wpmath0000003_14

Riemann–Liouville_integral.html

  1. I α f ( x ) = 1 Γ ( α ) a x f ( t ) ( x - t ) α - 1 d t I^{\alpha}f(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}f(t)(x-t)^{\alpha-1}\,dt
  2. D x - α a f ( x ) = 1 Γ ( α ) a x f ( t ) ( x - t ) α - 1 d t . {}_{a}D_{x}^{-\alpha}f(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}f(t)(x-t)^{% \alpha-1}\,dt.
  3. d d x I α + 1 f ( x ) = I α f ( x ) , I α ( I β f ) = I α + β f , \frac{d}{dx}I^{\alpha+1}f(x)=I^{\alpha}f(x),\quad I^{\alpha}(I^{\beta}f)=I^{% \alpha+\beta}f,
  4. I α : L 1 ( a , b ) L 1 ( a , b ) . I^{\alpha}:L^{1}(a,b)\to L^{1}(a,b).
  5. I α f 1 | b - a | re ( α ) re ( α ) | Γ ( α ) | f 1 . \|I^{\alpha}f\|_{1}\leq\frac{|b-a|^{\operatorname{re}(\alpha)}}{\operatorname{% re}(\alpha)|\Gamma(\alpha)|}\|f\|_{1}.
  6. 1 \|\cdot\|_{1}
  7. I α f p | b - a | re ( α ) / p re ( α ) | Γ ( α ) | f p \|I^{\alpha}f\|_{p}\leq\frac{|b-a|^{\operatorname{re}(\alpha)/p}}{% \operatorname{re}(\alpha)|\Gamma(\alpha)|}\|f\|_{p}
  8. p \|\cdot\|_{p}
  9. lim α 0 α > 0 I α f - f p = 0 \underset{\alpha>0}{\lim_{\alpha\to 0}}\|I^{\alpha}f-f\|_{p}=0
  10. f = - | f ( t ) | e - σ | t | d t \|f\|=\int_{-\infty}^{\infty}|f(t)|e^{-\sigma|t|}\,dt
  11. ( I α f ) ( s ) = s - α F ( s ) (\mathcal{L}I^{\alpha}f)(s)=s^{-\alpha}F(s)
  12. d α d x α f = d e f d α d x α I α - α f \frac{d^{\alpha}}{dx^{\alpha}}f\overset{def}{=}\frac{d^{\lceil\alpha\rceil}}{% dx^{\lceil\alpha\rceil}}I^{\lceil\alpha\rceil-\alpha}f
  13. \lceil\cdot\rceil
  14. D x α f ( x ) = { d α d x α I α - α f ( x ) α > 0 f ( x ) α = 0 I - α f ( x ) α < 0. D^{\alpha}_{x}f(x)=\begin{cases}\frac{d^{\lceil\alpha\rceil}}{dx^{\lceil\alpha% \rceil}}I^{\lceil\alpha\rceil-\alpha}f(x)&\alpha>0\\ f(x)&\alpha=0\\ I^{-\alpha}f(x)&\alpha<0.\end{cases}
  15. x x
  16. D x α f ( y ) = 1 Γ ( 1 - α ) x y f ( y - u ) ( u - x ) - α d u . D_{x}^{\alpha}f(y)=\frac{1}{\Gamma(1-\alpha)}\int_{x}^{y}f^{\prime}(y-u)(u-x)^% {-\alpha}du.
  17. D ~ x α a f ( x ) = I α - α ( d α f d x α ) . {}_{a}\tilde{D}^{\alpha}_{x}f(x)=I^{\lceil\alpha\rceil-\alpha}\left(\frac{d^{% \lceil\alpha\rceil}f}{dx^{\lceil\alpha\rceil}}\right).

Riemann–Stieltjes_integral.html

  1. a b f ( x ) d g ( x ) \int_{a}^{b}f(x)\,dg(x)
  2. P = { a = x 0 < x 1 < < x n = b } P=\{a=x_{0}<x_{1}<\cdots<x_{n}=b\}
  3. S ( P , f , g ) = i = 0 n - 1 f ( c i ) ( g ( x i + 1 ) - g ( x i ) ) S(P,f,g)=\sum_{i=0}^{n-1}f(c_{i})(g(x_{i+1})-g(x_{i}))
  4. | S ( P , f , g ) - A | < ε . |S(P,f,g)-A|<\varepsilon.\,
  5. | S ( P , f , g ) - A | < ε |S(P,f,g)-A|<\varepsilon\,
  6. U ( P , f , g ) = i = 1 n sup x [ x i , x i + 1 ] f ( x ) ( g ( x i + 1 ) - g ( x i ) ) U(P,f,g)=\sum_{i=1}^{n}\sup_{x\in[x_{i},x_{i+1}]}f(x)\,\,(g(x_{i+1})-g(x_{i}))
  7. L ( P , f , g ) = i = 1 n inf x [ x i , x i + 1 ] f ( x ) ( g ( x i + 1 ) - g ( x i ) ) . L(P,f,g)=\sum_{i=1}^{n}\inf_{x\in[x_{i},x_{i+1}]}f(x)\,\,(g(x_{i+1})-g(x_{i})).
  8. U ( P , f , g ) - L ( P , f , g ) < ε . U(P,f,g)-L(P,f,g)<\varepsilon.
  9. lim mesh ( P ) 0 [ U ( P , f , g ) - L ( P , f , g ) ] = 0. \lim_{\operatorname{mesh}(P)\to 0}[U(P,f,g)-L(P,f,g)]=0.
  10. f ( x ) g ( x ) f(x)g^{\prime}(x)
  11. a b f ( x ) g ( x ) d x , \int_{a}^{b}f(x)g^{\prime}(x)\,dx,
  12. a b f ( x ) d g ( x ) = f ( b ) g ( b ) - f ( a ) g ( a ) - a b g ( x ) d f ( x ) . \int_{a}^{b}f(x)\,dg(x)=f(b)g(b)-f(a)g(a)-\int_{a}^{b}g(x)\,df(x).
  13. E ( f ( X ) ) = - f ( x ) g ( x ) d x . E(f(X))=\int_{-\infty}^{\infty}f(x)g^{\prime}(x)\,dx.
  14. E ( f ( X ) ) = - f ( x ) d g ( x ) E(f(X))=\int_{-\infty}^{\infty}f(x)\,dg(x)
  15. E ( X n ) = - x n d g ( x ) . E(X^{n})=\int_{-\infty}^{\infty}x^{n}\,dg(x).
  16. sup i g ( t i ) - g ( t i + 1 ) X < \sup\sum_{i}\|g(t_{i})-g(t_{i+1})\|_{X}<\infty
  17. a = t 0 t 1 t n = b a=t_{0}\leq t_{1}\leq\cdots\leq t_{n}=b

Riesel_number.html

  1. { k 2 n - 1 : n } . \left\{\,k2^{n}-1:n\in\mathbb{N}\,\right\}.
  2. k * 2 n - 1 k*2^{n}-1

Rigged_Hilbert_space.html

  1. x e i x , x\mapsto e^{ix},
  2. - i d d x -i\frac{d}{dx}
  3. Φ H \Phi\subseteq H
  4. ϕ v , ϕ \phi\mapsto\langle v,\phi\rangle
  5. Φ H Φ * . \Phi\subseteq H\subseteq\Phi^{*}.
  6. n \mathbb{R}^{n}
  7. H = L 2 ( n ) , Φ = H s ( n ) , Φ * = H - s ( n ) H=L^{2}(\mathbb{R}^{n}),\ \Phi=H^{s}(\mathbb{R}^{n}),\ \Phi^{*}=H^{-s}(\mathbb% {R}^{n})
  8. s > 0 s>0
  9. i * : H = H * Φ * . i^{*}:H=H^{*}\to\Phi^{*}.
  10. u , v Φ × Φ * = ( u , v ) H \langle u,v\rangle_{\Phi\times\Phi^{*}}=(u,v)_{H}
  11. u Φ H u\in\Phi\subset H
  12. v H = H * Φ * v\in H=H^{*}\subset\Phi^{*}
  13. ( Φ , H , Φ * ) (\Phi,\,\,H,\,\,\Phi^{*})
  14. i * i : Φ H = H * Φ * . i^{*}i:\Phi\subset H=H^{*}\to\Phi^{*}.

Rigid_body.html

  1. b 3 = b 1 × b 2 b_{3}=b_{1}\times b_{2}
  2. s N y m b o l ω B = s N y m b o l ω D + s D y m b o l ω B . {}^{\mathrm{N}}\!symbol{\omega}^{\mathrm{B}}={}^{\mathrm{N}}\!symbol{\omega}^{% \mathrm{D}}+{}^{\mathrm{D}}\!symbol{\omega}^{\mathrm{B}}.
  3. 𝐫 PR = 𝐫 PQ + 𝐫 QR . \mathbf{r}^{\mathrm{PR}}=\mathbf{r}^{\mathrm{PQ}}+\mathbf{r}^{\mathrm{QR}}.
  4. 𝐯 P N = d N d t ( 𝐫 OP ) {}^{\mathrm{N}}\mathbf{v}^{\mathrm{P}}=\frac{{}^{\mathrm{N}}\mathrm{d}}{% \mathrm{d}t}(\mathbf{r}^{\mathrm{OP}})
  5. 𝐚 P N = d N d t ( 𝐯 P N ) . {}^{\mathrm{N}}\mathbf{a}^{\mathrm{P}}=\frac{{}^{\mathrm{N}}\mathrm{d}}{% \mathrm{d}t}({}^{\mathrm{N}}\mathbf{v}^{\mathrm{P}}).
  6. s N y m b o l ω B \scriptstyle{{}^{\mathrm{N}}symbol{\omega}^{\mathrm{B}}}
  7. 𝐯 Q N = 𝐯 P N + s N y m b o l ω B × 𝐫 PQ . {}^{\mathrm{N}}\mathbf{v}^{\mathrm{Q}}={}^{\mathrm{N}}\!\mathbf{v}^{\mathrm{P}% }+{}^{\mathrm{N}}symbol{\omega}^{\mathrm{B}}\times\mathbf{r}^{\mathrm{PQ}}.
  8. 𝐚 Q N = 𝐚 P N + s N y m b o l ω B × ( s N y m b o l ω B × 𝐫 PQ ) + s N y m b o l α B × 𝐫 PQ {}^{\mathrm{N}}\mathbf{a}^{\mathrm{Q}}={}^{\mathrm{N}}\mathbf{a}^{\mathrm{P}}+% {}^{\mathrm{N}}symbol{\omega}^{\mathrm{B}}\times\left({}^{\mathrm{N}}symbol{% \omega}^{\mathrm{B}}\times\mathbf{r}^{\mathrm{PQ}}\right)+{}^{\mathrm{N}}% symbol{\alpha}^{\mathrm{B}}\times\mathbf{r}^{\mathrm{PQ}}
  9. s N y m b o l α B \scriptstyle{{}^{\mathrm{N}}\!symbol{\alpha}^{\mathrm{B}}}
  10. s N y m b o l ω B {}^{\mathrm{N}}symbol{\omega}^{\mathrm{B}}
  11. s N y m b o l α B . {}^{\mathrm{N}}symbol{\alpha}^{\mathrm{B}}.
  12. 𝐯 R N = 𝐯 Q N + 𝐯 R B {}^{\mathrm{N}}\mathbf{v}^{\mathrm{R}}={}^{\mathrm{N}}\mathbf{v}^{\mathrm{Q}}+% {}^{\mathrm{B}}\mathbf{v}^{\mathrm{R}}
  13. 𝐚 R N = 𝐚 Q N + 𝐚 R B + 2 s N y m b o l ω B × 𝐯 R B {}^{\mathrm{N}}\mathbf{a}^{\mathrm{R}}={}^{\mathrm{N}}\mathbf{a}^{\mathrm{Q}}+% {}^{\mathrm{B}}\mathbf{a}^{\mathrm{R}}+2{}^{\mathrm{N}}symbol{\omega}^{\mathrm% {B}}\times{}^{\mathrm{B}}\mathbf{v}^{\mathrm{R}}
  14. s y m b o l ψ ( t , 𝐫 0 ) = 𝐚 ( t , 𝐫 0 ) - s y m b o l ω ( t ) × 𝐯 ( t , 𝐫 0 ) = s y m b o l ψ c ( t ) + s y m b o l α ( t ) × A ( t ) 𝐫 0 symbol\psi(t,\mathbf{r}_{0})=\mathbf{a}(t,\mathbf{r}_{0})-symbol\omega(t)% \times\mathbf{v}(t,\mathbf{r}_{0})=symbol\psi_{c}(t)+symbol\alpha(t)\times A(t% )\mathbf{r}_{0}
  15. 𝐫 0 \mathbf{r}_{0}
  16. A ( t ) A(t)\,
  17. s y m b o l ω ( t ) symbol\omega(t)
  18. 𝐯 ( t , 𝐫 0 ) \mathbf{v}(t,\mathbf{r}_{0})
  19. 𝐚 ( t , 𝐫 0 ) \mathbf{a}(t,\mathbf{r}_{0})
  20. s y m b o l α ( t ) symbol\alpha(t)
  21. s y m b o l ψ ( t , 𝐫 0 ) symbol\psi(t,\mathbf{r}_{0})
  22. s y m b o l ψ c ( t ) symbol\psi_{c}(t)

Rigid_body_dynamics.html

  1. i {}_{i}
  2. 𝐅 = i = 1 N m i 𝐀 i , 𝐓 = i = 1 N ( 𝐫 i - 𝐑 ) × ( m i 𝐀 i ) , \mathbf{F}=\sum_{i=1}^{N}m_{i}\mathbf{A}_{i},\quad\mathbf{T}=\sum_{i=1}^{N}(% \mathbf{r}_{i}-\mathbf{R})\times(m_{i}\mathbf{A}_{i}),
  3. i {}_{i}
  4. i {}_{i}
  5. 𝐀 i = α × ( 𝐫 i - 𝐑 ) + ω × ω × ( 𝐫 i - 𝐑 ) + 𝐀 . \mathbf{A}_{i}=\alpha\times(\mathbf{r}_{i}-\mathbf{R})+\omega\times\omega% \times(\mathbf{r}_{i}-\mathbf{R})+\mathbf{A}.
  6. i {}_{i}
  7. i {}_{i}
  8. i {}_{i}
  9. i {}_{i}
  10. 𝐀 i = α ( Δ r i 𝐭 i ) - ω 2 ( Δ r i 𝐞 i ) + 𝐀 . \mathbf{A}_{i}=\alpha(\Delta r_{i}\mathbf{t}_{i})-\omega^{2}(\Delta r_{i}% \mathbf{e}_{i})+\mathbf{A}.
  11. 𝐅 = α i = 1 N m i ( Δ r i 𝐭 i ) - ω 2 i = 1 N m i ( Δ r i 𝐞 i ) + ( i = 1 N m i ) 𝐀 , \mathbf{F}=\alpha\sum_{i=1}^{N}m_{i}(\Delta r_{i}\mathbf{t}_{i})-\omega^{2}% \sum_{i=1}^{N}m_{i}(\Delta r_{i}\mathbf{e}_{i})+(\sum_{i=1}^{N}m_{i})\mathbf{A},
  12. 𝐓 = i = 1 N ( m i Δ r i 𝐞 i ) × ( α ( Δ r i 𝐭 i ) - ω 2 ( Δ r i 𝐞 i ) + 𝐀 ) = ( i = 1 N m i Δ r i 2 ) α k + ( i = 1 N m i Δ r i 𝐞 i ) × 𝐀 , \mathbf{T}=\sum_{i=1}^{N}(m_{i}\Delta r_{i}\mathbf{e}_{i})\times(\alpha(\Delta r% _{i}\mathbf{t}_{i})-\omega^{2}(\Delta r_{i}\mathbf{e}_{i})+\mathbf{A})=(\sum_{% i=1}^{N}m_{i}\Delta r_{i}^{2})\alpha\vec{k}+(\sum_{i=1}^{N}m_{i}\Delta r_{i}% \mathbf{e}_{i})\times\mathbf{A},
  13. i {}_{i}
  14. i {}_{i}
  15. i {}_{i}
  16. i {}_{i}
  17. i {}_{i}
  18. 𝐅 = M 𝐀 , 𝐓 = I C α k , \mathbf{F}=M\mathbf{A},\quad\mathbf{T}=I_{C}\alpha\vec{k},
  19. C {}_{C}
  20. 𝐅 = m 𝐚 , \mathbf{F}=m\mathbf{a},
  21. 𝐅 i + j = 1 N 𝐅 i j = m i 𝐚 i , i = 1 , , N , \mathbf{F}_{i}+\sum_{j=1}^{N}\mathbf{F}_{ij}=m_{i}\mathbf{a}_{i},\quad i=1,% \ldots,N,
  22. 𝐅 = i = 1 N 𝐅 i , 𝐓 = i = 1 N ( 𝐑 i - 𝐑 ) × 𝐅 i , \mathbf{F}=\sum_{i=1}^{N}\mathbf{F}_{i},\quad\mathbf{T}=\sum_{i=1}^{N}(\mathbf% {R}_{i}-\mathbf{R})\times\mathbf{F}_{i},
  23. 𝐅 = i = 1 N m i 𝐚 i , 𝐓 = i = 1 N ( 𝐑 i - 𝐑 ) × ( m i 𝐚 i ) , \mathbf{F}=\sum_{i=1}^{N}m_{i}\mathbf{a}_{i},\quad\mathbf{T}=\sum_{i=1}^{N}(% \mathbf{R}_{i}-\mathbf{R})\times(m_{i}\mathbf{a}_{i}),
  24. 𝐚 i = α × ( 𝐑 i - 𝐑 ) + ω × ( ω × ( 𝐑 i - 𝐑 ) ) + 𝐚 . \mathbf{a}_{i}=\alpha\times(\mathbf{R}_{i}-\mathbf{R})+\omega\times(\omega% \times(\mathbf{R}_{i}-\mathbf{R}))+\mathbf{a}.
  25. i = 1 N m i ( 𝐑 i - 𝐑 ) = 0 , \sum_{i=1}^{N}m_{i}(\mathbf{R}_{i}-\mathbf{R})=0,
  26. [ I R ] = i = 1 N m i ( 𝐈 ( 𝐒 i T 𝐒 i ) - 𝐒 i 𝐒 i T ) , [I_{R}]=\sum_{i=1}^{N}m_{i}(\mathbf{I}(\mathbf{S}_{i}^{T}\mathbf{S}_{i})-% \mathbf{S}_{i}\mathbf{S}_{i}^{T}),
  27. 𝐒 i \mathbf{S}_{i}
  28. 𝐒 i T \mathbf{S}_{i}^{T}
  29. 𝐒 i T 𝐒 i \mathbf{S}_{i}^{T}\mathbf{S}_{i}
  30. 𝐒 i \mathbf{S}_{i}
  31. 𝐒 i 𝐒 i T \mathbf{S}_{i}\mathbf{S}_{i}^{T}
  32. 𝐒 i \mathbf{S}_{i}
  33. 𝐈 \mathbf{I}
  34. 𝐅 = m 𝐚 , 𝐓 = [ I R ] α + ω × [ I R ] ω , \mathbf{F}=m\mathbf{a},\quad\mathbf{T}=[I_{R}]\alpha+\omega\times[I_{R}]\omega,
  35. 𝐅 j = m j 𝐚 j , 𝐓 j = [ I R ] j α j + ω j × [ I R ] j ω j , j = 1 , , M . \mathbf{F}_{j}=m_{j}\mathbf{a}_{j},\quad\mathbf{T}_{j}=[I_{R}]_{j}\alpha_{j}+% \omega_{j}\times[I_{R}]_{j}\omega_{j},\quad j=1,\ldots,M.
  36. s y m b o l τ = D 𝐋 D t = d 𝐋 d t + s y m b o l ω × 𝐋 = d ( I s y m b o l ω ) d t + ω × I s y m b o l ω = I s y m b o l α + ω × I s y m b o l ω symbol\tau={{D\mathbf{L}}\over{Dt}}={{d\mathbf{L}}\over{dt}}+symbol\omega% \times\mathbf{L}={{d(Isymbol\omega)}\over{dt}}+\omega\times{Isymbol\omega}=% Isymbol\alpha+\omega\times{Isymbol\omega}
  37. s y m b o l τ = s y m b o l Ω P × 𝐋 . symbol\tau=symbol\Omega_{\mathrm{P}}\times\mathbf{L}.
  38. τ = Ω P L sin θ , \tau=\mathit{\Omega}_{\mathrm{P}}L\sin\theta,\!
  39. 𝐕 i = ω × ( 𝐑 i - 𝐑 ) + 𝐕 , \mathbf{V}_{i}=\vec{\omega}\times(\mathbf{R}_{i}-\mathbf{R})+\mathbf{V},
  40. δ W = i = 1 n 𝐅 i δ 𝐫 i . \delta W=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\delta\mathbf{r}_{i}.
  41. δ 𝐫 i = j = 1 m 𝐫 i q j δ q j = j = 1 m 𝐕 i q ˙ j δ q j . \delta\mathbf{r}_{i}=\sum_{j=1}^{m}\frac{\partial\mathbf{r}_{i}}{\partial q_{j% }}\delta q_{j}=\sum_{j=1}^{m}\frac{\partial\mathbf{V}_{i}}{\partial\dot{q}_{j}% }\delta q_{j}.
  42. δ W = 𝐅 1 ( j = 1 m 𝐕 1 q ˙ j δ q j ) + + 𝐅 n ( j = 1 m 𝐕 n q ˙ j δ q j ) \delta W=\mathbf{F}_{1}\cdot\left(\sum_{j=1}^{m}\frac{\partial\mathbf{V}_{1}}{% \partial\dot{q}_{j}}\delta q_{j}\right)+\ldots+\mathbf{F}_{n}\cdot(\sum_{j=1}^% {m}\frac{\partial\mathbf{V}_{n}}{\partial\dot{q}_{j}}\delta q_{j})
  43. δ W = ( i = 1 n 𝐅 i 𝐕 i q ˙ 1 ) δ q 1 + + ( 1 = 1 n 𝐅 i 𝐕 i q ˙ m ) δ q m . \delta W=\left(\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{% \partial\dot{q}_{1}}\right)\delta q_{1}+\ldots+(\sum_{1=1}^{n}\mathbf{F}_{i}% \cdot\frac{\partial\mathbf{V}_{i}}{\partial\dot{q}_{m}})\delta q_{m}.
  44. δ W = ( i = 1 n 𝐅 i 𝐕 i q ˙ ) δ q = ( i = 1 n 𝐅 i ( ω × ( 𝐑 i - 𝐑 ) + 𝐕 ) q ˙ ) δ q . \delta W=\left(\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{% \partial\dot{q}}\right)\delta q=\left(\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{% \partial(\vec{\omega}\times(\mathbf{R}_{i}-\mathbf{R})+\mathbf{V})}{\partial% \dot{q}}\right)\delta q.
  45. δ W = ( 𝐅 𝐕 q ˙ + 𝐓 ω q ˙ ) δ q . \delta W=\left(\mathbf{F}\cdot\frac{\partial\mathbf{V}}{\partial\dot{q}}+% \mathbf{T}\cdot\frac{\partial\vec{\omega}}{\partial\dot{q}}\right)\delta q.
  46. Q = 𝐅 𝐕 q ˙ + 𝐓 ω q ˙ , Q=\mathbf{F}\cdot\frac{\partial\mathbf{V}}{\partial\dot{q}}+\mathbf{T}\cdot% \frac{\partial\vec{\omega}}{\partial\dot{q}},
  47. δ W = j = 1 m Q j δ q j , \delta W=\sum_{j=1}^{m}Q_{j}\delta q_{j},
  48. Q j = 𝐅 𝐕 q ˙ j + 𝐓 ω q ˙ j , j = 1 , , m . Q_{j}=\mathbf{F}\cdot\frac{\partial\mathbf{V}}{\partial\dot{q}_{j}}+\mathbf{T}% \cdot\frac{\partial\vec{\omega}}{\partial\dot{q}_{j}},\quad j=1,\ldots,m.
  49. Q j = - V q j , j = 1 , , m . Q_{j}=-\frac{\partial V}{\partial q_{j}},\quad j=1,\ldots,m.
  50. δ W = i = 1 n ( 𝐅 i 𝐕 i q ˙ + 𝐓 i ω i q ˙ ) δ q = Q δ q , \delta W=\sum_{i=1}^{n}(\mathbf{F}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{% \partial\dot{q}}+\mathbf{T}_{i}\cdot\frac{\partial\vec{\omega}_{i}}{\partial% \dot{q}})\delta q=Q\delta q,
  51. Q = i = 1 n ( 𝐅 i 𝐕 i q ˙ + 𝐓 i ω i q ˙ ) , Q=\sum_{i=1}^{n}(\mathbf{F}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{\partial\dot% {q}}+\mathbf{T}_{i}\cdot\frac{\partial\vec{\omega}_{i}}{\partial\dot{q}}),
  52. δ W = j = 1 m Q j δ q j , \delta W=\sum_{j=1}^{m}Q_{j}\delta q_{j},
  53. Q j = i = 1 n ( 𝐅 i 𝐕 i q ˙ j + 𝐓 i ω i q ˙ j ) , j = 1 , , m . Q_{j}=\sum_{i=1}^{n}(\mathbf{F}_{i}\cdot\frac{\partial\mathbf{V}_{i}}{\partial% \dot{q}_{j}}+\mathbf{T}_{i}\cdot\frac{\partial\vec{\omega}_{i}}{\partial\dot{q% }_{j}}),\quad j=1,\ldots,m.
  54. Q j = 0 , j = 1 , , m . Q_{j}=0,\quad j=1,\ldots,m.
  55. Q * = - ( M 𝐀 ) 𝐕 q ˙ - ( [ I R ] α + ω × [ I R ] ω ) ω q ˙ . Q^{*}=-(M\mathbf{A})\cdot\frac{\partial\mathbf{V}}{\partial\dot{q}}-([I_{R}]% \alpha+\omega\times[I_{R}]\omega)\cdot\frac{\partial\vec{\omega}}{\partial\dot% {q}}.
  56. T = 1 2 M 𝐕 𝐕 + 1 2 ω [ I R ] ω , T=\frac{1}{2}M\mathbf{V}\cdot\mathbf{V}+\frac{1}{2}\vec{\omega}\cdot[I_{R}]% \vec{\omega},
  57. Q * = - ( d d t T q ˙ - T q ) . Q^{*}=-\left(\frac{d}{dt}\frac{\partial T}{\partial\dot{q}}-\frac{\partial T}{% \partial q}\right).
  58. T = i = 1 n ( 1 2 M 𝐕 i 𝐕 i + 1 2 ω i [ I R ] ω i ) , T=\sum_{i=1}^{n}(\frac{1}{2}M\mathbf{V}_{i}\cdot\mathbf{V}_{i}+\frac{1}{2}\vec% {\omega}_{i}\cdot[I_{R}]\vec{\omega}_{i}),
  59. Q j * = - ( d d t T q ˙ j - T q j ) , j = 1 , , m . Q^{*}_{j}=-\left(\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{% \partial T}{\partial q_{j}}\right),\quad j=1,\ldots,m.
  60. δ W = ( Q 1 + Q 1 * ) δ q 1 + + ( Q m + Q m * ) δ q m = 0 , \delta W=(Q_{1}+Q^{*}_{1})\delta q_{1}+\ldots+(Q_{m}+Q^{*}_{m})\delta q_{m}=0,
  61. Q j + Q j * = 0 , j = 1 , , m , Q_{j}+Q^{*}_{j}=0,\quad j=1,\ldots,m,
  62. d d t T q ˙ j - T q j = Q j , j = 1 , , m . \frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{\partial T}{\partial q% _{j}}=Q_{j},\quad j=1,\ldots,m.
  63. d d t T q ˙ j - T q j = - V q j , j = 1 , , m . \frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{\partial T}{\partial q% _{j}}=-\frac{\partial V}{\partial q_{j}},\quad j=1,\ldots,m.
  64. d d t L q ˙ j - L q j = 0 j = 1 , , m . \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{j}}-\frac{\partial L}{\partial q% _{j}}=0\quad j=1,\ldots,m.
  65. 𝐫 i = ( 𝐫 i - 𝐑 ) + 𝐑 , 𝐯 i = d d t ( 𝐫 i - 𝐑 ) + 𝐕 . \mathbf{r}_{i}=(\mathbf{r}_{i}-\mathbf{R})+\mathbf{R},\quad\mathbf{v}_{i}=% \frac{d}{dt}(\mathbf{r}_{i}-\mathbf{R})+\mathbf{V}.
  66. 𝐩 = d d t ( i = 1 n m i ( 𝐫 i - 𝐑 ) ) + ( i = 1 n m i ) 𝐕 , \mathbf{p}=\frac{d}{dt}(\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R}))+(\sum_% {i=1}^{n}m_{i})\mathbf{V},
  67. 𝐋 = i = 1 n m i ( 𝐫 i - 𝐑 ) × d d t ( 𝐫 i - 𝐑 ) + ( i = 1 n m i ( 𝐫 i - 𝐑 ) ) × 𝐕 . \mathbf{L}=\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R})\times\frac{d}{dt}(% \mathbf{r}_{i}-\mathbf{R})+(\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R}))% \times\mathbf{V}.
  68. 𝐩 = M 𝐕 , 𝐋 = i = 1 n m i ( 𝐫 i - 𝐑 ) × d d t ( 𝐫 i - 𝐑 ) . \mathbf{p}=M\mathbf{V},\quad\mathbf{L}=\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-% \mathbf{R})\times\frac{d}{dt}(\mathbf{r}_{i}-\mathbf{R}).
  69. i {}_{i}
  70. i {}_{i}
  71. i {}_{i}
  72. 𝐫 i = ( 𝐫 i - 𝐑 ) + 𝐑 , 𝐯 i = ω × ( 𝐫 i - 𝐑 ) + 𝐕 , \mathbf{r}_{i}=(\mathbf{r}_{i}-\mathbf{R})+\mathbf{R},\quad\mathbf{v}_{i}=% \omega\times(\mathbf{r}_{i}-\mathbf{R})+\mathbf{V},
  73. 𝐩 = ( i = 1 n m i ) 𝐕 , 𝐋 = i = 1 n m i ( 𝐫 i - 𝐑 ) × 𝐯 i = i = 1 n m i ( 𝐫 i - 𝐑 ) × ( ω × ( 𝐫 i - 𝐑 ) ) . \mathbf{p}=(\sum_{i=1}^{n}m_{i})\mathbf{V},\quad\mathbf{L}=\sum_{i=1}^{n}m_{i}% (\mathbf{r}_{i}-\mathbf{R})\times\mathbf{v}_{i}=\sum_{i=1}^{n}m_{i}(\mathbf{r}% _{i}-\mathbf{R})\times(\omega\times(\mathbf{r}_{i}-\mathbf{R})).
  74. 𝐩 = M 𝐕 , 𝐋 = [ I R ] ω , \mathbf{p}=M\mathbf{V},\quad\mathbf{L}=[I_{R}]\omega,
  75. R {}_{R}
  76. [ I R ] = - i = 1 n m i [ r i - R ] [ r i - R ] , [I_{R}]=-\sum_{i=1}^{n}m_{i}[r_{i}-R][r_{i}-R],

Ring_of_integers.html

  1. K K
  2. K K
  3. 𝒪 K \mathcal{O}_{K}
  4. K K
  5. 𝐙 \mathbf{Z}
  6. 𝐙 \mathbf{Z}
  7. 𝐐 \mathbf{Q}
  8. 𝐙 \mathbf{Z}
  9. 𝐙 \mathbf{Z}
  10. 𝐙 \mathbf{Z}
  11. 𝐐 \mathbf{Q}
  12. K K
  13. x x
  14. x = i = 1 n a i b i , x=\sum_{i=1}^{n}a_{i}b_{i},
  15. n n
  16. 𝐙 \mathbf{Z}
  17. K K
  18. 𝐐 \mathbf{Q}
  19. p p
  20. p p
  21. s i z e = 120 % K = 𝐐 ( ζ ) size=120\%K=\mathbf{Q}(ζ)
  22. d d
  23. s i z e = 120 % K = 𝐐 ( d ) size=120\%K=\mathbf{Q}(\sqrt{d})
  24. s i z e = 120 % ( 1 , ( 1 + d ) / 2 ) size=120\%(1, (1+\sqrt{d})/2)
  25. d 1 ( m o d 4 ) d≡1(mod4)
  26. s i z e = 120 % ( 1 , d ) size=120\%(1,\sqrt{d})
  27. d 2 , 3 ( m o d 4 ) d≡2, 3(mod4)
  28. 6 = 2 3 = ( 1 + - 5 ) ( 1 - - 5 ) . 6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})\ .
  29. p p
  30. s i z e = 120 % 𝐙 < s u b > p size=120\%\mathbf{Z}<sub>p
  31. 𝐙 \mathbf{Z}

Ring_of_symmetric_functions.html

  1. X 1 + X 2 + + X n , X_{1}+X_{2}+\cdots+X_{n},\,
  2. X 1 3 + X 2 3 + + X n 3 , X_{1}^{3}+X_{2}^{3}+\cdots+X_{n}^{3},\,
  3. X 1 X 2 X n . X_{1}X_{2}\cdots X_{n}.\,
  4. p 3 ( X 1 , , X n ) = e 1 ( X 1 , , X n ) 3 - 3 e 2 ( X 1 , , X n ) e 1 ( X 1 , , X n ) + 3 e 3 ( X 1 , , X n ) , p_{3}(X_{1},\ldots,X_{n})=e_{1}(X_{1},\ldots,X_{n})^{3}-3e_{2}(X_{1},\ldots,X_% {n})e_{1}(X_{1},\ldots,X_{n})+3e_{3}(X_{1},\ldots,X_{n}),
  5. e i e_{i}
  6. p 3 = e 1 3 - 3 e 2 e 1 + 3 e 3 p_{3}=e_{1}^{3}-3e_{2}e_{1}+3e_{3}
  7. n + 1 n+1
  8. e 2 = i < j X i X j e_{2}=\sum_{i<j}X_{i}X_{j}\,
  9. Π i = 1 n X i \Pi_{i=1}^{n}X_{i}
  10. Π i = 1 n ( X i + 1 ) \Pi_{i=1}^{n}(X_{i}+1)
  11. m α = β α X β . m_{\alpha}=\sum\nolimits_{\beta\sim\alpha}X^{\beta}.
  12. X α = Π i = 1 k X i X^{\alpha}=\Pi_{i=1}^{k}X_{i}
  13. p 0 ( X 1 , , X n ) = Σ i = 1 n X i 0 = n p_{0}(X_{1},\ldots,X_{n})=\Sigma_{i=1}^{n}X_{i}^{0}=n
  14. ( i < j ( X i - X j ) ) 2 \textstyle(\prod_{i<j}(X_{i}-X_{j}))^{2}
  15. P = Q P=Q
  16. i = 0 k ( - 1 ) i e i h k - i = 0 = i = 0 k ( - 1 ) i h i e k - i for all k > 0 , \sum_{i=0}^{k}(-1)^{i}e_{i}h_{k-i}=0=\sum_{i=0}^{k}(-1)^{i}h_{i}e_{k-i}\quad% \mbox{for all }~{}k>0,
  17. k e k = i = 1 k ( - 1 ) i - 1 p i e k - i for all k 0 , ke_{k}=\sum_{i=1}^{k}(-1)^{i-1}p_{i}e_{k-i}\quad\mbox{for all }~{}k\geq 0,
  18. k h k = i = 1 k p i h k - i for all k 0. kh_{k}=\sum_{i=1}^{k}p_{i}h_{k-i}\quad\mbox{for all }~{}k\geq 0.
  19. i = 1 1 1 - t i \textstyle\prod_{i=1}^{\infty}\frac{1}{1-t^{i}}
  20. E ( t ) = k 0 e k ( X ) t k = i = 1 ( 1 + X i t ) . E(t)=\sum_{k\geq 0}e_{k}(X)t^{k}=\prod_{i=1}^{\infty}(1+X_{i}t).
  21. H ( t ) = k 0 h k ( X ) t k = i = 1 ( k 0 ( X i t ) k ) = i = 1 1 1 - X i t . H(t)=\sum_{k\geq 0}h_{k}(X)t^{k}=\prod_{i=1}^{\infty}\left(\sum_{k\geq 0}(X_{i% }t)^{k}\right)=\prod_{i=1}^{\infty}\frac{1}{1-X_{i}t}.
  22. E ( - t ) H ( t ) = 1 = E ( t ) H ( - t ) E(-t)H(t)=1=E(t)H(-t)
  23. P ( t ) = k > 0 p k ( X ) t k = k > 0 i = 1 ( X i t ) k = i = 1 X i t 1 - X i t = t E ( - t ) E ( - t ) = t H ( t ) H ( t ) P(t)=\sum_{k>0}p_{k}(X)t^{k}=\sum_{k>0}\sum_{i=1}^{\infty}(X_{i}t)^{k}=\sum_{i% =1}^{\infty}\frac{X_{i}t}{1-X_{i}t}=\frac{tE^{\prime}(-t)}{E(-t)}=\frac{tH^{% \prime}(t)}{H(t)}
  24. P ( t ) = - t d d t log ( E ( - t ) ) = t d d t log ( H ( t ) ) , P(t)=-t\frac{d}{dt}\log(E(-t))=t\frac{d}{dt}\log(H(t)),
  25. log ( 1 - t S ) = - i > 0 1 i ( t S ) i \textstyle\log(1-tS)=-\sum_{i>0}\frac{1}{i}(tS)^{i}

Risk-neutral_measure.html

  1. 1 1 + R \frac{1}{1+R}
  2. 1 + R 1+R
  3. T T
  4. H T H_{T}
  5. H T H_{T}
  6. T T
  7. P ( 0 , T ) P(0,T)
  8. H 0 = P ( 0 , T ) E Q ( H T ) . H_{0}=P(0,T)\operatorname{E}_{Q}(H_{T}).
  9. Q Q
  10. H 0 = P ( 0 , T ) E P ( d Q d P H T ) H_{0}=P(0,T)\operatorname{E}_{P}\left(\frac{dQ}{dP}H_{T}\right)
  11. d Q d P \frac{dQ}{dP}
  12. Q Q
  13. P P
  14. ( Ω , 𝔉 , ) (\Omega,\mathfrak{F},\mathbb{P})
  15. * \mathbb{P^{*}}
  16. i { 0 , , d } i\in\{0,...,d\}
  17. π i = 𝔼 * ( S i / ( 1 + r ) ) \pi^{i}=\mathbb{E}_{\mathbb{P}^{*}}(S^{i}/(1+r))
  18. S S
  19. S u S^{u}
  20. S d S^{d}
  21. R > 0 R>0
  22. S d ( 1 + R ) S S u S^{d}\leq(1+R)S\leq S^{u}
  23. π = ( 1 + R ) S - S d S u - S d . \pi=\frac{(1+R)S-S^{d}}{S^{u}-S^{d}}.
  24. X u X^{u}
  25. X d X^{d}
  26. X = π X u + ( 1 - π ) X d 1 + R . X=\frac{\pi X^{u}+(1-\pi)X^{d}}{1+R}.
  27. d S t = μ S t d t + σ S t d W t dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}
  28. W t W_{t}
  29. W ~ t = W t + μ - r σ t , \tilde{W}_{t}=W_{t}+\frac{\mu-r}{\sigma}t,
  30. Q Q
  31. W ~ t \tilde{W}_{t}
  32. μ - r σ \frac{\mu-r}{\sigma}
  33. d W t = d W ~ t - μ - r σ d t . dW_{t}=d\tilde{W}_{t}-\frac{\mu-r}{\sigma}\,dt.
  34. d S t = r S t d t + σ S t d W ~ t . dS_{t}=rS_{t}\,dt+\sigma S_{t}\,d\tilde{W}_{t}.
  35. S ~ t \tilde{S}_{t}
  36. S ~ t = e - r t S t \tilde{S}_{t}=e^{-rt}S_{t}
  37. d S ~ t = σ S ~ t d W ~ t . d\tilde{S}_{t}=\sigma\tilde{S}_{t}\,d\tilde{W}_{t}.
  38. Q Q
  39. H t = E Q ( H T | F t ) H_{t}=\operatorname{E}_{Q}(H_{T}|F_{t})
  40. Q Q
  41. S ~ \tilde{S}
  42. H H
  43. Q Q
  44. H t H_{t}
  45. t T t\leq T

Risk_premium.html

  1. π \pi
  2. π \pi
  3. u ( r f ) = E u ( r f + π + x ) . u(r_{f})=Eu(r_{f}+\pi+x).
  4. u ( C ) = E u ( r ) ; u(C)=Eu(r);

Robert_Mills_(physicist).html

  1. μ F μ ν + 2 ϵ ( b μ × F μ ν ) = J ν \partial_{\mu}F^{\mu\nu}+2\epsilon(b_{\mu}\times F^{\mu\nu})=J^{\nu}

Roberts_cross.html

  1. y i , j = x i , j y_{i,j}=\sqrt{x_{i,j}}
  2. z i , j = ( y i , j - y i + 1 , j + 1 ) 2 + ( y i + 1 , j - y i , j + 1 ) 2 z_{i,j}=\sqrt{(y_{i,j}-y_{i+1,j+1})^{2}+(y_{i+1,j}-y_{i,j+1})^{2}}
  3. [ + 1 0 0 - 1 ] and [ 0 + 1 - 1 0 ] . \begin{bmatrix}+1&0\\ 0&-1\\ \end{bmatrix}\quad\mbox{and}~{}\quad\begin{bmatrix}0&+1\\ -1&0\\ \end{bmatrix}.
  4. I ( x , y ) I(x,y)
  5. G x ( x , y ) G_{x}(x,y)
  6. G y ( x , y ) G_{y}(x,y)
  7. I ( x , y ) = G ( x , y ) = G x 2 + G y 2 . \nabla I(x,y)=G(x,y)=\sqrt{G_{x}^{2}+G_{y}^{2}}.
  8. Θ ( x , y ) = arctan ( G y ( x , y ) G x ( x , y ) ) . \Theta(x,y)=\arctan{\left(\frac{G_{y}(x,y)}{G_{x}(x,y)}\right)}.

