,t)\in\mathbf{R}\times(0,\infty% )\\ v(x,0)=0,\,w(x,0)=g(x)&IC\end{cases}
  75. { u t = k u x x + f ( x , t ) [ 0 , ) × ( 0 , ) u ( x , 0 ) = g ( x ) I C u ( 0 , t ) = h ( t ) B C \begin{cases}u_{t}=ku_{xx}+f&(x,t)\in[0,\infty)\times(0,\infty)\\ u(x,0)=g(x)&IC\\ u(0,t)=h(t)&BC\end{cases}
  76. { v t = k v x x + f , w t = k w x x , r t = k r x x ( x , t ) [ 0 , ) × ( 0 , ) v ( x , 0 ) = 0 , w ( x , 0 ) = g ( x ) , r ( x , 0 ) = 0 I C v ( 0 , t ) = 0 , w ( 0 , t ) = 0 , r ( 0 , t ) = h ( t ) B C \begin{cases}v_{t}=kv_{xx}+f,\,w_{t}=kw_{xx},\,r_{t}=kr_{xx}&(x,t)\in[0,\infty% )\times(0,\infty)\\ v(x,0)=0,\;w(x,0)=g(x),\;r(x,0)=0&IC\\ v(0,t)=0,\;w(0,t)=0,\;r(0,t)=h(t)&BC\end{cases}
  77. ( t - Δ ) u = 0 (\partial_{t}-\Delta)u=0
  78. Δ u = 0 \Delta u=0
  79. ( t - Δ ) u = 0 (\partial_{t}-\Delta)u=0
  80. ( x , t ) + E λ dom ( u ) (x,t)+E_{\lambda}\subset\mathrm{dom}(u)
  81. u ( x , t ) = λ 4 E λ u ( x - y , t - s ) | y | 2 s 2 d s d y , u(x,t)=\frac{\lambda}{4}\int_{E_{\lambda}}u(x-y,t-s)\frac{|y|^{2}}{s^{2}}ds\,dy,
  82. E λ := { ( y , s ) : Φ ( y , s ) > λ } , E_{\lambda}:=\{(y,s)\,:\,\Phi(y,s)>\lambda\},
  83. Φ ( x , t ) := ( 4 t π ) - n 2 exp ( - | x | 2 4 t ) . \Phi(x,t):=(4t\pi)^{-\frac{n}{2}}\exp\left(-\frac{|x|^{2}}{4t}\right).
  84. diam ( E λ ) = o ( 1 ) \mathrm{diam}(E_{\lambda})=o(1)
  85. u t = 0 \frac{\partial u}{\partial t}=0
  86. u t = 0 \frac{\partial u}{\partial t}=0
  87. k 2 u = q k\nabla^{2}u=q
  88. 2 u = 0 \nabla^{2}u=0
  89. c t = D Δ c , c_{t}=D\Delta c,\quad
  90. P t = D Δ P . P_{t}=D\Delta P.\quad
  91. P ( R , t ) = G ( R , t ) = 1 ( 2 π ) 3 ( det ( D ) t ) exp ( - R T D - 1 R 2 t ) P(R,t)=G(R,t)=\frac{1}{\sqrt{(2\pi)^{3}(\det(D)t)}}\exp\left(-\frac{{R}^{T}D^{% -1}{R}}{2t}\right)
  92. ψ t = i 2 m Δ ψ \psi_{t}=\frac{i\hbar}{2m}\Delta\psi
  93. c ( R , t ) \displaystyle c(R,t)
  94. ψ ( R , t ) = ψ ( R 0 , t = 0 ) G ( R - R 0 , t ) d R x 0 d R y 0 d R z 0 , \psi(R,t)=\int\psi(R^{0},t=0)G(R-R^{0},t)dR_{x}^{0}\,dR_{y}^{0}\,dR_{z}^{0},
  95. G ( R , t ) = ( m 2 π i t ) 3 / 2 e - R 2 m 2 i t . G(R,t)=\bigg(\frac{m}{2\pi i\hbar t}\bigg)^{3/2}e^{-\frac{R^{2}m}{2i\hbar t}}.
  96. T C - T S T 0 - T S = 2 n = 1 ( - 1 ) n + 1 exp ( - n 2 π 2 α t L 2 ) \frac{T_{C}-T_{S}}{T_{0}-T_{S}}=2\sum_{n=1}^{\infty}(-1)^{n+1}\exp\left({-% \frac{n^{2}\pi^{2}\alpha t}{L^{2}}}\right)
  97. T < s u b > 0 T<sub>0

Heat_exchanger.html

  1. C i C_{i}
  2. j i j_{i}
  3. T 1 ( x ) T_{1}(x)
  4. T 2 ( x ) T_{2}(x)
  5. d u 1 d t = γ ( T 2 - T 1 ) \frac{du_{1}}{dt}=\gamma(T_{2}-T_{1})
  6. d u 2 d t = γ ( T 1 - T 2 ) \frac{du_{2}}{dt}=\gamma(T_{1}-T_{2})
  7. u i ( x ) u_{i}(x)
  8. d u 1 d t = J 1 d T 1 d x \frac{du_{1}}{dt}=J_{1}\frac{dT_{1}}{dx}
  9. d u 2 d t = J 2 d T 2 d x \frac{du_{2}}{dt}=J_{2}\frac{dT_{2}}{dx}
  10. J i = C i j i J_{i}={C_{i}}{j_{i}}
  11. J 1 T 1 x = γ ( T 2 - T 1 ) J_{1}\frac{\partial T_{1}}{\partial x}=\gamma(T_{2}-T_{1})
  12. J 2 T 2 x = γ ( T 1 - T 2 ) . J_{2}\frac{\partial T_{2}}{\partial x}=\gamma(T_{1}-T_{2}).
  13. T 1 = A - B k 1 k e - k x T_{1}=A-\frac{Bk_{1}}{k}\,e^{-kx}
  14. T 2 = A + B k 2 k e - k x T_{2}=A+\frac{Bk_{2}}{k}\,e^{-kx}
  15. k 1 = γ / J 1 k_{1}=\gamma/J_{1}
  16. k 2 = γ / J 2 k_{2}=\gamma/J_{2}
  17. k = k 1 + k 2 k=k_{1}+k_{2}
  18. T 10 T_{10}
  19. T 20 T_{20}
  20. T 1 L T_{1L}
  21. T 2 L T_{2L}
  22. T ¯ 1 = 1 L 0 L T 1 ( x ) d x \overline{T}_{1}=\frac{1}{L}\int_{0}^{L}T_{1}(x)dx
  23. T ¯ 2 = 1 L 0 L T 2 ( x ) d x . \overline{T}_{2}=\frac{1}{L}\int_{0}^{L}T_{2}(x)dx.
  24. T 10 = A - B k 1 k T_{10}=A-\frac{Bk_{1}}{k}
  25. T 20 = A + B k 2 k T_{20}=A+\frac{Bk_{2}}{k}
  26. T 1 L = A - B k 1 k e - k L T_{1L}=A-\frac{Bk_{1}}{k}e^{-kL}
  27. T 2 L = A + B k 2 k e - k L T_{2L}=A+\frac{Bk_{2}}{k}e^{-kL}
  28. T ¯ 1 = A - B k 1 k 2 L ( 1 - e - k L ) \overline{T}_{1}=A-\frac{Bk_{1}}{k^{2}L}(1-e^{-kL})
  29. T ¯ 2 = A + B k 2 k 2 L ( 1 - e - k L ) . \overline{T}_{2}=A+\frac{Bk_{2}}{k^{2}L}(1-e^{-kL}).
  30. d U 1 d t = 0 L d u 1 d t d x = J 1 ( T 1 L - T 10 ) = γ L ( T ¯ 2 - T ¯ 1 ) \frac{dU_{1}}{dt}=\int_{0}^{L}\frac{du_{1}}{dt}\,dx=J_{1}(T_{1L}-T_{10})=% \gamma L(\overline{T}_{2}-\overline{T}_{1})
  31. d U 2 d t = 0 L d u 2 d t d x = J 2 ( T 2 L - T 20 ) = γ L ( T ¯ 1 - T ¯ 2 ) . \frac{dU_{2}}{dt}=\int_{0}^{L}\frac{du_{2}}{dt}\,dx=J_{2}(T_{2L}-T_{20})=% \gamma L(\overline{T}_{1}-\overline{T}_{2}).
  32. T ¯ 2 - T ¯ 1 \overline{T}_{2}-\overline{T}_{1}

Heat_index.html

  1. HI = c 1 + c 2 T + c 3 R + c 4 T R + c 5 T 2 + c 6 R 2 + c 7 T 2 R + c 8 T R 2 + c 9 T 2 R 2 \mathrm{HI}=c_{1}+c_{2}T+c_{3}R+c_{4}TR+c_{5}T^{2}+c_{6}R^{2}+c_{7}T^{2}R+c_{8% }TR^{2}+c_{9}T^{2}R^{2}\ \,
  2. HI \mathrm{HI}\,\!
  3. T T\,\!
  4. R R\,\!
  5. c 1 = - 42.379 , c_{1}=-42.379,\,\!
  6. c 2 = 2.04901523 , c_{2}=2.04901523,\,\!
  7. c 3 = 10.14333127 , c_{3}=10.14333127,\,\!
  8. c 4 = - 0.22475541 , c_{4}=-0.22475541,\,\!
  9. c 5 = - 6.83783 × 10 - 3 , c_{5}=-6.83783\times 10^{-3},\,\!
  10. c 6 = - 5.481717 × 10 - 2 , c_{6}=-5.481717\times 10^{-2},\,\!
  11. c 7 = 1.22874 × 10 - 3 , c_{7}=1.22874\times 10^{-3},\,\!
  12. c 8 = 8.5282 × 10 - 4 , c_{8}=8.5282\times 10^{-4},\,\!
  13. c 9 = - 1.99 × 10 - 6 . c_{9}=-1.99\times 10^{-6}.\,\!
  14. c 2 = 0.988622465 , c_{2}=0.988622465,\,\!
  15. c 3 = 4.777114035 , c_{3}=4.777114035,\,\!
  16. c 4 = - 0.114037667 , c_{4}=-0.114037667,\,\!
  17. c 5 = - 0.000850208 , c_{5}=-0.000850208,\,\!
  18. c 6 = - 0.020716198 , c_{6}=-0.020716198,\,\!
  19. c 7 = 0.000687678 , c_{7}=0.000687678,\,\!
  20. c 8 = 0.000274954 , c_{8}=0.000274954,\,\!
  21. c 9 = 0 c_{9}=0\,\!
  22. ( c 9 (c_{9}\,\!
  23. u n u s e d ) . unused).
  24. HI = c 1 + c 2 T + c 3 R + c 4 T R + c 5 T 2 + c 6 R 2 + c 7 T 2 R + c 8 T R 2 + c 9 T 2 R 2 + c 10 T 3 + c 11 R 3 + c 12 T 3 R + \mathrm{HI}=c_{1}+c_{2}T+c_{3}R+c_{4}TR+c_{5}T^{2}+c_{6}R^{2}+c_{7}T^{2}R+c_{8% }TR^{2}+c_{9}T^{2}R^{2}+c_{10}T^{3}+c_{11}R^{3}+c_{12}T^{3}R+
  25. c 13 T R 3 + c 14 T 3 R 2 + c 15 T 2 R 3 + c 16 T 3 R 3 c_{13}TR^{3}+c_{14}T^{3}R^{2}+c_{15}T^{2}R^{3}+c_{16}T^{3}R^{3}\ \,
  26. c 1 = 16.923 , c_{1}=16.923,\,\!
  27. c 2 = 0.185212 , c_{2}=0.185212,\,\!
  28. c 3 = 5.37941 , c_{3}=5.37941,\,\!
  29. c 4 = - 0.100254 , c_{4}=-0.100254,\,\!
  30. c 5 = 9.41695 × 10 - 3 , c_{5}=9.41695\times 10^{-3},\,\!
  31. c 6 = 7.28898 × 10 - 3 , c_{6}=7.28898\times 10^{-3},\,\!
  32. c 7 = 3.45372 × 10 - 4 , c_{7}=3.45372\times 10^{-4},\,\!
  33. c 8 = - 8.14971 × 10 - 4 , c_{8}=-8.14971\times 10^{-4},\,\!
  34. c 9 = 1.02102 × 10 - 5 , c_{9}=1.02102\times 10^{-5},\,\!
  35. c 10 = - 3.8646 × 10 - 5 , c_{10}=-3.8646\times 10^{-5},\,\!
  36. c 11 = 2.91583 × 10 - 5 , c_{11}=2.91583\times 10^{-5},\,\!
  37. c 12 = 1.42721 × 10 - 6 , c_{12}=1.42721\times 10^{-6},\,\!
  38. c 13 = 1.97483 × 10 - 7 , c_{13}=1.97483\times 10^{-7},\,\!
  39. c 14 = - 2.18429 × 10 - 8 , c_{14}=-2.18429\times 10^{-8},\,\!
  40. c 15 = 8.43296 × 10 - 10 , c_{15}=8.43296\times 10^{-10},\,\!
  41. c 16 = - 4.81975 × 10 - 11 . c_{16}=-4.81975\times 10^{-11}.\,\!

Heat_transfer.html

  1. Q = v ρ c p Δ T Q=v\rho c_{p}\Delta T
  2. c p c_{p}
  3. v v
  4. R a Ra
  5. R a = g Δ ρ L 3 μ α = g β Δ T L 3 ν α Ra=\frac{g\Delta\rho L^{3}}{\mu\alpha}=\frac{g\beta\Delta TL^{3}}{\nu\alpha}
  6. Δ ρ \Delta\rho
  7. g Δ ρ L 3 g\Delta\rho L^{3}
  8. g Δ ρ L g\Delta\rho L
  9. μ V / L = μ / T c o n v \mu V/L=\mu/T_{conv}
  10. T c o n v T_{conv}
  11. T c o n d = L 2 / α T_{cond}=L^{2}/\alpha
  12. Q = ϵ σ T 4 Q=\epsilon\sigma T^{4}
  13. Q = ϵ σ ( T a 4 - T b 4 ) Q=\epsilon\sigma(T_{a}^{4}-T_{b}^{4})

Heaviside_step_function.html

  1. H ( x ) = - x δ ( s ) d s H(x)=\int_{-\infty}^{x}{\delta(s)}\,\mathrm{d}s
  2. H [ n ] = { 0 , n < 0 , 1 , n > 0 , H[n]=\begin{cases}0,&n<0,\\ 1,&n>0,\end{cases}
  3. H [ n ] = { 0 , n < 0 , 1 / 2 , n = 0 , 1 , n > 0 , H[n]=\begin{cases}0,&n<0,\\ 1/2,&n=0,\\ 1,&n>0,\end{cases}
  4. δ [ n ] = H [ n ] - H [ n - 1 ] . \delta\left[n\right]=H[n]-H[n-1].
  5. H [ n ] = k = - n δ [ k ] H[n]=\sum_{k=-\infty}^{n}\delta[k]\,
  6. δ [ k ] = δ k , 0 \delta[k]=\delta_{k,0}\,
  7. H ( x ) 1 2 + 1 2 tanh ( k x ) = 1 1 + e - 2 k x , H(x)\approx\frac{1}{2}+\frac{1}{2}\tanh(kx)=\frac{1}{1+\mathrm{e}^{-2kx}},
  8. H ( x ) = lim k 1 2 ( 1 + tanh k x ) = lim k 1 1 + e - 2 k x . H(x)=\lim_{k\rightarrow\infty}\frac{1}{2}(1+\tanh kx)=\lim_{k\rightarrow\infty% }\frac{1}{1+\mathrm{e}^{-2kx}}.
  9. H ( x ) \displaystyle H(x)
  10. H ( x ) = lim ϵ 0 + - 1 2 π i - 1 τ + i ϵ e - i x τ d τ = lim ϵ 0 + 1 2 π i - 1 τ - i ϵ e i x τ d τ . H(x)=\lim_{\epsilon\to 0^{+}}-{1\over 2\pi i}\int_{-\infty}^{\infty}{1\over% \tau+i\epsilon}\mathrm{e}^{-ix\tau}\mathrm{d}\tau=\lim_{\epsilon\to 0^{+}}{1% \over 2\pi i}\int_{-\infty}^{\infty}{1\over\tau-i\epsilon}\mathrm{e}^{ix\tau}% \mathrm{d}\tau.
  11. L L^{\infty}
  12. H ( x ) = 1 2 ( 1 + sgn ( x ) ) . H(x)=\tfrac{1}{2}(1+\operatorname{sgn}(x)).
  13. H ( x ) = 𝟏 [ 0 , ) ( x ) . H(x)=\mathbf{1}_{[0,\infty)}(x).\,
  14. H ( x ) = 𝟏 ( 0 , ) ( x ) . H(x)=\mathbf{1}_{(0,\infty)}(x).\,
  15. R ( x ) := - x H ( ξ ) d ξ = x H ( x ) . R(x):=\int_{-\infty}^{x}H(\xi)\mathrm{d}\xi=xH(x).
  16. d H ( x ) d x = δ ( x ) . \tfrac{dH(x)}{dx}=\delta(x).
  17. H ^ ( s ) = lim N - N N e - 2 π i x s H ( x ) d x = 1 2 ( δ ( s ) - i π p . v . 1 s ) . \hat{H}(s)=\lim_{N\to\infty}\int^{N}_{-N}\mathrm{e}^{-2\pi ixs}H(x)\,\mathrm{d% }x=\frac{1}{2}\left(\delta(s)-\frac{i}{\pi}\mathrm{p.v.}\frac{1}{s}\right).
  18. p . v . 1 s \mathrm{p.v.}\frac{1}{s}
  19. φ \varphi
  20. - φ ( s ) / s d s . \int^{\infty}_{-\infty}\varphi(s)/s\,\mathrm{d}s.
  21. H ^ ( s ) = lim N 0 N e - s x H ( x ) d x = = lim N 0 N e - s x d x = 1 s \hat{H}(s)=\lim_{N\to\infty}\int^{N}_{0}\mathrm{e}^{-sx}H(x)\,\mathrm{d}x==% \lim_{N\to\infty}\int^{N}_{0}\mathrm{e}^{-sx}\,\mathrm{d}x=\frac{1}{s}
  22. H ( x ) = ( 1 2 π i log ( z ) , 1 2 π i log ( z ) - 1 ) H(x)=\left(\frac{1}{2\pi i}\log(z),\frac{1}{2\pi i}\log(z)-1\right)

Hebron,_Connecticut.html

  1. \stackrel{\bigwedge}{\vee}

Heisenberg_picture.html

  1. A A
  2. H H
  3. H H
  4. A A
  5. A t = ψ ( t ) | A | ψ ( t ) . \langle A\rangle_{t}=\langle\psi(t)|A|\psi(t)\rangle.
  6. t t
  7. U ( t ) U(t)
  8. | ψ ( t ) = U ( t ) | ψ ( 0 ) . |\psi(t)\rangle=U(t)|\psi(0)\rangle.
  9. U ( t ) = e - i H t / , U(t)=e^{-iHt/\hbar},
  10. H H
  11. ħ ħ
  12. A t = ψ ( 0 ) | e i H t / A e - i H t / | ψ ( 0 ) . \langle A\rangle_{t}=\langle\psi(0)|e^{iHt/\hbar}Ae^{-iHt/\hbar}|\psi(0)\rangle.
  13. A ( t ) := e i H t / A e - i H t / . A(t):=e^{iHt/\hbar}Ae^{-iHt/\hbar}.
  14. d d t A ( t ) = i H e i H t / A e - i H t / + e i H t / ( A t ) e - i H t / + i e i H t / A ( - H ) e - i H t / = i e i H t / ( H A - A H ) e - i H t / + e i H t / ( A t ) e - i H t / = i ( H A ( t ) - A ( t ) H ) + e i H t / ( A t ) e - i H t / . \begin{aligned}\displaystyle{d\over dt}A(t)&\displaystyle={i\over\hbar}He^{iHt% /\hbar}Ae^{-iHt/\hbar}+e^{iHt/\hbar}\left(\frac{\partial A}{\partial t}\right)% e^{-iHt/\hbar}+{i\over\hbar}e^{iHt/\hbar}A\cdot(-H)e^{-iHt/\hbar}\\ &\displaystyle={i\over\hbar}e^{iHt/\hbar}\left(HA-AH\right)e^{-iHt/\hbar}+e^{% iHt/\hbar}\left(\frac{\partial A}{\partial t}\right)e^{-iHt/\hbar}\\ &\displaystyle={i\over\hbar}\left(HA(t)-A(t)H\right)+e^{iHt/\hbar}\left(\frac{% \partial A}{\partial t}\right)e^{-iHt/\hbar}.\end{aligned}
  15. e x p ( i H t / ħ ) exp(−iHt/ħ)
  16. H H
  17. d d t A ( t ) = i [ H , A ( t ) ] + e i H t / ( A t ) e - i H t / , {d\over dt}A(t)={i\over\hbar}[H,A(t)]+e^{iHt/\hbar}\left(\frac{\partial A}{% \partial t}\right)e^{-iHt/\hbar},
  18. e B A e - B = A + [ B , A ] + 1 2 ! [ B , [ B , A ] ] + 1 3 ! [ B , [ B , [ B , A ] ] ] + . {e^{B}Ae^{-B}}=A+[B,A]+\frac{1}{2!}[B,[B,A]]+\frac{1}{3!}[B,[B,[B,A]]]+\cdots.
  19. A ( t ) = A + i t [ H , A ] - t 2 2 ! 2 [ H , [ H , A ] ] - i t 3 3 ! 3 [ H , [ H , [ H , A ] ] ] + A(t)=A+\frac{it}{\hbar}[H,A]-\frac{t^{2}}{2!\hbar^{2}}[H,[H,A]]-\frac{it^{3}}{% 3!\hbar^{3}}[H,[H,[H,A]]]+\dots
  20. [ A , H ] i { A , H } [A,H]\leftrightarrow i\hbar\{A,H\}
  21. { A , H } = d d t A , \{A,H\}={d\over dt}A~{},
  22. H = p 2 2 m + m ω 2 x 2 2 H=\frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}
  23. d d t x ( t ) = i [ H , x ( t ) ] = p m {d\over dt}x(t)={i\over\hbar}[H,x(t)]=\frac{p}{m}
  24. d d t p ( t ) = i [ H , p ( t ) ] = - m ω 2 x {d\over dt}p(t)={i\over\hbar}[H,p(t)]=-m\omega^{2}x
  25. p ˙ ( 0 ) = - m ω 2 x 0 , \dot{p}(0)=-m\omega^{2}x_{0},
  26. x ˙ ( 0 ) = p 0 m , \dot{x}(0)=\frac{p_{0}}{m},
  27. x ( t ) = x 0 cos ( ω t ) + p 0 ω m sin ( ω t ) x(t)=x_{0}\cos(\omega t)+\frac{p_{0}}{\omega m}\sin(\omega t)
  28. p ( t ) = p 0 cos ( ω t ) - m ω x 0 sin ( ω t ) p(t)=p_{0}\cos(\omega t)-m\omega\!x_{0}\sin(\omega t)
  29. [ x ( t 1 ) , x ( t 2 ) ] = i m ω sin ( ω t 2 - ω t 1 ) [x(t_{1}),x(t_{2})]=\frac{i\hbar}{m\omega}\sin(\omega t_{2}-\omega t_{1})
  30. [ p ( t 1 ) , p ( t 2 ) ] = i m ω sin ( ω t 2 - ω t 1 ) [p(t_{1}),p(t_{2})]=i\hbar m\omega\sin(\omega t_{2}-\omega t_{1})
  31. [ x ( t 1 ) , p ( t 2 ) ] = i cos ( ω t 2 - ω t 1 ) [x(t_{1}),p(t_{2})]=i\hbar\cos(\omega t_{2}-\omega t_{1})
  32. t 1 = t 2 t_{1}=t_{2}
  33. | ψ I ( t ) = e i H 0 , S t / | ψ S ( t ) |\psi_{I}(t)\rangle=e^{iH_{0,S}~{}t/\hbar}|\psi_{S}(t)\rangle
  34. | ψ S ( t ) = e - i H S t / | ψ S ( 0 ) |\psi_{S}(t)\rangle=e^{-iH_{S}~{}t/\hbar}|\psi_{S}(0)\rangle
  35. A H ( t ) = e i H S t / A S e - i H S t / A_{H}(t)=e^{iH_{S}~{}t/\hbar}A_{S}e^{-iH_{S}~{}t/\hbar}
  36. A I ( t ) = e i H 0 , S t / A S e - i H 0 , S t / A_{I}(t)=e^{iH_{0,S}~{}t/\hbar}A_{S}e^{-iH_{0,S}~{}t/\hbar}
  37. ρ I ( t ) = e i H 0 , S t / ρ S ( t ) e - i H 0 , S t / \rho_{I}(t)=e^{iH_{0,S}~{}t/\hbar}\rho_{S}(t)e^{-iH_{0,S}~{}t/\hbar}
  38. ρ S ( t ) = e - i H S t / ρ S ( 0 ) e i H S t / \rho_{S}(t)=e^{-iH_{S}~{}t/\hbar}\rho_{S}(0)e^{iH_{S}~{}t/\hbar}

Helix.html

  1. x ( t ) = cos ( t ) , x(t)=\cos(t),\,
  2. y ( t ) = sin ( t ) , y(t)=\sin(t),\,
  3. z ( t ) = t . z(t)=t.\,
  4. r ( t ) = 1 , r(t)=1,\,
  5. θ ( t ) = t , \theta(t)=t,\,
  6. h ( t ) = t . h(t)=t.\,
  7. x ( t ) = a cos ( t ) , x(t)=a\cos(t),\,
  8. y ( t ) = a sin ( t ) , y(t)=a\sin(t),\,
  9. z ( t ) = b t . z(t)=bt.\,
  10. t ( a cos t , a sin t , b t ) , t [ 0 , T ] t\mapsto(a\cos t,a\sin t,bt),t\in[0,T]
  11. T a 2 + b 2 T\cdot\sqrt{a^{2}+b^{2}}
  12. | a | a 2 + b 2 \frac{|a|}{a^{2}+b^{2}}
  13. b a 2 + b 2 . \frac{b}{a^{2}+b^{2}}.

Helmholtz_free_energy.html

  1. A U - T S A\equiv U-TS\,
  2. d U = δ Q - δ W \mathrm{d}U=\delta Q\ -\delta W\,
  3. U U
  4. δ Q \delta Q
  5. δ W \delta W
  6. δ Q = T d S \delta Q=T\mathrm{d}S
  7. δ W = p d V \delta W=p\mathrm{d}V
  8. d U = T d S - p d V \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V\,
  9. d U = d ( T S ) - S d T - p d V \mathrm{d}U=d(TS)-S\mathrm{d}T-p\mathrm{d}V\,
  10. d ( U - T S ) = - S d T - p d V \mathrm{d}(U-TS)=-S\mathrm{d}T-p\mathrm{d}V\,
  11. d A = - S d T - p d V \mathrm{d}A=-S\mathrm{d}T-p\mathrm{d}V\,
  12. d U = δ Q - δ W \mathrm{d}U=\delta Q\ -\delta W\,
  13. δ Q T d S \delta Q\leq T\mathrm{d}S\,
  14. d U T d S - δ W \mathrm{d}U\leq T\mathrm{d}S-\delta W\,
  15. d U - T d S - δ W \mathrm{d}U-T\mathrm{d}S\leq-\delta W\,
  16. d A = d U - T d S \mathrm{d}A=\mathrm{d}U-T\mathrm{d}S\,
  17. δ W - d A , \delta W\leq-\mathrm{d}A,
  18. δ W \delta W\,
  19. A f A i , A_{f}\leq A_{i},
  20. Δ U \Delta U
  21. Δ S \Delta S
  22. W W
  23. Δ U bath + Δ U + W = 0 \Delta U_{\,\text{bath}}+\Delta U+W=0\,
  24. Q bath = Δ U bath = - ( Δ U + W ) Q_{\,\text{bath}}=\Delta U_{\,\text{bath}}=-\left(\Delta U+W\right)\,
  25. Δ S bath = Q bath T = - Δ U + W T \Delta S_{\,\text{bath}}=\frac{Q_{\,\text{bath}}}{T}=-\frac{\Delta U+W}{T}\,
  26. Δ S bath + Δ S = - Δ U - T Δ S + W T \Delta S_{\,\text{bath}}+\Delta S=-\frac{\Delta U-T\Delta S+W}{T}\,
  27. Δ S bath + Δ S = - Δ A + W T \Delta S_{\,\text{bath}}+\Delta S=-\frac{\Delta A+W}{T}\,
  28. W - Δ A W\leq-\Delta A\,
  29. Δ A 0 \Delta A\leq 0\,
  30. d A = - S d T - P d V dA=-SdT-PdV
  31. d A = 0 dA=0
  32. A = constant A=\,\text{ constant}
  33. Δ A = A 2 - A 1 0 \Delta A=A_{2}-A_{1}\leq 0
  34. A i A_{i}
  35. d A = - S d T - p d V + j μ j d N j dA=-SdT-pdV+\sum_{j}\mu_{j}dN_{j}\,
  36. N j N_{j}
  37. μ j \mu_{j}
  38. d A = - S d T - i X i d x i + j μ j d N j dA=-SdT-\sum_{i}X_{i}dx_{i}+\sum_{j}\mu_{j}dN_{j}\,
  39. x i x_{i}
  40. X i X_{i}
  41. P r = e - β E r Z P_{r}=\frac{e^{-\beta E_{r}}}{Z}\,
  42. β 1 k T \beta\equiv\frac{1}{kT}\,
  43. E r = energy of eigenstate r E_{r}=\,\text{ energy of eigenstate }r\,
  44. Z = r e - β E r Z=\sum_{r}e^{-\beta E_{r}}
  45. U E = r P r E r = - log Z β U\equiv\left\langle E\right\rangle=\sum_{r}P_{r}E_{r}=-\frac{\partial\log Z}{% \partial\beta}\,
  46. X r = - E r x X_{r}=-\frac{\partial E_{r}}{\partial x}\,
  47. X = r P r X r = 1 β log Z x X=\sum_{r}P_{r}X_{r}=\frac{1}{\beta}\frac{\partial\log Z}{\partial x}\,
  48. x x
  49. d β d\beta
  50. d x dx
  51. log Z \log Z
  52. d ( log Z ) = log Z β d β + log Z x d x = - U d β + β X d x d\left(\log Z\right)=\frac{\partial\log Z}{\partial\beta}d\beta+\frac{\partial% \log Z}{\partial x}dx=-U\,d\beta+\beta X\,dx\,
  53. U d β U\,d\beta
  54. U d β = d ( β U ) - β d U U\,d\beta=d\left(\beta U\right)-\beta\,dU\,
  55. d ( log Z ) = - d ( β U ) + β d U + β X d x d\left(\log Z\right)=-d\left(\beta U\right)+\beta\,dU+\beta X\,dx\,
  56. d U = 1 β d ( log Z + β U ) - X d x dU=\frac{1}{\beta}d\left(\log Z+\beta U\right)-X\,dx\,
  57. d U = T d S - X d x dU=T\,dS-X\,dx\,
  58. S = k log Z + U T + c S=k\log Z+\frac{U}{T}+c\,
  59. S = k log Ω 0 S=k\log\Omega_{0}
  60. Ω 0 \Omega_{0}
  61. Ω 0 e - β U 0 \Omega_{0}e^{-\beta U_{0}}
  62. U 0 U_{0}
  63. c = 0 c=0
  64. A = - k T log ( Z ) A=-kT\log\left(Z\right)\,
  65. H H
  66. H ~ \tilde{H}
  67. H ~ = H \left\langle\tilde{H}\right\rangle=\left\langle H\right\rangle\,
  68. H ~ \tilde{H}
  69. A A ~ A\leq\tilde{A}\,
  70. A A
  71. A ~ \tilde{A}
  72. H = H 0 + Δ H H=H_{0}+\Delta H\,
  73. H 0 H_{0}
  74. H ~ = H 0 + Δ H 0 \tilde{H}=H_{0}+\left\langle\Delta H\right\rangle_{0}\,
  75. X 0 \left\langle X\right\rangle_{0}
  76. H 0 H_{0}
  77. H ~ \tilde{H}
  78. H 0 H_{0}
  79. X 0 = X \left\langle X\right\rangle_{0}=\left\langle X\right\rangle\,
  80. H ~ = H 0 + Δ H = H \left\langle\tilde{H}\right\rangle=\left\langle H_{0}+\left\langle\Delta H% \right\rangle\right\rangle=\left\langle H\right\rangle\,
  81. A A ~ A\leq\tilde{A}\,
  82. A ~ \tilde{A}
  83. H 0 H_{0}
  84. Δ H \left\langle\Delta H\right\rangle
  85. A ~ = H 0 0 - T S 0 + Δ H 0 = H 0 - T S 0 \tilde{A}=\left\langle H_{0}\right\rangle_{0}-TS_{0}+\left\langle\Delta H% \right\rangle_{0}=\left\langle H\right\rangle_{0}-TS_{0}\,
  86. A H 0 - T S 0 A\leq\left\langle H\right\rangle_{0}-TS_{0}\,
  87. P r P_{r}
  88. P ~ r \tilde{P}_{r}
  89. r P ~ r log ( P ~ r ) r P ~ r log ( P r ) \sum_{r}\tilde{P}_{r}\log\left(\tilde{P}_{r}\right)\geq\sum_{r}\tilde{P}_{r}% \log\left(P_{r}\right)\,
  90. r P ~ r log ( P ~ r P r ) \sum_{r}\tilde{P}_{r}\log\left(\frac{\tilde{P}_{r}}{P_{r}}\right)\,
  91. log ( x ) 1 - 1 x \log\left(x\right)\geq 1-\frac{1}{x}\,
  92. r P ~ r log ( P ~ r P r ) r ( P ~ r - P r ) = 0 \sum_{r}\tilde{P}_{r}\log\left(\frac{\tilde{P}_{r}}{P_{r}}\right)\geq\sum_{r}% \left(\tilde{P}_{r}-P_{r}\right)=0\,
  93. log ( P ~ r ) log ( P r ) \left\langle\log\left(\tilde{P}_{r}\right)\right\rangle\geq\left\langle\log% \left(P_{r}\right)\right\rangle\,
  94. P ~ r \tilde{P}_{r}
  95. P r = exp [ - β H ( r ) ] Z P_{r}=\frac{\exp\left[-\beta H\left(r\right)\right]}{Z}\,
  96. P ~ r = exp [ - β H ~ ( r ) ] Z ~ \tilde{P}_{r}=\frac{\exp\left[-\beta\tilde{H}\left(r\right)\right]}{\tilde{Z}}\,
  97. - β H ~ - log ( Z ~ ) - β H - log ( Z ) \left\langle-\beta\tilde{H}-\log\left(\tilde{Z}\right)\right\rangle\geq\left% \langle-\beta H-\log\left(Z\right)\right\rangle
  98. H H
  99. H ~ \tilde{H}
  100. A A ~ A\leq\tilde{A}
  101. H ~ \tilde{H}
  102. | r \left|r\right\rangle
  103. H H
  104. H ~ \tilde{H}
  105. P r = r | exp [ - β H ] Z | r P_{r}=\left\langle r\left|\frac{\exp\left[-\beta H\right]}{Z}\right|r\right\rangle\,
  106. P ~ r = r | exp [ - β H ~ ] Z ~ | r = exp ( - β E ~ r ) Z ~ \tilde{P}_{r}=\left\langle r\left|\frac{\exp\left[-\beta\tilde{H}\right]}{% \tilde{Z}}\right|r\right\rangle=\frac{\exp\left(-\beta\tilde{E}_{r}\right)}{% \tilde{Z}}\,
  107. E ~ r \tilde{E}_{r}
  108. H ~ \tilde{H}
  109. H ~ \tilde{H}
  110. H ~ \tilde{H}
  111. H ~ = H \left\langle\tilde{H}\right\rangle=\left\langle H\right\rangle\,
  112. H = r P ~ r r | H | r \left\langle H\right\rangle=\sum_{r}\tilde{P}_{r}\left\langle r\left|H\right|r% \right\rangle\,
  113. r P ~ r log ( P ~ r ) r P ~ r log ( P r ) \sum_{r}\tilde{P}_{r}\log\left(\tilde{P}_{r}\right)\geq\sum_{r}\tilde{P}_{r}% \log\left(P_{r}\right)\,
  114. P r P_{r}
  115. P ~ r \tilde{P}_{r}
  116. log ( P ~ r ) = - β E ~ r - log ( Z ~ ) \log\left(\tilde{P}_{r}\right)=-\beta\tilde{E}_{r}-\log\left(\tilde{Z}\right)\,
  117. exp ( X ) r exp ( X r ) \left\langle\exp\left(X\right)\right\rangle_{r}\geq\exp\left(\left\langle X% \right\rangle_{r}\right)\,
  118. Y r r | Y | r \left\langle Y\right\rangle_{r}\equiv\left\langle r\left|Y\right|r\right\rangle\,
  119. log [ exp ( X ) r ] X r \log\left[\left\langle\exp\left(X\right)\right\rangle_{r}\right]\geq\left% \langle X\right\rangle_{r}\,
  120. log ( P r ) = log [ exp ( - β H - log ( Z ) ) r ] - β H - log ( Z ) r \log\left(P_{r}\right)=\log\left[\left\langle\exp\left(-\beta H-\log\left(Z% \right)\right)\right\rangle_{r}\right]\geq\left\langle-\beta H-\log\left(Z% \right)\right\rangle_{r}\,
  121. H ~ \tilde{H}
  122. A A ~ A\leq\tilde{A}
  123. p d V p{\rm d}V
  124. d A = V i j σ i j d ε i j - S d T + i μ i d N i {\rm d}A=V\sum_{ij}\sigma_{ij}\,{\rm d}\varepsilon_{ij}-S{\rm d}T+\sum_{i}\mu_% {i}\,{\rm d}N_{i}\,
  125. σ i j \sigma_{ij}
  126. ε i j \varepsilon_{ij}
  127. σ i j = C i j k l ϵ k l \sigma_{ij}=C_{ijkl}\epsilon_{kl}
  128. d A {\rm d}A
  129. A = 1 2 V C i j k l ϵ i j ϵ k l - S T + i μ i N i A=\frac{1}{2}VC_{ijkl}\epsilon_{ij}\epsilon_{kl}-ST+\sum_{i}\mu_{i}N_{i}\,
  130. = 1 2 V σ i j ϵ i j - S T + i μ i N i =\frac{1}{2}V\sigma_{ij}\epsilon_{ij}-ST+\sum_{i}\mu_{i}N_{i}\,

Hemodynamics.html

  1. 2 / 3 {2}/{3}
  2. 1 / 3 {1}/{3}
  3. Δ P = 8 μ l Q π r 4 \Delta P=\frac{8\mu lQ}{\pi r^{4}}
  4. p 1 - p 2 l = Δ P \frac{p_{1}-p_{2}}{l}=\Delta P
  5. N R = ρ v L μ NR=\frac{\rho vL}{\mu}
  6. σ θ = P r t \sigma_{\theta}=\dfrac{Pr}{t}
  7. σ θ \sigma_{\theta}\!
  8. σ θ = F t l \sigma_{\theta}=\dfrac{F}{tl}

Henry's_law.html

  1. p = k H c p=k_{\mathrm{H}}c
  2. k H , pc = p c aq k_{\mathrm{H,pc}}=\frac{p}{c_{\mathrm{aq}}}
  3. k H , cp = c aq p k_{\mathrm{H,cp}}=\frac{c_{\mathrm{aq}}}{p}
  4. k H , px = p x k_{\mathrm{H,px}}=\frac{p}{x}
  5. k H , cc = c aq c gas k_{\mathrm{H,cc}}=\frac{c_{\mathrm{aq}}}{c_{\mathrm{gas}}}
  6. L atm mol \frac{\mathrm{L}\cdot\mathrm{atm}}{\mathrm{mol}}
  7. mol L atm \frac{\mathrm{mol}}{\mathrm{L}\cdot\mathrm{atm}}
  8. atm \rm atm\,
  9. × 10 3 \times 10^{−}3
  10. × 10 4 \times 10^{4}
  11. × 10 2 \times 10^{−}2
  12. × 10 4 \times 10^{−}4
  13. × 10 4 \times 10^{4}
  14. × 10 2 \times 10^{−}2
  15. × 10 2 \times 10^{−}2
  16. × 10 4 \times 10^{4}
  17. × 10 4 \times 10^{−}4
  18. × 10 4 \times 10^{4}
  19. × 10 2 \times 10^{−}2
  20. × 10 4 \times 10^{−}4
  21. × 10 4 \times 10^{4}
  22. × 10 3 \times 10^{−}3
  23. × 10 4 \times 10^{−}4
  24. × 10 4 \times 10^{4}
  25. × 10 2 \times 10^{−}2
  26. × 10 3 \times 10^{−}3
  27. × 10 4 \times 10^{4}
  28. × 10 2 \times 10^{−}2
  29. × 10 4 \times 10^{−}4
  30. × 10 4 \times 10^{4}
  31. × 10 2 \times 10^{−}2
  32. k H , pc ( T ) = k H , pc ( T ) exp [ - C ( 1 T - 1 T ) ] k_{\rm H,pc}(T)=k_{\rm H,pc}(T^{\ominus})\,\exp{\left[-C\,\left(\frac{1}{T}-% \frac{1}{T^{\ominus}}\right)\right]}\,
  33. k H , cp ( T ) = k H , cp ( T ) exp [ C ( 1 T - 1 T ) ] k_{\rm H,cp}(T)=k_{\rm H,cp}(T^{\ominus})\,\exp{\left[C\,\left(\frac{1}{T}-% \frac{1}{T^{\ominus}}\right)\right]}\,
  34. C = - Δ solv H R = - d [ ln k H ( T ) ] d ( 1 / T ) C=-\frac{\Delta_{\rm solv}H}{R}=-\frac{{\rm d}\left[\ln k_{\rm H}(T)\right]}{{% \rm d}(1/T)}
  35. log ( * z 1 / z 1 ) = k s y z y \log(*z_{1}/z_{1})=k_{syz}y
  36. C melt / C gas = exp [ - β ( μ melt E - μ gas E ) ] C_{\rm melt}/C_{\rm gas}=\exp\left[-\beta(\mu^{\rm E}_{\rm melt}-\mu^{\rm E}_{% \rm gas})\right]\,
  37. p = k H x p=k_{\rm H}\,x
  38. p = p x p=p^{\star}\,x
  39. lim x 1 ( p x ) = p \lim_{x\to 1}\left(\frac{p}{x}\right)=p^{\star}
  40. lim x 0 ( p x ) = k H \lim_{x\to 0}\left(\frac{p}{x}\right)=k_{\rm H}
  41. μ = μ c + R T ln ( γ c c c ) \mu=\mu_{c}^{\ominus}+RT\ln{\left(\frac{\gamma_{c}c}{c^{\ominus}}\right)}\,
  42. γ c = k H , c p \gamma_{c}=\frac{k_{{\rm H,}c}}{p^{\star}}
  43. i n i d μ i = 0 \sum_{i}n_{i}\,{\rm d}\mu_{i}=0
  44. μ = μ b + R T ln ( γ b b b ) \mu=\mu_{b}^{\ominus}+RT\ln{\left(\frac{\gamma_{b}b}{b^{\ominus}}\right)}\,
  45. γ b = k H , b p \gamma_{b}=\frac{k_{{\rm H,}b}}{p^{\star}}

Henry_(unit).html

  1. v ( t ) = L d i d t \displaystyle v(t)=L\frac{di}{dt}
  2. H = kg m 2 s 2 A 2 = kg m 2 C 2 = J A 2 = T m 2 A = Wb A = V s A = s 2 F = 1 F Hz 2 = Ω s \mbox{H}~{}=\dfrac{\mbox{kg}~{}\cdot\mbox{m}~{}^{2}}{\mbox{s}~{}^{2}\cdot\mbox% {A}~{}^{2}}=\dfrac{\mbox{kg}~{}\cdot\mbox{m}~{}^{2}}{\mbox{C}~{}^{2}}=\dfrac{% \mbox{J}~{}}{\mbox{A}~{}^{2}}=\dfrac{\mbox{T}~{}\cdot\mbox{m}~{}^{2}}{\mbox{A}% ~{}}=\dfrac{\mbox{Wb}~{}}{\mbox{A}~{}}=\dfrac{\mbox{V}~{}\cdot\mbox{s}~{}}{% \mbox{A}~{}}=\dfrac{\mbox{s}~{}^{2}}{\mbox{F}~{}}=\dfrac{\mbox{1}~{}}{\mbox{F}% ~{}\cdot\mbox{Hz}~{}^{2}}=\Omega\cdot\mbox{s}~{}

Heptadecagon.html

  1. F n = 2 2 𝑛 + 1 \scriptstyle F_{n}=2^{2^{\overset{n}{}}}+1
  2. 2 π / 17 2\pi/17
  3. 16 cos 2 π 17 = \displaystyle 16\,\operatorname{cos}{2\pi\over 17}=

Heptathlon.html

  1. P = a ( b - T ) c P=a\cdot(b-T)^{c}
  2. P = a ( M - b ) c P=a\cdot(M-b)^{c}
  3. P = a ( D - b ) c P=a\cdot(D-b)^{c}

Herd_immunity.html

  1. R 0 S = 1. \ R_{0}\cdot S=1.
  2. R 0 ( 1 - p ) = 1 , \ R_{0}\cdot(1-p)=1,
  3. 1 - p = 1 R 0 , 1-p=\frac{1}{R_{0}},
  4. p c = 1 - 1 R 0 . p_{c}=1-\frac{1}{R_{0}}.
  5. V c = 1 - 1 R 0 E . V_{c}=\frac{1-\frac{1}{R_{0}}}{E}.
  6. ( 1 - 1 R 0 ) - ( E × p v ) . (1-\frac{1}{R_{0}})-(E\times p_{v}).