Roller_chain.html

  1. % = ( ( M - ( S * P ) ) / ( S * P ) ) * 100 \%=((M-(S*P))/(S*P))*100
  2. 1 / 4 {1}/{4}
  3. 𝟐 / 8 {\mathbf{2}}/{8}
  4. 1 / 8 {1}/{8}
  5. 3 / 8 {3}/{8}
  6. 𝟑 / 8 {\mathbf{3}}/{8}
  7. 3 / 16 {3}/{16}
  8. 1 / 2 {1}/{2}
  9. 𝟒 / 8 {\mathbf{4}}/{8}
  10. 1 / 4 {1}/{4}
  11. 1 / 2 {1}/{2}
  12. 𝟒 / 8 {\mathbf{4}}/{8}
  13. 5 / 16 {5}/{16}
  14. 5 / 8 {5}/{8}
  15. 𝟓 / 8 {\mathbf{5}}/{8}
  16. 3 / 8 {3}/{8}
  17. 3 / 4 {3}/{4}
  18. 𝟔 / 8 {\mathbf{6}}/{8}
  19. 1 / 2 {1}/{2}
  20. 𝟖 / 8 {\mathbf{8}}/{8}
  21. 5 / 8 {5}/{8}

Rolling-element_bearing.html

  1. L 10 = ( C / P ) p L_{10}=(C/P)^{p}
  2. L 10 L_{10}
  3. C C
  4. P P
  5. p p
  6. L 10 L_{10}
  7. L 10 L_{10}
  8. p p
  9. p p
  10. p p

Ronald_N._Bracewell.html

  1. λ 2 \lambda^{2}

Root_locus.html

  1. X ( s ) X(s)
  2. Y ( s ) Y(s)
  3. G ( s ) G(s)
  4. H ( s ) H(s)
  5. T ( s ) = Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) T(s)=\frac{Y(s)}{X(s)}=\frac{G(s)}{1+G(s)H(s)}
  6. 1 + G ( s ) H ( s ) = 0 1+G(s)H(s)=0
  7. G ( s ) H ( s ) = - 1 G(s)H(s)=-1
  8. G ( s ) H ( s ) = - 1 G(s)H(s)=-1
  9. G ( s ) H ( s ) = K ( s + z 1 ) ( s + z 2 ) ( s + z m ) ( s + p 1 ) ( s + p 2 ) ( s + p m + n ) G(s)H(s)=\frac{K(s+z_{1})(s+z_{2})\cdots(s+z_{m})}{(s+p_{1})(s+p_{2})\cdots(s+% p_{m+n})}
  10. m m
  11. m + n m+n
  12. K K
  13. K K
  14. s s
  15. G ( s ) H ( s ) = - 1 G(s)H(s)=-1
  16. K K
  17. K K
  18. G ( s ) H ( s ) G(s)H(s)
  19. ( s a ) (s−a)
  20. a a
  21. s s
  22. K K
  23. s s
  24. T ( s ) T(s)
  25. K K
  26. K K
  27. K K
  28. K K
  29. P - Z = number of asymptotes P-Z=\,\text{number of asymptotes}\,
  30. α \alpha
  31. ϕ \phi
  32. ϕ l = 180 + ( l - 1 ) 360 P - Z , l = 1 , 2 , , P - Z \phi_{l}=\frac{180^{\circ}+(l-1)360^{\circ}}{P-Z},l=1,2,\ldots,P-Z
  33. α = P - Z P - Z \alpha=\frac{\sum_{P}-\sum_{Z}}{P-Z}
  34. P \sum_{P}
  35. Z \sum_{Z}
  36. d G ( s ) H ( s ) d s = 0 or d G H ¯ ( z ) d z = 0 \frac{dG(s)H(s)}{ds}=0\,\text{ or }\frac{d\overline{GH}(z)}{dz}=0
  37. z = e < s u p > s T z=e<sup>sT

Rossby_number.html

  1. v v U 2 / L v\cdot\nabla v\sim U^{2}/L
  2. Ω × v U Ω \Omega\times v\sim U\Omega
  3. Ro = U L f \mathrm{Ro}=\frac{U}{Lf}

Rossby_wave.html

  1. u = < u , v Align g t ; \vec{u}=<u,v&gt;
  2. u = U + u ( t , x , y ) u=U+u^{\prime}(t,x,y)\!
  3. v = v ( t , x , y ) v=v^{\prime}(t,x,y)\!
  4. U u , v U\gg u^{\prime},v^{\prime}\!
  5. η \eta
  6. ψ \psi
  7. u = ψ y u^{\prime}=\frac{\partial\psi}{\partial y}
  8. v = - ψ x v^{\prime}=-\frac{\partial\psi}{\partial x}
  9. η = × ( u 𝐬𝐲𝐦𝐛𝐨𝐥 ı ^ + v 𝐬𝐲𝐦𝐛𝐨𝐥 ȷ ^ ) = - 2 ψ \eta=\nabla\times(u^{\prime}\mathbf{\hat{symbol{\imath}}}+v^{\prime}\mathbf{% \hat{symbol{\jmath}}})=-\nabla^{2}\psi
  10. d ( η + f ) d t = 0 = η t + U η x + β v \frac{d(\eta+f)}{dt}=0=\frac{\partial\eta}{\partial t}+U\frac{\partial\eta}{% \partial x}+\beta v^{\prime}
  11. β = f y \beta=\frac{\partial f}{\partial y}
  12. 0 = 2 ψ t + U 2 ψ x + β ψ x 0=\frac{\partial\nabla^{2}\psi}{\partial t}+U\frac{\partial\nabla^{2}\psi}{% \partial x}+\beta\frac{\partial\psi}{\partial x}
  13. ω \omega
  14. ψ = ψ 0 e i ( k x + l y - ω t ) \psi=\psi_{0}e^{i(kx+ly-\omega t)}\!
  15. ω = U k - β k k 2 + l 2 \omega=Uk-\beta\frac{k}{k^{2}+l^{2}}
  16. c ω k = U - β ( k 2 + l 2 ) , c\ \equiv\ \frac{\omega}{k}=U-\frac{\beta}{(k^{2}+l^{2})},
  17. c g ω k = U - β ( l 2 - k 2 ) ( k 2 + l 2 ) 2 , c_{g}\ \equiv\ \frac{\partial\omega}{\partial k}\ =U-\frac{\beta(l^{2}-k^{2})}% {(k^{2}+l^{2})^{2}},
  18. c g c_{g}
  19. β \beta
  20. β = f y = 1 a d d ϕ ( 2 ω sin ϕ ) = 2 ω cos ϕ a \beta=\frac{\partial f}{\partial y}=\frac{1}{a}\frac{d}{d\phi}(2\omega\sin\phi% )=\frac{2\omega\cos\phi}{a}
  21. ϕ \phi
  22. β = 0 \beta=0
  23. f = 0 f=0
  24. β > 0 \beta>0

Rotation_(mathematics).html

  1. ( n 1 ) (n− 1)
  2. n n
  3. n > 2 n>2
  4. U ( 1 ) U(1)
  5. θ θ
  6. n × n n×n
  7. n n
  8. ( x , y ) (x,y)
  9. x x
  10. y y
  11. θ θ
  12. [ x y ] = [ cos θ - sin θ sin θ cos θ ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  13. ( x , y ) (x′,y′)
  14. x x′
  15. y y′
  16. x = x cos θ - y sin θ y = x sin θ + y cos θ . \begin{aligned}\displaystyle x^{\prime}&\displaystyle=x\cos\theta-y\sin\theta% \\ \displaystyle y^{\prime}&\displaystyle=x\sin\theta+y\cos\theta.\end{aligned}
  17. [ x y ] \begin{bmatrix}x\\ y\end{bmatrix}
  18. [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}
  19. θ θ
  20. ( x , y ) (x,y)
  21. z = x + i y z=x+iy
  22. θ θ
  23. e i θ z \displaystyle e^{i\theta}z
  24. x \displaystyle x^{\prime}
  25. ( x , y , z ) (x,y,z)
  26. ( x , y , z ) (x′,y′,z′)
  27. 𝐀 = ( a b c d e f g h i ) \mathbf{A}=\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}
  28. 𝐀 ( x y z ) = ( a b c d e f g h i ) ( x y z ) = ( x y z ) \mathbf{A}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}a&b&c\\ d&e&f\\ g&h&i\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}=\begin{pmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}
  29. 𝐀 \mathbf{A}
  30. S O ( 3 ) SO(3)
  31. 𝐱 = 𝐪𝐱𝐪 - 1 , \mathbf{x^{\prime}}=\mathbf{qxq}^{-1},
  32. 𝐪 \mathbf{q}
  33. 𝐱 \mathbf{x}
  34. 𝐪 = e 𝐯 / 2 , \mathbf{q}=e^{\mathbf{v}/2},
  35. 𝐯 \mathbf{v}
  36. n n
  37. S O ( n ) SO(n)
  38. S O ( n ) SO(n)
  39. S p i n ( n ) Spin(n)
  40. S p i n ( 3 ) Spin(3)
  41. n n
  42. ( n + 1 ) (n+ 1)
  43. S O ( n + 1 ) SO(n+ 1)
  44. n n
  45. n n
  46. S O ( 3 ) SO(3)
  47. S O ( 3 ; 1 ) < s u p > + SO(3;1)<sup>+

Rotational_energy.html

  1. E rotational = 1 2 I ω 2 E_{\mathrm{rotational}}=\frac{1}{2}I\omega^{2}
  2. ω \omega
  3. I I
  4. E E
  5. E translational = 1 2 m v 2 E_{\mathrm{translational}}=\frac{1}{2}mv^{2}
  6. ω \omega

Rotational_spectroscopy.html

  1. J J
  2. K K
  3. B ν = B - α ( ν + 1 2 ) B_{\nu}=B-\alpha(\nu+{1\over 2})
  4. I I
  5. I A I_{A}
  6. I B I_{B}
  7. I C I_{C}
  8. I A I B I C I_{A}\leq I_{B}\leq I_{C}
  9. A A
  10. A A
  11. I A = I B = I C I_{A}=I_{B}=I_{C}
  12. I A I B = I C I_{A}<<I_{B}=I_{C}
  13. I A I_{A}
  14. I A = I B I_{A}=I_{B}
  15. I B = I C I_{B}=I_{C}
  16. I A = I B < I C I_{A}=I_{B}<I_{C}
  17. I A < I B = I C I_{A}<I_{B}=I_{C}
  18. ν ~ \tilde{\nu}
  19. ν × λ = c \nu\times\lambda=c
  20. ν ~ / cm - 1 = 1 λ / cm = ν / s - 1 c / cm s - 1 = ν / s - 1 2.99792458 × 10 10 \tilde{\nu}/\mathrm{cm}^{-1}=\frac{1}{\lambda/\mathrm{cm}}=\frac{\nu/\mathrm{s% }^{-1}}{c/\mathrm{cm}\ \mathrm{s}^{-1}}=\frac{\nu/\mathrm{s}^{-1}}{2.99792458% \times 10^{10}}
  21. ν ~ / cm - 1 ν / GHz 30 \tilde{\nu}/\mathrm{cm}^{-1}\approx\frac{\nu/\mathrm{GHz}}{30}
  22. ν ~ = ν ~ v i b ± B J ( J + 1 ) \tilde{\nu}=\tilde{\nu}_{vib}\pm BJ(J+1)
  23. F ( J ) = B J ( J + 1 ) J = 0 , 1 , 2 , F\left(J\right)=BJ\left(J+1\right)\qquad J=0,1,2,...
  24. B B
  25. I B = I C , I A = 0 I_{B}=I_{C},I_{A}=0
  26. B = h 8 π 2 c I B = h 8 π 2 c I C B={h\over{8\pi^{2}cI_{B}}}={h\over{8\pi^{2}cI_{C}}}
  27. I = m 1 m 2 m 1 + m 2 d 2 I=\frac{m_{1}m_{2}}{m_{1}+m_{2}}d^{2}
  28. Δ J = J - J ′′ = ± 1 \Delta J=J^{\prime}-J^{\prime\prime}=\pm 1
  29. ν ~ J J ′′ = F ( J ) - F ( J ′′ ) = 2 B ( J ′′ + 1 ) J ′′ = 0 , 1 , 2 , \tilde{\nu}_{J^{\prime}\leftrightarrow J^{\prime\prime}}=F\left(J^{\prime}% \right)-F\left(J^{\prime\prime}\right)=2B\left(J^{\prime\prime}+1\right)\qquad J% ^{\prime\prime}=0,1,2,...
  30. J ′′ J^{\prime\prime}
  31. J J^{\prime}
  32. Δ J \Delta J
  33. J ′′ J J^{\prime\prime}{\leftarrow}J^{\prime}
  34. N J N 0 = e - E J k T = e - B h c J ( J + 1 ) k T \frac{N_{J}}{N_{0}}=e^{-\frac{E_{J}}{kT}}=e^{-\frac{BhcJ(J+1)}{kT}}
  35. population ( 2 J + 1 ) e - E J k T \mathrm{population}\propto(2J+1)e^{-\frac{E_{J}}{kT}}
  36. J = k T 2 h c B - 1 / 2 J=\sqrt{\frac{kT}{2hcB}}-1/2
  37. B B
  38. F ( J ) = B J ( J + 1 ) - D J 2 ( J + 1 ) 2 J = 0 , 1 , 2 , F\left(J\right)=BJ\left(J+1\right)-DJ^{2}\left(J+1\right)^{2}\qquad J=0,1,2,...
  39. D D
  40. ν ~ J J ′′ = 2 B ( J ′′ + 1 ) - 4 D ( J ′′ + 1 ) 3 J ′′ = 0 , 1 , 2 , \tilde{\nu}_{J^{\prime}\leftrightarrow J^{\prime\prime}}=2B\left(J^{\prime% \prime}+1\right)-4D\left(J^{\prime\prime}+1\right)^{3}\qquad J^{\prime\prime}=% 0,1,2,...
  41. D D
  42. D = h 3 32 π 4 I 2 r 2 k c D=\frac{h^{3}}{32\pi^{4}I^{2}r^{2}kc}
  43. B B
  44. D D
  45. D = 4 B 3 ω ~ 2 D=\frac{4B^{3}}{\tilde{\omega}^{2}}
  46. ω ~ \tilde{\omega}
  47. F ( J , K ) = B J ( J + 1 ) + ( A - B ) K 2 J = 0 , 1 , 2 , and K = + J , 0... - J F\left(J,K\right)=BJ\left(J+1\right)+\left(A-B\right)K^{2}\qquad J=0,1,2,...% \quad\mbox{and}~{}\quad K=+J,...0...-J
  48. B = h 8 π 2 c I B B={h\over{8\pi^{2}cI_{B}}}
  49. A = h 8 π 2 c I A A={h\over{8\pi^{2}cI_{A}}}
  50. A = h 8 π 2 c I C A={h\over{8\pi^{2}cI_{C}}}
  51. ν ~ J J ′′ , K = F ( J , K ) - F ( J ′′ , K ) = 2 B ( J ′′ + 1 ) J ′′ = 0 , 1 , 2 , \tilde{\nu}_{J^{\prime}\leftrightarrow J^{\prime\prime},K}=F\left(J^{\prime},K% \right)-F\left(J^{\prime\prime},K\right)=2B\left(J^{\prime\prime}+1\right)% \qquad J^{\prime\prime}=0,1,2,...
  52. ν ~ J J ′′ , K = F ( J , K ) - F ( J ′′ , K ) = 2 ( B - 2 D J K K 2 ) ( J ′′ + 1 ) - 4 D J ( J ′′ + 1 ) 3 J ′′ = 0 , 1 , 2 , \tilde{\nu}_{J^{\prime}\leftrightarrow J^{\prime\prime},K}=F\left(J^{\prime},K% \right)-F\left(J^{\prime\prime},K\right)=2\left(B-2D_{JK}K^{2}\right)\left(J^{% \prime\prime}+1\right)-4D_{J}\left(J^{\prime\prime}+1\right)^{3}\qquad J^{% \prime\prime}=0,1,2,...
  53. Δ J = ± 1 , Δ F = 0 , ± 1 \Delta J=\pm 1,\Delta F=0,\pm 1
  54. B B
  55. B B
  56. B ¯ = B / h c \bar{B}=B/hc

Rotational–vibrational_spectroscopy.html

  1. B ′′ , B^{\prime\prime},
  2. B . B^{\prime}.
  3. B B
  4. B ¯ = B / h c \bar{B}=B/hc
  5. B ′′ B^{\prime\prime}
  6. B B^{\prime}
  7. Δ 2 \Delta_{2}
  8. Δ 2 ′′ F ( J ) = ν ¯ [ R ( J - 1 ) ] - ν ¯ [ P ( J + 1 ) ] = ( 2 B ′′ - 3 D ′′ ) ( 2 J + 1 ) - D ′′ ( 2 J + 1 ) 3 \Delta_{2}^{\prime\prime}F(J)=\bar{\nu}[R(J-1)]-\bar{\nu}[P(J+1)]=(2B^{\prime% \prime}-3D^{\prime\prime})\left(2J+1\right)-D^{\prime\prime}\left(2J+1\right)^% {3}
  9. Δ 2 F ( J ) = ν ¯ [ R ( J ) ] - ν ¯ [ P ( J ) ] = ( 2 B - 3 D ) ( 2 J + 1 ) - D ( 2 J + 1 ) 3 \Delta_{2}^{\prime}F(J)=\bar{\nu}[R(J)]-\bar{\nu}[P(J)]=(2B^{\prime}-3D^{% \prime})\left(2J+1\right)-D^{\prime}\left(2J+1\right)^{3}
  10. G ( v ) G(v)
  11. G ( v ) = ω e ( v + 1 2 ) - ω e χ e ( v + 1 2 ) 2 G(v)=\omega_{e}\left(v+{1\over 2}\right)-\omega_{e}\chi_{e}\left(v+{1\over 2}% \right)^{2}\,
  12. F v ( J ) = B v J ( J + 1 ) - D J 2 ( J + 1 ) 2 F_{v}(J)=B_{v}J\left(J+1\right)-DJ^{2}\left(J+1\right)^{2}
  13. B v = h 8 π 2 c I v ; I v = m A m B m A + m B d v 2 B_{v}={h\over{8\pi^{2}cI_{v}}};\quad I_{v}=\frac{m_{A}m_{B}}{m_{A}+m_{B}}d_{v}% ^{2}
  14. G ( v ) + F v ( J ) = [ ω e ( v + 1 2 ) + B v J ( J + 1 ) ] - [ ω e χ e ( v + 1 2 ) 2 + D J 2 ( J + 1 ) 2 ] G(v)+F_{v}(J)=\left[\omega_{e}\left(v+{1\over 2}\right)+B_{v}J(J+1)\right]-% \left[\omega_{e}\chi_{e}\left(v+{1\over 2}\right)^{2}+DJ^{2}(J+1)^{2}\right]
  15. Δ v = ± 1 ( ± 2 , ± 3 , etc. \Delta v=\pm 1\ (\pm 2,\pm 3,\,\textit{ etc.}
  16. ) , Δ J = ± 1 ),\Delta J=\pm 1
  17. Δ v = ± 1 , Δ J = 0 \Delta v=\pm 1,\Delta J=0
  18. D D^{\prime}
  19. D ′′ D^{\prime\prime}
  20. ν ¯ = ω 0 + ( B + B ′′ ) m + ( B - B ′′ ) m 2 - 2 ( D + D ′′ ) m 3 , ω 0 = ω e ( 1 - 2 χ e ) m = ± 1 , ± 2 e t c . \bar{\nu}=\omega_{0}+(B^{\prime}+B^{\prime\prime})m+(B^{\prime}-B^{\prime% \prime})m^{2}-2(D^{\prime}+D^{\prime\prime})m^{3},\quad\omega_{0}=\omega_{e}(1% -2\chi_{e})\quad m=\pm 1,\pm 2\ etc.
  21. ( B + B ′′ ) m (B^{\prime}+B^{\prime\prime})m
  22. ( B - B ′′ ) m 2 (B^{\prime}-B^{\prime\prime})m^{2}
  23. B ′′ B^{\prime\prime}
  24. B B^{\prime}
  25. B ′′ B^{\prime\prime}
  26. B B^{\prime}
  27. B ν = B e q - α ( ν + 1 2 ) B_{\nu}=B_{eq}-\alpha\left(\nu+{1\over 2}\right)
  28. J \mathrm{J\hbar}\,
  29. Δ J = 0 , ± 2 \Delta J=0,\pm 2
  30. ν ¯ ( S ( J ) ) = ω 0 + 4 B J + 6 B = ω 0 + 6 B , ω 0 + 10 B , ω 0 + 14 B , \bar{\nu}(S(J))=\omega_{0}+4BJ+6B=\omega_{0}+6B,\quad\omega_{0}+10B,\quad% \omega_{0}+14B,\quad...
  31. ν ¯ ( O ( J ) ) = ω 0 - 4 B J + 2 B = ω 0 - 6 B , ω 0 - 10 B , ω 0 - 14 B , \bar{\nu}(O(J))=\omega_{0}-4BJ+2B=\omega_{0}-6B,\quad\omega_{0}-10B,\quad% \omega_{0}-14B,\quad...
  32. Δ 4 ′′ F ( J ) = ν ¯ [ S ( J - 2 ) ] - ν ¯ [ O ( J + 2 ) ] = 4 B ′′ ( 2 J + 1 ) \Delta_{4}^{\prime\prime}F(J)=\bar{\nu}[S(J-2)]-\bar{\nu}[O(J+2)]=4B^{\prime% \prime}(2J+1)
  33. Δ 4 F ( J ) = ν ¯ [ S ( J ) ] - ν ¯ [ O ( J ) ] = 4 B ( 2 J + 1 ) \Delta_{4}^{\prime}F(J)=\bar{\nu}[S(J)]-\bar{\nu}[O(J)]=4B^{\prime}(2J+1)
  34. J ′′ J^{\prime\prime}
  35. J ′′ J^{\prime\prime}
  36. J ′′ J^{\prime\prime}
  37. J ′′ J^{\prime\prime}
  38. \parallel
  39. \perp
  40. F + = B ν J ( J + 1 ) + 2 B ν ζ r ( J + 1 ) F^{+}=B_{\nu}J(J+1)+2B_{\nu}\zeta_{r}(J+1)
  41. F 0 = B ν J ( J + 1 ) F^{0}=B_{\nu}J(J+1)
  42. F - = B ν J ( J + 1 ) - 2 B ν ζ r ( J + 1 ) F^{-}=B_{\nu}J(J+1)-2B_{\nu}\zeta_{r}(J+1)
  43. ζ r \zeta_{r}
  44. Δ J = 0 , ± 1 \Delta J=0,\pm 1
  45. J = J ′′ = 1 , 2... J^{\prime}=J^{\prime\prime}=1,2...
  46. I I_{\perp}
  47. B = h 8 π 2 c I B={h\over{8\pi^{2}cI_{\perp}}}
  48. I I_{\parallel}
  49. A = h 8 π 2 c I A={h\over{8\pi^{2}cI_{\parallel}}}
  50. \parallel
  51. \perp
  52. \parallel
  53. \perp
  54. ν ¯ = ν ¯ 0 + ( B + B ′′ ) m + ( B - B ′′ - D J + D J ′′ ) m 2 \bar{\nu}=\bar{\nu}_{0}+(B^{\prime}+B^{\prime\prime})m+(B^{\prime}-B^{\prime% \prime}-D_{J}^{\prime}+D_{J}^{\prime\prime})m^{2}
  55. - 2 ( D J + D J ′′ ) m 3 - ( D J - D J ′′ ) m 4 -2(D_{J}^{\prime}+D_{J}^{\prime\prime})m^{3}-(D_{J}^{\prime}-D_{J}^{\prime% \prime})m^{4}
  56. + { [ ( A - B ) - ( A ′′ - B ′′ ) ] - [ D J K + D J K ′′ ] m - [ D J K - D J K ′′ ] m 2 } K 2 - ( D K - D K ′′ ) K 4 +\left\{\left[(A^{\prime}-B^{\prime})-(A^{\prime\prime}-B^{\prime\prime})% \right]-\left[D_{JK}^{\prime}+D_{JK}^{\prime\prime}\right]m-\left[D_{JK}^{% \prime}-D_{JK}^{\prime\prime}\right]m^{2}\right\}K^{2}-(D_{K}^{\prime}-D_{K}^{% \prime\prime})K^{4}
  57. D J , D J K D_{J},D_{JK}
  58. D K D_{K}
  59. ν ¯ = ν ¯ s u b + ( B - B ′′ ) J ( J + 1 ) - ( D J - D J ′′ ) J 2 ( J + 1 ) 2 - ( D J K - D J K ′′ ) J ( J + 1 ) K 2 \bar{\nu}=\bar{\nu}_{sub}+(B^{\prime}-B^{\prime\prime})J(J+1)-(D_{J}^{\prime}-% D_{J}^{\prime\prime})J^{2}(J+1)^{2}-(D_{JK}^{\prime}-D_{JK}^{\prime\prime})J(J% +1)K^{2}
  60. ν ¯ s u b \bar{\nu}_{sub}
  61. ν ¯ s u b = ν ¯ 0 + [ ( A - B ) - ( A ′′ - B ′′ ) ] K 2 - ( D K - D K ′′ ) K 4 \bar{\nu}_{sub}=\bar{\nu}_{0}+\left[(A^{\prime}-B^{\prime})-(A^{\prime\prime}-% B^{\prime\prime})\right]K^{2}-(D_{K}^{\prime}-D_{K}^{\prime\prime})K^{4}
  62. Δ 2 F ( J ) o b s e r v e d = ν ¯ [ R ( J ) ] - ν ¯ [ P ( J ) ] \Delta_{2}^{\prime}F(J)^{observed}=\bar{\nu}[R(J)]-\bar{\nu}[P(J)]
  63. Δ 2 F ( J ) c a l c u l a t e d = 2 B ′′ ( 2 J + 1 ) \Delta_{2}^{\prime}F(J)^{calculated}=2B^{\prime\prime}\left(2J+1\right)
  64. Δ 2 F ( J ) c a l c u l a t e d = ( 2 B ′′ - 3 D ′′ ) ( 2 J + 1 ) - D ′′ ( 2 J + 1 ) 3 \Delta_{2}^{\prime}F(J)^{calculated}=(2B^{\prime\prime}-3D^{\prime\prime})% \left(2J+1\right)-D^{\prime\prime}\left(2J+1\right)^{3}
  65. Δ 2 F ( J ) \Delta_{2}^{\prime}F(J)
  66. E = h c G ( v ) E=hcG(v)

Routing_transit_number.html

  1. 3 ( d 1 + d 4 + d 7 ) + 7 ( d 2 + d 5 + d 8 ) + ( d 3 + d 6 + d 9 ) mod 10 = 0. 3(d_{1}+d_{4}+d_{7})+7(d_{2}+d_{5}+d_{8})+(d_{3}+d_{6}+d_{9})\mod 10=0.\,
  2. 5 0 = 5 2 = 5 4 = 5 6 = 5 8 = 0 mod 10 5\cdot 0=5\cdot 2=5\cdot 4=5\cdot 6=5\cdot 8=0\mod 10
  3. 3 ( 1 + 0 + 0 ) + 7 ( 1 + 0 + 2 ) + ( 1 + 0 + 5 ) mod 10 = 0. 3(1+0+0)+7(1+0+2)+(1+0+5)\mod 10=0.\,
  4. d 9 = 7 ( d 1 + d 4 + d 7 ) + 3 ( d 2 + d 5 + d 8 ) + 9 ( d 3 + d 6 ) mod 10. d_{9}=7(d_{1}+d_{4}+d_{7})+3(d_{2}+d_{5}+d_{8})+9(d_{3}+d_{6})\mod 10.\,
  5. d 9 d_{9}
  6. 3 ( 10 - 3 ) = 7 ; 7 ( 10 - 7 ) = 3 ; 1 ( 10 - 1 ) = 9 3\mapsto(10-3)=7;7\mapsto(10-7)=3;1\mapsto(10-1)=9
  7. 7 ( 1 + 0 + 0 ) + 3 ( 1 + 0 + 2 ) + 9 ( 1 + 0 ) = 25 mod 10 = 5. 7(1+0+0)+3(1+0+2)+9(1+0)=25\mod 10=5.\,

Row_echelon_form.html

  1. [ 1 a 0 a 1 a 2 a 3 0 0 2 a 4 a 5 0 0 0 1 a 6 ] \left[\begin{array}[]{ccccc}1&a_{0}&a_{1}&a_{2}&a_{3}\\ 0&0&2&a_{4}&a_{5}\\ 0&0&0&1&a_{6}\end{array}\right]
  2. [ 1 0 a 1 0 b 1 0 1 a 2 0 b 2 0 0 0 1 b 3 ] \left[\begin{array}[]{ccccc}1&0&a_{1}&0&b_{1}\\ 0&1&a_{2}&0&b_{2}\\ 0&0&0&1&b_{3}\end{array}\right]
  3. [ 1 3 - 1 0 1 7 ] add row 2 to row 1 [ 1 4 6 0 1 7 ] . \begin{bmatrix}1&3&-1\\ 0&1&7\\ \end{bmatrix}\xrightarrow{\,\text{add row 2 to row 1}}\begin{bmatrix}1&4&6\\ 0&1&7\\ \end{bmatrix}.
  4. [ 1 3 - 1 0 1 7 ] subtract 3 times row 2 from row 1 [ 1 0 - 22 0 1 7 ] . \begin{bmatrix}1&3&-1\\ 0&1&7\\ \end{bmatrix}\xrightarrow{\,\text{subtract 3 times row 2 from row 1}}\begin{% bmatrix}1&0&-22\\ 0&1&7\\ \end{bmatrix}.

Rule_of_72.html

  1. T = ln ( 2 ) ln ( 1 + r ) T=\frac{\ln(2)}{\ln(1+r)}
  2. t 72 + ( r - 8 ) / 3 r t\approx\frac{72+(r-8)/3}{r}
  3. t 70 + ( r - 2 ) / 3 r t\approx\frac{70+(r-2)/3}{r}
  4. t 69.3 r + 0.33 t\approx\frac{69.3}{r}+0.33
  5. t 69.3 r × 200 200 - r t\approx\frac{69.3}{r}\times\frac{200}{200-r}
  6. t 70 r × 198 200 - r t\approx\frac{70}{r}\times\frac{198}{200-r}
  7. t 72 r × 192 200 - r t\approx\frac{72}{r}\times\frac{192}{200-r}
  8. t 69.3 r × 600 + 4 r 600 + r t\approx\frac{69.3}{r}\times\frac{600+4r}{600+r}
  9. F V = P V ( 1 + r ) t FV=PV\cdot(1+r)^{t}
  10. P V PV
  11. t t
  12. r r
  13. ( 1 + r ) t = 2 (1+r)^{t}=2\,
  14. t t
  15. ln ( ( 1 + r ) t ) = ln 2 t ln ( 1 + r ) = ln 2 t = ln 2 ln ( 1 + r ) \begin{array}[]{ccc}\ln((1+r)^{t})&=&\ln 2\\ t\cdot\ln(1+r)&=&\ln 2\\ t&=&\frac{\ln 2}{\ln(1+r)}\end{array}
  16. l n ( 2 ) 0.693147 ln(2)\approx 0.693147
  17. t 0.693147 r t\approx\frac{0.693147}{r}
  18. t 0.693147 r = 0.693147 R % = 0.693147 R 1 100 = 0.693147 100 R = 69.3147 R 70 R \begin{array}[]{ccc}t&\approx&\frac{0.693147}{r}\\ &&\\ &=&\frac{0.693147}{R\%}\\ &&\\ &=&\frac{0.693147}{R\frac{1}{100}}\\ &&\\ &=&\frac{0.693147\cdot 100}{R}\\ &&\\ &=&\frac{69.3147}{R}\\ &&\\ &\approx&\frac{70}{R}\end{array}
  19. ln ( 1 + r ) \ln(1+r)\,
  20. r - r 2 2 r-\frac{r^{2}}{2}
  21. 0.693 r - r 2 / 2 \frac{0.693}{r-r^{2}/2}
  22. 0.693 r - r 2 / 2 = 69.3 R - R 2 / 200 = 69.3 R 1 1 - R / 200 69.3 ( 1 + R / 200 ) R = 69.3 R + 69.3 200 = 69.3 R + 0.34 \begin{array}[]{ccc}\frac{0.693}{r-r^{2}/2}&=&\frac{69.3}{R-R^{2}/200}\\ &&\\ &=&\frac{69.3}{R}\frac{1}{1-R/200}\\ &&\\ &\approx&\frac{69.3(1+R/200)}{R}\\ &&\\ &=&\frac{69.3}{R}+\frac{69.3}{200}\\ &&\\ &=&\frac{69.3}{R}+0.34\end{array}
  23. ( e r ) p = 2 e r p = 2 ln e r p = ln 2 r p = ln 2 p = ln 2 r p 0.693147 r \begin{array}[]{ccc}(e^{r})^{p}&=&2\\ e^{rp}&=&2\\ \ln e^{rp}&=&\ln 2\\ rp&=&\ln 2\\ p&=&\frac{\ln 2}{r}\\ &&\\ p&\approx&\frac{0.693147}{r}\end{array}

Running_key_cipher.html

  1. 2 40 2^{40}

S-duality.html

  1. g g
  2. A = A 0 + A 1 g + A 2 g 2 + A 3 g 3 + A=A_{0}+A_{1}g+A_{2}g^{2}+A_{3}g^{3}+\dots
  3. g g
  4. g g
  5. 1 / g 1/g
  6. g g
  7. 1 / g 1/g
  8. 𝐄 = 0 , 𝐁 = 0 , × 𝐄 = - 𝐁 t , × 𝐁 = 1 c 2 𝐄 t . \begin{aligned}\displaystyle\nabla\cdot\mathbf{E}&\displaystyle=0,\\ \displaystyle\nabla\cdot\mathbf{B}&\displaystyle=0,\\ \displaystyle\nabla\times\mathbf{E}&\displaystyle=-\frac{\partial\mathbf{B}}{% \partial t},\\ \displaystyle\nabla\times\mathbf{B}&\displaystyle=\frac{1}{c^{2}}\frac{% \partial\mathbf{E}}{\partial t}.\end{aligned}
  9. 𝐄 \mathbf{E}
  10. 𝐁 \mathbf{B}
  11. t t
  12. c c
  13. 𝐄 \mathbf{E}
  14. 𝐁 \mathbf{B}
  15. 𝐁 \mathbf{B}
  16. - 1 / c 2 𝐄 -1/c^{2}\mathbf{E}
  17. 𝐄 \displaystyle\mathbf{E}
  18. G G
  19. G L {{}^{L}}G
  20. G L {{}^{L}}G
  21. G G
  22. τ = θ 2 π + 4 π i g 2 \tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^{2}}
  23. θ \theta
  24. g g
  25. g g
  26. e e
  27. τ \tau
  28. - 1 / τ -1/\tau
  29. g g
  30. 1 / g 1/g
  31. g g
  32. 1 / g 1/g

S-matrix.html

  1. E E
  2. V V
  3. E E
  4. S = S ( E ) S=S(E)
  5. V V
  6. 2 × 2 2×2
  7. V V
  8. V ( x ) V(x)
  9. E E
  10. ψ L ( x ) = A e i k x + B e - i k x \psi_{L}(x)=Ae^{ikx}+Be^{-ikx}
  11. ψ R ( x ) = C e i k x + D e - i k x \psi_{R}(x)=Ce^{ikx}+De^{-ikx}
  12. k = 2 m E / 2 k=\sqrt{2mE/\hbar^{2}}
  13. A A
  14. D D
  15. B B
  16. D D
  17. ( B C ) = ( S 11 S 12 S 21 S 22 ) ( A D ) . \begin{pmatrix}B\\ C\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}A\\ D\end{pmatrix}.
  18. Ψ o u t = S Ψ i n \Psi_{out}=S\Psi_{in}
  19. Ψ o u t = ( B C ) , Ψ i n = ( A D ) , S = ( S 11 S 12 S 21 S 22 ) . \Psi_{out}=\begin{pmatrix}B\\ C\end{pmatrix},\quad\Psi_{in}=\begin{pmatrix}A\\ D\end{pmatrix},\qquad S=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}.
  20. S S
  21. V ( x ) V(x)
  22. J J
  23. ψ ( x ) ψ(x)
  24. J = 2 m i ( ψ * ψ x - ψ ψ * x ) J=\frac{\hbar}{2mi}\left(\psi^{*}\frac{\partial\psi}{\partial x}-\psi\frac{% \partial\psi^{*}}{\partial x}\right)
  25. J L = k m ( | A | 2 - | B | 2 ) J_{L}=\frac{\hbar k}{m}\left(|A|^{2}-|B|^{2}\right)
  26. J R = k m ( | C | 2 - | D | 2 ) J_{R}=\frac{\hbar k}{m}\left(|C|^{2}-|D|^{2}\right)
  27. J L = J R | A | 2 - | B | 2 = | C | 2 - | D | 2 | B | 2 + | C | 2 = | A | 2 + | D | 2 Ψ o u t Ψ o u t = Ψ i n Ψ i n Ψ i n S S Ψ i n = Ψ i n Ψ i n S S = I \begin{aligned}&\displaystyle J_{L}=J_{R}\\ &\displaystyle|A|^{2}-|B|^{2}=|C|^{2}-|D|^{2}\\ &\displaystyle|B|^{2}+|C|^{2}=|A|^{2}+|D|^{2}\\ &\displaystyle\Psi_{out}^{\dagger}\Psi_{out}=\Psi_{in}^{\dagger}\Psi_{in}\\ &\displaystyle\Psi_{in}^{\dagger}S^{\dagger}S\Psi_{in}=\Psi_{in}^{\dagger}\Psi% _{in}\\ &\displaystyle S^{\dagger}S=I\\ \end{aligned}
  28. V ( x ) V(x)
  29. ψ ( x ) ψ(x)
  30. ψ * ( x ) ψ*(x)
  31. ψ L * ( x ) = A * e - i k x + B * e i k x \psi^{*}_{L}(x)=A^{*}e^{-ikx}+B^{*}e^{ikx}
  32. ψ R * ( x ) = C * e - i k x + D * e i k x \psi^{*}_{R}(x)=C^{*}e^{-ikx}+D^{*}e^{ikx}
  33. B * B*
  34. C * C*
  35. A * A*
  36. D * D*
  37. ( A * D * ) = ( S 11 S 12 S 21 S 22 ) ( B * C * ) \begin{pmatrix}A^{*}\\ D^{*}\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}B^{*}\\ C^{*}\end{pmatrix}\,
  38. Ψ i n * = S Ψ o u t * . \Psi^{*}_{in}=S\Psi^{*}_{out}.
  39. Ψ i n * = S Ψ o u t * , Ψ o u t = S Ψ i n \Psi^{*}_{in}=S\Psi^{*}_{out},\quad\Psi_{out}=S\Psi_{in}
  40. S * S = I S^{*}S=I
  41. S T = S . S^{T}=S.
  42. D = 0 D=0
  43. T L = | C | 2 | A | 2 = | S 21 | 2 . T_{L}=\frac{|C|^{2}}{|A|^{2}}=|S_{21}|^{2}.
  44. D = 0 D=0
  45. R L = | B | 2 | A | 2 = | S 11 | 2 . R_{L}=\frac{|B|^{2}}{|A|^{2}}=|S_{11}|^{2}.
  46. A = 0 A=0
  47. T R = | B | 2 | D | 2 = | S 12 | 2 . T_{R}=\frac{|B|^{2}}{|D|^{2}}=|S_{12}|^{2}.
  48. A = 0 A=0
  49. R R = | C | 2 | D | 2 = | S 22 | 2 . R_{R}=\frac{|C|^{2}}{|D|^{2}}=|S_{22}|^{2}.
  50. T L + R L = 1 T_{L}+R_{L}=1
  51. T R + R R = 1. T_{R}+R_{R}=1.
  52. V ( x ) = 0 V(x)=0
  53. S = ( 0 1 1 0 ) . S=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}.
  54. V ( x ) V(x)
  55. S = ( 2 i r 1 + 2 i t 1 + 2 i t 2 i r * 1 + 2 i t 1 - 2 i t * ) . S=\begin{pmatrix}2ir&1+2it\\ 1+2it&2ir^{*}\frac{1+2it}{1-2it^{*}}\end{pmatrix}.
  56. r r
  57. t t
  58. | r | 2 + | t | 2 = Im ( t ) . |r|^{2}+|t|^{2}=\operatorname{Im}(t).
  59. H H
  60. V V
  61. V V
  62. | Ψ ( t ) \left|\Psi(t)\right\rangle
  63. | Φ i . \left|\Phi_{i}\right\rangle.
  64. Φ f | . \left\langle\Phi_{f}\right|.
  65. S f i lim t + Φ f | Ψ ( t ) Φ f | S | Φ i , S_{fi}\equiv\lim_{t\rightarrow+\infty}\left\langle\Phi_{f}|\Psi(t)\right% \rangle\equiv\left\langle\Phi_{f}\right|S\left|\Phi_{i}\right\rangle,
  66. S S
  67. U U
  68. U ( t , t 0 ) = T e - i t 0 t d τ V ( τ ) , U(t,t_{0})=Te^{-i\int_{t_{0}}^{t}d\tau V(\tau)},
  69. T T
  70. S f i = lim t 2 + lim t 1 - Φ f | U ( t 2 , t 1 ) | Φ i , S_{fi}=\lim_{t_{2}\rightarrow+\infty}\lim_{t_{1}\rightarrow-\infty}\left% \langle\Phi_{f}\right|U(t_{2},t_{1})\left|\Phi_{i}\right\rangle,
  71. S = U ( , - ) . S=U(\infty,-\infty).
  72. U U
  73. S = n = 0 ( - i ) n n ! - d t 1 - d t n T [ V ( t 1 ) V ( t n ) ] , S=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{-\infty}^{\infty}dt_{1}\cdots% \int_{-\infty}^{\infty}dt_{n}T\left[V(t_{1})\cdots V(t_{n})\right],
  74. V V
  75. S = n = 0 ( - i ) n n ! - d x 1 4 - d x n 4 T [ ( t 1 ) ( t n ) ] . S=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{-\infty}^{\infty}dx_{1}^{4}% \cdots\int_{-\infty}^{\infty}dx_{n}^{4}T\left[\mathcal{H}(t_{1})\cdots\mathcal% {H}(t_{n})\right].
  76. S S
  77. S f i = Φ f | S | Φ i = Φ f | n = 0 ( - i ) n n ! - d x 1 4 - d x n 4 T [ ( t 1 ) ( t n ) ] | Φ i . S_{fi}=\left\langle\Phi_{f}|S|\Phi_{i}\right\rangle=\left\langle\Phi_{f}\left|% \sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int_{-\infty}^{\infty}dx_{1}^{4}\cdots% \int_{-\infty}^{\infty}dx_{n}^{4}T\left[\mathcal{H}(t_{1})\cdots\mathcal{H}(t_% {n})\right]\right|\Phi_{i}\right\rangle.
  78. a ( k ) a(k)
  79. a ( k ) | * , 0 = 0. a(k)\left|*,0\right\rangle=0.
  80. | * , 0 |*,0\rangle
  81. 0
  82. P μ | * , 0 = 0 , M μ ν | * , 0 = | * , 0 , P^{\mu}|*,0\rangle=0,\quad M^{\mu\nu}|*,0\rangle=|*,0\rangle,
  83. ( 2 + m 2 ) ϕ i , o ( x ) = 0 , (\Box^{2}+m^{2})\phi_{i,o}(x)=0,
  84. [ ϕ i , o ( x ) , π i , o ( y ) ] x 0 = y 0 = i δ ( 𝐱 - 𝐲 ) , [ ϕ i , o ( x ) , ϕ i , o ( y ) ] x 0 = y 0 = [ π i , o ( x ) , π i , o ( y ) ] x 0 = y 0 = 0 , \begin{aligned}\displaystyle{[\phi_{i,o}(x),\pi_{i,o}(y)]}_{x_{0}=y_{0}}&% \displaystyle=i\delta(\mathbf{x}-\mathbf{y}),\\ \displaystyle{[\phi_{i,o}(x),\phi_{i,o}(y)]}_{x_{0}=y_{0}}&\displaystyle={[\pi% _{i,o}(x),\pi_{i,o}(y)]}_{x_{0}=y_{0}}=0\end{aligned},
  85. i i
  86. f f
  87. [ a i , o ( 𝐩 ) , a i , o ( 𝐩 ) ] = i δ ( 𝐩 - 𝐩 ) , [ a i , o ( 𝐩 ) , a i , o ( 𝐩 ) ] = [ a i , o ( 𝐩 ) , a i , o ( 𝐩 ) ] = 0. \begin{aligned}\displaystyle{[a_{i,o}(\mathbf{p}),a^{\dagger}_{i,o}(\mathbf{p}% ^{\prime})]}&\displaystyle=i\delta(\mathbf{p}-\mathbf{p^{\prime}}),\\ \displaystyle{[a_{i,o}(\mathbf{p}),a_{i,o}(\mathbf{p^{\prime}})]}&% \displaystyle={[a^{\dagger}_{i,o}(\mathbf{p}),a^{\dagger}_{i,o}(\mathbf{p^{% \prime}})]}=0.\end{aligned}
  88. | i , k 1 k n = a i ( k 1 ) a i ( k n ) | i , 0 , | f , p 1 p n = a f ( p 1 ) a f ( p n ) | f , 0 , \begin{aligned}\displaystyle\left|i,k_{1}\ldots k_{n}\right\rangle&% \displaystyle=a_{i}^{\dagger}(k_{1})\cdots a_{i}^{\dagger}(k_{n})\left|i,0% \right\rangle,\\ \displaystyle\left|f,p_{1}\ldots p_{n}\right\rangle&\displaystyle=a_{f}^{% \dagger}(p_{1})\cdots a_{f}^{\dagger}(p_{n})\left|f,0\right\rangle,\end{aligned}
  89. i = span { | i , k 1 k n = a i ( k 1 ) a i ( k n ) | i , 0 } , \mathcal{H}_{i}=\operatorname{span}\{\left|i,k_{1}\ldots k_{n}\right\rangle=a_% {i}^{\dagger}(k_{1})\cdots a_{i}^{\dagger}(k_{n})\left|i,0\right\rangle\},
  90. f = span { | f , p 1 p n = a f ( p 1 ) a f ( p n ) | f , 0 } . \mathcal{H}_{f}=\operatorname{span}\{\left|f,p_{1}\ldots p_{n}\right\rangle=a_% {f}^{\dagger}(p_{1})\cdots a_{f}^{\dagger}(p_{n})\left|f,0\right\rangle\}.
  91. P μ | i , k 1 k m = k 1 μ + + k m μ | i , k 1 k m , P μ | f , p 1 p n = p 1 μ + + p n μ | f , p 1 p n . P^{\mu}\left|i,k_{1}\ldots k_{m}\right\rangle=k_{1}^{\mu}+\cdots+k_{m}^{\mu}% \left|i,k_{1}\ldots k_{m}\right\rangle,\quad P^{\mu}\left|f,p_{1}\ldots p_{n}% \right\rangle=p_{1}^{\mu}+\cdots+p_{n}^{\mu}\left|f,p_{1}\ldots p_{n}\right\rangle.
  92. H = P 0 . H=P^{0}.
  93. | i , 0 = | f , 0 = | * , 0 | 0 . |i,0\rangle=|f,0\rangle=|*,0\rangle\equiv|0\rangle.
  94. t t→−∞
  95. α α
  96. β β
  97. t + t→+∞
  98. Ψ Ψ
  99. t = 0 t=0
  100. t = τ t=τ
  101. τ τ
  102. Ψ Ψ
  103. t = τ t=τ
  104. t = 0 t=0
  105. Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 ; , \Psi_{p_{1}\sigma_{1}n_{1};p_{2}\sigma_{2}n_{2};\cdots},
  106. p p
  107. σ σ
  108. n n
  109. ( Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 ; , Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 ; ) = δ 3 ( 𝐩 1 - 𝐩 1 ) δ σ 1 σ 1 δ n 1 n 1 δ 3 ( 𝐩 2 - 𝐩 2 ) δ σ 2 σ 2 δ n 2 n 2 ± permutations . \left(\Psi_{p_{1}^{\prime}\sigma_{1}^{\prime}n_{1}^{\prime};p_{2}^{\prime}% \sigma_{2}^{\prime}n_{2}^{\prime};\cdots},\Psi_{p_{1}\sigma_{1}n_{1};p_{2}% \sigma_{2}n_{2};\cdots}\right)=\delta^{3}(\mathbf{p}_{1}^{\prime}-\mathbf{p}_{% 1})\delta_{\sigma_{1}^{\prime}\sigma_{1}}\delta_{n_{1}^{\prime}n_{1}}\delta^{3% }(\mathbf{p}_{2}^{\prime}-\mathbf{p}_{2})\delta_{\sigma_{2}^{\prime}\sigma_{2}% }\delta_{n_{2}^{\prime}n_{2}}\cdots\quad\pm\,\text{ permutations}.
  110. k k
  111. n s ( i ) = n i , 1 i k , n_{s(i)}^{\prime}=n_{i},\quad 1\leq i\leq k,
  112. s s
  113. ( Ψ α , Ψ α ) = δ ( α - α ) . \left(\Psi_{\alpha^{\prime}},\Psi_{\alpha}\right)=\delta(\alpha^{\prime}-% \alpha).
  114. d α n 1 σ 1 n 2 σ 2 d 3 p 1 d 3 p 2 , \int d\alpha\cdots\equiv\sum_{n_{1}\sigma_{1}n_{2}\sigma_{2}\cdots}\int d^{3}p% _{1}d^{3}p_{2}\cdots,
  115. Ψ = d α Ψ α ( Ψ α , Ψ ) , \Psi=\int d\alpha\Psi_{\alpha}\left(\Psi_{\alpha},\Psi\right),
  116. d α | Ψ α Ψ α | = 1 , \int d\alpha\left|\Psi_{\alpha}\right\rangle\left\langle\Psi_{\alpha}\right|=1,
  117. α α
  118. α α
  119. ( Λ , a ) (Λ,a)
  120. U ( Λ , a ) Ψ p 1 σ 1 n 1 ; p 2 σ 2 n 2 = e - i a μ ( ( Λ p 1 ) μ + ( Λ p 2 ) μ + ) ( Λ p 1 ) 0 ( Λ p 2 ) 0 p 1 0 p 2 0 σ 1 σ 2 D σ 1 σ 1 ( j 1 ) ( W ( Λ , p 1 ) ) D σ 2 σ 2 ( j 2 ) ( W ( Λ , p 2 ) ) Ψ Λ p 1 σ 1 n 1 ; Λ p 2 σ 2 n 2 , U(\Lambda,a)\Psi_{p_{1}\sigma_{1}n_{1};p_{2}\sigma_{2}n_{2}\cdots}=e^{-ia_{\mu% }((\Lambda p_{1})^{\mu}+(\Lambda p_{2})^{\mu}+\cdots)}\sqrt{\frac{(\Lambda p_{% 1})^{0}(\Lambda p_{2})^{0}\cdots}{p_{1}^{0}p_{2}^{0}\cdots}}\sum_{\sigma_{1}^{% \prime}\sigma_{2}^{\prime}\cdots}D_{\sigma_{1}^{\prime}\sigma_{1}}^{(j_{1})}(W% (\Lambda,p_{1}))D_{\sigma_{2}^{\prime}\sigma_{2}}^{(j_{2})}(W(\Lambda,p_{2}))% \cdots\Psi_{\Lambda p_{1}\sigma_{1}^{\prime}n_{1};\Lambda p_{2}\sigma_{2}^{% \prime}n_{2}\cdots},
  121. W ( Λ , p ) W(Λ,p)
  122. S O ( 3 ) SO(3)
  123. Λ = 1 , a = ( τ , 0 , 0 , 0 ) Λ=1,a=(τ,0,0,0)
  124. U U
  125. e x p ( i H τ ) exp(iHτ)
  126. H Ψ = E α Ψ , E α = p 1 0 + p 2 0 + , H\Psi=E_{\alpha}\Psi,\quad E_{\alpha}=p_{1}^{0}+p_{2}^{0}+\cdots,
  127. e - i H τ d α g ( α ) Ψ α ± = d α e - i E α τ g ( α ) Ψ α ± e^{-iH\tau}\int d\alpha g(\alpha)\Psi_{\alpha}^{\pm}=\int d\alpha e^{-iE_{% \alpha}\tau}g(\alpha)\Psi_{\alpha}^{\pm}
  128. τ τ
  129. g g
  130. g g
  131. H H
  132. V V
  133. H 0 Φ α = E α Φ α , H_{0}\Phi_{\alpha}=E_{\alpha}\Phi_{\alpha},
  134. ( Φ α , Φ α ) = δ ( α - α ) . (\Phi_{\alpha}^{\prime},\Phi_{\alpha})=\delta(\alpha^{\prime}-\alpha).
  135. H Ψ α ± = E α Ψ α ± , H\Psi_{\alpha}^{\pm}=E_{\alpha}\Psi_{\alpha}^{\pm},
  136. e - i H τ d α g ( α ) Ψ α ± e - i H 0 τ d α g ( α ) Φ α . e^{-iH\tau}\int d\alpha g(\alpha)\Psi_{\alpha}^{\pm}\rightarrow e^{-iH_{0}\tau% }\int d\alpha g(\alpha)\Phi_{\alpha}.
  137. τ τ→−∞
  138. τ + τ→+∞
  139. Ω ( τ ) e + i H τ e - i H 0 τ , \Omega(\tau)\equiv e^{+iH\tau}e^{-iH_{0}\tau},
  140. Ψ α ± = Ω ( ) Φ α . \Psi_{\alpha}^{\pm}=\Omega(\mp\infty)\Phi_{\alpha}.
  141. ( Ψ β + , Ψ α + ) = ( Φ β , Φ α ) = ( Ψ β - , Ψ α - ) = δ ( β - α ) , (\Psi_{\beta}^{+},\Psi_{\alpha}^{+})=(\Phi_{\beta},\Phi_{\alpha})=(\Psi_{\beta% }^{-},\Psi_{\alpha}^{-})=\delta(\beta-\alpha),
  142. ( E α - H 0 ± i ϵ ) Ψ α ± = ± i ϵ Ψ α ± + V Ψ α ± , (E_{\alpha}-H_{0}\pm i\epsilon)\Psi_{\alpha}^{\pm}=\pm i\epsilon\Psi_{\alpha}^% {\pm}+V\Psi_{\alpha}^{\pm},
  143. ± i ε ±iε
  144. V 0 V→0
  145. i ϵ Ψ α ± = i ϵ Φ α i\epsilon\Psi_{\alpha}^{\pm}=i\epsilon\Phi_{\alpha}
  146. Ψ α ± = Φ α + ( E α - H 0 ± i ϵ ) - 1 V Ψ α ± . \Psi_{\alpha}^{\pm}=\Phi_{\alpha}+(E_{\alpha}-H_{0}\pm i\epsilon)^{-1}V\Psi_{% \alpha}^{\pm}.
  147. V Ψ α ± = d β ( Φ β , V Ψ α ± ) Φ β d β T β α ± Φ β , V\Psi_{\alpha}^{\pm}=\int d\beta(\Phi_{\beta},V\Psi_{\alpha}^{\pm})\Phi_{\beta% }\equiv\int d\beta T_{\beta\alpha}^{\pm}\Phi_{\beta},
  148. Ψ α ± = Φ α + d β T β α ± Φ β E α - E β ± i ϵ . \Psi_{\alpha}^{\pm}=\Phi_{\alpha}+\int d\beta\frac{T_{\beta\alpha}^{\pm}\Phi_{% \beta}}{E_{\alpha}-E_{\beta}\pm i\epsilon}.
  149. Ψ α - = d β ( Ψ β + , Ψ α - ) Ψ β + = d β | Ψ β + Ψ β + | Ψ α - = n 1 σ 1 n 2 σ 2 d 3 p 1 d 3 p 2 ( Ψ β + , Ψ α - ) Ψ β + , \Psi_{\alpha}^{-}=\int d\beta(\Psi_{\beta}^{+},\Psi_{\alpha}^{-})\Psi_{\beta}^% {+}=\int d\beta|\Psi_{\beta}^{+}\rangle\langle\Psi_{\beta}^{+}|\Psi_{\alpha}^{% -}\rangle=\sum_{n_{1}\sigma_{1}n_{2}\sigma_{2}\cdots}\int d^{3}p_{1}d^{3}p_{2}% \cdots(\Psi_{\beta}^{+},\Psi_{\alpha}^{-})\Psi_{\beta}^{+},
  150. Ψ α - = | i , k 1 k n = C 0 | f , 0 + m = 1 d 4 p 1 d 4 p m C m ( p 1 p m ) | f , p 1 p m , \Psi_{\alpha}^{-}=\left|i,k_{1}\ldots k_{n}\right\rangle=C_{0}\left|f,0\right% \rangle\ +\sum_{m=1}^{\infty}\int{d^{4}p_{1}\ldots d^{4}p_{m}C_{m}(p_{1}\ldots p% _{m})\left|f,p_{1}\ldots p_{m}\right\rangle}~{},
  151. | i , k 1 k n = Ψ α - \left|i,k_{1}\ldots k_{n}\right\rangle=\Psi_{\alpha}^{-}
  152. | f , p 1 p m = Ψ β + \left|f,p_{1}\ldots p_{m}\right\rangle=\Psi_{\beta}^{+}
  153. C m ( p 1 p m ) = f , p 1 p m | i , k 1 k n = ( Ψ β + , Ψ α - ) C_{m}(p_{1}\ldots p_{m})=\left\langle f,p_{1}\ldots p_{m}\right|i,k_{1}\ldots k% _{n}\rangle=(\Psi_{\beta}^{+},\Psi_{\alpha}^{-})
  154. | i , k 1 k n = C 0 | f , 0 + m = 1 d 4 p 1 d 4 p m | f , p 1 p m f , p 1 p m | i , k 1 k n . \left|i,k_{1}\ldots k_{n}\right\rangle=C_{0}\left|f,0\right\rangle\ +\sum_{m=1% }^{\infty}\int{d^{4}p_{1}\ldots d^{4}p_{m}\left|f,p_{1}\ldots p_{m}\right% \rangle}\left\langle f,p_{1}\ldots p_{m}\right|i,k_{1}\ldots k_{n}\rangle~{}.
  155. S β α = Ψ β - | Ψ α + = f , β | i , α , | f , β f , | i , α i . S_{\beta\alpha}=\langle\Psi_{\beta}^{-}|\Psi_{\alpha}^{+}\rangle=\langle f,% \beta|i,\alpha\rangle,\qquad|f,\beta\rangle\in\mathcal{H}_{f},\quad|i,\alpha% \rangle\in\mathcal{H}_{i}.
  156. α α
  157. β β
  158. S S
  159. Φ β | S | Φ α S β α , \langle\Phi_{\beta}|S|\Phi_{\alpha}\rangle\equiv S_{\beta\alpha},
  160. Ψ β + | S | Ψ α + = S β α = Ψ β - | S | Ψ α - . \langle\Psi_{\beta}^{+}|S|\Psi_{\alpha}^{+}\rangle=S_{\beta\alpha}=\langle\Psi% _{\beta}^{-}|S|\Psi_{\alpha}^{-}\rangle.
  161. S S
  162. Ψ β - | S | Ψ α - = S β α = Ψ β - | Ψ α + . \langle\Psi_{\beta}^{-}|S|\Psi_{\alpha}^{-}\rangle=S_{\beta\alpha}=\langle\Psi% _{\beta}^{-}|\Psi_{\alpha}^{+}\rangle.
  163. S | Ψ α - = | Ψ α + , S|\Psi_{\alpha}^{-}\rangle=|\Psi_{\alpha}^{+}\rangle,
  164. S S
  165. S | 0 = | 0 S\left|0\right\rangle=\left|0\right\rangle
  166. ϕ f = S ϕ i S - 1 . \phi_{f}=S\phi_{i}S^{-1}~{}.
  167. a f ( p ) = S a i ( p ) S - 1 , a f ( p ) = S a i ( p ) S - 1 , a_{f}(p)=Sa_{i}(p)S^{-1},a_{f}^{\dagger}(p)=Sa_{i}^{\dagger}(p)S^{-1},
  168. S | i , k 1 , k 2 , , k n = S a i ( k 1 ) a i ( k 2 ) a i ( k n ) | 0 = S a i ( k 1 ) S - 1 S a i ( k 2 ) S - 1 S a i ( k n ) S - 1 S | 0 = a o ( k 1 ) a o ( k 2 ) a o ( k n ) S | 0 = a o ( k 1 ) a o ( k 2 ) a o ( k n ) | 0 = | o , k 1 , k 2 , , k n . \begin{aligned}\displaystyle S|i,k_{1},k_{2},\ldots,k_{n}\rangle&\displaystyle% =Sa_{i}^{\dagger}(k_{1})a_{i}^{\dagger}(k_{2})\cdots a_{i}^{\dagger}(k_{n})|0% \rangle=Sa_{i}^{\dagger}(k_{1})S^{-1}Sa_{i}^{\dagger}(k_{2})S^{-1}\cdots Sa_{i% }^{\dagger}(k_{n})S^{-1}S|0\rangle\\ &\displaystyle=a_{o}^{\dagger}(k_{1})a_{o}^{\dagger}(k_{2})\cdots a_{o}^{% \dagger}(k_{n})S|0\rangle=a_{o}^{\dagger}(k_{1})a_{o}^{\dagger}(k_{2})\cdots a% _{o}^{\dagger}(k_{n})|0\rangle=|o,k_{1},k_{2},\ldots,k_{n}\rangle.\end{aligned}
  169. S S
  170. S β α = o , β | i , α = i , β | S | i , α = o , β | S | o , α . S_{\beta\alpha}=\langle o,\beta|i,\alpha\rangle=\langle i,\beta|S|i,\alpha% \rangle=\langle o,\beta|S|o,\alpha\rangle.
  171. S S
  172. | k |k⟩
  173. S | k = | k S|k⟩=|k⟩
  174. a ( k , t ) = U - 1 ( t ) a i ( k ) U ( t ) a^{\dagger}\left(k,t\right)=U^{-1}(t)a^{\dagger}_{i}\left(k\right)U\left(t\right)
  175. a ( k , t ) = U - 1 ( t ) a i ( k ) U ( t ) , a\left(k,t\right)=U^{-1}(t)a_{i}\left(k\right)U\left(t\right)~{},
  176. ϕ f = U - 1 ( ) ϕ i U ( ) = S - 1 ϕ i S , \phi_{f}=U^{-1}(\infty)\phi_{i}U(\infty)=S^{-1}\phi_{i}S~{},
  177. S = e i α U ( ) S=e^{i\alpha}\,U(\infty)
  178. e i α = 0 | U ( ) | 0 - 1 , e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1}~{},
  179. S S
  180. S | 0 = | 0 0 | S | 0 = 0 | 0 = 1 . S\left|0\right\rangle=\left|0\right\rangle\Longrightarrow\left\langle 0|S|0% \right\rangle=\left\langle 0|0\right\rangle=1~{}.
  181. U U
  182. S = 1 0 | U ( ) | 0 𝒯 e - i d τ H int ( τ ) , S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal{T}e^{-i\int{d\tau H% _{\rm{int}}(\tau)}}~{},
  183. H int H_{\rm{int}}
  184. 𝒯 \mathcal{T}
  185. S = n = 0 ( - i ) n n ! d 4 x 1 d 4 x 2 d 4 x n T [ int ( x 1 ) int ( x 2 ) int ( x n ) ] S=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int\cdots\int d^{4}x_{1}d^{4}x_{2}% \ldots d^{4}x_{n}T[\mathcal{H}_{\rm{int}}(x_{1})\mathcal{H}_{\rm{int}}(x_{2})% \cdots\mathcal{H}_{\rm{int}}(x_{n})]
  186. T [ ] T[\cdots]
  187. int ( x ) \;\mathcal{H}_{\rm{int}}(x)
  188. H H
  189. H < s u b > 0 H<sub>0
  190. Λ Λ
  191. U ( Λ ) U(Λ)
  192. H < s u b > i H<sub>i

Sallen–Key_topology.html

  1. v + = v - = v out . v_{+}=v_{-}=v_{\,\text{out}}.\,
  2. ( 1 ) (1)\,
  3. v in - v x Z 1 = v x - v out Z 3 + v x - v - Z 2 . \frac{v_{\,\text{in}}-v_{x}}{Z_{1}}=\frac{v_{x}-v_{\,\text{out}}}{Z_{3}}+\frac% {v_{x}-v_{-}}{Z_{2}}.
  4. ( 2 ) (2)\,
  5. v in - v x Z 1 = v x - v out Z 3 + v x - v out Z 2 \frac{v_{\,\text{in}}-v_{x}}{Z_{1}}=\frac{v_{x}-v_{\,\text{out}}}{Z_{3}}+\frac% {v_{x}-v_{\,\text{out}}}{Z_{2}}
  6. v x - v out Z 2 = v out Z 4 , \frac{v_{x}-v_{\,\text{out}}}{Z_{2}}=\frac{v_{\,\text{out}}}{Z_{4}},
  7. v x = v out ( Z 2 Z 4 + 1 ) . v_{x}=v_{\,\text{out}}\left(\frac{Z_{2}}{Z_{4}}+1\right).
  8. ( 3 ) (3)\,
  9. v in - v out ( Z 2 Z 4 + 1 ) Z 1 = v out ( Z 2 Z 4 + 1 ) - v out Z 3 + v out ( Z 2 Z 4 + 1 ) - v out Z 2 . \frac{v_{\,\text{in}}-v_{\,\text{out}}\left(\frac{Z_{2}}{Z_{4}}+1\right)}{Z_{1% }}=\frac{v_{\,\text{out}}\left(\frac{Z_{2}}{Z_{4}}+1\right)-v_{\,\text{out}}}{% Z_{3}}+\frac{v_{\,\text{out}}\left(\frac{Z_{2}}{Z_{4}}+1\right)-v_{\,\text{out% }}}{Z_{2}}.
  10. ( 4 ) (4)\,
  11. v out v in = Z 3 Z 4 Z 1 Z 2 + Z 3 ( Z 1 + Z 2 ) + Z 3 Z 4 , \frac{v_{\,\text{out}}}{v_{\,\text{in}}}=\frac{Z_{3}Z_{4}}{Z_{1}Z_{2}+Z_{3}(Z_% {1}+Z_{2})+Z_{3}Z_{4}},
  12. ( 5 ) (5)\,
  13. Z 3 Z_{3}\,
  14. Z 1 Z_{1}\,
  15. Z 3 Z_{3}\,
  16. Z 2 Z_{2}\,
  17. Z 4 Z_{4}\,
  18. Z 3 Z_{3}\,
  19. Z 1 Z_{1}\,
  20. Z 2 Z_{2}\,
  21. Z 4 Z_{4}\,
  22. Z 3 Z_{3}\,
  23. R R\,
  24. Z R Z_{\mathrm{R}}\,
  25. Z R = R , Z_{\mathrm{R}}=R\,,
  26. C C\,
  27. Z C Z_{\mathrm{C}}\,
  28. Z C = 1 s C , Z_{\mathrm{C}}=\frac{1}{sC}\,,
  29. s = j ω = ( - 1 ) 2 π f s=j\omega=\left(\sqrt{-1}\right)2\pi f\,
  30. f f\,
  31. Z 1 = R 1 , Z 2 = R 2 , Z 3 = 1 s C 1 , and Z 4 = 1 s C 2 . Z_{1}=R_{1},\quad Z_{2}=R_{2},\quad Z_{3}=\frac{1}{sC_{1}},\quad\,\text{and}% \quad Z_{4}=\frac{1}{sC_{2}}.\,
  32. H ( s ) = ω 0 2 s 2 + 2 α s + ω 0 2 H(s)={{\omega_{0}}^{2}\over s^{2}+2\alpha s+{\omega_{0}}^{2}}
  33. f 0 f_{0}\,
  34. α \alpha
  35. Q Q\,
  36. ζ \zeta
  37. ω 0 = 2 π f 0 = 1 R 1 R 2 C 1 C 2 \omega_{0}=2\pi f_{0}=\frac{1}{\sqrt{R_{1}R_{2}C_{1}C_{2}}}
  38. 2 α = 2 ζ ω 0 = ω 0 Q = 1 C 1 ( 1 R 1 + 1 R 2 ) = 1 C 1 ( R 1 + R 2 R 1 R 2 ) . 2\alpha=2\zeta\omega_{0}=\frac{\omega_{0}}{Q}=\frac{1}{C_{1}}\left(\frac{1}{R_% {1}}+\frac{1}{R_{2}}\right)={1\over C_{1}}\left({R_{1}+R_{2}\over R_{1}R_{2}}% \right).\,
  39. Q = ω 0 2 α = R 1 R 2 C 1 C 2 C 2 ( R 1 + R 2 ) Q={\omega_{0}\over 2\alpha}=\frac{\sqrt{R_{1}R_{2}C_{1}C_{2}}}{C_{2}\left(R_{1% }+R_{2}\right)}\qquad
  40. Q Q\,
  41. f 0 f_{0}\,
  42. s = - α ± j α 2 - ω 0 2 s=-\alpha\pm j\sqrt{\alpha^{2}-{\omega_{0}}^{2}}
  43. Q Q\,
  44. f 0 f_{0}\,
  45. Q Q
  46. Q Q\,
  47. 1 / 2 1/\sqrt{2}\,
  48. Q = 1 / 2 Q=1/2
  49. C 1 C_{1}\,
  50. C 2 C_{2}\,
  51. n n\,
  52. R 1 R_{1}\,
  53. R 2 R_{2}\,
  54. m m\,
  55. R 1 = m R , R_{1}=mR,\,
  56. R 2 = R , R_{2}=R,\,
  57. C 1 = n C , C_{1}=nC,\,
  58. C 2 = C . C_{2}=C.\,
  59. f 0 f_{0}\,
  60. Q Q\,
  61. ω 0 = 2 π f 0 = 1 R C m n , \omega_{0}=2\pi f_{0}=\frac{1}{RC\sqrt{mn}},\,
  62. Q = m n m + 1 . Q=\frac{\sqrt{mn}}{m+1}.
  63. f 0 f_{0}\,
  64. 15.9 kHz 15.9\,\,\text{kHz}\,
  65. Q Q\,
  66. 0.5 0.5\,
  67. H ( s ) = 1 1 + C 2 ( R 1 + R 2 ) 2 ζ ω 0 = 1 ω 0 Q s + C 1 C 2 R 1 R 2 1 ω 0 2 s 2 , H(s)=\frac{1}{1+\underbrace{C_{2}(R_{1}+R_{2})}_{\frac{2\zeta}{\omega_{0}}=% \frac{1}{\omega_{0}Q}}s+\underbrace{C_{1}C_{2}R_{1}R_{2}}_{\frac{1}{{\omega_{0% }}^{2}}}s^{2}},
  68. H ( s ) = 1 1 + R C ( m + 1 ) 2 ζ ω 0 = 1 ω 0 Q s + m n R 2 C 2 1 ω 0 2 s 2 H(s)=\frac{1}{1+\underbrace{RC(m+1)}_{\frac{2\zeta}{\omega_{0}}=\frac{1}{% \omega_{0}Q}}s+\underbrace{mnR^{2}C^{2}}_{\frac{1}{{\omega_{0}}^{2}}}s^{2}}
  69. ( R , C ) (R,C)\,
  70. ( m , n ) (m,n)\,
  71. f 0 f_{0}
  72. Q Q
  73. Z ( s ) = R 1 s 2 + s / Q + 1 s 2 + s k / Q , Z(s)=R_{1}\frac{s^{{}^{\prime}2}+s^{^{\prime}}/Q+1}{s^{{}^{\prime}2}+s^{^{% \prime}}k/Q},
  74. s = s ω o s^{^{\prime}}=\frac{s}{\omega_{o}}
  75. k = R 1 R 1 + R 2 = m m + 1 k=\frac{R_{1}}{R_{1}+R_{2}}=\frac{m}{m+1}
  76. Q > 1 - k 2 2 Q>\sqrt{\frac{1-k^{2}}{2}}
  77. | Z ( s ) | m i n = R 1 1 - ( 2 Q 2 + k 2 - 1 ) 2 2 Q 4 + k 2 ( 2 Q 2 + k 2 - 1 ) 2 + 2 Q 2 Q 4 + k 2 ( 2 Q 2 + k 2 - 1 ) |Z(s)|_{min}=R_{1}\sqrt{1-\frac{(2Q^{2}+k^{2}-1)^{2}}{2Q^{4}+k^{2}(2Q^{2}+k^{2% }-1)^{2}+2Q^{2}\sqrt{Q^{4}+k^{2}(2Q^{2}+k^{2}-1)}}}
  78. | Z ( s ) | m i n R 1 1 Q 2 + k 2 + 0.34 |Z(s)|_{min}\approx R_{1}\sqrt{\frac{1}{Q^{2}+k^{2}+0.34}}
  79. 0.25 k 0.75 0.25\leq k\leq 0.75
  80. k k
  81. ω m i n = ω o Q 2 + Q 4 + k 2 ( 2 Q 2 + k 2 - 1 ) 2 Q 2 + k 2 - 1 \omega_{min}=\omega_{o}\sqrt{\frac{Q^{2}+\sqrt{Q^{4}+k^{2}(2Q^{2}+k^{2}-1)}}{2% Q^{2}+k^{2}-1}}
  82. ω m i n ω o 2 Q 2 2 Q 2 + k 2 - 1 \omega_{min}\approx\omega_{o}\sqrt{\frac{2Q^{2}}{2Q^{2}+k^{2}-1}}
  83. f 0 f_{0}\,
  84. 72 Hz 72\,\,\text{Hz}\,
  85. Q Q\,
  86. 0.5 0.5\,
  87. H ( s ) = s 2 s 2 + 2 π ( f 0 Q ) 2 ζ ω 0 = ω 0 Q s + ( 2 π f 0 ) 2 ω 0 2 , H(s)=\frac{s^{2}}{s^{2}+\underbrace{2\pi\left(\frac{f_{0}}{Q}\right)}_{2\zeta% \omega_{0}=\frac{\omega_{0}}{Q}}s+\underbrace{(2\pi f_{0})^{2}}_{{\omega_{0}}^% {2}}},
  88. f 0 f_{0}\,
  89. Q Q\,
  90. ω 0 = 2 π f 0 = 1 R 1 R 2 C 1 C 2 \omega_{0}=2\pi f_{0}=\frac{1}{\sqrt{R_{1}R_{2}C_{1}C_{2}}}\,
  91. 1 2 ζ = Q = ω 0 2 α = R 1 R 2 C 1 C 2 R 1 ( C 1 + C 2 ) . \frac{1}{2\zeta}=Q={\omega_{0}\over 2\alpha}=\frac{\sqrt{R_{1}R_{2}C_{1}C_{2}}% }{R_{1}(C_{1}+C_{2})}.\,
  92. 2 α = 2 ζ ω 0 = ω 0 Q = C 1 + C 2 R 2 C 1 C 2 . 2\alpha=2\zeta\omega_{0}=\frac{\omega_{0}}{Q}=\frac{C_{1}+C_{2}}{R_{2}C_{1}C_{% 2}}.
  93. H ( s ) = ( 1 + R b R a ) G s R 1 C 1 s 2 + ( 1 R 1 C 1 + 1 R 2 C 1 + 1 R 2 C 2 - R b R a R f C 1 ) 2 ζ ω 0 = ω 0 Q s + R 1 + R f R 1 R f R 2 C 1 C 2 ω 0 2 = ( 2 π f 0 ) 2 H(s)=\frac{\overbrace{\left(1+\frac{R_{\mathrm{b}}}{R_{\mathrm{a}}}\right)}^{G% }\frac{s}{R_{1}C_{1}}}{s^{2}+\underbrace{\left(\frac{1}{R_{1}C_{1}}+\frac{1}{R% _{2}C_{1}}+\frac{1}{R_{2}C_{2}}-\frac{R_{\mathrm{b}}}{R_{\mathrm{a}}R_{\mathrm% {f}}C_{1}}\right)}_{2\zeta\omega_{0}=\frac{\omega_{0}}{Q}}s+\underbrace{\frac{% R_{1}+R_{\mathrm{f}}}{R_{1}R_{\mathrm{f}}R_{2}C_{1}C_{2}}}_{{\omega_{0}}^{2}=(% 2\pi f_{0})^{2}}}
  94. f 0 f_{0}
  95. f 0 = 1 2 π R f + R 1 C 1 C 2 R 1 R 2 R f f_{0}=\frac{1}{2\pi}\sqrt{\frac{R_{\mathrm{f}}+R_{1}}{C_{1}C_{2}R_{1}R_{2}R_{% \mathrm{f}}}}
  96. Q Q
  97. Q = ω 0 2 ζ ω 0 = ω 0 ω 0 Q = R 1 + R f R 1 R f R 2 C 1 C 2 1 R 1 C 1 + 1 R 2 C 1 + 1 R 2 C 2 - R b R a R f C 1 = ( R 1 + R f ) R 1 R f R 2 C 1 C 2 R 1 R f ( C 1 + C 2 ) + R 2 C 2 ( R f - R b R a R 1 ) \begin{aligned}\displaystyle Q&\displaystyle=\frac{\omega_{0}}{2\zeta\omega_{0% }}=\frac{\omega_{0}}{\frac{\omega_{0}}{Q}}\\ &\displaystyle=\frac{\sqrt{\frac{R_{1}+R_{\mathrm{f}}}{R_{1}R_{\mathrm{f}}R_{2% }C_{1}C_{2}}}}{\frac{1}{R_{1}C_{1}}+\frac{1}{R_{2}C_{1}}+\frac{1}{R_{2}C_{2}}-% \frac{R_{\mathrm{b}}}{R_{\mathrm{a}}R_{\mathrm{f}}C_{1}}}\\ &\displaystyle=\frac{\sqrt{(R_{1}+R_{\mathrm{f}})R_{1}R_{\mathrm{f}}R_{2}C_{1}% C_{2}}}{R_{1}R_{\mathrm{f}}(C_{1}+C_{2})+R_{2}C_{2}(R_{\mathrm{f}}-\frac{R_{% \mathrm{b}}}{R_{\mathrm{a}}}R_{1})}\end{aligned}
  98. G G
  99. G = 1 + R b R a G=1+\frac{R_{\mathrm{b}}}{R_{\mathrm{a}}}
  100. G G