Herfindahl_index.html

  1. H = i = 1 N s i 2 H=\sum_{i=1}^{N}s_{i}^{2}
  2. H * = ( H - 1 / N ) 1 - 1 / N H^{*}={\left(H-1/N\right)\over 1-1/N}
  3. H * = 1 H^{*}=1
  4. H = 1 N + N V H=\frac{1}{N}+NV
  5. V = i = 1 N ( s i - 1 / N ) 2 N V=\frac{\sum_{i=1}^{N}\left(s_{i}-1/N\right)^{2}}{N}

Heritability.html

  1. H 2 = Var ( G ) Var ( P ) H^{2}=\frac{\mathrm{Var}(G)}{\mathrm{Var}(P)}
  2. h 2 = Var ( A ) Var ( P ) h^{2}=\frac{\mathrm{Var}(A)}{\mathrm{Var}(P)}
  3. P i j = μ + α i + α j + d i j P_{ij}=\mu+\alpha_{i}+\alpha_{j}+d_{ij}
  4. a i j = α i + α j a_{ij}=\alpha_{i}+\alpha_{j}
  5. d i j d_{ij}
  6. Var ( A ) = f ( b b ) a b b 2 + f ( B b ) a B b 2 + f ( B B ) a B B 2 , \mathrm{Var}(A)=f(bb)a^{2}_{bb}+f(Bb)a^{2}_{Bb}+f(BB)a^{2}_{BB},
  7. f ( b b ) a b b + f ( B b ) a B b + f ( B B ) a B B = 0. f(bb)a_{bb}+f(Bb)a_{Bb}+f(BB)a_{BB}=0.
  8. Var ( D ) = f ( b b ) d b b 2 + f ( B b ) d B b 2 + f ( B B ) d B B 2 , \mathrm{Var}(D)=f(bb)d^{2}_{bb}+f(Bb)d^{2}_{Bb}+f(BB)d^{2}_{BB},
  9. f ( b b ) d b b + f ( B b ) d B b + f ( B B ) d B B = 0. f(bb)d_{bb}+f(Bb)d_{Bb}+f(BB)d_{BB}=0.
  10. h 2 = b r = t 2 r h^{2}=\frac{b}{r}=\frac{t^{2}}{r}
  11. y i = μ + g i + e y_{i}=\mu+g_{i}+e
  12. g i g_{i}
  13. e e
  14. z i = μ + 1 2 g i + e z_{i}=\mu+\frac{1}{2}g_{i}+e
  15. corr ( z , z ) = corr ( μ + 1 2 g + e , μ + 1 2 g + e ) = 1 4 V g \mathrm{corr}(z,z^{\prime})=\mathrm{corr}(\mu+\frac{1}{2}g+e,\mu+\frac{1}{2}g+% e^{\prime})=\frac{1}{4}V_{g}
  16. n n
  17. r r
  18. V g V_{g}
  19. V e V_{e}
  20. n - 1 n-1
  21. S S
  22. 3 4 V g + V e + r ( 1 4 V g ) \frac{3}{4}V_{g}+V_{e}+r({\frac{1}{4}V_{g}})
  23. n ( r - 1 ) n(r-1)
  24. W W
  25. 3 4 V g + V e \frac{3}{4}V_{g}+V_{e}
  26. 1 4 V g \frac{1}{4}V_{g}
  27. H 2 = V g V g + V e = 4 ( S - W ) S + ( r - 1 ) W H^{2}=\frac{V_{g}}{V_{g}+V_{e}}=\frac{4(S-W)}{S+(r-1)W}
  28. y i j = μ + α i + α j + d i j + e , y_{ij}=\mu+\alpha_{i}+\alpha_{j}+d_{ij}+e,
  29. α i \alpha_{i}
  30. α j \alpha_{j}
  31. d i j d_{ij}
  32. e e
  33. V a V_{a}
  34. V d V_{d}
  35. = r V a + θ V d , =rV_{a}+\theta V_{d},
  36. r r
  37. θ \theta
  38. r = r=
  39. θ = \theta=
  40. r r
  41. θ \theta
  42. 1 1
  43. 1 1
  44. 1 2 \frac{1}{2}
  45. 0
  46. 1 4 \frac{1}{4}
  47. 0
  48. 1 2 \frac{1}{2}
  49. 1 4 \frac{1}{4}
  50. 1 8 \frac{1}{8}
  51. 0
  52. 1 4 \frac{1}{4}
  53. 1 16 \frac{1}{16}
  54. R = h 2 S R=h^{2}S
  55. Var ( A ) / Var ( P ) \mathrm{Var}(A)/\mathrm{Var}(P)