Sampling_distribution.html

  1. x ¯ \scriptstyle\bar{x}
  2. 𝒩 ( μ , σ 2 / n ) \scriptstyle\mathcal{N}(\mu,\,\sigma^{2}/n)
  3. σ x ¯ = σ n \sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}
  4. σ \sigma
  5. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  6. X ¯ \bar{X}
  7. X ¯ 𝒩 ( μ , σ 2 n ) \bar{X}\sim\mathcal{N}\Big(\mu,\,\frac{\sigma^{2}}{n}\Big)
  8. Bernoulli ( p ) \operatorname{Bernoulli}(p)
  9. X ¯ \bar{X}
  10. n X ¯ Binomial ( n , p ) n\bar{X}\sim\operatorname{Binomial}(n,p)
  11. 𝒩 ( μ 1 , σ 1 2 ) \mathcal{N}(\mu_{1},\sigma_{1}^{2})
  12. 𝒩 ( μ 2 , σ 2 2 ) \mathcal{N}(\mu_{2},\sigma_{2}^{2})
  13. X ¯ 1 - X ¯ 2 \bar{X}_{1}-\bar{X}_{2}
  14. X ¯ 1 - X ¯ 2 𝒩 ( μ 1 - μ 2 , σ 1 2 n 1 + σ 2 2 n 2 ) \bar{X}_{1}-\bar{X}_{2}\sim\mathcal{N}\!\left(\mu_{1}-\mu_{2},\,\frac{\sigma_{% 1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}\right)
  15. X ( k ) X_{(k)}
  16. X ( 1 ) X_{(1)}
  17. X ( n ) X_{(n)}
  18. f X ( k ) ( x ) = ( 2 k - 1 ) ! ( k - 1 ) ! 2 f ( x ) ( F ( x ) ( 1 - F ( x ) ) ) k - 1 f_{X_{(k)}}(x)=\frac{(2k-1)!}{(k-1)!^{2}}f(x)\Big(F(x)(1-F(x))\Big)^{k-1}
  19. M = max X k M=\max\ X_{k}
  20. F M ( x ) = P ( M x ) = P ( X k x ) = ( F ( x ) ) n F_{M}(x)=P(M\leq x)=\prod P(X_{k}\leq x)=\left(F(x)\right)^{n}

Sato–Tate_conjecture.html

  1. N p / p = 1 + O ( 1 / p ) N_{p}/p=1+O(1/\sqrt{p})
  2. p + 1 - N p = 2 p cos θ p ( 0 θ p π ) . p+1-N_{p}=2\sqrt{p}\cos{\theta_{p}}~{}~{}(0\leq\theta_{p}\leq\pi).
  3. α \alpha
  4. β \beta
  5. 0 α < β π 0\leq\alpha<\beta\leq\pi
  6. lim N # { p N : α θ p β } # { p N } = 2 π α β sin 2 θ d θ . \lim_{N\to\infty}\frac{\#\{p\leq N:\alpha\leq\theta_{p}\leq\beta\}}{\#\{p\leq N% \}}=\frac{2}{\pi}\int_{\alpha}^{\beta}\sin^{2}\theta\,d\theta.
  7. ( ( p + 1 ) - N p ) 2 p = : a p 2 p \frac{((p+1)-N_{p})}{2\sqrt{p}}=:\frac{a_{p}}{2\sqrt{p}}
  8. sin 2 θ d θ . \sin^{2}\theta\,d\theta.
  9. c X / log X c\sqrt{X}/\log X

Saturated_model.html

  1. 1 \aleph_{1}
  2. 1 \aleph_{1}
  3. 0 \aleph_{0}
  4. x < 0 \textstyle{x<0}

Scalar_potential.html

  1. 𝐅 = - P = - ( P x , P y , P z ) , \mathbf{F}=-\nabla P=-\left(\frac{\partial P}{\partial x},\frac{\partial P}{% \partial y},\frac{\partial P}{\partial z}\right),
  2. - a b 𝐅 d 𝐥 = P ( 𝐛 ) - P ( 𝐚 ) -\int_{a}^{b}\mathbf{F}\cdot d\mathbf{l}=P(\mathbf{b})-P(\mathbf{a})
  3. 𝐅 d 𝐥 = 0 \oint\mathbf{F}\cdot d\mathbf{l}=0
  4. × 𝐅 = 0. {\nabla}\times{\mathbf{F}}=0.
  5. 𝐫 0 \mathbf{r}_{0}
  6. V ( 𝐫 ) = - C 𝐅 ( 𝐫 ) d 𝐫 = - a b 𝐅 ( 𝐫 ( t ) ) 𝐫 ( t ) d t , V(\mathbf{r})=-\int_{C}\mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}=-\int_{a}^{b}% \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t)\,dt,
  7. 𝐫 0 \mathbf{r}_{0}
  8. 𝐫 , \mathbf{r},
  9. 𝐫 ( t ) , a t b , 𝐫 ( a ) = 𝐫 𝟎 , 𝐫 ( b ) = 𝐫 . \mathbf{r}(t),a\leq t\leq b,\mathbf{r}(a)=\mathbf{r_{0}},\mathbf{r}(b)=\mathbf% {r}.
  10. 𝐫 0 \mathbf{r}_{0}
  11. 𝐫 \mathbf{r}
  12. 𝐅 = - V , \mathbf{F}=-\nabla V,
  13. 𝐫 0 . \mathbf{r}_{0}.
  14. U = m g h U=mgh
  15. F S = - m g sin θ F_{S}=-mg\ \sin\theta
  16. F P = - m g sin θ cos θ = - 1 2 m g sin 2 θ . F_{P}=-mg\ \sin\theta\ \cos\theta=-{1\over 2}mg\sin 2\theta.
  17. θ = tan - 1 Δ h Δ x \theta=\tan^{-1}\frac{\Delta h}{\Delta x}
  18. F P = - m g Δ x Δ h Δ x 2 + Δ h 2 . F_{P}=-mg{\Delta x\,\Delta h\over\Delta x^{2}+\Delta h^{2}}.
  19. 𝐟 𝐁 = - p . \mathbf{f_{B}}=-\nabla p.\,
  20. F B = - S p d 𝐒 . F_{B}=-\oint_{S}\nabla p\cdot\,d\mathbf{S}.
  21. ϕ ( r ) = 1 4 π r r E ( r ) r - r d τ \phi(\vec{r})={1\over 4\pi}\iiint_{\vec{r}^{\prime}}{\vec{\nabla}_{\vec{r}^{% \prime}}\cdot\vec{E}(\vec{r}^{\prime})\over\|\vec{r}-\vec{r}^{\prime}\|}\,d% \tau^{\prime}
  22. E = - ϕ = - 1 4 π r r E ( r ) r - r d τ \vec{E}=-\vec{\nabla}\phi=-{1\over 4\pi}\vec{\nabla}\iiint_{\vec{r}^{\prime}}{% \vec{\nabla}_{\vec{r}^{\prime}}\cdot\vec{E}(\vec{r}^{\prime})\over\|\vec{r}-% \vec{r}^{\prime}\|}\,d\tau^{\prime}
  23. ϕ = - E \vec{\nabla}\phi=-\vec{E}
  24. ϕ = - E 2 ϕ = - E \vec{\nabla}\cdot\vec{\nabla}\phi=-\vec{\nabla}\cdot\vec{E}\iff\nabla^{2}\phi=% -\vec{\nabla}\cdot\vec{E}
  25. ϕ ( r ) = 1 4 π r r E ( r ) r - r d τ \phi(\vec{r})=\frac{1}{4\pi}\iiint_{\vec{r}^{\prime}}\frac{\vec{\nabla}_{\vec{% r}^{\prime}}\cdot\vec{E}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r}^{\prime}\|}d\tau% ^{\prime}
  26. ϕ ( r ) = 1 4 π r r E ( r ) r - r d τ = 1 4 π ( r r E ( r ) r - r d τ - r ( E ( r ) r 1 r - r ) d τ ) \phi(\vec{r})=\frac{1}{4\pi}\iiint_{\vec{r}^{\prime}}\frac{\vec{\nabla}_{\vec{% r}^{\prime}}\cdot\vec{E}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r}^{\prime}\|}d\tau% ^{\prime}=\frac{1}{4\pi}\left(\iiint_{\vec{r}^{\prime}}\vec{\nabla}_{\vec{r}^{% \prime}}\cdot\frac{\vec{E}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r}^{\prime}\|}d% \tau^{\prime}-\iiint_{\vec{r}^{\prime}}\left(\vec{E}(\vec{r}^{\prime})\cdot% \vec{\nabla}_{\vec{r}^{\prime}}\frac{1}{\|\vec{r}-\vec{r}^{\prime}\|}\right)d% \tau^{\prime}\right)
  27. = 1 4 π ( r is at infinity E ( r ) r - r d A - r ( E ( r ) - r - r r - r 3 ) d τ ) =\frac{1}{4\pi}\left(\iint_{\vec{r}^{\prime}\;\,\text{is at infinity}}\frac{% \vec{E}(\vec{r}^{\prime})}{\|\vec{r}-\vec{r}^{\prime}\|}\cdot d\vec{A}^{\prime% }-\iiint_{\vec{r}^{\prime}}\left(\vec{E}(\vec{r}^{\prime})\cdot-\frac{\vec{r}^% {\prime}-\vec{r}}{\|\vec{r}-\vec{r}^{\prime}\|^{3}}\right)d\tau^{\prime}\right)
  28. = - 1 4 π r E ( r ) ( r - r ) r - r 3 d τ =-\frac{1}{4\pi}\iiint_{\vec{r}^{\prime}}\frac{\vec{E}(\vec{r}^{\prime})\cdot(% \vec{r}-\vec{r}^{\prime})}{\|\vec{r}-\vec{r}^{\prime}\|^{3}}d\tau^{\prime}
  29. ϕ ( r ) = - 1 4 π r E ( r ) ( r - r ) r - r 3 d τ \phi(\vec{r})=-\frac{1}{4\pi}\iiint_{\vec{r}^{\prime}}\frac{\vec{E}(\vec{r}^{% \prime})\cdot(\vec{r}-\vec{r}^{\prime})}{\|\vec{r}-\vec{r}^{\prime}\|^{3}}d% \tau^{\prime}
  30. n 3 n\geq 3
  31. A n A_{n}
  32. n \mathbb{R}^{n}
  33. F \vec{F}
  34. n \mathbb{R}^{n}
  35. i , j { 1 , 2 , , n } i,j\in\{1,2,\dots,n\}
  36. x i F j - x j F i = 0 \frac{\partial}{\partial x_{i}}F_{j}-\frac{\partial}{\partial x_{j}}F_{i}=0
  37. F ( r ) = r n δ n ( r - r ) F ( r ) d τ \vec{F}(\vec{r})=\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}\delta^{n}(\vec{r}-% \vec{r}^{\prime})\vec{F}(\vec{r}^{\prime})d\tau^{\prime}
  38. = 1 A n r n ( r r - r | r - r | n ) F ( r ) d τ =\frac{1}{A_{n}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}\left(\vec{\nabla}_{% \vec{r}}\cdot{}\frac{\vec{r}-\vec{r}^{\prime}}{|\vec{r}-\vec{r}^{\prime}|^{n}}% \right)\vec{F}(\vec{r}^{\prime})d\tau^{\prime}
  39. = - 1 ( n - 2 ) A n r n ( r 2 1 | r - r | n - 2 ) F ( r ) d τ =\frac{-1}{(n-2)A_{n}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}\left(\nabla^{2}% _{\vec{r}}\frac{1}{|\vec{r}-\vec{r}^{\prime}|^{n-2}}\right)\vec{F}(\vec{r}^{% \prime})d\tau^{\prime}
  40. = - 1 ( n - 2 ) A n r 2 r n 1 | r - r | n - 2 F ( r ) d τ =\frac{-1}{(n-2)A_{n}}\nabla^{2}_{\vec{r}}\int_{\vec{r}^{\prime}\in\mathbb{R}^% {n}}\frac{1}{|\vec{r}-\vec{r}^{\prime}|^{n-2}}\vec{F}(\vec{r}^{\prime})d\tau^{\prime}
  41. r n 1 | r - r | n - 2 F ( r ) d τ \int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|% ^{n-2}}\vec{F}(\vec{r}^{\prime})d\tau^{\prime}}
  42. i , j { 1 , 2 , , n } i,j\in\{1,2,\dots,n\}
  43. i j i\neq j
  44. x i r n 1 | r - r | n - 2 F j ( r ) d τ - x j r n 1 | r - r | n - 2 F i ( r ) d τ = 0 \frac{\partial}{\partial x_{i}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{% 1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{j}(\vec{r}^{\prime})d\tau^{\prime}}-% \frac{\partial}{\partial x_{j}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{% 1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{i}(\vec{r}^{\prime})d\tau^{\prime}}=0
  45. x i r n 1 | r - r | n - 2 F j ( r ) d τ - x j r n 1 | r - r | n - 2 F i ( r ) d τ \frac{\partial}{\partial x_{i}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{% 1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{j}(\vec{r}^{\prime})d\tau^{\prime}}-% \frac{\partial}{\partial x_{j}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{% 1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{i}(\vec{r}^{\prime})d\tau^{\prime}}
  46. = r n ( x i 1 | r - r | n - 2 ) F j ( r ) d τ - r n ( x j 1 | r - r | n - 2 ) F i ( r ) d τ =\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\left(\frac{\partial}{\partial x_{i}% }\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\right)F_{j}(\vec{r}^{\prime})d% \tau^{\prime}}-\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\left(\frac{\partial}{% \partial x_{j}}\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\right)F_{i}(\vec{r% }^{\prime})d\tau^{\prime}}
  47. = r n - ( < m t p l > x i 1 | r - r | n - 2 ) F j ( r ) d τ - r n - ( x j 1 | r - r | n - 2 ) F i ( r ) d τ =\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{-\left(\frac{\partial}{<}mtpl>{{% \partial x_{i}^{\prime}}}\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\right)F_% {j}(\vec{r}^{\prime})d\tau^{\prime}}-\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{% -\left(\frac{\partial}{{\partial x_{j}^{\prime}}}\frac{1}{|{\vec{r}-\vec{r}^{% \prime}}|^{n-2}}\right)F_{i}(\vec{r}^{\prime})d\tau^{\prime}}
  48. = r n ( - ( < m t p l > x i 1 | r - r | n - 2 ) F j ( r ) + ( x j 1 | r - r | n - 2 ) F i ( r ) ) d τ =\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\bigg(-\left(\frac{\partial}{<}mtpl>% {{\partial x_{i}^{\prime}}}\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\right)% F_{j}(\vec{r}^{\prime})+\left(\frac{\partial}{{\partial x_{j}^{\prime}}}\frac{% 1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\right)F_{i}(\vec{r}^{\prime})\bigg)d% \tau^{\prime}}
  49. = r n ( - < m t p l > x i ( 1 | r - r | n - 2 F j ( r ) ) + 1 | r - r | n - 2 x i F j ( r ) + x j ( 1 | r - r | n - 2 F i ( r ) ) - 1 | r - r | n - 2 x j F i ( r ) ) d τ =\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\bigg(-\frac{\partial}{<}mtpl>{{% \partial x_{i}^{\prime}}}\left(\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{% j}(\vec{r}^{\prime})\right)+\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\frac{% \partial}{{\partial x_{i}^{\prime}}}F_{j}(\vec{r}^{\prime})+\frac{\partial}{{% \partial x_{j}^{\prime}}}\left(\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{% i}(\vec{r}^{\prime})\right)-\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\frac{% \partial}{{\partial x_{j}^{\prime}}}F_{i}(\vec{r}^{\prime})\bigg)d\tau^{\prime}}
  50. = r n ( - < m t p l > x i ( 1 | r - r | n - 2 F j ( r ) ) + x j ( 1 | r - r | n - 2 F i ( r ) ) + 1 | r - r | n - 2 ( x i F j ( r ) - x j F i ( r ) ) ) d τ =\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\bigg(-\frac{\partial}{<}mtpl>{{% \partial x_{i}^{\prime}}}\left(\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{% j}(\vec{r}^{\prime})\right)+\frac{\partial}{{\partial x_{j}^{\prime}}}\left(% \frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{i}(\vec{r}^{\prime})\right)+% \frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}\left(\frac{\partial}{{\partial x_% {i}^{\prime}}}F_{j}(\vec{r}^{\prime})-\frac{\partial}{{\partial x_{j}^{\prime}% }}F_{i}(\vec{r}^{\prime})\right)\bigg)d\tau^{\prime}}
  51. F \vec{F}
  52. = r n ( - < m t p l > x i ( 1 | r - r | n - 2 F j ( r ) ) + x j ( 1 | r - r | n - 2 F i ( r ) ) ) d τ =\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\bigg(-\frac{\partial}{<}mtpl>{{% \partial x_{i}^{\prime}}}\left(\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{% j}(\vec{r}^{\prime})\right)+\frac{\partial}{{\partial x_{j}^{\prime}}}\left(% \frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{i}(\vec{r}^{\prime})\right)% \bigg)d\tau^{\prime}}
  53. = - r n ( < m t p l > x i ( 1 | r - r | n - 2 F j ( r ) ) - x j ( 1 | r - r | n - 2 F i ( r ) ) ) d τ =-\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\bigg(\frac{\partial}{<}mtpl>{{% \partial x_{i}^{\prime}}}\left(\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{% j}(\vec{r}^{\prime})\right)-\frac{\partial}{{\partial x_{j}^{\prime}}}\left(% \frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{i}(\vec{r}^{\prime})\right)% \bigg)d\tau^{\prime}}
  54. x i x_{i}
  55. x j x_{j}
  56. x i x_{i}
  57. x j x_{j}
  58. x i , x j x_{i},x_{j}
  59. - r is at infinity ( ( 1 | r - r | n - 2 F i ( r ) ) x ^ i + ( 1 | r - r | n - 2 F j ( r ) ) x ^ j ) d r -{\oint_{\vec{r}^{\prime}\,\text{ is at infinity}}{\bigg(\left(\frac{1}{|{\vec% {r}-\vec{r}^{\prime}}|^{n-2}}F_{i}(\vec{r}^{\prime})\right)\hat{x}_{i}+\left(% \frac{1}{|{\vec{r}-\vec{r}^{\prime}}|^{n-2}}F_{j}(\vec{r}^{\prime})\right)\hat% {x}_{j}\bigg)\cdot{}d\vec{r}^{\prime}}}
  60. o ( | r | - ( n - 2 ) ) o(|\vec{r}^{\prime}|^{-(n-2)})
  61. F ( r ) \vec{F}(\vec{r})
  62. o ( | r | - 1 ) o(|\vec{r}|^{-1})
  63. r \vec{r}
  64. r n 1 | r - r | n - 2 F ( r ) d τ \int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{1}{|{\vec{r}-\vec{r}^{\prime}}|% ^{n-2}}\vec{F}(\vec{r}^{\prime})d\tau^{\prime}}
  65. G \vec{G}
  66. 2 G = ( G ) \nabla^{2}\vec{G}=\vec{\nabla}(\vec{\nabla}\cdot{}\vec{G})
  67. F ( r ) = r ( - 1 ( n - 2 ) A n r r n 1 | r - r | n - 2 F ( r ) d τ ) \vec{F}(\vec{r})=\vec{\nabla}_{\vec{r}}\bigg(\frac{-1}{(n-2)A_{n}}\vec{\nabla}% _{\vec{r}}\cdot{}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{1}{|\vec{r}-% \vec{r}^{\prime}|^{n-2}}\vec{F}(\vec{r}^{\prime})d\tau^{\prime}}\bigg)
  68. = r ( - 1 ( n - 2 ) A n r n r ( 1 | r - r | n - 2 F ( r ) ) d τ ) =\vec{\nabla}_{\vec{r}}\bigg(\frac{-1}{(n-2)A_{n}}\int_{\vec{r}^{\prime}\in% \mathbb{R}^{n}}{\vec{\nabla}_{\vec{r}}\cdot{}\bigg(\frac{1}{|\vec{r}-\vec{r}^{% \prime}|^{n-2}}\vec{F}(\vec{r}^{\prime})\bigg)d\tau^{\prime}}\bigg)
  69. = r ( - 1 ( n - 2 ) A n r n ( r 1 | r - r | n - 2 ) F ( r ) d τ ) =\vec{\nabla}_{\vec{r}}\bigg(\frac{-1}{(n-2)A_{n}}\int_{\vec{r}^{\prime}\in% \mathbb{R}^{n}}{\bigg(\vec{\nabla}_{\vec{r}}\frac{1}{|\vec{r}-\vec{r}^{\prime}% |^{n-2}}\bigg)\cdot{}\vec{F}(\vec{r}^{\prime})d\tau^{\prime}}\bigg)
  70. = r ( 1 A n r n r - r | r - r | n F ( r ) d τ ) =\vec{\nabla}_{\vec{r}}\bigg(\frac{1}{A_{n}}\int_{\vec{r}^{\prime}\in\mathbb{R% }^{n}}{\frac{\vec{r}-\vec{r}^{\prime}}{|\vec{r}-\vec{r}^{\prime}|^{n}}\cdot{}% \vec{F}(\vec{r}^{\prime})d\tau^{\prime}}\bigg)
  71. = r ( 1 A n r n ( r - r ) F ( r ) | r - r | n d τ ) =\vec{\nabla}_{\vec{r}}\bigg(\frac{1}{A_{n}}\int_{\vec{r}^{\prime}\in\mathbb{R% }^{n}}{\frac{{(\vec{r}-\vec{r}^{\prime})\cdot{}\vec{F}(\vec{r}^{\prime})}}{|% \vec{r}-\vec{r}^{\prime}|^{n}}d\tau^{\prime}}\bigg)
  72. ϕ ( r ) = - 1 A n r n ( r - r ) F ( r ) | r - r | n d τ \phi(\vec{r})=-{\frac{1}{A_{n}}\int_{\vec{r}^{\prime}\in\mathbb{R}^{n}}{\frac{% {(\vec{r}-\vec{r}^{\prime})\cdot{}\vec{F}(\vec{r}^{\prime})}}{|\vec{r}-\vec{r}% ^{\prime}|^{n}}d\tau^{\prime}}}
  73. F \vec{F}

Schedule_(computer_science).html

  1. D = [ T 1 T 2 T 3 R ( X ) W ( X ) C o m . R ( Y ) W ( Y ) C o m . R ( Z ) W ( Z ) C o m . ] D=\begin{bmatrix}T1&T2&T3\\ R(X)&&\\ W(X)&&\\ Com.&&\\ &R(Y)&\\ &W(Y)&\\ &Com.&\\ &&R(Z)\\ &&W(Z)\\ &&Com.\end{bmatrix}
  2. E = [ T 1 T 2 T 3 R ( X ) R ( Y ) R ( Z ) W ( X ) W ( Y ) W ( Z ) C o m . C o m . C o m . ] E=\begin{bmatrix}T1&T2&T3\\ R(X)&&\\ &R(Y)&\\ &&R(Z)\\ W(X)&&\\ &W(Y)&\\ &&W(Z)\\ Com.&Com.&Com.\end{bmatrix}
  3. G = [ T 1 T 2 R ( A ) R ( A ) W ( B ) C o m . W ( A ) C o m . ] G=\begin{bmatrix}T1&T2\\ R(A)&\\ &R(A)\\ W(B)&\\ Com.&\\ &W(A)\\ &Com.\\ &\end{bmatrix}
  4. T i T_{i}
  5. T i T_{i}
  6. T i T_{i}
  7. T j T_{j}
  8. T i T_{i}
  9. T i T_{i}
  10. T i T_{i}
  11. G = [ T 1 T 2 R ( A ) R ( A ) W ( B ) ] G=\begin{bmatrix}T1&T2\\ R(A)&\\ &R(A)\\ W(B)&\\ \end{bmatrix}
  12. H = [ T 1 T 2 T 3 R ( A ) W ( A ) C o m . W ( A ) C o m . W ( A ) C o m . ] H=\begin{bmatrix}T1&T2&T3\\ R(A)&&\\ &W(A)&\\ &Com.&\\ W(A)&&\\ Com.&&\\ &&W(A)\\ &&Com.\\ &&\end{bmatrix}
  13. F = [ T 1 T 2 R ( A ) W ( A ) R ( A ) W ( A ) C o m . C o m . ] F 2 = [ T 1 T 2 R ( A ) W ( A ) R ( A ) W ( A ) A b o r t A b o r t ] F=\begin{bmatrix}T1&T2\\ R(A)&\\ W(A)&\\ &R(A)\\ &W(A)\\ Com.&\\ &Com.\\ &\end{bmatrix}F2=\begin{bmatrix}T1&T2\\ R(A)&\\ W(A)&\\ &R(A)\\ &W(A)\\ Abort&\\ &Abort\\ &\end{bmatrix}
  14. G = [ T 1 T 2 R ( A ) W ( A ) R ( A ) W ( A ) C o m . A b o r t ] G=\begin{bmatrix}T1&T2\\ R(A)&\\ W(A)&\\ &R(A)\\ &W(A)\\ &Com.\\ Abort&\\ &\end{bmatrix}
  15. F = [ T 1 T 2 R ( A ) W ( A ) R ( A ) W ( A ) C o m . C o m . ] F 2 = [ T 1 T 2 R ( A ) W ( A ) R ( A ) W ( A ) A b o r t A b o r t ] F=\begin{bmatrix}T1&T2\\ R(A)&\\ W(A)&\\ &R(A)\\ &W(A)\\ Com.&\\ &Com.\\ &\end{bmatrix}F2=\begin{bmatrix}T1&T2\\ R(A)&\\ W(A)&\\ &R(A)\\ &W(A)\\ Abort&\\ &Abort\\ &\end{bmatrix}
  16. F 3 = [ T 1 T 2 R ( A ) R ( A ) W ( A ) W ( A ) A b o r t C o m m i t ] F3=\begin{bmatrix}T1&T2\\ &R(A)\\ R(A)&\\ W(A)&\\ &W(A)\\ Abort&\\ &Commit\\ &\end{bmatrix}

Scheme_(mathematics).html

  1. Hom Schemes ( X , Spec ( A ) ) Hom CRing ( A , 𝒪 X ( X ) ) . \operatorname{Hom}_{\rm Schemes}(X,\operatorname{Spec}(A))\cong\operatorname{% Hom}_{\rm CRing}(A,{\mathcal{O}}_{X}(X)).
  2. S S
  3. S S
  4. S S

Schinzel's_hypothesis_H.html

  1. f ( n ) , g ( n ) , , f(n),g(n),\ldots,
  2. n n
  3. f ( x ) , g ( x ) , , f(x),g(x),\ldots,
  4. n n
  5. n n
  6. x + 4 x+4
  7. x + 7 x+7
  8. n > 0 n>0
  9. n + 4 n+4
  10. n + 7 n+7
  11. n > 2 n>2
  12. Q ( x ) Q(x)
  13. m m
  14. m > 1 m>1
  15. Q ( x ) m \frac{Q(x)}{m}
  16. ( x + 4 ) ( x + 7 ) (x+4)(x+7)
  17. Q ( x ) = i = 1 k f i ( x ) Q(x)=\prod_{i=1}^{k}f_{i}(x)
  18. f i f_{i}
  19. i = 1 , k i=1,\ldots k
  20. f i ( n ) f_{i}(n)
  21. n n
  22. Q Q
  23. f i ( n ) f_{i}(n)
  24. f i ( x ) f_{i}(x)
  25. x 2 + 1 x^{2}+1
  26. n 2 + 1 n^{2}+1
  27. n 2 + 1 n^{2}+1
  28. n n
  29. F ( x ) F(x)
  30. x x
  31. x 8 + u 3 x^{8}+u^{3}\,

Schismatic_temperament.html

  1. \approx
  2. \approx
  3. \approx
  4. \approx
  5. \approx
  6. \approx
  7. \approx
  8. \approx
  9. \approx

Schläfli_symbol.html

  1. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  2. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  3. t { q , p } t\begin{Bmatrix}q,p\end{Bmatrix}
  4. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  5. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  6. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}
  7. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  8. h { 2 p , q } h\begin{Bmatrix}2p,q\end{Bmatrix}
  9. s { p , 2 q } s\begin{Bmatrix}p,2q\end{Bmatrix}
  10. s { q , 2 p } s\begin{Bmatrix}q,2p\end{Bmatrix}
  11. h { 2 q , p } h\begin{Bmatrix}2q,p\end{Bmatrix}
  12. h { p q } h\begin{Bmatrix}p\\ q\end{Bmatrix}
  13. h r { p q } hr\begin{Bmatrix}p\\ q\end{Bmatrix}
  14. q { p q } q\begin{Bmatrix}p\\ q\end{Bmatrix}
  15. s { p q } s\begin{Bmatrix}p\\ q\end{Bmatrix}
  16. a { p , q } a\begin{Bmatrix}p,q\end{Bmatrix}
  17. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  18. { p , q , r } \begin{Bmatrix}p,q,r\end{Bmatrix}
  19. t { p , q , r } t\begin{Bmatrix}p,q,r\end{Bmatrix}
  20. { p q , r } \left\{\begin{array}[]{l}p\\ q,r\end{array}\right\}
  21. { q , p r } \left\{\begin{array}[]{l}q,p\\ r\end{array}\right\}
  22. t { r , q , p } t\begin{Bmatrix}r,q,p\end{Bmatrix}
  23. { r , q , p } \begin{Bmatrix}r,q,p\end{Bmatrix}
  24. r { p q , r } r\left\{\begin{array}[]{l}p\\ q,r\end{array}\right\}
  25. t { p q , r } t\left\{\begin{array}[]{l}p\\ q,r\end{array}\right\}
  26. e 3 { p , q , r } e_{3}\begin{Bmatrix}p,q,r\end{Bmatrix}
  27. h { p , q , r } h\begin{Bmatrix}p,q,r\end{Bmatrix}
  28. q { p , q , r } q\begin{Bmatrix}p,q,r\end{Bmatrix}
  29. s { p , q , r } s\begin{Bmatrix}p,q,r\end{Bmatrix}
  30. s { p q , r } s\left\{\begin{array}[]{l}p\\ q,r\end{array}\right\}
  31. { p , q q } \left\{p,{q\atop q}\right\}
  32. t { p , q q } t\left\{p,{q\atop q}\right\}
  33. { p q q } \left\{\begin{array}[]{l}p\\ q\\ q\end{array}\right\}
  34. r { p q q } r\left\{\begin{array}[]{l}p\\ q\\ q\end{array}\right\}
  35. t { p q q } t\left\{\begin{array}[]{l}p\\ q\\ q\end{array}\right\}
  36. s { p q q } s\left\{\begin{array}[]{l}p\\ q\\ q\end{array}\right\}
  37. { r , p q } \left\{r,{p\atop q}\right\}
  38. t { r , p q } t\left\{r,{p\atop q}\right\}
  39. { r p q } \left\{\begin{array}[]{l}r\\ p\\ q\end{array}\right\}
  40. r { r p q } r\left\{\begin{array}[]{l}r\\ p\\ q\end{array}\right\}
  41. t { r p q } t\left\{\begin{array}[]{l}r\\ p\\ q\end{array}\right\}
  42. s { p q r } s\left\{\begin{array}[]{l}p\\ q\\ r\end{array}\right\}

Schmitt_trigger.html

  1. V HT = R E R E + R C2 V + V_{\mathrm{HT}}=\frac{R_{\mathrm{E}}}{R_{\mathrm{E}}+R_{\mathrm{C2}}}{V_{+}}
  2. V LT = R E R E + R C1 V + V_{\mathrm{LT}}=\frac{R_{\mathrm{E}}}{R_{\mathrm{E}}+R_{\mathrm{C1}}}{V_{+}}
  3. V + = R 2 R 1 + R 2 V in + R 1 R 1 + R 2 V s V_{\mathrm{+}}=\frac{R_{2}}{R_{1}+R_{2}}\cdot V_{\mathrm{in}}+\frac{R_{1}}{R_{% 1}+R_{2}}\cdot V_{\mathrm{s}}
  4. R 2 V in = - R 1 V s {R_{2}}\cdot V_{\mathrm{in}}=-{R_{1}}\cdot V_{\mathrm{s}}
  5. V in V_{\,\text{in}}
  6. - R 1 R 2 V s -\frac{R_{1}}{R_{2}}{V_{s}}
  7. + R 1 R 2 V s +\frac{R_{1}}{R_{2}}{V_{s}}
  8. ± R 1 R 2 V s \pm\frac{R_{1}}{R_{2}}{V_{s}}
  9. R 1 R 2 V s \frac{R_{1}}{R_{2}}{V_{s}}
  10. V + = R 1 R 1 + R 2 V s V_{\mathrm{+}}=\frac{R_{1}}{R_{1}+R_{2}}\cdot V_{\mathrm{s}}
  11. V in V_{\,\text{in}}
  12. - R 1 R 1 + R 2 V s -\frac{R_{1}}{R_{1}+R_{2}}{V_{s}}
  13. ± R 1 R 1 + R 2 V s \pm\frac{R_{1}}{R_{1}+R_{2}}{V_{s}}

Schnirelmann_density.html

  1. σ A = inf n A ( n ) n , \sigma A=\inf_{n}\frac{A(n)}{n},
  2. σ A = 0 ϵ > 0 n A ( n ) < ϵ n . \sigma A=0\Rightarrow\forall\epsilon>0\ \exists n\ A(n)<\epsilon n.
  3. k k A σ A 1 - 1 / k \forall k\ k\notin A\Rightarrow\sigma A\leq 1-1/k
  4. 1 A σ A = 0 1\notin A\Rightarrow\sigma A=0
  5. 2 A σ A 1 2 . 2\notin A\Rightarrow\sigma A\leq\frac{1}{2}.
  6. 𝔊 2 = { k 2 } k = 1 \mathfrak{G}^{2}=\{k^{2}\}_{k=1}^{\infty}
  7. σ ( 𝔊 2 𝔊 2 𝔊 2 𝔊 2 ) = 1 \sigma(\mathfrak{G}^{2}\oplus\mathfrak{G}^{2}\oplus\mathfrak{G}^{2}\oplus% \mathfrak{G}^{2})=1
  8. A B A\oplus B
  9. A { 0 } A\cup\{0\}
  10. B { 0 } B\cup\{0\}
  11. σ 𝔊 2 = 0 \sigma\mathfrak{G}^{2}=0
  12. σ ( 𝔊 2 𝔊 2 ) = 0 \sigma(\mathfrak{G}^{2}\oplus\mathfrak{G}^{2})=0
  13. σ ( 𝔊 2 𝔊 2 𝔊 2 ) = 5 / 6 \sigma(\mathfrak{G}^{2}\oplus\mathfrak{G}^{2}\oplus\mathfrak{G}^{2})=5/6
  14. 𝔊 2 \mathfrak{G}^{2}
  15. 𝒩 \mathcal{N}
  16. A A
  17. B B
  18. 𝒩 \mathcal{N}
  19. σ ( A B ) σ A + σ B - σ A σ B . \sigma(A\oplus B)\geq\sigma A+\sigma B-\sigma A\cdot\sigma B.
  20. σ A + σ B - σ A σ B = 1 - ( 1 - σ A ) ( 1 - σ B ) \sigma A+\sigma B-\sigma A\cdot\sigma B=1-(1-\sigma A)(1-\sigma B)
  21. A i 𝒩 A_{i}\subseteq\mathcal{N}
  22. 𝒩 \mathcal{N}
  23. σ ( i A i ) 1 - i ( 1 - σ A i ) . \sigma(\bigoplus_{i}A_{i})\geq 1-\prod_{i}(1-\sigma A_{i}).
  24. σ \sigma
  25. A A
  26. B B
  27. 𝒩 \mathcal{N}
  28. σ A + σ B 1 \sigma A+\sigma B\geq 1
  29. A B = 𝒩 . A\oplus B=\mathcal{N}.
  30. A 𝒩 A\subseteq\mathcal{N}
  31. σ A > 0 \sigma A>0
  32. k k
  33. i = 1 k A = 𝒩 . \bigoplus^{k}_{i=1}A=\mathcal{N}.
  34. A 𝒩 A\subseteq\mathcal{N}
  35. A A A = 𝒩 A\oplus A\oplus\cdots\oplus A=\mathcal{N}
  36. 𝔊 2 = { k 2 } k = 1 \mathfrak{G}^{2}=\{k^{2}\}_{k=1}^{\infty}
  37. α + β \alpha+\beta
  38. A A
  39. B B
  40. 𝒩 \mathcal{N}
  41. A B 𝒩 A\oplus B\neq\mathcal{N}
  42. σ ( A B ) σ A + σ B . \sigma(A\oplus B)\geq\sigma A+\sigma B.
  43. k k
  44. N N
  45. 𝔊 k = { i k } i = 1 \mathfrak{G}^{k}=\{i^{k}\}_{i=1}^{\infty}
  46. r N k ( n ) r_{N}^{k}(n)
  47. x 1 k + x 2 k + + x N k = n x_{1}^{k}+x_{2}^{k}+\cdots+x_{N}^{k}=n\,
  48. R N k ( n ) R_{N}^{k}(n)
  49. 0 x 1 k + x 2 k + + x N k n , 0\leq x_{1}^{k}+x_{2}^{k}+\cdots+x_{N}^{k}\leq n,\,
  50. x i x_{i}
  51. R N k ( n ) = i = 0 n r N k ( i ) R_{N}^{k}(n)=\sum_{i=0}^{n}r_{N}^{k}(i)
  52. r N k ( n ) > 0 n N 𝔊 k , r_{N}^{k}(n)>0\leftrightarrow n\in N\mathfrak{G}^{k},
  53. R N k ( n ) ( n N ) N k . R_{N}^{k}(n)\geq\left(\frac{n}{N}\right)^{\frac{N}{k}}.
  54. N N
  55. 0 x 1 k + x 2 k + + x N k n 0\leq x_{1}^{k}+x_{2}^{k}+\cdots+x_{N}^{k}\leq n
  56. n 1 / k n^{1/k}
  57. R N k ( n ) = i = 0 n r N k ( i ) = n N / k R_{N}^{k}(n)=\sum_{i=0}^{n}r_{N}^{k}(i)=n^{N/k}
  58. k 𝒩 k\in\mathcal{N}
  59. N 𝒩 N\in\mathcal{N}
  60. c = c ( k ) c=c(k)
  61. k k
  62. n 𝒩 n\in\mathcal{N}
  63. r N k ( m ) < c n N k - 1 r_{N}^{k}(m)<cn^{\frac{N}{k}-1}
  64. 0 m n . 0\leq m\leq n.\,
  65. k k
  66. N N
  67. σ ( N 𝔊 k ) > 0 \sigma(N\mathfrak{G}^{k})>0
  68. k k
  69. N N
  70. k k
  71. n n
  72. N N
  73. k k
  74. σ ( A + 𝔊 2 ) > σ ( A ) for 0 < σ ( A ) < 1. \sigma(A+\mathfrak{G}^{2})>\sigma(A)\,\text{ for }0<\sigma(A)<1.\,
  75. σ ( A + B ) α + α ( 1 - α ) 2 k , \sigma(A+B)\geq\alpha+\frac{\alpha(1-\alpha)}{2k}\,,
  76. σ ( A + B ) α 1 1 - k . \sigma(A+B)\geq\alpha^{\frac{1}{1-k}}\ .
  77. e ( log x ) c e^{(\log x)^{c}}\,

Schnorr_signature.html

  1. G G
  2. g g
  3. q q
  4. H : { 0 , 1 } * q H:\{0,1\}^{*}\rightarrow\mathbb{Z}_{q}
  5. M { 0 , 1 } * M\in\{0,1\}^{*}
  6. s , e , e v q s,e,e_{v}\in\mathbb{Z}_{q}
  7. q q
  8. x , k q × x,k\in\mathbb{Z}_{q}^{\times}
  9. q q
  10. q q
  11. q × = q 0 ¯ q \mathbb{Z}_{q}^{\times}=\mathbb{Z}_{q}\setminus\overline{0}_{q}
  12. y , r , r v G y,r,r_{v}\in G
  13. x x
  14. y = g x y=g^{x}
  15. M M
  16. k k
  17. r = g k r=g^{k}
  18. e = H ( M r ) e=H(M\|r)
  19. \|
  20. r r
  21. s = ( k - x e ) s=(k-xe)
  22. ( s , e ) (s,e)
  23. s , e q s,e\in\mathbb{Z}_{q}
  24. q < 2 160 q<2^{160}
  25. r v = g s y e r_{v}=g^{s}y^{e}
  26. e v = H ( M r v ) e_{v}=H(M\|r_{v})
  27. e v = e e_{v}=e
  28. e v = e e_{v}=e
  29. r v = g s y e = g k - x e g x e = g k = r r_{v}=g^{s}y^{e}=g^{k-xe}g^{xe}=g^{k}=r
  30. e v = H ( M r v ) = H ( M r ) = e e_{v}=H(M\|r_{v})=H(M\|r)=e
  31. G G
  32. g g
  33. q q
  34. y y
  35. s s
  36. e e
  37. r r
  38. k k
  39. x x
  40. H H
  41. H H
  42. H H
  43. f ( ϵ F ) q h f({\epsilon}_{F})q_{h}
  44. f 1 f\leq 1
  45. ϵ F {\epsilon}_{F}
  46. ϵ F {\epsilon}_{F}
  47. q h q_{h}

Schrödinger_picture.html

  1. | ψ ( t 0 ) |\psi(t_{0})\rangle
  2. t 0 t_{0}
  3. | ψ ( t ) |\psi(t)\rangle
  4. t t
  5. U ( t , t 0 ) U(t,t_{0})
  6. | ψ ( t ) = U ( t , t 0 ) | ψ ( t 0 ) . |\psi(t)\rangle=U(t,t_{0})|\psi(t_{0})\rangle.
  7. U ( t , t 0 ) = e - i H ( t - t 0 ) / , U(t,t_{0})=e^{-iH(t-t_{0})/\hbar},
  8. t H = 0 \partial_{t}H=0
  9. | ψ |\psi\rangle
  10. | ψ |\psi\rangle
  11. | ψ |\psi^{\prime}\rangle
  12. | ψ |\psi\rangle
  13. ψ | p ^ | ψ \langle\psi|\hat{p}|\psi\rangle
  14. | ψ |\psi\rangle
  15. p ^ \hat{p}
  16. | ψ ( t ) = U ( t , t 0 ) | ψ ( t 0 ) . |\psi(t)\rangle=U(t,t_{0})|\psi(t_{0})\rangle.
  17. ψ ( t ) | = ψ ( t 0 ) | U ( t , t 0 ) . \langle\psi(t)|=\langle\psi(t_{0})|U^{\dagger}(t,t_{0}).
  18. ψ ( t ) | ψ ( t ) = ψ ( t 0 ) | U ( t , t 0 ) U ( t , t 0 ) | ψ ( t 0 ) = ψ ( t 0 ) | ψ ( t 0 ) . \langle\psi(t)|\psi(t)\rangle=\langle\psi(t_{0})|U^{\dagger}(t,t_{0})U(t,t_{0}% )|\psi(t_{0})\rangle=\langle\psi(t_{0})|\psi(t_{0})\rangle.
  19. U ( t , t 0 ) U ( t , t 0 ) = I . U^{\dagger}(t,t_{0})U(t,t_{0})=I.
  20. | ψ ( t 0 ) = U ( t 0 , t 0 ) | ψ ( t 0 ) . |\psi(t_{0})\rangle=U(t_{0},t_{0})|\psi(t_{0})\rangle.
  21. U ( t , t 0 ) = U ( t , t 1 ) U ( t 1 , t 0 ) . U(t,t_{0})=U(t,t_{1})U(t_{1},t_{0}).
  22. i t | ψ ( t ) = H | ψ ( t ) , i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=H|\psi(t)\rangle,
  23. | ψ ( t ) = U ( t ) | ψ ( 0 ) |\psi(t)\rangle=U(t)|\psi(0)\rangle
  24. i t U ( t ) | ψ ( 0 ) = H U ( t ) | ψ ( 0 ) . i\hbar{\partial\over\partial t}U(t)|\psi(0)\rangle=HU(t)|\psi(0)\rangle.
  25. | ψ ( 0 ) |\psi(0)\rangle
  26. i t U ( t ) = H U ( t ) . i\hbar{\partial\over\partial t}U(t)=HU(t).
  27. U ( t ) = e - i H t / . U(t)=e^{-iHt/\hbar}.
  28. e - i H t / = 1 - i H t - 1 2 ( H t ) 2 + . e^{-iHt/\hbar}=1-\frac{iHt}{\hbar}-\frac{1}{2}\left(\frac{Ht}{\hbar}\right)^{2% }+\cdots.
  29. | ψ ( t ) = e - i H t / | ψ ( 0 ) . |\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle.
  30. | ψ ( 0 ) |\psi(0)\rangle
  31. | ψ ( t ) = e - i E t / | ψ ( 0 ) . |\psi(t)\rangle=e^{-iEt/\hbar}|\psi(0)\rangle.
  32. U ( t ) = exp ( - i 0 t H ( t ) d t ) , U(t)=\exp\left({-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime})\,dt^{\prime}}\right),
  33. U ( t ) = T exp ( - i 0 t H ( t ) d t ) , U(t)=\mathrm{T}\exp\left({-\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime})\,dt^{% \prime}}\right),

Schur's_lemma.html

  1. f ( r m ) = r f ( m ) for all m M and r R . f(rm)=rf(m)\,\text{ for all }m\in M\,\text{ and }r\in R.\,
  2. if g V V for all g in G , then either V = 0 or V = n . \,\text{if }gV\subseteq V\,\text{ for all }g\,\text{ in }G,\,\text{ then % either }V=0\,\text{ or }V=\mathbb{C}^{n}.

Schur_complement.html

  1. M = [ A B C D ] M=\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]
  2. A - B D - 1 C . A-BD^{-1}C.\,
  3. L = [ I p 0 - D - 1 C I q ] . L=\left[\begin{matrix}I_{p}&0\\ -D^{-1}C&I_{q}\end{matrix}\right].
  4. M L = [ A B C D ] [ I p 0 - D - 1 C I q ] = [ A - B D - 1 C B 0 D ] = [ I p B D - 1 0 I q ] [ A - B D - 1 C 0 0 D ] . \begin{aligned}\displaystyle ML&\displaystyle=\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]\left[\begin{matrix}I_{p}&0\\ -D^{-1}C&I_{q}\end{matrix}\right]=\left[\begin{matrix}A-BD^{-1}C&B\\ 0&D\end{matrix}\right]\\ &\displaystyle=\left[\begin{matrix}I_{p}&BD^{-1}\\ 0&I_{q}\end{matrix}\right]\left[\begin{matrix}A-BD^{-1}C&0\\ 0&D\end{matrix}\right].\end{aligned}
  5. [ A B C D ] = [ I p B D - 1 0 I q ] [ A - B D - 1 C 0 0 D ] [ I p 0 D - 1 C I q ] , \begin{aligned}\displaystyle\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]&\displaystyle=\left[\begin{matrix}I_{p}&BD^{-1}\\ 0&I_{q}\end{matrix}\right]\left[\begin{matrix}A-BD^{-1}C&0\\ 0&D\end{matrix}\right]\left[\begin{matrix}I_{p}&0\\ D^{-1}C&I_{q}\end{matrix}\right],\end{aligned}
  6. [ A B C D ] - 1 = [ I p 0 - D - 1 C I q ] [ ( A - B D - 1 C ) - 1 0 0 D - 1 ] [ I p - B D - 1 0 I q ] = [ ( A - B D - 1 C ) - 1 - ( A - B D - 1 C ) - 1 B D - 1 - D - 1 C ( A - B D - 1 C ) - 1 D - 1 + D - 1 C ( A - B D - 1 C ) - 1 B D - 1 ] . \begin{aligned}&\displaystyle{}\quad\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]^{-1}=\left[\begin{matrix}I_{p}&0\\ -D^{-1}C&I_{q}\end{matrix}\right]\left[\begin{matrix}(A-BD^{-1}C)^{-1}&0\\ 0&D^{-1}\end{matrix}\right]\left[\begin{matrix}I_{p}&-BD^{-1}\\ 0&I_{q}\end{matrix}\right]\\ &\displaystyle=\left[\begin{matrix}\left(A-BD^{-1}C\right)^{-1}&-\left(A-BD^{-% 1}C\right)^{-1}BD^{-1}\\ -D^{-1}C\left(A-BD^{-1}C\right)^{-1}&D^{-1}+D^{-1}C\left(A-BD^{-1}C\right)^{-1% }BD^{-1}\end{matrix}\right].\end{aligned}
  7. M - 1 = 1 A D - B C [ D - B - C A ] M^{-1}=\frac{1}{AD-BC}\left[\begin{matrix}D&-B\\ -C&A\end{matrix}\right]
  8. det ( M ) = det ( D ) det ( A - B D - 1 C ) \det(M)=\det(D)\det(A-BD^{-1}C)
  9. A x + B y = a Ax+By=a\,
  10. C x + D y = b Cx+Dy=b\,
  11. B D - 1 BD^{-1}
  12. ( A - B D - 1 C ) x = a - B D - 1 b . (A-BD^{-1}C)x=a-BD^{-1}b.\,
  13. C x + D y = b Cx+Dy=b
  14. ( p + q ) × ( p + q ) (p+q)\times(p+q)
  15. Σ = [ A B B T C ] , \Sigma=\left[\begin{matrix}A&B\\ B^{T}&C\end{matrix}\right],
  16. A n × n A\in\mathbb{R}^{n\times n}
  17. C m × m C\in\mathbb{R}^{m\times m}
  18. B n × m B\in\mathbb{R}^{n\times m}
  19. Σ \Sigma
  20. Cov ( X Y ) = A - B C - 1 B T . \operatorname{Cov}(X\mid Y)=A-BC^{-1}B^{T}.
  21. E ( X Y ) = E ( X ) + B C - 1 ( Y - E ( Y ) ) . \operatorname{E}(X\mid Y)=\operatorname{E}(X)+BC^{-1}(Y-\operatorname{E}(Y)).
  22. Σ \Sigma
  23. Σ \Sigma
  24. X = [ A B B T C ] . X=\left[\begin{matrix}A&B\\ B^{T}&C\end{matrix}\right].
  25. S = C - B T A - 1 B . S=C-B^{T}A^{-1}B.\,
  26. X X
  27. A A
  28. S S
  29. X 0 A 0 , S = C - B T A - 1 B 0 X\succ 0\Leftrightarrow A\succ 0,S=C-B^{T}A^{-1}B\succ 0
  30. X X
  31. C C
  32. A - B C - 1 B T A-BC^{-1}B^{T}
  33. X 0 C 0 , A - B C - 1 B T 0 X\succ 0\Leftrightarrow C\succ 0,A-BC^{-1}B^{T}\succ 0
  34. A A
  35. X X
  36. S S
  37. If \,\text{If}
  38. A 0 A\succ 0
  39. then \,\text{then}
  40. X 0 S = C - B T A - 1 B 0 X\succeq 0\Leftrightarrow S=C-B^{T}A^{-1}B\succeq 0
  41. C C
  42. X X
  43. A - B C - 1 B T A-BC^{-1}B^{T}
  44. If \,\text{If}
  45. C 0 C\succ 0
  46. then \,\text{then}
  47. X 0 A - B C - 1 B T 0 X\succeq 0\Leftrightarrow A-BC^{-1}B^{T}\succeq 0
  48. u T A u + 2 v T B T u + v T C v , u^{T}Au+2v^{T}B^{T}u+v^{T}Cv,\,
  49. [ A B B T C ] 0 [ C B T B A ] 0 \left[\begin{matrix}A&B\\ B^{T}&C\end{matrix}\right]\succ 0\Longleftrightarrow\left[\begin{matrix}C&B^{T% }\\ B&A\end{matrix}\right]\succ 0

Schur_decomposition.html

  1. A = Q U Q - 1 A=QUQ^{-1}
  2. [ Z 1 Z 2 ] * A [ Z 1 Z 2 ] = [ λ I λ A 12 0 A 22 ] : V λ V λ V λ V λ \begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}^{*}A\begin{bmatrix}Z_{1}&Z_{2}\end{% bmatrix}=\begin{bmatrix}\lambda\,I_{\lambda}&A_{12}\\ 0&A_{22}\end{bmatrix}:\begin{matrix}V_{\lambda}\\ \oplus\\ V_{\lambda}^{\perp}\end{matrix}\rightarrow\begin{matrix}V_{\lambda}\\ \oplus\\ V_{\lambda}^{\perp}\end{matrix}
  3. A = Q S Z * A=QSZ^{*}
  4. B = Q T Z * B=QTZ^{*}
  5. λ \lambda
  6. A x = λ B x Ax=\lambda Bx
  7. λ i \lambda_{i}
  8. λ i = S i i / T i i \lambda_{i}=S_{ii}/T_{ii}

Schwarzian_derivative.html

  1. ( S f ) ( z ) = ( f ′′ ( z ) f ( z ) ) - 1 2 ( f ′′ ( z ) f ( z ) ) 2 = f ′′′ ( z ) f ( z ) - 3 2 ( f ′′ ( z ) f ( z ) ) 2 . (Sf)(z)=\left(\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\right)^{\prime}-\frac{% 1}{2}\left({f^{\prime\prime}(z)\over f^{\prime}(z)}\right)^{2}=\frac{f^{\prime% \prime\prime}(z)}{f^{\prime}(z)}-\frac{3}{2}\left({f^{\prime\prime}(z)\over f^% {\prime}(z)}\right)^{2}.
  2. { f , z } = ( S f ) ( z ) \{f,z\}=(Sf)(z)
  3. g ( z ) = a z + b c z + d g(z)=\frac{az+b}{cz+d}
  4. ( S ( f g ) ) ( z ) = ( S f ) ( g ( z ) ) g ( z ) 2 . (S(f\circ g))(z)=(Sf)(g(z))\cdot g^{\prime}(z)^{2}.
  5. S ( f g ) = ( S ( f ) g ) ( g ) 2 + S ( g ) . S(f\circ g)=\left(S(f)\circ g\right)\cdot(g^{\prime})^{2}+S(g).
  6. F ( z , w ) = log ( f ( z ) - f ( w ) z - w ) , F(z,w)=\log\left(\frac{f(z)-f(w)}{z-w}\right),
  7. 2 F ( z , w ) z w = f ( z ) f ( w ) ( f ( z ) - f ( w ) ) 2 - 1 ( z - w ) 2 , \frac{\partial^{2}F(z,w)}{\partial z\,\partial w}={f^{\prime}(z)f^{\prime}(w)% \over(f(z)-f(w))^{2}}-{1\over(z-w)^{2}},
  8. ( S f ) ( z ) = 6 2 F ( z , w ) z w | z = w . (Sf)(z)=\left.6\cdot{\partial^{2}F(z,w)\over\partial z\partial w}\right|_{z=w}.
  9. ( S w ) ( v ) = - ( d w d v ) 2 ( S v ) ( w ) (Sw)(v)=-\left(\frac{dw}{dv}\right)^{2}(Sv)(w)
  10. v ( w ) = 1 / w . v^{\prime}(w)=1/w^{\prime}.
  11. f 1 ( z ) f_{1}(z)
  12. f 2 ( z ) f_{2}(z)
  13. d 2 f d z 2 + Q ( z ) f ( z ) = 0. \frac{d^{2}f}{dz^{2}}+Q(z)f(z)=0.
  14. g ( z ) = f 1 ( z ) / f 2 ( z ) g(z)=f_{1}(z)/f_{2}(z)
  15. ( S g ) ( z ) = 2 Q ( z ) (Sg)(z)=2Q(z)
  16. f 1 ( z ) f_{1}(z)
  17. f 2 ( z ) f_{2}(z)
  18. f 2 ( z ) 0. f_{2}(z)\neq 0.
  19. f 1 f_{1}
  20. f 2 f_{2}
  21. | S ( f ) | 6. |S(f)|\leq 6.
  22. | S ( f ) ( z ) | 2 ( 1 - | z | 2 ) - 2 , |S(f)(z)|\leq 2(1-|z|^{2})^{-2},
  23. | S ( f ) | 2. |S(f)|\leq 2.
  24. p ( z ) = i = 1 n ( 1 - α i 2 ) 2 ( z - a i ) 2 + β i z - a i . p(z)=\sum_{i=1}^{n}\frac{(1-\alpha_{i}^{2})}{2(z-a_{i})^{2}}+\frac{\beta_{i}}{% z-a_{i}}.
  25. β i = 0 \sum\beta_{i}=0
  26. 2 a i β i + ( 1 - α i 2 ) = 0 \sum 2a_{i}\beta_{i}+\left(1-\alpha_{i}^{2}\right)=0
  27. a i 2 β i + a i ( 1 - α i 2 ) = 0 \sum a_{i}^{2}\beta_{i}+a_{i}\left(1-\alpha_{i}^{2}\right)=0
  28. z - 1 , z - 2 z^{-1},z^{-2}
  29. z - 3 z^{-3}
  30. f ( z ) = u 1 ( z ) u 2 ( z ) , f(z)={u_{1}(z)\over u_{2}(z)},
  31. u 1 ( z ) u_{1}(z)
  32. u 2 ( z ) u_{2}(z)
  33. u ′′ ( z ) + 1 2 p ( z ) u ( z ) = 0. u^{\prime\prime}(z)+\tfrac{1}{2}p(z)u(z)=0.
  34. a ( z ) U ′′ ( z ) + b ( z ) U ( z ) + ( c ( z ) + λ ) U ( z ) = 0. a(z)U^{\prime\prime}(z)+b(z)U^{\prime}(z)+(c(z)+\lambda)U(z)=0.
  35. q ( z ) u i ( z ) q(z)u_{i}(z)
  36. [ a i , a i + 1 ] [a_{i},a_{i+1}]
  37. u 2 ( z ) u_{2}(z)
  38. f ~ \tilde{f}
  39. f ~ \tilde{f}
  40. F z ¯ = μ ( z ) F z , \frac{\partial F}{\partial\overline{z}}=\mu(z)\frac{\partial F}{\partial z},
  41. μ ( z ) = f z ¯ f z \mu(z)={{\partial f\over\partial\overline{z}}\over{\partial f\over\partial z}}
  42. g = S ( f ~ ) , g=S(\tilde{f}),
  43. g ( z ) d z 2 g(z)dz^{2}
  44. f S ( f - 1 ) . f\to S(f^{-1}).
  45. H 1 ( Diff ( 𝐒 1 ) ; F 2 ) = 𝐑 H^{1}(\,\text{Diff}(\mathbf{S}^{1});F_{2})=\mathbf{R}
  46. Moeb ( 𝐒 1 ) Diff ( 𝐒 1 ) QS ( 𝐒 1 ) \,\text{Moeb}(\mathbf{S}^{1})\subset\,\text{Diff}(\mathbf{S}^{1})\subset\,% \text{QS}(\mathbf{S}^{1})
  47. d 2 d θ 2 + q ( θ ) , {d^{2}\over d\theta^{2}}+q(\theta),
  48. d 2 d θ 2 + f ( θ ) 2 q f ( θ ) + 1 2 S ( f ) ( θ ) . {d^{2}\over d\theta^{2}}+f^{\prime}(\theta)^{2}\,q\circ f(\theta)+\tfrac{1}{2}% S(f)(\theta).

Scintillator.html

  1. N = A exp ( - t τ f ) + B exp ( - t τ s ) N=A\exp\left(-\frac{t}{{\tau}_{f}}\right)+B\exp\left(-\frac{t}{{\tau}_{s}}\right)
  2. T 0 + T 0 S * + S 0 + phonons T_{0}+T_{0}\rightarrow S^{*}+S_{0}+\,\text{phonons}

Score_(statistics).html

  1. L ( θ ; X ) L(\theta;X)
  2. θ \theta
  3. θ \theta
  4. θ \theta
  5. θ \theta
  6. X X
  7. L ( θ ; X ) L(\theta;X)
  8. V V
  9. V V ( θ , X ) = θ log L ( θ ; X ) = 1 L ( θ ; X ) L ( θ ; X ) θ . V\equiv V(\theta,X)=\frac{\partial}{\partial\theta}\log L(\theta;X)=\frac{1}{L% (\theta;X)}\frac{\partial L(\theta;X)}{\partial\theta}.
  10. V V
  11. L ( θ ; X ) L(\theta;X)
  12. V V
  13. θ \theta
  14. X X
  15. θ \theta
  16. θ \theta
  17. L ( θ ; X ) = f ( X + θ ) L(\theta;X)=f(X+\theta)
  18. V linear = X log f ( X ) V_{\rm linear}=\frac{\partial}{\partial X}\log f(X)
  19. V V
  20. x x
  21. θ \theta
  22. 𝔼 ( V θ ) \mathbb{E}(V\mid\theta)
  23. L ( θ ; x ) = f ( x ; θ ) L(\theta;x)=f(x;\theta)
  24. 𝔼 ( V θ ) = - + f ( x ; θ ) θ log L ( θ ; X ) d x = - + θ log L ( θ ; X ) f ( x ; θ ) d x \mathbb{E}(V\mid\theta)=\int_{-\infty}^{+\infty}f(x;\theta)\frac{\partial}{% \partial\theta}\log L(\theta;X)\,dx=\int_{-\infty}^{+\infty}\frac{\partial}{% \partial\theta}\log L(\theta;X)f(x;\theta)\,dx
  25. = - + 1 f ( x ; θ ) f ( x ; θ ) θ f ( x ; θ ) d x = - + f ( x ; θ ) θ d x =\int_{-\infty}^{+\infty}\frac{1}{f(x;\theta)}\frac{\partial f(x;\theta)}{% \partial\theta}f(x;\theta)\,dx=\int_{-\infty}^{+\infty}\frac{\partial f(x;% \theta)}{\partial\theta}\,dx
  26. θ - + f ( x ; θ ) d x = θ 1 = 0. \frac{\partial}{\partial\theta}\int_{-\infty}^{+\infty}f(x;\theta)\,dx=\frac{% \partial}{\partial\theta}1=0.
  27. ( θ ) \mathcal{I}(\theta)
  28. ( θ ) = 𝔼 { [ θ log L ( θ ; X ) ] 2 | θ } . \mathcal{I}(\theta)=\mathbb{E}\left\{\left.\left[\frac{\partial}{\partial% \theta}\log L(\theta;X)\right]^{2}\right|\theta\right\}.
  29. X X
  30. L ( θ ; A , B ) = ( A + B ) ! A ! B ! θ A ( 1 - θ ) B , L(\theta;A,B)=\frac{(A+B)!}{A!B!}\theta^{A}(1-\theta)^{B},
  31. V = 1 L L θ = A θ - B 1 - θ . V=\frac{1}{L}\frac{\partial L}{\partial\theta}=\frac{A}{\theta}-\frac{B}{1-% \theta}.
  32. E ( V ) = n θ θ - n ( 1 - θ ) 1 - θ = n - n = 0. E(V)=\frac{n\theta}{\theta}-\frac{n(1-\theta)}{1-\theta}=n-n=0.
  33. V V
  34. var ( V ) = var ( A θ - n - A 1 - θ ) = var ( A ( 1 θ + 1 1 - θ ) ) = ( 1 θ + 1 1 - θ ) 2 var ( A ) = n θ ( 1 - θ ) . \begin{aligned}\displaystyle\operatorname{var}(V)&\displaystyle=\operatorname{% var}\left(\frac{A}{\theta}-\frac{n-A}{1-\theta}\right)=\operatorname{var}\left% (A\left(\frac{1}{\theta}+\frac{1}{1-\theta}\right)\right)\\ &\displaystyle=\left(\frac{1}{\theta}+\frac{1}{1-\theta}\right)^{2}% \operatorname{var}(A)=\frac{n}{\theta(1-\theta)}.\end{aligned}
  35. S = Y log ( p ) + ( Y - 1 ) ( log ( 1 - p ) ) S=Y\log(p)+(Y-1)(\log(1-p))

Screening_effect.html

  1. V ( r ) = Z e r exp [ - q r ] V(r)=\frac{Ze}{r}\exp[{-qr}]
  2. V ( k ) = 4 π Z e q 2 + k 2 V(k)=\frac{4\pi Ze}{q^{2}+k^{2}}

Second-order_logic.html

  1. P x ( x P x P ) \forall P\,\forall x(x\in P\lor x\notin P)
  2. Σ 0 1 \Sigma^{1}_{0}
  3. Π 0 1 \Pi^{1}_{0}
  4. Σ 1 1 \Sigma^{1}_{1}
  5. R 0 R m ϕ \exists R_{0}\ldots\exists R_{m}\phi
  6. ϕ \phi
  7. Σ 1 1 \Sigma^{1}_{1}
  8. Π 1 1 \Pi^{1}_{1}
  9. Σ k + 1 1 \Sigma^{1}_{k+1}
  10. R 0 R m ϕ \exists R_{0}\ldots\exists R_{m}\phi
  11. ϕ \phi
  12. Π k 1 \Pi^{1}_{k}
  13. Π k + 1 1 \Pi^{1}_{k+1}
  14. R 0 R m ϕ \forall R_{0}\ldots\forall R_{m}\phi
  15. ϕ \phi
  16. Σ k 1 \Sigma^{1}_{k}
  17. + , , \langle+,\cdot,\leq\rangle