Hermite_polynomials.html

  1. 𝐻𝑒 n ( x ) = ( - 1 ) n e x 2 2 d n d x n e - x 2 2 = ( x - d d x ) n 1 \mathit{He}_{n}(x)=(-1)^{n}e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}e^{-\frac{x^% {2}}{2}}=\left(x-\frac{d}{dx}\right)^{n}\cdot 1
  2. H n ( x ) = ( - 1 ) n e x 2 d n d x n e - x 2 = ( 2 x - d d x ) n 1 H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}=\left(2x-\frac{d}{dx}% \right)^{n}\cdot 1
  3. H n ( x ) = 2 n 2 𝐻𝑒 n ( 2 x ) , 𝐻𝑒 n ( x ) = 2 - n 2 H n ( x 2 ) . H_{n}(x)=2^{\tfrac{n}{2}}{\mathit{He}}_{n}(\sqrt{2}\,x),\qquad\mathit{He}_{n}(% x)=2^{-\tfrac{n}{2}}H_{n}\left(\frac{x}{\sqrt{2}}\right).
  4. 1 2 π e - x 2 2 \frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}
  5. 𝐻𝑒 0 ( x ) = 1 {\mathit{He}}_{0}(x)=1\,
  6. 𝐻𝑒 1 ( x ) = x {\mathit{He}}_{1}(x)=x\,
  7. 𝐻𝑒 2 ( x ) = x 2 - 1 {\mathit{He}}_{2}(x)=x^{2}-1\,
  8. 𝐻𝑒 3 ( x ) = x 3 - 3 x {\mathit{He}}_{3}(x)=x^{3}-3x\,
  9. 𝐻𝑒 4 ( x ) = x 4 - 6 x 2 + 3 {\mathit{He}}_{4}(x)=x^{4}-6x^{2}+3\,
  10. 𝐻𝑒 5 ( x ) = x 5 - 10 x 3 + 15 x {\mathit{He}}_{5}(x)=x^{5}-10x^{3}+15x\,
  11. 𝐻𝑒 6 ( x ) = x 6 - 15 x 4 + 45 x 2 - 15 {\mathit{He}}_{6}(x)=x^{6}-15x^{4}+45x^{2}-15\,
  12. 𝐻𝑒 7 ( x ) = x 7 - 21 x 5 + 105 x 3 - 105 x {\mathit{He}}_{7}(x)=x^{7}-21x^{5}+105x^{3}-105x\,
  13. 𝐻𝑒 8 ( x ) = x 8 - 28 x 6 + 210 x 4 - 420 x 2 + 105 {\mathit{He}}_{8}(x)=x^{8}-28x^{6}+210x^{4}-420x^{2}+105\,
  14. 𝐻𝑒 9 ( x ) = x 9 - 36 x 7 + 378 x 5 - 1260 x 3 + 945 x {\mathit{He}}_{9}(x)=x^{9}-36x^{7}+378x^{5}-1260x^{3}+945x\,
  15. 𝐻𝑒 10 ( x ) = x 10 - 45 x 8 + 630 x 6 - 3150 x 4 + 4725 x 2 - 945 {\mathit{He}}_{10}(x)=x^{10}-45x^{8}+630x^{6}-3150x^{4}+4725x^{2}-945\,
  16. H 0 ( x ) = 1 H_{0}(x)=1\,
  17. H 1 ( x ) = 2 x H_{1}(x)=2x\,
  18. H 2 ( x ) = 4 x 2 - 2 H_{2}(x)=4x^{2}-2\,
  19. H 3 ( x ) = 8 x 3 - 12 x H_{3}(x)=8x^{3}-12x\,
  20. H 4 ( x ) = 16 x 4 - 48 x 2 + 12 H_{4}(x)=16x^{4}-48x^{2}+12\,
  21. H 5 ( x ) = 32 x 5 - 160 x 3 + 120 x H_{5}(x)=32x^{5}-160x^{3}+120x\,
  22. H 6 ( x ) = 64 x 6 - 480 x 4 + 720 x 2 - 120 H_{6}(x)=64x^{6}-480x^{4}+720x^{2}-120\,
  23. H 7 ( x ) = 128 x 7 - 1344 x 5 + 3360 x 3 - 1680 x H_{7}(x)=128x^{7}-1344x^{5}+3360x^{3}-1680x\,
  24. H 8 ( x ) = 256 x 8 - 3584 x 6 + 13440 x 4 - 13440 x 2 + 1680 H_{8}(x)=256x^{8}-3584x^{6}+13440x^{4}-13440x^{2}+1680\,
  25. H 9 ( x ) = 512 x 9 - 9216 x 7 + 48384 x 5 - 80640 x 3 + 30240 x H_{9}(x)=512x^{9}-9216x^{7}+48384x^{5}-80640x^{3}+30240x\,
  26. H 10 ( x ) = 1024 x 10 - 23040 x 8 + 161280 x 6 - 403200 x 4 + 302400 x 2 - 30240 H_{10}(x)=1024x^{10}-23040x^{8}+161280x^{6}-403200x^{4}+302400x^{2}-30240\,
  27. w ( x ) = e - x 2 2 w(x)=e^{-\frac{x^{2}}{2}}
  28. w ( x ) = e - x 2 w(x)=e^{-x^{2}}
  29. - H m ( x ) H n ( x ) w ( x ) d x = 0 , m n . \int_{-\infty}^{\infty}H_{m}(x)H_{n}(x)\,w(x)\,\mathrm{d}x=0,\qquad m\neq n.
  30. - 𝐻𝑒 m ( x ) 𝐻𝑒 n ( x ) e - x 2 2 d x = 2 π n ! δ n m \int_{-\infty}^{\infty}{\mathit{He}}_{m}(x)\mathit{He}_{n}(x)\,e^{-\frac{x^{2}% }{2}}\,\mathrm{d}x=\sqrt{2\pi}n!\delta_{nm}
  31. - H m ( x ) H n ( x ) e - x 2 d x = π 2 n n ! δ n m \int_{-\infty}^{\infty}H_{m}(x)H_{n}(x)e^{-x^{2}}\,\mathrm{d}x=\sqrt{\pi}2^{n}% n!\delta_{nm}
  32. - | f ( x ) | 2 w ( x ) d x < , \int_{-\infty}^{\infty}|f(x)|^{2}\,w(x)\,\mathrm{d}x<\infty~{},
  33. w ( x ) w(x)
  34. f , g = - f ( x ) g ( x ) ¯ w ( x ) d x . \langle f,g\rangle=\int_{-\infty}^{\infty}f(x)\overline{g(x)}\,w(x)\,\mathrm{d% }x~{}.
  35. ƒ ƒ
  36. - f ( x ) x n e - x 2 d x = 0 \int_{-\infty}^{\infty}f(x)x^{n}e^{-x^{2}}\,\mathrm{d}x=0
  37. n n
  38. f f
  39. F ( z ) = - f ( x ) e z x - x 2 d x = n = 0 z n n ! f ( x ) x n e - x 2 d x = 0 F(z)=\int_{-\infty}^{\infty}f(x)e^{zx-x^{2}}\,\mathrm{d}x=\sum_{n=0}^{\infty}% \frac{z^{n}}{n!}\int f(x)x^{n}e^{-x^{2}}\,\mathrm{d}x=0
  40. F ( i t ) = 0 F(it)=0
  41. t t
  42. f f
  43. ( e - x 2 2 u ) + λ e - 1 2 x 2 u = 0 \left(e^{-\frac{x^{2}}{2}}u^{\prime}\right)^{\prime}+\lambda e^{-\frac{1}{2}x^% {2}}u=0
  44. λ λ
  45. u u
  46. L [ u ] = u ′′ - x u = - λ u , L[u]=u^{\prime\prime}-xu^{\prime}=-\lambda u~{},
  47. L L
  48. u ′′ - 2 x u = - 2 λ u u^{\prime\prime}-2xu^{\prime}=-2\lambda u
  49. λ λ
  50. 𝐻𝑒 n + 1 ( x ) = x 𝐻𝑒 n ( x ) - 𝐻𝑒 n ( x ) . {\mathit{He}}_{n+1}(x)=x{\mathit{He}}_{n}(x)-{\mathit{He}}_{n}^{\prime}(x).
  51. a n + 1 , k = a n , k - 1 - n a n - 1 , k k > 0 a_{n+1,k}=a_{n,k-1}-na_{n-1,k}\ \ k>0
  52. a n + 1 , k = - n a n - 1 , k k = 0 a_{n+1,k}=-na_{n-1,k}\ \ k=0
  53. a a
  54. a a
  55. a a
  56. H n ( x ) = k = 0 n a n , k x k H_{n}(x)=\sum\limits^{n}_{k=0}a_{n,k}\ x^{k}
  57. H n + 1 ( x ) = 2 x H n ( x ) - H n ( x ) . H_{n+1}(x)=2xH_{n}(x)-H_{n}^{\prime}(x).
  58. a n + 1 , k = 2 a n , k - 1 - 2 n a n - 1 , k k > 0 a_{n+1,k}=2a_{n,k-1}-2na_{n-1,k}\ \ k>0
  59. a n + 1 , k = - 2 n a n - 1 , k k = 0 a_{n+1,k}=-2na_{n-1,k}\ \ k=0
  60. 𝐻𝑒 n ( x ) = n 𝐻𝑒 n - 1 ( x ) , {\mathit{He}}_{n}^{\prime}(x)=n{\mathit{He}}_{n-1}(x),\,\!
  61. H n ( x ) = 2 n H n - 1 ( x ) , H_{n}^{\prime}(x)=2nH_{n-1}(x),\,\!
  62. 𝐻𝑒 n ( x + y ) = k = 0 n ( n k ) x n - k 𝐻𝑒 k ( y ) {\mathit{He}}_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}x^{n-k}{\mathit{He}}_{k}(y)
  63. H n ( x + y ) = k = 0 n ( n k ) H k ( x ) ( 2 y ) ( n - k ) = 2 - n 2 k = 0 n ( n k ) H n - k ( x 2 ) H k ( y 2 ) . H_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}H_{k}(x)(2y)^{(n-k)}=2^{-\frac{n}{2}}% \cdot\sum_{k=0}^{n}{n\choose k}H_{n-k}\left(x\sqrt{2}\right)H_{k}\left(y\sqrt{% 2}\right).
  64. H n ( x ) = 2 n e - D 2 / 4 x n H_{n}(x)=2^{n}e^{-D^{2}/4}x^{n}
  65. m m
  66. 𝐻𝑒 n ( m ) ( x ) = n ! ( n - m ) ! 𝐻𝑒 n - m ( x ) = m ! ( n m ) 𝐻𝑒 n - m ( x ) , {\mathit{He}}_{n}^{(m)}(x)=\frac{n!}{(n-m)!}\cdot{\mathit{He}}_{n-m}(x)=m!% \cdot{n\choose m}\cdot{\mathit{He}}_{n-m}(x),\,\!
  67. H n ( m ) ( x ) = 2 m n ! ( n - m ) ! H n - m ( x ) = 2 m m ! ( n m ) H n - m ( x ) . {\mathit{H}}_{n}^{(m)}(x)=2^{m}\cdot\frac{n!}{(n-m)!}\cdot{\mathit{H}}_{n-m}(x% )=2^{m}\cdot m!\cdot{n\choose m}\cdot{\mathit{H}}_{n-m}(x).\,\!
  68. 𝐻𝑒 n + 1 ( x ) = x 𝐻𝑒 n ( x ) - n 𝐻𝑒 n - 1 ( x ) , {\mathit{He}}_{n+1}(x)=x{\mathit{He}}_{n}(x)-n{\mathit{He}}_{n-1}(x),
  69. H n + 1 ( x ) = 2 x H n ( x ) - 2 n H n - 1 ( x ) . H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x).\,\!
  70. 𝐻𝑒 n ( x ) 2 - 𝐻𝑒 n - 1 ( x ) 𝐻𝑒 n + 1 ( x ) = ( n - 1 ) ! i = 0 n - 1 2 n - i i ! 𝐻𝑒 i ( x ) 2 > 0. \mathit{He}_{n}(x)^{2}-\mathit{He}_{n-1}(x){\mathit{He}}_{n+1}(x)=(n-1)!\cdot% \sum_{i=0}^{n-1}\frac{2^{n-i}}{i!}{\mathit{He}}_{i}(x)^{2}>0.
  71. H n ( γ x ) = i = 0 n 2 γ n - 2 i ( γ 2 - 1 ) i ( n 2 i ) ( 2 i ) ! i ! H n - 2 i ( x ) . {\mathit{H}}_{n}(\gamma x)=\sum_{i=0}^{\lfloor\tfrac{n}{2}\rfloor}\gamma^{n-2i% }(\gamma^{2}-1)^{i}{n\choose 2i}\frac{(2i)!}{i!}{\mathit{H}}_{n-2i}(x).
  72. H n ( x ) = n ! = 0 n 2 ( - 1 ) n 2 - ( 2 ) ! ( n 2 - ) ! ( 2 x ) 2 H_{n}(x)=n!\sum_{\ell=0}^{\tfrac{n}{2}}\frac{(-1)^{\tfrac{n}{2}-\ell}}{(2\ell)% !\left(\tfrac{n}{2}-\ell\right)!}(2x)^{2\ell}
  73. n n
  74. H n ( x ) = n ! = 0 n - 1 2 ( - 1 ) n - 1 2 - ( 2 + 1 ) ! ( n - 1 2 - ) ! ( 2 x ) 2 + 1 H_{n}(x)=n!\sum_{\ell=0}^{\frac{n-1}{2}}\frac{(-1)^{\frac{n-1}{2}-\ell}}{(2% \ell+1)!\left(\frac{n-1}{2}-\ell\right)!}(2x)^{2\ell+1}
  75. n n
  76. H n ( x ) = n ! m = 0 n 2 ( - 1 ) m m ! ( n - 2 m ) ! ( 2 x ) n - 2 m . H_{n}(x)=n!\sum_{m=0}^{\lfloor\tfrac{n}{2}\rfloor}\frac{(-1)^{m}}{m!(n-2m)!}(2% x)^{n-2m}.
  77. x x
  78. H e n ( x ) = n ! m = 0 n 2 ( - 1 ) m m ! ( n - 2 m ) ! x n - 2 m 2 m . He_{n}(x)=n!\sum_{m=0}^{\lfloor\tfrac{n}{2}\rfloor}\frac{(-1)^{m}}{m!(n-2m)!}% \frac{x^{n-2m}}{2^{m}}.
  79. H e He
  80. x n = n ! m = 0 n 2 1 2 m m ! ( n - 2 m ) ! H e n - 2 m ( x ) . x^{n}=n!\sum_{m=0}^{\lfloor\tfrac{n}{2}\rfloor}\frac{1}{2^{m}~{}m!(n-2m)!}~{}% He_{n-2m}(x)~{}.
  81. H H
  82. x n = n ! 2 n m = 0 n 2 1 m ! ( n - 2 m ) ! H n - 2 m ( x ) . x^{n}=\frac{n!}{2^{n}}\sum_{m=0}^{\lfloor\tfrac{n}{2}\rfloor}\frac{1}{m!(n-2m)% !}~{}H_{n-2m}(x)~{}.
  83. exp ( x t - t 2 2 ) = n = 0 𝐻𝑒 n ( x ) t n n ! \exp(xt-\frac{t^{2}}{2})=\sum_{n=0}^{\infty}{\mathit{He}}_{n}(x)\frac{t^{n}}{n!}
  84. exp ( 2 x t - t 2 ) = n = 0 H n ( x ) t n n ! \exp(2xt-t^{2})=\sum_{n=0}^{\infty}H_{n}(x)\frac{t^{n}}{n!}\,\!
  85. H n ( x ) = ( - 1 ) n e x 2 d n d x n e - x 2 = ( - 1 ) n e x 2 n ! 2 π i γ e - z 2 ( z - x ) n + 1 d z . H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}=(-1)^{n}e^{x^{2}}{n!% \over 2\pi i}\oint_{\gamma}{e^{-z^{2}}\over(z-x)^{n+1}}\,dz.
  86. n = 0 H n ( x ) t n n ! , \sum_{n=0}^{\infty}H_{n}(x)\frac{t^{n}}{n!},
  87. μ μ
  88. E ( 𝐻𝑒 n ( X ) ) = μ n . E({\mathit{He}}_{n}(X))=\mu^{n}.
  89. E ( X 2 n ) = ( - 1 ) n 𝐻𝑒 2 n ( 0 ) = ( 2 n - 1 ) ! ! E(X^{2n})=(-1)^{n}{\mathit{He}}_{2n}(0)=(2n-1)!!
  90. ( 2 n - 1 ) ! ! (2n-1)!!
  91. 𝐻𝑒 n ( x ) = - ( x + i y ) n e - y 2 2 d y / 2 π . \mathit{He}_{n}(x)=\int_{-\infty}^{\infty}(x+iy)^{n}e^{-\frac{y^{2}}{2}}\,% \mathrm{d}y/\sqrt{2\pi}.
  92. e - x 2 2 H n ( x ) 2 n π Γ ( n + 1 2 ) cos ( x 2 n - n π 2 ) e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)\sim\frac{2^{n}}{\sqrt{\pi}}\Gamma\left(% \frac{n+1}{2}\right)\cos\left(x\sqrt{2n}-n\frac{\pi}{2}\right)
  93. e - x 2 2 H n ( x ) 2 n π Γ ( n + 1 2 ) cos ( x 2 n - n π 2 ) ( 1 - x 2 2 n ) - 1 4 = 2 Γ ( n ) Γ ( n 2 ) cos ( x 2 n - n π 2 ) ( 1 - x 2 2 n ) - 1 4 e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)\sim\frac{2^{n}}{\sqrt{\pi}}\Gamma\left(% \frac{n+1}{2}\right)\cos\left(x\sqrt{2n}-n\frac{\pi}{2}\right)\left(1-\frac{x^% {2}}{2n}\right)^{-\frac{1}{4}}=\frac{2\Gamma\left(n\right)}{\Gamma\left(\frac{% n}{2}\right)}\cos\left(x\sqrt{2n}-n\frac{\pi}{2}\right)\left(1-\frac{x^{2}}{2n% }\right)^{-\frac{1}{4}}
  94. e - x 2 2 H n ( x ) ( 2 n e ) n 2 2 cos ( x 2 n - n π 2 ) ( 1 - x 2 2 n ) - 1 4 e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)\sim\left(\frac{2n}{e}\right)^{\frac{n}{2}}{% \sqrt{2}}\cos\left(x\sqrt{2n}-n\frac{\pi}{2}\right)\left(1-\frac{x^{2}}{2n}% \right)^{-\frac{1}{4}}
  95. e - x 2 2 H n ( x ) ( 2 n e ) n 2 2 cos ( x 2 n + 1 - x 2 3 - n π 2 ) ( 1 - x 2 2 n ) - 1 4 e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)\sim\left(\frac{2n}{e}\right)^{\frac{n}{2}}{% \sqrt{2}}\cos\left(x\sqrt{2n+1-\frac{x^{2}}{3}}-n\frac{\pi}{2}\right)\left(1-% \frac{x^{2}}{2n}\right)^{-\frac{1}{4}}
  96. x = 2 n + 1 cos ( ϕ ) , 0 < ϵ ϕ π - ϵ , x=\sqrt{2n+1}\cos(\phi),\qquad 0<\epsilon\leq\phi\leq\pi-\epsilon~{},
  97. e - x 2 2 H n ( x ) = 2 n 2 + 1 4 n ! ( π n ) - 1 4 ( sin ϕ ) - 1 2 [ sin ( ( n 2 + 1 4 ) ( sin ( 2 ϕ ) - 2 ϕ ) + 3 π 4 ) + O ( n - 1 ) ] . e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)=2^{\tfrac{n}{2}+\frac{1}{4}}\sqrt{n!}(\pi n% )^{-\frac{1}{4}}(\sin\phi)^{-\tfrac{1}{2}}\cdot\left[\sin\left(\left(\frac{n}{% 2}+\frac{1}{4}\right)\left(\sin(2\phi)-2\phi\right)+\frac{3\pi}{4}\right)+O(n^% {-1})\right]~{}.
  98. x = 2 n + 1 cosh ( ϕ ) , 0 < ϵ ϕ ω < , x=\sqrt{2n+1}\cosh(\phi),\qquad 0<\epsilon\leq\phi\leq\omega<\infty~{},
  99. e - x 2 2 H n ( x ) = 2 n 2 - 3 4 n ! ( π n ) - 1 4 ( sinh ϕ ) - 1 2 exp ( ( n 2 + 1 4 ) ( 2 ϕ - sinh ( 2 ϕ ) ) ) [ 1 + O ( n - 1 ) ] , e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)=2^{\tfrac{n}{2}-\frac{3}{4}}\sqrt{n!}(\pi n% )^{-\frac{1}{4}}(\sinh\phi)^{-\frac{1}{2}}\cdot\exp\left(\left(\frac{n}{2}+% \frac{1}{4}\right)\left(2\phi-\sinh(2\phi)\right)\right)\left[1+O(n^{-1})% \right],
  100. x = 2 n + 1 - 2 - 1 2 3 - 1 / 3 n - 1 / 6 t x=\sqrt{2n+1}-2^{-\frac{1}{2}}3^{-1/3}n^{-1/6}t
  101. t t
  102. e - x 2 2 H n ( x ) = π 1 4 2 n 2 + 1 4 n ! n - 1 / 12 [ Ai ( - 3 - 1 / 3 t ) + O ( n - 2 / 3 ) ] e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)=\pi^{\frac{1}{4}}2^{\tfrac{n}{2}+\frac{1}{4% }}\sqrt{n!}n^{-1/12}\left[\mathrm{Ai}(-3^{-1/3}t)+O(n^{-2/3})\right]
  103. H n ( 0 ) H_{n}(0)
  104. H n ( 0 ) = { 0 , if n is odd ( - 1 ) n 2 2 n 2 ( n - 1 ) ! ! , if n is even H_{n}(0)=\begin{cases}0,&\mbox{if }~{}n\mbox{ is odd}\\ (-1)^{\tfrac{n}{2}}2^{\tfrac{n}{2}}(n-1)!!,&\mbox{if }~{}n\mbox{ is even}\end{cases}
  105. H n ( 0 ) = - 2 ( n - 1 ) H n - 2 ( 0 ) H_{n}(0)=-2(n-1)H_{n-2}(0)
  106. H e n ( 0 ) = { 0 , if n is odd ( - 1 ) n 2 ( n - 1 ) ! ! if n is even . He_{n}(0)=\begin{cases}0,&\mbox{if }~{}n\mbox{ is odd}\\ (-1)^{\tfrac{n}{2}}(n-1)!!&\mbox{if }~{}n\mbox{ is even}~{}.\end{cases}
  107. H 2 n ( x ) = ( - 4 ) n n ! L n ( - 1 2 ) ( x 2 ) = 4 n n ! i = 0 n ( - 1 ) n - i ( n - 1 2 n - i ) x 2 i i ! H_{2n}(x)=(-4)^{n}\,n!\,L_{n}^{(-\frac{1}{2})}(x^{2})=4^{n}\,n!\sum_{i=0}^{n}(% -1)^{n-i}{n-\frac{1}{2}\choose n-i}\frac{x^{2i}}{i!}
  108. H 2 n + 1 ( x ) = 2 ( - 4 ) n n ! x L n ( 1 2 ) ( x 2 ) = 2 4 n n ! i = 0 n ( - 1 ) n - i ( n + 1 2 n - i ) x 2 i + 1 i ! H_{2n+1}(x)=2(-4)^{n}\,n!\,x\,L_{n}^{(\frac{1}{2})}(x^{2})=2\cdot 4^{n}\,n!% \sum_{i=0}^{n}(-1)^{n-i}{n+\frac{1}{2}\choose n-i}\frac{x^{2i+1}}{i!}
  109. H n ( x ) = 2 n U ( - n 2 , 1 2 , x 2 ) H_{n}(x)=2^{n}\,U\left(-\frac{n}{2},\frac{1}{2},x^{2}\right)
  110. U ( a , b , z ) U(a,b,z)
  111. H 2 n ( x ) = ( - 1 ) n ( 2 n ) ! n ! 1 F 1 ( - n , 1 2 ; x 2 ) H_{2n}(x)=(-1)^{n}\,\frac{(2n)!}{n!}\,_{1}F_{1}\left(-n,\frac{1}{2};x^{2}\right)
  112. H 2 n + 1 ( x ) = ( - 1 ) n ( 2 n + 1 ) ! n ! 2 x 1 F 1 ( - n , 3 2 ; x 2 ) H_{2n+1}(x)=(-1)^{n}\,\frac{(2n+1)!}{n!}\,2x\,_{1}F_{1}\left(-n,\frac{3}{2};x^% {2}\right)
  113. F 1 1 ( a , b ; z ) = M ( a , b ; z ) \,{}_{1}F_{1}(a,b;z)=M(a,b;z)
  114. 𝐻𝑒 n ( x ) = e - D 2 2 x n , {\mathit{He}}_{n}(x)=e^{-\frac{D^{2}}{2}}x^{n},
  115. D D
  116. x x
  117. W W
  118. g ( D ) g(D)
  119. W W
  120. 𝐻𝑒 n ( x ) = n ! 2 π i e t x - t 2 2 t n + 1 d t {\mathit{He}}_{n}(x)=\frac{n!}{2\pi i}\oint\frac{e^{tx-\frac{t^{2}}{2}}}{t^{n+% 1}}\,dt
  121. H n ( x ) = n ! 2 π i e 2 t x - t 2 t n + 1 d t H_{n}(x)=\frac{n!}{2\pi i}\oint\frac{e^{2tx-t^{2}}}{t^{n+1}}\,dt
  122. 1 2 π e - x 2 2 , \frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}~{},
  123. 𝐻𝑒 n [ α ] ( x ) {\mathit{He}}_{n}^{[\alpha]}(x)
  124. α α
  125. α α
  126. ( 2 π α ) - 1 2 e - x 2 / ( 2 α ) . (2\pi\alpha)^{-\frac{1}{2}}e^{-x^{2}/(2\alpha)}~{}.
  127. 𝐻𝑒 n [ α ] ( x ) = α n 2 𝐻𝑒 n ( x α ) = ( α 2 ) n 2 H n ( x 2 α ) = e - α D 2 / 2 ( x n ) . {\,\textit{He}}_{n}^{[\alpha]}(x)=\alpha^{\tfrac{n}{2}}\,\textit{He}_{n}\left(% \frac{x}{\sqrt{\alpha}}\right)=\left(\frac{\alpha}{2}\right)^{\tfrac{n}{2}}H_{% n}\left(\frac{x}{\sqrt{2\alpha}}\right)=e^{-\alpha D^{2}/2}(x^{n}).
  128. 𝐻𝑒 n [ α ] ( x ) = k = 0 n h n , k [ α ] x k , {\mathit{He}}_{n}^{[\alpha]}(x)=\sum_{k=0}^{n}h^{[\alpha]}_{n,k}x^{k}~{},
  129. n n
  130. ( 𝐻𝑒 n [ α ] 𝐻𝑒 [ β ] ) ( x ) k = 0 n h n , k [ α ] 𝐻𝑒 k [ β ] ( x ) \left({\mathit{He}}_{n}^{[\alpha]}\circ{\mathit{He}}^{[\beta]}\right)(x)\equiv% \sum_{k=0}^{n}h^{[\alpha]}_{n,k}\,{\mathit{He}}_{k}^{[\beta]}(x)
  131. ( 𝐻𝑒 n [ α ] 𝐻𝑒 [ β ] ) ( x ) = 𝐻𝑒 n [ α + β ] ( x ) \left({\mathit{He}}_{n}^{[\alpha]}\circ{\mathit{He}}^{[\beta]}\right)(x)={% \mathit{He}}_{n}^{[\alpha+\beta]}(x)
  132. 𝐻𝑒 n [ α + β ] ( x + y ) = k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ β ] ( y ) . {\mathit{He}}_{n}^{[\alpha+\beta]}(x+y)=\sum_{k=0}^{n}{n\choose k}{\mathit{He}% }_{k}^{[\alpha]}(x){\mathit{He}}_{n-k}^{[\beta]}(y)~{}.
  133. α α
  134. β β
  135. 𝐻𝑒 n [ - α ] ( x ) {\mathit{He}}_{n}^{[-\alpha]}(x)\,\!
  136. α α
  137. μ μ
  138. σ σ
  139. E ( X n ) = 𝐻𝑒 n [ - σ 2 ] ( μ ) E(X^{n})=\mathit{He}_{n}^{[-\sigma^{2}]}(\mu)
  140. X X
  141. k = 0 n ( n k ) 𝐻𝑒 k [ α ] ( x ) 𝐻𝑒 n - k [ - α ] ( y ) = 𝐻𝑒 n [ 0 ] ( x + y ) = ( x + y ) n . \sum_{k=0}^{n}{n\choose k}{\mathit{He}}_{k}^{[\alpha]}(x){\mathit{He}}_{n-k}^{% [-\alpha]}(y)={\mathit{He}}_{n}^{[0]}(x+y)=(x+y)^{n}.
  142. ψ n ( x ) = ( 2 n n ! π ) - 1 2 e - x 2 2 H n ( x ) = ( - 1 ) n ( 2 n n ! π ) - 1 2 e x 2 2 d n d x n e - x 2 . \psi_{n}(x)=\left(2^{n}n!\sqrt{\pi}\right)^{-\frac{1}{2}}e^{-\frac{x^{2}}{2}}H% _{n}(x)=(-1)^{n}\left(2^{n}n!\sqrt{\pi}\right)^{-\frac{1}{2}}e^{\frac{x^{2}}{2% }}\frac{d^{n}}{dx^{n}}e^{-x^{2}}~{}.
  143. - ψ n ( x ) ψ m ( x ) d x = δ n m \int_{-\infty}^{\infty}\psi_{n}(x)\psi_{m}(x)\,\mathrm{d}x=\delta_{n\,m}\,
  144. D n ( z ) = ( n ! π ) 1 2 ψ n ( z / 2 ) = π - 1 4 2 e z 2 / 4 d n d z n e - z 2 , D_{n}(z)=(n!\sqrt{\pi})^{\frac{1}{2}}\psi_{n}(z/\sqrt{2})=\pi^{-\frac{1}{4}}% \sqrt{2}e^{z^{2}/4}\frac{d^{n}}{dz^{n}}e^{-z^{2}}~{},
  145. ψ n ′′ ( x ) + ( 2 n + 1 - x 2 ) ψ n ( x ) = 0. \psi_{n}^{\prime\prime}(x)+(2n+1-x^{2})\psi_{n}(x)=0.
  146. ψ 0 ( x ) = π - 1 4 e - 1 2 x 2 \psi_{0}(x)=\pi^{-\frac{1}{4}}\,e^{-\frac{1}{2}x^{2}}
  147. ψ 1 ( x ) = 2 π - 1 4 x e - 1 2 x 2 \psi_{1}(x)=\sqrt{2}\,\pi^{-\frac{1}{4}}\,x\,e^{-\frac{1}{2}x^{2}}
  148. ψ 2 ( x ) = ( 2 π 1 4 ) - 1 ( 2 x 2 - 1 ) e - 1 2 x 2 \psi_{2}(x)=(\sqrt{2}\,\pi^{\frac{1}{4}})^{-1}\,(2x^{2}-1)\,e^{-\frac{1}{2}x^{% 2}}
  149. ψ 3 ( x ) = ( 3 π 1 4 ) - 1 ( 2 x 3 - 3 x ) e - 1 2 x 2 \psi_{3}(x)=(\sqrt{3}\,\pi^{\frac{1}{4}})^{-1}\,(2x^{3}-3x)\,e^{-\frac{1}{2}x^% {2}}
  150. ψ 4 ( x ) = ( 2 6 π 1 4 ) - 1 ( 4 x 4 - 12 x 2 + 3 ) e - 1 2 x 2 \psi_{4}(x)=(2\sqrt{6}\,\pi^{\frac{1}{4}})^{-1}\,(4x^{4}-12x^{2}+3)\,e^{-\frac% {1}{2}x^{2}}
  151. ψ 5 ( x ) = ( 2 15 π 1 4 ) - 1 ( 4 x 5 - 20 x 3 + 15 x ) e - 1 2 x 2 \psi_{5}(x)=(2\sqrt{15}\,\pi^{\frac{1}{4}})^{-1}\,(4x^{5}-20x^{3}+15x)\,e^{-% \frac{1}{2}x^{2}}
  152. ψ n ( x ) = n 2 ψ n - 1 ( x ) - n + 1 2 ψ n + 1 ( x ) , \psi_{n}^{\prime}(x)=\sqrt{\frac{n}{2}}\psi_{n-1}(x)-\sqrt{\frac{n+1}{2}}\psi_% {n+1}(x)~{},
  153. x ψ n ( x ) = n 2 ψ n - 1 ( x ) + n + 1 2 ψ n + 1 ( x ) . x\;\psi_{n}(x)=\sqrt{\frac{n}{2}}\psi_{n-1}(x)+\sqrt{\frac{n+1}{2}}\psi_{n+1}(% x).
  154. ψ n ( m ) ( x ) = k = 0 m ( m k ) ( - 1 ) k 2 ( m - k ) / 2 n ! ( n - m + k ) ! ψ n - m + k ( x ) 𝐻𝑒 k ( x ) . \psi_{n}^{(m)}(x)=\sum_{k=0}^{m}{m\choose k}(-1)^{k}2^{(m-k)/2}\sqrt{\frac{n!}% {(n-m+k)!}}\cdot\psi_{n-m+k}(x)\cdot{\mathit{He}}_{k}(x).
  155. | ψ n ( x ) | K π - 1 4 |\psi_{n}(x)|\leq K\pi^{-\frac{1}{4}}
  156. x x
  157. K K
  158. exp ( - x 2 / 2 + 2 x t - t 2 ) = n = 0 exp ( - x 2 / 2 ) H n ( x ) t n n ! . \exp(-x^{2}/2+2xt-t^{2})=\sum_{n=0}^{\infty}\exp(-x^{2}/2)H_{n}(x)\frac{t^{n}}% {n!}~{}.
  159. { exp ( - x 2 2 + 2 x t - t 2 ) } ( k ) = 1 2 π - exp ( - i x k ) exp ( - x 2 / 2 + 2 x t - t 2 ) d x = exp ( - k 2 / 2 - 2 k i t + t 2 ) = n = 0 exp ( - k 2 2 ) H n ( k ) ( - i t ) n n ! \begin{aligned}\displaystyle\mathcal{F}\left\{\exp\left(-\frac{x^{2}}{2}+2xt-t% ^{2}\right)\right\}(k)&\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{% \infty}\exp(-ixk)\exp(-x^{2}/2+2xt-t^{2})\,\mathrm{d}x\\ &\displaystyle=\exp(-k^{2}/2-2kit+t^{2})\\ &\displaystyle=\sum_{n=0}^{\infty}\exp\left(-\frac{k^{2}}{2}\right)H_{n}(k)% \frac{(-it)^{n}}{n!}\end{aligned}
  160. { n = 0 e - x 2 2 H n ( x ) t n n ! } = n = 0 { e - x 2 2 H n ( x ) } t n n ! . \mathcal{F}\left\{\sum_{n=0}^{\infty}e^{-\frac{x^{2}}{2}}H_{n}(x)\frac{t^{n}}{% n!}\right\}=\sum_{n=0}^{\infty}\mathcal{F}\left\{e^{-\frac{x^{2}}{2}}H_{n}(x)% \right\}\frac{t^{n}}{n!}.
  161. { e - x 2 2 H n ( x ) } = ( - i ) n e - k 2 2 H n ( k ) . \mathcal{F}\left\{e^{-\frac{x^{2}}{2}}H_{n}(x)\right\}=(-i)^{n}e^{-\frac{k^{2}% }{2}}H_{n}(k).\,\!
  162. L n ( x ) := k = 0 n ( n k ) ( - 1 ) k k ! x k , L_{n}(x):=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}\frac{(-1)^{k}}{k!}x^{k},
  163. l n ( x ) := e - x 2 L n ( x ) . l_{n}(x):=e^{-\frac{x}{2}}L_{n}(x)~{}.
  164. n n
  165. W ψ n ( t , f ) = ( - 1 ) n l n ( 4 π ( t 2 + f 2 ) ) , W_{\psi_{n}}(t,f)=(-1)^{n}l_{n}\left(4\pi(t^{2}+f^{2})\right),
  166. W x ( t , f ) = - x ( t + τ / 2 ) x ( t - τ / 2 ) * e - 2 π i τ f d τ . W_{x}(t,f)=\int_{-\infty}^{\infty}x(t+\tau/2)\,x(t-\tau/2)^{*}\,e^{-2\pi i\tau f% }\,d\tau~{}.
  167. T ( n ) = H e n ( i ) i n . T(n)=\frac{\mathop{He}_{n}(i)}{i^{n}}.
  168. k = 0 n H k ( x ) H k ( y ) k ! 2 k = 1 n ! 2 n + 1 H n ( y ) H n + 1 ( x ) - H n ( x ) H n + 1 ( y ) x - y . \sum_{k=0}^{n}\frac{H_{k}(x)H_{k}(y)}{k!2^{k}}=\frac{1}{n!2^{n+1}}~{}\frac{H_{% n}(y)H_{n+1}(x)-H_{n}(x)H_{n+1}(y)}{x-y}.
  169. n = 0 ψ n ( x ) ψ n ( y ) = δ ( x - y ) , \sum_{n=0}^{\infty}\psi_{n}(x)\psi_{n}(y)=\delta(x-y),
  170. δ δ
  171. u 1 u→ 1
  172. n = 0 H n ( x ) H n ( y ) n ! ( u 2 ) n = 1 1 - u 2 e 2 u 1 + u x y - u 2 1 - u 2 ( x - y ) 2 . \sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{n!}\left(\frac{u}{2}\right)^{n}=% \frac{1}{\sqrt{1-u^{2}}}e^{\frac{2u}{1+u}xy-\frac{u^{2}}{1-u^{2}}(x-y)^{2}}~{}.
  173. ( x , y ) E ( x , y ; u ) (x,y) →E(x,y;u)
  174. u u
  175. y y
  176. x x
  177. n = 0 u n f , ψ n ψ n , g = E ( x , y ; u ) f ( x ) g ( y ) ¯ d x d y f ( x ) g ( x ) ¯ d x = f , g , \left\langle\sum_{n=0}^{\infty}u^{n}\langle f,\psi_{n}\rangle\psi_{n},g\right% \rangle=\iint E(x,y;u)f(x)\overline{g(y)}\,\mathrm{d}x\,\mathrm{d}y\to\int f(x% )\overline{g(x)}\,\mathrm{d}x=\langle f,g\rangle~{},
  178. f , g f,g
  179. f f
  180. f = n = 0 f , ψ n ψ n . f=\sum_{n=0}^{\infty}\langle f,\psi_{n}\rangle\psi_{n}~{}.
  181. E ( x , y ; u ) E(x,y;u)
  182. ρ π e - ρ 2 x 2 4 = e i s x - s 2 / ρ 2 d s , ρ > 0. \rho\sqrt{\pi}e^{-\frac{\rho^{2}x^{2}}{4}}=\int e^{isx-s^{2}/\rho^{2}}\,% \mathrm{d}s,\qquad\rho>0.
  183. H n ( x ) = ( - 1 ) n e x 2 d n d x n ( 1 2 π e i s x - s 2 4 d s ) = ( - 1 ) n e x 2 1 2 π ( i s ) n e i s x - s 2 4 d s . H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\left(\frac{1}% {2\sqrt{\pi}}\int e^{isx-\frac{s^{2}}{4}}\,\mathrm{d}s\right)=(-1)^{n}e^{x^{2}% }\frac{1}{2\sqrt{\pi}}\int(is)^{n}e^{isx-\frac{s^{2}}{4}}\,\mathrm{d}s.
  184. E ( x , y ; u ) = n = 0 u n 2 n n ! π H n ( x ) H n ( y ) e - x 2 + y 2 2 = e x 2 + y 2 2 4 π π ( n = 0 1 2 n n ! ( - u s t ) n ) e i s x + i t y - s 2 4 - t 2 4 d s d t = e x 2 + y 2 2 4 π π e - u s t / 2 e i s x + i t y - s 2 4 - t 2 4 d s d t \begin{aligned}\displaystyle E(x,y;u)&\displaystyle=\sum_{n=0}^{\infty}\frac{u% ^{n}}{2^{n}n!\sqrt{\pi}}\,H_{n}(x)H_{n}(y)e^{-\frac{x^{2}+y^{2}}{2}}\\ &\displaystyle=\frac{e^{\frac{x^{2}+y^{2}}{2}}}{4\pi\sqrt{\pi}}\iint\left(\sum% _{n=0}^{\infty}\frac{1}{2^{n}n!}(-ust)^{n}\right)e^{isx+ity-\frac{s^{2}}{4}-% \frac{t^{2}}{4}}\,\mathrm{d}s\,\mathrm{d}t\\ &\displaystyle=\frac{e^{\frac{x^{2}+y^{2}}{2}}}{4\pi\sqrt{\pi}}\iint e^{-ust/2% }\,e^{isx+ity-\frac{s^{2}}{4}-\frac{t^{2}}{4}}\,\mathrm{d}s\,\mathrm{d}t\end{aligned}
  185. s = σ + τ 2 , t = σ - τ 2 . s=\frac{\sigma+\tau}{\sqrt{2}},\qquad\qquad t=\frac{\sigma-\tau}{\sqrt{2}}.

Hermitian_matrix.html

  1. i i
  2. j j
  3. j j
  4. i i
  5. i i
  6. j j
  7. a i j = a j i ¯ a_{ij}=\overline{a_{ji}}
  8. A = A T ¯ A=\overline{A\text{T}}
  9. A A
  10. A A^{\dagger}
  11. A = A . A=A^{\dagger}.
  12. [ 2 2 + i 4 2 - i 3 i 4 - i 1 ] \begin{bmatrix}2&2+i&4\\ 2-i&3&i\\ 4&-i&1\\ \end{bmatrix}
  13. A A
  14. A = B B A=BB^{\dagger}
  15. A A
  16. B B
  17. A A
  18. A A
  19. n n
  20. A A
  21. n n
  22. n n
  23. A A
  24. A A
  25. B B
  26. A B = B A AB=BA
  27. A A
  28. n n
  29. v v
  30. v A v v^{\dagger}Av
  31. v A v = ( v A v ) v^{\dagger}Av=(v^{\dagger}Av)^{\dagger}
  32. n n
  33. n n
  34. 𝐑 \mathbf{R}
  35. n × n n×n
  36. 𝐑 \mathbf{R}
  37. n n
  38. n n
  39. j , k j,k
  40. E j j \;E_{jj}
  41. 1 j n 1\leq j\leq n
  42. n n
  43. E j k + E k j \;E_{jk}+E_{kj}
  44. 1 j < k n 1\leq j<k\leq n
  45. n < s u p > 2 n n\frac{<sup>2−}{n}
  46. i ( E j k - E k j ) \;i(E_{jk}-E_{kj})
  47. 1 j < k n 1\leq j<k\leq n
  48. n < s u p > 2 n n\frac{<sup>2−}{n}
  49. i i
  50. - 1 \sqrt{-1}
  51. n n
  52. u 1 , , u n u_{1},\dots,u_{n}
  53. U U
  54. A A
  55. A = U Λ U A=U\Lambda U^{\dagger}
  56. U U = I = U U UU^{\dagger}=I=U^{\dagger}U
  57. A = j λ j u j u j A=\sum_{j}\lambda_{j}u_{j}u_{j}^{\dagger}
  58. λ j \lambda_{j}
  59. Λ \;\Lambda
  60. ( C + C ) (C+C^{\dagger})
  61. ( C - C ) (C-C^{\dagger})
  62. C C
  63. A A
  64. B B
  65. C = A + B with A = 1 2 ( C + C ) and B = 1 2 ( C - C ) . C=A+B\quad\mbox{with}~{}\quad A=\frac{1}{2}(C+C^{\dagger})\quad\mbox{and}~{}% \quad B=\frac{1}{2}(C-C^{\dagger}).
  66. det ( A ) = det ( A T ) det ( A ) = det ( A ) * \det(A)=\det(A^{\mathrm{T}})\quad\Rightarrow\quad\det(A^{\dagger})=\det(A)^{*}
  67. A = A det ( A ) = det ( A ) * . A=A^{\dagger}\quad\Rightarrow\quad\det(A)=\det(A)^{*}.

Heron's_formula.html

  1. A = s ( s - a ) ( s - b ) ( s - c ) , A=\sqrt{s(s-a)(s-b)(s-c)},
  2. s = a + b + c 2 . s=\frac{a+b+c}{2}.
  3. A = 1 4 ( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c ) A=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}
  4. A = 1 4 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) - ( a 4 + b 4 + c 4 ) A=\frac{1}{4}\sqrt{2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}
  5. A = 1 4 ( a 2 + b 2 + c 2 ) 2 - 2 ( a 4 + b 4 + c 4 ) A=\frac{1}{4}\sqrt{(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}
  6. A = 1 4 4 a 2 b 2 - ( a 2 + b 2 - c 2 ) 2 . A=\frac{1}{4}\sqrt{4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})^{2}}.
  7. s = 1 2 ( a + b + c ) = 1 2 ( 4 + 13 + 15 ) = 16 s=\tfrac{1}{2}(a+b+c)=\tfrac{1}{2}(4+13+15)=16
  8. A = s ( s - a ) ( s - b ) ( s - c ) = 16 ( 16 - 4 ) ( 16 - 13 ) ( 16 - 15 ) = 16 12 3 1 = 576 = 24. \begin{aligned}\displaystyle A&\displaystyle=\sqrt{s\left(s-a\right)\left(s-b% \right)\left(s-c\right)}=\sqrt{16\cdot(16-4)\cdot(16-13)\cdot(16-15)}\\ &\displaystyle=\sqrt{16\cdot 12\cdot 3\cdot 1}=\sqrt{576}=24.\end{aligned}
  9. A = 1 2 a 2 c 2 - ( a 2 + c 2 - b 2 2 ) 2 A=\frac{1}{2}\sqrt{a^{2}c^{2}-\left(\frac{a^{2}+c^{2}-b^{2}}{2}\right)^{2}}
  10. a b c a\geq b\geq c
  11. cos γ = a 2 + b 2 - c 2 2 a b \cos\gamma=\frac{a^{2}+b^{2}-c^{2}}{2ab}
  12. sin γ = 1 - cos 2 γ = 4 a 2 b 2 - ( a 2 + b 2 - c 2 ) 2 2 a b . \sin\gamma=\sqrt{1-\cos^{2}\gamma}=\frac{\sqrt{4a^{2}b^{2}-(a^{2}+b^{2}-c^{2})% ^{2}}}{2ab}.
  13. A \displaystyle A
  14. b 2 = h 2 + d 2 b^{2}=h^{2}+d^{2}
  15. a 2 = h 2 + ( c - d ) 2 a^{2}=h^{2}+(c-d)^{2}
  16. a 2 - b 2 = c 2 - 2 c d a^{2}-b^{2}=c^{2}-2cd
  17. d d
  18. d = - a 2 + b 2 + c 2 2 c d=\frac{-a^{2}+b^{2}+c^{2}}{2c}
  19. h 2 = b 2 - d 2 h^{2}=b^{2}-d^{2}
  20. d d
  21. h 2 = b 2 - ( - a 2 + b 2 + c 2 2 c ) 2 = ( 2 b c - a 2 + b 2 + c 2 ) ( 2 b c + a 2 - b 2 - c 2 ) 4 c 2 = ( ( b + c ) 2 - a 2 ) ( a 2 - ( b - c ) 2 ) 4 c 2 = ( b + c - a ) ( b + c + a ) ( a + b - c ) ( a - b + c ) 4 c 2 = 2 ( s - a ) 2 s 2 ( s - c ) 2 ( s - b ) 4 c 2 = 4 s ( s - a ) ( s - b ) ( s - c ) c 2 \begin{aligned}\displaystyle h^{2}&\displaystyle=b^{2}-\left(\frac{-a^{2}+b^{2% }+c^{2}}{2c}\right)^{2}\\ &\displaystyle=\frac{(2bc-a^{2}+b^{2}+c^{2})(2bc+a^{2}-b^{2}-c^{2})}{4c^{2}}\\ &\displaystyle=\frac{((b+c)^{2}-a^{2})(a^{2}-(b-c)^{2})}{4c^{2}}\\ &\displaystyle=\frac{(b+c-a)(b+c+a)(a+b-c)(a-b+c)}{4c^{2}}\\ &\displaystyle=\frac{2(s-a)\cdot 2s\cdot 2(s-c)\cdot 2(s-b)}{4c^{2}}\\ &\displaystyle=\frac{4s(s-a)(s-b)(s-c)}{c^{2}}\end{aligned}
  22. A \displaystyle A
  23. A = r ( ( s - a ) + ( s - b ) + ( s - c ) ) = r 2 ( s - a r + s - b r + s - c r ) = r 2 ( cot α 2 + cot β 2 + cot γ 2 ) \begin{aligned}\displaystyle A&\displaystyle=r\big((s-a)+(s-b)+(s-c)\big)=r^{2% }\left(\frac{s-a}{r}+\frac{s-b}{r}+\frac{s-c}{r}\right)\\ &\displaystyle=r^{2}\left(\cot{\frac{\alpha}{2}}+\cot{\frac{\beta}{2}}+\cot{% \frac{\gamma}{2}}\right)\\ \end{aligned}
  24. A = r s A=rs
  25. π 2 \tfrac{\pi}{2}
  26. A = r 2 ( cot α 2 cot β 2 cot γ 2 ) = r 2 ( s - a r s - b r s - c r ) = ( s - a ) ( s - b ) ( s - c ) r \begin{aligned}\displaystyle A&\displaystyle=r^{2}\left(\cot{\frac{\alpha}{2}}% \cot{\frac{\beta}{2}}\cot{\frac{\gamma}{2}}\right)=r^{2}\left(\frac{s-a}{r}% \cdot\frac{s-b}{r}\cdot\frac{s-c}{r}\right)\\ &\displaystyle=\frac{(s-a)(s-b)(s-c)}{r}\\ \end{aligned}
  27. A 2 = s ( s - a ) ( s - b ) ( s - c ) A^{2}=s(s-a)(s-b)(s-c)
  28. a b c a\geq b\geq c
  29. A = 1 4 ( a + ( b + c ) ) ( c - ( a - b ) ) ( c + ( a - b ) ) ( a + ( b - c ) ) . A=\frac{1}{4}\sqrt{(a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c))}.
  30. A = 4 3 σ ( σ - m a ) ( σ - m b ) ( σ - m c ) . A=\frac{4}{3}\sqrt{\sigma(\sigma-m_{a})(\sigma-m_{b})(\sigma-m_{c})}.
  31. H = ( h a - 1 + h b - 1 + h c - 1 ) / 2 H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2
  32. A - 1 = 4 H ( H - h a - 1 ) ( H - h b - 1 ) ( H - h c - 1 ) . A^{-1}=4\sqrt{H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}.
  33. A = D 2 S ( S - sin α ) ( S - sin β ) ( S - sin γ ) A=D^{2}\sqrt{S(S-\sin\alpha)(S-\sin\beta)(S-\sin\gamma)}
  34. D = a sin α = b sin β = c sin γ . D=\tfrac{a}{\sin\alpha}=\tfrac{b}{\sin\beta}=\tfrac{c}{\sin\gamma}.
  35. A = 1 4 - | 0 a 2 b 2 1 a 2 0 c 2 1 b 2 c 2 0 1 1 1 1 0 | A=\frac{1}{4}\sqrt{-\begin{vmatrix}0&a^{2}&b^{2}&1\\ a^{2}&0&c^{2}&1\\ b^{2}&c^{2}&0&1\\ 1&1&1&0\end{vmatrix}}
  36. volume = ( - a + b + c + d ) ( a - b + c + d ) ( a + b - c + d ) ( a + b + c - d ) 192 u v w \,\text{volume}=\frac{\sqrt{\,(-a+b+c+d)\,(a-b+c+d)\,(a+b-c+d)\,(a+b+c-d)}}{19% 2\,u\,v\,w}
  37. a \displaystyle a

Hess's_law.html

  1. Δ H f \Delta H_{f}
  2. Δ H = - Δ H f ( r e a c t a n t s ) \Delta H^{\ominus}=-\sum\Delta H_{\mathrm{f}\,(reactants)}^{\ominus}
  3. Δ H = Δ H f ( p r o d u c t s ) \Delta H^{\ominus}=\sum\Delta H_{\mathrm{f}\,(products)}^{\ominus}
  4. Δ G r e a c t i o n = Δ G f ( p r o d u c t s ) - Δ G f ( r e a c t a n t s ) . \Delta G_{reaction}^{\ominus}=\sum\Delta G_{\mathrm{f}\,(products)}^{\ominus}-% \sum\Delta G_{\mathrm{f}\,(reactants)}^{\ominus}.
  5. Δ S r e a c t i o n = S ( p r o d u c t s ) - S ( r e a c t a n t s ) . \Delta S_{reaction}^{\ominus}=\sum S_{(products)}^{\ominus}-\sum S_{(reactants% )}^{\ominus}.

Heyting_algebra.html

  1. a x b . a\wedge x\leq b.
  2. a ¬ a = 0 a\wedge\lnot a=0
  3. a ¬ a = 1 a\vee\lnot a=1
  4. { f a : H H f a ( x ) = a x \begin{cases}f_{a}\colon H\to H\\ f_{a}(x)=a\wedge x\end{cases}
  5. a a = 1 a\to a=1
  6. a ( a b ) = a b a\wedge(a\to b)=a\wedge b
  7. b ( a b ) = b b\wedge(a\to b)=b
  8. a ( b c ) = ( a b ) ( a c ) a\to(b\wedge c)=(a\to b)\wedge(a\to c)
  9. a b a\leq b
  10. If x y and y x then x = y , \mbox{If }~{}x\leq y\mbox{ and }~{}y\leq x\mbox{ then }~{}x=y,
  11. If 1 y , then y = 1 , \mbox{If }~{}1\leq y,\mbox{ then }~{}y=1,
  12. x y x , x\leq y\to x,
  13. x ( y z ) ( x y ) ( x z ) , x\to(y\to z)\leq(x\to y)\to(x\to z),
  14. x and y x , x\and y\leq x,
  15. x and y y , x\and y\leq y,
  16. x y ( x and y ) , x\leq y\to(x\and y),
  17. x x y , x\leq xy,
  18. y x y , y\leq xy,
  19. x z ( y z ) ( x y z ) , x\to z\leq(y\to z)\to(xy\to z),
  20. 0 x . 0\leq x.
  21. a b a\land b
  22. a b a\lor b
  23. \leq
  24. a b a\leq b
  25. F ( A 1 , A 2 , , A n ) F(A_{1},A_{2},\ldots,A_{n})
  26. , , ¬ , \land,\lor,\lnot,\to
  27. F ( a 1 , a 2 , , a n ) = 1 F(a_{1},a_{2},\ldots,a_{n})=1
  28. a 1 , a 2 , , a n H a_{1},a_{2},\ldots,a_{n}\in H
  29. a b a\leq b
  30. F ( a 1 , a 2 , , a n ) G ( a 1 , a 2 , , a n ) F(a_{1},a_{2},\ldots,a_{n})\leq G(a_{1},a_{2},\ldots,a_{n})
  31. a 1 , a 2 , , a n H a_{1},a_{2},\ldots,a_{n}\in H
  32. F ( a 1 , a 2 , , a n ) = G ( a 1 , a 2 , , a n ) F(a_{1},a_{2},\ldots,a_{n})=G(a_{1},a_{2},\ldots,a_{n})
  33. F ( A 1 , A 2 , , A n ) F(A_{1},A_{2},\ldots,A_{n})
  34. a 1 , a 2 , , a n H a_{1},a_{2},\ldots,a_{n}\in H
  35. F ( a 1 , a 2 , , a n ) 1 F(a_{1},a_{2},\ldots,a_{n})\neq 1
  36. ( a b ) c = a ( b c ) (a\land b)\to c=a\to(b\to c)
  37. a ( b c ) = ( a b ) ( a c ) a\wedge(b\vee c)=(a\wedge b)\vee(a\wedge c)
  38. a ( b c ) = ( a b ) ( a c ) a\vee(b\wedge c)=(a\vee b)\wedge(a\vee c)
  39. \wedge
  40. \wedge
  41. x Y = { x y y Y } x\wedge\bigvee Y=\bigvee\{x\wedge y\mid y\in Y\}
  42. a b = { c a c b } a\to b=\bigvee\{c\mid a\land c\leq b\}
  43. x , y H : ¬ ( x y ) = ¬ x ¬ y . \forall x,y\in H:\qquad\lnot(x\vee y)=\lnot x\wedge\lnot y.
  44. x , y H : ¬ ( x y ) = ¬ ¬ ( ¬ x ¬ y ) . \forall x,y\in H:\qquad\lnot(x\wedge y)=\lnot\lnot(\lnot x\vee\lnot y).
  45. ¬ ( x y ) = ¬ x ¬ y for all x , y H , \lnot(x\wedge y)=\lnot x\vee\lnot y\mbox{ for all }~{}x,y\in H,
  46. ¬ ( x y ) = ¬ x ¬ y for all regular x , y H , \lnot(x\wedge y)=\lnot x\vee\lnot y\mbox{ for all regular }~{}x,y\in H,
  47. ¬ ¬ ( x y ) = ¬ ¬ x ¬ ¬ y for all x , y H , \lnot\lnot(x\vee y)=\lnot\lnot x\vee\lnot\lnot y\mbox{ for all }~{}x,y\in H,
  48. ¬ ¬ ( x y ) = x y for all regular x , y H , \lnot\lnot(x\vee y)=x\vee y\mbox{ for all regular }~{}x,y\in H,
  49. ¬ ( ¬ x ¬ y ) = x y for all regular x , y H , \lnot(\lnot x\wedge\lnot y)=x\vee y\mbox{ for all regular }~{}x,y\in H,
  50. ¬ x ¬ ¬ x = 1 for all x H . \lnot x\vee\lnot\lnot x=1\mbox{ for all }~{}x\in H.
  51. f ( 0 ) = 0 , f(0)=0,
  52. f ( x and y ) = f ( x ) and f ( y ) , f(x\and y)=f(x)\and f(y),
  53. f ( x y ) = f ( x ) f ( y ) , f(xy)=f(x)f(y),
  54. f ( x y ) = f ( x ) f ( y ) , f(x\to y)=f(x)\to f(y),
  55. 1 F , 1\in F,
  56. If x , y F then x and y F , \mbox{If }~{}x,y\in F\mbox{ then }~{}x\and y\in F,
  57. If x F , y H , and x y then y F . \mbox{If }~{}x\in F,\ y\in H,\ \mbox{and }~{}x\leq y\mbox{ then }~{}y\in F.
  58. P Q x P P\land Q\land x\leq P
  59. P 1 Q P\land 1\leq Q
  60. 1 1 Q 1\land 1\leq Q