Second_quantization.html

  1. r i {r}_{i}
  2. r i {r}_{i}
  3. r i r j {r}_{i}\leftrightarrow{r}_{j}
  4. Ψ B ( , r i , , r j , ) = + Ψ B ( , r j , , r i , ) \Psi_{B}(\cdots,{r}_{i},\cdots,{r}_{j},\cdots)=+\Psi_{B}(\cdots,{r}_{j},\cdots% ,{r}_{i},\cdots)
  5. Ψ F ( , r i , , r j , ) = - Ψ F ( , r j , , r i , ) \Psi_{F}(\cdots,{r}_{i},\cdots,{r}_{j},\cdots)=-\Psi_{F}(\cdots,{r}_{j},\cdots% ,{r}_{i},\cdots)
  6. ψ α ( r ) \psi_{\alpha}({r})
  7. α \alpha
  8. Ψ [ r i ] = i = 1 N ψ α i ( r i ) ψ α 1 ψ α 2 ψ α N \Psi[{r}_{i}]=\prod_{i=1}^{N}\psi_{\alpha_{i}}({r}_{i})\equiv\psi_{\alpha_{1}}% \otimes\psi_{\alpha_{2}}\otimes\cdots\otimes\psi_{\alpha_{N}}
  9. | α i |{\alpha_{i}}\rangle
  10. Ψ \Psi
  11. 𝒮 \mathcal{S}
  12. 𝒜 \mathcal{A}
  13. Ψ B [ r i ] = 𝒩 𝒮 Ψ [ r i ] = 𝒩 π S N i = 1 N ψ α π ( i ) ( r i ) = 𝒩 π S N ψ α π ( 1 ) ψ α π ( 2 ) ψ α π ( N ) ; \Psi_{B}[{r}_{i}]=\mathcal{N}\mathcal{S}\Psi[{r}_{i}]=\mathcal{N}\sum_{\pi\in S% _{N}}\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}({r}_{i})=\mathcal{N}\sum_{\pi\in S_% {N}}\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}\otimes\cdots\otimes% \psi_{\alpha_{\pi(N)}};
  14. Ψ F [ r i ] = 𝒩 𝒜 Ψ [ r i ] = 𝒩 π S N ( - 1 ) π i = 1 N ψ α π ( i ) ( r i ) = 𝒩 π S N ( - 1 ) π ψ α π ( 1 ) ψ α π ( 2 ) ψ α π ( N ) . \Psi_{F}[{r}_{i}]=\mathcal{N}\mathcal{A}\Psi[{r}_{i}]=\mathcal{N}\sum_{\pi\in S% _{N}}(-1)^{\pi}\prod_{i=1}^{N}\psi_{\alpha_{\pi(i)}}({r}_{i})=\mathcal{N}\sum_% {\pi\in S_{N}}(-1)^{\pi}\psi_{\alpha_{\pi(1)}}\otimes\psi_{\alpha_{\pi(2)}}% \otimes\cdots\otimes\psi_{\alpha_{\pi(N)}}.
  15. π \pi
  16. S N S_{N}
  17. α i \alpha_{i}
  18. ( - 1 ) π (-1)^{\pi}
  19. 𝒩 \mathcal{N}
  20. U U
  21. U i j = ψ α j ( r i ) r i | α j U_{ij}=\psi_{\alpha_{j}}({r}_{i})\equiv\langle{r}_{i}|\alpha_{j}\rangle
  22. Ψ B = 𝒩 perm U \Psi_{B}=\mathcal{N}\operatorname{perm}U
  23. Ψ F = 𝒩 det U \Psi_{F}=\mathcal{N}\operatorname{det}U
  24. ψ 1 ψ 2 \psi_{1}\psi_{2}
  25. ψ 2 ψ 1 \psi_{2}\psi_{1}
  26. | [ n α ] | n 1 , n 2 , , n α , , |[n_{\alpha}]\rangle\equiv|n_{1},n_{2},\cdots,n_{\alpha},\cdots\rangle,
  27. n α n_{\alpha}
  28. | α |\alpha\rangle
  29. α n α = N \sum_{\alpha}n_{\alpha}=N
  30. n α n_{\alpha}
  31. n α = { 0 , 1 fermions, 0 , 1 , 2 , 3 , bosons. n_{\alpha}=\begin{cases}0,1&\,\text{fermions,}\\ 0,1,2,3,...&\,\text{bosons.}\end{cases}
  32. | [ n α ] |[n_{\alpha}]\rangle
  33. | 0 | , 0 α , |0\rangle\equiv|\cdots,0_{\alpha},\cdots\rangle
  34. | n α | , 0 , n α , 0 , |n_{\alpha}\rangle\equiv|\cdots,0,n_{\alpha},0,\cdots\rangle
  35. | 0 = 1 |0\rangle=1
  36. | 1 α = ψ α |1_{\alpha}\rangle=\psi_{\alpha}
  37. | 2 α = ψ α ψ α |2_{\alpha}\rangle=\psi_{\alpha}\otimes\psi_{\alpha}
  38. | n α = ψ α n |n_{\alpha}\rangle=\psi_{\alpha}^{\otimes n}
  39. | 1 1 , 1 2 = ( ψ 1 ψ 2 + ψ 2 ψ 1 ) / 2 |1_{1},1_{2}\rangle=(\psi_{1}\psi_{2}+\psi_{2}\psi_{1})/\sqrt{2}
  40. | 1 1 , 1 2 = ( ψ 1 ψ 2 - ψ 2 ψ 1 ) / 2 |1_{1},1_{2}\rangle=(\psi_{1}\psi_{2}-\psi_{2}\psi_{1})/\sqrt{2}
  41. \otimes
  42. ψ 1 \psi_{1}
  43. ψ 2 \psi_{2}
  44. α n α ! N ! \sqrt{\tfrac{\prod_{\alpha}n_{\alpha}!}{N!}}
  45. 1 N ! \tfrac{1}{\sqrt{N!}}
  46. n α n_{\alpha}
  47. | [ n α ] B = ( α n α ! N ! ) 1 / 2 𝒮 α ψ α n α |[n_{\alpha}]\rangle_{B}=\left(\frac{\prod_{\alpha}n_{\alpha}!}{N!}\right)^{1/% 2}\mathcal{S}\bigotimes\limits_{\alpha}\psi_{\alpha}^{\otimes n_{\alpha}}
  48. | [ n α ] F = 1 N ! 𝒜 α ψ α |[n_{\alpha}]\rangle_{F}=\frac{1}{\sqrt{N!}}\mathcal{A}\bigotimes\limits_{% \alpha}\psi_{\alpha}
  49. ψ α \psi_{\alpha}
  50. ψ α 1 ψ α ψ α 1 \psi_{\alpha}\equiv 1\otimes\psi_{\alpha}\equiv\psi_{\alpha}\otimes 1
  51. Ψ = ψ α 1 ψ α 2 \Psi=\psi_{\alpha_{1}}\otimes\psi_{\alpha_{2}}\otimes\cdots
  52. ± \otimes_{\pm}
  53. ± \oslash_{\pm}
  54. ψ α ± 1 = ψ α , ψ α ± ( ψ β Ψ ) = ψ α ψ β Ψ ± ψ β ( ψ α ± Ψ ) ; \psi_{\alpha}\otimes_{\pm}1=\psi_{\alpha},\quad\psi_{\alpha}\otimes_{\pm}(\psi% _{\beta}\otimes\Psi)=\psi_{\alpha}\otimes\psi_{\beta}\otimes\Psi\pm\psi_{\beta% }\otimes(\psi_{\alpha}\otimes_{\pm}\Psi);
  55. ψ α ± 1 = 0 , ψ α ± ( ψ β Ψ ) = δ α β Ψ ± ψ β ( ψ α ± Ψ ) . \psi_{\alpha}\oslash_{\pm}1=0,\quad\psi_{\alpha}\oslash_{\pm}(\psi_{\beta}% \otimes\Psi)=\delta_{\alpha\beta}\Psi\pm\psi_{\beta}\otimes(\psi_{\alpha}% \oslash_{\pm}\Psi).
  56. δ α β \delta_{\alpha\beta}
  57. α = β \alpha=\beta
  58. b α b_{\alpha}^{\dagger}
  59. b α b_{\alpha}
  60. b α b_{\alpha}^{\dagger}
  61. | α |\alpha\rangle
  62. b α b_{\alpha}
  63. | α |\alpha\rangle
  64. b α b α b_{\alpha}\neq b_{\alpha}^{\dagger}
  65. Ψ \Psi
  66. b α Ψ = 1 N + 1 ψ α + Ψ , b_{\alpha}^{\dagger}\Psi=\frac{1}{\sqrt{N+1}}\psi_{\alpha}\otimes_{+}\Psi,
  67. b α Ψ = 1 N ψ α + Ψ , b_{\alpha}\Psi=\frac{1}{\sqrt{N}}\psi_{\alpha}\oslash_{+}\Psi,
  68. ψ α + \psi_{\alpha}\otimes_{+}
  69. ψ α \psi_{\alpha}
  70. N + 1 N+1
  71. ψ α + \psi_{\alpha}\oslash_{+}
  72. ψ α \psi_{\alpha}
  73. N N
  74. \otimes
  75. | 1 1 , 1 2 = ( ψ 1 ψ 2 + ψ 2 ψ 1 ) / 2 |1_{1},1_{2}\rangle=(\psi_{1}\psi_{2}+\psi_{2}\psi_{1})/\sqrt{2}
  76. ψ 1 \psi_{1}
  77. b 1 | 1 1 , 1 2 = 1 2 ( b 1 ψ 1 ψ 2 + b 1 ψ 2 ψ 1 ) = 1 2 ( 1 3 ψ 1 + ψ 1 ψ 2 + 1 3 ψ 1 + ψ 2 ψ 1 ) = 1 2 ( 1 3 ( ψ 1 ψ 1 ψ 2 + ψ 1 ψ 1 ψ 2 + ψ 1 ψ 2 ψ 1 ) + 1 3 ( ψ 1 ψ 2 ψ 1 + ψ 2 ψ 1 ψ 1 + ψ 2 ψ 1 ψ 1 ) ) = 2 3 ( ψ 1 ψ 1 ψ 2 + ψ 1 ψ 2 ψ 1 + ψ 2 ψ 1 ψ 1 ) = 2 | 2 1 , 1 2 . \begin{array}[]{rl}b_{1}^{\dagger}|1_{1},1_{2}\rangle=&\frac{1}{\sqrt{2}}(b_{1% }^{\dagger}\psi_{1}\psi_{2}+b_{1}^{\dagger}\psi_{2}\psi_{1})\\ =&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}\psi_{1}\otimes_{+}\psi_{1}\psi_{2% }+\frac{1}{\sqrt{3}}\psi_{1}\otimes_{+}\psi_{2}\psi_{1}\right)\\ =&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}(\psi_{1}\psi_{1}\psi_{2}+\psi_{1}% \psi_{1}\psi_{2}+\psi_{1}\psi_{2}\psi_{1})+\frac{1}{\sqrt{3}}(\psi_{1}\psi_{2}% \psi_{1}+\psi_{2}\psi_{1}\psi_{1}+\psi_{2}\psi_{1}\psi_{1})\right)\\ =&\frac{\sqrt{2}}{\sqrt{3}}(\psi_{1}\psi_{1}\psi_{2}+\psi_{1}\psi_{2}\psi_{1}+% \psi_{2}\psi_{1}\psi_{1})\\ =&\sqrt{2}|2_{1},1_{2}\rangle.\end{array}
  78. ψ 1 \psi_{1}
  79. b 1 | 2 1 , 1 2 = 1 3 ( b 1 ψ 1 ψ 1 ψ 2 + b 1 ψ 1 ψ 2 ψ 1 + b 1 ψ 2 ψ 1 ψ 1 ) = 1 3 ( 1 3 ψ 1 + ψ 1 ψ 1 ψ 2 + 1 3 ψ 1 + ψ 1 ψ 2 ψ 1 + 1 3 ψ 1 + ψ 2 ψ 1 ψ 1 ) = 1 3 ( 1 3 ( ψ 1 ψ 2 + ψ 1 ψ 2 + 0 ) + 1 3 ( ψ 2 ψ 1 + 0 + ψ 1 ψ 2 ) + 1 3 ( 0 + ψ 2 ψ 1 + ψ 2 ψ 1 ) ) = ψ 1 ψ 2 + ψ 2 ψ 1 = 2 | 1 1 , 1 2 . \begin{array}[]{rl}b_{1}|2_{1},1_{2}\rangle=&\frac{1}{\sqrt{3}}(b_{1}\psi_{1}% \psi_{1}\psi_{2}+b_{1}\psi_{1}\psi_{2}\psi_{1}+b_{1}\psi_{2}\psi_{1}\psi_{1})% \\ =&\frac{1}{\sqrt{3}}\left(\frac{1}{\sqrt{3}}\psi_{1}\oslash_{+}\psi_{1}\psi_{1% }\psi_{2}+\frac{1}{\sqrt{3}}\psi_{1}\oslash_{+}\psi_{1}\psi_{2}\psi_{1}+\frac{% 1}{\sqrt{3}}\psi_{1}\oslash_{+}\psi_{2}\psi_{1}\psi_{1}\right)\\ =&\frac{1}{\sqrt{3}}\left(\frac{1}{\sqrt{3}}(\psi_{1}\psi_{2}+\psi_{1}\psi_{2}% +0)+\frac{1}{\sqrt{3}}(\psi_{2}\psi_{1}+0+\psi_{1}\psi_{2})+\frac{1}{\sqrt{3}}% (0+\psi_{2}\psi_{1}+\psi_{2}\psi_{1})\right)\\ =&\psi_{1}\psi_{2}+\psi_{2}\psi_{1}\\ =&\sqrt{2}|1_{1},1_{2}\rangle.\end{array}
  80. | 0 α = 1 |0_{\alpha}\rangle=1
  81. b α b_{\alpha}^{\dagger}
  82. b α | 0 α = ψ α + 1 = ψ α = | 1 α , b_{\alpha}^{\dagger}|0_{\alpha}\rangle=\psi_{\alpha}\otimes_{+}1=\psi_{\alpha}% =|1_{\alpha}\rangle,
  83. b α | n α = 1 n α + 1 ψ α + ψ α n α = n α + 1 ψ α ( n α + 1 ) = n α + 1 | n α + 1 . b_{\alpha}^{\dagger}|n_{\alpha}\rangle=\frac{1}{\sqrt{n_{\alpha}+1}}\psi_{% \alpha}\otimes_{+}\psi_{\alpha}^{\otimes n_{\alpha}}=\sqrt{n_{\alpha}+1}\psi_{% \alpha}^{\otimes(n_{\alpha}+1)}=\sqrt{n_{\alpha}+1}|n_{\alpha}+1\rangle.
  84. | n α = 1 n α ! ( b α ) n α | 0 α . |n_{\alpha}\rangle=\frac{1}{\sqrt{n_{\alpha}!}}(b_{\alpha}^{\dagger})^{n_{% \alpha}}|0_{\alpha}\rangle.
  85. b α b_{\alpha}
  86. b α | n α = 1 n α ψ α + ψ α n α = n α ψ α ( n α - 1 ) = n α | n α - 1 . b_{\alpha}|n_{\alpha}\rangle=\frac{1}{\sqrt{n_{\alpha}}}\psi_{\alpha}\oslash_{% +}\psi_{\alpha}^{\otimes n_{\alpha}}=\sqrt{n_{\alpha}}\psi_{\alpha}^{\otimes(n% _{\alpha}-1)}=\sqrt{n_{\alpha}}|n_{\alpha}-1\rangle.
  87. b α | 0 α = 0 b_{\alpha}|0_{\alpha}\rangle=0
  88. b α b α | n α = n α | n α , b_{\alpha}^{\dagger}b_{\alpha}|n_{\alpha}\rangle=n_{\alpha}|n_{\alpha}\rangle,
  89. n ^ α = b α b α \hat{n}_{\alpha}=b_{\alpha}^{\dagger}b_{\alpha}
  90. b α | , n β , n α , n γ , = n α + 1 | , n β , n α + 1 , n γ , . b_{\alpha}^{\dagger}|\cdots,n_{\beta},n_{\alpha},n_{\gamma},\cdots\rangle=% \sqrt{n_{\alpha}+1}|\cdots,n_{\beta},n_{\alpha}+1,n_{\gamma},\cdots\rangle.
  91. b α | , n β , n α , n γ , = n α | , n β , n α - 1 , n γ , . b_{\alpha}|\cdots,n_{\beta},n_{\alpha},n_{\gamma},\cdots\rangle=\sqrt{n_{% \alpha}}|\cdots,n_{\beta},n_{\alpha}-1,n_{\gamma},\cdots\rangle.
  92. [ b α , b β ] = [ b α , b β ] = 0 , [ b α , b β ] = δ α β . [b_{\alpha}^{\dagger},b_{\beta}^{\dagger}]=[b_{\alpha},b_{\beta}]=0,\quad[b_{% \alpha},b_{\beta}^{\dagger}]=\delta_{\alpha\beta}.
  93. c α c_{\alpha}^{\dagger}
  94. c α c_{\alpha}
  95. c α c_{\alpha}^{\dagger}
  96. | α |\alpha\rangle
  97. c α c_{\alpha}
  98. | α |\alpha\rangle
  99. c α c α c_{\alpha}\neq c_{\alpha}^{\dagger}
  100. χ α , Re = ( c α + c α ) / 2 , χ α , Im = ( c α - c α ) / ( 2 i ) , \chi_{\alpha,\,\text{Re}}=(c_{\alpha}+c_{\alpha}^{\dagger})/2,\quad\chi_{% \alpha,\,\text{Im}}=(c_{\alpha}-c_{\alpha}^{\dagger})/(2\mathrm{i}),
  101. Ψ \Psi
  102. c α Ψ = 1 N + 1 ψ α - Ψ , c_{\alpha}^{\dagger}\Psi=\frac{1}{\sqrt{N+1}}\psi_{\alpha}\otimes_{-}\Psi,
  103. c α Ψ = 1 N ψ α - Ψ , c_{\alpha}\Psi=\frac{1}{\sqrt{N}}\psi_{\alpha}\oslash_{-}\Psi,
  104. ψ α - \psi_{\alpha}\otimes_{-}
  105. ψ α \psi_{\alpha}
  106. N + 1 N+1
  107. ψ α - \psi_{\alpha}\oslash_{-}
  108. ψ α \psi_{\alpha}
  109. N N
  110. \otimes
  111. | 1 1 , 1 2 = ( ψ 1 ψ 2 - ψ 2 ψ 1 ) / 2 |1_{1},1_{2}\rangle=(\psi_{1}\psi_{2}-\psi_{2}\psi_{1})/\sqrt{2}
  112. ψ 1 \psi_{1}
  113. c 1 | 1 1 , 1 2 = 1 2 ( c 1 ψ 1 ψ 2 - c 1 ψ 2 ψ 1 ) = 1 2 ( 1 3 ψ 1 - ψ 1 ψ 2 - 1 3 ψ 1 - ψ 2 ψ 1 ) = 1 2 ( 1 3 ( ψ 1 ψ 1 ψ 2 - ψ 1 ψ 1 ψ 2 + ψ 1 ψ 2 ψ 1 ) - 1 3 ( ψ 1 ψ 2 ψ 1 - ψ 2 ψ 1 ψ 1 + ψ 2 ψ 1 ψ 1 ) ) = 0. \begin{array}[]{rl}c_{1}^{\dagger}|1_{1},1_{2}\rangle=&\frac{1}{\sqrt{2}}(c_{1% }^{\dagger}\psi_{1}\psi_{2}-c_{1}^{\dagger}\psi_{2}\psi_{1})\\ =&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}\psi_{1}\otimes_{-}\psi_{1}\psi_{2% }-\frac{1}{\sqrt{3}}\psi_{1}\otimes_{-}\psi_{2}\psi_{1}\right)\\ =&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{3}}(\psi_{1}\psi_{1}\psi_{2}-\psi_{1}% \psi_{1}\psi_{2}+\psi_{1}\psi_{2}\psi_{1})-\frac{1}{\sqrt{3}}(\psi_{1}\psi_{2}% \psi_{1}-\psi_{2}\psi_{1}\psi_{1}+\psi_{2}\psi_{1}\psi_{1})\right)\\ =&0.\end{array}
  114. ψ 2 \psi_{2}
  115. | 1 1 , 1 2 = ( ψ 1 ψ 2 - ψ 2 ψ 1 ) / 2 |1_{1},1_{2}\rangle=(\psi_{1}\psi_{2}-\psi_{2}\psi_{1})/\sqrt{2}
  116. c 2 | 1 1 , 1 2 = 1 2 ( c 2 ψ 1 ψ 2 - c 2 ψ 2 ψ 1 ) = 1 2 ( 1 2 ψ 2 - ψ 1 ψ 2 - 1 2 ψ 2 - ψ 2 ψ 1 ) = 1 2 ( 1 2 ( 0 - ψ 1 ) - 1 2 ( ψ 1 - 0 ) ) = - ψ 1 = - | 1 1 , 0 2 . \begin{array}[]{rl}c_{2}|1_{1},1_{2}\rangle=&\frac{1}{\sqrt{2}}(c_{2}\psi_{1}% \psi_{2}-c_{2}\psi_{2}\psi_{1})\\ =&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}\psi_{2}\oslash_{-}\psi_{1}\psi_{2% }-\frac{1}{\sqrt{2}}\psi_{2}\oslash_{-}\psi_{2}\psi_{1}\right)\\ =&\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}}(0-\psi_{1})-\frac{1}{\sqrt{2}}(% \psi_{1}-0)\right)\\ =&-\psi_{1}\\ =&-|1_{1},0_{2}\rangle.\end{array}
  117. | 0 α = 1 |0_{\alpha}\rangle=1
  118. c α c_{\alpha}^{\dagger}
  119. c α | 0 α = ψ α - 1 = ψ α = | 1 α , c_{\alpha}^{\dagger}|0_{\alpha}\rangle=\psi_{\alpha}\otimes_{-}1=\psi_{\alpha}% =|1_{\alpha}\rangle,
  120. c α | 1 α = 1 2 ψ α - ψ α = 0. c_{\alpha}^{\dagger}|1_{\alpha}\rangle=\frac{1}{\sqrt{2}}\psi_{\alpha}\otimes_% {-}\psi_{\alpha}=0.
  121. | α |\alpha\rangle
  122. c α c_{\alpha}
  123. c α | 1 α = ψ α - ψ α = 1 = | 0 α , c_{\alpha}|1_{\alpha}\rangle=\psi_{\alpha}\oslash_{-}\psi_{\alpha}=1=|0_{% \alpha}\rangle,
  124. c α | 0 α = 0. c_{\alpha}|0_{\alpha}\rangle=0.
  125. | n α = ( c α ) n α | 0 α . |n_{\alpha}\rangle=(c_{\alpha}^{\dagger})^{n_{\alpha}}|0_{\alpha}\rangle.
  126. c α c α | n α = n α | n α , c_{\alpha}^{\dagger}c_{\alpha}|n_{\alpha}\rangle=n_{\alpha}|n_{\alpha}\rangle,
  127. n ^ α = c α c α \hat{n}_{\alpha}=c_{\alpha}^{\dagger}c_{\alpha}
  128. c α | , n β , n α , n γ , = ( - 1 ) β < α n β ( 1 - n α ) | , n β , 1 - n α , n γ , . c_{\alpha}^{\dagger}|\cdots,n_{\beta},n_{\alpha},n_{\gamma},\cdots\rangle=(-1)% ^{\sum_{\beta<\alpha}n_{\beta}}(1-n_{\alpha})|\cdots,n_{\beta},1-n_{\alpha},n_% {\gamma},\cdots\rangle.
  129. c α | , n β , n α , n γ , = ( - 1 ) β < α n β n α | , n β , 1 - n α , n γ , . c_{\alpha}|\cdots,n_{\beta},n_{\alpha},n_{\gamma},\cdots\rangle=(-1)^{\sum_{% \beta<\alpha}n_{\beta}}n_{\alpha}|\cdots,n_{\beta},1-n_{\alpha},n_{\gamma},% \cdots\rangle.
  130. n α n_{\alpha}
  131. ( - 1 ) β < α n β (-1)^{\sum_{\beta<\alpha}n_{\beta}}
  132. { c α , c β } = { c α , c β } = 0 , { c α , c β } = δ α β . \{c_{\alpha}^{\dagger},c_{\beta}^{\dagger}\}=\{c_{\alpha},c_{\beta}\}=0,\quad% \{c_{\alpha},c_{\beta}^{\dagger}\}=\delta_{\alpha\beta}.
  133. a ν a^{\dagger}_{\nu}
  134. ( c ν ) (c^{\dagger}_{\nu})
  135. ( b ν ) (b^{\dagger}_{\nu})
  136. Ψ ( r ) \Psi({r})
  137. Ψ ( r ) \Psi^{\dagger}({r})
  138. Ψ ( r ) = ν ψ ν ( r ) a ν \Psi({r})=\sum_{\nu}\psi_{\nu}\left({r}\right)a_{\nu}
  139. Ψ ( r ) = ν ψ ν * ( r ) a ν \Psi^{\dagger}({r})=\sum_{\nu}\psi^{*}_{\nu}\left({r}\right)a^{\dagger}_{\nu}
  140. ψ ν ( r ) \psi_{\nu}\left({r}\right)
  141. ψ ν * ( r ) \psi^{*}_{\nu}\left({r}\right)
  142. Ψ ( r ) \Psi^{\dagger}({r})
  143. ψ ν ( r ) \psi_{\nu}\left({r}\right)
  144. Ψ ( r ) \Psi({r})
  145. Ψ ( r ) \Psi^{\dagger}({r})
  146. [ Ψ ( r 1 ) , Ψ ( r 2 ) ] = δ ( r 1 - r 2 ) \left[\Psi({r}_{1}),\Psi^{\dagger}({r}_{2})\right]=\delta({r}_{1}-{r}_{2})
  147. { Ψ ( r 1 ) , Ψ ( r 2 ) } = δ ( r 1 - r 2 ) \{\Psi({r}_{1}),\Psi^{\dagger}({r}_{2})\}=\delta({r}_{1}-{r}_{2})
  148. Ψ ( r ) = 1 V k e i k r a k \Psi({r})={1\over\sqrt{V}}\sum_{{k}}e^{i{k\cdot r}}a_{{k}}
  149. Ψ ( r ) = 1 V k e - i k r a k \Psi^{\dagger}({r})={1\over\sqrt{V}}\sum_{{k}}e^{-i{k\cdot r}}{a^{\dagger}}_{{% k}}

Secondary_average.html

  1. S e c A = B B + ( T B - H ) + ( S B - C S ) A B SecA=\frac{BB+(TB-H)+(SB-CS)}{AB}

Secret_sharing.html

  1. m 1 , m 2 , , m k m_{1},m_{2},...,m_{k}
  2. S < i = 1 k m i S<\prod_{i=1}^{k}m_{i}
  3. m i m_{i}

Section_(fiber_bundle).html

  1. π \pi
  2. π \pi
  3. E E
  4. B B
  5. π : E B \pi:E\mapsto B
  6. σ : B E \sigma:B\mapsto E
  7. π ( σ ( x ) ) = x \pi(\sigma(x))=x
  8. x B x\in B
  9. g : B Y g:B\mapsto Y
  10. E = B × Y E=B\times Y
  11. B B
  12. Y Y
  13. σ ( x ) = ( x , g ( x ) ) E , σ : B E \sigma(x)=(x,g(x))\in E,\ \sigma:B\mapsto E
  14. π : E X \pi:E\mapsto X
  15. π ( x , y ) = x \pi(x,y)=x
  16. σ \sigma
  17. π ( σ ( x ) ) = x \pi(\sigma(x))=x
  18. π : E B \pi:E\mapsto B
  19. σ ( x ) \sigma(x)
  20. π ( σ ( x ) ) = x \pi(\sigma(x))=x
  21. x x
  22. x x

Seiche.html

  1. T = 2 L g h T=\frac{2L}{\sqrt{gh}}
  2. T = 2 L c T=\frac{2L}{c}
  3. c 2 = g ρ 2 - ρ 1 ρ 2 h 1 h 2 h 1 + h 2 c^{2}=g\frac{\rho_{2}-\rho_{1}}{\rho_{2}}\frac{h_{1}h_{2}}{h_{1}+h_{2}}
  4. h 1 , h 2 h_{1},h_{2}
  5. ρ 1 , ρ 2 \rho_{1},\rho_{2}

Seifert–van_Kampen_theorem.html

  1. X X
  2. U U
  3. V V
  4. X X
  5. X X
  6. W W
  7. U U
  8. V V
  9. U U
  10. V V
  11. V V
  12. U U
  13. w w
  14. W W
  15. U U
  16. V V
  17. W W
  18. W W
  19. U U
  20. V V
  21. X X
  22. π 1 ( X , w ) \pi_{1}(X,w)
  23. π 1 ( U , w ) \pi_{1}(U,w)
  24. π 1 ( V , w ) \pi_{1}(V,w)
  25. I I
  26. J J
  27. π 1 ( U , w ) = u 1 , , u k α 1 , , α l \pi_{1}(U,w)=\langle u_{1},\ldots,u_{k}\mid\alpha_{1},\ldots,\alpha_{l}\rangle
  28. π 1 ( V , w ) = v 1 , , v m β 1 , , β n , \pi_{1}(V,w)=\langle v_{1},\ldots,v_{m}\mid\beta_{1},\ldots,\beta_{n}\rangle,
  29. π 1 ( W , w ) = w 1 , , w p γ 1 , , γ q \pi_{1}(W,w)=\langle w_{1},\ldots,w_{p}\mid\gamma_{1},\ldots,\gamma_{q}\rangle
  30. π 1 ( X , w ) = u 1 , , u k , v 1 , , v m α 1 , , α l , β 1 , , β n , I ( w 1 ) J ( w 1 ) - 1 , , I ( w p ) J ( w p ) - 1 . \pi_{1}(X,w)=\langle u_{1},\ldots,u_{k},v_{1},\ldots,v_{m}\mid\alpha_{1},% \ldots,\alpha_{l},\beta_{1},\ldots,\beta_{n},I(w_{1})J(w_{1})^{-1},\ldots,I(w_% {p})J(w_{p})^{-1}\rangle.
  31. π 1 ( X , w ) \pi_{1}(X,w)
  32. π 1 ( U , w ) π 1 ( W , w ) π 1 ( V , w ) . \pi_{1}(U,w)\leftarrow\pi_{1}(W,w)\to\pi_{1}(V,w).
  33. U 1 U_{1}
  34. U 2 U_{2}
  35. U 1 U 2 U_{1}\cap U_{2}
  36. U 1 U_{1}
  37. U 2 U_{2}
  38. π 1 ( U 1 U 2 , x 0 ) \pi_{1}(U_{1}\cap U_{2},x_{0})
  39. π 1 ( X , A ) \pi_{1}(X,A)
  40. A X A\cap X
  41. π 1 ( X , A ) \pi_{1}(X,A)
  42. \mathcal{I}
  43. \mathcal{I}
  44. X 1 , X 2 X_{1},X_{2}
  45. X 1 , X 2 X_{1},X_{2}
  46. X 0 := X 1 X 2 X_{0}:=X_{1}\cap X_{2}
  47. π 1 ( X , A ) \pi_{1}(X,A)
  48. \mathcal{I}
  49. { U λ : λ Λ } \{U_{\lambda}:\lambda\in\Lambda\}
  50. U λ U_{\lambda}
  51. ( λ , μ ) Λ 2 π 1 ( U λ U μ , A ) λ Λ π 1 ( U λ , A ) π 1 ( X , A ) \bigsqcup_{(\lambda,\mu)\in\Lambda^{2}}\pi_{1}(U_{\lambda}\cap U_{\mu},A)% \rightrightarrows\bigsqcup_{\lambda\in\Lambda}\pi_{1}(U_{\lambda},A)% \rightarrow\pi_{1}(X,A)
  52. S 2 S^{2}
  53. A = S 2 - n A=S^{2}-n
  54. B = S 2 - s B=S^{2}-s
  55. A B A\cap B
  56. A B A\cap B
  57. S 2 S^{2}
  58. π 1 ( S 2 ) = π 1 ( A ) π 1 ( B ) / ker ( Φ ) \pi_{1}(S^{2})=\pi_{1}(A)\cdot\pi_{1}(B)/\ker(\Phi)
  59. 2 \mathbb{R}^{2}
  60. S 2 S^{2}
  61. π 1 ( A B ) = π 1 ( S 1 ) \pi_{1}(A\cap B)=\pi_{1}(S^{1})
  62. π 1 ( A ) = π 1 ( D 2 ) = 1 \pi_{1}(A)=\pi_{1}(D^{2})={1}
  63. π 1 ( A B ) \pi_{1}(A\cap B)
  64. π 1 ( A ) \pi_{1}(A)
  65. π 1 ( A B ) \pi_{1}(A\cap B)
  66. π 1 ( B ) \pi_{1}(B)
  67. π 1 ( B ) \pi_{1}(B)
  68. { A 1 , B 1 , , A n , B n } \{A_{1},B_{1},\ldots,A_{n},B_{n}\}
  69. { A 1 , B 1 , , A n , B n } \{A_{1},B_{1},\ldots,A_{n},B_{n}\}
  70. A 1 , B 1 , , A n , B n A 1 B 1 A 1 - 1 B 1 - 1 A n B n A n - 1 B n - 1 . \langle A_{1},B_{1},\ldots,A_{n},B_{n}\mid A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}% \ldots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}\rangle.
  71. π 1 ( X , A ) \pi_{1}(X,A)

Seismic_moment.html

  1. M 0 M_{0}
  2. M 0 = μ A D M_{0}=\mu AD
  3. μ \mu
  4. A A
  5. D D
  6. A A
  7. M 0 M_{0}

Seismic_tomography.html

  1. t = L s t=Ls

Seki_Takakazu.html

  1. ( a b c ) (a\ b\ c)
  2. a x 2 + b x + c . ax^{2}+bx+c.
  3. a x + b ax+b
  4. a x + b = 0 ax+b=0

Selection_algorithm.html

  1. k n k\leq n
  2. 2 = 1 / ( 1 - ( 1 / 2 ) ) 2=1/(1-(1/2))
  3. W t ( n ) W_{t}(n)
  4. W t ( n ) n - t + n + 1 - t < j n log 2 j for n t W_{t}(n)\leq n-t+\sum_{n+1-t<j\leq n}\lceil{\log_{2}\,j}\rceil\quad\,\text{for% }\,n\geq t

Self-assembly.html

  1. π - π \pi-\pi

Self-balancing_binary_search_tree.html

  1. n 2 h + 1 - 1 n\leq 2^{h+1}-1
  2. h log 2 ( n + 1 ) - 1 log 2 n h\geq\lceil\log_{2}(n+1)-1\rceil\geq\lfloor\log_{2}n\rfloor
  3. log 2 n \lfloor\log_{2}n\rfloor
  4. log 2 ( 1 , 000 , 000 ) = 19 \lfloor\log_{2}(1,000,000)\rfloor=19
  5. log 2 ( n ) \lfloor\log_{2}(n)\rfloor

Self-information.html

  1. I ( ω n ) I(\omega_{n})
  2. ω n \omega_{n}
  3. P ( ω n ) P(\omega_{n})
  4. I ( ω n ) = log ( 1 P ( ω n ) ) = - log ( P ( ω n ) ) I(\omega_{n})=\log\left(\frac{1}{P(\omega_{n})}\right)=-\log(P(\omega_{n}))
  5. I ( ω n ) \displaystyle I(\omega_{n})
  6. e \displaystyle e
  7. C C
  8. P ( k ) P(k)
  9. k k
  10. I ( C ) = E ( - log ( P ( C ) ) ) = - k = 1 n P ( k ) log ( P ( k ) ) I(C)=E(-\log(P(C)))=-\sum_{k=1}^{n}P(k)\log(P(k))
  11. H ( X ) = I ( X ; X ) H(X)=I(X;X)
  12. I ( X ; X ) I(X;X)

Self-ionization_of_water.html

  1. a a
  2. K eq = a H 3 O + a OH - a H 2 O 2 K_{\rm eq}=\frac{a_{\rm{H_{3}O^{+}}}\cdot a_{\rm{OH^{-}}}}{a_{\rm{H_{2}O}}^{2}}
  3. K eq = a H + a OH - a H 2 O K_{\rm eq}=\frac{a_{\rm{H^{+}}}\cdot a_{\rm{OH^{-}}}}{a_{\rm{H_{2}O}}}
  4. K eq = a H 3 O + a OH - K_{\rm eq}=a_{\rm{H_{3}O^{+}}}\cdot a_{\rm{OH^{-}}}
  5. K w = [ H 3 O + ] [ OH - ] K_{\rm w}=[{\rm{H_{3}O^{+}}}][{\rm{OH^{-}}}]
  6. p K w = - log 10 K w = 14 \mathrm{p}K_{\rm w}=-\log_{10}K_{\rm w}=14
  7. p K w = pH + pOH \mathrm{p}K_{\rm w}=\mathrm{pH}+\mathrm{pOH}
  8. a a
  9. K eq = a D 3 O + a OD - a D 2 O 2 K_{\rm eq}=\frac{a_{\rm{D_{3}O^{+}}}\cdot a_{\rm{OD^{-}}}}{a_{\rm{D_{2}O}}^{2}}
  10. K w = [ D 3 O + ] [ OD - ] K_{\rm w}=[{\rm{D_{3}O^{+}}}][{\rm{OD^{-}}}]
  11. N N 0 = e - Δ E k T \frac{N}{N_{0}}=e^{-\frac{\Delta E^{\ddagger}}{kT}}
  12. k k

Semantic_memory.html

  1. 𝐌 t , d = ln ( 1 + 𝐌 t , d ) - i = 0 D P ( i | t ) ln P ( i | t ) \mathbf{M}_{t,d}^{\prime}=\frac{\ln{(1+\mathbf{M}_{t,d})}}{-\sum_{i=0}^{D}P(i|% t)\ln{P(i|t)}}
  2. P ( i | t ) P(i|t)
  3. i i
  4. t t
  5. 𝐌 t , d \mathbf{M}_{t,d}
  6. i = 0 D 𝐌 t , i \sum_{i=0}^{D}\mathbf{M}_{t,i}
  7. 𝐌 \mathbf{M}^{\prime}
  8. 𝐌 \mathbf{M}
  9. Δ = 11 - d \Delta=11-d
  10. d d

Semicircle.html

  1. π \pi
  2. ( x 0 , y 0 ) (x_{0},y_{0})
  3. y = y 0 + r 2 - ( x - x 0 ) 2 . y=y_{0}+\sqrt{r^{2}-(x-x_{0})^{2}}.
  4. y = y 0 - r 2 - ( x - x 0 ) 2 . y=y_{0}-\sqrt{r^{2}-(x-x_{0})^{2}}.

Semiprime.html

  1. n n
  2. n 3 \leq\sqrt[3]{n}
  3. Ω ( n ) = 2 1 n 2 0.1407604 \sum_{\Omega(n)=2}\frac{1}{n^{2}}\approx 0.1407604
  4. Ω ( n ) = 2 1 n ( n - 1 ) 0.17105 \sum_{\Omega(n)=2}\frac{1}{n(n-1)}\approx 0.17105
  5. Ω ( n ) = 2 ln n n 2 0.28360 \sum_{\Omega(n)=2}\frac{\ln n}{n^{2}}\approx 0.28360

Semiring.html

  1. a b = a + b + b a - a b a - b a b a\nabla b=a+b+ba-aba-bab
  2. x y = - log ( e - x + e - y ) , x\oplus y=-\log(e^{-x}+e^{-y})\ ,
  3. L 1 L 2 = { w 1 w 2 w 1 L 1 , w 2 L 2 } L_{1}\cdot L_{2}=\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\}
  4. U V = { u v : u U , v v } U\cdot V=\{u\cdot v:u\in U,\ v\in v\}
  5. i I ( a a i ) = a ( i I a i ) ; i I ( a i a ) = ( i I a i ) a . \sum_{i\in I}{(a\cdot a_{i})}=a\cdot(\sum_{i\in I}{a_{i}});\quad\sum_{i\in I}{% (a_{i}\cdot a)}=(\sum_{i\in I}{a_{i}})\cdot a.
  6. a * = 1 + a a * = 1 + a * a . a^{*}=1+aa^{*}=1+a^{*}a.\,
  7. R * = n 0 R n R^{*}=\bigcup_{n\geq 0}R^{n}
  8. R U × U R\subseteq U\times U
  9. ( a + b ) * = ( a * b ) * a * , (a+b)^{*}=(a^{*}b)^{*}a^{*},\,
  10. ( a b ) * = 1 + a ( b a ) * b . (ab)^{*}=1+a(ba)^{*}b.\,
  11. a * = j 0 a j a^{*}=\sum_{j\geq 0}{a^{j}}
  12. a 0 = 1 a^{0}=1
  13. a j + 1 = a a j = a j a a^{j+1}=a\cdot a^{j}=a^{j}\cdot a
  14. j 0 j\geq 0
  15. S \emptyset\in S
  16. E S E\in S
  17. F S F\in S
  18. E F S E\cap F\in S
  19. E S E\in S
  20. F S F\in S
  21. C i S C_{i}\in S
  22. i = 1 , , n i=1,\ldots,n
  23. E F = i = 1 n C i E\setminus F=\bigcup_{i=1}^{n}C_{i}
  24. [ a , b ) [a,b)\subset\mathbb{R}

Semitone.html

  1. ( 4 3 / ( 9 8 ) 2 = 256 243 ) \left(\begin{matrix}\frac{4}{3}\end{matrix}/{{\begin{matrix}(\frac{9}{8})\end{% matrix}}^{2}}=\begin{matrix}\frac{256}{243}\end{matrix}\right)
  2. 18 / 17 99.0 cents, 18/17\approx 99.0\,\text{ cents,}
  3. 2 3 - 2 4 100.4 cents, \sqrt[4]{\frac{2}{3-\sqrt{2}}}\approx 100.4\,\text{ cents,}
  4. ( 139 / 138 ) 8 99.9995 cents, (139/138)^{8}\approx 99.9995\,\text{ cents,}
  5. 256 243 = 2 8 3 5 90.2 cents \frac{256}{243}=\frac{2^{8}}{3^{5}}\approx 90.2\,\text{ cents}
  6. 2187 2048 = 3 7 2 11 113.7 cents \frac{2187}{2048}=\frac{3^{7}}{2^{11}}\approx 113.7\,\text{ cents}
  7. 4 3 ÷ 5 4 = 16 15 \tfrac{4}{3}\div\tfrac{5}{4}=\tfrac{16}{15}
  8. S 1 = 25 24 70.7 cents S_{1}={25\over 24}\approx 70.7\ \hbox{cents}
  9. S 2 = 135 128 92.2 cents S_{2}={135\over 128}\approx 92.2\ \hbox{cents}
  10. S 3 = 16 15 111.7 cents S_{3}={16\over 15}\approx 111.7\ \hbox{cents}
  11. S 4 = 27 25 133.2 cents S_{4}={27\over 25}\approx 133.2\ \hbox{cents}

Sensitivity_(electronics).html

  1. S i S_{i}
  2. S i = k ( T a + T r x ) B S o N o S_{i}=k(T_{a}+T_{rx})B{\cdot}\frac{S_{o}}{N_{o}}
  3. S i S_{i}
  4. T a T_{a}
  5. T r x T_{rx}
  6. S o N o \frac{S_{o}}{N_{o}}

Sensitivity_analysis.html

  1. | Y X i | 𝐱 0 \left|\frac{\partial Y}{\partial X_{i}}\right|_{\textbf{x}^{0}}
  2. Var X i ( E 𝐗 i ( Y X i ) ) \operatorname{Var}_{X_{i}}\left(E_{\,\textbf{X}_{\sim i}}\left(Y\mid X_{i}% \right)\right)

Separable_extension.html

  1. E F E\supset F
  2. α E \alpha\in E
  3. α \alpha
  4. E F E\supset F
  5. α E F \alpha\in E\setminus F
  6. α \alpha
  7. E F E\supseteq F
  8. ( X - α ) 2 (X-\alpha)^{2}
  9. α = 2 \alpha=2
  10. \mathbb{Q}
  11. \mathbb{R}
  12. E F E\supseteq F
  13. α E \alpha\in E
  14. ( X - α ) 2 (X-\alpha)^{2}
  15. K F K\supseteq F
  16. f ( X ) = g ( X p n ) f(X)=g(X^{p^{n}})
  17. g ( X ) = a i X i g(X)=\sum a_{i}X^{i}
  18. g ( X ) = b i p X i g(X)=\sum b_{i}^{p}X^{i}
  19. f ( X ) = g ( X p ) = b i p X p i = ( b i X i ) p f(X)=g(X^{p})=\sum b_{i}^{p}X^{pi}=(\sum b_{i}X^{i})^{p}
  20. E K ( X ) E\supseteq K(X)
  21. α \alpha
  22. α p = X \alpha^{p}=X
  23. f ( Y ) = Y p - X = Y p - α p = ( Y - α ) p f(Y)=Y^{p}-X=Y^{p}-\alpha^{p}=(Y-\alpha)^{p}
  24. E F E\supseteq F
  25. α , β E \alpha,\beta\in E
  26. α + β \alpha+\beta
  27. α β \alpha\beta
  28. E L F E\supseteq L\supseteq F
  29. E L E\supseteq L
  30. L F L\supseteq F
  31. E F E\supseteq F
  32. E F E\supseteq F
  33. E L E\supseteq L
  34. L F L\supseteq F
  35. E F E\supseteq F
  36. α E \alpha\in E
  37. E = F [ α ] E=F[\alpha]
  38. E F E\supseteq F
  39. S = { α E | α is separable over F } S=\{\alpha\in E|\alpha\mbox{ is separable over }~{}F\}
  40. E F E\supseteq F
  41. E F E\supseteq F
  42. E F E\supseteq F
  43. S = E S=E
  44. E F E\supseteq F
  45. E F E\supseteq F
  46. E F E\supseteq F
  47. K F K\supseteq F
  48. E U F E\supseteq U\supseteq F
  49. F / k F/k
  50. k k
  51. F L = L k F F_{L}=L\otimes_{k}F
  52. k k
  53. F p F^{p}
  54. k k
  55. k p k^{p}
  56. F k 1 / p F_{k^{1/p}}
  57. F L F_{L}
  58. F L F_{L}
  59. F F
  60. F L F_{L}
  61. k k
  62. F / k F/k
  63. F / k F/k
  64. F = k ( a 1 , , a r ) F=k(a_{1},...,a_{r})
  65. a 1 , , a r a_{1},...,a_{r}
  66. r = 1. r=1.
  67. [ F : k ] [F:k]
  68. [ F : k ] [F:k]
  69. k ¯ \overline{k}
  70. k s k_{s}
  71. k ¯ \overline{k}
  72. k s k_{s}
  73. k ¯ \overline{k}
  74. k s k_{s}
  75. k ¯ \overline{k}
  76. k ¯ \overline{k}
  77. k s k_{s}
  78. k ¯ = k s \overline{k}=k_{s}
  79. F F
  80. k k
  81. dim F Der k ( F , F ) tr . deg k F \dim_{F}\operatorname{Der}_{k}(F,F)\geq\operatorname{tr.deg}_{k}F
  82. F / k F/k
  83. Der k ( F , F ) = 0 \operatorname{Der}_{k}(F,F)=0
  84. F / k F/k
  85. D 1 , , D m D_{1},...,D_{m}
  86. Der k ( F , F ) \operatorname{Der}_{k}(F,F)
  87. a 1 , , a m F a_{1},...,a_{m}\in F
  88. F F
  89. k ( a 1 , , a m ) k(a_{1},...,a_{m})
  90. D i ( a j ) D_{i}(a_{j})
  91. m = tr . deg k F m=\operatorname{tr.deg}_{k}F
  92. { a 1 , , a m } \{a_{1},...,a_{m}\}