Hidden_Markov_model.html

  1. t t
  2. t - 1 t-1
  3. N N
  4. N N
  5. t t
  6. N N
  7. t + 1 t+1
  8. N 2 N^{2}
  9. N × N N\times N
  10. N ( N - 1 ) N(N-1)
  11. N N
  12. M M
  13. M - 1 M-1
  14. N ( M - 1 ) N(M-1)
  15. M M
  16. M M
  17. M ( M + 1 ) / 2 M(M+1)/2
  18. N ( M + M ( M + 1 ) 2 ) = N M ( M + 3 ) / 2 = O ( N M 2 ) N(M+\frac{M(M+1)}{2})=NM(M+3)/2=O(NM^{2})
  19. M M
  20. Y = y ( 0 ) , y ( 1 ) , , y ( L - 1 ) Y=y(0),y(1),\dots,y(L-1)\,
  21. P ( Y ) = X P ( Y X ) P ( X ) , P(Y)=\sum_{X}P(Y\mid X)P(X),\,
  22. X = x ( 0 ) , x ( 1 ) , , x ( L - 1 ) . X=x(0),x(1),\dots,x(L-1).\,
  23. y ( 1 ) , , y ( t ) . y(1),\dots,y(t).
  24. P ( x ( t ) | y ( 1 ) , , y ( t ) ) P(x(t)\ |\ y(1),\dots,y(t))
  25. P ( x ( k ) | y ( 1 ) , , y ( t ) ) P(x(k)\ |\ y(1),\dots,y(t))
  26. k < t k<t
  27. N N
  28. = =
  29. T T
  30. = =
  31. θ i = 1 N \theta_{i=1\dots N}
  32. = =
  33. i i
  34. ϕ i = 1 N , j = 1 N \phi_{i=1\dots N,j=1\dots N}
  35. = =
  36. i i
  37. j j
  38. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  39. = =
  40. N N
  41. ϕ i , 1 N \phi_{i,1\dots N}
  42. 1 1
  43. x t = 1 T x_{t=1\dots T}
  44. = =
  45. t t
  46. y t = 1 T y_{t=1\dots T}
  47. = =
  48. t t
  49. F ( y | θ ) F(y|\theta)
  50. = =
  51. θ \theta
  52. x t = 2 T x_{t=2\dots T}
  53. \sim
  54. Categorical ( s y m b o l ϕ x t - 1 ) \operatorname{Categorical}(symbol\phi_{x_{t-1}})
  55. y t = 1 T y_{t=1\dots T}
  56. \sim
  57. F ( θ x t ) F(\theta_{x_{t}})
  58. x 1 x_{1}
  59. N , T N,T
  60. = =
  61. θ i = 1 N , ϕ i = 1 N , j = 1 N , s y m b o l ϕ i = 1 N \theta_{i=1\dots N},\phi_{i=1\dots N,j=1\dots N},symbol\phi_{i=1\dots N}
  62. = =
  63. x t = 1 T , y t = 1 T , F ( y | θ ) x_{t=1\dots T},y_{t=1\dots T},F(y|\theta)
  64. = =
  65. α \alpha
  66. = =
  67. β \beta
  68. = =
  69. H ( θ | α ) H(\theta|\alpha)
  70. = =
  71. α \alpha
  72. θ i = 1 N \theta_{i=1\dots N}
  73. \sim
  74. H ( α ) H(\alpha)
  75. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  76. \sim
  77. Symmetric - Dirichlet N ( β ) \operatorname{Symmetric-Dirichlet}_{N}(\beta)
  78. x t = 2 T x_{t=2\dots T}
  79. \sim
  80. Categorical ( s y m b o l ϕ x t - 1 ) \operatorname{Categorical}(symbol\phi_{x_{t-1}})
  81. y t = 1 T y_{t=1\dots T}
  82. \sim
  83. F ( θ x t ) F(\theta_{x_{t}})
  84. F F
  85. H H
  86. H H
  87. F F
  88. F F
  89. N N
  90. = =
  91. T T
  92. = =
  93. θ i = 1 N \theta_{i=1\dots N}
  94. = =
  95. i i
  96. ϕ i = 1 N \phi_{i=1\dots N}
  97. = =
  98. i i
  99. s y m b o l ϕ symbol\phi
  100. = =
  101. N N
  102. ϕ 1 N \phi_{1\dots N}
  103. 1 1
  104. x t = 1 T x_{t=1\dots T}
  105. = =
  106. t t
  107. y t = 1 T y_{t=1\dots T}
  108. = =
  109. t t
  110. F ( y | θ ) F(y|\theta)
  111. = =
  112. θ \theta
  113. x t = 1 T x_{t=1\dots T}
  114. \sim
  115. Categorical ( s y m b o l ϕ ) \operatorname{Categorical}(symbol\phi)
  116. y t = 1 T y_{t=1\dots T}
  117. \sim
  118. F ( θ x t ) F(\theta_{x_{t}})
  119. N , T N,T
  120. = =
  121. θ i = 1 N , ϕ i = 1 N , s y m b o l ϕ \theta_{i=1\dots N},\phi_{i=1\dots N},symbol\phi
  122. = =
  123. x t = 1 T , y t = 1 T , F ( y | θ ) x_{t=1\dots T},y_{t=1\dots T},F(y|\theta)
  124. = =
  125. α \alpha
  126. = =
  127. β \beta
  128. = =
  129. H ( θ | α ) H(\theta|\alpha)
  130. = =
  131. α \alpha
  132. θ i = 1 N \theta_{i=1\dots N}
  133. \sim
  134. H ( α ) H(\alpha)
  135. s y m b o l ϕ symbol\phi
  136. \sim
  137. Symmetric - Dirichlet N ( β ) \operatorname{Symmetric-Dirichlet}_{N}(\beta)
  138. x t = 1 T x_{t=1\dots T}
  139. \sim
  140. Categorical ( s y m b o l ϕ ) \operatorname{Categorical}(symbol\phi)
  141. y t = 1 T y_{t=1\dots T}
  142. \sim
  143. F ( θ x t ) F(\theta_{x_{t}})
  144. N N
  145. = =
  146. T T
  147. = =
  148. ϕ i = 1 N , j = 1 N \phi_{i=1\dots N,j=1\dots N}
  149. = =
  150. i i
  151. j j
  152. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  153. = =
  154. N N
  155. ϕ i , 1 N \phi_{i,1\dots N}
  156. 1 1
  157. μ i = 1 N \mu_{i=1\dots N}
  158. = =
  159. i i
  160. σ i = 1 N 2 \sigma^{2}_{i=1\dots N}
  161. = =
  162. i i
  163. x t = 1 T x_{t=1\dots T}
  164. = =
  165. t t
  166. y t = 1 T y_{t=1\dots T}
  167. = =
  168. t t
  169. x t = 2 T x_{t=2\dots T}
  170. \sim
  171. Categorical ( s y m b o l ϕ x t - 1 ) \operatorname{Categorical}(symbol\phi_{x_{t-1}})
  172. y t = 1 T y_{t=1\dots T}
  173. \sim
  174. 𝒩 ( μ x t , σ x t 2 ) \mathcal{N}(\mu_{x_{t}},\sigma_{x_{t}}^{2})
  175. N N
  176. = =
  177. T T
  178. = =
  179. ϕ i = 1 N , j = 1 N \phi_{i=1\dots N,j=1\dots N}
  180. = =
  181. i i
  182. j j
  183. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  184. = =
  185. N N
  186. ϕ i , 1 N \phi_{i,1\dots N}
  187. 1 1
  188. μ i = 1 N \mu_{i=1\dots N}
  189. = =
  190. i i
  191. σ i = 1 N 2 \sigma^{2}_{i=1\dots N}
  192. = =
  193. i i
  194. x t = 1 T x_{t=1\dots T}
  195. = =
  196. t t
  197. y t = 1 T y_{t=1\dots T}
  198. = =
  199. t t
  200. β \beta
  201. = =
  202. μ 0 , λ \mu_{0},\lambda
  203. = =
  204. ν , σ 0 2 \nu,\sigma_{0}^{2}
  205. = =
  206. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  207. \sim
  208. Symmetric - Dirichlet N ( β ) \operatorname{Symmetric-Dirichlet}_{N}(\beta)
  209. x t = 2 T x_{t=2\dots T}
  210. \sim
  211. Categorical ( s y m b o l ϕ x t - 1 ) \operatorname{Categorical}(symbol\phi_{x_{t-1}})
  212. μ i = 1 N \mu_{i=1\dots N}
  213. \sim
  214. 𝒩 ( μ 0 , λ σ i 2 ) \mathcal{N}(\mu_{0},\lambda\sigma_{i}^{2})
  215. σ i = 1 N 2 \sigma_{i=1\dots N}^{2}
  216. \sim
  217. Inverse - Gamma ( ν , σ 0 2 ) \operatorname{Inverse-Gamma}(\nu,\sigma_{0}^{2})
  218. y t = 1 T y_{t=1\dots T}
  219. \sim
  220. 𝒩 ( μ x t , σ x t 2 ) \mathcal{N}(\mu_{x_{t}},\sigma_{x_{t}}^{2})
  221. N N
  222. = =
  223. T T
  224. = =
  225. ϕ i = 1 N , j = 1 N \phi_{i=1\dots N,j=1\dots N}
  226. = =
  227. i i
  228. j j
  229. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  230. = =
  231. N N
  232. ϕ i , 1 N \phi_{i,1\dots N}
  233. 1 1
  234. V V
  235. = =
  236. θ i = 1 N , j = 1 V \theta_{i=1\dots N,j=1\dots V}
  237. = =
  238. i i
  239. j j
  240. s y m b o l θ i = 1 N symbol\theta_{i=1\dots N}
  241. = =
  242. V V
  243. θ i , 1 V \theta_{i,1\dots V}
  244. 1 1
  245. x t = 1 T x_{t=1\dots T}
  246. = =
  247. t t
  248. y t = 1 T y_{t=1\dots T}
  249. = =
  250. t t
  251. x t = 2 T x_{t=2\dots T}
  252. \sim
  253. Categorical ( s y m b o l ϕ x t - 1 ) \operatorname{Categorical}(symbol\phi_{x_{t-1}})
  254. y t = 1 T y_{t=1\dots T}
  255. \sim
  256. Categorical ( s y m b o l θ x t ) \,\text{Categorical}(symbol\theta_{x_{t}})
  257. N N
  258. = =
  259. T T
  260. = =
  261. ϕ i = 1 N , j = 1 N \phi_{i=1\dots N,j=1\dots N}
  262. = =
  263. i i
  264. j j
  265. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  266. = =
  267. N N
  268. ϕ i , 1 N \phi_{i,1\dots N}
  269. 1 1
  270. V V
  271. = =
  272. θ i = 1 N , j = 1 V \theta_{i=1\dots N,j=1\dots V}
  273. = =
  274. i i
  275. j j
  276. s y m b o l θ i = 1 N symbol\theta_{i=1\dots N}
  277. = =
  278. V V
  279. θ i , 1 V \theta_{i,1\dots V}
  280. 1 1
  281. x t = 1 T x_{t=1\dots T}
  282. = =
  283. t t
  284. y t = 1 T y_{t=1\dots T}
  285. = =
  286. t t
  287. α \alpha
  288. = =
  289. s y m b o l θ symbol\theta
  290. β \beta
  291. = =
  292. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  293. \sim
  294. Symmetric - Dirichlet N ( β ) \operatorname{Symmetric-Dirichlet}_{N}(\beta)
  295. s y m b o l θ 1 V symbol\theta_{1\dots V}
  296. \sim
  297. Symmetric - Dirichlet V ( α ) \operatorname{Symmetric-Dirichlet}_{V}(\alpha)
  298. x t = 2 T x_{t=2\dots T}
  299. \sim
  300. Categorical ( s y m b o l ϕ x t - 1 ) \operatorname{Categorical}(symbol\phi_{x_{t-1}})
  301. y t = 1 T y_{t=1\dots T}
  302. \sim
  303. Categorical ( s y m b o l θ x t ) \operatorname{Categorical}(symbol\theta_{x_{t}})
  304. β \beta
  305. β \beta
  306. β \beta
  307. β \beta
  308. = =
  309. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  310. \sim
  311. Symmetric - Dirichlet N ( β ) \operatorname{Symmetric-Dirichlet}_{N}(\beta)
  312. γ \gamma
  313. = =
  314. β \beta
  315. = =
  316. s y m b o l η symbol\eta
  317. = =
  318. N N
  319. s y m b o l η symbol\eta
  320. \sim
  321. Symmetric - Dirichlet N ( γ ) \operatorname{Symmetric-Dirichlet}_{N}(\gamma)
  322. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  323. \sim
  324. Dirichlet N ( β N s y m b o l η ) \operatorname{Dirichlet}_{N}(\beta Nsymbol\eta)
  325. s y m b o l η symbol\eta
  326. γ \gamma
  327. s y m b o l η symbol\eta
  328. β \beta
  329. s y m b o l ϕ i = 1 N symbol\phi_{i=1\dots N}
  330. β \beta
  331. s y m b o l ϕ symbol\phi
  332. s y m b o l η symbol\eta
  333. β \beta
  334. s y m b o l ϕ symbol\phi
  335. s y m b o l ϕ symbol\phi
  336. s y m b o l η symbol\eta
  337. β \beta
  338. s y m b o l ϕ symbol\phi
  339. 𝐉 i \mathbf{J}_{i}
  340. s y m b o l η symbol\eta
  341. s y m b o l η symbol\eta
  342. 𝐉 i \mathbf{J}_{i}
  343. 𝐉 i \mathbf{J}_{i}
  344. s y m b o l η symbol\eta
  345. 𝐉 i \mathbf{J}_{i}
  346. x i x_{i}
  347. s y m b o l η symbol\eta
  348. s y m b o l η symbol\eta
  349. γ \gamma
  350. γ \gamma
  351. γ \gamma
  352. K K
  353. N K N^{K}
  354. N N
  355. T T
  356. O ( N 2 K T ) O(N^{2K}\,T)
  357. O ( N K + 1 K T ) O(N^{K+1}\,K\,T)
  358. K K
  359. O ( N K T ) O(N^{K}\,T)
  360. K K
  361. T T
  362. T T

Hidden_variable_theory.html

  1. | ψ |\psi\rangle
  2. | ψ ψ ( x , y , z ) |\psi\rangle\to\psi(x,y,z)

Higher-order_function.html

  1. ( τ 1 τ 2 ) τ 3 (\tau_{1}\to\tau_{2})\to\tau_{3}

Highest_averages_method.html

  1. n n
  2. n n
  3. n ( n + 1 ) \sqrt{n(n+1)}

Highly_composite_number.html

  1. 2 2
  2. 2 2
  3. 2 2 2^{2}
  4. 2 2 2^{2}
  5. 2 3 2\cdot 3
  6. 6 6
  7. 2 2 3 2^{2}\cdot 3
  8. 2 6 2\cdot 6
  9. 2 3 3 2^{3}\cdot 3
  10. 2 2 6 2^{2}\cdot 6
  11. 2 2 3 2 2^{2}\cdot 3^{2}
  12. 6 2 6^{2}
  13. 2 4 3 2^{4}\cdot 3
  14. 2 3 6 2^{3}\cdot 6
  15. 2 2 3 5 2^{2}\cdot 3\cdot 5
  16. 2 30 2\cdot 30
  17. 2 3 3 5 2^{3}\cdot 3\cdot 5
  18. 2 2 30 2^{2}\cdot 30
  19. 2 2 3 2 5 2^{2}\cdot 3^{2}\cdot 5
  20. 6 30 6\cdot 30
  21. 2 4 3 5 2^{4}\cdot 3\cdot 5
  22. 2 3 30 2^{3}\cdot 30
  23. 2 3 3 2 5 2^{3}\cdot 3^{2}\cdot 5
  24. 2 6 30 2\cdot 6\cdot 30
  25. 2 4 3 2 5 2^{4}\cdot 3^{2}\cdot 5
  26. 2 2 6 30 2^{2}\cdot 6\cdot 30
  27. 2 3 3 5 7 2^{3}\cdot 3\cdot 5\cdot 7
  28. 2 2 210 2^{2}\cdot 210
  29. 2 2 3 2 5 7 2^{2}\cdot 3^{2}\cdot 5\cdot 7
  30. 6 210 6\cdot 210
  31. 2 4 3 5 7 2^{4}\cdot 3\cdot 5\cdot 7
  32. 2 3 210 2^{3}\cdot 210
  33. 2 3 3 2 5 7 2^{3}\cdot 3^{2}\cdot 5\cdot 7
  34. 2 6 210 2\cdot 6\cdot 210
  35. 2 4 3 2 5 7 2^{4}\cdot 3^{2}\cdot 5\cdot 7
  36. 2 2 6 210 2^{2}\cdot 6\cdot 210
  37. 2 3 3 3 5 7 2^{3}\cdot 3^{3}\cdot 5\cdot 7
  38. 6 2 210 6^{2}\cdot 210
  39. 2 5 3 2 5 7 2^{5}\cdot 3^{2}\cdot 5\cdot 7
  40. 2 3 6 210 2^{3}\cdot 6\cdot 210
  41. 2 4 3 3 5 7 2^{4}\cdot 3^{3}\cdot 5\cdot 7
  42. 2 6 2 210 2\cdot 6^{2}\cdot 210
  43. 2 6 3 2 5 7 2^{6}\cdot 3^{2}\cdot 5\cdot 7
  44. 2 4 6 210 2^{4}\cdot 6\cdot 210
  45. 2 4 3 2 5 2 7 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7
  46. 2 2 30 210 2^{2}\cdot 30\cdot 210
  47. 2 3 3 2 5 7 11 2^{3}\cdot 3^{2}\cdot 5\cdot 7\cdot 11
  48. 2 6 2310 2\cdot 6\cdot 2310
  49. 2 4 3 4 5 7 2^{4}\cdot 3^{4}\cdot 5\cdot 7
  50. 6 3 210 6^{3}\cdot 210
  51. 2 5 3 2 5 2 7 2^{5}\cdot 3^{2}\cdot 5^{2}\cdot 7
  52. 2 3 30 210 2^{3}\cdot 30\cdot 210
  53. 2 4 3 2 5 7 11 2^{4}\cdot 3^{2}\cdot 5\cdot 7\cdot 11
  54. 2 2 6 2310 2^{2}\cdot 6\cdot 2310
  55. 2 3 3 3 5 7 11 2^{3}\cdot 3^{3}\cdot 5\cdot 7\cdot 11
  56. 6 2 2310 6^{2}\cdot 2310
  57. 2 5 3 2 5 7 11 2^{5}\cdot 3^{2}\cdot 5\cdot 7\cdot 11
  58. 2 3 6 2310 2^{3}\cdot 6\cdot 2310
  59. 2 4 3 3 5 7 11 2^{4}\cdot 3^{3}\cdot 5\cdot 7\cdot 11
  60. 2 6 2 2310 2\cdot 6^{2}\cdot 2310
  61. 2 6 3 2 5 7 11 2^{6}\cdot 3^{2}\cdot 5\cdot 7\cdot 11
  62. 2 4 6 2310 2^{4}\cdot 6\cdot 2310
  63. 2 4 3 2 5 2 7 11 2^{4}\cdot 3^{2}\cdot 5^{2}\cdot 7\cdot 11
  64. 2 2 30 2310 2^{2}\cdot 30\cdot 2310
  65. 2 5 3 3 5 7 11 2^{5}\cdot 3^{3}\cdot 5\cdot 7\cdot 11
  66. 2 2 6 2 2310 2^{2}\cdot 6^{2}\cdot 2310
  67. 2 4 3 4 5 7 11 2^{4}\cdot 3^{4}\cdot 5\cdot 7\cdot 11
  68. 6 3 2310 6^{3}\cdot 2310
  69. 2 5 3 2 5 2 7 11 2^{5}\cdot 3^{2}\cdot 5^{2}\cdot 7\cdot 11
  70. 2 3 30 2310 2^{3}\cdot 30\cdot 2310
  71. 2 6 3 3 5 7 11 2^{6}\cdot 3^{3}\cdot 5\cdot 7\cdot 11
  72. 2 3 6 2 2310 2^{3}\cdot 6^{2}\cdot 2310
  73. 2 4 3 2 5 7 11 13 2^{4}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13
  74. 2 2 6 30030 2^{2}\cdot 6\cdot 30030
  75. a 0 14 a 1 9 a 2 6 a 3 4 a 4 4 a 5 3 a 6 3 a 7 3 a 8 2 a 9 2 a 10 2 a 11 2 a 12 2 a 13 2 a 14 2 a 15 2 a 16 2 a 17 2 a 18 2 a 19 a 20 a 21 a 229 , a_{0}^{14}a_{1}^{9}a_{2}^{6}a_{3}^{4}a_{4}^{4}a_{5}^{3}a_{6}^{3}a_{7}^{3}a_{8}% ^{2}a_{9}^{2}a_{10}^{2}a_{11}^{2}a_{12}^{2}a_{13}^{2}a_{14}^{2}a_{15}^{2}a_{16% }^{2}a_{17}^{2}a_{18}^{2}a_{19}a_{20}a_{21}\cdots a_{229},
  76. a n a_{n}
  77. 2 14 × 3 9 × 5 6 × × 1451 2^{14}\times 3^{9}\times 5^{6}\times\cdots\times 1451
  78. n = p 1 c 1 × p 2 c 2 × × p k c k ( 1 ) n=p_{1}^{c_{1}}\times p_{2}^{c_{2}}\times\cdots\times p_{k}^{c_{k}}\qquad(1)
  79. p 1 < p 2 < < p k p_{1}<p_{2}<\cdots<p_{k}
  80. c i c_{i}
  81. p 1 d 1 × p 2 d 2 × × p k d k , 0 d i c i , 0 < i k p_{1}^{d_{1}}\times p_{2}^{d_{2}}\times\cdots\times p_{k}^{d_{k}},0\leq d_{i}% \leq c_{i},0<i\leq k
  82. d ( n ) = ( c 1 + 1 ) × ( c 2 + 1 ) × × ( c k + 1 ) . ( 2 ) d(n)=(c_{1}+1)\times(c_{2}+1)\times\cdots\times(c_{k}+1).\qquad(2)
  83. c 1 c 2 c k c_{1}\geq c_{2}\geq\cdots\geq c_{k}
  84. ln ( x ) a Q ( x ) ln ( x ) b . \ln(x)^{a}\leq Q(x)\leq\ln(x)^{b}\,.
  85. 1.13862 < lim inf log Q ( x ) log log x 1.44 1.13862<\liminf\frac{\log Q(x)}{\log\log x}\leq 1.44
  86. lim sup log Q ( x ) log log x 1.71 . \limsup\frac{\log Q(x)}{\log\log x}\leq 1.71\ .
  87. ( log x ) c log Q L ( x ) ( log x ) d (\log x)^{c}\leq\log Q_{L}(x)\leq(\log x)^{d}
  88. 0.2 c d 0.5 0.2\leq c\leq d\leq 0.5