Separable_polynomial.html

  1. K L K\subset L
  2. α L \alpha\in L
  3. α \alpha
  4. X 2 - D X^{2}-D

Separation_of_variables.html

  1. d d x f ( x ) = g ( x ) h ( f ( x ) ) \frac{d}{dx}f(x)=g(x)h(f(x))
  2. y = f ( x ) y=f(x)
  3. d y d x = g ( x ) h ( y ) . \frac{dy}{dx}=g(x)h(y).
  4. d y h ( y ) = g ( x ) d x , {dy\over h(y)}={g(x)dx},
  5. 1 h ( y ) d y d x = g ( x ) , \frac{1}{h(y)}\frac{dy}{dx}=g(x),
  6. x x
  7. 1 h ( y ) d y d x d x = g ( x ) d x , ( 1 ) \int\frac{1}{h(y)}\frac{dy}{dx}\,dx=\int g(x)\,dx,\qquad\qquad(1)
  8. 1 h ( y ) d y = g ( x ) d x \int\frac{1}{h(y)}\,dy=\int g(x)\,dx
  9. d y d x \frac{dy}{dx}
  10. 1 h ( y ) d y + C 1 = g ( x ) d x + C 2 , \int\frac{1}{h(y)}\,dy+C_{1}=\int g(x)\,dx+C_{2},
  11. C = C 2 - C 1 C=C_{2}-C_{1}
  12. d P d t = k P ( 1 - P K ) \frac{dP}{dt}=kP\left(1-\frac{P}{K}\right)
  13. P P
  14. t t
  15. k k
  16. K K
  17. d P d t = k P ( 1 - P K ) \frac{dP}{dt}=kP\left(1-\frac{P}{K}\right)
  18. d P P ( 1 - P K ) = k d t \int\frac{dP}{P\left(1-\frac{P}{K}\right)}=\int k\,dt
  19. 1 P ( 1 - P K ) = K P ( K - P ) \frac{1}{P\left(1-\frac{P}{K}\right)}=\frac{K}{P\left(K-P\right)}
  20. K P ( K - P ) = 1 P + 1 K - P \frac{K}{P\left(K-P\right)}=\frac{1}{P}+\frac{1}{K-P}
  21. ( 1 P + 1 K - P ) d P = k d t \int\left(\frac{1}{P}+\frac{1}{K-P}\right)\,dP=\int k\,dt
  22. ln | P | - ln | K - P | = k t + C \ln\begin{vmatrix}P\end{vmatrix}-\ln\begin{vmatrix}K-P\end{vmatrix}=kt+C
  23. ln | K - P | - ln | P | = - k t - C \ln\begin{vmatrix}K-P\end{vmatrix}-\ln\begin{vmatrix}P\end{vmatrix}=-kt-C
  24. ln | K - P P | = - k t - C \ln\begin{vmatrix}\cfrac{K-P}{P}\end{vmatrix}=-kt-C
  25. | K - P P | = e - k t - C \begin{vmatrix}\cfrac{K-P}{P}\end{vmatrix}=e^{-kt-C}
  26. | K - P P | = e - C e - k t \begin{vmatrix}\cfrac{K-P}{P}\end{vmatrix}=e^{-C}e^{-kt}
  27. K - P P = ± e - C e - k t \frac{K-P}{P}=\pm e^{-C}e^{-kt}
  28. A = ± e - C A=\pm e^{-C}
  29. K - P P = A e - k t \frac{K-P}{P}=Ae^{-kt}
  30. K P - 1 = A e - k t \frac{K}{P}-1=Ae^{-kt}
  31. K P = 1 + A e - k t \frac{K}{P}=1+Ae^{-kt}
  32. P K = 1 1 + A e - k t \frac{P}{K}=\frac{1}{1+Ae^{-kt}}
  33. P = K 1 + A e - k t P=\frac{K}{1+Ae^{-kt}}
  34. P ( t ) = K 1 + A e - k t P\left(t\right)=\frac{K}{1+Ae^{-kt}}
  35. A A
  36. t = 0 t=0
  37. P ( 0 ) = P 0 P\left(0\right)=P_{0}
  38. P 0 = K 1 + A e 0 P_{0}=\frac{K}{1+Ae^{0}}
  39. e 0 = 1 e^{0}=1
  40. A = K - P 0 P 0 A=\frac{K-P_{0}}{P_{0}}
  41. X ( x ) = B x + C . X(x)=Bx+C.
  42. T ( t ) = A e - λ α t , T(t)=Ae^{-\lambda\alpha t},
  43. X ( x ) = B sin ( λ x ) + C cos ( λ x ) . X(x)=B\sin(\sqrt{\lambda}\,x)+C\cos(\sqrt{\lambda}\,x).
  44. λ = n π L . \sqrt{\lambda}=n\frac{\pi}{L}.
  45. u ( x , t ) = n = 1 D n sin n π x L exp ( - n 2 π 2 α t L 2 ) , u(x,t)=\sum_{n=1}^{\infty}D_{n}\sin\frac{n\pi x}{L}\exp\left(-\frac{n^{2}\pi^{% 2}\alpha t}{L^{2}}\right),
  46. u | t = 0 = f ( x ) , u\big|_{t=0}=f(x),
  47. f ( x ) = n = 1 D n sin n π x L . f(x)=\sum_{n=1}^{\infty}D_{n}\sin\frac{n\pi x}{L}.
  48. sin n π x L \sin\frac{n\pi x}{L}
  49. 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\frac{n\pi x}{L}\,dx.
  50. { sin n π x L } n = 1 \left\{\sin\frac{n\pi x}{L}\right\}_{n=1}^{\infty}
  51. u n ( t ) + α n 2 π 2 L 2 u n ( t ) = h n ( t ) , u^{\prime}_{n}(t)+\alpha\frac{n^{2}\pi^{2}}{L^{2}}u_{n}(t)=h_{n}(t),
  52. u n ( t ) = e - α n 2 π 2 L 2 t ( b n + 0 t h n ( s ) e α n 2 π 2 L 2 s d s ) . u_{n}(t)=e^{-\alpha\frac{n^{2}\pi^{2}}{L^{2}}t}\left(b_{n}+\int_{0}^{t}h_{n}(s% )e^{\alpha\frac{n^{2}\pi^{2}}{L^{2}}s}\,ds\right).
  53. L = 𝐃 𝐱𝐱 𝐃 𝐲𝐲 = 𝐃 𝐱𝐱 𝐈 + 𝐈 𝐃 𝐲𝐲 , L=\mathbf{D_{xx}}\oplus\mathbf{D_{yy}}=\mathbf{D_{xx}}\otimes\mathbf{I}+% \mathbf{I}\otimes\mathbf{D_{yy}},\,
  54. 𝐃 𝐱𝐱 \mathbf{D_{xx}}
  55. 𝐃 𝐲𝐲 \mathbf{D_{yy}}
  56. 𝐈 \mathbf{I}

Serre's_multiplicity_conjectures.html

  1. χ ( R / P , R / Q ) := i = 0 ( - 1 ) i R ( Tor i R ( R / P , R / Q ) ) . \chi(R/P,R/Q):=\sum_{i=0}^{\infty}(-1)^{i}\ell_{R}(\mathrm{Tor}^{R}_{i}(R/P,R/% Q)).
  2. R ( ( R / P ) ( R / Q ) ) < . \ell_{R}((R/P)\otimes(R/Q))<\infty.
  3. dim ( R / P ) + dim ( R / Q ) dim ( R ) \dim(R/P)+\dim(R/Q)\leq\dim(R)
  4. χ ( R / P , R / Q ) 0 \chi(R/P,R/Q)\geq 0
  5. dim ( R / P ) + dim ( R / Q ) < dim ( R ) \dim(R/P)+\dim(R/Q)<\dim(R)
  6. χ ( R / P , R / Q ) = 0. \chi(R/P,R/Q)=0.
  7. dim ( R / P ) + dim ( R / Q ) = dim ( R ) \dim(R/P)+\dim(R/Q)=\dim(R)
  8. χ ( R / P , R / Q ) > 0. \chi(R/P,R/Q)>0.

Serre_duality.html

  1. H q ( V , E ) H n - q ( V , K E ) , H^{q}(V,E)\cong H^{n-q}(V,K\otimes E^{\ast})^{\ast},

Set-theoretic_definition_of_natural_numbers.html

  1. ${ }$
  2. n {n}
  3. 0 , 1 , , n 1 {0,1,...,n−1}
  4. ${ }$
  5. 0 {0}
  6. ${ { } }$
  7. 0 , 1 {0,1}
  8. , {{},{{}}}
  9. 0 , 1 , 2 {0,1,2}
  10. , , , {{},{{}},{{},{{}}}}
  11. n {n}
  12. { } \{\varnothing\}
  13. { x { y } x A y A } . \{x\cup\{y\}\mid x\in A\wedge y\notin A\}.
  14. σ \sigma

Set-theoretic_limit.html

  1. lim inf n A n = n 1 j n A j \liminf_{n\rightarrow\infty}A_{n}=\bigcup_{n\geq 1}\bigcap_{j\geq n}A_{j}
  2. lim sup n A n = n 1 j n A j . \limsup_{n\rightarrow\infty}A_{n}=\bigcap_{n\geq 1}\bigcup_{j\geq n}A_{j}.
  3. lim inf n A n = { x X : lim inf n 𝟏 A n ( x ) = 1 } \liminf_{n\rightarrow\infty}A_{n}=\{x\in X:\liminf_{n\rightarrow\infty}\mathbf% {1}_{A_{n}}(x)=1\}
  4. lim sup n A n = { x X : lim sup n 𝟏 A n ( x ) = 1 } , \limsup_{n\rightarrow\infty}A_{n}=\{x\in X:\limsup_{n\rightarrow\infty}\mathbf% {1}_{A_{n}}(x)=1\},
  5. x n 1 j n A n \scriptstyle x\ \in\ \bigcup_{n\geq 1}\ \bigcap_{j\geq n}A_{n}
  6. j n A j = j 1 A j and j n A j = A n . \bigcap_{j\geq n}A_{j}=\bigcap_{j\geq 1}A_{j}\,\text{ and }\bigcup_{j\geq n}A_% {j}=A_{n}.
  7. lim inf n A n = n 1 j n A j = j 1 A j = n 1 j n A j = lim sup n A n . \liminf_{n\rightarrow\infty}A_{n}=\bigcup_{n\geq 1}\bigcap_{j\geq n}A_{j}=% \bigcap_{j\geq 1}A_{j}=\bigcap_{n\geq 1}\bigcup_{j\geq n}A_{j}=\limsup_{n% \rightarrow\infty}A_{n}.
  8. lim n A n = j 1 A j . \lim_{n\rightarrow\infty}A_{n}=\bigcup_{j\geq 1}A_{j}.
  9. lim n A n = { x X : lim i 𝟏 A n ( x ) = 1 } . \lim_{n\rightarrow\infty}A_{n}=\{x\in X:\lim_{i\rightarrow\infty}\mathbf{1}_{A% _{n}}(x)=1\}.
  10. lim inf n A n lim sup n A n , \liminf_{n\to\infty}A_{n}\subset\limsup_{n\to\infty}A_{n},
  11. B n = j n A j \scriptstyle B_{n}=\bigcap_{j\geq n}A_{j}
  12. C n = j n A j \scriptstyle C_{n}=\bigcup_{j\geq n}A_{j}
  13. lim inf n A n = lim n j n A j and lim sup n A n = lim n j n A j . \liminf_{n\to\infty}A_{n}=\lim_{n\to\infty}\bigcap_{j\geq n}A_{j}\,\text{ and % }\limsup_{n\to\infty}A_{n}=\lim_{n\to\infty}\bigcup_{j\geq n}A_{j}.
  14. lim inf n A n = i ( j n A j c ) c = ( n j n A j c ) c = ( lim sup n A n c ) c . \liminf_{n\rightarrow\infty}A_{n}=\bigcup_{i}\Bigl(\bigcup_{j\geq n}A_{j}^{c}% \Bigr)^{c}=\Bigl(\bigcap_{n}\bigcup_{j\geq n}A_{j}^{c}\Bigr)^{c}=\Bigl(\limsup% _{n\rightarrow\infty}A_{n}^{c}\Bigr)^{c}.
  15. 𝟏 lim inf n A n ( x ) = lim inf n 𝟏 A j ( x ) = sup n 1 inf j n 𝟏 A j ( x ) \,\textbf{1}_{\liminf_{n\rightarrow\infty}A_{n}}(x)=\liminf_{n\rightarrow% \infty}\,\textbf{1}_{A_{j}}(x)=\sup_{n\geq 1}\inf_{j\geq n}\,\textbf{1}_{A_{j}% }(x)
  16. 𝟏 lim sup n A n ( x ) = lim sup n 𝟏 A j ( x ) = inf n 1 sup j n 𝟏 A j ( x ) . \,\textbf{1}_{\limsup_{n\rightarrow\infty}A_{n}}(x)=\limsup_{n\rightarrow% \infty}\,\textbf{1}_{A_{j}}(x)=\inf_{n\geq 1}\sup_{j\geq n}\,\textbf{1}_{A_{j}% }(x).
  17. \scriptstyle\mathcal{F}
  18. \scriptstyle\mathcal{F}
  19. \scriptstyle\mathcal{F}
  20. \scriptstyle\mathcal{F}
  21. lim inf n A n = n j n ( - 1 n , 1 - 1 n ] = n [ 0 , 1 - 1 n ] = [ 0 , 1 ) \liminf_{n\rightarrow\infty}A_{n}=\bigcup_{n}\bigcap_{j\geq n}\Bigl(-\frac{1}{% n},1-\frac{1}{n}\Bigr]=\bigcup_{n}\Bigl[0,1-\frac{1}{n}\Bigr]=[0,1)
  22. lim sup n A n = n j n ( - 1 n , 1 - 1 n ] = n ( - 1 n , 1 ) = [ 0 , 1 ) . \limsup_{n\rightarrow\infty}A_{n}=\bigcap_{n}\bigcup_{j\geq n}\Bigl(-\frac{1}{% n},1-\frac{1}{n}\Bigr]=\bigcap_{n}\Bigl(-\frac{1}{n},1\Bigr)=[0,1).
  23. l i m < s u b > n A n = ( 0 , 1 ] lim<sub>n→∞A_{n}=(0,1]
  24. lim inf n A n = n j n ( ( - 1 ) n n , 1 - ( - 1 ) n n ] = n ( 1 2 n , 1 - 1 2 n ] = ( 0 , 1 ) \liminf_{n\rightarrow\infty}A_{n}=\bigcup_{n}\bigcap_{j\geq n}\Bigl(\frac{(-1)% ^{n}}{n},1-\frac{(-1)^{n}}{n}\Bigr]=\bigcup_{n}\Bigl(\frac{1}{2n},1-\frac{1}{2% n}\Bigr]=(0,1)
  25. lim sup n A n = n j n ( ( - 1 ) n n , 1 - ( - 1 ) n n ] = n ( - 1 2 n - 1 , 1 + 1 2 n - 1 ] = [ 0 , 1 ] . \limsup_{n\rightarrow\infty}A_{n}=\bigcap_{n}\bigcup_{j\geq n}\Bigl(\frac{(-1)% ^{n}}{n},1-\frac{(-1)^{n}}{n}\Bigr]=\bigcap_{n}\Bigl(-\frac{1}{2n-1},1+\frac{1% }{2n-1}\Bigr]=[0,1].
  26. l i m < s u b > n A n lim<sub>n→∞A_{n}
  27. j n A j = [ 0 , 1 ] \bigcup_{j\geq n}A_{j}=\mathbb{Q}\cap[0,1]
  28. j n A j = { 0 , 1 } , \bigcap_{j\geq n}A_{j}=\{0,1\},
  29. lim inf n A n = { 0 , 1 } . \liminf_{n\rightarrow\infty}A_{n}=\{0,1\}.
  30. l i m s u p < s u b > n A n limsup<sub>n→∞A_{n}
  31. ( X , , ) \scriptstyle(X,\mathcal{F},\mathbb{P})
  32. \scriptstyle\mathcal{F}
  33. X \scriptstyle X
  34. \scriptstyle\mathbb{P}
  35. \scriptstyle\mathcal{F}
  36. ( lim n A n ) = lim n ( A n ) . \mathbb{P}(\lim_{n\rightarrow\infty}A_{n})=\lim_{n\rightarrow\infty}\mathbb{P}% (A_{n}).
  37. if n = 1 ( A n ) < then ( lim sup n A n ) = 0. \,\text{if }\sum_{n=1}^{\infty}\mathbb{P}(A_{n})<\infty\,\text{ then }\mathbb{% P}(\limsup_{n\rightarrow\infty}A_{n})=0.
  38. if A 1 , A 2 , are independent events and n = 1 ( A n ) = then ( lim sup n A n ) = 1. \,\text{if }A_{1},A_{2},\dots\,\text{ are independent events and }\sum_{n=1}^{% \infty}\mathbb{P}(A_{n})=\infty\,\text{ then }\mathbb{P}(\limsup_{n\rightarrow% \infty}A_{n})=1.
  39. { lim sup n | Y n - Y | = 0 } \scriptstyle\{\limsup_{n\to\infty}|Y_{n}-Y|=0\}
  40. { lim sup n | Y n - Y | 0 } = { lim sup n | Y n - Y | > 1 k for some k } \{\limsup_{n\to\infty}|Y_{n}-Y|\neq 0\}=\{\limsup_{n\to\infty}|Y_{n}-Y|>\frac{% 1}{k}\,\text{ for some }k\}
  41. = k 1 n 1 j n { | Y n - Y | > 1 k } = lim k lim sup n { | Y n - Y | > 1 k } . =\bigcup_{k\geq 1}\bigcap_{n\geq 1}\bigcup_{j\geq n}\{|Y_{n}-Y|>\frac{1}{k}\}=% \lim_{k\to\infty}\limsup_{n\to\infty}\{|Y_{n}-Y|>\frac{1}{k}\}.
  42. ( { lim sup n | Y n - Y | 0 } ) = lim k ( lim sup n { | Y n - Y | > 1 k } ) . \mathbb{P}(\{\limsup_{n\to\infty}|Y_{n}-Y|\neq 0\})=\lim_{k\to\infty}\mathbb{P% }(\limsup_{n\to\infty}\{|Y_{n}-Y|>\frac{1}{k}\}).

Set_theory_(music).html

  1. \langle\rangle
  2. 0 , 1 , 2 \langle 0,1,2\rangle

Several_complex_variables.html

  1. f ( z 1 , z 2 , , z n ) f(z_{1},z_{2},\ldots,z_{n})
  2. n n
  3. n = 1 n=1
  4. n n
  5. f : 𝐂 n 𝐂 f:\mathbf{C}^{n}\longrightarrow\mathbf{C}
  6. n > 1 n>1
  7. n = 2 n=2
  8. D D
  9. 𝐂 \mathbf{C}
  10. n > 1 n>1
  11. D D
  12. G L ( 2 ) GL(2)
  13. f ( z ) f(z)
  14. U n U\subset\mathbb{C}^{n}
  15. f ( z ) f(z)
  16. a = ( a 1 , , a n ) U n a=(a^{1},\dots,a^{n})\in U\subset\mathbb{C}^{n}
  17. f ( z ) f(z)
  18. U U
  19. f ( z ) = c k 1 , , k n ( z 1 - a 1 ) k 1 ( z n - a n ) k n , f(z)=\sum c_{k_{1},\dots,k_{n}}(z^{1}-a^{1})^{k_{1}}\cdots(z^{n}-a^{n})^{k_{n}% }\ ,
  20. f ( z ) f(z)
  21. U U
  22. z λ z^{\lambda}
  23. f ( z ) f(z)
  24. f z ¯ λ = 0 \frac{\partial f}{\partial\bar{z}^{\lambda}}=0
  25. z λ = x λ + i y λ , f ( x λ + i y λ ) = u λ + i v λ z^{\lambda}=x^{\lambda}+iy^{\lambda},\ \ \ \ f(x^{\lambda}+iy^{\lambda})=u^{% \lambda}+iv^{\lambda}
  26. d z λ = d x λ + i d y λ , d z ¯ λ = d x λ - i y λ z λ = 1 2 ( x λ - i y λ ) , z ¯ λ = 1 2 ( x λ + i y λ ) \begin{aligned}\displaystyle dz^{\lambda}&\displaystyle=dx^{\lambda}+idy^{% \lambda},&\displaystyle d\bar{z}^{\lambda}&\displaystyle=dx^{\lambda}-iy^{% \lambda}\\ \displaystyle\frac{\partial}{\partial z^{\lambda}}&\displaystyle=\frac{1}{2}% \biggl(\frac{\partial}{\partial x^{\lambda}}-i\frac{\partial}{\partial y^{% \lambda}}\biggr),&\displaystyle\frac{\partial}{\partial\bar{z}^{\lambda}}&% \displaystyle=\frac{1}{2}\biggl(\frac{\partial}{\partial x^{\lambda}}+i\frac{% \partial}{\partial y^{\lambda}}\biggr)\end{aligned}
  27. Re ( f z ¯ λ ) = u x λ - v y λ = 0 , Im ( f z ¯ λ ) = u y λ + u x λ = 0 \,\text{Re}\biggl(\frac{\partial f}{\partial\bar{z}^{\lambda}}\biggr)=\frac{% \partial u}{\partial x^{\lambda}}-\frac{\partial v}{\partial y^{\lambda}}=0,\ % \ \ \ \,\text{Im}\biggl(\frac{\partial f}{\partial\bar{z}^{\lambda}}\biggr)=% \frac{\partial u}{\partial y^{\lambda}}+\frac{\partial u}{\partial x^{\lambda}% }=0
  28. c k 1 , , k n c_{k_{1},\dots,k_{n}}
  29. r i r_{i}
  30. G 1 , G 2 G_{1},G_{2}\subset\mathbb{C}
  31. G 1 G 2 G_{1}\cap G_{2}
  32. f 1 f_{1}
  33. f 2 f_{2}
  34. G 1 , G 2 G_{1},G_{2}
  35. z 0 = x 0 + i y 0 G 1 G 2 z^{0}=x^{0}+iy^{0}\in G_{1}\cap G_{2}
  36. f 1 = f 2 f_{1}=f_{2}
  37. { z | | z j - z j 0 | < r j , y = y 0 , 1 j n } \{z\bigl||z_{j}-z_{j}^{0}|<r_{j},y=y^{0},1\leq j\leq n\}
  38. f f
  39. G 1 G 2 G_{1}\cup G_{2}
  40. f = f 1 f=f_{1}
  41. G 1 G_{1}
  42. f = f 2 f=f_{2}
  43. G 2 G_{2}
  44. U , V U,V
  45. n \mathbb{C}^{n}
  46. f 𝒪 ( U ) f\in\mathcal{O}(U)
  47. g 𝒪 ( V ) g\in\mathcal{O}(V)
  48. U V ϕ U\cap V\neq\phi
  49. W W
  50. U V U\cap V
  51. f | W = g | W f|_{W}=g|_{W}
  52. h h
  53. h ( z ) = { f ( z ) z U , g ( z ) z V . h(z)=\begin{cases}f(z)&z\in U,\\ g(z)&z\in V.\end{cases}
  54. h h
  55. f f
  56. g g
  57. h h
  58. n = 1 n=1
  59. U U\varsubsetneqq\mathbb{C}
  60. f f
  61. U U
  62. U U
  63. a U a\in\partial U
  64. f = 1 z - a f=\frac{1}{z-a}
  65. a a
  66. n 2 n\geq 2
  67. U ~ U \widetilde{U}\varsupsetneqq U
  68. f 𝒪 ( U ) f\in\mathcal{O}(U)
  69. f ~ U ~ \tilde{f}\in\widetilde{U}
  70. n n
  71. n n
  72. n n
  73. 2 n 2n
  74. 𝐑 \mathbf{R}
  75. 2 n 2n
  76. J J
  77. i i
  78. w = u + i v w=u+iv
  79. ( u - v v u ) , \begin{pmatrix}u&&-v\\ v&&u\end{pmatrix},
  80. u 2 + v 2 = | w | 2 . u^{2}+v^{2}=|w|^{2}\,.
  81. 𝐂 < s u p > n \mathbf{C}<sup>n

Shear_stress.html

  1. τ \tau\,
  2. τ = F A , \tau={F\over A},
  3. τ \tau
  4. F F
  5. A A
  6. γ \gamma
  7. τ = γ G \tau=\gamma G\,
  8. G G
  9. G = E 2 ( 1 + ν ) . G=\frac{E}{2(1+\nu)}.
  10. E E
  11. ν \nu
  12. τ = V Q I t , \tau={VQ\over It},
  13. τ = 2 ( U G V ) 1 2 , \tau=2\left({UG\over V}\right)^{1\over 2},
  14. U = U r o t a t i n g + U a p p l i e d ; U=U_{rotating}+U_{applied}\,;
  15. U r o t a t i n g = 1 2 I ω 2 ; U_{rotating}={1\over 2}I\omega^{2}\,;
  16. U a p p l i e d = T θ d i s p l a c e d ; U_{applied}=T\theta_{displaced}\,;
  17. I I\,
  18. ω \omega\,
  19. τ ( y ) = μ u y \tau(y)=\mu\frac{\partial u}{\partial y}
  20. μ \mu
  21. u u
  22. y y
  23. τ w τ ( y = 0 ) = μ u y | y = 0 . \tau_{\mathrm{w}}\equiv\tau(y=0)=\mu\left.\frac{\partial u}{\partial y}\right|% _{y=0}~{}~{}.

Sheila_Greibach.html

  1. L 0 L_{0}
  2. L 0 L_{0}
  3. L 0 - { e } L_{0}-\{e\}

Shelah_cardinal.html

  1. κ \kappa
  2. f : κ κ f:\kappa\rightarrow\kappa
  3. N N
  4. j : V N j:V\rightarrow N
  5. κ \kappa
  6. V j ( f ) ( κ ) N V_{j(f)(\kappa)}\subset N

Shell_integration.html

  1. 2 π a b x f ( x ) d x 2\pi\int_{a}^{b}xf(x)\mathrm{d}x
  2. 2 π a b y f ( y ) d y 2\pi\int_{a}^{b}yf(y)\mathrm{d}y
  3. y = ( x - 1 ) 2 ( x - 2 ) 2 y=(x-1)^{2}(x-2)^{2}
  4. 2 π 1 2 x ( x - 1 ) 2 ( x - 2 ) 2 d x 2\pi\int_{1}^{2}x(x-1)^{2}(x-2)^{2}\mathrm{d}x
  5. π 10 \frac{\pi}{10}

Sherwood_number.html

  1. Sh = K L D = Mass transfer rate Diffusion rate \mathrm{Sh}=\frac{KL}{D}=\frac{\mbox{Mass transfer rate}~{}}{\mbox{Diffusion % rate}~{}}
  2. Sh = f ( Re , Sc ) \mathrm{Sh}=f(\mathrm{Re},\mathrm{Sc})
  3. Sh = Sh 0 + C Re m Sc 1 3 \mathrm{Sh}=\mathrm{Sh}_{0}+C\,\mathrm{Re}^{m}\,\mathrm{Sc}^{\frac{1}{3}}
  4. Sh 0 \mathrm{Sh}_{0}
  5. Sh = 2 + 0.552 Re 1 2 Sc 1 3 \mathrm{Sh}=2+0.552\,\mathrm{Re}^{\frac{1}{2}}\,\mathrm{Sc}^{\frac{1}{3}}
  6. Nu = 2 + 0.6 Re 1 2 Pr 1 3 , 0 Re < 200 , 0 Pr < 250 \mathrm{Nu}=2+0.6\,\mathrm{Re}^{\frac{1}{2}}\,\mathrm{Pr}^{\frac{1}{3}},~{}0% \leq~{}\mathrm{Re}<200,~{}0\leq\mathrm{Pr}<250
  7. Sh = 2 + 0.6 Re 1 2 Sc 1 3 , 0 Re < 200 , 0 Sc < 250 \mathrm{Sh}=2+0.6\,\mathrm{Re}^{\frac{1}{2}}\,\mathrm{Sc}^{\frac{1}{3}},~{}0% \leq~{}\mathrm{Re}<200,~{}0\leq\mathrm{Sc}<250

Shielding_effect.html

  1. Z eff = Z - σ Z_{\mathrm{eff}}=Z-\sigma\,
  2. σ \sigma\,
  3. σ \sigma\,

Shift_JIS.html

  1. j 1 j 2 j_{1}j_{2}
  2. s 1 s 2 s_{1}s_{2}
  3. s 1 = { j 1 + 1 2 + 112 if 33 j 1 94 j 1 + 1 2 + 176 if 95 j 1 126 s_{1}=\begin{cases}\left\lfloor\frac{j_{1}+1}{2}\right\rfloor+112&\mbox{if }~{% }33\leq j_{1}\leq 94\\ \left\lfloor\frac{j_{1}+1}{2}\right\rfloor+176&\mbox{if }~{}95\leq j_{1}\leq 1% 26\end{cases}
  4. s 2 = { j 2 + 31 + j 2 96 if j 1 is odd j 2 + 126 if j 1 is even s_{2}=\begin{cases}j_{2}+31+\left\lfloor\frac{j_{2}}{96}\right\rfloor&\mbox{if% }~{}j_{1}\mbox{ is odd }\\ j_{2}+126&\mbox{if }~{}j_{1}\mbox{ is even }\end{cases}

Shift_operator.html

  1. f ( x ) f(x)
  2. f ( x + a ) f(x+a)
  3. t 𝐑 t∈\mathbf{R}
  4. f f
  5. f t ( x ) = f ( x + t ) . f_{t}(x)=f(x+t)~{}.
  6. T t = e t d d x , T^{t}=e^{t\frac{d}{dx}}~{},
  7. S * : ( a 1 , a 2 , a 3 , ) ( a 2 , a 3 , a 4 , ) S^{*}:(a_{1},a_{2},a_{3},\ldots)\mapsto(a_{2},a_{3},a_{4},\ldots)
  8. T : ( a k ) k = - ( a k + 1 ) k = - . T:(a_{k})_{k=-\infty}^{\infty}\mapsto(a_{k+1})_{k=-\infty}^{\infty}.
  9. S : ( a 1 , a 2 , a 3 , ) ( 0 , a 1 , a 2 , ) S:(a_{1},a_{2},a_{3},\ldots)\mapsto(0,a_{1},a_{2},\ldots)
  10. T - 1 : ( a k ) k = - ( a k - 1 ) k = - . T^{-1}:(a_{k})_{k=-\infty}^{\infty}\mapsto(a_{k-1})_{k=-\infty}^{\infty}.
  11. f f
  12. G G
  13. g g
  14. G G
  15. f f
  16. f g ( h ) = f ( g + h ) . f_{g}(h)=f(g+h).
  17. T t = M t , \mathcal{F}T^{t}=M^{t}\mathcal{F},
  18. e x p ( i t x ) exp(itx)
  19. S S
  20. T - 1 y = S x for each x 2 ( ) , T^{-1}y=Sx\,\text{ for each }x\in\ell^{2}(\mathbb{N}),\,
  21. y y
  22. l < s u b > 2 ( 𝐙 ) l<sub>2(\mathbf{Z})

Shifting_nth_root_algorithm.html

  1. y n + r = x y^{n}+r=x
  2. ( y + 1 ) n > x (y+1)^{n}>x
  3. x = B n x + α x^{\prime}=B^{n}x+\alpha
  4. y = B y + β y^{\prime}=By+\beta
  5. x = y n + r x^{\prime}=y^{\prime n}+r^{\prime}
  6. B n x + α = ( B y + β ) n + r . B^{n}x+\alpha=(By+\beta)^{n}+r^{\prime}.
  7. ( B y + β ) n B n x + α (By+\beta)^{n}\leq B^{n}x+\alpha
  8. r = B n x + α - ( B y + β ) n . r^{\prime}=B^{n}x+\alpha-(By+\beta)^{n}.
  9. β = 0 \beta=0
  10. B n y n B n x + α B^{n}y^{n}\leq B^{n}x+\alpha
  11. y n x y^{n}\leq x
  12. β = B \beta=B
  13. ( B ( y + 1 ) ) n B n x + α (B(y+1))^{n}\leq B^{n}x+\alpha\,
  14. B n x < B n ( y + 1 ) n B^{n}x<B^{n}(y+1)^{n}\,
  15. B n x B^{n}x
  16. B n ( y + 1 ) n B^{n}(y+1)^{n}
  17. B n B^{n}
  18. B n B^{n}
  19. B n x + B n B n ( y + 1 ) n B^{n}x+B^{n}\leq B^{n}(y+1)^{n}\,
  20. B n x + B n B n x + α B^{n}x+B^{n}\leq B^{n}x+\alpha\,
  21. B n α B^{n}\leq\alpha\,
  22. 0 α < B n 0\leq\alpha<B^{n}
  23. ( B y + β ) n (By+\beta)^{n}
  24. γ < B \gamma<B
  25. r 0 r^{\prime}\geq 0
  26. 0 β γ 0\leq\beta\leq\gamma
  27. ( y + 1 ) n > x (y^{\prime}+1)^{n}>x^{\prime}\,
  28. ( B y + β + 1 ) n > B n x + α (By+\beta+1)^{n}>B^{n}x+\alpha\,
  29. β + 1 \beta+1
  30. ( B y + β + 1 ) n B n x + α (By+\beta+1)^{n}\leq B^{n}x+\alpha\,
  31. x = B n x + α x^{\prime}=B^{n}x+\alpha
  32. ( B y + β ) n B n x + α (By+\beta)^{n}\leq B^{n}x+\alpha
  33. y = B y + β y^{\prime}=By+\beta
  34. r = x - y n r^{\prime}=x^{\prime}-y^{\prime n}
  35. x = y n + r x=y^{n}+r
  36. ( B y + β ) n B n x + α (By+\beta)^{n}\leq B^{n}x+\alpha
  37. ( B y + β ) n - B n y n B n r + α (By+\beta)^{n}-B^{n}y^{n}\leq B^{n}r+\alpha
  38. r = x - y n = B n x + α - ( B y + β ) n r^{\prime}=x^{\prime}-y^{\prime n}=B^{n}x+\alpha-(By+\beta)^{n}
  39. r = B n r + α - ( ( B y + β ) n - B n y n ) r^{\prime}=B^{n}r+\alpha-((By+\beta)^{n}-B^{n}y^{n})
  40. x x
  41. r = x - y n r=x-y^{n}
  42. x < ( y + 1 ) n x<(y+1)^{n}
  43. r < ( y + 1 ) n - y n r<(y+1)^{n}-y^{n}
  44. r < n y n - 1 + O ( y n - 2 ) r<ny^{n-1}+O(y^{n-2})
  45. r < n x n - 1 n + O ( x n - 2 n ) r<nx^{{n-1}\over n}+O(x^{{n-2}\over n})
  46. r r
  47. x x
  48. B n y n B^{n}y^{n}
  49. ( B y + β ) n (By+\beta)^{n}
  50. y n - 1 y^{n-1}
  51. y n y^{n}
  52. ( B y + β ) n - B n y n B n r + α . (By+\beta)^{n}-B^{n}y^{n}\leq B^{n}r+\alpha.
  53. y = B y + β y^{\prime}=By+\beta
  54. r = B n r + α - ( ( B y + β ) n - B n y n ) . r^{\prime}=B^{n}r+\alpha-((By+\beta)^{n}-B^{n}y^{n}).
  55. y y y\leftarrow y^{\prime}
  56. r r . r\leftarrow r^{\prime}.
  57. y y
  58. y n < x B k y^{n}<xB^{k}
  59. y n + r = x B k y^{n}+r=xB^{k}
  60. k k
  61. ( B y + β ) n - B n y n (By+\beta)^{n}-B^{n}y^{n}
  62. r + α r+\alpha
  63. n B n - 1 y n - 1 nB^{n-1}y^{n-1}
  64. O ( log ( B ) ) O(\log(B))
  65. ( B y + β ) n - B n y n (By+\beta)^{n}-B^{n}y^{n}
  66. 2 n - 4 2n-4
  67. k ( n - 1 ) k(n-1)
  68. n - 2 n-2
  69. k ( n - 1 ) k(n-1)
  70. n - 1 n-1
  71. n - 2 n-2
  72. k ( n - 1 ) k(n-1)
  73. O ( n 2 ) O(n^{2})
  74. O ( n ) O(n)
  75. O ( k 2 n 2 ) O(k^{2}n^{2})
  76. O ( k 2 n 2 log ( B ) ) O(k^{2}n^{2}\log(B))
  77. O ( k ) O(k)
  78. O ( k 2 n 2 log ( B ) ) O(k^{2}n^{2}\log(B))
  79. O ( k 3 n 2 log ( B ) ) O(k^{3}n^{2}\log(B))
  80. O ( k ) O(k)
  81. O ( log ( B ) ) O(\log(B))
  82. O ( log 2 ( B ) ) O(\log^{2}(B))