Hilary_Putnam.html

  1. n > 2 n>2
  2. x n + y n = z n x^{n}+y^{n}=z^{n}
  3. n > 2 n>2
  4. β 0 \beta_{0}
  5. β \beta
  6. L β L_{\beta}
  7. β 0 \beta_{0}

Hilbert's_fifth_problem.html

  1. e e
  2. G G
  3. U U
  4. e e
  5. V V
  6. U U
  7. F : V × V U F:V×V→U
  8. F F
  9. e e
  10. e e
  11. F F
  12. C < s u p > k C<sup> k

Hilbert's_Nullstellensatz.html

  1. I ( V ( J ) ) = J \hbox{I}(\hbox{V}(J))=\sqrt{J}
  2. J \sqrt{J}
  3. P = ( a 1 , , a n ) K n P=(a_{1},\dots,a_{n})\in K^{n}
  4. I ( P ) = ( X 1 - a 1 , , X n - a n ) I(P)=(X_{1}-a_{1},\dots,X_{n}-a_{n})
  5. I = ( a 1 , , a n ) V ( I ) ( X 1 - a 1 , , X n - a n ) . \sqrt{I}=\bigcap_{(a_{1},\dots,a_{n})\in V(I)}(X_{1}-a_{1},\dots,X_{n}-a_{n}).
  6. K [ X 1 , , X n ] K[X_{1},\dots,X_{n}]
  7. K K
  8. ( X 1 - a 1 , , X n - a n ) (X_{1}-a_{1},\dots,X_{n}-a_{n})
  9. a 1 , , a n K a_{1},\dots,a_{n}\in K
  10. I ( W ) I(W)
  11. A = k [ t 1 , , t n ] A=k[t_{1},...,t_{n}]
  12. k n k^{n}
  13. I I ( V ) \sqrt{I}\subseteq I(V)
  14. f I f\not\in\sqrt{I}
  15. f 𝔭 f\not\in\mathfrak{p}
  16. 𝔭 I \mathfrak{p}\supseteq I
  17. R = ( A / 𝔭 ) [ f - 1 ] R=(A/\mathfrak{p})[f^{-1}]
  18. 𝔪 \mathfrak{m}
  19. R R
  20. R / 𝔪 R/\mathfrak{m}
  21. x i x_{i}
  22. t i t_{i}
  23. A k A\to k
  24. x = ( x 1 , , x n ) V x=(x_{1},...,x_{n})\in V
  25. f ( x ) 0 f(x)\neq 0
  26. R R
  27. S S
  28. S S
  29. 𝔫 S \mathfrak{n}\subset S
  30. 𝔪 := 𝔫 R \mathfrak{m}:=\mathfrak{n}\cap R
  31. S / 𝔫 S/\mathfrak{n}
  32. R / 𝔪 R/\mathfrak{m}
  33. f : Y X f:Y\to X
  34. X X^{\prime}
  35. X Y X^{\prime}\to Y
  36. g g
  37. g = 1 g=1
  38. r r
  39. 0
  40. n 2 n≥2
  41. deg ( f i g i ) max ( d s , 3 ) j = 1 min ( n , s ) - 1 max ( d j , 3 ) . \deg(f_{i}g_{i})\leq\max(d_{s},3)\prod_{j=1}^{\min(n,s)-1}\max(d_{j},3).
  42. d d
  43. max ( 3 , d ) min ( n , s ) . \max(3,d)^{\min(n,s)}.
  44. deg ( f i g i ) 2 d s j = 1 min ( n , s ) - 1 d j . \deg(f_{i}g_{i})\leq 2d_{s}\prod_{j=1}^{\min(n,s)-1}d_{j}.
  45. R = k [ t 0 , , t n ] . R=k[t_{0},...,t_{n}].
  46. R + = d 1 R d R_{+}=\bigoplus_{d\geq 1}R_{d}
  47. S n S\subseteq\mathbb{P}^{n}
  48. I n ( S ) = { f R + | f = 0 on S } , V n ( I ) = { x n | f ( x ) = 0 for all f I } . \begin{aligned}\displaystyle\operatorname{I}_{\mathbb{P}^{n}}(S)&\displaystyle% =\{f\in R_{+}|f=0\,\text{ on }S\},\\ \displaystyle\operatorname{V}_{\mathbb{P}^{n}}(I)&\displaystyle=\{x\in\mathbb{% P}^{n}|f(x)=0\,\text{ for all }f\in I\}.\end{aligned}
  49. f = 0 on S f=0\,\text{ on }S
  50. ( a 0 : : a n ) (a_{0}:\cdots:a_{n})
  51. f ( a 0 , , a n ) = 0 f(a_{0},\ldots,a_{n})=0
  52. I n ( S ) \operatorname{I}_{\mathbb{P}^{n}}(S)
  53. I n ( S ) \operatorname{I}_{\mathbb{P}^{n}}(S)
  54. I R + I\subseteq R_{+}
  55. I = I n ( V n ( I ) ) , \sqrt{I}=\operatorname{I}_{\mathbb{P}^{n}}(\operatorname{V}_{\mathbb{P}^{n}}(I% )),
  56. n \mathbb{P}^{n}
  57. V n ( I ) . \operatorname{V}_{\mathbb{P}^{n}}(I).
  58. I n \operatorname{I}_{\mathbb{P}^{n}}
  59. V n . \operatorname{V}_{\mathbb{P}^{n}}.

Hilbert's_tenth_problem.html

  1. p ( x 1 , x 2 , , x n ) = 0. p(x_{1},x_{2},\ldots,x_{n})=0.
  2. n n
  3. p ( x 1 , x 2 , , x n ) = 0 , p(x_{1},x_{2},\ldots,x_{n})=0,\,
  4. p ( ± x 1 , ± x 2 , , ± x n ) = 0. p(\pm x_{1},\pm x_{2},\ldots,\pm x_{n})=0.\,
  5. p 1 = 0 , , p k = 0 p_{1}=0,\ldots,p_{k}=0\,
  6. p 1 2 + + p k 2 = 0. p_{1}^{2}+\cdots+p_{k}^{2}=0.\,
  7. p ( a , x 1 , , x n ) p(a,x_{1},\ldots,x_{n})
  8. a a
  9. p ( a , x 1 , , x n ) = 0 p(a,x_{1},\ldots,x_{n})=0
  10. { a y k y x 1 , , x n [ p ( a , k , y , x 1 , , x n ) = 0 ] } \{\,a\mid\exists y\,\forall k\!\leq y\,\exists x_{1},\ldots,x_{n}[p(a,k,y,x_{1% },\ldots,x_{n})=0]\,\}
  11. p p
  12. ( a , b , c ) (a,b,c)
  13. a = b c a=b^{c}
  14. D D
  15. ( a , b ) (a,b)
  16. ( a , b ) D b < a a (a,b)\in D\Rightarrow b<a^{a}
  17. k > 0 k>0
  18. ( a , b ) D \exists(a,b)\in D
  19. b > a k . b>a^{k}.
  20. p ( a , n , x 1 , , x k ) p(a,n,x_{1},\ldots,x_{k})
  21. S S
  22. n 0 n_{0}
  23. S = { a x 1 , , x k [ p ( a , n 0 , x 1 , , x k ) = 0 ] } . S=\{\,a\mid\exists x_{1},\ldots,x_{k}[p(a,n_{0},x_{1},\ldots,x_{k})=0]\,\}.
  24. S S
  25. q ( x 0 , x 1 , , x n ) q(x_{0},x_{1},\ldots,x_{n})\,
  26. S S
  27. q q
  28. x 0 , x 1 , , x n x_{0},x_{1},\ldots,x_{n}\,
  29. p ( a , y 1 , , y n ) = 0 p(a,y_{1},\ldots,y_{n})=0\,
  30. S S
  31. q ( x 0 , x 1 , , x n ) = x 0 [ 1 - p ( x 0 , x 1 , , x n ) 2 ] . q(x_{0},x_{1},\ldots,x_{n})=x_{0}[1-p(x_{0},x_{1},\ldots,x_{n})^{2}].\,
  32. Π 1 0 \Pi^{0}_{1}
  33. Π 1 0 \Pi^{0}_{1}
  34. Π 1 0 \Pi^{0}_{1}
  35. Π 1 0 \Pi^{0}_{1}
  36. p ( a , x 1 , , x k ) = 0 p(a,x_{1},\ldots,x_{k})=0\,
  37. A A
  38. n n
  39. p ( n , x 1 , , x k ) = 0 p(n,x_{1},\ldots,x_{k})=0\,
  40. n 0 n_{0}
  41. A A
  42. p ( n 0 , x 1 , , x k ) = 0 p(n_{0},x_{1},\ldots,x_{k})=0\,
  43. n 0 n_{0}
  44. n n
  45. A A
  46. n n
  47. k k
  48. p ( n , x 1 , , x k ) = 0. p(n,x_{1},\ldots,x_{k})=0.
  49. A A
  50. n n
  51. ¬ x 1 , , x k [ p ( n , x 1 , , x k ) = 0 ] \neg\exists x_{1},\ldots,x_{k}[p(n,x_{1},\ldots,x_{k})=0]\,
  52. A = { 0 , 1 , 2 , 3 , , 0 } A=\{0,1,2,3,\ldots,\aleph_{0}\}
  53. C C
  54. A A
  55. C C
  56. p ( x 1 , , x k ) = 0 p(x_{1},\ldots,x_{k})=0\,
  57. p p
  58. d d
  59. ( z + 1 ) d p ( x 1 z + 1 , , x k z + 1 ) = 0 (z+1)^{d}\;p\left(\frac{x_{1}}{z+1},\ldots,\frac{x_{k}}{z+1}\right)=0
  60. 0
  61. Π 1 0 \Pi^{0}_{1}
  62. n n
  63. 2 n + 4 2n+4

Hilbert's_third_problem.html

  1. base area × height 3 , \frac{\,\text{base area}\times\,\text{height}}{3},
  2. D ( P ) = e ( e ) ( θ ( e ) + π ) \operatorname{D}(P)=\sum_{e}\ell(e)\otimes(\theta(e)+\mathbb{Q}\pi)

Hilbert_cube.html

  1. { 1 / n } n \{1/n\}_{n\in\mathbb{N}}
  2. { a n } \{a_{n}\}
  3. 0 a n 1 / n 0\leq a_{n}\leq 1/n
  4. h : a n n a n h:a_{n}\rightarrow n\cdot a_{n}

Hilbert_matrix.html

  1. H i j = 1 i + j - 1 . H_{ij}=\frac{1}{i+j-1}.
  2. H = [ 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7 1 4 1 5 1 6 1 7 1 8 1 5 1 6 1 7 1 8 1 9 ] . H=\begin{bmatrix}1&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\frac{1}{5}\\ \frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\frac{1}{6}\\ \frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\frac{1}{6}&\frac{1}{7}\\ \frac{1}{4}&\frac{1}{5}&\frac{1}{6}&\frac{1}{7}&\frac{1}{8}\\ \frac{1}{5}&\frac{1}{6}&\frac{1}{7}&\frac{1}{8}&\frac{1}{9}\end{bmatrix}.
  3. H i j = 0 1 x i + j - 2 d x , H_{ij}=\int_{0}^{1}x^{i+j-2}\,dx,
  4. a b P ( x ) 2 d x \int_{a}^{b}P(x)^{2}dx
  5. det ( H ) = c n 4 c 2 n \det(H)={{c_{n}^{\;4}}\over{c_{2n}}}
  6. c n = i = 1 n - 1 i n - i = i = 1 n - 1 i ! . c_{n}=\prod_{i=1}^{n-1}i^{n-i}=\prod_{i=1}^{n-1}i!.\,
  7. 1 det ( H ) = c 2 n c n 4 = n ! i = 1 2 n - 1 ( i [ i / 2 ] ) . {1\over\det(H)}={{c_{2n}}\over{c_{n}^{\;4}}}=n!\cdot\prod_{i=1}^{2n-1}{i% \choose[i/2]}.
  8. det ( H ) = a n n - 1 / 4 ( 2 π ) n 4 - n 2 \det(H)=a_{n}\,n^{-1/4}(2\pi)^{n}\,4^{-n^{2}}
  9. e 1 / 4 2 1 / 12 A - 3 0.6450 e^{1/4}2^{1/12}A^{-3}\approx 0.6450
  10. n n\rightarrow\infty
  11. ( H - 1 ) i j = ( - 1 ) i + j ( i + j - 1 ) ( n + i - 1 n - j ) ( n + j - 1 n - i ) ( i + j - 2 i - 1 ) 2 (H^{-1})_{ij}=(-1)^{i+j}(i+j-1){n+i-1\choose n-j}{n+j-1\choose n-i}{i+j-2% \choose i-1}^{2}
  12. O ( ( 1 + 2 ) 4 n / n ) O((1+\sqrt{2})^{4n}/\sqrt{n})

History_of_geodesy.html

  1. h = d tan θ 1 tan θ 2 tan θ 2 - tan θ 1 h=\frac{d\tan\theta_{1}\tan\theta_{2}}{\tan\theta_{2}-\tan\theta_{1}}
  2. R = h cos θ 1 - cos θ R=\frac{h\cos\theta}{1-\cos\theta}

History_of_large_numbers.html

  1. 10 7 × 2 122 10^{7\times 2^{122}}
  2. 10 8 × 10 16 10^{8\times 10^{16}}
  3. M β M^{\!\!\!\!\!{}^{\beta}}

Hodge_conjecture.html

  1. H k ( X , 𝐂 ) = p + q = k H p , q ( X ) , H^{k}(X,\mathbf{C})=\bigoplus_{p+q=k}H^{p,q}(X),\,
  2. d z i 1 d z i p d z ¯ j 1 d z ¯ j q . dz_{i_{1}}\wedge\cdots\wedge dz_{i_{p}}\wedge d\bar{z}_{j_{1}}\wedge\cdots% \wedge d\bar{z}_{j_{q}}.
  3. : H p , q ( X ) × H p , q ( X ) H p + p , q + q ( X ) . \cup:H^{p,q}(X)\times H^{p^{\prime},q^{\prime}}(X)\rightarrow H^{p+p^{\prime},% q+q^{\prime}}(X).\,
  4. Z i * α . \int_{Z}i^{*}\alpha.\!\,
  5. Hdg k ( X ) = H 2 k ( X , 𝐐 ) H k , k ( X ) . \operatorname{Hdg}^{k}(X)=H^{2k}(X,\mathbf{Q})\cap H^{k,k}(X).
  6. i c i Z i . \sum_{i}c_{i}Z_{i}.\,
  7. i c i [ Z i ] . \sum_{i}c_{i}[Z_{i}].\,
  8. Hdg * ( X ) = k Hdg k ( X ) \operatorname{Hdg}^{*}(X)=\sum_{k}\operatorname{Hdg}^{k}(X)\,
  9. X 𝐏 2 m + 1 X\subset\mathbf{P}^{2m+1}
  10. d 5 d\leq 5
  11. N c H k ( X , 𝐙 ) H k ( X , 𝐙 ) ( H k - c , c ( X ) H c , k - c ( X ) ) . N^{c}H^{k}(X,\mathbf{Z})\subseteq H^{k}(X,\mathbf{Z})\cap(H^{k-c,c}(X)\oplus% \cdots\oplus H^{c,k-c}(X)).
  12. N c H k ( X , 𝐙 ) = H k ( X , 𝐙 ) ( H k - c , c ( X ) H c , k - c ( X ) ) . N^{c}H^{k}(X,\mathbf{Z})=H^{k}(X,\mathbf{Z})\cap(H^{k-c,c}(X)\oplus\cdots% \oplus H^{c,k-c}(X)).
  13. H k - c , c ( X ) H c , k - c ( X ) . H^{k-c,c}(X)\oplus\cdots\oplus H^{c,k-c}(X).