Shooting_method.html

  1. y ′′ ( t ) = f ( t , y ( t ) , y ( t ) ) , y ( t 0 ) = y 0 , y ( t 1 ) = y 1 y^{\prime\prime}(t)=f(t,y(t),y^{\prime}(t)),\quad y(t_{0})=y_{0},\quad y(t_{1}% )=y_{1}
  2. y ′′ ( t ) = f ( t , y ( t ) , y ( t ) ) , y ( t 0 ) = y 0 , y ( t 0 ) = a y^{\prime\prime}(t)=f(t,y(t),y^{\prime}(t)),\quad y(t_{0})=y_{0},\quad y^{% \prime}(t_{0})=a
  3. F ( a ) = y ( t 1 ; a ) - y 1 F(a)=y(t_{1};a)-y_{1}\,
  4. f ( t , y ( t ) , y ( t ) ) = p ( t ) y ( t ) + q ( t ) y ( t ) + r ( t ) . f(t,y(t),y^{\prime}(t))=p(t)y^{\prime}(t)+q(t)y(t)+r(t).\,
  5. y ( t ) = y ( 1 ) ( t ) + y ( 1 ) ( t ) - y ( 1 ) ( t 1 ) y ( 2 ) ( t 1 ) y ( 2 ) ( t ) y(t)=y_{(1)}(t)+\frac{y_{(1)}(t)-y_{(1)}(t_{1})}{y_{(2)}(t_{1})}y_{(2)}(t)
  6. y ( 1 ) ( t ) y_{(1)}(t)
  7. y ( 1 ) ′′ ( t ) = p ( t ) y ( 1 ) ( t ) + q ( t ) y ( 1 ) ( t ) + r ( t ) , y ( 1 ) ( t 0 ) = y 0 , y ( 1 ) ( t 0 ) = 0 , y_{(1)}^{\prime\prime}(t)=p(t)y_{(1)}^{\prime}(t)+q(t)y_{(1)}(t)+r(t),\quad y_% {(1)}(t_{0})=y_{0},\quad y_{(1)}^{\prime}(t_{0})=0,
  8. y ( 2 ) ( t ) y_{(2)}(t)
  9. y ( 2 ) ′′ ( t ) = p ( t ) y ( 2 ) ( t ) + q ( t ) y ( 2 ) ( t ) , y ( 2 ) ( t 0 ) = 0 , y ( 2 ) ( t 0 ) = 1. y_{(2)}^{\prime\prime}(t)=p(t)y_{(2)}^{\prime}(t)+q(t)y_{(2)}(t),\quad y_{(2)}% (t_{0})=0,\quad y_{(2)}^{\prime}(t_{0})=1.
  10. w ′′ ( t ) = 3 2 w 2 , w ( 0 ) = 4 , w ( 1 ) = 1 w^{\prime\prime}(t)=\frac{3}{2}w^{2},\quad w(0)=4,\quad w(1)=1
  11. w ′′ ( t ) = 3 2 w 2 , w ( 0 ) = 4 , w ( 0 ) = s w^{\prime\prime}(t)=\frac{3}{2}w^{2},\quad w(0)=4,\quad w^{\prime}(0)=s

Short-time_Fourier_transform.html

  1. 𝐒𝐓𝐅𝐓 { x ( t ) } ( τ , ω ) X ( τ , ω ) = - x ( t ) w ( t - τ ) e - j ω t d t \mathbf{STFT}\{x(t)\}(\tau,\omega)\equiv X(\tau,\omega)=\int_{-\infty}^{\infty% }x(t)w(t-\tau)e^{-j\omega t}\,dt
  2. 𝐒𝐓𝐅𝐓 { x [ n ] } ( m , ω ) X ( m , ω ) = n = - x [ n ] w [ n - m ] e - j ω n \mathbf{STFT}\{x[n]\}(m,\omega)\equiv X(m,\omega)=\sum_{n=-\infty}^{\infty}x[n% ]w[n-m]e^{-j\omega n}
  3. spectrogram { x ( t ) } ( τ , ω ) | X ( τ , ω ) | 2 \operatorname{spectrogram}\{x(t)\}(\tau,\omega)\equiv|X(\tau,\omega)|^{2}
  4. - w ( τ ) d τ = 1. \int_{-\infty}^{\infty}w(\tau)\,d\tau=1.
  5. - w ( t - τ ) d τ = 1 t \int_{-\infty}^{\infty}w(t-\tau)\,d\tau=1\quad\forall\ t
  6. x ( t ) = x ( t ) - w ( t - τ ) d τ = - x ( t ) w ( t - τ ) d τ . x(t)=x(t)\int_{-\infty}^{\infty}w(t-\tau)\,d\tau=\int_{-\infty}^{\infty}x(t)w(% t-\tau)\,d\tau.
  7. X ( ω ) = - x ( t ) e - j ω t d t . X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt.
  8. X ( ω ) = - [ - x ( t ) w ( t - τ ) d τ ] e - j ω t d t X(\omega)=\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}x(t)w(t-\tau)\,d% \tau\right]\,e^{-j\omega t}\,dt
  9. = - - x ( t ) w ( t - τ ) e - j ω t d τ d t . =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(t)w(t-\tau)\,e^{-j\omega t}\,% d\tau\,dt.
  10. X ( ω ) = - - x ( t ) w ( t - τ ) e - j ω t d t d τ X(\omega)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(t)w(t-\tau)\,e^{-j% \omega t}\,dt\,d\tau
  11. = - [ - x ( t ) w ( t - τ ) e - j ω t d t ] d τ =\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}x(t)w(t-\tau)\,e^{-j% \omega t}\,dt\right]\,d\tau
  12. = - X ( τ , ω ) d τ . =\int_{-\infty}^{\infty}X(\tau,\omega)\,d\tau.
  13. x ( t ) = 1 2 π - X ( ω ) e + j ω t d ω , x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{+j\omega t}\,d\omega,
  14. x ( t ) = 1 2 π - - X ( τ , ω ) e + j ω t d τ d ω . x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}X(\tau,\omega% )e^{+j\omega t}\,d\tau\,d\omega.
  15. x ( t ) = - [ 1 2 π - X ( τ , ω ) e + j ω t d ω ] d τ . x(t)=\int_{-\infty}^{\infty}\left[\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\tau,% \omega)e^{+j\omega t}\,d\omega\right]\,d\tau.
  16. x ( t ) w ( t - τ ) = 1 2 π - X ( τ , ω ) e + j ω t d ω . x(t)w(t-\tau)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\tau,\omega)e^{+j\omega t% }\,d\omega.
  17. x ( t ) x(t)
  18. x ( t ) x(t)
  19. x ( t ) = { cos ( 2 π 10 t ) 0 s t < 5 s cos ( 2 π 25 t ) 5 s t < 10 s cos ( 2 π 50 t ) 10 s t < 15 s cos ( 2 π 100 t ) 15 s t < 20 s x(t)=\begin{cases}\cos(2\pi 10t)&0\,\mathrm{s}\leq t<5\,\mathrm{s}\\ \cos(2\pi 25t)&5\,\mathrm{s}\leq t<10\,\mathrm{s}\\ \cos(2\pi 50t)&10\,\mathrm{s}\leq t<15\,\mathrm{s}\\ \cos(2\pi 100t)&15\,\mathrm{s}\leq t<20\,\mathrm{s}\\ \end{cases}

Sierpiński_space.html

  1. { , { 1 } , { 0 , 1 } } . \{\varnothing,\{1\},\{0,1\}\}.
  2. { , { 0 } , { 0 , 1 } } . \{\varnothing,\{0\},\{0,1\}\}.
  3. { 0 } ¯ = { 0 } , { 1 } ¯ = { 0 , 1 } . \overline{\{0\}}=\{0\},\qquad\overline{\{1\}}=\{0,1\}.
  4. 0 0 , 0 1 , 1 1. 0\leq 0,\qquad 0\leq 1,\qquad 1\leq 1.
  5. d ( 0 , 1 ) = 0 d(0,1)=0
  6. d ( 1 , 0 ) = 1 d(1,0)=1
  7. χ U ( x ) = { 1 x U 0 x U \chi_{U}(x)=\begin{cases}1&x\in U\\ 0&x\not\in U\end{cases}
  8. χ U - 1 ( 1 ) = U . \chi_{U}^{-1}(1)=U.
  9. C ( X , S ) 𝒯 ( X ) C(X,S)\cong\mathcal{T}(X)
  10. f - 1 : 𝒯 ( Y ) 𝒯 ( X ) . f^{-1}:\mathcal{T}(Y)\to\mathcal{T}(X).
  11. e : X U 𝒯 ( X ) S = S 𝒯 ( X ) e:X\to\prod_{U\in\mathcal{T}(X)}S=S^{\mathcal{T}(X)}
  12. e ( x ) U = χ U ( x ) . e(x)_{U}=\chi_{U}(x).\,

Sigma.html

  1. | Σ |\!\!\!\Sigma
  2. A A
  3. σ ( A ) \sigma(A)
  4. A A
  5. σ ( A ) \sigma(A)

Sign_function.html

  1. sgn ( x ) := { - 1 if x < 0 , 0 if x = 0 , 1 if x > 0. \operatorname{sgn}(x):=\begin{cases}-1&\,\text{if }x<0,\\ 0&\,\text{if }x=0,\\ 1&\,\text{if }x>0.\end{cases}
  2. x = sgn ( x ) | x | . x=\operatorname{sgn}(x)\cdot|x|\,.
  3. sgn ( x ) = x | x | = | x | x \operatorname{sgn}(x)={x\over|x|}={|x|\over x}
  4. | x | = sgn ( x ) x |x|=\operatorname{sgn}(x)\cdot x
  5. d | x | d x = sgn ( x ) for x 0 {d|x|\over dx}=\operatorname{sgn}(x)\mbox{ for }~{}x\neq 0
  6. sgn ( x ) = 2 H ( x ) - 1 \operatorname{sgn}(x)=2H(x)-1\,
  7. d sgn ( x ) d x = 2 d H ( x ) d x = 2 δ ( x ) . {d\operatorname{sgn}(x)\over dx}=2{dH(x)\over dx}=2\delta(x)\,.
  8. sgn ( x ) = - [ x < 0 ] + [ x > 0 ] . \ \operatorname{sgn}(x)=-[x<0]+[x>0]\,.
  9. sgn ( x ) = x | x | + 1 - - x | - x | + 1 . \ \operatorname{sgn}(x)=\Bigg\lfloor\frac{x}{|x|+1}\Bigg\rfloor-\Bigg\lfloor% \frac{-x}{|-x|+1}\Bigg\rfloor\,.
  10. k 1 k\gg 1
  11. sgn ( x ) tanh ( k x ) . \ \operatorname{sgn}(x)\approx\tanh(kx)\,.
  12. sgn ( x ) x x 2 + ϵ 2 . \ \operatorname{sgn}(x)\approx\frac{x}{\sqrt{x^{2}+\epsilon^{2}}}\,.
  13. ϵ 0 \epsilon\to 0
  14. x 2 + ϵ 2 \sqrt{x^{2}+\epsilon^{2}}
  15. ϵ = 0 \epsilon=0
  16. x 2 + y 2 \sqrt{x^{2}+y^{2}}
  17. sgn ( z ) = z | z | \operatorname{sgn}(z)=\frac{z}{|z|}
  18. \mathbb{C}
  19. sgn ( z ) = e i arg z , \operatorname{sgn}(z)=e^{i\arg z}\,,
  20. sgn ( 0 + 0 i ) = 0 \operatorname{sgn}(0+0i)=0
  21. csgn ( z ) = { 1 if ( z ) > 0 , - 1 if ( z ) < 0 , sgn ( ( z ) ) if ( z ) = 0 \operatorname{csgn}(z)=\begin{cases}1&\,\text{if }\Re(z)>0,\\ -1&\,\text{if }\Re(z)<0,\\ \operatorname{sgn}(\Im(z))&\,\text{if }\Re(z)=0\end{cases}
  22. ( z ) \Re(z)
  23. ( z ) \Im(z)
  24. csgn ( z ) = z z 2 = z 2 z . \operatorname{csgn}(z)=\frac{z}{\sqrt{z^{2}}}=\frac{\sqrt{z^{2}}}{z}.
  25. x ~{}x~{}
  26. ε ( x ) , \varepsilon(x),
  27. ε ( x ) 2 = 1 ~{}\varepsilon(x)^{2}=1~{}
  28. x = 0 ~{}x=0~{}
  29. sgn ~{}\operatorname{sgn}~{}
  30. sgn ( 0 ) 2 = 0 \operatorname{sgn}(0)^{2}=0~{}
  31. ε ( x ) δ ( x ) + δ ( x ) ε ( x ) = 0 ; \varepsilon(x)\delta(x)+\delta(x)\varepsilon(x)=0~{};
  32. ε ( x ) ~{}\varepsilon(x)~{}
  33. x = 0 ~{}x=0~{}
  34. ε \varepsilon
  35. sgn ~{}\operatorname{sgn}~{}
  36. ε ( 0 ) \varepsilon(0)
  37. sgn ( 0 ) = 0 ~{}\operatorname{sgn}(0)=0~{}

Signal_reconstruction.html

  1. L 2 L^{2}
  2. n \mathbb{C}^{n}
  3. n \mathbb{C}^{n}
  4. n \mathbb{C}^{n}
  5. L 2 L^{2}
  6. L 2 L^{2}
  7. d k := ( 0 , , 0 , 1 , 0 , , 0 ) d_{k}:=(0,...,0,1,0,...,0)
  8. n \mathbb{C}^{n}
  9. e k L 2 e_{k}\in L^{2}
  10. F ( e k ) = d k F(e_{k})=d_{k}
  11. { e k } \{e_{k}\}
  12. L 2 L^{2}
  13. e k ( t ) := e 2 π i k t e_{k}(t):=e^{2\pi ikt}\,
  14. R ( d k ) = e k R(d_{k})=e_{k}\,
  15. k = - n / 2 , , ( n - 1 ) / 2 k=\lfloor-n/2\rfloor,...,\lfloor(n-1)/2\rfloor
  16. ( d k ) (d_{k})
  17. n \mathbb{C}^{n}
  18. d k ( j ) = e 2 π i j k n d_{k}(j)=e^{2\pi ijk\over n}
  19. k = - n / 2 , , ( n - 1 ) / 2 k=\lfloor-n/2\rfloor,...,\lfloor(n-1)/2\rfloor

Signed_number_representations.html

  1. / 2 N \scriptstyle\mathbb{Z}/2^{N}\mathbb{Z}

Simple_algebra.html

  1. { [ 0 α 0 0 ] | α } \left\{\left.\begin{bmatrix}0&\alpha\\ 0&0\\ \end{bmatrix}\,\right|\,\alpha\in\mathbb{C}\right\}

Simplex_algorithm.html

  1. 𝐜 𝐱 \mathbf{c}\cdot\mathbf{x}
  2. 𝐀𝐱 = 𝐛 , x i 0 \mathbf{A}\mathbf{x}=\mathbf{b},\,x_{i}\geq 0
  3. x = ( x 1 , , x n ) x=(x_{1},\,\dots,\,x_{n})
  4. c = ( c 1 , , c n ) c=(c_{1},\,\dots,\,c_{n})
  5. b = ( b 1 , , b p ) b=(b_{1},\,\dots,\,b_{p})
  6. b j 0 b_{j}\geq 0
  7. 𝐀𝐱 𝐛 , x i 0 \mathbf{A}\mathbf{x}\leq\mathbf{b},\,x_{i}\geq 0
  8. x = ( x 1 , , x n ) x=(x_{1},\,\dots,\,x_{n})
  9. A i A_{i}
  10. x i 0 x_{i}\neq 0
  11. x 1 5 x_{1}\geq 5
  12. y 1 y_{1}
  13. y 1 = x 1 - 5 x 1 = y 1 + 5 \begin{aligned}\displaystyle y_{1}=x_{1}-5\\ \displaystyle x_{1}=y_{1}+5\end{aligned}
  14. x 1 x_{1}
  15. x 2 + 2 x 3 3 - x 4 + 3 x 5 2 \begin{aligned}\displaystyle x_{2}+2x_{3}&\displaystyle\leq 3\\ \displaystyle-x_{4}+3x_{5}&\displaystyle\geq 2\end{aligned}
  16. x 2 + 2 x 3 + s 1 = 3 - x 4 + 3 x 5 - s 2 = 2 s 1 , s 2 0 \begin{aligned}\displaystyle x_{2}+2x_{3}+s_{1}&\displaystyle=3\\ \displaystyle-x_{4}+3x_{5}-s_{2}&\displaystyle=2\\ \displaystyle s_{1},\,s_{2}&\displaystyle\geq 0\end{aligned}
  17. z 1 z_{1}
  18. z 1 = z 1 + - z 1 - z 1 + , z 1 - 0 \begin{aligned}&\displaystyle z_{1}=z_{1}^{+}-z_{1}^{-}\\ &\displaystyle z_{1}^{+},\,z_{1}^{-}\geq 0\end{aligned}
  19. z 1 z_{1}
  20. 𝐀𝐱 = 𝐛 , x i 0 \mathbf{A}\mathbf{x}=\mathbf{b},\,x_{i}\geq 0
  21. 𝐀 \mathbf{A}
  22. 𝐀𝐱 𝐛 \mathbf{A}\mathbf{x}\geq\mathbf{b}
  23. [ 1 - 𝐜 T 0 0 𝐀 𝐛 ] \begin{bmatrix}1&-\mathbf{c}^{T}&0\\ 0&\mathbf{A}&\mathbf{b}\end{bmatrix}
  24. [ 1 - 𝐜 B T - 𝐜 D T 0 0 I 𝐃 𝐛 ] \begin{bmatrix}1&-\mathbf{c}^{T}_{B}&-\mathbf{c}^{T}_{D}&0\\ 0&I&\mathbf{D}&\mathbf{b}\end{bmatrix}
  25. [ 1 0 - 𝐜 ¯ D T z B 0 I 𝐃 𝐛 ] \begin{bmatrix}1&0&-\bar{\mathbf{c}}^{T}_{D}&z_{B}\\ 0&I&\mathbf{D}&\mathbf{b}\end{bmatrix}
  26. z ( 𝐱 ) = z B + nonnegative terms corresponding to nonbasic variables z(\mathbf{x})=z_{B}+\,\text{nonnegative terms corresponding to nonbasic variables}
  27. b r / a r c b_{r}/a_{rc}\,
  28. Z = - 2 x - 3 y - 4 z Z=-2x-3y-4z\,
  29. 3 x + 2 y + z 10 2 x + 5 y + 3 z 15 x , y , z 0 \begin{aligned}\displaystyle 3x+2y+z&\displaystyle\leq 10\\ \displaystyle 2x+5y+3z&\displaystyle\leq 15\\ \displaystyle x,\,y,\,z&\displaystyle\geq 0\end{aligned}
  30. [ 1 2 3 4 0 0 0 0 3 2 1 1 0 10 0 2 5 3 0 1 15 ] \begin{bmatrix}1&2&3&4&0&0&0\\ 0&3&2&1&1&0&10\\ 0&2&5&3&0&1&15\end{bmatrix}
  31. x = y = z = 0 , s = 10 , t = 15. x=y=z=0,\,s=10,\,t=15.
  32. [ 1 - 2 3 - 11 3 0 0 - 4 3 - 20 0 7 3 1 3 0 1 - 1 3 5 0 2 3 5 3 1 0 1 3 5 ] \begin{bmatrix}1&-\tfrac{2}{3}&-\tfrac{11}{3}&0&0&-\tfrac{4}{3}&-20\\ 0&\tfrac{7}{3}&\tfrac{1}{3}&0&1&-\tfrac{1}{3}&5\\ 0&\tfrac{2}{3}&\tfrac{5}{3}&1&0&\tfrac{1}{3}&5\end{bmatrix}
  33. x = y = t = 0 , z = 5 , s = 5. x=y=t=0,\,z=5,\,s=5.
  34. Z = - 20 + 2 3 x + 11 3 y + 4 3 t Z=-20+\tfrac{2}{3}x+\tfrac{11}{3}y+\tfrac{4}{3}t
  35. Z = - 2 x - 3 y - 4 z Z=-2x-3y-4z\,
  36. 3 x + 2 y + z = 10 2 x + 5 y + 3 z = 15 x , y , z 0 \begin{aligned}\displaystyle 3x+2y+z&\displaystyle=10\\ \displaystyle 2x+5y+3z&\displaystyle=15\\ \displaystyle x,\,y,\,z&\displaystyle\geq 0\end{aligned}
  37. [ 1 2 3 4 0 0 3 2 1 10 0 2 5 3 15 ] \begin{bmatrix}1&2&3&4&0\\ 0&3&2&1&10\\ 0&2&5&3&15\end{bmatrix}
  38. [ 1 0 0 0 0 - 1 - 1 0 0 1 2 3 4 0 0 0 0 0 3 2 1 1 0 10 0 0 2 5 3 0 1 15 ] \begin{bmatrix}1&0&0&0&0&-1&-1&0\\ 0&1&2&3&4&0&0&0\\ 0&0&3&2&1&1&0&10\\ 0&0&2&5&3&0&1&15\end{bmatrix}
  39. [ 1 0 5 7 4 0 0 25 0 1 2 3 4 0 0 0 0 0 3 2 1 1 0 10 0 0 2 5 3 0 1 15 ] \begin{bmatrix}1&0&5&7&4&0&0&25\\ 0&1&2&3&4&0&0&0\\ 0&0&3&2&1&1&0&10\\ 0&0&2&5&3&0&1&15\end{bmatrix}
  40. [ 1 0 7 3 1 3 0 0 - 4 3 5 0 1 - 2 3 - 11 3 0 0 - 4 3 - 20 0 0 7 3 1 3 0 1 - 1 3 5 0 0 2 3 5 3 1 0 1 3 5 ] \begin{bmatrix}1&0&\tfrac{7}{3}&\tfrac{1}{3}&0&0&-\tfrac{4}{3}&5\\ 0&1&-\tfrac{2}{3}&-\tfrac{11}{3}&0&0&-\tfrac{4}{3}&-20\\ 0&0&\tfrac{7}{3}&\tfrac{1}{3}&0&1&-\tfrac{1}{3}&5\\ 0&0&\tfrac{2}{3}&\tfrac{5}{3}&1&0&\tfrac{1}{3}&5\end{bmatrix}
  41. [ 1 0 0 0 0 - 1 - 1 0 0 1 0 - 25 7 0 2 7 - 10 7 - 130 7 0 0 1 1 7 0 3 7 - 1 7 15 7 0 0 0 11 7 1 - 2 7 3 7 25 7 ] \begin{bmatrix}1&0&0&0&0&-1&-1&0\\ 0&1&0&-\tfrac{25}{7}&0&\tfrac{2}{7}&-\tfrac{10}{7}&-\tfrac{130}{7}\\ 0&0&1&\tfrac{1}{7}&0&\tfrac{3}{7}&-\tfrac{1}{7}&\tfrac{15}{7}\\ 0&0&0&\tfrac{11}{7}&1&-\tfrac{2}{7}&\tfrac{3}{7}&\tfrac{25}{7}\end{bmatrix}
  42. [ 1 0 - 25 7 0 - 130 7 0 1 1 7 0 15 7 0 0 11 7 1 25 7 ] \begin{bmatrix}1&0&-\tfrac{25}{7}&0&-\tfrac{130}{7}\\ 0&1&\tfrac{1}{7}&0&\tfrac{15}{7}\\ 0&0&\tfrac{11}{7}&1&\tfrac{25}{7}\end{bmatrix}

Simplicial_approximation_theorem.html

  1. K K
  2. L L
  3. f : K L f:K\to L
  4. F : | K | | L | F:|K|\to|L|
  5. x | K | x\in|K|
  6. | f | ( x ) |f|(x)
  7. L L
  8. F ( x ) F(x)
  9. f f
  10. F F
  11. f f
  12. | f | |f|
  13. F F
  14. F : | K | | L | F:|K|\to|L|
  15. n 0 n_{0}
  16. n n 0 n\geq n_{0}
  17. f : Bd n K L f:\mathrm{Bd}^{n}K\to L
  18. F F
  19. Bd K \mathrm{Bd}\;K
  20. K K
  21. Bd n K \mathrm{Bd}^{n}K
  22. n n

Simulation_preorder.html

  1. p 𝛼 p p\overset{\alpha}{\rightarrow}p^{\prime}
  2. q 𝛼 q q\overset{\alpha}{\rightarrow}q^{\prime}
  3. R - 1 ; 𝛼 𝛼 ; R - 1 R^{-1}\,;\overset{\alpha}{\rightarrow}\quad{\subseteq}\quad\overset{\alpha}{% \rightarrow}\,;R^{-1}

Sinc_function.html

  1. s i n c ( x ) sinc(x)
  2. sinc ( x ) = sin ( x ) x . \operatorname{sinc}(x)=\frac{\sin(x)}{x}~{}.
  3. sinc ( x ) = sin ( π x ) π x . \operatorname{sinc}(x)=\frac{\sin(\pi x)}{\pi x}~{}.
  4. x x
  5. s i n c ( 0 ) sinc(0)
  6. π π
  7. x x
  8. π π
  9. ξ ξ
  10. x n ( n + 1 2 ) π - 1 ( n + 1 2 ) π , x_{n}\approx(n+\tfrac{1}{2})\pi-\frac{1}{(n+\frac{1}{2})\pi}~{},
  11. sin ( π x ) π x = n = 1 ( 1 - x 2 n 2 ) \frac{\sin(\pi x)}{\pi x}=\prod_{n=1}^{\infty}\left(1-\frac{x^{2}}{n^{2}}\right)
  12. Γ ( x ) Γ(x)
  13. sin ( π x ) π x = 1 Γ ( 1 + x ) Γ ( 1 - x ) . \frac{\sin(\pi x)}{\pi x}=\frac{1}{\Gamma(1+x)\Gamma(1-x)}~{}.
  14. sin ( x ) x = n = 1 cos ( x 2 n ) . \frac{\sin(x)}{x}=\prod_{n=1}^{\infty}\cos\left(\frac{x}{2^{n}}\right)~{}.
  15. f f
  16. - sinc ( t ) e i 2 π f t d t = rect ( f ) , \int_{-\infty}^{\infty}\operatorname{sinc}(t)\,e^{i2\pi ft}\,dt=\operatorname{% rect}(f)~{},
  17. - sin ( π x ) π x d x = rect ( 0 ) = 1 \int_{-\infty}^{\infty}\frac{\sin(\pi x)}{\pi x}\,dx=\operatorname{rect}(0)=1\,\!
  18. - | sin ( π x ) π x | d x = + . \int_{-\infty}^{\infty}\left|\frac{\sin(\pi x)}{\pi x}\right|\,dx=+\infty~{}.
  19. 0 x sin ( θ ) θ d θ = Si ( x ) \int_{0}^{x}\frac{\sin(\theta)}{\theta}\,d\theta=\operatorname{Si}(x)\,\!
  20. λ s i n c ( λ x ) λsinc(λx)
  21. x d 2 y d x 2 + 2 d y d x + λ 2 x y = 0. x\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+\lambda^{2}xy=0.\,\!
  22. - sin 2 ( θ ) θ 2 d θ = π - sinc 2 ( x ) d x = 1 , \int_{-\infty}^{\infty}\frac{\sin^{2}(\theta)}{\theta^{2}}\,d\theta=\pi\,\!% \rightarrow\int_{-\infty}^{\infty}\operatorname{sinc}^{2}(x)\,dx=1~{},
  23. - sin 3 ( θ ) θ 3 d θ = 3 π 4 \int_{-\infty}^{\infty}\frac{\sin^{3}(\theta)}{\theta^{3}}\,d\theta=\frac{3\pi% }{4}\,\!
  24. - sin 4 ( θ ) θ 4 d θ = 2 π 3 . \int_{-\infty}^{\infty}\frac{\sin^{4}(\theta)}{\theta^{4}}\,d\theta=\frac{2\pi% }{3}~{}.
  25. lim a 0 sin ( π x / a ) π x = lim a 0 1 a sinc ( x / a ) = δ ( x ) . \lim_{a\rightarrow 0}\frac{\sin(\pi x/a)}{\pi x}=\lim_{a\rightarrow 0}\frac{1}% {a}\textrm{sinc}(x/a)=\delta(x)~{}.
  26. lim a 0 - 1 a sinc ( x / a ) φ ( x ) d x = φ ( 0 ) , \lim_{a\rightarrow 0}\int_{-\infty}^{\infty}\frac{1}{a}\textrm{sinc}(x/a)% \varphi(x)\,dx=\varphi(0)~{},
  27. φ ( x ) φ(x)
  28. a a
  29. ± 1 / ( π x ) ±1/(πx)
  30. a a
  31. δ ( x ) δ(x)
  32. x x
  33. x x
  34. n n
  35. n n
  36. ( π 1 ) / 2 (\pi− 1)/2
  37. n = 1 sinc ( n ) = sinc ( 1 ) + sinc ( 2 ) + sinc ( 3 ) + sinc ( 4 ) + = π - 1 2 \sum_{n=1}^{\infty}\operatorname{sinc}(n)=\operatorname{sinc}(1)+\operatorname% {sinc}(2)+\operatorname{sinc}(3)+\operatorname{sinc}(4)+\cdots=\frac{\pi-1}{2}
  38. ( π 1 ) / 2 (\pi− 1)/2
  39. n = 1 sinc 2 ( n ) = sinc 2 ( 1 ) + sinc 2 ( 2 ) + sinc 2 ( 3 ) + sinc 2 ( 4 ) + = π - 1 2 \sum_{n=1}^{\infty}\operatorname{sinc}^{2}(n)=\operatorname{sinc}^{2}(1)+% \operatorname{sinc}^{2}(2)+\operatorname{sinc}^{2}(3)+\operatorname{sinc}^{2}(% 4)+\cdots=\frac{\pi-1}{2}
  40. n = 1 ( - 1 ) n + 1 sinc ( n ) = sinc ( 1 ) - sinc ( 2 ) + sinc ( 3 ) - sinc ( 4 ) + = 1 2 \sum_{n=1}^{\infty}(-1)^{n+1}\,\operatorname{sinc}(n)=\operatorname{sinc}(1)-% \operatorname{sinc}(2)+\operatorname{sinc}(3)-\operatorname{sinc}(4)+\cdots=% \frac{1}{2}
  41. sinc ( x ) = sin ( x ) x = n = 0 ( - x 2 ) n ( 2 n + 1 ) ! \operatorname{sinc}(x)=\frac{\sin(x)}{x}=\sum_{n=0}^{\infty}\frac{\left(-x^{2}% \right)^{n}}{(2n+1)!}
  42. sinc C ( x , y ) = sinc ( x ) sinc ( y ) \operatorname{sinc}_{\operatorname{C}}(x,y)=\operatorname{sinc}(x)% \operatorname{sinc}(y)
  43. u 1 = [ 1 / 2 3 / 2 ] u_{1}=\left[\begin{array}[]{c}1/2\\ \sqrt{3}/2\end{array}\right]
  44. u 2 = [ 1 / 2 - 3 / 2 ] u_{2}=\left[\begin{array}[]{c}1/2\\ -\sqrt{3}/2\end{array}\right]
  45. ξ 1 = 2 / 3 u 1 , ξ 2 = 2 / 3 u 2 , ξ 3 = - 2 / 3 ( u 1 + u 2 ) \xi_{1}=2/3u_{1},\xi_{2}=2/3u_{2},\xi_{3}=-2/3(u_{1}+u_{2})
  46. 𝐱 = [ x y ] \mathbf{x}=\left[\begin{array}[]{c}x\\ y\end{array}\right]
  47. sinc H ( 𝐱 ) = 1 / 3 ( \displaystyle\operatorname{sinc}_{\rm H}(\mathbf{x})=1/3\big(

Sine_wave.html

  1. y ( t ) = A sin ( 2 π f t + φ ) = A sin ( ω t + φ ) y(t)=A\sin(2\pi ft+\varphi)=A\sin(\omega t+\varphi)
  2. φ \varphi
  3. φ \varphi
  4. φ \varphi
  5. y ( x , t ) = A sin ( k x - ω t + φ ) + D y(x,t)=A\sin(kx-\omega t+\varphi)+D\,
  6. y ( x , t ) = A sin ( k x + ω t + φ ) + D y(x,t)=A\sin(kx+\omega t+\varphi)+D\,
  7. k = ω v = 2 π f v = 2 π λ k={\omega\over v}={2\pi f\over v}={2\pi\over\lambda}
  8. cos ( x ) = sin ( x + π / 2 ) , \cos(x)=\sin(x+\pi/2),
  9. u ( t , x ) = A sin ( k x - ω t + φ ) u(t,x)=A\sin(kx-\omega t+\varphi)

Singleton_(mathematics).html

  1. S S
  2. b : X { 0 , 1 } b:X\to\{0,1\}
  3. S S
  4. y X y∈X
  5. x X x∈X
  6. b ( x ) = ( x = y ) b(x)=(x=y)\,
  7. 1 = def α ^ { ( x ) . α = ι ȷ x } 1\ \overset{\underset{\mathrm{def}}{}}{=}\ \hat{\alpha}\{(\exists x).\alpha=% \iota\jmath x\}
  8. ι ȷ x = def y ^ ( y = x ) \iota\jmath x\ \overset{\underset{\mathrm{def}}{}}{=}\ \hat{y}(y=x)

Singular_homology.html

  1. H n ( X ) H_{n}(X)
  2. σ n \sigma_{n}
  3. Δ n \Delta^{n}
  4. σ n : Δ n X \sigma_{n}:\Delta^{n}\to X
  5. σ n ( Δ n ) \sigma_{n}(\Delta^{n})
  6. n σ n ( Δ n ) \partial_{n}\sigma_{n}(\Delta^{n})
  7. σ \sigma
  8. σ n \sigma_{n}
  9. [ p 0 , p 1 , , p n ] = [ σ n ( e 0 ) , σ n ( e 1 ) , , σ n ( e n ) ] [p_{0},p_{1},\cdots,p_{n}]=[\sigma_{n}(e_{0}),\sigma_{n}(e_{1}),\cdots,\sigma_% {n}(e_{n})]
  10. e k e_{k}
  11. Δ n \Delta^{n}
  12. σ n \sigma_{n}
  13. n σ n ( Δ n ) = k = 0 n ( - 1 ) k [ p 0 , , p k - 1 , p k + 1 , p n ] \partial_{n}\sigma_{n}(\Delta^{n})=\sum_{k=0}^{n}(-1)^{k}[p_{0},\cdots,p_{k-1}% ,p_{k+1},\cdots p_{n}]
  14. σ n \sigma_{n}
  15. Δ n \Delta^{n}
  16. σ = [ p 0 , p 1 ] \sigma=[p_{0},p_{1}]
  17. p 0 p_{0}
  18. p 1 p_{1}
  19. [ p 1 ] - [ p 0 ] [p_{1}]-[p_{0}]
  20. σ n ( Δ n ) \sigma_{n}(\Delta^{n})
  21. σ n ( Δ n ) \sigma_{n}(\Delta^{n})
  22. C n ( X ) C_{n}(X)
  23. C n ( X ) C_{n}(X)
  24. \partial
  25. n : C n C n - 1 , \partial_{n}:C_{n}\to C_{n-1},
  26. C n C_{n}
  27. ( C ( X ) , ) (C_{\bullet}(X),\partial_{\bullet})
  28. C ( X ) C_{\bullet}(X)
  29. Z n ( X ) = ker ( n ) Z_{n}(X)=\ker(\partial_{n})
  30. B n ( X ) = im ( n + 1 ) B_{n}(X)=\operatorname{im}(\partial_{n+1})
  31. n n + 1 = 0 \partial_{n}\circ\partial_{n+1}=0
  32. n n
  33. X X
  34. H n ( X ) = Z n ( X ) / B n ( X ) . H_{n}(X)=Z_{n}(X)/B_{n}(X).
  35. H n ( X ) H_{n}(X)
  36. H n ( X ) = H n ( Y ) H_{n}(X)=H_{n}(Y)\,
  37. H 0 ( X ) = H_{0}(X)=\mathbb{Z}
  38. f : C n ( X ) C n ( Y ) . f_{\sharp}:C_{n}(X)\rightarrow C_{n}(Y).
  39. f = f , \partial f_{\sharp}=f_{\sharp}\partial,
  40. f * : H n ( X ) H n ( Y ) . f_{*}:H_{n}(X)\rightarrow H_{n}(Y).
  41. P : C n ( X ) C n + 1 ( Y ) P:C_{n}(X)\rightarrow C_{n+1}(Y)
  42. P ( σ ) = f ( σ ) - g ( σ ) + P ( σ ) . \partial P(\sigma)=f_{\sharp}(\sigma)-g_{\sharp}(\sigma)+P(\partial\sigma).
  43. f ( α ) - g ( α ) = P ( α ) , f_{\sharp}(\alpha)-g_{\sharp}(\alpha)=\partial P(\alpha),
  44. X C n ( X ) X\mapsto C_{n}(X)
  45. C n ( X ) C_{n}(X)
  46. f : X Y f:X\to Y
  47. f * : C n ( X ) C n ( Y ) f_{*}:C_{n}(X)\to C_{n}(Y)\,
  48. f * ( i a i σ i ) = i a i ( f σ i ) f_{*}\left(\sum_{i}a_{i}\sigma_{i}\right)=\sum_{i}a_{i}(f\circ\sigma_{i})
  49. σ i : Δ n X \sigma_{i}:\Delta^{n}\to X
  50. i a i σ i \sum_{i}a_{i}\sigma_{i}\,
  51. C n ( X ) C_{n}(X)
  52. C n C_{n}
  53. C n : T o p A b C_{n}:{Top}\to{Ab}
  54. n f * = f * n \partial_{n}f_{*}=f_{*}\partial_{n}
  55. X H n ( X ) X\mapsto H_{n}(X)
  56. H n : T o p A b H_{n}:{Top}\to{Ab}
  57. H n H_{n}
  58. H n : h T o p A b . H_{n}:{hTop}\to{Ab}.
  59. H n H_{n}
  60. C : T o p C o m p C_{\bullet}:{Top}\to{Comp}
  61. X ( C ( X ) , ) X\mapsto(C_{\bullet}(X),\partial_{\bullet})
  62. f f * f\mapsto f_{*}
  63. C C_{\bullet}
  64. H n : C o m p A b H_{n}:{Comp}\to{Ab}
  65. C H n ( C ) = Z n ( C ) / B n ( C ) C_{\bullet}\mapsto H_{n}(C_{\bullet})=Z_{n}(C_{\bullet})/B_{n}(C_{\bullet})
  66. H n ( X , R ) H_{n}(X,R)
  67. H n ( X , ) = H n ( X ) H_{n}(X,\mathbb{Z})=H_{n}(X)
  68. A X A\subset X
  69. H n ( X , A ) = H n ( C ( X ) / C ( A ) ) H_{n}(X,A)=H_{n}(C_{\bullet}(X)/C_{\bullet}(A))
  70. 0 C ( A ) C ( X ) C ( X ) / C ( A ) 0. 0\to C_{\bullet}(A)\to C_{\bullet}(X)\to C_{\bullet}(X)/C_{\bullet}(A)\to 0.
  71. δ \delta