Hodge_dual.html

  1. n n
  2. k k
  3. 0 k n 0≤k≤n
  4. k k
  5. ( n k ) (n−k)
  6. k k
  7. k k
  8. k k
  9. ( n k ) {n\choose k}
  10. ( n n - k ) , {n\choose n-k},
  11. k k
  12. V V
  13. 1 , 3 , 3 , 1 1,3,3,1
  14. V V
  15. V V
  16. k k
  17. k k
  18. k k
  19. V V
  20. V V
  21. k k
  22. ( n k ) (n−k)
  23. n = d i m V n=dimV
  24. 0 k n 0≤k≤n
  25. k k
  26. α , β α,β
  27. α ( β ) = α , β ω \alpha\wedge(\star\beta)=\langle\alpha,\beta\rangle\omega
  28. , \langle\cdot,\cdot\rangle
  29. k k
  30. ω ω
  31. n n
  32. , \langle\cdot,\cdot\rangle
  33. k k
  34. V V
  35. α , β = det ( α i , β j ) \langle\alpha,\beta\rangle=\det\left(\left\langle\alpha_{i},\beta_{j}\right% \rangle\right)
  36. k k
  37. α = α 1 α k \alpha=\alpha_{1}\wedge\dots\wedge\alpha_{k}
  38. β = β 1 β k \beta=\beta_{1}\wedge\dots\wedge\beta_{k}
  39. n n
  40. ω ω
  41. ω ω
  42. V V
  43. W W
  44. , W \langle\cdot,\cdot\rangle_{W}
  45. f W * f\in W^{*}
  46. v v
  47. W W
  48. f ( w ) = w , v W f(w)=\langle w,v\rangle_{W}
  49. w w
  50. W W
  51. W * W W^{*}\to W
  52. f v f\mapsto v
  53. V V
  54. n n
  55. { e 1 , , e n } \{e_{1},\ldots,e_{n}\}
  56. 0 k n 0≤k≤n
  57. k V \bigwedge^{k}V
  58. n - k V \bigwedge^{n-k}V
  59. λ k V , θ n - k V , \lambda\in\bigwedge^{k}V,\quad\theta\in\bigwedge^{n-k}V,
  60. λ θ n V . \lambda\wedge\theta\in\bigwedge^{n}V.
  61. n n
  62. e 1 e n e_{1}\wedge\ldots\wedge e_{n}
  63. λ θ \lambda\wedge\theta
  64. e 1 e n e_{1}\wedge\ldots\wedge e_{n}
  65. λ k V \lambda\in\bigwedge^{k}V
  66. θ n - k V \theta\in\bigwedge^{n-k}V
  67. λ k V \lambda\in\bigwedge^{k}V
  68. f λ ( n - k V ) * f_{\lambda}\in\left(\bigwedge^{n-k}V\right)^{\!*}
  69. θ n - k V : λ θ = f λ ( θ ) ( e 1 e n ) . \forall\theta\in\bigwedge^{n-k}V:\qquad\lambda\wedge\theta=f_{\lambda}(\theta)% \,(e_{1}\wedge\ldots\wedge e_{n}).
  70. f λ ( θ ) f_{\lambda}(\theta)
  71. , \langle\cdot,\cdot\rangle
  72. ( n k ) (n−k)
  73. ( n k ) (n−k)
  74. λ n - k V , \star\lambda\in\bigwedge^{n-k}V,
  75. θ n - k V : f λ ( θ ) = θ , λ . \forall\theta\in\bigwedge^{n-k}V:\qquad f_{\lambda}(\theta)=\langle\theta,% \star\lambda\rangle.
  76. ( n k ) (n−k)
  77. λ ★λ
  78. λ λ
  79. f λ f_{\lambda}
  80. ( n - k V ) * n - k V . \left(\bigwedge^{n-k}V\right)^{\!*}\cong\bigwedge^{n-k}V.
  81. : k V n - k V \star:\bigwedge^{k}V\to\bigwedge^{n-k}V
  82. ( e 1 , , e n ) (e_{1},\cdots,e_{n})
  83. ω = e 1 e n \omega=e_{1}\wedge\cdots\wedge e_{n}
  84. ( e i 1 e i 2 e i k ) = e i k + 1 e i k + 2 e i n , \star(e_{i_{1}}\wedge e_{i_{2}}\wedge\cdots\wedge e_{i_{k}})=e_{i_{k+1}}\wedge e% _{i_{k+2}}\wedge\cdots\wedge e_{i_{n}},
  85. ( i 1 , i 2 , , i n ) (i_{1},i_{2},\cdots,i_{n})
  86. n ! 2 n!\over 2
  87. ( n k ) n\choose k
  88. ( e 1 e 2 e k ) = e k + 1 e k + 2 e n . \star(e_{1}\wedge e_{2}\wedge\cdots\wedge e_{k})=e_{k+1}\wedge e_{k+2}\wedge% \cdots\wedge e_{n}.
  89. k k
  90. n n
  91. g g
  92. g g
  93. ( η ) i 1 , i 2 , , i n - k = 1 ( n - k ) ! η j 1 , , j k | det g | ϵ j 1 , , j k , i 1 , , i n - k (\star\eta)_{i_{1},i_{2},\ldots,i_{n-k}}=\frac{1}{(n-k)!}\eta^{j_{1},\ldots,j_% {k}}\,\sqrt{|\det g|}\,\epsilon_{j_{1},\ldots,j_{k},i_{1},\ldots,i_{n-k}}
  94. η η
  95. k k
  96. n = 3 n=3
  97. d x = d y d z \star\mathrm{d}x=\mathrm{d}y\wedge\mathrm{d}z
  98. d y = d z d x \star\mathrm{d}y=\mathrm{d}z\wedge\mathrm{d}x
  99. d z = d x d y \star\mathrm{d}z=\mathrm{d}x\wedge\mathrm{d}y
  100. 𝐀 \mathbf{A}
  101. 𝐀 = 𝐚 𝐚 = 𝐀 \mathbf{A}=\star\mathbf{a}\qquad\mathbf{a}=\star\mathbf{A}
  102. 𝐀 = 𝐚 i , 𝐚 = - 𝐀 i . \mathbf{A}=\mathbf{a}i\,,\quad\mathbf{a}=-\mathbf{A}i.
  103. i i
  104. 𝐚 i \displaystyle\mathbf{a}i
  105. 𝐀 i = ( A 1 𝐞 𝟐 𝐞 𝟑 + A 2 𝐞 𝟑 𝐞 𝟏 + A 3 𝐞 𝟏 𝐞 𝟐 ) 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 = A 1 𝐞 𝟏 ( 𝐞 𝟐 𝐞 𝟑 ) 2 + A 2 𝐞 𝟐 ( 𝐞 𝟑 𝐞 𝟏 ) 2 + A 3 𝐞 𝟑 ( 𝐞 𝟏 𝐞 𝟐 ) 2 = - ( A 1 𝐞 𝟏 + A 2 𝐞 𝟐 + A 3 𝐞 𝟑 ) = - ( 𝐀 ) \begin{aligned}\displaystyle\mathbf{A}i&\displaystyle=\left(A_{1}\mathbf{e_{2}% e_{3}}+A_{2}\mathbf{e_{3}e_{1}}+A_{3}\mathbf{e_{1}e_{2}}\right)\mathbf{e_{1}e_% {2}e_{3}}\\ &\displaystyle=A_{1}\mathbf{e_{1}}(\mathbf{e_{2}e_{3}})^{2}+A_{2}\mathbf{e_{2}% }(\mathbf{e_{3}e_{1}})^{2}+A_{3}\mathbf{e_{3}}(\mathbf{e_{1}e_{2}})^{2}\\ &\displaystyle=-\left(A_{1}\mathbf{e_{1}}+A_{2}\mathbf{e_{2}}+A_{3}\mathbf{e_{% 3}}\right)\\ &\displaystyle=-(\star\mathbf{A})\end{aligned}
  106. ( 𝐞 𝟏 𝐞 𝟐 ) 2 = 𝐞 𝟏 𝐞 𝟐 𝐞 𝟏 𝐞 𝟐 = - 𝐞 𝟏 𝐞 𝟐 𝐞 𝟐 𝐞 𝟏 = - 1 (\mathbf{e_{1}e_{2}})^{2}=\mathbf{e_{1}e_{2}e_{1}e_{2}}=-\mathbf{e_{1}e_{2}e_{% 2}e_{1}}=-1
  107. i 2 = ( 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 ) 2 = 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 = 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 𝐞 𝟑 𝐞 𝟏 𝐞 𝟐 = 𝐞 𝟏 𝐞 𝟐 𝐞 𝟏 𝐞 𝟐 = - 1. \mathit{i}^{2}=(\mathbf{e_{1}e_{2}e_{3}})^{2}=\mathbf{e_{1}e_{2}e_{3}e_{1}e_{2% }e_{3}}=\mathbf{e_{1}e_{2}e_{3}e_{3}e_{1}e_{2}}=\mathbf{e_{1}e_{2}e_{1}e_{2}}=% -1.
  108. i i
  109. 𝐚 = 𝐮 × 𝐯 \mathbf{a}=\mathbf{u}×\mathbf{v}
  110. 𝐀 = 𝐮 𝐯 \mathbf{A}=\mathbf{u}∧\mathbf{v}
  111. 𝐮 \mathbf{u}
  112. 𝐯 \mathbf{v}
  113. 𝐚 = 𝐮 × 𝐯 = | 𝐞 1 𝐞 2 𝐞 3 u 1 u 2 u 3 v 1 v 2 v 3 | , 𝐀 = 𝐮 𝐯 = | 𝐞 23 𝐞 31 𝐞 12 u 1 u 2 u 3 v 1 v 2 v 3 | , \mathbf{a}=\mathbf{u}\times\mathbf{v}=\begin{vmatrix}\mathbf{e}_{1}&\mathbf{e}% _{2}&\mathbf{e}_{3}\\ u_{1}&u_{2}&u_{3}\\ v_{1}&v_{2}&v_{3}\end{vmatrix}\,,\quad\mathbf{A}=\mathbf{u}\wedge\mathbf{v}=% \begin{vmatrix}\mathbf{e}_{23}&\mathbf{e}_{31}&\mathbf{e}_{12}\\ u_{1}&u_{2}&u_{3}\\ v_{1}&v_{2}&v_{3}\end{vmatrix},
  114. ( 𝐮 𝐯 ) = 𝐮 × 𝐯 , ( 𝐮 × 𝐯 ) = 𝐮 𝐯 , \star(\mathbf{u}\wedge\mathbf{v})=\mathbf{u\times v}\,,\quad\star(\mathbf{u}% \times\mathbf{v})=\mathbf{u}\wedge\mathbf{v},
  115. 𝐞 = 𝐞 i = 𝐞 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 = 𝐞 m 𝐞 n , \star\mathbf{e}_{\ell}=\mathbf{e}_{\ell}\mathit{i}=\mathbf{e}_{\ell}\mathbf{e_% {1}e_{2}e_{3}}=\mathbf{e}_{m}\mathbf{e}_{n}\,,
  116. , m , n ℓ,m,n
  117. ( 𝐞 𝐞 m ) = - ( 𝐞 𝐞 m ) i = - ( 𝐞 𝐞 m ) 𝐞 𝟏 𝐞 𝟐 𝐞 𝟑 = 𝐞 n \star(\mathbf{e}_{\ell}\mathbf{e}_{m})=-(\mathbf{e}_{\ell}\mathbf{e}_{m})% \mathit{i}=-\left(\mathbf{e}_{\ell}\mathbf{e}_{m}\right)\mathbf{e_{1}e_{2}e_{3% }}=\mathbf{e}_{n}
  118. , m , n ℓ,m,n
  119. i i
  120. 𝐮 × 𝐯 = - ( 𝐮 𝐯 ) i , 𝐮 𝐯 = ( 𝐮 × 𝐯 ) i . \mathbf{u\times v}=-(\mathbf{u}\wedge\mathbf{v})i\,,\quad\mathbf{u}\wedge% \mathbf{v}=(\mathbf{u\times v})i\ .
  121. n = 4 n=4
  122. n = 4 n=4
  123. ( + ) (+−−−)
  124. ( t , x , y , z ) (t,x,y,z)
  125. ε 0123 = 1 \varepsilon_{0123}=1
  126. d t = d x d y d z \star\mathrm{d}t=\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z
  127. d x = d t d y d z \star\mathrm{d}x=\mathrm{d}t\wedge\mathrm{d}y\wedge\mathrm{d}z
  128. d y = d t d z d x \star\mathrm{d}y=\mathrm{d}t\wedge\mathrm{d}z\wedge\mathrm{d}x
  129. d z = d t d x d y \star\mathrm{d}z=\mathrm{d}t\wedge\mathrm{d}x\wedge\mathrm{d}y
  130. ( d t d x ) = - d y d z \star(\mathrm{d}t\wedge\mathrm{d}x)=-\mathrm{d}y\wedge\mathrm{d}z
  131. ( d t d y ) = d x d z \star(\mathrm{d}t\wedge\mathrm{d}y)=\mathrm{d}x\wedge\mathrm{d}z
  132. ( d t d z ) = - d x d y \star(\mathrm{d}t\wedge\mathrm{d}z)=-\mathrm{d}x\wedge\mathrm{d}y
  133. ( d x d y ) = d t d z \star(\mathrm{d}x\wedge\mathrm{d}y)=\mathrm{d}t\wedge\mathrm{d}z
  134. ( d x d z ) = - d t d y \star(\mathrm{d}x\wedge\mathrm{d}z)=-\mathrm{d}t\wedge\mathrm{d}y
  135. ( d y d z ) = d t d x \star(\mathrm{d}y\wedge\mathrm{d}z)=\mathrm{d}t\wedge\mathrm{d}x
  136. k k
  137. k k
  138. V V
  139. k k
  140. η η
  141. ζ ζ
  142. ζ η = ζ , η ω \zeta\wedge\star\eta=\langle\zeta,\eta\rangle\;\omega
  143. ω ω
  144. n n
  145. ω ω = ω ω∧★ω=ω
  146. n n
  147. n n
  148. ω = | det [ g i j ] | d x 1 d x n \omega=\sqrt{\left|\det[g_{ij}]\right|}\;\mathrm{d}x^{1}\wedge\cdots\wedge% \mathrm{d}x^{n}
  149. [ g i j ] \left[g_{ij}\right]
  150. Λ k ( V ) \Lambda^{k}(V)
  151. V V
  152. V V
  153. k k
  154. η η
  155. n n
  156. V V
  157. η = ( - 1 ) k ( n - k ) s η \star{\star\eta}=(-1)^{k(n-k)}s\eta
  158. s s
  159. V V
  160. s s
  161. n = 4 n=4
  162. ( + ) (+−−−)
  163. ( + + + ) (−+++)
  164. s = 1 s=−1
  165. s = 1 s=1
  166. { - 1 : Λ k Λ n - k η ( - 1 ) k ( n - k ) s η \begin{cases}\star^{-1}:\Lambda^{k}\to\Lambda^{n-k}\\ \eta\mapsto(-1)^{k(n-k)}s{\star\eta}\end{cases}
  167. n n
  168. k ( n k ) k(n−k)
  169. k k
  170. n n
  171. k ( n k ) k(n−k)
  172. k k
  173. { - 1 = s n is odd - 1 = ( - 1 ) k s n is even \begin{cases}\star^{-1}=s\star&n\,\text{ is odd}\\ \star^{-1}=(-1)^{k}s\star&n\,\text{ is even}\end{cases}
  174. k k
  175. n n
  176. ( n k ) (n−k)
  177. k k
  178. ( η , ζ ) = M η ζ = M η , ζ d Vol (\eta,\zeta)=\int_{M}\eta\wedge\star\zeta=\int_{M}\langle\eta,\zeta\rangle\;% \mathrm{d}\,\text{Vol}
  179. η η
  180. ζ ζ
  181. Λ k ( T * M ) \Lambda^{k}(T^{*}M)
  182. Ω k ( M ) = Γ ( Λ k ( T * M ) ) \Omega^{k}(M)=\Gamma(\Lambda^{k}(T^{*}M))
  183. Ω k ( M ) \Omega^{k}(M)
  184. k k
  185. k k
  186. ( n k ) (n−k)
  187. δ δ
  188. k k
  189. δ = ( - 1 ) n ( k - 1 ) + 1 s d = ( - 1 ) k - 1 d \delta=(-1)^{n(k-1)+1}s\,{\star\mathrm{d}\star}=(-1)^{k}\,{\star^{-1}\mathrm{d% }\star}
  190. d d
  191. s = 1 s=1
  192. d : Ω k ( M ) Ω k + 1 ( M ) \mathrm{d}:\Omega^{k}(M)\to\Omega^{k+1}(M)
  193. δ : Ω k ( M ) Ω k - 1 ( M ) . \delta:\Omega^{k}(M)\to\Omega^{k-1}(M).
  194. ( η , δ ζ ) = ( d η , ζ ) . (\eta,\delta\zeta)=(\mathrm{d}\eta,\zeta).
  195. ζ ζ
  196. ( k + 1 ) (k+1)
  197. η η
  198. k k
  199. M d ( η ζ ) = 0 = M ( d η ζ - η ( - 1 ) k + 1 - 1 d ζ ) = ( d η , ζ ) - ( η , δ ζ ) \int_{M}\mathrm{d}(\eta\wedge\star\zeta)=0=\int_{M}(\mathrm{d}\eta\wedge\star% \zeta-\eta\wedge\star(-1)^{k+1}\,{\star^{-1}\mathrm{d}{\star\zeta}})=(\mathrm{% d}\eta,\zeta)-(\eta,\delta\zeta)
  200. M M
  201. η η
  202. ζ ★ζ
  203. δ 2 = s 2 d d = ( - 1 ) k ( n - k ) s 3 d 2 = 0 \!\delta^{2}=s^{2}{\star\mathrm{d}{\star{\star\mathrm{d}{\star}}}}=(-1)^{k(n-k% )}s^{3}{\star\mathrm{d}^{2}\star}=0
  204. Δ = ( δ + d ) 2 = δ d + d δ \!\Delta=(\delta+\mathrm{d})^{2}=\delta\mathrm{d}+\mathrm{d}\delta
  205. ( Δ ζ , η ) = ( ζ , Δ η ) (\Delta\zeta,\eta)=(\zeta,\Delta\eta)
  206. ( Δ η , η ) 0. (\Delta\eta,\eta)\geq 0.
  207. k k
  208. : H Δ k ( M ) H Δ n - k ( M ) , \star:H^{k}_{\Delta}(M)\to H^{n-k}_{\Delta}(M),
  209. d d
  210. d d
  211. ω = f ( x , y , z ) \omega=f(x,y,z)
  212. d ω = f x d x + f y d y + f z d z . \mathrm{d}\omega=\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{% \partial y}\mathrm{d}y+\frac{\partial f}{\partial z}\mathrm{d}z.
  213. η = A d x + B d y + C d z \eta=A\,\mathrm{d}x+B\,\mathrm{d}y+C\,\mathrm{d}z
  214. d η = ( C y - B z ) d y d z + ( C x - A z ) d x d z + ( B x - A y ) d x d y . \mathrm{d}\eta=\left({\partial C\over\partial y}-{\partial B\over\partial z}% \right)\mathrm{d}y\wedge\mathrm{d}z+\left({\partial C\over\partial x}-{% \partial A\over\partial z}\right)\mathrm{d}x\wedge\mathrm{d}z+\left({\partial B% \over\partial x}-{\partial A\over\partial y}\right)\mathrm{d}x\wedge\mathrm{d}y.
  215. d η = ( C y - B z ) d x - ( C x - A z ) d y + ( B x - A y ) d z . \star\mathrm{d}\eta=\left({\partial C\over\partial y}-{\partial B\over\partial z% }\right)\mathrm{d}x-\left({\partial C\over\partial x}-{\partial A\over\partial z% }\right)\mathrm{d}y+\left({\partial B\over\partial x}-{\partial A\over\partial y% }\right)\mathrm{d}z.
  216. η = A d x + B d y + C d z \eta=A\,\mathrm{d}x+B\,\mathrm{d}y+C\,\mathrm{d}z
  217. η \displaystyle\star\eta
  218. c u r l ( g r a d ( f ) ) = 0 curl(grad(f))=0
  219. d i v ( c u r l ( 𝐅 ) ) = 0 div(curl(\mathbf{F}))=0
  220. Δ f = d i v g r a d f Δf=div gradf
  221. ω = f ( x , y , z ) \omega=f(x,y,z)
  222. Δ ω = d d ω = 2 f x 2 + 2 f y 2 + 2 f z 2 \Delta\omega=\star\mathrm{d}{\star\mathrm{d}\omega}=\frac{\partial^{2}f}{% \partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{% \partial z^{2}}

Hohmann_transfer_orbit.html

  1. a a
  2. E = 1 2 m v 2 - G M m r = - G M m 2 a . E=\begin{matrix}\frac{1}{2}\end{matrix}mv^{2}-\frac{GMm}{r}=\frac{-GMm}{2a}.\,
  3. v 2 = μ ( 2 r - 1 a ) v^{2}=\mu\left(\frac{2}{r}-\frac{1}{a}\right)
  4. v v\,\!
  5. μ = G M \mu=GM\,\!
  6. M + m M+m
  7. M M
  8. v M v v_{M}\ll v
  9. r r\,\!
  10. a a\,\!
  11. Δ v 1 = μ r 1 ( 2 r 2 r 1 + r 2 - 1 ) , \Delta v_{1}=\sqrt{\frac{\mu}{r_{1}}}\left(\sqrt{\frac{2r_{2}}{r_{1}+r_{2}}}-1% \right),
  12. r = r 1 r=r_{1}
  13. r 1 r_{1}
  14. Δ v 2 = μ r 2 ( 1 - 2 r 1 r 1 + r 2 ) , \Delta v_{2}=\sqrt{\frac{\mu}{r_{2}}}\left(1-\sqrt{\frac{2r_{1}}{r_{1}+r_{2}}}% \,\,\right),
  15. r = r 2 r=r_{2}
  16. r 2 r_{2}
  17. r 1 r_{1}
  18. r 2 r_{2}
  19. r 1 r_{1}
  20. r 2 r_{2}
  21. Δ v \Delta v
  22. Δ v t o t a l = Δ v 1 + Δ v 2 . \Delta v_{total}=\Delta v_{1}+\Delta v_{2}.
  23. t H = 1 2 4 π 2 a H 3 μ = π ( r 1 + r 2 ) 3 8 μ t_{H}=\begin{matrix}\frac{1}{2}\end{matrix}\sqrt{\frac{4\pi^{2}a^{3}_{H}}{\mu}% }=\pi\sqrt{\frac{(r_{1}+r_{2})^{3}}{8\mu}}
  24. a H a_{H}\,\!
  25. ω 2 = μ r 2 3 \omega_{2}=\sqrt{\frac{\mu}{r_{2}^{3}}}
  26. α = π - ω 2 t H = π ( 1 - 1 2 2 ( r 1 r 2 + 1 ) 3 ) \alpha=\pi-\omega_{2}t_{H}=\pi\left(1-\frac{1}{2\sqrt{2}}\sqrt{\left(\frac{r_{% 1}}{r_{2}}+1\right)^{3}}\,\right)
  27. r p r_{p}
  28. r a r_{a}
  29. 2 ¯ \overline{2}
  30. 2 ¯ \overline{2}
  31. 5 + 4 7 cos ( 1 3 tan - 1 3 37 ) 5+4\sqrt{7}\cos\left({1\over 3}\tan^{-1}{\sqrt{3}\over 37}\right)